id	sid	tid	token	lemma	pos
ejpam-2634	1	1	compile	compile	NOUN
ejpam-2634	1	2	/	/	SYM
ejpam-2634	1	3	output.dvi	output.dvi	NOUN
ejpam-2634	1	4	european	european	ADJ
ejpam-2634	1	5	journal	journal	NOUN
ejpam-2634	1	6	of	of	ADP
ejpam-2634	1	7	pure	pure	ADJ
ejpam-2634	1	8	and	and	CCONJ
ejpam-2634	1	9	applied	apply	VERB
ejpam-2634	1	10	mathematics	mathematic	NOUN
ejpam-2634	1	11	vol	vol	NOUN
ejpam-2634	1	12	.	.	PROPN
ejpam-2634	2	1	9	9	NUM
ejpam-2634	2	2	,	,	PUNCT
ejpam-2634	2	3	no	no	INTJ
ejpam-2634	2	4	.	.	NOUN
ejpam-2634	2	5	3	3	NUM
ejpam-2634	2	6	,	,	PUNCT
ejpam-2634	2	7	2016	2016	NUM
ejpam-2634	2	8	,	,	PUNCT
ejpam-2634	2	9	244	244	NUM
ejpam-2634	2	10	-	-	SYM
ejpam-2634	2	11	249	249	NUM
ejpam-2634	2	12	issn	issn	PROPN
ejpam-2634	2	13	1307	1307	NUM
ejpam-2634	2	14	-	-	SYM
ejpam-2634	2	15	5543	5543	NUM
ejpam-2634	2	16	–	–	PUNCT
ejpam-2634	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2634	2	18	comultiplication	comultiplication	NOUN
ejpam-2634	2	19	modules	module	VERB
ejpam-2634	2	20	jafar	jafar	PROPN
ejpam-2634	2	21	a’zami	a’zami	PROPN
ejpam-2634	2	22	,	,	PUNCT
ejpam-2634	2	23	m.	m.	NOUN
ejpam-2634	2	24	khajepour∗	khajepour∗	PROPN
ejpam-2634	2	25	faculty	faculty	NOUN
ejpam-2634	2	26	of	of	ADP
ejpam-2634	2	27	mathematical	mathematical	ADJ
ejpam-2634	2	28	sciences	science	NOUN
ejpam-2634	2	29	,	,	PUNCT
ejpam-2634	2	30	department	department	NOUN
ejpam-2634	2	31	of	of	ADP
ejpam-2634	2	32	mathematics	mathematics	PROPN
ejpam-2634	2	33	,	,	PUNCT
ejpam-2634	2	34	university	university	NOUN
ejpam-2634	2	35	of	of	ADP
ejpam-2634	2	36	mohaghegh	mohaghegh	PROPN
ejpam-2634	2	37	ardabili	ardabili	PROPN
ejpam-2634	2	38	,	,	PUNCT
ejpam-2634	2	39	ardabil	ardabil	VERB
ejpam-2634	2	40	,	,	PUNCT
ejpam-2634	2	41	iran	iran	PROPN
ejpam-2634	2	42	abstract	abstract	NOUN
ejpam-2634	2	43	.	.	PUNCT
ejpam-2634	3	1	let	let	VERB
ejpam-2634	3	2	r	r	PRON
ejpam-2634	3	3	be	be	AUX
ejpam-2634	3	4	a	a	DET
ejpam-2634	3	5	commutative	commutative	ADJ
ejpam-2634	3	6	ring	ring	NOUN
ejpam-2634	3	7	.	.	PUNCT
ejpam-2634	4	1	an	an	DET
ejpam-2634	4	2	r	r	NOUN
ejpam-2634	4	3	-	-	PUNCT
ejpam-2634	4	4	module	module	NOUN
ejpam-2634	4	5	m	m	NOUN
ejpam-2634	4	6	is	be	AUX
ejpam-2634	4	7	comutiplication	comutiplication	NOUN
ejpam-2634	4	8	if	if	SCONJ
ejpam-2634	4	9	for	for	ADP
ejpam-2634	4	10	every	every	DET
ejpam-2634	4	11	submodule	submodule	NOUN
ejpam-2634	4	12	n	n	PROPN
ejpam-2634	4	13	of	of	ADP
ejpam-2634	4	14	m	m	VERB
ejpam-2634	4	15	there	there	PRON
ejpam-2634	4	16	exists	exist	VERB
ejpam-2634	4	17	an	an	DET
ejpam-2634	4	18	ideal	ideal	NOUN
ejpam-2634	4	19	i	i	PRON
ejpam-2634	4	20	of	of	ADP
ejpam-2634	4	21	r	r	NOUN
ejpam-2634	5	1	such	such	ADJ
ejpam-2634	5	2	that	that	SCONJ
ejpam-2634	5	3	n	n	NOUN
ejpam-2634	5	4	=	=	SYM
ejpam-2634	5	5	(	(	PUNCT
ejpam-2634	5	6	0	0	NUM
ejpam-2634	5	7	:	:	PUNCT
ejpam-2634	5	8	m	m	VERB
ejpam-2634	5	9	i	i	NOUN
ejpam-2634	5	10	)	)	PUNCT
ejpam-2634	5	11	.	.	PUNCT
ejpam-2634	6	1	this	this	DET
ejpam-2634	6	2	paper	paper	NOUN
ejpam-2634	6	3	is	be	AUX
ejpam-2634	6	4	devoted	devote	VERB
ejpam-2634	6	5	to	to	PART
ejpam-2634	6	6	study	study	VERB
ejpam-2634	6	7	some	some	DET
ejpam-2634	6	8	properties	property	NOUN
ejpam-2634	6	9	of	of	ADP
ejpam-2634	6	10	comultiplication	comultiplication	NOUN
ejpam-2634	6	11	rings	ring	NOUN
ejpam-2634	6	12	and	and	CCONJ
ejpam-2634	6	13	modules	module	NOUN
ejpam-2634	6	14	.	.	PUNCT
ejpam-2634	7	1	2010	2010	NUM
ejpam-2634	7	2	mathematics	mathematic	NOUN
ejpam-2634	7	3	subject	subject	NOUN
ejpam-2634	7	4	classifications	classification	NOUN
ejpam-2634	7	5	:	:	PUNCT
ejpam-2634	7	6	13c13	13c13	NUM
ejpam-2634	7	7	key	key	ADJ
ejpam-2634	7	8	words	word	NOUN
ejpam-2634	7	9	and	and	CCONJ
ejpam-2634	7	10	phrases	phrase	NOUN
ejpam-2634	7	11	:	:	PUNCT
ejpam-2634	7	12	multiplication	multiplication	NOUN
ejpam-2634	7	13	modules	module	NOUN
ejpam-2634	7	14	,	,	PUNCT
ejpam-2634	7	15	comultiplication	comultiplication	NOUN
ejpam-2634	7	16	modules	module	NOUN
ejpam-2634	7	17	1	1	NUM
ejpam-2634	7	18	.	.	PUNCT
ejpam-2634	8	1	introduction	introduction	NOUN
ejpam-2634	8	2	throughout	throughout	ADP
ejpam-2634	8	3	this	this	DET
ejpam-2634	8	4	paper	paper	NOUN
ejpam-2634	8	5	,	,	PUNCT
ejpam-2634	8	6	r	r	NOUN
ejpam-2634	8	7	will	will	AUX
ejpam-2634	8	8	denote	denote	VERB
ejpam-2634	8	9	a	a	DET
ejpam-2634	8	10	commutative	commutative	ADJ
ejpam-2634	8	11	ring	ring	NOUN
ejpam-2634	8	12	with	with	ADP
ejpam-2634	8	13	identity	identity	NOUN
ejpam-2634	8	14	.	.	PUNCT
ejpam-2634	9	1	we	we	PRON
ejpam-2634	9	2	recall	recall	VERB
ejpam-2634	9	3	that	that	SCONJ
ejpam-2634	9	4	r	r	NOUN
ejpam-2634	9	5	-	-	PUNCT
ejpam-2634	9	6	module	module	NOUN
ejpam-2634	9	7	m	m	NOUN
ejpam-2634	9	8	is	be	AUX
ejpam-2634	9	9	comutiplication	comutiplication	NOUN
ejpam-2634	9	10	if	if	SCONJ
ejpam-2634	9	11	for	for	ADP
ejpam-2634	9	12	every	every	DET
ejpam-2634	9	13	submodule	submodule	NOUN
ejpam-2634	9	14	n	n	PROPN
ejpam-2634	9	15	of	of	ADP
ejpam-2634	9	16	m	m	VERB
ejpam-2634	9	17	there	there	PRON
ejpam-2634	9	18	exists	exist	VERB
ejpam-2634	9	19	an	an	DET
ejpam-2634	9	20	ideal	ideal	NOUN
ejpam-2634	9	21	i	i	PRON
ejpam-2634	9	22	of	of	ADP
ejpam-2634	9	23	r	r	NOUN
ejpam-2634	10	1	such	such	ADJ
ejpam-2634	10	2	that	that	SCONJ
ejpam-2634	10	3	n	n	NOUN
ejpam-2634	10	4	=	=	SYM
ejpam-2634	10	5	(	(	PUNCT
ejpam-2634	10	6	0	0	NUM
ejpam-2634	10	7	:	:	PUNCT
ejpam-2634	10	8	m	m	VERB
ejpam-2634	10	9	i	i	NOUN
ejpam-2634	10	10	)	)	PUNCT
ejpam-2634	10	11	.	.	PUNCT
ejpam-2634	11	1	it	it	PRON
ejpam-2634	11	2	was	be	AUX
ejpam-2634	11	3	shown	show	VERB
ejpam-2634	11	4	that	that	SCONJ
ejpam-2634	11	5	m	m	NOUN
ejpam-2634	11	6	is	be	AUX
ejpam-2634	11	7	comultiplication	comultiplication	NOUN
ejpam-2634	11	8	if	if	SCONJ
ejpam-2634	11	9	and	and	CCONJ
ejpam-2634	11	10	only	only	ADV
ejpam-2634	11	11	if	if	SCONJ
ejpam-2634	11	12	for	for	ADP
ejpam-2634	11	13	each	each	DET
ejpam-2634	11	14	submodule	submodule	NOUN
ejpam-2634	11	15	n	n	PROPN
ejpam-2634	11	16	of	of	ADP
ejpam-2634	11	17	m	m	PROPN
ejpam-2634	11	18	,	,	PUNCT
ejpam-2634	11	19	n	n	PROPN
ejpam-2634	11	20	=	=	PUNCT
ejpam-2634	11	21	(	(	PUNCT
ejpam-2634	11	22	0	0	NUM
ejpam-2634	11	23	:	:	PUNCT
ejpam-2634	11	24	m	m	VERB
ejpam-2634	11	25	annr(n	annr(n	ADP
ejpam-2634	11	26	)	)	PUNCT
ejpam-2634	11	27	)	)	PUNCT
ejpam-2634	12	1	[	[	X
ejpam-2634	12	2	4	4	NUM
ejpam-2634	12	3	]	]	PUNCT
ejpam-2634	12	4	.	.	PUNCT
ejpam-2634	13	1	also	also	ADV
ejpam-2634	13	2	a	a	DET
ejpam-2634	13	3	noetherian	noetherian	ADJ
ejpam-2634	13	4	local	local	ADJ
ejpam-2634	13	5	ring	ring	NOUN
ejpam-2634	13	6	r	r	NOUN
ejpam-2634	13	7	is	be	AUX
ejpam-2634	13	8	a	a	DET
ejpam-2634	13	9	gorenstein	gorenstein	ADJ
ejpam-2634	13	10	ring	ring	NOUN
ejpam-2634	13	11	if	if	SCONJ
ejpam-2634	13	12	in	in	ADP
ejpam-2634	13	13	jdimr<∞	jdimr<∞	NOUN
ejpam-2634	13	14	[	[	X
ejpam-2634	13	15	6	6	NUM
ejpam-2634	13	16	]	]	PUNCT
ejpam-2634	13	17	.	.	PUNCT
ejpam-2634	14	1	in	in	ADP
ejpam-2634	14	2	this	this	DET
ejpam-2634	14	3	article	article	NOUN
ejpam-2634	14	4	,	,	PUNCT
ejpam-2634	14	5	among	among	ADP
ejpam-2634	14	6	other	other	ADJ
ejpam-2634	14	7	results	result	NOUN
ejpam-2634	14	8	,	,	PUNCT
ejpam-2634	14	9	we	we	PRON
ejpam-2634	14	10	will	will	AUX
ejpam-2634	14	11	show	show	VERB
ejpam-2634	14	12	that	that	SCONJ
ejpam-2634	14	13	if	if	SCONJ
ejpam-2634	14	14	r	r	NOUN
ejpam-2634	14	15	is	be	AUX
ejpam-2634	14	16	a	a	DET
ejpam-2634	14	17	local	local	ADJ
ejpam-2634	14	18	artinian	artinian	ADJ
ejpam-2634	14	19	ring	ring	NOUN
ejpam-2634	14	20	,	,	PUNCT
ejpam-2634	14	21	then	then	ADV
ejpam-2634	14	22	r	r	NOUN
ejpam-2634	14	23	is	be	AUX
ejpam-2634	14	24	comultiplication	comultiplication	NOUN
ejpam-2634	14	25	if	if	SCONJ
ejpam-2634	14	26	and	and	CCONJ
ejpam-2634	14	27	only	only	ADV
ejpam-2634	14	28	if	if	SCONJ
ejpam-2634	14	29	r	r	NOUN
ejpam-2634	14	30	is	be	AUX
ejpam-2634	14	31	gorenstein	gorenstein	ADJ
ejpam-2634	14	32	.	.	PUNCT
ejpam-2634	15	1	an	an	DET
ejpam-2634	15	2	r	r	NOUN
ejpam-2634	15	3	-	-	PUNCT
ejpam-2634	15	4	module	module	NOUN
ejpam-2634	15	5	m	m	NOUN
ejpam-2634	15	6	is	be	AUX
ejpam-2634	15	7	called	call	VERB
ejpam-2634	15	8	generalized	generalized	ADJ
ejpam-2634	15	9	hopfian	hopfian	NOUN
ejpam-2634	15	10	,	,	PUNCT
ejpam-2634	15	11	if	if	SCONJ
ejpam-2634	15	12	every	every	DET
ejpam-2634	15	13	surjective	surjective	ADJ
ejpam-2634	15	14	endomorphism	endomorphism	NOUN
ejpam-2634	15	15	of	of	ADP
ejpam-2634	15	16	m	m	PROPN
ejpam-2634	15	17	has	have	VERB
ejpam-2634	15	18	a	a	DET
ejpam-2634	15	19	small	small	ADJ
ejpam-2634	15	20	kernel	kernel	NOUN
ejpam-2634	15	21	.	.	PUNCT
ejpam-2634	16	1	it	it	PRON
ejpam-2634	16	2	is	be	AUX
ejpam-2634	16	3	proved	prove	VERB
ejpam-2634	16	4	that	that	SCONJ
ejpam-2634	16	5	every	every	DET
ejpam-2634	16	6	comultiplication	comultiplication	NOUN
ejpam-2634	16	7	module	module	NOUN
ejpam-2634	16	8	is	be	AUX
ejpam-2634	16	9	generalized	generalize	VERB
ejpam-2634	16	10	hopfian	hopfian	NOUN
ejpam-2634	16	11	.	.	PUNCT
ejpam-2634	17	1	at	at	ADP
ejpam-2634	17	2	last	last	ADJ
ejpam-2634	17	3	but	but	CCONJ
ejpam-2634	17	4	not	not	PART
ejpam-2634	17	5	at	at	ADP
ejpam-2634	17	6	least	least	ADJ
ejpam-2634	17	7	,	,	PUNCT
ejpam-2634	17	8	we	we	PRON
ejpam-2634	17	9	consider	consider	VERB
ejpam-2634	17	10	the	the	DET
ejpam-2634	17	11	direct	direct	ADJ
ejpam-2634	17	12	sum	sum	NOUN
ejpam-2634	17	13	of	of	ADP
ejpam-2634	17	14	comultiplication	comultiplication	NOUN
ejpam-2634	17	15	modules	module	NOUN
ejpam-2634	17	16	,	,	PUNCT
ejpam-2634	17	17	it	it	PRON
ejpam-2634	17	18	is	be	AUX
ejpam-2634	17	19	shown	show	VERB
ejpam-2634	17	20	that	that	SCONJ
ejpam-2634	17	21	m	m	NOUN
ejpam-2634	17	22	=	=	SYM
ejpam-2634	17	23	⊕	⊕	PROPN
ejpam-2634	17	24	i∈i	i∈i	PROPN
ejpam-2634	17	25	mi	mi	PROPN
ejpam-2634	17	26	,	,	PUNCT
ejpam-2634	17	27	is	be	AUX
ejpam-2634	17	28	comultiplication	comultiplication	NOUN
ejpam-2634	17	29	if	if	SCONJ
ejpam-2634	17	30	and	and	CCONJ
ejpam-2634	17	31	only	only	ADV
ejpam-2634	17	32	if	if	SCONJ
ejpam-2634	17	33	for	for	ADP
ejpam-2634	17	34	each	each	DET
ejpam-2634	17	35	i	i	PRON
ejpam-2634	17	36	∈	∈	PROPN
ejpam-2634	18	1	i	i	PRON
ejpam-2634	18	2	,	,	PUNCT
ejpam-2634	18	3	mi	mi	PROPN
ejpam-2634	18	4	is	be	AUX
ejpam-2634	18	5	a	a	DET
ejpam-2634	18	6	comultiplication	comultiplication	NOUN
ejpam-2634	18	7	module	module	NOUN
ejpam-2634	18	8	and	and	CCONJ
ejpam-2634	18	9	for	for	ADP
ejpam-2634	18	10	each	each	DET
ejpam-2634	18	11	submodule	submodule	NOUN
ejpam-2634	18	12	n	n	PROPN
ejpam-2634	18	13	of	of	ADP
ejpam-2634	18	14	m	m	PROPN
ejpam-2634	18	15	,	,	PUNCT
ejpam-2634	18	16	n	n	PROPN
ejpam-2634	18	17	=	=	PROPN
ejpam-2634	18	18	⊕	⊕	PROPN
ejpam-2634	18	19	i∈i(n	i∈i(n	PROPN
ejpam-2634	18	20	⋂	⋂	PROPN
ejpam-2634	18	21	mi	mi	PROPN
ejpam-2634	18	22	)	)	PUNCT
ejpam-2634	18	23	.	.	PUNCT
ejpam-2634	19	1	2	2	X
ejpam-2634	19	2	.	.	X
ejpam-2634	19	3	auxiliary	auxiliary	ADJ
ejpam-2634	19	4	results	result	NOUN
ejpam-2634	19	5	in	in	ADP
ejpam-2634	19	6	this	this	DET
ejpam-2634	19	7	section	section	NOUN
ejpam-2634	19	8	we	we	PRON
ejpam-2634	19	9	will	will	AUX
ejpam-2634	19	10	provide	provide	VERB
ejpam-2634	19	11	the	the	DET
ejpam-2634	19	12	definitions	definition	NOUN
ejpam-2634	19	13	and	and	CCONJ
ejpam-2634	19	14	results	result	NOUN
ejpam-2634	19	15	which	which	PRON
ejpam-2634	19	16	are	be	AUX
ejpam-2634	19	17	necessary	necessary	ADJ
ejpam-2634	19	18	in	in	ADP
ejpam-2634	19	19	the	the	DET
ejpam-2634	19	20	next	next	ADJ
ejpam-2634	19	21	section	section	NOUN
ejpam-2634	19	22	.	.	PUNCT
ejpam-2634	20	1	definition	definition	NOUN
ejpam-2634	20	2	1	1	NUM
ejpam-2634	20	3	.	.	PUNCT
ejpam-2634	20	4	∗corresponding	∗corresponde	VERB
ejpam-2634	20	5	author	author	NOUN
ejpam-2634	20	6	.	.	PUNCT
ejpam-2634	21	1	email	email	NOUN
ejpam-2634	21	2	addresses	address	NOUN
ejpam-2634	21	3	:	:	PUNCT
ejpam-2634	21	4	jafar.azami@gmail.com	jafar.azami@gmail.com	X
ejpam-2634	21	5	,	,	PUNCT
ejpam-2634	21	6	azami@uma.ac.ir	azami@uma.ac.ir	VERB
ejpam-2634	21	7	(	(	PUNCT
ejpam-2634	21	8	j.	j.	PROPN
ejpam-2634	21	9	a’zami	a’zami	PROPN
ejpam-2634	21	10	)	)	PUNCT
ejpam-2634	21	11	,	,	PUNCT
ejpam-2634	21	12	maryamkhajepour@uma.ac.ir	maryamkhajepour@uma.ac.ir	NOUN
ejpam-2634	21	13	(	(	PUNCT
ejpam-2634	21	14	m.	m.	NOUN
ejpam-2634	21	15	khajepour	khajepour	PROPN
ejpam-2634	21	16	)	)	PUNCT
ejpam-2634	21	17	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2634	22	1	244	244	NUM
ejpam-2634	23	1	c	c	X
ejpam-2634	23	2	©	©	PROPN
ejpam-2634	23	3	2016	2016	NUM
ejpam-2634	23	4	ejpam	ejpam	VERB
ejpam-2634	23	5	all	all	DET
ejpam-2634	23	6	rights	right	NOUN
ejpam-2634	23	7	reserved	reserve	VERB
ejpam-2634	23	8	.	.	PUNCT
ejpam-2634	24	1	j.	j.	PROPN
ejpam-2634	24	2	a’zami	a’zami	PROPN
ejpam-2634	24	3	,	,	PUNCT
ejpam-2634	24	4	m.	m.	NOUN
ejpam-2634	24	5	khajepour	khajepour	PROPN
ejpam-2634	24	6	/	/	SYM
ejpam-2634	24	7	eur	eur	PROPN
ejpam-2634	24	8	.	.	PUNCT
ejpam-2634	25	1	j.	j.	PROPN
ejpam-2634	25	2	pure	pure	PROPN
ejpam-2634	25	3	appl	appl	PROPN
ejpam-2634	25	4	.	.	PROPN
ejpam-2634	25	5	math	math	PROPN
ejpam-2634	25	6	,	,	PUNCT
ejpam-2634	25	7	9	9	NUM
ejpam-2634	25	8	(	(	PUNCT
ejpam-2634	25	9	2016	2016	NUM
ejpam-2634	25	10	)	)	PUNCT
ejpam-2634	25	11	,	,	PUNCT
ejpam-2634	25	12	244	244	NUM
ejpam-2634	25	13	-	-	SYM
ejpam-2634	25	14	249	249	NUM
ejpam-2634	25	15	245	245	NUM
ejpam-2634	25	16	(	(	PUNCT
ejpam-2634	25	17	1	1	NUM
ejpam-2634	25	18	)	)	PUNCT
ejpam-2634	25	19	let	let	VERB
ejpam-2634	25	20	m	m	PRON
ejpam-2634	25	21	be	be	AUX
ejpam-2634	25	22	an	an	DET
ejpam-2634	25	23	r−	r−	NOUN
ejpam-2634	25	24	module	module	NOUN
ejpam-2634	25	25	.	.	PUNCT
ejpam-2634	26	1	a	a	DET
ejpam-2634	26	2	submodule	submodule	PROPN
ejpam-2634	26	3	n	n	PROPN
ejpam-2634	26	4	of	of	ADP
ejpam-2634	26	5	m	m	PROPN
ejpam-2634	26	6	is	be	AUX
ejpam-2634	26	7	said	say	VERB
ejpam-2634	26	8	to	to	PART
ejpam-2634	26	9	be	be	AUX
ejpam-2634	26	10	large	large	ADJ
ejpam-2634	26	11	(	(	PUNCT
ejpam-2634	26	12	resp	resp	NOUN
ejpam-2634	26	13	.	.	PUNCT
ejpam-2634	27	1	small	small	ADJ
ejpam-2634	27	2	)	)	PUNCT
ejpam-2634	27	3	if	if	SCONJ
ejpam-2634	27	4	for	for	ADP
ejpam-2634	27	5	every	every	DET
ejpam-2634	27	6	non	non	ADJ
ejpam-2634	27	7	-	-	ADJ
ejpam-2634	27	8	zero	zero	NUM
ejpam-2634	27	9	submodule	submodule	NOUN
ejpam-2634	27	10	k	k	PROPN
ejpam-2634	27	11	of	of	ADP
ejpam-2634	27	12	m	m	PROPN
ejpam-2634	27	13	,	,	PUNCT
ejpam-2634	27	14	we	we	PRON
ejpam-2634	27	15	have	have	VERB
ejpam-2634	28	1	n	n	PRON
ejpam-2634	28	2	⋂	⋂	PROPN
ejpam-2634	28	3	k	k	X
ejpam-2634	28	4	6=	6=	PROPN
ejpam-2634	28	5	0	0	NUM
ejpam-2634	28	6	(	(	PUNCT
ejpam-2634	28	7	resp	resp	NOUN
ejpam-2634	28	8	.	.	PUNCT
ejpam-2634	29	1	n	n	PROPN
ejpam-2634	30	1	+	+	NUM
ejpam-2634	30	2	k	k	PROPN
ejpam-2634	30	3	6=	6=	PROPN
ejpam-2634	30	4	m	m	PROPN
ejpam-2634	30	5	)	)	PUNCT
ejpam-2634	30	6	.	.	PUNCT
ejpam-2634	31	1	(	(	PUNCT
ejpam-2634	31	2	2	2	X
ejpam-2634	31	3	)	)	PUNCT
ejpam-2634	31	4	an	an	DET
ejpam-2634	31	5	r	r	NOUN
ejpam-2634	31	6	-	-	PUNCT
ejpam-2634	31	7	module	module	NOUN
ejpam-2634	31	8	m	m	NOUN
ejpam-2634	31	9	is	be	AUX
ejpam-2634	31	10	called	call	VERB
ejpam-2634	31	11	generalized	generalized	ADJ
ejpam-2634	31	12	hopfian	hopfian	NOUN
ejpam-2634	31	13	,	,	PUNCT
ejpam-2634	31	14	if	if	SCONJ
ejpam-2634	31	15	every	every	DET
ejpam-2634	31	16	subjective	subjective	ADJ
ejpam-2634	31	17	endomorphism	endomorphism	NOUN
ejpam-2634	31	18	of	of	ADP
ejpam-2634	31	19	m	m	PROPN
ejpam-2634	31	20	has	have	VERB
ejpam-2634	31	21	a	a	DET
ejpam-2634	31	22	small	small	ADJ
ejpam-2634	31	23	kernel	kernel	NOUN
ejpam-2634	31	24	.	.	PUNCT
ejpam-2634	32	1	(	(	PUNCT
ejpam-2634	32	2	3	3	X
ejpam-2634	32	3	)	)	PUNCT
ejpam-2634	32	4	an	an	DET
ejpam-2634	32	5	r	r	NOUN
ejpam-2634	32	6	-	-	PUNCT
ejpam-2634	32	7	module	module	NOUN
ejpam-2634	32	8	m	m	NOUN
ejpam-2634	32	9	is	be	AUX
ejpam-2634	32	10	called	call	VERB
ejpam-2634	32	11	weakly	weakly	ADJ
ejpam-2634	32	12	co	co	NOUN
ejpam-2634	32	13	-	-	NOUN
ejpam-2634	32	14	hopfian	hopfian	ADJ
ejpam-2634	32	15	,	,	PUNCT
ejpam-2634	32	16	if	if	SCONJ
ejpam-2634	32	17	every	every	DET
ejpam-2634	32	18	injective	injective	ADJ
ejpam-2634	32	19	endomorphism	endomorphism	NOUN
ejpam-2634	32	20	of	of	ADP
ejpam-2634	32	21	m	m	PROPN
ejpam-2634	32	22	has	have	VERB
ejpam-2634	32	23	a	a	DET
ejpam-2634	32	24	large	large	ADJ
ejpam-2634	32	25	image	image	NOUN
ejpam-2634	32	26	.	.	PUNCT
ejpam-2634	33	1	(	(	PUNCT
ejpam-2634	33	2	4	4	X
ejpam-2634	33	3	)	)	PUNCT
ejpam-2634	33	4	let	let	VERB
ejpam-2634	33	5	i	i	PRON
ejpam-2634	33	6	be	be	AUX
ejpam-2634	33	7	an	an	DET
ejpam-2634	33	8	ideal	ideal	NOUN
ejpam-2634	33	9	of	of	ADP
ejpam-2634	33	10	r.	r.	PROPN
ejpam-2634	33	11	we	we	PRON
ejpam-2634	33	12	say	say	VERB
ejpam-2634	33	13	that	that	SCONJ
ejpam-2634	33	14	i	i	PRON
ejpam-2634	33	15	is	be	AUX
ejpam-2634	33	16	a	a	DET
ejpam-2634	33	17	second	second	ADJ
ejpam-2634	33	18	ideal	ideal	NOUN
ejpam-2634	33	19	of	of	ADP
ejpam-2634	33	20	r	r	NOUN
ejpam-2634	33	21	,	,	PUNCT
ejpam-2634	33	22	if	if	SCONJ
ejpam-2634	33	23	for	for	ADP
ejpam-2634	33	24	each	each	DET
ejpam-2634	33	25	r	r	NOUN
ejpam-2634	33	26	∈	∈	NOUN
ejpam-2634	33	27	r	r	NOUN
ejpam-2634	33	28	,	,	PUNCT
ejpam-2634	34	1	r	r	NOUN
ejpam-2634	34	2	i	i	NOUN
ejpam-2634	34	3	=	=	PUNCT
ejpam-2634	34	4	0	0	NUM
ejpam-2634	34	5	or	or	CCONJ
ejpam-2634	34	6	r	r	NOUN
ejpam-2634	35	1	i	i	NOUN
ejpam-2634	35	2	=	=	NOUN
ejpam-2634	36	1	i	i	PRON
ejpam-2634	36	2	.	.	PUNCT
ejpam-2634	37	1	(	(	PUNCT
ejpam-2634	37	2	5	5	X
ejpam-2634	37	3	)	)	PUNCT
ejpam-2634	37	4	an	an	DET
ejpam-2634	37	5	r	r	NOUN
ejpam-2634	37	6	-	-	PUNCT
ejpam-2634	37	7	module	module	NOUN
ejpam-2634	37	8	m	m	NOUN
ejpam-2634	37	9	is	be	AUX
ejpam-2634	37	10	called	call	VERB
ejpam-2634	37	11	uniform	uniform	ADJ
ejpam-2634	37	12	,	,	PUNCT
ejpam-2634	37	13	if	if	SCONJ
ejpam-2634	37	14	every	every	DET
ejpam-2634	37	15	submodule	submodule	NOUN
ejpam-2634	37	16	of	of	ADP
ejpam-2634	37	17	m	m	PROPN
ejpam-2634	37	18	is	be	AUX
ejpam-2634	37	19	large	large	ADJ
ejpam-2634	37	20	.	.	PUNCT
ejpam-2634	38	1	(	(	PUNCT
ejpam-2634	38	2	6	6	NUM
ejpam-2634	38	3	)	)	PUNCT
ejpam-2634	38	4	an	an	DET
ejpam-2634	38	5	ideal	ideal	NOUN
ejpam-2634	38	6	i	i	PRON
ejpam-2634	38	7	of	of	ADP
ejpam-2634	38	8	r	r	NOUN
ejpam-2634	38	9	is	be	AUX
ejpam-2634	38	10	a	a	DET
ejpam-2634	38	11	pure	pure	ADJ
ejpam-2634	38	12	ideal	ideal	NOUN
ejpam-2634	38	13	if	if	SCONJ
ejpam-2634	38	14	for	for	ADP
ejpam-2634	38	15	each	each	DET
ejpam-2634	38	16	ideal	ideal	ADJ
ejpam-2634	38	17	j	j	PROPN
ejpam-2634	38	18	of	of	ADP
ejpam-2634	38	19	r	r	PROPN
ejpam-2634	38	20	,	,	PUNCT
ejpam-2634	38	21	i	i	PRON
ejpam-2634	38	22	j	j	NOUN
ejpam-2634	39	1	=	=	PUNCT
ejpam-2634	39	2	i	i	PROPN
ejpam-2634	39	3	∩	∩	PROPN
ejpam-2634	39	4	j.	j.	PROPN
ejpam-2634	39	5	(	(	PUNCT
ejpam-2634	39	6	7	7	NUM
ejpam-2634	39	7	)	)	PUNCT
ejpam-2634	39	8	a	a	DET
ejpam-2634	39	9	submodule	submodule	NOUN
ejpam-2634	39	10	n	n	PROPN
ejpam-2634	39	11	of	of	ADP
ejpam-2634	39	12	m	m	PROPN
ejpam-2634	39	13	is	be	AUX
ejpam-2634	39	14	a	a	DET
ejpam-2634	39	15	copure	copure	NOUN
ejpam-2634	39	16	submodule	submodule	NOUN
ejpam-2634	39	17	if	if	SCONJ
ejpam-2634	39	18	for	for	ADP
ejpam-2634	39	19	each	each	DET
ejpam-2634	39	20	ideal	ideal	NOUN
ejpam-2634	39	21	i	i	PRON
ejpam-2634	39	22	of	of	ADP
ejpam-2634	39	23	r	r	PROPN
ejpam-2634	39	24	,	,	PUNCT
ejpam-2634	39	25	(	(	PUNCT
ejpam-2634	39	26	n	n	X
ejpam-2634	39	27	:	:	PUNCT
ejpam-2634	39	28	m	m	VERB
ejpam-2634	39	29	i	i	NOUN
ejpam-2634	39	30	)	)	PUNCT
ejpam-2634	39	31	=	=	PUNCT
ejpam-2634	40	1	n+(0	n+(0	NOUN
ejpam-2634	40	2	:	:	PUNCT
ejpam-2634	41	1	m	m	VERB
ejpam-2634	41	2	i	i	NOUN
ejpam-2634	41	3	)	)	PUNCT
ejpam-2634	41	4	.	.	PUNCT
ejpam-2634	42	1	(	(	PUNCT
ejpam-2634	42	2	8)	8)	NUM
ejpam-2634	42	3	an	an	DET
ejpam-2634	42	4	r	r	NOUN
ejpam-2634	42	5	-	-	PUNCT
ejpam-2634	42	6	module	module	NOUN
ejpam-2634	42	7	m	m	NOUN
ejpam-2634	42	8	is	be	AUX
ejpam-2634	42	9	called	call	VERB
ejpam-2634	42	10	weak	weak	ADJ
ejpam-2634	42	11	comultiplication	comultiplication	NOUN
ejpam-2634	42	12	if	if	SCONJ
ejpam-2634	42	13	for	for	ADP
ejpam-2634	42	14	every	every	DET
ejpam-2634	42	15	prime	prime	ADJ
ejpam-2634	42	16	submodule	submodule	PROPN
ejpam-2634	42	17	n	n	PROPN
ejpam-2634	42	18	of	of	ADP
ejpam-2634	42	19	m	m	PRON
ejpam-2634	42	20	,	,	PUNCT
ejpam-2634	42	21	there	there	PRON
ejpam-2634	42	22	exists	exist	VERB
ejpam-2634	42	23	an	an	DET
ejpam-2634	42	24	ideal	ideal	NOUN
ejpam-2634	42	25	i	i	PRON
ejpam-2634	42	26	of	of	ADP
ejpam-2634	42	27	r	r	NOUN
ejpam-2634	43	1	such	such	ADJ
ejpam-2634	43	2	that	that	SCONJ
ejpam-2634	43	3	n	n	NOUN
ejpam-2634	43	4	=	=	SYM
ejpam-2634	43	5	(	(	PUNCT
ejpam-2634	43	6	0	0	NUM
ejpam-2634	43	7	:	:	PUNCT
ejpam-2634	43	8	m	m	VERB
ejpam-2634	43	9	i	i	NOUN
ejpam-2634	43	10	)	)	PUNCT
ejpam-2634	43	11	theorem	theorem	VERB
ejpam-2634	43	12	1	1	NUM
ejpam-2634	43	13	.	.	PUNCT
ejpam-2634	44	1	let	let	VERB
ejpam-2634	44	2	r	r	PRON
ejpam-2634	44	3	be	be	AUX
ejpam-2634	44	4	a	a	DET
ejpam-2634	44	5	discrete	discrete	ADJ
ejpam-2634	44	6	valuation	valuation	NOUN
ejpam-2634	44	7	ring	ring	NOUN
ejpam-2634	44	8	with	with	ADP
ejpam-2634	44	9	the	the	DET
ejpam-2634	44	10	unique	unique	ADJ
ejpam-2634	44	11	maximal	maximal	ADJ
ejpam-2634	44	12	ideal	ideal	ADJ
ejpam-2634	44	13	m.	m.	NOUN
ejpam-2634	44	14	if	if	SCONJ
ejpam-2634	44	15	r	r	NOUN
ejpam-2634	44	16	-	-	PUNCT
ejpam-2634	44	17	module	module	NOUN
ejpam-2634	44	18	m	m	NOUN
ejpam-2634	44	19	is	be	AUX
ejpam-2634	44	20	comultiplication	comultiplication	NOUN
ejpam-2634	44	21	,	,	PUNCT
ejpam-2634	44	22	then	then	ADV
ejpam-2634	44	23	m	m	VERB
ejpam-2634	44	24	∼=	∼=	PROPN
ejpam-2634	44	25	e(r	e(r	NOUN
ejpam-2634	44	26	/	/	SYM
ejpam-2634	44	27	m	m	NOUN
ejpam-2634	44	28	)	)	PUNCT
ejpam-2634	44	29	or	or	CCONJ
ejpam-2634	44	30	m	m	VERB
ejpam-2634	44	31	∼=	∼=	PROPN
ejpam-2634	44	32	r	r	PROPN
ejpam-2634	44	33	/	/	SYM
ejpam-2634	44	34	mn	mn	PROPN
ejpam-2634	44	35	,	,	PUNCT
ejpam-2634	44	36	for	for	ADP
ejpam-2634	44	37	some	some	DET
ejpam-2634	44	38	n	n	PRON
ejpam-2634	44	39	∈	∈	NOUN
ejpam-2634	44	40	n.	n.	NOUN
ejpam-2634	44	41	proof	proof	NOUN
ejpam-2634	44	42	.	.	PUNCT
ejpam-2634	45	1	see	see	VERB
ejpam-2634	45	2	[	[	X
ejpam-2634	45	3	1	1	X
ejpam-2634	45	4	]	]	PUNCT
ejpam-2634	45	5	and	and	CCONJ
ejpam-2634	45	6	[	[	X
ejpam-2634	45	7	2	2	NUM
ejpam-2634	45	8	]	]	PUNCT
ejpam-2634	45	9	.	.	PUNCT
ejpam-2634	46	1	theorem	theorem	ADJ
ejpam-2634	46	2	2	2	NUM
ejpam-2634	46	3	(	(	PUNCT
ejpam-2634	46	4	[	[	X
ejpam-2634	46	5	4	4	NUM
ejpam-2634	46	6	]	]	NUM
ejpam-2634	46	7	)	)	PUNCT
ejpam-2634	46	8	.	.	PUNCT
ejpam-2634	47	1	let	let	VERB
ejpam-2634	47	2	r	r	PRON
ejpam-2634	47	3	be	be	AUX
ejpam-2634	47	4	a	a	DET
ejpam-2634	47	5	noetherian	noetherian	ADJ
ejpam-2634	47	6	ring	ring	NOUN
ejpam-2634	47	7	,	,	PUNCT
ejpam-2634	47	8	and	and	CCONJ
ejpam-2634	47	9	m	m	AUX
ejpam-2634	47	10	be	be	AUX
ejpam-2634	47	11	a	a	DET
ejpam-2634	47	12	comultiplication	comultiplication	NOUN
ejpam-2634	47	13	r	r	NOUN
ejpam-2634	47	14	-	-	PUNCT
ejpam-2634	47	15	module	module	NOUN
ejpam-2634	47	16	so	so	ADV
ejpam-2634	47	17	m	m	VERB
ejpam-2634	47	18	is	be	AUX
ejpam-2634	47	19	artinian	artinian	ADJ
ejpam-2634	47	20	.	.	PUNCT
ejpam-2634	48	1	theorem	theorem	VERB
ejpam-2634	48	2	3	3	NUM
ejpam-2634	48	3	(	(	PUNCT
ejpam-2634	48	4	[	[	X
ejpam-2634	48	5	3	3	NUM
ejpam-2634	48	6	]	]	PUNCT
ejpam-2634	48	7	)	)	PUNCT
ejpam-2634	48	8	.	.	PUNCT
ejpam-2634	49	1	let	let	VERB
ejpam-2634	49	2	r	r	PRON
ejpam-2634	49	3	be	be	AUX
ejpam-2634	49	4	a	a	DET
ejpam-2634	49	5	noetherian	noetherian	ADJ
ejpam-2634	49	6	ring	ring	NOUN
ejpam-2634	49	7	,	,	PUNCT
ejpam-2634	49	8	and	and	CCONJ
ejpam-2634	49	9	m	m	VERB
ejpam-2634	49	10	be	be	AUX
ejpam-2634	49	11	an	an	DET
ejpam-2634	49	12	injective	injective	ADJ
ejpam-2634	49	13	multiplication	multiplication	NOUN
ejpam-2634	49	14	r	r	NOUN
ejpam-2634	49	15	-	-	PUNCT
ejpam-2634	49	16	module	module	NOUN
ejpam-2634	49	17	,	,	PUNCT
ejpam-2634	49	18	so	so	SCONJ
ejpam-2634	49	19	m	m	NOUN
ejpam-2634	49	20	is	be	AUX
ejpam-2634	49	21	comultiplication	comultiplication	NOUN
ejpam-2634	49	22	.	.	PUNCT
ejpam-2634	50	1	lemma	lemma	PROPN
ejpam-2634	50	2	1	1	NUM
ejpam-2634	50	3	(	(	PUNCT
ejpam-2634	50	4	[	[	X
ejpam-2634	50	5	5	5	NUM
ejpam-2634	50	6	]	]	PUNCT
ejpam-2634	50	7	)	)	PUNCT
ejpam-2634	50	8	.	.	PUNCT
ejpam-2634	51	1	if	if	SCONJ
ejpam-2634	51	2	m	m	NOUN
ejpam-2634	51	3	is	be	AUX
ejpam-2634	51	4	a	a	DET
ejpam-2634	51	5	comultiplication	comultiplication	NOUN
ejpam-2634	51	6	r	r	NOUN
ejpam-2634	51	7	-module	-module	NOUN
ejpam-2634	51	8	,	,	PUNCT
ejpam-2634	51	9	then	then	ADV
ejpam-2634	51	10	for	for	SCONJ
ejpam-2634	51	11	each	each	DET
ejpam-2634	51	12	endomorphism	endomorphism	PROPN
ejpam-2634	51	13	f	f	PROPN
ejpam-2634	51	14	of	of	ADP
ejpam-2634	51	15	m	m	PROPN
ejpam-2634	51	16	,	,	PUNCT
ejpam-2634	51	17	imf	imf	PROPN
ejpam-2634	51	18	=	=	SYM
ejpam-2634	51	19	annr(ker	annr(ker	PROPN
ejpam-2634	51	20	f	f	PROPN
ejpam-2634	51	21	)	)	PUNCT
ejpam-2634	51	22	m.	m.	NOUN
ejpam-2634	51	23	3	3	NUM
ejpam-2634	51	24	.	.	X
ejpam-2634	51	25	main	main	ADJ
ejpam-2634	51	26	results	result	NOUN
ejpam-2634	51	27	lemma	lemma	PROPN
ejpam-2634	51	28	2	2	X
ejpam-2634	51	29	.	.	PUNCT
ejpam-2634	51	30	let	let	VERB
ejpam-2634	51	31	r	r	PRON
ejpam-2634	51	32	be	be	AUX
ejpam-2634	51	33	a	a	DET
ejpam-2634	51	34	noetherian	noetherian	ADJ
ejpam-2634	51	35	ring	ring	NOUN
ejpam-2634	51	36	.	.	PUNCT
ejpam-2634	52	1	then	then	ADV
ejpam-2634	52	2	the	the	DET
ejpam-2634	52	3	following	follow	VERB
ejpam-2634	52	4	statements	statement	NOUN
ejpam-2634	52	5	are	be	AUX
ejpam-2634	52	6	equivalent	equivalent	ADJ
ejpam-2634	52	7	:	:	PUNCT
ejpam-2634	52	8	(	(	PUNCT
ejpam-2634	52	9	1	1	X
ejpam-2634	52	10	)	)	PUNCT
ejpam-2634	52	11	r	r	NOUN
ejpam-2634	52	12	is	be	AUX
ejpam-2634	52	13	a	a	DET
ejpam-2634	52	14	comultiplication	comultiplication	NOUN
ejpam-2634	52	15	ring	ring	NOUN
ejpam-2634	52	16	;	;	PUNCT
ejpam-2634	52	17	(	(	PUNCT
ejpam-2634	52	18	2	2	X
ejpam-2634	52	19	)	)	PUNCT
ejpam-2634	52	20	for	for	ADP
ejpam-2634	52	21	all	all	PRON
ejpam-2634	52	22	p	p	PROPN
ejpam-2634	52	23	∈	∈	PROPN
ejpam-2634	52	24	spec(r	spec(r	PROPN
ejpam-2634	52	25	)	)	PUNCT
ejpam-2634	52	26	,	,	PUNCT
ejpam-2634	52	27	rp	rp	NOUN
ejpam-2634	52	28	is	be	AUX
ejpam-2634	52	29	a	a	DET
ejpam-2634	52	30	comultiplication	comultiplication	NOUN
ejpam-2634	52	31	ring	ring	NOUN
ejpam-2634	52	32	;	;	PUNCT
ejpam-2634	52	33	(	(	PUNCT
ejpam-2634	52	34	3	3	X
ejpam-2634	52	35	)	)	PUNCT
ejpam-2634	52	36	for	for	ADP
ejpam-2634	52	37	all	all	DET
ejpam-2634	52	38	p	p	PROPN
ejpam-2634	52	39	∈	∈	PROPN
ejpam-2634	52	40	max(r	max(r	PROPN
ejpam-2634	52	41	)	)	PUNCT
ejpam-2634	52	42	,	,	PUNCT
ejpam-2634	52	43	rp	rp	NOUN
ejpam-2634	52	44	is	be	AUX
ejpam-2634	52	45	a	a	DET
ejpam-2634	52	46	comultiplication	comultiplication	NOUN
ejpam-2634	52	47	ring	ring	NOUN
ejpam-2634	52	48	.	.	PUNCT
ejpam-2634	53	1	proof	proof	NOUN
ejpam-2634	53	2	.	.	PUNCT
ejpam-2634	54	1	(	(	PUNCT
ejpam-2634	54	2	1→2	1→2	ADV
ejpam-2634	54	3	)	)	PUNCT
ejpam-2634	54	4	let	let	VERB
ejpam-2634	54	5	j	j	PROPN
ejpam-2634	54	6	be	be	AUX
ejpam-2634	54	7	an	an	DET
ejpam-2634	54	8	ideal	ideal	NOUN
ejpam-2634	54	9	of	of	ADP
ejpam-2634	54	10	rp	rp	NOUN
ejpam-2634	54	11	,	,	PUNCT
ejpam-2634	54	12	then	then	ADV
ejpam-2634	54	13	there	there	PRON
ejpam-2634	54	14	exists	exist	VERB
ejpam-2634	54	15	an	an	DET
ejpam-2634	54	16	ideal	ideal	NOUN
ejpam-2634	54	17	i	i	PRON
ejpam-2634	54	18	of	of	ADP
ejpam-2634	54	19	r	r	NOUN
ejpam-2634	54	20	such	such	ADJ
ejpam-2634	54	21	that	that	DET
ejpam-2634	54	22	j	j	PROPN
ejpam-2634	54	23	=	=	NOUN
ejpam-2634	54	24	ip	ip	PROPN
ejpam-2634	54	25	.	.	PUNCT
ejpam-2634	55	1	now	now	ADV
ejpam-2634	55	2	i	i	PRON
ejpam-2634	55	3	=	=	PUNCT
ejpam-2634	55	4	annanni	annanni	PROPN
ejpam-2634	55	5	and	and	CCONJ
ejpam-2634	55	6	so	so	ADV
ejpam-2634	55	7	j	j	PROPN
ejpam-2634	56	1	=	=	NOUN
ejpam-2634	56	2	ip	ip	NOUN
ejpam-2634	56	3	=	=	NOUN
ejpam-2634	56	4	annannip	annannip	NOUN
ejpam-2634	56	5	=	=	X
ejpam-2634	56	6	annannj	annannj	NOUN
ejpam-2634	56	7	.	.	PUNCT
ejpam-2634	57	1	(	(	PUNCT
ejpam-2634	57	2	2→3	2→3	X
ejpam-2634	57	3	)	)	PUNCT
ejpam-2634	57	4	it	it	PRON
ejpam-2634	57	5	is	be	AUX
ejpam-2634	57	6	clear	clear	ADJ
ejpam-2634	57	7	.	.	PUNCT
ejpam-2634	58	1	(	(	PUNCT
ejpam-2634	58	2	3→1	3→1	NOUN
ejpam-2634	58	3	)	)	PUNCT
ejpam-2634	58	4	let	let	VERB
ejpam-2634	58	5	i	i	PRON
ejpam-2634	58	6	be	be	AUX
ejpam-2634	58	7	an	an	DET
ejpam-2634	58	8	ideal	ideal	NOUN
ejpam-2634	58	9	of	of	ADP
ejpam-2634	58	10	r.	r.	PROPN
ejpam-2634	58	11	for	for	ADP
ejpam-2634	58	12	all	all	DET
ejpam-2634	58	13	p	p	PROPN
ejpam-2634	58	14	∈	∈	PROPN
ejpam-2634	58	15	max(r	max(r	PROPN
ejpam-2634	58	16	)	)	PUNCT
ejpam-2634	58	17	,	,	PUNCT
ejpam-2634	58	18	we	we	PRON
ejpam-2634	58	19	have	have	VERB
ejpam-2634	58	20	ip	ip	ADJ
ejpam-2634	58	21	=	=	NOUN
ejpam-2634	58	22	annannip	annannip	NOUN
ejpam-2634	58	23	=	=	SYM
ejpam-2634	58	24	(	(	PUNCT
ejpam-2634	58	25	annanni)p	annanni)p	PROPN
ejpam-2634	59	1	and	and	CCONJ
ejpam-2634	59	2	so	so	ADV
ejpam-2634	59	3	i	i	PRON
ejpam-2634	59	4	=	=	PUNCT
ejpam-2634	59	5	annanni	annanni	PROPN
ejpam-2634	59	6	.	.	PUNCT
ejpam-2634	60	1	j.	j.	PROPN
ejpam-2634	60	2	a’zami	a’zami	PROPN
ejpam-2634	60	3	,	,	PUNCT
ejpam-2634	60	4	m.	m.	NOUN
ejpam-2634	60	5	khajepour	khajepour	PROPN
ejpam-2634	60	6	/	/	SYM
ejpam-2634	60	7	eur	eur	PROPN
ejpam-2634	60	8	.	.	PUNCT
ejpam-2634	61	1	j.	j.	PROPN
ejpam-2634	61	2	pure	pure	PROPN
ejpam-2634	61	3	appl	appl	PROPN
ejpam-2634	61	4	.	.	PROPN
ejpam-2634	61	5	math	math	PROPN
ejpam-2634	61	6	,	,	PUNCT
ejpam-2634	61	7	9	9	NUM
ejpam-2634	61	8	(	(	PUNCT
ejpam-2634	61	9	2016	2016	NUM
ejpam-2634	61	10	)	)	PUNCT
ejpam-2634	61	11	,	,	PUNCT
ejpam-2634	61	12	244	244	NUM
ejpam-2634	61	13	-	-	SYM
ejpam-2634	61	14	249	249	NUM
ejpam-2634	61	15	246	246	NUM
ejpam-2634	61	16	theorem	theorem	NOUN
ejpam-2634	61	17	4	4	NUM
ejpam-2634	61	18	.	.	PUNCT
ejpam-2634	62	1	let	let	VERB
ejpam-2634	62	2	m	m	PRON
ejpam-2634	62	3	be	be	AUX
ejpam-2634	62	4	a	a	DET
ejpam-2634	62	5	comultiplication	comultiplication	NOUN
ejpam-2634	62	6	r	r	NOUN
ejpam-2634	62	7	-	-	PUNCT
ejpam-2634	62	8	module	module	NOUN
ejpam-2634	62	9	then	then	ADV
ejpam-2634	62	10	(	(	PUNCT
ejpam-2634	62	11	1	1	X
ejpam-2634	62	12	)	)	PUNCT
ejpam-2634	62	13	if	if	SCONJ
ejpam-2634	62	14	i	i	PRON
ejpam-2634	62	15	is	be	AUX
ejpam-2634	62	16	a	a	DET
ejpam-2634	62	17	second	second	ADJ
ejpam-2634	62	18	ideal	ideal	NOUN
ejpam-2634	62	19	of	of	ADP
ejpam-2634	62	20	r	r	NOUN
ejpam-2634	62	21	,	,	PUNCT
ejpam-2634	62	22	then	then	ADV
ejpam-2634	62	23	n	n	NOUN
ejpam-2634	62	24	=	=	SYM
ejpam-2634	62	25	(	(	PUNCT
ejpam-2634	62	26	0	0	NUM
ejpam-2634	62	27	:	:	PUNCT
ejpam-2634	62	28	m	m	VERB
ejpam-2634	62	29	i	i	NOUN
ejpam-2634	62	30	)	)	PUNCT
ejpam-2634	62	31	is	be	AUX
ejpam-2634	62	32	a	a	DET
ejpam-2634	62	33	prime	prime	ADJ
ejpam-2634	62	34	submodule	submodule	NOUN
ejpam-2634	62	35	of	of	ADP
ejpam-2634	62	36	m.	m.	NOUN
ejpam-2634	62	37	(	(	PUNCT
ejpam-2634	62	38	2	2	NUM
ejpam-2634	62	39	)	)	PUNCT
ejpam-2634	62	40	if	if	SCONJ
ejpam-2634	62	41	n	n	PRON
ejpam-2634	62	42	is	be	AUX
ejpam-2634	62	43	a	a	DET
ejpam-2634	62	44	second	second	ADJ
ejpam-2634	62	45	submodule	submodule	NOUN
ejpam-2634	62	46	of	of	ADP
ejpam-2634	62	47	m	m	PROPN
ejpam-2634	62	48	,	,	PUNCT
ejpam-2634	62	49	then	then	ADV
ejpam-2634	62	50	annrn	annrn	NOUN
ejpam-2634	62	51	is	be	AUX
ejpam-2634	62	52	a	a	DET
ejpam-2634	62	53	prime	prime	ADJ
ejpam-2634	62	54	ideal	ideal	NOUN
ejpam-2634	62	55	of	of	ADP
ejpam-2634	62	56	r.	r.	PROPN
ejpam-2634	62	57	(	(	PUNCT
ejpam-2634	62	58	3	3	X
ejpam-2634	62	59	)	)	PUNCT
ejpam-2634	62	60	if	if	SCONJ
ejpam-2634	62	61	m	m	NOUN
ejpam-2634	62	62	is	be	AUX
ejpam-2634	62	63	faithful	faithful	ADJ
ejpam-2634	62	64	,	,	PUNCT
ejpam-2634	62	65	and	and	CCONJ
ejpam-2634	62	66	n	n	DET
ejpam-2634	62	67	a	a	DET
ejpam-2634	62	68	submodule	submodule	NOUN
ejpam-2634	62	69	of	of	ADP
ejpam-2634	62	70	m	m	PRON
ejpam-2634	62	71	such	such	ADJ
ejpam-2634	62	72	that	that	SCONJ
ejpam-2634	62	73	annrn	annrn	NOUN
ejpam-2634	62	74	is	be	AUX
ejpam-2634	62	75	a	a	DET
ejpam-2634	62	76	large	large	ADJ
ejpam-2634	62	77	ideal	ideal	NOUN
ejpam-2634	62	78	of	of	ADP
ejpam-2634	62	79	r	r	NOUN
ejpam-2634	62	80	,	,	PUNCT
ejpam-2634	62	81	then	then	ADV
ejpam-2634	62	82	n	n	PRON
ejpam-2634	62	83	is	be	AUX
ejpam-2634	62	84	a	a	DET
ejpam-2634	62	85	small	small	ADJ
ejpam-2634	62	86	submodule	submodule	NOUN
ejpam-2634	62	87	of	of	ADP
ejpam-2634	62	88	m.	m.	NOUN
ejpam-2634	62	89	(	(	PUNCT
ejpam-2634	62	90	4	4	NUM
ejpam-2634	62	91	)	)	PUNCT
ejpam-2634	62	92	if	if	SCONJ
ejpam-2634	62	93	n	n	PRON
ejpam-2634	62	94	be	be	VERB
ejpam-2634	62	95	a	a	DET
ejpam-2634	62	96	submodule	submodule	NOUN
ejpam-2634	62	97	of	of	ADP
ejpam-2634	62	98	m	m	PRON
ejpam-2634	62	99	such	such	ADJ
ejpam-2634	62	100	that	that	SCONJ
ejpam-2634	62	101	annn	annn	NOUN
ejpam-2634	62	102	is	be	AUX
ejpam-2634	62	103	a	a	DET
ejpam-2634	62	104	pure	pure	ADJ
ejpam-2634	62	105	ideal	ideal	NOUN
ejpam-2634	62	106	of	of	ADP
ejpam-2634	62	107	r	r	NOUN
ejpam-2634	62	108	,	,	PUNCT
ejpam-2634	62	109	then	then	ADV
ejpam-2634	62	110	n	n	PRON
ejpam-2634	62	111	is	be	AUX
ejpam-2634	62	112	a	a	DET
ejpam-2634	62	113	copure	copure	ADJ
ejpam-2634	62	114	submodule	submodule	NOUN
ejpam-2634	62	115	of	of	ADP
ejpam-2634	62	116	m.	m.	NOUN
ejpam-2634	62	117	proof	proof	NOUN
ejpam-2634	62	118	.	.	PUNCT
ejpam-2634	63	1	(	(	PUNCT
ejpam-2634	63	2	1	1	X
ejpam-2634	63	3	)	)	PUNCT
ejpam-2634	63	4	let	let	VERB
ejpam-2634	63	5	r	r	NOUN
ejpam-2634	63	6	∈	∈	NOUN
ejpam-2634	63	7	r	r	NOUN
ejpam-2634	63	8	and	and	CCONJ
ejpam-2634	63	9	m	m	PROPN
ejpam-2634	63	10	∈	∈	NOUN
ejpam-2634	64	1	m	m	VERB
ejpam-2634	64	2	be	be	VERB
ejpam-2634	64	3	elements	element	NOUN
ejpam-2634	64	4	such	such	ADJ
ejpam-2634	64	5	that	that	PRON
ejpam-2634	64	6	rm	rm	PROPN
ejpam-2634	64	7	∈	∈	PROPN
ejpam-2634	64	8	n	n	PROPN
ejpam-2634	65	1	and	and	CCONJ
ejpam-2634	65	2	r	r	PROPN
ejpam-2634	65	3	6∈	6∈	NOUN
ejpam-2634	65	4	(	(	PUNCT
ejpam-2634	65	5	n	n	NUM
ejpam-2634	65	6	:	:	PUNCT
ejpam-2634	65	7	r	r	NOUN
ejpam-2634	65	8	m	m	PROPN
ejpam-2634	65	9	)	)	PUNCT
ejpam-2634	65	10	.	.	PUNCT
ejpam-2634	66	1	therefore	therefore	ADV
ejpam-2634	66	2	rm	rm	PROPN
ejpam-2634	66	3	6⊆	6⊆	PROPN
ejpam-2634	66	4	n	n	PROPN
ejpam-2634	66	5	and	and	CCONJ
ejpam-2634	66	6	so	so	ADV
ejpam-2634	66	7	rm	rm	PROPN
ejpam-2634	66	8	i	i	PROPN
ejpam-2634	66	9	6=	6=	PROPN
ejpam-2634	66	10	0	0	X
ejpam-2634	66	11	.	.	PUNCT
ejpam-2634	67	1	this	this	PRON
ejpam-2634	67	2	shows	show	VERB
ejpam-2634	67	3	that	that	SCONJ
ejpam-2634	67	4	r	r	NOUN
ejpam-2634	67	5	i	i	PRON
ejpam-2634	67	6	6=	6=	PROPN
ejpam-2634	67	7	0	0	NUM
ejpam-2634	67	8	,	,	PUNCT
ejpam-2634	67	9	and	and	CCONJ
ejpam-2634	67	10	by	by	ADP
ejpam-2634	67	11	hypothesis	hypothesis	NOUN
ejpam-2634	67	12	r	r	NOUN
ejpam-2634	68	1	i	i	NOUN
ejpam-2634	69	1	=	=	NOUN
ejpam-2634	70	1	i	i	PROPN
ejpam-2634	70	2	.	.	PUNCT
ejpam-2634	71	1	since	since	SCONJ
ejpam-2634	71	2	rm	rm	PROPN
ejpam-2634	71	3	∈	∈	PROPN
ejpam-2634	71	4	n	n	NOUN
ejpam-2634	71	5	=	=	SYM
ejpam-2634	71	6	(	(	PUNCT
ejpam-2634	71	7	0	0	NUM
ejpam-2634	71	8	:	:	PUNCT
ejpam-2634	71	9	m	m	VERB
ejpam-2634	71	10	i	i	NOUN
ejpam-2634	71	11	)	)	PUNCT
ejpam-2634	71	12	,	,	PUNCT
ejpam-2634	71	13	it	it	PRON
ejpam-2634	71	14	follows	follow	VERB
ejpam-2634	71	15	that	that	DET
ejpam-2634	71	16	rmi	rmi	NOUN
ejpam-2634	71	17	=	=	NOUN
ejpam-2634	71	18	0	0	NUM
ejpam-2634	72	1	and	and	CCONJ
ejpam-2634	72	2	so	so	ADV
ejpam-2634	72	3	mi	mi	PROPN
ejpam-2634	72	4	=	=	PROPN
ejpam-2634	72	5	0	0	PROPN
ejpam-2634	72	6	,	,	PUNCT
ejpam-2634	72	7	that	that	PRON
ejpam-2634	72	8	implies	imply	VERB
ejpam-2634	72	9	m	m	PRON
ejpam-2634	72	10	∈	∈	NOUN
ejpam-2634	72	11	(	(	PUNCT
ejpam-2634	72	12	0	0	NUM
ejpam-2634	72	13	:	:	PUNCT
ejpam-2634	72	14	m	m	VERB
ejpam-2634	72	15	i	i	ADJ
ejpam-2634	72	16	)	)	PUNCT
ejpam-2634	72	17	=	=	SYM
ejpam-2634	73	1	n	n	PROPN
ejpam-2634	73	2	.	.	PUNCT
ejpam-2634	74	1	(	(	PUNCT
ejpam-2634	74	2	2	2	X
ejpam-2634	74	3	)	)	PUNCT
ejpam-2634	74	4	let	let	VERB
ejpam-2634	74	5	n	n	PRON
ejpam-2634	74	6	be	be	AUX
ejpam-2634	74	7	a	a	DET
ejpam-2634	74	8	second	second	ADJ
ejpam-2634	74	9	submodule	submodule	NOUN
ejpam-2634	74	10	of	of	ADP
ejpam-2634	74	11	m	m	PROPN
ejpam-2634	74	12	.	.	PUNCT
ejpam-2634	75	1	set	set	VERB
ejpam-2634	75	2	i	i	PRON
ejpam-2634	75	3	:	:	PUNCT
ejpam-2634	76	1	=	=	SYM
ejpam-2634	76	2	annrn	annrn	NOUN
ejpam-2634	76	3	and	and	CCONJ
ejpam-2634	76	4	so	so	ADV
ejpam-2634	76	5	n	n	ADV
ejpam-2634	76	6	=	=	SYM
ejpam-2634	76	7	(	(	PUNCT
ejpam-2634	76	8	0	0	NUM
ejpam-2634	76	9	:	:	PUNCT
ejpam-2634	76	10	m	m	VERB
ejpam-2634	76	11	i	i	NOUN
ejpam-2634	76	12	)	)	PUNCT
ejpam-2634	76	13	.	.	PUNCT
ejpam-2634	77	1	suppose	suppose	VERB
ejpam-2634	77	2	that	that	SCONJ
ejpam-2634	77	3	x	x	SYM
ejpam-2634	77	4	,	,	PUNCT
ejpam-2634	77	5	y	y	PROPN
ejpam-2634	77	6	be	be	VERB
ejpam-2634	77	7	two	two	NUM
ejpam-2634	77	8	elements	element	NOUN
ejpam-2634	77	9	of	of	ADP
ejpam-2634	77	10	r	r	NOUN
ejpam-2634	77	11	such	such	ADJ
ejpam-2634	77	12	that	that	SCONJ
ejpam-2634	77	13	x	x	X
ejpam-2634	77	14	y	y	PROPN
ejpam-2634	77	15	∈	∈	PROPN
ejpam-2634	78	1	i	i	PRON
ejpam-2634	78	2	but	but	CCONJ
ejpam-2634	78	3	x	x	SYM
ejpam-2634	78	4	6∈	6∈	NOUN
ejpam-2634	79	1	i	i	PRON
ejpam-2634	79	2	and	and	CCONJ
ejpam-2634	79	3	y	y	PROPN
ejpam-2634	79	4	6∈	6∈	PROPN
ejpam-2634	80	1	i	i	PRON
ejpam-2634	80	2	.	.	PUNCT
ejpam-2634	81	1	now	now	ADV
ejpam-2634	81	2	x	x	VERB
ejpam-2634	81	3	y	y	PROPN
ejpam-2634	81	4	∈	∈	PROPN
ejpam-2634	81	5	i	i	PRON
ejpam-2634	81	6	,	,	PUNCT
ejpam-2634	81	7	implies	imply	VERB
ejpam-2634	81	8	that	that	SCONJ
ejpam-2634	81	9	x	x	X
ejpam-2634	81	10	yn	yn	PROPN
ejpam-2634	81	11	=	=	PUNCT
ejpam-2634	81	12	0	0	PROPN
ejpam-2634	81	13	and	and	CCONJ
ejpam-2634	81	14	hence	hence	ADV
ejpam-2634	81	15	(	(	PUNCT
ejpam-2634	81	16	x	x	PROPN
ejpam-2634	81	17	y)nn	y)nn	PROPN
ejpam-2634	81	18	6=	6=	ADP
ejpam-2634	81	19	n	n	PROPN
ejpam-2634	81	20	for	for	ADP
ejpam-2634	81	21	each	each	DET
ejpam-2634	81	22	n	n	PRON
ejpam-2634	81	23	∈	∈	PROPN
ejpam-2634	81	24	n	n	NOUN
ejpam-2634	81	25	.	.	PUNCT
ejpam-2634	82	1	since	since	SCONJ
ejpam-2634	82	2	x	x	X
ejpam-2634	82	3	,	,	PUNCT
ejpam-2634	82	4	y	y	PROPN
ejpam-2634	82	5	6∈	6∈	PROPN
ejpam-2634	83	1	i	i	PRON
ejpam-2634	83	2	,	,	PUNCT
ejpam-2634	83	3	it	it	PRON
ejpam-2634	83	4	follows	follow	VERB
ejpam-2634	83	5	that	that	SCONJ
ejpam-2634	83	6	there	there	PRON
ejpam-2634	83	7	exists	exist	VERB
ejpam-2634	83	8	n	n	PRON
ejpam-2634	83	9	∈	∈	PROPN
ejpam-2634	83	10	n	n	PRON
ejpam-2634	83	11	such	such	ADJ
ejpam-2634	83	12	that	that	SCONJ
ejpam-2634	83	13	xnn	xnn	PROPN
ejpam-2634	83	14	=	=	SYM
ejpam-2634	83	15	n	n	PROPN
ejpam-2634	83	16	and	and	CCONJ
ejpam-2634	83	17	ynn	ynn	PROPN
ejpam-2634	83	18	=	=	PROPN
ejpam-2634	83	19	n	n	PROPN
ejpam-2634	83	20	.	.	PUNCT
ejpam-2634	84	1	consequently	consequently	ADV
ejpam-2634	84	2	(	(	PUNCT
ejpam-2634	84	3	x	x	PUNCT
ejpam-2634	84	4	y)nn	y)nn	NOUN
ejpam-2634	84	5	=	=	SYM
ejpam-2634	84	6	xn	xn	PROPN
ejpam-2634	84	7	ynn	ynn	PROPN
ejpam-2634	84	8	=	=	PROPN
ejpam-2634	84	9	n	n	PROPN
ejpam-2634	84	10	,	,	PUNCT
ejpam-2634	84	11	which	which	PRON
ejpam-2634	84	12	is	be	AUX
ejpam-2634	84	13	a	a	DET
ejpam-2634	84	14	contradiction	contradiction	NOUN
ejpam-2634	84	15	.	.	PUNCT
ejpam-2634	85	1	(	(	PUNCT
ejpam-2634	85	2	3	3	X
ejpam-2634	85	3	)	)	PUNCT
ejpam-2634	85	4	let	let	VERB
ejpam-2634	85	5	there	there	PRON
ejpam-2634	85	6	exists	exist	VERB
ejpam-2634	85	7	a	a	DET
ejpam-2634	85	8	submodule	submodule	NOUN
ejpam-2634	85	9	k	k	PROPN
ejpam-2634	85	10	of	of	ADP
ejpam-2634	85	11	m	m	PROPN
ejpam-2634	85	12	such	such	ADJ
ejpam-2634	85	13	that	that	SCONJ
ejpam-2634	85	14	m	m	VERB
ejpam-2634	85	15	=	=	SYM
ejpam-2634	85	16	n	n	PROPN
ejpam-2634	85	17	+	+	X
ejpam-2634	85	18	k	k	X
ejpam-2634	85	19	.	.	PUNCT
ejpam-2634	86	1	so	so	ADV
ejpam-2634	86	2	m	m	VERB
ejpam-2634	86	3	=	=	SYM
ejpam-2634	86	4	n	n	PROPN
ejpam-2634	86	5	+	+	CCONJ
ejpam-2634	86	6	k	k	NOUN
ejpam-2634	86	7	=	=	SYM
ejpam-2634	87	1	(	(	PUNCT
ejpam-2634	87	2	0	0	NUM
ejpam-2634	87	3	:	:	PUNCT
ejpam-2634	87	4	m	m	VERB
ejpam-2634	87	5	annn	annn	NOUN
ejpam-2634	87	6	)	)	PUNCT
ejpam-2634	88	1	+	+	CCONJ
ejpam-2634	88	2	(	(	PUNCT
ejpam-2634	88	3	0	0	NUM
ejpam-2634	88	4	:	:	PUNCT
ejpam-2634	88	5	m	m	VERB
ejpam-2634	88	6	annk	annk	NOUN
ejpam-2634	88	7	)	)	PUNCT
ejpam-2634	88	8	=	=	PUNCT
ejpam-2634	89	1	(	(	PUNCT
ejpam-2634	89	2	0	0	NUM
ejpam-2634	89	3	:	:	PUNCT
ejpam-2634	89	4	m	m	AUX
ejpam-2634	89	5	annn	annn	ADP
ejpam-2634	89	6	⋂	⋂	PROPN
ejpam-2634	89	7	annk	annk	NOUN
ejpam-2634	89	8	)	)	PUNCT
ejpam-2634	89	9	.	.	PUNCT
ejpam-2634	90	1	therefore	therefore	ADV
ejpam-2634	90	2	annn	annn	VERB
ejpam-2634	90	3	⋂	⋂	PROPN
ejpam-2634	90	4	annk	annk	NOUN
ejpam-2634	90	5	⊆	⊆	NUM
ejpam-2634	90	6	annm	annm	NOUN
ejpam-2634	90	7	=	=	SYM
ejpam-2634	90	8	0	0	NUM
ejpam-2634	90	9	,	,	PUNCT
ejpam-2634	90	10	and	and	CCONJ
ejpam-2634	90	11	consequently	consequently	ADV
ejpam-2634	90	12	annk	annk	NOUN
ejpam-2634	90	13	=	=	SYM
ejpam-2634	90	14	0	0	NUM
ejpam-2634	90	15	which	which	PRON
ejpam-2634	90	16	implies	imply	VERB
ejpam-2634	90	17	that	that	SCONJ
ejpam-2634	90	18	k	k	PROPN
ejpam-2634	90	19	=	=	PRON
ejpam-2634	90	20	(	(	PUNCT
ejpam-2634	90	21	0	0	NUM
ejpam-2634	90	22	:	:	PUNCT
ejpam-2634	90	23	m	m	VERB
ejpam-2634	90	24	annk	annk	NOUN
ejpam-2634	90	25	)	)	PUNCT
ejpam-2634	91	1	=	=	PUNCT
ejpam-2634	92	1	m	m	NOUN
ejpam-2634	92	2	.	.	PUNCT
ejpam-2634	93	1	(	(	PUNCT
ejpam-2634	93	2	4	4	X
ejpam-2634	93	3	)	)	PUNCT
ejpam-2634	93	4	we	we	PRON
ejpam-2634	93	5	show	show	VERB
ejpam-2634	93	6	that	that	SCONJ
ejpam-2634	93	7	for	for	ADP
ejpam-2634	93	8	each	each	DET
ejpam-2634	93	9	ideal	ideal	NOUN
ejpam-2634	93	10	i	i	PRON
ejpam-2634	93	11	of	of	ADP
ejpam-2634	93	12	r	r	PROPN
ejpam-2634	93	13	,	,	PUNCT
ejpam-2634	93	14	(	(	PUNCT
ejpam-2634	93	15	n	n	X
ejpam-2634	93	16	:	:	PUNCT
ejpam-2634	93	17	m	m	VERB
ejpam-2634	93	18	i	i	ADJ
ejpam-2634	93	19	)	)	PUNCT
ejpam-2634	93	20	=	=	SYM
ejpam-2634	93	21	n	n	PRON
ejpam-2634	93	22	+	+	ADJ
ejpam-2634	93	23	(	(	PUNCT
ejpam-2634	93	24	0	0	NUM
ejpam-2634	93	25	:	:	PUNCT
ejpam-2634	93	26	m	m	VERB
ejpam-2634	93	27	i	i	NOUN
ejpam-2634	93	28	)	)	PUNCT
ejpam-2634	93	29	.	.	PUNCT
ejpam-2634	94	1	note	note	VERB
ejpam-2634	94	2	that	that	SCONJ
ejpam-2634	94	3	for	for	ADP
ejpam-2634	94	4	each	each	DET
ejpam-2634	94	5	ideal	ideal	NOUN
ejpam-2634	94	6	i	i	PRON
ejpam-2634	94	7	of	of	ADP
ejpam-2634	94	8	r	r	NOUN
ejpam-2634	94	9	there	there	PRON
ejpam-2634	94	10	exists	exist	VERB
ejpam-2634	94	11	a	a	DET
ejpam-2634	94	12	submodule	submodule	NOUN
ejpam-2634	94	13	k	k	PROPN
ejpam-2634	94	14	of	of	ADP
ejpam-2634	94	15	m	m	PRON
ejpam-2634	94	16	such	such	ADJ
ejpam-2634	94	17	that	that	SCONJ
ejpam-2634	94	18	(	(	PUNCT
ejpam-2634	94	19	0	0	NUM
ejpam-2634	94	20	:	:	PUNCT
ejpam-2634	94	21	m	m	VERB
ejpam-2634	94	22	i	i	NOUN
ejpam-2634	94	23	)	)	PUNCT
ejpam-2634	94	24	=	=	PUNCT
ejpam-2634	95	1	(	(	PUNCT
ejpam-2634	95	2	0	0	NUM
ejpam-2634	95	3	:	:	PUNCT
ejpam-2634	95	4	m	m	VERB
ejpam-2634	95	5	annk	annk	NOUN
ejpam-2634	95	6	)	)	PUNCT
ejpam-2634	95	7	,	,	PUNCT
ejpam-2634	95	8	so	so	CCONJ
ejpam-2634	95	9	(	(	PUNCT
ejpam-2634	95	10	n	n	X
ejpam-2634	95	11	:	:	PUNCT
ejpam-2634	95	12	m	m	VERB
ejpam-2634	95	13	i	i	NOUN
ejpam-2634	95	14	)	)	PUNCT
ejpam-2634	95	15	=(	=(	NOUN
ejpam-2634	95	16	(	(	PUNCT
ejpam-2634	95	17	0	0	NUM
ejpam-2634	95	18	:	:	PUNCT
ejpam-2634	95	19	m	m	VERB
ejpam-2634	95	20	annn	annn	NOUN
ejpam-2634	95	21	)	)	PUNCT
ejpam-2634	95	22	:	:	PUNCT
ejpam-2634	95	23	m	m	VERB
ejpam-2634	95	24	i	i	NOUN
ejpam-2634	95	25	)	)	PUNCT
ejpam-2634	95	26	=	=	SYM
ejpam-2634	96	1	(	(	PUNCT
ejpam-2634	96	2	(	(	PUNCT
ejpam-2634	96	3	0	0	NUM
ejpam-2634	96	4	:	:	PUNCT
ejpam-2634	96	5	m	m	VERB
ejpam-2634	96	6	i	i	NOUN
ejpam-2634	96	7	)	)	PUNCT
ejpam-2634	96	8	:	:	PUNCT
ejpam-2634	96	9	m	m	VERB
ejpam-2634	96	10	annn	annn	NOUN
ejpam-2634	96	11	)	)	PUNCT
ejpam-2634	96	12	=	=	SYM
ejpam-2634	96	13	(	(	PUNCT
ejpam-2634	96	14	(	(	PUNCT
ejpam-2634	96	15	0	0	NUM
ejpam-2634	96	16	:	:	PUNCT
ejpam-2634	96	17	m	m	NOUN
ejpam-2634	96	18	annk	annk	NOUN
ejpam-2634	96	19	)	)	PUNCT
ejpam-2634	96	20	:	:	PUNCT
ejpam-2634	96	21	m	m	VERB
ejpam-2634	96	22	annn	annn	NOUN
ejpam-2634	96	23	)	)	PUNCT
ejpam-2634	96	24	=(	=(	NOUN
ejpam-2634	96	25	0	0	NUM
ejpam-2634	96	26	:	:	PUNCT
ejpam-2634	96	27	m	m	NOUN
ejpam-2634	96	28	annkannn	annkannn	NOUN
ejpam-2634	96	29	)	)	PUNCT
ejpam-2634	96	30	=	=	PUNCT
ejpam-2634	97	1	(	(	PUNCT
ejpam-2634	97	2	0	0	NUM
ejpam-2634	97	3	:	:	PUNCT
ejpam-2634	97	4	m	m	VERB
ejpam-2634	97	5	annk	annk	NOUN
ejpam-2634	97	6	⋂	⋂	PROPN
ejpam-2634	97	7	annn	annn	NOUN
ejpam-2634	97	8	)	)	PUNCT
ejpam-2634	97	9	=(	=(	NOUN
ejpam-2634	97	10	0	0	NUM
ejpam-2634	97	11	:	:	PUNCT
ejpam-2634	97	12	m	m	VERB
ejpam-2634	97	13	annn	annn	NOUN
ejpam-2634	97	14	)	)	PUNCT
ejpam-2634	97	15	+	+	CCONJ
ejpam-2634	97	16	(	(	PUNCT
ejpam-2634	97	17	0	0	NUM
ejpam-2634	97	18	:	:	PUNCT
ejpam-2634	97	19	m	m	VERB
ejpam-2634	97	20	annk	annk	NOUN
ejpam-2634	97	21	)	)	PUNCT
ejpam-2634	97	22	=	=	SYM
ejpam-2634	97	23	n	n	PROPN
ejpam-2634	97	24	+	+	CCONJ
ejpam-2634	97	25	(	(	PUNCT
ejpam-2634	97	26	0	0	NUM
ejpam-2634	97	27	:	:	PUNCT
ejpam-2634	97	28	m	m	VERB
ejpam-2634	97	29	i	i	NOUN
ejpam-2634	97	30	)	)	PUNCT
ejpam-2634	97	31	.	.	PUNCT
ejpam-2634	98	1	theorem	theorem	ADJ
ejpam-2634	98	2	5	5	NUM
ejpam-2634	98	3	.	.	PUNCT
ejpam-2634	99	1	every	every	DET
ejpam-2634	99	2	comultiplication	comultiplication	NOUN
ejpam-2634	99	3	module	module	NOUN
ejpam-2634	99	4	is	be	AUX
ejpam-2634	99	5	a	a	DET
ejpam-2634	99	6	generalized	generalized	ADJ
ejpam-2634	99	7	hopfian	hopfian	NOUN
ejpam-2634	99	8	and	and	CCONJ
ejpam-2634	99	9	weakly	weakly	ADJ
ejpam-2634	99	10	co	co	ADJ
ejpam-2634	99	11	-	-	ADJ
ejpam-2634	99	12	hopfian	hopfian	ADJ
ejpam-2634	99	13	module	module	NOUN
ejpam-2634	99	14	.	.	PUNCT
ejpam-2634	100	1	proof	proof	NOUN
ejpam-2634	100	2	.	.	PUNCT
ejpam-2634	101	1	let	let	VERB
ejpam-2634	101	2	m	m	PRON
ejpam-2634	101	3	be	be	AUX
ejpam-2634	101	4	a	a	DET
ejpam-2634	101	5	comultiplication	comultiplication	NOUN
ejpam-2634	101	6	module	module	NOUN
ejpam-2634	101	7	and	and	CCONJ
ejpam-2634	101	8	f	f	PROPN
ejpam-2634	101	9	be	be	AUX
ejpam-2634	101	10	a	a	DET
ejpam-2634	101	11	surjective	surjective	ADJ
ejpam-2634	101	12	endomorphism	endomorphism	NOUN
ejpam-2634	101	13	of	of	ADP
ejpam-2634	101	14	m	m	PROPN
ejpam-2634	101	15	.	.	PUNCT
ejpam-2634	102	1	suppose	suppose	VERB
ejpam-2634	102	2	that	that	SCONJ
ejpam-2634	102	3	there	there	PRON
ejpam-2634	102	4	exists	exist	VERB
ejpam-2634	102	5	a	a	DET
ejpam-2634	102	6	submodule	submodule	NOUN
ejpam-2634	102	7	n	n	PROPN
ejpam-2634	102	8	of	of	ADP
ejpam-2634	102	9	m	m	PRON
ejpam-2634	102	10	such	such	ADJ
ejpam-2634	102	11	that	that	SCONJ
ejpam-2634	102	12	m	m	VERB
ejpam-2634	102	13	=	=	NOUN
ejpam-2634	102	14	ker	ker	PROPN
ejpam-2634	102	15	f	f	PROPN
ejpam-2634	102	16	+	+	CCONJ
ejpam-2634	102	17	n	n	CCONJ
ejpam-2634	102	18	.	.	PUNCT
ejpam-2634	103	1	in	in	ADP
ejpam-2634	103	2	this	this	DET
ejpam-2634	103	3	case	case	NOUN
ejpam-2634	103	4	f	f	X
ejpam-2634	103	5	(	(	PUNCT
ejpam-2634	103	6	m	m	PROPN
ejpam-2634	103	7	)	)	PUNCT
ejpam-2634	103	8	=	=	SYM
ejpam-2634	103	9	f	f	PROPN
ejpam-2634	103	10	(	(	PUNCT
ejpam-2634	103	11	ker	ker	NOUN
ejpam-2634	103	12	f	f	PROPN
ejpam-2634	103	13	+	+	CCONJ
ejpam-2634	103	14	n	n	CCONJ
ejpam-2634	103	15	)	)	PUNCT
ejpam-2634	103	16	=	=	SYM
ejpam-2634	103	17	f	f	PROPN
ejpam-2634	103	18	(	(	PUNCT
ejpam-2634	103	19	n	n	CCONJ
ejpam-2634	103	20	)	)	PUNCT
ejpam-2634	103	21	.	.	PUNCT
ejpam-2634	104	1	therefore	therefore	ADV
ejpam-2634	104	2	m	m	VERB
ejpam-2634	104	3	=	=	SYM
ejpam-2634	104	4	f	f	X
ejpam-2634	104	5	(	(	PUNCT
ejpam-2634	104	6	n	n	CCONJ
ejpam-2634	104	7	)	)	PUNCT
ejpam-2634	104	8	=	=	SYM
ejpam-2634	105	1	(	(	PUNCT
ejpam-2634	105	2	0	0	NUM
ejpam-2634	105	3	:	:	PUNCT
ejpam-2634	105	4	m	m	VERB
ejpam-2634	105	5	(	(	PUNCT
ejpam-2634	105	6	0	0	NUM
ejpam-2634	105	7	:	:	PUNCT
ejpam-2634	105	8	r	r	NOUN
ejpam-2634	105	9	f	f	X
ejpam-2634	105	10	(	(	PUNCT
ejpam-2634	105	11	n	n	CCONJ
ejpam-2634	105	12	)	)	PUNCT
ejpam-2634	105	13	)	)	PUNCT
ejpam-2634	105	14	)	)	PUNCT
ejpam-2634	105	15	.	.	PUNCT
ejpam-2634	106	1	j.	j.	PROPN
ejpam-2634	106	2	a’zami	a’zami	PROPN
ejpam-2634	106	3	,	,	PUNCT
ejpam-2634	106	4	m.	m.	NOUN
ejpam-2634	106	5	khajepour	khajepour	PROPN
ejpam-2634	106	6	/	/	SYM
ejpam-2634	106	7	eur	eur	PROPN
ejpam-2634	106	8	.	.	PUNCT
ejpam-2634	107	1	j.	j.	PROPN
ejpam-2634	107	2	pure	pure	PROPN
ejpam-2634	107	3	appl	appl	PROPN
ejpam-2634	107	4	.	.	PROPN
ejpam-2634	107	5	math	math	PROPN
ejpam-2634	107	6	,	,	PUNCT
ejpam-2634	107	7	9	9	NUM
ejpam-2634	107	8	(	(	PUNCT
ejpam-2634	107	9	2016	2016	NUM
ejpam-2634	107	10	)	)	PUNCT
ejpam-2634	107	11	,	,	PUNCT
ejpam-2634	107	12	244	244	NUM
ejpam-2634	107	13	-	-	SYM
ejpam-2634	107	14	249	249	NUM
ejpam-2634	107	15	247	247	NUM
ejpam-2634	107	16	since	since	SCONJ
ejpam-2634	107	17	m	m	PROPN
ejpam-2634	107	18	is	be	AUX
ejpam-2634	107	19	a	a	DET
ejpam-2634	107	20	comultiplication	comultiplication	NOUN
ejpam-2634	107	21	r	r	NOUN
ejpam-2634	107	22	-	-	PUNCT
ejpam-2634	107	23	module	module	NOUN
ejpam-2634	108	1	,	,	PUNCT
ejpam-2634	108	2	it	it	PRON
ejpam-2634	108	3	follows	follow	VERB
ejpam-2634	108	4	that	that	SCONJ
ejpam-2634	108	5	f	f	PROPN
ejpam-2634	108	6	(	(	PUNCT
ejpam-2634	108	7	n	n	CCONJ
ejpam-2634	108	8	)	)	PUNCT
ejpam-2634	108	9	⊆	⊆	NUM
ejpam-2634	108	10	n	n	NOUN
ejpam-2634	108	11	and	and	CCONJ
ejpam-2634	108	12	so	so	ADV
ejpam-2634	108	13	we	we	PRON
ejpam-2634	108	14	have	have	VERB
ejpam-2634	108	15	:	:	PUNCT
ejpam-2634	108	16	m	m	VERB
ejpam-2634	108	17	=	=	SYM
ejpam-2634	108	18	(	(	PUNCT
ejpam-2634	108	19	0	0	NUM
ejpam-2634	108	20	:	:	PUNCT
ejpam-2634	108	21	m	m	VERB
ejpam-2634	108	22	(	(	PUNCT
ejpam-2634	108	23	0	0	NUM
ejpam-2634	108	24	:	:	PUNCT
ejpam-2634	108	25	r	r	NOUN
ejpam-2634	108	26	f	f	X
ejpam-2634	108	27	(	(	PUNCT
ejpam-2634	108	28	n	n	CCONJ
ejpam-2634	108	29	)	)	PUNCT
ejpam-2634	108	30	)	)	PUNCT
ejpam-2634	108	31	)	)	PUNCT
ejpam-2634	109	1	⊆	⊆	X
ejpam-2634	109	2	(	(	PUNCT
ejpam-2634	109	3	0	0	NUM
ejpam-2634	109	4	:	:	PUNCT
ejpam-2634	109	5	m	m	VERB
ejpam-2634	109	6	(	(	PUNCT
ejpam-2634	109	7	0	0	NUM
ejpam-2634	109	8	:	:	PUNCT
ejpam-2634	109	9	r	r	NOUN
ejpam-2634	109	10	n	n	CCONJ
ejpam-2634	109	11	)	)	PUNCT
ejpam-2634	109	12	)	)	PUNCT
ejpam-2634	110	1	=	=	SYM
ejpam-2634	110	2	n	n	NOUN
ejpam-2634	110	3	,	,	PUNCT
ejpam-2634	110	4	and	and	CCONJ
ejpam-2634	110	5	hence	hence	ADV
ejpam-2634	110	6	ker	ker	PROPN
ejpam-2634	111	1	f	f	PROPN
ejpam-2634	111	2	is	be	AUX
ejpam-2634	111	3	a	a	DET
ejpam-2634	111	4	small	small	ADJ
ejpam-2634	111	5	submodule	submodule	NOUN
ejpam-2634	111	6	of	of	ADP
ejpam-2634	111	7	m	m	PROPN
ejpam-2634	111	8	.	.	PUNCT
ejpam-2634	112	1	now	now	ADV
ejpam-2634	112	2	let	let	VERB
ejpam-2634	112	3	f	f	PRON
ejpam-2634	112	4	be	be	AUX
ejpam-2634	112	5	an	an	DET
ejpam-2634	112	6	injective	injective	ADJ
ejpam-2634	112	7	endomorphism	endomorphism	NOUN
ejpam-2634	112	8	of	of	ADP
ejpam-2634	112	9	m	m	PROPN
ejpam-2634	112	10	and	and	CCONJ
ejpam-2634	112	11	n	n	CCONJ
ejpam-2634	112	12	be	be	VERB
ejpam-2634	112	13	a	a	DET
ejpam-2634	112	14	submodule	submodule	NOUN
ejpam-2634	112	15	of	of	ADP
ejpam-2634	112	16	m	m	PRON
ejpam-2634	112	17	such	such	ADJ
ejpam-2634	112	18	that	that	SCONJ
ejpam-2634	112	19	imf	imf	PROPN
ejpam-2634	112	20	⋂	⋂	PROPN
ejpam-2634	112	21	n	n	PROPN
ejpam-2634	112	22	=	=	SYM
ejpam-2634	112	23	0	0	PROPN
ejpam-2634	112	24	.	.	PUNCT
ejpam-2634	113	1	by	by	ADP
ejpam-2634	113	2	the	the	DET
ejpam-2634	113	3	previous	previous	ADJ
ejpam-2634	113	4	lemma	lemma	PROPN
ejpam-2634	113	5	,	,	PUNCT
ejpam-2634	113	6	imf	imf	PROPN
ejpam-2634	113	7	=	=	SYM
ejpam-2634	113	8	annr(ker	annr(ker	PROPN
ejpam-2634	113	9	f	f	PROPN
ejpam-2634	113	10	)	)	PUNCT
ejpam-2634	113	11	m	m	PROPN
ejpam-2634	113	12	,	,	PUNCT
ejpam-2634	113	13	and	and	CCONJ
ejpam-2634	113	14	so	so	ADV
ejpam-2634	113	15	annr(ker	annr(ker	PROPN
ejpam-2634	113	16	f	f	PROPN
ejpam-2634	113	17	)	)	PUNCT
ejpam-2634	113	18	m	m	PROPN
ejpam-2634	113	19	⋂	⋂	PROPN
ejpam-2634	113	20	n	n	NOUN
ejpam-2634	113	21	=	=	SYM
ejpam-2634	113	22	0	0	PROPN
ejpam-2634	113	23	.	.	PUNCT
ejpam-2634	114	1	but	but	CCONJ
ejpam-2634	114	2	ker	ker	NOUN
ejpam-2634	114	3	f	f	PROPN
ejpam-2634	114	4	=	=	SYM
ejpam-2634	114	5	0	0	PROPN
ejpam-2634	114	6	and	and	CCONJ
ejpam-2634	114	7	hence	hence	ADV
ejpam-2634	114	8	n	n	NOUN
ejpam-2634	115	1	=	=	SYM
ejpam-2634	115	2	m	m	VERB
ejpam-2634	115	3	⋂	⋂	PROPN
ejpam-2634	115	4	n	n	NOUN
ejpam-2634	115	5	=	=	SYM
ejpam-2634	115	6	0	0	PROPN
ejpam-2634	115	7	.	.	PUNCT
ejpam-2634	115	8	theorem	theorem	NOUN
ejpam-2634	115	9	6	6	NUM
ejpam-2634	115	10	.	.	PUNCT
ejpam-2634	116	1	let	let	AUX
ejpam-2634	116	2	(	(	PUNCT
ejpam-2634	116	3	r	r	NOUN
ejpam-2634	116	4	,	,	PUNCT
ejpam-2634	116	5	m	m	VERB
ejpam-2634	116	6	)	)	PUNCT
ejpam-2634	116	7	be	be	VERB
ejpam-2634	116	8	a	a	DET
ejpam-2634	116	9	local	local	ADJ
ejpam-2634	116	10	artinian	artinian	ADJ
ejpam-2634	116	11	ring	ring	NOUN
ejpam-2634	116	12	,	,	PUNCT
ejpam-2634	116	13	then	then	ADV
ejpam-2634	116	14	the	the	DET
ejpam-2634	116	15	following	following	ADJ
ejpam-2634	116	16	statements	statement	NOUN
ejpam-2634	116	17	are	be	AUX
ejpam-2634	116	18	equivalent	equivalent	ADJ
ejpam-2634	116	19	:	:	PUNCT
ejpam-2634	116	20	(	(	PUNCT
ejpam-2634	116	21	1	1	X
ejpam-2634	116	22	)	)	PUNCT
ejpam-2634	116	23	r	r	NOUN
ejpam-2634	116	24	is	be	AUX
ejpam-2634	116	25	a	a	DET
ejpam-2634	116	26	comultiplication	comultiplication	NOUN
ejpam-2634	116	27	ring	ring	NOUN
ejpam-2634	116	28	;	;	PUNCT
ejpam-2634	116	29	(	(	PUNCT
ejpam-2634	116	30	2	2	X
ejpam-2634	116	31	)	)	PUNCT
ejpam-2634	116	32	r	r	NOUN
ejpam-2634	116	33	is	be	AUX
ejpam-2634	116	34	a	a	DET
ejpam-2634	116	35	gorenstein	gorenstein	ADJ
ejpam-2634	116	36	ring	ring	NOUN
ejpam-2634	116	37	;	;	PUNCT
ejpam-2634	116	38	(	(	PUNCT
ejpam-2634	116	39	3	3	X
ejpam-2634	116	40	)	)	PUNCT
ejpam-2634	116	41	soc(r)≈	soc(r)≈	VERB
ejpam-2634	116	42	r	r	NOUN
ejpam-2634	116	43	/	/	SYM
ejpam-2634	116	44	m	m	PROPN
ejpam-2634	116	45	;	;	PUNCT
ejpam-2634	116	46	(	(	PUNCT
ejpam-2634	116	47	4	4	X
ejpam-2634	116	48	)	)	PUNCT
ejpam-2634	116	49	e(r	e(r	NUM
ejpam-2634	116	50	/	/	SYM
ejpam-2634	116	51	m	m	NOUN
ejpam-2634	116	52	)	)	PUNCT
ejpam-2634	116	53	is	be	AUX
ejpam-2634	116	54	a	a	DET
ejpam-2634	116	55	multiplication	multiplication	NOUN
ejpam-2634	116	56	r	r	NOUN
ejpam-2634	116	57	module	module	NOUN
ejpam-2634	116	58	.	.	PUNCT
ejpam-2634	117	1	proof	proof	NOUN
ejpam-2634	117	2	.	.	PUNCT
ejpam-2634	118	1	(	(	PUNCT
ejpam-2634	118	2	1→	1→	NUM
ejpam-2634	118	3	2	2	NUM
ejpam-2634	118	4	)	)	PUNCT
ejpam-2634	118	5	it	it	PRON
ejpam-2634	118	6	is	be	AUX
ejpam-2634	118	7	enough	enough	ADJ
ejpam-2634	118	8	to	to	PART
ejpam-2634	118	9	show	show	VERB
ejpam-2634	118	10	that	that	SCONJ
ejpam-2634	118	11	r(r	r(r	NOUN
ejpam-2634	118	12	)	)	PUNCT
ejpam-2634	118	13	=	=	SYM
ejpam-2634	119	1	1	1	X
ejpam-2634	119	2	.	.	PUNCT
ejpam-2634	119	3	suppose	suppose	VERB
ejpam-2634	119	4	on	on	ADP
ejpam-2634	119	5	the	the	DET
ejpam-2634	119	6	contrary	contrary	NOUN
ejpam-2634	119	7	that	that	SCONJ
ejpam-2634	119	8	r(r	r(r	PROPN
ejpam-2634	119	9	)	)	PUNCT
ejpam-2634	119	10	6=	6=	ADP
ejpam-2634	119	11	1	1	NUM
ejpam-2634	119	12	,	,	PUNCT
ejpam-2634	119	13	so	so	ADV
ejpam-2634	119	14	r(r	r(r	NOUN
ejpam-2634	119	15	)	)	PUNCT
ejpam-2634	119	16	=	=	SYM
ejpam-2634	119	17	0	0	NUM
ejpam-2634	119	18	or	or	CCONJ
ejpam-2634	119	19	r(r	r(r	PROPN
ejpam-2634	119	20	)	)	PUNCT
ejpam-2634	119	21	>	>	X
ejpam-2634	120	1	1	1	X
ejpam-2634	120	2	.	.	PUNCT
ejpam-2634	121	1	if	if	SCONJ
ejpam-2634	121	2	r(r	r(r	PROPN
ejpam-2634	121	3	)	)	PUNCT
ejpam-2634	122	1	=	=	SYM
ejpam-2634	122	2	0	0	NUM
ejpam-2634	122	3	,	,	PUNCT
ejpam-2634	122	4	then	then	ADV
ejpam-2634	122	5	r(r	r(r	NOUN
ejpam-2634	122	6	)	)	PUNCT
ejpam-2634	123	1	=	=	PUNCT
ejpam-2634	123	2	dimkhomr(r	dimkhomr(r	NOUN
ejpam-2634	123	3	/	/	SYM
ejpam-2634	123	4	m	m	PROPN
ejpam-2634	123	5	,	,	PUNCT
ejpam-2634	123	6	r	r	NOUN
ejpam-2634	123	7	)	)	PUNCT
ejpam-2634	123	8	=	=	SYM
ejpam-2634	123	9	0	0	NUM
ejpam-2634	124	1	and	and	CCONJ
ejpam-2634	124	2	so	so	ADV
ejpam-2634	124	3	(	(	PUNCT
ejpam-2634	124	4	0	0	NUM
ejpam-2634	124	5	:	:	PUNCT
ejpam-2634	124	6	r	r	NOUN
ejpam-2634	124	7	m)≈	m)≈	PROPN
ejpam-2634	124	8	homr(r	homr(r	PROPN
ejpam-2634	124	9	/	/	SYM
ejpam-2634	124	10	m	m	PROPN
ejpam-2634	124	11	,	,	PUNCT
ejpam-2634	124	12	r	r	NOUN
ejpam-2634	124	13	)	)	PUNCT
ejpam-2634	124	14	=	=	SYM
ejpam-2634	125	1	0	0	X
ejpam-2634	125	2	.	.	NOUN
ejpam-2634	125	3	which	which	PRON
ejpam-2634	125	4	is	be	AUX
ejpam-2634	125	5	a	a	DET
ejpam-2634	125	6	contradiction	contradiction	NOUN
ejpam-2634	125	7	,	,	PUNCT
ejpam-2634	125	8	because	because	SCONJ
ejpam-2634	125	9	the	the	DET
ejpam-2634	125	10	annihilator	annihilator	NOUN
ejpam-2634	125	11	of	of	ADP
ejpam-2634	125	12	any	any	DET
ejpam-2634	125	13	proper	proper	ADJ
ejpam-2634	125	14	ideal	ideal	NOUN
ejpam-2634	125	15	of	of	ADP
ejpam-2634	125	16	an	an	DET
ejpam-2634	125	17	artinian	artinian	ADJ
ejpam-2634	125	18	local	local	ADJ
ejpam-2634	125	19	ring	ring	NOUN
ejpam-2634	125	20	is	be	AUX
ejpam-2634	125	21	non	non	ADJ
ejpam-2634	125	22	-	-	ADJ
ejpam-2634	125	23	zero	zero	NUM
ejpam-2634	125	24	.	.	PUNCT
ejpam-2634	126	1	now	now	ADV
ejpam-2634	126	2	suppose	suppose	VERB
ejpam-2634	126	3	thatr(r	thatr(r	NOUN
ejpam-2634	126	4	)	)	PUNCT
ejpam-2634	126	5	>	>	X
ejpam-2634	126	6	1	1	NUM
ejpam-2634	126	7	,	,	PUNCT
ejpam-2634	126	8	so	so	SCONJ
ejpam-2634	126	9	there	there	PRON
ejpam-2634	126	10	exist	exist	VERB
ejpam-2634	126	11	two	two	NUM
ejpam-2634	126	12	ideals	ideal	NOUN
ejpam-2634	126	13	i	i	PRON
ejpam-2634	126	14	and	and	CCONJ
ejpam-2634	126	15	j	j	PROPN
ejpam-2634	126	16	of	of	ADP
ejpam-2634	126	17	r	r	NOUN
ejpam-2634	126	18	such	such	ADJ
ejpam-2634	126	19	that	that	SCONJ
ejpam-2634	126	20	(	(	PUNCT
ejpam-2634	126	21	0	0	NUM
ejpam-2634	126	22	:	:	PUNCT
ejpam-2634	126	23	r	r	NOUN
ejpam-2634	126	24	m	m	NOUN
ejpam-2634	126	25	)	)	PUNCT
ejpam-2634	126	26	=	=	SYM
ejpam-2634	127	1	i	i	PRON
ejpam-2634	127	2	⊕	⊕	PROPN
ejpam-2634	127	3	j	j	PROPN
ejpam-2634	128	1	=	=	PUNCT
ejpam-2634	128	2	(	(	PUNCT
ejpam-2634	128	3	0	0	NUM
ejpam-2634	128	4	:	:	PUNCT
ejpam-2634	128	5	r	r	NOUN
ejpam-2634	128	6	anni	anni	PROPN
ejpam-2634	128	7	)	)	PUNCT
ejpam-2634	128	8	⊕	⊕	PROPN
ejpam-2634	128	9	(	(	PUNCT
ejpam-2634	128	10	0	0	NUM
ejpam-2634	128	11	:	:	PUNCT
ejpam-2634	128	12	r	r	NOUN
ejpam-2634	128	13	annj	annj	NOUN
ejpam-2634	128	14	)	)	PUNCT
ejpam-2634	128	15	=	=	PUNCT
ejpam-2634	129	1	(	(	PUNCT
ejpam-2634	129	2	0	0	NUM
ejpam-2634	129	3	:	:	PUNCT
ejpam-2634	129	4	r	r	NOUN
ejpam-2634	129	5	anni	anni	PROPN
ejpam-2634	129	6	⋂	⋂	PROPN
ejpam-2634	129	7	annj	annj	PROPN
ejpam-2634	129	8	)	)	PUNCT
ejpam-2634	130	1	,	,	PUNCT
ejpam-2634	130	2	this	this	PRON
ejpam-2634	130	3	means	mean	VERB
ejpam-2634	130	4	that	that	SCONJ
ejpam-2634	130	5	(	(	PUNCT
ejpam-2634	130	6	0	0	NUM
ejpam-2634	130	7	:	:	PUNCT
ejpam-2634	130	8	r	r	NOUN
ejpam-2634	130	9	anni	anni	PROPN
ejpam-2634	130	10	⋂	⋂	PROPN
ejpam-2634	130	11	annj	annj	PROPN
ejpam-2634	130	12	)	)	PUNCT
ejpam-2634	130	13	6=	6=	ADP
ejpam-2634	130	14	0	0	X
ejpam-2634	130	15	.	.	PUNCT
ejpam-2634	131	1	on	on	ADP
ejpam-2634	131	2	the	the	DET
ejpam-2634	131	3	other	other	ADJ
ejpam-2634	131	4	hand	hand	NOUN
ejpam-2634	131	5	i	i	PRON
ejpam-2634	131	6	⋂	⋂	PROPN
ejpam-2634	131	7	j	j	PROPN
ejpam-2634	131	8	=	=	SYM
ejpam-2634	131	9	annanni	annanni	PROPN
ejpam-2634	131	10	⋂	⋂	PROPN
ejpam-2634	131	11	annannj	annannj	NOUN
ejpam-2634	131	12	=	=	PUNCT
ejpam-2634	131	13	ann(anni	ann(anni	PUNCT
ejpam-2634	131	14	+	+	NUM
ejpam-2634	131	15	annj	annj	NOUN
ejpam-2634	131	16	)	)	PUNCT
ejpam-2634	131	17	6=	6=	ADP
ejpam-2634	131	18	0	0	NUM
ejpam-2634	131	19	,	,	PUNCT
ejpam-2634	131	20	which	which	PRON
ejpam-2634	131	21	is	be	AUX
ejpam-2634	131	22	a	a	DET
ejpam-2634	131	23	contradiction	contradiction	NOUN
ejpam-2634	131	24	.	.	PUNCT
ejpam-2634	132	1	(	(	PUNCT
ejpam-2634	132	2	2→3	2→3	NUM
ejpam-2634	132	3	)	)	PUNCT
ejpam-2634	132	4	since	since	SCONJ
ejpam-2634	132	5	r	r	NOUN
ejpam-2634	132	6	is	be	AUX
ejpam-2634	132	7	gorenstein	gorenstein	ADJ
ejpam-2634	132	8	,	,	PUNCT
ejpam-2634	132	9	it	it	PRON
ejpam-2634	132	10	follows	follow	VERB
ejpam-2634	132	11	from	from	ADP
ejpam-2634	132	12	[	[	X
ejpam-2634	132	13	6	6	NUM
ejpam-2634	132	14	]	]	PUNCT
ejpam-2634	132	15	,	,	PUNCT
ejpam-2634	132	16	for	for	ADP
ejpam-2634	132	17	all	all	DET
ejpam-2634	132	18	non	non	ADJ
ejpam-2634	132	19	-	-	ADJ
ejpam-2634	132	20	zero	zero	NUM
ejpam-2634	132	21	ideals	ideal	NOUN
ejpam-2634	133	1	i	i	PRON
ejpam-2634	133	2	and	and	CCONJ
ejpam-2634	133	3	j	j	PROPN
ejpam-2634	133	4	of	of	ADP
ejpam-2634	133	5	r	r	PROPN
ejpam-2634	133	6	,	,	PUNCT
ejpam-2634	133	7	i	i	PRON
ejpam-2634	133	8	⋂	⋂	PROPN
ejpam-2634	133	9	j	j	PROPN
ejpam-2634	133	10	6=	6=	PROPN
ejpam-2634	133	11	0	0	NUM
ejpam-2634	133	12	.	.	PUNCT
ejpam-2634	134	1	now	now	ADV
ejpam-2634	134	2	let	let	VERB
ejpam-2634	134	3	s1	s1	NOUN
ejpam-2634	134	4	and	and	CCONJ
ejpam-2634	134	5	s2	s2	NOUN
ejpam-2634	134	6	be	be	AUX
ejpam-2634	134	7	two	two	NUM
ejpam-2634	134	8	simple	simple	ADJ
ejpam-2634	134	9	submodules	submodule	NOUN
ejpam-2634	134	10	of	of	ADP
ejpam-2634	134	11	r	r	NOUN
ejpam-2634	134	12	,	,	PUNCT
ejpam-2634	134	13	then	then	ADV
ejpam-2634	134	14	s1	s1	PROPN
ejpam-2634	134	15	⋂	⋂	PROPN
ejpam-2634	134	16	s2	s2	PROPN
ejpam-2634	134	17	6=	6=	ADP
ejpam-2634	134	18	0	0	NUM
ejpam-2634	134	19	and	and	CCONJ
ejpam-2634	134	20	consequently	consequently	ADV
ejpam-2634	134	21	s1	s1	PROPN
ejpam-2634	134	22	=	=	SYM
ejpam-2634	134	23	s2	s2	PROPN
ejpam-2634	134	24	.	.	PUNCT
ejpam-2634	135	1	(	(	PUNCT
ejpam-2634	135	2	3→4	3→4	NOUN
ejpam-2634	135	3	)	)	PUNCT
ejpam-2634	135	4	since	since	SCONJ
ejpam-2634	135	5	soc(r	soc(r	PROPN
ejpam-2634	135	6	)	)	PUNCT
ejpam-2634	135	7	=	=	PUNCT
ejpam-2634	136	1	(	(	PUNCT
ejpam-2634	136	2	0	0	NUM
ejpam-2634	136	3	:	:	PUNCT
ejpam-2634	136	4	r	r	NOUN
ejpam-2634	136	5	m	m	NOUN
ejpam-2634	136	6	)	)	PUNCT
ejpam-2634	137	1	≈	≈	PROPN
ejpam-2634	137	2	r	r	X
ejpam-2634	137	3	/	/	SYM
ejpam-2634	137	4	m	m	PROPN
ejpam-2634	137	5	,	,	PUNCT
ejpam-2634	137	6	it	it	PRON
ejpam-2634	137	7	follows	follow	VERB
ejpam-2634	137	8	that	that	SCONJ
ejpam-2634	137	9	r(r	r(r	NOUN
ejpam-2634	137	10	)	)	PUNCT
ejpam-2634	137	11	=	=	SYM
ejpam-2634	138	1	1	1	NUM
ejpam-2634	139	1	and	and	CCONJ
ejpam-2634	139	2	so	so	ADV
ejpam-2634	139	3	r	r	NOUN
ejpam-2634	139	4	is	be	AUX
ejpam-2634	139	5	a	a	DET
ejpam-2634	139	6	gorenstein	gorenstein	ADJ
ejpam-2634	139	7	ring	ring	NOUN
ejpam-2634	139	8	.	.	PUNCT
ejpam-2634	140	1	on	on	ADP
ejpam-2634	140	2	the	the	DET
ejpam-2634	140	3	other	other	ADJ
ejpam-2634	140	4	hand	hand	NOUN
ejpam-2634	140	5	dimr	dimr	NOUN
ejpam-2634	140	6	=	=	PUNCT
ejpam-2634	140	7	in	in	ADP
ejpam-2634	140	8	jdimr	jdimr	NOUN
ejpam-2634	140	9	=	=	SYM
ejpam-2634	140	10	0	0	X
ejpam-2634	140	11	.	.	PUNCT
ejpam-2634	141	1	thus	thus	ADV
ejpam-2634	141	2	r	r	NOUN
ejpam-2634	141	3	is	be	AUX
ejpam-2634	141	4	an	an	DET
ejpam-2634	141	5	injective	injective	ADJ
ejpam-2634	141	6	r	r	NOUN
ejpam-2634	141	7	module	module	NOUN
ejpam-2634	141	8	and	and	CCONJ
ejpam-2634	141	9	so	so	ADV
ejpam-2634	141	10	r≈	r≈	NOUN
ejpam-2634	141	11	e(r	e(r	PROPN
ejpam-2634	141	12	/	/	SYM
ejpam-2634	141	13	m	m	NOUN
ejpam-2634	141	14	)	)	PUNCT
ejpam-2634	141	15	,	,	PUNCT
ejpam-2634	141	16	by	by	ADP
ejpam-2634	141	17	[	[	X
ejpam-2634	141	18	6	6	NUM
ejpam-2634	141	19	]	]	PUNCT
ejpam-2634	141	20	.	.	PUNCT
ejpam-2634	142	1	(	(	PUNCT
ejpam-2634	142	2	4→1	4→1	NOUN
ejpam-2634	142	3	)	)	PUNCT
ejpam-2634	142	4	e(r	e(r	NUM
ejpam-2634	142	5	/	/	SYM
ejpam-2634	142	6	m	m	NOUN
ejpam-2634	142	7	)	)	PUNCT
ejpam-2634	142	8	is	be	AUX
ejpam-2634	142	9	multiplication	multiplication	NOUN
ejpam-2634	142	10	and	and	CCONJ
ejpam-2634	142	11	artinian	artinian	ADJ
ejpam-2634	142	12	,	,	PUNCT
ejpam-2634	142	13	it	it	PRON
ejpam-2634	142	14	follows	follow	VERB
ejpam-2634	142	15	that	that	SCONJ
ejpam-2634	142	16	e(r	e(r	NUM
ejpam-2634	142	17	/	/	SYM
ejpam-2634	142	18	m	m	NOUN
ejpam-2634	142	19	)	)	PUNCT
ejpam-2634	142	20	is	be	AUX
ejpam-2634	142	21	cyclic	cyclic	ADJ
ejpam-2634	142	22	and	and	CCONJ
ejpam-2634	142	23	so	so	ADV
ejpam-2634	142	24	e(r	e(r	PROPN
ejpam-2634	142	25	/	/	SYM
ejpam-2634	142	26	m	m	NOUN
ejpam-2634	142	27	)	)	PUNCT
ejpam-2634	143	1	≈	≈	PROPN
ejpam-2634	143	2	r.	r.	PROPN
ejpam-2634	143	3	now	now	ADV
ejpam-2634	143	4	r	r	NOUN
ejpam-2634	143	5	is	be	AUX
ejpam-2634	143	6	an	an	DET
ejpam-2634	143	7	injective	injective	ADJ
ejpam-2634	143	8	and	and	CCONJ
ejpam-2634	143	9	multiplication	multiplication	NOUN
ejpam-2634	143	10	r	r	NOUN
ejpam-2634	143	11	module	module	NOUN
ejpam-2634	143	12	,	,	PUNCT
ejpam-2634	143	13	then	then	ADV
ejpam-2634	143	14	by	by	ADP
ejpam-2634	143	15	[	[	X
ejpam-2634	143	16	3	3	NUM
ejpam-2634	143	17	]	]	X
ejpam-2634	143	18	r	r	NOUN
ejpam-2634	143	19	is	be	AUX
ejpam-2634	143	20	comultiplication	comultiplication	NOUN
ejpam-2634	143	21	.	.	PUNCT
ejpam-2634	144	1	theorem	theorem	ADJ
ejpam-2634	144	2	7	7	NUM
ejpam-2634	144	3	.	.	PUNCT
ejpam-2634	145	1	let	let	AUX
ejpam-2634	145	2	(	(	PUNCT
ejpam-2634	145	3	r	r	NOUN
ejpam-2634	145	4	,	,	PUNCT
ejpam-2634	145	5	m	m	VERB
ejpam-2634	145	6	)	)	PUNCT
ejpam-2634	145	7	be	be	VERB
ejpam-2634	145	8	a	a	DET
ejpam-2634	145	9	local	local	ADJ
ejpam-2634	145	10	artinian	artinian	ADJ
ejpam-2634	145	11	ring	ring	NOUN
ejpam-2634	145	12	,	,	PUNCT
ejpam-2634	145	13	and	and	CCONJ
ejpam-2634	145	14	m	m	VERB
ejpam-2634	145	15	be	be	AUX
ejpam-2634	145	16	a	a	DET
ejpam-2634	145	17	faithful	faithful	ADJ
ejpam-2634	145	18	comultiplication	comultiplication	NOUN
ejpam-2634	145	19	r	r	NOUN
ejpam-2634	145	20	-	-	PUNCT
ejpam-2634	145	21	module	module	NOUN
ejpam-2634	145	22	.	.	PUNCT
ejpam-2634	146	1	then	then	ADV
ejpam-2634	146	2	m	m	PROPN
ejpam-2634	146	3	is	be	AUX
ejpam-2634	146	4	uniform	uniform	ADJ
ejpam-2634	146	5	.	.	PUNCT
ejpam-2634	147	1	proof	proof	NOUN
ejpam-2634	147	2	.	.	PUNCT
ejpam-2634	148	1	let	let	AUX
ejpam-2634	148	2	(	(	PUNCT
ejpam-2634	148	3	r	r	NOUN
ejpam-2634	148	4	,	,	PUNCT
ejpam-2634	148	5	m	m	VERB
ejpam-2634	148	6	)	)	PUNCT
ejpam-2634	148	7	be	be	VERB
ejpam-2634	148	8	a	a	DET
ejpam-2634	148	9	local	local	ADJ
ejpam-2634	148	10	artinian	artinian	ADJ
ejpam-2634	148	11	ring	ring	NOUN
ejpam-2634	148	12	,	,	PUNCT
ejpam-2634	148	13	and	and	CCONJ
ejpam-2634	148	14	m	m	VERB
ejpam-2634	148	15	be	be	AUX
ejpam-2634	148	16	a	a	DET
ejpam-2634	148	17	faithful	faithful	ADJ
ejpam-2634	148	18	comultiplication	comultiplication	NOUN
ejpam-2634	148	19	r	r	NOUN
ejpam-2634	148	20	-	-	PUNCT
ejpam-2634	148	21	module	module	NOUN
ejpam-2634	148	22	and	and	CCONJ
ejpam-2634	148	23	n	n	CCONJ
ejpam-2634	148	24	be	be	VERB
ejpam-2634	148	25	a	a	DET
ejpam-2634	148	26	submodule	submodule	NOUN
ejpam-2634	148	27	of	of	ADP
ejpam-2634	148	28	m	m	PRON
ejpam-2634	148	29	such	such	ADJ
ejpam-2634	149	1	that	that	SCONJ
ejpam-2634	149	2	n	n	ADV
ejpam-2634	149	3	∩k	∩k	NOUN
ejpam-2634	149	4	=	=	SYM
ejpam-2634	149	5	0	0	NUM
ejpam-2634	149	6	,	,	PUNCT
ejpam-2634	149	7	for	for	ADP
ejpam-2634	149	8	some	some	DET
ejpam-2634	149	9	submodule	submodule	NOUN
ejpam-2634	149	10	k	k	PROPN
ejpam-2634	149	11	of	of	ADP
ejpam-2634	149	12	m	m	PROPN
ejpam-2634	149	13	.	.	PUNCT
ejpam-2634	150	1	then	then	ADV
ejpam-2634	150	2	we	we	PRON
ejpam-2634	150	3	have	have	VERB
ejpam-2634	150	4	n	n	NOUN
ejpam-2634	150	5	∩	∩	ADJ
ejpam-2634	150	6	k	k	X
ejpam-2634	150	7	=	=	PUNCT
ejpam-2634	150	8	(	(	PUNCT
ejpam-2634	150	9	0	0	NUM
ejpam-2634	150	10	:	:	PUNCT
ejpam-2634	150	11	m	m	VERB
ejpam-2634	150	12	annr(n))∩	annr(n))∩	INTJ
ejpam-2634	150	13	(	(	PUNCT
ejpam-2634	150	14	0	0	NUM
ejpam-2634	150	15	:	:	PUNCT
ejpam-2634	150	16	m	m	PROPN
ejpam-2634	150	17	annr(k	annr(k	ADJ
ejpam-2634	150	18	)	)	PUNCT
ejpam-2634	150	19	)	)	PUNCT
ejpam-2634	151	1	=	=	PUNCT
ejpam-2634	151	2	(	(	PUNCT
ejpam-2634	151	3	0	0	NUM
ejpam-2634	151	4	:	:	PUNCT
ejpam-2634	151	5	m	m	VERB
ejpam-2634	151	6	annr(n	annr(n	PROPN
ejpam-2634	151	7	)	)	PUNCT
ejpam-2634	151	8	+	+	CCONJ
ejpam-2634	151	9	annr(k	annr(k	ADJ
ejpam-2634	151	10	)	)	PUNCT
ejpam-2634	151	11	)	)	PUNCT
ejpam-2634	151	12	.	.	PUNCT
ejpam-2634	152	1	j.	j.	PROPN
ejpam-2634	152	2	a’zami	a’zami	PROPN
ejpam-2634	152	3	,	,	PUNCT
ejpam-2634	152	4	m.	m.	NOUN
ejpam-2634	152	5	khajepour	khajepour	PROPN
ejpam-2634	152	6	/	/	SYM
ejpam-2634	152	7	eur	eur	PROPN
ejpam-2634	152	8	.	.	PUNCT
ejpam-2634	153	1	j.	j.	PROPN
ejpam-2634	153	2	pure	pure	PROPN
ejpam-2634	153	3	appl	appl	PROPN
ejpam-2634	153	4	.	.	PROPN
ejpam-2634	153	5	math	math	PROPN
ejpam-2634	153	6	,	,	PUNCT
ejpam-2634	153	7	9	9	NUM
ejpam-2634	153	8	(	(	PUNCT
ejpam-2634	153	9	2016	2016	NUM
ejpam-2634	153	10	)	)	PUNCT
ejpam-2634	153	11	,	,	PUNCT
ejpam-2634	153	12	244	244	NUM
ejpam-2634	153	13	-	-	SYM
ejpam-2634	153	14	249	249	NUM
ejpam-2634	153	15	248	248	NUM
ejpam-2634	153	16	now	now	ADV
ejpam-2634	153	17	if	if	SCONJ
ejpam-2634	153	18	annr(n	annr(n	VERB
ejpam-2634	153	19	)	)	PUNCT
ejpam-2634	153	20	+	+	CCONJ
ejpam-2634	153	21	annr(k	annr(k	ADJ
ejpam-2634	153	22	)	)	PUNCT
ejpam-2634	153	23	=	=	SYM
ejpam-2634	153	24	0	0	NUM
ejpam-2634	153	25	,	,	PUNCT
ejpam-2634	153	26	then	then	ADV
ejpam-2634	153	27	n	n	X
ejpam-2634	153	28	∩	∩	X
ejpam-2634	153	29	k	k	PROPN
ejpam-2634	153	30	=	=	PUNCT
ejpam-2634	153	31	(	(	PUNCT
ejpam-2634	153	32	0	0	NUM
ejpam-2634	153	33	:	:	PUNCT
ejpam-2634	153	34	m	m	PROPN
ejpam-2634	153	35	0	0	NUM
ejpam-2634	153	36	)	)	PUNCT
ejpam-2634	153	37	=	=	SYM
ejpam-2634	153	38	m	m	PROPN
ejpam-2634	153	39	6=	6=	NUM
ejpam-2634	153	40	0	0	NUM
ejpam-2634	154	1	so	so	CCONJ
ejpam-2634	154	2	suppose	suppose	VERB
ejpam-2634	154	3	that	that	SCONJ
ejpam-2634	154	4	annr(n	annr(n	VERB
ejpam-2634	154	5	)	)	PUNCT
ejpam-2634	154	6	+	+	CCONJ
ejpam-2634	154	7	annr(k	annr(k	PROPN
ejpam-2634	154	8	)	)	PUNCT
ejpam-2634	154	9	6=	6=	ADP
ejpam-2634	154	10	0	0	NUM
ejpam-2634	154	11	,	,	PUNCT
ejpam-2634	154	12	then	then	ADV
ejpam-2634	154	13	annr(n	annr(n	VERB
ejpam-2634	154	14	)	)	PUNCT
ejpam-2634	154	15	+	+	CCONJ
ejpam-2634	154	16	annr(k	annr(k	ADJ
ejpam-2634	154	17	)	)	PUNCT
ejpam-2634	154	18	⊆	⊆	NUM
ejpam-2634	154	19	m.	m.	NOUN
ejpam-2634	154	20	therefore	therefore	ADV
ejpam-2634	154	21	0=	0=	NUM
ejpam-2634	154	22	n	n	PROPN
ejpam-2634	154	23	∩	∩	X
ejpam-2634	154	24	k	k	X
ejpam-2634	154	25	=	=	PUNCT
ejpam-2634	154	26	(	(	PUNCT
ejpam-2634	154	27	0	0	NUM
ejpam-2634	154	28	:	:	PUNCT
ejpam-2634	154	29	m	m	VERB
ejpam-2634	154	30	annr(n	annr(n	PROPN
ejpam-2634	154	31	)	)	PUNCT
ejpam-2634	155	1	+	+	CCONJ
ejpam-2634	155	2	annr(k	annr(k	ADJ
ejpam-2634	155	3	)	)	PUNCT
ejpam-2634	155	4	)	)	PUNCT
ejpam-2634	155	5	.	.	PUNCT
ejpam-2634	156	1	(	(	PUNCT
ejpam-2634	156	2	1	1	X
ejpam-2634	156	3	)	)	PUNCT
ejpam-2634	156	4	hence	hence	ADV
ejpam-2634	156	5	ann(annr(n	ann(annr(n	ADV
ejpam-2634	156	6	)	)	PUNCT
ejpam-2634	157	1	+	+	CCONJ
ejpam-2634	157	2	annr(k	annr(k	ADJ
ejpam-2634	157	3	)	)	PUNCT
ejpam-2634	157	4	)	)	PUNCT
ejpam-2634	158	1	⊆	⊆	NUM
ejpam-2634	158	2	ann(m	ann(m	PROPN
ejpam-2634	158	3	)	)	PUNCT
ejpam-2634	158	4	=	=	SYM
ejpam-2634	158	5	0	0	PUNCT
ejpam-2634	158	6	(	(	PUNCT
ejpam-2634	158	7	2	2	NUM
ejpam-2634	158	8	)	)	PUNCT
ejpam-2634	158	9	which	which	PRON
ejpam-2634	158	10	is	be	AUX
ejpam-2634	158	11	a	a	DET
ejpam-2634	158	12	contradiction	contradiction	NOUN
ejpam-2634	158	13	,	,	PUNCT
ejpam-2634	158	14	because	because	SCONJ
ejpam-2634	158	15	r	r	NOUN
ejpam-2634	158	16	is	be	AUX
ejpam-2634	158	17	artinian	artinian	ADJ
ejpam-2634	158	18	.	.	PUNCT
ejpam-2634	159	1	lemma	lemma	PROPN
ejpam-2634	159	2	3	3	X
ejpam-2634	159	3	.	.	PUNCT
ejpam-2634	160	1	let	let	AUX
ejpam-2634	160	2	(	(	PUNCT
ejpam-2634	160	3	r	r	NOUN
ejpam-2634	160	4	,	,	PUNCT
ejpam-2634	160	5	m	m	VERB
ejpam-2634	160	6	)	)	PUNCT
ejpam-2634	160	7	be	be	VERB
ejpam-2634	160	8	an	an	DET
ejpam-2634	160	9	artinian	artinian	ADJ
ejpam-2634	160	10	local	local	ADJ
ejpam-2634	160	11	ring	ring	NOUN
ejpam-2634	160	12	.	.	PUNCT
ejpam-2634	161	1	then	then	ADV
ejpam-2634	161	2	r	r	NOUN
ejpam-2634	161	3	is	be	AUX
ejpam-2634	161	4	comultiplication	comultiplication	NOUN
ejpam-2634	161	5	if	if	SCONJ
ejpam-2634	161	6	and	and	CCONJ
ejpam-2634	161	7	only	only	ADV
ejpam-2634	161	8	if	if	SCONJ
ejpam-2634	161	9	(	(	PUNCT
ejpam-2634	161	10	0	0	NUM
ejpam-2634	161	11	:	:	PUNCT
ejpam-2634	161	12	r	r	NOUN
ejpam-2634	161	13	mn+1)/(0	mn+1)/(0	PROPN
ejpam-2634	161	14	:	:	PUNCT
ejpam-2634	161	15	r	r	NOUN
ejpam-2634	161	16	mn)≃	mn)≃	PROPN
ejpam-2634	161	17	mn	mn	PROPN
ejpam-2634	161	18	/	/	SYM
ejpam-2634	161	19	mn+1	mn+1	PROPN
ejpam-2634	161	20	for	for	ADP
ejpam-2634	161	21	all	all	DET
ejpam-2634	161	22	n≥	n≥	NOUN
ejpam-2634	161	23	0	0	NUM
ejpam-2634	161	24	.	.	PUNCT
ejpam-2634	162	1	proof	proof	NOUN
ejpam-2634	162	2	.	.	PUNCT
ejpam-2634	163	1	let	let	VERB
ejpam-2634	163	2	r	r	PRON
ejpam-2634	163	3	be	be	AUX
ejpam-2634	163	4	a	a	DET
ejpam-2634	163	5	comultiplication	comultiplication	NOUN
ejpam-2634	163	6	ring	ring	NOUN
ejpam-2634	163	7	,	,	PUNCT
ejpam-2634	163	8	then	then	ADV
ejpam-2634	163	9	by	by	ADP
ejpam-2634	163	10	theorem	theorem	NOUN
ejpam-2634	163	11	6	6	NUM
ejpam-2634	163	12	,	,	PUNCT
ejpam-2634	163	13	soc(r	soc(r	PROPN
ejpam-2634	163	14	)	)	PUNCT
ejpam-2634	164	1	≈	≈	PROPN
ejpam-2634	164	2	r	r	X
ejpam-2634	164	3	/	/	SYM
ejpam-2634	164	4	m	m	NOUN
ejpam-2634	165	1	and	and	CCONJ
ejpam-2634	165	2	so	so	ADV
ejpam-2634	165	3	we	we	PRON
ejpam-2634	165	4	have	have	VERB
ejpam-2634	165	5	(	(	PUNCT
ejpam-2634	165	6	0	0	NUM
ejpam-2634	165	7	:	:	PUNCT
ejpam-2634	165	8	r	r	NOUN
ejpam-2634	165	9	m)≈	m)≈	NOUN
ejpam-2634	165	10	r	r	NOUN
ejpam-2634	165	11	/	/	SYM
ejpam-2634	165	12	m.	m.	NOUN
ejpam-2634	165	13	now	now	ADV
ejpam-2634	165	14	let	let	VERB
ejpam-2634	165	15	n≥	n≥	PROPN
ejpam-2634	165	16	1	1	NUM
ejpam-2634	165	17	,	,	PUNCT
ejpam-2634	165	18	consider	consider	VERB
ejpam-2634	165	19	the	the	DET
ejpam-2634	165	20	following	follow	VERB
ejpam-2634	165	21	exact	exact	ADJ
ejpam-2634	165	22	sequence	sequence	NOUN
ejpam-2634	165	23	:	:	PUNCT
ejpam-2634	165	24	0→	0→	PROPN
ejpam-2634	165	25	mn	mn	PROPN
ejpam-2634	165	26	/	/	SYM
ejpam-2634	165	27	mn+1→	mn+1→	PROPN
ejpam-2634	165	28	r	r	PROPN
ejpam-2634	165	29	/	/	SYM
ejpam-2634	165	30	mn+1→	mn+1→	PROPN
ejpam-2634	165	31	r	r	NOUN
ejpam-2634	165	32	/	/	SYM
ejpam-2634	165	33	mn→	mn→	NOUN
ejpam-2634	165	34	0	0	PUNCT
ejpam-2634	165	35	since	since	SCONJ
ejpam-2634	165	36	r	r	NOUN
ejpam-2634	165	37	is	be	AUX
ejpam-2634	165	38	gorenstein	gorenstein	ADJ
ejpam-2634	165	39	,	,	PUNCT
ejpam-2634	165	40	it	it	PRON
ejpam-2634	165	41	follows	follow	VERB
ejpam-2634	165	42	that	that	PRON
ejpam-2634	165	43	0	0	NUM
ejpam-2634	165	44	=	=	SYM
ejpam-2634	165	45	dimr	dimr	NOUN
ejpam-2634	165	46	≤	≤	NOUN
ejpam-2634	165	47	in	in	ADP
ejpam-2634	165	48	jdimr	jdimr	NOUN
ejpam-2634	165	49	=	=	PUNCT
ejpam-2634	165	50	depthr	depthr	NOUN
ejpam-2634	165	51	≤	≤	NOUN
ejpam-2634	165	52	dimr	dimr	NOUN
ejpam-2634	165	53	=	=	SYM
ejpam-2634	165	54	0	0	NUM
ejpam-2634	165	55	,	,	PUNCT
ejpam-2634	165	56	and	and	CCONJ
ejpam-2634	165	57	so	so	ADV
ejpam-2634	165	58	r	r	NOUN
ejpam-2634	165	59	is	be	AUX
ejpam-2634	165	60	injective	injective	ADJ
ejpam-2634	165	61	r	r	NOUN
ejpam-2634	165	62	-	-	PUNCT
ejpam-2634	165	63	module	module	NOUN
ejpam-2634	165	64	.	.	PUNCT
ejpam-2634	166	1	therefore	therefore	ADV
ejpam-2634	166	2	we	we	PRON
ejpam-2634	166	3	have	have	VERB
ejpam-2634	166	4	the	the	DET
ejpam-2634	166	5	following	follow	VERB
ejpam-2634	166	6	exact	exact	ADJ
ejpam-2634	166	7	sequence	sequence	NOUN
ejpam-2634	166	8	.	.	PUNCT
ejpam-2634	167	1	0→	0→	NOUN
ejpam-2634	167	2	(	(	PUNCT
ejpam-2634	167	3	0	0	NUM
ejpam-2634	167	4	:	:	PUNCT
ejpam-2634	167	5	r	r	NOUN
ejpam-2634	167	6	mn)→	mn)→	NOUN
ejpam-2634	167	7	(	(	PUNCT
ejpam-2634	167	8	0	0	NUM
ejpam-2634	167	9	:	:	PUNCT
ejpam-2634	167	10	r	r	NOUN
ejpam-2634	167	11	mn+1)→	mn+1)→	NOUN
ejpam-2634	167	12	hom(mn	hom(mn	NOUN
ejpam-2634	167	13	/	/	SYM
ejpam-2634	167	14	mn+1,r)→	mn+1,r)→	NOUN
ejpam-2634	167	15	0	0	NUM
ejpam-2634	167	16	on	on	ADP
ejpam-2634	167	17	the	the	DET
ejpam-2634	167	18	other	other	ADJ
ejpam-2634	167	19	hand	hand	NOUN
ejpam-2634	167	20	r(r	r(r	PROPN
ejpam-2634	167	21	)	)	PUNCT
ejpam-2634	167	22	=	=	SYM
ejpam-2634	167	23	1	1	NUM
ejpam-2634	167	24	and	and	CCONJ
ejpam-2634	167	25	we	we	PRON
ejpam-2634	167	26	have	have	VERB
ejpam-2634	167	27	hom(mn	hom(mn	NOUN
ejpam-2634	167	28	/	/	SYM
ejpam-2634	167	29	mn+1,r)≈	mn+1,r)≈	VERB
ejpam-2634	167	30	hom	hom	INTJ
ejpam-2634	167	31	(	(	PUNCT
ejpam-2634	167	32	t⊕	t⊕	NOUN
ejpam-2634	167	33	i=1	i=1	X
ejpam-2634	167	34	r	r	PROPN
ejpam-2634	167	35	/	/	SYM
ejpam-2634	167	36	m	m	PROPN
ejpam-2634	167	37	,	,	PUNCT
ejpam-2634	167	38	r)≈	r)≈	X
ejpam-2634	167	39	t⊕	t⊕	NOUN
ejpam-2634	167	40	i=1	i=1	ADP
ejpam-2634	167	41	hom(r	hom(r	PROPN
ejpam-2634	167	42	/	/	SYM
ejpam-2634	167	43	m	m	PROPN
ejpam-2634	167	44	,	,	PUNCT
ejpam-2634	167	45	r)≈	r)≈	X
ejpam-2634	167	46	t⊕	t⊕	NOUN
ejpam-2634	167	47	i=1	i=1	X
ejpam-2634	167	48	r	r	X
ejpam-2634	167	49	/	/	SYM
ejpam-2634	167	50	m=	m=	X
ejpam-2634	167	51	mn	mn	PROPN
ejpam-2634	167	52	/	/	SYM
ejpam-2634	167	53	mn+1	mn+1	PROPN
ejpam-2634	167	54	.	.	PUNCT
ejpam-2634	168	1	therefore	therefore	ADV
ejpam-2634	168	2	by	by	ADP
ejpam-2634	168	3	the	the	DET
ejpam-2634	168	4	last	last	ADJ
ejpam-2634	168	5	exact	exact	ADJ
ejpam-2634	168	6	sequence	sequence	NOUN
ejpam-2634	168	7	(	(	PUNCT
ejpam-2634	168	8	0	0	NUM
ejpam-2634	168	9	:	:	PUNCT
ejpam-2634	168	10	r	r	NOUN
ejpam-2634	168	11	mn)/(0	mn)/(0	NOUN
ejpam-2634	168	12	:	:	PUNCT
ejpam-2634	168	13	r	r	NOUN
ejpam-2634	168	14	mn+1)≈	mn+1)≈	NUM
ejpam-2634	168	15	mn	mn	PROPN
ejpam-2634	168	16	/	/	SYM
ejpam-2634	168	17	mn+1	mn+1	PROPN
ejpam-2634	168	18	.	.	PUNCT
ejpam-2634	169	1	lemma	lemma	PROPN
ejpam-2634	169	2	4	4	X
ejpam-2634	169	3	.	.	PUNCT
ejpam-2634	170	1	let	let	VERB
ejpam-2634	170	2	m1	m1	PROPN
ejpam-2634	170	3	,	,	PUNCT
ejpam-2634	170	4	m2	m2	PROPN
ejpam-2634	170	5	be	be	VERB
ejpam-2634	170	6	two	two	NUM
ejpam-2634	170	7	submodules	submodule	NOUN
ejpam-2634	170	8	of	of	ADP
ejpam-2634	170	9	a	a	DET
ejpam-2634	170	10	comultiplication	comultiplication	NOUN
ejpam-2634	170	11	r	r	NOUN
ejpam-2634	170	12	-	-	PUNCT
ejpam-2634	170	13	module	module	NOUN
ejpam-2634	170	14	m	m	NOUN
ejpam-2634	170	15	such	such	ADJ
ejpam-2634	170	16	that	that	SCONJ
ejpam-2634	170	17	m	m	PROPN
ejpam-2634	170	18	=	=	SYM
ejpam-2634	170	19	m1	m1	PROPN
ejpam-2634	170	20	⊕	⊕	PROPN
ejpam-2634	170	21	m2	m2	PROPN
ejpam-2634	170	22	.	.	PUNCT
ejpam-2634	171	1	then	then	ADV
ejpam-2634	171	2	homr(m1	homr(m1	NOUN
ejpam-2634	171	3	,	,	PUNCT
ejpam-2634	171	4	m2	m2	PROPN
ejpam-2634	171	5	)	)	PUNCT
ejpam-2634	171	6	=	=	SYM
ejpam-2634	171	7	homr(m2	homr(m2	PROPN
ejpam-2634	171	8	,	,	PUNCT
ejpam-2634	171	9	m1	m1	NOUN
ejpam-2634	171	10	)	)	PUNCT
ejpam-2634	171	11	=	=	SYM
ejpam-2634	172	1	0	0	X
ejpam-2634	172	2	.	.	PUNCT
ejpam-2634	172	3	proof	proof	NOUN
ejpam-2634	172	4	.	.	PUNCT
ejpam-2634	173	1	let	let	VERB
ejpam-2634	173	2	f	f	NOUN
ejpam-2634	173	3	:	:	PUNCT
ejpam-2634	173	4	m1→	m1→	PROPN
ejpam-2634	173	5	m2	m2	PROPN
ejpam-2634	173	6	be	be	VERB
ejpam-2634	173	7	a	a	DET
ejpam-2634	173	8	homomorphism	homomorphism	NOUN
ejpam-2634	173	9	.	.	PUNCT
ejpam-2634	174	1	since	since	SCONJ
ejpam-2634	174	2	m	m	PROPN
ejpam-2634	174	3	is	be	AUX
ejpam-2634	174	4	comultiplication	comultiplication	NOUN
ejpam-2634	174	5	,	,	PUNCT
ejpam-2634	174	6	f	f	PROPN
ejpam-2634	174	7	(	(	PUNCT
ejpam-2634	174	8	m1	m1	PROPN
ejpam-2634	174	9	)	)	PUNCT
ejpam-2634	174	10	⊆	⊆	NUM
ejpam-2634	174	11	m1	m1	NOUN
ejpam-2634	174	12	,	,	PUNCT
ejpam-2634	174	13	by	by	ADP
ejpam-2634	174	14	[	[	X
ejpam-2634	174	15	2	2	NUM
ejpam-2634	174	16	]	]	PUNCT
ejpam-2634	174	17	.	.	PUNCT
ejpam-2634	175	1	on	on	ADP
ejpam-2634	175	2	the	the	DET
ejpam-2634	175	3	other	other	ADJ
ejpam-2634	175	4	hand	hand	NOUN
ejpam-2634	175	5	f	f	PROPN
ejpam-2634	175	6	(	(	PUNCT
ejpam-2634	175	7	m1	m1	PROPN
ejpam-2634	175	8	)	)	PUNCT
ejpam-2634	175	9	⊆	⊆	NUM
ejpam-2634	175	10	m2	m2	PROPN
ejpam-2634	175	11	and	and	CCONJ
ejpam-2634	175	12	so	so	ADV
ejpam-2634	175	13	f	f	PROPN
ejpam-2634	175	14	(	(	PUNCT
ejpam-2634	175	15	m1	m1	PROPN
ejpam-2634	175	16	)	)	PUNCT
ejpam-2634	175	17	⊆	⊆	NUM
ejpam-2634	175	18	m1	m1	PROPN
ejpam-2634	175	19	⋂	⋂	PROPN
ejpam-2634	175	20	m2	m2	PROPN
ejpam-2634	175	21	=	=	SYM
ejpam-2634	175	22	0	0	PROPN
ejpam-2634	175	23	.	.	PUNCT
ejpam-2634	176	1	this	this	PRON
ejpam-2634	176	2	shows	show	VERB
ejpam-2634	176	3	that	that	SCONJ
ejpam-2634	176	4	f	f	PROPN
ejpam-2634	176	5	=	=	SYM
ejpam-2634	176	6	0	0	PROPN
ejpam-2634	176	7	.	.	PUNCT
ejpam-2634	176	8	theorem	theorem	NOUN
ejpam-2634	176	9	8	8	NUM
ejpam-2634	176	10	.	.	PUNCT
ejpam-2634	177	1	let	let	VERB
ejpam-2634	177	2	r	r	PRON
ejpam-2634	177	3	be	be	AUX
ejpam-2634	177	4	a	a	DET
ejpam-2634	177	5	dedekind	dedekind	ADJ
ejpam-2634	177	6	domain	domain	NOUN
ejpam-2634	177	7	,	,	PUNCT
ejpam-2634	177	8	and	and	CCONJ
ejpam-2634	177	9	m	m	AUX
ejpam-2634	177	10	be	be	VERB
ejpam-2634	177	11	a	a	DET
ejpam-2634	177	12	comultiplication	comultiplication	NOUN
ejpam-2634	177	13	module	module	NOUN
ejpam-2634	177	14	,	,	PUNCT
ejpam-2634	177	15	then	then	ADV
ejpam-2634	177	16	there	there	PRON
ejpam-2634	177	17	exist	exist	VERB
ejpam-2634	177	18	distinct	distinct	ADJ
ejpam-2634	177	19	maximal	maximal	ADJ
ejpam-2634	177	20	ideals	ideal	NOUN
ejpam-2634	178	1	pi	pi	NOUN
ejpam-2634	178	2	i∈i	i∈i	ADV
ejpam-2634	178	3	of	of	ADP
ejpam-2634	178	4	r	r	NOUN
ejpam-2634	178	5	and	and	CCONJ
ejpam-2634	178	6	submodules	submodule	NOUN
ejpam-2634	178	7	mi	mi	PROPN
ejpam-2634	178	8	,	,	PUNCT
ejpam-2634	178	9	i	i	PRON
ejpam-2634	178	10	∈	∈	VERB
ejpam-2634	178	11	i	i	PRON
ejpam-2634	178	12	of	of	ADP
ejpam-2634	178	13	m	m	PROPN
ejpam-2634	178	14	,	,	PUNCT
ejpam-2634	179	1	such	such	ADJ
ejpam-2634	179	2	that	that	SCONJ
ejpam-2634	179	3	m	m	NOUN
ejpam-2634	179	4	=	=	SYM
ejpam-2634	179	5	⊕	⊕	PROPN
ejpam-2634	179	6	i∈i	i∈i	ADJ
ejpam-2634	179	7	mi	mi	PROPN
ejpam-2634	179	8	and	and	CCONJ
ejpam-2634	179	9	for	for	ADP
ejpam-2634	179	10	each	each	DET
ejpam-2634	179	11	i	i	PRON
ejpam-2634	179	12	∈	∈	PROPN
ejpam-2634	180	1	i	i	PRON
ejpam-2634	180	2	,	,	PUNCT
ejpam-2634	180	3	mi	mi	PROPN
ejpam-2634	180	4	∼=	∼=	PROPN
ejpam-2634	180	5	e(r	e(r	NOUN
ejpam-2634	180	6	/	/	SYM
ejpam-2634	180	7	pi	pi	NOUN
ejpam-2634	180	8	)	)	PUNCT
ejpam-2634	180	9	or	or	CCONJ
ejpam-2634	180	10	mi	mi	NOUN
ejpam-2634	180	11	∼=	∼=	PROPN
ejpam-2634	180	12	r	r	NOUN
ejpam-2634	180	13	/	/	SYM
ejpam-2634	180	14	p	p	NOUN
ejpam-2634	180	15	ni	ni	NOUN
ejpam-2634	180	16	i	i	PROPN
ejpam-2634	180	17	,	,	PUNCT
ejpam-2634	180	18	for	for	ADP
ejpam-2634	180	19	some	some	DET
ejpam-2634	180	20	ni	ni	PROPN
ejpam-2634	180	21	∈	∈	PROPN
ejpam-2634	180	22	n.	n.	NOUN
ejpam-2634	180	23	references	reference	VERB
ejpam-2634	180	24	249	249	NUM
ejpam-2634	180	25	proof	proof	NOUN
ejpam-2634	180	26	.	.	PUNCT
ejpam-2634	181	1	let	let	VERB
ejpam-2634	181	2	r	r	PRON
ejpam-2634	181	3	be	be	AUX
ejpam-2634	181	4	a	a	DET
ejpam-2634	181	5	dedekind	dedekind	ADJ
ejpam-2634	181	6	domain	domain	NOUN
ejpam-2634	181	7	and	and	CCONJ
ejpam-2634	181	8	m	m	AUX
ejpam-2634	181	9	be	be	VERB
ejpam-2634	181	10	comultiplication	comultiplication	ADJ
ejpam-2634	181	11	,	,	PUNCT
ejpam-2634	181	12	so	so	CCONJ
ejpam-2634	181	13	r	r	NOUN
ejpam-2634	181	14	is	be	AUX
ejpam-2634	181	15	noetherian	noetherian	ADJ
ejpam-2634	181	16	and	and	CCONJ
ejpam-2634	181	17	m	m	NOUN
ejpam-2634	181	18	is	be	AUX
ejpam-2634	181	19	artinian	artinian	ADJ
ejpam-2634	181	20	by	by	ADP
ejpam-2634	181	21	[	[	X
ejpam-2634	181	22	4	4	NUM
ejpam-2634	181	23	]	]	PUNCT
ejpam-2634	181	24	.	.	PUNCT
ejpam-2634	182	1	set	set	VERB
ejpam-2634	182	2	m(p	m(p	PROPN
ejpam-2634	182	3	)	)	PUNCT
ejpam-2634	182	4	:	:	PUNCT
ejpam-2634	183	1	=	=	SYM
ejpam-2634	183	2	{	{	PUNCT
ejpam-2634	183	3	m	m	VERB
ejpam-2634	183	4	∈	∈	ADJ
ejpam-2634	183	5	m	m	VERB
ejpam-2634	183	6	|	|	ADV
ejpam-2634	184	1	∃n	∃n	INTJ
ejpam-2634	184	2	∈	∈	PROPN
ejpam-2634	184	3	n	n	NOUN
ejpam-2634	184	4	,	,	PUNCT
ejpam-2634	184	5	pnm=	pnm=	NOUN
ejpam-2634	184	6	0	0	NUM
ejpam-2634	184	7	}	}	PUNCT
ejpam-2634	184	8	for	for	ADP
ejpam-2634	184	9	p	p	PROPN
ejpam-2634	184	10	∈	∈	PROPN
ejpam-2634	184	11	spec(r	spec(r	PROPN
ejpam-2634	184	12	)	)	PUNCT
ejpam-2634	184	13	.	.	PUNCT
ejpam-2634	185	1	there	there	PRON
ejpam-2634	185	2	exist	exist	VERB
ejpam-2634	185	3	distinct	distinct	ADJ
ejpam-2634	185	4	maximal	maximal	ADJ
ejpam-2634	185	5	ideal	ideal	NOUN
ejpam-2634	185	6	{	{	PUNCT
ejpam-2634	185	7	pi}i∈i	pi}i∈i	INTJ
ejpam-2634	185	8	such	such	ADJ
ejpam-2634	185	9	that	that	SCONJ
ejpam-2634	185	10	m	m	NOUN
ejpam-2634	185	11	=	=	SYM
ejpam-2634	185	12	⊕	⊕	PROPN
ejpam-2634	185	13	i∈i	i∈i	ADJ
ejpam-2634	185	14	m(pi	m(pi	NOUN
ejpam-2634	185	15	)	)	PUNCT
ejpam-2634	185	16	.	.	PUNCT
ejpam-2634	186	1	let	let	VERB
ejpam-2634	186	2	mi	mi	PROPN
ejpam-2634	186	3	=	=	PUNCT
ejpam-2634	186	4	m(pi	m(pi	PROPN
ejpam-2634	186	5	)	)	PUNCT
ejpam-2634	186	6	,	,	PUNCT
ejpam-2634	186	7	since	since	SCONJ
ejpam-2634	186	8	m	m	PROPN
ejpam-2634	186	9	is	be	AUX
ejpam-2634	186	10	comultiplication	comultiplication	NOUN
ejpam-2634	186	11	,	,	PUNCT
ejpam-2634	186	12	it	it	PRON
ejpam-2634	186	13	follows	follow	VERB
ejpam-2634	186	14	that	that	SCONJ
ejpam-2634	186	15	each	each	DET
ejpam-2634	186	16	mi	mi	PROPN
ejpam-2634	186	17	is	be	AUX
ejpam-2634	186	18	also	also	ADV
ejpam-2634	186	19	comultiplication	comultiplication	NOUN
ejpam-2634	186	20	.	.	PUNCT
ejpam-2634	187	1	on	on	ADP
ejpam-2634	187	2	the	the	DET
ejpam-2634	187	3	other	other	ADJ
ejpam-2634	187	4	hand	hand	NOUN
ejpam-2634	187	5	each	each	DET
ejpam-2634	187	6	mi	mi	PROPN
ejpam-2634	187	7	is	be	AUX
ejpam-2634	187	8	an	an	DET
ejpam-2634	187	9	rpi	rpi	ADJ
ejpam-2634	187	10	-module	-module	NOUN
ejpam-2634	187	11	.	.	PUNCT
ejpam-2634	188	1	so	so	ADV
ejpam-2634	188	2	by	by	ADP
ejpam-2634	188	3	theorem	theorem	NOUN
ejpam-2634	188	4	8	8	NUM
ejpam-2634	188	5	for	for	ADP
ejpam-2634	188	6	each	each	DET
ejpam-2634	188	7	i	i	PRON
ejpam-2634	188	8	∈	∈	PROPN
ejpam-2634	189	1	i	i	PRON
ejpam-2634	189	2	,	,	PUNCT
ejpam-2634	189	3	mi	mi	PROPN
ejpam-2634	189	4	∼=	∼=	PROPN
ejpam-2634	189	5	e(rpi	e(rpi	PROPN
ejpam-2634	189	6	/pirpi	/pirpi	PUNCT
ejpam-2634	189	7	)	)	PUNCT
ejpam-2634	189	8	or	or	CCONJ
ejpam-2634	189	9	mi	mi	NOUN
ejpam-2634	189	10	∼=	∼=	PROPN
ejpam-2634	189	11	rpi	rpi	PROPN
ejpam-2634	189	12	/	/	SYM
ejpam-2634	189	13	pir	pir	PROPN
ejpam-2634	189	14	ni	ni	PROPN
ejpam-2634	189	15	pi	pi	PROPN
ejpam-2634	189	16	,	,	PUNCT
ejpam-2634	189	17	since	since	SCONJ
ejpam-2634	189	18	pi	pi	PROPN
ejpam-2634	189	19	⋂	⋂	PROPN
ejpam-2634	189	20	(	(	PUNCT
ejpam-2634	189	21	r	r	NOUN
ejpam-2634	189	22	\	\	PROPN
ejpam-2634	189	23	pi	pi	NOUN
ejpam-2634	189	24	)	)	PUNCT
ejpam-2634	189	25	=	=	PUNCT
ejpam-2634	189	26	;	;	PUNCT
ejpam-2634	189	27	,	,	PUNCT
ejpam-2634	189	28	it	it	PRON
ejpam-2634	189	29	follows	follow	VERB
ejpam-2634	189	30	that	that	DET
ejpam-2634	189	31	e(rpi	e(rpi	NOUN
ejpam-2634	189	32	/pirpi	/pirpi	PUNCT
ejpam-2634	189	33	)	)	PUNCT
ejpam-2634	189	34	∼=	∼=	NOUN
ejpam-2634	189	35	e(r	e(r	NOUN
ejpam-2634	189	36	/	/	SYM
ejpam-2634	189	37	pi	pi	NOUN
ejpam-2634	189	38	)	)	PUNCT
ejpam-2634	189	39	,	,	PUNCT
ejpam-2634	189	40	also	also	ADV
ejpam-2634	189	41	rpi	rpi	VERB
ejpam-2634	189	42	/pir	/pir	PUNCT
ejpam-2634	189	43	ni	ni	PROPN
ejpam-2634	189	44	pi	pi	NOUN
ejpam-2634	189	45	∼=	∼=	PROPN
ejpam-2634	189	46	r	r	NOUN
ejpam-2634	189	47	/	/	SYM
ejpam-2634	189	48	p	p	NOUN
ejpam-2634	189	49	ni	ni	PROPN
ejpam-2634	189	50	i	i	PROPN
ejpam-2634	189	51	.	.	PUNCT
ejpam-2634	190	1	theorem	theorem	ADJ
ejpam-2634	190	2	9	9	NUM
ejpam-2634	190	3	.	.	PUNCT
ejpam-2634	191	1	let	let	VERB
ejpam-2634	191	2	r	r	NOUN
ejpam-2634	191	3	⊆	⊆	NUM
ejpam-2634	191	4	r	r	NOUN
ejpam-2634	191	5	be	be	VERB
ejpam-2634	191	6	an	an	DET
ejpam-2634	191	7	integral	integral	ADJ
ejpam-2634	191	8	extension	extension	NOUN
ejpam-2634	191	9	and	and	CCONJ
ejpam-2634	191	10	r	r	NOUN
ejpam-2634	191	11	be	be	AUX
ejpam-2634	191	12	weak	weak	ADJ
ejpam-2634	191	13	comultiplication	comultiplication	NOUN
ejpam-2634	191	14	,	,	PUNCT
ejpam-2634	191	15	then	then	ADV
ejpam-2634	191	16	r	r	NOUN
ejpam-2634	191	17	is	be	AUX
ejpam-2634	191	18	weak	weak	ADJ
ejpam-2634	191	19	comultiplication	comultiplication	NOUN
ejpam-2634	191	20	.	.	PUNCT
ejpam-2634	192	1	proof	proof	NOUN
ejpam-2634	192	2	.	.	PUNCT
ejpam-2634	193	1	let	let	VERB
ejpam-2634	193	2	p	p	PROPN
ejpam-2634	193	3	∈	∈	PROPN
ejpam-2634	193	4	spec(r	spec(r	PROPN
ejpam-2634	193	5	)	)	PUNCT
ejpam-2634	193	6	so	so	ADV
ejpam-2634	193	7	there	there	PRON
ejpam-2634	193	8	exists	exist	VERB
ejpam-2634	193	9	a	a	DET
ejpam-2634	193	10	prime	prime	ADJ
ejpam-2634	193	11	ideal	ideal	NOUN
ejpam-2634	193	12	q	q	NOUN
ejpam-2634	193	13	of	of	ADP
ejpam-2634	193	14	r	r	NOUN
ejpam-2634	194	1	such	such	ADJ
ejpam-2634	194	2	that	that	SCONJ
ejpam-2634	194	3	p	p	PROPN
ejpam-2634	194	4	=	=	X
ejpam-2634	194	5	qc	qc	PROPN
ejpam-2634	195	1	so	so	ADV
ejpam-2634	195	2	p	p	PROPN
ejpam-2634	195	3	=	=	X
ejpam-2634	195	4	qc	qc	PROPN
ejpam-2634	195	5	=	=	SYM
ejpam-2634	195	6	(	(	PUNCT
ejpam-2634	195	7	0	0	NUM
ejpam-2634	195	8	:	:	PUNCT
ejpam-2634	195	9	r	r	NOUN
ejpam-2634	195	10	annr(q	annr(q	NOUN
ejpam-2634	195	11	)	)	PUNCT
ejpam-2634	195	12	)	)	PUNCT
ejpam-2634	196	1	c	c	PROPN
ejpam-2634	196	2	⊇	⊇	X
ejpam-2634	196	3	(	(	PUNCT
ejpam-2634	196	4	0	0	NUM
ejpam-2634	196	5	:	:	PUNCT
ejpam-2634	196	6	r	r	NOUN
ejpam-2634	196	7	annr(q	annr(q	NOUN
ejpam-2634	196	8	c	c	NOUN
ejpam-2634	196	9	)	)	PUNCT
ejpam-2634	196	10	)	)	PUNCT
ejpam-2634	197	1	=	=	PUNCT
ejpam-2634	197	2	(	(	PUNCT
ejpam-2634	197	3	0	0	NUM
ejpam-2634	197	4	:	:	PUNCT
ejpam-2634	197	5	r	r	NOUN
ejpam-2634	197	6	annr(p	annr(p	NOUN
ejpam-2634	197	7	)	)	PUNCT
ejpam-2634	197	8	)	)	PUNCT
ejpam-2634	197	9	.	.	PUNCT
ejpam-2634	198	1	(	(	PUNCT
ejpam-2634	198	2	3	3	X
ejpam-2634	198	3	)	)	PUNCT
ejpam-2634	198	4	references	reference	NOUN
ejpam-2634	198	5	[	[	X
ejpam-2634	198	6	1	1	NUM
ejpam-2634	198	7	]	]	X
ejpam-2634	198	8	y	y	PROPN
ejpam-2634	198	9	al	al	PROPN
ejpam-2634	198	10	-	-	PUNCT
ejpam-2634	198	11	shaniafi	shaniafi	PROPN
ejpam-2634	198	12	and	and	CCONJ
ejpam-2634	198	13	p	p	PROPN
ejpam-2634	198	14	f	f	PROPN
ejpam-2634	198	15	smith	smith	PROPN
ejpam-2634	198	16	.	.	PUNCT
ejpam-2634	199	1	comultiplication	comultiplication	NOUN
ejpam-2634	199	2	modules	module	NOUN
ejpam-2634	199	3	over	over	ADP
ejpam-2634	199	4	commutative	commutative	ADJ
ejpam-2634	199	5	rings	ring	NOUN
ejpam-2634	199	6	.	.	PUNCT
ejpam-2634	200	1	journal	journal	PROPN
ejpam-2634	200	2	of	of	ADP
ejpam-2634	200	3	commutative	commutative	ADJ
ejpam-2634	200	4	algebra	algebra	NOUN
ejpam-2634	200	5	,	,	PUNCT
ejpam-2634	200	6	3:1–29	3:1–29	NUM
ejpam-2634	200	7	,	,	PUNCT
ejpam-2634	200	8	2011	2011	NUM
ejpam-2634	200	9	.	.	PUNCT
ejpam-2634	201	1	[	[	X
ejpam-2634	201	2	2	2	NUM
ejpam-2634	201	3	]	]	PUNCT
ejpam-2634	201	4	h	h	NOUN
ejpam-2634	201	5	ansari	ansari	ADJ
ejpam-2634	201	6	-	-	PUNCT
ejpam-2634	201	7	toroghy	toroghy	NOUN
ejpam-2634	201	8	and	and	CCONJ
ejpam-2634	201	9	f	f	PROPN
ejpam-2634	201	10	farshadifar	farshadifar	ADV
ejpam-2634	201	11	.	.	PUNCT
ejpam-2634	202	1	the	the	DET
ejpam-2634	202	2	dual	dual	ADJ
ejpam-2634	202	3	notion	notion	NOUN
ejpam-2634	202	4	of	of	ADP
ejpam-2634	202	5	multipication	multipication	NOUN
ejpam-2634	202	6	modules	module	NOUN
ejpam-2634	202	7	.	.	PUNCT
ejpam-2634	203	1	taiwanese	taiwanese	ADJ
ejpam-2634	203	2	journal	journal	NOUN
ejpam-2634	203	3	of	of	ADP
ejpam-2634	203	4	mathematics	mathematic	NOUN
ejpam-2634	203	5	,	,	PUNCT
ejpam-2634	203	6	11:1189–1201	11:1189–1201	NUM
ejpam-2634	203	7	,	,	PUNCT
ejpam-2634	203	8	2007	2007	NUM
ejpam-2634	203	9	.	.	PUNCT
ejpam-2634	204	1	[	[	X
ejpam-2634	204	2	3	3	X
ejpam-2634	204	3	]	]	X
ejpam-2634	204	4	h	h	NOUN
ejpam-2634	204	5	ansari	ansari	ADJ
ejpam-2634	204	6	-	-	PUNCT
ejpam-2634	204	7	toroghy	toroghy	NOUN
ejpam-2634	204	8	and	and	CCONJ
ejpam-2634	204	9	f	f	PROPN
ejpam-2634	204	10	farshadifar	farshadifar	PROPN
ejpam-2634	204	11	.	.	PUNCT
ejpam-2634	205	1	comultiplication	comultiplication	NOUN
ejpam-2634	205	2	modules	module	NOUN
ejpam-2634	205	3	and	and	CCONJ
ejpam-2634	205	4	related	related	ADJ
ejpam-2634	205	5	results	result	NOUN
ejpam-2634	205	6	.	.	PUNCT
ejpam-2634	206	1	honam	honam	PROPN
ejpam-2634	206	2	mathematical	mathematical	PROPN
ejpam-2634	206	3	journal	journal	PROPN
ejpam-2634	206	4	,	,	PUNCT
ejpam-2634	206	5	30:91–99	30:91–99	NUM
ejpam-2634	206	6	,	,	PUNCT
ejpam-2634	206	7	2008	2008	NUM
ejpam-2634	206	8	.	.	PUNCT
ejpam-2634	207	1	[	[	X
ejpam-2634	207	2	4	4	NUM
ejpam-2634	207	3	]	]	X
ejpam-2634	207	4	h	h	NOUN
ejpam-2634	207	5	ansari	ansari	ADJ
ejpam-2634	207	6	-	-	PUNCT
ejpam-2634	207	7	toroghy	toroghy	NOUN
ejpam-2634	207	8	and	and	CCONJ
ejpam-2634	207	9	f	f	PROPN
ejpam-2634	207	10	farshadifar	farshadifar	ADV
ejpam-2634	207	11	.	.	PUNCT
ejpam-2634	208	1	on	on	ADP
ejpam-2634	208	2	comultiplication	comultiplication	NOUN
ejpam-2634	208	3	modules	module	NOUN
ejpam-2634	208	4	.	.	PUNCT
ejpam-2634	209	1	korean	korean	PROPN
ejpam-2634	209	2	ann	ann	PROPN
ejpam-2634	209	3	math	math	PROPN
ejpam-2634	209	4	,	,	PUNCT
ejpam-2634	209	5	25:57–66	25:57–66	NUM
ejpam-2634	209	6	,	,	PUNCT
ejpam-2634	209	7	2008	2008	NUM
ejpam-2634	209	8	.	.	PUNCT
ejpam-2634	210	1	[	[	X
ejpam-2634	210	2	5	5	NUM
ejpam-2634	210	3	]	]	PUNCT
ejpam-2634	210	4	h	h	NOUN
ejpam-2634	210	5	ansari	ansari	ADJ
ejpam-2634	210	6	-	-	PUNCT
ejpam-2634	210	7	toroghy	toroghy	NOUN
ejpam-2634	210	8	and	and	CCONJ
ejpam-2634	210	9	f	f	PROPN
ejpam-2634	210	10	farshadifar	farshadifar	PROPN
ejpam-2634	210	11	.	.	PUNCT
ejpam-2634	210	12	multiplication	multiplication	NOUN
ejpam-2634	210	13	and	and	CCONJ
ejpam-2634	210	14	comultiplication	comultiplication	NOUN
ejpam-2634	210	15	modules	module	NOUN
ejpam-2634	210	16	.	.	PUNCT
ejpam-2634	211	1	novi	novi	PROPN
ejpam-2634	211	2	sad	sad	PROPN
ejpam-2634	211	3	journal	journal	PROPN
ejpam-2634	211	4	math	math	PROPN
ejpam-2634	211	5	,	,	PUNCT
ejpam-2634	211	6	41:117–122	41:117–122	PROPN
ejpam-2634	211	7	,	,	PUNCT
ejpam-2634	211	8	2011	2011	NUM
ejpam-2634	211	9	.	.	PUNCT
ejpam-2634	212	1	[	[	X
ejpam-2634	212	2	6	6	NUM
ejpam-2634	212	3	]	]	PUNCT
ejpam-2634	212	4	m	m	VERB
ejpam-2634	212	5	p	p	NOUN
ejpam-2634	212	6	brodmann	brodmann	NOUN
ejpam-2634	212	7	and	and	CCONJ
ejpam-2634	212	8	r	r	NOUN
ejpam-2634	212	9	y	y	PROPN
ejpam-2634	212	10	sharp	sharp	ADJ
ejpam-2634	212	11	.	.	PUNCT
ejpam-2634	213	1	local	local	ADJ
ejpam-2634	213	2	cohomology	cohomology	NOUN
ejpam-2634	213	3	;	;	PUNCT
ejpam-2634	213	4	an	an	DET
ejpam-2634	213	5	algebraic	algebraic	ADJ
ejpam-2634	213	6	introduction	introduction	NOUN
ejpam-2634	213	7	with	with	ADP
ejpam-2634	213	8	geometric	geometric	ADJ
ejpam-2634	213	9	applications	application	NOUN
ejpam-2634	213	10	.	.	PUNCT
ejpam-2634	214	1	cambridge	cambridge	PROPN
ejpam-2634	214	2	university	university	PROPN
ejpam-2634	214	3	press	press	PROPN
ejpam-2634	214	4	,	,	PUNCT
ejpam-2634	214	5	cambridge	cambridge	PROPN
ejpam-2634	214	6	,	,	PUNCT
ejpam-2634	214	7	1998	1998	NUM
ejpam-2634	214	8	.	.	PUNCT