id	sid	tid	token	lemma	pos
ejpam-3588	1	1	european	european	PROPN
ejpam-3588	1	2	journal	journal	PROPN
ejpam-3588	1	3	of	of	ADP
ejpam-3588	1	4	pure	pure	ADJ
ejpam-3588	1	5	and	and	CCONJ
ejpam-3588	1	6	applied	apply	VERB
ejpam-3588	1	7	mathematics	mathematic	NOUN
ejpam-3588	1	8	vol	vol	NOUN
ejpam-3588	1	9	.	.	PROPN
ejpam-3588	2	1	13	13	NUM
ejpam-3588	2	2	,	,	PUNCT
ejpam-3588	2	3	no	no	INTJ
ejpam-3588	2	4	.	.	NOUN
ejpam-3588	2	5	1	1	NUM
ejpam-3588	2	6	,	,	PUNCT
ejpam-3588	2	7	2020	2020	NUM
ejpam-3588	2	8	,	,	PUNCT
ejpam-3588	2	9	158	158	NUM
ejpam-3588	2	10	-	-	SYM
ejpam-3588	2	11	169	169	NUM
ejpam-3588	2	12	issn	issn	PROPN
ejpam-3588	2	13	1307	1307	NUM
ejpam-3588	2	14	-	-	SYM
ejpam-3588	2	15	5543	5543	NUM
ejpam-3588	2	16	–	–	PUNCT
ejpam-3588	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3588	2	18	published	publish	VERB
ejpam-3588	2	19	by	by	ADP
ejpam-3588	2	20	new	new	PROPN
ejpam-3588	2	21	york	york	PROPN
ejpam-3588	2	22	business	business	PROPN
ejpam-3588	2	23	global	global	PROPN
ejpam-3588	2	24	some	some	DET
ejpam-3588	2	25	results	result	NOUN
ejpam-3588	2	26	on	on	ADP
ejpam-3588	2	27	c	c	NOUN
ejpam-3588	2	28	-	-	PUNCT
ejpam-3588	2	29	retractable	retractable	ADJ
ejpam-3588	2	30	modules	module	NOUN
ejpam-3588	2	31	abdoul	abdoul	VERB
ejpam-3588	2	32	djibril	djibril	PROPN
ejpam-3588	2	33	diallo1	diallo1	PROPN
ejpam-3588	2	34	,	,	PUNCT
ejpam-3588	2	35	papa	papa	NOUN
ejpam-3588	2	36	cheikhou	cheikhou	PROPN
ejpam-3588	2	37	diop2,∗	diop2,∗	PROPN
ejpam-3588	2	38	,	,	PUNCT
ejpam-3588	2	39	mamadou	mamadou	PROPN
ejpam-3588	2	40	barry1	barry1	PROPN
ejpam-3588	2	41	1	1	NUM
ejpam-3588	2	42	département	département	PROPN
ejpam-3588	2	43	de	de	X
ejpam-3588	2	44	mathématiques	mathématiques	X
ejpam-3588	2	45	et	et	PROPN
ejpam-3588	2	46	informatique	informatique	PROPN
ejpam-3588	2	47	,	,	PUNCT
ejpam-3588	2	48	faculté	faculté	NOUN
ejpam-3588	2	49	des	des	PROPN
ejpam-3588	2	50	sciences	sciences	PROPN
ejpam-3588	2	51	et	et	PROPN
ejpam-3588	2	52	techniques	technique	NOUN
ejpam-3588	2	53	,	,	PUNCT
ejpam-3588	2	54	université	université	NOUN
ejpam-3588	2	55	cheikh	cheikh	PROPN
ejpam-3588	2	56	anta	anta	PROPN
ejpam-3588	2	57	diop	diop	PROPN
ejpam-3588	2	58	,	,	PUNCT
ejpam-3588	2	59	dakar	dakar	NOUN
ejpam-3588	2	60	,	,	PUNCT
ejpam-3588	2	61	sénégal	sénégal	ADJ
ejpam-3588	2	62	2	2	NUM
ejpam-3588	2	63	département	département	PROPN
ejpam-3588	2	64	de	de	X
ejpam-3588	2	65	mathématiques	mathématiques	X
ejpam-3588	2	66	,	,	PUNCT
ejpam-3588	2	67	ufr	ufr	PROPN
ejpam-3588	2	68	sciences	sciences	PROPN
ejpam-3588	2	69	et	et	PROPN
ejpam-3588	2	70	technologies	technology	NOUN
ejpam-3588	2	71	,	,	PUNCT
ejpam-3588	2	72	université	université	NOUN
ejpam-3588	2	73	de	de	ADP
ejpam-3588	2	74	thiès	thiès	PROPN
ejpam-3588	2	75	,	,	PUNCT
ejpam-3588	2	76	thiès	thiès	NOUN
ejpam-3588	2	77	,	,	PUNCT
ejpam-3588	2	78	sénégal	sénégal	ADJ
ejpam-3588	2	79	abstract	abstract	NOUN
ejpam-3588	2	80	.	.	PUNCT
ejpam-3588	3	1	an	an	DET
ejpam-3588	3	2	r	r	NOUN
ejpam-3588	3	3	-	-	PUNCT
ejpam-3588	3	4	module	module	NOUN
ejpam-3588	3	5	m	m	NOUN
ejpam-3588	3	6	is	be	AUX
ejpam-3588	3	7	called	call	VERB
ejpam-3588	3	8	c	c	ADJ
ejpam-3588	3	9	-	-	PUNCT
ejpam-3588	3	10	retractable	retractable	ADJ
ejpam-3588	3	11	if	if	SCONJ
ejpam-3588	3	12	there	there	PRON
ejpam-3588	3	13	exists	exist	VERB
ejpam-3588	3	14	a	a	DET
ejpam-3588	3	15	nonzero	nonzero	NOUN
ejpam-3588	3	16	homomorphism	homomorphism	NOUN
ejpam-3588	3	17	from	from	ADP
ejpam-3588	3	18	m	m	PRON
ejpam-3588	3	19	to	to	ADP
ejpam-3588	3	20	any	any	PRON
ejpam-3588	3	21	of	of	ADP
ejpam-3588	3	22	its	its	PRON
ejpam-3588	3	23	nonzero	nonzero	ADJ
ejpam-3588	3	24	complement	complement	NOUN
ejpam-3588	3	25	submodules	submodule	NOUN
ejpam-3588	3	26	.	.	PUNCT
ejpam-3588	4	1	in	in	ADP
ejpam-3588	4	2	this	this	DET
ejpam-3588	4	3	paper	paper	NOUN
ejpam-3588	4	4	,	,	PUNCT
ejpam-3588	4	5	we	we	PRON
ejpam-3588	4	6	provide	provide	VERB
ejpam-3588	4	7	some	some	DET
ejpam-3588	4	8	new	new	ADJ
ejpam-3588	4	9	results	result	NOUN
ejpam-3588	4	10	of	of	ADP
ejpam-3588	4	11	cretractable	cretractable	ADJ
ejpam-3588	4	12	modules	module	NOUN
ejpam-3588	4	13	.	.	PUNCT
ejpam-3588	5	1	it	it	PRON
ejpam-3588	5	2	is	be	AUX
ejpam-3588	5	3	shown	show	VERB
ejpam-3588	5	4	that	that	SCONJ
ejpam-3588	5	5	every	every	DET
ejpam-3588	5	6	projective	projective	ADJ
ejpam-3588	5	7	module	module	NOUN
ejpam-3588	5	8	over	over	ADP
ejpam-3588	5	9	a	a	DET
ejpam-3588	5	10	right	right	ADJ
ejpam-3588	5	11	si	si	NOUN
ejpam-3588	5	12	-	-	PUNCT
ejpam-3588	5	13	ring	ring	NOUN
ejpam-3588	5	14	is	be	AUX
ejpam-3588	5	15	c	c	NOUN
ejpam-3588	5	16	-	-	PUNCT
ejpam-3588	5	17	retractable	retractable	ADJ
ejpam-3588	5	18	.	.	PUNCT
ejpam-3588	6	1	a	a	DET
ejpam-3588	6	2	dual	dual	ADJ
ejpam-3588	6	3	baer	baer	PROPN
ejpam-3588	6	4	c	c	ADJ
ejpam-3588	6	5	-	-	PUNCT
ejpam-3588	6	6	retractable	retractable	ADJ
ejpam-3588	6	7	module	module	NOUN
ejpam-3588	6	8	is	be	AUX
ejpam-3588	6	9	a	a	DET
ejpam-3588	6	10	direct	direct	ADJ
ejpam-3588	6	11	sum	sum	NOUN
ejpam-3588	6	12	of	of	ADP
ejpam-3588	6	13	a	a	DET
ejpam-3588	6	14	z2	z2	ADJ
ejpam-3588	6	15	-	-	PUNCT
ejpam-3588	6	16	torsion	torsion	NOUN
ejpam-3588	6	17	module	module	NOUN
ejpam-3588	6	18	and	and	CCONJ
ejpam-3588	6	19	a	a	DET
ejpam-3588	6	20	module	module	NOUN
ejpam-3588	6	21	which	which	PRON
ejpam-3588	6	22	is	be	AUX
ejpam-3588	6	23	a	a	DET
ejpam-3588	6	24	direct	direct	ADJ
ejpam-3588	6	25	sum	sum	NOUN
ejpam-3588	6	26	of	of	ADP
ejpam-3588	6	27	nonsingular	nonsingular	ADJ
ejpam-3588	6	28	uniform	uniform	ADJ
ejpam-3588	6	29	quasi	quasi	PROPN
ejpam-3588	6	30	-	-	PROPN
ejpam-3588	6	31	baer	baer	ADJ
ejpam-3588	6	32	modules	module	NOUN
ejpam-3588	6	33	whose	whose	DET
ejpam-3588	6	34	endomorphism	endomorphism	PROPN
ejpam-3588	6	35	rings	ring	NOUN
ejpam-3588	6	36	are	be	AUX
ejpam-3588	6	37	semilocal	semilocal	ADJ
ejpam-3588	6	38	quasi	quasi	NOUN
ejpam-3588	6	39	-	-	NOUN
ejpam-3588	6	40	baer	baer	PROPN
ejpam-3588	6	41	.	.	PUNCT
ejpam-3588	7	1	conditions	condition	NOUN
ejpam-3588	7	2	are	be	AUX
ejpam-3588	7	3	found	find	VERB
ejpam-3588	7	4	under	under	ADP
ejpam-3588	7	5	which	which	PRON
ejpam-3588	7	6	,	,	PUNCT
ejpam-3588	7	7	a	a	DET
ejpam-3588	7	8	c	c	NOUN
ejpam-3588	7	9	-	-	PUNCT
ejpam-3588	7	10	retractable	retractable	ADJ
ejpam-3588	7	11	module	module	NOUN
ejpam-3588	7	12	is	be	AUX
ejpam-3588	7	13	extending	extend	VERB
ejpam-3588	7	14	,	,	PUNCT
ejpam-3588	7	15	quasicontinuous	quasicontinuous	ADJ
ejpam-3588	7	16	,	,	PUNCT
ejpam-3588	7	17	quasi	quasi	ADJ
ejpam-3588	7	18	-	-	ADJ
ejpam-3588	7	19	injective	injective	ADJ
ejpam-3588	7	20	and	and	CCONJ
ejpam-3588	7	21	retractable	retractable	ADJ
ejpam-3588	7	22	.	.	PUNCT
ejpam-3588	8	1	also	also	ADV
ejpam-3588	8	2	,	,	PUNCT
ejpam-3588	8	3	it	it	PRON
ejpam-3588	8	4	is	be	AUX
ejpam-3588	8	5	shown	show	VERB
ejpam-3588	8	6	that	that	SCONJ
ejpam-3588	8	7	a	a	DET
ejpam-3588	8	8	locally	locally	ADV
ejpam-3588	8	9	noetherian	noetherian	ADJ
ejpam-3588	8	10	c	c	NOUN
ejpam-3588	8	11	-	-	PUNCT
ejpam-3588	8	12	retractable	retractable	ADJ
ejpam-3588	8	13	module	module	NOUN
ejpam-3588	8	14	is	be	AUX
ejpam-3588	8	15	homo	homo	NOUN
ejpam-3588	8	16	-	-	PUNCT
ejpam-3588	8	17	related	relate	VERB
ejpam-3588	8	18	to	to	ADP
ejpam-3588	8	19	a	a	DET
ejpam-3588	8	20	direct	direct	ADJ
ejpam-3588	8	21	sum	sum	NOUN
ejpam-3588	8	22	of	of	ADP
ejpam-3588	8	23	uniform	uniform	ADJ
ejpam-3588	8	24	modules	module	NOUN
ejpam-3588	8	25	.	.	PUNCT
ejpam-3588	9	1	finally	finally	ADV
ejpam-3588	9	2	,	,	PUNCT
ejpam-3588	9	3	rings	ring	NOUN
ejpam-3588	9	4	over	over	ADP
ejpam-3588	9	5	which	which	PRON
ejpam-3588	9	6	every	every	DET
ejpam-3588	9	7	cretractable	cretractable	ADJ
ejpam-3588	9	8	is	be	AUX
ejpam-3588	9	9	a	a	DET
ejpam-3588	9	10	c4	c4	NOUN
ejpam-3588	9	11	-	-	PUNCT
ejpam-3588	9	12	module	module	NOUN
ejpam-3588	9	13	are	be	AUX
ejpam-3588	9	14	determined	determine	VERB
ejpam-3588	9	15	.	.	PUNCT
ejpam-3588	10	1	2020	2020	NUM
ejpam-3588	10	2	mathematics	mathematic	NOUN
ejpam-3588	10	3	subject	subject	NOUN
ejpam-3588	10	4	classifications	classification	NOUN
ejpam-3588	10	5	:	:	PUNCT
ejpam-3588	10	6	13b10,13c05,13c13	13b10,13c05,13c13	NUM
ejpam-3588	10	7	key	key	ADJ
ejpam-3588	10	8	words	word	NOUN
ejpam-3588	10	9	and	and	CCONJ
ejpam-3588	10	10	phrases	phrase	NOUN
ejpam-3588	10	11	:	:	PUNCT
ejpam-3588	10	12	retractable	retractable	ADJ
ejpam-3588	10	13	modules	module	NOUN
ejpam-3588	10	14	,	,	PUNCT
ejpam-3588	10	15	complement	complement	NOUN
ejpam-3588	10	16	submodules	submodule	NOUN
ejpam-3588	10	17	,	,	PUNCT
ejpam-3588	10	18	c	c	NOUN
ejpam-3588	10	19	-	-	PUNCT
ejpam-3588	10	20	retractable	retractable	ADJ
ejpam-3588	10	21	modules	module	NOUN
ejpam-3588	10	22	,	,	PUNCT
ejpam-3588	10	23	projective	projective	ADJ
ejpam-3588	10	24	modules	module	NOUN
ejpam-3588	10	25	1	1	NUM
ejpam-3588	10	26	.	.	PUNCT
ejpam-3588	11	1	introduction	introduction	NOUN
ejpam-3588	11	2	throughout	throughout	ADP
ejpam-3588	11	3	all	all	DET
ejpam-3588	11	4	rings	ring	NOUN
ejpam-3588	11	5	are	be	AUX
ejpam-3588	11	6	associative	associative	ADJ
ejpam-3588	11	7	with	with	ADP
ejpam-3588	11	8	identity	identity	NOUN
ejpam-3588	11	9	and	and	CCONJ
ejpam-3588	11	10	all	all	DET
ejpam-3588	11	11	modules	module	NOUN
ejpam-3588	11	12	are	be	AUX
ejpam-3588	11	13	unitary	unitary	ADJ
ejpam-3588	11	14	right	right	ADJ
ejpam-3588	11	15	module	module	NOUN
ejpam-3588	11	16	.	.	PUNCT
ejpam-3588	12	1	let	let	VERB
ejpam-3588	12	2	r	r	PRON
ejpam-3588	12	3	be	be	AUX
ejpam-3588	12	4	a	a	DET
ejpam-3588	12	5	ring	ring	NOUN
ejpam-3588	12	6	.	.	PUNCT
ejpam-3588	13	1	following	follow	VERB
ejpam-3588	13	2	[	[	X
ejpam-3588	13	3	19	19	NUM
ejpam-3588	13	4	]	]	PUNCT
ejpam-3588	13	5	,	,	PUNCT
ejpam-3588	13	6	we	we	PRON
ejpam-3588	13	7	say	say	VERB
ejpam-3588	13	8	that	that	SCONJ
ejpam-3588	13	9	an	an	DET
ejpam-3588	13	10	r	r	NOUN
ejpam-3588	13	11	-	-	PUNCT
ejpam-3588	13	12	module	module	NOUN
ejpam-3588	13	13	m	m	NOUN
ejpam-3588	13	14	is	be	AUX
ejpam-3588	13	15	retractable	retractable	ADJ
ejpam-3588	13	16	if	if	SCONJ
ejpam-3588	13	17	homr(m	homr(m	PROPN
ejpam-3588	13	18	,	,	PUNCT
ejpam-3588	13	19	n	n	CCONJ
ejpam-3588	13	20	)	)	PUNCT
ejpam-3588	13	21	6=	6=	X
ejpam-3588	13	22	{	{	PUNCT
ejpam-3588	13	23	0	0	NUM
ejpam-3588	13	24	}	}	PUNCT
ejpam-3588	13	25	for	for	ADP
ejpam-3588	13	26	any	any	DET
ejpam-3588	13	27	nonzero	nonzero	PROPN
ejpam-3588	13	28	submodules	submodule	NOUN
ejpam-3588	13	29	n	n	PROPN
ejpam-3588	13	30	of	of	ADP
ejpam-3588	13	31	m	m	PROPN
ejpam-3588	13	32	.	.	PUNCT
ejpam-3588	14	1	it	it	PRON
ejpam-3588	14	2	is	be	AUX
ejpam-3588	14	3	shown	show	VERB
ejpam-3588	14	4	in	in	ADP
ejpam-3588	14	5	[	[	X
ejpam-3588	14	6	19	19	NUM
ejpam-3588	14	7	]	]	PUNCT
ejpam-3588	14	8	that	that	SCONJ
ejpam-3588	14	9	every	every	DET
ejpam-3588	14	10	projective	projective	ADJ
ejpam-3588	14	11	module	module	NOUN
ejpam-3588	14	12	over	over	ADP
ejpam-3588	14	13	a	a	DET
ejpam-3588	14	14	right	right	ADJ
ejpam-3588	14	15	v	v	NOUN
ejpam-3588	14	16	-ring	-ring	NOUN
ejpam-3588	14	17	is	be	AUX
ejpam-3588	14	18	retractable	retractable	ADJ
ejpam-3588	14	19	.	.	PUNCT
ejpam-3588	15	1	in	in	ADP
ejpam-3588	15	2	[	[	X
ejpam-3588	15	3	19	19	NUM
ejpam-3588	15	4	]	]	PUNCT
ejpam-3588	15	5	again	again	ADV
ejpam-3588	15	6	,	,	PUNCT
ejpam-3588	15	7	the	the	DET
ejpam-3588	15	8	semisimplicity	semisimplicity	NOUN
ejpam-3588	15	9	of	of	ADP
ejpam-3588	15	10	retractable	retractable	ADJ
ejpam-3588	15	11	modules	module	NOUN
ejpam-3588	15	12	is	be	AUX
ejpam-3588	15	13	studied	study	VERB
ejpam-3588	15	14	.	.	PUNCT
ejpam-3588	16	1	m.	m.	PROPN
ejpam-3588	16	2	r.	r.	PROPN
ejpam-3588	16	3	vedadi	vedadi	PROPN
ejpam-3588	17	1	[	[	X
ejpam-3588	17	2	23	23	NUM
ejpam-3588	17	3	]	]	PUNCT
ejpam-3588	17	4	,	,	PUNCT
ejpam-3588	17	5	introduced	introduce	VERB
ejpam-3588	17	6	the	the	DET
ejpam-3588	17	7	concept	concept	NOUN
ejpam-3588	17	8	of	of	ADP
ejpam-3588	17	9	essentially	essentially	ADV
ejpam-3588	17	10	retractable	retractable	ADJ
ejpam-3588	17	11	modules	module	NOUN
ejpam-3588	17	12	and	and	CCONJ
ejpam-3588	17	13	proved	prove	VERB
ejpam-3588	17	14	that	that	SCONJ
ejpam-3588	17	15	over	over	ADP
ejpam-3588	17	16	semiprime	semiprime	NOUN
ejpam-3588	17	17	right	right	ADJ
ejpam-3588	17	18	nonsingular	nonsingular	PROPN
ejpam-3588	17	19	rings	ring	NOUN
ejpam-3588	17	20	,	,	PUNCT
ejpam-3588	17	21	a	a	DET
ejpam-3588	17	22	nonsingular	nonsingular	ADJ
ejpam-3588	17	23	essentially	essentially	ADV
ejpam-3588	17	24	retractable	retractable	ADJ
ejpam-3588	17	25	module	module	NOUN
ejpam-3588	17	26	is	be	AUX
ejpam-3588	17	27	precisely	precisely	ADV
ejpam-3588	17	28	a	a	DET
ejpam-3588	17	29	module	module	NOUN
ejpam-3588	17	30	with	with	ADP
ejpam-3588	17	31	non	non	ADJ
ejpam-3588	17	32	-	-	ADJ
ejpam-3588	17	33	zero	zero	ADJ
ejpam-3588	17	34	dual	dual	NOUN
ejpam-3588	17	35	.	.	PUNCT
ejpam-3588	18	1	in	in	ADP
ejpam-3588	18	2	[	[	X
ejpam-3588	18	3	7	7	NUM
ejpam-3588	18	4	]	]	PUNCT
ejpam-3588	18	5	,	,	PUNCT
ejpam-3588	18	6	a.	a.	NOUN
ejpam-3588	18	7	ghorbani	ghorbani	NOUN
ejpam-3588	18	8	and	and	CCONJ
ejpam-3588	18	9	m.	m.	PROPN
ejpam-3588	18	10	r.	r.	PROPN
ejpam-3588	18	11	vedadi	vedadi	PROPN
ejpam-3588	18	12	introduced	introduce	VERB
ejpam-3588	18	13	and	and	CCONJ
ejpam-3588	18	14	studied	study	VERB
ejpam-3588	18	15	the	the	DET
ejpam-3588	18	16	notion	notion	NOUN
ejpam-3588	18	17	of	of	ADP
ejpam-3588	18	18	epi	epi	NOUN
ejpam-3588	18	19	-	-	ADJ
ejpam-3588	18	20	retractable	retractable	ADJ
ejpam-3588	18	21	module	module	NOUN
ejpam-3588	18	22	,	,	PUNCT
ejpam-3588	18	23	where	where	SCONJ
ejpam-3588	18	24	a	a	DET
ejpam-3588	18	25	module	module	NOUN
ejpam-3588	18	26	m	m	VERB
ejpam-3588	18	27	is	be	AUX
ejpam-3588	18	28	called	call	VERB
ejpam-3588	18	29	epi	epi	NOUN
ejpam-3588	18	30	-	-	NOUN
ejpam-3588	18	31	retractable	retractable	ADJ
ejpam-3588	18	32	if	if	SCONJ
ejpam-3588	18	33	every	every	DET
ejpam-3588	18	34	submodule	submodule	NOUN
ejpam-3588	18	35	of	of	ADP
ejpam-3588	18	36	m	m	PROPN
ejpam-3588	18	37	is	be	AUX
ejpam-3588	18	38	a	a	DET
ejpam-3588	18	39	homomorphic	homomorphic	ADJ
ejpam-3588	18	40	image	image	NOUN
ejpam-3588	18	41	of	of	ADP
ejpam-3588	18	42	m	m	PROPN
ejpam-3588	18	43	.	.	PUNCT
ejpam-3588	19	1	they	they	PRON
ejpam-3588	19	2	reveal	reveal	VERB
ejpam-3588	19	3	some	some	DET
ejpam-3588	19	4	applications	application	NOUN
ejpam-3588	19	5	of	of	ADP
ejpam-3588	19	6	projective	projective	ADJ
ejpam-3588	19	7	,	,	PUNCT
ejpam-3588	19	8	nonsingular	nonsingular	ADJ
ejpam-3588	19	9	,	,	PUNCT
ejpam-3588	19	10	injective	injective	ADJ
ejpam-3588	19	11	epi	epi	NOUN
ejpam-3588	19	12	-	-	ADJ
ejpam-3588	19	13	retractable	retractable	ADJ
ejpam-3588	19	14	∗corresponding	∗corresponde	VERB
ejpam-3588	19	15	author	author	NOUN
ejpam-3588	19	16	.	.	PUNCT
ejpam-3588	20	1	doi	doi	NOUN
ejpam-3588	20	2	:	:	PUNCT
ejpam-3588	20	3	https://doi.org/10.29020/nybg.ejpam.v13i1.3588	https://doi.org/10.29020/nybg.ejpam.v13i1.3588	ADP
ejpam-3588	20	4	email	email	NOUN
ejpam-3588	20	5	addresses	address	NOUN
ejpam-3588	20	6	:	:	PUNCT
ejpam-3588	20	7	cheikpapa@yahoo.fr	cheikpapa@yahoo.fr	PROPN
ejpam-3588	20	8	(	(	PUNCT
ejpam-3588	20	9	p.	p.	NOUN
ejpam-3588	20	10	c.	c.	PROPN
ejpam-3588	20	11	diop	diop	PROPN
ejpam-3588	20	12	)	)	PUNCT
ejpam-3588	20	13	,	,	PUNCT
ejpam-3588	20	14	dialloabdoulaziz58@yahoo.fr	dialloabdoulaziz58@yahoo.fr	PROPN
ejpam-3588	20	15	(	(	PUNCT
ejpam-3588	20	16	a.	a.	PROPN
ejpam-3588	20	17	d.	d.	PROPN
ejpam-3588	20	18	diallo	diallo	PROPN
ejpam-3588	20	19	)	)	PUNCT
ejpam-3588	20	20	,	,	PUNCT
ejpam-3588	20	21	mansabadion1@hotmail.com	mansabadion1@hotmail.com	X
ejpam-3588	20	22	(	(	PUNCT
ejpam-3588	20	23	m.	m.	NOUN
ejpam-3588	20	24	barry	barry	PROPN
ejpam-3588	20	25	)	)	PUNCT
ejpam-3588	20	26	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3588	21	1	158	158	NUM
ejpam-3588	21	2	c	c	X
ejpam-3588	21	3	©	©	NOUN
ejpam-3588	21	4	2020	2020	NUM
ejpam-3588	21	5	ejpam	ejpam	VERB
ejpam-3588	21	6	all	all	DET
ejpam-3588	21	7	rights	right	NOUN
ejpam-3588	21	8	reserved	reserve	VERB
ejpam-3588	21	9	.	.	PUNCT
ejpam-3588	22	1	a.	a.	PROPN
ejpam-3588	22	2	d.	d.	PROPN
ejpam-3588	22	3	diallo	diallo	PROPN
ejpam-3588	22	4	,	,	PUNCT
ejpam-3588	22	5	p.	p.	PROPN
ejpam-3588	22	6	c.	c.	PROPN
ejpam-3588	22	7	diop	diop	PROPN
ejpam-3588	22	8	,	,	PUNCT
ejpam-3588	22	9	m.	m.	NOUN
ejpam-3588	22	10	barry	barry	PROPN
ejpam-3588	22	11	/	/	SYM
ejpam-3588	22	12	eur	eur	PROPN
ejpam-3588	22	13	.	.	PUNCT
ejpam-3588	23	1	j.	j.	PROPN
ejpam-3588	23	2	pure	pure	PROPN
ejpam-3588	23	3	appl	appl	PROPN
ejpam-3588	23	4	.	.	PROPN
ejpam-3588	23	5	math	math	PROPN
ejpam-3588	23	6	,	,	PUNCT
ejpam-3588	23	7	13	13	NUM
ejpam-3588	23	8	(	(	PUNCT
ejpam-3588	23	9	1	1	NUM
ejpam-3588	23	10	)	)	PUNCT
ejpam-3588	23	11	(	(	PUNCT
ejpam-3588	23	12	2020	2020	NUM
ejpam-3588	23	13	)	)	PUNCT
ejpam-3588	23	14	,	,	PUNCT
ejpam-3588	23	15	158	158	NUM
ejpam-3588	23	16	-	-	SYM
ejpam-3588	23	17	169	169	NUM
ejpam-3588	23	18	159	159	NUM
ejpam-3588	23	19	modules	module	NOUN
ejpam-3588	23	20	regarding	regard	VERB
ejpam-3588	23	21	the	the	DET
ejpam-3588	23	22	characterization	characterization	NOUN
ejpam-3588	23	23	of	of	ADP
ejpam-3588	23	24	bezout	bezout	NOUN
ejpam-3588	23	25	,	,	PUNCT
ejpam-3588	23	26	pri	pri	NOUN
ejpam-3588	23	27	,	,	PUNCT
ejpam-3588	23	28	quasi	quasi	ADJ
ejpam-3588	23	29	-	-	ADJ
ejpam-3588	23	30	frobenius	frobenius	ADJ
ejpam-3588	23	31	rings	ring	NOUN
ejpam-3588	23	32	.	.	PUNCT
ejpam-3588	24	1	note	note	VERB
ejpam-3588	24	2	that	that	SCONJ
ejpam-3588	24	3	epi	epi	NOUN
ejpam-3588	24	4	-	-	ADJ
ejpam-3588	24	5	retractable	retractable	ADJ
ejpam-3588	24	6	modules	module	NOUN
ejpam-3588	24	7	are	be	AUX
ejpam-3588	24	8	retractable	retractable	ADJ
ejpam-3588	24	9	.	.	PUNCT
ejpam-3588	25	1	earlier	early	ADV
ejpam-3588	25	2	,	,	PUNCT
ejpam-3588	25	3	p.	p.	PROPN
ejpam-3588	25	4	f.	f.	PROPN
ejpam-3588	25	5	smith	smith	PROPN
ejpam-3588	25	6	and	and	CCONJ
ejpam-3588	25	7	a.	a.	NOUN
ejpam-3588	25	8	tercan	tercan	PROPN
ejpam-3588	26	1	[	[	X
ejpam-3588	26	2	20	20	NUM
ejpam-3588	26	3	]	]	PUNCT
ejpam-3588	26	4	introduced	introduce	VERB
ejpam-3588	26	5	c11	c11	NOUN
ejpam-3588	26	6	-	-	PUNCT
ejpam-3588	26	7	module	module	NOUN
ejpam-3588	26	8	as	as	ADP
ejpam-3588	26	9	a	a	DET
ejpam-3588	26	10	generalization	generalization	NOUN
ejpam-3588	26	11	of	of	ADP
ejpam-3588	26	12	extending	extend	VERB
ejpam-3588	26	13	modules	module	NOUN
ejpam-3588	26	14	,	,	PUNCT
ejpam-3588	26	15	where	where	SCONJ
ejpam-3588	26	16	a	a	DET
ejpam-3588	26	17	module	module	NOUN
ejpam-3588	26	18	m	m	VERB
ejpam-3588	26	19	is	be	AUX
ejpam-3588	26	20	said	say	VERB
ejpam-3588	26	21	to	to	PART
ejpam-3588	26	22	be	be	AUX
ejpam-3588	26	23	satisfy	satisfy	VERB
ejpam-3588	26	24	c11	c11	NOUN
ejpam-3588	26	25	-	-	PUNCT
ejpam-3588	26	26	condition	condition	NOUN
ejpam-3588	26	27	if	if	SCONJ
ejpam-3588	26	28	every	every	DET
ejpam-3588	26	29	submodule	submodule	NOUN
ejpam-3588	26	30	of	of	ADP
ejpam-3588	26	31	m	m	PROPN
ejpam-3588	26	32	has	have	VERB
ejpam-3588	26	33	a	a	DET
ejpam-3588	26	34	complement	complement	NOUN
ejpam-3588	26	35	which	which	PRON
ejpam-3588	26	36	is	be	AUX
ejpam-3588	26	37	a	a	DET
ejpam-3588	26	38	direct	direct	ADJ
ejpam-3588	26	39	summand	summand	NOUN
ejpam-3588	26	40	.	.	PUNCT
ejpam-3588	27	1	it	it	PRON
ejpam-3588	27	2	is	be	AUX
ejpam-3588	27	3	shown	show	VERB
ejpam-3588	27	4	in	in	ADP
ejpam-3588	27	5	(	(	PUNCT
ejpam-3588	27	6	[	[	X
ejpam-3588	27	7	20	20	NUM
ejpam-3588	27	8	]	]	PUNCT
ejpam-3588	27	9	,	,	PUNCT
ejpam-3588	27	10	theorem	theorem	VERB
ejpam-3588	27	11	2.7	2.7	NUM
ejpam-3588	27	12	)	)	PUNCT
ejpam-3588	27	13	that	that	SCONJ
ejpam-3588	27	14	a	a	DET
ejpam-3588	27	15	module	module	NOUN
ejpam-3588	27	16	satisfies	satisfie	NOUN
ejpam-3588	27	17	(	(	PUNCT
ejpam-3588	27	18	c11	c11	NOUN
ejpam-3588	27	19	if	if	SCONJ
ejpam-3588	27	20	and	and	CCONJ
ejpam-3588	27	21	only	only	ADV
ejpam-3588	27	22	if	if	SCONJ
ejpam-3588	27	23	m	m	VERB
ejpam-3588	27	24	=	=	SYM
ejpam-3588	27	25	z2(m	z2(m	X
ejpam-3588	27	26	)	)	PUNCT
ejpam-3588	27	27	⊕k	⊕k	NOUN
ejpam-3588	27	28	for	for	ADP
ejpam-3588	27	29	some	some	PRON
ejpam-3588	27	30	(	(	PUNCT
ejpam-3588	27	31	nonsingular	nonsingular	ADJ
ejpam-3588	27	32	)	)	PUNCT
ejpam-3588	27	33	k	k	PROPN
ejpam-3588	27	34	of	of	ADP
ejpam-3588	27	35	m	m	PROPN
ejpam-3588	27	36	and	and	CCONJ
ejpam-3588	27	37	z2(m	z2(m	NOUN
ejpam-3588	27	38	)	)	PUNCT
ejpam-3588	27	39	and	and	CCONJ
ejpam-3588	27	40	k	k	PROPN
ejpam-3588	27	41	both	both	PRON
ejpam-3588	27	42	satisfy	satisfy	VERB
ejpam-3588	27	43	(	(	PUNCT
ejpam-3588	27	44	c11	c11	NOUN
ejpam-3588	27	45	)	)	PUNCT
ejpam-3588	27	46	.	.	PUNCT
ejpam-3588	28	1	later	later	ADV
ejpam-3588	28	2	,	,	PUNCT
ejpam-3588	28	3	the	the	DET
ejpam-3588	28	4	same	same	ADJ
ejpam-3588	28	5	authors	author	NOUN
ejpam-3588	28	6	investigated	investigate	VERB
ejpam-3588	28	7	when	when	SCONJ
ejpam-3588	28	8	a	a	DET
ejpam-3588	28	9	direct	direct	ADJ
ejpam-3588	28	10	summand	summand	NOUN
ejpam-3588	28	11	of	of	ADP
ejpam-3588	28	12	a	a	DET
ejpam-3588	28	13	c11	c11	NOUN
ejpam-3588	28	14	-	-	PUNCT
ejpam-3588	28	15	module	module	NOUN
ejpam-3588	28	16	inherits	inherit	VERB
ejpam-3588	28	17	the	the	DET
ejpam-3588	28	18	property	property	NOUN
ejpam-3588	28	19	[	[	X
ejpam-3588	28	20	21	21	NUM
ejpam-3588	28	21	]	]	PUNCT
ejpam-3588	28	22	.	.	PUNCT
ejpam-3588	29	1	recently	recently	ADV
ejpam-3588	29	2	,	,	PUNCT
ejpam-3588	29	3	t	t	PROPN
ejpam-3588	29	4	-	-	PUNCT
ejpam-3588	29	5	closed	close	VERB
ejpam-3588	29	6	submodules	submodule	NOUN
ejpam-3588	29	7	of	of	ADP
ejpam-3588	29	8	a	a	DET
ejpam-3588	29	9	module	module	NOUN
ejpam-3588	29	10	m	m	NOUN
ejpam-3588	29	11	are	be	AUX
ejpam-3588	29	12	defined	define	VERB
ejpam-3588	29	13	in	in	ADP
ejpam-3588	29	14	[	[	X
ejpam-3588	29	15	2	2	NUM
ejpam-3588	29	16	]	]	PUNCT
ejpam-3588	29	17	as	as	ADP
ejpam-3588	29	18	closed	close	VERB
ejpam-3588	29	19	submodules	submodule	NOUN
ejpam-3588	29	20	of	of	ADP
ejpam-3588	29	21	m	m	PRON
ejpam-3588	29	22	which	which	PRON
ejpam-3588	29	23	contain	contain	VERB
ejpam-3588	29	24	z2(m	z2(m	NOUN
ejpam-3588	29	25	)	)	PUNCT
ejpam-3588	29	26	.	.	PUNCT
ejpam-3588	30	1	in	in	ADP
ejpam-3588	30	2	[	[	X
ejpam-3588	30	3	3	3	NUM
ejpam-3588	30	4	]	]	PUNCT
ejpam-3588	30	5	,	,	PUNCT
ejpam-3588	30	6	s.	s.	PROPN
ejpam-3588	30	7	h.	h.	PROPN
ejpam-3588	30	8	asgari	asgari	PROPN
ejpam-3588	30	9	,	,	PUNCT
ejpam-3588	30	10	a.	a.	PROPN
ejpam-3588	30	11	haghany	haghany	PROPN
ejpam-3588	30	12	and	and	CCONJ
ejpam-3588	30	13	a.r	a.r	PROPN
ejpam-3588	30	14	.	.	PROPN
ejpam-3588	30	15	rezaei	rezaei	PROPN
ejpam-3588	30	16	studied	study	VERB
ejpam-3588	30	17	the	the	DET
ejpam-3588	30	18	modules	module	NOUN
ejpam-3588	30	19	m	m	VERB
ejpam-3588	30	20	for	for	ADP
ejpam-3588	30	21	which	which	PRON
ejpam-3588	30	22	c11	c11	NOUN
ejpam-3588	30	23	-	-	PUNCT
ejpam-3588	30	24	condition	condition	NOUN
ejpam-3588	30	25	holds	hold	VERB
ejpam-3588	30	26	for	for	ADP
ejpam-3588	30	27	t	t	NOUN
ejpam-3588	30	28	-	-	PUNCT
ejpam-3588	30	29	closed	close	VERB
ejpam-3588	30	30	submodules	submodule	NOUN
ejpam-3588	30	31	(	(	PUNCT
ejpam-3588	30	32	t11	t11	NOUN
ejpam-3588	30	33	-	-	PUNCT
ejpam-3588	30	34	type	type	NOUN
ejpam-3588	30	35	,	,	PUNCT
ejpam-3588	30	36	for	for	ADP
ejpam-3588	30	37	short	short	ADJ
ejpam-3588	30	38	)	)	PUNCT
ejpam-3588	30	39	.	.	PUNCT
ejpam-3588	31	1	they	they	PRON
ejpam-3588	31	2	showed	show	VERB
ejpam-3588	31	3	among	among	ADP
ejpam-3588	31	4	others	other	NOUN
ejpam-3588	31	5	the	the	DET
ejpam-3588	31	6	following	follow	VERB
ejpam-3588	31	7	results	result	NOUN
ejpam-3588	31	8	:	:	PUNCT
ejpam-3588	31	9	i	i	X
ejpam-3588	31	10	)	)	PUNCT
ejpam-3588	31	11	a	a	DET
ejpam-3588	31	12	t11	t11	NOUN
ejpam-3588	31	13	-	-	PUNCT
ejpam-3588	31	14	type	type	NOUN
ejpam-3588	31	15	module	module	NOUN
ejpam-3588	31	16	is	be	AUX
ejpam-3588	31	17	exactly	exactly	ADV
ejpam-3588	31	18	a	a	DET
ejpam-3588	31	19	direct	direct	ADJ
ejpam-3588	31	20	sum	sum	NOUN
ejpam-3588	31	21	of	of	ADP
ejpam-3588	31	22	a	a	DET
ejpam-3588	31	23	z2	z2	ADJ
ejpam-3588	31	24	-	-	PUNCT
ejpam-3588	31	25	torsion	torsion	NOUN
ejpam-3588	31	26	module	module	NOUN
ejpam-3588	31	27	and	and	CCONJ
ejpam-3588	31	28	a	a	DET
ejpam-3588	31	29	nonsingular	nonsingular	ADJ
ejpam-3588	31	30	c11	c11	NOUN
ejpam-3588	31	31	-	-	PUNCT
ejpam-3588	31	32	modules	module	NOUN
ejpam-3588	31	33	.	.	PUNCT
ejpam-3588	32	1	(	(	PUNCT
ejpam-3588	32	2	ii	ii	NOUN
ejpam-3588	32	3	)	)	PUNCT
ejpam-3588	32	4	a	a	DET
ejpam-3588	32	5	t+	t+	PUNCT
ejpam-3588	32	6	11	11	NUM
ejpam-3588	32	7	-	-	PUNCT
ejpam-3588	32	8	module	module	NOUN
ejpam-3588	32	9	(	(	PUNCT
ejpam-3588	32	10	modules	module	NOUN
ejpam-3588	32	11	for	for	ADP
ejpam-3588	32	12	which	which	PRON
ejpam-3588	32	13	direct	direct	ADJ
ejpam-3588	32	14	summands	summand	NOUN
ejpam-3588	32	15	are	be	AUX
ejpam-3588	32	16	t11	t11	NOUN
ejpam-3588	32	17	-	-	PUNCT
ejpam-3588	32	18	type	type	NOUN
ejpam-3588	32	19	)	)	PUNCT
ejpam-3588	32	20	is	be	AUX
ejpam-3588	32	21	precisely	precisely	ADV
ejpam-3588	32	22	a	a	DET
ejpam-3588	32	23	direct	direct	ADJ
ejpam-3588	32	24	sum	sum	NOUN
ejpam-3588	32	25	of	of	ADP
ejpam-3588	32	26	z2	z2	NOUN
ejpam-3588	32	27	-	-	PUNCT
ejpam-3588	32	28	torsion	torsion	NOUN
ejpam-3588	32	29	and	and	CCONJ
ejpam-3588	32	30	nonsingular	nonsingular	ADJ
ejpam-3588	32	31	c+	c+	VERB
ejpam-3588	32	32	11	11	NUM
ejpam-3588	32	33	-	-	PUNCT
ejpam-3588	32	34	module	module	NOUN
ejpam-3588	32	35	(	(	PUNCT
ejpam-3588	32	36	modules	module	NOUN
ejpam-3588	32	37	for	for	ADP
ejpam-3588	32	38	which	which	PRON
ejpam-3588	32	39	direct	direct	ADJ
ejpam-3588	32	40	summands	summand	VERB
ejpam-3588	32	41	satisly	satisly	ADV
ejpam-3588	32	42	c11	c11	NOUN
ejpam-3588	32	43	)	)	PUNCT
ejpam-3588	32	44	.	.	PUNCT
ejpam-3588	33	1	a.	a.	PROPN
ejpam-3588	33	2	w.	w.	PROPN
ejpam-3588	33	3	chatters	chatter	VERB
ejpam-3588	33	4	and	and	CCONJ
ejpam-3588	33	5	s.	s.	PROPN
ejpam-3588	33	6	m.	m.	PROPN
ejpam-3588	33	7	kheuri	kheuri	PROPN
ejpam-3588	34	1	[	[	X
ejpam-3588	34	2	4	4	X
ejpam-3588	34	3	]	]	PUNCT
ejpam-3588	34	4	defined	define	VERB
ejpam-3588	34	5	the	the	DET
ejpam-3588	34	6	concept	concept	NOUN
ejpam-3588	34	7	of	of	ADP
ejpam-3588	34	8	c	c	NOUN
ejpam-3588	34	9	-	-	PUNCT
ejpam-3588	34	10	retractable	retractable	ADJ
ejpam-3588	34	11	module	module	NOUN
ejpam-3588	34	12	,	,	PUNCT
ejpam-3588	34	13	where	where	SCONJ
ejpam-3588	34	14	an	an	DET
ejpam-3588	34	15	r	r	NOUN
ejpam-3588	34	16	-	-	PUNCT
ejpam-3588	34	17	module	module	NOUN
ejpam-3588	34	18	m	m	NOUN
ejpam-3588	34	19	is	be	AUX
ejpam-3588	34	20	called	call	VERB
ejpam-3588	34	21	c	c	ADJ
ejpam-3588	34	22	-	-	NOUN
ejpam-3588	34	23	retractable	retractable	ADJ
ejpam-3588	34	24	if	if	SCONJ
ejpam-3588	34	25	homr(m	homr(m	PROPN
ejpam-3588	34	26	,	,	PUNCT
ejpam-3588	34	27	c	c	NOUN
ejpam-3588	34	28	)	)	PUNCT
ejpam-3588	34	29	6=	6=	ADP
ejpam-3588	34	30	{	{	PUNCT
ejpam-3588	34	31	0	0	NUM
ejpam-3588	34	32	}	}	PUNCT
ejpam-3588	34	33	for	for	ADP
ejpam-3588	34	34	any	any	DET
ejpam-3588	34	35	nonzero	nonzero	NOUN
ejpam-3588	34	36	complement	complement	NOUN
ejpam-3588	34	37	submodules	submodule	NOUN
ejpam-3588	34	38	c	c	PROPN
ejpam-3588	34	39	of	of	ADP
ejpam-3588	34	40	m	m	PROPN
ejpam-3588	34	41	.	.	PUNCT
ejpam-3588	35	1	this	this	DET
ejpam-3588	35	2	notion	notion	NOUN
ejpam-3588	35	3	is	be	AUX
ejpam-3588	35	4	a	a	DET
ejpam-3588	35	5	generalization	generalization	NOUN
ejpam-3588	35	6	of	of	ADP
ejpam-3588	35	7	both	both	CCONJ
ejpam-3588	35	8	the	the	DET
ejpam-3588	35	9	retractable	retractable	ADJ
ejpam-3588	35	10	modules	module	NOUN
ejpam-3588	35	11	and	and	CCONJ
ejpam-3588	35	12	the	the	DET
ejpam-3588	35	13	extending	extend	VERB
ejpam-3588	35	14	modules	module	NOUN
ejpam-3588	35	15	.	.	PUNCT
ejpam-3588	36	1	they	they	PRON
ejpam-3588	36	2	have	have	AUX
ejpam-3588	36	3	shown	show	VERB
ejpam-3588	36	4	that	that	SCONJ
ejpam-3588	36	5	if	if	SCONJ
ejpam-3588	36	6	m	m	NOUN
ejpam-3588	36	7	is	be	AUX
ejpam-3588	36	8	a	a	DET
ejpam-3588	36	9	nonsingular	nonsingular	ADJ
ejpam-3588	36	10	c	c	NOUN
ejpam-3588	36	11	-	-	PUNCT
ejpam-3588	36	12	retractable	retractable	ADJ
ejpam-3588	36	13	module	module	NOUN
ejpam-3588	36	14	such	such	ADJ
ejpam-3588	36	15	that	that	SCONJ
ejpam-3588	36	16	ss	ss	PROPN
ejpam-3588	36	17	is	be	AUX
ejpam-3588	36	18	extending	extend	VERB
ejpam-3588	36	19	,	,	PUNCT
ejpam-3588	36	20	then	then	ADV
ejpam-3588	36	21	m	m	VERB
ejpam-3588	36	22	is	be	AUX
ejpam-3588	36	23	extending	extend	VERB
ejpam-3588	36	24	.	.	PUNCT
ejpam-3588	37	1	but	but	CCONJ
ejpam-3588	37	2	the	the	DET
ejpam-3588	37	3	converse	converse	NOUN
ejpam-3588	37	4	is	be	AUX
ejpam-3588	37	5	not	not	PART
ejpam-3588	37	6	true	true	ADJ
ejpam-3588	37	7	in	in	ADP
ejpam-3588	37	8	general	general	ADJ
ejpam-3588	37	9	.	.	PUNCT
ejpam-3588	38	1	on	on	ADP
ejpam-3588	38	2	the	the	DET
ejpam-3588	38	3	other	other	ADJ
ejpam-3588	38	4	hand	hand	NOUN
ejpam-3588	38	5	it	it	PRON
ejpam-3588	38	6	is	be	AUX
ejpam-3588	38	7	shown	show	VERB
ejpam-3588	38	8	in	in	ADP
ejpam-3588	38	9	[	[	X
ejpam-3588	38	10	22	22	NUM
ejpam-3588	38	11	]	]	PUNCT
ejpam-3588	38	12	that	that	SCONJ
ejpam-3588	38	13	if	if	SCONJ
ejpam-3588	38	14	m	m	NOUN
ejpam-3588	38	15	is	be	AUX
ejpam-3588	38	16	a	a	DET
ejpam-3588	38	17	retractable	retractable	ADJ
ejpam-3588	38	18	wdrickart	wdrickart	NOUN
ejpam-3588	38	19	module	module	NOUN
ejpam-3588	38	20	,	,	PUNCT
ejpam-3588	38	21	then	then	ADV
ejpam-3588	38	22	every	every	DET
ejpam-3588	38	23	indecomposable	indecomposable	ADJ
ejpam-3588	38	24	submodule	submodule	NOUN
ejpam-3588	38	25	of	of	ADP
ejpam-3588	38	26	m	m	PROPN
ejpam-3588	38	27	is	be	AUX
ejpam-3588	38	28	a	a	DET
ejpam-3588	38	29	simple	simple	ADJ
ejpam-3588	38	30	direct	direct	ADJ
ejpam-3588	38	31	summand	summand	NOUN
ejpam-3588	38	32	.	.	PUNCT
ejpam-3588	39	1	motivated	motivate	VERB
ejpam-3588	39	2	by	by	ADP
ejpam-3588	39	3	the	the	DET
ejpam-3588	39	4	definition	definition	NOUN
ejpam-3588	39	5	of	of	ADP
ejpam-3588	39	6	the	the	DET
ejpam-3588	39	7	modules	module	NOUN
ejpam-3588	39	8	mentioned	mention	VERB
ejpam-3588	39	9	above	above	ADV
ejpam-3588	39	10	and	and	CCONJ
ejpam-3588	39	11	the	the	DET
ejpam-3588	39	12	results	result	NOUN
ejpam-3588	39	13	on	on	ADP
ejpam-3588	39	14	retractable	retractable	ADJ
ejpam-3588	39	15	and	and	CCONJ
ejpam-3588	39	16	c	c	NOUN
ejpam-3588	39	17	-	-	PUNCT
ejpam-3588	39	18	retractable	retractable	ADJ
ejpam-3588	39	19	modules	module	NOUN
ejpam-3588	39	20	,	,	PUNCT
ejpam-3588	39	21	we	we	PRON
ejpam-3588	39	22	investigate	investigate	VERB
ejpam-3588	39	23	the	the	DET
ejpam-3588	39	24	c	c	NOUN
ejpam-3588	39	25	-	-	NOUN
ejpam-3588	39	26	retractibility	retractibility	NOUN
ejpam-3588	39	27	.	.	PUNCT
ejpam-3588	40	1	our	our	PRON
ejpam-3588	40	2	aim	aim	NOUN
ejpam-3588	40	3	in	in	ADP
ejpam-3588	40	4	this	this	DET
ejpam-3588	40	5	paper	paper	NOUN
ejpam-3588	40	6	is	be	AUX
ejpam-3588	40	7	to	to	PART
ejpam-3588	40	8	give	give	VERB
ejpam-3588	40	9	some	some	DET
ejpam-3588	40	10	new	new	ADJ
ejpam-3588	40	11	results	result	NOUN
ejpam-3588	40	12	on	on	ADP
ejpam-3588	40	13	c	c	NOUN
ejpam-3588	40	14	-	-	PUNCT
ejpam-3588	40	15	retractable	retractable	ADJ
ejpam-3588	40	16	modules	module	NOUN
ejpam-3588	40	17	.	.	PUNCT
ejpam-3588	41	1	in	in	ADP
ejpam-3588	41	2	general	general	ADJ
ejpam-3588	41	3	,	,	PUNCT
ejpam-3588	41	4	c	c	NOUN
ejpam-3588	41	5	-	-	PUNCT
ejpam-3588	41	6	retractable	retractable	ADJ
ejpam-3588	41	7	modules	module	NOUN
ejpam-3588	41	8	need	need	AUX
ejpam-3588	41	9	not	not	PART
ejpam-3588	41	10	be	be	AUX
ejpam-3588	41	11	projective	projective	ADJ
ejpam-3588	41	12	and	and	CCONJ
ejpam-3588	41	13	vice	vice	ADV
ejpam-3588	41	14	versa	versa	ADV
ejpam-3588	41	15	.	.	PUNCT
ejpam-3588	42	1	connections	connection	NOUN
ejpam-3588	42	2	between	between	ADP
ejpam-3588	42	3	projectivity	projectivity	NOUN
ejpam-3588	42	4	and	and	CCONJ
ejpam-3588	42	5	c	c	NOUN
ejpam-3588	42	6	-	-	PUNCT
ejpam-3588	42	7	retractibility	retractibility	NOUN
ejpam-3588	42	8	are	be	AUX
ejpam-3588	42	9	investigated	investigate	VERB
ejpam-3588	42	10	.	.	PUNCT
ejpam-3588	43	1	conditions	condition	NOUN
ejpam-3588	43	2	are	be	AUX
ejpam-3588	43	3	found	find	VERB
ejpam-3588	43	4	under	under	ADP
ejpam-3588	43	5	which	which	PRON
ejpam-3588	43	6	,	,	PUNCT
ejpam-3588	43	7	a	a	DET
ejpam-3588	43	8	c	c	NOUN
ejpam-3588	43	9	-	-	PUNCT
ejpam-3588	43	10	retractable	retractable	ADJ
ejpam-3588	43	11	module	module	NOUN
ejpam-3588	43	12	is	be	AUX
ejpam-3588	43	13	extending	extend	VERB
ejpam-3588	43	14	,	,	PUNCT
ejpam-3588	43	15	quasi	quasi	ADJ
ejpam-3588	43	16	-	-	ADJ
ejpam-3588	43	17	continuous	continuous	ADJ
ejpam-3588	43	18	,	,	PUNCT
ejpam-3588	43	19	quasi	quasi	ADJ
ejpam-3588	43	20	-	-	ADJ
ejpam-3588	43	21	injective	injective	ADJ
ejpam-3588	43	22	and	and	CCONJ
ejpam-3588	43	23	retractable	retractable	ADJ
ejpam-3588	43	24	.	.	PUNCT
ejpam-3588	44	1	with	with	ADP
ejpam-3588	44	2	the	the	DET
ejpam-3588	44	3	help	help	NOUN
ejpam-3588	44	4	of	of	ADP
ejpam-3588	44	5	c	c	NOUN
ejpam-3588	44	6	-	-	PUNCT
ejpam-3588	44	7	retractability	retractability	NOUN
ejpam-3588	44	8	,	,	PUNCT
ejpam-3588	44	9	we	we	PRON
ejpam-3588	44	10	investigated	investigate	VERB
ejpam-3588	44	11	when	when	SCONJ
ejpam-3588	44	12	the	the	DET
ejpam-3588	44	13	notions	notion	NOUN
ejpam-3588	44	14	of	of	ADP
ejpam-3588	44	15	k	k	NOUN
ejpam-3588	44	16	-	-	PUNCT
ejpam-3588	44	17	nonsingularity	nonsingularity	NOUN
ejpam-3588	44	18	and	and	CCONJ
ejpam-3588	44	19	baer	baer	PROPN
ejpam-3588	44	20	modules	module	NOUN
ejpam-3588	44	21	are	be	AUX
ejpam-3588	44	22	equivalent	equivalent	ADJ
ejpam-3588	44	23	.	.	PUNCT
ejpam-3588	45	1	also	also	ADV
ejpam-3588	45	2	,	,	PUNCT
ejpam-3588	45	3	we	we	PRON
ejpam-3588	45	4	characterize	characterize	VERB
ejpam-3588	45	5	semisimple	semisimple	NOUN
ejpam-3588	45	6	artinian	artinian	ADJ
ejpam-3588	45	7	rings	ring	NOUN
ejpam-3588	45	8	in	in	ADP
ejpam-3588	45	9	termes	terme	NOUN
ejpam-3588	45	10	of	of	ADP
ejpam-3588	45	11	c	c	NOUN
ejpam-3588	45	12	-	-	PUNCT
ejpam-3588	45	13	retractable	retractable	ADJ
ejpam-3588	45	14	modules	module	NOUN
ejpam-3588	45	15	.	.	PUNCT
ejpam-3588	46	1	our	our	PRON
ejpam-3588	46	2	paper	paper	NOUN
ejpam-3588	46	3	is	be	AUX
ejpam-3588	46	4	structured	structure	VERB
ejpam-3588	46	5	as	as	SCONJ
ejpam-3588	46	6	follows	follow	VERB
ejpam-3588	46	7	:	:	PUNCT
ejpam-3588	46	8	in	in	ADP
ejpam-3588	46	9	the	the	DET
ejpam-3588	46	10	second	second	ADJ
ejpam-3588	46	11	section	section	NOUN
ejpam-3588	46	12	,	,	PUNCT
ejpam-3588	46	13	we	we	PRON
ejpam-3588	46	14	are	be	AUX
ejpam-3588	46	15	going	go	VERB
ejpam-3588	46	16	to	to	PART
ejpam-3588	46	17	give	give	VERB
ejpam-3588	46	18	preliminary	preliminary	ADJ
ejpam-3588	46	19	definitions	definition	NOUN
ejpam-3588	46	20	which	which	PRON
ejpam-3588	46	21	we	we	PRON
ejpam-3588	46	22	will	will	AUX
ejpam-3588	46	23	use	use	VERB
ejpam-3588	46	24	throughout	throughout	ADP
ejpam-3588	46	25	this	this	DET
ejpam-3588	46	26	paper	paper	NOUN
ejpam-3588	46	27	.	.	PUNCT
ejpam-3588	47	1	in	in	ADP
ejpam-3588	47	2	the	the	DET
ejpam-3588	47	3	third	third	ADJ
ejpam-3588	47	4	section	section	NOUN
ejpam-3588	47	5	,	,	PUNCT
ejpam-3588	47	6	we	we	PRON
ejpam-3588	47	7	are	be	AUX
ejpam-3588	47	8	going	go	VERB
ejpam-3588	47	9	to	to	PART
ejpam-3588	47	10	show	show	VERB
ejpam-3588	47	11	among	among	ADP
ejpam-3588	47	12	others	other	NOUN
ejpam-3588	47	13	,	,	PUNCT
ejpam-3588	47	14	the	the	DET
ejpam-3588	47	15	following	following	ADJ
ejpam-3588	47	16	results	result	NOUN
ejpam-3588	47	17	:	:	PUNCT
ejpam-3588	47	18	(	(	PUNCT
ejpam-3588	47	19	1	1	X
ejpam-3588	47	20	)	)	PUNCT
ejpam-3588	47	21	every	every	DET
ejpam-3588	47	22	projective	projective	ADJ
ejpam-3588	47	23	module	module	NOUN
ejpam-3588	47	24	over	over	ADP
ejpam-3588	47	25	a	a	DET
ejpam-3588	47	26	right	right	ADJ
ejpam-3588	47	27	si	si	NOUN
ejpam-3588	47	28	-	-	PUNCT
ejpam-3588	47	29	ring	ring	NOUN
ejpam-3588	47	30	is	be	AUX
ejpam-3588	47	31	c	c	NOUN
ejpam-3588	47	32	-	-	PUNCT
ejpam-3588	47	33	retractable	retractable	ADJ
ejpam-3588	47	34	.	.	PUNCT
ejpam-3588	48	1	(	(	PUNCT
ejpam-3588	48	2	2	2	X
ejpam-3588	48	3	)	)	PUNCT
ejpam-3588	48	4	let	let	VERB
ejpam-3588	48	5	m	m	PRON
ejpam-3588	48	6	be	be	AUX
ejpam-3588	48	7	a	a	DET
ejpam-3588	48	8	wd	wd	ADJ
ejpam-3588	48	9	-	-	PUNCT
ejpam-3588	48	10	rickart	rickart	NOUN
ejpam-3588	48	11	module	module	NOUN
ejpam-3588	48	12	in	in	ADP
ejpam-3588	48	13	which	which	PRON
ejpam-3588	48	14	local	local	ADJ
ejpam-3588	48	15	summands	summand	NOUN
ejpam-3588	48	16	are	be	AUX
ejpam-3588	48	17	summand	summand	NOUN
ejpam-3588	48	18	.	.	PUNCT
ejpam-3588	49	1	then	then	ADV
ejpam-3588	49	2	m	m	PROPN
ejpam-3588	49	3	is	be	AUX
ejpam-3588	49	4	uniform	uniform	ADJ
ejpam-3588	49	5	-	-	PUNCT
ejpam-3588	49	6	extending	extend	VERB
ejpam-3588	49	7	and	and	CCONJ
ejpam-3588	49	8	c	c	NOUN
ejpam-3588	49	9	-	-	NOUN
ejpam-3588	49	10	retractable	retractable	ADJ
ejpam-3588	49	11	if	if	SCONJ
ejpam-3588	49	12	and	and	CCONJ
ejpam-3588	49	13	only	only	ADV
ejpam-3588	49	14	if	if	SCONJ
ejpam-3588	49	15	m	m	NOUN
ejpam-3588	49	16	is	be	AUX
ejpam-3588	49	17	extending	extend	VERB
ejpam-3588	49	18	.	.	PUNCT
ejpam-3588	50	1	(	(	PUNCT
ejpam-3588	50	2	3	3	X
ejpam-3588	50	3	)	)	PUNCT
ejpam-3588	50	4	let	let	VERB
ejpam-3588	50	5	m	m	PRON
ejpam-3588	50	6	be	be	AUX
ejpam-3588	50	7	a	a	DET
ejpam-3588	50	8	dual	dual	ADJ
ejpam-3588	50	9	baer	baer	PROPN
ejpam-3588	50	10	c	c	NOUN
ejpam-3588	50	11	-	-	PUNCT
ejpam-3588	50	12	retractable	retractable	ADJ
ejpam-3588	50	13	r	r	NOUN
ejpam-3588	50	14	-	-	PUNCT
ejpam-3588	50	15	module	module	NOUN
ejpam-3588	50	16	.	.	PUNCT
ejpam-3588	51	1	then	then	ADV
ejpam-3588	51	2	the	the	DET
ejpam-3588	51	3	following	follow	VERB
ejpam-3588	51	4	hold	hold	NOUN
ejpam-3588	51	5	:	:	PUNCT
ejpam-3588	51	6	(	(	PUNCT
ejpam-3588	51	7	i	i	NOUN
ejpam-3588	51	8	)	)	PUNCT
ejpam-3588	51	9	m	m	VERB
ejpam-3588	51	10	is	be	AUX
ejpam-3588	51	11	a	a	DET
ejpam-3588	51	12	direct	direct	ADJ
ejpam-3588	51	13	sum	sum	NOUN
ejpam-3588	51	14	of	of	ADP
ejpam-3588	51	15	uniform	uniform	ADJ
ejpam-3588	51	16	submodules	submodule	NOUN
ejpam-3588	51	17	.	.	PUNCT
ejpam-3588	52	1	(	(	PUNCT
ejpam-3588	52	2	ii	ii	X
ejpam-3588	52	3	)	)	PUNCT
ejpam-3588	52	4	m	m	PROPN
ejpam-3588	52	5	=	=	SYM
ejpam-3588	52	6	z2(m	z2(m	X
ejpam-3588	52	7	)	)	PUNCT
ejpam-3588	52	8	⊕	⊕	PROPN
ejpam-3588	52	9	(	(	PUNCT
ejpam-3588	52	10	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	52	11	)	)	PUNCT
ejpam-3588	52	12	with	with	ADP
ejpam-3588	52	13	all	all	DET
ejpam-3588	52	14	mi	mi	PROPN
ejpam-3588	52	15	nonsingular	nonsingular	PROPN
ejpam-3588	52	16	uniform	uniform	PROPN
ejpam-3588	52	17	quasi	quasi	PROPN
ejpam-3588	52	18	-	-	PROPN
ejpam-3588	52	19	baer	baer	PROPN
ejpam-3588	52	20	and	and	CCONJ
ejpam-3588	52	21	end(mi	end(mi	NOUN
ejpam-3588	52	22	)	)	PUNCT
ejpam-3588	52	23	semi	semi	ADJ
ejpam-3588	52	24	-	-	ADJ
ejpam-3588	52	25	local	local	ADJ
ejpam-3588	52	26	quasi	quasi	NOUN
ejpam-3588	52	27	-	-	NOUN
ejpam-3588	52	28	baer	baer	PROPN
ejpam-3588	52	29	.	.	PUNCT
ejpam-3588	53	1	(	(	PUNCT
ejpam-3588	53	2	iii	iii	X
ejpam-3588	53	3	)	)	PUNCT
ejpam-3588	53	4	m	m	VERB
ejpam-3588	53	5	is	be	AUX
ejpam-3588	53	6	ads	ad	NOUN
ejpam-3588	53	7	if	if	SCONJ
ejpam-3588	53	8	and	and	CCONJ
ejpam-3588	53	9	only	only	ADV
ejpam-3588	53	10	if	if	SCONJ
ejpam-3588	53	11	m	m	NOUN
ejpam-3588	53	12	is	be	AUX
ejpam-3588	53	13	quasi	quasi	ADJ
ejpam-3588	53	14	-	-	ADJ
ejpam-3588	53	15	continuous	continuous	ADJ
ejpam-3588	53	16	.	.	PUNCT
ejpam-3588	54	1	(	(	PUNCT
ejpam-3588	54	2	iv	iv	X
ejpam-3588	54	3	)	)	PUNCT
ejpam-3588	54	4	m	m	VERB
ejpam-3588	54	5	is	be	AUX
ejpam-3588	54	6	auto	auto	NOUN
ejpam-3588	54	7	-	-	PUNCT
ejpam-3588	54	8	invariant	invariant	ADJ
ejpam-3588	54	9	if	if	SCONJ
ejpam-3588	55	1	and	and	CCONJ
ejpam-3588	55	2	only	only	ADV
ejpam-3588	55	3	if	if	SCONJ
ejpam-3588	55	4	m	m	NOUN
ejpam-3588	55	5	is	be	AUX
ejpam-3588	55	6	quasi	quasi	ADJ
ejpam-3588	55	7	-	-	ADJ
ejpam-3588	55	8	injective	injective	ADJ
ejpam-3588	55	9	.	.	PUNCT
ejpam-3588	56	1	(	(	PUNCT
ejpam-3588	56	2	5	5	X
ejpam-3588	56	3	)	)	PUNCT
ejpam-3588	56	4	the	the	DET
ejpam-3588	56	5	following	follow	VERB
ejpam-3588	56	6	conditions	condition	NOUN
ejpam-3588	56	7	are	be	AUX
ejpam-3588	56	8	equivalent	equivalent	ADJ
ejpam-3588	56	9	for	for	ADP
ejpam-3588	56	10	a	a	DET
ejpam-3588	56	11	ring	ring	NOUN
ejpam-3588	56	12	r	r	NOUN
ejpam-3588	56	13	:	:	PUNCT
ejpam-3588	56	14	a.	a.	PROPN
ejpam-3588	56	15	d.	d.	PROPN
ejpam-3588	56	16	diallo	diallo	PROPN
ejpam-3588	56	17	,	,	PUNCT
ejpam-3588	56	18	p.	p.	PROPN
ejpam-3588	56	19	c.	c.	PROPN
ejpam-3588	56	20	diop	diop	PROPN
ejpam-3588	56	21	,	,	PUNCT
ejpam-3588	56	22	m.	m.	NOUN
ejpam-3588	56	23	barry	barry	PROPN
ejpam-3588	56	24	/	/	SYM
ejpam-3588	56	25	eur	eur	PROPN
ejpam-3588	56	26	.	.	PUNCT
ejpam-3588	57	1	j.	j.	PROPN
ejpam-3588	57	2	pure	pure	PROPN
ejpam-3588	57	3	appl	appl	PROPN
ejpam-3588	57	4	.	.	PROPN
ejpam-3588	57	5	math	math	PROPN
ejpam-3588	57	6	,	,	PUNCT
ejpam-3588	57	7	13	13	NUM
ejpam-3588	57	8	(	(	PUNCT
ejpam-3588	57	9	1	1	NUM
ejpam-3588	57	10	)	)	PUNCT
ejpam-3588	57	11	(	(	PUNCT
ejpam-3588	57	12	2020	2020	NUM
ejpam-3588	57	13	)	)	PUNCT
ejpam-3588	57	14	,	,	PUNCT
ejpam-3588	57	15	158	158	NUM
ejpam-3588	57	16	-	-	SYM
ejpam-3588	57	17	169	169	NUM
ejpam-3588	57	18	160	160	NUM
ejpam-3588	57	19	(	(	PUNCT
ejpam-3588	57	20	a	a	X
ejpam-3588	57	21	)	)	PUNCT
ejpam-3588	57	22	r	r	NOUN
ejpam-3588	57	23	is	be	AUX
ejpam-3588	57	24	semisimple	semisimple	NOUN
ejpam-3588	57	25	artinian	artinian	ADJ
ejpam-3588	57	26	.	.	PUNCT
ejpam-3588	58	1	(	(	PUNCT
ejpam-3588	58	2	b	b	X
ejpam-3588	58	3	)	)	PUNCT
ejpam-3588	58	4	every	every	DET
ejpam-3588	58	5	c	c	NOUN
ejpam-3588	58	6	-	-	PUNCT
ejpam-3588	58	7	retractable	retractable	ADJ
ejpam-3588	58	8	r	r	NOUN
ejpam-3588	58	9	-	-	PUNCT
ejpam-3588	58	10	module	module	NOUN
ejpam-3588	58	11	is	be	AUX
ejpam-3588	58	12	a	a	DET
ejpam-3588	58	13	c4	c4	NOUN
ejpam-3588	58	14	-	-	PUNCT
ejpam-3588	58	15	module	module	NOUN
ejpam-3588	58	16	.	.	PUNCT
ejpam-3588	59	1	(	(	PUNCT
ejpam-3588	59	2	c	c	X
ejpam-3588	59	3	)	)	PUNCT
ejpam-3588	59	4	every	every	DET
ejpam-3588	59	5	c	c	NOUN
ejpam-3588	59	6	-	-	PUNCT
ejpam-3588	59	7	retractable	retractable	ADJ
ejpam-3588	59	8	r	r	NOUN
ejpam-3588	59	9	-	-	PUNCT
ejpam-3588	59	10	module	module	NOUN
ejpam-3588	59	11	is	be	AUX
ejpam-3588	59	12	pseudo	pseudo	NOUN
ejpam-3588	59	13	-	-	NOUN
ejpam-3588	59	14	projective	projective	ADJ
ejpam-3588	59	15	.	.	PUNCT
ejpam-3588	60	1	(	(	PUNCT
ejpam-3588	60	2	6	6	X
ejpam-3588	60	3	)	)	PUNCT
ejpam-3588	60	4	let	let	VERB
ejpam-3588	60	5	m	m	PRON
ejpam-3588	60	6	be	be	AUX
ejpam-3588	60	7	a	a	DET
ejpam-3588	60	8	locally	locally	ADV
ejpam-3588	60	9	noetherian	noetherian	ADJ
ejpam-3588	60	10	c	c	NOUN
ejpam-3588	60	11	-	-	PUNCT
ejpam-3588	60	12	retractable	retractable	ADJ
ejpam-3588	60	13	r	r	NOUN
ejpam-3588	60	14	-	-	PUNCT
ejpam-3588	60	15	module	module	NOUN
ejpam-3588	60	16	.	.	PUNCT
ejpam-3588	61	1	then	then	ADV
ejpam-3588	61	2	m	m	PROPN
ejpam-3588	61	3	is	be	AUX
ejpam-3588	61	4	homo	homo	NOUN
ejpam-3588	61	5	-	-	PUNCT
ejpam-3588	61	6	related	relate	VERB
ejpam-3588	61	7	to	to	ADP
ejpam-3588	61	8	a	a	DET
ejpam-3588	61	9	direct	direct	ADJ
ejpam-3588	61	10	sum	sum	NOUN
ejpam-3588	61	11	⊕i∈iui	⊕i∈iui	PROPN
ejpam-3588	61	12	of	of	ADP
ejpam-3588	61	13	uniform	uniform	ADJ
ejpam-3588	61	14	submodules	submodule	NOUN
ejpam-3588	61	15	of	of	ADP
ejpam-3588	61	16	m	m	PROPN
ejpam-3588	61	17	.	.	PUNCT
ejpam-3588	62	1	for	for	ADP
ejpam-3588	62	2	an	an	DET
ejpam-3588	62	3	r	r	NOUN
ejpam-3588	62	4	-	-	PUNCT
ejpam-3588	62	5	module	module	NOUN
ejpam-3588	62	6	m	m	NOUN
ejpam-3588	62	7	,	,	PUNCT
ejpam-3588	62	8	s	s	PART
ejpam-3588	62	9	=	=	ADJ
ejpam-3588	62	10	endr(m	endr(m	PROPN
ejpam-3588	62	11	)	)	PUNCT
ejpam-3588	62	12	denotes	denote	VERB
ejpam-3588	62	13	the	the	DET
ejpam-3588	62	14	endomorphism	endomorphism	PROPN
ejpam-3588	62	15	ring	ring	NOUN
ejpam-3588	62	16	of	of	ADP
ejpam-3588	62	17	m	m	PROPN
ejpam-3588	62	18	.	.	PUNCT
ejpam-3588	63	1	for	for	ADP
ejpam-3588	63	2	φ	φ	PROPN
ejpam-3588	63	3	∈	∈	PROPN
ejpam-3588	63	4	s	s	PROPN
ejpam-3588	63	5	,	,	PUNCT
ejpam-3588	63	6	imφ	imφ	PROPN
ejpam-3588	63	7	stands	stand	VERB
ejpam-3588	63	8	for	for	ADP
ejpam-3588	63	9	image	image	NOUN
ejpam-3588	63	10	of	of	ADP
ejpam-3588	63	11	φ	φ	PROPN
ejpam-3588	63	12	.	.	PUNCT
ejpam-3588	64	1	the	the	DET
ejpam-3588	64	2	notations	notation	NOUN
ejpam-3588	64	3	n	n	PRON
ejpam-3588	64	4	≤m	≤m	NOUN
ejpam-3588	64	5	,	,	PUNCT
ejpam-3588	64	6	n	n	CCONJ
ejpam-3588	64	7	≤e	≤e	VERB
ejpam-3588	64	8	m	m	PROPN
ejpam-3588	64	9	and	and	CCONJ
ejpam-3588	64	10	n	n	PRON
ejpam-3588	64	11	≤⊕	≤⊕	AUX
ejpam-3588	64	12	m	m	AUX
ejpam-3588	64	13	mean	mean	VERB
ejpam-3588	64	14	that	that	SCONJ
ejpam-3588	64	15	n	n	PRON
ejpam-3588	64	16	is	be	AUX
ejpam-3588	64	17	a	a	DET
ejpam-3588	64	18	submodule	submodule	NOUN
ejpam-3588	64	19	of	of	ADP
ejpam-3588	64	20	m	m	PROPN
ejpam-3588	64	21	,	,	PUNCT
ejpam-3588	64	22	an	an	DET
ejpam-3588	64	23	essential	essential	ADJ
ejpam-3588	64	24	submodule	submodule	NOUN
ejpam-3588	64	25	and	and	CCONJ
ejpam-3588	64	26	a	a	DET
ejpam-3588	64	27	direct	direct	ADJ
ejpam-3588	64	28	summand	summand	NOUN
ejpam-3588	64	29	of	of	ADP
ejpam-3588	64	30	m	m	PROPN
ejpam-3588	64	31	,	,	PUNCT
ejpam-3588	64	32	respectively	respectively	ADV
ejpam-3588	64	33	.	.	PUNCT
ejpam-3588	65	1	2	2	X
ejpam-3588	65	2	.	.	X
ejpam-3588	65	3	preliminaries	preliminary	NOUN
ejpam-3588	65	4	in	in	ADP
ejpam-3588	65	5	this	this	DET
ejpam-3588	65	6	section	section	NOUN
ejpam-3588	65	7	,	,	PUNCT
ejpam-3588	65	8	we	we	PRON
ejpam-3588	65	9	are	be	AUX
ejpam-3588	65	10	going	go	VERB
ejpam-3588	65	11	to	to	PART
ejpam-3588	65	12	give	give	VERB
ejpam-3588	65	13	preliminary	preliminary	ADJ
ejpam-3588	65	14	definitions	definition	NOUN
ejpam-3588	65	15	which	which	PRON
ejpam-3588	65	16	we	we	PRON
ejpam-3588	65	17	will	will	AUX
ejpam-3588	65	18	use	use	VERB
ejpam-3588	65	19	throughout	throughout	ADP
ejpam-3588	65	20	this	this	DET
ejpam-3588	65	21	paper	paper	NOUN
ejpam-3588	65	22	.	.	PUNCT
ejpam-3588	66	1	definition	definition	NOUN
ejpam-3588	66	2	1	1	NUM
ejpam-3588	66	3	.	.	PUNCT
ejpam-3588	67	1	let	let	VERB
ejpam-3588	67	2	s	s	PRON
ejpam-3588	67	3	be	be	AUX
ejpam-3588	67	4	a	a	DET
ejpam-3588	67	5	submodule	submodule	NOUN
ejpam-3588	67	6	of	of	ADP
ejpam-3588	67	7	an	an	DET
ejpam-3588	67	8	r	r	NOUN
ejpam-3588	67	9	-	-	PUNCT
ejpam-3588	67	10	module	module	NOUN
ejpam-3588	67	11	m	m	NOUN
ejpam-3588	67	12	.	.	PUNCT
ejpam-3588	68	1	a	a	DET
ejpam-3588	68	2	submodule	submodule	PROPN
ejpam-3588	68	3	c	c	PROPN
ejpam-3588	68	4	of	of	ADP
ejpam-3588	68	5	m	m	PROPN
ejpam-3588	68	6	is	be	AUX
ejpam-3588	68	7	said	say	VERB
ejpam-3588	68	8	to	to	PART
ejpam-3588	68	9	be	be	AUX
ejpam-3588	68	10	complement	complement	NOUN
ejpam-3588	68	11	to	to	ADP
ejpam-3588	68	12	s	s	PROPN
ejpam-3588	68	13	in	in	ADP
ejpam-3588	68	14	m	m	PROPN
ejpam-3588	68	15	if	if	SCONJ
ejpam-3588	68	16	c	c	NOUN
ejpam-3588	68	17	is	be	AUX
ejpam-3588	68	18	maximal	maximal	ADJ
ejpam-3588	68	19	with	with	ADP
ejpam-3588	68	20	respect	respect	NOUN
ejpam-3588	68	21	to	to	ADP
ejpam-3588	68	22	the	the	DET
ejpam-3588	68	23	property	property	NOUN
ejpam-3588	68	24	that	that	PRON
ejpam-3588	68	25	c	c	AUX
ejpam-3588	68	26	∩	∩	PROPN
ejpam-3588	68	27	s	s	PART
ejpam-3588	68	28	=	=	X
ejpam-3588	68	29	{	{	PUNCT
ejpam-3588	68	30	0	0	NUM
ejpam-3588	68	31	}	}	PUNCT
ejpam-3588	68	32	.	.	PUNCT
ejpam-3588	69	1	definition	definition	NOUN
ejpam-3588	69	2	2	2	NUM
ejpam-3588	69	3	.	.	PUNCT
ejpam-3588	70	1	a	a	DET
ejpam-3588	70	2	submodule	submodule	NOUN
ejpam-3588	70	3	c	c	NOUN
ejpam-3588	70	4	of	of	ADP
ejpam-3588	70	5	an	an	DET
ejpam-3588	70	6	r	r	NOUN
ejpam-3588	70	7	-	-	PUNCT
ejpam-3588	70	8	module	module	NOUN
ejpam-3588	70	9	is	be	AUX
ejpam-3588	70	10	a	a	DET
ejpam-3588	70	11	complement	complement	NOUN
ejpam-3588	70	12	in	in	ADP
ejpam-3588	70	13	m	m	PROPN
ejpam-3588	70	14	(	(	PUNCT
ejpam-3588	70	15	c	c	PROPN
ejpam-3588	70	16	⊆c	⊆c	NOUN
ejpam-3588	70	17	m	m	PROPN
ejpam-3588	70	18	,	,	PUNCT
ejpam-3588	70	19	for	for	ADP
ejpam-3588	70	20	short	short	ADJ
ejpam-3588	70	21	)	)	PUNCT
ejpam-3588	70	22	if	if	SCONJ
ejpam-3588	70	23	there	there	PRON
ejpam-3588	70	24	exists	exist	VERB
ejpam-3588	70	25	a	a	DET
ejpam-3588	70	26	submodule	submodule	NOUN
ejpam-3588	70	27	s	s	PROPN
ejpam-3588	70	28	of	of	ADP
ejpam-3588	70	29	m	m	PRON
ejpam-3588	70	30	such	such	ADJ
ejpam-3588	70	31	that	that	SCONJ
ejpam-3588	70	32	c	c	PROPN
ejpam-3588	70	33	is	be	AUX
ejpam-3588	70	34	complement	complement	VERB
ejpam-3588	70	35	to	to	ADP
ejpam-3588	70	36	s	s	PROPN
ejpam-3588	70	37	in	in	ADP
ejpam-3588	70	38	m	m	PROPN
ejpam-3588	70	39	.	.	PUNCT
ejpam-3588	71	1	definition	definition	NOUN
ejpam-3588	71	2	3	3	NUM
ejpam-3588	71	3	.	.	NOUN
ejpam-3588	71	4	1	1	NUM
ejpam-3588	71	5	.	.	PUNCT
ejpam-3588	72	1	an	an	DET
ejpam-3588	72	2	r	r	NOUN
ejpam-3588	72	3	-	-	PUNCT
ejpam-3588	72	4	module	module	NOUN
ejpam-3588	72	5	m	m	NOUN
ejpam-3588	72	6	is	be	AUX
ejpam-3588	72	7	called	call	VERB
ejpam-3588	72	8	extending	extend	VERB
ejpam-3588	72	9	module	module	NOUN
ejpam-3588	72	10	if	if	SCONJ
ejpam-3588	72	11	every	every	DET
ejpam-3588	72	12	complement	complement	NOUN
ejpam-3588	72	13	submodule	submodule	NOUN
ejpam-3588	72	14	of	of	ADP
ejpam-3588	72	15	m	m	PROPN
ejpam-3588	72	16	is	be	AUX
ejpam-3588	72	17	a	a	DET
ejpam-3588	72	18	direct	direct	ADJ
ejpam-3588	72	19	summand	summand	NOUN
ejpam-3588	72	20	.	.	PUNCT
ejpam-3588	73	1	2	2	X
ejpam-3588	73	2	.	.	X
ejpam-3588	73	3	an	an	DET
ejpam-3588	73	4	r	r	NOUN
ejpam-3588	73	5	-	-	PUNCT
ejpam-3588	73	6	module	module	NOUN
ejpam-3588	73	7	m	m	NOUN
ejpam-3588	73	8	is	be	AUX
ejpam-3588	73	9	called	call	VERB
ejpam-3588	73	10	continuous	continuous	ADJ
ejpam-3588	73	11	if	if	SCONJ
ejpam-3588	73	12	it	it	PRON
ejpam-3588	73	13	is	be	AUX
ejpam-3588	73	14	extending	extend	VERB
ejpam-3588	73	15	and	and	CCONJ
ejpam-3588	73	16	satisfies	satisfy	VERB
ejpam-3588	73	17	the	the	DET
ejpam-3588	73	18	following	follow	VERB
ejpam-3588	73	19	condition	condition	NOUN
ejpam-3588	73	20	:	:	PUNCT
ejpam-3588	73	21	(	(	PUNCT
ejpam-3588	73	22	c2	c2	PROPN
ejpam-3588	73	23	)	)	PUNCT
ejpam-3588	73	24	every	every	DET
ejpam-3588	73	25	submodule	submodule	NOUN
ejpam-3588	73	26	of	of	ADP
ejpam-3588	73	27	m	m	PRON
ejpam-3588	73	28	that	that	PRON
ejpam-3588	73	29	is	be	AUX
ejpam-3588	73	30	isomorphic	isomorphic	ADJ
ejpam-3588	73	31	to	to	ADP
ejpam-3588	73	32	a	a	DET
ejpam-3588	73	33	direct	direct	ADJ
ejpam-3588	73	34	summand	summand	NOUN
ejpam-3588	73	35	m	m	VERB
ejpam-3588	73	36	is	be	AUX
ejpam-3588	73	37	itself	itself	PRON
ejpam-3588	73	38	a	a	DET
ejpam-3588	73	39	direct	direct	ADJ
ejpam-3588	73	40	summand	summand	NOUN
ejpam-3588	73	41	of	of	ADP
ejpam-3588	73	42	m	m	PROPN
ejpam-3588	73	43	.	.	PUNCT
ejpam-3588	74	1	3	3	NUM
ejpam-3588	74	2	an	an	DET
ejpam-3588	74	3	r	r	NOUN
ejpam-3588	74	4	-	-	PUNCT
ejpam-3588	74	5	module	module	NOUN
ejpam-3588	74	6	m	m	NOUN
ejpam-3588	74	7	is	be	AUX
ejpam-3588	74	8	called	call	VERB
ejpam-3588	74	9	quasi	quasi	ADJ
ejpam-3588	74	10	-	-	ADJ
ejpam-3588	74	11	continuous	continuous	ADJ
ejpam-3588	74	12	if	if	SCONJ
ejpam-3588	74	13	it	it	PRON
ejpam-3588	74	14	is	be	AUX
ejpam-3588	74	15	extending	extend	VERB
ejpam-3588	74	16	and	and	CCONJ
ejpam-3588	74	17	satisfies	satisfy	VERB
ejpam-3588	74	18	the	the	DET
ejpam-3588	74	19	following	follow	VERB
ejpam-3588	74	20	condition	condition	NOUN
ejpam-3588	74	21	:	:	PUNCT
ejpam-3588	74	22	(	(	PUNCT
ejpam-3588	74	23	c3	c3	NOUN
ejpam-3588	74	24	)	)	PUNCT
ejpam-3588	74	25	if	if	SCONJ
ejpam-3588	74	26	n	n	PROPN
ejpam-3588	74	27	and	and	CCONJ
ejpam-3588	74	28	k	k	PROPN
ejpam-3588	74	29	are	be	AUX
ejpam-3588	74	30	direct	direct	ADJ
ejpam-3588	74	31	summands	summand	NOUN
ejpam-3588	74	32	of	of	ADP
ejpam-3588	74	33	m	m	PROPN
ejpam-3588	74	34	with	with	ADP
ejpam-3588	74	35	n	n	NOUN
ejpam-3588	74	36	∩k	∩k	NOUN
ejpam-3588	74	37	=	=	SYM
ejpam-3588	74	38	0	0	NUM
ejpam-3588	74	39	,	,	PUNCT
ejpam-3588	74	40	then	then	ADV
ejpam-3588	74	41	n	n	CCONJ
ejpam-3588	74	42	⊕k	⊕k	NOUN
ejpam-3588	74	43	is	be	AUX
ejpam-3588	74	44	a	a	DET
ejpam-3588	74	45	direct	direct	ADJ
ejpam-3588	74	46	summand	summand	NOUN
ejpam-3588	74	47	of	of	ADP
ejpam-3588	74	48	m	m	PROPN
ejpam-3588	74	49	.	.	PUNCT
ejpam-3588	75	1	definition	definition	NOUN
ejpam-3588	75	2	4	4	NUM
ejpam-3588	75	3	.	.	PUNCT
ejpam-3588	76	1	let	let	VERB
ejpam-3588	76	2	m	m	PRON
ejpam-3588	76	3	be	be	AUX
ejpam-3588	76	4	an	an	DET
ejpam-3588	76	5	r	r	NOUN
ejpam-3588	76	6	-	-	PUNCT
ejpam-3588	76	7	module	module	NOUN
ejpam-3588	76	8	,	,	PUNCT
ejpam-3588	76	9	put	put	VERB
ejpam-3588	76	10	z(m	z(m	NOUN
ejpam-3588	76	11	)	)	PUNCT
ejpam-3588	77	1	=	=	PRON
ejpam-3588	77	2	{	{	PUNCT
ejpam-3588	77	3	m	m	VERB
ejpam-3588	77	4	∈	∈	ADJ
ejpam-3588	77	5	m	m	NOUN
ejpam-3588	77	6	:	:	PUNCT
ejpam-3588	77	7	annr(m	annr(m	NOUN
ejpam-3588	77	8	)	)	PUNCT
ejpam-3588	77	9	≤e	≤e	VERB
ejpam-3588	77	10	r	r	NOUN
ejpam-3588	77	11	}	}	PUNCT
ejpam-3588	77	12	.	.	PUNCT
ejpam-3588	78	1	m	m	PROPN
ejpam-3588	78	2	is	be	AUX
ejpam-3588	78	3	called	call	VERB
ejpam-3588	78	4	nonsingular	nonsingular	ADJ
ejpam-3588	78	5	if	if	SCONJ
ejpam-3588	78	6	z(m	z(m	NOUN
ejpam-3588	78	7	)	)	PUNCT
ejpam-3588	78	8	=	=	PRON
ejpam-3588	78	9	{	{	PUNCT
ejpam-3588	78	10	0	0	NUM
ejpam-3588	78	11	}	}	PUNCT
ejpam-3588	78	12	,	,	PUNCT
ejpam-3588	78	13	and	and	CCONJ
ejpam-3588	78	14	singular	singular	ADJ
ejpam-3588	78	15	if	if	SCONJ
ejpam-3588	78	16	z(m	z(m	NOUN
ejpam-3588	78	17	)	)	PUNCT
ejpam-3588	79	1	=	=	PUNCT
ejpam-3588	79	2	m	m	NOUN
ejpam-3588	79	3	.	.	PUNCT
ejpam-3588	80	1	the	the	DET
ejpam-3588	80	2	goldie	goldie	PROPN
ejpam-3588	80	3	torsion	torsion	PROPN
ejpam-3588	80	4	submodule	submodule	PROPN
ejpam-3588	80	5	z2(m	z2(m	NOUN
ejpam-3588	80	6	)	)	PUNCT
ejpam-3588	80	7	of	of	ADP
ejpam-3588	80	8	m	m	PROPN
ejpam-3588	80	9	is	be	AUX
ejpam-3588	80	10	defined	define	VERB
ejpam-3588	80	11	by	by	ADP
ejpam-3588	80	12	z(m	z(m	PROPN
ejpam-3588	80	13	/	/	SYM
ejpam-3588	80	14	z(m	z(m	NUM
ejpam-3588	80	15	)	)	PUNCT
ejpam-3588	80	16	)	)	PUNCT
ejpam-3588	81	1	=	=	PUNCT
ejpam-3588	81	2	z2(m)/z(m	z2(m)/z(m	PROPN
ejpam-3588	81	3	)	)	PUNCT
ejpam-3588	81	4	.	.	PUNCT
ejpam-3588	82	1	m	m	PROPN
ejpam-3588	82	2	is	be	AUX
ejpam-3588	82	3	z2	z2	NUM
ejpam-3588	82	4	-	-	PUNCT
ejpam-3588	82	5	torsion	torsion	NOUN
ejpam-3588	82	6	if	if	SCONJ
ejpam-3588	82	7	,	,	PUNCT
ejpam-3588	82	8	z2(m	z2(m	X
ejpam-3588	82	9	)	)	PUNCT
ejpam-3588	82	10	=	=	NOUN
ejpam-3588	82	11	m	m	NOUN
ejpam-3588	82	12	.	.	PUNCT
ejpam-3588	83	1	definition	definition	NOUN
ejpam-3588	83	2	5	5	NUM
ejpam-3588	83	3	.	.	PUNCT
ejpam-3588	84	1	a	a	DET
ejpam-3588	84	2	module	module	NOUN
ejpam-3588	84	3	m	m	VERB
ejpam-3588	84	4	has	have	VERB
ejpam-3588	84	5	finite	finite	ADJ
ejpam-3588	84	6	uniform	uniform	ADJ
ejpam-3588	84	7	dimension	dimension	PROPN
ejpam-3588	84	8	n	n	CCONJ
ejpam-3588	84	9	(	(	PUNCT
ejpam-3588	84	10	written	write	VERB
ejpam-3588	84	11	udim(m	udim(m	PROPN
ejpam-3588	84	12	)	)	PUNCT
ejpam-3588	84	13	=	=	SYM
ejpam-3588	84	14	n	n	CCONJ
ejpam-3588	84	15	)	)	PUNCT
ejpam-3588	84	16	if	if	SCONJ
ejpam-3588	84	17	there	there	PRON
ejpam-3588	84	18	is	be	VERB
ejpam-3588	84	19	an	an	DET
ejpam-3588	84	20	essential	essential	ADJ
ejpam-3588	84	21	submodule	submodule	NOUN
ejpam-3588	84	22	v	v	ADP
ejpam-3588	84	23	≤e	≤e	NOUN
ejpam-3588	84	24	m	m	VERB
ejpam-3588	84	25	that	that	PRON
ejpam-3588	84	26	is	be	AUX
ejpam-3588	84	27	a	a	DET
ejpam-3588	84	28	direct	direct	ADJ
ejpam-3588	84	29	sum	sum	NOUN
ejpam-3588	84	30	of	of	ADP
ejpam-3588	84	31	n	n	CCONJ
ejpam-3588	84	32	uniform	uniform	ADJ
ejpam-3588	84	33	submodules	submodule	NOUN
ejpam-3588	84	34	.	.	PUNCT
ejpam-3588	85	1	3	3	X
ejpam-3588	85	2	.	.	X
ejpam-3588	85	3	main	main	ADJ
ejpam-3588	85	4	results	result	NOUN
ejpam-3588	85	5	definition	definition	NOUN
ejpam-3588	85	6	6	6	NUM
ejpam-3588	85	7	.	.	PUNCT
ejpam-3588	86	1	an	an	DET
ejpam-3588	86	2	r	r	NOUN
ejpam-3588	86	3	-	-	PUNCT
ejpam-3588	86	4	module	module	NOUN
ejpam-3588	86	5	is	be	AUX
ejpam-3588	86	6	called	call	VERB
ejpam-3588	86	7	c	c	NOUN
ejpam-3588	86	8	-	-	NOUN
ejpam-3588	86	9	retractable	retractable	ADJ
ejpam-3588	86	10	if	if	SCONJ
ejpam-3588	86	11	homr(m	homr(m	PROPN
ejpam-3588	86	12	,	,	PUNCT
ejpam-3588	86	13	c	c	NOUN
ejpam-3588	86	14	)	)	PUNCT
ejpam-3588	86	15	6=	6=	ADP
ejpam-3588	86	16	0	0	NUM
ejpam-3588	86	17	for	for	ADP
ejpam-3588	86	18	each	each	DET
ejpam-3588	86	19	0	0	NUM
ejpam-3588	86	20	6=	6=	ADP
ejpam-3588	86	21	c	c	PROPN
ejpam-3588	86	22	⊆c	⊆c	PROPN
ejpam-3588	86	23	m	m	PROPN
ejpam-3588	86	24	.	.	PUNCT
ejpam-3588	87	1	remark	remark	PROPN
ejpam-3588	87	2	1	1	NUM
ejpam-3588	87	3	.	.	PUNCT
ejpam-3588	88	1	cleary	cleary	PROPN
ejpam-3588	88	2	,	,	PUNCT
ejpam-3588	88	3	every	every	DET
ejpam-3588	88	4	retractable	retractable	ADJ
ejpam-3588	88	5	module	module	NOUN
ejpam-3588	88	6	is	be	AUX
ejpam-3588	88	7	c	c	NOUN
ejpam-3588	88	8	-	-	PUNCT
ejpam-3588	88	9	retractable	retractable	ADJ
ejpam-3588	88	10	.	.	PUNCT
ejpam-3588	89	1	the	the	DET
ejpam-3588	89	2	converse	converse	NOUN
ejpam-3588	89	3	is	be	AUX
ejpam-3588	89	4	not	not	PART
ejpam-3588	89	5	true	true	ADJ
ejpam-3588	89	6	in	in	ADP
ejpam-3588	89	7	general	general	ADJ
ejpam-3588	89	8	.	.	PUNCT
ejpam-3588	90	1	for	for	ADP
ejpam-3588	90	2	example	example	NOUN
ejpam-3588	90	3	:	:	PUNCT
ejpam-3588	90	4	q	q	X
ejpam-3588	90	5	as	as	SCONJ
ejpam-3588	90	6	a	a	DET
ejpam-3588	90	7	z	z	NOUN
ejpam-3588	90	8	-	-	PUNCT
ejpam-3588	90	9	module	module	NOUN
ejpam-3588	90	10	is	be	AUX
ejpam-3588	90	11	c	c	NOUN
ejpam-3588	90	12	-	-	PUNCT
ejpam-3588	90	13	retractable	retractable	ADJ
ejpam-3588	90	14	while	while	SCONJ
ejpam-3588	90	15	it	it	PRON
ejpam-3588	90	16	is	be	AUX
ejpam-3588	90	17	not	not	PART
ejpam-3588	90	18	retractable	retractable	ADJ
ejpam-3588	90	19	..	..	PUNCT
ejpam-3588	90	20	a.	a.	PROPN
ejpam-3588	90	21	d.	d.	PROPN
ejpam-3588	90	22	diallo	diallo	PROPN
ejpam-3588	90	23	,	,	PUNCT
ejpam-3588	90	24	p.	p.	PROPN
ejpam-3588	90	25	c.	c.	PROPN
ejpam-3588	90	26	diop	diop	PROPN
ejpam-3588	90	27	,	,	PUNCT
ejpam-3588	90	28	m.	m.	NOUN
ejpam-3588	90	29	barry	barry	PROPN
ejpam-3588	90	30	/	/	SYM
ejpam-3588	90	31	eur	eur	PROPN
ejpam-3588	90	32	.	.	PUNCT
ejpam-3588	91	1	j.	j.	PROPN
ejpam-3588	91	2	pure	pure	PROPN
ejpam-3588	91	3	appl	appl	PROPN
ejpam-3588	91	4	.	.	PROPN
ejpam-3588	91	5	math	math	PROPN
ejpam-3588	91	6	,	,	PUNCT
ejpam-3588	91	7	13	13	NUM
ejpam-3588	91	8	(	(	PUNCT
ejpam-3588	91	9	1	1	NUM
ejpam-3588	91	10	)	)	PUNCT
ejpam-3588	91	11	(	(	PUNCT
ejpam-3588	91	12	2020	2020	NUM
ejpam-3588	91	13	)	)	PUNCT
ejpam-3588	91	14	,	,	PUNCT
ejpam-3588	91	15	158	158	NUM
ejpam-3588	91	16	-	-	SYM
ejpam-3588	91	17	169	169	NUM
ejpam-3588	91	18	161	161	NUM
ejpam-3588	91	19	example	example	NOUN
ejpam-3588	91	20	1	1	NUM
ejpam-3588	91	21	.	.	PUNCT
ejpam-3588	92	1	every	every	DET
ejpam-3588	92	2	extending	extend	VERB
ejpam-3588	92	3	module	module	NOUN
ejpam-3588	92	4	is	be	AUX
ejpam-3588	92	5	c	c	NOUN
ejpam-3588	92	6	-	-	PUNCT
ejpam-3588	92	7	retractable	retractable	ADJ
ejpam-3588	92	8	.	.	PUNCT
ejpam-3588	93	1	remark	remark	NOUN
ejpam-3588	93	2	2	2	NUM
ejpam-3588	93	3	.	.	PUNCT
ejpam-3588	94	1	if	if	SCONJ
ejpam-3588	94	2	r	r	NOUN
ejpam-3588	94	3	=	=	SYM
ejpam-3588	94	4	z[x	z[x	NOUN
ejpam-3588	94	5	]	]	PUNCT
ejpam-3588	94	6	,	,	PUNCT
ejpam-3588	94	7	then	then	ADV
ejpam-3588	94	8	r	r	NOUN
ejpam-3588	94	9	is	be	AUX
ejpam-3588	94	10	c	c	NOUN
ejpam-3588	94	11	-	-	NOUN
ejpam-3588	94	12	retractable	retractable	ADJ
ejpam-3588	94	13	by	by	ADP
ejpam-3588	94	14	(	(	PUNCT
ejpam-3588	94	15	[	[	X
ejpam-3588	94	16	4	4	NUM
ejpam-3588	94	17	]	]	PUNCT
ejpam-3588	94	18	,	,	PUNCT
ejpam-3588	94	19	example	example	NOUN
ejpam-3588	94	20	2.4	2.4	NUM
ejpam-3588	94	21	)	)	PUNCT
ejpam-3588	94	22	.	.	PUNCT
ejpam-3588	95	1	clearly	clearly	ADV
ejpam-3588	95	2	,	,	PUNCT
ejpam-3588	95	3	r⊕r	r⊕r	NOUN
ejpam-3588	95	4	is	be	AUX
ejpam-3588	95	5	a	a	DET
ejpam-3588	95	6	c	c	NOUN
ejpam-3588	95	7	-	-	PUNCT
ejpam-3588	95	8	retractable	retractable	ADJ
ejpam-3588	95	9	r	r	NOUN
ejpam-3588	95	10	-	-	PUNCT
ejpam-3588	95	11	module	module	NOUN
ejpam-3588	95	12	.	.	PUNCT
ejpam-3588	96	1	however	however	ADV
ejpam-3588	96	2	,	,	PUNCT
ejpam-3588	96	3	r⊕r	r⊕r	NOUN
ejpam-3588	96	4	is	be	AUX
ejpam-3588	96	5	not	not	PART
ejpam-3588	96	6	extending	extend	VERB
ejpam-3588	96	7	.	.	PUNCT
ejpam-3588	97	1	(	(	PUNCT
ejpam-3588	97	2	see	see	VERB
ejpam-3588	97	3	[	[	X
ejpam-3588	97	4	4	4	NUM
ejpam-3588	97	5	]	]	PUNCT
ejpam-3588	97	6	,	,	PUNCT
ejpam-3588	97	7	example	example	NOUN
ejpam-3588	97	8	2.4	2.4	NUM
ejpam-3588	97	9	)	)	PUNCT
ejpam-3588	97	10	.	.	PUNCT
ejpam-3588	98	1	remark	remark	PROPN
ejpam-3588	98	2	3	3	NUM
ejpam-3588	98	3	.	.	PUNCT
ejpam-3588	99	1	(	(	PUNCT
ejpam-3588	99	2	[	[	X
ejpam-3588	99	3	4	4	NUM
ejpam-3588	99	4	]	]	PUNCT
ejpam-3588	99	5	,	,	PUNCT
ejpam-3588	99	6	example	example	NOUN
ejpam-3588	99	7	3.2	3.2	NUM
ejpam-3588	99	8	)	)	PUNCT
ejpam-3588	99	9	let	let	VERB
ejpam-3588	99	10	r	r	NOUN
ejpam-3588	99	11	be	be	AUX
ejpam-3588	99	12	the	the	DET
ejpam-3588	99	13	ring	ring	NOUN
ejpam-3588	99	14	of	of	ADP
ejpam-3588	99	15	all	all	DET
ejpam-3588	99	16	2	2	NUM
ejpam-3588	99	17	by	by	ADP
ejpam-3588	99	18	2	2	NUM
ejpam-3588	99	19	upper	upper	ADJ
ejpam-3588	99	20	triangular	triangular	NOUN
ejpam-3588	99	21	matrices	matrix	NOUN
ejpam-3588	99	22	which	which	PRON
ejpam-3588	99	23	have	have	VERB
ejpam-3588	99	24	arbitrary	arbitrary	ADJ
ejpam-3588	99	25	real	real	ADJ
ejpam-3588	99	26	numbers	number	NOUN
ejpam-3588	99	27	on	on	ADP
ejpam-3588	99	28	the	the	DET
ejpam-3588	99	29	diagonal	diagonal	ADJ
ejpam-3588	99	30	and	and	CCONJ
ejpam-3588	99	31	an	an	DET
ejpam-3588	99	32	arbitrary	arbitrary	ADJ
ejpam-3588	99	33	complex	complex	ADJ
ejpam-3588	99	34	number	number	NOUN
ejpam-3588	99	35	in	in	ADP
ejpam-3588	99	36	the	the	DET
ejpam-3588	99	37	(	(	PUNCT
ejpam-3588	99	38	1	1	NUM
ejpam-3588	99	39	,	,	PUNCT
ejpam-3588	99	40	2)-position	2)-position	NUM
ejpam-3588	99	41	and	and	CCONJ
ejpam-3588	99	42	let	let	VERB
ejpam-3588	99	43	eij	eij	PROPN
ejpam-3588	99	44	be	be	AUX
ejpam-3588	99	45	the	the	DET
ejpam-3588	99	46	element	element	NOUN
ejpam-3588	99	47	of	of	ADP
ejpam-3588	99	48	r	r	NOUN
ejpam-3588	99	49	with	with	ADP
ejpam-3588	99	50	1	1	NUM
ejpam-3588	99	51	in	in	ADP
ejpam-3588	99	52	the	the	DET
ejpam-3588	99	53	(	(	PUNCT
ejpam-3588	99	54	i	i	PROPN
ejpam-3588	99	55	,	,	PUNCT
ejpam-3588	99	56	j)-position	j)-position	NOUN
ejpam-3588	99	57	and	and	CCONJ
ejpam-3588	99	58	0	0	NUM
ejpam-3588	99	59	elsewhere	elsewhere	ADV
ejpam-3588	99	60	.	.	PUNCT
ejpam-3588	100	1	set	set	VERB
ejpam-3588	100	2	p	p	NOUN
ejpam-3588	100	3	=	=	PUNCT
ejpam-3588	100	4	e11r	e11r	PROPN
ejpam-3588	100	5	and	and	CCONJ
ejpam-3588	100	6	let	let	VERB
ejpam-3588	100	7	k	k	PROPN
ejpam-3588	100	8	denote	denote	VERB
ejpam-3588	100	9	the	the	DET
ejpam-3588	100	10	field	field	NOUN
ejpam-3588	100	11	of	of	ADP
ejpam-3588	100	12	real	real	ADJ
ejpam-3588	100	13	numbers	number	NOUN
ejpam-3588	100	14	.	.	PUNCT
ejpam-3588	101	1	hence	hence	ADV
ejpam-3588	101	2	,	,	PUNCT
ejpam-3588	101	3	p	p	PROPN
ejpam-3588	101	4	is	be	AUX
ejpam-3588	101	5	a	a	DET
ejpam-3588	101	6	nonsingular	nonsingular	ADJ
ejpam-3588	101	7	projective	projective	ADJ
ejpam-3588	101	8	r	r	NOUN
ejpam-3588	101	9	-	-	PUNCT
ejpam-3588	101	10	module	module	NOUN
ejpam-3588	101	11	which	which	PRON
ejpam-3588	101	12	is	be	AUX
ejpam-3588	101	13	not	not	PART
ejpam-3588	101	14	a	a	DET
ejpam-3588	101	15	c	c	NOUN
ejpam-3588	101	16	-	-	PUNCT
ejpam-3588	101	17	retractable	retractable	ADJ
ejpam-3588	101	18	r	r	NOUN
ejpam-3588	101	19	-	-	PUNCT
ejpam-3588	101	20	module	module	NOUN
ejpam-3588	101	21	while	while	SCONJ
ejpam-3588	101	22	the	the	DET
ejpam-3588	101	23	r	r	NOUN
ejpam-3588	101	24	-	-	PUNCT
ejpam-3588	101	25	module	module	NOUN
ejpam-3588	101	26	m	m	NOUN
ejpam-3588	101	27	=	=	SYM
ejpam-3588	101	28	r	r	NOUN
ejpam-3588	101	29	⊕	⊕	PROPN
ejpam-3588	101	30	p	p	NOUN
ejpam-3588	101	31	is	be	AUX
ejpam-3588	101	32	c	c	NOUN
ejpam-3588	101	33	-	-	PUNCT
ejpam-3588	101	34	retractable	retractable	ADJ
ejpam-3588	101	35	.	.	PUNCT
ejpam-3588	102	1	this	this	PRON
ejpam-3588	102	2	shows	show	VERB
ejpam-3588	102	3	that	that	SCONJ
ejpam-3588	102	4	a	a	DET
ejpam-3588	102	5	direct	direct	ADJ
ejpam-3588	102	6	summand	summand	NOUN
ejpam-3588	102	7	(	(	PUNCT
ejpam-3588	102	8	hence	hence	ADV
ejpam-3588	102	9	a	a	DET
ejpam-3588	102	10	submodule	submodule	NOUN
ejpam-3588	102	11	or	or	CCONJ
ejpam-3588	102	12	a	a	DET
ejpam-3588	102	13	factor	factor	NOUN
ejpam-3588	102	14	module	module	NOUN
ejpam-3588	102	15	)	)	PUNCT
ejpam-3588	102	16	of	of	ADP
ejpam-3588	102	17	a	a	DET
ejpam-3588	102	18	c	c	NOUN
ejpam-3588	102	19	-	-	PUNCT
ejpam-3588	102	20	retractable	retractable	ADJ
ejpam-3588	102	21	module	module	NOUN
ejpam-3588	102	22	need	need	AUX
ejpam-3588	102	23	not	not	PART
ejpam-3588	102	24	be	be	AUX
ejpam-3588	102	25	c	c	NOUN
ejpam-3588	102	26	-	-	PUNCT
ejpam-3588	102	27	retractable	retractable	ADJ
ejpam-3588	102	28	.	.	PUNCT
ejpam-3588	103	1	proposition	proposition	NOUN
ejpam-3588	103	2	1	1	NUM
ejpam-3588	103	3	.	.	PUNCT
ejpam-3588	104	1	let	let	VERB
ejpam-3588	104	2	m	m	PRON
ejpam-3588	104	3	be	be	AUX
ejpam-3588	104	4	a	a	DET
ejpam-3588	104	5	c	c	NOUN
ejpam-3588	104	6	-	-	PUNCT
ejpam-3588	104	7	retractable	retractable	ADJ
ejpam-3588	104	8	r	r	NOUN
ejpam-3588	104	9	-	-	PUNCT
ejpam-3588	104	10	module	module	NOUN
ejpam-3588	104	11	.	.	PUNCT
ejpam-3588	105	1	then	then	ADV
ejpam-3588	105	2	m	m	PROPN
ejpam-3588	105	3	/	/	SYM
ejpam-3588	105	4	n	n	PROPN
ejpam-3588	105	5	is	be	AUX
ejpam-3588	105	6	c	c	NOUN
ejpam-3588	105	7	-	-	NOUN
ejpam-3588	105	8	retractable	retractable	ADJ
ejpam-3588	105	9	for	for	ADP
ejpam-3588	105	10	any	any	DET
ejpam-3588	105	11	fully	fully	ADV
ejpam-3588	105	12	invariant	invariant	ADJ
ejpam-3588	105	13	complement	complement	NOUN
ejpam-3588	105	14	submodule	submodule	NOUN
ejpam-3588	105	15	n	n	PROPN
ejpam-3588	105	16	≤m	≤m	NOUN
ejpam-3588	105	17	.	.	PUNCT
ejpam-3588	106	1	proof	proof	NOUN
ejpam-3588	106	2	.	.	PUNCT
ejpam-3588	107	1	let	let	VERB
ejpam-3588	107	2	k	k	X
ejpam-3588	107	3	/	/	SYM
ejpam-3588	107	4	n	n	PROPN
ejpam-3588	107	5	⊆c	⊆c	NOUN
ejpam-3588	107	6	m	m	PROPN
ejpam-3588	107	7	/	/	SYM
ejpam-3588	107	8	n	n	PROPN
ejpam-3588	107	9	where	where	SCONJ
ejpam-3588	107	10	n	n	DET
ejpam-3588	107	11	≤	≤	NOUN
ejpam-3588	107	12	k	k	PROPN
ejpam-3588	107	13	≤m	≤m	PROPN
ejpam-3588	107	14	and	and	CCONJ
ejpam-3588	107	15	n	n	PROPN
ejpam-3588	107	16	is	be	AUX
ejpam-3588	107	17	a	a	DET
ejpam-3588	107	18	fully	fully	ADV
ejpam-3588	107	19	invariant	invariant	ADJ
ejpam-3588	107	20	complement	complement	NOUN
ejpam-3588	107	21	submodule	submodule	NOUN
ejpam-3588	107	22	of	of	ADP
ejpam-3588	107	23	m	m	PROPN
ejpam-3588	107	24	.	.	PUNCT
ejpam-3588	108	1	then	then	ADV
ejpam-3588	108	2	,	,	PUNCT
ejpam-3588	108	3	k	k	PROPN
ejpam-3588	108	4	⊆c	⊆c	X
ejpam-3588	108	5	m	m	VERB
ejpam-3588	108	6	by	by	ADP
ejpam-3588	108	7	proposition	proposition	NOUN
ejpam-3588	108	8	6.28	6.28	NUM
ejpam-3588	108	9	in	in	ADP
ejpam-3588	108	10	[	[	X
ejpam-3588	108	11	11	11	NUM
ejpam-3588	108	12	]	]	PUNCT
ejpam-3588	108	13	.	.	PUNCT
ejpam-3588	109	1	thus	thus	ADV
ejpam-3588	109	2	,	,	PUNCT
ejpam-3588	109	3	there	there	PRON
ejpam-3588	109	4	exists	exist	VERB
ejpam-3588	109	5	a	a	DET
ejpam-3588	109	6	nonzero	nonzero	NOUN
ejpam-3588	109	7	homomorphism	homomorphism	PROPN
ejpam-3588	109	8	f	f	X
ejpam-3588	109	9	:	:	PUNCT
ejpam-3588	109	10	m	m	VERB
ejpam-3588	109	11	−→	−→	ADJ
ejpam-3588	109	12	k.	k.	PROPN
ejpam-3588	109	13	now	now	ADV
ejpam-3588	109	14	,	,	PUNCT
ejpam-3588	109	15	f(n	f(n	PROPN
ejpam-3588	109	16	)	)	PUNCT
ejpam-3588	109	17	⊆	⊆	NUM
ejpam-3588	109	18	n	n	NUM
ejpam-3588	109	19	by	by	ADP
ejpam-3588	109	20	hypothesis	hypothesis	NOUN
ejpam-3588	109	21	,	,	PUNCT
ejpam-3588	109	22	and	and	CCONJ
ejpam-3588	109	23	so	so	ADV
ejpam-3588	109	24	f	f	X
ejpam-3588	109	25	:	:	PUNCT
ejpam-3588	109	26	m	m	X
ejpam-3588	109	27	/	/	SYM
ejpam-3588	109	28	n	n	ADP
ejpam-3588	109	29	−→	−→	NOUN
ejpam-3588	109	30	k	k	NOUN
ejpam-3588	109	31	/	/	SYM
ejpam-3588	109	32	n	n	PRON
ejpam-3588	109	33	defined	define	VERB
ejpam-3588	109	34	by	by	ADP
ejpam-3588	109	35	f(m+n	f(m+n	PROPN
ejpam-3588	109	36	)	)	PUNCT
ejpam-3588	109	37	=	=	SYM
ejpam-3588	109	38	f(m	f(m	PROPN
ejpam-3588	109	39	)	)	PUNCT
ejpam-3588	109	40	+	+	NOUN
ejpam-3588	109	41	n	n	NUM
ejpam-3588	109	42	for	for	ADP
ejpam-3588	109	43	all	all	DET
ejpam-3588	109	44	m	m	NOUN
ejpam-3588	109	45	∈m	∈m	NOUN
ejpam-3588	109	46	is	be	AUX
ejpam-3588	109	47	a	a	DET
ejpam-3588	109	48	nonzero	nonzero	ADJ
ejpam-3588	109	49	homomorphism	homomorphism	NOUN
ejpam-3588	109	50	.	.	PUNCT
ejpam-3588	110	1	proposition	proposition	NOUN
ejpam-3588	110	2	2	2	NUM
ejpam-3588	110	3	.	.	PUNCT
ejpam-3588	111	1	let	let	VERB
ejpam-3588	111	2	m	m	PRON
ejpam-3588	111	3	be	be	AUX
ejpam-3588	111	4	a	a	DET
ejpam-3588	111	5	c	c	NOUN
ejpam-3588	111	6	-	-	PUNCT
ejpam-3588	111	7	retractable	retractable	ADJ
ejpam-3588	111	8	r	r	NOUN
ejpam-3588	111	9	-	-	PUNCT
ejpam-3588	111	10	module	module	NOUN
ejpam-3588	111	11	such	such	ADJ
ejpam-3588	111	12	that	that	DET
ejpam-3588	111	13	homr(m	homr(m	PROPN
ejpam-3588	111	14	/	/	SYM
ejpam-3588	111	15	c	c	NOUN
ejpam-3588	111	16	,	,	PUNCT
ejpam-3588	111	17	c	c	NOUN
ejpam-3588	111	18	)	)	PUNCT
ejpam-3588	111	19	contains	contain	VERB
ejpam-3588	111	20	a	a	DET
ejpam-3588	111	21	monomorphism	monomorphism	NOUN
ejpam-3588	111	22	for	for	ADP
ejpam-3588	111	23	any	any	DET
ejpam-3588	111	24	c	c	NOUN
ejpam-3588	111	25	⊆c	⊆c	NOUN
ejpam-3588	111	26	m	m	PROPN
ejpam-3588	111	27	.	.	PUNCT
ejpam-3588	112	1	then	then	ADV
ejpam-3588	112	2	m	m	PROPN
ejpam-3588	112	3	/	/	SYM
ejpam-3588	112	4	c	c	PROPN
ejpam-3588	112	5	is	be	AUX
ejpam-3588	112	6	c	c	NOUN
ejpam-3588	112	7	-	-	PUNCT
ejpam-3588	112	8	retractable	retractable	ADJ
ejpam-3588	112	9	.	.	PUNCT
ejpam-3588	113	1	proof	proof	NOUN
ejpam-3588	113	2	.	.	PUNCT
ejpam-3588	114	1	letn	letn	PROPN
ejpam-3588	114	2	/	/	SYM
ejpam-3588	114	3	c	c	PROPN
ejpam-3588	114	4	⊆c	⊆c	NOUN
ejpam-3588	114	5	m	m	PROPN
ejpam-3588	114	6	/	/	SYM
ejpam-3588	114	7	c.	c.	PROPN
ejpam-3588	114	8	by	by	ADP
ejpam-3588	114	9	the	the	DET
ejpam-3588	114	10	c	c	NOUN
ejpam-3588	114	11	-	-	PUNCT
ejpam-3588	114	12	retractable	retractable	ADJ
ejpam-3588	114	13	condition	condition	NOUN
ejpam-3588	114	14	onm	onm	NOUN
ejpam-3588	114	15	,	,	PUNCT
ejpam-3588	114	16	there	there	PRON
ejpam-3588	114	17	is	be	VERB
ejpam-3588	114	18	a	a	DET
ejpam-3588	114	19	nonzero	nonzero	NOUN
ejpam-3588	114	20	homomorphism	homomorphism	NOUN
ejpam-3588	114	21	g	g	NOUN
ejpam-3588	114	22	:	:	PUNCT
ejpam-3588	114	23	m	m	VERB
ejpam-3588	114	24	−→	−→	ADJ
ejpam-3588	114	25	n	n	ADV
ejpam-3588	114	26	.	.	PUNCT
ejpam-3588	115	1	from	from	ADP
ejpam-3588	115	2	this	this	PRON
ejpam-3588	115	3	and	and	CCONJ
ejpam-3588	115	4	by	by	ADP
ejpam-3588	115	5	our	our	PRON
ejpam-3588	115	6	assumption	assumption	NOUN
ejpam-3588	115	7	,	,	PUNCT
ejpam-3588	115	8	homr(m	homr(m	PROPN
ejpam-3588	115	9	/	/	SYM
ejpam-3588	115	10	c	c	NOUN
ejpam-3588	115	11	,	,	PUNCT
ejpam-3588	115	12	n	n	CCONJ
ejpam-3588	115	13	/	/	SYM
ejpam-3588	115	14	c	c	NOUN
ejpam-3588	115	15	)	)	PUNCT
ejpam-3588	115	16	6=	6=	ADP
ejpam-3588	115	17	0	0	X
ejpam-3588	115	18	.	.	PUNCT
ejpam-3588	115	19	proposition	proposition	NOUN
ejpam-3588	115	20	3	3	X
ejpam-3588	115	21	.	.	PUNCT
ejpam-3588	116	1	let	let	VERB
ejpam-3588	116	2	m	m	PRON
ejpam-3588	116	3	be	be	AUX
ejpam-3588	116	4	a	a	DET
ejpam-3588	116	5	c	c	NOUN
ejpam-3588	116	6	-	-	PUNCT
ejpam-3588	116	7	retractable	retractable	ADJ
ejpam-3588	116	8	r	r	NOUN
ejpam-3588	116	9	-	-	PUNCT
ejpam-3588	116	10	module	module	NOUN
ejpam-3588	116	11	.	.	PUNCT
ejpam-3588	117	1	if	if	SCONJ
ejpam-3588	117	2	m	m	NOUN
ejpam-3588	117	3	=	=	VERB
ejpam-3588	117	4	l⊕n	l⊕n	VERB
ejpam-3588	117	5	such	such	ADJ
ejpam-3588	117	6	that	that	SCONJ
ejpam-3588	117	7	homr(l	homr(l	PROPN
ejpam-3588	117	8	,	,	PUNCT
ejpam-3588	117	9	n	n	CCONJ
ejpam-3588	117	10	)	)	PUNCT
ejpam-3588	117	11	=	=	SYM
ejpam-3588	117	12	0	0	NUM
ejpam-3588	117	13	,	,	PUNCT
ejpam-3588	117	14	then	then	ADV
ejpam-3588	117	15	n	n	PRON
ejpam-3588	117	16	is	be	AUX
ejpam-3588	117	17	a	a	DET
ejpam-3588	117	18	c	c	NOUN
ejpam-3588	117	19	-	-	PUNCT
ejpam-3588	117	20	retractable	retractable	ADJ
ejpam-3588	117	21	r	r	NOUN
ejpam-3588	117	22	-	-	PUNCT
ejpam-3588	117	23	module	module	NOUN
ejpam-3588	117	24	.	.	PUNCT
ejpam-3588	118	1	proof	proof	NOUN
ejpam-3588	118	2	.	.	PUNCT
ejpam-3588	119	1	note	note	VERB
ejpam-3588	119	2	that	that	SCONJ
ejpam-3588	119	3	endr(m	endr(m	VERB
ejpam-3588	119	4	)	)	PUNCT
ejpam-3588	119	5	=	=	SYM
ejpam-3588	119	6	[	[	PUNCT
ejpam-3588	119	7	endr(l	endr(l	NOUN
ejpam-3588	119	8	)	)	PUNCT
ejpam-3588	119	9	homr(n	homr(n	PROPN
ejpam-3588	119	10	,	,	PUNCT
ejpam-3588	119	11	l	l	NOUN
ejpam-3588	119	12	)	)	PUNCT
ejpam-3588	119	13	0	0	PUNCT
ejpam-3588	120	1	endr(n	endr(n	NOUN
ejpam-3588	120	2	)	)	PUNCT
ejpam-3588	120	3	]	]	PUNCT
ejpam-3588	120	4	.	.	PUNCT
ejpam-3588	121	1	hence	hence	ADV
ejpam-3588	121	2	,	,	PUNCT
ejpam-3588	121	3	endr(m	endr(m	PROPN
ejpam-3588	121	4	)	)	PUNCT
ejpam-3588	121	5	[	[	PUNCT
ejpam-3588	121	6	l	l	NOUN
ejpam-3588	121	7	0	0	NUM
ejpam-3588	121	8	]	]	PUNCT
ejpam-3588	122	1	⊆	⊆	NUM
ejpam-3588	122	2	[	[	PUNCT
ejpam-3588	122	3	l	l	NOUN
ejpam-3588	122	4	0	0	NUM
ejpam-3588	122	5	]	]	PUNCT
ejpam-3588	122	6	.	.	PUNCT
ejpam-3588	123	1	it	it	PRON
ejpam-3588	123	2	follows	follow	VERB
ejpam-3588	123	3	that	that	SCONJ
ejpam-3588	123	4	(	(	PUNCT
ejpam-3588	123	5	l⊕0	l⊕0	PROPN
ejpam-3588	123	6	)	)	PUNCT
ejpam-3588	123	7	is	be	AUX
ejpam-3588	123	8	a	a	DET
ejpam-3588	123	9	fully	fully	ADV
ejpam-3588	123	10	invariant	invariant	ADJ
ejpam-3588	123	11	complement	complement	NOUN
ejpam-3588	123	12	submodule	submodule	NOUN
ejpam-3588	123	13	of	of	ADP
ejpam-3588	123	14	m	m	PROPN
ejpam-3588	123	15	.	.	PUNCT
ejpam-3588	124	1	now	now	ADV
ejpam-3588	124	2	,	,	PUNCT
ejpam-3588	124	3	an	an	DET
ejpam-3588	124	4	application	application	NOUN
ejpam-3588	124	5	of	of	ADP
ejpam-3588	124	6	proposition	proposition	NOUN
ejpam-3588	124	7	1	1	NUM
ejpam-3588	124	8	shows	show	VERB
ejpam-3588	124	9	that	that	SCONJ
ejpam-3588	124	10	n	n	PRON
ejpam-3588	124	11	is	be	AUX
ejpam-3588	124	12	c	c	NOUN
ejpam-3588	124	13	-	-	PUNCT
ejpam-3588	124	14	retractable	retractable	ADJ
ejpam-3588	124	15	.	.	PUNCT
ejpam-3588	125	1	proposition	proposition	NOUN
ejpam-3588	125	2	4	4	NUM
ejpam-3588	125	3	.	.	PUNCT
ejpam-3588	126	1	let	let	VERB
ejpam-3588	126	2	m	m	PRON
ejpam-3588	126	3	be	be	AUX
ejpam-3588	126	4	a	a	DET
ejpam-3588	126	5	c	c	NOUN
ejpam-3588	126	6	-	-	PUNCT
ejpam-3588	126	7	retractable	retractable	ADJ
ejpam-3588	126	8	r	r	NOUN
ejpam-3588	126	9	-	-	PUNCT
ejpam-3588	126	10	module	module	NOUN
ejpam-3588	126	11	and	and	CCONJ
ejpam-3588	126	12	0	0	NUM
ejpam-3588	126	13	6=	6=	NUM
ejpam-3588	126	14	c	c	PROPN
ejpam-3588	126	15	⊆c	⊆c	NOUN
ejpam-3588	126	16	m	m	VERB
ejpam-3588	126	17	.	.	PUNCT
ejpam-3588	127	1	if	if	SCONJ
ejpam-3588	127	2	homr(m	homr(m	PROPN
ejpam-3588	127	3	/	/	SYM
ejpam-3588	127	4	c	c	NOUN
ejpam-3588	127	5	,	,	PUNCT
ejpam-3588	127	6	c	c	NOUN
ejpam-3588	127	7	)	)	PUNCT
ejpam-3588	127	8	=	=	SYM
ejpam-3588	127	9	0	0	NUM
ejpam-3588	127	10	,	,	PUNCT
ejpam-3588	127	11	then	then	ADV
ejpam-3588	127	12	c	c	PROPN
ejpam-3588	127	13	is	be	AUX
ejpam-3588	127	14	c	c	NOUN
ejpam-3588	127	15	-	-	PUNCT
ejpam-3588	127	16	retractable	retractable	ADJ
ejpam-3588	127	17	.	.	PUNCT
ejpam-3588	128	1	proof	proof	NOUN
ejpam-3588	128	2	.	.	PUNCT
ejpam-3588	129	1	let	let	VERB
ejpam-3588	129	2	0	0	NUM
ejpam-3588	129	3	6=	6=	NUM
ejpam-3588	129	4	k	k	PROPN
ejpam-3588	129	5	⊆c	⊆c	PROPN
ejpam-3588	129	6	c.	c.	PROPN
ejpam-3588	130	1	thus	thus	ADV
ejpam-3588	130	2	,	,	PUNCT
ejpam-3588	130	3	there	there	PRON
ejpam-3588	130	4	exists	exist	VERB
ejpam-3588	130	5	0	0	PUNCT
ejpam-3588	131	1	6=	6=	NUM
ejpam-3588	131	2	f	f	PROPN
ejpam-3588	131	3	∈	∈	PROPN
ejpam-3588	131	4	s	s	VERB
ejpam-3588	131	5	such	such	ADJ
ejpam-3588	131	6	that	that	SCONJ
ejpam-3588	131	7	imf	imf	PROPN
ejpam-3588	131	8	⊆	⊆	NUM
ejpam-3588	131	9	k.	k.	NOUN
ejpam-3588	131	10	if	if	SCONJ
ejpam-3588	131	11	f(c	f(c	PROPN
ejpam-3588	131	12	)	)	PUNCT
ejpam-3588	132	1	=	=	SYM
ejpam-3588	132	2	0	0	NUM
ejpam-3588	132	3	,	,	PUNCT
ejpam-3588	132	4	then	then	ADV
ejpam-3588	132	5	the	the	DET
ejpam-3588	132	6	rule	rule	NOUN
ejpam-3588	132	7	m+	m+	NOUN
ejpam-3588	132	8	n	n	CCONJ
ejpam-3588	132	9	−→	−→	NOUN
ejpam-3588	132	10	m+kerf	m+kerf	PROPN
ejpam-3588	132	11	yelds	yeld	NOUN
ejpam-3588	132	12	a	a	DET
ejpam-3588	132	13	nonzero	nonzero	NOUN
ejpam-3588	132	14	homomorphism	homomorphism	PROPN
ejpam-3588	132	15	m	m	PROPN
ejpam-3588	132	16	/	/	SYM
ejpam-3588	132	17	c	c	NOUN
ejpam-3588	132	18	−→m	−→m	NOUN
ejpam-3588	132	19	/	/	SYM
ejpam-3588	132	20	kerf	kerf	NOUN
ejpam-3588	132	21	∼=	∼=	NOUN
ejpam-3588	132	22	imf	imf	NOUN
ejpam-3588	132	23	which	which	PRON
ejpam-3588	132	24	is	be	AUX
ejpam-3588	132	25	in	in	ADP
ejpam-3588	132	26	contradiction	contradiction	NOUN
ejpam-3588	132	27	with	with	ADP
ejpam-3588	132	28	our	our	PRON
ejpam-3588	132	29	assumption	assumption	NOUN
ejpam-3588	132	30	homr(m	homr(m	PROPN
ejpam-3588	132	31	/	/	SYM
ejpam-3588	132	32	c	c	NOUN
ejpam-3588	132	33	,	,	PUNCT
ejpam-3588	132	34	c	c	NOUN
ejpam-3588	132	35	)	)	PUNCT
ejpam-3588	132	36	=	=	SYM
ejpam-3588	133	1	0	0	X
ejpam-3588	133	2	.	.	PUNCT
ejpam-3588	134	1	thus	thus	ADV
ejpam-3588	134	2	,	,	PUNCT
ejpam-3588	134	3	f(c	f(c	PROPN
ejpam-3588	134	4	)	)	PUNCT
ejpam-3588	134	5	6=	6=	ADP
ejpam-3588	134	6	0	0	NUM
ejpam-3588	134	7	,	,	PUNCT
ejpam-3588	134	8	hence	hence	ADV
ejpam-3588	134	9	f	f	PROPN
ejpam-3588	134	10	|c	|c	VERB
ejpam-3588	134	11	is	be	AUX
ejpam-3588	134	12	a	a	DET
ejpam-3588	134	13	nonzero	nonzero	ADJ
ejpam-3588	134	14	endomorphism	endomorphism	NOUN
ejpam-3588	134	15	of	of	ADP
ejpam-3588	134	16	c	c	PROPN
ejpam-3588	134	17	with	with	ADP
ejpam-3588	134	18	image	image	NOUN
ejpam-3588	134	19	in	in	ADP
ejpam-3588	134	20	k.	k.	PROPN
ejpam-3588	134	21	a.	a.	PROPN
ejpam-3588	134	22	d.	d.	PROPN
ejpam-3588	134	23	diallo	diallo	PROPN
ejpam-3588	134	24	,	,	PUNCT
ejpam-3588	134	25	p.	p.	PROPN
ejpam-3588	134	26	c.	c.	PROPN
ejpam-3588	134	27	diop	diop	PROPN
ejpam-3588	134	28	,	,	PUNCT
ejpam-3588	134	29	m.	m.	NOUN
ejpam-3588	134	30	barry	barry	PROPN
ejpam-3588	134	31	/	/	SYM
ejpam-3588	134	32	eur	eur	PROPN
ejpam-3588	134	33	.	.	PUNCT
ejpam-3588	135	1	j.	j.	PROPN
ejpam-3588	135	2	pure	pure	PROPN
ejpam-3588	135	3	appl	appl	PROPN
ejpam-3588	135	4	.	.	PROPN
ejpam-3588	135	5	math	math	PROPN
ejpam-3588	135	6	,	,	PUNCT
ejpam-3588	135	7	13	13	NUM
ejpam-3588	135	8	(	(	PUNCT
ejpam-3588	135	9	1	1	NUM
ejpam-3588	135	10	)	)	PUNCT
ejpam-3588	135	11	(	(	PUNCT
ejpam-3588	135	12	2020	2020	NUM
ejpam-3588	135	13	)	)	PUNCT
ejpam-3588	135	14	,	,	PUNCT
ejpam-3588	135	15	158	158	NUM
ejpam-3588	135	16	-	-	SYM
ejpam-3588	135	17	169	169	NUM
ejpam-3588	135	18	162	162	NUM
ejpam-3588	135	19	proposition	proposition	NOUN
ejpam-3588	135	20	5	5	NUM
ejpam-3588	135	21	.	.	PUNCT
ejpam-3588	136	1	if	if	SCONJ
ejpam-3588	136	2	an	an	DET
ejpam-3588	136	3	arbitrary	arbitrary	ADJ
ejpam-3588	136	4	direct	direct	ADJ
ejpam-3588	136	5	sum	sum	NOUN
ejpam-3588	136	6	of	of	ADP
ejpam-3588	136	7	copies	copy	NOUN
ejpam-3588	136	8	of	of	ADP
ejpam-3588	136	9	m	m	PROPN
ejpam-3588	136	10	is	be	AUX
ejpam-3588	136	11	c	c	NOUN
ejpam-3588	136	12	-	-	ADJ
ejpam-3588	136	13	retracatable	retracatable	ADJ
ejpam-3588	136	14	,	,	PUNCT
ejpam-3588	136	15	then	then	ADV
ejpam-3588	136	16	m	m	VERB
ejpam-3588	136	17	is	be	AUX
ejpam-3588	136	18	c	c	NOUN
ejpam-3588	136	19	-	-	PUNCT
ejpam-3588	136	20	retractable	retractable	ADJ
ejpam-3588	136	21	.	.	PUNCT
ejpam-3588	137	1	proof	proof	NOUN
ejpam-3588	137	2	.	.	PUNCT
ejpam-3588	138	1	this	this	PRON
ejpam-3588	138	2	follows	follow	VERB
ejpam-3588	138	3	from	from	ADP
ejpam-3588	138	4	(	(	PUNCT
ejpam-3588	138	5	[	[	X
ejpam-3588	138	6	17	17	NUM
ejpam-3588	138	7	]	]	PUNCT
ejpam-3588	138	8	,	,	PUNCT
ejpam-3588	138	9	proposition	proposition	NOUN
ejpam-3588	138	10	2.10	2.10	NUM
ejpam-3588	138	11	)	)	PUNCT
ejpam-3588	138	12	.	.	PUNCT
ejpam-3588	139	1	remark	remark	PROPN
ejpam-3588	139	2	4	4	NUM
ejpam-3588	139	3	.	.	PUNCT
ejpam-3588	140	1	a	a	DET
ejpam-3588	140	2	projective	projective	ADJ
ejpam-3588	140	3	module	module	NOUN
ejpam-3588	140	4	need	need	AUX
ejpam-3588	140	5	not	not	PART
ejpam-3588	140	6	be	be	AUX
ejpam-3588	140	7	c	c	NOUN
ejpam-3588	140	8	-	-	PUNCT
ejpam-3588	140	9	retracatble	retracatble	ADJ
ejpam-3588	140	10	and	and	CCONJ
ejpam-3588	140	11	vice	vice	NOUN
ejpam-3588	140	12	-	-	NOUN
ejpam-3588	140	13	versa	versa	NOUN
ejpam-3588	140	14	.	.	PUNCT
ejpam-3588	141	1	in	in	ADP
ejpam-3588	141	2	fact	fact	NOUN
ejpam-3588	141	3	a	a	DET
ejpam-3588	141	4	simple	simple	NOUN
ejpam-3588	141	5	is	be	AUX
ejpam-3588	141	6	c	c	NOUN
ejpam-3588	141	7	-	-	NOUN
ejpam-3588	141	8	retractable	retractable	ADJ
ejpam-3588	141	9	but	but	CCONJ
ejpam-3588	141	10	not	not	PART
ejpam-3588	141	11	be	be	AUX
ejpam-3588	141	12	projective	projective	ADJ
ejpam-3588	141	13	.	.	PUNCT
ejpam-3588	142	1	moreover	moreover	ADV
ejpam-3588	142	2	,	,	PUNCT
ejpam-3588	142	3	by	by	ADP
ejpam-3588	142	4	remark	remark	NOUN
ejpam-3588	142	5	3	3	NUM
ejpam-3588	142	6	,	,	PUNCT
ejpam-3588	142	7	there	there	PRON
ejpam-3588	142	8	is	be	VERB
ejpam-3588	142	9	a	a	DET
ejpam-3588	142	10	projective	projective	ADJ
ejpam-3588	142	11	module	module	NOUN
ejpam-3588	142	12	which	which	PRON
ejpam-3588	142	13	is	be	AUX
ejpam-3588	142	14	not	not	PART
ejpam-3588	142	15	c	c	NOUN
ejpam-3588	142	16	-	-	NOUN
ejpam-3588	142	17	retractable	retractable	ADJ
ejpam-3588	142	18	.	.	PUNCT
ejpam-3588	143	1	in	in	ADP
ejpam-3588	143	2	the	the	DET
ejpam-3588	143	3	following	following	NOUN
ejpam-3588	143	4	,	,	PUNCT
ejpam-3588	143	5	we	we	PRON
ejpam-3588	143	6	show	show	VERB
ejpam-3588	143	7	that	that	SCONJ
ejpam-3588	143	8	certains	certain	NOUN
ejpam-3588	143	9	classes	class	NOUN
ejpam-3588	143	10	of	of	ADP
ejpam-3588	143	11	projective	projective	ADJ
ejpam-3588	143	12	modules	module	NOUN
ejpam-3588	143	13	are	be	AUX
ejpam-3588	143	14	c	c	NOUN
ejpam-3588	143	15	-	-	PUNCT
ejpam-3588	143	16	retractable	retractable	ADJ
ejpam-3588	143	17	.	.	PUNCT
ejpam-3588	144	1	following	follow	VERB
ejpam-3588	144	2	[	[	X
ejpam-3588	144	3	24	24	NUM
ejpam-3588	144	4	]	]	PUNCT
ejpam-3588	144	5	,	,	PUNCT
ejpam-3588	144	6	we	we	PRON
ejpam-3588	144	7	call	call	VERB
ejpam-3588	144	8	an	an	DET
ejpam-3588	144	9	r	r	NOUN
ejpam-3588	144	10	-	-	PUNCT
ejpam-3588	144	11	module	module	NOUN
ejpam-3588	144	12	si	si	NOUN
ejpam-3588	144	13	if	if	SCONJ
ejpam-3588	144	14	every	every	DET
ejpam-3588	144	15	singular	singular	ADJ
ejpam-3588	144	16	module	module	NOUN
ejpam-3588	144	17	is	be	AUX
ejpam-3588	144	18	m	m	PRON
ejpam-3588	144	19	-injective	-injective	ADJ
ejpam-3588	144	20	.	.	PUNCT
ejpam-3588	145	1	recall	recall	VERB
ejpam-3588	145	2	that	that	SCONJ
ejpam-3588	145	3	a	a	DET
ejpam-3588	145	4	ring	ring	NOUN
ejpam-3588	145	5	r	r	NOUN
ejpam-3588	145	6	is	be	AUX
ejpam-3588	145	7	called	call	VERB
ejpam-3588	145	8	right	right	ADJ
ejpam-3588	145	9	si	si	ADP
ejpam-3588	145	10	,	,	PUNCT
ejpam-3588	145	11	if	if	SCONJ
ejpam-3588	145	12	every	every	DET
ejpam-3588	145	13	singular	singular	ADJ
ejpam-3588	145	14	r	r	NOUN
ejpam-3588	145	15	-	-	PUNCT
ejpam-3588	145	16	module	module	NOUN
ejpam-3588	145	17	is	be	AUX
ejpam-3588	145	18	injective	injective	ADJ
ejpam-3588	145	19	.	.	PUNCT
ejpam-3588	146	1	lemma	lemma	PROPN
ejpam-3588	146	2	1	1	NUM
ejpam-3588	146	3	.	.	PUNCT
ejpam-3588	147	1	(	(	PUNCT
ejpam-3588	147	2	[	[	X
ejpam-3588	147	3	24	24	NUM
ejpam-3588	147	4	]	]	PUNCT
ejpam-3588	147	5	,	,	PUNCT
ejpam-3588	147	6	proposition	proposition	NOUN
ejpam-3588	147	7	2.2	2.2	NUM
ejpam-3588	147	8	)	)	PUNCT
ejpam-3588	147	9	every	every	DET
ejpam-3588	147	10	homomorphic	homomorphic	ADJ
ejpam-3588	147	11	image	image	NOUN
ejpam-3588	147	12	of	of	ADP
ejpam-3588	147	13	a	a	DET
ejpam-3588	147	14	si	si	NOUN
ejpam-3588	147	15	-	-	PUNCT
ejpam-3588	147	16	module	module	NOUN
ejpam-3588	147	17	is	be	AUX
ejpam-3588	147	18	a	a	DET
ejpam-3588	147	19	si	si	NOUN
ejpam-3588	147	20	-	-	NOUN
ejpam-3588	147	21	module	module	NOUN
ejpam-3588	147	22	.	.	PUNCT
ejpam-3588	148	1	lemma	lemma	PROPN
ejpam-3588	148	2	2	2	NUM
ejpam-3588	148	3	.	.	PUNCT
ejpam-3588	149	1	(	(	PUNCT
ejpam-3588	149	2	[	[	X
ejpam-3588	149	3	24	24	NUM
ejpam-3588	149	4	]	]	PUNCT
ejpam-3588	149	5	,	,	PUNCT
ejpam-3588	149	6	proposition	proposition	NOUN
ejpam-3588	149	7	2.7	2.7	NUM
ejpam-3588	149	8	)	)	PUNCT
ejpam-3588	149	9	the	the	DET
ejpam-3588	149	10	following	follow	VERB
ejpam-3588	149	11	conditions	condition	NOUN
ejpam-3588	149	12	are	be	AUX
ejpam-3588	149	13	equivalent	equivalent	ADJ
ejpam-3588	149	14	for	for	ADP
ejpam-3588	149	15	a	a	DET
ejpam-3588	149	16	ring	ring	NOUN
ejpam-3588	149	17	r.	r.	NOUN
ejpam-3588	149	18	(	(	PUNCT
ejpam-3588	149	19	1	1	X
ejpam-3588	149	20	)	)	PUNCT
ejpam-3588	149	21	r	r	NOUN
ejpam-3588	149	22	is	be	AUX
ejpam-3588	149	23	a	a	DET
ejpam-3588	149	24	right	right	ADJ
ejpam-3588	149	25	si	si	NOUN
ejpam-3588	149	26	-	-	NOUN
ejpam-3588	149	27	ring	ring	NOUN
ejpam-3588	149	28	.	.	PUNCT
ejpam-3588	150	1	(	(	PUNCT
ejpam-3588	150	2	2	2	X
ejpam-3588	150	3	)	)	PUNCT
ejpam-3588	150	4	every	every	DET
ejpam-3588	150	5	r	r	NOUN
ejpam-3588	150	6	-	-	PUNCT
ejpam-3588	150	7	module	module	NOUN
ejpam-3588	150	8	is	be	AUX
ejpam-3588	150	9	a	a	DET
ejpam-3588	150	10	si	si	NOUN
ejpam-3588	150	11	-	-	PUNCT
ejpam-3588	150	12	module	module	NOUN
ejpam-3588	150	13	.	.	PUNCT
ejpam-3588	151	1	theorem	theorem	NOUN
ejpam-3588	151	2	1	1	NUM
ejpam-3588	151	3	.	.	PUNCT
ejpam-3588	152	1	let	let	VERB
ejpam-3588	152	2	r	r	NOUN
ejpam-3588	152	3	be	be	AUX
ejpam-3588	152	4	any	any	DET
ejpam-3588	152	5	ring	ring	NOUN
ejpam-3588	152	6	.	.	PUNCT
ejpam-3588	153	1	then	then	ADV
ejpam-3588	153	2	every	every	DET
ejpam-3588	153	3	projective	projective	NOUN
ejpam-3588	153	4	si	si	X
ejpam-3588	153	5	r	r	NOUN
ejpam-3588	153	6	-	-	PUNCT
ejpam-3588	153	7	module	module	NOUN
ejpam-3588	153	8	is	be	AUX
ejpam-3588	153	9	retractable	retractable	ADJ
ejpam-3588	153	10	and	and	CCONJ
ejpam-3588	153	11	hence	hence	ADV
ejpam-3588	153	12	c	c	NOUN
ejpam-3588	153	13	-	-	PUNCT
ejpam-3588	153	14	retractable	retractable	ADJ
ejpam-3588	153	15	.	.	PUNCT
ejpam-3588	154	1	proof	proof	NOUN
ejpam-3588	154	2	.	.	PUNCT
ejpam-3588	155	1	let	let	VERB
ejpam-3588	155	2	m	m	PRON
ejpam-3588	155	3	be	be	AUX
ejpam-3588	155	4	a	a	DET
ejpam-3588	155	5	nonzero	nonzero	X
ejpam-3588	155	6	projective	projective	ADJ
ejpam-3588	155	7	si	si	X
ejpam-3588	155	8	r	r	NOUN
ejpam-3588	155	9	-	-	PUNCT
ejpam-3588	155	10	module	module	NOUN
ejpam-3588	155	11	.	.	PUNCT
ejpam-3588	156	1	let	let	VERB
ejpam-3588	156	2	0	0	NUM
ejpam-3588	156	3	6=	6=	ADP
ejpam-3588	156	4	m	m	NOUN
ejpam-3588	156	5	∈m	∈m	NOUN
ejpam-3588	156	6	.	.	PUNCT
ejpam-3588	157	1	for	for	ADP
ejpam-3588	157	2	a	a	DET
ejpam-3588	157	3	given	give	VERB
ejpam-3588	157	4	submodule	submodule	NOUN
ejpam-3588	157	5	a	a	PRON
ejpam-3588	157	6	of	of	ADP
ejpam-3588	157	7	mr	mr	PROPN
ejpam-3588	157	8	,	,	PUNCT
ejpam-3588	157	9	there	there	PRON
ejpam-3588	157	10	exists	exist	VERB
ejpam-3588	157	11	a	a	DET
ejpam-3588	157	12	submodule	submodule	NOUN
ejpam-3588	157	13	c	c	PROPN
ejpam-3588	157	14	of	of	ADP
ejpam-3588	157	15	mr	mr	PROPN
ejpam-3588	157	16	such	such	ADJ
ejpam-3588	157	17	that	that	SCONJ
ejpam-3588	157	18	c	c	PROPN
ejpam-3588	157	19	⊕	⊕	PROPN
ejpam-3588	157	20	a	a	DET
ejpam-3588	157	21	≤e	≤e	NOUN
ejpam-3588	157	22	mr	mr	PROPN
ejpam-3588	157	23	.	.	PROPN
ejpam-3588	157	24	thus	thus	ADV
ejpam-3588	157	25	,	,	PUNCT
ejpam-3588	157	26	mr/(c	mr/(c	PROPN
ejpam-3588	157	27	⊕	⊕	PROPN
ejpam-3588	157	28	a	a	PRON
ejpam-3588	157	29	)	)	PUNCT
ejpam-3588	157	30	is	be	AUX
ejpam-3588	157	31	singular	singular	ADJ
ejpam-3588	157	32	.	.	PUNCT
ejpam-3588	158	1	since	since	SCONJ
ejpam-3588	158	2	m	m	PROPN
ejpam-3588	158	3	is	be	AUX
ejpam-3588	158	4	a	a	DET
ejpam-3588	158	5	si	si	NOUN
ejpam-3588	158	6	-	-	NOUN
ejpam-3588	158	7	module	module	NOUN
ejpam-3588	158	8	,	,	PUNCT
ejpam-3588	158	9	m/(c	m/(c	PROPN
ejpam-3588	158	10	⊕	⊕	PROPN
ejpam-3588	158	11	a	a	PRON
ejpam-3588	158	12	)	)	PUNCT
ejpam-3588	158	13	is	be	AUX
ejpam-3588	158	14	a	a	DET
ejpam-3588	158	15	si	si	NOUN
ejpam-3588	158	16	-	-	NOUN
ejpam-3588	158	17	module	module	NOUN
ejpam-3588	158	18	by	by	ADP
ejpam-3588	158	19	lemma	lemma	PROPN
ejpam-3588	158	20	1	1	NUM
ejpam-3588	158	21	.	.	PUNCT
ejpam-3588	159	1	hence	hence	ADV
ejpam-3588	159	2	,	,	PUNCT
ejpam-3588	159	3	mr/(c⊕a	mr/(c⊕a	PROPN
ejpam-3588	159	4	)	)	PUNCT
ejpam-3588	159	5	is	be	AUX
ejpam-3588	159	6	m/(c⊕a)-injective	m/(c⊕a)-injective	ADJ
ejpam-3588	159	7	and	and	CCONJ
ejpam-3588	159	8	hence	hence	ADV
ejpam-3588	159	9	a	a	DET
ejpam-3588	159	10	direct	direct	ADJ
ejpam-3588	159	11	summand	summand	NOUN
ejpam-3588	159	12	of	of	ADP
ejpam-3588	159	13	m/(c⊕a	m/(c⊕a	NOUN
ejpam-3588	159	14	)	)	PUNCT
ejpam-3588	159	15	.	.	PUNCT
ejpam-3588	160	1	it	it	PRON
ejpam-3588	160	2	follows	follow	VERB
ejpam-3588	160	3	that	that	SCONJ
ejpam-3588	160	4	m	m	PROPN
ejpam-3588	160	5	has	have	VERB
ejpam-3588	160	6	a	a	DET
ejpam-3588	160	7	submodule	submodule	NOUN
ejpam-3588	160	8	b	b	NOUN
ejpam-3588	160	9	such	such	ADJ
ejpam-3588	160	10	that	that	SCONJ
ejpam-3588	160	11	m	m	PROPN
ejpam-3588	160	12	/	/	SYM
ejpam-3588	160	13	b	b	PROPN
ejpam-3588	160	14	is	be	AUX
ejpam-3588	160	15	isomorphic	isomorphic	ADJ
ejpam-3588	160	16	to	to	ADP
ejpam-3588	160	17	mr/(c	mr/(c	PROPN
ejpam-3588	160	18	⊕	⊕	PROPN
ejpam-3588	160	19	a	a	PRON
ejpam-3588	160	20	)	)	PUNCT
ejpam-3588	160	21	.	.	PUNCT
ejpam-3588	161	1	hence	hence	ADV
ejpam-3588	161	2	there	there	PRON
ejpam-3588	161	3	exists	exist	VERB
ejpam-3588	161	4	a	a	DET
ejpam-3588	161	5	nonzero	nonzero	NOUN
ejpam-3588	161	6	homomorphism	homomorphism	PROPN
ejpam-3588	161	7	f	f	X
ejpam-3588	161	8	:	:	PUNCT
ejpam-3588	161	9	m	m	VERB
ejpam-3588	161	10	−→	−→	ADJ
ejpam-3588	161	11	mr/(c	mr/(c	PROPN
ejpam-3588	161	12	⊕	⊕	PROPN
ejpam-3588	161	13	a	a	PRON
ejpam-3588	161	14	)	)	PUNCT
ejpam-3588	161	15	.	.	PUNCT
ejpam-3588	162	1	by	by	ADP
ejpam-3588	162	2	the	the	DET
ejpam-3588	162	3	projective	projective	ADJ
ejpam-3588	162	4	condition	condition	NOUN
ejpam-3588	162	5	on	on	ADP
ejpam-3588	162	6	m	m	PROPN
ejpam-3588	162	7	,	,	PUNCT
ejpam-3588	162	8	f	f	PROPN
ejpam-3588	162	9	can	can	AUX
ejpam-3588	162	10	be	be	AUX
ejpam-3588	162	11	lifted	lift	VERB
ejpam-3588	162	12	to	to	ADP
ejpam-3588	162	13	a	a	DET
ejpam-3588	162	14	nonzero	nonzero	NOUN
ejpam-3588	162	15	of	of	ADP
ejpam-3588	162	16	homomorpism	homomorpism	NOUN
ejpam-3588	162	17	g	g	NOUN
ejpam-3588	162	18	:	:	PUNCT
ejpam-3588	162	19	m	m	VERB
ejpam-3588	162	20	−→	−→	ADJ
ejpam-3588	162	21	mr	mr	PROPN
ejpam-3588	162	22	.	.	PROPN
ejpam-3588	162	23	therefore	therefore	ADV
ejpam-3588	162	24	,	,	PUNCT
ejpam-3588	162	25	m	m	VERB
ejpam-3588	162	26	is	be	AUX
ejpam-3588	162	27	c	c	NOUN
ejpam-3588	162	28	-	-	PUNCT
ejpam-3588	162	29	retractable	retractable	ADJ
ejpam-3588	162	30	.	.	PUNCT
ejpam-3588	163	1	corollary	corollary	ADJ
ejpam-3588	163	2	1	1	NUM
ejpam-3588	163	3	.	.	PUNCT
ejpam-3588	164	1	let	let	VERB
ejpam-3588	164	2	r	r	PRON
ejpam-3588	164	3	be	be	AUX
ejpam-3588	164	4	a	a	DET
ejpam-3588	164	5	right	right	ADJ
ejpam-3588	164	6	si	si	NOUN
ejpam-3588	164	7	-	-	NOUN
ejpam-3588	164	8	ring	ring	NOUN
ejpam-3588	164	9	.	.	PUNCT
ejpam-3588	165	1	then	then	ADV
ejpam-3588	165	2	every	every	DET
ejpam-3588	165	3	projective	projective	ADJ
ejpam-3588	165	4	r	r	NOUN
ejpam-3588	165	5	-	-	PUNCT
ejpam-3588	165	6	module	module	NOUN
ejpam-3588	165	7	is	be	AUX
ejpam-3588	165	8	c	c	NOUN
ejpam-3588	165	9	-	-	PUNCT
ejpam-3588	165	10	retractable	retractable	ADJ
ejpam-3588	165	11	.	.	PUNCT
ejpam-3588	166	1	theorem	theorem	NOUN
ejpam-3588	166	2	2	2	NUM
ejpam-3588	166	3	.	.	PUNCT
ejpam-3588	167	1	let	let	VERB
ejpam-3588	167	2	r	r	PRON
ejpam-3588	167	3	be	be	AUX
ejpam-3588	167	4	a	a	DET
ejpam-3588	167	5	right	right	ADJ
ejpam-3588	167	6	perfect	perfect	ADJ
ejpam-3588	167	7	ring	ring	NOUN
ejpam-3588	167	8	.	.	PUNCT
ejpam-3588	168	1	then	then	ADV
ejpam-3588	168	2	the	the	DET
ejpam-3588	168	3	following	follow	VERB
ejpam-3588	168	4	statements	statement	NOUN
ejpam-3588	168	5	are	be	AUX
ejpam-3588	168	6	equivalent	equivalent	ADJ
ejpam-3588	168	7	for	for	ADP
ejpam-3588	168	8	a	a	DET
ejpam-3588	168	9	hereditary	hereditary	ADJ
ejpam-3588	168	10	r	r	NOUN
ejpam-3588	168	11	-	-	PUNCT
ejpam-3588	168	12	module	module	NOUN
ejpam-3588	168	13	:	:	PUNCT
ejpam-3588	168	14	(	(	PUNCT
ejpam-3588	168	15	1	1	X
ejpam-3588	168	16	)	)	PUNCT
ejpam-3588	168	17	m	m	VERB
ejpam-3588	168	18	is	be	AUX
ejpam-3588	168	19	c	c	NOUN
ejpam-3588	168	20	-	-	PUNCT
ejpam-3588	168	21	retractable	retractable	ADJ
ejpam-3588	168	22	.	.	PUNCT
ejpam-3588	169	1	(	(	PUNCT
ejpam-3588	169	2	2	2	X
ejpam-3588	169	3	)	)	PUNCT
ejpam-3588	169	4	homr(m	homr(m	PROPN
ejpam-3588	169	5	,	,	PUNCT
ejpam-3588	169	6	c	c	NOUN
ejpam-3588	169	7	)	)	PUNCT
ejpam-3588	169	8	contains	contain	VERB
ejpam-3588	169	9	an	an	DET
ejpam-3588	169	10	epimorphism	epimorphism	NOUN
ejpam-3588	169	11	for	for	ADP
ejpam-3588	169	12	any	any	DET
ejpam-3588	169	13	0	0	NUM
ejpam-3588	169	14	6=	6=	NUM
ejpam-3588	169	15	c	c	PROPN
ejpam-3588	169	16	⊆c	⊆c	NOUN
ejpam-3588	169	17	m	m	PROPN
ejpam-3588	169	18	.	.	PUNCT
ejpam-3588	170	1	(	(	PUNCT
ejpam-3588	170	2	3	3	X
ejpam-3588	170	3	)	)	PUNCT
ejpam-3588	170	4	m	m	VERB
ejpam-3588	170	5	is	be	AUX
ejpam-3588	170	6	extending	extend	VERB
ejpam-3588	170	7	.	.	PUNCT
ejpam-3588	171	1	proof	proof	NOUN
ejpam-3588	171	2	.	.	PUNCT
ejpam-3588	172	1	(	(	PUNCT
ejpam-3588	172	2	1)⇒	1)⇒	NUM
ejpam-3588	172	3	(	(	PUNCT
ejpam-3588	172	4	2	2	NUM
ejpam-3588	172	5	)	)	PUNCT
ejpam-3588	172	6	follows	follow	VERB
ejpam-3588	172	7	a	a	DET
ejpam-3588	172	8	similar	similar	ADJ
ejpam-3588	172	9	argument	argument	NOUN
ejpam-3588	172	10	to	to	ADP
ejpam-3588	172	11	the	the	DET
ejpam-3588	172	12	one	one	NOUN
ejpam-3588	172	13	used	use	VERB
ejpam-3588	172	14	in	in	ADP
ejpam-3588	172	15	(	(	PUNCT
ejpam-3588	172	16	[	[	X
ejpam-3588	172	17	14	14	NUM
ejpam-3588	172	18	]	]	PUNCT
ejpam-3588	172	19	,	,	PUNCT
ejpam-3588	172	20	theorem	theorem	VERB
ejpam-3588	172	21	2.2	2.2	NUM
ejpam-3588	172	22	)	)	PUNCT
ejpam-3588	172	23	.	.	PUNCT
ejpam-3588	173	1	(	(	PUNCT
ejpam-3588	173	2	2	2	X
ejpam-3588	173	3	)	)	PUNCT
ejpam-3588	173	4	⇒	⇒	NOUN
ejpam-3588	173	5	(	(	PUNCT
ejpam-3588	173	6	3	3	NUM
ejpam-3588	173	7	)	)	PUNCT
ejpam-3588	173	8	.	.	PUNCT
ejpam-3588	174	1	let	let	VERB
ejpam-3588	174	2	0	0	NUM
ejpam-3588	174	3	6=	6=	NUM
ejpam-3588	174	4	c	c	PROPN
ejpam-3588	174	5	⊆c	⊆c	NOUN
ejpam-3588	174	6	m	m	VERB
ejpam-3588	174	7	.	.	PUNCT
ejpam-3588	175	1	by	by	ADP
ejpam-3588	175	2	(	(	PUNCT
ejpam-3588	175	3	2	2	NUM
ejpam-3588	175	4	)	)	PUNCT
ejpam-3588	175	5	,	,	PUNCT
ejpam-3588	175	6	there	there	PRON
ejpam-3588	175	7	exists	exist	VERB
ejpam-3588	175	8	an	an	DET
ejpam-3588	175	9	epimorhism	epimorhism	NOUN
ejpam-3588	175	10	f	f	NOUN
ejpam-3588	175	11	:	:	PUNCT
ejpam-3588	175	12	m	m	AUX
ejpam-3588	175	13	−→	−→	ADJ
ejpam-3588	175	14	c.	c.	NOUN
ejpam-3588	176	1	then	then	ADV
ejpam-3588	176	2	ic	ic	X
ejpam-3588	176	3	:	:	PUNCT
ejpam-3588	176	4	c	c	AUX
ejpam-3588	176	5	−→	−→	NOUN
ejpam-3588	176	6	c	c	NOUN
ejpam-3588	176	7	can	can	AUX
ejpam-3588	176	8	be	be	AUX
ejpam-3588	176	9	lifted	lift	VERB
ejpam-3588	176	10	to	to	ADP
ejpam-3588	176	11	a	a	DET
ejpam-3588	176	12	nozero	nozero	NOUN
ejpam-3588	176	13	homomorphism	homomorphism	NOUN
ejpam-3588	176	14	g	g	NOUN
ejpam-3588	176	15	:	:	PUNCT
ejpam-3588	176	16	c	c	PROPN
ejpam-3588	176	17	−→m	−→m	INTJ
ejpam-3588	176	18	,	,	PUNCT
ejpam-3588	176	19	and	and	CCONJ
ejpam-3588	176	20	hence	hence	ADV
ejpam-3588	176	21	c	c	X
ejpam-3588	176	22	≤⊕	≤⊕	NOUN
ejpam-3588	176	23	m	m	VERB
ejpam-3588	176	24	.	.	PUNCT
ejpam-3588	177	1	a.	a.	PROPN
ejpam-3588	177	2	d.	d.	PROPN
ejpam-3588	177	3	diallo	diallo	PROPN
ejpam-3588	177	4	,	,	PUNCT
ejpam-3588	177	5	p.	p.	PROPN
ejpam-3588	177	6	c.	c.	PROPN
ejpam-3588	177	7	diop	diop	PROPN
ejpam-3588	177	8	,	,	PUNCT
ejpam-3588	177	9	m.	m.	NOUN
ejpam-3588	177	10	barry	barry	PROPN
ejpam-3588	177	11	/	/	SYM
ejpam-3588	177	12	eur	eur	PROPN
ejpam-3588	177	13	.	.	PUNCT
ejpam-3588	178	1	j.	j.	PROPN
ejpam-3588	178	2	pure	pure	PROPN
ejpam-3588	178	3	appl	appl	PROPN
ejpam-3588	178	4	.	.	PROPN
ejpam-3588	178	5	math	math	PROPN
ejpam-3588	178	6	,	,	PUNCT
ejpam-3588	178	7	13	13	NUM
ejpam-3588	178	8	(	(	PUNCT
ejpam-3588	178	9	1	1	NUM
ejpam-3588	178	10	)	)	PUNCT
ejpam-3588	178	11	(	(	PUNCT
ejpam-3588	178	12	2020	2020	NUM
ejpam-3588	178	13	)	)	PUNCT
ejpam-3588	178	14	,	,	PUNCT
ejpam-3588	178	15	158	158	NUM
ejpam-3588	178	16	-	-	SYM
ejpam-3588	178	17	169	169	NUM
ejpam-3588	178	18	163	163	NUM
ejpam-3588	178	19	therefore	therefore	ADV
ejpam-3588	178	20	,	,	PUNCT
ejpam-3588	178	21	m	m	VERB
ejpam-3588	178	22	is	be	AUX
ejpam-3588	178	23	extending	extend	VERB
ejpam-3588	178	24	.	.	PUNCT
ejpam-3588	179	1	(	(	PUNCT
ejpam-3588	179	2	3)⇒	3)⇒	NUM
ejpam-3588	179	3	(	(	PUNCT
ejpam-3588	179	4	1	1	NUM
ejpam-3588	179	5	)	)	PUNCT
ejpam-3588	179	6	it	it	PRON
ejpam-3588	179	7	is	be	AUX
ejpam-3588	179	8	easy	easy	ADJ
ejpam-3588	179	9	to	to	PART
ejpam-3588	179	10	see	see	VERB
ejpam-3588	179	11	.	.	PUNCT
ejpam-3588	180	1	recall	recall	VERB
ejpam-3588	180	2	that	that	SCONJ
ejpam-3588	180	3	a	a	DET
ejpam-3588	180	4	family	family	NOUN
ejpam-3588	180	5	{	{	PUNCT
ejpam-3588	180	6	ni}i	ni}i	PROPN
ejpam-3588	180	7	of	of	ADP
ejpam-3588	180	8	independent	independent	ADJ
ejpam-3588	180	9	submodules	submodule	NOUN
ejpam-3588	180	10	of	of	ADP
ejpam-3588	180	11	a	a	DET
ejpam-3588	180	12	module	module	NOUN
ejpam-3588	180	13	m	m	NOUN
ejpam-3588	180	14	is	be	AUX
ejpam-3588	180	15	said	say	VERB
ejpam-3588	180	16	to	to	PART
ejpam-3588	180	17	be	be	AUX
ejpam-3588	180	18	a	a	DET
ejpam-3588	180	19	local	local	ADJ
ejpam-3588	180	20	summand	summand	NOUN
ejpam-3588	180	21	,	,	PUNCT
ejpam-3588	180	22	if	if	SCONJ
ejpam-3588	180	23	for	for	ADP
ejpam-3588	180	24	any	any	DET
ejpam-3588	180	25	finite	finite	NOUN
ejpam-3588	180	26	subset	subset	VERB
ejpam-3588	180	27	a	a	DET
ejpam-3588	180	28	⊂	⊂	PROPN
ejpam-3588	180	29	i	i	PROPN
ejpam-3588	180	30	,	,	PUNCT
ejpam-3588	180	31	⊕anα	⊕anα	PROPN
ejpam-3588	180	32	is	be	AUX
ejpam-3588	180	33	a	a	DET
ejpam-3588	180	34	direct	direct	ADJ
ejpam-3588	180	35	summand	summand	NOUN
ejpam-3588	180	36	of	of	ADP
ejpam-3588	180	37	n	n	PROPN
ejpam-3588	180	38	.	.	PUNCT
ejpam-3588	181	1	an	an	DET
ejpam-3588	181	2	r	r	NOUN
ejpam-3588	181	3	-	-	PUNCT
ejpam-3588	181	4	module	module	NOUN
ejpam-3588	181	5	m	m	NOUN
ejpam-3588	181	6	is	be	AUX
ejpam-3588	181	7	called	call	VERB
ejpam-3588	181	8	uniform	uniform	ADV
ejpam-3588	181	9	-	-	PUNCT
ejpam-3588	181	10	extending	extend	VERB
ejpam-3588	181	11	if	if	SCONJ
ejpam-3588	181	12	every	every	DET
ejpam-3588	181	13	uniform	uniform	ADJ
ejpam-3588	181	14	submodule	submodule	PROPN
ejpam-3588	181	15	is	be	AUX
ejpam-3588	181	16	essential	essential	ADJ
ejpam-3588	181	17	in	in	ADP
ejpam-3588	181	18	a	a	DET
ejpam-3588	181	19	direct	direct	ADJ
ejpam-3588	181	20	summand	summand	NOUN
ejpam-3588	181	21	of	of	ADP
ejpam-3588	181	22	m	m	PROPN
ejpam-3588	181	23	.	.	PUNCT
ejpam-3588	182	1	recall	recall	VERB
ejpam-3588	182	2	that	that	SCONJ
ejpam-3588	182	3	a	a	DET
ejpam-3588	182	4	module	module	NOUN
ejpam-3588	182	5	m	m	VERB
ejpam-3588	182	6	is	be	AUX
ejpam-3588	182	7	called	call	VERB
ejpam-3588	182	8	wd	wd	ADJ
ejpam-3588	182	9	-	-	NOUN
ejpam-3588	182	10	rickart	rickart	NOUN
ejpam-3588	182	11	if	if	SCONJ
ejpam-3588	182	12	the	the	DET
ejpam-3588	182	13	image	image	NOUN
ejpam-3588	182	14	any	any	DET
ejpam-3588	182	15	endomorphism	endomorphism	NOUN
ejpam-3588	182	16	of	of	ADP
ejpam-3588	182	17	m	m	PROPN
ejpam-3588	182	18	contains	contain	VERB
ejpam-3588	182	19	a	a	DET
ejpam-3588	182	20	nonzero	nonzero	ADJ
ejpam-3588	182	21	direct	direct	ADJ
ejpam-3588	182	22	summand	summand	NOUN
ejpam-3588	182	23	.	.	PUNCT
ejpam-3588	183	1	lemma	lemma	PROPN
ejpam-3588	184	1	3	3	X
ejpam-3588	184	2	.	.	PUNCT
ejpam-3588	185	1	if	if	SCONJ
ejpam-3588	185	2	m	m	NOUN
ejpam-3588	185	3	is	be	AUX
ejpam-3588	185	4	an	an	DET
ejpam-3588	185	5	r	r	NOUN
ejpam-3588	185	6	-	-	PUNCT
ejpam-3588	185	7	module	module	NOUN
ejpam-3588	185	8	such	such	ADJ
ejpam-3588	185	9	that	that	SCONJ
ejpam-3588	185	10	every	every	DET
ejpam-3588	185	11	nonzero	nonzero	NOUN
ejpam-3588	185	12	complement	complement	NOUN
ejpam-3588	185	13	submodule	submodule	NOUN
ejpam-3588	185	14	contains	contain	VERB
ejpam-3588	185	15	a	a	DET
ejpam-3588	185	16	nonzero	nonzero	ADJ
ejpam-3588	185	17	direct	direct	ADJ
ejpam-3588	185	18	summand	summand	NOUN
ejpam-3588	185	19	,	,	PUNCT
ejpam-3588	185	20	then	then	ADV
ejpam-3588	185	21	m	m	VERB
ejpam-3588	185	22	is	be	AUX
ejpam-3588	185	23	c	c	NOUN
ejpam-3588	185	24	-	-	PUNCT
ejpam-3588	185	25	retractable	retractable	ADJ
ejpam-3588	185	26	.	.	PUNCT
ejpam-3588	186	1	proof	proof	NOUN
ejpam-3588	186	2	.	.	PUNCT
ejpam-3588	187	1	this	this	PRON
ejpam-3588	187	2	is	be	AUX
ejpam-3588	187	3	clear	clear	ADJ
ejpam-3588	187	4	.	.	PUNCT
ejpam-3588	188	1	lemma	lemma	PROPN
ejpam-3588	188	2	4	4	X
ejpam-3588	188	3	.	.	PUNCT
ejpam-3588	189	1	let	let	VERB
ejpam-3588	189	2	m	m	PRON
ejpam-3588	189	3	be	be	AUX
ejpam-3588	189	4	a	a	DET
ejpam-3588	189	5	wd	wd	ADJ
ejpam-3588	189	6	-	-	PUNCT
ejpam-3588	189	7	rickart	rickart	NOUN
ejpam-3588	189	8	r	r	NOUN
ejpam-3588	189	9	-	-	PUNCT
ejpam-3588	189	10	module	module	NOUN
ejpam-3588	189	11	.	.	PUNCT
ejpam-3588	190	1	then	then	ADV
ejpam-3588	190	2	m	m	PROPN
ejpam-3588	190	3	is	be	AUX
ejpam-3588	190	4	c	c	NOUN
ejpam-3588	190	5	-	-	PUNCT
ejpam-3588	190	6	retractable	retractable	ADJ
ejpam-3588	190	7	if	if	SCONJ
ejpam-3588	190	8	and	and	CCONJ
ejpam-3588	190	9	only	only	ADV
ejpam-3588	190	10	if	if	SCONJ
ejpam-3588	190	11	every	every	DET
ejpam-3588	190	12	nonzero	nonzero	NOUN
ejpam-3588	190	13	complement	complement	NOUN
ejpam-3588	190	14	submodule	submodule	NOUN
ejpam-3588	190	15	of	of	ADP
ejpam-3588	190	16	m	m	PROPN
ejpam-3588	190	17	contains	contain	VERB
ejpam-3588	190	18	a	a	DET
ejpam-3588	190	19	nonzero	nonzero	ADJ
ejpam-3588	190	20	direct	direct	ADJ
ejpam-3588	190	21	summand	summand	NOUN
ejpam-3588	190	22	.	.	PUNCT
ejpam-3588	191	1	proof	proof	NOUN
ejpam-3588	191	2	.	.	PUNCT
ejpam-3588	192	1	the	the	DET
ejpam-3588	192	2	suffiency	suffiency	NOUN
ejpam-3588	192	3	follows	follow	VERB
ejpam-3588	192	4	from	from	ADP
ejpam-3588	192	5	lemma	lemma	PROPN
ejpam-3588	192	6	3	3	NUM
ejpam-3588	192	7	.	.	PUNCT
ejpam-3588	192	8	conversely	conversely	ADV
ejpam-3588	192	9	,	,	PUNCT
ejpam-3588	192	10	assume	assume	VERB
ejpam-3588	192	11	that	that	SCONJ
ejpam-3588	192	12	m	m	PROPN
ejpam-3588	192	13	is	be	AUX
ejpam-3588	192	14	any	any	DET
ejpam-3588	192	15	wd	wd	PROPN
ejpam-3588	192	16	-	-	PUNCT
ejpam-3588	192	17	rickart	rickart	ADJ
ejpam-3588	192	18	cretractable	cretractable	ADJ
ejpam-3588	192	19	module	module	NOUN
ejpam-3588	192	20	.	.	PUNCT
ejpam-3588	193	1	let	let	VERB
ejpam-3588	193	2	0	0	NUM
ejpam-3588	193	3	6=	6=	NUM
ejpam-3588	193	4	c	c	PROPN
ejpam-3588	193	5	⊆c	⊆c	NOUN
ejpam-3588	193	6	m	m	PROPN
ejpam-3588	193	7	.	.	PUNCT
ejpam-3588	194	1	since	since	SCONJ
ejpam-3588	194	2	m	m	PROPN
ejpam-3588	194	3	is	be	AUX
ejpam-3588	194	4	c	c	NOUN
ejpam-3588	194	5	-	-	PUNCT
ejpam-3588	194	6	retractable	retractable	ADJ
ejpam-3588	194	7	,	,	PUNCT
ejpam-3588	194	8	there	there	PRON
ejpam-3588	194	9	is	be	VERB
ejpam-3588	194	10	a	a	DET
ejpam-3588	194	11	nonzero	nonzero	NOUN
ejpam-3588	194	12	endomorphism	endomorphism	PROPN
ejpam-3588	194	13	ϕ	ϕ	PROPN
ejpam-3588	194	14	of	of	ADP
ejpam-3588	194	15	m	m	PRON
ejpam-3588	194	16	such	such	ADJ
ejpam-3588	194	17	that	that	PRON
ejpam-3588	194	18	imϕ	imϕ	VERB
ejpam-3588	194	19	⊆	⊆	NUM
ejpam-3588	194	20	c.	c.	NOUN
ejpam-3588	194	21	thus	thus	ADV
ejpam-3588	194	22	,	,	PUNCT
ejpam-3588	194	23	the	the	DET
ejpam-3588	194	24	wd	wd	PROPN
ejpam-3588	194	25	-	-	PUNCT
ejpam-3588	194	26	rickart	rickart	NOUN
ejpam-3588	194	27	property	property	NOUN
ejpam-3588	194	28	of	of	ADP
ejpam-3588	194	29	m	m	PROPN
ejpam-3588	194	30	implies	imply	VERB
ejpam-3588	194	31	that	that	SCONJ
ejpam-3588	194	32	c	c	PROPN
ejpam-3588	194	33	contains	contain	VERB
ejpam-3588	194	34	a	a	DET
ejpam-3588	194	35	nonzero	nonzero	ADJ
ejpam-3588	194	36	direct	direct	ADJ
ejpam-3588	194	37	summand	summand	NOUN
ejpam-3588	194	38	.	.	PUNCT
ejpam-3588	195	1	lemma	lemma	PROPN
ejpam-3588	195	2	5	5	NUM
ejpam-3588	195	3	.	.	PUNCT
ejpam-3588	196	1	if	if	SCONJ
ejpam-3588	196	2	m	m	NOUN
ejpam-3588	196	3	is	be	AUX
ejpam-3588	196	4	any	any	DET
ejpam-3588	196	5	wd	wd	PROPN
ejpam-3588	196	6	-	-	PUNCT
ejpam-3588	196	7	rickart	rickart	NOUN
ejpam-3588	196	8	c	c	NOUN
ejpam-3588	196	9	-	-	PUNCT
ejpam-3588	196	10	retractable	retractable	ADJ
ejpam-3588	196	11	r	r	NOUN
ejpam-3588	196	12	-	-	PUNCT
ejpam-3588	196	13	module	module	NOUN
ejpam-3588	196	14	,	,	PUNCT
ejpam-3588	196	15	then	then	ADV
ejpam-3588	196	16	every	every	DET
ejpam-3588	196	17	indecomposable	indecomposable	ADJ
ejpam-3588	196	18	complement	complement	NOUN
ejpam-3588	196	19	submodule	submodule	NOUN
ejpam-3588	196	20	of	of	ADP
ejpam-3588	196	21	m	m	PROPN
ejpam-3588	196	22	is	be	AUX
ejpam-3588	196	23	uniform	uniform	ADJ
ejpam-3588	196	24	.	.	PUNCT
ejpam-3588	197	1	proof	proof	NOUN
ejpam-3588	197	2	.	.	PUNCT
ejpam-3588	198	1	let	let	VERB
ejpam-3588	198	2	m	m	PRON
ejpam-3588	198	3	be	be	AUX
ejpam-3588	198	4	any	any	DET
ejpam-3588	198	5	wd	wd	PROPN
ejpam-3588	198	6	-	-	PUNCT
ejpam-3588	198	7	rickart	rickart	ADJ
ejpam-3588	198	8	c	c	ADJ
ejpam-3588	198	9	-	-	PUNCT
ejpam-3588	198	10	retractable	retractable	ADJ
ejpam-3588	198	11	module	module	NOUN
ejpam-3588	198	12	.	.	PUNCT
ejpam-3588	199	1	let	let	VERB
ejpam-3588	199	2	c	c	PRON
ejpam-3588	199	3	be	be	AUX
ejpam-3588	199	4	an	an	DET
ejpam-3588	199	5	indecompsable	indecompsable	ADJ
ejpam-3588	199	6	complement	complement	NOUN
ejpam-3588	199	7	submodule	submodule	NOUN
ejpam-3588	199	8	of	of	ADP
ejpam-3588	199	9	m	m	PROPN
ejpam-3588	199	10	.	.	PUNCT
ejpam-3588	200	1	let	let	VERB
ejpam-3588	200	2	d	d	NOUN
ejpam-3588	200	3	any	any	DET
ejpam-3588	200	4	nonzero	nonzero	NOUN
ejpam-3588	200	5	complement	complement	NOUN
ejpam-3588	200	6	submodule	submodule	NOUN
ejpam-3588	200	7	of	of	ADP
ejpam-3588	200	8	c.	c.	PROPN
ejpam-3588	200	9	since	since	SCONJ
ejpam-3588	200	10	d	d	PROPN
ejpam-3588	200	11	⊆c	⊆c	ADV
ejpam-3588	200	12	m	m	VERB
ejpam-3588	200	13	,	,	PUNCT
ejpam-3588	200	14	we	we	PRON
ejpam-3588	200	15	infer	infer	VERB
ejpam-3588	200	16	from	from	ADP
ejpam-3588	200	17	lemma	lemma	PROPN
ejpam-3588	200	18	4	4	NUM
ejpam-3588	200	19	that	that	PRON
ejpam-3588	200	20	d	d	NOUN
ejpam-3588	200	21	contains	contain	VERB
ejpam-3588	200	22	a	a	DET
ejpam-3588	200	23	nonzero	nonzero	ADJ
ejpam-3588	200	24	direct	direct	ADJ
ejpam-3588	200	25	summand	summand	NOUN
ejpam-3588	200	26	e	e	PROPN
ejpam-3588	200	27	of	of	ADP
ejpam-3588	200	28	m	m	PROPN
ejpam-3588	200	29	.	.	PUNCT
ejpam-3588	201	1	as	as	SCONJ
ejpam-3588	201	2	e	e	PROPN
ejpam-3588	201	3	≤	≤	X
ejpam-3588	201	4	c	c	NOUN
ejpam-3588	201	5	≤m	≤m	PROPN
ejpam-3588	201	6	and	and	CCONJ
ejpam-3588	201	7	e	e	NOUN
ejpam-3588	201	8	≤⊕	≤⊕	ADV
ejpam-3588	201	9	m	m	VERB
ejpam-3588	201	10	,	,	PUNCT
ejpam-3588	201	11	e	e	X
ejpam-3588	201	12	≤⊕	≤⊕	NUM
ejpam-3588	201	13	c.	c.	PROPN
ejpam-3588	201	14	since	since	SCONJ
ejpam-3588	201	15	c	c	PROPN
ejpam-3588	201	16	is	be	AUX
ejpam-3588	201	17	indecomposable	indecomposable	ADJ
ejpam-3588	201	18	,	,	PUNCT
ejpam-3588	201	19	c	c	NOUN
ejpam-3588	201	20	=	=	SYM
ejpam-3588	201	21	e	e	PROPN
ejpam-3588	201	22	=	=	PUNCT
ejpam-3588	201	23	d.	d.	PROPN
ejpam-3588	201	24	it	it	PRON
ejpam-3588	201	25	follows	follow	VERB
ejpam-3588	201	26	that	that	SCONJ
ejpam-3588	201	27	d	d	NOUN
ejpam-3588	201	28	is	be	AUX
ejpam-3588	201	29	a	a	DET
ejpam-3588	201	30	direct	direct	ADJ
ejpam-3588	201	31	summand	summand	NOUN
ejpam-3588	201	32	of	of	ADP
ejpam-3588	201	33	c	c	NOUN
ejpam-3588	201	34	,	,	PUNCT
ejpam-3588	201	35	and	and	CCONJ
ejpam-3588	201	36	hence	hence	ADV
ejpam-3588	201	37	c	c	PROPN
ejpam-3588	201	38	is	be	AUX
ejpam-3588	201	39	an	an	DET
ejpam-3588	201	40	extending	extend	VERB
ejpam-3588	201	41	module	module	NOUN
ejpam-3588	201	42	.	.	PUNCT
ejpam-3588	202	1	since	since	SCONJ
ejpam-3588	202	2	c	c	PROPN
ejpam-3588	202	3	is	be	AUX
ejpam-3588	202	4	indecomposable	indecomposable	ADJ
ejpam-3588	202	5	,	,	PUNCT
ejpam-3588	202	6	c	c	PROPN
ejpam-3588	202	7	is	be	AUX
ejpam-3588	202	8	uniform	uniform	ADJ
ejpam-3588	202	9	.	.	PUNCT
ejpam-3588	203	1	theorem	theorem	NOUN
ejpam-3588	203	2	3	3	X
ejpam-3588	203	3	.	.	PUNCT
ejpam-3588	204	1	let	let	VERB
ejpam-3588	204	2	m	m	PRON
ejpam-3588	204	3	be	be	AUX
ejpam-3588	204	4	a	a	DET
ejpam-3588	204	5	wd	wd	ADJ
ejpam-3588	204	6	-	-	PUNCT
ejpam-3588	204	7	rickart	rickart	NOUN
ejpam-3588	204	8	r	r	NOUN
ejpam-3588	204	9	-	-	NOUN
ejpam-3588	204	10	module	module	NOUN
ejpam-3588	204	11	for	for	ADP
ejpam-3588	204	12	which	which	PRON
ejpam-3588	204	13	local	local	ADJ
ejpam-3588	204	14	summands	summand	NOUN
ejpam-3588	204	15	are	be	AUX
ejpam-3588	204	16	summand	summand	NOUN
ejpam-3588	204	17	.	.	PUNCT
ejpam-3588	205	1	then	then	ADV
ejpam-3588	205	2	m	m	PROPN
ejpam-3588	205	3	is	be	AUX
ejpam-3588	205	4	uniform	uniform	ADJ
ejpam-3588	205	5	-	-	PUNCT
ejpam-3588	205	6	extending	extend	VERB
ejpam-3588	205	7	and	and	CCONJ
ejpam-3588	205	8	c	c	NOUN
ejpam-3588	205	9	-	-	NOUN
ejpam-3588	205	10	retractable	retractable	ADJ
ejpam-3588	205	11	if	if	SCONJ
ejpam-3588	205	12	and	and	CCONJ
ejpam-3588	205	13	only	only	ADV
ejpam-3588	205	14	if	if	SCONJ
ejpam-3588	205	15	m	m	NOUN
ejpam-3588	205	16	is	be	AUX
ejpam-3588	205	17	extending	extend	VERB
ejpam-3588	205	18	.	.	PUNCT
ejpam-3588	206	1	proof	proof	NOUN
ejpam-3588	206	2	.	.	PUNCT
ejpam-3588	207	1	suppose	suppose	VERB
ejpam-3588	207	2	that	that	SCONJ
ejpam-3588	207	3	m	m	PROPN
ejpam-3588	207	4	is	be	AUX
ejpam-3588	207	5	a	a	DET
ejpam-3588	207	6	uniform	uniform	NOUN
ejpam-3588	207	7	-	-	PUNCT
ejpam-3588	207	8	extending	extend	VERB
ejpam-3588	207	9	c	c	NOUN
ejpam-3588	207	10	-	-	PUNCT
ejpam-3588	207	11	retractable	retractable	ADJ
ejpam-3588	207	12	module	module	NOUN
ejpam-3588	207	13	.	.	PUNCT
ejpam-3588	208	1	since	since	SCONJ
ejpam-3588	208	2	local	local	ADJ
ejpam-3588	208	3	summands	summand	NOUN
ejpam-3588	208	4	of	of	ADP
ejpam-3588	208	5	m	m	PROPN
ejpam-3588	208	6	are	be	AUX
ejpam-3588	208	7	summand	summand	NOUN
ejpam-3588	208	8	,	,	PUNCT
ejpam-3588	208	9	m	m	VERB
ejpam-3588	208	10	is	be	AUX
ejpam-3588	208	11	a	a	DET
ejpam-3588	208	12	direct	direct	ADJ
ejpam-3588	208	13	sum	sum	NOUN
ejpam-3588	208	14	of	of	ADP
ejpam-3588	208	15	indecomposable	indecomposable	ADJ
ejpam-3588	208	16	modules	module	NOUN
ejpam-3588	208	17	(	(	PUNCT
ejpam-3588	208	18	see	see	VERB
ejpam-3588	208	19	[	[	X
ejpam-3588	208	20	14	14	NUM
ejpam-3588	208	21	]	]	PUNCT
ejpam-3588	208	22	,	,	PUNCT
ejpam-3588	208	23	theorem	theorem	VERB
ejpam-3588	208	24	2.17	2.17	NUM
ejpam-3588	208	25	)	)	PUNCT
ejpam-3588	208	26	.	.	PUNCT
ejpam-3588	209	1	thus	thus	ADV
ejpam-3588	209	2	by	by	ADP
ejpam-3588	209	3	lemma	lemma	PROPN
ejpam-3588	209	4	5	5	NUM
ejpam-3588	209	5	,	,	PUNCT
ejpam-3588	209	6	m	m	VERB
ejpam-3588	209	7	is	be	AUX
ejpam-3588	209	8	direct	direct	ADJ
ejpam-3588	209	9	sum	sum	NOUN
ejpam-3588	209	10	of	of	ADP
ejpam-3588	209	11	uniform	uniform	ADJ
ejpam-3588	209	12	modules	module	NOUN
ejpam-3588	209	13	.	.	PUNCT
ejpam-3588	210	1	therefore	therefore	ADV
ejpam-3588	210	2	,	,	PUNCT
ejpam-3588	210	3	by	by	ADP
ejpam-3588	210	4	(	(	PUNCT
ejpam-3588	210	5	[	[	X
ejpam-3588	210	6	6	6	NUM
ejpam-3588	210	7	]	]	PUNCT
ejpam-3588	210	8	,	,	PUNCT
ejpam-3588	210	9	8.5	8.5	NUM
ejpam-3588	210	10	)	)	PUNCT
ejpam-3588	210	11	,	,	PUNCT
ejpam-3588	210	12	m	m	VERB
ejpam-3588	210	13	is	be	AUX
ejpam-3588	210	14	extending	extend	VERB
ejpam-3588	210	15	.	.	PUNCT
ejpam-3588	211	1	the	the	DET
ejpam-3588	211	2	converse	converse	PROPN
ejpam-3588	211	3	implication	implication	NOUN
ejpam-3588	211	4	is	be	AUX
ejpam-3588	211	5	clear	clear	ADJ
ejpam-3588	211	6	.	.	PUNCT
ejpam-3588	212	1	a.	a.	PROPN
ejpam-3588	212	2	d.	d.	PROPN
ejpam-3588	212	3	diallo	diallo	PROPN
ejpam-3588	212	4	,	,	PUNCT
ejpam-3588	212	5	p.	p.	PROPN
ejpam-3588	212	6	c.	c.	PROPN
ejpam-3588	212	7	diop	diop	PROPN
ejpam-3588	212	8	,	,	PUNCT
ejpam-3588	212	9	m.	m.	NOUN
ejpam-3588	212	10	barry	barry	PROPN
ejpam-3588	212	11	/	/	SYM
ejpam-3588	212	12	eur	eur	PROPN
ejpam-3588	212	13	.	.	PUNCT
ejpam-3588	213	1	j.	j.	PROPN
ejpam-3588	213	2	pure	pure	PROPN
ejpam-3588	213	3	appl	appl	PROPN
ejpam-3588	213	4	.	.	PROPN
ejpam-3588	213	5	math	math	PROPN
ejpam-3588	213	6	,	,	PUNCT
ejpam-3588	213	7	13	13	NUM
ejpam-3588	213	8	(	(	PUNCT
ejpam-3588	213	9	1	1	NUM
ejpam-3588	213	10	)	)	PUNCT
ejpam-3588	213	11	(	(	PUNCT
ejpam-3588	213	12	2020	2020	NUM
ejpam-3588	213	13	)	)	PUNCT
ejpam-3588	213	14	,	,	PUNCT
ejpam-3588	213	15	158	158	NUM
ejpam-3588	213	16	-	-	SYM
ejpam-3588	213	17	169	169	NUM
ejpam-3588	213	18	164	164	NUM
ejpam-3588	213	19	corollary	corollary	NOUN
ejpam-3588	213	20	2	2	NUM
ejpam-3588	213	21	.	.	PUNCT
ejpam-3588	214	1	let	let	VERB
ejpam-3588	214	2	m	m	PRON
ejpam-3588	214	3	be	be	AUX
ejpam-3588	214	4	a	a	DET
ejpam-3588	214	5	wd	wd	ADJ
ejpam-3588	214	6	-	-	PUNCT
ejpam-3588	214	7	rickart	rickart	NOUN
ejpam-3588	214	8	quasi	quasi	NOUN
ejpam-3588	214	9	-	-	NOUN
ejpam-3588	214	10	discrete	discrete	ADJ
ejpam-3588	214	11	r	r	NOUN
ejpam-3588	214	12	-	-	PUNCT
ejpam-3588	214	13	module	module	NOUN
ejpam-3588	214	14	.	.	PUNCT
ejpam-3588	215	1	then	then	ADV
ejpam-3588	215	2	m	m	PROPN
ejpam-3588	215	3	is	be	AUX
ejpam-3588	215	4	uniformextending	uniformextende	VERB
ejpam-3588	215	5	and	and	CCONJ
ejpam-3588	215	6	c	c	NOUN
ejpam-3588	215	7	-	-	NOUN
ejpam-3588	215	8	retractable	retractable	ADJ
ejpam-3588	215	9	if	if	SCONJ
ejpam-3588	215	10	and	and	CCONJ
ejpam-3588	215	11	only	only	ADV
ejpam-3588	215	12	if	if	SCONJ
ejpam-3588	215	13	m	m	NOUN
ejpam-3588	215	14	is	be	AUX
ejpam-3588	215	15	extending	extend	VERB
ejpam-3588	215	16	.	.	PUNCT
ejpam-3588	216	1	proof	proof	NOUN
ejpam-3588	216	2	.	.	PUNCT
ejpam-3588	217	1	this	this	PRON
ejpam-3588	217	2	follows	follow	VERB
ejpam-3588	217	3	from	from	ADP
ejpam-3588	217	4	theorem	theorem	ADJ
ejpam-3588	217	5	3	3	NUM
ejpam-3588	217	6	and	and	CCONJ
ejpam-3588	217	7	the	the	DET
ejpam-3588	217	8	fact	fact	NOUN
ejpam-3588	217	9	that	that	SCONJ
ejpam-3588	217	10	any	any	DET
ejpam-3588	217	11	local	local	ADJ
ejpam-3588	217	12	summand	summand	NOUN
ejpam-3588	217	13	of	of	ADP
ejpam-3588	217	14	a	a	DET
ejpam-3588	217	15	quasi	quasi	ADJ
ejpam-3588	217	16	-	-	ADJ
ejpam-3588	217	17	discrete	discrete	ADJ
ejpam-3588	217	18	module	module	NOUN
ejpam-3588	217	19	is	be	AUX
ejpam-3588	217	20	a	a	DET
ejpam-3588	217	21	summand	summand	NOUN
ejpam-3588	217	22	(	(	PUNCT
ejpam-3588	217	23	see	see	VERB
ejpam-3588	217	24	[	[	X
ejpam-3588	217	25	6	6	NUM
ejpam-3588	217	26	]	]	PUNCT
ejpam-3588	217	27	,	,	PUNCT
ejpam-3588	217	28	corollary	corollary	ADJ
ejpam-3588	217	29	4.13	4.13	NUM
ejpam-3588	217	30	)	)	PUNCT
ejpam-3588	217	31	.	.	PUNCT
ejpam-3588	218	1	remark	remark	PROPN
ejpam-3588	218	2	5	5	NUM
ejpam-3588	218	3	.	.	PUNCT
ejpam-3588	218	4	by	by	ADP
ejpam-3588	218	5	lemma	lemma	PROPN
ejpam-3588	218	6	5	5	NUM
ejpam-3588	218	7	,	,	PUNCT
ejpam-3588	218	8	an	an	DET
ejpam-3588	218	9	indecomposable	indecomposable	ADJ
ejpam-3588	218	10	wd	wd	ADJ
ejpam-3588	218	11	-	-	PUNCT
ejpam-3588	218	12	rickart	rickart	ADJ
ejpam-3588	218	13	c	c	ADJ
ejpam-3588	218	14	-	-	PUNCT
ejpam-3588	218	15	retractable	retractable	ADJ
ejpam-3588	218	16	module	module	NOUN
ejpam-3588	218	17	is	be	AUX
ejpam-3588	218	18	uniform	uniform	ADJ
ejpam-3588	218	19	.	.	PUNCT
ejpam-3588	219	1	recall	recall	VERB
ejpam-3588	219	2	that	that	SCONJ
ejpam-3588	219	3	a	a	DET
ejpam-3588	219	4	module	module	NOUN
ejpam-3588	219	5	m	m	VERB
ejpam-3588	219	6	is	be	AUX
ejpam-3588	219	7	called	call	VERB
ejpam-3588	219	8	simple	simple	ADJ
ejpam-3588	219	9	radical	radical	NOUN
ejpam-3588	219	10	,	,	PUNCT
ejpam-3588	219	11	if	if	SCONJ
ejpam-3588	219	12	m	m	VERB
ejpam-3588	219	13	6=	6=	NUM
ejpam-3588	219	14	0	0	NUM
ejpam-3588	219	15	such	such	ADJ
ejpam-3588	219	16	that	that	DET
ejpam-3588	219	17	rad(m	rad(m	NOUN
ejpam-3588	219	18	)	)	PUNCT
ejpam-3588	220	1	=	=	SYM
ejpam-3588	220	2	m	m	NOUN
ejpam-3588	220	3	and	and	CCONJ
ejpam-3588	220	4	m	m	VERB
ejpam-3588	220	5	has	have	VERB
ejpam-3588	220	6	no	no	DET
ejpam-3588	220	7	proper	proper	ADJ
ejpam-3588	220	8	nonzero	nonzero	ADJ
ejpam-3588	220	9	submodules	submodule	NOUN
ejpam-3588	220	10	n	n	NOUN
ejpam-3588	220	11	with	with	ADP
ejpam-3588	220	12	rad(n	rad(n	NOUN
ejpam-3588	220	13	)	)	PUNCT
ejpam-3588	220	14	=	=	SYM
ejpam-3588	221	1	n	n	NOUN
ejpam-3588	221	2	.	.	PUNCT
ejpam-3588	222	1	hence	hence	ADV
ejpam-3588	222	2	a	a	DET
ejpam-3588	222	3	simple	simple	ADJ
ejpam-3588	222	4	radical	radical	ADJ
ejpam-3588	222	5	c	c	NOUN
ejpam-3588	222	6	-	-	PUNCT
ejpam-3588	222	7	retractable	retractable	ADJ
ejpam-3588	222	8	module	module	NOUN
ejpam-3588	222	9	is	be	AUX
ejpam-3588	222	10	uniform	uniform	ADJ
ejpam-3588	222	11	.	.	PUNCT
ejpam-3588	223	1	let	let	VERB
ejpam-3588	223	2	m	m	PRON
ejpam-3588	223	3	be	be	AUX
ejpam-3588	223	4	an	an	DET
ejpam-3588	223	5	r	r	NOUN
ejpam-3588	223	6	-	-	PUNCT
ejpam-3588	223	7	module	module	NOUN
ejpam-3588	223	8	and	and	CCONJ
ejpam-3588	223	9	n	n	PRON
ejpam-3588	223	10	≤	≤	NOUN
ejpam-3588	223	11	m	m	VERB
ejpam-3588	223	12	.	.	PUNCT
ejpam-3588	224	1	put	put	VERB
ejpam-3588	224	2	d(n	d(n	NOUN
ejpam-3588	224	3	)	)	PUNCT
ejpam-3588	225	1	=	=	PRON
ejpam-3588	225	2	{	{	PUNCT
ejpam-3588	225	3	ϕ	ϕ	NOUN
ejpam-3588	225	4	∈	∈	PROPN
ejpam-3588	225	5	s	s	PART
ejpam-3588	225	6	:	:	PUNCT
ejpam-3588	225	7	imϕ	imϕ	VERB
ejpam-3588	225	8	⊆	⊆	NUM
ejpam-3588	225	9	n	n	CCONJ
ejpam-3588	225	10	}	}	PUNCT
ejpam-3588	225	11	.	.	PUNCT
ejpam-3588	226	1	m	m	PROPN
ejpam-3588	226	2	is	be	AUX
ejpam-3588	226	3	called	call	VERB
ejpam-3588	226	4	dual	dual	ADJ
ejpam-3588	226	5	baer	baer	PROPN
ejpam-3588	226	6	if	if	SCONJ
ejpam-3588	226	7	for	for	ADP
ejpam-3588	226	8	every	every	DET
ejpam-3588	226	9	n	n	DET
ejpam-3588	226	10	≤m	≤m	NOUN
ejpam-3588	226	11	,	,	PUNCT
ejpam-3588	226	12	there	there	PRON
ejpam-3588	226	13	is	be	VERB
ejpam-3588	226	14	e2	e2	NOUN
ejpam-3588	226	15	=	=	PUNCT
ejpam-3588	226	16	e	e	PROPN
ejpam-3588	226	17	∈	∈	PROPN
ejpam-3588	226	18	s	s	VERB
ejpam-3588	226	19	such	such	ADJ
ejpam-3588	226	20	that	that	SCONJ
ejpam-3588	226	21	d(n	d(n	NOUN
ejpam-3588	226	22	)	)	PUNCT
ejpam-3588	226	23	=	=	SYM
ejpam-3588	226	24	es	es	PROPN
ejpam-3588	226	25	.	.	NOUN
ejpam-3588	226	26	recall	recall	VERB
ejpam-3588	226	27	that	that	SCONJ
ejpam-3588	226	28	an	an	DET
ejpam-3588	226	29	r	r	NOUN
ejpam-3588	226	30	-	-	PUNCT
ejpam-3588	226	31	module	module	NOUN
ejpam-3588	226	32	m	m	NOUN
ejpam-3588	226	33	is	be	AUX
ejpam-3588	226	34	said	say	VERB
ejpam-3588	226	35	to	to	PART
ejpam-3588	226	36	be	be	AUX
ejpam-3588	226	37	ads	ad	NOUN
ejpam-3588	226	38	if	if	SCONJ
ejpam-3588	226	39	for	for	ADP
ejpam-3588	226	40	every	every	DET
ejpam-3588	226	41	decomposition	decomposition	NOUN
ejpam-3588	226	42	m	m	VERB
ejpam-3588	226	43	=	=	SYM
ejpam-3588	226	44	s	s	PROPN
ejpam-3588	226	45	⊕	⊕	PROPN
ejpam-3588	226	46	t	t	PROPN
ejpam-3588	226	47	and	and	CCONJ
ejpam-3588	226	48	every	every	DET
ejpam-3588	226	49	complement	complement	NOUN
ejpam-3588	226	50	t	t	X
ejpam-3588	226	51	′	′	NUM
ejpam-3588	226	52	of	of	ADP
ejpam-3588	226	53	s	s	PROPN
ejpam-3588	226	54	,	,	PUNCT
ejpam-3588	226	55	we	we	PRON
ejpam-3588	226	56	have	have	VERB
ejpam-3588	226	57	m	m	NOUN
ejpam-3588	226	58	=	=	SYM
ejpam-3588	226	59	s	s	PROPN
ejpam-3588	226	60	⊕	⊕	PROPN
ejpam-3588	226	61	t	t	PROPN
ejpam-3588	226	62	′.recall	′.recall	PROPN
ejpam-3588	226	63	that	that	SCONJ
ejpam-3588	226	64	an	an	DET
ejpam-3588	226	65	r	r	NOUN
ejpam-3588	226	66	-	-	PUNCT
ejpam-3588	226	67	module	module	NOUN
ejpam-3588	226	68	m	m	NOUN
ejpam-3588	226	69	is	be	AUX
ejpam-3588	226	70	called	call	VERB
ejpam-3588	226	71	quasi	quasi	NOUN
ejpam-3588	226	72	-	-	NOUN
ejpam-3588	226	73	baer	baer	PROPN
ejpam-3588	226	74	if	if	SCONJ
ejpam-3588	226	75	,	,	PUNCT
ejpam-3588	226	76	for	for	ADP
ejpam-3588	226	77	all	all	DET
ejpam-3588	226	78	fully	fully	ADV
ejpam-3588	226	79	invariant	invariant	ADJ
ejpam-3588	226	80	submodules	submodule	NOUN
ejpam-3588	226	81	n	n	PRON
ejpam-3588	226	82	≤m	≤m	NOUN
ejpam-3588	226	83	,	,	PUNCT
ejpam-3588	226	84	ls(n	ls(n	NUM
ejpam-3588	226	85	)	)	PUNCT
ejpam-3588	226	86	=	=	PUNCT
ejpam-3588	226	87	se	se	X
ejpam-3588	226	88	,	,	PUNCT
ejpam-3588	226	89	with	with	ADP
ejpam-3588	226	90	e2	e2	PROPN
ejpam-3588	226	91	=	=	PUNCT
ejpam-3588	226	92	e	e	PROPN
ejpam-3588	226	93	∈	∈	PROPN
ejpam-3588	226	94	s.	s.	PROPN
ejpam-3588	226	95	proposition	proposition	NOUN
ejpam-3588	226	96	6	6	NUM
ejpam-3588	226	97	.	.	PUNCT
ejpam-3588	227	1	let	let	VERB
ejpam-3588	227	2	m	m	PRON
ejpam-3588	227	3	be	be	AUX
ejpam-3588	227	4	a	a	DET
ejpam-3588	227	5	dual	dual	ADJ
ejpam-3588	227	6	baer	baer	PROPN
ejpam-3588	227	7	c	c	NOUN
ejpam-3588	227	8	-	-	PUNCT
ejpam-3588	227	9	retractable	retractable	ADJ
ejpam-3588	227	10	r	r	NOUN
ejpam-3588	227	11	-	-	PUNCT
ejpam-3588	227	12	module	module	NOUN
ejpam-3588	227	13	.	.	PUNCT
ejpam-3588	228	1	then	then	ADV
ejpam-3588	228	2	the	the	DET
ejpam-3588	228	3	following	follow	VERB
ejpam-3588	228	4	statements	statement	NOUN
ejpam-3588	228	5	hold	hold	VERB
ejpam-3588	228	6	:	:	PUNCT
ejpam-3588	228	7	(	(	PUNCT
ejpam-3588	228	8	1	1	X
ejpam-3588	228	9	)	)	PUNCT
ejpam-3588	229	1	m	m	VERB
ejpam-3588	229	2	is	be	AUX
ejpam-3588	229	3	a	a	DET
ejpam-3588	229	4	direct	direct	ADJ
ejpam-3588	229	5	sum	sum	NOUN
ejpam-3588	229	6	of	of	ADP
ejpam-3588	229	7	uniform	uniform	ADJ
ejpam-3588	229	8	submodules	submodule	NOUN
ejpam-3588	229	9	.	.	PUNCT
ejpam-3588	230	1	(	(	PUNCT
ejpam-3588	230	2	2	2	X
ejpam-3588	230	3	)	)	PUNCT
ejpam-3588	230	4	m	m	PROPN
ejpam-3588	230	5	=	=	SYM
ejpam-3588	230	6	z2(m	z2(m	X
ejpam-3588	230	7	)	)	PUNCT
ejpam-3588	230	8	⊕	⊕	PROPN
ejpam-3588	230	9	(	(	PUNCT
ejpam-3588	230	10	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	230	11	)	)	PUNCT
ejpam-3588	230	12	with	with	ADP
ejpam-3588	230	13	all	all	DET
ejpam-3588	230	14	mi	mi	PROPN
ejpam-3588	230	15	nonsingular	nonsingular	PROPN
ejpam-3588	230	16	uniform	uniform	PROPN
ejpam-3588	230	17	quasi	quasi	PROPN
ejpam-3588	230	18	-	-	PROPN
ejpam-3588	230	19	baer	baer	PROPN
ejpam-3588	230	20	and	and	CCONJ
ejpam-3588	230	21	end(mi	end(mi	NOUN
ejpam-3588	230	22	)	)	PUNCT
ejpam-3588	230	23	semi	semi	ADJ
ejpam-3588	230	24	-	-	ADJ
ejpam-3588	230	25	local	local	ADJ
ejpam-3588	230	26	quasi	quasi	NOUN
ejpam-3588	230	27	-	-	NOUN
ejpam-3588	230	28	baer	baer	PROPN
ejpam-3588	230	29	.	.	PUNCT
ejpam-3588	231	1	(	(	PUNCT
ejpam-3588	231	2	1	1	X
ejpam-3588	231	3	)	)	PUNCT
ejpam-3588	231	4	if	if	SCONJ
ejpam-3588	231	5	r	r	NOUN
ejpam-3588	231	6	is	be	AUX
ejpam-3588	231	7	a	a	DET
ejpam-3588	231	8	right	right	ADJ
ejpam-3588	231	9	self	self	NOUN
ejpam-3588	231	10	-	-	PUNCT
ejpam-3588	231	11	injective	injective	ADJ
ejpam-3588	231	12	ring	ring	NOUN
ejpam-3588	231	13	,	,	PUNCT
ejpam-3588	231	14	then	then	ADV
ejpam-3588	231	15	m	m	VERB
ejpam-3588	231	16	=	=	SYM
ejpam-3588	231	17	z2(m	z2(m	NOUN
ejpam-3588	231	18	)	)	PUNCT
ejpam-3588	231	19	⊕m	⊕m	NOUN
ejpam-3588	232	1	′	′	NUM
ejpam-3588	232	2	where	where	SCONJ
ejpam-3588	232	3	m	m	VERB
ejpam-3588	232	4	′	′	VERB
ejpam-3588	232	5	is	be	AUX
ejpam-3588	232	6	nonsingular	nonsingular	ADJ
ejpam-3588	232	7	semisimple	semisimple	NOUN
ejpam-3588	232	8	.	.	PUNCT
ejpam-3588	233	1	proof	proof	NOUN
ejpam-3588	233	2	.	.	PUNCT
ejpam-3588	234	1	(	(	PUNCT
ejpam-3588	234	2	1	1	X
ejpam-3588	234	3	)	)	PUNCT
ejpam-3588	234	4	suppose	suppose	VERB
ejpam-3588	234	5	m	m	NOUN
ejpam-3588	234	6	is	be	AUX
ejpam-3588	234	7	dual	dual	ADJ
ejpam-3588	234	8	-	-	PUNCT
ejpam-3588	234	9	baer	baer	NOUN
ejpam-3588	234	10	c	c	NOUN
ejpam-3588	234	11	-	-	PUNCT
ejpam-3588	234	12	retractable	retractable	ADJ
ejpam-3588	234	13	.	.	PUNCT
ejpam-3588	235	1	by	by	ADP
ejpam-3588	235	2	corollary	corollary	ADJ
ejpam-3588	235	3	2.6(i	2.6(i	NUM
ejpam-3588	235	4	)	)	PUNCT
ejpam-3588	235	5	in	in	ADP
ejpam-3588	235	6	[	[	X
ejpam-3588	235	7	10	10	NUM
ejpam-3588	235	8	]	]	PUNCT
ejpam-3588	235	9	,	,	PUNCT
ejpam-3588	235	10	m	m	VERB
ejpam-3588	235	11	is	be	AUX
ejpam-3588	235	12	a	a	DET
ejpam-3588	235	13	direct	direct	ADJ
ejpam-3588	235	14	sum	sum	NOUN
ejpam-3588	235	15	of	of	ADP
ejpam-3588	235	16	indecomposable	indecomposable	ADJ
ejpam-3588	235	17	submodules	submodule	NOUN
ejpam-3588	235	18	.	.	PUNCT
ejpam-3588	236	1	by	by	ADP
ejpam-3588	236	2	(	(	PUNCT
ejpam-3588	236	3	[	[	X
ejpam-3588	236	4	24	24	NUM
ejpam-3588	236	5	]	]	PUNCT
ejpam-3588	236	6	,	,	PUNCT
ejpam-3588	236	7	theorem	theorem	VERB
ejpam-3588	236	8	3.1	3.1	NUM
ejpam-3588	236	9	)	)	PUNCT
ejpam-3588	236	10	,	,	PUNCT
ejpam-3588	236	11	m	m	PROPN
ejpam-3588	236	12	is	be	AUX
ejpam-3588	236	13	wd	wd	PROPN
ejpam-3588	236	14	-	-	PUNCT
ejpam-3588	236	15	rickart	rickart	NOUN
ejpam-3588	236	16	.	.	PUNCT
ejpam-3588	237	1	thus	thus	ADV
ejpam-3588	237	2	,	,	PUNCT
ejpam-3588	237	3	according	accord	VERB
ejpam-3588	237	4	to	to	ADP
ejpam-3588	237	5	lemma	lemma	PROPN
ejpam-3588	237	6	5	5	NUM
ejpam-3588	237	7	,	,	PUNCT
ejpam-3588	237	8	m	m	VERB
ejpam-3588	237	9	is	be	AUX
ejpam-3588	237	10	a	a	DET
ejpam-3588	237	11	direct	direct	ADJ
ejpam-3588	237	12	sum	sum	NOUN
ejpam-3588	237	13	of	of	ADP
ejpam-3588	237	14	uniform	uniform	ADJ
ejpam-3588	237	15	submodules	submodule	NOUN
ejpam-3588	237	16	.	.	PUNCT
ejpam-3588	238	1	(	(	PUNCT
ejpam-3588	238	2	2	2	X
ejpam-3588	238	3	)	)	PUNCT
ejpam-3588	238	4	suppose	suppose	VERB
ejpam-3588	238	5	m	m	NOUN
ejpam-3588	238	6	has	have	VERB
ejpam-3588	238	7	the	the	DET
ejpam-3588	238	8	stated	state	VERB
ejpam-3588	238	9	condition	condition	NOUN
ejpam-3588	238	10	.	.	PUNCT
ejpam-3588	239	1	then	then	ADV
ejpam-3588	239	2	by	by	ADP
ejpam-3588	239	3	(	(	PUNCT
ejpam-3588	239	4	1	1	NUM
ejpam-3588	239	5	)	)	PUNCT
ejpam-3588	239	6	,	,	PUNCT
ejpam-3588	239	7	m	m	VERB
ejpam-3588	239	8	is	be	AUX
ejpam-3588	239	9	a	a	DET
ejpam-3588	239	10	direct	direct	ADJ
ejpam-3588	239	11	sum	sum	NOUN
ejpam-3588	239	12	of	of	ADP
ejpam-3588	239	13	uniform	uniform	ADJ
ejpam-3588	239	14	modules	module	NOUN
ejpam-3588	239	15	.	.	PUNCT
ejpam-3588	240	1	hence	hence	ADV
ejpam-3588	240	2	by	by	ADP
ejpam-3588	240	3	(	(	PUNCT
ejpam-3588	240	4	[	[	X
ejpam-3588	240	5	2	2	NUM
ejpam-3588	240	6	]	]	PUNCT
ejpam-3588	240	7	,	,	PUNCT
ejpam-3588	240	8	corollary	corollary	ADJ
ejpam-3588	240	9	2.3	2.3	NUM
ejpam-3588	240	10	,	,	PUNCT
ejpam-3588	240	11	theorems	theorem	VERB
ejpam-3588	240	12	3.2	3.2	NUM
ejpam-3588	240	13	and	and	CCONJ
ejpam-3588	240	14	3.9	3.9	NUM
ejpam-3588	240	15	)	)	PUNCT
ejpam-3588	240	16	,	,	PUNCT
ejpam-3588	240	17	m	m	VERB
ejpam-3588	240	18	=	=	SYM
ejpam-3588	240	19	z2(m	z2(m	NOUN
ejpam-3588	240	20	)	)	PUNCT
ejpam-3588	240	21	⊕m	⊕m	NOUN
ejpam-3588	241	1	′	′	NUM
ejpam-3588	241	2	where	where	SCONJ
ejpam-3588	241	3	m	m	VERB
ejpam-3588	241	4	′	′	VERB
ejpam-3588	241	5	is	be	AUX
ejpam-3588	241	6	quasi	quasi	ADJ
ejpam-3588	241	7	-	-	NOUN
ejpam-3588	241	8	baer	baer	PROPN
ejpam-3588	241	9	.	.	PUNCT
ejpam-3588	242	1	since	since	SCONJ
ejpam-3588	242	2	m	m	PROPN
ejpam-3588	242	3	is	be	AUX
ejpam-3588	242	4	dual	dual	ADJ
ejpam-3588	242	5	baer	baer	PROPN
ejpam-3588	242	6	,	,	PUNCT
ejpam-3588	242	7	we	we	PRON
ejpam-3588	242	8	infer	infer	VERB
ejpam-3588	242	9	from	from	ADP
ejpam-3588	242	10	corollaries	corollary	NOUN
ejpam-3588	242	11	2.5	2.5	NUM
ejpam-3588	242	12	and	and	CCONJ
ejpam-3588	242	13	2.6	2.6	NUM
ejpam-3588	242	14	in	in	ADP
ejpam-3588	242	15	[	[	X
ejpam-3588	242	16	2	2	X
ejpam-3588	242	17	]	]	PUNCT
ejpam-3588	242	18	that	that	PRON
ejpam-3588	242	19	m	m	VERB
ejpam-3588	242	20	′	′	NUM
ejpam-3588	243	1	=	=	PUNCT
ejpam-3588	243	2	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	243	3	with	with	ADP
ejpam-3588	243	4	all	all	DET
ejpam-3588	243	5	mi	mi	PROPN
ejpam-3588	243	6	indecomposable	indecomposable	ADJ
ejpam-3588	243	7	.	.	PUNCT
ejpam-3588	244	1	thus	thus	ADV
ejpam-3588	244	2	,	,	PUNCT
ejpam-3588	244	3	m	m	VERB
ejpam-3588	244	4	=	=	ADJ
ejpam-3588	244	5	z2(m	z2(m	X
ejpam-3588	244	6	)	)	PUNCT
ejpam-3588	244	7	⊕	⊕	PROPN
ejpam-3588	244	8	(	(	PUNCT
ejpam-3588	244	9	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	244	10	)	)	PUNCT
ejpam-3588	244	11	where	where	SCONJ
ejpam-3588	244	12	each	each	DET
ejpam-3588	244	13	mi	mi	PROPN
ejpam-3588	244	14	is	be	AUX
ejpam-3588	244	15	indecompsable	indecompsable	ADJ
ejpam-3588	244	16	.	.	PUNCT
ejpam-3588	245	1	consequently	consequently	ADV
ejpam-3588	245	2	,	,	PUNCT
ejpam-3588	245	3	each	each	DET
ejpam-3588	245	4	mi	mi	PROPN
ejpam-3588	245	5	is	be	AUX
ejpam-3588	245	6	nonsingular	nonsingular	ADJ
ejpam-3588	245	7	uniform	uniform	NOUN
ejpam-3588	245	8	by	by	ADP
ejpam-3588	245	9	lemma	lemma	PROPN
ejpam-3588	245	10	5	5	NUM
ejpam-3588	245	11	.	.	PUNCT
ejpam-3588	246	1	on	on	ADP
ejpam-3588	246	2	the	the	DET
ejpam-3588	246	3	other	other	ADJ
ejpam-3588	246	4	hand	hand	NOUN
ejpam-3588	246	5	since	since	SCONJ
ejpam-3588	246	6	m	m	PROPN
ejpam-3588	246	7	′	′	VERB
ejpam-3588	246	8	is	be	AUX
ejpam-3588	246	9	quasi	quasi	ADJ
ejpam-3588	246	10	-	-	NOUN
ejpam-3588	246	11	baer	baer	PROPN
ejpam-3588	246	12	,	,	PUNCT
ejpam-3588	246	13	it	it	PRON
ejpam-3588	246	14	follows	follow	VERB
ejpam-3588	246	15	from	from	ADP
ejpam-3588	246	16	(	(	PUNCT
ejpam-3588	246	17	[	[	X
ejpam-3588	246	18	18	18	NUM
ejpam-3588	246	19	]	]	PUNCT
ejpam-3588	246	20	,	,	PUNCT
ejpam-3588	246	21	theorem	theorem	VERB
ejpam-3588	246	22	3.17	3.17	NUM
ejpam-3588	246	23	)	)	PUNCT
ejpam-3588	246	24	that	that	PRON
ejpam-3588	246	25	each	each	DET
ejpam-3588	246	26	mi	mi	PROPN
ejpam-3588	246	27	is	be	AUX
ejpam-3588	246	28	quasi	quasi	ADJ
ejpam-3588	246	29	-	-	NOUN
ejpam-3588	246	30	baer	baer	PROPN
ejpam-3588	246	31	for	for	ADP
ejpam-3588	246	32	each	each	DET
ejpam-3588	246	33	i	i	PROPN
ejpam-3588	246	34	∈	∈	PROPN
ejpam-3588	246	35	i.	i.	NOUN
ejpam-3588	246	36	the	the	DET
ejpam-3588	246	37	last	last	ADJ
ejpam-3588	246	38	part	part	NOUN
ejpam-3588	246	39	follows	follow	VERB
ejpam-3588	246	40	from	from	ADP
ejpam-3588	246	41	(	(	PUNCT
ejpam-3588	246	42	[	[	X
ejpam-3588	246	43	10	10	NUM
ejpam-3588	246	44	]	]	PUNCT
ejpam-3588	246	45	,	,	PUNCT
ejpam-3588	246	46	corollary	corollary	ADJ
ejpam-3588	246	47	2.5	2.5	NUM
ejpam-3588	246	48	and	and	CCONJ
ejpam-3588	246	49	proposition	proposition	NOUN
ejpam-3588	246	50	2.17	2.17	NUM
ejpam-3588	246	51	)	)	PUNCT
ejpam-3588	246	52	and	and	CCONJ
ejpam-3588	246	53	(	(	PUNCT
ejpam-3588	246	54	[	[	X
ejpam-3588	246	55	18	18	NUM
ejpam-3588	246	56	]	]	PUNCT
ejpam-3588	246	57	,	,	PUNCT
ejpam-3588	246	58	theorem	theorem	VERB
ejpam-3588	246	59	4.1	4.1	NUM
ejpam-3588	246	60	)	)	PUNCT
ejpam-3588	246	61	.	.	PUNCT
ejpam-3588	247	1	(	(	PUNCT
ejpam-3588	247	2	3	3	X
ejpam-3588	247	3	)	)	PUNCT
ejpam-3588	247	4	by	by	ADP
ejpam-3588	247	5	(	(	PUNCT
ejpam-3588	247	6	2	2	NUM
ejpam-3588	247	7	)	)	PUNCT
ejpam-3588	247	8	,	,	PUNCT
ejpam-3588	247	9	m	m	VERB
ejpam-3588	247	10	=	=	ADJ
ejpam-3588	247	11	z2(m)⊕	z2(m)⊕	X
ejpam-3588	247	12	(	(	PUNCT
ejpam-3588	247	13	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	247	14	)	)	PUNCT
ejpam-3588	247	15	with	with	ADP
ejpam-3588	247	16	all	all	DET
ejpam-3588	247	17	mi	mi	PROPN
ejpam-3588	247	18	nonsingular	nonsingular	ADJ
ejpam-3588	247	19	uniform	uniform	NOUN
ejpam-3588	247	20	.	.	PUNCT
ejpam-3588	248	1	let	let	VERB
ejpam-3588	248	2	m	m	AUX
ejpam-3588	248	3	′	′	VERB
ejpam-3588	249	1	=	=	PUNCT
ejpam-3588	249	2	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	249	3	.	.	PUNCT
ejpam-3588	250	1	thus	thus	ADV
ejpam-3588	250	2	,	,	PUNCT
ejpam-3588	250	3	since	since	SCONJ
ejpam-3588	250	4	r	r	NOUN
ejpam-3588	250	5	is	be	AUX
ejpam-3588	250	6	right	right	ADJ
ejpam-3588	250	7	self	self	NOUN
ejpam-3588	250	8	-	-	PUNCT
ejpam-3588	250	9	injective	injective	ADJ
ejpam-3588	250	10	,	,	PUNCT
ejpam-3588	250	11	all	all	DET
ejpam-3588	250	12	mi	mi	NOUN
ejpam-3588	250	13	are	be	AUX
ejpam-3588	250	14	simple	simple	ADJ
ejpam-3588	250	15	,	,	PUNCT
ejpam-3588	250	16	proving	prove	VERB
ejpam-3588	250	17	the	the	DET
ejpam-3588	250	18	result	result	NOUN
ejpam-3588	250	19	.	.	PUNCT
ejpam-3588	251	1	theorem	theorem	ADJ
ejpam-3588	251	2	4	4	NUM
ejpam-3588	251	3	.	.	PUNCT
ejpam-3588	252	1	let	let	VERB
ejpam-3588	252	2	m	m	PRON
ejpam-3588	252	3	be	be	AUX
ejpam-3588	252	4	a	a	DET
ejpam-3588	252	5	dual	dual	ADJ
ejpam-3588	252	6	baer	baer	PROPN
ejpam-3588	252	7	r	r	NOUN
ejpam-3588	252	8	-	-	PUNCT
ejpam-3588	252	9	module	module	NOUN
ejpam-3588	252	10	.	.	PUNCT
ejpam-3588	253	1	then	then	ADV
ejpam-3588	253	2	the	the	DET
ejpam-3588	253	3	following	follow	VERB
ejpam-3588	253	4	statements	statement	NOUN
ejpam-3588	253	5	are	be	AUX
ejpam-3588	253	6	equivalent	equivalent	ADJ
ejpam-3588	253	7	:	:	PUNCT
ejpam-3588	253	8	(	(	PUNCT
ejpam-3588	253	9	1	1	X
ejpam-3588	253	10	)	)	PUNCT
ejpam-3588	253	11	m	m	VERB
ejpam-3588	253	12	is	be	AUX
ejpam-3588	253	13	ads	ad	NOUN
ejpam-3588	253	14	and	and	CCONJ
ejpam-3588	253	15	c	c	NOUN
ejpam-3588	253	16	-	-	PUNCT
ejpam-3588	253	17	retractable	retractable	ADJ
ejpam-3588	253	18	.	.	PUNCT
ejpam-3588	254	1	(	(	PUNCT
ejpam-3588	254	2	2	2	X
ejpam-3588	254	3	)	)	PUNCT
ejpam-3588	254	4	m	m	VERB
ejpam-3588	254	5	is	be	AUX
ejpam-3588	254	6	continuous	continuous	ADJ
ejpam-3588	254	7	.	.	PUNCT
ejpam-3588	255	1	(	(	PUNCT
ejpam-3588	255	2	3	3	X
ejpam-3588	255	3	)	)	PUNCT
ejpam-3588	255	4	m	m	VERB
ejpam-3588	255	5	is	be	AUX
ejpam-3588	255	6	quasi	quasi	ADJ
ejpam-3588	255	7	-	-	ADJ
ejpam-3588	255	8	continuous	continuous	ADJ
ejpam-3588	255	9	.	.	PUNCT
ejpam-3588	255	10	a.	a.	PROPN
ejpam-3588	255	11	d.	d.	PROPN
ejpam-3588	255	12	diallo	diallo	PROPN
ejpam-3588	255	13	,	,	PUNCT
ejpam-3588	255	14	p.	p.	PROPN
ejpam-3588	255	15	c.	c.	PROPN
ejpam-3588	255	16	diop	diop	PROPN
ejpam-3588	255	17	,	,	PUNCT
ejpam-3588	255	18	m.	m.	NOUN
ejpam-3588	255	19	barry	barry	PROPN
ejpam-3588	255	20	/	/	SYM
ejpam-3588	255	21	eur	eur	PROPN
ejpam-3588	255	22	.	.	PUNCT
ejpam-3588	256	1	j.	j.	PROPN
ejpam-3588	256	2	pure	pure	PROPN
ejpam-3588	256	3	appl	appl	PROPN
ejpam-3588	256	4	.	.	PROPN
ejpam-3588	256	5	math	math	PROPN
ejpam-3588	256	6	,	,	PUNCT
ejpam-3588	256	7	13	13	NUM
ejpam-3588	256	8	(	(	PUNCT
ejpam-3588	256	9	1	1	NUM
ejpam-3588	256	10	)	)	PUNCT
ejpam-3588	256	11	(	(	PUNCT
ejpam-3588	256	12	2020	2020	NUM
ejpam-3588	256	13	)	)	PUNCT
ejpam-3588	256	14	,	,	PUNCT
ejpam-3588	256	15	158	158	NUM
ejpam-3588	256	16	-	-	SYM
ejpam-3588	256	17	169	169	NUM
ejpam-3588	256	18	165	165	NUM
ejpam-3588	256	19	proof	proof	NOUN
ejpam-3588	256	20	.	.	PUNCT
ejpam-3588	257	1	(	(	PUNCT
ejpam-3588	257	2	1	1	X
ejpam-3588	257	3	)	)	PUNCT
ejpam-3588	257	4	⇒	⇒	NOUN
ejpam-3588	257	5	(	(	PUNCT
ejpam-3588	257	6	2	2	X
ejpam-3588	257	7	)	)	PUNCT
ejpam-3588	257	8	suppose	suppose	VERB
ejpam-3588	257	9	m	m	NOUN
ejpam-3588	257	10	is	be	AUX
ejpam-3588	257	11	ads	ad	NOUN
ejpam-3588	257	12	and	and	CCONJ
ejpam-3588	257	13	c	c	NOUN
ejpam-3588	257	14	-	-	PUNCT
ejpam-3588	257	15	retractable	retractable	ADJ
ejpam-3588	257	16	.	.	PUNCT
ejpam-3588	258	1	since	since	SCONJ
ejpam-3588	258	2	m	m	PROPN
ejpam-3588	258	3	is	be	AUX
ejpam-3588	258	4	dual	dual	ADJ
ejpam-3588	258	5	baer	baer	PROPN
ejpam-3588	258	6	,	,	PUNCT
ejpam-3588	258	7	we	we	PRON
ejpam-3588	258	8	infer	infer	VERB
ejpam-3588	258	9	from	from	ADP
ejpam-3588	258	10	proposition	proposition	NOUN
ejpam-3588	258	11	6(1	6(1	NUM
ejpam-3588	258	12	)	)	PUNCT
ejpam-3588	258	13	that	that	SCONJ
ejpam-3588	258	14	m	m	VERB
ejpam-3588	258	15	=	=	ADJ
ejpam-3588	258	16	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	258	17	is	be	AUX
ejpam-3588	258	18	a	a	DET
ejpam-3588	258	19	direct	direct	ADJ
ejpam-3588	258	20	sum	sum	NOUN
ejpam-3588	258	21	of	of	ADP
ejpam-3588	258	22	uniform	uniform	ADJ
ejpam-3588	258	23	modules	module	NOUN
ejpam-3588	258	24	.	.	PUNCT
ejpam-3588	259	1	thus	thus	ADV
ejpam-3588	259	2	,	,	PUNCT
ejpam-3588	259	3	every	every	DET
ejpam-3588	259	4	mi	mi	NOUN
ejpam-3588	259	5	is	be	AUX
ejpam-3588	259	6	quasi	quasi	ADJ
ejpam-3588	259	7	-	-	ADJ
ejpam-3588	259	8	continuous	continuous	ADJ
ejpam-3588	259	9	for	for	ADP
ejpam-3588	259	10	every	every	DET
ejpam-3588	259	11	i	i	PROPN
ejpam-3588	259	12	∈	∈	PROPN
ejpam-3588	259	13	i.	i.	NOUN
ejpam-3588	259	14	on	on	ADP
ejpam-3588	259	15	the	the	DET
ejpam-3588	259	16	other	other	ADJ
ejpam-3588	259	17	hand	hand	NOUN
ejpam-3588	259	18	since	since	SCONJ
ejpam-3588	259	19	m	m	PROPN
ejpam-3588	259	20	is	be	AUX
ejpam-3588	259	21	ads	ad	NOUN
ejpam-3588	259	22	,	,	PUNCT
ejpam-3588	259	23	we	we	PRON
ejpam-3588	259	24	infer	infer	VERB
ejpam-3588	259	25	from	from	ADP
ejpam-3588	259	26	lemma	lemma	PROPN
ejpam-3588	259	27	3.1	3.1	NUM
ejpam-3588	259	28	in	in	ADP
ejpam-3588	259	29	[	[	X
ejpam-3588	259	30	1	1	X
ejpam-3588	259	31	]	]	PUNCT
ejpam-3588	259	32	that	that	PRON
ejpam-3588	259	33	⊕i	⊕i	VERB
ejpam-3588	259	34	6	6	NUM
ejpam-3588	259	35	=	=	NOUN
ejpam-3588	259	36	j∈imj	j∈imj	PROPN
ejpam-3588	259	37	is	be	AUX
ejpam-3588	259	38	mi	mi	NOUN
ejpam-3588	259	39	-	-	ADJ
ejpam-3588	259	40	injective	injective	ADJ
ejpam-3588	259	41	for	for	ADP
ejpam-3588	259	42	every	every	DET
ejpam-3588	259	43	i	i	PROPN
ejpam-3588	259	44	∈	∈	PROPN
ejpam-3588	259	45	i.	i.	NOUN
ejpam-3588	259	46	therefore	therefore	ADV
ejpam-3588	259	47	m	m	PROPN
ejpam-3588	259	48	is	be	AUX
ejpam-3588	259	49	quasi	quasi	ADJ
ejpam-3588	259	50	-	-	ADJ
ejpam-3588	259	51	continuous	continuous	ADJ
ejpam-3588	259	52	by	by	ADP
ejpam-3588	259	53	(	(	PUNCT
ejpam-3588	259	54	[	[	X
ejpam-3588	259	55	14	14	NUM
ejpam-3588	259	56	]	]	PUNCT
ejpam-3588	259	57	,	,	PUNCT
ejpam-3588	259	58	theorem	theorem	VERB
ejpam-3588	259	59	2.13	2.13	NUM
ejpam-3588	259	60	)	)	PUNCT
ejpam-3588	259	61	.	.	PUNCT
ejpam-3588	260	1	now	now	ADV
ejpam-3588	260	2	,	,	PUNCT
ejpam-3588	260	3	let	let	VERB
ejpam-3588	260	4	ϕ	ϕ	NOUN
ejpam-3588	260	5	be	be	AUX
ejpam-3588	260	6	an	an	DET
ejpam-3588	260	7	essential	essential	ADJ
ejpam-3588	260	8	monomorphism	monomorphism	NOUN
ejpam-3588	260	9	of	of	ADP
ejpam-3588	260	10	m	m	PROPN
ejpam-3588	260	11	.	.	PUNCT
ejpam-3588	261	1	then	then	ADV
ejpam-3588	261	2	imϕ	imϕ	VERB
ejpam-3588	261	3	≤e	≤e	VERB
ejpam-3588	261	4	m	m	PROPN
ejpam-3588	261	5	.	.	PUNCT
ejpam-3588	262	1	since	since	SCONJ
ejpam-3588	262	2	m	m	PROPN
ejpam-3588	262	3	is	be	AUX
ejpam-3588	262	4	dual	dual	ADJ
ejpam-3588	262	5	baer	baer	PROPN
ejpam-3588	262	6	,	,	PUNCT
ejpam-3588	262	7	imϕ	imϕ	VERB
ejpam-3588	262	8	≤⊕	≤⊕	NUM
ejpam-3588	262	9	m	m	VERB
ejpam-3588	262	10	.	.	PUNCT
ejpam-3588	263	1	hence	hence	ADV
ejpam-3588	263	2	,	,	PUNCT
ejpam-3588	263	3	imϕ	imϕ	VERB
ejpam-3588	263	4	=	=	NOUN
ejpam-3588	263	5	m	m	VERB
ejpam-3588	263	6	.	.	PUNCT
ejpam-3588	264	1	therefore	therefore	ADV
ejpam-3588	264	2	,	,	PUNCT
ejpam-3588	264	3	according	accord	VERB
ejpam-3588	264	4	to	to	ADP
ejpam-3588	264	5	(	(	PUNCT
ejpam-3588	264	6	[	[	X
ejpam-3588	264	7	14	14	NUM
ejpam-3588	264	8	]	]	X
ejpam-3588	264	9	,	,	PUNCT
ejpam-3588	264	10	lemma	lemma	PROPN
ejpam-3588	264	11	3.14	3.14	NUM
ejpam-3588	264	12	)	)	PUNCT
ejpam-3588	264	13	,	,	PUNCT
ejpam-3588	264	14	m	m	VERB
ejpam-3588	264	15	is	be	AUX
ejpam-3588	264	16	continuous	continuous	ADJ
ejpam-3588	264	17	.	.	PUNCT
ejpam-3588	265	1	(	(	PUNCT
ejpam-3588	265	2	3)⇒	3)⇒	NUM
ejpam-3588	265	3	(	(	PUNCT
ejpam-3588	265	4	1	1	NUM
ejpam-3588	265	5	)	)	PUNCT
ejpam-3588	265	6	this	this	DET
ejpam-3588	265	7	implication	implication	NOUN
ejpam-3588	265	8	is	be	AUX
ejpam-3588	265	9	clear	clear	ADJ
ejpam-3588	265	10	.	.	PUNCT
ejpam-3588	266	1	corollary	corollary	ADJ
ejpam-3588	266	2	3	3	X
ejpam-3588	266	3	.	.	PUNCT
ejpam-3588	267	1	let	let	VERB
ejpam-3588	267	2	m	m	PRON
ejpam-3588	267	3	be	be	AUX
ejpam-3588	267	4	a	a	DET
ejpam-3588	267	5	dual	dual	ADJ
ejpam-3588	267	6	baer	baer	PROPN
ejpam-3588	267	7	c	c	NOUN
ejpam-3588	267	8	-	-	PUNCT
ejpam-3588	267	9	retractable	retractable	ADJ
ejpam-3588	267	10	r	r	NOUN
ejpam-3588	267	11	-	-	PUNCT
ejpam-3588	267	12	module	module	NOUN
ejpam-3588	267	13	such	such	ADJ
ejpam-3588	267	14	that	that	SCONJ
ejpam-3588	267	15	every	every	DET
ejpam-3588	267	16	nonsingular	nonsingular	ADJ
ejpam-3588	267	17	summand	summand	NOUN
ejpam-3588	267	18	is	be	AUX
ejpam-3588	267	19	ads	ad	NOUN
ejpam-3588	267	20	.	.	PUNCT
ejpam-3588	268	1	then	then	ADV
ejpam-3588	268	2	m	m	VERB
ejpam-3588	268	3	=	=	ADJ
ejpam-3588	268	4	z2(m)⊕m	z2(m)⊕m	NOUN
ejpam-3588	268	5	′	′	NUM
ejpam-3588	268	6	where	where	SCONJ
ejpam-3588	268	7	m	m	VERB
ejpam-3588	268	8	′	′	VERB
ejpam-3588	268	9	is	be	AUX
ejpam-3588	268	10	nonsingular	nonsingular	ADJ
ejpam-3588	268	11	quasi	quasi	ADJ
ejpam-3588	268	12	-	-	ADJ
ejpam-3588	268	13	continuous	continuous	ADJ
ejpam-3588	268	14	.	.	PUNCT
ejpam-3588	269	1	proof	proof	NOUN
ejpam-3588	269	2	.	.	PUNCT
ejpam-3588	270	1	by	by	ADP
ejpam-3588	270	2	proposition	proposition	NOUN
ejpam-3588	270	3	6(2	6(2	NUM
ejpam-3588	270	4	)	)	PUNCT
ejpam-3588	270	5	,	,	PUNCT
ejpam-3588	270	6	m	m	VERB
ejpam-3588	270	7	=	=	SYM
ejpam-3588	270	8	z2(m	z2(m	X
ejpam-3588	270	9	)	)	PUNCT
ejpam-3588	270	10	⊕	⊕	PROPN
ejpam-3588	270	11	(	(	PUNCT
ejpam-3588	270	12	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	270	13	)	)	PUNCT
ejpam-3588	270	14	with	with	ADP
ejpam-3588	270	15	all	all	DET
ejpam-3588	270	16	mi	mi	PROPN
ejpam-3588	270	17	nonsingular	nonsingular	ADJ
ejpam-3588	270	18	uniform	uniform	NOUN
ejpam-3588	270	19	.	.	PUNCT
ejpam-3588	271	1	let	let	VERB
ejpam-3588	271	2	m	m	AUX
ejpam-3588	271	3	′	′	VERB
ejpam-3588	272	1	=	=	PUNCT
ejpam-3588	272	2	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	272	3	.	.	PUNCT
ejpam-3588	273	1	thus	thus	ADV
ejpam-3588	273	2	,	,	PUNCT
ejpam-3588	273	3	by	by	ADP
ejpam-3588	273	4	our	our	PRON
ejpam-3588	273	5	assumption	assumption	NOUN
ejpam-3588	273	6	,	,	PUNCT
ejpam-3588	273	7	m	m	VERB
ejpam-3588	273	8	′	′	NOUN
ejpam-3588	273	9	is	be	AUX
ejpam-3588	273	10	ads	ad	NOUN
ejpam-3588	273	11	.	.	PUNCT
ejpam-3588	274	1	therefore	therefore	ADV
ejpam-3588	274	2	,	,	PUNCT
ejpam-3588	274	3	applying	apply	VERB
ejpam-3588	274	4	the	the	DET
ejpam-3588	274	5	same	same	ADJ
ejpam-3588	274	6	techniques	technique	NOUN
ejpam-3588	274	7	as	as	ADP
ejpam-3588	274	8	in	in	ADP
ejpam-3588	274	9	the	the	DET
ejpam-3588	274	10	proof	proof	NOUN
ejpam-3588	274	11	of	of	ADP
ejpam-3588	274	12	theorem	theorem	ADJ
ejpam-3588	274	13	4	4	NUM
ejpam-3588	274	14	,	,	PUNCT
ejpam-3588	274	15	one	one	PRON
ejpam-3588	274	16	can	can	AUX
ejpam-3588	274	17	show	show	VERB
ejpam-3588	274	18	easily	easily	ADV
ejpam-3588	274	19	that	that	SCONJ
ejpam-3588	274	20	m	m	VERB
ejpam-3588	274	21	′	′	VERB
ejpam-3588	274	22	is	be	AUX
ejpam-3588	274	23	quasi	quasi	ADJ
ejpam-3588	274	24	-	-	ADJ
ejpam-3588	274	25	continuous	continuous	ADJ
ejpam-3588	274	26	.	.	PUNCT
ejpam-3588	275	1	proposition	proposition	NOUN
ejpam-3588	275	2	7	7	NUM
ejpam-3588	275	3	.	.	PUNCT
ejpam-3588	276	1	let	let	VERB
ejpam-3588	276	2	m	m	PRON
ejpam-3588	276	3	be	be	AUX
ejpam-3588	276	4	a	a	DET
ejpam-3588	276	5	d	d	NOUN
ejpam-3588	276	6	-	-	PUNCT
ejpam-3588	276	7	rickart	rickart	NOUN
ejpam-3588	276	8	r	r	NOUN
ejpam-3588	276	9	-	-	NOUN
ejpam-3588	276	10	module	module	NOUN
ejpam-3588	276	11	with	with	ADP
ejpam-3588	276	12	s	s	PROPN
ejpam-3588	276	13	is	be	AUX
ejpam-3588	276	14	left	leave	VERB
ejpam-3588	276	15	t	t	PROPN
ejpam-3588	276	16	-nilpotent	-nilpotent	PROPN
ejpam-3588	276	17	.	.	PUNCT
ejpam-3588	277	1	then	then	ADV
ejpam-3588	277	2	the	the	DET
ejpam-3588	277	3	following	follow	VERB
ejpam-3588	277	4	statements	statement	NOUN
ejpam-3588	277	5	are	be	AUX
ejpam-3588	277	6	equivalent	equivalent	ADJ
ejpam-3588	277	7	:	:	PUNCT
ejpam-3588	277	8	(	(	PUNCT
ejpam-3588	277	9	1	1	X
ejpam-3588	277	10	)	)	PUNCT
ejpam-3588	277	11	m	m	VERB
ejpam-3588	277	12	is	be	AUX
ejpam-3588	277	13	ads	ad	NOUN
ejpam-3588	277	14	and	and	CCONJ
ejpam-3588	277	15	c	c	NOUN
ejpam-3588	277	16	-	-	PUNCT
ejpam-3588	277	17	retractable	retractable	ADJ
ejpam-3588	277	18	.	.	PUNCT
ejpam-3588	278	1	(	(	PUNCT
ejpam-3588	278	2	2	2	X
ejpam-3588	278	3	)	)	PUNCT
ejpam-3588	278	4	m	m	VERB
ejpam-3588	278	5	is	be	AUX
ejpam-3588	278	6	quasi	quasi	ADJ
ejpam-3588	278	7	-	-	ADJ
ejpam-3588	278	8	continuous	continuous	ADJ
ejpam-3588	278	9	.	.	PUNCT
ejpam-3588	279	1	proof	proof	NOUN
ejpam-3588	279	2	.	.	PUNCT
ejpam-3588	280	1	(	(	PUNCT
ejpam-3588	280	2	1	1	X
ejpam-3588	280	3	)	)	PUNCT
ejpam-3588	280	4	⇒	⇒	NOUN
ejpam-3588	280	5	(	(	PUNCT
ejpam-3588	280	6	2	2	NUM
ejpam-3588	280	7	)	)	PUNCT
ejpam-3588	280	8	since	since	SCONJ
ejpam-3588	280	9	m	m	PROPN
ejpam-3588	280	10	is	be	AUX
ejpam-3588	280	11	d	d	NOUN
ejpam-3588	280	12	-	-	PUNCT
ejpam-3588	280	13	rickart	rickart	NOUN
ejpam-3588	280	14	and	and	CCONJ
ejpam-3588	280	15	s	s	NOUN
ejpam-3588	280	16	is	be	AUX
ejpam-3588	280	17	left	leave	VERB
ejpam-3588	280	18	t	t	PROPN
ejpam-3588	280	19	-nilpotent	-nilpotent	PROPN
ejpam-3588	280	20	,	,	PUNCT
ejpam-3588	280	21	it	it	PRON
ejpam-3588	280	22	follows	follow	VERB
ejpam-3588	280	23	from	from	ADP
ejpam-3588	280	24	proposition	proposition	NOUN
ejpam-3588	280	25	3.4.11	3.4.11	NUM
ejpam-3588	280	26	in	in	ADP
ejpam-3588	280	27	[	[	X
ejpam-3588	280	28	12	12	NUM
ejpam-3588	280	29	]	]	PUNCT
ejpam-3588	280	30	that	that	SCONJ
ejpam-3588	280	31	m	m	VERB
ejpam-3588	280	32	=	=	VERB
ejpam-3588	280	33	⊕nimi	⊕nimi	NOUN
ejpam-3588	280	34	with	with	ADP
ejpam-3588	280	35	all	all	DET
ejpam-3588	280	36	mi	mi	PROPN
ejpam-3588	280	37	indecomposable	indecomposable	ADJ
ejpam-3588	280	38	.	.	PUNCT
ejpam-3588	281	1	since	since	SCONJ
ejpam-3588	281	2	d	d	NOUN
ejpam-3588	281	3	-	-	PUNCT
ejpam-3588	281	4	rickart	rickart	NOUN
ejpam-3588	281	5	modules	module	NOUN
ejpam-3588	281	6	are	be	AUX
ejpam-3588	281	7	wd	wd	ADJ
ejpam-3588	281	8	-	-	NOUN
ejpam-3588	281	9	rickart	rickart	NOUN
ejpam-3588	281	10	,	,	PUNCT
ejpam-3588	281	11	we	we	PRON
ejpam-3588	281	12	infer	infer	VERB
ejpam-3588	281	13	from	from	ADP
ejpam-3588	281	14	lemma	lemma	PROPN
ejpam-3588	281	15	5	5	NUM
ejpam-3588	281	16	that	that	PRON
ejpam-3588	281	17	m	m	VERB
ejpam-3588	281	18	=	=	VERB
ejpam-3588	281	19	⊕nimi	⊕nimi	NOUN
ejpam-3588	281	20	with	with	ADP
ejpam-3588	281	21	all	all	DET
ejpam-3588	281	22	mi	mi	PROPN
ejpam-3588	281	23	uniform	uniform	NOUN
ejpam-3588	281	24	.	.	PUNCT
ejpam-3588	282	1	on	on	ADP
ejpam-3588	282	2	the	the	DET
ejpam-3588	282	3	other	other	ADJ
ejpam-3588	282	4	hand	hand	NOUN
ejpam-3588	282	5	since	since	SCONJ
ejpam-3588	282	6	m	m	PROPN
ejpam-3588	282	7	is	be	AUX
ejpam-3588	282	8	ads	ad	NOUN
ejpam-3588	282	9	,	,	PUNCT
ejpam-3588	282	10	we	we	PRON
ejpam-3588	282	11	infer	infer	VERB
ejpam-3588	282	12	from	from	ADP
ejpam-3588	282	13	lemma	lemma	PROPN
ejpam-3588	282	14	3.1	3.1	NUM
ejpam-3588	282	15	in	in	ADP
ejpam-3588	282	16	[	[	X
ejpam-3588	282	17	1	1	X
ejpam-3588	282	18	]	]	PUNCT
ejpam-3588	282	19	that	that	PRON
ejpam-3588	282	20	⊕i	⊕i	VERB
ejpam-3588	282	21	6	6	NUM
ejpam-3588	282	22	=	=	SYM
ejpam-3588	282	23	jmj	jmj	PROPN
ejpam-3588	282	24	is	be	AUX
ejpam-3588	282	25	mi	mi	NOUN
ejpam-3588	282	26	-	-	ADJ
ejpam-3588	282	27	injective	injective	ADJ
ejpam-3588	282	28	for	for	ADP
ejpam-3588	282	29	every	every	DET
ejpam-3588	282	30	1	1	NUM
ejpam-3588	282	31	≤	≤	NUM
ejpam-3588	282	32	i	i	PRON
ejpam-3588	282	33	≤	≤	ADJ
ejpam-3588	282	34	n.	n.	NOUN
ejpam-3588	282	35	thus	thus	ADV
ejpam-3588	282	36	m	m	VERB
ejpam-3588	282	37	is	be	AUX
ejpam-3588	282	38	quasi	quasi	ADJ
ejpam-3588	282	39	-	-	ADJ
ejpam-3588	282	40	continuous	continuous	ADJ
ejpam-3588	282	41	by	by	ADP
ejpam-3588	282	42	(	(	PUNCT
ejpam-3588	282	43	[	[	X
ejpam-3588	282	44	14	14	NUM
ejpam-3588	282	45	]	]	X
ejpam-3588	282	46	,	,	PUNCT
ejpam-3588	282	47	lemma	lemma	PROPN
ejpam-3588	282	48	2.14	2.14	NUM
ejpam-3588	282	49	)	)	PUNCT
ejpam-3588	282	50	.	.	PUNCT
ejpam-3588	283	1	(	(	PUNCT
ejpam-3588	283	2	2)⇒	2)⇒	NUM
ejpam-3588	283	3	(	(	PUNCT
ejpam-3588	283	4	1	1	X
ejpam-3588	283	5	)	)	PUNCT
ejpam-3588	283	6	this	this	DET
ejpam-3588	283	7	implication	implication	NOUN
ejpam-3588	283	8	is	be	AUX
ejpam-3588	283	9	clear	clear	ADJ
ejpam-3588	283	10	.	.	PUNCT
ejpam-3588	284	1	let	let	VERB
ejpam-3588	284	2	m	m	PRON
ejpam-3588	284	3	be	be	AUX
ejpam-3588	284	4	an	an	DET
ejpam-3588	284	5	r	r	NOUN
ejpam-3588	284	6	-	-	PUNCT
ejpam-3588	284	7	module	module	NOUN
ejpam-3588	284	8	.	.	PUNCT
ejpam-3588	285	1	the	the	DET
ejpam-3588	285	2	left	left	ADJ
ejpam-3588	285	3	annihilator	annihilator	NOUN
ejpam-3588	285	4	of	of	ADP
ejpam-3588	285	5	n	n	PRON
ejpam-3588	285	6	≤	≤	NOUN
ejpam-3588	285	7	m	m	VERB
ejpam-3588	285	8	in	in	ADP
ejpam-3588	285	9	s	s	NOUN
ejpam-3588	285	10	=	=	SYM
ejpam-3588	285	11	endr(m	endr(m	PROPN
ejpam-3588	285	12	)	)	PUNCT
ejpam-3588	285	13	is	be	AUX
ejpam-3588	285	14	denoted	denote	VERB
ejpam-3588	285	15	by	by	ADP
ejpam-3588	285	16	ls(n	ls(n	NOUN
ejpam-3588	285	17	)	)	PUNCT
ejpam-3588	285	18	=	=	SYM
ejpam-3588	285	19	{	{	PUNCT
ejpam-3588	285	20	φ	φ	PROPN
ejpam-3588	285	21	∈	∈	PROPN
ejpam-3588	285	22	s	s	PART
ejpam-3588	285	23	:	:	PUNCT
ejpam-3588	285	24	φn	φn	NOUN
ejpam-3588	285	25	=	=	PUNCT
ejpam-3588	285	26	{	{	PUNCT
ejpam-3588	285	27	0	0	NUM
ejpam-3588	285	28	}	}	PUNCT
ejpam-3588	285	29	}	}	PUNCT
ejpam-3588	285	30	.	.	PUNCT
ejpam-3588	286	1	let	let	VERB
ejpam-3588	286	2	m	m	PRON
ejpam-3588	286	3	be	be	AUX
ejpam-3588	286	4	a	a	DET
ejpam-3588	286	5	module	module	NOUN
ejpam-3588	286	6	.	.	PUNCT
ejpam-3588	287	1	a	a	DET
ejpam-3588	287	2	submodule	submodule	NOUN
ejpam-3588	287	3	n	n	PROPN
ejpam-3588	287	4	of	of	ADP
ejpam-3588	287	5	m	m	PROPN
ejpam-3588	287	6	is	be	AUX
ejpam-3588	287	7	said	say	VERB
ejpam-3588	287	8	to	to	PART
ejpam-3588	287	9	be	be	AUX
ejpam-3588	287	10	an	an	DET
ejpam-3588	287	11	automorphism	automorphism	NOUN
ejpam-3588	287	12	-	-	PUNCT
ejpam-3588	287	13	invariant	invariant	ADJ
ejpam-3588	287	14	submodule	submodule	NOUN
ejpam-3588	287	15	if	if	SCONJ
ejpam-3588	287	16	ϕn	ϕn	PRON
ejpam-3588	287	17	⊆	⊆	NUM
ejpam-3588	287	18	n	n	NOUN
ejpam-3588	287	19	for	for	ADP
ejpam-3588	287	20	automorphism	automorphism	NOUN
ejpam-3588	287	21	ϕ	ϕ	NOUN
ejpam-3588	287	22	of	of	ADP
ejpam-3588	287	23	m	m	PROPN
ejpam-3588	287	24	.	.	PUNCT
ejpam-3588	288	1	m	m	PROPN
ejpam-3588	288	2	is	be	AUX
ejpam-3588	288	3	called	call	VERB
ejpam-3588	288	4	auto	auto	NOUN
ejpam-3588	288	5	-	-	PUNCT
ejpam-3588	288	6	invariant	invariant	ADJ
ejpam-3588	288	7	if	if	SCONJ
ejpam-3588	288	8	it	it	PRON
ejpam-3588	288	9	is	be	AUX
ejpam-3588	288	10	an	an	DET
ejpam-3588	288	11	automorphism	automorphism	NOUN
ejpam-3588	288	12	-	-	PUNCT
ejpam-3588	288	13	invariant	invariant	ADJ
ejpam-3588	288	14	submodule	submodule	NOUN
ejpam-3588	288	15	of	of	ADP
ejpam-3588	288	16	its	its	PRON
ejpam-3588	288	17	injective	injective	ADJ
ejpam-3588	288	18	hull	hull	NOUN
ejpam-3588	288	19	.	.	PUNCT
ejpam-3588	289	1	proposition	proposition	NOUN
ejpam-3588	289	2	8	8	NUM
ejpam-3588	289	3	.	.	PUNCT
ejpam-3588	290	1	let	let	VERB
ejpam-3588	290	2	m	m	PRON
ejpam-3588	290	3	be	be	AUX
ejpam-3588	290	4	a	a	DET
ejpam-3588	290	5	dual	dual	ADJ
ejpam-3588	290	6	baer	baer	PROPN
ejpam-3588	290	7	r	r	NOUN
ejpam-3588	290	8	-	-	PUNCT
ejpam-3588	290	9	module	module	NOUN
ejpam-3588	290	10	.	.	PUNCT
ejpam-3588	291	1	then	then	ADV
ejpam-3588	291	2	m	m	PROPN
ejpam-3588	291	3	is	be	AUX
ejpam-3588	291	4	auto	auto	NOUN
ejpam-3588	291	5	-	-	PUNCT
ejpam-3588	291	6	invariant	invariant	ADJ
ejpam-3588	291	7	and	and	CCONJ
ejpam-3588	291	8	cretractable	cretractable	ADJ
ejpam-3588	291	9	if	if	SCONJ
ejpam-3588	291	10	and	and	CCONJ
ejpam-3588	291	11	only	only	ADV
ejpam-3588	291	12	if	if	SCONJ
ejpam-3588	291	13	m	m	NOUN
ejpam-3588	291	14	is	be	AUX
ejpam-3588	291	15	quasi	quasi	ADJ
ejpam-3588	291	16	-	-	ADJ
ejpam-3588	291	17	injective	injective	ADJ
ejpam-3588	291	18	.	.	PUNCT
ejpam-3588	292	1	proof	proof	NOUN
ejpam-3588	292	2	.	.	PUNCT
ejpam-3588	293	1	suppose	suppose	VERB
ejpam-3588	293	2	m	m	NOUN
ejpam-3588	293	3	is	be	AUX
ejpam-3588	293	4	auto	auto	NOUN
ejpam-3588	293	5	-	-	PUNCT
ejpam-3588	293	6	invariant	invariant	ADJ
ejpam-3588	293	7	and	and	CCONJ
ejpam-3588	293	8	c	c	NOUN
ejpam-3588	293	9	-	-	PUNCT
ejpam-3588	293	10	retractable	retractable	ADJ
ejpam-3588	293	11	.	.	PUNCT
ejpam-3588	294	1	since	since	SCONJ
ejpam-3588	294	2	m	m	PROPN
ejpam-3588	294	3	is	be	AUX
ejpam-3588	294	4	dual	dual	ADJ
ejpam-3588	294	5	baer	baer	PROPN
ejpam-3588	294	6	,	,	PUNCT
ejpam-3588	294	7	we	we	PRON
ejpam-3588	294	8	infer	infer	VERB
ejpam-3588	294	9	from	from	ADP
ejpam-3588	294	10	proposition	proposition	NOUN
ejpam-3588	294	11	6(1	6(1	NUM
ejpam-3588	294	12	)	)	PUNCT
ejpam-3588	294	13	that	that	SCONJ
ejpam-3588	294	14	m	m	VERB
ejpam-3588	294	15	=	=	ADJ
ejpam-3588	294	16	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	294	17	is	be	AUX
ejpam-3588	294	18	a	a	DET
ejpam-3588	294	19	direct	direct	ADJ
ejpam-3588	294	20	sum	sum	NOUN
ejpam-3588	294	21	of	of	ADP
ejpam-3588	294	22	extending	extend	VERB
ejpam-3588	294	23	modules	module	NOUN
ejpam-3588	294	24	.	.	PUNCT
ejpam-3588	295	1	thus	thus	ADV
ejpam-3588	295	2	,	,	PUNCT
ejpam-3588	295	3	by	by	ADP
ejpam-3588	295	4	corollary	corollary	ADJ
ejpam-3588	295	5	15	15	NUM
ejpam-3588	295	6	in	in	ADP
ejpam-3588	295	7	[	[	X
ejpam-3588	295	8	13	13	NUM
ejpam-3588	295	9	]	]	PUNCT
ejpam-3588	295	10	,	,	PUNCT
ejpam-3588	295	11	m	m	VERB
ejpam-3588	295	12	is	be	AUX
ejpam-3588	295	13	quasi	quasi	ADJ
ejpam-3588	295	14	-	-	ADJ
ejpam-3588	295	15	injective	injective	ADJ
ejpam-3588	295	16	.	.	PUNCT
ejpam-3588	296	1	the	the	DET
ejpam-3588	296	2	converse	converse	PROPN
ejpam-3588	296	3	implication	implication	NOUN
ejpam-3588	296	4	is	be	AUX
ejpam-3588	296	5	clear	clear	ADJ
ejpam-3588	296	6	.	.	PUNCT
ejpam-3588	297	1	a.	a.	PROPN
ejpam-3588	297	2	d.	d.	PROPN
ejpam-3588	297	3	diallo	diallo	PROPN
ejpam-3588	297	4	,	,	PUNCT
ejpam-3588	297	5	p.	p.	PROPN
ejpam-3588	297	6	c.	c.	PROPN
ejpam-3588	297	7	diop	diop	PROPN
ejpam-3588	297	8	,	,	PUNCT
ejpam-3588	297	9	m.	m.	NOUN
ejpam-3588	297	10	barry	barry	PROPN
ejpam-3588	297	11	/	/	SYM
ejpam-3588	297	12	eur	eur	PROPN
ejpam-3588	297	13	.	.	PUNCT
ejpam-3588	298	1	j.	j.	PROPN
ejpam-3588	298	2	pure	pure	PROPN
ejpam-3588	298	3	appl	appl	PROPN
ejpam-3588	298	4	.	.	PROPN
ejpam-3588	298	5	math	math	PROPN
ejpam-3588	298	6	,	,	PUNCT
ejpam-3588	298	7	13	13	NUM
ejpam-3588	298	8	(	(	PUNCT
ejpam-3588	298	9	1	1	NUM
ejpam-3588	298	10	)	)	PUNCT
ejpam-3588	298	11	(	(	PUNCT
ejpam-3588	298	12	2020	2020	NUM
ejpam-3588	298	13	)	)	PUNCT
ejpam-3588	298	14	,	,	PUNCT
ejpam-3588	298	15	158	158	NUM
ejpam-3588	298	16	-	-	SYM
ejpam-3588	298	17	169	169	NUM
ejpam-3588	298	18	166	166	NUM
ejpam-3588	298	19	recall	recall	NOUN
ejpam-3588	298	20	that	that	SCONJ
ejpam-3588	298	21	an	an	DET
ejpam-3588	298	22	r	r	NOUN
ejpam-3588	298	23	-	-	PUNCT
ejpam-3588	298	24	module	module	NOUN
ejpam-3588	298	25	m	m	NOUN
ejpam-3588	298	26	is	be	AUX
ejpam-3588	298	27	called	call	VERB
ejpam-3588	298	28	c4	c4	NOUN
ejpam-3588	298	29	if	if	SCONJ
ejpam-3588	298	30	,	,	PUNCT
ejpam-3588	298	31	whenever	whenever	SCONJ
ejpam-3588	298	32	a	a	PRON
ejpam-3588	298	33	and	and	CCONJ
ejpam-3588	298	34	b	b	NOUN
ejpam-3588	298	35	are	be	AUX
ejpam-3588	298	36	submodules	submodule	NOUN
ejpam-3588	298	37	of	of	ADP
ejpam-3588	298	38	m	m	PROPN
ejpam-3588	298	39	with	with	ADP
ejpam-3588	298	40	m	m	PROPN
ejpam-3588	298	41	=	=	PUNCT
ejpam-3588	298	42	a	a	DET
ejpam-3588	298	43	⊕	⊕	PROPN
ejpam-3588	298	44	b	b	PROPN
ejpam-3588	298	45	and	and	CCONJ
ejpam-3588	298	46	f	f	PROPN
ejpam-3588	298	47	:	:	PUNCT
ejpam-3588	298	48	a	a	DET
ejpam-3588	298	49	−→	−→	NOUN
ejpam-3588	298	50	b	b	NOUN
ejpam-3588	298	51	is	be	AUX
ejpam-3588	298	52	an	an	DET
ejpam-3588	298	53	homomorphism	homomorphism	NOUN
ejpam-3588	298	54	with	with	ADP
ejpam-3588	298	55	kerf	kerf	NOUN
ejpam-3588	298	56	≤⊕	≤⊕	PRON
ejpam-3588	298	57	a	a	X
ejpam-3588	298	58	,	,	PUNCT
ejpam-3588	298	59	we	we	PRON
ejpam-3588	298	60	have	have	VERB
ejpam-3588	298	61	imf	imf	PROPN
ejpam-3588	298	62	≤⊕	≤⊕	NUM
ejpam-3588	298	63	b.	b.	PROPN
ejpam-3588	298	64	proposition	proposition	PROPN
ejpam-3588	298	65	9	9	NUM
ejpam-3588	298	66	.	.	PUNCT
ejpam-3588	299	1	if	if	SCONJ
ejpam-3588	299	2	every	every	DET
ejpam-3588	299	3	2	2	NUM
ejpam-3588	299	4	-	-	PUNCT
ejpam-3588	299	5	generated	generate	VERB
ejpam-3588	299	6	r	r	NOUN
ejpam-3588	299	7	-	-	PUNCT
ejpam-3588	299	8	module	module	NOUN
ejpam-3588	299	9	is	be	AUX
ejpam-3588	299	10	a	a	DET
ejpam-3588	299	11	c4	c4	NOUN
ejpam-3588	299	12	-	-	PUNCT
ejpam-3588	299	13	module	module	NOUN
ejpam-3588	299	14	,	,	PUNCT
ejpam-3588	299	15	then	then	ADV
ejpam-3588	299	16	every	every	DET
ejpam-3588	299	17	dual	dual	ADJ
ejpam-3588	299	18	baer	baer	PROPN
ejpam-3588	299	19	c	c	NOUN
ejpam-3588	299	20	-	-	PUNCT
ejpam-3588	299	21	retractable	retractable	ADJ
ejpam-3588	299	22	r	r	NOUN
ejpam-3588	299	23	-	-	PUNCT
ejpam-3588	299	24	module	module	NOUN
ejpam-3588	299	25	is	be	AUX
ejpam-3588	299	26	semisimple	semisimple	ADJ
ejpam-3588	299	27	.	.	PUNCT
ejpam-3588	300	1	proof	proof	NOUN
ejpam-3588	300	2	.	.	PUNCT
ejpam-3588	301	1	let	let	VERB
ejpam-3588	301	2	m	m	PRON
ejpam-3588	301	3	be	be	AUX
ejpam-3588	301	4	any	any	DET
ejpam-3588	301	5	dual	dual	ADJ
ejpam-3588	301	6	baer	baer	PROPN
ejpam-3588	301	7	c	c	NOUN
ejpam-3588	301	8	-	-	PUNCT
ejpam-3588	301	9	retractable	retractable	ADJ
ejpam-3588	301	10	r	r	NOUN
ejpam-3588	301	11	-	-	PUNCT
ejpam-3588	301	12	module	module	NOUN
ejpam-3588	301	13	.	.	PUNCT
ejpam-3588	302	1	thus	thus	ADV
ejpam-3588	302	2	,	,	PUNCT
ejpam-3588	302	3	as	as	ADP
ejpam-3588	302	4	in	in	ADP
ejpam-3588	302	5	the	the	DET
ejpam-3588	302	6	proof	proof	NOUN
ejpam-3588	302	7	of	of	ADP
ejpam-3588	302	8	theorem	theorem	ADJ
ejpam-3588	302	9	4	4	NUM
ejpam-3588	302	10	,	,	PUNCT
ejpam-3588	302	11	m	m	VERB
ejpam-3588	302	12	=	=	ADJ
ejpam-3588	302	13	⊕i∈imi	⊕i∈imi	PROPN
ejpam-3588	302	14	where	where	SCONJ
ejpam-3588	302	15	each	each	DET
ejpam-3588	302	16	mi	mi	PROPN
ejpam-3588	302	17	is	be	AUX
ejpam-3588	302	18	uniform	uniform	ADJ
ejpam-3588	302	19	.	.	PUNCT
ejpam-3588	303	1	now	now	ADV
ejpam-3588	303	2	,	,	PUNCT
ejpam-3588	303	3	we	we	PRON
ejpam-3588	303	4	have	have	VERB
ejpam-3588	303	5	to	to	PART
ejpam-3588	303	6	show	show	VERB
ejpam-3588	303	7	that	that	SCONJ
ejpam-3588	303	8	each	each	DET
ejpam-3588	303	9	mi	mi	PROPN
ejpam-3588	303	10	is	be	AUX
ejpam-3588	303	11	semisimple	semisimple	ADJ
ejpam-3588	303	12	.	.	PUNCT
ejpam-3588	304	1	for	for	ADP
ejpam-3588	304	2	any	any	DET
ejpam-3588	304	3	0	0	NUM
ejpam-3588	304	4	6=	6=	ADP
ejpam-3588	304	5	m	m	PROPN
ejpam-3588	304	6	∈	∈	PROPN
ejpam-3588	304	7	e(mi	e(mi	NOUN
ejpam-3588	304	8	)	)	PUNCT
ejpam-3588	304	9	,	,	PUNCT
ejpam-3588	304	10	let	let	VERB
ejpam-3588	304	11	0	0	NUM
ejpam-3588	304	12	6=	6=	NUM
ejpam-3588	305	1	n	n	CCONJ
ejpam-3588	305	2	≤	≤	NOUN
ejpam-3588	305	3	mr	mr	PROPN
ejpam-3588	305	4	and	and	CCONJ
ejpam-3588	305	5	take	take	VERB
ejpam-3588	305	6	0	0	NUM
ejpam-3588	305	7	6=	6=	NUM
ejpam-3588	305	8	n	n	ADP
ejpam-3588	305	9	∈	∈	PROPN
ejpam-3588	305	10	n	n	ADV
ejpam-3588	305	11	.	.	PUNCT
ejpam-3588	306	1	by	by	ADP
ejpam-3588	306	2	our	our	PRON
ejpam-3588	306	3	assumption	assumption	NOUN
ejpam-3588	306	4	,	,	PUNCT
ejpam-3588	306	5	mr⊕	mr⊕	PROPN
ejpam-3588	306	6	nr	nr	PROPN
ejpam-3588	306	7	is	be	AUX
ejpam-3588	306	8	a	a	DET
ejpam-3588	306	9	c4	c4	NOUN
ejpam-3588	306	10	-	-	PUNCT
ejpam-3588	306	11	module	module	NOUN
ejpam-3588	306	12	.	.	PUNCT
ejpam-3588	307	1	consider	consider	VERB
ejpam-3588	307	2	the	the	DET
ejpam-3588	307	3	inclusion	inclusion	NOUN
ejpam-3588	307	4	map	map	NOUN
ejpam-3588	307	5	i	i	PRON
ejpam-3588	307	6	:	:	PUNCT
ejpam-3588	307	7	nr	nr	PROPN
ejpam-3588	307	8	−→	−→	ADJ
ejpam-3588	307	9	mr	mr	PROPN
ejpam-3588	307	10	.	.	PROPN
ejpam-3588	307	11	thus	thus	ADV
ejpam-3588	307	12	i(nr	i(nr	NUM
ejpam-3588	307	13	)	)	PUNCT
ejpam-3588	308	1	=	=	SYM
ejpam-3588	308	2	nr	nr	PROPN
ejpam-3588	308	3	≤⊕	≤⊕	NUM
ejpam-3588	308	4	mr	mr	PROPN
ejpam-3588	308	5	.	.	PROPN
ejpam-3588	308	6	since	since	SCONJ
ejpam-3588	308	7	mr	mr	PROPN
ejpam-3588	308	8	is	be	AUX
ejpam-3588	308	9	indecomposable	indecomposable	ADJ
ejpam-3588	308	10	,	,	PUNCT
ejpam-3588	308	11	nr	nr	PROPN
ejpam-3588	308	12	=	=	SYM
ejpam-3588	308	13	mr	mr	PROPN
ejpam-3588	308	14	,	,	PUNCT
ejpam-3588	308	15	and	and	CCONJ
ejpam-3588	308	16	hence	hence	ADV
ejpam-3588	308	17	n	n	NOUN
ejpam-3588	308	18	=	=	SYM
ejpam-3588	308	19	mr	mr	PROPN
ejpam-3588	308	20	.	.	PROPN
ejpam-3588	308	21	thus	thus	ADV
ejpam-3588	308	22	,	,	PUNCT
ejpam-3588	308	23	every	every	DET
ejpam-3588	308	24	cyclic	cyclic	ADJ
ejpam-3588	308	25	submodule	submodule	NOUN
ejpam-3588	308	26	of	of	ADP
ejpam-3588	308	27	mr	mr	PROPN
ejpam-3588	308	28	is	be	AUX
ejpam-3588	308	29	a	a	DET
ejpam-3588	308	30	direct	direct	ADJ
ejpam-3588	308	31	summand	summand	NOUN
ejpam-3588	308	32	.	.	PUNCT
ejpam-3588	309	1	it	it	PRON
ejpam-3588	309	2	follows	follow	VERB
ejpam-3588	309	3	that	that	SCONJ
ejpam-3588	309	4	mr	mr	PROPN
ejpam-3588	309	5	is	be	AUX
ejpam-3588	309	6	semisimple	semisimple	ADJ
ejpam-3588	309	7	.	.	PUNCT
ejpam-3588	310	1	hence	hence	ADV
ejpam-3588	310	2	,	,	PUNCT
ejpam-3588	310	3	e(mi	e(mi	NOUN
ejpam-3588	310	4	)	)	PUNCT
ejpam-3588	310	5	is	be	AUX
ejpam-3588	310	6	semisimple	semisimple	ADJ
ejpam-3588	310	7	.	.	PUNCT
ejpam-3588	311	1	consequently	consequently	ADV
ejpam-3588	311	2	,	,	PUNCT
ejpam-3588	311	3	mi	mi	PROPN
ejpam-3588	311	4	is	be	AUX
ejpam-3588	311	5	semisimple	semisimple	ADJ
ejpam-3588	311	6	.	.	PUNCT
ejpam-3588	312	1	therefore	therefore	ADV
ejpam-3588	312	2	,	,	PUNCT
ejpam-3588	312	3	m	m	VERB
ejpam-3588	312	4	is	be	AUX
ejpam-3588	312	5	semisimple	semisimple	ADJ
ejpam-3588	312	6	.	.	PUNCT
ejpam-3588	313	1	theorem	theorem	ADJ
ejpam-3588	313	2	5	5	NUM
ejpam-3588	313	3	.	.	PUNCT
ejpam-3588	314	1	the	the	DET
ejpam-3588	314	2	following	follow	VERB
ejpam-3588	314	3	conditions	condition	NOUN
ejpam-3588	314	4	are	be	AUX
ejpam-3588	314	5	equivalentes	equivalente	NOUN
ejpam-3588	314	6	for	for	ADP
ejpam-3588	314	7	a	a	DET
ejpam-3588	314	8	ring	ring	NOUN
ejpam-3588	314	9	r	r	NOUN
ejpam-3588	314	10	:	:	PUNCT
ejpam-3588	314	11	(	(	PUNCT
ejpam-3588	314	12	1	1	X
ejpam-3588	314	13	)	)	PUNCT
ejpam-3588	314	14	r	r	NOUN
ejpam-3588	314	15	is	be	AUX
ejpam-3588	314	16	semisimple	semisimple	NOUN
ejpam-3588	314	17	artinian	artinian	ADJ
ejpam-3588	314	18	.	.	PUNCT
ejpam-3588	315	1	(	(	PUNCT
ejpam-3588	315	2	2	2	X
ejpam-3588	315	3	)	)	PUNCT
ejpam-3588	315	4	every	every	DET
ejpam-3588	315	5	c	c	NOUN
ejpam-3588	315	6	-	-	PUNCT
ejpam-3588	315	7	retractable	retractable	ADJ
ejpam-3588	315	8	r	r	NOUN
ejpam-3588	315	9	-	-	PUNCT
ejpam-3588	315	10	module	module	NOUN
ejpam-3588	315	11	is	be	AUX
ejpam-3588	315	12	a	a	DET
ejpam-3588	315	13	c4	c4	NOUN
ejpam-3588	315	14	-	-	PUNCT
ejpam-3588	315	15	module	module	NOUN
ejpam-3588	315	16	.	.	PUNCT
ejpam-3588	316	1	(	(	PUNCT
ejpam-3588	316	2	3	3	X
ejpam-3588	316	3	)	)	PUNCT
ejpam-3588	316	4	every	every	DET
ejpam-3588	316	5	c	c	NOUN
ejpam-3588	316	6	-	-	PUNCT
ejpam-3588	316	7	retractable	retractable	ADJ
ejpam-3588	316	8	r	r	NOUN
ejpam-3588	316	9	-	-	PUNCT
ejpam-3588	316	10	module	module	NOUN
ejpam-3588	316	11	is	be	AUX
ejpam-3588	316	12	pseudo	pseudo	NOUN
ejpam-3588	316	13	-	-	NOUN
ejpam-3588	316	14	projective	projective	ADJ
ejpam-3588	316	15	.	.	PUNCT
ejpam-3588	317	1	proof	proof	NOUN
ejpam-3588	317	2	.	.	PUNCT
ejpam-3588	318	1	(	(	PUNCT
ejpam-3588	318	2	1)⇒	1)⇒	NUM
ejpam-3588	318	3	(	(	PUNCT
ejpam-3588	318	4	2	2	NUM
ejpam-3588	318	5	)	)	PUNCT
ejpam-3588	318	6	is	be	AUX
ejpam-3588	318	7	clear	clear	ADJ
ejpam-3588	318	8	.	.	PUNCT
ejpam-3588	319	1	(	(	PUNCT
ejpam-3588	319	2	2)⇒	2)⇒	NUM
ejpam-3588	319	3	(	(	PUNCT
ejpam-3588	319	4	1	1	X
ejpam-3588	319	5	)	)	PUNCT
ejpam-3588	319	6	let	let	VERB
ejpam-3588	319	7	i	i	PRON
ejpam-3588	319	8	be	be	AUX
ejpam-3588	319	9	a	a	DET
ejpam-3588	319	10	right	right	ADJ
ejpam-3588	319	11	ideal	ideal	NOUN
ejpam-3588	319	12	of	of	ADP
ejpam-3588	319	13	r.	r.	PROPN
ejpam-3588	319	14	clearly	clearly	ADV
ejpam-3588	319	15	,	,	PUNCT
ejpam-3588	319	16	i⊕r	i⊕r	NOUN
ejpam-3588	319	17	is	be	AUX
ejpam-3588	319	18	c	c	NOUN
ejpam-3588	319	19	-	-	PUNCT
ejpam-3588	319	20	retractable	retractable	ADJ
ejpam-3588	319	21	,	,	PUNCT
ejpam-3588	319	22	and	and	CCONJ
ejpam-3588	319	23	hence	hence	ADV
ejpam-3588	319	24	a	a	DET
ejpam-3588	319	25	c4	c4	NOUN
ejpam-3588	319	26	-	-	PUNCT
ejpam-3588	319	27	module	module	NOUN
ejpam-3588	319	28	by	by	ADP
ejpam-3588	319	29	(	(	PUNCT
ejpam-3588	319	30	2	2	NUM
ejpam-3588	319	31	)	)	PUNCT
ejpam-3588	319	32	.	.	PUNCT
ejpam-3588	320	1	consider	consider	VERB
ejpam-3588	321	1	the	the	DET
ejpam-3588	321	2	inclusion	inclusion	NOUN
ejpam-3588	321	3	map	map	NOUN
ejpam-3588	321	4	i	i	PRON
ejpam-3588	321	5	:	:	PUNCT
ejpam-3588	321	6	i	i	PROPN
ejpam-3588	321	7	−→	−→	PROPN
ejpam-3588	321	8	r.	r.	PROPN
ejpam-3588	321	9	therefore	therefore	ADV
ejpam-3588	321	10	,	,	PUNCT
ejpam-3588	321	11	i(i	i(i	PROPN
ejpam-3588	321	12	)	)	PUNCT
ejpam-3588	321	13	=	=	SYM
ejpam-3588	322	1	i	i	PRON
ejpam-3588	322	2	≤⊕	≤⊕	AUX
ejpam-3588	322	3	r.	r.	PROPN
ejpam-3588	322	4	hence	hence	ADV
ejpam-3588	322	5	,	,	PUNCT
ejpam-3588	322	6	rr	rr	PROPN
ejpam-3588	322	7	is	be	AUX
ejpam-3588	322	8	semisimple	semisimple	ADJ
ejpam-3588	322	9	.	.	PUNCT
ejpam-3588	323	1	thus	thus	ADV
ejpam-3588	323	2	,	,	PUNCT
ejpam-3588	323	3	r	r	NOUN
ejpam-3588	323	4	is	be	AUX
ejpam-3588	323	5	semisimple	semisimple	NOUN
ejpam-3588	323	6	artinian	artinian	ADJ
ejpam-3588	323	7	.	.	PUNCT
ejpam-3588	324	1	(	(	PUNCT
ejpam-3588	324	2	1)⇒	1)⇒	NUM
ejpam-3588	324	3	(	(	PUNCT
ejpam-3588	324	4	3	3	NUM
ejpam-3588	324	5	)	)	PUNCT
ejpam-3588	324	6	is	be	AUX
ejpam-3588	324	7	clear	clear	ADJ
ejpam-3588	324	8	.	.	PUNCT
ejpam-3588	325	1	(	(	PUNCT
ejpam-3588	325	2	3	3	X
ejpam-3588	325	3	)	)	PUNCT
ejpam-3588	325	4	⇒	⇒	NOUN
ejpam-3588	325	5	(	(	PUNCT
ejpam-3588	325	6	1	1	X
ejpam-3588	325	7	)	)	PUNCT
ejpam-3588	325	8	let	let	VERB
ejpam-3588	325	9	s	s	PRON
ejpam-3588	325	10	be	be	AUX
ejpam-3588	325	11	a	a	DET
ejpam-3588	325	12	simple	simple	ADJ
ejpam-3588	325	13	r	r	NOUN
ejpam-3588	325	14	-	-	PUNCT
ejpam-3588	325	15	module	module	NOUN
ejpam-3588	325	16	.	.	PUNCT
ejpam-3588	326	1	then	then	ADV
ejpam-3588	326	2	there	there	PRON
ejpam-3588	326	3	is	be	VERB
ejpam-3588	326	4	a	a	DET
ejpam-3588	326	5	free	free	ADJ
ejpam-3588	326	6	r	r	NOUN
ejpam-3588	326	7	-	-	PUNCT
ejpam-3588	326	8	module	module	NOUN
ejpam-3588	326	9	f	f	NOUN
ejpam-3588	326	10	and	and	CCONJ
ejpam-3588	326	11	an	an	DET
ejpam-3588	326	12	epimorphism	epimorphism	NOUN
ejpam-3588	326	13	f	f	NOUN
ejpam-3588	326	14	:	:	PUNCT
ejpam-3588	326	15	f	f	PROPN
ejpam-3588	326	16	−→	−→	PROPN
ejpam-3588	326	17	s.	s.	PROPN
ejpam-3588	326	18	hence	hence	ADV
ejpam-3588	326	19	,	,	PUNCT
ejpam-3588	326	20	s	s	PROPN
ejpam-3588	326	21	⊕	⊕	PROPN
ejpam-3588	326	22	f	f	PROPN
ejpam-3588	326	23	is	be	AUX
ejpam-3588	326	24	c	c	NOUN
ejpam-3588	326	25	-	-	NOUN
ejpam-3588	326	26	retractable	retractable	ADJ
ejpam-3588	326	27	by	by	ADP
ejpam-3588	326	28	(	(	PUNCT
ejpam-3588	326	29	[	[	X
ejpam-3588	326	30	19	19	NUM
ejpam-3588	326	31	]	]	PUNCT
ejpam-3588	326	32	,	,	PUNCT
ejpam-3588	326	33	proposition	proposition	NOUN
ejpam-3588	326	34	1.4	1.4	NUM
ejpam-3588	326	35	)	)	PUNCT
ejpam-3588	326	36	.	.	PUNCT
ejpam-3588	327	1	by	by	ADP
ejpam-3588	327	2	our	our	PRON
ejpam-3588	327	3	assumption	assumption	NOUN
ejpam-3588	327	4	,	,	PUNCT
ejpam-3588	327	5	s	s	PROPN
ejpam-3588	327	6	⊕	⊕	PROPN
ejpam-3588	327	7	f	f	PROPN
ejpam-3588	327	8	is	be	AUX
ejpam-3588	327	9	pseudo	pseudo	NOUN
ejpam-3588	327	10	-	-	NOUN
ejpam-3588	327	11	projective	projective	ADJ
ejpam-3588	327	12	.	.	PUNCT
ejpam-3588	328	1	now	now	ADV
ejpam-3588	328	2	,	,	PUNCT
ejpam-3588	328	3	consider	consider	VERB
ejpam-3588	328	4	the	the	DET
ejpam-3588	328	5	exact	exact	ADJ
ejpam-3588	328	6	sequence	sequence	NOUN
ejpam-3588	328	7	0−→kerf	0−→kerf	NOUN
ejpam-3588	328	8	g−→	g−→	NOUN
ejpam-3588	328	9	m	m	NOUN
ejpam-3588	328	10	f−→	f−→	NOUN
ejpam-3588	328	11	0	0	NUM
ejpam-3588	328	12	.	.	PUNCT
ejpam-3588	329	1	so	so	ADV
ejpam-3588	329	2	,	,	PUNCT
ejpam-3588	329	3	by	by	ADP
ejpam-3588	329	4	the	the	DET
ejpam-3588	329	5	proof	proof	NOUN
ejpam-3588	329	6	of	of	ADP
ejpam-3588	329	7	(	(	PUNCT
ejpam-3588	329	8	[	[	X
ejpam-3588	329	9	15	15	NUM
ejpam-3588	329	10	]	]	PUNCT
ejpam-3588	329	11	,	,	PUNCT
ejpam-3588	329	12	proposition	proposition	NOUN
ejpam-3588	329	13	3.9	3.9	NUM
ejpam-3588	329	14	)	)	PUNCT
ejpam-3588	329	15	,	,	PUNCT
ejpam-3588	329	16	this	this	DET
ejpam-3588	329	17	sequence	sequence	NOUN
ejpam-3588	329	18	splits	split	VERB
ejpam-3588	329	19	.	.	PUNCT
ejpam-3588	330	1	consequently	consequently	ADV
ejpam-3588	330	2	,	,	PUNCT
ejpam-3588	330	3	s	s	VERB
ejpam-3588	330	4	≤⊕	≤⊕	NOUN
ejpam-3588	330	5	f	f	X
ejpam-3588	330	6	,	,	PUNCT
ejpam-3588	330	7	and	and	CCONJ
ejpam-3588	330	8	hence	hence	ADV
ejpam-3588	330	9	s	s	VERB
ejpam-3588	330	10	is	be	AUX
ejpam-3588	330	11	projective	projective	ADJ
ejpam-3588	330	12	.	.	PUNCT
ejpam-3588	331	1	therefore	therefore	ADV
ejpam-3588	331	2	,	,	PUNCT
ejpam-3588	331	3	r	r	NOUN
ejpam-3588	331	4	is	be	AUX
ejpam-3588	331	5	semisimple	semisimple	ADJ
ejpam-3588	331	6	.	.	PUNCT
ejpam-3588	332	1	remark	remark	PROPN
ejpam-3588	332	2	6	6	NUM
ejpam-3588	332	3	.	.	PUNCT
ejpam-3588	333	1	theorem	theorem	VERB
ejpam-3588	333	2	5	5	NUM
ejpam-3588	333	3	shows	show	VERB
ejpam-3588	333	4	that	that	SCONJ
ejpam-3588	333	5	the	the	DET
ejpam-3588	333	6	condition	condition	NOUN
ejpam-3588	333	7	”	"	PUNCT
ejpam-3588	333	8	right	right	ADV
ejpam-3588	333	9	v	v	X
ejpam-3588	333	10	-ring	-ring	NOUN
ejpam-3588	333	11	”	"	PUNCT
ejpam-3588	333	12	in	in	ADP
ejpam-3588	333	13	(	(	PUNCT
ejpam-3588	333	14	[	[	X
ejpam-3588	333	15	15	15	NUM
ejpam-3588	333	16	]	]	PUNCT
ejpam-3588	333	17	,	,	PUNCT
ejpam-3588	333	18	proposition	proposition	NOUN
ejpam-3588	333	19	3.9	3.9	NUM
ejpam-3588	333	20	)	)	PUNCT
ejpam-3588	333	21	is	be	AUX
ejpam-3588	333	22	superfluous	superfluous	ADJ
ejpam-3588	333	23	.	.	PUNCT
ejpam-3588	334	1	recall	recall	VERB
ejpam-3588	334	2	that	that	SCONJ
ejpam-3588	334	3	an	an	DET
ejpam-3588	334	4	r	r	NOUN
ejpam-3588	334	5	-	-	PUNCT
ejpam-3588	334	6	module	module	NOUN
ejpam-3588	334	7	is	be	AUX
ejpam-3588	334	8	called	call	VERB
ejpam-3588	334	9	baer	baer	PROPN
ejpam-3588	334	10	if	if	SCONJ
ejpam-3588	334	11	,	,	PUNCT
ejpam-3588	334	12	for	for	ADP
ejpam-3588	334	13	all	all	DET
ejpam-3588	334	14	n	n	DET
ejpam-3588	334	15	≤m	≤m	NOUN
ejpam-3588	334	16	,	,	PUNCT
ejpam-3588	334	17	ls(n	ls(n	NUM
ejpam-3588	334	18	)	)	PUNCT
ejpam-3588	334	19	=	=	PUNCT
ejpam-3588	335	1	se	se	X
ejpam-3588	335	2	,	,	PUNCT
ejpam-3588	335	3	with	with	ADP
ejpam-3588	335	4	e2	e2	PROPN
ejpam-3588	335	5	=	=	PUNCT
ejpam-3588	335	6	e	e	PROPN
ejpam-3588	335	7	∈	∈	PROPN
ejpam-3588	335	8	s.	s.	PROPN
ejpam-3588	335	9	a	a	DET
ejpam-3588	335	10	module	module	NOUN
ejpam-3588	335	11	m	m	VERB
ejpam-3588	335	12	is	be	AUX
ejpam-3588	335	13	called	call	VERB
ejpam-3588	335	14	k	k	ADJ
ejpam-3588	335	15	-	-	PUNCT
ejpam-3588	335	16	nonsingular	nonsingular	ADJ
ejpam-3588	335	17	if	if	SCONJ
ejpam-3588	335	18	,	,	PUNCT
ejpam-3588	335	19	∀ϕ	∀ϕ	PROPN
ejpam-3588	335	20	∈	∈	PROPN
ejpam-3588	335	21	end(m	end(m	PROPN
ejpam-3588	335	22	)	)	PUNCT
ejpam-3588	335	23	,	,	PUNCT
ejpam-3588	335	24	kerϕ	kerϕ	PROPN
ejpam-3588	335	25	≤e	≤e	PROPN
ejpam-3588	335	26	m	m	VERB
ejpam-3588	335	27	implies	imply	VERB
ejpam-3588	335	28	ϕ	ϕ	PROPN
ejpam-3588	335	29	=	=	SYM
ejpam-3588	335	30	0	0	PROPN
ejpam-3588	335	31	.	.	PUNCT
ejpam-3588	336	1	proposition	proposition	NOUN
ejpam-3588	336	2	10	10	NUM
ejpam-3588	336	3	.	.	PUNCT
ejpam-3588	337	1	let	let	VERB
ejpam-3588	337	2	m	m	PRON
ejpam-3588	337	3	be	be	AUX
ejpam-3588	337	4	a	a	DET
ejpam-3588	337	5	k	k	NOUN
ejpam-3588	337	6	-	-	ADJ
ejpam-3588	337	7	nonsingular	nonsingular	ADJ
ejpam-3588	337	8	c	c	NOUN
ejpam-3588	337	9	-	-	PUNCT
ejpam-3588	337	10	retractable	retractable	ADJ
ejpam-3588	337	11	r	r	NOUN
ejpam-3588	337	12	-	-	PUNCT
ejpam-3588	337	13	module	module	NOUN
ejpam-3588	337	14	.	.	PUNCT
ejpam-3588	338	1	then	then	ADV
ejpam-3588	338	2	s	s	VERB
ejpam-3588	338	3	is	be	AUX
ejpam-3588	338	4	right	right	ADV
ejpam-3588	338	5	nonsingular	nonsingular	ADJ
ejpam-3588	338	6	.	.	PUNCT
ejpam-3588	339	1	proof	proof	NOUN
ejpam-3588	339	2	.	.	PUNCT
ejpam-3588	340	1	see	see	VERB
ejpam-3588	340	2	proof	proof	NOUN
ejpam-3588	340	3	of	of	ADP
ejpam-3588	340	4	(	(	PUNCT
ejpam-3588	340	5	[	[	X
ejpam-3588	340	6	16	16	NUM
ejpam-3588	340	7	]	]	PUNCT
ejpam-3588	340	8	,	,	PUNCT
ejpam-3588	340	9	proposition	proposition	NOUN
ejpam-3588	340	10	3.6	3.6	NUM
ejpam-3588	340	11	)	)	PUNCT
ejpam-3588	340	12	.	.	PUNCT
ejpam-3588	341	1	proposition	proposition	NOUN
ejpam-3588	341	2	11	11	NUM
ejpam-3588	341	3	.	.	PUNCT
ejpam-3588	342	1	let	let	VERB
ejpam-3588	342	2	m	m	PRON
ejpam-3588	342	3	be	be	AUX
ejpam-3588	342	4	a	a	DET
ejpam-3588	342	5	c	c	NOUN
ejpam-3588	342	6	-	-	PUNCT
ejpam-3588	342	7	retractable	retractable	ADJ
ejpam-3588	342	8	r	r	NOUN
ejpam-3588	342	9	-	-	PUNCT
ejpam-3588	342	10	module	module	NOUN
ejpam-3588	342	11	such	such	ADJ
ejpam-3588	342	12	that	that	SCONJ
ejpam-3588	342	13	ss	ss	PROPN
ejpam-3588	342	14	is	be	AUX
ejpam-3588	342	15	extending	extend	VERB
ejpam-3588	342	16	.	.	PUNCT
ejpam-3588	343	1	then	then	ADV
ejpam-3588	343	2	m	m	PROPN
ejpam-3588	343	3	is	be	AUX
ejpam-3588	343	4	k	k	ADJ
ejpam-3588	343	5	-	-	ADJ
ejpam-3588	343	6	nonsingular	nonsingular	ADJ
ejpam-3588	343	7	if	if	SCONJ
ejpam-3588	344	1	and	and	CCONJ
ejpam-3588	344	2	only	only	ADV
ejpam-3588	344	3	if	if	SCONJ
ejpam-3588	344	4	m	m	NOUN
ejpam-3588	344	5	is	be	AUX
ejpam-3588	344	6	baer	baer	PROPN
ejpam-3588	344	7	.	.	PUNCT
ejpam-3588	345	1	a.	a.	PROPN
ejpam-3588	345	2	d.	d.	PROPN
ejpam-3588	345	3	diallo	diallo	PROPN
ejpam-3588	345	4	,	,	PUNCT
ejpam-3588	345	5	p.	p.	PROPN
ejpam-3588	345	6	c.	c.	PROPN
ejpam-3588	345	7	diop	diop	PROPN
ejpam-3588	345	8	,	,	PUNCT
ejpam-3588	345	9	m.	m.	NOUN
ejpam-3588	345	10	barry	barry	PROPN
ejpam-3588	345	11	/	/	SYM
ejpam-3588	345	12	eur	eur	PROPN
ejpam-3588	345	13	.	.	PUNCT
ejpam-3588	346	1	j.	j.	PROPN
ejpam-3588	346	2	pure	pure	PROPN
ejpam-3588	346	3	appl	appl	PROPN
ejpam-3588	346	4	.	.	PROPN
ejpam-3588	346	5	math	math	PROPN
ejpam-3588	346	6	,	,	PUNCT
ejpam-3588	346	7	13	13	NUM
ejpam-3588	346	8	(	(	PUNCT
ejpam-3588	346	9	1	1	NUM
ejpam-3588	346	10	)	)	PUNCT
ejpam-3588	346	11	(	(	PUNCT
ejpam-3588	346	12	2020	2020	NUM
ejpam-3588	346	13	)	)	PUNCT
ejpam-3588	346	14	,	,	PUNCT
ejpam-3588	346	15	158	158	NUM
ejpam-3588	346	16	-	-	SYM
ejpam-3588	346	17	169	169	NUM
ejpam-3588	346	18	167	167	NUM
ejpam-3588	346	19	proof	proof	NOUN
ejpam-3588	346	20	.	.	PUNCT
ejpam-3588	347	1	suppose	suppose	VERB
ejpam-3588	347	2	m	m	PRON
ejpam-3588	347	3	is	be	AUX
ejpam-3588	347	4	k	k	NOUN
ejpam-3588	347	5	-	-	ADJ
ejpam-3588	347	6	nonsingular	nonsingular	ADJ
ejpam-3588	347	7	.	.	PUNCT
ejpam-3588	348	1	by	by	ADP
ejpam-3588	348	2	proposition	proposition	NOUN
ejpam-3588	348	3	10	10	NUM
ejpam-3588	348	4	,	,	PUNCT
ejpam-3588	348	5	s	s	VERB
ejpam-3588	348	6	is	be	AUX
ejpam-3588	348	7	right	right	ADV
ejpam-3588	348	8	nonsingular	nonsingular	ADJ
ejpam-3588	348	9	.	.	PUNCT
ejpam-3588	349	1	let	let	VERB
ejpam-3588	349	2	n	n	PRON
ejpam-3588	349	3	be	be	AUX
ejpam-3588	349	4	a	a	DET
ejpam-3588	349	5	submodule	submodule	NOUN
ejpam-3588	349	6	of	of	ADP
ejpam-3588	349	7	m	m	PROPN
ejpam-3588	349	8	.	.	PUNCT
ejpam-3588	350	1	thus	thus	ADV
ejpam-3588	350	2	,	,	PUNCT
ejpam-3588	350	3	ls(n	ls(n	NUM
ejpam-3588	350	4	)	)	PUNCT
ejpam-3588	350	5	is	be	AUX
ejpam-3588	350	6	a	a	DET
ejpam-3588	350	7	complement	complement	NOUN
ejpam-3588	350	8	right	right	ADJ
ejpam-3588	350	9	ideal	ideal	NOUN
ejpam-3588	350	10	in	in	ADP
ejpam-3588	350	11	s.	s.	PROPN
ejpam-3588	350	12	because	because	SCONJ
ejpam-3588	350	13	ss	ss	PROPN
ejpam-3588	350	14	is	be	AUX
ejpam-3588	350	15	extending	extend	VERB
ejpam-3588	350	16	,	,	PUNCT
ejpam-3588	350	17	then	then	ADV
ejpam-3588	350	18	ls(n	ls(n	X
ejpam-3588	350	19	)	)	PUNCT
ejpam-3588	351	1	=	=	SYM
ejpam-3588	351	2	s(1	s(1	PROPN
ejpam-3588	351	3	−	−	PROPN
ejpam-3588	351	4	e	e	NOUN
ejpam-3588	351	5	)	)	PUNCT
ejpam-3588	351	6	for	for	ADP
ejpam-3588	351	7	some	some	DET
ejpam-3588	351	8	e	e	NOUN
ejpam-3588	351	9	=	=	PROPN
ejpam-3588	351	10	e2	e2	PROPN
ejpam-3588	351	11	∈	∈	PROPN
ejpam-3588	351	12	s	s	NOUN
ejpam-3588	351	13	,	,	PUNCT
ejpam-3588	351	14	and	and	CCONJ
ejpam-3588	351	15	hence	hence	ADV
ejpam-3588	351	16	m	m	VERB
ejpam-3588	351	17	is	be	AUX
ejpam-3588	351	18	baer	baer	PROPN
ejpam-3588	351	19	.	.	PUNCT
ejpam-3588	352	1	the	the	DET
ejpam-3588	352	2	converse	converse	PROPN
ejpam-3588	352	3	implication	implication	NOUN
ejpam-3588	352	4	follows	follow	VERB
ejpam-3588	352	5	from	from	ADP
ejpam-3588	352	6	(	(	PUNCT
ejpam-3588	352	7	[	[	X
ejpam-3588	352	8	18	18	NUM
ejpam-3588	352	9	]	]	PUNCT
ejpam-3588	352	10	,	,	PUNCT
ejpam-3588	352	11	lemma	lemma	PROPN
ejpam-3588	352	12	2.15	2.15	NUM
ejpam-3588	352	13	)	)	PUNCT
ejpam-3588	352	14	.	.	PUNCT
ejpam-3588	353	1	recall	recall	VERB
ejpam-3588	353	2	that	that	SCONJ
ejpam-3588	353	3	a	a	DET
ejpam-3588	353	4	module	module	NOUN
ejpam-3588	353	5	is	be	AUX
ejpam-3588	353	6	locally	locally	ADV
ejpam-3588	353	7	noetherian	noetherian	ADJ
ejpam-3588	353	8	if	if	SCONJ
ejpam-3588	353	9	any	any	PRON
ejpam-3588	353	10	of	of	ADP
ejpam-3588	353	11	its	its	PRON
ejpam-3588	353	12	finitely	finitely	ADV
ejpam-3588	353	13	generated	generate	VERB
ejpam-3588	353	14	submodules	submodule	NOUN
ejpam-3588	353	15	is	be	AUX
ejpam-3588	353	16	noetherian	noetherian	ADJ
ejpam-3588	353	17	.	.	PUNCT
ejpam-3588	354	1	an	an	DET
ejpam-3588	354	2	r	r	NOUN
ejpam-3588	354	3	-	-	PUNCT
ejpam-3588	354	4	module	module	NOUN
ejpam-3588	354	5	m	m	NOUN
ejpam-3588	354	6	is	be	AUX
ejpam-3588	354	7	said	say	VERB
ejpam-3588	354	8	to	to	PART
ejpam-3588	354	9	be	be	AUX
ejpam-3588	354	10	homo	homo	NOUN
ejpam-3588	354	11	-	-	PUNCT
ejpam-3588	354	12	related	relate	VERB
ejpam-3588	354	13	to	to	ADP
ejpam-3588	354	14	an	an	DET
ejpam-3588	354	15	r	r	NOUN
ejpam-3588	354	16	-	-	PUNCT
ejpam-3588	354	17	module	module	NOUN
ejpam-3588	354	18	l	l	NOUN
ejpam-3588	354	19	if	if	SCONJ
ejpam-3588	354	20	there	there	PRON
ejpam-3588	354	21	are	be	VERB
ejpam-3588	354	22	α	α	NOUN
ejpam-3588	354	23	:	:	PUNCT
ejpam-3588	354	24	m	m	VERB
ejpam-3588	354	25	−→	−→	ADJ
ejpam-3588	354	26	l	l	NOUN
ejpam-3588	354	27	and	and	CCONJ
ejpam-3588	354	28	β	β	X
ejpam-3588	354	29	:	:	PUNCT
ejpam-3588	355	1	l	l	PUNCT
ejpam-3588	355	2	−→m	−→m	PUNCT
ejpam-3588	355	3	such	such	ADJ
ejpam-3588	355	4	that	that	PRON
ejpam-3588	355	5	βα	βα	PRON
ejpam-3588	355	6	6=	6=	ADP
ejpam-3588	355	7	0	0	X
ejpam-3588	355	8	.	.	PUNCT
ejpam-3588	355	9	theorem	theorem	NOUN
ejpam-3588	355	10	6	6	NUM
ejpam-3588	355	11	.	.	PUNCT
ejpam-3588	356	1	let	let	VERB
ejpam-3588	356	2	m	m	PRON
ejpam-3588	356	3	be	be	AUX
ejpam-3588	356	4	a	a	DET
ejpam-3588	356	5	locally	locally	ADV
ejpam-3588	356	6	noetherian	noetherian	ADJ
ejpam-3588	356	7	c	c	NOUN
ejpam-3588	356	8	-	-	PUNCT
ejpam-3588	356	9	retractable	retractable	ADJ
ejpam-3588	356	10	r	r	NOUN
ejpam-3588	356	11	-	-	PUNCT
ejpam-3588	356	12	module	module	NOUN
ejpam-3588	356	13	.	.	PUNCT
ejpam-3588	357	1	then	then	ADV
ejpam-3588	357	2	m	m	VERB
ejpam-3588	357	3	is	be	AUX
ejpam-3588	357	4	homorelated	homorelate	VERB
ejpam-3588	357	5	to	to	ADP
ejpam-3588	357	6	a	a	DET
ejpam-3588	357	7	direct	direct	ADJ
ejpam-3588	357	8	sum	sum	NOUN
ejpam-3588	357	9	⊕i∈iui	⊕i∈iui	PROPN
ejpam-3588	357	10	of	of	ADP
ejpam-3588	357	11	uniform	uniform	ADJ
ejpam-3588	357	12	submodules	submodule	NOUN
ejpam-3588	357	13	of	of	ADP
ejpam-3588	357	14	m	m	PROPN
ejpam-3588	357	15	.	.	PUNCT
ejpam-3588	358	1	proof	proof	NOUN
ejpam-3588	358	2	.	.	PUNCT
ejpam-3588	359	1	suppose	suppose	VERB
ejpam-3588	359	2	m	m	PRON
ejpam-3588	359	3	is	be	AUX
ejpam-3588	359	4	a	a	DET
ejpam-3588	359	5	c	c	NOUN
ejpam-3588	359	6	-	-	PUNCT
ejpam-3588	359	7	retractable	retractable	ADJ
ejpam-3588	359	8	locally	locally	ADV
ejpam-3588	359	9	noetherian	noetherian	ADJ
ejpam-3588	359	10	module	module	NOUN
ejpam-3588	359	11	.	.	PUNCT
ejpam-3588	360	1	hence	hence	ADV
ejpam-3588	360	2	,	,	PUNCT
ejpam-3588	360	3	every	every	DET
ejpam-3588	360	4	submodule	submodule	NOUN
ejpam-3588	360	5	of	of	ADP
ejpam-3588	360	6	m	m	PROPN
ejpam-3588	360	7	contains	contain	VERB
ejpam-3588	360	8	a	a	DET
ejpam-3588	360	9	uniform	uniform	ADJ
ejpam-3588	360	10	submodule	submodule	NOUN
ejpam-3588	360	11	.	.	PUNCT
ejpam-3588	361	1	thus	thus	ADV
ejpam-3588	361	2	,	,	PUNCT
ejpam-3588	361	3	by	by	ADP
ejpam-3588	361	4	zorn	zorn	PROPN
ejpam-3588	361	5	’s	’s	PART
ejpam-3588	361	6	lemma	lemma	PROPN
ejpam-3588	361	7	,	,	PUNCT
ejpam-3588	361	8	m	m	PROPN
ejpam-3588	361	9	conains	conain	VERB
ejpam-3588	361	10	a	a	DET
ejpam-3588	361	11	maximal	maximal	ADJ
ejpam-3588	361	12	local	local	ADJ
ejpam-3588	361	13	direct	direct	ADJ
ejpam-3588	361	14	summand	summand	NOUN
ejpam-3588	361	15	n	n	PROPN
ejpam-3588	361	16	=	=	PROPN
ejpam-3588	361	17	⊕i∈iui	⊕i∈iui	PROPN
ejpam-3588	361	18	where	where	SCONJ
ejpam-3588	361	19	each	each	DET
ejpam-3588	361	20	ui	ui	NOUN
ejpam-3588	361	21	is	be	AUX
ejpam-3588	361	22	uniform	uniform	ADJ
ejpam-3588	361	23	.	.	PUNCT
ejpam-3588	362	1	also	also	ADV
ejpam-3588	362	2	by	by	ADP
ejpam-3588	362	3	the	the	DET
ejpam-3588	362	4	locally	locally	ADV
ejpam-3588	362	5	noetherian	noetherian	ADJ
ejpam-3588	362	6	condition	condition	NOUN
ejpam-3588	362	7	on	on	ADP
ejpam-3588	362	8	m	m	PRON
ejpam-3588	362	9	again	again	ADV
ejpam-3588	362	10	,	,	PUNCT
ejpam-3588	362	11	r	r	NOUN
ejpam-3588	362	12	/	/	SYM
ejpam-3588	362	13	r(m	r(m	NOUN
ejpam-3588	362	14	)	)	PUNCT
ejpam-3588	362	15	∼=	∼=	PROPN
ejpam-3588	362	16	mr	mr	NOUN
ejpam-3588	362	17	is	be	AUX
ejpam-3588	362	18	noetherian	noetherian	ADJ
ejpam-3588	362	19	for	for	ADP
ejpam-3588	362	20	any	any	DET
ejpam-3588	362	21	element	element	NOUN
ejpam-3588	362	22	m	m	VERB
ejpam-3588	362	23	in	in	ADP
ejpam-3588	362	24	m	m	PROPN
ejpam-3588	362	25	.	.	PUNCT
ejpam-3588	363	1	hence	hence	ADV
ejpam-3588	363	2	,	,	PUNCT
ejpam-3588	363	3	r	r	NOUN
ejpam-3588	363	4	satisfies	satisfy	VERB
ejpam-3588	363	5	acc	acc	PROPN
ejpam-3588	363	6	on	on	ADP
ejpam-3588	363	7	right	right	ADJ
ejpam-3588	363	8	ideals	ideal	NOUN
ejpam-3588	363	9	of	of	ADP
ejpam-3588	363	10	the	the	DET
ejpam-3588	363	11	form	form	NOUN
ejpam-3588	363	12	r(m	r(m	PROPN
ejpam-3588	363	13	)	)	PUNCT
ejpam-3588	363	14	where	where	SCONJ
ejpam-3588	363	15	m	m	VERB
ejpam-3588	363	16	∈	∈	PROPN
ejpam-3588	363	17	m	m	VERB
ejpam-3588	363	18	.	.	PUNCT
ejpam-3588	364	1	thus	thus	ADV
ejpam-3588	364	2	,	,	PUNCT
ejpam-3588	364	3	according	accord	VERB
ejpam-3588	364	4	to	to	ADP
ejpam-3588	364	5	(	(	PUNCT
ejpam-3588	364	6	[	[	X
ejpam-3588	364	7	6	6	NUM
ejpam-3588	364	8	]	]	PUNCT
ejpam-3588	364	9	,	,	PUNCT
ejpam-3588	364	10	8.1	8.1	NUM
ejpam-3588	364	11	)	)	PUNCT
ejpam-3588	364	12	,	,	PUNCT
ejpam-3588	364	13	n	n	PRON
ejpam-3588	364	14	is	be	AUX
ejpam-3588	364	15	a	a	DET
ejpam-3588	364	16	complement	complement	NOUN
ejpam-3588	364	17	submodule	submodule	NOUN
ejpam-3588	364	18	of	of	ADP
ejpam-3588	364	19	m	m	PROPN
ejpam-3588	364	20	.	.	PUNCT
ejpam-3588	365	1	since	since	SCONJ
ejpam-3588	365	2	m	m	PROPN
ejpam-3588	365	3	is	be	AUX
ejpam-3588	365	4	c	c	NOUN
ejpam-3588	365	5	-	-	PUNCT
ejpam-3588	365	6	retractable	retractable	ADJ
ejpam-3588	365	7	,	,	PUNCT
ejpam-3588	365	8	there	there	PRON
ejpam-3588	365	9	exists	exist	VERB
ejpam-3588	365	10	a	a	DET
ejpam-3588	365	11	nonzero	nonzero	NOUN
ejpam-3588	365	12	homomorphism	homomorphism	PROPN
ejpam-3588	365	13	f	f	X
ejpam-3588	365	14	:	:	PUNCT
ejpam-3588	365	15	m	m	VERB
ejpam-3588	365	16	−→	−→	ADJ
ejpam-3588	365	17	n	n	ADV
ejpam-3588	365	18	.	.	PUNCT
ejpam-3588	366	1	it	it	PRON
ejpam-3588	366	2	follows	follow	VERB
ejpam-3588	366	3	that	that	SCONJ
ejpam-3588	366	4	m	m	PROPN
ejpam-3588	366	5	is	be	AUX
ejpam-3588	366	6	homo	homo	NOUN
ejpam-3588	366	7	-	-	PUNCT
ejpam-3588	366	8	related	relate	VERB
ejpam-3588	366	9	to	to	ADP
ejpam-3588	366	10	n	n	PROPN
ejpam-3588	366	11	.	.	PUNCT
ejpam-3588	367	1	corollary	corollary	ADJ
ejpam-3588	367	2	4	4	NUM
ejpam-3588	367	3	.	.	PUNCT
ejpam-3588	368	1	let	let	VERB
ejpam-3588	368	2	r	r	PRON
ejpam-3588	368	3	be	be	AUX
ejpam-3588	368	4	a	a	DET
ejpam-3588	368	5	right	right	ADJ
ejpam-3588	368	6	noetherian	noetherian	ADJ
ejpam-3588	368	7	ring	ring	NOUN
ejpam-3588	368	8	.	.	PUNCT
ejpam-3588	369	1	then	then	ADV
ejpam-3588	369	2	every	every	DET
ejpam-3588	369	3	c	c	NOUN
ejpam-3588	369	4	-	-	PUNCT
ejpam-3588	369	5	retractable	retractable	ADJ
ejpam-3588	369	6	r	r	NOUN
ejpam-3588	369	7	-	-	PUNCT
ejpam-3588	369	8	module	module	NOUN
ejpam-3588	369	9	is	be	AUX
ejpam-3588	369	10	homo	homo	NOUN
ejpam-3588	369	11	-	-	PUNCT
ejpam-3588	369	12	related	relate	VERB
ejpam-3588	369	13	to	to	ADP
ejpam-3588	369	14	a	a	DET
ejpam-3588	369	15	direct	direct	ADJ
ejpam-3588	369	16	sum	sum	NOUN
ejpam-3588	369	17	⊕i∈iui	⊕i∈iui	PROPN
ejpam-3588	369	18	of	of	ADP
ejpam-3588	369	19	uniform	uniform	ADJ
ejpam-3588	369	20	submodules	submodule	NOUN
ejpam-3588	369	21	of	of	ADP
ejpam-3588	369	22	m	m	PROPN
ejpam-3588	369	23	.	.	PUNCT
ejpam-3588	370	1	theorem	theorem	ADJ
ejpam-3588	370	2	7	7	NUM
ejpam-3588	370	3	.	.	PUNCT
ejpam-3588	371	1	let	let	VERB
ejpam-3588	371	2	m	m	PRON
ejpam-3588	371	3	be	be	AUX
ejpam-3588	371	4	a	a	DET
ejpam-3588	371	5	nonsingular	nonsingular	ADJ
ejpam-3588	371	6	c	c	NOUN
ejpam-3588	371	7	-	-	PUNCT
ejpam-3588	371	8	retractable	retractable	ADJ
ejpam-3588	371	9	r	r	NOUN
ejpam-3588	371	10	-	-	PUNCT
ejpam-3588	371	11	module	module	NOUN
ejpam-3588	371	12	such	such	ADJ
ejpam-3588	371	13	that	that	SCONJ
ejpam-3588	371	14	every	every	DET
ejpam-3588	371	15	udim(mr	udim(mr	NOUN
ejpam-3588	371	16	)	)	PUNCT
ejpam-3588	371	17	<	<	X
ejpam-3588	371	18	∞	∞	PROPN
ejpam-3588	371	19	for	for	ADP
ejpam-3588	371	20	every	every	DET
ejpam-3588	371	21	element	element	NOUN
ejpam-3588	371	22	m	m	NOUN
ejpam-3588	371	23	∈	∈	NOUN
ejpam-3588	371	24	m	m	NOUN
ejpam-3588	371	25	.	.	PUNCT
ejpam-3588	372	1	then	then	ADV
ejpam-3588	372	2	m	m	PROPN
ejpam-3588	372	3	is	be	AUX
ejpam-3588	372	4	homo	homo	NOUN
ejpam-3588	372	5	-	-	PUNCT
ejpam-3588	372	6	related	relate	VERB
ejpam-3588	372	7	to	to	ADP
ejpam-3588	372	8	a	a	DET
ejpam-3588	372	9	direct	direct	ADJ
ejpam-3588	372	10	sum	sum	NOUN
ejpam-3588	372	11	⊕i∈iui	⊕i∈iui	PROPN
ejpam-3588	372	12	of	of	ADP
ejpam-3588	372	13	indecomposable	indecomposable	ADJ
ejpam-3588	372	14	nonsingular	nonsingular	ADJ
ejpam-3588	372	15	submodules	submodule	NOUN
ejpam-3588	372	16	of	of	ADP
ejpam-3588	372	17	m	m	PROPN
ejpam-3588	372	18	.	.	PUNCT
ejpam-3588	373	1	proof	proof	NOUN
ejpam-3588	373	2	.	.	PUNCT
ejpam-3588	374	1	suppose	suppose	VERB
ejpam-3588	374	2	m	m	NOUN
ejpam-3588	374	3	has	have	VERB
ejpam-3588	374	4	the	the	DET
ejpam-3588	374	5	stated	state	VERB
ejpam-3588	374	6	condition	condition	NOUN
ejpam-3588	374	7	.	.	PUNCT
ejpam-3588	375	1	by	by	ADP
ejpam-3588	375	2	zorn	zorn	PROPN
ejpam-3588	375	3	’s	’s	PART
ejpam-3588	375	4	lemma	lemma	PROPN
ejpam-3588	375	5	,	,	PUNCT
ejpam-3588	375	6	m	m	PROPN
ejpam-3588	375	7	conains	conain	VERB
ejpam-3588	375	8	a	a	DET
ejpam-3588	375	9	maximal	maximal	ADJ
ejpam-3588	375	10	local	local	ADJ
ejpam-3588	375	11	direct	direct	ADJ
ejpam-3588	375	12	summand	summand	NOUN
ejpam-3588	375	13	n	n	PROPN
ejpam-3588	375	14	=	=	PROPN
ejpam-3588	375	15	⊕i∈iui	⊕i∈iui	PROPN
ejpam-3588	375	16	where	where	SCONJ
ejpam-3588	375	17	each	each	DET
ejpam-3588	375	18	ui	ui	NOUN
ejpam-3588	375	19	is	be	AUX
ejpam-3588	375	20	indecomposable	indecomposable	ADJ
ejpam-3588	375	21	nonsingular	nonsingular	ADJ
ejpam-3588	375	22	.	.	PUNCT
ejpam-3588	376	1	let	let	VERB
ejpam-3588	376	2	m	m	PRON
ejpam-3588	376	3	∈	∈	VERB
ejpam-3588	376	4	m	m	NOUN
ejpam-3588	376	5	.	.	PUNCT
ejpam-3588	377	1	then	then	ADV
ejpam-3588	377	2	r	r	X
ejpam-3588	377	3	/	/	SYM
ejpam-3588	377	4	r(m	r(m	PROPN
ejpam-3588	377	5	)	)	PUNCT
ejpam-3588	377	6	is	be	AUX
ejpam-3588	377	7	a	a	DET
ejpam-3588	377	8	nonsingular	nonsingular	ADJ
ejpam-3588	377	9	r	r	NOUN
ejpam-3588	377	10	-	-	PUNCT
ejpam-3588	377	11	module	module	NOUN
ejpam-3588	377	12	which	which	PRON
ejpam-3588	377	13	has	have	VERB
ejpam-3588	377	14	finite	finite	ADJ
ejpam-3588	377	15	uniform	uniform	ADJ
ejpam-3588	377	16	dimension	dimension	NOUN
ejpam-3588	377	17	.	.	PUNCT
ejpam-3588	378	1	by	by	ADP
ejpam-3588	378	2	(	(	PUNCT
ejpam-3588	378	3	[	[	X
ejpam-3588	378	4	6	6	NUM
ejpam-3588	378	5	]	]	PUNCT
ejpam-3588	378	6	,	,	PUNCT
ejpam-3588	378	7	section	section	NOUN
ejpam-3588	378	8	5.10	5.10	NUM
ejpam-3588	378	9	)	)	PUNCT
ejpam-3588	378	10	,	,	PUNCT
ejpam-3588	378	11	r	r	NOUN
ejpam-3588	378	12	has	have	VERB
ejpam-3588	378	13	acc	acc	PROPN
ejpam-3588	378	14	on	on	ADP
ejpam-3588	378	15	right	right	ADJ
ejpam-3588	378	16	ideals	ideal	NOUN
ejpam-3588	378	17	of	of	ADP
ejpam-3588	378	18	the	the	DET
ejpam-3588	378	19	form	form	NOUN
ejpam-3588	378	20	r(m	r(m	PROPN
ejpam-3588	378	21	)	)	PUNCT
ejpam-3588	378	22	where	where	SCONJ
ejpam-3588	378	23	m	m	VERB
ejpam-3588	378	24	∈	∈	PROPN
ejpam-3588	378	25	m	m	VERB
ejpam-3588	378	26	.	.	PUNCT
ejpam-3588	379	1	thus	thus	ADV
ejpam-3588	379	2	,	,	PUNCT
ejpam-3588	379	3	according	accord	VERB
ejpam-3588	379	4	to	to	ADP
ejpam-3588	379	5	(	(	PUNCT
ejpam-3588	379	6	[	[	X
ejpam-3588	379	7	6	6	NUM
ejpam-3588	379	8	]	]	PUNCT
ejpam-3588	379	9	,	,	PUNCT
ejpam-3588	379	10	8.1	8.1	NUM
ejpam-3588	379	11	)	)	PUNCT
ejpam-3588	379	12	,	,	PUNCT
ejpam-3588	379	13	n	n	PRON
ejpam-3588	379	14	is	be	AUX
ejpam-3588	379	15	a	a	DET
ejpam-3588	379	16	complement	complement	NOUN
ejpam-3588	379	17	submodule	submodule	NOUN
ejpam-3588	379	18	of	of	ADP
ejpam-3588	379	19	m	m	PROPN
ejpam-3588	379	20	.	.	PUNCT
ejpam-3588	380	1	since	since	SCONJ
ejpam-3588	380	2	m	m	PROPN
ejpam-3588	380	3	is	be	AUX
ejpam-3588	380	4	c	c	NOUN
ejpam-3588	380	5	-	-	PUNCT
ejpam-3588	380	6	retractable	retractable	ADJ
ejpam-3588	380	7	,	,	PUNCT
ejpam-3588	380	8	there	there	PRON
ejpam-3588	380	9	exists	exist	VERB
ejpam-3588	380	10	a	a	DET
ejpam-3588	380	11	nonzero	nonzero	NOUN
ejpam-3588	380	12	homomorphism	homomorphism	PROPN
ejpam-3588	380	13	f	f	X
ejpam-3588	380	14	:	:	PUNCT
ejpam-3588	380	15	m	m	VERB
ejpam-3588	380	16	−→	−→	ADJ
ejpam-3588	380	17	n	n	ADV
ejpam-3588	380	18	.	.	PUNCT
ejpam-3588	381	1	it	it	PRON
ejpam-3588	381	2	follows	follow	VERB
ejpam-3588	381	3	that	that	SCONJ
ejpam-3588	381	4	m	m	PROPN
ejpam-3588	381	5	is	be	AUX
ejpam-3588	381	6	homo	homo	NOUN
ejpam-3588	381	7	-	-	PUNCT
ejpam-3588	381	8	related	relate	VERB
ejpam-3588	381	9	to	to	ADP
ejpam-3588	381	10	n	n	PROPN
ejpam-3588	381	11	.	.	PUNCT
ejpam-3588	382	1	proposition	proposition	NOUN
ejpam-3588	382	2	12	12	NUM
ejpam-3588	382	3	.	.	PUNCT
ejpam-3588	383	1	let	let	VERB
ejpam-3588	383	2	m	m	PRON
ejpam-3588	383	3	be	be	AUX
ejpam-3588	383	4	a	a	DET
ejpam-3588	383	5	c	c	NOUN
ejpam-3588	383	6	-	-	PUNCT
ejpam-3588	383	7	retractable	retractable	ADJ
ejpam-3588	383	8	r	r	NOUN
ejpam-3588	383	9	-	-	PUNCT
ejpam-3588	383	10	module	module	NOUN
ejpam-3588	383	11	with	with	ADP
ejpam-3588	383	12	udim(m	udim(m	PROPN
ejpam-3588	383	13	)	)	PUNCT
ejpam-3588	383	14	≥	≥	NOUN
ejpam-3588	383	15	2	2	NUM
ejpam-3588	383	16	.	.	PUNCT
ejpam-3588	384	1	then	then	ADV
ejpam-3588	384	2	m	m	VERB
ejpam-3588	384	3	is	be	AUX
ejpam-3588	384	4	retractable	retractable	ADJ
ejpam-3588	384	5	.	.	PUNCT
ejpam-3588	385	1	proof	proof	NOUN
ejpam-3588	385	2	.	.	PUNCT
ejpam-3588	386	1	suppose	suppose	VERB
ejpam-3588	386	2	m	m	NOUN
ejpam-3588	386	3	has	have	VERB
ejpam-3588	386	4	the	the	DET
ejpam-3588	386	5	stated	state	VERB
ejpam-3588	386	6	condition	condition	NOUN
ejpam-3588	386	7	.	.	PUNCT
ejpam-3588	387	1	let	let	VERB
ejpam-3588	387	2	0	0	NUM
ejpam-3588	387	3	6=	6=	NUM
ejpam-3588	387	4	n	n	DET
ejpam-3588	387	5	≤m	≤m	NOUN
ejpam-3588	387	6	.	.	PUNCT
ejpam-3588	388	1	since	since	SCONJ
ejpam-3588	388	2	udim(n	udim(n	PROPN
ejpam-3588	388	3	)	)	PUNCT
ejpam-3588	388	4	<	<	X
ejpam-3588	388	5	∞	∞	PROPN
ejpam-3588	388	6	,	,	PUNCT
ejpam-3588	388	7	n	n	PRON
ejpam-3588	388	8	contains	contain	VERB
ejpam-3588	388	9	a	a	DET
ejpam-3588	388	10	uniform	uniform	ADJ
ejpam-3588	388	11	submodule	submodule	PROPN
ejpam-3588	388	12	u	u	PROPN
ejpam-3588	388	13	.	.	PUNCT
ejpam-3588	389	1	after	after	ADP
ejpam-3588	389	2	replacing	replace	VERB
ejpam-3588	389	3	u	u	NOUN
ejpam-3588	389	4	by	by	ADP
ejpam-3588	389	5	an	an	DET
ejpam-3588	389	6	essential	essential	ADJ
ejpam-3588	389	7	closure	closure	NOUN
ejpam-3588	389	8	,	,	PUNCT
ejpam-3588	389	9	we	we	PRON
ejpam-3588	389	10	may	may	AUX
ejpam-3588	389	11	assume	assume	VERB
ejpam-3588	389	12	that	that	SCONJ
ejpam-3588	389	13	u	u	PRON
ejpam-3588	389	14	is	be	AUX
ejpam-3588	389	15	a	a	DET
ejpam-3588	389	16	complement	complement	NOUN
ejpam-3588	389	17	submodule	submodule	NOUN
ejpam-3588	389	18	of	of	ADP
ejpam-3588	389	19	m	m	PROPN
ejpam-3588	389	20	.	.	PUNCT
ejpam-3588	390	1	by	by	ADP
ejpam-3588	390	2	our	our	PRON
ejpam-3588	390	3	assumption	assumption	NOUN
ejpam-3588	390	4	,	,	PUNCT
ejpam-3588	390	5	there	there	PRON
ejpam-3588	390	6	is	be	VERB
ejpam-3588	390	7	a	a	DET
ejpam-3588	390	8	nonzero	nonzero	NOUN
ejpam-3588	390	9	homomorphism	homomorphism	NOUN
ejpam-3588	390	10	m	m	VERB
ejpam-3588	390	11	−→	−→	ADJ
ejpam-3588	390	12	u	u	NOUN
ejpam-3588	390	13	.	.	PUNCT
ejpam-3588	391	1	therefore	therefore	ADV
ejpam-3588	391	2	,	,	PUNCT
ejpam-3588	391	3	m	m	VERB
ejpam-3588	391	4	is	be	AUX
ejpam-3588	391	5	retractable	retractable	ADJ
ejpam-3588	391	6	.	.	PUNCT
ejpam-3588	392	1	references	reference	NOUN
ejpam-3588	392	2	168	168	NUM
ejpam-3588	392	3	corollary	corollary	ADJ
ejpam-3588	392	4	5	5	NUM
ejpam-3588	392	5	.	.	PUNCT
ejpam-3588	393	1	let	let	VERB
ejpam-3588	393	2	m	m	PRON
ejpam-3588	393	3	be	be	AUX
ejpam-3588	393	4	an	an	DET
ejpam-3588	393	5	r	r	NOUN
ejpam-3588	393	6	-	-	PUNCT
ejpam-3588	393	7	module	module	NOUN
ejpam-3588	393	8	with	with	ADP
ejpam-3588	393	9	udim(m	udim(m	PROPN
ejpam-3588	393	10	)	)	PUNCT
ejpam-3588	393	11	≥	≥	NOUN
ejpam-3588	393	12	2	2	NUM
ejpam-3588	393	13	.	.	PUNCT
ejpam-3588	394	1	then	then	ADV
ejpam-3588	394	2	m	m	PROPN
ejpam-3588	394	3	is	be	AUX
ejpam-3588	394	4	wd	wd	ADJ
ejpam-3588	394	5	-	-	PUNCT
ejpam-3588	394	6	rickart	rickart	NOUN
ejpam-3588	394	7	cretractable	cretractable	ADJ
ejpam-3588	394	8	if	if	SCONJ
ejpam-3588	395	1	and	and	CCONJ
ejpam-3588	395	2	only	only	ADV
ejpam-3588	395	3	if	if	SCONJ
ejpam-3588	395	4	m	m	NOUN
ejpam-3588	395	5	is	be	AUX
ejpam-3588	395	6	semisimple	semisimple	ADJ
ejpam-3588	395	7	.	.	PUNCT
ejpam-3588	396	1	proof	proof	NOUN
ejpam-3588	396	2	.	.	PUNCT
ejpam-3588	397	1	suppose	suppose	VERB
ejpam-3588	397	2	m	m	NOUN
ejpam-3588	397	3	is	be	AUX
ejpam-3588	397	4	wd	wd	ADJ
ejpam-3588	397	5	-	-	PUNCT
ejpam-3588	397	6	rickart	rickart	ADJ
ejpam-3588	397	7	c	c	NOUN
ejpam-3588	397	8	-	-	PUNCT
ejpam-3588	397	9	retractable	retractable	ADJ
ejpam-3588	397	10	.	.	PUNCT
ejpam-3588	398	1	since	since	SCONJ
ejpam-3588	398	2	udim(m	udim(m	PROPN
ejpam-3588	398	3	)	)	PUNCT
ejpam-3588	398	4	≥	≥	NOUN
ejpam-3588	398	5	2	2	NUM
ejpam-3588	398	6	,	,	PUNCT
ejpam-3588	398	7	m	m	VERB
ejpam-3588	398	8	is	be	AUX
ejpam-3588	398	9	a	a	DET
ejpam-3588	398	10	finite	finite	ADJ
ejpam-3588	398	11	direct	direct	ADJ
ejpam-3588	398	12	sum	sum	NOUN
ejpam-3588	398	13	of	of	ADP
ejpam-3588	398	14	indecomposable	indecomposable	ADJ
ejpam-3588	398	15	submodules	submodule	NOUN
ejpam-3588	398	16	.	.	PUNCT
ejpam-3588	399	1	by	by	ADP
ejpam-3588	399	2	proposition	proposition	NOUN
ejpam-3588	399	3	12	12	NUM
ejpam-3588	399	4	,	,	PUNCT
ejpam-3588	399	5	m	m	VERB
ejpam-3588	399	6	is	be	AUX
ejpam-3588	399	7	retractable	retractable	ADJ
ejpam-3588	399	8	.	.	PUNCT
ejpam-3588	400	1	therefore	therefore	ADV
ejpam-3588	400	2	,	,	PUNCT
ejpam-3588	400	3	according	accord	VERB
ejpam-3588	400	4	to	to	ADP
ejpam-3588	400	5	(	(	PUNCT
ejpam-3588	400	6	[	[	X
ejpam-3588	400	7	23	23	NUM
ejpam-3588	400	8	]	]	PUNCT
ejpam-3588	400	9	,	,	PUNCT
ejpam-3588	400	10	proposition	proposition	NOUN
ejpam-3588	400	11	2.17	2.17	NUM
ejpam-3588	400	12	)	)	PUNCT
ejpam-3588	400	13	,	,	PUNCT
ejpam-3588	400	14	m	m	PROPN
ejpam-3588	400	15	is	be	AUX
ejpam-3588	400	16	semisimple	semisimple	ADJ
ejpam-3588	400	17	.	.	PUNCT
ejpam-3588	401	1	the	the	DET
ejpam-3588	401	2	converse	converse	PROPN
ejpam-3588	401	3	implication	implication	NOUN
ejpam-3588	401	4	is	be	AUX
ejpam-3588	401	5	clear	clear	ADJ
ejpam-3588	401	6	.	.	PUNCT
ejpam-3588	402	1	proposition	proposition	NOUN
ejpam-3588	402	2	13	13	NUM
ejpam-3588	402	3	.	.	PUNCT
ejpam-3588	403	1	the	the	DET
ejpam-3588	403	2	following	follow	VERB
ejpam-3588	403	3	statements	statement	NOUN
ejpam-3588	403	4	are	be	AUX
ejpam-3588	403	5	equivalent	equivalent	ADJ
ejpam-3588	403	6	for	for	ADP
ejpam-3588	403	7	an	an	DET
ejpam-3588	403	8	r	r	NOUN
ejpam-3588	403	9	-	-	PUNCT
ejpam-3588	403	10	module	module	NOUN
ejpam-3588	403	11	m	m	NOUN
ejpam-3588	403	12	with	with	ADP
ejpam-3588	403	13	udim(m	udim(m	NOUN
ejpam-3588	403	14	)	)	PUNCT
ejpam-3588	403	15	=	=	SYM
ejpam-3588	403	16	n	n	X
ejpam-3588	403	17	≥	≥	NOUN
ejpam-3588	403	18	2	2	NUM
ejpam-3588	403	19	.	.	PUNCT
ejpam-3588	404	1	(	(	PUNCT
ejpam-3588	404	2	1	1	X
ejpam-3588	404	3	)	)	PUNCT
ejpam-3588	404	4	m	m	VERB
ejpam-3588	404	5	is	be	AUX
ejpam-3588	404	6	c	c	NOUN
ejpam-3588	404	7	-	-	PUNCT
ejpam-3588	404	8	retractable	retractable	ADJ
ejpam-3588	404	9	.	.	PUNCT
ejpam-3588	405	1	(	(	PUNCT
ejpam-3588	405	2	2	2	X
ejpam-3588	405	3	)	)	PUNCT
ejpam-3588	405	4	homr(m	homr(m	PROPN
ejpam-3588	405	5	,	,	PUNCT
ejpam-3588	405	6	u	u	NOUN
ejpam-3588	405	7	)	)	PUNCT
ejpam-3588	405	8	6=	6=	ADP
ejpam-3588	405	9	0	0	NUM
ejpam-3588	405	10	for	for	ADP
ejpam-3588	405	11	every	every	DET
ejpam-3588	405	12	uniform	uniform	ADJ
ejpam-3588	405	13	submodule	submodule	PROPN
ejpam-3588	405	14	u	u	PROPN
ejpam-3588	405	15	of	of	ADP
ejpam-3588	405	16	m	m	PROPN
ejpam-3588	405	17	.	.	PUNCT
ejpam-3588	406	1	(	(	PUNCT
ejpam-3588	406	2	3	3	X
ejpam-3588	406	3	)	)	PUNCT
ejpam-3588	406	4	homr(m	homr(m	PROPN
ejpam-3588	406	5	,	,	PUNCT
ejpam-3588	406	6	u	u	NOUN
ejpam-3588	406	7	)	)	PUNCT
ejpam-3588	406	8	6=	6=	ADP
ejpam-3588	406	9	0	0	NUM
ejpam-3588	406	10	for	for	ADP
ejpam-3588	406	11	every	every	DET
ejpam-3588	406	12	cyclic	cyclic	ADJ
ejpam-3588	406	13	uniform	uniform	NOUN
ejpam-3588	406	14	submodule	submodule	PROPN
ejpam-3588	406	15	u	u	PROPN
ejpam-3588	406	16	of	of	ADP
ejpam-3588	406	17	m	m	PROPN
ejpam-3588	406	18	.	.	PUNCT
ejpam-3588	407	1	(	(	PUNCT
ejpam-3588	407	2	4	4	X
ejpam-3588	407	3	)	)	PUNCT
ejpam-3588	407	4	m	m	VERB
ejpam-3588	407	5	is	be	AUX
ejpam-3588	407	6	retractble	retractble	ADJ
ejpam-3588	407	7	.	.	PUNCT
ejpam-3588	408	1	proof	proof	NOUN
ejpam-3588	408	2	.	.	PUNCT
ejpam-3588	409	1	(	(	PUNCT
ejpam-3588	409	2	1)⇒	1)⇒	NUM
ejpam-3588	409	3	(	(	PUNCT
ejpam-3588	409	4	2	2	NUM
ejpam-3588	409	5	)	)	PUNCT
ejpam-3588	409	6	follows	follow	VERB
ejpam-3588	409	7	from	from	ADP
ejpam-3588	409	8	proposition	proposition	NOUN
ejpam-3588	409	9	12	12	NUM
ejpam-3588	409	10	.	.	PUNCT
ejpam-3588	410	1	(	(	PUNCT
ejpam-3588	410	2	2)⇒	2)⇒	NUM
ejpam-3588	410	3	(	(	PUNCT
ejpam-3588	410	4	3	3	X
ejpam-3588	410	5	)	)	PUNCT
ejpam-3588	410	6	clear	clear	ADJ
ejpam-3588	410	7	.	.	PUNCT
ejpam-3588	411	1	(	(	PUNCT
ejpam-3588	411	2	3	3	X
ejpam-3588	411	3	)	)	PUNCT
ejpam-3588	411	4	⇒	⇒	NOUN
ejpam-3588	411	5	(	(	PUNCT
ejpam-3588	411	6	4	4	X
ejpam-3588	411	7	)	)	PUNCT
ejpam-3588	411	8	let	let	VERB
ejpam-3588	411	9	0	0	NUM
ejpam-3588	411	10	6=	6=	NUM
ejpam-3588	411	11	n	n	PRON
ejpam-3588	411	12	≤	≤	NOUN
ejpam-3588	411	13	m	m	VERB
ejpam-3588	411	14	.	.	PUNCT
ejpam-3588	412	1	let	let	VERB
ejpam-3588	412	2	0	0	NUM
ejpam-3588	413	1	6=	6=	ADP
ejpam-3588	413	2	m	m	PROPN
ejpam-3588	413	3	∈	∈	PROPN
ejpam-3588	413	4	n	n	NOUN
ejpam-3588	413	5	.	.	PUNCT
ejpam-3588	414	1	by	by	ADP
ejpam-3588	414	2	hypothesis	hypothesis	NOUN
ejpam-3588	414	3	,	,	PUNCT
ejpam-3588	414	4	mr	mr	PROPN
ejpam-3588	414	5	has	have	VERB
ejpam-3588	414	6	finite	finite	VERB
ejpam-3588	414	7	uniform	uniform	ADJ
ejpam-3588	414	8	dimension	dimension	NOUN
ejpam-3588	414	9	and	and	CCONJ
ejpam-3588	414	10	hence	hence	ADV
ejpam-3588	414	11	mr	mr	PROPN
ejpam-3588	414	12	contains	contain	VERB
ejpam-3588	414	13	a	a	DET
ejpam-3588	414	14	uniform	uniform	ADJ
ejpam-3588	414	15	submodule	submodule	PROPN
ejpam-3588	414	16	u	u	PROPN
ejpam-3588	414	17	.	.	PUNCT
ejpam-3588	415	1	let	let	VERB
ejpam-3588	415	2	0	0	NUM
ejpam-3588	415	3	6=	6=	NUM
ejpam-3588	415	4	u	u	PROPN
ejpam-3588	415	5	∈	∈	PROPN
ejpam-3588	415	6	u	u	NOUN
ejpam-3588	415	7	.	.	PUNCT
ejpam-3588	416	1	by	by	ADP
ejpam-3588	416	2	(	(	PUNCT
ejpam-3588	416	3	3	3	NUM
ejpam-3588	416	4	)	)	PUNCT
ejpam-3588	416	5	,	,	PUNCT
ejpam-3588	416	6	homr(m	homr(m	PROPN
ejpam-3588	416	7	,	,	PUNCT
ejpam-3588	416	8	ur	ur	INTJ
ejpam-3588	416	9	)	)	PUNCT
ejpam-3588	416	10	6=	6=	ADP
ejpam-3588	416	11	0	0	NUM
ejpam-3588	416	12	.	.	PUNCT
ejpam-3588	417	1	hence	hence	ADV
ejpam-3588	417	2	,	,	PUNCT
ejpam-3588	417	3	m	m	VERB
ejpam-3588	417	4	is	be	AUX
ejpam-3588	417	5	retractable	retractable	ADJ
ejpam-3588	417	6	.	.	PUNCT
ejpam-3588	418	1	(	(	PUNCT
ejpam-3588	418	2	4)⇒	4)⇒	X
ejpam-3588	418	3	(	(	PUNCT
ejpam-3588	418	4	1	1	NUM
ejpam-3588	418	5	)	)	PUNCT
ejpam-3588	418	6	is	be	AUX
ejpam-3588	418	7	clear	clear	ADJ
ejpam-3588	418	8	.	.	PUNCT
ejpam-3588	419	1	remark	remark	PROPN
ejpam-3588	419	2	7	7	NUM
ejpam-3588	419	3	.	.	PUNCT
ejpam-3588	419	4	by	by	ADP
ejpam-3588	419	5	proposition	proposition	NOUN
ejpam-3588	419	6	12	12	NUM
ejpam-3588	419	7	,	,	PUNCT
ejpam-3588	419	8	every	every	DET
ejpam-3588	419	9	c	c	NOUN
ejpam-3588	419	10	-	-	PUNCT
ejpam-3588	419	11	retractable	retractable	ADJ
ejpam-3588	419	12	module	module	NOUN
ejpam-3588	419	13	m	m	VERB
ejpam-3588	419	14	with	with	ADP
ejpam-3588	419	15	udim(m	udim(m	NOUN
ejpam-3588	419	16	)	)	PUNCT
ejpam-3588	419	17	=	=	SYM
ejpam-3588	420	1	n	n	X
ejpam-3588	420	2	≥	≥	NOUN
ejpam-3588	420	3	2	2	NUM
ejpam-3588	420	4	is	be	AUX
ejpam-3588	420	5	retractable	retractable	ADJ
ejpam-3588	420	6	.	.	PUNCT
ejpam-3588	421	1	note	note	VERB
ejpam-3588	421	2	that	that	SCONJ
ejpam-3588	421	3	the	the	DET
ejpam-3588	421	4	condition	condition	NOUN
ejpam-3588	421	5	udim(m	udim(m	NOUN
ejpam-3588	421	6	)	)	PUNCT
ejpam-3588	421	7	=	=	SYM
ejpam-3588	421	8	n	n	X
ejpam-3588	421	9	≥	≥	NOUN
ejpam-3588	421	10	2	2	NUM
ejpam-3588	421	11	can	can	AUX
ejpam-3588	421	12	not	not	PART
ejpam-3588	421	13	be	be	AUX
ejpam-3588	421	14	dropped	drop	VERB
ejpam-3588	421	15	.	.	PUNCT
ejpam-3588	422	1	in	in	ADP
ejpam-3588	422	2	fact	fact	NOUN
ejpam-3588	422	3	,	,	PUNCT
ejpam-3588	422	4	q	q	X
ejpam-3588	422	5	as	as	SCONJ
ejpam-3588	422	6	a	a	DET
ejpam-3588	422	7	z	z	NOUN
ejpam-3588	422	8	-	-	PUNCT
ejpam-3588	422	9	module	module	NOUN
ejpam-3588	422	10	is	be	AUX
ejpam-3588	422	11	c	c	NOUN
ejpam-3588	422	12	-	-	PUNCT
ejpam-3588	422	13	retractable	retractable	ADJ
ejpam-3588	422	14	uniform	uniform	NOUN
ejpam-3588	422	15	but	but	CCONJ
ejpam-3588	422	16	it	it	PRON
ejpam-3588	422	17	is	be	AUX
ejpam-3588	422	18	not	not	PART
ejpam-3588	422	19	retractable	retractable	ADJ
ejpam-3588	422	20	.	.	PUNCT
ejpam-3588	423	1	references	reference	NOUN
ejpam-3588	423	2	[	[	X
ejpam-3588	423	3	1	1	NUM
ejpam-3588	423	4	]	]	PUNCT
ejpam-3588	423	5	a.	a.	NOUN
ejpam-3588	423	6	alahmadi	alahmadi	PROPN
ejpam-3588	423	7	,	,	PUNCT
ejpam-3588	423	8	s.	s.	PROPN
ejpam-3588	423	9	k	k	PROPN
ejpam-3588	423	10	jain	jain	PROPN
ejpam-3588	423	11	and	and	CCONJ
ejpam-3588	423	12	a.	a.	PROPN
ejpam-3588	423	13	leroy	leroy	PROPN
ejpam-3588	423	14	(	(	PUNCT
ejpam-3588	423	15	2012	2012	NUM
ejpam-3588	423	16	)	)	PUNCT
ejpam-3588	423	17	,	,	PUNCT
ejpam-3588	423	18	ads	ad	NOUN
ejpam-3588	423	19	modules	module	NOUN
ejpam-3588	423	20	,	,	PUNCT
ejpam-3588	423	21	j.	j.	PROPN
ejpam-3588	423	22	algebra	algebra	PROPN
ejpam-3588	423	23	352:215	352:215	PROPN
ejpam-3588	423	24	-	-	SYM
ejpam-3588	423	25	22	22	NUM
ejpam-3588	423	26	.	.	PUNCT
ejpam-3588	424	1	[	[	X
ejpam-3588	424	2	2	2	X
ejpam-3588	424	3	]	]	PUNCT
ejpam-3588	424	4	s.	s.	PROPN
ejpam-3588	424	5	asgari	asgari	PROPN
ejpam-3588	424	6	and	and	CCONJ
ejpam-3588	424	7	a.	a.	NOUN
ejpam-3588	424	8	haghany	haghany	NOUN
ejpam-3588	424	9	,	,	PUNCT
ejpam-3588	424	10	generalizations	generalization	NOUN
ejpam-3588	424	11	of	of	ADP
ejpam-3588	424	12	t	t	NOUN
ejpam-3588	424	13	-extending	-extending	NOUN
ejpam-3588	424	14	modules	module	NOUN
ejpam-3588	424	15	relative	relative	ADJ
ejpam-3588	424	16	to	to	ADP
ejpam-3588	424	17	fully	fully	ADV
ejpam-3588	424	18	invariant	invariant	ADJ
ejpam-3588	424	19	submodules	submodule	NOUN
ejpam-3588	424	20	,	,	PUNCT
ejpam-3588	424	21	j.	j.	PROPN
ejpam-3588	424	22	korean	korean	PROPN
ejpam-3588	424	23	math	math	PROPN
ejpam-3588	424	24	.	.	PUNCT
ejpam-3588	425	1	soc	soc	PROPN
ejpam-3588	425	2	.	.	PUNCT
ejpam-3588	426	1	49(2012	49(2012	NOUN
ejpam-3588	426	2	)	)	PUNCT
ejpam-3588	426	3	,	,	PUNCT
ejpam-3588	427	1	no	no	INTJ
ejpam-3588	427	2	.	.	NOUN
ejpam-3588	427	3	3	3	NUM
ejpam-3588	427	4	,	,	PUNCT
ejpam-3588	427	5	pp	pp	ADJ
ejpam-3588	427	6	.	.	PUNCT
ejpam-3588	428	1	503	503	NUM
ejpam-3588	428	2	-	-	SYM
ejpam-3588	428	3	514	514	NUM
ejpam-3588	428	4	.	.	PUNCT
ejpam-3588	429	1	[	[	X
ejpam-3588	429	2	3	3	X
ejpam-3588	429	3	]	]	X
ejpam-3588	429	4	sh	sh	PROPN
ejpam-3588	429	5	.	.	PROPN
ejpam-3588	429	6	asgari	asgari	PROPN
ejpam-3588	429	7	and	and	CCONJ
ejpam-3588	429	8	a.	a.	NOUN
ejpam-3588	429	9	haghany	haghany	PROPN
ejpam-3588	429	10	and	and	CCONJ
ejpam-3588	429	11	a.	a.	PROPN
ejpam-3588	429	12	r.	r.	PROPN
ejpam-3588	429	13	rezaei	rezaei	PROPN
ejpam-3588	429	14	,	,	PUNCT
ejpam-3588	429	15	(	(	PUNCT
ejpam-3588	429	16	2014	2014	NUM
ejpam-3588	429	17	)	)	PUNCT
ejpam-3588	429	18	.	.	PUNCT
ejpam-3588	430	1	modules	module	NOUN
ejpam-3588	430	2	whose	whose	DET
ejpam-3588	430	3	t	t	NOUN
ejpam-3588	430	4	-	-	PUNCT
ejpam-3588	430	5	closed	close	VERB
ejpam-3588	430	6	submodule	submodule	NOUN
ejpam-3588	430	7	have	have	VERB
ejpam-3588	430	8	a	a	DET
ejpam-3588	430	9	summand	summand	NOUN
ejpam-3588	430	10	as	as	ADP
ejpam-3588	430	11	a	a	DET
ejpam-3588	430	12	complement	complement	NOUN
ejpam-3588	430	13	,	,	PUNCT
ejpam-3588	430	14	comm	comm	NOUN
ejpam-3588	430	15	.	.	PUNCT
ejpam-3588	431	1	algebra	algebra	PROPN
ejpam-3588	431	2	,	,	PUNCT
ejpam-3588	431	3	42:5299	42:5299	NUM
ejpam-3588	431	4	-	-	NOUN
ejpam-3588	431	5	5318	5318	NUM
ejpam-3588	431	6	.	.	PUNCT
ejpam-3588	432	1	[	[	X
ejpam-3588	432	2	4	4	NUM
ejpam-3588	432	3	]	]	PUNCT
ejpam-3588	432	4	a.	a.	NOUN
ejpam-3588	432	5	w.	w.	PROPN
ejpam-3588	432	6	chatters	chatter	VERB
ejpam-3588	432	7	and	and	CCONJ
ejpam-3588	432	8	s.	s.	PROPN
ejpam-3588	432	9	m.	m.	PROPN
ejpam-3588	432	10	kheuri	kheuri	PROPN
ejpam-3588	432	11	,	,	PUNCT
ejpam-3588	432	12	endomorphism	endomorphism	PROPN
ejpam-3588	432	13	rings	ring	NOUN
ejpam-3588	432	14	of	of	ADP
ejpam-3588	432	15	modules	module	NOUN
ejpam-3588	432	16	over	over	ADP
ejpam-3588	432	17	nonsingular	nonsingular	PROPN
ejpam-3588	432	18	cs	cs	PROPN
ejpam-3588	432	19	rings	rings	PROPN
ejpam-3588	432	20	,	,	PUNCT
ejpam-3588	432	21	j.	j.	PROPN
ejpam-3588	432	22	london	london	PROPN
ejpam-3588	432	23	math	math	PROPN
ejpam-3588	432	24	.	.	PUNCT
ejpam-3588	433	1	soc	soc	PROPN
ejpam-3588	433	2	.	.	PUNCT
ejpam-3588	434	1	21(1980	21(1980	NUM
ejpam-3588	434	2	)	)	PUNCT
ejpam-3588	434	3	,	,	PUNCT
ejpam-3588	434	4	434	434	NUM
ejpam-3588	434	5	-	-	SYM
ejpam-3588	434	6	444	444	NUM
ejpam-3588	434	7	.	.	PUNCT
ejpam-3588	435	1	[	[	X
ejpam-3588	435	2	5	5	NUM
ejpam-3588	435	3	]	]	X
ejpam-3588	435	4	n.	n.	NOUN
ejpam-3588	435	5	ding	ding	PROPN
ejpam-3588	435	6	,	,	PUNCT
ejpam-3588	435	7	y.	y.	PROPN
ejpam-3588	435	8	ibrahim	ibrahim	PROPN
ejpam-3588	435	9	,	,	PUNCT
ejpam-3588	435	10	m.	m.	NOUN
ejpam-3588	435	11	yousif	yousif	PROPN
ejpam-3588	435	12	and	and	CCONJ
ejpam-3588	435	13	y.	y.	PROPN
ejpam-3588	435	14	zhou	zhou	PROPN
ejpam-3588	435	15	(	(	PUNCT
ejpam-3588	435	16	2017	2017	NUM
ejpam-3588	435	17	)	)	PUNCT
ejpam-3588	435	18	,	,	PUNCT
ejpam-3588	435	19	c4	c4	NOUN
ejpam-3588	435	20	-	-	PUNCT
ejpam-3588	435	21	modules	module	NOUN
ejpam-3588	435	22	,	,	PUNCT
ejpam-3588	435	23	comm	comm	NOUN
ejpam-3588	435	24	.	.	PUNCT
ejpam-3588	436	1	algebra	algebra	PROPN
ejpam-3588	436	2	45(4):1727	45(4):1727	NUM
ejpam-3588	436	3	-	-	PUNCT
ejpam-3588	436	4	1740	1740	NUM
ejpam-3588	436	5	.	.	PUNCT
ejpam-3588	437	1	[	[	X
ejpam-3588	437	2	6	6	NUM
ejpam-3588	437	3	]	]	X
ejpam-3588	437	4	n.	n.	NOUN
ejpam-3588	437	5	v.	v.	ADP
ejpam-3588	437	6	dung	dung	PROPN
ejpam-3588	437	7	,	,	PUNCT
ejpam-3588	437	8	d.	d.	PROPN
ejpam-3588	437	9	v.	v.	PROPN
ejpam-3588	437	10	huynh	huynh	PROPN
ejpam-3588	437	11	,	,	PUNCT
ejpam-3588	437	12	p.	p.	PROPN
ejpam-3588	437	13	f.	f.	PROPN
ejpam-3588	437	14	smith	smith	PROPN
ejpam-3588	437	15	and	and	CCONJ
ejpam-3588	437	16	r.	r.	PROPN
ejpam-3588	437	17	wisbauer	wisbauer	PROPN
ejpam-3588	437	18	(	(	PUNCT
ejpam-3588	437	19	1994	1994	NUM
ejpam-3588	437	20	)	)	PUNCT
ejpam-3588	437	21	,	,	PUNCT
ejpam-3588	437	22	extending	extend	VERB
ejpam-3588	437	23	modules	module	NOUN
ejpam-3588	437	24	,	,	PUNCT
ejpam-3588	437	25	pitman	pitman	NOUN
ejpam-3588	437	26	research	research	NOUN
ejpam-3588	437	27	notes	note	NOUN
ejpam-3588	437	28	in	in	ADP
ejpam-3588	437	29	mathematics	mathematics	PROPN
ejpam-3588	437	30	313	313	NUM
ejpam-3588	437	31	.	.	PUNCT
ejpam-3588	438	1	harlow	harlow	PROPN
ejpam-3588	438	2	:	:	PUNCT
ejpam-3588	439	1	longman	longman	NOUN
ejpam-3588	439	2	.	.	PUNCT
ejpam-3588	440	1	references	reference	NOUN
ejpam-3588	440	2	169	169	NUM
ejpam-3588	440	3	[	[	X
ejpam-3588	440	4	7	7	NUM
ejpam-3588	440	5	]	]	PUNCT
ejpam-3588	440	6	a.	a.	NOUN
ejpam-3588	440	7	ghorbani	ghorbani	NOUN
ejpam-3588	440	8	and	and	CCONJ
ejpam-3588	440	9	m.	m.	PROPN
ejpam-3588	440	10	r.	r.	PROPN
ejpam-3588	440	11	vedadi	vedadi	PROPN
ejpam-3588	440	12	,	,	PUNCT
ejpam-3588	440	13	epi	epi	NOUN
ejpam-3588	440	14	-	-	ADJ
ejpam-3588	440	15	retractable	retractable	ADJ
ejpam-3588	440	16	modules	module	NOUN
ejpam-3588	440	17	and	and	CCONJ
ejpam-3588	440	18	some	some	DET
ejpam-3588	440	19	application	application	NOUN
ejpam-3588	440	20	,	,	PUNCT
ejpam-3588	440	21	bull	bull	NOUN
ejpam-3588	440	22	.	.	PUNCT
ejpam-3588	441	1	iranian	iranian	ADJ
ejpam-3588	441	2	math	math	PROPN
ejpam-3588	441	3	.	.	PUNCT
ejpam-3588	442	1	soc	soc	PROPN
ejpam-3588	442	2	.	.	PUNCT
ejpam-3588	443	1	35	35	NUM
ejpam-3588	443	2	(	(	PUNCT
ejpam-3588	443	3	2009	2009	NUM
ejpam-3588	443	4	)	)	PUNCT
ejpam-3588	443	5	,	,	PUNCT
ejpam-3588	443	6	no	no	INTJ
ejpam-3588	443	7	.	.	NOUN
ejpam-3588	443	8	1	1	NUM
ejpam-3588	443	9	,	,	PUNCT
ejpam-3588	443	10	155	155	NUM
ejpam-3588	443	11	-	-	SYM
ejpam-3588	443	12	166	166	NUM
ejpam-3588	443	13	.	.	PUNCT
ejpam-3588	444	1	[	[	X
ejpam-3588	444	2	8	8	X
ejpam-3588	444	3	]	]	PUNCT
ejpam-3588	444	4	k.	k.	PROPN
ejpam-3588	444	5	r.	r.	PROPN
ejpam-3588	444	6	goodearl	goodearl	PROPN
ejpam-3588	444	7	,	,	PUNCT
ejpam-3588	444	8	ring	ring	NOUN
ejpam-3588	444	9	theory	theory	NOUN
ejpam-3588	444	10	,	,	PUNCT
ejpam-3588	444	11	nonsingular	nonsingular	ADJ
ejpam-3588	444	12	rings	ring	NOUN
ejpam-3588	444	13	and	and	CCONJ
ejpam-3588	444	14	modules	module	NOUN
ejpam-3588	444	15	.	.	PUNCT
ejpam-3588	445	1	marcel	marcel	PROPN
ejpam-3588	445	2	dekker	dekker	PROPN
ejpam-3588	445	3	,	,	PUNCT
ejpam-3588	445	4	inc	inc	PROPN
ejpam-3588	445	5	new	new	PROPN
ejpam-3588	445	6	york	york	PROPN
ejpam-3588	445	7	and	and	CCONJ
ejpam-3588	445	8	basel	basel	PROPN
ejpam-3588	445	9	(	(	PUNCT
ejpam-3588	445	10	1976	1976	NUM
ejpam-3588	445	11	)	)	PUNCT
ejpam-3588	445	12	.	.	PUNCT
ejpam-3588	446	1	[	[	X
ejpam-3588	446	2	9	9	X
ejpam-3588	446	3	]	]	X
ejpam-3588	446	4	y.	y.	PROPN
ejpam-3588	446	5	ibrahim	ibrahim	PROPN
ejpam-3588	446	6	,	,	PUNCT
ejpam-3588	446	7	x.	x.	PROPN
ejpam-3588	446	8	h.	h.	PROPN
ejpam-3588	446	9	nguyen	nguyen	PROPN
ejpam-3588	446	10	,	,	PUNCT
ejpam-3588	446	11	m.	m.	NOUN
ejpam-3588	446	12	yousif	yousif	PROPN
ejpam-3588	446	13	and	and	CCONJ
ejpam-3588	446	14	y.	y.	PROPN
ejpam-3588	446	15	zhou	zhou	PROPN
ejpam-3588	446	16	,	,	PUNCT
ejpam-3588	446	17	rings	ring	NOUN
ejpam-3588	446	18	whose	whose	DET
ejpam-3588	446	19	cyclics	cyclic	NOUN
ejpam-3588	446	20	are	be	AUX
ejpam-3588	446	21	c3module	c3module	PROPN
ejpam-3588	446	22	,	,	PUNCT
ejpam-3588	446	23	j.	j.	PROPN
ejpam-3588	446	24	algebra	algebra	PROPN
ejpam-3588	446	25	.	.	PUNCT
ejpam-3588	447	1	appl	appl	PROPN
ejpam-3588	447	2	.	.	PROPN
ejpam-3588	448	1	15	15	NUM
ejpam-3588	448	2	(	(	PUNCT
ejpam-3588	448	3	8)	8)	NUM
ejpam-3588	448	4	(	(	PUNCT
ejpam-3588	448	5	2016)(18	2016)(18	NUM
ejpam-3588	448	6	pages	page	NOUN
ejpam-3588	448	7	)	)	PUNCT
ejpam-3588	448	8	.	.	PUNCT
ejpam-3588	449	1	[	[	X
ejpam-3588	449	2	10	10	NUM
ejpam-3588	449	3	]	]	X
ejpam-3588	449	4	d.	d.	PROPN
ejpam-3588	449	5	keskin	keskin	PROPN
ejpam-3588	449	6	tutunu	tutunu	PROPN
ejpam-3588	449	7	and	and	CCONJ
ejpam-3588	449	8	r.	r.	PROPN
ejpam-3588	449	9	tribak	tribak	PROPN
ejpam-3588	449	10	(	(	PUNCT
ejpam-3588	449	11	2010	2010	NUM
ejpam-3588	449	12	)	)	PUNCT
ejpam-3588	449	13	,	,	PUNCT
ejpam-3588	449	14	on	on	ADP
ejpam-3588	449	15	dual	dual	ADJ
ejpam-3588	449	16	baer	baer	PROPN
ejpam-3588	449	17	modules	module	NOUN
ejpam-3588	449	18	52:261	52:261	PROPN
ejpam-3588	449	19	-	-	SYM
ejpam-3588	449	20	269	269	NUM
ejpam-3588	449	21	.	.	PUNCT
ejpam-3588	450	1	[	[	X
ejpam-3588	450	2	11	11	NUM
ejpam-3588	450	3	]	]	PUNCT
ejpam-3588	450	4	t.	t.	PROPN
ejpam-3588	450	5	y.	y.	PROPN
ejpam-3588	450	6	lam	lam	PROPN
ejpam-3588	450	7	,	,	PUNCT
ejpam-3588	450	8	lectures	lecture	VERB
ejpam-3588	450	9	on	on	ADP
ejpam-3588	450	10	modules	module	NOUN
ejpam-3588	450	11	and	and	CCONJ
ejpam-3588	450	12	rings	ring	NOUN
ejpam-3588	450	13	,	,	PUNCT
ejpam-3588	450	14	g.t.m.(189	g.t.m.(189	NOUN
ejpam-3588	450	15	)	)	PUNCT
ejpam-3588	450	16	,	,	PUNCT
ejpam-3588	450	17	springer	springer	NOUN
ejpam-3588	450	18	-	-	PUNCT
ejpam-3588	450	19	verlag	verlag	PROPN
ejpam-3588	450	20	,	,	PUNCT
ejpam-3588	450	21	berlinheidelber	berlinheidelber	PROPN
ejpam-3588	450	22	,	,	PUNCT
ejpam-3588	450	23	new	new	PROPN
ejpam-3588	450	24	york	york	PROPN
ejpam-3588	450	25	,	,	PUNCT
ejpam-3588	450	26	1999	1999	NUM
ejpam-3588	450	27	.	.	PUNCT
ejpam-3588	451	1	[	[	X
ejpam-3588	451	2	12	12	NUM
ejpam-3588	451	3	]	]	X
ejpam-3588	451	4	g.	g.	PROPN
ejpam-3588	451	5	lee	lee	PROPN
ejpam-3588	451	6	,	,	PUNCT
ejpam-3588	451	7	theory	theory	NOUN
ejpam-3588	451	8	of	of	ADP
ejpam-3588	451	9	rickart	rickart	NOUN
ejpam-3588	451	10	modules	module	NOUN
ejpam-3588	451	11	,	,	PUNCT
ejpam-3588	451	12	ph	ph	PROPN
ejpam-3588	451	13	.	.	PROPN
ejpam-3588	451	14	d.	d.	PROPN
ejpam-3588	451	15	thesis	thesis	PROPN
ejpam-3588	451	16	,	,	PUNCT
ejpam-3588	451	17	m.s	m.s	PROPN
ejpam-3588	451	18	.	.	PROPN
ejpam-3588	451	19	,	,	PUNCT
ejpam-3588	451	20	graduate	graduate	NOUN
ejpam-3588	451	21	,	,	PUNCT
ejpam-3588	451	22	school	school	NOUN
ejpam-3588	451	23	of	of	ADP
ejpam-3588	451	24	the	the	DET
ejpam-3588	451	25	ohio	ohio	PROPN
ejpam-3588	451	26	state	state	PROPN
ejpam-3588	451	27	university	university	PROPN
ejpam-3588	451	28	(	(	PUNCT
ejpam-3588	451	29	2010	2010	NUM
ejpam-3588	451	30	)	)	PUNCT
ejpam-3588	451	31	.	.	PUNCT
ejpam-3588	452	1	[	[	X
ejpam-3588	452	2	13	13	NUM
ejpam-3588	452	3	]	]	PUNCT
ejpam-3588	452	4	t.	t.	PROPN
ejpam-3588	452	5	k	k	PROPN
ejpam-3588	452	6	lee	lee	PROPN
ejpam-3588	452	7	and	and	CCONJ
ejpam-3588	452	8	y.	y.	PROPN
ejpam-3588	452	9	zhou	zhou	PROPN
ejpam-3588	452	10	(	(	PUNCT
ejpam-3588	452	11	2013	2013	NUM
ejpam-3588	452	12	)	)	PUNCT
ejpam-3588	452	13	,	,	PUNCT
ejpam-3588	452	14	modules	module	NOUN
ejpam-3588	452	15	which	which	PRON
ejpam-3588	452	16	are	be	AUX
ejpam-3588	452	17	invariant	invariant	ADJ
ejpam-3588	452	18	under	under	ADP
ejpam-3588	452	19	automorphisms	automorphisms	PROPN
ejpam-3588	452	20	injective	injective	ADJ
ejpam-3588	452	21	hulls	hull	NOUN
ejpam-3588	452	22	,	,	PUNCT
ejpam-3588	452	23	j.	j.	PROPN
ejpam-3588	452	24	algebra	algebra	PROPN
ejpam-3588	452	25	appl	appl	PROPN
ejpam-3588	452	26	.	.	PUNCT
ejpam-3588	453	1	12(2):9pp	12(2):9pp	NUM
ejpam-3588	453	2	.	.	PUNCT
ejpam-3588	454	1	[	[	X
ejpam-3588	454	2	14	14	NUM
ejpam-3588	454	3	]	]	PUNCT
ejpam-3588	454	4	s.	s.	PROPN
ejpam-3588	454	5	h.	h.	PROPN
ejpam-3588	454	6	mohamed	mohamed	PROPN
ejpam-3588	454	7	and	and	CCONJ
ejpam-3588	454	8	b.	b.	PROPN
ejpam-3588	454	9	j.	j.	PROPN
ejpam-3588	454	10	muller	muller	PROPN
ejpam-3588	454	11	,	,	PUNCT
ejpam-3588	454	12	continuous	continuous	ADJ
ejpam-3588	454	13	and	and	CCONJ
ejpam-3588	454	14	discrete	discrete	ADJ
ejpam-3588	454	15	modules	module	NOUN
ejpam-3588	454	16	,	,	PUNCT
ejpam-3588	454	17	lms	lm	NOUN
ejpam-3588	454	18	lecture	lecture	NOUN
ejpam-3588	454	19	note	note	NOUN
ejpam-3588	454	20	series	series	NOUN
ejpam-3588	454	21	,	,	PUNCT
ejpam-3588	454	22	147	147	NUM
ejpam-3588	454	23	.	.	PUNCT
ejpam-3588	455	1	cambridge	cambridge	PROPN
ejpam-3588	455	2	university	university	PROPN
ejpam-3588	455	3	press	press	PROPN
ejpam-3588	455	4	,	,	PUNCT
ejpam-3588	455	5	cambridge	cambridge	PROPN
ejpam-3588	455	6	,	,	PUNCT
ejpam-3588	455	7	1990	1990	NUM
ejpam-3588	455	8	.	.	PUNCT
ejpam-3588	456	1	[	[	X
ejpam-3588	456	2	15	15	NUM
ejpam-3588	456	3	]	]	X
ejpam-3588	456	4	b.	b.	PROPN
ejpam-3588	456	5	m.	m.	PROPN
ejpam-3588	456	6	pandeya	pandeya	PROPN
ejpam-3588	456	7	,	,	PUNCT
ejpam-3588	456	8	a.	a.	PROPN
ejpam-3588	456	9	k	k	PROPN
ejpam-3588	456	10	chaturvedi	chaturvedi	PROPN
ejpam-3588	456	11	and	and	CCONJ
ejpam-3588	456	12	a.	a.	PROPN
ejpam-3588	456	13	j.	j.	PROPN
ejpam-3588	456	14	gupta	gupta	PROPN
ejpam-3588	456	15	,	,	PUNCT
ejpam-3588	456	16	applications	application	NOUN
ejpam-3588	456	17	of	of	ADP
ejpam-3588	456	18	epi	epi	NOUN
ejpam-3588	456	19	-	-	ADJ
ejpam-3588	456	20	retractable	retractable	ADJ
ejpam-3588	456	21	modules	module	NOUN
ejpam-3588	456	22	,	,	PUNCT
ejpam-3588	456	23	bull	bull	NOUN
ejpam-3588	456	24	.	.	PUNCT
ejpam-3588	457	1	iranian	iranian	ADJ
ejpam-3588	457	2	math	math	PROPN
ejpam-3588	457	3	.	.	PUNCT
ejpam-3588	458	1	soc	soc	PROPN
ejpam-3588	458	2	.	.	PUNCT
ejpam-3588	459	1	35	35	NUM
ejpam-3588	459	2	no	no	NOUN
ejpam-3588	459	3	.	.	PUNCT
ejpam-3588	460	1	2(2012	2(2012	NUM
ejpam-3588	460	2	)	)	PUNCT
ejpam-3588	460	3	,	,	PUNCT
ejpam-3588	460	4	pp	pp	ADP
ejpam-3588	460	5	469	469	NUM
ejpam-3588	460	6	-	-	NOUN
ejpam-3588	460	7	477	477	NUM
ejpam-3588	460	8	.	.	PUNCT
ejpam-3588	461	1	[	[	X
ejpam-3588	461	2	16	16	NUM
ejpam-3588	461	3	]	]	PUNCT
ejpam-3588	461	4	s.	s.	PROPN
ejpam-3588	461	5	t.	t.	PROPN
ejpam-3588	461	6	rizvi	rizvi	PROPN
ejpam-3588	461	7	and	and	CCONJ
ejpam-3588	461	8	c.	c.	PROPN
ejpam-3588	461	9	s.	s.	PROPN
ejpam-3588	461	10	roman	roman	PROPN
ejpam-3588	461	11	,	,	PUNCT
ejpam-3588	461	12	on	on	ADP
ejpam-3588	461	13	k	k	ADJ
ejpam-3588	461	14	-	-	ADJ
ejpam-3588	461	15	nonsinglar	nonsinglar	ADJ
ejpam-3588	461	16	modules	module	NOUN
ejpam-3588	461	17	and	and	CCONJ
ejpam-3588	461	18	applications	application	NOUN
ejpam-3588	461	19	.	.	PUNCT
ejpam-3588	462	1	comm	comm	NOUN
ejpam-3588	462	2	.	.	PUNCT
ejpam-3588	463	1	algebra	algebra	NOUN
ejpam-3588	463	2	,	,	PUNCT
ejpam-3588	463	3	35	35	NUM
ejpam-3588	463	4	(	(	PUNCT
ejpam-3588	463	5	2007	2007	NUM
ejpam-3588	463	6	):	):	PUNCT
ejpam-3588	463	7	2960	2960	NUM
ejpam-3588	463	8	-	-	SYM
ejpam-3588	463	9	2980	2980	NUM
ejpam-3588	463	10	.	.	PUNCT
ejpam-3588	464	1	[	[	X
ejpam-3588	464	2	17	17	NUM
ejpam-3588	464	3	]	]	PUNCT
ejpam-3588	464	4	s.	s.	PROPN
ejpam-3588	464	5	t.	t.	PROPN
ejpam-3588	464	6	rizvi	rizvi	PROPN
ejpam-3588	464	7	and	and	CCONJ
ejpam-3588	464	8	c.	c.	PROPN
ejpam-3588	464	9	s.	s.	PROPN
ejpam-3588	464	10	roman	roman	PROPN
ejpam-3588	464	11	,	,	PUNCT
ejpam-3588	464	12	on	on	ADP
ejpam-3588	464	13	direct	direct	ADJ
ejpam-3588	464	14	sum	sum	NOUN
ejpam-3588	464	15	of	of	ADP
ejpam-3588	464	16	baer	baer	PROPN
ejpam-3588	464	17	modules	modules	PROPN
ejpam-3588	464	18	,	,	PUNCT
ejpam-3588	464	19	j.	j.	PROPN
ejpam-3588	464	20	algebra	algebra	PROPN
ejpam-3588	464	21	,	,	PUNCT
ejpam-3588	464	22	59	59	NUM
ejpam-3588	464	23	(	(	PUNCT
ejpam-3588	464	24	2009	2009	NUM
ejpam-3588	464	25	):	):	PUNCT
ejpam-3588	464	26	632	632	NUM
ejpam-3588	464	27	-	-	SYM
ejpam-3588	464	28	696	696	NUM
ejpam-3588	464	29	.	.	PUNCT
ejpam-3588	465	1	[	[	X
ejpam-3588	465	2	18	18	NUM
ejpam-3588	465	3	]	]	PUNCT
ejpam-3588	465	4	s.	s.	PROPN
ejpam-3588	465	5	t.	t.	PROPN
ejpam-3588	465	6	rizvi	rizvi	PROPN
ejpam-3588	465	7	and	and	CCONJ
ejpam-3588	465	8	c.	c.	PROPN
ejpam-3588	465	9	s.	s.	PROPN
ejpam-3588	465	10	roman	roman	PROPN
ejpam-3588	465	11	,	,	PUNCT
ejpam-3588	465	12	baer	baer	PROPN
ejpam-3588	465	13	and	and	CCONJ
ejpam-3588	465	14	quasi	quasi	PROPN
ejpam-3588	465	15	-	-	ADJ
ejpam-3588	465	16	baer	baer	ADJ
ejpam-3588	465	17	modules	module	NOUN
ejpam-3588	465	18	.	.	PUNCT
ejpam-3588	466	1	comm	comm	NOUN
ejpam-3588	466	2	.	.	PUNCT
ejpam-3588	467	1	algebra	algebra	NOUN
ejpam-3588	467	2	,	,	PUNCT
ejpam-3588	467	3	32	32	NUM
ejpam-3588	467	4	(	(	PUNCT
ejpam-3588	467	5	2004	2004	NUM
ejpam-3588	467	6	):	):	PUNCT
ejpam-3588	467	7	103	103	NUM
ejpam-3588	467	8	-	-	SYM
ejpam-3588	467	9	123	123	NUM
ejpam-3588	467	10	.	.	PUNCT
ejpam-3588	468	1	[	[	X
ejpam-3588	468	2	19	19	NUM
ejpam-3588	468	3	]	]	PUNCT
ejpam-3588	468	4	p.	p.	PROPN
ejpam-3588	468	5	f.	f.	PROPN
ejpam-3588	468	6	smith	smith	PROPN
ejpam-3588	468	7	,	,	PUNCT
ejpam-3588	468	8	modules	module	NOUN
ejpam-3588	468	9	with	with	ADP
ejpam-3588	468	10	many	many	ADJ
ejpam-3588	468	11	homomorphisms	homomorphism	NOUN
ejpam-3588	468	12	,	,	PUNCT
ejpam-3588	468	13	j.	j.	PROPN
ejpam-3588	468	14	of	of	ADP
ejpam-3588	468	15	pure	pure	ADJ
ejpam-3588	468	16	and	and	CCONJ
ejpam-3588	468	17	appl	appl	NOUN
ejpam-3588	468	18	.	.	PUNCT
ejpam-3588	469	1	algebra	algebra	NOUN
ejpam-3588	469	2	,	,	PUNCT
ejpam-3588	469	3	197	197	NUM
ejpam-3588	469	4	:	:	SYM
ejpam-3588	469	5	305	305	NUM
ejpam-3588	469	6	-	-	SYM
ejpam-3588	469	7	321	321	NUM
ejpam-3588	469	8	(	(	PUNCT
ejpam-3588	469	9	2005	2005	NUM
ejpam-3588	469	10	)	)	PUNCT
ejpam-3588	469	11	.	.	PUNCT
ejpam-3588	470	1	[	[	X
ejpam-3588	470	2	20	20	NUM
ejpam-3588	470	3	]	]	PUNCT
ejpam-3588	470	4	p.	p.	PROPN
ejpam-3588	470	5	f.	f.	PROPN
ejpam-3588	470	6	smith	smith	PROPN
ejpam-3588	470	7	and	and	CCONJ
ejpam-3588	470	8	a.	a.	NOUN
ejpam-3588	470	9	tercan	tercan	PROPN
ejpam-3588	470	10	(	(	PUNCT
ejpam-3588	470	11	1993	1993	NUM
ejpam-3588	470	12	)	)	PUNCT
ejpam-3588	470	13	.	.	PUNCT
ejpam-3588	471	1	generalizations	generalization	NOUN
ejpam-3588	471	2	of	of	ADP
ejpam-3588	471	3	cs	c	NOUN
ejpam-3588	471	4	-	-	NOUN
ejpam-3588	471	5	modules	module	NOUN
ejpam-3588	471	6	.	.	PUNCT
ejpam-3588	472	1	comm	comm	NOUN
ejpam-3588	472	2	.	.	PUNCT
ejpam-3588	473	1	algebra	algebra	PROPN
ejpam-3588	473	2	21:1809	21:1809	PROPN
ejpam-3588	473	3	-	-	NOUN
ejpam-3588	473	4	1847	1847	NUM
ejpam-3588	473	5	.	.	PUNCT
ejpam-3588	474	1	[	[	X
ejpam-3588	474	2	21	21	NUM
ejpam-3588	474	3	]	]	PUNCT
ejpam-3588	474	4	p.	p.	PROPN
ejpam-3588	474	5	f.	f.	PROPN
ejpam-3588	474	6	smith	smith	PROPN
ejpam-3588	474	7	and	and	CCONJ
ejpam-3588	474	8	a.	a.	NOUN
ejpam-3588	474	9	tercan	tercan	PROPN
ejpam-3588	474	10	(	(	PUNCT
ejpam-3588	474	11	1993	1993	NUM
ejpam-3588	474	12	)	)	PUNCT
ejpam-3588	474	13	.	.	PUNCT
ejpam-3588	475	1	direct	direct	ADJ
ejpam-3588	475	2	summands	summand	NOUN
ejpam-3588	475	3	of	of	ADP
ejpam-3588	475	4	modules	module	NOUN
ejpam-3588	475	5	which	which	PRON
ejpam-3588	475	6	satisfy	satisfy	VERB
ejpam-3588	475	7	(	(	PUNCT
ejpam-3588	475	8	c11	c11	NOUN
ejpam-3588	475	9	)	)	PUNCT
ejpam-3588	475	10	.	.	PUNCT
ejpam-3588	476	1	algebra	algebra	PROPN
ejpam-3588	476	2	colloq	colloq	PROPN
ejpam-3588	476	3	.	.	PUNCT
ejpam-3588	477	1	11:231	11:231	NUM
ejpam-3588	477	2	-	-	SYM
ejpam-3588	477	3	237	237	NUM
ejpam-3588	477	4	.	.	PUNCT
ejpam-3588	478	1	[	[	X
ejpam-3588	478	2	22	22	NUM
ejpam-3588	478	3	]	]	PUNCT
ejpam-3588	478	4	r.	r.	PROPN
ejpam-3588	478	5	tribak	tribak	PROPN
ejpam-3588	478	6	(	(	PUNCT
ejpam-3588	478	7	2015	2015	NUM
ejpam-3588	478	8	)	)	PUNCT
ejpam-3588	478	9	,	,	PUNCT
ejpam-3588	478	10	on	on	ADP
ejpam-3588	478	11	weakly	weakly	ADJ
ejpam-3588	478	12	dual	dual	ADJ
ejpam-3588	478	13	rickart	rickart	NOUN
ejpam-3588	478	14	module	module	NOUN
ejpam-3588	478	15	and	and	CCONJ
ejpam-3588	478	16	dual	dual	ADJ
ejpam-3588	478	17	baer	baer	PROPN
ejpam-3588	478	18	modules	module	NOUN
ejpam-3588	478	19	,	,	PUNCT
ejpam-3588	478	20	comm	comm	NOUN
ejpam-3588	478	21	.	.	PUNCT
ejpam-3588	479	1	algebra	algebra	NOUN
ejpam-3588	479	2	43:8	43:8	NUM
ejpam-3588	479	3	,	,	PUNCT
ejpam-3588	479	4	3190	3190	NUM
ejpam-3588	479	5	-	-	SYM
ejpam-3588	479	6	3206	3206	NUM
ejpam-3588	479	7	.	.	PUNCT
ejpam-3588	480	1	[	[	X
ejpam-3588	480	2	23	23	NUM
ejpam-3588	480	3	]	]	PUNCT
ejpam-3588	480	4	m.	m.	PROPN
ejpam-3588	480	5	r.	r.	PROPN
ejpam-3588	480	6	vedadi	vedadi	PROPN
ejpam-3588	480	7	,	,	PUNCT
ejpam-3588	480	8	essentially	essentially	ADV
ejpam-3588	480	9	retractable	retractable	ADJ
ejpam-3588	480	10	modules	module	NOUN
ejpam-3588	480	11	,	,	PUNCT
ejpam-3588	480	12	journal	journal	NOUN
ejpam-3588	480	13	of	of	ADP
ejpam-3588	480	14	science	science	NOUN
ejpam-3588	480	15	,	,	PUNCT
ejpam-3588	480	16	islamic	islamic	PROPN
ejpam-3588	480	17	republic	republic	PROPN
ejpam-3588	480	18	of	of	ADP
ejpam-3588	480	19	iran	iran	PROPN
ejpam-3588	480	20	18(4	18(4	NUM
ejpam-3588	480	21	)	)	PUNCT
ejpam-3588	480	22	,	,	PUNCT
ejpam-3588	480	23	2007	2007	NUM
ejpam-3588	480	24	,	,	PUNCT
ejpam-3588	480	25	355	355	NUM
ejpam-3588	480	26	-	-	SYM
ejpam-3588	480	27	360	360	NUM
ejpam-3588	480	28	.	.	PUNCT
ejpam-3588	481	1	[	[	X
ejpam-3588	481	2	24	24	NUM
ejpam-3588	481	3	]	]	PUNCT
ejpam-3588	481	4	m.	m.	PROPN
ejpam-3588	481	5	f.	f.	PROPN
ejpam-3588	481	6	yousif	yousif	PROPN
ejpam-3588	481	7	,	,	PUNCT
ejpam-3588	481	8	si	si	NOUN
ejpam-3588	481	9	-	-	PUNCT
ejpam-3588	481	10	modules	module	NOUN
ejpam-3588	481	11	,	,	PUNCT
ejpam-3588	481	12	math	math	NOUN
ejpam-3588	481	13	.	.	PUNCT
ejpam-3588	482	1	j.	j.	PROPN
ejpam-3588	482	2	okayama	okayama	PROPN
ejpam-3588	482	3	univ	univ	PROPN
ejpam-3588	482	4	.	.	PROPN
ejpam-3588	483	1	28	28	NUM
ejpam-3588	483	2	,	,	PUNCT
ejpam-3588	483	3	133	133	NUM
ejpam-3588	483	4	-	-	SYM
ejpam-3588	483	5	146(1986	146(1986	NUM
ejpam-3588	483	6	)	)	PUNCT
ejpam-3588	483	7	.	.	PUNCT