id	sid	tid	token	lemma	pos
ejpam-2803	1	1	european	european	PROPN
ejpam-2803	1	2	journal	journal	PROPN
ejpam-2803	1	3	of	of	ADP
ejpam-2803	1	4	pure	pure	ADJ
ejpam-2803	1	5	and	and	CCONJ
ejpam-2803	1	6	applied	apply	VERB
ejpam-2803	1	7	mathematics	mathematic	NOUN
ejpam-2803	1	8	vol	vol	NOUN
ejpam-2803	1	9	.	.	PROPN
ejpam-2803	2	1	10	10	NUM
ejpam-2803	2	2	,	,	PUNCT
ejpam-2803	2	3	no	no	INTJ
ejpam-2803	2	4	.	.	NOUN
ejpam-2803	2	5	2	2	NUM
ejpam-2803	2	6	,	,	PUNCT
ejpam-2803	2	7	2017	2017	NUM
ejpam-2803	2	8	,	,	PUNCT
ejpam-2803	2	9	211	211	NUM
ejpam-2803	2	10	-	-	SYM
ejpam-2803	2	11	230	230	NUM
ejpam-2803	2	12	issn	issn	PROPN
ejpam-2803	2	13	1307	1307	NUM
ejpam-2803	2	14	-	-	SYM
ejpam-2803	2	15	5543	5543	NUM
ejpam-2803	2	16	–	–	PUNCT
ejpam-2803	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2803	2	18	published	publish	VERB
ejpam-2803	2	19	by	by	ADP
ejpam-2803	2	20	new	new	PROPN
ejpam-2803	2	21	york	york	PROPN
ejpam-2803	2	22	business	business	PROPN
ejpam-2803	2	23	global	global	PROPN
ejpam-2803	3	1	a	a	DET
ejpam-2803	3	2	module	module	NOUN
ejpam-2803	3	3	whose	whose	DET
ejpam-2803	3	4	second	second	ADJ
ejpam-2803	3	5	spectrum	spectrum	NOUN
ejpam-2803	3	6	has	have	VERB
ejpam-2803	3	7	the	the	DET
ejpam-2803	3	8	surjective	surjective	ADJ
ejpam-2803	3	9	or	or	CCONJ
ejpam-2803	3	10	injective	injective	ADJ
ejpam-2803	3	11	natural	natural	ADJ
ejpam-2803	3	12	map	map	NOUN
ejpam-2803	3	13	h.	h.	PROPN
ejpam-2803	3	14	ansari	ansari	PROPN
ejpam-2803	3	15	-	-	PUNCT
ejpam-2803	3	16	toroghy1	toroghy1	PROPN
ejpam-2803	3	17	,	,	PUNCT
ejpam-2803	3	18	s.	s.	PROPN
ejpam-2803	3	19	s.	s.	PROPN
ejpam-2803	3	20	pourmortazavi	pourmortazavi	VERB
ejpam-2803	3	21	1,∗	1,∗	PROPN
ejpam-2803	3	22	1	1	NUM
ejpam-2803	3	23	department	department	NOUN
ejpam-2803	3	24	of	of	ADP
ejpam-2803	3	25	pure	pure	ADJ
ejpam-2803	3	26	mathematics	mathematic	NOUN
ejpam-2803	3	27	,	,	PUNCT
ejpam-2803	3	28	faculty	faculty	NOUN
ejpam-2803	3	29	of	of	ADP
ejpam-2803	3	30	mathematical	mathematical	ADJ
ejpam-2803	3	31	science	science	NOUN
ejpam-2803	3	32	,	,	PUNCT
ejpam-2803	3	33	university	university	NOUN
ejpam-2803	3	34	of	of	ADP
ejpam-2803	3	35	guilan	guilan	PROPN
ejpam-2803	3	36	,	,	PUNCT
ejpam-2803	3	37	rasht	rasht	NOUN
ejpam-2803	3	38	,	,	PUNCT
ejpam-2803	3	39	iran	iran	PROPN
ejpam-2803	3	40	abstract	abstract	NOUN
ejpam-2803	3	41	.	.	PUNCT
ejpam-2803	4	1	let	let	VERB
ejpam-2803	4	2	r	r	PRON
ejpam-2803	4	3	be	be	AUX
ejpam-2803	4	4	a	a	DET
ejpam-2803	4	5	commutative	commutative	ADJ
ejpam-2803	4	6	ring	ring	NOUN
ejpam-2803	4	7	and	and	CCONJ
ejpam-2803	4	8	m	m	AUX
ejpam-2803	4	9	be	be	AUX
ejpam-2803	4	10	an	an	DET
ejpam-2803	4	11	r	r	NOUN
ejpam-2803	4	12	-	-	PUNCT
ejpam-2803	4	13	module	module	NOUN
ejpam-2803	4	14	.	.	PUNCT
ejpam-2803	5	1	let	let	VERB
ejpam-2803	5	2	specs(m	specs(m	NOUN
ejpam-2803	5	3	)	)	PUNCT
ejpam-2803	5	4	be	be	VERB
ejpam-2803	5	5	the	the	DET
ejpam-2803	5	6	set	set	NOUN
ejpam-2803	5	7	of	of	ADP
ejpam-2803	5	8	all	all	DET
ejpam-2803	5	9	second	second	ADJ
ejpam-2803	5	10	submodules	submodule	NOUN
ejpam-2803	5	11	of	of	ADP
ejpam-2803	5	12	m	m	PROPN
ejpam-2803	5	13	.	.	PUNCT
ejpam-2803	6	1	in	in	ADP
ejpam-2803	6	2	this	this	DET
ejpam-2803	6	3	article	article	NOUN
ejpam-2803	6	4	,	,	PUNCT
ejpam-2803	6	5	we	we	PRON
ejpam-2803	6	6	topologize	topologize	VERB
ejpam-2803	6	7	specs(m	specs(m	PROPN
ejpam-2803	6	8	)	)	PUNCT
ejpam-2803	6	9	with	with	ADP
ejpam-2803	6	10	zariski	zariski	ADJ
ejpam-2803	6	11	and	and	CCONJ
ejpam-2803	6	12	classical	classical	ADJ
ejpam-2803	6	13	zariski	zariski	NOUN
ejpam-2803	6	14	topologies	topology	NOUN
ejpam-2803	6	15	and	and	CCONJ
ejpam-2803	6	16	study	study	VERB
ejpam-2803	6	17	the	the	DET
ejpam-2803	6	18	classes	class	NOUN
ejpam-2803	6	19	of	of	ADP
ejpam-2803	6	20	all	all	DET
ejpam-2803	6	21	modules	module	NOUN
ejpam-2803	6	22	whose	whose	DET
ejpam-2803	6	23	second	second	ADJ
ejpam-2803	6	24	spectrum	spectrum	NOUN
ejpam-2803	6	25	have	have	VERB
ejpam-2803	6	26	the	the	DET
ejpam-2803	6	27	surjective	surjective	ADJ
ejpam-2803	6	28	or	or	CCONJ
ejpam-2803	6	29	injective	injective	ADJ
ejpam-2803	6	30	natural	natural	ADJ
ejpam-2803	6	31	map	map	NOUN
ejpam-2803	6	32	.	.	PUNCT
ejpam-2803	7	1	moreover	moreover	ADV
ejpam-2803	7	2	,	,	PUNCT
ejpam-2803	7	3	we	we	PRON
ejpam-2803	7	4	investigate	investigate	VERB
ejpam-2803	7	5	the	the	DET
ejpam-2803	7	6	interplay	interplay	NOUN
ejpam-2803	7	7	between	between	ADP
ejpam-2803	7	8	the	the	DET
ejpam-2803	7	9	algebraic	algebraic	ADJ
ejpam-2803	7	10	properties	property	NOUN
ejpam-2803	7	11	of	of	ADP
ejpam-2803	7	12	m	m	PRON
ejpam-2803	7	13	and	and	CCONJ
ejpam-2803	7	14	the	the	DET
ejpam-2803	7	15	topological	topological	ADJ
ejpam-2803	7	16	properties	property	NOUN
ejpam-2803	7	17	of	of	ADP
ejpam-2803	7	18	specs(m	specs(m	NOUN
ejpam-2803	7	19	)	)	PUNCT
ejpam-2803	7	20	.	.	PUNCT
ejpam-2803	8	1	2010	2010	NUM
ejpam-2803	8	2	mathematics	mathematic	NOUN
ejpam-2803	8	3	subject	subject	NOUN
ejpam-2803	8	4	classifications	classification	NOUN
ejpam-2803	8	5	:	:	PUNCT
ejpam-2803	8	6	13c13	13c13	NUM
ejpam-2803	8	7	,	,	PUNCT
ejpam-2803	8	8	13c99	13c99	NUM
ejpam-2803	8	9	key	key	ADJ
ejpam-2803	8	10	words	word	NOUN
ejpam-2803	8	11	and	and	CCONJ
ejpam-2803	8	12	phrases	phrase	NOUN
ejpam-2803	8	13	:	:	PUNCT
ejpam-2803	8	14	cotop	cotop	NOUN
ejpam-2803	8	15	module	module	NOUN
ejpam-2803	8	16	,	,	PUNCT
ejpam-2803	8	17	second	second	ADJ
ejpam-2803	8	18	submodule	submodule	NOUN
ejpam-2803	8	19	,	,	PUNCT
ejpam-2803	8	20	xs	xs	NOUN
ejpam-2803	8	21	-	-	PUNCT
ejpam-2803	8	22	injective	injective	ADJ
ejpam-2803	8	23	module	module	NOUN
ejpam-2803	8	24	,	,	PUNCT
ejpam-2803	8	25	zariski	zariski	NOUN
ejpam-2803	8	26	topology	topology	NOUN
ejpam-2803	8	27	1	1	NUM
ejpam-2803	8	28	.	.	PUNCT
ejpam-2803	9	1	introduction	introduction	NOUN
ejpam-2803	9	2	throughout	throughout	ADP
ejpam-2803	9	3	this	this	DET
ejpam-2803	9	4	article	article	NOUN
ejpam-2803	9	5	,	,	PUNCT
ejpam-2803	9	6	r	r	NOUN
ejpam-2803	9	7	denotes	denote	VERB
ejpam-2803	9	8	a	a	DET
ejpam-2803	9	9	commutative	commutative	ADJ
ejpam-2803	9	10	ring	ring	NOUN
ejpam-2803	9	11	with	with	ADP
ejpam-2803	9	12	identity	identity	NOUN
ejpam-2803	9	13	and	and	CCONJ
ejpam-2803	9	14	all	all	DET
ejpam-2803	9	15	modules	module	NOUN
ejpam-2803	9	16	are	be	AUX
ejpam-2803	9	17	unitary	unitary	ADJ
ejpam-2803	9	18	.	.	PUNCT
ejpam-2803	10	1	also	also	ADV
ejpam-2803	10	2	the	the	DET
ejpam-2803	10	3	notation	notation	NOUN
ejpam-2803	10	4	z	z	PROPN
ejpam-2803	10	5	(	(	PUNCT
ejpam-2803	10	6	resp	resp	NOUN
ejpam-2803	10	7	.	.	PUNCT
ejpam-2803	11	1	q	q	X
ejpam-2803	11	2	)	)	PUNCT
ejpam-2803	11	3	will	will	AUX
ejpam-2803	11	4	denote	denote	VERB
ejpam-2803	11	5	the	the	DET
ejpam-2803	11	6	ring	ring	NOUN
ejpam-2803	11	7	of	of	ADP
ejpam-2803	11	8	integers	integer	NOUN
ejpam-2803	11	9	(	(	PUNCT
ejpam-2803	11	10	resp	resp	NOUN
ejpam-2803	11	11	.	.	PUNCT
ejpam-2803	12	1	the	the	DET
ejpam-2803	12	2	field	field	NOUN
ejpam-2803	12	3	of	of	ADP
ejpam-2803	12	4	fractions	fraction	NOUN
ejpam-2803	12	5	of	of	ADP
ejpam-2803	12	6	z	z	NOUN
ejpam-2803	12	7	)	)	PUNCT
ejpam-2803	12	8	.	.	PUNCT
ejpam-2803	13	1	if	if	SCONJ
ejpam-2803	13	2	n	n	PRON
ejpam-2803	13	3	is	be	AUX
ejpam-2803	13	4	a	a	DET
ejpam-2803	13	5	subset	subset	NOUN
ejpam-2803	13	6	of	of	ADP
ejpam-2803	13	7	an	an	DET
ejpam-2803	13	8	r	r	NOUN
ejpam-2803	13	9	-	-	PUNCT
ejpam-2803	13	10	module	module	NOUN
ejpam-2803	13	11	m	m	NOUN
ejpam-2803	13	12	,	,	PUNCT
ejpam-2803	13	13	then	then	ADV
ejpam-2803	13	14	n	n	PRON
ejpam-2803	13	15	≤	≤	NOUN
ejpam-2803	13	16	m	m	VERB
ejpam-2803	13	17	denotes	denote	NOUN
ejpam-2803	13	18	n	n	PART
ejpam-2803	13	19	is	be	AUX
ejpam-2803	13	20	an	an	DET
ejpam-2803	13	21	r	r	NOUN
ejpam-2803	13	22	-	-	PUNCT
ejpam-2803	13	23	submodule	submodule	NOUN
ejpam-2803	13	24	of	of	ADP
ejpam-2803	13	25	m	m	PROPN
ejpam-2803	13	26	.	.	PUNCT
ejpam-2803	14	1	for	for	ADP
ejpam-2803	14	2	any	any	DET
ejpam-2803	14	3	ideal	ideal	NOUN
ejpam-2803	14	4	i	i	PRON
ejpam-2803	14	5	of	of	ADP
ejpam-2803	14	6	r	r	NOUN
ejpam-2803	14	7	containing	contain	VERB
ejpam-2803	14	8	annr(m	annr(m	NOUN
ejpam-2803	14	9	)	)	PUNCT
ejpam-2803	14	10	,	,	PUNCT
ejpam-2803	14	11	r̄	r̄	NOUN
ejpam-2803	14	12	and	and	CCONJ
ejpam-2803	14	13	ī	ī	NOUN
ejpam-2803	14	14	denote	denote	NOUN
ejpam-2803	14	15	r	r	NOUN
ejpam-2803	14	16	/	/	SYM
ejpam-2803	14	17	annr(m	annr(m	NOUN
ejpam-2803	14	18	)	)	PUNCT
ejpam-2803	14	19	and	and	CCONJ
ejpam-2803	14	20	i	i	PRON
ejpam-2803	14	21	/	/	SYM
ejpam-2803	14	22	annr(m	annr(m	PROPN
ejpam-2803	14	23	)	)	PUNCT
ejpam-2803	14	24	,	,	PUNCT
ejpam-2803	14	25	respectively	respectively	ADV
ejpam-2803	14	26	.	.	PUNCT
ejpam-2803	15	1	the	the	DET
ejpam-2803	15	2	colon	colon	NOUN
ejpam-2803	15	3	ideal	ideal	NOUN
ejpam-2803	15	4	of	of	ADP
ejpam-2803	15	5	m	m	PROPN
ejpam-2803	15	6	into	into	ADP
ejpam-2803	15	7	n	n	PROPN
ejpam-2803	15	8	is	be	AUX
ejpam-2803	15	9	defined	define	VERB
ejpam-2803	15	10	to	to	PART
ejpam-2803	15	11	be	be	AUX
ejpam-2803	15	12	(	(	PUNCT
ejpam-2803	15	13	n	n	NOUN
ejpam-2803	15	14	:	:	PUNCT
ejpam-2803	15	15	m	m	X
ejpam-2803	15	16	)	)	PUNCT
ejpam-2803	16	1	=	=	PRON
ejpam-2803	16	2	{	{	PUNCT
ejpam-2803	16	3	r	r	NOUN
ejpam-2803	16	4	∈	∈	PROPN
ejpam-2803	16	5	r	r	NOUN
ejpam-2803	16	6	:	:	PUNCT
ejpam-2803	16	7	rm	rm	PROPN
ejpam-2803	16	8	⊆	⊆	NUM
ejpam-2803	16	9	n	n	CCONJ
ejpam-2803	16	10	}	}	PUNCT
ejpam-2803	16	11	=	=	SYM
ejpam-2803	16	12	annr(m	annr(m	PROPN
ejpam-2803	16	13	/	/	SYM
ejpam-2803	16	14	n	n	CCONJ
ejpam-2803	16	15	)	)	PUNCT
ejpam-2803	16	16	.	.	PUNCT
ejpam-2803	17	1	let	let	VERB
ejpam-2803	17	2	m	m	PRON
ejpam-2803	17	3	be	be	AUX
ejpam-2803	17	4	an	an	DET
ejpam-2803	17	5	r	r	NOUN
ejpam-2803	17	6	-	-	PUNCT
ejpam-2803	17	7	module	module	NOUN
ejpam-2803	17	8	.	.	PUNCT
ejpam-2803	18	1	a	a	DET
ejpam-2803	18	2	proper	proper	ADJ
ejpam-2803	18	3	submodule	submodule	NOUN
ejpam-2803	18	4	n	n	PROPN
ejpam-2803	18	5	of	of	ADP
ejpam-2803	18	6	m	m	PROPN
ejpam-2803	18	7	is	be	AUX
ejpam-2803	18	8	said	say	VERB
ejpam-2803	18	9	to	to	PART
ejpam-2803	18	10	be	be	AUX
ejpam-2803	18	11	prime	prime	ADJ
ejpam-2803	18	12	if	if	SCONJ
ejpam-2803	18	13	for	for	ADP
ejpam-2803	18	14	any	any	DET
ejpam-2803	18	15	r	r	NOUN
ejpam-2803	18	16	∈	∈	NOUN
ejpam-2803	18	17	r	r	NOUN
ejpam-2803	18	18	and	and	CCONJ
ejpam-2803	18	19	m	m	PROPN
ejpam-2803	18	20	∈	∈	NOUN
ejpam-2803	18	21	m	m	NOUN
ejpam-2803	18	22	with	with	ADP
ejpam-2803	18	23	rm	rm	PROPN
ejpam-2803	18	24	∈	∈	PROPN
ejpam-2803	19	1	n	n	X
ejpam-2803	19	2	,	,	PUNCT
ejpam-2803	19	3	we	we	PRON
ejpam-2803	19	4	have	have	VERB
ejpam-2803	19	5	m	m	PROPN
ejpam-2803	19	6	∈	∈	NOUN
ejpam-2803	19	7	n	n	ADJ
ejpam-2803	19	8	or	or	CCONJ
ejpam-2803	19	9	r	r	NOUN
ejpam-2803	19	10	∈	∈	PROPN
ejpam-2803	19	11	(	(	PUNCT
ejpam-2803	19	12	n	n	NOUN
ejpam-2803	19	13	:	:	PUNCT
ejpam-2803	19	14	r	r	NOUN
ejpam-2803	19	15	m	m	PROPN
ejpam-2803	19	16	)	)	PUNCT
ejpam-2803	19	17	.	.	PUNCT
ejpam-2803	20	1	this	this	PRON
ejpam-2803	20	2	implies	imply	VERB
ejpam-2803	20	3	that	that	SCONJ
ejpam-2803	20	4	(	(	PUNCT
ejpam-2803	20	5	n	n	X
ejpam-2803	20	6	:	:	PUNCT
ejpam-2803	20	7	r	r	NOUN
ejpam-2803	20	8	m	m	NOUN
ejpam-2803	20	9	)	)	PUNCT
ejpam-2803	21	1	=	=	SYM
ejpam-2803	21	2	p	p	NOUN
ejpam-2803	21	3	is	be	AUX
ejpam-2803	21	4	prime	prime	ADJ
ejpam-2803	21	5	ideal	ideal	NOUN
ejpam-2803	21	6	of	of	ADP
ejpam-2803	21	7	r	r	NOUN
ejpam-2803	21	8	and	and	CCONJ
ejpam-2803	21	9	we	we	PRON
ejpam-2803	21	10	say	say	VERB
ejpam-2803	21	11	that	that	SCONJ
ejpam-2803	21	12	n	n	PRON
ejpam-2803	21	13	is	be	AUX
ejpam-2803	21	14	a	a	DET
ejpam-2803	21	15	p	p	ADJ
ejpam-2803	21	16	-	-	PUNCT
ejpam-2803	21	17	prime	prime	NOUN
ejpam-2803	21	18	submodule	submodule	NOUN
ejpam-2803	21	19	of	of	ADP
ejpam-2803	21	20	m	m	PROPN
ejpam-2803	21	21	.	.	PUNCT
ejpam-2803	22	1	the	the	DET
ejpam-2803	22	2	dual	dual	ADJ
ejpam-2803	22	3	notion	notion	NOUN
ejpam-2803	22	4	of	of	ADP
ejpam-2803	22	5	prime	prime	ADJ
ejpam-2803	22	6	submodules	submodule	NOUN
ejpam-2803	22	7	(	(	PUNCT
ejpam-2803	22	8	i.e.	i.e.	X
ejpam-2803	22	9	,	,	PUNCT
ejpam-2803	22	10	second	second	ADJ
ejpam-2803	22	11	submodules	submodule	NOUN
ejpam-2803	22	12	)	)	PUNCT
ejpam-2803	22	13	was	be	AUX
ejpam-2803	22	14	introduced	introduce	VERB
ejpam-2803	22	15	and	and	CCONJ
ejpam-2803	22	16	studied	study	VERB
ejpam-2803	22	17	in	in	ADP
ejpam-2803	22	18	[	[	X
ejpam-2803	22	19	30	30	NUM
ejpam-2803	22	20	]	]	PUNCT
ejpam-2803	22	21	.	.	PUNCT
ejpam-2803	23	1	a	a	DET
ejpam-2803	23	2	non	non	ADJ
ejpam-2803	23	3	-	-	ADJ
ejpam-2803	23	4	zero	zero	NUM
ejpam-2803	23	5	submodule	submodule	NOUN
ejpam-2803	23	6	s	s	PROPN
ejpam-2803	23	7	of	of	ADP
ejpam-2803	23	8	m	m	PROPN
ejpam-2803	23	9	is	be	AUX
ejpam-2803	23	10	said	say	VERB
ejpam-2803	23	11	to	to	PART
ejpam-2803	23	12	be	be	AUX
ejpam-2803	23	13	second	second	ADJ
ejpam-2803	23	14	if	if	SCONJ
ejpam-2803	23	15	for	for	ADP
ejpam-2803	23	16	each	each	DET
ejpam-2803	23	17	a	a	DET
ejpam-2803	23	18	∈	∈	NOUN
ejpam-2803	23	19	r	r	NOUN
ejpam-2803	23	20	the	the	DET
ejpam-2803	23	21	homomorphism	homomorphism	NOUN
ejpam-2803	23	22	s	s	X
ejpam-2803	23	23	a→	a→	X
ejpam-2803	23	24	s	s	X
ejpam-2803	23	25	is	be	AUX
ejpam-2803	23	26	either	either	CCONJ
ejpam-2803	23	27	surjective	surjective	ADJ
ejpam-2803	23	28	or	or	CCONJ
ejpam-2803	23	29	zero	zero	NUM
ejpam-2803	23	30	.	.	PUNCT
ejpam-2803	24	1	this	this	PRON
ejpam-2803	24	2	implies	imply	VERB
ejpam-2803	24	3	that	that	SCONJ
ejpam-2803	24	4	annr(s	annr(s	NOUN
ejpam-2803	24	5	)	)	PUNCT
ejpam-2803	24	6	=	=	PUNCT
ejpam-2803	25	1	p	p	NOUN
ejpam-2803	25	2	is	be	AUX
ejpam-2803	25	3	a	a	DET
ejpam-2803	25	4	prime	prime	ADJ
ejpam-2803	25	5	ideal	ideal	NOUN
ejpam-2803	25	6	of	of	ADP
ejpam-2803	25	7	r	r	NOUN
ejpam-2803	25	8	and	and	CCONJ
ejpam-2803	25	9	s	s	NOUN
ejpam-2803	25	10	is	be	AUX
ejpam-2803	25	11	said	say	VERB
ejpam-2803	25	12	to	to	PART
ejpam-2803	25	13	be	be	AUX
ejpam-2803	25	14	p	p	NOUN
ejpam-2803	25	15	-	-	PUNCT
ejpam-2803	25	16	second	second	NOUN
ejpam-2803	25	17	.	.	PUNCT
ejpam-2803	26	1	more	more	ADJ
ejpam-2803	26	2	information	information	NOUN
ejpam-2803	26	3	about	about	ADP
ejpam-2803	26	4	this	this	DET
ejpam-2803	26	5	class	class	NOUN
ejpam-2803	26	6	of	of	ADP
ejpam-2803	26	7	modules	module	NOUN
ejpam-2803	26	8	can	can	AUX
ejpam-2803	26	9	be	be	AUX
ejpam-2803	26	10	seen	see	VERB
ejpam-2803	26	11	in	in	ADP
ejpam-2803	26	12	[	[	X
ejpam-2803	26	13	4	4	NUM
ejpam-2803	26	14	,	,	PUNCT
ejpam-2803	26	15	5	5	NUM
ejpam-2803	26	16	,	,	PUNCT
ejpam-2803	26	17	7	7	NUM
ejpam-2803	26	18	,	,	PUNCT
ejpam-2803	26	19	8	8	NUM
ejpam-2803	26	20	,	,	PUNCT
ejpam-2803	26	21	14	14	NUM
ejpam-2803	26	22	,	,	PUNCT
ejpam-2803	26	23	16	16	NUM
ejpam-2803	26	24	,	,	PUNCT
ejpam-2803	26	25	15	15	NUM
ejpam-2803	26	26	]	]	PUNCT
ejpam-2803	26	27	.	.	PUNCT
ejpam-2803	27	1	∗corresponding	∗corresponde	VERB
ejpam-2803	27	2	author	author	NOUN
ejpam-2803	27	3	.	.	PUNCT
ejpam-2803	28	1	email	email	NOUN
ejpam-2803	28	2	addresses	address	NOUN
ejpam-2803	28	3	:	:	PUNCT
ejpam-2803	28	4	ansari@guilan.ac.irr	ansari@guilan.ac.irr	INTJ
ejpam-2803	28	5	(	(	PUNCT
ejpam-2803	28	6	h.	h.	PROPN
ejpam-2803	28	7	ansari	ansari	PROPN
ejpam-2803	28	8	-	-	PUNCT
ejpam-2803	28	9	toroghy	toroghy	NOUN
ejpam-2803	28	10	)	)	PUNCT
ejpam-2803	28	11	,	,	PUNCT
ejpam-2803	28	12	mortazavi@phd.guilan.ac.ir	mortazavi@phd.guilan.ac.ir	PROPN
ejpam-2803	28	13	(	(	PUNCT
ejpam-2803	28	14	s.	s.	PROPN
ejpam-2803	28	15	s.	s.	PROPN
ejpam-2803	28	16	pourmortazavi	pourmortazavi	VERB
ejpam-2803	28	17	)	)	PUNCT
ejpam-2803	28	18	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2803	29	1	211	211	NUM
ejpam-2803	29	2	c	c	X
ejpam-2803	29	3	©	©	PROPN
ejpam-2803	29	4	2017	2017	NUM
ejpam-2803	29	5	ejpam	ejpam	NOUN
ejpam-2803	29	6	all	all	DET
ejpam-2803	29	7	rights	right	NOUN
ejpam-2803	29	8	reserved	reserve	VERB
ejpam-2803	29	9	.	.	PUNCT
ejpam-2803	30	1	h.	h.	PROPN
ejpam-2803	30	2	ansari	ansari	PROPN
ejpam-2803	30	3	-	-	PUNCT
ejpam-2803	30	4	toroghy	toroghy	NOUN
ejpam-2803	30	5	,	,	PUNCT
ejpam-2803	30	6	s.	s.	PROPN
ejpam-2803	30	7	s.	s.	PROPN
ejpam-2803	30	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	30	9	/	/	SYM
ejpam-2803	30	10	eur	eur	PROPN
ejpam-2803	30	11	.	.	PUNCT
ejpam-2803	31	1	j.	j.	PROPN
ejpam-2803	31	2	pure	pure	PROPN
ejpam-2803	31	3	appl	appl	PROPN
ejpam-2803	31	4	.	.	PROPN
ejpam-2803	31	5	math	math	PROPN
ejpam-2803	31	6	,	,	PUNCT
ejpam-2803	31	7	10	10	NUM
ejpam-2803	31	8	(	(	PUNCT
ejpam-2803	31	9	2	2	NUM
ejpam-2803	31	10	)	)	PUNCT
ejpam-2803	31	11	(	(	PUNCT
ejpam-2803	31	12	2017	2017	NUM
ejpam-2803	31	13	)	)	PUNCT
ejpam-2803	31	14	,	,	PUNCT
ejpam-2803	31	15	211	211	NUM
ejpam-2803	31	16	-	-	SYM
ejpam-2803	31	17	230	230	NUM
ejpam-2803	31	18	212	212	NUM
ejpam-2803	31	19	the	the	DET
ejpam-2803	31	20	concept	concept	NOUN
ejpam-2803	31	21	of	of	ADP
ejpam-2803	31	22	prime	prime	ADJ
ejpam-2803	31	23	submodule	submodule	PROPN
ejpam-2803	31	24	has	have	AUX
ejpam-2803	31	25	led	lead	VERB
ejpam-2803	31	26	to	to	ADP
ejpam-2803	31	27	the	the	DET
ejpam-2803	31	28	development	development	NOUN
ejpam-2803	31	29	of	of	ADP
ejpam-2803	31	30	topologies	topology	NOUN
ejpam-2803	31	31	on	on	ADP
ejpam-2803	31	32	the	the	DET
ejpam-2803	31	33	spectrum	spectrum	NOUN
ejpam-2803	31	34	of	of	ADP
ejpam-2803	31	35	modules	module	NOUN
ejpam-2803	31	36	.	.	PUNCT
ejpam-2803	32	1	a	a	DET
ejpam-2803	32	2	brief	brief	ADJ
ejpam-2803	32	3	history	history	NOUN
ejpam-2803	32	4	of	of	ADP
ejpam-2803	32	5	this	this	DET
ejpam-2803	32	6	development	development	NOUN
ejpam-2803	32	7	can	can	AUX
ejpam-2803	32	8	be	be	AUX
ejpam-2803	32	9	seen	see	VERB
ejpam-2803	32	10	in	in	ADP
ejpam-2803	32	11	[	[	X
ejpam-2803	32	12	23	23	NUM
ejpam-2803	32	13	,	,	PUNCT
ejpam-2803	32	14	p.	p.	NOUN
ejpam-2803	32	15	808	808	NUM
ejpam-2803	32	16	]	]	PUNCT
ejpam-2803	32	17	.	.	PUNCT
ejpam-2803	33	1	more	more	ADJ
ejpam-2803	33	2	information	information	NOUN
ejpam-2803	33	3	concerning	concern	VERB
ejpam-2803	33	4	the	the	DET
ejpam-2803	33	5	spectrum	spectrum	NOUN
ejpam-2803	33	6	of	of	ADP
ejpam-2803	33	7	rings	ring	NOUN
ejpam-2803	33	8	,	,	PUNCT
ejpam-2803	33	9	posets	poset	NOUN
ejpam-2803	33	10	,	,	PUNCT
ejpam-2803	33	11	and	and	CCONJ
ejpam-2803	33	12	modules	module	NOUN
ejpam-2803	33	13	can	can	AUX
ejpam-2803	33	14	be	be	AUX
ejpam-2803	33	15	found	find	VERB
ejpam-2803	33	16	in	in	ADP
ejpam-2803	33	17	[	[	X
ejpam-2803	33	18	1	1	NUM
ejpam-2803	33	19	,	,	PUNCT
ejpam-2803	33	20	2	2	NUM
ejpam-2803	33	21	,	,	PUNCT
ejpam-2803	33	22	7	7	NUM
ejpam-2803	33	23	,	,	PUNCT
ejpam-2803	33	24	8	8	NUM
ejpam-2803	33	25	,	,	PUNCT
ejpam-2803	33	26	12	12	NUM
ejpam-2803	33	27	,	,	PUNCT
ejpam-2803	33	28	14	14	NUM
ejpam-2803	33	29	,	,	PUNCT
ejpam-2803	33	30	20	20	NUM
ejpam-2803	33	31	,	,	PUNCT
ejpam-2803	33	32	22	22	NUM
ejpam-2803	33	33	,	,	PUNCT
ejpam-2803	33	34	26	26	NUM
ejpam-2803	33	35	]	]	PUNCT
ejpam-2803	33	36	.	.	PUNCT
ejpam-2803	34	1	the	the	DET
ejpam-2803	34	2	second	second	ADJ
ejpam-2803	34	3	spectrum	spectrum	NOUN
ejpam-2803	34	4	of	of	ADP
ejpam-2803	34	5	m	m	PROPN
ejpam-2803	34	6	is	be	AUX
ejpam-2803	34	7	defined	define	VERB
ejpam-2803	34	8	as	as	ADP
ejpam-2803	34	9	the	the	DET
ejpam-2803	34	10	set	set	NOUN
ejpam-2803	34	11	of	of	ADP
ejpam-2803	34	12	all	all	DET
ejpam-2803	34	13	second	second	ADJ
ejpam-2803	34	14	submodules	submodule	NOUN
ejpam-2803	34	15	of	of	ADP
ejpam-2803	34	16	m	m	PRON
ejpam-2803	34	17	and	and	CCONJ
ejpam-2803	34	18	denoted	denote	VERB
ejpam-2803	34	19	by	by	ADP
ejpam-2803	34	20	specs(m	specs(m	PROPN
ejpam-2803	34	21	)	)	PUNCT
ejpam-2803	34	22	or	or	CCONJ
ejpam-2803	34	23	xs	xs	PROPN
ejpam-2803	34	24	.	.	PUNCT
ejpam-2803	35	1	we	we	PRON
ejpam-2803	35	2	call	call	VERB
ejpam-2803	35	3	the	the	DET
ejpam-2803	35	4	map	map	NOUN
ejpam-2803	35	5	ψ	ψ	X
ejpam-2803	35	6	:	:	PUNCT
ejpam-2803	35	7	xs	xs	PROPN
ejpam-2803	35	8	→	→	SYM
ejpam-2803	35	9	spec(r	spec(r	PROPN
ejpam-2803	35	10	)	)	PUNCT
ejpam-2803	35	11	given	give	VERB
ejpam-2803	35	12	by	by	ADP
ejpam-2803	35	13	s	s	PROPN
ejpam-2803	35	14	7→	7→	NUM
ejpam-2803	35	15	annr(s	annr(s	NOUN
ejpam-2803	35	16	)	)	PUNCT
ejpam-2803	35	17	as	as	ADP
ejpam-2803	35	18	the	the	DET
ejpam-2803	35	19	natural	natural	ADJ
ejpam-2803	35	20	map	map	NOUN
ejpam-2803	35	21	of	of	ADP
ejpam-2803	35	22	xs	xs	PROPN
ejpam-2803	35	23	.	.	PUNCT
ejpam-2803	36	1	let	let	VERB
ejpam-2803	36	2	n	n	PRON
ejpam-2803	36	3	be	be	AUX
ejpam-2803	36	4	a	a	DET
ejpam-2803	36	5	submodule	submodule	NOUN
ejpam-2803	36	6	of	of	ADP
ejpam-2803	36	7	m	m	PROPN
ejpam-2803	36	8	.	.	PUNCT
ejpam-2803	37	1	define	define	VERB
ejpam-2803	37	2	v	v	ADP
ejpam-2803	37	3	s(n	s(n	NOUN
ejpam-2803	37	4	)	)	PUNCT
ejpam-2803	38	1	:	:	PUNCT
ejpam-2803	38	2	=	=	PUNCT
ejpam-2803	38	3	{	{	PUNCT
ejpam-2803	38	4	s	s	NOUN
ejpam-2803	38	5	∈	∈	PROPN
ejpam-2803	38	6	specs(m	specs(m	PROPN
ejpam-2803	38	7	)	)	PUNCT
ejpam-2803	38	8	:	:	PUNCT
ejpam-2803	38	9	annr(n	annr(n	VERB
ejpam-2803	38	10	)	)	PUNCT
ejpam-2803	38	11	⊆	⊆	NUM
ejpam-2803	38	12	annr(s	annr(s	NOUN
ejpam-2803	38	13	)	)	PUNCT
ejpam-2803	38	14	}	}	PUNCT
ejpam-2803	38	15	and	and	CCONJ
ejpam-2803	38	16	set	set	VERB
ejpam-2803	38	17	ζs(m	ζs(m	PUNCT
ejpam-2803	38	18	)	)	PUNCT
ejpam-2803	38	19	:	:	PUNCT
ejpam-2803	38	20	=	=	SYM
ejpam-2803	38	21	{	{	PUNCT
ejpam-2803	38	22	v	v	NOUN
ejpam-2803	38	23	s(n	s(n	NOUN
ejpam-2803	38	24	)	)	PUNCT
ejpam-2803	38	25	:	:	PUNCT
ejpam-2803	38	26	n	n	CCONJ
ejpam-2803	38	27	≤m	≤m	NOUN
ejpam-2803	38	28	}	}	PUNCT
ejpam-2803	38	29	.	.	PUNCT
ejpam-2803	39	1	then	then	ADV
ejpam-2803	39	2	there	there	PRON
ejpam-2803	39	3	exists	exist	VERB
ejpam-2803	39	4	a	a	DET
ejpam-2803	39	5	topology	topology	NOUN
ejpam-2803	39	6	,	,	PUNCT
ejpam-2803	39	7	τ	τ	PROPN
ejpam-2803	39	8	s	s	PART
ejpam-2803	39	9	say	say	VERB
ejpam-2803	39	10	,	,	PUNCT
ejpam-2803	39	11	on	on	ADP
ejpam-2803	39	12	specs(m	specs(m	NOUN
ejpam-2803	39	13	)	)	PUNCT
ejpam-2803	39	14	having	have	VERB
ejpam-2803	39	15	ζs	ζs	ADP
ejpam-2803	39	16	as	as	ADP
ejpam-2803	39	17	the	the	DET
ejpam-2803	39	18	family	family	NOUN
ejpam-2803	39	19	of	of	ADP
ejpam-2803	39	20	all	all	DET
ejpam-2803	39	21	its	its	PRON
ejpam-2803	39	22	closed	closed	ADJ
ejpam-2803	39	23	sets	set	NOUN
ejpam-2803	39	24	.	.	PUNCT
ejpam-2803	40	1	this	this	DET
ejpam-2803	40	2	topology	topology	NOUN
ejpam-2803	40	3	is	be	AUX
ejpam-2803	40	4	called	call	VERB
ejpam-2803	40	5	the	the	DET
ejpam-2803	40	6	zariski	zariski	ADJ
ejpam-2803	40	7	topology	topology	NOUN
ejpam-2803	40	8	on	on	ADP
ejpam-2803	40	9	specs(m	specs(m	PROPN
ejpam-2803	40	10	)	)	PUNCT
ejpam-2803	40	11	(	(	PUNCT
ejpam-2803	40	12	see	see	VERB
ejpam-2803	40	13	[	[	X
ejpam-2803	40	14	7	7	NUM
ejpam-2803	40	15	]	]	NUM
ejpam-2803	40	16	)	)	PUNCT
ejpam-2803	40	17	.	.	PUNCT
ejpam-2803	41	1	for	for	ADP
ejpam-2803	41	2	any	any	DET
ejpam-2803	41	3	submodule	submodule	NOUN
ejpam-2803	41	4	n	n	PROPN
ejpam-2803	41	5	of	of	ADP
ejpam-2803	41	6	m	m	PRON
ejpam-2803	41	7	,	,	PUNCT
ejpam-2803	41	8	define	define	VERB
ejpam-2803	41	9	v	v	ADP
ejpam-2803	41	10	s∗(n	s∗(n	NOUN
ejpam-2803	41	11	)	)	PUNCT
ejpam-2803	41	12	=	=	PRON
ejpam-2803	41	13	{	{	PUNCT
ejpam-2803	41	14	s	s	NOUN
ejpam-2803	41	15	∈	∈	PROPN
ejpam-2803	41	16	specs(m	specs(m	PROPN
ejpam-2803	41	17	)	)	PUNCT
ejpam-2803	41	18	:	:	PUNCT
ejpam-2803	41	19	s	s	VERB
ejpam-2803	41	20	⊆	⊆	NUM
ejpam-2803	41	21	n	n	CCONJ
ejpam-2803	41	22	}	}	PUNCT
ejpam-2803	41	23	.	.	PUNCT
ejpam-2803	42	1	set	set	VERB
ejpam-2803	42	2	ζs∗(m	ζs∗(m	PROPN
ejpam-2803	42	3	)	)	PUNCT
ejpam-2803	43	1	=	=	PRON
ejpam-2803	43	2	{	{	PUNCT
ejpam-2803	43	3	v	v	NUM
ejpam-2803	43	4	s∗(n	s∗(n	PROPN
ejpam-2803	43	5	)	)	PUNCT
ejpam-2803	43	6	:	:	PUNCT
ejpam-2803	43	7	n	n	DET
ejpam-2803	43	8	⊆m	⊆m	NOUN
ejpam-2803	43	9	}	}	PUNCT
ejpam-2803	43	10	.	.	PUNCT
ejpam-2803	44	1	then	then	ADV
ejpam-2803	44	2	ζs∗(m	ζs∗(m	PROPN
ejpam-2803	44	3	)	)	PUNCT
ejpam-2803	44	4	contains	contain	VERB
ejpam-2803	44	5	the	the	DET
ejpam-2803	44	6	empty	empty	ADJ
ejpam-2803	44	7	set	set	NOUN
ejpam-2803	44	8	and	and	CCONJ
ejpam-2803	44	9	specs(m	specs(m	NOUN
ejpam-2803	44	10	)	)	PUNCT
ejpam-2803	44	11	,	,	PUNCT
ejpam-2803	44	12	and	and	CCONJ
ejpam-2803	44	13	it	it	PRON
ejpam-2803	44	14	is	be	AUX
ejpam-2803	44	15	closed	close	VERB
ejpam-2803	44	16	under	under	ADP
ejpam-2803	44	17	arbitrary	arbitrary	ADJ
ejpam-2803	44	18	intersections	intersection	NOUN
ejpam-2803	44	19	.	.	PUNCT
ejpam-2803	45	1	in	in	ADP
ejpam-2803	45	2	general	general	ADJ
ejpam-2803	45	3	ζs∗(m	ζs∗(m	PROPN
ejpam-2803	45	4	)	)	PUNCT
ejpam-2803	45	5	is	be	AUX
ejpam-2803	45	6	not	not	PART
ejpam-2803	45	7	closed	close	VERB
ejpam-2803	45	8	under	under	ADP
ejpam-2803	45	9	finite	finite	ADJ
ejpam-2803	45	10	unions	union	NOUN
ejpam-2803	45	11	.	.	PUNCT
ejpam-2803	46	1	a	a	DET
ejpam-2803	46	2	module	module	NOUN
ejpam-2803	46	3	m	m	VERB
ejpam-2803	46	4	is	be	AUX
ejpam-2803	46	5	called	call	VERB
ejpam-2803	46	6	a	a	DET
ejpam-2803	46	7	cotop	cotop	NOUN
ejpam-2803	46	8	module	module	NOUN
ejpam-2803	46	9	if	if	SCONJ
ejpam-2803	46	10	ζs∗(m	ζs∗(m	PROPN
ejpam-2803	46	11	)	)	PUNCT
ejpam-2803	46	12	is	be	AUX
ejpam-2803	46	13	closed	close	VERB
ejpam-2803	46	14	under	under	ADP
ejpam-2803	46	15	finite	finite	ADJ
ejpam-2803	46	16	unions	union	NOUN
ejpam-2803	46	17	.	.	PUNCT
ejpam-2803	47	1	in	in	ADP
ejpam-2803	47	2	this	this	DET
ejpam-2803	47	3	case	case	NOUN
ejpam-2803	47	4	,	,	PUNCT
ejpam-2803	47	5	ζs∗(m	ζs∗(m	PROPN
ejpam-2803	47	6	)	)	PUNCT
ejpam-2803	47	7	is	be	AUX
ejpam-2803	47	8	called	call	VERB
ejpam-2803	47	9	the	the	DET
ejpam-2803	47	10	quasi	quasi	ADJ
ejpam-2803	47	11	zariski	zariski	NOUN
ejpam-2803	47	12	topology	topology	NOUN
ejpam-2803	47	13	(	(	PUNCT
ejpam-2803	47	14	see	see	VERB
ejpam-2803	47	15	[	[	X
ejpam-2803	47	16	7	7	NUM
ejpam-2803	47	17	]	]	NUM
ejpam-2803	47	18	)	)	PUNCT
ejpam-2803	47	19	.	.	PUNCT
ejpam-2803	48	1	now	now	ADV
ejpam-2803	48	2	for	for	ADP
ejpam-2803	48	3	a	a	DET
ejpam-2803	48	4	submodule	submodule	NOUN
ejpam-2803	48	5	n	n	PROPN
ejpam-2803	48	6	of	of	ADP
ejpam-2803	48	7	m	m	PRON
ejpam-2803	48	8	,	,	PUNCT
ejpam-2803	48	9	define	define	VERB
ejpam-2803	48	10	w	w	ADP
ejpam-2803	48	11	s(n	s(n	PROPN
ejpam-2803	48	12	)	)	PUNCT
ejpam-2803	48	13	=	=	PUNCT
ejpam-2803	48	14	specs(m)−v	specs(m)−v	NOUN
ejpam-2803	48	15	s∗(n	s∗(n	PROPN
ejpam-2803	48	16	)	)	PUNCT
ejpam-2803	48	17	and	and	CCONJ
ejpam-2803	48	18	set	set	VERB
ejpam-2803	48	19	ωs(m	ωs(m	NUM
ejpam-2803	48	20	)	)	PUNCT
ejpam-2803	48	21	=	=	PRON
ejpam-2803	48	22	{	{	PUNCT
ejpam-2803	48	23	w	w	PROPN
ejpam-2803	48	24	s(n	s(n	PROPN
ejpam-2803	48	25	)	)	PUNCT
ejpam-2803	48	26	:	:	PUNCT
ejpam-2803	49	1	n	n	CCONJ
ejpam-2803	49	2	≤m	≤m	NOUN
ejpam-2803	49	3	}	}	PUNCT
ejpam-2803	49	4	.	.	PUNCT
ejpam-2803	50	1	let	let	AUX
ejpam-2803	50	2	ηs(m	ηs(m	PUNCT
ejpam-2803	50	3	)	)	PUNCT
ejpam-2803	50	4	be	be	AUX
ejpam-2803	50	5	the	the	DET
ejpam-2803	50	6	topology	topology	NOUN
ejpam-2803	50	7	on	on	ADP
ejpam-2803	50	8	specs(m	specs(m	PROPN
ejpam-2803	50	9	)	)	PUNCT
ejpam-2803	50	10	by	by	ADP
ejpam-2803	50	11	the	the	DET
ejpam-2803	50	12	sub	sub	NOUN
ejpam-2803	50	13	-	-	NOUN
ejpam-2803	50	14	basis	basis	NOUN
ejpam-2803	50	15	ωs(m	ωs(m	NUM
ejpam-2803	50	16	)	)	PUNCT
ejpam-2803	50	17	.	.	PUNCT
ejpam-2803	51	1	in	in	ADP
ejpam-2803	51	2	fact	fact	NOUN
ejpam-2803	51	3	ηs(m	ηs(m	PUNCT
ejpam-2803	51	4	)	)	PUNCT
ejpam-2803	51	5	is	be	AUX
ejpam-2803	51	6	the	the	DET
ejpam-2803	51	7	collection	collection	NOUN
ejpam-2803	51	8	u	u	NOUN
ejpam-2803	51	9	of	of	ADP
ejpam-2803	51	10	all	all	DET
ejpam-2803	51	11	unions	union	NOUN
ejpam-2803	51	12	of	of	ADP
ejpam-2803	51	13	finite	finite	ADJ
ejpam-2803	51	14	intersections	intersection	NOUN
ejpam-2803	51	15	of	of	ADP
ejpam-2803	51	16	elements	element	NOUN
ejpam-2803	51	17	of	of	ADP
ejpam-2803	51	18	ωs(m	ωs(m	NUM
ejpam-2803	51	19	)	)	PUNCT
ejpam-2803	51	20	.	.	PUNCT
ejpam-2803	52	1	we	we	PRON
ejpam-2803	52	2	call	call	VERB
ejpam-2803	52	3	this	this	DET
ejpam-2803	52	4	topology	topology	NOUN
ejpam-2803	52	5	the	the	DET
ejpam-2803	52	6	classical	classical	ADJ
ejpam-2803	52	7	zariski	zariski	NOUN
ejpam-2803	52	8	topology	topology	NOUN
ejpam-2803	52	9	on	on	ADP
ejpam-2803	52	10	specs(m	specs(m	PROPN
ejpam-2803	52	11	)	)	PUNCT
ejpam-2803	52	12	(	(	PUNCT
ejpam-2803	52	13	or	or	CCONJ
ejpam-2803	52	14	second	second	ADJ
ejpam-2803	52	15	classical	classical	ADJ
ejpam-2803	52	16	zariski	zariski	NOUN
ejpam-2803	52	17	topology	topology	NOUN
ejpam-2803	52	18	)	)	PUNCT
ejpam-2803	52	19	(	(	PUNCT
ejpam-2803	52	20	see	see	VERB
ejpam-2803	52	21	[	[	X
ejpam-2803	52	22	8	8	NUM
ejpam-2803	52	23	]	]	NUM
ejpam-2803	52	24	)	)	PUNCT
ejpam-2803	52	25	.	.	PUNCT
ejpam-2803	53	1	it	it	PRON
ejpam-2803	53	2	is	be	AUX
ejpam-2803	53	3	clear	clear	ADJ
ejpam-2803	53	4	that	that	SCONJ
ejpam-2803	53	5	if	if	SCONJ
ejpam-2803	53	6	m	m	NOUN
ejpam-2803	53	7	is	be	AUX
ejpam-2803	53	8	a	a	DET
ejpam-2803	53	9	cotop	cotop	NOUN
ejpam-2803	53	10	module	module	NOUN
ejpam-2803	53	11	,	,	PUNCT
ejpam-2803	53	12	then	then	ADV
ejpam-2803	53	13	its	its	PRON
ejpam-2803	53	14	related	related	ADJ
ejpam-2803	53	15	topology	topology	NOUN
ejpam-2803	53	16	,	,	PUNCT
ejpam-2803	53	17	as	as	SCONJ
ejpam-2803	53	18	it	it	PRON
ejpam-2803	53	19	was	be	AUX
ejpam-2803	53	20	mentioned	mention	VERB
ejpam-2803	53	21	in	in	ADP
ejpam-2803	53	22	the	the	DET
ejpam-2803	53	23	above	above	ADJ
ejpam-2803	53	24	paragraph	paragraph	NOUN
ejpam-2803	53	25	,	,	PUNCT
ejpam-2803	53	26	coincide	coincide	VERB
ejpam-2803	53	27	with	with	ADP
ejpam-2803	53	28	the	the	DET
ejpam-2803	53	29	second	second	ADJ
ejpam-2803	53	30	classical	classical	ADJ
ejpam-2803	53	31	zariski	zariski	NOUN
ejpam-2803	53	32	topology	topology	NOUN
ejpam-2803	53	33	.	.	PUNCT
ejpam-2803	54	1	the	the	DET
ejpam-2803	54	2	dual	dual	ADJ
ejpam-2803	54	3	notion	notion	NOUN
ejpam-2803	54	4	of	of	ADP
ejpam-2803	54	5	the	the	DET
ejpam-2803	54	6	prime	prime	ADJ
ejpam-2803	54	7	radical	radical	NOUN
ejpam-2803	54	8	of	of	ADP
ejpam-2803	54	9	a	a	DET
ejpam-2803	54	10	submodule	submodule	NOUN
ejpam-2803	54	11	(	(	PUNCT
ejpam-2803	54	12	i.e	i.e	PROPN
ejpam-2803	54	13	,	,	PUNCT
ejpam-2803	54	14	second	second	ADJ
ejpam-2803	54	15	socle	socle	NOUN
ejpam-2803	54	16	or	or	CCONJ
ejpam-2803	54	17	second	second	ADJ
ejpam-2803	54	18	radical	radical	ADJ
ejpam-2803	54	19	)	)	PUNCT
ejpam-2803	54	20	have	have	AUX
ejpam-2803	54	21	been	be	AUX
ejpam-2803	54	22	introduced	introduce	VERB
ejpam-2803	54	23	by	by	ADP
ejpam-2803	54	24	h.	h.	PROPN
ejpam-2803	54	25	ansari	ansari	PROPN
ejpam-2803	54	26	-	-	PUNCT
ejpam-2803	54	27	toroghy	toroghy	ADJ
ejpam-2803	54	28	and	and	CCONJ
ejpam-2803	54	29	f.	f.	PROPN
ejpam-2803	54	30	farshadifar	farshadifar	ADV
ejpam-2803	54	31	in	in	ADP
ejpam-2803	54	32	[	[	X
ejpam-2803	54	33	4	4	NUM
ejpam-2803	54	34	]	]	PUNCT
ejpam-2803	54	35	.	.	PUNCT
ejpam-2803	55	1	for	for	ADP
ejpam-2803	55	2	a	a	DET
ejpam-2803	55	3	submodule	submodule	NOUN
ejpam-2803	55	4	n	n	PROPN
ejpam-2803	55	5	of	of	ADP
ejpam-2803	55	6	m	m	PRON
ejpam-2803	55	7	,	,	PUNCT
ejpam-2803	55	8	the	the	DET
ejpam-2803	55	9	second	second	ADJ
ejpam-2803	55	10	radical	radical	NOUN
ejpam-2803	55	11	of	of	ADP
ejpam-2803	55	12	n	n	NUM
ejpam-2803	55	13	is	be	AUX
ejpam-2803	55	14	defined	define	VERB
ejpam-2803	55	15	as	as	ADP
ejpam-2803	55	16	the	the	DET
ejpam-2803	55	17	sum	sum	NOUN
ejpam-2803	55	18	of	of	ADP
ejpam-2803	55	19	all	all	DET
ejpam-2803	55	20	second	second	ADJ
ejpam-2803	55	21	submodules	submodule	NOUN
ejpam-2803	55	22	of	of	ADP
ejpam-2803	55	23	m	m	PRON
ejpam-2803	55	24	contained	contain	VERB
ejpam-2803	55	25	in	in	ADP
ejpam-2803	55	26	n	n	PRON
ejpam-2803	55	27	and	and	CCONJ
ejpam-2803	55	28	denoted	denote	VERB
ejpam-2803	55	29	by	by	ADP
ejpam-2803	55	30	sec(n	sec(n	PROPN
ejpam-2803	55	31	)	)	PUNCT
ejpam-2803	55	32	(	(	PUNCT
ejpam-2803	55	33	or	or	CCONJ
ejpam-2803	55	34	soc(n	soc(n	NOUN
ejpam-2803	55	35	)	)	PUNCT
ejpam-2803	55	36	)	)	PUNCT
ejpam-2803	55	37	.	.	PUNCT
ejpam-2803	56	1	in	in	ADP
ejpam-2803	56	2	case	case	NOUN
ejpam-2803	56	3	n	n	PRON
ejpam-2803	56	4	does	do	AUX
ejpam-2803	56	5	not	not	PART
ejpam-2803	56	6	contain	contain	VERB
ejpam-2803	56	7	any	any	DET
ejpam-2803	56	8	second	second	ADJ
ejpam-2803	56	9	submodule	submodule	NOUN
ejpam-2803	56	10	,	,	PUNCT
ejpam-2803	56	11	the	the	DET
ejpam-2803	56	12	second	second	ADJ
ejpam-2803	56	13	radical	radical	NOUN
ejpam-2803	56	14	of	of	ADP
ejpam-2803	56	15	n	n	NUM
ejpam-2803	56	16	is	be	AUX
ejpam-2803	56	17	defined	define	VERB
ejpam-2803	56	18	to	to	PART
ejpam-2803	56	19	be	be	AUX
ejpam-2803	56	20	(	(	PUNCT
ejpam-2803	56	21	0	0	NUM
ejpam-2803	56	22	)	)	PUNCT
ejpam-2803	56	23	.	.	PUNCT
ejpam-2803	57	1	also	also	ADV
ejpam-2803	57	2	,	,	PUNCT
ejpam-2803	57	3	n	n	PROPN
ejpam-2803	57	4	6=	6=	NUM
ejpam-2803	57	5	(	(	PUNCT
ejpam-2803	57	6	0	0	NUM
ejpam-2803	57	7	)	)	PUNCT
ejpam-2803	57	8	is	be	AUX
ejpam-2803	57	9	said	say	VERB
ejpam-2803	57	10	to	to	PART
ejpam-2803	57	11	be	be	AUX
ejpam-2803	57	12	a	a	DET
ejpam-2803	57	13	socle	socle	NOUN
ejpam-2803	57	14	submodule	submodule	NOUN
ejpam-2803	57	15	of	of	ADP
ejpam-2803	57	16	m	m	PROPN
ejpam-2803	57	17	if	if	SCONJ
ejpam-2803	57	18	sec(n	sec(n	PROPN
ejpam-2803	57	19	)	)	PUNCT
ejpam-2803	57	20	=	=	SYM
ejpam-2803	58	1	n	n	PROPN
ejpam-2803	58	2	.	.	PUNCT
ejpam-2803	59	1	one	one	PRON
ejpam-2803	59	2	can	can	AUX
ejpam-2803	59	3	see	see	VERB
ejpam-2803	59	4	that	that	SCONJ
ejpam-2803	59	5	m	m	NOUN
ejpam-2803	59	6	is	be	AUX
ejpam-2803	59	7	cotop	cotop	ADJ
ejpam-2803	59	8	if	if	SCONJ
ejpam-2803	59	9	and	and	CCONJ
ejpam-2803	59	10	only	only	ADV
ejpam-2803	59	11	if	if	SCONJ
ejpam-2803	59	12	for	for	ADP
ejpam-2803	59	13	every	every	DET
ejpam-2803	59	14	two	two	NUM
ejpam-2803	59	15	socle	socle	NOUN
ejpam-2803	59	16	submodules	submodule	NOUN
ejpam-2803	59	17	n	n	PROPN
ejpam-2803	59	18	and	and	CCONJ
ejpam-2803	59	19	k	k	PROPN
ejpam-2803	59	20	of	of	ADP
ejpam-2803	59	21	m	m	PROPN
ejpam-2803	59	22	and	and	CCONJ
ejpam-2803	59	23	every	every	DET
ejpam-2803	59	24	second	second	ADJ
ejpam-2803	59	25	submodule	submodule	NOUN
ejpam-2803	59	26	s	s	PROPN
ejpam-2803	59	27	of	of	ADP
ejpam-2803	59	28	m	m	PROPN
ejpam-2803	59	29	with	with	ADP
ejpam-2803	59	30	s	s	PRON
ejpam-2803	59	31	⊆	⊆	NUM
ejpam-2803	59	32	n	n	NOUN
ejpam-2803	59	33	+	+	NOUN
ejpam-2803	59	34	k	k	NOUN
ejpam-2803	59	35	,	,	PUNCT
ejpam-2803	59	36	we	we	PRON
ejpam-2803	59	37	have	have	VERB
ejpam-2803	59	38	s	s	VERB
ejpam-2803	59	39	⊆	⊆	NUM
ejpam-2803	59	40	n	n	NOUN
ejpam-2803	59	41	or	or	CCONJ
ejpam-2803	59	42	s	s	PRON
ejpam-2803	59	43	⊆	⊆	NUM
ejpam-2803	59	44	k.	k.	NOUN
ejpam-2803	59	45	in	in	ADP
ejpam-2803	59	46	this	this	DET
ejpam-2803	59	47	article	article	NOUN
ejpam-2803	59	48	,	,	PUNCT
ejpam-2803	59	49	we	we	PRON
ejpam-2803	59	50	introduce	introduce	VERB
ejpam-2803	59	51	the	the	DET
ejpam-2803	59	52	concept	concept	NOUN
ejpam-2803	59	53	of	of	ADP
ejpam-2803	59	54	xs	xs	NOUN
ejpam-2803	59	55	-	-	PUNCT
ejpam-2803	59	56	injective	injective	ADJ
ejpam-2803	59	57	modules	module	NOUN
ejpam-2803	59	58	and	and	CCONJ
ejpam-2803	59	59	investigate	investigate	VERB
ejpam-2803	59	60	some	some	PRON
ejpam-2803	59	61	of	of	ADP
ejpam-2803	59	62	their	their	PRON
ejpam-2803	59	63	basic	basic	ADJ
ejpam-2803	59	64	properties	property	NOUN
ejpam-2803	59	65	.	.	PUNCT
ejpam-2803	60	1	we	we	PRON
ejpam-2803	60	2	say	say	VERB
ejpam-2803	60	3	that	that	SCONJ
ejpam-2803	60	4	m	m	PROPN
ejpam-2803	60	5	is	be	AUX
ejpam-2803	60	6	xs	xs	NOUN
ejpam-2803	60	7	-	-	PUNCT
ejpam-2803	60	8	injective	injective	ADJ
ejpam-2803	60	9	if	if	SCONJ
ejpam-2803	60	10	the	the	DET
ejpam-2803	60	11	natural	natural	ADJ
ejpam-2803	60	12	map	map	NOUN
ejpam-2803	60	13	of	of	ADP
ejpam-2803	60	14	xs	xs	PROPN
ejpam-2803	60	15	is	be	AUX
ejpam-2803	60	16	injective	injective	ADJ
ejpam-2803	60	17	(	(	PUNCT
ejpam-2803	60	18	see	see	VERB
ejpam-2803	60	19	definition	definition	NOUN
ejpam-2803	60	20	3.9	3.9	NUM
ejpam-2803	60	21	)	)	PUNCT
ejpam-2803	60	22	.	.	PUNCT
ejpam-2803	61	1	the	the	DET
ejpam-2803	61	2	primeful	primeful	ADJ
ejpam-2803	61	3	r	r	NOUN
ejpam-2803	61	4	-	-	PUNCT
ejpam-2803	61	5	modules	module	NOUN
ejpam-2803	61	6	was	be	AUX
ejpam-2803	61	7	introduced	introduce	VERB
ejpam-2803	61	8	and	and	CCONJ
ejpam-2803	61	9	studied	study	VERB
ejpam-2803	61	10	by	by	ADP
ejpam-2803	61	11	c.p	c.p	PROPN
ejpam-2803	61	12	.	.	PROPN
ejpam-2803	61	13	lu	lu	PROPN
ejpam-2803	61	14	in	in	ADP
ejpam-2803	61	15	several	several	ADJ
ejpam-2803	61	16	papers	paper	NOUN
ejpam-2803	61	17	.	.	PUNCT
ejpam-2803	62	1	the	the	DET
ejpam-2803	62	2	dual	dual	ADJ
ejpam-2803	62	3	of	of	ADP
ejpam-2803	62	4	this	this	DET
ejpam-2803	62	5	notion	notion	NOUN
ejpam-2803	62	6	(	(	PUNCT
ejpam-2803	62	7	i.e.	i.e.	X
ejpam-2803	62	8	,	,	PUNCT
ejpam-2803	62	9	secondful	secondful	ADJ
ejpam-2803	62	10	modules	module	NOUN
ejpam-2803	62	11	)	)	PUNCT
ejpam-2803	62	12	was	be	AUX
ejpam-2803	62	13	studied	study	VERB
ejpam-2803	62	14	by	by	ADP
ejpam-2803	62	15	h.	h.	PROPN
ejpam-2803	62	16	ansari	ansari	PROPN
ejpam-2803	62	17	-	-	PUNCT
ejpam-2803	62	18	toroghy	toroghy	ADJ
ejpam-2803	62	19	and	and	CCONJ
ejpam-2803	62	20	f.	f.	PROPN
ejpam-2803	62	21	farshadifar	farshadifar	ADV
ejpam-2803	62	22	in	in	ADP
ejpam-2803	62	23	[	[	X
ejpam-2803	62	24	3	3	NUM
ejpam-2803	62	25	]	]	PUNCT
ejpam-2803	62	26	and	and	CCONJ
ejpam-2803	62	27	[	[	X
ejpam-2803	62	28	17	17	NUM
ejpam-2803	62	29	]	]	PUNCT
ejpam-2803	62	30	.	.	PUNCT
ejpam-2803	63	1	an	an	DET
ejpam-2803	63	2	r	r	NOUN
ejpam-2803	63	3	-	-	PUNCT
ejpam-2803	63	4	module	module	NOUN
ejpam-2803	63	5	m	m	NOUN
ejpam-2803	63	6	is	be	AUX
ejpam-2803	63	7	secondful	secondful	ADJ
ejpam-2803	63	8	if	if	SCONJ
ejpam-2803	63	9	the	the	DET
ejpam-2803	63	10	natural	natural	ADJ
ejpam-2803	63	11	map	map	NOUN
ejpam-2803	63	12	of	of	ADP
ejpam-2803	63	13	xs	xs	PROPN
ejpam-2803	63	14	is	be	AUX
ejpam-2803	63	15	surjective	surjective	ADJ
ejpam-2803	63	16	.	.	PUNCT
ejpam-2803	64	1	in	in	ADP
ejpam-2803	64	2	section	section	NOUN
ejpam-2803	64	3	two	two	NUM
ejpam-2803	64	4	,	,	PUNCT
ejpam-2803	64	5	we	we	PRON
ejpam-2803	64	6	explore	explore	VERB
ejpam-2803	64	7	more	more	ADJ
ejpam-2803	64	8	properties	property	NOUN
ejpam-2803	64	9	of	of	ADP
ejpam-2803	64	10	this	this	DET
ejpam-2803	64	11	class	class	NOUN
ejpam-2803	64	12	of	of	ADP
ejpam-2803	64	13	modules	module	NOUN
ejpam-2803	64	14	.	.	PUNCT
ejpam-2803	65	1	in	in	ADP
ejpam-2803	65	2	theorem	theorem	NOUN
ejpam-2803	65	3	2.3	2.3	NUM
ejpam-2803	65	4	,	,	PUNCT
ejpam-2803	65	5	we	we	PRON
ejpam-2803	65	6	provide	provide	VERB
ejpam-2803	65	7	a	a	DET
ejpam-2803	65	8	useful	useful	ADJ
ejpam-2803	65	9	characterization	characterization	NOUN
ejpam-2803	65	10	for	for	ADP
ejpam-2803	65	11	artinian	artinian	ADJ
ejpam-2803	65	12	secondful	secondful	ADJ
ejpam-2803	65	13	modules	module	NOUN
ejpam-2803	65	14	and	and	CCONJ
ejpam-2803	65	15	by	by	ADP
ejpam-2803	65	16	using	use	VERB
ejpam-2803	65	17	this	this	PRON
ejpam-2803	65	18	,	,	PUNCT
ejpam-2803	65	19	we	we	PRON
ejpam-2803	65	20	prove	prove	VERB
ejpam-2803	65	21	that	that	SCONJ
ejpam-2803	65	22	if	if	SCONJ
ejpam-2803	65	23	m	m	NOUN
ejpam-2803	65	24	is	be	AUX
ejpam-2803	65	25	an	an	DET
ejpam-2803	65	26	artinian	artinian	ADJ
ejpam-2803	65	27	secondful	secondful	ADJ
ejpam-2803	65	28	module	module	NOUN
ejpam-2803	65	29	with	with	ADP
ejpam-2803	65	30	noetherian	noetherian	ADJ
ejpam-2803	65	31	spectrum	spectrum	NOUN
ejpam-2803	65	32	,	,	PUNCT
ejpam-2803	65	33	then	then	ADV
ejpam-2803	65	34	homr(rp	homr(rp	NOUN
ejpam-2803	65	35	,	,	PUNCT
ejpam-2803	65	36	m	m	NOUN
ejpam-2803	65	37	)	)	PUNCT
ejpam-2803	65	38	has	have	VERB
ejpam-2803	65	39	also	also	ADV
ejpam-2803	65	40	a	a	DET
ejpam-2803	65	41	noetherian	noetherian	ADJ
ejpam-2803	65	42	second	second	ADJ
ejpam-2803	65	43	spectrum	spectrum	NOUN
ejpam-2803	65	44	with	with	ADP
ejpam-2803	65	45	zariski	zariski	NOUN
ejpam-2803	65	46	topology	topology	NOUN
ejpam-2803	65	47	.	.	PUNCT
ejpam-2803	66	1	moreover	moreover	ADV
ejpam-2803	66	2	,	,	PUNCT
ejpam-2803	66	3	in	in	ADP
ejpam-2803	66	4	theorem	theorem	NOUN
ejpam-2803	66	5	2.9	2.9	NUM
ejpam-2803	66	6	we	we	PRON
ejpam-2803	66	7	provide	provide	VERB
ejpam-2803	66	8	a	a	DET
ejpam-2803	66	9	useful	useful	ADJ
ejpam-2803	66	10	characterization	characterization	NOUN
ejpam-2803	66	11	for	for	ADP
ejpam-2803	66	12	secondful	secondful	ADJ
ejpam-2803	66	13	modules	module	NOUN
ejpam-2803	66	14	.	.	PUNCT
ejpam-2803	67	1	in	in	ADP
ejpam-2803	67	2	section	section	NOUN
ejpam-2803	67	3	three	three	NUM
ejpam-2803	67	4	,	,	PUNCT
ejpam-2803	67	5	among	among	ADP
ejpam-2803	67	6	other	other	ADJ
ejpam-2803	67	7	results	result	NOUN
ejpam-2803	67	8	,	,	PUNCT
ejpam-2803	67	9	we	we	PRON
ejpam-2803	67	10	show	show	VERB
ejpam-2803	67	11	that	that	SCONJ
ejpam-2803	67	12	if	if	SCONJ
ejpam-2803	67	13	every	every	DET
ejpam-2803	67	14	closed	closed	ADJ
ejpam-2803	67	15	subset	subset	NOUN
ejpam-2803	67	16	of	of	ADP
ejpam-2803	67	17	(	(	PUNCT
ejpam-2803	67	18	specs(m	specs(m	PROPN
ejpam-2803	67	19	)	)	PUNCT
ejpam-2803	67	20	,	,	PUNCT
ejpam-2803	67	21	τ	τ	PROPN
ejpam-2803	67	22	s∗	s∗	PROPN
ejpam-2803	67	23	)	)	PUNCT
ejpam-2803	67	24	has	have	VERB
ejpam-2803	67	25	a	a	DET
ejpam-2803	67	26	finite	finite	ADJ
ejpam-2803	67	27	number	number	NOUN
ejpam-2803	67	28	of	of	ADP
ejpam-2803	67	29	irreducible	irreducible	ADJ
ejpam-2803	67	30	components	component	NOUN
ejpam-2803	67	31	,	,	PUNCT
ejpam-2803	67	32	then	then	ADV
ejpam-2803	67	33	every	every	DET
ejpam-2803	67	34	submodule	submodule	NOUN
ejpam-2803	67	35	of	of	ADP
ejpam-2803	67	36	m	m	PROPN
ejpam-2803	67	37	has	have	VERB
ejpam-2803	67	38	a	a	DET
ejpam-2803	67	39	finite	finite	ADJ
ejpam-2803	67	40	h.	h.	PROPN
ejpam-2803	67	41	ansari	ansari	PROPN
ejpam-2803	67	42	-	-	PUNCT
ejpam-2803	67	43	toroghy	toroghy	NOUN
ejpam-2803	67	44	,	,	PUNCT
ejpam-2803	67	45	s.	s.	PROPN
ejpam-2803	67	46	s.	s.	PROPN
ejpam-2803	67	47	pourmortazavi	pourmortazavi	VERB
ejpam-2803	67	48	/	/	SYM
ejpam-2803	67	49	eur	eur	PROPN
ejpam-2803	67	50	.	.	PUNCT
ejpam-2803	68	1	j.	j.	PROPN
ejpam-2803	68	2	pure	pure	PROPN
ejpam-2803	68	3	appl	appl	PROPN
ejpam-2803	68	4	.	.	PROPN
ejpam-2803	68	5	math	math	PROPN
ejpam-2803	68	6	,	,	PUNCT
ejpam-2803	68	7	10	10	NUM
ejpam-2803	68	8	(	(	PUNCT
ejpam-2803	68	9	2	2	NUM
ejpam-2803	68	10	)	)	PUNCT
ejpam-2803	68	11	(	(	PUNCT
ejpam-2803	68	12	2017	2017	NUM
ejpam-2803	68	13	)	)	PUNCT
ejpam-2803	68	14	,	,	PUNCT
ejpam-2803	68	15	211	211	NUM
ejpam-2803	68	16	-	-	SYM
ejpam-2803	68	17	230	230	NUM
ejpam-2803	68	18	213	213	NUM
ejpam-2803	68	19	number	number	NOUN
ejpam-2803	68	20	of	of	ADP
ejpam-2803	68	21	maximal	maximal	ADJ
ejpam-2803	68	22	second	second	ADJ
ejpam-2803	68	23	submodules	submodule	NOUN
ejpam-2803	68	24	.	.	PUNCT
ejpam-2803	69	1	we	we	PRON
ejpam-2803	69	2	provide	provide	VERB
ejpam-2803	69	3	an	an	DET
ejpam-2803	69	4	example	example	NOUN
ejpam-2803	69	5	which	which	PRON
ejpam-2803	69	6	shows	show	VERB
ejpam-2803	69	7	the	the	DET
ejpam-2803	69	8	converse	converse	NOUN
ejpam-2803	69	9	is	be	AUX
ejpam-2803	69	10	not	not	PART
ejpam-2803	69	11	true	true	ADJ
ejpam-2803	69	12	in	in	ADP
ejpam-2803	69	13	general	general	ADJ
ejpam-2803	69	14	.	.	PUNCT
ejpam-2803	70	1	in	in	ADP
ejpam-2803	70	2	[	[	X
ejpam-2803	70	3	8	8	NUM
ejpam-2803	70	4	,	,	PUNCT
ejpam-2803	70	5	proposition	proposition	NOUN
ejpam-2803	70	6	3.13	3.13	NUM
ejpam-2803	70	7	]	]	PUNCT
ejpam-2803	70	8	,	,	PUNCT
ejpam-2803	70	9	it	it	PRON
ejpam-2803	70	10	is	be	AUX
ejpam-2803	70	11	proved	prove	VERB
ejpam-2803	70	12	that	that	SCONJ
ejpam-2803	70	13	if	if	SCONJ
ejpam-2803	70	14	m	m	PROPN
ejpam-2803	70	15	has	have	AUX
ejpam-2803	70	16	dcc	dcc	PROPN
ejpam-2803	70	17	condition	condition	NOUN
ejpam-2803	70	18	on	on	ADP
ejpam-2803	70	19	its	its	PRON
ejpam-2803	70	20	socle	socle	NOUN
ejpam-2803	70	21	submodules	submodule	NOUN
ejpam-2803	70	22	,	,	PUNCT
ejpam-2803	70	23	then	then	ADV
ejpam-2803	70	24	every	every	DET
ejpam-2803	70	25	irreducible	irreducible	ADJ
ejpam-2803	70	26	closed	closed	ADJ
ejpam-2803	70	27	subset	subset	NOUN
ejpam-2803	70	28	of	of	ADP
ejpam-2803	70	29	specs(m	specs(m	PROPN
ejpam-2803	70	30	)	)	PUNCT
ejpam-2803	70	31	(	(	PUNCT
ejpam-2803	70	32	with	with	ADP
ejpam-2803	70	33	second	second	ADJ
ejpam-2803	70	34	classical	classical	ADJ
ejpam-2803	70	35	zariski	zariski	NOUN
ejpam-2803	70	36	topology	topology	NOUN
ejpam-2803	70	37	)	)	PUNCT
ejpam-2803	70	38	has	have	VERB
ejpam-2803	70	39	a	a	DET
ejpam-2803	70	40	generic	generic	ADJ
ejpam-2803	70	41	point	point	NOUN
ejpam-2803	70	42	.	.	PUNCT
ejpam-2803	71	1	in	in	ADP
ejpam-2803	71	2	theorem	theorem	NOUN
ejpam-2803	71	3	3.2	3.2	NUM
ejpam-2803	71	4	,	,	PUNCT
ejpam-2803	71	5	we	we	PRON
ejpam-2803	71	6	remove	remove	VERB
ejpam-2803	71	7	this	this	DET
ejpam-2803	71	8	restriction	restriction	NOUN
ejpam-2803	71	9	and	and	CCONJ
ejpam-2803	71	10	prove	prove	VERB
ejpam-2803	71	11	that	that	SCONJ
ejpam-2803	71	12	every	every	DET
ejpam-2803	71	13	irreducible	irreducible	ADJ
ejpam-2803	71	14	closed	closed	ADJ
ejpam-2803	71	15	subset	subset	NOUN
ejpam-2803	71	16	of	of	ADP
ejpam-2803	71	17	specs(m	specs(m	PROPN
ejpam-2803	71	18	)	)	PUNCT
ejpam-2803	71	19	has	have	VERB
ejpam-2803	71	20	a	a	DET
ejpam-2803	71	21	generic	generic	ADJ
ejpam-2803	71	22	point	point	NOUN
ejpam-2803	71	23	.	.	PUNCT
ejpam-2803	72	1	further	far	ADV
ejpam-2803	72	2	it	it	PRON
ejpam-2803	72	3	is	be	AUX
ejpam-2803	72	4	shown	show	VERB
ejpam-2803	72	5	that	that	SCONJ
ejpam-2803	72	6	if	if	SCONJ
ejpam-2803	72	7	specs(m	specs(m	PROPN
ejpam-2803	72	8	)	)	PUNCT
ejpam-2803	72	9	is	be	AUX
ejpam-2803	72	10	a	a	DET
ejpam-2803	72	11	noetherian	noetherian	ADJ
ejpam-2803	72	12	space	space	NOUN
ejpam-2803	72	13	,	,	PUNCT
ejpam-2803	72	14	then	then	ADV
ejpam-2803	72	15	it	it	PRON
ejpam-2803	72	16	is	be	AUX
ejpam-2803	72	17	a	a	DET
ejpam-2803	72	18	spectral	spectral	ADJ
ejpam-2803	72	19	space	space	NOUN
ejpam-2803	72	20	(	(	PUNCT
ejpam-2803	72	21	see	see	VERB
ejpam-2803	72	22	corollary	corollary	ADJ
ejpam-2803	72	23	3.3	3.3	NUM
ejpam-2803	72	24	)	)	PUNCT
ejpam-2803	72	25	.	.	PUNCT
ejpam-2803	73	1	proposition	proposition	NOUN
ejpam-2803	73	2	3.13	3.13	NUM
ejpam-2803	73	3	states	state	VERB
ejpam-2803	73	4	that	that	SCONJ
ejpam-2803	73	5	if	if	SCONJ
ejpam-2803	73	6	(	(	PUNCT
ejpam-2803	73	7	mi)i∈i	mi)i∈i	NUM
ejpam-2803	73	8	is	be	AUX
ejpam-2803	73	9	a	a	DET
ejpam-2803	73	10	family	family	NOUN
ejpam-2803	73	11	of	of	ADP
ejpam-2803	73	12	r	r	NOUN
ejpam-2803	73	13	-	-	PUNCT
ejpam-2803	73	14	modules	module	NOUN
ejpam-2803	73	15	and	and	CCONJ
ejpam-2803	73	16	m	m	NOUN
ejpam-2803	73	17	=	=	PROPN
ejpam-2803	73	18	⊕	⊕	PROPN
ejpam-2803	73	19	i∈imi	i∈imi	PROPN
ejpam-2803	73	20	is	be	AUX
ejpam-2803	73	21	an	an	DET
ejpam-2803	73	22	xs	xs	NOUN
ejpam-2803	73	23	-	-	PUNCT
ejpam-2803	73	24	injective	injective	ADJ
ejpam-2803	73	25	module	module	NOUN
ejpam-2803	73	26	,	,	PUNCT
ejpam-2803	73	27	then	then	ADV
ejpam-2803	73	28	specs(m	specs(m	NOUN
ejpam-2803	73	29	)	)	PUNCT
ejpam-2803	73	30	can	can	AUX
ejpam-2803	73	31	be	be	AUX
ejpam-2803	73	32	specified	specify	VERB
ejpam-2803	73	33	in	in	ADP
ejpam-2803	73	34	terms	term	NOUN
ejpam-2803	73	35	of	of	ADP
ejpam-2803	73	36	second	second	ADJ
ejpam-2803	73	37	submodules	submodule	NOUN
ejpam-2803	73	38	of	of	ADP
ejpam-2803	73	39	mi	mi	PROPN
ejpam-2803	73	40	.	.	PROPN
ejpam-2803	73	41	theorem	theorem	PROPN
ejpam-2803	73	42	3.15	3.15	NUM
ejpam-2803	73	43	says	say	VERB
ejpam-2803	73	44	that	that	SCONJ
ejpam-2803	73	45	m	m	PROPN
ejpam-2803	73	46	=	=	SYM
ejpam-2803	73	47	⊕	⊕	PROPN
ejpam-2803	73	48	i∈imi	i∈imi	PROPN
ejpam-2803	73	49	is	be	AUX
ejpam-2803	73	50	an	an	DET
ejpam-2803	73	51	xs	xs	NOUN
ejpam-2803	73	52	-	-	PUNCT
ejpam-2803	73	53	injective	injective	ADJ
ejpam-2803	73	54	(	(	PUNCT
ejpam-2803	73	55	resp	resp	NOUN
ejpam-2803	73	56	.	.	PUNCT
ejpam-2803	74	1	a	a	DET
ejpam-2803	74	2	cotop	cotop	NOUN
ejpam-2803	74	3	)	)	PUNCT
ejpam-2803	74	4	r	r	NOUN
ejpam-2803	74	5	-	-	PUNCT
ejpam-2803	74	6	module	module	NOUN
ejpam-2803	74	7	if	if	SCONJ
ejpam-2803	74	8	and	and	CCONJ
ejpam-2803	74	9	only	only	ADV
ejpam-2803	74	10	if	if	SCONJ
ejpam-2803	74	11	(	(	PUNCT
ejpam-2803	74	12	mi)i∈i	mi)i∈i	NUM
ejpam-2803	74	13	is	be	AUX
ejpam-2803	74	14	a	a	DET
ejpam-2803	74	15	family	family	NOUN
ejpam-2803	74	16	of	of	ADP
ejpam-2803	74	17	second	second	ADV
ejpam-2803	74	18	-	-	PUNCT
ejpam-2803	74	19	compatible	compatible	ADJ
ejpam-2803	74	20	xs	xs	NOUN
ejpam-2803	74	21	-	-	PUNCT
ejpam-2803	74	22	injective	injective	ADJ
ejpam-2803	74	23	(	(	PUNCT
ejpam-2803	74	24	resp	resp	NOUN
ejpam-2803	74	25	.	.	PUNCT
ejpam-2803	75	1	cotop	cotop	NOUN
ejpam-2803	75	2	)	)	PUNCT
ejpam-2803	75	3	r	r	NOUN
ejpam-2803	75	4	-	-	PUNCT
ejpam-2803	75	5	modules	module	NOUN
ejpam-2803	75	6	.	.	PUNCT
ejpam-2803	76	1	moreover	moreover	ADV
ejpam-2803	76	2	,	,	PUNCT
ejpam-2803	76	3	in	in	ADP
ejpam-2803	76	4	theorem	theorem	NOUN
ejpam-2803	76	5	3.18	3.18	NUM
ejpam-2803	76	6	,	,	PUNCT
ejpam-2803	76	7	we	we	PRON
ejpam-2803	76	8	prove	prove	VERB
ejpam-2803	76	9	that	that	SCONJ
ejpam-2803	76	10	if	if	SCONJ
ejpam-2803	76	11	r	r	NOUN
ejpam-2803	76	12	is	be	AUX
ejpam-2803	76	13	a	a	DET
ejpam-2803	76	14	perfect	perfect	ADJ
ejpam-2803	76	15	ring	ring	NOUN
ejpam-2803	76	16	,	,	PUNCT
ejpam-2803	76	17	then	then	ADV
ejpam-2803	76	18	the	the	DET
ejpam-2803	76	19	classes	class	NOUN
ejpam-2803	76	20	of	of	ADP
ejpam-2803	76	21	cotop	cotop	NOUN
ejpam-2803	76	22	,	,	PUNCT
ejpam-2803	76	23	weak	weak	ADJ
ejpam-2803	76	24	comultiplication	comultiplication	NOUN
ejpam-2803	76	25	,	,	PUNCT
ejpam-2803	76	26	and	and	CCONJ
ejpam-2803	76	27	xs	xs	NOUN
ejpam-2803	76	28	-	-	PUNCT
ejpam-2803	76	29	injective	injective	ADJ
ejpam-2803	76	30	modules	module	NOUN
ejpam-2803	76	31	are	be	AUX
ejpam-2803	76	32	all	all	ADV
ejpam-2803	76	33	equal	equal	ADJ
ejpam-2803	76	34	.	.	PUNCT
ejpam-2803	77	1	in	in	ADP
ejpam-2803	77	2	the	the	DET
ejpam-2803	77	3	rest	rest	NOUN
ejpam-2803	77	4	of	of	ADP
ejpam-2803	77	5	this	this	DET
ejpam-2803	77	6	article	article	NOUN
ejpam-2803	77	7	,	,	PUNCT
ejpam-2803	77	8	xs	xs	PROPN
ejpam-2803	77	9	:	:	PUNCT
ejpam-2803	77	10	=	=	SYM
ejpam-2803	77	11	specs(m	specs(m	PROPN
ejpam-2803	77	12	)	)	PUNCT
ejpam-2803	77	13	will	will	AUX
ejpam-2803	77	14	denote	denote	VERB
ejpam-2803	77	15	the	the	DET
ejpam-2803	77	16	set	set	NOUN
ejpam-2803	77	17	of	of	ADP
ejpam-2803	77	18	all	all	DET
ejpam-2803	77	19	second	second	ADJ
ejpam-2803	77	20	submodules	submodule	NOUN
ejpam-2803	77	21	of	of	ADP
ejpam-2803	77	22	m	m	PROPN
ejpam-2803	77	23	.	.	PUNCT
ejpam-2803	78	1	also	also	ADV
ejpam-2803	78	2	the	the	DET
ejpam-2803	78	3	map	map	NOUN
ejpam-2803	78	4	ψ	ψ	X
ejpam-2803	78	5	:	:	PUNCT
ejpam-2803	78	6	xs	xs	PROPN
ejpam-2803	78	7	→	→	SYM
ejpam-2803	78	8	spec(r	spec(r	PROPN
ejpam-2803	78	9	)	)	PUNCT
ejpam-2803	78	10	given	give	VERB
ejpam-2803	78	11	by	by	ADP
ejpam-2803	78	12	s	s	PROPN
ejpam-2803	78	13	7→	7→	NUM
ejpam-2803	78	14	annr(s	annr(s	NOUN
ejpam-2803	78	15	)	)	PUNCT
ejpam-2803	78	16	is	be	AUX
ejpam-2803	78	17	called	call	VERB
ejpam-2803	78	18	the	the	DET
ejpam-2803	78	19	natural	natural	ADJ
ejpam-2803	78	20	map	map	NOUN
ejpam-2803	78	21	of	of	ADP
ejpam-2803	78	22	xs	xs	PROPN
ejpam-2803	78	23	.	.	PUNCT
ejpam-2803	79	1	2	2	X
ejpam-2803	79	2	.	.	X
ejpam-2803	79	3	secondful	secondful	ADJ
ejpam-2803	79	4	modules	module	NOUN
ejpam-2803	79	5	we	we	PRON
ejpam-2803	79	6	recall	recall	VERB
ejpam-2803	79	7	that	that	SCONJ
ejpam-2803	79	8	an	an	DET
ejpam-2803	79	9	r	r	NOUN
ejpam-2803	79	10	-	-	PUNCT
ejpam-2803	79	11	module	module	NOUN
ejpam-2803	79	12	m	m	NOUN
ejpam-2803	79	13	is	be	AUX
ejpam-2803	79	14	secondful	secondful	ADJ
ejpam-2803	79	15	if	if	SCONJ
ejpam-2803	79	16	the	the	DET
ejpam-2803	79	17	natural	natural	ADJ
ejpam-2803	79	18	map	map	NOUN
ejpam-2803	79	19	xs	xs	PROPN
ejpam-2803	79	20	is	be	AUX
ejpam-2803	79	21	surjective	surjective	ADJ
ejpam-2803	79	22	.	.	PUNCT
ejpam-2803	80	1	for	for	ADP
ejpam-2803	80	2	example	example	NOUN
ejpam-2803	80	3	,	,	PUNCT
ejpam-2803	80	4	for	for	ADP
ejpam-2803	80	5	each	each	DET
ejpam-2803	80	6	positive	positive	ADJ
ejpam-2803	80	7	integer	integer	NOUN
ejpam-2803	80	8	n	n	PROPN
ejpam-2803	80	9	(	(	PUNCT
ejpam-2803	80	10	n	n	CCONJ
ejpam-2803	80	11	>	>	X
ejpam-2803	80	12	1	1	NUM
ejpam-2803	80	13	)	)	PUNCT
ejpam-2803	80	14	,	,	PUNCT
ejpam-2803	80	15	zn	zn	INTJ
ejpam-2803	80	16	as	as	SCONJ
ejpam-2803	80	17	z	z	NOUN
ejpam-2803	80	18	-	-	PUNCT
ejpam-2803	80	19	module	module	NOUN
ejpam-2803	80	20	is	be	AUX
ejpam-2803	80	21	secondful	secondful	ADJ
ejpam-2803	80	22	.	.	PUNCT
ejpam-2803	81	1	throughout	throughout	ADP
ejpam-2803	81	2	this	this	DET
ejpam-2803	81	3	section	section	NOUN
ejpam-2803	81	4	,	,	PUNCT
ejpam-2803	81	5	we	we	PRON
ejpam-2803	81	6	assume	assume	VERB
ejpam-2803	81	7	that	that	SCONJ
ejpam-2803	81	8	specs(m	specs(m	NOUN
ejpam-2803	81	9	)	)	PUNCT
ejpam-2803	81	10	is	be	AUX
ejpam-2803	81	11	topologized	topologize	VERB
ejpam-2803	81	12	with	with	ADP
ejpam-2803	81	13	zariski	zariski	NOUN
ejpam-2803	81	14	topology	topology	NOUN
ejpam-2803	81	15	.	.	PUNCT
ejpam-2803	82	1	remark	remark	VERB
ejpam-2803	82	2	2.1	2.1	NUM
ejpam-2803	82	3	.	.	PUNCT
ejpam-2803	83	1	let	let	VERB
ejpam-2803	83	2	m	m	PRON
ejpam-2803	83	3	be	be	AUX
ejpam-2803	83	4	an	an	DET
ejpam-2803	83	5	r	r	NOUN
ejpam-2803	83	6	-	-	PUNCT
ejpam-2803	83	7	module	module	NOUN
ejpam-2803	83	8	.	.	PUNCT
ejpam-2803	84	1	then	then	ADV
ejpam-2803	84	2	we	we	PRON
ejpam-2803	84	3	have	have	VERB
ejpam-2803	84	4	the	the	DET
ejpam-2803	84	5	following	following	NOUN
ejpam-2803	84	6	.	.	PUNCT
ejpam-2803	85	1	(	(	PUNCT
ejpam-2803	85	2	a	a	X
ejpam-2803	85	3	)	)	PUNCT
ejpam-2803	85	4	let	let	VERB
ejpam-2803	85	5	m	m	PRON
ejpam-2803	85	6	be	be	AUX
ejpam-2803	85	7	an	an	DET
ejpam-2803	85	8	artinian	artinian	ADJ
ejpam-2803	85	9	module	module	NOUN
ejpam-2803	85	10	.	.	PUNCT
ejpam-2803	86	1	then	then	ADV
ejpam-2803	86	2	m	m	PROPN
ejpam-2803	86	3	is	be	AUX
ejpam-2803	86	4	secondful	secondful	ADJ
ejpam-2803	86	5	⇔	⇔	PROPN
ejpam-2803	86	6	homr(rp	homr(rp	PROPN
ejpam-2803	86	7	,	,	PUNCT
ejpam-2803	86	8	(	(	PUNCT
ejpam-2803	86	9	0	0	NUM
ejpam-2803	86	10	:	:	PUNCT
ejpam-2803	86	11	m	m	VERB
ejpam-2803	86	12	p	p	NOUN
ejpam-2803	86	13	)	)	PUNCT
ejpam-2803	86	14	)	)	PUNCT
ejpam-2803	87	1	6=	6=	X
ejpam-2803	87	2	(	(	PUNCT
ejpam-2803	87	3	0	0	NUM
ejpam-2803	87	4	)	)	PUNCT
ejpam-2803	87	5	for	for	ADP
ejpam-2803	87	6	every	every	DET
ejpam-2803	87	7	p	p	PROPN
ejpam-2803	87	8	∈	∈	PROPN
ejpam-2803	87	9	v	v	NOUN
ejpam-2803	87	10	(	(	PUNCT
ejpam-2803	87	11	annr(m	annr(m	PROPN
ejpam-2803	87	12	)	)	PUNCT
ejpam-2803	87	13	)	)	PUNCT
ejpam-2803	88	1	[	[	X
ejpam-2803	88	2	7	7	NUM
ejpam-2803	88	3	,	,	PUNCT
ejpam-2803	88	4	theorem	theorem	VERB
ejpam-2803	88	5	3.8	3.8	NUM
ejpam-2803	88	6	]	]	PUNCT
ejpam-2803	88	7	.	.	PUNCT
ejpam-2803	89	1	(	(	PUNCT
ejpam-2803	89	2	b	b	X
ejpam-2803	89	3	)	)	PUNCT
ejpam-2803	89	4	let	let	VERB
ejpam-2803	89	5	m	m	PRON
ejpam-2803	89	6	be	be	AUX
ejpam-2803	89	7	a	a	DET
ejpam-2803	89	8	secondful	secondful	ADJ
ejpam-2803	89	9	r	r	NOUN
ejpam-2803	89	10	-	-	PUNCT
ejpam-2803	89	11	module	module	NOUN
ejpam-2803	89	12	.	.	PUNCT
ejpam-2803	90	1	then	then	ADV
ejpam-2803	90	2	m	m	PROPN
ejpam-2803	90	3	has	have	VERB
ejpam-2803	90	4	a	a	DET
ejpam-2803	90	5	noetherian	noetherian	ADJ
ejpam-2803	90	6	second	second	ADJ
ejpam-2803	90	7	spectrum	spectrum	NOUN
ejpam-2803	90	8	if	if	SCONJ
ejpam-2803	90	9	and	and	CCONJ
ejpam-2803	90	10	only	only	ADV
ejpam-2803	90	11	if	if	SCONJ
ejpam-2803	90	12	the	the	DET
ejpam-2803	90	13	ring	ring	NOUN
ejpam-2803	90	14	r	r	NOUN
ejpam-2803	90	15	has	have	VERB
ejpam-2803	90	16	noetherian	noetherian	ADJ
ejpam-2803	90	17	spectrum	spectrum	NOUN
ejpam-2803	91	1	[	[	X
ejpam-2803	91	2	17	17	NUM
ejpam-2803	91	3	,	,	PUNCT
ejpam-2803	91	4	theorem	theorem	VERB
ejpam-2803	91	5	4.3	4.3	NUM
ejpam-2803	91	6	]	]	PUNCT
ejpam-2803	91	7	.	.	PUNCT
ejpam-2803	92	1	lemma	lemma	PROPN
ejpam-2803	92	2	2.2	2.2	NUM
ejpam-2803	92	3	.	.	PUNCT
ejpam-2803	93	1	let	let	VERB
ejpam-2803	93	2	m	m	PRON
ejpam-2803	93	3	be	be	AUX
ejpam-2803	93	4	an	an	DET
ejpam-2803	93	5	r	r	NOUN
ejpam-2803	93	6	-	-	PUNCT
ejpam-2803	93	7	module	module	NOUN
ejpam-2803	93	8	and	and	CCONJ
ejpam-2803	93	9	p	p	NOUN
ejpam-2803	93	10	,	,	PUNCT
ejpam-2803	93	11	q	q	ADJ
ejpam-2803	93	12	be	be	AUX
ejpam-2803	93	13	two	two	NUM
ejpam-2803	93	14	prime	prime	ADJ
ejpam-2803	93	15	ideals	ideal	NOUN
ejpam-2803	93	16	of	of	ADP
ejpam-2803	93	17	r	r	NOUN
ejpam-2803	93	18	such	such	ADJ
ejpam-2803	93	19	that	that	DET
ejpam-2803	93	20	q	q	PROPN
ejpam-2803	93	21	⊆	⊆	NUM
ejpam-2803	93	22	p.	p.	NOUN
ejpam-2803	93	23	then	then	ADV
ejpam-2803	93	24	homr(rq	homr(rq	PROPN
ejpam-2803	93	25	,	,	PUNCT
ejpam-2803	93	26	(	(	PUNCT
ejpam-2803	93	27	0	0	NUM
ejpam-2803	93	28	:	:	PUNCT
ejpam-2803	93	29	m	m	VERB
ejpam-2803	93	30	q	q	NOUN
ejpam-2803	93	31	)	)	PUNCT
ejpam-2803	93	32	)	)	PUNCT
ejpam-2803	94	1	∼=	∼=	VERB
ejpam-2803	94	2	homrp((rp)qrp	homrp((rp)qrp	NOUN
ejpam-2803	94	3	,	,	PUNCT
ejpam-2803	94	4	(	(	PUNCT
ejpam-2803	94	5	0	0	NUM
ejpam-2803	94	6	:	:	PUNCT
ejpam-2803	94	7	homr(rp	homr(rp	NOUN
ejpam-2803	94	8	,	,	PUNCT
ejpam-2803	94	9	m	m	NOUN
ejpam-2803	94	10	)	)	PUNCT
ejpam-2803	94	11	qrp	qrp	PROPN
ejpam-2803	94	12	)	)	PUNCT
ejpam-2803	94	13	)	)	PUNCT
ejpam-2803	94	14	.	.	PUNCT
ejpam-2803	95	1	proof	proof	NOUN
ejpam-2803	95	2	.	.	PUNCT
ejpam-2803	96	1	homr(rq	homr(rq	INTJ
ejpam-2803	96	2	,	,	PUNCT
ejpam-2803	96	3	(	(	PUNCT
ejpam-2803	96	4	0	0	NUM
ejpam-2803	96	5	:	:	PUNCT
ejpam-2803	96	6	m	m	VERB
ejpam-2803	96	7	q	q	NOUN
ejpam-2803	96	8	)	)	PUNCT
ejpam-2803	96	9	)	)	PUNCT
ejpam-2803	97	1	∼=	∼=	NOUN
ejpam-2803	97	2	homr(rp	homr(rp	NOUN
ejpam-2803	97	3	⊗rp	⊗rp	NUM
ejpam-2803	97	4	rq	rq	NOUN
ejpam-2803	97	5	,	,	PUNCT
ejpam-2803	97	6	(	(	PUNCT
ejpam-2803	97	7	0	0	NUM
ejpam-2803	97	8	:	:	PUNCT
ejpam-2803	97	9	m	m	VERB
ejpam-2803	97	10	q	q	NOUN
ejpam-2803	97	11	)	)	PUNCT
ejpam-2803	97	12	)	)	PUNCT
ejpam-2803	98	1	∼=	∼=	VERB
ejpam-2803	98	2	homrp(rq	homrp(rq	ADJ
ejpam-2803	98	3	,	,	PUNCT
ejpam-2803	98	4	homr(rp	homr(rp	NOUN
ejpam-2803	98	5	,	,	PUNCT
ejpam-2803	98	6	homr(r	homr(r	PROPN
ejpam-2803	98	7	/	/	SYM
ejpam-2803	98	8	q	q	NOUN
ejpam-2803	98	9	,	,	PUNCT
ejpam-2803	98	10	m	m	NOUN
ejpam-2803	98	11	)	)	PUNCT
ejpam-2803	98	12	)	)	PUNCT
ejpam-2803	98	13	)	)	PUNCT
ejpam-2803	99	1	∼=	∼=	VERB
ejpam-2803	99	2	homrp(rq	homrp(rq	ADV
ejpam-2803	99	3	,	,	PUNCT
ejpam-2803	99	4	homr(rp	homr(rp	NOUN
ejpam-2803	99	5	⊗r	⊗r	NOUN
ejpam-2803	99	6	r	r	NOUN
ejpam-2803	99	7	/	/	SYM
ejpam-2803	99	8	q	q	NOUN
ejpam-2803	99	9	,	,	PUNCT
ejpam-2803	99	10	m	m	NOUN
ejpam-2803	99	11	)	)	PUNCT
ejpam-2803	99	12	)	)	PUNCT
ejpam-2803	100	1	∼=	∼=	VERB
ejpam-2803	100	2	homrp(rq	homrp(rq	ADV
ejpam-2803	100	3	,	,	PUNCT
ejpam-2803	100	4	(	(	PUNCT
ejpam-2803	100	5	0	0	NUM
ejpam-2803	100	6	:	:	PUNCT
ejpam-2803	100	7	homr(rp	homr(rp	NOUN
ejpam-2803	100	8	,	,	PUNCT
ejpam-2803	100	9	m	m	NOUN
ejpam-2803	100	10	)	)	PUNCT
ejpam-2803	100	11	qrp	qrp	PROPN
ejpam-2803	100	12	)	)	PUNCT
ejpam-2803	100	13	)	)	PUNCT
ejpam-2803	101	1	∼=	∼=	VERB
ejpam-2803	101	2	homrp((rp)qrp	homrp((rp)qrp	NOUN
ejpam-2803	101	3	,	,	PUNCT
ejpam-2803	101	4	(	(	PUNCT
ejpam-2803	101	5	0	0	NUM
ejpam-2803	101	6	:	:	PUNCT
ejpam-2803	101	7	homr(rp	homr(rp	NOUN
ejpam-2803	101	8	,	,	PUNCT
ejpam-2803	101	9	m	m	NOUN
ejpam-2803	101	10	)	)	PUNCT
ejpam-2803	101	11	qrp	qrp	PROPN
ejpam-2803	101	12	)	)	PUNCT
ejpam-2803	101	13	)	)	PUNCT
ejpam-2803	101	14	.	.	PUNCT
ejpam-2803	102	1	h.	h.	PROPN
ejpam-2803	102	2	ansari	ansari	PROPN
ejpam-2803	102	3	-	-	PUNCT
ejpam-2803	102	4	toroghy	toroghy	NOUN
ejpam-2803	102	5	,	,	PUNCT
ejpam-2803	102	6	s.	s.	PROPN
ejpam-2803	102	7	s.	s.	PROPN
ejpam-2803	102	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	102	9	/	/	SYM
ejpam-2803	102	10	eur	eur	PROPN
ejpam-2803	102	11	.	.	PUNCT
ejpam-2803	103	1	j.	j.	PROPN
ejpam-2803	103	2	pure	pure	PROPN
ejpam-2803	103	3	appl	appl	PROPN
ejpam-2803	103	4	.	.	PROPN
ejpam-2803	103	5	math	math	PROPN
ejpam-2803	103	6	,	,	PUNCT
ejpam-2803	103	7	10	10	NUM
ejpam-2803	103	8	(	(	PUNCT
ejpam-2803	103	9	2	2	NUM
ejpam-2803	103	10	)	)	PUNCT
ejpam-2803	103	11	(	(	PUNCT
ejpam-2803	103	12	2017	2017	NUM
ejpam-2803	103	13	)	)	PUNCT
ejpam-2803	103	14	,	,	PUNCT
ejpam-2803	103	15	211	211	NUM
ejpam-2803	103	16	-	-	SYM
ejpam-2803	103	17	230	230	NUM
ejpam-2803	103	18	214	214	NUM
ejpam-2803	103	19	theorem	theorem	NOUN
ejpam-2803	103	20	2.3	2.3	NUM
ejpam-2803	103	21	.	.	PUNCT
ejpam-2803	104	1	let	let	VERB
ejpam-2803	104	2	m	m	PRON
ejpam-2803	104	3	be	be	AUX
ejpam-2803	104	4	an	an	DET
ejpam-2803	104	5	artinian	artinian	ADJ
ejpam-2803	104	6	r	r	NOUN
ejpam-2803	104	7	-	-	PUNCT
ejpam-2803	104	8	module	module	NOUN
ejpam-2803	104	9	.	.	PUNCT
ejpam-2803	105	1	then	then	ADV
ejpam-2803	105	2	the	the	DET
ejpam-2803	105	3	following	follow	VERB
ejpam-2803	105	4	statements	statement	NOUN
ejpam-2803	105	5	are	be	AUX
ejpam-2803	105	6	equivalent	equivalent	ADJ
ejpam-2803	105	7	:	:	PUNCT
ejpam-2803	105	8	(	(	PUNCT
ejpam-2803	105	9	a	a	X
ejpam-2803	105	10	)	)	PUNCT
ejpam-2803	105	11	m	m	VERB
ejpam-2803	105	12	is	be	AUX
ejpam-2803	105	13	a	a	DET
ejpam-2803	105	14	non	non	ADJ
ejpam-2803	105	15	-	-	ADJ
ejpam-2803	105	16	zero	zero	ADJ
ejpam-2803	105	17	secondful	secondful	ADJ
ejpam-2803	105	18	r	r	NOUN
ejpam-2803	105	19	-	-	PUNCT
ejpam-2803	105	20	module	module	NOUN
ejpam-2803	105	21	;	;	PUNCT
ejpam-2803	105	22	(	(	PUNCT
ejpam-2803	105	23	b	b	X
ejpam-2803	105	24	)	)	PUNCT
ejpam-2803	105	25	homr(rp	homr(rp	NOUN
ejpam-2803	105	26	,	,	PUNCT
ejpam-2803	105	27	m	m	NOUN
ejpam-2803	105	28	)	)	PUNCT
ejpam-2803	105	29	is	be	AUX
ejpam-2803	105	30	non	non	ADJ
ejpam-2803	105	31	-	-	ADJ
ejpam-2803	105	32	zero	zero	ADJ
ejpam-2803	105	33	secondful	secondful	ADJ
ejpam-2803	105	34	rp	rp	NOUN
ejpam-2803	105	35	-	-	PUNCT
ejpam-2803	105	36	module	module	NOUN
ejpam-2803	105	37	for	for	ADP
ejpam-2803	105	38	every	every	DET
ejpam-2803	105	39	p	p	PROPN
ejpam-2803	105	40	∈	∈	PROPN
ejpam-2803	105	41	v	v	NOUN
ejpam-2803	105	42	(	(	PUNCT
ejpam-2803	105	43	annr(m	annr(m	PROPN
ejpam-2803	105	44	)	)	PUNCT
ejpam-2803	105	45	)	)	PUNCT
ejpam-2803	105	46	.	.	PUNCT
ejpam-2803	106	1	proof	proof	NOUN
ejpam-2803	106	2	.	.	PUNCT
ejpam-2803	107	1	(	(	PUNCT
ejpam-2803	107	2	a	a	X
ejpam-2803	107	3	)	)	PUNCT
ejpam-2803	107	4	⇒	⇒	NOUN
ejpam-2803	107	5	(	(	PUNCT
ejpam-2803	107	6	b	b	X
ejpam-2803	107	7	)	)	PUNCT
ejpam-2803	107	8	let	let	VERB
ejpam-2803	107	9	p	p	PRON
ejpam-2803	107	10	be	be	AUX
ejpam-2803	107	11	a	a	DET
ejpam-2803	107	12	prime	prime	ADJ
ejpam-2803	107	13	ideal	ideal	NOUN
ejpam-2803	107	14	of	of	ADP
ejpam-2803	107	15	r	r	NOUN
ejpam-2803	107	16	such	such	ADJ
ejpam-2803	107	17	that	that	SCONJ
ejpam-2803	107	18	p	p	PROPN
ejpam-2803	107	19	⊇	⊇	PROPN
ejpam-2803	107	20	annr(m	annr(m	NOUN
ejpam-2803	107	21	)	)	PUNCT
ejpam-2803	107	22	.	.	PUNCT
ejpam-2803	108	1	since	since	SCONJ
ejpam-2803	108	2	m	m	PROPN
ejpam-2803	108	3	is	be	AUX
ejpam-2803	108	4	a	a	DET
ejpam-2803	108	5	non	non	ADJ
ejpam-2803	108	6	-	-	ADJ
ejpam-2803	108	7	zero	zero	ADJ
ejpam-2803	108	8	secondful	secondful	ADJ
ejpam-2803	108	9	module	module	NOUN
ejpam-2803	108	10	,	,	PUNCT
ejpam-2803	108	11	homr(rp	homr(rp	NOUN
ejpam-2803	108	12	,	,	PUNCT
ejpam-2803	108	13	(	(	PUNCT
ejpam-2803	108	14	0	0	NUM
ejpam-2803	108	15	:	:	PUNCT
ejpam-2803	108	16	m	m	VERB
ejpam-2803	108	17	p	p	NOUN
ejpam-2803	108	18	)	)	PUNCT
ejpam-2803	108	19	)	)	PUNCT
ejpam-2803	108	20	6=	6=	X
ejpam-2803	109	1	(	(	PUNCT
ejpam-2803	109	2	0	0	NUM
ejpam-2803	109	3	)	)	PUNCT
ejpam-2803	109	4	by	by	ADP
ejpam-2803	109	5	remark	remark	NOUN
ejpam-2803	109	6	2.1	2.1	NUM
ejpam-2803	109	7	(	(	PUNCT
ejpam-2803	109	8	a	a	NOUN
ejpam-2803	109	9	)	)	PUNCT
ejpam-2803	109	10	.	.	PUNCT
ejpam-2803	110	1	hence	hence	ADV
ejpam-2803	110	2	we	we	PRON
ejpam-2803	110	3	have	have	VERB
ejpam-2803	110	4	homr(rp	homr(rp	NOUN
ejpam-2803	110	5	,	,	PUNCT
ejpam-2803	110	6	m	m	NOUN
ejpam-2803	110	7	)	)	PUNCT
ejpam-2803	110	8	6=	6=	PUNCT
ejpam-2803	110	9	(	(	PUNCT
ejpam-2803	110	10	0	0	NUM
ejpam-2803	110	11	)	)	PUNCT
ejpam-2803	110	12	.	.	PUNCT
ejpam-2803	111	1	to	to	PART
ejpam-2803	111	2	prove	prove	VERB
ejpam-2803	111	3	that	that	SCONJ
ejpam-2803	111	4	homr(rp	homr(rp	NOUN
ejpam-2803	111	5	,	,	PUNCT
ejpam-2803	111	6	m	m	VERB
ejpam-2803	111	7	)	)	PUNCT
ejpam-2803	111	8	is	be	AUX
ejpam-2803	111	9	secondful	secondful	ADJ
ejpam-2803	111	10	rp	rp	NOUN
ejpam-2803	111	11	-	-	PUNCT
ejpam-2803	111	12	module	module	NOUN
ejpam-2803	111	13	,	,	PUNCT
ejpam-2803	111	14	let	let	VERB
ejpam-2803	111	15	qrp	qrp	PROPN
ejpam-2803	111	16	∈	∈	PROPN
ejpam-2803	111	17	v	v	PROPN
ejpam-2803	111	18	(	(	PUNCT
ejpam-2803	111	19	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	111	20	,	,	PUNCT
ejpam-2803	111	21	m	m	NOUN
ejpam-2803	111	22	)	)	PUNCT
ejpam-2803	111	23	)	)	PUNCT
ejpam-2803	111	24	)	)	PUNCT
ejpam-2803	111	25	.	.	PUNCT
ejpam-2803	112	1	since	since	SCONJ
ejpam-2803	112	2	rp	rp	NOUN
ejpam-2803	112	3	is	be	AUX
ejpam-2803	112	4	quasi	quasi	ADJ
ejpam-2803	112	5	-	-	ADJ
ejpam-2803	112	6	local	local	ADJ
ejpam-2803	112	7	ring	ring	NOUN
ejpam-2803	112	8	,	,	PUNCT
ejpam-2803	112	9	prp	prp	PROPN
ejpam-2803	112	10	⊇	⊇	PROPN
ejpam-2803	112	11	qrp	qrp	PROPN
ejpam-2803	112	12	⊇	⊇	PROPN
ejpam-2803	112	13	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	112	14	,	,	PUNCT
ejpam-2803	112	15	m	m	NOUN
ejpam-2803	112	16	)	)	PUNCT
ejpam-2803	112	17	)	)	PUNCT
ejpam-2803	113	1	⊇	⊇	PROPN
ejpam-2803	113	2	annr(m)rp	annr(m)rp	PROPN
ejpam-2803	113	3	.	.	PUNCT
ejpam-2803	114	1	taking	take	VERB
ejpam-2803	114	2	the	the	DET
ejpam-2803	114	3	contraction	contraction	NOUN
ejpam-2803	114	4	of	of	ADP
ejpam-2803	114	5	each	each	DET
ejpam-2803	114	6	term	term	NOUN
ejpam-2803	114	7	of	of	ADP
ejpam-2803	114	8	this	this	DET
ejpam-2803	114	9	sequence	sequence	NOUN
ejpam-2803	114	10	,	,	PUNCT
ejpam-2803	114	11	we	we	PRON
ejpam-2803	114	12	have	have	VERB
ejpam-2803	114	13	that	that	DET
ejpam-2803	114	14	p	p	PROPN
ejpam-2803	114	15	⊇	⊇	PROPN
ejpam-2803	114	16	q	q	PROPN
ejpam-2803	114	17	⊇	⊇	PROPN
ejpam-2803	114	18	sp(annr(m	sp(annr(m	NOUN
ejpam-2803	114	19	)	)	PUNCT
ejpam-2803	114	20	)	)	PUNCT
ejpam-2803	114	21	⊇	⊇	PROPN
ejpam-2803	114	22	annr(m	annr(m	NOUN
ejpam-2803	114	23	)	)	PUNCT
ejpam-2803	114	24	.	.	PUNCT
ejpam-2803	115	1	hence	hence	ADV
ejpam-2803	115	2	homr(rq	homr(rq	PROPN
ejpam-2803	115	3	,	,	PUNCT
ejpam-2803	115	4	(	(	PUNCT
ejpam-2803	115	5	0	0	NUM
ejpam-2803	115	6	:	:	PUNCT
ejpam-2803	115	7	m	m	VERB
ejpam-2803	115	8	q	q	NOUN
ejpam-2803	115	9	)	)	PUNCT
ejpam-2803	115	10	)	)	PUNCT
ejpam-2803	115	11	6=	6=	X
ejpam-2803	116	1	(	(	PUNCT
ejpam-2803	116	2	0	0	NUM
ejpam-2803	116	3	)	)	PUNCT
ejpam-2803	116	4	by	by	ADP
ejpam-2803	116	5	remark	remark	NOUN
ejpam-2803	116	6	2.1	2.1	NUM
ejpam-2803	116	7	(	(	PUNCT
ejpam-2803	116	8	a	a	NOUN
ejpam-2803	116	9	)	)	PUNCT
ejpam-2803	116	10	.	.	PUNCT
ejpam-2803	117	1	it	it	PRON
ejpam-2803	117	2	follows	follow	VERB
ejpam-2803	117	3	that	that	SCONJ
ejpam-2803	117	4	homr(rp	homr(rp	NOUN
ejpam-2803	117	5	,	,	PUNCT
ejpam-2803	117	6	m	m	NOUN
ejpam-2803	117	7	)	)	PUNCT
ejpam-2803	117	8	is	be	AUX
ejpam-2803	117	9	a	a	DET
ejpam-2803	117	10	secondful	secondful	ADJ
ejpam-2803	117	11	rp	rp	NOUN
ejpam-2803	117	12	-	-	PUNCT
ejpam-2803	117	13	module	module	NOUN
ejpam-2803	117	14	by	by	ADP
ejpam-2803	117	15	remark	remark	NOUN
ejpam-2803	117	16	2.1	2.1	NUM
ejpam-2803	117	17	(	(	PUNCT
ejpam-2803	117	18	a	a	NOUN
ejpam-2803	117	19	)	)	PUNCT
ejpam-2803	117	20	and	and	CCONJ
ejpam-2803	117	21	lemma	lemma	PROPN
ejpam-2803	117	22	2.2	2.2	NUM
ejpam-2803	117	23	.	.	PUNCT
ejpam-2803	118	1	(	(	PUNCT
ejpam-2803	118	2	b)⇒	b)⇒	PROPN
ejpam-2803	118	3	(	(	PUNCT
ejpam-2803	118	4	a	a	X
ejpam-2803	118	5	)	)	PUNCT
ejpam-2803	118	6	let	let	VERB
ejpam-2803	118	7	p	p	PRON
ejpam-2803	118	8	∈	∈	PROPN
ejpam-2803	118	9	v	v	NOUN
ejpam-2803	118	10	(	(	PUNCT
ejpam-2803	118	11	annr(m	annr(m	PROPN
ejpam-2803	118	12	)	)	PUNCT
ejpam-2803	118	13	)	)	PUNCT
ejpam-2803	118	14	.	.	PUNCT
ejpam-2803	119	1	by	by	ADP
ejpam-2803	119	2	part	part	NOUN
ejpam-2803	119	3	(	(	PUNCT
ejpam-2803	119	4	b	b	NOUN
ejpam-2803	119	5	)	)	PUNCT
ejpam-2803	119	6	,	,	PUNCT
ejpam-2803	119	7	homr(rp	homr(rp	NOUN
ejpam-2803	119	8	,	,	PUNCT
ejpam-2803	119	9	m	m	NOUN
ejpam-2803	119	10	)	)	PUNCT
ejpam-2803	119	11	6=	6=	PUNCT
ejpam-2803	119	12	(	(	PUNCT
ejpam-2803	119	13	0	0	NUM
ejpam-2803	119	14	)	)	PUNCT
ejpam-2803	119	15	.	.	PUNCT
ejpam-2803	120	1	thusannrp(homr(rp	thusannrp(homr(rp	PROPN
ejpam-2803	120	2	,	,	PUNCT
ejpam-2803	120	3	m	m	NOUN
ejpam-2803	120	4	)	)	PUNCT
ejpam-2803	120	5	)	)	PUNCT
ejpam-2803	121	1	6=	6=	X
ejpam-2803	122	1	rp	rp	NOUN
ejpam-2803	122	2	.	.	PUNCT
ejpam-2803	123	1	therefore	therefore	ADV
ejpam-2803	123	2	,	,	PUNCT
ejpam-2803	123	3	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	123	4	,	,	PUNCT
ejpam-2803	123	5	m	m	NOUN
ejpam-2803	123	6	)	)	PUNCT
ejpam-2803	123	7	)	)	PUNCT
ejpam-2803	124	1	⊆	⊆	NUM
ejpam-2803	124	2	prp	prp	NOUN
ejpam-2803	124	3	and	and	CCONJ
ejpam-2803	124	4	so	so	ADV
ejpam-2803	124	5	prp	prp	PROPN
ejpam-2803	124	6	∈	∈	PROPN
ejpam-2803	124	7	v	v	PROPN
ejpam-2803	124	8	(	(	PUNCT
ejpam-2803	124	9	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	124	10	,	,	PUNCT
ejpam-2803	124	11	m	m	NOUN
ejpam-2803	124	12	)	)	PUNCT
ejpam-2803	124	13	)	)	PUNCT
ejpam-2803	124	14	)	)	PUNCT
ejpam-2803	124	15	.	.	PUNCT
ejpam-2803	125	1	now	now	ADV
ejpam-2803	125	2	by	by	ADP
ejpam-2803	125	3	remark	remark	NOUN
ejpam-2803	125	4	2.1	2.1	NUM
ejpam-2803	125	5	(	(	PUNCT
ejpam-2803	125	6	a	a	NOUN
ejpam-2803	125	7	)	)	PUNCT
ejpam-2803	125	8	,	,	PUNCT
ejpam-2803	125	9	homrp((rp)p	homrp((rp)p	PROPN
ejpam-2803	125	10	,	,	PUNCT
ejpam-2803	125	11	(	(	PUNCT
ejpam-2803	125	12	0	0	NUM
ejpam-2803	125	13	:	:	PUNCT
ejpam-2803	125	14	homr(rp	homr(rp	NOUN
ejpam-2803	125	15	,	,	PUNCT
ejpam-2803	125	16	m	m	NOUN
ejpam-2803	125	17	)	)	PUNCT
ejpam-2803	125	18	prp	prp	NOUN
ejpam-2803	125	19	)	)	PUNCT
ejpam-2803	125	20	)	)	PUNCT
ejpam-2803	126	1	6=	6=	X
ejpam-2803	126	2	(	(	PUNCT
ejpam-2803	126	3	0	0	NUM
ejpam-2803	126	4	)	)	PUNCT
ejpam-2803	126	5	,	,	PUNCT
ejpam-2803	126	6	and	and	CCONJ
ejpam-2803	126	7	so	so	ADV
ejpam-2803	126	8	homr(rp	homr(rp	ADJ
ejpam-2803	126	9	,	,	PUNCT
ejpam-2803	126	10	(	(	PUNCT
ejpam-2803	126	11	0	0	NUM
ejpam-2803	126	12	:	:	PUNCT
ejpam-2803	126	13	m	m	VERB
ejpam-2803	126	14	p	p	NOUN
ejpam-2803	126	15	)	)	PUNCT
ejpam-2803	126	16	)	)	PUNCT
ejpam-2803	126	17	6=	6=	X
ejpam-2803	127	1	(	(	PUNCT
ejpam-2803	127	2	0	0	NUM
ejpam-2803	127	3	)	)	PUNCT
ejpam-2803	127	4	.	.	PUNCT
ejpam-2803	128	1	thus	thus	ADV
ejpam-2803	128	2	m	m	NOUN
ejpam-2803	128	3	is	be	AUX
ejpam-2803	128	4	a	a	DET
ejpam-2803	128	5	secondful	secondful	ADJ
ejpam-2803	128	6	r	r	NOUN
ejpam-2803	128	7	-	-	PUNCT
ejpam-2803	128	8	module	module	NOUN
ejpam-2803	128	9	.	.	PUNCT
ejpam-2803	129	1	corollary	corollary	ADJ
ejpam-2803	129	2	2.4	2.4	NUM
ejpam-2803	129	3	.	.	PUNCT
ejpam-2803	130	1	let	let	VERB
ejpam-2803	130	2	m	m	PRON
ejpam-2803	130	3	be	be	AUX
ejpam-2803	130	4	a	a	DET
ejpam-2803	130	5	non	non	ADJ
ejpam-2803	130	6	-	-	ADJ
ejpam-2803	130	7	zero	zero	ADJ
ejpam-2803	130	8	artinian	artinian	ADJ
ejpam-2803	130	9	secondful	secondful	ADJ
ejpam-2803	130	10	r	r	NOUN
ejpam-2803	130	11	-	-	PUNCT
ejpam-2803	130	12	module	module	NOUN
ejpam-2803	130	13	and	and	CCONJ
ejpam-2803	130	14	suppose	suppose	VERB
ejpam-2803	130	15	p	p	PROPN
ejpam-2803	130	16	∈	∈	PROPN
ejpam-2803	130	17	v	v	NOUN
ejpam-2803	130	18	(	(	PUNCT
ejpam-2803	130	19	annr(m	annr(m	PROPN
ejpam-2803	130	20	)	)	PUNCT
ejpam-2803	130	21	)	)	PUNCT
ejpam-2803	130	22	.	.	PUNCT
ejpam-2803	131	1	then	then	ADV
ejpam-2803	131	2	if	if	SCONJ
ejpam-2803	131	3	m	m	NOUN
ejpam-2803	131	4	has	have	VERB
ejpam-2803	131	5	noetherian	noetherian	ADJ
ejpam-2803	131	6	second	second	ADJ
ejpam-2803	131	7	spectrum	spectrum	NOUN
ejpam-2803	131	8	,	,	PUNCT
ejpam-2803	131	9	so	so	ADV
ejpam-2803	131	10	does	do	AUX
ejpam-2803	131	11	the	the	DET
ejpam-2803	131	12	rp	rp	NOUN
ejpam-2803	131	13	-	-	PUNCT
ejpam-2803	131	14	module	module	NOUN
ejpam-2803	131	15	homr(rp	homr(rp	NOUN
ejpam-2803	131	16	,	,	PUNCT
ejpam-2803	131	17	m	m	NOUN
ejpam-2803	131	18	)	)	PUNCT
ejpam-2803	131	19	.	.	PUNCT
ejpam-2803	132	1	proof	proof	NOUN
ejpam-2803	132	2	.	.	PUNCT
ejpam-2803	133	1	let	let	VERB
ejpam-2803	133	2	p	p	PROPN
ejpam-2803	133	3	∈	∈	PROPN
ejpam-2803	133	4	spec(r	spec(r	PROPN
ejpam-2803	133	5	)	)	PUNCT
ejpam-2803	133	6	and	and	CCONJ
ejpam-2803	133	7	annr(m	annr(m	PROPN
ejpam-2803	133	8	)	)	PUNCT
ejpam-2803	133	9	⊆	⊆	NUM
ejpam-2803	133	10	p.	p.	NOUN
ejpam-2803	133	11	by	by	ADP
ejpam-2803	133	12	theorem	theorem	ADJ
ejpam-2803	133	13	2.3	2.3	NUM
ejpam-2803	133	14	,	,	PUNCT
ejpam-2803	133	15	homr(rp	homr(rp	NOUN
ejpam-2803	133	16	,	,	PUNCT
ejpam-2803	133	17	m	m	NOUN
ejpam-2803	133	18	)	)	PUNCT
ejpam-2803	133	19	is	be	AUX
ejpam-2803	133	20	a	a	DET
ejpam-2803	133	21	secondful	secondful	ADJ
ejpam-2803	133	22	rp	rp	NOUN
ejpam-2803	133	23	-	-	PUNCT
ejpam-2803	133	24	module	module	NOUN
ejpam-2803	133	25	.	.	PUNCT
ejpam-2803	134	1	hence	hence	ADV
ejpam-2803	134	2	,	,	PUNCT
ejpam-2803	134	3	in	in	ADP
ejpam-2803	134	4	order	order	NOUN
ejpam-2803	134	5	to	to	PART
ejpam-2803	134	6	show	show	VERB
ejpam-2803	134	7	that	that	SCONJ
ejpam-2803	134	8	homr(rp	homr(rp	NOUN
ejpam-2803	134	9	,	,	PUNCT
ejpam-2803	134	10	m	m	VERB
ejpam-2803	134	11	)	)	PUNCT
ejpam-2803	134	12	has	have	VERB
ejpam-2803	134	13	noetherian	noetherian	ADJ
ejpam-2803	134	14	second	second	ADJ
ejpam-2803	134	15	spectrum	spectrum	NOUN
ejpam-2803	134	16	,	,	PUNCT
ejpam-2803	134	17	we	we	PRON
ejpam-2803	134	18	need	need	VERB
ejpam-2803	134	19	to	to	PART
ejpam-2803	134	20	prove	prove	VERB
ejpam-2803	134	21	that	that	PRON
ejpam-2803	134	22	rp	rp	NOUN
ejpam-2803	134	23	/	/	SYM
ejpam-2803	134	24	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	134	25	,	,	PUNCT
ejpam-2803	134	26	m	m	NOUN
ejpam-2803	134	27	)	)	PUNCT
ejpam-2803	134	28	)	)	PUNCT
ejpam-2803	134	29	has	have	VERB
ejpam-2803	134	30	noetherian	noetherian	ADJ
ejpam-2803	134	31	spectrum	spectrum	NOUN
ejpam-2803	134	32	by	by	ADP
ejpam-2803	134	33	remark	remark	NOUN
ejpam-2803	134	34	2.1	2.1	NUM
ejpam-2803	134	35	(	(	PUNCT
ejpam-2803	134	36	b	b	NOUN
ejpam-2803	134	37	)	)	PUNCT
ejpam-2803	134	38	.	.	PUNCT
ejpam-2803	135	1	let	let	VERB
ejpam-2803	135	2	s	s	PRON
ejpam-2803	135	3	=	=	NOUN
ejpam-2803	135	4	r\p	r\p	PROPN
ejpam-2803	135	5	.	.	PUNCT
ejpam-2803	136	1	then	then	ADV
ejpam-2803	136	2	s	s	VERB
ejpam-2803	136	3	=	=	PUNCT
ejpam-2803	136	4	{	{	PUNCT
ejpam-2803	136	5	s	s	NOUN
ejpam-2803	136	6	+	+	NUM
ejpam-2803	136	7	ann(m	ann(m	PROPN
ejpam-2803	136	8	)	)	PUNCT
ejpam-2803	136	9	:	:	PUNCT
ejpam-2803	136	10	s	s	VERB
ejpam-2803	136	11	∈	∈	PROPN
ejpam-2803	136	12	s	s	PART
ejpam-2803	136	13	}	}	PUNCT
ejpam-2803	136	14	is	be	AUX
ejpam-2803	136	15	a	a	DET
ejpam-2803	136	16	multiplicative	multiplicative	ADJ
ejpam-2803	136	17	closed	close	VERB
ejpam-2803	136	18	subset	subset	NOUN
ejpam-2803	136	19	of	of	ADP
ejpam-2803	136	20	r	r	NOUN
ejpam-2803	136	21	=	=	SYM
ejpam-2803	136	22	r	r	NOUN
ejpam-2803	136	23	/	/	SYM
ejpam-2803	136	24	annr(m	annr(m	NOUN
ejpam-2803	136	25	)	)	PUNCT
ejpam-2803	136	26	.	.	PUNCT
ejpam-2803	137	1	let	let	VERB
ejpam-2803	137	2	φ	φ	NOUN
ejpam-2803	137	3	:	:	PUNCT
ejpam-2803	137	4	r→	r→	X
ejpam-2803	137	5	(	(	PUNCT
ejpam-2803	137	6	r)s	r)s	NOUN
ejpam-2803	137	7	be	be	AUX
ejpam-2803	137	8	the	the	DET
ejpam-2803	137	9	natural	natural	ADJ
ejpam-2803	137	10	homomorphism	homomorphism	NOUN
ejpam-2803	137	11	and	and	CCONJ
ejpam-2803	137	12	φ∗	φ∗	NOUN
ejpam-2803	137	13	:	:	PUNCT
ejpam-2803	137	14	spec((r)s)→	spec((r)s)→	NOUN
ejpam-2803	137	15	spec(r	spec(r	PROPN
ejpam-2803	137	16	)	)	PUNCT
ejpam-2803	137	17	the	the	DET
ejpam-2803	137	18	associated	associated	ADJ
ejpam-2803	137	19	mapping	mapping	NOUN
ejpam-2803	137	20	.	.	PUNCT
ejpam-2803	138	1	set	set	VERB
ejpam-2803	138	2	σ	σ	PROPN
ejpam-2803	138	3	=	=	PUNCT
ejpam-2803	138	4	{	{	PUNCT
ejpam-2803	138	5	p	p	NOUN
ejpam-2803	138	6	∈	∈	PROPN
ejpam-2803	138	7	spec(r	spec(r	PROPN
ejpam-2803	138	8	)	)	PUNCT
ejpam-2803	138	9	:	:	PUNCT
ejpam-2803	139	1	p	p	X
ejpam-2803	139	2	∩	∩	NOUN
ejpam-2803	139	3	s	s	PART
ejpam-2803	139	4	=	=	NOUN
ejpam-2803	139	5	∅	∅	NOUN
ejpam-2803	139	6	}	}	PUNCT
ejpam-2803	139	7	.	.	PUNCT
ejpam-2803	140	1	since	since	SCONJ
ejpam-2803	140	2	m	m	PROPN
ejpam-2803	140	3	is	be	AUX
ejpam-2803	140	4	secondful	secondful	ADJ
ejpam-2803	140	5	and	and	CCONJ
ejpam-2803	140	6	has	have	VERB
ejpam-2803	140	7	noetherian	noetherian	ADJ
ejpam-2803	140	8	spectrum	spectrum	NOUN
ejpam-2803	140	9	,	,	PUNCT
ejpam-2803	140	10	spec(r	spec(r	PROPN
ejpam-2803	140	11	)	)	PUNCT
ejpam-2803	140	12	is	be	AUX
ejpam-2803	140	13	noetherian	noetherian	ADJ
ejpam-2803	140	14	by	by	ADP
ejpam-2803	140	15	remark	remark	NOUN
ejpam-2803	140	16	2.1	2.1	NUM
ejpam-2803	140	17	(	(	PUNCT
ejpam-2803	140	18	b	b	NOUN
ejpam-2803	140	19	)	)	PUNCT
ejpam-2803	140	20	.	.	PUNCT
ejpam-2803	141	1	further	further	PROPN
ejpam-2803	141	2	σ	σ	PROPN
ejpam-2803	141	3	as	as	ADP
ejpam-2803	141	4	a	a	DET
ejpam-2803	141	5	subspace	subspace	NOUN
ejpam-2803	141	6	of	of	ADP
ejpam-2803	141	7	spec(r	spec(r	PROPN
ejpam-2803	141	8	)	)	PUNCT
ejpam-2803	141	9	is	be	AUX
ejpam-2803	141	10	a	a	DET
ejpam-2803	141	11	noetherian	noetherian	ADJ
ejpam-2803	141	12	space	space	NOUN
ejpam-2803	141	13	and	and	CCONJ
ejpam-2803	141	14	it	it	PRON
ejpam-2803	141	15	is	be	AUX
ejpam-2803	141	16	homeomorphic	homeomorphic	ADJ
ejpam-2803	141	17	to	to	ADP
ejpam-2803	141	18	spec((r)s	spec((r)s	PROPN
ejpam-2803	141	19	)	)	PUNCT
ejpam-2803	141	20	by	by	ADP
ejpam-2803	141	21	[	[	X
ejpam-2803	141	22	21	21	NUM
ejpam-2803	141	23	,	,	PUNCT
ejpam-2803	141	24	p.	p.	NOUN
ejpam-2803	141	25	81	81	NUM
ejpam-2803	141	26	,	,	PUNCT
ejpam-2803	141	27	proposition	proposition	NOUN
ejpam-2803	141	28	4.12	4.12	NUM
ejpam-2803	141	29	(	(	PUNCT
ejpam-2803	141	30	b	b	NOUN
ejpam-2803	141	31	)	)	PUNCT
ejpam-2803	141	32	]	]	PUNCT
ejpam-2803	141	33	.	.	PUNCT
ejpam-2803	142	1	hence	hence	ADV
ejpam-2803	142	2	spec((r)s	spec((r)s	PROPN
ejpam-2803	142	3	)	)	PUNCT
ejpam-2803	142	4	is	be	AUX
ejpam-2803	142	5	a	a	DET
ejpam-2803	142	6	noetherian	noetherian	ADJ
ejpam-2803	142	7	space	space	NOUN
ejpam-2803	142	8	.	.	PUNCT
ejpam-2803	143	1	also	also	ADV
ejpam-2803	143	2	,	,	PUNCT
ejpam-2803	143	3	we	we	PRON
ejpam-2803	143	4	have	have	VERB
ejpam-2803	143	5	(	(	PUNCT
ejpam-2803	143	6	r)s	r)s	NOUN
ejpam-2803	143	7	∼=	∼=	PART
ejpam-2803	143	8	rp/(annr(m))p	rp/(annr(m))p	PROPN
ejpam-2803	143	9	.	.	PUNCT
ejpam-2803	144	1	thus	thus	ADV
ejpam-2803	144	2	spec(rp/(annr(m))p	spec(rp/(annr(m))p	PROPN
ejpam-2803	144	3	)	)	PUNCT
ejpam-2803	144	4	is	be	AUX
ejpam-2803	144	5	a	a	DET
ejpam-2803	144	6	noetherian	noetherian	ADJ
ejpam-2803	144	7	space	space	NOUN
ejpam-2803	144	8	.	.	PUNCT
ejpam-2803	145	1	on	on	ADP
ejpam-2803	145	2	other	other	ADJ
ejpam-2803	145	3	hand	hand	NOUN
ejpam-2803	145	4	,	,	PUNCT
ejpam-2803	145	5	we	we	PRON
ejpam-2803	145	6	have	have	VERB
ejpam-2803	145	7	(	(	PUNCT
ejpam-2803	145	8	annr(m))p	annr(m))p	NOUN
ejpam-2803	145	9	⊆	⊆	NUM
ejpam-2803	145	10	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	145	11	,	,	PUNCT
ejpam-2803	145	12	m	m	NOUN
ejpam-2803	145	13	)	)	PUNCT
ejpam-2803	145	14	)	)	PUNCT
ejpam-2803	145	15	.	.	PUNCT
ejpam-2803	146	1	now	now	ADV
ejpam-2803	146	2	rp	rp	PROPN
ejpam-2803	146	3	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	146	4	,	,	PUNCT
ejpam-2803	146	5	m	m	NOUN
ejpam-2803	146	6	)	)	PUNCT
ejpam-2803	146	7	)	)	PUNCT
ejpam-2803	147	1	∼=	∼=	PROPN
ejpam-2803	147	2	(	(	PUNCT
ejpam-2803	147	3	rp	rp	NOUN
ejpam-2803	147	4	(	(	PUNCT
ejpam-2803	147	5	annr(m))p	annr(m))p	NOUN
ejpam-2803	147	6	)	)	PUNCT
ejpam-2803	147	7	/	/	PUNCT
ejpam-2803	147	8	(	(	PUNCT
ejpam-2803	147	9	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	147	10	,	,	PUNCT
ejpam-2803	147	11	m	m	NOUN
ejpam-2803	147	12	)	)	PUNCT
ejpam-2803	147	13	)	)	PUNCT
ejpam-2803	147	14	(	(	PUNCT
ejpam-2803	147	15	annr(m))p	annr(m))p	NOUN
ejpam-2803	147	16	)	)	PUNCT
ejpam-2803	147	17	.	.	PUNCT
ejpam-2803	148	1	h.	h.	PROPN
ejpam-2803	148	2	ansari	ansari	PROPN
ejpam-2803	148	3	-	-	PUNCT
ejpam-2803	148	4	toroghy	toroghy	NOUN
ejpam-2803	148	5	,	,	PUNCT
ejpam-2803	148	6	s.	s.	PROPN
ejpam-2803	148	7	s.	s.	PROPN
ejpam-2803	148	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	148	9	/	/	SYM
ejpam-2803	148	10	eur	eur	PROPN
ejpam-2803	148	11	.	.	PUNCT
ejpam-2803	149	1	j.	j.	PROPN
ejpam-2803	149	2	pure	pure	PROPN
ejpam-2803	149	3	appl	appl	PROPN
ejpam-2803	149	4	.	.	PROPN
ejpam-2803	149	5	math	math	PROPN
ejpam-2803	149	6	,	,	PUNCT
ejpam-2803	149	7	10	10	NUM
ejpam-2803	149	8	(	(	PUNCT
ejpam-2803	149	9	2	2	NUM
ejpam-2803	149	10	)	)	PUNCT
ejpam-2803	149	11	(	(	PUNCT
ejpam-2803	149	12	2017	2017	NUM
ejpam-2803	149	13	)	)	PUNCT
ejpam-2803	149	14	,	,	PUNCT
ejpam-2803	149	15	211	211	NUM
ejpam-2803	149	16	-	-	SYM
ejpam-2803	149	17	230	230	NUM
ejpam-2803	149	18	215	215	NUM
ejpam-2803	149	19	from	from	ADP
ejpam-2803	149	20	this	this	PRON
ejpam-2803	149	21	,	,	PUNCT
ejpam-2803	149	22	we	we	PRON
ejpam-2803	149	23	deduce	deduce	VERB
ejpam-2803	149	24	that	that	DET
ejpam-2803	149	25	rp	rp	PROPN
ejpam-2803	149	26	/	/	SYM
ejpam-2803	149	27	annrp(homr(rp	annrp(homr(rp	PROPN
ejpam-2803	149	28	,	,	PUNCT
ejpam-2803	149	29	m	m	NOUN
ejpam-2803	149	30	)	)	PUNCT
ejpam-2803	149	31	)	)	PUNCT
ejpam-2803	149	32	has	have	VERB
ejpam-2803	149	33	noetherian	noetherian	ADJ
ejpam-2803	149	34	spectrum	spectrum	NOUN
ejpam-2803	149	35	.	.	PUNCT
ejpam-2803	150	1	let	let	VERB
ejpam-2803	150	2	t	t	NOUN
ejpam-2803	150	3	be	be	AUX
ejpam-2803	150	4	a	a	DET
ejpam-2803	150	5	topological	topological	ADJ
ejpam-2803	150	6	space	space	NOUN
ejpam-2803	150	7	and	and	CCONJ
ejpam-2803	150	8	t	t	NOUN
ejpam-2803	150	9	′	′	NUM
ejpam-2803	150	10	⊆	⊆	NUM
ejpam-2803	150	11	t	t	NOUN
ejpam-2803	150	12	.	.	PUNCT
ejpam-2803	151	1	then	then	ADV
ejpam-2803	151	2	t	t	PROPN
ejpam-2803	151	3	is	be	AUX
ejpam-2803	151	4	irreducible	irreducible	ADJ
ejpam-2803	151	5	if	if	SCONJ
ejpam-2803	151	6	t	t	PROPN
ejpam-2803	151	7	6=	6=	NOUN
ejpam-2803	151	8	∅	∅	NOUN
ejpam-2803	151	9	and	and	CCONJ
ejpam-2803	151	10	for	for	ADP
ejpam-2803	151	11	every	every	DET
ejpam-2803	151	12	decomposition	decomposition	NOUN
ejpam-2803	151	13	t	t	NOUN
ejpam-2803	151	14	=	=	NOUN
ejpam-2803	151	15	a1	a1	PROPN
ejpam-2803	151	16	∪	∪	NOUN
ejpam-2803	151	17	a2	a2	PROPN
ejpam-2803	151	18	with	with	ADP
ejpam-2803	151	19	closed	closed	ADJ
ejpam-2803	151	20	subsets	subset	NOUN
ejpam-2803	151	21	ai	ai	VERB
ejpam-2803	151	22	⊆	⊆	NUM
ejpam-2803	151	23	t	t	NOUN
ejpam-2803	151	24	,	,	PUNCT
ejpam-2803	151	25	i	i	PRON
ejpam-2803	151	26	=	=	NOUN
ejpam-2803	151	27	1	1	NUM
ejpam-2803	151	28	,	,	PUNCT
ejpam-2803	151	29	2	2	NUM
ejpam-2803	151	30	,	,	PUNCT
ejpam-2803	151	31	we	we	PRON
ejpam-2803	151	32	have	have	AUX
ejpam-2803	151	33	a1	a1	NOUN
ejpam-2803	151	34	=	=	PROPN
ejpam-2803	151	35	t	t	PROPN
ejpam-2803	151	36	or	or	CCONJ
ejpam-2803	151	37	a2	a2	PROPN
ejpam-2803	151	38	=	=	SYM
ejpam-2803	151	39	t	t	PROPN
ejpam-2803	151	40	.	.	PUNCT
ejpam-2803	152	1	the	the	DET
ejpam-2803	152	2	subset	subset	NOUN
ejpam-2803	152	3	t	t	PROPN
ejpam-2803	152	4	′	′	NOUN
ejpam-2803	152	5	of	of	ADP
ejpam-2803	152	6	t	t	PROPN
ejpam-2803	152	7	is	be	AUX
ejpam-2803	152	8	irreducible	irreducible	ADJ
ejpam-2803	152	9	if	if	SCONJ
ejpam-2803	152	10	it	it	PRON
ejpam-2803	152	11	is	be	AUX
ejpam-2803	152	12	irreducible	irreducible	ADJ
ejpam-2803	152	13	as	as	ADP
ejpam-2803	152	14	a	a	DET
ejpam-2803	152	15	space	space	NOUN
ejpam-2803	152	16	with	with	ADP
ejpam-2803	152	17	the	the	DET
ejpam-2803	152	18	relative	relative	ADJ
ejpam-2803	152	19	topology	topology	NOUN
ejpam-2803	152	20	.	.	PUNCT
ejpam-2803	153	1	equivalently	equivalently	ADV
ejpam-2803	153	2	,	,	PUNCT
ejpam-2803	153	3	t	t	PROPN
ejpam-2803	153	4	′	′	NUM
ejpam-2803	153	5	is	be	AUX
ejpam-2803	153	6	irreducible	irreducible	ADJ
ejpam-2803	153	7	if	if	SCONJ
ejpam-2803	153	8	and	and	CCONJ
ejpam-2803	153	9	only	only	ADV
ejpam-2803	153	10	if	if	SCONJ
ejpam-2803	153	11	for	for	ADP
ejpam-2803	153	12	every	every	DET
ejpam-2803	153	13	pair	pair	NOUN
ejpam-2803	153	14	of	of	ADP
ejpam-2803	153	15	sets	set	NOUN
ejpam-2803	153	16	f	f	X
ejpam-2803	153	17	,	,	PUNCT
ejpam-2803	153	18	g	g	NOUN
ejpam-2803	153	19	which	which	PRON
ejpam-2803	153	20	are	be	AUX
ejpam-2803	153	21	closed	close	VERB
ejpam-2803	153	22	in	in	ADP
ejpam-2803	153	23	t	t	PROPN
ejpam-2803	153	24	,	,	PUNCT
ejpam-2803	153	25	it	it	PRON
ejpam-2803	153	26	holds	hold	VERB
ejpam-2803	153	27	that	that	SCONJ
ejpam-2803	153	28	t	t	NOUN
ejpam-2803	154	1	′	′	NUM
ejpam-2803	155	1	(	(	PUNCT
ejpam-2803	155	2	f	f	PROPN
ejpam-2803	155	3	∪g	∪g	PROPN
ejpam-2803	155	4	,	,	PUNCT
ejpam-2803	155	5	t	t	NOUN
ejpam-2803	155	6	′	′	NUM
ejpam-2803	156	1	(	(	PUNCT
ejpam-2803	156	2	f	f	PROPN
ejpam-2803	156	3	or	or	CCONJ
ejpam-2803	156	4	t	t	PROPN
ejpam-2803	156	5	′	′	NUM
ejpam-2803	157	1	(	(	PUNCT
ejpam-2803	157	2	g	g	PROPN
ejpam-2803	157	3	[	[	X
ejpam-2803	157	4	13	13	NUM
ejpam-2803	157	5	,	,	PUNCT
ejpam-2803	157	6	p.	p.	NOUN
ejpam-2803	157	7	94	94	NUM
ejpam-2803	157	8	]	]	PUNCT
ejpam-2803	157	9	.	.	PUNCT
ejpam-2803	158	1	an	an	DET
ejpam-2803	158	2	irreducible	irreducible	ADJ
ejpam-2803	158	3	component	component	NOUN
ejpam-2803	158	4	of	of	ADP
ejpam-2803	158	5	t	t	PROPN
ejpam-2803	158	6	is	be	AUX
ejpam-2803	158	7	a	a	DET
ejpam-2803	158	8	maximal	maximal	ADJ
ejpam-2803	158	9	irreducible	irreducible	ADJ
ejpam-2803	158	10	subset	subset	NOUN
ejpam-2803	158	11	of	of	ADP
ejpam-2803	158	12	t	t	PROPN
ejpam-2803	158	13	.	.	PUNCT
ejpam-2803	159	1	every	every	DET
ejpam-2803	159	2	irreducible	irreducible	ADJ
ejpam-2803	159	3	subset	subset	NOUN
ejpam-2803	159	4	of	of	ADP
ejpam-2803	159	5	t	t	PROPN
ejpam-2803	159	6	is	be	AUX
ejpam-2803	159	7	contained	contain	VERB
ejpam-2803	159	8	in	in	ADP
ejpam-2803	159	9	an	an	DET
ejpam-2803	159	10	irreducible	irreducible	ADJ
ejpam-2803	159	11	component	component	NOUN
ejpam-2803	159	12	of	of	ADP
ejpam-2803	159	13	t	t	PROPN
ejpam-2803	159	14	,	,	PUNCT
ejpam-2803	159	15	and	and	CCONJ
ejpam-2803	159	16	t	t	PROPN
ejpam-2803	159	17	is	be	AUX
ejpam-2803	159	18	the	the	DET
ejpam-2803	159	19	union	union	NOUN
ejpam-2803	159	20	of	of	ADP
ejpam-2803	159	21	its	its	PRON
ejpam-2803	159	22	irreducible	irreducible	ADJ
ejpam-2803	159	23	components	component	NOUN
ejpam-2803	159	24	.	.	PUNCT
ejpam-2803	160	1	let	let	VERB
ejpam-2803	160	2	z	z	PRON
ejpam-2803	160	3	be	be	AUX
ejpam-2803	160	4	a	a	DET
ejpam-2803	160	5	subset	subset	NOUN
ejpam-2803	160	6	of	of	ADP
ejpam-2803	160	7	a	a	DET
ejpam-2803	160	8	topological	topological	ADJ
ejpam-2803	160	9	space	space	NOUN
ejpam-2803	160	10	w	w	NOUN
ejpam-2803	160	11	.	.	PUNCT
ejpam-2803	161	1	then	then	ADV
ejpam-2803	161	2	the	the	DET
ejpam-2803	161	3	notion	notion	NOUN
ejpam-2803	161	4	cl(z	cl(z	NOUN
ejpam-2803	161	5	)	)	PUNCT
ejpam-2803	161	6	will	will	AUX
ejpam-2803	161	7	denote	denote	VERB
ejpam-2803	161	8	the	the	DET
ejpam-2803	161	9	closure	closure	NOUN
ejpam-2803	161	10	of	of	ADP
ejpam-2803	161	11	z	z	PROPN
ejpam-2803	161	12	in	in	ADP
ejpam-2803	161	13	w	w	PROPN
ejpam-2803	161	14	.	.	PUNCT
ejpam-2803	162	1	let	let	VERB
ejpam-2803	162	2	y	y	PRON
ejpam-2803	162	3	be	be	AUX
ejpam-2803	162	4	a	a	DET
ejpam-2803	162	5	closed	closed	ADJ
ejpam-2803	162	6	subset	subset	NOUN
ejpam-2803	162	7	of	of	ADP
ejpam-2803	162	8	a	a	DET
ejpam-2803	162	9	topological	topological	ADJ
ejpam-2803	162	10	space	space	NOUN
ejpam-2803	162	11	.	.	PUNCT
ejpam-2803	163	1	an	an	DET
ejpam-2803	163	2	element	element	NOUN
ejpam-2803	163	3	y	y	PROPN
ejpam-2803	163	4	∈	∈	PROPN
ejpam-2803	163	5	y	y	PROPN
ejpam-2803	163	6	is	be	AUX
ejpam-2803	163	7	called	call	VERB
ejpam-2803	163	8	a	a	DET
ejpam-2803	163	9	generic	generic	ADJ
ejpam-2803	163	10	point	point	NOUN
ejpam-2803	163	11	of	of	ADP
ejpam-2803	163	12	y	y	PROPN
ejpam-2803	163	13	if	if	SCONJ
ejpam-2803	163	14	y	y	PROPN
ejpam-2803	163	15	=	=	PUNCT
ejpam-2803	163	16	cl({y	cl({y	PROPN
ejpam-2803	163	17	}	}	PUNCT
ejpam-2803	163	18	)	)	PUNCT
ejpam-2803	163	19	.	.	PUNCT
ejpam-2803	164	1	if	if	SCONJ
ejpam-2803	164	2	the	the	DET
ejpam-2803	164	3	topological	topological	ADJ
ejpam-2803	164	4	space	space	NOUN
ejpam-2803	164	5	is	be	AUX
ejpam-2803	164	6	a	a	DET
ejpam-2803	164	7	t0	t0	NOUN
ejpam-2803	164	8	-	-	NOUN
ejpam-2803	164	9	space	space	NOUN
ejpam-2803	164	10	,	,	PUNCT
ejpam-2803	164	11	then	then	ADV
ejpam-2803	164	12	a	a	DET
ejpam-2803	164	13	generic	generic	ADJ
ejpam-2803	164	14	point	point	NOUN
ejpam-2803	164	15	of	of	ADP
ejpam-2803	164	16	every	every	DET
ejpam-2803	164	17	closed	closed	ADJ
ejpam-2803	164	18	subset	subset	NOUN
ejpam-2803	164	19	is	be	AUX
ejpam-2803	164	20	unique	unique	ADJ
ejpam-2803	164	21	.	.	PUNCT
ejpam-2803	165	1	proposition	proposition	NOUN
ejpam-2803	165	2	2.5	2.5	NUM
ejpam-2803	165	3	.	.	PUNCT
ejpam-2803	166	1	let	let	VERB
ejpam-2803	166	2	m	m	PRON
ejpam-2803	166	3	be	be	AUX
ejpam-2803	166	4	a	a	DET
ejpam-2803	166	5	secondful	secondful	ADJ
ejpam-2803	166	6	module	module	NOUN
ejpam-2803	166	7	over	over	ADP
ejpam-2803	166	8	r	r	NOUN
ejpam-2803	166	9	and	and	CCONJ
ejpam-2803	166	10	n	n	CCONJ
ejpam-2803	166	11	≤m	≤m	NOUN
ejpam-2803	166	12	.	.	PUNCT
ejpam-2803	167	1	(	(	PUNCT
ejpam-2803	167	2	a	a	X
ejpam-2803	167	3	)	)	PUNCT
ejpam-2803	167	4	let	let	VERB
ejpam-2803	167	5	y	y	PRON
ejpam-2803	167	6	be	be	AUX
ejpam-2803	167	7	a	a	DET
ejpam-2803	167	8	nonempty	nonempty	ADJ
ejpam-2803	167	9	subset	subset	NOUN
ejpam-2803	167	10	of	of	ADP
ejpam-2803	167	11	v	v	NOUN
ejpam-2803	167	12	s(n	s(n	NOUN
ejpam-2803	167	13	)	)	PUNCT
ejpam-2803	167	14	.	.	PUNCT
ejpam-2803	168	1	then	then	ADV
ejpam-2803	168	2	y	y	PROPN
ejpam-2803	168	3	is	be	AUX
ejpam-2803	168	4	an	an	DET
ejpam-2803	168	5	irreducible	irreducible	ADJ
ejpam-2803	168	6	closed	closed	ADJ
ejpam-2803	168	7	subset	subset	NOUN
ejpam-2803	168	8	of	of	ADP
ejpam-2803	168	9	v	v	NOUN
ejpam-2803	168	10	s(n	s(n	NOUN
ejpam-2803	168	11	)	)	PUNCT
ejpam-2803	169	1	if	if	SCONJ
ejpam-2803	169	2	and	and	CCONJ
ejpam-2803	169	3	only	only	ADV
ejpam-2803	169	4	if	if	SCONJ
ejpam-2803	169	5	y	y	PROPN
ejpam-2803	169	6	has	have	VERB
ejpam-2803	169	7	a	a	DET
ejpam-2803	169	8	generic	generic	ADJ
ejpam-2803	169	9	point	point	NOUN
ejpam-2803	169	10	in	in	ADP
ejpam-2803	169	11	v	v	ADP
ejpam-2803	169	12	s(n	s(n	NOUN
ejpam-2803	169	13	)	)	PUNCT
ejpam-2803	169	14	.	.	PUNCT
ejpam-2803	170	1	(	(	PUNCT
ejpam-2803	170	2	b	b	X
ejpam-2803	170	3	)	)	PUNCT
ejpam-2803	170	4	the	the	DET
ejpam-2803	170	5	mapping	mapping	NOUN
ejpam-2803	170	6	ρ	ρ	NOUN
ejpam-2803	170	7	:	:	PUNCT
ejpam-2803	170	8	s	s	AUX
ejpam-2803	170	9	7→	7→	NUM
ejpam-2803	170	10	v	v	ADP
ejpam-2803	170	11	s(s	s(s	PROPN
ejpam-2803	170	12	)	)	PUNCT
ejpam-2803	170	13	is	be	AUX
ejpam-2803	170	14	a	a	DET
ejpam-2803	170	15	surjection	surjection	NOUN
ejpam-2803	170	16	of	of	ADP
ejpam-2803	170	17	v	v	NOUN
ejpam-2803	170	18	s(n	s(n	NOUN
ejpam-2803	170	19	)	)	PUNCT
ejpam-2803	170	20	onto	onto	ADP
ejpam-2803	170	21	the	the	DET
ejpam-2803	170	22	set	set	NOUN
ejpam-2803	170	23	of	of	ADP
ejpam-2803	170	24	irreducible	irreducible	ADJ
ejpam-2803	170	25	closed	closed	ADJ
ejpam-2803	170	26	subsets	subset	NOUN
ejpam-2803	170	27	of	of	ADP
ejpam-2803	170	28	v	v	NOUN
ejpam-2803	170	29	s(n	s(n	NOUN
ejpam-2803	170	30	)	)	PUNCT
ejpam-2803	170	31	.	.	PUNCT
ejpam-2803	171	1	(	(	PUNCT
ejpam-2803	171	2	c	c	X
ejpam-2803	171	3	)	)	PUNCT
ejpam-2803	171	4	the	the	DET
ejpam-2803	171	5	mapping	mapping	NOUN
ejpam-2803	171	6	φ	φ	PROPN
ejpam-2803	171	7	:	:	PUNCT
ejpam-2803	171	8	v	v	X
ejpam-2803	171	9	s(s	s(s	PROPN
ejpam-2803	171	10	)	)	PUNCT
ejpam-2803	171	11	7→	7→	NUM
ejpam-2803	171	12	annr(s	annr(s	NOUN
ejpam-2803	171	13	)	)	PUNCT
ejpam-2803	171	14	∈	∈	PROPN
ejpam-2803	171	15	spec(r	spec(r	PROPN
ejpam-2803	171	16	)	)	PUNCT
ejpam-2803	171	17	is	be	AUX
ejpam-2803	171	18	a	a	DET
ejpam-2803	171	19	bijection	bijection	NOUN
ejpam-2803	171	20	of	of	ADP
ejpam-2803	171	21	the	the	DET
ejpam-2803	171	22	set	set	NOUN
ejpam-2803	171	23	of	of	ADP
ejpam-2803	171	24	irreducible	irreducible	ADJ
ejpam-2803	171	25	components	component	NOUN
ejpam-2803	171	26	of	of	ADP
ejpam-2803	171	27	v	v	PRON
ejpam-2803	171	28	s(n	s(n	NOUN
ejpam-2803	171	29	)	)	PUNCT
ejpam-2803	171	30	onto	onto	ADP
ejpam-2803	171	31	the	the	DET
ejpam-2803	171	32	set	set	NOUN
ejpam-2803	171	33	of	of	ADP
ejpam-2803	171	34	minimal	minimal	ADJ
ejpam-2803	171	35	prime	prime	ADJ
ejpam-2803	171	36	divisors	divisor	NOUN
ejpam-2803	171	37	of	of	ADP
ejpam-2803	171	38	annr(n	annr(n	PROPN
ejpam-2803	171	39	)	)	PUNCT
ejpam-2803	171	40	in	in	ADP
ejpam-2803	171	41	r	r	NOUN
ejpam-2803	171	42	=	=	SYM
ejpam-2803	171	43	r	r	NOUN
ejpam-2803	171	44	/	/	SYM
ejpam-2803	171	45	annr(m	annr(m	NOUN
ejpam-2803	171	46	)	)	PUNCT
ejpam-2803	171	47	.	.	PUNCT
ejpam-2803	172	1	proof	proof	NOUN
ejpam-2803	172	2	.	.	PUNCT
ejpam-2803	173	1	(	(	PUNCT
ejpam-2803	173	2	a	a	X
ejpam-2803	173	3	)	)	PUNCT
ejpam-2803	173	4	this	this	PRON
ejpam-2803	173	5	is	be	AUX
ejpam-2803	173	6	clear	clear	ADJ
ejpam-2803	173	7	.	.	PUNCT
ejpam-2803	174	1	(	(	PUNCT
ejpam-2803	174	2	b	b	X
ejpam-2803	174	3	)	)	PUNCT
ejpam-2803	174	4	this	this	PRON
ejpam-2803	174	5	follows	follow	VERB
ejpam-2803	174	6	directly	directly	ADV
ejpam-2803	174	7	from	from	ADP
ejpam-2803	174	8	part	part	NOUN
ejpam-2803	174	9	(	(	PUNCT
ejpam-2803	174	10	a	a	NOUN
ejpam-2803	174	11	)	)	PUNCT
ejpam-2803	174	12	.	.	PUNCT
ejpam-2803	175	1	(	(	PUNCT
ejpam-2803	175	2	c	c	X
ejpam-2803	175	3	)	)	PUNCT
ejpam-2803	175	4	let	let	VERB
ejpam-2803	175	5	φ	φ	PROPN
ejpam-2803	175	6	:	:	PUNCT
ejpam-2803	175	7	v	v	X
ejpam-2803	175	8	s(s	s(s	PROPN
ejpam-2803	175	9	)	)	PUNCT
ejpam-2803	175	10	7→	7→	NUM
ejpam-2803	175	11	annr(s	annr(s	NOUN
ejpam-2803	175	12	)	)	PUNCT
ejpam-2803	175	13	.	.	PUNCT
ejpam-2803	176	1	then	then	ADV
ejpam-2803	176	2	φ	φ	PROPN
ejpam-2803	176	3	is	be	AUX
ejpam-2803	176	4	a	a	DET
ejpam-2803	176	5	well	well	ADV
ejpam-2803	176	6	defined	define	VERB
ejpam-2803	176	7	injective	injective	ADJ
ejpam-2803	176	8	mapping	mapping	NOUN
ejpam-2803	176	9	by	by	ADP
ejpam-2803	176	10	part	part	NOUN
ejpam-2803	176	11	(	(	PUNCT
ejpam-2803	176	12	a	a	NOUN
ejpam-2803	176	13	)	)	PUNCT
ejpam-2803	176	14	.	.	PUNCT
ejpam-2803	177	1	we	we	PRON
ejpam-2803	177	2	show	show	VERB
ejpam-2803	177	3	that	that	SCONJ
ejpam-2803	177	4	φ	φ	PROPN
ejpam-2803	177	5	is	be	AUX
ejpam-2803	177	6	surjective	surjective	ADJ
ejpam-2803	177	7	.	.	PUNCT
ejpam-2803	178	1	let	let	VERB
ejpam-2803	178	2	p	p	PRON
ejpam-2803	178	3	be	be	AUX
ejpam-2803	178	4	a	a	DET
ejpam-2803	178	5	minimal	minimal	ADJ
ejpam-2803	178	6	prime	prime	ADJ
ejpam-2803	178	7	divisor	divisor	NOUN
ejpam-2803	178	8	of	of	ADP
ejpam-2803	178	9	annr(n	annr(n	PROPN
ejpam-2803	178	10	)	)	PUNCT
ejpam-2803	178	11	in	in	ADP
ejpam-2803	178	12	r	r	NOUN
ejpam-2803	178	13	and	and	CCONJ
ejpam-2803	178	14	let	let	VERB
ejpam-2803	178	15	p	p	PRON
ejpam-2803	178	16	be	be	AUX
ejpam-2803	178	17	the	the	DET
ejpam-2803	178	18	prime	prime	ADJ
ejpam-2803	178	19	ideal	ideal	NOUN
ejpam-2803	178	20	of	of	ADP
ejpam-2803	178	21	r	r	NOUN
ejpam-2803	178	22	such	such	ADJ
ejpam-2803	178	23	that	that	SCONJ
ejpam-2803	178	24	p	p	X
ejpam-2803	178	25	/	/	SYM
ejpam-2803	178	26	annr(m	annr(m	NOUN
ejpam-2803	178	27	)	)	PUNCT
ejpam-2803	178	28	=	=	VERB
ejpam-2803	179	1	p.	p.	NOUN
ejpam-2803	179	2	then	then	ADV
ejpam-2803	179	3	annr(m	annr(m	PROPN
ejpam-2803	179	4	)	)	PUNCT
ejpam-2803	179	5	⊆	⊆	NUM
ejpam-2803	179	6	annr(n	annr(n	NOUN
ejpam-2803	179	7	)	)	PUNCT
ejpam-2803	179	8	⊆	⊆	NUM
ejpam-2803	179	9	p.	p.	NOUN
ejpam-2803	179	10	since	since	SCONJ
ejpam-2803	179	11	m	m	PROPN
ejpam-2803	179	12	is	be	AUX
ejpam-2803	179	13	secondful	secondful	ADJ
ejpam-2803	179	14	,	,	PUNCT
ejpam-2803	179	15	there	there	PRON
ejpam-2803	179	16	exists	exist	VERB
ejpam-2803	179	17	a	a	DET
ejpam-2803	179	18	p	p	ADJ
ejpam-2803	179	19	-	-	PUNCT
ejpam-2803	179	20	second	second	NOUN
ejpam-2803	179	21	submodule	submodule	NOUN
ejpam-2803	179	22	s	s	PROPN
ejpam-2803	179	23	∈	∈	PROPN
ejpam-2803	179	24	xs	xs	PROPN
ejpam-2803	179	25	.	.	PUNCT
ejpam-2803	180	1	now	now	ADV
ejpam-2803	180	2	annr(n	annr(n	VERB
ejpam-2803	180	3	)	)	PUNCT
ejpam-2803	180	4	⊆	⊆	NUM
ejpam-2803	180	5	p	p	NOUN
ejpam-2803	180	6	=	=	PROPN
ejpam-2803	180	7	annr(s	annr(s	PROPN
ejpam-2803	180	8	)	)	PUNCT
ejpam-2803	180	9	implies	imply	VERB
ejpam-2803	180	10	that	that	SCONJ
ejpam-2803	180	11	s	s	VERB
ejpam-2803	180	12	∈	∈	NOUN
ejpam-2803	180	13	v	v	ADP
ejpam-2803	180	14	s(n	s(n	NOUN
ejpam-2803	180	15	)	)	PUNCT
ejpam-2803	180	16	,	,	PUNCT
ejpam-2803	180	17	and	and	CCONJ
ejpam-2803	180	18	so	so	ADV
ejpam-2803	180	19	v	v	ADP
ejpam-2803	180	20	s(s	s(s	PROPN
ejpam-2803	180	21	)	)	PUNCT
ejpam-2803	180	22	⊆	⊆	NUM
ejpam-2803	180	23	v	v	ADP
ejpam-2803	180	24	s(n	s(n	NOUN
ejpam-2803	180	25	)	)	PUNCT
ejpam-2803	180	26	.	.	PUNCT
ejpam-2803	181	1	thus	thus	ADV
ejpam-2803	181	2	v	v	ADP
ejpam-2803	181	3	s(s	s(s	PROPN
ejpam-2803	181	4	)	)	PUNCT
ejpam-2803	181	5	is	be	AUX
ejpam-2803	181	6	an	an	DET
ejpam-2803	181	7	irreducible	irreducible	ADJ
ejpam-2803	181	8	closed	closed	ADJ
ejpam-2803	181	9	subset	subset	NOUN
ejpam-2803	181	10	of	of	ADP
ejpam-2803	181	11	v	v	NOUN
ejpam-2803	181	12	s(n	s(n	NOUN
ejpam-2803	181	13	)	)	PUNCT
ejpam-2803	181	14	.	.	PUNCT
ejpam-2803	182	1	note	note	VERB
ejpam-2803	182	2	that	that	SCONJ
ejpam-2803	182	3	the	the	DET
ejpam-2803	182	4	minimality	minimality	NOUN
ejpam-2803	182	5	of	of	ADP
ejpam-2803	182	6	p	p	PROPN
ejpam-2803	182	7	∈	∈	PROPN
ejpam-2803	182	8	v	v	NOUN
ejpam-2803	182	9	(	(	PUNCT
ejpam-2803	182	10	annr(n	annr(n	PROPN
ejpam-2803	182	11	)	)	PUNCT
ejpam-2803	182	12	)	)	PUNCT
ejpam-2803	182	13	implies	imply	VERB
ejpam-2803	182	14	the	the	DET
ejpam-2803	182	15	maximality	maximality	NOUN
ejpam-2803	182	16	of	of	ADP
ejpam-2803	182	17	v	v	PROPN
ejpam-2803	182	18	s(s	s(s	PROPN
ejpam-2803	182	19	)	)	PUNCT
ejpam-2803	182	20	among	among	ADP
ejpam-2803	182	21	all	all	DET
ejpam-2803	182	22	irreducible	irreducible	ADJ
ejpam-2803	182	23	closed	close	VERB
ejpam-2803	182	24	subsets	subset	NOUN
ejpam-2803	182	25	v	v	ADP
ejpam-2803	182	26	s(s′	s(s′	NOUN
ejpam-2803	182	27	)	)	PUNCT
ejpam-2803	182	28	,	,	PUNCT
ejpam-2803	182	29	s′	s′	ADJ
ejpam-2803	182	30	∈	∈	PROPN
ejpam-2803	182	31	v	v	ADP
ejpam-2803	182	32	s(n	s(n	NOUN
ejpam-2803	182	33	)	)	PUNCT
ejpam-2803	182	34	,	,	PUNCT
ejpam-2803	182	35	as	as	ADP
ejpam-2803	182	36	annr(n	annr(n	VERB
ejpam-2803	182	37	)	)	PUNCT
ejpam-2803	182	38	⊆	⊆	NUM
ejpam-2803	182	39	annr(s′	annr(s′	NUM
ejpam-2803	182	40	)	)	PUNCT
ejpam-2803	182	41	.	.	PUNCT
ejpam-2803	183	1	therefore	therefore	ADV
ejpam-2803	183	2	,	,	PUNCT
ejpam-2803	183	3	v	v	PROPN
ejpam-2803	183	4	s(s	s(s	PROPN
ejpam-2803	183	5	)	)	PUNCT
ejpam-2803	183	6	is	be	AUX
ejpam-2803	183	7	an	an	DET
ejpam-2803	183	8	irreducible	irreducible	ADJ
ejpam-2803	183	9	component	component	NOUN
ejpam-2803	183	10	of	of	ADP
ejpam-2803	183	11	v	v	NOUN
ejpam-2803	183	12	s(n	s(n	NOUN
ejpam-2803	183	13	)	)	PUNCT
ejpam-2803	183	14	.	.	PUNCT
ejpam-2803	184	1	this	this	PRON
ejpam-2803	184	2	proves	prove	VERB
ejpam-2803	184	3	that	that	SCONJ
ejpam-2803	184	4	φ	φ	PROPN
ejpam-2803	184	5	is	be	AUX
ejpam-2803	184	6	surjective	surjective	ADJ
ejpam-2803	184	7	.	.	PUNCT
ejpam-2803	185	1	h.	h.	PROPN
ejpam-2803	185	2	ansari	ansari	PROPN
ejpam-2803	185	3	-	-	PUNCT
ejpam-2803	185	4	toroghy	toroghy	NOUN
ejpam-2803	185	5	,	,	PUNCT
ejpam-2803	185	6	s.	s.	PROPN
ejpam-2803	185	7	s.	s.	PROPN
ejpam-2803	185	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	185	9	/	/	SYM
ejpam-2803	185	10	eur	eur	PROPN
ejpam-2803	185	11	.	.	PUNCT
ejpam-2803	186	1	j.	j.	PROPN
ejpam-2803	186	2	pure	pure	PROPN
ejpam-2803	186	3	appl	appl	PROPN
ejpam-2803	186	4	.	.	PROPN
ejpam-2803	186	5	math	math	PROPN
ejpam-2803	186	6	,	,	PUNCT
ejpam-2803	186	7	10	10	NUM
ejpam-2803	186	8	(	(	PUNCT
ejpam-2803	186	9	2	2	NUM
ejpam-2803	186	10	)	)	PUNCT
ejpam-2803	186	11	(	(	PUNCT
ejpam-2803	186	12	2017	2017	NUM
ejpam-2803	186	13	)	)	PUNCT
ejpam-2803	186	14	,	,	PUNCT
ejpam-2803	186	15	211	211	NUM
ejpam-2803	186	16	-	-	SYM
ejpam-2803	186	17	230	230	NUM
ejpam-2803	186	18	216	216	NUM
ejpam-2803	186	19	proposition	proposition	NOUN
ejpam-2803	186	20	2.6	2.6	NUM
ejpam-2803	186	21	.	.	PUNCT
ejpam-2803	187	1	let	let	VERB
ejpam-2803	187	2	m	m	PRON
ejpam-2803	187	3	be	be	AUX
ejpam-2803	187	4	a	a	DET
ejpam-2803	187	5	secondful	secondful	ADJ
ejpam-2803	187	6	r	r	NOUN
ejpam-2803	187	7	-	-	PUNCT
ejpam-2803	187	8	module	module	NOUN
ejpam-2803	187	9	.	.	PUNCT
ejpam-2803	188	1	then	then	ADV
ejpam-2803	188	2	specs(m	specs(m	PROPN
ejpam-2803	188	3	)	)	PUNCT
ejpam-2803	188	4	has	have	VERB
ejpam-2803	188	5	a	a	DET
ejpam-2803	188	6	chain	chain	NOUN
ejpam-2803	188	7	of	of	ADP
ejpam-2803	188	8	irreducible	irreducible	ADJ
ejpam-2803	188	9	closed	close	VERB
ejpam-2803	188	10	subsets	subset	NOUN
ejpam-2803	188	11	of	of	ADP
ejpam-2803	188	12	length	length	NOUN
ejpam-2803	188	13	r	r	NOUN
ejpam-2803	188	14	if	if	SCONJ
ejpam-2803	189	1	and	and	CCONJ
ejpam-2803	189	2	only	only	ADV
ejpam-2803	189	3	if	if	SCONJ
ejpam-2803	189	4	r	r	NOUN
ejpam-2803	189	5	has	have	VERB
ejpam-2803	189	6	a	a	DET
ejpam-2803	189	7	chain	chain	NOUN
ejpam-2803	189	8	of	of	ADP
ejpam-2803	189	9	prime	prime	ADJ
ejpam-2803	189	10	ideals	ideal	NOUN
ejpam-2803	189	11	of	of	ADP
ejpam-2803	189	12	length	length	NOUN
ejpam-2803	189	13	r.	r.	PROPN
ejpam-2803	189	14	proof	proof	PROPN
ejpam-2803	189	15	.	.	PUNCT
ejpam-2803	190	1	let	let	VERB
ejpam-2803	190	2	z0	z0	PROPN
ejpam-2803	190	3	(	(	PUNCT
ejpam-2803	190	4	z1	z1	PROPN
ejpam-2803	190	5	(	(	PUNCT
ejpam-2803	190	6	...	...	PUNCT
ejpam-2803	190	7	(	(	PUNCT
ejpam-2803	190	8	zr	zr	INTJ
ejpam-2803	190	9	be	be	AUX
ejpam-2803	190	10	a	a	DET
ejpam-2803	190	11	strictly	strictly	ADV
ejpam-2803	190	12	increasing	increase	VERB
ejpam-2803	190	13	chain	chain	NOUN
ejpam-2803	190	14	of	of	ADP
ejpam-2803	190	15	irreducible	irreducible	ADJ
ejpam-2803	190	16	closed	close	VERB
ejpam-2803	190	17	subset	subset	NOUN
ejpam-2803	190	18	zi	zi	NOUN
ejpam-2803	190	19	of	of	ADP
ejpam-2803	190	20	specs(m	specs(m	PROPN
ejpam-2803	190	21	)	)	PUNCT
ejpam-2803	190	22	of	of	ADP
ejpam-2803	190	23	length	length	NOUN
ejpam-2803	190	24	r.	r.	PROPN
ejpam-2803	190	25	by	by	ADP
ejpam-2803	190	26	[	[	X
ejpam-2803	190	27	7	7	NUM
ejpam-2803	190	28	,	,	PUNCT
ejpam-2803	190	29	theorem	theorem	VERB
ejpam-2803	190	30	5.8	5.8	NUM
ejpam-2803	190	31	]	]	PUNCT
ejpam-2803	190	32	,	,	PUNCT
ejpam-2803	190	33	each	each	DET
ejpam-2803	190	34	zi	zi	NOUN
ejpam-2803	190	35	has	have	VERB
ejpam-2803	190	36	a	a	DET
ejpam-2803	190	37	generic	generic	ADJ
ejpam-2803	190	38	point	point	NOUN
ejpam-2803	190	39	,	,	PUNCT
ejpam-2803	190	40	so	so	SCONJ
ejpam-2803	190	41	that	that	SCONJ
ejpam-2803	190	42	zi	zi	NOUN
ejpam-2803	190	43	=	=	PROPN
ejpam-2803	190	44	v	v	PROPN
ejpam-2803	190	45	s(si	s(si	PROPN
ejpam-2803	190	46	)	)	PUNCT
ejpam-2803	190	47	for	for	ADP
ejpam-2803	190	48	some	some	DET
ejpam-2803	190	49	si	si	PROPN
ejpam-2803	190	50	∈	∈	PROPN
ejpam-2803	190	51	specs(m	specs(m	PROPN
ejpam-2803	190	52	)	)	PUNCT
ejpam-2803	190	53	.	.	PUNCT
ejpam-2803	191	1	hence	hence	ADV
ejpam-2803	191	2	,	,	PUNCT
ejpam-2803	191	3	we	we	PRON
ejpam-2803	191	4	have	have	VERB
ejpam-2803	191	5	v	v	NUM
ejpam-2803	191	6	s(s0	s(s0	NOUN
ejpam-2803	191	7	)	)	PUNCT
ejpam-2803	191	8	(	(	PUNCT
ejpam-2803	191	9	v	v	NOUN
ejpam-2803	191	10	s(s1	s(s1	NOUN
ejpam-2803	191	11	)	)	PUNCT
ejpam-2803	191	12	(	(	PUNCT
ejpam-2803	191	13	...	...	PUNCT
ejpam-2803	191	14	v	v	ADP
ejpam-2803	191	15	s(sr	s(sr	NUM
ejpam-2803	191	16	)	)	PUNCT
ejpam-2803	191	17	.	.	PUNCT
ejpam-2803	192	1	thus	thus	ADV
ejpam-2803	192	2	annr(s0	annr(s0	ADJ
ejpam-2803	192	3	)	)	PUNCT
ejpam-2803	192	4	)	)	PUNCT
ejpam-2803	193	1	annr(s1	annr(s1	PROPN
ejpam-2803	193	2	)	)	PUNCT
ejpam-2803	193	3	)	)	PUNCT
ejpam-2803	193	4	...	...	PUNCT
ejpam-2803	193	5	)	)	PUNCT
ejpam-2803	194	1	annr(sr	annr(sr	ADJ
ejpam-2803	194	2	)	)	PUNCT
ejpam-2803	194	3	,	,	PUNCT
ejpam-2803	194	4	a	a	DET
ejpam-2803	194	5	strictly	strictly	ADV
ejpam-2803	194	6	decreasing	decrease	VERB
ejpam-2803	194	7	chain	chain	NOUN
ejpam-2803	194	8	of	of	ADP
ejpam-2803	194	9	prime	prime	ADJ
ejpam-2803	194	10	ideals	ideal	NOUN
ejpam-2803	194	11	r	r	NOUN
ejpam-2803	194	12	of	of	ADP
ejpam-2803	194	13	length	length	NOUN
ejpam-2803	194	14	r.	r.	PROPN
ejpam-2803	194	15	conversely	conversely	ADV
ejpam-2803	194	16	,	,	PUNCT
ejpam-2803	194	17	let	let	VERB
ejpam-2803	194	18	q0	q0	PROPN
ejpam-2803	194	19	)	)	PUNCT
ejpam-2803	194	20	q1	q1	PROPN
ejpam-2803	194	21	)	)	PUNCT
ejpam-2803	194	22	...	...	PUNCT
ejpam-2803	194	23	)	)	PUNCT
ejpam-2803	195	1	qr	qr	INTJ
ejpam-2803	195	2	be	be	AUX
ejpam-2803	195	3	a	a	DET
ejpam-2803	195	4	strictly	strictly	ADV
ejpam-2803	195	5	decreasing	decrease	VERB
ejpam-2803	195	6	chain	chain	NOUN
ejpam-2803	195	7	of	of	ADP
ejpam-2803	195	8	prime	prime	ADJ
ejpam-2803	195	9	ideals	ideal	NOUN
ejpam-2803	195	10	in	in	ADP
ejpam-2803	195	11	r	r	NOUN
ejpam-2803	195	12	of	of	ADP
ejpam-2803	195	13	length	length	NOUN
ejpam-2803	195	14	r	r	NOUN
ejpam-2803	195	15	and	and	CCONJ
ejpam-2803	195	16	,	,	PUNCT
ejpam-2803	195	17	for	for	ADP
ejpam-2803	195	18	each	each	DET
ejpam-2803	195	19	i	i	PRON
ejpam-2803	195	20	(	(	PUNCT
ejpam-2803	195	21	1	1	NUM
ejpam-2803	195	22	≤	≤	NUM
ejpam-2803	195	23	i	i	NOUN
ejpam-2803	195	24	≤	≤	ADJ
ejpam-2803	195	25	r	r	NOUN
ejpam-2803	195	26	)	)	PUNCT
ejpam-2803	195	27	,	,	PUNCT
ejpam-2803	195	28	let	let	VERB
ejpam-2803	195	29	qi	qi	PRON
ejpam-2803	195	30	be	be	AUX
ejpam-2803	195	31	a	a	DET
ejpam-2803	195	32	prime	prime	ADJ
ejpam-2803	195	33	ideal	ideal	NOUN
ejpam-2803	195	34	of	of	ADP
ejpam-2803	195	35	r	r	NOUN
ejpam-2803	195	36	containing	contain	VERB
ejpam-2803	195	37	annr(m	annr(m	NOUN
ejpam-2803	195	38	)	)	PUNCT
ejpam-2803	195	39	such	such	ADJ
ejpam-2803	195	40	that	that	SCONJ
ejpam-2803	195	41	qi	qi	PROPN
ejpam-2803	195	42	=	=	SYM
ejpam-2803	195	43	qi	qi	PROPN
ejpam-2803	195	44	/	/	SYM
ejpam-2803	195	45	annr(m	annr(m	NOUN
ejpam-2803	195	46	)	)	PUNCT
ejpam-2803	195	47	.	.	PUNCT
ejpam-2803	196	1	since	since	SCONJ
ejpam-2803	196	2	m	m	PROPN
ejpam-2803	196	3	is	be	AUX
ejpam-2803	196	4	a	a	DET
ejpam-2803	196	5	secondful	secondful	ADJ
ejpam-2803	196	6	r	r	NOUN
ejpam-2803	196	7	-	-	PUNCT
ejpam-2803	196	8	module	module	NOUN
ejpam-2803	196	9	,	,	PUNCT
ejpam-2803	196	10	there	there	PRON
ejpam-2803	196	11	exists	exist	VERB
ejpam-2803	196	12	a	a	DET
ejpam-2803	196	13	qi	qi	NOUN
ejpam-2803	196	14	-	-	PUNCT
ejpam-2803	196	15	second	second	NOUN
ejpam-2803	196	16	submodule	submodule	NOUN
ejpam-2803	196	17	qi	qi	PROPN
ejpam-2803	196	18	of	of	ADP
ejpam-2803	196	19	m	m	PROPN
ejpam-2803	196	20	for	for	ADP
ejpam-2803	196	21	each	each	DET
ejpam-2803	196	22	i	i	PRON
ejpam-2803	196	23	(	(	PUNCT
ejpam-2803	196	24	1	1	NUM
ejpam-2803	196	25	≤	≤	NUM
ejpam-2803	196	26	i	i	NOUN
ejpam-2803	196	27	≤	≤	ADJ
ejpam-2803	196	28	r	r	NOUN
ejpam-2803	196	29	)	)	PUNCT
ejpam-2803	196	30	.	.	PUNCT
ejpam-2803	197	1	hence	hence	ADV
ejpam-2803	197	2	we	we	PRON
ejpam-2803	197	3	have	have	VERB
ejpam-2803	197	4	annr(q0	annr(q0	ADJ
ejpam-2803	197	5	)	)	PUNCT
ejpam-2803	197	6	)	)	PUNCT
ejpam-2803	198	1	annr(q1	annr(q1	PROPN
ejpam-2803	198	2	)	)	PUNCT
ejpam-2803	198	3	)	)	PUNCT
ejpam-2803	198	4	...	...	PUNCT
ejpam-2803	198	5	)	)	PUNCT
ejpam-2803	199	1	annr(qr	annr(qr	PROPN
ejpam-2803	199	2	)	)	PUNCT
ejpam-2803	199	3	so	so	SCONJ
ejpam-2803	199	4	that	that	SCONJ
ejpam-2803	199	5	v	v	NOUN
ejpam-2803	199	6	s(q0	s(q0	NOUN
ejpam-2803	199	7	)	)	PUNCT
ejpam-2803	199	8	(	(	PUNCT
ejpam-2803	199	9	v	v	NUM
ejpam-2803	199	10	s(q1	s(q1	NOUN
ejpam-2803	199	11	)	)	PUNCT
ejpam-2803	199	12	(	(	PUNCT
ejpam-2803	199	13	...	...	PUNCT
ejpam-2803	199	14	(	(	PUNCT
ejpam-2803	199	15	v	v	NUM
ejpam-2803	199	16	s(qr	s(qr	PROPN
ejpam-2803	199	17	)	)	PUNCT
ejpam-2803	199	18	.	.	PUNCT
ejpam-2803	200	1	thus	thus	ADV
ejpam-2803	200	2	,	,	PUNCT
ejpam-2803	200	3	we	we	PRON
ejpam-2803	200	4	obtain	obtain	VERB
ejpam-2803	200	5	a	a	DET
ejpam-2803	200	6	strictly	strictly	ADV
ejpam-2803	200	7	increasing	increase	VERB
ejpam-2803	200	8	chain	chain	NOUN
ejpam-2803	200	9	of	of	ADP
ejpam-2803	200	10	length	length	NOUN
ejpam-2803	200	11	r	r	NOUN
ejpam-2803	200	12	of	of	ADP
ejpam-2803	200	13	irreducible	irreducible	ADJ
ejpam-2803	200	14	closed	closed	ADJ
ejpam-2803	200	15	subsets	subset	NOUN
ejpam-2803	200	16	of	of	ADP
ejpam-2803	200	17	specs(m	specs(m	NOUN
ejpam-2803	200	18	)	)	PUNCT
ejpam-2803	200	19	.	.	PUNCT
ejpam-2803	201	1	we	we	PRON
ejpam-2803	201	2	consider	consider	VERB
ejpam-2803	201	3	strictly	strictly	ADV
ejpam-2803	201	4	decreasing	decrease	VERB
ejpam-2803	201	5	(	(	PUNCT
ejpam-2803	201	6	or	or	CCONJ
ejpam-2803	201	7	strictly	strictly	ADV
ejpam-2803	201	8	increasing	increase	VERB
ejpam-2803	201	9	)	)	PUNCT
ejpam-2803	201	10	chain	chain	NOUN
ejpam-2803	201	11	z0	z0	PROPN
ejpam-2803	201	12	,	,	PUNCT
ejpam-2803	201	13	z1	z1	PROPN
ejpam-2803	201	14	,	,	PUNCT
ejpam-2803	201	15	...	...	PUNCT
ejpam-2803	201	16	,	,	PUNCT
ejpam-2803	201	17	zr	zr	NOUN
ejpam-2803	201	18	of	of	ADP
ejpam-2803	201	19	length	length	NOUN
ejpam-2803	201	20	r	r	NOUN
ejpam-2803	201	21	of	of	ADP
ejpam-2803	201	22	irreducible	irreducible	ADJ
ejpam-2803	201	23	closed	close	VERB
ejpam-2803	201	24	subsets	subset	NOUN
ejpam-2803	201	25	zi	zi	PROPN
ejpam-2803	201	26	of	of	ADP
ejpam-2803	201	27	t	t	PROPN
ejpam-2803	201	28	.	.	PUNCT
ejpam-2803	202	1	the	the	DET
ejpam-2803	202	2	supremum	supremum	NOUN
ejpam-2803	202	3	of	of	ADP
ejpam-2803	202	4	the	the	DET
ejpam-2803	202	5	lengths	length	NOUN
ejpam-2803	202	6	,	,	PUNCT
ejpam-2803	202	7	taken	take	VERB
ejpam-2803	202	8	over	over	ADP
ejpam-2803	202	9	all	all	DET
ejpam-2803	202	10	such	such	ADJ
ejpam-2803	202	11	chains	chain	NOUN
ejpam-2803	202	12	,	,	PUNCT
ejpam-2803	202	13	is	be	AUX
ejpam-2803	202	14	called	call	VERB
ejpam-2803	202	15	the	the	DET
ejpam-2803	202	16	combinatorial	combinatorial	ADJ
ejpam-2803	202	17	dimension	dimension	NOUN
ejpam-2803	202	18	of	of	ADP
ejpam-2803	202	19	t	t	PROPN
ejpam-2803	202	20	and	and	CCONJ
ejpam-2803	202	21	denoted	denote	VERB
ejpam-2803	202	22	by	by	ADP
ejpam-2803	202	23	dimt	dimt	ADJ
ejpam-2803	202	24	(	(	PUNCT
ejpam-2803	202	25	for	for	ADP
ejpam-2803	202	26	t	t	NOUN
ejpam-2803	202	27	=	=	SYM
ejpam-2803	202	28	∅	∅	NOUN
ejpam-2803	202	29	,	,	PUNCT
ejpam-2803	202	30	define	define	VERB
ejpam-2803	202	31	dimt	dimt	ADJ
ejpam-2803	202	32	=	=	SYM
ejpam-2803	202	33	−1	−1	NOUN
ejpam-2803	202	34	)	)	PUNCT
ejpam-2803	202	35	.	.	PUNCT
ejpam-2803	203	1	a	a	DET
ejpam-2803	203	2	submodule	submodule	PROPN
ejpam-2803	203	3	n	n	PROPN
ejpam-2803	203	4	of	of	ADP
ejpam-2803	203	5	m	m	PROPN
ejpam-2803	203	6	is	be	AUX
ejpam-2803	203	7	said	say	VERB
ejpam-2803	203	8	to	to	PART
ejpam-2803	203	9	be	be	AUX
ejpam-2803	203	10	cocyclic	cocyclic	ADJ
ejpam-2803	203	11	if	if	SCONJ
ejpam-2803	203	12	n	n	PRON
ejpam-2803	203	13	⊆	⊆	NUM
ejpam-2803	203	14	e(r	e(r	NUM
ejpam-2803	203	15	/	/	SYM
ejpam-2803	203	16	m	m	NOUN
ejpam-2803	203	17	)	)	PUNCT
ejpam-2803	203	18	for	for	ADP
ejpam-2803	203	19	some	some	DET
ejpam-2803	203	20	maximal	maximal	ADJ
ejpam-2803	203	21	ideal	ideal	NOUN
ejpam-2803	203	22	m	m	NOUN
ejpam-2803	203	23	of	of	ADP
ejpam-2803	203	24	r	r	NOUN
ejpam-2803	203	25	(	(	PUNCT
ejpam-2803	203	26	here	here	ADV
ejpam-2803	203	27	e(r	e(r	NUM
ejpam-2803	203	28	/	/	SYM
ejpam-2803	203	29	m	m	NOUN
ejpam-2803	203	30	)	)	PUNCT
ejpam-2803	203	31	denote	denote	VERB
ejpam-2803	203	32	the	the	DET
ejpam-2803	203	33	injective	injective	ADJ
ejpam-2803	203	34	envelope	envelope	NOUN
ejpam-2803	203	35	of	of	ADP
ejpam-2803	203	36	r	r	PROPN
ejpam-2803	203	37	/	/	SYM
ejpam-2803	203	38	m	m	NOUN
ejpam-2803	203	39	)	)	PUNCT
ejpam-2803	203	40	(	(	PUNCT
ejpam-2803	203	41	see	see	VERB
ejpam-2803	203	42	[	[	X
ejpam-2803	203	43	29	29	NUM
ejpam-2803	203	44	]	]	NUM
ejpam-2803	203	45	)	)	PUNCT
ejpam-2803	203	46	.	.	PUNCT
ejpam-2803	204	1	the	the	DET
ejpam-2803	204	2	cosupport	cosupport	NOUN
ejpam-2803	204	3	of	of	ADP
ejpam-2803	204	4	m	m	PROPN
ejpam-2803	204	5	,	,	PUNCT
ejpam-2803	204	6	denoted	denote	VERB
ejpam-2803	204	7	by	by	ADP
ejpam-2803	204	8	cosupp(m	cosupp(m	PROPN
ejpam-2803	204	9	)	)	PUNCT
ejpam-2803	204	10	,	,	PUNCT
ejpam-2803	204	11	is	be	AUX
ejpam-2803	204	12	defined	define	VERB
ejpam-2803	204	13	as	as	ADP
ejpam-2803	204	14	the	the	DET
ejpam-2803	204	15	set	set	NOUN
ejpam-2803	204	16	of	of	ADP
ejpam-2803	204	17	all	all	DET
ejpam-2803	204	18	prime	prime	ADJ
ejpam-2803	204	19	ideals	ideal	NOUN
ejpam-2803	204	20	p	p	NOUN
ejpam-2803	204	21	of	of	ADP
ejpam-2803	204	22	r	r	NOUN
ejpam-2803	204	23	such	such	ADJ
ejpam-2803	204	24	that	that	SCONJ
ejpam-2803	204	25	p	p	PROPN
ejpam-2803	204	26	⊇	⊇	PROPN
ejpam-2803	204	27	annr(n	annr(n	PROPN
ejpam-2803	204	28	)	)	PUNCT
ejpam-2803	204	29	for	for	ADP
ejpam-2803	204	30	some	some	DET
ejpam-2803	204	31	cocyclic	cocyclic	ADJ
ejpam-2803	204	32	homomorphic	homomorphic	ADJ
ejpam-2803	204	33	image	image	NOUN
ejpam-2803	204	34	n	n	PROPN
ejpam-2803	204	35	of	of	ADP
ejpam-2803	204	36	m	m	PROPN
ejpam-2803	204	37	(	(	PUNCT
ejpam-2803	204	38	see	see	VERB
ejpam-2803	204	39	[	[	X
ejpam-2803	204	40	28	28	NUM
ejpam-2803	204	41	]	]	NUM
ejpam-2803	204	42	)	)	PUNCT
ejpam-2803	204	43	.	.	PUNCT
ejpam-2803	205	1	for	for	ADP
ejpam-2803	205	2	every	every	DET
ejpam-2803	205	3	finitely	finitely	ADV
ejpam-2803	205	4	cogenerated	cogenerate	VERB
ejpam-2803	205	5	module	module	NOUN
ejpam-2803	205	6	m	m	PROPN
ejpam-2803	205	7	,	,	PUNCT
ejpam-2803	205	8	cosupp(m	cosupp(m	PROPN
ejpam-2803	205	9	)	)	PUNCT
ejpam-2803	205	10	=	=	SYM
ejpam-2803	205	11	v	v	NOUN
ejpam-2803	205	12	(	(	PUNCT
ejpam-2803	205	13	annr(m	annr(m	PROPN
ejpam-2803	205	14	)	)	PUNCT
ejpam-2803	205	15	)	)	PUNCT
ejpam-2803	206	1	[	[	X
ejpam-2803	206	2	28	28	NUM
ejpam-2803	206	3	,	,	PUNCT
ejpam-2803	206	4	lemma	lemma	PROPN
ejpam-2803	206	5	2.3	2.3	NUM
ejpam-2803	206	6	]	]	PUNCT
ejpam-2803	206	7	.	.	PUNCT
ejpam-2803	207	1	the	the	DET
ejpam-2803	207	2	equality	equality	NOUN
ejpam-2803	207	3	holds	hold	VERB
ejpam-2803	207	4	also	also	ADV
ejpam-2803	207	5	if	if	SCONJ
ejpam-2803	207	6	m	m	NOUN
ejpam-2803	207	7	is	be	AUX
ejpam-2803	207	8	a	a	DET
ejpam-2803	207	9	secondful	secondful	ADJ
ejpam-2803	207	10	module	module	NOUN
ejpam-2803	207	11	by	by	ADP
ejpam-2803	207	12	[	[	X
ejpam-2803	207	13	17	17	NUM
ejpam-2803	207	14	,	,	PUNCT
ejpam-2803	207	15	theorem	theorem	VERB
ejpam-2803	207	16	2.5	2.5	NUM
ejpam-2803	207	17	]	]	PUNCT
ejpam-2803	207	18	.	.	PUNCT
ejpam-2803	208	1	using	use	VERB
ejpam-2803	208	2	this	this	DET
ejpam-2803	208	3	fact	fact	NOUN
ejpam-2803	208	4	and	and	CCONJ
ejpam-2803	208	5	proposition	proposition	NOUN
ejpam-2803	208	6	2.6	2.6	NUM
ejpam-2803	208	7	,	,	PUNCT
ejpam-2803	208	8	we	we	PRON
ejpam-2803	208	9	can	can	AUX
ejpam-2803	208	10	establish	establish	VERB
ejpam-2803	208	11	the	the	DET
ejpam-2803	208	12	next	next	ADJ
ejpam-2803	208	13	theorem	theorem	PROPN
ejpam-2803	208	14	.	.	PUNCT
ejpam-2803	208	15	theorem	theorem	VERB
ejpam-2803	208	16	2.7	2.7	NUM
ejpam-2803	208	17	.	.	PUNCT
ejpam-2803	209	1	let	let	VERB
ejpam-2803	209	2	m	m	PRON
ejpam-2803	209	3	be	be	AUX
ejpam-2803	209	4	a	a	DET
ejpam-2803	209	5	secondful	secondful	ADJ
ejpam-2803	209	6	r	r	NOUN
ejpam-2803	209	7	-	-	PUNCT
ejpam-2803	209	8	module	module	NOUN
ejpam-2803	209	9	and	and	CCONJ
ejpam-2803	209	10	equip	equip	NOUN
ejpam-2803	209	11	specs(m	specs(m	PROPN
ejpam-2803	209	12	)	)	PUNCT
ejpam-2803	209	13	and	and	CCONJ
ejpam-2803	209	14	spec(r	spec(r	PROPN
ejpam-2803	209	15	)	)	PUNCT
ejpam-2803	209	16	with	with	ADP
ejpam-2803	209	17	their	their	PRON
ejpam-2803	209	18	zariski	zariski	ADJ
ejpam-2803	209	19	topologies	topology	NOUN
ejpam-2803	209	20	.	.	PUNCT
ejpam-2803	210	1	then	then	ADV
ejpam-2803	210	2	the	the	DET
ejpam-2803	210	3	combinatorial	combinatorial	ADJ
ejpam-2803	210	4	dimension	dimension	NOUN
ejpam-2803	210	5	of	of	ADP
ejpam-2803	210	6	specs(m	specs(m	PROPN
ejpam-2803	210	7	)	)	PUNCT
ejpam-2803	210	8	,	,	PUNCT
ejpam-2803	210	9	the	the	DET
ejpam-2803	210	10	krull	krull	PROPN
ejpam-2803	210	11	dimension	dimension	NOUN
ejpam-2803	210	12	of	of	ADP
ejpam-2803	210	13	r	r	NOUN
ejpam-2803	210	14	=	=	SYM
ejpam-2803	210	15	r	r	NOUN
ejpam-2803	210	16	/	/	SYM
ejpam-2803	210	17	annr(m	annr(m	NOUN
ejpam-2803	210	18	)	)	PUNCT
ejpam-2803	210	19	,	,	PUNCT
ejpam-2803	210	20	and	and	CCONJ
ejpam-2803	210	21	the	the	DET
ejpam-2803	210	22	combinatorial	combinatorial	ADJ
ejpam-2803	210	23	dimension	dimension	NOUN
ejpam-2803	210	24	of	of	ADP
ejpam-2803	210	25	the	the	DET
ejpam-2803	210	26	closed	closed	ADJ
ejpam-2803	210	27	subspace	subspace	NOUN
ejpam-2803	210	28	cosupp(m	cosupp(m	PROPN
ejpam-2803	210	29	)	)	PUNCT
ejpam-2803	210	30	=	=	SYM
ejpam-2803	210	31	v	v	NOUN
ejpam-2803	210	32	(	(	PUNCT
ejpam-2803	210	33	annr(m	annr(m	PROPN
ejpam-2803	210	34	)	)	PUNCT
ejpam-2803	210	35	)	)	PUNCT
ejpam-2803	210	36	of	of	ADP
ejpam-2803	210	37	spec(r	spec(r	PROPN
ejpam-2803	210	38	)	)	PUNCT
ejpam-2803	210	39	are	be	AUX
ejpam-2803	210	40	all	all	ADV
ejpam-2803	210	41	equal	equal	ADJ
ejpam-2803	210	42	.	.	PUNCT
ejpam-2803	211	1	as	as	ADP
ejpam-2803	211	2	a	a	DET
ejpam-2803	211	3	result	result	NOUN
ejpam-2803	211	4	of	of	ADP
ejpam-2803	211	5	theorem	theorem	NOUN
ejpam-2803	211	6	2.7	2.7	NUM
ejpam-2803	211	7	,	,	PUNCT
ejpam-2803	211	8	the	the	DET
ejpam-2803	211	9	definition	definition	NOUN
ejpam-2803	211	10	of	of	ADP
ejpam-2803	211	11	the	the	DET
ejpam-2803	211	12	classical	classical	ADJ
ejpam-2803	211	13	krull	krull	PROPN
ejpam-2803	211	14	dimension	dimension	NOUN
ejpam-2803	211	15	of	of	ADP
ejpam-2803	211	16	modules	module	NOUN
ejpam-2803	211	17	,	,	PUNCT
ejpam-2803	211	18	which	which	PRON
ejpam-2803	211	19	is	be	AUX
ejpam-2803	211	20	independent	independent	ADJ
ejpam-2803	211	21	of	of	ADP
ejpam-2803	211	22	specs(m	specs(m	PROPN
ejpam-2803	211	23	)	)	PUNCT
ejpam-2803	211	24	,	,	PUNCT
ejpam-2803	211	25	becomes	become	VERB
ejpam-2803	211	26	more	more	ADV
ejpam-2803	211	27	significant	significant	ADJ
ejpam-2803	211	28	for	for	ADP
ejpam-2803	211	29	secondful	secondful	ADJ
ejpam-2803	211	30	modules	module	NOUN
ejpam-2803	211	31	m	m	VERB
ejpam-2803	211	32	.	.	PUNCT
ejpam-2803	212	1	let	let	VERB
ejpam-2803	212	2	p	p	PRON
ejpam-2803	212	3	be	be	AUX
ejpam-2803	212	4	a	a	DET
ejpam-2803	212	5	prime	prime	ADJ
ejpam-2803	212	6	ideal	ideal	NOUN
ejpam-2803	212	7	of	of	ADP
ejpam-2803	212	8	r.	r.	PROPN
ejpam-2803	212	9	for	for	ADP
ejpam-2803	212	10	an	an	DET
ejpam-2803	212	11	r	r	NOUN
ejpam-2803	212	12	-	-	PUNCT
ejpam-2803	212	13	module	module	NOUN
ejpam-2803	212	14	m	m	NOUN
ejpam-2803	212	15	,	,	PUNCT
ejpam-2803	212	16	specsp(m	specsp(m	PROPN
ejpam-2803	212	17	)	)	PUNCT
ejpam-2803	212	18	denotes	denote	VERB
ejpam-2803	212	19	the	the	DET
ejpam-2803	212	20	set	set	NOUN
ejpam-2803	212	21	of	of	ADP
ejpam-2803	212	22	all	all	DET
ejpam-2803	212	23	p	p	ADJ
ejpam-2803	212	24	-	-	PUNCT
ejpam-2803	212	25	second	second	NOUN
ejpam-2803	212	26	submodules	submodule	NOUN
ejpam-2803	212	27	of	of	ADP
ejpam-2803	212	28	m	m	PROPN
ejpam-2803	212	29	.	.	PUNCT
ejpam-2803	213	1	corollary	corollary	ADJ
ejpam-2803	213	2	2.8	2.8	NUM
ejpam-2803	213	3	.	.	PUNCT
ejpam-2803	214	1	let	let	VERB
ejpam-2803	214	2	m	m	PRON
ejpam-2803	214	3	be	be	AUX
ejpam-2803	214	4	a	a	DET
ejpam-2803	214	5	secondful	secondful	ADJ
ejpam-2803	214	6	r	r	NOUN
ejpam-2803	214	7	-	-	PUNCT
ejpam-2803	214	8	module	module	NOUN
ejpam-2803	214	9	such	such	ADJ
ejpam-2803	214	10	that	that	DET
ejpam-2803	214	11	specs(m	specs(m	NOUN
ejpam-2803	214	12	)	)	PUNCT
ejpam-2803	214	13	has	have	VERB
ejpam-2803	214	14	zero	zero	NUM
ejpam-2803	214	15	combinatorial	combinatorial	ADJ
ejpam-2803	214	16	dimension	dimension	NOUN
ejpam-2803	214	17	.	.	PUNCT
ejpam-2803	215	1	then	then	ADV
ejpam-2803	215	2	we	we	PRON
ejpam-2803	215	3	have	have	VERB
ejpam-2803	215	4	the	the	DET
ejpam-2803	215	5	following	following	NOUN
ejpam-2803	215	6	.	.	PUNCT
ejpam-2803	216	1	(	(	PUNCT
ejpam-2803	216	2	a	a	X
ejpam-2803	216	3	)	)	PUNCT
ejpam-2803	216	4	every	every	DET
ejpam-2803	216	5	irreducible	irreducible	ADJ
ejpam-2803	216	6	closed	closed	ADJ
ejpam-2803	216	7	subset	subset	NOUN
ejpam-2803	216	8	of	of	ADP
ejpam-2803	216	9	specs(m	specs(m	PROPN
ejpam-2803	216	10	)	)	PUNCT
ejpam-2803	216	11	is	be	AUX
ejpam-2803	216	12	an	an	DET
ejpam-2803	216	13	irreducible	irreducible	ADJ
ejpam-2803	216	14	component	component	NOUN
ejpam-2803	216	15	of	of	ADP
ejpam-2803	216	16	specs(m	specs(m	NOUN
ejpam-2803	216	17	)	)	PUNCT
ejpam-2803	216	18	.	.	PUNCT
ejpam-2803	217	1	(	(	PUNCT
ejpam-2803	217	2	b	b	X
ejpam-2803	217	3	)	)	PUNCT
ejpam-2803	217	4	for	for	ADP
ejpam-2803	217	5	every	every	DET
ejpam-2803	217	6	p	p	PROPN
ejpam-2803	217	7	∈	∈	PROPN
ejpam-2803	217	8	v	v	NOUN
ejpam-2803	217	9	(	(	PUNCT
ejpam-2803	217	10	annr(m	annr(m	PROPN
ejpam-2803	217	11	)	)	PUNCT
ejpam-2803	217	12	)	)	PUNCT
ejpam-2803	217	13	and	and	CCONJ
ejpam-2803	217	14	for	for	ADP
ejpam-2803	217	15	every	every	DET
ejpam-2803	217	16	p	p	ADJ
ejpam-2803	217	17	-	-	PUNCT
ejpam-2803	217	18	second	second	NOUN
ejpam-2803	217	19	submodule	submodule	NOUN
ejpam-2803	217	20	s	s	PROPN
ejpam-2803	217	21	of	of	ADP
ejpam-2803	217	22	m	m	PROPN
ejpam-2803	217	23	,	,	PUNCT
ejpam-2803	217	24	specsp(m	specsp(m	PROPN
ejpam-2803	217	25	)	)	PUNCT
ejpam-2803	217	26	=	=	PROPN
ejpam-2803	217	27	v	v	ADP
ejpam-2803	217	28	s(s	s(s	PROPN
ejpam-2803	217	29	)	)	PUNCT
ejpam-2803	217	30	.	.	PUNCT
ejpam-2803	218	1	h.	h.	PROPN
ejpam-2803	218	2	ansari	ansari	PROPN
ejpam-2803	218	3	-	-	PUNCT
ejpam-2803	218	4	toroghy	toroghy	NOUN
ejpam-2803	218	5	,	,	PUNCT
ejpam-2803	218	6	s.	s.	PROPN
ejpam-2803	218	7	s.	s.	PROPN
ejpam-2803	218	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	218	9	/	/	SYM
ejpam-2803	218	10	eur	eur	PROPN
ejpam-2803	218	11	.	.	PUNCT
ejpam-2803	219	1	j.	j.	PROPN
ejpam-2803	219	2	pure	pure	PROPN
ejpam-2803	219	3	appl	appl	PROPN
ejpam-2803	219	4	.	.	PROPN
ejpam-2803	219	5	math	math	PROPN
ejpam-2803	219	6	,	,	PUNCT
ejpam-2803	219	7	10	10	NUM
ejpam-2803	219	8	(	(	PUNCT
ejpam-2803	219	9	2	2	NUM
ejpam-2803	219	10	)	)	PUNCT
ejpam-2803	219	11	(	(	PUNCT
ejpam-2803	219	12	2017	2017	NUM
ejpam-2803	219	13	)	)	PUNCT
ejpam-2803	219	14	,	,	PUNCT
ejpam-2803	219	15	211	211	NUM
ejpam-2803	219	16	-	-	SYM
ejpam-2803	219	17	230	230	NUM
ejpam-2803	219	18	217	217	NUM
ejpam-2803	219	19	(	(	PUNCT
ejpam-2803	219	20	c	c	X
ejpam-2803	219	21	)	)	PUNCT
ejpam-2803	219	22	if	if	SCONJ
ejpam-2803	219	23	m	m	NOUN
ejpam-2803	219	24	has	have	VERB
ejpam-2803	219	25	noetherian	noetherian	ADJ
ejpam-2803	219	26	second	second	ADJ
ejpam-2803	219	27	spectrum	spectrum	NOUN
ejpam-2803	219	28	,	,	PUNCT
ejpam-2803	219	29	then	then	ADV
ejpam-2803	219	30	the	the	DET
ejpam-2803	219	31	set	set	NOUN
ejpam-2803	219	32	of	of	ADP
ejpam-2803	219	33	irreducible	irreducible	ADJ
ejpam-2803	219	34	components	component	NOUN
ejpam-2803	219	35	of	of	ADP
ejpam-2803	219	36	specs(m	specs(m	NOUN
ejpam-2803	219	37	)	)	PUNCT
ejpam-2803	219	38	is	be	AUX
ejpam-2803	219	39	{	{	PUNCT
ejpam-2803	219	40	v	v	NUM
ejpam-2803	219	41	s((0	s((0	PROPN
ejpam-2803	219	42	:	:	PUNCT
ejpam-2803	219	43	m	m	NOUN
ejpam-2803	219	44	m1	m1	NOUN
ejpam-2803	219	45	)	)	PUNCT
ejpam-2803	219	46	)	)	PUNCT
ejpam-2803	219	47	,	,	PUNCT
ejpam-2803	219	48	v	v	X
ejpam-2803	219	49	s((0	s((0	PROPN
ejpam-2803	219	50	:	:	PUNCT
ejpam-2803	219	51	m	m	NOUN
ejpam-2803	219	52	m2	m2	PROPN
ejpam-2803	219	53	)	)	PUNCT
ejpam-2803	219	54	)	)	PUNCT
ejpam-2803	219	55	,	,	PUNCT
ejpam-2803	219	56	...	...	PUNCT
ejpam-2803	219	57	,	,	PUNCT
ejpam-2803	219	58	v	v	X
ejpam-2803	219	59	s((0	s((0	PROPN
ejpam-2803	219	60	:	:	PUNCT
ejpam-2803	219	61	m	m	PROPN
ejpam-2803	219	62	mk	mk	PROPN
ejpam-2803	219	63	)	)	PUNCT
ejpam-2803	219	64	)	)	PUNCT
ejpam-2803	219	65	}	}	PUNCT
ejpam-2803	219	66	for	for	ADP
ejpam-2803	219	67	some	some	DET
ejpam-2803	219	68	positive	positive	ADJ
ejpam-2803	219	69	integer	integer	NOUN
ejpam-2803	219	70	k	k	PROPN
ejpam-2803	219	71	,	,	PUNCT
ejpam-2803	219	72	where	where	SCONJ
ejpam-2803	219	73	mi	mi	NOUN
ejpam-2803	219	74	,	,	PUNCT
ejpam-2803	219	75	for	for	ADP
ejpam-2803	219	76	i	i	PROPN
ejpam-2803	219	77	=	=	SYM
ejpam-2803	219	78	1	1	NUM
ejpam-2803	219	79	,	,	PUNCT
ejpam-2803	219	80	2	2	NUM
ejpam-2803	219	81	,	,	PUNCT
ejpam-2803	219	82	...	...	PUNCT
ejpam-2803	219	83	,	,	PUNCT
ejpam-2803	219	84	k	k	NOUN
ejpam-2803	219	85	,	,	PUNCT
ejpam-2803	219	86	are	be	AUX
ejpam-2803	219	87	all	all	DET
ejpam-2803	219	88	the	the	DET
ejpam-2803	219	89	minimal	minimal	ADJ
ejpam-2803	219	90	prime	prime	ADJ
ejpam-2803	219	91	divisors	divisor	NOUN
ejpam-2803	219	92	of	of	ADP
ejpam-2803	219	93	annr(m	annr(m	NOUN
ejpam-2803	219	94	)	)	PUNCT
ejpam-2803	219	95	.	.	PUNCT
ejpam-2803	220	1	proof	proof	NOUN
ejpam-2803	220	2	.	.	PUNCT
ejpam-2803	221	1	(	(	PUNCT
ejpam-2803	221	2	a	a	X
ejpam-2803	221	3	)	)	PUNCT
ejpam-2803	221	4	let	let	AUX
ejpam-2803	221	5	v	v	ADP
ejpam-2803	221	6	s(n	s(n	NOUN
ejpam-2803	221	7	)	)	PUNCT
ejpam-2803	221	8	be	be	VERB
ejpam-2803	221	9	an	an	DET
ejpam-2803	221	10	irreducible	irreducible	ADJ
ejpam-2803	221	11	closed	closed	ADJ
ejpam-2803	221	12	subset	subset	NOUN
ejpam-2803	221	13	of	of	ADP
ejpam-2803	221	14	specs(m	specs(m	PROPN
ejpam-2803	221	15	)	)	PUNCT
ejpam-2803	221	16	.	.	PUNCT
ejpam-2803	222	1	by	by	ADP
ejpam-2803	222	2	proposition	proposition	NOUN
ejpam-2803	222	3	2.5	2.5	NUM
ejpam-2803	222	4	(	(	PUNCT
ejpam-2803	222	5	a	a	X
ejpam-2803	222	6	)	)	PUNCT
ejpam-2803	222	7	,	,	PUNCT
ejpam-2803	222	8	there	there	PRON
ejpam-2803	222	9	exists	exist	VERB
ejpam-2803	222	10	s	s	PROPN
ejpam-2803	222	11	∈	∈	PROPN
ejpam-2803	222	12	specs(m	specs(m	NOUN
ejpam-2803	222	13	)	)	PUNCT
ejpam-2803	222	14	such	such	ADJ
ejpam-2803	222	15	that	that	PRON
ejpam-2803	222	16	v	v	ADP
ejpam-2803	222	17	s(n	s(n	PROPN
ejpam-2803	222	18	)	)	PUNCT
ejpam-2803	222	19	=	=	PROPN
ejpam-2803	222	20	v	v	NUM
ejpam-2803	222	21	s(s	s(s	PROPN
ejpam-2803	222	22	)	)	PUNCT
ejpam-2803	222	23	.	.	PUNCT
ejpam-2803	223	1	now	now	ADV
ejpam-2803	223	2	if	if	SCONJ
ejpam-2803	223	3	v	v	ADP
ejpam-2803	223	4	s(s	s(s	PROPN
ejpam-2803	223	5	)	)	PUNCT
ejpam-2803	223	6	is	be	AUX
ejpam-2803	223	7	not	not	PART
ejpam-2803	223	8	component	component	NOUN
ejpam-2803	223	9	,	,	PUNCT
ejpam-2803	223	10	then	then	ADV
ejpam-2803	223	11	v	v	ADP
ejpam-2803	223	12	s(s	s(s	PROPN
ejpam-2803	223	13	)	)	PUNCT
ejpam-2803	223	14	is	be	AUX
ejpam-2803	223	15	contained	contain	VERB
ejpam-2803	223	16	properly	properly	ADV
ejpam-2803	223	17	in	in	ADP
ejpam-2803	223	18	some	some	DET
ejpam-2803	223	19	component	component	NOUN
ejpam-2803	223	20	v	v	ADP
ejpam-2803	223	21	s(s′	s(s′	NOUN
ejpam-2803	223	22	)	)	PUNCT
ejpam-2803	223	23	for	for	ADP
ejpam-2803	223	24	some	some	DET
ejpam-2803	223	25	s′	s′	ADJ
ejpam-2803	223	26	∈	∈	PROPN
ejpam-2803	223	27	specs(m	specs(m	NOUN
ejpam-2803	223	28	)	)	PUNCT
ejpam-2803	223	29	by	by	ADP
ejpam-2803	223	30	[	[	X
ejpam-2803	223	31	13	13	NUM
ejpam-2803	223	32	,	,	PUNCT
ejpam-2803	223	33	p.	p.	NOUN
ejpam-2803	223	34	95	95	NUM
ejpam-2803	223	35	,	,	PUNCT
ejpam-2803	223	36	proposition	proposition	NOUN
ejpam-2803	223	37	5	5	NUM
ejpam-2803	223	38	]	]	PUNCT
ejpam-2803	223	39	.	.	PUNCT
ejpam-2803	224	1	it	it	PRON
ejpam-2803	224	2	follows	follow	VERB
ejpam-2803	224	3	that	that	PRON
ejpam-2803	224	4	annr(s	annr(s	NOUN
ejpam-2803	224	5	)	)	PUNCT
ejpam-2803	224	6	)	)	PUNCT
ejpam-2803	224	7	annr(s′	annr(s′	NUM
ejpam-2803	224	8	)	)	PUNCT
ejpam-2803	224	9	and	and	CCONJ
ejpam-2803	224	10	hence	hence	ADV
ejpam-2803	224	11	dim(specs(m	dim(specs(m	AUX
ejpam-2803	224	12	)	)	PUNCT
ejpam-2803	224	13	)	)	PUNCT
ejpam-2803	224	14	≥	≥	NOUN
ejpam-2803	224	15	1	1	NUM
ejpam-2803	224	16	,	,	PUNCT
ejpam-2803	224	17	a	a	DET
ejpam-2803	224	18	contradiction	contradiction	NOUN
ejpam-2803	224	19	.	.	PUNCT
ejpam-2803	225	1	(	(	PUNCT
ejpam-2803	225	2	b	b	X
ejpam-2803	225	3	)	)	PUNCT
ejpam-2803	225	4	let	let	VERB
ejpam-2803	225	5	p	p	PRON
ejpam-2803	225	6	∈	∈	PROPN
ejpam-2803	225	7	v	v	NOUN
ejpam-2803	225	8	(	(	PUNCT
ejpam-2803	225	9	annr(m	annr(m	PROPN
ejpam-2803	225	10	)	)	PUNCT
ejpam-2803	225	11	)	)	PUNCT
ejpam-2803	225	12	and	and	CCONJ
ejpam-2803	225	13	let	let	VERB
ejpam-2803	225	14	s	s	PRON
ejpam-2803	225	15	be	be	AUX
ejpam-2803	225	16	a	a	DET
ejpam-2803	225	17	p	p	ADJ
ejpam-2803	225	18	-	-	PUNCT
ejpam-2803	225	19	second	second	NOUN
ejpam-2803	225	20	submodule	submodule	NOUN
ejpam-2803	225	21	ofm	ofm	PROPN
ejpam-2803	225	22	.	.	PUNCT
ejpam-2803	226	1	since	since	SCONJ
ejpam-2803	226	2	dim(specs(m	dim(specs(m	PROPN
ejpam-2803	226	3	)	)	PUNCT
ejpam-2803	226	4	)	)	PUNCT
ejpam-2803	227	1	=	=	SYM
ejpam-2803	227	2	0	0	NUM
ejpam-2803	227	3	,	,	PUNCT
ejpam-2803	227	4	we	we	PRON
ejpam-2803	227	5	have	have	VERB
ejpam-2803	227	6	dim(r	dim(r	PROPN
ejpam-2803	227	7	)	)	PUNCT
ejpam-2803	228	1	=	=	SYM
ejpam-2803	229	1	0	0	X
ejpam-2803	229	2	.	.	PUNCT
ejpam-2803	229	3	hence	hence	ADV
ejpam-2803	229	4	spec(r	spec(r	PROPN
ejpam-2803	229	5	)	)	PUNCT
ejpam-2803	229	6	=	=	PUNCT
ejpam-2803	229	7	max(r	max(r	PROPN
ejpam-2803	229	8	)	)	PUNCT
ejpam-2803	229	9	.	.	PUNCT
ejpam-2803	230	1	now	now	ADV
ejpam-2803	230	2	if	if	SCONJ
ejpam-2803	230	3	s′	s′	ADJ
ejpam-2803	230	4	∈	∈	PROPN
ejpam-2803	230	5	specs(m	specs(m	NOUN
ejpam-2803	230	6	)	)	PUNCT
ejpam-2803	230	7	,	,	PUNCT
ejpam-2803	230	8	then	then	ADV
ejpam-2803	230	9	annr(s′	annr(s′	NUM
ejpam-2803	230	10	)	)	PUNCT
ejpam-2803	230	11	is	be	AUX
ejpam-2803	230	12	also	also	ADV
ejpam-2803	230	13	a	a	DET
ejpam-2803	230	14	maximal	maximal	ADJ
ejpam-2803	230	15	ideal	ideal	NOUN
ejpam-2803	230	16	.	.	PUNCT
ejpam-2803	231	1	therefore	therefore	ADV
ejpam-2803	231	2	,	,	PUNCT
ejpam-2803	231	3	we	we	PRON
ejpam-2803	231	4	have	have	VERB
ejpam-2803	231	5	s′	s′	VERB
ejpam-2803	231	6	∈	∈	PROPN
ejpam-2803	231	7	v	v	ADP
ejpam-2803	231	8	s(s	s(s	PROPN
ejpam-2803	231	9	)	)	PUNCT
ejpam-2803	231	10	⇔	⇔	PROPN
ejpam-2803	231	11	annr(s′	annr(s′	NUM
ejpam-2803	231	12	)	)	PUNCT
ejpam-2803	231	13	=	=	SYM
ejpam-2803	231	14	annr(s	annr(s	NOUN
ejpam-2803	231	15	)	)	PUNCT
ejpam-2803	231	16	=	=	SYM
ejpam-2803	232	1	p⇔	p⇔	ADJ
ejpam-2803	232	2	s′	s′	ADJ
ejpam-2803	232	3	∈	∈	PROPN
ejpam-2803	232	4	specsp(m	specsp(m	NOUN
ejpam-2803	232	5	)	)	PUNCT
ejpam-2803	232	6	.	.	PUNCT
ejpam-2803	233	1	thus	thus	ADV
ejpam-2803	233	2	v	v	ADP
ejpam-2803	233	3	s(s	s(s	PROPN
ejpam-2803	233	4	)	)	PUNCT
ejpam-2803	233	5	=	=	SYM
ejpam-2803	233	6	specsp(m	specsp(m	PROPN
ejpam-2803	233	7	)	)	PUNCT
ejpam-2803	233	8	.	.	PUNCT
ejpam-2803	234	1	(	(	PUNCT
ejpam-2803	234	2	c	c	X
ejpam-2803	234	3	)	)	PUNCT
ejpam-2803	234	4	since	since	SCONJ
ejpam-2803	234	5	specs(m	specs(m	NOUN
ejpam-2803	234	6	)	)	PUNCT
ejpam-2803	234	7	is	be	AUX
ejpam-2803	234	8	a	a	DET
ejpam-2803	234	9	noetherian	noetherian	ADJ
ejpam-2803	234	10	space	space	NOUN
ejpam-2803	234	11	with	with	ADP
ejpam-2803	234	12	dim(specs(m	dim(specs(m	PROPN
ejpam-2803	234	13	)	)	PUNCT
ejpam-2803	234	14	)	)	PUNCT
ejpam-2803	235	1	=	=	SYM
ejpam-2803	235	2	0	0	NUM
ejpam-2803	235	3	,	,	PUNCT
ejpam-2803	235	4	we	we	PRON
ejpam-2803	235	5	have	have	VERB
ejpam-2803	235	6	dim(r	dim(r	PROPN
ejpam-2803	235	7	)	)	PUNCT
ejpam-2803	236	1	=	=	SYM
ejpam-2803	236	2	0	0	PUNCT
ejpam-2803	237	1	and	and	CCONJ
ejpam-2803	237	2	r	r	NOUN
ejpam-2803	237	3	has	have	VERB
ejpam-2803	237	4	a	a	DET
ejpam-2803	237	5	noetherian	noetherian	ADJ
ejpam-2803	237	6	spectrum	spectrum	NOUN
ejpam-2803	237	7	.	.	PUNCT
ejpam-2803	238	1	now	now	ADV
ejpam-2803	238	2	spec(r	spec(r	ADP
ejpam-2803	238	3	)	)	PUNCT
ejpam-2803	238	4	=	=	SYM
ejpam-2803	238	5	max(r	max(r	PROPN
ejpam-2803	238	6	)	)	PUNCT
ejpam-2803	238	7	and	and	CCONJ
ejpam-2803	238	8	spec(r	spec(r	PROPN
ejpam-2803	238	9	)	)	PUNCT
ejpam-2803	238	10	has	have	VERB
ejpam-2803	238	11	only	only	ADV
ejpam-2803	238	12	finitely	finitely	ADV
ejpam-2803	238	13	many	many	ADJ
ejpam-2803	238	14	elements	element	NOUN
ejpam-2803	238	15	m1,m2	m1,m2	PROPN
ejpam-2803	238	16	,	,	PUNCT
ejpam-2803	238	17	...	...	PUNCT
ejpam-2803	238	18	,	,	PUNCT
ejpam-2803	238	19	mk	mk	PROPN
ejpam-2803	238	20	,	,	PUNCT
ejpam-2803	238	21	each	each	PRON
ejpam-2803	238	22	of	of	ADP
ejpam-2803	238	23	which	which	PRON
ejpam-2803	238	24	is	be	AUX
ejpam-2803	238	25	both	both	PRON
ejpam-2803	238	26	maximal	maximal	ADJ
ejpam-2803	238	27	and	and	CCONJ
ejpam-2803	238	28	minimal	minimal	ADJ
ejpam-2803	238	29	prime	prime	ADJ
ejpam-2803	238	30	ideal	ideal	NOUN
ejpam-2803	238	31	of	of	ADP
ejpam-2803	238	32	r	r	NOUN
ejpam-2803	238	33	by	by	ADP
ejpam-2803	238	34	[	[	X
ejpam-2803	238	35	21	21	NUM
ejpam-2803	238	36	,	,	PUNCT
ejpam-2803	238	37	p.	p.	NOUN
ejpam-2803	238	38	41	41	NUM
ejpam-2803	238	39	,	,	PUNCT
ejpam-2803	238	40	examples	example	NOUN
ejpam-2803	238	41	1.4	1.4	NUM
ejpam-2803	238	42	(	(	PUNCT
ejpam-2803	238	43	c	c	NOUN
ejpam-2803	238	44	)	)	PUNCT
ejpam-2803	238	45	and	and	CCONJ
ejpam-2803	238	46	(	(	PUNCT
ejpam-2803	238	47	d	d	NOUN
ejpam-2803	238	48	)	)	PUNCT
ejpam-2803	238	49	]	]	PUNCT
ejpam-2803	238	50	.	.	PUNCT
ejpam-2803	239	1	then	then	ADV
ejpam-2803	239	2	mi	mi	PROPN
ejpam-2803	239	3	is	be	AUX
ejpam-2803	239	4	a	a	DET
ejpam-2803	239	5	minimal	minimal	ADJ
ejpam-2803	239	6	prime	prime	ADJ
ejpam-2803	239	7	divisor	divisor	NOUN
ejpam-2803	239	8	of	of	ADP
ejpam-2803	239	9	annr(m	annr(m	PROPN
ejpam-2803	239	10	)	)	PUNCT
ejpam-2803	239	11	,	,	PUNCT
ejpam-2803	239	12	which	which	PRON
ejpam-2803	239	13	is	be	AUX
ejpam-2803	239	14	also	also	ADV
ejpam-2803	239	15	a	a	DET
ejpam-2803	239	16	maximal	maximal	ADJ
ejpam-2803	239	17	ideal	ideal	NOUN
ejpam-2803	239	18	of	of	ADP
ejpam-2803	239	19	r	r	NOUN
ejpam-2803	239	20	for	for	ADP
ejpam-2803	239	21	every	every	DET
ejpam-2803	239	22	i	i	NOUN
ejpam-2803	239	23	(	(	PUNCT
ejpam-2803	239	24	1	1	NUM
ejpam-2803	239	25	≤	≤	NUM
ejpam-2803	239	26	i	i	NOUN
ejpam-2803	239	27	≤	≤	NUM
ejpam-2803	239	28	k	k	X
ejpam-2803	239	29	)	)	PUNCT
ejpam-2803	239	30	.	.	PUNCT
ejpam-2803	240	1	thus	thus	ADV
ejpam-2803	240	2	{	{	PUNCT
ejpam-2803	240	3	m1,m2	m1,m2	PROPN
ejpam-2803	240	4	,	,	PUNCT
ejpam-2803	240	5	...	...	PUNCT
ejpam-2803	240	6	,	,	PUNCT
ejpam-2803	240	7	mk	mk	PROPN
ejpam-2803	240	8	}	}	PUNCT
ejpam-2803	240	9	is	be	AUX
ejpam-2803	240	10	the	the	DET
ejpam-2803	240	11	set	set	NOUN
ejpam-2803	240	12	of	of	ADP
ejpam-2803	240	13	all	all	DET
ejpam-2803	240	14	minimal	minimal	ADJ
ejpam-2803	240	15	prime	prime	ADJ
ejpam-2803	240	16	divisor	divisor	NOUN
ejpam-2803	240	17	of	of	ADP
ejpam-2803	240	18	annr(m	annr(m	NOUN
ejpam-2803	240	19	)	)	PUNCT
ejpam-2803	240	20	.	.	PUNCT
ejpam-2803	241	1	since	since	SCONJ
ejpam-2803	241	2	m	m	PROPN
ejpam-2803	241	3	is	be	AUX
ejpam-2803	241	4	a	a	DET
ejpam-2803	241	5	secondful	secondful	ADJ
ejpam-2803	241	6	r	r	NOUN
ejpam-2803	241	7	-	-	PUNCT
ejpam-2803	241	8	module	module	NOUN
ejpam-2803	241	9	,	,	PUNCT
ejpam-2803	241	10	(	(	PUNCT
ejpam-2803	241	11	0	0	NUM
ejpam-2803	241	12	:	:	PUNCT
ejpam-2803	241	13	m	m	PROPN
ejpam-2803	241	14	mi	mi	PROPN
ejpam-2803	241	15	)	)	PUNCT
ejpam-2803	241	16	6=	6=	ADP
ejpam-2803	241	17	(	(	PUNCT
ejpam-2803	241	18	0	0	NUM
ejpam-2803	241	19	)	)	PUNCT
ejpam-2803	241	20	and	and	CCONJ
ejpam-2803	241	21	annr((0	annr((0	VERB
ejpam-2803	241	22	:	:	PUNCT
ejpam-2803	241	23	m	m	PROPN
ejpam-2803	241	24	mi	mi	NOUN
ejpam-2803	241	25	)	)	PUNCT
ejpam-2803	241	26	)	)	PUNCT
ejpam-2803	242	1	=	=	SYM
ejpam-2803	242	2	mi	mi	PROPN
ejpam-2803	242	3	is	be	AUX
ejpam-2803	242	4	a	a	DET
ejpam-2803	242	5	maximal	maximal	ADJ
ejpam-2803	242	6	ideal	ideal	NOUN
ejpam-2803	242	7	of	of	ADP
ejpam-2803	242	8	r.	r.	PROPN
ejpam-2803	242	9	thus	thus	ADV
ejpam-2803	242	10	(	(	PUNCT
ejpam-2803	242	11	0	0	NUM
ejpam-2803	242	12	:	:	PUNCT
ejpam-2803	242	13	m	m	PROPN
ejpam-2803	242	14	mi	mi	NOUN
ejpam-2803	242	15	)	)	PUNCT
ejpam-2803	242	16	is	be	AUX
ejpam-2803	242	17	an	an	DET
ejpam-2803	242	18	mi	mi	ADJ
ejpam-2803	242	19	-	-	ADJ
ejpam-2803	242	20	second	second	ADJ
ejpam-2803	242	21	submodule	submodule	NOUN
ejpam-2803	242	22	and	and	CCONJ
ejpam-2803	242	23	v	v	ADP
ejpam-2803	242	24	s((0	s((0	PROPN
ejpam-2803	242	25	:	:	PUNCT
ejpam-2803	242	26	m	m	PROPN
ejpam-2803	242	27	mi	mi	NOUN
ejpam-2803	242	28	)	)	PUNCT
ejpam-2803	242	29	)	)	PUNCT
ejpam-2803	243	1	is	be	AUX
ejpam-2803	243	2	an	an	DET
ejpam-2803	243	3	irreducible	irreducible	ADJ
ejpam-2803	243	4	component	component	NOUN
ejpam-2803	243	5	of	of	ADP
ejpam-2803	243	6	specs(m	specs(m	NOUN
ejpam-2803	243	7	)	)	PUNCT
ejpam-2803	243	8	for	for	ADP
ejpam-2803	243	9	every	every	DET
ejpam-2803	243	10	i	i	NOUN
ejpam-2803	243	11	by	by	ADP
ejpam-2803	243	12	part	part	NOUN
ejpam-2803	243	13	(	(	PUNCT
ejpam-2803	243	14	a	a	NOUN
ejpam-2803	243	15	)	)	PUNCT
ejpam-2803	243	16	.	.	PUNCT
ejpam-2803	244	1	applying	apply	VERB
ejpam-2803	244	2	proposition	proposition	NOUN
ejpam-2803	244	3	2.5	2.5	NUM
ejpam-2803	244	4	(	(	PUNCT
ejpam-2803	244	5	c	c	NOUN
ejpam-2803	244	6	)	)	PUNCT
ejpam-2803	244	7	,	,	PUNCT
ejpam-2803	244	8	we	we	PRON
ejpam-2803	244	9	can	can	AUX
ejpam-2803	244	10	conclude	conclude	VERB
ejpam-2803	244	11	that	that	SCONJ
ejpam-2803	244	12	{	{	PUNCT
ejpam-2803	244	13	v	v	NUM
ejpam-2803	244	14	s((0	s((0	PROPN
ejpam-2803	244	15	:	:	PUNCT
ejpam-2803	244	16	m	m	NOUN
ejpam-2803	244	17	m1	m1	NOUN
ejpam-2803	244	18	)	)	PUNCT
ejpam-2803	244	19	)	)	PUNCT
ejpam-2803	244	20	,	,	PUNCT
ejpam-2803	244	21	v	v	X
ejpam-2803	244	22	s((0	s((0	PROPN
ejpam-2803	244	23	:	:	PUNCT
ejpam-2803	244	24	m	m	NOUN
ejpam-2803	244	25	m2	m2	PROPN
ejpam-2803	244	26	)	)	PUNCT
ejpam-2803	244	27	)	)	PUNCT
ejpam-2803	244	28	,	,	PUNCT
ejpam-2803	244	29	...	...	PUNCT
ejpam-2803	244	30	,	,	PUNCT
ejpam-2803	244	31	v	v	X
ejpam-2803	244	32	s((0	s((0	PROPN
ejpam-2803	244	33	:	:	PUNCT
ejpam-2803	244	34	m	m	PROPN
ejpam-2803	244	35	mk	mk	PROPN
ejpam-2803	244	36	)	)	PUNCT
ejpam-2803	244	37	)	)	PUNCT
ejpam-2803	244	38	}	}	PUNCT
ejpam-2803	244	39	is	be	AUX
ejpam-2803	244	40	the	the	DET
ejpam-2803	244	41	set	set	NOUN
ejpam-2803	244	42	of	of	ADP
ejpam-2803	244	43	all	all	DET
ejpam-2803	244	44	irreducible	irreducible	ADJ
ejpam-2803	244	45	components	component	NOUN
ejpam-2803	244	46	of	of	ADP
ejpam-2803	244	47	specs(m	specs(m	PROPN
ejpam-2803	244	48	)	)	PUNCT
ejpam-2803	244	49	.	.	PUNCT
ejpam-2803	245	1	theorem	theorem	VERB
ejpam-2803	245	2	2.9	2.9	NUM
ejpam-2803	245	3	.	.	PUNCT
ejpam-2803	246	1	let	let	VERB
ejpam-2803	246	2	m	m	PRON
ejpam-2803	246	3	be	be	AUX
ejpam-2803	246	4	a	a	DET
ejpam-2803	246	5	non	non	ADJ
ejpam-2803	246	6	-	-	ADJ
ejpam-2803	246	7	zero	zero	ADJ
ejpam-2803	246	8	secondful	secondful	ADJ
ejpam-2803	246	9	r	r	NOUN
ejpam-2803	246	10	-	-	PUNCT
ejpam-2803	246	11	module	module	NOUN
ejpam-2803	246	12	.	.	PUNCT
ejpam-2803	247	1	then	then	ADV
ejpam-2803	247	2	the	the	DET
ejpam-2803	247	3	following	follow	VERB
ejpam-2803	247	4	are	be	AUX
ejpam-2803	247	5	equivalent	equivalent	ADJ
ejpam-2803	247	6	:	:	PUNCT
ejpam-2803	247	7	(	(	PUNCT
ejpam-2803	247	8	a	a	X
ejpam-2803	247	9	)	)	PUNCT
ejpam-2803	247	10	specs(m	specs(m	NOUN
ejpam-2803	247	11	)	)	PUNCT
ejpam-2803	248	1	=	=	SYM
ejpam-2803	248	2	min(m	min(m	PROPN
ejpam-2803	248	3	)	)	PUNCT
ejpam-2803	248	4	(	(	PUNCT
ejpam-2803	248	5	here	here	ADV
ejpam-2803	248	6	min(m	min(m	PROPN
ejpam-2803	248	7	)	)	PUNCT
ejpam-2803	248	8	denotes	denote	VERB
ejpam-2803	248	9	the	the	DET
ejpam-2803	248	10	set	set	NOUN
ejpam-2803	248	11	of	of	ADP
ejpam-2803	248	12	all	all	DET
ejpam-2803	248	13	minimal	minimal	ADJ
ejpam-2803	248	14	submodules	submodule	NOUN
ejpam-2803	248	15	of	of	ADP
ejpam-2803	248	16	m	m	PROPN
ejpam-2803	248	17	)	)	PUNCT
ejpam-2803	248	18	;	;	PUNCT
ejpam-2803	248	19	(	(	PUNCT
ejpam-2803	248	20	b	b	X
ejpam-2803	248	21	)	)	PUNCT
ejpam-2803	248	22	(	(	PUNCT
ejpam-2803	248	23	specs(m	specs(m	PROPN
ejpam-2803	248	24	)	)	PUNCT
ejpam-2803	248	25	,	,	PUNCT
ejpam-2803	248	26	τ	τ	PROPN
ejpam-2803	248	27	s	s	PART
ejpam-2803	248	28	)	)	PUNCT
ejpam-2803	248	29	is	be	AUX
ejpam-2803	248	30	a	a	DET
ejpam-2803	248	31	t1	t1	NOUN
ejpam-2803	248	32	-	-	PUNCT
ejpam-2803	248	33	space	space	NOUN
ejpam-2803	248	34	;	;	PUNCT
ejpam-2803	248	35	(	(	PUNCT
ejpam-2803	248	36	c	c	X
ejpam-2803	248	37	)	)	PUNCT
ejpam-2803	248	38	(	(	PUNCT
ejpam-2803	248	39	specs(m	specs(m	NOUN
ejpam-2803	248	40	)	)	PUNCT
ejpam-2803	248	41	,	,	PUNCT
ejpam-2803	248	42	τ	τ	PROPN
ejpam-2803	248	43	s	s	PART
ejpam-2803	248	44	)	)	PUNCT
ejpam-2803	248	45	is	be	AUX
ejpam-2803	248	46	a	a	DET
ejpam-2803	248	47	t4	t4	PROPN
ejpam-2803	248	48	-	-	PUNCT
ejpam-2803	248	49	space	space	NOUN
ejpam-2803	248	50	;	;	PUNCT
ejpam-2803	248	51	(	(	PUNCT
ejpam-2803	248	52	d	d	X
ejpam-2803	248	53	)	)	PUNCT
ejpam-2803	248	54	(	(	PUNCT
ejpam-2803	248	55	specs(m	specs(m	NOUN
ejpam-2803	248	56	)	)	PUNCT
ejpam-2803	248	57	,	,	PUNCT
ejpam-2803	248	58	τ	τ	PROPN
ejpam-2803	248	59	s	s	PART
ejpam-2803	248	60	)	)	PUNCT
ejpam-2803	248	61	is	be	AUX
ejpam-2803	248	62	a	a	DET
ejpam-2803	248	63	t0	t0	NOUN
ejpam-2803	248	64	-	-	NOUN
ejpam-2803	248	65	space	space	NOUN
ejpam-2803	248	66	and	and	CCONJ
ejpam-2803	248	67	cosupp(m	cosupp(m	PROPN
ejpam-2803	248	68	)	)	PUNCT
ejpam-2803	249	1	≈	≈	PROPN
ejpam-2803	249	2	spec(r	spec(r	PROPN
ejpam-2803	249	3	)	)	PUNCT
ejpam-2803	249	4	=	=	SYM
ejpam-2803	249	5	max(r	max(r	PROPN
ejpam-2803	249	6	)	)	PUNCT
ejpam-2803	249	7	;	;	PUNCT
ejpam-2803	249	8	(	(	PUNCT
ejpam-2803	249	9	e	e	NOUN
ejpam-2803	249	10	)	)	PUNCT
ejpam-2803	249	11	specs(m	specs(m	NOUN
ejpam-2803	249	12	)	)	PUNCT
ejpam-2803	249	13	=	=	PRON
ejpam-2803	249	14	{	{	PUNCT
ejpam-2803	249	15	(	(	PUNCT
ejpam-2803	249	16	0	0	NUM
ejpam-2803	249	17	:	:	PUNCT
ejpam-2803	249	18	m	m	VERB
ejpam-2803	249	19	p	p	X
ejpam-2803	249	20	)	)	PUNCT
ejpam-2803	250	1	|	|	ADV
ejpam-2803	250	2	p	p	PROPN
ejpam-2803	250	3	∈	∈	PROPN
ejpam-2803	250	4	v	v	NOUN
ejpam-2803	250	5	(	(	PUNCT
ejpam-2803	250	6	annr(m	annr(m	PROPN
ejpam-2803	250	7	)	)	PUNCT
ejpam-2803	250	8	)	)	PUNCT
ejpam-2803	250	9	∩max(r	∩max(r	PUNCT
ejpam-2803	250	10	)	)	PUNCT
ejpam-2803	250	11	}	}	PUNCT
ejpam-2803	250	12	.	.	PUNCT
ejpam-2803	251	1	proof	proof	NOUN
ejpam-2803	251	2	.	.	PUNCT
ejpam-2803	252	1	(	(	PUNCT
ejpam-2803	252	2	a	a	X
ejpam-2803	252	3	)	)	PUNCT
ejpam-2803	252	4	⇒	⇒	NOUN
ejpam-2803	252	5	(	(	PUNCT
ejpam-2803	252	6	b	b	NOUN
ejpam-2803	252	7	)	)	PUNCT
ejpam-2803	252	8	.	.	PUNCT
ejpam-2803	253	1	let	let	VERB
ejpam-2803	253	2	{	{	PUNCT
ejpam-2803	253	3	s	s	AUX
ejpam-2803	253	4	}	}	PUNCT
ejpam-2803	253	5	be	be	AUX
ejpam-2803	253	6	a	a	DET
ejpam-2803	253	7	singleton	singleton	NOUN
ejpam-2803	253	8	subset	subset	NOUN
ejpam-2803	253	9	of	of	ADP
ejpam-2803	253	10	specs(m	specs(m	PROPN
ejpam-2803	253	11	)	)	PUNCT
ejpam-2803	253	12	.	.	PUNCT
ejpam-2803	254	1	let	let	VERB
ejpam-2803	254	2	s′	s′	ADJ
ejpam-2803	254	3	∈	∈	PROPN
ejpam-2803	254	4	v	v	ADP
ejpam-2803	254	5	s(s	s(s	PROPN
ejpam-2803	254	6	)	)	PUNCT
ejpam-2803	254	7	with	with	ADP
ejpam-2803	254	8	annr(s′	annr(s′	NUM
ejpam-2803	254	9	)	)	PUNCT
ejpam-2803	254	10	=	=	VERB
ejpam-2803	255	1	q.	q.	NOUN
ejpam-2803	255	2	then	then	ADV
ejpam-2803	255	3	by	by	ADP
ejpam-2803	255	4	[	[	X
ejpam-2803	255	5	30	30	NUM
ejpam-2803	255	6	,	,	PUNCT
ejpam-2803	255	7	proposition	proposition	NOUN
ejpam-2803	255	8	1.6	1.6	NUM
ejpam-2803	255	9	]	]	PUNCT
ejpam-2803	255	10	,	,	PUNCT
ejpam-2803	255	11	q	q	PROPN
ejpam-2803	255	12	∈	∈	PROPN
ejpam-2803	255	13	max(r	max(r	PROPN
ejpam-2803	255	14	)	)	PUNCT
ejpam-2803	255	15	and	and	CCONJ
ejpam-2803	255	16	(	(	PUNCT
ejpam-2803	255	17	0	0	NUM
ejpam-2803	255	18	:	:	PUNCT
ejpam-2803	255	19	m	m	VERB
ejpam-2803	255	20	q	q	NOUN
ejpam-2803	255	21	)	)	PUNCT
ejpam-2803	255	22	6=	6=	X
ejpam-2803	255	23	(	(	PUNCT
ejpam-2803	255	24	0	0	NUM
ejpam-2803	255	25	)	)	PUNCT
ejpam-2803	255	26	,	,	PUNCT
ejpam-2803	255	27	so	so	ADV
ejpam-2803	255	28	h.	h.	PROPN
ejpam-2803	255	29	ansari	ansari	PROPN
ejpam-2803	255	30	-	-	PUNCT
ejpam-2803	255	31	toroghy	toroghy	NOUN
ejpam-2803	255	32	,	,	PUNCT
ejpam-2803	255	33	s.	s.	PROPN
ejpam-2803	255	34	s.	s.	PROPN
ejpam-2803	255	35	pourmortazavi	pourmortazavi	VERB
ejpam-2803	255	36	/	/	SYM
ejpam-2803	255	37	eur	eur	PROPN
ejpam-2803	255	38	.	.	PUNCT
ejpam-2803	256	1	j.	j.	PROPN
ejpam-2803	256	2	pure	pure	PROPN
ejpam-2803	256	3	appl	appl	PROPN
ejpam-2803	256	4	.	.	PROPN
ejpam-2803	256	5	math	math	PROPN
ejpam-2803	256	6	,	,	PUNCT
ejpam-2803	256	7	10	10	NUM
ejpam-2803	256	8	(	(	PUNCT
ejpam-2803	256	9	2	2	NUM
ejpam-2803	256	10	)	)	PUNCT
ejpam-2803	256	11	(	(	PUNCT
ejpam-2803	256	12	2017	2017	NUM
ejpam-2803	256	13	)	)	PUNCT
ejpam-2803	256	14	,	,	PUNCT
ejpam-2803	256	15	211	211	NUM
ejpam-2803	256	16	-	-	SYM
ejpam-2803	256	17	230	230	NUM
ejpam-2803	256	18	218	218	NUM
ejpam-2803	256	19	annr(0	annr(0	PROPN
ejpam-2803	256	20	:	:	PUNCT
ejpam-2803	256	21	m	m	VERB
ejpam-2803	256	22	q	q	NOUN
ejpam-2803	256	23	)	)	PUNCT
ejpam-2803	256	24	=	=	VERB
ejpam-2803	257	1	q.	q.	VERB
ejpam-2803	257	2	by	by	ADP
ejpam-2803	257	3	[	[	X
ejpam-2803	257	4	30	30	NUM
ejpam-2803	257	5	,	,	PUNCT
ejpam-2803	257	6	proposition	proposition	NOUN
ejpam-2803	257	7	1.4	1.4	NUM
ejpam-2803	257	8	]	]	PUNCT
ejpam-2803	257	9	,	,	PUNCT
ejpam-2803	257	10	it	it	PRON
ejpam-2803	257	11	follows	follow	VERB
ejpam-2803	257	12	that	that	SCONJ
ejpam-2803	257	13	(	(	PUNCT
ejpam-2803	257	14	0	0	NUM
ejpam-2803	257	15	:	:	PUNCT
ejpam-2803	257	16	m	m	VERB
ejpam-2803	257	17	q	q	ADJ
ejpam-2803	257	18	)	)	PUNCT
ejpam-2803	257	19	∈	∈	PROPN
ejpam-2803	257	20	specs(m	specs(m	PROPN
ejpam-2803	257	21	)	)	PUNCT
ejpam-2803	257	22	and	and	CCONJ
ejpam-2803	257	23	hence	hence	ADV
ejpam-2803	257	24	(	(	PUNCT
ejpam-2803	257	25	0	0	NUM
ejpam-2803	257	26	:	:	PUNCT
ejpam-2803	257	27	m	m	VERB
ejpam-2803	257	28	q	q	ADJ
ejpam-2803	257	29	)	)	PUNCT
ejpam-2803	257	30	∈min(m	∈min(m	PROPN
ejpam-2803	257	31	)	)	PUNCT
ejpam-2803	257	32	.	.	PUNCT
ejpam-2803	258	1	but	but	CCONJ
ejpam-2803	258	2	annr(s	annr(s	NOUN
ejpam-2803	258	3	)	)	PUNCT
ejpam-2803	258	4	=	=	SYM
ejpam-2803	258	5	annr(s′	annr(s′	NUM
ejpam-2803	258	6	)	)	PUNCT
ejpam-2803	258	7	=	=	PRON
ejpam-2803	259	1	q	q	PROPN
ejpam-2803	259	2	implies	imply	VERB
ejpam-2803	259	3	that	that	SCONJ
ejpam-2803	259	4	s	s	VERB
ejpam-2803	259	5	⊆	⊆	NUM
ejpam-2803	259	6	(	(	PUNCT
ejpam-2803	259	7	0	0	NUM
ejpam-2803	259	8	:	:	PUNCT
ejpam-2803	259	9	m	m	VERB
ejpam-2803	259	10	q	q	NOUN
ejpam-2803	259	11	)	)	PUNCT
ejpam-2803	259	12	and	and	CCONJ
ejpam-2803	259	13	s′	s′	ADJ
ejpam-2803	259	14	⊆	⊆	NUM
ejpam-2803	259	15	(	(	PUNCT
ejpam-2803	259	16	0	0	NUM
ejpam-2803	259	17	:	:	PUNCT
ejpam-2803	259	18	m	m	VERB
ejpam-2803	259	19	q	q	NOUN
ejpam-2803	259	20	)	)	PUNCT
ejpam-2803	259	21	.	.	PUNCT
ejpam-2803	260	1	hence	hence	ADV
ejpam-2803	260	2	s	s	PART
ejpam-2803	260	3	=	=	NOUN
ejpam-2803	260	4	s′	s′	VERB
ejpam-2803	260	5	=	=	PUNCT
ejpam-2803	260	6	(	(	PUNCT
ejpam-2803	260	7	0	0	NUM
ejpam-2803	260	8	:	:	PUNCT
ejpam-2803	260	9	m	m	VERB
ejpam-2803	260	10	q	q	NOUN
ejpam-2803	260	11	)	)	PUNCT
ejpam-2803	260	12	.	.	PUNCT
ejpam-2803	261	1	it	it	PRON
ejpam-2803	261	2	turns	turn	VERB
ejpam-2803	261	3	out	out	ADP
ejpam-2803	261	4	that	that	SCONJ
ejpam-2803	261	5	{	{	PUNCT
ejpam-2803	261	6	s	s	NOUN
ejpam-2803	261	7	}	}	PUNCT
ejpam-2803	261	8	=	=	SYM
ejpam-2803	261	9	v	v	NUM
ejpam-2803	261	10	s(s	s(s	PROPN
ejpam-2803	261	11	)	)	PUNCT
ejpam-2803	261	12	is	be	AUX
ejpam-2803	261	13	a	a	DET
ejpam-2803	261	14	closed	closed	ADJ
ejpam-2803	261	15	subset	subset	NOUN
ejpam-2803	261	16	,	,	PUNCT
ejpam-2803	261	17	as	as	SCONJ
ejpam-2803	261	18	required	require	VERB
ejpam-2803	261	19	.	.	PUNCT
ejpam-2803	262	1	(	(	PUNCT
ejpam-2803	262	2	b)⇒	b)⇒	PROPN
ejpam-2803	262	3	(	(	PUNCT
ejpam-2803	262	4	c	c	NOUN
ejpam-2803	262	5	)	)	PUNCT
ejpam-2803	262	6	.	.	PUNCT
ejpam-2803	263	1	let	let	VERB
ejpam-2803	263	2	specs(m	specs(m	NOUN
ejpam-2803	263	3	)	)	PUNCT
ejpam-2803	263	4	be	be	AUX
ejpam-2803	263	5	a	a	DET
ejpam-2803	263	6	t1	t1	NOUN
ejpam-2803	263	7	-	-	PUNCT
ejpam-2803	263	8	space	space	NOUN
ejpam-2803	263	9	.	.	PUNCT
ejpam-2803	264	1	then	then	ADV
ejpam-2803	264	2	specs(m	specs(m	PROPN
ejpam-2803	264	3	)	)	PUNCT
ejpam-2803	264	4	is	be	AUX
ejpam-2803	264	5	homeomorphic	homeomorphic	ADJ
ejpam-2803	264	6	to	to	ADP
ejpam-2803	264	7	spec(r	spec(r	PROPN
ejpam-2803	264	8	)	)	PUNCT
ejpam-2803	264	9	by	by	ADP
ejpam-2803	264	10	[	[	X
ejpam-2803	264	11	7	7	NUM
ejpam-2803	264	12	,	,	PUNCT
ejpam-2803	264	13	theorem	theorem	VERB
ejpam-2803	264	14	6.3	6.3	NUM
ejpam-2803	264	15	]	]	PUNCT
ejpam-2803	264	16	.	.	PUNCT
ejpam-2803	265	1	thus	thus	ADV
ejpam-2803	265	2	spec(r	spec(r	VERB
ejpam-2803	265	3	)	)	PUNCT
ejpam-2803	265	4	is	be	AUX
ejpam-2803	265	5	a	a	DET
ejpam-2803	265	6	t1	t1	NOUN
ejpam-2803	265	7	-	-	PUNCT
ejpam-2803	265	8	space	space	NOUN
ejpam-2803	265	9	.	.	PUNCT
ejpam-2803	266	1	now	now	ADV
ejpam-2803	266	2	by	by	ADP
ejpam-2803	266	3	[	[	X
ejpam-2803	266	4	10	10	NUM
ejpam-2803	266	5	,	,	PUNCT
ejpam-2803	266	6	p.44	p.44	NOUN
ejpam-2803	266	7	,	,	PUNCT
ejpam-2803	266	8	exer	exer	NOUN
ejpam-2803	266	9	.	.	PUNCT
ejpam-2803	267	1	11	11	NUM
ejpam-2803	267	2	]	]	PUNCT
ejpam-2803	267	3	,	,	PUNCT
ejpam-2803	267	4	spec(r	spec(r	PROPN
ejpam-2803	267	5	)	)	PUNCT
ejpam-2803	267	6	is	be	AUX
ejpam-2803	267	7	a	a	DET
ejpam-2803	267	8	t1	t1	NOUN
ejpam-2803	267	9	-	-	PUNCT
ejpam-2803	267	10	space	space	NOUN
ejpam-2803	267	11	if	if	SCONJ
ejpam-2803	267	12	and	and	CCONJ
ejpam-2803	267	13	only	only	ADV
ejpam-2803	267	14	if	if	SCONJ
ejpam-2803	267	15	it	it	PRON
ejpam-2803	267	16	is	be	AUX
ejpam-2803	267	17	a	a	DET
ejpam-2803	267	18	hausdorff	hausdorff	NOUN
ejpam-2803	267	19	topological	topological	ADJ
ejpam-2803	267	20	space	space	NOUN
ejpam-2803	267	21	(	(	PUNCT
ejpam-2803	267	22	i.e.	i.e.	X
ejpam-2803	267	23	t2	t2	NOUN
ejpam-2803	267	24	-	-	PUNCT
ejpam-2803	267	25	space	space	NOUN
ejpam-2803	267	26	)	)	PUNCT
ejpam-2803	267	27	.	.	PUNCT
ejpam-2803	268	1	hence	hence	ADV
ejpam-2803	268	2	by	by	ADP
ejpam-2803	268	3	the	the	DET
ejpam-2803	268	4	above	above	ADJ
ejpam-2803	268	5	arguments	argument	NOUN
ejpam-2803	268	6	,	,	PUNCT
ejpam-2803	268	7	specs(m	specs(m	NOUN
ejpam-2803	268	8	)	)	PUNCT
ejpam-2803	268	9	is	be	AUX
ejpam-2803	268	10	a	a	DET
ejpam-2803	268	11	hausdorff	hausdorff	NOUN
ejpam-2803	268	12	topological	topological	ADJ
ejpam-2803	268	13	space	space	NOUN
ejpam-2803	268	14	.	.	PUNCT
ejpam-2803	269	1	on	on	ADP
ejpam-2803	269	2	the	the	DET
ejpam-2803	269	3	other	other	ADJ
ejpam-2803	269	4	hand	hand	NOUN
ejpam-2803	269	5	,	,	PUNCT
ejpam-2803	269	6	by	by	ADP
ejpam-2803	269	7	[	[	PUNCT
ejpam-2803	269	8	27	27	NUM
ejpam-2803	269	9	,	,	PUNCT
ejpam-2803	269	10	chap	chap	NOUN
ejpam-2803	269	11	.	.	PUNCT
ejpam-2803	270	1	3	3	NUM
ejpam-2803	270	2	,	,	PUNCT
ejpam-2803	270	3	exer	exer	NOUN
ejpam-2803	270	4	.	.	PUNCT
ejpam-2803	271	1	3.5	3.5	NUM
ejpam-2803	271	2	]	]	PUNCT
ejpam-2803	271	3	,	,	PUNCT
ejpam-2803	271	4	every	every	DET
ejpam-2803	271	5	hausdorff	hausdorff	NOUN
ejpam-2803	271	6	quasi	quasi	ADJ
ejpam-2803	271	7	-	-	ADJ
ejpam-2803	271	8	compact	compact	ADJ
ejpam-2803	271	9	topological	topological	ADJ
ejpam-2803	271	10	space	space	NOUN
ejpam-2803	271	11	is	be	AUX
ejpam-2803	271	12	a	a	DET
ejpam-2803	271	13	t4	t4	PROPN
ejpam-2803	271	14	-	-	PUNCT
ejpam-2803	271	15	space	space	NOUN
ejpam-2803	271	16	.	.	PUNCT
ejpam-2803	272	1	therefore	therefore	ADV
ejpam-2803	272	2	,	,	PUNCT
ejpam-2803	272	3	the	the	DET
ejpam-2803	272	4	result	result	NOUN
ejpam-2803	272	5	follows	follow	VERB
ejpam-2803	272	6	from	from	ADP
ejpam-2803	272	7	the	the	DET
ejpam-2803	272	8	fact	fact	NOUN
ejpam-2803	272	9	that	that	SCONJ
ejpam-2803	272	10	(	(	PUNCT
ejpam-2803	272	11	specs(m	specs(m	NOUN
ejpam-2803	272	12	)	)	PUNCT
ejpam-2803	272	13	,	,	PUNCT
ejpam-2803	272	14	τ	τ	PROPN
ejpam-2803	272	15	s	s	PART
ejpam-2803	272	16	)	)	PUNCT
ejpam-2803	272	17	is	be	AUX
ejpam-2803	272	18	a	a	DET
ejpam-2803	272	19	compact	compact	ADJ
ejpam-2803	272	20	topological	topological	ADJ
ejpam-2803	272	21	space	space	NOUN
ejpam-2803	272	22	[	[	X
ejpam-2803	272	23	7	7	NUM
ejpam-2803	272	24	,	,	PUNCT
ejpam-2803	272	25	theorem	theorem	VERB
ejpam-2803	272	26	4.4	4.4	NUM
ejpam-2803	272	27	]	]	PUNCT
ejpam-2803	272	28	.	.	PUNCT
ejpam-2803	273	1	(	(	PUNCT
ejpam-2803	273	2	c	c	X
ejpam-2803	273	3	)	)	PUNCT
ejpam-2803	273	4	⇒	⇒	NOUN
ejpam-2803	273	5	(	(	PUNCT
ejpam-2803	273	6	d	d	NOUN
ejpam-2803	273	7	)	)	PUNCT
ejpam-2803	273	8	.	.	PUNCT
ejpam-2803	274	1	since	since	SCONJ
ejpam-2803	274	2	(	(	PUNCT
ejpam-2803	274	3	specs(m	specs(m	PROPN
ejpam-2803	274	4	)	)	PUNCT
ejpam-2803	274	5	,	,	PUNCT
ejpam-2803	274	6	τ	τ	PROPN
ejpam-2803	274	7	s	s	PART
ejpam-2803	274	8	)	)	PUNCT
ejpam-2803	274	9	is	be	AUX
ejpam-2803	274	10	a	a	DET
ejpam-2803	274	11	t0	t0	NOUN
ejpam-2803	274	12	-	-	NOUN
ejpam-2803	274	13	space	space	NOUN
ejpam-2803	274	14	,	,	PUNCT
ejpam-2803	274	15	it	it	PRON
ejpam-2803	274	16	is	be	AUX
ejpam-2803	274	17	homeomorphic	homeomorphic	ADJ
ejpam-2803	274	18	to	to	ADP
ejpam-2803	274	19	spec(r	spec(r	PROPN
ejpam-2803	274	20	)	)	PUNCT
ejpam-2803	274	21	by	by	ADP
ejpam-2803	274	22	[	[	X
ejpam-2803	274	23	7	7	NUM
ejpam-2803	274	24	,	,	PUNCT
ejpam-2803	274	25	theorem	theorem	VERB
ejpam-2803	274	26	6.3	6.3	NUM
ejpam-2803	274	27	]	]	PUNCT
ejpam-2803	274	28	.	.	PUNCT
ejpam-2803	275	1	thus	thus	ADV
ejpam-2803	275	2	spec(r	spec(r	VERB
ejpam-2803	275	3	)	)	PUNCT
ejpam-2803	275	4	is	be	AUX
ejpam-2803	275	5	t4	t4	PROPN
ejpam-2803	275	6	-	-	PUNCT
ejpam-2803	275	7	space	space	NOUN
ejpam-2803	275	8	.	.	PUNCT
ejpam-2803	276	1	this	this	PRON
ejpam-2803	276	2	implies	imply	VERB
ejpam-2803	276	3	that	that	SCONJ
ejpam-2803	276	4	spec(r	spec(r	PROPN
ejpam-2803	276	5	)	)	PUNCT
ejpam-2803	276	6	=	=	SYM
ejpam-2803	276	7	max(r	max(r	PROPN
ejpam-2803	276	8	)	)	PUNCT
ejpam-2803	276	9	by	by	ADP
ejpam-2803	276	10	[	[	X
ejpam-2803	276	11	10	10	NUM
ejpam-2803	276	12	,	,	PUNCT
ejpam-2803	276	13	p.	p.	NOUN
ejpam-2803	276	14	44	44	NUM
ejpam-2803	276	15	,	,	PUNCT
ejpam-2803	276	16	exer	exer	NOUN
ejpam-2803	276	17	.	.	PUNCT
ejpam-2803	277	1	11	11	NUM
ejpam-2803	277	2	]	]	PUNCT
ejpam-2803	277	3	,	,	PUNCT
ejpam-2803	277	4	but	but	CCONJ
ejpam-2803	277	5	spec(r	spec(r	X
ejpam-2803	277	6	)	)	PUNCT
ejpam-2803	277	7	is	be	AUX
ejpam-2803	277	8	homeomorphic	homeomorphic	ADJ
ejpam-2803	277	9	to	to	ADP
ejpam-2803	277	10	v	v	PROPN
ejpam-2803	277	11	(	(	PUNCT
ejpam-2803	277	12	annr(m	annr(m	PROPN
ejpam-2803	277	13	)	)	PUNCT
ejpam-2803	277	14	)	)	PUNCT
ejpam-2803	277	15	by	by	ADP
ejpam-2803	277	16	[	[	X
ejpam-2803	277	17	10	10	NUM
ejpam-2803	277	18	,	,	PUNCT
ejpam-2803	277	19	p.	p.	NOUN
ejpam-2803	277	20	13	13	NUM
ejpam-2803	277	21	,	,	PUNCT
ejpam-2803	277	22	exer	exer	NOUN
ejpam-2803	277	23	.	.	PUNCT
ejpam-2803	278	1	21	21	NUM
ejpam-2803	278	2	]	]	PUNCT
ejpam-2803	278	3	.	.	PUNCT
ejpam-2803	279	1	on	on	ADP
ejpam-2803	279	2	the	the	DET
ejpam-2803	279	3	other	other	ADJ
ejpam-2803	279	4	hand	hand	NOUN
ejpam-2803	279	5	,	,	PUNCT
ejpam-2803	279	6	v	v	NOUN
ejpam-2803	279	7	(	(	PUNCT
ejpam-2803	279	8	annr(m	annr(m	NOUN
ejpam-2803	279	9	)	)	PUNCT
ejpam-2803	279	10	)	)	PUNCT
ejpam-2803	280	1	=	=	SYM
ejpam-2803	280	2	cosupp(m	cosupp(m	PROPN
ejpam-2803	280	3	)	)	PUNCT
ejpam-2803	280	4	by	by	ADP
ejpam-2803	280	5	[	[	X
ejpam-2803	280	6	17	17	NUM
ejpam-2803	280	7	,	,	PUNCT
ejpam-2803	280	8	theorem	theorem	VERB
ejpam-2803	280	9	2.5	2.5	NUM
ejpam-2803	280	10	]	]	PUNCT
ejpam-2803	280	11	.	.	PUNCT
ejpam-2803	281	1	it	it	PRON
ejpam-2803	281	2	follows	follow	VERB
ejpam-2803	281	3	that	that	PRON
ejpam-2803	281	4	cosupp(m	cosupp(m	PROPN
ejpam-2803	281	5	)	)	PUNCT
ejpam-2803	282	1	≈	≈	PROPN
ejpam-2803	282	2	spec(r	spec(r	PROPN
ejpam-2803	282	3	)	)	PUNCT
ejpam-2803	282	4	=	=	PUNCT
ejpam-2803	282	5	max(r	max(r	PROPN
ejpam-2803	282	6	)	)	PUNCT
ejpam-2803	282	7	,	,	PUNCT
ejpam-2803	282	8	as	as	SCONJ
ejpam-2803	282	9	desired	desire	VERB
ejpam-2803	282	10	.	.	PUNCT
ejpam-2803	283	1	(	(	PUNCT
ejpam-2803	283	2	d)⇒	d)⇒	NOUN
ejpam-2803	283	3	(	(	PUNCT
ejpam-2803	283	4	e	e	NOUN
ejpam-2803	283	5	)	)	PUNCT
ejpam-2803	283	6	.	.	PUNCT
ejpam-2803	284	1	set	set	VERB
ejpam-2803	284	2	t	t	PROPN
ejpam-2803	284	3	=	=	SYM
ejpam-2803	284	4	{	{	PUNCT
ejpam-2803	284	5	(	(	PUNCT
ejpam-2803	284	6	0	0	NUM
ejpam-2803	284	7	:	:	PUNCT
ejpam-2803	284	8	m	m	VERB
ejpam-2803	284	9	p	p	X
ejpam-2803	284	10	)	)	PUNCT
ejpam-2803	285	1	|	|	ADV
ejpam-2803	285	2	p	p	PROPN
ejpam-2803	285	3	∈	∈	PROPN
ejpam-2803	285	4	v	v	ADP
ejpam-2803	285	5	(	(	PUNCT
ejpam-2803	285	6	annr(m))∩max(r	annr(m))∩max(r	NOUN
ejpam-2803	285	7	)	)	PUNCT
ejpam-2803	285	8	}	}	PUNCT
ejpam-2803	285	9	.	.	PUNCT
ejpam-2803	286	1	let	let	VERB
ejpam-2803	286	2	p	p	PRON
ejpam-2803	286	3	∈	∈	PROPN
ejpam-2803	286	4	v	v	NOUN
ejpam-2803	286	5	(	(	PUNCT
ejpam-2803	286	6	annr(m))∩	annr(m))∩	PROPN
ejpam-2803	286	7	max(r	max(r	PROPN
ejpam-2803	286	8	)	)	PUNCT
ejpam-2803	286	9	.	.	PUNCT
ejpam-2803	287	1	since	since	SCONJ
ejpam-2803	287	2	m	m	PROPN
ejpam-2803	287	3	is	be	AUX
ejpam-2803	287	4	secondful	secondful	ADJ
ejpam-2803	287	5	,	,	PUNCT
ejpam-2803	287	6	(	(	PUNCT
ejpam-2803	287	7	0	0	NUM
ejpam-2803	287	8	:	:	PUNCT
ejpam-2803	287	9	m	m	VERB
ejpam-2803	287	10	p	p	NOUN
ejpam-2803	287	11	)	)	PUNCT
ejpam-2803	287	12	6=	6=	X
ejpam-2803	287	13	(	(	PUNCT
ejpam-2803	287	14	0	0	NUM
ejpam-2803	287	15	)	)	PUNCT
ejpam-2803	287	16	.	.	PUNCT
ejpam-2803	288	1	thus	thus	ADV
ejpam-2803	288	2	(	(	PUNCT
ejpam-2803	288	3	0	0	NUM
ejpam-2803	288	4	:	:	PUNCT
ejpam-2803	288	5	m	m	VERB
ejpam-2803	288	6	p	p	X
ejpam-2803	288	7	)	)	PUNCT
ejpam-2803	288	8	∈	∈	PROPN
ejpam-2803	288	9	specs(m	specs(m	NOUN
ejpam-2803	288	10	)	)	PUNCT
ejpam-2803	288	11	by	by	ADP
ejpam-2803	288	12	[	[	X
ejpam-2803	288	13	30	30	NUM
ejpam-2803	288	14	,	,	PUNCT
ejpam-2803	288	15	proposition	proposition	NOUN
ejpam-2803	288	16	1.4	1.4	NUM
ejpam-2803	288	17	]	]	PUNCT
ejpam-2803	288	18	.	.	PUNCT
ejpam-2803	289	1	therefore	therefore	ADV
ejpam-2803	289	2	,	,	PUNCT
ejpam-2803	289	3	t	t	PROPN
ejpam-2803	289	4	⊆	⊆	NUM
ejpam-2803	289	5	specs(m	specs(m	PROPN
ejpam-2803	289	6	)	)	PUNCT
ejpam-2803	289	7	.	.	PUNCT
ejpam-2803	290	1	conversely	conversely	ADV
ejpam-2803	290	2	,	,	PUNCT
ejpam-2803	290	3	let	let	VERB
ejpam-2803	290	4	s	s	PRON
ejpam-2803	290	5	∈	∈	PROPN
ejpam-2803	290	6	specs(m	specs(m	PROPN
ejpam-2803	290	7	)	)	PUNCT
ejpam-2803	290	8	.	.	PUNCT
ejpam-2803	291	1	then	then	ADV
ejpam-2803	291	2	annr(s	annr(s	PROPN
ejpam-2803	291	3	)	)	PUNCT
ejpam-2803	291	4	∈	∈	PROPN
ejpam-2803	291	5	spec(r	spec(r	PROPN
ejpam-2803	291	6	)	)	PUNCT
ejpam-2803	291	7	.	.	PUNCT
ejpam-2803	292	1	this	this	PRON
ejpam-2803	292	2	implies	imply	VERB
ejpam-2803	292	3	that	that	SCONJ
ejpam-2803	292	4	annr(s	annr(s	NOUN
ejpam-2803	292	5	)	)	PUNCT
ejpam-2803	292	6	∈	∈	PROPN
ejpam-2803	292	7	v	v	NOUN
ejpam-2803	292	8	(	(	PUNCT
ejpam-2803	292	9	annr(m	annr(m	NOUN
ejpam-2803	292	10	)	)	PUNCT
ejpam-2803	292	11	)	)	PUNCT
ejpam-2803	292	12	∩	∩	PROPN
ejpam-2803	292	13	max(r	max(r	PROPN
ejpam-2803	292	14	)	)	PUNCT
ejpam-2803	292	15	.	.	PUNCT
ejpam-2803	293	1	set	set	VERB
ejpam-2803	293	2	p	p	NOUN
ejpam-2803	293	3	=	=	PROPN
ejpam-2803	293	4	annr(s	annr(s	PROPN
ejpam-2803	293	5	)	)	PUNCT
ejpam-2803	293	6	.	.	PUNCT
ejpam-2803	294	1	then	then	ADV
ejpam-2803	294	2	s	s	VERB
ejpam-2803	294	3	⊆	⊆	NUM
ejpam-2803	294	4	(	(	PUNCT
ejpam-2803	294	5	0	0	NUM
ejpam-2803	294	6	:	:	PUNCT
ejpam-2803	294	7	m	m	VERB
ejpam-2803	294	8	p	p	NOUN
ejpam-2803	294	9	)	)	PUNCT
ejpam-2803	294	10	,	,	PUNCT
ejpam-2803	294	11	so	so	ADV
ejpam-2803	294	12	annr(0	annr(0	PROPN
ejpam-2803	294	13	:	:	PUNCT
ejpam-2803	294	14	m	m	VERB
ejpam-2803	294	15	p	p	ADJ
ejpam-2803	294	16	)	)	PUNCT
ejpam-2803	294	17	=	=	SYM
ejpam-2803	294	18	annr(s	annr(s	NOUN
ejpam-2803	294	19	)	)	PUNCT
ejpam-2803	294	20	=	=	PUNCT
ejpam-2803	295	1	p	p	X
ejpam-2803	295	2	∈	∈	PROPN
ejpam-2803	295	3	max(r	max(r	PROPN
ejpam-2803	295	4	)	)	PUNCT
ejpam-2803	295	5	.	.	PUNCT
ejpam-2803	296	1	thus	thus	ADV
ejpam-2803	296	2	(	(	PUNCT
ejpam-2803	296	3	0	0	NUM
ejpam-2803	296	4	:	:	PUNCT
ejpam-2803	296	5	m	m	VERB
ejpam-2803	296	6	p	p	X
ejpam-2803	296	7	)	)	PUNCT
ejpam-2803	296	8	∈	∈	PROPN
ejpam-2803	296	9	specs(m	specs(m	PROPN
ejpam-2803	296	10	)	)	PUNCT
ejpam-2803	296	11	.	.	PUNCT
ejpam-2803	297	1	now	now	ADV
ejpam-2803	297	2	since	since	SCONJ
ejpam-2803	297	3	(	(	PUNCT
ejpam-2803	297	4	specs(m	specs(m	PROPN
ejpam-2803	297	5	)	)	PUNCT
ejpam-2803	297	6	,	,	PUNCT
ejpam-2803	297	7	τ	τ	PROPN
ejpam-2803	297	8	s	s	PART
ejpam-2803	297	9	)	)	PUNCT
ejpam-2803	297	10	is	be	AUX
ejpam-2803	297	11	a	a	DET
ejpam-2803	297	12	t0	t0	NOUN
ejpam-2803	297	13	-	-	NOUN
ejpam-2803	297	14	space	space	NOUN
ejpam-2803	297	15	,	,	PUNCT
ejpam-2803	297	16	the	the	DET
ejpam-2803	297	17	natural	natural	ADJ
ejpam-2803	297	18	map	map	NOUN
ejpam-2803	297	19	of	of	ADP
ejpam-2803	297	20	specs(m	specs(m	NOUN
ejpam-2803	297	21	)	)	PUNCT
ejpam-2803	297	22	is	be	AUX
ejpam-2803	297	23	injective	injective	ADJ
ejpam-2803	297	24	by	by	ADP
ejpam-2803	297	25	[	[	X
ejpam-2803	297	26	7	7	NUM
ejpam-2803	297	27	,	,	PUNCT
ejpam-2803	297	28	theorem	theorem	VERB
ejpam-2803	297	29	6.3	6.3	NUM
ejpam-2803	297	30	]	]	PUNCT
ejpam-2803	297	31	.	.	PUNCT
ejpam-2803	298	1	it	it	PRON
ejpam-2803	298	2	follows	follow	VERB
ejpam-2803	298	3	that	that	PRON
ejpam-2803	298	4	s	s	VERB
ejpam-2803	298	5	=	=	X
ejpam-2803	298	6	(	(	PUNCT
ejpam-2803	298	7	0	0	NUM
ejpam-2803	298	8	:	:	PUNCT
ejpam-2803	298	9	m	m	VERB
ejpam-2803	298	10	p	p	X
ejpam-2803	298	11	)	)	PUNCT
ejpam-2803	298	12	as	as	SCONJ
ejpam-2803	298	13	required	require	VERB
ejpam-2803	298	14	.	.	PUNCT
ejpam-2803	299	1	(	(	PUNCT
ejpam-2803	299	2	e	e	NOUN
ejpam-2803	299	3	)	)	PUNCT
ejpam-2803	299	4	⇒	⇒	NOUN
ejpam-2803	299	5	(	(	PUNCT
ejpam-2803	299	6	a	a	X
ejpam-2803	299	7	)	)	PUNCT
ejpam-2803	299	8	.	.	PUNCT
ejpam-2803	300	1	let	let	VERB
ejpam-2803	300	2	s	s	PRON
ejpam-2803	300	3	∈	∈	PROPN
ejpam-2803	300	4	specs(m	specs(m	PROPN
ejpam-2803	300	5	)	)	PUNCT
ejpam-2803	300	6	.	.	PUNCT
ejpam-2803	301	1	then	then	ADV
ejpam-2803	301	2	there	there	PRON
ejpam-2803	301	3	is	be	VERB
ejpam-2803	301	4	p	p	PROPN
ejpam-2803	301	5	∈	∈	PROPN
ejpam-2803	301	6	v	v	NOUN
ejpam-2803	301	7	(	(	PUNCT
ejpam-2803	301	8	annr(m	annr(m	NOUN
ejpam-2803	301	9	)	)	PUNCT
ejpam-2803	301	10	)	)	PUNCT
ejpam-2803	301	11	∩	∩	PROPN
ejpam-2803	301	12	max(r	max(r	PROPN
ejpam-2803	301	13	)	)	PUNCT
ejpam-2803	301	14	such	such	ADJ
ejpam-2803	301	15	that	that	SCONJ
ejpam-2803	301	16	(	(	PUNCT
ejpam-2803	301	17	0	0	NUM
ejpam-2803	301	18	:	:	PUNCT
ejpam-2803	301	19	m	m	VERB
ejpam-2803	301	20	p	p	ADJ
ejpam-2803	301	21	)	)	PUNCT
ejpam-2803	301	22	=	=	VERB
ejpam-2803	302	1	s.	s.	PROPN
ejpam-2803	302	2	now	now	ADV
ejpam-2803	302	3	let	let	VERB
ejpam-2803	302	4	n	n	PRON
ejpam-2803	302	5	be	be	AUX
ejpam-2803	302	6	a	a	DET
ejpam-2803	302	7	non	non	ADJ
ejpam-2803	302	8	-	-	ADJ
ejpam-2803	302	9	zero	zero	NUM
ejpam-2803	302	10	submodule	submodule	NOUN
ejpam-2803	302	11	of	of	ADP
ejpam-2803	302	12	m	m	PROPN
ejpam-2803	302	13	with	with	ADP
ejpam-2803	302	14	n	n	PROPN
ejpam-2803	302	15	⊆	⊆	NUM
ejpam-2803	302	16	s.	s.	PROPN
ejpam-2803	302	17	then	then	ADV
ejpam-2803	302	18	annr(s	annr(s	PROPN
ejpam-2803	302	19	)	)	PUNCT
ejpam-2803	302	20	=	=	PUNCT
ejpam-2803	302	21	annr(n	annr(n	PROPN
ejpam-2803	302	22	)	)	PUNCT
ejpam-2803	302	23	=	=	PUNCT
ejpam-2803	302	24	p	p	NOUN
ejpam-2803	302	25	∈max(r	∈max(r	PROPN
ejpam-2803	302	26	)	)	PUNCT
ejpam-2803	302	27	.	.	PUNCT
ejpam-2803	303	1	hence	hence	ADV
ejpam-2803	303	2	we	we	PRON
ejpam-2803	303	3	have	have	VERB
ejpam-2803	303	4	n	n	NUM
ejpam-2803	303	5	∈	∈	NOUN
ejpam-2803	303	6	specsp(m	specsp(m	NOUN
ejpam-2803	303	7	)	)	PUNCT
ejpam-2803	303	8	by	by	ADP
ejpam-2803	303	9	[	[	X
ejpam-2803	303	10	30	30	NUM
ejpam-2803	303	11	,	,	PUNCT
ejpam-2803	303	12	proposition	proposition	NOUN
ejpam-2803	303	13	1.4	1.4	NUM
ejpam-2803	303	14	]	]	PUNCT
ejpam-2803	303	15	,	,	PUNCT
ejpam-2803	303	16	so	so	ADV
ejpam-2803	303	17	n	n	NOUN
ejpam-2803	303	18	=	=	SYM
ejpam-2803	303	19	(	(	PUNCT
ejpam-2803	303	20	0	0	NUM
ejpam-2803	303	21	:	:	PUNCT
ejpam-2803	303	22	m	m	VERB
ejpam-2803	303	23	p	p	ADJ
ejpam-2803	303	24	)	)	PUNCT
ejpam-2803	303	25	=	=	VERB
ejpam-2803	304	1	s.	s.	PROPN
ejpam-2803	304	2	it	it	PRON
ejpam-2803	304	3	follows	follow	VERB
ejpam-2803	304	4	that	that	SCONJ
ejpam-2803	304	5	s	s	VERB
ejpam-2803	304	6	is	be	AUX
ejpam-2803	304	7	a	a	DET
ejpam-2803	304	8	minimal	minimal	ADJ
ejpam-2803	304	9	submodule	submodule	NOUN
ejpam-2803	304	10	of	of	ADP
ejpam-2803	304	11	m	m	PROPN
ejpam-2803	304	12	and	and	CCONJ
ejpam-2803	304	13	the	the	DET
ejpam-2803	304	14	proof	proof	NOUN
ejpam-2803	304	15	is	be	AUX
ejpam-2803	304	16	completed	complete	VERB
ejpam-2803	304	17	.	.	PUNCT
ejpam-2803	305	1	corollary	corollary	ADJ
ejpam-2803	305	2	2.10	2.10	NUM
ejpam-2803	305	3	.	.	PUNCT
ejpam-2803	306	1	let	let	VERB
ejpam-2803	306	2	m	m	PRON
ejpam-2803	306	3	be	be	AUX
ejpam-2803	306	4	a	a	DET
ejpam-2803	306	5	non	non	ADJ
ejpam-2803	306	6	-	-	ADJ
ejpam-2803	306	7	zero	zero	ADJ
ejpam-2803	306	8	secondful	secondful	ADJ
ejpam-2803	306	9	r	r	NOUN
ejpam-2803	306	10	-	-	PUNCT
ejpam-2803	306	11	module	module	NOUN
ejpam-2803	306	12	.	.	PUNCT
ejpam-2803	307	1	then	then	ADV
ejpam-2803	307	2	specs(m	specs(m	PROPN
ejpam-2803	307	3	)	)	PUNCT
ejpam-2803	308	1	=	=	SYM
ejpam-2803	308	2	min(m	min(m	PROPN
ejpam-2803	308	3	)	)	PUNCT
ejpam-2803	308	4	if	if	SCONJ
ejpam-2803	308	5	and	and	CCONJ
ejpam-2803	308	6	only	only	ADV
ejpam-2803	308	7	if	if	SCONJ
ejpam-2803	308	8	specs(m	specs(m	PROPN
ejpam-2803	308	9	)	)	PUNCT
ejpam-2803	308	10	is	be	AUX
ejpam-2803	308	11	a	a	DET
ejpam-2803	308	12	singleton	singleton	NOUN
ejpam-2803	308	13	set	set	NOUN
ejpam-2803	308	14	provided	provide	VERB
ejpam-2803	308	15	that	that	SCONJ
ejpam-2803	308	16	r	r	NOUN
ejpam-2803	308	17	contains	contain	VERB
ejpam-2803	308	18	no	no	DET
ejpam-2803	308	19	idempotent	idempotent	NOUN
ejpam-2803	308	20	other	other	ADJ
ejpam-2803	308	21	than	than	ADP
ejpam-2803	308	22	0	0	NUM
ejpam-2803	308	23	and	and	CCONJ
ejpam-2803	308	24	1	1	NUM
ejpam-2803	308	25	;	;	PUNCT
ejpam-2803	308	26	r	r	NOUN
ejpam-2803	308	27	is	be	AUX
ejpam-2803	308	28	such	such	DET
ejpam-2803	308	29	a	a	DET
ejpam-2803	308	30	ring	ring	NOUN
ejpam-2803	308	31	if	if	SCONJ
ejpam-2803	308	32	r	r	NOUN
ejpam-2803	308	33	is	be	AUX
ejpam-2803	308	34	a	a	DET
ejpam-2803	308	35	quasi	quasi	ADJ
ejpam-2803	308	36	-	-	ADJ
ejpam-2803	308	37	local	local	ADJ
ejpam-2803	308	38	ring	ring	NOUN
ejpam-2803	308	39	or	or	CCONJ
ejpam-2803	308	40	annr(m	annr(m	NOUN
ejpam-2803	308	41	)	)	PUNCT
ejpam-2803	308	42	∈	∈	PROPN
ejpam-2803	308	43	spec(r	spec(r	PROPN
ejpam-2803	308	44	)	)	PUNCT
ejpam-2803	308	45	.	.	PUNCT
ejpam-2803	309	1	(	(	PUNCT
ejpam-2803	309	2	we	we	PRON
ejpam-2803	309	3	recall	recall	VERB
ejpam-2803	309	4	that	that	SCONJ
ejpam-2803	309	5	r	r	NOUN
ejpam-2803	309	6	is	be	AUX
ejpam-2803	309	7	a	a	DET
ejpam-2803	309	8	quasi	quasi	ADJ
ejpam-2803	309	9	-	-	ADJ
ejpam-2803	309	10	local	local	ADJ
ejpam-2803	309	11	ring	ring	NOUN
ejpam-2803	309	12	if	if	SCONJ
ejpam-2803	309	13	|max(r)|	|max(r)|	NOUN
ejpam-2803	309	14	=	=	NOUN
ejpam-2803	309	15	1	1	NUM
ejpam-2803	309	16	.	.	PUNCT
ejpam-2803	309	17	)	)	PUNCT
ejpam-2803	310	1	proof	proof	NOUN
ejpam-2803	310	2	.	.	PUNCT
ejpam-2803	311	1	let	let	VERB
ejpam-2803	311	2	specs(m	specs(m	NOUN
ejpam-2803	311	3	)	)	PUNCT
ejpam-2803	311	4	be	be	AUX
ejpam-2803	311	5	a	a	DET
ejpam-2803	311	6	singleton	singleton	NOUN
ejpam-2803	311	7	set	set	NOUN
ejpam-2803	311	8	.	.	PUNCT
ejpam-2803	312	1	then	then	ADV
ejpam-2803	312	2	specs(m	specs(m	PROPN
ejpam-2803	312	3	)	)	PUNCT
ejpam-2803	312	4	is	be	AUX
ejpam-2803	312	5	a	a	DET
ejpam-2803	312	6	t1	t1	NOUN
ejpam-2803	312	7	-	-	PUNCT
ejpam-2803	312	8	space	space	NOUN
ejpam-2803	312	9	.	.	PUNCT
ejpam-2803	313	1	therefore	therefore	ADV
ejpam-2803	313	2	,	,	PUNCT
ejpam-2803	313	3	specs(m	specs(m	PROPN
ejpam-2803	313	4	)	)	PUNCT
ejpam-2803	313	5	=	=	SYM
ejpam-2803	313	6	min(m	min(m	PROPN
ejpam-2803	313	7	)	)	PUNCT
ejpam-2803	313	8	by	by	ADP
ejpam-2803	313	9	theorem	theorem	NOUN
ejpam-2803	313	10	2.9	2.9	NUM
ejpam-2803	313	11	.	.	PUNCT
ejpam-2803	314	1	conversely	conversely	ADV
ejpam-2803	314	2	if	if	SCONJ
ejpam-2803	314	3	specs(m	specs(m	NOUN
ejpam-2803	314	4	)	)	PUNCT
ejpam-2803	314	5	=	=	SYM
ejpam-2803	314	6	min(m	min(m	PROPN
ejpam-2803	314	7	)	)	PUNCT
ejpam-2803	314	8	,	,	PUNCT
ejpam-2803	314	9	then	then	ADV
ejpam-2803	314	10	we	we	PRON
ejpam-2803	314	11	have	have	VERB
ejpam-2803	314	12	spec(r	spec(r	PROPN
ejpam-2803	314	13	)	)	PUNCT
ejpam-2803	314	14	=	=	SYM
ejpam-2803	314	15	max(r	max(r	PROPN
ejpam-2803	314	16	)	)	PUNCT
ejpam-2803	314	17	by	by	ADP
ejpam-2803	314	18	theorem	theorem	NOUN
ejpam-2803	314	19	2.9	2.9	NUM
ejpam-2803	314	20	.	.	PUNCT
ejpam-2803	315	1	but	but	CCONJ
ejpam-2803	315	2	by	by	ADP
ejpam-2803	315	3	[	[	X
ejpam-2803	315	4	10	10	NUM
ejpam-2803	315	5	,	,	PUNCT
ejpam-2803	315	6	p.	p.	NOUN
ejpam-2803	315	7	44	44	NUM
ejpam-2803	315	8	,	,	PUNCT
ejpam-2803	315	9	exer	exer	NOUN
ejpam-2803	315	10	.	.	PUNCT
ejpam-2803	316	1	11	11	NUM
ejpam-2803	316	2	]	]	PUNCT
ejpam-2803	316	3	,	,	PUNCT
ejpam-2803	316	4	spec(r	spec(r	PROPN
ejpam-2803	316	5	)	)	PUNCT
ejpam-2803	316	6	is	be	AUX
ejpam-2803	316	7	totally	totally	ADV
ejpam-2803	316	8	disconnected	disconnected	ADJ
ejpam-2803	316	9	topological	topological	ADJ
ejpam-2803	316	10	space	space	NOUN
ejpam-2803	316	11	.	.	PUNCT
ejpam-2803	317	1	thus	thus	ADV
ejpam-2803	317	2	specs(m	specs(m	PROPN
ejpam-2803	317	3	)	)	PUNCT
ejpam-2803	318	1	≈	≈	PROPN
ejpam-2803	318	2	spec(r	spec(r	PROPN
ejpam-2803	318	3	)	)	PUNCT
ejpam-2803	318	4	is	be	AUX
ejpam-2803	318	5	a	a	DET
ejpam-2803	318	6	totally	totally	ADV
ejpam-2803	318	7	disconnected	disconnected	ADJ
ejpam-2803	318	8	space	space	NOUN
ejpam-2803	318	9	.	.	PUNCT
ejpam-2803	319	1	on	on	ADP
ejpam-2803	319	2	the	the	DET
ejpam-2803	319	3	other	other	ADJ
ejpam-2803	319	4	hand	hand	NOUN
ejpam-2803	319	5	,	,	PUNCT
ejpam-2803	319	6	specs(m	specs(m	PROPN
ejpam-2803	319	7	)	)	PUNCT
ejpam-2803	319	8	is	be	AUX
ejpam-2803	319	9	a	a	DET
ejpam-2803	319	10	connected	connected	ADJ
ejpam-2803	319	11	space	space	NOUN
ejpam-2803	319	12	by	by	ADP
ejpam-2803	319	13	[	[	X
ejpam-2803	319	14	7	7	NUM
ejpam-2803	319	15	,	,	PUNCT
ejpam-2803	319	16	corollary	corollary	ADJ
ejpam-2803	319	17	3.13	3.13	NUM
ejpam-2803	319	18	]	]	PUNCT
ejpam-2803	319	19	.	.	PUNCT
ejpam-2803	320	1	thus	thus	ADV
ejpam-2803	320	2	specs(m	specs(m	PROPN
ejpam-2803	320	3	)	)	PUNCT
ejpam-2803	320	4	is	be	AUX
ejpam-2803	320	5	a	a	DET
ejpam-2803	320	6	single	single	ADJ
ejpam-2803	320	7	set	set	NOUN
ejpam-2803	320	8	and	and	CCONJ
ejpam-2803	320	9	the	the	DET
ejpam-2803	320	10	proof	proof	NOUN
ejpam-2803	320	11	is	be	AUX
ejpam-2803	320	12	completed	complete	VERB
ejpam-2803	320	13	.	.	PUNCT
ejpam-2803	321	1	h.	h.	PROPN
ejpam-2803	321	2	ansari	ansari	PROPN
ejpam-2803	321	3	-	-	PUNCT
ejpam-2803	321	4	toroghy	toroghy	NOUN
ejpam-2803	321	5	,	,	PUNCT
ejpam-2803	321	6	s.	s.	PROPN
ejpam-2803	321	7	s.	s.	PROPN
ejpam-2803	321	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	321	9	/	/	SYM
ejpam-2803	321	10	eur	eur	PROPN
ejpam-2803	321	11	.	.	PUNCT
ejpam-2803	322	1	j.	j.	PROPN
ejpam-2803	322	2	pure	pure	PROPN
ejpam-2803	322	3	appl	appl	PROPN
ejpam-2803	322	4	.	.	PROPN
ejpam-2803	322	5	math	math	PROPN
ejpam-2803	322	6	,	,	PUNCT
ejpam-2803	322	7	10	10	NUM
ejpam-2803	322	8	(	(	PUNCT
ejpam-2803	322	9	2	2	NUM
ejpam-2803	322	10	)	)	PUNCT
ejpam-2803	322	11	(	(	PUNCT
ejpam-2803	322	12	2017	2017	NUM
ejpam-2803	322	13	)	)	PUNCT
ejpam-2803	322	14	,	,	PUNCT
ejpam-2803	322	15	211	211	NUM
ejpam-2803	322	16	-	-	SYM
ejpam-2803	322	17	230	230	NUM
ejpam-2803	322	18	219	219	NUM
ejpam-2803	322	19	3	3	NUM
ejpam-2803	322	20	.	.	PUNCT
ejpam-2803	323	1	on	on	ADP
ejpam-2803	323	2	the	the	DET
ejpam-2803	323	3	second	second	ADJ
ejpam-2803	323	4	classical	classical	ADJ
ejpam-2803	323	5	zariski	zariski	NOUN
ejpam-2803	323	6	topology	topology	NOUN
ejpam-2803	323	7	throughout	throughout	ADP
ejpam-2803	323	8	this	this	DET
ejpam-2803	323	9	section	section	NOUN
ejpam-2803	323	10	,	,	PUNCT
ejpam-2803	323	11	specs(m	specs(m	PROPN
ejpam-2803	323	12	)	)	PUNCT
ejpam-2803	323	13	is	be	AUX
ejpam-2803	323	14	equipped	equip	VERB
ejpam-2803	323	15	with	with	ADP
ejpam-2803	323	16	classical	classical	ADJ
ejpam-2803	323	17	zariski	zariski	NOUN
ejpam-2803	323	18	topology	topology	NOUN
ejpam-2803	323	19	.	.	PUNCT
ejpam-2803	324	1	we	we	PRON
ejpam-2803	324	2	recall	recall	VERB
ejpam-2803	324	3	that	that	SCONJ
ejpam-2803	324	4	if	if	SCONJ
ejpam-2803	324	5	m	m	NOUN
ejpam-2803	324	6	is	be	AUX
ejpam-2803	324	7	a	a	DET
ejpam-2803	324	8	cotop	cotop	NOUN
ejpam-2803	324	9	module	module	NOUN
ejpam-2803	324	10	,	,	PUNCT
ejpam-2803	324	11	then	then	ADV
ejpam-2803	324	12	its	its	PRON
ejpam-2803	324	13	related	related	ADJ
ejpam-2803	324	14	topology	topology	NOUN
ejpam-2803	324	15	(	(	PUNCT
ejpam-2803	324	16	i.e.	i.e.	X
ejpam-2803	324	17	quasi	quasi	ADJ
ejpam-2803	324	18	-	-	ADJ
ejpam-2803	324	19	zariski	zariski	ADJ
ejpam-2803	324	20	topology	topology	NOUN
ejpam-2803	324	21	)	)	PUNCT
ejpam-2803	324	22	coincide	coincide	NOUN
ejpam-2803	324	23	with	with	ADP
ejpam-2803	324	24	classical	classical	ADJ
ejpam-2803	324	25	zariski	zariski	NOUN
ejpam-2803	324	26	topology	topology	NOUN
ejpam-2803	324	27	.	.	PUNCT
ejpam-2803	325	1	a	a	DET
ejpam-2803	325	2	spectral	spectral	ADJ
ejpam-2803	325	3	space	space	NOUN
ejpam-2803	325	4	is	be	AUX
ejpam-2803	325	5	a	a	DET
ejpam-2803	325	6	topological	topological	ADJ
ejpam-2803	325	7	space	space	NOUN
ejpam-2803	325	8	homeomorphic	homeomorphic	NOUN
ejpam-2803	325	9	to	to	ADP
ejpam-2803	325	10	the	the	DET
ejpam-2803	325	11	prime	prime	ADJ
ejpam-2803	325	12	spectrum	spectrum	NOUN
ejpam-2803	325	13	of	of	ADP
ejpam-2803	325	14	a	a	DET
ejpam-2803	325	15	commutative	commutative	ADJ
ejpam-2803	325	16	ring	ring	NOUN
ejpam-2803	325	17	equipped	equip	VERB
ejpam-2803	325	18	with	with	ADP
ejpam-2803	325	19	the	the	DET
ejpam-2803	325	20	zariski	zariski	NOUN
ejpam-2803	325	21	topology	topology	NOUN
ejpam-2803	325	22	.	.	PUNCT
ejpam-2803	326	1	spectral	spectral	ADJ
ejpam-2803	326	2	spaces	space	NOUN
ejpam-2803	326	3	have	have	AUX
ejpam-2803	326	4	been	be	AUX
ejpam-2803	326	5	characterized	characterize	VERB
ejpam-2803	326	6	by	by	ADP
ejpam-2803	326	7	m.	m.	NOUN
ejpam-2803	326	8	hochster	hochster	NOUN
ejpam-2803	326	9	as	as	ADP
ejpam-2803	326	10	quasi	quasi	ADJ
ejpam-2803	326	11	-	-	ADJ
ejpam-2803	326	12	compact	compact	ADJ
ejpam-2803	326	13	t0	t0	NOUN
ejpam-2803	326	14	-	-	NOUN
ejpam-2803	326	15	space	space	NOUN
ejpam-2803	326	16	having	have	VERB
ejpam-2803	326	17	a	a	DET
ejpam-2803	326	18	quasi	quasi	ADJ
ejpam-2803	326	19	-	-	ADJ
ejpam-2803	326	20	compact	compact	ADJ
ejpam-2803	326	21	open	open	ADJ
ejpam-2803	326	22	base	base	NOUN
ejpam-2803	326	23	closed	close	VERB
ejpam-2803	326	24	under	under	ADP
ejpam-2803	326	25	finite	finite	ADJ
ejpam-2803	326	26	intersections	intersection	NOUN
ejpam-2803	326	27	and	and	CCONJ
ejpam-2803	326	28	each	each	DET
ejpam-2803	326	29	irreducible	irreducible	ADJ
ejpam-2803	326	30	closed	close	VERB
ejpam-2803	326	31	subset	subset	NOUN
ejpam-2803	326	32	has	have	VERB
ejpam-2803	326	33	a	a	DET
ejpam-2803	326	34	generic	generic	ADJ
ejpam-2803	326	35	point	point	NOUN
ejpam-2803	326	36	(	(	PUNCT
ejpam-2803	326	37	see	see	VERB
ejpam-2803	326	38	[	[	X
ejpam-2803	326	39	19	19	NUM
ejpam-2803	326	40	]	]	NUM
ejpam-2803	326	41	)	)	PUNCT
ejpam-2803	326	42	.	.	PUNCT
ejpam-2803	327	1	lemma	lemma	PROPN
ejpam-2803	327	2	3.1	3.1	NUM
ejpam-2803	327	3	.	.	PUNCT
ejpam-2803	328	1	let	let	VERB
ejpam-2803	328	2	m	m	PRON
ejpam-2803	328	3	be	be	AUX
ejpam-2803	328	4	an	an	DET
ejpam-2803	328	5	r	r	NOUN
ejpam-2803	328	6	-	-	PUNCT
ejpam-2803	328	7	module	module	NOUN
ejpam-2803	328	8	and	and	CCONJ
ejpam-2803	328	9	y	y	PROPN
ejpam-2803	328	10	be	be	AUX
ejpam-2803	328	11	a	a	DET
ejpam-2803	328	12	nonempty	nonempty	ADJ
ejpam-2803	328	13	subset	subset	NOUN
ejpam-2803	328	14	of	of	ADP
ejpam-2803	328	15	specsr(m	specsr(m	NOUN
ejpam-2803	328	16	)	)	PUNCT
ejpam-2803	328	17	.	.	PUNCT
ejpam-2803	329	1	then	then	ADV
ejpam-2803	329	2	cl(y	cl(y	PUNCT
ejpam-2803	329	3	)	)	PUNCT
ejpam-2803	329	4	=	=	SYM
ejpam-2803	329	5	cl	cl	NOUN
ejpam-2803	329	6	(	(	PUNCT
ejpam-2803	329	7	⋃	⋃	ADV
ejpam-2803	329	8	s∈y	s∈y	ADJ
ejpam-2803	329	9	v	v	PROPN
ejpam-2803	329	10	s∗(s	s∗(s	NOUN
ejpam-2803	329	11	)	)	PUNCT
ejpam-2803	329	12	)	)	PUNCT
ejpam-2803	329	13	.	.	PUNCT
ejpam-2803	330	1	in	in	ADP
ejpam-2803	330	2	particular	particular	ADJ
ejpam-2803	330	3	,	,	PUNCT
ejpam-2803	330	4	when	when	SCONJ
ejpam-2803	330	5	y	y	PROPN
ejpam-2803	330	6	is	be	AUX
ejpam-2803	330	7	closed	closed	ADJ
ejpam-2803	330	8	we	we	PRON
ejpam-2803	330	9	have	have	VERB
ejpam-2803	330	10	y	y	NOUN
ejpam-2803	330	11	=	=	PUNCT
ejpam-2803	330	12	⋃	⋃	PROPN
ejpam-2803	330	13	s∈y	s∈y	VERB
ejpam-2803	330	14	v	v	ADP
ejpam-2803	330	15	s∗(s	s∗(s	NOUN
ejpam-2803	330	16	)	)	PUNCT
ejpam-2803	330	17	.	.	PUNCT
ejpam-2803	331	1	proof	proof	NOUN
ejpam-2803	331	2	.	.	PUNCT
ejpam-2803	332	1	this	this	PRON
ejpam-2803	332	2	is	be	AUX
ejpam-2803	332	3	straightforward	straightforward	ADJ
ejpam-2803	332	4	.	.	PUNCT
ejpam-2803	333	1	theorem	theorem	ADJ
ejpam-2803	333	2	3.2	3.2	NUM
ejpam-2803	333	3	.	.	PUNCT
ejpam-2803	334	1	for	for	ADP
ejpam-2803	334	2	any	any	DET
ejpam-2803	334	3	r	r	NOUN
ejpam-2803	334	4	-	-	PUNCT
ejpam-2803	334	5	module	module	NOUN
ejpam-2803	334	6	m	m	NOUN
ejpam-2803	334	7	,	,	PUNCT
ejpam-2803	334	8	every	every	DET
ejpam-2803	334	9	irreducible	irreducible	ADJ
ejpam-2803	334	10	closed	closed	ADJ
ejpam-2803	334	11	subset	subset	NOUN
ejpam-2803	334	12	of	of	ADP
ejpam-2803	334	13	specsr(m	specsr(m	NOUN
ejpam-2803	334	14	)	)	PUNCT
ejpam-2803	334	15	has	have	VERB
ejpam-2803	334	16	a	a	DET
ejpam-2803	334	17	generic	generic	ADJ
ejpam-2803	334	18	point	point	NOUN
ejpam-2803	334	19	.	.	PUNCT
ejpam-2803	335	1	in	in	ADP
ejpam-2803	335	2	particular	particular	ADJ
ejpam-2803	335	3	,	,	PUNCT
ejpam-2803	335	4	this	this	PRON
ejpam-2803	335	5	is	be	AUX
ejpam-2803	335	6	true	true	ADJ
ejpam-2803	335	7	when	when	SCONJ
ejpam-2803	335	8	m	m	PROPN
ejpam-2803	335	9	is	be	AUX
ejpam-2803	335	10	a	a	DET
ejpam-2803	335	11	cotop	cotop	NOUN
ejpam-2803	335	12	module	module	NOUN
ejpam-2803	335	13	.	.	PUNCT
ejpam-2803	336	1	proof	proof	NOUN
ejpam-2803	336	2	.	.	PUNCT
ejpam-2803	337	1	let	let	VERB
ejpam-2803	337	2	y	y	PRON
ejpam-2803	337	3	be	be	AUX
ejpam-2803	337	4	an	an	DET
ejpam-2803	337	5	irreducible	irreducible	ADJ
ejpam-2803	337	6	closed	closed	ADJ
ejpam-2803	337	7	subset	subset	NOUN
ejpam-2803	337	8	of	of	ADP
ejpam-2803	337	9	specs(m	specs(m	PROPN
ejpam-2803	337	10	)	)	PUNCT
ejpam-2803	337	11	and	and	CCONJ
ejpam-2803	337	12	∑	∑	ADP
ejpam-2803	337	13	s∈y	s∈y	NOUN
ejpam-2803	337	14	s	s	X
ejpam-2803	337	15	=	=	NOUN
ejpam-2803	337	16	s1	s1	PROPN
ejpam-2803	337	17	.	.	PUNCT
ejpam-2803	338	1	then	then	ADV
ejpam-2803	338	2	by	by	ADP
ejpam-2803	338	3	[	[	X
ejpam-2803	338	4	8	8	NUM
ejpam-2803	338	5	,	,	PUNCT
ejpam-2803	338	6	theorem	theorem	VERB
ejpam-2803	338	7	3.5	3.5	NUM
ejpam-2803	338	8	(	(	PUNCT
ejpam-2803	338	9	a	a	NOUN
ejpam-2803	338	10	)	)	PUNCT
ejpam-2803	338	11	]	]	PUNCT
ejpam-2803	338	12	,	,	PUNCT
ejpam-2803	338	13	s1	s1	PROPN
ejpam-2803	338	14	is	be	AUX
ejpam-2803	338	15	a	a	DET
ejpam-2803	338	16	second	second	ADJ
ejpam-2803	338	17	submodule	submodule	NOUN
ejpam-2803	338	18	of	of	ADP
ejpam-2803	338	19	m	m	PROPN
ejpam-2803	338	20	.	.	PUNCT
ejpam-2803	339	1	we	we	PRON
ejpam-2803	339	2	claim	claim	VERB
ejpam-2803	339	3	that	that	SCONJ
ejpam-2803	339	4	y	y	PROPN
ejpam-2803	339	5	=	=	PUNCT
ejpam-2803	339	6	v	v	NUM
ejpam-2803	339	7	s∗(s1	s∗(s1	NOUN
ejpam-2803	339	8	)	)	PUNCT
ejpam-2803	339	9	.	.	PUNCT
ejpam-2803	340	1	by	by	ADP
ejpam-2803	340	2	lemma	lemma	PROPN
ejpam-2803	340	3	3.1	3.1	NUM
ejpam-2803	340	4	,	,	PUNCT
ejpam-2803	340	5	it	it	PRON
ejpam-2803	340	6	is	be	AUX
ejpam-2803	340	7	enough	enough	ADJ
ejpam-2803	340	8	to	to	PART
ejpam-2803	340	9	show	show	VERB
ejpam-2803	340	10	that	that	SCONJ
ejpam-2803	340	11	⋃	⋃	PUNCT
ejpam-2803	340	12	s∈y	s∈y	ADJ
ejpam-2803	340	13	v	v	NOUN
ejpam-2803	340	14	s∗(s	s∗(s	NOUN
ejpam-2803	340	15	)	)	PUNCT
ejpam-2803	340	16	=	=	NOUN
ejpam-2803	340	17	v	v	ADP
ejpam-2803	340	18	s∗(s1	s∗(s1	NOUN
ejpam-2803	340	19	)	)	PUNCT
ejpam-2803	340	20	.	.	PUNCT
ejpam-2803	341	1	clearly	clearly	ADV
ejpam-2803	341	2	,	,	PUNCT
ejpam-2803	341	3	⋃	⋃	NOUN
ejpam-2803	341	4	s∈y	s∈y	VERB
ejpam-2803	341	5	v	v	ADP
ejpam-2803	341	6	s∗(s	s∗(s	NOUN
ejpam-2803	341	7	)	)	PUNCT
ejpam-2803	341	8	⊆	⊆	NUM
ejpam-2803	341	9	v	v	ADP
ejpam-2803	341	10	s∗(s1	s∗(s1	NOUN
ejpam-2803	341	11	)	)	PUNCT
ejpam-2803	341	12	.	.	PUNCT
ejpam-2803	342	1	to	to	PART
ejpam-2803	342	2	see	see	VERB
ejpam-2803	342	3	the	the	DET
ejpam-2803	342	4	reverse	reverse	ADJ
ejpam-2803	342	5	inclusion	inclusion	NOUN
ejpam-2803	342	6	,	,	PUNCT
ejpam-2803	342	7	let	let	VERB
ejpam-2803	342	8	f	f	PRON
ejpam-2803	342	9	be	be	AUX
ejpam-2803	342	10	a	a	DET
ejpam-2803	342	11	closed	closed	ADJ
ejpam-2803	342	12	subset	subset	NOUN
ejpam-2803	342	13	of	of	ADP
ejpam-2803	342	14	specs(m	specs(m	NOUN
ejpam-2803	342	15	)	)	PUNCT
ejpam-2803	342	16	containing⋃	containing⋃	PROPN
ejpam-2803	342	17	s∈y	s∈y	VERB
ejpam-2803	342	18	v	v	NOUN
ejpam-2803	342	19	s∗(s	s∗(s	PROPN
ejpam-2803	342	20	)	)	PUNCT
ejpam-2803	342	21	.	.	PUNCT
ejpam-2803	343	1	since	since	SCONJ
ejpam-2803	343	2	f	f	PROPN
ejpam-2803	343	3	is	be	AUX
ejpam-2803	343	4	closed	close	VERB
ejpam-2803	343	5	,	,	PUNCT
ejpam-2803	343	6	f	f	PROPN
ejpam-2803	343	7	=	=	SYM
ejpam-2803	343	8	⋂	⋂	PROPN
ejpam-2803	343	9	i∈λ	i∈λ	NOUN
ejpam-2803	343	10	⋃ni	⋃ni	PUNCT
ejpam-2803	343	11	j=1	j=1	PROPN
ejpam-2803	343	12	v	v	ADP
ejpam-2803	343	13	s∗(ni	s∗(ni	PROPN
ejpam-2803	343	14	,	,	PUNCT
ejpam-2803	343	15	j	j	NOUN
ejpam-2803	343	16	)	)	PUNCT
ejpam-2803	343	17	for	for	ADP
ejpam-2803	343	18	some	some	DET
ejpam-2803	343	19	submodules	submodule	NOUN
ejpam-2803	343	20	ni	ni	PROPN
ejpam-2803	343	21	,	,	PUNCT
ejpam-2803	343	22	j	j	PROPN
ejpam-2803	343	23	of	of	ADP
ejpam-2803	343	24	m	m	PROPN
ejpam-2803	343	25	.	.	PUNCT
ejpam-2803	344	1	without	without	ADP
ejpam-2803	344	2	loss	loss	NOUN
ejpam-2803	344	3	of	of	ADP
ejpam-2803	344	4	generality	generality	NOUN
ejpam-2803	344	5	,	,	PUNCT
ejpam-2803	344	6	we	we	PRON
ejpam-2803	344	7	can	can	AUX
ejpam-2803	344	8	assume	assume	VERB
ejpam-2803	344	9	f	f	X
ejpam-2803	344	10	=	=	SYM
ejpam-2803	344	11	v	v	PROPN
ejpam-2803	344	12	s∗(n1)∪	s∗(n1)∪	NOUN
ejpam-2803	344	13	v	v	ADP
ejpam-2803	344	14	s∗(n2	s∗(n2	PROPN
ejpam-2803	344	15	)	)	PUNCT
ejpam-2803	344	16	,	,	PUNCT
ejpam-2803	344	17	where	where	SCONJ
ejpam-2803	344	18	n1	n1	PROPN
ejpam-2803	344	19	and	and	CCONJ
ejpam-2803	344	20	n2	n2	NOUN
ejpam-2803	344	21	are	be	AUX
ejpam-2803	344	22	submodules	submodule	NOUN
ejpam-2803	344	23	of	of	ADP
ejpam-2803	344	24	m	m	PROPN
ejpam-2803	344	25	.	.	PUNCT
ejpam-2803	345	1	now	now	ADV
ejpam-2803	345	2	by	by	ADP
ejpam-2803	345	3	lemma	lemma	PROPN
ejpam-2803	345	4	3.1	3.1	NUM
ejpam-2803	345	5	,	,	PUNCT
ejpam-2803	345	6	we	we	PRON
ejpam-2803	345	7	have	have	AUX
ejpam-2803	345	8	⋃	⋃	ADV
ejpam-2803	345	9	s∈y	s∈y	ADJ
ejpam-2803	345	10	v	v	NOUN
ejpam-2803	345	11	s∗(s	s∗(s	NOUN
ejpam-2803	345	12	)	)	PUNCT
ejpam-2803	346	1	=	=	SYM
ejpam-2803	346	2	y	y	PROPN
ejpam-2803	346	3	is	be	AUX
ejpam-2803	346	4	irreducible	irreducible	ADJ
ejpam-2803	346	5	.	.	PUNCT
ejpam-2803	347	1	since⋃	since⋃	PROPN
ejpam-2803	347	2	s∈y	s∈y	PROPN
ejpam-2803	347	3	v	v	PROPN
ejpam-2803	347	4	s∗(s	s∗(s	PROPN
ejpam-2803	347	5	)	)	PUNCT
ejpam-2803	347	6	⊆	⊆	NUM
ejpam-2803	347	7	v	v	ADP
ejpam-2803	347	8	s∗(n1	s∗(n1	NOUN
ejpam-2803	347	9	)	)	PUNCT
ejpam-2803	347	10	∪	∪	ADP
ejpam-2803	347	11	v	v	ADP
ejpam-2803	347	12	s∗(n2	s∗(n2	PROPN
ejpam-2803	347	13	)	)	PUNCT
ejpam-2803	347	14	,	,	PUNCT
ejpam-2803	347	15	we	we	PRON
ejpam-2803	347	16	have	have	AUX
ejpam-2803	347	17	⋃	⋃	ADV
ejpam-2803	347	18	s∈y	s∈y	ADJ
ejpam-2803	347	19	v	v	ADP
ejpam-2803	347	20	s∗(s	s∗(s	NOUN
ejpam-2803	347	21	)	)	PUNCT
ejpam-2803	347	22	⊆	⊆	NUM
ejpam-2803	347	23	v	v	ADP
ejpam-2803	347	24	s∗(n1	s∗(n1	NOUN
ejpam-2803	347	25	)	)	PUNCT
ejpam-2803	347	26	or	or	CCONJ
ejpam-2803	348	1	⋃	⋃	ADP
ejpam-2803	348	2	s∈y	s∈y	ADJ
ejpam-2803	348	3	v	v	ADP
ejpam-2803	348	4	s∗(s	s∗(s	NOUN
ejpam-2803	348	5	)	)	PUNCT
ejpam-2803	348	6	⊆	⊆	PROPN
ejpam-2803	348	7	v	v	ADP
ejpam-2803	348	8	s∗(n2	s∗(n2	PROPN
ejpam-2803	348	9	)	)	PUNCT
ejpam-2803	348	10	.	.	PUNCT
ejpam-2803	349	1	it	it	PRON
ejpam-2803	349	2	follows	follow	VERB
ejpam-2803	349	3	that	that	SCONJ
ejpam-2803	349	4	s1	s1	NOUN
ejpam-2803	349	5	=	=	PUNCT
ejpam-2803	349	6	∑	∑	PUNCT
ejpam-2803	349	7	s∈y	s∈y	PROPN
ejpam-2803	349	8	s	s	PROPN
ejpam-2803	349	9	⊆	⊆	NUM
ejpam-2803	349	10	n1	n1	NOUN
ejpam-2803	349	11	or	or	CCONJ
ejpam-2803	349	12	s1	s1	NOUN
ejpam-2803	349	13	=	=	PUNCT
ejpam-2803	349	14	∑	∑	PUNCT
ejpam-2803	349	15	s∈y	s∈y	PROPN
ejpam-2803	349	16	s	s	PROPN
ejpam-2803	349	17	⊆	⊆	NUM
ejpam-2803	349	18	n2	n2	NOUN
ejpam-2803	349	19	.	.	PUNCT
ejpam-2803	350	1	thus	thus	ADV
ejpam-2803	350	2	v	v	ADP
ejpam-2803	350	3	s∗(s1	s∗(s1	NOUN
ejpam-2803	350	4	)	)	PUNCT
ejpam-2803	350	5	⊆	⊆	NUM
ejpam-2803	350	6	n1	n1	NOUN
ejpam-2803	350	7	or	or	CCONJ
ejpam-2803	350	8	v	v	ADP
ejpam-2803	350	9	s∗(s1	s∗(s1	NOUN
ejpam-2803	350	10	)	)	PUNCT
ejpam-2803	350	11	⊆	⊆	NUM
ejpam-2803	350	12	n2	n2	NOUN
ejpam-2803	350	13	.	.	PUNCT
ejpam-2803	351	1	therefore	therefore	ADV
ejpam-2803	351	2	v	v	X
ejpam-2803	351	3	s∗(s1	s∗(s1	NOUN
ejpam-2803	351	4	)	)	PUNCT
ejpam-2803	351	5	⊆	⊆	NUM
ejpam-2803	351	6	f	f	NOUN
ejpam-2803	351	7	.	.	PUNCT
ejpam-2803	352	1	this	this	PRON
ejpam-2803	352	2	in	in	ADP
ejpam-2803	352	3	turn	turn	NOUN
ejpam-2803	352	4	implies	imply	VERB
ejpam-2803	352	5	that	that	SCONJ
ejpam-2803	352	6	v	v	ADP
ejpam-2803	352	7	s∗(s1	s∗(s1	NOUN
ejpam-2803	352	8	)	)	PUNCT
ejpam-2803	352	9	=	=	NOUN
ejpam-2803	352	10	⋃	⋃	NOUN
ejpam-2803	352	11	s∈y	s∈y	ADJ
ejpam-2803	352	12	v	v	NOUN
ejpam-2803	352	13	s∗(s	s∗(s	NOUN
ejpam-2803	352	14	)	)	PUNCT
ejpam-2803	352	15	.	.	PUNCT
ejpam-2803	353	1	therefore	therefore	ADV
ejpam-2803	353	2	y	y	PROPN
ejpam-2803	353	3	=	=	PUNCT
ejpam-2803	353	4	v	v	NUM
ejpam-2803	353	5	s∗(s1	s∗(s1	PROPN
ejpam-2803	353	6	)	)	PUNCT
ejpam-2803	353	7	=	=	SYM
ejpam-2803	354	1	⋃	⋃	NOUN
ejpam-2803	354	2	s∈y	s∈y	VERB
ejpam-2803	354	3	v	v	ADP
ejpam-2803	354	4	s∗(s	s∗(s	NOUN
ejpam-2803	354	5	)	)	PUNCT
ejpam-2803	354	6	.	.	PUNCT
ejpam-2803	355	1	corollary	corollary	ADJ
ejpam-2803	355	2	3.3	3.3	NUM
ejpam-2803	355	3	.	.	PUNCT
ejpam-2803	356	1	let	let	VERB
ejpam-2803	356	2	m	m	PRON
ejpam-2803	356	3	be	be	AUX
ejpam-2803	356	4	an	an	DET
ejpam-2803	356	5	r	r	NOUN
ejpam-2803	356	6	-	-	PUNCT
ejpam-2803	356	7	module	module	NOUN
ejpam-2803	356	8	and	and	CCONJ
ejpam-2803	356	9	suppose	suppose	VERB
ejpam-2803	356	10	specs(m	specs(m	PROPN
ejpam-2803	356	11	)	)	PUNCT
ejpam-2803	356	12	is	be	AUX
ejpam-2803	356	13	a	a	DET
ejpam-2803	356	14	noetherian	noetherian	ADJ
ejpam-2803	356	15	space	space	NOUN
ejpam-2803	356	16	.	.	PUNCT
ejpam-2803	357	1	then	then	ADV
ejpam-2803	357	2	specs(m	specs(m	PROPN
ejpam-2803	357	3	)	)	PUNCT
ejpam-2803	357	4	is	be	AUX
ejpam-2803	357	5	a	a	DET
ejpam-2803	357	6	spectral	spectral	ADJ
ejpam-2803	357	7	space	space	NOUN
ejpam-2803	357	8	.	.	PUNCT
ejpam-2803	358	1	in	in	ADP
ejpam-2803	358	2	particular	particular	ADJ
ejpam-2803	358	3	,	,	PUNCT
ejpam-2803	358	4	this	this	PRON
ejpam-2803	358	5	is	be	AUX
ejpam-2803	358	6	true	true	ADJ
ejpam-2803	358	7	when	when	SCONJ
ejpam-2803	358	8	m	m	PROPN
ejpam-2803	358	9	is	be	AUX
ejpam-2803	358	10	a	a	DET
ejpam-2803	358	11	cotop	cotop	NOUN
ejpam-2803	358	12	module	module	NOUN
ejpam-2803	358	13	.	.	PUNCT
ejpam-2803	359	1	proof	proof	NOUN
ejpam-2803	359	2	.	.	PUNCT
ejpam-2803	360	1	since	since	SCONJ
ejpam-2803	360	2	specs(m	specs(m	PROPN
ejpam-2803	360	3	)	)	PUNCT
ejpam-2803	360	4	is	be	AUX
ejpam-2803	360	5	noetherian	noetherian	ADJ
ejpam-2803	360	6	,	,	PUNCT
ejpam-2803	360	7	it	it	PRON
ejpam-2803	360	8	is	be	AUX
ejpam-2803	360	9	quasi	quasi	ADJ
ejpam-2803	360	10	-	-	ADJ
ejpam-2803	360	11	compact	compact	ADJ
ejpam-2803	360	12	and	and	CCONJ
ejpam-2803	360	13	the	the	DET
ejpam-2803	360	14	quasi	quasi	ADJ
ejpam-2803	360	15	-	-	ADJ
ejpam-2803	360	16	compact	compact	ADJ
ejpam-2803	360	17	open	open	ADJ
ejpam-2803	360	18	subsets	subset	NOUN
ejpam-2803	360	19	of	of	ADP
ejpam-2803	360	20	specs(m	specs(m	NOUN
ejpam-2803	360	21	)	)	PUNCT
ejpam-2803	360	22	are	be	AUX
ejpam-2803	360	23	closed	close	VERB
ejpam-2803	360	24	under	under	ADP
ejpam-2803	360	25	finite	finite	ADJ
ejpam-2803	360	26	intersection	intersection	NOUN
ejpam-2803	360	27	and	and	CCONJ
ejpam-2803	360	28	form	form	VERB
ejpam-2803	360	29	an	an	DET
ejpam-2803	360	30	open	open	ADJ
ejpam-2803	360	31	base	base	NOUN
ejpam-2803	360	32	by	by	ADP
ejpam-2803	360	33	[	[	X
ejpam-2803	360	34	10	10	NUM
ejpam-2803	360	35	,	,	PUNCT
ejpam-2803	360	36	p.	p.	NOUN
ejpam-2803	360	37	79	79	NUM
ejpam-2803	360	38	,	,	PUNCT
ejpam-2803	360	39	exer	exer	NOUN
ejpam-2803	360	40	.	.	PUNCT
ejpam-2803	361	1	6	6	NUM
ejpam-2803	361	2	]	]	PUNCT
ejpam-2803	361	3	.	.	PUNCT
ejpam-2803	362	1	also	also	ADV
ejpam-2803	362	2	specs(m	specs(m	PROPN
ejpam-2803	362	3	)	)	PUNCT
ejpam-2803	362	4	is	be	AUX
ejpam-2803	362	5	a	a	DET
ejpam-2803	362	6	t0	t0	NOUN
ejpam-2803	362	7	space	space	NOUN
ejpam-2803	362	8	by	by	ADP
ejpam-2803	362	9	[	[	X
ejpam-2803	362	10	8	8	NUM
ejpam-2803	362	11	,	,	PUNCT
ejpam-2803	362	12	lemma	lemma	PROPN
ejpam-2803	362	13	3.7	3.7	NUM
ejpam-2803	362	14	(	(	PUNCT
ejpam-2803	362	15	a	a	NOUN
ejpam-2803	362	16	)	)	PUNCT
ejpam-2803	362	17	]	]	PUNCT
ejpam-2803	362	18	.	.	PUNCT
ejpam-2803	363	1	now	now	ADV
ejpam-2803	363	2	the	the	DET
ejpam-2803	363	3	result	result	NOUN
ejpam-2803	363	4	follows	follow	VERB
ejpam-2803	363	5	from	from	ADP
ejpam-2803	363	6	theorem	theorem	ADJ
ejpam-2803	363	7	3.2	3.2	NUM
ejpam-2803	363	8	and	and	CCONJ
ejpam-2803	363	9	hochster	hochster	NOUN
ejpam-2803	363	10	’s	’s	PART
ejpam-2803	363	11	characterizations	characterization	NOUN
ejpam-2803	363	12	.	.	PUNCT
ejpam-2803	364	1	let	let	VERB
ejpam-2803	364	2	n	n	PRON
ejpam-2803	364	3	be	be	AUX
ejpam-2803	364	4	a	a	DET
ejpam-2803	364	5	non	non	ADJ
ejpam-2803	364	6	-	-	ADJ
ejpam-2803	364	7	zero	zero	NUM
ejpam-2803	364	8	submodule	submodule	NOUN
ejpam-2803	364	9	of	of	ADP
ejpam-2803	364	10	an	an	DET
ejpam-2803	364	11	r	r	NOUN
ejpam-2803	364	12	-	-	PUNCT
ejpam-2803	364	13	module	module	NOUN
ejpam-2803	364	14	m	m	NOUN
ejpam-2803	364	15	and	and	CCONJ
ejpam-2803	364	16	let	let	VERB
ejpam-2803	364	17	s	s	PRON
ejpam-2803	364	18	be	be	AUX
ejpam-2803	364	19	a	a	DET
ejpam-2803	364	20	second	second	ADJ
ejpam-2803	364	21	submodule	submodule	NOUN
ejpam-2803	364	22	of	of	ADP
ejpam-2803	364	23	m	m	PRON
ejpam-2803	364	24	such	such	ADJ
ejpam-2803	364	25	that	that	PRON
ejpam-2803	364	26	s	s	VERB
ejpam-2803	364	27	≤	≤	NOUN
ejpam-2803	364	28	n	n	NOUN
ejpam-2803	364	29	.	.	PUNCT
ejpam-2803	365	1	s	s	PART
ejpam-2803	365	2	is	be	AUX
ejpam-2803	365	3	said	say	VERB
ejpam-2803	365	4	to	to	PART
ejpam-2803	365	5	be	be	AUX
ejpam-2803	365	6	a	a	DET
ejpam-2803	365	7	maximal	maximal	ADJ
ejpam-2803	365	8	second	second	ADJ
ejpam-2803	365	9	submodule	submodule	NOUN
ejpam-2803	365	10	of	of	ADP
ejpam-2803	365	11	n	n	PRON
ejpam-2803	365	12	if	if	SCONJ
ejpam-2803	365	13	there	there	PRON
ejpam-2803	365	14	does	do	AUX
ejpam-2803	365	15	n’t	not	PART
ejpam-2803	365	16	exist	exist	VERB
ejpam-2803	365	17	s′	s′	ADJ
ejpam-2803	365	18	∈	∈	PROPN
ejpam-2803	365	19	specs(m	specs(m	NOUN
ejpam-2803	365	20	)	)	PUNCT
ejpam-2803	365	21	with	with	ADP
ejpam-2803	365	22	s	s	PROPN
ejpam-2803	365	23	�	�	PROPN
ejpam-2803	365	24	s′	s′	ADJ
ejpam-2803	365	25	�	�	PROPN
ejpam-2803	365	26	n	n	CCONJ
ejpam-2803	365	27	(	(	PUNCT
ejpam-2803	365	28	see	see	VERB
ejpam-2803	365	29	[	[	X
ejpam-2803	365	30	4	4	NUM
ejpam-2803	365	31	]	]	NUM
ejpam-2803	365	32	)	)	PUNCT
ejpam-2803	365	33	.	.	PUNCT
ejpam-2803	366	1	definition	definition	NOUN
ejpam-2803	366	2	3.4	3.4	NUM
ejpam-2803	366	3	.	.	PUNCT
ejpam-2803	367	1	let	let	VERB
ejpam-2803	367	2	n	n	PRON
ejpam-2803	367	3	be	be	AUX
ejpam-2803	367	4	a	a	DET
ejpam-2803	367	5	non	non	ADJ
ejpam-2803	367	6	-	-	ADJ
ejpam-2803	367	7	zero	zero	NUM
ejpam-2803	367	8	submodule	submodule	NOUN
ejpam-2803	367	9	of	of	ADP
ejpam-2803	367	10	an	an	DET
ejpam-2803	367	11	r	r	NOUN
ejpam-2803	367	12	-	-	PUNCT
ejpam-2803	367	13	module	module	NOUN
ejpam-2803	367	14	m	m	NOUN
ejpam-2803	367	15	and	and	CCONJ
ejpam-2803	367	16	let	let	VERB
ejpam-2803	367	17	s	s	PRON
ejpam-2803	367	18	be	be	AUX
ejpam-2803	367	19	a	a	DET
ejpam-2803	367	20	second	second	ADJ
ejpam-2803	367	21	submodule	submodule	NOUN
ejpam-2803	367	22	of	of	ADP
ejpam-2803	367	23	m	m	PROPN
ejpam-2803	367	24	.	.	PUNCT
ejpam-2803	368	1	we	we	PRON
ejpam-2803	368	2	say	say	VERB
ejpam-2803	368	3	that	that	PRON
ejpam-2803	368	4	s	s	VERB
ejpam-2803	368	5	is	be	AUX
ejpam-2803	368	6	a	a	DET
ejpam-2803	368	7	second	second	ADJ
ejpam-2803	368	8	summand	summand	NOUN
ejpam-2803	368	9	(	(	PUNCT
ejpam-2803	368	10	resp	resp	NOUN
ejpam-2803	368	11	.	.	PUNCT
ejpam-2803	369	1	maximal	maximal	ADJ
ejpam-2803	369	2	second	second	ADJ
ejpam-2803	369	3	summand	summand	NOUN
ejpam-2803	369	4	)	)	PUNCT
ejpam-2803	369	5	of	of	ADP
ejpam-2803	369	6	n	n	PRON
ejpam-2803	369	7	if	if	SCONJ
ejpam-2803	369	8	s	s	X
ejpam-2803	369	9	∈	∈	PROPN
ejpam-2803	369	10	v	v	ADP
ejpam-2803	369	11	s∗(n	s∗(n	NOUN
ejpam-2803	369	12	)	)	PUNCT
ejpam-2803	369	13	(	(	PUNCT
ejpam-2803	369	14	resp	resp	NOUN
ejpam-2803	369	15	.	.	PUNCT
ejpam-2803	370	1	s	s	PART
ejpam-2803	370	2	∈	∈	PROPN
ejpam-2803	370	3	max(v	max(v	NOUN
ejpam-2803	370	4	s∗(n	s∗(n	NOUN
ejpam-2803	370	5	)	)	PUNCT
ejpam-2803	370	6	)	)	PUNCT
ejpam-2803	370	7	)	)	PUNCT
ejpam-2803	370	8	.	.	PUNCT
ejpam-2803	371	1	if	if	SCONJ
ejpam-2803	371	2	v	v	ADP
ejpam-2803	371	3	s∗(n	s∗(n	PROPN
ejpam-2803	371	4	)	)	PUNCT
ejpam-2803	371	5	6=	6=	ADP
ejpam-2803	371	6	∅	∅	NOUN
ejpam-2803	371	7	,	,	PUNCT
ejpam-2803	371	8	then	then	ADV
ejpam-2803	371	9	by	by	ADP
ejpam-2803	371	10	using	use	VERB
ejpam-2803	371	11	zorn	zorn	PROPN
ejpam-2803	371	12	’s	’s	PART
ejpam-2803	371	13	lemma	lemma	PROPN
ejpam-2803	371	14	,	,	PUNCT
ejpam-2803	371	15	one	one	PRON
ejpam-2803	371	16	can	can	AUX
ejpam-2803	371	17	see	see	VERB
ejpam-2803	371	18	that	that	SCONJ
ejpam-2803	371	19	n	n	PROPN
ejpam-2803	371	20	contains	contain	VERB
ejpam-2803	371	21	a	a	DET
ejpam-2803	371	22	maximal	maximal	ADJ
ejpam-2803	371	23	second	second	ADJ
ejpam-2803	371	24	summand	summand	NOUN
ejpam-2803	371	25	.	.	PUNCT
ejpam-2803	372	1	h.	h.	PROPN
ejpam-2803	372	2	ansari	ansari	PROPN
ejpam-2803	372	3	-	-	PUNCT
ejpam-2803	372	4	toroghy	toroghy	NOUN
ejpam-2803	372	5	,	,	PUNCT
ejpam-2803	372	6	s.	s.	PROPN
ejpam-2803	372	7	s.	s.	PROPN
ejpam-2803	372	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	372	9	/	/	SYM
ejpam-2803	372	10	eur	eur	PROPN
ejpam-2803	372	11	.	.	PUNCT
ejpam-2803	373	1	j.	j.	PROPN
ejpam-2803	373	2	pure	pure	PROPN
ejpam-2803	373	3	appl	appl	PROPN
ejpam-2803	373	4	.	.	PROPN
ejpam-2803	373	5	math	math	PROPN
ejpam-2803	373	6	,	,	PUNCT
ejpam-2803	373	7	10	10	NUM
ejpam-2803	373	8	(	(	PUNCT
ejpam-2803	373	9	2	2	NUM
ejpam-2803	373	10	)	)	PUNCT
ejpam-2803	373	11	(	(	PUNCT
ejpam-2803	373	12	2017	2017	NUM
ejpam-2803	373	13	)	)	PUNCT
ejpam-2803	373	14	,	,	PUNCT
ejpam-2803	373	15	211	211	NUM
ejpam-2803	373	16	-	-	SYM
ejpam-2803	373	17	230	230	NUM
ejpam-2803	373	18	220	220	NUM
ejpam-2803	373	19	the	the	DET
ejpam-2803	373	20	second	second	ADJ
ejpam-2803	373	21	submodule	submodule	NOUN
ejpam-2803	373	22	dimension	dimension	NOUN
ejpam-2803	373	23	of	of	ADP
ejpam-2803	373	24	an	an	DET
ejpam-2803	373	25	r	r	NOUN
ejpam-2803	373	26	-	-	PUNCT
ejpam-2803	373	27	module	module	NOUN
ejpam-2803	373	28	m	m	NOUN
ejpam-2803	373	29	,	,	PUNCT
ejpam-2803	373	30	denoted	denote	VERB
ejpam-2803	373	31	by	by	ADP
ejpam-2803	373	32	s.dimm	s.dimm	PRON
ejpam-2803	373	33	,	,	PUNCT
ejpam-2803	373	34	is	be	AUX
ejpam-2803	373	35	defined	define	VERB
ejpam-2803	373	36	to	to	PART
ejpam-2803	373	37	be	be	AUX
ejpam-2803	373	38	the	the	DET
ejpam-2803	373	39	supremum	supremum	NOUN
ejpam-2803	373	40	of	of	ADP
ejpam-2803	373	41	the	the	DET
ejpam-2803	373	42	length	length	NOUN
ejpam-2803	373	43	of	of	ADP
ejpam-2803	373	44	chains	chain	NOUN
ejpam-2803	373	45	of	of	ADP
ejpam-2803	373	46	second	second	ADJ
ejpam-2803	373	47	submodules	submodule	NOUN
ejpam-2803	373	48	of	of	ADP
ejpam-2803	373	49	m	m	PROPN
ejpam-2803	373	50	if	if	SCONJ
ejpam-2803	373	51	specs(m	specs(m	PROPN
ejpam-2803	373	52	)	)	PUNCT
ejpam-2803	373	53	6=	6=	ADP
ejpam-2803	373	54	∅	∅	NOUN
ejpam-2803	373	55	and	and	CCONJ
ejpam-2803	373	56	−1	−1	NOUN
ejpam-2803	373	57	otherwise	otherwise	ADV
ejpam-2803	373	58	(	(	PUNCT
ejpam-2803	373	59	see	see	VERB
ejpam-2803	373	60	[	[	X
ejpam-2803	373	61	6	6	NUM
ejpam-2803	373	62	]	]	NUM
ejpam-2803	373	63	)	)	PUNCT
ejpam-2803	373	64	.	.	PUNCT
ejpam-2803	374	1	theorem	theorem	VERB
ejpam-2803	374	2	3.5	3.5	NUM
ejpam-2803	374	3	.	.	PUNCT
ejpam-2803	375	1	let	let	VERB
ejpam-2803	375	2	m	m	PRON
ejpam-2803	375	3	be	be	AUX
ejpam-2803	375	4	an	an	DET
ejpam-2803	375	5	r	r	NOUN
ejpam-2803	375	6	-	-	PUNCT
ejpam-2803	375	7	module	module	NOUN
ejpam-2803	375	8	.	.	PUNCT
ejpam-2803	376	1	(	(	PUNCT
ejpam-2803	376	2	a	a	X
ejpam-2803	376	3	)	)	PUNCT
ejpam-2803	376	4	let	let	VERB
ejpam-2803	376	5	y	y	PRON
ejpam-2803	376	6	be	be	AUX
ejpam-2803	376	7	a	a	DET
ejpam-2803	376	8	closed	closed	ADJ
ejpam-2803	376	9	subset	subset	NOUN
ejpam-2803	376	10	of	of	ADP
ejpam-2803	376	11	specs(m	specs(m	PROPN
ejpam-2803	376	12	)	)	PUNCT
ejpam-2803	376	13	.	.	PUNCT
ejpam-2803	377	1	then	then	ADV
ejpam-2803	377	2	z	z	PROPN
ejpam-2803	377	3	is	be	AUX
ejpam-2803	377	4	an	an	DET
ejpam-2803	377	5	irreducible	irreducible	ADJ
ejpam-2803	377	6	component	component	NOUN
ejpam-2803	377	7	of	of	ADP
ejpam-2803	377	8	y	y	PRON
ejpam-2803	377	9	if	if	SCONJ
ejpam-2803	378	1	and	and	CCONJ
ejpam-2803	378	2	only	only	ADV
ejpam-2803	378	3	if	if	SCONJ
ejpam-2803	378	4	z	z	NOUN
ejpam-2803	378	5	=	=	SYM
ejpam-2803	378	6	v	v	NOUN
ejpam-2803	378	7	s∗(s	s∗(s	PROPN
ejpam-2803	378	8	)	)	PUNCT
ejpam-2803	378	9	for	for	ADP
ejpam-2803	378	10	some	some	DET
ejpam-2803	378	11	maximal	maximal	ADJ
ejpam-2803	378	12	element	element	NOUN
ejpam-2803	378	13	s	s	PROPN
ejpam-2803	378	14	of	of	ADP
ejpam-2803	378	15	y	y	PROPN
ejpam-2803	378	16	.	.	PUNCT
ejpam-2803	379	1	(	(	PUNCT
ejpam-2803	379	2	b	b	X
ejpam-2803	379	3	)	)	PUNCT
ejpam-2803	379	4	if	if	SCONJ
ejpam-2803	379	5	every	every	DET
ejpam-2803	379	6	closed	closed	ADJ
ejpam-2803	379	7	subset	subset	NOUN
ejpam-2803	379	8	of	of	ADP
ejpam-2803	379	9	specs(m	specs(m	PROPN
ejpam-2803	379	10	)	)	PUNCT
ejpam-2803	379	11	has	have	VERB
ejpam-2803	379	12	a	a	DET
ejpam-2803	379	13	finite	finite	ADJ
ejpam-2803	379	14	number	number	NOUN
ejpam-2803	379	15	of	of	ADP
ejpam-2803	379	16	irreducible	irreducible	ADJ
ejpam-2803	379	17	components	component	NOUN
ejpam-2803	379	18	,	,	PUNCT
ejpam-2803	379	19	then	then	ADV
ejpam-2803	379	20	every	every	DET
ejpam-2803	379	21	submodule	submodule	NOUN
ejpam-2803	379	22	of	of	ADP
ejpam-2803	379	23	m	m	PROPN
ejpam-2803	379	24	has	have	VERB
ejpam-2803	379	25	a	a	DET
ejpam-2803	379	26	finite	finite	ADJ
ejpam-2803	379	27	number	number	NOUN
ejpam-2803	379	28	of	of	ADP
ejpam-2803	379	29	maximal	maximal	ADJ
ejpam-2803	379	30	second	second	ADJ
ejpam-2803	379	31	summands	summand	NOUN
ejpam-2803	379	32	.	.	PUNCT
ejpam-2803	380	1	however	however	ADV
ejpam-2803	380	2	,	,	PUNCT
ejpam-2803	380	3	the	the	DET
ejpam-2803	380	4	converse	converse	NOUN
ejpam-2803	380	5	is	be	AUX
ejpam-2803	380	6	not	not	PART
ejpam-2803	380	7	true	true	ADJ
ejpam-2803	380	8	in	in	ADP
ejpam-2803	380	9	general	general	ADJ
ejpam-2803	380	10	.	.	PUNCT
ejpam-2803	381	1	(	(	PUNCT
ejpam-2803	381	2	c	c	X
ejpam-2803	381	3	)	)	PUNCT
ejpam-2803	381	4	dim(specs(m	dim(specs(m	NOUN
ejpam-2803	381	5	)	)	PUNCT
ejpam-2803	381	6	)	)	PUNCT
ejpam-2803	382	1	=	=	NOUN
ejpam-2803	382	2	s.dimm	s.dimm	NOUN
ejpam-2803	382	3	.	.	PUNCT
ejpam-2803	383	1	proof	proof	NOUN
ejpam-2803	383	2	.	.	PUNCT
ejpam-2803	384	1	(	(	PUNCT
ejpam-2803	384	2	a	a	X
ejpam-2803	384	3	)	)	PUNCT
ejpam-2803	384	4	let	let	VERB
ejpam-2803	384	5	y	y	PRON
ejpam-2803	384	6	be	be	AUX
ejpam-2803	384	7	a	a	DET
ejpam-2803	384	8	closed	closed	ADJ
ejpam-2803	384	9	subset	subset	NOUN
ejpam-2803	384	10	of	of	ADP
ejpam-2803	384	11	specs(m	specs(m	PROPN
ejpam-2803	384	12	)	)	PUNCT
ejpam-2803	384	13	and	and	CCONJ
ejpam-2803	384	14	let	let	VERB
ejpam-2803	384	15	z	z	PRON
ejpam-2803	384	16	be	be	AUX
ejpam-2803	384	17	an	an	DET
ejpam-2803	384	18	irreducible	irreducible	ADJ
ejpam-2803	384	19	component	component	NOUN
ejpam-2803	384	20	of	of	ADP
ejpam-2803	384	21	y	y	PROPN
ejpam-2803	384	22	.	.	PUNCT
ejpam-2803	385	1	since	since	SCONJ
ejpam-2803	385	2	every	every	DET
ejpam-2803	385	3	irreducible	irreducible	ADJ
ejpam-2803	385	4	component	component	NOUN
ejpam-2803	385	5	is	be	AUX
ejpam-2803	385	6	closed	closed	ADJ
ejpam-2803	385	7	,	,	PUNCT
ejpam-2803	385	8	z	z	NOUN
ejpam-2803	385	9	is	be	AUX
ejpam-2803	385	10	closed	close	VERB
ejpam-2803	385	11	in	in	ADP
ejpam-2803	385	12	y	y	PROPN
ejpam-2803	385	13	.	.	PUNCT
ejpam-2803	386	1	but	but	CCONJ
ejpam-2803	386	2	every	every	DET
ejpam-2803	386	3	irreducible	irreducible	ADJ
ejpam-2803	386	4	closed	closed	ADJ
ejpam-2803	386	5	subset	subset	NOUN
ejpam-2803	386	6	of	of	ADP
ejpam-2803	386	7	y	y	PROPN
ejpam-2803	386	8	is	be	AUX
ejpam-2803	386	9	an	an	DET
ejpam-2803	386	10	irreducible	irreducible	ADJ
ejpam-2803	386	11	closed	closed	ADJ
ejpam-2803	386	12	subset	subset	NOUN
ejpam-2803	386	13	of	of	ADP
ejpam-2803	386	14	specs(m	specs(m	PROPN
ejpam-2803	386	15	)	)	PUNCT
ejpam-2803	386	16	.	.	PUNCT
ejpam-2803	387	1	therefore	therefore	ADV
ejpam-2803	387	2	z	z	X
ejpam-2803	387	3	=	=	SYM
ejpam-2803	387	4	v	v	NUM
ejpam-2803	387	5	s∗(s1	s∗(s1	NOUN
ejpam-2803	387	6	)	)	PUNCT
ejpam-2803	387	7	for	for	ADP
ejpam-2803	387	8	some	some	DET
ejpam-2803	387	9	second	second	ADJ
ejpam-2803	387	10	submodule	submodule	NOUN
ejpam-2803	387	11	s1	s1	NOUN
ejpam-2803	387	12	of	of	ADP
ejpam-2803	387	13	m	m	PRON
ejpam-2803	387	14	by	by	ADP
ejpam-2803	387	15	theorem	theorem	NOUN
ejpam-2803	387	16	3.2	3.2	NUM
ejpam-2803	387	17	.	.	PUNCT
ejpam-2803	388	1	since	since	SCONJ
ejpam-2803	388	2	z	z	NOUN
ejpam-2803	388	3	=	=	SYM
ejpam-2803	388	4	v	v	NUM
ejpam-2803	388	5	s∗(s1	s∗(s1	NOUN
ejpam-2803	388	6	)	)	PUNCT
ejpam-2803	388	7	⊆	⊆	NUM
ejpam-2803	388	8	y	y	PROPN
ejpam-2803	388	9	is	be	AUX
ejpam-2803	388	10	an	an	DET
ejpam-2803	388	11	irreducible	irreducible	ADJ
ejpam-2803	388	12	component	component	NOUN
ejpam-2803	388	13	of	of	ADP
ejpam-2803	388	14	y	y	PROPN
ejpam-2803	388	15	,	,	PUNCT
ejpam-2803	388	16	it	it	PRON
ejpam-2803	388	17	follows	follow	VERB
ejpam-2803	388	18	that	that	SCONJ
ejpam-2803	388	19	s1	s1	NOUN
ejpam-2803	388	20	is	be	AUX
ejpam-2803	388	21	a	a	DET
ejpam-2803	388	22	maximal	maximal	ADJ
ejpam-2803	388	23	element	element	NOUN
ejpam-2803	388	24	in	in	ADP
ejpam-2803	388	25	y	y	PROPN
ejpam-2803	388	26	(	(	PUNCT
ejpam-2803	388	27	note	note	VERB
ejpam-2803	388	28	that	that	SCONJ
ejpam-2803	388	29	if	if	SCONJ
ejpam-2803	388	30	s1	s1	NOUN
ejpam-2803	388	31	,	,	PUNCT
ejpam-2803	388	32	s2	s2	PROPN
ejpam-2803	388	33	∈	∈	PROPN
ejpam-2803	388	34	specs(m	specs(m	PROPN
ejpam-2803	388	35	)	)	PUNCT
ejpam-2803	388	36	,	,	PUNCT
ejpam-2803	388	37	then	then	ADV
ejpam-2803	388	38	s1	s1	PROPN
ejpam-2803	388	39	≤	≤	PROPN
ejpam-2803	388	40	s2	s2	PROPN
ejpam-2803	388	41	⇔	⇔	X
ejpam-2803	388	42	v	v	ADP
ejpam-2803	388	43	s∗(s1	s∗(s1	PROPN
ejpam-2803	388	44	)	)	PUNCT
ejpam-2803	388	45	⊆	⊆	NUM
ejpam-2803	388	46	v	v	NOUN
ejpam-2803	388	47	s∗(s2	s∗(s2	NOUN
ejpam-2803	388	48	)	)	PUNCT
ejpam-2803	388	49	by	by	ADP
ejpam-2803	388	50	[	[	X
ejpam-2803	388	51	8	8	NUM
ejpam-2803	388	52	,	,	PUNCT
ejpam-2803	388	53	corollary	corollary	ADJ
ejpam-2803	388	54	3.2	3.2	NUM
ejpam-2803	388	55	(	(	PUNCT
ejpam-2803	388	56	b	b	NOUN
ejpam-2803	388	57	)	)	PUNCT
ejpam-2803	388	58	]	]	PUNCT
ejpam-2803	388	59	)	)	PUNCT
ejpam-2803	388	60	.	.	PUNCT
ejpam-2803	389	1	conversely	conversely	ADV
ejpam-2803	389	2	,	,	PUNCT
ejpam-2803	389	3	suppose	suppose	VERB
ejpam-2803	389	4	s	s	NOUN
ejpam-2803	389	5	is	be	AUX
ejpam-2803	389	6	a	a	DET
ejpam-2803	389	7	maximal	maximal	ADJ
ejpam-2803	389	8	element	element	NOUN
ejpam-2803	389	9	of	of	ADP
ejpam-2803	389	10	y	y	PROPN
ejpam-2803	389	11	.	.	PUNCT
ejpam-2803	390	1	then	then	ADV
ejpam-2803	390	2	v	v	X
ejpam-2803	390	3	s∗(s	s∗(s	NOUN
ejpam-2803	390	4	)	)	PUNCT
ejpam-2803	390	5	is	be	AUX
ejpam-2803	390	6	an	an	DET
ejpam-2803	390	7	irreducible	irreducible	ADJ
ejpam-2803	390	8	closed	closed	ADJ
ejpam-2803	390	9	subset	subset	NOUN
ejpam-2803	390	10	of	of	ADP
ejpam-2803	390	11	specs(m	specs(m	PROPN
ejpam-2803	390	12	)	)	PUNCT
ejpam-2803	390	13	.	.	PUNCT
ejpam-2803	391	1	since	since	SCONJ
ejpam-2803	391	2	s	s	PROPN
ejpam-2803	391	3	∈	∈	PROPN
ejpam-2803	391	4	y	y	PROPN
ejpam-2803	391	5	,	,	PUNCT
ejpam-2803	391	6	v	v	NOUN
ejpam-2803	391	7	s∗(s	s∗(s	PROPN
ejpam-2803	391	8	)	)	PUNCT
ejpam-2803	391	9	⊆	⊆	NUM
ejpam-2803	391	10	cl(y	cl(y	NOUN
ejpam-2803	391	11	)	)	PUNCT
ejpam-2803	391	12	=	=	SYM
ejpam-2803	391	13	y	y	PROPN
ejpam-2803	391	14	by	by	ADP
ejpam-2803	391	15	lemma	lemma	PROPN
ejpam-2803	391	16	3.1	3.1	NUM
ejpam-2803	391	17	.	.	PUNCT
ejpam-2803	392	1	now	now	ADV
ejpam-2803	392	2	let	let	VERB
ejpam-2803	392	3	v	v	NOUN
ejpam-2803	392	4	s∗(s	s∗(s	NOUN
ejpam-2803	392	5	)	)	PUNCT
ejpam-2803	392	6	⊆	⊆	NUM
ejpam-2803	392	7	t	t	NOUN
ejpam-2803	392	8	,	,	PUNCT
ejpam-2803	392	9	where	where	SCONJ
ejpam-2803	392	10	t	t	PROPN
ejpam-2803	392	11	is	be	AUX
ejpam-2803	392	12	an	an	DET
ejpam-2803	392	13	irreducible	irreducible	ADJ
ejpam-2803	392	14	subset	subset	NOUN
ejpam-2803	392	15	of	of	ADP
ejpam-2803	392	16	y	y	PROPN
ejpam-2803	392	17	.	.	PUNCT
ejpam-2803	393	1	this	this	PRON
ejpam-2803	393	2	implies	imply	VERB
ejpam-2803	393	3	that	that	SCONJ
ejpam-2803	393	4	cl(t	cl(t	NOUN
ejpam-2803	393	5	)	)	PUNCT
ejpam-2803	393	6	be	be	AUX
ejpam-2803	393	7	an	an	DET
ejpam-2803	393	8	irreducible	irreducible	ADJ
ejpam-2803	393	9	closed	closed	ADJ
ejpam-2803	393	10	subset	subset	NOUN
ejpam-2803	393	11	of	of	ADP
ejpam-2803	393	12	specs(m	specs(m	PROPN
ejpam-2803	393	13	)	)	PUNCT
ejpam-2803	393	14	and	and	CCONJ
ejpam-2803	393	15	so	so	ADV
ejpam-2803	393	16	(	(	PUNCT
ejpam-2803	393	17	t	t	PROPN
ejpam-2803	393	18	)	)	PUNCT
ejpam-2803	393	19	=	=	PUNCT
ejpam-2803	393	20	v	v	NUM
ejpam-2803	393	21	s∗(s1	s∗(s1	NOUN
ejpam-2803	393	22	)	)	PUNCT
ejpam-2803	393	23	for	for	ADP
ejpam-2803	393	24	some	some	DET
ejpam-2803	393	25	second	second	ADJ
ejpam-2803	393	26	submodule	submodule	NOUN
ejpam-2803	393	27	of	of	ADP
ejpam-2803	393	28	m	m	PRON
ejpam-2803	393	29	by	by	ADP
ejpam-2803	393	30	theorem	theorem	NOUN
ejpam-2803	393	31	3.2	3.2	NUM
ejpam-2803	393	32	.	.	PUNCT
ejpam-2803	394	1	it	it	PRON
ejpam-2803	394	2	follows	follow	VERB
ejpam-2803	394	3	that	that	PRON
ejpam-2803	394	4	s	s	VERB
ejpam-2803	394	5	≤	≤	ADJ
ejpam-2803	394	6	s1	s1	NOUN
ejpam-2803	394	7	so	so	SCONJ
ejpam-2803	394	8	that	that	PRON
ejpam-2803	394	9	s	s	VERB
ejpam-2803	394	10	=	=	SYM
ejpam-2803	394	11	s1	s1	PROPN
ejpam-2803	394	12	.	.	PUNCT
ejpam-2803	395	1	hence	hence	ADV
ejpam-2803	395	2	t	t	PROPN
ejpam-2803	395	3	=	=	PUNCT
ejpam-2803	395	4	v	v	PROPN
ejpam-2803	395	5	s∗(s	s∗(s	PROPN
ejpam-2803	395	6	)	)	PUNCT
ejpam-2803	395	7	is	be	AUX
ejpam-2803	395	8	a	a	DET
ejpam-2803	395	9	maximal	maximal	ADJ
ejpam-2803	395	10	irreducible	irreducible	ADJ
ejpam-2803	395	11	subset	subset	NOUN
ejpam-2803	395	12	of	of	ADP
ejpam-2803	395	13	y	y	PROPN
ejpam-2803	395	14	.	.	PUNCT
ejpam-2803	396	1	(	(	PUNCT
ejpam-2803	396	2	b	b	X
ejpam-2803	396	3	)	)	PUNCT
ejpam-2803	396	4	let	let	VERB
ejpam-2803	396	5	n	n	PRON
ejpam-2803	396	6	be	be	AUX
ejpam-2803	396	7	a	a	DET
ejpam-2803	396	8	non	non	ADJ
ejpam-2803	396	9	-	-	ADJ
ejpam-2803	396	10	zero	zero	NUM
ejpam-2803	396	11	submodule	submodule	NOUN
ejpam-2803	396	12	of	of	ADP
ejpam-2803	396	13	m	m	PRON
ejpam-2803	396	14	and	and	CCONJ
ejpam-2803	396	15	let	let	VERB
ejpam-2803	396	16	s	s	PRON
ejpam-2803	396	17	be	be	AUX
ejpam-2803	396	18	a	a	DET
ejpam-2803	396	19	maximal	maximal	ADJ
ejpam-2803	396	20	second	second	ADJ
ejpam-2803	396	21	summand	summand	NOUN
ejpam-2803	396	22	of	of	ADP
ejpam-2803	396	23	n	n	PROPN
ejpam-2803	396	24	.	.	PUNCT
ejpam-2803	397	1	then	then	ADV
ejpam-2803	397	2	v	v	X
ejpam-2803	397	3	s∗(s	s∗(s	NOUN
ejpam-2803	397	4	)	)	PUNCT
ejpam-2803	397	5	is	be	AUX
ejpam-2803	397	6	an	an	DET
ejpam-2803	397	7	irreducible	irreducible	ADJ
ejpam-2803	397	8	component	component	NOUN
ejpam-2803	397	9	of	of	ADP
ejpam-2803	397	10	v	v	NUM
ejpam-2803	397	11	s∗(n	s∗(n	NOUN
ejpam-2803	397	12	)	)	PUNCT
ejpam-2803	397	13	by	by	ADP
ejpam-2803	397	14	part	part	NOUN
ejpam-2803	397	15	(	(	PUNCT
ejpam-2803	397	16	a	a	NOUN
ejpam-2803	397	17	)	)	PUNCT
ejpam-2803	397	18	.	.	PUNCT
ejpam-2803	398	1	hence	hence	ADV
ejpam-2803	398	2	every	every	DET
ejpam-2803	398	3	submodule	submodule	NOUN
ejpam-2803	398	4	of	of	ADP
ejpam-2803	398	5	m	m	PROPN
ejpam-2803	398	6	has	have	VERB
ejpam-2803	398	7	a	a	DET
ejpam-2803	398	8	finite	finite	ADJ
ejpam-2803	398	9	number	number	NOUN
ejpam-2803	398	10	of	of	ADP
ejpam-2803	398	11	maximal	maximal	ADJ
ejpam-2803	398	12	second	second	ADJ
ejpam-2803	398	13	submodules	submodule	NOUN
ejpam-2803	398	14	by	by	ADP
ejpam-2803	398	15	hypothesis	hypothesis	NOUN
ejpam-2803	398	16	.	.	PUNCT
ejpam-2803	399	1	to	to	PART
ejpam-2803	399	2	see	see	VERB
ejpam-2803	399	3	the	the	DET
ejpam-2803	399	4	second	second	ADJ
ejpam-2803	399	5	assertion	assertion	NOUN
ejpam-2803	399	6	,	,	PUNCT
ejpam-2803	399	7	set	set	VERB
ejpam-2803	399	8	m	m	PROPN
ejpam-2803	399	9	=	=	PUNCT
ejpam-2803	399	10	q⊕q⊕q⊕	q⊕q⊕q⊕	NOUN
ejpam-2803	399	11	·	·	PUNCT
ejpam-2803	399	12	·	·	PUNCT
ejpam-2803	399	13	·	·	PUNCT
ejpam-2803	399	14	and	and	CCONJ
ejpam-2803	399	15	regard	regard	VERB
ejpam-2803	399	16	m	m	PROPN
ejpam-2803	399	17	as	as	ADP
ejpam-2803	399	18	q	q	NOUN
ejpam-2803	399	19	-	-	NOUN
ejpam-2803	399	20	module	module	NOUN
ejpam-2803	399	21	.	.	PUNCT
ejpam-2803	400	1	the	the	DET
ejpam-2803	400	2	second	second	ADJ
ejpam-2803	400	3	submodules	submodule	NOUN
ejpam-2803	400	4	of	of	ADP
ejpam-2803	400	5	a	a	DET
ejpam-2803	400	6	vector	vector	NOUN
ejpam-2803	400	7	space	space	NOUN
ejpam-2803	400	8	are	be	AUX
ejpam-2803	400	9	just	just	ADV
ejpam-2803	400	10	the	the	DET
ejpam-2803	400	11	non	non	ADJ
ejpam-2803	400	12	-	-	ADJ
ejpam-2803	400	13	zero	zero	NUM
ejpam-2803	400	14	submodules	submodule	NOUN
ejpam-2803	400	15	,	,	PUNCT
ejpam-2803	400	16	so	so	CCONJ
ejpam-2803	400	17	every	every	DET
ejpam-2803	400	18	submodule	submodule	NOUN
ejpam-2803	400	19	of	of	ADP
ejpam-2803	400	20	m	m	PROPN
ejpam-2803	400	21	has	have	VERB
ejpam-2803	400	22	a	a	DET
ejpam-2803	400	23	finite	finite	ADJ
ejpam-2803	400	24	number	number	NOUN
ejpam-2803	400	25	of	of	ADP
ejpam-2803	400	26	maximal	maximal	ADJ
ejpam-2803	400	27	second	second	ADJ
ejpam-2803	400	28	summands	summand	NOUN
ejpam-2803	400	29	.	.	PUNCT
ejpam-2803	401	1	now	now	ADV
ejpam-2803	401	2	let	let	VERB
ejpam-2803	401	3	s1	s1	PROPN
ejpam-2803	401	4	=	=	SYM
ejpam-2803	401	5	(	(	PUNCT
ejpam-2803	401	6	0	0	X
ejpam-2803	401	7	)	)	PUNCT
ejpam-2803	401	8	⊕	⊕	PROPN
ejpam-2803	401	9	q	q	PROPN
ejpam-2803	401	10	⊕	⊕	PROPN
ejpam-2803	401	11	q	q	PROPN
ejpam-2803	401	12	⊕	⊕	PROPN
ejpam-2803	401	13	·	·	PUNCT
ejpam-2803	401	14	·	·	PUNCT
ejpam-2803	401	15	·	·	PUNCT
ejpam-2803	401	16	,	,	PUNCT
ejpam-2803	401	17	s2	s2	VERB
ejpam-2803	401	18	=	=	SYM
ejpam-2803	401	19	q	q	PROPN
ejpam-2803	401	20	⊕	⊕	PROPN
ejpam-2803	401	21	(	(	PUNCT
ejpam-2803	401	22	0	0	NUM
ejpam-2803	401	23	)	)	PUNCT
ejpam-2803	401	24	⊕	⊕	PROPN
ejpam-2803	401	25	q	q	PROPN
ejpam-2803	401	26	⊕	⊕	PROPN
ejpam-2803	401	27	·	·	PUNCT
ejpam-2803	401	28	·	·	PUNCT
ejpam-2803	401	29	·	·	PUNCT
ejpam-2803	401	30	,	,	PUNCT
ejpam-2803	401	31	s3	s3	PROPN
ejpam-2803	401	32	=	=	PROPN
ejpam-2803	401	33	q	q	PROPN
ejpam-2803	401	34	⊕	⊕	PROPN
ejpam-2803	401	35	q	q	PROPN
ejpam-2803	401	36	⊕	⊕	PROPN
ejpam-2803	401	37	(	(	PUNCT
ejpam-2803	401	38	0	0	NUM
ejpam-2803	401	39	)	)	PUNCT
ejpam-2803	401	40	⊕	⊕	PROPN
ejpam-2803	401	41	q	q	PROPN
ejpam-2803	401	42	⊕	⊕	PROPN
ejpam-2803	401	43	·	·	PUNCT
ejpam-2803	401	44	·	·	PUNCT
ejpam-2803	401	45	·	·	PUNCT
ejpam-2803	401	46	,	,	PUNCT
ejpam-2803	401	47	·	·	PUNCT
ejpam-2803	401	48	·	·	PUNCT
ejpam-2803	401	49	·	·	PUNCT
ejpam-2803	401	50	,	,	PUNCT
ejpam-2803	401	51	s′1	s′1	NOUN
ejpam-2803	401	52	=	=	SYM
ejpam-2803	401	53	q⊕(0)⊕(0)⊕	q⊕(0)⊕(0)⊕	X
ejpam-2803	401	54	·	·	PUNCT
ejpam-2803	401	55	·	·	PUNCT
ejpam-2803	401	56	·	·	PUNCT
ejpam-2803	401	57	,	,	PUNCT
ejpam-2803	401	58	s′2	s′2	NOUN
ejpam-2803	401	59	=	=	SYM
ejpam-2803	401	60	(	(	PUNCT
ejpam-2803	401	61	0)⊕q⊕(0)⊕	0)⊕q⊕(0)⊕	NUM
ejpam-2803	401	62	·	·	PUNCT
ejpam-2803	401	63	·	·	PUNCT
ejpam-2803	401	64	·	·	PUNCT
ejpam-2803	401	65	,	,	PUNCT
ejpam-2803	402	1	s′3	s′3	ADJ
ejpam-2803	402	2	=	=	SYM
ejpam-2803	402	3	(	(	PUNCT
ejpam-2803	402	4	0)⊕(0)⊕q⊕(0)⊕	0)⊕(0)⊕q⊕(0)⊕	PROPN
ejpam-2803	402	5	·	·	PUNCT
ejpam-2803	402	6	·	·	PUNCT
ejpam-2803	402	7	·	·	PUNCT
ejpam-2803	402	8	,	,	PUNCT
ejpam-2803	402	9	·	·	PUNCT
ejpam-2803	402	10	·	·	PUNCT
ejpam-2803	402	11	·	·	PUNCT
ejpam-2803	402	12	,	,	PUNCT
ejpam-2803	402	13	and	and	CCONJ
ejpam-2803	402	14	y	y	PROPN
ejpam-2803	402	15	=	=	SYM
ejpam-2803	402	16	⋂	⋂	PROPN
ejpam-2803	402	17	i∈n(v	i∈n(v	NUM
ejpam-2803	402	18	s∗(si	s∗(si	PROPN
ejpam-2803	402	19	)	)	PUNCT
ejpam-2803	402	20	∪	∪	ADP
ejpam-2803	402	21	v	v	ADP
ejpam-2803	402	22	s∗(s′i	s∗(s′i	PROPN
ejpam-2803	402	23	)	)	PUNCT
ejpam-2803	402	24	)	)	PUNCT
ejpam-2803	402	25	.	.	PUNCT
ejpam-2803	403	1	one	one	PRON
ejpam-2803	403	2	can	can	AUX
ejpam-2803	403	3	see	see	VERB
ejpam-2803	403	4	that	that	DET
ejpam-2803	403	5	s′1	s′1	NOUN
ejpam-2803	403	6	,	,	PUNCT
ejpam-2803	403	7	s	s	VERB
ejpam-2803	403	8	′	′	NOUN
ejpam-2803	403	9	2	2	NUM
ejpam-2803	403	10	,	,	PUNCT
ejpam-2803	403	11	·	·	PUNCT
ejpam-2803	403	12	·	·	PUNCT
ejpam-2803	403	13	·	·	PUNCT
ejpam-2803	403	14	are	be	AUX
ejpam-2803	403	15	maximal	maximal	ADJ
ejpam-2803	403	16	elements	element	NOUN
ejpam-2803	403	17	of	of	ADP
ejpam-2803	403	18	y	y	PROPN
ejpam-2803	403	19	.	.	PUNCT
ejpam-2803	404	1	hence	hence	ADV
ejpam-2803	404	2	y	y	PROPN
ejpam-2803	404	3	is	be	AUX
ejpam-2803	404	4	a	a	DET
ejpam-2803	404	5	closed	closed	ADJ
ejpam-2803	404	6	subset	subset	NOUN
ejpam-2803	404	7	of	of	ADP
ejpam-2803	404	8	specs(m	specs(m	PROPN
ejpam-2803	404	9	)	)	PUNCT
ejpam-2803	404	10	with	with	ADP
ejpam-2803	404	11	infinitely	infinitely	ADV
ejpam-2803	404	12	many	many	ADJ
ejpam-2803	404	13	irreducible	irreducible	ADJ
ejpam-2803	404	14	components	component	NOUN
ejpam-2803	404	15	by	by	ADP
ejpam-2803	404	16	part	part	NOUN
ejpam-2803	404	17	(	(	PUNCT
ejpam-2803	404	18	a	a	NOUN
ejpam-2803	404	19	)	)	PUNCT
ejpam-2803	404	20	.	.	PUNCT
ejpam-2803	405	1	(	(	PUNCT
ejpam-2803	405	2	c	c	X
ejpam-2803	405	3	)	)	PUNCT
ejpam-2803	405	4	let	let	VERB
ejpam-2803	405	5	z0	z0	PROPN
ejpam-2803	405	6	(	(	PUNCT
ejpam-2803	405	7	z1	z1	PROPN
ejpam-2803	405	8	(	(	PUNCT
ejpam-2803	405	9	...	...	PUNCT
ejpam-2803	405	10	(	(	PUNCT
ejpam-2803	405	11	zt	zt	INTJ
ejpam-2803	405	12	be	be	AUX
ejpam-2803	405	13	a	a	DET
ejpam-2803	405	14	strictly	strictly	ADV
ejpam-2803	405	15	increasing	increase	VERB
ejpam-2803	405	16	chain	chain	NOUN
ejpam-2803	405	17	of	of	ADP
ejpam-2803	405	18	irreducible	irreducible	ADJ
ejpam-2803	405	19	closed	close	VERB
ejpam-2803	405	20	subsets	subset	NOUN
ejpam-2803	405	21	zi	zi	PROPN
ejpam-2803	405	22	of	of	ADP
ejpam-2803	405	23	specs(m	specs(m	PROPN
ejpam-2803	405	24	)	)	PUNCT
ejpam-2803	405	25	of	of	ADP
ejpam-2803	405	26	length	length	NOUN
ejpam-2803	405	27	t.	t.	PROPN
ejpam-2803	405	28	by	by	ADP
ejpam-2803	405	29	theorem	theorem	NOUN
ejpam-2803	405	30	3.2	3.2	NUM
ejpam-2803	405	31	,	,	PUNCT
ejpam-2803	405	32	for	for	ADP
ejpam-2803	405	33	each	each	DET
ejpam-2803	405	34	i	i	PRON
ejpam-2803	405	35	,	,	PUNCT
ejpam-2803	405	36	0	0	NUM
ejpam-2803	405	37	≤	≤	NUM
ejpam-2803	405	38	i	i	PRON
ejpam-2803	405	39	≤	≤	PROPN
ejpam-2803	405	40	t	t	PROPN
ejpam-2803	405	41	,	,	PUNCT
ejpam-2803	405	42	we	we	PRON
ejpam-2803	405	43	have	have	VERB
ejpam-2803	405	44	zi	zi	NOUN
ejpam-2803	405	45	=	=	SYM
ejpam-2803	405	46	v	v	ADP
ejpam-2803	405	47	s∗(si	s∗(si	PROPN
ejpam-2803	405	48	)	)	PUNCT
ejpam-2803	405	49	for	for	ADP
ejpam-2803	405	50	some	some	DET
ejpam-2803	405	51	si	si	PROPN
ejpam-2803	405	52	∈	∈	PROPN
ejpam-2803	405	53	specs(m	specs(m	PROPN
ejpam-2803	405	54	)	)	PUNCT
ejpam-2803	405	55	.	.	PUNCT
ejpam-2803	406	1	on	on	ADP
ejpam-2803	406	2	the	the	DET
ejpam-2803	406	3	other	other	ADJ
ejpam-2803	406	4	hand	hand	NOUN
ejpam-2803	406	5	v	v	ADP
ejpam-2803	406	6	s∗(si	s∗(si	PROPN
ejpam-2803	406	7	)	)	PUNCT
ejpam-2803	406	8	(	(	PUNCT
ejpam-2803	406	9	v	v	X
ejpam-2803	406	10	s∗(sj	s∗(sj	NOUN
ejpam-2803	406	11	)	)	PUNCT
ejpam-2803	407	1	if	if	SCONJ
ejpam-2803	407	2	and	and	CCONJ
ejpam-2803	407	3	only	only	ADV
ejpam-2803	407	4	if	if	SCONJ
ejpam-2803	407	5	si	si	X
ejpam-2803	407	6	(	(	PUNCT
ejpam-2803	407	7	sj	sj	INTJ
ejpam-2803	407	8	.	.	PUNCT
ejpam-2803	407	9	h.	h.	PROPN
ejpam-2803	407	10	ansari	ansari	PROPN
ejpam-2803	407	11	-	-	PUNCT
ejpam-2803	407	12	toroghy	toroghy	NOUN
ejpam-2803	407	13	,	,	PUNCT
ejpam-2803	407	14	s.	s.	PROPN
ejpam-2803	407	15	s.	s.	PROPN
ejpam-2803	407	16	pourmortazavi	pourmortazavi	VERB
ejpam-2803	407	17	/	/	SYM
ejpam-2803	407	18	eur	eur	PROPN
ejpam-2803	407	19	.	.	PUNCT
ejpam-2803	408	1	j.	j.	PROPN
ejpam-2803	408	2	pure	pure	PROPN
ejpam-2803	408	3	appl	appl	PROPN
ejpam-2803	408	4	.	.	PROPN
ejpam-2803	408	5	math	math	PROPN
ejpam-2803	408	6	,	,	PUNCT
ejpam-2803	408	7	10	10	NUM
ejpam-2803	408	8	(	(	PUNCT
ejpam-2803	408	9	2	2	NUM
ejpam-2803	408	10	)	)	PUNCT
ejpam-2803	408	11	(	(	PUNCT
ejpam-2803	408	12	2017	2017	NUM
ejpam-2803	408	13	)	)	PUNCT
ejpam-2803	408	14	,	,	PUNCT
ejpam-2803	408	15	211	211	NUM
ejpam-2803	408	16	-	-	SYM
ejpam-2803	408	17	230	230	NUM
ejpam-2803	408	18	221	221	NUM
ejpam-2803	408	19	hence	hence	ADV
ejpam-2803	408	20	s0	s0	NOUN
ejpam-2803	408	21	)	)	PUNCT
ejpam-2803	408	22	s1	s1	PROPN
ejpam-2803	408	23	)	)	PUNCT
ejpam-2803	408	24	...	...	PUNCT
ejpam-2803	408	25	)	)	PUNCT
ejpam-2803	409	1	st	st	PROPN
ejpam-2803	409	2	,	,	PUNCT
ejpam-2803	409	3	is	be	AUX
ejpam-2803	409	4	an	an	DET
ejpam-2803	409	5	strictly	strictly	ADV
ejpam-2803	409	6	decreasing	decrease	VERB
ejpam-2803	409	7	chain	chain	NOUN
ejpam-2803	409	8	of	of	ADP
ejpam-2803	409	9	second	second	ADJ
ejpam-2803	409	10	submodules	submodule	NOUN
ejpam-2803	409	11	of	of	ADP
ejpam-2803	409	12	m	m	NOUN
ejpam-2803	409	13	of	of	ADP
ejpam-2803	409	14	length	length	NOUN
ejpam-2803	409	15	t.	t.	NOUN
ejpam-2803	409	16	conversely	conversely	ADV
ejpam-2803	409	17	,	,	PUNCT
ejpam-2803	409	18	for	for	ADP
ejpam-2803	409	19	every	every	DET
ejpam-2803	409	20	strictly	strictly	ADV
ejpam-2803	409	21	decreasing	decrease	VERB
ejpam-2803	409	22	chain	chain	NOUN
ejpam-2803	409	23	s0	s0	NOUN
ejpam-2803	409	24	)	)	PUNCT
ejpam-2803	409	25	s1	s1	PROPN
ejpam-2803	409	26	)	)	PUNCT
ejpam-2803	409	27	...	...	PUNCT
ejpam-2803	409	28	)	)	PUNCT
ejpam-2803	410	1	st	st	PROPN
ejpam-2803	410	2	of	of	ADP
ejpam-2803	410	3	second	second	ADJ
ejpam-2803	410	4	submodules	submodule	NOUN
ejpam-2803	410	5	of	of	ADP
ejpam-2803	410	6	m	m	NOUN
ejpam-2803	410	7	of	of	ADP
ejpam-2803	410	8	length	length	NOUN
ejpam-2803	410	9	t	t	PROPN
ejpam-2803	410	10	,	,	PUNCT
ejpam-2803	410	11	v	v	X
ejpam-2803	410	12	s∗(s0	s∗(s0	PROPN
ejpam-2803	410	13	)	)	PUNCT
ejpam-2803	410	14	(	(	PUNCT
ejpam-2803	410	15	v	v	NOUN
ejpam-2803	410	16	s∗(s1	s∗(s1	NOUN
ejpam-2803	410	17	)	)	PUNCT
ejpam-2803	410	18	(	(	PUNCT
ejpam-2803	410	19	...	...	PUNCT
ejpam-2803	410	20	(	(	PUNCT
ejpam-2803	410	21	v	v	NUM
ejpam-2803	410	22	s∗(st	s∗(st	NOUN
ejpam-2803	410	23	)	)	PUNCT
ejpam-2803	410	24	is	be	AUX
ejpam-2803	410	25	a	a	DET
ejpam-2803	410	26	strictly	strictly	ADV
ejpam-2803	410	27	increasing	increase	VERB
ejpam-2803	410	28	chain	chain	NOUN
ejpam-2803	410	29	of	of	ADP
ejpam-2803	410	30	irreducible	irreducible	ADJ
ejpam-2803	410	31	closed	closed	ADJ
ejpam-2803	410	32	subsets	subset	NOUN
ejpam-2803	410	33	of	of	ADP
ejpam-2803	410	34	specs(m	specs(m	NOUN
ejpam-2803	410	35	)	)	PUNCT
ejpam-2803	410	36	of	of	ADP
ejpam-2803	410	37	length	length	NOUN
ejpam-2803	410	38	t.	t.	PROPN
ejpam-2803	411	1	this	this	PRON
ejpam-2803	411	2	in	in	ADP
ejpam-2803	411	3	turn	turn	NOUN
ejpam-2803	411	4	implies	imply	VERB
ejpam-2803	411	5	that	that	SCONJ
ejpam-2803	411	6	dim(specs(m	dim(specs(m	VERB
ejpam-2803	411	7	)	)	PUNCT
ejpam-2803	411	8	)	)	PUNCT
ejpam-2803	412	1	=	=	NOUN
ejpam-2803	412	2	s.dimm	s.dimm	NOUN
ejpam-2803	412	3	and	and	CCONJ
ejpam-2803	412	4	the	the	DET
ejpam-2803	412	5	proof	proof	NOUN
ejpam-2803	412	6	is	be	AUX
ejpam-2803	412	7	completed	complete	VERB
ejpam-2803	412	8	.	.	PUNCT
ejpam-2803	413	1	a	a	DET
ejpam-2803	413	2	proper	proper	ADJ
ejpam-2803	413	3	submodule	submodule	NOUN
ejpam-2803	413	4	n	n	PROPN
ejpam-2803	413	5	of	of	ADP
ejpam-2803	413	6	an	an	DET
ejpam-2803	413	7	r	r	NOUN
ejpam-2803	413	8	-	-	PUNCT
ejpam-2803	413	9	module	module	NOUN
ejpam-2803	413	10	m	m	NOUN
ejpam-2803	413	11	is	be	AUX
ejpam-2803	413	12	said	say	VERB
ejpam-2803	413	13	to	to	PART
ejpam-2803	413	14	be	be	AUX
ejpam-2803	413	15	completely	completely	ADV
ejpam-2803	413	16	irreducible	irreducible	ADJ
ejpam-2803	413	17	if	if	SCONJ
ejpam-2803	413	18	n	n	PROPN
ejpam-2803	413	19	=	=	SYM
ejpam-2803	413	20	⋂	⋂	PROPN
ejpam-2803	413	21	i∈i	i∈i	PROPN
ejpam-2803	413	22	ni	ni	PROPN
ejpam-2803	413	23	,	,	PUNCT
ejpam-2803	413	24	where	where	SCONJ
ejpam-2803	413	25	{	{	PUNCT
ejpam-2803	413	26	ni}i∈i	ni}i∈i	INTJ
ejpam-2803	413	27	is	be	AUX
ejpam-2803	413	28	a	a	DET
ejpam-2803	413	29	family	family	NOUN
ejpam-2803	413	30	of	of	ADP
ejpam-2803	413	31	submodules	submodule	NOUN
ejpam-2803	413	32	of	of	ADP
ejpam-2803	413	33	m	m	PROPN
ejpam-2803	413	34	,	,	PUNCT
ejpam-2803	413	35	implies	imply	VERB
ejpam-2803	413	36	that	that	SCONJ
ejpam-2803	413	37	n	n	PROPN
ejpam-2803	413	38	=	=	SYM
ejpam-2803	413	39	ni	ni	PROPN
ejpam-2803	413	40	for	for	ADP
ejpam-2803	413	41	some	some	DET
ejpam-2803	413	42	i	i	PRON
ejpam-2803	413	43	∈	∈	PROPN
ejpam-2803	413	44	i	i	PRON
ejpam-2803	413	45	(	(	PUNCT
ejpam-2803	413	46	see	see	VERB
ejpam-2803	413	47	[	[	X
ejpam-2803	413	48	18	18	NUM
ejpam-2803	413	49	]	]	NUM
ejpam-2803	413	50	)	)	PUNCT
ejpam-2803	413	51	.	.	PUNCT
ejpam-2803	414	1	let	let	VERB
ejpam-2803	414	2	p	p	PRON
ejpam-2803	414	3	be	be	AUX
ejpam-2803	414	4	a	a	DET
ejpam-2803	414	5	prime	prime	ADJ
ejpam-2803	414	6	ideal	ideal	NOUN
ejpam-2803	414	7	of	of	ADP
ejpam-2803	414	8	r	r	NOUN
ejpam-2803	414	9	and	and	CCONJ
ejpam-2803	414	10	let	let	VERB
ejpam-2803	414	11	n	n	PRON
ejpam-2803	414	12	be	be	AUX
ejpam-2803	414	13	a	a	DET
ejpam-2803	414	14	submodule	submodule	NOUN
ejpam-2803	414	15	of	of	ADP
ejpam-2803	414	16	an	an	DET
ejpam-2803	414	17	r	r	NOUN
ejpam-2803	414	18	-	-	PUNCT
ejpam-2803	414	19	module	module	NOUN
ejpam-2803	414	20	m	m	NOUN
ejpam-2803	414	21	.	.	PUNCT
ejpam-2803	415	1	then	then	ADV
ejpam-2803	415	2	n	n	ADV
ejpam-2803	415	3	ec	ec	NOUN
ejpam-2803	415	4	=	=	PUNCT
ejpam-2803	415	5	{	{	PUNCT
ejpam-2803	415	6	m	m	VERB
ejpam-2803	415	7	∈	∈	NOUN
ejpam-2803	415	8	m	m	VERB
ejpam-2803	415	9	:	:	PUNCT
ejpam-2803	415	10	cm	cm	NOUN
ejpam-2803	415	11	∈	∈	PROPN
ejpam-2803	415	12	n	n	NOUN
ejpam-2803	415	13	for	for	ADP
ejpam-2803	415	14	some	some	DET
ejpam-2803	415	15	c	c	PROPN
ejpam-2803	415	16	∈	∈	PROPN
ejpam-2803	415	17	r\p	r\p	NOUN
ejpam-2803	415	18	}	}	PUNCT
ejpam-2803	415	19	and	and	CCONJ
ejpam-2803	415	20	it	it	PRON
ejpam-2803	415	21	is	be	AUX
ejpam-2803	415	22	called	call	VERB
ejpam-2803	415	23	the	the	DET
ejpam-2803	415	24	p	p	NOUN
ejpam-2803	415	25	-	-	PUNCT
ejpam-2803	415	26	closure	closure	NOUN
ejpam-2803	415	27	of	of	ADP
ejpam-2803	415	28	n	n	NUM
ejpam-2803	415	29	and	and	CCONJ
ejpam-2803	415	30	denoted	denote	VERB
ejpam-2803	415	31	by	by	ADP
ejpam-2803	415	32	clp(n	clp(n	PROPN
ejpam-2803	415	33	)	)	PUNCT
ejpam-2803	415	34	(	(	PUNCT
ejpam-2803	415	35	see	see	VERB
ejpam-2803	415	36	[	[	X
ejpam-2803	415	37	24	24	NUM
ejpam-2803	415	38	,	,	PUNCT
ejpam-2803	415	39	p.	p.	NOUN
ejpam-2803	415	40	92	92	NUM
ejpam-2803	415	41	]	]	PUNCT
ejpam-2803	415	42	)	)	PUNCT
ejpam-2803	415	43	.	.	PUNCT
ejpam-2803	416	1	the	the	DET
ejpam-2803	416	2	dual	dual	ADJ
ejpam-2803	416	3	of	of	ADP
ejpam-2803	416	4	this	this	DET
ejpam-2803	416	5	notion	notion	NOUN
ejpam-2803	416	6	,	,	PUNCT
ejpam-2803	416	7	i.e.	i.e.	X
ejpam-2803	416	8	,	,	PUNCT
ejpam-2803	416	9	p	p	NOUN
ejpam-2803	416	10	-	-	NOUN
ejpam-2803	416	11	interior	interior	NOUN
ejpam-2803	416	12	of	of	ADP
ejpam-2803	416	13	n	n	PROPN
ejpam-2803	416	14	relative	relative	ADJ
ejpam-2803	416	15	to	to	ADP
ejpam-2803	416	16	m	m	PROPN
ejpam-2803	416	17	is	be	AUX
ejpam-2803	416	18	defined	define	VERB
ejpam-2803	416	19	as	as	ADP
ejpam-2803	416	20	the	the	DET
ejpam-2803	416	21	set	set	NOUN
ejpam-2803	416	22	imp	imp	X
ejpam-2803	416	23	(	(	PUNCT
ejpam-2803	416	24	n	n	CCONJ
ejpam-2803	416	25	)	)	PUNCT
ejpam-2803	416	26	:	:	PUNCT
ejpam-2803	416	27	=	=	SYM
ejpam-2803	416	28	⋂	⋂	PROPN
ejpam-2803	416	29	{	{	PUNCT
ejpam-2803	416	30	l|l	l|l	X
ejpam-2803	416	31	is	be	AUX
ejpam-2803	416	32	a	a	DET
ejpam-2803	416	33	completely	completely	ADV
ejpam-2803	416	34	irreducible	irreducible	ADJ
ejpam-2803	416	35	submodule	submodule	NOUN
ejpam-2803	416	36	of	of	ADP
ejpam-2803	416	37	m	m	PROPN
ejpam-2803	416	38	and	and	CCONJ
ejpam-2803	416	39	rn	rn	PROPN
ejpam-2803	416	40	⊆	⊆	NUM
ejpam-2803	416	41	l	l	NOUN
ejpam-2803	416	42	for	for	ADP
ejpam-2803	416	43	some	some	DET
ejpam-2803	416	44	r	r	NOUN
ejpam-2803	416	45	∈	∈	NOUN
ejpam-2803	416	46	r\p	r\p	NOUN
ejpam-2803	416	47	}	}	PUNCT
ejpam-2803	416	48	(	(	PUNCT
ejpam-2803	416	49	see	see	VERB
ejpam-2803	416	50	[	[	X
ejpam-2803	416	51	4	4	NUM
ejpam-2803	416	52	]	]	NUM
ejpam-2803	416	53	)	)	PUNCT
ejpam-2803	416	54	.	.	PUNCT
ejpam-2803	417	1	it	it	PRON
ejpam-2803	417	2	is	be	AUX
ejpam-2803	417	3	easy	easy	ADJ
ejpam-2803	417	4	to	to	PART
ejpam-2803	417	5	see	see	VERB
ejpam-2803	417	6	that	that	DET
ejpam-2803	417	7	imp	imp	NOUN
ejpam-2803	417	8	(	(	PUNCT
ejpam-2803	417	9	n	n	CCONJ
ejpam-2803	417	10	)	)	PUNCT
ejpam-2803	417	11	=	=	SYM
ejpam-2803	418	1	⋂	⋂	PROPN
ejpam-2803	418	2	r∈r\p	r∈r\p	PROPN
ejpam-2803	418	3	rn	rn	PROPN
ejpam-2803	418	4	.	.	PUNCT
ejpam-2803	419	1	let	let	VERB
ejpam-2803	419	2	r	r	PRON
ejpam-2803	419	3	be	be	AUX
ejpam-2803	419	4	an	an	DET
ejpam-2803	419	5	integral	integral	ADJ
ejpam-2803	419	6	domain	domain	NOUN
ejpam-2803	419	7	.	.	PUNCT
ejpam-2803	420	1	a	a	DET
ejpam-2803	420	2	submodule	submodule	NOUN
ejpam-2803	420	3	n	n	PROPN
ejpam-2803	420	4	of	of	ADP
ejpam-2803	420	5	an	an	DET
ejpam-2803	420	6	r	r	NOUN
ejpam-2803	420	7	-	-	PUNCT
ejpam-2803	420	8	module	module	NOUN
ejpam-2803	420	9	m	m	NOUN
ejpam-2803	420	10	is	be	AUX
ejpam-2803	420	11	said	say	VERB
ejpam-2803	420	12	to	to	PART
ejpam-2803	420	13	be	be	AUX
ejpam-2803	420	14	cotorsion	cotorsion	NOUN
ejpam-2803	420	15	-	-	PUNCT
ejpam-2803	420	16	free	free	ADJ
ejpam-2803	420	17	(	(	PUNCT
ejpam-2803	420	18	resp	resp	NOUN
ejpam-2803	420	19	.	.	PUNCT
ejpam-2803	421	1	cotorsion	cotorsion	NOUN
ejpam-2803	421	2	)	)	PUNCT
ejpam-2803	422	1	if	if	SCONJ
ejpam-2803	422	2	im0	im0	X
ejpam-2803	422	3	(	(	PUNCT
ejpam-2803	422	4	n	n	CCONJ
ejpam-2803	422	5	)	)	PUNCT
ejpam-2803	422	6	=	=	SYM
ejpam-2803	422	7	n	n	PROPN
ejpam-2803	422	8	(	(	PUNCT
ejpam-2803	422	9	resp	resp	NOUN
ejpam-2803	422	10	.	.	PUNCT
ejpam-2803	422	11	im0	im0	PROPN
ejpam-2803	422	12	(	(	PUNCT
ejpam-2803	422	13	n	n	CCONJ
ejpam-2803	422	14	)	)	PUNCT
ejpam-2803	422	15	=	=	SYM
ejpam-2803	422	16	(	(	PUNCT
ejpam-2803	422	17	0	0	NUM
ejpam-2803	422	18	)	)	PUNCT
ejpam-2803	422	19	)	)	PUNCT
ejpam-2803	422	20	(	(	PUNCT
ejpam-2803	422	21	see	see	VERB
ejpam-2803	422	22	[	[	X
ejpam-2803	422	23	5	5	NUM
ejpam-2803	422	24	]	]	NUM
ejpam-2803	422	25	)	)	PUNCT
ejpam-2803	422	26	.	.	PUNCT
ejpam-2803	423	1	we	we	PRON
ejpam-2803	423	2	need	need	VERB
ejpam-2803	423	3	the	the	DET
ejpam-2803	423	4	following	follow	VERB
ejpam-2803	423	5	lemma	lemma	PROPN
ejpam-2803	423	6	.	.	PUNCT
ejpam-2803	424	1	lemma	lemma	PROPN
ejpam-2803	424	2	3.6	3.6	NUM
ejpam-2803	424	3	.	.	PUNCT
ejpam-2803	425	1	let	let	VERB
ejpam-2803	425	2	p	p	NOUN
ejpam-2803	425	3	and	and	CCONJ
ejpam-2803	425	4	q	q	NOUN
ejpam-2803	425	5	be	be	AUX
ejpam-2803	425	6	prime	prime	ADJ
ejpam-2803	425	7	ideals	ideal	NOUN
ejpam-2803	425	8	of	of	ADP
ejpam-2803	425	9	r	r	NOUN
ejpam-2803	425	10	with	with	ADP
ejpam-2803	425	11	q	q	NOUN
ejpam-2803	425	12	⊆	⊆	NUM
ejpam-2803	425	13	p.	p.	NOUN
ejpam-2803	425	14	let	let	VERB
ejpam-2803	425	15	m	m	PRON
ejpam-2803	425	16	be	be	AUX
ejpam-2803	425	17	an	an	DET
ejpam-2803	425	18	r	r	NOUN
ejpam-2803	425	19	-	-	PUNCT
ejpam-2803	425	20	module	module	NOUN
ejpam-2803	425	21	with	with	ADP
ejpam-2803	425	22	clp(0	clp(0	NOUN
ejpam-2803	425	23	)	)	PUNCT
ejpam-2803	425	24	=	=	SYM
ejpam-2803	426	1	(	(	PUNCT
ejpam-2803	426	2	0	0	NUM
ejpam-2803	426	3	)	)	PUNCT
ejpam-2803	427	1	and	and	CCONJ
ejpam-2803	427	2	let	let	VERB
ejpam-2803	427	3	φ	φ	PROPN
ejpam-2803	427	4	:	:	PUNCT
ejpam-2803	427	5	homr(rp	homr(rp	PROPN
ejpam-2803	427	6	,	,	PUNCT
ejpam-2803	427	7	m	m	NOUN
ejpam-2803	427	8	)	)	PUNCT
ejpam-2803	427	9	→	→	PUNCT
ejpam-2803	427	10	m	m	AUX
ejpam-2803	427	11	be	be	VERB
ejpam-2803	427	12	the	the	DET
ejpam-2803	427	13	natural	natural	ADJ
ejpam-2803	427	14	homomorphism	homomorphism	NOUN
ejpam-2803	427	15	given	give	VERB
ejpam-2803	427	16	by	by	ADP
ejpam-2803	427	17	f	f	PROPN
ejpam-2803	427	18	7→	7→	PROPN
ejpam-2803	427	19	f(1/1	f(1/1	PROPN
ejpam-2803	427	20	)	)	PUNCT
ejpam-2803	427	21	.	.	PUNCT
ejpam-2803	428	1	then	then	ADV
ejpam-2803	428	2	we	we	PRON
ejpam-2803	428	3	have	have	VERB
ejpam-2803	428	4	the	the	DET
ejpam-2803	428	5	following	following	NOUN
ejpam-2803	428	6	.	.	PUNCT
ejpam-2803	429	1	(	(	PUNCT
ejpam-2803	429	2	i	i	NOUN
ejpam-2803	429	3	)	)	PUNCT
ejpam-2803	429	4	if	if	SCONJ
ejpam-2803	429	5	s	s	NOUN
ejpam-2803	429	6	is	be	AUX
ejpam-2803	429	7	an	an	DET
ejpam-2803	429	8	rp	rp	NOUN
ejpam-2803	429	9	-	-	PUNCT
ejpam-2803	429	10	submodule	submodule	NOUN
ejpam-2803	429	11	of	of	ADP
ejpam-2803	429	12	homr(rp	homr(rp	PROPN
ejpam-2803	429	13	,	,	PUNCT
ejpam-2803	429	14	m	m	NOUN
ejpam-2803	429	15	)	)	PUNCT
ejpam-2803	429	16	,	,	PUNCT
ejpam-2803	429	17	then	then	ADV
ejpam-2803	429	18	we	we	PRON
ejpam-2803	429	19	have	have	VERB
ejpam-2803	429	20	sec	sec	PROPN
ejpam-2803	429	21	=	=	SYM
ejpam-2803	429	22	s	s	PART
ejpam-2803	429	23	=	=	X
ejpam-2803	429	24	homr(rp	homr(rp	PROPN
ejpam-2803	429	25	,	,	PUNCT
ejpam-2803	429	26	l	l	NOUN
ejpam-2803	429	27	)	)	PUNCT
ejpam-2803	429	28	,	,	PUNCT
ejpam-2803	429	29	where	where	SCONJ
ejpam-2803	429	30	l	l	NOUN
ejpam-2803	429	31	=	=	PUNCT
ejpam-2803	429	32	se	se	X
ejpam-2803	429	33	.	.	PUNCT
ejpam-2803	430	1	(	(	PUNCT
ejpam-2803	430	2	here	here	ADV
ejpam-2803	430	3	t	t	X
ejpam-2803	430	4	e	e	NOUN
ejpam-2803	430	5	,	,	PUNCT
ejpam-2803	430	6	where	where	SCONJ
ejpam-2803	430	7	t	t	PROPN
ejpam-2803	430	8	⊆	⊆	NUM
ejpam-2803	430	9	homr(rp	homr(rp	NOUN
ejpam-2803	430	10	,	,	PUNCT
ejpam-2803	430	11	m	m	NOUN
ejpam-2803	430	12	)	)	PUNCT
ejpam-2803	430	13	,	,	PUNCT
ejpam-2803	430	14	and	and	CCONJ
ejpam-2803	430	15	n	n	PRON
ejpam-2803	430	16	c	c	NOUN
ejpam-2803	430	17	,	,	PUNCT
ejpam-2803	430	18	where	where	SCONJ
ejpam-2803	430	19	n	n	NUM
ejpam-2803	430	20	⊆m	⊆m	NOUN
ejpam-2803	430	21	,	,	PUNCT
ejpam-2803	430	22	denote	denote	VERB
ejpam-2803	430	23	φ(t	φ(t	PROPN
ejpam-2803	430	24	)	)	PUNCT
ejpam-2803	430	25	and	and	CCONJ
ejpam-2803	430	26	φ−1(n	φ−1(n	NOUN
ejpam-2803	430	27	)	)	PUNCT
ejpam-2803	430	28	,	,	PUNCT
ejpam-2803	430	29	respectively	respectively	ADV
ejpam-2803	430	30	.	.	PUNCT
ejpam-2803	430	31	)	)	PUNCT
ejpam-2803	431	1	(	(	PUNCT
ejpam-2803	431	2	ii	ii	X
ejpam-2803	431	3	)	)	PUNCT
ejpam-2803	431	4	ifm	ifm	PROPN
ejpam-2803	431	5	is	be	AUX
ejpam-2803	431	6	an	an	DET
ejpam-2803	431	7	artinianr	artinianr	NOUN
ejpam-2803	431	8	-	-	PUNCT
ejpam-2803	431	9	module	module	NOUN
ejpam-2803	431	10	andk	andk	NOUN
ejpam-2803	431	11	is	be	AUX
ejpam-2803	431	12	a	a	DET
ejpam-2803	431	13	q	q	ADJ
ejpam-2803	431	14	-	-	PUNCT
ejpam-2803	431	15	second	second	ADJ
ejpam-2803	431	16	submodule	submodule	NOUN
ejpam-2803	431	17	ofm	ofm	PROPN
ejpam-2803	431	18	,	,	PUNCT
ejpam-2803	431	19	thenhomr(rp	thenhomr(rp	PROPN
ejpam-2803	431	20	,	,	PUNCT
ejpam-2803	431	21	k	k	NOUN
ejpam-2803	431	22	)	)	PUNCT
ejpam-2803	431	23	is	be	AUX
ejpam-2803	431	24	a	a	DET
ejpam-2803	431	25	qrp	qrp	PROPN
ejpam-2803	431	26	-	-	PUNCT
ejpam-2803	431	27	second	second	ADJ
ejpam-2803	431	28	submodule	submodule	NOUN
ejpam-2803	431	29	of	of	ADP
ejpam-2803	431	30	homr(rp	homr(rp	PROPN
ejpam-2803	431	31	,	,	PUNCT
ejpam-2803	431	32	m	m	NOUN
ejpam-2803	431	33	)	)	PUNCT
ejpam-2803	431	34	and	and	CCONJ
ejpam-2803	431	35	kc	kc	PROPN
ejpam-2803	431	36	=	=	PUNCT
ejpam-2803	431	37	homr(rp	homr(rp	PROPN
ejpam-2803	431	38	,	,	PUNCT
ejpam-2803	431	39	k	k	NOUN
ejpam-2803	431	40	)	)	PUNCT
ejpam-2803	431	41	.	.	PUNCT
ejpam-2803	432	1	further	far	ADV
ejpam-2803	432	2	we	we	PRON
ejpam-2803	432	3	have	have	VERB
ejpam-2803	432	4	(	(	PUNCT
ejpam-2803	432	5	homr(rp	homr(rp	NOUN
ejpam-2803	432	6	,	,	PUNCT
ejpam-2803	432	7	k))e	k))e	NOUN
ejpam-2803	432	8	=	=	SYM
ejpam-2803	432	9	φ(homr(rp	φ(homr(rp	PROPN
ejpam-2803	432	10	,	,	PUNCT
ejpam-2803	432	11	k	k	NOUN
ejpam-2803	432	12	)	)	PUNCT
ejpam-2803	432	13	)	)	PUNCT
ejpam-2803	433	1	=	=	SYM
ejpam-2803	433	2	ip(k	ip(k	NOUN
ejpam-2803	433	3	)	)	PUNCT
ejpam-2803	433	4	=	=	SYM
ejpam-2803	433	5	k	k	PROPN
ejpam-2803	433	6	and	and	CCONJ
ejpam-2803	433	7	kce	kce	PROPN
ejpam-2803	433	8	=	=	SYM
ejpam-2803	433	9	k.	k.	PROPN
ejpam-2803	433	10	proof	proof	NOUN
ejpam-2803	433	11	.	.	PUNCT
ejpam-2803	434	1	(	(	PUNCT
ejpam-2803	434	2	i	i	NOUN
ejpam-2803	434	3	)	)	PUNCT
ejpam-2803	434	4	clearly	clearly	ADV
ejpam-2803	434	5	,	,	PUNCT
ejpam-2803	434	6	s	s	VERB
ejpam-2803	434	7	⊆	⊆	NUM
ejpam-2803	434	8	sec	sec	PROPN
ejpam-2803	434	9	.	.	PUNCT
ejpam-2803	435	1	so	so	ADV
ejpam-2803	435	2	let	let	VERB
ejpam-2803	435	3	g	g	PROPN
ejpam-2803	435	4	∈	∈	PROPN
ejpam-2803	435	5	sec	sec	PROPN
ejpam-2803	435	6	.	.	PUNCT
ejpam-2803	436	1	then	then	ADV
ejpam-2803	436	2	there	there	PRON
ejpam-2803	436	3	exists	exist	VERB
ejpam-2803	436	4	f	f	PROPN
ejpam-2803	436	5	∈	∈	PROPN
ejpam-2803	436	6	s	s	VERB
ejpam-2803	436	7	such	such	ADJ
ejpam-2803	436	8	that	that	PRON
ejpam-2803	436	9	g(1/1	g(1/1	NOUN
ejpam-2803	436	10	)	)	PUNCT
ejpam-2803	436	11	=	=	SYM
ejpam-2803	436	12	f(1/1	f(1/1	X
ejpam-2803	436	13	)	)	PUNCT
ejpam-2803	436	14	∈	∈	PROPN
ejpam-2803	436	15	se	se	X
ejpam-2803	437	1	and	and	CCONJ
ejpam-2803	437	2	so	so	ADV
ejpam-2803	437	3	f(r/1	f(r/1	ADJ
ejpam-2803	437	4	)	)	PUNCT
ejpam-2803	437	5	=	=	SYM
ejpam-2803	437	6	g(r/1	g(r/1	NOUN
ejpam-2803	437	7	)	)	PUNCT
ejpam-2803	437	8	for	for	ADP
ejpam-2803	437	9	every	every	DET
ejpam-2803	437	10	r	r	NOUN
ejpam-2803	437	11	∈	∈	PROPN
ejpam-2803	437	12	r.	r.	NOUN
ejpam-2803	437	13	now	now	ADV
ejpam-2803	437	14	let	let	VERB
ejpam-2803	437	15	λ	λ	X
ejpam-2803	437	16	∈	∈	PROPN
ejpam-2803	437	17	rp	rp	NOUN
ejpam-2803	437	18	,	,	PUNCT
ejpam-2803	437	19	then	then	ADV
ejpam-2803	437	20	λ	λ	X
ejpam-2803	437	21	=	=	SYM
ejpam-2803	437	22	r	r	X
ejpam-2803	437	23	/	/	SYM
ejpam-2803	437	24	s	s	NOUN
ejpam-2803	437	25	for	for	ADP
ejpam-2803	437	26	some	some	DET
ejpam-2803	437	27	r	r	NOUN
ejpam-2803	437	28	∈	∈	NOUN
ejpam-2803	437	29	r	r	NOUN
ejpam-2803	437	30	and	and	CCONJ
ejpam-2803	437	31	s	s	NOUN
ejpam-2803	437	32	∈	∈	PROPN
ejpam-2803	437	33	r	r	NOUN
ejpam-2803	437	34	−	−	PROPN
ejpam-2803	438	1	p.	p.	NOUN
ejpam-2803	438	2	then	then	ADV
ejpam-2803	438	3	we	we	PRON
ejpam-2803	438	4	have	have	VERB
ejpam-2803	438	5	s(f(λ)−	s(f(λ)−	VERB
ejpam-2803	438	6	g(λ	g(λ	PROPN
ejpam-2803	438	7	)	)	PUNCT
ejpam-2803	438	8	)	)	PUNCT
ejpam-2803	439	1	=	=	PUNCT
ejpam-2803	439	2	0	0	X
ejpam-2803	439	3	.	.	PUNCT
ejpam-2803	439	4	since	since	SCONJ
ejpam-2803	439	5	clp(0	clp(0	NOUN
ejpam-2803	439	6	)	)	PUNCT
ejpam-2803	439	7	=	=	SYM
ejpam-2803	439	8	0	0	NUM
ejpam-2803	439	9	,	,	PUNCT
ejpam-2803	439	10	f(λ	f(λ	NOUN
ejpam-2803	439	11	)	)	PUNCT
ejpam-2803	439	12	=	=	SYM
ejpam-2803	439	13	g(λ	g(λ	PROPN
ejpam-2803	439	14	)	)	PUNCT
ejpam-2803	439	15	,	,	PUNCT
ejpam-2803	439	16	so	so	SCONJ
ejpam-2803	439	17	g	g	PROPN
ejpam-2803	439	18	∈	∈	PROPN
ejpam-2803	439	19	s.	s.	PROPN
ejpam-2803	439	20	now	now	ADV
ejpam-2803	439	21	homr(rp	homr(rp	VERB
ejpam-2803	439	22	,	,	PUNCT
ejpam-2803	439	23	l	l	NOUN
ejpam-2803	439	24	)	)	PUNCT
ejpam-2803	439	25	=	=	SYM
ejpam-2803	439	26	sec	sec	PROPN
ejpam-2803	439	27	,	,	PUNCT
ejpam-2803	439	28	where	where	SCONJ
ejpam-2803	439	29	l	l	NOUN
ejpam-2803	439	30	=	=	SYM
ejpam-2803	439	31	se	se	X
ejpam-2803	439	32	,	,	PUNCT
ejpam-2803	439	33	follows	follow	VERB
ejpam-2803	439	34	directly	directly	ADV
ejpam-2803	439	35	from	from	ADP
ejpam-2803	439	36	the	the	DET
ejpam-2803	439	37	above	above	ADJ
ejpam-2803	439	38	arguments	argument	NOUN
ejpam-2803	439	39	.	.	PUNCT
ejpam-2803	440	1	(	(	PUNCT
ejpam-2803	440	2	ii	ii	NOUN
ejpam-2803	440	3	)	)	PUNCT
ejpam-2803	440	4	letk	letk	NOUN
ejpam-2803	440	5	be	be	VERB
ejpam-2803	440	6	a	a	DET
ejpam-2803	440	7	q	q	ADJ
ejpam-2803	440	8	-	-	PUNCT
ejpam-2803	440	9	second	second	ADJ
ejpam-2803	440	10	submodule	submodule	NOUN
ejpam-2803	440	11	ofm	ofm	PROPN
ejpam-2803	440	12	.	.	PUNCT
ejpam-2803	441	1	then	then	ADV
ejpam-2803	441	2	by	by	ADP
ejpam-2803	441	3	[	[	PUNCT
ejpam-2803	441	4	25	25	NUM
ejpam-2803	441	5	,	,	PUNCT
ejpam-2803	441	6	theorem	theorem	VERB
ejpam-2803	441	7	3.1	3.1	NUM
ejpam-2803	441	8	(	(	PUNCT
ejpam-2803	441	9	2	2	NUM
ejpam-2803	441	10	)	)	PUNCT
ejpam-2803	441	11	]	]	PUNCT
ejpam-2803	441	12	,	,	PUNCT
ejpam-2803	441	13	φ	φ	PROPN
ejpam-2803	441	14	:	:	PUNCT
ejpam-2803	441	15	homr(rp	homr(rp	PROPN
ejpam-2803	441	16	,	,	PUNCT
ejpam-2803	441	17	k	k	NOUN
ejpam-2803	441	18	)	)	PUNCT
ejpam-2803	441	19	→	→	SYM
ejpam-2803	441	20	k	k	PROPN
ejpam-2803	441	21	given	give	VERB
ejpam-2803	441	22	by	by	ADP
ejpam-2803	441	23	f	f	PROPN
ejpam-2803	441	24	7→	7→	PROPN
ejpam-2803	441	25	f(1/1	f(1/1	PROPN
ejpam-2803	441	26	)	)	PUNCT
ejpam-2803	441	27	is	be	AUX
ejpam-2803	441	28	a	a	DET
ejpam-2803	441	29	surjective	surjective	ADJ
ejpam-2803	441	30	homomorphism	homomorphism	NOUN
ejpam-2803	441	31	,	,	PUNCT
ejpam-2803	441	32	so	so	ADV
ejpam-2803	441	33	homr(rp	homr(rp	NOUN
ejpam-2803	441	34	,	,	PUNCT
ejpam-2803	441	35	k	k	NOUN
ejpam-2803	441	36	)	)	PUNCT
ejpam-2803	441	37	6=	6=	ADP
ejpam-2803	441	38	(	(	PUNCT
ejpam-2803	441	39	0	0	NUM
ejpam-2803	441	40	)	)	PUNCT
ejpam-2803	441	41	.	.	PUNCT
ejpam-2803	442	1	to	to	PART
ejpam-2803	442	2	see	see	VERB
ejpam-2803	442	3	homr(rp	homr(rp	NOUN
ejpam-2803	442	4	,	,	PUNCT
ejpam-2803	442	5	k	k	NOUN
ejpam-2803	442	6	)	)	PUNCT
ejpam-2803	442	7	is	be	AUX
ejpam-2803	442	8	a	a	DET
ejpam-2803	442	9	qrp	qrp	PROPN
ejpam-2803	442	10	-	-	PUNCT
ejpam-2803	442	11	second	second	NOUN
ejpam-2803	442	12	,	,	PUNCT
ejpam-2803	442	13	let	let	VERB
ejpam-2803	442	14	r	r	NOUN
ejpam-2803	442	15	s	s	NOUN
ejpam-2803	442	16	∈	∈	NOUN
ejpam-2803	442	17	rp	rp	NOUN
ejpam-2803	442	18	\qrp	\qrp	PROPN
ejpam-2803	443	1	and	and	CCONJ
ejpam-2803	443	2	so	so	ADV
ejpam-2803	443	3	r	r	NOUN
ejpam-2803	443	4	∈	∈	PROPN
ejpam-2803	443	5	r\q	r\q	NOUN
ejpam-2803	443	6	.	.	PUNCT
ejpam-2803	444	1	thus	thus	ADV
ejpam-2803	444	2	rk	rk	VERB
ejpam-2803	444	3	=	=	SYM
ejpam-2803	444	4	k	k	PROPN
ejpam-2803	444	5	because	because	SCONJ
ejpam-2803	444	6	k	k	PROPN
ejpam-2803	444	7	is	be	AUX
ejpam-2803	444	8	q	q	NOUN
ejpam-2803	444	9	-	-	PUNCT
ejpam-2803	444	10	second	second	NOUN
ejpam-2803	444	11	.	.	PUNCT
ejpam-2803	445	1	since	since	SCONJ
ejpam-2803	445	2	clp(0	clp(0	NOUN
ejpam-2803	445	3	)	)	PUNCT
ejpam-2803	445	4	=	=	SYM
ejpam-2803	445	5	(	(	PUNCT
ejpam-2803	445	6	0	0	NUM
ejpam-2803	445	7	)	)	PUNCT
ejpam-2803	445	8	and	and	CCONJ
ejpam-2803	445	9	k	k	PROPN
ejpam-2803	445	10	is	be	AUX
ejpam-2803	445	11	an	an	DET
ejpam-2803	445	12	artinian	artinian	ADJ
ejpam-2803	445	13	r	r	NOUN
ejpam-2803	445	14	-	-	PUNCT
ejpam-2803	445	15	module	module	NOUN
ejpam-2803	445	16	,	,	PUNCT
ejpam-2803	445	17	we	we	PRON
ejpam-2803	445	18	see	see	VERB
ejpam-2803	445	19	that	that	DET
ejpam-2803	445	20	homr(rp	homr(rp	NOUN
ejpam-2803	445	21	,	,	PUNCT
ejpam-2803	445	22	rk	rk	NOUN
ejpam-2803	445	23	)	)	PUNCT
ejpam-2803	445	24	=	=	SYM
ejpam-2803	445	25	rhomr(rp	rhomr(rp	PROPN
ejpam-2803	445	26	,	,	PUNCT
ejpam-2803	445	27	k	k	NOUN
ejpam-2803	445	28	)	)	PUNCT
ejpam-2803	445	29	by	by	ADP
ejpam-2803	445	30	[	[	X
ejpam-2803	445	31	25	25	NUM
ejpam-2803	445	32	,	,	PUNCT
ejpam-2803	445	33	proposition	proposition	NOUN
ejpam-2803	445	34	2.4	2.4	NUM
ejpam-2803	445	35	]	]	PUNCT
ejpam-2803	445	36	.	.	PUNCT
ejpam-2803	446	1	this	this	PRON
ejpam-2803	446	2	implies	imply	VERB
ejpam-2803	446	3	that	that	SCONJ
ejpam-2803	446	4	r	r	NOUN
ejpam-2803	446	5	s	s	X
ejpam-2803	446	6	(	(	PUNCT
ejpam-2803	446	7	homr(rp	homr(rp	NOUN
ejpam-2803	446	8	,	,	PUNCT
ejpam-2803	446	9	k	k	NOUN
ejpam-2803	446	10	)	)	PUNCT
ejpam-2803	446	11	)	)	PUNCT
ejpam-2803	446	12	=	=	PUNCT
ejpam-2803	446	13	1	1	NUM
ejpam-2803	446	14	s	s	NOUN
ejpam-2803	446	15	(	(	PUNCT
ejpam-2803	446	16	homr(rp	homr(rp	NOUN
ejpam-2803	446	17	,	,	PUNCT
ejpam-2803	446	18	rk	rk	NOUN
ejpam-2803	446	19	)	)	PUNCT
ejpam-2803	446	20	)	)	PUNCT
ejpam-2803	447	1	=	=	SYM
ejpam-2803	447	2	homr(rp	homr(rp	PROPN
ejpam-2803	447	3	,	,	PUNCT
ejpam-2803	447	4	k	k	NOUN
ejpam-2803	447	5	)	)	PUNCT
ejpam-2803	447	6	.	.	PUNCT
ejpam-2803	448	1	h.	h.	PROPN
ejpam-2803	448	2	ansari	ansari	PROPN
ejpam-2803	448	3	-	-	PUNCT
ejpam-2803	448	4	toroghy	toroghy	NOUN
ejpam-2803	448	5	,	,	PUNCT
ejpam-2803	448	6	s.	s.	PROPN
ejpam-2803	448	7	s.	s.	PROPN
ejpam-2803	448	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	448	9	/	/	SYM
ejpam-2803	448	10	eur	eur	PROPN
ejpam-2803	448	11	.	.	PUNCT
ejpam-2803	449	1	j.	j.	PROPN
ejpam-2803	449	2	pure	pure	PROPN
ejpam-2803	449	3	appl	appl	PROPN
ejpam-2803	449	4	.	.	PROPN
ejpam-2803	449	5	math	math	PROPN
ejpam-2803	449	6	,	,	PUNCT
ejpam-2803	449	7	10	10	NUM
ejpam-2803	449	8	(	(	PUNCT
ejpam-2803	449	9	2	2	NUM
ejpam-2803	449	10	)	)	PUNCT
ejpam-2803	449	11	(	(	PUNCT
ejpam-2803	449	12	2017	2017	NUM
ejpam-2803	449	13	)	)	PUNCT
ejpam-2803	449	14	,	,	PUNCT
ejpam-2803	449	15	211	211	NUM
ejpam-2803	449	16	-	-	SYM
ejpam-2803	449	17	230	230	NUM
ejpam-2803	449	18	222	222	NUM
ejpam-2803	449	19	if	if	SCONJ
ejpam-2803	449	20	r	r	NOUN
ejpam-2803	449	21	s	s	X
ejpam-2803	449	22	∈	∈	PROPN
ejpam-2803	449	23	qrp	qrp	PROPN
ejpam-2803	449	24	,	,	PUNCT
ejpam-2803	449	25	then	then	ADV
ejpam-2803	449	26	r	r	PROPN
ejpam-2803	449	27	s(homr(rp	s(homr(rp	PROPN
ejpam-2803	449	28	,	,	PUNCT
ejpam-2803	449	29	k	k	NOUN
ejpam-2803	449	30	)	)	PUNCT
ejpam-2803	449	31	)	)	PUNCT
ejpam-2803	449	32	=	=	PUNCT
ejpam-2803	450	1	1	1	NUM
ejpam-2803	450	2	s	s	NOUN
ejpam-2803	450	3	(	(	PUNCT
ejpam-2803	450	4	homr(rp	homr(rp	NOUN
ejpam-2803	450	5	,	,	PUNCT
ejpam-2803	450	6	rk	rk	NOUN
ejpam-2803	450	7	)	)	PUNCT
ejpam-2803	450	8	)	)	PUNCT
ejpam-2803	450	9	=	=	PUNCT
ejpam-2803	450	10	(	(	PUNCT
ejpam-2803	450	11	0	0	NUM
ejpam-2803	450	12	)	)	PUNCT
ejpam-2803	450	13	.	.	PUNCT
ejpam-2803	451	1	further	further	ADJ
ejpam-2803	451	2	clp(0	clp(0	NOUN
ejpam-2803	451	3	)	)	PUNCT
ejpam-2803	451	4	=	=	SYM
ejpam-2803	451	5	(	(	PUNCT
ejpam-2803	451	6	0	0	NUM
ejpam-2803	451	7	)	)	PUNCT
ejpam-2803	451	8	implies	imply	VERB
ejpam-2803	451	9	that	that	SCONJ
ejpam-2803	451	10	kc	kc	PROPN
ejpam-2803	451	11	=	=	PUNCT
ejpam-2803	451	12	homr(rp	homr(rp	PROPN
ejpam-2803	451	13	,	,	PUNCT
ejpam-2803	451	14	k	k	NOUN
ejpam-2803	451	15	)	)	PUNCT
ejpam-2803	451	16	.	.	PUNCT
ejpam-2803	452	1	the	the	DET
ejpam-2803	452	2	last	last	ADJ
ejpam-2803	452	3	assertion	assertion	NOUN
ejpam-2803	452	4	follows	follow	VERB
ejpam-2803	452	5	from	from	ADP
ejpam-2803	452	6	[	[	X
ejpam-2803	452	7	7	7	NUM
ejpam-2803	452	8	,	,	PUNCT
ejpam-2803	452	9	lemma	lemma	PROPN
ejpam-2803	452	10	2.15	2.15	NUM
ejpam-2803	452	11	]	]	PUNCT
ejpam-2803	452	12	.	.	PUNCT
ejpam-2803	453	1	proposition	proposition	NOUN
ejpam-2803	453	2	3.7	3.7	NUM
ejpam-2803	453	3	.	.	PUNCT
ejpam-2803	454	1	let	let	VERB
ejpam-2803	454	2	p	p	PRON
ejpam-2803	454	3	be	be	AUX
ejpam-2803	454	4	a	a	DET
ejpam-2803	454	5	prime	prime	ADJ
ejpam-2803	454	6	ideal	ideal	NOUN
ejpam-2803	454	7	of	of	ADP
ejpam-2803	454	8	r	r	NOUN
ejpam-2803	454	9	and	and	CCONJ
ejpam-2803	454	10	let	let	VERB
ejpam-2803	454	11	m	m	PRON
ejpam-2803	454	12	be	be	AUX
ejpam-2803	454	13	a	a	DET
ejpam-2803	454	14	cotop	cotop	NOUN
ejpam-2803	454	15	rmodule	rmodule	NOUN
ejpam-2803	454	16	with	with	ADP
ejpam-2803	454	17	clp(0	clp(0	NOUN
ejpam-2803	454	18	)	)	PUNCT
ejpam-2803	455	1	=	=	SYM
ejpam-2803	455	2	0	0	X
ejpam-2803	455	3	.	.	PUNCT
ejpam-2803	456	1	then	then	ADV
ejpam-2803	456	2	homr(rp	homr(rp	PROPN
ejpam-2803	456	3	,	,	PUNCT
ejpam-2803	456	4	m	m	NOUN
ejpam-2803	456	5	)	)	PUNCT
ejpam-2803	456	6	is	be	AUX
ejpam-2803	456	7	a	a	DET
ejpam-2803	456	8	cotop	cotop	NOUN
ejpam-2803	456	9	rp	rp	NOUN
ejpam-2803	456	10	-	-	PUNCT
ejpam-2803	456	11	module	module	NOUN
ejpam-2803	456	12	.	.	PUNCT
ejpam-2803	457	1	proof	proof	NOUN
ejpam-2803	457	2	.	.	PUNCT
ejpam-2803	458	1	let	let	VERB
ejpam-2803	458	2	s	s	PRON
ejpam-2803	458	3	be	be	AUX
ejpam-2803	458	4	a	a	DET
ejpam-2803	458	5	second	second	ADJ
ejpam-2803	458	6	submodule	submodule	NOUN
ejpam-2803	458	7	of	of	ADP
ejpam-2803	458	8	the	the	DET
ejpam-2803	458	9	rp	rp	NOUN
ejpam-2803	458	10	-	-	PUNCT
ejpam-2803	458	11	module	module	NOUN
ejpam-2803	458	12	homr(rp	homr(rp	NOUN
ejpam-2803	458	13	,	,	PUNCT
ejpam-2803	458	14	m	m	NOUN
ejpam-2803	458	15	)	)	PUNCT
ejpam-2803	458	16	.	.	PUNCT
ejpam-2803	459	1	then	then	ADV
ejpam-2803	459	2	we	we	PRON
ejpam-2803	459	3	have	have	VERB
ejpam-2803	459	4	se	se	X
ejpam-2803	459	5	=	=	SYM
ejpam-2803	459	6	φ(s	φ(s	NOUN
ejpam-2803	459	7	)	)	PUNCT
ejpam-2803	459	8	6=	6=	PUNCT
ejpam-2803	459	9	(	(	PUNCT
ejpam-2803	459	10	0	0	NUM
ejpam-2803	459	11	)	)	PUNCT
ejpam-2803	459	12	.	.	PUNCT
ejpam-2803	460	1	clearly	clearly	ADV
ejpam-2803	460	2	,	,	PUNCT
ejpam-2803	460	3	se	se	X
ejpam-2803	460	4	is	be	AUX
ejpam-2803	460	5	a	a	DET
ejpam-2803	460	6	second	second	ADJ
ejpam-2803	460	7	submodule	submodule	NOUN
ejpam-2803	460	8	of	of	ADP
ejpam-2803	460	9	m	m	PROPN
ejpam-2803	460	10	.	.	PUNCT
ejpam-2803	461	1	now	now	ADV
ejpam-2803	461	2	let	let	VERB
ejpam-2803	461	3	s1	s1	NOUN
ejpam-2803	461	4	and	and	CCONJ
ejpam-2803	461	5	s2	s2	PROPN
ejpam-2803	461	6	be	be	AUX
ejpam-2803	461	7	socle	socle	NOUN
ejpam-2803	461	8	submodules	submodule	NOUN
ejpam-2803	461	9	of	of	ADP
ejpam-2803	461	10	rp	rp	NOUN
ejpam-2803	461	11	-	-	PUNCT
ejpam-2803	461	12	module	module	NOUN
ejpam-2803	461	13	homr(rp	homr(rp	NOUN
ejpam-2803	461	14	,	,	PUNCT
ejpam-2803	461	15	m	m	NOUN
ejpam-2803	461	16	)	)	PUNCT
ejpam-2803	461	17	with	with	ADP
ejpam-2803	461	18	s	s	PRON
ejpam-2803	461	19	⊆	⊆	NUM
ejpam-2803	461	20	s1	s1	NOUN
ejpam-2803	461	21	+	+	CCONJ
ejpam-2803	461	22	s2	s2	PROPN
ejpam-2803	461	23	.	.	PUNCT
ejpam-2803	462	1	then	then	ADV
ejpam-2803	462	2	se1	se1	PROPN
ejpam-2803	462	3	and	and	CCONJ
ejpam-2803	462	4	se2	se2	PROPN
ejpam-2803	462	5	are	be	AUX
ejpam-2803	462	6	socle	socle	NOUN
ejpam-2803	462	7	submodules	submodule	NOUN
ejpam-2803	462	8	of	of	ADP
ejpam-2803	462	9	m	m	PROPN
ejpam-2803	462	10	with	with	ADP
ejpam-2803	462	11	se	se	PROPN
ejpam-2803	462	12	⊆	⊆	NUM
ejpam-2803	462	13	(	(	PUNCT
ejpam-2803	462	14	s1	s1	NOUN
ejpam-2803	462	15	+	+	PROPN
ejpam-2803	462	16	s2)e	s2)e	NOUN
ejpam-2803	462	17	=	=	PUNCT
ejpam-2803	462	18	se1	se1	PROPN
ejpam-2803	462	19	+	+	PROPN
ejpam-2803	462	20	se2	se2	PROPN
ejpam-2803	462	21	.	.	PUNCT
ejpam-2803	463	1	since	since	SCONJ
ejpam-2803	463	2	m	m	PROPN
ejpam-2803	463	3	is	be	AUX
ejpam-2803	463	4	a	a	DET
ejpam-2803	463	5	cotop	cotop	NOUN
ejpam-2803	463	6	r	r	NOUN
ejpam-2803	463	7	-	-	PUNCT
ejpam-2803	463	8	module	module	NOUN
ejpam-2803	463	9	,	,	PUNCT
ejpam-2803	463	10	we	we	PRON
ejpam-2803	463	11	have	have	VERB
ejpam-2803	463	12	se	se	PROPN
ejpam-2803	463	13	⊆	⊆	NUM
ejpam-2803	463	14	se1	se1	NOUN
ejpam-2803	463	15	or	or	CCONJ
ejpam-2803	463	16	se	se	X
ejpam-2803	463	17	⊆	⊆	NUM
ejpam-2803	463	18	se2	se2	NOUN
ejpam-2803	463	19	.	.	PUNCT
ejpam-2803	464	1	thus	thus	ADV
ejpam-2803	464	2	we	we	PRON
ejpam-2803	464	3	have	have	VERB
ejpam-2803	464	4	s	s	NOUN
ejpam-2803	464	5	=	=	PUNCT
ejpam-2803	464	6	sec	sec	PROPN
ejpam-2803	464	7	⊆	⊆	NUM
ejpam-2803	464	8	sec1	sec1	PROPN
ejpam-2803	464	9	=	=	SYM
ejpam-2803	464	10	s1	s1	PROPN
ejpam-2803	464	11	or	or	CCONJ
ejpam-2803	464	12	s	s	NOUN
ejpam-2803	464	13	=	=	PROPN
ejpam-2803	464	14	sec	sec	PROPN
ejpam-2803	464	15	⊆	⊆	NUM
ejpam-2803	464	16	sec2	sec2	NOUN
ejpam-2803	464	17	=	=	SYM
ejpam-2803	464	18	s2	s2	PROPN
ejpam-2803	464	19	by	by	ADP
ejpam-2803	464	20	lemma	lemma	PROPN
ejpam-2803	464	21	3.6	3.6	NUM
ejpam-2803	464	22	(	(	PUNCT
ejpam-2803	464	23	i	i	NOUN
ejpam-2803	464	24	)	)	PUNCT
ejpam-2803	464	25	.	.	PUNCT
ejpam-2803	465	1	this	this	PRON
ejpam-2803	465	2	implies	imply	VERB
ejpam-2803	465	3	that	that	SCONJ
ejpam-2803	465	4	homr(rp	homr(rp	NOUN
ejpam-2803	465	5	,	,	PUNCT
ejpam-2803	465	6	m	m	NOUN
ejpam-2803	465	7	)	)	PUNCT
ejpam-2803	465	8	is	be	AUX
ejpam-2803	465	9	a	a	DET
ejpam-2803	465	10	cotop	cotop	NOUN
ejpam-2803	465	11	rp	rp	NOUN
ejpam-2803	465	12	-	-	PUNCT
ejpam-2803	465	13	module	module	NOUN
ejpam-2803	465	14	.	.	PUNCT
ejpam-2803	465	15	example	example	NOUN
ejpam-2803	465	16	3.8	3.8	NUM
ejpam-2803	465	17	.	.	PUNCT
ejpam-2803	466	1	let	let	VERB
ejpam-2803	466	2	r	r	PRON
ejpam-2803	466	3	be	be	AUX
ejpam-2803	466	4	an	an	DET
ejpam-2803	466	5	integral	integral	ADJ
ejpam-2803	466	6	domain	domain	NOUN
ejpam-2803	466	7	and	and	CCONJ
ejpam-2803	466	8	let	let	VERB
ejpam-2803	466	9	q	q	NOUN
ejpam-2803	466	10	be	be	AUX
ejpam-2803	466	11	the	the	DET
ejpam-2803	466	12	field	field	NOUN
ejpam-2803	466	13	of	of	ADP
ejpam-2803	466	14	fractions	fraction	NOUN
ejpam-2803	466	15	of	of	ADP
ejpam-2803	466	16	r.	r.	PROPN
ejpam-2803	466	17	then	then	ADV
ejpam-2803	466	18	clearly	clearly	ADV
ejpam-2803	466	19	,	,	PUNCT
ejpam-2803	466	20	q	q	X
ejpam-2803	466	21	is	be	AUX
ejpam-2803	466	22	a	a	DET
ejpam-2803	466	23	torsion	torsion	NOUN
ejpam-2803	466	24	-	-	PUNCT
ejpam-2803	466	25	free	free	ADJ
ejpam-2803	466	26	cotop	cotop	NOUN
ejpam-2803	466	27	r	r	NOUN
ejpam-2803	466	28	-	-	PUNCT
ejpam-2803	466	29	module	module	NOUN
ejpam-2803	466	30	and	and	CCONJ
ejpam-2803	466	31	hence	hence	ADV
ejpam-2803	466	32	for	for	SCONJ
ejpam-2803	466	33	every	every	DET
ejpam-2803	466	34	prime	prime	ADJ
ejpam-2803	466	35	ideal	ideal	NOUN
ejpam-2803	466	36	p	p	NOUN
ejpam-2803	466	37	of	of	ADP
ejpam-2803	466	38	r	r	NOUN
ejpam-2803	466	39	,	,	PUNCT
ejpam-2803	466	40	homr(rp	homr(rp	NOUN
ejpam-2803	466	41	,	,	PUNCT
ejpam-2803	466	42	q	q	NOUN
ejpam-2803	466	43	)	)	PUNCT
ejpam-2803	466	44	is	be	AUX
ejpam-2803	466	45	a	a	DET
ejpam-2803	466	46	cotop	cotop	NOUN
ejpam-2803	466	47	rp	rp	NOUN
ejpam-2803	466	48	-	-	PUNCT
ejpam-2803	466	49	module	module	NOUN
ejpam-2803	466	50	by	by	ADP
ejpam-2803	466	51	proposition	proposition	NOUN
ejpam-2803	466	52	3.7	3.7	NUM
ejpam-2803	466	53	.	.	PUNCT
ejpam-2803	467	1	note	note	VERB
ejpam-2803	467	2	that	that	SCONJ
ejpam-2803	467	3	since	since	SCONJ
ejpam-2803	467	4	q	q	NOUN
ejpam-2803	467	5	is	be	AUX
ejpam-2803	467	6	a	a	DET
ejpam-2803	467	7	torsion	torsion	NOUN
ejpam-2803	467	8	-	-	PUNCT
ejpam-2803	467	9	free	free	ADJ
ejpam-2803	467	10	r	r	NOUN
ejpam-2803	467	11	-	-	PUNCT
ejpam-2803	467	12	module	module	NOUN
ejpam-2803	467	13	,	,	PUNCT
ejpam-2803	467	14	clp(0	clp(0	NOUN
ejpam-2803	467	15	)	)	PUNCT
ejpam-2803	467	16	=	=	SYM
ejpam-2803	468	1	(	(	PUNCT
ejpam-2803	468	2	0	0	NUM
ejpam-2803	468	3	)	)	PUNCT
ejpam-2803	468	4	for	for	ADP
ejpam-2803	468	5	every	every	DET
ejpam-2803	468	6	prime	prime	ADJ
ejpam-2803	468	7	ideal	ideal	NOUN
ejpam-2803	468	8	p	p	PROPN
ejpam-2803	468	9	of	of	ADP
ejpam-2803	468	10	r.	r.	PROPN
ejpam-2803	468	11	let	let	VERB
ejpam-2803	468	12	m	m	PRON
ejpam-2803	468	13	be	be	AUX
ejpam-2803	468	14	an	an	DET
ejpam-2803	468	15	r	r	NOUN
ejpam-2803	468	16	-	-	PUNCT
ejpam-2803	468	17	module	module	NOUN
ejpam-2803	468	18	.	.	PUNCT
ejpam-2803	469	1	the	the	DET
ejpam-2803	469	2	class	class	NOUN
ejpam-2803	469	3	of	of	ADP
ejpam-2803	469	4	x	x	ADJ
ejpam-2803	469	5	-	-	ADJ
ejpam-2803	469	6	injective	injective	ADJ
ejpam-2803	469	7	modules	module	NOUN
ejpam-2803	469	8	,	,	PUNCT
ejpam-2803	469	9	where	where	SCONJ
ejpam-2803	469	10	x	x	PRON
ejpam-2803	469	11	is	be	AUX
ejpam-2803	469	12	the	the	DET
ejpam-2803	469	13	set	set	NOUN
ejpam-2803	469	14	of	of	ADP
ejpam-2803	469	15	all	all	DET
ejpam-2803	469	16	prime	prime	ADJ
ejpam-2803	469	17	submodules	submodule	NOUN
ejpam-2803	469	18	of	of	ADP
ejpam-2803	469	19	m	m	PRON
ejpam-2803	469	20	,	,	PUNCT
ejpam-2803	469	21	was	be	AUX
ejpam-2803	469	22	studied	study	VERB
ejpam-2803	469	23	by	by	ADP
ejpam-2803	469	24	h.	h.	PROPN
ejpam-2803	469	25	ansari	ansari	PROPN
ejpam-2803	469	26	-	-	PUNCT
ejpam-2803	469	27	toroghy	toroghy	ADJ
ejpam-2803	469	28	and	and	CCONJ
ejpam-2803	469	29	r.	r.	PROPN
ejpam-2803	469	30	ovlyaee	ovlyaee	PROPN
ejpam-2803	469	31	-	-	PUNCT
ejpam-2803	469	32	sarmazdeh	sarmazdeh	NOUN
ejpam-2803	469	33	in	in	ADP
ejpam-2803	469	34	[	[	X
ejpam-2803	469	35	9	9	NUM
ejpam-2803	469	36	]	]	PUNCT
ejpam-2803	469	37	.	.	PUNCT
ejpam-2803	470	1	m	m	PROPN
ejpam-2803	470	2	is	be	AUX
ejpam-2803	470	3	x	x	X
ejpam-2803	470	4	-	-	NOUN
ejpam-2803	470	5	injective	injective	ADJ
ejpam-2803	470	6	if	if	SCONJ
ejpam-2803	470	7	the	the	DET
ejpam-2803	470	8	map	map	NOUN
ejpam-2803	470	9	φ	φ	X
ejpam-2803	470	10	:	:	PUNCT
ejpam-2803	471	1	x	x	X
ejpam-2803	471	2	→	→	SYM
ejpam-2803	471	3	r̄	r̄	NOUN
ejpam-2803	471	4	given	give	VERB
ejpam-2803	471	5	by	by	ADP
ejpam-2803	471	6	p	p	PROPN
ejpam-2803	471	7	7→	7→	PROPN
ejpam-2803	471	8	(	(	PUNCT
ejpam-2803	471	9	p	p	X
ejpam-2803	471	10	:	:	PUNCT
ejpam-2803	471	11	r	r	NOUN
ejpam-2803	471	12	m	m	NOUN
ejpam-2803	471	13	)	)	PUNCT
ejpam-2803	471	14	=	=	SYM
ejpam-2803	472	1	(	(	PUNCT
ejpam-2803	472	2	p	p	X
ejpam-2803	472	3	:	:	PUNCT
ejpam-2803	472	4	r	r	NOUN
ejpam-2803	472	5	m)/annr(m	m)/annr(m	NOUN
ejpam-2803	472	6	)	)	PUNCT
ejpam-2803	472	7	is	be	AUX
ejpam-2803	472	8	injective	injective	ADJ
ejpam-2803	472	9	.	.	PUNCT
ejpam-2803	473	1	definition	definition	NOUN
ejpam-2803	473	2	3.9	3.9	NUM
ejpam-2803	473	3	.	.	PUNCT
ejpam-2803	474	1	let	let	VERB
ejpam-2803	474	2	m	m	PRON
ejpam-2803	474	3	be	be	AUX
ejpam-2803	474	4	an	an	DET
ejpam-2803	474	5	r	r	NOUN
ejpam-2803	474	6	-	-	PUNCT
ejpam-2803	474	7	module	module	NOUN
ejpam-2803	474	8	.	.	PUNCT
ejpam-2803	475	1	we	we	PRON
ejpam-2803	475	2	say	say	VERB
ejpam-2803	475	3	that	that	SCONJ
ejpam-2803	475	4	m	m	PROPN
ejpam-2803	475	5	is	be	AUX
ejpam-2803	475	6	an	an	DET
ejpam-2803	475	7	xs	xs	NOUN
ejpam-2803	475	8	-	-	PUNCT
ejpam-2803	475	9	injective	injective	ADJ
ejpam-2803	475	10	if	if	SCONJ
ejpam-2803	475	11	the	the	DET
ejpam-2803	475	12	natural	natural	ADJ
ejpam-2803	475	13	map	map	NOUN
ejpam-2803	475	14	of	of	ADP
ejpam-2803	475	15	xs	xs	PROPN
ejpam-2803	475	16	is	be	AUX
ejpam-2803	475	17	injective	injective	ADJ
ejpam-2803	475	18	.	.	PUNCT
ejpam-2803	476	1	equivalently	equivalently	ADV
ejpam-2803	476	2	,	,	PUNCT
ejpam-2803	476	3	m	m	VERB
ejpam-2803	476	4	is	be	AUX
ejpam-2803	476	5	xs	xs	NOUN
ejpam-2803	476	6	-	-	PUNCT
ejpam-2803	476	7	injective	injective	ADJ
ejpam-2803	476	8	if	if	SCONJ
ejpam-2803	476	9	and	and	CCONJ
ejpam-2803	476	10	only	only	ADV
ejpam-2803	476	11	if	if	SCONJ
ejpam-2803	476	12	annr(s1	annr(s1	NOUN
ejpam-2803	476	13	)	)	PUNCT
ejpam-2803	476	14	=	=	SYM
ejpam-2803	476	15	annr(s2	annr(s2	NOUN
ejpam-2803	476	16	)	)	PUNCT
ejpam-2803	476	17	,	,	PUNCT
ejpam-2803	476	18	s1	s1	NOUN
ejpam-2803	476	19	,	,	PUNCT
ejpam-2803	476	20	s2	s2	PROPN
ejpam-2803	476	21	∈	∈	PROPN
ejpam-2803	476	22	xs	xs	PROPN
ejpam-2803	476	23	,	,	PUNCT
ejpam-2803	476	24	implies	imply	VERB
ejpam-2803	476	25	that	that	SCONJ
ejpam-2803	476	26	s1	s1	NOUN
ejpam-2803	476	27	=	=	PUNCT
ejpam-2803	476	28	s2	s2	VERB
ejpam-2803	476	29	if	if	SCONJ
ejpam-2803	476	30	and	and	CCONJ
ejpam-2803	476	31	only	only	ADV
ejpam-2803	476	32	if	if	SCONJ
ejpam-2803	476	33	for	for	ADP
ejpam-2803	476	34	every	every	DET
ejpam-2803	476	35	p	p	PROPN
ejpam-2803	476	36	∈	∈	PROPN
ejpam-2803	476	37	spec(r	spec(r	PROPN
ejpam-2803	476	38	)	)	PUNCT
ejpam-2803	476	39	,	,	PUNCT
ejpam-2803	476	40	|specsp(m)|	|specsp(m)|	PROPN
ejpam-2803	476	41	≤	≤	NUM
ejpam-2803	476	42	1	1	NUM
ejpam-2803	476	43	.	.	PUNCT
ejpam-2803	477	1	let	let	VERB
ejpam-2803	477	2	m	m	PRON
ejpam-2803	477	3	be	be	AUX
ejpam-2803	477	4	an	an	DET
ejpam-2803	477	5	r	r	NOUN
ejpam-2803	477	6	-	-	PUNCT
ejpam-2803	477	7	module	module	NOUN
ejpam-2803	477	8	.	.	PUNCT
ejpam-2803	478	1	m	m	PROPN
ejpam-2803	478	2	is	be	AUX
ejpam-2803	478	3	said	say	VERB
ejpam-2803	478	4	to	to	PART
ejpam-2803	478	5	be	be	AUX
ejpam-2803	478	6	a	a	DET
ejpam-2803	478	7	comultiplication	comultiplication	NOUN
ejpam-2803	478	8	module	module	NOUN
ejpam-2803	478	9	if	if	SCONJ
ejpam-2803	478	10	for	for	ADP
ejpam-2803	478	11	every	every	DET
ejpam-2803	478	12	submodule	submodule	NOUN
ejpam-2803	478	13	n	n	PROPN
ejpam-2803	478	14	of	of	ADP
ejpam-2803	478	15	m	m	VERB
ejpam-2803	478	16	there	there	PRON
ejpam-2803	478	17	exists	exist	VERB
ejpam-2803	478	18	an	an	DET
ejpam-2803	478	19	ideal	ideal	NOUN
ejpam-2803	478	20	i	i	PRON
ejpam-2803	478	21	of	of	ADP
ejpam-2803	478	22	r	r	NOUN
ejpam-2803	479	1	such	such	ADJ
ejpam-2803	479	2	that	that	SCONJ
ejpam-2803	479	3	n	n	NOUN
ejpam-2803	479	4	=	=	SYM
ejpam-2803	479	5	(	(	PUNCT
ejpam-2803	479	6	0	0	NUM
ejpam-2803	479	7	:	:	PUNCT
ejpam-2803	479	8	m	m	VERB
ejpam-2803	479	9	i	i	NOUN
ejpam-2803	479	10	)	)	PUNCT
ejpam-2803	479	11	(	(	PUNCT
ejpam-2803	479	12	see	see	VERB
ejpam-2803	479	13	[	[	X
ejpam-2803	479	14	3	3	NUM
ejpam-2803	479	15	]	]	NUM
ejpam-2803	479	16	)	)	PUNCT
ejpam-2803	479	17	.	.	PUNCT
ejpam-2803	480	1	example	example	NOUN
ejpam-2803	481	1	3.10	3.10	NUM
ejpam-2803	481	2	.	.	PUNCT
ejpam-2803	482	1	let	let	VERB
ejpam-2803	482	2	p	p	PRON
ejpam-2803	482	3	be	be	AUX
ejpam-2803	482	4	a	a	DET
ejpam-2803	482	5	prime	prime	ADJ
ejpam-2803	482	6	integer	integer	NOUN
ejpam-2803	482	7	.	.	PUNCT
ejpam-2803	483	1	(	(	PUNCT
ejpam-2803	483	2	a	a	X
ejpam-2803	483	3	)	)	PUNCT
ejpam-2803	483	4	every	every	DET
ejpam-2803	483	5	comultiplication	comultiplication	NOUN
ejpam-2803	483	6	module	module	NOUN
ejpam-2803	483	7	is	be	AUX
ejpam-2803	483	8	xs	xs	NOUN
ejpam-2803	483	9	-	-	PUNCT
ejpam-2803	483	10	injective	injective	ADJ
ejpam-2803	483	11	.	.	PUNCT
ejpam-2803	484	1	in	in	ADP
ejpam-2803	484	2	particular	particular	ADJ
ejpam-2803	484	3	zp∞	zp∞	PROPN
ejpam-2803	484	4	=	=	SYM
ejpam-2803	484	5	e(z	e(z	PROPN
ejpam-2803	484	6	/	/	SYM
ejpam-2803	484	7	pz	pz	NOUN
ejpam-2803	484	8	)	)	PUNCT
ejpam-2803	484	9	is	be	AUX
ejpam-2803	484	10	an	an	DET
ejpam-2803	484	11	xs	xs	NOUN
ejpam-2803	484	12	-	-	PUNCT
ejpam-2803	484	13	injective	injective	ADJ
ejpam-2803	484	14	z	z	NOUN
ejpam-2803	484	15	-	-	PUNCT
ejpam-2803	484	16	module	module	NOUN
ejpam-2803	484	17	.	.	PUNCT
ejpam-2803	485	1	note	note	VERB
ejpam-2803	485	2	that	that	SCONJ
ejpam-2803	485	3	specs(zp∞	specs(zp∞	PROPN
ejpam-2803	485	4	)	)	PUNCT
ejpam-2803	486	1	=	=	PRON
ejpam-2803	486	2	{	{	PUNCT
ejpam-2803	486	3	〈	〈	NOUN
ejpam-2803	486	4	1	1	NUM
ejpam-2803	486	5	/	/	SYM
ejpam-2803	486	6	p+	p+	NOUN
ejpam-2803	486	7	z〉	z〉	PROPN
ejpam-2803	486	8	,	,	PUNCT
ejpam-2803	486	9	zp∞	zp∞	PROPN
ejpam-2803	486	10	}	}	PUNCT
ejpam-2803	486	11	.	.	PUNCT
ejpam-2803	487	1	(	(	PUNCT
ejpam-2803	487	2	b	b	X
ejpam-2803	487	3	)	)	PUNCT
ejpam-2803	487	4	let	let	VERB
ejpam-2803	487	5	m	m	NOUN
ejpam-2803	487	6	=	=	VERB
ejpam-2803	487	7	q	q	PROPN
ejpam-2803	487	8	⊕	⊕	PROPN
ejpam-2803	488	1	zp	zp	PROPN
ejpam-2803	488	2	.	.	PUNCT
ejpam-2803	489	1	then	then	ADV
ejpam-2803	489	2	m	m	PROPN
ejpam-2803	489	3	is	be	AUX
ejpam-2803	489	4	xs	xs	NOUN
ejpam-2803	489	5	-	-	PUNCT
ejpam-2803	489	6	injective	injective	ADJ
ejpam-2803	489	7	z	z	NOUN
ejpam-2803	489	8	-	-	PUNCT
ejpam-2803	489	9	module	module	NOUN
ejpam-2803	489	10	.	.	PUNCT
ejpam-2803	490	1	note	note	VERB
ejpam-2803	490	2	that	that	DET
ejpam-2803	490	3	specs(m	specs(m	NOUN
ejpam-2803	490	4	)	)	PUNCT
ejpam-2803	490	5	=	=	PRON
ejpam-2803	491	1	{	{	PUNCT
ejpam-2803	491	2	q⊕	q⊕	PROPN
ejpam-2803	491	3	(	(	PUNCT
ejpam-2803	491	4	0	0	NUM
ejpam-2803	491	5	)	)	PUNCT
ejpam-2803	491	6	,	,	PUNCT
ejpam-2803	491	7	(	(	PUNCT
ejpam-2803	491	8	0)⊕	0)⊕	NUM
ejpam-2803	491	9	zp	zp	NOUN
ejpam-2803	491	10	}	}	PUNCT
ejpam-2803	491	11	.	.	PUNCT
ejpam-2803	492	1	(	(	PUNCT
ejpam-2803	492	2	c	c	X
ejpam-2803	492	3	)	)	PUNCT
ejpam-2803	492	4	every	every	DET
ejpam-2803	492	5	submodule	submodule	NOUN
ejpam-2803	492	6	of	of	ADP
ejpam-2803	492	7	an	an	DET
ejpam-2803	492	8	xs	xs	NOUN
ejpam-2803	492	9	-	-	PUNCT
ejpam-2803	492	10	injective	injective	ADJ
ejpam-2803	492	11	r	r	NOUN
ejpam-2803	492	12	-	-	PUNCT
ejpam-2803	492	13	module	module	NOUN
ejpam-2803	492	14	is	be	AUX
ejpam-2803	492	15	an	an	DET
ejpam-2803	492	16	xs	xs	NOUN
ejpam-2803	492	17	-	-	PUNCT
ejpam-2803	492	18	injective	injective	ADJ
ejpam-2803	492	19	module	module	NOUN
ejpam-2803	492	20	.	.	PUNCT
ejpam-2803	493	1	the	the	DET
ejpam-2803	493	2	following	follow	VERB
ejpam-2803	493	3	example	example	NOUN
ejpam-2803	493	4	shows	show	VERB
ejpam-2803	493	5	that	that	SCONJ
ejpam-2803	493	6	not	not	PART
ejpam-2803	493	7	every	every	DET
ejpam-2803	493	8	homomorphic	homomorphic	ADJ
ejpam-2803	493	9	image	image	NOUN
ejpam-2803	493	10	of	of	ADP
ejpam-2803	493	11	an	an	DET
ejpam-2803	493	12	xs	xs	NOUN
ejpam-2803	493	13	-	-	PUNCT
ejpam-2803	493	14	injective	injective	ADJ
ejpam-2803	493	15	r	r	NOUN
ejpam-2803	493	16	-	-	PUNCT
ejpam-2803	493	17	module	module	NOUN
ejpam-2803	493	18	is	be	AUX
ejpam-2803	493	19	xs	xs	NOUN
ejpam-2803	493	20	-	-	PUNCT
ejpam-2803	493	21	injective	injective	ADJ
ejpam-2803	493	22	.	.	PUNCT
ejpam-2803	494	1	h.	h.	PROPN
ejpam-2803	494	2	ansari	ansari	PROPN
ejpam-2803	494	3	-	-	PUNCT
ejpam-2803	494	4	toroghy	toroghy	NOUN
ejpam-2803	494	5	,	,	PUNCT
ejpam-2803	494	6	s.	s.	PROPN
ejpam-2803	494	7	s.	s.	PROPN
ejpam-2803	494	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	494	9	/	/	SYM
ejpam-2803	494	10	eur	eur	PROPN
ejpam-2803	494	11	.	.	PUNCT
ejpam-2803	495	1	j.	j.	PROPN
ejpam-2803	495	2	pure	pure	PROPN
ejpam-2803	495	3	appl	appl	PROPN
ejpam-2803	495	4	.	.	PROPN
ejpam-2803	495	5	math	math	PROPN
ejpam-2803	495	6	,	,	PUNCT
ejpam-2803	495	7	10	10	NUM
ejpam-2803	495	8	(	(	PUNCT
ejpam-2803	495	9	2	2	NUM
ejpam-2803	495	10	)	)	PUNCT
ejpam-2803	495	11	(	(	PUNCT
ejpam-2803	495	12	2017	2017	NUM
ejpam-2803	495	13	)	)	PUNCT
ejpam-2803	495	14	,	,	PUNCT
ejpam-2803	495	15	211	211	NUM
ejpam-2803	495	16	-	-	SYM
ejpam-2803	495	17	230	230	NUM
ejpam-2803	495	18	223	223	NUM
ejpam-2803	495	19	example	example	NOUN
ejpam-2803	495	20	3.11	3.11	NUM
ejpam-2803	495	21	.	.	PUNCT
ejpam-2803	496	1	let	let	VERB
ejpam-2803	496	2	p	p	PRON
ejpam-2803	496	3	be	be	AUX
ejpam-2803	496	4	a	a	DET
ejpam-2803	496	5	positive	positive	ADJ
ejpam-2803	496	6	prime	prime	ADJ
ejpam-2803	496	7	integer	integer	NOUN
ejpam-2803	496	8	and	and	CCONJ
ejpam-2803	496	9	let	let	VERB
ejpam-2803	496	10	m	m	VERB
ejpam-2803	496	11	=	=	VERB
ejpam-2803	496	12	q	q	PROPN
ejpam-2803	496	13	⊕	⊕	PROPN
ejpam-2803	496	14	zp	zp	PROPN
ejpam-2803	496	15	.	.	PUNCT
ejpam-2803	497	1	then	then	ADV
ejpam-2803	497	2	m	m	PROPN
ejpam-2803	497	3	is	be	AUX
ejpam-2803	497	4	an	an	DET
ejpam-2803	497	5	xs	xs	NOUN
ejpam-2803	497	6	-	-	PUNCT
ejpam-2803	497	7	injective	injective	ADJ
ejpam-2803	497	8	z	z	NOUN
ejpam-2803	497	9	-	-	PUNCT
ejpam-2803	497	10	module	module	NOUN
ejpam-2803	497	11	but	but	CCONJ
ejpam-2803	497	12	its	its	PRON
ejpam-2803	497	13	homomorphic	homomorphic	ADJ
ejpam-2803	497	14	image	image	NOUN
ejpam-2803	497	15	z∞p	z∞p	NOUN
ejpam-2803	497	16	⊕	⊕	PROPN
ejpam-2803	497	17	zp	zp	PROPN
ejpam-2803	497	18	is	be	AUX
ejpam-2803	497	19	not	not	PART
ejpam-2803	497	20	.	.	PUNCT
ejpam-2803	498	1	note	note	VERB
ejpam-2803	498	2	that	that	SCONJ
ejpam-2803	498	3	specs(z∞p	specs(z∞p	PROPN
ejpam-2803	498	4	⊕	⊕	PROPN
ejpam-2803	498	5	zp	zp	PROPN
ejpam-2803	498	6	)	)	PUNCT
ejpam-2803	498	7	=	=	PRON
ejpam-2803	498	8	{	{	PUNCT
ejpam-2803	498	9	z∞p	z∞p	PROPN
ejpam-2803	498	10	⊕	⊕	PROPN
ejpam-2803	498	11	(	(	PUNCT
ejpam-2803	498	12	0	0	NUM
ejpam-2803	498	13	)	)	PUNCT
ejpam-2803	498	14	,	,	PUNCT
ejpam-2803	498	15	(	(	PUNCT
ejpam-2803	498	16	0)⊕	0)⊕	NUM
ejpam-2803	498	17	zp	zp	NOUN
ejpam-2803	498	18	,	,	PUNCT
ejpam-2803	498	19	<	<	X
ejpam-2803	498	20	1	1	NUM
ejpam-2803	498	21	/	/	SYM
ejpam-2803	498	22	p+	p+	NOUN
ejpam-2803	498	23	z	z	NOUN
ejpam-2803	498	24	>	>	PUNCT
ejpam-2803	498	25	}	}	PUNCT
ejpam-2803	498	26	.	.	PUNCT
ejpam-2803	499	1	remark	remark	PROPN
ejpam-2803	499	2	3.12	3.12	NUM
ejpam-2803	499	3	.	.	PUNCT
ejpam-2803	500	1	let	let	VERB
ejpam-2803	500	2	s	s	PRON
ejpam-2803	500	3	be	be	AUX
ejpam-2803	500	4	a	a	DET
ejpam-2803	500	5	commutative	commutative	ADJ
ejpam-2803	500	6	ring	ring	NOUN
ejpam-2803	500	7	with	with	ADP
ejpam-2803	500	8	identity	identity	NOUN
ejpam-2803	500	9	.	.	PUNCT
ejpam-2803	501	1	s	s	PART
ejpam-2803	501	2	is	be	AUX
ejpam-2803	501	3	said	say	VERB
ejpam-2803	501	4	to	to	PART
ejpam-2803	501	5	be	be	AUX
ejpam-2803	501	6	a	a	DET
ejpam-2803	501	7	perfect	perfect	ADJ
ejpam-2803	501	8	ring	ring	NOUN
ejpam-2803	501	9	if	if	SCONJ
ejpam-2803	501	10	it	it	PRON
ejpam-2803	501	11	satisfies	satisfy	VERB
ejpam-2803	501	12	dcc	dcc	PROPN
ejpam-2803	501	13	on	on	ADP
ejpam-2803	501	14	principal	principal	ADJ
ejpam-2803	501	15	ideals	ideal	NOUN
ejpam-2803	501	16	.	.	PUNCT
ejpam-2803	502	1	clearly	clearly	ADV
ejpam-2803	502	2	,	,	PUNCT
ejpam-2803	502	3	every	every	DET
ejpam-2803	502	4	artinian	artinian	ADJ
ejpam-2803	502	5	ring	ring	NOUN
ejpam-2803	502	6	is	be	AUX
ejpam-2803	502	7	perfect	perfect	ADJ
ejpam-2803	502	8	.	.	PUNCT
ejpam-2803	503	1	note	note	VERB
ejpam-2803	503	2	that	that	SCONJ
ejpam-2803	503	3	if	if	SCONJ
ejpam-2803	503	4	s	s	NOUN
ejpam-2803	503	5	is	be	AUX
ejpam-2803	503	6	a	a	DET
ejpam-2803	503	7	perfect	perfect	ADJ
ejpam-2803	503	8	ring	ring	NOUN
ejpam-2803	503	9	and	and	CCONJ
ejpam-2803	503	10	p	p	NOUN
ejpam-2803	503	11	∈	∈	PROPN
ejpam-2803	503	12	spec(s	spec(s	PROPN
ejpam-2803	503	13	)	)	PUNCT
ejpam-2803	503	14	,	,	PUNCT
ejpam-2803	503	15	then	then	ADV
ejpam-2803	503	16	by	by	ADP
ejpam-2803	503	17	[	[	PUNCT
ejpam-2803	503	18	11	11	NUM
ejpam-2803	503	19	,	,	PUNCT
ejpam-2803	503	20	lemma	lemma	PROPN
ejpam-2803	503	21	2.2	2.2	NUM
ejpam-2803	503	22	]	]	PUNCT
ejpam-2803	503	23	,	,	PUNCT
ejpam-2803	503	24	r	r	X
ejpam-2803	503	25	/	/	SYM
ejpam-2803	503	26	p	p	NOUN
ejpam-2803	503	27	is	be	AUX
ejpam-2803	503	28	a	a	DET
ejpam-2803	503	29	perfect	perfect	ADJ
ejpam-2803	503	30	domain	domain	NOUN
ejpam-2803	503	31	so	so	SCONJ
ejpam-2803	503	32	that	that	SCONJ
ejpam-2803	503	33	it	it	PRON
ejpam-2803	503	34	is	be	AUX
ejpam-2803	503	35	a	a	DET
ejpam-2803	503	36	field	field	NOUN
ejpam-2803	503	37	.	.	PUNCT
ejpam-2803	504	1	hence	hence	ADV
ejpam-2803	504	2	dim(s	dim(s	PROPN
ejpam-2803	504	3	)	)	PUNCT
ejpam-2803	505	1	=	=	SYM
ejpam-2803	505	2	0	0	X
ejpam-2803	505	3	.	.	PUNCT
ejpam-2803	506	1	furthermore	furthermore	ADV
ejpam-2803	506	2	,	,	PUNCT
ejpam-2803	506	3	every	every	DET
ejpam-2803	506	4	perfect	perfect	ADJ
ejpam-2803	506	5	ring	ring	NOUN
ejpam-2803	506	6	is	be	AUX
ejpam-2803	506	7	a	a	DET
ejpam-2803	506	8	semilocal	semilocal	ADJ
ejpam-2803	506	9	ring	ring	NOUN
ejpam-2803	506	10	by	by	ADP
ejpam-2803	506	11	[	[	X
ejpam-2803	506	12	11	11	NUM
ejpam-2803	506	13	,	,	PUNCT
ejpam-2803	506	14	theorem	theorem	VERB
ejpam-2803	506	15	p	p	NOUN
ejpam-2803	506	16	or	or	CCONJ
ejpam-2803	506	17	p.	p.	NOUN
ejpam-2803	506	18	475	475	NUM
ejpam-2803	506	19	,	,	PUNCT
ejpam-2803	506	20	examples	example	NOUN
ejpam-2803	506	21	(	(	PUNCT
ejpam-2803	506	22	6	6	NUM
ejpam-2803	506	23	)	)	PUNCT
ejpam-2803	506	24	]	]	PUNCT
ejpam-2803	506	25	.	.	PUNCT
ejpam-2803	507	1	proposition	proposition	NOUN
ejpam-2803	507	2	3.13	3.13	NUM
ejpam-2803	507	3	.	.	PUNCT
ejpam-2803	508	1	(	(	PUNCT
ejpam-2803	508	2	i	i	NOUN
ejpam-2803	508	3	)	)	PUNCT
ejpam-2803	508	4	if	if	SCONJ
ejpam-2803	508	5	m	m	NOUN
ejpam-2803	508	6	is	be	AUX
ejpam-2803	508	7	a	a	DET
ejpam-2803	508	8	cotop	cotop	NOUN
ejpam-2803	508	9	module	module	NOUN
ejpam-2803	508	10	over	over	ADP
ejpam-2803	508	11	a	a	DET
ejpam-2803	508	12	semilocal	semilocal	ADJ
ejpam-2803	508	13	(	(	PUNCT
ejpam-2803	508	14	for	for	ADP
ejpam-2803	508	15	example	example	NOUN
ejpam-2803	508	16	a	a	DET
ejpam-2803	508	17	perfect	perfect	ADJ
ejpam-2803	508	18	)	)	PUNCT
ejpam-2803	508	19	ring	ring	NOUN
ejpam-2803	508	20	r.	r.	PROPN
ejpam-2803	508	21	then	then	ADV
ejpam-2803	508	22	(	(	PUNCT
ejpam-2803	508	23	0	0	NUM
ejpam-2803	508	24	:	:	PUNCT
ejpam-2803	508	25	m	m	VERB
ejpam-2803	508	26	jac(r	jac(r	ADJ
ejpam-2803	508	27	)	)	PUNCT
ejpam-2803	508	28	)	)	PUNCT
ejpam-2803	508	29	is	be	AUX
ejpam-2803	508	30	cyclic	cyclic	ADJ
ejpam-2803	508	31	.	.	PUNCT
ejpam-2803	509	1	(	(	PUNCT
ejpam-2803	509	2	ii	ii	NOUN
ejpam-2803	509	3	)	)	PUNCT
ejpam-2803	509	4	let	let	VERB
ejpam-2803	509	5	(	(	PUNCT
ejpam-2803	509	6	mi)i∈i	mi)i∈i	NUM
ejpam-2803	509	7	be	be	AUX
ejpam-2803	509	8	a	a	DET
ejpam-2803	509	9	family	family	NOUN
ejpam-2803	509	10	of	of	ADP
ejpam-2803	509	11	r	r	NOUN
ejpam-2803	509	12	-	-	PUNCT
ejpam-2803	509	13	modules	module	NOUN
ejpam-2803	509	14	and	and	CCONJ
ejpam-2803	509	15	let	let	VERB
ejpam-2803	509	16	m	m	NOUN
ejpam-2803	509	17	=	=	PROPN
ejpam-2803	509	18	⊕	⊕	PROPN
ejpam-2803	509	19	i∈imi	i∈imi	PROPN
ejpam-2803	509	20	.	.	PUNCT
ejpam-2803	510	1	if	if	SCONJ
ejpam-2803	510	2	m	m	NOUN
ejpam-2803	510	3	is	be	AUX
ejpam-2803	510	4	an	an	DET
ejpam-2803	510	5	xs	xs	NOUN
ejpam-2803	510	6	-	-	PUNCT
ejpam-2803	510	7	injective	injective	ADJ
ejpam-2803	510	8	module	module	NOUN
ejpam-2803	510	9	,	,	PUNCT
ejpam-2803	510	10	then	then	ADV
ejpam-2803	510	11	specs(m	specs(m	NOUN
ejpam-2803	510	12	)	)	PUNCT
ejpam-2803	510	13	=	=	PRON
ejpam-2803	510	14	{	{	PUNCT
ejpam-2803	511	1	s	s	PROPN
ejpam-2803	511	2	⊕	⊕	PROPN
ejpam-2803	511	3	(	(	PUNCT
ejpam-2803	511	4	⊕	⊕	PROPN
ejpam-2803	511	5	j	j	PROPN
ejpam-2803	511	6	6	6	NUM
ejpam-2803	511	7	=	=	NOUN
ejpam-2803	511	8	i∈i	i∈i	ADJ
ejpam-2803	511	9	(	(	PUNCT
ejpam-2803	511	10	0	0	NUM
ejpam-2803	511	11	)	)	PUNCT
ejpam-2803	511	12	)	)	PUNCT
ejpam-2803	512	1	|	|	ADV
ejpam-2803	512	2	j	j	PROPN
ejpam-2803	512	3	∈	∈	PROPN
ejpam-2803	512	4	i	i	PRON
ejpam-2803	512	5	,	,	PUNCT
ejpam-2803	512	6	s	s	PROPN
ejpam-2803	512	7	∈	∈	PROPN
ejpam-2803	512	8	specs(mj	specs(mj	NOUN
ejpam-2803	512	9	)	)	PUNCT
ejpam-2803	512	10	}	}	PUNCT
ejpam-2803	512	11	.	.	PUNCT
ejpam-2803	513	1	proof	proof	NOUN
ejpam-2803	513	2	.	.	PUNCT
ejpam-2803	514	1	(	(	PUNCT
ejpam-2803	514	2	i	i	NOUN
ejpam-2803	514	3	)	)	PUNCT
ejpam-2803	514	4	suppose	suppose	VERB
ejpam-2803	514	5	that	that	SCONJ
ejpam-2803	514	6	m	m	PROPN
ejpam-2803	514	7	is	be	AUX
ejpam-2803	514	8	a	a	DET
ejpam-2803	514	9	cotop	cotop	NOUN
ejpam-2803	514	10	module	module	NOUN
ejpam-2803	514	11	.	.	PUNCT
ejpam-2803	515	1	let	let	VERB
ejpam-2803	515	2	m1	m1	PROPN
ejpam-2803	515	3	,	,	PUNCT
ejpam-2803	515	4	m2	m2	PROPN
ejpam-2803	515	5	,	,	PUNCT
ejpam-2803	515	6	...	...	PUNCT
ejpam-2803	515	7	,	,	PUNCT
ejpam-2803	515	8	mn	mn	PROPN
ejpam-2803	515	9	denote	denote	VERB
ejpam-2803	515	10	the	the	DET
ejpam-2803	515	11	distinct	distinct	ADJ
ejpam-2803	515	12	maximal	maximal	ADJ
ejpam-2803	515	13	ideals	ideal	NOUN
ejpam-2803	515	14	of	of	ADP
ejpam-2803	515	15	r	r	NOUN
ejpam-2803	515	16	,	,	PUNCT
ejpam-2803	515	17	where	where	SCONJ
ejpam-2803	515	18	n	n	PRON
ejpam-2803	515	19	is	be	AUX
ejpam-2803	515	20	a	a	DET
ejpam-2803	515	21	positive	positive	ADJ
ejpam-2803	515	22	integer	integer	NOUN
ejpam-2803	515	23	.	.	PUNCT
ejpam-2803	516	1	by	by	ADP
ejpam-2803	516	2	[	[	X
ejpam-2803	516	3	7	7	NUM
ejpam-2803	516	4	,	,	PUNCT
ejpam-2803	516	5	corollary	corollary	ADJ
ejpam-2803	516	6	2.6	2.6	NUM
ejpam-2803	516	7	(	(	PUNCT
ejpam-2803	516	8	e	e	NOUN
ejpam-2803	516	9	)	)	PUNCT
ejpam-2803	516	10	]	]	PUNCT
ejpam-2803	516	11	,	,	PUNCT
ejpam-2803	516	12	(	(	PUNCT
ejpam-2803	516	13	0	0	NUM
ejpam-2803	516	14	:	:	PUNCT
ejpam-2803	516	15	m	m	PROPN
ejpam-2803	516	16	mi	mi	NOUN
ejpam-2803	516	17	)	)	PUNCT
ejpam-2803	516	18	is	be	AUX
ejpam-2803	516	19	cyclic	cyclic	ADJ
ejpam-2803	516	20	for	for	ADP
ejpam-2803	516	21	each	each	DET
ejpam-2803	516	22	1	1	NUM
ejpam-2803	516	23	≤	≤	NUM
ejpam-2803	516	24	i	i	PRON
ejpam-2803	516	25	≤	≤	ADJ
ejpam-2803	516	26	n.	n.	NOUN
ejpam-2803	516	27	we	we	PRON
ejpam-2803	516	28	show	show	VERB
ejpam-2803	516	29	that	that	SCONJ
ejpam-2803	516	30	(	(	PUNCT
ejpam-2803	516	31	0	0	NUM
ejpam-2803	516	32	:	:	PUNCT
ejpam-2803	516	33	m	m	VERB
ejpam-2803	516	34	jac(r	jac(r	ADJ
ejpam-2803	516	35	)	)	PUNCT
ejpam-2803	516	36	)	)	PUNCT
ejpam-2803	517	1	=	=	PUNCT
ejpam-2803	517	2	⊕n	⊕n	NOUN
ejpam-2803	517	3	i=1(0	i=1(0	NOUN
ejpam-2803	517	4	:	:	PUNCT
ejpam-2803	517	5	m	m	PROPN
ejpam-2803	517	6	mi	mi	NOUN
ejpam-2803	517	7	)	)	PUNCT
ejpam-2803	517	8	.	.	PUNCT
ejpam-2803	518	1	set	set	VERB
ejpam-2803	518	2	i	i	NOUN
ejpam-2803	518	3	=	=	NOUN
ejpam-2803	518	4	m2	m2	PROPN
ejpam-2803	518	5	∩m3	∩m3	PROPN
ejpam-2803	518	6	∩	∩	PROPN
ejpam-2803	518	7	...	...	PUNCT
ejpam-2803	519	1	∩mn	∩mn	ADJ
ejpam-2803	519	2	,	,	PUNCT
ejpam-2803	519	3	so	so	SCONJ
ejpam-2803	519	4	that	that	SCONJ
ejpam-2803	519	5	jac(r	jac(r	PROPN
ejpam-2803	519	6	)	)	PUNCT
ejpam-2803	519	7	=	=	SYM
ejpam-2803	519	8	m1	m1	PROPN
ejpam-2803	519	9	∩	∩	PROPN
ejpam-2803	519	10	i.	i.	NOUN
ejpam-2803	519	11	note	note	VERB
ejpam-2803	519	12	that	that	SCONJ
ejpam-2803	519	13	m1	m1	PROPN
ejpam-2803	520	1	+	+	CCONJ
ejpam-2803	520	2	i	i	NOUN
ejpam-2803	520	3	=	=	NOUN
ejpam-2803	520	4	r	r	NOUN
ejpam-2803	520	5	implies	imply	VERB
ejpam-2803	520	6	that	that	SCONJ
ejpam-2803	520	7	0	0	X
ejpam-2803	520	8	=	=	SYM
ejpam-2803	520	9	(	(	PUNCT
ejpam-2803	520	10	0	0	NUM
ejpam-2803	520	11	:	:	PUNCT
ejpam-2803	520	12	m	m	VERB
ejpam-2803	520	13	r	r	NOUN
ejpam-2803	520	14	)	)	PUNCT
ejpam-2803	520	15	=	=	SYM
ejpam-2803	521	1	(	(	PUNCT
ejpam-2803	521	2	0	0	NUM
ejpam-2803	521	3	:	:	PUNCT
ejpam-2803	521	4	m	m	VERB
ejpam-2803	521	5	m1	m1	PROPN
ejpam-2803	521	6	+	+	CCONJ
ejpam-2803	521	7	i	i	NOUN
ejpam-2803	521	8	)	)	PUNCT
ejpam-2803	522	1	=	=	PUNCT
ejpam-2803	522	2	(	(	PUNCT
ejpam-2803	522	3	0	0	NUM
ejpam-2803	522	4	:	:	PUNCT
ejpam-2803	522	5	m	m	VERB
ejpam-2803	522	6	m1	m1	NOUN
ejpam-2803	522	7	)	)	PUNCT
ejpam-2803	522	8	∩	∩	NOUN
ejpam-2803	522	9	(	(	PUNCT
ejpam-2803	522	10	0	0	NUM
ejpam-2803	522	11	:	:	PUNCT
ejpam-2803	522	12	m	m	VERB
ejpam-2803	522	13	i	i	NOUN
ejpam-2803	522	14	)	)	PUNCT
ejpam-2803	522	15	.	.	PUNCT
ejpam-2803	523	1	moreover	moreover	ADV
ejpam-2803	523	2	,	,	PUNCT
ejpam-2803	523	3	(	(	PUNCT
ejpam-2803	523	4	0	0	NUM
ejpam-2803	523	5	:	:	PUNCT
ejpam-2803	523	6	m	m	NOUN
ejpam-2803	523	7	m1	m1	NOUN
ejpam-2803	523	8	)	)	PUNCT
ejpam-2803	524	1	+	+	CCONJ
ejpam-2803	524	2	(	(	PUNCT
ejpam-2803	524	3	0	0	NUM
ejpam-2803	524	4	:	:	PUNCT
ejpam-2803	524	5	m	m	VERB
ejpam-2803	524	6	i	i	NOUN
ejpam-2803	524	7	)	)	PUNCT
ejpam-2803	525	1	=	=	SYM
ejpam-2803	525	2	(	(	PUNCT
ejpam-2803	525	3	(	(	PUNCT
ejpam-2803	525	4	0	0	NUM
ejpam-2803	525	5	:	:	PUNCT
ejpam-2803	525	6	m	m	NOUN
ejpam-2803	525	7	m1	m1	NOUN
ejpam-2803	525	8	)	)	PUNCT
ejpam-2803	526	1	+	+	CCONJ
ejpam-2803	526	2	(	(	PUNCT
ejpam-2803	526	3	0	0	NUM
ejpam-2803	526	4	:	:	PUNCT
ejpam-2803	526	5	m	m	VERB
ejpam-2803	526	6	i	i	NOUN
ejpam-2803	526	7	)	)	PUNCT
ejpam-2803	526	8	:	:	PUNCT
ejpam-2803	527	1	m	m	AUX
ejpam-2803	527	2	m1	m1	PROPN
ejpam-2803	527	3	+	+	CCONJ
ejpam-2803	527	4	i	i	PROPN
ejpam-2803	527	5	)	)	PUNCT
ejpam-2803	527	6	⊇	⊇	NOUN
ejpam-2803	527	7	(	(	PUNCT
ejpam-2803	527	8	(	(	PUNCT
ejpam-2803	527	9	0	0	NUM
ejpam-2803	527	10	:	:	PUNCT
ejpam-2803	527	11	m	m	NOUN
ejpam-2803	527	12	m1	m1	NOUN
ejpam-2803	527	13	)	)	PUNCT
ejpam-2803	527	14	:	:	PUNCT
ejpam-2803	527	15	m	m	VERB
ejpam-2803	527	16	i	i	NOUN
ejpam-2803	527	17	)	)	PUNCT
ejpam-2803	527	18	∩	∩	NOUN
ejpam-2803	527	19	(	(	PUNCT
ejpam-2803	527	20	(	(	PUNCT
ejpam-2803	527	21	0	0	NUM
ejpam-2803	527	22	:	:	PUNCT
ejpam-2803	527	23	m	m	VERB
ejpam-2803	527	24	i	i	NOUN
ejpam-2803	527	25	)	)	PUNCT
ejpam-2803	527	26	:	:	PUNCT
ejpam-2803	527	27	m	m	VERB
ejpam-2803	527	28	m1	m1	NOUN
ejpam-2803	527	29	)	)	PUNCT
ejpam-2803	527	30	=	=	PUNCT
ejpam-2803	527	31	(	(	PUNCT
ejpam-2803	527	32	0	0	NUM
ejpam-2803	527	33	:	:	PUNCT
ejpam-2803	527	34	m	m	VERB
ejpam-2803	527	35	m1i	m1i	X
ejpam-2803	527	36	)	)	PUNCT
ejpam-2803	527	37	=	=	SYM
ejpam-2803	527	38	(	(	PUNCT
ejpam-2803	527	39	0	0	NUM
ejpam-2803	527	40	:	:	PUNCT
ejpam-2803	527	41	m	m	VERB
ejpam-2803	527	42	jac(r	jac(r	ADJ
ejpam-2803	527	43	)	)	PUNCT
ejpam-2803	527	44	)	)	PUNCT
ejpam-2803	527	45	⊇	⊇	NOUN
ejpam-2803	527	46	(	(	PUNCT
ejpam-2803	527	47	0	0	NUM
ejpam-2803	527	48	:	:	PUNCT
ejpam-2803	527	49	m	m	NOUN
ejpam-2803	527	50	m1	m1	NOUN
ejpam-2803	527	51	)	)	PUNCT
ejpam-2803	528	1	+	+	CCONJ
ejpam-2803	528	2	(	(	PUNCT
ejpam-2803	528	3	0	0	NUM
ejpam-2803	528	4	:	:	PUNCT
ejpam-2803	528	5	m	m	VERB
ejpam-2803	528	6	i	i	NOUN
ejpam-2803	528	7	)	)	PUNCT
ejpam-2803	528	8	.	.	PUNCT
ejpam-2803	529	1	therefore	therefore	ADV
ejpam-2803	529	2	(	(	PUNCT
ejpam-2803	529	3	0	0	NUM
ejpam-2803	529	4	:	:	PUNCT
ejpam-2803	529	5	m	m	NOUN
ejpam-2803	529	6	m1	m1	NOUN
ejpam-2803	529	7	)	)	PUNCT
ejpam-2803	530	1	+	+	CCONJ
ejpam-2803	530	2	(	(	PUNCT
ejpam-2803	530	3	0	0	NUM
ejpam-2803	530	4	:	:	PUNCT
ejpam-2803	530	5	m	m	VERB
ejpam-2803	530	6	i	i	NOUN
ejpam-2803	530	7	)	)	PUNCT
ejpam-2803	530	8	=	=	PUNCT
ejpam-2803	531	1	(	(	PUNCT
ejpam-2803	531	2	0	0	NUM
ejpam-2803	531	3	:	:	PUNCT
ejpam-2803	531	4	m	m	VERB
ejpam-2803	531	5	jac(r	jac(r	ADJ
ejpam-2803	531	6	)	)	PUNCT
ejpam-2803	531	7	)	)	PUNCT
ejpam-2803	531	8	.	.	PUNCT
ejpam-2803	532	1	by	by	ADP
ejpam-2803	532	2	induction	induction	NOUN
ejpam-2803	532	3	,	,	PUNCT
ejpam-2803	532	4	this	this	PRON
ejpam-2803	532	5	implies	imply	VERB
ejpam-2803	532	6	that	that	SCONJ
ejpam-2803	532	7	(	(	PUNCT
ejpam-2803	532	8	0	0	NUM
ejpam-2803	532	9	:	:	PUNCT
ejpam-2803	532	10	m	m	VERB
ejpam-2803	532	11	jac(r	jac(r	ADJ
ejpam-2803	532	12	)	)	PUNCT
ejpam-2803	532	13	)	)	PUNCT
ejpam-2803	533	1	=	=	PUNCT
ejpam-2803	533	2	⊕n	⊕n	NOUN
ejpam-2803	533	3	i=1(0	i=1(0	NOUN
ejpam-2803	533	4	:	:	PUNCT
ejpam-2803	533	5	m	m	PROPN
ejpam-2803	533	6	mi	mi	PROPN
ejpam-2803	533	7	)	)	PUNCT
ejpam-2803	533	8	.	.	PUNCT
ejpam-2803	534	1	without	without	ADP
ejpam-2803	534	2	loss	loss	NOUN
ejpam-2803	534	3	of	of	ADP
ejpam-2803	534	4	generality	generality	NOUN
ejpam-2803	534	5	,	,	PUNCT
ejpam-2803	534	6	we	we	PRON
ejpam-2803	534	7	may	may	AUX
ejpam-2803	534	8	assume	assume	VERB
ejpam-2803	534	9	that	that	SCONJ
ejpam-2803	534	10	(	(	PUNCT
ejpam-2803	534	11	0	0	NUM
ejpam-2803	534	12	:	:	PUNCT
ejpam-2803	534	13	m	m	PROPN
ejpam-2803	534	14	mi	mi	PROPN
ejpam-2803	534	15	)	)	PUNCT
ejpam-2803	534	16	6=	6=	ADP
ejpam-2803	534	17	0	0	NUM
ejpam-2803	534	18	.	.	PUNCT
ejpam-2803	535	1	thus	thus	ADV
ejpam-2803	535	2	we	we	PRON
ejpam-2803	535	3	have	have	VERB
ejpam-2803	535	4	(	(	PUNCT
ejpam-2803	535	5	0	0	NUM
ejpam-2803	535	6	:	:	PUNCT
ejpam-2803	535	7	m	m	VERB
ejpam-2803	535	8	jac(r	jac(r	ADJ
ejpam-2803	535	9	)	)	PUNCT
ejpam-2803	535	10	)	)	PUNCT
ejpam-2803	536	1	=	=	SYM
ejpam-2803	536	2	n⊕	n⊕	PROPN
ejpam-2803	536	3	i=1	i=1	PROPN
ejpam-2803	537	1	(	(	PUNCT
ejpam-2803	537	2	0	0	NUM
ejpam-2803	537	3	:	:	PUNCT
ejpam-2803	537	4	m	m	PROPN
ejpam-2803	537	5	mi	mi	ADJ
ejpam-2803	537	6	)	)	PUNCT
ejpam-2803	537	7	=	=	SYM
ejpam-2803	537	8	n⊕	n⊕	NOUN
ejpam-2803	537	9	i=1	i=1	X
ejpam-2803	537	10	r	r	NOUN
ejpam-2803	537	11	/	/	SYM
ejpam-2803	537	12	mi	mi	NOUN
ejpam-2803	537	13	∼=	∼=	PROPN
ejpam-2803	537	14	r	r	PROPN
ejpam-2803	537	15	/	/	SYM
ejpam-2803	537	16	jac(r	jac(r	PROPN
ejpam-2803	537	17	)	)	PUNCT
ejpam-2803	537	18	.	.	PUNCT
ejpam-2803	538	1	hence	hence	ADV
ejpam-2803	538	2	(	(	PUNCT
ejpam-2803	538	3	0	0	NUM
ejpam-2803	538	4	:	:	PUNCT
ejpam-2803	538	5	m	m	VERB
ejpam-2803	538	6	jac(r	jac(r	ADJ
ejpam-2803	538	7	)	)	PUNCT
ejpam-2803	538	8	)	)	PUNCT
ejpam-2803	538	9	is	be	AUX
ejpam-2803	538	10	cyclic	cyclic	ADJ
ejpam-2803	538	11	.	.	PUNCT
ejpam-2803	539	1	(	(	PUNCT
ejpam-2803	539	2	ii	ii	X
ejpam-2803	539	3	)	)	PUNCT
ejpam-2803	539	4	the	the	DET
ejpam-2803	539	5	inclusion	inclusion	NOUN
ejpam-2803	539	6	⊇	⊇	NOUN
ejpam-2803	539	7	is	be	AUX
ejpam-2803	539	8	trivial	trivial	ADJ
ejpam-2803	539	9	.	.	PUNCT
ejpam-2803	540	1	conversely	conversely	ADV
ejpam-2803	540	2	,	,	PUNCT
ejpam-2803	540	3	let	let	VERB
ejpam-2803	540	4	s	s	PRON
ejpam-2803	540	5	be	be	AUX
ejpam-2803	540	6	a	a	DET
ejpam-2803	540	7	p	p	ADJ
ejpam-2803	540	8	-	-	PUNCT
ejpam-2803	540	9	second	second	NOUN
ejpam-2803	540	10	submodule	submodule	NOUN
ejpam-2803	540	11	of	of	ADP
ejpam-2803	540	12	m	m	PROPN
ejpam-2803	540	13	.	.	PUNCT
ejpam-2803	541	1	then	then	ADV
ejpam-2803	541	2	there	there	PRON
ejpam-2803	541	3	exists	exist	VERB
ejpam-2803	541	4	j	j	PROPN
ejpam-2803	541	5	∈	∈	PROPN
ejpam-2803	541	6	i	i	PRON
ejpam-2803	541	7	such	such	VERB
ejpam-2803	542	1	that	that	PRON
ejpam-2803	542	2	s	s	VERB
ejpam-2803	542	3	*	*	X
ejpam-2803	542	4	(	(	PUNCT
ejpam-2803	542	5	0	0	NUM
ejpam-2803	542	6	)	)	PUNCT
ejpam-2803	542	7	⊕	⊕	PROPN
ejpam-2803	542	8	(	(	PUNCT
ejpam-2803	542	9	⊕	⊕	NOUN
ejpam-2803	542	10	j	j	PROPN
ejpam-2803	542	11	6	6	NUM
ejpam-2803	542	12	=	=	NUM
ejpam-2803	542	13	i∈i(mi	i∈i(mi	NUM
ejpam-2803	542	14	)	)	PUNCT
ejpam-2803	542	15	)	)	PUNCT
ejpam-2803	542	16	.	.	PUNCT
ejpam-2803	543	1	by	by	ADP
ejpam-2803	543	2	[	[	X
ejpam-2803	543	3	7	7	NUM
ejpam-2803	543	4	,	,	PUNCT
ejpam-2803	543	5	lemma	lemma	PROPN
ejpam-2803	543	6	2.2	2.2	NUM
ejpam-2803	543	7	(	(	PUNCT
ejpam-2803	543	8	b	b	NOUN
ejpam-2803	543	9	)	)	PUNCT
ejpam-2803	543	10	]	]	PUNCT
ejpam-2803	543	11	,	,	PUNCT
ejpam-2803	543	12	s+((0)⊕	s+((0)⊕	PROPN
ejpam-2803	543	13	(	(	PUNCT
ejpam-2803	543	14	⊕	⊕	PROPN
ejpam-2803	543	15	j	j	PROPN
ejpam-2803	543	16	6	6	NUM
ejpam-2803	543	17	=	=	NUM
ejpam-2803	543	18	i∈i(mi	i∈i(mi	NUM
ejpam-2803	543	19	)	)	PUNCT
ejpam-2803	543	20	)	)	PUNCT
ejpam-2803	543	21	)	)	PUNCT
ejpam-2803	543	22	(	(	PUNCT
ejpam-2803	543	23	0)⊕	0)⊕	NUM
ejpam-2803	543	24	(	(	PUNCT
ejpam-2803	543	25	⊕	⊕	PROPN
ejpam-2803	543	26	j	j	PROPN
ejpam-2803	543	27	6	6	NUM
ejpam-2803	543	28	=	=	NUM
ejpam-2803	543	29	i∈i(mi	i∈i(mi	NUM
ejpam-2803	543	30	)	)	PUNCT
ejpam-2803	543	31	)	)	PUNCT
ejpam-2803	543	32	∈	∈	PROPN
ejpam-2803	543	33	specsp(mj	specsp(mj	NOUN
ejpam-2803	543	34	)	)	PUNCT
ejpam-2803	543	35	.	.	PUNCT
ejpam-2803	544	1	this	this	PRON
ejpam-2803	544	2	implies	imply	VERB
ejpam-2803	544	3	that	that	PRON
ejpam-2803	544	4	s	s	VERB
ejpam-2803	544	5	=	=	X
ejpam-2803	544	6	sj	sj	PROPN
ejpam-2803	544	7	⊕	⊕	PROPN
ejpam-2803	544	8	(	(	PUNCT
ejpam-2803	544	9	⊕	⊕	PROPN
ejpam-2803	544	10	j	j	PROPN
ejpam-2803	544	11	6	6	NUM
ejpam-2803	544	12	=	=	NOUN
ejpam-2803	544	13	i∈i(0	i∈i(0	NUM
ejpam-2803	544	14	)	)	PUNCT
ejpam-2803	544	15	)	)	PUNCT
ejpam-2803	544	16	,	,	PUNCT
ejpam-2803	544	17	where	where	SCONJ
ejpam-2803	544	18	sj	sj	PROPN
ejpam-2803	544	19	∈	∈	PROPN
ejpam-2803	544	20	specs(mj	specs(mj	PROPN
ejpam-2803	544	21	)	)	PUNCT
ejpam-2803	544	22	.	.	PUNCT
ejpam-2803	545	1	h.	h.	PROPN
ejpam-2803	545	2	ansari	ansari	PROPN
ejpam-2803	545	3	-	-	PUNCT
ejpam-2803	545	4	toroghy	toroghy	NOUN
ejpam-2803	545	5	,	,	PUNCT
ejpam-2803	545	6	s.	s.	PROPN
ejpam-2803	545	7	s.	s.	PROPN
ejpam-2803	545	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	545	9	/	/	SYM
ejpam-2803	545	10	eur	eur	PROPN
ejpam-2803	545	11	.	.	PUNCT
ejpam-2803	546	1	j.	j.	PROPN
ejpam-2803	546	2	pure	pure	PROPN
ejpam-2803	546	3	appl	appl	PROPN
ejpam-2803	546	4	.	.	PROPN
ejpam-2803	546	5	math	math	PROPN
ejpam-2803	546	6	,	,	PUNCT
ejpam-2803	546	7	10	10	NUM
ejpam-2803	546	8	(	(	PUNCT
ejpam-2803	546	9	2	2	NUM
ejpam-2803	546	10	)	)	PUNCT
ejpam-2803	546	11	(	(	PUNCT
ejpam-2803	546	12	2017	2017	NUM
ejpam-2803	546	13	)	)	PUNCT
ejpam-2803	546	14	,	,	PUNCT
ejpam-2803	546	15	211	211	NUM
ejpam-2803	546	16	-	-	SYM
ejpam-2803	546	17	230	230	NUM
ejpam-2803	546	18	224	224	NUM
ejpam-2803	546	19	definition	definition	NOUN
ejpam-2803	546	20	3.14	3.14	NUM
ejpam-2803	546	21	.	.	PUNCT
ejpam-2803	547	1	a	a	DET
ejpam-2803	547	2	family	family	NOUN
ejpam-2803	547	3	(	(	PUNCT
ejpam-2803	547	4	mi)i∈i	mi)i∈i	NUM
ejpam-2803	547	5	of	of	ADP
ejpam-2803	547	6	r	r	NOUN
ejpam-2803	547	7	-	-	PUNCT
ejpam-2803	547	8	modules	module	NOUN
ejpam-2803	547	9	is	be	AUX
ejpam-2803	547	10	said	say	VERB
ejpam-2803	547	11	to	to	PART
ejpam-2803	547	12	be	be	AUX
ejpam-2803	547	13	second	second	ADV
ejpam-2803	547	14	-	-	PUNCT
ejpam-2803	547	15	compatible	compatible	ADJ
ejpam-2803	547	16	if	if	SCONJ
ejpam-2803	547	17	for	for	ADP
ejpam-2803	547	18	all	all	DET
ejpam-2803	547	19	i	i	PRON
ejpam-2803	547	20	6=	6=	PROPN
ejpam-2803	547	21	j	j	PROPN
ejpam-2803	547	22	in	in	ADP
ejpam-2803	547	23	i	i	PRON
ejpam-2803	547	24	,	,	PUNCT
ejpam-2803	547	25	there	there	PRON
ejpam-2803	547	26	does	do	AUX
ejpam-2803	547	27	n’t	not	PART
ejpam-2803	547	28	exist	exist	VERB
ejpam-2803	547	29	a	a	DET
ejpam-2803	547	30	prime	prime	ADJ
ejpam-2803	547	31	ideal	ideal	NOUN
ejpam-2803	547	32	p	p	NOUN
ejpam-2803	547	33	in	in	ADP
ejpam-2803	547	34	r	r	NOUN
ejpam-2803	547	35	with	with	ADP
ejpam-2803	547	36	specsp(mi	specsp(mi	PROPN
ejpam-2803	547	37	)	)	PUNCT
ejpam-2803	547	38	and	and	CCONJ
ejpam-2803	547	39	specsp(mj	specsp(mj	NOUN
ejpam-2803	547	40	)	)	PUNCT
ejpam-2803	547	41	both	both	CCONJ
ejpam-2803	547	42	nonempty	nonempty	ADJ
ejpam-2803	547	43	.	.	PUNCT
ejpam-2803	548	1	theorem	theorem	VERB
ejpam-2803	548	2	3.15	3.15	NUM
ejpam-2803	548	3	.	.	PUNCT
ejpam-2803	549	1	let	let	VERB
ejpam-2803	549	2	(	(	PUNCT
ejpam-2803	549	3	mi)i∈i	mi)i∈i	NUM
ejpam-2803	549	4	be	be	AUX
ejpam-2803	549	5	a	a	DET
ejpam-2803	549	6	family	family	NOUN
ejpam-2803	549	7	of	of	ADP
ejpam-2803	549	8	r	r	NOUN
ejpam-2803	549	9	-	-	PUNCT
ejpam-2803	549	10	modules	module	NOUN
ejpam-2803	549	11	and	and	CCONJ
ejpam-2803	549	12	let	let	VERB
ejpam-2803	549	13	m	m	NOUN
ejpam-2803	549	14	=	=	PROPN
ejpam-2803	549	15	⊕	⊕	PROPN
ejpam-2803	549	16	i∈imi	i∈imi	PROPN
ejpam-2803	549	17	.	.	PROPN
ejpam-2803	550	1	then	then	ADV
ejpam-2803	550	2	m	m	PROPN
ejpam-2803	550	3	is	be	AUX
ejpam-2803	550	4	an	an	DET
ejpam-2803	550	5	xs	xs	NOUN
ejpam-2803	550	6	-	-	PUNCT
ejpam-2803	550	7	injective	injective	ADJ
ejpam-2803	550	8	(	(	PUNCT
ejpam-2803	550	9	resp	resp	NOUN
ejpam-2803	550	10	.	.	PUNCT
ejpam-2803	551	1	a	a	DET
ejpam-2803	551	2	cotop	cotop	NOUN
ejpam-2803	551	3	)	)	PUNCT
ejpam-2803	551	4	r	r	NOUN
ejpam-2803	551	5	-	-	PUNCT
ejpam-2803	551	6	module	module	NOUN
ejpam-2803	551	7	if	if	SCONJ
ejpam-2803	551	8	and	and	CCONJ
ejpam-2803	551	9	only	only	ADV
ejpam-2803	551	10	if	if	SCONJ
ejpam-2803	551	11	(	(	PUNCT
ejpam-2803	551	12	mi)i∈i	mi)i∈i	NUM
ejpam-2803	551	13	is	be	AUX
ejpam-2803	551	14	a	a	DET
ejpam-2803	551	15	family	family	NOUN
ejpam-2803	551	16	of	of	ADP
ejpam-2803	551	17	secondcompatible	secondcompatible	ADJ
ejpam-2803	551	18	xs	xs	PROPN
ejpam-2803	551	19	-	-	PUNCT
ejpam-2803	551	20	injective	injective	ADJ
ejpam-2803	551	21	(	(	PUNCT
ejpam-2803	551	22	resp	resp	NOUN
ejpam-2803	551	23	.	.	PUNCT
ejpam-2803	552	1	cotop	cotop	NOUN
ejpam-2803	552	2	)	)	PUNCT
ejpam-2803	552	3	r	r	NOUN
ejpam-2803	552	4	-	-	PUNCT
ejpam-2803	552	5	modules	module	NOUN
ejpam-2803	552	6	.	.	PUNCT
ejpam-2803	553	1	proof	proof	NOUN
ejpam-2803	553	2	.	.	PUNCT
ejpam-2803	554	1	we	we	PRON
ejpam-2803	554	2	consider	consider	VERB
ejpam-2803	554	3	the	the	DET
ejpam-2803	554	4	proof	proof	NOUN
ejpam-2803	554	5	for	for	ADP
ejpam-2803	554	6	two	two	NUM
ejpam-2803	554	7	cases	case	NOUN
ejpam-2803	554	8	xs	xs	NOUN
ejpam-2803	554	9	-	-	PUNCT
ejpam-2803	554	10	injective	injective	ADJ
ejpam-2803	554	11	and	and	CCONJ
ejpam-2803	554	12	cotop	cotop	NOUN
ejpam-2803	554	13	modules	module	NOUN
ejpam-2803	554	14	.	.	PUNCT
ejpam-2803	555	1	case(i	case(i	PROPN
ejpam-2803	555	2	)	)	PUNCT
ejpam-2803	555	3	.	.	PUNCT
ejpam-2803	556	1	xs	xs	PROPN
ejpam-2803	556	2	-	-	PUNCT
ejpam-2803	556	3	injective	injective	ADJ
ejpam-2803	556	4	modules	module	NOUN
ejpam-2803	556	5	.	.	PUNCT
ejpam-2803	557	1	(	(	PUNCT
ejpam-2803	557	2	⇒	⇒	PROPN
ejpam-2803	557	3	)	)	PUNCT
ejpam-2803	557	4	.	.	PUNCT
ejpam-2803	558	1	since	since	SCONJ
ejpam-2803	558	2	every	every	DET
ejpam-2803	558	3	submodule	submodule	NOUN
ejpam-2803	558	4	of	of	ADP
ejpam-2803	558	5	an	an	DET
ejpam-2803	558	6	xs	xs	NOUN
ejpam-2803	558	7	-	-	PUNCT
ejpam-2803	558	8	injective	injective	ADJ
ejpam-2803	558	9	module	module	NOUN
ejpam-2803	558	10	is	be	AUX
ejpam-2803	558	11	xs	xs	NOUN
ejpam-2803	558	12	-	-	PUNCT
ejpam-2803	558	13	injective	injective	ADJ
ejpam-2803	558	14	,	,	PUNCT
ejpam-2803	558	15	mi	mi	PROPN
ejpam-2803	558	16	’s	’s	PART
ejpam-2803	558	17	are	be	AUX
ejpam-2803	558	18	xsinjective	xsinjective	ADJ
ejpam-2803	558	19	.	.	PUNCT
ejpam-2803	559	1	let	let	VERB
ejpam-2803	559	2	i	i	PRON
ejpam-2803	559	3	6=	6=	ADP
ejpam-2803	559	4	j	j	PROPN
ejpam-2803	559	5	and	and	CCONJ
ejpam-2803	559	6	let	let	VERB
ejpam-2803	559	7	s	s	PRON
ejpam-2803	559	8	∈	∈	PROPN
ejpam-2803	559	9	specsp(mi	specsp(mi	NOUN
ejpam-2803	559	10	)	)	PUNCT
ejpam-2803	559	11	andk	andk	NOUN
ejpam-2803	559	12	∈	∈	PROPN
ejpam-2803	559	13	specsp(mj	specsp(mj	NOUN
ejpam-2803	559	14	)	)	PUNCT
ejpam-2803	559	15	.	.	PUNCT
ejpam-2803	560	1	then	then	ADV
ejpam-2803	560	2	s⊕	s⊕	NOUN
ejpam-2803	560	3	(	(	PUNCT
ejpam-2803	560	4	⊕	⊕	NOUN
ejpam-2803	560	5	i	i	NOUN
ejpam-2803	560	6	6	6	NUM
ejpam-2803	560	7	=	=	X
ejpam-2803	560	8	k∈i(0)),k⊕	k∈i(0)),k⊕	X
ejpam-2803	560	9	(	(	PUNCT
ejpam-2803	560	10	⊕	⊕	NOUN
ejpam-2803	560	11	j	j	PROPN
ejpam-2803	560	12	6	6	NUM
ejpam-2803	560	13	=	=	NOUN
ejpam-2803	560	14	k∈i(0	k∈i(0	NOUN
ejpam-2803	560	15	)	)	PUNCT
ejpam-2803	560	16	)	)	PUNCT
ejpam-2803	560	17	∈	∈	PROPN
ejpam-2803	560	18	specsp(m	specsp(m	NOUN
ejpam-2803	560	19	)	)	PUNCT
ejpam-2803	560	20	.	.	PUNCT
ejpam-2803	561	1	since	since	SCONJ
ejpam-2803	561	2	m	m	PROPN
ejpam-2803	561	3	is	be	AUX
ejpam-2803	561	4	xs	xs	NOUN
ejpam-2803	561	5	-	-	PUNCT
ejpam-2803	561	6	injective	injective	ADJ
ejpam-2803	561	7	,	,	PUNCT
ejpam-2803	561	8	s	s	PROPN
ejpam-2803	561	9	⊕	⊕	PROPN
ejpam-2803	561	10	(	(	PUNCT
ejpam-2803	561	11	⊕	⊕	NOUN
ejpam-2803	561	12	i	i	PRON
ejpam-2803	561	13	6	6	NUM
ejpam-2803	561	14	=	=	SYM
ejpam-2803	561	15	j∈i(0	j∈i(0	NOUN
ejpam-2803	561	16	)	)	PUNCT
ejpam-2803	561	17	)	)	PUNCT
ejpam-2803	562	1	=	=	SYM
ejpam-2803	562	2	k	k	PROPN
ejpam-2803	562	3	⊕	⊕	PROPN
ejpam-2803	562	4	(	(	PUNCT
ejpam-2803	562	5	⊕	⊕	PROPN
ejpam-2803	562	6	j	j	PROPN
ejpam-2803	562	7	6	6	NUM
ejpam-2803	562	8	=	=	NOUN
ejpam-2803	562	9	i∈i(0	i∈i(0	NUM
ejpam-2803	562	10	)	)	PUNCT
ejpam-2803	562	11	)	)	PUNCT
ejpam-2803	562	12	,	,	PUNCT
ejpam-2803	562	13	a	a	DET
ejpam-2803	562	14	contradiction	contradiction	NOUN
ejpam-2803	562	15	.	.	PUNCT
ejpam-2803	563	1	hence	hence	ADV
ejpam-2803	563	2	mi	mi	PROPN
ejpam-2803	563	3	’s	’s	PART
ejpam-2803	563	4	are	be	AUX
ejpam-2803	563	5	second	second	ADV
ejpam-2803	563	6	-	-	PUNCT
ejpam-2803	563	7	compatible	compatible	ADJ
ejpam-2803	563	8	.	.	PUNCT
ejpam-2803	564	1	(	(	PUNCT
ejpam-2803	564	2	⇐	⇐	NOUN
ejpam-2803	564	3	)	)	PUNCT
ejpam-2803	564	4	.	.	PUNCT
ejpam-2803	565	1	let	let	VERB
ejpam-2803	565	2	s	s	PRON
ejpam-2803	565	3	be	be	AUX
ejpam-2803	565	4	a	a	DET
ejpam-2803	565	5	second	second	ADJ
ejpam-2803	565	6	submodule	submodule	NOUN
ejpam-2803	565	7	of	of	ADP
ejpam-2803	565	8	m	m	PRON
ejpam-2803	565	9	and	and	CCONJ
ejpam-2803	565	10	let	let	VERB
ejpam-2803	565	11	p	p	NOUN
ejpam-2803	565	12	=	=	PROPN
ejpam-2803	565	13	annr(s	annr(s	PROPN
ejpam-2803	565	14	)	)	PUNCT
ejpam-2803	565	15	.	.	PUNCT
ejpam-2803	566	1	since	since	SCONJ
ejpam-2803	566	2	s	s	PRON
ejpam-2803	566	3	6=	6=	NUM
ejpam-2803	566	4	0	0	NUM
ejpam-2803	566	5	,	,	PUNCT
ejpam-2803	566	6	it	it	PRON
ejpam-2803	566	7	follows	follow	VERB
ejpam-2803	566	8	that	that	SCONJ
ejpam-2803	566	9	there	there	PRON
ejpam-2803	566	10	exists	exist	VERB
ejpam-2803	566	11	j	j	PROPN
ejpam-2803	566	12	∈	∈	PROPN
ejpam-2803	566	13	i	i	PRON
ejpam-2803	566	14	with	with	ADP
ejpam-2803	566	15	s	s	PRON
ejpam-2803	566	16	*	*	PUNCT
ejpam-2803	566	17	(	(	PUNCT
ejpam-2803	566	18	⊕	⊕	NOUN
ejpam-2803	566	19	j	j	PROPN
ejpam-2803	566	20	6	6	NUM
ejpam-2803	566	21	=	=	SYM
ejpam-2803	566	22	i∈imi	i∈imi	PROPN
ejpam-2803	566	23	)	)	PUNCT
ejpam-2803	566	24	⊕	⊕	PROPN
ejpam-2803	566	25	(	(	PUNCT
ejpam-2803	566	26	0	0	NUM
ejpam-2803	566	27	)	)	PUNCT
ejpam-2803	566	28	and	and	CCONJ
ejpam-2803	566	29	so	so	ADV
ejpam-2803	566	30	s+	s+	ADV
ejpam-2803	566	31	(	(	PUNCT
ejpam-2803	566	32	(	(	PUNCT
ejpam-2803	566	33	⊕	⊕	NOUN
ejpam-2803	566	34	j	j	NOUN
ejpam-2803	567	1	6	6	NUM
ejpam-2803	567	2	=	=	NOUN
ejpam-2803	567	3	i∈i	i∈i	ADJ
ejpam-2803	567	4	mi)⊕(0	mi)⊕(0	NOUN
ejpam-2803	567	5	)	)	PUNCT
ejpam-2803	567	6	)	)	PUNCT
ejpam-2803	568	1	(	(	PUNCT
ejpam-2803	568	2	⊕	⊕	NOUN
ejpam-2803	568	3	j	j	PROPN
ejpam-2803	568	4	6	6	NUM
ejpam-2803	568	5	=	=	NOUN
ejpam-2803	568	6	i∈i	i∈i	ADJ
ejpam-2803	568	7	mi)⊕(0	mi)⊕(0	NOUN
ejpam-2803	568	8	)	)	PUNCT
ejpam-2803	568	9	∈	∈	PROPN
ejpam-2803	568	10	specsp(mj	specsp(mj	NOUN
ejpam-2803	568	11	)	)	PUNCT
ejpam-2803	568	12	.	.	PUNCT
ejpam-2803	569	1	by	by	ADP
ejpam-2803	569	2	hypothesis	hypothesis	NOUN
ejpam-2803	569	3	,	,	PUNCT
ejpam-2803	569	4	specsp(mi	specsp(mi	PROPN
ejpam-2803	569	5	)	)	PUNCT
ejpam-2803	569	6	is	be	AUX
ejpam-2803	569	7	empty	empty	ADJ
ejpam-2803	569	8	for	for	ADP
ejpam-2803	569	9	all	all	DET
ejpam-2803	569	10	j	j	PROPN
ejpam-2803	569	11	6=	6=	NUM
ejpam-2803	570	1	i	i	PRON
ejpam-2803	570	2	∈	∈	PROPN
ejpam-2803	571	1	i	i	PRON
ejpam-2803	571	2	and	and	CCONJ
ejpam-2803	571	3	hence	hence	ADV
ejpam-2803	571	4	s	s	VERB
ejpam-2803	571	5	⊆	⊆	NUM
ejpam-2803	571	6	(	(	PUNCT
ejpam-2803	571	7	⊕	⊕	NOUN
ejpam-2803	571	8	i	i	PRON
ejpam-2803	571	9	6	6	NUM
ejpam-2803	571	10	=	=	SYM
ejpam-2803	571	11	k∈imk	k∈imk	PROPN
ejpam-2803	571	12	)	)	PUNCT
ejpam-2803	571	13	⊕	⊕	PROPN
ejpam-2803	571	14	(	(	PUNCT
ejpam-2803	571	15	0	0	NUM
ejpam-2803	571	16	)	)	PUNCT
ejpam-2803	571	17	.	.	PUNCT
ejpam-2803	572	1	this	this	PRON
ejpam-2803	572	2	implies	imply	VERB
ejpam-2803	572	3	that	that	SCONJ
ejpam-2803	572	4	s	s	VERB
ejpam-2803	572	5	⊆	⊆	NUM
ejpam-2803	572	6	⋂	⋂	PROPN
ejpam-2803	572	7	j	j	PROPN
ejpam-2803	572	8	6	6	NUM
ejpam-2803	572	9	=	=	NOUN
ejpam-2803	572	10	i∈i	i∈i	ADJ
ejpam-2803	572	11	(	(	PUNCT
ejpam-2803	572	12	(	(	PUNCT
ejpam-2803	572	13	⊕	⊕	NOUN
ejpam-2803	572	14	i	i	PRON
ejpam-2803	572	15	6	6	NUM
ejpam-2803	572	16	=	=	NOUN
ejpam-2803	572	17	k∈i	k∈i	PROPN
ejpam-2803	572	18	mk	mk	NOUN
ejpam-2803	572	19	)	)	PUNCT
ejpam-2803	572	20	⊕	⊕	PROPN
ejpam-2803	572	21	(	(	PUNCT
ejpam-2803	572	22	0	0	NUM
ejpam-2803	572	23	)	)	PUNCT
ejpam-2803	572	24	)	)	PUNCT
ejpam-2803	573	1	=	=	SYM
ejpam-2803	573	2	mj	mj	PROPN
ejpam-2803	573	3	⊕	⊕	PROPN
ejpam-2803	573	4	(	(	PUNCT
ejpam-2803	573	5	⊕	⊕	PROPN
ejpam-2803	573	6	j	j	PROPN
ejpam-2803	573	7	6	6	NUM
ejpam-2803	573	8	=	=	NOUN
ejpam-2803	573	9	i∈i	i∈i	ADJ
ejpam-2803	573	10	(	(	PUNCT
ejpam-2803	573	11	0	0	NUM
ejpam-2803	573	12	)	)	PUNCT
ejpam-2803	573	13	)	)	PUNCT
ejpam-2803	573	14	.	.	PUNCT
ejpam-2803	574	1	thus	thus	ADV
ejpam-2803	574	2	there	there	PRON
ejpam-2803	574	3	exist	exist	VERB
ejpam-2803	574	4	sj	sj	NOUN
ejpam-2803	574	5	∈	∈	PROPN
ejpam-2803	574	6	specsp(mj	specsp(mj	NOUN
ejpam-2803	574	7	)	)	PUNCT
ejpam-2803	574	8	such	such	ADJ
ejpam-2803	574	9	that	that	DET
ejpam-2803	574	10	s	s	PART
ejpam-2803	574	11	=	=	X
ejpam-2803	574	12	sj	sj	PROPN
ejpam-2803	574	13	⊕	⊕	PROPN
ejpam-2803	574	14	(	(	PUNCT
ejpam-2803	574	15	⊕	⊕	PROPN
ejpam-2803	574	16	j	j	PROPN
ejpam-2803	574	17	6	6	NUM
ejpam-2803	574	18	=	=	NOUN
ejpam-2803	574	19	i∈i(0	i∈i(0	NUM
ejpam-2803	574	20	)	)	PUNCT
ejpam-2803	574	21	)	)	PUNCT
ejpam-2803	574	22	.	.	PUNCT
ejpam-2803	575	1	now	now	ADV
ejpam-2803	575	2	let	let	VERB
ejpam-2803	575	3	s	s	NOUN
ejpam-2803	575	4	,	,	PUNCT
ejpam-2803	575	5	k	k	PROPN
ejpam-2803	575	6	∈	∈	PROPN
ejpam-2803	575	7	specsp(m	specsp(m	PROPN
ejpam-2803	575	8	)	)	PUNCT
ejpam-2803	575	9	.	.	PUNCT
ejpam-2803	576	1	by	by	ADP
ejpam-2803	576	2	the	the	DET
ejpam-2803	576	3	above	above	ADJ
ejpam-2803	576	4	arguments	argument	NOUN
ejpam-2803	576	5	,	,	PUNCT
ejpam-2803	576	6	there	there	PRON
ejpam-2803	576	7	exist	exist	VERB
ejpam-2803	576	8	si	si	PROPN
ejpam-2803	576	9	∈	∈	PROPN
ejpam-2803	576	10	specsp(mi	specsp(mi	PROPN
ejpam-2803	576	11	)	)	PUNCT
ejpam-2803	576	12	and	and	CCONJ
ejpam-2803	576	13	sj	sj	PROPN
ejpam-2803	576	14	∈	∈	PROPN
ejpam-2803	576	15	specsp(mj	specsp(mj	NOUN
ejpam-2803	576	16	)	)	PUNCT
ejpam-2803	576	17	such	such	ADJ
ejpam-2803	576	18	that	that	DET
ejpam-2803	576	19	s	s	PART
ejpam-2803	576	20	=	=	SYM
ejpam-2803	576	21	si	si	PROPN
ejpam-2803	576	22	⊕	⊕	PROPN
ejpam-2803	576	23	(	(	PUNCT
ejpam-2803	576	24	⊕	⊕	NOUN
ejpam-2803	576	25	i	i	PRON
ejpam-2803	576	26	6	6	NUM
ejpam-2803	576	27	=	=	SYM
ejpam-2803	576	28	j∈i(0	j∈i(0	NOUN
ejpam-2803	576	29	)	)	PUNCT
ejpam-2803	576	30	)	)	PUNCT
ejpam-2803	576	31	and	and	CCONJ
ejpam-2803	576	32	k	k	X
ejpam-2803	576	33	=	=	X
ejpam-2803	576	34	sj	sj	PROPN
ejpam-2803	576	35	⊕	⊕	PROPN
ejpam-2803	576	36	(	(	PUNCT
ejpam-2803	576	37	⊕	⊕	PROPN
ejpam-2803	576	38	j	j	PROPN
ejpam-2803	576	39	6	6	NUM
ejpam-2803	576	40	=	=	NOUN
ejpam-2803	576	41	i∈i(0	i∈i(0	NUM
ejpam-2803	576	42	)	)	PUNCT
ejpam-2803	576	43	)	)	PUNCT
ejpam-2803	576	44	.	.	PUNCT
ejpam-2803	577	1	this	this	PRON
ejpam-2803	577	2	implies	imply	VERB
ejpam-2803	577	3	that	that	SCONJ
ejpam-2803	577	4	i	i	PRON
ejpam-2803	577	5	=	=	SYM
ejpam-2803	577	6	j	j	PROPN
ejpam-2803	577	7	because	because	SCONJ
ejpam-2803	577	8	mi	mi	PROPN
ejpam-2803	577	9	’s	’s	PART
ejpam-2803	577	10	are	be	AUX
ejpam-2803	577	11	second	second	ADV
ejpam-2803	577	12	compatible	compatible	ADJ
ejpam-2803	577	13	.	.	PUNCT
ejpam-2803	578	1	hence	hence	ADV
ejpam-2803	578	2	k	k	PROPN
ejpam-2803	578	3	=	=	SYM
ejpam-2803	578	4	s	s	PROPN
ejpam-2803	578	5	,	,	PUNCT
ejpam-2803	578	6	i.e.	i.e.	X
ejpam-2803	578	7	,	,	PUNCT
ejpam-2803	578	8	m	m	VERB
ejpam-2803	578	9	is	be	AUX
ejpam-2803	578	10	xs	xs	NOUN
ejpam-2803	578	11	-	-	PUNCT
ejpam-2803	578	12	injective	injective	ADJ
ejpam-2803	578	13	.	.	PUNCT
ejpam-2803	579	1	case(ii	case(ii	ADJ
ejpam-2803	579	2	)	)	PUNCT
ejpam-2803	579	3	.	.	PUNCT
ejpam-2803	580	1	cotop	cotop	NOUN
ejpam-2803	580	2	modules	module	NOUN
ejpam-2803	580	3	.	.	PUNCT
ejpam-2803	581	1	(	(	PUNCT
ejpam-2803	581	2	⇒	⇒	PROPN
ejpam-2803	581	3	)	)	PUNCT
ejpam-2803	581	4	.	.	PUNCT
ejpam-2803	582	1	since	since	SCONJ
ejpam-2803	582	2	every	every	DET
ejpam-2803	582	3	submodule	submodule	NOUN
ejpam-2803	582	4	of	of	ADP
ejpam-2803	582	5	a	a	DET
ejpam-2803	582	6	cotop	cotop	NOUN
ejpam-2803	582	7	module	module	NOUN
ejpam-2803	582	8	is	be	AUX
ejpam-2803	582	9	cotop	cotop	NOUN
ejpam-2803	582	10	,	,	PUNCT
ejpam-2803	582	11	mi	mi	PROPN
ejpam-2803	582	12	’s	’s	PART
ejpam-2803	582	13	are	be	AUX
ejpam-2803	582	14	cotop	cotop	NOUN
ejpam-2803	582	15	.	.	PUNCT
ejpam-2803	583	1	let	let	VERB
ejpam-2803	583	2	i	i	PRON
ejpam-2803	583	3	6=	6=	ADP
ejpam-2803	583	4	j	j	PROPN
ejpam-2803	583	5	and	and	CCONJ
ejpam-2803	583	6	let	let	VERB
ejpam-2803	583	7	s	s	PRON
ejpam-2803	583	8	∈	∈	PROPN
ejpam-2803	583	9	specsp(mi	specsp(mi	PROPN
ejpam-2803	583	10	)	)	PUNCT
ejpam-2803	583	11	and	and	CCONJ
ejpam-2803	583	12	k	k	PROPN
ejpam-2803	583	13	∈	∈	PROPN
ejpam-2803	583	14	specsp(mj	specsp(mj	NOUN
ejpam-2803	583	15	)	)	PUNCT
ejpam-2803	583	16	.	.	PUNCT
ejpam-2803	584	1	set	set	VERB
ejpam-2803	584	2	s1	s1	NOUN
ejpam-2803	584	3	=	=	SYM
ejpam-2803	584	4	s	s	PROPN
ejpam-2803	584	5	⊕	⊕	PROPN
ejpam-2803	584	6	(	(	PUNCT
ejpam-2803	584	7	⊕	⊕	NOUN
ejpam-2803	584	8	i	i	PRON
ejpam-2803	584	9	6	6	NUM
ejpam-2803	584	10	=	=	NOUN
ejpam-2803	584	11	k∈i(0	k∈i(0	NOUN
ejpam-2803	584	12	)	)	PUNCT
ejpam-2803	584	13	)	)	PUNCT
ejpam-2803	584	14	and	and	CCONJ
ejpam-2803	584	15	k1	k1	NOUN
ejpam-2803	584	16	=	=	SYM
ejpam-2803	584	17	k	k	PROPN
ejpam-2803	584	18	⊕	⊕	PROPN
ejpam-2803	584	19	(	(	PUNCT
ejpam-2803	584	20	⊕	⊕	PROPN
ejpam-2803	584	21	j	j	PROPN
ejpam-2803	584	22	6	6	NUM
ejpam-2803	584	23	=	=	NOUN
ejpam-2803	584	24	k∈i(0	k∈i(0	NOUN
ejpam-2803	584	25	)	)	PUNCT
ejpam-2803	584	26	)	)	PUNCT
ejpam-2803	584	27	.	.	PUNCT
ejpam-2803	585	1	thus	thus	ADV
ejpam-2803	585	2	s1	s1	NOUN
ejpam-2803	585	3	+	+	CCONJ
ejpam-2803	585	4	k1	k1	PROPN
ejpam-2803	585	5	∈	∈	PROPN
ejpam-2803	585	6	specsp(m	specsp(m	NOUN
ejpam-2803	585	7	)	)	PUNCT
ejpam-2803	585	8	.	.	PUNCT
ejpam-2803	586	1	since	since	SCONJ
ejpam-2803	586	2	m	m	PROPN
ejpam-2803	586	3	is	be	AUX
ejpam-2803	586	4	cotop	cotop	NOUN
ejpam-2803	586	5	,	,	PUNCT
ejpam-2803	586	6	k1	k1	VERB
ejpam-2803	586	7	⊆	⊆	NUM
ejpam-2803	586	8	s1	s1	NOUN
ejpam-2803	586	9	or	or	CCONJ
ejpam-2803	586	10	s1	s1	PROPN
ejpam-2803	586	11	⊆	⊆	NUM
ejpam-2803	586	12	k1	k1	NOUN
ejpam-2803	586	13	,	,	PUNCT
ejpam-2803	586	14	which	which	PRON
ejpam-2803	586	15	is	be	AUX
ejpam-2803	586	16	a	a	DET
ejpam-2803	586	17	contradiction	contradiction	NOUN
ejpam-2803	586	18	.	.	PUNCT
ejpam-2803	587	1	therefore	therefore	ADV
ejpam-2803	587	2	mi	mi	PROPN
ejpam-2803	587	3	’s	’s	PART
ejpam-2803	587	4	are	be	AUX
ejpam-2803	587	5	second	second	ADV
ejpam-2803	587	6	-	-	PUNCT
ejpam-2803	587	7	compatible	compatible	ADJ
ejpam-2803	587	8	.	.	PUNCT
ejpam-2803	588	1	(	(	PUNCT
ejpam-2803	588	2	⇐	⇐	NOUN
ejpam-2803	588	3	)	)	PUNCT
ejpam-2803	588	4	.	.	PUNCT
ejpam-2803	589	1	let	let	VERB
ejpam-2803	589	2	s	s	PRON
ejpam-2803	589	3	be	be	AUX
ejpam-2803	589	4	p	p	NOUN
ejpam-2803	589	5	-	-	PUNCT
ejpam-2803	589	6	second	second	NOUN
ejpam-2803	589	7	and	and	CCONJ
ejpam-2803	589	8	let	let	VERB
ejpam-2803	589	9	s1	s1	NOUN
ejpam-2803	589	10	and	and	CCONJ
ejpam-2803	589	11	s2	s2	PROPN
ejpam-2803	589	12	be	be	AUX
ejpam-2803	589	13	socle	socle	NOUN
ejpam-2803	589	14	submodules	submodule	NOUN
ejpam-2803	589	15	of	of	ADP
ejpam-2803	589	16	m	m	PRON
ejpam-2803	589	17	such	such	ADJ
ejpam-2803	589	18	that	that	PRON
ejpam-2803	589	19	s	s	VERB
ejpam-2803	589	20	⊆	⊆	NUM
ejpam-2803	589	21	s1+s2	s1+s2	NOUN
ejpam-2803	589	22	.	.	PUNCT
ejpam-2803	590	1	then	then	ADV
ejpam-2803	590	2	by	by	ADP
ejpam-2803	590	3	similar	similar	ADJ
ejpam-2803	590	4	arguments	argument	NOUN
ejpam-2803	590	5	in	in	ADP
ejpam-2803	590	6	case(i	case(i	PROPN
ejpam-2803	590	7	)	)	PUNCT
ejpam-2803	590	8	,	,	PUNCT
ejpam-2803	590	9	for	for	ADP
ejpam-2803	590	10	each	each	DET
ejpam-2803	590	11	i	i	PRON
ejpam-2803	590	12	∈	∈	PROPN
ejpam-2803	590	13	i	i	PRON
ejpam-2803	590	14	,	,	PUNCT
ejpam-2803	590	15	there	there	PRON
ejpam-2803	590	16	exist	exist	VERB
ejpam-2803	590	17	submodules	submodule	NOUN
ejpam-2803	590	18	s1i	s1i	PROPN
ejpam-2803	590	19	and	and	CCONJ
ejpam-2803	590	20	s2i	s2i	NOUN
ejpam-2803	590	21	of	of	ADP
ejpam-2803	590	22	mi	mi	PROPN
ejpam-2803	590	23	,	,	PUNCT
ejpam-2803	590	24	such	such	ADJ
ejpam-2803	590	25	that	that	SCONJ
ejpam-2803	590	26	sk	sk	PROPN
ejpam-2803	590	27	=	=	PROPN
ejpam-2803	590	28	⊕	⊕	PROPN
ejpam-2803	590	29	i∈i	i∈i	ADJ
ejpam-2803	590	30	ski	ski	NOUN
ejpam-2803	591	1	(	(	PUNCT
ejpam-2803	591	2	k	k	NOUN
ejpam-2803	591	3	=	=	SYM
ejpam-2803	591	4	1	1	NUM
ejpam-2803	591	5	,	,	PUNCT
ejpam-2803	591	6	2	2	NUM
ejpam-2803	591	7	)	)	PUNCT
ejpam-2803	591	8	and	and	CCONJ
ejpam-2803	591	9	each	each	DET
ejpam-2803	591	10	submodule	submodule	NOUN
ejpam-2803	591	11	ski	ski	NOUN
ejpam-2803	591	12	is	be	AUX
ejpam-2803	591	13	either	either	DET
ejpam-2803	591	14	socle	socle	NOUN
ejpam-2803	591	15	submodule	submodule	NOUN
ejpam-2803	591	16	or	or	CCONJ
ejpam-2803	591	17	equals	equal	VERB
ejpam-2803	591	18	(	(	PUNCT
ejpam-2803	591	19	0	0	NUM
ejpam-2803	591	20	)	)	PUNCT
ejpam-2803	591	21	.	.	PUNCT
ejpam-2803	592	1	also	also	ADV
ejpam-2803	592	2	there	there	PRON
ejpam-2803	592	3	exists	exist	VERB
ejpam-2803	592	4	a	a	DET
ejpam-2803	592	5	second	second	ADJ
ejpam-2803	592	6	submodule	submodule	NOUN
ejpam-2803	592	7	sj	sj	NOUN
ejpam-2803	592	8	of	of	ADP
ejpam-2803	592	9	mj	mj	PROPN
ejpam-2803	592	10	such	such	ADJ
ejpam-2803	592	11	that	that	DET
ejpam-2803	592	12	s	s	PART
ejpam-2803	592	13	=	=	PUNCT
ejpam-2803	592	14	sj⊕	sj⊕	PROPN
ejpam-2803	592	15	(	(	PUNCT
ejpam-2803	592	16	⊕	⊕	PROPN
ejpam-2803	592	17	j	j	PROPN
ejpam-2803	592	18	6	6	NUM
ejpam-2803	592	19	=	=	NOUN
ejpam-2803	592	20	i∈i(0	i∈i(0	NUM
ejpam-2803	592	21	)	)	PUNCT
ejpam-2803	592	22	)	)	PUNCT
ejpam-2803	592	23	.	.	PUNCT
ejpam-2803	593	1	therefore	therefore	ADV
ejpam-2803	593	2	sj	sj	VERB
ejpam-2803	593	3	⊆	⊆	NUM
ejpam-2803	593	4	s1j	s1j	PROPN
ejpam-2803	593	5	+	+	PROPN
ejpam-2803	593	6	s2j	s2j	NOUN
ejpam-2803	593	7	.	.	PUNCT
ejpam-2803	594	1	since	since	SCONJ
ejpam-2803	594	2	mj	mj	PROPN
ejpam-2803	594	3	is	be	AUX
ejpam-2803	594	4	cotop	cotop	NOUN
ejpam-2803	594	5	,	,	PUNCT
ejpam-2803	594	6	sj	sj	ADV
ejpam-2803	594	7	⊆	⊆	NUM
ejpam-2803	594	8	s1j	s1j	PROPN
ejpam-2803	594	9	or	or	CCONJ
ejpam-2803	594	10	sj	sj	VERB
ejpam-2803	594	11	⊆	⊆	NUM
ejpam-2803	594	12	s2j	s2j	NOUN
ejpam-2803	594	13	.	.	PUNCT
ejpam-2803	595	1	it	it	PRON
ejpam-2803	595	2	follows	follow	VERB
ejpam-2803	595	3	that	that	PRON
ejpam-2803	595	4	s	s	VERB
ejpam-2803	595	5	⊆	⊆	NUM
ejpam-2803	595	6	s1	s1	NOUN
ejpam-2803	595	7	or	or	CCONJ
ejpam-2803	595	8	s	s	NOUN
ejpam-2803	595	9	⊆	⊆	NUM
ejpam-2803	595	10	s2	s2	NOUN
ejpam-2803	595	11	and	and	CCONJ
ejpam-2803	595	12	hence	hence	ADV
ejpam-2803	595	13	m	m	VERB
ejpam-2803	595	14	is	be	AUX
ejpam-2803	595	15	a	a	DET
ejpam-2803	595	16	cotop	cotop	NOUN
ejpam-2803	595	17	module	module	NOUN
ejpam-2803	595	18	.	.	PUNCT
ejpam-2803	596	1	an	an	DET
ejpam-2803	596	2	r	r	NOUN
ejpam-2803	596	3	-	-	PUNCT
ejpam-2803	596	4	module	module	NOUN
ejpam-2803	596	5	m	m	NOUN
ejpam-2803	596	6	is	be	AUX
ejpam-2803	596	7	said	say	VERB
ejpam-2803	596	8	to	to	PART
ejpam-2803	596	9	have	have	VERB
ejpam-2803	596	10	the	the	DET
ejpam-2803	596	11	double	double	ADJ
ejpam-2803	596	12	annihilator	annihilator	NOUN
ejpam-2803	596	13	conditions	condition	NOUN
ejpam-2803	596	14	if	if	SCONJ
ejpam-2803	596	15	for	for	ADP
ejpam-2803	596	16	each	each	DET
ejpam-2803	596	17	ideal	ideal	NOUN
ejpam-2803	596	18	i	i	PRON
ejpam-2803	596	19	of	of	ADP
ejpam-2803	596	20	r	r	PROPN
ejpam-2803	596	21	,	,	PUNCT
ejpam-2803	596	22	we	we	PRON
ejpam-2803	596	23	have	have	VERB
ejpam-2803	596	24	i	i	PRON
ejpam-2803	596	25	=	=	PUNCT
ejpam-2803	596	26	annr(0	annr(0	PRON
ejpam-2803	596	27	:	:	PUNCT
ejpam-2803	596	28	m	m	VERB
ejpam-2803	596	29	i	i	NOUN
ejpam-2803	596	30	)	)	PUNCT
ejpam-2803	596	31	.	.	PUNCT
ejpam-2803	597	1	corollary	corollary	ADJ
ejpam-2803	597	2	3.16	3.16	NUM
ejpam-2803	597	3	.	.	PUNCT
ejpam-2803	598	1	(	(	PUNCT
ejpam-2803	598	2	a	a	X
ejpam-2803	598	3	)	)	PUNCT
ejpam-2803	598	4	let	let	VERB
ejpam-2803	598	5	r	r	PRON
ejpam-2803	598	6	be	be	AUX
ejpam-2803	598	7	a	a	DET
ejpam-2803	598	8	domain	domain	NOUN
ejpam-2803	598	9	with	with	ADP
ejpam-2803	598	10	field	field	NOUN
ejpam-2803	598	11	of	of	ADP
ejpam-2803	598	12	fractions	fraction	NOUN
ejpam-2803	598	13	q	q	PUNCT
ejpam-2803	598	14	and	and	CCONJ
ejpam-2803	598	15	suppose	suppose	VERB
ejpam-2803	598	16	m	m	PRON
ejpam-2803	598	17	is	be	AUX
ejpam-2803	598	18	an	an	DET
ejpam-2803	598	19	r	r	NOUN
ejpam-2803	598	20	-	-	PUNCT
ejpam-2803	598	21	module	module	NOUN
ejpam-2803	598	22	such	such	ADJ
ejpam-2803	598	23	that	that	SCONJ
ejpam-2803	598	24	m	m	PROPN
ejpam-2803	598	25	/	/	SYM
ejpam-2803	598	26	i0(m	i0(m	NOUN
ejpam-2803	598	27	)	)	PUNCT
ejpam-2803	598	28	is	be	AUX
ejpam-2803	598	29	finitely	finitely	ADV
ejpam-2803	598	30	cogenerated	cogenerate	VERB
ejpam-2803	598	31	.	.	PUNCT
ejpam-2803	599	1	then	then	ADV
ejpam-2803	599	2	the	the	DET
ejpam-2803	599	3	r	r	NOUN
ejpam-2803	599	4	-	-	PUNCT
ejpam-2803	599	5	module	module	NOUN
ejpam-2803	599	6	q⊕m	q⊕m	PROPN
ejpam-2803	599	7	is	be	AUX
ejpam-2803	599	8	h.	h.	PROPN
ejpam-2803	599	9	ansari	ansari	PROPN
ejpam-2803	599	10	-	-	PUNCT
ejpam-2803	599	11	toroghy	toroghy	NOUN
ejpam-2803	599	12	,	,	PUNCT
ejpam-2803	599	13	s.	s.	PROPN
ejpam-2803	599	14	s.	s.	PROPN
ejpam-2803	599	15	pourmortazavi	pourmortazavi	VERB
ejpam-2803	599	16	/	/	SYM
ejpam-2803	599	17	eur	eur	PROPN
ejpam-2803	599	18	.	.	PUNCT
ejpam-2803	600	1	j.	j.	PROPN
ejpam-2803	600	2	pure	pure	PROPN
ejpam-2803	600	3	appl	appl	PROPN
ejpam-2803	600	4	.	.	PROPN
ejpam-2803	600	5	math	math	PROPN
ejpam-2803	600	6	,	,	PUNCT
ejpam-2803	600	7	10	10	NUM
ejpam-2803	600	8	(	(	PUNCT
ejpam-2803	600	9	2	2	NUM
ejpam-2803	600	10	)	)	PUNCT
ejpam-2803	600	11	(	(	PUNCT
ejpam-2803	600	12	2017	2017	NUM
ejpam-2803	600	13	)	)	PUNCT
ejpam-2803	600	14	,	,	PUNCT
ejpam-2803	600	15	211	211	NUM
ejpam-2803	600	16	-	-	SYM
ejpam-2803	600	17	230	230	NUM
ejpam-2803	600	18	225	225	NUM
ejpam-2803	600	19	a	a	DET
ejpam-2803	600	20	cotop	cotop	NOUN
ejpam-2803	600	21	(	(	PUNCT
ejpam-2803	600	22	resp	resp	NOUN
ejpam-2803	600	23	.	.	PUNCT
ejpam-2803	601	1	an	an	DET
ejpam-2803	601	2	xs	xs	NOUN
ejpam-2803	601	3	-	-	PUNCT
ejpam-2803	601	4	injective	injective	ADJ
ejpam-2803	601	5	)	)	PUNCT
ejpam-2803	601	6	module	module	NOUN
ejpam-2803	601	7	if	if	SCONJ
ejpam-2803	601	8	and	and	CCONJ
ejpam-2803	601	9	only	only	ADV
ejpam-2803	601	10	if	if	SCONJ
ejpam-2803	601	11	m	m	NOUN
ejpam-2803	601	12	is	be	AUX
ejpam-2803	601	13	a	a	DET
ejpam-2803	601	14	cotorsion	cotorsion	NOUN
ejpam-2803	601	15	cotop	cotop	NOUN
ejpam-2803	601	16	(	(	PUNCT
ejpam-2803	601	17	resp	resp	NOUN
ejpam-2803	601	18	.	.	PUNCT
ejpam-2803	602	1	xs	xs	NOUN
ejpam-2803	602	2	-	-	PUNCT
ejpam-2803	602	3	injective	injective	ADJ
ejpam-2803	602	4	)	)	PUNCT
ejpam-2803	602	5	module	module	NOUN
ejpam-2803	602	6	.	.	PUNCT
ejpam-2803	603	1	(	(	PUNCT
ejpam-2803	603	2	b	b	X
ejpam-2803	603	3	)	)	PUNCT
ejpam-2803	603	4	let	let	VERB
ejpam-2803	603	5	(	(	PUNCT
ejpam-2803	603	6	r	r	NOUN
ejpam-2803	603	7	,	,	PUNCT
ejpam-2803	603	8	m	m	VERB
ejpam-2803	603	9	)	)	PUNCT
ejpam-2803	603	10	be	be	AUX
ejpam-2803	603	11	a	a	DET
ejpam-2803	603	12	local	local	ADJ
ejpam-2803	603	13	ring	ring	NOUN
ejpam-2803	603	14	,	,	PUNCT
ejpam-2803	603	15	iλ	iλ	PUNCT
ejpam-2803	603	16	(	(	PUNCT
ejpam-2803	603	17	λ	λ	PROPN
ejpam-2803	603	18	∈	∈	PROPN
ejpam-2803	603	19	λ	λ	PROPN
ejpam-2803	603	20	)	)	PUNCT
ejpam-2803	603	21	a	a	DET
ejpam-2803	603	22	family	family	NOUN
ejpam-2803	603	23	of	of	ADP
ejpam-2803	603	24	ideals	ideal	NOUN
ejpam-2803	603	25	of	of	ADP
ejpam-2803	603	26	r	r	NOUN
ejpam-2803	603	27	,	,	PUNCT
ejpam-2803	603	28	and	and	CCONJ
ejpam-2803	603	29	(	(	PUNCT
ejpam-2803	603	30	0	0	NUM
ejpam-2803	603	31	:	:	PUNCT
ejpam-2803	603	32	m	m	VERB
ejpam-2803	603	33	m	m	VERB
ejpam-2803	603	34	)	)	PUNCT
ejpam-2803	603	35	6=	6=	ADP
ejpam-2803	604	1	0	0	X
ejpam-2803	604	2	.	.	PUNCT
ejpam-2803	605	1	if	if	SCONJ
ejpam-2803	605	2	m	m	NOUN
ejpam-2803	605	3	=	=	SYM
ejpam-2803	605	4	⊕	⊕	PROPN
ejpam-2803	605	5	λ∈λ(0	λ∈λ(0	PROPN
ejpam-2803	605	6	:	:	PUNCT
ejpam-2803	605	7	m	m	VERB
ejpam-2803	605	8	iλ	iλ	NOUN
ejpam-2803	605	9	)	)	PUNCT
ejpam-2803	605	10	is	be	AUX
ejpam-2803	605	11	a	a	DET
ejpam-2803	605	12	cotop	cotop	NOUN
ejpam-2803	605	13	(	(	PUNCT
ejpam-2803	605	14	resp	resp	NOUN
ejpam-2803	605	15	.	.	PUNCT
ejpam-2803	606	1	an	an	DET
ejpam-2803	606	2	xs	xs	NOUN
ejpam-2803	606	3	-	-	PUNCT
ejpam-2803	606	4	injective	injective	ADJ
ejpam-2803	606	5	)	)	PUNCT
ejpam-2803	606	6	r	r	NOUN
ejpam-2803	606	7	-	-	PUNCT
ejpam-2803	606	8	module	module	NOUN
ejpam-2803	606	9	,	,	PUNCT
ejpam-2803	606	10	then	then	ADV
ejpam-2803	606	11	the	the	DET
ejpam-2803	606	12	ideals	ideal	NOUN
ejpam-2803	606	13	iλ	iλ	VERB
ejpam-2803	606	14	(	(	PUNCT
ejpam-2803	606	15	λ	λ	PROPN
ejpam-2803	606	16	∈	∈	PROPN
ejpam-2803	606	17	λ	λ	NOUN
ejpam-2803	606	18	)	)	PUNCT
ejpam-2803	606	19	are	be	AUX
ejpam-2803	606	20	comaximal	comaximal	ADJ
ejpam-2803	606	21	.	.	PUNCT
ejpam-2803	607	1	(	(	PUNCT
ejpam-2803	607	2	c	c	X
ejpam-2803	607	3	)	)	PUNCT
ejpam-2803	607	4	let	let	VERB
ejpam-2803	607	5	m	m	PRON
ejpam-2803	607	6	be	be	AUX
ejpam-2803	607	7	a	a	DET
ejpam-2803	607	8	weak	weak	ADJ
ejpam-2803	607	9	comultiplication	comultiplication	NOUN
ejpam-2803	607	10	module	module	NOUN
ejpam-2803	607	11	and	and	CCONJ
ejpam-2803	607	12	let	let	VERB
ejpam-2803	607	13	(	(	PUNCT
ejpam-2803	607	14	iλ)λ∈λ	iλ)λ∈λ	NOUN
ejpam-2803	607	15	be	be	AUX
ejpam-2803	607	16	a	a	DET
ejpam-2803	607	17	family	family	NOUN
ejpam-2803	607	18	of	of	ADP
ejpam-2803	607	19	ideals	ideal	NOUN
ejpam-2803	607	20	of	of	ADP
ejpam-2803	607	21	r.	r.	PROPN
ejpam-2803	607	22	if	if	SCONJ
ejpam-2803	607	23	m	m	NOUN
ejpam-2803	607	24	=	=	SYM
ejpam-2803	607	25	⊕	⊕	PROPN
ejpam-2803	607	26	λ∈λ(0	λ∈λ(0	PROPN
ejpam-2803	607	27	:	:	PUNCT
ejpam-2803	607	28	m	m	VERB
ejpam-2803	607	29	iλ	iλ	NOUN
ejpam-2803	607	30	)	)	PUNCT
ejpam-2803	607	31	and	and	CCONJ
ejpam-2803	607	32	the	the	DET
ejpam-2803	607	33	ideals	ideal	NOUN
ejpam-2803	607	34	iλ	iλ	VERB
ejpam-2803	607	35	(	(	PUNCT
ejpam-2803	607	36	λ	λ	PROPN
ejpam-2803	607	37	∈	∈	PROPN
ejpam-2803	607	38	λ	λ	NOUN
ejpam-2803	607	39	)	)	PUNCT
ejpam-2803	607	40	are	be	AUX
ejpam-2803	607	41	comaximal	comaximal	ADJ
ejpam-2803	607	42	,	,	PUNCT
ejpam-2803	607	43	then	then	ADV
ejpam-2803	607	44	m	m	VERB
ejpam-2803	607	45	is	be	AUX
ejpam-2803	607	46	a	a	DET
ejpam-2803	607	47	cotop	cotop	NOUN
ejpam-2803	607	48	(	(	PUNCT
ejpam-2803	607	49	resp	resp	NOUN
ejpam-2803	607	50	.	.	PUNCT
ejpam-2803	608	1	an	an	DET
ejpam-2803	608	2	xs	xs	NOUN
ejpam-2803	608	3	-	-	PUNCT
ejpam-2803	608	4	injective	injective	ADJ
ejpam-2803	608	5	)	)	PUNCT
ejpam-2803	608	6	r	r	NOUN
ejpam-2803	608	7	-	-	PUNCT
ejpam-2803	608	8	module	module	NOUN
ejpam-2803	608	9	.	.	PUNCT
ejpam-2803	608	10	proof	proof	NOUN
ejpam-2803	608	11	.	.	PUNCT
ejpam-2803	609	1	we	we	PRON
ejpam-2803	609	2	just	just	ADV
ejpam-2803	609	3	consider	consider	VERB
ejpam-2803	609	4	the	the	DET
ejpam-2803	609	5	proof	proof	NOUN
ejpam-2803	609	6	for	for	ADP
ejpam-2803	609	7	cotop	cotop	NOUN
ejpam-2803	609	8	modules	module	NOUN
ejpam-2803	609	9	.	.	PUNCT
ejpam-2803	610	1	we	we	PRON
ejpam-2803	610	2	have	have	VERB
ejpam-2803	610	3	similar	similar	ADJ
ejpam-2803	610	4	arguments	argument	NOUN
ejpam-2803	610	5	when	when	SCONJ
ejpam-2803	610	6	m	m	PROPN
ejpam-2803	610	7	is	be	AUX
ejpam-2803	610	8	an	an	DET
ejpam-2803	610	9	xs	xs	NOUN
ejpam-2803	610	10	-	-	PUNCT
ejpam-2803	610	11	injective	injective	ADJ
ejpam-2803	610	12	module	module	NOUN
ejpam-2803	610	13	.	.	PUNCT
ejpam-2803	611	1	(	(	PUNCT
ejpam-2803	611	2	a	a	X
ejpam-2803	611	3	)	)	PUNCT
ejpam-2803	611	4	(	(	PUNCT
ejpam-2803	611	5	⇐	⇐	NOUN
ejpam-2803	611	6	)	)	PUNCT
ejpam-2803	611	7	.	.	PUNCT
ejpam-2803	612	1	we	we	PRON
ejpam-2803	612	2	can	can	AUX
ejpam-2803	612	3	see	see	VERB
ejpam-2803	612	4	that	that	PRON
ejpam-2803	612	5	specs(q	specs(q	NOUN
ejpam-2803	612	6	)	)	PUNCT
ejpam-2803	612	7	=	=	SYM
ejpam-2803	612	8	specs0(q	specs0(q	NOUN
ejpam-2803	612	9	)	)	PUNCT
ejpam-2803	612	10	=	=	PRON
ejpam-2803	612	11	{	{	PUNCT
ejpam-2803	612	12	q	q	X
ejpam-2803	612	13	}	}	PUNCT
ejpam-2803	612	14	.	.	PUNCT
ejpam-2803	613	1	we	we	PRON
ejpam-2803	613	2	show	show	VERB
ejpam-2803	613	3	that	that	SCONJ
ejpam-2803	613	4	r	r	NOUN
ejpam-2803	613	5	-	-	PUNCT
ejpam-2803	613	6	module	module	NOUN
ejpam-2803	613	7	m	m	NOUN
ejpam-2803	613	8	and	and	CCONJ
ejpam-2803	613	9	r	r	NOUN
ejpam-2803	613	10	-	-	PUNCT
ejpam-2803	613	11	module	module	NOUN
ejpam-2803	613	12	q	q	NOUN
ejpam-2803	613	13	are	be	AUX
ejpam-2803	613	14	second	second	ADV
ejpam-2803	613	15	-	-	PUNCT
ejpam-2803	613	16	compatible	compatible	ADJ
ejpam-2803	613	17	.	.	PUNCT
ejpam-2803	614	1	let	let	VERB
ejpam-2803	614	2	s	s	PRON
ejpam-2803	614	3	∈	∈	PROPN
ejpam-2803	614	4	specs0(m	specs0(m	NOUN
ejpam-2803	614	5	)	)	PUNCT
ejpam-2803	614	6	.	.	PUNCT
ejpam-2803	615	1	by	by	ADP
ejpam-2803	615	2	[	[	X
ejpam-2803	615	3	5	5	NUM
ejpam-2803	615	4	,	,	PUNCT
ejpam-2803	615	5	theorem	theorem	VERB
ejpam-2803	615	6	2.10	2.10	NUM
ejpam-2803	615	7	]	]	PUNCT
ejpam-2803	615	8	,	,	PUNCT
ejpam-2803	615	9	im0	im0	PROPN
ejpam-2803	615	10	(	(	PUNCT
ejpam-2803	615	11	s	s	X
ejpam-2803	615	12	)	)	PUNCT
ejpam-2803	615	13	=	=	VERB
ejpam-2803	616	1	s.	s.	PROPN
ejpam-2803	616	2	since	since	SCONJ
ejpam-2803	616	3	im0	im0	PROPN
ejpam-2803	616	4	(	(	PUNCT
ejpam-2803	616	5	s	s	NOUN
ejpam-2803	616	6	)	)	PUNCT
ejpam-2803	616	7	⊆	⊆	NUM
ejpam-2803	616	8	im0	im0	PROPN
ejpam-2803	616	9	(	(	PUNCT
ejpam-2803	616	10	m	m	NOUN
ejpam-2803	616	11	)	)	PUNCT
ejpam-2803	616	12	and	and	CCONJ
ejpam-2803	616	13	m	m	PROPN
ejpam-2803	616	14	is	be	AUX
ejpam-2803	616	15	cotorsion	cotorsion	NOUN
ejpam-2803	616	16	,	,	PUNCT
ejpam-2803	616	17	we	we	PRON
ejpam-2803	616	18	have	have	VERB
ejpam-2803	616	19	s	s	NOUN
ejpam-2803	616	20	=	=	X
ejpam-2803	616	21	(	(	PUNCT
ejpam-2803	616	22	0	0	NUM
ejpam-2803	616	23	)	)	PUNCT
ejpam-2803	616	24	,	,	PUNCT
ejpam-2803	616	25	a	a	DET
ejpam-2803	616	26	contradiction	contradiction	NOUN
ejpam-2803	616	27	.	.	PUNCT
ejpam-2803	617	1	therefore	therefore	ADV
ejpam-2803	617	2	q⊕m	q⊕m	PROPN
ejpam-2803	617	3	is	be	AUX
ejpam-2803	617	4	cotop	cotop	VERB
ejpam-2803	617	5	by	by	ADP
ejpam-2803	617	6	theorem	theorem	NOUN
ejpam-2803	617	7	3.15	3.15	NUM
ejpam-2803	617	8	.	.	PUNCT
ejpam-2803	618	1	(	(	PUNCT
ejpam-2803	618	2	⇒	⇒	NOUN
ejpam-2803	618	3	)	)	PUNCT
ejpam-2803	618	4	.	.	PUNCT
ejpam-2803	619	1	by	by	ADP
ejpam-2803	619	2	theorem	theorem	NOUN
ejpam-2803	619	3	3.15	3.15	NUM
ejpam-2803	619	4	,	,	PUNCT
ejpam-2803	619	5	q	q	PUNCT
ejpam-2803	619	6	and	and	CCONJ
ejpam-2803	619	7	m	m	PROPN
ejpam-2803	619	8	are	be	AUX
ejpam-2803	619	9	cotop	cotop	VERB
ejpam-2803	619	10	modules	module	NOUN
ejpam-2803	619	11	and	and	CCONJ
ejpam-2803	619	12	second	second	ADV
ejpam-2803	619	13	-	-	PUNCT
ejpam-2803	619	14	compatible	compatible	ADJ
ejpam-2803	619	15	.	.	PUNCT
ejpam-2803	620	1	if	if	SCONJ
ejpam-2803	620	2	m	m	NOUN
ejpam-2803	620	3	is	be	AUX
ejpam-2803	620	4	not	not	PART
ejpam-2803	620	5	cotorsion	cotorsion	NOUN
ejpam-2803	620	6	,	,	PUNCT
ejpam-2803	620	7	then	then	ADV
ejpam-2803	620	8	im0	im0	PROPN
ejpam-2803	620	9	(	(	PUNCT
ejpam-2803	620	10	m	m	NOUN
ejpam-2803	620	11	)	)	PUNCT
ejpam-2803	620	12	belongs	belong	VERB
ejpam-2803	620	13	to	to	ADP
ejpam-2803	620	14	specs0(m	specs0(m	NOUN
ejpam-2803	620	15	)	)	PUNCT
ejpam-2803	620	16	by	by	ADP
ejpam-2803	620	17	[	[	X
ejpam-2803	620	18	4	4	NUM
ejpam-2803	620	19	,	,	PUNCT
ejpam-2803	620	20	corollary	corollary	ADJ
ejpam-2803	620	21	2.10	2.10	NUM
ejpam-2803	620	22	]	]	PUNCT
ejpam-2803	620	23	which	which	PRON
ejpam-2803	620	24	is	be	AUX
ejpam-2803	620	25	a	a	DET
ejpam-2803	620	26	contradiction	contradiction	NOUN
ejpam-2803	620	27	.	.	PUNCT
ejpam-2803	621	1	thus	thus	ADV
ejpam-2803	621	2	m	m	PROPN
ejpam-2803	621	3	is	be	AUX
ejpam-2803	621	4	cotorsion	cotorsion	NOUN
ejpam-2803	621	5	.	.	PUNCT
ejpam-2803	622	1	(	(	PUNCT
ejpam-2803	622	2	b	b	X
ejpam-2803	622	3	)	)	PUNCT
ejpam-2803	622	4	let	let	VERB
ejpam-2803	622	5	m	m	PRON
ejpam-2803	622	6	be	be	AUX
ejpam-2803	622	7	a	a	DET
ejpam-2803	622	8	cotop	cotop	NOUN
ejpam-2803	622	9	r	r	NOUN
ejpam-2803	622	10	-	-	PUNCT
ejpam-2803	622	11	module	module	NOUN
ejpam-2803	622	12	,	,	PUNCT
ejpam-2803	622	13	and	and	CCONJ
ejpam-2803	622	14	λ	λ	NOUN
ejpam-2803	622	15	,	,	PUNCT
ejpam-2803	622	16	λ′	λ′	X
ejpam-2803	622	17	∈	∈	PROPN
ejpam-2803	622	18	λ	λ	PROPN
ejpam-2803	622	19	.	.	PUNCT
ejpam-2803	623	1	if	if	SCONJ
ejpam-2803	623	2	iλ	iλ	ADP
ejpam-2803	623	3	+	+	NUM
ejpam-2803	623	4	iλ′	iλ′	NOUN
ejpam-2803	623	5	6=	6=	ADP
ejpam-2803	623	6	r	r	NOUN
ejpam-2803	623	7	,	,	PUNCT
ejpam-2803	623	8	then	then	ADV
ejpam-2803	623	9	iλ	iλ	VERB
ejpam-2803	623	10	+	+	NUM
ejpam-2803	623	11	iλ′	iλ′	NOUN
ejpam-2803	623	12	⊆	⊆	NUM
ejpam-2803	623	13	m.	m.	NOUN
ejpam-2803	623	14	therefore	therefore	ADV
ejpam-2803	623	15	(	(	PUNCT
ejpam-2803	623	16	0	0	NUM
ejpam-2803	623	17	:	:	PUNCT
ejpam-2803	623	18	m	m	VERB
ejpam-2803	623	19	m	m	VERB
ejpam-2803	623	20	)	)	PUNCT
ejpam-2803	624	1	⊆	⊆	NUM
ejpam-2803	624	2	(	(	PUNCT
ejpam-2803	624	3	0	0	NUM
ejpam-2803	624	4	:	:	PUNCT
ejpam-2803	624	5	m	m	VERB
ejpam-2803	624	6	iλ	iλ	ADJ
ejpam-2803	624	7	+	+	NUM
ejpam-2803	624	8	iλ′	iλ′	NOUN
ejpam-2803	624	9	)	)	PUNCT
ejpam-2803	624	10	=	=	SYM
ejpam-2803	624	11	(	(	PUNCT
ejpam-2803	624	12	0	0	NUM
ejpam-2803	624	13	:	:	PUNCT
ejpam-2803	624	14	m	m	VERB
ejpam-2803	624	15	iλ	iλ	ADJ
ejpam-2803	624	16	)	)	PUNCT
ejpam-2803	624	17	∩	∩	NOUN
ejpam-2803	624	18	(	(	PUNCT
ejpam-2803	624	19	0	0	NUM
ejpam-2803	624	20	:	:	PUNCT
ejpam-2803	624	21	m	m	NOUN
ejpam-2803	624	22	iλ′	iλ′	NOUN
ejpam-2803	624	23	)	)	PUNCT
ejpam-2803	624	24	.	.	PUNCT
ejpam-2803	625	1	but	but	CCONJ
ejpam-2803	625	2	(	(	PUNCT
ejpam-2803	625	3	0	0	NUM
ejpam-2803	625	4	:	:	PUNCT
ejpam-2803	625	5	m	m	VERB
ejpam-2803	625	6	m	m	VERB
ejpam-2803	625	7	)	)	PUNCT
ejpam-2803	625	8	is	be	AUX
ejpam-2803	625	9	an	an	DET
ejpam-2803	625	10	m	m	ADJ
ejpam-2803	625	11	-	-	ADJ
ejpam-2803	625	12	second	second	ADJ
ejpam-2803	625	13	submodule	submodule	NOUN
ejpam-2803	625	14	of	of	ADP
ejpam-2803	625	15	m	m	PROPN
ejpam-2803	625	16	.	.	PUNCT
ejpam-2803	626	1	it	it	PRON
ejpam-2803	626	2	follows	follow	VERB
ejpam-2803	626	3	that	that	PRON
ejpam-2803	626	4	specsm((0	specsm((0	ADP
ejpam-2803	626	5	:	:	PUNCT
ejpam-2803	626	6	m	m	VERB
ejpam-2803	626	7	iλ	iλ	NOUN
ejpam-2803	626	8	)	)	PUNCT
ejpam-2803	626	9	)	)	PUNCT
ejpam-2803	626	10	and	and	CCONJ
ejpam-2803	626	11	specsm((0	specsm((0	ADP
ejpam-2803	626	12	:	:	PUNCT
ejpam-2803	626	13	m	m	NUM
ejpam-2803	626	14	iλ′	iλ′	NOUN
ejpam-2803	626	15	)	)	PUNCT
ejpam-2803	626	16	)	)	PUNCT
ejpam-2803	626	17	are	be	AUX
ejpam-2803	626	18	both	both	PRON
ejpam-2803	626	19	nonempty	nonempty	ADJ
ejpam-2803	626	20	sets	set	NOUN
ejpam-2803	626	21	which	which	PRON
ejpam-2803	626	22	is	be	AUX
ejpam-2803	626	23	a	a	DET
ejpam-2803	626	24	contradiction	contradiction	NOUN
ejpam-2803	626	25	by	by	ADP
ejpam-2803	626	26	theorem	theorem	NOUN
ejpam-2803	626	27	3.15	3.15	NUM
ejpam-2803	626	28	.	.	PUNCT
ejpam-2803	627	1	(	(	PUNCT
ejpam-2803	627	2	c	c	X
ejpam-2803	627	3	)	)	PUNCT
ejpam-2803	627	4	we	we	PRON
ejpam-2803	627	5	show	show	VERB
ejpam-2803	627	6	that	that	SCONJ
ejpam-2803	627	7	for	for	ADP
ejpam-2803	627	8	every	every	DET
ejpam-2803	627	9	λ	λ	PROPN
ejpam-2803	627	10	∈	∈	PROPN
ejpam-2803	627	11	λ	λ	PROPN
ejpam-2803	627	12	,	,	PUNCT
ejpam-2803	627	13	(	(	PUNCT
ejpam-2803	627	14	0	0	NUM
ejpam-2803	627	15	:	:	PUNCT
ejpam-2803	627	16	m	m	VERB
ejpam-2803	627	17	iλ	iλ	NOUN
ejpam-2803	627	18	)	)	PUNCT
ejpam-2803	627	19	is	be	AUX
ejpam-2803	627	20	a	a	DET
ejpam-2803	627	21	cotop	cotop	NOUN
ejpam-2803	627	22	r	r	NOUN
ejpam-2803	627	23	-	-	PUNCT
ejpam-2803	627	24	module	module	NOUN
ejpam-2803	627	25	.	.	PUNCT
ejpam-2803	628	1	let	let	VERB
ejpam-2803	628	2	s	s	PRON
ejpam-2803	628	3	∈	∈	NOUN
ejpam-2803	628	4	specsp((0	specsp((0	ADP
ejpam-2803	628	5	:	:	PUNCT
ejpam-2803	628	6	m	m	VERB
ejpam-2803	628	7	iλ	iλ	NOUN
ejpam-2803	628	8	)	)	PUNCT
ejpam-2803	628	9	)	)	PUNCT
ejpam-2803	628	10	and	and	CCONJ
ejpam-2803	628	11	s1	s1	NOUN
ejpam-2803	628	12	,	,	PUNCT
ejpam-2803	628	13	s2	s2	PROPN
ejpam-2803	628	14	are	be	AUX
ejpam-2803	628	15	socle	socle	NOUN
ejpam-2803	628	16	submodules	submodule	NOUN
ejpam-2803	628	17	of	of	ADP
ejpam-2803	628	18	(	(	PUNCT
ejpam-2803	628	19	0	0	NUM
ejpam-2803	628	20	:	:	PUNCT
ejpam-2803	628	21	m	m	VERB
ejpam-2803	628	22	iλ	iλ	NOUN
ejpam-2803	628	23	)	)	PUNCT
ejpam-2803	628	24	such	such	ADJ
ejpam-2803	628	25	that	that	PRON
ejpam-2803	628	26	s	s	VERB
ejpam-2803	628	27	⊆	⊆	NUM
ejpam-2803	628	28	s1	s1	NOUN
ejpam-2803	628	29	+	+	CCONJ
ejpam-2803	628	30	s2	s2	PROPN
ejpam-2803	628	31	.	.	PUNCT
ejpam-2803	629	1	thus	thus	ADV
ejpam-2803	629	2	annr(s1+s2	annr(s1+s2	NOUN
ejpam-2803	629	3	)	)	PUNCT
ejpam-2803	629	4	=	=	SYM
ejpam-2803	629	5	annr(s1)∩annr(s2	annr(s1)∩annr(s2	X
ejpam-2803	629	6	)	)	PUNCT
ejpam-2803	629	7	⊆	⊆	NUM
ejpam-2803	629	8	annr(s	annr(s	NOUN
ejpam-2803	629	9	)	)	PUNCT
ejpam-2803	629	10	∈	∈	PROPN
ejpam-2803	629	11	spec(r	spec(r	PROPN
ejpam-2803	629	12	)	)	PUNCT
ejpam-2803	629	13	.	.	PUNCT
ejpam-2803	630	1	thereforeannr(s1	thereforeannr(s1	PROPN
ejpam-2803	630	2	)	)	PUNCT
ejpam-2803	631	1	⊆	⊆	NUM
ejpam-2803	631	2	p	p	NOUN
ejpam-2803	631	3	or	or	CCONJ
ejpam-2803	631	4	annr(s2	annr(s2	NOUN
ejpam-2803	631	5	)	)	PUNCT
ejpam-2803	631	6	⊆	⊆	NUM
ejpam-2803	631	7	p.	p.	NOUN
ejpam-2803	631	8	since	since	SCONJ
ejpam-2803	631	9	m	m	PROPN
ejpam-2803	631	10	is	be	AUX
ejpam-2803	631	11	weak	weak	ADJ
ejpam-2803	631	12	comultiplication	comultiplication	NOUN
ejpam-2803	631	13	,	,	PUNCT
ejpam-2803	631	14	s	s	PART
ejpam-2803	631	15	=	=	PUNCT
ejpam-2803	631	16	(	(	PUNCT
ejpam-2803	631	17	0	0	NUM
ejpam-2803	631	18	:	:	PUNCT
ejpam-2803	631	19	m	m	VERB
ejpam-2803	631	20	p	p	X
ejpam-2803	631	21	)	)	PUNCT
ejpam-2803	631	22	⊆	⊆	NUM
ejpam-2803	631	23	(	(	PUNCT
ejpam-2803	631	24	0	0	NUM
ejpam-2803	631	25	:	:	PUNCT
ejpam-2803	631	26	m	m	NOUN
ejpam-2803	631	27	annr(s1	annr(s1	ADJ
ejpam-2803	631	28	)	)	PUNCT
ejpam-2803	631	29	)	)	PUNCT
ejpam-2803	632	1	=	=	SYM
ejpam-2803	632	2	s1	s1	PROPN
ejpam-2803	632	3	or	or	CCONJ
ejpam-2803	632	4	s	s	NOUN
ejpam-2803	632	5	=	=	X
ejpam-2803	632	6	(	(	PUNCT
ejpam-2803	632	7	0	0	NUM
ejpam-2803	632	8	:	:	PUNCT
ejpam-2803	632	9	m	m	VERB
ejpam-2803	632	10	p	p	X
ejpam-2803	632	11	)	)	PUNCT
ejpam-2803	632	12	⊆	⊆	NUM
ejpam-2803	632	13	(	(	PUNCT
ejpam-2803	632	14	0	0	NUM
ejpam-2803	632	15	:	:	PUNCT
ejpam-2803	632	16	m	m	NOUN
ejpam-2803	632	17	annr(s1	annr(s1	ADJ
ejpam-2803	632	18	)	)	PUNCT
ejpam-2803	632	19	)	)	PUNCT
ejpam-2803	633	1	=	=	SYM
ejpam-2803	633	2	s2	s2	PROPN
ejpam-2803	633	3	.	.	PUNCT
ejpam-2803	634	1	we	we	PRON
ejpam-2803	634	2	show	show	VERB
ejpam-2803	634	3	that	that	SCONJ
ejpam-2803	634	4	(	(	PUNCT
ejpam-2803	634	5	0	0	NUM
ejpam-2803	634	6	:	:	PUNCT
ejpam-2803	634	7	m	m	VERB
ejpam-2803	634	8	iλ	iλ	ADJ
ejpam-2803	634	9	)	)	PUNCT
ejpam-2803	634	10	’s	’	VERB
ejpam-2803	634	11	are	be	AUX
ejpam-2803	634	12	second	second	ADV
ejpam-2803	634	13	-	-	PUNCT
ejpam-2803	634	14	compatible	compatible	ADJ
ejpam-2803	634	15	.	.	PUNCT
ejpam-2803	635	1	let	let	VERB
ejpam-2803	635	2	λ1	λ1	ADJ
ejpam-2803	635	3	,	,	PUNCT
ejpam-2803	635	4	λ2	λ2	PROPN
ejpam-2803	635	5	∈	∈	PROPN
ejpam-2803	635	6	λ	λ	NOUN
ejpam-2803	635	7	and	and	CCONJ
ejpam-2803	635	8	q	q	PROPN
ejpam-2803	635	9	∈	∈	PROPN
ejpam-2803	635	10	spec(r	spec(r	PROPN
ejpam-2803	635	11	)	)	PUNCT
ejpam-2803	635	12	and	and	CCONJ
ejpam-2803	635	13	let	let	VERB
ejpam-2803	635	14	s1	s1	PROPN
ejpam-2803	635	15	∈	∈	PROPN
ejpam-2803	635	16	specsq((0	specsq((0	NOUN
ejpam-2803	635	17	:	:	PUNCT
ejpam-2803	635	18	m	m	VERB
ejpam-2803	635	19	iλ1	iλ1	X
ejpam-2803	635	20	)	)	PUNCT
ejpam-2803	635	21	)	)	PUNCT
ejpam-2803	635	22	and	and	CCONJ
ejpam-2803	635	23	s2	s2	PROPN
ejpam-2803	635	24	∈	∈	PROPN
ejpam-2803	635	25	specsq((0	specsq((0	NOUN
ejpam-2803	635	26	:	:	PUNCT
ejpam-2803	635	27	m	m	PROPN
ejpam-2803	635	28	iλ2	iλ2	NOUN
ejpam-2803	635	29	)	)	PUNCT
ejpam-2803	635	30	)	)	PUNCT
ejpam-2803	635	31	.	.	PUNCT
ejpam-2803	636	1	since	since	SCONJ
ejpam-2803	636	2	si	si	PROPN
ejpam-2803	636	3	⊆	⊆	NUM
ejpam-2803	636	4	(	(	PUNCT
ejpam-2803	636	5	0	0	NUM
ejpam-2803	636	6	:	:	PUNCT
ejpam-2803	636	7	m	m	VERB
ejpam-2803	636	8	iλi	iλi	PROPN
ejpam-2803	636	9	)	)	PUNCT
ejpam-2803	636	10	,	,	PUNCT
ejpam-2803	636	11	we	we	PRON
ejpam-2803	636	12	have	have	VERB
ejpam-2803	636	13	iλi	iλi	PROPN
ejpam-2803	636	14	⊆	⊆	NUM
ejpam-2803	636	15	annr((0	annr((0	PROPN
ejpam-2803	636	16	:	:	PUNCT
ejpam-2803	636	17	m	m	VERB
ejpam-2803	636	18	iλi	iλi	PROPN
ejpam-2803	636	19	)	)	PUNCT
ejpam-2803	636	20	)	)	PUNCT
ejpam-2803	637	1	⊆	⊆	NUM
ejpam-2803	637	2	annr(si	annr(si	PROPN
ejpam-2803	637	3	)	)	PUNCT
ejpam-2803	637	4	=	=	PUNCT
ejpam-2803	637	5	q	q	X
ejpam-2803	637	6	(	(	PUNCT
ejpam-2803	637	7	i	i	NOUN
ejpam-2803	637	8	=	=	NOUN
ejpam-2803	637	9	1	1	NUM
ejpam-2803	637	10	,	,	PUNCT
ejpam-2803	637	11	2	2	NUM
ejpam-2803	637	12	)	)	PUNCT
ejpam-2803	637	13	.	.	PUNCT
ejpam-2803	638	1	this	this	PRON
ejpam-2803	638	2	implies	imply	VERB
ejpam-2803	638	3	that	that	SCONJ
ejpam-2803	638	4	r	r	NOUN
ejpam-2803	638	5	=	=	SYM
ejpam-2803	638	6	iλ1	iλ1	PROPN
ejpam-2803	639	1	+	+	CCONJ
ejpam-2803	639	2	iλ2	iλ2	VERB
ejpam-2803	639	3	⊆	⊆	NUM
ejpam-2803	639	4	q	q	NOUN
ejpam-2803	639	5	,	,	PUNCT
ejpam-2803	639	6	a	a	DET
ejpam-2803	639	7	contradiction	contradiction	NOUN
ejpam-2803	639	8	by	by	ADP
ejpam-2803	639	9	hypothesis	hypothesis	NOUN
ejpam-2803	639	10	.	.	PUNCT
ejpam-2803	640	1	theorem	theorem	VERB
ejpam-2803	640	2	3.17	3.17	NUM
ejpam-2803	640	3	.	.	PUNCT
ejpam-2803	641	1	let	let	VERB
ejpam-2803	641	2	m	m	PRON
ejpam-2803	641	3	be	be	AUX
ejpam-2803	641	4	a	a	DET
ejpam-2803	641	5	non	non	ADJ
ejpam-2803	641	6	-	-	ADJ
ejpam-2803	641	7	zero	zero	NUM
ejpam-2803	641	8	xs	xs	NOUN
ejpam-2803	641	9	-	-	PUNCT
ejpam-2803	641	10	injective	injective	ADJ
ejpam-2803	641	11	artinian	artinian	ADJ
ejpam-2803	641	12	r	r	NOUN
ejpam-2803	641	13	-	-	PUNCT
ejpam-2803	641	14	module	module	NOUN
ejpam-2803	641	15	.	.	PUNCT
ejpam-2803	642	1	(	(	PUNCT
ejpam-2803	642	2	a	a	PRON
ejpam-2803	642	3	)	)	PUNCT
ejpam-2803	642	4	specs(m	specs(m	NOUN
ejpam-2803	642	5	)	)	PUNCT
ejpam-2803	642	6	=	=	PRON
ejpam-2803	642	7	{	{	PUNCT
ejpam-2803	642	8	imp	imp	X
ejpam-2803	642	9	(	(	PUNCT
ejpam-2803	642	10	(	(	PUNCT
ejpam-2803	642	11	0	0	NUM
ejpam-2803	642	12	:	:	PUNCT
ejpam-2803	642	13	m	m	VERB
ejpam-2803	642	14	p	p	NOUN
ejpam-2803	642	15	)	)	PUNCT
ejpam-2803	642	16	)	)	PUNCT
ejpam-2803	643	1	|	|	ADV
ejpam-2803	643	2	p	p	PROPN
ejpam-2803	643	3	∈	∈	PROPN
ejpam-2803	643	4	v	v	NOUN
ejpam-2803	643	5	(	(	PUNCT
ejpam-2803	643	6	annr(m	annr(m	PROPN
ejpam-2803	643	7	)	)	PUNCT
ejpam-2803	643	8	)	)	PUNCT
ejpam-2803	643	9	,	,	PUNCT
ejpam-2803	643	10	imp	imp	X
ejpam-2803	643	11	(	(	PUNCT
ejpam-2803	643	12	0	0	NUM
ejpam-2803	643	13	:	:	PUNCT
ejpam-2803	643	14	m	m	VERB
ejpam-2803	643	15	p	p	NOUN
ejpam-2803	643	16	)	)	PUNCT
ejpam-2803	643	17	6=	6=	X
ejpam-2803	643	18	(	(	PUNCT
ejpam-2803	643	19	0	0	NUM
ejpam-2803	643	20	)	)	PUNCT
ejpam-2803	643	21	}	}	PUNCT
ejpam-2803	643	22	,	,	PUNCT
ejpam-2803	643	23	min(m	min(m	PROPN
ejpam-2803	643	24	)	)	PUNCT
ejpam-2803	643	25	=	=	PRON
ejpam-2803	643	26	{	{	PUNCT
ejpam-2803	643	27	(	(	PUNCT
ejpam-2803	643	28	0	0	NUM
ejpam-2803	643	29	:	:	PUNCT
ejpam-2803	643	30	m	m	VERB
ejpam-2803	643	31	p	p	X
ejpam-2803	643	32	)	)	PUNCT
ejpam-2803	643	33	|	|	ADV
ejpam-2803	643	34	p	p	NOUN
ejpam-2803	643	35	∈max(r	∈max(r	PROPN
ejpam-2803	643	36	)	)	PUNCT
ejpam-2803	643	37	,	,	PUNCT
ejpam-2803	643	38	(	(	PUNCT
ejpam-2803	643	39	0	0	NUM
ejpam-2803	643	40	:	:	PUNCT
ejpam-2803	643	41	m	m	VERB
ejpam-2803	643	42	p	p	NOUN
ejpam-2803	643	43	)	)	PUNCT
ejpam-2803	643	44	6=	6=	X
ejpam-2803	643	45	(	(	PUNCT
ejpam-2803	643	46	0	0	NUM
ejpam-2803	643	47	)	)	PUNCT
ejpam-2803	643	48	}	}	PUNCT
ejpam-2803	643	49	.	.	PUNCT
ejpam-2803	644	1	(	(	PUNCT
ejpam-2803	644	2	b	b	X
ejpam-2803	644	3	)	)	PUNCT
ejpam-2803	644	4	if	if	SCONJ
ejpam-2803	644	5	m	m	NOUN
ejpam-2803	644	6	is	be	AUX
ejpam-2803	644	7	secondful	secondful	ADJ
ejpam-2803	644	8	,	,	PUNCT
ejpam-2803	644	9	then	then	ADV
ejpam-2803	644	10	specs(m	specs(m	NOUN
ejpam-2803	644	11	)	)	PUNCT
ejpam-2803	644	12	=	=	PRON
ejpam-2803	644	13	{	{	PUNCT
ejpam-2803	644	14	imp	imp	X
ejpam-2803	644	15	(	(	PUNCT
ejpam-2803	644	16	(	(	PUNCT
ejpam-2803	644	17	0	0	NUM
ejpam-2803	644	18	:	:	PUNCT
ejpam-2803	644	19	m	m	VERB
ejpam-2803	644	20	p	p	NOUN
ejpam-2803	644	21	)	)	PUNCT
ejpam-2803	644	22	)	)	PUNCT
ejpam-2803	645	1	|	|	ADV
ejpam-2803	645	2	p	p	PROPN
ejpam-2803	645	3	∈	∈	PROPN
ejpam-2803	645	4	v	v	NOUN
ejpam-2803	645	5	(	(	PUNCT
ejpam-2803	645	6	annr(m	annr(m	PROPN
ejpam-2803	645	7	)	)	PUNCT
ejpam-2803	645	8	)	)	PUNCT
ejpam-2803	645	9	}	}	PUNCT
ejpam-2803	645	10	,	,	PUNCT
ejpam-2803	645	11	min(m	min(m	PROPN
ejpam-2803	645	12	)	)	PUNCT
ejpam-2803	645	13	=	=	PRON
ejpam-2803	645	14	{	{	PUNCT
ejpam-2803	645	15	(	(	PUNCT
ejpam-2803	645	16	0	0	NUM
ejpam-2803	645	17	:	:	PUNCT
ejpam-2803	645	18	m	m	VERB
ejpam-2803	645	19	p	p	X
ejpam-2803	645	20	)	)	PUNCT
ejpam-2803	645	21	|	|	ADV
ejpam-2803	645	22	p	p	PROPN
ejpam-2803	645	23	∈	∈	PROPN
ejpam-2803	645	24	v	v	NOUN
ejpam-2803	645	25	(	(	PUNCT
ejpam-2803	645	26	annr(m	annr(m	PROPN
ejpam-2803	645	27	)	)	PUNCT
ejpam-2803	645	28	)	)	PUNCT
ejpam-2803	645	29	∩max(r	∩max(r	PUNCT
ejpam-2803	645	30	)	)	PUNCT
ejpam-2803	645	31	}	}	PUNCT
ejpam-2803	645	32	.	.	PUNCT
ejpam-2803	646	1	h.	h.	PROPN
ejpam-2803	646	2	ansari	ansari	PROPN
ejpam-2803	646	3	-	-	PUNCT
ejpam-2803	646	4	toroghy	toroghy	NOUN
ejpam-2803	646	5	,	,	PUNCT
ejpam-2803	646	6	s.	s.	PROPN
ejpam-2803	646	7	s.	s.	PROPN
ejpam-2803	646	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	646	9	/	/	SYM
ejpam-2803	646	10	eur	eur	PROPN
ejpam-2803	646	11	.	.	PUNCT
ejpam-2803	647	1	j.	j.	PROPN
ejpam-2803	647	2	pure	pure	PROPN
ejpam-2803	647	3	appl	appl	PROPN
ejpam-2803	647	4	.	.	PROPN
ejpam-2803	647	5	math	math	PROPN
ejpam-2803	647	6	,	,	PUNCT
ejpam-2803	647	7	10	10	NUM
ejpam-2803	647	8	(	(	PUNCT
ejpam-2803	647	9	2	2	NUM
ejpam-2803	647	10	)	)	PUNCT
ejpam-2803	647	11	(	(	PUNCT
ejpam-2803	647	12	2017	2017	NUM
ejpam-2803	647	13	)	)	PUNCT
ejpam-2803	647	14	,	,	PUNCT
ejpam-2803	647	15	211	211	NUM
ejpam-2803	647	16	-	-	SYM
ejpam-2803	647	17	230	230	NUM
ejpam-2803	647	18	226	226	NUM
ejpam-2803	647	19	(	(	PUNCT
ejpam-2803	647	20	c	c	NOUN
ejpam-2803	647	21	)	)	PUNCT
ejpam-2803	647	22	if	if	SCONJ
ejpam-2803	647	23	r	r	NOUN
ejpam-2803	647	24	is	be	AUX
ejpam-2803	647	25	pid	pid	NOUN
ejpam-2803	647	26	and	and	CCONJ
ejpam-2803	647	27	m	m	PROPN
ejpam-2803	647	28	is	be	AUX
ejpam-2803	647	29	faithful	faithful	ADJ
ejpam-2803	647	30	secondful	secondful	ADJ
ejpam-2803	647	31	,	,	PUNCT
ejpam-2803	647	32	then	then	ADV
ejpam-2803	647	33	specs(m	specs(m	NOUN
ejpam-2803	647	34	)	)	PUNCT
ejpam-2803	647	35	=	=	SYM
ejpam-2803	647	36	min(m	min(m	PROPN
ejpam-2803	647	37	)	)	PUNCT
ejpam-2803	647	38	∪	∪	NOUN
ejpam-2803	647	39	{	{	PUNCT
ejpam-2803	647	40	im0	im0	PROPN
ejpam-2803	647	41	(	(	PUNCT
ejpam-2803	647	42	m	m	NOUN
ejpam-2803	647	43	)	)	PUNCT
ejpam-2803	647	44	}	}	PUNCT
ejpam-2803	647	45	,	,	PUNCT
ejpam-2803	647	46	where	where	SCONJ
ejpam-2803	647	47	min(m	min(m	NOUN
ejpam-2803	647	48	)	)	PUNCT
ejpam-2803	647	49	=	=	PRON
ejpam-2803	647	50	{	{	PUNCT
ejpam-2803	647	51	(	(	PUNCT
ejpam-2803	647	52	0	0	NUM
ejpam-2803	647	53	:	:	PUNCT
ejpam-2803	647	54	m	m	VERB
ejpam-2803	647	55	p	p	X
ejpam-2803	647	56	)	)	PUNCT
ejpam-2803	647	57	|	|	ADV
ejpam-2803	647	58	p	p	NOUN
ejpam-2803	647	59	∈max(r	∈max(r	NOUN
ejpam-2803	647	60	)	)	PUNCT
ejpam-2803	647	61	}	}	PUNCT
ejpam-2803	647	62	.	.	PUNCT
ejpam-2803	648	1	proof	proof	NOUN
ejpam-2803	648	2	.	.	PUNCT
ejpam-2803	649	1	(	(	PUNCT
ejpam-2803	649	2	a	a	X
ejpam-2803	649	3	)	)	PUNCT
ejpam-2803	649	4	put	put	VERB
ejpam-2803	649	5	t	t	NOUN
ejpam-2803	649	6	=	=	SYM
ejpam-2803	649	7	{	{	PUNCT
ejpam-2803	649	8	imp	imp	X
ejpam-2803	649	9	(	(	PUNCT
ejpam-2803	649	10	(	(	PUNCT
ejpam-2803	649	11	0	0	NUM
ejpam-2803	649	12	:	:	PUNCT
ejpam-2803	649	13	m	m	VERB
ejpam-2803	649	14	p	p	NOUN
ejpam-2803	649	15	)	)	PUNCT
ejpam-2803	649	16	)	)	PUNCT
ejpam-2803	650	1	|	|	ADV
ejpam-2803	650	2	p	p	PROPN
ejpam-2803	650	3	∈	∈	PROPN
ejpam-2803	650	4	v	v	NOUN
ejpam-2803	650	5	(	(	PUNCT
ejpam-2803	650	6	annr(m	annr(m	PROPN
ejpam-2803	650	7	)	)	PUNCT
ejpam-2803	650	8	)	)	PUNCT
ejpam-2803	650	9	,	,	PUNCT
ejpam-2803	650	10	imp	imp	X
ejpam-2803	650	11	(	(	PUNCT
ejpam-2803	650	12	(	(	PUNCT
ejpam-2803	650	13	0	0	NUM
ejpam-2803	650	14	:	:	PUNCT
ejpam-2803	650	15	m	m	VERB
ejpam-2803	650	16	p	p	NOUN
ejpam-2803	650	17	)	)	PUNCT
ejpam-2803	650	18	)	)	PUNCT
ejpam-2803	650	19	6=	6=	X
ejpam-2803	651	1	(	(	PUNCT
ejpam-2803	651	2	0	0	NUM
ejpam-2803	651	3	)	)	PUNCT
ejpam-2803	651	4	}	}	PUNCT
ejpam-2803	651	5	.	.	PUNCT
ejpam-2803	652	1	then	then	ADV
ejpam-2803	652	2	t	t	PROPN
ejpam-2803	652	3	⊆	⊆	NUM
ejpam-2803	652	4	specs(m	specs(m	PROPN
ejpam-2803	652	5	)	)	PUNCT
ejpam-2803	652	6	by	by	ADP
ejpam-2803	652	7	[	[	X
ejpam-2803	652	8	4	4	NUM
ejpam-2803	652	9	,	,	PUNCT
ejpam-2803	652	10	lemma	lemma	PROPN
ejpam-2803	652	11	2.9	2.9	NUM
ejpam-2803	652	12	]	]	PUNCT
ejpam-2803	652	13	.	.	PUNCT
ejpam-2803	653	1	let	let	VERB
ejpam-2803	653	2	s	s	PRON
ejpam-2803	653	3	∈	∈	PROPN
ejpam-2803	653	4	specs(m	specs(m	PROPN
ejpam-2803	653	5	)	)	PUNCT
ejpam-2803	653	6	.	.	PUNCT
ejpam-2803	654	1	then	then	ADV
ejpam-2803	654	2	specs(m	specs(m	PROPN
ejpam-2803	654	3	)	)	PUNCT
ejpam-2803	654	4	6=	6=	ADP
ejpam-2803	654	5	∅	∅	NOUN
ejpam-2803	654	6	for	for	ADP
ejpam-2803	654	7	p	p	PROPN
ejpam-2803	654	8	=	=	PROPN
ejpam-2803	654	9	annr(s	annr(s	PROPN
ejpam-2803	654	10	)	)	PUNCT
ejpam-2803	654	11	.	.	PUNCT
ejpam-2803	655	1	thus	thus	ADV
ejpam-2803	655	2	imp	imp	X
ejpam-2803	655	3	(	(	PUNCT
ejpam-2803	655	4	(	(	PUNCT
ejpam-2803	655	5	0	0	NUM
ejpam-2803	655	6	:	:	PUNCT
ejpam-2803	655	7	m	m	VERB
ejpam-2803	655	8	p	p	NOUN
ejpam-2803	655	9	)	)	PUNCT
ejpam-2803	655	10	)	)	PUNCT
ejpam-2803	656	1	∈	∈	PROPN
ejpam-2803	656	2	specsp(m	specsp(m	NOUN
ejpam-2803	656	3	)	)	PUNCT
ejpam-2803	656	4	by	by	ADP
ejpam-2803	656	5	[	[	X
ejpam-2803	656	6	4	4	NUM
ejpam-2803	656	7	,	,	PUNCT
ejpam-2803	656	8	lemma	lemma	PROPN
ejpam-2803	656	9	2.9	2.9	NUM
ejpam-2803	656	10	]	]	PUNCT
ejpam-2803	656	11	as	as	ADP
ejpam-2803	656	12	imp	imp	X
ejpam-2803	656	13	(	(	PUNCT
ejpam-2803	656	14	(	(	PUNCT
ejpam-2803	656	15	0	0	NUM
ejpam-2803	656	16	:	:	PUNCT
ejpam-2803	656	17	m	m	VERB
ejpam-2803	656	18	p	p	NOUN
ejpam-2803	656	19	)	)	PUNCT
ejpam-2803	656	20	)	)	PUNCT
ejpam-2803	656	21	6=	6=	X
ejpam-2803	656	22	(	(	PUNCT
ejpam-2803	656	23	0	0	NUM
ejpam-2803	656	24	)	)	PUNCT
ejpam-2803	656	25	.	.	PUNCT
ejpam-2803	657	1	since	since	SCONJ
ejpam-2803	657	2	m	m	PROPN
ejpam-2803	657	3	is	be	AUX
ejpam-2803	657	4	xs	xs	NOUN
ejpam-2803	657	5	-	-	PUNCT
ejpam-2803	657	6	injective	injective	ADJ
ejpam-2803	657	7	,	,	PUNCT
ejpam-2803	657	8	s	s	PART
ejpam-2803	657	9	=	=	X
ejpam-2803	657	10	imp	imp	X
ejpam-2803	657	11	(	(	PUNCT
ejpam-2803	657	12	(	(	PUNCT
ejpam-2803	657	13	0	0	NUM
ejpam-2803	657	14	:	:	PUNCT
ejpam-2803	657	15	m	m	VERB
ejpam-2803	657	16	p	p	NOUN
ejpam-2803	657	17	)	)	PUNCT
ejpam-2803	657	18	)	)	PUNCT
ejpam-2803	657	19	.	.	PUNCT
ejpam-2803	658	1	this	this	PRON
ejpam-2803	658	2	implies	imply	VERB
ejpam-2803	658	3	that	that	SCONJ
ejpam-2803	658	4	s	s	VERB
ejpam-2803	658	5	∈	∈	PROPN
ejpam-2803	658	6	t	t	NOUN
ejpam-2803	658	7	.	.	PUNCT
ejpam-2803	659	1	to	to	PART
ejpam-2803	659	2	prove	prove	VERB
ejpam-2803	659	3	the	the	DET
ejpam-2803	659	4	second	second	ADJ
ejpam-2803	659	5	assertion	assertion	NOUN
ejpam-2803	659	6	,	,	PUNCT
ejpam-2803	659	7	put	put	VERB
ejpam-2803	659	8	ω	ω	NUM
ejpam-2803	659	9	=	=	SYM
ejpam-2803	659	10	{	{	PUNCT
ejpam-2803	659	11	(	(	PUNCT
ejpam-2803	659	12	0	0	NUM
ejpam-2803	659	13	:	:	PUNCT
ejpam-2803	659	14	m	m	VERB
ejpam-2803	659	15	p	p	X
ejpam-2803	659	16	)	)	PUNCT
ejpam-2803	659	17	|	|	ADV
ejpam-2803	659	18	p	p	PROPN
ejpam-2803	659	19	∈	∈	PROPN
ejpam-2803	659	20	max(r	max(r	PROPN
ejpam-2803	659	21	)	)	PUNCT
ejpam-2803	659	22	,	,	PUNCT
ejpam-2803	659	23	(	(	PUNCT
ejpam-2803	659	24	0	0	NUM
ejpam-2803	659	25	:	:	PUNCT
ejpam-2803	659	26	m	m	VERB
ejpam-2803	659	27	p	p	NOUN
ejpam-2803	659	28	)	)	PUNCT
ejpam-2803	659	29	6=	6=	X
ejpam-2803	659	30	(	(	PUNCT
ejpam-2803	659	31	0	0	NUM
ejpam-2803	659	32	)	)	PUNCT
ejpam-2803	659	33	}	}	PUNCT
ejpam-2803	659	34	and	and	CCONJ
ejpam-2803	659	35	let	let	VERB
ejpam-2803	659	36	(	(	PUNCT
ejpam-2803	659	37	0	0	NUM
ejpam-2803	659	38	:	:	PUNCT
ejpam-2803	660	1	m	m	VERB
ejpam-2803	660	2	p	p	X
ejpam-2803	660	3	)	)	PUNCT
ejpam-2803	660	4	∈	∈	PROPN
ejpam-2803	660	5	ω	ω	PROPN
ejpam-2803	660	6	.	.	PUNCT
ejpam-2803	661	1	then	then	ADV
ejpam-2803	661	2	(	(	PUNCT
ejpam-2803	661	3	0	0	NUM
ejpam-2803	661	4	:	:	PUNCT
ejpam-2803	661	5	m	m	VERB
ejpam-2803	661	6	p	p	X
ejpam-2803	661	7	)	)	PUNCT
ejpam-2803	661	8	∈	∈	PROPN
ejpam-2803	661	9	specs(m	specs(m	NOUN
ejpam-2803	661	10	)	)	PUNCT
ejpam-2803	661	11	by	by	ADP
ejpam-2803	661	12	[	[	X
ejpam-2803	661	13	30	30	NUM
ejpam-2803	661	14	,	,	PUNCT
ejpam-2803	661	15	proposition	proposition	NOUN
ejpam-2803	661	16	1.4	1.4	NUM
ejpam-2803	661	17	]	]	PUNCT
ejpam-2803	661	18	.	.	PUNCT
ejpam-2803	662	1	moreover	moreover	ADV
ejpam-2803	662	2	,	,	PUNCT
ejpam-2803	662	3	(	(	PUNCT
ejpam-2803	662	4	0	0	NUM
ejpam-2803	662	5	:	:	PUNCT
ejpam-2803	662	6	m	m	VERB
ejpam-2803	662	7	p	p	X
ejpam-2803	662	8	)	)	PUNCT
ejpam-2803	662	9	∈min(m	∈min(m	PROPN
ejpam-2803	662	10	)	)	PUNCT
ejpam-2803	662	11	because	because	SCONJ
ejpam-2803	662	12	if	if	SCONJ
ejpam-2803	662	13	k	k	PROPN
ejpam-2803	662	14	⊆	⊆	NUM
ejpam-2803	662	15	(	(	PUNCT
ejpam-2803	662	16	0	0	NUM
ejpam-2803	662	17	:	:	PUNCT
ejpam-2803	662	18	m	m	VERB
ejpam-2803	662	19	p	p	X
ejpam-2803	662	20	)	)	PUNCT
ejpam-2803	662	21	for	for	ADP
ejpam-2803	662	22	some	some	DET
ejpam-2803	662	23	non	non	ADJ
ejpam-2803	662	24	-	-	ADJ
ejpam-2803	662	25	zero	zero	NUM
ejpam-2803	662	26	submodule	submodule	NOUN
ejpam-2803	662	27	k	k	PROPN
ejpam-2803	662	28	of	of	ADP
ejpam-2803	662	29	m	m	PROPN
ejpam-2803	662	30	,	,	PUNCT
ejpam-2803	662	31	then	then	ADV
ejpam-2803	662	32	annr((0	annr((0	VERB
ejpam-2803	662	33	:	:	PUNCT
ejpam-2803	662	34	m	m	VERB
ejpam-2803	662	35	p	p	NOUN
ejpam-2803	662	36	)	)	PUNCT
ejpam-2803	662	37	)	)	PUNCT
ejpam-2803	663	1	=	=	PUNCT
ejpam-2803	663	2	annr(k	annr(k	ADJ
ejpam-2803	663	3	)	)	PUNCT
ejpam-2803	663	4	=	=	SYM
ejpam-2803	664	1	p	p	NOUN
ejpam-2803	664	2	and	and	CCONJ
ejpam-2803	664	3	hence	hence	ADV
ejpam-2803	664	4	(	(	PUNCT
ejpam-2803	664	5	0	0	NUM
ejpam-2803	664	6	:	:	PUNCT
ejpam-2803	664	7	m	m	VERB
ejpam-2803	664	8	p	p	ADJ
ejpam-2803	664	9	)	)	PUNCT
ejpam-2803	664	10	=	=	SYM
ejpam-2803	665	1	k	k	PROPN
ejpam-2803	665	2	because	because	SCONJ
ejpam-2803	665	3	m	m	PROPN
ejpam-2803	665	4	is	be	AUX
ejpam-2803	665	5	xsinjective	xsinjective	ADJ
ejpam-2803	665	6	.	.	PUNCT
ejpam-2803	666	1	thus	thus	ADV
ejpam-2803	666	2	ω	ω	NUM
ejpam-2803	666	3	⊆	⊆	NUM
ejpam-2803	666	4	min(m	min(m	PROPN
ejpam-2803	666	5	)	)	PUNCT
ejpam-2803	666	6	.	.	PUNCT
ejpam-2803	667	1	conversely	conversely	ADV
ejpam-2803	667	2	,	,	PUNCT
ejpam-2803	667	3	let	let	VERB
ejpam-2803	667	4	s	s	PRON
ejpam-2803	667	5	∈	∈	PROPN
ejpam-2803	667	6	min(m	min(m	PROPN
ejpam-2803	667	7	)	)	PUNCT
ejpam-2803	667	8	.	.	PUNCT
ejpam-2803	668	1	then	then	ADV
ejpam-2803	668	2	s	s	VERB
ejpam-2803	668	3	∈	∈	PROPN
ejpam-2803	668	4	specs(m	specs(m	PROPN
ejpam-2803	668	5	)	)	PUNCT
ejpam-2803	668	6	and	and	CCONJ
ejpam-2803	668	7	we	we	PRON
ejpam-2803	668	8	have	have	VERB
ejpam-2803	668	9	p	p	NOUN
ejpam-2803	668	10	=	=	NOUN
ejpam-2803	668	11	annr(s	annr(s	PROPN
ejpam-2803	668	12	)	)	PUNCT
ejpam-2803	668	13	∈	∈	PROPN
ejpam-2803	668	14	max(r	max(r	PROPN
ejpam-2803	668	15	)	)	PUNCT
ejpam-2803	668	16	.	.	PUNCT
ejpam-2803	669	1	whence	whence	ADP
ejpam-2803	669	2	p	p	NOUN
ejpam-2803	669	3	=	=	PROPN
ejpam-2803	669	4	annr(s	annr(s	PROPN
ejpam-2803	669	5	)	)	PUNCT
ejpam-2803	669	6	=	=	PRON
ejpam-2803	669	7	annr((0	annr((0	VERB
ejpam-2803	669	8	:	:	PUNCT
ejpam-2803	669	9	m	m	VERB
ejpam-2803	669	10	p	p	NOUN
ejpam-2803	669	11	)	)	PUNCT
ejpam-2803	669	12	)	)	PUNCT
ejpam-2803	669	13	.	.	PUNCT
ejpam-2803	670	1	it	it	PRON
ejpam-2803	670	2	follows	follow	VERB
ejpam-2803	670	3	that	that	PRON
ejpam-2803	670	4	s	s	VERB
ejpam-2803	670	5	=	=	X
ejpam-2803	670	6	(	(	PUNCT
ejpam-2803	670	7	0	0	NUM
ejpam-2803	670	8	:	:	PUNCT
ejpam-2803	670	9	m	m	VERB
ejpam-2803	670	10	p	p	NOUN
ejpam-2803	670	11	)	)	PUNCT
ejpam-2803	670	12	6=	6=	X
ejpam-2803	670	13	(	(	PUNCT
ejpam-2803	670	14	0	0	NUM
ejpam-2803	670	15	)	)	PUNCT
ejpam-2803	670	16	.	.	PUNCT
ejpam-2803	671	1	thus	thus	ADV
ejpam-2803	671	2	s	s	X
ejpam-2803	671	3	∈	∈	PROPN
ejpam-2803	671	4	ω	ω	NUM
ejpam-2803	671	5	,	,	PUNCT
ejpam-2803	671	6	and	and	CCONJ
ejpam-2803	671	7	we	we	PRON
ejpam-2803	671	8	can	can	AUX
ejpam-2803	671	9	conclude	conclude	VERB
ejpam-2803	671	10	that	that	PRON
ejpam-2803	671	11	min(m	min(m	PROPN
ejpam-2803	671	12	)	)	PUNCT
ejpam-2803	671	13	=	=	SYM
ejpam-2803	671	14	ω	ω	PROPN
ejpam-2803	671	15	.	.	PUNCT
ejpam-2803	672	1	(	(	PUNCT
ejpam-2803	672	2	b	b	X
ejpam-2803	672	3	)	)	PUNCT
ejpam-2803	672	4	let	let	VERB
ejpam-2803	672	5	p	p	PRON
ejpam-2803	672	6	∈	∈	PROPN
ejpam-2803	672	7	v	v	NOUN
ejpam-2803	672	8	(	(	PUNCT
ejpam-2803	672	9	annr(m	annr(m	PROPN
ejpam-2803	672	10	)	)	PUNCT
ejpam-2803	672	11	)	)	PUNCT
ejpam-2803	672	12	.	.	PUNCT
ejpam-2803	673	1	since	since	SCONJ
ejpam-2803	673	2	m	m	PROPN
ejpam-2803	673	3	is	be	AUX
ejpam-2803	673	4	secondful	secondful	ADJ
ejpam-2803	673	5	,	,	PUNCT
ejpam-2803	673	6	there	there	PRON
ejpam-2803	673	7	exists	exist	VERB
ejpam-2803	673	8	s	s	PROPN
ejpam-2803	673	9	∈	∈	PROPN
ejpam-2803	673	10	specs(m	specs(m	NOUN
ejpam-2803	673	11	)	)	PUNCT
ejpam-2803	673	12	such	such	ADJ
ejpam-2803	673	13	that	that	DET
ejpam-2803	673	14	annr(s	annr(s	NOUN
ejpam-2803	673	15	)	)	PUNCT
ejpam-2803	673	16	=	=	VERB
ejpam-2803	674	1	p.	p.	NOUN
ejpam-2803	674	2	thus	thus	ADV
ejpam-2803	674	3	imp	imp	X
ejpam-2803	674	4	(	(	PUNCT
ejpam-2803	674	5	(	(	PUNCT
ejpam-2803	674	6	0	0	NUM
ejpam-2803	674	7	:	:	PUNCT
ejpam-2803	674	8	m	m	VERB
ejpam-2803	674	9	p	p	NOUN
ejpam-2803	674	10	)	)	PUNCT
ejpam-2803	674	11	)	)	PUNCT
ejpam-2803	674	12	6=	6=	X
ejpam-2803	674	13	(	(	PUNCT
ejpam-2803	674	14	0	0	NUM
ejpam-2803	674	15	)	)	PUNCT
ejpam-2803	674	16	by	by	ADP
ejpam-2803	674	17	[	[	X
ejpam-2803	674	18	4	4	NUM
ejpam-2803	674	19	,	,	PUNCT
ejpam-2803	674	20	corollary	corollary	ADJ
ejpam-2803	674	21	2.10	2.10	NUM
ejpam-2803	674	22	]	]	PUNCT
ejpam-2803	674	23	.	.	PUNCT
ejpam-2803	675	1	now	now	ADV
ejpam-2803	675	2	the	the	DET
ejpam-2803	675	3	proof	proof	NOUN
ejpam-2803	675	4	follows	follow	VERB
ejpam-2803	675	5	from	from	ADP
ejpam-2803	675	6	part	part	NOUN
ejpam-2803	675	7	(	(	PUNCT
ejpam-2803	675	8	a	a	NOUN
ejpam-2803	675	9	)	)	PUNCT
ejpam-2803	675	10	.	.	PUNCT
ejpam-2803	676	1	(	(	PUNCT
ejpam-2803	676	2	c	c	X
ejpam-2803	676	3	)	)	PUNCT
ejpam-2803	676	4	use	use	VERB
ejpam-2803	676	5	part	part	NOUN
ejpam-2803	676	6	(	(	PUNCT
ejpam-2803	676	7	b	b	NOUN
ejpam-2803	676	8	)	)	PUNCT
ejpam-2803	676	9	,	,	PUNCT
ejpam-2803	676	10	the	the	DET
ejpam-2803	676	11	fact	fact	NOUN
ejpam-2803	676	12	that	that	SCONJ
ejpam-2803	676	13	annr(m	annr(m	NOUN
ejpam-2803	676	14	)	)	PUNCT
ejpam-2803	676	15	=	=	SYM
ejpam-2803	676	16	(	(	PUNCT
ejpam-2803	676	17	0	0	NUM
ejpam-2803	676	18	)	)	PUNCT
ejpam-2803	676	19	,	,	PUNCT
ejpam-2803	676	20	and	and	CCONJ
ejpam-2803	676	21	that	that	SCONJ
ejpam-2803	676	22	if	if	SCONJ
ejpam-2803	676	23	m	m	NOUN
ejpam-2803	676	24	is	be	AUX
ejpam-2803	676	25	secondful	secondful	ADJ
ejpam-2803	676	26	,	,	PUNCT
ejpam-2803	676	27	then	then	ADV
ejpam-2803	676	28	for	for	ADP
ejpam-2803	676	29	every	every	DET
ejpam-2803	676	30	maximal	maximal	ADJ
ejpam-2803	676	31	ideal	ideal	NOUN
ejpam-2803	676	32	p	p	NOUN
ejpam-2803	676	33	of	of	ADP
ejpam-2803	676	34	r	r	NOUN
ejpam-2803	676	35	,	,	PUNCT
ejpam-2803	676	36	imp	imp	X
ejpam-2803	676	37	(	(	PUNCT
ejpam-2803	676	38	(	(	PUNCT
ejpam-2803	676	39	0	0	NUM
ejpam-2803	676	40	:	:	PUNCT
ejpam-2803	676	41	m	m	VERB
ejpam-2803	676	42	p	p	NOUN
ejpam-2803	676	43	)	)	PUNCT
ejpam-2803	676	44	)	)	PUNCT
ejpam-2803	676	45	=	=	PUNCT
ejpam-2803	677	1	(	(	PUNCT
ejpam-2803	677	2	0	0	NUM
ejpam-2803	677	3	:	:	PUNCT
ejpam-2803	677	4	m	m	VERB
ejpam-2803	677	5	p	p	NOUN
ejpam-2803	677	6	)	)	PUNCT
ejpam-2803	677	7	.	.	PUNCT
ejpam-2803	678	1	let	let	VERB
ejpam-2803	678	2	m	m	PRON
ejpam-2803	678	3	be	be	AUX
ejpam-2803	678	4	an	an	DET
ejpam-2803	678	5	r	r	NOUN
ejpam-2803	678	6	-	-	PUNCT
ejpam-2803	678	7	module	module	NOUN
ejpam-2803	678	8	.	.	PUNCT
ejpam-2803	679	1	m	m	PROPN
ejpam-2803	679	2	said	say	VERB
ejpam-2803	679	3	to	to	PART
ejpam-2803	679	4	be	be	AUX
ejpam-2803	679	5	a	a	DET
ejpam-2803	679	6	weak	weak	ADJ
ejpam-2803	679	7	comultiplication	comultiplication	NOUN
ejpam-2803	679	8	module	module	NOUN
ejpam-2803	679	9	if	if	SCONJ
ejpam-2803	679	10	m	m	NOUN
ejpam-2803	679	11	does	do	AUX
ejpam-2803	679	12	not	not	PART
ejpam-2803	679	13	have	have	VERB
ejpam-2803	679	14	any	any	DET
ejpam-2803	679	15	second	second	ADJ
ejpam-2803	679	16	submodule	submodule	NOUN
ejpam-2803	679	17	or	or	CCONJ
ejpam-2803	679	18	for	for	ADP
ejpam-2803	679	19	every	every	DET
ejpam-2803	679	20	second	second	ADJ
ejpam-2803	679	21	submodule	submodule	NOUN
ejpam-2803	679	22	s	s	PROPN
ejpam-2803	679	23	of	of	ADP
ejpam-2803	679	24	m	m	PRON
ejpam-2803	679	25	,	,	PUNCT
ejpam-2803	679	26	s	s	PART
ejpam-2803	679	27	=	=	PUNCT
ejpam-2803	679	28	(	(	PUNCT
ejpam-2803	679	29	0	0	NUM
ejpam-2803	679	30	:	:	PUNCT
ejpam-2803	679	31	m	m	VERB
ejpam-2803	679	32	i	i	NOUN
ejpam-2803	679	33	)	)	PUNCT
ejpam-2803	679	34	for	for	ADP
ejpam-2803	679	35	some	some	DET
ejpam-2803	679	36	ideal	ideal	ADJ
ejpam-2803	679	37	i	i	PRON
ejpam-2803	679	38	of	of	ADP
ejpam-2803	679	39	r	r	NOUN
ejpam-2803	679	40	(	(	PUNCT
ejpam-2803	679	41	see	see	VERB
ejpam-2803	679	42	[	[	X
ejpam-2803	679	43	5	5	NUM
ejpam-2803	679	44	]	]	PUNCT
ejpam-2803	679	45	)	)	PUNCT
ejpam-2803	679	46	.	.	PUNCT
ejpam-2803	680	1	assume	assume	VERB
ejpam-2803	680	2	that	that	SCONJ
ejpam-2803	680	3	v	v	ADP
ejpam-2803	680	4	s(sec(n	s(sec(n	PROPN
ejpam-2803	680	5	)	)	PUNCT
ejpam-2803	680	6	)	)	PUNCT
ejpam-2803	681	1	=	=	PUNCT
ejpam-2803	681	2	v	v	ADP
ejpam-2803	681	3	s∗(sec(n	s∗(sec(n	NOUN
ejpam-2803	681	4	)	)	PUNCT
ejpam-2803	681	5	)	)	PUNCT
ejpam-2803	682	1	for	for	ADP
ejpam-2803	682	2	every	every	DET
ejpam-2803	682	3	n	n	NOUN
ejpam-2803	682	4	≤	≤	NOUN
ejpam-2803	682	5	m	m	NOUN
ejpam-2803	682	6	.	.	PUNCT
ejpam-2803	683	1	then	then	ADV
ejpam-2803	683	2	clearly	clearly	ADV
ejpam-2803	683	3	,	,	PUNCT
ejpam-2803	683	4	τ	τ	PROPN
ejpam-2803	683	5	s∗m	s∗m	NUM
ejpam-2803	683	6	⊆	⊆	NUM
ejpam-2803	683	7	τ	τ	X
ejpam-2803	683	8	sm	sm	INTJ
ejpam-2803	683	9	and	and	CCONJ
ejpam-2803	683	10	hence	hence	ADV
ejpam-2803	683	11	m	m	VERB
ejpam-2803	683	12	is	be	AUX
ejpam-2803	683	13	a	a	DET
ejpam-2803	683	14	cotop	cotop	NOUN
ejpam-2803	683	15	module	module	NOUN
ejpam-2803	683	16	.	.	PUNCT
ejpam-2803	684	1	theorem	theorem	VERB
ejpam-2803	684	2	3.18	3.18	NUM
ejpam-2803	684	3	.	.	PUNCT
ejpam-2803	685	1	(	(	PUNCT
ejpam-2803	685	2	a	a	X
ejpam-2803	685	3	)	)	PUNCT
ejpam-2803	685	4	let	let	VERB
ejpam-2803	685	5	r	r	PRON
ejpam-2803	685	6	be	be	AUX
ejpam-2803	685	7	a	a	DET
ejpam-2803	685	8	perfect	perfect	ADJ
ejpam-2803	685	9	ring	ring	NOUN
ejpam-2803	685	10	.	.	PUNCT
ejpam-2803	686	1	then	then	ADV
ejpam-2803	686	2	the	the	DET
ejpam-2803	686	3	three	three	NUM
ejpam-2803	686	4	classes	class	NOUN
ejpam-2803	686	5	of	of	ADP
ejpam-2803	686	6	cotop	cotop	NOUN
ejpam-2803	686	7	,	,	PUNCT
ejpam-2803	686	8	weak	weak	ADJ
ejpam-2803	686	9	comultiplication	comultiplication	NOUN
ejpam-2803	686	10	,	,	PUNCT
ejpam-2803	686	11	and	and	CCONJ
ejpam-2803	686	12	xs	xs	NOUN
ejpam-2803	686	13	-	-	PUNCT
ejpam-2803	686	14	injective	injective	ADJ
ejpam-2803	686	15	modules	module	NOUN
ejpam-2803	686	16	are	be	AUX
ejpam-2803	686	17	all	all	ADV
ejpam-2803	686	18	equal	equal	ADJ
ejpam-2803	686	19	.	.	PUNCT
ejpam-2803	687	1	(	(	PUNCT
ejpam-2803	687	2	b	b	X
ejpam-2803	687	3	)	)	PUNCT
ejpam-2803	687	4	let	let	VERB
ejpam-2803	687	5	r	r	NOUN
ejpam-2803	687	6	be	be	AUX
ejpam-2803	687	7	a	a	DET
ejpam-2803	687	8	one	one	NUM
ejpam-2803	687	9	-	-	PUNCT
ejpam-2803	687	10	dimensional	dimensional	ADJ
ejpam-2803	687	11	integral	integral	ADJ
ejpam-2803	687	12	domain	domain	NOUN
ejpam-2803	687	13	and	and	CCONJ
ejpam-2803	687	14	let	let	VERB
ejpam-2803	687	15	m	m	PRON
ejpam-2803	687	16	be	be	AUX
ejpam-2803	687	17	an	an	DET
ejpam-2803	687	18	artinian	artinian	ADJ
ejpam-2803	687	19	r	r	NOUN
ejpam-2803	687	20	-	-	PUNCT
ejpam-2803	687	21	module	module	NOUN
ejpam-2803	687	22	.	.	PUNCT
ejpam-2803	688	1	then	then	ADV
ejpam-2803	688	2	m	m	PROPN
ejpam-2803	688	3	is	be	AUX
ejpam-2803	688	4	cotorsion	cotorsion	NOUN
ejpam-2803	688	5	or	or	CCONJ
ejpam-2803	688	6	cotorsion	cotorsion	NOUN
ejpam-2803	688	7	-	-	PUNCT
ejpam-2803	688	8	free	free	ADJ
ejpam-2803	688	9	xs	xs	NOUN
ejpam-2803	688	10	-	-	PUNCT
ejpam-2803	688	11	injective	injective	ADJ
ejpam-2803	688	12	if	if	SCONJ
ejpam-2803	688	13	and	and	CCONJ
ejpam-2803	688	14	only	only	ADV
ejpam-2803	688	15	if	if	SCONJ
ejpam-2803	688	16	m	m	NOUN
ejpam-2803	688	17	is	be	AUX
ejpam-2803	688	18	weak	weak	ADJ
ejpam-2803	688	19	comultiplication	comultiplication	NOUN
ejpam-2803	688	20	.	.	PUNCT
ejpam-2803	689	1	(	(	PUNCT
ejpam-2803	689	2	c	c	X
ejpam-2803	689	3	)	)	PUNCT
ejpam-2803	689	4	let	let	VERB
ejpam-2803	689	5	m	m	PRON
ejpam-2803	689	6	be	be	AUX
ejpam-2803	689	7	a	a	DET
ejpam-2803	689	8	secondful	secondful	ADJ
ejpam-2803	689	9	xs	xs	NOUN
ejpam-2803	689	10	-	-	PUNCT
ejpam-2803	689	11	injective	injective	ADJ
ejpam-2803	689	12	r	r	NOUN
ejpam-2803	689	13	-	-	PUNCT
ejpam-2803	689	14	module	module	NOUN
ejpam-2803	689	15	.	.	PUNCT
ejpam-2803	690	1	if	if	SCONJ
ejpam-2803	690	2	v	v	ADP
ejpam-2803	690	3	s(sec(n	s(sec(n	PROPN
ejpam-2803	690	4	)	)	PUNCT
ejpam-2803	690	5	)	)	PUNCT
ejpam-2803	691	1	=	=	PUNCT
ejpam-2803	691	2	v	v	ADP
ejpam-2803	691	3	s∗(sec(n	s∗(sec(n	NOUN
ejpam-2803	691	4	)	)	PUNCT
ejpam-2803	691	5	)	)	PUNCT
ejpam-2803	691	6	for	for	ADP
ejpam-2803	691	7	every	every	DET
ejpam-2803	691	8	submodule	submodule	NOUN
ejpam-2803	691	9	n	n	PROPN
ejpam-2803	691	10	of	of	ADP
ejpam-2803	691	11	m	m	PRON
ejpam-2803	691	12	,	,	PUNCT
ejpam-2803	691	13	then	then	ADV
ejpam-2803	691	14	(	(	PUNCT
ejpam-2803	691	15	specs(m	specs(m	NOUN
ejpam-2803	691	16	)	)	PUNCT
ejpam-2803	691	17	,	,	PUNCT
ejpam-2803	691	18	τ	τ	PROPN
ejpam-2803	691	19	s∗	s∗	PROPN
ejpam-2803	691	20	)	)	PUNCT
ejpam-2803	691	21	is	be	AUX
ejpam-2803	691	22	homeomorphic	homeomorphic	ADJ
ejpam-2803	691	23	to	to	ADP
ejpam-2803	691	24	spec(r	spec(r	PROPN
ejpam-2803	691	25	)	)	PUNCT
ejpam-2803	691	26	.	.	PUNCT
ejpam-2803	692	1	therefore	therefore	ADV
ejpam-2803	692	2	,	,	PUNCT
ejpam-2803	692	3	(	(	PUNCT
ejpam-2803	692	4	specs(m	specs(m	NOUN
ejpam-2803	692	5	)	)	PUNCT
ejpam-2803	692	6	,	,	PUNCT
ejpam-2803	692	7	τ	τ	PROPN
ejpam-2803	692	8	s∗	s∗	PROPN
ejpam-2803	692	9	)	)	PUNCT
ejpam-2803	692	10	is	be	AUX
ejpam-2803	692	11	a	a	DET
ejpam-2803	692	12	spectral	spectral	ADJ
ejpam-2803	692	13	space	space	NOUN
ejpam-2803	692	14	.	.	PUNCT
ejpam-2803	693	1	h.	h.	PROPN
ejpam-2803	693	2	ansari	ansari	PROPN
ejpam-2803	693	3	-	-	PUNCT
ejpam-2803	693	4	toroghy	toroghy	NOUN
ejpam-2803	693	5	,	,	PUNCT
ejpam-2803	693	6	s.	s.	PROPN
ejpam-2803	693	7	s.	s.	PROPN
ejpam-2803	693	8	pourmortazavi	pourmortazavi	VERB
ejpam-2803	693	9	/	/	SYM
ejpam-2803	693	10	eur	eur	PROPN
ejpam-2803	693	11	.	.	PUNCT
ejpam-2803	694	1	j.	j.	PROPN
ejpam-2803	694	2	pure	pure	PROPN
ejpam-2803	694	3	appl	appl	PROPN
ejpam-2803	694	4	.	.	PROPN
ejpam-2803	694	5	math	math	PROPN
ejpam-2803	694	6	,	,	PUNCT
ejpam-2803	694	7	10	10	NUM
ejpam-2803	694	8	(	(	PUNCT
ejpam-2803	694	9	2	2	NUM
ejpam-2803	694	10	)	)	PUNCT
ejpam-2803	694	11	(	(	PUNCT
ejpam-2803	694	12	2017	2017	NUM
ejpam-2803	694	13	)	)	PUNCT
ejpam-2803	694	14	,	,	PUNCT
ejpam-2803	694	15	211	211	NUM
ejpam-2803	694	16	-	-	SYM
ejpam-2803	694	17	230	230	NUM
ejpam-2803	694	18	227	227	NUM
ejpam-2803	694	19	(	(	PUNCT
ejpam-2803	694	20	d	d	X
ejpam-2803	694	21	)	)	PUNCT
ejpam-2803	694	22	let	let	VERB
ejpam-2803	694	23	m	m	PRON
ejpam-2803	694	24	be	be	AUX
ejpam-2803	694	25	an	an	DET
ejpam-2803	694	26	xs	xs	NOUN
ejpam-2803	694	27	-	-	PUNCT
ejpam-2803	694	28	injective	injective	ADJ
ejpam-2803	694	29	artinian	artinian	ADJ
ejpam-2803	694	30	r	r	NOUN
ejpam-2803	694	31	-	-	PUNCT
ejpam-2803	694	32	module	module	NOUN
ejpam-2803	694	33	.	.	PUNCT
ejpam-2803	695	1	if	if	SCONJ
ejpam-2803	695	2	for	for	ADP
ejpam-2803	695	3	any	any	DET
ejpam-2803	695	4	p	p	NOUN
ejpam-2803	695	5	∈	∈	PROPN
ejpam-2803	695	6	v	v	NOUN
ejpam-2803	695	7	(	(	PUNCT
ejpam-2803	695	8	annr(m	annr(m	PROPN
ejpam-2803	695	9	)	)	PUNCT
ejpam-2803	695	10	)	)	PUNCT
ejpam-2803	695	11	and	and	CCONJ
ejpam-2803	695	12	every	every	DET
ejpam-2803	695	13	family	family	NOUN
ejpam-2803	695	14	{	{	PUNCT
ejpam-2803	695	15	pi}i∈i	pi}i∈i	INTJ
ejpam-2803	695	16	,	,	PUNCT
ejpam-2803	695	17	where	where	SCONJ
ejpam-2803	695	18	pi	pi	NOUN
ejpam-2803	695	19	∈	∈	PROPN
ejpam-2803	695	20	v	v	NOUN
ejpam-2803	695	21	(	(	PUNCT
ejpam-2803	695	22	annr(m	annr(m	PROPN
ejpam-2803	695	23	)	)	PUNCT
ejpam-2803	695	24	)	)	PUNCT
ejpam-2803	695	25	,	,	PUNCT
ejpam-2803	695	26	⋂	⋂	PROPN
ejpam-2803	695	27	i∈i	i∈i	ADJ
ejpam-2803	695	28	pi	pi	NOUN
ejpam-2803	695	29	⊆	⊆	NUM
ejpam-2803	695	30	p	p	NOUN
ejpam-2803	695	31	implies	imply	VERB
ejpam-2803	695	32	that	that	SCONJ
ejpam-2803	695	33	imp	imp	X
ejpam-2803	695	34	(	(	PUNCT
ejpam-2803	695	35	(	(	PUNCT
ejpam-2803	695	36	0	0	NUM
ejpam-2803	695	37	:	:	PUNCT
ejpam-2803	695	38	m	m	VERB
ejpam-2803	695	39	p	p	NOUN
ejpam-2803	695	40	)	)	PUNCT
ejpam-2803	695	41	)	)	PUNCT
ejpam-2803	696	1	⊆	⊆	NUM
ejpam-2803	696	2	∑	∑	PUNCT
ejpam-2803	696	3	i∈i	i∈i	ADJ
ejpam-2803	696	4	i	i	PRON
ejpam-2803	696	5	m	m	VERB
ejpam-2803	696	6	pi	pi	NOUN
ejpam-2803	696	7	(	(	PUNCT
ejpam-2803	696	8	(	(	PUNCT
ejpam-2803	696	9	0	0	NUM
ejpam-2803	696	10	:	:	PUNCT
ejpam-2803	696	11	m	m	NOUN
ejpam-2803	696	12	pi	pi	NOUN
ejpam-2803	696	13	)	)	PUNCT
ejpam-2803	696	14	)	)	PUNCT
ejpam-2803	696	15	,	,	PUNCT
ejpam-2803	696	16	then	then	ADV
ejpam-2803	696	17	we	we	PRON
ejpam-2803	696	18	have	have	VERB
ejpam-2803	696	19	v	v	ADP
ejpam-2803	696	20	s(sec(n	s(sec(n	NOUN
ejpam-2803	696	21	)	)	PUNCT
ejpam-2803	696	22	)	)	PUNCT
ejpam-2803	697	1	=	=	PUNCT
ejpam-2803	697	2	v	v	ADP
ejpam-2803	697	3	s∗(sec(n	s∗(sec(n	NOUN
ejpam-2803	697	4	)	)	PUNCT
ejpam-2803	697	5	)	)	PUNCT
ejpam-2803	697	6	.	.	PUNCT
ejpam-2803	698	1	proof	proof	NOUN
ejpam-2803	698	2	.	.	PUNCT
ejpam-2803	699	1	(	(	PUNCT
ejpam-2803	699	2	a	a	X
ejpam-2803	699	3	)	)	PUNCT
ejpam-2803	699	4	first	first	ADV
ejpam-2803	699	5	we	we	PRON
ejpam-2803	699	6	assume	assume	VERB
ejpam-2803	699	7	that	that	SCONJ
ejpam-2803	699	8	m	m	PROPN
ejpam-2803	699	9	is	be	AUX
ejpam-2803	699	10	a	a	DET
ejpam-2803	699	11	cotop	cotop	NOUN
ejpam-2803	699	12	r	r	NOUN
ejpam-2803	699	13	-	-	PUNCT
ejpam-2803	699	14	module	module	NOUN
ejpam-2803	699	15	and	and	CCONJ
ejpam-2803	699	16	show	show	VERB
ejpam-2803	699	17	that	that	SCONJ
ejpam-2803	699	18	it	it	PRON
ejpam-2803	699	19	is	be	AUX
ejpam-2803	699	20	weak	weak	ADJ
ejpam-2803	699	21	comultiplication	comultiplication	NOUN
ejpam-2803	699	22	.	.	PUNCT
ejpam-2803	700	1	to	to	PART
ejpam-2803	700	2	see	see	VERB
ejpam-2803	700	3	this	this	PRON
ejpam-2803	700	4	,	,	PUNCT
ejpam-2803	700	5	let	let	VERB
ejpam-2803	700	6	s	s	PRON
ejpam-2803	700	7	∈	∈	PROPN
ejpam-2803	700	8	specs(m	specs(m	PROPN
ejpam-2803	700	9	)	)	PUNCT
ejpam-2803	700	10	.	.	PUNCT
ejpam-2803	701	1	clearly	clearly	ADV
ejpam-2803	701	2	,	,	PUNCT
ejpam-2803	701	3	s	s	VERB
ejpam-2803	701	4	⊆	⊆	NUM
ejpam-2803	701	5	(	(	PUNCT
ejpam-2803	701	6	0	0	NUM
ejpam-2803	701	7	:	:	PUNCT
ejpam-2803	701	8	m	m	PROPN
ejpam-2803	701	9	annr(s	annr(s	NOUN
ejpam-2803	701	10	)	)	PUNCT
ejpam-2803	701	11	)	)	PUNCT
ejpam-2803	701	12	.	.	PUNCT
ejpam-2803	702	1	since	since	SCONJ
ejpam-2803	702	2	r	r	NOUN
ejpam-2803	702	3	is	be	AUX
ejpam-2803	702	4	perfect	perfect	ADJ
ejpam-2803	702	5	,	,	PUNCT
ejpam-2803	702	6	(	(	PUNCT
ejpam-2803	702	7	0	0	NUM
ejpam-2803	702	8	:	:	PUNCT
ejpam-2803	702	9	m	m	PROPN
ejpam-2803	702	10	annr(s	annr(s	NOUN
ejpam-2803	702	11	)	)	PUNCT
ejpam-2803	702	12	)	)	PUNCT
ejpam-2803	702	13	a	a	DET
ejpam-2803	702	14	simple	simple	ADJ
ejpam-2803	702	15	r	r	NOUN
ejpam-2803	702	16	/	/	SYM
ejpam-2803	702	17	annr(s)-module	annr(s)-module	NOUN
ejpam-2803	702	18	by	by	ADP
ejpam-2803	702	19	[	[	X
ejpam-2803	702	20	7	7	NUM
ejpam-2803	702	21	,	,	PUNCT
ejpam-2803	702	22	corollary	corollary	ADJ
ejpam-2803	702	23	2.6	2.6	NUM
ejpam-2803	702	24	(	(	PUNCT
ejpam-2803	702	25	a	a	NOUN
ejpam-2803	702	26	)	)	PUNCT
ejpam-2803	702	27	and	and	CCONJ
ejpam-2803	702	28	(	(	PUNCT
ejpam-2803	702	29	d	d	NOUN
ejpam-2803	702	30	)	)	PUNCT
ejpam-2803	702	31	]	]	PUNCT
ejpam-2803	702	32	and	and	CCONJ
ejpam-2803	702	33	hence	hence	ADV
ejpam-2803	702	34	a	a	DET
ejpam-2803	702	35	simple	simple	ADJ
ejpam-2803	702	36	r	r	NOUN
ejpam-2803	702	37	-	-	PUNCT
ejpam-2803	702	38	module	module	NOUN
ejpam-2803	702	39	.	.	PUNCT
ejpam-2803	703	1	this	this	PRON
ejpam-2803	703	2	implies	imply	VERB
ejpam-2803	703	3	that	that	PRON
ejpam-2803	703	4	s	s	VERB
ejpam-2803	703	5	=	=	X
ejpam-2803	703	6	(	(	PUNCT
ejpam-2803	703	7	0	0	NUM
ejpam-2803	703	8	:	:	PUNCT
ejpam-2803	703	9	m	m	PROPN
ejpam-2803	703	10	annr(s	annr(s	NOUN
ejpam-2803	703	11	)	)	PUNCT
ejpam-2803	703	12	)	)	PUNCT
ejpam-2803	703	13	,	,	PUNCT
ejpam-2803	703	14	as	as	SCONJ
ejpam-2803	703	15	desired	desire	VERB
ejpam-2803	703	16	.	.	PUNCT
ejpam-2803	704	1	also	also	ADV
ejpam-2803	704	2	,	,	PUNCT
ejpam-2803	704	3	it	it	PRON
ejpam-2803	704	4	is	be	AUX
ejpam-2803	704	5	clear	clear	ADJ
ejpam-2803	704	6	every	every	DET
ejpam-2803	704	7	weak	weak	ADJ
ejpam-2803	704	8	comultiplication	comultiplication	NOUN
ejpam-2803	704	9	module	module	NOUN
ejpam-2803	704	10	is	be	AUX
ejpam-2803	704	11	an	an	DET
ejpam-2803	704	12	xs	xs	NOUN
ejpam-2803	704	13	-	-	PUNCT
ejpam-2803	704	14	injective	injective	ADJ
ejpam-2803	704	15	module	module	NOUN
ejpam-2803	704	16	.	.	PUNCT
ejpam-2803	705	1	to	to	PART
ejpam-2803	705	2	complete	complete	VERB
ejpam-2803	705	3	the	the	DET
ejpam-2803	705	4	proof	proof	NOUN
ejpam-2803	705	5	,	,	PUNCT
ejpam-2803	705	6	we	we	PRON
ejpam-2803	705	7	assume	assume	VERB
ejpam-2803	705	8	that	that	SCONJ
ejpam-2803	705	9	m	m	PROPN
ejpam-2803	705	10	is	be	AUX
ejpam-2803	705	11	an	an	DET
ejpam-2803	705	12	xs	xs	NOUN
ejpam-2803	705	13	-	-	PUNCT
ejpam-2803	705	14	injective	injective	ADJ
ejpam-2803	705	15	module	module	NOUN
ejpam-2803	705	16	and	and	CCONJ
ejpam-2803	705	17	show	show	VERB
ejpam-2803	705	18	that	that	SCONJ
ejpam-2803	705	19	m	m	PROPN
ejpam-2803	705	20	is	be	AUX
ejpam-2803	705	21	a	a	DET
ejpam-2803	705	22	cotop	cotop	NOUN
ejpam-2803	705	23	module	module	NOUN
ejpam-2803	705	24	.	.	PUNCT
ejpam-2803	706	1	to	to	PART
ejpam-2803	706	2	see	see	VERB
ejpam-2803	706	3	this	this	PRON
ejpam-2803	706	4	,	,	PUNCT
ejpam-2803	706	5	let	let	VERB
ejpam-2803	706	6	s	s	PRON
ejpam-2803	706	7	be	be	AUX
ejpam-2803	706	8	a	a	DET
ejpam-2803	706	9	second	second	ADJ
ejpam-2803	706	10	submodule	submodule	NOUN
ejpam-2803	706	11	and	and	CCONJ
ejpam-2803	706	12	let	let	VERB
ejpam-2803	706	13	n	n	PRON
ejpam-2803	706	14	,	,	PUNCT
ejpam-2803	706	15	l	l	NOUN
ejpam-2803	706	16	are	be	AUX
ejpam-2803	706	17	socle	socle	NOUN
ejpam-2803	706	18	submodules	submodule	NOUN
ejpam-2803	706	19	of	of	ADP
ejpam-2803	706	20	m	m	PRON
ejpam-2803	706	21	such	such	ADJ
ejpam-2803	706	22	that	that	PRON
ejpam-2803	706	23	s	s	VERB
ejpam-2803	706	24	⊆	⊆	NUM
ejpam-2803	706	25	n	n	PROPN
ejpam-2803	706	26	+	+	CCONJ
ejpam-2803	706	27	l.	l.	NOUN
ejpam-2803	706	28	put	put	VERB
ejpam-2803	706	29	n	n	NOUN
ejpam-2803	706	30	=	=	PUNCT
ejpam-2803	706	31	∑	∑	PUNCT
ejpam-2803	706	32	α∈i	α∈i	NOUN
ejpam-2803	706	33	nα	nα	VERB
ejpam-2803	706	34	and	and	CCONJ
ejpam-2803	706	35	l	l	NOUN
ejpam-2803	706	36	=	=	PUNCT
ejpam-2803	706	37	∑	∑	PUNCT
ejpam-2803	706	38	β∈j	β∈j	PROPN
ejpam-2803	706	39	lβ	lβ	PROPN
ejpam-2803	706	40	,	,	PUNCT
ejpam-2803	706	41	where	where	SCONJ
ejpam-2803	706	42	nα	nα	NOUN
ejpam-2803	706	43	and	and	CCONJ
ejpam-2803	706	44	lβ	lβ	PROPN
ejpam-2803	706	45	are	be	AUX
ejpam-2803	706	46	second	second	ADJ
ejpam-2803	706	47	submodules	submodule	NOUN
ejpam-2803	706	48	of	of	ADP
ejpam-2803	706	49	m	m	PRON
ejpam-2803	706	50	for	for	ADP
ejpam-2803	706	51	each	each	DET
ejpam-2803	706	52	α	α	NOUN
ejpam-2803	706	53	∈	∈	PROPN
ejpam-2803	707	1	i	i	PRON
ejpam-2803	707	2	and	and	CCONJ
ejpam-2803	707	3	β	β	X
ejpam-2803	707	4	∈	∈	PROPN
ejpam-2803	707	5	j	j	PROPN
ejpam-2803	707	6	.	.	PUNCT
ejpam-2803	708	1	we	we	PRON
ejpam-2803	708	2	have	have	AUX
ejpam-2803	708	3	annr(n	annr(n	VERB
ejpam-2803	708	4	+	+	NOUN
ejpam-2803	708	5	l	l	NOUN
ejpam-2803	708	6	)	)	PUNCT
ejpam-2803	708	7	=	=	SYM
ejpam-2803	708	8	annr(n	annr(n	VERB
ejpam-2803	708	9	)	)	PUNCT
ejpam-2803	708	10	∩	∩	NOUN
ejpam-2803	708	11	annr(l	annr(l	ADJ
ejpam-2803	708	12	)	)	PUNCT
ejpam-2803	708	13	⊆	⊆	NUM
ejpam-2803	708	14	annr(s	annr(s	NOUN
ejpam-2803	708	15	)	)	PUNCT
ejpam-2803	708	16	and	and	CCONJ
ejpam-2803	708	17	so	so	ADV
ejpam-2803	708	18	annr(n	annr(n	ADJ
ejpam-2803	708	19	)	)	PUNCT
ejpam-2803	708	20	⊆	⊆	NUM
ejpam-2803	708	21	annr(s	annr(s	NOUN
ejpam-2803	708	22	)	)	PUNCT
ejpam-2803	708	23	or	or	CCONJ
ejpam-2803	708	24	annr(l	annr(l	PRON
ejpam-2803	708	25	)	)	PUNCT
ejpam-2803	708	26	⊆	⊆	NUM
ejpam-2803	708	27	annr(s	annr(s	NOUN
ejpam-2803	708	28	)	)	PUNCT
ejpam-2803	708	29	.	.	PUNCT
ejpam-2803	709	1	thus	thus	ADV
ejpam-2803	709	2	⋂	⋂	PROPN
ejpam-2803	709	3	α∈i	α∈i	NUM
ejpam-2803	709	4	annr(nα	annr(nα	NOUN
ejpam-2803	709	5	)	)	PUNCT
ejpam-2803	709	6	⊆	⊆	NUM
ejpam-2803	709	7	annr(s	annr(s	NOUN
ejpam-2803	709	8	)	)	PUNCT
ejpam-2803	709	9	or	or	CCONJ
ejpam-2803	709	10	⋂	⋂	PROPN
ejpam-2803	709	11	β∈j	β∈j	ADJ
ejpam-2803	709	12	annr(lβ	annr(lβ	NOUN
ejpam-2803	709	13	)	)	PUNCT
ejpam-2803	709	14	⊆	⊆	NUM
ejpam-2803	709	15	annr(s	annr(s	NOUN
ejpam-2803	709	16	)	)	PUNCT
ejpam-2803	709	17	.	.	PUNCT
ejpam-2803	710	1	now	now	ADV
ejpam-2803	710	2	since	since	SCONJ
ejpam-2803	710	3	r	r	NOUN
ejpam-2803	710	4	is	be	AUX
ejpam-2803	710	5	a	a	DET
ejpam-2803	710	6	perfect	perfect	ADJ
ejpam-2803	710	7	ring	ring	NOUN
ejpam-2803	710	8	,	,	PUNCT
ejpam-2803	710	9	both	both	CCONJ
ejpam-2803	710	10	i	i	PROPN
ejpam-2803	710	11	and	and	CCONJ
ejpam-2803	710	12	j	j	PROPN
ejpam-2803	710	13	are	be	AUX
ejpam-2803	710	14	finite	finite	ADJ
ejpam-2803	710	15	index	index	NOUN
ejpam-2803	710	16	sets	set	NOUN
ejpam-2803	710	17	.	.	PUNCT
ejpam-2803	711	1	consequently	consequently	ADV
ejpam-2803	711	2	,	,	PUNCT
ejpam-2803	711	3	there	there	PRON
ejpam-2803	711	4	exists	exist	VERB
ejpam-2803	711	5	α0	α0	PROPN
ejpam-2803	711	6	∈	∈	PROPN
ejpam-2803	711	7	i	i	PRON
ejpam-2803	711	8	or	or	CCONJ
ejpam-2803	711	9	β0	β0	PROPN
ejpam-2803	711	10	∈	∈	PROPN
ejpam-2803	711	11	j	j	NOUN
ejpam-2803	711	12	such	such	ADJ
ejpam-2803	711	13	that	that	SCONJ
ejpam-2803	711	14	annr(nα0	annr(nα0	PROPN
ejpam-2803	711	15	)	)	PUNCT
ejpam-2803	711	16	=	=	SYM
ejpam-2803	711	17	annr(s	annr(s	NOUN
ejpam-2803	711	18	)	)	PUNCT
ejpam-2803	711	19	or	or	CCONJ
ejpam-2803	711	20	annr(lβ0	annr(lβ0	NOUN
ejpam-2803	711	21	)	)	PUNCT
ejpam-2803	711	22	=	=	SYM
ejpam-2803	711	23	annr(s	annr(s	NOUN
ejpam-2803	711	24	)	)	PUNCT
ejpam-2803	711	25	.	.	PUNCT
ejpam-2803	712	1	as	as	SCONJ
ejpam-2803	712	2	m	m	PROPN
ejpam-2803	712	3	is	be	AUX
ejpam-2803	712	4	an	an	DET
ejpam-2803	712	5	xs	xs	NOUN
ejpam-2803	712	6	-	-	PUNCT
ejpam-2803	712	7	injective	injective	ADJ
ejpam-2803	712	8	module	module	NOUN
ejpam-2803	712	9	,	,	PUNCT
ejpam-2803	712	10	we	we	PRON
ejpam-2803	712	11	have	have	VERB
ejpam-2803	712	12	nα0	nα0	NOUN
ejpam-2803	712	13	=	=	SYM
ejpam-2803	712	14	s	s	NOUN
ejpam-2803	712	15	or	or	CCONJ
ejpam-2803	712	16	lβ0	lβ0	X
ejpam-2803	712	17	=	=	PUNCT
ejpam-2803	713	1	s.	s.	PROPN
ejpam-2803	713	2	this	this	PRON
ejpam-2803	713	3	implies	imply	VERB
ejpam-2803	713	4	that	that	SCONJ
ejpam-2803	713	5	s	s	VERB
ejpam-2803	713	6	⊆	⊆	NUM
ejpam-2803	713	7	n	n	NOUN
ejpam-2803	713	8	or	or	CCONJ
ejpam-2803	713	9	s	s	PRON
ejpam-2803	713	10	⊆	⊆	NUM
ejpam-2803	713	11	l.	l.	NOUN
ejpam-2803	713	12	hence	hence	ADV
ejpam-2803	713	13	m	m	PROPN
ejpam-2803	713	14	is	be	AUX
ejpam-2803	713	15	a	a	DET
ejpam-2803	713	16	cotop	cotop	NOUN
ejpam-2803	713	17	module	module	NOUN
ejpam-2803	713	18	.	.	PUNCT
ejpam-2803	714	1	(	(	PUNCT
ejpam-2803	714	2	b	b	X
ejpam-2803	714	3	)	)	PUNCT
ejpam-2803	714	4	let	let	VERB
ejpam-2803	714	5	m	m	PRON
ejpam-2803	714	6	be	be	AUX
ejpam-2803	714	7	a	a	DET
ejpam-2803	714	8	cotorsion	cotorsion	NOUN
ejpam-2803	714	9	-	-	PUNCT
ejpam-2803	714	10	free	free	ADJ
ejpam-2803	714	11	xs	xs	NOUN
ejpam-2803	714	12	-	-	PUNCT
ejpam-2803	714	13	injective	injective	ADJ
ejpam-2803	714	14	,	,	PUNCT
ejpam-2803	714	15	and	and	CCONJ
ejpam-2803	714	16	let	let	VERB
ejpam-2803	714	17	s	s	PRON
ejpam-2803	714	18	be	be	AUX
ejpam-2803	714	19	a	a	DET
ejpam-2803	714	20	second	second	ADJ
ejpam-2803	714	21	submodule	submodule	NOUN
ejpam-2803	714	22	of	of	ADP
ejpam-2803	714	23	m	m	PROPN
ejpam-2803	714	24	.	.	PUNCT
ejpam-2803	715	1	then	then	ADV
ejpam-2803	715	2	we	we	PRON
ejpam-2803	715	3	have	have	VERB
ejpam-2803	715	4	s	s	NOUN
ejpam-2803	715	5	=	=	X
ejpam-2803	715	6	imp	imp	X
ejpam-2803	715	7	(	(	PUNCT
ejpam-2803	715	8	s	s	NOUN
ejpam-2803	715	9	)	)	PUNCT
ejpam-2803	715	10	=	=	SYM
ejpam-2803	715	11	imp	imp	X
ejpam-2803	715	12	(	(	PUNCT
ejpam-2803	715	13	0	0	NUM
ejpam-2803	715	14	:	:	PUNCT
ejpam-2803	715	15	m	m	VERB
ejpam-2803	715	16	p	p	X
ejpam-2803	715	17	)	)	PUNCT
ejpam-2803	715	18	for	for	ADP
ejpam-2803	715	19	p	p	NOUN
ejpam-2803	715	20	=	=	PROPN
ejpam-2803	715	21	annr(s	annr(s	PROPN
ejpam-2803	715	22	)	)	PUNCT
ejpam-2803	715	23	by	by	ADP
ejpam-2803	715	24	[	[	X
ejpam-2803	715	25	4	4	NUM
ejpam-2803	715	26	,	,	PUNCT
ejpam-2803	715	27	corollary	corollary	ADJ
ejpam-2803	715	28	2.10	2.10	NUM
ejpam-2803	715	29	]	]	PUNCT
ejpam-2803	715	30	.	.	PUNCT
ejpam-2803	716	1	if	if	SCONJ
ejpam-2803	716	2	p	p	X
ejpam-2803	716	3	=	=	X
ejpam-2803	716	4	(	(	PUNCT
ejpam-2803	716	5	0	0	NUM
ejpam-2803	716	6	)	)	PUNCT
ejpam-2803	716	7	,	,	PUNCT
ejpam-2803	716	8	then	then	ADV
ejpam-2803	716	9	s	s	VERB
ejpam-2803	716	10	=	=	SYM
ejpam-2803	716	11	im0	im0	X
ejpam-2803	716	12	(	(	PUNCT
ejpam-2803	716	13	s	s	NOUN
ejpam-2803	716	14	)	)	PUNCT
ejpam-2803	716	15	=	=	SYM
ejpam-2803	716	16	im0	im0	X
ejpam-2803	716	17	(	(	PUNCT
ejpam-2803	716	18	m	m	NOUN
ejpam-2803	716	19	)	)	PUNCT
ejpam-2803	716	20	=	=	PUNCT
ejpam-2803	717	1	m	m	NOUN
ejpam-2803	717	2	=	=	SYM
ejpam-2803	717	3	(	(	PUNCT
ejpam-2803	717	4	0	0	NUM
ejpam-2803	717	5	:	:	PUNCT
ejpam-2803	717	6	m	m	PROPN
ejpam-2803	717	7	0	0	NUM
ejpam-2803	717	8	)	)	PUNCT
ejpam-2803	717	9	.	.	PUNCT
ejpam-2803	718	1	if	if	SCONJ
ejpam-2803	718	2	p	p	PROPN
ejpam-2803	718	3	6=	6=	PROPN
ejpam-2803	718	4	(	(	PUNCT
ejpam-2803	718	5	0	0	NUM
ejpam-2803	718	6	)	)	PUNCT
ejpam-2803	718	7	,	,	PUNCT
ejpam-2803	718	8	then	then	ADV
ejpam-2803	718	9	p	p	NOUN
ejpam-2803	718	10	is	be	AUX
ejpam-2803	718	11	a	a	DET
ejpam-2803	718	12	maximal	maximal	ADJ
ejpam-2803	718	13	ideal	ideal	NOUN
ejpam-2803	718	14	so	so	SCONJ
ejpam-2803	718	15	that	that	PRON
ejpam-2803	718	16	s	s	VERB
ejpam-2803	718	17	=	=	X
ejpam-2803	718	18	imp	imp	X
ejpam-2803	718	19	(	(	PUNCT
ejpam-2803	718	20	0	0	NUM
ejpam-2803	718	21	:	:	PUNCT
ejpam-2803	718	22	m	m	VERB
ejpam-2803	718	23	p	p	ADJ
ejpam-2803	718	24	)	)	PUNCT
ejpam-2803	718	25	=	=	SYM
ejpam-2803	719	1	(	(	PUNCT
ejpam-2803	719	2	0	0	NUM
ejpam-2803	719	3	:	:	PUNCT
ejpam-2803	719	4	m	m	AUX
ejpam-2803	719	5	p	p	NOUN
ejpam-2803	719	6	)	)	PUNCT
ejpam-2803	719	7	by	by	ADP
ejpam-2803	719	8	[	[	X
ejpam-2803	719	9	30	30	NUM
ejpam-2803	719	10	,	,	PUNCT
ejpam-2803	719	11	proposition	proposition	NOUN
ejpam-2803	719	12	1.4	1.4	NUM
ejpam-2803	719	13	]	]	PUNCT
ejpam-2803	719	14	.	.	PUNCT
ejpam-2803	720	1	now	now	ADV
ejpam-2803	720	2	let	let	VERB
ejpam-2803	720	3	m	m	PRON
ejpam-2803	720	4	be	be	AUX
ejpam-2803	720	5	a	a	DET
ejpam-2803	720	6	cotorsion	cotorsion	NOUN
ejpam-2803	720	7	xs	xs	NOUN
ejpam-2803	720	8	-	-	PUNCT
ejpam-2803	720	9	injective	injective	ADJ
ejpam-2803	720	10	r	r	NOUN
ejpam-2803	720	11	-	-	PUNCT
ejpam-2803	720	12	module	module	NOUN
ejpam-2803	720	13	and	and	CCONJ
ejpam-2803	720	14	s	s	VERB
ejpam-2803	720	15	a	a	DET
ejpam-2803	720	16	p	p	ADJ
ejpam-2803	720	17	-	-	PUNCT
ejpam-2803	720	18	second	second	NOUN
ejpam-2803	720	19	submodule	submodule	NOUN
ejpam-2803	720	20	of	of	ADP
ejpam-2803	720	21	m	m	PROPN
ejpam-2803	720	22	.	.	PUNCT
ejpam-2803	721	1	if	if	SCONJ
ejpam-2803	721	2	p	p	X
ejpam-2803	721	3	=	=	X
ejpam-2803	721	4	(	(	PUNCT
ejpam-2803	721	5	0	0	NUM
ejpam-2803	721	6	)	)	PUNCT
ejpam-2803	721	7	,	,	PUNCT
ejpam-2803	721	8	then	then	ADV
ejpam-2803	721	9	s	s	VERB
ejpam-2803	721	10	=	=	SYM
ejpam-2803	721	11	im0	im0	X
ejpam-2803	721	12	(	(	PUNCT
ejpam-2803	721	13	s	s	NOUN
ejpam-2803	721	14	)	)	PUNCT
ejpam-2803	721	15	=	=	SYM
ejpam-2803	721	16	im0	im0	X
ejpam-2803	721	17	(	(	PUNCT
ejpam-2803	721	18	m	m	NOUN
ejpam-2803	721	19	)	)	PUNCT
ejpam-2803	721	20	=	=	SYM
ejpam-2803	721	21	(	(	PUNCT
ejpam-2803	721	22	0	0	NUM
ejpam-2803	721	23	)	)	PUNCT
ejpam-2803	721	24	,	,	PUNCT
ejpam-2803	721	25	a	a	DET
ejpam-2803	721	26	contradiction	contradiction	NOUN
ejpam-2803	721	27	.	.	PUNCT
ejpam-2803	722	1	this	this	PRON
ejpam-2803	722	2	implies	imply	VERB
ejpam-2803	722	3	that	that	SCONJ
ejpam-2803	722	4	p	p	PROPN
ejpam-2803	722	5	6=	6=	PROPN
ejpam-2803	722	6	(	(	PUNCT
ejpam-2803	722	7	0	0	NUM
ejpam-2803	722	8	)	)	PUNCT
ejpam-2803	722	9	and	and	CCONJ
ejpam-2803	722	10	hence	hence	ADV
ejpam-2803	722	11	s	s	PART
ejpam-2803	722	12	=	=	X
ejpam-2803	722	13	imp	imp	X
ejpam-2803	722	14	(	(	PUNCT
ejpam-2803	722	15	s	s	NOUN
ejpam-2803	722	16	)	)	PUNCT
ejpam-2803	722	17	=	=	SYM
ejpam-2803	722	18	imp	imp	X
ejpam-2803	722	19	(	(	PUNCT
ejpam-2803	722	20	0	0	NUM
ejpam-2803	722	21	:	:	PUNCT
ejpam-2803	722	22	m	m	VERB
ejpam-2803	722	23	p	p	ADJ
ejpam-2803	722	24	)	)	PUNCT
ejpam-2803	722	25	=	=	SYM
ejpam-2803	722	26	(	(	PUNCT
ejpam-2803	722	27	0	0	NUM
ejpam-2803	722	28	:	:	PUNCT
ejpam-2803	722	29	m	m	VERB
ejpam-2803	722	30	p	p	NOUN
ejpam-2803	722	31	)	)	PUNCT
ejpam-2803	722	32	.	.	PUNCT
ejpam-2803	723	1	thus	thus	ADV
ejpam-2803	723	2	m	m	NOUN
ejpam-2803	723	3	is	be	AUX
ejpam-2803	723	4	a	a	DET
ejpam-2803	723	5	weak	weak	ADJ
ejpam-2803	723	6	comultiplication	comultiplication	NOUN
ejpam-2803	723	7	module	module	NOUN
ejpam-2803	723	8	.	.	PUNCT
ejpam-2803	724	1	the	the	DET
ejpam-2803	724	2	reverse	reverse	ADJ
ejpam-2803	724	3	implication	implication	NOUN
ejpam-2803	724	4	is	be	AUX
ejpam-2803	724	5	clear	clear	ADJ
ejpam-2803	724	6	.	.	PUNCT
ejpam-2803	725	1	(	(	PUNCT
ejpam-2803	725	2	c	c	AUX
ejpam-2803	725	3	)	)	PUNCT
ejpam-2803	725	4	let	let	VERB
ejpam-2803	725	5	ψ	ψ	X
ejpam-2803	725	6	:	:	PUNCT
ejpam-2803	725	7	specs(m)→	specs(m)→	VERB
ejpam-2803	725	8	spec(r	spec(r	VERB
ejpam-2803	725	9	)	)	PUNCT
ejpam-2803	725	10	be	be	AUX
ejpam-2803	725	11	the	the	DET
ejpam-2803	725	12	natural	natural	ADJ
ejpam-2803	725	13	map	map	NOUN
ejpam-2803	725	14	of	of	ADP
ejpam-2803	725	15	specs(m	specs(m	NOUN
ejpam-2803	725	16	)	)	PUNCT
ejpam-2803	725	17	.	.	PUNCT
ejpam-2803	726	1	as	as	SCONJ
ejpam-2803	726	2	m	m	PROPN
ejpam-2803	726	3	is	be	AUX
ejpam-2803	726	4	a	a	DET
ejpam-2803	726	5	secondful	secondful	ADJ
ejpam-2803	726	6	xs	xs	NOUN
ejpam-2803	726	7	-	-	PUNCT
ejpam-2803	726	8	injective	injective	ADJ
ejpam-2803	726	9	r	r	NOUN
ejpam-2803	726	10	-	-	PUNCT
ejpam-2803	726	11	module	module	NOUN
ejpam-2803	726	12	,	,	PUNCT
ejpam-2803	726	13	ψ	ψ	NOUN
ejpam-2803	726	14	is	be	AUX
ejpam-2803	726	15	a	a	DET
ejpam-2803	726	16	bijective	bijective	ADJ
ejpam-2803	726	17	map	map	NOUN
ejpam-2803	726	18	.	.	PUNCT
ejpam-2803	727	1	now	now	ADV
ejpam-2803	727	2	let	let	VERB
ejpam-2803	727	3	i	i	PRON
ejpam-2803	727	4	be	be	AUX
ejpam-2803	727	5	an	an	DET
ejpam-2803	727	6	ideal	ideal	NOUN
ejpam-2803	727	7	of	of	ADP
ejpam-2803	727	8	r.	r.	PROPN
ejpam-2803	727	9	then	then	ADV
ejpam-2803	727	10	by	by	ADP
ejpam-2803	727	11	[	[	X
ejpam-2803	727	12	7	7	NUM
ejpam-2803	727	13	,	,	PUNCT
ejpam-2803	727	14	lemma	lemma	X
ejpam-2803	727	15	3.3	3.3	NUM
ejpam-2803	727	16	(	(	PUNCT
ejpam-2803	727	17	c	c	NOUN
ejpam-2803	727	18	)	)	PUNCT
ejpam-2803	727	19	and	and	CCONJ
ejpam-2803	727	20	proposition	proposition	NOUN
ejpam-2803	727	21	3.6	3.6	NUM
ejpam-2803	727	22	]	]	PUNCT
ejpam-2803	727	23	,	,	PUNCT
ejpam-2803	727	24	(	(	PUNCT
ejpam-2803	727	25	ψ)−1(v	ψ)−1(v	PROPN
ejpam-2803	727	26	(	(	PUNCT
ejpam-2803	727	27	i	i	NOUN
ejpam-2803	727	28	)	)	PUNCT
ejpam-2803	727	29	)	)	PUNCT
ejpam-2803	728	1	=	=	SYM
ejpam-2803	728	2	v	v	ADP
ejpam-2803	728	3	s(0	s(0	PROPN
ejpam-2803	728	4	:	:	PUNCT
ejpam-2803	728	5	m	m	VERB
ejpam-2803	728	6	i	i	NOUN
ejpam-2803	728	7	)	)	PUNCT
ejpam-2803	728	8	=	=	SYM
ejpam-2803	728	9	v	v	NUM
ejpam-2803	728	10	s∗(0	s∗(0	NOUN
ejpam-2803	728	11	:	:	PUNCT
ejpam-2803	728	12	m	m	VERB
ejpam-2803	728	13	i	i	NOUN
ejpam-2803	728	14	)	)	PUNCT
ejpam-2803	728	15	,	,	PUNCT
ejpam-2803	728	16	so	so	CCONJ
ejpam-2803	728	17	ψ	ψ	NOUN
ejpam-2803	728	18	is	be	AUX
ejpam-2803	728	19	continuous	continuous	ADJ
ejpam-2803	728	20	.	.	PUNCT
ejpam-2803	729	1	now	now	ADV
ejpam-2803	729	2	let	let	VERB
ejpam-2803	729	3	n	n	PRON
ejpam-2803	729	4	be	be	AUX
ejpam-2803	729	5	a	a	DET
ejpam-2803	729	6	submodule	submodule	NOUN
ejpam-2803	729	7	of	of	ADP
ejpam-2803	729	8	m	m	PROPN
ejpam-2803	729	9	.	.	PUNCT
ejpam-2803	730	1	then	then	ADV
ejpam-2803	730	2	we	we	PRON
ejpam-2803	730	3	have	have	VERB
ejpam-2803	730	4	ψ(v	ψ(v	PROPN
ejpam-2803	730	5	s∗(n	s∗(n	NOUN
ejpam-2803	730	6	)	)	PUNCT
ejpam-2803	730	7	)	)	PUNCT
ejpam-2803	731	1	=	=	PUNCT
ejpam-2803	732	1	ψ(v	ψ(v	PROPN
ejpam-2803	732	2	s∗(sec(n	s∗(sec(n	PROPN
ejpam-2803	732	3	)	)	PUNCT
ejpam-2803	732	4	)	)	PUNCT
ejpam-2803	732	5	)	)	PUNCT
ejpam-2803	733	1	=	=	PUNCT
ejpam-2803	733	2	ψ(v	ψ(v	PROPN
ejpam-2803	733	3	s(sec(n	s(sec(n	PROPN
ejpam-2803	733	4	)	)	PUNCT
ejpam-2803	733	5	)	)	PUNCT
ejpam-2803	733	6	)	)	PUNCT
ejpam-2803	734	1	=	=	SYM
ejpam-2803	734	2	v	v	X
ejpam-2803	734	3	(	(	PUNCT
ejpam-2803	734	4	annr(sec(n	annr(sec(n	ADJ
ejpam-2803	734	5	)	)	PUNCT
ejpam-2803	734	6	)	)	PUNCT
ejpam-2803	734	7	)	)	PUNCT
ejpam-2803	734	8	.	.	PUNCT
ejpam-2803	735	1	it	it	PRON
ejpam-2803	735	2	follows	follow	VERB
ejpam-2803	735	3	that	that	SCONJ
ejpam-2803	735	4	ψ	ψ	NOUN
ejpam-2803	735	5	is	be	AUX
ejpam-2803	735	6	a	a	DET
ejpam-2803	735	7	closed	closed	ADJ
ejpam-2803	735	8	map	map	NOUN
ejpam-2803	735	9	and	and	CCONJ
ejpam-2803	735	10	the	the	DET
ejpam-2803	735	11	proof	proof	NOUN
ejpam-2803	735	12	is	be	AUX
ejpam-2803	735	13	completed	complete	VERB
ejpam-2803	735	14	.	.	PUNCT
ejpam-2803	736	1	(	(	PUNCT
ejpam-2803	736	2	d	d	X
ejpam-2803	736	3	)	)	PUNCT
ejpam-2803	736	4	if	if	SCONJ
ejpam-2803	736	5	m	m	VERB
ejpam-2803	736	6	=	=	SYM
ejpam-2803	736	7	(	(	PUNCT
ejpam-2803	736	8	0	0	NUM
ejpam-2803	736	9	)	)	PUNCT
ejpam-2803	736	10	,	,	PUNCT
ejpam-2803	736	11	there	there	PRON
ejpam-2803	736	12	is	be	VERB
ejpam-2803	736	13	nothing	nothing	PRON
ejpam-2803	736	14	to	to	PART
ejpam-2803	736	15	prove	prove	VERB
ejpam-2803	736	16	.	.	PUNCT
ejpam-2803	737	1	hence	hence	ADV
ejpam-2803	737	2	we	we	PRON
ejpam-2803	737	3	assume	assume	VERB
ejpam-2803	737	4	that	that	SCONJ
ejpam-2803	737	5	m	m	VERB
ejpam-2803	737	6	6=	6=	NUM
ejpam-2803	737	7	(	(	PUNCT
ejpam-2803	737	8	0	0	NUM
ejpam-2803	737	9	)	)	PUNCT
ejpam-2803	737	10	.	.	PUNCT
ejpam-2803	738	1	let	let	VERB
ejpam-2803	738	2	n	n	PRON
ejpam-2803	738	3	be	be	AUX
ejpam-2803	738	4	a	a	DET
ejpam-2803	738	5	submodule	submodule	NOUN
ejpam-2803	738	6	of	of	ADP
ejpam-2803	738	7	m	m	PROPN
ejpam-2803	738	8	.	.	PUNCT
ejpam-2803	739	1	clearly	clearly	ADV
ejpam-2803	739	2	,	,	PUNCT
ejpam-2803	739	3	v	v	ADP
ejpam-2803	739	4	s∗(sec(n	s∗(sec(n	NOUN
ejpam-2803	739	5	)	)	PUNCT
ejpam-2803	739	6	)	)	PUNCT
ejpam-2803	740	1	⊆	⊆	NUM
ejpam-2803	740	2	v	v	ADP
ejpam-2803	740	3	s(sec(n	s(sec(n	PROPN
ejpam-2803	740	4	)	)	PUNCT
ejpam-2803	740	5	)	)	PUNCT
ejpam-2803	740	6	.	.	PUNCT
ejpam-2803	741	1	so	so	ADV
ejpam-2803	741	2	we	we	PRON
ejpam-2803	741	3	assume	assume	VERB
ejpam-2803	741	4	that	that	SCONJ
ejpam-2803	741	5	s	s	VERB
ejpam-2803	741	6	∈	∈	PROPN
ejpam-2803	741	7	v	v	ADP
ejpam-2803	741	8	s(sec(n	s(sec(n	PROPN
ejpam-2803	741	9	)	)	PUNCT
ejpam-2803	741	10	)	)	PUNCT
ejpam-2803	741	11	.	.	PUNCT
ejpam-2803	742	1	then	then	ADV
ejpam-2803	742	2	by	by	ADP
ejpam-2803	742	3	theorem	theorem	NOUN
ejpam-2803	742	4	3.17	3.17	NUM
ejpam-2803	742	5	(	(	PUNCT
ejpam-2803	742	6	a	a	NOUN
ejpam-2803	742	7	)	)	PUNCT
ejpam-2803	742	8	,	,	PUNCT
ejpam-2803	742	9	s	s	NOUN
ejpam-2803	742	10	=	=	X
ejpam-2803	742	11	imp	imp	X
ejpam-2803	742	12	(	(	PUNCT
ejpam-2803	742	13	(	(	PUNCT
ejpam-2803	742	14	0	0	NUM
ejpam-2803	742	15	:	:	PUNCT
ejpam-2803	742	16	m	m	VERB
ejpam-2803	742	17	p	p	NOUN
ejpam-2803	742	18	)	)	PUNCT
ejpam-2803	742	19	)	)	PUNCT
ejpam-2803	742	20	,	,	PUNCT
ejpam-2803	742	21	where	where	SCONJ
ejpam-2803	742	22	p	p	PROPN
ejpam-2803	742	23	=	=	PROPN
ejpam-2803	742	24	annr(s	annr(s	PROPN
ejpam-2803	742	25	)	)	PUNCT
ejpam-2803	742	26	∈	∈	PROPN
ejpam-2803	742	27	v	v	NOUN
ejpam-2803	742	28	(	(	PUNCT
ejpam-2803	742	29	annr(m	annr(m	PROPN
ejpam-2803	742	30	)	)	PUNCT
ejpam-2803	742	31	)	)	PUNCT
ejpam-2803	742	32	.	.	PUNCT
ejpam-2803	743	1	if	if	SCONJ
ejpam-2803	743	2	set	set	VERB
ejpam-2803	743	3	i	i	PRON
ejpam-2803	743	4	=	=	PUNCT
ejpam-2803	743	5	{	{	PUNCT
ejpam-2803	743	6	q	q	NOUN
ejpam-2803	743	7	∈	∈	PROPN
ejpam-2803	743	8	v	v	NOUN
ejpam-2803	743	9	(	(	PUNCT
ejpam-2803	743	10	annr(m	annr(m	PROPN
ejpam-2803	743	11	)	)	PUNCT
ejpam-2803	743	12	)	)	PUNCT
ejpam-2803	744	1	|	|	ADV
ejpam-2803	744	2	(	(	PUNCT
ejpam-2803	744	3	0	0	NUM
ejpam-2803	744	4	)	)	PUNCT
ejpam-2803	744	5	6=	6=	NUM
ejpam-2803	744	6	imq	imq	NOUN
ejpam-2803	744	7	(	(	PUNCT
ejpam-2803	744	8	(	(	PUNCT
ejpam-2803	744	9	0	0	NUM
ejpam-2803	744	10	:	:	PUNCT
ejpam-2803	744	11	m	m	VERB
ejpam-2803	744	12	q	q	NOUN
ejpam-2803	744	13	)	)	PUNCT
ejpam-2803	744	14	)	)	PUNCT
ejpam-2803	744	15	⊆	⊆	NUM
ejpam-2803	744	16	n	n	CCONJ
ejpam-2803	744	17	}	}	PUNCT
ejpam-2803	744	18	,	,	PUNCT
ejpam-2803	744	19	references	reference	NOUN
ejpam-2803	744	20	228	228	NUM
ejpam-2803	744	21	then	then	ADV
ejpam-2803	744	22	sec(n	sec(n	PROPN
ejpam-2803	744	23	)	)	PUNCT
ejpam-2803	744	24	=	=	PUNCT
ejpam-2803	745	1	∑	∑	PUNCT
ejpam-2803	745	2	q∈i	q∈i	NOUN
ejpam-2803	745	3	i	i	PRON
ejpam-2803	745	4	m	m	VERB
ejpam-2803	745	5	q	q	X
ejpam-2803	745	6	(	(	PUNCT
ejpam-2803	745	7	(	(	PUNCT
ejpam-2803	745	8	0	0	NUM
ejpam-2803	745	9	:	:	PUNCT
ejpam-2803	745	10	m	m	VERB
ejpam-2803	745	11	q	q	NOUN
ejpam-2803	745	12	)	)	PUNCT
ejpam-2803	745	13	)	)	PUNCT
ejpam-2803	745	14	.	.	PUNCT
ejpam-2803	746	1	it	it	PRON
ejpam-2803	746	2	follows	follow	VERB
ejpam-2803	746	3	that	that	SCONJ
ejpam-2803	746	4	annr(sec(n	annr(sec(n	ADJ
ejpam-2803	746	5	)	)	PUNCT
ejpam-2803	746	6	)	)	PUNCT
ejpam-2803	747	1	=	=	SYM
ejpam-2803	747	2	annr	annr	NOUN
ejpam-2803	747	3	(	(	PUNCT
ejpam-2803	747	4	∑	∑	ADV
ejpam-2803	747	5	q∈i	q∈i	NOUN
ejpam-2803	747	6	imq	imq	NOUN
ejpam-2803	747	7	(	(	PUNCT
ejpam-2803	747	8	(	(	PUNCT
ejpam-2803	747	9	0	0	NUM
ejpam-2803	747	10	:	:	PUNCT
ejpam-2803	747	11	m	m	VERB
ejpam-2803	747	12	q	q	NOUN
ejpam-2803	747	13	)	)	PUNCT
ejpam-2803	747	14	)	)	PUNCT
ejpam-2803	747	15	)	)	PUNCT
ejpam-2803	748	1	=	=	PUNCT
ejpam-2803	748	2	⋂	⋂	PROPN
ejpam-2803	748	3	q∈i	q∈i	NOUN
ejpam-2803	748	4	q	q	PROPN
ejpam-2803	748	5	⊆	⊆	NUM
ejpam-2803	748	6	p.	p.	NOUN
ejpam-2803	748	7	now	now	ADV
ejpam-2803	748	8	by	by	ADP
ejpam-2803	748	9	using	use	VERB
ejpam-2803	748	10	the	the	DET
ejpam-2803	748	11	assumption	assumption	NOUN
ejpam-2803	748	12	,	,	PUNCT
ejpam-2803	748	13	we	we	PRON
ejpam-2803	748	14	have	have	VERB
ejpam-2803	748	15	s	s	NOUN
ejpam-2803	748	16	=	=	X
ejpam-2803	748	17	imp	imp	X
ejpam-2803	748	18	(	(	PUNCT
ejpam-2803	748	19	(	(	PUNCT
ejpam-2803	748	20	0	0	NUM
ejpam-2803	748	21	:	:	PUNCT
ejpam-2803	748	22	m	m	VERB
ejpam-2803	748	23	p	p	NOUN
ejpam-2803	748	24	)	)	PUNCT
ejpam-2803	748	25	)	)	PUNCT
ejpam-2803	749	1	⊆	⊆	NUM
ejpam-2803	749	2	∑	∑	ADP
ejpam-2803	749	3	q∈i	q∈i	NOUN
ejpam-2803	749	4	imq	imq	NOUN
ejpam-2803	749	5	(	(	PUNCT
ejpam-2803	749	6	(	(	PUNCT
ejpam-2803	749	7	0	0	NUM
ejpam-2803	749	8	:	:	PUNCT
ejpam-2803	749	9	m	m	VERB
ejpam-2803	749	10	q	q	NOUN
ejpam-2803	749	11	)	)	PUNCT
ejpam-2803	749	12	)	)	PUNCT
ejpam-2803	750	1	=	=	SYM
ejpam-2803	750	2	sec(n	sec(n	PROPN
ejpam-2803	750	3	)	)	PUNCT
ejpam-2803	750	4	.	.	PUNCT
ejpam-2803	751	1	so	so	ADV
ejpam-2803	751	2	s	s	VERB
ejpam-2803	751	3	∈	∈	PROPN
ejpam-2803	751	4	v	v	ADP
ejpam-2803	751	5	s∗(sec(n	s∗(sec(n	PROPN
ejpam-2803	751	6	)	)	PUNCT
ejpam-2803	751	7	)	)	PUNCT
ejpam-2803	751	8	,	,	PUNCT
ejpam-2803	751	9	as	as	SCONJ
ejpam-2803	751	10	desired	desire	VERB
ejpam-2803	751	11	.	.	PUNCT
ejpam-2803	752	1	example	example	NOUN
ejpam-2803	752	2	3.19	3.19	NUM
ejpam-2803	752	3	.	.	PUNCT
ejpam-2803	753	1	set	set	VERB
ejpam-2803	753	2	m	m	NOUN
ejpam-2803	753	3	=	=	VERB
ejpam-2803	753	4	⊕n	⊕n	NOUN
ejpam-2803	753	5	i=1	i=1	PROPN
ejpam-2803	753	6	zpi	zpi	NOUN
ejpam-2803	753	7	,	,	PUNCT
ejpam-2803	753	8	where	where	SCONJ
ejpam-2803	753	9	pi	pi	NOUN
ejpam-2803	753	10	are	be	AUX
ejpam-2803	753	11	distinct	distinct	ADJ
ejpam-2803	753	12	positive	positive	ADJ
ejpam-2803	753	13	prime	prime	ADJ
ejpam-2803	753	14	integers	integer	NOUN
ejpam-2803	753	15	.	.	PUNCT
ejpam-2803	754	1	then	then	ADV
ejpam-2803	754	2	by	by	ADP
ejpam-2803	754	3	using	use	VERB
ejpam-2803	754	4	proposition	proposition	NOUN
ejpam-2803	754	5	3.13	3.13	NUM
ejpam-2803	754	6	(	(	PUNCT
ejpam-2803	754	7	ii	ii	NOUN
ejpam-2803	754	8	)	)	PUNCT
ejpam-2803	754	9	and	and	CCONJ
ejpam-2803	754	10	theorem	theorem	VERB
ejpam-2803	754	11	3.15	3.15	NUM
ejpam-2803	754	12	,	,	PUNCT
ejpam-2803	754	13	we	we	PRON
ejpam-2803	754	14	see	see	VERB
ejpam-2803	754	15	that	that	SCONJ
ejpam-2803	754	16	m	m	PROPN
ejpam-2803	754	17	is	be	AUX
ejpam-2803	754	18	a	a	DET
ejpam-2803	754	19	secondful	secondful	ADJ
ejpam-2803	754	20	xs	xs	NOUN
ejpam-2803	754	21	-	-	PUNCT
ejpam-2803	754	22	injective	injective	ADJ
ejpam-2803	754	23	zmodule	zmodule	NOUN
ejpam-2803	754	24	and	and	CCONJ
ejpam-2803	754	25	we	we	PRON
ejpam-2803	754	26	have	have	VERB
ejpam-2803	754	27	specs(m	specs(m	NOUN
ejpam-2803	754	28	)	)	PUNCT
ejpam-2803	754	29	=	=	PRON
ejpam-2803	754	30	{	{	PUNCT
ejpam-2803	754	31	zpj	zpj	PROPN
ejpam-2803	754	32	⊕	⊕	PROPN
ejpam-2803	754	33	(	(	PUNCT
ejpam-2803	754	34	⊕	⊕	NOUN
ejpam-2803	754	35	1≤i	1≤i	NUM
ejpam-2803	754	36	6	6	NUM
ejpam-2803	754	37	=	=	NOUN
ejpam-2803	754	38	j≤n	j≤n	NOUN
ejpam-2803	754	39	(	(	PUNCT
ejpam-2803	754	40	0	0	NUM
ejpam-2803	754	41	)	)	PUNCT
ejpam-2803	754	42	)	)	PUNCT
ejpam-2803	755	1	|	|	ADV
ejpam-2803	755	2	1	1	NUM
ejpam-2803	755	3	≤	≤	NUM
ejpam-2803	755	4	j	j	PROPN
ejpam-2803	755	5	≤	≤	PROPN
ejpam-2803	755	6	n	n	CCONJ
ejpam-2803	755	7	}	}	PUNCT
ejpam-2803	755	8	.	.	PUNCT
ejpam-2803	756	1	moreover	moreover	ADV
ejpam-2803	756	2	,	,	PUNCT
ejpam-2803	756	3	v	v	ADJ
ejpam-2803	756	4	s(sec(n	s(sec(n	NOUN
ejpam-2803	756	5	)	)	PUNCT
ejpam-2803	756	6	)	)	PUNCT
ejpam-2803	757	1	=	=	PUNCT
ejpam-2803	757	2	v	v	ADP
ejpam-2803	757	3	s∗(sec(n	s∗(sec(n	NOUN
ejpam-2803	757	4	)	)	PUNCT
ejpam-2803	757	5	)	)	PUNCT
ejpam-2803	757	6	for	for	ADP
ejpam-2803	757	7	every	every	DET
ejpam-2803	757	8	submodule	submodule	NOUN
ejpam-2803	757	9	n	n	PROPN
ejpam-2803	757	10	of	of	ADP
ejpam-2803	757	11	m	m	PROPN
ejpam-2803	757	12	.	.	PUNCT
ejpam-2803	758	1	hence	hence	ADV
ejpam-2803	758	2	(	(	PUNCT
ejpam-2803	758	3	specs(m	specs(m	PROPN
ejpam-2803	758	4	)	)	PUNCT
ejpam-2803	758	5	,	,	PUNCT
ejpam-2803	758	6	τ	τ	PROPN
ejpam-2803	758	7	s∗	s∗	PROPN
ejpam-2803	758	8	)	)	PUNCT
ejpam-2803	758	9	is	be	AUX
ejpam-2803	758	10	a	a	DET
ejpam-2803	758	11	spectral	spectral	ADJ
ejpam-2803	758	12	space	space	NOUN
ejpam-2803	758	13	by	by	ADP
ejpam-2803	758	14	theorem	theorem	NOUN
ejpam-2803	758	15	3.18	3.18	NUM
ejpam-2803	758	16	.	.	PUNCT
ejpam-2803	759	1	this	this	DET
ejpam-2803	759	2	example	example	NOUN
ejpam-2803	759	3	shows	show	VERB
ejpam-2803	759	4	that	that	SCONJ
ejpam-2803	759	5	for	for	ADP
ejpam-2803	759	6	each	each	DET
ejpam-2803	759	7	n	n	NOUN
ejpam-2803	759	8	>	>	X
ejpam-2803	759	9	1	1	NUM
ejpam-2803	759	10	,	,	PUNCT
ejpam-2803	759	11	when	when	SCONJ
ejpam-2803	759	12	n	n	PRON
ejpam-2803	759	13	is	be	AUX
ejpam-2803	759	14	square	square	ADV
ejpam-2803	759	15	free	free	ADJ
ejpam-2803	759	16	,	,	PUNCT
ejpam-2803	759	17	(	(	PUNCT
ejpam-2803	759	18	specs(zn	specs(zn	NOUN
ejpam-2803	759	19	)	)	PUNCT
ejpam-2803	759	20	,	,	PUNCT
ejpam-2803	759	21	τ	τ	PROPN
ejpam-2803	759	22	s∗	s∗	PROPN
ejpam-2803	759	23	)	)	PUNCT
ejpam-2803	759	24	is	be	AUX
ejpam-2803	759	25	a	a	DET
ejpam-2803	759	26	spectral	spectral	ADJ
ejpam-2803	759	27	space	space	NOUN
ejpam-2803	759	28	.	.	PUNCT
ejpam-2803	760	1	remark	remark	VERB
ejpam-2803	760	2	3.20	3.20	NUM
ejpam-2803	760	3	.	.	PUNCT
ejpam-2803	761	1	(	(	PUNCT
ejpam-2803	761	2	a	a	X
ejpam-2803	761	3	)	)	PUNCT
ejpam-2803	761	4	let	let	VERB
ejpam-2803	761	5	m	m	VERB
ejpam-2803	761	6	=	=	SYM
ejpam-2803	761	7	zp∞	zp∞	PROPN
ejpam-2803	761	8	⊕	⊕	PROPN
ejpam-2803	761	9	zq	zq	PROPN
ejpam-2803	761	10	,	,	PUNCT
ejpam-2803	761	11	where	where	SCONJ
ejpam-2803	761	12	p	p	NOUN
ejpam-2803	761	13	and	and	CCONJ
ejpam-2803	761	14	q	q	NOUN
ejpam-2803	761	15	are	be	AUX
ejpam-2803	761	16	distinct	distinct	ADJ
ejpam-2803	761	17	positive	positive	ADJ
ejpam-2803	761	18	prime	prime	ADJ
ejpam-2803	761	19	integers	integer	NOUN
ejpam-2803	761	20	.	.	PUNCT
ejpam-2803	762	1	then	then	ADV
ejpam-2803	762	2	m	m	PROPN
ejpam-2803	762	3	is	be	AUX
ejpam-2803	762	4	an	an	DET
ejpam-2803	762	5	artinian	artinian	ADJ
ejpam-2803	762	6	xs	xs	NOUN
ejpam-2803	762	7	-	-	PUNCT
ejpam-2803	762	8	injective	injective	ADJ
ejpam-2803	762	9	over	over	ADP
ejpam-2803	762	10	the	the	DET
ejpam-2803	762	11	one	one	NUM
ejpam-2803	762	12	dimensional	dimensional	ADJ
ejpam-2803	762	13	integral	integral	ADJ
ejpam-2803	762	14	domain	domain	NOUN
ejpam-2803	762	15	z	z	NOUN
ejpam-2803	762	16	;	;	PUNCT
ejpam-2803	762	17	but	but	CCONJ
ejpam-2803	762	18	it	it	PRON
ejpam-2803	762	19	is	be	AUX
ejpam-2803	762	20	not	not	PART
ejpam-2803	762	21	a	a	DET
ejpam-2803	762	22	weak	weak	ADJ
ejpam-2803	762	23	comultiplication	comultiplication	NOUN
ejpam-2803	762	24	z	z	NOUN
ejpam-2803	762	25	-	-	PUNCT
ejpam-2803	762	26	module	module	NOUN
ejpam-2803	762	27	.	.	PUNCT
ejpam-2803	763	1	this	this	PRON
ejpam-2803	763	2	shows	show	VERB
ejpam-2803	763	3	that	that	SCONJ
ejpam-2803	763	4	the	the	DET
ejpam-2803	763	5	condition	condition	NOUN
ejpam-2803	763	6	“	"	PUNCT
ejpam-2803	763	7	m	m	NOUN
ejpam-2803	763	8	is	be	AUX
ejpam-2803	763	9	cotorsion	cotorsion	NOUN
ejpam-2803	763	10	or	or	CCONJ
ejpam-2803	763	11	cotorsion	cotorsion	NOUN
ejpam-2803	763	12	-	-	PUNCT
ejpam-2803	763	13	free	free	ADJ
ejpam-2803	763	14	”	"	PUNCT
ejpam-2803	763	15	in	in	ADP
ejpam-2803	763	16	part	part	NOUN
ejpam-2803	763	17	(	(	PUNCT
ejpam-2803	763	18	b	b	NOUN
ejpam-2803	763	19	)	)	PUNCT
ejpam-2803	763	20	of	of	ADP
ejpam-2803	763	21	theorem	theorem	ADJ
ejpam-2803	763	22	3.18	3.18	NUM
ejpam-2803	763	23	is	be	AUX
ejpam-2803	763	24	a	a	DET
ejpam-2803	763	25	necessary	necessary	ADJ
ejpam-2803	763	26	condition	condition	NOUN
ejpam-2803	763	27	and	and	CCONJ
ejpam-2803	763	28	can	can	AUX
ejpam-2803	763	29	not	not	PART
ejpam-2803	763	30	be	be	AUX
ejpam-2803	763	31	omitted	omit	VERB
ejpam-2803	763	32	.	.	PUNCT
ejpam-2803	764	1	(	(	PUNCT
ejpam-2803	764	2	b	b	X
ejpam-2803	764	3	)	)	PUNCT
ejpam-2803	764	4	set	set	NOUN
ejpam-2803	764	5	m	m	NOUN
ejpam-2803	764	6	=	=	PUNCT
ejpam-2803	764	7	(	(	PUNCT
ejpam-2803	764	8	⊕	⊕	PROPN
ejpam-2803	764	9	p	p	PROPN
ejpam-2803	764	10	zp)⊕q	zp)⊕q	PROPN
ejpam-2803	764	11	,	,	PUNCT
ejpam-2803	764	12	where	where	SCONJ
ejpam-2803	764	13	p	p	NOUN
ejpam-2803	764	14	runs	run	VERB
ejpam-2803	764	15	over	over	ADP
ejpam-2803	764	16	all	all	DET
ejpam-2803	764	17	distinct	distinct	ADJ
ejpam-2803	764	18	positive	positive	ADJ
ejpam-2803	764	19	prime	prime	ADJ
ejpam-2803	764	20	integers	integer	NOUN
ejpam-2803	764	21	.	.	PUNCT
ejpam-2803	765	1	then	then	ADV
ejpam-2803	765	2	by	by	ADP
ejpam-2803	765	3	using	use	VERB
ejpam-2803	765	4	proposition	proposition	NOUN
ejpam-2803	765	5	3.13	3.13	NUM
ejpam-2803	765	6	(	(	PUNCT
ejpam-2803	765	7	ii	ii	NOUN
ejpam-2803	765	8	)	)	PUNCT
ejpam-2803	765	9	and	and	CCONJ
ejpam-2803	765	10	theorem	theorem	VERB
ejpam-2803	765	11	3.15	3.15	NUM
ejpam-2803	765	12	,	,	PUNCT
ejpam-2803	765	13	m	m	VERB
ejpam-2803	765	14	is	be	AUX
ejpam-2803	765	15	a	a	DET
ejpam-2803	765	16	secondful	secondful	ADJ
ejpam-2803	765	17	xs	xs	NOUN
ejpam-2803	765	18	-	-	PUNCT
ejpam-2803	765	19	injective	injective	ADJ
ejpam-2803	765	20	zmodule	zmodule	NOUN
ejpam-2803	765	21	.	.	PUNCT
ejpam-2803	766	1	moreover	moreover	ADV
ejpam-2803	766	2	,	,	PUNCT
ejpam-2803	766	3	we	we	PRON
ejpam-2803	766	4	see	see	VERB
ejpam-2803	766	5	that	that	SCONJ
ejpam-2803	766	6	(	(	PUNCT
ejpam-2803	766	7	specs(m	specs(m	NOUN
ejpam-2803	766	8	)	)	PUNCT
ejpam-2803	766	9	,	,	PUNCT
ejpam-2803	766	10	τ	τ	PROPN
ejpam-2803	766	11	s∗	s∗	PROPN
ejpam-2803	766	12	)	)	PUNCT
ejpam-2803	766	13	is	be	AUX
ejpam-2803	766	14	not	not	PART
ejpam-2803	766	15	a	a	DET
ejpam-2803	766	16	quasi	quasi	ADJ
ejpam-2803	766	17	-	-	ADJ
ejpam-2803	766	18	compact	compact	ADJ
ejpam-2803	766	19	space	space	NOUN
ejpam-2803	766	20	and	and	CCONJ
ejpam-2803	766	21	hence	hence	ADV
ejpam-2803	766	22	not	not	PART
ejpam-2803	766	23	a	a	DET
ejpam-2803	766	24	spectral	spectral	ADJ
ejpam-2803	766	25	space	space	NOUN
ejpam-2803	766	26	by	by	ADP
ejpam-2803	766	27	hochster	hochster	PROPN
ejpam-2803	766	28	’s	’s	PART
ejpam-2803	766	29	characterizations	characterization	NOUN
ejpam-2803	766	30	.	.	PUNCT
ejpam-2803	767	1	this	this	PRON
ejpam-2803	767	2	shows	show	VERB
ejpam-2803	767	3	that	that	SCONJ
ejpam-2803	767	4	the	the	DET
ejpam-2803	767	5	condition	condition	NOUN
ejpam-2803	767	6	“	"	PUNCT
ejpam-2803	767	7	v	v	ADP
ejpam-2803	767	8	s(sec(n	s(sec(n	PROPN
ejpam-2803	767	9	)	)	PUNCT
ejpam-2803	767	10	)	)	PUNCT
ejpam-2803	767	11	=	=	PUNCT
ejpam-2803	767	12	v	v	ADP
ejpam-2803	767	13	s∗(sec(n	s∗(sec(n	NOUN
ejpam-2803	767	14	)	)	PUNCT
ejpam-2803	767	15	)	)	PUNCT
ejpam-2803	767	16	for	for	ADP
ejpam-2803	767	17	every	every	DET
ejpam-2803	767	18	submodule	submodule	NOUN
ejpam-2803	767	19	n	n	PROPN
ejpam-2803	767	20	of	of	ADP
ejpam-2803	767	21	m	m	PROPN
ejpam-2803	767	22	”	"	PUNCT
ejpam-2803	767	23	in	in	ADP
ejpam-2803	767	24	theorem	theorem	ADJ
ejpam-2803	767	25	3.18	3.18	NUM
ejpam-2803	767	26	(	(	PUNCT
ejpam-2803	767	27	d	d	NOUN
ejpam-2803	767	28	)	)	PUNCT
ejpam-2803	767	29	is	be	AUX
ejpam-2803	767	30	a	a	DET
ejpam-2803	767	31	necessary	necessary	ADJ
ejpam-2803	767	32	condition	condition	NOUN
ejpam-2803	767	33	and	and	CCONJ
ejpam-2803	767	34	can	can	AUX
ejpam-2803	767	35	not	not	PART
ejpam-2803	767	36	be	be	AUX
ejpam-2803	767	37	omitted	omit	VERB
ejpam-2803	767	38	.	.	PUNCT
ejpam-2803	768	1	we	we	PRON
ejpam-2803	768	2	end	end	VERB
ejpam-2803	768	3	this	this	DET
ejpam-2803	768	4	section	section	NOUN
ejpam-2803	768	5	with	with	ADP
ejpam-2803	768	6	the	the	DET
ejpam-2803	768	7	following	follow	VERB
ejpam-2803	768	8	question	question	NOUN
ejpam-2803	768	9	.	.	PUNCT
ejpam-2803	769	1	question	question	NOUN
ejpam-2803	769	2	3.21	3.21	NUM
ejpam-2803	769	3	.	.	PUNCT
ejpam-2803	770	1	is	be	AUX
ejpam-2803	770	2	every	every	DET
ejpam-2803	770	3	cotop	cotop	NOUN
ejpam-2803	770	4	r	r	NOUN
ejpam-2803	770	5	-	-	PUNCT
ejpam-2803	770	6	module	module	NOUN
ejpam-2803	770	7	an	an	DET
ejpam-2803	770	8	xs	xs	NOUN
ejpam-2803	770	9	-	-	PUNCT
ejpam-2803	770	10	injective	injective	ADJ
ejpam-2803	770	11	r	r	NOUN
ejpam-2803	770	12	-	-	PUNCT
ejpam-2803	770	13	module	module	NOUN
ejpam-2803	770	14	?	?	PUNCT
ejpam-2803	771	1	references	reference	NOUN
ejpam-2803	771	2	[	[	X
ejpam-2803	771	3	1	1	NUM
ejpam-2803	771	4	]	]	PUNCT
ejpam-2803	771	5	j	j	PROPN
ejpam-2803	771	6	abuhlail	abuhlail	NOUN
ejpam-2803	771	7	.	.	PUNCT
ejpam-2803	772	1	a	a	DET
ejpam-2803	772	2	dual	dual	ADJ
ejpam-2803	772	3	zariski	zariski	ADJ
ejpam-2803	772	4	topology	topology	NOUN
ejpam-2803	772	5	for	for	ADP
ejpam-2803	772	6	modules	module	NOUN
ejpam-2803	772	7	.	.	PUNCT
ejpam-2803	773	1	topology	topology	NOUN
ejpam-2803	773	2	appl	appl	NOUN
ejpam-2803	773	3	,	,	PUNCT
ejpam-2803	773	4	158(3):457–467	158(3):457–467	NUM
ejpam-2803	773	5	,	,	PUNCT
ejpam-2803	773	6	2011	2011	NUM
ejpam-2803	773	7	.	.	PUNCT
ejpam-2803	774	1	references	reference	NOUN
ejpam-2803	774	2	229	229	NUM
ejpam-2803	775	1	[	[	X
ejpam-2803	775	2	2	2	NUM
ejpam-2803	775	3	]	]	PUNCT
ejpam-2803	775	4	j	j	PROPN
ejpam-2803	775	5	abuhlail	abuhlail	NOUN
ejpam-2803	775	6	.	.	PUNCT
ejpam-2803	776	1	zariski	zariski	NOUN
ejpam-2803	776	2	topologies	topology	NOUN
ejpam-2803	776	3	for	for	ADP
ejpam-2803	776	4	coprime	coprime	NOUN
ejpam-2803	776	5	and	and	CCONJ
ejpam-2803	776	6	second	second	ADJ
ejpam-2803	776	7	submodules	submodule	NOUN
ejpam-2803	776	8	.	.	PUNCT
ejpam-2803	777	1	22(01):47–72	22(01):47–72	NUM
ejpam-2803	777	2	,	,	PUNCT
ejpam-2803	777	3	2015	2015	NUM
ejpam-2803	777	4	.	.	PUNCT
ejpam-2803	778	1	[	[	X
ejpam-2803	778	2	3	3	X
ejpam-2803	778	3	]	]	X
ejpam-2803	778	4	h	h	NOUN
ejpam-2803	778	5	ansari	ansari	ADJ
ejpam-2803	778	6	-	-	PUNCT
ejpam-2803	778	7	toroghy	toroghy	NOUN
ejpam-2803	778	8	and	and	CCONJ
ejpam-2803	778	9	f	f	PROPN
ejpam-2803	778	10	farshadifar	farshadifar	ADV
ejpam-2803	778	11	.	.	PUNCT
ejpam-2803	779	1	the	the	DET
ejpam-2803	779	2	dual	dual	ADJ
ejpam-2803	779	3	notion	notion	NOUN
ejpam-2803	779	4	of	of	ADP
ejpam-2803	779	5	multiplication	multiplication	NOUN
ejpam-2803	779	6	modules	module	NOUN
ejpam-2803	779	7	.	.	PUNCT
ejpam-2803	780	1	taiwanese	taiwanese	ADJ
ejpam-2803	780	2	j.	j.	PROPN
ejpam-2803	780	3	math	math	PROPN
ejpam-2803	780	4	,	,	PUNCT
ejpam-2803	780	5	11(4):1189–1201	11(4):1189–1201	NUM
ejpam-2803	780	6	,	,	PUNCT
ejpam-2803	780	7	2007	2007	NUM
ejpam-2803	780	8	.	.	PUNCT
ejpam-2803	781	1	[	[	X
ejpam-2803	781	2	4	4	X
ejpam-2803	781	3	]	]	X
ejpam-2803	781	4	h	h	NOUN
ejpam-2803	781	5	ansari	ansari	ADJ
ejpam-2803	781	6	-	-	PUNCT
ejpam-2803	781	7	toroghy	toroghy	NOUN
ejpam-2803	781	8	and	and	CCONJ
ejpam-2803	781	9	f	f	PROPN
ejpam-2803	781	10	farshadifar	farshadifar	ADV
ejpam-2803	781	11	.	.	PUNCT
ejpam-2803	782	1	on	on	ADP
ejpam-2803	782	2	the	the	DET
ejpam-2803	782	3	dual	dual	ADJ
ejpam-2803	782	4	notion	notion	NOUN
ejpam-2803	782	5	of	of	ADP
ejpam-2803	782	6	prime	prime	ADJ
ejpam-2803	782	7	submodules	submodule	NOUN
ejpam-2803	782	8	.	.	PUNCT
ejpam-2803	783	1	19(spec01):1109–1116	19(spec01):1109–1116	NUM
ejpam-2803	783	2	,	,	PUNCT
ejpam-2803	783	3	2012	2012	NUM
ejpam-2803	783	4	.	.	PUNCT
ejpam-2803	784	1	[	[	X
ejpam-2803	784	2	5	5	NUM
ejpam-2803	784	3	]	]	PUNCT
ejpam-2803	784	4	h	h	NOUN
ejpam-2803	784	5	ansari	ansari	ADJ
ejpam-2803	784	6	-	-	PUNCT
ejpam-2803	784	7	toroghy	toroghy	NOUN
ejpam-2803	784	8	and	and	CCONJ
ejpam-2803	784	9	f	f	PROPN
ejpam-2803	784	10	farshadifar	farshadifar	ADV
ejpam-2803	784	11	.	.	PUNCT
ejpam-2803	785	1	on	on	ADP
ejpam-2803	785	2	the	the	DET
ejpam-2803	785	3	dual	dual	ADJ
ejpam-2803	785	4	notion	notion	NOUN
ejpam-2803	785	5	of	of	ADP
ejpam-2803	785	6	prime	prime	ADJ
ejpam-2803	785	7	submodules	submodule	NOUN
ejpam-2803	785	8	(	(	PUNCT
ejpam-2803	785	9	ii	ii	NOUN
ejpam-2803	785	10	)	)	PUNCT
ejpam-2803	785	11	.	.	PUNCT
ejpam-2803	786	1	mediterr	mediterr	PROPN
ejpam-2803	786	2	.	.	PUNCT
ejpam-2803	787	1	j.	j.	PROPN
ejpam-2803	787	2	math	math	PROPN
ejpam-2803	787	3	,	,	PUNCT
ejpam-2803	787	4	9(2):327–336	9(2):327–336	NUM
ejpam-2803	787	5	,	,	PUNCT
ejpam-2803	787	6	2012	2012	NUM
ejpam-2803	787	7	.	.	PUNCT
ejpam-2803	788	1	[	[	X
ejpam-2803	788	2	6	6	NUM
ejpam-2803	788	3	]	]	PUNCT
ejpam-2803	788	4	h	h	NOUN
ejpam-2803	788	5	ansari	ansari	ADJ
ejpam-2803	788	6	-	-	PUNCT
ejpam-2803	788	7	toroghy	toroghy	NOUN
ejpam-2803	788	8	and	and	CCONJ
ejpam-2803	788	9	f	f	PROPN
ejpam-2803	788	10	farshadifar	farshadifar	ADV
ejpam-2803	788	11	.	.	PUNCT
ejpam-2803	789	1	on	on	ADP
ejpam-2803	789	2	the	the	DET
ejpam-2803	789	3	dual	dual	ADJ
ejpam-2803	789	4	notion	notion	NOUN
ejpam-2803	789	5	of	of	ADP
ejpam-2803	789	6	prime	prime	ADJ
ejpam-2803	789	7	radicals	radical	NOUN
ejpam-2803	789	8	of	of	ADP
ejpam-2803	789	9	submodules	submodule	NOUN
ejpam-2803	789	10	.	.	PUNCT
ejpam-2803	790	1	asian	asian	ADJ
ejpam-2803	790	2	-	-	PUNCT
ejpam-2803	790	3	european	european	ADJ
ejpam-2803	790	4	journal	journal	NOUN
ejpam-2803	790	5	of	of	ADP
ejpam-2803	790	6	mathematics	mathematic	NOUN
ejpam-2803	790	7	,	,	PUNCT
ejpam-2803	790	8	6(02):1350024	6(02):1350024	NOUN
ejpam-2803	790	9	(	(	PUNCT
ejpam-2803	790	10	11	11	NUM
ejpam-2803	790	11	pages	page	NOUN
ejpam-2803	790	12	)	)	PUNCT
ejpam-2803	790	13	,	,	PUNCT
ejpam-2803	790	14	2013	2013	NUM
ejpam-2803	790	15	.	.	PUNCT
ejpam-2803	791	1	[	[	X
ejpam-2803	791	2	7	7	X
ejpam-2803	791	3	]	]	X
ejpam-2803	791	4	h	h	NOUN
ejpam-2803	791	5	ansari	ansari	ADJ
ejpam-2803	791	6	-	-	PUNCT
ejpam-2803	791	7	toroghy	toroghy	NOUN
ejpam-2803	791	8	and	and	CCONJ
ejpam-2803	791	9	f	f	PROPN
ejpam-2803	791	10	farshadifar	farshadifar	ADV
ejpam-2803	791	11	.	.	PUNCT
ejpam-2803	792	1	the	the	DET
ejpam-2803	792	2	zariski	zariski	NOUN
ejpam-2803	792	3	topology	topology	NOUN
ejpam-2803	792	4	on	on	ADP
ejpam-2803	792	5	the	the	DET
ejpam-2803	792	6	second	second	ADJ
ejpam-2803	792	7	spectrum	spectrum	NOUN
ejpam-2803	792	8	of	of	ADP
ejpam-2803	792	9	a	a	DET
ejpam-2803	792	10	module	module	NOUN
ejpam-2803	792	11	.	.	PUNCT
ejpam-2803	793	1	algebra	algebra	PROPN
ejpam-2803	793	2	colloq	colloq	PROPN
ejpam-2803	793	3	,	,	PUNCT
ejpam-2803	793	4	21(04):671–688	21(04):671–688	PROPN
ejpam-2803	793	5	,	,	PUNCT
ejpam-2803	793	6	2014	2014	NUM
ejpam-2803	793	7	.	.	PUNCT
ejpam-2803	794	1	[	[	X
ejpam-2803	794	2	8	8	NUM
ejpam-2803	794	3	]	]	X
ejpam-2803	794	4	h	h	NOUN
ejpam-2803	794	5	ansari	ansari	ADJ
ejpam-2803	794	6	-	-	PUNCT
ejpam-2803	794	7	toroghy	toroghy	ADJ
ejpam-2803	794	8	,	,	PUNCT
ejpam-2803	794	9	s	s	NOUN
ejpam-2803	794	10	keyvani	keyvani	NOUN
ejpam-2803	794	11	,	,	PUNCT
ejpam-2803	794	12	and	and	CCONJ
ejpam-2803	794	13	f	f	PROPN
ejpam-2803	794	14	farshadifar	farshadifar	ADV
ejpam-2803	794	15	.	.	PUNCT
ejpam-2803	795	1	the	the	DET
ejpam-2803	795	2	zariski	zariski	NOUN
ejpam-2803	795	3	topology	topology	NOUN
ejpam-2803	795	4	on	on	ADP
ejpam-2803	795	5	the	the	DET
ejpam-2803	795	6	second	second	ADJ
ejpam-2803	795	7	spectrum	spectrum	NOUN
ejpam-2803	795	8	of	of	ADP
ejpam-2803	795	9	a	a	DET
ejpam-2803	795	10	module	module	NOUN
ejpam-2803	795	11	(	(	PUNCT
ejpam-2803	795	12	ii	ii	NOUN
ejpam-2803	795	13	)	)	PUNCT
ejpam-2803	795	14	.	.	PUNCT
ejpam-2803	796	1	to	to	PART
ejpam-2803	796	2	appear	appear	VERB
ejpam-2803	796	3	in	in	ADP
ejpam-2803	796	4	bull	bull	NOUN
ejpam-2803	796	5	.	.	PUNCT
ejpam-2803	797	1	malays	malays	PROPN
ejpam-2803	797	2	.	.	PUNCT
ejpam-2803	798	1	math	math	NOUN
ejpam-2803	798	2	.	.	PUNCT
ejpam-2803	799	1	sci	sci	PROPN
ejpam-2803	799	2	.	.	PROPN
ejpam-2803	799	3	soc	soc	PROPN
ejpam-2803	799	4	.	.	PUNCT
ejpam-2803	800	1	[	[	X
ejpam-2803	800	2	9	9	NUM
ejpam-2803	800	3	]	]	SYM
ejpam-2803	800	4	h	h	NOUN
ejpam-2803	800	5	ansari	ansari	ADJ
ejpam-2803	800	6	-	-	PUNCT
ejpam-2803	800	7	toroghy	toroghy	ADJ
ejpam-2803	800	8	and	and	CCONJ
ejpam-2803	800	9	r	r	NOUN
ejpam-2803	800	10	ovlyaee	ovlyaee	ADJ
ejpam-2803	800	11	-	-	PUNCT
ejpam-2803	800	12	sarmazdeh	sarmazdeh	NOUN
ejpam-2803	800	13	.	.	PUNCT
ejpam-2803	801	1	on	on	ADP
ejpam-2803	801	2	the	the	DET
ejpam-2803	801	3	prime	prime	ADJ
ejpam-2803	801	4	spectrum	spectrum	NOUN
ejpam-2803	801	5	of	of	ADP
ejpam-2803	801	6	x	x	ADJ
ejpam-2803	801	7	-	-	ADJ
ejpam-2803	801	8	injective	injective	ADJ
ejpam-2803	801	9	modules	module	NOUN
ejpam-2803	801	10	.	.	PUNCT
ejpam-2803	802	1	comm	comm	NOUN
ejpam-2803	802	2	.	.	PUNCT
ejpam-2803	803	1	algebra	algebra	PROPN
ejpam-2803	803	2	,	,	PUNCT
ejpam-2803	803	3	38(7):2606–2621	38(7):2606–2621	NUM
ejpam-2803	803	4	,	,	PUNCT
ejpam-2803	803	5	2010	2010	NUM
ejpam-2803	803	6	.	.	PUNCT
ejpam-2803	804	1	[	[	X
ejpam-2803	804	2	10	10	NUM
ejpam-2803	804	3	]	]	X
ejpam-2803	804	4	m	m	VERB
ejpam-2803	804	5	f	f	X
ejpam-2803	804	6	atiyah	atiyah	NOUN
ejpam-2803	804	7	and	and	CCONJ
ejpam-2803	804	8	i	i	PRON
ejpam-2803	804	9	g	g	PROPN
ejpam-2803	804	10	macdonald	macdonald	PROPN
ejpam-2803	804	11	.	.	PUNCT
ejpam-2803	805	1	introduction	introduction	NOUN
ejpam-2803	805	2	to	to	ADP
ejpam-2803	805	3	commutative	commutative	ADJ
ejpam-2803	805	4	algebra	algebra	NOUN
ejpam-2803	805	5	.	.	PUNCT
ejpam-2803	806	1	1969	1969	NUM
ejpam-2803	806	2	.	.	PUNCT
ejpam-2803	807	1	[	[	X
ejpam-2803	807	2	11	11	NUM
ejpam-2803	807	3	]	]	PUNCT
ejpam-2803	807	4	h	h	NOUN
ejpam-2803	807	5	bass	bass	NOUN
ejpam-2803	807	6	.	.	PUNCT
ejpam-2803	808	1	finitistic	finitistic	ADJ
ejpam-2803	808	2	dimension	dimension	NOUN
ejpam-2803	808	3	and	and	CCONJ
ejpam-2803	808	4	a	a	DET
ejpam-2803	808	5	homological	homological	ADJ
ejpam-2803	808	6	generalization	generalization	NOUN
ejpam-2803	808	7	of	of	ADP
ejpam-2803	808	8	semi	semi	ADJ
ejpam-2803	808	9	-	-	ADJ
ejpam-2803	808	10	primary	primary	ADJ
ejpam-2803	808	11	rings	ring	NOUN
ejpam-2803	808	12	.	.	PUNCT
ejpam-2803	809	1	trans	trans	PROPN
ejpam-2803	809	2	.	.	PUNCT
ejpam-2803	810	1	amer	amer	PROPN
ejpam-2803	810	2	.	.	PUNCT
ejpam-2803	810	3	math	math	PROPN
ejpam-2803	810	4	.	.	PUNCT
ejpam-2803	811	1	soc	soc	PROPN
ejpam-2803	811	2	,	,	PUNCT
ejpam-2803	811	3	95(3):466–488	95(3):466–488	PROPN
ejpam-2803	811	4	,	,	PUNCT
ejpam-2803	811	5	1960	1960	NUM
ejpam-2803	811	6	.	.	PUNCT
ejpam-2803	812	1	[	[	X
ejpam-2803	812	2	12	12	NUM
ejpam-2803	812	3	]	]	X
ejpam-2803	812	4	m	m	VERB
ejpam-2803	812	5	behboodi	behboodi	NOUN
ejpam-2803	812	6	and	and	CCONJ
ejpam-2803	812	7	m	m	PROPN
ejpam-2803	812	8	r	r	NOUN
ejpam-2803	812	9	haddadi	haddadi	NOUN
ejpam-2803	812	10	.	.	PUNCT
ejpam-2803	813	1	classical	classical	ADJ
ejpam-2803	813	2	zariski	zariski	NOUN
ejpam-2803	813	3	topology	topology	NOUN
ejpam-2803	813	4	of	of	ADP
ejpam-2803	813	5	modules	module	NOUN
ejpam-2803	813	6	and	and	CCONJ
ejpam-2803	813	7	spectral	spectral	ADJ
ejpam-2803	813	8	spaces	space	NOUN
ejpam-2803	813	9	i.	i.	PROPN
ejpam-2803	813	10	int	int	PROPN
ejpam-2803	813	11	.	.	PUNCT
ejpam-2803	814	1	electron	electron	PROPN
ejpam-2803	814	2	.	.	PUNCT
ejpam-2803	815	1	j.	j.	PROPN
ejpam-2803	815	2	algebra	algebra	PROPN
ejpam-2803	815	3	,	,	PUNCT
ejpam-2803	815	4	4:104–130	4:104–130	NOUN
ejpam-2803	815	5	,	,	PUNCT
ejpam-2803	815	6	2008	2008	NUM
ejpam-2803	815	7	.	.	PUNCT
ejpam-2803	816	1	[	[	X
ejpam-2803	816	2	13	13	NUM
ejpam-2803	816	3	]	]	PUNCT
ejpam-2803	816	4	n	n	X
ejpam-2803	816	5	bourbaki	bourbaki	VERB
ejpam-2803	816	6	.	.	PUNCT
ejpam-2803	817	1	commutative	commutative	ADJ
ejpam-2803	817	2	algebra	algebra	NOUN
ejpam-2803	817	3	.	.	PUNCT
ejpam-2803	818	1	1972	1972	NUM
ejpam-2803	818	2	.	.	PUNCT
ejpam-2803	819	1	[	[	X
ejpam-2803	819	2	14	14	NUM
ejpam-2803	819	3	]	]	X
ejpam-2803	819	4	s	s	AUX
ejpam-2803	819	5	çeken	çeken	VERB
ejpam-2803	819	6	and	and	CCONJ
ejpam-2803	819	7	m	m	NOUN
ejpam-2803	819	8	alkan	alkan	PROPN
ejpam-2803	819	9	.	.	PUNCT
ejpam-2803	820	1	on	on	ADP
ejpam-2803	820	2	the	the	DET
ejpam-2803	820	3	second	second	ADJ
ejpam-2803	820	4	spectrum	spectrum	NOUN
ejpam-2803	820	5	and	and	CCONJ
ejpam-2803	820	6	the	the	DET
ejpam-2803	820	7	second	second	ADJ
ejpam-2803	820	8	classical	classical	ADJ
ejpam-2803	820	9	zariski	zariski	NOUN
ejpam-2803	820	10	topology	topology	NOUN
ejpam-2803	820	11	of	of	ADP
ejpam-2803	820	12	a	a	DET
ejpam-2803	820	13	module	module	NOUN
ejpam-2803	820	14	.	.	PUNCT
ejpam-2803	821	1	j.	j.	PROPN
ejpam-2803	821	2	algebra	algebra	PROPN
ejpam-2803	821	3	appl	appl	PROPN
ejpam-2803	821	4	,	,	PUNCT
ejpam-2803	821	5	14(10):1550150	14(10):1550150	NUM
ejpam-2803	821	6	(	(	PUNCT
ejpam-2803	821	7	13	13	NUM
ejpam-2803	821	8	pages	page	NOUN
ejpam-2803	821	9	)	)	PUNCT
ejpam-2803	821	10	,	,	PUNCT
ejpam-2803	821	11	2015	2015	NUM
ejpam-2803	821	12	.	.	PUNCT
ejpam-2803	822	1	[	[	X
ejpam-2803	822	2	15	15	NUM
ejpam-2803	822	3	]	]	X
ejpam-2803	822	4	s	s	AUX
ejpam-2803	822	5	çeken	çeken	VERB
ejpam-2803	822	6	,	,	PUNCT
ejpam-2803	822	7	m	m	PROPN
ejpam-2803	822	8	alkan	alkan	PROPN
ejpam-2803	822	9	,	,	PUNCT
ejpam-2803	822	10	and	and	CCONJ
ejpam-2803	822	11	p	p	PROPN
ejpam-2803	822	12	f	f	PROPN
ejpam-2803	822	13	smith	smith	PROPN
ejpam-2803	822	14	.	.	PUNCT
ejpam-2803	823	1	the	the	DET
ejpam-2803	823	2	dual	dual	ADJ
ejpam-2803	823	3	notion	notion	NOUN
ejpam-2803	823	4	of	of	ADP
ejpam-2803	823	5	the	the	DET
ejpam-2803	823	6	prime	prime	ADJ
ejpam-2803	823	7	radical	radical	NOUN
ejpam-2803	823	8	of	of	ADP
ejpam-2803	823	9	a	a	DET
ejpam-2803	823	10	module	module	NOUN
ejpam-2803	823	11	.	.	PUNCT
ejpam-2803	824	1	j.	j.	PROPN
ejpam-2803	824	2	algebra	algebra	PROPN
ejpam-2803	824	3	,	,	PUNCT
ejpam-2803	824	4	392:265–275	392:265–275	NUM
ejpam-2803	824	5	,	,	PUNCT
ejpam-2803	824	6	2013	2013	NUM
ejpam-2803	824	7	.	.	PUNCT
ejpam-2803	825	1	[	[	X
ejpam-2803	825	2	16	16	NUM
ejpam-2803	825	3	]	]	X
ejpam-2803	825	4	s	s	AUX
ejpam-2803	825	5	çeken	çeken	VERB
ejpam-2803	825	6	,	,	PUNCT
ejpam-2803	825	7	m	m	PROPN
ejpam-2803	825	8	alkan	alkan	PROPN
ejpam-2803	825	9	,	,	PUNCT
ejpam-2803	825	10	and	and	CCONJ
ejpam-2803	825	11	p	p	PROPN
ejpam-2803	825	12	f	f	PROPN
ejpam-2803	825	13	smith	smith	PROPN
ejpam-2803	825	14	.	.	PUNCT
ejpam-2803	826	1	second	second	ADJ
ejpam-2803	826	2	modules	module	NOUN
ejpam-2803	826	3	over	over	ADP
ejpam-2803	826	4	noncommutative	noncommutative	ADJ
ejpam-2803	826	5	rings	ring	NOUN
ejpam-2803	826	6	.	.	PUNCT
ejpam-2803	827	1	comm	comm	NOUN
ejpam-2803	827	2	.	.	PUNCT
ejpam-2803	828	1	algebra	algebra	NOUN
ejpam-2803	828	2	,	,	PUNCT
ejpam-2803	828	3	41(1):83–98	41(1):83–98	NUM
ejpam-2803	828	4	,	,	PUNCT
ejpam-2803	828	5	2013	2013	NUM
ejpam-2803	828	6	.	.	PUNCT
ejpam-2803	829	1	[	[	X
ejpam-2803	829	2	17	17	NUM
ejpam-2803	829	3	]	]	X
ejpam-2803	829	4	f	f	PROPN
ejpam-2803	829	5	farshadifar	farshadifar	ADV
ejpam-2803	829	6	.	.	PUNCT
ejpam-2803	830	1	modules	module	NOUN
ejpam-2803	830	2	with	with	ADP
ejpam-2803	830	3	noetherian	noetherian	ADJ
ejpam-2803	830	4	second	second	ADJ
ejpam-2803	830	5	spectrum	spectrum	NOUN
ejpam-2803	830	6	.	.	PUNCT
ejpam-2803	831	1	journal	journal	NOUN
ejpam-2803	831	2	of	of	ADP
ejpam-2803	831	3	algebra	algebra	PROPN
ejpam-2803	831	4	and	and	CCONJ
ejpam-2803	831	5	related	related	ADJ
ejpam-2803	831	6	topics	topic	NOUN
ejpam-2803	831	7	,	,	PUNCT
ejpam-2803	831	8	1(1):19–30	1(1):19–30	NUM
ejpam-2803	831	9	,	,	PUNCT
ejpam-2803	831	10	2013	2013	NUM
ejpam-2803	831	11	.	.	PUNCT
ejpam-2803	832	1	[	[	X
ejpam-2803	832	2	18	18	NUM
ejpam-2803	832	3	]	]	PUNCT
ejpam-2803	832	4	l	l	NOUN
ejpam-2803	832	5	fuchs	fuchs	PROPN
ejpam-2803	832	6	,	,	PUNCT
ejpam-2803	832	7	w	w	NOUN
ejpam-2803	832	8	heinzer	heinzer	NOUN
ejpam-2803	832	9	,	,	PUNCT
ejpam-2803	832	10	and	and	CCONJ
ejpam-2803	832	11	b	b	NOUN
ejpam-2803	832	12	olberding	olberding	NOUN
ejpam-2803	832	13	.	.	PUNCT
ejpam-2803	833	1	commutative	commutative	ADJ
ejpam-2803	833	2	ideal	ideal	PROPN
ejpam-2803	833	3	theory	theory	NOUN
ejpam-2803	833	4	without	without	ADP
ejpam-2803	833	5	finiteness	finiteness	ADJ
ejpam-2803	833	6	conditions	condition	NOUN
ejpam-2803	833	7	:	:	PUNCT
ejpam-2803	833	8	irreducibility	irreducibility	NOUN
ejpam-2803	833	9	in	in	ADP
ejpam-2803	833	10	the	the	DET
ejpam-2803	833	11	quotient	quotient	NOUN
ejpam-2803	833	12	filed	file	VERB
ejpam-2803	833	13	.	.	PUNCT
ejpam-2803	834	1	abelian	abelian	PROPN
ejpam-2803	834	2	groups	group	NOUN
ejpam-2803	834	3	,	,	PUNCT
ejpam-2803	834	4	rings	ring	NOUN
ejpam-2803	834	5	,	,	PUNCT
ejpam-2803	834	6	modules	module	NOUN
ejpam-2803	834	7	,	,	PUNCT
ejpam-2803	834	8	and	and	CCONJ
ejpam-2803	834	9	homological	homological	ADJ
ejpam-2803	834	10	algebra	algebra	NOUN
ejpam-2803	834	11	,	,	PUNCT
ejpam-2803	834	12	lect	lect	PROPN
ejpam-2803	834	13	.	.	PUNCT
ejpam-2803	834	14	notes	note	VERB
ejpam-2803	834	15	pure	pure	ADJ
ejpam-2803	834	16	appl	appl	PROPN
ejpam-2803	834	17	.	.	PUNCT
ejpam-2803	835	1	math	math	PROPN
ejpam-2803	835	2	,	,	PUNCT
ejpam-2803	835	3	249:121–145	249:121–145	NUM
ejpam-2803	835	4	,	,	PUNCT
ejpam-2803	835	5	2006	2006	NUM
ejpam-2803	835	6	.	.	PUNCT
ejpam-2803	836	1	references	reference	NOUN
ejpam-2803	836	2	230	230	NUM
ejpam-2803	836	3	[	[	SYM
ejpam-2803	836	4	19	19	NUM
ejpam-2803	836	5	]	]	X
ejpam-2803	836	6	m	m	VERB
ejpam-2803	836	7	hochster	hochster	ADJ
ejpam-2803	836	8	.	.	PUNCT
ejpam-2803	837	1	prime	prime	ADJ
ejpam-2803	837	2	ideal	ideal	ADJ
ejpam-2803	837	3	structure	structure	NOUN
ejpam-2803	837	4	in	in	ADP
ejpam-2803	837	5	commutative	commutative	ADJ
ejpam-2803	837	6	rings	ring	NOUN
ejpam-2803	837	7	.	.	PUNCT
ejpam-2803	838	1	trans	trans	PROPN
ejpam-2803	838	2	.	.	PUNCT
ejpam-2803	839	1	amer	amer	PROPN
ejpam-2803	839	2	.	.	PUNCT
ejpam-2803	839	3	math	math	PROPN
ejpam-2803	839	4	.	.	PUNCT
ejpam-2803	840	1	soc	soc	PROPN
ejpam-2803	840	2	,	,	PUNCT
ejpam-2803	840	3	142:43–60	142:43–60	PROPN
ejpam-2803	840	4	,	,	PUNCT
ejpam-2803	840	5	1969	1969	NUM
ejpam-2803	840	6	.	.	PUNCT
ejpam-2803	841	1	[	[	X
ejpam-2803	841	2	20	20	NUM
ejpam-2803	841	3	]	]	X
ejpam-2803	841	4	m	m	VERB
ejpam-2803	841	5	hochster	hochster	ADJ
ejpam-2803	841	6	.	.	PUNCT
ejpam-2803	842	1	the	the	DET
ejpam-2803	842	2	minimal	minimal	ADJ
ejpam-2803	842	3	prime	prime	ADJ
ejpam-2803	842	4	spectrum	spectrum	NOUN
ejpam-2803	842	5	of	of	ADP
ejpam-2803	842	6	a	a	DET
ejpam-2803	842	7	commutative	commutative	ADJ
ejpam-2803	842	8	ring	ring	NOUN
ejpam-2803	842	9	.	.	PUNCT
ejpam-2803	843	1	canad	canad	PROPN
ejpam-2803	843	2	.	.	PUNCT
ejpam-2803	844	1	j.	j.	PROPN
ejpam-2803	844	2	math	math	PROPN
ejpam-2803	844	3	,	,	PUNCT
ejpam-2803	844	4	23(5):749–758	23(5):749–758	PROPN
ejpam-2803	844	5	,	,	PUNCT
ejpam-2803	844	6	1971	1971	NUM
ejpam-2803	844	7	.	.	PUNCT
ejpam-2803	845	1	[	[	X
ejpam-2803	845	2	21	21	NUM
ejpam-2803	845	3	]	]	X
ejpam-2803	845	4	e	e	X
ejpam-2803	845	5	kunz	kunz	PROPN
ejpam-2803	845	6	.	.	PUNCT
ejpam-2803	846	1	introduction	introduction	NOUN
ejpam-2803	846	2	to	to	ADP
ejpam-2803	846	3	commutative	commutative	ADJ
ejpam-2803	846	4	algebra	algebra	NOUN
ejpam-2803	846	5	and	and	CCONJ
ejpam-2803	846	6	algebraic	algebraic	ADJ
ejpam-2803	846	7	geometry	geometry	NOUN
ejpam-2803	846	8	.	.	PUNCT
ejpam-2803	847	1	modern	modern	ADJ
ejpam-2803	847	2	birkhäuser	birkhäuser	NOUN
ejpam-2803	847	3	classics	classic	NOUN
ejpam-2803	847	4	,	,	PUNCT
ejpam-2803	847	5	2013	2013	NUM
ejpam-2803	847	6	.	.	PUNCT
ejpam-2803	848	1	[	[	X
ejpam-2803	848	2	22	22	NUM
ejpam-2803	848	3	]	]	X
ejpam-2803	848	4	h	h	NOUN
ejpam-2803	848	5	li	li	PROPN
ejpam-2803	848	6	and	and	CCONJ
ejpam-2803	848	7	k	k	PROPN
ejpam-2803	848	8	shum	shum	PROPN
ejpam-2803	848	9	.	.	PUNCT
ejpam-2803	849	1	on	on	ADP
ejpam-2803	849	2	a	a	DET
ejpam-2803	849	3	problem	problem	NOUN
ejpam-2803	849	4	of	of	ADP
ejpam-2803	849	5	spectral	spectral	ADJ
ejpam-2803	849	6	posets	poset	NOUN
ejpam-2803	849	7	.	.	PUNCT
ejpam-2803	850	1	j.	j.	PROPN
ejpam-2803	850	2	appl	appl	PROPN
ejpam-2803	850	3	.	.	PUNCT
ejpam-2803	851	1	algebra	algebra	NOUN
ejpam-2803	851	2	discrete	discrete	ADJ
ejpam-2803	851	3	struct	struct	NOUN
ejpam-2803	851	4	,	,	PUNCT
ejpam-2803	851	5	1(3):203–209	1(3):203–209	NUM
ejpam-2803	851	6	,	,	PUNCT
ejpam-2803	851	7	2003	2003	NUM
ejpam-2803	851	8	.	.	PUNCT
ejpam-2803	852	1	[	[	X
ejpam-2803	852	2	23	23	NUM
ejpam-2803	852	3	]	]	X
ejpam-2803	852	4	c	c	PROPN
ejpam-2803	852	5	p	p	PROPN
ejpam-2803	852	6	lu	lu	PROPN
ejpam-2803	852	7	.	.	PUNCT
ejpam-2803	853	1	modules	module	NOUN
ejpam-2803	853	2	with	with	ADP
ejpam-2803	853	3	noetherian	noetherian	ADJ
ejpam-2803	853	4	spectrum	spectrum	NOUN
ejpam-2803	853	5	.	.	PUNCT
ejpam-2803	854	1	comm	comm	NOUN
ejpam-2803	854	2	.	.	PUNCT
ejpam-2803	855	1	algebra	algebra	NOUN
ejpam-2803	855	2	,	,	PUNCT
ejpam-2803	855	3	38(3):807–828	38(3):807–828	PROPN
ejpam-2803	855	4	,	,	PUNCT
ejpam-2803	855	5	2010	2010	NUM
ejpam-2803	855	6	.	.	PUNCT
ejpam-2803	856	1	[	[	X
ejpam-2803	856	2	24	24	NUM
ejpam-2803	856	3	]	]	SYM
ejpam-2803	856	4	r	r	NOUN
ejpam-2803	856	5	l	l	NOUN
ejpam-2803	856	6	mccasland	mccasland	NOUN
ejpam-2803	856	7	,	,	PUNCT
ejpam-2803	856	8	m	m	PROPN
ejpam-2803	856	9	e	e	NOUN
ejpam-2803	856	10	moore	moore	NOUN
ejpam-2803	856	11	,	,	PUNCT
ejpam-2803	856	12	and	and	CCONJ
ejpam-2803	856	13	p	p	PROPN
ejpam-2803	856	14	f	f	PROPN
ejpam-2803	856	15	smith	smith	PROPN
ejpam-2803	856	16	.	.	PUNCT
ejpam-2803	857	1	on	on	ADP
ejpam-2803	857	2	the	the	DET
ejpam-2803	857	3	spectrum	spectrum	NOUN
ejpam-2803	857	4	of	of	ADP
ejpam-2803	857	5	a	a	DET
ejpam-2803	857	6	module	module	NOUN
ejpam-2803	857	7	over	over	ADP
ejpam-2803	857	8	a	a	DET
ejpam-2803	857	9	commutative	commutative	ADJ
ejpam-2803	857	10	ring	ring	NOUN
ejpam-2803	857	11	.	.	PUNCT
ejpam-2803	858	1	comm	comm	NOUN
ejpam-2803	858	2	.	.	PUNCT
ejpam-2803	859	1	algebra	algebra	PROPN
ejpam-2803	859	2	,	,	PUNCT
ejpam-2803	859	3	25(1):79–103	25(1):79–103	NUM
ejpam-2803	859	4	,	,	PUNCT
ejpam-2803	859	5	1997	1997	NUM
ejpam-2803	859	6	.	.	PUNCT
ejpam-2803	860	1	[	[	X
ejpam-2803	860	2	25	25	NUM
ejpam-2803	860	3	]	]	PUNCT
ejpam-2803	860	4	l	l	NOUN
ejpam-2803	860	5	melkersson	melkersson	NOUN
ejpam-2803	860	6	and	and	CCONJ
ejpam-2803	860	7	p	p	NOUN
ejpam-2803	860	8	schenzel	schenzel	NOUN
ejpam-2803	860	9	.	.	PUNCT
ejpam-2803	861	1	the	the	DET
ejpam-2803	861	2	co	co	NOUN
ejpam-2803	861	3	-	-	NOUN
ejpam-2803	861	4	localization	localization	NOUN
ejpam-2803	861	5	of	of	ADP
ejpam-2803	861	6	an	an	DET
ejpam-2803	861	7	artinian	artinian	ADJ
ejpam-2803	861	8	module	module	NOUN
ejpam-2803	861	9	.	.	PUNCT
ejpam-2803	862	1	proc	proc	PROPN
ejpam-2803	862	2	.	.	PUNCT
ejpam-2803	863	1	edinburgh	edinburgh	PROPN
ejpam-2803	863	2	math	math	PROPN
ejpam-2803	863	3	,	,	PUNCT
ejpam-2803	863	4	38(01):121–131	38(01):121–131	NUM
ejpam-2803	863	5	,	,	PUNCT
ejpam-2803	863	6	1995	1995	NUM
ejpam-2803	863	7	.	.	PUNCT
ejpam-2803	864	1	[	[	X
ejpam-2803	864	2	26	26	NUM
ejpam-2803	864	3	]	]	PUNCT
ejpam-2803	864	4	n	n	PRON
ejpam-2803	864	5	v	v	X
ejpam-2803	864	6	sanh	sanh	NOUN
ejpam-2803	864	7	,	,	PUNCT
ejpam-2803	864	8	l	l	PROPN
ejpam-2803	864	9	p	p	PROPN
ejpam-2803	864	10	thao	thao	PROPN
ejpam-2803	864	11	,	,	PUNCT
ejpam-2803	864	12	n	n	PROPN
ejpam-2803	864	13	f	f	PROPN
ejpam-2803	864	14	al	al	PROPN
ejpam-2803	864	15	-	-	PUNCT
ejpam-2803	864	16	mayahi	mayahi	NOUN
ejpam-2803	864	17	,	,	PUNCT
ejpam-2803	864	18	and	and	CCONJ
ejpam-2803	864	19	k	k	PROPN
ejpam-2803	864	20	p	p	X
ejpam-2803	864	21	shum	shum	NOUN
ejpam-2803	864	22	.	.	PUNCT
ejpam-2803	865	1	zariski	zariski	NOUN
ejpam-2803	865	2	topology	topology	NOUN
ejpam-2803	865	3	of	of	ADP
ejpam-2803	865	4	prime	prime	ADJ
ejpam-2803	865	5	spectrum	spectrum	NOUN
ejpam-2803	865	6	of	of	ADP
ejpam-2803	865	7	a	a	DET
ejpam-2803	865	8	module	module	NOUN
ejpam-2803	865	9	.	.	PUNCT
ejpam-2803	866	1	proceedings	proceeding	NOUN
ejpam-2803	866	2	of	of	ADP
ejpam-2803	866	3	the	the	DET
ejpam-2803	866	4	international	international	ADJ
ejpam-2803	866	5	conference	conference	NOUN
ejpam-2803	866	6	on	on	ADP
ejpam-2803	866	7	algebra	algebra	NOUN
ejpam-2803	866	8	2010	2010	NUM
ejpam-2803	866	9	:	:	PUNCT
ejpam-2803	866	10	advances	advance	NOUN
ejpam-2803	866	11	in	in	ADP
ejpam-2803	866	12	algebraic	algebraic	ADJ
ejpam-2803	866	13	structures	structure	NOUN
ejpam-2803	866	14	,	,	PUNCT
ejpam-2803	866	15	pages	page	NOUN
ejpam-2803	866	16	461–477	461–477	NUM
ejpam-2803	866	17	,	,	PUNCT
ejpam-2803	866	18	2011	2011	NUM
ejpam-2803	866	19	.	.	PUNCT
ejpam-2803	867	1	[	[	X
ejpam-2803	867	2	27	27	NUM
ejpam-2803	867	3	]	]	SYM
ejpam-2803	867	4	b	b	PROPN
ejpam-2803	867	5	t	t	NOUN
ejpam-2803	867	6	sims	sim	NOUN
ejpam-2803	867	7	.	.	PUNCT
ejpam-2803	868	1	fundamentals	fundamental	NOUN
ejpam-2803	868	2	of	of	ADP
ejpam-2803	868	3	topology	topology	NOUN
ejpam-2803	868	4	.	.	PUNCT
ejpam-2803	869	1	macmillan	macmillan	PROPN
ejpam-2803	869	2	publishing	publishing	PROPN
ejpam-2803	869	3	co.	co.	PROPN
ejpam-2803	869	4	inc	inc	PROPN
ejpam-2803	869	5	,	,	PUNCT
ejpam-2803	869	6	1976	1976	NUM
ejpam-2803	869	7	.	.	PUNCT
ejpam-2803	870	1	[	[	X
ejpam-2803	870	2	28	28	NUM
ejpam-2803	870	3	]	]	X
ejpam-2803	870	4	s	s	PART
ejpam-2803	870	5	yassemi	yassemi	NOUN
ejpam-2803	870	6	.	.	PUNCT
ejpam-2803	871	1	maximal	maximal	ADJ
ejpam-2803	871	2	elements	element	NOUN
ejpam-2803	871	3	of	of	ADP
ejpam-2803	871	4	support	support	NOUN
ejpam-2803	871	5	and	and	CCONJ
ejpam-2803	871	6	cosupport	cosupport	NOUN
ejpam-2803	871	7	.	.	PUNCT
ejpam-2803	872	1	http://streaming.ictp.it	http://streaming.ictp.it	X
ejpam-2803	872	2	/preprints/	/preprints/	NOUN
ejpam-2803	872	3	p/97/051.pdf	p/97/051.pdf	NOUN
ejpam-2803	872	4	,	,	PUNCT
ejpam-2803	872	5	1997	1997	NUM
ejpam-2803	872	6	.	.	PUNCT
ejpam-2803	873	1	[	[	X
ejpam-2803	873	2	29	29	NUM
ejpam-2803	873	3	]	]	SYM
ejpam-2803	873	4	s	s	PART
ejpam-2803	873	5	yassemi	yassemi	NOUN
ejpam-2803	873	6	.	.	PUNCT
ejpam-2803	874	1	the	the	DET
ejpam-2803	874	2	dual	dual	ADJ
ejpam-2803	874	3	notion	notion	NOUN
ejpam-2803	874	4	of	of	ADP
ejpam-2803	874	5	the	the	DET
ejpam-2803	874	6	cyclic	cyclic	ADJ
ejpam-2803	874	7	modules	module	NOUN
ejpam-2803	874	8	.	.	PUNCT
ejpam-2803	875	1	kobe	kobe	PROPN
ejpam-2803	875	2	.	.	PUNCT
ejpam-2803	876	1	j.	j.	PROPN
ejpam-2803	876	2	math	math	PROPN
ejpam-2803	876	3	,	,	PUNCT
ejpam-2803	876	4	15(1):41–46	15(1):41–46	NUM
ejpam-2803	876	5	,	,	PUNCT
ejpam-2803	876	6	1998	1998	NUM
ejpam-2803	876	7	.	.	PUNCT
ejpam-2803	877	1	[	[	X
ejpam-2803	877	2	30	30	NUM
ejpam-2803	877	3	]	]	SYM
ejpam-2803	877	4	s	s	PART
ejpam-2803	877	5	yassemi	yassemi	NOUN
ejpam-2803	877	6	.	.	PUNCT
ejpam-2803	878	1	the	the	DET
ejpam-2803	878	2	dual	dual	ADJ
ejpam-2803	878	3	notion	notion	NOUN
ejpam-2803	878	4	of	of	ADP
ejpam-2803	878	5	prime	prime	ADJ
ejpam-2803	878	6	submodules	submodule	NOUN
ejpam-2803	878	7	.	.	PUNCT
ejpam-2803	879	1	arch	arch	NOUN
ejpam-2803	879	2	.	.	PUNCT
ejpam-2803	880	1	math	math	NOUN
ejpam-2803	880	2	.	.	PUNCT
ejpam-2803	881	1	(	(	PUNCT
ejpam-2803	881	2	brno	brno	NOUN
ejpam-2803	881	3	)	)	PUNCT
ejpam-2803	881	4	,	,	PUNCT
ejpam-2803	881	5	37(4):273–278	37(4):273–278	NOUN
ejpam-2803	881	6	,	,	PUNCT
ejpam-2803	881	7	2001	2001	NUM
ejpam-2803	881	8	.	.	PUNCT