id	sid	tid	token	lemma	pos
ejpam-4215	1	1	european	european	PROPN
ejpam-4215	1	2	journal	journal	PROPN
ejpam-4215	1	3	of	of	ADP
ejpam-4215	1	4	pure	pure	ADJ
ejpam-4215	1	5	and	and	CCONJ
ejpam-4215	1	6	applied	apply	VERB
ejpam-4215	1	7	mathematics	mathematic	NOUN
ejpam-4215	1	8	vol	vol	NOUN
ejpam-4215	1	9	.	.	PROPN
ejpam-4215	2	1	15	15	NUM
ejpam-4215	2	2	,	,	PUNCT
ejpam-4215	2	3	no	no	INTJ
ejpam-4215	2	4	.	.	NOUN
ejpam-4215	2	5	1	1	NUM
ejpam-4215	2	6	,	,	PUNCT
ejpam-4215	2	7	2022	2022	NUM
ejpam-4215	2	8	,	,	PUNCT
ejpam-4215	2	9	224	224	NUM
ejpam-4215	2	10	-	-	SYM
ejpam-4215	2	11	228	228	NUM
ejpam-4215	2	12	issn	issn	PROPN
ejpam-4215	2	13	1307	1307	NUM
ejpam-4215	2	14	-	-	SYM
ejpam-4215	2	15	5543	5543	NUM
ejpam-4215	2	16	–	–	PUNCT
ejpam-4215	2	17	ejpam.com	ejpam.com	X
ejpam-4215	2	18	published	publish	VERB
ejpam-4215	2	19	by	by	ADP
ejpam-4215	2	20	new	new	PROPN
ejpam-4215	2	21	york	york	PROPN
ejpam-4215	2	22	business	business	PROPN
ejpam-4215	2	23	global	global	PROPN
ejpam-4215	2	24	e∗-essential	e∗-essential	PROPN
ejpam-4215	2	25	submodule	submodule	PROPN
ejpam-4215	2	26	hiba	hiba	PROPN
ejpam-4215	2	27	r.	r.	PROPN
ejpam-4215	2	28	baanoon1,∗	baanoon1,∗	PROPN
ejpam-4215	2	29	,	,	PUNCT
ejpam-4215	2	30	wasan	wasan	PROPN
ejpam-4215	2	31	khalid	khalid	PROPN
ejpam-4215	2	32	2	2	NUM
ejpam-4215	2	33	college	college	NOUN
ejpam-4215	2	34	of	of	ADP
ejpam-4215	2	35	science	science	NOUN
ejpam-4215	2	36	,	,	PUNCT
ejpam-4215	2	37	university	university	NOUN
ejpam-4215	2	38	of	of	ADP
ejpam-4215	2	39	baghdad	baghdad	PROPN
ejpam-4215	2	40	,	,	PUNCT
ejpam-4215	2	41	baghdad	baghdad	PROPN
ejpam-4215	2	42	,	,	PUNCT
ejpam-4215	2	43	iraq	iraq	PROPN
ejpam-4215	2	44	abstract	abstract	NOUN
ejpam-4215	2	45	.	.	PUNCT
ejpam-4215	3	1	the	the	DET
ejpam-4215	3	2	purpose	purpose	NOUN
ejpam-4215	3	3	of	of	ADP
ejpam-4215	3	4	this	this	DET
ejpam-4215	3	5	paper	paper	NOUN
ejpam-4215	3	6	is	be	AUX
ejpam-4215	3	7	to	to	PART
ejpam-4215	3	8	introduce	introduce	VERB
ejpam-4215	3	9	a	a	DET
ejpam-4215	3	10	new	new	ADJ
ejpam-4215	3	11	concept	concept	NOUN
ejpam-4215	3	12	in	in	ADP
ejpam-4215	3	13	a	a	DET
ejpam-4215	3	14	module	module	NOUN
ejpam-4215	3	15	m	m	NOUN
ejpam-4215	3	16	over	over	ADP
ejpam-4215	3	17	a	a	DET
ejpam-4215	3	18	ring	ring	NOUN
ejpam-4215	3	19	r	r	NOUN
ejpam-4215	3	20	,	,	PUNCT
ejpam-4215	3	21	this	this	DET
ejpam-4215	3	22	concept	concept	NOUN
ejpam-4215	3	23	is	be	AUX
ejpam-4215	3	24	called	call	VERB
ejpam-4215	3	25	e∗-essential	e∗-essential	PROPN
ejpam-4215	3	26	submodule	submodule	NOUN
ejpam-4215	3	27	,	,	PUNCT
ejpam-4215	3	28	which	which	PRON
ejpam-4215	3	29	is	be	AUX
ejpam-4215	3	30	a	a	DET
ejpam-4215	3	31	generalization	generalization	NOUN
ejpam-4215	3	32	of	of	ADP
ejpam-4215	3	33	an	an	DET
ejpam-4215	3	34	essential	essential	ADJ
ejpam-4215	3	35	submodule	submodule	NOUN
ejpam-4215	3	36	.	.	PUNCT
ejpam-4215	4	1	we	we	PRON
ejpam-4215	4	2	will	will	AUX
ejpam-4215	4	3	introduce	introduce	VERB
ejpam-4215	4	4	some	some	DET
ejpam-4215	4	5	examples	example	NOUN
ejpam-4215	4	6	and	and	CCONJ
ejpam-4215	4	7	properties	property	NOUN
ejpam-4215	4	8	about	about	ADP
ejpam-4215	4	9	this	this	DET
ejpam-4215	4	10	concept	concept	NOUN
ejpam-4215	4	11	such	such	ADJ
ejpam-4215	4	12	that	that	SCONJ
ejpam-4215	4	13	,	,	PUNCT
ejpam-4215	4	14	what	what	PRON
ejpam-4215	4	15	is	be	AUX
ejpam-4215	4	16	the	the	DET
ejpam-4215	4	17	inverse	inverse	ADJ
ejpam-4215	4	18	image	image	NOUN
ejpam-4215	4	19	of	of	ADP
ejpam-4215	4	20	e∗-essential	e∗-essential	PROPN
ejpam-4215	4	21	submodule	submodule	NOUN
ejpam-4215	4	22	,	,	PUNCT
ejpam-4215	4	23	the	the	DET
ejpam-4215	4	24	intersection	intersection	NOUN
ejpam-4215	4	25	of	of	ADP
ejpam-4215	4	26	e∗-essential	e∗-essential	PROPN
ejpam-4215	4	27	submodules	submodule	NOUN
ejpam-4215	4	28	and	and	CCONJ
ejpam-4215	4	29	direct	direct	ADJ
ejpam-4215	4	30	sum	sum	NOUN
ejpam-4215	4	31	of	of	ADP
ejpam-4215	4	32	e∗-essential	e∗-essential	ADJ
ejpam-4215	4	33	submodules	submodule	NOUN
ejpam-4215	4	34	.	.	PUNCT
ejpam-4215	5	1	we	we	PRON
ejpam-4215	5	2	will	will	AUX
ejpam-4215	5	3	show	show	VERB
ejpam-4215	5	4	the	the	DET
ejpam-4215	5	5	relationship	relationship	NOUN
ejpam-4215	5	6	between	between	ADP
ejpam-4215	5	7	e∗-essential	e∗-essential	PROPN
ejpam-4215	5	8	submodule	submodule	NOUN
ejpam-4215	5	9	and	and	CCONJ
ejpam-4215	5	10	noetherian	noetherian	ADJ
ejpam-4215	5	11	r	r	NOUN
ejpam-4215	5	12	-	-	PUNCT
ejpam-4215	5	13	module	module	NOUN
ejpam-4215	5	14	.	.	PUNCT
ejpam-4215	6	1	also	also	ADV
ejpam-4215	6	2	we	we	PRON
ejpam-4215	6	3	will	will	AUX
ejpam-4215	6	4	define	define	VERB
ejpam-4215	6	5	e∗-closed	e∗-close	VERB
ejpam-4215	6	6	submodule	submodule	NOUN
ejpam-4215	6	7	with	with	ADP
ejpam-4215	6	8	some	some	DET
ejpam-4215	6	9	properties	property	NOUN
ejpam-4215	6	10	2020	2020	NUM
ejpam-4215	6	11	mathematics	mathematic	NOUN
ejpam-4215	6	12	subject	subject	NOUN
ejpam-4215	6	13	classifications	classification	NOUN
ejpam-4215	6	14	:	:	PUNCT
ejpam-4215	6	15	16d90	16d90	NUM
ejpam-4215	6	16	,	,	PUNCT
ejpam-4215	6	17	16d99	16d99	NUM
ejpam-4215	6	18	,	,	PUNCT
ejpam-4215	6	19	16p40	16p40	NUM
ejpam-4215	6	20	key	key	ADJ
ejpam-4215	6	21	words	word	NOUN
ejpam-4215	6	22	and	and	CCONJ
ejpam-4215	6	23	phrases	phrase	NOUN
ejpam-4215	6	24	:	:	PUNCT
ejpam-4215	6	25	essential	essential	ADJ
ejpam-4215	6	26	submodule	submodule	NOUN
ejpam-4215	6	27	,	,	PUNCT
ejpam-4215	6	28	small	small	ADJ
ejpam-4215	6	29	submodule	submodule	NOUN
ejpam-4215	6	30	,	,	PUNCT
ejpam-4215	6	31	e∗-essential	e∗-essential	PROPN
ejpam-4215	6	32	submodule	submodule	NOUN
ejpam-4215	6	33	,	,	PUNCT
ejpam-4215	6	34	e∗closed	e∗close	VERB
ejpam-4215	6	35	submodule	submodule	NOUN
ejpam-4215	6	36	,	,	PUNCT
ejpam-4215	6	37	noetherian	noetherian	ADJ
ejpam-4215	6	38	r	r	NOUN
ejpam-4215	6	39	-	-	PUNCT
ejpam-4215	6	40	module	module	NOUN
ejpam-4215	6	41	1	1	NUM
ejpam-4215	6	42	.	.	PUNCT
ejpam-4215	7	1	introduction	introduction	NOUN
ejpam-4215	7	2	let	let	VERB
ejpam-4215	7	3	r	r	PRON
ejpam-4215	7	4	be	be	AUX
ejpam-4215	7	5	a	a	DET
ejpam-4215	7	6	ring	ring	NOUN
ejpam-4215	7	7	with	with	ADP
ejpam-4215	7	8	identity	identity	NOUN
ejpam-4215	7	9	,	,	PUNCT
ejpam-4215	7	10	m	m	VERB
ejpam-4215	7	11	be	be	VERB
ejpam-4215	7	12	a	a	DET
ejpam-4215	7	13	right	right	ADJ
ejpam-4215	7	14	r	r	NOUN
ejpam-4215	7	15	-	-	PUNCT
ejpam-4215	7	16	module	module	NOUN
ejpam-4215	7	17	and	and	CCONJ
ejpam-4215	7	18	e(m	e(m	PROPN
ejpam-4215	7	19	)	)	PUNCT
ejpam-4215	7	20	be	be	VERB
ejpam-4215	7	21	the	the	DET
ejpam-4215	7	22	injective	injective	ADJ
ejpam-4215	7	23	hull	hull	NOUN
ejpam-4215	7	24	of	of	ADP
ejpam-4215	7	25	m	m	PROPN
ejpam-4215	7	26	.	.	PUNCT
ejpam-4215	8	1	a	a	DET
ejpam-4215	8	2	submodule	submodule	PROPN
ejpam-4215	8	3	n	n	PROPN
ejpam-4215	8	4	of	of	ADP
ejpam-4215	8	5	an	an	DET
ejpam-4215	8	6	r	r	NOUN
ejpam-4215	8	7	-	-	PUNCT
ejpam-4215	8	8	module	module	NOUN
ejpam-4215	8	9	m	m	NOUN
ejpam-4215	8	10	is	be	AUX
ejpam-4215	8	11	called	call	VERB
ejpam-4215	8	12	a	a	DET
ejpam-4215	8	13	small	small	ADJ
ejpam-4215	8	14	submodule	submodule	NOUN
ejpam-4215	8	15	of	of	ADP
ejpam-4215	8	16	m	m	PROPN
ejpam-4215	8	17	(	(	PUNCT
ejpam-4215	8	18	n	n	CCONJ
ejpam-4215	8	19	≪	≪	ADJ
ejpam-4215	8	20	m	m	NOUN
ejpam-4215	8	21	)	)	PUNCT
ejpam-4215	8	22	if	if	SCONJ
ejpam-4215	8	23	for	for	ADP
ejpam-4215	8	24	any	any	DET
ejpam-4215	8	25	submodule	submodule	NOUN
ejpam-4215	8	26	a	a	PRON
ejpam-4215	8	27	of	of	ADP
ejpam-4215	8	28	m	m	PRON
ejpam-4215	8	29	such	such	ADJ
ejpam-4215	8	30	that	that	SCONJ
ejpam-4215	8	31	m	m	VERB
ejpam-4215	8	32	=	=	SYM
ejpam-4215	8	33	n	n	PROPN
ejpam-4215	8	34	+	+	CCONJ
ejpam-4215	8	35	a	a	X
ejpam-4215	8	36	,	,	PUNCT
ejpam-4215	8	37	then	then	ADV
ejpam-4215	8	38	a	a	DET
ejpam-4215	8	39	=	=	NOUN
ejpam-4215	8	40	m	m	NOUN
ejpam-4215	8	41	[	[	X
ejpam-4215	8	42	5	5	NUM
ejpam-4215	8	43	]	]	PUNCT
ejpam-4215	8	44	.	.	PUNCT
ejpam-4215	9	1	leonard	leonard	PROPN
ejpam-4215	9	2	defines	define	VERB
ejpam-4215	9	3	a	a	DET
ejpam-4215	9	4	module	module	NOUN
ejpam-4215	9	5	m	m	PRON
ejpam-4215	9	6	to	to	PART
ejpam-4215	9	7	be	be	AUX
ejpam-4215	9	8	small	small	ADJ
ejpam-4215	9	9	if	if	SCONJ
ejpam-4215	9	10	it	it	PRON
ejpam-4215	9	11	is	be	AUX
ejpam-4215	9	12	a	a	DET
ejpam-4215	9	13	small	small	ADJ
ejpam-4215	9	14	submodule	submodule	NOUN
ejpam-4215	9	15	of	of	ADP
ejpam-4215	9	16	some	some	DET
ejpam-4215	9	17	r	r	NOUN
ejpam-4215	9	18	-	-	PUNCT
ejpam-4215	9	19	module	module	NOUN
ejpam-4215	9	20	and	and	CCONJ
ejpam-4215	9	21	he	he	PRON
ejpam-4215	9	22	shows	show	VERB
ejpam-4215	9	23	that	that	SCONJ
ejpam-4215	9	24	m	m	NOUN
ejpam-4215	9	25	is	be	AUX
ejpam-4215	9	26	small	small	ADJ
ejpam-4215	9	27	if	if	SCONJ
ejpam-4215	9	28	and	and	CCONJ
ejpam-4215	9	29	only	only	ADV
ejpam-4215	9	30	if	if	SCONJ
ejpam-4215	9	31	m	m	NOUN
ejpam-4215	9	32	is	be	AUX
ejpam-4215	9	33	small	small	ADJ
ejpam-4215	9	34	in	in	ADP
ejpam-4215	9	35	its	its	PRON
ejpam-4215	9	36	injective	injective	ADJ
ejpam-4215	9	37	hull	hull	NOUN
ejpam-4215	9	38	[	[	X
ejpam-4215	9	39	1	1	NUM
ejpam-4215	9	40	]	]	PUNCT
ejpam-4215	9	41	.	.	PUNCT
ejpam-4215	10	1	recall	recall	VERB
ejpam-4215	10	2	that	that	SCONJ
ejpam-4215	10	3	a	a	DET
ejpam-4215	10	4	submodule	submodule	NOUN
ejpam-4215	10	5	a	a	PRON
ejpam-4215	10	6	of	of	ADP
ejpam-4215	10	7	r	r	NOUN
ejpam-4215	10	8	-	-	PUNCT
ejpam-4215	10	9	module	module	NOUN
ejpam-4215	10	10	b	b	NOUN
ejpam-4215	10	11	is	be	AUX
ejpam-4215	10	12	called	call	VERB
ejpam-4215	10	13	essential	essential	ADJ
ejpam-4215	10	14	in	in	ADP
ejpam-4215	10	15	b	b	NOUN
ejpam-4215	10	16	if	if	SCONJ
ejpam-4215	10	17	every	every	DET
ejpam-4215	10	18	nonzero	nonzero	PROPN
ejpam-4215	10	19	submodule	submodule	PROPN
ejpam-4215	10	20	of	of	ADP
ejpam-4215	10	21	b	b	PROPN
ejpam-4215	10	22	has	have	VERB
ejpam-4215	10	23	nonzero	nonzero	ADJ
ejpam-4215	10	24	intersection	intersection	NOUN
ejpam-4215	10	25	with	with	ADP
ejpam-4215	10	26	a	a	DET
ejpam-4215	10	27	[	[	X
ejpam-4215	10	28	5	5	NUM
ejpam-4215	10	29	]	]	PUNCT
ejpam-4215	10	30	,	,	PUNCT
ejpam-4215	10	31	[	[	X
ejpam-4215	10	32	3	3	NUM
ejpam-4215	10	33	]	]	PUNCT
ejpam-4215	10	34	and	and	CCONJ
ejpam-4215	10	35	[	[	X
ejpam-4215	10	36	4	4	NUM
ejpam-4215	10	37	]	]	PUNCT
ejpam-4215	10	38	.	.	PUNCT
ejpam-4215	11	1	oscan	oscan	PROPN
ejpam-4215	11	2	in	in	ADP
ejpam-4215	11	3	[	[	X
ejpam-4215	11	4	2	2	NUM
ejpam-4215	11	5	]	]	PUNCT
ejpam-4215	11	6	,	,	PUNCT
ejpam-4215	11	7	introduced	introduce	VERB
ejpam-4215	11	8	the	the	DET
ejpam-4215	11	9	concept	concept	NOUN
ejpam-4215	11	10	of	of	ADP
ejpam-4215	11	11	cosingular	cosingular	ADJ
ejpam-4215	11	12	submodule	submodule	NOUN
ejpam-4215	11	13	as	as	ADP
ejpam-4215	11	14	the	the	DET
ejpam-4215	11	15	following	following	NOUN
ejpam-4215	11	16	:	:	PUNCT
ejpam-4215	11	17	z∗(m	z∗(m	X
ejpam-4215	11	18	)	)	PUNCT
ejpam-4215	11	19	=	=	PRON
ejpam-4215	11	20	{	{	PUNCT
ejpam-4215	12	1	m	m	VERB
ejpam-4215	12	2	∈	∈	ADJ
ejpam-4215	12	3	m	m	VERB
ejpam-4215	12	4	|mr	|mr	PRON
ejpam-4215	12	5	≪	≪	ADJ
ejpam-4215	12	6	e(m	e(m	NOUN
ejpam-4215	12	7	)	)	PUNCT
ejpam-4215	12	8	}	}	PUNCT
ejpam-4215	12	9	.	.	PUNCT
ejpam-4215	13	1	an	an	DET
ejpam-4215	13	2	r	r	NOUN
ejpam-4215	13	3	-	-	PUNCT
ejpam-4215	13	4	module	module	NOUN
ejpam-4215	13	5	m	m	NOUN
ejpam-4215	13	6	is	be	AUX
ejpam-4215	13	7	called	call	VERB
ejpam-4215	13	8	cosingular	cosingular	ADJ
ejpam-4215	13	9	z∗(m	z∗(m	PROPN
ejpam-4215	13	10	)	)	PUNCT
ejpam-4215	14	1	=	=	PUNCT
ejpam-4215	15	1	m	m	PROPN
ejpam-4215	15	2	.	.	PUNCT
ejpam-4215	16	1	as	as	SCONJ
ejpam-4215	16	2	in	in	ADP
ejpam-4215	16	3	[	[	X
ejpam-4215	16	4	6	6	NUM
ejpam-4215	16	5	]	]	PUNCT
ejpam-4215	16	6	,	,	PUNCT
ejpam-4215	16	7	we	we	PRON
ejpam-4215	16	8	will	will	AUX
ejpam-4215	16	9	used	use	VERB
ejpam-4215	16	10	the	the	DET
ejpam-4215	16	11	oscan	oscan	NOUN
ejpam-4215	16	12	presented	present	VERB
ejpam-4215	16	13	to	to	PART
ejpam-4215	16	14	generalize	generalize	VERB
ejpam-4215	16	15	the	the	DET
ejpam-4215	16	16	essential	essential	ADJ
ejpam-4215	16	17	submodule	submodule	NOUN
ejpam-4215	16	18	,	,	PUNCT
ejpam-4215	16	19	to	to	PART
ejpam-4215	16	20	introduce	introduce	VERB
ejpam-4215	16	21	the	the	DET
ejpam-4215	16	22	concepte	concepte	NOUN
ejpam-4215	16	23	e∗-essential	e∗-essential	PROPN
ejpam-4215	16	24	and	and	CCONJ
ejpam-4215	16	25	investigate	investigate	VERB
ejpam-4215	16	26	some	some	DET
ejpam-4215	16	27	properties	property	NOUN
ejpam-4215	16	28	.	.	PUNCT
ejpam-4215	17	1	∗corresponding	∗corresponde	VERB
ejpam-4215	17	2	author	author	NOUN
ejpam-4215	17	3	.	.	PUNCT
ejpam-4215	18	1	doi	doi	NOUN
ejpam-4215	18	2	:	:	PUNCT
ejpam-4215	18	3	https://doi.org/10.29020/nybg.ejpam.v15i1.4215	https://doi.org/10.29020/nybg.ejpam.v15i1.4215	VERB
ejpam-4215	18	4	email	email	NOUN
ejpam-4215	18	5	addresses	address	NOUN
ejpam-4215	18	6	:	:	PUNCT
ejpam-4215	18	7	hibabaanoon@uomisan.edu.iq	hibabaanoon@uomisan.edu.iq	NOUN
ejpam-4215	18	8	(	(	PUNCT
ejpam-4215	18	9	h.r	h.r	PROPN
ejpam-4215	18	10	.	.	PROPN
ejpam-4215	18	11	baanoon	baanoon	PROPN
ejpam-4215	18	12	)	)	PUNCT
ejpam-4215	18	13	,	,	PUNCT
ejpam-4215	18	14	wasan.hasan@sc.uobaghdad.edu.iq	wasan.hasan@sc.uobaghdad.edu.iq	NOUN
ejpam-4215	18	15	(	(	PUNCT
ejpam-4215	18	16	w.	w.	PROPN
ejpam-4215	18	17	khalid	khalid	PROPN
ejpam-4215	18	18	)	)	PUNCT
ejpam-4215	18	19	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4215	19	1	224	224	NUM
ejpam-4215	19	2	©	©	PROPN
ejpam-4215	19	3	2022	2022	NUM
ejpam-4215	19	4	ejpam	ejpam	VERB
ejpam-4215	19	5	all	all	DET
ejpam-4215	19	6	rights	right	NOUN
ejpam-4215	19	7	reserved	reserve	VERB
ejpam-4215	19	8	.	.	PUNCT
ejpam-4215	20	1	h.r	h.r	PROPN
ejpam-4215	20	2	.	.	PROPN
ejpam-4215	20	3	baanoon	baanoon	PROPN
ejpam-4215	20	4	,	,	PUNCT
ejpam-4215	20	5	w.	w.	PROPN
ejpam-4215	20	6	khalid	khalid	PROPN
ejpam-4215	20	7	/	/	PUNCT
ejpam-4215	20	8	eur	eur	PROPN
ejpam-4215	20	9	.	.	PUNCT
ejpam-4215	21	1	j.	j.	PROPN
ejpam-4215	21	2	pure	pure	PROPN
ejpam-4215	21	3	appl	appl	PROPN
ejpam-4215	21	4	.	.	PROPN
ejpam-4215	21	5	math	math	PROPN
ejpam-4215	21	6	,	,	PUNCT
ejpam-4215	21	7	15	15	NUM
ejpam-4215	21	8	(	(	PUNCT
ejpam-4215	21	9	1	1	NUM
ejpam-4215	21	10	)	)	PUNCT
ejpam-4215	21	11	(	(	PUNCT
ejpam-4215	21	12	2022	2022	NUM
ejpam-4215	21	13	)	)	PUNCT
ejpam-4215	21	14	,	,	PUNCT
ejpam-4215	21	15	224	224	NUM
ejpam-4215	21	16	-	-	SYM
ejpam-4215	21	17	228	228	NUM
ejpam-4215	21	18	225	225	NUM
ejpam-4215	21	19	2	2	NUM
ejpam-4215	21	20	.	.	PUNCT
ejpam-4215	22	1	e∗-essential	e∗-essential	PROPN
ejpam-4215	22	2	submodule	submodule	PROPN
ejpam-4215	22	3	definition	definition	NOUN
ejpam-4215	22	4	1	1	NUM
ejpam-4215	22	5	.	.	X
ejpam-4215	22	6	•	•	NUM
ejpam-4215	22	7	let	let	VERB
ejpam-4215	22	8	m	m	PRON
ejpam-4215	22	9	be	be	AUX
ejpam-4215	22	10	r	r	NOUN
ejpam-4215	22	11	-	-	PUNCT
ejpam-4215	22	12	module	module	NOUN
ejpam-4215	22	13	,	,	PUNCT
ejpam-4215	22	14	a	a	DET
ejpam-4215	22	15	submodule	submodule	NOUN
ejpam-4215	22	16	a	a	PRON
ejpam-4215	22	17	of	of	ADP
ejpam-4215	22	18	m	m	PROPN
ejpam-4215	22	19	is	be	AUX
ejpam-4215	22	20	said	say	VERB
ejpam-4215	22	21	to	to	PART
ejpam-4215	22	22	be	be	AUX
ejpam-4215	22	23	e∗-essential	e∗-essential	PROPN
ejpam-4215	22	24	if	if	SCONJ
ejpam-4215	22	25	a	a	DET
ejpam-4215	22	26	∩b	∩b	NOUN
ejpam-4215	22	27	̸=	̸=	PROPN
ejpam-4215	22	28	0	0	NUM
ejpam-4215	22	29	for	for	ADP
ejpam-4215	22	30	each	each	DET
ejpam-4215	22	31	nonzero	nonzero	NOUN
ejpam-4215	22	32	cosingular	cosingular	PROPN
ejpam-4215	22	33	submodule	submodule	PROPN
ejpam-4215	22	34	b	b	PROPN
ejpam-4215	22	35	of	of	ADP
ejpam-4215	22	36	m	m	PROPN
ejpam-4215	22	37	.	.	PUNCT
ejpam-4215	23	1	denoted	denote	VERB
ejpam-4215	23	2	by	by	ADP
ejpam-4215	23	3	a	a	DET
ejpam-4215	23	4	≤e∗	≤e∗	PROPN
ejpam-4215	23	5	b.	b.	PROPN
ejpam-4215	23	6	•	•	ADP
ejpam-4215	23	7	a	a	DET
ejpam-4215	23	8	right	right	ADJ
ejpam-4215	23	9	ideal	ideal	NOUN
ejpam-4215	23	10	b	b	PROPN
ejpam-4215	23	11	of	of	ADP
ejpam-4215	23	12	a	a	DET
ejpam-4215	23	13	ring	ring	NOUN
ejpam-4215	23	14	r	r	NOUN
ejpam-4215	23	15	is	be	AUX
ejpam-4215	23	16	e∗-essential	e∗-essential	ADJ
ejpam-4215	23	17	in	in	ADP
ejpam-4215	23	18	r	r	NOUN
ejpam-4215	23	19	if	if	SCONJ
ejpam-4215	24	1	and	and	CCONJ
ejpam-4215	24	2	only	only	ADV
ejpam-4215	24	3	if	if	SCONJ
ejpam-4215	24	4	b	b	NOUN
ejpam-4215	24	5	is	be	AUX
ejpam-4215	24	6	e∗-essential	e∗-essential	PROPN
ejpam-4215	24	7	submodule	submodule	NOUN
ejpam-4215	24	8	of	of	ADP
ejpam-4215	24	9	rr	rr	PROPN
ejpam-4215	24	10	.	.	PROPN
ejpam-4215	25	1	•	•	NUM
ejpam-4215	25	2	an	an	DET
ejpam-4215	25	3	r	r	NOUN
ejpam-4215	25	4	-	-	PUNCT
ejpam-4215	25	5	homomorphism	homomorphism	ADJ
ejpam-4215	25	6	f	f	NOUN
ejpam-4215	25	7	:	:	PUNCT
ejpam-4215	25	8	a	a	DET
ejpam-4215	25	9	→	→	SYM
ejpam-4215	25	10	b	b	PROPN
ejpam-4215	25	11	is	be	AUX
ejpam-4215	25	12	said	say	VERB
ejpam-4215	25	13	e∗-essential	e∗-essential	PROPN
ejpam-4215	25	14	if	if	SCONJ
ejpam-4215	25	15	and	and	CCONJ
ejpam-4215	25	16	only	only	ADV
ejpam-4215	25	17	if	if	SCONJ
ejpam-4215	25	18	,	,	PUNCT
ejpam-4215	25	19	im(f	im(f	NOUN
ejpam-4215	25	20	)	)	PUNCT
ejpam-4215	25	21	is	be	AUX
ejpam-4215	25	22	e∗essential	e∗essential	ADJ
ejpam-4215	25	23	submodule	submodule	NOUN
ejpam-4215	25	24	in	in	ADP
ejpam-4215	25	25	b.	b.	PROPN
ejpam-4215	25	26	•	•	ADP
ejpam-4215	25	27	we	we	PRON
ejpam-4215	25	28	may	may	AUX
ejpam-4215	25	29	deduce	deduce	VERB
ejpam-4215	25	30	the	the	DET
ejpam-4215	25	31	following	following	NOUN
ejpam-4215	25	32	from	from	ADP
ejpam-4215	25	33	the	the	DET
ejpam-4215	25	34	definition	definition	NOUN
ejpam-4215	25	35	:	:	PUNCT
ejpam-4215	25	36	1	1	X
ejpam-4215	25	37	.	.	X
ejpam-4215	25	38	a	a	DET
ejpam-4215	25	39	≤e∗	≤e∗	PROPN
ejpam-4215	25	40	m	m	NOUN
ejpam-4215	25	41	if	if	SCONJ
ejpam-4215	25	42	a	a	DET
ejpam-4215	25	43	∩k	∩k	NOUN
ejpam-4215	26	1	=	=	SYM
ejpam-4215	26	2	0	0	NUM
ejpam-4215	26	3	,	,	PUNCT
ejpam-4215	26	4	then	then	ADV
ejpam-4215	26	5	k	k	PROPN
ejpam-4215	27	1	=	=	PUNCT
ejpam-4215	27	2	0	0	NUM
ejpam-4215	27	3	where	where	SCONJ
ejpam-4215	27	4	k	k	PROPN
ejpam-4215	27	5	is	be	AUX
ejpam-4215	27	6	cosingular	cosingular	ADJ
ejpam-4215	27	7	submodule	submodule	NOUN
ejpam-4215	27	8	in	in	ADP
ejpam-4215	27	9	m	m	PROPN
ejpam-4215	27	10	.	.	PUNCT
ejpam-4215	28	1	2	2	X
ejpam-4215	28	2	.	.	X
ejpam-4215	29	1	if	if	SCONJ
ejpam-4215	29	2	m	m	PRON
ejpam-4215	29	3	̸=	̸=	NOUN
ejpam-4215	29	4	0	0	NUM
ejpam-4215	29	5	and	and	CCONJ
ejpam-4215	29	6	l	l	PROPN
ejpam-4215	29	7	≤e∗	≤e∗	PROPN
ejpam-4215	30	1	m	m	VERB
ejpam-4215	30	2	then	then	ADV
ejpam-4215	30	3	l	l	NOUN
ejpam-4215	30	4	̸=	̸=	PROPN
ejpam-4215	30	5	0	0	NUM
ejpam-4215	30	6	.	.	PUNCT
ejpam-4215	31	1	examples	example	NOUN
ejpam-4215	31	2	and	and	CCONJ
ejpam-4215	31	3	remarks	remark	VERB
ejpam-4215	31	4	1	1	NUM
ejpam-4215	31	5	.	.	NOUN
ejpam-4215	31	6	1	1	NUM
ejpam-4215	31	7	.	.	X
ejpam-4215	32	1	every	every	DET
ejpam-4215	32	2	essential	essential	ADJ
ejpam-4215	32	3	submodule	submodule	NOUN
ejpam-4215	32	4	is	be	AUX
ejpam-4215	32	5	e∗-essential	e∗-essential	PROPN
ejpam-4215	32	6	,	,	PUNCT
ejpam-4215	32	7	but	but	CCONJ
ejpam-4215	32	8	the	the	DET
ejpam-4215	32	9	converse	converse	NOUN
ejpam-4215	32	10	need	need	VERB
ejpam-4215	32	11	not	not	PART
ejpam-4215	32	12	to	to	PART
ejpam-4215	32	13	be	be	AUX
ejpam-4215	32	14	true	true	ADJ
ejpam-4215	32	15	in	in	ADP
ejpam-4215	32	16	general	general	ADJ
ejpam-4215	32	17	.	.	PUNCT
ejpam-4215	33	1	for	for	ADP
ejpam-4215	33	2	example	example	NOUN
ejpam-4215	33	3	,	,	PUNCT
ejpam-4215	33	4	in	in	ADP
ejpam-4215	33	5	z6	z6	PROPN
ejpam-4215	33	6	as	as	ADP
ejpam-4215	33	7	z6	z6	NOUN
ejpam-4215	33	8	-	-	PUNCT
ejpam-4215	33	9	module	module	NOUN
ejpam-4215	33	10	,	,	PUNCT
ejpam-4215	33	11	the	the	DET
ejpam-4215	33	12	only	only	ADJ
ejpam-4215	33	13	cosingular	cosingular	ADJ
ejpam-4215	33	14	submodle	submodle	NOUN
ejpam-4215	33	15	of	of	ADP
ejpam-4215	33	16	z6	z6	PROPN
ejpam-4215	33	17	is	be	AUX
ejpam-4215	33	18	{	{	PUNCT
ejpam-4215	33	19	0	0	NUM
ejpam-4215	33	20	}	}	PUNCT
ejpam-4215	33	21	.	.	PUNCT
ejpam-4215	34	1	hence	hence	ADV
ejpam-4215	34	2	every	every	DET
ejpam-4215	34	3	submodule	submodule	NOUN
ejpam-4215	34	4	k	k	PROPN
ejpam-4215	34	5	of	of	ADP
ejpam-4215	34	6	z6	z6	PROPN
ejpam-4215	34	7	is	be	AUX
ejpam-4215	34	8	e∗-essential	e∗-essential	PROPN
ejpam-4215	34	9	,	,	PUNCT
ejpam-4215	34	10	since	since	SCONJ
ejpam-4215	34	11	k	k	PROPN
ejpam-4215	34	12	∩	∩	X
ejpam-4215	34	13	{	{	PUNCT
ejpam-4215	34	14	0	0	NUM
ejpam-4215	34	15	}	}	PUNCT
ejpam-4215	34	16	=	=	SYM
ejpam-4215	34	17	0	0	X
ejpam-4215	34	18	.	.	PUNCT
ejpam-4215	35	1	therefore	therefore	ADV
ejpam-4215	35	2	,	,	PUNCT
ejpam-4215	35	3	{	{	PUNCT
ejpam-4215	35	4	0	0	NUM
ejpam-4215	35	5	,	,	PUNCT
ejpam-4215	35	6	2	2	NUM
ejpam-4215	35	7	,	,	PUNCT
ejpam-4215	35	8	4	4	NUM
ejpam-4215	35	9	}	}	PUNCT
ejpam-4215	35	10	is	be	AUX
ejpam-4215	35	11	e∗-essential	e∗-essential	PROPN
ejpam-4215	35	12	which	which	PRON
ejpam-4215	35	13	is	be	AUX
ejpam-4215	35	14	not	not	PART
ejpam-4215	35	15	essential	essential	ADJ
ejpam-4215	35	16	submodule	submodule	NOUN
ejpam-4215	35	17	in	in	ADP
ejpam-4215	35	18	z6	z6	PROPN
ejpam-4215	35	19	as	as	ADP
ejpam-4215	35	20	z6	z6	NOUN
ejpam-4215	35	21	-	-	PUNCT
ejpam-4215	35	22	module	module	NOUN
ejpam-4215	35	23	since	since	SCONJ
ejpam-4215	35	24	there	there	PRON
ejpam-4215	35	25	is	be	VERB
ejpam-4215	35	26	a	a	DET
ejpam-4215	35	27	nonzero	nonzero	NOUN
ejpam-4215	35	28	submodule	submodule	NOUN
ejpam-4215	35	29	{	{	PUNCT
ejpam-4215	35	30	0	0	NUM
ejpam-4215	35	31	,	,	PUNCT
ejpam-4215	35	32	3	3	NUM
ejpam-4215	35	33	}	}	PUNCT
ejpam-4215	35	34	but	but	CCONJ
ejpam-4215	35	35	{	{	PUNCT
ejpam-4215	35	36	0	0	NUM
ejpam-4215	35	37	,	,	PUNCT
ejpam-4215	35	38	2	2	NUM
ejpam-4215	35	39	,	,	PUNCT
ejpam-4215	35	40	4	4	NUM
ejpam-4215	35	41	}	}	PUNCT
ejpam-4215	35	42	∩	∩	NOUN
ejpam-4215	35	43	{	{	PUNCT
ejpam-4215	35	44	0	0	NUM
ejpam-4215	35	45	,	,	PUNCT
ejpam-4215	35	46	3	3	NUM
ejpam-4215	35	47	}	}	PUNCT
ejpam-4215	35	48	=	=	SYM
ejpam-4215	35	49	0	0	NUM
ejpam-4215	35	50	.	.	NOUN
ejpam-4215	36	1	2	2	NUM
ejpam-4215	36	2	.	.	X
ejpam-4215	36	3	for	for	ADP
ejpam-4215	36	4	any	any	DET
ejpam-4215	36	5	r	r	NOUN
ejpam-4215	36	6	-	-	PUNCT
ejpam-4215	36	7	module	module	NOUN
ejpam-4215	36	8	m	m	NOUN
ejpam-4215	36	9	,	,	PUNCT
ejpam-4215	36	10	we	we	PRON
ejpam-4215	36	11	have	have	VERB
ejpam-4215	36	12	m	m	PROPN
ejpam-4215	36	13	≤e∗	≤e∗	PROPN
ejpam-4215	36	14	m	m	NOUN
ejpam-4215	36	15	.	.	PUNCT
ejpam-4215	37	1	3	3	X
ejpam-4215	37	2	.	.	X
ejpam-4215	38	1	every	every	DET
ejpam-4215	38	2	nonzero	nonzero	PROPN
ejpam-4215	38	3	submodule	submodule	NOUN
ejpam-4215	38	4	of	of	ADP
ejpam-4215	38	5	z	z	PROPN
ejpam-4215	38	6	as	as	ADP
ejpam-4215	38	7	z	z	NOUN
ejpam-4215	38	8	-	-	PUNCT
ejpam-4215	38	9	module	module	NOUN
ejpam-4215	38	10	is	be	AUX
ejpam-4215	38	11	cosingular	cosingular	ADJ
ejpam-4215	38	12	[	[	X
ejpam-4215	38	13	2	2	NUM
ejpam-4215	38	14	]	]	PUNCT
ejpam-4215	38	15	.	.	PUNCT
ejpam-4215	39	1	hence	hence	ADV
ejpam-4215	39	2	,	,	PUNCT
ejpam-4215	39	3	nz	nz	PROPN
ejpam-4215	39	4	∩	∩	NOUN
ejpam-4215	39	5	mz	mz	PROPN
ejpam-4215	39	6	=	=	PROPN
ejpam-4215	39	7	nmz	nmz	PROPN
ejpam-4215	39	8	̸=	̸=	PROPN
ejpam-4215	39	9	0	0	NUM
ejpam-4215	39	10	for	for	ADP
ejpam-4215	39	11	each	each	DET
ejpam-4215	39	12	n	n	DET
ejpam-4215	39	13	̸=	̸=	PROPN
ejpam-4215	39	14	0	0	NUM
ejpam-4215	39	15	and	and	CCONJ
ejpam-4215	39	16	m	m	PROPN
ejpam-4215	39	17	̸=	̸=	PROPN
ejpam-4215	39	18	0	0	NUM
ejpam-4215	39	19	.	.	PUNCT
ejpam-4215	40	1	so	so	SCONJ
ejpam-4215	40	2	that	that	SCONJ
ejpam-4215	40	3	every	every	DET
ejpam-4215	40	4	submodule	submodule	NOUN
ejpam-4215	40	5	of	of	ADP
ejpam-4215	40	6	z	z	PROPN
ejpam-4215	40	7	is	be	AUX
ejpam-4215	40	8	e∗-essential	e∗-essential	PROPN
ejpam-4215	40	9	.	.	PUNCT
ejpam-4215	41	1	4	4	X
ejpam-4215	41	2	.	.	X
ejpam-4215	41	3	in	in	ADP
ejpam-4215	41	4	z6	z6	PROPN
ejpam-4215	41	5	as	as	ADP
ejpam-4215	41	6	z	z	NOUN
ejpam-4215	41	7	-	-	PUNCT
ejpam-4215	41	8	module	module	NOUN
ejpam-4215	41	9	every	every	DET
ejpam-4215	41	10	submodule	submodule	NOUN
ejpam-4215	41	11	is	be	AUX
ejpam-4215	41	12	cosingular	cosingular	ADJ
ejpam-4215	41	13	[	[	X
ejpam-4215	41	14	2	2	NUM
ejpam-4215	41	15	]	]	PUNCT
ejpam-4215	41	16	,	,	PUNCT
ejpam-4215	41	17	but	but	CCONJ
ejpam-4215	41	18	{	{	PUNCT
ejpam-4215	41	19	0	0	NUM
ejpam-4215	41	20	,	,	PUNCT
ejpam-4215	41	21	2	2	NUM
ejpam-4215	41	22	,	,	PUNCT
ejpam-4215	41	23	4	4	NUM
ejpam-4215	41	24	}	}	PUNCT
ejpam-4215	41	25	is	be	AUX
ejpam-4215	41	26	not	not	PART
ejpam-4215	41	27	e∗-essential	e∗-essential	ADJ
ejpam-4215	41	28	since	since	SCONJ
ejpam-4215	41	29	{	{	PUNCT
ejpam-4215	41	30	0	0	NUM
ejpam-4215	41	31	,	,	PUNCT
ejpam-4215	41	32	2	2	NUM
ejpam-4215	41	33	,	,	PUNCT
ejpam-4215	41	34	4	4	NUM
ejpam-4215	41	35	}	}	PUNCT
ejpam-4215	41	36	∩	∩	NOUN
ejpam-4215	41	37	{	{	PUNCT
ejpam-4215	41	38	0	0	NUM
ejpam-4215	41	39	,	,	PUNCT
ejpam-4215	41	40	3	3	NUM
ejpam-4215	41	41	}	}	PUNCT
ejpam-4215	42	1	=	=	SYM
ejpam-4215	42	2	0	0	NUM
ejpam-4215	43	1	where	where	SCONJ
ejpam-4215	43	2	{	{	PUNCT
ejpam-4215	43	3	0	0	NUM
ejpam-4215	43	4	,	,	PUNCT
ejpam-4215	43	5	3	3	X
ejpam-4215	43	6	}	}	PUNCT
ejpam-4215	43	7	a	a	DET
ejpam-4215	43	8	nonzero	nonzero	ADJ
ejpam-4215	43	9	cosingular	cosingular	ADJ
ejpam-4215	43	10	submodule	submodule	NOUN
ejpam-4215	43	11	.	.	PUNCT
ejpam-4215	44	1	5	5	X
ejpam-4215	44	2	.	.	X
ejpam-4215	44	3	the	the	DET
ejpam-4215	44	4	image	image	NOUN
ejpam-4215	44	5	of	of	ADP
ejpam-4215	44	6	e∗-essential	e∗-essential	PROPN
ejpam-4215	44	7	need	need	AUX
ejpam-4215	44	8	not	not	PART
ejpam-4215	44	9	be	be	AUX
ejpam-4215	44	10	e∗-essential	e∗-essential	ADJ
ejpam-4215	44	11	for	for	ADP
ejpam-4215	44	12	example	example	NOUN
ejpam-4215	44	13	.	.	PUNCT
ejpam-4215	45	1	let	let	VERB
ejpam-4215	45	2	f	f	NOUN
ejpam-4215	45	3	:	:	PUNCT
ejpam-4215	45	4	z	z	X
ejpam-4215	45	5	→	→	SYM
ejpam-4215	45	6	z2	z2	PROPN
ejpam-4215	45	7	be	be	AUX
ejpam-4215	45	8	a	a	DET
ejpam-4215	45	9	z	z	NOUN
ejpam-4215	45	10	-	-	PUNCT
ejpam-4215	45	11	homomorphism	homomorphism	NOUN
ejpam-4215	45	12	defined	define	VERB
ejpam-4215	45	13	by	by	ADP
ejpam-4215	45	14	f(x	f(x	PROPN
ejpam-4215	45	15	)	)	PUNCT
ejpam-4215	45	16	=	=	PRON
ejpam-4215	46	1	{	{	PUNCT
ejpam-4215	46	2	0	0	NUM
ejpam-4215	46	3	if	if	SCONJ
ejpam-4215	46	4	xeven	xeven	PROPN
ejpam-4215	46	5	1	1	NUM
ejpam-4215	46	6	if	if	SCONJ
ejpam-4215	46	7	xodd	xodd	VERB
ejpam-4215	46	8	so	so	ADV
ejpam-4215	46	9	f(2z	f(2z	NOUN
ejpam-4215	46	10	)	)	PUNCT
ejpam-4215	46	11	=	=	PUNCT
ejpam-4215	46	12	{	{	PUNCT
ejpam-4215	46	13	0	0	NUM
ejpam-4215	46	14	}	}	PUNCT
ejpam-4215	46	15	.	.	PUNCT
ejpam-4215	47	1	hence	hence	ADV
ejpam-4215	47	2	,	,	PUNCT
ejpam-4215	47	3	2z	2z	NUM
ejpam-4215	47	4	is	be	AUX
ejpam-4215	47	5	e∗-essential	e∗-essential	ADJ
ejpam-4215	47	6	in	in	ADP
ejpam-4215	47	7	z	z	PROPN
ejpam-4215	47	8	but	but	CCONJ
ejpam-4215	47	9	{	{	PUNCT
ejpam-4215	47	10	0	0	X
ejpam-4215	47	11	}	}	PUNCT
ejpam-4215	47	12	is	be	AUX
ejpam-4215	47	13	not	not	PART
ejpam-4215	47	14	e∗-essential	e∗-essential	PROPN
ejpam-4215	47	15	in	in	ADP
ejpam-4215	47	16	z2	z2	PROPN
ejpam-4215	47	17	,	,	PUNCT
ejpam-4215	47	18	since	since	SCONJ
ejpam-4215	47	19	{	{	PUNCT
ejpam-4215	47	20	0	0	NUM
ejpam-4215	47	21	}	}	PUNCT
ejpam-4215	47	22	∩	∩	ADJ
ejpam-4215	47	23	z2	z2	NOUN
ejpam-4215	47	24	=	=	SYM
ejpam-4215	47	25	0	0	NUM
ejpam-4215	47	26	where	where	SCONJ
ejpam-4215	47	27	z2	z2	PROPN
ejpam-4215	47	28	is	be	AUX
ejpam-4215	47	29	nonzero	nonzero	NOUN
ejpam-4215	47	30	cosingular	cosingular	ADJ
ejpam-4215	47	31	.	.	PUNCT
ejpam-4215	48	1	6	6	X
ejpam-4215	48	2	.	.	X
ejpam-4215	49	1	the	the	DET
ejpam-4215	49	2	quotient	quotient	NOUN
ejpam-4215	49	3	submodule	submodule	NOUN
ejpam-4215	49	4	of	of	ADP
ejpam-4215	49	5	e∗-essential	e∗-essential	PROPN
ejpam-4215	49	6	submodule	submodule	NOUN
ejpam-4215	49	7	need	need	AUX
ejpam-4215	49	8	not	not	PART
ejpam-4215	49	9	to	to	PART
ejpam-4215	49	10	be	be	AUX
ejpam-4215	49	11	e∗-essential	e∗-essential	VERB
ejpam-4215	49	12	,	,	PUNCT
ejpam-4215	49	13	for	for	ADP
ejpam-4215	49	14	example	example	NOUN
ejpam-4215	49	15	:	:	PUNCT
ejpam-4215	49	16	2zz	2zz	ADJ
ejpam-4215	49	17	is	be	AUX
ejpam-4215	49	18	e∗-essential	e∗-essential	PROPN
ejpam-4215	49	19	submodule	submodule	NOUN
ejpam-4215	49	20	of	of	ADP
ejpam-4215	49	21	zz	zz	PROPN
ejpam-4215	49	22	,	,	PUNCT
ejpam-4215	49	23	but	but	CCONJ
ejpam-4215	49	24	2z	2z	NUM
ejpam-4215	49	25	2z	2z	NUM
ejpam-4215	49	26	=	=	SYM
ejpam-4215	49	27	0	0	PUNCT
ejpam-4215	49	28	not	not	PART
ejpam-4215	49	29	e∗-essential	e∗-essential	PROPN
ejpam-4215	49	30	submodule	submodule	NOUN
ejpam-4215	49	31	of	of	ADP
ejpam-4215	49	32	z	z	PROPN
ejpam-4215	49	33	2z	2z	NUM
ejpam-4215	49	34	∼=	∼=	PROPN
ejpam-4215	49	35	z2	z2	NOUN
ejpam-4215	49	36	.	.	PUNCT
ejpam-4215	50	1	in	in	ADP
ejpam-4215	50	2	the	the	DET
ejpam-4215	50	3	following	follow	VERB
ejpam-4215	50	4	lemma	lemma	PROPN
ejpam-4215	50	5	,	,	PUNCT
ejpam-4215	50	6	gives	give	VERB
ejpam-4215	50	7	a	a	DET
ejpam-4215	50	8	property	property	NOUN
ejpam-4215	50	9	of	of	ADP
ejpam-4215	50	10	cosingular	cosingular	ADJ
ejpam-4215	50	11	submodule	submodule	PROPN
ejpam-4215	50	12	h.r	h.r	PROPN
ejpam-4215	50	13	.	.	PROPN
ejpam-4215	50	14	baanoon	baanoon	PROPN
ejpam-4215	50	15	,	,	PUNCT
ejpam-4215	50	16	w.	w.	PROPN
ejpam-4215	50	17	khalid	khalid	PROPN
ejpam-4215	50	18	/	/	PUNCT
ejpam-4215	50	19	eur	eur	PROPN
ejpam-4215	50	20	.	.	PUNCT
ejpam-4215	51	1	j.	j.	PROPN
ejpam-4215	51	2	pure	pure	PROPN
ejpam-4215	51	3	appl	appl	PROPN
ejpam-4215	51	4	.	.	PROPN
ejpam-4215	51	5	math	math	PROPN
ejpam-4215	51	6	,	,	PUNCT
ejpam-4215	51	7	15	15	NUM
ejpam-4215	51	8	(	(	PUNCT
ejpam-4215	51	9	1	1	NUM
ejpam-4215	51	10	)	)	PUNCT
ejpam-4215	51	11	(	(	PUNCT
ejpam-4215	51	12	2022	2022	NUM
ejpam-4215	51	13	)	)	PUNCT
ejpam-4215	51	14	,	,	PUNCT
ejpam-4215	51	15	224	224	NUM
ejpam-4215	51	16	-	-	SYM
ejpam-4215	51	17	228	228	NUM
ejpam-4215	51	18	226	226	NUM
ejpam-4215	51	19	lemma	lemma	PROPN
ejpam-4215	51	20	1	1	NUM
ejpam-4215	51	21	.	.	PUNCT
ejpam-4215	52	1	if	if	SCONJ
ejpam-4215	52	2	k	k	PROPN
ejpam-4215	52	3	is	be	AUX
ejpam-4215	52	4	cosingular	cosingular	ADJ
ejpam-4215	52	5	submodule	submodule	NOUN
ejpam-4215	52	6	of	of	ADP
ejpam-4215	52	7	b	b	PROPN
ejpam-4215	52	8	and	and	CCONJ
ejpam-4215	52	9	b	b	NOUN
ejpam-4215	52	10	≤	≤	NOUN
ejpam-4215	52	11	a	a	DET
ejpam-4215	52	12	≤	≤	NUM
ejpam-4215	52	13	m	m	PROPN
ejpam-4215	52	14	,	,	PUNCT
ejpam-4215	52	15	then	then	ADV
ejpam-4215	52	16	k	k	PROPN
ejpam-4215	52	17	is	be	AUX
ejpam-4215	52	18	cosingular	cosingular	ADJ
ejpam-4215	52	19	in	in	ADP
ejpam-4215	52	20	a.	a.	NOUN
ejpam-4215	52	21	proof	proof	NOUN
ejpam-4215	52	22	.	.	PUNCT
ejpam-4215	53	1	since	since	SCONJ
ejpam-4215	53	2	k	k	PROPN
ejpam-4215	53	3	is	be	AUX
ejpam-4215	53	4	cosingular	cosingular	ADJ
ejpam-4215	53	5	submodule	submodule	NOUN
ejpam-4215	53	6	of	of	ADP
ejpam-4215	53	7	b	b	PROPN
ejpam-4215	53	8	by	by	ADP
ejpam-4215	53	9	lemma	lemma	PROPN
ejpam-4215	53	10	2.2	2.2	NUM
ejpam-4215	53	11	in	in	ADP
ejpam-4215	53	12	[	[	X
ejpam-4215	53	13	2	2	NUM
ejpam-4215	53	14	]	]	PUNCT
ejpam-4215	53	15	z∗(k	z∗(k	NOUN
ejpam-4215	53	16	)	)	PUNCT
ejpam-4215	53	17	=	=	SYM
ejpam-4215	53	18	k∩z∗(b	k∩z∗(b	PROPN
ejpam-4215	53	19	)	)	PUNCT
ejpam-4215	53	20	and	and	CCONJ
ejpam-4215	53	21	again	again	ADV
ejpam-4215	53	22	since	since	SCONJ
ejpam-4215	53	23	b	b	PROPN
ejpam-4215	53	24	is	be	AUX
ejpam-4215	53	25	a	a	DET
ejpam-4215	53	26	submodule	submodule	NOUN
ejpam-4215	53	27	of	of	ADP
ejpam-4215	53	28	a.	a.	NOUN
ejpam-4215	53	29	so	so	SCONJ
ejpam-4215	53	30	that	that	SCONJ
ejpam-4215	53	31	,	,	PUNCT
ejpam-4215	53	32	z∗(k	z∗(k	NOUN
ejpam-4215	53	33	)	)	PUNCT
ejpam-4215	53	34	=	=	SYM
ejpam-4215	53	35	k∩(b∩z∗(a	k∩(b∩z∗(a	PROPN
ejpam-4215	53	36	)	)	PUNCT
ejpam-4215	53	37	)	)	PUNCT
ejpam-4215	53	38	,	,	PUNCT
ejpam-4215	53	39	from	from	ADP
ejpam-4215	53	40	hypothesis	hypothesis	NOUN
ejpam-4215	53	41	z∗(k	z∗(k	NOUN
ejpam-4215	53	42	)	)	PUNCT
ejpam-4215	53	43	=	=	PUNCT
ejpam-4215	54	1	k.	k.	PROPN
ejpam-4215	54	2	hence	hence	ADV
ejpam-4215	54	3	,	,	PUNCT
ejpam-4215	54	4	k	k	PROPN
ejpam-4215	54	5	≤	≤	X
ejpam-4215	54	6	z∗(a	z∗(a	NUM
ejpam-4215	54	7	)	)	PUNCT
ejpam-4215	54	8	.	.	PUNCT
ejpam-4215	55	1	therefore	therefore	ADV
ejpam-4215	55	2	,	,	PUNCT
ejpam-4215	55	3	z∗(k	z∗(k	NOUN
ejpam-4215	55	4	)	)	PUNCT
ejpam-4215	55	5	=	=	SYM
ejpam-4215	55	6	k	k	PROPN
ejpam-4215	55	7	∩	∩	X
ejpam-4215	55	8	z∗(a	z∗(a	X
ejpam-4215	55	9	)	)	PUNCT
ejpam-4215	55	10	=	=	SYM
ejpam-4215	56	1	k	k	NOUN
ejpam-4215	56	2	,	,	PUNCT
ejpam-4215	56	3	i.e.	i.e.	X
ejpam-4215	56	4	k	k	X
ejpam-4215	56	5	is	be	AUX
ejpam-4215	56	6	cosingular	cosingular	ADJ
ejpam-4215	56	7	in	in	ADP
ejpam-4215	56	8	a.	a.	NOUN
ejpam-4215	56	9	now	now	ADV
ejpam-4215	56	10	,	,	PUNCT
ejpam-4215	56	11	we	we	PRON
ejpam-4215	56	12	will	will	AUX
ejpam-4215	56	13	prove	prove	VERB
ejpam-4215	56	14	some	some	DET
ejpam-4215	56	15	properties	property	NOUN
ejpam-4215	56	16	which	which	PRON
ejpam-4215	56	17	e∗-essential	e∗-essential	PROPN
ejpam-4215	56	18	submodule	submodule	NOUN
ejpam-4215	56	19	satisfied	satisfied	ADJ
ejpam-4215	56	20	:	:	PUNCT
ejpam-4215	56	21	proposition	proposition	NOUN
ejpam-4215	56	22	1	1	NUM
ejpam-4215	56	23	.	.	PUNCT
ejpam-4215	56	24	let	let	VERB
ejpam-4215	56	25	a	a	DET
ejpam-4215	56	26	≤	≤	NUM
ejpam-4215	56	27	b	b	NOUN
ejpam-4215	56	28	≤	≤	NUM
ejpam-4215	56	29	m	m	PROPN
ejpam-4215	56	30	,	,	PUNCT
ejpam-4215	56	31	then	then	ADV
ejpam-4215	56	32	a	a	DET
ejpam-4215	56	33	≤e∗	≤e∗	PROPN
ejpam-4215	56	34	m	m	VERB
ejpam-4215	56	35	if	if	SCONJ
ejpam-4215	56	36	and	and	CCONJ
ejpam-4215	56	37	only	only	ADV
ejpam-4215	56	38	if	if	SCONJ
ejpam-4215	56	39	a	a	DET
ejpam-4215	56	40	≤e∗	≤e∗	PROPN
ejpam-4215	56	41	b	b	PROPN
ejpam-4215	56	42	≤e∗	≤e∗	PROPN
ejpam-4215	56	43	m	m	PROPN
ejpam-4215	56	44	proof	proof	NOUN
ejpam-4215	56	45	.	.	PUNCT
ejpam-4215	57	1	⇒	⇒	NOUN
ejpam-4215	57	2	)	)	PUNCT
ejpam-4215	57	3	let	let	VERB
ejpam-4215	57	4	k	k	PROPN
ejpam-4215	57	5	̸=	̸=	PROPN
ejpam-4215	57	6	0	0	NUM
ejpam-4215	57	7	be	be	AUX
ejpam-4215	57	8	a	a	DET
ejpam-4215	57	9	cosingular	cosingular	ADJ
ejpam-4215	57	10	submodule	submodule	NOUN
ejpam-4215	57	11	of	of	ADP
ejpam-4215	57	12	b	b	PROPN
ejpam-4215	57	13	,	,	PUNCT
ejpam-4215	57	14	hence	hence	ADV
ejpam-4215	57	15	k	k	PROPN
ejpam-4215	57	16	≤	≤	PROPN
ejpam-4215	57	17	m	m	VERB
ejpam-4215	57	18	since	since	SCONJ
ejpam-4215	57	19	a	a	DET
ejpam-4215	57	20	≤e∗	≤e∗	PROPN
ejpam-4215	57	21	m	m	NOUN
ejpam-4215	57	22	.	.	PUNCT
ejpam-4215	58	1	therefore	therefore	ADV
ejpam-4215	58	2	a	a	DET
ejpam-4215	58	3	∩	∩	ADJ
ejpam-4215	58	4	k	k	PROPN
ejpam-4215	58	5	̸=	̸=	PROPN
ejpam-4215	58	6	0	0	NUM
ejpam-4215	58	7	.	.	PUNCT
ejpam-4215	59	1	hence	hence	ADV
ejpam-4215	59	2	,	,	PUNCT
ejpam-4215	59	3	a	a	DET
ejpam-4215	59	4	≤e∗	≤e∗	PROPN
ejpam-4215	59	5	b.	b.	NOUN
ejpam-4215	59	6	now	now	ADV
ejpam-4215	59	7	,	,	PUNCT
ejpam-4215	59	8	for	for	ADP
ejpam-4215	59	9	b	b	PROPN
ejpam-4215	59	10	≤e∗	≤e∗	PROPN
ejpam-4215	59	11	m	m	VERB
ejpam-4215	59	12	,	,	PUNCT
ejpam-4215	59	13	let	let	VERB
ejpam-4215	59	14	0	0	NUM
ejpam-4215	59	15	̸=	̸=	PROPN
ejpam-4215	59	16	l	l	NOUN
ejpam-4215	59	17	be	be	AUX
ejpam-4215	59	18	a	a	DET
ejpam-4215	59	19	consigular	consigular	ADJ
ejpam-4215	59	20	submodule	submodule	NOUN
ejpam-4215	59	21	of	of	ADP
ejpam-4215	59	22	m	m	PROPN
ejpam-4215	59	23	.	.	PUNCT
ejpam-4215	60	1	hence	hence	ADV
ejpam-4215	60	2	,	,	PUNCT
ejpam-4215	60	3	a	a	DET
ejpam-4215	60	4	∩	∩	ADJ
ejpam-4215	60	5	l	l	NOUN
ejpam-4215	60	6	̸=	̸=	PROPN
ejpam-4215	60	7	0	0	PUNCT
ejpam-4215	60	8	and	and	CCONJ
ejpam-4215	60	9	since	since	SCONJ
ejpam-4215	60	10	a	a	DET
ejpam-4215	60	11	≤	≤	NUM
ejpam-4215	60	12	b	b	NOUN
ejpam-4215	60	13	so	so	SCONJ
ejpam-4215	60	14	that	that	SCONJ
ejpam-4215	60	15	,	,	PUNCT
ejpam-4215	60	16	b	b	PROPN
ejpam-4215	60	17	∩	∩	NOUN
ejpam-4215	60	18	l	l	PROPN
ejpam-4215	60	19	̸=	̸=	PROPN
ejpam-4215	60	20	0	0	NUM
ejpam-4215	60	21	.	.	PUNCT
ejpam-4215	61	1	⇐	⇐	NOUN
ejpam-4215	61	2	)	)	PUNCT
ejpam-4215	61	3	let	let	VERB
ejpam-4215	61	4	n	n	PRON
ejpam-4215	61	5	be	be	AUX
ejpam-4215	61	6	a	a	DET
ejpam-4215	61	7	nonzero	nonzero	ADJ
ejpam-4215	61	8	cosingular	cosingular	ADJ
ejpam-4215	61	9	submodule	submodule	NOUN
ejpam-4215	61	10	of	of	ADP
ejpam-4215	61	11	m	m	PROPN
ejpam-4215	61	12	.	.	PUNCT
ejpam-4215	62	1	since	since	SCONJ
ejpam-4215	62	2	b	b	PROPN
ejpam-4215	62	3	≤e∗	≤e∗	PROPN
ejpam-4215	62	4	m	m	VERB
ejpam-4215	62	5	.	.	PUNCT
ejpam-4215	63	1	hence	hence	ADV
ejpam-4215	63	2	,	,	PUNCT
ejpam-4215	63	3	b	b	PROPN
ejpam-4215	63	4	∩n	∩n	NOUN
ejpam-4215	63	5	̸=	̸=	PROPN
ejpam-4215	63	6	0	0	NUM
ejpam-4215	63	7	,	,	PUNCT
ejpam-4215	63	8	so	so	SCONJ
ejpam-4215	63	9	that	that	SCONJ
ejpam-4215	63	10	b	b	PROPN
ejpam-4215	63	11	∩n	∩n	PROPN
ejpam-4215	63	12	is	be	AUX
ejpam-4215	63	13	a	a	DET
ejpam-4215	63	14	nonzero	nonzero	ADJ
ejpam-4215	63	15	cosingular	cosingular	ADJ
ejpam-4215	63	16	submodule	submodule	NOUN
ejpam-4215	63	17	of	of	ADP
ejpam-4215	63	18	b	b	PROPN
ejpam-4215	63	19	(	(	PUNCT
ejpam-4215	63	20	since	since	SCONJ
ejpam-4215	63	21	b	b	PROPN
ejpam-4215	63	22	∩n	∩n	PROPN
ejpam-4215	63	23	≤	≤	NOUN
ejpam-4215	63	24	n	n	CCONJ
ejpam-4215	63	25	and	and	CCONJ
ejpam-4215	63	26	by	by	ADP
ejpam-4215	63	27	lemma	lemma	PROPN
ejpam-4215	63	28	2.2	2.2	NUM
ejpam-4215	63	29	in	in	ADP
ejpam-4215	63	30	[	[	X
ejpam-4215	63	31	2	2	NUM
ejpam-4215	63	32	]	]	PUNCT
ejpam-4215	63	33	z∗(b	z∗(b	PROPN
ejpam-4215	63	34	∩n	∩n	NOUN
ejpam-4215	63	35	)	)	PUNCT
ejpam-4215	64	1	=	=	PRON
ejpam-4215	64	2	(	(	PUNCT
ejpam-4215	64	3	b	b	PROPN
ejpam-4215	64	4	∩n	∩n	NOUN
ejpam-4215	64	5	)	)	PUNCT
ejpam-4215	64	6	∩	∩	NOUN
ejpam-4215	64	7	z∗(n	z∗(n	NUM
ejpam-4215	64	8	)	)	PUNCT
ejpam-4215	64	9	=	=	PUNCT
ejpam-4215	65	1	(	(	PUNCT
ejpam-4215	65	2	b	b	NOUN
ejpam-4215	65	3	∩n	∩n	NOUN
ejpam-4215	65	4	)	)	PUNCT
ejpam-4215	65	5	∩n	∩n	NOUN
ejpam-4215	65	6	=	=	SYM
ejpam-4215	65	7	b	b	PROPN
ejpam-4215	65	8	∩n	∩n	PROPN
ejpam-4215	65	9	.	.	PUNCT
ejpam-4215	66	1	since	since	SCONJ
ejpam-4215	66	2	a	a	DET
ejpam-4215	66	3	≤e∗	≤e∗	PROPN
ejpam-4215	66	4	b	b	NOUN
ejpam-4215	66	5	then	then	ADV
ejpam-4215	66	6	a	a	DET
ejpam-4215	66	7	∩b	∩b	NOUN
ejpam-4215	66	8	∩n	∩n	NOUN
ejpam-4215	66	9	̸=	̸=	PROPN
ejpam-4215	66	10	0	0	NUM
ejpam-4215	66	11	and	and	CCONJ
ejpam-4215	66	12	a	a	DET
ejpam-4215	66	13	∩n	∩n	NOUN
ejpam-4215	66	14	̸=	̸=	PROPN
ejpam-4215	66	15	0	0	NUM
ejpam-4215	66	16	.	.	PUNCT
ejpam-4215	67	1	therefore	therefore	ADV
ejpam-4215	67	2	,	,	PUNCT
ejpam-4215	67	3	a	a	DET
ejpam-4215	67	4	≤e∗	≤e∗	PROPN
ejpam-4215	67	5	m	m	NOUN
ejpam-4215	67	6	.	.	PUNCT
ejpam-4215	68	1	corollary	corollary	ADJ
ejpam-4215	68	2	1	1	NUM
ejpam-4215	68	3	.	.	PUNCT
ejpam-4215	69	1	if	if	SCONJ
ejpam-4215	69	2	a1	a1	NOUN
ejpam-4215	69	3	≤	≤	PROPN
ejpam-4215	69	4	a2	a2	PROPN
ejpam-4215	69	5	≤	≤	PROPN
ejpam-4215	69	6	a3	a3	NOUN
ejpam-4215	69	7	≤	≤	PUNCT
ejpam-4215	69	8	m	m	PROPN
ejpam-4215	69	9	and	and	CCONJ
ejpam-4215	69	10	a1	a1	VERB
ejpam-4215	69	11	≤e∗	≤e∗	PROPN
ejpam-4215	69	12	m	m	NOUN
ejpam-4215	69	13	,	,	PUNCT
ejpam-4215	69	14	then	then	ADV
ejpam-4215	69	15	a2	a2	PROPN
ejpam-4215	69	16	≤e∗	≤e∗	PROPN
ejpam-4215	69	17	a3	a3	NOUN
ejpam-4215	69	18	.	.	PUNCT
ejpam-4215	70	1	proof	proof	NOUN
ejpam-4215	70	2	.	.	PUNCT
ejpam-4215	71	1	let	let	VERB
ejpam-4215	71	2	l	l	NOUN
ejpam-4215	71	3	be	be	AUX
ejpam-4215	71	4	a	a	DET
ejpam-4215	71	5	nonzero	nonzero	NOUN
ejpam-4215	71	6	cosingular	cosingular	ADJ
ejpam-4215	71	7	in	in	ADP
ejpam-4215	71	8	a3	a3	NOUN
ejpam-4215	71	9	.	.	PUNCT
ejpam-4215	72	1	by	by	ADP
ejpam-4215	72	2	lemma	lemma	PROPN
ejpam-4215	72	3	1	1	NUM
ejpam-4215	72	4	,	,	PUNCT
ejpam-4215	72	5	we	we	PRON
ejpam-4215	72	6	have	have	VERB
ejpam-4215	72	7	that	that	DET
ejpam-4215	72	8	l	l	NOUN
ejpam-4215	72	9	is	be	AUX
ejpam-4215	72	10	cosingular	cosingular	ADJ
ejpam-4215	72	11	in	in	ADP
ejpam-4215	72	12	m	m	PROPN
ejpam-4215	72	13	and	and	CCONJ
ejpam-4215	72	14	since	since	SCONJ
ejpam-4215	72	15	a1	a1	PROPN
ejpam-4215	72	16	≤e∗	≤e∗	PROPN
ejpam-4215	72	17	m	m	PROPN
ejpam-4215	72	18	.	.	PUNCT
ejpam-4215	73	1	thus	thus	ADV
ejpam-4215	73	2	,	,	PUNCT
ejpam-4215	73	3	a1	a1	NOUN
ejpam-4215	73	4	∩l	∩l	ADP
ejpam-4215	73	5	̸=	̸=	PROPN
ejpam-4215	73	6	0	0	NUM
ejpam-4215	73	7	and	and	CCONJ
ejpam-4215	73	8	since	since	SCONJ
ejpam-4215	73	9	a1	a1	NOUN
ejpam-4215	73	10	≤	≤	NOUN
ejpam-4215	73	11	a2	a2	PROPN
ejpam-4215	73	12	.	.	PUNCT
ejpam-4215	74	1	therefore	therefore	ADV
ejpam-4215	74	2	,	,	PUNCT
ejpam-4215	74	3	a2	a2	PROPN
ejpam-4215	74	4	∩l	∩l	VERB
ejpam-4215	74	5	̸=	̸=	PROPN
ejpam-4215	74	6	0	0	NUM
ejpam-4215	74	7	,	,	PUNCT
ejpam-4215	74	8	i.e.	i.e.	X
ejpam-4215	74	9	a2	a2	PROPN
ejpam-4215	74	10	≤e∗	≤e∗	PROPN
ejpam-4215	74	11	a3	a3	NOUN
ejpam-4215	74	12	.	.	PUNCT
ejpam-4215	75	1	proposition	proposition	NOUN
ejpam-4215	75	2	2	2	NUM
ejpam-4215	75	3	.	.	PUNCT
ejpam-4215	76	1	let	let	VERB
ejpam-4215	76	2	f	f	NOUN
ejpam-4215	76	3	:	:	PUNCT
ejpam-4215	76	4	m	m	VERB
ejpam-4215	76	5	→	→	SYM
ejpam-4215	76	6	m	m	AUX
ejpam-4215	76	7	′	′	NUM
ejpam-4215	76	8	be	be	VERB
ejpam-4215	76	9	r	r	NOUN
ejpam-4215	76	10	-	-	PUNCT
ejpam-4215	76	11	homomorphism	homomorphism	NOUN
ejpam-4215	76	12	,	,	PUNCT
ejpam-4215	76	13	if	if	SCONJ
ejpam-4215	76	14	a	a	DET
ejpam-4215	76	15	≤e∗	≤e∗	NOUN
ejpam-4215	76	16	m	m	VERB
ejpam-4215	76	17	′	′	NOUN
ejpam-4215	76	18	,	,	PUNCT
ejpam-4215	76	19	then	then	ADV
ejpam-4215	76	20	f−1(a	f−1(a	PROPN
ejpam-4215	76	21	)	)	PUNCT
ejpam-4215	76	22	≤e∗	≤e∗	PROPN
ejpam-4215	76	23	m	m	NOUN
ejpam-4215	76	24	.	.	PUNCT
ejpam-4215	77	1	proof	proof	NOUN
ejpam-4215	77	2	.	.	PUNCT
ejpam-4215	78	1	let	let	VERB
ejpam-4215	78	2	a	a	DET
ejpam-4215	78	3	≤e∗	≤e∗	NOUN
ejpam-4215	78	4	m	m	NOUN
ejpam-4215	78	5	′	′	NOUN
ejpam-4215	78	6	.	.	PUNCT
ejpam-4215	79	1	hence	hence	ADV
ejpam-4215	79	2	,	,	PUNCT
ejpam-4215	79	3	f−1(a	f−1(a	PROPN
ejpam-4215	79	4	)	)	PUNCT
ejpam-4215	79	5	≤	≤	PUNCT
ejpam-4215	79	6	m	m	VERB
ejpam-4215	79	7	,	,	PUNCT
ejpam-4215	79	8	suppose	suppose	VERB
ejpam-4215	79	9	that	that	SCONJ
ejpam-4215	79	10	f−1(a	f−1(a	PROPN
ejpam-4215	79	11	)	)	PUNCT
ejpam-4215	79	12	is	be	AUX
ejpam-4215	79	13	not	not	PART
ejpam-4215	79	14	e∗-essential	e∗-essential	PROPN
ejpam-4215	79	15	submodule	submodule	NOUN
ejpam-4215	79	16	of	of	ADP
ejpam-4215	79	17	m	m	PROPN
ejpam-4215	79	18	,	,	PUNCT
ejpam-4215	79	19	i.e.	i.e.	X
ejpam-4215	79	20	there	there	PRON
ejpam-4215	79	21	exists	exist	VERB
ejpam-4215	79	22	a	a	DET
ejpam-4215	79	23	nonzero	nonzero	PROPN
ejpam-4215	79	24	cosingular	cosingular	PROPN
ejpam-4215	79	25	submodule	submodule	PROPN
ejpam-4215	79	26	b	b	PROPN
ejpam-4215	79	27	of	of	ADP
ejpam-4215	79	28	m	m	PRON
ejpam-4215	79	29	such	such	ADJ
ejpam-4215	79	30	that	that	DET
ejpam-4215	79	31	f−1(a	f−1(a	NOUN
ejpam-4215	79	32	)	)	PUNCT
ejpam-4215	79	33	∩	∩	PROPN
ejpam-4215	79	34	b	b	X
ejpam-4215	79	35	=	=	SYM
ejpam-4215	79	36	0	0	PROPN
ejpam-4215	79	37	.	.	PUNCT
ejpam-4215	80	1	since	since	SCONJ
ejpam-4215	80	2	ker(f	ker(f	PROPN
ejpam-4215	80	3	|b	|b	PROPN
ejpam-4215	80	4	)	)	PUNCT
ejpam-4215	80	5	=	=	SYM
ejpam-4215	80	6	f−1(a	f−1(a	NOUN
ejpam-4215	80	7	)	)	PUNCT
ejpam-4215	80	8	∩	∩	NOUN
ejpam-4215	80	9	b	b	X
ejpam-4215	80	10	=	=	SYM
ejpam-4215	80	11	0	0	PROPN
ejpam-4215	80	12	.	.	PUNCT
ejpam-4215	81	1	thus	thus	ADV
ejpam-4215	81	2	,	,	PUNCT
ejpam-4215	81	3	b	b	X
ejpam-4215	81	4	∼=	∼=	ADP
ejpam-4215	81	5	f(b	f(b	NOUN
ejpam-4215	81	6	)	)	PUNCT
ejpam-4215	81	7	.	.	PUNCT
ejpam-4215	82	1	also	also	ADV
ejpam-4215	82	2	,	,	PUNCT
ejpam-4215	82	3	we	we	PRON
ejpam-4215	82	4	have	have	VERB
ejpam-4215	82	5	that	that	PRON
ejpam-4215	82	6	a	a	DET
ejpam-4215	82	7	∩	∩	ADJ
ejpam-4215	82	8	f(b	f(b	X
ejpam-4215	82	9	)	)	PUNCT
ejpam-4215	82	10	=	=	SYM
ejpam-4215	82	11	0	0	PUNCT
ejpam-4215	82	12	since	since	SCONJ
ejpam-4215	82	13	if	if	SCONJ
ejpam-4215	82	14	not	not	PART
ejpam-4215	82	15	,	,	PUNCT
ejpam-4215	82	16	i.e.	i.e.	X
ejpam-4215	82	17	there	there	PRON
ejpam-4215	82	18	exists	exist	VERB
ejpam-4215	82	19	0	0	NUM
ejpam-4215	83	1	̸=	̸=	NOUN
ejpam-4215	83	2	x	x	SYM
ejpam-4215	83	3	=	=	SYM
ejpam-4215	83	4	f(b	f(b	X
ejpam-4215	83	5	)	)	PUNCT
ejpam-4215	83	6	∈	∈	PROPN
ejpam-4215	83	7	a	a	DET
ejpam-4215	83	8	∩	∩	ADJ
ejpam-4215	83	9	f(b	f(b	NOUN
ejpam-4215	83	10	)	)	PUNCT
ejpam-4215	83	11	.	.	PUNCT
ejpam-4215	84	1	hence	hence	ADV
ejpam-4215	84	2	,	,	PUNCT
ejpam-4215	84	3	0	0	NUM
ejpam-4215	84	4	̸=	̸=	PROPN
ejpam-4215	84	5	b	b	PROPN
ejpam-4215	84	6	∈	∈	PROPN
ejpam-4215	84	7	f−1(a	f−1(a	NOUN
ejpam-4215	84	8	)	)	PUNCT
ejpam-4215	84	9	∩	∩	PROPN
ejpam-4215	84	10	b	b	X
ejpam-4215	84	11	which	which	PRON
ejpam-4215	84	12	is	be	AUX
ejpam-4215	84	13	contradiction	contradiction	NOUN
ejpam-4215	84	14	.	.	PUNCT
ejpam-4215	85	1	since	since	SCONJ
ejpam-4215	85	2	b	b	NOUN
ejpam-4215	85	3	is	be	AUX
ejpam-4215	85	4	cosingular	cosingular	ADJ
ejpam-4215	85	5	by	by	ADP
ejpam-4215	85	6	lemma	lemma	PROPN
ejpam-4215	85	7	2.6	2.6	NUM
ejpam-4215	85	8	in	in	ADP
ejpam-4215	85	9	[	[	PUNCT
ejpam-4215	85	10	2	2	NUM
ejpam-4215	85	11	]	]	PUNCT
ejpam-4215	85	12	,	,	PUNCT
ejpam-4215	85	13	f(b	f(b	PROPN
ejpam-4215	85	14	)	)	PUNCT
ejpam-4215	85	15	is	be	AUX
ejpam-4215	85	16	cosingular	cosingular	ADJ
ejpam-4215	85	17	also	also	ADV
ejpam-4215	85	18	a	a	DET
ejpam-4215	85	19	≤e∗	≤e∗	PROPN
ejpam-4215	85	20	m	m	NOUN
ejpam-4215	85	21	′	′	NOUN
ejpam-4215	85	22	.	.	PUNCT
ejpam-4215	86	1	hence	hence	ADV
ejpam-4215	86	2	,	,	PUNCT
ejpam-4215	86	3	f(b	f(b	PROPN
ejpam-4215	86	4	)	)	PUNCT
ejpam-4215	86	5	=	=	SYM
ejpam-4215	86	6	0	0	NUM
ejpam-4215	86	7	which	which	PRON
ejpam-4215	86	8	is	be	AUX
ejpam-4215	86	9	contradiction	contradiction	NOUN
ejpam-4215	86	10	.	.	PUNCT
ejpam-4215	87	1	therefore	therefore	ADV
ejpam-4215	87	2	,	,	PUNCT
ejpam-4215	87	3	f−1(a	f−1(a	PROPN
ejpam-4215	87	4	)	)	PUNCT
ejpam-4215	87	5	≤e∗	≤e∗	PROPN
ejpam-4215	87	6	m	m	PROPN
ejpam-4215	87	7	.	.	PUNCT
ejpam-4215	88	1	proposition	proposition	NOUN
ejpam-4215	88	2	3	3	NUM
ejpam-4215	88	3	.	.	PUNCT
ejpam-4215	89	1	if	if	SCONJ
ejpam-4215	89	2	a	a	DET
ejpam-4215	89	3	≤e∗	≤e∗	PROPN
ejpam-4215	89	4	b	b	PROPN
ejpam-4215	89	5	≤	≤	NUM
ejpam-4215	89	6	m	m	PROPN
ejpam-4215	89	7	and	and	CCONJ
ejpam-4215	89	8	a	a	DET
ejpam-4215	89	9	′	′	NUM
ejpam-4215	89	10	≤e∗	≤e∗	PROPN
ejpam-4215	89	11	b	b	NOUN
ejpam-4215	89	12	′	′	NOUN
ejpam-4215	89	13	≤	≤	NUM
ejpam-4215	89	14	m	m	VERB
ejpam-4215	89	15	,	,	PUNCT
ejpam-4215	89	16	then	then	ADV
ejpam-4215	89	17	a	a	DET
ejpam-4215	89	18	∩a	∩a	PROPN
ejpam-4215	89	19	′	′	NUM
ejpam-4215	90	1	≤e∗	≤e∗	PROPN
ejpam-4215	90	2	b	b	X
ejpam-4215	91	1	∩b	∩b	NOUN
ejpam-4215	91	2	′	′	NOUN
ejpam-4215	91	3	.	.	PUNCT
ejpam-4215	92	1	proof	proof	NOUN
ejpam-4215	92	2	.	.	PUNCT
ejpam-4215	93	1	let	let	VERB
ejpam-4215	93	2	k	k	PRON
ejpam-4215	93	3	be	be	AUX
ejpam-4215	93	4	a	a	DET
ejpam-4215	93	5	nonzero	nonzero	ADJ
ejpam-4215	93	6	cosingular	cosingular	ADJ
ejpam-4215	93	7	submodule	submodule	NOUN
ejpam-4215	93	8	of	of	ADP
ejpam-4215	93	9	b	b	PROPN
ejpam-4215	93	10	∩	∩	NOUN
ejpam-4215	93	11	b	b	NOUN
ejpam-4215	93	12	′	′	NUM
ejpam-4215	93	13	.	.	PUNCT
ejpam-4215	94	1	by	by	ADP
ejpam-4215	94	2	lemma	lemma	PROPN
ejpam-4215	94	3	1	1	NUM
ejpam-4215	94	4	k	k	NOUN
ejpam-4215	94	5	be	be	AUX
ejpam-4215	94	6	a	a	DET
ejpam-4215	94	7	nonzero	nonzero	ADJ
ejpam-4215	94	8	cosingular	cosingular	ADJ
ejpam-4215	94	9	submodule	submodule	NOUN
ejpam-4215	94	10	of	of	ADP
ejpam-4215	94	11	b	b	PROPN
ejpam-4215	94	12	and	and	CCONJ
ejpam-4215	94	13	b	b	NOUN
ejpam-4215	94	14	′	′	NOUN
ejpam-4215	94	15	.	.	PUNCT
ejpam-4215	95	1	since	since	SCONJ
ejpam-4215	95	2	a	a	DET
ejpam-4215	95	3	≤e∗	≤e∗	PROPN
ejpam-4215	95	4	b.	b.	NOUN
ejpam-4215	96	1	so	so	SCONJ
ejpam-4215	96	2	that	that	SCONJ
ejpam-4215	96	3	,	,	PUNCT
ejpam-4215	96	4	a	a	DET
ejpam-4215	96	5	∩	∩	NOUN
ejpam-4215	96	6	k	k	PROPN
ejpam-4215	96	7	̸=	̸=	PROPN
ejpam-4215	96	8	0	0	PUNCT
ejpam-4215	97	1	and	and	CCONJ
ejpam-4215	97	2	since	since	SCONJ
ejpam-4215	97	3	a	a	DET
ejpam-4215	97	4	∩k	∩k	NOUN
ejpam-4215	97	5	is	be	AUX
ejpam-4215	97	6	a	a	DET
ejpam-4215	97	7	nonzero	nonzero	ADJ
ejpam-4215	97	8	submodule	submodule	NOUN
ejpam-4215	97	9	of	of	ADP
ejpam-4215	97	10	cosingular	cosingular	PROPN
ejpam-4215	97	11	k.	k.	PROPN
ejpam-4215	97	12	hence	hence	ADV
ejpam-4215	97	13	,	,	PUNCT
ejpam-4215	97	14	a	a	DET
ejpam-4215	97	15	∩k	∩k	NOUN
ejpam-4215	97	16	is	be	AUX
ejpam-4215	97	17	cosingular	cosingular	ADJ
ejpam-4215	97	18	.	.	PUNCT
ejpam-4215	98	1	but	but	CCONJ
ejpam-4215	98	2	,	,	PUNCT
ejpam-4215	98	3	a	a	DET
ejpam-4215	98	4	′	′	NUM
ejpam-4215	98	5	≤e∗	≤e∗	PROPN
ejpam-4215	98	6	b	b	NOUN
ejpam-4215	98	7	′	′	NOUN
ejpam-4215	98	8	.	.	PUNCT
ejpam-4215	99	1	hence	hence	ADV
ejpam-4215	99	2	,	,	PUNCT
ejpam-4215	99	3	a	a	DET
ejpam-4215	99	4	′	′	NUM
ejpam-4215	99	5	∩	∩	NOUN
ejpam-4215	99	6	(	(	PUNCT
ejpam-4215	99	7	a	a	DET
ejpam-4215	99	8	∩k	∩k	NOUN
ejpam-4215	99	9	)	)	PUNCT
ejpam-4215	99	10	̸=	̸=	PROPN
ejpam-4215	99	11	0	0	NUM
ejpam-4215	99	12	.	.	PUNCT
ejpam-4215	100	1	therefore	therefore	ADV
ejpam-4215	100	2	,	,	PUNCT
ejpam-4215	100	3	a	a	DET
ejpam-4215	100	4	∩a	∩a	PROPN
ejpam-4215	100	5	′	′	NUM
ejpam-4215	100	6	≤e∗	≤e∗	PROPN
ejpam-4215	100	7	b	b	X
ejpam-4215	100	8	∩b	∩b	NOUN
ejpam-4215	100	9	′	′	NOUN
ejpam-4215	100	10	.	.	PUNCT
ejpam-4215	101	1	corollary	corollary	ADJ
ejpam-4215	101	2	2	2	NUM
ejpam-4215	101	3	.	.	PUNCT
ejpam-4215	102	1	let	let	AUX
ejpam-4215	102	2	bj	bj	VERB
ejpam-4215	102	3	≤e∗	≤e∗	PROPN
ejpam-4215	102	4	m	m	NOUN
ejpam-4215	102	5	for	for	ADP
ejpam-4215	102	6	each	each	PRON
ejpam-4215	102	7	j	j	PROPN
ejpam-4215	103	1	=	=	SYM
ejpam-4215	103	2	1	1	NUM
ejpam-4215	103	3	,	,	PUNCT
ejpam-4215	103	4	...	...	PUNCT
ejpam-4215	103	5	,	,	PUNCT
ejpam-4215	103	6	n	n	CCONJ
ejpam-4215	103	7	,	,	PUNCT
ejpam-4215	104	1	then	then	ADV
ejpam-4215	104	2	∩n	∩n	PROPN
ejpam-4215	104	3	i=1	i=1	PROPN
ejpam-4215	104	4	≤e∗	≤e∗	PROPN
ejpam-4215	104	5	m	m	VERB
ejpam-4215	104	6	proof	proof	NOUN
ejpam-4215	104	7	.	.	PUNCT
ejpam-4215	105	1	the	the	DET
ejpam-4215	105	2	prove	prove	NOUN
ejpam-4215	105	3	by	by	ADP
ejpam-4215	105	4	induction	induction	NOUN
ejpam-4215	105	5	on	on	ADP
ejpam-4215	105	6	n.	n.	NOUN
ejpam-4215	105	7	proposition	proposition	NOUN
ejpam-4215	105	8	4	4	NUM
ejpam-4215	105	9	.	.	PUNCT
ejpam-4215	105	10	let	let	VERB
ejpam-4215	105	11	m	m	NOUN
ejpam-4215	105	12	=	=	VERB
ejpam-4215	105	13	m1	m1	PROPN
ejpam-4215	105	14	⊕m2	⊕m2	PROPN
ejpam-4215	105	15	with	with	ADP
ejpam-4215	105	16	k1	k1	NOUN
ejpam-4215	105	17	≤	≤	ADJ
ejpam-4215	105	18	m1	m1	PROPN
ejpam-4215	105	19	and	and	CCONJ
ejpam-4215	105	20	k2	k2	PROPN
ejpam-4215	105	21	≤	≤	PROPN
ejpam-4215	105	22	m2	m2	PROPN
ejpam-4215	105	23	,	,	PUNCT
ejpam-4215	105	24	then	then	ADV
ejpam-4215	105	25	k1	k1	PROPN
ejpam-4215	105	26	≤e∗	≤e∗	PROPN
ejpam-4215	105	27	m1	m1	PROPN
ejpam-4215	105	28	and	and	CCONJ
ejpam-4215	105	29	k2	k2	PROPN
ejpam-4215	105	30	≤e∗	≤e∗	PROPN
ejpam-4215	105	31	m2	m2	PROPN
ejpam-4215	105	32	if	if	SCONJ
ejpam-4215	105	33	and	and	CCONJ
ejpam-4215	105	34	only	only	ADV
ejpam-4215	105	35	if	if	SCONJ
ejpam-4215	105	36	,	,	PUNCT
ejpam-4215	105	37	k1	k1	PROPN
ejpam-4215	105	38	⊕	⊕	PROPN
ejpam-4215	105	39	k2	k2	PROPN
ejpam-4215	105	40	≤e∗	≤e∗	PROPN
ejpam-4215	105	41	m	m	PROPN
ejpam-4215	105	42	h.r	h.r	PROPN
ejpam-4215	105	43	.	.	PROPN
ejpam-4215	105	44	baanoon	baanoon	PROPN
ejpam-4215	105	45	,	,	PUNCT
ejpam-4215	105	46	w.	w.	PROPN
ejpam-4215	105	47	khalid	khalid	PROPN
ejpam-4215	105	48	/	/	PUNCT
ejpam-4215	105	49	eur	eur	PROPN
ejpam-4215	105	50	.	.	PUNCT
ejpam-4215	106	1	j.	j.	PROPN
ejpam-4215	106	2	pure	pure	PROPN
ejpam-4215	106	3	appl	appl	PROPN
ejpam-4215	106	4	.	.	PROPN
ejpam-4215	106	5	math	math	PROPN
ejpam-4215	106	6	,	,	PUNCT
ejpam-4215	106	7	15	15	NUM
ejpam-4215	106	8	(	(	PUNCT
ejpam-4215	106	9	1	1	NUM
ejpam-4215	106	10	)	)	PUNCT
ejpam-4215	106	11	(	(	PUNCT
ejpam-4215	106	12	2022	2022	NUM
ejpam-4215	106	13	)	)	PUNCT
ejpam-4215	106	14	,	,	PUNCT
ejpam-4215	106	15	224	224	NUM
ejpam-4215	106	16	-	-	SYM
ejpam-4215	106	17	228	228	NUM
ejpam-4215	106	18	227	227	NUM
ejpam-4215	106	19	proof	proof	NOUN
ejpam-4215	106	20	.	.	PUNCT
ejpam-4215	107	1	⇒	⇒	NOUN
ejpam-4215	107	2	)	)	PUNCT
ejpam-4215	107	3	there	there	PRON
ejpam-4215	107	4	exists	exist	VERB
ejpam-4215	107	5	an	an	DET
ejpam-4215	107	6	r	r	NOUN
ejpam-4215	107	7	-	-	PUNCT
ejpam-4215	107	8	homomorphism	homomorphism	NOUN
ejpam-4215	107	9	ρ1	ρ1	NOUN
ejpam-4215	107	10	:	:	PUNCT
ejpam-4215	107	11	m1	m1	PROPN
ejpam-4215	107	12	⊕	⊕	PROPN
ejpam-4215	107	13	m2	m2	PROPN
ejpam-4215	107	14	→	→	SYM
ejpam-4215	107	15	m1	m1	PROPN
ejpam-4215	107	16	and	and	CCONJ
ejpam-4215	107	17	ρ2	ρ2	PROPN
ejpam-4215	107	18	:	:	PUNCT
ejpam-4215	107	19	m1	m1	PROPN
ejpam-4215	107	20	⊕	⊕	PROPN
ejpam-4215	107	21	m2	m2	PROPN
ejpam-4215	107	22	→	→	SYM
ejpam-4215	107	23	m2	m2	PROPN
ejpam-4215	107	24	which	which	PRON
ejpam-4215	107	25	define	define	VERB
ejpam-4215	107	26	by	by	ADP
ejpam-4215	107	27	ρ1(m1,m2	ρ1(m1,m2	NOUN
ejpam-4215	107	28	)	)	PUNCT
ejpam-4215	108	1	=	=	SYM
ejpam-4215	108	2	m1	m1	PROPN
ejpam-4215	108	3	and	and	CCONJ
ejpam-4215	108	4	ρ2(m1,m2	ρ2(m1,m2	PROPN
ejpam-4215	108	5	)	)	PUNCT
ejpam-4215	108	6	=	=	SYM
ejpam-4215	108	7	m2	m2	PROPN
ejpam-4215	108	8	.	.	PUNCT
ejpam-4215	109	1	by	by	ADP
ejpam-4215	109	2	proposition	proposition	NOUN
ejpam-4215	109	3	2	2	NUM
ejpam-4215	109	4	ρ−1	ρ−1	PROPN
ejpam-4215	109	5	1	1	NUM
ejpam-4215	109	6	(	(	PUNCT
ejpam-4215	109	7	k1	k1	NOUN
ejpam-4215	109	8	)	)	PUNCT
ejpam-4215	109	9	=	=	SYM
ejpam-4215	109	10	k1	k1	PROPN
ejpam-4215	109	11	⊕	⊕	PROPN
ejpam-4215	109	12	m2	m2	PROPN
ejpam-4215	109	13	≤e∗	≤e∗	PROPN
ejpam-4215	109	14	m1	m1	PROPN
ejpam-4215	109	15	⊕	⊕	PROPN
ejpam-4215	109	16	m2	m2	PROPN
ejpam-4215	109	17	and	and	CCONJ
ejpam-4215	109	18	ρ−1	ρ−1	PROPN
ejpam-4215	109	19	2	2	NUM
ejpam-4215	109	20	(	(	PUNCT
ejpam-4215	109	21	k2	k2	NOUN
ejpam-4215	109	22	)	)	PUNCT
ejpam-4215	109	23	=	=	SYM
ejpam-4215	109	24	m1	m1	PROPN
ejpam-4215	109	25	⊕	⊕	PROPN
ejpam-4215	109	26	k2	k2	PROPN
ejpam-4215	109	27	≤e∗	≤e∗	PROPN
ejpam-4215	109	28	m1	m1	PROPN
ejpam-4215	109	29	⊕	⊕	PROPN
ejpam-4215	109	30	m2	m2	PROPN
ejpam-4215	109	31	.	.	PUNCT
ejpam-4215	110	1	hence	hence	ADV
ejpam-4215	110	2	,	,	PUNCT
ejpam-4215	110	3	by	by	ADP
ejpam-4215	110	4	proposition	proposition	NOUN
ejpam-4215	110	5	3	3	NUM
ejpam-4215	110	6	k1	k1	PROPN
ejpam-4215	110	7	⊕m2	⊕m2	PROPN
ejpam-4215	110	8	∩m1	∩m1	PROPN
ejpam-4215	110	9	⊕k2	⊕k2	NOUN
ejpam-4215	110	10	=	=	PROPN
ejpam-4215	110	11	k1	k1	PROPN
ejpam-4215	110	12	⊕	⊕	PROPN
ejpam-4215	110	13	k2	k2	PROPN
ejpam-4215	110	14	≤e∗	≤e∗	PROPN
ejpam-4215	110	15	m	m	NUM
ejpam-4215	110	16	⇐	⇐	PROPN
ejpam-4215	110	17	)	)	PUNCT
ejpam-4215	110	18	there	there	PRON
ejpam-4215	110	19	exists	exist	VERB
ejpam-4215	110	20	an	an	DET
ejpam-4215	110	21	r	r	NOUN
ejpam-4215	110	22	-	-	PUNCT
ejpam-4215	110	23	homomorphism	homomorphism	NOUN
ejpam-4215	110	24	j1	j1	NOUN
ejpam-4215	110	25	:	:	PUNCT
ejpam-4215	110	26	m1	m1	PROPN
ejpam-4215	110	27	→	→	SYM
ejpam-4215	110	28	m1	m1	PROPN
ejpam-4215	110	29	⊕m2	⊕m2	PROPN
ejpam-4215	110	30	and	and	CCONJ
ejpam-4215	110	31	j2	j2	PROPN
ejpam-4215	110	32	:	:	PUNCT
ejpam-4215	110	33	m2	m2	PROPN
ejpam-4215	110	34	→	→	SYM
ejpam-4215	110	35	m1	m1	PROPN
ejpam-4215	110	36	⊕m2	⊕m2	NUM
ejpam-4215	110	37	which	which	PRON
ejpam-4215	110	38	define	define	VERB
ejpam-4215	110	39	by	by	ADP
ejpam-4215	110	40	j1(m1	j1(m1	NOUN
ejpam-4215	110	41	)	)	PUNCT
ejpam-4215	110	42	=	=	SYM
ejpam-4215	110	43	(	(	PUNCT
ejpam-4215	110	44	m1	m1	PROPN
ejpam-4215	110	45	,	,	PUNCT
ejpam-4215	110	46	0	0	NUM
ejpam-4215	110	47	)	)	PUNCT
ejpam-4215	110	48	and	and	CCONJ
ejpam-4215	110	49	j2(m2	j2(m2	NOUN
ejpam-4215	110	50	)	)	PUNCT
ejpam-4215	110	51	=	=	SYM
ejpam-4215	110	52	(	(	PUNCT
ejpam-4215	110	53	0,m2	0,m2	NOUN
ejpam-4215	110	54	)	)	PUNCT
ejpam-4215	110	55	.	.	PUNCT
ejpam-4215	111	1	by	by	ADP
ejpam-4215	111	2	proposition	proposition	NOUN
ejpam-4215	111	3	2	2	NUM
ejpam-4215	111	4	j−1	j−1	NOUN
ejpam-4215	111	5	1	1	NUM
ejpam-4215	111	6	(	(	PUNCT
ejpam-4215	111	7	k1	k1	PROPN
ejpam-4215	111	8	⊕	⊕	PROPN
ejpam-4215	111	9	k2	k2	PROPN
ejpam-4215	111	10	)	)	PUNCT
ejpam-4215	111	11	=	=	SYM
ejpam-4215	111	12	k1	k1	PROPN
ejpam-4215	111	13	≤e∗	≤e∗	PROPN
ejpam-4215	111	14	m1	m1	PROPN
ejpam-4215	111	15	and	and	CCONJ
ejpam-4215	111	16	j−1	j−1	PROPN
ejpam-4215	111	17	2	2	NUM
ejpam-4215	111	18	(	(	PUNCT
ejpam-4215	111	19	k1	k1	PROPN
ejpam-4215	111	20	⊕	⊕	PROPN
ejpam-4215	111	21	k2	k2	PROPN
ejpam-4215	111	22	)	)	PUNCT
ejpam-4215	111	23	=	=	SYM
ejpam-4215	111	24	k2	k2	PROPN
ejpam-4215	111	25	≤e∗	≤e∗	PROPN
ejpam-4215	111	26	m2	m2	PROPN
ejpam-4215	111	27	.	.	PUNCT
ejpam-4215	112	1	in	in	ADP
ejpam-4215	112	2	the	the	DET
ejpam-4215	112	3	following	follow	VERB
ejpam-4215	112	4	proposition	proposition	NOUN
ejpam-4215	112	5	we	we	PRON
ejpam-4215	112	6	will	will	AUX
ejpam-4215	112	7	give	give	VERB
ejpam-4215	112	8	a	a	DET
ejpam-4215	112	9	characterization	characterization	NOUN
ejpam-4215	112	10	of	of	ADP
ejpam-4215	112	11	e∗-essential	e∗-essential	PROPN
ejpam-4215	112	12	submodule	submodule	NOUN
ejpam-4215	112	13	.	.	PUNCT
ejpam-4215	113	1	proposition	proposition	NOUN
ejpam-4215	113	2	5	5	NUM
ejpam-4215	113	3	.	.	PUNCT
ejpam-4215	114	1	let	let	VERB
ejpam-4215	114	2	m	m	PRON
ejpam-4215	114	3	be	be	AUX
ejpam-4215	114	4	r	r	NOUN
ejpam-4215	114	5	-	-	PUNCT
ejpam-4215	114	6	module	module	NOUN
ejpam-4215	114	7	and	and	CCONJ
ejpam-4215	114	8	n	n	PRON
ejpam-4215	114	9	≤	≤	NOUN
ejpam-4215	114	10	m	m	VERB
ejpam-4215	114	11	,	,	PUNCT
ejpam-4215	114	12	then	then	ADV
ejpam-4215	114	13	n	n	CCONJ
ejpam-4215	114	14	is	be	AUX
ejpam-4215	114	15	e∗-essential	e∗-essential	PROPN
ejpam-4215	114	16	submodule	submodule	NOUN
ejpam-4215	114	17	of	of	ADP
ejpam-4215	114	18	m	m	PROPN
ejpam-4215	114	19	if	if	SCONJ
ejpam-4215	115	1	and	and	CCONJ
ejpam-4215	115	2	only	only	ADV
ejpam-4215	115	3	if	if	SCONJ
ejpam-4215	115	4	n	n	NOUN
ejpam-4215	115	5	∩	∩	NOUN
ejpam-4215	115	6	xr	xr	PROPN
ejpam-4215	115	7	̸=	̸=	PROPN
ejpam-4215	115	8	0	0	NUM
ejpam-4215	115	9	for	for	ADP
ejpam-4215	115	10	each	each	DET
ejpam-4215	115	11	nonzero	nonzero	PROPN
ejpam-4215	115	12	cyclic	cyclic	ADJ
ejpam-4215	115	13	cosingular	cosingular	ADJ
ejpam-4215	115	14	submodule	submodule	NOUN
ejpam-4215	115	15	of	of	ADP
ejpam-4215	115	16	m	m	PROPN
ejpam-4215	115	17	.	.	PUNCT
ejpam-4215	116	1	proof	proof	NOUN
ejpam-4215	116	2	.	.	PUNCT
ejpam-4215	117	1	⇒	⇒	NOUN
ejpam-4215	117	2	)	)	PUNCT
ejpam-4215	117	3	clear	clear	ADJ
ejpam-4215	117	4	.	.	PUNCT
ejpam-4215	118	1	⇐	⇐	ADJ
ejpam-4215	118	2	)	)	PUNCT
ejpam-4215	118	3	let	let	VERB
ejpam-4215	118	4	n	n	PRON
ejpam-4215	118	5	be	be	AUX
ejpam-4215	118	6	a	a	DET
ejpam-4215	118	7	submodule	submodule	NOUN
ejpam-4215	118	8	of	of	ADP
ejpam-4215	118	9	m	m	PROPN
ejpam-4215	118	10	and	and	CCONJ
ejpam-4215	118	11	k	k	PROPN
ejpam-4215	118	12	be	be	AUX
ejpam-4215	118	13	a	a	DET
ejpam-4215	118	14	nonzero	nonzero	ADJ
ejpam-4215	118	15	cosingular	cosingular	ADJ
ejpam-4215	118	16	submodule	submodule	NOUN
ejpam-4215	118	17	of	of	ADP
ejpam-4215	118	18	m	m	PROPN
ejpam-4215	118	19	.	.	PUNCT
ejpam-4215	119	1	hence	hence	ADV
ejpam-4215	119	2	,	,	PUNCT
ejpam-4215	119	3	there	there	PRON
ejpam-4215	119	4	exists	exist	VERB
ejpam-4215	119	5	0	0	NUM
ejpam-4215	119	6	̸=	̸=	PROPN
ejpam-4215	119	7	x	x	SYM
ejpam-4215	119	8	∈	∈	PROPN
ejpam-4215	119	9	k	k	PROPN
ejpam-4215	119	10	with	with	ADP
ejpam-4215	119	11	xr	xr	PROPN
ejpam-4215	119	12	≤	≤	PROPN
ejpam-4215	119	13	k	k	PROPN
ejpam-4215	119	14	,	,	PUNCT
ejpam-4215	119	15	also	also	ADV
ejpam-4215	119	16	z∗(xr	z∗(xr	NOUN
ejpam-4215	119	17	)	)	PUNCT
ejpam-4215	119	18	=	=	PUNCT
ejpam-4215	120	1	xr	xr	PROPN
ejpam-4215	120	2	.	.	PUNCT
ejpam-4215	121	1	so	so	ADV
ejpam-4215	121	2	by	by	ADP
ejpam-4215	121	3	hypothesis	hypothesis	NOUN
ejpam-4215	121	4	n	n	PROPN
ejpam-4215	121	5	∩	∩	NOUN
ejpam-4215	121	6	xr	xr	PROPN
ejpam-4215	121	7	̸=	̸=	PROPN
ejpam-4215	121	8	0	0	NUM
ejpam-4215	121	9	.	.	PUNCT
ejpam-4215	122	1	hence	hence	ADV
ejpam-4215	122	2	,	,	PUNCT
ejpam-4215	122	3	n	n	PRON
ejpam-4215	122	4	∩k	∩k	NOUN
ejpam-4215	122	5	̸=	̸=	PROPN
ejpam-4215	122	6	0	0	NUM
ejpam-4215	122	7	.	.	PUNCT
ejpam-4215	123	1	therefore	therefore	ADV
ejpam-4215	123	2	,	,	PUNCT
ejpam-4215	123	3	n	n	PRON
ejpam-4215	123	4	≤e∗	≤e∗	PROPN
ejpam-4215	123	5	m	m	VERB
ejpam-4215	123	6	.	.	PUNCT
ejpam-4215	124	1	in	in	ADP
ejpam-4215	124	2	the	the	DET
ejpam-4215	124	3	following	follow	VERB
ejpam-4215	124	4	proposition	proposition	NOUN
ejpam-4215	124	5	shows	show	VERB
ejpam-4215	124	6	that	that	SCONJ
ejpam-4215	124	7	,	,	PUNCT
ejpam-4215	124	8	the	the	DET
ejpam-4215	124	9	composition	composition	NOUN
ejpam-4215	124	10	of	of	ADP
ejpam-4215	124	11	e∗-essentialr	e∗-essentialr	NOUN
ejpam-4215	124	12	-	-	NOUN
ejpam-4215	124	13	monomorphism	monomorphism	NOUN
ejpam-4215	124	14	is	be	AUX
ejpam-4215	124	15	also	also	ADV
ejpam-4215	124	16	e∗-essential	e∗-essential	ADJ
ejpam-4215	124	17	r	r	NOUN
ejpam-4215	124	18	-	-	PUNCT
ejpam-4215	124	19	monomorphism	monomorphism	NOUN
ejpam-4215	124	20	.	.	PUNCT
ejpam-4215	125	1	proposition	proposition	NOUN
ejpam-4215	125	2	6	6	NUM
ejpam-4215	125	3	.	.	PUNCT
ejpam-4215	126	1	let	let	VERB
ejpam-4215	126	2	f	f	NOUN
ejpam-4215	126	3	:	:	PUNCT
ejpam-4215	126	4	a	a	DET
ejpam-4215	126	5	→	→	SYM
ejpam-4215	126	6	b	b	PROPN
ejpam-4215	126	7	and	and	CCONJ
ejpam-4215	126	8	g	g	PROPN
ejpam-4215	126	9	:	:	PUNCT
ejpam-4215	126	10	b	b	X
ejpam-4215	126	11	→	→	SYM
ejpam-4215	126	12	c	c	X
ejpam-4215	126	13	are	be	AUX
ejpam-4215	126	14	e∗-essential	e∗-essential	PROPN
ejpam-4215	126	15	r	r	NOUN
ejpam-4215	126	16	-	-	PUNCT
ejpam-4215	126	17	monomorphism	monomorphism	NOUN
ejpam-4215	126	18	.	.	PUNCT
ejpam-4215	127	1	then	then	ADV
ejpam-4215	127	2	,	,	PUNCT
ejpam-4215	127	3	g	g	PROPN
ejpam-4215	127	4	◦	◦	NOUN
ejpam-4215	127	5	f	f	X
ejpam-4215	127	6	:	:	PUNCT
ejpam-4215	127	7	a	a	DET
ejpam-4215	127	8	→	→	X
ejpam-4215	127	9	c	c	NOUN
ejpam-4215	127	10	is	be	AUX
ejpam-4215	127	11	also	also	ADV
ejpam-4215	127	12	e∗-essential	e∗-essential	ADJ
ejpam-4215	127	13	r	r	NOUN
ejpam-4215	127	14	-	-	PUNCT
ejpam-4215	127	15	monomorphism	monomorphism	NOUN
ejpam-4215	127	16	.	.	PUNCT
ejpam-4215	128	1	proof	proof	NOUN
ejpam-4215	128	2	.	.	PUNCT
ejpam-4215	129	1	let	let	VERB
ejpam-4215	129	2	l	l	NOUN
ejpam-4215	129	3	be	be	AUX
ejpam-4215	129	4	cosingular	cosingular	ADJ
ejpam-4215	129	5	submodule	submodule	NOUN
ejpam-4215	129	6	of	of	ADP
ejpam-4215	129	7	c	c	PROPN
ejpam-4215	129	8	such	such	ADJ
ejpam-4215	129	9	that	that	PRON
ejpam-4215	129	10	im(g	im(g	PUNCT
ejpam-4215	129	11	◦	◦	NOUN
ejpam-4215	129	12	f	f	X
ejpam-4215	129	13	)	)	PUNCT
ejpam-4215	129	14	∩	∩	NOUN
ejpam-4215	129	15	l	l	NOUN
ejpam-4215	130	1	=	=	SYM
ejpam-4215	131	1	0	0	X
ejpam-4215	131	2	.	.	PUNCT
ejpam-4215	132	1	since	since	SCONJ
ejpam-4215	132	2	g	g	PROPN
ejpam-4215	132	3	is	be	AUX
ejpam-4215	132	4	monomorphism	monomorphism	NOUN
ejpam-4215	132	5	0	0	X
ejpam-4215	132	6	=	=	SYM
ejpam-4215	132	7	kerg	kerg	NOUN
ejpam-4215	132	8	=	=	SYM
ejpam-4215	132	9	g−1(0	g−1(0	NOUN
ejpam-4215	132	10	)	)	PUNCT
ejpam-4215	132	11	=	=	SYM
ejpam-4215	133	1	g−1	g−1	X
ejpam-4215	133	2	(	(	PUNCT
ejpam-4215	133	3	im(g	im(g	PUNCT
ejpam-4215	133	4	◦	◦	NOUN
ejpam-4215	133	5	f	f	X
ejpam-4215	133	6	)	)	PUNCT
ejpam-4215	133	7	∩	∩	ADJ
ejpam-4215	133	8	l	l	NOUN
ejpam-4215	133	9	)	)	PUNCT
ejpam-4215	133	10	.	.	PUNCT
ejpam-4215	134	1	hence	hence	ADV
ejpam-4215	134	2	g−1	g−1	PROPN
ejpam-4215	134	3	(	(	PUNCT
ejpam-4215	134	4	im(g	im(g	PUNCT
ejpam-4215	134	5	◦	◦	NOUN
ejpam-4215	134	6	f	f	NOUN
ejpam-4215	134	7	)	)	PUNCT
ejpam-4215	134	8	)	)	PUNCT
ejpam-4215	134	9	∩	∩	PROPN
ejpam-4215	134	10	g−1(l	g−1(l	PROPN
ejpam-4215	134	11	)	)	PUNCT
ejpam-4215	134	12	=	=	SYM
ejpam-4215	134	13	im(f	im(f	NOUN
ejpam-4215	134	14	)	)	PUNCT
ejpam-4215	134	15	∩	∩	PROPN
ejpam-4215	134	16	g−1(l	g−1(l	PROPN
ejpam-4215	134	17	)	)	PUNCT
ejpam-4215	134	18	=	=	PUNCT
ejpam-4215	135	1	0	0	X
ejpam-4215	135	2	.	.	PUNCT
ejpam-4215	136	1	since	since	SCONJ
ejpam-4215	136	2	g−1	g−1	PROPN
ejpam-4215	136	3	is	be	AUX
ejpam-4215	136	4	r	r	NOUN
ejpam-4215	136	5	-	-	PUNCT
ejpam-4215	136	6	homomorphism	homomorphism	NOUN
ejpam-4215	136	7	and	and	CCONJ
ejpam-4215	136	8	l	l	NOUN
ejpam-4215	136	9	is	be	AUX
ejpam-4215	136	10	cosingular	cosingular	ADJ
ejpam-4215	136	11	submodule	submodule	NOUN
ejpam-4215	136	12	of	of	ADP
ejpam-4215	136	13	c.	c.	PROPN
ejpam-4215	136	14	hence	hence	ADV
ejpam-4215	136	15	g−1(l	g−1(l	PROPN
ejpam-4215	136	16	)	)	PUNCT
ejpam-4215	136	17	is	be	AUX
ejpam-4215	136	18	cosingular	cosingular	ADJ
ejpam-4215	136	19	submodule	submodule	NOUN
ejpam-4215	136	20	of	of	ADP
ejpam-4215	136	21	b	b	PROPN
ejpam-4215	136	22	and	and	CCONJ
ejpam-4215	136	23	since	since	SCONJ
ejpam-4215	136	24	im(f	im(f	NOUN
ejpam-4215	136	25	)	)	PUNCT
ejpam-4215	136	26	≤e∗	≤e∗	PROPN
ejpam-4215	136	27	b.	b.	PROPN
ejpam-4215	137	1	thus	thus	ADV
ejpam-4215	137	2	,	,	PUNCT
ejpam-4215	137	3	g−1(l	g−1(l	PROPN
ejpam-4215	137	4	)	)	PUNCT
ejpam-4215	137	5	=	=	SYM
ejpam-4215	137	6	0	0	NUM
ejpam-4215	137	7	and	and	CCONJ
ejpam-4215	137	8	im(g	im(g	ADJ
ejpam-4215	137	9	)	)	PUNCT
ejpam-4215	137	10	∩	∩	NOUN
ejpam-4215	137	11	l	l	NOUN
ejpam-4215	137	12	=	=	SYM
ejpam-4215	137	13	0	0	X
ejpam-4215	137	14	.	.	PUNCT
ejpam-4215	138	1	since	since	SCONJ
ejpam-4215	138	2	g	g	NOUN
ejpam-4215	138	3	:	:	PUNCT
ejpam-4215	138	4	b	b	X
ejpam-4215	138	5	→	→	SYM
ejpam-4215	138	6	c	c	PROPN
ejpam-4215	138	7	is	be	AUX
ejpam-4215	138	8	e∗-essential	e∗-essential	PROPN
ejpam-4215	138	9	.	.	PUNCT
ejpam-4215	139	1	therefore	therefore	ADV
ejpam-4215	139	2	,	,	PUNCT
ejpam-4215	139	3	l	l	NOUN
ejpam-4215	139	4	=	=	SYM
ejpam-4215	139	5	0	0	NUM
ejpam-4215	139	6	i.e.	i.e.	X
ejpam-4215	139	7	g	g	PROPN
ejpam-4215	139	8	◦	◦	PROPN
ejpam-4215	139	9	f	f	PROPN
ejpam-4215	139	10	is	be	AUX
ejpam-4215	139	11	e∗-essential	e∗-essential	PROPN
ejpam-4215	139	12	r	r	NOUN
ejpam-4215	139	13	-	-	PUNCT
ejpam-4215	139	14	monomorphism	monomorphism	NOUN
ejpam-4215	139	15	.	.	PUNCT
ejpam-4215	140	1	in	in	ADP
ejpam-4215	140	2	the	the	DET
ejpam-4215	140	3	following	follow	VERB
ejpam-4215	140	4	proposition	proposition	NOUN
ejpam-4215	140	5	we	we	PRON
ejpam-4215	140	6	will	will	AUX
ejpam-4215	140	7	give	give	VERB
ejpam-4215	140	8	another	another	DET
ejpam-4215	140	9	characterization	characterization	NOUN
ejpam-4215	140	10	of	of	ADP
ejpam-4215	140	11	noetherian	noetherian	ADJ
ejpam-4215	140	12	rmodule	rmodule	NOUN
ejpam-4215	140	13	.	.	PUNCT
ejpam-4215	141	1	also	also	ADV
ejpam-4215	141	2	,	,	PUNCT
ejpam-4215	141	3	it	it	PRON
ejpam-4215	141	4	is	be	AUX
ejpam-4215	141	5	show	show	VERB
ejpam-4215	141	6	the	the	DET
ejpam-4215	141	7	relationship	relationship	NOUN
ejpam-4215	141	8	between	between	ADP
ejpam-4215	141	9	e∗-essential	e∗-essential	PROPN
ejpam-4215	141	10	submodule	submodule	NOUN
ejpam-4215	141	11	and	and	CCONJ
ejpam-4215	141	12	noetherian	noetherian	ADJ
ejpam-4215	141	13	r	r	NOUN
ejpam-4215	141	14	-	-	PUNCT
ejpam-4215	141	15	module	module	NOUN
ejpam-4215	141	16	.	.	PUNCT
ejpam-4215	142	1	proposition	proposition	NOUN
ejpam-4215	142	2	7	7	NUM
ejpam-4215	142	3	.	.	PUNCT
ejpam-4215	143	1	an	an	DET
ejpam-4215	143	2	r	r	NOUN
ejpam-4215	143	3	-	-	PUNCT
ejpam-4215	143	4	module	module	NOUN
ejpam-4215	143	5	m	m	NOUN
ejpam-4215	143	6	is	be	AUX
ejpam-4215	143	7	noetherian	noetherian	ADJ
ejpam-4215	143	8	if	if	SCONJ
ejpam-4215	143	9	and	and	CCONJ
ejpam-4215	143	10	only	only	ADV
ejpam-4215	143	11	if	if	SCONJ
ejpam-4215	143	12	,	,	PUNCT
ejpam-4215	143	13	every	every	DET
ejpam-4215	143	14	e∗-essential	e∗-essential	PROPN
ejpam-4215	143	15	submodule	submodule	NOUN
ejpam-4215	143	16	of	of	ADP
ejpam-4215	143	17	m	m	PROPN
ejpam-4215	143	18	is	be	AUX
ejpam-4215	143	19	finitely	finitely	ADV
ejpam-4215	143	20	generated	generate	VERB
ejpam-4215	143	21	.	.	PUNCT
ejpam-4215	144	1	proof	proof	NOUN
ejpam-4215	144	2	.	.	PUNCT
ejpam-4215	145	1	⇒	⇒	NOUN
ejpam-4215	145	2	)	)	PUNCT
ejpam-4215	145	3	clear	clear	ADJ
ejpam-4215	145	4	.	.	PUNCT
ejpam-4215	146	1	⇐	⇐	ADJ
ejpam-4215	146	2	)	)	PUNCT
ejpam-4215	146	3	let	let	VERB
ejpam-4215	146	4	a	a	PRON
ejpam-4215	146	5	be	be	AUX
ejpam-4215	146	6	an	an	DET
ejpam-4215	146	7	essential	essential	ADJ
ejpam-4215	146	8	submodule	submodule	NOUN
ejpam-4215	146	9	of	of	ADP
ejpam-4215	146	10	m	m	PROPN
ejpam-4215	146	11	.	.	PUNCT
ejpam-4215	147	1	hence	hence	ADV
ejpam-4215	147	2	,	,	PUNCT
ejpam-4215	147	3	a	a	PRON
ejpam-4215	147	4	is	be	AUX
ejpam-4215	147	5	e∗-essential	e∗-essential	PROPN
ejpam-4215	147	6	and	and	CCONJ
ejpam-4215	147	7	by	by	ADP
ejpam-4215	147	8	the	the	DET
ejpam-4215	147	9	hypothsis	hypothsis	NOUN
ejpam-4215	147	10	a	a	PRON
ejpam-4215	147	11	is	be	AUX
ejpam-4215	147	12	a	a	DET
ejpam-4215	147	13	finitely	finitely	ADV
ejpam-4215	147	14	generated	generate	VERB
ejpam-4215	147	15	.	.	PUNCT
ejpam-4215	148	1	hence	hence	ADV
ejpam-4215	148	2	,	,	PUNCT
ejpam-4215	148	3	every	every	DET
ejpam-4215	148	4	essential	essential	ADJ
ejpam-4215	148	5	submodule	submodule	NOUN
ejpam-4215	148	6	is	be	AUX
ejpam-4215	148	7	finitely	finitely	ADV
ejpam-4215	148	8	generated	generate	VERB
ejpam-4215	148	9	by	by	ADP
ejpam-4215	148	10	[	[	X
ejpam-4215	148	11	3	3	NUM
ejpam-4215	148	12	]	]	PUNCT
ejpam-4215	148	13	.	.	PUNCT
ejpam-4215	149	1	therefore	therefore	ADV
ejpam-4215	149	2	,	,	PUNCT
ejpam-4215	149	3	m	m	VERB
ejpam-4215	149	4	is	be	AUX
ejpam-4215	149	5	noetherian	noetherian	ADJ
ejpam-4215	149	6	.	.	PUNCT
ejpam-4215	150	1	references	reference	NOUN
ejpam-4215	150	2	228	228	NUM
ejpam-4215	150	3	3	3	NUM
ejpam-4215	150	4	.	.	PUNCT
ejpam-4215	151	1	e∗-closed	e∗-close	VERB
ejpam-4215	151	2	submodule	submodule	NOUN
ejpam-4215	151	3	definition	definition	NOUN
ejpam-4215	151	4	2	2	NUM
ejpam-4215	151	5	.	.	PUNCT
ejpam-4215	151	6	a	a	DET
ejpam-4215	151	7	submodule	submodule	NOUN
ejpam-4215	151	8	a	a	PRON
ejpam-4215	151	9	of	of	ADP
ejpam-4215	151	10	r	r	NOUN
ejpam-4215	151	11	-	-	PUNCT
ejpam-4215	151	12	module	module	NOUN
ejpam-4215	151	13	c	c	NOUN
ejpam-4215	151	14	is	be	AUX
ejpam-4215	151	15	said	say	VERB
ejpam-4215	151	16	to	to	PART
ejpam-4215	151	17	be	be	AUX
ejpam-4215	151	18	e∗-closed	e∗-close	VERB
ejpam-4215	151	19	submodule	submodule	NOUN
ejpam-4215	151	20	of	of	ADP
ejpam-4215	151	21	c	c	PROPN
ejpam-4215	151	22	,	,	PUNCT
ejpam-4215	151	23	if	if	SCONJ
ejpam-4215	151	24	a	a	PRON
ejpam-4215	151	25	has	have	VERB
ejpam-4215	151	26	no	no	DET
ejpam-4215	151	27	proper	proper	ADJ
ejpam-4215	151	28	e∗-essential	e∗-essential	ADJ
ejpam-4215	151	29	extension	extension	NOUN
ejpam-4215	151	30	inside	inside	ADP
ejpam-4215	151	31	c.	c.	NOUN
ejpam-4215	151	32	denoted	denote	VERB
ejpam-4215	151	33	by	by	ADP
ejpam-4215	151	34	a	a	DET
ejpam-4215	151	35	≤e∗c	≤e∗c	PROPN
ejpam-4215	151	36	c.	c.	NOUN
ejpam-4215	151	37	examples	example	NOUN
ejpam-4215	151	38	and	and	CCONJ
ejpam-4215	151	39	remarks	remark	VERB
ejpam-4215	151	40	2	2	NUM
ejpam-4215	151	41	.	.	NOUN
ejpam-4215	151	42	1	1	NUM
ejpam-4215	151	43	.	.	X
ejpam-4215	152	1	for	for	ADP
ejpam-4215	152	2	any	any	DET
ejpam-4215	152	3	module	module	NOUN
ejpam-4215	152	4	m	m	PROPN
ejpam-4215	152	5	.	.	PUNCT
ejpam-4215	152	6	0	0	PUNCT
ejpam-4215	153	1	and	and	CCONJ
ejpam-4215	153	2	m	m	VERB
ejpam-4215	153	3	always	always	ADV
ejpam-4215	153	4	e∗-closed	e∗-close	VERB
ejpam-4215	153	5	.	.	PUNCT
ejpam-4215	154	1	2	2	X
ejpam-4215	154	2	.	.	X
ejpam-4215	154	3	in	in	ADP
ejpam-4215	154	4	z6	z6	PROPN
ejpam-4215	154	5	as	as	ADP
ejpam-4215	154	6	z6	z6	NOUN
ejpam-4215	154	7	-	-	PUNCT
ejpam-4215	154	8	module	module	NOUN
ejpam-4215	154	9	,	,	PUNCT
ejpam-4215	154	10	{	{	PUNCT
ejpam-4215	154	11	0	0	NUM
ejpam-4215	154	12	,	,	PUNCT
ejpam-4215	154	13	2	2	NUM
ejpam-4215	154	14	,	,	PUNCT
ejpam-4215	154	15	4	4	NUM
ejpam-4215	154	16	}	}	PUNCT
ejpam-4215	154	17	is	be	AUX
ejpam-4215	154	18	not	not	PART
ejpam-4215	154	19	e∗-closed	e∗-close	VERB
ejpam-4215	154	20	submodule	submodule	NOUN
ejpam-4215	154	21	since	since	SCONJ
ejpam-4215	154	22	{	{	PUNCT
ejpam-4215	154	23	0	0	NUM
ejpam-4215	154	24	,	,	PUNCT
ejpam-4215	154	25	2	2	NUM
ejpam-4215	154	26	,	,	PUNCT
ejpam-4215	154	27	4	4	NUM
ejpam-4215	154	28	}	}	PUNCT
ejpam-4215	154	29	is	be	AUX
ejpam-4215	154	30	e∗-essential	e∗-essential	PROPN
ejpam-4215	154	31	in	in	ADP
ejpam-4215	154	32	z6	z6	PROPN
ejpam-4215	154	33	.	.	PUNCT
ejpam-4215	155	1	in	in	ADP
ejpam-4215	155	2	the	the	DET
ejpam-4215	155	3	following	follow	VERB
ejpam-4215	155	4	proposition	proposition	NOUN
ejpam-4215	155	5	shows	show	VERB
ejpam-4215	155	6	that	that	SCONJ
ejpam-4215	155	7	when	when	SCONJ
ejpam-4215	155	8	the	the	DET
ejpam-4215	155	9	quotient	quotient	NOUN
ejpam-4215	155	10	submodule	submodule	NOUN
ejpam-4215	155	11	of	of	ADP
ejpam-4215	155	12	e∗-essential	e∗-essential	PROPN
ejpam-4215	155	13	submodule	submodule	NOUN
ejpam-4215	155	14	is	be	AUX
ejpam-4215	155	15	e∗-essential	e∗-essential	PROPN
ejpam-4215	155	16	:	:	PUNCT
ejpam-4215	155	17	proposition	proposition	NOUN
ejpam-4215	155	18	8	8	NUM
ejpam-4215	155	19	.	.	PUNCT
ejpam-4215	156	1	let	let	VERB
ejpam-4215	156	2	m	m	PRON
ejpam-4215	156	3	be	be	AUX
ejpam-4215	156	4	r	r	NOUN
ejpam-4215	156	5	-	-	PUNCT
ejpam-4215	156	6	module	module	NOUN
ejpam-4215	156	7	,	,	PUNCT
ejpam-4215	156	8	if	if	SCONJ
ejpam-4215	156	9	b	b	X
ejpam-4215	156	10	≤e∗c	≤e∗c	PROPN
ejpam-4215	156	11	m	m	PROPN
ejpam-4215	156	12	and	and	CCONJ
ejpam-4215	156	13	b	b	PROPN
ejpam-4215	156	14	≤	≤	NUM
ejpam-4215	156	15	k	k	PRON
ejpam-4215	156	16	≤e∗	≤e∗	PROPN
ejpam-4215	157	1	m	m	VERB
ejpam-4215	157	2	then	then	ADV
ejpam-4215	157	3	k	k	PROPN
ejpam-4215	157	4	b	b	PROPN
ejpam-4215	157	5	≤e∗	≤e∗	PROPN
ejpam-4215	157	6	m	m	PROPN
ejpam-4215	157	7	b	b	PROPN
ejpam-4215	157	8	.	.	PUNCT
ejpam-4215	158	1	proof	proof	NOUN
ejpam-4215	158	2	.	.	PUNCT
ejpam-4215	159	1	let	let	VERB
ejpam-4215	159	2	l	l	NOUN
ejpam-4215	159	3	b	b	PROPN
ejpam-4215	159	4	be	be	AUX
ejpam-4215	159	5	cosingular	cosingular	ADJ
ejpam-4215	159	6	submodule	submodule	NOUN
ejpam-4215	159	7	of	of	ADP
ejpam-4215	159	8	m	m	PROPN
ejpam-4215	159	9	b	b	PROPN
ejpam-4215	159	10	with	with	ADP
ejpam-4215	159	11	k	k	PROPN
ejpam-4215	159	12	b	b	PROPN
ejpam-4215	159	13	∩	∩	PROPN
ejpam-4215	159	14	l	l	NOUN
ejpam-4215	159	15	b	b	X
ejpam-4215	159	16	=	=	SYM
ejpam-4215	159	17	0	0	PROPN
ejpam-4215	159	18	.	.	PUNCT
ejpam-4215	160	1	hence	hence	ADV
ejpam-4215	160	2	,	,	PUNCT
ejpam-4215	160	3	k	k	X
ejpam-4215	160	4	∩	∩	ADJ
ejpam-4215	160	5	l	l	NOUN
ejpam-4215	160	6	=	=	SYM
ejpam-4215	160	7	b	b	PROPN
ejpam-4215	160	8	since	since	SCONJ
ejpam-4215	160	9	k	k	PROPN
ejpam-4215	160	10	≤e∗	≤e∗	PROPN
ejpam-4215	160	11	m	m	PROPN
ejpam-4215	160	12	.	.	PUNCT
ejpam-4215	161	1	thus	thus	ADV
ejpam-4215	161	2	,	,	PUNCT
ejpam-4215	161	3	k	k	PROPN
ejpam-4215	161	4	∩	∩	PROPN
ejpam-4215	161	5	l	l	PROPN
ejpam-4215	161	6	≤e∗	≤e∗	PROPN
ejpam-4215	161	7	m	m	PROPN
ejpam-4215	161	8	∩	∩	ADJ
ejpam-4215	161	9	l	l	NOUN
ejpam-4215	161	10	=	=	PUNCT
ejpam-4215	161	11	l.	l.	PROPN
ejpam-4215	162	1	hence	hence	ADV
ejpam-4215	162	2	b	b	PROPN
ejpam-4215	162	3	≤e∗	≤e∗	PROPN
ejpam-4215	162	4	l	l	NOUN
ejpam-4215	162	5	≤	≤	NUM
ejpam-4215	162	6	m	m	PROPN
ejpam-4215	162	7	but	but	CCONJ
ejpam-4215	162	8	b	b	X
ejpam-4215	162	9	≤e∗c	≤e∗c	PROPN
ejpam-4215	162	10	m	m	PROPN
ejpam-4215	162	11	,	,	PUNCT
ejpam-4215	162	12	b	b	X
ejpam-4215	162	13	=	=	SYM
ejpam-4215	162	14	l.	l.	PROPN
ejpam-4215	163	1	hence	hence	ADV
ejpam-4215	163	2	,	,	PUNCT
ejpam-4215	163	3	l	l	PROPN
ejpam-4215	163	4	b	b	X
ejpam-4215	163	5	=	=	SYM
ejpam-4215	163	6	0	0	PROPN
ejpam-4215	163	7	.	.	PUNCT
ejpam-4215	164	1	therefore	therefore	ADV
ejpam-4215	164	2	,	,	PUNCT
ejpam-4215	164	3	k	k	PROPN
ejpam-4215	164	4	b	b	PROPN
ejpam-4215	164	5	≤e∗	≤e∗	PROPN
ejpam-4215	164	6	m	m	PROPN
ejpam-4215	164	7	b	b	PROPN
ejpam-4215	164	8	.	.	PUNCT
ejpam-4215	165	1	acknowledgements	acknowledgement	NOUN
ejpam-4215	165	2	the	the	DET
ejpam-4215	165	3	authors	author	NOUN
ejpam-4215	165	4	would	would	AUX
ejpam-4215	165	5	like	like	VERB
ejpam-4215	165	6	to	to	PART
ejpam-4215	165	7	thank	thank	VERB
ejpam-4215	165	8	the	the	DET
ejpam-4215	165	9	reviewers	reviewer	NOUN
ejpam-4215	165	10	for	for	ADP
ejpam-4215	165	11	their	their	PRON
ejpam-4215	165	12	invaluable	invaluable	ADJ
ejpam-4215	165	13	comments	comment	NOUN
ejpam-4215	165	14	and	and	CCONJ
ejpam-4215	165	15	suggestions	suggestion	NOUN
ejpam-4215	165	16	that	that	PRON
ejpam-4215	165	17	led	lead	VERB
ejpam-4215	165	18	to	to	ADP
ejpam-4215	165	19	this	this	DET
ejpam-4215	165	20	improved	improve	VERB
ejpam-4215	165	21	version	version	NOUN
ejpam-4215	165	22	of	of	ADP
ejpam-4215	165	23	the	the	DET
ejpam-4215	165	24	paper	paper	NOUN
ejpam-4215	165	25	.	.	PUNCT
ejpam-4215	166	1	references	reference	NOUN
ejpam-4215	166	2	[	[	X
ejpam-4215	166	3	1	1	NUM
ejpam-4215	166	4	]	]	PUNCT
ejpam-4215	166	5	f.	f.	NOUN
ejpam-4215	166	6	auslander	auslander	PROPN
ejpam-4215	166	7	and	and	CCONJ
ejpam-4215	166	8	k.	k.	PROPN
ejpam-4215	166	9	fuller	fuller	PROPN
ejpam-4215	166	10	.	.	PUNCT
ejpam-4215	167	1	rings	ring	NOUN
ejpam-4215	167	2	and	and	CCONJ
ejpam-4215	167	3	categories	category	NOUN
ejpam-4215	167	4	of	of	ADP
ejpam-4215	167	5	modules	module	NOUN
ejpam-4215	167	6	,	,	PUNCT
ejpam-4215	167	7	graduate	graduate	NOUN
ejpam-4215	167	8	texts	text	NOUN
ejpam-4215	167	9	in	in	ADP
ejpam-4215	167	10	mathematicx	mathematicx	NOUN
ejpam-4215	167	11	,	,	PUNCT
ejpam-4215	167	12	1974	1974	NUM
ejpam-4215	167	13	.	.	PUNCT
ejpam-4215	168	1	[	[	X
ejpam-4215	168	2	2	2	NUM
ejpam-4215	168	3	]	]	PUNCT
ejpam-4215	168	4	a.	a.	NOUN
ejpam-4215	168	5	ç.	ç.	ADP
ejpam-4215	168	6	özcan	özcan	PROPN
ejpam-4215	168	7	.	.	PUNCT
ejpam-4215	169	1	modules	module	NOUN
ejpam-4215	169	2	with	with	ADP
ejpam-4215	169	3	small	small	ADJ
ejpam-4215	169	4	cyclic	cyclic	ADJ
ejpam-4215	169	5	submodules	submodule	NOUN
ejpam-4215	169	6	in	in	ADP
ejpam-4215	169	7	their	their	PRON
ejpam-4215	169	8	injective	injective	ADJ
ejpam-4215	169	9	hulls	hull	NOUN
ejpam-4215	169	10	.	.	PUNCT
ejpam-4215	170	1	2002	2002	NUM
ejpam-4215	170	2	.	.	PUNCT
ejpam-4215	171	1	[	[	X
ejpam-4215	171	2	3	3	X
ejpam-4215	171	3	]	]	X
ejpam-4215	171	4	k.	k.	PROPN
ejpam-4215	171	5	goodearl	goodearl	PROPN
ejpam-4215	171	6	.	.	PROPN
ejpam-4215	171	7	ring	ring	PROPN
ejpam-4215	171	8	theory	theory	PROPN
ejpam-4215	171	9	:	:	PUNCT
ejpam-4215	171	10	nonsingular	nonsingular	ADJ
ejpam-4215	171	11	rings	ring	NOUN
ejpam-4215	171	12	and	and	CCONJ
ejpam-4215	171	13	modules	module	NOUN
ejpam-4215	171	14	,	,	PUNCT
ejpam-4215	171	15	volume	volume	NOUN
ejpam-4215	171	16	33	33	NUM
ejpam-4215	171	17	.	.	PUNCT
ejpam-4215	172	1	crc	crc	PROPN
ejpam-4215	172	2	press	press	PROPN
ejpam-4215	172	3	,	,	PUNCT
ejpam-4215	172	4	1976	1976	NUM
ejpam-4215	172	5	.	.	PUNCT
ejpam-4215	173	1	[	[	X
ejpam-4215	173	2	4	4	X
ejpam-4215	173	3	]	]	PUNCT
ejpam-4215	173	4	m.	m.	NOUN
ejpam-4215	173	5	hazewinkel	hazewinkel	PROPN
ejpam-4215	173	6	,	,	PUNCT
ejpam-4215	173	7	n.	n.	PROPN
ejpam-4215	173	8	gubareni	gubareni	PROPN
ejpam-4215	173	9	,	,	PUNCT
ejpam-4215	173	10	and	and	CCONJ
ejpam-4215	173	11	vladimir	vladimir	PROPN
ejpam-4215	173	12	v.	v.	PROPN
ejpam-4215	173	13	kirichenko	kirichenko	PROPN
ejpam-4215	173	14	.	.	PUNCT
ejpam-4215	174	1	algebras	algebras	PROPN
ejpam-4215	174	2	,	,	PUNCT
ejpam-4215	174	3	rings	ring	NOUN
ejpam-4215	174	4	and	and	CCONJ
ejpam-4215	174	5	modules	module	NOUN
ejpam-4215	174	6	,	,	PUNCT
ejpam-4215	174	7	volume	volume	NOUN
ejpam-4215	174	8	1	1	NUM
ejpam-4215	174	9	.	.	PUNCT
ejpam-4215	174	10	springer	springer	PROPN
ejpam-4215	174	11	science	science	PROPN
ejpam-4215	174	12	&	&	CCONJ
ejpam-4215	174	13	business	business	NOUN
ejpam-4215	174	14	media	medium	NOUN
ejpam-4215	174	15	,	,	PUNCT
ejpam-4215	174	16	2004	2004	NUM
ejpam-4215	174	17	.	.	PUNCT
ejpam-4215	175	1	[	[	X
ejpam-4215	175	2	5	5	X
ejpam-4215	175	3	]	]	PUNCT
ejpam-4215	175	4	f.	f.	PROPN
ejpam-4215	175	5	kasch	kasch	PROPN
ejpam-4215	175	6	.	.	PUNCT
ejpam-4215	175	7	modules	module	NOUN
ejpam-4215	175	8	and	and	CCONJ
ejpam-4215	175	9	rings	ring	NOUN
ejpam-4215	175	10	,	,	PUNCT
ejpam-4215	175	11	volume	volume	NOUN
ejpam-4215	175	12	17	17	NUM
ejpam-4215	175	13	.	.	PUNCT
ejpam-4215	176	1	academic	academic	ADJ
ejpam-4215	176	2	press	press	NOUN
ejpam-4215	176	3	,	,	PUNCT
ejpam-4215	176	4	1982	1982	NUM
ejpam-4215	176	5	.	.	PUNCT
ejpam-4215	177	1	[	[	X
ejpam-4215	177	2	6	6	NUM
ejpam-4215	177	3	]	]	X
ejpam-4215	177	4	d.x	d.x	PROPN
ejpam-4215	177	5	.	.	PROPN
ejpam-4215	177	6	zhou	zhou	PROPN
ejpam-4215	177	7	and	and	CCONJ
ejpam-4215	177	8	x.r	x.r	PROPN
ejpam-4215	177	9	.	.	PUNCT
ejpam-4215	178	1	zhang	zhang	PROPN
ejpam-4215	178	2	.	.	PUNCT
ejpam-4215	178	3	small	small	ADJ
ejpam-4215	178	4	-	-	PUNCT
ejpam-4215	178	5	essential	essential	ADJ
ejpam-4215	178	6	submodules	submodule	NOUN
ejpam-4215	178	7	and	and	CCONJ
ejpam-4215	178	8	morita	morita	PROPN
ejpam-4215	178	9	duality	duality	PROPN
ejpam-4215	178	10	.	.	PUNCT
ejpam-4215	179	1	southeast	southeast	ADJ
ejpam-4215	179	2	asian	asian	ADJ
ejpam-4215	179	3	bulletin	bulletin	NOUN
ejpam-4215	179	4	of	of	ADP
ejpam-4215	179	5	mathematics	mathematic	NOUN
ejpam-4215	179	6	,	,	PUNCT
ejpam-4215	179	7	35(6	35(6	NUM
ejpam-4215	179	8	)	)	PUNCT
ejpam-4215	179	9	,	,	PUNCT
ejpam-4215	179	10	2011	2011	NUM
ejpam-4215	179	11	.	.	PUNCT