id	sid	tid	token	lemma	pos
ejpam-3461	1	1	european	european	PROPN
ejpam-3461	1	2	journal	journal	PROPN
ejpam-3461	1	3	of	of	ADP
ejpam-3461	1	4	pure	pure	ADJ
ejpam-3461	1	5	and	and	CCONJ
ejpam-3461	1	6	applied	apply	VERB
ejpam-3461	1	7	mathematics	mathematic	NOUN
ejpam-3461	1	8	vol	vol	NOUN
ejpam-3461	1	9	.	.	PROPN
ejpam-3461	2	1	12	12	NUM
ejpam-3461	2	2	,	,	PUNCT
ejpam-3461	2	3	no	no	INTJ
ejpam-3461	2	4	.	.	NOUN
ejpam-3461	2	5	3	3	NUM
ejpam-3461	2	6	,	,	PUNCT
ejpam-3461	2	7	2019	2019	NUM
ejpam-3461	2	8	,	,	PUNCT
ejpam-3461	2	9	1187	1187	NUM
ejpam-3461	2	10	-	-	SYM
ejpam-3461	2	11	1198	1198	NUM
ejpam-3461	2	12	issn	issn	PROPN
ejpam-3461	2	13	1307	1307	NUM
ejpam-3461	2	14	-	-	SYM
ejpam-3461	2	15	5543	5543	NUM
ejpam-3461	2	16	–	–	PUNCT
ejpam-3461	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3461	2	18	published	publish	VERB
ejpam-3461	2	19	by	by	ADP
ejpam-3461	2	20	new	new	PROPN
ejpam-3461	2	21	york	york	PROPN
ejpam-3461	2	22	business	business	PROPN
ejpam-3461	2	23	global	global	PROPN
ejpam-3461	2	24	on	on	ADP
ejpam-3461	2	25	c	c	NOUN
ejpam-3461	2	26	-	-	PUNCT
ejpam-3461	2	27	co	co	ADJ
ejpam-3461	2	28	-	-	ADJ
ejpam-3461	2	29	epi	epi	ADJ
ejpam-3461	2	30	-	-	ADJ
ejpam-3461	2	31	retractable	retractable	ADJ
ejpam-3461	2	32	modules	module	NOUN
ejpam-3461	2	33	abdoul	abdoul	VERB
ejpam-3461	2	34	djibril	djibril	PROPN
ejpam-3461	2	35	diallo1	diallo1	PROPN
ejpam-3461	2	36	,	,	PUNCT
ejpam-3461	2	37	papa	papa	NOUN
ejpam-3461	2	38	cheikhou	cheikhou	PROPN
ejpam-3461	2	39	diop2,∗	diop2,∗	PROPN
ejpam-3461	2	40	,	,	PUNCT
ejpam-3461	2	41	mamadou	mamadou	PROPN
ejpam-3461	2	42	barry1	barry1	PROPN
ejpam-3461	2	43	1	1	NUM
ejpam-3461	2	44	département	département	PROPN
ejpam-3461	2	45	de	de	X
ejpam-3461	2	46	mathématiques	mathématiques	X
ejpam-3461	2	47	et	et	PROPN
ejpam-3461	2	48	informatique	informatique	PROPN
ejpam-3461	2	49	,	,	PUNCT
ejpam-3461	2	50	faculté	faculté	NOUN
ejpam-3461	2	51	des	des	PROPN
ejpam-3461	2	52	sciences	sciences	PROPN
ejpam-3461	2	53	et	et	PROPN
ejpam-3461	2	54	techniques	technique	NOUN
ejpam-3461	2	55	,	,	PUNCT
ejpam-3461	2	56	université	université	NOUN
ejpam-3461	2	57	cheikh	cheikh	PROPN
ejpam-3461	2	58	anta	anta	PROPN
ejpam-3461	2	59	diop	diop	PROPN
ejpam-3461	2	60	,	,	PUNCT
ejpam-3461	2	61	dakar	dakar	NOUN
ejpam-3461	2	62	,	,	PUNCT
ejpam-3461	2	63	sénégal	sénégal	ADJ
ejpam-3461	2	64	2	2	NUM
ejpam-3461	2	65	département	département	PROPN
ejpam-3461	2	66	de	de	X
ejpam-3461	2	67	mathématiques	mathématiques	X
ejpam-3461	2	68	,	,	PUNCT
ejpam-3461	2	69	ufr	ufr	PROPN
ejpam-3461	2	70	sciences	sciences	PROPN
ejpam-3461	2	71	et	et	PROPN
ejpam-3461	2	72	technologies	technology	NOUN
ejpam-3461	2	73	,	,	PUNCT
ejpam-3461	2	74	université	université	NOUN
ejpam-3461	2	75	de	de	ADP
ejpam-3461	2	76	thiès	thiès	PROPN
ejpam-3461	2	77	,	,	PUNCT
ejpam-3461	2	78	thiès	thiès	NOUN
ejpam-3461	2	79	,	,	PUNCT
ejpam-3461	2	80	sénégal	sénégal	ADJ
ejpam-3461	2	81	abstract	abstract	NOUN
ejpam-3461	2	82	.	.	PUNCT
ejpam-3461	3	1	in	in	ADP
ejpam-3461	3	2	this	this	DET
ejpam-3461	3	3	paper	paper	NOUN
ejpam-3461	3	4	,	,	PUNCT
ejpam-3461	3	5	we	we	PRON
ejpam-3461	3	6	introduce	introduce	VERB
ejpam-3461	3	7	the	the	DET
ejpam-3461	3	8	notion	notion	NOUN
ejpam-3461	3	9	of	of	ADP
ejpam-3461	3	10	c	c	NOUN
ejpam-3461	3	11	-	-	PUNCT
ejpam-3461	3	12	co	co	ADJ
ejpam-3461	3	13	-	-	ADJ
ejpam-3461	3	14	epi	epi	ADJ
ejpam-3461	3	15	-	-	ADJ
ejpam-3461	3	16	retractable	retractable	ADJ
ejpam-3461	3	17	modules	module	NOUN
ejpam-3461	3	18	.	.	PUNCT
ejpam-3461	4	1	an	an	DET
ejpam-3461	4	2	r	r	NOUN
ejpam-3461	4	3	-	-	PUNCT
ejpam-3461	4	4	modulem	modulem	NOUN
ejpam-3461	4	5	is	be	AUX
ejpam-3461	4	6	called	call	VERB
ejpam-3461	4	7	c	c	AUX
ejpam-3461	4	8	-	-	PUNCT
ejpam-3461	4	9	co	co	NOUN
ejpam-3461	4	10	-	-	ADJ
ejpam-3461	4	11	epi	epi	ADJ
ejpam-3461	4	12	-	-	NOUN
ejpam-3461	4	13	retractable	retractable	ADJ
ejpam-3461	4	14	if	if	SCONJ
ejpam-3461	4	15	it	it	PRON
ejpam-3461	4	16	contains	contain	VERB
ejpam-3461	4	17	a	a	DET
ejpam-3461	4	18	copy	copy	NOUN
ejpam-3461	4	19	of	of	ADP
ejpam-3461	4	20	its	its	PRON
ejpam-3461	4	21	factor	factor	NOUN
ejpam-3461	4	22	module	module	NOUN
ejpam-3461	4	23	by	by	ADP
ejpam-3461	4	24	a	a	DET
ejpam-3461	4	25	complement	complement	NOUN
ejpam-3461	4	26	submodule	submodule	NOUN
ejpam-3461	4	27	.	.	PUNCT
ejpam-3461	5	1	the	the	DET
ejpam-3461	5	2	ring	ring	NOUN
ejpam-3461	5	3	r	r	NOUN
ejpam-3461	5	4	is	be	AUX
ejpam-3461	5	5	called	call	VERB
ejpam-3461	5	6	c	c	NOUN
ejpam-3461	5	7	-	-	PUNCT
ejpam-3461	5	8	co	co	NOUN
ejpam-3461	5	9	-	-	NOUN
ejpam-3461	5	10	pri	pri	NOUN
ejpam-3461	5	11	if	if	SCONJ
ejpam-3461	5	12	rr	rr	PROPN
ejpam-3461	5	13	is	be	AUX
ejpam-3461	5	14	c	c	NOUN
ejpam-3461	5	15	-	-	PUNCT
ejpam-3461	5	16	co	co	ADJ
ejpam-3461	5	17	-	-	ADJ
ejpam-3461	5	18	epi	epi	NOUN
ejpam-3461	5	19	-	-	NOUN
ejpam-3461	5	20	retractable	retractable	ADJ
ejpam-3461	5	21	.	.	PUNCT
ejpam-3461	6	1	conditions	condition	NOUN
ejpam-3461	6	2	are	be	AUX
ejpam-3461	6	3	found	find	VERB
ejpam-3461	6	4	under	under	ADP
ejpam-3461	6	5	which	which	PRON
ejpam-3461	6	6	,	,	PUNCT
ejpam-3461	6	7	a	a	DET
ejpam-3461	6	8	c	c	NOUN
ejpam-3461	6	9	-	-	PUNCT
ejpam-3461	6	10	coepi	coepi	NOUN
ejpam-3461	6	11	-	-	PUNCT
ejpam-3461	6	12	retractable	retractable	ADJ
ejpam-3461	6	13	module	module	NOUN
ejpam-3461	6	14	is	be	AUX
ejpam-3461	6	15	extending	extend	VERB
ejpam-3461	6	16	,	,	PUNCT
ejpam-3461	6	17	retractable	retractable	ADJ
ejpam-3461	6	18	,	,	PUNCT
ejpam-3461	6	19	semi	semi	ADJ
ejpam-3461	6	20	-	-	ADJ
ejpam-3461	6	21	simple	simple	ADJ
ejpam-3461	6	22	,	,	PUNCT
ejpam-3461	6	23	quasi	quasi	ADJ
ejpam-3461	6	24	-	-	ADJ
ejpam-3461	6	25	injective	injective	ADJ
ejpam-3461	6	26	,	,	PUNCT
ejpam-3461	6	27	injective	injective	ADJ
ejpam-3461	6	28	and	and	CCONJ
ejpam-3461	6	29	simple	simple	ADJ
ejpam-3461	6	30	.	.	PUNCT
ejpam-3461	7	1	also	also	ADV
ejpam-3461	7	2	,	,	PUNCT
ejpam-3461	7	3	we	we	PRON
ejpam-3461	7	4	investigate	investigate	VERB
ejpam-3461	7	5	when	when	SCONJ
ejpam-3461	7	6	c	c	NOUN
ejpam-3461	7	7	-	-	PUNCT
ejpam-3461	7	8	co	co	NOUN
ejpam-3461	7	9	-	-	ADJ
ejpam-3461	7	10	epi	epi	ADJ
ejpam-3461	7	11	-	-	ADJ
ejpam-3461	7	12	retractable	retractable	ADJ
ejpam-3461	7	13	modules	module	NOUN
ejpam-3461	7	14	have	have	VERB
ejpam-3461	7	15	finite	finite	NOUN
ejpam-3461	7	16	uniform	uniform	ADJ
ejpam-3461	7	17	dimension	dimension	NOUN
ejpam-3461	7	18	.	.	PUNCT
ejpam-3461	8	1	finally	finally	ADV
ejpam-3461	8	2	,	,	PUNCT
ejpam-3461	8	3	right	right	ADJ
ejpam-3461	8	4	si	si	NOUN
ejpam-3461	8	5	-	-	PUNCT
ejpam-3461	8	6	rings	ring	NOUN
ejpam-3461	8	7	,	,	PUNCT
ejpam-3461	8	8	semi	semi	ADJ
ejpam-3461	8	9	-	-	ADJ
ejpam-3461	8	10	simple	simple	ADJ
ejpam-3461	8	11	artinian	artinian	ADJ
ejpam-3461	8	12	rings	ring	NOUN
ejpam-3461	8	13	and	and	CCONJ
ejpam-3461	8	14	quasi	quasi	ADJ
ejpam-3461	8	15	-	-	ADJ
ejpam-3461	8	16	frobenius	frobenius	ADJ
ejpam-3461	8	17	rings	ring	NOUN
ejpam-3461	8	18	are	be	AUX
ejpam-3461	8	19	characterized	characterize	VERB
ejpam-3461	8	20	in	in	ADP
ejpam-3461	8	21	termes	terme	NOUN
ejpam-3461	8	22	of	of	ADP
ejpam-3461	8	23	c	c	NOUN
ejpam-3461	8	24	-	-	PUNCT
ejpam-3461	8	25	co	co	NOUN
ejpam-3461	8	26	-	-	ADJ
ejpam-3461	8	27	epi	epi	ADJ
ejpam-3461	8	28	-	-	ADJ
ejpam-3461	8	29	retractable	retractable	ADJ
ejpam-3461	8	30	modules	module	NOUN
ejpam-3461	8	31	.	.	PUNCT
ejpam-3461	9	1	2010	2010	NUM
ejpam-3461	9	2	mathematics	mathematic	NOUN
ejpam-3461	9	3	subject	subject	NOUN
ejpam-3461	9	4	classifications	classification	NOUN
ejpam-3461	9	5	:	:	PUNCT
ejpam-3461	9	6	13b10,13c05,13c13	13b10,13c05,13c13	NUM
ejpam-3461	9	7	key	key	ADJ
ejpam-3461	9	8	words	word	NOUN
ejpam-3461	9	9	and	and	CCONJ
ejpam-3461	9	10	phrases	phrase	NOUN
ejpam-3461	9	11	:	:	PUNCT
ejpam-3461	9	12	co	co	ADJ
ejpam-3461	9	13	-	-	ADJ
ejpam-3461	9	14	epi	epi	ADJ
ejpam-3461	9	15	-	-	ADJ
ejpam-3461	9	16	retractable	retractable	ADJ
ejpam-3461	9	17	modules	module	NOUN
ejpam-3461	9	18	,	,	PUNCT
ejpam-3461	9	19	c	c	X
ejpam-3461	9	20	-	-	PUNCT
ejpam-3461	9	21	co	co	NOUN
ejpam-3461	9	22	-	-	ADJ
ejpam-3461	9	23	epi	epi	ADJ
ejpam-3461	9	24	-	-	ADJ
ejpam-3461	9	25	retactable	retactable	ADJ
ejpam-3461	9	26	modules	module	NOUN
ejpam-3461	9	27	,	,	PUNCT
ejpam-3461	9	28	extending	extend	VERB
ejpam-3461	9	29	modules	module	NOUN
ejpam-3461	9	30	,	,	PUNCT
ejpam-3461	9	31	rickart	rickart	NOUN
ejpam-3461	9	32	modules	module	NOUN
ejpam-3461	9	33	1	1	NUM
ejpam-3461	9	34	.	.	PUNCT
ejpam-3461	10	1	introduction	introduction	NOUN
ejpam-3461	10	2	throughout	throughout	ADP
ejpam-3461	10	3	all	all	DET
ejpam-3461	10	4	rings	ring	NOUN
ejpam-3461	10	5	are	be	AUX
ejpam-3461	10	6	associative	associative	ADJ
ejpam-3461	10	7	with	with	ADP
ejpam-3461	10	8	identity	identity	NOUN
ejpam-3461	10	9	and	and	CCONJ
ejpam-3461	10	10	all	all	DET
ejpam-3461	10	11	modules	module	NOUN
ejpam-3461	10	12	are	be	AUX
ejpam-3461	10	13	unitary	unitary	ADJ
ejpam-3461	10	14	right	right	ADJ
ejpam-3461	10	15	module	module	NOUN
ejpam-3461	10	16	.	.	PUNCT
ejpam-3461	11	1	in	in	ADP
ejpam-3461	11	2	[	[	X
ejpam-3461	11	3	10	10	NUM
ejpam-3461	11	4	]	]	PUNCT
ejpam-3461	11	5	,	,	PUNCT
ejpam-3461	11	6	ghorbani	ghorbani	NOUN
ejpam-3461	11	7	introduced	introduce	VERB
ejpam-3461	11	8	the	the	DET
ejpam-3461	11	9	co	co	NOUN
ejpam-3461	11	10	-	-	ADJ
ejpam-3461	11	11	epi	epi	ADJ
ejpam-3461	11	12	-	-	ADJ
ejpam-3461	11	13	retractable	retractable	ADJ
ejpam-3461	11	14	modules	module	NOUN
ejpam-3461	11	15	.	.	PUNCT
ejpam-3461	12	1	an	an	DET
ejpam-3461	12	2	r	r	NOUN
ejpam-3461	12	3	-	-	PUNCT
ejpam-3461	12	4	module	module	NOUN
ejpam-3461	12	5	m	m	NOUN
ejpam-3461	12	6	is	be	AUX
ejpam-3461	12	7	called	call	VERB
ejpam-3461	12	8	co	co	ADJ
ejpam-3461	12	9	-	-	ADJ
ejpam-3461	12	10	epi	epi	ADJ
ejpam-3461	12	11	-	-	NOUN
ejpam-3461	12	12	retractable	retractable	ADJ
ejpam-3461	12	13	if	if	SCONJ
ejpam-3461	12	14	it	it	PRON
ejpam-3461	12	15	contains	contain	VERB
ejpam-3461	12	16	any	any	PRON
ejpam-3461	12	17	of	of	ADP
ejpam-3461	12	18	its	its	PRON
ejpam-3461	12	19	factor	factor	NOUN
ejpam-3461	12	20	modules	module	NOUN
ejpam-3461	12	21	.	.	PUNCT
ejpam-3461	13	1	a	a	DET
ejpam-3461	13	2	ring	ring	NOUN
ejpam-3461	13	3	r	r	NOUN
ejpam-3461	13	4	is	be	AUX
ejpam-3461	13	5	called	call	VERB
ejpam-3461	13	6	co	co	NOUN
ejpam-3461	13	7	-	-	NOUN
ejpam-3461	13	8	pri	pri	NOUN
ejpam-3461	13	9	if	if	SCONJ
ejpam-3461	13	10	rr	rr	PROPN
ejpam-3461	13	11	is	be	AUX
ejpam-3461	13	12	a	a	DET
ejpam-3461	13	13	co	co	NOUN
ejpam-3461	13	14	-	-	ADJ
ejpam-3461	13	15	epi	epi	ADJ
ejpam-3461	13	16	-	-	ADJ
ejpam-3461	13	17	retractable	retractable	ADJ
ejpam-3461	13	18	module	module	NOUN
ejpam-3461	13	19	.	.	PUNCT
ejpam-3461	14	1	it	it	PRON
ejpam-3461	14	2	is	be	AUX
ejpam-3461	14	3	was	be	AUX
ejpam-3461	14	4	shown	show	VERB
ejpam-3461	14	5	in	in	ADP
ejpam-3461	14	6	[	[	X
ejpam-3461	14	7	10	10	NUM
ejpam-3461	14	8	]	]	PUNCT
ejpam-3461	14	9	,	,	PUNCT
ejpam-3461	14	10	that	that	SCONJ
ejpam-3461	14	11	a	a	DET
ejpam-3461	14	12	ring	ring	NOUN
ejpam-3461	14	13	r	r	NOUN
ejpam-3461	14	14	is	be	AUX
ejpam-3461	14	15	copri	copri	NOUN
ejpam-3461	14	16	iff	iff	VERB
ejpam-3461	14	17	its	its	PRON
ejpam-3461	14	18	right	right	ADJ
ejpam-3461	14	19	ideals	ideal	NOUN
ejpam-3461	14	20	is	be	AUX
ejpam-3461	14	21	the	the	DET
ejpam-3461	14	22	right	right	ADJ
ejpam-3461	14	23	annihilator	annihilator	NOUN
ejpam-3461	14	24	of	of	ADP
ejpam-3461	14	25	an	an	DET
ejpam-3461	14	26	element	element	NOUN
ejpam-3461	14	27	of	of	ADP
ejpam-3461	14	28	r.	r.	PROPN
ejpam-3461	14	29	also	also	ADV
ejpam-3461	14	30	co	co	ADJ
ejpam-3461	14	31	-	-	ADJ
ejpam-3461	14	32	pi	pi	ADV
ejpam-3461	14	33	-	-	PUNCT
ejpam-3461	14	34	retractable	retractable	ADJ
ejpam-3461	14	35	modules	module	NOUN
ejpam-3461	14	36	have	have	AUX
ejpam-3461	14	37	been	be	AUX
ejpam-3461	14	38	investigated	investigate	VERB
ejpam-3461	14	39	by	by	ADP
ejpam-3461	14	40	mostafanasab	mostafanasab	PROPN
ejpam-3461	15	1	[	[	X
ejpam-3461	15	2	15	15	NUM
ejpam-3461	15	3	]	]	PUNCT
ejpam-3461	15	4	.	.	PUNCT
ejpam-3461	16	1	he	he	PRON
ejpam-3461	16	2	studied	study	VERB
ejpam-3461	16	3	the	the	DET
ejpam-3461	16	4	simplicity	simplicity	NOUN
ejpam-3461	16	5	and	and	CCONJ
ejpam-3461	16	6	the	the	DET
ejpam-3461	16	7	semi	semi	ADJ
ejpam-3461	16	8	-	-	NOUN
ejpam-3461	16	9	simplicity	simplicity	NOUN
ejpam-3461	16	10	of	of	ADP
ejpam-3461	16	11	co	co	ADJ
ejpam-3461	16	12	-	-	ADJ
ejpam-3461	16	13	pi	pi	ADV
ejpam-3461	16	14	-	-	PUNCT
ejpam-3461	16	15	retractable	retractable	ADJ
ejpam-3461	16	16	modules	module	NOUN
ejpam-3461	16	17	.	.	PUNCT
ejpam-3461	17	1	recall	recall	VERB
ejpam-3461	17	2	that	that	SCONJ
ejpam-3461	17	3	a	a	DET
ejpam-3461	17	4	module	module	NOUN
ejpam-3461	17	5	m	m	VERB
ejpam-3461	17	6	is	be	AUX
ejpam-3461	17	7	called	call	VERB
ejpam-3461	17	8	extending	extend	VERB
ejpam-3461	17	9	if	if	SCONJ
ejpam-3461	17	10	every	every	DET
ejpam-3461	17	11	complement	complement	NOUN
ejpam-3461	17	12	submodule	submodule	NOUN
ejpam-3461	17	13	is	be	AUX
ejpam-3461	17	14	a	a	DET
ejpam-3461	17	15	direct	direct	ADJ
ejpam-3461	17	16	summand	summand	NOUN
ejpam-3461	17	17	.	.	PUNCT
ejpam-3461	18	1	motivated	motivate	VERB
ejpam-3461	18	2	by	by	ADP
ejpam-3461	18	3	the	the	DET
ejpam-3461	18	4	definition	definition	NOUN
ejpam-3461	18	5	of	of	ADP
ejpam-3461	18	6	a	a	DET
ejpam-3461	18	7	co	co	NOUN
ejpam-3461	18	8	-	-	ADJ
ejpam-3461	18	9	epi	epi	ADJ
ejpam-3461	18	10	-	-	ADJ
ejpam-3461	18	11	retractable	retractable	ADJ
ejpam-3461	18	12	module	module	NOUN
ejpam-3461	18	13	and	and	CCONJ
ejpam-3461	18	14	the	the	DET
ejpam-3461	18	15	definition	definition	NOUN
ejpam-3461	18	16	of	of	ADP
ejpam-3461	18	17	a	a	DET
ejpam-3461	18	18	extending	extend	VERB
ejpam-3461	18	19	module	module	NOUN
ejpam-3461	18	20	,	,	PUNCT
ejpam-3461	18	21	we	we	PRON
ejpam-3461	18	22	say	say	VERB
ejpam-3461	18	23	that	that	SCONJ
ejpam-3461	18	24	a	a	DET
ejpam-3461	18	25	module	module	NOUN
ejpam-3461	18	26	is	be	AUX
ejpam-3461	18	27	c	c	NOUN
ejpam-3461	18	28	-	-	PUNCT
ejpam-3461	18	29	co	co	NOUN
ejpam-3461	18	30	-	-	ADJ
ejpam-3461	18	31	epi	epi	ADJ
ejpam-3461	18	32	-	-	NOUN
ejpam-3461	18	33	retractable	retractable	ADJ
ejpam-3461	18	34	if	if	SCONJ
ejpam-3461	18	35	it	it	PRON
ejpam-3461	18	36	contains	contain	VERB
ejpam-3461	18	37	a	a	DET
ejpam-3461	18	38	copy	copy	NOUN
ejpam-3461	18	39	of	of	ADP
ejpam-3461	18	40	its	its	PRON
ejpam-3461	18	41	factor	factor	NOUN
ejpam-3461	18	42	modules	module	NOUN
ejpam-3461	18	43	by	by	ADP
ejpam-3461	18	44	a	a	DET
ejpam-3461	18	45	complement	complement	NOUN
ejpam-3461	18	46	submodule	submodule	NOUN
ejpam-3461	18	47	.	.	PUNCT
ejpam-3461	19	1	every	every	DET
ejpam-3461	19	2	co	co	NOUN
ejpam-3461	19	3	-	-	ADJ
ejpam-3461	19	4	epi	epi	ADJ
ejpam-3461	19	5	-	-	ADJ
ejpam-3461	19	6	retractable	retractable	ADJ
ejpam-3461	19	7	module	module	NOUN
ejpam-3461	19	8	and	and	CCONJ
ejpam-3461	19	9	every	every	DET
ejpam-3461	19	10	extending	extend	VERB
ejpam-3461	19	11	module	module	NOUN
ejpam-3461	19	12	is	be	AUX
ejpam-3461	19	13	c	c	NOUN
ejpam-3461	19	14	-	-	PUNCT
ejpam-3461	19	15	co	co	ADJ
ejpam-3461	19	16	-	-	ADJ
ejpam-3461	19	17	epi	epi	NOUN
ejpam-3461	19	18	-	-	NOUN
ejpam-3461	19	19	retractable	retractable	ADJ
ejpam-3461	19	20	.	.	PUNCT
ejpam-3461	20	1	in	in	ADP
ejpam-3461	20	2	particular	particular	ADJ
ejpam-3461	20	3	uniform	uniform	ADJ
ejpam-3461	20	4	modules	module	NOUN
ejpam-3461	20	5	and	and	CCONJ
ejpam-3461	20	6	semi	semi	ADJ
ejpam-3461	20	7	-	-	ADJ
ejpam-3461	20	8	simple	simple	ADJ
ejpam-3461	20	9	modules	module	NOUN
ejpam-3461	20	10	are	be	AUX
ejpam-3461	20	11	c	c	NOUN
ejpam-3461	20	12	-	-	PUNCT
ejpam-3461	20	13	co	co	ADJ
ejpam-3461	20	14	-	-	ADJ
ejpam-3461	20	15	epi	epi	NOUN
ejpam-3461	20	16	-	-	NOUN
ejpam-3461	20	17	retractable	retractable	ADJ
ejpam-3461	20	18	.	.	PUNCT
ejpam-3461	21	1	in	in	ADP
ejpam-3461	21	2	this	this	DET
ejpam-3461	21	3	∗corresponding	∗corresponding	NOUN
ejpam-3461	21	4	author	author	NOUN
ejpam-3461	21	5	.	.	PUNCT
ejpam-3461	22	1	doi	doi	NOUN
ejpam-3461	22	2	:	:	PUNCT
ejpam-3461	22	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3461	https://doi.org/10.29020/nybg.ejpam.v12i3.3461	DET
ejpam-3461	22	4	email	email	NOUN
ejpam-3461	22	5	addresses	address	VERB
ejpam-3461	22	6	:	:	PUNCT
ejpam-3461	22	7	cheikpapa@yahoo.fr	cheikpapa@yahoo.fr	PROPN
ejpam-3461	22	8	(	(	PUNCT
ejpam-3461	22	9	p.	p.	NOUN
ejpam-3461	22	10	c.	c.	PROPN
ejpam-3461	22	11	diop	diop	PROPN
ejpam-3461	22	12	)	)	PUNCT
ejpam-3461	22	13	,	,	PUNCT
ejpam-3461	22	14	dialloabdoulaziz58@yahoo.fr	dialloabdoulaziz58@yahoo.fr	PROPN
ejpam-3461	22	15	(	(	PUNCT
ejpam-3461	22	16	a.	a.	PROPN
ejpam-3461	22	17	d.	d.	PROPN
ejpam-3461	22	18	diallo	diallo	PROPN
ejpam-3461	22	19	)	)	PUNCT
ejpam-3461	22	20	,	,	PUNCT
ejpam-3461	22	21	mansabadion1@hotmail.com	mansabadion1@hotmail.com	X
ejpam-3461	22	22	(	(	PUNCT
ejpam-3461	22	23	m.	m.	NOUN
ejpam-3461	22	24	barry	barry	PROPN
ejpam-3461	22	25	)	)	PUNCT
ejpam-3461	22	26	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3461	23	1	1187	1187	NUM
ejpam-3461	23	2	c	c	X
ejpam-3461	23	3	©	©	PROPN
ejpam-3461	23	4	2019	2019	NUM
ejpam-3461	23	5	ejpam	ejpam	NOUN
ejpam-3461	23	6	all	all	DET
ejpam-3461	23	7	rights	right	NOUN
ejpam-3461	23	8	reserved	reserve	VERB
ejpam-3461	23	9	.	.	PUNCT
ejpam-3461	24	1	a.	a.	PROPN
ejpam-3461	24	2	d.	d.	PROPN
ejpam-3461	24	3	diallo	diallo	PROPN
ejpam-3461	24	4	,	,	PUNCT
ejpam-3461	24	5	p.	p.	PROPN
ejpam-3461	24	6	c.	c.	PROPN
ejpam-3461	24	7	diop	diop	PROPN
ejpam-3461	24	8	,	,	PUNCT
ejpam-3461	24	9	m.	m.	NOUN
ejpam-3461	24	10	barry	barry	PROPN
ejpam-3461	24	11	/	/	SYM
ejpam-3461	24	12	eur	eur	PROPN
ejpam-3461	24	13	.	.	PUNCT
ejpam-3461	25	1	j.	j.	PROPN
ejpam-3461	25	2	pure	pure	PROPN
ejpam-3461	25	3	appl	appl	PROPN
ejpam-3461	25	4	.	.	PROPN
ejpam-3461	25	5	math	math	PROPN
ejpam-3461	25	6	,	,	PUNCT
ejpam-3461	25	7	12	12	NUM
ejpam-3461	25	8	(	(	PUNCT
ejpam-3461	25	9	3	3	NUM
ejpam-3461	25	10	)	)	PUNCT
ejpam-3461	25	11	(	(	PUNCT
ejpam-3461	25	12	2019	2019	NUM
ejpam-3461	25	13	)	)	PUNCT
ejpam-3461	25	14	,	,	PUNCT
ejpam-3461	25	15	1187	1187	NUM
ejpam-3461	25	16	-	-	SYM
ejpam-3461	25	17	1198	1198	NUM
ejpam-3461	25	18	1188	1188	NUM
ejpam-3461	25	19	paper	paper	NOUN
ejpam-3461	25	20	,	,	PUNCT
ejpam-3461	25	21	we	we	PRON
ejpam-3461	25	22	investigate	investigate	VERB
ejpam-3461	25	23	when	when	SCONJ
ejpam-3461	25	24	c	c	NOUN
ejpam-3461	25	25	-	-	PUNCT
ejpam-3461	25	26	co	co	NOUN
ejpam-3461	25	27	-	-	ADJ
ejpam-3461	25	28	epi	epi	ADJ
ejpam-3461	25	29	-	-	ADJ
ejpam-3461	25	30	retractable	retractable	ADJ
ejpam-3461	25	31	modules	module	NOUN
ejpam-3461	25	32	are	be	AUX
ejpam-3461	25	33	extending	extend	VERB
ejpam-3461	25	34	,	,	PUNCT
ejpam-3461	25	35	continuous	continuous	ADJ
ejpam-3461	25	36	,	,	PUNCT
ejpam-3461	25	37	quasicontinuous	quasicontinuous	ADJ
ejpam-3461	25	38	,	,	PUNCT
ejpam-3461	25	39	semi	semi	ADJ
ejpam-3461	25	40	-	-	ADJ
ejpam-3461	25	41	simple	simple	ADJ
ejpam-3461	25	42	,	,	PUNCT
ejpam-3461	25	43	retractable	retractable	ADJ
ejpam-3461	25	44	,	,	PUNCT
ejpam-3461	25	45	quasi	quasi	ADJ
ejpam-3461	25	46	-	-	ADJ
ejpam-3461	25	47	injective	injective	ADJ
ejpam-3461	25	48	,	,	PUNCT
ejpam-3461	25	49	injective	injective	ADJ
ejpam-3461	25	50	and	and	CCONJ
ejpam-3461	25	51	simple	simple	ADJ
ejpam-3461	25	52	.	.	PUNCT
ejpam-3461	26	1	also	also	ADV
ejpam-3461	26	2	,	,	PUNCT
ejpam-3461	26	3	we	we	PRON
ejpam-3461	26	4	prove	prove	VERB
ejpam-3461	26	5	under	under	ADP
ejpam-3461	26	6	certain	certain	ADJ
ejpam-3461	26	7	conditions	condition	NOUN
ejpam-3461	26	8	that	that	SCONJ
ejpam-3461	26	9	a	a	DET
ejpam-3461	26	10	c	c	NOUN
ejpam-3461	26	11	-	-	PUNCT
ejpam-3461	26	12	co	co	NOUN
ejpam-3461	26	13	-	-	ADJ
ejpam-3461	26	14	epi	epi	ADJ
ejpam-3461	26	15	-	-	ADJ
ejpam-3461	26	16	retractable	retractable	ADJ
ejpam-3461	26	17	module	module	NOUN
ejpam-3461	26	18	has	have	VERB
ejpam-3461	26	19	finite	finite	ADJ
ejpam-3461	26	20	uniform	uniform	ADJ
ejpam-3461	26	21	dimension	dimension	NOUN
ejpam-3461	26	22	.	.	PUNCT
ejpam-3461	27	1	finally	finally	ADV
ejpam-3461	27	2	,	,	PUNCT
ejpam-3461	27	3	we	we	PRON
ejpam-3461	27	4	characterize	characterize	VERB
ejpam-3461	27	5	some	some	DET
ejpam-3461	27	6	well	well	ADV
ejpam-3461	27	7	-	-	PUNCT
ejpam-3461	27	8	known	know	VERB
ejpam-3461	27	9	rings	ring	NOUN
ejpam-3461	27	10	with	with	ADP
ejpam-3461	27	11	the	the	DET
ejpam-3461	27	12	help	help	NOUN
ejpam-3461	27	13	of	of	ADP
ejpam-3461	27	14	c	c	NOUN
ejpam-3461	27	15	-	-	PUNCT
ejpam-3461	27	16	co	co	NOUN
ejpam-3461	27	17	-	-	ADJ
ejpam-3461	27	18	epi	epi	ADJ
ejpam-3461	27	19	-	-	ADJ
ejpam-3461	27	20	retratacble	retratacble	ADJ
ejpam-3461	27	21	modules	module	NOUN
ejpam-3461	27	22	.	.	PUNCT
ejpam-3461	28	1	our	our	PRON
ejpam-3461	28	2	paper	paper	NOUN
ejpam-3461	28	3	is	be	AUX
ejpam-3461	28	4	structured	structure	VERB
ejpam-3461	28	5	as	as	SCONJ
ejpam-3461	28	6	follows	follow	VERB
ejpam-3461	28	7	:	:	PUNCT
ejpam-3461	28	8	in	in	ADP
ejpam-3461	28	9	the	the	DET
ejpam-3461	28	10	second	second	ADJ
ejpam-3461	28	11	section	section	NOUN
ejpam-3461	28	12	,	,	PUNCT
ejpam-3461	28	13	we	we	PRON
ejpam-3461	28	14	give	give	VERB
ejpam-3461	28	15	preliminary	preliminary	ADJ
ejpam-3461	28	16	definitions	definition	NOUN
ejpam-3461	28	17	and	and	CCONJ
ejpam-3461	28	18	results	result	NOUN
ejpam-3461	28	19	which	which	PRON
ejpam-3461	28	20	we	we	PRON
ejpam-3461	28	21	will	will	AUX
ejpam-3461	28	22	use	use	VERB
ejpam-3461	28	23	throughout	throughout	ADP
ejpam-3461	28	24	this	this	DET
ejpam-3461	28	25	paper	paper	NOUN
ejpam-3461	28	26	.	.	PUNCT
ejpam-3461	29	1	in	in	ADP
ejpam-3461	29	2	the	the	DET
ejpam-3461	29	3	third	third	ADJ
ejpam-3461	29	4	section	section	NOUN
ejpam-3461	29	5	,	,	PUNCT
ejpam-3461	29	6	we	we	PRON
ejpam-3461	29	7	define	define	VERB
ejpam-3461	29	8	the	the	DET
ejpam-3461	29	9	c	c	NOUN
ejpam-3461	29	10	-	-	PUNCT
ejpam-3461	29	11	co	co	NOUN
ejpam-3461	29	12	-	-	ADJ
ejpam-3461	29	13	epi	epi	ADJ
ejpam-3461	29	14	-	-	ADJ
ejpam-3461	29	15	retractable	retractable	ADJ
ejpam-3461	29	16	modules	module	NOUN
ejpam-3461	29	17	.	.	PUNCT
ejpam-3461	30	1	our	our	PRON
ejpam-3461	30	2	aim	aim	NOUN
ejpam-3461	30	3	in	in	ADP
ejpam-3461	30	4	this	this	DET
ejpam-3461	30	5	section	section	NOUN
ejpam-3461	30	6	is	be	AUX
ejpam-3461	30	7	to	to	PART
ejpam-3461	30	8	work	work	VERB
ejpam-3461	30	9	on	on	ADP
ejpam-3461	30	10	the	the	DET
ejpam-3461	30	11	concept	concept	NOUN
ejpam-3461	30	12	of	of	ADP
ejpam-3461	30	13	c	c	NOUN
ejpam-3461	30	14	-	-	PUNCT
ejpam-3461	30	15	co	co	NOUN
ejpam-3461	30	16	-	-	ADJ
ejpam-3461	30	17	epi	epi	ADJ
ejpam-3461	30	18	-	-	ADJ
ejpam-3461	30	19	retractable	retractable	ADJ
ejpam-3461	30	20	modules	module	NOUN
ejpam-3461	30	21	.	.	PUNCT
ejpam-3461	31	1	we	we	PRON
ejpam-3461	31	2	show	show	VERB
ejpam-3461	31	3	,	,	PUNCT
ejpam-3461	31	4	among	among	ADP
ejpam-3461	31	5	others	other	NOUN
ejpam-3461	31	6	,	,	PUNCT
ejpam-3461	31	7	the	the	DET
ejpam-3461	31	8	following	follow	VERB
ejpam-3461	31	9	results	result	NOUN
ejpam-3461	31	10	.	.	PUNCT
ejpam-3461	32	1	(	(	PUNCT
ejpam-3461	32	2	1	1	X
ejpam-3461	32	3	)	)	PUNCT
ejpam-3461	32	4	for	for	ADP
ejpam-3461	32	5	an	an	DET
ejpam-3461	32	6	r	r	NOUN
ejpam-3461	32	7	-	-	PUNCT
ejpam-3461	32	8	module	module	NOUN
ejpam-3461	32	9	with	with	ADP
ejpam-3461	32	10	regular	regular	ADJ
ejpam-3461	32	11	endomorphism	endomorphism	NOUN
ejpam-3461	32	12	ring	ring	NOUN
ejpam-3461	32	13	,	,	PUNCT
ejpam-3461	32	14	the	the	DET
ejpam-3461	32	15	properties	property	NOUN
ejpam-3461	32	16	,	,	PUNCT
ejpam-3461	32	17	c	c	X
ejpam-3461	32	18	-	-	PUNCT
ejpam-3461	32	19	co	co	NOUN
ejpam-3461	32	20	-	-	ADJ
ejpam-3461	32	21	epi	epi	NOUN
ejpam-3461	32	22	-	-	NOUN
ejpam-3461	32	23	retractable	retractable	ADJ
ejpam-3461	32	24	,	,	PUNCT
ejpam-3461	32	25	extending	extend	VERB
ejpam-3461	32	26	,	,	PUNCT
ejpam-3461	32	27	continuous	continuous	ADJ
ejpam-3461	32	28	and	and	CCONJ
ejpam-3461	32	29	quasi	quasi	ADJ
ejpam-3461	32	30	-	-	ADJ
ejpam-3461	32	31	continuous	continuous	ADJ
ejpam-3461	32	32	are	be	AUX
ejpam-3461	32	33	all	all	ADV
ejpam-3461	32	34	equivalent	equivalent	ADJ
ejpam-3461	32	35	.	.	PUNCT
ejpam-3461	33	1	(	(	PUNCT
ejpam-3461	33	2	2	2	X
ejpam-3461	33	3	)	)	PUNCT
ejpam-3461	33	4	if	if	SCONJ
ejpam-3461	33	5	m	m	NOUN
ejpam-3461	33	6	is	be	AUX
ejpam-3461	33	7	a	a	DET
ejpam-3461	33	8	c	c	NOUN
ejpam-3461	33	9	-	-	PUNCT
ejpam-3461	33	10	co	co	NOUN
ejpam-3461	33	11	-	-	ADJ
ejpam-3461	33	12	epi	epi	ADJ
ejpam-3461	33	13	-	-	ADJ
ejpam-3461	33	14	retractable	retractable	ADJ
ejpam-3461	33	15	module	module	NOUN
ejpam-3461	33	16	with	with	ADP
ejpam-3461	33	17	udim(m	udim(m	PROPN
ejpam-3461	33	18	)	)	PUNCT
ejpam-3461	33	19	=	=	SYM
ejpam-3461	33	20	n	n	X
ejpam-3461	33	21	≥	≥	NOUN
ejpam-3461	33	22	2	2	NUM
ejpam-3461	33	23	,	,	PUNCT
ejpam-3461	33	24	then	then	ADV
ejpam-3461	33	25	m	m	VERB
ejpam-3461	33	26	is	be	AUX
ejpam-3461	33	27	retractable	retractable	ADJ
ejpam-3461	33	28	and	and	CCONJ
ejpam-3461	33	29	for	for	ADP
ejpam-3461	33	30	any	any	DET
ejpam-3461	33	31	0	0	NUM
ejpam-3461	33	32	6=	6=	NUM
ejpam-3461	33	33	c	c	PROPN
ejpam-3461	33	34	⊆c	⊆c	NOUN
ejpam-3461	33	35	m	m	PROPN
ejpam-3461	33	36	,	,	PUNCT
ejpam-3461	33	37	m	m	VERB
ejpam-3461	33	38	/	/	SYM
ejpam-3461	33	39	c	c	PROPN
ejpam-3461	33	40	is	be	AUX
ejpam-3461	33	41	uniform	uniform	ADJ
ejpam-3461	33	42	.	.	PUNCT
ejpam-3461	34	1	(	(	PUNCT
ejpam-3461	34	2	3	3	X
ejpam-3461	34	3	)	)	PUNCT
ejpam-3461	34	4	let	let	VERB
ejpam-3461	34	5	r	r	PRON
ejpam-3461	34	6	be	be	AUX
ejpam-3461	34	7	a	a	DET
ejpam-3461	34	8	right	right	ADJ
ejpam-3461	34	9	self	self	NOUN
ejpam-3461	34	10	-	-	PUNCT
ejpam-3461	34	11	injective	injective	ADJ
ejpam-3461	34	12	ring	ring	NOUN
ejpam-3461	34	13	and	and	CCONJ
ejpam-3461	34	14	m	m	AUX
ejpam-3461	34	15	be	be	AUX
ejpam-3461	34	16	a	a	DET
ejpam-3461	34	17	self	self	NOUN
ejpam-3461	34	18	-	-	PUNCT
ejpam-3461	34	19	hereditary	hereditary	ADJ
ejpam-3461	34	20	r	r	NOUN
ejpam-3461	34	21	-	-	PUNCT
ejpam-3461	34	22	module	module	NOUN
ejpam-3461	34	23	.	.	PUNCT
ejpam-3461	35	1	then	then	ADV
ejpam-3461	35	2	m	m	PROPN
ejpam-3461	35	3	is	be	AUX
ejpam-3461	35	4	c	c	NOUN
ejpam-3461	35	5	-	-	PUNCT
ejpam-3461	35	6	co	co	ADJ
ejpam-3461	35	7	-	-	ADJ
ejpam-3461	35	8	epi	epi	ADJ
ejpam-3461	35	9	-	-	ADJ
ejpam-3461	35	10	retractable	retractable	ADJ
ejpam-3461	35	11	iff	iff	NOUN
ejpam-3461	35	12	it	it	PRON
ejpam-3461	35	13	is	be	AUX
ejpam-3461	35	14	finitely	finitely	ADV
ejpam-3461	35	15	generated	generate	VERB
ejpam-3461	35	16	semi	semi	ADJ
ejpam-3461	35	17	-	-	ADJ
ejpam-3461	35	18	simple	simple	ADJ
ejpam-3461	35	19	injective	injective	NOUN
ejpam-3461	35	20	.	.	PUNCT
ejpam-3461	36	1	(	(	PUNCT
ejpam-3461	36	2	4	4	X
ejpam-3461	36	3	)	)	PUNCT
ejpam-3461	36	4	let	let	VERB
ejpam-3461	36	5	m	m	PRON
ejpam-3461	36	6	be	be	AUX
ejpam-3461	36	7	a	a	DET
ejpam-3461	36	8	c	c	NOUN
ejpam-3461	36	9	-	-	PUNCT
ejpam-3461	36	10	co	co	NOUN
ejpam-3461	36	11	-	-	ADJ
ejpam-3461	36	12	epi	epi	ADJ
ejpam-3461	36	13	-	-	ADJ
ejpam-3461	36	14	retractable	retractable	ADJ
ejpam-3461	36	15	r	r	NOUN
ejpam-3461	36	16	-	-	PUNCT
ejpam-3461	36	17	module	module	NOUN
ejpam-3461	36	18	such	such	ADJ
ejpam-3461	36	19	that	that	DET
ejpam-3461	36	20	s	s	PART
ejpam-3461	36	21	satisfies	satisfy	VERB
ejpam-3461	36	22	dcc	dcc	PROPN
ejpam-3461	36	23	for	for	ADP
ejpam-3461	36	24	cyclic	cyclic	ADJ
ejpam-3461	36	25	right	right	ADJ
ejpam-3461	36	26	ideals	ideal	NOUN
ejpam-3461	36	27	.	.	PUNCT
ejpam-3461	37	1	if	if	SCONJ
ejpam-3461	37	2	for	for	ADP
ejpam-3461	37	3	any	any	DET
ejpam-3461	37	4	finitely	finitely	ADV
ejpam-3461	37	5	generated	generate	VERB
ejpam-3461	37	6	right	right	ADJ
ejpam-3461	37	7	ideal	ideal	NOUN
ejpam-3461	37	8	i	i	PRON
ejpam-3461	37	9	⊆	⊆	NUM
ejpam-3461	37	10	s	s	NOUN
ejpam-3461	37	11	,	,	PUNCT
ejpam-3461	37	12	r(keri	r(keri	NOUN
ejpam-3461	37	13	)	)	PUNCT
ejpam-3461	38	1	=	=	SYM
ejpam-3461	39	1	i	i	PRON
ejpam-3461	39	2	then	then	ADV
ejpam-3461	39	3	m	m	VERB
ejpam-3461	39	4	has	have	VERB
ejpam-3461	39	5	finite	finite	ADJ
ejpam-3461	39	6	uniform	uniform	ADJ
ejpam-3461	39	7	dimension	dimension	NOUN
ejpam-3461	39	8	.	.	PUNCT
ejpam-3461	40	1	(	(	PUNCT
ejpam-3461	40	2	5	5	X
ejpam-3461	40	3	)	)	PUNCT
ejpam-3461	40	4	the	the	DET
ejpam-3461	40	5	following	follow	VERB
ejpam-3461	40	6	conditions	condition	NOUN
ejpam-3461	40	7	are	be	AUX
ejpam-3461	40	8	euivalent	euivalent	NOUN
ejpam-3461	40	9	for	for	ADP
ejpam-3461	40	10	a	a	DET
ejpam-3461	40	11	right	right	ADJ
ejpam-3461	40	12	si	si	NOUN
ejpam-3461	40	13	-	-	ADJ
ejpam-3461	40	14	ring	ring	NOUN
ejpam-3461	40	15	r	r	NOUN
ejpam-3461	40	16	:	:	PUNCT
ejpam-3461	40	17	(	(	PUNCT
ejpam-3461	40	18	a	a	X
ejpam-3461	40	19	)	)	PUNCT
ejpam-3461	40	20	r	r	NOUN
ejpam-3461	40	21	(	(	PUNCT
ejpam-3461	40	22	n	n	CCONJ
ejpam-3461	40	23	)	)	PUNCT
ejpam-3461	40	24	r	r	NOUN
ejpam-3461	40	25	is	be	AUX
ejpam-3461	40	26	extending	extend	VERB
ejpam-3461	40	27	.	.	PUNCT
ejpam-3461	41	1	(	(	PUNCT
ejpam-3461	41	2	b	b	X
ejpam-3461	41	3	)	)	PUNCT
ejpam-3461	41	4	every	every	DET
ejpam-3461	41	5	r	r	NOUN
ejpam-3461	41	6	-	-	PUNCT
ejpam-3461	41	7	module	module	NOUN
ejpam-3461	41	8	is	be	AUX
ejpam-3461	41	9	c	c	NOUN
ejpam-3461	41	10	-	-	PUNCT
ejpam-3461	41	11	co	co	ADJ
ejpam-3461	41	12	-	-	ADJ
ejpam-3461	41	13	epi	epi	NOUN
ejpam-3461	41	14	-	-	NOUN
ejpam-3461	41	15	retractable	retractable	ADJ
ejpam-3461	41	16	.	.	PUNCT
ejpam-3461	42	1	(	(	PUNCT
ejpam-3461	42	2	c	c	X
ejpam-3461	42	3	)	)	PUNCT
ejpam-3461	42	4	every	every	DET
ejpam-3461	42	5	r	r	NOUN
ejpam-3461	42	6	-	-	PUNCT
ejpam-3461	42	7	module	module	NOUN
ejpam-3461	42	8	is	be	AUX
ejpam-3461	42	9	extending	extend	VERB
ejpam-3461	42	10	.	.	PUNCT
ejpam-3461	43	1	for	for	ADP
ejpam-3461	43	2	an	an	DET
ejpam-3461	43	3	r	r	NOUN
ejpam-3461	43	4	-	-	PUNCT
ejpam-3461	43	5	module	module	NOUN
ejpam-3461	43	6	m	m	NOUN
ejpam-3461	43	7	,	,	PUNCT
ejpam-3461	43	8	s	s	PART
ejpam-3461	43	9	=	=	ADJ
ejpam-3461	43	10	endr(m	endr(m	PROPN
ejpam-3461	43	11	)	)	PUNCT
ejpam-3461	43	12	denotes	denote	VERB
ejpam-3461	43	13	the	the	DET
ejpam-3461	43	14	endomorphism	endomorphism	PROPN
ejpam-3461	43	15	ring	ring	NOUN
ejpam-3461	43	16	of	of	ADP
ejpam-3461	43	17	m	m	PROPN
ejpam-3461	43	18	.	.	PUNCT
ejpam-3461	44	1	for	for	ADP
ejpam-3461	44	2	φ	φ	PROPN
ejpam-3461	44	3	∈	∈	PROPN
ejpam-3461	44	4	s	s	PROPN
ejpam-3461	44	5	,	,	PUNCT
ejpam-3461	44	6	kerφ	kerφ	PROPN
ejpam-3461	44	7	and	and	CCONJ
ejpam-3461	44	8	imφ	imφ	VERB
ejpam-3461	44	9	stand	stand	NOUN
ejpam-3461	44	10	for	for	ADP
ejpam-3461	44	11	kernel	kernel	NOUN
ejpam-3461	44	12	and	and	CCONJ
ejpam-3461	44	13	image	image	NOUN
ejpam-3461	44	14	of	of	ADP
ejpam-3461	44	15	φ	φ	PROPN
ejpam-3461	44	16	,	,	PUNCT
ejpam-3461	44	17	respectively	respectively	ADV
ejpam-3461	44	18	.	.	PUNCT
ejpam-3461	45	1	the	the	DET
ejpam-3461	45	2	notations	notation	NOUN
ejpam-3461	45	3	n	n	X
ejpam-3461	45	4	≤	≤	NOUN
ejpam-3461	45	5	m	m	VERB
ejpam-3461	45	6	,	,	PUNCT
ejpam-3461	45	7	n	n	CCONJ
ejpam-3461	45	8	≤e	≤e	VERB
ejpam-3461	45	9	m	m	PROPN
ejpam-3461	45	10	and	and	CCONJ
ejpam-3461	45	11	n	n	PRON
ejpam-3461	45	12	≤⊕	≤⊕	AUX
ejpam-3461	45	13	m	m	VERB
ejpam-3461	45	14	mean	mean	VERB
ejpam-3461	45	15	that	that	SCONJ
ejpam-3461	45	16	n	n	PRON
ejpam-3461	45	17	is	be	AUX
ejpam-3461	45	18	a	a	DET
ejpam-3461	45	19	submodule	submodule	NOUN
ejpam-3461	45	20	of	of	ADP
ejpam-3461	45	21	m	m	PROPN
ejpam-3461	45	22	,	,	PUNCT
ejpam-3461	45	23	an	an	DET
ejpam-3461	45	24	essential	essential	ADJ
ejpam-3461	45	25	submodule	submodule	NOUN
ejpam-3461	45	26	and	and	CCONJ
ejpam-3461	45	27	a	a	DET
ejpam-3461	45	28	direct	direct	ADJ
ejpam-3461	45	29	summand	summand	NOUN
ejpam-3461	45	30	of	of	ADP
ejpam-3461	45	31	m	m	PROPN
ejpam-3461	45	32	,	,	PUNCT
ejpam-3461	45	33	respectively	respectively	ADV
ejpam-3461	45	34	.	.	PUNCT
ejpam-3461	46	1	also	also	ADV
ejpam-3461	46	2	e(m	e(m	PROPN
ejpam-3461	46	3	)	)	PUNCT
ejpam-3461	46	4	denotes	denote	VERB
ejpam-3461	46	5	the	the	DET
ejpam-3461	46	6	injective	injective	ADJ
ejpam-3461	46	7	hull	hull	NOUN
ejpam-3461	46	8	of	of	ADP
ejpam-3461	46	9	m	m	PROPN
ejpam-3461	46	10	.	.	PUNCT
ejpam-3461	47	1	2	2	X
ejpam-3461	47	2	.	.	X
ejpam-3461	47	3	preliminaries	preliminary	NOUN
ejpam-3461	47	4	in	in	ADP
ejpam-3461	47	5	this	this	DET
ejpam-3461	47	6	section	section	NOUN
ejpam-3461	47	7	,	,	PUNCT
ejpam-3461	47	8	we	we	PRON
ejpam-3461	47	9	are	be	AUX
ejpam-3461	47	10	going	go	VERB
ejpam-3461	47	11	to	to	PART
ejpam-3461	47	12	give	give	VERB
ejpam-3461	47	13	preliminary	preliminary	ADJ
ejpam-3461	47	14	definitions	definition	NOUN
ejpam-3461	47	15	and	and	CCONJ
ejpam-3461	47	16	results	result	NOUN
ejpam-3461	47	17	which	which	PRON
ejpam-3461	47	18	we	we	PRON
ejpam-3461	47	19	will	will	AUX
ejpam-3461	47	20	use	use	VERB
ejpam-3461	47	21	throughout	throughout	ADP
ejpam-3461	47	22	this	this	DET
ejpam-3461	47	23	paper	paper	NOUN
ejpam-3461	47	24	.	.	PUNCT
ejpam-3461	48	1	definition	definition	NOUN
ejpam-3461	48	2	1	1	NUM
ejpam-3461	48	3	.	.	NOUN
ejpam-3461	48	4	1	1	NUM
ejpam-3461	48	5	.	.	PUNCT
ejpam-3461	49	1	an	an	DET
ejpam-3461	49	2	r	r	NOUN
ejpam-3461	49	3	-	-	PUNCT
ejpam-3461	49	4	module	module	NOUN
ejpam-3461	49	5	m	m	NOUN
ejpam-3461	49	6	is	be	AUX
ejpam-3461	49	7	called	call	VERB
ejpam-3461	49	8	cs	cs	PROPN
ejpam-3461	49	9	module	module	NOUN
ejpam-3461	49	10	if	if	SCONJ
ejpam-3461	49	11	every	every	DET
ejpam-3461	49	12	complement	complement	NOUN
ejpam-3461	49	13	submodule	submodule	NOUN
ejpam-3461	49	14	of	of	ADP
ejpam-3461	49	15	m	m	PROPN
ejpam-3461	49	16	is	be	AUX
ejpam-3461	49	17	a	a	DET
ejpam-3461	49	18	direct	direct	ADJ
ejpam-3461	49	19	summand	summand	NOUN
ejpam-3461	49	20	.	.	PUNCT
ejpam-3461	50	1	2	2	X
ejpam-3461	50	2	.	.	X
ejpam-3461	50	3	an	an	DET
ejpam-3461	50	4	r	r	NOUN
ejpam-3461	50	5	-	-	PUNCT
ejpam-3461	50	6	module	module	NOUN
ejpam-3461	50	7	m	m	NOUN
ejpam-3461	50	8	is	be	AUX
ejpam-3461	50	9	called	call	VERB
ejpam-3461	50	10	continuous	continuous	ADJ
ejpam-3461	50	11	if	if	SCONJ
ejpam-3461	50	12	it	it	PRON
ejpam-3461	50	13	is	be	AUX
ejpam-3461	50	14	a	a	DET
ejpam-3461	50	15	cs	cs	NOUN
ejpam-3461	50	16	module	module	NOUN
ejpam-3461	50	17	and	and	CCONJ
ejpam-3461	50	18	satisfies	satisfy	VERB
ejpam-3461	50	19	the	the	DET
ejpam-3461	50	20	following	follow	VERB
ejpam-3461	50	21	condition	condition	NOUN
ejpam-3461	50	22	:	:	PUNCT
ejpam-3461	50	23	(	(	PUNCT
ejpam-3461	50	24	c2	c2	PROPN
ejpam-3461	50	25	)	)	PUNCT
ejpam-3461	50	26	every	every	DET
ejpam-3461	50	27	submodule	submodule	NOUN
ejpam-3461	50	28	of	of	ADP
ejpam-3461	50	29	m	m	PRON
ejpam-3461	50	30	that	that	PRON
ejpam-3461	50	31	is	be	AUX
ejpam-3461	50	32	isomorphic	isomorphic	ADJ
ejpam-3461	50	33	to	to	ADP
ejpam-3461	50	34	a	a	DET
ejpam-3461	50	35	direct	direct	ADJ
ejpam-3461	50	36	summand	summand	NOUN
ejpam-3461	51	1	m	m	VERB
ejpam-3461	51	2	is	be	AUX
ejpam-3461	51	3	itself	itself	PRON
ejpam-3461	51	4	a	a	DET
ejpam-3461	51	5	direct	direct	ADJ
ejpam-3461	51	6	summand	summand	NOUN
ejpam-3461	51	7	of	of	ADP
ejpam-3461	51	8	m	m	PROPN
ejpam-3461	51	9	.	.	PUNCT
ejpam-3461	52	1	3	3	X
ejpam-3461	52	2	.	.	X
ejpam-3461	52	3	an	an	DET
ejpam-3461	52	4	r	r	NOUN
ejpam-3461	52	5	-	-	PUNCT
ejpam-3461	52	6	module	module	NOUN
ejpam-3461	52	7	m	m	NOUN
ejpam-3461	52	8	is	be	AUX
ejpam-3461	52	9	called	call	VERB
ejpam-3461	52	10	quasi	quasi	ADJ
ejpam-3461	52	11	-	-	ADJ
ejpam-3461	52	12	continuous	continuous	ADJ
ejpam-3461	52	13	if	if	SCONJ
ejpam-3461	52	14	it	it	PRON
ejpam-3461	52	15	is	be	AUX
ejpam-3461	52	16	a	a	DET
ejpam-3461	52	17	cs	cs	NOUN
ejpam-3461	52	18	module	module	NOUN
ejpam-3461	52	19	and	and	CCONJ
ejpam-3461	52	20	satisfies	satisfy	VERB
ejpam-3461	52	21	the	the	DET
ejpam-3461	52	22	following	follow	VERB
ejpam-3461	52	23	condition	condition	NOUN
ejpam-3461	52	24	:	:	PUNCT
ejpam-3461	52	25	(	(	PUNCT
ejpam-3461	52	26	c3	c3	NOUN
ejpam-3461	52	27	)	)	PUNCT
ejpam-3461	52	28	if	if	SCONJ
ejpam-3461	52	29	n	n	PROPN
ejpam-3461	52	30	and	and	CCONJ
ejpam-3461	52	31	k	k	PROPN
ejpam-3461	52	32	are	be	AUX
ejpam-3461	52	33	direct	direct	ADJ
ejpam-3461	52	34	summands	summand	NOUN
ejpam-3461	52	35	of	of	ADP
ejpam-3461	52	36	m	m	PROPN
ejpam-3461	52	37	with	with	ADP
ejpam-3461	52	38	n	n	PRON
ejpam-3461	52	39	∩k	∩k	NOUN
ejpam-3461	52	40	=	=	PUNCT
ejpam-3461	52	41	{	{	PUNCT
ejpam-3461	52	42	0	0	NUM
ejpam-3461	52	43	}	}	PUNCT
ejpam-3461	52	44	,	,	PUNCT
ejpam-3461	52	45	then	then	ADV
ejpam-3461	52	46	n	n	DET
ejpam-3461	52	47	⊕k	⊕k	NOUN
ejpam-3461	52	48	is	be	AUX
ejpam-3461	52	49	a	a	DET
ejpam-3461	52	50	direct	direct	ADJ
ejpam-3461	52	51	summand	summand	NOUN
ejpam-3461	52	52	of	of	ADP
ejpam-3461	52	53	m	m	PROPN
ejpam-3461	52	54	.	.	PUNCT
ejpam-3461	53	1	a.	a.	PROPN
ejpam-3461	53	2	d.	d.	PROPN
ejpam-3461	53	3	diallo	diallo	PROPN
ejpam-3461	53	4	,	,	PUNCT
ejpam-3461	53	5	p.	p.	PROPN
ejpam-3461	53	6	c.	c.	PROPN
ejpam-3461	53	7	diop	diop	PROPN
ejpam-3461	53	8	,	,	PUNCT
ejpam-3461	53	9	m.	m.	NOUN
ejpam-3461	53	10	barry	barry	PROPN
ejpam-3461	53	11	/	/	SYM
ejpam-3461	53	12	eur	eur	PROPN
ejpam-3461	53	13	.	.	PUNCT
ejpam-3461	54	1	j.	j.	PROPN
ejpam-3461	54	2	pure	pure	PROPN
ejpam-3461	54	3	appl	appl	PROPN
ejpam-3461	54	4	.	.	PROPN
ejpam-3461	54	5	math	math	PROPN
ejpam-3461	54	6	,	,	PUNCT
ejpam-3461	54	7	12	12	NUM
ejpam-3461	54	8	(	(	PUNCT
ejpam-3461	54	9	3	3	NUM
ejpam-3461	54	10	)	)	PUNCT
ejpam-3461	54	11	(	(	PUNCT
ejpam-3461	54	12	2019	2019	NUM
ejpam-3461	54	13	)	)	PUNCT
ejpam-3461	54	14	,	,	PUNCT
ejpam-3461	54	15	1187	1187	NUM
ejpam-3461	54	16	-	-	SYM
ejpam-3461	54	17	1198	1198	NUM
ejpam-3461	54	18	1189	1189	NUM
ejpam-3461	54	19	definition	definition	NOUN
ejpam-3461	54	20	2	2	NUM
ejpam-3461	54	21	.	.	PUNCT
ejpam-3461	55	1	let	let	VERB
ejpam-3461	55	2	m	m	PRON
ejpam-3461	55	3	be	be	AUX
ejpam-3461	55	4	an	an	DET
ejpam-3461	55	5	r	r	NOUN
ejpam-3461	55	6	-	-	PUNCT
ejpam-3461	55	7	module	module	NOUN
ejpam-3461	55	8	,	,	PUNCT
ejpam-3461	55	9	put	put	VERB
ejpam-3461	55	10	z(m	z(m	NOUN
ejpam-3461	55	11	)	)	PUNCT
ejpam-3461	56	1	=	=	PRON
ejpam-3461	56	2	{	{	PUNCT
ejpam-3461	56	3	m	m	VERB
ejpam-3461	56	4	∈	∈	ADJ
ejpam-3461	56	5	m	m	NOUN
ejpam-3461	56	6	:	:	PUNCT
ejpam-3461	56	7	annr(m	annr(m	NOUN
ejpam-3461	56	8	)	)	PUNCT
ejpam-3461	56	9	≤e	≤e	VERB
ejpam-3461	56	10	r	r	NOUN
ejpam-3461	56	11	}	}	PUNCT
ejpam-3461	56	12	.	.	PUNCT
ejpam-3461	57	1	m	m	PROPN
ejpam-3461	57	2	is	be	AUX
ejpam-3461	57	3	called	call	VERB
ejpam-3461	57	4	nonsingular	nonsingular	ADJ
ejpam-3461	57	5	if	if	SCONJ
ejpam-3461	57	6	z(m	z(m	NOUN
ejpam-3461	57	7	)	)	PUNCT
ejpam-3461	57	8	=	=	PRON
ejpam-3461	57	9	{	{	PUNCT
ejpam-3461	57	10	0	0	NUM
ejpam-3461	57	11	}	}	PUNCT
ejpam-3461	57	12	,	,	PUNCT
ejpam-3461	57	13	and	and	CCONJ
ejpam-3461	57	14	singular	singular	ADJ
ejpam-3461	57	15	if	if	SCONJ
ejpam-3461	57	16	z(m	z(m	NOUN
ejpam-3461	57	17	)	)	PUNCT
ejpam-3461	58	1	=	=	PUNCT
ejpam-3461	58	2	m	m	NOUN
ejpam-3461	58	3	.	.	PUNCT
ejpam-3461	59	1	the	the	DET
ejpam-3461	59	2	goldie	goldie	PROPN
ejpam-3461	59	3	torsion	torsion	PROPN
ejpam-3461	59	4	submodule	submodule	PROPN
ejpam-3461	59	5	z2(m	z2(m	NOUN
ejpam-3461	59	6	)	)	PUNCT
ejpam-3461	59	7	of	of	ADP
ejpam-3461	59	8	m	m	PROPN
ejpam-3461	59	9	is	be	AUX
ejpam-3461	59	10	defined	define	VERB
ejpam-3461	59	11	by	by	ADP
ejpam-3461	59	12	z(m	z(m	PROPN
ejpam-3461	59	13	/	/	SYM
ejpam-3461	59	14	z(m	z(m	NUM
ejpam-3461	59	15	)	)	PUNCT
ejpam-3461	59	16	)	)	PUNCT
ejpam-3461	60	1	=	=	PUNCT
ejpam-3461	60	2	z2(m)/z(m	z2(m)/z(m	PROPN
ejpam-3461	60	3	)	)	PUNCT
ejpam-3461	60	4	.	.	PUNCT
ejpam-3461	61	1	m	m	PROPN
ejpam-3461	61	2	is	be	AUX
ejpam-3461	61	3	z2	z2	NUM
ejpam-3461	61	4	-	-	PUNCT
ejpam-3461	61	5	torsion	torsion	NOUN
ejpam-3461	61	6	if	if	SCONJ
ejpam-3461	61	7	,	,	PUNCT
ejpam-3461	61	8	z2(m	z2(m	X
ejpam-3461	61	9	)	)	PUNCT
ejpam-3461	61	10	=	=	NOUN
ejpam-3461	61	11	m	m	NOUN
ejpam-3461	61	12	.	.	PUNCT
ejpam-3461	62	1	3	3	X
ejpam-3461	62	2	.	.	X
ejpam-3461	62	3	c	c	X
ejpam-3461	62	4	-	-	PUNCT
ejpam-3461	62	5	co	co	NOUN
ejpam-3461	62	6	-	-	ADJ
ejpam-3461	62	7	epi	epi	ADJ
ejpam-3461	62	8	-	-	ADJ
ejpam-3461	62	9	retractable	retractable	ADJ
ejpam-3461	62	10	modules	module	NOUN
ejpam-3461	62	11	and	and	CCONJ
ejpam-3461	62	12	some	some	DET
ejpam-3461	62	13	applications	application	NOUN
ejpam-3461	62	14	definition	definition	NOUN
ejpam-3461	62	15	3	3	NUM
ejpam-3461	62	16	.	.	PUNCT
ejpam-3461	63	1	an	an	DET
ejpam-3461	63	2	r	r	NOUN
ejpam-3461	63	3	-	-	PUNCT
ejpam-3461	63	4	module	module	NOUN
ejpam-3461	63	5	m	m	NOUN
ejpam-3461	63	6	is	be	AUX
ejpam-3461	63	7	called	call	VERB
ejpam-3461	63	8	c	c	NOUN
ejpam-3461	63	9	-	-	PUNCT
ejpam-3461	63	10	co	co	NOUN
ejpam-3461	63	11	-	-	ADJ
ejpam-3461	63	12	epi	epi	ADJ
ejpam-3461	63	13	-	-	NOUN
ejpam-3461	63	14	retractable	retractable	ADJ
ejpam-3461	63	15	if	if	SCONJ
ejpam-3461	63	16	,	,	PUNCT
ejpam-3461	63	17	for	for	ADP
ejpam-3461	63	18	every	every	DET
ejpam-3461	63	19	complement	complement	NOUN
ejpam-3461	63	20	submodule	submodule	NOUN
ejpam-3461	63	21	n	n	PROPN
ejpam-3461	63	22	of	of	ADP
ejpam-3461	63	23	m	m	PROPN
ejpam-3461	63	24	,	,	PUNCT
ejpam-3461	63	25	there	there	PRON
ejpam-3461	63	26	exists	exist	VERB
ejpam-3461	63	27	a	a	DET
ejpam-3461	63	28	monomorphism	monomorphism	NOUN
ejpam-3461	63	29	f	f	X
ejpam-3461	63	30	:	:	PUNCT
ejpam-3461	63	31	m	m	X
ejpam-3461	63	32	/	/	SYM
ejpam-3461	63	33	n	n	PROPN
ejpam-3461	63	34	−→	−→	NOUN
ejpam-3461	63	35	m	m	NOUN
ejpam-3461	63	36	.	.	PUNCT
ejpam-3461	64	1	the	the	DET
ejpam-3461	64	2	ring	ring	NOUN
ejpam-3461	64	3	r	r	NOUN
ejpam-3461	64	4	is	be	AUX
ejpam-3461	64	5	called	call	VERB
ejpam-3461	64	6	c	c	NOUN
ejpam-3461	64	7	-	-	PUNCT
ejpam-3461	64	8	co	co	NOUN
ejpam-3461	64	9	-	-	NOUN
ejpam-3461	64	10	pri	pri	NOUN
ejpam-3461	64	11	if	if	SCONJ
ejpam-3461	64	12	rr	rr	PROPN
ejpam-3461	64	13	is	be	AUX
ejpam-3461	64	14	c	c	NOUN
ejpam-3461	64	15	-	-	PUNCT
ejpam-3461	64	16	co	co	ADJ
ejpam-3461	64	17	-	-	ADJ
ejpam-3461	64	18	epi	epi	NOUN
ejpam-3461	64	19	-	-	NOUN
ejpam-3461	64	20	retractable	retractable	ADJ
ejpam-3461	64	21	.	.	PUNCT
ejpam-3461	65	1	remark	remark	NOUN
ejpam-3461	65	2	1	1	NUM
ejpam-3461	65	3	.	.	PUNCT
ejpam-3461	66	1	clearly	clearly	ADV
ejpam-3461	66	2	,	,	PUNCT
ejpam-3461	66	3	every	every	DET
ejpam-3461	66	4	co	co	NOUN
ejpam-3461	66	5	-	-	ADJ
ejpam-3461	66	6	epi	epi	ADJ
ejpam-3461	66	7	-	-	ADJ
ejpam-3461	66	8	retractable	retractable	ADJ
ejpam-3461	66	9	module	module	NOUN
ejpam-3461	66	10	is	be	AUX
ejpam-3461	66	11	c	c	NOUN
ejpam-3461	66	12	-	-	PUNCT
ejpam-3461	66	13	co	co	NOUN
ejpam-3461	66	14	-	-	ADJ
ejpam-3461	66	15	epi	epi	ADJ
ejpam-3461	66	16	-	-	NOUN
ejpam-3461	66	17	retractable	retractable	ADJ
ejpam-3461	66	18	while	while	SCONJ
ejpam-3461	66	19	the	the	DET
ejpam-3461	66	20	converse	converse	NOUN
ejpam-3461	66	21	is	be	AUX
ejpam-3461	66	22	not	not	PART
ejpam-3461	66	23	true	true	ADJ
ejpam-3461	66	24	.	.	PUNCT
ejpam-3461	67	1	for	for	ADP
ejpam-3461	67	2	example	example	NOUN
ejpam-3461	67	3	q	q	PROPN
ejpam-3461	67	4	as	as	SCONJ
ejpam-3461	67	5	z	z	NOUN
ejpam-3461	67	6	-	-	PUNCT
ejpam-3461	67	7	module	module	NOUN
ejpam-3461	67	8	is	be	AUX
ejpam-3461	67	9	c	c	NOUN
ejpam-3461	67	10	-	-	PUNCT
ejpam-3461	67	11	co	co	NOUN
ejpam-3461	67	12	-	-	ADJ
ejpam-3461	67	13	epi	epi	ADJ
ejpam-3461	67	14	-	-	NOUN
ejpam-3461	67	15	retractable	retractable	ADJ
ejpam-3461	67	16	but	but	CCONJ
ejpam-3461	67	17	it	it	PRON
ejpam-3461	67	18	is	be	AUX
ejpam-3461	67	19	not	not	PART
ejpam-3461	67	20	co	co	ADJ
ejpam-3461	67	21	-	-	ADJ
ejpam-3461	67	22	epiretractable	epiretractable	ADJ
ejpam-3461	67	23	.	.	PUNCT
ejpam-3461	68	1	lemma	lemma	PROPN
ejpam-3461	68	2	1	1	NUM
ejpam-3461	68	3	.	.	PUNCT
ejpam-3461	69	1	the	the	DET
ejpam-3461	69	2	following	follow	VERB
ejpam-3461	69	3	statements	statement	NOUN
ejpam-3461	69	4	are	be	AUX
ejpam-3461	69	5	equivalent	equivalent	ADJ
ejpam-3461	69	6	for	for	ADP
ejpam-3461	69	7	an	an	DET
ejpam-3461	69	8	r	r	NOUN
ejpam-3461	69	9	-	-	PUNCT
ejpam-3461	69	10	module	module	NOUN
ejpam-3461	69	11	m	m	NOUN
ejpam-3461	69	12	:	:	PUNCT
ejpam-3461	69	13	(	(	PUNCT
ejpam-3461	69	14	1	1	X
ejpam-3461	69	15	)	)	PUNCT
ejpam-3461	69	16	m	m	VERB
ejpam-3461	69	17	is	be	AUX
ejpam-3461	69	18	a	a	DET
ejpam-3461	69	19	c	c	NOUN
ejpam-3461	69	20	-	-	PUNCT
ejpam-3461	69	21	co	co	NOUN
ejpam-3461	69	22	-	-	ADJ
ejpam-3461	69	23	epi	epi	ADJ
ejpam-3461	69	24	-	-	ADJ
ejpam-3461	69	25	retractable	retractable	ADJ
ejpam-3461	69	26	module	module	NOUN
ejpam-3461	69	27	.	.	PUNCT
ejpam-3461	70	1	(	(	PUNCT
ejpam-3461	70	2	2	2	X
ejpam-3461	70	3	)	)	PUNCT
ejpam-3461	70	4	there	there	PRON
ejpam-3461	70	5	exists	exist	VERB
ejpam-3461	70	6	ϕ	ϕ	PROPN
ejpam-3461	70	7	∈	∈	PROPN
ejpam-3461	70	8	endr(m	endr(m	PROPN
ejpam-3461	70	9	)	)	PUNCT
ejpam-3461	70	10	such	such	ADJ
ejpam-3461	70	11	that	that	DET
ejpam-3461	70	12	kerϕ	kerϕ	NOUN
ejpam-3461	70	13	=	=	PUNCT
ejpam-3461	70	14	n	n	PROPN
ejpam-3461	70	15	for	for	ADP
ejpam-3461	70	16	any	any	DET
ejpam-3461	70	17	nonzero	nonzero	NOUN
ejpam-3461	70	18	n	n	CCONJ
ejpam-3461	70	19	⊆c	⊆c	NOUN
ejpam-3461	70	20	m	m	VERB
ejpam-3461	70	21	.	.	PUNCT
ejpam-3461	71	1	proposition	proposition	NOUN
ejpam-3461	71	2	1	1	NUM
ejpam-3461	71	3	.	.	PUNCT
ejpam-3461	72	1	let	let	VERB
ejpam-3461	72	2	m	m	PRON
ejpam-3461	72	3	be	be	AUX
ejpam-3461	72	4	a	a	DET
ejpam-3461	72	5	c	c	NOUN
ejpam-3461	72	6	-	-	PUNCT
ejpam-3461	72	7	co	co	NOUN
ejpam-3461	72	8	-	-	ADJ
ejpam-3461	72	9	epi	epi	ADJ
ejpam-3461	72	10	-	-	ADJ
ejpam-3461	72	11	retractable	retractable	ADJ
ejpam-3461	72	12	r	r	NOUN
ejpam-3461	72	13	-	-	PUNCT
ejpam-3461	72	14	module	module	NOUN
ejpam-3461	72	15	.	.	PUNCT
ejpam-3461	73	1	then	then	ADV
ejpam-3461	73	2	a	a	DET
ejpam-3461	73	3	fully	fully	ADV
ejpam-3461	73	4	invariant	invariant	ADJ
ejpam-3461	73	5	complement	complement	NOUN
ejpam-3461	73	6	submodule	submodule	NOUN
ejpam-3461	73	7	of	of	ADP
ejpam-3461	73	8	a	a	DET
ejpam-3461	73	9	m	m	NOUN
ejpam-3461	73	10	is	be	AUX
ejpam-3461	73	11	also	also	ADV
ejpam-3461	73	12	c	c	NOUN
ejpam-3461	73	13	-	-	PUNCT
ejpam-3461	73	14	co	co	ADJ
ejpam-3461	73	15	-	-	ADJ
ejpam-3461	73	16	epi	epi	NOUN
ejpam-3461	73	17	-	-	NOUN
ejpam-3461	73	18	retractable	retractable	ADJ
ejpam-3461	73	19	.	.	PUNCT
ejpam-3461	74	1	proof	proof	NOUN
ejpam-3461	74	2	.	.	PUNCT
ejpam-3461	75	1	let	let	VERB
ejpam-3461	75	2	m	m	PRON
ejpam-3461	75	3	be	be	AUX
ejpam-3461	75	4	a	a	DET
ejpam-3461	75	5	c	c	NOUN
ejpam-3461	75	6	-	-	PUNCT
ejpam-3461	75	7	co	co	NOUN
ejpam-3461	75	8	-	-	ADJ
ejpam-3461	75	9	epi	epi	ADJ
ejpam-3461	75	10	-	-	ADJ
ejpam-3461	75	11	retractable	retractable	ADJ
ejpam-3461	75	12	module	module	NOUN
ejpam-3461	75	13	and	and	CCONJ
ejpam-3461	75	14	n	n	NOUN
ejpam-3461	75	15	⊆c	⊆c	NOUN
ejpam-3461	75	16	m	m	VERB
ejpam-3461	75	17	with	with	ADP
ejpam-3461	75	18	n	n	ADV
ejpam-3461	75	19	fully	fully	ADV
ejpam-3461	75	20	invariant	invariant	ADJ
ejpam-3461	75	21	.	.	PUNCT
ejpam-3461	76	1	let	let	VERB
ejpam-3461	76	2	k	k	PROPN
ejpam-3461	76	3	⊆c	⊆c	VERB
ejpam-3461	76	4	n	n	PROPN
ejpam-3461	76	5	.	.	PUNCT
ejpam-3461	77	1	then	then	ADV
ejpam-3461	77	2	,	,	PUNCT
ejpam-3461	77	3	k	k	PROPN
ejpam-3461	77	4	⊆c	⊆c	PROPN
ejpam-3461	77	5	m	m	VERB
ejpam-3461	77	6	.	.	PUNCT
ejpam-3461	78	1	so	so	ADV
ejpam-3461	78	2	,	,	PUNCT
ejpam-3461	78	3	there	there	PRON
ejpam-3461	78	4	is	be	VERB
ejpam-3461	78	5	an	an	DET
ejpam-3461	78	6	endomorphism	endomorphism	NOUN
ejpam-3461	78	7	f	f	NOUN
ejpam-3461	78	8	:	:	PUNCT
ejpam-3461	78	9	m	m	VERB
ejpam-3461	78	10	−→m	−→m	NUM
ejpam-3461	78	11	such	such	ADJ
ejpam-3461	78	12	that	that	SCONJ
ejpam-3461	78	13	k	k	NOUN
ejpam-3461	78	14	=	=	NOUN
ejpam-3461	78	15	kerf	kerf	NOUN
ejpam-3461	78	16	.	.	PUNCT
ejpam-3461	79	1	then	then	ADV
ejpam-3461	79	2	f	f	X
ejpam-3461	79	3	|n	|n	NOUN
ejpam-3461	79	4	:	:	PUNCT
ejpam-3461	79	5	n	n	CCONJ
ejpam-3461	79	6	−→	−→	NOUN
ejpam-3461	79	7	n	n	NOUN
ejpam-3461	79	8	and	and	CCONJ
ejpam-3461	79	9	k	k	PROPN
ejpam-3461	79	10	=	=	SYM
ejpam-3461	79	11	ker(f	ker(f	PROPN
ejpam-3461	79	12	|n	|n	NOUN
ejpam-3461	79	13	)	)	PUNCT
ejpam-3461	79	14	.	.	PUNCT
ejpam-3461	80	1	therefore	therefore	ADV
ejpam-3461	80	2	,	,	PUNCT
ejpam-3461	80	3	n	n	PRON
ejpam-3461	80	4	is	be	AUX
ejpam-3461	80	5	a	a	DET
ejpam-3461	80	6	c	c	NOUN
ejpam-3461	80	7	-	-	PUNCT
ejpam-3461	80	8	co	co	NOUN
ejpam-3461	80	9	-	-	ADJ
ejpam-3461	80	10	epi	epi	ADJ
ejpam-3461	80	11	-	-	ADJ
ejpam-3461	80	12	retractable	retractable	ADJ
ejpam-3461	80	13	module	module	NOUN
ejpam-3461	80	14	.	.	PUNCT
ejpam-3461	81	1	corollary	corollary	ADJ
ejpam-3461	81	2	1	1	NUM
ejpam-3461	81	3	.	.	PUNCT
ejpam-3461	82	1	every	every	DET
ejpam-3461	82	2	fully	fully	ADV
ejpam-3461	82	3	invariant	invariant	ADJ
ejpam-3461	82	4	direct	direct	ADJ
ejpam-3461	82	5	summand	summand	NOUN
ejpam-3461	82	6	of	of	ADP
ejpam-3461	82	7	a	a	DET
ejpam-3461	82	8	c	c	NOUN
ejpam-3461	82	9	-	-	PUNCT
ejpam-3461	82	10	co	co	NOUN
ejpam-3461	82	11	-	-	ADJ
ejpam-3461	82	12	epi	epi	ADJ
ejpam-3461	82	13	-	-	ADJ
ejpam-3461	82	14	retractable	retractable	ADJ
ejpam-3461	82	15	module	module	NOUN
ejpam-3461	82	16	is	be	AUX
ejpam-3461	82	17	c	c	NOUN
ejpam-3461	82	18	-	-	PUNCT
ejpam-3461	82	19	co	co	ADJ
ejpam-3461	82	20	-	-	ADJ
ejpam-3461	82	21	epi	epi	NOUN
ejpam-3461	82	22	-	-	NOUN
ejpam-3461	82	23	retractable	retractable	ADJ
ejpam-3461	82	24	.	.	PUNCT
ejpam-3461	83	1	proposition	proposition	NOUN
ejpam-3461	83	2	2	2	NUM
ejpam-3461	83	3	.	.	PUNCT
ejpam-3461	84	1	let	let	VERB
ejpam-3461	84	2	m	m	PRON
ejpam-3461	84	3	be	be	AUX
ejpam-3461	84	4	an	an	DET
ejpam-3461	84	5	r	r	NOUN
ejpam-3461	84	6	-	-	PUNCT
ejpam-3461	84	7	module	module	NOUN
ejpam-3461	84	8	with	with	ADP
ejpam-3461	84	9	s	s	NOUN
ejpam-3461	84	10	=	=	PUNCT
ejpam-3461	84	11	endr(m	endr(m	NOUN
ejpam-3461	84	12	)	)	PUNCT
ejpam-3461	84	13	regular	regular	ADV
ejpam-3461	84	14	.	.	PUNCT
ejpam-3461	85	1	then	then	ADV
ejpam-3461	85	2	the	the	DET
ejpam-3461	85	3	following	follow	VERB
ejpam-3461	85	4	conditions	condition	NOUN
ejpam-3461	85	5	are	be	AUX
ejpam-3461	85	6	equivalent	equivalent	ADJ
ejpam-3461	85	7	:	:	PUNCT
ejpam-3461	85	8	(	(	PUNCT
ejpam-3461	85	9	1	1	X
ejpam-3461	85	10	)	)	PUNCT
ejpam-3461	85	11	m	m	VERB
ejpam-3461	85	12	is	be	AUX
ejpam-3461	85	13	a	a	DET
ejpam-3461	85	14	c	c	NOUN
ejpam-3461	85	15	-	-	PUNCT
ejpam-3461	85	16	co	co	NOUN
ejpam-3461	85	17	-	-	ADJ
ejpam-3461	85	18	epi	epi	ADJ
ejpam-3461	85	19	-	-	ADJ
ejpam-3461	85	20	retractable	retractable	ADJ
ejpam-3461	85	21	module	module	NOUN
ejpam-3461	85	22	.	.	PUNCT
ejpam-3461	86	1	(	(	PUNCT
ejpam-3461	86	2	2	2	X
ejpam-3461	86	3	)	)	PUNCT
ejpam-3461	86	4	m	m	VERB
ejpam-3461	86	5	is	be	AUX
ejpam-3461	86	6	an	an	DET
ejpam-3461	86	7	extending	extend	VERB
ejpam-3461	86	8	module	module	NOUN
ejpam-3461	86	9	.	.	PUNCT
ejpam-3461	87	1	proof	proof	NOUN
ejpam-3461	87	2	.	.	PUNCT
ejpam-3461	88	1	(	(	PUNCT
ejpam-3461	88	2	1	1	X
ejpam-3461	88	3	)	)	PUNCT
ejpam-3461	88	4	⇒	⇒	NOUN
ejpam-3461	88	5	(	(	PUNCT
ejpam-3461	88	6	2	2	X
ejpam-3461	88	7	)	)	PUNCT
ejpam-3461	88	8	suppose	suppose	VERB
ejpam-3461	88	9	m	m	NOUN
ejpam-3461	88	10	is	be	AUX
ejpam-3461	88	11	a	a	DET
ejpam-3461	88	12	c	c	NOUN
ejpam-3461	88	13	-	-	PUNCT
ejpam-3461	88	14	co	co	NOUN
ejpam-3461	88	15	-	-	ADJ
ejpam-3461	88	16	epi	epi	ADJ
ejpam-3461	88	17	-	-	ADJ
ejpam-3461	88	18	retractable	retractable	ADJ
ejpam-3461	88	19	module	module	NOUN
ejpam-3461	88	20	and	and	CCONJ
ejpam-3461	88	21	k	k	X
ejpam-3461	88	22	a	a	DET
ejpam-3461	88	23	complement	complement	NOUN
ejpam-3461	88	24	submodule	submodule	NOUN
ejpam-3461	88	25	of	of	ADP
ejpam-3461	88	26	m	m	PROPN
ejpam-3461	88	27	.	.	PUNCT
ejpam-3461	89	1	then	then	ADV
ejpam-3461	89	2	,	,	PUNCT
ejpam-3461	89	3	there	there	PRON
ejpam-3461	89	4	is	be	VERB
ejpam-3461	89	5	0	0	NUM
ejpam-3461	89	6	6=	6=	NUM
ejpam-3461	89	7	g	g	PROPN
ejpam-3461	89	8	∈	∈	PROPN
ejpam-3461	89	9	endr(m	endr(m	PROPN
ejpam-3461	89	10	)	)	PUNCT
ejpam-3461	89	11	such	such	ADJ
ejpam-3461	89	12	that	that	DET
ejpam-3461	89	13	kerg	kerg	PROPN
ejpam-3461	89	14	=	=	PUNCT
ejpam-3461	89	15	k.	k.	PROPN
ejpam-3461	89	16	by	by	ADP
ejpam-3461	89	17	our	our	PRON
ejpam-3461	89	18	assumption	assumption	NOUN
ejpam-3461	89	19	,	,	PUNCT
ejpam-3461	89	20	kerg	kerg	PROPN
ejpam-3461	89	21	=	=	SYM
ejpam-3461	89	22	k	k	PROPN
ejpam-3461	89	23	is	be	AUX
ejpam-3461	89	24	a	a	DET
ejpam-3461	89	25	direct	direct	ADJ
ejpam-3461	89	26	summand	summand	NOUN
ejpam-3461	89	27	of	of	ADP
ejpam-3461	89	28	m	m	PROPN
ejpam-3461	89	29	.	.	PUNCT
ejpam-3461	90	1	therefore	therefore	ADV
ejpam-3461	90	2	,	,	PUNCT
ejpam-3461	90	3	m	m	VERB
ejpam-3461	90	4	is	be	AUX
ejpam-3461	90	5	an	an	DET
ejpam-3461	90	6	extending	extend	VERB
ejpam-3461	90	7	module	module	NOUN
ejpam-3461	90	8	.	.	PUNCT
ejpam-3461	91	1	(	(	PUNCT
ejpam-3461	91	2	2)⇒	2)⇒	NUM
ejpam-3461	91	3	(	(	PUNCT
ejpam-3461	91	4	1	1	NUM
ejpam-3461	91	5	)	)	PUNCT
ejpam-3461	91	6	is	be	AUX
ejpam-3461	91	7	obvious	obvious	ADJ
ejpam-3461	91	8	.	.	PUNCT
ejpam-3461	92	1	corollary	corollary	ADJ
ejpam-3461	92	2	2	2	NUM
ejpam-3461	92	3	.	.	PUNCT
ejpam-3461	93	1	a	a	DET
ejpam-3461	93	2	ring	ring	NOUN
ejpam-3461	93	3	r	r	NOUN
ejpam-3461	93	4	is	be	AUX
ejpam-3461	93	5	regular	regular	ADJ
ejpam-3461	93	6	c	c	NOUN
ejpam-3461	93	7	-	-	PUNCT
ejpam-3461	93	8	co	co	NOUN
ejpam-3461	93	9	-	-	NOUN
ejpam-3461	93	10	pri	pri	NOUN
ejpam-3461	93	11	if	if	SCONJ
ejpam-3461	93	12	and	and	CCONJ
ejpam-3461	93	13	only	only	ADV
ejpam-3461	93	14	if	if	SCONJ
ejpam-3461	93	15	it	it	PRON
ejpam-3461	93	16	is	be	AUX
ejpam-3461	93	17	right	right	ADV
ejpam-3461	93	18	nonsingular	nonsingular	ADJ
ejpam-3461	93	19	right	right	ADJ
ejpam-3461	93	20	continuous	continuous	ADJ
ejpam-3461	93	21	.	.	PUNCT
ejpam-3461	94	1	proposition	proposition	NOUN
ejpam-3461	94	2	3	3	NUM
ejpam-3461	94	3	.	.	PUNCT
ejpam-3461	95	1	a	a	DET
ejpam-3461	95	2	ring	ring	NOUN
ejpam-3461	95	3	r	r	NOUN
ejpam-3461	95	4	is	be	AUX
ejpam-3461	95	5	c	c	NOUN
ejpam-3461	95	6	-	-	PUNCT
ejpam-3461	95	7	co	co	NOUN
ejpam-3461	95	8	-	-	NOUN
ejpam-3461	95	9	pri	pri	NOUN
ejpam-3461	95	10	if	if	SCONJ
ejpam-3461	96	1	and	and	CCONJ
ejpam-3461	96	2	only	only	ADV
ejpam-3461	96	3	if	if	SCONJ
ejpam-3461	96	4	every	every	DET
ejpam-3461	96	5	complement	complement	NOUN
ejpam-3461	96	6	right	right	ADJ
ejpam-3461	96	7	ideal	ideal	NOUN
ejpam-3461	96	8	of	of	ADP
ejpam-3461	96	9	r	r	NOUN
ejpam-3461	96	10	is	be	AUX
ejpam-3461	96	11	the	the	DET
ejpam-3461	96	12	right	right	ADJ
ejpam-3461	96	13	annihilator	annihilator	NOUN
ejpam-3461	96	14	of	of	ADP
ejpam-3461	96	15	an	an	DET
ejpam-3461	96	16	element	element	NOUN
ejpam-3461	96	17	of	of	ADP
ejpam-3461	96	18	r.	r.	PROPN
ejpam-3461	96	19	a.	a.	PROPN
ejpam-3461	96	20	d.	d.	PROPN
ejpam-3461	96	21	diallo	diallo	PROPN
ejpam-3461	96	22	,	,	PUNCT
ejpam-3461	96	23	p.	p.	PROPN
ejpam-3461	96	24	c.	c.	PROPN
ejpam-3461	96	25	diop	diop	PROPN
ejpam-3461	96	26	,	,	PUNCT
ejpam-3461	96	27	m.	m.	NOUN
ejpam-3461	96	28	barry	barry	PROPN
ejpam-3461	96	29	/	/	SYM
ejpam-3461	96	30	eur	eur	PROPN
ejpam-3461	96	31	.	.	PUNCT
ejpam-3461	97	1	j.	j.	PROPN
ejpam-3461	97	2	pure	pure	PROPN
ejpam-3461	97	3	appl	appl	PROPN
ejpam-3461	97	4	.	.	PROPN
ejpam-3461	97	5	math	math	PROPN
ejpam-3461	97	6	,	,	PUNCT
ejpam-3461	97	7	12	12	NUM
ejpam-3461	97	8	(	(	PUNCT
ejpam-3461	97	9	3	3	NUM
ejpam-3461	97	10	)	)	PUNCT
ejpam-3461	97	11	(	(	PUNCT
ejpam-3461	97	12	2019	2019	NUM
ejpam-3461	97	13	)	)	PUNCT
ejpam-3461	97	14	,	,	PUNCT
ejpam-3461	97	15	1187	1187	NUM
ejpam-3461	97	16	-	-	SYM
ejpam-3461	97	17	1198	1198	NUM
ejpam-3461	97	18	1190	1190	NUM
ejpam-3461	97	19	proof	proof	NOUN
ejpam-3461	97	20	.	.	PUNCT
ejpam-3461	98	1	let	let	VERB
ejpam-3461	98	2	i	i	PRON
ejpam-3461	98	3	be	be	AUX
ejpam-3461	98	4	a	a	DET
ejpam-3461	98	5	right	right	ADJ
ejpam-3461	98	6	complement	complement	NOUN
ejpam-3461	98	7	ideal	ideal	NOUN
ejpam-3461	98	8	of	of	ADP
ejpam-3461	98	9	r.	r.	PROPN
ejpam-3461	98	10	if	if	SCONJ
ejpam-3461	98	11	r	r	NOUN
ejpam-3461	98	12	is	be	AUX
ejpam-3461	98	13	c	c	NOUN
ejpam-3461	98	14	-	-	PUNCT
ejpam-3461	98	15	co	co	NOUN
ejpam-3461	98	16	-	-	NOUN
ejpam-3461	98	17	pri	pri	ADJ
ejpam-3461	98	18	,	,	PUNCT
ejpam-3461	98	19	there	there	PRON
ejpam-3461	98	20	is	be	VERB
ejpam-3461	98	21	a	a	DET
ejpam-3461	98	22	monomorphism	monomorphism	NOUN
ejpam-3461	98	23	f	f	X
ejpam-3461	98	24	:	:	PUNCT
ejpam-3461	98	25	r	r	AUX
ejpam-3461	98	26	/	/	SYM
ejpam-3461	98	27	i	i	PRON
ejpam-3461	98	28	−→	−→	PROPN
ejpam-3461	98	29	r.	r.	PROPN
ejpam-3461	98	30	set	set	VERB
ejpam-3461	98	31	x	x	X
ejpam-3461	99	1	=	=	PUNCT
ejpam-3461	99	2	f(1	f(1	PROPN
ejpam-3461	100	1	+	+	NUM
ejpam-3461	100	2	i	i	NOUN
ejpam-3461	100	3	)	)	PUNCT
ejpam-3461	100	4	,	,	PUNCT
ejpam-3461	100	5	then	then	ADV
ejpam-3461	100	6	i	i	PRON
ejpam-3461	100	7	=	=	SYM
ejpam-3461	100	8	r(x	r(x	PROPN
ejpam-3461	100	9	)	)	PUNCT
ejpam-3461	100	10	,	,	PUNCT
ejpam-3461	100	11	where	where	SCONJ
ejpam-3461	100	12	r(x	r(x	NOUN
ejpam-3461	100	13	)	)	PUNCT
ejpam-3461	100	14	denotes	denote	VERB
ejpam-3461	100	15	the	the	DET
ejpam-3461	100	16	right	right	ADJ
ejpam-3461	100	17	annihilator	annihilator	NOUN
ejpam-3461	100	18	of	of	ADP
ejpam-3461	100	19	x.	x.	PROPN
ejpam-3461	100	20	on	on	ADP
ejpam-3461	100	21	the	the	DET
ejpam-3461	100	22	other	other	ADJ
ejpam-3461	100	23	hand	hand	NOUN
ejpam-3461	100	24	,	,	PUNCT
ejpam-3461	100	25	if	if	SCONJ
ejpam-3461	100	26	i	i	PRON
ejpam-3461	100	27	=	=	SYM
ejpam-3461	100	28	r(x	r(x	PROPN
ejpam-3461	100	29	)	)	PUNCT
ejpam-3461	100	30	is	be	AUX
ejpam-3461	100	31	a	a	DET
ejpam-3461	100	32	right	right	ADJ
ejpam-3461	100	33	complement	complement	NOUN
ejpam-3461	100	34	ideal	ideal	NOUN
ejpam-3461	100	35	of	of	ADP
ejpam-3461	100	36	r	r	NOUN
ejpam-3461	100	37	for	for	ADP
ejpam-3461	100	38	an	an	DET
ejpam-3461	100	39	element	element	NOUN
ejpam-3461	100	40	x	x	SYM
ejpam-3461	100	41	∈	∈	PROPN
ejpam-3461	100	42	r	r	NOUN
ejpam-3461	100	43	,	,	PUNCT
ejpam-3461	100	44	then	then	ADV
ejpam-3461	100	45	r	r	AUX
ejpam-3461	100	46	/	/	SYM
ejpam-3461	100	47	i	i	PRON
ejpam-3461	100	48	∼=	∼=	PROPN
ejpam-3461	100	49	xr	xr	PROPN
ejpam-3461	100	50	.	.	PUNCT
ejpam-3461	101	1	let	let	VERB
ejpam-3461	101	2	m	m	PRON
ejpam-3461	101	3	be	be	AUX
ejpam-3461	101	4	an	an	DET
ejpam-3461	101	5	r	r	NOUN
ejpam-3461	101	6	-	-	PUNCT
ejpam-3461	101	7	module	module	NOUN
ejpam-3461	101	8	.	.	PUNCT
ejpam-3461	102	1	the	the	DET
ejpam-3461	102	2	left	left	ADJ
ejpam-3461	102	3	annihilator	annihilator	NOUN
ejpam-3461	102	4	of	of	ADP
ejpam-3461	102	5	n	n	PRON
ejpam-3461	102	6	≤	≤	NOUN
ejpam-3461	102	7	m	m	VERB
ejpam-3461	102	8	in	in	ADP
ejpam-3461	102	9	s	s	NOUN
ejpam-3461	102	10	=	=	SYM
ejpam-3461	102	11	endr(m	endr(m	PROPN
ejpam-3461	102	12	)	)	PUNCT
ejpam-3461	102	13	is	be	AUX
ejpam-3461	102	14	denoted	denote	VERB
ejpam-3461	102	15	by	by	ADP
ejpam-3461	102	16	ls(n	ls(n	NOUN
ejpam-3461	102	17	)	)	PUNCT
ejpam-3461	102	18	=	=	SYM
ejpam-3461	102	19	{	{	PUNCT
ejpam-3461	102	20	φ	φ	PROPN
ejpam-3461	102	21	∈	∈	PROPN
ejpam-3461	102	22	s	s	PART
ejpam-3461	102	23	:	:	PUNCT
ejpam-3461	102	24	φn	φn	NOUN
ejpam-3461	102	25	=	=	PUNCT
ejpam-3461	102	26	{	{	PUNCT
ejpam-3461	102	27	0	0	NUM
ejpam-3461	102	28	}	}	PUNCT
ejpam-3461	102	29	}	}	PUNCT
ejpam-3461	102	30	and	and	CCONJ
ejpam-3461	102	31	the	the	DET
ejpam-3461	102	32	right	right	ADJ
ejpam-3461	102	33	annihilator	annihilator	NOUN
ejpam-3461	102	34	of	of	ADP
ejpam-3461	102	35	a	a	DET
ejpam-3461	102	36	left	left	ADJ
ejpam-3461	102	37	ideal	ideal	NOUN
ejpam-3461	102	38	i	i	PRON
ejpam-3461	102	39	of	of	ADP
ejpam-3461	102	40	s	s	PROPN
ejpam-3461	102	41	is	be	AUX
ejpam-3461	102	42	rm	rm	NOUN
ejpam-3461	102	43	(	(	PUNCT
ejpam-3461	102	44	i	i	NOUN
ejpam-3461	102	45	)	)	PUNCT
ejpam-3461	102	46	=	=	PRON
ejpam-3461	103	1	{	{	PUNCT
ejpam-3461	103	2	m	m	NOUN
ejpam-3461	103	3	∈m	∈m	NOUN
ejpam-3461	103	4	:	:	PUNCT
ejpam-3461	103	5	i	i	PRON
ejpam-3461	103	6	m	m	VERB
ejpam-3461	103	7	=	=	PUNCT
ejpam-3461	103	8	{	{	PUNCT
ejpam-3461	103	9	0	0	NUM
ejpam-3461	103	10	}	}	PUNCT
ejpam-3461	103	11	}	}	PUNCT
ejpam-3461	103	12	recall	recall	VERB
ejpam-3461	103	13	that	that	SCONJ
ejpam-3461	103	14	an	an	DET
ejpam-3461	103	15	r	r	NOUN
ejpam-3461	103	16	-	-	PUNCT
ejpam-3461	103	17	module	module	NOUN
ejpam-3461	103	18	is	be	AUX
ejpam-3461	103	19	called	call	VERB
ejpam-3461	103	20	baer	baer	PROPN
ejpam-3461	103	21	if	if	SCONJ
ejpam-3461	103	22	,	,	PUNCT
ejpam-3461	103	23	for	for	ADP
ejpam-3461	103	24	all	all	DET
ejpam-3461	103	25	n	n	DET
ejpam-3461	103	26	≤m	≤m	NOUN
ejpam-3461	103	27	,	,	PUNCT
ejpam-3461	103	28	ls(n	ls(n	NUM
ejpam-3461	103	29	)	)	PUNCT
ejpam-3461	103	30	=	=	PUNCT
ejpam-3461	103	31	se	se	X
ejpam-3461	103	32	,	,	PUNCT
ejpam-3461	103	33	with	with	ADP
ejpam-3461	103	34	e2	e2	PROPN
ejpam-3461	103	35	=	=	PUNCT
ejpam-3461	103	36	e	e	PROPN
ejpam-3461	103	37	∈	∈	PROPN
ejpam-3461	103	38	s.	s.	PROPN
ejpam-3461	103	39	equivalenly	equivalenly	PROPN
ejpam-3461	103	40	,	,	PUNCT
ejpam-3461	103	41	m	m	PROPN
ejpam-3461	103	42	is	be	AUX
ejpam-3461	103	43	baer	baer	PROPN
ejpam-3461	103	44	if	if	SCONJ
ejpam-3461	103	45	,	,	PUNCT
ejpam-3461	103	46	for	for	ADP
ejpam-3461	103	47	all	all	DET
ejpam-3461	103	48	ideal	ideal	NOUN
ejpam-3461	103	49	i	i	PRON
ejpam-3461	103	50	≤s	≤s	PROPN
ejpam-3461	103	51	s	s	PROPN
ejpam-3461	103	52	,	,	PUNCT
ejpam-3461	103	53	rm	rm	PROPN
ejpam-3461	103	54	(	(	PUNCT
ejpam-3461	103	55	i	i	NOUN
ejpam-3461	103	56	)	)	PUNCT
ejpam-3461	104	1	=	=	VERB
ejpam-3461	104	2	em	em	PRON
ejpam-3461	104	3	with	with	ADP
ejpam-3461	104	4	e2	e2	PROPN
ejpam-3461	104	5	=	=	PUNCT
ejpam-3461	104	6	e	e	PROPN
ejpam-3461	104	7	∈	∈	PROPN
ejpam-3461	104	8	s.	s.	PROPN
ejpam-3461	104	9	an	an	DET
ejpam-3461	104	10	r	r	NOUN
ejpam-3461	104	11	-	-	PUNCT
ejpam-3461	104	12	module	module	NOUN
ejpam-3461	104	13	m	m	NOUN
ejpam-3461	104	14	is	be	AUX
ejpam-3461	104	15	called	call	VERB
ejpam-3461	104	16	rickart	rickart	NOUN
ejpam-3461	104	17	if	if	SCONJ
ejpam-3461	104	18	any	any	DET
ejpam-3461	104	19	endomorphism	endomorphism	NOUN
ejpam-3461	104	20	of	of	ADP
ejpam-3461	104	21	m	m	PROPN
ejpam-3461	104	22	has	have	VERB
ejpam-3461	104	23	a	a	DET
ejpam-3461	104	24	direct	direct	ADJ
ejpam-3461	104	25	summand	summand	NOUN
ejpam-3461	104	26	kernel	kernel	NOUN
ejpam-3461	104	27	.	.	PUNCT
ejpam-3461	105	1	a	a	DET
ejpam-3461	105	2	module	module	NOUN
ejpam-3461	105	3	m	m	VERB
ejpam-3461	105	4	is	be	AUX
ejpam-3461	105	5	called	call	VERB
ejpam-3461	105	6	k	k	ADJ
ejpam-3461	105	7	-	-	PUNCT
ejpam-3461	105	8	nonsingular	nonsingular	ADJ
ejpam-3461	105	9	if	if	SCONJ
ejpam-3461	105	10	,	,	PUNCT
ejpam-3461	105	11	∀ϕ	∀ϕ	PROPN
ejpam-3461	105	12	∈	∈	PROPN
ejpam-3461	105	13	end(m	end(m	PROPN
ejpam-3461	105	14	)	)	PUNCT
ejpam-3461	105	15	,	,	PUNCT
ejpam-3461	105	16	kerϕ	kerϕ	PROPN
ejpam-3461	105	17	≤e	≤e	PROPN
ejpam-3461	105	18	m	m	VERB
ejpam-3461	105	19	implies	imply	VERB
ejpam-3461	105	20	ϕ	ϕ	PROPN
ejpam-3461	105	21	=	=	SYM
ejpam-3461	105	22	0	0	X
ejpam-3461	105	23	.	.	PUNCT
ejpam-3461	106	1	lemma	lemma	PROPN
ejpam-3461	106	2	2	2	NUM
ejpam-3461	106	3	.	.	PUNCT
ejpam-3461	107	1	(	(	PUNCT
ejpam-3461	107	2	[	[	X
ejpam-3461	107	3	16	16	NUM
ejpam-3461	107	4	]	]	PUNCT
ejpam-3461	107	5	,	,	PUNCT
ejpam-3461	107	6	lemma	lemma	PROPN
ejpam-3461	107	7	2.14	2.14	NUM
ejpam-3461	107	8	)	)	PUNCT
ejpam-3461	107	9	any	any	DET
ejpam-3461	107	10	k	k	ADJ
ejpam-3461	107	11	-	-	PUNCT
ejpam-3461	107	12	nonsingular	nonsingular	ADJ
ejpam-3461	107	13	extending	extend	VERB
ejpam-3461	107	14	module	module	NOUN
ejpam-3461	107	15	is	be	AUX
ejpam-3461	107	16	baer	baer	PROPN
ejpam-3461	107	17	.	.	PUNCT
ejpam-3461	108	1	in	in	ADP
ejpam-3461	108	2	the	the	DET
ejpam-3461	108	3	two	two	NUM
ejpam-3461	108	4	following	follow	VERB
ejpam-3461	108	5	results	result	NOUN
ejpam-3461	108	6	,	,	PUNCT
ejpam-3461	108	7	we	we	PRON
ejpam-3461	108	8	show	show	VERB
ejpam-3461	108	9	that	that	SCONJ
ejpam-3461	108	10	for	for	ADP
ejpam-3461	108	11	a	a	DET
ejpam-3461	108	12	c	c	NOUN
ejpam-3461	108	13	-	-	PUNCT
ejpam-3461	108	14	co	co	NOUN
ejpam-3461	108	15	-	-	ADJ
ejpam-3461	108	16	epi	epi	ADJ
ejpam-3461	108	17	-	-	ADJ
ejpam-3461	108	18	retractable	retractable	ADJ
ejpam-3461	108	19	r	r	NOUN
ejpam-3461	108	20	-	-	PUNCT
ejpam-3461	108	21	module	module	NOUN
ejpam-3461	108	22	or	or	CCONJ
ejpam-3461	108	23	a	a	DET
ejpam-3461	108	24	module	module	NOUN
ejpam-3461	108	25	with	with	ADP
ejpam-3461	108	26	c	c	NOUN
ejpam-3461	108	27	-	-	PUNCT
ejpam-3461	108	28	co	co	NOUN
ejpam-3461	108	29	-	-	ADJ
ejpam-3461	108	30	pri	pri	ADJ
ejpam-3461	108	31	endomorphism	endomorphism	NOUN
ejpam-3461	108	32	ring	ring	NOUN
ejpam-3461	108	33	the	the	DET
ejpam-3461	108	34	properties	property	NOUN
ejpam-3461	108	35	rickart	rickart	NOUN
ejpam-3461	108	36	and	and	CCONJ
ejpam-3461	108	37	baer	baer	PROPN
ejpam-3461	108	38	are	be	AUX
ejpam-3461	108	39	equivalent	equivalent	ADJ
ejpam-3461	108	40	.	.	PUNCT
ejpam-3461	109	1	proposition	proposition	NOUN
ejpam-3461	109	2	4	4	NUM
ejpam-3461	109	3	.	.	PUNCT
ejpam-3461	110	1	let	let	VERB
ejpam-3461	110	2	m	m	PRON
ejpam-3461	110	3	be	be	AUX
ejpam-3461	110	4	a	a	DET
ejpam-3461	110	5	c	c	NOUN
ejpam-3461	110	6	-	-	PUNCT
ejpam-3461	110	7	co	co	NOUN
ejpam-3461	110	8	-	-	ADJ
ejpam-3461	110	9	epi	epi	ADJ
ejpam-3461	110	10	-	-	ADJ
ejpam-3461	110	11	retractable	retractable	ADJ
ejpam-3461	110	12	r	r	NOUN
ejpam-3461	110	13	-	-	PUNCT
ejpam-3461	110	14	module	module	NOUN
ejpam-3461	110	15	.	.	PUNCT
ejpam-3461	111	1	then	then	ADV
ejpam-3461	111	2	m	m	PROPN
ejpam-3461	111	3	is	be	AUX
ejpam-3461	111	4	rickart	rickart	NOUN
ejpam-3461	111	5	if	if	SCONJ
ejpam-3461	112	1	and	and	CCONJ
ejpam-3461	112	2	only	only	ADV
ejpam-3461	112	3	if	if	SCONJ
ejpam-3461	112	4	m	m	NOUN
ejpam-3461	112	5	is	be	AUX
ejpam-3461	112	6	baer	baer	PROPN
ejpam-3461	112	7	.	.	PUNCT
ejpam-3461	113	1	proof	proof	NOUN
ejpam-3461	113	2	.	.	PUNCT
ejpam-3461	114	1	suppose	suppose	VERB
ejpam-3461	114	2	m	m	NOUN
ejpam-3461	114	3	is	be	AUX
ejpam-3461	114	4	rickart	rickart	NOUN
ejpam-3461	114	5	.	.	PUNCT
ejpam-3461	115	1	since	since	ADV
ejpam-3461	115	2	,	,	PUNCT
ejpam-3461	115	3	m	m	VERB
ejpam-3461	115	4	is	be	AUX
ejpam-3461	115	5	c	c	NOUN
ejpam-3461	115	6	-	-	PUNCT
ejpam-3461	115	7	co	co	ADJ
ejpam-3461	115	8	-	-	ADJ
ejpam-3461	115	9	epi	epi	NOUN
ejpam-3461	115	10	-	-	NOUN
ejpam-3461	115	11	retractable	retractable	ADJ
ejpam-3461	115	12	,	,	PUNCT
ejpam-3461	115	13	it	it	PRON
ejpam-3461	115	14	is	be	AUX
ejpam-3461	115	15	also	also	ADV
ejpam-3461	115	16	extending	extend	VERB
ejpam-3461	115	17	by	by	ADP
ejpam-3461	115	18	proposition	proposition	NOUN
ejpam-3461	115	19	2	2	NUM
ejpam-3461	115	20	.	.	PUNCT
ejpam-3461	116	1	now	now	ADV
ejpam-3461	116	2	,	,	PUNCT
ejpam-3461	116	3	suppose	suppose	VERB
ejpam-3461	116	4	kerf	kerf	NOUN
ejpam-3461	116	5	≤e	≤e	VERB
ejpam-3461	116	6	m	m	NOUN
ejpam-3461	116	7	for	for	ADP
ejpam-3461	116	8	some	some	DET
ejpam-3461	116	9	f	f	PROPN
ejpam-3461	116	10	∈	∈	PROPN
ejpam-3461	116	11	endr(m	endr(m	PROPN
ejpam-3461	116	12	)	)	PUNCT
ejpam-3461	116	13	.	.	PUNCT
ejpam-3461	117	1	the	the	DET
ejpam-3461	117	2	property	property	NOUN
ejpam-3461	117	3	of	of	ADP
ejpam-3461	117	4	rickart	rickart	NOUN
ejpam-3461	117	5	implies	imply	VERB
ejpam-3461	117	6	that	that	SCONJ
ejpam-3461	117	7	kerf	kerf	NOUN
ejpam-3461	117	8	≤⊕	≤⊕	ADV
ejpam-3461	117	9	m	m	VERB
ejpam-3461	117	10	,	,	PUNCT
ejpam-3461	117	11	and	and	CCONJ
ejpam-3461	117	12	so	so	ADV
ejpam-3461	117	13	kerf	kerf	NOUN
ejpam-3461	117	14	=	=	NOUN
ejpam-3461	117	15	m	m	VERB
ejpam-3461	117	16	.	.	PUNCT
ejpam-3461	118	1	consequently	consequently	ADV
ejpam-3461	118	2	,	,	PUNCT
ejpam-3461	118	3	f	f	PROPN
ejpam-3461	118	4	=	=	SYM
ejpam-3461	118	5	0	0	PROPN
ejpam-3461	118	6	.	.	PUNCT
ejpam-3461	119	1	therefore	therefore	ADV
ejpam-3461	119	2	,	,	PUNCT
ejpam-3461	119	3	according	accord	VERB
ejpam-3461	119	4	to	to	ADP
ejpam-3461	119	5	lemma	lemma	PROPN
ejpam-3461	119	6	2	2	NUM
ejpam-3461	119	7	,	,	PUNCT
ejpam-3461	119	8	m	m	VERB
ejpam-3461	119	9	is	be	AUX
ejpam-3461	119	10	baer	baer	PROPN
ejpam-3461	119	11	.	.	PUNCT
ejpam-3461	120	1	the	the	DET
ejpam-3461	120	2	converse	converse	PROPN
ejpam-3461	120	3	implication	implication	NOUN
ejpam-3461	120	4	is	be	AUX
ejpam-3461	120	5	clear	clear	ADJ
ejpam-3461	120	6	.	.	PUNCT
ejpam-3461	121	1	corollary	corollary	ADJ
ejpam-3461	121	2	3	3	X
ejpam-3461	121	3	.	.	PUNCT
ejpam-3461	122	1	let	let	VERB
ejpam-3461	122	2	r	r	PRON
ejpam-3461	122	3	be	be	AUX
ejpam-3461	122	4	a	a	DET
ejpam-3461	122	5	c	c	NOUN
ejpam-3461	122	6	-	-	PUNCT
ejpam-3461	122	7	co	co	ADJ
ejpam-3461	122	8	-	-	ADJ
ejpam-3461	122	9	pri	pri	ADJ
ejpam-3461	122	10	ring	ring	NOUN
ejpam-3461	122	11	.	.	PUNCT
ejpam-3461	123	1	then	then	ADV
ejpam-3461	123	2	r	r	NOUN
ejpam-3461	123	3	is	be	AUX
ejpam-3461	123	4	baer	baer	PROPN
ejpam-3461	123	5	if	if	SCONJ
ejpam-3461	123	6	and	and	CCONJ
ejpam-3461	123	7	only	only	ADV
ejpam-3461	123	8	if	if	SCONJ
ejpam-3461	123	9	r	r	NOUN
ejpam-3461	123	10	is	be	AUX
ejpam-3461	123	11	right	right	ADJ
ejpam-3461	123	12	rickart	rickart	NOUN
ejpam-3461	123	13	.	.	PUNCT
ejpam-3461	124	1	proposition	proposition	NOUN
ejpam-3461	124	2	5	5	NUM
ejpam-3461	124	3	.	.	PUNCT
ejpam-3461	125	1	let	let	VERB
ejpam-3461	125	2	m	m	PRON
ejpam-3461	125	3	be	be	AUX
ejpam-3461	125	4	an	an	DET
ejpam-3461	125	5	r	r	NOUN
ejpam-3461	125	6	-	-	PUNCT
ejpam-3461	125	7	module	module	NOUN
ejpam-3461	125	8	for	for	ADP
ejpam-3461	125	9	which	which	PRON
ejpam-3461	125	10	s	s	VERB
ejpam-3461	125	11	is	be	AUX
ejpam-3461	125	12	c	c	NOUN
ejpam-3461	125	13	-	-	PUNCT
ejpam-3461	125	14	co	co	NOUN
ejpam-3461	125	15	-	-	NOUN
ejpam-3461	125	16	pri	pri	NOUN
ejpam-3461	125	17	.	.	PUNCT
ejpam-3461	126	1	then	then	ADV
ejpam-3461	126	2	the	the	DET
ejpam-3461	126	3	following	follow	VERB
ejpam-3461	126	4	statements	statement	NOUN
ejpam-3461	126	5	are	be	AUX
ejpam-3461	126	6	equivalent	equivalent	ADJ
ejpam-3461	126	7	:	:	PUNCT
ejpam-3461	126	8	(	(	PUNCT
ejpam-3461	126	9	1	1	X
ejpam-3461	126	10	)	)	PUNCT
ejpam-3461	126	11	m	m	VERB
ejpam-3461	126	12	is	be	AUX
ejpam-3461	126	13	baer	baer	PROPN
ejpam-3461	126	14	.	.	PUNCT
ejpam-3461	127	1	(	(	PUNCT
ejpam-3461	127	2	2	2	X
ejpam-3461	127	3	)	)	PUNCT
ejpam-3461	127	4	m	m	VERB
ejpam-3461	127	5	is	be	AUX
ejpam-3461	127	6	rickart	rickart	NOUN
ejpam-3461	127	7	.	.	PUNCT
ejpam-3461	128	1	(	(	PUNCT
ejpam-3461	128	2	3	3	X
ejpam-3461	128	3	)	)	PUNCT
ejpam-3461	128	4	s	s	VERB
ejpam-3461	128	5	is	be	AUX
ejpam-3461	128	6	right	right	ADJ
ejpam-3461	128	7	rickart	rickart	NOUN
ejpam-3461	128	8	.	.	PUNCT
ejpam-3461	129	1	proof	proof	NOUN
ejpam-3461	129	2	.	.	PUNCT
ejpam-3461	130	1	(	(	PUNCT
ejpam-3461	130	2	1)⇒	1)⇒	NUM
ejpam-3461	130	3	(	(	PUNCT
ejpam-3461	130	4	2	2	NUM
ejpam-3461	130	5	)	)	PUNCT
ejpam-3461	130	6	this	this	PRON
ejpam-3461	130	7	is	be	AUX
ejpam-3461	130	8	clear	clear	ADJ
ejpam-3461	130	9	.	.	PUNCT
ejpam-3461	131	1	(	(	PUNCT
ejpam-3461	131	2	2)⇒	2)⇒	NUM
ejpam-3461	131	3	(	(	PUNCT
ejpam-3461	131	4	3	3	NUM
ejpam-3461	131	5	)	)	PUNCT
ejpam-3461	131	6	follows	follow	VERB
ejpam-3461	131	7	from	from	ADP
ejpam-3461	131	8	proposition	proposition	NOUN
ejpam-3461	131	9	2.2.1	2.2.1	NUM
ejpam-3461	131	10	in	in	ADP
ejpam-3461	131	11	[	[	X
ejpam-3461	131	12	13	13	NUM
ejpam-3461	131	13	]	]	PUNCT
ejpam-3461	131	14	.	.	PUNCT
ejpam-3461	132	1	(	(	PUNCT
ejpam-3461	132	2	3)⇒	3)⇒	NUM
ejpam-3461	132	3	(	(	PUNCT
ejpam-3461	132	4	1	1	NUM
ejpam-3461	132	5	)	)	PUNCT
ejpam-3461	132	6	let	let	VERB
ejpam-3461	132	7	n	n	PRON
ejpam-3461	132	8	be	be	AUX
ejpam-3461	132	9	a	a	DET
ejpam-3461	132	10	submodule	submodule	NOUN
ejpam-3461	132	11	of	of	ADP
ejpam-3461	132	12	m	m	PROPN
ejpam-3461	132	13	.	.	PUNCT
ejpam-3461	133	1	since	since	SCONJ
ejpam-3461	133	2	s	s	NOUN
ejpam-3461	133	3	is	be	AUX
ejpam-3461	133	4	right	right	ADJ
ejpam-3461	133	5	rickart	rickart	NOUN
ejpam-3461	133	6	,	,	PUNCT
ejpam-3461	133	7	it	it	PRON
ejpam-3461	133	8	is	be	AUX
ejpam-3461	133	9	also	also	ADV
ejpam-3461	133	10	right	right	ADV
ejpam-3461	133	11	nonsingular	nonsingular	ADJ
ejpam-3461	133	12	.	.	PUNCT
ejpam-3461	134	1	thus	thus	ADV
ejpam-3461	134	2	,	,	PUNCT
ejpam-3461	134	3	ls(n	ls(n	NUM
ejpam-3461	134	4	)	)	PUNCT
ejpam-3461	134	5	is	be	AUX
ejpam-3461	134	6	a	a	DET
ejpam-3461	134	7	complement	complement	NOUN
ejpam-3461	134	8	right	right	ADJ
ejpam-3461	134	9	ideal	ideal	NOUN
ejpam-3461	134	10	in	in	ADP
ejpam-3461	134	11	s.	s.	PROPN
ejpam-3461	134	12	because	because	SCONJ
ejpam-3461	134	13	s	s	PROPN
ejpam-3461	134	14	is	be	AUX
ejpam-3461	134	15	c	c	NOUN
ejpam-3461	134	16	-	-	PUNCT
ejpam-3461	134	17	co	co	NOUN
ejpam-3461	134	18	-	-	NOUN
ejpam-3461	134	19	pri	pri	NOUN
ejpam-3461	134	20	,	,	PUNCT
ejpam-3461	134	21	it	it	PRON
ejpam-3461	134	22	follows	follow	VERB
ejpam-3461	134	23	from	from	ADP
ejpam-3461	134	24	proposition	proposition	NOUN
ejpam-3461	134	25	2	2	NUM
ejpam-3461	134	26	that	that	PRON
ejpam-3461	134	27	s	s	VERB
ejpam-3461	134	28	is	be	AUX
ejpam-3461	134	29	right	right	ADJ
ejpam-3461	134	30	extending	extend	VERB
ejpam-3461	134	31	.	.	PUNCT
ejpam-3461	135	1	therefore	therefore	ADV
ejpam-3461	135	2	,	,	PUNCT
ejpam-3461	135	3	ls(n	ls(n	NOUN
ejpam-3461	135	4	)	)	PUNCT
ejpam-3461	135	5	=	=	SYM
ejpam-3461	135	6	s(1−e	s(1−e	NOUN
ejpam-3461	135	7	)	)	PUNCT
ejpam-3461	135	8	for	for	ADP
ejpam-3461	135	9	some	some	DET
ejpam-3461	135	10	e	e	NOUN
ejpam-3461	135	11	=	=	PROPN
ejpam-3461	135	12	e2	e2	PROPN
ejpam-3461	135	13	∈	∈	PROPN
ejpam-3461	135	14	s	s	PROPN
ejpam-3461	135	15	,	,	PUNCT
ejpam-3461	135	16	a.	a.	PROPN
ejpam-3461	135	17	d.	d.	PROPN
ejpam-3461	135	18	diallo	diallo	PROPN
ejpam-3461	135	19	,	,	PUNCT
ejpam-3461	135	20	p.	p.	PROPN
ejpam-3461	135	21	c.	c.	PROPN
ejpam-3461	135	22	diop	diop	PROPN
ejpam-3461	135	23	,	,	PUNCT
ejpam-3461	135	24	m.	m.	NOUN
ejpam-3461	135	25	barry	barry	PROPN
ejpam-3461	135	26	/	/	SYM
ejpam-3461	135	27	eur	eur	PROPN
ejpam-3461	135	28	.	.	PUNCT
ejpam-3461	136	1	j.	j.	PROPN
ejpam-3461	136	2	pure	pure	PROPN
ejpam-3461	136	3	appl	appl	PROPN
ejpam-3461	136	4	.	.	PROPN
ejpam-3461	136	5	math	math	PROPN
ejpam-3461	136	6	,	,	PUNCT
ejpam-3461	136	7	12	12	NUM
ejpam-3461	136	8	(	(	PUNCT
ejpam-3461	136	9	3	3	NUM
ejpam-3461	136	10	)	)	PUNCT
ejpam-3461	136	11	(	(	PUNCT
ejpam-3461	136	12	2019	2019	NUM
ejpam-3461	136	13	)	)	PUNCT
ejpam-3461	136	14	,	,	PUNCT
ejpam-3461	136	15	1187	1187	NUM
ejpam-3461	136	16	-	-	SYM
ejpam-3461	136	17	1198	1198	NUM
ejpam-3461	136	18	1191	1191	NUM
ejpam-3461	136	19	and	and	CCONJ
ejpam-3461	136	20	hence	hence	ADV
ejpam-3461	136	21	m	m	VERB
ejpam-3461	136	22	is	be	AUX
ejpam-3461	136	23	baer	baer	PROPN
ejpam-3461	136	24	.	.	PUNCT
ejpam-3461	137	1	recall	recall	VERB
ejpam-3461	137	2	that	that	SCONJ
ejpam-3461	137	3	an	an	DET
ejpam-3461	137	4	r	r	NOUN
ejpam-3461	137	5	-	-	PUNCT
ejpam-3461	137	6	module	module	NOUN
ejpam-3461	137	7	n	n	NOUN
ejpam-3461	137	8	is	be	AUX
ejpam-3461	137	9	said	say	VERB
ejpam-3461	137	10	to	to	PART
ejpam-3461	137	11	subgenerated	subgenerate	VERB
ejpam-3461	137	12	by	by	ADP
ejpam-3461	137	13	m	m	PROPN
ejpam-3461	137	14	if	if	SCONJ
ejpam-3461	137	15	n	n	NOUN
ejpam-3461	137	16	is	be	AUX
ejpam-3461	137	17	isomorphic	isomorphic	ADJ
ejpam-3461	137	18	to	to	ADP
ejpam-3461	137	19	a	a	DET
ejpam-3461	137	20	submodule	submodule	NOUN
ejpam-3461	137	21	of	of	ADP
ejpam-3461	137	22	an	an	DET
ejpam-3461	137	23	m	m	ADV
ejpam-3461	137	24	-generated	-generate	VERB
ejpam-3461	137	25	module	module	NOUN
ejpam-3461	137	26	,	,	PUNCT
ejpam-3461	137	27	i.e	i.e	PROPN
ejpam-3461	137	28	n	n	PRON
ejpam-3461	137	29	is	be	AUX
ejpam-3461	137	30	a	a	DET
ejpam-3461	137	31	kernel	kernel	NOUN
ejpam-3461	137	32	of	of	ADP
ejpam-3461	137	33	a	a	DET
ejpam-3461	137	34	morphism	morphism	NOUN
ejpam-3461	137	35	between	between	ADP
ejpam-3461	137	36	m	m	NOUN
ejpam-3461	137	37	generated	generate	VERB
ejpam-3461	137	38	modules	module	NOUN
ejpam-3461	137	39	.	.	PUNCT
ejpam-3461	138	1	we	we	PRON
ejpam-3461	138	2	denote	denote	VERB
ejpam-3461	138	3	by	by	ADP
ejpam-3461	138	4	σ[m	σ[m	ADJ
ejpam-3461	138	5	]	]	PUNCT
ejpam-3461	138	6	,	,	PUNCT
ejpam-3461	138	7	the	the	DET
ejpam-3461	138	8	full	full	ADJ
ejpam-3461	138	9	subcategory	subcategory	NOUN
ejpam-3461	138	10	of	of	ADP
ejpam-3461	138	11	mod	mod	PROPN
ejpam-3461	138	12	-	-	PUNCT
ejpam-3461	138	13	r	r	NOUN
ejpam-3461	138	14	whose	whose	DET
ejpam-3461	138	15	objects	object	NOUN
ejpam-3461	138	16	are	be	AUX
ejpam-3461	138	17	all	all	PRON
ejpam-3461	138	18	r	r	NOUN
ejpam-3461	138	19	-	-	PUNCT
ejpam-3461	138	20	modules	module	NOUN
ejpam-3461	138	21	subgenerated	subgenerate	VERB
ejpam-3461	138	22	by	by	ADP
ejpam-3461	138	23	m	m	PROPN
ejpam-3461	138	24	.	.	PUNCT
ejpam-3461	139	1	recall	recall	VERB
ejpam-3461	139	2	that	that	SCONJ
ejpam-3461	139	3	an	an	DET
ejpam-3461	139	4	r	r	NOUN
ejpam-3461	139	5	-	-	PUNCT
ejpam-3461	139	6	module	module	NOUN
ejpam-3461	139	7	m	m	NOUN
ejpam-3461	139	8	is	be	AUX
ejpam-3461	139	9	self	self	NOUN
ejpam-3461	139	10	-	-	PUNCT
ejpam-3461	139	11	hereditary	hereditary	ADJ
ejpam-3461	139	12	if	if	SCONJ
ejpam-3461	139	13	every	every	DET
ejpam-3461	139	14	submodule	submodule	NOUN
ejpam-3461	139	15	of	of	ADP
ejpam-3461	139	16	m	m	PROPN
ejpam-3461	139	17	is	be	AUX
ejpam-3461	139	18	projective	projective	ADJ
ejpam-3461	139	19	in	in	ADP
ejpam-3461	139	20	σ[m	σ[m	ADJ
ejpam-3461	139	21	]	]	PUNCT
ejpam-3461	139	22	.	.	PUNCT
ejpam-3461	140	1	theorem	theorem	NOUN
ejpam-3461	140	2	1	1	X
ejpam-3461	140	3	.	.	PUNCT
ejpam-3461	141	1	let	let	VERB
ejpam-3461	141	2	r	r	PRON
ejpam-3461	141	3	be	be	AUX
ejpam-3461	141	4	a	a	DET
ejpam-3461	141	5	right	right	ADJ
ejpam-3461	141	6	self	self	NOUN
ejpam-3461	141	7	-	-	PUNCT
ejpam-3461	141	8	injective	injective	ADJ
ejpam-3461	141	9	ring	ring	NOUN
ejpam-3461	141	10	and	and	CCONJ
ejpam-3461	141	11	m	m	AUX
ejpam-3461	141	12	be	be	AUX
ejpam-3461	141	13	a	a	DET
ejpam-3461	141	14	self	self	NOUN
ejpam-3461	141	15	-	-	PUNCT
ejpam-3461	141	16	hereditary	hereditary	ADJ
ejpam-3461	141	17	r	r	NOUN
ejpam-3461	141	18	-	-	PUNCT
ejpam-3461	141	19	module	module	NOUN
ejpam-3461	141	20	.	.	PUNCT
ejpam-3461	142	1	then	then	ADV
ejpam-3461	142	2	the	the	DET
ejpam-3461	142	3	following	follow	VERB
ejpam-3461	142	4	conditions	condition	NOUN
ejpam-3461	142	5	are	be	AUX
ejpam-3461	142	6	equivalent	equivalent	ADJ
ejpam-3461	142	7	:	:	PUNCT
ejpam-3461	142	8	(	(	PUNCT
ejpam-3461	142	9	1	1	X
ejpam-3461	142	10	)	)	PUNCT
ejpam-3461	142	11	m	m	VERB
ejpam-3461	142	12	is	be	AUX
ejpam-3461	142	13	c	c	NOUN
ejpam-3461	142	14	-	-	PUNCT
ejpam-3461	142	15	co	co	ADJ
ejpam-3461	142	16	-	-	ADJ
ejpam-3461	142	17	epi	epi	NOUN
ejpam-3461	142	18	-	-	NOUN
ejpam-3461	142	19	retractable	retractable	ADJ
ejpam-3461	142	20	.	.	PUNCT
ejpam-3461	143	1	(	(	PUNCT
ejpam-3461	143	2	2	2	X
ejpam-3461	143	3	)	)	PUNCT
ejpam-3461	143	4	m	m	VERB
ejpam-3461	143	5	is	be	AUX
ejpam-3461	143	6	extending	extend	VERB
ejpam-3461	143	7	.	.	PUNCT
ejpam-3461	144	1	(	(	PUNCT
ejpam-3461	144	2	3	3	X
ejpam-3461	144	3	)	)	PUNCT
ejpam-3461	144	4	m	m	VERB
ejpam-3461	144	5	is	be	AUX
ejpam-3461	144	6	continuous	continuous	ADJ
ejpam-3461	144	7	.	.	PUNCT
ejpam-3461	145	1	(	(	PUNCT
ejpam-3461	145	2	4	4	X
ejpam-3461	145	3	)	)	PUNCT
ejpam-3461	145	4	m	m	VERB
ejpam-3461	145	5	is	be	AUX
ejpam-3461	145	6	finitely	finitely	ADV
ejpam-3461	145	7	generated	generate	VERB
ejpam-3461	145	8	semi	semi	ADJ
ejpam-3461	145	9	-	-	ADJ
ejpam-3461	145	10	simple	simple	ADJ
ejpam-3461	145	11	injective	injective	NOUN
ejpam-3461	145	12	.	.	PUNCT
ejpam-3461	146	1	proof	proof	NOUN
ejpam-3461	146	2	.	.	PUNCT
ejpam-3461	147	1	(	(	PUNCT
ejpam-3461	147	2	1	1	X
ejpam-3461	147	3	)	)	PUNCT
ejpam-3461	147	4	⇒	⇒	NOUN
ejpam-3461	147	5	(	(	PUNCT
ejpam-3461	147	6	2	2	X
ejpam-3461	147	7	)	)	PUNCT
ejpam-3461	147	8	suppose	suppose	VERB
ejpam-3461	147	9	m	m	NOUN
ejpam-3461	147	10	is	be	AUX
ejpam-3461	147	11	c	c	NOUN
ejpam-3461	147	12	-	-	PUNCT
ejpam-3461	147	13	co	co	ADJ
ejpam-3461	147	14	-	-	ADJ
ejpam-3461	147	15	epi	epi	NOUN
ejpam-3461	147	16	-	-	NOUN
ejpam-3461	147	17	retractable	retractable	ADJ
ejpam-3461	147	18	.	.	PUNCT
ejpam-3461	148	1	thus	thus	ADV
ejpam-3461	148	2	for	for	ADP
ejpam-3461	148	3	any	any	DET
ejpam-3461	148	4	complement	complement	NOUN
ejpam-3461	148	5	submodule	submodule	NOUN
ejpam-3461	148	6	c	c	PROPN
ejpam-3461	148	7	of	of	ADP
ejpam-3461	148	8	m	m	PROPN
ejpam-3461	148	9	,	,	PUNCT
ejpam-3461	148	10	there	there	PRON
ejpam-3461	148	11	exists	exist	VERB
ejpam-3461	148	12	a	a	DET
ejpam-3461	148	13	submodule	submodule	NOUN
ejpam-3461	148	14	n	n	PROPN
ejpam-3461	148	15	of	of	ADP
ejpam-3461	148	16	m	m	PRON
ejpam-3461	148	17	such	such	ADJ
ejpam-3461	148	18	that	that	SCONJ
ejpam-3461	148	19	m	m	NOUN
ejpam-3461	148	20	/	/	SYM
ejpam-3461	148	21	c	c	NOUN
ejpam-3461	148	22	∼=	∼=	PROPN
ejpam-3461	148	23	n	n	NOUN
ejpam-3461	148	24	.	.	PUNCT
ejpam-3461	149	1	consequently	consequently	ADV
ejpam-3461	149	2	,	,	PUNCT
ejpam-3461	149	3	the	the	DET
ejpam-3461	149	4	property	property	NOUN
ejpam-3461	149	5	of	of	ADP
ejpam-3461	149	6	self	self	NOUN
ejpam-3461	149	7	-	-	PUNCT
ejpam-3461	149	8	hereditary	hereditary	ADJ
ejpam-3461	149	9	implies	imply	VERB
ejpam-3461	149	10	that	that	SCONJ
ejpam-3461	149	11	c	c	PROPN
ejpam-3461	149	12	is	be	AUX
ejpam-3461	149	13	a	a	DET
ejpam-3461	149	14	direct	direct	ADJ
ejpam-3461	149	15	summand	summand	NOUN
ejpam-3461	149	16	of	of	ADP
ejpam-3461	149	17	m	m	PROPN
ejpam-3461	149	18	.	.	PUNCT
ejpam-3461	150	1	hence	hence	ADV
ejpam-3461	150	2	m	m	PROPN
ejpam-3461	150	3	is	be	AUX
ejpam-3461	150	4	extending	extend	VERB
ejpam-3461	150	5	.	.	PUNCT
ejpam-3461	151	1	(	(	PUNCT
ejpam-3461	151	2	2)⇒	2)⇒	NUM
ejpam-3461	151	3	(	(	PUNCT
ejpam-3461	151	4	3	3	X
ejpam-3461	151	5	)	)	PUNCT
ejpam-3461	151	6	suppose	suppose	VERB
ejpam-3461	151	7	m	m	NOUN
ejpam-3461	151	8	is	be	AUX
ejpam-3461	151	9	extending	extend	VERB
ejpam-3461	151	10	.	.	PUNCT
ejpam-3461	152	1	thus	thus	ADV
ejpam-3461	152	2	,	,	PUNCT
ejpam-3461	152	3	according	accord	VERB
ejpam-3461	152	4	to	to	ADP
ejpam-3461	152	5	theorem	theorem	NOUN
ejpam-3461	152	6	10.5	10.5	NUM
ejpam-3461	152	7	in	in	ADP
ejpam-3461	152	8	[	[	X
ejpam-3461	152	9	7	7	NUM
ejpam-3461	152	10	]	]	PUNCT
ejpam-3461	152	11	,	,	PUNCT
ejpam-3461	152	12	m	m	VERB
ejpam-3461	152	13	is	be	AUX
ejpam-3461	152	14	nonsingular	nonsingular	ADJ
ejpam-3461	152	15	and	and	CCONJ
ejpam-3461	152	16	has	have	VERB
ejpam-3461	152	17	finite	finite	ADJ
ejpam-3461	152	18	uniform	uniform	ADJ
ejpam-3461	152	19	dimension	dimension	NOUN
ejpam-3461	152	20	.	.	PUNCT
ejpam-3461	153	1	hence	hence	ADV
ejpam-3461	153	2	,	,	PUNCT
ejpam-3461	153	3	there	there	PRON
ejpam-3461	153	4	exists	exist	VERB
ejpam-3461	153	5	uniform	uniform	ADJ
ejpam-3461	153	6	independent	independent	ADJ
ejpam-3461	153	7	submodules	submodule	NOUN
ejpam-3461	153	8	u1	u1	NOUN
ejpam-3461	153	9	,	,	PUNCT
ejpam-3461	153	10	....	....	PUNCT
ejpam-3461	153	11	,	,	PUNCT
ejpam-3461	153	12	un	un	PROPN
ejpam-3461	153	13	of	of	ADP
ejpam-3461	153	14	m	m	PROPN
ejpam-3461	153	15	such	such	ADJ
ejpam-3461	153	16	that	that	DET
ejpam-3461	153	17	v	v	NOUN
ejpam-3461	153	18	=	=	SYM
ejpam-3461	153	19	u1	u1	NOUN
ejpam-3461	153	20	⊕	⊕	PROPN
ejpam-3461	153	21	u2	u2	PROPN
ejpam-3461	153	22	⊕	⊕	PROPN
ejpam-3461	153	23	...	...	PUNCT
ejpam-3461	154	1	⊕	⊕	PROPN
ejpam-3461	154	2	un	un	PROPN
ejpam-3461	154	3	is	be	AUX
ejpam-3461	154	4	an	an	DET
ejpam-3461	154	5	essential	essential	ADJ
ejpam-3461	154	6	submodule	submodule	NOUN
ejpam-3461	154	7	of	of	ADP
ejpam-3461	154	8	m	m	PROPN
ejpam-3461	154	9	.	.	PUNCT
ejpam-3461	155	1	set	set	VERB
ejpam-3461	155	2	0	0	NUM
ejpam-3461	155	3	6=	6=	NUM
ejpam-3461	155	4	ui	ui	PROPN
ejpam-3461	155	5	∈	∈	PROPN
ejpam-3461	155	6	ui	ui	PROPN
ejpam-3461	155	7	,	,	PUNCT
ejpam-3461	155	8	1	1	NUM
ejpam-3461	155	9	≤	≤	NUM
ejpam-3461	155	10	i	i	PRON
ejpam-3461	155	11	≤	≤	PROPN
ejpam-3461	156	1	n.	n.	NOUN
ejpam-3461	156	2	then	then	ADV
ejpam-3461	156	3	,	,	PUNCT
ejpam-3461	156	4	ui	ui	PROPN
ejpam-3461	156	5	=	=	SYM
ejpam-3461	156	6	uir	uir	PROPN
ejpam-3461	156	7	.	.	PUNCT
ejpam-3461	157	1	it	it	PRON
ejpam-3461	157	2	is	be	AUX
ejpam-3461	157	3	easy	easy	ADJ
ejpam-3461	157	4	to	to	PART
ejpam-3461	157	5	see	see	VERB
ejpam-3461	157	6	that	that	PRON
ejpam-3461	157	7	m	m	PROPN
ejpam-3461	157	8	=	=	ADJ
ejpam-3461	157	9	v	v	NOUN
ejpam-3461	157	10	.	.	PUNCT
ejpam-3461	158	1	therefore	therefore	ADV
ejpam-3461	158	2	,	,	PUNCT
ejpam-3461	158	3	m	m	VERB
ejpam-3461	158	4	is	be	AUX
ejpam-3461	158	5	finitely	finitely	ADV
ejpam-3461	158	6	generated	generate	VERB
ejpam-3461	158	7	semi	semi	ADJ
ejpam-3461	158	8	-	-	ADJ
ejpam-3461	158	9	simple	simple	ADJ
ejpam-3461	158	10	injective	injective	NOUN
ejpam-3461	158	11	.	.	PUNCT
ejpam-3461	159	1	this	this	PRON
ejpam-3461	159	2	means	mean	VERB
ejpam-3461	159	3	that	that	SCONJ
ejpam-3461	159	4	m	m	NOUN
ejpam-3461	159	5	is	be	AUX
ejpam-3461	159	6	continuous	continuous	ADJ
ejpam-3461	159	7	.	.	PUNCT
ejpam-3461	160	1	(	(	PUNCT
ejpam-3461	160	2	3)⇒	3)⇒	NUM
ejpam-3461	160	3	(	(	PUNCT
ejpam-3461	160	4	4	4	NUM
ejpam-3461	160	5	)	)	PUNCT
ejpam-3461	160	6	follows	follow	VERB
ejpam-3461	160	7	from	from	ADP
ejpam-3461	160	8	an	an	DET
ejpam-3461	160	9	argument	argument	NOUN
ejpam-3461	160	10	similar	similar	ADJ
ejpam-3461	160	11	to	to	ADP
ejpam-3461	160	12	the	the	DET
ejpam-3461	160	13	one	one	NUM
ejpam-3461	160	14	in	in	ADP
ejpam-3461	160	15	(	(	PUNCT
ejpam-3461	160	16	2)⇒	2)⇒	NUM
ejpam-3461	160	17	(	(	PUNCT
ejpam-3461	160	18	3	3	NUM
ejpam-3461	160	19	)	)	PUNCT
ejpam-3461	160	20	.	.	PUNCT
ejpam-3461	161	1	(	(	PUNCT
ejpam-3461	161	2	4)⇔	4)⇔	NUM
ejpam-3461	161	3	(	(	PUNCT
ejpam-3461	161	4	1	1	X
ejpam-3461	161	5	)	)	PUNCT
ejpam-3461	161	6	it	it	PRON
ejpam-3461	161	7	is	be	AUX
ejpam-3461	161	8	easy	easy	ADJ
ejpam-3461	161	9	to	to	PART
ejpam-3461	161	10	see	see	VERB
ejpam-3461	161	11	.	.	PUNCT
ejpam-3461	162	1	corollary	corollary	ADJ
ejpam-3461	162	2	4	4	NUM
ejpam-3461	162	3	.	.	PUNCT
ejpam-3461	163	1	a	a	DET
ejpam-3461	163	2	right	right	ADJ
ejpam-3461	163	3	self	self	NOUN
ejpam-3461	163	4	-	-	PUNCT
ejpam-3461	163	5	injective	injective	ADJ
ejpam-3461	163	6	right	right	ADJ
ejpam-3461	163	7	hereditary	hereditary	ADJ
ejpam-3461	163	8	ring	ring	NOUN
ejpam-3461	163	9	is	be	AUX
ejpam-3461	163	10	semi	semi	ADJ
ejpam-3461	163	11	-	-	ADJ
ejpam-3461	163	12	simple	simple	ADJ
ejpam-3461	163	13	artinian	artinian	NOUN
ejpam-3461	163	14	.	.	PUNCT
ejpam-3461	164	1	recall	recall	VERB
ejpam-3461	164	2	that	that	SCONJ
ejpam-3461	164	3	a	a	DET
ejpam-3461	164	4	module	module	NOUN
ejpam-3461	164	5	m	m	VERB
ejpam-3461	164	6	is	be	AUX
ejpam-3461	164	7	said	say	VERB
ejpam-3461	164	8	to	to	PART
ejpam-3461	164	9	be	be	AUX
ejpam-3461	164	10	retractable	retractable	ADJ
ejpam-3461	164	11	if	if	SCONJ
ejpam-3461	164	12	for	for	ADP
ejpam-3461	164	13	any	any	DET
ejpam-3461	164	14	0	0	NUM
ejpam-3461	164	15	6=	6=	NUM
ejpam-3461	164	16	n	n	CCONJ
ejpam-3461	164	17	≤	≤	NOUN
ejpam-3461	164	18	m	m	VERB
ejpam-3461	164	19	,	,	PUNCT
ejpam-3461	164	20	there	there	PRON
ejpam-3461	164	21	exists	exist	VERB
ejpam-3461	164	22	a	a	DET
ejpam-3461	164	23	nonzero	nonzero	NOUN
ejpam-3461	164	24	homomorphism	homomorphism	PROPN
ejpam-3461	164	25	form	form	NOUN
ejpam-3461	164	26	m	m	VERB
ejpam-3461	164	27	to	to	ADP
ejpam-3461	164	28	n	n	PROPN
ejpam-3461	164	29	.	.	PUNCT
ejpam-3461	165	1	a	a	DET
ejpam-3461	165	2	module	module	NOUN
ejpam-3461	165	3	m	m	VERB
ejpam-3461	165	4	has	have	VERB
ejpam-3461	165	5	finite	finite	ADJ
ejpam-3461	165	6	uniform	uniform	ADJ
ejpam-3461	165	7	dimension	dimension	PROPN
ejpam-3461	165	8	n	n	CCONJ
ejpam-3461	165	9	(	(	PUNCT
ejpam-3461	165	10	written	write	VERB
ejpam-3461	165	11	udim(m	udim(m	PROPN
ejpam-3461	165	12	)	)	PUNCT
ejpam-3461	165	13	=	=	SYM
ejpam-3461	165	14	n	n	CCONJ
ejpam-3461	165	15	)	)	PUNCT
ejpam-3461	165	16	if	if	SCONJ
ejpam-3461	165	17	there	there	PRON
ejpam-3461	165	18	is	be	VERB
ejpam-3461	165	19	an	an	DET
ejpam-3461	165	20	essential	essential	ADJ
ejpam-3461	165	21	submodule	submodule	NOUN
ejpam-3461	165	22	v	v	ADP
ejpam-3461	165	23	≤e	≤e	NOUN
ejpam-3461	165	24	m	m	VERB
ejpam-3461	165	25	that	that	PRON
ejpam-3461	165	26	is	be	AUX
ejpam-3461	165	27	a	a	DET
ejpam-3461	165	28	direct	direct	ADJ
ejpam-3461	165	29	sum	sum	NOUN
ejpam-3461	165	30	of	of	ADP
ejpam-3461	165	31	n	n	CCONJ
ejpam-3461	165	32	uniform	uniform	ADJ
ejpam-3461	165	33	submodules	submodule	NOUN
ejpam-3461	165	34	.	.	PUNCT
ejpam-3461	166	1	remark	remark	NOUN
ejpam-3461	166	2	2	2	NUM
ejpam-3461	166	3	.	.	PUNCT
ejpam-3461	167	1	a	a	DET
ejpam-3461	167	2	c	c	NOUN
ejpam-3461	167	3	-	-	PUNCT
ejpam-3461	167	4	co	co	NOUN
ejpam-3461	167	5	-	-	ADJ
ejpam-3461	167	6	epi	epi	ADJ
ejpam-3461	167	7	-	-	ADJ
ejpam-3461	167	8	retractable	retractable	ADJ
ejpam-3461	167	9	module	module	NOUN
ejpam-3461	167	10	with	with	ADP
ejpam-3461	167	11	finite	finite	PROPN
ejpam-3461	167	12	uniform	uniform	PROPN
ejpam-3461	167	13	dimension	dimension	NOUN
ejpam-3461	167	14	need	need	AUX
ejpam-3461	167	15	not	not	PART
ejpam-3461	167	16	be	be	AUX
ejpam-3461	167	17	retractable	retractable	ADJ
ejpam-3461	167	18	.	.	PUNCT
ejpam-3461	168	1	in	in	ADP
ejpam-3461	168	2	fact	fact	NOUN
ejpam-3461	168	3	,	,	PUNCT
ejpam-3461	168	4	the	the	DET
ejpam-3461	168	5	z	z	NOUN
ejpam-3461	168	6	-	-	PUNCT
ejpam-3461	168	7	module	module	NOUN
ejpam-3461	168	8	q	q	NOUN
ejpam-3461	168	9	is	be	AUX
ejpam-3461	168	10	c	c	NOUN
ejpam-3461	168	11	-	-	PUNCT
ejpam-3461	168	12	co	co	ADJ
ejpam-3461	168	13	-	-	ADJ
ejpam-3461	168	14	epi	epi	NOUN
ejpam-3461	168	15	-	-	NOUN
ejpam-3461	168	16	retractable	retractable	ADJ
ejpam-3461	168	17	with	with	ADP
ejpam-3461	168	18	finite	finite	ADJ
ejpam-3461	168	19	uniform	uniform	ADJ
ejpam-3461	168	20	dimension	dimension	NOUN
ejpam-3461	168	21	but	but	CCONJ
ejpam-3461	168	22	it	it	PRON
ejpam-3461	168	23	is	be	AUX
ejpam-3461	168	24	not	not	PART
ejpam-3461	168	25	retractable	retractable	ADJ
ejpam-3461	168	26	.	.	PUNCT
ejpam-3461	169	1	clearly	clearly	ADV
ejpam-3461	169	2	,	,	PUNCT
ejpam-3461	169	3	a	a	DET
ejpam-3461	169	4	c	c	NOUN
ejpam-3461	169	5	-	-	PUNCT
ejpam-3461	169	6	co	co	NOUN
ejpam-3461	169	7	-	-	ADJ
ejpam-3461	169	8	epi	epi	ADJ
ejpam-3461	169	9	-	-	NOUN
ejpam-3461	169	10	retractable	retractable	ADJ
ejpam-3461	169	11	need	need	VERB
ejpam-3461	169	12	not	not	PART
ejpam-3461	169	13	to	to	PART
ejpam-3461	169	14	have	have	VERB
ejpam-3461	169	15	a	a	DET
ejpam-3461	169	16	finite	finite	ADJ
ejpam-3461	169	17	uniform	uniform	ADJ
ejpam-3461	169	18	dimension	dimension	NOUN
ejpam-3461	169	19	.	.	PUNCT
ejpam-3461	170	1	for	for	ADP
ejpam-3461	170	2	example	example	NOUN
ejpam-3461	170	3	extending	extend	VERB
ejpam-3461	170	4	modules	module	NOUN
ejpam-3461	170	5	are	be	AUX
ejpam-3461	170	6	c	c	NOUN
ejpam-3461	170	7	-	-	PUNCT
ejpam-3461	170	8	co	co	ADJ
ejpam-3461	170	9	-	-	ADJ
ejpam-3461	170	10	epi	epi	ADJ
ejpam-3461	170	11	-	-	NOUN
ejpam-3461	170	12	retractable	retractable	ADJ
ejpam-3461	170	13	which	which	PRON
ejpam-3461	170	14	need	need	AUX
ejpam-3461	170	15	not	not	PART
ejpam-3461	170	16	have	have	VERB
ejpam-3461	170	17	finite	finite	ADJ
ejpam-3461	170	18	uniform	uniform	ADJ
ejpam-3461	170	19	dimension	dimension	NOUN
ejpam-3461	170	20	.	.	PUNCT
ejpam-3461	171	1	proposition	proposition	NOUN
ejpam-3461	171	2	6	6	NUM
ejpam-3461	171	3	.	.	PUNCT
ejpam-3461	172	1	if	if	SCONJ
ejpam-3461	172	2	m	m	NOUN
ejpam-3461	172	3	is	be	AUX
ejpam-3461	172	4	a	a	DET
ejpam-3461	172	5	c	c	NOUN
ejpam-3461	172	6	-	-	PUNCT
ejpam-3461	172	7	co	co	NOUN
ejpam-3461	172	8	-	-	ADJ
ejpam-3461	172	9	epi	epi	ADJ
ejpam-3461	172	10	-	-	ADJ
ejpam-3461	172	11	retractable	retractable	ADJ
ejpam-3461	172	12	r	r	NOUN
ejpam-3461	172	13	-	-	PUNCT
ejpam-3461	172	14	module	module	NOUN
ejpam-3461	172	15	with	with	ADP
ejpam-3461	172	16	udim(m	udim(m	PROPN
ejpam-3461	172	17	)	)	PUNCT
ejpam-3461	172	18	=	=	SYM
ejpam-3461	172	19	n	n	X
ejpam-3461	172	20	≥	≥	NOUN
ejpam-3461	172	21	2	2	NUM
ejpam-3461	172	22	,	,	PUNCT
ejpam-3461	172	23	then	then	ADV
ejpam-3461	172	24	the	the	DET
ejpam-3461	172	25	following	follow	VERB
ejpam-3461	172	26	assertions	assertion	NOUN
ejpam-3461	172	27	are	be	AUX
ejpam-3461	172	28	verified	verify	VERB
ejpam-3461	172	29	:	:	PUNCT
ejpam-3461	172	30	(	(	PUNCT
ejpam-3461	172	31	1	1	X
ejpam-3461	172	32	)	)	PUNCT
ejpam-3461	172	33	m	m	VERB
ejpam-3461	172	34	is	be	AUX
ejpam-3461	172	35	retractable	retractable	ADJ
ejpam-3461	172	36	.	.	PUNCT
ejpam-3461	173	1	(	(	PUNCT
ejpam-3461	173	2	2	2	X
ejpam-3461	173	3	)	)	PUNCT
ejpam-3461	173	4	for	for	ADP
ejpam-3461	173	5	every	every	DET
ejpam-3461	173	6	0	0	NUM
ejpam-3461	173	7	6=	6=	ADP
ejpam-3461	173	8	c	c	PROPN
ejpam-3461	173	9	⊆c	⊆c	NOUN
ejpam-3461	173	10	m	m	PROPN
ejpam-3461	173	11	,	,	PUNCT
ejpam-3461	173	12	m	m	VERB
ejpam-3461	173	13	/	/	SYM
ejpam-3461	173	14	c	c	PROPN
ejpam-3461	173	15	is	be	AUX
ejpam-3461	173	16	uniform	uniform	ADJ
ejpam-3461	173	17	.	.	PUNCT
ejpam-3461	174	1	a.	a.	PROPN
ejpam-3461	174	2	d.	d.	PROPN
ejpam-3461	174	3	diallo	diallo	PROPN
ejpam-3461	174	4	,	,	PUNCT
ejpam-3461	174	5	p.	p.	PROPN
ejpam-3461	174	6	c.	c.	PROPN
ejpam-3461	174	7	diop	diop	PROPN
ejpam-3461	174	8	,	,	PUNCT
ejpam-3461	174	9	m.	m.	NOUN
ejpam-3461	174	10	barry	barry	PROPN
ejpam-3461	174	11	/	/	SYM
ejpam-3461	174	12	eur	eur	PROPN
ejpam-3461	174	13	.	.	PUNCT
ejpam-3461	175	1	j.	j.	PROPN
ejpam-3461	175	2	pure	pure	PROPN
ejpam-3461	175	3	appl	appl	PROPN
ejpam-3461	175	4	.	.	PROPN
ejpam-3461	175	5	math	math	PROPN
ejpam-3461	175	6	,	,	PUNCT
ejpam-3461	175	7	12	12	NUM
ejpam-3461	175	8	(	(	PUNCT
ejpam-3461	175	9	3	3	NUM
ejpam-3461	175	10	)	)	PUNCT
ejpam-3461	175	11	(	(	PUNCT
ejpam-3461	175	12	2019	2019	NUM
ejpam-3461	175	13	)	)	PUNCT
ejpam-3461	175	14	,	,	PUNCT
ejpam-3461	175	15	1187	1187	NUM
ejpam-3461	175	16	-	-	SYM
ejpam-3461	175	17	1198	1198	NUM
ejpam-3461	175	18	1192	1192	NUM
ejpam-3461	175	19	proof	proof	NOUN
ejpam-3461	175	20	.	.	PUNCT
ejpam-3461	176	1	(	(	PUNCT
ejpam-3461	176	2	1	1	X
ejpam-3461	176	3	)	)	PUNCT
ejpam-3461	176	4	let	let	VERB
ejpam-3461	176	5	0	0	NUM
ejpam-3461	176	6	6=	6=	NUM
ejpam-3461	176	7	n	n	DET
ejpam-3461	176	8	≤m	≤m	NOUN
ejpam-3461	176	9	.	.	PUNCT
ejpam-3461	177	1	since	since	SCONJ
ejpam-3461	177	2	udim(n	udim(n	PROPN
ejpam-3461	177	3	)	)	PUNCT
ejpam-3461	177	4	<	<	X
ejpam-3461	177	5	∞	∞	PROPN
ejpam-3461	177	6	,	,	PUNCT
ejpam-3461	177	7	n	n	PRON
ejpam-3461	177	8	contains	contain	VERB
ejpam-3461	177	9	a	a	DET
ejpam-3461	177	10	uniform	uniform	ADJ
ejpam-3461	177	11	submodule	submodule	NOUN
ejpam-3461	177	12	,	,	PUNCT
ejpam-3461	177	13	say	say	VERB
ejpam-3461	177	14	u	u	NOUN
ejpam-3461	177	15	.	.	PUNCT
ejpam-3461	178	1	after	after	ADP
ejpam-3461	178	2	replacing	replace	VERB
ejpam-3461	178	3	u	u	NOUN
ejpam-3461	178	4	by	by	ADP
ejpam-3461	178	5	an	an	DET
ejpam-3461	178	6	essential	essential	ADJ
ejpam-3461	178	7	closure	closure	NOUN
ejpam-3461	178	8	,	,	PUNCT
ejpam-3461	178	9	we	we	PRON
ejpam-3461	178	10	may	may	AUX
ejpam-3461	178	11	assume	assume	VERB
ejpam-3461	178	12	that	that	SCONJ
ejpam-3461	178	13	u	u	PRON
ejpam-3461	178	14	is	be	AUX
ejpam-3461	178	15	a	a	DET
ejpam-3461	178	16	complement	complement	NOUN
ejpam-3461	178	17	submodule	submodule	NOUN
ejpam-3461	178	18	of	of	ADP
ejpam-3461	178	19	m	m	PROPN
ejpam-3461	178	20	.	.	PUNCT
ejpam-3461	179	1	by	by	ADP
ejpam-3461	179	2	the	the	DET
ejpam-3461	179	3	c	c	PROPN
ejpam-3461	179	4	-	-	PUNCT
ejpam-3461	179	5	co	co	NOUN
ejpam-3461	179	6	-	-	ADJ
ejpam-3461	179	7	epi	epi	ADJ
ejpam-3461	179	8	-	-	ADJ
ejpam-3461	179	9	retractable	retractable	ADJ
ejpam-3461	179	10	condition	condition	NOUN
ejpam-3461	179	11	on	on	ADP
ejpam-3461	179	12	m	m	PROPN
ejpam-3461	179	13	,	,	PUNCT
ejpam-3461	179	14	there	there	PRON
ejpam-3461	179	15	exists	exist	VERB
ejpam-3461	179	16	a	a	DET
ejpam-3461	179	17	monomorphism	monomorphism	NOUN
ejpam-3461	179	18	f	f	X
ejpam-3461	179	19	:	:	PUNCT
ejpam-3461	179	20	m	m	PROPN
ejpam-3461	179	21	/	/	SYM
ejpam-3461	179	22	u	u	SYM
ejpam-3461	179	23	−→m	−→m	PROPN
ejpam-3461	179	24	.	.	PUNCT
ejpam-3461	180	1	consider	consider	VERB
ejpam-3461	180	2	the	the	DET
ejpam-3461	180	3	inclusion	inclusion	NOUN
ejpam-3461	180	4	map	map	NOUN
ejpam-3461	180	5	i	i	PRON
ejpam-3461	180	6	:	:	PUNCT
ejpam-3461	180	7	u	u	NOUN
ejpam-3461	180	8	−→m	−→m	X
ejpam-3461	180	9	.	.	PUNCT
ejpam-3461	181	1	thus	thus	ADV
ejpam-3461	181	2	,	,	PUNCT
ejpam-3461	181	3	f	f	PROPN
ejpam-3461	181	4	=	=	PRON
ejpam-3461	181	5	ij	ij	NOUN
ejpam-3461	181	6	is	be	AUX
ejpam-3461	181	7	a	a	DET
ejpam-3461	181	8	monomorphism	monomorphism	NOUN
ejpam-3461	181	9	where	where	SCONJ
ejpam-3461	181	10	j	j	NOUN
ejpam-3461	181	11	:	:	PUNCT
ejpam-3461	181	12	m	m	X
ejpam-3461	181	13	/	/	SYM
ejpam-3461	181	14	u	u	PROPN
ejpam-3461	181	15	−→	−→	NOUN
ejpam-3461	181	16	u	u	NOUN
ejpam-3461	181	17	.	.	PUNCT
ejpam-3461	182	1	consequently	consequently	ADV
ejpam-3461	182	2	,	,	PUNCT
ejpam-3461	182	3	j	j	PROPN
ejpam-3461	182	4	is	be	AUX
ejpam-3461	182	5	a	a	DET
ejpam-3461	182	6	monomorphism	monomorphism	NOUN
ejpam-3461	182	7	.	.	PUNCT
ejpam-3461	183	1	now	now	ADV
ejpam-3461	183	2	,	,	PUNCT
ejpam-3461	183	3	consider	consider	VERB
ejpam-3461	183	4	the	the	DET
ejpam-3461	183	5	inclusion	inclusion	NOUN
ejpam-3461	183	6	map	map	NOUN
ejpam-3461	183	7	i1	i1	PROPN
ejpam-3461	183	8	:	:	PUNCT
ejpam-3461	183	9	u	u	NOUN
ejpam-3461	183	10	−→	−→	NOUN
ejpam-3461	183	11	n	n	ADV
ejpam-3461	183	12	.	.	PUNCT
ejpam-3461	184	1	so	so	ADV
ejpam-3461	184	2	,	,	PUNCT
ejpam-3461	184	3	i1jπ	i1jπ	X
ejpam-3461	184	4	:	:	PUNCT
ejpam-3461	184	5	m	m	VERB
ejpam-3461	184	6	−→	−→	NOUN
ejpam-3461	185	1	n	n	NOUN
ejpam-3461	185	2	is	be	AUX
ejpam-3461	185	3	a	a	DET
ejpam-3461	185	4	nonzero	nonzero	NOUN
ejpam-3461	185	5	homomorphism	homomorphism	NOUN
ejpam-3461	185	6	where	where	SCONJ
ejpam-3461	185	7	π	π	NOUN
ejpam-3461	185	8	:	:	PUNCT
ejpam-3461	185	9	m	m	PROPN
ejpam-3461	185	10	−→m	−→m	NUM
ejpam-3461	185	11	/	/	SYM
ejpam-3461	185	12	u	u	NOUN
ejpam-3461	185	13	is	be	AUX
ejpam-3461	185	14	the	the	DET
ejpam-3461	185	15	natural	natural	ADJ
ejpam-3461	185	16	surjection	surjection	NOUN
ejpam-3461	185	17	.	.	PUNCT
ejpam-3461	186	1	therefore	therefore	ADV
ejpam-3461	186	2	,	,	PUNCT
ejpam-3461	186	3	m	m	VERB
ejpam-3461	186	4	is	be	AUX
ejpam-3461	186	5	retractable	retractable	ADJ
ejpam-3461	186	6	.	.	PUNCT
ejpam-3461	187	1	(	(	PUNCT
ejpam-3461	187	2	2	2	NUM
ejpam-3461	187	3	)	)	PUNCT
ejpam-3461	187	4	since	since	SCONJ
ejpam-3461	187	5	udim(m	udim(m	PROPN
ejpam-3461	187	6	)	)	PUNCT
ejpam-3461	187	7	=	=	SYM
ejpam-3461	187	8	n	n	X
ejpam-3461	187	9	≥	≥	NOUN
ejpam-3461	187	10	2	2	NUM
ejpam-3461	187	11	,	,	PUNCT
ejpam-3461	187	12	there	there	PRON
ejpam-3461	187	13	exist	exist	VERB
ejpam-3461	187	14	complements	complement	NOUN
ejpam-3461	187	15	submodules	submodule	NOUN
ejpam-3461	187	16	ci	ci	PROPN
ejpam-3461	187	17	⊆c	⊆c	NOUN
ejpam-3461	187	18	m(1	m(1	NOUN
ejpam-3461	187	19	≤	≤	PUNCT
ejpam-3461	187	20	i	i	NOUN
ejpam-3461	187	21	≤	≤	NOUN
ejpam-3461	187	22	n	n	CCONJ
ejpam-3461	187	23	)	)	PUNCT
ejpam-3461	187	24	such	such	ADJ
ejpam-3461	187	25	that	that	SCONJ
ejpam-3461	187	26	each	each	DET
ejpam-3461	187	27	m	m	PROPN
ejpam-3461	187	28	/	/	SYM
ejpam-3461	187	29	ci	ci	PROPN
ejpam-3461	187	30	is	be	AUX
ejpam-3461	187	31	uniform	uniform	ADJ
ejpam-3461	187	32	and	and	CCONJ
ejpam-3461	187	33	c1∩	c1∩	DET
ejpam-3461	187	34	...	...	PUNCT
ejpam-3461	187	35	∩cn	∩cn	NOUN
ejpam-3461	187	36	=	=	SYM
ejpam-3461	187	37	0	0	X
ejpam-3461	187	38	.	.	PUNCT
ejpam-3461	188	1	thus	thus	ADV
ejpam-3461	188	2	,	,	PUNCT
ejpam-3461	188	3	there	there	PRON
ejpam-3461	188	4	exists	exist	VERB
ejpam-3461	188	5	a	a	DET
ejpam-3461	188	6	monomorphism	monomorphism	NOUN
ejpam-3461	188	7	f	f	X
ejpam-3461	188	8	:	:	PUNCT
ejpam-3461	188	9	m	m	AUX
ejpam-3461	188	10	−→	−→	VERB
ejpam-3461	188	11	⊕n	⊕n	NOUN
ejpam-3461	188	12	im	im	PROPN
ejpam-3461	188	13	/	/	SYM
ejpam-3461	188	14	ci	ci	NOUN
ejpam-3461	188	15	.	.	PUNCT
ejpam-3461	189	1	let	let	VERB
ejpam-3461	189	2	0	0	NUM
ejpam-3461	190	1	6=	6=	NUM
ejpam-3461	190	2	c	c	PROPN
ejpam-3461	190	3	⊆c	⊆c	NOUN
ejpam-3461	190	4	m	m	PROPN
ejpam-3461	190	5	.	.	PUNCT
ejpam-3461	191	1	since	since	SCONJ
ejpam-3461	191	2	m	m	PROPN
ejpam-3461	191	3	is	be	AUX
ejpam-3461	191	4	c	c	NOUN
ejpam-3461	191	5	-	-	PUNCT
ejpam-3461	191	6	co	co	ADJ
ejpam-3461	191	7	-	-	ADJ
ejpam-3461	191	8	epi	epi	NOUN
ejpam-3461	191	9	-	-	NOUN
ejpam-3461	191	10	retractable	retractable	ADJ
ejpam-3461	191	11	,	,	PUNCT
ejpam-3461	191	12	there	there	PRON
ejpam-3461	191	13	exists	exist	VERB
ejpam-3461	191	14	a	a	DET
ejpam-3461	191	15	monomorphism	monomorphism	NOUN
ejpam-3461	191	16	g	g	NOUN
ejpam-3461	191	17	:	:	PUNCT
ejpam-3461	191	18	m	m	PROPN
ejpam-3461	191	19	/	/	SYM
ejpam-3461	191	20	c	c	PROPN
ejpam-3461	191	21	−→m	−→m	PROPN
ejpam-3461	191	22	.	.	PUNCT
ejpam-3461	192	1	hence	hence	ADV
ejpam-3461	192	2	,	,	PUNCT
ejpam-3461	192	3	h	h	NOUN
ejpam-3461	192	4	=	=	SYM
ejpam-3461	192	5	fg	fg	PROPN
ejpam-3461	192	6	:	:	PUNCT
ejpam-3461	192	7	m	m	X
ejpam-3461	192	8	/	/	SYM
ejpam-3461	192	9	c	c	VERB
ejpam-3461	192	10	−→	−→	ADJ
ejpam-3461	192	11	⊕n	⊕n	NOUN
ejpam-3461	192	12	im	im	NOUN
ejpam-3461	192	13	/	/	SYM
ejpam-3461	192	14	ci	ci	PROPN
ejpam-3461	192	15	is	be	AUX
ejpam-3461	192	16	a	a	DET
ejpam-3461	192	17	monomorphism	monomorphism	NOUN
ejpam-3461	192	18	.	.	PUNCT
ejpam-3461	193	1	consider	consider	VERB
ejpam-3461	193	2	the	the	DET
ejpam-3461	193	3	inclusion	inclusion	NOUN
ejpam-3461	193	4	map	map	NOUN
ejpam-3461	193	5	i	i	PRON
ejpam-3461	193	6	:	:	PUNCT
ejpam-3461	193	7	m	m	X
ejpam-3461	193	8	/	/	SYM
ejpam-3461	193	9	ci	ci	VERB
ejpam-3461	193	10	−→	−→	ADJ
ejpam-3461	193	11	⊕n	⊕n	NOUN
ejpam-3461	193	12	im	im	PROPN
ejpam-3461	193	13	/	/	SYM
ejpam-3461	193	14	ci	ci	NOUN
ejpam-3461	193	15	.	.	PUNCT
ejpam-3461	194	1	thus	thus	ADV
ejpam-3461	194	2	,	,	PUNCT
ejpam-3461	194	3	h	h	NOUN
ejpam-3461	194	4	=	=	NOUN
ejpam-3461	194	5	ii1	ii1	NOUN
ejpam-3461	194	6	is	be	AUX
ejpam-3461	194	7	a	a	DET
ejpam-3461	194	8	monomorphism	monomorphism	NOUN
ejpam-3461	194	9	where	where	SCONJ
ejpam-3461	194	10	i1	i1	PROPN
ejpam-3461	194	11	:	:	PUNCT
ejpam-3461	194	12	m	m	PROPN
ejpam-3461	194	13	/	/	SYM
ejpam-3461	194	14	c	c	VERB
ejpam-3461	194	15	−→	−→	NOUN
ejpam-3461	194	16	m	m	PROPN
ejpam-3461	194	17	/	/	SYM
ejpam-3461	194	18	ci	ci	PROPN
ejpam-3461	194	19	.	.	PUNCT
ejpam-3461	195	1	therefore	therefore	ADV
ejpam-3461	195	2	,	,	PUNCT
ejpam-3461	195	3	i1	i1	PROPN
ejpam-3461	195	4	is	be	AUX
ejpam-3461	195	5	a	a	DET
ejpam-3461	195	6	monomorphism	monomorphism	NOUN
ejpam-3461	195	7	.	.	PUNCT
ejpam-3461	196	1	it	it	PRON
ejpam-3461	196	2	follows	follow	VERB
ejpam-3461	196	3	that	that	SCONJ
ejpam-3461	196	4	m	m	PROPN
ejpam-3461	196	5	/	/	SYM
ejpam-3461	196	6	c	c	PROPN
ejpam-3461	196	7	is	be	AUX
ejpam-3461	196	8	uniform	uniform	ADJ
ejpam-3461	196	9	.	.	PUNCT
ejpam-3461	197	1	corollary	corollary	ADJ
ejpam-3461	197	2	5	5	NUM
ejpam-3461	197	3	.	.	PUNCT
ejpam-3461	198	1	an	an	DET
ejpam-3461	198	2	r	r	NOUN
ejpam-3461	198	3	-	-	PUNCT
ejpam-3461	198	4	module	module	NOUN
ejpam-3461	198	5	m	m	NOUN
ejpam-3461	198	6	is	be	AUX
ejpam-3461	198	7	simple	simple	ADJ
ejpam-3461	198	8	if	if	SCONJ
ejpam-3461	199	1	and	and	CCONJ
ejpam-3461	199	2	only	only	ADV
ejpam-3461	199	3	if	if	SCONJ
ejpam-3461	199	4	m	m	NOUN
ejpam-3461	199	5	is	be	AUX
ejpam-3461	199	6	artinian	artinian	ADJ
ejpam-3461	199	7	c	c	X
ejpam-3461	199	8	-	-	PUNCT
ejpam-3461	199	9	co	co	NOUN
ejpam-3461	199	10	-	-	ADJ
ejpam-3461	199	11	epi	epi	ADJ
ejpam-3461	199	12	-	-	NOUN
ejpam-3461	199	13	retractable	retractable	ADJ
ejpam-3461	199	14	and	and	CCONJ
ejpam-3461	199	15	every	every	DET
ejpam-3461	199	16	endomorphisme	endomorphisme	NOUN
ejpam-3461	199	17	of	of	ADP
ejpam-3461	199	18	m	m	PROPN
ejpam-3461	199	19	is	be	AUX
ejpam-3461	199	20	a	a	DET
ejpam-3461	199	21	monomorphism	monomorphism	NOUN
ejpam-3461	199	22	.	.	PUNCT
ejpam-3461	200	1	let	let	VERB
ejpam-3461	200	2	n	n	NOUN
ejpam-3461	200	3	and	and	CCONJ
ejpam-3461	200	4	m	m	AUX
ejpam-3461	200	5	be	be	VERB
ejpam-3461	200	6	r	r	NOUN
ejpam-3461	200	7	-	-	PUNCT
ejpam-3461	200	8	modules	module	NOUN
ejpam-3461	200	9	and	and	CCONJ
ejpam-3461	200	10	s	s	NOUN
ejpam-3461	200	11	=	=	ADJ
ejpam-3461	200	12	endr(m	endr(m	PROPN
ejpam-3461	200	13	)	)	PUNCT
ejpam-3461	200	14	.	.	PUNCT
ejpam-3461	201	1	we	we	PRON
ejpam-3461	201	2	denote	denote	VERB
ejpam-3461	201	3	by	by	ADP
ejpam-3461	201	4	n	n	DET
ejpam-3461	201	5	the	the	DET
ejpam-3461	201	6	set	set	NOUN
ejpam-3461	201	7	of	of	ADP
ejpam-3461	201	8	rsubmodules	rsubmodule	NOUN
ejpam-3461	201	9	of	of	ADP
ejpam-3461	201	10	n	n	PRON
ejpam-3461	201	11	and	and	CCONJ
ejpam-3461	201	12	by	by	ADP
ejpam-3461	201	13	h	h	NOUN
ejpam-3461	201	14	the	the	DET
ejpam-3461	201	15	set	set	NOUN
ejpam-3461	201	16	of	of	ADP
ejpam-3461	201	17	s	s	NOUN
ejpam-3461	201	18	-	-	PUNCT
ejpam-3461	201	19	submodules	submodule	NOUN
ejpam-3461	201	20	of	of	ADP
ejpam-3461	201	21	homr(n	homr(n	NOUN
ejpam-3461	201	22	,	,	PUNCT
ejpam-3461	201	23	m)s	m)s	ADJ
ejpam-3461	201	24	.	.	PUNCT
ejpam-3461	202	1	for	for	ADP
ejpam-3461	202	2	x	x	PROPN
ejpam-3461	202	3	∈	∈	PROPN
ejpam-3461	202	4	h	h	NOUN
ejpam-3461	202	5	we	we	PRON
ejpam-3461	202	6	put	put	VERB
ejpam-3461	202	7	:	:	PUNCT
ejpam-3461	202	8	ker(x	ker(x	X
ejpam-3461	202	9	)	)	PUNCT
ejpam-3461	202	10	=	=	PUNCT
ejpam-3461	202	11	∩{kerg|g	∩{kerg|g	ADP
ejpam-3461	202	12	∈	∈	NOUN
ejpam-3461	202	13	x	x	X
ejpam-3461	202	14	}	}	PUNCT
ejpam-3461	202	15	∈	∈	PROPN
ejpam-3461	202	16	n	n	NOUN
ejpam-3461	202	17	.	.	PUNCT
ejpam-3461	203	1	in	in	ADP
ejpam-3461	203	2	the	the	DET
ejpam-3461	203	3	next	next	ADJ
ejpam-3461	203	4	result	result	NOUN
ejpam-3461	203	5	,	,	PUNCT
ejpam-3461	203	6	we	we	PRON
ejpam-3461	203	7	investigate	investigate	VERB
ejpam-3461	203	8	when	when	SCONJ
ejpam-3461	203	9	c	c	AUX
ejpam-3461	203	10	-	-	PUNCT
ejpam-3461	203	11	co	co	NOUN
ejpam-3461	203	12	-	-	ADJ
ejpam-3461	203	13	epi	epi	ADJ
ejpam-3461	203	14	-	-	ADJ
ejpam-3461	203	15	retractable	retractable	ADJ
ejpam-3461	203	16	r	r	NOUN
ejpam-3461	203	17	-	-	PUNCT
ejpam-3461	203	18	modules	module	NOUN
ejpam-3461	203	19	have	have	VERB
ejpam-3461	203	20	finite	finite	NOUN
ejpam-3461	203	21	uniform	uniform	ADJ
ejpam-3461	203	22	dimension	dimension	NOUN
ejpam-3461	203	23	.	.	PUNCT
ejpam-3461	204	1	proposition	proposition	NOUN
ejpam-3461	204	2	7	7	NUM
ejpam-3461	204	3	.	.	PUNCT
ejpam-3461	205	1	let	let	VERB
ejpam-3461	205	2	m	m	PRON
ejpam-3461	205	3	be	be	AUX
ejpam-3461	205	4	a	a	DET
ejpam-3461	205	5	c	c	NOUN
ejpam-3461	205	6	-	-	PUNCT
ejpam-3461	205	7	co	co	NOUN
ejpam-3461	205	8	-	-	ADJ
ejpam-3461	205	9	epi	epi	ADJ
ejpam-3461	205	10	-	-	ADJ
ejpam-3461	205	11	retractable	retractable	ADJ
ejpam-3461	205	12	r	r	NOUN
ejpam-3461	205	13	-	-	PUNCT
ejpam-3461	205	14	module	module	NOUN
ejpam-3461	205	15	such	such	ADJ
ejpam-3461	205	16	that	that	DET
ejpam-3461	205	17	s	s	PART
ejpam-3461	205	18	satisfies	satisfy	VERB
ejpam-3461	205	19	dcc	dcc	PROPN
ejpam-3461	205	20	for	for	ADP
ejpam-3461	205	21	cyclic	cyclic	ADJ
ejpam-3461	205	22	right	right	ADJ
ejpam-3461	205	23	ideals	ideal	NOUN
ejpam-3461	205	24	.	.	PUNCT
ejpam-3461	206	1	if	if	SCONJ
ejpam-3461	206	2	for	for	ADP
ejpam-3461	206	3	any	any	DET
ejpam-3461	206	4	finitely	finitely	ADV
ejpam-3461	206	5	generated	generate	VERB
ejpam-3461	206	6	right	right	ADJ
ejpam-3461	206	7	ideal	ideal	NOUN
ejpam-3461	206	8	i	i	PRON
ejpam-3461	206	9	⊆	⊆	NUM
ejpam-3461	206	10	s	s	NOUN
ejpam-3461	206	11	,	,	PUNCT
ejpam-3461	206	12	r(keri	r(keri	NOUN
ejpam-3461	206	13	)	)	PUNCT
ejpam-3461	207	1	=	=	SYM
ejpam-3461	207	2	i	i	PROPN
ejpam-3461	207	3	,	,	PUNCT
ejpam-3461	207	4	then	then	ADV
ejpam-3461	207	5	has	have	VERB
ejpam-3461	207	6	finite	finite	ADJ
ejpam-3461	207	7	uniform	uniform	ADJ
ejpam-3461	207	8	dimension	dimension	NOUN
ejpam-3461	207	9	.	.	PUNCT
ejpam-3461	208	1	proof	proof	NOUN
ejpam-3461	208	2	.	.	PUNCT
ejpam-3461	209	1	in	in	ADP
ejpam-3461	209	2	view	view	NOUN
ejpam-3461	209	3	of	of	ADP
ejpam-3461	209	4	proposition	proposition	NOUN
ejpam-3461	209	5	6.30	6.30	NUM
ejpam-3461	209	6	in	in	ADP
ejpam-3461	209	7	[	[	X
ejpam-3461	209	8	12	12	NUM
ejpam-3461	209	9	]	]	PUNCT
ejpam-3461	209	10	,	,	PUNCT
ejpam-3461	209	11	we	we	PRON
ejpam-3461	209	12	need	need	VERB
ejpam-3461	209	13	to	to	PART
ejpam-3461	209	14	show	show	VERB
ejpam-3461	209	15	that	that	SCONJ
ejpam-3461	209	16	the	the	DET
ejpam-3461	209	17	complements	complement	NOUN
ejpam-3461	209	18	in	in	ADP
ejpam-3461	209	19	m	m	PROPN
ejpam-3461	209	20	satisfy	satisfy	PROPN
ejpam-3461	209	21	acc	acc	PROPN
ejpam-3461	209	22	.	.	PUNCT
ejpam-3461	210	1	now	now	ADV
ejpam-3461	210	2	,	,	PUNCT
ejpam-3461	210	3	let	let	VERB
ejpam-3461	210	4	c1	c1	PROPN
ejpam-3461	210	5	⊆	⊆	NUM
ejpam-3461	210	6	c2	c2	PROPN
ejpam-3461	210	7	⊆	⊆	NUM
ejpam-3461	210	8	....	....	PUNCT
ejpam-3461	210	9	be	be	AUX
ejpam-3461	210	10	an	an	DET
ejpam-3461	210	11	ascending	ascend	VERB
ejpam-3461	210	12	chain	chain	NOUN
ejpam-3461	210	13	of	of	ADP
ejpam-3461	210	14	complement	complement	NOUN
ejpam-3461	210	15	submodules	submodule	NOUN
ejpam-3461	210	16	of	of	ADP
ejpam-3461	210	17	m	m	PROPN
ejpam-3461	210	18	.	.	PUNCT
ejpam-3461	211	1	by	by	ADP
ejpam-3461	211	2	the	the	DET
ejpam-3461	211	3	c	c	PROPN
ejpam-3461	211	4	-	-	PUNCT
ejpam-3461	211	5	co	co	NOUN
ejpam-3461	211	6	-	-	ADJ
ejpam-3461	211	7	epi	epi	ADJ
ejpam-3461	211	8	-	-	ADJ
ejpam-3461	211	9	retractable	retractable	ADJ
ejpam-3461	211	10	condition	condition	NOUN
ejpam-3461	211	11	on	on	ADP
ejpam-3461	211	12	m	m	PROPN
ejpam-3461	211	13	,	,	PUNCT
ejpam-3461	211	14	there	there	PRON
ejpam-3461	211	15	is	be	VERB
ejpam-3461	211	16	fi	fi	NOUN
ejpam-3461	211	17	∈	∈	NOUN
ejpam-3461	211	18	s	s	NOUN
ejpam-3461	211	19	such	such	ADJ
ejpam-3461	211	20	that	that	SCONJ
ejpam-3461	211	21	each	each	DET
ejpam-3461	211	22	ci	ci	NOUN
ejpam-3461	211	23	is	be	AUX
ejpam-3461	211	24	of	of	ADP
ejpam-3461	211	25	the	the	DET
ejpam-3461	211	26	form	form	NOUN
ejpam-3461	211	27	kerfi	kerfi	NOUN
ejpam-3461	211	28	=	=	SYM
ejpam-3461	211	29	kerfis	kerfis	PROPN
ejpam-3461	211	30	.	.	PUNCT
ejpam-3461	212	1	with	with	ADP
ejpam-3461	212	2	applying	apply	VERB
ejpam-3461	212	3	r(−	r(−	PROPN
ejpam-3461	212	4	)	)	PUNCT
ejpam-3461	212	5	to	to	ADP
ejpam-3461	212	6	this	this	DET
ejpam-3461	212	7	chain	chain	NOUN
ejpam-3461	212	8	,	,	PUNCT
ejpam-3461	212	9	we	we	PRON
ejpam-3461	212	10	see	see	VERB
ejpam-3461	212	11	that	that	SCONJ
ejpam-3461	212	12	f1s	f1s	ADJ
ejpam-3461	212	13	⊇	⊇	PROPN
ejpam-3461	212	14	f2s	f2s	PROPN
ejpam-3461	212	15	⊇	⊇	NOUN
ejpam-3461	212	16	.....	.....	PUNCT
ejpam-3461	212	17	by	by	ADP
ejpam-3461	212	18	our	our	PRON
ejpam-3461	212	19	assumption	assumption	NOUN
ejpam-3461	212	20	,	,	PUNCT
ejpam-3461	212	21	there	there	PRON
ejpam-3461	212	22	is	be	VERB
ejpam-3461	212	23	some	some	PRON
ejpam-3461	212	24	n	n	ADP
ejpam-3461	212	25	such	such	ADJ
ejpam-3461	212	26	that	that	DET
ejpam-3461	212	27	fis	fis	PROPN
ejpam-3461	212	28	=	=	SYM
ejpam-3461	212	29	fns	fns	PROPN
ejpam-3461	212	30	for	for	ADP
ejpam-3461	212	31	all	all	PRON
ejpam-3461	212	32	i	i	PRON
ejpam-3461	212	33	≥	≥	VERB
ejpam-3461	212	34	n.	n.	NOUN
ejpam-3461	212	35	hence	hence	ADV
ejpam-3461	212	36	,	,	PUNCT
ejpam-3461	212	37	ci	ci	PROPN
ejpam-3461	212	38	=	=	PUNCT
ejpam-3461	212	39	cn	cn	PROPN
ejpam-3461	212	40	for	for	ADP
ejpam-3461	212	41	all	all	PRON
ejpam-3461	212	42	i	i	PRON
ejpam-3461	212	43	≥	≥	VERB
ejpam-3461	212	44	n.	n.	NOUN
ejpam-3461	212	45	therefore	therefore	ADV
ejpam-3461	212	46	,	,	PUNCT
ejpam-3461	212	47	m	m	PROPN
ejpam-3461	212	48	has	have	VERB
ejpam-3461	212	49	finite	finite	ADJ
ejpam-3461	212	50	uniform	uniform	ADJ
ejpam-3461	212	51	dimension	dimension	NOUN
ejpam-3461	212	52	.	.	PUNCT
ejpam-3461	213	1	recall	recall	VERB
ejpam-3461	213	2	that	that	SCONJ
ejpam-3461	213	3	an	an	DET
ejpam-3461	213	4	r	r	NOUN
ejpam-3461	213	5	-	-	PUNCT
ejpam-3461	213	6	module	module	NOUN
ejpam-3461	213	7	m	m	NOUN
ejpam-3461	213	8	is	be	AUX
ejpam-3461	213	9	said	say	VERB
ejpam-3461	213	10	to	to	PART
ejpam-3461	213	11	be	be	AUX
ejpam-3461	213	12	have	have	VERB
ejpam-3461	213	13	the	the	DET
ejpam-3461	213	14	summand	summand	NOUN
ejpam-3461	213	15	sum	sum	NOUN
ejpam-3461	213	16	property	property	NOUN
ejpam-3461	213	17	(	(	PUNCT
ejpam-3461	213	18	ssp	ssp	NOUN
ejpam-3461	213	19	,	,	PUNCT
ejpam-3461	213	20	for	for	ADP
ejpam-3461	213	21	short	short	ADJ
ejpam-3461	213	22	)	)	PUNCT
ejpam-3461	213	23	if	if	SCONJ
ejpam-3461	213	24	,	,	PUNCT
ejpam-3461	213	25	the	the	DET
ejpam-3461	213	26	sum	sum	NOUN
ejpam-3461	213	27	of	of	ADP
ejpam-3461	213	28	any	any	DET
ejpam-3461	213	29	two	two	NUM
ejpam-3461	213	30	direct	direct	ADJ
ejpam-3461	213	31	summands	summand	NOUN
ejpam-3461	213	32	of	of	ADP
ejpam-3461	213	33	m	m	VERB
ejpam-3461	213	34	is	be	AUX
ejpam-3461	213	35	again	again	ADV
ejpam-3461	213	36	a	a	DET
ejpam-3461	213	37	direct	direct	ADJ
ejpam-3461	213	38	summand	summand	NOUN
ejpam-3461	213	39	of	of	ADP
ejpam-3461	213	40	m	m	PROPN
ejpam-3461	213	41	.	.	PUNCT
ejpam-3461	214	1	corollary	corollary	ADJ
ejpam-3461	214	2	6	6	NUM
ejpam-3461	214	3	.	.	PUNCT
ejpam-3461	215	1	let	let	VERB
ejpam-3461	215	2	m	m	PRON
ejpam-3461	215	3	be	be	AUX
ejpam-3461	215	4	a	a	DET
ejpam-3461	215	5	quasi	quasi	ADJ
ejpam-3461	215	6	-	-	ADJ
ejpam-3461	215	7	injective	injective	ADJ
ejpam-3461	215	8	r	r	NOUN
ejpam-3461	215	9	-	-	PUNCT
ejpam-3461	215	10	module	module	NOUN
ejpam-3461	215	11	such	such	ADJ
ejpam-3461	215	12	that	that	SCONJ
ejpam-3461	215	13	r⊕m	r⊕m	NOUN
ejpam-3461	215	14	has	have	VERB
ejpam-3461	215	15	the	the	DET
ejpam-3461	215	16	ssp	ssp	NOUN
ejpam-3461	215	17	.	.	PUNCT
ejpam-3461	216	1	assume	assume	VERB
ejpam-3461	216	2	that	that	SCONJ
ejpam-3461	216	3	s	s	AUX
ejpam-3461	216	4	satisfies	satisfy	VERB
ejpam-3461	216	5	dcc	dcc	PROPN
ejpam-3461	216	6	for	for	ADP
ejpam-3461	216	7	cyclic	cyclic	ADJ
ejpam-3461	216	8	right	right	ADJ
ejpam-3461	216	9	ideals	ideal	NOUN
ejpam-3461	216	10	.	.	PUNCT
ejpam-3461	217	1	then	then	ADV
ejpam-3461	217	2	m	m	PROPN
ejpam-3461	217	3	is	be	AUX
ejpam-3461	217	4	finitely	finitely	ADV
ejpam-3461	217	5	generated	generate	VERB
ejpam-3461	217	6	semi	semi	ADJ
ejpam-3461	217	7	-	-	ADJ
ejpam-3461	217	8	simple	simple	ADJ
ejpam-3461	217	9	.	.	PUNCT
ejpam-3461	218	1	a.	a.	PROPN
ejpam-3461	218	2	d.	d.	PROPN
ejpam-3461	218	3	diallo	diallo	PROPN
ejpam-3461	218	4	,	,	PUNCT
ejpam-3461	218	5	p.	p.	PROPN
ejpam-3461	218	6	c.	c.	PROPN
ejpam-3461	218	7	diop	diop	PROPN
ejpam-3461	218	8	,	,	PUNCT
ejpam-3461	218	9	m.	m.	NOUN
ejpam-3461	218	10	barry	barry	PROPN
ejpam-3461	218	11	/	/	SYM
ejpam-3461	218	12	eur	eur	PROPN
ejpam-3461	218	13	.	.	PUNCT
ejpam-3461	219	1	j.	j.	PROPN
ejpam-3461	219	2	pure	pure	PROPN
ejpam-3461	219	3	appl	appl	PROPN
ejpam-3461	219	4	.	.	PROPN
ejpam-3461	219	5	math	math	PROPN
ejpam-3461	219	6	,	,	PUNCT
ejpam-3461	219	7	12	12	NUM
ejpam-3461	219	8	(	(	PUNCT
ejpam-3461	219	9	3	3	NUM
ejpam-3461	219	10	)	)	PUNCT
ejpam-3461	219	11	(	(	PUNCT
ejpam-3461	219	12	2019	2019	NUM
ejpam-3461	219	13	)	)	PUNCT
ejpam-3461	219	14	,	,	PUNCT
ejpam-3461	219	15	1187	1187	NUM
ejpam-3461	219	16	-	-	SYM
ejpam-3461	219	17	1198	1198	NUM
ejpam-3461	219	18	1193	1193	NUM
ejpam-3461	219	19	proof	proof	NOUN
ejpam-3461	219	20	.	.	PUNCT
ejpam-3461	220	1	since	since	SCONJ
ejpam-3461	220	2	r⊕m	r⊕m	NOUN
ejpam-3461	220	3	has	have	VERB
ejpam-3461	220	4	the	the	DET
ejpam-3461	220	5	ssp	ssp	NOUN
ejpam-3461	220	6	,	,	PUNCT
ejpam-3461	220	7	we	we	PRON
ejpam-3461	220	8	infer	infer	VERB
ejpam-3461	220	9	from	from	ADP
ejpam-3461	220	10	proposition	proposition	NOUN
ejpam-3461	220	11	3.4	3.4	NUM
ejpam-3461	220	12	in	in	ADP
ejpam-3461	220	13	[	[	X
ejpam-3461	220	14	9	9	NUM
ejpam-3461	220	15	]	]	PUNCT
ejpam-3461	220	16	that	that	SCONJ
ejpam-3461	220	17	every	every	DET
ejpam-3461	220	18	cyclic	cyclic	ADJ
ejpam-3461	220	19	submodule	submodule	NOUN
ejpam-3461	220	20	of	of	ADP
ejpam-3461	220	21	m	m	PROPN
ejpam-3461	220	22	is	be	AUX
ejpam-3461	220	23	a	a	DET
ejpam-3461	220	24	direct	direct	ADJ
ejpam-3461	220	25	summand	summand	NOUN
ejpam-3461	220	26	of	of	ADP
ejpam-3461	220	27	m	m	PROPN
ejpam-3461	220	28	.	.	PUNCT
ejpam-3461	221	1	since	since	SCONJ
ejpam-3461	221	2	m	m	PROPN
ejpam-3461	221	3	is	be	AUX
ejpam-3461	221	4	quasi	quasi	ADJ
ejpam-3461	221	5	-	-	ADJ
ejpam-3461	221	6	injective	injective	ADJ
ejpam-3461	221	7	,	,	PUNCT
ejpam-3461	221	8	r(keri	r(keri	NOUN
ejpam-3461	221	9	)	)	PUNCT
ejpam-3461	222	1	=	=	PUNCT
ejpam-3461	222	2	i	i	PRON
ejpam-3461	222	3	for	for	ADP
ejpam-3461	222	4	any	any	DET
ejpam-3461	222	5	finitely	finitely	ADV
ejpam-3461	222	6	generated	generate	VERB
ejpam-3461	222	7	right	right	ADJ
ejpam-3461	222	8	ideal	ideal	NOUN
ejpam-3461	222	9	i	i	PRON
ejpam-3461	222	10	⊆	⊆	NUM
ejpam-3461	222	11	s	s	X
ejpam-3461	222	12	by	by	X
ejpam-3461	222	13	(	(	PUNCT
ejpam-3461	222	14	[	[	X
ejpam-3461	222	15	19	19	NUM
ejpam-3461	222	16	]	]	X
ejpam-3461	222	17	,	,	PUNCT
ejpam-3461	222	18	28.1	28.1	NUM
ejpam-3461	222	19	)	)	PUNCT
ejpam-3461	222	20	.	.	PUNCT
ejpam-3461	223	1	but	but	CCONJ
ejpam-3461	223	2	m	m	PROPN
ejpam-3461	223	3	is	be	AUX
ejpam-3461	223	4	c	c	NOUN
ejpam-3461	223	5	-	-	PUNCT
ejpam-3461	223	6	co	co	ADJ
ejpam-3461	223	7	-	-	ADJ
ejpam-3461	223	8	epi	epi	NOUN
ejpam-3461	223	9	-	-	NOUN
ejpam-3461	223	10	retractable	retractable	ADJ
ejpam-3461	223	11	.	.	PUNCT
ejpam-3461	224	1	then	then	ADV
ejpam-3461	224	2	,	,	PUNCT
ejpam-3461	224	3	according	accord	VERB
ejpam-3461	224	4	to	to	ADP
ejpam-3461	224	5	proposition	proposition	NOUN
ejpam-3461	224	6	7	7	NUM
ejpam-3461	224	7	,	,	PUNCT
ejpam-3461	224	8	m	m	VERB
ejpam-3461	224	9	has	have	VERB
ejpam-3461	224	10	finite	finite	ADJ
ejpam-3461	224	11	uniform	uniform	ADJ
ejpam-3461	224	12	dimension	dimension	NOUN
ejpam-3461	224	13	.	.	PUNCT
ejpam-3461	225	1	therefore	therefore	ADV
ejpam-3461	225	2	,	,	PUNCT
ejpam-3461	225	3	m	m	VERB
ejpam-3461	225	4	is	be	AUX
ejpam-3461	225	5	finitely	finitely	ADV
ejpam-3461	225	6	generated	generate	VERB
ejpam-3461	225	7	semisimple	semisimple	NOUN
ejpam-3461	225	8	.	.	PUNCT
ejpam-3461	226	1	corollary	corollary	ADJ
ejpam-3461	226	2	7	7	NUM
ejpam-3461	226	3	.	.	PUNCT
ejpam-3461	227	1	if	if	SCONJ
ejpam-3461	227	2	m	m	NOUN
ejpam-3461	227	3	is	be	AUX
ejpam-3461	227	4	a	a	DET
ejpam-3461	227	5	quasi	quasi	ADJ
ejpam-3461	227	6	-	-	ADJ
ejpam-3461	227	7	injective	injective	ADJ
ejpam-3461	227	8	r	r	NOUN
ejpam-3461	227	9	-	-	PUNCT
ejpam-3461	227	10	module	module	NOUN
ejpam-3461	227	11	such	such	ADJ
ejpam-3461	227	12	that	that	DET
ejpam-3461	227	13	s	s	PART
ejpam-3461	227	14	satisfies	satisfy	VERB
ejpam-3461	227	15	dcc	dcc	PROPN
ejpam-3461	227	16	for	for	ADP
ejpam-3461	227	17	cyclic	cyclic	ADJ
ejpam-3461	227	18	right	right	ADJ
ejpam-3461	227	19	ideals	ideal	NOUN
ejpam-3461	227	20	,	,	PUNCT
ejpam-3461	227	21	then	then	ADV
ejpam-3461	227	22	m	m	VERB
ejpam-3461	227	23	is	be	AUX
ejpam-3461	227	24	a	a	DET
ejpam-3461	227	25	finite	finite	ADJ
ejpam-3461	227	26	direct	direct	ADJ
ejpam-3461	227	27	sum	sum	NOUN
ejpam-3461	227	28	of	of	ADP
ejpam-3461	227	29	uniform	uniform	ADJ
ejpam-3461	227	30	submodules	submodule	NOUN
ejpam-3461	227	31	.	.	PUNCT
ejpam-3461	228	1	proof	proof	NOUN
ejpam-3461	228	2	.	.	PUNCT
ejpam-3461	229	1	suppose	suppose	VERB
ejpam-3461	229	2	m	m	PRON
ejpam-3461	229	3	is	be	AUX
ejpam-3461	229	4	quasi	quasi	ADJ
ejpam-3461	229	5	-	-	ADJ
ejpam-3461	229	6	injective	injective	ADJ
ejpam-3461	229	7	such	such	ADJ
ejpam-3461	229	8	that	that	SCONJ
ejpam-3461	229	9	s	s	PROPN
ejpam-3461	229	10	satisfies	satisfy	VERB
ejpam-3461	229	11	dcc	dcc	PROPN
ejpam-3461	229	12	for	for	ADP
ejpam-3461	229	13	cyclic	cyclic	ADJ
ejpam-3461	229	14	right	right	ADJ
ejpam-3461	229	15	ideals	ideal	NOUN
ejpam-3461	229	16	.	.	PUNCT
ejpam-3461	230	1	thus	thus	ADV
ejpam-3461	230	2	by	by	ADP
ejpam-3461	230	3	(	(	PUNCT
ejpam-3461	230	4	[	[	X
ejpam-3461	230	5	19	19	NUM
ejpam-3461	230	6	]	]	PUNCT
ejpam-3461	230	7	,	,	PUNCT
ejpam-3461	230	8	28.1	28.1	NUM
ejpam-3461	230	9	)	)	PUNCT
ejpam-3461	230	10	,	,	PUNCT
ejpam-3461	230	11	r(keri	r(keri	NOUN
ejpam-3461	230	12	)	)	PUNCT
ejpam-3461	231	1	=	=	PUNCT
ejpam-3461	231	2	i	i	PRON
ejpam-3461	231	3	for	for	ADP
ejpam-3461	231	4	any	any	DET
ejpam-3461	231	5	finitely	finitely	ADV
ejpam-3461	231	6	generated	generate	VERB
ejpam-3461	231	7	right	right	ADJ
ejpam-3461	231	8	ideal	ideal	NOUN
ejpam-3461	231	9	i	i	PRON
ejpam-3461	231	10	⊆	⊆	NUM
ejpam-3461	231	11	s.	s.	PROPN
ejpam-3461	231	12	therefore	therefore	ADV
ejpam-3461	231	13	,	,	PUNCT
ejpam-3461	231	14	according	accord	VERB
ejpam-3461	231	15	to	to	ADP
ejpam-3461	231	16	proposition	proposition	NOUN
ejpam-3461	231	17	7	7	NUM
ejpam-3461	231	18	,	,	PUNCT
ejpam-3461	231	19	m	m	VERB
ejpam-3461	231	20	is	be	AUX
ejpam-3461	231	21	a	a	DET
ejpam-3461	231	22	finite	finite	ADJ
ejpam-3461	231	23	direct	direct	ADJ
ejpam-3461	231	24	sum	sum	NOUN
ejpam-3461	231	25	of	of	ADP
ejpam-3461	231	26	uniform	uniform	ADJ
ejpam-3461	231	27	submodules	submodule	NOUN
ejpam-3461	231	28	.	.	PUNCT
ejpam-3461	232	1	corollary	corollary	ADJ
ejpam-3461	232	2	8	8	NUM
ejpam-3461	232	3	.	.	PUNCT
ejpam-3461	233	1	let	let	VERB
ejpam-3461	233	2	r	r	PRON
ejpam-3461	233	3	be	be	AUX
ejpam-3461	233	4	a	a	DET
ejpam-3461	233	5	right	right	ADJ
ejpam-3461	233	6	self	self	NOUN
ejpam-3461	233	7	-	-	PUNCT
ejpam-3461	233	8	injective	injective	ADJ
ejpam-3461	233	9	ring	ring	NOUN
ejpam-3461	233	10	and	and	CCONJ
ejpam-3461	233	11	m	m	VERB
ejpam-3461	233	12	a	a	DET
ejpam-3461	233	13	nonsingular	nonsingular	ADJ
ejpam-3461	233	14	r	r	NOUN
ejpam-3461	233	15	-	-	PUNCT
ejpam-3461	233	16	module	module	NOUN
ejpam-3461	233	17	such	such	ADJ
ejpam-3461	233	18	that	that	DET
ejpam-3461	233	19	s	s	PART
ejpam-3461	233	20	satisfies	satisfy	VERB
ejpam-3461	233	21	dcc	dcc	PROPN
ejpam-3461	233	22	for	for	ADP
ejpam-3461	233	23	cyclic	cyclic	ADJ
ejpam-3461	233	24	right	right	ADJ
ejpam-3461	233	25	ideals	ideal	NOUN
ejpam-3461	233	26	.	.	PUNCT
ejpam-3461	234	1	then	then	ADV
ejpam-3461	234	2	the	the	DET
ejpam-3461	234	3	following	follow	VERB
ejpam-3461	234	4	conditions	condition	NOUN
ejpam-3461	234	5	are	be	AUX
ejpam-3461	234	6	equivalent	equivalent	ADJ
ejpam-3461	234	7	.	.	PUNCT
ejpam-3461	235	1	(	(	PUNCT
ejpam-3461	235	2	1	1	X
ejpam-3461	235	3	)	)	PUNCT
ejpam-3461	235	4	m	m	VERB
ejpam-3461	235	5	is	be	AUX
ejpam-3461	235	6	quasi	quasi	ADJ
ejpam-3461	235	7	-	-	ADJ
ejpam-3461	235	8	injective	injective	ADJ
ejpam-3461	235	9	.	.	PUNCT
ejpam-3461	236	1	(	(	PUNCT
ejpam-3461	236	2	2	2	X
ejpam-3461	236	3	)	)	PUNCT
ejpam-3461	236	4	m	m	VERB
ejpam-3461	236	5	is	be	AUX
ejpam-3461	236	6	semi	semi	ADJ
ejpam-3461	236	7	-	-	ADJ
ejpam-3461	236	8	simple	simple	ADJ
ejpam-3461	236	9	injective	injective	NOUN
ejpam-3461	236	10	.	.	PUNCT
ejpam-3461	237	1	recall	recall	VERB
ejpam-3461	237	2	that	that	SCONJ
ejpam-3461	237	3	an	an	DET
ejpam-3461	237	4	r	r	NOUN
ejpam-3461	237	5	-	-	PUNCT
ejpam-3461	237	6	module	module	NOUN
ejpam-3461	237	7	m	m	NOUN
ejpam-3461	237	8	is	be	AUX
ejpam-3461	237	9	called	call	VERB
ejpam-3461	237	10	compressible	compressible	ADJ
ejpam-3461	237	11	if	if	SCONJ
ejpam-3461	237	12	for	for	SCONJ
ejpam-3461	237	13	every	every	DET
ejpam-3461	237	14	nonzero	nonzero	PROPN
ejpam-3461	237	15	submodule	submodule	PROPN
ejpam-3461	237	16	n	n	PROPN
ejpam-3461	237	17	of	of	ADP
ejpam-3461	237	18	m	m	PRON
ejpam-3461	237	19	there	there	PRON
ejpam-3461	237	20	is	be	VERB
ejpam-3461	237	21	a	a	DET
ejpam-3461	237	22	monomorphism	monomorphism	NOUN
ejpam-3461	237	23	f	f	X
ejpam-3461	237	24	:	:	PUNCT
ejpam-3461	237	25	m	m	VERB
ejpam-3461	237	26	−→	−→	ADJ
ejpam-3461	237	27	n	n	ADV
ejpam-3461	237	28	.	.	PUNCT
ejpam-3461	238	1	proposition	proposition	NOUN
ejpam-3461	238	2	8	8	NUM
ejpam-3461	238	3	.	.	PUNCT
ejpam-3461	239	1	an	an	DET
ejpam-3461	239	2	r	r	NOUN
ejpam-3461	239	3	-	-	PUNCT
ejpam-3461	239	4	module	module	NOUN
ejpam-3461	239	5	is	be	AUX
ejpam-3461	239	6	simple	simple	ADJ
ejpam-3461	239	7	if	if	SCONJ
ejpam-3461	240	1	and	and	CCONJ
ejpam-3461	240	2	only	only	ADV
ejpam-3461	240	3	if	if	SCONJ
ejpam-3461	240	4	it	it	PRON
ejpam-3461	240	5	is	be	AUX
ejpam-3461	240	6	compressible	compressible	ADJ
ejpam-3461	240	7	c	c	NOUN
ejpam-3461	240	8	-	-	PUNCT
ejpam-3461	240	9	co	co	NOUN
ejpam-3461	240	10	-	-	ADJ
ejpam-3461	240	11	epi	epi	ADJ
ejpam-3461	240	12	-	-	NOUN
ejpam-3461	240	13	retractable	retractable	ADJ
ejpam-3461	240	14	and	and	CCONJ
ejpam-3461	240	15	contains	contain	VERB
ejpam-3461	240	16	a	a	DET
ejpam-3461	240	17	maximal	maximal	ADJ
ejpam-3461	240	18	complement	complement	NOUN
ejpam-3461	240	19	submodule	submodule	NOUN
ejpam-3461	240	20	.	.	PUNCT
ejpam-3461	241	1	proof	proof	NOUN
ejpam-3461	241	2	.	.	PUNCT
ejpam-3461	242	1	the	the	DET
ejpam-3461	242	2	necessity	necessity	NOUN
ejpam-3461	242	3	is	be	AUX
ejpam-3461	242	4	clear	clear	ADJ
ejpam-3461	242	5	.	.	PUNCT
ejpam-3461	243	1	conversely	conversely	ADV
ejpam-3461	243	2	,	,	PUNCT
ejpam-3461	243	3	assume	assume	VERB
ejpam-3461	243	4	that	that	SCONJ
ejpam-3461	243	5	m	m	NOUN
ejpam-3461	243	6	is	be	AUX
ejpam-3461	243	7	compressible	compressible	ADJ
ejpam-3461	243	8	c	c	X
ejpam-3461	243	9	-	-	ADJ
ejpam-3461	243	10	co	co	NOUN
ejpam-3461	243	11	-	-	ADJ
ejpam-3461	243	12	epi	epi	ADJ
ejpam-3461	243	13	-	-	NOUN
ejpam-3461	243	14	retractable	retractable	ADJ
ejpam-3461	243	15	and	and	CCONJ
ejpam-3461	243	16	contains	contain	VERB
ejpam-3461	243	17	a	a	DET
ejpam-3461	243	18	maximal	maximal	ADJ
ejpam-3461	243	19	complement	complement	NOUN
ejpam-3461	243	20	submodule	submodule	NOUN
ejpam-3461	243	21	c.	c.	PROPN
ejpam-3461	243	22	then	then	ADV
ejpam-3461	243	23	there	there	PRON
ejpam-3461	243	24	is	be	VERB
ejpam-3461	243	25	a	a	DET
ejpam-3461	243	26	submodule	submodule	NOUN
ejpam-3461	243	27	n	n	PROPN
ejpam-3461	243	28	of	of	ADP
ejpam-3461	243	29	m	m	PRON
ejpam-3461	243	30	such	such	ADJ
ejpam-3461	243	31	that	that	SCONJ
ejpam-3461	243	32	m	m	NOUN
ejpam-3461	243	33	/	/	SYM
ejpam-3461	243	34	c	c	NOUN
ejpam-3461	243	35	∼=	∼=	PROPN
ejpam-3461	243	36	n	n	NOUN
ejpam-3461	243	37	.	.	PUNCT
ejpam-3461	244	1	hence	hence	ADV
ejpam-3461	244	2	,	,	PUNCT
ejpam-3461	244	3	n	n	PRON
ejpam-3461	244	4	is	be	AUX
ejpam-3461	244	5	simple	simple	ADJ
ejpam-3461	244	6	.	.	PUNCT
ejpam-3461	245	1	by	by	ADP
ejpam-3461	245	2	the	the	DET
ejpam-3461	245	3	compressible	compressible	ADJ
ejpam-3461	245	4	condition	condition	NOUN
ejpam-3461	245	5	on	on	ADP
ejpam-3461	245	6	m	m	PROPN
ejpam-3461	245	7	,	,	PUNCT
ejpam-3461	245	8	there	there	PRON
ejpam-3461	245	9	is	be	VERB
ejpam-3461	245	10	a	a	DET
ejpam-3461	245	11	monomorphism	monomorphism	NOUN
ejpam-3461	245	12	f	f	X
ejpam-3461	245	13	:	:	PUNCT
ejpam-3461	245	14	m	m	VERB
ejpam-3461	245	15	−→	−→	ADJ
ejpam-3461	245	16	n	n	ADV
ejpam-3461	245	17	.	.	PUNCT
ejpam-3461	246	1	thus	thus	ADV
ejpam-3461	246	2	,	,	PUNCT
ejpam-3461	246	3	m	m	VERB
ejpam-3461	246	4	is	be	AUX
ejpam-3461	246	5	isomorphic	isomorphic	ADJ
ejpam-3461	246	6	to	to	ADP
ejpam-3461	246	7	a	a	DET
ejpam-3461	246	8	submodule	submodule	NOUN
ejpam-3461	246	9	of	of	ADP
ejpam-3461	246	10	m	m	PROPN
ejpam-3461	246	11	.	.	PUNCT
ejpam-3461	247	1	as	as	SCONJ
ejpam-3461	247	2	f	f	PROPN
ejpam-3461	247	3	6=	6=	PROPN
ejpam-3461	247	4	0	0	NUM
ejpam-3461	247	5	,	,	PUNCT
ejpam-3461	247	6	m	m	VERB
ejpam-3461	247	7	=	=	SYM
ejpam-3461	247	8	n	n	ADJ
ejpam-3461	247	9	,	,	PUNCT
ejpam-3461	247	10	and	and	CCONJ
ejpam-3461	247	11	so	so	ADV
ejpam-3461	247	12	m	m	VERB
ejpam-3461	247	13	is	be	AUX
ejpam-3461	247	14	simple	simple	ADJ
ejpam-3461	247	15	.	.	PUNCT
ejpam-3461	248	1	corollary	corollary	ADJ
ejpam-3461	248	2	9	9	NUM
ejpam-3461	248	3	.	.	PUNCT
ejpam-3461	249	1	an	an	DET
ejpam-3461	249	2	r	r	NOUN
ejpam-3461	249	3	-	-	PUNCT
ejpam-3461	249	4	module	module	NOUN
ejpam-3461	249	5	is	be	AUX
ejpam-3461	249	6	simple	simple	ADJ
ejpam-3461	249	7	if	if	SCONJ
ejpam-3461	250	1	and	and	CCONJ
ejpam-3461	250	2	only	only	ADV
ejpam-3461	250	3	if	if	SCONJ
ejpam-3461	250	4	it	it	PRON
ejpam-3461	250	5	is	be	AUX
ejpam-3461	250	6	compressible	compressible	ADJ
ejpam-3461	250	7	finitely	finitely	ADV
ejpam-3461	250	8	generayed	generaye	VERB
ejpam-3461	250	9	c	c	NOUN
ejpam-3461	250	10	-	-	PUNCT
ejpam-3461	250	11	co	co	NOUN
ejpam-3461	250	12	-	-	ADJ
ejpam-3461	250	13	epi	epi	NOUN
ejpam-3461	250	14	-	-	NOUN
ejpam-3461	250	15	retractable	retractable	ADJ
ejpam-3461	250	16	.	.	PUNCT
ejpam-3461	251	1	recall	recall	VERB
ejpam-3461	251	2	that	that	SCONJ
ejpam-3461	251	3	an	an	DET
ejpam-3461	251	4	r	r	NOUN
ejpam-3461	251	5	-	-	PUNCT
ejpam-3461	251	6	module	module	NOUN
ejpam-3461	251	7	m	m	NOUN
ejpam-3461	251	8	is	be	AUX
ejpam-3461	251	9	cohereditary	cohereditary	ADJ
ejpam-3461	251	10	if	if	SCONJ
ejpam-3461	251	11	every	every	DET
ejpam-3461	251	12	factor	factor	NOUN
ejpam-3461	251	13	module	module	NOUN
ejpam-3461	251	14	of	of	ADP
ejpam-3461	251	15	m	m	PROPN
ejpam-3461	251	16	is	be	AUX
ejpam-3461	251	17	injective	injective	ADJ
ejpam-3461	251	18	.	.	PUNCT
ejpam-3461	252	1	now	now	ADV
ejpam-3461	252	2	,	,	PUNCT
ejpam-3461	252	3	let	let	VERB
ejpam-3461	252	4	us	we	PRON
ejpam-3461	252	5	introduce	introduce	VERB
ejpam-3461	252	6	the	the	DET
ejpam-3461	252	7	following	following	ADJ
ejpam-3461	252	8	notion	notion	NOUN
ejpam-3461	252	9	.	.	PUNCT
ejpam-3461	253	1	definition	definition	NOUN
ejpam-3461	253	2	4	4	NUM
ejpam-3461	253	3	.	.	PUNCT
ejpam-3461	254	1	an	an	DET
ejpam-3461	254	2	r	r	NOUN
ejpam-3461	254	3	-	-	PUNCT
ejpam-3461	254	4	module	module	NOUN
ejpam-3461	254	5	module	module	NOUN
ejpam-3461	254	6	is	be	AUX
ejpam-3461	254	7	called	call	VERB
ejpam-3461	254	8	c	c	NOUN
ejpam-3461	254	9	-	-	ADJ
ejpam-3461	254	10	cohereditary	cohereditary	ADJ
ejpam-3461	254	11	if	if	SCONJ
ejpam-3461	254	12	m	m	PROPN
ejpam-3461	254	13	/	/	SYM
ejpam-3461	254	14	c	c	PROPN
ejpam-3461	254	15	is	be	AUX
ejpam-3461	254	16	injective	injective	ADJ
ejpam-3461	254	17	for	for	ADP
ejpam-3461	254	18	each	each	DET
ejpam-3461	254	19	nonzero	nonzero	ADJ
ejpam-3461	254	20	proper	proper	ADJ
ejpam-3461	254	21	complement	complement	NOUN
ejpam-3461	254	22	submodule	submodule	NOUN
ejpam-3461	254	23	c	c	PROPN
ejpam-3461	254	24	of	of	ADP
ejpam-3461	254	25	m	m	PROPN
ejpam-3461	254	26	.	.	PUNCT
ejpam-3461	255	1	the	the	DET
ejpam-3461	255	2	ring	ring	NOUN
ejpam-3461	255	3	r	r	NOUN
ejpam-3461	255	4	is	be	AUX
ejpam-3461	255	5	called	call	VERB
ejpam-3461	255	6	c	c	NOUN
ejpam-3461	255	7	-	-	ADJ
ejpam-3461	255	8	cohereditary	cohereditary	ADJ
ejpam-3461	255	9	if	if	SCONJ
ejpam-3461	255	10	rr	rr	PROPN
ejpam-3461	255	11	is	be	AUX
ejpam-3461	255	12	c	c	NOUN
ejpam-3461	255	13	-	-	ADJ
ejpam-3461	255	14	cohereditary	cohereditary	ADJ
ejpam-3461	255	15	.	.	PUNCT
ejpam-3461	256	1	proposition	proposition	NOUN
ejpam-3461	256	2	9	9	NUM
ejpam-3461	256	3	.	.	PUNCT
ejpam-3461	257	1	a	a	DET
ejpam-3461	257	2	c	c	NOUN
ejpam-3461	257	3	-	-	ADJ
ejpam-3461	257	4	cohereditary	cohereditary	ADJ
ejpam-3461	257	5	c	c	NOUN
ejpam-3461	257	6	-	-	PUNCT
ejpam-3461	257	7	co	co	NOUN
ejpam-3461	257	8	-	-	ADJ
ejpam-3461	257	9	epi	epi	ADJ
ejpam-3461	257	10	-	-	ADJ
ejpam-3461	257	11	retractable	retractable	ADJ
ejpam-3461	257	12	r	r	NOUN
ejpam-3461	257	13	-	-	PUNCT
ejpam-3461	257	14	module	module	NOUN
ejpam-3461	257	15	m	m	NOUN
ejpam-3461	257	16	is	be	AUX
ejpam-3461	257	17	injective	injective	ADJ
ejpam-3461	257	18	.	.	PUNCT
ejpam-3461	258	1	moreover	moreover	ADV
ejpam-3461	258	2	,	,	PUNCT
ejpam-3461	258	3	m	m	PROPN
ejpam-3461	258	4	is	be	AUX
ejpam-3461	258	5	a	a	DET
ejpam-3461	258	6	direct	direct	ADJ
ejpam-3461	258	7	sum	sum	NOUN
ejpam-3461	258	8	of	of	ADP
ejpam-3461	258	9	a	a	DET
ejpam-3461	258	10	nonsingular	nonsingular	ADJ
ejpam-3461	258	11	module	module	NOUN
ejpam-3461	258	12	and	and	CCONJ
ejpam-3461	258	13	an	an	DET
ejpam-3461	258	14	injective	injective	ADJ
ejpam-3461	258	15	module	module	NOUN
ejpam-3461	258	16	.	.	PUNCT
ejpam-3461	259	1	a.	a.	PROPN
ejpam-3461	259	2	d.	d.	PROPN
ejpam-3461	259	3	diallo	diallo	PROPN
ejpam-3461	259	4	,	,	PUNCT
ejpam-3461	259	5	p.	p.	PROPN
ejpam-3461	259	6	c.	c.	PROPN
ejpam-3461	259	7	diop	diop	PROPN
ejpam-3461	259	8	,	,	PUNCT
ejpam-3461	259	9	m.	m.	NOUN
ejpam-3461	259	10	barry	barry	PROPN
ejpam-3461	259	11	/	/	SYM
ejpam-3461	259	12	eur	eur	PROPN
ejpam-3461	259	13	.	.	PUNCT
ejpam-3461	260	1	j.	j.	PROPN
ejpam-3461	260	2	pure	pure	PROPN
ejpam-3461	260	3	appl	appl	PROPN
ejpam-3461	260	4	.	.	PROPN
ejpam-3461	260	5	math	math	PROPN
ejpam-3461	260	6	,	,	PUNCT
ejpam-3461	260	7	12	12	NUM
ejpam-3461	260	8	(	(	PUNCT
ejpam-3461	260	9	3	3	NUM
ejpam-3461	260	10	)	)	PUNCT
ejpam-3461	260	11	(	(	PUNCT
ejpam-3461	260	12	2019	2019	NUM
ejpam-3461	260	13	)	)	PUNCT
ejpam-3461	260	14	,	,	PUNCT
ejpam-3461	260	15	1187	1187	NUM
ejpam-3461	260	16	-	-	SYM
ejpam-3461	260	17	1198	1198	NUM
ejpam-3461	260	18	1194	1194	NUM
ejpam-3461	260	19	proof	proof	NOUN
ejpam-3461	260	20	.	.	PUNCT
ejpam-3461	261	1	supposem	supposem	PROPN
ejpam-3461	261	2	is	be	AUX
ejpam-3461	261	3	c	c	NOUN
ejpam-3461	261	4	-	-	PUNCT
ejpam-3461	261	5	co	co	ADJ
ejpam-3461	261	6	-	-	ADJ
ejpam-3461	261	7	epi	epi	NOUN
ejpam-3461	261	8	-	-	NOUN
ejpam-3461	261	9	retractable	retractable	ADJ
ejpam-3461	261	10	.	.	PUNCT
ejpam-3461	262	1	it	it	PRON
ejpam-3461	262	2	is	be	AUX
ejpam-3461	262	3	well	well	ADV
ejpam-3461	262	4	known	know	VERB
ejpam-3461	262	5	that	that	SCONJ
ejpam-3461	262	6	z2(m	z2(m	X
ejpam-3461	262	7	)	)	PUNCT
ejpam-3461	262	8	is	be	AUX
ejpam-3461	262	9	a	a	DET
ejpam-3461	262	10	complement	complement	NOUN
ejpam-3461	262	11	submodule	submodule	NOUN
ejpam-3461	262	12	of	of	ADP
ejpam-3461	262	13	m	m	PROPN
ejpam-3461	262	14	.	.	PUNCT
ejpam-3461	263	1	by	by	ADP
ejpam-3461	263	2	the	the	DET
ejpam-3461	263	3	c	c	PROPN
ejpam-3461	263	4	-	-	PUNCT
ejpam-3461	263	5	co	co	NOUN
ejpam-3461	263	6	-	-	ADJ
ejpam-3461	263	7	epi	epi	ADJ
ejpam-3461	263	8	-	-	ADJ
ejpam-3461	263	9	retractable	retractable	ADJ
ejpam-3461	263	10	condition	condition	NOUN
ejpam-3461	263	11	on	on	ADP
ejpam-3461	263	12	m	m	PROPN
ejpam-3461	263	13	,	,	PUNCT
ejpam-3461	263	14	there	there	PRON
ejpam-3461	263	15	exists	exist	VERB
ejpam-3461	263	16	a	a	DET
ejpam-3461	263	17	nonzero	nonzero	PROPN
ejpam-3461	263	18	endomorphism	endomorphism	PROPN
ejpam-3461	263	19	f	f	PROPN
ejpam-3461	263	20	of	of	ADP
ejpam-3461	263	21	m	m	PRON
ejpam-3461	263	22	such	such	ADJ
ejpam-3461	263	23	that	that	DET
ejpam-3461	263	24	kerf	kerf	NOUN
ejpam-3461	263	25	=	=	SYM
ejpam-3461	263	26	z2(m	z2(m	NOUN
ejpam-3461	263	27	)	)	PUNCT
ejpam-3461	263	28	.	.	PUNCT
ejpam-3461	264	1	hence	hence	ADV
ejpam-3461	264	2	,	,	PUNCT
ejpam-3461	264	3	m	m	NOUN
ejpam-3461	264	4	/	/	SYM
ejpam-3461	264	5	kerf	kerf	NOUN
ejpam-3461	264	6	∼=	∼=	NOUN
ejpam-3461	264	7	imf	imf	NOUN
ejpam-3461	264	8	.	.	PUNCT
ejpam-3461	265	1	by	by	ADP
ejpam-3461	265	2	our	our	PRON
ejpam-3461	265	3	assumption	assumption	NOUN
ejpam-3461	265	4	,	,	PUNCT
ejpam-3461	265	5	imf	imf	PROPN
ejpam-3461	265	6	is	be	AUX
ejpam-3461	265	7	injective	injective	ADJ
ejpam-3461	265	8	,	,	PUNCT
ejpam-3461	265	9	and	and	CCONJ
ejpam-3461	265	10	so	so	ADV
ejpam-3461	265	11	a	a	DET
ejpam-3461	265	12	direct	direct	ADJ
ejpam-3461	265	13	summand	summand	NOUN
ejpam-3461	265	14	of	of	ADP
ejpam-3461	265	15	m	m	PROPN
ejpam-3461	265	16	.	.	PUNCT
ejpam-3461	266	1	thus	thus	ADV
ejpam-3461	266	2	,	,	PUNCT
ejpam-3461	266	3	there	there	PRON
ejpam-3461	266	4	exists	exist	VERB
ejpam-3461	266	5	a	a	DET
ejpam-3461	266	6	submodule	submodule	NOUN
ejpam-3461	266	7	k	k	PROPN
ejpam-3461	266	8	of	of	ADP
ejpam-3461	266	9	m	m	PROPN
ejpam-3461	266	10	such	such	ADJ
ejpam-3461	266	11	that	that	SCONJ
ejpam-3461	266	12	m	m	PROPN
ejpam-3461	266	13	=	=	SYM
ejpam-3461	266	14	imf	imf	PROPN
ejpam-3461	266	15	⊕	⊕	PROPN
ejpam-3461	266	16	k.	k.	PROPN
ejpam-3461	266	17	by	by	ADP
ejpam-3461	266	18	hypthesis	hypthesis	PROPN
ejpam-3461	266	19	again	again	ADV
ejpam-3461	266	20	,	,	PUNCT
ejpam-3461	266	21	m	m	NOUN
ejpam-3461	266	22	/	/	SYM
ejpam-3461	266	23	imf	imf	PROPN
ejpam-3461	266	24	∼=	∼=	PROPN
ejpam-3461	266	25	k	k	NOUN
ejpam-3461	266	26	is	be	AUX
ejpam-3461	266	27	injective	injective	ADJ
ejpam-3461	266	28	.	.	PUNCT
ejpam-3461	267	1	therefore	therefore	ADV
ejpam-3461	267	2	,	,	PUNCT
ejpam-3461	267	3	m	m	VERB
ejpam-3461	267	4	is	be	AUX
ejpam-3461	267	5	injective	injective	ADJ
ejpam-3461	267	6	.	.	PUNCT
ejpam-3461	268	1	the	the	DET
ejpam-3461	268	2	last	last	ADJ
ejpam-3461	268	3	part	part	NOUN
ejpam-3461	268	4	is	be	AUX
ejpam-3461	268	5	clear	clear	ADJ
ejpam-3461	268	6	since	since	SCONJ
ejpam-3461	268	7	imf	imf	PROPN
ejpam-3461	268	8	is	be	AUX
ejpam-3461	268	9	nonsingular	nonsingular	ADJ
ejpam-3461	268	10	.	.	PUNCT
ejpam-3461	269	1	corollary	corollary	ADJ
ejpam-3461	269	2	10	10	NUM
ejpam-3461	269	3	.	.	PUNCT
ejpam-3461	270	1	a	a	DET
ejpam-3461	270	2	c	c	NOUN
ejpam-3461	270	3	-	-	PUNCT
ejpam-3461	270	4	co	co	NOUN
ejpam-3461	270	5	-	-	NOUN
ejpam-3461	270	6	pri	pri	ADJ
ejpam-3461	270	7	right	right	ADJ
ejpam-3461	270	8	c	c	NOUN
ejpam-3461	270	9	-	-	ADJ
ejpam-3461	270	10	cohereditary	cohereditary	ADJ
ejpam-3461	270	11	ring	ring	NOUN
ejpam-3461	270	12	is	be	AUX
ejpam-3461	270	13	right	right	ADJ
ejpam-3461	270	14	self	self	NOUN
ejpam-3461	270	15	-	-	PUNCT
ejpam-3461	270	16	injective	injective	ADJ
ejpam-3461	270	17	.	.	PUNCT
ejpam-3461	271	1	corollary	corollary	ADJ
ejpam-3461	271	2	11	11	NUM
ejpam-3461	271	3	.	.	PUNCT
ejpam-3461	272	1	a	a	DET
ejpam-3461	272	2	right	right	NOUN
ejpam-3461	272	3	extending	extend	VERB
ejpam-3461	272	4	right	right	NOUN
ejpam-3461	272	5	c	c	NOUN
ejpam-3461	272	6	-	-	ADJ
ejpam-3461	272	7	cohereditary	cohereditary	ADJ
ejpam-3461	272	8	ring	ring	NOUN
ejpam-3461	272	9	is	be	AUX
ejpam-3461	272	10	right	right	ADJ
ejpam-3461	272	11	self	self	NOUN
ejpam-3461	272	12	-	-	PUNCT
ejpam-3461	272	13	injective	injective	ADJ
ejpam-3461	272	14	.	.	PUNCT
ejpam-3461	273	1	corollary	corollary	ADJ
ejpam-3461	273	2	12	12	NUM
ejpam-3461	273	3	.	.	PUNCT
ejpam-3461	274	1	any	any	DET
ejpam-3461	274	2	extending	extend	VERB
ejpam-3461	274	3	c	c	NOUN
ejpam-3461	274	4	-	-	ADJ
ejpam-3461	274	5	cohereditary	cohereditary	ADJ
ejpam-3461	274	6	r	r	NOUN
ejpam-3461	274	7	-	-	PUNCT
ejpam-3461	274	8	module	module	NOUN
ejpam-3461	274	9	is	be	AUX
ejpam-3461	274	10	injective	injective	ADJ
ejpam-3461	274	11	we	we	PRON
ejpam-3461	274	12	end	end	VERB
ejpam-3461	274	13	this	this	DET
ejpam-3461	274	14	section	section	NOUN
ejpam-3461	274	15	with	with	ADP
ejpam-3461	274	16	some	some	DET
ejpam-3461	274	17	applications	application	NOUN
ejpam-3461	274	18	of	of	ADP
ejpam-3461	274	19	c	c	NOUN
ejpam-3461	274	20	-	-	PUNCT
ejpam-3461	274	21	co	co	NOUN
ejpam-3461	274	22	-	-	ADJ
ejpam-3461	274	23	epi	epi	ADJ
ejpam-3461	274	24	-	-	ADJ
ejpam-3461	274	25	retractable	retractable	ADJ
ejpam-3461	274	26	modules	module	NOUN
ejpam-3461	274	27	regarding	regard	VERB
ejpam-3461	274	28	the	the	DET
ejpam-3461	274	29	characterization	characterization	NOUN
ejpam-3461	274	30	of	of	ADP
ejpam-3461	274	31	right	right	ADJ
ejpam-3461	274	32	si	si	NOUN
ejpam-3461	274	33	,	,	PUNCT
ejpam-3461	274	34	semi	semi	ADJ
ejpam-3461	274	35	-	-	ADJ
ejpam-3461	274	36	simple	simple	ADJ
ejpam-3461	274	37	artinian	artinian	NOUN
ejpam-3461	274	38	and	and	CCONJ
ejpam-3461	274	39	quasi	quasi	ADJ
ejpam-3461	274	40	-	-	ADJ
ejpam-3461	274	41	frobenius	frobenius	ADJ
ejpam-3461	274	42	rings	ring	NOUN
ejpam-3461	274	43	.	.	PUNCT
ejpam-3461	275	1	recall	recall	VERB
ejpam-3461	275	2	that	that	SCONJ
ejpam-3461	275	3	a	a	DET
ejpam-3461	275	4	ring	ring	NOUN
ejpam-3461	275	5	r	r	NOUN
ejpam-3461	275	6	is	be	AUX
ejpam-3461	275	7	said	say	VERB
ejpam-3461	275	8	to	to	PART
ejpam-3461	275	9	be	be	AUX
ejpam-3461	275	10	right	right	ADJ
ejpam-3461	275	11	si	si	INTJ
ejpam-3461	275	12	if	if	SCONJ
ejpam-3461	275	13	every	every	DET
ejpam-3461	275	14	singular	singular	ADJ
ejpam-3461	275	15	r	r	NOUN
ejpam-3461	275	16	-	-	PUNCT
ejpam-3461	275	17	module	module	NOUN
ejpam-3461	275	18	is	be	AUX
ejpam-3461	275	19	injective	injective	ADJ
ejpam-3461	275	20	.	.	PUNCT
ejpam-3461	276	1	lemma	lemma	PROPN
ejpam-3461	276	2	3	3	NUM
ejpam-3461	276	3	.	.	PUNCT
ejpam-3461	277	1	(	(	PUNCT
ejpam-3461	277	2	(	(	PUNCT
ejpam-3461	277	3	[	[	X
ejpam-3461	277	4	17	17	NUM
ejpam-3461	277	5	]	]	PUNCT
ejpam-3461	277	6	,	,	PUNCT
ejpam-3461	277	7	lemma	lemma	PROPN
ejpam-3461	277	8	3.1	3.1	NUM
ejpam-3461	277	9	)	)	PUNCT
ejpam-3461	277	10	and	and	CCONJ
ejpam-3461	277	11	(	(	PUNCT
ejpam-3461	277	12	[	[	X
ejpam-3461	277	13	11	11	NUM
ejpam-3461	277	14	]	]	PUNCT
ejpam-3461	277	15	,	,	PUNCT
ejpam-3461	277	16	theorem	theorem	VERB
ejpam-3461	277	17	3	3	NUM
ejpam-3461	277	18	)	)	PUNCT
ejpam-3461	277	19	)	)	PUNCT
ejpam-3461	278	1	if	if	SCONJ
ejpam-3461	278	2	r	r	NOUN
ejpam-3461	278	3	is	be	AUX
ejpam-3461	278	4	a	a	DET
ejpam-3461	278	5	right	right	ADJ
ejpam-3461	278	6	si	si	NOUN
ejpam-3461	278	7	-	-	NOUN
ejpam-3461	278	8	ring	ring	NOUN
ejpam-3461	278	9	,	,	PUNCT
ejpam-3461	278	10	then	then	ADV
ejpam-3461	278	11	r	r	NOUN
ejpam-3461	278	12	is	be	AUX
ejpam-3461	278	13	right	right	ADV
ejpam-3461	278	14	nonsingular	nonsingular	ADJ
ejpam-3461	278	15	right	right	ADJ
ejpam-3461	278	16	hereditary	hereditary	NOUN
ejpam-3461	278	17	and	and	CCONJ
ejpam-3461	278	18	every	every	DET
ejpam-3461	278	19	singular	singular	ADJ
ejpam-3461	278	20	r	r	NOUN
ejpam-3461	278	21	-	-	PUNCT
ejpam-3461	278	22	module	module	NOUN
ejpam-3461	278	23	is	be	AUX
ejpam-3461	278	24	semi	semi	ADJ
ejpam-3461	278	25	-	-	ADJ
ejpam-3461	278	26	simple	simple	ADJ
ejpam-3461	278	27	.	.	PUNCT
ejpam-3461	279	1	lemma	lemma	PROPN
ejpam-3461	279	2	4	4	NUM
ejpam-3461	279	3	.	.	PUNCT
ejpam-3461	280	1	(	(	PUNCT
ejpam-3461	280	2	[	[	X
ejpam-3461	280	3	4	4	NUM
ejpam-3461	280	4	]	]	PUNCT
ejpam-3461	280	5	,	,	PUNCT
ejpam-3461	280	6	corollary	corollary	ADJ
ejpam-3461	280	7	3.2	3.2	NUM
ejpam-3461	280	8	)	)	PUNCT
ejpam-3461	280	9	let	let	VERB
ejpam-3461	280	10	m	m	PRON
ejpam-3461	280	11	be	be	AUX
ejpam-3461	280	12	an	an	DET
ejpam-3461	280	13	r	r	NOUN
ejpam-3461	280	14	-	-	PUNCT
ejpam-3461	280	15	module	module	NOUN
ejpam-3461	280	16	having	have	VERB
ejpam-3461	280	17	c3	c3	NOUN
ejpam-3461	280	18	-	-	NOUN
ejpam-3461	280	19	condition	condition	NOUN
ejpam-3461	280	20	.	.	PUNCT
ejpam-3461	281	1	if	if	SCONJ
ejpam-3461	281	2	m	m	NOUN
ejpam-3461	281	3	=	=	VERB
ejpam-3461	281	4	m1	m1	PROPN
ejpam-3461	281	5	⊕m2	⊕m2	PROPN
ejpam-3461	281	6	and	and	CCONJ
ejpam-3461	281	7	f	f	PROPN
ejpam-3461	281	8	:	:	PUNCT
ejpam-3461	281	9	m1	m1	PROPN
ejpam-3461	281	10	−→	−→	NOUN
ejpam-3461	281	11	m2	m2	PROPN
ejpam-3461	281	12	is	be	AUX
ejpam-3461	281	13	a	a	DET
ejpam-3461	281	14	monomorphism	monomorphism	NOUN
ejpam-3461	281	15	,	,	PUNCT
ejpam-3461	281	16	then	then	ADV
ejpam-3461	281	17	imf	imf	PROPN
ejpam-3461	281	18	≤⊕	≤⊕	NUM
ejpam-3461	281	19	m2	m2	PROPN
ejpam-3461	281	20	.	.	PROPN
ejpam-3461	281	21	theorem	theorem	PROPN
ejpam-3461	281	22	2	2	NUM
ejpam-3461	281	23	.	.	PUNCT
ejpam-3461	282	1	the	the	DET
ejpam-3461	282	2	following	follow	VERB
ejpam-3461	282	3	conditions	condition	NOUN
ejpam-3461	282	4	are	be	AUX
ejpam-3461	282	5	equivalent	equivalent	ADJ
ejpam-3461	282	6	for	for	ADP
ejpam-3461	282	7	a	a	DET
ejpam-3461	282	8	ring	ring	NOUN
ejpam-3461	282	9	r.	r.	NOUN
ejpam-3461	282	10	(	(	PUNCT
ejpam-3461	282	11	1	1	X
ejpam-3461	282	12	)	)	PUNCT
ejpam-3461	282	13	r	r	NOUN
ejpam-3461	282	14	is	be	AUX
ejpam-3461	282	15	a	a	DET
ejpam-3461	282	16	right	right	ADJ
ejpam-3461	282	17	si	si	NOUN
ejpam-3461	282	18	-	-	NOUN
ejpam-3461	282	19	ring	ring	NOUN
ejpam-3461	282	20	.	.	PUNCT
ejpam-3461	283	1	(	(	PUNCT
ejpam-3461	283	2	2	2	X
ejpam-3461	283	3	)	)	PUNCT
ejpam-3461	283	4	every	every	DET
ejpam-3461	283	5	z2	z2	NUM
ejpam-3461	283	6	-	-	PUNCT
ejpam-3461	283	7	torsion	torsion	NOUN
ejpam-3461	283	8	c	c	NOUN
ejpam-3461	283	9	-	-	PUNCT
ejpam-3461	283	10	co	co	NOUN
ejpam-3461	283	11	-	-	ADJ
ejpam-3461	283	12	epi	epi	ADJ
ejpam-3461	283	13	-	-	ADJ
ejpam-3461	283	14	retractable	retractable	ADJ
ejpam-3461	283	15	r	r	NOUN
ejpam-3461	283	16	-	-	PUNCT
ejpam-3461	283	17	module	module	NOUN
ejpam-3461	283	18	is	be	AUX
ejpam-3461	283	19	injective	injective	ADJ
ejpam-3461	283	20	.	.	PUNCT
ejpam-3461	284	1	(	(	PUNCT
ejpam-3461	284	2	3	3	X
ejpam-3461	284	3	)	)	PUNCT
ejpam-3461	284	4	every	every	DET
ejpam-3461	284	5	goldie	goldie	PROPN
ejpam-3461	284	6	-	-	PUNCT
ejpam-3461	284	7	torsion	torsion	NOUN
ejpam-3461	284	8	r	r	NOUN
ejpam-3461	284	9	-	-	PUNCT
ejpam-3461	284	10	module	module	NOUN
ejpam-3461	284	11	has	have	VERB
ejpam-3461	284	12	c3	c3	NOUN
ejpam-3461	284	13	-	-	NOUN
ejpam-3461	284	14	condition	condition	NOUN
ejpam-3461	284	15	.	.	PUNCT
ejpam-3461	285	1	proof	proof	NOUN
ejpam-3461	285	2	.	.	PUNCT
ejpam-3461	286	1	(	(	PUNCT
ejpam-3461	286	2	1)⇔	1)⇔	NUM
ejpam-3461	286	3	(	(	PUNCT
ejpam-3461	286	4	2	2	NUM
ejpam-3461	286	5	)	)	PUNCT
ejpam-3461	286	6	follows	follow	VERB
ejpam-3461	286	7	from	from	ADP
ejpam-3461	286	8	theorem	theorem	ADJ
ejpam-3461	286	9	3	3	NUM
ejpam-3461	286	10	in	in	ADP
ejpam-3461	286	11	[	[	X
ejpam-3461	286	12	11	11	NUM
ejpam-3461	286	13	]	]	PUNCT
ejpam-3461	286	14	.	.	PUNCT
ejpam-3461	287	1	the	the	DET
ejpam-3461	287	2	implication	implication	NOUN
ejpam-3461	287	3	(	(	PUNCT
ejpam-3461	287	4	1)⇒	1)⇒	NUM
ejpam-3461	287	5	(	(	PUNCT
ejpam-3461	287	6	3	3	NUM
ejpam-3461	287	7	)	)	PUNCT
ejpam-3461	287	8	is	be	AUX
ejpam-3461	287	9	clear	clear	ADJ
ejpam-3461	287	10	.	.	PUNCT
ejpam-3461	288	1	(	(	PUNCT
ejpam-3461	288	2	3	3	X
ejpam-3461	288	3	)	)	PUNCT
ejpam-3461	288	4	⇒	⇒	NOUN
ejpam-3461	288	5	(	(	PUNCT
ejpam-3461	288	6	1	1	X
ejpam-3461	288	7	)	)	PUNCT
ejpam-3461	288	8	let	let	VERB
ejpam-3461	288	9	m	m	PRON
ejpam-3461	288	10	be	be	AUX
ejpam-3461	288	11	a	a	DET
ejpam-3461	288	12	cyclic	cyclic	ADJ
ejpam-3461	288	13	z2	z2	ADJ
ejpam-3461	288	14	-	-	PUNCT
ejpam-3461	288	15	torsion	torsion	NOUN
ejpam-3461	288	16	r	r	NOUN
ejpam-3461	288	17	-	-	PUNCT
ejpam-3461	288	18	module	module	NOUN
ejpam-3461	288	19	.	.	PUNCT
ejpam-3461	289	1	it	it	PRON
ejpam-3461	289	2	is	be	AUX
ejpam-3461	289	3	clear	clear	ADJ
ejpam-3461	289	4	that	that	SCONJ
ejpam-3461	289	5	m	m	PROPN
ejpam-3461	289	6	⊕	⊕	PROPN
ejpam-3461	289	7	e(m	e(m	PROPN
ejpam-3461	289	8	)	)	PUNCT
ejpam-3461	289	9	is	be	AUX
ejpam-3461	289	10	goldietorsion	goldietorsion	NOUN
ejpam-3461	289	11	and	and	CCONJ
ejpam-3461	289	12	has	have	VERB
ejpam-3461	289	13	the	the	DET
ejpam-3461	289	14	c3	c3	NOUN
ejpam-3461	289	15	-	-	NOUN
ejpam-3461	289	16	condition	condition	NOUN
ejpam-3461	289	17	by	by	ADP
ejpam-3461	289	18	(	(	PUNCT
ejpam-3461	289	19	4	4	NUM
ejpam-3461	289	20	)	)	PUNCT
ejpam-3461	289	21	.	.	PUNCT
ejpam-3461	290	1	consider	consider	VERB
ejpam-3461	290	2	the	the	DET
ejpam-3461	290	3	inclusion	inclusion	NOUN
ejpam-3461	290	4	map	map	NOUN
ejpam-3461	290	5	i	i	PRON
ejpam-3461	290	6	:	:	PUNCT
ejpam-3461	290	7	m	m	AUX
ejpam-3461	290	8	−→	−→	ADJ
ejpam-3461	290	9	e(m	e(m	PROPN
ejpam-3461	290	10	)	)	PUNCT
ejpam-3461	290	11	.	.	PUNCT
ejpam-3461	291	1	hence	hence	ADV
ejpam-3461	291	2	,	,	PUNCT
ejpam-3461	291	3	i(m	i(m	NOUN
ejpam-3461	291	4	)	)	PUNCT
ejpam-3461	291	5	=	=	SYM
ejpam-3461	291	6	m	m	VERB
ejpam-3461	291	7	≤⊕	≤⊕	NUM
ejpam-3461	291	8	e(m	e(m	ADJ
ejpam-3461	291	9	)	)	PUNCT
ejpam-3461	291	10	by	by	ADP
ejpam-3461	291	11	lemma	lemma	PROPN
ejpam-3461	291	12	4	4	NUM
ejpam-3461	291	13	.	.	PUNCT
ejpam-3461	292	1	it	it	PRON
ejpam-3461	292	2	follows	follow	VERB
ejpam-3461	292	3	that	that	SCONJ
ejpam-3461	292	4	m	m	NOUN
ejpam-3461	292	5	is	be	AUX
ejpam-3461	292	6	injective	injective	ADJ
ejpam-3461	292	7	.	.	PUNCT
ejpam-3461	293	1	this	this	PRON
ejpam-3461	293	2	means	mean	VERB
ejpam-3461	293	3	that	that	SCONJ
ejpam-3461	293	4	every	every	DET
ejpam-3461	293	5	cyclic	cyclic	ADJ
ejpam-3461	293	6	singular	singular	ADJ
ejpam-3461	293	7	r	r	NOUN
ejpam-3461	293	8	-	-	PUNCT
ejpam-3461	293	9	module	module	NOUN
ejpam-3461	293	10	is	be	AUX
ejpam-3461	293	11	injective	injective	ADJ
ejpam-3461	293	12	.	.	PUNCT
ejpam-3461	294	1	therefore	therefore	ADV
ejpam-3461	294	2	,	,	PUNCT
ejpam-3461	294	3	according	accord	VERB
ejpam-3461	294	4	to	to	ADP
ejpam-3461	294	5	(	(	PUNCT
ejpam-3461	294	6	[	[	X
ejpam-3461	294	7	7	7	NUM
ejpam-3461	294	8	]	]	PUNCT
ejpam-3461	294	9	,	,	PUNCT
ejpam-3461	294	10	17.4	17.4	NUM
ejpam-3461	294	11	)	)	PUNCT
ejpam-3461	294	12	,	,	PUNCT
ejpam-3461	294	13	r	r	NOUN
ejpam-3461	294	14	is	be	AUX
ejpam-3461	294	15	a	a	DET
ejpam-3461	294	16	right	right	ADJ
ejpam-3461	294	17	si	si	NOUN
ejpam-3461	294	18	-	-	NOUN
ejpam-3461	294	19	ring	ring	NOUN
ejpam-3461	294	20	.	.	PUNCT
ejpam-3461	295	1	the	the	DET
ejpam-3461	295	2	following	follow	VERB
ejpam-3461	295	3	lemmas	lemma	NOUN
ejpam-3461	295	4	are	be	AUX
ejpam-3461	295	5	crucial	crucial	ADJ
ejpam-3461	295	6	in	in	ADP
ejpam-3461	295	7	the	the	DET
ejpam-3461	295	8	establishement	establishement	NOUN
ejpam-3461	295	9	of	of	ADP
ejpam-3461	295	10	the	the	DET
ejpam-3461	295	11	next	next	ADJ
ejpam-3461	295	12	theorem	theorem	NOUN
ejpam-3461	295	13	.	.	PUNCT
ejpam-3461	296	1	lemma	lemma	PROPN
ejpam-3461	296	2	5	5	NUM
ejpam-3461	296	3	.	.	PUNCT
ejpam-3461	297	1	(	(	PUNCT
ejpam-3461	297	2	[	[	X
ejpam-3461	297	3	7	7	NUM
ejpam-3461	297	4	]	]	PUNCT
ejpam-3461	297	5	,	,	PUNCT
ejpam-3461	297	6	corollary	corollary	NOUN
ejpam-3461	297	7	11.4	11.4	NUM
ejpam-3461	297	8	)	)	PUNCT
ejpam-3461	297	9	let	let	VERB
ejpam-3461	297	10	r	r	NOUN
ejpam-3461	297	11	be	be	AUX
ejpam-3461	297	12	ring	re	VERB
ejpam-3461	297	13	such	such	ADJ
ejpam-3461	297	14	that	that	DET
ejpam-3461	297	15	r	r	NOUN
ejpam-3461	297	16	(	(	PUNCT
ejpam-3461	297	17	a	a	X
ejpam-3461	297	18	)	)	PUNCT
ejpam-3461	297	19	r	r	NOUN
ejpam-3461	297	20	is	be	AUX
ejpam-3461	297	21	extending	extend	VERB
ejpam-3461	297	22	,	,	PUNCT
ejpam-3461	297	23	then	then	ADV
ejpam-3461	297	24	the	the	DET
ejpam-3461	297	25	following	follow	VERB
ejpam-3461	297	26	statements	statement	NOUN
ejpam-3461	297	27	hold	hold	VERB
ejpam-3461	297	28	true	true	ADJ
ejpam-3461	297	29	:	:	PUNCT
ejpam-3461	297	30	(	(	PUNCT
ejpam-3461	297	31	1	1	X
ejpam-3461	297	32	)	)	PUNCT
ejpam-3461	297	33	every	every	DET
ejpam-3461	297	34	nonsingular	nonsingular	ADJ
ejpam-3461	297	35	r	r	NOUN
ejpam-3461	297	36	-	-	PUNCT
ejpam-3461	297	37	module	module	NOUN
ejpam-3461	297	38	is	be	AUX
ejpam-3461	297	39	extending	extend	VERB
ejpam-3461	297	40	.	.	PUNCT
ejpam-3461	298	1	(	(	PUNCT
ejpam-3461	298	2	2	2	X
ejpam-3461	298	3	)	)	PUNCT
ejpam-3461	298	4	every	every	DET
ejpam-3461	298	5	nonsingular	nonsingular	ADJ
ejpam-3461	298	6	r	r	NOUN
ejpam-3461	298	7	-	-	PUNCT
ejpam-3461	298	8	module	module	NOUN
ejpam-3461	298	9	is	be	AUX
ejpam-3461	298	10	projective	projective	ADJ
ejpam-3461	298	11	.	.	PUNCT
ejpam-3461	299	1	a.	a.	PROPN
ejpam-3461	299	2	d.	d.	PROPN
ejpam-3461	299	3	diallo	diallo	PROPN
ejpam-3461	299	4	,	,	PUNCT
ejpam-3461	299	5	p.	p.	PROPN
ejpam-3461	299	6	c.	c.	PROPN
ejpam-3461	299	7	diop	diop	PROPN
ejpam-3461	299	8	,	,	PUNCT
ejpam-3461	299	9	m.	m.	NOUN
ejpam-3461	299	10	barry	barry	PROPN
ejpam-3461	299	11	/	/	SYM
ejpam-3461	299	12	eur	eur	PROPN
ejpam-3461	299	13	.	.	PUNCT
ejpam-3461	300	1	j.	j.	PROPN
ejpam-3461	300	2	pure	pure	PROPN
ejpam-3461	300	3	appl	appl	PROPN
ejpam-3461	300	4	.	.	PROPN
ejpam-3461	300	5	math	math	PROPN
ejpam-3461	300	6	,	,	PUNCT
ejpam-3461	300	7	12	12	NUM
ejpam-3461	300	8	(	(	PUNCT
ejpam-3461	300	9	3	3	NUM
ejpam-3461	300	10	)	)	PUNCT
ejpam-3461	300	11	(	(	PUNCT
ejpam-3461	300	12	2019	2019	NUM
ejpam-3461	300	13	)	)	PUNCT
ejpam-3461	300	14	,	,	PUNCT
ejpam-3461	300	15	1187	1187	NUM
ejpam-3461	300	16	-	-	SYM
ejpam-3461	300	17	1198	1198	NUM
ejpam-3461	300	18	1195	1195	NUM
ejpam-3461	300	19	lemma	lemma	PROPN
ejpam-3461	300	20	6	6	NUM
ejpam-3461	300	21	.	.	PUNCT
ejpam-3461	301	1	(	(	PUNCT
ejpam-3461	301	2	[	[	X
ejpam-3461	301	3	7	7	NUM
ejpam-3461	301	4	]	]	PUNCT
ejpam-3461	301	5	,	,	PUNCT
ejpam-3461	301	6	7.11	7.11	NUM
ejpam-3461	301	7	)	)	PUNCT
ejpam-3461	301	8	an	an	DET
ejpam-3461	301	9	r	r	NOUN
ejpam-3461	301	10	-	-	PUNCT
ejpam-3461	301	11	module	module	NOUN
ejpam-3461	301	12	m	m	NOUN
ejpam-3461	301	13	is	be	AUX
ejpam-3461	301	14	extending	extend	VERB
ejpam-3461	301	15	if	if	SCONJ
ejpam-3461	301	16	and	and	CCONJ
ejpam-3461	301	17	only	only	ADV
ejpam-3461	301	18	if	if	SCONJ
ejpam-3461	301	19	m	m	VERB
ejpam-3461	301	20	=	=	VERB
ejpam-3461	301	21	z2(m)⊕m	z2(m)⊕m	NOUN
ejpam-3461	301	22	′	′	NOUN
ejpam-3461	301	23	,	,	PUNCT
ejpam-3461	301	24	for	for	ADP
ejpam-3461	301	25	some	some	DET
ejpam-3461	301	26	submodule	submodule	NOUN
ejpam-3461	302	1	m	m	PROPN
ejpam-3461	302	2	′	′	NOUN
ejpam-3461	302	3	of	of	ADP
ejpam-3461	302	4	m	m	PRON
ejpam-3461	302	5	,	,	PUNCT
ejpam-3461	302	6	such	such	ADJ
ejpam-3461	302	7	that	that	SCONJ
ejpam-3461	302	8	m	m	VERB
ejpam-3461	302	9	′	′	NOUN
ejpam-3461	302	10	and	and	CCONJ
ejpam-3461	302	11	z2(m	z2(m	X
ejpam-3461	302	12	)	)	PUNCT
ejpam-3461	302	13	are	be	AUX
ejpam-3461	302	14	both	both	PRON
ejpam-3461	302	15	extending	extend	VERB
ejpam-3461	302	16	and	and	CCONJ
ejpam-3461	302	17	z2(m	z2(m	NOUN
ejpam-3461	302	18	)	)	PUNCT
ejpam-3461	302	19	is	be	AUX
ejpam-3461	302	20	m	m	VERB
ejpam-3461	302	21	′-injective	′-injective	ADJ
ejpam-3461	302	22	.	.	PUNCT
ejpam-3461	303	1	now	now	ADV
ejpam-3461	303	2	,	,	PUNCT
ejpam-3461	303	3	we	we	PRON
ejpam-3461	303	4	are	be	AUX
ejpam-3461	303	5	able	able	ADJ
ejpam-3461	303	6	to	to	PART
ejpam-3461	303	7	prove	prove	VERB
ejpam-3461	303	8	the	the	DET
ejpam-3461	303	9	following	follow	VERB
ejpam-3461	303	10	result	result	NOUN
ejpam-3461	303	11	.	.	PUNCT
ejpam-3461	304	1	theorem	theorem	NOUN
ejpam-3461	304	2	3	3	NUM
ejpam-3461	304	3	.	.	X
ejpam-3461	304	4	for	for	ADP
ejpam-3461	304	5	a	a	DET
ejpam-3461	304	6	right	right	ADJ
ejpam-3461	304	7	si	si	NOUN
ejpam-3461	304	8	-	-	ADJ
ejpam-3461	304	9	ring	ring	NOUN
ejpam-3461	304	10	r	r	NOUN
ejpam-3461	304	11	,	,	PUNCT
ejpam-3461	304	12	the	the	DET
ejpam-3461	304	13	following	follow	VERB
ejpam-3461	304	14	conditions	condition	NOUN
ejpam-3461	304	15	are	be	AUX
ejpam-3461	304	16	equivalent	equivalent	ADJ
ejpam-3461	304	17	:	:	PUNCT
ejpam-3461	304	18	(	(	PUNCT
ejpam-3461	304	19	1	1	X
ejpam-3461	304	20	)	)	PUNCT
ejpam-3461	304	21	r	r	NOUN
ejpam-3461	304	22	(	(	PUNCT
ejpam-3461	304	23	n	n	CCONJ
ejpam-3461	304	24	)	)	PUNCT
ejpam-3461	304	25	r	r	NOUN
ejpam-3461	304	26	is	be	AUX
ejpam-3461	304	27	extending	extend	VERB
ejpam-3461	304	28	.	.	PUNCT
ejpam-3461	305	1	(	(	PUNCT
ejpam-3461	305	2	2	2	X
ejpam-3461	305	3	)	)	PUNCT
ejpam-3461	305	4	r	r	NOUN
ejpam-3461	305	5	(	(	PUNCT
ejpam-3461	305	6	n	n	CCONJ
ejpam-3461	305	7	)	)	PUNCT
ejpam-3461	305	8	r	r	NOUN
ejpam-3461	305	9	is	be	AUX
ejpam-3461	305	10	c	c	NOUN
ejpam-3461	305	11	-	-	PUNCT
ejpam-3461	305	12	co	co	ADJ
ejpam-3461	305	13	-	-	ADJ
ejpam-3461	305	14	epi	epi	NOUN
ejpam-3461	305	15	-	-	NOUN
ejpam-3461	305	16	retractable	retractable	ADJ
ejpam-3461	305	17	.	.	PUNCT
ejpam-3461	306	1	(	(	PUNCT
ejpam-3461	306	2	3	3	X
ejpam-3461	306	3	)	)	PUNCT
ejpam-3461	306	4	every	every	DET
ejpam-3461	306	5	r	r	NOUN
ejpam-3461	306	6	-	-	PUNCT
ejpam-3461	306	7	module	module	NOUN
ejpam-3461	306	8	is	be	AUX
ejpam-3461	306	9	extending	extend	VERB
ejpam-3461	306	10	.	.	PUNCT
ejpam-3461	307	1	(	(	PUNCT
ejpam-3461	307	2	4	4	X
ejpam-3461	307	3	)	)	PUNCT
ejpam-3461	307	4	every	every	DET
ejpam-3461	307	5	r	r	NOUN
ejpam-3461	307	6	-	-	PUNCT
ejpam-3461	307	7	module	module	NOUN
ejpam-3461	307	8	is	be	AUX
ejpam-3461	307	9	c	c	NOUN
ejpam-3461	307	10	-	-	PUNCT
ejpam-3461	307	11	co	co	ADJ
ejpam-3461	307	12	-	-	ADJ
ejpam-3461	307	13	epi	epi	NOUN
ejpam-3461	307	14	-	-	NOUN
ejpam-3461	307	15	retractable	retractable	ADJ
ejpam-3461	307	16	.	.	PUNCT
ejpam-3461	308	1	proof	proof	NOUN
ejpam-3461	308	2	.	.	PUNCT
ejpam-3461	309	1	(	(	PUNCT
ejpam-3461	309	2	1)⇒	1)⇒	NUM
ejpam-3461	309	3	(	(	PUNCT
ejpam-3461	309	4	2	2	NUM
ejpam-3461	309	5	)	)	PUNCT
ejpam-3461	309	6	it	it	PRON
ejpam-3461	309	7	is	be	AUX
ejpam-3461	309	8	easy	easy	ADJ
ejpam-3461	309	9	to	to	PART
ejpam-3461	309	10	see	see	VERB
ejpam-3461	309	11	.	.	PUNCT
ejpam-3461	310	1	(	(	PUNCT
ejpam-3461	310	2	2)⇒	2)⇒	NUM
ejpam-3461	310	3	(	(	PUNCT
ejpam-3461	310	4	3	3	X
ejpam-3461	310	5	)	)	PUNCT
ejpam-3461	310	6	suppose	suppose	VERB
ejpam-3461	310	7	r	r	NOUN
ejpam-3461	310	8	(	(	PUNCT
ejpam-3461	310	9	n	n	CCONJ
ejpam-3461	310	10	)	)	PUNCT
ejpam-3461	310	11	r	r	NOUN
ejpam-3461	310	12	is	be	AUX
ejpam-3461	310	13	c	c	NOUN
ejpam-3461	310	14	-	-	PUNCT
ejpam-3461	310	15	co	co	NOUN
ejpam-3461	310	16	-	-	ADJ
ejpam-3461	310	17	epi	epi	NOUN
ejpam-3461	310	18	-	-	NOUN
ejpam-3461	310	19	retracatble	retracatble	NOUN
ejpam-3461	310	20	.	.	PUNCT
ejpam-3461	311	1	hence	hence	ADV
ejpam-3461	311	2	,	,	PUNCT
ejpam-3461	311	3	for	for	ADP
ejpam-3461	311	4	every	every	DET
ejpam-3461	311	5	complement	complement	NOUN
ejpam-3461	311	6	right	right	ADJ
ejpam-3461	311	7	ideal	ideal	NOUN
ejpam-3461	311	8	i	i	PRON
ejpam-3461	311	9	of	of	ADP
ejpam-3461	311	10	r(n	r(n	PROPN
ejpam-3461	311	11	)	)	PUNCT
ejpam-3461	311	12	,	,	PUNCT
ejpam-3461	311	13	there	there	PRON
ejpam-3461	311	14	exists	exist	VERB
ejpam-3461	311	15	a	a	DET
ejpam-3461	311	16	right	right	ADJ
ejpam-3461	311	17	ideal	ideal	NOUN
ejpam-3461	311	18	j	j	PROPN
ejpam-3461	311	19	of	of	ADP
ejpam-3461	311	20	r(n	r(n	PROPN
ejpam-3461	311	21	)	)	PUNCT
ejpam-3461	311	22	such	such	ADJ
ejpam-3461	311	23	that	that	SCONJ
ejpam-3461	311	24	r(n)/i	r(n)/i	PROPN
ejpam-3461	311	25	∼=	∼=	PROPN
ejpam-3461	311	26	j	j	PROPN
ejpam-3461	311	27	.	.	PUNCT
ejpam-3461	312	1	then	then	ADV
ejpam-3461	312	2	it	it	PRON
ejpam-3461	312	3	follows	follow	VERB
ejpam-3461	312	4	from	from	ADP
ejpam-3461	312	5	lemma	lemma	PROPN
ejpam-3461	312	6	3	3	NUM
ejpam-3461	312	7	that	that	PRON
ejpam-3461	312	8	r(n	r(n	VERB
ejpam-3461	312	9	)	)	PUNCT
ejpam-3461	312	10	is	be	AUX
ejpam-3461	312	11	an	an	DET
ejpam-3461	312	12	extending	extend	VERB
ejpam-3461	312	13	r	r	NOUN
ejpam-3461	312	14	-	-	PUNCT
ejpam-3461	312	15	module	module	NOUN
ejpam-3461	312	16	.	.	PUNCT
ejpam-3461	313	1	thus	thus	ADV
ejpam-3461	313	2	,	,	PUNCT
ejpam-3461	313	3	by	by	ADP
ejpam-3461	313	4	(	(	PUNCT
ejpam-3461	313	5	[	[	X
ejpam-3461	313	6	17	17	NUM
ejpam-3461	313	7	]	]	PUNCT
ejpam-3461	313	8	,	,	PUNCT
ejpam-3461	313	9	propositions	proposition	NOUN
ejpam-3461	313	10	3.4	3.4	NUM
ejpam-3461	313	11	and	and	CCONJ
ejpam-3461	313	12	3.9	3.9	NUM
ejpam-3461	313	13	)	)	PUNCT
ejpam-3461	313	14	,	,	PUNCT
ejpam-3461	313	15	r	r	NOUN
ejpam-3461	313	16	is	be	AUX
ejpam-3461	313	17	right	right	ADJ
ejpam-3461	313	18	noetherian	noetherian	NOUN
ejpam-3461	313	19	.	.	PUNCT
ejpam-3461	314	1	so	so	ADV
ejpam-3461	314	2	,	,	PUNCT
ejpam-3461	314	3	according	accord	VERB
ejpam-3461	314	4	to	to	ADP
ejpam-3461	314	5	corollary	corollary	ADJ
ejpam-3461	314	6	11.12	11.12	NUM
ejpam-3461	314	7	in	in	ADP
ejpam-3461	314	8	[	[	X
ejpam-3461	314	9	7	7	NUM
ejpam-3461	314	10	]	]	PUNCT
ejpam-3461	314	11	,	,	PUNCT
ejpam-3461	314	12	r	r	NOUN
ejpam-3461	314	13	(	(	PUNCT
ejpam-3461	314	14	a	a	X
ejpam-3461	314	15	)	)	PUNCT
ejpam-3461	314	16	r	r	NOUN
ejpam-3461	314	17	is	be	AUX
ejpam-3461	314	18	extending	extend	VERB
ejpam-3461	314	19	for	for	ADP
ejpam-3461	314	20	any	any	DET
ejpam-3461	314	21	index	index	NOUN
ejpam-3461	314	22	set	set	VERB
ejpam-3461	314	23	a.	a.	NOUN
ejpam-3461	314	24	now	now	ADV
ejpam-3461	314	25	,	,	PUNCT
ejpam-3461	314	26	let	let	VERB
ejpam-3461	314	27	m	m	PRON
ejpam-3461	314	28	be	be	AUX
ejpam-3461	314	29	any	any	DET
ejpam-3461	314	30	r	r	NOUN
ejpam-3461	314	31	-	-	PUNCT
ejpam-3461	314	32	module	module	NOUN
ejpam-3461	314	33	.	.	PUNCT
ejpam-3461	315	1	thus	thus	ADV
ejpam-3461	315	2	,	,	PUNCT
ejpam-3461	315	3	by	by	ADP
ejpam-3461	315	4	lemma	lemma	PROPN
ejpam-3461	315	5	5	5	NUM
ejpam-3461	315	6	,	,	PUNCT
ejpam-3461	315	7	m	m	VERB
ejpam-3461	315	8	=	=	SYM
ejpam-3461	315	9	z2(m	z2(m	X
ejpam-3461	315	10	)	)	PUNCT
ejpam-3461	315	11	⊕	⊕	PROPN
ejpam-3461	315	12	n	n	PROPN
ejpam-3461	315	13	for	for	ADP
ejpam-3461	315	14	some	some	DET
ejpam-3461	315	15	submodule	submodule	NOUN
ejpam-3461	315	16	n	n	PROPN
ejpam-3461	315	17	of	of	ADP
ejpam-3461	315	18	m	m	PROPN
ejpam-3461	315	19	and	and	CCONJ
ejpam-3461	315	20	clearly	clearly	ADV
ejpam-3461	315	21	n	n	PRON
ejpam-3461	315	22	is	be	AUX
ejpam-3461	315	23	nonsingular	nonsingular	ADJ
ejpam-3461	315	24	.	.	PUNCT
ejpam-3461	316	1	in	in	ADP
ejpam-3461	316	2	view	view	NOUN
ejpam-3461	316	3	of	of	ADP
ejpam-3461	316	4	lemma	lemma	PROPN
ejpam-3461	316	5	5	5	NUM
ejpam-3461	316	6	again	again	ADV
ejpam-3461	316	7	,	,	PUNCT
ejpam-3461	316	8	n	n	PRON
ejpam-3461	316	9	is	be	AUX
ejpam-3461	316	10	an	an	DET
ejpam-3461	316	11	extending	extend	VERB
ejpam-3461	316	12	module	module	NOUN
ejpam-3461	316	13	.	.	PUNCT
ejpam-3461	317	1	moroever	moroever	PROPN
ejpam-3461	317	2	,	,	PUNCT
ejpam-3461	317	3	since	since	SCONJ
ejpam-3461	317	4	r	r	NOUN
ejpam-3461	317	5	is	be	AUX
ejpam-3461	317	6	right	right	ADV
ejpam-3461	317	7	nonsingular	nonsingular	ADJ
ejpam-3461	317	8	,	,	PUNCT
ejpam-3461	317	9	z2(m	z2(m	X
ejpam-3461	317	10	)	)	PUNCT
ejpam-3461	317	11	=	=	SYM
ejpam-3461	317	12	z(m	z(m	X
ejpam-3461	317	13	)	)	PUNCT
ejpam-3461	317	14	is	be	AUX
ejpam-3461	317	15	singular	singular	ADJ
ejpam-3461	317	16	.	.	PUNCT
ejpam-3461	318	1	hence	hence	ADV
ejpam-3461	318	2	,	,	PUNCT
ejpam-3461	318	3	z2(m	z2(m	X
ejpam-3461	318	4	)	)	PUNCT
ejpam-3461	318	5	is	be	AUX
ejpam-3461	318	6	injective	injective	ADJ
ejpam-3461	318	7	.	.	PUNCT
ejpam-3461	319	1	therefore	therefore	ADV
ejpam-3461	319	2	,	,	PUNCT
ejpam-3461	319	3	according	accord	VERB
ejpam-3461	319	4	to	to	ADP
ejpam-3461	319	5	lemma	lemma	PROPN
ejpam-3461	319	6	6	6	NUM
ejpam-3461	319	7	,	,	PUNCT
ejpam-3461	319	8	m	m	VERB
ejpam-3461	319	9	is	be	AUX
ejpam-3461	319	10	extending	extend	VERB
ejpam-3461	319	11	,	,	PUNCT
ejpam-3461	319	12	as	as	SCONJ
ejpam-3461	319	13	desired	desire	VERB
ejpam-3461	319	14	.	.	PUNCT
ejpam-3461	320	1	(	(	PUNCT
ejpam-3461	320	2	3)⇒	3)⇒	NUM
ejpam-3461	320	3	(	(	PUNCT
ejpam-3461	320	4	4	4	NUM
ejpam-3461	320	5	)	)	PUNCT
ejpam-3461	320	6	is	be	AUX
ejpam-3461	320	7	clear	clear	ADJ
ejpam-3461	320	8	.	.	PUNCT
ejpam-3461	321	1	(	(	PUNCT
ejpam-3461	321	2	4	4	X
ejpam-3461	321	3	)	)	PUNCT
ejpam-3461	321	4	⇒	⇒	NOUN
ejpam-3461	321	5	(	(	PUNCT
ejpam-3461	321	6	1	1	NUM
ejpam-3461	321	7	)	)	PUNCT
ejpam-3461	321	8	by	by	ADP
ejpam-3461	321	9	(	(	PUNCT
ejpam-3461	321	10	4	4	NUM
ejpam-3461	321	11	)	)	PUNCT
ejpam-3461	321	12	,	,	PUNCT
ejpam-3461	321	13	r	r	NOUN
ejpam-3461	321	14	(	(	PUNCT
ejpam-3461	321	15	n	n	CCONJ
ejpam-3461	321	16	)	)	PUNCT
ejpam-3461	321	17	r	r	NOUN
ejpam-3461	321	18	is	be	AUX
ejpam-3461	321	19	c	c	NOUN
ejpam-3461	321	20	-	-	PUNCT
ejpam-3461	321	21	co	co	NOUN
ejpam-3461	321	22	-	-	ADJ
ejpam-3461	321	23	epi	epi	NOUN
ejpam-3461	321	24	-	-	NOUN
ejpam-3461	321	25	retracatble	retracatble	NOUN
ejpam-3461	321	26	.	.	PUNCT
ejpam-3461	322	1	but	but	CCONJ
ejpam-3461	322	2	r	r	NOUN
ejpam-3461	322	3	is	be	AUX
ejpam-3461	322	4	right	right	ADV
ejpam-3461	322	5	hereditary	hereditary	ADJ
ejpam-3461	322	6	.	.	PUNCT
ejpam-3461	323	1	thus	thus	ADV
ejpam-3461	323	2	,	,	PUNCT
ejpam-3461	323	3	by	by	ADP
ejpam-3461	323	4	the	the	DET
ejpam-3461	323	5	proof	proof	NOUN
ejpam-3461	323	6	of	of	ADP
ejpam-3461	323	7	(	(	PUNCT
ejpam-3461	323	8	2)⇒	2)⇒	NUM
ejpam-3461	323	9	(	(	PUNCT
ejpam-3461	323	10	3	3	NUM
ejpam-3461	323	11	)	)	PUNCT
ejpam-3461	323	12	,	,	PUNCT
ejpam-3461	323	13	r	r	NOUN
ejpam-3461	323	14	(	(	PUNCT
ejpam-3461	323	15	n	n	CCONJ
ejpam-3461	323	16	)	)	PUNCT
ejpam-3461	323	17	r	r	NOUN
ejpam-3461	323	18	is	be	AUX
ejpam-3461	323	19	extending	extend	VERB
ejpam-3461	323	20	.	.	PUNCT
ejpam-3461	324	1	corollary	corollary	ADJ
ejpam-3461	324	2	13	13	NUM
ejpam-3461	324	3	.	.	PUNCT
ejpam-3461	325	1	the	the	DET
ejpam-3461	325	2	the	the	DET
ejpam-3461	325	3	following	follow	VERB
ejpam-3461	325	4	conditions	condition	NOUN
ejpam-3461	325	5	are	be	AUX
ejpam-3461	325	6	equivalent	equivalent	ADJ
ejpam-3461	325	7	for	for	ADP
ejpam-3461	325	8	a	a	DET
ejpam-3461	325	9	ring	ring	NOUN
ejpam-3461	325	10	r	r	NOUN
ejpam-3461	325	11	:	:	PUNCT
ejpam-3461	325	12	(	(	PUNCT
ejpam-3461	325	13	1	1	X
ejpam-3461	325	14	)	)	PUNCT
ejpam-3461	325	15	r	r	NOUN
ejpam-3461	325	16	is	be	AUX
ejpam-3461	325	17	semi	semi	ADJ
ejpam-3461	325	18	-	-	ADJ
ejpam-3461	325	19	simple	simple	ADJ
ejpam-3461	325	20	artinian	artinian	NOUN
ejpam-3461	325	21	.	.	PUNCT
ejpam-3461	326	1	(	(	PUNCT
ejpam-3461	326	2	2	2	X
ejpam-3461	326	3	)	)	PUNCT
ejpam-3461	326	4	r	r	NOUN
ejpam-3461	326	5	is	be	AUX
ejpam-3461	326	6	a	a	DET
ejpam-3461	326	7	regular	regular	ADJ
ejpam-3461	326	8	right	right	ADJ
ejpam-3461	326	9	si	si	NOUN
ejpam-3461	326	10	-	-	ADJ
ejpam-3461	326	11	ring	ring	NOUN
ejpam-3461	326	12	and	and	CCONJ
ejpam-3461	326	13	r	r	NOUN
ejpam-3461	326	14	(	(	PUNCT
ejpam-3461	326	15	n	n	CCONJ
ejpam-3461	326	16	)	)	PUNCT
ejpam-3461	326	17	r	r	NOUN
ejpam-3461	326	18	is	be	AUX
ejpam-3461	326	19	c	c	NOUN
ejpam-3461	326	20	-	-	PUNCT
ejpam-3461	326	21	co	co	ADJ
ejpam-3461	326	22	-	-	ADJ
ejpam-3461	326	23	epi	epi	NOUN
ejpam-3461	326	24	-	-	NOUN
ejpam-3461	326	25	retractable	retractable	ADJ
ejpam-3461	326	26	.	.	PUNCT
ejpam-3461	327	1	(	(	PUNCT
ejpam-3461	327	2	3	3	X
ejpam-3461	327	3	)	)	PUNCT
ejpam-3461	327	4	r	r	NOUN
ejpam-3461	327	5	is	be	AUX
ejpam-3461	327	6	a	a	DET
ejpam-3461	327	7	regular	regular	ADJ
ejpam-3461	327	8	right	right	ADJ
ejpam-3461	327	9	si	si	NOUN
ejpam-3461	327	10	-	-	ADJ
ejpam-3461	327	11	ring	ring	NOUN
ejpam-3461	327	12	and	and	CCONJ
ejpam-3461	327	13	r	r	NOUN
ejpam-3461	327	14	(	(	PUNCT
ejpam-3461	327	15	n	n	CCONJ
ejpam-3461	327	16	)	)	PUNCT
ejpam-3461	327	17	r	r	NOUN
ejpam-3461	327	18	is	be	AUX
ejpam-3461	327	19	extending	extend	VERB
ejpam-3461	327	20	.	.	PUNCT
ejpam-3461	328	1	(	(	PUNCT
ejpam-3461	328	2	4	4	X
ejpam-3461	328	3	)	)	PUNCT
ejpam-3461	328	4	r	r	NOUN
ejpam-3461	328	5	is	be	AUX
ejpam-3461	328	6	right	right	ADJ
ejpam-3461	328	7	si	si	NOUN
ejpam-3461	328	8	-	-	ADJ
ejpam-3461	328	9	ring	ring	NOUN
ejpam-3461	328	10	and	and	CCONJ
ejpam-3461	328	11	r	r	NOUN
ejpam-3461	328	12	(	(	PUNCT
ejpam-3461	328	13	n	n	CCONJ
ejpam-3461	328	14	)	)	PUNCT
ejpam-3461	328	15	r	r	NOUN
ejpam-3461	328	16	is	be	AUX
ejpam-3461	328	17	continuous	continuous	ADJ
ejpam-3461	328	18	.	.	PUNCT
ejpam-3461	329	1	(	(	PUNCT
ejpam-3461	329	2	5	5	X
ejpam-3461	329	3	)	)	PUNCT
ejpam-3461	329	4	r	r	NOUN
ejpam-3461	329	5	is	be	AUX
ejpam-3461	329	6	right	right	ADJ
ejpam-3461	329	7	si	si	NOUN
ejpam-3461	329	8	-	-	ADJ
ejpam-3461	329	9	ring	ring	NOUN
ejpam-3461	329	10	and	and	CCONJ
ejpam-3461	329	11	r	r	NOUN
ejpam-3461	329	12	(	(	PUNCT
ejpam-3461	329	13	n	n	CCONJ
ejpam-3461	329	14	)	)	PUNCT
ejpam-3461	329	15	r	r	NOUN
ejpam-3461	329	16	is	be	AUX
ejpam-3461	329	17	quasi	quasi	ADJ
ejpam-3461	329	18	-	-	ADJ
ejpam-3461	329	19	continuous	continuous	ADJ
ejpam-3461	329	20	.	.	PUNCT
ejpam-3461	330	1	proof	proof	NOUN
ejpam-3461	330	2	.	.	PUNCT
ejpam-3461	331	1	(	(	PUNCT
ejpam-3461	331	2	1)⇒	1)⇒	NUM
ejpam-3461	331	3	(	(	PUNCT
ejpam-3461	331	4	2)⇒	2)⇒	NUM
ejpam-3461	331	5	(	(	PUNCT
ejpam-3461	331	6	3)⇒	3)⇒	NUM
ejpam-3461	331	7	(	(	PUNCT
ejpam-3461	331	8	4)⇒	4)⇒	NUM
ejpam-3461	331	9	(	(	PUNCT
ejpam-3461	331	10	5	5	NUM
ejpam-3461	331	11	)	)	PUNCT
ejpam-3461	331	12	are	be	AUX
ejpam-3461	331	13	clear	clear	ADJ
ejpam-3461	331	14	.	.	PUNCT
ejpam-3461	332	1	(	(	PUNCT
ejpam-3461	332	2	5)⇒	5)⇒	NUM
ejpam-3461	332	3	(	(	PUNCT
ejpam-3461	332	4	1	1	NUM
ejpam-3461	332	5	)	)	PUNCT
ejpam-3461	332	6	assume	assume	VERB
ejpam-3461	332	7	that	that	SCONJ
ejpam-3461	332	8	r	r	NOUN
ejpam-3461	332	9	is	be	AUX
ejpam-3461	332	10	a	a	DET
ejpam-3461	332	11	right	right	ADJ
ejpam-3461	332	12	si	si	NOUN
ejpam-3461	332	13	-	-	NOUN
ejpam-3461	332	14	ring	ring	NOUN
ejpam-3461	332	15	such	such	ADJ
ejpam-3461	332	16	that	that	PRON
ejpam-3461	332	17	r	r	NOUN
ejpam-3461	332	18	(	(	PUNCT
ejpam-3461	332	19	n	n	CCONJ
ejpam-3461	332	20	)	)	PUNCT
ejpam-3461	332	21	r	r	NOUN
ejpam-3461	332	22	is	be	AUX
ejpam-3461	332	23	quasi	quasi	ADJ
ejpam-3461	332	24	-	-	ADJ
ejpam-3461	332	25	continuous	continuous	ADJ
ejpam-3461	332	26	.	.	PUNCT
ejpam-3461	333	1	in	in	ADP
ejpam-3461	333	2	particular	particular	ADJ
ejpam-3461	333	3	,	,	PUNCT
ejpam-3461	333	4	r	r	NOUN
ejpam-3461	333	5	(	(	PUNCT
ejpam-3461	333	6	n	n	CCONJ
ejpam-3461	333	7	)	)	PUNCT
ejpam-3461	333	8	r	r	NOUN
ejpam-3461	333	9	is	be	AUX
ejpam-3461	333	10	extending	extend	VERB
ejpam-3461	333	11	.	.	PUNCT
ejpam-3461	334	1	then	then	ADV
ejpam-3461	334	2	,	,	PUNCT
ejpam-3461	334	3	by	by	ADP
ejpam-3461	334	4	theorem	theorem	NOUN
ejpam-3461	334	5	3	3	NUM
ejpam-3461	334	6	every	every	DET
ejpam-3461	334	7	r	r	NOUN
ejpam-3461	334	8	-	-	PUNCT
ejpam-3461	334	9	module	module	NOUN
ejpam-3461	334	10	is	be	AUX
ejpam-3461	334	11	extending	extend	VERB
ejpam-3461	334	12	.	.	PUNCT
ejpam-3461	335	1	so	so	ADV
ejpam-3461	335	2	by	by	ADP
ejpam-3461	335	3	(	(	PUNCT
ejpam-3461	335	4	[	[	X
ejpam-3461	335	5	7	7	NUM
ejpam-3461	335	6	]	]	PUNCT
ejpam-3461	335	7	,	,	PUNCT
ejpam-3461	335	8	13.5	13.5	NUM
ejpam-3461	335	9	)	)	PUNCT
ejpam-3461	335	10	,	,	PUNCT
ejpam-3461	335	11	r	r	NOUN
ejpam-3461	335	12	is	be	AUX
ejpam-3461	335	13	an	an	DET
ejpam-3461	335	14	artinian	artinian	ADJ
ejpam-3461	335	15	serial	serial	ADJ
ejpam-3461	335	16	ring	ring	NOUN
ejpam-3461	335	17	.	.	PUNCT
ejpam-3461	336	1	thus	thus	ADV
ejpam-3461	336	2	,	,	PUNCT
ejpam-3461	336	3	by	by	ADP
ejpam-3461	336	4	(	(	PUNCT
ejpam-3461	336	5	[	[	X
ejpam-3461	336	6	6	6	NUM
ejpam-3461	336	7	]	]	PUNCT
ejpam-3461	336	8	,	,	PUNCT
ejpam-3461	336	9	proposition	proposition	NOUN
ejpam-3461	336	10	6.1	6.1	NUM
ejpam-3461	336	11	(	(	PUNCT
ejpam-3461	336	12	4	4	NUM
ejpam-3461	336	13	)	)	PUNCT
ejpam-3461	336	14	)	)	PUNCT
ejpam-3461	336	15	,	,	PUNCT
ejpam-3461	336	16	r	r	NOUN
ejpam-3461	336	17	(	(	PUNCT
ejpam-3461	336	18	n	n	CCONJ
ejpam-3461	336	19	)	)	PUNCT
ejpam-3461	336	20	r	r	NOUN
ejpam-3461	336	21	is	be	AUX
ejpam-3461	336	22	quasi	quasi	ADJ
ejpam-3461	336	23	-	-	ADJ
ejpam-3461	336	24	injective	injective	ADJ
ejpam-3461	336	25	,	,	PUNCT
ejpam-3461	336	26	and	and	CCONJ
ejpam-3461	336	27	hence	hence	ADV
ejpam-3461	336	28	rr	rr	PROPN
ejpam-3461	336	29	is	be	AUX
ejpam-3461	336	30	quasi	quasi	ADJ
ejpam-3461	336	31	-	-	ADJ
ejpam-3461	336	32	injective	injective	ADJ
ejpam-3461	336	33	.	.	PUNCT
ejpam-3461	337	1	consequently	consequently	ADV
ejpam-3461	337	2	,	,	PUNCT
ejpam-3461	337	3	r	r	NOUN
ejpam-3461	337	4	is	be	AUX
ejpam-3461	337	5	right	right	ADJ
ejpam-3461	337	6	self	self	NOUN
ejpam-3461	337	7	-	-	PUNCT
ejpam-3461	337	8	injective	injective	ADJ
ejpam-3461	337	9	by	by	ADP
ejpam-3461	337	10	(	(	PUNCT
ejpam-3461	337	11	[	[	X
ejpam-3461	337	12	12	12	NUM
ejpam-3461	337	13	]	]	PUNCT
ejpam-3461	337	14	,	,	PUNCT
ejpam-3461	337	15	remark	remark	NOUN
ejpam-3461	337	16	6.71(2b	6.71(2b	NOUN
ejpam-3461	337	17	)	)	PUNCT
ejpam-3461	337	18	)	)	PUNCT
ejpam-3461	337	19	.	.	PUNCT
ejpam-3461	338	1	since	since	SCONJ
ejpam-3461	338	2	r	r	NOUN
ejpam-3461	338	3	is	be	AUX
ejpam-3461	338	4	right	right	ADV
ejpam-3461	338	5	artinian	artinian	ADJ
ejpam-3461	338	6	,	,	PUNCT
ejpam-3461	338	7	rr	rr	PROPN
ejpam-3461	338	8	has	have	VERB
ejpam-3461	338	9	finite	finite	PROPN
ejpam-3461	338	10	uniform	uniform	ADJ
ejpam-3461	338	11	dimension	dimension	NOUN
ejpam-3461	338	12	.	.	PUNCT
ejpam-3461	339	1	because	because	SCONJ
ejpam-3461	339	2	r	r	NOUN
ejpam-3461	339	3	is	be	AUX
ejpam-3461	339	4	right	right	ADJ
ejpam-3461	339	5	si	si	INTJ
ejpam-3461	339	6	,	,	PUNCT
ejpam-3461	339	7	it	it	PRON
ejpam-3461	339	8	is	be	AUX
ejpam-3461	339	9	right	right	ADV
ejpam-3461	339	10	nonsingular	nonsingular	ADJ
ejpam-3461	339	11	by	by	ADP
ejpam-3461	339	12	lemma	lemma	PROPN
ejpam-3461	339	13	3	3	NUM
ejpam-3461	339	14	.	.	PUNCT
ejpam-3461	340	1	now	now	ADV
ejpam-3461	340	2	,	,	PUNCT
ejpam-3461	340	3	rr	rr	PROPN
ejpam-3461	340	4	is	be	AUX
ejpam-3461	340	5	nonsingular	nonsingular	ADJ
ejpam-3461	340	6	extending	extend	VERB
ejpam-3461	340	7	and	and	CCONJ
ejpam-3461	340	8	has	have	VERB
ejpam-3461	340	9	finite	finite	ADJ
ejpam-3461	340	10	uniform	uniform	ADJ
ejpam-3461	340	11	dimension	dimension	NOUN
ejpam-3461	340	12	.	.	PUNCT
ejpam-3461	341	1	thus	thus	ADV
ejpam-3461	341	2	,	,	PUNCT
ejpam-3461	341	3	rr	rr	PROPN
ejpam-3461	341	4	is	be	AUX
ejpam-3461	341	5	a	a	DET
ejpam-3461	341	6	finite	finite	ADJ
ejpam-3461	341	7	direct	direct	ADJ
ejpam-3461	341	8	sum	sum	NOUN
ejpam-3461	341	9	of	of	ADP
ejpam-3461	341	10	uniform	uniform	ADJ
ejpam-3461	341	11	submodules	submodule	NOUN
ejpam-3461	341	12	.	.	PUNCT
ejpam-3461	342	1	but	but	CCONJ
ejpam-3461	342	2	r	r	NOUN
ejpam-3461	342	3	is	be	AUX
ejpam-3461	342	4	right	right	ADJ
ejpam-3461	342	5	self	self	NOUN
ejpam-3461	342	6	-	-	PUNCT
ejpam-3461	342	7	injective	injective	ADJ
ejpam-3461	342	8	.	.	PUNCT
ejpam-3461	343	1	then	then	ADV
ejpam-3461	343	2	as	as	ADP
ejpam-3461	343	3	in	in	ADP
ejpam-3461	343	4	the	the	DET
ejpam-3461	343	5	proof	proof	NOUN
ejpam-3461	343	6	of	of	ADP
ejpam-3461	343	7	(	(	PUNCT
ejpam-3461	343	8	2	2	NUM
ejpam-3461	343	9	)	)	PUNCT
ejpam-3461	343	10	⇒	⇒	NOUN
ejpam-3461	343	11	(	(	PUNCT
ejpam-3461	343	12	3	3	NUM
ejpam-3461	343	13	)	)	PUNCT
ejpam-3461	343	14	in	in	ADP
ejpam-3461	343	15	theorem	theorem	NOUN
ejpam-3461	343	16	1	1	NUM
ejpam-3461	343	17	,	,	PUNCT
ejpam-3461	343	18	rr	rr	X
ejpam-3461	343	19	is	be	AUX
ejpam-3461	343	20	semi	semi	ADJ
ejpam-3461	343	21	-	-	ADJ
ejpam-3461	343	22	simple	simple	ADJ
ejpam-3461	343	23	.	.	PUNCT
ejpam-3461	344	1	therefore	therefore	ADV
ejpam-3461	344	2	,	,	PUNCT
ejpam-3461	344	3	r	r	NOUN
ejpam-3461	344	4	is	be	AUX
ejpam-3461	344	5	semi	semi	ADJ
ejpam-3461	344	6	-	-	ADJ
ejpam-3461	344	7	simple	simple	ADJ
ejpam-3461	344	8	arinian	arinian	NOUN
ejpam-3461	344	9	.	.	PUNCT
ejpam-3461	345	1	a.	a.	PROPN
ejpam-3461	345	2	d.	d.	PROPN
ejpam-3461	345	3	diallo	diallo	PROPN
ejpam-3461	345	4	,	,	PUNCT
ejpam-3461	345	5	p.	p.	PROPN
ejpam-3461	345	6	c.	c.	PROPN
ejpam-3461	345	7	diop	diop	PROPN
ejpam-3461	345	8	,	,	PUNCT
ejpam-3461	345	9	m.	m.	NOUN
ejpam-3461	345	10	barry	barry	PROPN
ejpam-3461	345	11	/	/	SYM
ejpam-3461	345	12	eur	eur	PROPN
ejpam-3461	345	13	.	.	PUNCT
ejpam-3461	346	1	j.	j.	PROPN
ejpam-3461	346	2	pure	pure	PROPN
ejpam-3461	346	3	appl	appl	PROPN
ejpam-3461	346	4	.	.	PROPN
ejpam-3461	346	5	math	math	PROPN
ejpam-3461	346	6	,	,	PUNCT
ejpam-3461	346	7	12	12	NUM
ejpam-3461	346	8	(	(	PUNCT
ejpam-3461	346	9	3	3	NUM
ejpam-3461	346	10	)	)	PUNCT
ejpam-3461	346	11	(	(	PUNCT
ejpam-3461	346	12	2019	2019	NUM
ejpam-3461	346	13	)	)	PUNCT
ejpam-3461	346	14	,	,	PUNCT
ejpam-3461	346	15	1187	1187	NUM
ejpam-3461	346	16	-	-	SYM
ejpam-3461	346	17	1198	1198	NUM
ejpam-3461	346	18	1196	1196	NUM
ejpam-3461	346	19	corollary	corollary	NOUN
ejpam-3461	346	20	14	14	NUM
ejpam-3461	346	21	.	.	PUNCT
ejpam-3461	347	1	if	if	SCONJ
ejpam-3461	347	2	r	r	NOUN
ejpam-3461	347	3	is	be	AUX
ejpam-3461	347	4	a	a	DET
ejpam-3461	347	5	right	right	ADJ
ejpam-3461	347	6	si	si	NOUN
ejpam-3461	347	7	-	-	NOUN
ejpam-3461	347	8	ring	ring	NOUN
ejpam-3461	347	9	such	such	ADJ
ejpam-3461	347	10	that	that	PRON
ejpam-3461	347	11	r	r	NOUN
ejpam-3461	347	12	(	(	PUNCT
ejpam-3461	347	13	n	n	CCONJ
ejpam-3461	347	14	)	)	PUNCT
ejpam-3461	347	15	r	r	NOUN
ejpam-3461	347	16	is	be	AUX
ejpam-3461	347	17	c	c	NOUN
ejpam-3461	347	18	-	-	PUNCT
ejpam-3461	347	19	co	co	ADJ
ejpam-3461	347	20	-	-	ADJ
ejpam-3461	347	21	epi	epi	NOUN
ejpam-3461	347	22	-	-	NOUN
ejpam-3461	347	23	retractable	retractable	ADJ
ejpam-3461	347	24	,	,	PUNCT
ejpam-3461	347	25	then	then	ADV
ejpam-3461	347	26	all	all	DET
ejpam-3461	347	27	r	r	NOUN
ejpam-3461	347	28	-	-	PUNCT
ejpam-3461	347	29	modules	module	NOUN
ejpam-3461	347	30	with	with	ADP
ejpam-3461	347	31	a	a	DET
ejpam-3461	347	32	regular	regular	ADJ
ejpam-3461	347	33	endomorphism	endomorphism	NOUN
ejpam-3461	347	34	ring	ring	NOUN
ejpam-3461	347	35	are	be	AUX
ejpam-3461	347	36	quasi	quasi	ADJ
ejpam-3461	347	37	-	-	ADJ
ejpam-3461	347	38	injective	injective	ADJ
ejpam-3461	347	39	.	.	PUNCT
ejpam-3461	348	1	proof	proof	NOUN
ejpam-3461	348	2	.	.	PUNCT
ejpam-3461	349	1	let	let	VERB
ejpam-3461	349	2	m	m	PRON
ejpam-3461	349	3	be	be	AUX
ejpam-3461	349	4	an	an	DET
ejpam-3461	349	5	r	r	NOUN
ejpam-3461	349	6	-	-	PUNCT
ejpam-3461	349	7	module	module	NOUN
ejpam-3461	349	8	with	with	ADP
ejpam-3461	349	9	a	a	DET
ejpam-3461	349	10	regular	regular	ADJ
ejpam-3461	349	11	endomorphism	endomorphism	NOUN
ejpam-3461	349	12	ring	ring	NOUN
ejpam-3461	349	13	.	.	PUNCT
ejpam-3461	350	1	thus	thus	ADV
ejpam-3461	350	2	,	,	PUNCT
ejpam-3461	350	3	by	by	ADP
ejpam-3461	350	4	theorem	theorem	NOUN
ejpam-3461	350	5	3	3	NUM
ejpam-3461	350	6	,	,	PUNCT
ejpam-3461	350	7	m	m	VERB
ejpam-3461	350	8	is	be	AUX
ejpam-3461	350	9	extending	extend	VERB
ejpam-3461	350	10	.	.	PUNCT
ejpam-3461	351	1	it	it	PRON
ejpam-3461	351	2	follows	follow	VERB
ejpam-3461	351	3	that	that	SCONJ
ejpam-3461	351	4	m	m	NOUN
ejpam-3461	351	5	is	be	AUX
ejpam-3461	351	6	quasi	quasi	ADJ
ejpam-3461	351	7	-	-	ADJ
ejpam-3461	351	8	continuous	continuous	ADJ
ejpam-3461	351	9	.	.	PUNCT
ejpam-3461	352	1	on	on	ADP
ejpam-3461	352	2	the	the	DET
ejpam-3461	352	3	other	other	ADJ
ejpam-3461	352	4	hand	hand	NOUN
ejpam-3461	352	5	,	,	PUNCT
ejpam-3461	352	6	r	r	NOUN
ejpam-3461	352	7	is	be	AUX
ejpam-3461	352	8	artinian	artinian	ADJ
ejpam-3461	352	9	serial	serial	NOUN
ejpam-3461	352	10	by	by	ADP
ejpam-3461	352	11	(	(	PUNCT
ejpam-3461	352	12	[	[	X
ejpam-3461	352	13	7	7	NUM
ejpam-3461	352	14	]	]	PUNCT
ejpam-3461	352	15	,	,	PUNCT
ejpam-3461	352	16	13.5	13.5	NUM
ejpam-3461	352	17	)	)	PUNCT
ejpam-3461	352	18	.	.	PUNCT
ejpam-3461	353	1	hence	hence	ADV
ejpam-3461	353	2	by	by	ADP
ejpam-3461	353	3	(	(	PUNCT
ejpam-3461	353	4	[	[	X
ejpam-3461	353	5	6	6	NUM
ejpam-3461	353	6	]	]	PUNCT
ejpam-3461	353	7	,	,	PUNCT
ejpam-3461	353	8	proposition	proposition	NOUN
ejpam-3461	353	9	6.1	6.1	NUM
ejpam-3461	353	10	(	(	PUNCT
ejpam-3461	353	11	4	4	NUM
ejpam-3461	353	12	)	)	PUNCT
ejpam-3461	353	13	)	)	PUNCT
ejpam-3461	353	14	,	,	PUNCT
ejpam-3461	353	15	m	m	VERB
ejpam-3461	353	16	is	be	AUX
ejpam-3461	353	17	quasi	quasi	ADJ
ejpam-3461	353	18	-	-	ADJ
ejpam-3461	353	19	injective	injective	ADJ
ejpam-3461	353	20	.	.	PUNCT
ejpam-3461	354	1	note	note	VERB
ejpam-3461	354	2	that	that	SCONJ
ejpam-3461	354	3	along	along	ADP
ejpam-3461	354	4	the	the	DET
ejpam-3461	354	5	lines	line	NOUN
ejpam-3461	354	6	of	of	ADP
ejpam-3461	354	7	the	the	DET
ejpam-3461	354	8	proof	proof	NOUN
ejpam-3461	354	9	of	of	ADP
ejpam-3461	354	10	the	the	DET
ejpam-3461	354	11	above	above	ADJ
ejpam-3461	354	12	theorem	theorem	NOUN
ejpam-3461	354	13	we	we	PRON
ejpam-3461	354	14	have	have	AUX
ejpam-3461	354	15	shown	show	VERB
ejpam-3461	354	16	that	that	SCONJ
ejpam-3461	354	17	if	if	SCONJ
ejpam-3461	354	18	r	r	NOUN
ejpam-3461	354	19	is	be	AUX
ejpam-3461	354	20	a	a	DET
ejpam-3461	354	21	right	right	ADJ
ejpam-3461	354	22	si	si	NOUN
ejpam-3461	354	23	-	-	NOUN
ejpam-3461	354	24	ring	ring	NOUN
ejpam-3461	354	25	such	such	ADJ
ejpam-3461	354	26	that	that	SCONJ
ejpam-3461	354	27	r(n	r(n	PROPN
ejpam-3461	354	28	)	)	PUNCT
ejpam-3461	354	29	is	be	AUX
ejpam-3461	354	30	right	right	ADJ
ejpam-3461	354	31	extending	extend	VERB
ejpam-3461	354	32	,	,	PUNCT
ejpam-3461	354	33	then	then	ADV
ejpam-3461	354	34	r(a	r(a	PROPN
ejpam-3461	354	35	)	)	PUNCT
ejpam-3461	354	36	is	be	AUX
ejpam-3461	354	37	right	right	ADV
ejpam-3461	354	38	extending	extend	VERB
ejpam-3461	354	39	for	for	ADP
ejpam-3461	354	40	any	any	DET
ejpam-3461	354	41	index	index	NOUN
ejpam-3461	354	42	set	set	VERB
ejpam-3461	354	43	a.	a.	NOUN
ejpam-3461	354	44	theorem	theorem	NOUN
ejpam-3461	354	45	4	4	NUM
ejpam-3461	354	46	.	.	PUNCT
ejpam-3461	355	1	the	the	DET
ejpam-3461	355	2	following	follow	VERB
ejpam-3461	355	3	conditions	condition	NOUN
ejpam-3461	355	4	are	be	AUX
ejpam-3461	355	5	equivalent	equivalent	ADJ
ejpam-3461	355	6	for	for	ADP
ejpam-3461	355	7	a	a	DET
ejpam-3461	355	8	ring	ring	NOUN
ejpam-3461	355	9	r	r	NOUN
ejpam-3461	355	10	with	with	ADP
ejpam-3461	355	11	r	r	NOUN
ejpam-3461	355	12	=	=	SYM
ejpam-3461	355	13	r	r	NOUN
ejpam-3461	355	14	/	/	SYM
ejpam-3461	355	15	z2(rr	z2(rr	NUM
ejpam-3461	355	16	)	)	PUNCT
ejpam-3461	355	17	.	.	PUNCT
ejpam-3461	356	1	(	(	PUNCT
ejpam-3461	356	2	1	1	X
ejpam-3461	356	3	)	)	PUNCT
ejpam-3461	356	4	r	r	NOUN
ejpam-3461	356	5	is	be	AUX
ejpam-3461	356	6	semi	semi	ADJ
ejpam-3461	356	7	-	-	ADJ
ejpam-3461	356	8	simple	simple	ADJ
ejpam-3461	356	9	.	.	PUNCT
ejpam-3461	357	1	(	(	PUNCT
ejpam-3461	357	2	2	2	X
ejpam-3461	357	3	)	)	PUNCT
ejpam-3461	357	4	every	every	DET
ejpam-3461	357	5	c	c	NOUN
ejpam-3461	357	6	-	-	PUNCT
ejpam-3461	357	7	co	co	NOUN
ejpam-3461	357	8	-	-	ADJ
ejpam-3461	357	9	epi	epi	ADJ
ejpam-3461	357	10	-	-	ADJ
ejpam-3461	357	11	retractable	retractable	ADJ
ejpam-3461	357	12	r	r	NOUN
ejpam-3461	357	13	-	-	PUNCT
ejpam-3461	357	14	module	module	NOUN
ejpam-3461	357	15	is	be	AUX
ejpam-3461	357	16	r	r	NOUN
ejpam-3461	357	17	-	-	PUNCT
ejpam-3461	357	18	injective	injective	ADJ
ejpam-3461	357	19	.	.	PUNCT
ejpam-3461	358	1	(	(	PUNCT
ejpam-3461	358	2	3	3	X
ejpam-3461	358	3	)	)	PUNCT
ejpam-3461	358	4	every	every	DET
ejpam-3461	358	5	nonsingular	nonsingular	ADJ
ejpam-3461	358	6	r	r	NOUN
ejpam-3461	358	7	-	-	PUNCT
ejpam-3461	358	8	module	module	NOUN
ejpam-3461	358	9	is	be	AUX
ejpam-3461	358	10	quasi	quasi	ADJ
ejpam-3461	358	11	-	-	ADJ
ejpam-3461	358	12	injective	injective	ADJ
ejpam-3461	358	13	.	.	PUNCT
ejpam-3461	359	1	(	(	PUNCT
ejpam-3461	359	2	4	4	X
ejpam-3461	359	3	)	)	PUNCT
ejpam-3461	359	4	every	every	DET
ejpam-3461	359	5	nonsingular	nonsingular	ADJ
ejpam-3461	359	6	r	r	NOUN
ejpam-3461	359	7	-	-	PUNCT
ejpam-3461	359	8	module	module	NOUN
ejpam-3461	359	9	is	be	AUX
ejpam-3461	359	10	quasi	quasi	ADJ
ejpam-3461	359	11	-	-	ADJ
ejpam-3461	359	12	continuous	continuous	ADJ
ejpam-3461	359	13	.	.	PUNCT
ejpam-3461	360	1	(	(	PUNCT
ejpam-3461	360	2	5	5	NUM
ejpam-3461	360	3	)	)	PUNCT
ejpam-3461	360	4	every	every	DET
ejpam-3461	360	5	nonsingular	nonsingular	ADJ
ejpam-3461	360	6	r	r	NOUN
ejpam-3461	360	7	-	-	PUNCT
ejpam-3461	360	8	module	module	NOUN
ejpam-3461	360	9	has	have	VERB
ejpam-3461	360	10	c3	c3	NOUN
ejpam-3461	360	11	-	-	NOUN
ejpam-3461	360	12	condition	condition	NOUN
ejpam-3461	360	13	.	.	PUNCT
ejpam-3461	361	1	(	(	PUNCT
ejpam-3461	361	2	6	6	X
ejpam-3461	361	3	)	)	PUNCT
ejpam-3461	361	4	every	every	DET
ejpam-3461	361	5	submodule	submodule	NOUN
ejpam-3461	361	6	of	of	ADP
ejpam-3461	361	7	a	a	DET
ejpam-3461	361	8	nonsingular	nonsingular	ADJ
ejpam-3461	361	9	r	r	NOUN
ejpam-3461	361	10	-	-	PUNCT
ejpam-3461	361	11	module	module	NOUN
ejpam-3461	361	12	is	be	AUX
ejpam-3461	361	13	a	a	DET
ejpam-3461	361	14	c3	c3	NOUN
ejpam-3461	361	15	-	-	NOUN
ejpam-3461	361	16	module	module	NOUN
ejpam-3461	361	17	.	.	PUNCT
ejpam-3461	362	1	(	(	PUNCT
ejpam-3461	362	2	7	7	X
ejpam-3461	362	3	)	)	PUNCT
ejpam-3461	362	4	every	every	DET
ejpam-3461	362	5	submodule	submodule	NOUN
ejpam-3461	362	6	of	of	ADP
ejpam-3461	362	7	r⊕r	r⊕r	NOUN
ejpam-3461	362	8	is	be	AUX
ejpam-3461	362	9	a	a	DET
ejpam-3461	362	10	c3	c3	NOUN
ejpam-3461	362	11	-	-	NOUN
ejpam-3461	362	12	module	module	NOUN
ejpam-3461	362	13	.	.	PUNCT
ejpam-3461	363	1	proof	proof	NOUN
ejpam-3461	363	2	.	.	PUNCT
ejpam-3461	364	1	the	the	DET
ejpam-3461	364	2	implication	implication	NOUN
ejpam-3461	364	3	(	(	PUNCT
ejpam-3461	364	4	1)⇔	1)⇔	NUM
ejpam-3461	364	5	(	(	PUNCT
ejpam-3461	364	6	2	2	NUM
ejpam-3461	364	7	)	)	PUNCT
ejpam-3461	364	8	follows	follow	VERB
ejpam-3461	364	9	from	from	ADP
ejpam-3461	364	10	a	a	DET
ejpam-3461	364	11	similar	similar	ADJ
ejpam-3461	364	12	proof	proof	NOUN
ejpam-3461	364	13	to	to	ADP
ejpam-3461	364	14	(	(	PUNCT
ejpam-3461	364	15	[	[	X
ejpam-3461	364	16	1	1	NUM
ejpam-3461	364	17	]	]	PUNCT
ejpam-3461	364	18	,	,	PUNCT
ejpam-3461	364	19	theorem	theorem	VERB
ejpam-3461	364	20	4.5	4.5	NUM
ejpam-3461	364	21	)	)	PUNCT
ejpam-3461	364	22	.	.	PUNCT
ejpam-3461	365	1	the	the	DET
ejpam-3461	365	2	implication	implication	NOUN
ejpam-3461	365	3	(	(	PUNCT
ejpam-3461	365	4	1)⇒	1)⇒	NUM
ejpam-3461	365	5	(	(	PUNCT
ejpam-3461	365	6	3	3	NUM
ejpam-3461	365	7	)	)	PUNCT
ejpam-3461	365	8	is	be	AUX
ejpam-3461	365	9	clear	clear	ADJ
ejpam-3461	365	10	by	by	ADP
ejpam-3461	365	11	(	(	PUNCT
ejpam-3461	365	12	[	[	X
ejpam-3461	365	13	3	3	NUM
ejpam-3461	365	14	]	]	PUNCT
ejpam-3461	365	15	,	,	PUNCT
ejpam-3461	365	16	theorem	theorem	VERB
ejpam-3461	365	17	3.2	3.2	NUM
ejpam-3461	365	18	)	)	PUNCT
ejpam-3461	365	19	.	.	PUNCT
ejpam-3461	366	1	implications	implication	NOUN
ejpam-3461	366	2	(	(	PUNCT
ejpam-3461	366	3	3)⇒	3)⇒	NUM
ejpam-3461	366	4	(	(	PUNCT
ejpam-3461	366	5	4)⇒	4)⇒	NUM
ejpam-3461	366	6	(	(	PUNCT
ejpam-3461	366	7	5)⇒	5)⇒	NUM
ejpam-3461	366	8	(	(	PUNCT
ejpam-3461	366	9	6)⇒	6)⇒	NUM
ejpam-3461	366	10	(	(	PUNCT
ejpam-3461	366	11	7	7	NUM
ejpam-3461	366	12	)	)	PUNCT
ejpam-3461	366	13	are	be	AUX
ejpam-3461	366	14	easy	easy	ADJ
ejpam-3461	366	15	to	to	PART
ejpam-3461	366	16	see	see	VERB
ejpam-3461	366	17	.	.	PUNCT
ejpam-3461	367	1	(	(	PUNCT
ejpam-3461	367	2	7)⇒	7)⇒	NUM
ejpam-3461	367	3	(	(	PUNCT
ejpam-3461	367	4	1	1	NUM
ejpam-3461	367	5	)	)	PUNCT
ejpam-3461	367	6	by	by	ADP
ejpam-3461	367	7	(	(	PUNCT
ejpam-3461	367	8	[	[	X
ejpam-3461	367	9	3	3	NUM
ejpam-3461	367	10	]	]	PUNCT
ejpam-3461	367	11	,	,	PUNCT
ejpam-3461	367	12	theorem	theorem	VERB
ejpam-3461	367	13	3.2	3.2	NUM
ejpam-3461	367	14	)	)	PUNCT
ejpam-3461	367	15	,	,	PUNCT
ejpam-3461	367	16	we	we	PRON
ejpam-3461	367	17	need	need	VERB
ejpam-3461	367	18	to	to	PART
ejpam-3461	367	19	show	show	VERB
ejpam-3461	367	20	that	that	SCONJ
ejpam-3461	367	21	r	r	NOUN
ejpam-3461	367	22	is	be	AUX
ejpam-3461	367	23	semi	semi	ADJ
ejpam-3461	367	24	-	-	ADJ
ejpam-3461	367	25	simple	simple	ADJ
ejpam-3461	367	26	.	.	PUNCT
ejpam-3461	368	1	let	let	VERB
ejpam-3461	368	2	i	i	PRON
ejpam-3461	368	3	be	be	AUX
ejpam-3461	368	4	a	a	DET
ejpam-3461	368	5	right	right	ADJ
ejpam-3461	368	6	ideal	ideal	NOUN
ejpam-3461	368	7	of	of	ADP
ejpam-3461	368	8	r.	r.	PROPN
ejpam-3461	368	9	thus	thus	ADV
ejpam-3461	368	10	i	i	PROPN
ejpam-3461	368	11	⊕	⊕	PROPN
ejpam-3461	368	12	r	r	NOUN
ejpam-3461	368	13	,	,	PUNCT
ejpam-3461	368	14	being	be	AUX
ejpam-3461	368	15	a	a	DET
ejpam-3461	368	16	submodule	submodule	NOUN
ejpam-3461	368	17	of	of	ADP
ejpam-3461	368	18	r	r	PROPN
ejpam-3461	368	19	⊕	⊕	PROPN
ejpam-3461	368	20	r	r	NOUN
ejpam-3461	368	21	is	be	AUX
ejpam-3461	368	22	a	a	DET
ejpam-3461	368	23	c3	c3	NOUN
ejpam-3461	368	24	-	-	NOUN
ejpam-3461	368	25	module	module	NOUN
ejpam-3461	368	26	by	by	ADP
ejpam-3461	368	27	(	(	PUNCT
ejpam-3461	368	28	7	7	NUM
ejpam-3461	368	29	)	)	PUNCT
ejpam-3461	368	30	.	.	PUNCT
ejpam-3461	369	1	now	now	ADV
ejpam-3461	369	2	,	,	PUNCT
ejpam-3461	369	3	let	let	VERB
ejpam-3461	369	4	i	i	PRON
ejpam-3461	369	5	:	:	PUNCT
ejpam-3461	369	6	i	i	PRON
ejpam-3461	369	7	−→	−→	VERB
ejpam-3461	369	8	r	r	NOUN
ejpam-3461	369	9	be	be	VERB
ejpam-3461	369	10	the	the	DET
ejpam-3461	369	11	inclusion	inclusion	NOUN
ejpam-3461	369	12	map	map	NOUN
ejpam-3461	369	13	.	.	PUNCT
ejpam-3461	370	1	by	by	ADP
ejpam-3461	370	2	lemma	lemma	PROPN
ejpam-3461	370	3	4	4	NUM
ejpam-3461	370	4	,	,	PUNCT
ejpam-3461	370	5	i	i	PRON
ejpam-3461	370	6	is	be	AUX
ejpam-3461	370	7	a	a	DET
ejpam-3461	370	8	direct	direct	ADJ
ejpam-3461	370	9	summand	summand	NOUN
ejpam-3461	370	10	of	of	ADP
ejpam-3461	370	11	r.	r.	PROPN
ejpam-3461	370	12	therfore	therfore	PROPN
ejpam-3461	370	13	,	,	PUNCT
ejpam-3461	370	14	r	r	NOUN
ejpam-3461	370	15	is	be	AUX
ejpam-3461	370	16	semi	semi	ADJ
ejpam-3461	370	17	-	-	ADJ
ejpam-3461	370	18	simple	simple	ADJ
ejpam-3461	370	19	.	.	PUNCT
ejpam-3461	371	1	corollary	corollary	ADJ
ejpam-3461	371	2	15	15	NUM
ejpam-3461	371	3	.	.	PUNCT
ejpam-3461	372	1	the	the	DET
ejpam-3461	372	2	following	follow	VERB
ejpam-3461	372	3	conditions	condition	NOUN
ejpam-3461	372	4	are	be	AUX
ejpam-3461	372	5	equivalent	equivalent	ADJ
ejpam-3461	372	6	for	for	ADP
ejpam-3461	372	7	a	a	DET
ejpam-3461	372	8	ring	ring	NOUN
ejpam-3461	372	9	r	r	NOUN
ejpam-3461	372	10	with	with	ADP
ejpam-3461	372	11	r	r	NOUN
ejpam-3461	372	12	/	/	SYM
ejpam-3461	372	13	z2(r	z2(r	NUM
ejpam-3461	372	14	)	)	PUNCT
ejpam-3461	372	15	=	=	SYM
ejpam-3461	372	16	r.	r.	NOUN
ejpam-3461	372	17	(	(	PUNCT
ejpam-3461	372	18	1	1	X
ejpam-3461	372	19	)	)	PUNCT
ejpam-3461	372	20	r	r	NOUN
ejpam-3461	372	21	is	be	AUX
ejpam-3461	372	22	quasi	quasi	ADJ
ejpam-3461	372	23	-	-	ADJ
ejpam-3461	372	24	frobenius	frobenius	ADJ
ejpam-3461	372	25	.	.	PUNCT
ejpam-3461	373	1	(	(	PUNCT
ejpam-3461	373	2	2	2	X
ejpam-3461	373	3	)	)	PUNCT
ejpam-3461	373	4	every	every	DET
ejpam-3461	373	5	c	c	NOUN
ejpam-3461	373	6	-	-	PUNCT
ejpam-3461	373	7	co	co	NOUN
ejpam-3461	373	8	-	-	ADJ
ejpam-3461	373	9	epi	epi	ADJ
ejpam-3461	373	10	-	-	ADJ
ejpam-3461	373	11	retractable	retractable	ADJ
ejpam-3461	373	12	r	r	NOUN
ejpam-3461	373	13	-	-	PUNCT
ejpam-3461	373	14	module	module	NOUN
ejpam-3461	373	15	is	be	AUX
ejpam-3461	373	16	r	r	NOUN
ejpam-3461	373	17	-	-	PUNCT
ejpam-3461	373	18	injective	injective	ADJ
ejpam-3461	373	19	and	and	CCONJ
ejpam-3461	373	20	z2(rr	z2(rr	NUM
ejpam-3461	373	21	)	)	PUNCT
ejpam-3461	373	22	is	be	AUX
ejpam-3461	373	23	an	an	DET
ejpam-3461	373	24	artinian	artinian	ADJ
ejpam-3461	373	25	injective	injective	ADJ
ejpam-3461	373	26	r	r	NOUN
ejpam-3461	373	27	-	-	PUNCT
ejpam-3461	373	28	module	module	NOUN
ejpam-3461	373	29	.	.	PUNCT
ejpam-3461	374	1	(	(	PUNCT
ejpam-3461	374	2	3	3	X
ejpam-3461	374	3	)	)	PUNCT
ejpam-3461	374	4	every	every	DET
ejpam-3461	374	5	c	c	PROPN
ejpam-3461	374	6	-	-	PUNCT
ejpam-3461	374	7	co	co	NOUN
ejpam-3461	374	8	-	-	ADJ
ejpam-3461	374	9	epi	epi	ADJ
ejpam-3461	374	10	-	-	ADJ
ejpam-3461	374	11	retractable	retractable	ADJ
ejpam-3461	374	12	r	r	NOUN
ejpam-3461	374	13	-	-	PUNCT
ejpam-3461	374	14	module	module	NOUN
ejpam-3461	374	15	is	be	AUX
ejpam-3461	374	16	r	r	NOUN
ejpam-3461	374	17	-	-	PUNCT
ejpam-3461	374	18	injective	injective	ADJ
ejpam-3461	374	19	and	and	CCONJ
ejpam-3461	374	20	z2(rr	z2(rr	NUM
ejpam-3461	374	21	)	)	PUNCT
ejpam-3461	374	22	is	be	AUX
ejpam-3461	374	23	a	a	DET
ejpam-3461	374	24	noetherian	noetherian	ADJ
ejpam-3461	374	25	injective	injective	ADJ
ejpam-3461	374	26	r	r	NOUN
ejpam-3461	374	27	-	-	PUNCT
ejpam-3461	374	28	module	module	NOUN
ejpam-3461	374	29	.	.	PUNCT
ejpam-3461	375	1	proof	proof	NOUN
ejpam-3461	375	2	.	.	PUNCT
ejpam-3461	376	1	(	(	PUNCT
ejpam-3461	376	2	1)⇒	1)⇒	NUM
ejpam-3461	376	3	(	(	PUNCT
ejpam-3461	376	4	2	2	NUM
ejpam-3461	376	5	)	)	PUNCT
ejpam-3461	376	6	since	since	SCONJ
ejpam-3461	376	7	r	r	NOUN
ejpam-3461	376	8	is	be	AUX
ejpam-3461	376	9	quasi	quasi	ADJ
ejpam-3461	376	10	-	-	ADJ
ejpam-3461	376	11	frobenius	frobenius	ADJ
ejpam-3461	376	12	,	,	PUNCT
ejpam-3461	376	13	it	it	PRON
ejpam-3461	376	14	is	be	AUX
ejpam-3461	376	15	right	right	ADV
ejpam-3461	376	16	continuous	continuous	ADJ
ejpam-3461	376	17	.	.	PUNCT
ejpam-3461	377	1	hence	hence	ADV
ejpam-3461	377	2	,	,	PUNCT
ejpam-3461	377	3	r	r	NOUN
ejpam-3461	377	4	is	be	AUX
ejpam-3461	377	5	a	a	DET
ejpam-3461	377	6	continuous	continuous	ADJ
ejpam-3461	377	7	rmodule	rmodule	NOUN
ejpam-3461	377	8	.	.	PUNCT
ejpam-3461	378	1	thus	thus	ADV
ejpam-3461	378	2	,	,	PUNCT
ejpam-3461	378	3	rr	rr	NOUN
ejpam-3461	378	4	=	=	PUNCT
ejpam-3461	378	5	z2(rr)⊕r′	z2(rr)⊕r′	PROPN
ejpam-3461	378	6	for	for	ADP
ejpam-3461	378	7	a	a	DET
ejpam-3461	378	8	continuous	continuous	ADJ
ejpam-3461	378	9	r	r	NOUN
ejpam-3461	378	10	-	-	PUNCT
ejpam-3461	378	11	module	module	NOUN
ejpam-3461	378	12	r′.	r′.	NOUN
ejpam-3461	378	13	it	it	PRON
ejpam-3461	378	14	follows	follow	VERB
ejpam-3461	378	15	that	that	SCONJ
ejpam-3461	378	16	r	r	NOUN
ejpam-3461	378	17	is	be	AUX
ejpam-3461	378	18	right	right	ADV
ejpam-3461	378	19	nonsingular	nonsingular	ADJ
ejpam-3461	378	20	right	right	ADJ
ejpam-3461	378	21	continuous	continuous	ADJ
ejpam-3461	378	22	.	.	PUNCT
ejpam-3461	379	1	consequently	consequently	ADV
ejpam-3461	379	2	,	,	PUNCT
ejpam-3461	379	3	r	r	NOUN
ejpam-3461	379	4	is	be	AUX
ejpam-3461	379	5	regular	regular	ADJ
ejpam-3461	379	6	.	.	PUNCT
ejpam-3461	380	1	since	since	SCONJ
ejpam-3461	380	2	r	r	NOUN
ejpam-3461	380	3	is	be	AUX
ejpam-3461	380	4	right	right	ADJ
ejpam-3461	380	5	noetherian	noetherian	ADJ
ejpam-3461	380	6	,	,	PUNCT
ejpam-3461	380	7	r	r	NOUN
ejpam-3461	380	8	,	,	PUNCT
ejpam-3461	380	9	also	also	ADV
ejpam-3461	380	10	,	,	PUNCT
ejpam-3461	380	11	is	be	AUX
ejpam-3461	380	12	right	right	ADJ
ejpam-3461	380	13	noetherian	noetherian	NOUN
ejpam-3461	380	14	.	.	PUNCT
ejpam-3461	381	1	thus	thus	ADV
ejpam-3461	381	2	,	,	PUNCT
ejpam-3461	381	3	the	the	DET
ejpam-3461	381	4	property	property	NOUN
ejpam-3461	381	5	of	of	ADP
ejpam-3461	381	6	regular	regular	ADJ
ejpam-3461	381	7	implies	implie	NOUN
ejpam-3461	381	8	that	that	SCONJ
ejpam-3461	381	9	r	r	NOUN
ejpam-3461	381	10	is	be	AUX
ejpam-3461	381	11	semi	semi	ADJ
ejpam-3461	381	12	-	-	ADJ
ejpam-3461	381	13	simple	simple	ADJ
ejpam-3461	381	14	.	.	PUNCT
ejpam-3461	382	1	thus	thus	ADV
ejpam-3461	382	2	,	,	PUNCT
ejpam-3461	382	3	in	in	ADP
ejpam-3461	382	4	view	view	NOUN
ejpam-3461	382	5	of	of	ADP
ejpam-3461	382	6	theorem	theorem	NOUN
ejpam-3461	382	7	4	4	NUM
ejpam-3461	382	8	,	,	PUNCT
ejpam-3461	382	9	every	every	DET
ejpam-3461	382	10	c	c	NOUN
ejpam-3461	382	11	-	-	PUNCT
ejpam-3461	382	12	co	co	NOUN
ejpam-3461	382	13	-	-	ADJ
ejpam-3461	382	14	epi	epi	ADJ
ejpam-3461	382	15	-	-	ADJ
ejpam-3461	382	16	retractable	retractable	ADJ
ejpam-3461	382	17	r	r	NOUN
ejpam-3461	382	18	-	-	PUNCT
ejpam-3461	382	19	module	module	NOUN
ejpam-3461	382	20	is	be	AUX
ejpam-3461	382	21	r	r	NOUN
ejpam-3461	382	22	-	-	PUNCT
ejpam-3461	382	23	injective	injective	ADJ
ejpam-3461	382	24	.	.	PUNCT
ejpam-3461	383	1	the	the	DET
ejpam-3461	383	2	last	last	ADJ
ejpam-3461	383	3	part	part	NOUN
ejpam-3461	383	4	is	be	AUX
ejpam-3461	383	5	clear	clear	ADJ
ejpam-3461	383	6	since	since	SCONJ
ejpam-3461	383	7	rr	rr	PROPN
ejpam-3461	383	8	is	be	AUX
ejpam-3461	383	9	injective	injective	ADJ
ejpam-3461	383	10	and	and	CCONJ
ejpam-3461	383	11	artinian	artinian	ADJ
ejpam-3461	383	12	.	.	PUNCT
ejpam-3461	384	1	references	reference	NOUN
ejpam-3461	384	2	1197	1197	NUM
ejpam-3461	384	3	(	(	PUNCT
ejpam-3461	384	4	2	2	NUM
ejpam-3461	384	5	)	)	PUNCT
ejpam-3461	384	6	⇒	⇒	NOUN
ejpam-3461	384	7	(	(	PUNCT
ejpam-3461	384	8	1	1	X
ejpam-3461	384	9	)	)	PUNCT
ejpam-3461	384	10	assume	assume	VERB
ejpam-3461	384	11	that	that	SCONJ
ejpam-3461	384	12	every	every	DET
ejpam-3461	384	13	c	c	NOUN
ejpam-3461	384	14	-	-	PUNCT
ejpam-3461	384	15	co	co	NOUN
ejpam-3461	384	16	-	-	ADJ
ejpam-3461	384	17	epi	epi	ADJ
ejpam-3461	384	18	-	-	ADJ
ejpam-3461	384	19	retractable	retractable	ADJ
ejpam-3461	384	20	r	r	NOUN
ejpam-3461	384	21	-	-	PUNCT
ejpam-3461	384	22	module	module	NOUN
ejpam-3461	384	23	is	be	AUX
ejpam-3461	384	24	r	r	NOUN
ejpam-3461	384	25	-	-	PUNCT
ejpam-3461	384	26	injective	injective	ADJ
ejpam-3461	384	27	and	and	CCONJ
ejpam-3461	384	28	z2(rr	z2(rr	NUM
ejpam-3461	384	29	)	)	PUNCT
ejpam-3461	384	30	is	be	AUX
ejpam-3461	384	31	an	an	DET
ejpam-3461	384	32	artinian	artinian	ADJ
ejpam-3461	384	33	injective	injective	ADJ
ejpam-3461	384	34	ring	ring	NOUN
ejpam-3461	384	35	.	.	PUNCT
ejpam-3461	385	1	since	since	SCONJ
ejpam-3461	385	2	every	every	DET
ejpam-3461	385	3	c	c	NOUN
ejpam-3461	385	4	-	-	PUNCT
ejpam-3461	385	5	co	co	NOUN
ejpam-3461	385	6	-	-	ADJ
ejpam-3461	385	7	epi	epi	ADJ
ejpam-3461	385	8	-	-	ADJ
ejpam-3461	385	9	retractable	retractable	ADJ
ejpam-3461	385	10	r	r	NOUN
ejpam-3461	385	11	-	-	PUNCT
ejpam-3461	385	12	module	module	NOUN
ejpam-3461	385	13	is	be	AUX
ejpam-3461	385	14	r	r	NOUN
ejpam-3461	385	15	-	-	PUNCT
ejpam-3461	385	16	injective	injective	ADJ
ejpam-3461	385	17	,	,	PUNCT
ejpam-3461	385	18	we	we	PRON
ejpam-3461	385	19	infer	infer	VERB
ejpam-3461	385	20	from	from	ADP
ejpam-3461	385	21	theorem	theorem	ADJ
ejpam-3461	385	22	4	4	NUM
ejpam-3461	385	23	that	that	SCONJ
ejpam-3461	385	24	r	r	NOUN
ejpam-3461	385	25	is	be	AUX
ejpam-3461	385	26	a	a	DET
ejpam-3461	385	27	semi	semi	ADJ
ejpam-3461	385	28	-	-	ADJ
ejpam-3461	385	29	simple	simple	ADJ
ejpam-3461	385	30	ring	ring	NOUN
ejpam-3461	385	31	.	.	PUNCT
ejpam-3461	386	1	thus	thus	ADV
ejpam-3461	386	2	,	,	PUNCT
ejpam-3461	386	3	r	r	NOUN
ejpam-3461	386	4	is	be	AUX
ejpam-3461	386	5	semi	semi	ADJ
ejpam-3461	386	6	-	-	ADJ
ejpam-3461	386	7	simple	simple	ADJ
ejpam-3461	386	8	as	as	ADP
ejpam-3461	386	9	an	an	DET
ejpam-3461	386	10	r	r	NOUN
ejpam-3461	386	11	-	-	PUNCT
ejpam-3461	386	12	module	module	NOUN
ejpam-3461	386	13	.	.	PUNCT
ejpam-3461	387	1	since	since	SCONJ
ejpam-3461	387	2	r	r	NOUN
ejpam-3461	387	3	is	be	AUX
ejpam-3461	387	4	a	a	DET
ejpam-3461	387	5	nonsingular	nonsingular	ADJ
ejpam-3461	387	6	r	r	NOUN
ejpam-3461	387	7	-	-	PUNCT
ejpam-3461	387	8	module	module	NOUN
ejpam-3461	387	9	,	,	PUNCT
ejpam-3461	387	10	r	r	NOUN
ejpam-3461	387	11	is	be	AUX
ejpam-3461	387	12	a	a	DET
ejpam-3461	387	13	projective	projective	ADJ
ejpam-3461	387	14	r	r	NOUN
ejpam-3461	387	15	-	-	PUNCT
ejpam-3461	387	16	module	module	NOUN
ejpam-3461	387	17	.	.	PUNCT
ejpam-3461	388	1	so	so	ADV
ejpam-3461	388	2	,	,	PUNCT
ejpam-3461	388	3	z2(rr	z2(rr	NUM
ejpam-3461	388	4	)	)	PUNCT
ejpam-3461	388	5	≤⊕	≤⊕	NOUN
ejpam-3461	389	1	r	r	NOUN
ejpam-3461	389	2	,	,	PUNCT
ejpam-3461	389	3	say	say	VERB
ejpam-3461	389	4	r	r	NOUN
ejpam-3461	389	5	=	=	PUNCT
ejpam-3461	389	6	z2(rr	z2(rr	NUM
ejpam-3461	389	7	)	)	PUNCT
ejpam-3461	389	8	⊕	⊕	PROPN
ejpam-3461	389	9	r′	r′	PROPN
ejpam-3461	389	10	where	where	SCONJ
ejpam-3461	389	11	r′	r′	PROPN
ejpam-3461	389	12	is	be	AUX
ejpam-3461	389	13	semi	semi	ADJ
ejpam-3461	389	14	-	-	ADJ
ejpam-3461	389	15	simple	simple	ADJ
ejpam-3461	389	16	ring	ring	NOUN
ejpam-3461	389	17	.	.	PUNCT
ejpam-3461	390	1	by	by	ADP
ejpam-3461	390	2	our	our	PRON
ejpam-3461	390	3	assumption	assumption	NOUN
ejpam-3461	390	4	,	,	PUNCT
ejpam-3461	390	5	r	r	NOUN
ejpam-3461	390	6	is	be	AUX
ejpam-3461	390	7	right	right	ADJ
ejpam-3461	390	8	artinian	artinian	ADJ
ejpam-3461	390	9	right	right	ADJ
ejpam-3461	390	10	self	self	NOUN
ejpam-3461	390	11	-	-	PUNCT
ejpam-3461	390	12	injective	injective	ADJ
ejpam-3461	390	13	.	.	PUNCT
ejpam-3461	391	1	consequently	consequently	ADV
ejpam-3461	391	2	,	,	PUNCT
ejpam-3461	391	3	r	r	NOUN
ejpam-3461	391	4	is	be	AUX
ejpam-3461	391	5	quasi	quasi	ADJ
ejpam-3461	391	6	-	-	ADJ
ejpam-3461	391	7	frobenius	frobenius	ADJ
ejpam-3461	391	8	.	.	PUNCT
ejpam-3461	392	1	similarly	similarly	ADV
ejpam-3461	392	2	,	,	PUNCT
ejpam-3461	392	3	(	(	PUNCT
ejpam-3461	392	4	3	3	X
ejpam-3461	392	5	)	)	PUNCT
ejpam-3461	392	6	is	be	AUX
ejpam-3461	392	7	equivalent	equivalent	ADJ
ejpam-3461	392	8	to	to	ADP
ejpam-3461	392	9	(	(	PUNCT
ejpam-3461	392	10	1	1	NUM
ejpam-3461	392	11	)	)	PUNCT
ejpam-3461	392	12	.	.	PUNCT
ejpam-3461	393	1	proposition	proposition	NOUN
ejpam-3461	393	2	10	10	NUM
ejpam-3461	393	3	.	.	PUNCT
ejpam-3461	394	1	the	the	DET
ejpam-3461	394	2	following	follow	VERB
ejpam-3461	394	3	statements	statement	NOUN
ejpam-3461	394	4	are	be	AUX
ejpam-3461	394	5	equivalent	equivalent	ADJ
ejpam-3461	394	6	for	for	ADP
ejpam-3461	394	7	a	a	DET
ejpam-3461	394	8	ring	ring	NOUN
ejpam-3461	394	9	r.	r.	NOUN
ejpam-3461	394	10	(	(	PUNCT
ejpam-3461	394	11	1	1	X
ejpam-3461	394	12	)	)	PUNCT
ejpam-3461	394	13	r	r	NOUN
ejpam-3461	394	14	is	be	AUX
ejpam-3461	394	15	an	an	DET
ejpam-3461	394	16	artinian	artinian	ADJ
ejpam-3461	394	17	serial	serial	ADJ
ejpam-3461	394	18	ring	ring	NOUN
ejpam-3461	394	19	with	with	ADP
ejpam-3461	394	20	j2(r	j2(r	PROPN
ejpam-3461	394	21	)	)	PUNCT
ejpam-3461	394	22	=	=	SYM
ejpam-3461	395	1	0	0	X
ejpam-3461	395	2	.	.	PUNCT
ejpam-3461	396	1	(	(	PUNCT
ejpam-3461	396	2	2	2	X
ejpam-3461	396	3	)	)	PUNCT
ejpam-3461	396	4	every	every	DET
ejpam-3461	396	5	submodule	submodule	NOUN
ejpam-3461	396	6	of	of	ADP
ejpam-3461	396	7	a	a	DET
ejpam-3461	396	8	co	co	NOUN
ejpam-3461	396	9	-	-	ADJ
ejpam-3461	396	10	c	c	NOUN
ejpam-3461	396	11	-	-	PUNCT
ejpam-3461	396	12	epi	epi	NOUN
ejpam-3461	396	13	-	-	NOUN
ejpam-3461	396	14	retractable	retractable	ADJ
ejpam-3461	396	15	r	r	NOUN
ejpam-3461	396	16	-	-	PUNCT
ejpam-3461	396	17	module	module	NOUN
ejpam-3461	396	18	is	be	AUX
ejpam-3461	396	19	extending	extend	VERB
ejpam-3461	396	20	.	.	PUNCT
ejpam-3461	397	1	(	(	PUNCT
ejpam-3461	397	2	3	3	X
ejpam-3461	397	3	)	)	PUNCT
ejpam-3461	397	4	every	every	DET
ejpam-3461	397	5	submodule	submodule	NOUN
ejpam-3461	397	6	of	of	ADP
ejpam-3461	397	7	an	an	DET
ejpam-3461	397	8	extending	extending	ADJ
ejpam-3461	397	9	r	r	NOUN
ejpam-3461	397	10	-	-	PUNCT
ejpam-3461	397	11	module	module	NOUN
ejpam-3461	397	12	is	be	AUX
ejpam-3461	397	13	extending	extend	VERB
ejpam-3461	397	14	.	.	PUNCT
ejpam-3461	398	1	proof	proof	NOUN
ejpam-3461	398	2	.	.	PUNCT
ejpam-3461	399	1	(	(	PUNCT
ejpam-3461	399	2	1)⇒	1)⇒	NUM
ejpam-3461	399	3	(	(	PUNCT
ejpam-3461	399	4	2	2	NUM
ejpam-3461	399	5	)	)	PUNCT
ejpam-3461	399	6	follows	follow	VERB
ejpam-3461	399	7	from	from	ADP
ejpam-3461	399	8	(	(	PUNCT
ejpam-3461	399	9	[	[	X
ejpam-3461	399	10	7	7	NUM
ejpam-3461	399	11	]	]	PUNCT
ejpam-3461	399	12	,	,	PUNCT
ejpam-3461	399	13	13.5	13.5	NUM
ejpam-3461	399	14	)	)	PUNCT
ejpam-3461	399	15	.	.	PUNCT
ejpam-3461	400	1	(	(	PUNCT
ejpam-3461	400	2	2)⇒	2)⇒	NUM
ejpam-3461	400	3	(	(	PUNCT
ejpam-3461	400	4	1	1	X
ejpam-3461	400	5	)	)	PUNCT
ejpam-3461	400	6	let	let	VERB
ejpam-3461	400	7	m	m	PRON
ejpam-3461	400	8	be	be	AUX
ejpam-3461	400	9	any	any	DET
ejpam-3461	400	10	r	r	NOUN
ejpam-3461	400	11	-	-	PUNCT
ejpam-3461	400	12	module	module	NOUN
ejpam-3461	400	13	.	.	PUNCT
ejpam-3461	401	1	then	then	ADV
ejpam-3461	401	2	m⊕e(m	m⊕e(m	PROPN
ejpam-3461	401	3	)	)	PUNCT
ejpam-3461	401	4	,	,	PUNCT
ejpam-3461	401	5	being	be	AUX
ejpam-3461	401	6	a	a	DET
ejpam-3461	401	7	submodule	submodule	NOUN
ejpam-3461	401	8	of	of	ADP
ejpam-3461	401	9	e(m)⊕e(m	e(m)⊕e(m	PROPN
ejpam-3461	401	10	)	)	PUNCT
ejpam-3461	401	11	is	be	AUX
ejpam-3461	401	12	extending	extend	VERB
ejpam-3461	401	13	by	by	ADP
ejpam-3461	401	14	(	(	PUNCT
ejpam-3461	401	15	2	2	NUM
ejpam-3461	401	16	)	)	PUNCT
ejpam-3461	401	17	.	.	PUNCT
ejpam-3461	402	1	in	in	ADP
ejpam-3461	402	2	view	view	NOUN
ejpam-3461	402	3	of	of	ADP
ejpam-3461	402	4	proposition	proposition	NOUN
ejpam-3461	402	5	2.7	2.7	NUM
ejpam-3461	402	6	in	in	ADP
ejpam-3461	402	7	[	[	X
ejpam-3461	402	8	14	14	NUM
ejpam-3461	402	9	]	]	PUNCT
ejpam-3461	402	10	,	,	PUNCT
ejpam-3461	402	11	m	m	PROPN
ejpam-3461	402	12	is	be	AUX
ejpam-3461	402	13	cs	cs	PROPN
ejpam-3461	402	14	.	.	PROPN
ejpam-3461	402	15	therefore	therefore	ADV
ejpam-3461	402	16	,	,	PUNCT
ejpam-3461	402	17	r	r	NOUN
ejpam-3461	402	18	is	be	AUX
ejpam-3461	402	19	a	a	DET
ejpam-3461	402	20	artinian	artinian	ADJ
ejpam-3461	402	21	serial	serial	ADJ
ejpam-3461	402	22	ring	ring	NOUN
ejpam-3461	402	23	with	with	ADP
ejpam-3461	402	24	j2	j2	PROPN
ejpam-3461	402	25	=	=	SYM
ejpam-3461	402	26	0	0	NUM
ejpam-3461	402	27	by	by	ADP
ejpam-3461	402	28	(	(	PUNCT
ejpam-3461	402	29	[	[	X
ejpam-3461	402	30	7	7	NUM
ejpam-3461	402	31	]	]	PUNCT
ejpam-3461	402	32	,	,	PUNCT
ejpam-3461	402	33	13.5	13.5	NUM
ejpam-3461	402	34	)	)	PUNCT
ejpam-3461	402	35	.	.	PUNCT
ejpam-3461	403	1	similarly	similarly	ADV
ejpam-3461	403	2	,	,	PUNCT
ejpam-3461	403	3	(	(	PUNCT
ejpam-3461	403	4	1	1	X
ejpam-3461	403	5	)	)	PUNCT
ejpam-3461	403	6	and	and	CCONJ
ejpam-3461	403	7	(	(	PUNCT
ejpam-3461	403	8	3	3	X
ejpam-3461	403	9	)	)	PUNCT
ejpam-3461	403	10	are	be	AUX
ejpam-3461	403	11	equivalent	equivalent	ADJ
ejpam-3461	403	12	.	.	PUNCT
ejpam-3461	404	1	proposition	proposition	NOUN
ejpam-3461	404	2	11	11	NUM
ejpam-3461	404	3	.	.	PUNCT
ejpam-3461	405	1	the	the	DET
ejpam-3461	405	2	following	follow	VERB
ejpam-3461	405	3	conditions	condition	NOUN
ejpam-3461	405	4	are	be	AUX
ejpam-3461	405	5	equivalent	equivalent	ADJ
ejpam-3461	405	6	for	for	ADP
ejpam-3461	405	7	a	a	DET
ejpam-3461	405	8	ring	ring	NOUN
ejpam-3461	405	9	r.	r.	NOUN
ejpam-3461	405	10	(	(	PUNCT
ejpam-3461	405	11	1	1	X
ejpam-3461	405	12	)	)	PUNCT
ejpam-3461	405	13	r	r	NOUN
ejpam-3461	405	14	is	be	AUX
ejpam-3461	405	15	semi	semi	ADJ
ejpam-3461	405	16	-	-	ADJ
ejpam-3461	405	17	simple	simple	ADJ
ejpam-3461	405	18	artinian	artinian	NOUN
ejpam-3461	405	19	.	.	PUNCT
ejpam-3461	406	1	(	(	PUNCT
ejpam-3461	406	2	2	2	X
ejpam-3461	406	3	)	)	PUNCT
ejpam-3461	406	4	every	every	DET
ejpam-3461	406	5	c	c	NOUN
ejpam-3461	406	6	-	-	PUNCT
ejpam-3461	406	7	co	co	NOUN
ejpam-3461	406	8	-	-	ADJ
ejpam-3461	406	9	epi	epi	ADJ
ejpam-3461	406	10	-	-	ADJ
ejpam-3461	406	11	retractable	retractable	ADJ
ejpam-3461	406	12	r	r	NOUN
ejpam-3461	406	13	-	-	PUNCT
ejpam-3461	406	14	module	module	NOUN
ejpam-3461	406	15	is	be	AUX
ejpam-3461	406	16	semi	semi	ADJ
ejpam-3461	406	17	-	-	ADJ
ejpam-3461	406	18	simple	simple	ADJ
ejpam-3461	406	19	.	.	PUNCT
ejpam-3461	407	1	(	(	PUNCT
ejpam-3461	407	2	3	3	X
ejpam-3461	407	3	)	)	PUNCT
ejpam-3461	407	4	every	every	DET
ejpam-3461	407	5	c	c	PROPN
ejpam-3461	407	6	-	-	PUNCT
ejpam-3461	407	7	co	co	NOUN
ejpam-3461	407	8	-	-	ADJ
ejpam-3461	407	9	epi	epi	ADJ
ejpam-3461	407	10	-	-	ADJ
ejpam-3461	407	11	retractable	retractable	ADJ
ejpam-3461	407	12	r	r	NOUN
ejpam-3461	407	13	-	-	PUNCT
ejpam-3461	407	14	module	module	NOUN
ejpam-3461	407	15	is	be	AUX
ejpam-3461	407	16	injective	injective	ADJ
ejpam-3461	407	17	.	.	PUNCT
ejpam-3461	408	1	(	(	PUNCT
ejpam-3461	408	2	4	4	X
ejpam-3461	408	3	)	)	PUNCT
ejpam-3461	408	4	every	every	DET
ejpam-3461	408	5	submodule	submodule	NOUN
ejpam-3461	408	6	of	of	ADP
ejpam-3461	408	7	a	a	DET
ejpam-3461	408	8	c	c	NOUN
ejpam-3461	408	9	-	-	PUNCT
ejpam-3461	408	10	co	co	NOUN
ejpam-3461	408	11	-	-	ADJ
ejpam-3461	408	12	epi	epi	ADJ
ejpam-3461	408	13	-	-	ADJ
ejpam-3461	408	14	retractable	retractable	ADJ
ejpam-3461	408	15	r	r	NOUN
ejpam-3461	408	16	-	-	PUNCT
ejpam-3461	408	17	module	module	NOUN
ejpam-3461	408	18	is	be	AUX
ejpam-3461	408	19	quasi	quasi	ADJ
ejpam-3461	408	20	-	-	ADJ
ejpam-3461	408	21	continuous	continuous	ADJ
ejpam-3461	408	22	.	.	PUNCT
ejpam-3461	409	1	proof	proof	NOUN
ejpam-3461	409	2	.	.	PUNCT
ejpam-3461	410	1	(	(	PUNCT
ejpam-3461	410	2	1)⇒	1)⇒	NUM
ejpam-3461	410	3	(	(	PUNCT
ejpam-3461	410	4	2	2	NUM
ejpam-3461	410	5	)	)	PUNCT
ejpam-3461	410	6	this	this	PRON
ejpam-3461	410	7	is	be	AUX
ejpam-3461	410	8	clear	clear	ADJ
ejpam-3461	410	9	.	.	PUNCT
ejpam-3461	411	1	(	(	PUNCT
ejpam-3461	411	2	2)⇒	2)⇒	NUM
ejpam-3461	411	3	(	(	PUNCT
ejpam-3461	411	4	1	1	X
ejpam-3461	411	5	)	)	PUNCT
ejpam-3461	411	6	let	let	VERB
ejpam-3461	411	7	m	m	PRON
ejpam-3461	411	8	be	be	AUX
ejpam-3461	411	9	any	any	DET
ejpam-3461	411	10	r	r	NOUN
ejpam-3461	411	11	-	-	PUNCT
ejpam-3461	411	12	module	module	NOUN
ejpam-3461	411	13	.	.	PUNCT
ejpam-3461	412	1	by	by	ADP
ejpam-3461	412	2	(	(	PUNCT
ejpam-3461	412	3	2	2	NUM
ejpam-3461	412	4	)	)	PUNCT
ejpam-3461	412	5	,	,	PUNCT
ejpam-3461	412	6	e(m	e(m	PROPN
ejpam-3461	412	7	)	)	PUNCT
ejpam-3461	412	8	is	be	AUX
ejpam-3461	412	9	semi	semi	ADJ
ejpam-3461	412	10	-	-	ADJ
ejpam-3461	412	11	simple	simple	ADJ
ejpam-3461	412	12	,	,	PUNCT
ejpam-3461	412	13	and	and	CCONJ
ejpam-3461	412	14	hence	hence	ADV
ejpam-3461	412	15	m	m	VERB
ejpam-3461	412	16	=	=	SYM
ejpam-3461	412	17	e(m	e(m	PROPN
ejpam-3461	412	18	)	)	PUNCT
ejpam-3461	412	19	.	.	PUNCT
ejpam-3461	413	1	therefore	therefore	ADV
ejpam-3461	413	2	,	,	PUNCT
ejpam-3461	413	3	r	r	NOUN
ejpam-3461	413	4	is	be	AUX
ejpam-3461	413	5	semi	semi	ADJ
ejpam-3461	413	6	-	-	ADJ
ejpam-3461	413	7	simple	simple	ADJ
ejpam-3461	413	8	artinian	artinian	NOUN
ejpam-3461	413	9	.	.	PUNCT
ejpam-3461	414	1	(	(	PUNCT
ejpam-3461	414	2	1)⇒	1)⇒	NUM
ejpam-3461	414	3	(	(	PUNCT
ejpam-3461	414	4	4	4	NUM
ejpam-3461	414	5	)	)	PUNCT
ejpam-3461	414	6	is	be	AUX
ejpam-3461	414	7	clear	clear	ADJ
ejpam-3461	414	8	.	.	PUNCT
ejpam-3461	415	1	(	(	PUNCT
ejpam-3461	415	2	4)⇒	4)⇒	X
ejpam-3461	415	3	(	(	PUNCT
ejpam-3461	415	4	1	1	NUM
ejpam-3461	415	5	)	)	PUNCT
ejpam-3461	415	6	let	let	VERB
ejpam-3461	415	7	m	m	PRON
ejpam-3461	415	8	be	be	AUX
ejpam-3461	415	9	any	any	DET
ejpam-3461	415	10	r	r	NOUN
ejpam-3461	415	11	-	-	PUNCT
ejpam-3461	415	12	module	module	NOUN
ejpam-3461	415	13	.	.	PUNCT
ejpam-3461	416	1	then	then	ADV
ejpam-3461	416	2	m⊕e(m	m⊕e(m	PROPN
ejpam-3461	416	3	)	)	PUNCT
ejpam-3461	416	4	,	,	PUNCT
ejpam-3461	416	5	being	be	AUX
ejpam-3461	416	6	a	a	DET
ejpam-3461	416	7	submodule	submodule	NOUN
ejpam-3461	416	8	of	of	ADP
ejpam-3461	416	9	e(m)⊕e(m	e(m)⊕e(m	PROPN
ejpam-3461	416	10	)	)	PUNCT
ejpam-3461	416	11	is	be	AUX
ejpam-3461	416	12	quasi	quasi	ADJ
ejpam-3461	416	13	-	-	ADJ
ejpam-3461	416	14	continuous	continuous	ADJ
ejpam-3461	416	15	by	by	ADP
ejpam-3461	416	16	(	(	PUNCT
ejpam-3461	416	17	2	2	NUM
ejpam-3461	416	18	)	)	PUNCT
ejpam-3461	416	19	.	.	PUNCT
ejpam-3461	417	1	consequently	consequently	ADV
ejpam-3461	417	2	,	,	PUNCT
ejpam-3461	417	3	m	m	PROPN
ejpam-3461	417	4	⊕e(m	⊕e(m	NOUN
ejpam-3461	417	5	)	)	PUNCT
ejpam-3461	417	6	has	have	VERB
ejpam-3461	417	7	c3	c3	NOUN
ejpam-3461	417	8	-	-	NOUN
ejpam-3461	417	9	condition	condition	NOUN
ejpam-3461	417	10	.	.	PUNCT
ejpam-3461	418	1	by	by	ADP
ejpam-3461	418	2	lemma	lemma	PROPN
ejpam-3461	418	3	4	4	NUM
ejpam-3461	418	4	,	,	PUNCT
ejpam-3461	418	5	m	m	VERB
ejpam-3461	418	6	injective	injective	ADJ
ejpam-3461	418	7	and	and	CCONJ
ejpam-3461	418	8	so	so	ADV
ejpam-3461	418	9	r	r	NOUN
ejpam-3461	418	10	is	be	AUX
ejpam-3461	418	11	semi	semi	ADJ
ejpam-3461	418	12	-	-	ADJ
ejpam-3461	418	13	simple	simple	ADJ
ejpam-3461	418	14	artinian	artinian	NOUN
ejpam-3461	418	15	.	.	PUNCT
ejpam-3461	419	1	(	(	PUNCT
ejpam-3461	419	2	1)⇔	1)⇔	NUM
ejpam-3461	419	3	(	(	PUNCT
ejpam-3461	419	4	3	3	NUM
ejpam-3461	419	5	)	)	PUNCT
ejpam-3461	419	6	follows	follow	VERB
ejpam-3461	419	7	from	from	ADP
ejpam-3461	419	8	corollary	corollary	ADJ
ejpam-3461	419	9	2	2	NUM
ejpam-3461	419	10	in	in	ADP
ejpam-3461	419	11	[	[	X
ejpam-3461	419	12	11	11	NUM
ejpam-3461	419	13	]	]	PUNCT
ejpam-3461	419	14	.	.	PUNCT
ejpam-3461	420	1	references	reference	NOUN
ejpam-3461	420	2	[	[	X
ejpam-3461	420	3	1	1	NUM
ejpam-3461	420	4	]	]	X
ejpam-3461	420	5	sh	sh	PROPN
ejpam-3461	420	6	.	.	PROPN
ejpam-3461	420	7	asgari	asgari	PROPN
ejpam-3461	420	8	,	,	PUNCT
ejpam-3461	420	9	t	t	PROPN
ejpam-3461	420	10	-continuous	-continuous	ADJ
ejpam-3461	420	11	modules	module	NOUN
ejpam-3461	420	12	,	,	PUNCT
ejpam-3461	420	13	comm	comm	NOUN
ejpam-3461	420	14	.	.	PUNCT
ejpam-3461	421	1	algebra	algebra	PROPN
ejpam-3461	421	2	,	,	PUNCT
ejpam-3461	421	3	(	(	PUNCT
ejpam-3461	421	4	2017	2017	NUM
ejpam-3461	421	5	)	)	PUNCT
ejpam-3461	421	6	,	,	PUNCT
ejpam-3461	421	7	(	(	PUNCT
ejpam-3461	421	8	45	45	NUM
ejpam-3461	421	9	)	)	PUNCT
ejpam-3461	421	10	1941	1941	NUM
ejpam-3461	421	11	-	-	SYM
ejpam-3461	421	12	1952	1952	NUM
ejpam-3461	421	13	.	.	PUNCT
ejpam-3461	422	1	[	[	X
ejpam-3461	422	2	2	2	NUM
ejpam-3461	422	3	]	]	X
ejpam-3461	422	4	sh	sh	PROPN
ejpam-3461	422	5	.	.	PROPN
ejpam-3461	422	6	asgari	asgari	PROPN
ejpam-3461	422	7	and	and	CCONJ
ejpam-3461	422	8	a.	a.	NOUN
ejpam-3461	422	9	haghany	haghany	NOUN
ejpam-3461	422	10	,	,	PUNCT
ejpam-3461	422	11	t	t	NOUN
ejpam-3461	422	12	-extending	-extending	NOUN
ejpam-3461	422	13	modules	module	NOUN
ejpam-3461	422	14	and	and	CCONJ
ejpam-3461	422	15	t	t	NOUN
ejpam-3461	422	16	-baer	-baer	PROPN
ejpam-3461	422	17	modules	module	NOUN
ejpam-3461	422	18	,	,	PUNCT
ejpam-3461	422	19	comm	comm	NOUN
ejpam-3461	422	20	.	.	PUNCT
ejpam-3461	423	1	algebra	algebra	PROPN
ejpam-3461	423	2	,	,	PUNCT
ejpam-3461	423	3	(	(	PUNCT
ejpam-3461	423	4	2011	2011	NUM
ejpam-3461	423	5	)	)	PUNCT
ejpam-3461	423	6	,	,	PUNCT
ejpam-3461	423	7	(	(	PUNCT
ejpam-3461	423	8	39	39	NUM
ejpam-3461	423	9	)	)	PUNCT
ejpam-3461	423	10	1605	1605	NUM
ejpam-3461	423	11	-	-	SYM
ejpam-3461	423	12	1023	1023	NUM
ejpam-3461	423	13	.	.	PUNCT
ejpam-3461	424	1	[	[	X
ejpam-3461	424	2	3	3	X
ejpam-3461	424	3	]	]	X
ejpam-3461	424	4	sh	sh	PROPN
ejpam-3461	424	5	.	.	PROPN
ejpam-3461	424	6	asgari	asgari	PROPN
ejpam-3461	424	7	,	,	PUNCT
ejpam-3461	424	8	a.	a.	NOUN
ejpam-3461	424	9	haghany	haghany	PROPN
ejpam-3461	424	10	and	and	CCONJ
ejpam-3461	424	11	y.	y.	PROPN
ejpam-3461	424	12	tolooei	tolooei	PROPN
ejpam-3461	424	13	t	t	PROPN
ejpam-3461	424	14	-semi	-semi	PROPN
ejpam-3461	424	15	-	-	PUNCT
ejpam-3461	424	16	simple	simple	ADJ
ejpam-3461	424	17	modules	module	NOUN
ejpam-3461	424	18	and	and	CCONJ
ejpam-3461	424	19	t	t	NOUN
ejpam-3461	424	20	-semi	-semi	PROPN
ejpam-3461	424	21	-	-	PUNCT
ejpam-3461	424	22	simple	simple	ADJ
ejpam-3461	424	23	rings	ring	NOUN
ejpam-3461	424	24	,	,	PUNCT
ejpam-3461	424	25	comm	comm	NOUN
ejpam-3461	424	26	.	.	PUNCT
ejpam-3461	425	1	algebra	algebra	PROPN
ejpam-3461	425	2	,	,	PUNCT
ejpam-3461	425	3	(	(	PUNCT
ejpam-3461	425	4	2013	2013	NUM
ejpam-3461	425	5	)	)	PUNCT
ejpam-3461	425	6	,	,	PUNCT
ejpam-3461	425	7	(	(	PUNCT
ejpam-3461	425	8	41	41	NUM
ejpam-3461	425	9	)	)	PUNCT
ejpam-3461	425	10	1882	1882	NUM
ejpam-3461	425	11	-	-	SYM
ejpam-3461	425	12	1902	1902	NUM
ejpam-3461	425	13	.	.	PUNCT
ejpam-3461	426	1	references	reference	NOUN
ejpam-3461	426	2	1198	1198	NUM
ejpam-3461	426	3	[	[	X
ejpam-3461	426	4	4	4	X
ejpam-3461	426	5	]	]	PUNCT
ejpam-3461	426	6	s.	s.	PROPN
ejpam-3461	426	7	e.	e.	PROPN
ejpam-3461	426	8	atani	atani	PROPN
ejpam-3461	426	9	,	,	PUNCT
ejpam-3461	426	10	m.	m.	NOUN
ejpam-3461	426	11	khoramdel	khoramdel	PROPN
ejpam-3461	426	12	and	and	CCONJ
ejpam-3461	426	13	s.	s.	PROPN
ejpam-3461	426	14	d.	d.	PROPN
ejpam-3461	426	15	p.	p.	PROPN
ejpam-3461	426	16	hesari	hesari	PROPN
ejpam-3461	426	17	,	,	PUNCT
ejpam-3461	426	18	c3	c3	PROPN
ejpam-3461	426	19	-	-	PUNCT
ejpam-3461	426	20	modules	module	NOUN
ejpam-3461	426	21	,	,	PUNCT
ejpam-3461	426	22	demostratio	demostratio	PROPN
ejpam-3461	426	23	mathematica	mathematica	PROPN
ejpam-3461	426	24	,	,	PUNCT
ejpam-3461	426	25	49(2016	49(2016	NUM
ejpam-3461	426	26	)	)	PUNCT
ejpam-3461	426	27	,	,	PUNCT
ejpam-3461	426	28	no	no	INTJ
ejpam-3461	426	29	.	.	NOUN
ejpam-3461	426	30	3	3	NUM
ejpam-3461	426	31	,	,	PUNCT
ejpam-3461	426	32	282	282	NUM
ejpam-3461	426	33	-	-	SYM
ejpam-3461	426	34	292	292	NUM
ejpam-3461	426	35	.	.	PUNCT
ejpam-3461	427	1	[	[	X
ejpam-3461	427	2	5	5	X
ejpam-3461	427	3	]	]	PUNCT
ejpam-3461	427	4	k.	k.	PROPN
ejpam-3461	427	5	a.	a.	PROPN
ejpam-3461	427	6	byrd	byrd	PROPN
ejpam-3461	427	7	,	,	PUNCT
ejpam-3461	427	8	rings	ring	VERB
ejpam-3461	427	9	whose	whose	DET
ejpam-3461	427	10	quasi	quasi	ADJ
ejpam-3461	427	11	-	-	ADJ
ejpam-3461	427	12	injective	injective	ADJ
ejpam-3461	427	13	modules	module	NOUN
ejpam-3461	427	14	are	be	AUX
ejpam-3461	427	15	injective	injective	ADJ
ejpam-3461	427	16	,	,	PUNCT
ejpam-3461	427	17	proc	proc	NOUN
ejpam-3461	427	18	.	.	PUNCT
ejpam-3461	428	1	amer	amer	PROPN
ejpam-3461	428	2	.	.	PUNCT
ejpam-3461	428	3	math	math	PROPN
ejpam-3461	428	4	.	.	PUNCT
ejpam-3461	429	1	soc	soc	PROPN
ejpam-3461	429	2	.	.	PUNCT
ejpam-3461	430	1	33(1972	33(1972	NUM
ejpam-3461	430	2	)	)	PUNCT
ejpam-3461	430	3	,	,	PUNCT
ejpam-3461	431	1	no	no	INTJ
ejpam-3461	431	2	.	.	NOUN
ejpam-3461	431	3	2	2	NUM
ejpam-3461	431	4	:	:	SYM
ejpam-3461	431	5	235	235	NUM
ejpam-3461	431	6	-	-	SYM
ejpam-3461	431	7	240	240	NUM
ejpam-3461	431	8	.	.	PUNCT
ejpam-3461	432	1	[	[	X
ejpam-3461	432	2	6	6	NUM
ejpam-3461	432	3	]	]	X
ejpam-3461	432	4	n.	n.	NOUN
ejpam-3461	432	5	ding	ding	PROPN
ejpam-3461	432	6	,	,	PUNCT
ejpam-3461	432	7	y.	y.	PROPN
ejpam-3461	432	8	ibrahim	ibrahim	PROPN
ejpam-3461	432	9	,	,	PUNCT
ejpam-3461	432	10	m.	m.	NOUN
ejpam-3461	432	11	yousif	yousif	PROPN
ejpam-3461	432	12	and	and	CCONJ
ejpam-3461	432	13	y.	y.	PROPN
ejpam-3461	432	14	zhou	zhou	PROPN
ejpam-3461	432	15	,	,	PUNCT
ejpam-3461	432	16	d4	d4	NOUN
ejpam-3461	432	17	-	-	PUNCT
ejpam-3461	432	18	modules	module	NOUN
ejpam-3461	432	19	,	,	PUNCT
ejpam-3461	432	20	j.	j.	PROPN
ejpam-3461	432	21	algebra	algebra	PROPN
ejpam-3461	432	22	appl	appl	PROPN
ejpam-3461	432	23	.	.	PROPN
ejpam-3461	432	24	16(9	16(9	NUM
ejpam-3461	432	25	)	)	PUNCT
ejpam-3461	432	26	,	,	PUNCT
ejpam-3461	432	27	1750166	1750166	NUM
ejpam-3461	432	28	(	(	PUNCT
ejpam-3461	432	29	25	25	NUM
ejpam-3461	432	30	pages	page	NOUN
ejpam-3461	432	31	)	)	PUNCT
ejpam-3461	432	32	,	,	PUNCT
ejpam-3461	432	33	2017	2017	NUM
ejpam-3461	432	34	.	.	PUNCT
ejpam-3461	433	1	[	[	X
ejpam-3461	433	2	7	7	X
ejpam-3461	433	3	]	]	X
ejpam-3461	433	4	n.	n.	NOUN
ejpam-3461	433	5	v.	v.	ADP
ejpam-3461	433	6	dung	dung	PROPN
ejpam-3461	433	7	,	,	PUNCT
ejpam-3461	433	8	d.	d.	PROPN
ejpam-3461	433	9	v.	v.	PROPN
ejpam-3461	433	10	huynh	huynh	PROPN
ejpam-3461	433	11	,	,	PUNCT
ejpam-3461	433	12	p.	p.	PROPN
ejpam-3461	433	13	f.	f.	PROPN
ejpam-3461	433	14	smith	smith	PROPN
ejpam-3461	433	15	and	and	CCONJ
ejpam-3461	433	16	r.	r.	PROPN
ejpam-3461	433	17	wisbauer	wisbauer	PROPN
ejpam-3461	433	18	(	(	PUNCT
ejpam-3461	433	19	1994	1994	NUM
ejpam-3461	433	20	)	)	PUNCT
ejpam-3461	433	21	,	,	PUNCT
ejpam-3461	433	22	extending	extend	VERB
ejpam-3461	433	23	modules	module	NOUN
ejpam-3461	433	24	,	,	PUNCT
ejpam-3461	433	25	pitman	pitman	NOUN
ejpam-3461	433	26	research	research	NOUN
ejpam-3461	433	27	notes	note	NOUN
ejpam-3461	433	28	in	in	ADP
ejpam-3461	433	29	mathematics	mathematics	PROPN
ejpam-3461	433	30	313	313	NUM
ejpam-3461	433	31	.	.	PUNCT
ejpam-3461	434	1	harlow	harlow	PROPN
ejpam-3461	434	2	:	:	PUNCT
ejpam-3461	434	3	longman	longman	NOUN
ejpam-3461	434	4	.	.	PUNCT
ejpam-3461	435	1	[	[	X
ejpam-3461	435	2	8	8	NUM
ejpam-3461	435	3	]	]	PUNCT
ejpam-3461	435	4	a.	a.	NOUN
ejpam-3461	435	5	ghorbani	ghorbani	NOUN
ejpam-3461	435	6	,	,	PUNCT
ejpam-3461	435	7	co	co	ADJ
ejpam-3461	435	8	-	-	ADJ
ejpam-3461	435	9	epi	epi	ADJ
ejpam-3461	435	10	-	-	ADJ
ejpam-3461	435	11	retractable	retractable	ADJ
ejpam-3461	435	12	modules	module	NOUN
ejpam-3461	435	13	and	and	CCONJ
ejpam-3461	435	14	co	co	NOUN
ejpam-3461	435	15	-	-	NOUN
ejpam-3461	435	16	pri	pri	ADJ
ejpam-3461	435	17	rings	ring	NOUN
ejpam-3461	435	18	,	,	PUNCT
ejpam-3461	435	19	comm	comm	NOUN
ejpam-3461	435	20	.	.	PUNCT
ejpam-3461	436	1	algebra	algebra	NOUN
ejpam-3461	436	2	38(2010	38(2010	NUM
ejpam-3461	436	3	)	)	PUNCT
ejpam-3461	436	4	,	,	PUNCT
ejpam-3461	436	5	no	no	INTJ
ejpam-3461	436	6	.	.	NOUN
ejpam-3461	436	7	10	10	NUM
ejpam-3461	436	8	,	,	PUNCT
ejpam-3461	436	9	3589	3589	NUM
ejpam-3461	436	10	-	-	SYM
ejpam-3461	436	11	3596	3596	NUM
ejpam-3461	436	12	.	.	PUNCT
ejpam-3461	437	1	[	[	X
ejpam-3461	437	2	9	9	NUM
ejpam-3461	437	3	]	]	PUNCT
ejpam-3461	437	4	a.	a.	NOUN
ejpam-3461	437	5	hamdouni	hamdouni	PROPN
ejpam-3461	437	6	,	,	PUNCT
ejpam-3461	437	7	a.	a.	PROPN
ejpam-3461	437	8	c.	c.	PROPN
ejpam-3461	437	9	ozcan	ozcan	PROPN
ejpam-3461	437	10	and	and	CCONJ
ejpam-3461	437	11	a.	a.	NOUN
ejpam-3461	437	12	harmanci	harmanci	PROPN
ejpam-3461	437	13	,	,	PUNCT
ejpam-3461	437	14	charcterizations	charcterization	NOUN
ejpam-3461	437	15	of	of	ADP
ejpam-3461	437	16	modules	module	NOUN
ejpam-3461	437	17	and	and	CCONJ
ejpam-3461	437	18	rings	ring	NOUN
ejpam-3461	437	19	by	by	ADP
ejpam-3461	437	20	the	the	DET
ejpam-3461	437	21	summand	summand	NOUN
ejpam-3461	437	22	intersection	intersection	NOUN
ejpam-3461	437	23	property	property	NOUN
ejpam-3461	437	24	and	and	CCONJ
ejpam-3461	437	25	the	the	DET
ejpam-3461	437	26	summand	summand	NOUN
ejpam-3461	437	27	sum	sum	NOUN
ejpam-3461	437	28	property	property	NOUN
ejpam-3461	437	29	,	,	PUNCT
ejpam-3461	437	30	jp	jp	NOUN
ejpam-3461	437	31	j.	j.	PROPN
ejpam-3461	437	32	algebra	algebra	PROPN
ejpam-3461	437	33	number	number	NOUN
ejpam-3461	437	34	theory	theory	NOUN
ejpam-3461	437	35	appl	appl	NOUN
ejpam-3461	437	36	.	.	PUNCT
ejpam-3461	438	1	5(2005	5(2005	NUM
ejpam-3461	438	2	)	)	PUNCT
ejpam-3461	438	3	,	,	PUNCT
ejpam-3461	439	1	no	no	INTJ
ejpam-3461	439	2	.	.	NOUN
ejpam-3461	439	3	3	3	NUM
ejpam-3461	439	4	,	,	PUNCT
ejpam-3461	439	5	469	469	NUM
ejpam-3461	439	6	-	-	NUM
ejpam-3461	439	7	490	490	NUM
ejpam-3461	439	8	.	.	PUNCT
ejpam-3461	440	1	[	[	X
ejpam-3461	440	2	10	10	NUM
ejpam-3461	440	3	]	]	X
ejpam-3461	440	4	d.v	d.v	PROPN
ejpam-3461	440	5	huynh	huynh	PROPN
ejpam-3461	440	6	,	,	PUNCT
ejpam-3461	440	7	some	some	DET
ejpam-3461	440	8	remarks	remark	NOUN
ejpam-3461	440	9	on	on	ADP
ejpam-3461	440	10	cs	cs	ADJ
ejpam-3461	440	11	modules	module	NOUN
ejpam-3461	440	12	and	and	CCONJ
ejpam-3461	440	13	si	si	PROPN
ejpam-3461	440	14	rings	ring	NOUN
ejpam-3461	440	15	,	,	PUNCT
ejpam-3461	440	16	aust	aust	PROPN
ejpam-3461	440	17	.	.	PUNCT
ejpam-3461	440	18	math	math	PROPN
ejpam-3461	440	19	.	.	PUNCT
ejpam-3461	441	1	soc	soc	PROPN
ejpam-3461	441	2	.	.	PUNCT
ejpam-3461	442	1	65(2002	65(2002	NUM
ejpam-3461	442	2	)	)	PUNCT
ejpam-3461	442	3	,	,	PUNCT
ejpam-3461	442	4	461	461	NUM
ejpam-3461	442	5	-	-	SYM
ejpam-3461	442	6	466	466	NUM
ejpam-3461	442	7	.	.	PUNCT
ejpam-3461	443	1	[	[	X
ejpam-3461	443	2	11	11	NUM
ejpam-3461	443	3	]	]	X
ejpam-3461	443	4	d.	d.	PROPN
ejpam-3461	443	5	v.	v.	PROPN
ejpam-3461	443	6	huynh	huynh	PROPN
ejpam-3461	443	7	and	and	CCONJ
ejpam-3461	443	8	s.	s.	PROPN
ejpam-3461	443	9	t.	t.	PROPN
ejpam-3461	443	10	rizvi	rizvi	PROPN
ejpam-3461	443	11	,	,	PUNCT
ejpam-3461	443	12	an	an	DET
ejpam-3461	443	13	approche	approche	NOUN
ejpam-3461	443	14	to	to	ADP
ejpam-3461	443	15	boyle	boyle	PROPN
ejpam-3461	443	16	’s	’s	PART
ejpam-3461	443	17	conjecture	conjecture	NOUN
ejpam-3461	443	18	,	,	PUNCT
ejpam-3461	443	19	proceeding	proceeding	NOUN
ejpam-3461	443	20	of	of	ADP
ejpam-3461	443	21	the	the	DET
ejpam-3461	443	22	edinburg	edinburg	PROPN
ejpam-3461	443	23	.	.	PUNCT
ejpam-3461	444	1	math	math	PROPN
ejpam-3461	444	2	.	.	PUNCT
ejpam-3461	445	1	society	society	PROPN
ejpam-3461	445	2	40(1997	40(1997	PROPN
ejpam-3461	445	3	)	)	PUNCT
ejpam-3461	445	4	,	,	PUNCT
ejpam-3461	445	5	267	267	NUM
ejpam-3461	445	6	-	-	SYM
ejpam-3461	445	7	273	273	NUM
ejpam-3461	445	8	.	.	PUNCT
ejpam-3461	446	1	[	[	X
ejpam-3461	446	2	12	12	NUM
ejpam-3461	446	3	]	]	PUNCT
ejpam-3461	446	4	t.	t.	PROPN
ejpam-3461	446	5	y.	y.	PROPN
ejpam-3461	446	6	lam	lam	PROPN
ejpam-3461	446	7	,	,	PUNCT
ejpam-3461	446	8	lectures	lecture	VERB
ejpam-3461	446	9	on	on	ADP
ejpam-3461	446	10	modules	module	NOUN
ejpam-3461	446	11	and	and	CCONJ
ejpam-3461	446	12	rings	ring	NOUN
ejpam-3461	446	13	,	,	PUNCT
ejpam-3461	446	14	g.t.m.(189	g.t.m.(189	NOUN
ejpam-3461	446	15	)	)	PUNCT
ejpam-3461	446	16	,	,	PUNCT
ejpam-3461	446	17	springer	springer	NOUN
ejpam-3461	446	18	-	-	PUNCT
ejpam-3461	446	19	verlag	verlag	PROPN
ejpam-3461	446	20	,	,	PUNCT
ejpam-3461	446	21	berlinheidelber	berlinheidelber	PROPN
ejpam-3461	446	22	,	,	PUNCT
ejpam-3461	446	23	new	new	PROPN
ejpam-3461	446	24	york	york	PROPN
ejpam-3461	446	25	,	,	PUNCT
ejpam-3461	446	26	1999	1999	NUM
ejpam-3461	446	27	.	.	PUNCT
ejpam-3461	447	1	[	[	X
ejpam-3461	447	2	13	13	NUM
ejpam-3461	447	3	]	]	X
ejpam-3461	447	4	g.	g.	PROPN
ejpam-3461	447	5	lee	lee	PROPN
ejpam-3461	447	6	,	,	PUNCT
ejpam-3461	447	7	theory	theory	NOUN
ejpam-3461	447	8	of	of	ADP
ejpam-3461	447	9	rickart	rickart	NOUN
ejpam-3461	447	10	modules	module	NOUN
ejpam-3461	447	11	,	,	PUNCT
ejpam-3461	447	12	ph	ph	PROPN
ejpam-3461	447	13	.	.	PROPN
ejpam-3461	447	14	d.	d.	PROPN
ejpam-3461	447	15	thesis	thesis	PROPN
ejpam-3461	447	16	,	,	PUNCT
ejpam-3461	447	17	m.s	m.s	PROPN
ejpam-3461	447	18	.	.	PROPN
ejpam-3461	447	19	,	,	PUNCT
ejpam-3461	447	20	graduate	graduate	NOUN
ejpam-3461	447	21	,	,	PUNCT
ejpam-3461	447	22	school	school	NOUN
ejpam-3461	447	23	of	of	ADP
ejpam-3461	447	24	the	the	DET
ejpam-3461	447	25	ohio	ohio	PROPN
ejpam-3461	447	26	state	state	PROPN
ejpam-3461	447	27	university	university	PROPN
ejpam-3461	447	28	(	(	PUNCT
ejpam-3461	447	29	2010	2010	NUM
ejpam-3461	447	30	)	)	PUNCT
ejpam-3461	447	31	.	.	PUNCT
ejpam-3461	448	1	[	[	X
ejpam-3461	448	2	14	14	NUM
ejpam-3461	448	3	]	]	PUNCT
ejpam-3461	448	4	s.	s.	PROPN
ejpam-3461	448	5	h.	h.	PROPN
ejpam-3461	448	6	mohamed	mohamed	PROPN
ejpam-3461	448	7	and	and	CCONJ
ejpam-3461	448	8	b.	b.	PROPN
ejpam-3461	448	9	j.	j.	PROPN
ejpam-3461	448	10	muller	muller	PROPN
ejpam-3461	448	11	,	,	PUNCT
ejpam-3461	448	12	continuous	continuous	ADJ
ejpam-3461	448	13	and	and	CCONJ
ejpam-3461	448	14	discrete	discrete	ADJ
ejpam-3461	448	15	modules	module	NOUN
ejpam-3461	448	16	,	,	PUNCT
ejpam-3461	448	17	lms	lm	NOUN
ejpam-3461	448	18	lecture	lecture	NOUN
ejpam-3461	448	19	note	note	NOUN
ejpam-3461	448	20	series	series	NOUN
ejpam-3461	448	21	,	,	PUNCT
ejpam-3461	448	22	147	147	NUM
ejpam-3461	448	23	.	.	PUNCT
ejpam-3461	449	1	cambridge	cambridge	PROPN
ejpam-3461	449	2	university	university	PROPN
ejpam-3461	449	3	press	press	PROPN
ejpam-3461	449	4	,	,	PUNCT
ejpam-3461	449	5	cambridge	cambridge	PROPN
ejpam-3461	449	6	,	,	PUNCT
ejpam-3461	449	7	1990	1990	NUM
ejpam-3461	449	8	.	.	PUNCT
ejpam-3461	450	1	[	[	X
ejpam-3461	450	2	15	15	NUM
ejpam-3461	450	3	]	]	X
ejpam-3461	450	4	h.	h.	PROPN
ejpam-3461	450	5	mostafanasab	mostafanasab	VERB
ejpam-3461	450	6	,	,	PUNCT
ejpam-3461	450	7	applications	application	NOUN
ejpam-3461	450	8	of	of	ADP
ejpam-3461	450	9	epi	epi	NOUN
ejpam-3461	450	10	-	-	NOUN
ejpam-3461	450	11	retractable	retractable	ADJ
ejpam-3461	450	12	and	and	CCONJ
ejpam-3461	450	13	co	co	NOUN
ejpam-3461	450	14	-	-	ADJ
ejpam-3461	450	15	epi	epi	ADJ
ejpam-3461	450	16	-	-	ADJ
ejpam-3461	450	17	retractable	retractable	ADJ
ejpam-3461	450	18	modules	module	NOUN
ejpam-3461	450	19	,	,	PUNCT
ejpam-3461	450	20	bull	bull	NOUN
ejpam-3461	450	21	.	.	PUNCT
ejpam-3461	451	1	iranian	iranian	ADJ
ejpam-3461	451	2	math	math	PROPN
ejpam-3461	451	3	.	.	PUNCT
ejpam-3461	452	1	soc	soc	PROPN
ejpam-3461	452	2	.	.	PUNCT
ejpam-3461	453	1	38	38	NUM
ejpam-3461	453	2	(	(	PUNCT
ejpam-3461	453	3	2013	2013	NUM
ejpam-3461	453	4	)	)	PUNCT
ejpam-3461	453	5	,	,	PUNCT
ejpam-3461	453	6	no	no	INTJ
ejpam-3461	453	7	.	.	NOUN
ejpam-3461	453	8	5	5	NUM
ejpam-3461	453	9	,	,	PUNCT
ejpam-3461	453	10	903	903	NUM
ejpam-3461	453	11	-	-	SYM
ejpam-3461	453	12	917	917	NUM
ejpam-3461	453	13	.	.	PUNCT
ejpam-3461	454	1	[	[	X
ejpam-3461	454	2	16	16	NUM
ejpam-3461	454	3	]	]	PUNCT
ejpam-3461	454	4	s.	s.	PROPN
ejpam-3461	454	5	t.	t.	PROPN
ejpam-3461	454	6	rizvi	rizvi	PROPN
ejpam-3461	454	7	and	and	CCONJ
ejpam-3461	454	8	c.	c.	PROPN
ejpam-3461	454	9	s.	s.	PROPN
ejpam-3461	454	10	roman	roman	PROPN
ejpam-3461	454	11	,	,	PUNCT
ejpam-3461	454	12	baer	baer	PROPN
ejpam-3461	454	13	and	and	CCONJ
ejpam-3461	454	14	quasi	quasi	PROPN
ejpam-3461	454	15	-	-	ADJ
ejpam-3461	454	16	baer	baer	ADJ
ejpam-3461	454	17	modules	module	NOUN
ejpam-3461	454	18	.	.	PUNCT
ejpam-3461	455	1	comm	comm	NOUN
ejpam-3461	455	2	.	.	PUNCT
ejpam-3461	456	1	algebra	algebra	NOUN
ejpam-3461	456	2	,	,	PUNCT
ejpam-3461	456	3	32	32	NUM
ejpam-3461	456	4	(	(	PUNCT
ejpam-3461	456	5	2004	2004	NUM
ejpam-3461	456	6	):	):	PUNCT
ejpam-3461	456	7	103	103	NUM
ejpam-3461	456	8	-	-	SYM
ejpam-3461	456	9	123	123	NUM
ejpam-3461	456	10	.	.	PUNCT
ejpam-3461	457	1	[	[	X
ejpam-3461	457	2	17	17	NUM
ejpam-3461	457	3	]	]	PUNCT
ejpam-3461	457	4	s.	s.	PROPN
ejpam-3461	457	5	t.	t.	PROPN
ejpam-3461	457	6	rizvi	rizvi	PROPN
ejpam-3461	457	7	and	and	CCONJ
ejpam-3461	457	8	m.	m.	PROPN
ejpam-3461	457	9	f.	f.	PROPN
ejpam-3461	457	10	yousif	yousif	PROPN
ejpam-3461	457	11	,	,	PUNCT
ejpam-3461	457	12	on	on	ADP
ejpam-3461	457	13	continuous	continuous	ADJ
ejpam-3461	457	14	and	and	CCONJ
ejpam-3461	457	15	singular	singular	ADJ
ejpam-3461	457	16	modules	module	NOUN
ejpam-3461	457	17	,	,	PUNCT
ejpam-3461	457	18	non	non	X
ejpam-3461	457	19	commutative	commutative	ADJ
ejpam-3461	457	20	ring	ring	NOUN
ejpam-3461	457	21	theory	theory	NOUN
ejpam-3461	457	22	,	,	PUNCT
ejpam-3461	457	23	s.	s.	PROPN
ejpam-3461	457	24	k.	k.	PROPN
ejpam-3461	457	25	jain	jain	PROPN
ejpam-3461	457	26	and	and	CCONJ
ejpam-3461	457	27	s.	s.	PROPN
ejpam-3461	457	28	r.	r.	PROPN
ejpam-3461	457	29	lópez	lópez	PROPN
ejpam-3461	457	30	-	-	PUNCT
ejpam-3461	457	31	permouth	permouth	NOUN
ejpam-3461	457	32	,	,	PUNCT
ejpam-3461	457	33	eds	ed	NOUN
ejpam-3461	457	34	,	,	PUNCT
ejpam-3461	457	35	lecture	lecture	NOUN
ejpam-3461	457	36	notes	note	NOUN
ejpam-3461	457	37	in	in	ADP
ejpam-3461	457	38	math	math	NOUN
ejpam-3461	457	39	.	.	PUNCT
ejpam-3461	458	1	1448	1448	NUM
ejpam-3461	458	2	,	,	PUNCT
ejpam-3461	458	3	springer	springer	NOUN
ejpam-3461	458	4	-	-	PUNCT
ejpam-3461	458	5	verlag	verlag	PROPN
ejpam-3461	458	6	,	,	PUNCT
ejpam-3461	458	7	berlin	berlin	PROPN
ejpam-3461	458	8	(	(	PUNCT
ejpam-3461	458	9	1990	1990	NUM
ejpam-3461	458	10	)	)	PUNCT
ejpam-3461	458	11	,	,	PUNCT
ejpam-3461	458	12	pp:116	pp:116	PROPN
ejpam-3461	458	13	-	-	X
ejpam-3461	458	14	124	124	NUM
ejpam-3461	458	15	.	.	PUNCT
ejpam-3461	459	1	[	[	X
ejpam-3461	459	2	18	18	NUM
ejpam-3461	459	3	]	]	PUNCT
ejpam-3461	459	4	p.	p.	PROPN
ejpam-3461	459	5	smith	smith	PROPN
ejpam-3461	459	6	,	,	PUNCT
ejpam-3461	459	7	modules	module	NOUN
ejpam-3461	459	8	with	with	ADP
ejpam-3461	459	9	many	many	ADJ
ejpam-3461	459	10	homomorphisms	homomorphism	NOUN
ejpam-3461	459	11	,	,	PUNCT
ejpam-3461	459	12	j.	j.	PROPN
ejpam-3461	459	13	pure	pure	ADJ
ejpam-3461	459	14	and	and	CCONJ
ejpam-3461	459	15	appl	appl	NOUN
ejpam-3461	459	16	.	.	PUNCT
ejpam-3461	460	1	algebra	algebra	PROPN
ejpam-3461	460	2	197(2005	197(2005	PROPN
ejpam-3461	460	3	)	)	PUNCT
ejpam-3461	460	4	305	305	NUM
ejpam-3461	460	5	-	-	SYM
ejpam-3461	460	6	321	321	NUM
ejpam-3461	460	7	.	.	PUNCT
ejpam-3461	461	1	[	[	X
ejpam-3461	461	2	19	19	NUM
ejpam-3461	461	3	]	]	X
ejpam-3461	461	4	r.	r.	PROPN
ejpam-3461	461	5	wisbauer	wisbauer	NOUN
ejpam-3461	461	6	,	,	PUNCT
ejpam-3461	461	7	foundations	foundation	NOUN
ejpam-3461	461	8	of	of	ADP
ejpam-3461	461	9	module	module	NOUN
ejpam-3461	461	10	and	and	CCONJ
ejpam-3461	461	11	ring	ring	NOUN
ejpam-3461	461	12	theory	theory	NOUN
ejpam-3461	461	13	,	,	PUNCT
ejpam-3461	461	14	gordon	gordon	PROPN
ejpam-3461	461	15	and	and	CCONJ
ejpam-3461	461	16	breach	breach	VERB
ejpam-3461	461	17	sciences	science	NOUN
ejpam-3461	461	18	publishers	publisher	NOUN
ejpam-3461	461	19	,	,	PUNCT
ejpam-3461	461	20	philadelphia	philadelphia	PROPN
ejpam-3461	461	21	,	,	PUNCT
ejpam-3461	461	22	1991	1991	NUM
ejpam-3461	461	23	.	.	PUNCT