id	sid	tid	token	lemma	pos
cana-3972	1	1	communications	communication	NOUN
cana-3972	1	2	on	on	ADP
cana-3972	1	3	applied	apply	VERB
cana-3972	1	4	nonlinear	nonlinear	ADJ
cana-3972	1	5	analysis	analysis	NOUN
cana-3972	1	6	issn	issn	NOUN
cana-3972	1	7	:	:	PUNCT
cana-3972	1	8	1074	1074	NUM
cana-3972	1	9	-	-	PUNCT
cana-3972	1	10	133x	133x	NUM
cana-3972	1	11	vol	vol	NOUN
cana-3972	1	12	32	32	NUM
cana-3972	1	13	no	no	NOUN
cana-3972	1	14	.	.	PUNCT
cana-3972	2	1	9s	9s	NUM
cana-3972	2	2	(	(	PUNCT
cana-3972	2	3	2025	2025	NUM
cana-3972	2	4	)	)	PUNCT
cana-3972	2	5	675	675	NUM
cana-3972	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	2	7	on	on	ADP
cana-3972	2	8	modules	module	NOUN
cana-3972	2	9	for	for	ADP
cana-3972	2	10	which	which	PRON
cana-3972	2	11	every	every	DET
cana-3972	2	12	cosingular	cosingular	PROPN
cana-3972	2	13	veri	veri	PROPN
cana-3972	2	14	fy	fy	PROPN
cana-3972	3	1	the	the	DET
cana-3972	3	2	d4condition	d4condition	NOUN
cana-3972	3	3	papa	papa	PROPN
cana-3972	3	4	cheikhou	cheikhou	PROPN
cana-3972	3	5	diop	diop	PROPN
cana-3972	3	6	1	1	NUM
cana-3972	3	7	and	and	CCONJ
cana-3972	3	8	modou	modou	NOUN
cana-3972	3	9	seye	seye	NOUN
cana-3972	3	10	1	1	NUM
cana-3972	3	11	1département	1département	NUM
cana-3972	3	12	de	de	X
cana-3972	3	13	mathématiques	mathématique	NOUN
cana-3972	3	14	,	,	PUNCT
cana-3972	3	15	ufr	ufr	PROPN
cana-3972	3	16	sciences	sciences	PROPN
cana-3972	3	17	et	et	PROPN
cana-3972	3	18	technologies	technology	NOUN
cana-3972	3	19	,	,	PUNCT
cana-3972	3	20	université	université	PROPN
cana-3972	3	21	iba	iba	PROPN
cana-3972	3	22	der	der	PROPN
cana-3972	3	23	thiam	thiam	PROPN
cana-3972	3	24	de	de	PROPN
cana-3972	3	25	thiès	thiès	PROPN
cana-3972	3	26	,	,	PUNCT
cana-3972	3	27	sénégal	sénégal	ADJ
cana-3972	3	28	email	email	NOUN
cana-3972	3	29	i	i	PROPN
cana-3972	3	30	d	d	PROPN
cana-3972	3	31	:	:	PUNCT
cana-3972	3	32	cheikh.diop@univ-thies.sn	cheikh.diop@univ-thies.sn	NOUN
cana-3972	3	33	,	,	PUNCT
cana-3972	3	34	modou.seye@univ-thies.sn	modou.seye@univ-thies.sn	PROPN
cana-3972	3	35	corresponding	correspond	VERB
cana-3972	3	36	author	author	NOUN
cana-3972	3	37	:	:	PUNCT
cana-3972	3	38	cheikh.diop@univ-thies.sn	cheikh.diop@univ-thies.sn	PROPN
cana-3972	3	39	article	article	NOUN
cana-3972	3	40	history	history	NOUN
cana-3972	3	41	:	:	PUNCT
cana-3972	3	42	received	receive	VERB
cana-3972	3	43	:	:	PUNCT
cana-3972	3	44	14	14	NUM
cana-3972	3	45	-	-	SYM
cana-3972	3	46	11	11	NUM
cana-3972	3	47	-	-	PUNCT
cana-3972	3	48	2024	2024	NUM
cana-3972	3	49	revised	revise	VERB
cana-3972	3	50	:	:	PUNCT
cana-3972	3	51	22	22	NUM
cana-3972	3	52	-	-	SYM
cana-3972	3	53	12	12	NUM
cana-3972	3	54	-	-	PUNCT
cana-3972	3	55	2024	2024	NUM
cana-3972	3	56	accepted	accept	VERB
cana-3972	3	57	:	:	PUNCT
cana-3972	3	58	17	17	NUM
cana-3972	3	59	-	-	SYM
cana-3972	3	60	01	01	NUM
cana-3972	3	61	-	-	PUNCT
cana-3972	3	62	2025	2025	NUM
cana-3972	3	63	abstract	abstract	NOUN
cana-3972	3	64	let	let	VERB
cana-3972	3	65	r	r	PRON
cana-3972	3	66	be	be	AUX
cana-3972	3	67	an	an	DET
cana-3972	3	68	associative	associative	ADJ
cana-3972	3	69	ring	ring	NOUN
cana-3972	3	70	with	with	ADP
cana-3972	3	71	unity	unity	NOUN
cana-3972	3	72	and	and	CCONJ
cana-3972	3	73	m	m	VERB
cana-3972	3	74	an	an	DET
cana-3972	3	75	unital	unital	ADJ
cana-3972	3	76	left	left	ADJ
cana-3972	3	77	r	r	NOUN
cana-3972	3	78	-	-	PUNCT
cana-3972	3	79	module	module	NOUN
cana-3972	3	80	.	.	PUNCT
cana-3972	4	1	in	in	ADP
cana-3972	4	2	this	this	DET
cana-3972	4	3	paper	paper	NOUN
cana-3972	4	4	we	we	PRON
cana-3972	4	5	introduce	introduce	VERB
cana-3972	4	6	d41	d41	NOUN
cana-3972	4	7	-	-	PUNCT
cana-3972	4	8	module	module	NOUN
cana-3972	4	9	which	which	PRON
cana-3972	4	10	is	be	AUX
cana-3972	4	11	a	a	DET
cana-3972	4	12	generalization	generalization	NOUN
cana-3972	4	13	of	of	ADP
cana-3972	4	14	d4	d4	PROPN
cana-3972	4	15	-	-	PUNCT
cana-3972	4	16	module	module	NOUN
cana-3972	4	17	.	.	PUNCT
cana-3972	5	1	a	a	DET
cana-3972	5	2	module	module	NOUN
cana-3972	5	3	m	m	VERB
cana-3972	5	4	is	be	AUX
cana-3972	5	5	called	call	VERB
cana-3972	5	6	d41	d41	PROPN
cana-3972	5	7	if	if	SCONJ
cana-3972	5	8	m	m	NOUN
cana-3972	5	9	=	=	SYM
cana-3972	5	10	n	n	PROPN
cana-3972	5	11	⊕	⊕	PROPN
cana-3972	5	12	k	k	PROPN
cana-3972	5	13	with	with	ADP
cana-3972	5	14	n	n	CCONJ
cana-3972	5	15	,	,	PUNCT
cana-3972	5	16	k	k	PROPN
cana-3972	5	17	≤	≤	PROPN
cana-3972	5	18	m	m	PROPN
cana-3972	5	19	,	,	PUNCT
cana-3972	5	20	k	k	PROPN
cana-3972	5	21	is	be	AUX
cana-3972	5	22	cosingular	cosingular	ADJ
cana-3972	5	23	and	and	CCONJ
cana-3972	5	24	f	f	PROPN
cana-3972	5	25	∶	∶	NOUN
cana-3972	5	26	n	n	PROPN
cana-3972	5	27	→	→	SYM
cana-3972	5	28	k	k	X
cana-3972	5	29	is	be	AUX
cana-3972	5	30	an	an	DET
cana-3972	5	31	epimorphism	epimorphism	NOUN
cana-3972	5	32	,	,	PUNCT
cana-3972	5	33	then	then	ADV
cana-3972	5	34	ker(𝑓	ker(𝑓	PROPN
cana-3972	5	35	)	)	PUNCT
cana-3972	5	36	is	be	AUX
cana-3972	5	37	a	a	DET
cana-3972	5	38	direct	direct	ADJ
cana-3972	5	39	summand	summand	NOUN
cana-3972	5	40	of	of	ADP
cana-3972	5	41	n	n	PROPN
cana-3972	5	42	.	.	PUNCT
cana-3972	6	1	some	some	DET
cana-3972	6	2	basic	basic	ADJ
cana-3972	6	3	properties	property	NOUN
cana-3972	6	4	of	of	ADP
cana-3972	6	5	these	these	DET
cana-3972	6	6	modules	module	NOUN
cana-3972	6	7	are	be	AUX
cana-3972	6	8	investigated	investigate	VERB
cana-3972	6	9	.	.	PUNCT
cana-3972	7	1	i	i	PRON
cana-3972	7	2	t	t	PROPN
cana-3972	7	3	is	be	AUX
cana-3972	7	4	shown	show	VERB
cana-3972	7	5	that	that	SCONJ
cana-3972	7	6	the	the	DET
cana-3972	7	7	class	class	NOUN
cana-3972	7	8	of	of	ADP
cana-3972	7	9	rings	ring	NOUN
cana-3972	7	10	r	r	NOUN
cana-3972	7	11	over	over	ADP
cana-3972	7	12	which	which	PRON
cana-3972	7	13	a	a	DET
cana-3972	7	14	d41	d41	NOUN
cana-3972	7	15	-	-	PUNCT
cana-3972	7	16	module	module	NOUN
cana-3972	7	17	is	be	AUX
cana-3972	7	18	a	a	DET
cana-3972	7	19	d4module	d4module	NOUN
cana-3972	7	20	is	be	AUX
cana-3972	7	21	exactly	exactly	ADV
cana-3972	7	22	that	that	PRON
cana-3972	7	23	of	of	ADP
cana-3972	7	24	cosp	cosp	PROPN
cana-3972	7	25	-rings	-ring	NOUN
cana-3972	7	26	.	.	PUNCT
cana-3972	8	1	also	also	ADV
cana-3972	8	2	,	,	PUNCT
cana-3972	8	3	we	we	PRON
cana-3972	8	4	study	study	VERB
cana-3972	8	5	the	the	DET
cana-3972	8	6	relations	relation	NOUN
cana-3972	8	7	between	between	ADP
cana-3972	8	8	d41	d41	NOUN
cana-3972	8	9	-	-	PUNCT
cana-3972	8	10	module	module	NOUN
cana-3972	8	11	and	and	CCONJ
cana-3972	8	12	other	other	ADJ
cana-3972	8	13	related	related	ADJ
cana-3972	8	14	modules	module	NOUN
cana-3972	8	15	.	.	PUNCT
cana-3972	9	1	2010	2010	NUM
cana-3972	9	2	mathematics	mathematic	NOUN
cana-3972	9	3	subject	subject	NOUN
cana-3972	9	4	classifications	classification	NOUN
cana-3972	9	5	:	:	PUNCT
cana-3972	9	6	16d10	16d10	NUM
cana-3972	9	7	,	,	PUNCT
cana-3972	9	8	16d40	16d40	NUM
cana-3972	9	9	,	,	PUNCT
cana-3972	9	10	16d60	16d60	NUM
cana-3972	9	11	,	,	PUNCT
cana-3972	9	12	16l30	16l30	NUM
cana-3972	9	13	.	.	PUNCT
cana-3972	10	1	keywords	keyword	NOUN
cana-3972	10	2	:	:	PUNCT
cana-3972	10	3	small	small	ADJ
cana-3972	10	4	submodules	submodule	NOUN
cana-3972	10	5	,	,	PUNCT
cana-3972	10	6	cosingular	cosingular	ADJ
cana-3972	10	7	modules	module	NOUN
cana-3972	10	8	,	,	PUNCT
cana-3972	10	9	projectivemodules	projectivemodule	NOUN
cana-3972	10	10	,	,	PUNCT
cana-3972	10	11	d4	d4	NOUN
cana-3972	10	12	-	-	PUNCT
cana-3972	10	13	modules	module	NOUN
cana-3972	10	14	,	,	PUNCT
cana-3972	10	15	d41modules	d41module	NOUN
cana-3972	10	16	.	.	PUNCT
cana-3972	11	1	1	1	X
cana-3972	11	2	.	.	X
cana-3972	11	3	introduction	introduction	NOUN
cana-3972	11	4	in	in	ADP
cana-3972	11	5	this	this	DET
cana-3972	11	6	paper	paper	NOUN
cana-3972	11	7	,	,	PUNCT
cana-3972	11	8	we	we	PRON
cana-3972	11	9	focus	focus	VERB
cana-3972	11	10	on	on	ADP
cana-3972	11	11	rings	ring	NOUN
cana-3972	11	12	that	that	PRON
cana-3972	11	13	are	be	AUX
cana-3972	11	14	associative	associative	ADJ
cana-3972	11	15	and	and	CCONJ
cana-3972	11	16	unitary	unitary	ADJ
cana-3972	11	17	,	,	PUNCT
cana-3972	11	18	and	and	CCONJ
cana-3972	11	19	we	we	PRON
cana-3972	11	20	consider	consider	VERB
cana-3972	11	21	all	all	DET
cana-3972	11	22	modules	module	NOUN
cana-3972	11	23	as	as	ADP
cana-3972	11	24	left	leave	VERB
cana-3972	11	25	modules	module	NOUN
cana-3972	11	26	,	,	PUNCT
cana-3972	11	27	unless	unless	SCONJ
cana-3972	11	28	otherwise	otherwise	ADV
cana-3972	11	29	specified	specify	VERB
cana-3972	11	30	.	.	PUNCT
cana-3972	12	1	we	we	PRON
cana-3972	12	2	will	will	AUX
cana-3972	12	3	introduce	introduce	VERB
cana-3972	12	4	some	some	DET
cana-3972	12	5	important	important	ADJ
cana-3972	12	6	notations	notation	NOUN
cana-3972	12	7	.	.	PUNCT
cana-3972	13	1	let	let	VERB
cana-3972	13	2	m	m	PRON
cana-3972	13	3	be	be	AUX
cana-3972	13	4	a	a	DET
cana-3972	13	5	module	module	NOUN
cana-3972	13	6	;	;	PUNCT
cana-3972	13	7	we	we	PRON
cana-3972	13	8	denote	denote	VERB
cana-3972	13	9	n	n	ADV
cana-3972	13	10	≤	≤	NOUN
cana-3972	13	11	m	m	VERB
cana-3972	13	12	to	to	PART
cana-3972	13	13	indicate	indicate	VERB
cana-3972	13	14	that	that	SCONJ
cana-3972	13	15	n	n	PRON
cana-3972	13	16	is	be	AUX
cana-3972	13	17	a	a	DET
cana-3972	13	18	submodule	submodule	NOUN
cana-3972	13	19	of	of	ADP
cana-3972	13	20	m	m	PROPN
cana-3972	13	21	.	.	PUNCT
cana-3972	14	1	the	the	DET
cana-3972	14	2	notation	notation	NOUN
cana-3972	14	3	n	n	CCONJ
cana-3972	14	4	≤	≤	PROPN
cana-3972	14	5	⊕	⊕	PROPN
cana-3972	14	6	m	m	NOUN
cana-3972	14	7	signifies	signifie	NOUN
cana-3972	14	8	that	that	SCONJ
cana-3972	14	9	n	n	VERB
cana-3972	14	10	is	be	AUX
cana-3972	14	11	a	a	DET
cana-3972	14	12	direct	direct	ADJ
cana-3972	14	13	summand	summand	NOUN
cana-3972	14	14	of	of	ADP
cana-3972	14	15	m	m	PROPN
cana-3972	14	16	.	.	PUNCT
cana-3972	15	1	for	for	ADP
cana-3972	15	2	a	a	DET
cana-3972	15	3	module	module	NOUN
cana-3972	15	4	m	m	NOUN
cana-3972	15	5	,	,	PUNCT
cana-3972	15	6	we	we	PRON
cana-3972	15	7	denote	denote	VERB
cana-3972	15	8	its	its	PRON
cana-3972	15	9	injective	injective	ADJ
cana-3972	15	10	envelope	envelope	NOUN
cana-3972	15	11	as	as	ADP
cana-3972	15	12	e(m	e(m	PROPN
cana-3972	15	13	)	)	PUNCT
cana-3972	15	14	.	.	PUNCT
cana-3972	16	1	the	the	DET
cana-3972	16	2	socle	socle	NOUN
cana-3972	16	3	and	and	CCONJ
cana-3972	16	4	radical	radical	NOUN
cana-3972	16	5	of	of	ADP
cana-3972	16	6	the	the	DET
cana-3972	16	7	module	module	NOUN
cana-3972	16	8	m	m	NOUN
cana-3972	16	9	are	be	AUX
cana-3972	16	10	represented	represent	VERB
cana-3972	16	11	by	by	ADP
cana-3972	16	12	soc(m	soc(m	PROPN
cana-3972	16	13	)	)	PUNCT
cana-3972	16	14	and	and	CCONJ
cana-3972	16	15	rad(m	rad(m	NUM
cana-3972	16	16	)	)	PUNCT
cana-3972	16	17	,	,	PUNCT
cana-3972	16	18	respectively	respectively	ADV
cana-3972	16	19	.	.	PUNCT
cana-3972	17	1	we	we	PRON
cana-3972	17	2	use	use	VERB
cana-3972	17	3	n	n	PRON
cana-3972	17	4	⊆	⊆	NUM
cana-3972	17	5	m	m	NOUN
cana-3972	17	6	to	to	PART
cana-3972	17	7	indicate	indicate	VERB
cana-3972	17	8	that	that	SCONJ
cana-3972	17	9	n	n	PRON
cana-3972	17	10	is	be	AUX
cana-3972	17	11	a	a	DET
cana-3972	17	12	subset	subset	NOUN
cana-3972	17	13	of	of	ADP
cana-3972	17	14	m	m	PROPN
cana-3972	17	15	.	.	PUNCT
cana-3972	18	1	we	we	PRON
cana-3972	18	2	say	say	VERB
cana-3972	18	3	that	that	SCONJ
cana-3972	18	4	m	m	PROPN
cana-3972	18	5	satisfies	satisfy	VERB
cana-3972	18	6	the	the	DET
cana-3972	18	7	following	follow	VERB
cana-3972	18	8	conditions	condition	NOUN
cana-3972	18	9	:	:	PUNCT
cana-3972	18	10	(	(	PUNCT
cana-3972	18	11	d1	d1	NOUN
cana-3972	18	12	-	-	PUNCT
cana-3972	18	13	condition	condition	NOUN
cana-3972	18	14	):	):	PUNCT
cana-3972	18	15	for	for	ADP
cana-3972	18	16	every	every	DET
cana-3972	18	17	submodule	submodule	NOUN
cana-3972	18	18	n	n	CCONJ
cana-3972	18	19	≤	≤	NOUN
cana-3972	18	20	m	m	VERB
cana-3972	18	21	,	,	PUNCT
cana-3972	18	22	there	there	PRON
cana-3972	18	23	exists	exist	VERB
cana-3972	18	24	a	a	DET
cana-3972	18	25	decomposition	decomposition	NOUN
cana-3972	18	26	m	m	NOUN
cana-3972	18	27	=	=	SYM
cana-3972	18	28	m1	m1	PROPN
cana-3972	18	29	⊕	⊕	PROPN
cana-3972	18	30	m2	m2	PROPN
cana-3972	18	31	such	such	ADJ
cana-3972	18	32	that	that	DET
cana-3972	18	33	m1	m1	PROPN
cana-3972	18	34	≤	≤	NOUN
cana-3972	18	35	n	n	CCONJ
cana-3972	18	36	and	and	CCONJ
cana-3972	18	37	n	n	PROPN
cana-3972	18	38	∩	∩	NOUN
cana-3972	18	39	m2	m2	PROPN
cana-3972	18	40	is	be	AUX
cana-3972	18	41	small	small	ADJ
cana-3972	18	42	in	in	ADP
cana-3972	18	43	m	m	PROPN
cana-3972	18	44	.	.	PUNCT
cana-3972	19	1	(	(	PUNCT
cana-3972	19	2	d2	d2	NOUN
cana-3972	19	3	-	-	PUNCT
cana-3972	19	4	condition	condition	NOUN
cana-3972	19	5	):	):	PUNCT
cana-3972	19	6	for	for	ADP
cana-3972	19	7	every	every	DET
cana-3972	19	8	submodule	submodule	NOUN
cana-3972	19	9	n	n	CCONJ
cana-3972	19	10	≤	≤	NOUN
cana-3972	19	11	m	m	VERB
cana-3972	19	12	such	such	ADJ
cana-3972	19	13	that	that	SCONJ
cana-3972	19	14	m	m	NOUN
cana-3972	19	15	/	/	SYM
cana-3972	19	16	n	n	PROPN
cana-3972	19	17	is	be	AUX
cana-3972	19	18	isomorphic	isomorphic	ADJ
cana-3972	19	19	to	to	ADP
cana-3972	19	20	a	a	DET
cana-3972	19	21	direct	direct	ADJ
cana-3972	19	22	summand	summand	NOUN
cana-3972	19	23	of	of	ADP
cana-3972	19	24	m	m	PROPN
cana-3972	19	25	,	,	PUNCT
cana-3972	19	26	n	n	PRON
cana-3972	19	27	is	be	AUX
cana-3972	19	28	also	also	ADV
cana-3972	19	29	a	a	DET
cana-3972	19	30	direct	direct	ADJ
cana-3972	19	31	summand	summand	NOUN
cana-3972	19	32	of	of	ADP
cana-3972	19	33	m	m	PROPN
cana-3972	19	34	.	.	PUNCT
cana-3972	20	1	(	(	PUNCT
cana-3972	20	2	d3	d3	NOUN
cana-3972	20	3	-	-	PUNCT
cana-3972	20	4	condition	condition	NOUN
cana-3972	20	5	):	):	PUNCT
cana-3972	20	6	if	if	SCONJ
cana-3972	20	7	m1	m1	PROPN
cana-3972	20	8	and	and	CCONJ
cana-3972	20	9	m2	m2	PROPN
cana-3972	20	10	are	be	AUX
cana-3972	20	11	direct	direct	ADJ
cana-3972	20	12	summands	summand	NOUN
cana-3972	20	13	of	of	ADP
cana-3972	20	14	m	m	PROPN
cana-3972	20	15	and	and	CCONJ
cana-3972	20	16	m	m	PROPN
cana-3972	20	17	=	=	ADJ
cana-3972	20	18	m1	m1	PROPN
cana-3972	20	19	+	+	CCONJ
cana-3972	20	20	m2	m2	PROPN
cana-3972	20	21	,	,	PUNCT
cana-3972	20	22	then	then	ADV
cana-3972	20	23	m1	m1	PROPN
cana-3972	20	24	∩	∩	NOUN
cana-3972	20	25	m2	m2	PROPN
cana-3972	20	26	is	be	AUX
cana-3972	20	27	a	a	DET
cana-3972	20	28	direct	direct	ADJ
cana-3972	20	29	summand	summand	NOUN
cana-3972	20	30	of	of	ADP
cana-3972	20	31	m	m	PROPN
cana-3972	20	32	.	.	PUNCT
cana-3972	21	1	(	(	PUNCT
cana-3972	21	2	d4	d4	NOUN
cana-3972	21	3	-	-	PUNCT
cana-3972	21	4	condition	condition	NOUN
cana-3972	21	5	):	):	PUNCT
cana-3972	21	6	if	if	SCONJ
cana-3972	21	7	𝑀	𝑀	PROPN
cana-3972	21	8	=	=	SYM
cana-3972	21	9	𝑁	𝑁	PROPN
cana-3972	21	10	⊕	⊕	PROPN
cana-3972	21	11	𝐾	𝐾	PROPN
cana-3972	21	12	with	with	ADP
cana-3972	21	13	𝑁	𝑁	PROPN
cana-3972	21	14	,	,	PUNCT
cana-3972	21	15	𝐾	𝐾	PROPN
cana-3972	21	16	≤	≤	PROPN
cana-3972	21	17	𝑀	𝑀	PROPN
cana-3972	21	18	and	and	CCONJ
cana-3972	21	19	𝑓	𝑓	DET
cana-3972	21	20	∶	∶	NOUN
cana-3972	21	21	𝑁	𝑁	PROPN
cana-3972	21	22	→	→	SYM
cana-3972	21	23	𝐾	𝐾	PROPN
cana-3972	21	24	is	be	AUX
cana-3972	21	25	an	an	DET
cana-3972	21	26	epimorphism	epimorphism	NOUN
cana-3972	21	27	,	,	PUNCT
cana-3972	21	28	then	then	ADV
cana-3972	21	29	ker(𝑓	ker(𝑓	PROPN
cana-3972	21	30	)	)	PUNCT
cana-3972	21	31	≤⊕	≤⊕	AUX
cana-3972	22	1	𝑁.	𝑁.	PROPN
cana-3972	22	2	a	a	DET
cana-3972	22	3	module	module	NOUN
cana-3972	22	4	that	that	PRON
cana-3972	22	5	satisfies	satisfy	VERB
cana-3972	22	6	the	the	DET
cana-3972	22	7	di	di	NOUN
cana-3972	22	8	-	-	NOUN
cana-3972	22	9	condition	condition	NOUN
cana-3972	22	10	will	will	AUX
cana-3972	22	11	be	be	AUX
cana-3972	22	12	referred	refer	VERB
cana-3972	22	13	to	to	ADP
cana-3972	22	14	as	as	ADP
cana-3972	22	15	a	a	DET
cana-3972	22	16	di	di	NOUN
cana-3972	22	17	-	-	NOUN
cana-3972	22	18	module	module	NOUN
cana-3972	22	19	.	.	PUNCT
cana-3972	23	1	every	every	DET
cana-3972	23	2	quasi	quasi	ADJ
cana-3972	23	3	-	-	ADJ
cana-3972	23	4	projective	projective	ADJ
cana-3972	23	5	left	leave	VERB
cana-3972	23	6	r	r	NOUN
cana-3972	23	7	-	-	PUNCT
cana-3972	23	8	module	module	NOUN
cana-3972	23	9	is	be	AUX
cana-3972	23	10	a	a	DET
cana-3972	23	11	d2	d2	NOUN
cana-3972	23	12	-	-	PUNCT
cana-3972	23	13	module	module	NOUN
cana-3972	23	14	,	,	PUNCT
cana-3972	23	15	every	every	DET
cana-3972	23	16	d2	d2	NOUN
cana-3972	23	17	-	-	PUNCT
cana-3972	23	18	module	module	NOUN
cana-3972	23	19	is	be	AUX
cana-3972	23	20	a	a	DET
cana-3972	23	21	d3	d3	NOUN
cana-3972	23	22	-	-	PUNCT
cana-3972	23	23	module	module	NOUN
cana-3972	23	24	,	,	PUNCT
cana-3972	23	25	and	and	CCONJ
cana-3972	23	26	every	every	DET
cana-3972	23	27	d3	d3	PROPN
cana-3972	23	28	-	-	PUNCT
cana-3972	23	29	module	module	NOUN
cana-3972	23	30	is	be	AUX
cana-3972	23	31	a	a	DET
cana-3972	23	32	d4module	d4module	NOUN
cana-3972	23	33	.	.	PUNCT
cana-3972	24	1	however	however	ADV
cana-3972	24	2	,	,	PUNCT
cana-3972	24	3	there	there	PRON
cana-3972	24	4	exist	exist	VERB
cana-3972	24	5	examples	example	NOUN
cana-3972	24	6	of	of	ADP
cana-3972	24	7	d3	d3	PROPN
cana-3972	24	8	-	-	PUNCT
cana-3972	24	9	modules	module	NOUN
cana-3972	24	10	that	that	PRON
cana-3972	24	11	are	be	AUX
cana-3972	24	12	not	not	PART
cana-3972	24	13	d2	d2	NOUN
cana-3972	24	14	-	-	PUNCT
cana-3972	24	15	modules	module	NOUN
cana-3972	24	16	,	,	PUNCT
cana-3972	24	17	as	as	ADV
cana-3972	24	18	well	well	ADV
cana-3972	24	19	as	as	ADP
cana-3972	24	20	d2	d2	NOUN
cana-3972	24	21	-	-	PUNCT
cana-3972	24	22	modules	module	NOUN
cana-3972	24	23	that	that	PRON
cana-3972	24	24	are	be	AUX
cana-3972	24	25	not	not	PART
cana-3972	24	26	quasi	quasi	ADJ
cana-3972	24	27	-	-	NOUN
cana-3972	24	28	projective	projective	ADJ
cana-3972	24	29	.	.	PUNCT
cana-3972	25	1	d1	d1	NOUN
cana-3972	25	2	-	-	PUNCT
cana-3972	25	3	modules	module	NOUN
cana-3972	25	4	are	be	AUX
cana-3972	25	5	referred	refer	VERB
cana-3972	25	6	to	to	ADP
cana-3972	25	7	as	as	ADP
cana-3972	25	8	lifting	lift	VERB
cana-3972	25	9	modules	module	NOUN
cana-3972	25	10	by	by	ADP
cana-3972	25	11	oshiro	oshiro	PROPN
cana-3972	25	12	[	[	X
cana-3972	25	13	16	16	NUM
cana-3972	25	14	]	]	PUNCT
cana-3972	25	15	,	,	PUNCT
cana-3972	25	16	d2	d2	NOUN
cana-3972	25	17	-	-	PUNCT
cana-3972	25	18	modules	module	NOUN
cana-3972	25	19	are	be	AUX
cana-3972	25	20	called	call	VERB
cana-3972	25	21	direct	direct	ADJ
cana-3972	25	22	-	-	PUNCT
cana-3972	25	23	projective	projective	NOUN
cana-3972	25	24	by	by	ADP
cana-3972	25	25	w.k	w.k	PROPN
cana-3972	25	26	.	.	PROPN
cana-3972	25	27	nicholson	nicholson	PROPN
cana-3972	25	28	in	in	ADP
cana-3972	25	29	[	[	X
cana-3972	25	30	15	15	NUM
cana-3972	25	31	]	]	PUNCT
cana-3972	25	32	,	,	PUNCT
cana-3972	25	33	and	and	CCONJ
cana-3972	25	34	d3	d3	PROPN
cana-3972	25	35	-	-	PUNCT
cana-3972	25	36	modules	module	NOUN
cana-3972	25	37	are	be	AUX
cana-3972	25	38	termed	term	VERB
cana-3972	25	39	∩-direct	∩-direct	ADJ
cana-3972	25	40	-	-	PUNCT
cana-3972	25	41	projective	projective	NOUN
cana-3972	25	42	in	in	ADP
cana-3972	25	43	[	[	X
cana-3972	25	44	2	2	NUM
cana-3972	25	45	]	]	PUNCT
cana-3972	25	46	.	.	PUNCT
cana-3972	26	1	mailto:cheikh.diop@univ-thies.sn	mailto:cheikh.diop@univ-thies.sn	NOUN
cana-3972	26	2	mailto:modou.seye@univ-thies.sn	mailto:modou.seye@univ-thies.sn	PROPN
cana-3972	26	3	mailto:cheikh.diop@univ-thies.sn	mailto:cheikh.diop@univ-thies.sn	PROPN
cana-3972	26	4	communications	communication	NOUN
cana-3972	26	5	on	on	ADP
cana-3972	26	6	applied	apply	VERB
cana-3972	26	7	nonlinear	nonlinear	ADJ
cana-3972	26	8	analysis	analysis	NOUN
cana-3972	26	9	issn	issn	NOUN
cana-3972	26	10	:	:	PUNCT
cana-3972	26	11	1074	1074	NUM
cana-3972	26	12	-	-	PUNCT
cana-3972	26	13	133x	133x	NUM
cana-3972	26	14	vol	vol	NOUN
cana-3972	26	15	32	32	NUM
cana-3972	26	16	no	no	NOUN
cana-3972	26	17	.	.	PUNCT
cana-3972	27	1	9s	9s	NUM
cana-3972	27	2	(	(	PUNCT
cana-3972	27	3	2025	2025	NUM
cana-3972	27	4	)	)	PUNCT
cana-3972	27	5	676	676	NUM
cana-3972	27	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	27	7	in	in	ADP
cana-3972	27	8	2016	2016	NUM
cana-3972	27	9	,	,	PUNCT
cana-3972	27	10	n.	n.	NOUN
cana-3972	27	11	ding	ding	PROPN
cana-3972	27	12	,	,	PUNCT
cana-3972	27	13	y.	y.	PROPN
cana-3972	27	14	ibrahim	ibrahim	PROPN
cana-3972	27	15	,	,	PUNCT
cana-3972	27	16	m.	m.	NOUN
cana-3972	27	17	yousif	yousif	PROPN
cana-3972	27	18	,	,	PUNCT
cana-3972	27	19	and	and	CCONJ
cana-3972	27	20	y.	y.	NOUN
cana-3972	27	21	zhou	zhou	PROPN
cana-3972	28	1	[	[	X
cana-3972	28	2	6	6	NUM
cana-3972	28	3	]	]	PUNCT
cana-3972	28	4	defined	define	VERB
cana-3972	28	5	a	a	DET
cana-3972	28	6	module	module	NOUN
cana-3972	28	7	as	as	ADP
cana-3972	28	8	a	a	DET
cana-3972	28	9	c4	c4	NOUN
cana-3972	28	10	-	-	PUNCT
cana-3972	28	11	module	module	NOUN
cana-3972	28	12	if	if	SCONJ
cana-3972	28	13	,	,	PUNCT
cana-3972	28	14	for	for	ADP
cana-3972	28	15	any	any	DET
cana-3972	28	16	submodules	submodule	NOUN
cana-3972	28	17	a	a	PRON
cana-3972	28	18	and	and	CCONJ
cana-3972	28	19	b	b	NOUN
cana-3972	28	20	of	of	ADP
cana-3972	28	21	m	m	PRON
cana-3972	28	22	,	,	PUNCT
cana-3972	28	23	when	when	SCONJ
cana-3972	28	24	𝑀	𝑀	PROPN
cana-3972	28	25	=	=	PROPN
cana-3972	28	26	𝐴	𝐴	PROPN
cana-3972	28	27	⊕	⊕	PROPN
cana-3972	28	28	𝐵	𝐵	PROPN
cana-3972	28	29	and	and	CCONJ
cana-3972	28	30	𝑓	𝑓	DET
cana-3972	28	31	∶	∶	NOUN
cana-3972	28	32	𝐴	𝐴	PROPN
cana-3972	28	33	→	→	PUNCT
cana-3972	28	34	𝐵	𝐵	PROPN
cana-3972	28	35	is	be	AUX
cana-3972	28	36	a	a	DET
cana-3972	28	37	homomorphism	homomorphism	NOUN
cana-3972	28	38	with	with	ADP
cana-3972	28	39	ker(𝑓	ker(𝑓	PROPN
cana-3972	28	40	)	)	PUNCT
cana-3972	28	41	≤⊕	≤⊕	NUM
cana-3972	28	42	𝐴	𝐴	PROPN
cana-3972	28	43	,	,	PUNCT
cana-3972	28	44	then	then	ADV
cana-3972	28	45	𝐼𝑚(𝑓	𝐼𝑚(𝑓	NOUN
cana-3972	28	46	)	)	PUNCT
cana-3972	28	47	≤⊕	≤⊕	PUNCT
cana-3972	28	48	𝐵.	𝐵.	PROPN
cana-3972	28	49	d4	d4	PROPN
cana-3972	28	50	-	-	PUNCT
cana-3972	28	51	modules	module	NOUN
cana-3972	28	52	were	be	AUX
cana-3972	28	53	studied	study	VERB
cana-3972	28	54	by	by	ADP
cana-3972	28	55	n.	n.	NOUN
cana-3972	28	56	ding	ding	NOUN
cana-3972	28	57	and	and	CCONJ
cana-3972	28	58	others	other	NOUN
cana-3972	28	59	in	in	ADP
cana-3972	28	60	2017	2017	NUM
cana-3972	29	1	[	[	X
cana-3972	29	2	4	4	NUM
cana-3972	29	3	]	]	PUNCT
cana-3972	29	4	.	.	PUNCT
cana-3972	30	1	motivated	motivate	VERB
cana-3972	30	2	by	by	ADP
cana-3972	30	3	these	these	DET
cana-3972	30	4	developments	development	NOUN
cana-3972	30	5	and	and	CCONJ
cana-3972	30	6	the	the	DET
cana-3972	30	7	findings	finding	NOUN
cana-3972	30	8	in	in	ADP
cana-3972	30	9	[	[	X
cana-3972	30	10	4	4	NUM
cana-3972	30	11	]	]	PUNCT
cana-3972	30	12	and	and	CCONJ
cana-3972	30	13	[	[	X
cana-3972	30	14	6	6	NUM
cana-3972	30	15	]	]	PUNCT
cana-3972	30	16	,	,	PUNCT
cana-3972	30	17	we	we	PRON
cana-3972	30	18	introduce	introduce	VERB
cana-3972	30	19	the	the	DET
cana-3972	30	20	concept	concept	NOUN
cana-3972	30	21	of	of	ADP
cana-3972	30	22	cosingular	cosingular	ADJ
cana-3972	30	23	d4	d4	PROPN
cana-3972	30	24	-	-	PUNCT
cana-3972	30	25	modules	module	NOUN
cana-3972	30	26	,	,	PUNCT
cana-3972	30	27	which	which	PRON
cana-3972	30	28	generalizes	generalize	VERB
cana-3972	30	29	d4	d4	NOUN
cana-3972	30	30	-	-	PUNCT
cana-3972	30	31	modules	module	NOUN
cana-3972	30	32	and	and	CCONJ
cana-3972	30	33	a	a	DET
cana-3972	30	34	dual	dual	ADJ
cana-3972	30	35	notion	notion	NOUN
cana-3972	30	36	of	of	ADP
cana-3972	30	37	c41	c41	NOUN
cana-3972	30	38	-	-	PUNCT
cana-3972	30	39	modules	module	NOUN
cana-3972	30	40	[	[	X
cana-3972	30	41	3	3	NUM
cana-3972	30	42	]	]	PUNCT
cana-3972	30	43	.	.	PUNCT
cana-3972	31	1	a	a	DET
cana-3972	31	2	module	module	NOUN
cana-3972	31	3	is	be	AUX
cana-3972	31	4	defined	define	VERB
cana-3972	31	5	as	as	ADP
cana-3972	31	6	a	a	DET
cana-3972	31	7	cosingular	cosingular	ADJ
cana-3972	31	8	d4	d4	NOUN
cana-3972	31	9	-	-	PUNCT
cana-3972	31	10	module	module	NOUN
cana-3972	31	11	if	if	SCONJ
cana-3972	31	12	,	,	PUNCT
cana-3972	31	13	for	for	ADP
cana-3972	31	14	𝑀	𝑀	PROPN
cana-3972	31	15	=	=	SYM
cana-3972	31	16	𝑁	𝑁	PROPN
cana-3972	31	17	⊕	⊕	PROPN
cana-3972	31	18	𝐾	𝐾	PROPN
cana-3972	31	19	with	with	ADP
cana-3972	31	20	𝑁	𝑁	PROPN
cana-3972	31	21	,	,	PUNCT
cana-3972	31	22	𝐾	𝐾	PROPN
cana-3972	31	23	≤	≤	PROPN
cana-3972	31	24	𝑀	𝑀	PROPN
cana-3972	31	25	,	,	PUNCT
cana-3972	31	26	k	k	PROPN
cana-3972	31	27	is	be	AUX
cana-3972	31	28	cosingular	cosingular	ADJ
cana-3972	31	29	and	and	CCONJ
cana-3972	31	30	𝑓	𝑓	DET
cana-3972	31	31	∶	∶	NOUN
cana-3972	31	32	𝑁	𝑁	PROPN
cana-3972	31	33	→	→	SYM
cana-3972	31	34	𝐾	𝐾	PROPN
cana-3972	31	35	as	as	ADP
cana-3972	31	36	an	an	DET
cana-3972	31	37	epimorphism	epimorphism	NOUN
cana-3972	31	38	,	,	PUNCT
cana-3972	31	39	i	i	PRON
cana-3972	31	40	t	t	PROPN
cana-3972	31	41	follows	follow	VERB
cana-3972	31	42	that	that	SCONJ
cana-3972	31	43	ker(𝑓	ker(𝑓	PROPN
cana-3972	31	44	)	)	PUNCT
cana-3972	31	45	≤⊕	≤⊕	AUX
cana-3972	31	46	𝑁.	𝑁.	PROPN
cana-3972	31	47	we	we	PRON
cana-3972	31	48	refer	refer	VERB
cana-3972	31	49	to	to	ADP
cana-3972	31	50	these	these	PRON
cana-3972	31	51	as	as	ADP
cana-3972	31	52	d41	d41	NOUN
cana-3972	31	53	-	-	PUNCT
cana-3972	31	54	modules	module	NOUN
cana-3972	31	55	.	.	PUNCT
cana-3972	32	1	in	in	ADP
cana-3972	32	2	section	section	NOUN
cana-3972	32	3	3	3	NUM
cana-3972	32	4	,	,	PUNCT
cana-3972	32	5	we	we	PRON
cana-3972	32	6	explore	explore	VERB
cana-3972	32	7	some	some	DET
cana-3972	32	8	properties	property	NOUN
cana-3972	32	9	of	of	ADP
cana-3972	32	10	d41	d41	NOUN
cana-3972	32	11	-	-	PUNCT
cana-3972	32	12	modules	module	NOUN
cana-3972	32	13	.	.	PUNCT
cana-3972	33	1	we	we	PRON
cana-3972	33	2	demonstrate	demonstrate	VERB
cana-3972	33	3	that	that	SCONJ
cana-3972	33	4	every	every	DET
cana-3972	33	5	direct	direct	ADJ
cana-3972	33	6	summand	summand	NOUN
cana-3972	33	7	of	of	ADP
cana-3972	33	8	a	a	DET
cana-3972	33	9	d41	d41	NOUN
cana-3972	33	10	-	-	PUNCT
cana-3972	33	11	module	module	NOUN
cana-3972	33	12	also	also	ADV
cana-3972	33	13	inherits	inherit	VERB
cana-3972	33	14	this	this	DET
cana-3972	33	15	property	property	NOUN
cana-3972	33	16	.	.	PUNCT
cana-3972	34	1	we	we	PRON
cana-3972	34	2	also	also	ADV
cana-3972	34	3	show	show	VERB
cana-3972	34	4	that	that	SCONJ
cana-3972	34	5	if	if	SCONJ
cana-3972	34	6	m	m	PROPN
cana-3972	34	7	⊕	⊕	NOUN
cana-3972	34	8	m	m	VERB
cana-3972	34	9	is	be	AUX
cana-3972	34	10	a	a	DET
cana-3972	34	11	d41	d41	NOUN
cana-3972	34	12	,	,	PUNCT
cana-3972	34	13	then	then	ADV
cana-3972	34	14	m	m	VERB
cana-3972	34	15	is	be	AUX
cana-3972	34	16	a	a	DET
cana-3972	34	17	d2	d2	NOUN
cana-3972	34	18	.	.	PUNCT
cana-3972	35	1	it	it	PRON
cana-3972	35	2	is	be	AUX
cana-3972	35	3	proved	prove	VERB
cana-3972	35	4	in	in	ADP
cana-3972	35	5	this	this	DET
cana-3972	35	6	section	section	NOUN
cana-3972	35	7	that	that	SCONJ
cana-3972	35	8	any	any	DET
cana-3972	35	9	d41	d41	NOUN
cana-3972	35	10	-	-	PUNCT
cana-3972	35	11	module	module	NOUN
cana-3972	35	12	satisfying	satisfy	VERB
cana-3972	35	13	the	the	DET
cana-3972	35	14	ssp	ssp	ADJ
cana-3972	35	15	satisfies	satisfie	NOUN
cana-3972	35	16	also	also	ADV
cana-3972	35	17	the	the	DET
cana-3972	35	18	si	si	PROPN
cana-3972	35	19	p	p	PROPN
cana-3972	35	20	.	.	PUNCT
cana-3972	36	1	additionally	additionally	ADV
cana-3972	36	2	,	,	PUNCT
cana-3972	36	3	we	we	PRON
cana-3972	36	4	show	show	VERB
cana-3972	36	5	that	that	SCONJ
cana-3972	36	6	if	if	SCONJ
cana-3972	36	7	m	m	NOUN
cana-3972	36	8	is	be	AUX
cana-3972	36	9	a	a	DET
cana-3972	36	10	cosingular	cosingular	ADJ
cana-3972	36	11	d41	d41	NOUN
cana-3972	36	12	-	-	PUNCT
cana-3972	36	13	module	module	NOUN
cana-3972	36	14	,	,	PUNCT
cana-3972	36	15	then	then	ADV
cana-3972	36	16	m	m	PROPN
cana-3972	36	17	/	/	SYM
cana-3972	36	18	n	n	PROPN
cana-3972	36	19	is	be	AUX
cana-3972	36	20	also	also	ADV
cana-3972	36	21	a	a	DET
cana-3972	36	22	d41	d41	NOUN
cana-3972	36	23	-	-	PUNCT
cana-3972	36	24	module	module	NOUN
cana-3972	36	25	and	and	CCONJ
cana-3972	36	26	if	if	SCONJ
cana-3972	36	27	𝑀	𝑀	PROPN
cana-3972	36	28	�	�	PROPN
cana-3972	36	29	̅	̅	NOUN
cana-3972	36	30	�	�	NOUN
cana-3972	36	31	(𝑀	(𝑀	NUM
cana-3972	36	32	)	)	PUNCT
cana-3972	36	33	is	be	AUX
cana-3972	36	34	a	a	DET
cana-3972	36	35	d41	d41	NOUN
cana-3972	36	36	,	,	PUNCT
cana-3972	36	37	then	then	ADV
cana-3972	36	38	m	m	VERB
cana-3972	36	39	is	be	AUX
cana-3972	36	40	a	a	DET
cana-3972	36	41	d41	d41	NOUN
cana-3972	36	42	.	.	PUNCT
cana-3972	37	1	it	it	PRON
cana-3972	37	2	is	be	AUX
cana-3972	37	3	given	give	VERB
cana-3972	37	4	some	some	DET
cana-3972	37	5	properties	property	NOUN
cana-3972	37	6	of	of	ADP
cana-3972	37	7	d41	d41	NOUN
cana-3972	37	8	-	-	PUNCT
cana-3972	37	9	modules	module	NOUN
cana-3972	37	10	related	relate	VERB
cana-3972	37	11	to	to	ADP
cana-3972	37	12	direct	direct	VERB
cana-3972	37	13	finite	finite	ADJ
cana-3972	37	14	modules	module	NOUN
cana-3972	37	15	,	,	PUNCT
cana-3972	37	16	square	square	ADJ
cana-3972	37	17	free	free	ADJ
cana-3972	37	18	modules	module	NOUN
cana-3972	37	19	,	,	PUNCT
cana-3972	37	20	summand	summand	PROPN
cana-3972	37	21	square	square	ADJ
cana-3972	37	22	-	-	PUNCT
cana-3972	37	23	free	free	ADJ
cana-3972	37	24	modules	module	NOUN
cana-3972	37	25	,	,	PUNCT
cana-3972	37	26	summand	summand	NOUN
cana-3972	37	27	dual	dual	ADJ
cana-3972	37	28	-	-	PUNCT
cana-3972	37	29	quare	quare	NOUN
cana-3972	37	30	-	-	PUNCT
cana-3972	37	31	free	free	ADJ
cana-3972	37	32	modules	module	NOUN
cana-3972	37	33	,	,	PUNCT
cana-3972	37	34	hopfian	hopfian	ADJ
cana-3972	37	35	modules	module	NOUN
cana-3972	37	36	,	,	PUNCT
cana-3972	37	37	...	...	PUNCT
cana-3972	37	38	we	we	PRON
cana-3972	37	39	begin	begin	VERB
cana-3972	37	40	section	section	NOUN
cana-3972	37	41	4	4	NUM
cana-3972	37	42	by	by	ADP
cana-3972	37	43	providing	provide	VERB
cana-3972	37	44	a	a	DET
cana-3972	37	45	characterization	characterization	NOUN
cana-3972	37	46	of	of	ADP
cana-3972	37	47	d41	d41	NOUN
cana-3972	37	48	-	-	PUNCT
cana-3972	37	49	modules	module	NOUN
cana-3972	37	50	via	via	ADP
cana-3972	37	51	cosp	cosp	PROPN
cana-3972	37	52	-rings	-rings	PROPN
cana-3972	37	53	.	.	PUNCT
cana-3972	38	1	we	we	PRON
cana-3972	38	2	also	also	ADV
cana-3972	38	3	show	show	VERB
cana-3972	38	4	that	that	SCONJ
cana-3972	38	5	a	a	DET
cana-3972	38	6	ring	ring	NOUN
cana-3972	38	7	r	r	NOUN
cana-3972	38	8	is	be	AUX
cana-3972	38	9	semiregular	semiregular	PROPN
cana-3972	38	10	iff	iff	PROPN
cana-3972	38	11	every	every	DET
cana-3972	38	12	finitely	finitely	ADV
cana-3972	38	13	presented	present	VERB
cana-3972	38	14	r	r	NOUN
cana-3972	38	15	-	-	PUNCT
cana-3972	38	16	module	module	NOUN
cana-3972	38	17	has	have	VERB
cana-3972	38	18	a	a	DET
cana-3972	38	19	d41	d41	NOUN
cana-3972	38	20	-	-	PUNCT
cana-3972	38	21	cover	cover	NOUN
cana-3972	38	22	.	.	PUNCT
cana-3972	39	1	2	2	X
cana-3972	39	2	.	.	X
cana-3972	39	3	preliminaries	preliminary	NOUN
cana-3972	39	4	lemma	lemma	PROPN
cana-3972	39	5	2.1	2.1	NUM
cana-3972	39	6	.	.	PUNCT
cana-3972	40	1	lemma	lemma	PROPN
cana-3972	41	1	[	[	X
cana-3972	41	2	20	20	NUM
cana-3972	41	3	,	,	PUNCT
cana-3972	41	4	proposition	proposition	NOUN
cana-3972	41	5	2.1	2.1	NUM
cana-3972	41	6	]	]	PUNCT
cana-3972	41	7	let	let	VERB
cana-3972	41	8	𝑀	𝑀	PROPN
cana-3972	41	9	,	,	PUNCT
cana-3972	41	10	𝑁	𝑁	PROPN
cana-3972	41	11	and	and	CCONJ
cana-3972	41	12	(	(	PUNCT
cana-3972	41	13	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	41	14	)	)	PUNCT
cana-3972	41	15	,	,	PUNCT
cana-3972	41	16	𝑖	𝑖	PUNCT
cana-3972	42	1	∈	∈	NOUN
cana-3972	43	1	𝐼	𝐼	ADP
cana-3972	43	2	be	be	VERB
cana-3972	43	3	r	r	NOUN
cana-3972	43	4	-	-	PUNCT
cana-3972	43	5	modules	module	NOUN
cana-3972	43	6	.	.	PUNCT
cana-3972	44	1	then	then	ADV
cana-3972	44	2	:	:	PUNCT
cana-3972	44	3	a	a	X
cana-3972	44	4	)	)	PUNCT
cana-3972	44	5	if	if	SCONJ
cana-3972	44	6	𝑁	𝑁	PROPN
cana-3972	44	7	≤	≤	PROPN
cana-3972	44	8	𝑀	𝑀	PROPN
cana-3972	44	9	,	,	PUNCT
cana-3972	44	10	then	then	ADV
cana-3972	44	11	𝑍(𝑁	𝑍(𝑁	PROPN
cana-3972	44	12	)	)	PUNCT
cana-3972	44	13	≤	≤	NUM
cana-3972	44	14	�	�	PROPN
cana-3972	44	15	̅	̅	NOUN
cana-3972	44	16	�	�	NOUN
cana-3972	44	17	(𝑀	(𝑀	NUM
cana-3972	44	18	)	)	PUNCT
cana-3972	44	19	and	and	CCONJ
cana-3972	44	20	(	(	PUNCT
cana-3972	44	21	𝑁+	𝑁+	NOUN
cana-3972	44	22	�	�	PROPN
cana-3972	44	23	̅	̅	NOUN
cana-3972	44	24	�	�	NOUN
cana-3972	44	25	(𝑀	(𝑀	NUM
cana-3972	44	26	)	)	PUNCT
cana-3972	44	27	)	)	PUNCT
cana-3972	45	1	𝑁	𝑁	PROPN
cana-3972	45	2	≤	≤	NUM
cana-3972	45	3	𝑍(𝑀	𝑍(𝑀	NOUN
cana-3972	45	4	𝑁⁄	𝑁⁄	PROPN
cana-3972	45	5	)	)	PUNCT
cana-3972	45	6	b	b	X
cana-3972	45	7	)	)	PUNCT
cana-3972	45	8	if	if	SCONJ
cana-3972	45	9	𝑓	𝑓	PROPN
cana-3972	45	10	:	:	PUNCT
cana-3972	45	11	𝑀	𝑀	PROPN
cana-3972	45	12	→	→	PUNCT
cana-3972	45	13	𝑁	𝑁	PROPN
cana-3972	45	14	is	be	AUX
cana-3972	45	15	a	a	DET
cana-3972	45	16	homomorphism	homomorphism	NOUN
cana-3972	45	17	,	,	PUNCT
cana-3972	45	18	then	then	ADV
cana-3972	45	19	𝑓(𝑍(𝑀	𝑓(𝑍(𝑀	NOUN
cana-3972	45	20	)	)	PUNCT
cana-3972	45	21	)	)	PUNCT
cana-3972	45	22	≤	≤	NUM
cana-3972	45	23	𝑍(𝑁	𝑍(𝑁	X
cana-3972	45	24	)	)	PUNCT
cana-3972	45	25	c	c	X
cana-3972	45	26	)	)	PUNCT
cana-3972	45	27	𝑍	𝑍	PROPN
cana-3972	45	28	(	(	PUNCT
cana-3972	45	29	𝑀	𝑀	PROPN
cana-3972	45	30	𝑍(𝑀)⁄	𝑍(𝑀)⁄	PROPN
cana-3972	45	31	)	)	PUNCT
cana-3972	46	1	=	=	PUNCT
cana-3972	46	2	0	0	NUM
cana-3972	46	3	d	d	NOUN
cana-3972	46	4	)	)	PUNCT
cana-3972	46	5	𝑍(⊕𝑖∈𝐼	𝑍(⊕𝑖∈𝐼	PROPN
cana-3972	46	6	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	46	7	)	)	PUNCT
cana-3972	46	8	=	=	SYM
cana-3972	46	9	⊕𝑖∈𝐼	⊕𝑖∈𝐼	NUM
cana-3972	46	10	𝑍(𝑀𝑖	𝑍(𝑀𝑖	PROPN
cana-3972	46	11	)	)	PUNCT
cana-3972	46	12	e	e	NOUN
cana-3972	46	13	)	)	PUNCT
cana-3972	46	14	𝑍(∏𝑖∈𝐼𝑀𝑖	𝑍(∏𝑖∈𝐼𝑀𝑖	PROPN
cana-3972	46	15	)	)	PUNCT
cana-3972	46	16	≤	≤	NOUN
cana-3972	46	17	∏𝑖∈𝐼	∏𝑖∈𝐼	ADJ
cana-3972	47	1	𝑍(𝑀𝑖	𝑍(𝑀𝑖	NOUN
cana-3972	47	2	)	)	PUNCT
cana-3972	47	3	f	f	X
cana-3972	47	4	)	)	PUNCT
cana-3972	47	5	if	if	SCONJ
cana-3972	47	6	m	m	VERB
cana-3972	47	7	=	=	VERB
cana-3972	47	8	n	n	PROPN
cana-3972	47	9	+	+	X
cana-3972	47	10	s	s	VERB
cana-3972	47	11	where	where	SCONJ
cana-3972	47	12	s	s	NOUN
cana-3972	47	13	is	be	AUX
cana-3972	47	14	a	a	DET
cana-3972	47	15	small	small	ADJ
cana-3972	47	16	module	module	NOUN
cana-3972	47	17	,	,	PUNCT
cana-3972	47	18	then	then	ADV
cana-3972	47	19	𝑍(𝑀	𝑍(𝑀	NOUN
cana-3972	47	20	)	)	PUNCT
cana-3972	47	21	=	=	SYM
cana-3972	47	22	𝑍(𝑁	𝑍(𝑁	PROPN
cana-3972	47	23	)	)	PUNCT
cana-3972	47	24	g	g	NOUN
cana-3972	47	25	)	)	PUNCT
cana-3972	47	26	𝑍(𝑀	𝑍(𝑀	NOUN
cana-3972	47	27	)	)	PUNCT
cana-3972	47	28	is	be	AUX
cana-3972	47	29	the	the	DET
cana-3972	47	30	smallest	small	ADJ
cana-3972	47	31	submodule	submodule	NOUN
cana-3972	47	32	such	such	ADJ
cana-3972	47	33	that	that	DET
cana-3972	47	34	𝑍	𝑍	PROPN
cana-3972	47	35	(	(	PUNCT
cana-3972	47	36	𝑀	𝑀	PROPN
cana-3972	47	37	�	�	PROPN
cana-3972	47	38	̅	̅	NOUN
cana-3972	47	39	�	�	NOUN
cana-3972	47	40	(𝑀)⁄	(𝑀)⁄	NOUN
cana-3972	47	41	)	)	PUNCT
cana-3972	48	1	=	=	SYM
cana-3972	48	2	0	0	X
cana-3972	48	3	.	.	PUNCT
cana-3972	48	4	lemma	lemma	PROPN
cana-3972	48	5	2.2	2.2	NUM
cana-3972	48	6	.	.	PUNCT
cana-3972	49	1	[	[	X
cana-3972	49	2	20	20	NUM
cana-3972	49	3	,	,	PUNCT
cana-3972	49	4	corol	corol	PROPN
cana-3972	49	5	lary	lary	PROPN
cana-3972	49	6	2.2	2.2	NUM
cana-3972	49	7	]	]	PUNCT
cana-3972	49	8	the	the	DET
cana-3972	49	9	class	class	NOUN
cana-3972	49	10	of	of	ADP
cana-3972	49	11	all	all	DET
cana-3972	49	12	cosingular	cosingular	ADJ
cana-3972	49	13	modules	module	NOUN
cana-3972	49	14	is	be	AUX
cana-3972	49	15	closed	close	VERB
cana-3972	49	16	under	under	ADP
cana-3972	49	17	submodules	submodule	NOUN
cana-3972	49	18	,	,	PUNCT
cana-3972	49	19	directs	direct	VERB
cana-3972	49	20	sums	sum	NOUN
cana-3972	49	21	and	and	CCONJ
cana-3972	49	22	directs	direct	VERB
cana-3972	49	23	products	product	NOUN
cana-3972	49	24	.	.	PUNCT
cana-3972	50	1	lemma	lemma	PROPN
cana-3972	50	2	2.3	2.3	NUM
cana-3972	50	3	.	.	PUNCT
cana-3972	51	1	[	[	X
cana-3972	51	2	5	5	NUM
cana-3972	51	3	,	,	PUNCT
cana-3972	51	4	proposition	proposition	NOUN
cana-3972	51	5	2.1	2.1	NUM
cana-3972	51	6	]	]	PUNCT
cana-3972	51	7	let	let	VERB
cana-3972	51	8	m	m	PRON
cana-3972	51	9	a	a	DET
cana-3972	51	10	d3	d3	PROPN
cana-3972	51	11	-	-	PUNCT
cana-3972	51	12	module	module	NOUN
cana-3972	51	13	such	such	ADJ
cana-3972	51	14	that	that	DET
cana-3972	51	15	𝑀	𝑀	PROPN
cana-3972	51	16	=	=	PUNCT
cana-3972	51	17	𝐴1	𝐴1	PROPN
cana-3972	51	18	⊕	⊕	PROPN
cana-3972	51	19	𝐴2	𝐴2	PROPN
cana-3972	51	20	for	for	ADP
cana-3972	51	21	submodules	submodules	NOUN
cana-3972	51	22	a1	a1	NOUN
cana-3972	51	23	and	and	CCONJ
cana-3972	51	24	a2	a2	NOUN
cana-3972	51	25	.	.	PUNCT
cana-3972	52	1	𝐼𝑓	𝐼𝑓	VERB
cana-3972	52	2	𝑓	𝑓	DET
cana-3972	52	3	∶	∶	NOUN
cana-3972	52	4	𝐴1	𝐴1	PROPN
cana-3972	52	5	→	→	PUNCT
cana-3972	52	6	𝐴2	𝐴2	PROPN
cana-3972	52	7	is	be	AUX
cana-3972	52	8	a	a	DET
cana-3972	52	9	homomorphism	homomorphism	NOUN
cana-3972	52	10	such	such	ADJ
cana-3972	52	11	that	that	SCONJ
cana-3972	52	12	𝐼𝑚𝑓	𝐼𝑚𝑓	PROPN
cana-3972	52	13	≤⊕	≤⊕	PRON
cana-3972	52	14	𝐴2	𝐴2	PROPN
cana-3972	52	15	,	,	PUNCT
cana-3972	52	16	then	then	ADV
cana-3972	52	17	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	PROPN
cana-3972	52	18	≤⊕	≤⊕	AUX
cana-3972	52	19	𝐴1	𝐴1	PROPN
cana-3972	52	20	.	.	PUNCT
cana-3972	53	1	proposition	proposition	NOUN
cana-3972	53	2	2.4	2.4	NUM
cana-3972	53	3	.	.	PUNCT
cana-3972	54	1	let	let	VERB
cana-3972	54	2	m	m	PRON
cana-3972	54	3	be	be	AUX
cana-3972	54	4	a	a	DET
cana-3972	54	5	module	module	NOUN
cana-3972	54	6	,	,	PUNCT
cana-3972	54	7	n	n	PROPN
cana-3972	54	8	and	and	CCONJ
cana-3972	54	9	k	k	PROPN
cana-3972	54	10	two	two	NUM
cana-3972	54	11	submodules	submodule	NOUN
cana-3972	54	12	of	of	ADP
cana-3972	54	13	m	m	NOUN
cana-3972	54	14	such	such	ADJ
cana-3972	54	15	that	that	SCONJ
cana-3972	54	16	k	k	PROPN
cana-3972	54	17	is	be	AUX
cana-3972	54	18	cosingular	cosingular	ADJ
cana-3972	54	19	.	.	PUNCT
cana-3972	55	1	then	then	ADV
cana-3972	55	2	the	the	DET
cana-3972	55	3	following	follow	VERB
cana-3972	55	4	statements	statement	NOUN
cana-3972	55	5	are	be	AUX
cana-3972	55	6	equivalent	equivalent	ADJ
cana-3972	55	7	:	:	PUNCT
cana-3972	55	8	1	1	X
cana-3972	55	9	)	)	PUNCT
cana-3972	55	10	if	if	SCONJ
cana-3972	55	11	𝑀	𝑀	PROPN
cana-3972	55	12	=	=	SYM
cana-3972	55	13	𝑁	𝑁	PROPN
cana-3972	55	14	⊕	⊕	PROPN
cana-3972	55	15	𝐾	𝐾	PROPN
cana-3972	55	16	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
cana-3972	55	17	𝑓	𝑓	DET
cana-3972	55	18	∶	∶	NOUN
cana-3972	55	19	𝑁	𝑁	PROPN
cana-3972	55	20	→	→	SYM
cana-3972	55	21	𝐾	𝐾	PROPN
cana-3972	55	22	an	an	DET
cana-3972	55	23	epimorphism	epimorphism	NOUN
cana-3972	55	24	,	,	PUNCT
cana-3972	55	25	then	then	ADV
cana-3972	55	26	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	55	27	≤⊕	≤⊕	ADJ
cana-3972	55	28	𝑁.	𝑁.	PROPN
cana-3972	55	29	2	2	X
cana-3972	55	30	)	)	PUNCT
cana-3972	55	31	if	if	SCONJ
cana-3972	55	32	m	m	NOUN
cana-3972	55	33	=	=	SYM
cana-3972	55	34	n	n	PROPN
cana-3972	55	35	⊕	⊕	PROPN
cana-3972	55	36	k	k	PROPN
cana-3972	55	37	and	and	CCONJ
cana-3972	55	38	𝑓	𝑓	DET
cana-3972	55	39	∶	∶	NOUN
cana-3972	55	40	𝑁	𝑁	PROPN
cana-3972	55	41	→	→	SYM
cana-3972	55	42	𝐾	𝐾	PROPN
cana-3972	55	43	a	a	DET
cana-3972	55	44	homomorphism	homomorphism	NOUN
cana-3972	55	45	with	with	ADP
cana-3972	55	46	𝐼𝑚𝑓	𝐼𝑚𝑓	PROPN
cana-3972	55	47	≤⊕	≤⊕	NUM
cana-3972	55	48	𝐾	𝐾	PROPN
cana-3972	55	49	,	,	PUNCT
cana-3972	55	50	then	then	ADV
cana-3972	55	51	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	55	52	≤⊕	≤⊕	ADJ
cana-3972	55	53	𝑁.	𝑁.	PROPN
cana-3972	55	54	3	3	NUM
cana-3972	55	55	)	)	PUNCT
cana-3972	55	56	if	if	SCONJ
cana-3972	55	57	n	n	PRON
cana-3972	55	58	≤	≤	X
cana-3972	55	59	k	k	PROPN
cana-3972	55	60	and	and	CCONJ
cana-3972	55	61	𝑀	𝑀	PROPN
cana-3972	55	62	𝐾	𝐾	PROPN
cana-3972	55	63	≅	≅	PROPN
cana-3972	55	64	𝑁	𝑁	PROPN
cana-3972	55	65	≤⊕	≤⊕	NUM
cana-3972	55	66	𝑀	𝑀	PROPN
cana-3972	55	67	,	,	PUNCT
cana-3972	55	68	then	then	ADV
cana-3972	55	69	𝐾	𝐾	PROPN
cana-3972	55	70	≤⊕	≤⊕	ADV
cana-3972	55	71	𝑀.	𝑀.	PROPN
cana-3972	55	72	4	4	NUM
cana-3972	55	73	)	)	PUNCT
cana-3972	55	74	if	if	SCONJ
cana-3972	55	75	𝑀	𝑀	PROPN
cana-3972	55	76	=	=	SYM
cana-3972	55	77	𝑁	𝑁	PROPN
cana-3972	55	78	+	+	PROPN
cana-3972	55	79	𝐾	𝐾	PROPN
cana-3972	55	80	,	,	PUNCT
cana-3972	55	81	𝑁	𝑁	PROPN
cana-3972	55	82	≤⊕	≤⊕	NUM
cana-3972	55	83	𝑀	𝑀	PROPN
cana-3972	55	84	and	and	CCONJ
cana-3972	55	85	𝑀/𝑁	𝑀/𝑁	PROPN
cana-3972	55	86	≤⊕	≤⊕	NUM
cana-3972	55	87	𝑀/𝐾	𝑀/𝐾	NOUN
cana-3972	55	88	,	,	PUNCT
cana-3972	55	89	then	then	ADV
cana-3972	55	90	𝑁	𝑁	PROPN
cana-3972	55	91	∩	∩	ADJ
cana-3972	55	92	𝐾	𝐾	PROPN
cana-3972	55	93	≤⊕	≤⊕	ADV
cana-3972	55	94	𝑀.	𝑀.	PROPN
cana-3972	55	95	5	5	NUM
cana-3972	55	96	)	)	PUNCT
cana-3972	55	97	if	if	SCONJ
cana-3972	55	98	n	n	PROPN
cana-3972	55	99	and	and	CCONJ
cana-3972	55	100	k	k	PROPN
cana-3972	55	101	are	be	AUX
cana-3972	55	102	direct	direct	ADJ
cana-3972	55	103	summands	summand	NOUN
cana-3972	55	104	of	of	ADP
cana-3972	55	105	m	m	PROPN
cana-3972	55	106	with	with	ADP
cana-3972	55	107	𝑀	𝑀	PROPN
cana-3972	55	108	=	=	SYM
cana-3972	55	109	𝑁	𝑁	PROPN
cana-3972	55	110	+	+	CCONJ
cana-3972	55	111	𝐾	𝐾	PROPN
cana-3972	55	112	and	and	CCONJ
cana-3972	55	113	𝑀/𝑁	𝑀/𝑁	PROPN
cana-3972	55	114	≅	≅	PROPN
cana-3972	55	115	𝑀/𝐾	𝑀/𝐾	PROPN
cana-3972	55	116	,	,	PUNCT
cana-3972	55	117	then	then	ADV
cana-3972	55	118	communications	communication	NOUN
cana-3972	55	119	on	on	ADP
cana-3972	55	120	applied	apply	VERB
cana-3972	55	121	nonlinear	nonlinear	ADJ
cana-3972	55	122	analysis	analysis	NOUN
cana-3972	55	123	issn	issn	NOUN
cana-3972	55	124	:	:	PUNCT
cana-3972	55	125	1074	1074	NUM
cana-3972	55	126	-	-	PUNCT
cana-3972	55	127	133x	133x	NUM
cana-3972	55	128	vol	vol	NOUN
cana-3972	55	129	32	32	NUM
cana-3972	55	130	no	no	NOUN
cana-3972	55	131	.	.	PUNCT
cana-3972	56	1	9s	9s	NUM
cana-3972	56	2	(	(	PUNCT
cana-3972	56	3	2025	2025	NUM
cana-3972	56	4	)	)	PUNCT
cana-3972	56	5	677	677	NUM
cana-3972	57	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	57	2	𝑁	𝑁	PROPN
cana-3972	57	3	∩	∩	ADJ
cana-3972	57	4	𝐾	𝐾	PROPN
cana-3972	57	5	≤⊕	≤⊕	NUM
cana-3972	57	6	𝑀	𝑀	PROPN
cana-3972	57	7	.	.	PUNCT
cana-3972	58	1	6	6	X
cana-3972	58	2	)	)	PUNCT
cana-3972	58	3	if	if	SCONJ
cana-3972	58	4	𝑀	𝑀	PROPN
cana-3972	58	5	=	=	SYM
cana-3972	58	6	𝑁	𝑁	PROPN
cana-3972	58	7	+	+	PROPN
cana-3972	58	8	𝐾	𝐾	PROPN
cana-3972	58	9	,	,	PUNCT
cana-3972	58	10	𝑁	𝑁	PROPN
cana-3972	58	11	≤⊕	≤⊕	NUM
cana-3972	58	12	𝑀	𝑀	PROPN
cana-3972	58	13	and	and	CCONJ
cana-3972	58	14	𝑀/𝑁	𝑀/𝑁	PROPN
cana-3972	58	15	≅	≅	PROPN
cana-3972	58	16	𝑀/𝐾	𝑀/𝐾	PROPN
cana-3972	58	17	,	,	PUNCT
cana-3972	58	18	then	then	ADV
cana-3972	58	19	𝐾	𝐾	PROPN
cana-3972	58	20	≤⊕	≤⊕	ADV
cana-3972	58	21	𝑀.	𝑀.	PROPN
cana-3972	58	22	7	7	NUM
cana-3972	58	23	)	)	PUNCT
cana-3972	58	24	if	if	SCONJ
cana-3972	58	25	𝑀	𝑀	PROPN
cana-3972	58	26	=	=	SYM
cana-3972	58	27	𝑁	𝑁	PROPN
cana-3972	58	28	⊕	⊕	PROPN
cana-3972	58	29	𝑁′	𝑁′	X
cana-3972	58	30	=	=	SYM
cana-3972	58	31	𝐾	𝐾	PROPN
cana-3972	58	32	⊕	⊕	PROPN
cana-3972	58	33	𝐾′	𝐾′	NOUN
cana-3972	59	1	=	=	PUNCT
cana-3972	59	2	𝑁	𝑁	PROPN
cana-3972	59	3	+	+	PROPN
cana-3972	59	4	𝐾	𝐾	NOUN
cana-3972	59	5	=	=	SYM
cana-3972	59	6	𝑁	𝑁	PROPN
cana-3972	59	7	+	+	CCONJ
cana-3972	59	8	𝐾′	𝐾′	NOUN
cana-3972	59	9	,	,	PUNCT
cana-3972	59	10	where	where	SCONJ
cana-3972	59	11	𝑁′𝑎𝑛𝑑	𝑁′𝑎𝑛𝑑	NOUN
cana-3972	59	12	𝐾′	𝐾′	X
cana-3972	59	13	are	be	AUX
cana-3972	59	14	submodules	submodule	NOUN
cana-3972	59	15	of	of	ADP
cana-3972	59	16	m	m	PRON
cana-3972	59	17	,	,	PUNCT
cana-3972	59	18	then	then	ADV
cana-3972	59	19	𝑁	𝑁	PROPN
cana-3972	59	20	∩	∩	ADJ
cana-3972	59	21	𝐾	𝐾	PROPN
cana-3972	59	22	≤⊕	≤⊕	NUM
cana-3972	59	23	𝑀	𝑀	PROPN
cana-3972	59	24	.	.	PUNCT
cana-3972	60	1	8)	8)	NUM
cana-3972	60	2	if	if	SCONJ
cana-3972	60	3	n	n	PROPN
cana-3972	60	4	and	and	CCONJ
cana-3972	60	5	k	k	PROPN
cana-3972	60	6	are	be	AUX
cana-3972	60	7	direct	direct	ADJ
cana-3972	60	8	summands	summand	NOUN
cana-3972	60	9	of	of	ADP
cana-3972	60	10	m	m	PROPN
cana-3972	60	11	with	with	ADP
cana-3972	60	12	m	m	PROPN
cana-3972	60	13	=	=	SYM
cana-3972	60	14	n	n	PROPN
cana-3972	60	15	+	+	CCONJ
cana-3972	60	16	k	k	NOUN
cana-3972	60	17	and	and	CCONJ
cana-3972	60	18	𝑁	𝑁	PROPN
cana-3972	60	19	≅	≅	PROPN
cana-3972	60	20	𝐾	𝐾	PROPN
cana-3972	60	21	,	,	PUNCT
cana-3972	60	22	then	then	ADV
cana-3972	60	23	𝑁	𝑁	PROPN
cana-3972	60	24	∩	∩	ADJ
cana-3972	60	25	𝐾	𝐾	PROPN
cana-3972	60	26	≤⊕	≤⊕	ADV
cana-3972	60	27	𝑀.	𝑀.	PROPN
cana-3972	60	28	proof	proof	NOUN
cana-3972	60	29	:	:	PUNCT
cana-3972	60	30	the	the	DET
cana-3972	60	31	proof	proof	NOUN
cana-3972	60	32	follows	follow	VERB
cana-3972	60	33	by	by	ADP
cana-3972	60	34	the	the	DET
cana-3972	60	35	same	same	ADJ
cana-3972	60	36	method	method	NOUN
cana-3972	60	37	as	as	ADP
cana-3972	60	38	in	in	ADP
cana-3972	60	39	[	[	X
cana-3972	60	40	5	5	NUM
cana-3972	60	41	,	,	PUNCT
cana-3972	60	42	theorem	theorem	VERB
cana-3972	60	43	2.2	2.2	NUM
cana-3972	60	44	]	]	PUNCT
cana-3972	60	45	.	.	PUNCT
cana-3972	61	1	3	3	X
cana-3972	61	2	.	.	X
cana-3972	62	1	some	some	DET
cana-3972	62	2	properties	property	NOUN
cana-3972	62	3	of	of	ADP
cana-3972	62	4	d41	d41	NOUN
cana-3972	62	5	-	-	PUNCT
cana-3972	62	6	modules	module	NOUN
cana-3972	62	7	throughout	throughout	ADP
cana-3972	62	8	this	this	DET
cana-3972	62	9	section	section	NOUN
cana-3972	62	10	,	,	PUNCT
cana-3972	62	11	we	we	PRON
cana-3972	62	12	shall	shall	AUX
cana-3972	62	13	investigate	investigate	VERB
cana-3972	62	14	some	some	DET
cana-3972	62	15	general	general	ADJ
cana-3972	62	16	properties	property	NOUN
cana-3972	62	17	of	of	ADP
cana-3972	62	18	d41	d41	NOUN
cana-3972	62	19	-	-	PUNCT
cana-3972	62	20	modules	module	NOUN
cana-3972	62	21	.	.	PUNCT
cana-3972	63	1	definition	definition	NOUN
cana-3972	63	2	3.1	3.1	NUM
cana-3972	63	3	.	.	PUNCT
cana-3972	64	1	let	let	VERB
cana-3972	64	2	m	m	PRON
cana-3972	64	3	be	be	AUX
cana-3972	64	4	an	an	DET
cana-3972	64	5	r	r	NOUN
cana-3972	64	6	-	-	PUNCT
cana-3972	64	7	module	module	NOUN
cana-3972	64	8	.	.	PUNCT
cana-3972	65	1	we	we	PRON
cana-3972	65	2	say	say	VERB
cana-3972	65	3	that	that	SCONJ
cana-3972	65	4	m	m	PROPN
cana-3972	65	5	is	be	AUX
cana-3972	65	6	a	a	DET
cana-3972	65	7	d41	d41	NOUN
cana-3972	65	8	if	if	SCONJ
cana-3972	65	9	,	,	PUNCT
cana-3972	65	10	𝑀	𝑀	PROPN
cana-3972	65	11	=	=	SYM
cana-3972	65	12	𝑁	𝑁	PROPN
cana-3972	65	13	⊕	⊕	PROPN
cana-3972	65	14	𝐾	𝐾	PROPN
cana-3972	65	15	for	for	ADP
cana-3972	65	16	𝑁	𝑁	PROPN
cana-3972	65	17	,	,	PUNCT
cana-3972	65	18	𝐾	𝐾	PROPN
cana-3972	65	19	≤	≤	PUNCT
cana-3972	65	20	𝑀	𝑀	PROPN
cana-3972	65	21	such	such	ADJ
cana-3972	65	22	that	that	SCONJ
cana-3972	65	23	k	k	PROPN
cana-3972	65	24	is	be	AUX
cana-3972	65	25	cosingular	cosingular	ADJ
cana-3972	65	26	and	and	CCONJ
cana-3972	65	27	f	f	NOUN
cana-3972	65	28	:	:	PUNCT
cana-3972	65	29	n	n	PROPN
cana-3972	65	30	→	→	SYM
cana-3972	65	31	k	k	X
cana-3972	65	32	is	be	AUX
cana-3972	65	33	an	an	DET
cana-3972	65	34	epimorphism	epimorphism	NOUN
cana-3972	65	35	,	,	PUNCT
cana-3972	65	36	then	then	ADV
cana-3972	65	37	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	65	38	)	)	PUNCT
cana-3972	65	39	≤⊕	≤⊕	VERB
cana-3972	66	1	𝑁.	𝑁.	PROPN
cana-3972	66	2	the	the	DET
cana-3972	66	3	ring	ring	NOUN
cana-3972	66	4	r	r	NOUN
cana-3972	66	5	is	be	AUX
cana-3972	66	6	a	a	DET
cana-3972	66	7	right	right	NOUN
cana-3972	66	8	(	(	PUNCT
cana-3972	66	9	left	left	ADJ
cana-3972	66	10	)	)	PUNCT
cana-3972	66	11	d41	d41	NOUN
cana-3972	66	12	-	-	PUNCT
cana-3972	66	13	ring	ring	NOUN
cana-3972	66	14	if	if	SCONJ
cana-3972	66	15	the	the	DET
cana-3972	66	16	right	right	ADJ
cana-3972	66	17	r	r	NOUN
cana-3972	66	18	-	-	PUNCT
cana-3972	66	19	module	module	NOUN
cana-3972	66	20	rr	rr	NOUN
cana-3972	66	21	(	(	PUNCT
cana-3972	66	22	left	leave	VERB
cana-3972	66	23	rr	rr	NOUN
cana-3972	66	24	)	)	PUNCT
cana-3972	66	25	is	be	AUX
cana-3972	66	26	d41	d41	PROPN
cana-3972	66	27	.	.	PUNCT
cana-3972	66	28	example	example	NOUN
cana-3972	67	1	3.2	3.2	NUM
cana-3972	67	2	.	.	NOUN
cana-3972	67	3	1	1	NUM
cana-3972	67	4	)	)	PUNCT
cana-3972	67	5	every	every	DET
cana-3972	67	6	d4	d4	PROPN
cana-3972	67	7	-	-	PUNCT
cana-3972	67	8	module	module	NOUN
cana-3972	67	9	has	have	VERB
cana-3972	67	10	d41	d41	PROPN
cana-3972	67	11	.	.	PUNCT
cana-3972	68	1	particularly	particularly	ADV
cana-3972	68	2	,	,	PUNCT
cana-3972	68	3	each	each	DET
cana-3972	68	4	projective	projective	ADJ
cana-3972	68	5	module	module	NOUN
cana-3972	68	6	has	have	VERB
cana-3972	68	7	d41	d41	PROPN
cana-3972	68	8	.	.	NOUN
cana-3972	69	1	2	2	NUM
cana-3972	69	2	)	)	PUNCT
cana-3972	69	3	each	each	DET
cana-3972	69	4	hereditary	hereditary	ADJ
cana-3972	69	5	module	module	NOUN
cana-3972	69	6	is	be	AUX
cana-3972	69	7	d41	d41	NOUN
cana-3972	69	8	as	as	ADP
cana-3972	69	9	any	any	DET
cana-3972	69	10	submodule	submodule	NOUN
cana-3972	69	11	of	of	ADP
cana-3972	69	12	such	such	ADJ
cana-3972	69	13	module	module	NOUN
cana-3972	69	14	is	be	AUX
cana-3972	69	15	projective	projective	ADJ
cana-3972	69	16	.	.	PUNCT
cana-3972	70	1	3	3	X
cana-3972	70	2	)	)	PUNCT
cana-3972	70	3	every	every	DET
cana-3972	70	4	semisimple	semisimple	NOUN
cana-3972	70	5	module	module	NOUN
cana-3972	70	6	is	be	AUX
cana-3972	70	7	a	a	DET
cana-3972	70	8	d41	d41	NOUN
cana-3972	70	9	-	-	PUNCT
cana-3972	70	10	module	module	NOUN
cana-3972	70	11	.	.	PUNCT
cana-3972	71	1	4	4	X
cana-3972	71	2	)	)	PUNCT
cana-3972	71	3	each	each	DET
cana-3972	71	4	module	module	NOUN
cana-3972	71	5	with	with	ADP
cana-3972	71	6	the	the	DET
cana-3972	71	7	summand	summand	NOUN
cana-3972	71	8	intersection	intersection	NOUN
cana-3972	71	9	property	property	NOUN
cana-3972	71	10	(	(	PUNCT
cana-3972	71	11	sip	sip	NOUN
cana-3972	71	12	)	)	PUNCT
cana-3972	71	13	,	,	PUNCT
cana-3972	71	14	has	have	VERB
cana-3972	71	15	d41	d41	PROPN
cana-3972	71	16	.	.	PUNCT
cana-3972	72	1	next	next	ADV
cana-3972	72	2	,	,	PUNCT
cana-3972	72	3	we	we	PRON
cana-3972	72	4	give	give	VERB
cana-3972	72	5	an	an	DET
cana-3972	72	6	example	example	NOUN
cana-3972	72	7	of	of	ADP
cana-3972	72	8	a	a	DET
cana-3972	72	9	d41	d41	NOUN
cana-3972	72	10	-	-	PUNCT
cana-3972	72	11	module	module	NOUN
cana-3972	72	12	that	that	PRON
cana-3972	72	13	is	be	AUX
cana-3972	72	14	not	not	PART
cana-3972	72	15	a	a	DET
cana-3972	72	16	d4	d4	NOUN
cana-3972	72	17	-	-	PUNCT
cana-3972	72	18	module	module	NOUN
cana-3972	72	19	.	.	PUNCT
cana-3972	72	20	example	example	NOUN
cana-3972	72	21	3.3	3.3	NUM
cana-3972	72	22	.	.	PUNCT
cana-3972	73	1	consider	consider	VERB
cana-3972	73	2	the	the	DET
cana-3972	73	3	module	module	NOUN
cana-3972	73	4	𝑀	𝑀	PROPN
cana-3972	73	5	=	=	SYM
cana-3972	73	6	ℚ	ℚ	PROPN
cana-3972	73	7	⊕	⊕	PROPN
cana-3972	73	8	ℤ	ℤ	PROPN
cana-3972	73	9	,	,	PUNCT
cana-3972	73	10	where	where	SCONJ
cana-3972	73	11	ℚ	ℚ	PROPN
cana-3972	73	12	is	be	AUX
cana-3972	73	13	the	the	DET
cana-3972	73	14	rational	rational	ADJ
cana-3972	73	15	as	as	ADP
cana-3972	73	16	a	a	DET
cana-3972	73	17	ℤ	ℤ	PROPN
cana-3972	73	18	-module	-module	NOUN
cana-3972	73	19	and	and	CCONJ
cana-3972	73	20	ℤ	ℤ	PROPN
cana-3972	73	21	is	be	AUX
cana-3972	73	22	the	the	DET
cana-3972	73	23	integers	integer	NOUN
cana-3972	73	24	as	as	ADP
cana-3972	73	25	a	a	DET
cana-3972	73	26	ℤ	ℤ	PROPN
cana-3972	73	27	-module	-module	NOUN
cana-3972	73	28	.	.	PUNCT
cana-3972	74	1	we	we	PRON
cana-3972	74	2	decompose	decompose	VERB
cana-3972	74	3	𝑀	𝑀	NOUN
cana-3972	74	4	=	=	PUNCT
cana-3972	74	5	ℚ	ℚ	PROPN
cana-3972	74	6	⊕	⊕	PROPN
cana-3972	74	7	ℤ.	ℤ.	PROPN
cana-3972	74	8	ℤ	ℤ	PROPN
cana-3972	74	9	is	be	AUX
cana-3972	74	10	cosingular	cosingular	ADJ
cana-3972	74	11	as	as	ADP
cana-3972	74	12	a	a	DET
cana-3972	74	13	ℤ	ℤ	NOUN
cana-3972	74	14	-module	-module	NOUN
cana-3972	74	15	.	.	PUNCT
cana-3972	75	1	now	now	ADV
cana-3972	75	2	,	,	PUNCT
cana-3972	75	3	consider	consider	VERB
cana-3972	75	4	an	an	DET
cana-3972	75	5	epimorphism	epimorphism	NOUN
cana-3972	75	6	𝑓	𝑓	DET
cana-3972	75	7	∶	∶	NOUN
cana-3972	75	8	ℚ	ℚ	PROPN
cana-3972	75	9	→	→	SYM
cana-3972	75	10	ℤ.	ℤ.	PROPN
cana-3972	75	11	a	a	DET
cana-3972	75	12	natural	natural	ADJ
cana-3972	75	13	choice	choice	NOUN
cana-3972	75	14	for	for	ADP
cana-3972	75	15	such	such	DET
cana-3972	75	16	an	an	DET
cana-3972	75	17	epimorphism	epimorphism	NOUN
cana-3972	75	18	is	be	AUX
cana-3972	75	19	the	the	DET
cana-3972	75	20	inclusion	inclusion	NOUN
cana-3972	75	21	map	map	NOUN
cana-3972	75	22	:	:	PUNCT
cana-3972	75	23	𝑓	𝑓	DET
cana-3972	75	24	∶	∶	NOUN
cana-3972	75	25	ℚ	ℚ	PROPN
cana-3972	75	26	→	→	SYM
cana-3972	75	27	ℤ	ℤ	PROPN
cana-3972	75	28	,	,	PUNCT
cana-3972	75	29	with	with	ADP
cana-3972	75	30	𝑓(𝑞	𝑓(𝑞	NOUN
cana-3972	75	31	)	)	PUNCT
cana-3972	75	32	=	=	SYM
cana-3972	75	33	⌊𝑞⌋	⌊𝑞⌋	PUNCT
cana-3972	75	34	for	for	ADP
cana-3972	75	35	𝑞	𝑞	PROPN
cana-3972	75	36	∈	∈	PROPN
cana-3972	75	37	ℚ	ℚ	PROPN
cana-3972	75	38	,	,	PUNCT
cana-3972	75	39	where	where	SCONJ
cana-3972	75	40	the	the	DET
cana-3972	75	41	notation	notation	NOUN
cana-3972	75	42	⌊𝑞⌋	⌊𝑞⌋	PUNCT
cana-3972	75	43	represents	represent	VERB
cana-3972	75	44	the	the	DET
cana-3972	75	45	floor	floor	NOUN
cana-3972	75	46	function	function	NOUN
cana-3972	75	47	of	of	ADP
cana-3972	75	48	a	a	DET
cana-3972	75	49	real	real	ADJ
cana-3972	75	50	number	number	NOUN
cana-3972	75	51	𝑞.	𝑞.	NOUN
cana-3972	75	52	it	it	PRON
cana-3972	75	53	is	be	AUX
cana-3972	75	54	defined	define	VERB
cana-3972	75	55	as	as	ADP
cana-3972	75	56	the	the	DET
cana-3972	75	57	greatest	great	ADJ
cana-3972	75	58	integer	integer	NOUN
cana-3972	75	59	less	less	ADJ
cana-3972	75	60	than	than	ADP
cana-3972	75	61	or	or	CCONJ
cana-3972	75	62	equal	equal	ADJ
cana-3972	75	63	to	to	ADP
cana-3972	75	64	q.	q.	NOUN
cana-3972	75	65	here	here	ADV
cana-3972	75	66	,	,	PUNCT
cana-3972	75	67	the	the	DET
cana-3972	75	68	kernel	kernel	NOUN
cana-3972	75	69	of	of	ADP
cana-3972	75	70	this	this	DET
cana-3972	75	71	map	map	NOUN
cana-3972	75	72	is	be	AUX
cana-3972	75	73	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	75	74	)	)	PUNCT
cana-3972	75	75	=	=	SYM
cana-3972	75	76	ℚ	ℚ	PROPN
cana-3972	75	77	,	,	PUNCT
cana-3972	75	78	which	which	PRON
cana-3972	75	79	is	be	AUX
cana-3972	75	80	clearly	clearly	ADV
cana-3972	75	81	a	a	DET
cana-3972	75	82	direct	direct	ADJ
cana-3972	75	83	summand	summand	NOUN
cana-3972	75	84	of	of	ADP
cana-3972	75	85	ℚ	ℚ	PROPN
cana-3972	75	86	since	since	SCONJ
cana-3972	75	87	ℚ	ℚ	PROPN
cana-3972	75	88	=	=	SYM
cana-3972	75	89	ℚ	ℚ	PROPN
cana-3972	75	90	⊕	⊕	PROPN
cana-3972	75	91	{	{	PUNCT
cana-3972	75	92	0	0	NUM
cana-3972	75	93	}	}	PUNCT
cana-3972	75	94	.	.	PUNCT
cana-3972	76	1	thus	thus	ADV
cana-3972	76	2	,	,	PUNCT
cana-3972	76	3	the	the	DET
cana-3972	76	4	d41	d41	NOUN
cana-3972	76	5	-	-	PUNCT
cana-3972	76	6	condition	condition	NOUN
cana-3972	76	7	holds	hold	VERB
cana-3972	76	8	for	for	ADP
cana-3972	76	9	this	this	DET
cana-3972	76	10	map	map	NOUN
cana-3972	76	11	.	.	PUNCT
cana-3972	77	1	the	the	DET
cana-3972	77	2	𝐷4	𝐷4	NOUN
cana-3972	77	3	-	-	PUNCT
cana-3972	77	4	condition	condition	NOUN
cana-3972	77	5	requires	require	VERB
cana-3972	77	6	that	that	SCONJ
cana-3972	77	7	for	for	ADP
cana-3972	77	8	every	every	DET
cana-3972	77	9	decomposition	decomposition	NOUN
cana-3972	77	10	𝑀	𝑀	NOUN
cana-3972	77	11	=	=	SYM
cana-3972	77	12	𝑁	𝑁	PROPN
cana-3972	77	13	⊕	⊕	PROPN
cana-3972	77	14	𝐾	𝐾	PROPN
cana-3972	77	15	and	and	CCONJ
cana-3972	77	16	for	for	ADP
cana-3972	77	17	any	any	DET
cana-3972	77	18	epimorphism	epimorphism	NOUN
cana-3972	77	19	𝑓	𝑓	DET
cana-3972	77	20	∶	∶	NOUN
cana-3972	77	21	𝑁	𝑁	PROPN
cana-3972	77	22	→	→	SYM
cana-3972	77	23	𝐾	𝐾	PROPN
cana-3972	77	24	,	,	PUNCT
cana-3972	77	25	the	the	DET
cana-3972	77	26	kernel	kernel	PROPN
cana-3972	77	27	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	77	28	)	)	PUNCT
cana-3972	77	29	must	must	AUX
cana-3972	77	30	be	be	AUX
cana-3972	77	31	a	a	DET
cana-3972	77	32	direct	direct	ADJ
cana-3972	77	33	summand	summand	NOUN
cana-3972	77	34	of	of	ADP
cana-3972	77	35	n	n	CCONJ
cana-3972	77	36	,	,	PUNCT
cana-3972	77	37	without	without	ADP
cana-3972	77	38	assuming	assume	VERB
cana-3972	77	39	that	that	SCONJ
cana-3972	77	40	k	k	PROPN
cana-3972	77	41	is	be	AUX
cana-3972	77	42	cosingular	cosingular	ADJ
cana-3972	77	43	.	.	PUNCT
cana-3972	78	1	in	in	ADP
cana-3972	78	2	this	this	DET
cana-3972	78	3	case	case	NOUN
cana-3972	78	4	,	,	PUNCT
cana-3972	78	5	the	the	DET
cana-3972	78	6	map	map	NOUN
cana-3972	78	7	𝑓	𝑓	DET
cana-3972	78	8	∶	∶	NOUN
cana-3972	78	9	ℚ	ℚ	PROPN
cana-3972	78	10	→	→	SYM
cana-3972	78	11	ℤ	ℤ	PROPN
cana-3972	78	12	(	(	PUNCT
cana-3972	78	13	in	in	ADP
cana-3972	78	14	the	the	DET
cana-3972	78	15	previous	previous	ADJ
cana-3972	78	16	step	step	NOUN
cana-3972	78	17	)	)	PUNCT
cana-3972	78	18	satisfies	satisfy	VERB
cana-3972	78	19	the	the	DET
cana-3972	78	20	𝐷4	𝐷4	NOUN
cana-3972	78	21	-	-	PUNCT
cana-3972	78	22	condition	condition	NOUN
cana-3972	78	23	,	,	PUNCT
cana-3972	78	24	because	because	SCONJ
cana-3972	78	25	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	78	26	)	)	PUNCT
cana-3972	78	27	=	=	SYM
cana-3972	78	28	ℚ	ℚ	PROPN
cana-3972	78	29	and	and	CCONJ
cana-3972	78	30	it	it	PRON
cana-3972	78	31	is	be	AUX
cana-3972	78	32	a	a	DET
cana-3972	78	33	direct	direct	ADJ
cana-3972	78	34	summand	summand	NOUN
cana-3972	78	35	of	of	ADP
cana-3972	78	36	𝑁	𝑁	PROPN
cana-3972	78	37	=	=	PROPN
cana-3972	78	38	ℚ	ℚ	PROPN
cana-3972	78	39	.	.	PUNCT
cana-3972	79	1	however	however	ADV
cana-3972	79	2	,	,	PUNCT
cana-3972	79	3	if	if	SCONJ
cana-3972	79	4	𝐾	𝐾	PROPN
cana-3972	79	5	=	=	SYM
cana-3972	79	6	ℤ	ℤ	PROPN
cana-3972	79	7	which	which	PRON
cana-3972	79	8	is	be	AUX
cana-3972	79	9	not	not	PART
cana-3972	79	10	cosingular	cosingular	ADJ
cana-3972	79	11	,	,	PUNCT
cana-3972	79	12	the	the	DET
cana-3972	79	13	kernel	kernel	NOUN
cana-3972	79	14	might	might	AUX
cana-3972	79	15	fail	fail	VERB
cana-3972	79	16	to	to	PART
cana-3972	79	17	be	be	AUX
cana-3972	79	18	a	a	DET
cana-3972	79	19	direct	direct	ADJ
cana-3972	79	20	summand	summand	NOUN
cana-3972	79	21	of	of	ADP
cana-3972	79	22	n.	n.	PROPN
cana-3972	79	23	therefore	therefore	ADV
cana-3972	79	24	,	,	PUNCT
cana-3972	79	25	the	the	DET
cana-3972	79	26	d4	d4	PROPN
cana-3972	79	27	-	-	PUNCT
cana-3972	79	28	condition	condition	NOUN
cana-3972	79	29	does	do	AUX
cana-3972	79	30	not	not	PART
cana-3972	79	31	automatical	automatical	VERB
cana-3972	79	32	ly	ly	ADP
cana-3972	79	33	hold	hold	NOUN
cana-3972	79	34	unless	unless	SCONJ
cana-3972	79	35	we	we	PRON
cana-3972	79	36	impose	impose	VERB
cana-3972	79	37	the	the	DET
cana-3972	79	38	cosingularity	cosingularity	NOUN
cana-3972	79	39	condition	condition	NOUN
cana-3972	79	40	on	on	ADP
cana-3972	79	41	k	k	PROPN
cana-3972	79	42	,	,	PUNCT
cana-3972	79	43	which	which	PRON
cana-3972	79	44	is	be	AUX
cana-3972	79	45	present	present	ADJ
cana-3972	79	46	in	in	ADP
cana-3972	79	47	the	the	DET
cana-3972	79	48	d41	d41	NOUN
cana-3972	79	49	-	-	PUNCT
cana-3972	79	50	condition	condition	NOUN
cana-3972	79	51	.	.	PUNCT
cana-3972	80	1	thus	thus	ADV
cana-3972	80	2	,	,	PUNCT
cana-3972	80	3	𝑀	𝑀	PROPN
cana-3972	80	4	=	=	PROPN
cana-3972	80	5	ℚ	ℚ	PROPN
cana-3972	80	6	⊕	⊕	PROPN
cana-3972	80	7	ℤ	ℤ	PROPN
cana-3972	80	8	is	be	AUX
cana-3972	80	9	an	an	DET
cana-3972	80	10	example	example	NOUN
cana-3972	80	11	of	of	ADP
cana-3972	80	12	a	a	DET
cana-3972	80	13	d41	d41	NOUN
cana-3972	80	14	-	-	PUNCT
cana-3972	80	15	module	module	NOUN
cana-3972	80	16	that	that	PRON
cana-3972	80	17	is	be	AUX
cana-3972	80	18	not	not	PART
cana-3972	80	19	a	a	DET
cana-3972	80	20	d4	d4	NOUN
cana-3972	80	21	-	-	PUNCT
cana-3972	80	22	module	module	NOUN
cana-3972	80	23	.	.	PUNCT
cana-3972	81	1	the	the	DET
cana-3972	81	2	cosingularity	cosingularity	NOUN
cana-3972	81	3	of	of	ADP
cana-3972	81	4	k	k	PROPN
cana-3972	81	5	is	be	AUX
cana-3972	81	6	what	what	PRON
cana-3972	81	7	ensures	ensure	VERB
cana-3972	81	8	the	the	DET
cana-3972	81	9	kernel	kernel	NOUN
cana-3972	81	10	is	be	AUX
cana-3972	81	11	a	a	DET
cana-3972	81	12	direct	direct	ADJ
cana-3972	81	13	summand	summand	NOUN
cana-3972	81	14	in	in	ADP
cana-3972	81	15	the	the	DET
cana-3972	81	16	d41	d41	NOUN
cana-3972	81	17	-	-	PUNCT
cana-3972	81	18	case	case	NOUN
cana-3972	81	19	,	,	PUNCT
cana-3972	81	20	but	but	CCONJ
cana-3972	81	21	this	this	DET
cana-3972	81	22	condition	condition	NOUN
cana-3972	81	23	is	be	AUX
cana-3972	81	24	not	not	PART
cana-3972	81	25	guaranteed	guarantee	VERB
cana-3972	81	26	in	in	ADP
cana-3972	81	27	the	the	DET
cana-3972	81	28	d4	d4	NOUN
cana-3972	81	29	-	-	PUNCT
cana-3972	81	30	case	case	NOUN
cana-3972	81	31	.	.	PUNCT
cana-3972	82	1	proposition	proposition	NOUN
cana-3972	82	2	3.4	3.4	NUM
cana-3972	82	3	.	.	PUNCT
cana-3972	83	1	let	let	VERB
cana-3972	83	2	m	m	PRON
cana-3972	83	3	be	be	AUX
cana-3972	83	4	a	a	DET
cana-3972	83	5	d41	d41	NOUN
cana-3972	83	6	-	-	PUNCT
cana-3972	83	7	module	module	NOUN
cana-3972	83	8	.	.	PUNCT
cana-3972	84	1	then	then	ADV
cana-3972	84	2	any	any	DET
cana-3972	84	3	direct	direct	ADJ
cana-3972	84	4	summand	summand	NOUN
cana-3972	84	5	of	of	ADP
cana-3972	84	6	m	m	PROPN
cana-3972	84	7	is	be	AUX
cana-3972	84	8	a	a	DET
cana-3972	84	9	d41	d41	NOUN
cana-3972	84	10	-	-	PUNCT
cana-3972	84	11	module	module	NOUN
cana-3972	84	12	.	.	PUNCT
cana-3972	85	1	proof	proof	NOUN
cana-3972	85	2	:	:	PUNCT
cana-3972	85	3	let	let	VERB
cana-3972	85	4	m	m	PRON
cana-3972	85	5	be	be	AUX
cana-3972	85	6	a	a	DET
cana-3972	85	7	d41	d41	NOUN
cana-3972	85	8	-	-	PUNCT
cana-3972	85	9	module	module	NOUN
cana-3972	85	10	and	and	CCONJ
cana-3972	85	11	n	n	PRON
cana-3972	85	12	≤	≤	NOUN
cana-3972	85	13	⊕	⊕	PROPN
cana-3972	85	14	m	m	PROPN
cana-3972	85	15	.	.	PUNCT
cana-3972	86	1	we	we	PRON
cana-3972	86	2	want	want	VERB
cana-3972	86	3	to	to	PART
cana-3972	86	4	demonstrate	demonstrate	VERB
cana-3972	86	5	that	that	SCONJ
cana-3972	86	6	if	if	SCONJ
cana-3972	86	7	𝑁	𝑁	PROPN
cana-3972	86	8	=	=	SYM
cana-3972	86	9	𝐾	𝐾	PROPN
cana-3972	86	10	⊕	⊕	PROPN
cana-3972	86	11	𝐾′	𝐾′	PUNCT
cana-3972	86	12	such	such	ADJ
cana-3972	86	13	that	that	SCONJ
cana-3972	86	14	𝐾′	𝐾′	NOUN
cana-3972	86	15	is	be	AUX
cana-3972	86	16	cosingular	cosingular	ADJ
cana-3972	86	17	and	and	CCONJ
cana-3972	86	18	𝑓	𝑓	DET
cana-3972	86	19	∶	∶	NOUN
cana-3972	86	20	𝐾	𝐾	PROPN
cana-3972	86	21	→	→	SYM
cana-3972	86	22	𝐾′	𝐾′	X
cana-3972	86	23	is	be	AUX
cana-3972	86	24	an	an	DET
cana-3972	86	25	epimorphism	epimorphism	NOUN
cana-3972	86	26	,	,	PUNCT
cana-3972	86	27	then	then	ADV
cana-3972	86	28	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	86	29	≤⊕	≤⊕	AUX
cana-3972	86	30	𝐾.	𝐾.	PROPN
cana-3972	86	31	communications	communication	NOUN
cana-3972	86	32	on	on	ADP
cana-3972	86	33	applied	apply	VERB
cana-3972	86	34	nonlinear	nonlinear	ADJ
cana-3972	86	35	analysis	analysis	NOUN
cana-3972	86	36	issn	issn	NOUN
cana-3972	86	37	:	:	PUNCT
cana-3972	86	38	1074	1074	NUM
cana-3972	86	39	-	-	PUNCT
cana-3972	86	40	133x	133x	NUM
cana-3972	86	41	vol	vol	NOUN
cana-3972	86	42	32	32	NUM
cana-3972	86	43	no	no	NOUN
cana-3972	86	44	.	.	PUNCT
cana-3972	87	1	9s	9s	NUM
cana-3972	87	2	(	(	PUNCT
cana-3972	87	3	2025	2025	NUM
cana-3972	87	4	)	)	PUNCT
cana-3972	87	5	678	678	NUM
cana-3972	88	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	88	2	consider	consider	VERB
cana-3972	88	3	𝑀	𝑀	PROPN
cana-3972	88	4	=	=	SYM
cana-3972	88	5	𝑁	𝑁	PROPN
cana-3972	88	6	⊕	⊕	PROPN
cana-3972	88	7	𝑁′	𝑁′	X
cana-3972	88	8	=	=	SYM
cana-3972	88	9	𝐾	𝐾	PROPN
cana-3972	88	10	⊕	⊕	PROPN
cana-3972	88	11	𝐾′	𝐾′	PROPN
cana-3972	88	12	⊕	⊕	PROPN
cana-3972	88	13	𝑁′	𝑁′	PROPN
cana-3972	88	14	,	,	PUNCT
cana-3972	88	15	where	where	SCONJ
cana-3972	88	16	𝑁′	𝑁′	NOUN
cana-3972	88	17	≤	≤	ADJ
cana-3972	88	18	𝑀.	𝑀.	PROPN
cana-3972	88	19	define	define	VERB
cana-3972	88	20	the	the	DET
cana-3972	88	21	canonical	canonical	ADJ
cana-3972	88	22	projection	projection	NOUN
cana-3972	88	23	𝜋	𝜋	PROPN
cana-3972	88	24	∶	∶	NOUN
cana-3972	88	25	𝐾	𝐾	PROPN
cana-3972	88	26	⊕	⊕	PROPN
cana-3972	88	27	𝑁′	𝑁′	X
cana-3972	89	1	→	→	PUNCT
cana-3972	89	2	𝐾.	𝐾.	PROPN
cana-3972	89	3	it	it	PRON
cana-3972	89	4	follows	follow	VERB
cana-3972	89	5	that	that	SCONJ
cana-3972	89	6	the	the	DET
cana-3972	89	7	composition	composition	NOUN
cana-3972	89	8	𝑓	𝑓	ADV
cana-3972	89	9	∘	∘	X
cana-3972	89	10	𝜋	𝜋	NOUN
cana-3972	89	11	∶	∶	NOUN
cana-3972	89	12	𝐾	𝐾	PROPN
cana-3972	89	13	⊕	⊕	PROPN
cana-3972	89	14	𝑁′	𝑁′	PROPN
cana-3972	89	15	→	→	SYM
cana-3972	89	16	𝐾′	𝐾′	X
cana-3972	89	17	is	be	AUX
cana-3972	89	18	also	also	ADV
cana-3972	89	19	an	an	DET
cana-3972	89	20	epimorphism	epimorphism	NOUN
cana-3972	89	21	,	,	PUNCT
cana-3972	89	22	with	with	ADP
cana-3972	89	23	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	89	24	∘	∘	ADJ
cana-3972	89	25	𝜋	𝜋	NOUN
cana-3972	89	26	)	)	PUNCT
cana-3972	89	27	=	=	VERB
cana-3972	89	28	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	89	29	⊕	⊕	PROPN
cana-3972	89	30	𝑁′.	𝑁′.	VERB
cana-3972	89	31	since	since	SCONJ
cana-3972	89	32	𝑀	𝑀	PROPN
cana-3972	89	33	=	=	PUNCT
cana-3972	89	34	(	(	PUNCT
cana-3972	89	35	𝐾	𝐾	PROPN
cana-3972	89	36	⊕	⊕	PROPN
cana-3972	89	37	𝑁′	𝑁′	PROPN
cana-3972	89	38	)	)	PUNCT
cana-3972	89	39	⊕	⊕	PROPN
cana-3972	89	40	𝐾′	𝐾′	PROPN
cana-3972	89	41	is	be	AUX
cana-3972	89	42	a	a	DET
cana-3972	89	43	d41	d41	NOUN
cana-3972	89	44	-	-	PUNCT
cana-3972	89	45	module	module	NOUN
cana-3972	89	46	,	,	PUNCT
cana-3972	89	47	we	we	PRON
cana-3972	89	48	conclude	conclude	VERB
cana-3972	89	49	that	that	DET
cana-3972	89	50	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	PROPN
cana-3972	89	51	⊕	⊕	PROPN
cana-3972	89	52	𝑁′	𝑁′	PROPN
cana-3972	89	53	≤⊕	≤⊕	NUM
cana-3972	89	54	𝐾	𝐾	PROPN
cana-3972	89	55	⊕	⊕	PROPN
cana-3972	89	56	𝑁′	𝑁′	PROPN
cana-3972	89	57	≤⊕	≤⊕	ADV
cana-3972	89	58	𝑀.	𝑀.	PROPN
cana-3972	89	59	therefore	therefore	ADV
cana-3972	89	60	,	,	PUNCT
cana-3972	89	61	it	it	PRON
cana-3972	89	62	follows	follow	VERB
cana-3972	89	63	that	that	DET
cana-3972	89	64	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	89	65	≤⊕	≤⊕	PRON
cana-3972	89	66	𝑁.	𝑁.	PROPN
cana-3972	89	67	consequently	consequently	ADV
cana-3972	89	68	,	,	PUNCT
cana-3972	89	69	n	n	PRON
cana-3972	89	70	is	be	AUX
cana-3972	89	71	a	a	DET
cana-3972	89	72	𝐷41	𝐷41	NOUN
cana-3972	89	73	-	-	PUNCT
cana-3972	89	74	module	module	NOUN
cana-3972	89	75	.	.	PUNCT
cana-3972	90	1	proposition	proposition	NOUN
cana-3972	90	2	3.5	3.5	NUM
cana-3972	90	3	.	.	PUNCT
cana-3972	91	1	if	if	SCONJ
cana-3972	91	2	𝑀	𝑀	PROPN
cana-3972	91	3	⊕	⊕	PROPN
cana-3972	91	4	𝑀	𝑀	PROPN
cana-3972	91	5	is	be	AUX
cana-3972	91	6	a	a	DET
cana-3972	91	7	d41	d41	NOUN
cana-3972	91	8	-	-	PUNCT
cana-3972	91	9	module	module	NOUN
cana-3972	91	10	with	with	ADP
cana-3972	91	11	m	m	NOUN
cana-3972	91	12	cosingular	cosingular	ADJ
cana-3972	91	13	,	,	PUNCT
cana-3972	91	14	then	then	ADV
cana-3972	91	15	m	m	VERB
cana-3972	91	16	is	be	AUX
cana-3972	91	17	a	a	DET
cana-3972	91	18	d2	d2	NOUN
cana-3972	91	19	-	-	PUNCT
cana-3972	91	20	module	module	NOUN
cana-3972	91	21	.	.	PUNCT
cana-3972	92	1	proof	proof	NOUN
cana-3972	92	2	:	:	PUNCT
cana-3972	92	3	we	we	PRON
cana-3972	92	4	need	need	VERB
cana-3972	92	5	to	to	PART
cana-3972	92	6	demonstrate	demonstrate	VERB
cana-3972	93	1	that	that	SCONJ
cana-3972	93	2	if	if	SCONJ
cana-3972	93	3	𝑁	𝑁	PROPN
cana-3972	93	4	,	,	PUNCT
cana-3972	93	5	𝐾	𝐾	PROPN
cana-3972	93	6	≤	≤	PROPN
cana-3972	93	7	𝑀	𝑀	PROPN
cana-3972	93	8	with	with	ADP
cana-3972	93	9	𝑀	𝑀	PROPN
cana-3972	93	10	𝐾⁄	𝐾⁄	PROPN
cana-3972	93	11	≅	≅	PROPN
cana-3972	93	12	𝑁	𝑁	PROPN
cana-3972	93	13	≤⊕	≤⊕	NUM
cana-3972	93	14	𝑀	𝑀	PROPN
cana-3972	93	15	,	,	PUNCT
cana-3972	93	16	t	t	PROPN
cana-3972	93	17	he	he	PRON
cana-3972	93	18	n	n	PROPN
cana-3972	93	19	𝐾	𝐾	PROPN
cana-3972	93	20	≤⊕	≤⊕	ADV
cana-3972	93	21	𝑀.	𝑀.	PROPN
cana-3972	93	22	to	to	PART
cana-3972	93	23	show	show	VERB
cana-3972	93	24	this	this	PRON
cana-3972	93	25	,	,	PUNCT
cana-3972	93	26	express	express	VERB
cana-3972	93	27	m	m	PRON
cana-3972	93	28	as	as	ADP
cana-3972	93	29	𝑀	𝑀	PROPN
cana-3972	93	30	=	=	SYM
cana-3972	93	31	𝑁	𝑁	PROPN
cana-3972	93	32	⊕	⊕	PROPN
cana-3972	93	33	𝐿	𝐿	PROPN
cana-3972	93	34	for	for	ADP
cana-3972	93	35	some	some	DET
cana-3972	93	36	l	l	NOUN
cana-3972	93	37	≤	≤	NUM
cana-3972	93	38	m	m	VERB
cana-3972	93	39	.	.	PUNCT
cana-3972	94	1	then	then	ADV
cana-3972	94	2	we	we	PRON
cana-3972	94	3	have	have	VERB
cana-3972	94	4	𝑀	𝑀	PROPN
cana-3972	94	5	⊕	⊕	PROPN
cana-3972	94	6	𝑀	𝑀	PROPN
cana-3972	94	7	≅	≅	PROPN
cana-3972	94	8	𝑀	𝑀	PROPN
cana-3972	94	9	⊕	⊕	PROPN
cana-3972	94	10	𝑁	𝑁	PROPN
cana-3972	94	11	⊕	⊕	PROPN
cana-3972	94	12	𝐿	𝐿	PROPN
cana-3972	94	13	≅	≅	PROPN
cana-3972	94	14	𝑀	𝑀	PROPN
cana-3972	94	15	⊕	⊕	PROPN
cana-3972	94	16	(	(	PUNCT
cana-3972	94	17	𝑀/𝐾	𝑀/𝐾	NOUN
cana-3972	94	18	)	)	PUNCT
cana-3972	94	19	⊕	⊕	PROPN
cana-3972	94	20	𝐿.	𝐿.	VERB
cana-3972	94	21	from	from	ADP
cana-3972	94	22	this	this	PRON
cana-3972	94	23	,	,	PUNCT
cana-3972	94	24	we	we	PRON
cana-3972	94	25	see	see	VERB
cana-3972	94	26	that	that	SCONJ
cana-3972	94	27	𝑀	𝑀	PROPN
cana-3972	94	28	⊕	⊕	PROPN
cana-3972	94	29	(	(	PUNCT
cana-3972	94	30	𝑀/𝐾	𝑀/𝐾	NOUN
cana-3972	94	31	)	)	PUNCT
cana-3972	94	32	is	be	AUX
cana-3972	94	33	a	a	DET
cana-3972	94	34	d41	d41	NOUN
cana-3972	94	35	-	-	PUNCT
cana-3972	94	36	module	module	NOUN
cana-3972	94	37	,	,	PUNCT
cana-3972	94	38	which	which	PRON
cana-3972	94	39	implies	imply	VERB
cana-3972	94	40	that	that	SCONJ
cana-3972	94	41	the	the	DET
cana-3972	94	42	natural	natural	ADJ
cana-3972	94	43	epimorphism	epimorphism	NOUN
cana-3972	94	44	m	m	PROPN
cana-3972	94	45	→	→	SYM
cana-3972	94	46	m	m	NOUN
cana-3972	94	47	/k	/k	ADJ
cana-3972	94	48	splits	split	NOUN
cana-3972	94	49	.	.	PUNCT
cana-3972	95	1	consequently	consequently	ADV
cana-3972	95	2	,	,	PUNCT
cana-3972	95	3	it	it	PRON
cana-3972	95	4	follows	follow	VERB
cana-3972	95	5	that	that	SCONJ
cana-3972	95	6	k	k	PROPN
cana-3972	95	7	≤	≤	PROPN
cana-3972	95	8	⊕	⊕	PROPN
cana-3972	95	9	m	m	PROPN
cana-3972	95	10	.	.	PUNCT
cana-3972	96	1	proposition	proposition	NOUN
cana-3972	96	2	3.6	3.6	NUM
cana-3972	96	3	.	.	PUNCT
cana-3972	97	1	if	if	SCONJ
cana-3972	97	2	m1	m1	PROPN
cana-3972	97	3	⊕	⊕	PROPN
cana-3972	97	4	m2	m2	PROPN
cana-3972	97	5	is	be	AUX
cana-3972	97	6	a	a	DET
cana-3972	97	7	d41	d41	NOUN
cana-3972	97	8	-	-	PUNCT
cana-3972	97	9	module	module	NOUN
cana-3972	97	10	and	and	CCONJ
cana-3972	97	11	there	there	PRON
cana-3972	97	12	is	be	VERB
cana-3972	97	13	an	an	DET
cana-3972	97	14	epimorphism	epimorphism	NOUN
cana-3972	97	15	𝑓	𝑓	DET
cana-3972	97	16	∶	∶	NOUN
cana-3972	97	17	𝑀1	𝑀1	NOUN
cana-3972	97	18	→	→	SYM
cana-3972	97	19	𝑀2	𝑀2	PROPN
cana-3972	97	20	with	with	ADP
cana-3972	97	21	m2	m2	PROPN
cana-3972	97	22	is	be	AUX
cana-3972	97	23	cosingular	cosingular	ADJ
cana-3972	97	24	,	,	PUNCT
cana-3972	97	25	then	then	ADV
cana-3972	97	26	m2	m2	PROPN
cana-3972	97	27	is	be	AUX
cana-3972	97	28	a	a	DET
cana-3972	97	29	d2	d2	NOUN
cana-3972	97	30	-	-	PUNCT
cana-3972	97	31	module	module	NOUN
cana-3972	97	32	.	.	PUNCT
cana-3972	98	1	proof	proof	NOUN
cana-3972	98	2	:	:	PUNCT
cana-3972	98	3	since	since	SCONJ
cana-3972	98	4	𝑀1	𝑀1	PROPN
cana-3972	98	5	⊕	⊕	PROPN
cana-3972	98	6	𝑀2	𝑀2	PROPN
cana-3972	98	7	is	be	AUX
cana-3972	98	8	a	a	DET
cana-3972	98	9	d41	d41	NOUN
cana-3972	98	10	-	-	PUNCT
cana-3972	98	11	module	module	NOUN
cana-3972	98	12	and	and	CCONJ
cana-3972	98	13	𝑓	𝑓	DET
cana-3972	98	14	∶	∶	NOUN
cana-3972	98	15	𝑀1	𝑀1	PROPN
cana-3972	98	16	→	→	SYM
cana-3972	98	17	𝑀2	𝑀2	PROPN
cana-3972	98	18	is	be	AUX
cana-3972	98	19	an	an	DET
cana-3972	98	20	epimorphism	epimorphism	NOUN
cana-3972	98	21	,	,	PUNCT
cana-3972	98	22	we	we	PRON
cana-3972	98	23	have	have	VERB
cana-3972	98	24	ker	ker	PROPN
cana-3972	98	25	f	f	PROPN
cana-3972	98	26	≤	≤	PROPN
cana-3972	98	27	⊕	⊕	PROPN
cana-3972	98	28	m1	m1	PROPN
cana-3972	98	29	.	.	PUNCT
cana-3972	99	1	we	we	PRON
cana-3972	99	2	can	can	AUX
cana-3972	99	3	express	express	VERB
cana-3972	99	4	m1	m1	PROPN
cana-3972	99	5	as	as	ADP
cana-3972	99	6	m1	m1	PROPN
cana-3972	99	7	=	=	SYM
cana-3972	99	8	n	n	PROPN
cana-3972	99	9	⊕	⊕	PROPN
cana-3972	99	10	ker	ker	PROPN
cana-3972	99	11	f	f	X
cana-3972	99	12	,	,	PUNCT
cana-3972	99	13	where	where	SCONJ
cana-3972	99	14	n	n	PRON
cana-3972	99	15	≤	≤	NOUN
cana-3972	99	16	m1	m1	NOUN
cana-3972	99	17	.	.	PUNCT
cana-3972	100	1	thus	thus	ADV
cana-3972	100	2	,	,	PUNCT
cana-3972	100	3	𝑁	𝑁	PROPN
cana-3972	100	4	≅	≅	PROPN
cana-3972	100	5	𝑀1	𝑀1	PROPN
cana-3972	100	6	ker(𝑓)⁄	ker(𝑓)⁄	PROPN
cana-3972	100	7	≅	≅	PROPN
cana-3972	100	8	𝑀2	𝑀2	PROPN
cana-3972	100	9	.	.	PUNCT
cana-3972	101	1	now	now	ADV
cana-3972	101	2	,	,	PUNCT
cana-3972	101	3	we	we	PRON
cana-3972	101	4	have	have	VERB
cana-3972	101	5	𝑀2	𝑀2	PROPN
cana-3972	101	6	⊕	⊕	PROPN
cana-3972	101	7	𝑀2	𝑀2	PROPN
cana-3972	101	8	≅	≅	PROPN
cana-3972	101	9	(	(	PUNCT
cana-3972	101	10	𝑁	𝑁	PROPN
cana-3972	101	11	⊕	⊕	PROPN
cana-3972	101	12	𝑀2	𝑀2	PROPN
cana-3972	101	13	)	)	PUNCT
cana-3972	101	14	≤	≤	PROPN
cana-3972	101	15	⊕	⊕	PROPN
cana-3972	101	16	(	(	PUNCT
cana-3972	101	17	m1	m1	PROPN
cana-3972	101	18	⊕	⊕	PROPN
cana-3972	101	19	m2	m2	PROPN
cana-3972	101	20	)	)	PUNCT
cana-3972	101	21	.	.	PUNCT
cana-3972	102	1	since	since	SCONJ
cana-3972	102	2	m2	m2	PROPN
cana-3972	102	3	⊕	⊕	PROPN
cana-3972	102	4	m2	m2	PROPN
cana-3972	102	5	is	be	AUX
cana-3972	102	6	a	a	DET
cana-3972	102	7	𝐷41	𝐷41	NOUN
cana-3972	102	8	-	-	PUNCT
cana-3972	102	9	module	module	NOUN
cana-3972	102	10	(	(	PUNCT
cana-3972	102	11	by	by	ADP
cana-3972	102	12	proposition	proposition	NOUN
cana-3972	102	13	3.4	3.4	NUM
cana-3972	102	14	)	)	PUNCT
cana-3972	102	15	,	,	PUNCT
cana-3972	102	16	i	i	PRON
cana-3972	102	17	t	t	PROPN
cana-3972	102	18	follows	follow	VERB
cana-3972	102	19	that	that	SCONJ
cana-3972	102	20	m2	m2	PROPN
cana-3972	102	21	is	be	AUX
cana-3972	102	22	a	a	DET
cana-3972	102	23	d2	d2	NOUN
cana-3972	102	24	-	-	PUNCT
cana-3972	102	25	module	module	NOUN
cana-3972	102	26	(	(	PUNCT
cana-3972	102	27	by	by	ADP
cana-3972	102	28	proposition	proposition	NOUN
cana-3972	102	29	3.5	3.5	NUM
cana-3972	102	30	)	)	PUNCT
cana-3972	102	31	.	.	PUNCT
cana-3972	103	1	recall	recall	VERB
cana-3972	103	2	that	that	SCONJ
cana-3972	103	3	a	a	DET
cana-3972	103	4	module	module	NOUN
cana-3972	103	5	m	m	VERB
cana-3972	103	6	has	have	VERB
cana-3972	103	7	the	the	DET
cana-3972	103	8	summand	summand	NOUN
cana-3972	103	9	intersection	intersection	NOUN
cana-3972	103	10	property	property	NOUN
cana-3972	103	11	(	(	PUNCT
cana-3972	103	12	sip	sip	NOUN
cana-3972	103	13	)	)	PUNCT
cana-3972	103	14	if	if	SCONJ
cana-3972	103	15	the	the	DET
cana-3972	103	16	intersection	intersection	NOUN
cana-3972	103	17	of	of	ADP
cana-3972	103	18	any	any	DET
cana-3972	103	19	two	two	NUM
cana-3972	103	20	direct	direct	ADJ
cana-3972	103	21	summands	summand	NOUN
cana-3972	103	22	of	of	ADP
cana-3972	103	23	m	m	PROPN
cana-3972	103	24	is	be	AUX
cana-3972	103	25	a	a	DET
cana-3972	103	26	direct	direct	ADJ
cana-3972	103	27	summand	summand	NOUN
cana-3972	103	28	of	of	ADP
cana-3972	103	29	m	m	PROPN
cana-3972	103	30	.	.	PUNCT
cana-3972	104	1	lemma	lemma	PROPN
cana-3972	104	2	3.7	3.7	NUM
cana-3972	104	3	.	.	PUNCT
cana-3972	105	1	[	[	X
cana-3972	105	2	9	9	NUM
cana-3972	105	3	,	,	PUNCT
cana-3972	105	4	theorem	theorem	VERB
cana-3972	105	5	1.2	1.2	NUM
cana-3972	105	6	]	]	PUNCT
cana-3972	105	7	a	a	DET
cana-3972	105	8	module	module	NOUN
cana-3972	105	9	m	m	VERB
cana-3972	105	10	has	have	VERB
cana-3972	105	11	the	the	DET
cana-3972	105	12	summand	summand	NOUN
cana-3972	105	13	intersection	intersection	NOUN
cana-3972	105	14	property	property	NOUN
cana-3972	105	15	(	(	PUNCT
cana-3972	105	16	sip	sip	NOUN
cana-3972	105	17	)	)	PUNCT
cana-3972	106	1	if	if	SCONJ
cana-3972	106	2	and	and	CCONJ
cana-3972	106	3	only	only	ADV
cana-3972	106	4	if	if	SCONJ
cana-3972	106	5	,	,	PUNCT
cana-3972	106	6	for	for	ADP
cana-3972	106	7	every	every	DET
cana-3972	106	8	decomposition	decomposition	NOUN
cana-3972	106	9	m	m	VERB
cana-3972	106	10	=	=	PUNCT
cana-3972	106	11	a	a	DET
cana-3972	106	12	⊕	⊕	PROPN
cana-3972	106	13	b	b	PROPN
cana-3972	106	14	and	and	CCONJ
cana-3972	106	15	every	every	PRON
cana-3972	106	16	𝛾	𝛾	ADP
cana-3972	106	17	∶	∶	NOUN
cana-3972	106	18	𝐴	𝐴	PROPN
cana-3972	106	19	→	→	SYM
cana-3972	106	20	𝐵	𝐵	PROPN
cana-3972	106	21	,	,	PUNCT
cana-3972	106	22	the	the	DET
cana-3972	106	23	kernel	kernel	NOUN
cana-3972	106	24	of	of	ADP
cana-3972	106	25	γ	γ	PROPN
cana-3972	106	26	is	be	AUX
cana-3972	106	27	a	a	DET
cana-3972	106	28	direct	direct	ADJ
cana-3972	106	29	summand	summand	NOUN
cana-3972	106	30	of	of	ADP
cana-3972	106	31	a	a	DET
cana-3972	106	32	corollary	corollary	ADJ
cana-3972	106	33	3.8	3.8	NUM
cana-3972	106	34	.	.	PUNCT
cana-3972	107	1	the	the	DET
cana-3972	107	2	fol	fol	NOUN
cana-3972	107	3	lowing	low	VERB
cana-3972	107	4	conditions	condition	NOUN
cana-3972	107	5	are	be	AUX
cana-3972	107	6	equivalent	equivalent	ADJ
cana-3972	107	7	for	for	ADP
cana-3972	107	8	a	a	DET
cana-3972	107	9	d41	d41	NOUN
cana-3972	107	10	-	-	PUNCT
cana-3972	107	11	module	module	NOUN
cana-3972	107	12	m.	m.	NOUN
cana-3972	107	13	1	1	NUM
cana-3972	107	14	)	)	PUNCT
cana-3972	108	1	m	m	VERB
cana-3972	108	2	has	have	VERB
cana-3972	108	3	the	the	DET
cana-3972	108	4	si	si	PROPN
cana-3972	108	5	p	p	X
cana-3972	108	6	property	property	NOUN
cana-3972	108	7	2	2	NUM
cana-3972	108	8	)	)	PUNCT
cana-3972	108	9	for	for	ADP
cana-3972	108	10	every	every	DET
cana-3972	108	11	decomposition	decomposition	NOUN
cana-3972	108	12	m	m	NOUN
cana-3972	108	13	=	=	SYM
cana-3972	108	14	n	n	PRON
cana-3972	108	15	⊕	⊕	PROPN
cana-3972	108	16	n	n	PROPN
cana-3972	108	17	and	and	CCONJ
cana-3972	108	18	for	for	ADP
cana-3972	108	19	every	every	DET
cana-3972	108	20	epimorphism	epimorphism	NOUN
cana-3972	108	21	𝑓	𝑓	DET
cana-3972	108	22	∶	∶	NOUN
cana-3972	108	23	𝑁	𝑁	PROPN
cana-3972	108	24	→	→	SYM
cana-3972	108	25	𝐾	𝐾	PROPN
cana-3972	108	26	such	such	ADJ
cana-3972	108	27	that	that	SCONJ
cana-3972	108	28	k	k	PROPN
cana-3972	108	29	is	be	AUX
cana-3972	108	30	cosingular	cosingular	ADJ
cana-3972	108	31	and	and	CCONJ
cana-3972	108	32	𝐼𝑚(𝑓	𝐼𝑚(𝑓	NOUN
cana-3972	108	33	)	)	PUNCT
cana-3972	108	34	≤	≤	PROPN
cana-3972	109	1	⊕	⊕	PROPN
cana-3972	109	2	k	k	PROPN
cana-3972	109	3	,	,	PUNCT
cana-3972	109	4	then	then	ADV
cana-3972	109	5	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	109	6	)	)	PUNCT
cana-3972	109	7	≤	≤	NUM
cana-3972	109	8	⊕	⊕	PROPN
cana-3972	109	9	n.	n.	PROPN
cana-3972	109	10	proposition	proposition	NOUN
cana-3972	109	11	3.9	3.9	NUM
cana-3972	109	12	.	.	PUNCT
cana-3972	110	1	if	if	SCONJ
cana-3972	110	2	every	every	DET
cana-3972	110	3	submodule	submodule	NOUN
cana-3972	110	4	of	of	ADP
cana-3972	110	5	a	a	DET
cana-3972	110	6	d41	d41	NOUN
cana-3972	110	7	-	-	PUNCT
cana-3972	110	8	module	module	NOUN
cana-3972	110	9	m	m	NOUN
cana-3972	110	10	is	be	AUX
cana-3972	110	11	a	a	DET
cana-3972	110	12	d41	d41	NOUN
cana-3972	110	13	-	-	PUNCT
cana-3972	110	14	module	module	NOUN
cana-3972	110	15	,	,	PUNCT
cana-3972	110	16	then	then	ADV
cana-3972	110	17	m	m	PROPN
cana-3972	110	18	has	have	VERB
cana-3972	110	19	the	the	DET
cana-3972	110	20	summand	summand	NOUN
cana-3972	110	21	intersection	intersection	NOUN
cana-3972	110	22	property	property	NOUN
cana-3972	110	23	.	.	PUNCT
cana-3972	111	1	proof	proof	NOUN
cana-3972	111	2	:	:	PUNCT
cana-3972	111	3	let	let	VERB
cana-3972	111	4	m	m	PRON
cana-3972	111	5	be	be	AUX
cana-3972	111	6	d41	d41	NOUN
cana-3972	111	7	-	-	PUNCT
cana-3972	111	8	module	module	NOUN
cana-3972	111	9	such	such	ADJ
cana-3972	111	10	that	that	DET
cana-3972	111	11	𝑀	𝑀	PROPN
cana-3972	111	12	=	=	PUNCT
cana-3972	111	13	𝑁	𝑁	PROPN
cana-3972	111	14	⊕	⊕	PROPN
cana-3972	111	15	𝐾	𝐾	PROPN
cana-3972	111	16	with	with	ADP
cana-3972	111	17	k	k	PROPN
cana-3972	111	18	cosingular	cosingular	ADJ
cana-3972	111	19	and	and	CCONJ
cana-3972	111	20	𝜆	𝜆	DET
cana-3972	111	21	∶	∶	NOUN
cana-3972	111	22	𝑁	𝑁	PROPN
cana-3972	111	23	→	→	SYM
cana-3972	111	24	𝐾.	𝐾.	PROPN
cana-3972	111	25	then	then	ADV
cana-3972	111	26	𝑁	𝑁	PROPN
cana-3972	111	27	⊕	⊕	PROPN
cana-3972	111	28	𝐼𝑚(𝜆	𝐼𝑚(𝜆	NOUN
cana-3972	111	29	)	)	PUNCT
cana-3972	111	30	is	be	AUX
cana-3972	111	31	a	a	DET
cana-3972	111	32	submodule	submodule	NOUN
cana-3972	111	33	of	of	ADP
cana-3972	111	34	m	m	PROPN
cana-3972	111	35	.	.	PUNCT
cana-3972	112	1	by	by	ADP
cana-3972	112	2	hypothesis	hypothesis	NOUN
cana-3972	112	3	,	,	PUNCT
cana-3972	112	4	𝑁	𝑁	PROPN
cana-3972	112	5	⊕	⊕	PROPN
cana-3972	112	6	𝐼𝑚(𝜆	𝐼𝑚(𝜆	NOUN
cana-3972	112	7	)	)	PUNCT
cana-3972	112	8	is	be	AUX
cana-3972	112	9	a	a	DET
cana-3972	112	10	d41	d41	NOUN
cana-3972	112	11	-	-	PUNCT
cana-3972	112	12	module	module	NOUN
cana-3972	112	13	.	.	PUNCT
cana-3972	113	1	so	so	ADV
cana-3972	113	2	,	,	PUNCT
cana-3972	113	3	the	the	DET
cana-3972	113	4	epimorphism	epimorphism	NOUN
cana-3972	113	5	𝛼	𝛼	NOUN
cana-3972	113	6	∶	∶	NOUN
cana-3972	113	7	𝑁	𝑁	PROPN
cana-3972	113	8	→	→	SYM
cana-3972	113	9	𝐼𝑚(𝜆	𝐼𝑚(𝜆	NOUN
cana-3972	113	10	)	)	PUNCT
cana-3972	113	11	splits	split	VERB
cana-3972	113	12	.	.	PUNCT
cana-3972	114	1	thus	thus	ADV
cana-3972	114	2	𝑘𝑒𝑟(𝛼	𝑘𝑒𝑟(𝛼	X
cana-3972	114	3	)	)	PUNCT
cana-3972	114	4	is	be	AUX
cana-3972	114	5	a	a	DET
cana-3972	114	6	direct	direct	ADJ
cana-3972	114	7	summand	summand	NOUN
cana-3972	114	8	of	of	ADP
cana-3972	114	9	n	n	PROPN
cana-3972	114	10	.	.	PUNCT
cana-3972	115	1	consequently	consequently	ADV
cana-3972	115	2	,	,	PUNCT
cana-3972	115	3	m	m	VERB
cana-3972	115	4	has	have	VERB
cana-3972	115	5	the	the	DET
cana-3972	115	6	summand	summand	NOUN
cana-3972	115	7	intersection	intersection	NOUN
cana-3972	115	8	property	property	NOUN
cana-3972	115	9	by	by	ADP
cana-3972	115	10	lemma	lemma	PROPN
cana-3972	115	11	3.7	3.7	NUM
cana-3972	115	12	.	.	PUNCT
cana-3972	116	1	definition	definition	NOUN
cana-3972	116	2	3.10	3.10	NUM
cana-3972	116	3	.	.	PUNCT
cana-3972	117	1	let	let	VERB
cana-3972	117	2	m	m	PRON
cana-3972	117	3	be	be	AUX
cana-3972	117	4	a	a	DET
cana-3972	117	5	left	left	ADJ
cana-3972	117	6	r	r	NOUN
cana-3972	117	7	-	-	PUNCT
cana-3972	117	8	module	module	NOUN
cana-3972	117	9	.	.	PUNCT
cana-3972	118	1	we	we	PRON
cana-3972	118	2	say	say	VERB
cana-3972	118	3	that	that	SCONJ
cana-3972	118	4	m	m	VERB
cana-3972	118	5	satisfies	satisfy	VERB
cana-3972	118	6	the	the	DET
cana-3972	118	7	condition	condition	NOUN
cana-3972	118	8	(	(	PUNCT
cana-3972	118	9	⋆	⋆	NOUN
cana-3972	118	10	)	)	PUNCT
cana-3972	118	11	if	if	SCONJ
cana-3972	118	12	m	m	NOUN
cana-3972	118	13	=	=	SYM
cana-3972	118	14	n	n	PROPN
cana-3972	118	15	⊕	⊕	PROPN
cana-3972	118	16	k	k	PROPN
cana-3972	118	17	for	for	ADP
cana-3972	118	18	some	some	DET
cana-3972	118	19	submodules	submodule	NOUN
cana-3972	118	20	n	n	PRON
cana-3972	118	21	and	and	CCONJ
cana-3972	118	22	k	k	PROPN
cana-3972	118	23	of	of	ADP
cana-3972	118	24	m	m	PROPN
cana-3972	118	25	with	with	ADP
cana-3972	118	26	k	k	PROPN
cana-3972	118	27	being	be	AUX
cana-3972	118	28	cosingular	cosingular	ADJ
cana-3972	118	29	,	,	PUNCT
cana-3972	118	30	and	and	CCONJ
cana-3972	118	31	𝑓	𝑓	DET
cana-3972	118	32	∶	∶	NOUN
cana-3972	118	33	𝑁	𝑁	PROPN
cana-3972	118	34	→	→	SYM
cana-3972	118	35	𝐾	𝐾	PROPN
cana-3972	118	36	is	be	AUX
cana-3972	118	37	an	an	DET
cana-3972	118	38	r	r	NOUN
cana-3972	118	39	-	-	PUNCT
cana-3972	118	40	epimorphism	epimorphism	NOUN
cana-3972	118	41	while	while	SCONJ
cana-3972	118	42	𝑖𝑑	𝑖𝑑	SCONJ
cana-3972	118	43	∶	∶	NOUN
cana-3972	118	44	𝐾	𝐾	PROPN
cana-3972	118	45	→	→	SYM
cana-3972	118	46	𝐾	𝐾	PROPN
cana-3972	118	47	is	be	AUX
cana-3972	118	48	the	the	DET
cana-3972	118	49	identity	identity	NOUN
cana-3972	118	50	map	map	NOUN
cana-3972	118	51	,	,	PUNCT
cana-3972	118	52	then	then	ADV
cana-3972	118	53	there	there	PRON
cana-3972	118	54	exists	exist	VERB
cana-3972	118	55	an	an	DET
cana-3972	118	56	r	r	NOUN
cana-3972	118	57	-	-	PUNCT
cana-3972	118	58	homomorphism	homomorphism	NOUN
cana-3972	118	59	f	f	NOUN
cana-3972	119	1	′	′	NUM
cana-3972	119	2	:	:	PUNCT
cana-3972	120	1	k	k	X
cana-3972	120	2	→	→	PUNCT
cana-3972	120	3	n	n	X
cana-3972	120	4	such	such	ADJ
cana-3972	120	5	that	that	SCONJ
cana-3972	120	6	f	f	PROPN
cana-3972	120	7	◦	◦	NOUN
cana-3972	120	8	f	f	X
cana-3972	121	1	′	′	NUM
cana-3972	122	1	=	=	PUNCT
cana-3972	123	1	i	i	PRON
cana-3972	123	2	d.	d.	NOUN
cana-3972	123	3	it	it	PRON
cana-3972	123	4	is	be	AUX
cana-3972	123	5	clear	clear	ADJ
cana-3972	123	6	that	that	SCONJ
cana-3972	123	7	every	every	DET
cana-3972	123	8	d41	d41	NOUN
cana-3972	123	9	-	-	PUNCT
cana-3972	123	10	module	module	NOUN
cana-3972	123	11	satisfies	satisfie	NOUN
cana-3972	123	12	condition	condition	NOUN
cana-3972	123	13	(	(	PUNCT
cana-3972	123	14	⋆	⋆	NOUN
cana-3972	123	15	)	)	PUNCT
cana-3972	123	16	.	.	PUNCT
cana-3972	124	1	communications	communication	NOUN
cana-3972	124	2	on	on	ADP
cana-3972	124	3	applied	apply	VERB
cana-3972	124	4	nonlinear	nonlinear	ADJ
cana-3972	124	5	analysis	analysis	NOUN
cana-3972	124	6	issn	issn	NOUN
cana-3972	124	7	:	:	PUNCT
cana-3972	124	8	1074	1074	NUM
cana-3972	124	9	-	-	PUNCT
cana-3972	124	10	133x	133x	NUM
cana-3972	124	11	vol	vol	NOUN
cana-3972	124	12	32	32	NUM
cana-3972	124	13	no	no	NOUN
cana-3972	124	14	.	.	PUNCT
cana-3972	125	1	9s	9s	NUM
cana-3972	125	2	(	(	PUNCT
cana-3972	125	3	2025	2025	NUM
cana-3972	125	4	)	)	PUNCT
cana-3972	125	5	679	679	NUM
cana-3972	126	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	126	2	remark	remark	NOUN
cana-3972	126	3	3.11	3.11	NUM
cana-3972	126	4	.	.	PUNCT
cana-3972	127	1	given	give	VERB
cana-3972	127	2	that	that	DET
cana-3972	127	3	morita	morita	PROPN
cana-3972	127	4	equivalence	equivalence	NOUN
cana-3972	127	5	preserves	preserve	VERB
cana-3972	127	6	summands	summand	VERB
cana-3972	127	7	,	,	PUNCT
cana-3972	127	8	epimorphisms	epimorphism	NOUN
cana-3972	127	9	,	,	PUNCT
cana-3972	127	10	cosingularity	cosingularity	NOUN
cana-3972	127	11	,	,	PUNCT
cana-3972	127	12	and	and	CCONJ
cana-3972	127	13	isomorphisms	isomorphism	NOUN
cana-3972	127	14	,	,	PUNCT
cana-3972	127	15	it	it	PRON
cana-3972	127	16	follows	follow	VERB
cana-3972	127	17	that	that	SCONJ
cana-3972	127	18	if	if	SCONJ
cana-3972	127	19	r	r	NOUN
cana-3972	127	20	and	and	CCONJ
cana-3972	127	21	s	s	NOUN
cana-3972	127	22	are	be	AUX
cana-3972	127	23	morita	morita	NOUN
cana-3972	127	24	-	-	PUNCT
cana-3972	127	25	equivalent	equivalent	ADJ
cana-3972	127	26	rings	ring	NOUN
cana-3972	127	27	related	relate	VERB
cana-3972	127	28	by	by	ADP
cana-3972	127	29	a	a	DET
cana-3972	127	30	category	category	NOUN
cana-3972	127	31	equivalence	equivalence	NOUN
cana-3972	127	32	ϕ	ϕ	NOUN
cana-3972	127	33	:	:	PUNCT
cana-3972	127	34	mod	mod	ADJ
cana-3972	127	35	-	-	PUNCT
cana-3972	127	36	r	r	NOUN
cana-3972	127	37	→	→	SYM
cana-3972	127	38	mod	mod	PROPN
cana-3972	127	39	-	-	PUNCT
cana-3972	127	40	s	s	NOUN
cana-3972	127	41	,	,	PUNCT
cana-3972	127	42	then	then	ADV
cana-3972	127	43	a	a	DET
cana-3972	127	44	left	left	ADJ
cana-3972	127	45	r	r	NOUN
cana-3972	127	46	-	-	PUNCT
cana-3972	127	47	module	module	NOUN
cana-3972	127	48	𝑀𝑅	𝑀𝑅	PROPN
cana-3972	127	49	satisfies	satisfy	VERB
cana-3972	127	50	condition	condition	NOUN
cana-3972	127	51	(	(	PUNCT
cana-3972	127	52	⋆	⋆	NOUN
cana-3972	127	53	)	)	PUNCT
cana-3972	128	1	if	if	SCONJ
cana-3972	129	1	and	and	CCONJ
cana-3972	129	2	only	only	ADV
cana-3972	129	3	if	if	SCONJ
cana-3972	129	4	ϕ(mr)s	ϕ(mr)s	NOUN
cana-3972	129	5	also	also	ADV
cana-3972	129	6	satisfies	satisfy	VERB
cana-3972	129	7	condition	condition	NOUN
cana-3972	129	8	(	(	PUNCT
cana-3972	129	9	⋆	⋆	NOUN
cana-3972	129	10	)	)	PUNCT
cana-3972	129	11	.	.	PUNCT
cana-3972	130	1	proposition	proposition	NOUN
cana-3972	130	2	3.12	3.12	NUM
cana-3972	130	3	.	.	PUNCT
cana-3972	131	1	let	let	VERB
cana-3972	131	2	m	m	PROPN
cana-3972	131	3	⊕	⊕	PROPN
cana-3972	131	4	m	m	VERB
cana-3972	131	5	be	be	AUX
cana-3972	131	6	a	a	DET
cana-3972	131	7	cosingular	cosingular	ADJ
cana-3972	131	8	d41	d41	NOUN
cana-3972	131	9	-	-	PUNCT
cana-3972	131	10	module	module	NOUN
cana-3972	131	11	.	.	PUNCT
cana-3972	132	1	if	if	SCONJ
cana-3972	132	2	m	m	NOUN
cana-3972	132	3	is	be	AUX
cana-3972	132	4	a	a	DET
cana-3972	132	5	dual	dual	ADJ
cana-3972	132	6	-	-	PUNCT
cana-3972	132	7	rickart	rickart	NOUN
cana-3972	132	8	module	module	NOUN
cana-3972	132	9	,	,	PUNCT
cana-3972	132	10	then	then	ADV
cana-3972	132	11	endr(m	endr(m	PROPN
cana-3972	132	12	)	)	PUNCT
cana-3972	132	13	is	be	AUX
cana-3972	132	14	a	a	DET
cana-3972	132	15	von	von	PROPN
cana-3972	132	16	neumann	neumann	PROPN
cana-3972	132	17	regular	regular	ADJ
cana-3972	132	18	ring	ring	NOUN
cana-3972	132	19	.	.	PUNCT
cana-3972	133	1	proof	proof	NOUN
cana-3972	133	2	:	:	PUNCT
cana-3972	133	3	let	let	VERB
cana-3972	133	4	m	m	PROPN
cana-3972	133	5	⊕	⊕	PROPN
cana-3972	133	6	m	m	VERB
cana-3972	133	7	be	be	AUX
cana-3972	133	8	a	a	DET
cana-3972	133	9	d41	d41	NOUN
cana-3972	133	10	-	-	PUNCT
cana-3972	133	11	module	module	NOUN
cana-3972	133	12	and	and	CCONJ
cana-3972	133	13	f	f	NOUN
cana-3972	133	14	:	:	PUNCT
cana-3972	133	15	m	m	AUX
cana-3972	133	16	→	→	SYM
cana-3972	133	17	m	m	VERB
cana-3972	133	18	an	an	DET
cana-3972	133	19	endomorphism	endomorphism	NOUN
cana-3972	133	20	.	.	PUNCT
cana-3972	134	1	we	we	PRON
cana-3972	134	2	have	have	VERB
cana-3972	134	3	𝑀	𝑀	PROPN
cana-3972	134	4	ker	ker	NOUN
cana-3972	134	5	(	(	PUNCT
cana-3972	134	6	𝑓	𝑓	X
cana-3972	134	7	)	)	PUNCT
cana-3972	134	8	≅	≅	PROPN
cana-3972	134	9	𝐼𝑚(𝑓	𝐼𝑚(𝑓	NUM
cana-3972	134	10	)	)	PUNCT
cana-3972	134	11	.	.	PUNCT
cana-3972	135	1	since	since	SCONJ
cana-3972	135	2	𝐼𝑚(𝑓	𝐼𝑚(𝑓	NOUN
cana-3972	135	3	)	)	PUNCT
cana-3972	135	4	is	be	AUX
cana-3972	135	5	a	a	DET
cana-3972	135	6	direct	direct	ADJ
cana-3972	135	7	summand	summand	NOUN
cana-3972	135	8	of	of	ADP
cana-3972	135	9	m	m	PROPN
cana-3972	135	10	(	(	PUNCT
cana-3972	135	11	due	due	ADP
cana-3972	135	12	to	to	ADP
cana-3972	135	13	the	the	DET
cana-3972	135	14	dual	dual	ADJ
cana-3972	135	15	-	-	PUNCT
cana-3972	135	16	rickart	rickart	NOUN
cana-3972	135	17	condition	condition	NOUN
cana-3972	135	18	)	)	PUNCT
cana-3972	135	19	,	,	PUNCT
cana-3972	135	20	we	we	PRON
cana-3972	135	21	can	can	AUX
cana-3972	135	22	express	express	VERB
cana-3972	135	23	m	m	PRON
cana-3972	135	24	as	as	ADP
cana-3972	135	25	𝐼𝑚(𝑓	𝐼𝑚(𝑓	NOUN
cana-3972	135	26	)	)	PUNCT
cana-3972	135	27	⊕	⊕	PROPN
cana-3972	136	1	𝐾	𝐾	PROPN
cana-3972	136	2	≅	≅	NUM
cana-3972	136	3	𝑀/𝑘𝑒𝑟(𝑓	𝑀/𝑘𝑒𝑟(𝑓	NOUN
cana-3972	136	4	)	)	PUNCT
cana-3972	136	5	⊕	⊕	PROPN
cana-3972	136	6	𝐾	𝐾	PROPN
cana-3972	136	7	for	for	ADP
cana-3972	136	8	some	some	DET
cana-3972	136	9	k	k	PROPN
cana-3972	136	10	≤	≤	NUM
cana-3972	136	11	m	m	VERB
cana-3972	136	12	.	.	PUNCT
cana-3972	137	1	therefore	therefore	ADV
cana-3972	137	2	,	,	PUNCT
cana-3972	137	3	we	we	PRON
cana-3972	137	4	obtain	obtain	VERB
cana-3972	137	5	:	:	PUNCT
cana-3972	137	6	𝑀	𝑀	PROPN
cana-3972	137	7	⊕	⊕	PROPN
cana-3972	137	8	𝑀	𝑀	PROPN
cana-3972	137	9	=	=	SYM
cana-3972	137	10	𝑀	𝑀	PROPN
cana-3972	137	11	⊕	⊕	PROPN
cana-3972	137	12	𝐼𝑚(𝑓	𝐼𝑚(𝑓	NOUN
cana-3972	138	1	)	)	PUNCT
cana-3972	138	2	⊕	⊕	PROPN
cana-3972	138	3	𝐾	𝐾	PROPN
cana-3972	138	4	≅	≅	PROPN
cana-3972	138	5	𝑀	𝑀	PROPN
cana-3972	138	6	⊕	⊕	PROPN
cana-3972	138	7	𝑀/𝑘𝑒𝑟(𝑓	𝑀/𝑘𝑒𝑟(𝑓	NOUN
cana-3972	138	8	)	)	PUNCT
cana-3972	138	9	⊕	⊕	PROPN
cana-3972	138	10	𝐾.	𝐾.	PROPN
cana-3972	138	11	since	since	SCONJ
cana-3972	138	12	m	m	PROPN
cana-3972	138	13	⊕	⊕	PROPN
cana-3972	138	14	m	m	PROPN
cana-3972	138	15	is	be	AUX
cana-3972	138	16	a	a	DET
cana-3972	138	17	d41	d41	NOUN
cana-3972	138	18	-	-	PUNCT
cana-3972	138	19	module	module	NOUN
cana-3972	138	20	,	,	PUNCT
cana-3972	138	21	i	i	PRON
cana-3972	138	22	t	t	PROPN
cana-3972	138	23	follows	follow	VERB
cana-3972	138	24	that	that	SCONJ
cana-3972	138	25	𝑀	𝑀	PROPN
cana-3972	138	26	⊕	⊕	PROPN
cana-3972	138	27	𝑀/𝑘𝑒𝑟(𝑓	𝑀/𝑘𝑒𝑟(𝑓	NOUN
cana-3972	138	28	)	)	PUNCT
cana-3972	138	29	is	be	AUX
cana-3972	138	30	also	also	ADV
cana-3972	138	31	a	a	DET
cana-3972	138	32	d41	d41	NOUN
cana-3972	138	33	-	-	PUNCT
cana-3972	138	34	module	module	NOUN
cana-3972	138	35	,	,	PUNCT
cana-3972	138	36	which	which	PRON
cana-3972	138	37	implies	imply	VERB
cana-3972	138	38	that	that	SCONJ
cana-3972	138	39	the	the	DET
cana-3972	138	40	epimorphism	epimorphism	NOUN
cana-3972	138	41	𝑔	𝑔	PROPN
cana-3972	138	42	∶	∶	PROPN
cana-3972	138	43	𝑀	𝑀	PROPN
cana-3972	138	44	→	→	SYM
cana-3972	138	45	𝑀/𝑘𝑒𝑟(𝑓	𝑀/𝑘𝑒𝑟(𝑓	NOUN
cana-3972	138	46	)	)	PUNCT
cana-3972	138	47	splits	split	VERB
cana-3972	138	48	.	.	PUNCT
cana-3972	139	1	as	as	ADP
cana-3972	139	2	a	a	DET
cana-3972	139	3	result	result	NOUN
cana-3972	139	4	,	,	PUNCT
cana-3972	139	5	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	139	6	)	)	PUNCT
cana-3972	139	7	is	be	AUX
cana-3972	139	8	a	a	DET
cana-3972	139	9	direct	direct	ADJ
cana-3972	139	10	summand	summand	NOUN
cana-3972	139	11	of	of	ADP
cana-3972	139	12	m	m	PROPN
cana-3972	139	13	.	.	PUNCT
cana-3972	140	1	hence	hence	ADV
cana-3972	140	2	,	,	PUNCT
cana-3972	140	3	endr(m	endr(m	PROPN
cana-3972	140	4	)	)	PUNCT
cana-3972	140	5	is	be	AUX
cana-3972	140	6	a	a	DET
cana-3972	140	7	von	von	PROPN
cana-3972	140	8	neumann	neumann	PROPN
cana-3972	140	9	regular	regular	ADJ
cana-3972	140	10	ring	ring	NOUN
cana-3972	140	11	.	.	PUNCT
cana-3972	141	1	proposition	proposition	NOUN
cana-3972	141	2	3.13	3.13	NUM
cana-3972	141	3	.	.	PUNCT
cana-3972	142	1	the	the	DET
cana-3972	142	2	fol	fol	NOUN
cana-3972	142	3	lowing	lowing	NOUN
cana-3972	142	4	statements	statement	NOUN
cana-3972	142	5	are	be	AUX
cana-3972	142	6	equivalent	equivalent	ADJ
cana-3972	142	7	for	for	ADP
cana-3972	142	8	a	a	DET
cana-3972	142	9	ring	ring	NOUN
cana-3972	142	10	r	r	NOUN
cana-3972	142	11	and	and	CCONJ
cana-3972	142	12	any	any	DET
cana-3972	142	13	finitely	finitely	ADV
cana-3972	142	14	generated	generate	VERB
cana-3972	142	15	module	module	NOUN
cana-3972	142	16	m	m	PROPN
cana-3972	142	17	:	:	PUNCT
cana-3972	142	18	1	1	X
cana-3972	142	19	)	)	PUNCT
cana-3972	142	20	every	every	DET
cana-3972	142	21	r	r	NOUN
cana-3972	142	22	-	-	PUNCT
cana-3972	142	23	module	module	NOUN
cana-3972	142	24	is	be	AUX
cana-3972	142	25	rickart	rickart	NOUN
cana-3972	142	26	;	;	PUNCT
cana-3972	142	27	2	2	X
cana-3972	142	28	)	)	PUNCT
cana-3972	142	29	every	every	DET
cana-3972	142	30	r	r	NOUN
cana-3972	142	31	-	-	PUNCT
cana-3972	142	32	module	module	NOUN
cana-3972	142	33	has	have	VERB
cana-3972	142	34	the	the	DET
cana-3972	142	35	si	si	PROPN
cana-3972	142	36	p	p	X
cana-3972	142	37	property	property	NOUN
cana-3972	142	38	;	;	PUNCT
cana-3972	142	39	3	3	X
cana-3972	142	40	)	)	PUNCT
cana-3972	142	41	every	every	DET
cana-3972	142	42	submodule	submodule	NOUN
cana-3972	142	43	of	of	ADP
cana-3972	142	44	a	a	DET
cana-3972	142	45	cosingular	cosingular	ADJ
cana-3972	142	46	and	and	CCONJ
cana-3972	142	47	finitely	finitely	ADV
cana-3972	142	48	generated	generate	VERB
cana-3972	142	49	r	r	NOUN
cana-3972	142	50	-	-	PUNCT
cana-3972	142	51	module	module	NOUN
cana-3972	142	52	is	be	AUX
cana-3972	142	53	a	a	DET
cana-3972	142	54	summand	summand	NOUN
cana-3972	142	55	;	;	PUNCT
cana-3972	142	56	4	4	X
cana-3972	142	57	)	)	PUNCT
cana-3972	142	58	every	every	DET
cana-3972	142	59	r	r	NOUN
cana-3972	142	60	-	-	PUNCT
cana-3972	142	61	module	module	NOUN
cana-3972	142	62	is	be	AUX
cana-3972	142	63	a	a	DET
cana-3972	142	64	d41	d41	NOUN
cana-3972	142	65	-	-	PUNCT
cana-3972	142	66	module	module	NOUN
cana-3972	142	67	.	.	PUNCT
cana-3972	143	1	proof	proof	NOUN
cana-3972	143	2	:	:	PUNCT
cana-3972	143	3	1	1	X
cana-3972	143	4	)	)	PUNCT
cana-3972	143	5	⇒	⇒	NOUN
cana-3972	143	6	2	2	NUM
cana-3972	143	7	)	)	PUNCT
cana-3972	143	8	.	.	PUNCT
cana-3972	144	1	every	every	DET
cana-3972	144	2	rickart	rickart	NOUN
cana-3972	144	3	module	module	NOUN
cana-3972	144	4	has	have	VERB
cana-3972	144	5	the	the	DET
cana-3972	144	6	si	si	PROPN
cana-3972	144	7	p	p	PROPN
cana-3972	144	8	property	property	NOUN
cana-3972	144	9	(	(	PUNCT
cana-3972	144	10	see	see	VERB
cana-3972	144	11	[	[	X
cana-3972	144	12	12	12	NUM
cana-3972	144	13	,	,	PUNCT
cana-3972	144	14	proposition	proposition	NOUN
cana-3972	144	15	2.16	2.16	NUM
cana-3972	144	16	]	]	PUNCT
cana-3972	144	17	)	)	PUNCT
cana-3972	144	18	.	.	PUNCT
cana-3972	145	1	2	2	X
cana-3972	145	2	)	)	PUNCT
cana-3972	145	3	⇒	⇒	NOUN
cana-3972	145	4	3	3	NUM
cana-3972	145	5	)	)	PUNCT
cana-3972	145	6	.	.	PUNCT
cana-3972	146	1	let	let	VERB
cana-3972	146	2	n	n	PRON
cana-3972	147	1	and	and	CCONJ
cana-3972	147	2	k	k	PROPN
cana-3972	147	3	be	be	AUX
cana-3972	147	4	finitely	finitely	ADV
cana-3972	147	5	generated	generate	VERB
cana-3972	147	6	with	with	ADP
cana-3972	147	7	n	n	PRON
cana-3972	147	8	≤	≤	NOUN
cana-3972	147	9	m	m	VERB
cana-3972	147	10	.	.	PUNCT
cana-3972	148	1	by	by	ADP
cana-3972	148	2	hypothesis	hypothesis	NOUN
cana-3972	148	3	,	,	PUNCT
cana-3972	148	4	m	m	PROPN
cana-3972	148	5	⊕	⊕	PROPN
cana-3972	148	6	n	n	PRON
cana-3972	148	7	has	have	VERB
cana-3972	148	8	the	the	DET
cana-3972	148	9	si	si	PROPN
cana-3972	148	10	p	p	NOUN
cana-3972	148	11	property	property	NOUN
cana-3972	148	12	.	.	PUNCT
cana-3972	149	1	thus	thus	ADV
cana-3972	149	2	,	,	PUNCT
cana-3972	149	3	m	m	NOUN
cana-3972	149	4	∩	∩	NOUN
cana-3972	149	5	n	n	PRON
cana-3972	149	6	is	be	AUX
cana-3972	149	7	a	a	DET
cana-3972	149	8	direct	direct	ADJ
cana-3972	149	9	summand	summand	NOUN
cana-3972	149	10	of	of	ADP
cana-3972	149	11	m	m	PROPN
cana-3972	149	12	⊕	⊕	PROPN
cana-3972	149	13	n	n	PROPN
cana-3972	149	14	,	,	PUNCT
cana-3972	149	15	leading	lead	VERB
cana-3972	149	16	to	to	ADP
cana-3972	149	17	n	n	PRON
cana-3972	149	18	≤	≤	NOUN
cana-3972	149	19	⊕	⊕	PROPN
cana-3972	149	20	m	m	PROPN
cana-3972	149	21	.	.	PUNCT
cana-3972	150	1	3	3	X
cana-3972	150	2	)	)	PUNCT
cana-3972	150	3	⇒	⇒	NOUN
cana-3972	150	4	4	4	NUM
cana-3972	150	5	)	)	PUNCT
cana-3972	150	6	.	.	PUNCT
cana-3972	151	1	let	let	VERB
cana-3972	151	2	n	n	PRON
cana-3972	151	3	and	and	CCONJ
cana-3972	151	4	k	k	PROPN
cana-3972	151	5	be	be	AUX
cana-3972	151	6	direct	direct	ADJ
cana-3972	151	7	summands	summand	NOUN
cana-3972	151	8	of	of	ADP
cana-3972	151	9	a	a	DET
cana-3972	151	10	finitely	finitely	ADV
cana-3972	151	11	generated	generate	VERB
cana-3972	151	12	r	r	NOUN
cana-3972	151	13	-	-	PUNCT
cana-3972	151	14	module	module	NOUN
cana-3972	151	15	m	m	NOUN
cana-3972	151	16	with	with	ADP
cana-3972	151	17	n	n	PROPN
cana-3972	152	1	+	+	CCONJ
cana-3972	152	2	k	k	X
cana-3972	152	3	=	=	VERB
cana-3972	152	4	m	m	PROPN
cana-3972	152	5	and	and	CCONJ
cana-3972	152	6	k	k	PROPN
cana-3972	152	7	cosingular	cosingular	ADJ
cana-3972	152	8	.	.	PUNCT
cana-3972	153	1	then	then	ADV
cana-3972	153	2	,	,	PUNCT
cana-3972	153	3	n	n	CCONJ
cana-3972	153	4	∩	∩	NOUN
cana-3972	153	5	k	k	PROPN
cana-3972	153	6	is	be	AUX
cana-3972	153	7	finitely	finitely	ADV
cana-3972	153	8	generated	generate	VERB
cana-3972	153	9	,	,	PUNCT
cana-3972	153	10	which	which	PRON
cana-3972	153	11	implies	imply	VERB
cana-3972	153	12	that	that	SCONJ
cana-3972	153	13	n	n	ADP
cana-3972	153	14	∩	∩	NOUN
cana-3972	153	15	k	k	PROPN
cana-3972	153	16	≤	≤	PROPN
cana-3972	153	17	⊕	⊕	PROPN
cana-3972	153	18	m	m	PROPN
cana-3972	153	19	.	.	PUNCT
cana-3972	154	1	hence	hence	ADV
cana-3972	154	2	,	,	PUNCT
cana-3972	154	3	m	m	VERB
cana-3972	154	4	is	be	AUX
cana-3972	154	5	a	a	DET
cana-3972	154	6	d4	d4	NOUN
cana-3972	154	7	-	-	PUNCT
cana-3972	154	8	module	module	NOUN
cana-3972	154	9	,	,	PUNCT
cana-3972	154	10	i.e.	i.e.	X
cana-3972	154	11	,	,	PUNCT
cana-3972	154	12	a	a	DET
cana-3972	154	13	d41	d41	NOUN
cana-3972	154	14	-	-	PUNCT
cana-3972	154	15	module	module	NOUN
cana-3972	154	16	.	.	PUNCT
cana-3972	155	1	4	4	NUM
cana-3972	155	2	)	)	PUNCT
cana-3972	155	3	⇒	⇒	NOUN
cana-3972	155	4	1	1	NUM
cana-3972	155	5	)	)	PUNCT
cana-3972	155	6	.	.	PUNCT
cana-3972	156	1	let	let	VERB
cana-3972	156	2	m	m	PRON
cana-3972	156	3	be	be	AUX
cana-3972	156	4	a	a	DET
cana-3972	156	5	finitely	finitely	ADV
cana-3972	156	6	generated	generate	VERB
cana-3972	156	7	r	r	NOUN
cana-3972	156	8	-	-	PUNCT
cana-3972	156	9	module	module	NOUN
cana-3972	156	10	and	and	CCONJ
cana-3972	156	11	f	f	PROPN
cana-3972	156	12	an	an	DET
cana-3972	156	13	endomorphism	endomorphism	NOUN
cana-3972	156	14	of	of	ADP
cana-3972	156	15	m	m	PROPN
cana-3972	156	16	.	.	PUNCT
cana-3972	157	1	since	since	SCONJ
cana-3972	157	2	𝑀	𝑀	PROPN
cana-3972	157	3	+	+	CCONJ
cana-3972	157	4	(	(	PUNCT
cana-3972	157	5	𝑀/𝑘𝑒𝑟(𝑓	𝑀/𝑘𝑒𝑟(𝑓	NOUN
cana-3972	157	6	)	)	PUNCT
cana-3972	157	7	)	)	PUNCT
cana-3972	157	8	is	be	AUX
cana-3972	157	9	finitely	finitely	ADV
cana-3972	157	10	generated	generate	VERB
cana-3972	157	11	and	and	CCONJ
cana-3972	157	12	,	,	PUNCT
cana-3972	157	13	by	by	ADP
cana-3972	157	14	hypothesis	hypothesis	NOUN
cana-3972	157	15	,	,	PUNCT
cana-3972	157	16	is	be	AUX
cana-3972	157	17	a	a	DET
cana-3972	157	18	d41	d41	NOUN
cana-3972	157	19	-	-	PUNCT
cana-3972	157	20	module	module	NOUN
cana-3972	157	21	,	,	PUNCT
cana-3972	157	22	it	it	PRON
cana-3972	157	23	follows	follow	VERB
cana-3972	157	24	that	that	SCONJ
cana-3972	157	25	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	157	26	)	)	PUNCT
cana-3972	157	27	is	be	AUX
cana-3972	157	28	a	a	DET
cana-3972	157	29	direct	direct	ADJ
cana-3972	157	30	summand	summand	NOUN
cana-3972	157	31	of	of	ADP
cana-3972	157	32	m	m	PROPN
cana-3972	157	33	.	.	PUNCT
cana-3972	158	1	therefore	therefore	ADV
cana-3972	158	2	,	,	PUNCT
cana-3972	158	3	m	m	VERB
cana-3972	158	4	is	be	AUX
cana-3972	158	5	a	a	DET
cana-3972	158	6	dual	dual	ADJ
cana-3972	158	7	-	-	PUNCT
cana-3972	158	8	rickart	rickart	NOUN
cana-3972	158	9	module	module	NOUN
cana-3972	158	10	.	.	PUNCT
cana-3972	159	1	recall	recall	VERB
cana-3972	159	2	that	that	SCONJ
cana-3972	159	3	a	a	DET
cana-3972	159	4	module	module	NOUN
cana-3972	159	5	m	m	VERB
cana-3972	159	6	has	have	VERB
cana-3972	159	7	the	the	DET
cana-3972	159	8	summand	summand	NOUN
cana-3972	159	9	sum	sum	NOUN
cana-3972	159	10	property	property	NOUN
cana-3972	159	11	(	(	PUNCT
cana-3972	159	12	ssp	ssp	NOUN
cana-3972	159	13	)	)	PUNCT
cana-3972	159	14	if	if	SCONJ
cana-3972	159	15	the	the	DET
cana-3972	159	16	sum	sum	NOUN
cana-3972	159	17	of	of	ADP
cana-3972	159	18	any	any	DET
cana-3972	159	19	two	two	NUM
cana-3972	159	20	direct	direct	ADJ
cana-3972	159	21	summands	summand	NOUN
cana-3972	159	22	of	of	ADP
cana-3972	159	23	m	m	VERB
cana-3972	159	24	is	be	AUX
cana-3972	159	25	also	also	ADV
cana-3972	159	26	a	a	DET
cana-3972	159	27	direct	direct	ADJ
cana-3972	159	28	summand	summand	NOUN
cana-3972	159	29	of	of	ADP
cana-3972	159	30	m	m	PROPN
cana-3972	159	31	.	.	PUNCT
cana-3972	160	1	proposition	proposition	NOUN
cana-3972	160	2	3.14	3.14	NUM
cana-3972	160	3	.	.	PUNCT
cana-3972	161	1	let	let	VERB
cana-3972	161	2	m	m	PRON
cana-3972	161	3	be	be	AUX
cana-3972	161	4	a	a	DET
cana-3972	161	5	d41	d41	NOUN
cana-3972	161	6	-	-	PUNCT
cana-3972	161	7	module	module	NOUN
cana-3972	161	8	.	.	PUNCT
cana-3972	162	1	if	if	SCONJ
cana-3972	162	2	m	m	PROPN
cana-3972	162	3	possesses	possess	VERB
cana-3972	162	4	the	the	DET
cana-3972	162	5	ssp	ssp	NOUN
cana-3972	162	6	,	,	PUNCT
cana-3972	162	7	then	then	ADV
cana-3972	162	8	m	m	VERB
cana-3972	162	9	also	also	ADV
cana-3972	162	10	has	have	VERB
cana-3972	162	11	the	the	DET
cana-3972	162	12	sip	sip	NOUN
cana-3972	162	13	.	.	PUNCT
cana-3972	163	1	proof	proof	NOUN
cana-3972	163	2	:	:	PUNCT
cana-3972	163	3	let	let	VERB
cana-3972	163	4	m	m	PRON
cana-3972	163	5	be	be	AUX
cana-3972	163	6	a	a	DET
cana-3972	163	7	d41	d41	NOUN
cana-3972	163	8	-	-	PUNCT
cana-3972	163	9	module	module	NOUN
cana-3972	163	10	with	with	ADP
cana-3972	163	11	the	the	DET
cana-3972	163	12	ssp	ssp	ADJ
cana-3972	163	13	property	property	NOUN
cana-3972	163	14	.	.	PUNCT
cana-3972	164	1	consider	consider	VERB
cana-3972	164	2	direct	direct	ADJ
cana-3972	164	3	summands	summand	NOUN
cana-3972	164	4	n	n	PRON
cana-3972	164	5	and	and	CCONJ
cana-3972	164	6	k	k	X
cana-3972	164	7	of	of	ADP
cana-3972	164	8	m	m	PROPN
cana-3972	164	9	.	.	PUNCT
cana-3972	165	1	since	since	SCONJ
cana-3972	165	2	m	m	PROPN
cana-3972	165	3	has	have	VERB
cana-3972	165	4	the	the	DET
cana-3972	165	5	ssp	ssp	NOUN
cana-3972	166	1	,	,	PUNCT
cana-3972	166	2	then	then	ADV
cana-3972	166	3	n	n	PROPN
cana-3972	166	4	+	+	CCONJ
cana-3972	166	5	k	k	PROPN
cana-3972	166	6	is	be	AUX
cana-3972	166	7	a	a	DET
cana-3972	166	8	direct	direct	ADJ
cana-3972	166	9	summand	summand	NOUN
cana-3972	166	10	of	of	ADP
cana-3972	166	11	m	m	PROPN
cana-3972	166	12	.	.	PUNCT
cana-3972	167	1	by	by	ADP
cana-3972	167	2	proposition	proposition	NOUN
cana-3972	167	3	3.4	3.4	NUM
cana-3972	167	4	,	,	PUNCT
cana-3972	167	5	𝑃	𝑃	NOUN
cana-3972	167	6	=	=	SYM
cana-3972	167	7	𝑁	𝑁	PROPN
cana-3972	167	8	+	+	CCONJ
cana-3972	167	9	𝐾	𝐾	PROPN
cana-3972	167	10	is	be	AUX
cana-3972	167	11	a	a	DET
cana-3972	167	12	d41	d41	NOUN
cana-3972	167	13	-	-	PUNCT
cana-3972	167	14	module	module	NOUN
cana-3972	167	15	.	.	PUNCT
cana-3972	168	1	applying	apply	VERB
cana-3972	168	2	the	the	DET
cana-3972	168	3	isomorphisms	isomorphism	NOUN
cana-3972	168	4	theorems	theorem	NOUN
cana-3972	168	5	,	,	PUNCT
cana-3972	168	6	we	we	PRON
cana-3972	168	7	have	have	VERB
cana-3972	168	8	𝐾	𝐾	PROPN
cana-3972	168	9	≅	≅	NOUN
cana-3972	168	10	𝑃	𝑃	NOUN
cana-3972	168	11	𝑁	𝑁	PROPN
cana-3972	168	12	≅	≅	NUM
cana-3972	168	13	𝑁+𝐾	𝑁+𝐾	NOUN
cana-3972	168	14	𝑁	𝑁	PROPN
cana-3972	168	15	≅	≅	NOUN
cana-3972	168	16	𝑁/𝑁	𝑁/𝑁	NOUN
cana-3972	168	17	∩	∩	ADJ
cana-3972	168	18	𝐾	𝐾	PROPN
cana-3972	168	19	.	.	PUNCT
cana-3972	169	1	communications	communication	NOUN
cana-3972	169	2	on	on	ADP
cana-3972	169	3	applied	apply	VERB
cana-3972	169	4	nonlinear	nonlinear	ADJ
cana-3972	169	5	analysis	analysis	NOUN
cana-3972	169	6	issn	issn	NOUN
cana-3972	169	7	:	:	PUNCT
cana-3972	169	8	1074	1074	NUM
cana-3972	169	9	-	-	PUNCT
cana-3972	169	10	133x	133x	NUM
cana-3972	169	11	vol	vol	NOUN
cana-3972	169	12	32	32	NUM
cana-3972	169	13	no	no	NOUN
cana-3972	169	14	.	.	PUNCT
cana-3972	170	1	9s	9s	NUM
cana-3972	170	2	(	(	PUNCT
cana-3972	170	3	2025	2025	NUM
cana-3972	170	4	)	)	PUNCT
cana-3972	171	1	680	680	NUM
cana-3972	171	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	171	3	since	since	SCONJ
cana-3972	171	4	p	p	NOUN
cana-3972	171	5	is	be	AUX
cana-3972	171	6	a	a	DET
cana-3972	171	7	d41	d41	NOUN
cana-3972	171	8	-	-	PUNCT
cana-3972	171	9	module	module	NOUN
cana-3972	171	10	,	,	PUNCT
cana-3972	171	11	then	then	ADV
cana-3972	171	12	the	the	DET
cana-3972	171	13	epimorphism	epimorphism	NOUN
cana-3972	171	14	𝑓	𝑓	DET
cana-3972	171	15	∶	∶	NOUN
cana-3972	171	16	𝐾	𝐾	PROPN
cana-3972	171	17	→	→	SYM
cana-3972	171	18	𝑁/(𝑁	𝑁/(𝑁	ADJ
cana-3972	171	19	∩	∩	ADJ
cana-3972	171	20	𝐾	𝐾	NOUN
cana-3972	171	21	)	)	PUNCT
cana-3972	171	22	splits	split	VERB
cana-3972	171	23	.	.	PUNCT
cana-3972	172	1	so	so	ADV
cana-3972	172	2	,	,	PUNCT
cana-3972	172	3	n	n	CCONJ
cana-3972	172	4	∩	∩	X
cana-3972	172	5	k	k	PROPN
cana-3972	172	6	≤	≤	PROPN
cana-3972	172	7	⊕	⊕	PROPN
cana-3972	172	8	k	k	PROPN
cana-3972	173	1	and	and	CCONJ
cana-3972	173	2	hence	hence	ADV
cana-3972	173	3	a	a	DET
cana-3972	173	4	direct	direct	ADJ
cana-3972	173	5	summand	summand	NOUN
cana-3972	173	6	of	of	ADP
cana-3972	173	7	m	m	PRON
cana-3972	173	8	proving	prove	VERB
cana-3972	173	9	that	that	SCONJ
cana-3972	173	10	m	m	PROPN
cana-3972	173	11	has	have	VERB
cana-3972	173	12	the	the	DET
cana-3972	173	13	sip	sip	NOUN
cana-3972	173	14	.	.	PUNCT
cana-3972	174	1	proposition	proposition	NOUN
cana-3972	174	2	3.15	3.15	NUM
cana-3972	174	3	.	.	PUNCT
cana-3972	175	1	let	let	VERB
cana-3972	175	2	m	m	PRON
cana-3972	175	3	be	be	AUX
cana-3972	175	4	a	a	DET
cana-3972	175	5	module	module	NOUN
cana-3972	175	6	such	such	ADJ
cana-3972	175	7	that	that	DET
cana-3972	175	8	𝑀/𝑍(𝑀	𝑀/𝑍(𝑀	PROPN
cana-3972	175	9	)	)	PUNCT
cana-3972	175	10	is	be	AUX
cana-3972	175	11	a	a	DET
cana-3972	175	12	d41	d41	NOUN
cana-3972	175	13	-	-	PUNCT
cana-3972	175	14	module	module	NOUN
cana-3972	175	15	.	.	PUNCT
cana-3972	176	1	then	then	ADV
cana-3972	176	2	m	m	PROPN
cana-3972	176	3	is	be	AUX
cana-3972	176	4	a	a	DET
cana-3972	176	5	d41	d41	NOUN
cana-3972	176	6	-	-	PUNCT
cana-3972	176	7	module	module	NOUN
cana-3972	176	8	.	.	PUNCT
cana-3972	177	1	proof	proof	NOUN
cana-3972	177	2	:	:	PUNCT
cana-3972	177	3	let	let	VERB
cana-3972	177	4	n	n	PRON
cana-3972	177	5	be	be	AUX
cana-3972	177	6	a	a	DET
cana-3972	177	7	cosingular	cosingular	ADJ
cana-3972	177	8	submodule	submodule	NOUN
cana-3972	177	9	of	of	ADP
cana-3972	177	10	m	m	AUX
cana-3972	177	11	containing	contain	VERB
cana-3972	177	12	𝑍(𝑀	𝑍(𝑀	NOUN
cana-3972	177	13	)	)	PUNCT
cana-3972	177	14	such	such	ADJ
cana-3972	177	15	that	that	SCONJ
cana-3972	177	16	𝑀	𝑀	PROPN
cana-3972	177	17	𝑁	𝑁	PROPN
cana-3972	177	18	≅	≅	PROPN
cana-3972	177	19	𝐾	𝐾	PROPN
cana-3972	177	20	,	,	PUNCT
cana-3972	177	21	for	for	ADP
cana-3972	177	22	𝐾	𝐾	PROPN
cana-3972	177	23	a	a	DET
cana-3972	177	24	direct	direct	ADJ
cana-3972	177	25	summand	summand	NOUN
cana-3972	177	26	of	of	ADP
cana-3972	177	27	𝑀	𝑀	PROPN
cana-3972	177	28	with	with	ADP
cana-3972	177	29	𝐾	𝐾	PROPN
cana-3972	177	30	≤	≤	NOUN
cana-3972	177	31	𝑁.	𝑁.	PROPN
cana-3972	177	32	since	since	SCONJ
cana-3972	177	33	𝑀/𝑍(𝑀	𝑀/𝑍(𝑀	PROPN
cana-3972	177	34	)	)	PUNCT
cana-3972	177	35	is	be	AUX
cana-3972	177	36	a	a	DET
cana-3972	177	37	d41	d41	NOUN
cana-3972	177	38	-	-	PUNCT
cana-3972	177	39	module	module	NOUN
cana-3972	177	40	,	,	PUNCT
cana-3972	177	41	there	there	PRON
cana-3972	177	42	exists	exist	VERB
cana-3972	177	43	a	a	DET
cana-3972	177	44	direct	direct	ADJ
cana-3972	177	45	summand	summand	NOUN
cana-3972	177	46	𝐿	𝐿	PROPN
cana-3972	177	47	of	of	ADP
cana-3972	177	48	𝑀/𝑍(𝑀	𝑀/𝑍(𝑀	PROPN
cana-3972	177	49	)	)	PUNCT
cana-3972	177	50	such	such	ADJ
cana-3972	177	51	that	that	SCONJ
cana-3972	177	52	(	(	PUNCT
cana-3972	177	53	𝑀	𝑀	PROPN
cana-3972	177	54	�	�	PROPN
cana-3972	177	55	̅	̅	NOUN
cana-3972	177	56	�	�	NOUN
cana-3972	177	57	(𝑀	(𝑀	NUM
cana-3972	177	58	)	)	PUNCT
cana-3972	177	59	)	)	PUNCT
cana-3972	178	1	(	(	PUNCT
cana-3972	178	2	𝑁	𝑁	PROPN
cana-3972	178	3	�	�	PROPN
cana-3972	178	4	̅	̅	NOUN
cana-3972	178	5	�	�	NOUN
cana-3972	178	6	(𝑀	(𝑀	NUM
cana-3972	178	7	)	)	PUNCT
cana-3972	178	8	)	)	PUNCT
cana-3972	179	1	⁄	⁄	PROPN
cana-3972	179	2	≅	≅	PROPN
cana-3972	179	3	𝐿	𝐿	PROPN
cana-3972	179	4	and	and	CCONJ
cana-3972	179	5	𝐿	𝐿	PROPN
cana-3972	179	6	≤	≤	NOUN
cana-3972	179	7	𝑁	𝑁	PROPN
cana-3972	179	8	�	�	NOUN
cana-3972	179	9	̅	̅	NOUN
cana-3972	179	10	�	�	NOUN
cana-3972	179	11	(𝑀	(𝑀	NUM
cana-3972	179	12	)	)	PUNCT
cana-3972	179	13	.	.	PUNCT
cana-3972	180	1	applying	apply	VERB
cana-3972	180	2	the	the	DET
cana-3972	180	3	isomorphism	isomorphism	NOUN
cana-3972	180	4	theorem	theorem	NOUN
cana-3972	180	5	,	,	PUNCT
cana-3972	180	6	we	we	PRON
cana-3972	180	7	have	have	VERB
cana-3972	180	8	𝑀	𝑀	PROPN
cana-3972	180	9	𝑁	𝑁	PROPN
cana-3972	180	10	≅	≅	PROPN
cana-3972	180	11	(	(	PUNCT
cana-3972	180	12	𝑀	𝑀	PROPN
cana-3972	180	13	�	�	PROPN
cana-3972	180	14	̅	̅	NOUN
cana-3972	180	15	�	�	NOUN
cana-3972	180	16	(𝑀	(𝑀	NUM
cana-3972	180	17	)	)	PUNCT
cana-3972	180	18	)	)	PUNCT
cana-3972	181	1	(	(	PUNCT
cana-3972	181	2	𝑁	𝑁	PROPN
cana-3972	181	3	�	�	PROPN
cana-3972	181	4	̅	̅	NOUN
cana-3972	181	5	�	�	NOUN
cana-3972	181	6	(𝑀	(𝑀	NUM
cana-3972	181	7	)	)	PUNCT
cana-3972	181	8	)	)	PUNCT
cana-3972	182	1	⁄	⁄	PROPN
cana-3972	182	2	≅	≅	NUM
cana-3972	182	3	𝐿	𝐿	PROPN
cana-3972	182	4	≅	≅	PROPN
cana-3972	182	5	𝐾	𝐾	PROPN
cana-3972	182	6	≤⊕	≤⊕	ADV
cana-3972	182	7	𝑀.	𝑀.	PROPN
cana-3972	182	8	as	as	ADP
cana-3972	182	9	𝑁	𝑁	PROPN
cana-3972	182	10	�	�	PROPN
cana-3972	182	11	̅	̅	NOUN
cana-3972	182	12	�	�	NOUN
cana-3972	182	13	(𝑀	(𝑀	NUM
cana-3972	182	14	)	)	PUNCT
cana-3972	182	15	≤⊕	≤⊕	NOUN
cana-3972	182	16	𝑀/𝑁	𝑀/𝑁	PROPN
cana-3972	182	17	,	,	PUNCT
cana-3972	182	18	thus	thus	ADV
cana-3972	182	19	𝑁	𝑁	PROPN
cana-3972	182	20	≤⊕	≤⊕	NOUN
cana-3972	182	21	𝑀.	𝑀.	PROPN
cana-3972	182	22	hence	hence	ADV
cana-3972	182	23	,	,	PUNCT
cana-3972	182	24	𝑀	𝑀	PROPN
cana-3972	182	25	is	be	AUX
cana-3972	182	26	a	a	DET
cana-3972	182	27	d41	d41	NOUN
cana-3972	182	28	-	-	PUNCT
cana-3972	182	29	module	module	NOUN
cana-3972	182	30	.	.	PUNCT
cana-3972	183	1	the	the	DET
cana-3972	183	2	converse	converse	NOUN
cana-3972	183	3	of	of	ADP
cana-3972	183	4	the	the	DET
cana-3972	183	5	previous	previous	ADJ
cana-3972	183	6	proposition	proposition	NOUN
cana-3972	183	7	holds	hold	VERB
cana-3972	183	8	(	(	PUNCT
cana-3972	183	9	see	see	VERB
cana-3972	183	10	3.17	3.17	NUM
cana-3972	183	11	)	)	PUNCT
cana-3972	183	12	.	.	PUNCT
cana-3972	184	1	corollary	corollary	NOUN
cana-3972	184	2	3.16	3.16	NUM
cana-3972	184	3	.	.	PUNCT
cana-3972	185	1	let	let	VERB
cana-3972	185	2	m	m	NOUN
cana-3972	185	3	=	=	VERB
cana-3972	185	4	m1	m1	PROPN
cana-3972	185	5	⊕	⊕	PROPN
cana-3972	185	6	m2	m2	PROPN
cana-3972	185	7	a	a	DET
cana-3972	185	8	direct	direct	ADJ
cana-3972	185	9	sum	sum	NOUN
cana-3972	185	10	of	of	ADP
cana-3972	185	11	submodules	submodule	NOUN
cana-3972	185	12	m1	m1	PROPN
cana-3972	185	13	and	and	CCONJ
cana-3972	185	14	m2	m2	PROPN
cana-3972	185	15	such	such	ADJ
cana-3972	185	16	that	that	SCONJ
cana-3972	185	17	�	�	PROPN
cana-3972	185	18	̅	̅	NOUN
cana-3972	185	19	�	�	NOUN
cana-3972	185	20	(𝑀1	(𝑀1	NOUN
cana-3972	185	21	)	)	PUNCT
cana-3972	185	22	=	=	PUNCT
cana-3972	186	1	𝑀1	𝑀1	PROPN
cana-3972	186	2	and	and	CCONJ
cana-3972	186	3	m2	m2	PROPN
cana-3972	186	4	a	a	DET
cana-3972	186	5	cosingular	cosingular	ADJ
cana-3972	186	6	d41	d41	NOUN
cana-3972	186	7	-	-	PUNCT
cana-3972	186	8	module	module	NOUN
cana-3972	186	9	.	.	PUNCT
cana-3972	187	1	then	then	ADV
cana-3972	187	2	m	m	PROPN
cana-3972	187	3	is	be	AUX
cana-3972	187	4	a	a	DET
cana-3972	187	5	d41	d41	NOUN
cana-3972	187	6	-	-	PUNCT
cana-3972	187	7	module	module	NOUN
cana-3972	187	8	.	.	PUNCT
cana-3972	188	1	proof	proof	NOUN
cana-3972	188	2	:	:	PUNCT
cana-3972	188	3	since	since	SCONJ
cana-3972	188	4	�	�	PROPN
cana-3972	188	5	̅	̅	NOUN
cana-3972	188	6	�	�	NOUN
cana-3972	188	7	(𝑀2	(𝑀2	NUM
cana-3972	188	8	)	)	PUNCT
cana-3972	188	9	=	=	SYM
cana-3972	188	10	0	0	NUM
cana-3972	188	11	,	,	PUNCT
cana-3972	188	12	then	then	ADV
cana-3972	188	13	�	�	PROPN
cana-3972	188	14	̅	̅	NOUN
cana-3972	188	15	�	�	NOUN
cana-3972	188	16	(𝑀	(𝑀	NUM
cana-3972	188	17	)	)	PUNCT
cana-3972	188	18	=	=	SYM
cana-3972	189	1	𝑀1	𝑀1	PROPN
cana-3972	189	2	.	.	PUNCT
cana-3972	190	1	thus	thus	ADV
cana-3972	190	2	,	,	PUNCT
cana-3972	190	3	𝑀	𝑀	PROPN
cana-3972	190	4	�	�	PROPN
cana-3972	190	5	̅	̅	NOUN
cana-3972	190	6	�	�	NOUN
cana-3972	190	7	(𝑀	(𝑀	NUM
cana-3972	190	8	)	)	PUNCT
cana-3972	190	9	≅	≅	PROPN
cana-3972	190	10	𝑀2	𝑀2	PROPN
cana-3972	190	11	which	which	PRON
cana-3972	190	12	is	be	AUX
cana-3972	190	13	a	a	DET
cana-3972	190	14	d41	d41	NOUN
cana-3972	190	15	-	-	PUNCT
cana-3972	190	16	module	module	NOUN
cana-3972	190	17	.	.	PUNCT
cana-3972	191	1	hence	hence	ADV
cana-3972	191	2	m	m	PROPN
cana-3972	191	3	is	be	AUX
cana-3972	191	4	a	a	DET
cana-3972	191	5	d41	d41	NOUN
cana-3972	191	6	-	-	PUNCT
cana-3972	191	7	module	module	NOUN
cana-3972	191	8	by	by	ADP
cana-3972	191	9	the	the	DET
cana-3972	191	10	proposition	proposition	NOUN
cana-3972	191	11	3.15	3.15	NUM
cana-3972	191	12	.	.	PUNCT
cana-3972	192	1	proposition	proposition	NOUN
cana-3972	192	2	3.17	3.17	NUM
cana-3972	192	3	.	.	PUNCT
cana-3972	193	1	let	let	VERB
cana-3972	193	2	m	m	PRON
cana-3972	193	3	be	be	AUX
cana-3972	193	4	a	a	DET
cana-3972	193	5	d41	d41	NOUN
cana-3972	193	6	-	-	PUNCT
cana-3972	193	7	module	module	NOUN
cana-3972	193	8	.	.	PUNCT
cana-3972	194	1	then	then	ADV
cana-3972	194	2	,	,	PUNCT
cana-3972	194	3	for	for	ADP
cana-3972	194	4	every	every	DET
cana-3972	194	5	submodule	submodule	NOUN
cana-3972	194	6	n	n	PROPN
cana-3972	194	7	of	of	ADP
cana-3972	194	8	m	m	PROPN
cana-3972	194	9	,	,	PUNCT
cana-3972	194	10	m	m	PROPN
cana-3972	194	11	/	/	SYM
cana-3972	194	12	n	n	PROPN
cana-3972	194	13	is	be	AUX
cana-3972	194	14	a	a	DET
cana-3972	194	15	d41	d41	NOUN
cana-3972	194	16	-	-	PUNCT
cana-3972	194	17	module	module	NOUN
cana-3972	194	18	.	.	PUNCT
cana-3972	195	1	proof	proof	NOUN
cana-3972	195	2	:	:	PUNCT
cana-3972	195	3	let	let	VERB
cana-3972	195	4	m	m	PRON
cana-3972	195	5	be	be	AUX
cana-3972	195	6	a	a	DET
cana-3972	195	7	d41	d41	NOUN
cana-3972	195	8	-	-	PUNCT
cana-3972	195	9	module	module	NOUN
cana-3972	195	10	and	and	CCONJ
cana-3972	195	11	n	n	NOUN
cana-3972	195	12	a	a	DET
cana-3972	195	13	submodule	submodule	NOUN
cana-3972	195	14	of	of	ADP
cana-3972	195	15	m	m	PRON
cana-3972	195	16	and	and	CCONJ
cana-3972	195	17	let	let	VERB
cana-3972	195	18	k	k	NOUN
cana-3972	195	19	=	=	PUNCT
cana-3972	195	20	p	p	X
cana-3972	195	21	/n	/n	PUNCT
cana-3972	195	22	be	be	AUX
cana-3972	195	23	a	a	DET
cana-3972	195	24	co	co	ADJ
cana-3972	195	25	-	-	ADJ
cana-3972	195	26	singular	singular	ADJ
cana-3972	195	27	submodule	submodule	NOUN
cana-3972	195	28	of	of	ADP
cana-3972	195	29	m	m	PROPN
cana-3972	195	30	/	/	SYM
cana-3972	195	31	n	n	PROPN
cana-3972	195	32	with	with	ADP
cana-3972	195	33	p	p	PRON
cana-3972	195	34	a	a	DET
cana-3972	195	35	submodule	submodule	NOUN
cana-3972	195	36	of	of	ADP
cana-3972	195	37	m	m	PROPN
cana-3972	195	38	containing	contain	VERB
cana-3972	195	39	n	n	PRON
cana-3972	195	40	such	such	ADJ
cana-3972	195	41	that	that	SCONJ
cana-3972	195	42	(	(	PUNCT
cana-3972	195	43	𝑀/𝑁)/(𝑃/𝑁	𝑀/𝑁)/(𝑃/𝑁	PROPN
cana-3972	195	44	)	)	PUNCT
cana-3972	195	45	≅	≅	PROPN
cana-3972	195	46	𝐿	𝐿	PROPN
cana-3972	195	47	,	,	PUNCT
cana-3972	195	48	l	l	PROPN
cana-3972	195	49	≤	≤	PROPN
cana-3972	196	1	⊕	⊕	PROPN
cana-3972	196	2	m	m	PROPN
cana-3972	196	3	/	/	SYM
cana-3972	196	4	n	n	PROPN
cana-3972	196	5	and	and	CCONJ
cana-3972	196	6	l	l	NOUN
cana-3972	197	1	≤	≤	NOUN
cana-3972	197	2	p	p	X
cana-3972	197	3	/n	/n	PUNCT
cana-3972	197	4	.	.	PUNCT
cana-3972	198	1	by	by	ADP
cana-3972	198	2	the	the	DET
cana-3972	198	3	second	second	ADJ
cana-3972	198	4	theorem	theorem	NOUN
cana-3972	198	5	of	of	ADP
cana-3972	198	6	isomorphism	isomorphism	NOUN
cana-3972	198	7	,	,	PUNCT
cana-3972	198	8	we	we	PRON
cana-3972	198	9	have	have	VERB
cana-3972	198	10	:	:	PUNCT
cana-3972	198	11	𝑀/𝑃	𝑀/𝑃	PROPN
cana-3972	198	12	≅	≅	PROPN
cana-3972	198	13	(	(	PUNCT
cana-3972	198	14	𝑀/𝑁)/(𝑃/𝑁	𝑀/𝑁)/(𝑃/𝑁	PROPN
cana-3972	198	15	)	)	PUNCT
cana-3972	198	16	≅	≅	PROPN
cana-3972	198	17	𝐿.	𝐿.	VERB
cana-3972	198	18	as	as	SCONJ
cana-3972	198	19	m	m	PROPN
cana-3972	198	20	is	be	AUX
cana-3972	198	21	a	a	DET
cana-3972	198	22	d41	d41	NOUN
cana-3972	198	23	-	-	PUNCT
cana-3972	198	24	module	module	NOUN
cana-3972	198	25	,	,	PUNCT
cana-3972	198	26	then	then	ADV
cana-3972	198	27	p	p	PROPN
cana-3972	198	28	is	be	AUX
cana-3972	198	29	a	a	DET
cana-3972	198	30	direct	direct	ADJ
cana-3972	198	31	summand	summand	NOUN
cana-3972	198	32	of	of	ADP
cana-3972	198	33	m	m	PROPN
cana-3972	198	34	.	.	PUNCT
cana-3972	199	1	hence	hence	ADV
cana-3972	199	2	k	k	PROPN
cana-3972	200	1	=	=	PUNCT
cana-3972	200	2	p	p	X
cana-3972	200	3	/n	/n	PUNCT
cana-3972	200	4	is	be	AUX
cana-3972	200	5	a	a	DET
cana-3972	200	6	direct	direct	ADJ
cana-3972	200	7	summand	summand	NOUN
cana-3972	200	8	of	of	ADP
cana-3972	200	9	m	m	PROPN
cana-3972	200	10	/	/	SYM
cana-3972	200	11	n	n	PROPN
cana-3972	200	12	.	.	PUNCT
cana-3972	201	1	thus	thus	ADV
cana-3972	201	2	m	m	X
cana-3972	201	3	/	/	SYM
cana-3972	201	4	n	n	PROPN
cana-3972	201	5	is	be	AUX
cana-3972	201	6	a	a	DET
cana-3972	201	7	d41	d41	NOUN
cana-3972	201	8	-	-	PUNCT
cana-3972	201	9	module	module	NOUN
cana-3972	201	10	.	.	PUNCT
cana-3972	202	1	recall	recall	VERB
cana-3972	202	2	that	that	SCONJ
cana-3972	202	3	a	a	DET
cana-3972	202	4	module	module	NOUN
cana-3972	202	5	is	be	AUX
cana-3972	202	6	called	call	VERB
cana-3972	202	7	directly	directly	ADV
cana-3972	202	8	finite	finite	ADJ
cana-3972	202	9	if	if	SCONJ
cana-3972	202	10	it	it	PRON
cana-3972	202	11	is	be	AUX
cana-3972	202	12	not	not	PART
cana-3972	202	13	isomorphic	isomorphic	ADJ
cana-3972	202	14	to	to	ADP
cana-3972	202	15	a	a	DET
cana-3972	202	16	proper	proper	ADJ
cana-3972	202	17	summand	summand	NOUN
cana-3972	202	18	of	of	ADP
cana-3972	202	19	itself	itself	PRON
cana-3972	202	20	.	.	PUNCT
cana-3972	203	1	a	a	DET
cana-3972	203	2	module	module	NOUN
cana-3972	203	3	is	be	AUX
cana-3972	203	4	called	call	VERB
cana-3972	203	5	square	square	ADV
cana-3972	203	6	-	-	PUNCT
cana-3972	203	7	free	free	ADJ
cana-3972	203	8	if	if	SCONJ
cana-3972	203	9	it	it	PRON
cana-3972	203	10	contains	contain	VERB
cana-3972	203	11	no	no	DET
cana-3972	203	12	nonzero	nonzero	NOUN
cana-3972	203	13	submodules	submodule	NOUN
cana-3972	203	14	isomorphic	isomorphic	ADJ
cana-3972	203	15	to	to	ADP
cana-3972	203	16	a	a	DET
cana-3972	203	17	square	square	ADJ
cana-3972	203	18	𝑁	𝑁	PROPN
cana-3972	203	19	⊕	⊕	PROPN
cana-3972	203	20	𝑁.	𝑁.	PROPN
cana-3972	203	21	proposition	proposition	NOUN
cana-3972	203	22	3.18	3.18	NUM
cana-3972	203	23	.	.	PUNCT
cana-3972	204	1	let	let	VERB
cana-3972	204	2	m	m	VERB
cana-3972	204	3	=	=	SYM
cana-3972	204	4	n	n	PROPN
cana-3972	204	5	⊕	⊕	PROPN
cana-3972	204	6	k	k	PROPN
cana-3972	204	7	be	be	AUX
cana-3972	204	8	a	a	DET
cana-3972	204	9	d41	d41	NOUN
cana-3972	204	10	-	-	PUNCT
cana-3972	204	11	module	module	NOUN
cana-3972	204	12	,	,	PUNCT
cana-3972	204	13	with	with	ADP
cana-3972	204	14	epimorphisms	epimorphism	NOUN
cana-3972	204	15	𝑓	𝑓	DET
cana-3972	204	16	∶	∶	NOUN
cana-3972	204	17	𝑁	𝑁	PROPN
cana-3972	204	18	→	→	SYM
cana-3972	204	19	𝐾	𝐾	PROPN
cana-3972	204	20	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-3972	204	21	ℎ	ℎ	PROPN
cana-3972	204	22	∶	∶	NOUN
cana-3972	204	23	𝐾	𝐾	PROPN
cana-3972	204	24	→	→	PUNCT
cana-3972	204	25	𝑁	𝑁	PROPN
cana-3972	204	26	such	such	ADJ
cana-3972	204	27	that	that	DET
cana-3972	204	28	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	204	29	)	)	PUNCT
cana-3972	204	30	is	be	AUX
cana-3972	204	31	cosingular	cosingular	ADJ
cana-3972	204	32	.	.	PUNCT
cana-3972	205	1	1	1	X
cana-3972	205	2	)	)	PUNCT
cana-3972	205	3	if	if	SCONJ
cana-3972	205	4	k	k	PROPN
cana-3972	205	5	is	be	AUX
cana-3972	205	6	cosingular	cosingular	ADJ
cana-3972	205	7	and	and	CCONJ
cana-3972	205	8	directly	directly	ADV
cana-3972	205	9	finite	finite	VERB
cana-3972	205	10	,	,	PUNCT
cana-3972	205	11	then	then	ADV
cana-3972	205	12	𝑁	𝑁	PROPN
cana-3972	205	13	≅	≅	PROPN
cana-3972	205	14	𝐾.	𝐾.	PROPN
cana-3972	205	15	2	2	NUM
cana-3972	205	16	)	)	PUNCT
cana-3972	205	17	if	if	SCONJ
cana-3972	205	18	n	n	PRON
cana-3972	205	19	is	be	AUX
cana-3972	205	20	square	square	ADV
cana-3972	205	21	-	-	PUNCT
cana-3972	205	22	free	free	ADJ
cana-3972	205	23	and	and	CCONJ
cana-3972	205	24	k	k	PROPN
cana-3972	205	25	is	be	AUX
cana-3972	205	26	cosingular	cosingular	ADJ
cana-3972	205	27	,	,	PUNCT
cana-3972	205	28	then	then	ADV
cana-3972	205	29	𝑁	𝑁	PROPN
cana-3972	205	30	≅	≅	PROPN
cana-3972	205	31	𝐾.	𝐾.	PROPN
cana-3972	205	32	proof	proof	NOUN
cana-3972	205	33	:	:	PUNCT
cana-3972	205	34	1	1	X
cana-3972	205	35	)	)	PUNCT
cana-3972	205	36	assume	assume	VERB
cana-3972	205	37	k	k	PROPN
cana-3972	205	38	is	be	AUX
cana-3972	205	39	directly	directly	ADV
cana-3972	205	40	finite	finite	ADJ
cana-3972	205	41	and	and	CCONJ
cana-3972	205	42	cosingular	cosingular	ADJ
cana-3972	205	43	.	.	PUNCT
cana-3972	206	1	since	since	SCONJ
cana-3972	206	2	m	m	PROPN
cana-3972	206	3	is	be	AUX
cana-3972	206	4	a	a	DET
cana-3972	206	5	d41	d41	NOUN
cana-3972	206	6	-	-	PUNCT
cana-3972	206	7	module	module	NOUN
cana-3972	206	8	,	,	PUNCT
cana-3972	206	9	the	the	DET
cana-3972	206	10	epi	epi	NOUN
cana-3972	206	11	-	-	NOUN
cana-3972	206	12	morphism	morphism	NOUN
cana-3972	206	13	f	f	PROPN
cana-3972	206	14	splits	split	VERB
cana-3972	206	15	,	,	PUNCT
cana-3972	206	16	allowing	allow	VERB
cana-3972	206	17	us	we	PRON
cana-3972	206	18	to	to	PART
cana-3972	206	19	express	express	VERB
cana-3972	206	20	k	k	PROPN
cana-3972	206	21	as	as	ADP
cana-3972	206	22	𝐾	𝐾	PROPN
cana-3972	206	23	≅	≅	PROPN
cana-3972	206	24	𝐿	𝐿	PROPN
cana-3972	206	25	,	,	PUNCT
cana-3972	206	26	where	where	SCONJ
cana-3972	206	27	l	l	NOUN
cana-3972	206	28	≤	≤	PROPN
cana-3972	206	29	⊕	⊕	PROPN
cana-3972	206	30	n	n	PROPN
cana-3972	206	31	.	.	PUNCT
cana-3972	207	1	write	write	VERB
cana-3972	207	2	n	n	NOUN
cana-3972	207	3	=	=	SYM
cana-3972	207	4	l	l	NOUN
cana-3972	207	5	⊕	⊕	PROPN
cana-3972	208	1	k	k	PROPN
cana-3972	208	2	′	′	VERB
cana-3972	208	3	for	for	ADP
cana-3972	208	4	some	some	DET
cana-3972	208	5	submodule	submodule	NOUN
cana-3972	209	1	k	k	PROPN
cana-3972	209	2	′	′	PROPN
cana-3972	209	3	≤	≤	NUM
cana-3972	209	4	m	m	VERB
cana-3972	209	5	.	.	PUNCT
cana-3972	210	1	since	since	SCONJ
cana-3972	210	2	direct	direct	ADJ
cana-3972	210	3	summands	summand	NOUN
cana-3972	210	4	of	of	ADP
cana-3972	210	5	a	a	DET
cana-3972	210	6	d41	d41	NOUN
cana-3972	210	7	-	-	PUNCT
cana-3972	210	8	module	module	NOUN
cana-3972	210	9	remain	remain	VERB
cana-3972	210	10	d41	d41	NOUN
cana-3972	210	11	-	-	PUNCT
cana-3972	210	12	modules	module	NOUN
cana-3972	210	13	,	,	PUNCT
cana-3972	210	14	l	l	PROPN
cana-3972	210	15	⊕	⊕	PROPN
cana-3972	210	16	k	k	PROPN
cana-3972	210	17	is	be	AUX
cana-3972	210	18	also	also	ADV
cana-3972	210	19	a	a	DET
cana-3972	210	20	d41	d41	NOUN
cana-3972	210	21	-	-	PUNCT
cana-3972	210	22	module	module	NOUN
cana-3972	210	23	.	.	PUNCT
cana-3972	211	1	𝐿𝑒𝑡	𝐿𝑒𝑡	NOUN
cana-3972	211	2	𝜋	𝜋	NOUN
cana-3972	211	3	∶	∶	NOUN
cana-3972	211	4	𝑁	𝑁	PROPN
cana-3972	211	5	→	→	SYM
cana-3972	211	6	𝐿	𝐿	PROPN
cana-3972	211	7	be	be	AUX
cana-3972	211	8	the	the	DET
cana-3972	211	9	natural	natural	ADJ
cana-3972	211	10	projection	projection	NOUN
cana-3972	211	11	.	.	PUNCT
cana-3972	212	1	the	the	DET
cana-3972	212	2	composition	composition	NOUN
cana-3972	212	3	𝜋	𝜋	NOUN
cana-3972	212	4	∘	∘	NOUN
cana-3972	212	5	ℎ	ℎ	ADP
cana-3972	212	6	∶	∶	NOUN
cana-3972	212	7	𝐾	𝐾	PROPN
cana-3972	212	8	→	→	SYM
cana-3972	212	9	𝐿	𝐿	PROPN
cana-3972	212	10	is	be	AUX
cana-3972	212	11	an	an	DET
cana-3972	212	12	epimorphism	epimorphism	NOUN
cana-3972	212	13	that	that	PRON
cana-3972	212	14	splits	split	VERB
cana-3972	212	15	by	by	ADP
cana-3972	212	16	lemma	lemma	PROPN
cana-3972	212	17	2.4	2.4	NUM
cana-3972	212	18	.	.	PUNCT
cana-3972	213	1	therefore	therefore	ADV
cana-3972	213	2	,	,	PUNCT
cana-3972	213	3	we	we	PRON
cana-3972	213	4	can	can	AUX
cana-3972	213	5	express	express	VERB
cana-3972	213	6	k	k	PROPN
cana-3972	213	7	as	as	SCONJ
cana-3972	213	8	𝐾	𝐾	PROPN
cana-3972	213	9	=	=	SYM
cana-3972	213	10	𝑘𝑒𝑟(𝜋	𝑘𝑒𝑟(𝜋	PROPN
cana-3972	213	11	∘	∘	NUM
cana-3972	213	12	ℎ	ℎ	SYM
cana-3972	213	13	)	)	PUNCT
cana-3972	213	14	⊕	⊕	PROPN
cana-3972	213	15	𝑃	𝑃	VERB
cana-3972	213	16	for	for	ADP
cana-3972	213	17	some	some	DET
cana-3972	213	18	submodule	submodule	NOUN
cana-3972	213	19	p	p	PROPN
cana-3972	213	20	≤	≤	NOUN
cana-3972	213	21	m	m	VERB
cana-3972	213	22	.	.	PUNCT
cana-3972	214	1	given	give	VERB
cana-3972	214	2	𝐾	𝐾	PROPN
cana-3972	214	3	≅	≅	PROPN
cana-3972	214	4	𝐿	𝐿	PROPN
cana-3972	214	5	≅	≅	PROPN
cana-3972	214	6	𝐾/𝑘𝑒𝑟(𝜋	𝐾/𝑘𝑒𝑟(𝜋	NOUN
cana-3972	214	7	∘	∘	ADJ
cana-3972	214	8	ℎ	ℎ	NOUN
cana-3972	214	9	)	)	PUNCT
cana-3972	214	10	≅	≅	NOUN
cana-3972	214	11	𝑃	𝑃	PROPN
cana-3972	214	12	and	and	CCONJ
cana-3972	214	13	that	that	SCONJ
cana-3972	214	14	k	k	PROPN
cana-3972	214	15	is	be	AUX
cana-3972	214	16	directly	directly	ADV
cana-3972	214	17	finite	finite	ADJ
cana-3972	214	18	,	,	PUNCT
cana-3972	214	19	we	we	PRON
cana-3972	214	20	have	have	VERB
cana-3972	214	21	𝑘𝑒𝑟(𝜋	𝑘𝑒𝑟(𝜋	PROPN
cana-3972	214	22	∘	∘	NUM
cana-3972	214	23	ℎ	ℎ	NOUN
cana-3972	214	24	)	)	PUNCT
cana-3972	214	25	=	=	SYM
cana-3972	214	26	0	0	X
cana-3972	214	27	.	.	PUNCT
cana-3972	215	1	thus	thus	ADV
cana-3972	215	2	,	,	PUNCT
cana-3972	215	3	h	h	NOUN
cana-3972	215	4	is	be	AUX
cana-3972	215	5	an	an	DET
cana-3972	215	6	isomorphism	isomorphism	NOUN
cana-3972	215	7	since	since	SCONJ
cana-3972	215	8	𝑘𝑒𝑟ℎ	𝑘𝑒𝑟ℎ	NOUN
cana-3972	215	9	≤	≤	NUM
cana-3972	215	10	𝑘𝑒𝑟(𝜋	𝑘𝑒𝑟(𝜋	PROPN
cana-3972	215	11	∘	∘	NUM
cana-3972	215	12	ℎ	ℎ	NOUN
cana-3972	215	13	)	)	PUNCT
cana-3972	215	14	,	,	PUNCT
cana-3972	215	15	leading	lead	VERB
cana-3972	215	16	to	to	ADP
cana-3972	215	17	the	the	DET
cana-3972	215	18	conclusion	conclusion	NOUN
cana-3972	215	19	that	that	SCONJ
cana-3972	215	20	𝑁	𝑁	PROPN
cana-3972	215	21	≅	≅	PROPN
cana-3972	215	22	𝐾.	𝐾.	PROPN
cana-3972	215	23	communications	communication	NOUN
cana-3972	215	24	on	on	ADP
cana-3972	215	25	applied	apply	VERB
cana-3972	215	26	nonlinear	nonlinear	ADJ
cana-3972	215	27	analysis	analysis	NOUN
cana-3972	215	28	issn	issn	NOUN
cana-3972	215	29	:	:	PUNCT
cana-3972	215	30	1074	1074	NUM
cana-3972	215	31	-	-	PUNCT
cana-3972	215	32	133x	133x	NUM
cana-3972	215	33	vol	vol	NOUN
cana-3972	215	34	32	32	NUM
cana-3972	215	35	no	no	NOUN
cana-3972	215	36	.	.	PUNCT
cana-3972	216	1	9s	9s	NUM
cana-3972	216	2	(	(	PUNCT
cana-3972	216	3	2025	2025	NUM
cana-3972	216	4	)	)	PUNCT
cana-3972	217	1	681	681	NUM
cana-3972	217	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	217	3	=	=	SYM
cana-3972	217	4	2	2	X
cana-3972	217	5	)	)	PUNCT
cana-3972	217	6	assume	assume	VERB
cana-3972	217	7	n	n	PRON
cana-3972	217	8	is	be	AUX
cana-3972	217	9	square	square	ADV
cana-3972	217	10	-	-	PUNCT
cana-3972	217	11	free	free	ADJ
cana-3972	217	12	.	.	PUNCT
cana-3972	218	1	since	since	SCONJ
cana-3972	218	2	m	m	PROPN
cana-3972	218	3	is	be	AUX
cana-3972	218	4	a	a	DET
cana-3972	218	5	d41	d41	NOUN
cana-3972	218	6	-	-	PUNCT
cana-3972	218	7	module	module	NOUN
cana-3972	218	8	,	,	PUNCT
cana-3972	218	9	the	the	DET
cana-3972	218	10	epimorphism	epimorphism	NOUN
cana-3972	218	11	f	f	PROPN
cana-3972	218	12	splits	split	VERB
cana-3972	218	13	,	,	PUNCT
cana-3972	218	14	allowing	allow	VERB
cana-3972	218	15	us	we	PRON
cana-3972	218	16	to	to	PART
cana-3972	218	17	write	write	VERB
cana-3972	218	18	n	n	PRON
cana-3972	218	19	=	=	SYM
cana-3972	218	20	ker	ker	PROPN
cana-3972	218	21	f	f	PROPN
cana-3972	218	22	⊕	⊕	PROPN
cana-3972	218	23	l	l	PROPN
cana-3972	218	24	for	for	ADP
cana-3972	218	25	a	a	DET
cana-3972	218	26	submodule	submodule	NOUN
cana-3972	218	27	l	l	NOUN
cana-3972	218	28	isomorphic	isomorphic	ADJ
cana-3972	218	29	to	to	ADP
cana-3972	218	30	k	k	PROPN
cana-3972	218	31	.	.	PUNCT
cana-3972	219	1	consider	consider	VERB
cana-3972	219	2	the	the	DET
cana-3972	219	3	epimorphism	epimorphism	NOUN
cana-3972	219	4	𝜋	𝜋	NOUN
cana-3972	219	5	∘	∘	NOUN
cana-3972	219	6	ℎ	ℎ	ADP
cana-3972	219	7	∶	∶	NOUN
cana-3972	219	8	𝐾	𝐾	PROPN
cana-3972	219	9	→	→	SYM
cana-3972	219	10	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	219	11	with	with	ADP
cana-3972	219	12	the	the	DET
cana-3972	219	13	natural	natural	ADJ
cana-3972	219	14	projection	projection	NOUN
cana-3972	219	15	𝜋	𝜋	NOUN
cana-3972	219	16	∶	∶	NOUN
cana-3972	220	1	𝑁	𝑁	PROPN
cana-3972	220	2	→	→	SYM
cana-3972	220	3	𝑘𝑒𝑟𝑓.	𝑘𝑒𝑟𝑓.	NOUN
cana-3972	220	4	the	the	DET
cana-3972	220	5	direct	direct	ADJ
cana-3972	220	6	summands	summand	NOUN
cana-3972	220	7	of	of	ADP
cana-3972	220	8	d41	d41	NOUN
cana-3972	220	9	-	-	PUNCT
cana-3972	220	10	modules	module	NOUN
cana-3972	220	11	remain	remain	VERB
cana-3972	220	12	d41modules	d41module	NOUN
cana-3972	220	13	,	,	PUNCT
cana-3972	220	14	so	so	ADV
cana-3972	220	15	ker	ker	PROPN
cana-3972	220	16	f	f	PROPN
cana-3972	220	17	⊕	⊕	PROPN
cana-3972	220	18	k	k	PROPN
cana-3972	220	19	is	be	AUX
cana-3972	220	20	a	a	DET
cana-3972	220	21	d41	d41	NOUN
cana-3972	220	22	-	-	PUNCT
cana-3972	220	23	module	module	NOUN
cana-3972	220	24	.	.	PUNCT
cana-3972	221	1	by	by	ADP
cana-3972	221	2	lemma	lemma	PROPN
cana-3972	221	3	2.4	2.4	NUM
cana-3972	221	4	,	,	PUNCT
cana-3972	221	5	the	the	DET
cana-3972	221	6	epimorphism	epimorphism	NOUN
cana-3972	221	7	𝜋	𝜋	NOUN
cana-3972	221	8	∘	∘	VERB
cana-3972	221	9	ℎ	ℎ	PART
cana-3972	221	10	also	also	ADV
cana-3972	221	11	splits	split	VERB
cana-3972	221	12	,	,	PUNCT
cana-3972	221	13	yielding	yield	VERB
cana-3972	221	14	𝐾	𝐾	PROPN
cana-3972	221	15	=	=	SYM
cana-3972	221	16	𝑘𝑒𝑟(𝜋	𝑘𝑒𝑟(𝜋	PROPN
cana-3972	221	17	∘	∘	NUM
cana-3972	221	18	ℎ	ℎ	NOUN
cana-3972	221	19	)	)	PUNCT
cana-3972	221	20	⊕	⊕	PROPN
cana-3972	221	21	𝑄	𝑄	PROPN
cana-3972	221	22	for	for	ADP
cana-3972	221	23	some	some	DET
cana-3972	221	24	𝑄	𝑄	PROPN
cana-3972	221	25	≤	≤	PUNCT
cana-3972	221	26	𝑀	𝑀	PROPN
cana-3972	221	27	with	with	ADP
cana-3972	221	28	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	PROPN
cana-3972	221	29	≅	≅	PROPN
cana-3972	221	30	𝑄.	𝑄.	PROPN
cana-3972	221	31	let	let	VERB
cana-3972	221	32	𝜙	𝜙	PRON
cana-3972	221	33	∶	∶	VERB
cana-3972	221	34	𝐾	𝐾	PROPN
cana-3972	221	35	→	→	PUNCT
cana-3972	221	36	𝑄	𝑄	PROPN
cana-3972	221	37	be	be	VERB
cana-3972	221	38	the	the	DET
cana-3972	221	39	natural	natural	ADJ
cana-3972	221	40	projection	projection	NOUN
cana-3972	221	41	.	.	PUNCT
cana-3972	222	1	as	as	ADP
cana-3972	222	2	a	a	DET
cana-3972	222	3	direct	direct	ADJ
cana-3972	222	4	summand	summand	NOUN
cana-3972	222	5	of	of	ADP
cana-3972	222	6	m	m	PROPN
cana-3972	222	7	,	,	PUNCT
cana-3972	222	8	𝐿	𝐿	PROPN
cana-3972	222	9	⊕	⊕	PROPN
cana-3972	222	10	𝑄	𝑄	PROPN
cana-3972	222	11	is	be	AUX
cana-3972	222	12	a	a	DET
cana-3972	222	13	d41	d41	NOUN
cana-3972	222	14	-	-	PUNCT
cana-3972	222	15	module	module	NOUN
cana-3972	222	16	,	,	PUNCT
cana-3972	222	17	so	so	SCONJ
cana-3972	222	18	the	the	DET
cana-3972	222	19	epimorphism	epimorphism	NOUN
cana-3972	222	20	𝜙	𝜙	NOUN
cana-3972	222	21	∘	∘	X
cana-3972	222	22	𝜓	𝜓	PROPN
cana-3972	222	23	∶	∶	NOUN
cana-3972	222	24	𝐿	𝐿	PROPN
cana-3972	222	25	→	→	SYM
cana-3972	222	26	𝑄	𝑄	PRON
cana-3972	222	27	splits	split	VERB
cana-3972	222	28	by	by	ADP
cana-3972	222	29	lemma	lemma	PROPN
cana-3972	222	30	2.4	2.4	NUM
cana-3972	222	31	,	,	PUNCT
cana-3972	222	32	where	where	SCONJ
cana-3972	222	33	𝜓	𝜓	PROPN
cana-3972	222	34	∶	∶	NOUN
cana-3972	222	35	𝐿	𝐿	PROPN
cana-3972	222	36	→	→	SYM
cana-3972	222	37	𝐾	𝐾	PROPN
cana-3972	222	38	is	be	AUX
cana-3972	222	39	an	an	DET
cana-3972	222	40	isomorphism	isomorphism	NOUN
cana-3972	222	41	.	.	PUNCT
cana-3972	223	1	hence	hence	ADV
cana-3972	223	2	,	,	PUNCT
cana-3972	223	3	we	we	PRON
cana-3972	223	4	have	have	VERB
cana-3972	223	5	𝐿	𝐿	NOUN
cana-3972	223	6	=	=	SYM
cana-3972	223	7	𝑘𝑒𝑟(𝜙	𝑘𝑒𝑟(𝜙	PROPN
cana-3972	223	8	∘	∘	PROPN
cana-3972	223	9	𝜓	𝜓	PROPN
cana-3972	223	10	)	)	PUNCT
cana-3972	223	11	⊕	⊕	PROPN
cana-3972	223	12	𝐿′	𝐿′	NOUN
cana-3972	223	13	where	where	SCONJ
cana-3972	223	14	𝑄	𝑄	PROPN
cana-3972	223	15	≅	≅	VERB
cana-3972	223	16	𝐿′	𝐿′	NOUN
cana-3972	223	17	≤	≤	ADV
cana-3972	224	1	𝑀.	𝑀.	NOUN
cana-3972	224	2	this	this	PRON
cana-3972	224	3	leads	lead	VERB
cana-3972	224	4	to	to	ADP
cana-3972	224	5	the	the	DET
cana-3972	224	6	expression	expression	NOUN
cana-3972	224	7	,	,	PUNCT
cana-3972	224	8	𝑁	𝑁	PROPN
cana-3972	224	9	=	=	PUNCT
cana-3972	224	10	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	224	11	⊕	⊕	PROPN
cana-3972	224	12	𝐿	𝐿	PROPN
cana-3972	224	13	=	=	PROPN
cana-3972	224	14	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	224	15	⊕	⊕	PROPN
cana-3972	224	16	𝑘𝑒𝑟(𝜙	𝑘𝑒𝑟(𝜙	PROPN
cana-3972	224	17	∘	∘	PROPN
cana-3972	224	18	𝜓	𝜓	PROPN
cana-3972	224	19	)	)	PUNCT
cana-3972	224	20	⊕	⊕	PROPN
cana-3972	224	21	𝐿′	𝐿′	NOUN
cana-3972	224	22	with	with	ADP
cana-3972	224	23	𝐿′	𝐿′	NOUN
cana-3972	224	24	≅	≅	NOUN
cana-3972	224	25	𝑄	𝑄	PROPN
cana-3972	225	1	≅	≅	NOUN
cana-3972	225	2	𝑘𝑒𝑟𝑓.	𝑘𝑒𝑟𝑓.	NOUN
cana-3972	225	3	since	since	SCONJ
cana-3972	225	4	n	n	PROPN
cana-3972	225	5	is	be	AUX
cana-3972	225	6	square	square	ADV
cana-3972	225	7	-	-	PUNCT
cana-3972	225	8	free	free	ADJ
cana-3972	225	9	,	,	PUNCT
cana-3972	225	10	i	i	PRON
cana-3972	225	11	t	t	PROPN
cana-3972	225	12	follows	follow	VERB
cana-3972	225	13	that	that	DET
cana-3972	225	14	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	225	15	=	=	SYM
cana-3972	225	16	0	0	NUM
cana-3972	225	17	,	,	PUNCT
cana-3972	225	18	thus	thus	ADV
cana-3972	225	19	𝑁	𝑁	PROPN
cana-3972	225	20	≅	≅	PROPN
cana-3972	225	21	𝐾.	𝐾.	PROPN
cana-3972	225	22	recall	recall	VERB
cana-3972	225	23	that	that	SCONJ
cana-3972	225	24	two	two	NUM
cana-3972	225	25	direct	direct	ADJ
cana-3972	225	26	summands	summand	NOUN
cana-3972	225	27	n	n	PRON
cana-3972	225	28	and	and	CCONJ
cana-3972	225	29	k	k	PROPN
cana-3972	225	30	of	of	ADP
cana-3972	225	31	a	a	DET
cana-3972	225	32	module	module	NOUN
cana-3972	225	33	m	m	NOUN
cana-3972	225	34	are	be	AUX
cana-3972	225	35	called	call	VERB
cana-3972	225	36	perspective	perspective	NOUN
cana-3972	225	37	in	in	ADP
cana-3972	225	38	[	[	X
cana-3972	225	39	13	13	NUM
cana-3972	225	40	]	]	PUNCT
cana-3972	225	41	exactly	exactly	ADV
cana-3972	225	42	when	when	SCONJ
cana-3972	225	43	they	they	PRON
cana-3972	225	44	have	have	VERB
cana-3972	225	45	a	a	DET
cana-3972	225	46	common	common	ADJ
cana-3972	225	47	(	(	PUNCT
cana-3972	225	48	direct	direct	ADJ
cana-3972	225	49	sum	sum	NOUN
cana-3972	225	50	)	)	PUNCT
cana-3972	226	1	complement	complement	NOUN
cana-3972	226	2	l	l	NOUN
cana-3972	226	3	,	,	PUNCT
cana-3972	226	4	i.e.	i.e.	X
cana-3972	226	5	m	m	NOUN
cana-3972	226	6	=	=	SYM
cana-3972	226	7	n	n	PROPN
cana-3972	226	8	⊕	⊕	PROPN
cana-3972	226	9	l	l	NOUN
cana-3972	226	10	=	=	PUNCT
cana-3972	226	11	k	k	PROPN
cana-3972	226	12	⊕	⊕	PROPN
cana-3972	226	13	l	l	PROPN
cana-3972	226	14	.	.	PUNCT
cana-3972	227	1	proposition	proposition	NOUN
cana-3972	227	2	3.19	3.19	NUM
cana-3972	227	3	.	.	PUNCT
cana-3972	228	1	the	the	DET
cana-3972	228	2	following	follow	VERB
cana-3972	228	3	conditions	condition	NOUN
cana-3972	228	4	for	for	ADP
cana-3972	228	5	a	a	DET
cana-3972	228	6	module	module	NOUN
cana-3972	228	7	m	m	NOUN
cana-3972	228	8	are	be	AUX
cana-3972	228	9	equivalent	equivalent	ADJ
cana-3972	228	10	:	:	PUNCT
cana-3972	228	11	1	1	X
cana-3972	228	12	)	)	PUNCT
cana-3972	228	13	m	m	VERB
cana-3972	228	14	is	be	AUX
cana-3972	228	15	a	a	DET
cana-3972	228	16	d41	d41	NOUN
cana-3972	228	17	-	-	PUNCT
cana-3972	228	18	module	module	NOUN
cana-3972	228	19	.	.	PUNCT
cana-3972	229	1	2	2	X
cana-3972	229	2	)	)	PUNCT
cana-3972	229	3	if	if	SCONJ
cana-3972	229	4	n	n	PROPN
cana-3972	229	5	and	and	CCONJ
cana-3972	229	6	k	k	PROPN
cana-3972	229	7	are	be	AUX
cana-3972	229	8	perspective	perspective	ADJ
cana-3972	229	9	direct	direct	ADJ
cana-3972	229	10	summands	summand	NOUN
cana-3972	229	11	of	of	ADP
cana-3972	229	12	m	m	PROPN
cana-3972	229	13	with	with	ADP
cana-3972	229	14	common	common	ADJ
cana-3972	229	15	direct	direct	ADJ
cana-3972	229	16	sum	sum	NOUN
cana-3972	229	17	complement	complement	NOUN
cana-3972	229	18	is	be	AUX
cana-3972	229	19	cosingular	cosingular	ADJ
cana-3972	229	20	and	and	CCONJ
cana-3972	229	21	n	n	PROPN
cana-3972	230	1	+	+	CCONJ
cana-3972	230	2	k	k	X
cana-3972	230	3	=	=	SYM
cana-3972	230	4	m	m	PROPN
cana-3972	230	5	,	,	PUNCT
cana-3972	230	6	then	then	ADV
cana-3972	230	7	n	n	X
cana-3972	230	8	∩	∩	NOUN
cana-3972	230	9	k	k	PROPN
cana-3972	230	10	is	be	AUX
cana-3972	230	11	a	a	DET
cana-3972	230	12	direct	direct	ADJ
cana-3972	230	13	summand	summand	NOUN
cana-3972	230	14	of	of	ADP
cana-3972	230	15	m.	m.	NOUN
cana-3972	230	16	3	3	NUM
cana-3972	230	17	)	)	PUNCT
cana-3972	230	18	if	if	SCONJ
cana-3972	230	19	n	n	PROPN
cana-3972	230	20	and	and	CCONJ
cana-3972	230	21	k	k	PROPN
cana-3972	230	22	are	be	AUX
cana-3972	230	23	perspective	perspective	ADJ
cana-3972	230	24	direct	direct	ADJ
cana-3972	230	25	summands	summand	NOUN
cana-3972	230	26	of	of	ADP
cana-3972	230	27	m	m	PROPN
cana-3972	230	28	with	with	ADP
cana-3972	230	29	common	common	ADJ
cana-3972	230	30	direct	direct	ADJ
cana-3972	230	31	sum	sum	NOUN
cana-3972	230	32	complement	complement	NOUN
cana-3972	230	33	is	be	AUX
cana-3972	230	34	cosingular	cosingular	ADJ
cana-3972	230	35	and	and	CCONJ
cana-3972	230	36	n	n	PROPN
cana-3972	231	1	+	+	CCONJ
cana-3972	231	2	k	k	PROPN
cana-3972	231	3	is	be	AUX
cana-3972	231	4	a	a	DET
cana-3972	231	5	direct	direct	ADJ
cana-3972	231	6	summand	summand	NOUN
cana-3972	231	7	of	of	ADP
cana-3972	231	8	m	m	PROPN
cana-3972	231	9	,	,	PUNCT
cana-3972	231	10	then	then	ADV
cana-3972	231	11	n	n	X
cana-3972	231	12	∩	∩	NOUN
cana-3972	231	13	k	k	PROPN
cana-3972	231	14	is	be	AUX
cana-3972	231	15	also	also	ADV
cana-3972	231	16	a	a	DET
cana-3972	231	17	direct	direct	ADJ
cana-3972	231	18	summand	summand	NOUN
cana-3972	231	19	of	of	ADP
cana-3972	231	20	m.	m.	NOUN
cana-3972	231	21	proof	proof	NOUN
cana-3972	231	22	:	:	PUNCT
cana-3972	231	23	1	1	X
cana-3972	231	24	)	)	PUNCT
cana-3972	231	25	⇒	⇒	NOUN
cana-3972	231	26	2	2	NUM
cana-3972	231	27	)	)	PUNCT
cana-3972	231	28	let	let	VERB
cana-3972	231	29	n	n	PRON
cana-3972	231	30	and	and	CCONJ
cana-3972	231	31	k	k	PROPN
cana-3972	231	32	be	be	AUX
cana-3972	231	33	perspective	perspective	ADJ
cana-3972	231	34	direct	direct	ADJ
cana-3972	231	35	summands	summand	NOUN
cana-3972	231	36	with	with	ADP
cana-3972	231	37	a	a	DET
cana-3972	231	38	common	common	ADJ
cana-3972	231	39	direct	direct	ADJ
cana-3972	231	40	sum	sum	NOUN
cana-3972	231	41	complement	complement	NOUN
cana-3972	231	42	l	l	NOUN
cana-3972	231	43	and	and	CCONJ
cana-3972	231	44	l	l	NOUN
cana-3972	231	45	is	be	AUX
cana-3972	231	46	cosingular	cosingular	ADJ
cana-3972	231	47	,	,	PUNCT
cana-3972	231	48	and	and	CCONJ
cana-3972	231	49	suppose	suppose	VERB
cana-3972	231	50	n	n	X
cana-3972	231	51	+	+	CCONJ
cana-3972	232	1	k	k	X
cana-3972	232	2	=	=	NOUN
cana-3972	232	3	m	m	VERB
cana-3972	232	4	.	.	PUNCT
cana-3972	233	1	define	define	VERB
cana-3972	233	2	the	the	DET
cana-3972	233	3	projection	projection	NOUN
cana-3972	233	4	𝜋	𝜋	PROPN
cana-3972	233	5	∶	∶	NOUN
cana-3972	233	6	𝑀	𝑀	PROPN
cana-3972	233	7	→	→	SYM
cana-3972	233	8	𝐿	𝐿	PROPN
cana-3972	233	9	with	with	ADP
cana-3972	233	10	𝑘𝑒𝑟𝜋	𝑘𝑒𝑟𝜋	ADJ
cana-3972	233	11	=	=	SYM
cana-3972	233	12	𝐾.	𝐾.	NOUN
cana-3972	233	13	consider	consider	VERB
cana-3972	233	14	the	the	DET
cana-3972	233	15	restricted	restricted	ADJ
cana-3972	233	16	map	map	NOUN
cana-3972	233	17	𝜋|𝑁	𝜋|𝑁	NOUN
cana-3972	233	18	∶	∶	NOUN
cana-3972	233	19	𝑁	𝑁	PROPN
cana-3972	233	20	→	→	SYM
cana-3972	233	21	𝐿.	𝐿.	VERB
cana-3972	233	22	since	since	SCONJ
cana-3972	233	23	𝑁	𝑁	PROPN
cana-3972	233	24	+	+	PROPN
cana-3972	233	25	𝐾	𝐾	PROPN
cana-3972	233	26	=	=	SYM
cana-3972	233	27	𝑀	𝑀	PROPN
cana-3972	233	28	,	,	PUNCT
cana-3972	233	29	i	i	PRON
cana-3972	233	30	t	t	PROPN
cana-3972	233	31	follows	follow	VERB
cana-3972	233	32	that	that	SCONJ
cana-3972	233	33	π(n	π(n	PROPN
cana-3972	233	34	)	)	PUNCT
cana-3972	234	1	=	=	PUNCT
cana-3972	234	2	l	l	NOUN
cana-3972	234	3	≤	≤	NUM
cana-3972	234	4	⊕	⊕	PROPN
cana-3972	234	5	l	l	PROPN
cana-3972	234	6	,	,	PUNCT
cana-3972	234	7	which	which	PRON
cana-3972	234	8	implies	imply	VERB
cana-3972	234	9	𝑘𝑒𝑟𝜋|𝑁	𝑘𝑒𝑟𝜋|𝑁	NOUN
cana-3972	234	10	≤	≤	NUM
cana-3972	234	11	⊕	⊕	PROPN
cana-3972	234	12	n	n	PROPN
cana-3972	234	13	.	.	PUNCT
cana-3972	235	1	thus	thus	ADV
cana-3972	235	2	,	,	PUNCT
cana-3972	235	3	since	since	SCONJ
cana-3972	235	4	𝑘𝑒𝑟𝜋|𝑁	𝑘𝑒𝑟𝜋|𝑁	INTJ
cana-3972	235	5	=	=	SYM
cana-3972	235	6	𝑁	𝑁	PROPN
cana-3972	235	7	∩	∩	ADJ
cana-3972	235	8	𝐾	𝐾	PROPN
cana-3972	235	9	≤	≤	PROPN
cana-3972	235	10	⊕	⊕	PROPN
cana-3972	235	11	n	n	CCONJ
cana-3972	235	12	≤	≤	PROPN
cana-3972	235	13	⊕	⊕	PROPN
cana-3972	235	14	m	m	VERB
cana-3972	235	15	,	,	PUNCT
cana-3972	235	16	we	we	PRON
cana-3972	235	17	have	have	VERB
cana-3972	235	18	n	n	NOUN
cana-3972	235	19	∩	∩	NOUN
cana-3972	235	20	k	k	PROPN
cana-3972	235	21	≤	≤	PROPN
cana-3972	235	22	⊕	⊕	PROPN
cana-3972	235	23	m	m	PROPN
cana-3972	235	24	.	.	PUNCT
cana-3972	236	1	2	2	X
cana-3972	236	2	)	)	PUNCT
cana-3972	236	3	⇒	⇒	NOUN
cana-3972	236	4	3	3	NUM
cana-3972	236	5	)	)	PUNCT
cana-3972	236	6	let	let	VERB
cana-3972	236	7	n	n	PRON
cana-3972	236	8	and	and	CCONJ
cana-3972	236	9	k	k	PROPN
cana-3972	236	10	be	be	AUX
cana-3972	236	11	perspective	perspective	ADJ
cana-3972	236	12	direct	direct	ADJ
cana-3972	236	13	summands	summand	NOUN
cana-3972	236	14	of	of	ADP
cana-3972	236	15	m	m	VERB
cana-3972	236	16	such	such	ADJ
cana-3972	236	17	that	that	SCONJ
cana-3972	236	18	n	n	PROPN
cana-3972	236	19	+	+	CCONJ
cana-3972	236	20	k	k	PROPN
cana-3972	236	21	≤	≤	PROPN
cana-3972	236	22	⊕	⊕	PROPN
cana-3972	236	23	m	m	PROPN
cana-3972	236	24	.	.	PUNCT
cana-3972	237	1	there	there	PRON
cana-3972	237	2	exist	exist	VERB
cana-3972	237	3	l	l	NOUN
cana-3972	237	4	and	and	CCONJ
cana-3972	237	5	p	p	PROPN
cana-3972	237	6	≤	≤	PROPN
cana-3972	237	7	⊕	⊕	PROPN
cana-3972	237	8	m	m	VERB
cana-3972	237	9	such	such	ADJ
cana-3972	237	10	that	that	SCONJ
cana-3972	237	11	m	m	VERB
cana-3972	237	12	=	=	SYM
cana-3972	237	13	n	n	PROPN
cana-3972	237	14	⊕	⊕	PROPN
cana-3972	237	15	l	l	NOUN
cana-3972	238	1	=	=	PUNCT
cana-3972	238	2	k	k	PROPN
cana-3972	238	3	⊕	⊕	PROPN
cana-3972	238	4	l	l	NOUN
cana-3972	239	1	=	=	PUNCT
cana-3972	239	2	(	(	PUNCT
cana-3972	239	3	n	n	PROPN
cana-3972	239	4	+	+	CCONJ
cana-3972	239	5	k	k	X
cana-3972	239	6	)	)	PUNCT
cana-3972	239	7	⊕	⊕	PROPN
cana-3972	239	8	p	p	NOUN
cana-3972	239	9	with	with	ADP
cana-3972	239	10	l	l	NOUN
cana-3972	239	11	is	be	AUX
cana-3972	239	12	cosingular	cosingular	ADJ
cana-3972	239	13	.	.	PUNCT
cana-3972	240	1	by	by	ADP
cana-3972	240	2	modularity	modularity	NOUN
cana-3972	240	3	,	,	PUNCT
cana-3972	240	4	we	we	PRON
cana-3972	240	5	can	can	AUX
cana-3972	240	6	express	express	VERB
cana-3972	240	7	n	n	NOUN
cana-3972	240	8	+	+	CCONJ
cana-3972	240	9	k	k	X
cana-3972	240	10	=	=	SYM
cana-3972	240	11	n	n	PROPN
cana-3972	240	12	⊕	⊕	PROPN
cana-3972	240	13	(	(	PUNCT
cana-3972	240	14	l	l	NOUN
cana-3972	240	15	∩	∩	X
cana-3972	240	16	(	(	PUNCT
cana-3972	240	17	n	n	PROPN
cana-3972	240	18	+	+	CCONJ
cana-3972	240	19	k	k	NOUN
cana-3972	240	20	)	)	PUNCT
cana-3972	240	21	)	)	PUNCT
cana-3972	240	22	and	and	CCONJ
cana-3972	240	23	n	n	PROPN
cana-3972	240	24	+	+	CCONJ
cana-3972	240	25	k	k	PROPN
cana-3972	240	26	=	=	SYM
cana-3972	240	27	k	k	PROPN
cana-3972	240	28	⊕	⊕	PROPN
cana-3972	240	29	(	(	PUNCT
cana-3972	240	30	l	l	PROPN
cana-3972	240	31	∩	∩	X
cana-3972	240	32	(	(	PUNCT
cana-3972	240	33	n	n	PROPN
cana-3972	240	34	+	+	CCONJ
cana-3972	240	35	k	k	NOUN
cana-3972	240	36	)	)	PUNCT
cana-3972	240	37	)	)	PUNCT
cana-3972	240	38	.	.	PUNCT
cana-3972	241	1	the	the	DET
cana-3972	241	2	modules	module	NOUN
cana-3972	241	3	n	n	PROPN
cana-3972	241	4	⊕	⊕	PROPN
cana-3972	241	5	p	p	PROPN
cana-3972	241	6	and	and	CCONJ
cana-3972	241	7	k	k	PROPN
cana-3972	241	8	⊕	⊕	PROPN
cana-3972	241	9	p	p	PROPN
cana-3972	241	10	are	be	AUX
cana-3972	241	11	then	then	ADV
cana-3972	241	12	perspective	perspective	ADJ
cana-3972	241	13	direct	direct	ADJ
cana-3972	241	14	summands	summand	NOUN
cana-3972	241	15	of	of	ADP
cana-3972	241	16	m	m	PRON
cana-3972	241	17	,	,	PUNCT
cana-3972	241	18	satisfying	satisfy	VERB
cana-3972	241	19	(	(	PUNCT
cana-3972	241	20	n	n	PROPN
cana-3972	241	21	⊕	⊕	PROPN
cana-3972	241	22	p	p	NOUN
cana-3972	241	23	)	)	PUNCT
cana-3972	242	1	+	+	CCONJ
cana-3972	242	2	(	(	PUNCT
cana-3972	242	3	k	k	PROPN
cana-3972	242	4	⊕	⊕	PROPN
cana-3972	242	5	p	p	NOUN
cana-3972	242	6	)	)	PUNCT
cana-3972	243	1	=	=	PUNCT
cana-3972	244	1	m	m	NOUN
cana-3972	244	2	.	.	PUNCT
cana-3972	245	1	by	by	ADP
cana-3972	245	2	the	the	DET
cana-3972	245	3	hypothesis	hypothesis	NOUN
cana-3972	245	4	,	,	PUNCT
cana-3972	245	5	this	this	PRON
cana-3972	245	6	implies	imply	VERB
cana-3972	245	7	that	that	SCONJ
cana-3972	245	8	(	(	PUNCT
cana-3972	245	9	n	n	PROPN
cana-3972	245	10	⊕	⊕	PROPN
cana-3972	245	11	p	p	NOUN
cana-3972	245	12	)	)	PUNCT
cana-3972	245	13	∩	∩	NOUN
cana-3972	245	14	(	(	PUNCT
cana-3972	245	15	k	k	PROPN
cana-3972	245	16	⊕	⊕	PROPN
cana-3972	245	17	p	p	NOUN
cana-3972	245	18	)	)	PUNCT
cana-3972	245	19	=	=	SYM
cana-3972	245	20	(	(	PUNCT
cana-3972	245	21	n	n	X
cana-3972	245	22	∩	∩	X
cana-3972	245	23	k	k	X
cana-3972	245	24	)	)	PUNCT
cana-3972	245	25	⊕	⊕	PROPN
cana-3972	245	26	p	p	NOUN
cana-3972	245	27	≤	≤	PROPN
cana-3972	245	28	⊕	⊕	PROPN
cana-3972	245	29	m	m	PROPN
cana-3972	245	30	.	.	PUNCT
cana-3972	246	1	thus	thus	ADV
cana-3972	246	2	,	,	PUNCT
cana-3972	246	3	we	we	PRON
cana-3972	246	4	conclude	conclude	VERB
cana-3972	246	5	that	that	SCONJ
cana-3972	246	6	n	n	ADP
cana-3972	246	7	∩	∩	NOUN
cana-3972	246	8	k	k	PROPN
cana-3972	246	9	≤	≤	PROPN
cana-3972	246	10	⊕	⊕	PROPN
cana-3972	246	11	m	m	PROPN
cana-3972	246	12	.	.	PUNCT
cana-3972	247	1	3	3	X
cana-3972	247	2	)	)	PUNCT
cana-3972	247	3	⇒	⇒	NOUN
cana-3972	247	4	1	1	NUM
cana-3972	247	5	)	)	PUNCT
cana-3972	247	6	let	let	VERB
cana-3972	247	7	m	m	NOUN
cana-3972	247	8	=	=	SYM
cana-3972	247	9	n	n	PROPN
cana-3972	247	10	⊕	⊕	PROPN
cana-3972	247	11	k	k	PROPN
cana-3972	247	12	and	and	CCONJ
cana-3972	247	13	let	let	VERB
cana-3972	247	14	𝑓	𝑓	DET
cana-3972	247	15	∶	∶	NOUN
cana-3972	247	16	𝑁	𝑁	PROPN
cana-3972	247	17	→	→	SYM
cana-3972	247	18	𝐾	𝐾	PROPN
cana-3972	247	19	be	be	AUX
cana-3972	247	20	an	an	DET
cana-3972	247	21	epimorphism	epimorphism	NOUN
cana-3972	247	22	with	with	ADP
cana-3972	247	23	k	k	PROPN
cana-3972	247	24	is	be	AUX
cana-3972	247	25	cosingular	cosingular	ADJ
cana-3972	247	26	.	.	PUNCT
cana-3972	248	1	define	define	VERB
cana-3972	248	2	the	the	DET
cana-3972	248	3	graph	graph	NOUN
cana-3972	248	4	submodule	submodule	NOUN
cana-3972	248	5	𝐺	𝐺	PROPN
cana-3972	248	6	=	=	PUNCT
cana-3972	248	7	{	{	PUNCT
cana-3972	248	8	𝑎	𝑎	NOUN
cana-3972	248	9	+	+	CCONJ
cana-3972	248	10	𝑓(𝑎	𝑓(𝑎	ADJ
cana-3972	248	11	)	)	PUNCT
cana-3972	248	12	∶	∶	NOUN
cana-3972	248	13	𝑎	𝑎	PROPN
cana-3972	248	14	∈	∈	PROPN
cana-3972	248	15	𝐴	𝐴	PROPN
cana-3972	248	16	}	}	PUNCT
cana-3972	248	17	within	within	ADP
cana-3972	248	18	m	m	PROPN
cana-3972	248	19	.	.	PUNCT
cana-3972	249	1	clearly	clearly	ADV
cana-3972	249	2	,	,	PUNCT
cana-3972	249	3	m	m	VERB
cana-3972	249	4	=	=	SYM
cana-3972	249	5	g	g	PROPN
cana-3972	249	6	+	+	CCONJ
cana-3972	249	7	k	k	PROPN
cana-3972	249	8	and	and	CCONJ
cana-3972	249	9	g	g	PROPN
cana-3972	249	10	∩	∩	ADJ
cana-3972	249	11	k	k	NOUN
cana-3972	249	12	=	=	SYM
cana-3972	249	13	0	0	PROPN
cana-3972	249	14	.	.	PUNCT
cana-3972	250	1	this	this	PRON
cana-3972	250	2	shows	show	VERB
cana-3972	250	3	that	that	SCONJ
cana-3972	250	4	m	m	VERB
cana-3972	250	5	=	=	SYM
cana-3972	250	6	g	g	PROPN
cana-3972	250	7	⊕	⊕	PROPN
cana-3972	250	8	k	k	PROPN
cana-3972	251	1	=	=	PUNCT
cana-3972	251	2	n	n	PROPN
cana-3972	251	3	⊕	⊕	PROPN
cana-3972	251	4	k	k	PROPN
cana-3972	251	5	,	,	PUNCT
cana-3972	251	6	meaning	mean	VERB
cana-3972	251	7	n	n	NOUN
cana-3972	251	8	and	and	CCONJ
cana-3972	251	9	g	g	PROPN
cana-3972	251	10	are	be	AUX
cana-3972	251	11	perspective	perspective	ADJ
cana-3972	251	12	direct	direct	ADJ
cana-3972	251	13	summands	summand	NOUN
cana-3972	251	14	of	of	ADP
cana-3972	251	15	m	m	PRON
cana-3972	251	16	.	.	PUNCT
cana-3972	252	1	since	since	SCONJ
cana-3972	252	2	f	f	PROPN
cana-3972	252	3	is	be	AUX
cana-3972	252	4	an	an	DET
cana-3972	252	5	epimorphism	epimorphism	NOUN
cana-3972	252	6	,	,	PUNCT
cana-3972	252	7	we	we	PRON
cana-3972	252	8	have	have	VERB
cana-3972	252	9	n	n	NOUN
cana-3972	252	10	+	+	CCONJ
cana-3972	252	11	g	g	NOUN
cana-3972	252	12	=	=	NOUN
cana-3972	252	13	m	m	PROPN
cana-3972	252	14	.	.	PUNCT
cana-3972	253	1	by	by	ADP
cana-3972	253	2	the	the	DET
cana-3972	253	3	hypothesis	hypothesis	NOUN
cana-3972	253	4	,	,	PUNCT
cana-3972	253	5	i	i	PRON
cana-3972	253	6	t	t	PROPN
cana-3972	253	7	follows	follow	VERB
cana-3972	253	8	that	that	SCONJ
cana-3972	253	9	n	n	ADP
cana-3972	253	10	∩	∩	NOUN
cana-3972	253	11	g	g	PROPN
cana-3972	253	12	≤	≤	PROPN
cana-3972	253	13	⊕	⊕	PROPN
cana-3972	253	14	m	m	PROPN
cana-3972	253	15	.	.	PUNCT
cana-3972	254	1	it	it	PRON
cana-3972	254	2	can	can	AUX
cana-3972	254	3	be	be	AUX
cana-3972	254	4	readily	readily	ADV
cana-3972	254	5	demonstrated	demonstrate	VERB
cana-3972	254	6	that	that	SCONJ
cana-3972	254	7	n	n	ADP
cana-3972	254	8	∩	∩	NOUN
cana-3972	254	9	g	g	NOUN
cana-3972	254	10	=	=	SYM
cana-3972	254	11	ker	ker	PROPN
cana-3972	254	12	f	f	PROPN
cana-3972	254	13	,	,	PUNCT
cana-3972	254	14	leading	lead	VERB
cana-3972	254	15	to	to	ADP
cana-3972	254	16	the	the	DET
cana-3972	254	17	conclusion	conclusion	NOUN
cana-3972	254	18	that	that	SCONJ
cana-3972	254	19	ker	ker	PROPN
cana-3972	254	20	f	f	PROPN
cana-3972	254	21	≤	≤	PROPN
cana-3972	254	22	⊕	⊕	PROPN
cana-3972	254	23	m	m	PROPN
cana-3972	254	24	.	.	PUNCT
cana-3972	255	1	therefore	therefore	ADV
cana-3972	255	2	,	,	PUNCT
cana-3972	255	3	we	we	PRON
cana-3972	255	4	find	find	VERB
cana-3972	255	5	that	that	DET
cana-3972	255	6	ker	ker	PROPN
cana-3972	255	7	f	f	PROPN
cana-3972	255	8	≤	≤	PROPN
cana-3972	255	9	⊕	⊕	PROPN
cana-3972	255	10	n	n	PROPN
cana-3972	255	11	.	.	PUNCT
cana-3972	256	1	corollary	corollary	ADJ
cana-3972	256	2	3.20	3.20	NUM
cana-3972	256	3	.	.	PUNCT
cana-3972	257	1	the	the	DET
cana-3972	257	2	following	follow	VERB
cana-3972	257	3	conditions	condition	NOUN
cana-3972	257	4	for	for	ADP
cana-3972	257	5	a	a	DET
cana-3972	257	6	module	module	NOUN
cana-3972	257	7	m	m	NOUN
cana-3972	257	8	are	be	AUX
cana-3972	257	9	equivalent	equivalent	ADJ
cana-3972	257	10	:	:	PUNCT
cana-3972	257	11	1	1	X
cana-3972	257	12	)	)	PUNCT
cana-3972	257	13	m	m	VERB
cana-3972	257	14	is	be	AUX
cana-3972	257	15	a	a	DET
cana-3972	257	16	d41	d41	NOUN
cana-3972	257	17	-	-	PUNCT
cana-3972	257	18	module	module	NOUN
cana-3972	257	19	.	.	PUNCT
cana-3972	258	1	2	2	X
cana-3972	258	2	)	)	PUNCT
cana-3972	258	3	if	if	SCONJ
cana-3972	258	4	m	m	NOUN
cana-3972	258	5	=	=	SYM
cana-3972	258	6	n	n	PROPN
cana-3972	258	7	+	+	CCONJ
cana-3972	258	8	k	k	PROPN
cana-3972	258	9	for	for	ADP
cana-3972	258	10	any	any	DET
cana-3972	258	11	perspective	perspective	NOUN
cana-3972	258	12	direct	direct	ADJ
cana-3972	258	13	summands	summand	NOUN
cana-3972	258	14	n	n	PRON
cana-3972	258	15	and	and	CCONJ
cana-3972	258	16	k	k	PROPN
cana-3972	258	17	of	of	ADP
cana-3972	258	18	m	m	PROPN
cana-3972	258	19	with	with	ADP
cana-3972	258	20	common	common	ADJ
cana-3972	258	21	direct	direct	ADJ
cana-3972	258	22	sum	sum	NOUN
cana-3972	258	23	complement	complement	NOUN
cana-3972	258	24	is	be	AUX
cana-3972	258	25	cosingular	cosingular	ADJ
cana-3972	258	26	,	,	PUNCT
cana-3972	258	27	then	then	ADV
cana-3972	258	28	there	there	PRON
cana-3972	258	29	exists	exist	VERB
cana-3972	258	30	a	a	DET
cana-3972	258	31	submodule	submodule	NOUN
cana-3972	259	1	k	k	NOUN
cana-3972	260	1	′	′	NOUN
cana-3972	261	1	⊆	⊆	NUM
cana-3972	261	2	k	k	X
cana-3972	261	3	such	such	ADJ
cana-3972	261	4	that	that	SCONJ
cana-3972	261	5	m	m	VERB
cana-3972	261	6	=	=	SYM
cana-3972	261	7	n	n	PROPN
cana-3972	261	8	⊕	⊕	PROPN
cana-3972	261	9	k	k	PROPN
cana-3972	262	1	′	′	NUM
cana-3972	262	2	.	.	PUNCT
cana-3972	263	1	communications	communication	NOUN
cana-3972	263	2	on	on	ADP
cana-3972	263	3	applied	apply	VERB
cana-3972	263	4	nonlinear	nonlinear	ADJ
cana-3972	263	5	analysis	analysis	NOUN
cana-3972	263	6	issn	issn	NOUN
cana-3972	263	7	:	:	PUNCT
cana-3972	263	8	1074	1074	NUM
cana-3972	263	9	-	-	PUNCT
cana-3972	263	10	133x	133x	NUM
cana-3972	263	11	vol	vol	NOUN
cana-3972	263	12	32	32	NUM
cana-3972	263	13	no	no	NOUN
cana-3972	263	14	.	.	PUNCT
cana-3972	264	1	9s	9s	NUM
cana-3972	264	2	(	(	PUNCT
cana-3972	264	3	2025	2025	NUM
cana-3972	264	4	)	)	PUNCT
cana-3972	265	1	682	682	NUM
cana-3972	265	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	265	3	proof	proof	NOUN
cana-3972	265	4	:	:	PUNCT
cana-3972	265	5	1	1	X
cana-3972	265	6	)	)	PUNCT
cana-3972	265	7	⇒	⇒	NOUN
cana-3972	265	8	2	2	NUM
cana-3972	265	9	)	)	PUNCT
cana-3972	265	10	let	let	VERB
cana-3972	265	11	m	m	VERB
cana-3972	265	12	=	=	SYM
cana-3972	265	13	n	n	PROPN
cana-3972	265	14	+	+	CCONJ
cana-3972	265	15	k	k	PROPN
cana-3972	265	16	,	,	PUNCT
cana-3972	265	17	where	where	SCONJ
cana-3972	265	18	n	n	NUM
cana-3972	265	19	and	and	CCONJ
cana-3972	265	20	k	k	PROPN
cana-3972	265	21	are	be	AUX
cana-3972	265	22	perspective	perspective	ADJ
cana-3972	265	23	direct	direct	ADJ
cana-3972	265	24	summands	summand	NOUN
cana-3972	265	25	of	of	ADP
cana-3972	265	26	m	m	PROPN
cana-3972	265	27	with	with	ADP
cana-3972	265	28	common	common	ADJ
cana-3972	265	29	direct	direct	ADJ
cana-3972	265	30	sum	sum	NOUN
cana-3972	265	31	complement	complement	NOUN
cana-3972	265	32	is	be	AUX
cana-3972	265	33	cosingular	cosingular	ADJ
cana-3972	265	34	.	.	PUNCT
cana-3972	266	1	by	by	ADP
cana-3972	266	2	proposition	proposition	NOUN
cana-3972	266	3	3.19	3.19	NUM
cana-3972	266	4	,	,	PUNCT
cana-3972	266	5	we	we	PRON
cana-3972	266	6	can	can	AUX
cana-3972	266	7	express	express	VERB
cana-3972	266	8	m	m	PRON
cana-3972	266	9	as	as	ADP
cana-3972	266	10	m	m	PROPN
cana-3972	266	11	=	=	SYM
cana-3972	266	12	(	(	PUNCT
cana-3972	266	13	n	n	X
cana-3972	266	14	∩	∩	X
cana-3972	266	15	k	k	PROPN
cana-3972	266	16	)	)	PUNCT
cana-3972	266	17	⊕	⊕	PROPN
cana-3972	266	18	l	l	NOUN
cana-3972	266	19	for	for	ADP
cana-3972	266	20	some	some	DET
cana-3972	266	21	submodule	submodule	NOUN
cana-3972	266	22	l.	l.	NOUN
cana-3972	266	23	according	accord	VERB
cana-3972	266	24	to	to	ADP
cana-3972	266	25	the	the	DET
cana-3972	266	26	modular	modular	ADJ
cana-3972	266	27	law	law	NOUN
cana-3972	266	28	,	,	PUNCT
cana-3972	266	29	we	we	PRON
cana-3972	266	30	have	have	VERB
cana-3972	266	31	k	k	NOUN
cana-3972	266	32	=	=	SYM
cana-3972	266	33	(	(	PUNCT
cana-3972	266	34	n	n	X
cana-3972	266	35	∩	∩	X
cana-3972	266	36	k	k	PROPN
cana-3972	266	37	)	)	PUNCT
cana-3972	266	38	⊕	⊕	PROPN
cana-3972	266	39	(	(	PUNCT
cana-3972	266	40	l	l	PROPN
cana-3972	266	41	∩	∩	X
cana-3972	266	42	k	k	PROPN
cana-3972	266	43	)	)	PUNCT
cana-3972	266	44	.	.	PUNCT
cana-3972	267	1	thus	thus	ADV
cana-3972	267	2	,	,	PUNCT
cana-3972	267	3	we	we	PRON
cana-3972	267	4	can	can	AUX
cana-3972	267	5	write	write	VERB
cana-3972	267	6	m	m	PROPN
cana-3972	267	7	=	=	SYM
cana-3972	267	8	n	n	PROPN
cana-3972	267	9	⊕	⊕	PROPN
cana-3972	267	10	(	(	PUNCT
cana-3972	267	11	l	l	PROPN
cana-3972	267	12	∩	∩	X
cana-3972	267	13	k	k	PROPN
cana-3972	267	14	)	)	PUNCT
cana-3972	267	15	,	,	PUNCT
cana-3972	267	16	leading	lead	VERB
cana-3972	267	17	us	we	PRON
cana-3972	267	18	to	to	PART
cana-3972	267	19	define	define	VERB
cana-3972	267	20	k	k	PROPN
cana-3972	267	21	′	′	NUM
cana-3972	267	22	:	:	PUNCT
cana-3972	267	23	=	=	SYM
cana-3972	267	24	l	l	NOUN
cana-3972	267	25	∩	∩	X
cana-3972	267	26	k	k	X
cana-3972	267	27	.	.	PUNCT
cana-3972	267	28	2	2	X
cana-3972	267	29	)	)	PUNCT
cana-3972	267	30	⇒	⇒	NOUN
cana-3972	267	31	1	1	NUM
cana-3972	267	32	)	)	PUNCT
cana-3972	267	33	now	now	ADV
cana-3972	267	34	let	let	VERB
cana-3972	267	35	m	m	NOUN
cana-3972	267	36	=	=	SYM
cana-3972	267	37	n	n	PROPN
cana-3972	267	38	+	+	CCONJ
cana-3972	267	39	k	k	PROPN
cana-3972	267	40	,	,	PUNCT
cana-3972	267	41	where	where	SCONJ
cana-3972	267	42	n	n	NUM
cana-3972	267	43	and	and	CCONJ
cana-3972	267	44	k	k	PROPN
cana-3972	267	45	are	be	AUX
cana-3972	267	46	perspective	perspective	ADJ
cana-3972	267	47	direct	direct	ADJ
cana-3972	267	48	summands	summand	NOUN
cana-3972	267	49	of	of	ADP
cana-3972	267	50	m	m	PROPN
cana-3972	267	51	with	with	ADP
cana-3972	267	52	common	common	ADJ
cana-3972	267	53	direct	direct	ADJ
cana-3972	267	54	sum	sum	NOUN
cana-3972	267	55	complement	complement	NOUN
cana-3972	267	56	is	be	AUX
cana-3972	267	57	cosingular	cosingular	ADJ
cana-3972	267	58	.	.	PUNCT
cana-3972	268	1	by	by	ADP
cana-3972	268	2	the	the	DET
cana-3972	268	3	assumption	assumption	NOUN
cana-3972	268	4	,	,	PUNCT
cana-3972	268	5	there	there	PRON
cana-3972	268	6	exists	exist	VERB
cana-3972	268	7	a	a	DET
cana-3972	268	8	submodule	submodule	NOUN
cana-3972	268	9	𝐾′	𝐾′	NOUN
cana-3972	268	10	⊆	⊆	NUM
cana-3972	268	11	𝐾	𝐾	PROPN
cana-3972	268	12	such	such	ADJ
cana-3972	268	13	that	that	SCONJ
cana-3972	268	14	m	m	VERB
cana-3972	268	15	=	=	SYM
cana-3972	268	16	n	n	PROPN
cana-3972	268	17	⊕	⊕	PROPN
cana-3972	268	18	k	k	PROPN
cana-3972	268	19	′	′	NUM
cana-3972	268	20	.	.	PUNCT
cana-3972	269	1	this	this	PRON
cana-3972	269	2	means	mean	VERB
cana-3972	269	3	we	we	PRON
cana-3972	269	4	can	can	AUX
cana-3972	269	5	express	express	VERB
cana-3972	269	6	k	k	PROPN
cana-3972	269	7	as	as	ADP
cana-3972	269	8	k	k	PROPN
cana-3972	269	9	=	=	PUNCT
cana-3972	269	10	(	(	PUNCT
cana-3972	270	1	n	n	X
cana-3972	270	2	∩	∩	X
cana-3972	270	3	k	k	PROPN
cana-3972	270	4	)	)	PUNCT
cana-3972	270	5	⊕	⊕	PROPN
cana-3972	270	6	k	k	PROPN
cana-3972	271	1	′	′	NUM
cana-3972	271	2	.	.	PUNCT
cana-3972	272	1	consequently	consequently	ADV
cana-3972	272	2	,	,	PUNCT
cana-3972	272	3	n	n	PRON
cana-3972	272	4	∩	∩	NOUN
cana-3972	272	5	k	k	PROPN
cana-3972	272	6	is	be	AUX
cana-3972	272	7	a	a	DET
cana-3972	272	8	direct	direct	ADJ
cana-3972	272	9	summand	summand	NOUN
cana-3972	272	10	of	of	ADP
cana-3972	272	11	k	k	PROPN
cana-3972	272	12	,	,	PUNCT
cana-3972	272	13	and	and	CCONJ
cana-3972	272	14	therefore	therefore	ADV
cana-3972	272	15	also	also	ADV
cana-3972	272	16	a	a	DET
cana-3972	272	17	direct	direct	ADJ
cana-3972	272	18	summand	summand	NOUN
cana-3972	272	19	of	of	ADP
cana-3972	272	20	m	m	PROPN
cana-3972	272	21	.	.	PUNCT
cana-3972	273	1	recall	recall	VERB
cana-3972	273	2	that	that	SCONJ
cana-3972	273	3	a	a	DET
cana-3972	273	4	module	module	NOUN
cana-3972	273	5	m	m	NOUN
cana-3972	273	6	is	be	AUX
cana-3972	273	7	a	a	DET
cana-3972	273	8	c4	c4	NOUN
cana-3972	273	9	-	-	PUNCT
cana-3972	273	10	module	module	NOUN
cana-3972	273	11	if	if	SCONJ
cana-3972	273	12	m	m	NOUN
cana-3972	273	13	=	=	SYM
cana-3972	274	1	n	n	PROPN
cana-3972	274	2	⊕	⊕	PROPN
cana-3972	274	3	k	k	PROPN
cana-3972	274	4	,	,	PUNCT
cana-3972	274	5	then	then	ADV
cana-3972	274	6	every	every	DET
cana-3972	274	7	monomorphism	monomorphism	NOUN
cana-3972	274	8	f	f	X
cana-3972	274	9	:	:	PUNCT
cana-3972	274	10	n	n	PROPN
cana-3972	274	11	→	→	SYM
cana-3972	274	12	k	k	X
cana-3972	274	13	splits	split	VERB
cana-3972	274	14	[	[	X
cana-3972	274	15	6	6	NUM
cana-3972	274	16	]	]	PUNCT
cana-3972	274	17	.	.	PUNCT
cana-3972	275	1	we	we	PRON
cana-3972	275	2	next	next	ADV
cana-3972	275	3	give	give	VERB
cana-3972	275	4	a	a	DET
cana-3972	275	5	relationship	relationship	NOUN
cana-3972	275	6	between	between	ADP
cana-3972	275	7	c4	c4	NOUN
cana-3972	275	8	-	-	PUNCT
cana-3972	275	9	modules	module	NOUN
cana-3972	275	10	and	and	CCONJ
cana-3972	275	11	d41	d41	NOUN
cana-3972	275	12	-	-	PUNCT
cana-3972	275	13	modules	module	NOUN
cana-3972	275	14	under	under	ADP
cana-3972	275	15	certain	certain	ADJ
cana-3972	275	16	conditions	condition	NOUN
cana-3972	275	17	and	and	CCONJ
cana-3972	275	18	having	have	VERB
cana-3972	275	19	perspective	perspective	NOUN
cana-3972	275	20	direct	direct	ADJ
cana-3972	275	21	summand	summand	NOUN
cana-3972	275	22	.	.	PUNCT
cana-3972	276	1	a	a	DET
cana-3972	276	2	module	module	NOUN
cana-3972	276	3	m	m	VERB
cana-3972	276	4	is	be	AUX
cana-3972	276	5	referred	refer	VERB
cana-3972	276	6	to	to	ADP
cana-3972	276	7	as	as	ADP
cana-3972	276	8	a	a	DET
cana-3972	276	9	dsf	dsf	NOUN
cana-3972	276	10	-	-	PUNCT
cana-3972	276	11	module	module	NOUN
cana-3972	276	12	(	(	PUNCT
cana-3972	276	13	dual	dual	ADJ
cana-3972	276	14	square	square	ADJ
cana-3972	276	15	-	-	PUNCT
cana-3972	276	16	free	free	ADJ
cana-3972	276	17	module	module	NOUN
cana-3972	276	18	)	)	PUNCT
cana-3972	276	19	if	if	SCONJ
cana-3972	276	20	i	i	PRON
cana-3972	276	21	t	t	NOUN
cana-3972	276	22	does	do	AUX
cana-3972	276	23	not	not	PART
cana-3972	276	24	contain	contain	VERB
cana-3972	276	25	any	any	DET
cana-3972	276	26	proper	proper	ADJ
cana-3972	276	27	submodules	submodule	NOUN
cana-3972	276	28	n	n	NOUN
cana-3972	276	29	and	and	CCONJ
cana-3972	276	30	k	k	PROPN
cana-3972	277	1	such	such	ADJ
cana-3972	277	2	that	that	SCONJ
cana-3972	277	3	m	m	VERB
cana-3972	277	4	=	=	SYM
cana-3972	277	5	n	n	PROPN
cana-3972	277	6	+	+	CCONJ
cana-3972	277	7	k	k	PROPN
cana-3972	277	8	and	and	CCONJ
cana-3972	277	9	m	m	PROPN
cana-3972	277	10	/	/	SYM
cana-3972	277	11	n	n	PROPN
cana-3972	277	12	≅	≅	NUM
cana-3972	277	13	m	m	PROPN
cana-3972	277	14	/	/	SYM
cana-3972	277	15	k	k	PROPN
cana-3972	278	1	[	[	X
cana-3972	278	2	4	4	NUM
cana-3972	278	3	]	]	PUNCT
cana-3972	278	4	.	.	PUNCT
cana-3972	279	1	additionally	additionally	ADV
cana-3972	279	2	,	,	PUNCT
cana-3972	279	3	m	m	VERB
cana-3972	279	4	is	be	AUX
cana-3972	279	5	called	call	VERB
cana-3972	279	6	summand	summand	NOUN
cana-3972	279	7	-	-	PUNCT
cana-3972	279	8	dual	dual	ADJ
cana-3972	279	9	-	-	PUNCT
cana-3972	279	10	square	square	NOUN
cana-3972	279	11	-	-	PUNCT
cana-3972	279	12	free	free	ADJ
cana-3972	279	13	(	(	PUNCT
cana-3972	279	14	sdsf	sdsf	NOUN
cana-3972	279	15	)	)	PUNCT
cana-3972	279	16	if	if	SCONJ
cana-3972	279	17	the	the	DET
cana-3972	279	18	submodules	submodule	NOUN
cana-3972	279	19	n	n	PROPN
cana-3972	279	20	and	and	CCONJ
cana-3972	279	21	k	k	PROPN
cana-3972	279	22	are	be	AUX
cana-3972	279	23	direct	direct	ADJ
cana-3972	279	24	summands	summand	NOUN
cana-3972	279	25	of	of	ADP
cana-3972	279	26	m	m	PROPN
cana-3972	279	27	.	.	PUNCT
cana-3972	280	1	proposition	proposition	NOUN
cana-3972	280	2	3.21	3.21	NUM
cana-3972	280	3	.	.	PUNCT
cana-3972	281	1	the	the	DET
cana-3972	281	2	following	follow	VERB
cana-3972	281	3	conditions	condition	NOUN
cana-3972	281	4	on	on	ADP
cana-3972	281	5	a	a	DET
cana-3972	281	6	module	module	NOUN
cana-3972	281	7	m	m	NOUN
cana-3972	281	8	are	be	AUX
cana-3972	281	9	equivalent	equivalent	ADJ
cana-3972	281	10	:	:	PUNCT
cana-3972	281	11	1	1	X
cana-3972	281	12	)	)	PUNCT
cana-3972	281	13	m	m	VERB
cana-3972	281	14	is	be	AUX
cana-3972	281	15	a	a	DET
cana-3972	281	16	d41and	d41and	NOUN
cana-3972	281	17	summand	summand	NOUN
cana-3972	281	18	-	-	PUNCT
cana-3972	281	19	square	square	ADJ
cana-3972	281	20	-	-	PUNCT
cana-3972	281	21	free	free	ADJ
cana-3972	281	22	module	module	NOUN
cana-3972	281	23	.	.	PUNCT
cana-3972	282	1	2	2	X
cana-3972	282	2	)	)	PUNCT
cana-3972	282	3	m	m	VERB
cana-3972	282	4	is	be	AUX
cana-3972	282	5	a	a	DET
cana-3972	282	6	c4and	c4and	NOUN
cana-3972	282	7	summand	summand	NOUN
cana-3972	282	8	-	-	PUNCT
cana-3972	282	9	dual	dual	ADJ
cana-3972	282	10	-	-	PUNCT
cana-3972	282	11	square	square	ADJ
cana-3972	282	12	-	-	PUNCT
cana-3972	282	13	free	free	ADJ
cana-3972	282	14	module	module	NOUN
cana-3972	282	15	.	.	PUNCT
cana-3972	283	1	proof	proof	NOUN
cana-3972	283	2	:	:	PUNCT
cana-3972	283	3	1	1	X
cana-3972	283	4	)	)	PUNCT
cana-3972	283	5	⇒	⇒	NOUN
cana-3972	283	6	2	2	NUM
cana-3972	283	7	)	)	PUNCT
cana-3972	283	8	clearly	clearly	ADV
cana-3972	283	9	,	,	PUNCT
cana-3972	283	10	m	m	VERB
cana-3972	283	11	is	be	AUX
cana-3972	283	12	a	a	DET
cana-3972	283	13	c4	c4	NOUN
cana-3972	283	14	-	-	PUNCT
cana-3972	283	15	module	module	NOUN
cana-3972	283	16	by	by	ADP
cana-3972	283	17	[	[	X
cana-3972	283	18	6	6	NUM
cana-3972	283	19	]	]	PUNCT
cana-3972	283	20	.	.	PUNCT
cana-3972	284	1	next	next	ADV
cana-3972	284	2	,	,	PUNCT
cana-3972	284	3	we	we	PRON
cana-3972	284	4	show	show	VERB
cana-3972	284	5	that	that	SCONJ
cana-3972	284	6	m	m	PROPN
cana-3972	284	7	is	be	AUX
cana-3972	284	8	summand	summand	NOUN
cana-3972	284	9	-	-	PUNCT
cana-3972	284	10	dual	dual	ADJ
cana-3972	284	11	-	-	PUNCT
cana-3972	284	12	square	square	NOUN
cana-3972	284	13	-	-	PUNCT
cana-3972	284	14	free	free	ADJ
cana-3972	284	15	.	.	PUNCT
cana-3972	285	1	suppose	suppose	VERB
cana-3972	285	2	m	m	NOUN
cana-3972	285	3	is	be	AUX
cana-3972	285	4	not	not	PART
cana-3972	285	5	summand	summand	NOUN
cana-3972	285	6	-	-	PUNCT
cana-3972	285	7	dual	dual	ADJ
cana-3972	285	8	-	-	PUNCT
cana-3972	285	9	square	square	NOUN
cana-3972	285	10	-	-	PUNCT
cana-3972	285	11	free	free	ADJ
cana-3972	285	12	;	;	PUNCT
cana-3972	285	13	then	then	ADV
cana-3972	285	14	there	there	PRON
cana-3972	285	15	exist	exist	VERB
cana-3972	285	16	two	two	NUM
cana-3972	285	17	non	non	ADJ
cana-3972	285	18	-	-	ADJ
cana-3972	285	19	zero	zero	ADJ
cana-3972	285	20	proper	proper	ADJ
cana-3972	285	21	direct	direct	ADJ
cana-3972	285	22	summands	summand	NOUN
cana-3972	285	23	n	n	PRON
cana-3972	285	24	and	and	CCONJ
cana-3972	285	25	k	k	X
cana-3972	285	26	of	of	ADP
cana-3972	285	27	m	m	PRON
cana-3972	285	28	such	such	ADJ
cana-3972	285	29	that	that	SCONJ
cana-3972	285	30	n	n	PROPN
cana-3972	286	1	+	+	NOUN
cana-3972	286	2	k	k	X
cana-3972	286	3	=	=	X
cana-3972	286	4	m	m	VERB
cana-3972	286	5	with	with	ADP
cana-3972	286	6	k	k	PROPN
cana-3972	286	7	cosingular	cosingular	ADJ
cana-3972	286	8	and	and	CCONJ
cana-3972	286	9	m	m	PROPN
cana-3972	286	10	/	/	SYM
cana-3972	286	11	n	n	PROPN
cana-3972	286	12	≅	≅	NUM
cana-3972	286	13	m	m	NOUN
cana-3972	286	14	/k	/k	PUNCT
cana-3972	286	15	.	.	PUNCT
cana-3972	287	1	since	since	SCONJ
cana-3972	287	2	m	m	PROPN
cana-3972	287	3	is	be	AUX
cana-3972	287	4	a	a	DET
cana-3972	287	5	d41	d41	NOUN
cana-3972	287	6	-	-	PUNCT
cana-3972	287	7	module	module	NOUN
cana-3972	287	8	,	,	PUNCT
cana-3972	287	9	we	we	PRON
cana-3972	287	10	have	have	VERB
cana-3972	287	11	n	n	NOUN
cana-3972	287	12	∩	∩	NOUN
cana-3972	287	13	k	k	PROPN
cana-3972	287	14	≤	≤	PROPN
cana-3972	287	15	⊕	⊕	PROPN
cana-3972	287	16	m	m	PROPN
cana-3972	287	17	.	.	PUNCT
cana-3972	288	1	we	we	PRON
cana-3972	288	2	can	can	AUX
cana-3972	288	3	write	write	VERB
cana-3972	288	4	m	m	PROPN
cana-3972	288	5	=	=	PUNCT
cana-3972	288	6	(	(	PUNCT
cana-3972	288	7	n	n	X
cana-3972	288	8	∩	∩	X
cana-3972	288	9	k	k	PROPN
cana-3972	288	10	)	)	PUNCT
cana-3972	288	11	⊕	⊕	PROPN
cana-3972	288	12	l	l	PROPN
cana-3972	288	13	,	,	PUNCT
cana-3972	288	14	leading	lead	VERB
cana-3972	288	15	to	to	ADP
cana-3972	288	16	n	n	NOUN
cana-3972	288	17	=	=	SYM
cana-3972	288	18	(	(	PUNCT
cana-3972	288	19	n	n	X
cana-3972	288	20	∩	∩	X
cana-3972	288	21	k	k	PROPN
cana-3972	288	22	)	)	PUNCT
cana-3972	288	23	⊕	⊕	PROPN
cana-3972	288	24	(	(	PUNCT
cana-3972	288	25	n	n	X
cana-3972	288	26	∩	∩	X
cana-3972	288	27	l	l	NOUN
cana-3972	288	28	)	)	PUNCT
cana-3972	288	29	and	and	CCONJ
cana-3972	288	30	k	k	NOUN
cana-3972	288	31	=	=	SYM
cana-3972	288	32	(	(	PUNCT
cana-3972	288	33	n	n	X
cana-3972	288	34	∩	∩	X
cana-3972	288	35	k	k	PROPN
cana-3972	288	36	)	)	PUNCT
cana-3972	288	37	⊕	⊕	PROPN
cana-3972	288	38	(	(	PUNCT
cana-3972	288	39	k	k	X
cana-3972	288	40	∩	∩	PROPN
cana-3972	288	41	l	l	NOUN
cana-3972	288	42	)	)	PUNCT
cana-3972	288	43	.	.	PUNCT
cana-3972	289	1	this	this	PRON
cana-3972	289	2	gives	give	VERB
cana-3972	289	3	us	we	PRON
cana-3972	289	4	n	n	PRON
cana-3972	289	5	∩	∩	ADJ
cana-3972	289	6	l	l	NOUN
cana-3972	289	7	≅	≅	PROPN
cana-3972	289	8	n/(n	n/(n	PROPN
cana-3972	289	9	∩	∩	NOUN
cana-3972	289	10	k	k	PROPN
cana-3972	289	11	)	)	PUNCT
cana-3972	290	1	≅	≅	PROPN
cana-3972	290	2	m	m	PROPN
cana-3972	290	3	/	/	SYM
cana-3972	290	4	k	k	PROPN
cana-3972	291	1	≅	≅	PROPN
cana-3972	291	2	m	m	PROPN
cana-3972	291	3	/	/	SYM
cana-3972	291	4	n	n	PROPN
cana-3972	291	5	≅	≅	PROPN
cana-3972	291	6	k/(n	k/(n	X
cana-3972	291	7	∩	∩	X
cana-3972	291	8	k	k	PROPN
cana-3972	291	9	)	)	PUNCT
cana-3972	292	1	≅	≅	PROPN
cana-3972	292	2	k	k	PROPN
cana-3972	292	3	∩	∩	PROPN
cana-3972	292	4	l	l	NOUN
cana-3972	292	5	with	with	ADP
cana-3972	292	6	(	(	PUNCT
cana-3972	292	7	n	n	CCONJ
cana-3972	292	8	∩	∩	ADJ
cana-3972	292	9	l	l	NOUN
cana-3972	292	10	)	)	PUNCT
cana-3972	292	11	∩	∩	NOUN
cana-3972	292	12	(	(	PUNCT
cana-3972	292	13	k	k	X
cana-3972	292	14	∩	∩	ADJ
cana-3972	292	15	l	l	NOUN
cana-3972	292	16	)	)	PUNCT
cana-3972	292	17	=	=	SYM
cana-3972	292	18	(	(	PUNCT
cana-3972	292	19	n	n	X
cana-3972	292	20	∩	∩	X
cana-3972	292	21	k	k	ADJ
cana-3972	292	22	)	)	PUNCT
cana-3972	292	23	∩	∩	NOUN
cana-3972	292	24	l	l	NOUN
cana-3972	292	25	=	=	SYM
cana-3972	292	26	0	0	NUM
cana-3972	292	27	,	,	PUNCT
cana-3972	292	28	meaning	mean	VERB
cana-3972	292	29	both	both	CCONJ
cana-3972	292	30	n	n	CCONJ
cana-3972	292	31	∩	∩	ADJ
cana-3972	292	32	l	l	NOUN
cana-3972	292	33	and	and	CCONJ
cana-3972	292	34	k	k	PROPN
cana-3972	292	35	∩	∩	NOUN
cana-3972	292	36	l	l	NOUN
cana-3972	292	37	are	be	AUX
cana-3972	292	38	direct	direct	ADJ
cana-3972	292	39	summands	summand	NOUN
cana-3972	292	40	of	of	ADP
cana-3972	292	41	m	m	PRON
cana-3972	292	42	.	.	PUNCT
cana-3972	293	1	since	since	SCONJ
cana-3972	293	2	m	m	PROPN
cana-3972	293	3	is	be	AUX
cana-3972	293	4	summand	summand	NOUN
cana-3972	293	5	-	-	PUNCT
cana-3972	293	6	square	square	NOUN
cana-3972	293	7	-	-	PUNCT
cana-3972	293	8	free	free	ADJ
cana-3972	293	9	,	,	PUNCT
cana-3972	293	10	we	we	PRON
cana-3972	293	11	have	have	VERB
cana-3972	293	12	n	n	PRON
cana-3972	293	13	∩	∩	ADJ
cana-3972	293	14	l	l	NOUN
cana-3972	293	15	=	=	SYM
cana-3972	293	16	k	k	X
cana-3972	293	17	∩	∩	X
cana-3972	293	18	l	l	NOUN
cana-3972	293	19	=	=	SYM
cana-3972	293	20	0	0	NUM
cana-3972	293	21	.	.	PUNCT
cana-3972	294	1	thus	thus	ADV
cana-3972	294	2	,	,	PUNCT
cana-3972	294	3	n	n	PROPN
cana-3972	294	4	=	=	SYM
cana-3972	294	5	(	(	PUNCT
cana-3972	294	6	n	n	X
cana-3972	294	7	∩	∩	X
cana-3972	294	8	k	k	NOUN
cana-3972	294	9	)	)	PUNCT
cana-3972	294	10	=	=	SYM
cana-3972	294	11	k	k	PROPN
cana-3972	294	12	,	,	PUNCT
cana-3972	294	13	leading	lead	VERB
cana-3972	294	14	to	to	ADP
cana-3972	294	15	the	the	DET
cana-3972	294	16	contradiction	contradiction	NOUN
cana-3972	294	17	m	m	NOUN
cana-3972	294	18	=	=	SYM
cana-3972	294	19	n	n	PROPN
cana-3972	294	20	+	+	CCONJ
cana-3972	294	21	k	k	NOUN
cana-3972	294	22	=	=	PUNCT
cana-3972	294	23	n	n	PROPN
cana-3972	294	24	=	=	SYM
cana-3972	294	25	k	k	PROPN
cana-3972	294	26	.	.	PUNCT
cana-3972	295	1	hence	hence	ADV
cana-3972	295	2	,	,	PUNCT
cana-3972	295	3	m	m	PROPN
cana-3972	295	4	is	be	AUX
cana-3972	295	5	summand	summand	NOUN
cana-3972	295	6	-	-	PUNCT
cana-3972	295	7	dual	dual	ADJ
cana-3972	295	8	-	-	PUNCT
cana-3972	295	9	square	square	NOUN
cana-3972	295	10	-	-	PUNCT
cana-3972	295	11	free	free	ADJ
cana-3972	295	12	.	.	NOUN
cana-3972	295	13	2	2	X
cana-3972	295	14	)	)	PUNCT
cana-3972	295	15	⇒	⇒	NOUN
cana-3972	295	16	1	1	NUM
cana-3972	295	17	)	)	PUNCT
cana-3972	295	18	clearly	clearly	ADV
cana-3972	295	19	,	,	PUNCT
cana-3972	295	20	m	m	PROPN
cana-3972	295	21	is	be	AUX
cana-3972	295	22	a	a	DET
cana-3972	295	23	d41	d41	NOUN
cana-3972	295	24	-	-	PUNCT
cana-3972	295	25	module	module	NOUN
cana-3972	295	26	by	by	ADP
cana-3972	295	27	[	[	X
cana-3972	295	28	4	4	NUM
cana-3972	295	29	]	]	PUNCT
cana-3972	295	30	.	.	PUNCT
cana-3972	296	1	now	now	ADV
cana-3972	296	2	,	,	PUNCT
cana-3972	296	3	we	we	PRON
cana-3972	296	4	demonstrate	demonstrate	VERB
cana-3972	296	5	that	that	SCONJ
cana-3972	296	6	m	m	PROPN
cana-3972	296	7	is	be	AUX
cana-3972	296	8	summand	summand	NOUN
cana-3972	296	9	-	-	PUNCT
cana-3972	296	10	squarefree	squarefree	NOUN
cana-3972	296	11	.	.	PUNCT
cana-3972	297	1	assume	assume	VERB
cana-3972	297	2	m	m	NOUN
cana-3972	297	3	is	be	AUX
cana-3972	297	4	not	not	PART
cana-3972	297	5	summand	summand	NOUN
cana-3972	297	6	-	-	PUNCT
cana-3972	297	7	square	square	NOUN
cana-3972	297	8	-	-	PUNCT
cana-3972	297	9	free	free	ADJ
cana-3972	297	10	,	,	PUNCT
cana-3972	297	11	and	and	CCONJ
cana-3972	297	12	let	let	VERB
cana-3972	297	13	n	n	PRON
cana-3972	297	14	and	and	CCONJ
cana-3972	297	15	k	k	PROPN
cana-3972	297	16	be	be	AUX
cana-3972	297	17	non	non	ADJ
cana-3972	297	18	-	-	ADJ
cana-3972	297	19	zero	zero	ADJ
cana-3972	297	20	direct	direct	ADJ
cana-3972	297	21	summands	summand	NOUN
cana-3972	297	22	of	of	ADP
cana-3972	297	23	m	m	PROPN
cana-3972	297	24	with	with	ADP
cana-3972	297	25	n	n	PROPN
cana-3972	297	26	≅	≅	PROPN
cana-3972	297	27	k	k	PROPN
cana-3972	297	28	and	and	CCONJ
cana-3972	297	29	n	n	PROPN
cana-3972	297	30	∩	∩	X
cana-3972	297	31	k	k	NOUN
cana-3972	297	32	=	=	SYM
cana-3972	297	33	0	0	X
cana-3972	297	34	.	.	PUNCT
cana-3972	298	1	since	since	SCONJ
cana-3972	298	2	m	m	PROPN
cana-3972	298	3	is	be	AUX
cana-3972	298	4	a	a	DET
cana-3972	298	5	c4	c4	NOUN
cana-3972	298	6	-	-	PUNCT
cana-3972	298	7	module	module	NOUN
cana-3972	298	8	,	,	PUNCT
cana-3972	298	9	we	we	PRON
cana-3972	298	10	have	have	VERB
cana-3972	298	11	n	n	NUM
cana-3972	298	12	⊕	⊕	PROPN
cana-3972	298	13	k	k	PROPN
cana-3972	298	14	≤	≤	PROPN
cana-3972	298	15	⊕	⊕	PROPN
cana-3972	298	16	m	m	PROPN
cana-3972	298	17	.	.	PUNCT
cana-3972	299	1	we	we	PRON
cana-3972	299	2	can	can	AUX
cana-3972	299	3	write	write	VERB
cana-3972	299	4	m	m	PROPN
cana-3972	299	5	=	=	SYM
cana-3972	299	6	n	n	PROPN
cana-3972	299	7	⊕	⊕	PROPN
cana-3972	299	8	k	k	PROPN
cana-3972	299	9	⊕	⊕	PROPN
cana-3972	299	10	l	l	PROPN
cana-3972	299	11	for	for	ADP
cana-3972	299	12	some	some	DET
cana-3972	299	13	submodule	submodule	NOUN
cana-3972	299	14	l	l	NOUN
cana-3972	299	15	≤	≤	NUM
cana-3972	299	16	m	m	VERB
cana-3972	299	17	.	.	PUNCT
cana-3972	300	1	now	now	ADV
cana-3972	300	2	,	,	PUNCT
cana-3972	300	3	we	we	PRON
cana-3972	300	4	have	have	VERB
cana-3972	300	5	m/(n	m/(n	PROPN
cana-3972	300	6	⊕	⊕	PROPN
cana-3972	300	7	l	l	PROPN
cana-3972	300	8	)	)	PUNCT
cana-3972	301	1	≅	≅	PROPN
cana-3972	301	2	k	k	PROPN
cana-3972	302	1	≅	≅	PROPN
cana-3972	302	2	n	n	PROPN
cana-3972	302	3	≅	≅	PROPN
cana-3972	302	4	m	m	PROPN
cana-3972	302	5	/(k	/(k	PROPN
cana-3972	302	6	⊕	⊕	PROPN
cana-3972	302	7	l	l	PROPN
cana-3972	302	8	)	)	PUNCT
cana-3972	302	9	with	with	ADP
cana-3972	302	10	𝑀	𝑀	PROPN
cana-3972	302	11	=	=	PUNCT
cana-3972	302	12	𝑁	𝑁	PROPN
cana-3972	302	13	⊕	⊕	PROPN
cana-3972	302	14	𝐾	𝐾	PROPN
cana-3972	302	15	⊕	⊕	PROPN
cana-3972	302	16	𝐿	𝐿	PROPN
cana-3972	302	17	=	=	SYM
cana-3972	302	18	(	(	PUNCT
cana-3972	302	19	𝑁	𝑁	PROPN
cana-3972	302	20	⊕	⊕	PROPN
cana-3972	302	21	𝐿	𝐿	PROPN
cana-3972	302	22	)	)	PUNCT
cana-3972	302	23	+	+	CCONJ
cana-3972	302	24	(	(	PUNCT
cana-3972	302	25	𝐾	𝐾	PROPN
cana-3972	302	26	⊕	⊕	PROPN
cana-3972	302	27	𝐿	𝐿	PROPN
cana-3972	302	28	)	)	PUNCT
cana-3972	302	29	,	,	PUNCT
cana-3972	302	30	where	where	SCONJ
cana-3972	302	31	both	both	PRON
cana-3972	302	32	𝑁	𝑁	PROPN
cana-3972	302	33	⊕	⊕	PROPN
cana-3972	302	34	𝐿	𝐿	PROPN
cana-3972	302	35	and	and	CCONJ
cana-3972	302	36	𝐾	𝐾	PROPN
cana-3972	302	37	⊕	⊕	PROPN
cana-3972	302	38	𝐿	𝐿	PROPN
cana-3972	302	39	are	be	AUX
cana-3972	302	40	direct	direct	ADJ
cana-3972	302	41	summands	summand	NOUN
cana-3972	302	42	of	of	ADP
cana-3972	302	43	m	m	PRON
cana-3972	302	44	.	.	PUNCT
cana-3972	303	1	since	since	SCONJ
cana-3972	303	2	m	m	PROPN
cana-3972	303	3	is	be	AUX
cana-3972	303	4	summand	summand	NOUN
cana-3972	303	5	-	-	PUNCT
cana-3972	303	6	dual	dual	ADJ
cana-3972	303	7	-	-	PUNCT
cana-3972	303	8	square	square	NOUN
cana-3972	303	9	-	-	PUNCT
cana-3972	303	10	free	free	ADJ
cana-3972	303	11	,	,	PUNCT
cana-3972	303	12	it	it	PRON
cana-3972	303	13	follows	follow	VERB
cana-3972	303	14	that	that	SCONJ
cana-3972	303	15	𝑀	𝑀	PROPN
cana-3972	303	16	=	=	PUNCT
cana-3972	303	17	𝑁	𝑁	PROPN
cana-3972	303	18	⊕	⊕	PROPN
cana-3972	303	19	𝐿	𝐿	NOUN
cana-3972	303	20	=	=	SYM
cana-3972	303	21	𝐾	𝐾	PROPN
cana-3972	303	22	⊕	⊕	PROPN
cana-3972	303	23	𝐿	𝐿	PROPN
cana-3972	303	24	,	,	PUNCT
cana-3972	303	25	implying	imply	VERB
cana-3972	303	26	n	n	NOUN
cana-3972	303	27	=	=	SYM
cana-3972	303	28	k	k	PROPN
cana-3972	303	29	=	=	SYM
cana-3972	303	30	0	0	PROPN
cana-3972	303	31	,	,	PUNCT
cana-3972	303	32	which	which	PRON
cana-3972	303	33	leads	lead	VERB
cana-3972	303	34	to	to	ADP
cana-3972	303	35	a	a	DET
cana-3972	303	36	contradiction	contradiction	NOUN
cana-3972	303	37	.	.	PUNCT
cana-3972	304	1	therefore	therefore	ADV
cana-3972	304	2	,	,	PUNCT
cana-3972	304	3	m	m	PROPN
cana-3972	304	4	is	be	AUX
cana-3972	304	5	summand	summand	NOUN
cana-3972	304	6	-	-	PUNCT
cana-3972	304	7	square	square	NOUN
cana-3972	304	8	-	-	PUNCT
cana-3972	304	9	free	free	ADJ
cana-3972	304	10	.	.	PUNCT
cana-3972	305	1	proposition	proposition	NOUN
cana-3972	305	2	3.22	3.22	NUM
cana-3972	305	3	.	.	PUNCT
cana-3972	306	1	let	let	VERB
cana-3972	306	2	𝑀	𝑀	PROPN
cana-3972	307	1	=	=	NOUN
cana-3972	307	2	⊕𝑖∈𝐼	⊕𝑖∈𝐼	NOUN
cana-3972	308	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	308	2	be	be	AUX
cana-3972	308	3	a	a	DET
cana-3972	308	4	direct	direct	ADJ
cana-3972	308	5	sum	sum	NOUN
cana-3972	308	6	of	of	ADP
cana-3972	308	7	submodules	submodule	NOUN
cana-3972	308	8	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	308	9	.	.	PUNCT
cana-3972	309	1	if	if	SCONJ
cana-3972	309	2	for	for	ADP
cana-3972	309	3	every	every	DET
cana-3972	309	4	submodule	submodule	NOUN
cana-3972	309	5	n	n	PROPN
cana-3972	309	6	of	of	ADP
cana-3972	309	7	m	m	PROPN
cana-3972	309	8	,	,	PUNCT
cana-3972	309	9	we	we	PRON
cana-3972	309	10	have	have	VERB
cana-3972	309	11	𝑁	𝑁	PROPN
cana-3972	309	12	=	=	NOUN
cana-3972	309	13	⊕𝑖∈𝐼	⊕𝑖∈𝐼	X
cana-3972	309	14	(	(	PUNCT
cana-3972	309	15	𝑁	𝑁	PROPN
cana-3972	309	16	∩	∩	NOUN
cana-3972	309	17	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	309	18	)	)	PUNCT
cana-3972	309	19	,	,	PUNCT
cana-3972	309	20	then	then	ADV
cana-3972	309	21	m	m	PROPN
cana-3972	309	22	is	be	AUX
cana-3972	309	23	a	a	DET
cana-3972	309	24	d41	d41	NOUN
cana-3972	309	25	-	-	PUNCT
cana-3972	309	26	module	module	NOUN
cana-3972	309	27	if	if	SCONJ
cana-3972	309	28	and	and	CCONJ
cana-3972	309	29	only	only	ADV
cana-3972	309	30	if	if	SCONJ
cana-3972	309	31	each	each	DET
cana-3972	309	32	𝑀	𝑀	PROPN
cana-3972	309	33	𝑖	𝑖	X
cana-3972	309	34	(	(	PUNCT
cana-3972	309	35	for	for	ADP
cana-3972	309	36	i	i	PRON
cana-3972	309	37	∈	∈	PROPN
cana-3972	309	38	i	i	X
cana-3972	309	39	)	)	PUNCT
cana-3972	309	40	is	be	AUX
cana-3972	309	41	also	also	ADV
cana-3972	309	42	a	a	DET
cana-3972	309	43	d41module	d41module	PROPN
cana-3972	309	44	.	.	PUNCT
cana-3972	310	1	proof	proof	NOUN
cana-3972	310	2	:	:	PUNCT
cana-3972	310	3	assume	assume	VERB
cana-3972	310	4	that	that	SCONJ
cana-3972	310	5	each	each	DET
cana-3972	310	6	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	310	7	is	be	AUX
cana-3972	310	8	a	a	DET
cana-3972	310	9	d41	d41	NOUN
cana-3972	310	10	-	-	PUNCT
cana-3972	310	11	module	module	NOUN
cana-3972	310	12	for	for	ADP
cana-3972	310	13	every	every	DET
cana-3972	310	14	i	i	PROPN
cana-3972	310	15	∈	∈	PROPN
cana-3972	310	16	i.	i.	NOUN
cana-3972	310	17	communications	communication	NOUN
cana-3972	310	18	on	on	ADP
cana-3972	310	19	applied	apply	VERB
cana-3972	310	20	nonlinear	nonlinear	ADJ
cana-3972	310	21	analysis	analysis	NOUN
cana-3972	310	22	issn	issn	NOUN
cana-3972	310	23	:	:	PUNCT
cana-3972	310	24	1074	1074	NUM
cana-3972	310	25	-	-	PUNCT
cana-3972	310	26	133x	133x	NUM
cana-3972	310	27	vol	vol	NOUN
cana-3972	310	28	32	32	NUM
cana-3972	310	29	no	no	NOUN
cana-3972	310	30	.	.	PUNCT
cana-3972	311	1	9s	9s	NUM
cana-3972	311	2	(	(	PUNCT
cana-3972	311	3	2025	2025	NUM
cana-3972	311	4	)	)	PUNCT
cana-3972	311	5	683	683	NUM
cana-3972	312	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	312	2	l	l	NOUN
cana-3972	312	3	l	l	NOUN
cana-3972	312	4	let	let	VERB
cana-3972	312	5	m	m	VERB
cana-3972	312	6	=	=	SYM
cana-3972	312	7	n	n	PROPN
cana-3972	312	8	⊕	⊕	PROPN
cana-3972	312	9	l	l	NOUN
cana-3972	313	1	=	=	PUNCT
cana-3972	313	2	k	k	PROPN
cana-3972	313	3	⊕	⊕	PROPN
cana-3972	313	4	l	l	PROPN
cana-3972	313	5	with	with	ADP
cana-3972	313	6	n	n	PROPN
cana-3972	314	1	+	+	CCONJ
cana-3972	314	2	k	k	X
cana-3972	314	3	=	=	VERB
cana-3972	314	4	m	m	PROPN
cana-3972	314	5	and	and	CCONJ
cana-3972	314	6	k	k	PROPN
cana-3972	314	7	cosingular	cosingular	ADJ
cana-3972	314	8	.	.	PUNCT
cana-3972	315	1	according	accord	VERB
cana-3972	315	2	to	to	ADP
cana-3972	315	3	the	the	DET
cana-3972	315	4	hypothesis	hypothesis	NOUN
cana-3972	315	5	,	,	PUNCT
cana-3972	315	6	we	we	PRON
cana-3972	315	7	can	can	AUX
cana-3972	315	8	express	express	VERB
cana-3972	315	9	𝑁	𝑁	PROPN
cana-3972	315	10	=	=	NOUN
cana-3972	315	11	⊕𝑖∈𝐼	⊕𝑖∈𝐼	X
cana-3972	315	12	(	(	PUNCT
cana-3972	315	13	𝑁	𝑁	PROPN
cana-3972	315	14	∩	∩	NOUN
cana-3972	315	15	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	315	16	)	)	PUNCT
cana-3972	315	17	,	,	PUNCT
cana-3972	315	18	𝐾	𝐾	PROPN
cana-3972	315	19	=	=	PROPN
cana-3972	315	20	⊕𝑖∈𝐼	⊕𝑖∈𝐼	X
cana-3972	315	21	(	(	PUNCT
cana-3972	315	22	𝐾	𝐾	PROPN
cana-3972	315	23	∩	∩	NOUN
cana-3972	315	24	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	315	25	)	)	PUNCT
cana-3972	315	26	and	and	CCONJ
cana-3972	315	27	𝐿	𝐿	PROPN
cana-3972	315	28	=	=	PROPN
cana-3972	315	29	⊕𝑖∈𝐼	⊕𝑖∈𝐼	X
cana-3972	315	30	(	(	PUNCT
cana-3972	315	31	𝐿	𝐿	PROPN
cana-3972	315	32	∩	∩	NOUN
cana-3972	315	33	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	315	34	)	)	PUNCT
cana-3972	315	35	since	since	SCONJ
cana-3972	315	36	m	m	PROPN
cana-3972	315	37	=	=	SYM
cana-3972	315	38	n	n	PROPN
cana-3972	315	39	⊕	⊕	PROPN
cana-3972	315	40	l	l	NOUN
cana-3972	316	1	=	=	PUNCT
cana-3972	316	2	k	k	PROPN
cana-3972	316	3	⊕	⊕	PROPN
cana-3972	316	4	l	l	PROPN
cana-3972	316	5	,	,	PUNCT
cana-3972	316	6	we	we	PRON
cana-3972	316	7	have	have	VERB
cana-3972	316	8	:	:	PUNCT
cana-3972	316	9	𝑀	𝑀	PROPN
cana-3972	316	10	=	=	PUNCT
cana-3972	316	11	⊕𝑖∈𝐼	⊕𝑖∈𝐼	PROPN
cana-3972	317	1	[	[	X
cana-3972	317	2	(	(	PUNCT
cana-3972	317	3	𝑁	𝑁	PROPN
cana-3972	317	4	∩	∩	ADJ
cana-3972	317	5	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	317	6	)	)	PUNCT
cana-3972	317	7	⊕	⊕	PROPN
cana-3972	317	8	(	(	PUNCT
cana-3972	317	9	𝐿	𝐿	PROPN
cana-3972	317	10	∩	∩	NOUN
cana-3972	317	11	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	317	12	)	)	PUNCT
cana-3972	317	13	]	]	PUNCT
cana-3972	318	1	=	=	SYM
cana-3972	318	2	⊕𝑖∈𝐼	⊕𝑖∈𝐼	NOUN
cana-3972	318	3	[	[	X
cana-3972	318	4	(	(	PUNCT
cana-3972	318	5	𝐾	𝐾	PROPN
cana-3972	318	6	∩	∩	NOUN
cana-3972	318	7	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	318	8	)	)	PUNCT
cana-3972	318	9	⊕	⊕	PROPN
cana-3972	318	10	(	(	PUNCT
cana-3972	318	11	𝐿	𝐿	PROPN
cana-3972	318	12	∩	∩	NOUN
cana-3972	318	13	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	318	14	)	)	PUNCT
cana-3972	318	15	]	]	PUNCT
cana-3972	319	1	thus	thus	ADV
cana-3972	319	2	,	,	PUNCT
cana-3972	319	3	i	i	PRON
cana-3972	319	4	t	t	PROPN
cana-3972	319	5	follows	follow	VERB
cana-3972	319	6	that	that	PRON
cana-3972	319	7	mi	mi	PROPN
cana-3972	319	8	=	=	SYM
cana-3972	319	9	(	(	PUNCT
cana-3972	319	10	n	n	X
cana-3972	319	11	∩	∩	X
cana-3972	319	12	mi	mi	PROPN
cana-3972	319	13	)	)	PUNCT
cana-3972	319	14	⊕	⊕	PROPN
cana-3972	319	15	(	(	PUNCT
cana-3972	319	16	l	l	PROPN
cana-3972	319	17	∩	∩	X
cana-3972	319	18	mi	mi	PROPN
cana-3972	319	19	)	)	PUNCT
cana-3972	319	20	=	=	SYM
cana-3972	320	1	(	(	PUNCT
cana-3972	320	2	k	k	PROPN
cana-3972	320	3	∩	∩	PROPN
cana-3972	320	4	mi	mi	PROPN
cana-3972	320	5	)	)	PUNCT
cana-3972	320	6	⊕	⊕	PROPN
cana-3972	320	7	(	(	PUNCT
cana-3972	320	8	l	l	PROPN
cana-3972	320	9	∩	∩	X
cana-3972	320	10	mi	mi	PROPN
cana-3972	320	11	)	)	PUNCT
cana-3972	320	12	for	for	ADP
cana-3972	320	13	every	every	DET
cana-3972	320	14	i	i	NOUN
cana-3972	320	15	∈	∈	PROPN
cana-3972	321	1	i	i	PRON
cana-3972	321	2	.	.	PUNCT
cana-3972	322	1	additionally	additionally	ADV
cana-3972	322	2	,	,	PUNCT
cana-3972	322	3	since	since	SCONJ
cana-3972	322	4	m	m	PROPN
cana-3972	322	5	=	=	SYM
cana-3972	322	6	n	n	PROPN
cana-3972	322	7	+	+	CCONJ
cana-3972	322	8	k	k	PROPN
cana-3972	322	9	,	,	PUNCT
cana-3972	322	10	we	we	PRON
cana-3972	322	11	have	have	VERB
cana-3972	322	12	:	:	PUNCT
cana-3972	322	13	𝑀	𝑀	PROPN
cana-3972	322	14	=	=	PROPN
cana-3972	322	15	⊕𝑖∈𝐼	⊕𝑖∈𝐼	PROPN
cana-3972	323	1	[	[	X
cana-3972	323	2	(	(	PUNCT
cana-3972	323	3	𝑁	𝑁	PROPN
cana-3972	323	4	∩	∩	NOUN
cana-3972	323	5	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	323	6	)	)	PUNCT
cana-3972	324	1	+	+	CCONJ
cana-3972	324	2	(	(	PUNCT
cana-3972	324	3	𝐾	𝐾	PROPN
cana-3972	324	4	∩	∩	NOUN
cana-3972	324	5	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	324	6	)	)	PUNCT
cana-3972	324	7	]	]	PUNCT
cana-3972	324	8	,	,	PUNCT
cana-3972	324	9	which	which	PRON
cana-3972	324	10	implies	imply	VERB
cana-3972	324	11	mi	mi	PROPN
cana-3972	324	12	=	=	SYM
cana-3972	324	13	(	(	PUNCT
cana-3972	324	14	n	n	X
cana-3972	324	15	∩	∩	X
cana-3972	324	16	mi	mi	NOUN
cana-3972	324	17	)	)	PUNCT
cana-3972	324	18	+	+	CCONJ
cana-3972	324	19	(	(	PUNCT
cana-3972	324	20	k	k	PROPN
cana-3972	324	21	∩	∩	PROPN
cana-3972	324	22	mi	mi	PROPN
cana-3972	324	23	)	)	PUNCT
cana-3972	324	24	.	.	PUNCT
cana-3972	325	1	given	give	VERB
cana-3972	325	2	that	that	PRON
cana-3972	325	3	n	n	NOUN
cana-3972	325	4	∩	∩	X
cana-3972	325	5	mi	mi	PROPN
cana-3972	325	6	and	and	CCONJ
cana-3972	325	7	k	k	PROPN
cana-3972	325	8	∩	∩	PROPN
cana-3972	325	9	mi	mi	PROPN
cana-3972	325	10	are	be	AUX
cana-3972	325	11	perspective	perspective	ADJ
cana-3972	325	12	direct	direct	ADJ
cana-3972	325	13	summands	summand	NOUN
cana-3972	325	14	of	of	ADP
cana-3972	325	15	mi	mi	PROPN
cana-3972	325	16	with	with	ADP
cana-3972	325	17	(	(	PUNCT
cana-3972	325	18	n	n	X
cana-3972	325	19	∩	∩	X
cana-3972	325	20	mi	mi	NOUN
cana-3972	325	21	)	)	PUNCT
cana-3972	325	22	+	+	CCONJ
cana-3972	325	23	(	(	PUNCT
cana-3972	325	24	k	k	PROPN
cana-3972	325	25	∩	∩	PROPN
cana-3972	325	26	mi	mi	PROPN
cana-3972	325	27	)	)	PUNCT
cana-3972	325	28	=	=	SYM
cana-3972	325	29	mi	mi	PROPN
cana-3972	325	30	,	,	PUNCT
cana-3972	325	31	i	i	PRON
cana-3972	325	32	t	t	PROPN
cana-3972	325	33	follows	follow	VERB
cana-3972	325	34	that	that	SCONJ
cana-3972	325	35	(	(	PUNCT
cana-3972	325	36	n	n	X
cana-3972	325	37	∩	∩	X
cana-3972	325	38	mi	mi	NOUN
cana-3972	325	39	)	)	PUNCT
cana-3972	325	40	∩	∩	NOUN
cana-3972	325	41	(	(	PUNCT
cana-3972	325	42	k	k	PROPN
cana-3972	325	43	∩	∩	PROPN
cana-3972	325	44	mi	mi	PROPN
cana-3972	325	45	)	)	PUNCT
cana-3972	325	46	≤	≤	PROPN
cana-3972	325	47	⊕	⊕	PROPN
cana-3972	325	48	mi	mi	PROPN
cana-3972	325	49	for	for	ADP
cana-3972	325	50	every	every	DET
cana-3972	325	51	i	i	NOUN
cana-3972	325	52	∈	∈	PROPN
cana-3972	325	53	i	i	PRON
cana-3972	325	54	.	.	PUNCT
cana-3972	326	1	consequently	consequently	ADV
cana-3972	326	2	,	,	PUNCT
cana-3972	326	3	we	we	PRON
cana-3972	326	4	have	have	VERB
cana-3972	326	5	n	n	NOUN
cana-3972	326	6	∩	∩	ADJ
cana-3972	326	7	k	k	PROPN
cana-3972	326	8	=	=	SYM
cana-3972	326	9	⊕𝑖∈𝐼	⊕𝑖∈𝐼	PROPN
cana-3972	326	10	(	(	PUNCT
cana-3972	326	11	𝑁	𝑁	PROPN
cana-3972	326	12	∩	∩	ADJ
cana-3972	326	13	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	326	14	)	)	PUNCT
cana-3972	326	15	∩	∩	NOUN
cana-3972	326	16	⊕𝑖∈𝐼	⊕𝑖∈𝐼	NOUN
cana-3972	326	17	(	(	PUNCT
cana-3972	326	18	𝐾	𝐾	PROPN
cana-3972	326	19	∩	∩	NOUN
cana-3972	326	20	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	326	21	)	)	PUNCT
cana-3972	326	22	=	=	SYM
cana-3972	327	1	⊕𝑖∈𝐼	⊕𝑖∈𝐼	NUM
cana-3972	328	1	[	[	X
cana-3972	328	2	(	(	PUNCT
cana-3972	328	3	n	n	X
cana-3972	328	4	∩	∩	X
cana-3972	328	5	mi	mi	NOUN
cana-3972	328	6	)	)	PUNCT
cana-3972	328	7	∩	∩	NOUN
cana-3972	328	8	(	(	PUNCT
cana-3972	328	9	k	k	PROPN
cana-3972	328	10	∩	∩	PROPN
cana-3972	328	11	mi	mi	PROPN
cana-3972	328	12	)	)	PUNCT
cana-3972	328	13	]	]	PUNCT
cana-3972	328	14	≤	≤	PROPN
cana-3972	328	15	⊕	⊕	PROPN
cana-3972	328	16	m	m	PROPN
cana-3972	328	17	,	,	PUNCT
cana-3972	328	18	which	which	PRON
cana-3972	328	19	demonstrates	demonstrate	VERB
cana-3972	328	20	that	that	SCONJ
cana-3972	328	21	m	m	PROPN
cana-3972	328	22	is	be	AUX
cana-3972	328	23	a	a	DET
cana-3972	328	24	d41	d41	NOUN
cana-3972	328	25	-	-	PUNCT
cana-3972	328	26	module	module	NOUN
cana-3972	328	27	.	.	PUNCT
cana-3972	329	1	the	the	DET
cana-3972	329	2	converse	converse	NOUN
cana-3972	329	3	is	be	AUX
cana-3972	329	4	straightforward	straightforward	ADJ
cana-3972	329	5	,	,	PUNCT
cana-3972	329	6	as	as	ADP
cana-3972	329	7	a	a	DET
cana-3972	329	8	direct	direct	ADJ
cana-3972	329	9	summand	summand	NOUN
cana-3972	329	10	of	of	ADP
cana-3972	329	11	a	a	DET
cana-3972	329	12	d41	d41	NOUN
cana-3972	329	13	-	-	PUNCT
cana-3972	329	14	module	module	NOUN
cana-3972	329	15	is	be	AUX
cana-3972	329	16	itself	itself	PRON
cana-3972	329	17	a	a	DET
cana-3972	329	18	d41	d41	NOUN
cana-3972	329	19	-	-	PUNCT
cana-3972	329	20	module	module	NOUN
cana-3972	329	21	by	by	ADP
cana-3972	329	22	proposition	proposition	NOUN
cana-3972	329	23	3.4	3.4	NUM
cana-3972	329	24	.	.	PUNCT
cana-3972	330	1	recall	recall	VERB
cana-3972	330	2	that	that	SCONJ
cana-3972	330	3	a	a	DET
cana-3972	330	4	module	module	NOUN
cana-3972	330	5	m	m	VERB
cana-3972	330	6	is	be	AUX
cana-3972	330	7	called	call	VERB
cana-3972	330	8	hopfian	hopfian	ADJ
cana-3972	330	9	if	if	SCONJ
cana-3972	330	10	every	every	DET
cana-3972	330	11	surjective	surjective	ADJ
cana-3972	330	12	r	r	X
cana-3972	330	13	-	-	PUNCT
cana-3972	330	14	homomorphism	homomorphism	NOUN
cana-3972	330	15	𝑓	𝑓	DET
cana-3972	330	16	∶	∶	NOUN
cana-3972	330	17	𝑀	𝑀	PROPN
cana-3972	330	18	→	→	SYM
cana-3972	330	19	𝑀	𝑀	PROPN
cana-3972	330	20	is	be	AUX
cana-3972	330	21	an	an	DET
cana-3972	330	22	automorphism	automorphism	NOUN
cana-3972	330	23	.	.	PUNCT
cana-3972	331	1	definition	definition	NOUN
cana-3972	331	2	3.23	3.23	NUM
cana-3972	331	3	.	.	PUNCT
cana-3972	332	1	an	an	DET
cana-3972	332	2	r	r	NOUN
cana-3972	332	3	-	-	PUNCT
cana-3972	332	4	module	module	NOUN
cana-3972	332	5	m	m	NOUN
cana-3972	332	6	is	be	AUX
cana-3972	332	7	termed	term	VERB
cana-3972	332	8	cosingular	cosingular	ADJ
cana-3972	332	9	hopfian	hopfian	NOUN
cana-3972	332	10	if	if	SCONJ
cana-3972	332	11	every	every	DET
cana-3972	332	12	cosingular	cosingular	ADJ
cana-3972	332	13	epimorphism	epimorphism	NOUN
cana-3972	332	14	𝜉	𝜉	ADP
cana-3972	332	15	∶	∶	NOUN
cana-3972	332	16	𝑀	𝑀	PROPN
cana-3972	332	17	→	→	SYM
cana-3972	332	18	𝑀	𝑀	PROPN
cana-3972	332	19	is	be	AUX
cana-3972	332	20	an	an	DET
cana-3972	332	21	automorphism	automorphism	NOUN
cana-3972	332	22	.	.	PUNCT
cana-3972	333	1	proposition	proposition	NOUN
cana-3972	333	2	3.24	3.24	NUM
cana-3972	333	3	.	.	PUNCT
cana-3972	334	1	every	every	DET
cana-3972	334	2	indecomposable	indecomposable	ADJ
cana-3972	334	3	cosingular	cosingular	ADJ
cana-3972	334	4	hopfian	hopfian	ADJ
cana-3972	334	5	module	module	NOUN
cana-3972	334	6	is	be	AUX
cana-3972	334	7	a	a	DET
cana-3972	334	8	d41	d41	NOUN
cana-3972	334	9	-	-	PUNCT
cana-3972	334	10	module	module	NOUN
cana-3972	334	11	.	.	PUNCT
cana-3972	335	1	proof	proof	NOUN
cana-3972	335	2	:	:	PUNCT
cana-3972	335	3	let	let	VERB
cana-3972	335	4	m	m	PRON
cana-3972	335	5	be	be	AUX
cana-3972	335	6	an	an	DET
cana-3972	335	7	indecomposable	indecomposable	ADJ
cana-3972	335	8	cosingular	cosingular	ADJ
cana-3972	335	9	hopfian	hopfian	NOUN
cana-3972	335	10	left	leave	VERB
cana-3972	335	11	r	r	NOUN
cana-3972	335	12	-	-	PUNCT
cana-3972	335	13	module	module	NOUN
cana-3972	335	14	,	,	PUNCT
cana-3972	335	15	and	and	CCONJ
cana-3972	335	16	let	let	VERB
cana-3972	335	17	𝑓	𝑓	DET
cana-3972	335	18	∶	∶	NOUN
cana-3972	335	19	𝑀	𝑀	PROPN
cana-3972	335	20	→	→	PUNCT
cana-3972	335	21	𝑁	𝑁	PROPN
cana-3972	335	22	be	be	AUX
cana-3972	335	23	a	a	DET
cana-3972	335	24	cosingular	cosingular	ADJ
cana-3972	335	25	epimorphism	epimorphism	NOUN
cana-3972	335	26	where	where	SCONJ
cana-3972	335	27	n	n	PRON
cana-3972	335	28	is	be	AUX
cana-3972	335	29	a	a	DET
cana-3972	335	30	direct	direct	ADJ
cana-3972	335	31	summand	summand	NOUN
cana-3972	335	32	of	of	ADP
cana-3972	335	33	m	m	PROPN
cana-3972	335	34	.	.	PUNCT
cana-3972	336	1	in	in	ADP
cana-3972	336	2	this	this	DET
cana-3972	336	3	case	case	NOUN
cana-3972	336	4	,	,	PUNCT
cana-3972	336	5	n	n	PRON
cana-3972	336	6	must	must	AUX
cana-3972	336	7	either	either	CCONJ
cana-3972	336	8	be	be	AUX
cana-3972	336	9	0	0	NUM
cana-3972	336	10	or	or	CCONJ
cana-3972	336	11	m	m	VERB
cana-3972	336	12	.	.	PUNCT
cana-3972	337	1	in	in	ADP
cana-3972	337	2	the	the	DET
cana-3972	337	3	first	first	ADJ
cana-3972	337	4	scenario	scenario	NOUN
cana-3972	337	5	,	,	PUNCT
cana-3972	337	6	f	f	PROPN
cana-3972	337	7	trivially	trivially	ADV
cana-3972	337	8	splits	split	VERB
cana-3972	337	9	.	.	PUNCT
cana-3972	338	1	in	in	ADP
cana-3972	338	2	the	the	DET
cana-3972	338	3	second	second	ADJ
cana-3972	338	4	scenario	scenario	NOUN
cana-3972	338	5	,	,	PUNCT
cana-3972	338	6	where	where	SCONJ
cana-3972	338	7	n	n	X
cana-3972	338	8	=	=	SYM
cana-3972	338	9	m	m	PROPN
cana-3972	338	10	,	,	PUNCT
cana-3972	338	11	f	f	PROPN
cana-3972	338	12	is	be	AUX
cana-3972	338	13	an	an	DET
cana-3972	338	14	automorphism	automorphism	NOUN
cana-3972	338	15	and	and	CCONJ
cana-3972	338	16	thus	thus	ADV
cana-3972	338	17	also	also	ADV
cana-3972	338	18	splits	split	VERB
cana-3972	338	19	.	.	PUNCT
cana-3972	339	1	a	a	DET
cana-3972	339	2	left	left	ADJ
cana-3972	339	3	r	r	NOUN
cana-3972	339	4	-	-	PUNCT
cana-3972	339	5	module	module	NOUN
cana-3972	339	6	m	m	NOUN
cana-3972	339	7	is	be	AUX
cana-3972	339	8	said	say	VERB
cana-3972	339	9	to	to	PART
cana-3972	339	10	be	be	AUX
cana-3972	339	11	generalized	generalize	VERB
cana-3972	339	12	hopfian	hopfian	NOUN
cana-3972	339	13	if	if	SCONJ
cana-3972	339	14	every	every	DET
cana-3972	339	15	surjective	surjective	ADJ
cana-3972	339	16	r	r	NOUN
cana-3972	339	17	-	-	PUNCT
cana-3972	339	18	endomorphism	endomorphism	PROPN
cana-3972	339	19	f	f	PROPN
cana-3972	339	20	of	of	ADP
cana-3972	339	21	m	m	PROPN
cana-3972	339	22	is	be	AUX
cana-3972	339	23	superfluous	superfluous	ADJ
cana-3972	339	24	,	,	PUNCT
cana-3972	339	25	meaning	mean	VERB
cana-3972	339	26	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	339	27	)	)	PUNCT
cana-3972	339	28	≪	≪	PUNCT
cana-3972	339	29	𝑀.	𝑀.	NOUN
cana-3972	339	30	proposition	proposition	NOUN
cana-3972	339	31	3.25	3.25	NUM
cana-3972	339	32	.	.	PUNCT
cana-3972	340	1	every	every	DET
cana-3972	340	2	generalized	generalize	VERB
cana-3972	340	3	hopfian	hopfian	ADJ
cana-3972	340	4	d41	d41	NOUN
cana-3972	340	5	-	-	PUNCT
cana-3972	340	6	module	module	NOUN
cana-3972	340	7	has	have	VERB
cana-3972	340	8	a	a	DET
cana-3972	340	9	d41	d41	NOUN
cana-3972	340	10	-	-	PUNCT
cana-3972	340	11	cover	cover	NOUN
cana-3972	340	12	.	.	PUNCT
cana-3972	341	1	proof	proof	NOUN
cana-3972	341	2	:	:	PUNCT
cana-3972	341	3	the	the	DET
cana-3972	341	4	proof	proof	NOUN
cana-3972	341	5	is	be	AUX
cana-3972	341	6	straightforward	straightforward	ADJ
cana-3972	341	7	.	.	PUNCT
cana-3972	342	1	proposition	proposition	NOUN
cana-3972	342	2	3.26	3.26	NUM
cana-3972	342	3	.	.	PUNCT
cana-3972	343	1	let	let	VERB
cana-3972	343	2	m	m	PRON
cana-3972	343	3	be	be	AUX
cana-3972	343	4	a	a	DET
cana-3972	343	5	lifting	lifting	NOUN
cana-3972	343	6	module	module	NOUN
cana-3972	343	7	and	and	CCONJ
cana-3972	343	8	consider	consider	VERB
cana-3972	343	9	the	the	DET
cana-3972	343	10	following	follow	VERB
cana-3972	343	11	conditions	condition	NOUN
cana-3972	343	12	:	:	PUNCT
cana-3972	343	13	1	1	X
cana-3972	343	14	)	)	PUNCT
cana-3972	343	15	m	m	VERB
cana-3972	343	16	is	be	AUX
cana-3972	343	17	hopfian	hopfian	ADJ
cana-3972	343	18	;	;	PUNCT
cana-3972	343	19	2	2	X
cana-3972	343	20	)	)	PUNCT
cana-3972	343	21	m	m	VERB
cana-3972	343	22	is	be	AUX
cana-3972	343	23	generalized	generalized	ADJ
cana-3972	343	24	hopfian	hopfian	NOUN
cana-3972	343	25	;	;	PUNCT
cana-3972	343	26	3	3	X
cana-3972	343	27	)	)	PUNCT
cana-3972	343	28	m	m	VERB
cana-3972	343	29	is	be	AUX
cana-3972	343	30	dedekind	dedekind	NOUN
cana-3972	343	31	finite	finite	NOUN
cana-3972	343	32	.	.	PUNCT
cana-3972	344	1	then	then	ADV
cana-3972	344	2	,	,	PUNCT
cana-3972	344	3	1	1	X
cana-3972	344	4	)	)	PUNCT
cana-3972	344	5	⇒	⇒	NOUN
cana-3972	344	6	2	2	NUM
cana-3972	344	7	)	)	PUNCT
cana-3972	344	8	⇒	⇒	NOUN
cana-3972	344	9	3	3	NUM
cana-3972	344	10	)	)	PUNCT
cana-3972	344	11	(	(	PUNCT
cana-3972	344	12	by	by	ADP
cana-3972	344	13	[	[	X
cana-3972	344	14	8	8	NUM
cana-3972	344	15	,	,	PUNCT
cana-3972	344	16	corollary	corollary	ADJ
cana-3972	344	17	1.4	1.4	NUM
cana-3972	344	18	]	]	PUNCT
cana-3972	344	19	)	)	PUNCT
cana-3972	344	20	.	.	PUNCT
cana-3972	345	1	if	if	SCONJ
cana-3972	345	2	in	in	ADP
cana-3972	345	3	addition	addition	NOUN
cana-3972	345	4	,	,	PUNCT
cana-3972	345	5	m	m	VERB
cana-3972	345	6	⊕	⊕	PROPN
cana-3972	345	7	m	m	PROPN
cana-3972	345	8	is	be	AUX
cana-3972	345	9	a	a	DET
cana-3972	345	10	d41	d41	NOUN
cana-3972	345	11	-	-	PUNCT
cana-3972	345	12	module	module	NOUN
cana-3972	345	13	,	,	PUNCT
cana-3972	345	14	then	then	ADV
cana-3972	345	15	we	we	PRON
cana-3972	345	16	have	have	VERB
cana-3972	345	17	the	the	DET
cana-3972	345	18	equivalence	equivalence	NOUN
cana-3972	345	19	.	.	PUNCT
cana-3972	346	1	proof	proof	NOUN
cana-3972	346	2	:	:	PUNCT
cana-3972	346	3	for	for	ADP
cana-3972	346	4	3	3	NUM
cana-3972	346	5	)	)	PUNCT
cana-3972	346	6	⇒	⇒	NOUN
cana-3972	346	7	1	1	NUM
cana-3972	346	8	):	):	PUNCT
cana-3972	346	9	since	since	SCONJ
cana-3972	346	10	m	m	PROPN
cana-3972	346	11	⊕	⊕	PROPN
cana-3972	346	12	m	m	PROPN
cana-3972	346	13	is	be	AUX
cana-3972	346	14	a	a	DET
cana-3972	346	15	d41	d41	NOUN
cana-3972	346	16	-	-	PUNCT
cana-3972	346	17	module	module	NOUN
cana-3972	346	18	,	,	PUNCT
cana-3972	346	19	then	then	ADV
cana-3972	346	20	m	m	NOUN
cana-3972	346	21	is	be	AUX
cana-3972	346	22	a	a	DET
cana-3972	346	23	d2	d2	NOUN
cana-3972	346	24	-	-	PUNCT
cana-3972	346	25	module	module	NOUN
cana-3972	346	26	.	.	PUNCT
cana-3972	347	1	if	if	SCONJ
cana-3972	347	2	𝑓	𝑓	PRON
cana-3972	347	3	∶	∶	NOUN
cana-3972	347	4	𝑀	𝑀	PROPN
cana-3972	347	5	→	→	SYM
cana-3972	347	6	𝑀	𝑀	PROPN
cana-3972	347	7	is	be	AUX
cana-3972	347	8	an	an	DET
cana-3972	347	9	epimorphism	epimorphism	NOUN
cana-3972	347	10	,	,	PUNCT
cana-3972	347	11	then	then	ADV
cana-3972	347	12	there	there	PRON
cana-3972	347	13	exists	exist	VERB
cana-3972	347	14	an	an	DET
cana-3972	347	15	endomorphism	endomorphism	PROPN
cana-3972	347	16	g	g	NOUN
cana-3972	347	17	of	of	ADP
cana-3972	347	18	m	m	PRON
cana-3972	347	19	such	such	ADJ
cana-3972	347	20	that	that	PRON
cana-3972	347	21	𝑓𝑔	𝑓𝑔	ADV
cana-3972	347	22	=	=	ADJ
cana-3972	347	23	1	1	X
cana-3972	347	24	.	.	PUNCT
cana-3972	347	25	given	give	VERB
cana-3972	347	26	that	that	DET
cana-3972	347	27	end(m	end(m	PROPN
cana-3972	347	28	)	)	PUNCT
cana-3972	347	29	is	be	AUX
cana-3972	347	30	a	a	DET
cana-3972	347	31	directly	directly	ADV
cana-3972	347	32	finite	finite	ADJ
cana-3972	347	33	ring	ring	NOUN
cana-3972	347	34	,	,	PUNCT
cana-3972	347	35	we	we	PRON
cana-3972	347	36	can	can	AUX
cana-3972	347	37	conclude	conclude	VERB
cana-3972	347	38	that	that	PRON
cana-3972	347	39	𝑔𝑓	𝑔𝑓	VERB
cana-3972	347	40	=	=	SYM
cana-3972	347	41	1	1	X
cana-3972	347	42	.	.	PUNCT
cana-3972	348	1	therefore	therefore	ADV
cana-3972	348	2	,	,	PUNCT
cana-3972	348	3	f	f	PROPN
cana-3972	348	4	is	be	AUX
cana-3972	348	5	an	an	DET
cana-3972	348	6	isomorphism	isomorphism	NOUN
cana-3972	348	7	,	,	PUNCT
cana-3972	348	8	which	which	PRON
cana-3972	348	9	implies	imply	VERB
cana-3972	348	10	that	that	SCONJ
cana-3972	348	11	m	m	PROPN
cana-3972	348	12	is	be	AUX
cana-3972	348	13	hopfian	hopfian	ADJ
cana-3972	348	14	.	.	PUNCT
cana-3972	349	1	in	in	ADP
cana-3972	349	2	this	this	DET
cana-3972	349	3	section	section	NOUN
cana-3972	349	4	we	we	PRON
cana-3972	349	5	introduce	introduce	VERB
cana-3972	349	6	the	the	DET
cana-3972	349	7	concept	concept	NOUN
cana-3972	349	8	of	of	ADP
cana-3972	349	9	t	t	PROPN
cana-3972	349	10	-	-	PUNCT
cana-3972	349	11	cosingular	cosingular	ADJ
cana-3972	349	12	d4	d4	NOUN
cana-3972	349	13	-	-	PUNCT
cana-3972	349	14	module	module	NOUN
cana-3972	349	15	.	.	PUNCT
cana-3972	350	1	the	the	DET
cana-3972	350	2	notion	notion	NOUN
cana-3972	350	3	of	of	ADP
cana-3972	350	4	t	t	PROPN
cana-3972	350	5	-	-	PUNCT
cana-3972	350	6	cosingular	cosingular	ADJ
cana-3972	350	7	module	module	NOUN
cana-3972	350	8	was	be	AUX
cana-3972	350	9	introduced	introduce	VERB
cana-3972	350	10	by	by	ADP
cana-3972	350	11	y.	y.	PROPN
cana-3972	350	12	talebi	talebi	PROPN
cana-3972	350	13	and	and	CCONJ
cana-3972	350	14	a.	a.	PROPN
cana-3972	350	15	r.	r.	PROPN
cana-3972	350	16	m.	m.	PROPN
cana-3972	350	17	hamzekolaee	hamzekolaee	PROPN
cana-3972	350	18	in	in	ADP
cana-3972	350	19	2013	2013	NUM
cana-3972	350	20	(	(	PUNCT
cana-3972	350	21	see	see	VERB
cana-3972	350	22	[	[	X
cana-3972	350	23	17	17	NUM
cana-3972	350	24	]	]	NUM
cana-3972	350	25	)	)	PUNCT
cana-3972	350	26	.	.	PUNCT
cana-3972	351	1	a	a	DET
cana-3972	351	2	module	module	NOUN
cana-3972	351	3	m	m	VERB
cana-3972	351	4	is	be	AUX
cana-3972	351	5	communications	communication	NOUN
cana-3972	351	6	on	on	ADP
cana-3972	351	7	applied	apply	VERB
cana-3972	351	8	nonlinear	nonlinear	ADJ
cana-3972	351	9	analysis	analysis	NOUN
cana-3972	351	10	issn	issn	NOUN
cana-3972	351	11	:	:	PUNCT
cana-3972	351	12	1074	1074	NUM
cana-3972	351	13	-	-	PUNCT
cana-3972	351	14	133x	133x	NUM
cana-3972	351	15	vol	vol	NOUN
cana-3972	351	16	32	32	NUM
cana-3972	351	17	no	no	NOUN
cana-3972	351	18	.	.	PUNCT
cana-3972	352	1	9s	9s	NUM
cana-3972	352	2	(	(	PUNCT
cana-3972	352	3	2025	2025	NUM
cana-3972	352	4	)	)	PUNCT
cana-3972	352	5	684	684	NUM
cana-3972	353	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	353	2	called	call	VERB
cana-3972	353	3	t	t	NOUN
cana-3972	353	4	-	-	PUNCT
cana-3972	353	5	cosingular	cosingular	ADJ
cana-3972	353	6	if	if	SCONJ
cana-3972	353	7	𝑍𝑡̅̅	𝑍𝑡̅̅	ADV
cana-3972	353	8	̅(𝑀	̅(𝑀	NUM
cana-3972	353	9	)	)	PUNCT
cana-3972	354	1	=	=	SYM
cana-3972	354	2	0	0	NUM
cana-3972	354	3	where	where	SCONJ
cana-3972	354	4	𝑍𝑡̅̅	𝑍𝑡̅̅	ADV
cana-3972	354	5	̅(𝑀	̅(𝑀	PUNCT
cana-3972	354	6	)	)	PUNCT
cana-3972	354	7	=	=	SYM
cana-3972	355	1	𝑅𝑒𝑗(𝑀	𝑅𝑒𝑗(𝑀	NOUN
cana-3972	355	2	,	,	PUNCT
cana-3972	355	3	𝑇𝑆	𝑇𝑆	PROPN
cana-3972	355	4	)	)	PUNCT
cana-3972	355	5	=	=	PRON
cana-3972	355	6	{	{	PUNCT
cana-3972	356	1	ker	ker	NOUN
cana-3972	356	2	f	f	PROPN
cana-3972	357	1	|	|	ADV
cana-3972	357	2	f	f	X
cana-3972	357	3	:	:	PUNCT
cana-3972	358	1	m	m	VERB
cana-3972	358	2	→	→	SYM
cana-3972	358	3	l	l	NOUN
cana-3972	358	4	,	,	PUNCT
cana-3972	358	5	l	l	PROPN
cana-3972	358	6	∈	∈	PROPN
cana-3972	358	7	ts	t	VERB
cana-3972	358	8	}	}	PUNCT
cana-3972	358	9	with	with	ADP
cana-3972	358	10	ts	ts	ADP
cana-3972	358	11	the	the	DET
cana-3972	358	12	class	class	NOUN
cana-3972	358	13	of	of	ADP
cana-3972	358	14	t	t	PROPN
cana-3972	358	15	-	-	PUNCT
cana-3972	358	16	small	small	ADJ
cana-3972	358	17	modules	module	NOUN
cana-3972	358	18	.	.	PUNCT
cana-3972	359	1	definition	definition	NOUN
cana-3972	359	2	3.27	3.27	NUM
cana-3972	359	3	.	.	PUNCT
cana-3972	360	1	let	let	VERB
cana-3972	360	2	m	m	PRON
cana-3972	360	3	be	be	AUX
cana-3972	360	4	an	an	DET
cana-3972	360	5	r	r	NOUN
cana-3972	360	6	-	-	PUNCT
cana-3972	360	7	module	module	NOUN
cana-3972	360	8	.	.	PUNCT
cana-3972	361	1	we	we	PRON
cana-3972	361	2	say	say	VERB
cana-3972	361	3	that	that	SCONJ
cana-3972	361	4	m	m	PROPN
cana-3972	361	5	is	be	AUX
cana-3972	361	6	a	a	DET
cana-3972	361	7	t	t	NOUN
cana-3972	361	8	-	-	PUNCT
cana-3972	361	9	cosingular	cosingular	ADJ
cana-3972	361	10	d4	d4	NOUN
cana-3972	361	11	-	-	PUNCT
cana-3972	361	12	module	module	NOUN
cana-3972	361	13	if	if	SCONJ
cana-3972	361	14	,	,	PUNCT
cana-3972	361	15	m	m	PROPN
cana-3972	361	16	=	=	SYM
cana-3972	361	17	n	n	PROPN
cana-3972	361	18	⊕	⊕	PROPN
cana-3972	361	19	k	k	PROPN
cana-3972	361	20	for	for	ADP
cana-3972	361	21	n	n	PRON
cana-3972	361	22	,	,	PUNCT
cana-3972	361	23	k	k	PROPN
cana-3972	361	24	≤	≤	NUM
cana-3972	361	25	m	m	VERB
cana-3972	361	26	such	such	ADJ
cana-3972	361	27	that	that	SCONJ
cana-3972	361	28	k	k	PROPN
cana-3972	361	29	is	be	AUX
cana-3972	361	30	t	t	NOUN
cana-3972	361	31	-	-	PUNCT
cana-3972	361	32	cosingular	cosingular	ADJ
cana-3972	361	33	and	and	CCONJ
cana-3972	361	34	f	f	NOUN
cana-3972	361	35	:	:	PUNCT
cana-3972	361	36	n	n	PROPN
cana-3972	361	37	→	→	SYM
cana-3972	361	38	k	k	X
cana-3972	361	39	is	be	AUX
cana-3972	361	40	an	an	DET
cana-3972	361	41	epimorphism	epimorphism	NOUN
cana-3972	361	42	,	,	PUNCT
cana-3972	361	43	then	then	ADV
cana-3972	361	44	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	361	45	)	)	PUNCT
cana-3972	361	46	≤	≤	PROPN
cana-3972	361	47	⊕	⊕	PROPN
cana-3972	361	48	n.	n.	VERB
cana-3972	361	49	the	the	DET
cana-3972	361	50	ring	ring	NOUN
cana-3972	361	51	r	r	NOUN
cana-3972	361	52	is	be	AUX
cana-3972	361	53	a	a	DET
cana-3972	361	54	right	right	NOUN
cana-3972	361	55	(	(	PUNCT
cana-3972	361	56	left	left	ADJ
cana-3972	361	57	)	)	PUNCT
cana-3972	361	58	t	t	PROPN
cana-3972	361	59	-	-	PUNCT
cana-3972	361	60	cosingular	cosingular	ADJ
cana-3972	361	61	d4	d4	NOUN
cana-3972	361	62	-	-	PUNCT
cana-3972	361	63	ring	ring	NOUN
cana-3972	361	64	if	if	SCONJ
cana-3972	361	65	the	the	DET
cana-3972	361	66	right	right	ADJ
cana-3972	361	67	r	r	NOUN
cana-3972	361	68	-	-	PUNCT
cana-3972	361	69	module	module	NOUN
cana-3972	361	70	rr	rr	NOUN
cana-3972	361	71	(	(	PUNCT
cana-3972	361	72	left	leave	VERB
cana-3972	361	73	rr	rr	NOUN
cana-3972	361	74	)	)	PUNCT
cana-3972	361	75	is	be	AUX
cana-3972	361	76	tcosingular	tcosingular	PROPN
cana-3972	361	77	d4	d4	PROPN
cana-3972	361	78	.	.	PUNCT
cana-3972	362	1	remark	remark	PROPN
cana-3972	362	2	3.28	3.28	NUM
cana-3972	362	3	.	.	PUNCT
cana-3972	363	1	it	it	PRON
cana-3972	363	2	has	have	AUX
cana-3972	363	3	been	be	AUX
cana-3972	363	4	proved	prove	VERB
cana-3972	363	5	in	in	ADP
cana-3972	363	6	[	[	X
cana-3972	363	7	17	17	NUM
cana-3972	363	8	,	,	PUNCT
cana-3972	363	9	remark	remark	NOUN
cana-3972	363	10	1	1	NUM
cana-3972	363	11	]	]	PUNCT
cana-3972	363	12	that	that	PRON
cana-3972	363	13	𝑍𝑡̅̅	𝑍𝑡̅̅	ADP
cana-3972	363	14	̅(𝑀	̅(𝑀	NUM
cana-3972	363	15	)	)	PUNCT
cana-3972	363	16	⊆	⊆	NUM
cana-3972	363	17	�	�	NOUN
cana-3972	363	18	̅	̅	NOUN
cana-3972	363	19	�	�	NOUN
cana-3972	363	20	(𝑀	(𝑀	NUM
cana-3972	363	21	)	)	PUNCT
cana-3972	363	22	.	.	PUNCT
cana-3972	364	1	from	from	ADP
cana-3972	364	2	this	this	PRON
cana-3972	364	3	,	,	PUNCT
cana-3972	364	4	we	we	PRON
cana-3972	364	5	can	can	AUX
cana-3972	364	6	say	say	VERB
cana-3972	364	7	that	that	SCONJ
cana-3972	364	8	every	every	DET
cana-3972	364	9	d41	d41	NOUN
cana-3972	364	10	-	-	PUNCT
cana-3972	364	11	module	module	NOUN
cana-3972	364	12	is	be	AUX
cana-3972	364	13	also	also	ADV
cana-3972	364	14	a	a	DET
cana-3972	364	15	t	t	NOUN
cana-3972	364	16	-	-	PUNCT
cana-3972	364	17	cosingular	cosingular	ADJ
cana-3972	364	18	d4	d4	NOUN
cana-3972	364	19	-	-	PUNCT
cana-3972	364	20	module	module	NOUN
cana-3972	364	21	.	.	PUNCT
cana-3972	365	1	proposition	proposition	NOUN
cana-3972	365	2	3.29	3.29	NUM
cana-3972	365	3	.	.	PUNCT
cana-3972	366	1	any	any	DET
cana-3972	366	2	direct	direct	ADJ
cana-3972	366	3	summand	summand	NOUN
cana-3972	366	4	of	of	ADP
cana-3972	366	5	a	a	DET
cana-3972	366	6	t	t	NOUN
cana-3972	366	7	-	-	PUNCT
cana-3972	366	8	cosingular	cosingular	ADJ
cana-3972	366	9	d4	d4	NOUN
cana-3972	366	10	-	-	PUNCT
cana-3972	366	11	module	module	NOUN
cana-3972	366	12	is	be	AUX
cana-3972	366	13	again	again	ADV
cana-3972	366	14	a	a	DET
cana-3972	366	15	t	t	NOUN
cana-3972	366	16	-	-	PUNCT
cana-3972	366	17	cosingular	cosingular	ADJ
cana-3972	366	18	d4module	d4module	NOUN
cana-3972	366	19	.	.	PUNCT
cana-3972	367	1	proof	proof	NOUN
cana-3972	367	2	:	:	PUNCT
cana-3972	367	3	let	let	VERB
cana-3972	367	4	m	m	PRON
cana-3972	367	5	be	be	AUX
cana-3972	367	6	a	a	DET
cana-3972	367	7	t	t	NOUN
cana-3972	367	8	-	-	PUNCT
cana-3972	367	9	cosingular	cosingular	ADJ
cana-3972	367	10	d4	d4	NOUN
cana-3972	367	11	-	-	PUNCT
cana-3972	367	12	module	module	NOUN
cana-3972	367	13	,	,	PUNCT
cana-3972	367	14	and	and	CCONJ
cana-3972	367	15	suppose	suppose	VERB
cana-3972	367	16	n	n	PRON
cana-3972	367	17	≤	≤	PROPN
cana-3972	367	18	⊕	⊕	PROPN
cana-3972	367	19	m	m	PROPN
cana-3972	367	20	.	.	PUNCT
cana-3972	368	1	we	we	PRON
cana-3972	368	2	aim	aim	VERB
cana-3972	368	3	to	to	PART
cana-3972	368	4	show	show	VERB
cana-3972	368	5	that	that	SCONJ
cana-3972	368	6	if	if	SCONJ
cana-3972	368	7	n	n	PROPN
cana-3972	368	8	=	=	SYM
cana-3972	368	9	k	k	PROPN
cana-3972	368	10	⊕	⊕	PROPN
cana-3972	368	11	k	k	PROPN
cana-3972	368	12	′	′	INTJ
cana-3972	368	13	,	,	PUNCT
cana-3972	368	14	where	where	SCONJ
cana-3972	368	15	k	k	PROPN
cana-3972	368	16	′	′	PROPN
cana-3972	368	17	is	be	AUX
cana-3972	368	18	t	t	NOUN
cana-3972	368	19	-	-	PUNCT
cana-3972	368	20	cosingular	cosingular	ADJ
cana-3972	368	21	and	and	CCONJ
cana-3972	368	22	f	f	NOUN
cana-3972	368	23	:	:	PUNCT
cana-3972	368	24	k	k	X
cana-3972	368	25	→	→	PUNCT
cana-3972	368	26	k	k	ADJ
cana-3972	368	27	′	′	NOUN
cana-3972	368	28	is	be	AUX
cana-3972	368	29	an	an	DET
cana-3972	368	30	epimorphism	epimorphism	NOUN
cana-3972	368	31	,	,	PUNCT
cana-3972	368	32	then	then	ADV
cana-3972	368	33	ker	ker	PROPN
cana-3972	368	34	f	f	PROPN
cana-3972	368	35	≤	≤	PROPN
cana-3972	368	36	⊕	⊕	PROPN
cana-3972	369	1	k	k	PROPN
cana-3972	369	2	.	.	PUNCT
cana-3972	370	1	assume	assume	VERB
cana-3972	370	2	𝑀	𝑀	PROPN
cana-3972	370	3	=	=	SYM
cana-3972	370	4	𝑁	𝑁	PROPN
cana-3972	370	5	⊕	⊕	PROPN
cana-3972	370	6	𝑁′	𝑁′	X
cana-3972	370	7	=	=	SYM
cana-3972	370	8	𝐾	𝐾	PROPN
cana-3972	370	9	⊕	⊕	PROPN
cana-3972	370	10	𝐾′	𝐾′	PROPN
cana-3972	370	11	⊕	⊕	PROPN
cana-3972	370	12	𝑁′	𝑁′	PROPN
cana-3972	370	13	,	,	PUNCT
cana-3972	370	14	where	where	SCONJ
cana-3972	370	15	𝑁′	𝑁′	NOUN
cana-3972	370	16	≤	≤	ADJ
cana-3972	370	17	𝑀.	𝑀.	PROPN
cana-3972	370	18	define	define	VERB
cana-3972	370	19	the	the	DET
cana-3972	370	20	canonical	canonical	ADJ
cana-3972	370	21	projection	projection	NOUN
cana-3972	370	22	𝜋	𝜋	PROPN
cana-3972	370	23	∶	∶	NOUN
cana-3972	370	24	𝐾	𝐾	PROPN
cana-3972	370	25	⊕	⊕	PROPN
cana-3972	370	26	𝑁′	𝑁′	X
cana-3972	371	1	→	→	SYM
cana-3972	371	2	𝐾.	𝐾.	PROPN
cana-3972	371	3	then	then	ADV
cana-3972	371	4	the	the	DET
cana-3972	371	5	composition	composition	NOUN
cana-3972	371	6	𝑓	𝑓	ADV
cana-3972	371	7	∘	∘	X
cana-3972	371	8	𝜋	𝜋	NOUN
cana-3972	371	9	∶	∶	NOUN
cana-3972	371	10	𝐾	𝐾	PROPN
cana-3972	371	11	⊕	⊕	PROPN
cana-3972	371	12	𝑁′	𝑁′	PROPN
cana-3972	371	13	→	→	SYM
cana-3972	371	14	𝐾′	𝐾′	X
cana-3972	371	15	is	be	AUX
cana-3972	371	16	an	an	DET
cana-3972	371	17	epimorphism	epimorphism	NOUN
cana-3972	371	18	,	,	PUNCT
cana-3972	371	19	and	and	CCONJ
cana-3972	371	20	its	its	PRON
cana-3972	371	21	kernel	kernel	NOUN
cana-3972	371	22	is	be	AUX
cana-3972	371	23	given	give	VERB
cana-3972	371	24	by	by	ADP
cana-3972	371	25	𝑘𝑒𝑟(𝑓	𝑘𝑒𝑟(𝑓	PROPN
cana-3972	371	26	∘	∘	NUM
cana-3972	371	27	𝜋	𝜋	NOUN
cana-3972	371	28	)	)	PUNCT
cana-3972	371	29	=	=	VERB
cana-3972	371	30	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	371	31	⊕	⊕	PROPN
cana-3972	371	32	𝑁′.	𝑁′.	VERB
cana-3972	371	33	since	since	SCONJ
cana-3972	371	34	𝑀	𝑀	PROPN
cana-3972	372	1	=	=	PUNCT
cana-3972	372	2	(	(	PUNCT
cana-3972	372	3	𝐾	𝐾	PROPN
cana-3972	372	4	⊕	⊕	PROPN
cana-3972	372	5	𝑁′	𝑁′	PROPN
cana-3972	372	6	)	)	PUNCT
cana-3972	372	7	⊕	⊕	PROPN
cana-3972	372	8	𝐾′	𝐾′	PROPN
cana-3972	372	9	is	be	AUX
cana-3972	372	10	a	a	DET
cana-3972	372	11	t	t	NOUN
cana-3972	372	12	-	-	PUNCT
cana-3972	372	13	cosingular	cosingular	ADJ
cana-3972	372	14	d4	d4	NOUN
cana-3972	372	15	-	-	PUNCT
cana-3972	372	16	module	module	NOUN
cana-3972	372	17	,	,	PUNCT
cana-3972	372	18	we	we	PRON
cana-3972	372	19	deduce	deduce	VERB
cana-3972	372	20	that	that	DET
cana-3972	372	21	𝑘𝑒𝑟𝑓	𝑘𝑒𝑟𝑓	NOUN
cana-3972	372	22	⊕	⊕	PROPN
cana-3972	372	23	n	n	CCONJ
cana-3972	372	24	′	′	NOUN
cana-3972	372	25	≤	≤	NUM
cana-3972	373	1	⊕	⊕	PROPN
cana-3972	373	2	k	k	PROPN
cana-3972	373	3	⊕	⊕	PROPN
cana-3972	373	4	n	n	CCONJ
cana-3972	373	5	′	′	NOUN
cana-3972	373	6	≤	≤	NUM
cana-3972	374	1	⊕	⊕	PROPN
cana-3972	374	2	m	m	PROPN
cana-3972	374	3	.	.	PUNCT
cana-3972	375	1	thus	thus	ADV
cana-3972	375	2	,	,	PUNCT
cana-3972	375	3	ker	ker	PROPN
cana-3972	375	4	f	f	PROPN
cana-3972	375	5	≤	≤	PROPN
cana-3972	375	6	⊕	⊕	PROPN
cana-3972	375	7	n	n	PROPN
cana-3972	375	8	.	.	PUNCT
cana-3972	376	1	consequently	consequently	ADV
cana-3972	376	2	,	,	PUNCT
cana-3972	376	3	n	n	PRON
cana-3972	376	4	is	be	AUX
cana-3972	376	5	a	a	DET
cana-3972	376	6	t	t	NOUN
cana-3972	376	7	-	-	PUNCT
cana-3972	376	8	cosingular	cosingular	ADJ
cana-3972	376	9	d4module	d4module	NOUN
cana-3972	376	10	.	.	PUNCT
cana-3972	377	1	proposition	proposition	NOUN
cana-3972	377	2	3.30	3.30	NUM
cana-3972	377	3	.	.	PUNCT
cana-3972	378	1	let	let	VERB
cana-3972	378	2	m	m	PRON
cana-3972	378	3	be	be	AUX
cana-3972	378	4	a	a	DET
cana-3972	378	5	t	t	NOUN
cana-3972	378	6	-	-	PUNCT
cana-3972	378	7	cosingular	cosingular	ADJ
cana-3972	378	8	d4	d4	NOUN
cana-3972	378	9	-	-	PUNCT
cana-3972	378	10	module	module	NOUN
cana-3972	378	11	.	.	PUNCT
cana-3972	379	1	then	then	ADV
cana-3972	379	2	,	,	PUNCT
cana-3972	379	3	for	for	ADP
cana-3972	379	4	every	every	DET
cana-3972	379	5	submodule	submodule	NOUN
cana-3972	379	6	n	n	PROPN
cana-3972	379	7	of	of	ADP
cana-3972	379	8	m	m	PROPN
cana-3972	379	9	,	,	PUNCT
cana-3972	379	10	the	the	DET
cana-3972	379	11	quotient	quotient	NOUN
cana-3972	379	12	module	module	NOUN
cana-3972	379	13	m	m	PROPN
cana-3972	379	14	/	/	SYM
cana-3972	379	15	n	n	PROPN
cana-3972	379	16	is	be	AUX
cana-3972	379	17	also	also	ADV
cana-3972	379	18	a	a	DET
cana-3972	379	19	t	t	NOUN
cana-3972	379	20	-	-	PUNCT
cana-3972	379	21	cosingular	cosingular	ADJ
cana-3972	379	22	d4	d4	NOUN
cana-3972	379	23	-	-	PUNCT
cana-3972	379	24	module	module	NOUN
cana-3972	379	25	.	.	PUNCT
cana-3972	380	1	proof	proof	NOUN
cana-3972	380	2	:	:	PUNCT
cana-3972	380	3	let	let	VERB
cana-3972	380	4	m	m	PRON
cana-3972	380	5	be	be	AUX
cana-3972	380	6	a	a	DET
cana-3972	380	7	t	t	NOUN
cana-3972	380	8	-	-	PUNCT
cana-3972	380	9	cosingular	cosingular	ADJ
cana-3972	380	10	d4	d4	NOUN
cana-3972	380	11	-	-	PUNCT
cana-3972	380	12	module	module	NOUN
cana-3972	380	13	,	,	PUNCT
cana-3972	380	14	and	and	CCONJ
cana-3972	380	15	let	let	VERB
cana-3972	380	16	n	n	PRON
cana-3972	380	17	be	be	AUX
cana-3972	380	18	a	a	DET
cana-3972	380	19	submodule	submodule	NOUN
cana-3972	380	20	of	of	ADP
cana-3972	380	21	m	m	PROPN
cana-3972	380	22	.	.	PUNCT
cana-3972	381	1	consider	consider	VERB
cana-3972	381	2	a	a	DET
cana-3972	381	3	t	t	NOUN
cana-3972	381	4	-	-	PUNCT
cana-3972	381	5	cosingular	cosingular	ADJ
cana-3972	381	6	submodule	submodule	NOUN
cana-3972	381	7	k	k	PROPN
cana-3972	382	1	=	=	PUNCT
cana-3972	382	2	p	p	X
cana-3972	382	3	/n	/n	PUNCT
cana-3972	382	4	of	of	ADP
cana-3972	382	5	m	m	PROPN
cana-3972	382	6	/	/	SYM
cana-3972	382	7	n	n	PROPN
cana-3972	382	8	,	,	PUNCT
cana-3972	382	9	where	where	SCONJ
cana-3972	382	10	p	p	NOUN
cana-3972	382	11	is	be	AUX
cana-3972	382	12	a	a	DET
cana-3972	382	13	submodule	submodule	NOUN
cana-3972	382	14	of	of	ADP
cana-3972	382	15	m	m	PROPN
cana-3972	382	16	containing	contain	VERB
cana-3972	382	17	n	n	X
cana-3972	382	18	.	.	PUNCT
cana-3972	383	1	assume	assume	VERB
cana-3972	383	2	that	that	SCONJ
cana-3972	383	3	(	(	PUNCT
cana-3972	383	4	m	m	NOUN
cana-3972	383	5	/	/	SYM
cana-3972	383	6	n	n	PROPN
cana-3972	383	7	)	)	PUNCT
cana-3972	383	8	/(p	/(p	PROPN
cana-3972	383	9	/n	/n	PUNCT
cana-3972	383	10	)	)	PUNCT
cana-3972	384	1	≅	≅	PROPN
cana-3972	384	2	l	l	NOUN
cana-3972	384	3	,	,	PUNCT
cana-3972	384	4	where	where	SCONJ
cana-3972	384	5	l	l	NOUN
cana-3972	384	6	is	be	AUX
cana-3972	384	7	a	a	DET
cana-3972	384	8	direct	direct	ADJ
cana-3972	384	9	summand	summand	NOUN
cana-3972	384	10	of	of	ADP
cana-3972	384	11	m	m	PROPN
cana-3972	384	12	/	/	SYM
cana-3972	384	13	n	n	PROPN
cana-3972	384	14	and	and	CCONJ
cana-3972	384	15	l	l	NOUN
cana-3972	384	16	≤	≤	NOUN
cana-3972	384	17	p	p	X
cana-3972	384	18	/n	/n	PUNCT
cana-3972	384	19	.	.	PUNCT
cana-3972	385	1	by	by	ADP
cana-3972	385	2	the	the	DET
cana-3972	385	3	second	second	ADJ
cana-3972	385	4	isomorphism	isomorphism	NOUN
cana-3972	385	5	theorem	theorem	VERB
cana-3972	385	6	,	,	PUNCT
cana-3972	385	7	we	we	PRON
cana-3972	385	8	have	have	VERB
cana-3972	385	9	:	:	PUNCT
cana-3972	385	10	m	m	X
cana-3972	385	11	/	/	SYM
cana-3972	385	12	p	p	PRON
cana-3972	385	13	≅(m	≅(m	PROPN
cana-3972	385	14	/	/	SYM
cana-3972	385	15	n	n	NOUN
cana-3972	385	16	)	)	PUNCT
cana-3972	385	17	/(p	/(p	PROPN
cana-3972	385	18	/n	/n	PUNCT
cana-3972	385	19	)	)	PUNCT
cana-3972	386	1	≅	≅	PROPN
cana-3972	386	2	l.	l.	PROPN
cana-3972	386	3	since	since	SCONJ
cana-3972	386	4	m	m	PROPN
cana-3972	386	5	is	be	AUX
cana-3972	386	6	a	a	DET
cana-3972	386	7	t	t	NOUN
cana-3972	386	8	-	-	PUNCT
cana-3972	386	9	cosingular	cosingular	ADJ
cana-3972	386	10	d4	d4	NOUN
cana-3972	386	11	-	-	PUNCT
cana-3972	386	12	module	module	NOUN
cana-3972	386	13	,	,	PUNCT
cana-3972	386	14	p	p	PRON
cana-3972	386	15	is	be	AUX
cana-3972	386	16	a	a	DET
cana-3972	386	17	direct	direct	ADJ
cana-3972	386	18	summand	summand	NOUN
cana-3972	386	19	of	of	ADP
cana-3972	386	20	m	m	PROPN
cana-3972	386	21	.	.	PUNCT
cana-3972	387	1	consequently	consequently	ADV
cana-3972	387	2	,	,	PUNCT
cana-3972	387	3	k	k	PROPN
cana-3972	387	4	=	=	PUNCT
cana-3972	387	5	p	p	X
cana-3972	387	6	/n	/n	PUNCT
cana-3972	387	7	is	be	AUX
cana-3972	387	8	a	a	DET
cana-3972	387	9	direct	direct	ADJ
cana-3972	387	10	summand	summand	NOUN
cana-3972	387	11	of	of	ADP
cana-3972	387	12	m	m	PROPN
cana-3972	387	13	/	/	SYM
cana-3972	387	14	n	n	PROPN
cana-3972	387	15	.	.	PUNCT
cana-3972	388	1	therefore	therefore	ADV
cana-3972	388	2	,	,	PUNCT
cana-3972	388	3	m	m	PROPN
cana-3972	388	4	/	/	SYM
cana-3972	388	5	n	n	PROPN
cana-3972	388	6	is	be	AUX
cana-3972	388	7	a	a	DET
cana-3972	388	8	t	t	NOUN
cana-3972	388	9	-	-	PUNCT
cana-3972	388	10	cosingular	cosingular	ADJ
cana-3972	388	11	d4	d4	NOUN
cana-3972	388	12	-	-	PUNCT
cana-3972	388	13	module	module	NOUN
cana-3972	388	14	.	.	PUNCT
cana-3972	389	1	recall	recall	VERB
cana-3972	389	2	that	that	SCONJ
cana-3972	389	3	a	a	DET
cana-3972	389	4	module	module	NOUN
cana-3972	389	5	m	m	VERB
cana-3972	389	6	has	have	VERB
cana-3972	389	7	c∗	c∗	ADJ
cana-3972	389	8	if	if	SCONJ
cana-3972	389	9	every	every	DET
cana-3972	389	10	submodule	submodule	NOUN
cana-3972	389	11	n	n	PROPN
cana-3972	389	12	of	of	ADP
cana-3972	389	13	m	m	PROPN
cana-3972	389	14	contains	contain	VERB
cana-3972	389	15	a	a	DET
cana-3972	389	16	direct	direct	ADJ
cana-3972	389	17	summand	summand	NOUN
cana-3972	389	18	k	k	PROPN
cana-3972	389	19	of	of	ADP
cana-3972	389	20	m	m	PRON
cana-3972	389	21	such	such	ADJ
cana-3972	389	22	that	that	SCONJ
cana-3972	389	23	n	n	PROPN
cana-3972	389	24	/	/	SYM
cana-3972	389	25	k	k	PROPN
cana-3972	389	26	is	be	AUX
cana-3972	389	27	cosingular	cosingular	ADJ
cana-3972	389	28	[	[	X
cana-3972	389	29	19	19	NUM
cana-3972	389	30	]	]	PUNCT
cana-3972	389	31	and	and	CCONJ
cana-3972	389	32	a	a	DET
cana-3972	389	33	module	module	NOUN
cana-3972	389	34	m	m	VERB
cana-3972	389	35	is	be	AUX
cana-3972	389	36	lifting	lift	VERB
cana-3972	389	37	if	if	SCONJ
cana-3972	389	38	every	every	DET
cana-3972	389	39	submodule	submodule	NOUN
cana-3972	389	40	n	n	PROPN
cana-3972	389	41	of	of	ADP
cana-3972	389	42	m	m	PROPN
cana-3972	389	43	contains	contain	VERB
cana-3972	389	44	a	a	DET
cana-3972	389	45	direct	direct	ADJ
cana-3972	389	46	summand	summand	NOUN
cana-3972	389	47	k	k	PROPN
cana-3972	389	48	of	of	ADP
cana-3972	389	49	m	m	PRON
cana-3972	389	50	such	such	ADJ
cana-3972	389	51	that	that	SCONJ
cana-3972	389	52	n	n	PROPN
cana-3972	389	53	/	/	SYM
cana-3972	389	54	k	k	PROPN
cana-3972	389	55	≪	≪	PROPN
cana-3972	389	56	m	m	PROPN
cana-3972	389	57	/	/	SYM
cana-3972	389	58	k	k	X
cana-3972	389	59	.	.	PUNCT
cana-3972	390	1	we	we	PRON
cana-3972	390	2	now	now	ADV
cana-3972	390	3	define	define	VERB
cana-3972	390	4	a	a	DET
cana-3972	390	5	module	module	NOUN
cana-3972	390	6	having	have	VERB
cana-3972	390	7	t	t	PROPN
cana-3972	390	8	-	-	PUNCT
cana-3972	390	9	c∗	c∗	PROPN
cana-3972	390	10	if	if	SCONJ
cana-3972	390	11	every	every	DET
cana-3972	390	12	submodule	submodule	NOUN
cana-3972	390	13	n	n	PROPN
cana-3972	390	14	of	of	ADP
cana-3972	390	15	m	m	PROPN
cana-3972	390	16	contains	contain	VERB
cana-3972	390	17	a	a	DET
cana-3972	390	18	direct	direct	ADJ
cana-3972	390	19	summand	summand	NOUN
cana-3972	390	20	k	k	PROPN
cana-3972	390	21	of	of	ADP
cana-3972	390	22	m	m	PRON
cana-3972	390	23	such	such	ADJ
cana-3972	390	24	that	that	SCONJ
cana-3972	390	25	n	n	PROPN
cana-3972	390	26	/	/	SYM
cana-3972	390	27	k	k	PROPN
cana-3972	390	28	is	be	AUX
cana-3972	390	29	t	t	NOUN
cana-3972	390	30	-	-	PUNCT
cana-3972	390	31	cosingular	cosingular	ADJ
cana-3972	390	32	.	.	PUNCT
cana-3972	391	1	we	we	PRON
cana-3972	391	2	recall	recall	VERB
cana-3972	391	3	a	a	DET
cana-3972	391	4	module	module	NOUN
cana-3972	391	5	is	be	AUX
cana-3972	391	6	tlifting	tlifte	VERB
cana-3972	391	7	if	if	SCONJ
cana-3972	391	8	every	every	DET
cana-3972	391	9	submodule	submodule	NOUN
cana-3972	391	10	n	n	PROPN
cana-3972	391	11	of	of	ADP
cana-3972	391	12	m	m	PROPN
cana-3972	391	13	contains	contain	VERB
cana-3972	391	14	a	a	DET
cana-3972	391	15	direct	direct	ADJ
cana-3972	391	16	summand	summand	NOUN
cana-3972	391	17	k	k	PROPN
cana-3972	391	18	of	of	ADP
cana-3972	391	19	m	m	PRON
cana-3972	391	20	such	such	ADJ
cana-3972	391	21	that	that	SCONJ
cana-3972	391	22	n	n	PROPN
cana-3972	391	23	/	/	SYM
cana-3972	391	24	k	k	PROPN
cana-3972	391	25	is	be	AUX
cana-3972	391	26	t	t	NOUN
cana-3972	391	27	-	-	PUNCT
cana-3972	391	28	small	small	ADJ
cana-3972	391	29	in	in	ADP
cana-3972	391	30	m	m	PROPN
cana-3972	391	31	/k	/k	PUNCT
cana-3972	391	32	.	.	PUNCT
cana-3972	392	1	proposition	proposition	NOUN
cana-3972	392	2	3.31	3.31	NUM
cana-3972	392	3	.	.	PUNCT
cana-3972	393	1	every	every	DET
cana-3972	393	2	t	t	PROPN
cana-3972	393	3	-	-	PUNCT
cana-3972	393	4	cosingular	cosingular	ADJ
cana-3972	393	5	module	module	NOUN
cana-3972	393	6	(	(	PUNCT
cana-3972	393	7	and	and	CCONJ
cana-3972	393	8	consequently	consequently	ADV
cana-3972	393	9	,	,	PUNCT
cana-3972	393	10	every	every	DET
cana-3972	393	11	t	t	NOUN
cana-3972	393	12	-	-	PUNCT
cana-3972	393	13	small	small	ADJ
cana-3972	393	14	module	module	NOUN
cana-3972	393	15	)	)	PUNCT
cana-3972	393	16	satisfies	satisfy	VERB
cana-3972	393	17	the	the	DET
cana-3972	393	18	property	property	NOUN
cana-3972	393	19	t	t	PROPN
cana-3972	393	20	-	-	PUNCT
cana-3972	393	21	c∗.	c∗.	NOUN
cana-3972	393	22	communications	communication	NOUN
cana-3972	393	23	on	on	ADP
cana-3972	393	24	applied	apply	VERB
cana-3972	393	25	nonlinear	nonlinear	ADJ
cana-3972	393	26	analysis	analysis	NOUN
cana-3972	393	27	issn	issn	NOUN
cana-3972	393	28	:	:	PUNCT
cana-3972	393	29	1074	1074	NUM
cana-3972	393	30	-	-	PUNCT
cana-3972	393	31	133x	133x	NUM
cana-3972	393	32	vol	vol	NOUN
cana-3972	393	33	32	32	NUM
cana-3972	393	34	no	no	NOUN
cana-3972	393	35	.	.	PUNCT
cana-3972	394	1	9s	9s	NUM
cana-3972	394	2	(	(	PUNCT
cana-3972	394	3	2025	2025	NUM
cana-3972	394	4	)	)	PUNCT
cana-3972	394	5	685	685	NUM
cana-3972	394	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	394	7	proof	proof	NOUN
cana-3972	394	8	:	:	PUNCT
cana-3972	394	9	since	since	SCONJ
cana-3972	394	10	a	a	DET
cana-3972	394	11	submodule	submodule	NOUN
cana-3972	394	12	of	of	ADP
cana-3972	394	13	a	a	DET
cana-3972	394	14	t	t	NOUN
cana-3972	394	15	-	-	PUNCT
cana-3972	394	16	cosingular	cosingular	ADJ
cana-3972	394	17	module	module	NOUN
cana-3972	394	18	is	be	AUX
cana-3972	394	19	t	t	NOUN
cana-3972	394	20	-	-	PUNCT
cana-3972	394	21	cosingular	cosingular	ADJ
cana-3972	394	22	,	,	PUNCT
cana-3972	394	23	the	the	DET
cana-3972	394	24	proposition	proposition	NOUN
cana-3972	394	25	follows	follow	VERB
cana-3972	394	26	easily	easily	ADV
cana-3972	394	27	.	.	PUNCT
cana-3972	395	1	proposition	proposition	NOUN
cana-3972	395	2	3.32	3.32	NUM
cana-3972	395	3	.	.	PUNCT
cana-3972	396	1	for	for	ADP
cana-3972	396	2	an	an	DET
cana-3972	396	3	r	r	NOUN
cana-3972	396	4	-	-	PUNCT
cana-3972	396	5	module	module	NOUN
cana-3972	396	6	m	m	NOUN
cana-3972	396	7	,	,	PUNCT
cana-3972	396	8	the	the	DET
cana-3972	396	9	fol	fol	NOUN
cana-3972	396	10	lowing	lowing	NOUN
cana-3972	396	11	statements	statement	NOUN
cana-3972	396	12	are	be	AUX
cana-3972	396	13	equivalent	equivalent	ADJ
cana-3972	396	14	:	:	PUNCT
cana-3972	396	15	1	1	X
cana-3972	396	16	)	)	PUNCT
cana-3972	396	17	m	m	VERB
cana-3972	396	18	satisfies	satisfy	VERB
cana-3972	396	19	t	t	PROPN
cana-3972	396	20	-	-	PUNCT
cana-3972	396	21	c∗.	c∗.	NOUN
cana-3972	396	22	2	2	NUM
cana-3972	396	23	)	)	PUNCT
cana-3972	396	24	for	for	ADP
cana-3972	396	25	every	every	DET
cana-3972	396	26	submodule	submodule	NOUN
cana-3972	396	27	n	n	PROPN
cana-3972	396	28	of	of	ADP
cana-3972	396	29	m	m	PRON
cana-3972	396	30	,	,	PUNCT
cana-3972	396	31	there	there	PRON
cana-3972	396	32	exists	exist	VERB
cana-3972	396	33	a	a	DET
cana-3972	396	34	decomposition	decomposition	NOUN
cana-3972	396	35	m	m	NOUN
cana-3972	396	36	=	=	SYM
cana-3972	396	37	m1	m1	PROPN
cana-3972	396	38	⊕	⊕	PROPN
cana-3972	396	39	m2	m2	PROPN
cana-3972	396	40	such	such	ADJ
cana-3972	396	41	that	that	DET
cana-3972	396	42	m1	m1	PROPN
cana-3972	396	43	≤	≤	NOUN
cana-3972	396	44	n	n	CCONJ
cana-3972	396	45	and	and	CCONJ
cana-3972	396	46	n	n	PROPN
cana-3972	396	47	∩	∩	NOUN
cana-3972	396	48	m2	m2	PROPN
cana-3972	396	49	is	be	AUX
cana-3972	396	50	t	t	NOUN
cana-3972	396	51	-	-	PUNCT
cana-3972	396	52	cosingular	cosingular	ADJ
cana-3972	396	53	.	.	PUNCT
cana-3972	397	1	3	3	X
cana-3972	397	2	)	)	PUNCT
cana-3972	397	3	for	for	ADP
cana-3972	397	4	every	every	DET
cana-3972	397	5	submodule	submodule	NOUN
cana-3972	397	6	n	n	PROPN
cana-3972	397	7	of	of	ADP
cana-3972	397	8	m	m	PROPN
cana-3972	397	9	,	,	PUNCT
cana-3972	397	10	n	n	PRON
cana-3972	397	11	can	can	AUX
cana-3972	397	12	be	be	AUX
cana-3972	397	13	decomposed	decompose	VERB
cana-3972	397	14	as	as	ADP
cana-3972	397	15	n	n	NOUN
cana-3972	397	16	=	=	SYM
cana-3972	397	17	n1	n1	PROPN
cana-3972	397	18	⊕	⊕	PROPN
cana-3972	397	19	n2	n2	NOUN
cana-3972	397	20	,	,	PUNCT
cana-3972	397	21	where	where	SCONJ
cana-3972	397	22	n1	n1	PROPN
cana-3972	397	23	is	be	AUX
cana-3972	397	24	a	a	DET
cana-3972	397	25	direct	direct	ADJ
cana-3972	397	26	summand	summand	NOUN
cana-3972	397	27	of	of	ADP
cana-3972	397	28	m	m	PROPN
cana-3972	397	29	,	,	PUNCT
cana-3972	397	30	and	and	CCONJ
cana-3972	397	31	n2	n2	PROPN
cana-3972	397	32	is	be	AUX
cana-3972	397	33	t	t	NOUN
cana-3972	397	34	-	-	PUNCT
cana-3972	397	35	cosingular	cosingular	ADJ
cana-3972	397	36	.	.	PUNCT
cana-3972	398	1	proof	proof	NOUN
cana-3972	398	2	:	:	PUNCT
cana-3972	398	3	1	1	X
cana-3972	398	4	)	)	PUNCT
cana-3972	398	5	⇒	⇒	NOUN
cana-3972	398	6	2	2	NUM
cana-3972	398	7	)	)	PUNCT
cana-3972	398	8	let	let	VERB
cana-3972	398	9	n	n	PRON
cana-3972	398	10	≤	≤	NOUN
cana-3972	398	11	m	m	VERB
cana-3972	398	12	.	.	PUNCT
cana-3972	399	1	by	by	ADP
cana-3972	399	2	definition	definition	NOUN
cana-3972	399	3	,	,	PUNCT
cana-3972	399	4	there	there	PRON
cana-3972	399	5	exists	exist	VERB
cana-3972	399	6	a	a	DET
cana-3972	399	7	decomposition	decomposition	NOUN
cana-3972	399	8	m	m	NOUN
cana-3972	399	9	=	=	SYM
cana-3972	399	10	m1	m1	PROPN
cana-3972	399	11	⊕	⊕	PROPN
cana-3972	399	12	m2	m2	PROPN
cana-3972	399	13	such	such	ADJ
cana-3972	399	14	that	that	DET
cana-3972	399	15	m1	m1	PROPN
cana-3972	399	16	≤	≤	NOUN
cana-3972	399	17	n	n	CCONJ
cana-3972	399	18	and	and	CCONJ
cana-3972	399	19	n	n	CCONJ
cana-3972	399	20	/	/	SYM
cana-3972	399	21	m1	m1	PROPN
cana-3972	399	22	is	be	AUX
cana-3972	399	23	t	t	NOUN
cana-3972	399	24	-	-	PUNCT
cana-3972	399	25	cosingular	cosingular	ADJ
cana-3972	399	26	.	.	PUNCT
cana-3972	400	1	this	this	PRON
cana-3972	400	2	gives	give	VERB
cana-3972	400	3	n	n	NOUN
cana-3972	400	4	=	=	SYM
cana-3972	400	5	m1	m1	PROPN
cana-3972	400	6	⊕	⊕	PROPN
cana-3972	400	7	(	(	PUNCT
cana-3972	400	8	n	n	X
cana-3972	400	9	∩	∩	X
cana-3972	400	10	m2	m2	PROPN
cana-3972	400	11	)	)	PUNCT
cana-3972	400	12	,	,	PUNCT
cana-3972	400	13	and	and	CCONJ
cana-3972	400	14	since	since	SCONJ
cana-3972	400	15	n	n	NOUN
cana-3972	400	16	∩	∩	NOUN
cana-3972	400	17	m2	m2	PROPN
cana-3972	400	18	≅	≅	PROPN
cana-3972	400	19	n	n	CCONJ
cana-3972	400	20	/	/	SYM
cana-3972	400	21	m1	m1	NOUN
cana-3972	400	22	,	,	PUNCT
cana-3972	400	23	it	it	PRON
cana-3972	400	24	follows	follow	VERB
cana-3972	400	25	that	that	SCONJ
cana-3972	400	26	n	n	ADP
cana-3972	400	27	∩	∩	NOUN
cana-3972	400	28	m2	m2	PROPN
cana-3972	400	29	is	be	AUX
cana-3972	400	30	t	t	NOUN
cana-3972	400	31	-	-	PUNCT
cana-3972	400	32	cosingular	cosingular	ADJ
cana-3972	400	33	.	.	PUNCT
cana-3972	401	1	2	2	NUM
cana-3972	401	2	)	)	PUNCT
cana-3972	401	3	⇒	⇒	NOUN
cana-3972	401	4	3	3	NUM
cana-3972	401	5	)	)	PUNCT
cana-3972	401	6	assume	assume	VERB
cana-3972	401	7	m	m	NOUN
cana-3972	401	8	=	=	SYM
cana-3972	401	9	m1	m1	PROPN
cana-3972	401	10	⊕	⊕	PROPN
cana-3972	401	11	m2	m2	PROPN
cana-3972	401	12	with	with	ADP
cana-3972	401	13	m1	m1	PROPN
cana-3972	401	14	≤	≤	NOUN
cana-3972	401	15	n	n	ADV
cana-3972	401	16	.	.	PUNCT
cana-3972	402	1	then	then	ADV
cana-3972	402	2	n	n	PROPN
cana-3972	402	3	=	=	SYM
cana-3972	402	4	m1	m1	PROPN
cana-3972	402	5	⊕	⊕	PROPN
cana-3972	402	6	(	(	PUNCT
cana-3972	402	7	n	n	X
cana-3972	402	8	∩m2	∩m2	PROPN
cana-3972	402	9	)	)	PUNCT
cana-3972	402	10	.	.	PUNCT
cana-3972	403	1	define	define	VERB
cana-3972	403	2	n1	n1	PROPN
cana-3972	403	3	=	=	SYM
cana-3972	403	4	m1	m1	PROPN
cana-3972	403	5	and	and	CCONJ
cana-3972	403	6	n2	n2	ADJ
cana-3972	403	7	=	=	SYM
cana-3972	403	8	n	n	X
cana-3972	403	9	∩	∩	X
cana-3972	403	10	m2	m2	PROPN
cana-3972	403	11	,	,	PUNCT
cana-3972	403	12	satisfying	satisfy	VERB
cana-3972	403	13	the	the	DET
cana-3972	403	14	required	required	ADJ
cana-3972	403	15	decomposition	decomposition	NOUN
cana-3972	403	16	.	.	PUNCT
cana-3972	404	1	3	3	X
cana-3972	404	2	)	)	PUNCT
cana-3972	404	3	⇒	⇒	NOUN
cana-3972	404	4	1	1	NUM
cana-3972	404	5	)	)	PUNCT
cana-3972	404	6	suppose	suppose	VERB
cana-3972	404	7	n	n	PRON
cana-3972	404	8	≤	≤	NOUN
cana-3972	404	9	m	m	VERB
cana-3972	404	10	.	.	PUNCT
cana-3972	405	1	by	by	ADP
cana-3972	405	2	assumption	assumption	NOUN
cana-3972	405	3	,	,	PUNCT
cana-3972	405	4	n	n	PROPN
cana-3972	405	5	=	=	SYM
cana-3972	405	6	n1	n1	PROPN
cana-3972	405	7	⊕	⊕	PROPN
cana-3972	405	8	n2	n2	PROPN
cana-3972	405	9	,	,	PUNCT
cana-3972	405	10	where	where	SCONJ
cana-3972	405	11	n1	n1	PROPN
cana-3972	405	12	is	be	AUX
cana-3972	405	13	a	a	DET
cana-3972	405	14	direct	direct	ADJ
cana-3972	405	15	summand	summand	NOUN
cana-3972	405	16	of	of	ADP
cana-3972	405	17	m	m	PROPN
cana-3972	405	18	,	,	PUNCT
cana-3972	405	19	and	and	CCONJ
cana-3972	405	20	n2	n2	PROPN
cana-3972	405	21	is	be	AUX
cana-3972	405	22	t	t	NOUN
cana-3972	405	23	-	-	PUNCT
cana-3972	405	24	cosingular	cosingular	ADJ
cana-3972	405	25	.	.	PUNCT
cana-3972	406	1	since	since	SCONJ
cana-3972	406	2	𝑁/𝑁1	𝑁/𝑁1	PROPN
cana-3972	406	3	≅	≅	PROPN
cana-3972	406	4	𝑁2	𝑁2	NOUN
cana-3972	406	5	and	and	CCONJ
cana-3972	406	6	n2	n2	PROPN
cana-3972	406	7	is	be	AUX
cana-3972	406	8	t	t	NOUN
cana-3972	406	9	-	-	PUNCT
cana-3972	406	10	cosingular	cosingular	ADJ
cana-3972	406	11	,	,	PUNCT
cana-3972	406	12	n	n	CCONJ
cana-3972	406	13	/	/	SYM
cana-3972	406	14	n1	n1	NOUN
cana-3972	406	15	is	be	AUX
cana-3972	406	16	also	also	ADV
cana-3972	406	17	t	t	NOUN
cana-3972	406	18	-	-	PUNCT
cana-3972	406	19	cosingular	cosingular	ADJ
cana-3972	406	20	.	.	PUNCT
cana-3972	407	1	hence	hence	ADV
cana-3972	407	2	,	,	PUNCT
cana-3972	407	3	m	m	VERB
cana-3972	407	4	satisfies	satisfy	VERB
cana-3972	407	5	t	t	PROPN
cana-3972	407	6	-	-	PUNCT
cana-3972	407	7	c∗	c∗	PROPN
cana-3972	407	8	.	.	PUNCT
cana-3972	408	1	proposition	proposition	NOUN
cana-3972	408	2	3.33	3.33	NUM
cana-3972	408	3	.	.	PUNCT
cana-3972	409	1	the	the	DET
cana-3972	409	2	following	follow	VERB
cana-3972	409	3	statements	statement	NOUN
cana-3972	409	4	are	be	AUX
cana-3972	409	5	equivalent	equivalent	ADJ
cana-3972	409	6	for	for	ADP
cana-3972	409	7	a	a	DET
cana-3972	409	8	ring	ring	NOUN
cana-3972	409	9	r	r	NOUN
cana-3972	409	10	:	:	PUNCT
cana-3972	409	11	1	1	NUM
cana-3972	409	12	)	)	PUNCT
cana-3972	409	13	every	every	DET
cana-3972	409	14	right	right	ADJ
cana-3972	409	15	r	r	NOUN
cana-3972	409	16	-	-	PUNCT
cana-3972	409	17	module	module	NOUN
cana-3972	409	18	satisfies	satisfie	NOUN
cana-3972	409	19	t	t	NOUN
cana-3972	409	20	-	-	PUNCT
cana-3972	409	21	c∗.	c∗.	NOUN
cana-3972	409	22	2	2	NUM
cana-3972	409	23	)	)	PUNCT
cana-3972	409	24	every	every	DET
cana-3972	409	25	injective	injective	ADJ
cana-3972	409	26	right	right	ADJ
cana-3972	409	27	r	r	NOUN
cana-3972	409	28	-	-	PUNCT
cana-3972	409	29	module	module	NOUN
cana-3972	409	30	satisfies	satisfie	NOUN
cana-3972	409	31	t	t	NOUN
cana-3972	409	32	-	-	PUNCT
cana-3972	409	33	c∗.	c∗.	NOUN
cana-3972	409	34	3	3	NUM
cana-3972	409	35	)	)	PUNCT
cana-3972	410	1	every	every	DET
cana-3972	410	2	right	right	ADJ
cana-3972	410	3	r	r	NOUN
cana-3972	410	4	-	-	PUNCT
cana-3972	410	5	module	module	NOUN
cana-3972	410	6	can	can	AUX
cana-3972	410	7	be	be	AUX
cana-3972	410	8	expressed	express	VERB
cana-3972	410	9	as	as	ADP
cana-3972	410	10	a	a	DET
cana-3972	410	11	direct	direct	ADJ
cana-3972	410	12	sum	sum	NOUN
cana-3972	410	13	of	of	ADP
cana-3972	410	14	an	an	DET
cana-3972	410	15	injective	injective	ADJ
cana-3972	410	16	module	module	NOUN
cana-3972	410	17	and	and	CCONJ
cana-3972	410	18	a	a	DET
cana-3972	410	19	t	t	NOUN
cana-3972	410	20	-	-	PUNCT
cana-3972	410	21	cosingular	cosingular	ADJ
cana-3972	410	22	module	module	NOUN
cana-3972	410	23	.	.	PUNCT
cana-3972	411	1	proof	proof	NOUN
cana-3972	411	2	:	:	PUNCT
cana-3972	411	3	1	1	X
cana-3972	411	4	)	)	PUNCT
cana-3972	411	5	⇔	⇔	X
cana-3972	411	6	2	2	NUM
cana-3972	411	7	)	)	PUNCT
cana-3972	411	8	this	this	DET
cana-3972	411	9	equivalence	equivalence	NOUN
cana-3972	411	10	is	be	AUX
cana-3972	411	11	evident	evident	ADJ
cana-3972	411	12	,	,	PUNCT
cana-3972	411	13	as	as	SCONJ
cana-3972	411	14	any	any	DET
cana-3972	411	15	submodule	submodule	NOUN
cana-3972	411	16	of	of	ADP
cana-3972	411	17	a	a	DET
cana-3972	411	18	module	module	NOUN
cana-3972	411	19	that	that	PRON
cana-3972	411	20	satisfies	satisfy	VERB
cana-3972	411	21	t	t	PROPN
cana-3972	411	22	-	-	PUNCT
cana-3972	411	23	c∗	c∗	PROPN
cana-3972	411	24	also	also	ADV
cana-3972	411	25	satisfies	satisfy	VERB
cana-3972	411	26	tc∗	tc∗	NOUN
cana-3972	411	27	.	.	PUNCT
cana-3972	412	1	2	2	X
cana-3972	412	2	)	)	PUNCT
cana-3972	412	3	⇒	⇒	NOUN
cana-3972	412	4	3	3	NUM
cana-3972	412	5	)	)	PUNCT
cana-3972	412	6	let	let	VERB
cana-3972	412	7	m	m	PRON
cana-3972	412	8	be	be	AUX
cana-3972	412	9	an	an	DET
cana-3972	412	10	injective	injective	ADJ
cana-3972	412	11	module	module	NOUN
cana-3972	412	12	that	that	PRON
cana-3972	412	13	satisfies	satisfy	VERB
cana-3972	412	14	t	t	NOUN
cana-3972	412	15	-	-	PUNCT
cana-3972	412	16	c∗.	c∗.	NOUN
cana-3972	412	17	by	by	ADP
cana-3972	412	18	proposition	proposition	NOUN
cana-3972	412	19	3.32	3.32	NUM
cana-3972	412	20	,	,	PUNCT
cana-3972	412	21	every	every	DET
cana-3972	412	22	submodule	submodule	NOUN
cana-3972	412	23	of	of	ADP
cana-3972	412	24	m	m	PROPN
cana-3972	412	25	can	can	AUX
cana-3972	412	26	be	be	AUX
cana-3972	412	27	expressed	express	VERB
cana-3972	412	28	as	as	ADP
cana-3972	412	29	a	a	DET
cana-3972	412	30	direct	direct	ADJ
cana-3972	412	31	sum	sum	NOUN
cana-3972	412	32	of	of	ADP
cana-3972	412	33	an	an	DET
cana-3972	412	34	injective	injective	ADJ
cana-3972	412	35	module	module	NOUN
cana-3972	412	36	and	and	CCONJ
cana-3972	412	37	a	a	DET
cana-3972	412	38	t	t	NOUN
cana-3972	412	39	-	-	PUNCT
cana-3972	412	40	cosingular	cosingular	ADJ
cana-3972	412	41	module	module	NOUN
cana-3972	412	42	.	.	PUNCT
cana-3972	413	1	3	3	X
cana-3972	413	2	)	)	PUNCT
cana-3972	413	3	⇒	⇒	NOUN
cana-3972	413	4	1	1	NUM
cana-3972	413	5	)	)	PUNCT
cana-3972	413	6	if	if	SCONJ
cana-3972	413	7	every	every	DET
cana-3972	413	8	submodule	submodule	NOUN
cana-3972	413	9	of	of	ADP
cana-3972	413	10	m	m	PROPN
cana-3972	413	11	can	can	AUX
cana-3972	413	12	be	be	AUX
cana-3972	413	13	expressed	express	VERB
cana-3972	413	14	as	as	ADP
cana-3972	413	15	a	a	DET
cana-3972	413	16	direct	direct	ADJ
cana-3972	413	17	sum	sum	NOUN
cana-3972	413	18	of	of	ADP
cana-3972	413	19	an	an	DET
cana-3972	413	20	injective	injective	ADJ
cana-3972	413	21	module	module	NOUN
cana-3972	413	22	and	and	CCONJ
cana-3972	413	23	a	a	DET
cana-3972	413	24	tcosingular	tcosingular	NOUN
cana-3972	413	25	module	module	NOUN
cana-3972	413	26	,	,	PUNCT
cana-3972	413	27	then	then	ADV
cana-3972	413	28	m	m	VERB
cana-3972	413	29	satisfies	satisfie	NOUN
cana-3972	413	30	t	t	PROPN
cana-3972	413	31	-	-	PUNCT
cana-3972	413	32	c∗.	c∗.	NOUN
cana-3972	413	33	this	this	PRON
cana-3972	413	34	follows	follow	VERB
cana-3972	413	35	from	from	ADP
cana-3972	413	36	the	the	DET
cana-3972	413	37	fact	fact	NOUN
cana-3972	413	38	that	that	SCONJ
cana-3972	413	39	injective	injective	ADJ
cana-3972	413	40	submodules	submodule	NOUN
cana-3972	413	41	are	be	AUX
cana-3972	413	42	direct	direct	ADJ
cana-3972	413	43	summands	summand	NOUN
cana-3972	413	44	,	,	PUNCT
cana-3972	413	45	as	as	SCONJ
cana-3972	413	46	established	establish	VERB
cana-3972	413	47	in	in	ADP
cana-3972	413	48	proposition	proposition	NOUN
cana-3972	413	49	3.32	3.32	NUM
cana-3972	413	50	.	.	PUNCT
cana-3972	414	1	proposition	proposition	NOUN
cana-3972	414	2	3.34	3.34	NUM
cana-3972	414	3	.	.	PUNCT
cana-3972	415	1	let	let	AUX
cana-3972	415	2	m	m	NOUN
cana-3972	415	3	=	=	SYM
cana-3972	415	4	m1	m1	PROPN
cana-3972	415	5	⊕	⊕	PROPN
cana-3972	415	6	m2	m2	PROPN
cana-3972	415	7	,	,	PUNCT
cana-3972	415	8	where	where	SCONJ
cana-3972	415	9	m1	m1	PROPN
cana-3972	415	10	is	be	AUX
cana-3972	415	11	semisimple	semisimple	NOUN
cana-3972	415	12	and	and	CCONJ
cana-3972	415	13	m2	m2	PROPN
cana-3972	415	14	satisfies	satisfie	NOUN
cana-3972	415	15	t	t	PROPN
cana-3972	415	16	-	-	PUNCT
cana-3972	415	17	c∗.	c∗.	NOUN
cana-3972	415	18	then	then	ADV
cana-3972	415	19	m	m	VERB
cana-3972	415	20	also	also	ADV
cana-3972	415	21	satisfies	satisfy	VERB
cana-3972	415	22	t	t	NOUN
cana-3972	415	23	-	-	PUNCT
cana-3972	415	24	c∗.	c∗.	NOUN
cana-3972	415	25	proof	proof	NOUN
cana-3972	415	26	:	:	PUNCT
cana-3972	415	27	let	let	VERB
cana-3972	415	28	m	m	NOUN
cana-3972	415	29	=	=	SYM
cana-3972	415	30	m1	m1	PROPN
cana-3972	415	31	⊕	⊕	PROPN
cana-3972	415	32	m2	m2	PROPN
cana-3972	415	33	,	,	PUNCT
cana-3972	415	34	where	where	SCONJ
cana-3972	415	35	m1	m1	PROPN
cana-3972	415	36	is	be	AUX
cana-3972	415	37	semisimple	semisimple	NOUN
cana-3972	415	38	and	and	CCONJ
cana-3972	415	39	m2	m2	PROPN
cana-3972	415	40	satisfies	satisfie	NOUN
cana-3972	415	41	t	t	PROPN
cana-3972	415	42	-	-	PUNCT
cana-3972	415	43	c∗	c∗	PROPN
cana-3972	415	44	.	.	PUNCT
cana-3972	416	1	let	let	VERB
cana-3972	416	2	n	n	PRON
cana-3972	416	3	≤	≤	NOUN
cana-3972	416	4	m	m	VERB
cana-3972	416	5	.	.	PUNCT
cana-3972	417	1	then	then	ADV
cana-3972	417	2	m1	m1	PROPN
cana-3972	417	3	can	can	AUX
cana-3972	417	4	be	be	AUX
cana-3972	417	5	decomposed	decompose	VERB
cana-3972	417	6	as	as	ADP
cana-3972	417	7	m1	m1	PROPN
cana-3972	417	8	=	=	SYM
cana-3972	417	9	(	(	PUNCT
cana-3972	417	10	n	n	CCONJ
cana-3972	417	11	∩	∩	ADJ
cana-3972	417	12	m1	m1	NOUN
cana-3972	417	13	)	)	PUNCT
cana-3972	418	1	⊕	⊕	PROPN
cana-3972	418	2	m	m	VERB
cana-3972	418	3	′	′	NUM
cana-3972	418	4	for	for	ADP
cana-3972	418	5	some	some	DET
cana-3972	418	6	submodule	submodule	NOUN
cana-3972	418	7	m	m	PROPN
cana-3972	418	8	′	′	ADJ
cana-3972	418	9	≤	≤	NUM
cana-3972	418	10	m1	m1	NOUN
cana-3972	418	11	.	.	PUNCT
cana-3972	419	1	consequently	consequently	ADV
cana-3972	419	2	,	,	PUNCT
cana-3972	419	3	m	m	VERB
cana-3972	419	4	=	=	SYM
cana-3972	419	5	(	(	PUNCT
cana-3972	419	6	n	n	CCONJ
cana-3972	419	7	∩	∩	ADJ
cana-3972	419	8	m1	m1	NOUN
cana-3972	419	9	)	)	PUNCT
cana-3972	419	10	⊕	⊕	PROPN
cana-3972	419	11	m	m	VERB
cana-3972	419	12	′	′	NUM
cana-3972	419	13	⊕	⊕	PROPN
cana-3972	419	14	m2	m2	PROPN
cana-3972	419	15	and	and	CCONJ
cana-3972	419	16	n	n	NOUN
cana-3972	419	17	=	=	SYM
cana-3972	419	18	(	(	PUNCT
cana-3972	419	19	n	n	CCONJ
cana-3972	419	20	∩	∩	ADJ
cana-3972	419	21	m1	m1	NOUN
cana-3972	419	22	)	)	PUNCT
cana-3972	419	23	⊕	⊕	PROPN
cana-3972	419	24	a	a	X
cana-3972	419	25	,	,	PUNCT
cana-3972	419	26	where	where	SCONJ
cana-3972	419	27	a	a	DET
cana-3972	419	28	=	=	SYM
cana-3972	419	29	n	n	NOUN
cana-3972	419	30	∩	∩	NOUN
cana-3972	419	31	(	(	PUNCT
cana-3972	419	32	m	m	VERB
cana-3972	419	33	′	′	NUM
cana-3972	419	34	⊕	⊕	PROPN
cana-3972	419	35	m2	m2	PROPN
cana-3972	419	36	)	)	PUNCT
cana-3972	419	37	.	.	PUNCT
cana-3972	420	1	communications	communication	NOUN
cana-3972	420	2	on	on	ADP
cana-3972	420	3	applied	apply	VERB
cana-3972	420	4	nonlinear	nonlinear	ADJ
cana-3972	420	5	analysis	analysis	NOUN
cana-3972	420	6	issn	issn	NOUN
cana-3972	420	7	:	:	PUNCT
cana-3972	420	8	1074	1074	NUM
cana-3972	420	9	-	-	PUNCT
cana-3972	420	10	133x	133x	NUM
cana-3972	420	11	vol	vol	NOUN
cana-3972	420	12	32	32	NUM
cana-3972	420	13	no	no	NOUN
cana-3972	420	14	.	.	PUNCT
cana-3972	421	1	9s	9s	NUM
cana-3972	421	2	(	(	PUNCT
cana-3972	421	3	2025	2025	NUM
cana-3972	421	4	)	)	PUNCT
cana-3972	421	5	686	686	NUM
cana-3972	421	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	421	7	since	since	SCONJ
cana-3972	421	8	(	(	PUNCT
cana-3972	421	9	m2	m2	PROPN
cana-3972	421	10	⊕	⊕	PROPN
cana-3972	421	11	m	m	VERB
cana-3972	421	12	′	′	NUM
cana-3972	421	13	)	)	PUNCT
cana-3972	422	1	/m	/m	PUNCT
cana-3972	423	1	′	′	NUM
cana-3972	423	2	satisfies	satisfie	NOUN
cana-3972	423	3	t	t	PROPN
cana-3972	423	4	-	-	PUNCT
cana-3972	423	5	c∗	c∗	PROPN
cana-3972	423	6	,	,	PUNCT
cana-3972	423	7	i	i	PRON
cana-3972	423	8	t	t	PROPN
cana-3972	423	9	follows	follow	VERB
cana-3972	423	10	that	that	SCONJ
cana-3972	423	11	(	(	PUNCT
cana-3972	423	12	a	a	DET
cana-3972	423	13	+	+	NOUN
cana-3972	423	14	m	m	VERB
cana-3972	423	15	′)/m	′)/m	NOUN
cana-3972	423	16	′	′	NOUN
cana-3972	424	1	=	=	PUNCT
cana-3972	424	2	k	k	X
cana-3972	424	3	/	/	SYM
cana-3972	424	4	m	m	VERB
cana-3972	424	5	′	′	NUM
cana-3972	424	6	⊕	⊕	PROPN
cana-3972	424	7	l	l	PROPN
cana-3972	424	8	/	/	SYM
cana-3972	424	9	m	m	AUX
cana-3972	424	10	′	′	NOUN
cana-3972	424	11	for	for	ADP
cana-3972	424	12	some	some	DET
cana-3972	424	13	submodules	submodule	NOUN
cana-3972	425	1	k	k	PROPN
cana-3972	425	2	and	and	CCONJ
cana-3972	425	3	l	l	NOUN
cana-3972	425	4	containing	contain	VERB
cana-3972	425	5	m	m	NOUN
cana-3972	425	6	′	′	NOUN
cana-3972	425	7	,	,	PUNCT
cana-3972	425	8	where	where	SCONJ
cana-3972	425	9	k	k	X
cana-3972	425	10	/	/	SYM
cana-3972	425	11	m	m	AUX
cana-3972	425	12	′	′	NUM
cana-3972	425	13	is	be	AUX
cana-3972	425	14	a	a	DET
cana-3972	425	15	direct	direct	ADJ
cana-3972	425	16	summand	summand	NOUN
cana-3972	425	17	of	of	ADP
cana-3972	425	18	(	(	PUNCT
cana-3972	425	19	m2	m2	PROPN
cana-3972	425	20	⊕	⊕	PROPN
cana-3972	425	21	m	m	VERB
cana-3972	425	22	′	′	NUM
cana-3972	425	23	)	)	PUNCT
cana-3972	425	24	/m	/m	PUNCT
cana-3972	426	1	′	′	NUM
cana-3972	426	2	and	and	CCONJ
cana-3972	426	3	l	l	NOUN
cana-3972	426	4	/	/	SYM
cana-3972	426	5	m	m	VERB
cana-3972	426	6	′	′	NOUN
cana-3972	426	7	is	be	AUX
cana-3972	426	8	t	t	NOUN
cana-3972	426	9	-	-	PUNCT
cana-3972	426	10	cosingular	cosingular	ADJ
cana-3972	426	11	.	.	PUNCT
cana-3972	427	1	thus	thus	ADV
cana-3972	427	2	,	,	PUNCT
cana-3972	427	3	k	k	PROPN
cana-3972	427	4	is	be	AUX
cana-3972	427	5	a	a	DET
cana-3972	427	6	direct	direct	ADJ
cana-3972	427	7	summand	summand	NOUN
cana-3972	427	8	of	of	ADP
cana-3972	427	9	m	m	PROPN
cana-3972	427	10	.	.	PUNCT
cana-3972	428	1	moreover	moreover	ADV
cana-3972	428	2	,	,	PUNCT
cana-3972	428	3	since	since	SCONJ
cana-3972	428	4	k	k	PROPN
cana-3972	428	5	=	=	NOUN
cana-3972	428	6	m	m	VERB
cana-3972	428	7	′	′	NUM
cana-3972	428	8	⊕	⊕	PROPN
cana-3972	428	9	(	(	PUNCT
cana-3972	428	10	k	k	X
cana-3972	428	11	∩	∩	PROPN
cana-3972	428	12	a	a	X
cana-3972	428	13	)	)	PUNCT
cana-3972	428	14	,	,	PUNCT
cana-3972	428	15	k	k	PROPN
cana-3972	428	16	∩	∩	PROPN
cana-3972	428	17	a	a	PRON
cana-3972	428	18	is	be	AUX
cana-3972	428	19	also	also	ADV
cana-3972	428	20	a	a	DET
cana-3972	428	21	direct	direct	ADJ
cana-3972	428	22	summand	summand	NOUN
cana-3972	428	23	of	of	ADP
cana-3972	428	24	m	m	PROPN
cana-3972	428	25	.	.	PUNCT
cana-3972	429	1	therefore	therefore	ADV
cana-3972	429	2	,	,	PUNCT
cana-3972	429	3	(	(	PUNCT
cana-3972	429	4	n	n	CCONJ
cana-3972	429	5	∩	∩	NOUN
cana-3972	429	6	m	m	PROPN
cana-3972	429	7	⊕	⊕	PROPN
cana-3972	429	8	(	(	PUNCT
cana-3972	429	9	k	k	X
cana-3972	429	10	∩	∩	PROPN
cana-3972	429	11	a	a	X
cana-3972	429	12	)	)	PUNCT
cana-3972	429	13	is	be	AUX
cana-3972	429	14	a	a	DET
cana-3972	429	15	direct	direct	ADJ
cana-3972	429	16	summand	summand	NOUN
cana-3972	429	17	of	of	ADP
cana-3972	429	18	m	m	PROPN
cana-3972	429	19	.	.	PUNCT
cana-3972	430	1	finally	finally	ADV
cana-3972	430	2	,	,	PUNCT
cana-3972	430	3	n	n	PROPN
cana-3972	430	4	/	/	SYM
cana-3972	430	5	(	(	PUNCT
cana-3972	430	6	(	(	PUNCT
cana-3972	430	7	n	n	CCONJ
cana-3972	430	8	∩	∩	ADJ
cana-3972	430	9	m1	m1	NOUN
cana-3972	430	10	)	)	PUNCT
cana-3972	430	11	⊕	⊕	PROPN
cana-3972	430	12	(	(	PUNCT
cana-3972	430	13	k	k	X
cana-3972	430	14	∩	∩	PROPN
cana-3972	430	15	a))≅	a))≅	PROPN
cana-3972	430	16	a/(k	a/(k	PROPN
cana-3972	430	17	∩	∩	NOUN
cana-3972	430	18	a	a	PRON
cana-3972	430	19	)	)	PUNCT
cana-3972	430	20	≅	≅	NOUN
cana-3972	430	21	(	(	PUNCT
cana-3972	430	22	a	a	DET
cana-3972	430	23	+	+	X
cana-3972	430	24	k	k	NOUN
cana-3972	430	25	)	)	PUNCT
cana-3972	430	26	/k	/k	PUNCT
cana-3972	431	1	≅	≅	PROPN
cana-3972	431	2	(	(	PUNCT
cana-3972	431	3	a	a	DET
cana-3972	431	4	+	+	NOUN
cana-3972	431	5	m	m	NOUN
cana-3972	431	6	′	′	NUM
cana-3972	431	7	)	)	PUNCT
cana-3972	431	8	/k≅	/k≅	X
cana-3972	432	1	l	l	NOUN
cana-3972	432	2	/	/	SYM
cana-3972	432	3	m	m	AUX
cana-3972	432	4	′	′	NUM
cana-3972	432	5	is	be	AUX
cana-3972	432	6	tcosingular	tcosingular	NOUN
cana-3972	432	7	.	.	PUNCT
cana-3972	433	1	hence	hence	ADV
cana-3972	433	2	,	,	PUNCT
cana-3972	433	3	m	m	VERB
cana-3972	433	4	satisfies	satisfy	VERB
cana-3972	433	5	t	t	PROPN
cana-3972	433	6	-	-	PUNCT
cana-3972	433	7	c∗.	c∗.	NOUN
cana-3972	433	8	proposition	proposition	NOUN
cana-3972	433	9	3.35	3.35	NUM
cana-3972	433	10	.	.	PUNCT
cana-3972	434	1	every	every	DET
cana-3972	434	2	t	t	NOUN
cana-3972	434	3	-	-	PUNCT
cana-3972	434	4	lifting	lift	VERB
cana-3972	434	5	module	module	NOUN
cana-3972	434	6	satisfies	satisfy	VERB
cana-3972	434	7	the	the	DET
cana-3972	434	8	t	t	PROPN
cana-3972	434	9	-	-	PUNCT
cana-3972	434	10	c∗	c∗	NOUN
cana-3972	434	11	property	property	NOUN
cana-3972	434	12	.	.	PUNCT
cana-3972	435	1	proof	proof	NOUN
cana-3972	435	2	it	it	PRON
cana-3972	435	3	’s	’	VERB
cana-3972	435	4	obvious	obvious	ADJ
cana-3972	435	5	.	.	PUNCT
cana-3972	436	1	4	4	X
cana-3972	436	2	.	.	X
cana-3972	436	3	rings	ring	NOUN
cana-3972	436	4	over	over	ADP
cana-3972	436	5	which	which	PRON
cana-3972	436	6	certain	certain	ADJ
cana-3972	436	7	modules	module	NOUN
cana-3972	436	8	have	have	VERB
cana-3972	436	9	d41	d41	NOUN
cana-3972	436	10	in	in	ADP
cana-3972	436	11	this	this	DET
cana-3972	436	12	section	section	NOUN
cana-3972	436	13	,	,	PUNCT
cana-3972	436	14	we	we	PRON
cana-3972	436	15	provide	provide	VERB
cana-3972	436	16	a	a	DET
cana-3972	436	17	characterization	characterization	NOUN
cana-3972	436	18	of	of	ADP
cana-3972	436	19	certain	certain	ADJ
cana-3972	436	20	classes	class	NOUN
cana-3972	436	21	of	of	ADP
cana-3972	436	22	rings	ring	NOUN
cana-3972	436	23	in	in	ADP
cana-3972	436	24	relation	relation	NOUN
cana-3972	436	25	to	to	ADP
cana-3972	436	26	d41	d41	NOUN
cana-3972	436	27	-	-	PUNCT
cana-3972	436	28	modules	module	NOUN
cana-3972	436	29	.	.	PUNCT
cana-3972	437	1	we	we	PRON
cana-3972	437	2	start	start	VERB
cana-3972	437	3	with	with	ADP
cana-3972	437	4	a	a	DET
cana-3972	437	5	characterization	characterization	NOUN
cana-3972	437	6	of	of	ADP
cana-3972	437	7	the	the	DET
cana-3972	437	8	class	class	NOUN
cana-3972	437	9	of	of	ADP
cana-3972	437	10	rings	ring	NOUN
cana-3972	437	11	r	r	NOUN
cana-3972	437	12	for	for	ADP
cana-3972	437	13	which	which	PRON
cana-3972	437	14	every	every	DET
cana-3972	437	15	d41	d41	NOUN
cana-3972	437	16	-	-	PUNCT
cana-3972	437	17	module	module	NOUN
cana-3972	437	18	is	be	AUX
cana-3972	437	19	also	also	ADV
cana-3972	437	20	a	a	DET
cana-3972	437	21	d4	d4	NOUN
cana-3972	437	22	-	-	PUNCT
cana-3972	437	23	module	module	NOUN
cana-3972	437	24	.	.	PUNCT
cana-3972	438	1	recall	recall	VERB
cana-3972	438	2	that	that	SCONJ
cana-3972	438	3	a	a	DET
cana-3972	438	4	ring	ring	NOUN
cana-3972	438	5	is	be	AUX
cana-3972	438	6	cosingular	cosingular	ADJ
cana-3972	438	7	projective	projective	NOUN
cana-3972	438	8	(	(	PUNCT
cana-3972	438	9	cosp	cosp	PROPN
cana-3972	438	10	)	)	PUNCT
cana-3972	438	11	if	if	SCONJ
cana-3972	438	12	every	every	DET
cana-3972	438	13	cosingular	cosingular	ADJ
cana-3972	438	14	module	module	NOUN
cana-3972	438	15	over	over	ADP
cana-3972	438	16	r	r	NOUN
cana-3972	438	17	is	be	AUX
cana-3972	438	18	projective	projective	ADJ
cana-3972	438	19	(	(	PUNCT
cana-3972	438	20	see	see	VERB
cana-3972	438	21	[	[	X
cana-3972	438	22	18	18	NUM
cana-3972	438	23	,	,	PUNCT
cana-3972	438	24	3	3	NUM
cana-3972	438	25	]	]	NUM
cana-3972	438	26	)	)	PUNCT
cana-3972	438	27	.	.	PUNCT
cana-3972	439	1	theorem	theorem	VERB
cana-3972	439	2	4.1	4.1	NUM
cana-3972	439	3	.	.	PUNCT
cana-3972	440	1	the	the	DET
cana-3972	440	2	following	follow	VERB
cana-3972	440	3	conditions	condition	NOUN
cana-3972	440	4	are	be	AUX
cana-3972	440	5	equivalent	equivalent	ADJ
cana-3972	440	6	for	for	ADP
cana-3972	440	7	a	a	DET
cana-3972	440	8	semisimple	semisimple	NOUN
cana-3972	440	9	ring	ring	NOUN
cana-3972	440	10	r	r	NOUN
cana-3972	440	11	:	:	PUNCT
cana-3972	440	12	1	1	NUM
cana-3972	440	13	)	)	PUNCT
cana-3972	440	14	r	r	NOUN
cana-3972	440	15	is	be	AUX
cana-3972	440	16	a	a	DET
cana-3972	440	17	left	left	ADJ
cana-3972	440	18	cosingular	cosingular	ADJ
cana-3972	440	19	projective	projective	NOUN
cana-3972	440	20	-	-	PUNCT
cana-3972	440	21	ring	ring	NOUN
cana-3972	440	22	;	;	PUNCT
cana-3972	440	23	2	2	X
cana-3972	440	24	)	)	PUNCT
cana-3972	440	25	every	every	DET
cana-3972	440	26	d41	d41	NOUN
cana-3972	440	27	-	-	PUNCT
cana-3972	440	28	r	r	NOUN
cana-3972	440	29	-	-	PUNCT
cana-3972	440	30	module	module	NOUN
cana-3972	440	31	is	be	AUX
cana-3972	440	32	a	a	DET
cana-3972	440	33	d4	d4	NOUN
cana-3972	440	34	-	-	PUNCT
cana-3972	440	35	module	module	NOUN
cana-3972	440	36	;	;	PUNCT
cana-3972	440	37	proof	proof	NOUN
cana-3972	440	38	:	:	PUNCT
cana-3972	440	39	1	1	X
cana-3972	440	40	)	)	PUNCT
cana-3972	440	41	⇒	⇒	NOUN
cana-3972	440	42	2	2	NUM
cana-3972	440	43	)	)	PUNCT
cana-3972	440	44	it	it	PRON
cana-3972	440	45	’s	’	VERB
cana-3972	440	46	obvious	obvious	ADJ
cana-3972	440	47	.	.	PUNCT
cana-3972	441	1	2	2	X
cana-3972	441	2	)	)	PUNCT
cana-3972	441	3	⇒	⇒	NOUN
cana-3972	441	4	1	1	NUM
cana-3972	441	5	)	)	PUNCT
cana-3972	441	6	since	since	SCONJ
cana-3972	441	7	r	r	NOUN
cana-3972	441	8	is	be	AUX
cana-3972	441	9	a	a	DET
cana-3972	441	10	semisimple	semisimple	NOUN
cana-3972	441	11	ring	ring	NOUN
cana-3972	441	12	then	then	ADV
cana-3972	441	13	every	every	DET
cana-3972	441	14	module	module	NOUN
cana-3972	441	15	of	of	ADP
cana-3972	441	16	r	r	NOUN
cana-3972	441	17	is	be	AUX
cana-3972	441	18	projective	projective	ADJ
cana-3972	441	19	.	.	PUNCT
cana-3972	442	1	in	in	ADP
cana-3972	442	2	particular	particular	ADJ
cana-3972	442	3	every	every	DET
cana-3972	442	4	cosingular	cosingular	ADJ
cana-3972	442	5	module	module	NOUN
cana-3972	442	6	is	be	AUX
cana-3972	442	7	projective	projective	ADJ
cana-3972	442	8	.	.	PUNCT
cana-3972	443	1	thus	thus	ADV
cana-3972	443	2	,	,	PUNCT
cana-3972	443	3	r	r	NOUN
cana-3972	443	4	is	be	AUX
cana-3972	443	5	a	a	DET
cana-3972	443	6	cosingular	cosingular	ADJ
cana-3972	443	7	projective	projective	NOUN
cana-3972	443	8	-	-	PUNCT
cana-3972	443	9	ring	ring	NOUN
cana-3972	443	10	.	.	PUNCT
cana-3972	444	1	remark	remark	PROPN
cana-3972	444	2	4.2	4.2	NUM
cana-3972	444	3	.	.	PUNCT
cana-3972	445	1	let	let	VERB
cana-3972	445	2	r	r	PRON
cana-3972	445	3	be	be	AUX
cana-3972	445	4	a	a	DET
cana-3972	445	5	ring	ring	NOUN
cana-3972	445	6	which	which	PRON
cana-3972	445	7	is	be	AUX
cana-3972	445	8	not	not	PART
cana-3972	445	9	a	a	DET
cana-3972	445	10	left	left	ADJ
cana-3972	445	11	cosingular	cosingular	ADJ
cana-3972	445	12	projective	projective	NOUN
cana-3972	445	13	-	-	PUNCT
cana-3972	445	14	ring	ring	NOUN
cana-3972	445	15	.	.	PUNCT
cana-3972	446	1	from	from	ADP
cana-3972	446	2	theorem	theorem	ADJ
cana-3972	446	3	4.1	4.1	NUM
cana-3972	446	4	,	,	PUNCT
cana-3972	446	5	it	it	PRON
cana-3972	446	6	follows	follow	VERB
cana-3972	446	7	that	that	SCONJ
cana-3972	446	8	r	r	NOUN
cana-3972	446	9	has	have	VERB
cana-3972	446	10	a	a	DET
cana-3972	446	11	d41	d41	NOUN
cana-3972	446	12	-	-	PUNCT
cana-3972	446	13	module	module	NOUN
cana-3972	446	14	that	that	PRON
cana-3972	446	15	is	be	AUX
cana-3972	446	16	not	not	PART
cana-3972	446	17	a	a	DET
cana-3972	446	18	cosingular	cosingular	ADJ
cana-3972	446	19	projective	projective	NOUN
cana-3972	446	20	-	-	PUNCT
cana-3972	446	21	module	module	NOUN
cana-3972	446	22	.	.	PUNCT
cana-3972	447	1	theorem	theorem	VERB
cana-3972	447	2	4.3	4.3	NUM
cana-3972	447	3	.	.	PUNCT
cana-3972	448	1	the	the	DET
cana-3972	448	2	following	follow	VERB
cana-3972	448	3	statements	statement	NOUN
cana-3972	448	4	are	be	AUX
cana-3972	448	5	equivalent	equivalent	ADJ
cana-3972	448	6	:	:	PUNCT
cana-3972	448	7	1	1	X
cana-3972	448	8	)	)	PUNCT
cana-3972	448	9	r	r	NOUN
cana-3972	448	10	is	be	AUX
cana-3972	448	11	a	a	DET
cana-3972	448	12	left	left	ADJ
cana-3972	448	13	cosingular	cosingular	ADJ
cana-3972	448	14	semisimple	semisimple	NOUN
cana-3972	448	15	ring	ring	NOUN
cana-3972	448	16	.	.	PUNCT
cana-3972	449	1	2	2	NUM
cana-3972	449	2	)	)	PUNCT
cana-3972	449	3	every	every	DET
cana-3972	449	4	left	left	ADJ
cana-3972	449	5	r	r	NOUN
cana-3972	449	6	-	-	PUNCT
cana-3972	449	7	module	module	NOUN
cana-3972	449	8	is	be	AUX
cana-3972	449	9	cosingular	cosingular	ADJ
cana-3972	449	10	projective	projective	NOUN
cana-3972	449	11	.	.	PUNCT
cana-3972	450	1	3	3	X
cana-3972	450	2	)	)	PUNCT
cana-3972	450	3	every	every	DET
cana-3972	450	4	left	left	ADJ
cana-3972	450	5	r	r	NOUN
cana-3972	450	6	-	-	PUNCT
cana-3972	450	7	module	module	NOUN
cana-3972	450	8	is	be	AUX
cana-3972	450	9	cosingular	cosingular	ADJ
cana-3972	450	10	direct	direct	ADJ
cana-3972	450	11	-	-	PUNCT
cana-3972	450	12	projective	projective	NOUN
cana-3972	450	13	.	.	PUNCT
cana-3972	451	1	4	4	NUM
cana-3972	451	2	)	)	PUNCT
cana-3972	451	3	every	every	DET
cana-3972	451	4	left	left	ADJ
cana-3972	451	5	r	r	NOUN
cana-3972	451	6	-	-	PUNCT
cana-3972	451	7	module	module	NOUN
cana-3972	451	8	is	be	AUX
cana-3972	451	9	a	a	DET
cana-3972	451	10	d41	d41	NOUN
cana-3972	451	11	-	-	PUNCT
cana-3972	451	12	module	module	NOUN
cana-3972	451	13	.	.	PUNCT
cana-3972	452	1	5	5	NUM
cana-3972	452	2	)	)	PUNCT
cana-3972	452	3	every	every	DET
cana-3972	452	4	submodule	submodule	NOUN
cana-3972	452	5	of	of	ADP
cana-3972	452	6	a	a	DET
cana-3972	452	7	d41	d41	NOUN
cana-3972	452	8	-	-	PUNCT
cana-3972	452	9	module	module	NOUN
cana-3972	452	10	module	module	NOUN
cana-3972	452	11	is	be	AUX
cana-3972	452	12	a	a	DET
cana-3972	452	13	d41	d41	NOUN
cana-3972	452	14	-	-	PUNCT
cana-3972	452	15	module	module	NOUN
cana-3972	452	16	.	.	PUNCT
cana-3972	453	1	6	6	NUM
cana-3972	453	2	)	)	PUNCT
cana-3972	453	3	every	every	DET
cana-3972	453	4	direct	direct	ADJ
cana-3972	453	5	sum	sum	NOUN
cana-3972	453	6	of	of	ADP
cana-3972	453	7	d41	d41	NOUN
cana-3972	453	8	-	-	PUNCT
cana-3972	453	9	modules	module	NOUN
cana-3972	453	10	is	be	AUX
cana-3972	453	11	a	a	DET
cana-3972	453	12	d41	d41	NOUN
cana-3972	453	13	-	-	PUNCT
cana-3972	453	14	module	module	NOUN
cana-3972	453	15	.	.	PUNCT
cana-3972	454	1	7	7	X
cana-3972	454	2	)	)	PUNCT
cana-3972	454	3	every	every	DET
cana-3972	454	4	direct	direct	ADJ
cana-3972	454	5	sum	sum	NOUN
cana-3972	454	6	of	of	ADP
cana-3972	454	7	two	two	NUM
cana-3972	454	8	cyclic	cyclic	ADJ
cana-3972	454	9	d41	d41	NOUN
cana-3972	454	10	-	-	PUNCT
cana-3972	454	11	modules	module	NOUN
cana-3972	454	12	is	be	AUX
cana-3972	454	13	a	a	DET
cana-3972	454	14	d41	d41	NOUN
cana-3972	454	15	-	-	PUNCT
cana-3972	454	16	module	module	NOUN
cana-3972	454	17	.	.	PUNCT
cana-3972	455	1	proof	proof	NOUN
cana-3972	455	2	:	:	PUNCT
cana-3972	455	3	the	the	DET
cana-3972	455	4	implications	implication	NOUN
cana-3972	455	5	1	1	NUM
cana-3972	455	6	)	)	PUNCT
cana-3972	455	7	⇒	⇒	NOUN
cana-3972	455	8	2	2	NUM
cana-3972	455	9	)	)	PUNCT
cana-3972	455	10	⇒	⇒	NOUN
cana-3972	455	11	3	3	NUM
cana-3972	455	12	)	)	PUNCT
cana-3972	455	13	⇒	⇒	NOUN
cana-3972	455	14	4	4	NUM
cana-3972	455	15	)	)	PUNCT
cana-3972	455	16	⇒	⇒	NOUN
cana-3972	455	17	5	5	NUM
cana-3972	455	18	)	)	PUNCT
cana-3972	455	19	⇒	⇒	NOUN
cana-3972	455	20	6	6	NUM
cana-3972	455	21	)	)	PUNCT
cana-3972	455	22	⇒	⇒	NOUN
cana-3972	455	23	7	7	NUM
cana-3972	455	24	)	)	PUNCT
cana-3972	455	25	are	be	AUX
cana-3972	455	26	obvious	obvious	ADJ
cana-3972	455	27	.	.	PUNCT
cana-3972	456	1	6	6	NUM
cana-3972	456	2	)	)	PUNCT
cana-3972	456	3	⇒	⇒	NOUN
cana-3972	456	4	1	1	NUM
cana-3972	456	5	)	)	PUNCT
cana-3972	456	6	let	let	VERB
cana-3972	456	7	v	v	PART
cana-3972	456	8	be	be	AUX
cana-3972	456	9	a	a	DET
cana-3972	456	10	simple	simple	ADJ
cana-3972	456	11	r	r	NOUN
cana-3972	456	12	-	-	PUNCT
cana-3972	456	13	module	module	NOUN
cana-3972	456	14	and	and	CCONJ
cana-3972	456	15	𝑓	𝑓	DET
cana-3972	456	16	∶	∶	PROPN
cana-3972	456	17	𝑅	𝑅	PROPN
cana-3972	456	18	→	→	SYM
cana-3972	456	19	𝑉	𝑉	PROPN
cana-3972	456	20	an	an	DET
cana-3972	456	21	r	r	NOUN
cana-3972	456	22	-	-	PUNCT
cana-3972	456	23	epimorphism	epimorphism	NOUN
cana-3972	456	24	.	.	PUNCT
cana-3972	457	1	since	since	SCONJ
cana-3972	457	2	r	r	PROPN
cana-3972	457	3	⊕	⊕	PROPN
cana-3972	457	4	v	v	NOUN
cana-3972	457	5	is	be	AUX
cana-3972	457	6	a	a	DET
cana-3972	457	7	d41module	d41module	PROPN
cana-3972	457	8	,	,	PUNCT
cana-3972	457	9	by	by	ADP
cana-3972	457	10	proposition	proposition	NOUN
cana-3972	457	11	2.4	2.4	NUM
cana-3972	457	12	,	,	PUNCT
cana-3972	457	13	v	v	NOUN
cana-3972	457	14	is	be	AUX
cana-3972	457	15	isomorphic	isomorphic	ADJ
cana-3972	457	16	to	to	ADP
cana-3972	457	17	a	a	DET
cana-3972	457	18	direct	direct	ADJ
cana-3972	457	19	summand	summand	NOUN
cana-3972	457	20	of	of	ADP
cana-3972	457	21	r	r	NOUN
cana-3972	457	22	,	,	PUNCT
cana-3972	457	23	implying	imply	VERB
cana-3972	457	24	that	that	SCONJ
cana-3972	457	25	v	v	NOUN
cana-3972	457	26	is	be	AUX
cana-3972	457	27	projective	projective	ADJ
cana-3972	457	28	.	.	PUNCT
cana-3972	458	1	therefore	therefore	ADV
cana-3972	458	2	,	,	PUNCT
cana-3972	458	3	r	r	NOUN
cana-3972	458	4	is	be	AUX
cana-3972	458	5	semisimple	semisimple	NOUN
cana-3972	458	6	.	.	PUNCT
cana-3972	459	1	communications	communication	NOUN
cana-3972	459	2	on	on	ADP
cana-3972	459	3	applied	apply	VERB
cana-3972	459	4	nonlinear	nonlinear	ADJ
cana-3972	459	5	analysis	analysis	NOUN
cana-3972	459	6	issn	issn	NOUN
cana-3972	459	7	:	:	PUNCT
cana-3972	459	8	1074	1074	NUM
cana-3972	459	9	-	-	PUNCT
cana-3972	459	10	133x	133x	NUM
cana-3972	459	11	vol	vol	NOUN
cana-3972	459	12	32	32	NUM
cana-3972	459	13	no	no	NOUN
cana-3972	459	14	.	.	PUNCT
cana-3972	460	1	9s	9s	NUM
cana-3972	460	2	(	(	PUNCT
cana-3972	460	3	2025	2025	NUM
cana-3972	460	4	)	)	PUNCT
cana-3972	460	5	687	687	NUM
cana-3972	460	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	460	7	a	a	DET
cana-3972	460	8	module	module	NOUN
cana-3972	460	9	m	m	VERB
cana-3972	460	10	is	be	AUX
cana-3972	460	11	termed	term	VERB
cana-3972	460	12	regular	regular	ADJ
cana-3972	460	13	if	if	SCONJ
cana-3972	460	14	every	every	DET
cana-3972	460	15	cyclic	cyclic	ADJ
cana-3972	460	16	submodule	submodule	NOUN
cana-3972	460	17	of	of	ADP
cana-3972	460	18	m	m	PROPN
cana-3972	460	19	is	be	AUX
cana-3972	460	20	a	a	DET
cana-3972	460	21	direct	direct	ADJ
cana-3972	460	22	summand	summand	NOUN
cana-3972	460	23	of	of	ADP
cana-3972	460	24	m	m	PROPN
cana-3972	460	25	.	.	PUNCT
cana-3972	461	1	this	this	PRON
cana-3972	461	2	is	be	AUX
cana-3972	461	3	equivalent	equivalent	ADJ
cana-3972	461	4	to	to	ADP
cana-3972	461	5	stating	state	VERB
cana-3972	461	6	that	that	SCONJ
cana-3972	461	7	every	every	DET
cana-3972	461	8	finitely	finitely	ADV
cana-3972	461	9	generated	generate	VERB
cana-3972	461	10	submodule	submodule	NOUN
cana-3972	461	11	of	of	ADP
cana-3972	461	12	m	m	PROPN
cana-3972	461	13	is	be	AUX
cana-3972	461	14	also	also	ADV
cana-3972	461	15	a	a	DET
cana-3972	461	16	direct	direct	ADJ
cana-3972	461	17	summand	summand	NOUN
cana-3972	461	18	of	of	ADP
cana-3972	461	19	m	m	PROPN
cana-3972	461	20	(	(	PUNCT
cana-3972	461	21	see	see	VERB
cana-3972	461	22	[	[	X
cana-3972	461	23	21	21	NUM
cana-3972	461	24	,	,	PUNCT
cana-3972	461	25	p.	p.	NOUN
cana-3972	461	26	67	67	NUM
cana-3972	461	27	]	]	PUNCT
cana-3972	461	28	)	)	PUNCT
cana-3972	461	29	.	.	PUNCT
cana-3972	462	1	according	accord	VERB
cana-3972	462	2	to	to	ADP
cana-3972	462	3	[	[	X
cana-3972	462	4	11	11	NUM
cana-3972	462	5	]	]	PUNCT
cana-3972	462	6	,	,	PUNCT
cana-3972	462	7	a	a	DET
cana-3972	462	8	module	module	NOUN
cana-3972	462	9	m	m	VERB
cana-3972	462	10	is	be	AUX
cana-3972	462	11	defined	define	VERB
cana-3972	462	12	as	as	ADP
cana-3972	462	13	d	d	NOUN
cana-3972	462	14	-	-	NOUN
cana-3972	462	15	rickart	rickart	NOUN
cana-3972	462	16	if	if	SCONJ
cana-3972	462	17	for	for	ADP
cana-3972	462	18	every	every	DET
cana-3972	462	19	endomorphism	endomorphism	PROPN
cana-3972	462	20	φ	φ	PROPN
cana-3972	462	21	of	of	ADP
cana-3972	462	22	m	m	PROPN
cana-3972	462	23	,	,	PUNCT
cana-3972	462	24	the	the	DET
cana-3972	462	25	image	image	NOUN
cana-3972	462	26	𝐼𝑚(𝜑	𝐼𝑚(𝜑	NOUN
cana-3972	462	27	)	)	PUNCT
cana-3972	462	28	is	be	AUX
cana-3972	462	29	a	a	DET
cana-3972	462	30	direct	direct	ADJ
cana-3972	462	31	summand	summand	NOUN
cana-3972	462	32	of	of	ADP
cana-3972	462	33	m	m	PROPN
cana-3972	462	34	.	.	PUNCT
cana-3972	463	1	next	next	ADV
cana-3972	463	2	,	,	PUNCT
cana-3972	463	3	we	we	PRON
cana-3972	463	4	provide	provide	VERB
cana-3972	463	5	a	a	DET
cana-3972	463	6	characterization	characterization	NOUN
cana-3972	463	7	in	in	ADP
cana-3972	463	8	terms	term	NOUN
cana-3972	463	9	of	of	ADP
cana-3972	463	10	d41	d41	NOUN
cana-3972	463	11	-	-	PUNCT
cana-3972	463	12	modules	module	NOUN
cana-3972	463	13	for	for	ADP
cana-3972	463	14	a	a	DET
cana-3972	463	15	left	left	ADJ
cana-3972	463	16	semi	semi	ADJ
cana-3972	463	17	-	-	ADJ
cana-3972	463	18	hereditary	hereditary	ADJ
cana-3972	463	19	ring	ring	NOUN
cana-3972	463	20	to	to	PART
cana-3972	463	21	be	be	AUX
cana-3972	463	22	von	von	PROPN
cana-3972	463	23	neumann	neumann	PROPN
cana-3972	463	24	regular	regular	PROPN
cana-3972	463	25	.	.	PUNCT
cana-3972	464	1	proposition	proposition	NOUN
cana-3972	464	2	4.4	4.4	NUM
cana-3972	464	3	.	.	PUNCT
cana-3972	465	1	the	the	DET
cana-3972	465	2	fol	fol	NOUN
cana-3972	465	3	lowing	low	VERB
cana-3972	465	4	conditions	condition	NOUN
cana-3972	465	5	are	be	AUX
cana-3972	465	6	equivalent	equivalent	ADJ
cana-3972	465	7	for	for	ADP
cana-3972	465	8	a	a	DET
cana-3972	465	9	left	left	ADJ
cana-3972	465	10	semi	semi	ADJ
cana-3972	465	11	-	-	ADJ
cana-3972	465	12	hereditary	hereditary	ADJ
cana-3972	465	13	ring	ring	NOUN
cana-3972	465	14	r	r	NOUN
cana-3972	465	15	:	:	PUNCT
cana-3972	465	16	1	1	NUM
cana-3972	465	17	)	)	PUNCT
cana-3972	465	18	every	every	DET
cana-3972	465	19	finitely	finitely	ADV
cana-3972	465	20	generated	generate	VERB
cana-3972	465	21	r	r	NOUN
cana-3972	465	22	-	-	PUNCT
cana-3972	465	23	module	module	NOUN
cana-3972	465	24	is	be	AUX
cana-3972	465	25	a	a	DET
cana-3972	465	26	d4	d4	NOUN
cana-3972	465	27	-	-	PUNCT
cana-3972	465	28	module	module	NOUN
cana-3972	465	29	;	;	PUNCT
cana-3972	465	30	2	2	X
cana-3972	465	31	)	)	PUNCT
cana-3972	465	32	every	every	DET
cana-3972	465	33	finitely	finitely	ADV
cana-3972	465	34	generated	generate	VERB
cana-3972	465	35	r	r	NOUN
cana-3972	465	36	-	-	PUNCT
cana-3972	465	37	module	module	NOUN
cana-3972	465	38	is	be	AUX
cana-3972	465	39	a	a	DET
cana-3972	465	40	d41	d41	NOUN
cana-3972	465	41	-	-	PUNCT
cana-3972	465	42	module	module	NOUN
cana-3972	465	43	;	;	PUNCT
cana-3972	465	44	3	3	X
cana-3972	465	45	)	)	PUNCT
cana-3972	465	46	every	every	DET
cana-3972	465	47	finitely	finitely	ADV
cana-3972	465	48	generated	generate	VERB
cana-3972	465	49	projective	projective	ADJ
cana-3972	465	50	r	r	NOUN
cana-3972	465	51	-	-	PUNCT
cana-3972	465	52	module	module	NOUN
cana-3972	465	53	is	be	AUX
cana-3972	465	54	a	a	DET
cana-3972	465	55	d	d	ADJ
cana-3972	465	56	-	-	PUNCT
cana-3972	465	57	rickart	rickart	NOUN
cana-3972	465	58	module	module	NOUN
cana-3972	465	59	;	;	PUNCT
cana-3972	465	60	4	4	X
cana-3972	465	61	)	)	PUNCT
cana-3972	465	62	every	every	DET
cana-3972	465	63	finitely	finitely	ADV
cana-3972	465	64	generated	generate	VERB
cana-3972	465	65	projective	projective	ADJ
cana-3972	465	66	r	r	NOUN
cana-3972	465	67	-	-	PUNCT
cana-3972	465	68	module	module	NOUN
cana-3972	465	69	is	be	AUX
cana-3972	465	70	a	a	DET
cana-3972	465	71	regular	regular	ADJ
cana-3972	465	72	module	module	NOUN
cana-3972	465	73	;	;	PUNCT
cana-3972	465	74	5	5	X
cana-3972	465	75	)	)	PUNCT
cana-3972	465	76	r	r	NOUN
cana-3972	465	77	is	be	AUX
cana-3972	465	78	a	a	DET
cana-3972	465	79	von	von	PROPN
cana-3972	465	80	neumann	neumann	PROPN
cana-3972	465	81	regular	regular	ADJ
cana-3972	465	82	ring	ring	NOUN
cana-3972	465	83	.	.	PUNCT
cana-3972	466	1	proof	proof	NOUN
cana-3972	466	2	:	:	PUNCT
cana-3972	466	3	this	this	PRON
cana-3972	466	4	is	be	AUX
cana-3972	466	5	immediate	immediate	ADJ
cana-3972	466	6	.	.	PUNCT
cana-3972	467	1	proposition	proposition	NOUN
cana-3972	467	2	4.5	4.5	NUM
cana-3972	467	3	.	.	PUNCT
cana-3972	468	1	let	let	VERB
cana-3972	468	2	a	a	PRON
cana-3972	468	3	be	be	AUX
cana-3972	468	4	a	a	DET
cana-3972	468	5	class	class	NOUN
cana-3972	468	6	of	of	ADP
cana-3972	468	7	cosingular	cosingular	ADJ
cana-3972	468	8	r	r	NOUN
cana-3972	468	9	-	-	PUNCT
cana-3972	468	10	module	module	NOUN
cana-3972	468	11	and	and	CCONJ
cana-3972	468	12	closed	close	VERB
cana-3972	468	13	under	under	ADP
cana-3972	468	14	iso	iso	NOUN
cana-3972	468	15	-	-	PUNCT
cana-3972	468	16	morphisms	morphism	NOUN
cana-3972	468	17	and	and	CCONJ
cana-3972	468	18	directs	direct	VERB
cana-3972	468	19	summand	summand	NOUN
cana-3972	468	20	.	.	PUNCT
cana-3972	469	1	the	the	DET
cana-3972	469	2	following	follow	VERB
cana-3972	469	3	conditions	condition	NOUN
cana-3972	469	4	are	be	AUX
cana-3972	469	5	equivalent	equivalent	ADJ
cana-3972	469	6	:	:	PUNCT
cana-3972	469	7	1	1	X
cana-3972	469	8	)	)	PUNCT
cana-3972	469	9	all	all	DET
cana-3972	469	10	𝐴	𝐴	PROPN
cana-3972	469	11	∈	∈	PROPN
cana-3972	469	12	𝐴	𝐴	PROPN
cana-3972	469	13	is	be	AUX
cana-3972	469	14	a	a	DET
cana-3972	469	15	-	-	PUNCT
cana-3972	469	16	projective	projective	ADJ
cana-3972	469	17	.	.	PUNCT
cana-3972	470	1	2	2	NUM
cana-3972	470	2	)	)	PUNCT
cana-3972	470	3	every	every	DET
cana-3972	470	4	left	leave	VERB
cana-3972	470	5	r	r	NOUN
cana-3972	470	6	-	-	PUNCT
cana-3972	470	7	module	module	NOUN
cana-3972	470	8	in	in	ADP
cana-3972	470	9	a	a	PRON
cana-3972	470	10	is	be	AUX
cana-3972	470	11	a	a	DET
cana-3972	470	12	d2	d2	NOUN
cana-3972	470	13	-	-	PUNCT
cana-3972	470	14	module	module	NOUN
cana-3972	470	15	.	.	PUNCT
cana-3972	471	1	3	3	X
cana-3972	471	2	)	)	PUNCT
cana-3972	471	3	every	every	PRON
cana-3972	471	4	left	leave	VERB
cana-3972	471	5	r	r	NOUN
cana-3972	471	6	-	-	PUNCT
cana-3972	471	7	module	module	NOUN
cana-3972	471	8	in	in	ADP
cana-3972	471	9	a	a	PRON
cana-3972	471	10	is	be	AUX
cana-3972	471	11	a	a	DET
cana-3972	471	12	cosingular	cosingular	ADJ
cana-3972	471	13	direct	direct	ADJ
cana-3972	471	14	-	-	PUNCT
cana-3972	471	15	projective	projective	NOUN
cana-3972	471	16	module	module	NOUN
cana-3972	471	17	.	.	PUNCT
cana-3972	472	1	4	4	NUM
cana-3972	472	2	)	)	PUNCT
cana-3972	472	3	every	every	DET
cana-3972	472	4	left	leave	VERB
cana-3972	472	5	r	r	NOUN
cana-3972	472	6	-	-	PUNCT
cana-3972	472	7	module	module	NOUN
cana-3972	472	8	in	in	ADP
cana-3972	472	9	a	a	PRON
cana-3972	472	10	is	be	AUX
cana-3972	472	11	a	a	DET
cana-3972	472	12	d4module	d4module	NOUN
cana-3972	472	13	.	.	PUNCT
cana-3972	472	14	5	5	NUM
cana-3972	472	15	)	)	PUNCT
cana-3972	472	16	every	every	PRON
cana-3972	472	17	left	leave	VERB
cana-3972	472	18	r	r	NOUN
cana-3972	472	19	-	-	PUNCT
cana-3972	472	20	module	module	NOUN
cana-3972	472	21	in	in	ADP
cana-3972	472	22	a	a	DET
cana-3972	472	23	satisfies	satisfie	NOUN
cana-3972	472	24	(	(	PUNCT
cana-3972	472	25	d41	d41	NOUN
cana-3972	472	26	)	)	PUNCT
cana-3972	472	27	.	.	PUNCT
cana-3972	473	1	proof	proof	NOUN
cana-3972	473	2	:	:	PUNCT
cana-3972	473	3	the	the	DET
cana-3972	473	4	implications	implication	NOUN
cana-3972	473	5	1	1	NUM
cana-3972	473	6	)	)	PUNCT
cana-3972	473	7	⇒	⇒	NOUN
cana-3972	473	8	2	2	NUM
cana-3972	473	9	)	)	PUNCT
cana-3972	473	10	⇒	⇒	NOUN
cana-3972	473	11	3	3	NUM
cana-3972	473	12	)	)	PUNCT
cana-3972	473	13	⇒	⇒	NOUN
cana-3972	473	14	4	4	NUM
cana-3972	473	15	)	)	PUNCT
cana-3972	473	16	⇒	⇒	NOUN
cana-3972	473	17	5	5	NUM
cana-3972	473	18	)	)	PUNCT
cana-3972	473	19	are	be	AUX
cana-3972	473	20	obvious	obvious	ADJ
cana-3972	473	21	.	.	PUNCT
cana-3972	474	1	the	the	DET
cana-3972	474	2	implication	implication	NOUN
cana-3972	474	3	5	5	NUM
cana-3972	474	4	)	)	PUNCT
cana-3972	474	5	⇒	⇒	NOUN
cana-3972	474	6	1	1	NUM
cana-3972	474	7	)	)	PUNCT
cana-3972	474	8	is	be	AUX
cana-3972	474	9	demonstrated	demonstrate	VERB
cana-3972	474	10	in	in	ADP
cana-3972	474	11	[	[	X
cana-3972	474	12	14	14	NUM
cana-3972	474	13	,	,	PUNCT
cana-3972	474	14	proposition	proposition	NOUN
cana-3972	474	15	3.6	3.6	NUM
cana-3972	474	16	]	]	PUNCT
cana-3972	474	17	,	,	PUNCT
cana-3972	474	18	and	and	CCONJ
cana-3972	474	19	we	we	PRON
cana-3972	474	20	restate	restate	VERB
cana-3972	474	21	it	it	PRON
cana-3972	474	22	here	here	ADV
cana-3972	474	23	for	for	ADP
cana-3972	474	24	clarity	clarity	NOUN
cana-3972	474	25	.	.	PUNCT
cana-3972	475	1	let	let	VERB
cana-3972	475	2	n	n	PRON
cana-3972	475	3	∈	∈	VERB
cana-3972	475	4	a	a	PRON
cana-3972	475	5	and	and	CCONJ
cana-3972	475	6	consider	consider	VERB
cana-3972	475	7	the	the	DET
cana-3972	475	8	epimorphism	epimorphism	NOUN
cana-3972	475	9	r(i	r(i	X
cana-3972	475	10	)	)	PUNCT
cana-3972	475	11	→	→	SYM
cana-3972	476	1	n	n	CCONJ
cana-3972	476	2	.	.	PUNCT
cana-3972	477	1	this	this	PRON
cana-3972	477	2	implies	imply	VERB
cana-3972	477	3	that	that	SCONJ
cana-3972	477	4	r(i	r(i	NOUN
cana-3972	477	5	)	)	PUNCT
cana-3972	477	6	⊕	⊕	PROPN
cana-3972	478	1	n	n	PART
cana-3972	478	2	is	be	AUX
cana-3972	478	3	an	an	DET
cana-3972	478	4	a	a	DET
cana-3972	478	5	-	-	PUNCT
cana-3972	478	6	d41	d41	NOUN
cana-3972	478	7	module	module	NOUN
cana-3972	478	8	.	.	PUNCT
cana-3972	479	1	as	as	SCONJ
cana-3972	479	2	noted	note	VERB
cana-3972	479	3	in	in	ADP
cana-3972	479	4	[	[	NOUN
cana-3972	479	5	14	14	NUM
cana-3972	479	6	,	,	PUNCT
cana-3972	479	7	proposition	proposition	NOUN
cana-3972	479	8	2.6	2.6	NUM
cana-3972	479	9	]	]	PUNCT
cana-3972	479	10	,	,	PUNCT
cana-3972	479	11	n	n	X
cana-3972	479	12	is	be	AUX
cana-3972	479	13	isomorphic	isomorphic	ADJ
cana-3972	479	14	to	to	ADP
cana-3972	479	15	a	a	DET
cana-3972	479	16	direct	direct	ADJ
cana-3972	479	17	summand	summand	NOUN
cana-3972	479	18	of	of	ADP
cana-3972	479	19	r(i	r(i	PROPN
cana-3972	479	20	)	)	PUNCT
cana-3972	479	21	.	.	PUNCT
cana-3972	480	1	therefore	therefore	ADV
cana-3972	480	2	,	,	PUNCT
cana-3972	480	3	n	n	PRON
cana-3972	480	4	is	be	AUX
cana-3972	480	5	a	a	DET
cana-3972	480	6	projective	projective	ADJ
cana-3972	480	7	module	module	NOUN
cana-3972	480	8	.	.	PUNCT
cana-3972	481	1	recall	recall	VERB
cana-3972	481	2	that	that	SCONJ
cana-3972	481	3	a	a	DET
cana-3972	481	4	submodule	submodule	NOUN
cana-3972	481	5	a	a	PRON
cana-3972	481	6	of	of	ADP
cana-3972	481	7	an	an	DET
cana-3972	481	8	r	r	NOUN
cana-3972	481	9	-	-	PUNCT
cana-3972	481	10	module	module	NOUN
cana-3972	481	11	b	b	NOUN
cana-3972	481	12	is	be	AUX
cana-3972	481	13	termed	term	VERB
cana-3972	481	14	a	a	DET
cana-3972	481	15	pure	pure	ADJ
cana-3972	481	16	submodule	submodule	NOUN
cana-3972	481	17	if	if	SCONJ
cana-3972	481	18	,	,	PUNCT
cana-3972	481	19	for	for	ADP
cana-3972	481	20	any	any	DET
cana-3972	481	21	left	left	ADJ
cana-3972	481	22	r	r	NOUN
cana-3972	481	23	-	-	PUNCT
cana-3972	481	24	module	module	NOUN
cana-3972	481	25	x	x	SYM
cana-3972	481	26	,	,	PUNCT
cana-3972	481	27	the	the	DET
cana-3972	481	28	natural	natural	ADJ
cana-3972	481	29	homomorphism	homomorphism	PROPN
cana-3972	481	30	a	a	DET
cana-3972	481	31	⊗	⊗	PROPN
cana-3972	481	32	x	x	PUNCT
cana-3972	481	33	→	→	SYM
cana-3972	481	34	b	b	X
cana-3972	481	35	⊗	⊗	PROPN
cana-3972	481	36	x	x	PROPN
cana-3972	481	37	is	be	AUX
cana-3972	481	38	injective	injective	ADJ
cana-3972	481	39	.	.	PUNCT
cana-3972	482	1	a	a	DET
cana-3972	482	2	module	module	NOUN
cana-3972	482	3	m	m	VERB
cana-3972	482	4	is	be	AUX
cana-3972	482	5	called	call	VERB
cana-3972	482	6	pure	pure	ADJ
cana-3972	482	7	-	-	PUNCT
cana-3972	482	8	injective	injective	ADJ
cana-3972	482	9	if	if	SCONJ
cana-3972	482	10	every	every	DET
cana-3972	482	11	homomorphism	homomorphism	NOUN
cana-3972	482	12	from	from	ADP
cana-3972	482	13	a	a	DET
cana-3972	482	14	pure	pure	ADJ
cana-3972	482	15	submodule	submodule	NOUN
cana-3972	482	16	a	a	PRON
cana-3972	482	17	of	of	ADP
cana-3972	482	18	any	any	DET
cana-3972	482	19	module	module	NOUN
cana-3972	482	20	b	b	NOUN
cana-3972	482	21	can	can	AUX
cana-3972	482	22	be	be	AUX
cana-3972	482	23	extended	extend	VERB
cana-3972	482	24	to	to	ADP
cana-3972	482	25	a	a	DET
cana-3972	482	26	homomorphism	homomorphism	NOUN
cana-3972	482	27	from	from	ADP
cana-3972	482	28	b	b	PROPN
cana-3972	482	29	into	into	ADP
cana-3972	482	30	m	m	PRON
cana-3972	482	31	.	.	PUNCT
cana-3972	483	1	it	it	PRON
cana-3972	483	2	is	be	AUX
cana-3972	483	3	established	establish	VERB
cana-3972	483	4	that	that	SCONJ
cana-3972	483	5	both	both	PRON
cana-3972	483	6	direct	direct	ADJ
cana-3972	483	7	summands	summand	NOUN
cana-3972	483	8	and	and	CCONJ
cana-3972	483	9	direct	direct	ADJ
cana-3972	483	10	products	product	NOUN
cana-3972	483	11	of	of	ADP
cana-3972	483	12	pure	pure	ADJ
cana-3972	483	13	-	-	PUNCT
cana-3972	483	14	injective	injective	ADJ
cana-3972	483	15	modules	module	NOUN
cana-3972	483	16	are	be	AUX
cana-3972	483	17	also	also	ADV
cana-3972	483	18	pure	pure	ADJ
cana-3972	483	19	-	-	PUNCT
cana-3972	483	20	injective	injective	ADJ
cana-3972	483	21	.	.	PUNCT
cana-3972	484	1	proposition	proposition	NOUN
cana-3972	484	2	4.6	4.6	NUM
cana-3972	484	3	.	.	PUNCT
cana-3972	485	1	the	the	DET
cana-3972	485	2	following	follow	VERB
cana-3972	485	3	conditions	condition	NOUN
cana-3972	485	4	are	be	AUX
cana-3972	485	5	equivalent	equivalent	ADJ
cana-3972	485	6	for	for	SCONJ
cana-3972	485	7	a	a	DET
cana-3972	485	8	ring	ring	NOUN
cana-3972	485	9	r	r	NOUN
cana-3972	485	10	:	:	PUNCT
cana-3972	485	11	1	1	X
cana-3972	485	12	)	)	PUNCT
cana-3972	485	13	r	r	NOUN
cana-3972	485	14	is	be	AUX
cana-3972	485	15	a	a	DET
cana-3972	485	16	semisimple	semisimple	ADJ
cana-3972	485	17	artinian	artinian	ADJ
cana-3972	485	18	ring	ring	NOUN
cana-3972	485	19	.	.	PUNCT
cana-3972	486	1	2	2	NUM
cana-3972	486	2	)	)	PUNCT
cana-3972	486	3	every	every	DET
cana-3972	486	4	r	r	NOUN
cana-3972	486	5	-	-	PUNCT
cana-3972	486	6	module	module	NOUN
cana-3972	486	7	is	be	AUX
cana-3972	486	8	a	a	DET
cana-3972	486	9	d41	d41	NOUN
cana-3972	486	10	-	-	PUNCT
cana-3972	486	11	module	module	NOUN
cana-3972	486	12	.	.	PUNCT
cana-3972	487	1	3	3	X
cana-3972	487	2	)	)	PUNCT
cana-3972	487	3	every	every	DET
cana-3972	487	4	direct	direct	ADJ
cana-3972	487	5	sum	sum	NOUN
cana-3972	487	6	of	of	ADP
cana-3972	487	7	two	two	NUM
cana-3972	487	8	cyclic	cyclic	ADJ
cana-3972	487	9	d41	d41	NOUN
cana-3972	487	10	-	-	PUNCT
cana-3972	487	11	modules	module	NOUN
cana-3972	487	12	is	be	AUX
cana-3972	487	13	a	a	DET
cana-3972	487	14	d41	d41	NOUN
cana-3972	487	15	-	-	PUNCT
cana-3972	487	16	module	module	NOUN
cana-3972	487	17	.	.	PUNCT
cana-3972	488	1	4	4	X
cana-3972	488	2	)	)	PUNCT
cana-3972	488	3	every	every	DET
cana-3972	488	4	2	2	NUM
cana-3972	488	5	-	-	PUNCT
cana-3972	488	6	generated	generate	VERB
cana-3972	488	7	r	r	NOUN
cana-3972	488	8	-	-	PUNCT
cana-3972	488	9	module	module	NOUN
cana-3972	488	10	is	be	AUX
cana-3972	488	11	a	a	DET
cana-3972	488	12	d41	d41	NOUN
cana-3972	488	13	-	-	PUNCT
cana-3972	488	14	module	module	NOUN
cana-3972	488	15	.	.	PUNCT
cana-3972	489	1	communications	communication	NOUN
cana-3972	489	2	on	on	ADP
cana-3972	489	3	applied	apply	VERB
cana-3972	489	4	nonlinear	nonlinear	ADJ
cana-3972	489	5	analysis	analysis	NOUN
cana-3972	489	6	issn	issn	NOUN
cana-3972	489	7	:	:	PUNCT
cana-3972	489	8	1074	1074	NUM
cana-3972	489	9	-	-	PUNCT
cana-3972	489	10	133x	133x	NUM
cana-3972	489	11	vol	vol	NOUN
cana-3972	489	12	32	32	NUM
cana-3972	489	13	no	no	NOUN
cana-3972	489	14	.	.	PUNCT
cana-3972	490	1	9s	9s	NUM
cana-3972	490	2	(	(	PUNCT
cana-3972	490	3	2025	2025	NUM
cana-3972	490	4	)	)	PUNCT
cana-3972	490	5	688	688	NUM
cana-3972	490	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	490	7	5	5	NUM
cana-3972	490	8	)	)	PUNCT
cana-3972	490	9	every	every	DET
cana-3972	490	10	factor	factor	NOUN
cana-3972	490	11	module	module	NOUN
cana-3972	490	12	of	of	ADP
cana-3972	490	13	an	an	DET
cana-3972	490	14	injective	injective	ADJ
cana-3972	490	15	r	r	NOUN
cana-3972	490	16	-	-	PUNCT
cana-3972	490	17	module	module	NOUN
cana-3972	490	18	is	be	AUX
cana-3972	490	19	a	a	DET
cana-3972	490	20	d41	d41	NOUN
cana-3972	490	21	-	-	PUNCT
cana-3972	490	22	module	module	NOUN
cana-3972	490	23	.	.	PUNCT
cana-3972	491	1	6	6	NUM
cana-3972	491	2	)	)	PUNCT
cana-3972	491	3	r	r	NOUN
cana-3972	491	4	is	be	AUX
cana-3972	491	5	a	a	DET
cana-3972	491	6	von	von	PROPN
cana-3972	491	7	neumann	neumann	PROPN
cana-3972	491	8	regular	regular	ADJ
cana-3972	491	9	ring	ring	NOUN
cana-3972	491	10	in	in	ADP
cana-3972	491	11	which	which	PRON
cana-3972	491	12	every	every	DET
cana-3972	491	13	injective	injective	ADJ
cana-3972	491	14	module	module	NOUN
cana-3972	491	15	is	be	AUX
cana-3972	491	16	a	a	DET
cana-3972	491	17	d41	d41	NOUN
cana-3972	491	18	-	-	PUNCT
cana-3972	491	19	module	module	NOUN
cana-3972	491	20	.	.	PUNCT
cana-3972	492	1	7	7	X
cana-3972	492	2	)	)	PUNCT
cana-3972	492	3	every	every	DET
cana-3972	492	4	pure	pure	ADJ
cana-3972	492	5	-	-	PUNCT
cana-3972	492	6	injective	injective	ADJ
cana-3972	492	7	r	r	NOUN
cana-3972	492	8	-	-	PUNCT
cana-3972	492	9	module	module	NOUN
cana-3972	492	10	is	be	AUX
cana-3972	492	11	a	a	DET
cana-3972	492	12	d41	d41	NOUN
cana-3972	492	13	-	-	PUNCT
cana-3972	492	14	module	module	NOUN
cana-3972	492	15	.	.	PUNCT
cana-3972	493	1	proof	proof	NOUN
cana-3972	493	2	:	:	PUNCT
cana-3972	493	3	1	1	X
cana-3972	493	4	)	)	PUNCT
cana-3972	493	5	⇒	⇒	NOUN
cana-3972	493	6	𝑖	𝑖	X
cana-3972	493	7	)	)	PUNCT
cana-3972	493	8	for	for	ADP
cana-3972	493	9	𝑖	𝑖	NOUN
cana-3972	493	10	=	=	SYM
cana-3972	493	11	2	2	NUM
cana-3972	493	12	,	,	PUNCT
cana-3972	493	13	.	.	PUNCT
cana-3972	493	14	.	.	PUNCT
cana-3972	493	15	.	.	PUNCT
cana-3972	494	1	,	,	PUNCT
cana-3972	494	2	7	7	NUM
cana-3972	494	3	is	be	AUX
cana-3972	494	4	straightforward	straightforward	ADJ
cana-3972	494	5	.	.	PUNCT
cana-3972	495	1	the	the	DET
cana-3972	495	2	implications	implication	NOUN
cana-3972	495	3	2	2	NUM
cana-3972	495	4	)	)	PUNCT
cana-3972	495	5	⇒	⇒	NOUN
cana-3972	495	6	3	3	NUM
cana-3972	495	7	)	)	PUNCT
cana-3972	495	8	and	and	CCONJ
cana-3972	495	9	4	4	X
cana-3972	495	10	)	)	PUNCT
cana-3972	495	11	⇒	⇒	NOUN
cana-3972	495	12	3	3	NUM
cana-3972	495	13	)	)	PUNCT
cana-3972	495	14	are	be	AUX
cana-3972	495	15	also	also	ADV
cana-3972	495	16	clear	clear	ADJ
cana-3972	495	17	.	.	PUNCT
cana-3972	496	1	to	to	PART
cana-3972	496	2	show	show	VERB
cana-3972	496	3	3	3	NUM
cana-3972	496	4	)	)	PUNCT
cana-3972	496	5	⇒	⇒	NOUN
cana-3972	496	6	1	1	NUM
cana-3972	496	7	)	)	PUNCT
cana-3972	496	8	,	,	PUNCT
cana-3972	496	9	let	let	VERB
cana-3972	496	10	s	s	PRON
cana-3972	496	11	be	be	AUX
cana-3972	496	12	a	a	DET
cana-3972	496	13	simple	simple	ADJ
cana-3972	496	14	r	r	NOUN
cana-3972	496	15	-	-	PUNCT
cana-3972	496	16	module	module	NOUN
cana-3972	496	17	and	and	CCONJ
cana-3972	496	18	consider	consider	VERB
cana-3972	496	19	the	the	DET
cana-3972	496	20	r	r	NOUN
cana-3972	496	21	-	-	PUNCT
cana-3972	496	22	epimorphism	epimorphism	NOUN
cana-3972	496	23	𝑓	𝑓	DET
cana-3972	496	24	∶	∶	PROPN
cana-3972	496	25	𝑅	𝑅	PROPN
cana-3972	496	26	→	→	SYM
cana-3972	496	27	𝑆.	𝑆.	PROPN
cana-3972	496	28	since	since	SCONJ
cana-3972	496	29	r	r	PROPN
cana-3972	496	30	⊕	⊕	PROPN
cana-3972	496	31	s	s	PART
cana-3972	496	32	is	be	AUX
cana-3972	496	33	a	a	DET
cana-3972	496	34	d41	d41	NOUN
cana-3972	496	35	-	-	PUNCT
cana-3972	496	36	module	module	NOUN
cana-3972	496	37	,	,	PUNCT
cana-3972	496	38	lemma	lemma	PROPN
cana-3972	496	39	2.4	2.4	NUM
cana-3972	496	40	implies	imply	VERB
cana-3972	496	41	that	that	SCONJ
cana-3972	496	42	s	s	VERB
cana-3972	496	43	is	be	AUX
cana-3972	496	44	isomorphic	isomorphic	ADJ
cana-3972	496	45	to	to	ADP
cana-3972	496	46	a	a	DET
cana-3972	496	47	direct	direct	ADJ
cana-3972	496	48	summand	summand	NOUN
cana-3972	496	49	of	of	ADP
cana-3972	496	50	r	r	NOUN
cana-3972	496	51	,	,	PUNCT
cana-3972	496	52	which	which	PRON
cana-3972	496	53	indicates	indicate	VERB
cana-3972	496	54	that	that	SCONJ
cana-3972	496	55	s	s	VERB
cana-3972	496	56	is	be	AUX
cana-3972	496	57	projective	projective	ADJ
cana-3972	496	58	.	.	PUNCT
cana-3972	497	1	thus	thus	ADV
cana-3972	497	2	,	,	PUNCT
cana-3972	497	3	r	r	NOUN
cana-3972	497	4	must	must	AUX
cana-3972	497	5	be	be	AUX
cana-3972	497	6	semisimple	semisimple	NOUN
cana-3972	497	7	artinian	artinian	ADJ
cana-3972	497	8	.	.	PUNCT
cana-3972	498	1	for	for	ADP
cana-3972	498	2	5	5	NUM
cana-3972	498	3	)	)	PUNCT
cana-3972	498	4	⇒	⇒	NOUN
cana-3972	498	5	1	1	NUM
cana-3972	498	6	)	)	PUNCT
cana-3972	498	7	,	,	PUNCT
cana-3972	498	8	if	if	SCONJ
cana-3972	498	9	n	n	PRON
cana-3972	498	10	is	be	AUX
cana-3972	498	11	a	a	DET
cana-3972	498	12	right	right	ADJ
cana-3972	498	13	ideal	ideal	NOUN
cana-3972	498	14	of	of	ADP
cana-3972	498	15	r	r	NOUN
cana-3972	498	16	,	,	PUNCT
cana-3972	498	17	then	then	ADV
cana-3972	498	18	𝐸(𝑅	𝐸(𝑅	NOUN
cana-3972	498	19	)	)	PUNCT
cana-3972	498	20	⊕	⊕	PROPN
cana-3972	499	1	𝐸(𝑅)/𝑁	𝐸(𝑅)/𝑁	NOUN
cana-3972	499	2	is	be	AUX
cana-3972	499	3	a	a	DET
cana-3972	499	4	d41	d41	NOUN
cana-3972	499	5	-	-	PUNCT
cana-3972	499	6	module	module	NOUN
cana-3972	499	7	by	by	ADP
cana-3972	499	8	5	5	NUM
cana-3972	499	9	)	)	PUNCT
cana-3972	499	10	.	.	PUNCT
cana-3972	500	1	therefore	therefore	ADV
cana-3972	500	2	,	,	PUNCT
cana-3972	500	3	the	the	DET
cana-3972	500	4	canonical	canonical	ADJ
cana-3972	500	5	homomorphism	homomorphism	NOUN
cana-3972	500	6	𝜂	𝜂	X
cana-3972	500	7	∶	∶	NOUN
cana-3972	500	8	𝐸(𝑅	𝐸(𝑅	NOUN
cana-3972	500	9	)	)	PUNCT
cana-3972	500	10	→	→	PUNCT
cana-3972	500	11	𝐸(𝑅)/𝑁	𝐸(𝑅)/𝑁	NOUN
cana-3972	500	12	splits	split	VERB
cana-3972	500	13	by	by	ADP
cana-3972	500	14	lemma	lemma	PROPN
cana-3972	500	15	2.4	2.4	NUM
cana-3972	500	16	,	,	PUNCT
cana-3972	500	17	which	which	PRON
cana-3972	500	18	means	mean	VERB
cana-3972	500	19	that	that	SCONJ
cana-3972	500	20	n	n	PRON
cana-3972	500	21	is	be	AUX
cana-3972	500	22	a	a	DET
cana-3972	500	23	direct	direct	ADJ
cana-3972	500	24	summand	summand	NOUN
cana-3972	500	25	of	of	ADP
cana-3972	500	26	e(r	e(r	NOUN
cana-3972	500	27	)	)	PUNCT
cana-3972	500	28	and	and	CCONJ
cana-3972	500	29	hence	hence	ADV
cana-3972	500	30	a	a	DET
cana-3972	500	31	direct	direct	ADJ
cana-3972	500	32	summand	summand	NOUN
cana-3972	500	33	of	of	ADP
cana-3972	500	34	r.	r.	PROPN
cana-3972	500	35	this	this	PRON
cana-3972	500	36	shows	show	VERB
cana-3972	500	37	that	that	SCONJ
cana-3972	500	38	r	r	NOUN
cana-3972	500	39	is	be	AUX
cana-3972	500	40	semisimple	semisimple	NOUN
cana-3972	500	41	artinian	artinian	ADJ
cana-3972	500	42	.	.	PUNCT
cana-3972	501	1	now	now	ADV
cana-3972	501	2	,	,	PUNCT
cana-3972	501	3	for	for	ADP
cana-3972	501	4	6	6	NUM
cana-3972	501	5	⟹	⟹	NUM
cana-3972	501	6	1	1	NUM
cana-3972	501	7	)	)	PUNCT
cana-3972	501	8	,	,	PUNCT
cana-3972	501	9	it	it	PRON
cana-3972	501	10	suffices	suffice	VERB
cana-3972	501	11	to	to	PART
cana-3972	501	12	demonstrate	demonstrate	VERB
cana-3972	501	13	that	that	SCONJ
cana-3972	501	14	r	r	NOUN
cana-3972	501	15	is	be	AUX
cana-3972	501	16	right	right	ADJ
cana-3972	501	17	noetherian	noetherian	NOUN
cana-3972	501	18	.	.	PUNCT
cana-3972	502	1	for	for	ADP
cana-3972	502	2	a	a	DET
cana-3972	502	3	countable	countable	ADJ
cana-3972	502	4	family	family	NOUN
cana-3972	502	5	of	of	ADP
cana-3972	502	6	injective	injective	ADJ
cana-3972	502	7	r	r	NOUN
cana-3972	502	8	-	-	PUNCT
cana-3972	502	9	modules	module	NOUN
cana-3972	502	10	{	{	PUNCT
cana-3972	502	11	𝑀𝑖	𝑀𝑖	NOUN
cana-3972	502	12	:	:	PUNCT
cana-3972	502	13	𝑖	𝑖	SYM
cana-3972	502	14	=	=	SYM
cana-3972	502	15	1	1	NUM
cana-3972	502	16	,	,	PUNCT
cana-3972	502	17	2	2	NUM
cana-3972	502	18	,	,	PUNCT
cana-3972	502	19	…	…	PUNCT
cana-3972	502	20	}	}	PUNCT
cana-3972	502	21	,	,	PUNCT
cana-3972	502	22	the	the	DET
cana-3972	502	23	module	module	NOUN
cana-3972	502	24	(	(	PUNCT
cana-3972	502	25	∏	∏	PROPN
cana-3972	502	26	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	502	27	)	)	PUNCT
cana-3972	502	28	∞	∞	PROPN
cana-3972	502	29	𝑖=1	𝑖=1	PUNCT
cana-3972	502	30	/⊕∞	/⊕∞	PUNCT
cana-3972	502	31	𝑖=1	𝑖=1	PUNCT
cana-3972	503	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	503	2	is	be	AUX
cana-3972	503	3	pure	pure	ADJ
cana-3972	503	4	-	-	PUNCT
cana-3972	503	5	injective	injective	ADJ
cana-3972	503	6	by	by	ADP
cana-3972	503	7	[	[	X
cana-3972	503	8	7	7	NUM
cana-3972	503	9	,	,	PUNCT
cana-3972	503	10	theorem	theorem	VERB
cana-3972	503	11	38.1	38.1	NUM
cana-3972	503	12	and	and	CCONJ
cana-3972	503	13	corollary	corollary	ADJ
cana-3972	503	14	42.2	42.2	NUM
cana-3972	503	15	]	]	PUNCT
cana-3972	503	16	.	.	PUNCT
cana-3972	504	1	since	since	SCONJ
cana-3972	504	2	r	r	NOUN
cana-3972	504	3	is	be	AUX
cana-3972	504	4	von	von	PROPN
cana-3972	504	5	neumann	neumann	PROPN
cana-3972	504	6	regular	regular	PROPN
cana-3972	504	7	,	,	PUNCT
cana-3972	504	8	this	this	DET
cana-3972	504	9	module	module	NOUN
cana-3972	504	10	is	be	AUX
cana-3972	504	11	also	also	ADV
cana-3972	504	12	injective	injective	ADJ
cana-3972	504	13	by	by	ADP
cana-3972	504	14	[	[	X
cana-3972	504	15	22	22	NUM
cana-3972	504	16	,	,	PUNCT
cana-3972	504	17	37.6	37.6	NUM
cana-3972	504	18	]	]	PUNCT
cana-3972	504	19	.	.	PUNCT
cana-3972	505	1	therefore	therefore	ADV
cana-3972	505	2	,	,	PUNCT
cana-3972	505	3	by	by	ADP
cana-3972	505	4	6	6	NUM
cana-3972	505	5	)	)	PUNCT
cana-3972	505	6	,	,	PUNCT
cana-3972	505	7	(	(	PUNCT
cana-3972	505	8	∏	∏	PROPN
cana-3972	505	9	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	505	10	)	)	PUNCT
cana-3972	505	11	∞	∞	PROPN
cana-3972	505	12	𝑖=1	𝑖=1	PROPN
cana-3972	505	13	⊕	⊕	PROPN
cana-3972	505	14	[	[	PUNCT
cana-3972	505	15	∏	∏	PROPN
cana-3972	505	16	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	505	17	∞	∞	PROPN
cana-3972	505	18	𝑖=1	𝑖=1	PROPN
cana-3972	505	19	⊕∞	⊕∞	PROPN
cana-3972	505	20	𝑖=1𝑀𝑖	𝑖=1𝑀𝑖	NOUN
cana-3972	505	21	]	]	PUNCT
cana-3972	505	22	is	be	AUX
cana-3972	505	23	a	a	DET
cana-3972	505	24	d41	d41	NOUN
cana-3972	505	25	-	-	PUNCT
cana-3972	505	26	module	module	NOUN
cana-3972	505	27	.	.	PUNCT
cana-3972	506	1	this	this	PRON
cana-3972	506	2	implies	imply	VERB
cana-3972	506	3	that	that	SCONJ
cana-3972	506	4	the	the	DET
cana-3972	506	5	natural	natural	ADJ
cana-3972	506	6	epimorphism	epimorphism	NOUN
cana-3972	506	7	∏	∏	PROPN
cana-3972	507	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	507	2	∞	∞	PROPN
cana-3972	507	3	𝑖=1	𝑖=1	PUNCT
cana-3972	507	4	→	→	SYM
cana-3972	507	5	∏	∏	PROPN
cana-3972	508	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	508	2	∞	∞	PROPN
cana-3972	508	3	𝑖=1	𝑖=1	PROPN
cana-3972	508	4	⊕∞	⊕∞	NOUN
cana-3972	508	5	𝑖=1𝑀𝑖	𝑖=1𝑀𝑖	PRON
cana-3972	508	6	splits	split	VERB
cana-3972	508	7	,	,	PUNCT
cana-3972	508	8	leading	lead	VERB
cana-3972	508	9	to	to	ADP
cana-3972	508	10	⊕∞	⊕∞	X
cana-3972	508	11	𝑖=1	𝑖=1	PUNCT
cana-3972	509	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	509	2	≤⊕	≤⊕	NUM
cana-3972	509	3	∏	∏	PROPN
cana-3972	510	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	510	2	∞	∞	PROPN
cana-3972	510	3	𝑖=1	𝑖=1	PROPN
cana-3972	510	4	.	.	PUNCT
cana-3972	511	1	thus	thus	ADV
cana-3972	511	2	⊕∞	⊕∞	X
cana-3972	511	3	𝑖=1	𝑖=1	PUNCT
cana-3972	512	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	512	2	is	be	AUX
cana-3972	512	3	injective	injective	ADJ
cana-3972	512	4	,	,	PUNCT
cana-3972	512	5	demonstrating	demonstrate	VERB
cana-3972	512	6	that	that	SCONJ
cana-3972	512	7	r	r	NOUN
cana-3972	512	8	is	be	AUX
cana-3972	512	9	right	right	ADJ
cana-3972	512	10	noetherian	noetherian	NOUN
cana-3972	512	11	.	.	PUNCT
cana-3972	513	1	finally	finally	ADV
cana-3972	513	2	,	,	PUNCT
cana-3972	513	3	for	for	ADP
cana-3972	513	4	7	7	NUM
cana-3972	513	5	)	)	PUNCT
cana-3972	513	6	⟹	⟹	NUM
cana-3972	513	7	2	2	NUM
cana-3972	513	8	)	)	PUNCT
cana-3972	513	9	,	,	PUNCT
cana-3972	513	10	consider	consider	VERB
cana-3972	513	11	a	a	DET
cana-3972	513	12	countable	countable	ADJ
cana-3972	513	13	family	family	NOUN
cana-3972	513	14	of	of	ADP
cana-3972	513	15	pure	pure	ADJ
cana-3972	513	16	-	-	PUNCT
cana-3972	513	17	injective	injective	ADJ
cana-3972	513	18	r	r	NOUN
cana-3972	513	19	-	-	PUNCT
cana-3972	513	20	modules	module	NOUN
cana-3972	513	21	{	{	PUNCT
cana-3972	513	22	𝑀𝑖	𝑀𝑖	NOUN
cana-3972	513	23	:	:	PUNCT
cana-3972	513	24	𝑖	𝑖	SYM
cana-3972	513	25	=	=	SYM
cana-3972	513	26	1	1	NUM
cana-3972	513	27	,	,	PUNCT
cana-3972	513	28	2	2	NUM
cana-3972	513	29	,	,	PUNCT
cana-3972	513	30	…	…	PUNCT
cana-3972	513	31	}	}	PUNCT
cana-3972	513	32	.	.	PUNCT
cana-3972	514	1	the	the	DET
cana-3972	514	2	module	module	NOUN
cana-3972	514	3	(	(	PUNCT
cana-3972	514	4	∏	∏	PROPN
cana-3972	514	5	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	514	6	)	)	PUNCT
cana-3972	514	7	∞	∞	PROPN
cana-3972	514	8	𝑖=1	𝑖=1	PUNCT
cana-3972	514	9	/⊕∞	/⊕∞	PUNCT
cana-3972	514	10	𝑖=1	𝑖=1	PUNCT
cana-3972	515	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	515	2	is	be	AUX
cana-3972	515	3	pure	pure	ADJ
cana-3972	515	4	-	-	PUNCT
cana-3972	515	5	injective	injective	ADJ
cana-3972	515	6	by	by	ADP
cana-3972	515	7	[	[	X
cana-3972	515	8	7	7	NUM
cana-3972	515	9	,	,	PUNCT
cana-3972	515	10	theorem	theorem	VERB
cana-3972	515	11	38.1	38.1	NUM
cana-3972	515	12	and	and	CCONJ
cana-3972	515	13	corollary	corollary	ADJ
cana-3972	515	14	42.2	42.2	NUM
cana-3972	515	15	]	]	PUNCT
cana-3972	515	16	,	,	PUNCT
cana-3972	515	17	and	and	CCONJ
cana-3972	515	18	hence	hence	ADV
cana-3972	515	19	it	it	PRON
cana-3972	515	20	is	be	AUX
cana-3972	515	21	a	a	DET
cana-3972	515	22	d41	d41	NOUN
cana-3972	515	23	-	-	PUNCT
cana-3972	515	24	module	module	NOUN
cana-3972	515	25	according	accord	VERB
cana-3972	515	26	to	to	ADP
cana-3972	515	27	7	7	NUM
cana-3972	515	28	)	)	PUNCT
cana-3972	515	29	.	.	PUNCT
cana-3972	516	1	consequently	consequently	ADV
cana-3972	516	2	,	,	PUNCT
cana-3972	516	3	the	the	DET
cana-3972	516	4	natural	natural	ADJ
cana-3972	516	5	epimorphism	epimorphism	NOUN
cana-3972	516	6	∏	∏	PROPN
cana-3972	517	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	517	2	∞	∞	PROPN
cana-3972	517	3	𝑖=1	𝑖=1	PUNCT
cana-3972	517	4	→	→	SYM
cana-3972	517	5	∏	∏	PROPN
cana-3972	518	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	518	2	∞	∞	PROPN
cana-3972	518	3	𝑖=1	𝑖=1	PROPN
cana-3972	518	4	⊕∞	⊕∞	NOUN
cana-3972	518	5	𝑖=1𝑀𝑖	𝑖=1𝑀𝑖	PRON
cana-3972	518	6	splits	split	VERB
cana-3972	518	7	,	,	PUNCT
cana-3972	518	8	leading	lead	VERB
cana-3972	518	9	to	to	ADP
cana-3972	518	10	⊕∞	⊕∞	X
cana-3972	518	11	𝑖=1	𝑖=1	PUNCT
cana-3972	519	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	519	2	≤⊕	≤⊕	NUM
cana-3972	519	3	∏	∏	PROPN
cana-3972	520	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	520	2	∞	∞	PROPN
cana-3972	520	3	𝑖=1	𝑖=1	PROPN
cana-3972	520	4	.	.	PUNCT
cana-3972	521	1	therefore	therefore	ADV
cana-3972	521	2	⊕∞	⊕∞	X
cana-3972	521	3	𝑖=1	𝑖=1	PUNCT
cana-3972	522	1	𝑀𝑖	𝑀𝑖	PROPN
cana-3972	522	2	is	be	AUX
cana-3972	522	3	pure	pure	ADJ
cana-3972	522	4	-	-	PUNCT
cana-3972	522	5	injective	injective	ADJ
cana-3972	522	6	[	[	X
cana-3972	522	7	22	22	NUM
cana-3972	522	8	,	,	PUNCT
cana-3972	522	9	53.7	53.7	NUM
cana-3972	522	10	]	]	PUNCT
cana-3972	522	11	,	,	PUNCT
cana-3972	522	12	this	this	PRON
cana-3972	522	13	means	mean	VERB
cana-3972	522	14	every	every	DET
cana-3972	522	15	r	r	NOUN
cana-3972	522	16	-	-	PUNCT
cana-3972	522	17	modules	module	NOUN
cana-3972	522	18	is	be	AUX
cana-3972	522	19	pure	pure	ADJ
cana-3972	522	20	-	-	PUNCT
cana-3972	522	21	injective	injective	ADJ
cana-3972	522	22	,	,	PUNCT
cana-3972	522	23	thus	thus	ADV
cana-3972	522	24	every	every	DET
cana-3972	522	25	r	r	NOUN
cana-3972	522	26	-	-	PUNCT
cana-3972	522	27	module	module	NOUN
cana-3972	522	28	is	be	AUX
cana-3972	522	29	a	a	DET
cana-3972	522	30	d41	d41	NOUN
cana-3972	522	31	-	-	PUNCT
cana-3972	522	32	module	module	NOUN
cana-3972	522	33	by	by	ADP
cana-3972	522	34	7	7	NUM
cana-3972	522	35	)	)	PUNCT
cana-3972	522	36	.	.	PUNCT
cana-3972	523	1	recall	recall	VERB
cana-3972	523	2	that	that	SCONJ
cana-3972	523	3	a	a	DET
cana-3972	523	4	ring	ring	NOUN
cana-3972	523	5	is	be	AUX
cana-3972	523	6	called	call	VERB
cana-3972	523	7	perfect	perfect	ADJ
cana-3972	523	8	if	if	SCONJ
cana-3972	523	9	r	r	NOUN
cana-3972	523	10	/	/	SYM
cana-3972	523	11	rad(r	rad(r	NOUN
cana-3972	523	12	)	)	PUNCT
cana-3972	523	13	is	be	AUX
cana-3972	523	14	semisimple	semisimple	NOUN
cana-3972	523	15	and	and	CCONJ
cana-3972	523	16	rad(r	rad(r	NOUN
cana-3972	523	17	)	)	PUNCT
cana-3972	523	18	is	be	AUX
cana-3972	523	19	t	t	PROPN
cana-3972	523	20	-nilpotent	-nilpotent	PROPN
cana-3972	523	21	(	(	PUNCT
cana-3972	523	22	see	see	VERB
cana-3972	523	23	[	[	X
cana-3972	523	24	10	10	NUM
cana-3972	523	25	,	,	PUNCT
cana-3972	523	26	definition	definition	NOUN
cana-3972	523	27	23.18	23.18	NUM
cana-3972	523	28	]	]	PUNCT
cana-3972	523	29	)	)	PUNCT
cana-3972	523	30	.	.	PUNCT
cana-3972	524	1	an	an	DET
cana-3972	524	2	r	r	NOUN
cana-3972	524	3	-	-	PUNCT
cana-3972	524	4	module	module	NOUN
cana-3972	524	5	m	m	NOUN
cana-3972	524	6	is	be	AUX
cana-3972	524	7	termed	term	VERB
cana-3972	524	8	finitely	finitely	ADV
cana-3972	524	9	presented	present	VERB
cana-3972	524	10	(	(	PUNCT
cana-3972	524	11	or	or	CCONJ
cana-3972	524	12	finitely	finitely	ADV
cana-3972	524	13	related	relate	VERB
cana-3972	524	14	)	)	PUNCT
cana-3972	524	15	if	if	SCONJ
cana-3972	524	16	there	there	PRON
cana-3972	524	17	exists	exist	VERB
cana-3972	524	18	an	an	DET
cana-3972	524	19	exact	exact	ADJ
cana-3972	524	20	sequence	sequence	NOUN
cana-3972	524	21	0	0	NUM
cana-3972	524	22	→	→	SYM
cana-3972	524	23	𝐾	𝐾	PROPN
cana-3972	524	24	→	→	SYM
cana-3972	524	25	𝐹	𝐹	PROPN
cana-3972	524	26	→	→	SYM
cana-3972	524	27	𝑀	𝑀	PROPN
cana-3972	524	28	→	→	SYM
cana-3972	524	29	0	0	NUM
cana-3972	524	30	of	of	ADP
cana-3972	524	31	r	r	NOUN
cana-3972	524	32	-	-	PUNCT
cana-3972	524	33	modules	module	NOUN
cana-3972	524	34	,	,	PUNCT
cana-3972	524	35	where	where	SCONJ
cana-3972	524	36	f	f	PROPN
cana-3972	524	37	is	be	AUX
cana-3972	524	38	a	a	DET
cana-3972	524	39	free	free	ADJ
cana-3972	524	40	module	module	NOUN
cana-3972	524	41	and	and	CCONJ
cana-3972	524	42	both	both	DET
cana-3972	524	43	f	f	PROPN
cana-3972	524	44	and	and	CCONJ
cana-3972	524	45	k	k	PROPN
cana-3972	524	46	are	be	AUX
cana-3972	524	47	finitely	finitely	ADV
cana-3972	524	48	generated	generate	VERB
cana-3972	524	49	.	.	PUNCT
cana-3972	525	1	a	a	DET
cana-3972	525	2	ring	ring	NOUN
cana-3972	525	3	r	r	NOUN
cana-3972	525	4	is	be	AUX
cana-3972	525	5	referred	refer	VERB
cana-3972	525	6	to	to	ADP
cana-3972	525	7	as	as	ADP
cana-3972	525	8	semiregular	semiregular	PROPN
cana-3972	525	9	if	if	SCONJ
cana-3972	525	10	every	every	PRON
cana-3972	525	11	finitely	finitely	ADV
cana-3972	525	12	presented	present	VERB
cana-3972	525	13	r	r	NOUN
cana-3972	525	14	-	-	PUNCT
cana-3972	525	15	module	module	NOUN
cana-3972	525	16	possesses	possess	VERB
cana-3972	525	17	a	a	DET
cana-3972	525	18	projective	projective	ADJ
cana-3972	525	19	cover	cover	NOUN
cana-3972	525	20	(	(	PUNCT
cana-3972	525	21	see	see	VERB
cana-3972	525	22	[	[	X
cana-3972	525	23	15	15	NUM
cana-3972	525	24	,	,	PUNCT
cana-3972	525	25	theorem	theorem	VERB
cana-3972	525	26	2.9	2.9	NUM
cana-3972	525	27	]	]	PUNCT
cana-3972	525	28	)	)	PUNCT
cana-3972	525	29	.	.	PUNCT
cana-3972	526	1	proposition	proposition	NOUN
cana-3972	526	2	4.7	4.7	NUM
cana-3972	526	3	.	.	PUNCT
cana-3972	527	1	the	the	DET
cana-3972	527	2	following	follow	VERB
cana-3972	527	3	statements	statement	NOUN
cana-3972	527	4	are	be	AUX
cana-3972	527	5	equivalent	equivalent	ADJ
cana-3972	527	6	for	for	ADP
cana-3972	527	7	a	a	DET
cana-3972	527	8	ring	ring	NOUN
cana-3972	527	9	r	r	NOUN
cana-3972	527	10	:	:	PUNCT
cana-3972	527	11	1	1	NUM
cana-3972	527	12	)	)	PUNCT
cana-3972	527	13	r	r	NOUN
cana-3972	527	14	is	be	AUX
cana-3972	527	15	semiregular	semiregular	ADJ
cana-3972	527	16	.	.	PUNCT
cana-3972	528	1	2	2	NUM
cana-3972	528	2	)	)	PUNCT
cana-3972	528	3	every	every	PRON
cana-3972	528	4	finitely	finitely	ADV
cana-3972	528	5	presented	present	VERB
cana-3972	528	6	r	r	NOUN
cana-3972	528	7	-	-	PUNCT
cana-3972	528	8	module	module	NOUN
cana-3972	528	9	has	have	VERB
cana-3972	528	10	a	a	DET
cana-3972	528	11	d41	d41	NOUN
cana-3972	528	12	-	-	PUNCT
cana-3972	528	13	cover	cover	NOUN
cana-3972	528	14	.	.	PUNCT
cana-3972	529	1	proof	proof	NOUN
cana-3972	529	2	:	:	PUNCT
cana-3972	529	3	1	1	X
cana-3972	529	4	)	)	PUNCT
cana-3972	529	5	⇒	⇒	NOUN
cana-3972	529	6	2	2	NUM
cana-3972	529	7	):	):	PUNCT
cana-3972	529	8	this	this	PRON
cana-3972	529	9	is	be	AUX
cana-3972	529	10	evident	evident	ADJ
cana-3972	529	11	.	.	PUNCT
cana-3972	530	1	1	1	X
cana-3972	530	2	)	)	PUNCT
cana-3972	530	3	⇒	⇒	NOUN
cana-3972	530	4	2	2	NUM
cana-3972	530	5	):	):	PUNCT
cana-3972	530	6	let	let	VERB
cana-3972	530	7	m	m	PRON
cana-3972	530	8	be	be	AUX
cana-3972	530	9	a	a	DET
cana-3972	530	10	finitely	finitely	ADV
cana-3972	530	11	presented	present	VERB
cana-3972	530	12	r	r	NOUN
cana-3972	530	13	-	-	PUNCT
cana-3972	530	14	module	module	NOUN
cana-3972	530	15	and	and	CCONJ
cana-3972	530	16	𝑔	𝑔	NOUN
cana-3972	530	17	:	:	PUNCT
cana-3972	530	18	𝐹	𝐹	PROPN
cana-3972	530	19	→	→	SYM
cana-3972	530	20	𝑀	𝑀	PROPN
cana-3972	530	21	be	be	VERB
cana-3972	530	22	a	a	DET
cana-3972	530	23	homomorphism	homomorphism	NOUN
cana-3972	530	24	from	from	ADP
cana-3972	530	25	a	a	DET
cana-3972	530	26	free	free	ADJ
cana-3972	530	27	r	r	NOUN
cana-3972	530	28	-	-	PUNCT
cana-3972	530	29	module	module	NOUN
cana-3972	530	30	f	f	NOUN
cana-3972	530	31	to	to	ADP
cana-3972	530	32	m	m	PROPN
cana-3972	530	33	.	.	PUNCT
cana-3972	531	1	according	accord	VERB
cana-3972	531	2	to	to	ADP
cana-3972	531	3	2	2	NUM
cana-3972	531	4	)	)	PUNCT
cana-3972	531	5	,	,	PUNCT
cana-3972	531	6	f	f	PROPN
cana-3972	531	7	⊕	⊕	PROPN
cana-3972	531	8	m	m	AUX
cana-3972	531	9	has	have	VERB
cana-3972	531	10	a	a	DET
cana-3972	531	11	d41	d41	NOUN
cana-3972	531	12	-	-	PUNCT
cana-3972	531	13	cover	cover	NOUN
cana-3972	531	14	,	,	PUNCT
cana-3972	531	15	and	and	CCONJ
cana-3972	531	16	by	by	ADP
cana-3972	531	17	[	[	X
cana-3972	531	18	4	4	NUM
cana-3972	531	19	,	,	PUNCT
cana-3972	531	20	theorem	theorem	VERB
cana-3972	531	21	3.2	3.2	NUM
cana-3972	531	22	]	]	PUNCT
cana-3972	531	23	,	,	PUNCT
cana-3972	531	24	m	m	VERB
cana-3972	531	25	has	have	VERB
cana-3972	531	26	a	a	DET
cana-3972	531	27	projective	projective	ADJ
cana-3972	531	28	cover	cover	NOUN
cana-3972	531	29	.	.	PUNCT
cana-3972	532	1	therefore	therefore	ADV
cana-3972	532	2	,	,	PUNCT
cana-3972	532	3	r	r	NOUN
cana-3972	532	4	is	be	AUX
cana-3972	532	5	a	a	DET
cana-3972	532	6	semiregular	semiregular	PROPN
cana-3972	532	7	ring	ring	NOUN
cana-3972	532	8	.	.	PUNCT
cana-3972	533	1	proposition	proposition	NOUN
cana-3972	533	2	4.8	4.8	NUM
cana-3972	533	3	.	.	PUNCT
cana-3972	534	1	the	the	DET
cana-3972	534	2	following	follow	VERB
cana-3972	534	3	statements	statement	NOUN
cana-3972	534	4	are	be	AUX
cana-3972	534	5	equivalent	equivalent	ADJ
cana-3972	534	6	:	:	PUNCT
cana-3972	534	7	1	1	X
cana-3972	534	8	)	)	PUNCT
cana-3972	534	9	r	r	NOUN
cana-3972	534	10	is	be	AUX
cana-3972	534	11	left	leave	VERB
cana-3972	534	12	perfect	perfect	ADJ
cana-3972	534	13	;	;	PUNCT
cana-3972	534	14	2	2	X
cana-3972	534	15	)	)	PUNCT
cana-3972	534	16	every	every	DET
cana-3972	534	17	flat	flat	ADJ
cana-3972	534	18	left	leave	VERB
cana-3972	534	19	r	r	NOUN
cana-3972	534	20	-	-	PUNCT
cana-3972	534	21	module	module	NOUN
cana-3972	534	22	is	be	AUX
cana-3972	534	23	quasi	quasi	ADJ
cana-3972	534	24	-	-	NOUN
cana-3972	534	25	projective	projective	ADJ
cana-3972	534	26	;	;	PUNCT
cana-3972	534	27	communications	communication	NOUN
cana-3972	534	28	on	on	ADP
cana-3972	534	29	applied	apply	VERB
cana-3972	534	30	nonlinear	nonlinear	ADJ
cana-3972	534	31	analysis	analysis	NOUN
cana-3972	534	32	issn	issn	NOUN
cana-3972	534	33	:	:	PUNCT
cana-3972	534	34	1074	1074	NUM
cana-3972	534	35	-	-	PUNCT
cana-3972	534	36	133x	133x	NUM
cana-3972	534	37	vol	vol	NOUN
cana-3972	534	38	32	32	NUM
cana-3972	534	39	no	no	NOUN
cana-3972	534	40	.	.	PUNCT
cana-3972	535	1	9s	9s	NUM
cana-3972	535	2	(	(	PUNCT
cana-3972	535	3	2025	2025	NUM
cana-3972	535	4	)	)	PUNCT
cana-3972	535	5	689	689	NUM
cana-3972	535	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	535	7	3	3	X
cana-3972	535	8	)	)	PUNCT
cana-3972	535	9	every	every	DET
cana-3972	535	10	flat	flat	ADJ
cana-3972	535	11	left	leave	VERB
cana-3972	535	12	r	r	NOUN
cana-3972	535	13	-	-	PUNCT
cana-3972	535	14	module	module	NOUN
cana-3972	535	15	is	be	AUX
cana-3972	535	16	a	a	DET
cana-3972	535	17	d41	d41	NOUN
cana-3972	535	18	-	-	PUNCT
cana-3972	535	19	module	module	NOUN
cana-3972	535	20	;	;	PUNCT
cana-3972	535	21	4	4	X
cana-3972	535	22	)	)	PUNCT
cana-3972	535	23	every	every	DET
cana-3972	535	24	left	left	ADJ
cana-3972	535	25	r	r	NOUN
cana-3972	535	26	-	-	PUNCT
cana-3972	535	27	module	module	NOUN
cana-3972	535	28	has	have	VERB
cana-3972	535	29	a	a	DET
cana-3972	535	30	d41	d41	NOUN
cana-3972	535	31	-	-	PUNCT
cana-3972	535	32	cover	cover	NOUN
cana-3972	535	33	.	.	PUNCT
cana-3972	536	1	5	5	NUM
cana-3972	536	2	)	)	PUNCT
cana-3972	536	3	every	every	DET
cana-3972	536	4	countably	countably	ADV
cana-3972	536	5	generated	generate	VERB
cana-3972	536	6	r	r	NOUN
cana-3972	536	7	-	-	PUNCT
cana-3972	536	8	module	module	NOUN
cana-3972	536	9	has	have	VERB
cana-3972	536	10	a	a	DET
cana-3972	536	11	d41	d41	NOUN
cana-3972	536	12	-	-	PUNCT
cana-3972	536	13	cover	cover	NOUN
cana-3972	536	14	.	.	PUNCT
cana-3972	537	1	proof	proof	NOUN
cana-3972	537	2	:	:	PUNCT
cana-3972	537	3	the	the	DET
cana-3972	537	4	implications	implication	NOUN
cana-3972	537	5	1	1	NUM
cana-3972	537	6	)	)	PUNCT
cana-3972	537	7	⇒	⇒	NOUN
cana-3972	537	8	2	2	NUM
cana-3972	537	9	)	)	PUNCT
cana-3972	537	10	⇒	⇒	NOUN
cana-3972	537	11	3	3	NUM
cana-3972	537	12	)	)	PUNCT
cana-3972	537	13	and	and	CCONJ
cana-3972	537	14	1	1	X
cana-3972	537	15	)	)	PUNCT
cana-3972	537	16	⇒	⇒	NOUN
cana-3972	537	17	4	4	NUM
cana-3972	537	18	)	)	PUNCT
cana-3972	537	19	⇒	⇒	NOUN
cana-3972	537	20	5	5	NUM
cana-3972	537	21	)	)	PUNCT
cana-3972	537	22	are	be	AUX
cana-3972	537	23	straightforward	straightforward	ADJ
cana-3972	537	24	.	.	PUNCT
cana-3972	538	1	3	3	X
cana-3972	538	2	)	)	PUNCT
cana-3972	538	3	⇒	⇒	NOUN
cana-3972	538	4	1	1	NUM
cana-3972	538	5	)	)	PUNCT
cana-3972	538	6	.	.	PUNCT
cana-3972	539	1	let	let	VERB
cana-3972	539	2	m	m	PRON
cana-3972	539	3	be	be	AUX
cana-3972	539	4	a	a	DET
cana-3972	539	5	flat	flat	ADJ
cana-3972	539	6	left	leave	VERB
cana-3972	539	7	r	r	NOUN
cana-3972	539	8	-	-	PUNCT
cana-3972	539	9	module	module	NOUN
cana-3972	539	10	,	,	PUNCT
cana-3972	539	11	n	n	CCONJ
cana-3972	539	12	a	a	DET
cana-3972	539	13	free	free	ADJ
cana-3972	539	14	left	left	ADJ
cana-3972	539	15	r	r	NOUN
cana-3972	539	16	-	-	PUNCT
cana-3972	539	17	module	module	NOUN
cana-3972	539	18	,	,	PUNCT
cana-3972	539	19	and	and	CCONJ
cana-3972	539	20	𝑔	𝑔	PART
cana-3972	539	21	∶	∶	NOUN
cana-3972	539	22	𝑁	𝑁	PROPN
cana-3972	539	23	→	→	SYM
cana-3972	539	24	𝑀	𝑀	PROPN
cana-3972	539	25	an	an	DET
cana-3972	539	26	r	r	NOUN
cana-3972	539	27	-	-	PUNCT
cana-3972	539	28	epimorphism	epimorphism	NOUN
cana-3972	539	29	.	.	PUNCT
cana-3972	540	1	since	since	SCONJ
cana-3972	540	2	k	k	PROPN
cana-3972	540	3	=	=	PROPN
cana-3972	540	4	m	m	PROPN
cana-3972	540	5	⊕	⊕	PROPN
cana-3972	540	6	n	n	PRON
cana-3972	540	7	is	be	AUX
cana-3972	540	8	flat	flat	ADJ
cana-3972	540	9	and	and	CCONJ
cana-3972	540	10	,	,	PUNCT
cana-3972	540	11	by	by	ADP
cana-3972	540	12	assumption	assumption	NOUN
cana-3972	540	13	,	,	PUNCT
cana-3972	540	14	a	a	DET
cana-3972	540	15	d41	d41	NOUN
cana-3972	540	16	-	-	PUNCT
cana-3972	540	17	module	module	NOUN
cana-3972	540	18	,	,	PUNCT
cana-3972	540	19	then	then	ADV
cana-3972	540	20	𝑔	𝑔	PROPN
cana-3972	540	21	splits	split	VERB
cana-3972	540	22	,	,	PUNCT
cana-3972	540	23	which	which	PRON
cana-3972	540	24	implies	imply	VERB
cana-3972	540	25	that	that	SCONJ
cana-3972	540	26	m	m	PROPN
cana-3972	540	27	is	be	AUX
cana-3972	540	28	projective	projective	ADJ
cana-3972	540	29	.	.	PUNCT
cana-3972	541	1	consequently	consequently	ADV
cana-3972	541	2	,	,	PUNCT
cana-3972	541	3	by	by	ADP
cana-3972	541	4	[	[	X
cana-3972	541	5	24	24	NUM
cana-3972	541	6	,	,	PUNCT
cana-3972	541	7	lemma	lemma	PROPN
cana-3972	541	8	10	10	NUM
cana-3972	541	9	]	]	PUNCT
cana-3972	541	10	,	,	PUNCT
cana-3972	541	11	we	we	PRON
cana-3972	541	12	conclude	conclude	VERB
cana-3972	541	13	that	that	SCONJ
cana-3972	541	14	r	r	NOUN
cana-3972	541	15	is	be	AUX
cana-3972	541	16	a	a	DET
cana-3972	541	17	left	left	ADJ
cana-3972	541	18	perfect	perfect	ADJ
cana-3972	541	19	ring	ring	NOUN
cana-3972	541	20	.	.	PUNCT
cana-3972	542	1	5	5	NUM
cana-3972	542	2	)	)	PUNCT
cana-3972	542	3	⇒	⇒	NOUN
cana-3972	542	4	1	1	NUM
cana-3972	542	5	)	)	PUNCT
cana-3972	542	6	.	.	PUNCT
cana-3972	543	1	by	by	ADP
cana-3972	543	2	[	[	X
cana-3972	543	3	4	4	NUM
cana-3972	543	4	,	,	PUNCT
cana-3972	543	5	corollary	corollary	ADJ
cana-3972	543	6	3.4	3.4	NUM
cana-3972	543	7	]	]	PUNCT
cana-3972	543	8	,	,	PUNCT
cana-3972	543	9	rn	rn	PROPN
cana-3972	543	10	is	be	AUX
cana-3972	543	11	a	a	DET
cana-3972	543	12	lifting	lifting	NOUN
cana-3972	543	13	module	module	NOUN
cana-3972	543	14	.	.	PUNCT
cana-3972	544	1	so	so	ADV
cana-3972	544	2	,	,	PUNCT
cana-3972	544	3	by	by	ADP
cana-3972	544	4	[	[	X
cana-3972	544	5	1	1	NUM
cana-3972	544	6	,	,	PUNCT
cana-3972	544	7	theorem	theorem	VERB
cana-3972	544	8	1.2.17	1.2.17	NUM
cana-3972	544	9	]	]	PUNCT
cana-3972	544	10	,	,	PUNCT
cana-3972	544	11	r	r	NOUN
cana-3972	544	12	is	be	AUX
cana-3972	544	13	a	a	DET
cana-3972	544	14	right	right	ADJ
cana-3972	544	15	perfect	perfect	ADJ
cana-3972	544	16	ring	ring	NOUN
cana-3972	544	17	.	.	PUNCT
cana-3972	545	1	definition	definition	NOUN
cana-3972	545	2	4.9	4.9	NUM
cana-3972	545	3	.	.	PUNCT
cana-3972	546	1	a	a	DET
cana-3972	546	2	ring	ring	NOUN
cana-3972	546	3	r	r	NOUN
cana-3972	546	4	is	be	AUX
cana-3972	546	5	called	call	VERB
cana-3972	546	6	strongly	strongly	ADV
cana-3972	546	7	left	leave	VERB
cana-3972	546	8	d41	d41	NOUN
cana-3972	546	9	if	if	SCONJ
cana-3972	546	10	(	(	PUNCT
cana-3972	546	11	rr	rr	NOUN
cana-3972	546	12	)	)	PUNCT
cana-3972	547	1	n	n	NOUN
cana-3972	547	2	satisfies	satisfy	VERB
cana-3972	547	3	the	the	DET
cana-3972	547	4	d41	d41	NOUN
cana-3972	547	5	condition	condition	NOUN
cana-3972	547	6	for	for	ADP
cana-3972	547	7	every	every	DET
cana-3972	547	8	positive	positive	ADJ
cana-3972	547	9	integer	integer	NOUN
cana-3972	547	10	n	n	NOUN
cana-3972	547	11	;	;	PUNCT
cana-3972	547	12	that	that	PRON
cana-3972	547	13	is	is	ADV
cana-3972	547	14	,	,	PUNCT
cana-3972	547	15	every	every	DET
cana-3972	547	16	finitely	finitely	ADV
cana-3972	547	17	generated	generate	VERB
cana-3972	547	18	free	free	ADJ
cana-3972	547	19	left	left	ADJ
cana-3972	547	20	r	r	NOUN
cana-3972	547	21	-	-	PUNCT
cana-3972	547	22	module	module	NOUN
cana-3972	547	23	is	be	AUX
cana-3972	547	24	a	a	DET
cana-3972	547	25	d41	d41	NOUN
cana-3972	547	26	-	-	PUNCT
cana-3972	547	27	module	module	NOUN
cana-3972	547	28	.	.	PUNCT
cana-3972	548	1	a	a	DET
cana-3972	548	2	left	left	ADJ
cana-3972	548	3	rmodule	rmodule	NOUN
cana-3972	548	4	m	m	VERB
cana-3972	548	5	is	be	AUX
cana-3972	548	6	called	call	VERB
cana-3972	548	7	strongly	strongly	ADV
cana-3972	548	8	left	leave	VERB
cana-3972	548	9	d41	d41	PROPN
cana-3972	548	10	if	if	SCONJ
cana-3972	548	11	mn	mn	PROPN
cana-3972	548	12	is	be	AUX
cana-3972	548	13	a	a	DET
cana-3972	548	14	d41	d41	NOUN
cana-3972	548	15	-	-	PUNCT
cana-3972	548	16	module	module	NOUN
cana-3972	548	17	for	for	ADP
cana-3972	548	18	every	every	DET
cana-3972	548	19	positive	positive	ADJ
cana-3972	548	20	integer	integer	NOUN
cana-3972	548	21	n.	n.	NOUN
cana-3972	548	22	proposition	proposition	NOUN
cana-3972	548	23	4.10	4.10	NUM
cana-3972	548	24	.	.	PUNCT
cana-3972	549	1	the	the	DET
cana-3972	549	2	following	follow	VERB
cana-3972	549	3	conditions	condition	NOUN
cana-3972	549	4	are	be	AUX
cana-3972	549	5	equivalent	equivalent	ADJ
cana-3972	549	6	for	for	ADP
cana-3972	549	7	a	a	DET
cana-3972	549	8	ring	ring	NOUN
cana-3972	549	9	r	r	NOUN
cana-3972	549	10	:	:	PUNCT
cana-3972	549	11	1	1	NUM
cana-3972	549	12	)	)	PUNCT
cana-3972	549	13	r	r	NOUN
cana-3972	549	14	is	be	AUX
cana-3972	549	15	artinian	artinian	ADJ
cana-3972	549	16	semisimple	semisimple	NOUN
cana-3972	549	17	;	;	PUNCT
cana-3972	549	18	2	2	X
cana-3972	549	19	)	)	PUNCT
cana-3972	549	20	every	every	PRON
cana-3972	549	21	(	(	PUNCT
cana-3972	549	22	finitely	finitely	ADV
cana-3972	549	23	generated	generate	VERB
cana-3972	549	24	)	)	PUNCT
cana-3972	549	25	left	leave	VERB
cana-3972	549	26	r	r	NOUN
cana-3972	549	27	-	-	PUNCT
cana-3972	549	28	module	module	NOUN
cana-3972	549	29	is	be	AUX
cana-3972	549	30	a	a	DET
cana-3972	549	31	d41	d41	NOUN
cana-3972	549	32	-	-	PUNCT
cana-3972	549	33	module	module	NOUN
cana-3972	549	34	;	;	PUNCT
cana-3972	549	35	3	3	X
cana-3972	549	36	)	)	PUNCT
cana-3972	549	37	every	every	PRON
cana-3972	549	38	(	(	PUNCT
cana-3972	549	39	finitely	finitely	ADV
cana-3972	549	40	generated	generate	VERB
cana-3972	549	41	)	)	PUNCT
cana-3972	549	42	left	leave	VERB
cana-3972	549	43	r	r	NOUN
cana-3972	549	44	-	-	PUNCT
cana-3972	549	45	module	module	NOUN
cana-3972	549	46	is	be	AUX
cana-3972	549	47	a	a	DET
cana-3972	549	48	strongly	strongly	ADV
cana-3972	549	49	d41	d41	NOUN
cana-3972	549	50	-	-	PUNCT
cana-3972	549	51	module	module	NOUN
cana-3972	549	52	;	;	PUNCT
cana-3972	549	53	4	4	X
cana-3972	549	54	)	)	PUNCT
cana-3972	549	55	every	every	DET
cana-3972	549	56	2	2	NUM
cana-3972	549	57	-	-	PUNCT
cana-3972	549	58	generated	generate	VERB
cana-3972	549	59	left	left	ADJ
cana-3972	549	60	rmodule	rmodule	NOUN
cana-3972	549	61	is	be	AUX
cana-3972	549	62	a	a	DET
cana-3972	549	63	strongly	strongly	ADV
cana-3972	549	64	d41	d41	NOUN
cana-3972	549	65	-	-	PUNCT
cana-3972	549	66	module	module	NOUN
cana-3972	549	67	;	;	PUNCT
cana-3972	549	68	5	5	X
cana-3972	549	69	)	)	PUNCT
cana-3972	549	70	the	the	DET
cana-3972	549	71	class	class	NOUN
cana-3972	549	72	of	of	ADP
cana-3972	549	73	all	all	PRON
cana-3972	549	74	left	leave	VERB
cana-3972	549	75	d41	d41	NOUN
cana-3972	549	76	-	-	PUNCT
cana-3972	549	77	modules	module	NOUN
cana-3972	549	78	is	be	AUX
cana-3972	549	79	closed	close	VERB
cana-3972	549	80	under	under	ADP
cana-3972	549	81	(	(	PUNCT
cana-3972	549	82	finite	finite	ADJ
cana-3972	549	83	)	)	PUNCT
cana-3972	549	84	direct	direct	ADJ
cana-3972	549	85	sums	sum	NOUN
cana-3972	549	86	;	;	PUNCT
cana-3972	549	87	6	6	X
cana-3972	549	88	)	)	PUNCT
cana-3972	549	89	the	the	DET
cana-3972	549	90	class	class	NOUN
cana-3972	549	91	of	of	ADP
cana-3972	549	92	all	all	PRON
cana-3972	549	93	strongly	strongly	ADV
cana-3972	549	94	left	leave	VERB
cana-3972	549	95	d41	d41	NOUN
cana-3972	549	96	-	-	PUNCT
cana-3972	549	97	modules	module	NOUN
cana-3972	549	98	is	be	AUX
cana-3972	549	99	closed	close	VERB
cana-3972	549	100	under	under	ADP
cana-3972	549	101	(	(	PUNCT
cana-3972	549	102	finite	finite	ADJ
cana-3972	549	103	)	)	PUNCT
cana-3972	549	104	direct	direct	ADJ
cana-3972	549	105	sums	sum	NOUN
cana-3972	549	106	.	.	PUNCT
cana-3972	550	1	proof	proof	NOUN
cana-3972	550	2	:	:	PUNCT
cana-3972	550	3	the	the	DET
cana-3972	550	4	implications	implication	NOUN
cana-3972	550	5	1	1	NUM
cana-3972	550	6	)	)	PUNCT
cana-3972	550	7	⇔	⇔	X
cana-3972	550	8	2	2	NUM
cana-3972	550	9	)	)	PUNCT
cana-3972	550	10	⇔	⇔	X
cana-3972	550	11	5	5	NUM
cana-3972	550	12	)	)	PUNCT
cana-3972	550	13	follow	follow	VERB
cana-3972	550	14	from	from	ADP
cana-3972	550	15	[	[	X
cana-3972	550	16	23	23	NUM
cana-3972	550	17	,	,	PUNCT
cana-3972	550	18	theorem	theorem	VERB
cana-3972	550	19	9	9	NUM
cana-3972	550	20	]	]	PUNCT
cana-3972	550	21	.	.	PUNCT
cana-3972	551	1	6	6	X
cana-3972	551	2	)	)	PUNCT
cana-3972	551	3	⇒	⇒	NOUN
cana-3972	551	4	4	4	NUM
cana-3972	551	5	,	,	PUNCT
cana-3972	551	6	3	3	X
cana-3972	551	7	)	)	PUNCT
cana-3972	551	8	⇒	⇒	NOUN
cana-3972	551	9	4	4	NUM
cana-3972	551	10	)	)	PUNCT
cana-3972	551	11	and	and	CCONJ
cana-3972	551	12	1	1	X
cana-3972	551	13	)	)	PUNCT
cana-3972	551	14	⇒	⇒	NOUN
cana-3972	551	15	3	3	NUM
cana-3972	551	16	)	)	PUNCT
cana-3972	551	17	⇒	⇒	NOUN
cana-3972	551	18	2	2	NUM
cana-3972	551	19	)	)	PUNCT
cana-3972	551	20	are	be	AUX
cana-3972	551	21	obvious	obvious	ADJ
cana-3972	551	22	.	.	PUNCT
cana-3972	552	1	6	6	NUM
cana-3972	552	2	)	)	PUNCT
cana-3972	552	3	⇒	⇒	NOUN
cana-3972	552	4	1	1	NUM
cana-3972	552	5	)	)	PUNCT
cana-3972	552	6	let	let	VERB
cana-3972	552	7	m	m	PRON
cana-3972	552	8	be	be	AUX
cana-3972	552	9	a	a	DET
cana-3972	552	10	simple	simple	ADJ
cana-3972	552	11	r	r	NOUN
cana-3972	552	12	-	-	PUNCT
cana-3972	552	13	module	module	NOUN
cana-3972	552	14	.	.	PUNCT
cana-3972	553	1	this	this	PRON
cana-3972	553	2	implies	imply	VERB
cana-3972	553	3	that	that	SCONJ
cana-3972	553	4	m	m	PROPN
cana-3972	553	5	is	be	AUX
cana-3972	553	6	a	a	DET
cana-3972	553	7	strongly	strongly	ADV
cana-3972	553	8	d41	d41	NOUN
cana-3972	553	9	-	-	PUNCT
cana-3972	553	10	module	module	NOUN
cana-3972	553	11	.	.	PUNCT
cana-3972	554	1	given	give	VERB
cana-3972	554	2	the	the	DET
cana-3972	554	3	hypothesis	hypothesis	NOUN
cana-3972	554	4	,	,	PUNCT
cana-3972	554	5	r	r	PROPN
cana-3972	554	6	⊕	⊕	PROPN
cana-3972	554	7	m	m	VERB
cana-3972	554	8	is	be	AUX
cana-3972	554	9	also	also	ADV
cana-3972	554	10	strongly	strongly	ADV
cana-3972	554	11	d41	d41	PROPN
cana-3972	554	12	.	.	PUNCT
cana-3972	555	1	we	we	PRON
cana-3972	555	2	note	note	VERB
cana-3972	555	3	that	that	SCONJ
cana-3972	555	4	m	m	NOUN
cana-3972	555	5	is	be	AUX
cana-3972	555	6	isomorphic	isomorphic	ADJ
cana-3972	555	7	to	to	ADP
cana-3972	555	8	r	r	PROPN
cana-3972	555	9	/	/	SYM
cana-3972	555	10	j	j	NOUN
cana-3972	555	11	for	for	ADP
cana-3972	555	12	some	some	DET
cana-3972	555	13	maximal	maximal	ADJ
cana-3972	555	14	left	leave	VERB
cana-3972	555	15	ideal	ideal	PROPN
cana-3972	555	16	j	j	PROPN
cana-3972	555	17	of	of	ADP
cana-3972	555	18	r.	r.	PROPN
cana-3972	555	19	thus	thus	ADV
cana-3972	555	20	,	,	PUNCT
cana-3972	555	21	there	there	PRON
cana-3972	555	22	exists	exist	VERB
cana-3972	555	23	an	an	DET
cana-3972	555	24	epi	epi	NOUN
cana-3972	555	25	-	-	NOUN
cana-3972	555	26	morphism	morphism	NOUN
cana-3972	555	27	𝑅	𝑅	PROPN
cana-3972	555	28	→	→	PROPN
cana-3972	555	29	𝑀	𝑀	PROPN
cana-3972	555	30	→	→	SYM
cana-3972	555	31	0	0	NUM
cana-3972	555	32	.	.	PUNCT
cana-3972	556	1	by	by	ADP
cana-3972	556	2	proposition	proposition	NOUN
cana-3972	556	3	3.5	3.5	NUM
cana-3972	556	4	,	,	PUNCT
cana-3972	556	5	we	we	PRON
cana-3972	556	6	conclude	conclude	VERB
cana-3972	556	7	that	that	SCONJ
cana-3972	556	8	m	m	PROPN
cana-3972	556	9	is	be	AUX
cana-3972	556	10	a	a	DET
cana-3972	556	11	d41	d41	NOUN
cana-3972	556	12	-	-	PUNCT
cana-3972	556	13	module	module	NOUN
cana-3972	556	14	.	.	PUNCT
cana-3972	557	1	therefore	therefore	ADV
cana-3972	557	2	,	,	PUNCT
cana-3972	557	3	r	r	NOUN
cana-3972	557	4	is	be	AUX
cana-3972	557	5	semisimple	semisimple	ADJ
cana-3972	557	6	.	.	PUNCT
cana-3972	558	1	4	4	X
cana-3972	558	2	)	)	PUNCT
cana-3972	558	3	⇒	⇒	NOUN
cana-3972	558	4	1	1	NUM
cana-3972	558	5	)	)	PUNCT
cana-3972	558	6	since	since	SCONJ
cana-3972	558	7	every	every	DET
cana-3972	558	8	simple	simple	ADJ
cana-3972	558	9	r	r	NOUN
cana-3972	558	10	-	-	PUNCT
cana-3972	558	11	module	module	NOUN
cana-3972	558	12	m	m	NOUN
cana-3972	558	13	is	be	AUX
cana-3972	558	14	a	a	DET
cana-3972	558	15	strongly	strongly	ADV
cana-3972	558	16	d41	d41	NOUN
cana-3972	558	17	-	-	PUNCT
cana-3972	558	18	module	module	NOUN
cana-3972	558	19	and	and	CCONJ
cana-3972	558	20	r	r	PROPN
cana-3972	558	21	⊕	⊕	PROPN
cana-3972	558	22	m	m	VERB
cana-3972	558	23	is	be	AUX
cana-3972	558	24	2	2	NUM
cana-3972	558	25	-	-	PUNCT
cana-3972	558	26	generated	generate	VERB
cana-3972	558	27	,	,	PUNCT
cana-3972	558	28	the	the	DET
cana-3972	558	29	claim	claim	NOUN
cana-3972	558	30	can	can	AUX
cana-3972	558	31	be	be	AUX
cana-3972	558	32	proved	prove	VERB
cana-3972	558	33	using	use	VERB
cana-3972	558	34	an	an	DET
cana-3972	558	35	argument	argument	NOUN
cana-3972	558	36	similar	similar	ADJ
cana-3972	558	37	to	to	ADP
cana-3972	558	38	that	that	PRON
cana-3972	558	39	in	in	ADP
cana-3972	558	40	the	the	DET
cana-3972	558	41	proof	proof	NOUN
cana-3972	558	42	of	of	ADP
cana-3972	558	43	6	6	NUM
cana-3972	558	44	)	)	PUNCT
cana-3972	558	45	⇒	⇒	NOUN
cana-3972	558	46	1	1	NUM
cana-3972	558	47	)	)	PUNCT
cana-3972	558	48	.	.	PUNCT
cana-3972	559	1	proposition	proposition	NOUN
cana-3972	559	2	4.11	4.11	NUM
cana-3972	559	3	.	.	PUNCT
cana-3972	560	1	the	the	DET
cana-3972	560	2	following	follow	VERB
cana-3972	560	3	conditions	condition	NOUN
cana-3972	560	4	are	be	AUX
cana-3972	560	5	equivalent	equivalent	ADJ
cana-3972	560	6	for	for	ADP
cana-3972	560	7	a	a	DET
cana-3972	560	8	ring	ring	NOUN
cana-3972	560	9	r	r	NOUN
cana-3972	560	10	:	:	PUNCT
cana-3972	560	11	1	1	NUM
cana-3972	560	12	)	)	PUNCT
cana-3972	560	13	r	r	NOUN
cana-3972	560	14	is	be	AUX
cana-3972	560	15	a	a	DET
cana-3972	560	16	semisimple	semisimple	ADJ
cana-3972	560	17	artinian	artinian	ADJ
cana-3972	560	18	ring	ring	NOUN
cana-3972	560	19	.	.	PUNCT
cana-3972	561	1	2	2	NUM
cana-3972	561	2	)	)	PUNCT
cana-3972	561	3	every	every	DET
cana-3972	561	4	left	left	ADJ
cana-3972	561	5	r	r	NOUN
cana-3972	561	6	-	-	PUNCT
cana-3972	561	7	module	module	NOUN
cana-3972	561	8	has	have	VERB
cana-3972	561	9	a	a	DET
cana-3972	561	10	d41	d41	NOUN
cana-3972	561	11	-	-	PUNCT
cana-3972	561	12	cover	cover	NOUN
cana-3972	561	13	.	.	PUNCT
cana-3972	562	1	3	3	X
cana-3972	562	2	)	)	PUNCT
cana-3972	562	3	every	every	DET
cana-3972	562	4	2	2	NUM
cana-3972	562	5	-	-	PUNCT
cana-3972	562	6	generated	generate	VERB
cana-3972	562	7	left	leave	VERB
cana-3972	562	8	r	r	NOUN
cana-3972	562	9	-	-	PUNCT
cana-3972	562	10	module	module	NOUN
cana-3972	562	11	has	have	VERB
cana-3972	562	12	a	a	DET
cana-3972	562	13	d41	d41	NOUN
cana-3972	562	14	-	-	PUNCT
cana-3972	562	15	cover	cover	NOUN
cana-3972	562	16	.	.	PUNCT
cana-3972	563	1	4	4	X
cana-3972	563	2	)	)	PUNCT
cana-3972	563	3	every	every	DET
cana-3972	563	4	left	left	ADJ
cana-3972	563	5	r	r	NOUN
cana-3972	563	6	-	-	PUNCT
cana-3972	563	7	module	module	NOUN
cana-3972	563	8	has	have	VERB
cana-3972	563	9	a	a	DET
cana-3972	563	10	d41	d41	NOUN
cana-3972	563	11	-	-	PUNCT
cana-3972	563	12	envelope	envelope	NOUN
cana-3972	563	13	.	.	PUNCT
cana-3972	564	1	5	5	X
cana-3972	564	2	)	)	PUNCT
cana-3972	564	3	every	every	DET
cana-3972	564	4	2	2	NUM
cana-3972	564	5	-	-	PUNCT
cana-3972	564	6	generated	generate	VERB
cana-3972	564	7	left	leave	VERB
cana-3972	564	8	r	r	NOUN
cana-3972	564	9	-	-	PUNCT
cana-3972	564	10	module	module	NOUN
cana-3972	564	11	has	have	VERB
cana-3972	564	12	a	a	DET
cana-3972	564	13	d41	d41	NOUN
cana-3972	564	14	-	-	PUNCT
cana-3972	564	15	envelope	envelope	NOUN
cana-3972	564	16	.	.	PUNCT
cana-3972	565	1	proof	proof	NOUN
cana-3972	565	2	:	:	PUNCT
cana-3972	565	3	the	the	DET
cana-3972	565	4	implications	implication	NOUN
cana-3972	565	5	1	1	NUM
cana-3972	565	6	)	)	PUNCT
cana-3972	565	7	⇒	⇒	NOUN
cana-3972	565	8	2	2	NUM
cana-3972	565	9	)	)	PUNCT
cana-3972	565	10	⇒	⇒	NOUN
cana-3972	565	11	3	3	NUM
cana-3972	565	12	)	)	PUNCT
cana-3972	565	13	and	and	CCONJ
cana-3972	565	14	1	1	X
cana-3972	565	15	)	)	PUNCT
cana-3972	565	16	⇒	⇒	NOUN
cana-3972	565	17	4	4	NUM
cana-3972	565	18	)	)	PUNCT
cana-3972	565	19	⇒	⇒	NOUN
cana-3972	565	20	5	5	NUM
cana-3972	565	21	)	)	PUNCT
cana-3972	565	22	are	be	AUX
cana-3972	565	23	straightforward	straightforward	ADJ
cana-3972	565	24	.	.	PUNCT
cana-3972	566	1	3	3	X
cana-3972	566	2	)	)	PUNCT
cana-3972	566	3	⇒	⇒	NOUN
cana-3972	566	4	1	1	NUM
cana-3972	566	5	)	)	PUNCT
cana-3972	566	6	.	.	PUNCT
cana-3972	567	1	let	let	VERB
cana-3972	567	2	𝑆	𝑆	PROPN
cana-3972	567	3	be	be	AUX
cana-3972	567	4	a	a	DET
cana-3972	567	5	simple	simple	ADJ
cana-3972	567	6	left	leave	VERB
cana-3972	567	7	r	r	NOUN
cana-3972	567	8	-	-	PUNCT
cana-3972	567	9	module	module	NOUN
cana-3972	567	10	,	,	PUNCT
cana-3972	567	11	and	and	CCONJ
cana-3972	567	12	denote	denote	VERB
cana-3972	567	13	𝜓	𝜓	PROPN
cana-3972	567	14	∶	∶	PROPN
cana-3972	567	15	𝑅	𝑅	PROPN
cana-3972	567	16	𝑅	𝑅	PROPN
cana-3972	567	17	→	→	SYM
cana-3972	567	18	𝑆	𝑆	PROPN
cana-3972	567	19	as	as	ADP
cana-3972	567	20	an	an	DET
cana-3972	567	21	epimorphism	epimorphism	NOUN
cana-3972	567	22	.	.	PUNCT
cana-3972	568	1	by	by	ADP
cana-3972	568	2	condition	condition	NOUN
cana-3972	568	3	3	3	NUM
cana-3972	568	4	)	)	PUNCT
cana-3972	568	5	,	,	PUNCT
cana-3972	568	6	m	m	PROPN
cana-3972	568	7	=	=	NOUN
cana-3972	568	8	rr	rr	PROPN
cana-3972	568	9	⊕	⊕	PROPN
cana-3972	568	10	s	s	PROPN
cana-3972	568	11	has	have	VERB
cana-3972	568	12	a	a	DET
cana-3972	568	13	d41	d41	NOUN
cana-3972	568	14	-	-	PUNCT
cana-3972	568	15	cover	cover	NOUN
cana-3972	568	16	,	,	PUNCT
cana-3972	568	17	which	which	PRON
cana-3972	568	18	we	we	PRON
cana-3972	568	19	can	can	AUX
cana-3972	568	20	denote	denote	VERB
cana-3972	568	21	as	as	ADP
cana-3972	568	22	𝜆	𝜆	DET
cana-3972	568	23	∶	∶	PROPN
cana-3972	568	24	𝐶	𝐶	PROPN
cana-3972	568	25	→	→	SYM
cana-3972	568	26	𝑀	𝑀	PROPN
cana-3972	568	27	,	,	PUNCT
cana-3972	568	28	where	where	SCONJ
cana-3972	568	29	c	c	PROPN
cana-3972	568	30	is	be	AUX
cana-3972	568	31	a	a	DET
cana-3972	568	32	d41	d41	NOUN
cana-3972	568	33	-	-	PUNCT
cana-3972	568	34	module	module	NOUN
cana-3972	568	35	.	.	PUNCT
cana-3972	569	1	let	let	VERB
cana-3972	569	2	communications	communication	NOUN
cana-3972	569	3	on	on	ADP
cana-3972	569	4	applied	apply	VERB
cana-3972	569	5	nonlinear	nonlinear	ADJ
cana-3972	569	6	analysis	analysis	NOUN
cana-3972	569	7	issn	issn	NOUN
cana-3972	569	8	:	:	PUNCT
cana-3972	569	9	1074	1074	NUM
cana-3972	569	10	-	-	PUNCT
cana-3972	569	11	133x	133x	NUM
cana-3972	569	12	vol	vol	NOUN
cana-3972	569	13	32	32	NUM
cana-3972	570	1	no	no	NOUN
cana-3972	570	2	.	.	PUNCT
cana-3972	571	1	9s	9s	NUM
cana-3972	571	2	(	(	PUNCT
cana-3972	571	3	2025	2025	NUM
cana-3972	571	4	)	)	PUNCT
cana-3972	571	5	690	690	NUM
cana-3972	572	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	572	2	𝛿1	𝛿1	NOUN
cana-3972	572	3	∶	∶	NOUN
cana-3972	572	4	𝑆	𝑆	PROPN
cana-3972	572	5	→	→	SYM
cana-3972	572	6	𝑀	𝑀	PROPN
cana-3972	572	7	and	and	CCONJ
cana-3972	572	8	𝛿2	𝛿2	PROPN
cana-3972	572	9	:	:	PUNCT
cana-3972	572	10	rr	rr	X
cana-3972	572	11	→	→	X
cana-3972	572	12	m	m	AUX
cana-3972	572	13	be	be	AUX
cana-3972	572	14	the	the	DET
cana-3972	572	15	inclusion	inclusion	NOUN
cana-3972	572	16	maps	map	NOUN
cana-3972	572	17	for	for	ADP
cana-3972	572	18	i	i	PRON
cana-3972	572	19	=	=	NOUN
cana-3972	572	20	1	1	NUM
cana-3972	572	21	,	,	PUNCT
cana-3972	572	22	2	2	NUM
cana-3972	572	23	.	.	PUNCT
cana-3972	572	24	notably	notably	ADV
cana-3972	572	25	,	,	PUNCT
cana-3972	572	26	both	both	CCONJ
cana-3972	572	27	𝑆	𝑆	PROPN
cana-3972	572	28	and	and	CCONJ
cana-3972	572	29	rr	rr	PROPN
cana-3972	572	30	are	be	AUX
cana-3972	572	31	d41modules	d41module	NOUN
cana-3972	572	32	,	,	PUNCT
cana-3972	572	33	and	and	CCONJ
cana-3972	572	34	there	there	PRON
cana-3972	572	35	exist	exist	VERB
cana-3972	572	36	homomorphisms	homomorphism	NOUN
cana-3972	572	37	𝛾1	𝛾1	PROPN
cana-3972	572	38	∶	∶	PROPN
cana-3972	572	39	𝑆	𝑆	PROPN
cana-3972	572	40	→	→	SYM
cana-3972	572	41	𝐶	𝐶	PROPN
cana-3972	572	42	and	and	CCONJ
cana-3972	572	43	𝛾2	𝛾2	VERB
cana-3972	572	44	:	:	PUNCT
cana-3972	572	45	rr	rr	X
cana-3972	572	46	→	→	SYM
cana-3972	572	47	c	c	NOUN
cana-3972	572	48	such	such	ADJ
cana-3972	572	49	that	that	SCONJ
cana-3972	572	50	𝜆	𝜆	DET
cana-3972	572	51	∘	∘	ADJ
cana-3972	572	52	𝛾𝑖	𝛾𝑖	NOUN
cana-3972	572	53	=	=	NUM
cana-3972	572	54	𝛿𝑖.	𝛿𝑖.	NOUN
cana-3972	572	55	clearly	clearly	ADV
cana-3972	572	56	,	,	PUNCT
cana-3972	572	57	we	we	PRON
cana-3972	572	58	have	have	VERB
cana-3972	572	59	idm	idm	NOUN
cana-3972	572	60	=	=	SYM
cana-3972	572	61	δ1	δ1	NOUN
cana-3972	572	62	⊕	⊕	PROPN
cana-3972	572	63	δ2	δ2	VERB
cana-3972	572	64	=	=	PUNCT
cana-3972	572	65	λ	λ	PART
cana-3972	572	66	◦	◦	NOUN
cana-3972	572	67	(	(	PUNCT
cana-3972	572	68	γ1	γ1	PROPN
cana-3972	572	69	⊕	⊕	PROPN
cana-3972	572	70	γ2	γ2	PROPN
cana-3972	572	71	)	)	PUNCT
cana-3972	572	72	.	.	PUNCT
cana-3972	573	1	this	this	PRON
cana-3972	573	2	shows	show	VERB
cana-3972	573	3	that	that	SCONJ
cana-3972	573	4	m	m	NOUN
cana-3972	573	5	is	be	AUX
cana-3972	573	6	isomorphic	isomorphic	ADJ
cana-3972	573	7	to	to	ADP
cana-3972	573	8	a	a	DET
cana-3972	573	9	direct	direct	ADJ
cana-3972	573	10	summand	summand	NOUN
cana-3972	573	11	of	of	ADP
cana-3972	573	12	c	c	PROPN
cana-3972	573	13	,	,	PUNCT
cana-3972	573	14	implying	imply	VERB
cana-3972	573	15	that	that	SCONJ
cana-3972	573	16	m	m	NOUN
cana-3972	573	17	is	be	AUX
cana-3972	573	18	a	a	DET
cana-3972	573	19	d41	d41	NOUN
cana-3972	573	20	-	-	PUNCT
cana-3972	573	21	module	module	NOUN
cana-3972	573	22	.	.	PUNCT
cana-3972	574	1	consequently	consequently	ADV
cana-3972	574	2	,	,	PUNCT
cana-3972	574	3	we	we	PRON
cana-3972	574	4	conclude	conclude	VERB
cana-3972	574	5	that	that	SCONJ
cana-3972	574	6	𝑘𝑒𝑟(𝜓	𝑘𝑒𝑟(𝜓	PROPN
cana-3972	574	7	)	)	PUNCT
cana-3972	574	8	is	be	AUX
cana-3972	574	9	a	a	DET
cana-3972	574	10	direct	direct	ADJ
cana-3972	574	11	summand	summand	NOUN
cana-3972	574	12	of	of	ADP
cana-3972	574	13	rr	rr	NOUN
cana-3972	574	14	by	by	ADP
cana-3972	574	15	[	[	X
cana-3972	574	16	24	24	NUM
cana-3972	574	17	,	,	PUNCT
cana-3972	574	18	proposition	proposition	NOUN
cana-3972	574	19	4	4	NUM
cana-3972	574	20	]	]	PUNCT
cana-3972	574	21	.	.	PUNCT
cana-3972	575	1	therefore	therefore	ADV
cana-3972	575	2	,	,	PUNCT
cana-3972	575	3	s	s	VERB
cana-3972	575	4	is	be	AUX
cana-3972	575	5	a	a	DET
cana-3972	575	6	projective	projective	ADJ
cana-3972	575	7	module	module	NOUN
cana-3972	575	8	,	,	PUNCT
cana-3972	575	9	leading	lead	VERB
cana-3972	575	10	us	we	PRON
cana-3972	575	11	to	to	PART
cana-3972	575	12	conclude	conclude	VERB
cana-3972	575	13	that	that	SCONJ
cana-3972	575	14	r	r	NOUN
cana-3972	575	15	is	be	AUX
cana-3972	575	16	semisimple	semisimple	ADJ
cana-3972	575	17	.	.	PUNCT
cana-3972	576	1	5	5	X
cana-3972	576	2	)	)	PUNCT
cana-3972	576	3	⇒	⇒	NOUN
cana-3972	576	4	1	1	NUM
cana-3972	576	5	)	)	PUNCT
cana-3972	576	6	.	.	PUNCT
cana-3972	577	1	let	let	VERB
cana-3972	577	2	s	s	PRON
cana-3972	577	3	be	be	AUX
cana-3972	577	4	a	a	DET
cana-3972	577	5	simple	simple	ADJ
cana-3972	577	6	right	right	ADJ
cana-3972	577	7	r	r	NOUN
cana-3972	577	8	-	-	NOUN
cana-3972	577	9	module	module	NOUN
cana-3972	577	10	,	,	PUNCT
cana-3972	577	11	and	and	CCONJ
cana-3972	577	12	let	let	VERB
cana-3972	577	13	ψ	ψ	X
cana-3972	577	14	:	:	PUNCT
cana-3972	577	15	rr	rr	X
cana-3972	577	16	→	→	SYM
cana-3972	577	17	s	s	AUX
cana-3972	577	18	be	be	AUX
cana-3972	577	19	an	an	DET
cana-3972	577	20	epimorphism	epimorphism	NOUN
cana-3972	577	21	.	.	PUNCT
cana-3972	578	1	by	by	ADP
cana-3972	578	2	condition	condition	NOUN
cana-3972	578	3	5	5	NUM
cana-3972	578	4	)	)	PUNCT
cana-3972	578	5	,	,	PUNCT
cana-3972	578	6	m	m	VERB
cana-3972	578	7	=	=	NOUN
cana-3972	578	8	r	r	NOUN
cana-3972	578	9	r	r	NOUN
cana-3972	578	10	⊕	⊕	PROPN
cana-3972	578	11	s	s	PART
cana-3972	578	12	has	have	VERB
cana-3972	578	13	a	a	DET
cana-3972	578	14	d41	d41	NOUN
cana-3972	578	15	-	-	PUNCT
cana-3972	578	16	envelope	envelope	NOUN
cana-3972	578	17	,	,	PUNCT
cana-3972	578	18	denoted	denote	VERB
cana-3972	578	19	by	by	ADP
cana-3972	578	20	α	α	NOUN
cana-3972	578	21	:	:	PUNCT
cana-3972	578	22	m	m	VERB
cana-3972	578	23	→	→	SYM
cana-3972	578	24	k	k	X
cana-3972	578	25	,	,	PUNCT
cana-3972	578	26	where	where	SCONJ
cana-3972	578	27	k	k	PROPN
cana-3972	578	28	is	be	AUX
cana-3972	578	29	a	a	DET
cana-3972	578	30	d41	d41	NOUN
cana-3972	578	31	-	-	PUNCT
cana-3972	578	32	module	module	NOUN
cana-3972	578	33	.	.	PUNCT
cana-3972	579	1	since	since	SCONJ
cana-3972	579	2	both	both	DET
cana-3972	579	3	s	s	X
cana-3972	579	4	and	and	CCONJ
cana-3972	579	5	r	r	NOUN
cana-3972	579	6	are	be	AUX
cana-3972	579	7	d41	d41	NOUN
cana-3972	579	8	-	-	PUNCT
cana-3972	579	9	modules	module	NOUN
cana-3972	579	10	,	,	PUNCT
cana-3972	579	11	there	there	PRON
cana-3972	579	12	exist	exist	VERB
cana-3972	579	13	homo	homo	NOUN
cana-3972	579	14	-	-	PUNCT
cana-3972	579	15	morphisms	morphism	NOUN
cana-3972	579	16	𝑓1	𝑓1	PROPN
cana-3972	579	17	∶	∶	PROPN
cana-3972	579	18	𝐾	𝐾	PROPN
cana-3972	579	19	→	→	SYM
cana-3972	579	20	𝑆	𝑆	PROPN
cana-3972	579	21	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3972	579	22	𝑓2	𝑓2	PROPN
cana-3972	579	23	∶	∶	NOUN
cana-3972	579	24	𝐾	𝐾	PROPN
cana-3972	579	25	→	→	SYM
cana-3972	579	26	𝑅	𝑅	PROPN
cana-3972	579	27	such	such	ADJ
cana-3972	579	28	that	that	DET
cana-3972	579	29	𝑓𝑖𝛼	𝑓𝑖𝛼	NOUN
cana-3972	579	30	=	=	SYM
cana-3972	579	31	𝜋𝑖	𝜋𝑖	NOUN
cana-3972	579	32	,	,	PUNCT
cana-3972	579	33	where	where	SCONJ
cana-3972	579	34	𝜋1	𝜋1	NOUN
cana-3972	579	35	∶	∶	VERB
cana-3972	579	36	𝑀	𝑀	PROPN
cana-3972	579	37	→	→	SYM
cana-3972	579	38	𝑆	𝑆	PROPN
cana-3972	579	39	and	and	CCONJ
cana-3972	579	40	𝜋2	𝜋2	PROPN
cana-3972	579	41	∶	∶	PROPN
cana-3972	579	42	𝑀	𝑀	PROPN
cana-3972	579	43	→	→	SYM
cana-3972	579	44	𝑅	𝑅	PROPN
cana-3972	579	45	are	be	AUX
cana-3972	579	46	the	the	DET
cana-3972	579	47	projection	projection	NOUN
cana-3972	579	48	maps	map	NOUN
cana-3972	579	49	.	.	PUNCT
cana-3972	580	1	additionally	additionally	ADV
cana-3972	580	2	,	,	PUNCT
cana-3972	580	3	there	there	PRON
cana-3972	580	4	exists	exist	VERB
cana-3972	580	5	a	a	DET
cana-3972	580	6	homomorphism	homomorphism	NOUN
cana-3972	580	7	𝜙	𝜙	X
cana-3972	580	8	∶	∶	NOUN
cana-3972	580	9	𝐾	𝐾	PROPN
cana-3972	580	10	→	→	SYM
cana-3972	580	11	𝑀	𝑀	PROPN
cana-3972	580	12	such	such	ADJ
cana-3972	580	13	that	that	DET
cana-3972	580	14	𝜋𝑖𝜙	𝜋𝑖𝜙	NOUN
cana-3972	580	15	=	=	SYM
cana-3972	580	16	𝑓𝑖	𝑓𝑖	PROPN
cana-3972	580	17	for	for	ADP
cana-3972	580	18	i	i	PRON
cana-3972	580	19	=	=	NOUN
cana-3972	580	20	1	1	NUM
cana-3972	580	21	,	,	PUNCT
cana-3972	580	22	2	2	NUM
cana-3972	580	23	.	.	PUNCT
cana-3972	581	1	this	this	PRON
cana-3972	581	2	implies	imply	VERB
cana-3972	581	3	that	that	SCONJ
cana-3972	581	4	ϕα	ϕα	ADV
cana-3972	581	5	=	=	SYM
cana-3972	581	6	idm	idm	NOUN
cana-3972	581	7	,	,	PUNCT
cana-3972	581	8	making	make	VERB
cana-3972	581	9	α	α	PRON
cana-3972	581	10	a	a	DET
cana-3972	581	11	split	split	ADJ
cana-3972	581	12	monomorphism	monomorphism	NOUN
cana-3972	581	13	.	.	PUNCT
cana-3972	582	1	thus	thus	ADV
cana-3972	582	2	,	,	PUNCT
cana-3972	582	3	s	s	VERB
cana-3972	582	4	⊕e	⊕e	NOUN
cana-3972	582	5	(	(	PUNCT
cana-3972	582	6	s	s	X
cana-3972	582	7	)	)	PUNCT
cana-3972	582	8	is	be	AUX
cana-3972	582	9	isomorphic	isomorphic	ADJ
cana-3972	582	10	to	to	ADP
cana-3972	582	11	a	a	DET
cana-3972	582	12	direct	direct	ADJ
cana-3972	582	13	summand	summand	NOUN
cana-3972	582	14	of	of	ADP
cana-3972	582	15	k	k	PROPN
cana-3972	582	16	,	,	PUNCT
cana-3972	582	17	indicating	indicate	VERB
cana-3972	582	18	that	that	SCONJ
cana-3972	582	19	s	s	VERB
cana-3972	582	20	⊕r	⊕r	NOUN
cana-3972	582	21	is	be	AUX
cana-3972	582	22	also	also	ADV
cana-3972	582	23	a	a	DET
cana-3972	582	24	d41	d41	NOUN
cana-3972	582	25	-	-	PUNCT
cana-3972	582	26	module	module	NOUN
cana-3972	582	27	.	.	PUNCT
cana-3972	583	1	from	from	ADP
cana-3972	583	2	this	this	PRON
cana-3972	583	3	,	,	PUNCT
cana-3972	583	4	we	we	PRON
cana-3972	583	5	conclude	conclude	VERB
cana-3972	583	6	that	that	SCONJ
cana-3972	583	7	𝑘𝑒𝑟(𝜓	𝑘𝑒𝑟(𝜓	PROPN
cana-3972	583	8	)	)	PUNCT
cana-3972	583	9	is	be	AUX
cana-3972	583	10	a	a	DET
cana-3972	583	11	direct	direct	ADJ
cana-3972	583	12	summand	summand	NOUN
cana-3972	583	13	of	of	ADP
cana-3972	583	14	rr	rr	PROPN
cana-3972	583	15	.	.	PUNCT
cana-3972	584	1	therefore	therefore	ADV
cana-3972	584	2	,	,	PUNCT
cana-3972	584	3	s	s	VERB
cana-3972	584	4	is	be	AUX
cana-3972	584	5	a	a	DET
cana-3972	584	6	projective	projective	ADJ
cana-3972	584	7	module	module	NOUN
cana-3972	584	8	,	,	PUNCT
cana-3972	584	9	leading	lead	VERB
cana-3972	584	10	us	we	PRON
cana-3972	584	11	to	to	ADP
cana-3972	584	12	the	the	DET
cana-3972	584	13	conclusion	conclusion	NOUN
cana-3972	584	14	that	that	SCONJ
cana-3972	584	15	r	r	NOUN
cana-3972	584	16	is	be	AUX
cana-3972	584	17	semisimple	semisimple	NOUN
cana-3972	584	18	.	.	PUNCT
cana-3972	585	1	proposition	proposition	NOUN
cana-3972	585	2	4.12	4.12	NUM
cana-3972	585	3	.	.	PUNCT
cana-3972	586	1	the	the	DET
cana-3972	586	2	following	follow	VERB
cana-3972	586	3	conditions	condition	NOUN
cana-3972	586	4	are	be	AUX
cana-3972	586	5	equivalent	equivalent	ADJ
cana-3972	586	6	for	for	ADP
cana-3972	586	7	a	a	DET
cana-3972	586	8	ring	ring	NOUN
cana-3972	586	9	r	r	NOUN
cana-3972	586	10	:	:	PUNCT
cana-3972	586	11	1	1	NUM
cana-3972	586	12	)	)	PUNCT
cana-3972	586	13	r	r	NOUN
cana-3972	586	14	is	be	AUX
cana-3972	586	15	hereditary	hereditary	ADJ
cana-3972	586	16	(	(	PUNCT
cana-3972	586	17	or	or	CCONJ
cana-3972	586	18	semihereditary	semihereditary	ADJ
cana-3972	586	19	)	)	PUNCT
cana-3972	586	20	.	.	PUNCT
cana-3972	587	1	2	2	X
cana-3972	587	2	)	)	PUNCT
cana-3972	587	3	every	every	DET
cana-3972	587	4	submodule	submodule	NOUN
cana-3972	587	5	(	(	PUNCT
cana-3972	587	6	or	or	CCONJ
cana-3972	587	7	finitely	finitely	ADV
cana-3972	587	8	generated	generate	VERB
cana-3972	587	9	submodule	submodule	NOUN
cana-3972	587	10	)	)	PUNCT
cana-3972	587	11	of	of	ADP
cana-3972	587	12	a	a	DET
cana-3972	587	13	projective	projective	ADJ
cana-3972	587	14	r	r	NOUN
cana-3972	587	15	-	-	PUNCT
cana-3972	587	16	module	module	NOUN
cana-3972	587	17	is	be	AUX
cana-3972	587	18	a	a	DET
cana-3972	587	19	d41	d41	NOUN
cana-3972	587	20	-	-	PUNCT
cana-3972	587	21	module	module	NOUN
cana-3972	587	22	.	.	PUNCT
cana-3972	588	1	3	3	X
cana-3972	588	2	)	)	PUNCT
cana-3972	588	3	every	every	DET
cana-3972	588	4	submodule	submodule	NOUN
cana-3972	588	5	(	(	PUNCT
cana-3972	588	6	or	or	CCONJ
cana-3972	588	7	finitely	finitely	ADV
cana-3972	588	8	generated	generate	VERB
cana-3972	588	9	submodule	submodule	NOUN
cana-3972	588	10	)	)	PUNCT
cana-3972	588	11	of	of	ADP
cana-3972	588	12	a	a	DET
cana-3972	588	13	projective	projective	ADJ
cana-3972	588	14	r	r	NOUN
cana-3972	588	15	-	-	PUNCT
cana-3972	588	16	module	module	NOUN
cana-3972	588	17	is	be	AUX
cana-3972	588	18	a	a	DET
cana-3972	588	19	strongly	strongly	ADV
cana-3972	588	20	d41module	d41module	PROPN
cana-3972	588	21	.	.	PUNCT
cana-3972	589	1	proof	proof	NOUN
cana-3972	589	2	:	:	PUNCT
cana-3972	589	3	the	the	DET
cana-3972	589	4	implications	implication	NOUN
cana-3972	589	5	1	1	NUM
cana-3972	589	6	)	)	PUNCT
cana-3972	589	7	⇒	⇒	NOUN
cana-3972	589	8	3	3	NUM
cana-3972	589	9	)	)	PUNCT
cana-3972	589	10	⇒	⇒	NOUN
cana-3972	589	11	2	2	NUM
cana-3972	589	12	)	)	PUNCT
cana-3972	589	13	and	and	CCONJ
cana-3972	589	14	1	1	X
cana-3972	589	15	)	)	PUNCT
cana-3972	589	16	⇒	⇒	NOUN
cana-3972	589	17	2	2	NUM
cana-3972	589	18	)	)	PUNCT
cana-3972	589	19	are	be	AUX
cana-3972	589	20	evident	evident	ADJ
cana-3972	589	21	.	.	PUNCT
cana-3972	590	1	2	2	X
cana-3972	590	2	)	)	PUNCT
cana-3972	590	3	⇒	⇒	NOUN
cana-3972	590	4	1	1	NUM
cana-3972	590	5	)	)	PUNCT
cana-3972	590	6	.	.	PUNCT
cana-3972	591	1	let	let	VERB
cana-3972	591	2	k	k	PRON
cana-3972	591	3	be	be	AUX
cana-3972	591	4	a	a	DET
cana-3972	591	5	projective	projective	ADJ
cana-3972	591	6	submodule	submodule	NOUN
cana-3972	591	7	of	of	ADP
cana-3972	591	8	an	an	DET
cana-3972	591	9	r	r	NOUN
cana-3972	591	10	-	-	PUNCT
cana-3972	591	11	module	module	NOUN
cana-3972	591	12	m	m	NOUN
cana-3972	591	13	.	.	PUNCT
cana-3972	592	1	consider	consider	VERB
cana-3972	592	2	a	a	DET
cana-3972	592	3	free	free	ADJ
cana-3972	592	4	r	r	NOUN
cana-3972	592	5	-	-	PUNCT
cana-3972	592	6	module	module	NOUN
cana-3972	592	7	f	f	NOUN
cana-3972	592	8	and	and	CCONJ
cana-3972	592	9	an	an	DET
cana-3972	592	10	epimorphism	epimorphism	NOUN
cana-3972	592	11	𝑔	𝑔	PROPN
cana-3972	592	12	∶	∶	PROPN
cana-3972	592	13	𝐹	𝐹	PROPN
cana-3972	592	14	→	→	SYM
cana-3972	592	15	𝐾.	𝐾.	PROPN
cana-3972	592	16	then	then	ADV
cana-3972	592	17	f	f	PROPN
cana-3972	592	18	⊕	⊕	PROPN
cana-3972	592	19	k	k	PROPN
cana-3972	592	20	is	be	AUX
cana-3972	592	21	a	a	DET
cana-3972	592	22	projective	projective	ADJ
cana-3972	592	23	submodule	submodule	NOUN
cana-3972	592	24	of	of	ADP
cana-3972	592	25	f	f	PROPN
cana-3972	592	26	⊕	⊕	PROPN
cana-3972	592	27	m	m	PROPN
cana-3972	592	28	,	,	PUNCT
cana-3972	592	29	which	which	PRON
cana-3972	592	30	implies	imply	VERB
cana-3972	592	31	that	that	SCONJ
cana-3972	592	32	f	f	PROPN
cana-3972	592	33	⊕	⊕	PROPN
cana-3972	592	34	k	k	PROPN
cana-3972	592	35	is	be	AUX
cana-3972	592	36	a	a	DET
cana-3972	592	37	d41	d41	NOUN
cana-3972	592	38	-	-	PUNCT
cana-3972	592	39	module	module	NOUN
cana-3972	592	40	.	.	PUNCT
cana-3972	593	1	consequently	consequently	ADV
cana-3972	593	2	,	,	PUNCT
cana-3972	593	3	k	k	PROPN
cana-3972	593	4	is	be	AUX
cana-3972	593	5	a	a	DET
cana-3972	593	6	pure	pure	ADJ
cana-3972	593	7	direct	direct	ADJ
cana-3972	593	8	-	-	PUNCT
cana-3972	593	9	projective	projective	NOUN
cana-3972	593	10	module	module	NOUN
cana-3972	593	11	,	,	PUNCT
cana-3972	593	12	leading	lead	VERB
cana-3972	593	13	to	to	ADP
cana-3972	593	14	the	the	DET
cana-3972	593	15	conclusion	conclusion	NOUN
cana-3972	593	16	that	that	SCONJ
cana-3972	593	17	the	the	DET
cana-3972	593	18	epimorphism	epimorphism	NOUN
cana-3972	593	19	𝑔	𝑔	PROPN
cana-3972	593	20	∶	∶	PROPN
cana-3972	593	21	𝐹	𝐹	PROPN
cana-3972	593	22	→	→	SYM
cana-3972	593	23	𝐾	𝐾	PROPN
cana-3972	593	24	→	→	SYM
cana-3972	593	25	0	0	NUM
cana-3972	593	26	splits	split	NOUN
cana-3972	593	27	.	.	PUNCT
cana-3972	594	1	thus	thus	ADV
cana-3972	594	2	,	,	PUNCT
cana-3972	594	3	k	k	PROPN
cana-3972	594	4	is	be	AUX
cana-3972	594	5	projective	projective	ADJ
cana-3972	594	6	,	,	PUNCT
cana-3972	594	7	which	which	PRON
cana-3972	594	8	implies	imply	VERB
cana-3972	594	9	that	that	SCONJ
cana-3972	594	10	r	r	NOUN
cana-3972	594	11	is	be	AUX
cana-3972	594	12	hereditary	hereditary	ADJ
cana-3972	594	13	.	.	PUNCT
cana-3972	595	1	proposition	proposition	NOUN
cana-3972	595	2	4.13	4.13	NUM
cana-3972	595	3	.	.	PUNCT
cana-3972	596	1	for	for	ADP
cana-3972	596	2	a	a	DET
cana-3972	596	3	ring	ring	NOUN
cana-3972	596	4	r	r	NOUN
cana-3972	596	5	,	,	PUNCT
cana-3972	596	6	the	the	DET
cana-3972	596	7	following	follow	VERB
cana-3972	596	8	conditions	condition	NOUN
cana-3972	596	9	are	be	AUX
cana-3972	596	10	equivalent	equivalent	ADJ
cana-3972	596	11	:	:	PUNCT
cana-3972	596	12	1	1	X
cana-3972	596	13	)	)	PUNCT
cana-3972	596	14	r	r	NOUN
cana-3972	596	15	is	be	AUX
cana-3972	596	16	a	a	DET
cana-3972	596	17	semisimple	semisimple	NOUN
cana-3972	596	18	ring	ring	NOUN
cana-3972	596	19	.	.	PUNCT
cana-3972	597	1	2	2	NUM
cana-3972	597	2	)	)	PUNCT
cana-3972	597	3	every	every	DET
cana-3972	597	4	d41	d41	NOUN
cana-3972	597	5	-	-	PUNCT
cana-3972	597	6	module	module	NOUN
cana-3972	597	7	over	over	ADP
cana-3972	597	8	r	r	NOUN
cana-3972	597	9	is	be	AUX
cana-3972	597	10	projective	projective	ADJ
cana-3972	597	11	.	.	PUNCT
cana-3972	598	1	3	3	X
cana-3972	598	2	)	)	PUNCT
cana-3972	598	3	every	every	PRON
cana-3972	598	4	cosingular	cosingular	ADJ
cana-3972	598	5	pure	pure	ADJ
cana-3972	598	6	direct	direct	ADJ
cana-3972	598	7	-	-	PUNCT
cana-3972	598	8	projective	projective	ADJ
cana-3972	598	9	r	r	NOUN
cana-3972	598	10	-	-	PUNCT
cana-3972	598	11	module	module	NOUN
cana-3972	598	12	is	be	AUX
cana-3972	598	13	projective	projective	ADJ
cana-3972	598	14	.	.	PUNCT
cana-3972	599	1	4	4	X
cana-3972	599	2	)	)	PUNCT
cana-3972	599	3	every	every	DET
cana-3972	599	4	quasi	quasi	ADJ
cana-3972	599	5	-	-	ADJ
cana-3972	599	6	pure	pure	ADJ
cana-3972	599	7	-	-	PUNCT
cana-3972	599	8	injective	injective	ADJ
cana-3972	599	9	rmodule	rmodule	NOUN
cana-3972	599	10	is	be	AUX
cana-3972	599	11	projective	projective	ADJ
cana-3972	599	12	.	.	PUNCT
cana-3972	600	1	5	5	NUM
cana-3972	600	2	)	)	PUNCT
cana-3972	600	3	every	every	DET
cana-3972	600	4	pure	pure	ADJ
cana-3972	600	5	-	-	PUNCT
cana-3972	600	6	injective	injective	ADJ
cana-3972	600	7	r	r	NOUN
cana-3972	600	8	-	-	PUNCT
cana-3972	600	9	module	module	NOUN
cana-3972	600	10	is	be	AUX
cana-3972	600	11	projective	projective	ADJ
cana-3972	600	12	.	.	PUNCT
cana-3972	601	1	proof	proof	NOUN
cana-3972	601	2	:	:	PUNCT
cana-3972	601	3	1	1	X
cana-3972	601	4	)	)	PUNCT
cana-3972	601	5	⇒	⇒	NOUN
cana-3972	601	6	2	2	NUM
cana-3972	601	7	)	)	PUNCT
cana-3972	601	8	.	.	PUNCT
cana-3972	602	1	since	since	SCONJ
cana-3972	602	2	r	r	NOUN
cana-3972	602	3	is	be	AUX
cana-3972	602	4	a	a	DET
cana-3972	602	5	semisimple	semisimple	NOUN
cana-3972	602	6	ring	ring	NOUN
cana-3972	602	7	,	,	PUNCT
cana-3972	602	8	every	every	DET
cana-3972	602	9	r	r	NOUN
cana-3972	602	10	-	-	PUNCT
cana-3972	602	11	module	module	NOUN
cana-3972	602	12	m	m	NOUN
cana-3972	602	13	is	be	AUX
cana-3972	602	14	projective	projective	ADJ
cana-3972	603	1	[	[	X
cana-3972	603	2	22	22	NUM
cana-3972	603	3	,	,	PUNCT
cana-3972	603	4	proposition	proposition	NOUN
cana-3972	603	5	20.7	20.7	NUM
cana-3972	603	6	]	]	PUNCT
cana-3972	603	7	.	.	PUNCT
cana-3972	604	1	the	the	DET
cana-3972	604	2	implications	implication	NOUN
cana-3972	604	3	2	2	NUM
cana-3972	604	4	)	)	PUNCT
cana-3972	604	5	⇒	⇒	NOUN
cana-3972	604	6	3	3	NUM
cana-3972	604	7	)	)	PUNCT
cana-3972	604	8	⇒	⇒	NOUN
cana-3972	604	9	4	4	NUM
cana-3972	604	10	)	)	PUNCT
cana-3972	604	11	⇒	⇒	NOUN
cana-3972	604	12	5	5	NUM
cana-3972	604	13	)	)	PUNCT
cana-3972	604	14	are	be	AUX
cana-3972	604	15	straightforward	straightforward	ADJ
cana-3972	604	16	.	.	PUNCT
cana-3972	605	1	4	4	X
cana-3972	605	2	)	)	PUNCT
cana-3972	605	3	⇒	⇒	NOUN
cana-3972	605	4	1	1	NUM
cana-3972	605	5	)	)	PUNCT
cana-3972	605	6	.	.	PUNCT
cana-3972	606	1	if	if	SCONJ
cana-3972	606	2	every	every	DET
cana-3972	606	3	pure	pure	ADJ
cana-3972	606	4	-	-	PUNCT
cana-3972	606	5	injective	injective	ADJ
cana-3972	606	6	left	leave	VERB
cana-3972	606	7	r	r	NOUN
cana-3972	606	8	-	-	PUNCT
cana-3972	606	9	module	module	NOUN
cana-3972	606	10	is	be	AUX
cana-3972	606	11	projective	projective	ADJ
cana-3972	606	12	,	,	PUNCT
cana-3972	606	13	i	i	PRON
cana-3972	606	14	t	t	PROPN
cana-3972	606	15	follows	follow	VERB
cana-3972	606	16	that	that	SCONJ
cana-3972	606	17	every	every	DET
cana-3972	606	18	injective	injective	ADJ
cana-3972	606	19	module	module	NOUN
cana-3972	606	20	is	be	AUX
cana-3972	606	21	projective	projective	ADJ
cana-3972	606	22	.	.	PUNCT
cana-3972	607	1	thus	thus	ADV
cana-3972	607	2	,	,	PUNCT
cana-3972	607	3	condition	condition	NOUN
cana-3972	607	4	1	1	NUM
cana-3972	607	5	)	)	PUNCT
cana-3972	607	6	holds	hold	VERB
cana-3972	607	7	according	accord	VERB
cana-3972	607	8	to	to	ADP
cana-3972	607	9	[	[	X
cana-3972	607	10	22	22	NUM
cana-3972	607	11	,	,	PUNCT
cana-3972	607	12	proposition	proposition	NOUN
cana-3972	607	13	20.7	20.7	NUM
cana-3972	607	14	]	]	PUNCT
cana-3972	607	15	.	.	PUNCT
cana-3972	608	1	communications	communication	NOUN
cana-3972	608	2	on	on	ADP
cana-3972	608	3	applied	apply	VERB
cana-3972	608	4	nonlinear	nonlinear	ADJ
cana-3972	608	5	analysis	analysis	NOUN
cana-3972	608	6	issn	issn	NOUN
cana-3972	608	7	:	:	PUNCT
cana-3972	608	8	1074	1074	NUM
cana-3972	608	9	-	-	PUNCT
cana-3972	608	10	133x	133x	NUM
cana-3972	608	11	vol	vol	NOUN
cana-3972	608	12	32	32	NUM
cana-3972	608	13	no	no	NOUN
cana-3972	608	14	.	.	PUNCT
cana-3972	609	1	9s	9s	NUM
cana-3972	609	2	(	(	PUNCT
cana-3972	609	3	2025	2025	NUM
cana-3972	609	4	)	)	PUNCT
cana-3972	610	1	691	691	NUM
cana-3972	610	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3972	610	3	references	reference	NOUN
cana-3972	610	4	[	[	X
cana-3972	610	5	1	1	NUM
cana-3972	610	6	]	]	X
cana-3972	610	7	y.	y.	NOUN
cana-3972	610	8	baba	baba	PROPN
cana-3972	610	9	and	and	CCONJ
cana-3972	610	10	k.	k.	PROPN
cana-3972	610	11	oshiro	oshiro	PROPN
cana-3972	610	12	.	.	PUNCT
cana-3972	611	1	classical	classical	ADJ
cana-3972	611	2	artinian	artinian	ADJ
cana-3972	611	3	rings	ring	NOUN
cana-3972	611	4	and	and	CCONJ
cana-3972	611	5	related	related	ADJ
cana-3972	611	6	topics	topic	NOUN
cana-3972	611	7	.	.	PUNCT
cana-3972	612	1	world	world	NOUN
cana-3972	612	2	scientific	scientific	PROPN
cana-3972	612	3	publishing	publishing	PROPN
cana-3972	612	4	co.	co.	PROPN
cana-3972	612	5	pte	pte	PROPN
cana-3972	612	6	.	.	PROPN
cana-3972	612	7	ltd	ltd	PROPN
cana-3972	612	8	.	.	PROPN
cana-3972	612	9	,	,	PUNCT
cana-3972	612	10	london	london	PROPN
cana-3972	612	11	,	,	PUNCT
cana-3972	612	12	2009	2009	NUM
cana-3972	612	13	.	.	PUNCT
cana-3972	613	1	[	[	X
cana-3972	613	2	2	2	X
cana-3972	613	3	]	]	PUNCT
cana-3972	613	4	j.	j.	PROPN
cana-3972	613	5	clark	clark	PROPN
cana-3972	613	6	,	,	PUNCT
cana-3972	613	7	c.	c.	PROPN
cana-3972	613	8	lomp	lomp	PROPN
cana-3972	613	9	,	,	PUNCT
cana-3972	613	10	n.	n.	NOUN
cana-3972	613	11	vanaja	vanaja	PROPN
cana-3972	613	12	,	,	PUNCT
cana-3972	613	13	and	and	CCONJ
cana-3972	613	14	r.	r.	PROPN
cana-3972	613	15	wisbauer	wisbauer	NOUN
cana-3972	613	16	.	.	PUNCT
cana-3972	614	1	lifting	lift	VERB
cana-3972	614	2	modules	module	NOUN
cana-3972	614	3	.	.	PUNCT
cana-3972	615	1	supplements	supplement	NOUN
cana-3972	615	2	and	and	CCONJ
cana-3972	615	3	projectivity	projectivity	NOUN
cana-3972	615	4	in	in	ADP
cana-3972	615	5	module	module	NOUN
cana-3972	615	6	theory	theory	NOUN
cana-3972	615	7	.	.	PUNCT
cana-3972	616	1	frontiers	frontier	NOUN
cana-3972	616	2	in	in	ADP
cana-3972	616	3	mathematics	mathematics	PROPN
cana-3972	616	4	,	,	PUNCT
cana-3972	616	5	berlin	berlin	PROPN
cana-3972	616	6	,	,	PUNCT
cana-3972	616	7	2000	2000	NUM
cana-3972	616	8	.	.	PUNCT
cana-3972	617	1	[	[	X
cana-3972	617	2	3	3	NUM
cana-3972	617	3	]	]	PUNCT
cana-3972	617	4	a.	a.	PROPN
cana-3972	617	5	d.	d.	PROPN
cana-3972	617	6	diallo	diallo	PROPN
cana-3972	617	7	,	,	PUNCT
cana-3972	617	8	p.	p.	PROPN
cana-3972	617	9	c.	c.	PROPN
cana-3972	617	10	diop	diop	PROPN
cana-3972	617	11	,	,	PUNCT
cana-3972	617	12	f.	f.	PROPN
cana-3972	617	13	kourki	kourki	PROPN
cana-3972	617	14	,	,	PUNCT
cana-3972	617	15	and	and	CCONJ
cana-3972	617	16	r.	r.	PROPN
cana-3972	617	17	tribak	tribak	PROPN
cana-3972	617	18	.	.	PUNCT
cana-3972	618	1	on	on	ADP
cana-3972	618	2	a	a	DET
cana-3972	618	3	generalization	generalization	NOUN
cana-3972	618	4	of	of	ADP
cana-3972	618	5	c4	c4	NOUN
cana-3972	618	6	-	-	PUNCT
cana-3972	618	7	modules	module	NOUN
cana-3972	618	8	.	.	PUNCT
cana-3972	619	1	algebra	algebra	NOUN
cana-3972	619	2	and	and	CCONJ
cana-3972	619	3	its	its	PRON
cana-3972	619	4	applications	application	NOUN
cana-3972	619	5	.	.	PUNCT
cana-3972	620	1	icaa	icaa	NOUN
cana-3972	620	2	2023	2023	NUM
cana-3972	620	3	.	.	PUNCT
cana-3972	621	1	springer	springer	NOUN
cana-3972	621	2	proceedings	proceeding	NOUN
cana-3972	621	3	in	in	ADP
cana-3972	621	4	mathematics	mathematics	PROPN
cana-3972	621	5	&	&	CCONJ
cana-3972	621	6	statistics	statistic	NOUN
cana-3972	621	7	,	,	PUNCT
cana-3972	621	8	474	474	NUM
cana-3972	621	9	:	:	PUNCT
cana-3972	621	10	387–404	387–404	NUM
cana-3972	621	11	,	,	PUNCT
cana-3972	621	12	2025	2025	NUM
cana-3972	621	13	.	.	PUNCT
cana-3972	622	1	[	[	X
cana-3972	622	2	4	4	NUM
cana-3972	622	3	]	]	X
cana-3972	622	4	n.	n.	NOUN
cana-3972	622	5	ding	ding	PROPN
cana-3972	622	6	,	,	PUNCT
cana-3972	622	7	y.	y.	PROPN
cana-3972	622	8	ibrahim	ibrahim	PROPN
cana-3972	622	9	,	,	PUNCT
cana-3972	622	10	m.	m.	NOUN
cana-3972	622	11	yousif	yousif	PROPN
cana-3972	622	12	,	,	PUNCT
cana-3972	622	13	and	and	CCONJ
cana-3972	622	14	y.	y.	PROPN
cana-3972	622	15	zhou	zhou	PROPN
cana-3972	622	16	.	.	PUNCT
cana-3972	622	17	d4	d4	NOUN
cana-3972	622	18	-	-	PUNCT
cana-3972	622	19	modules	module	NOUN
cana-3972	622	20	.	.	PUNCT
cana-3972	622	21	journal	journal	NOUN
cana-3972	622	22	of	of	ADP
cana-3972	622	23	algebra	algebra	PROPN
cana-3972	622	24	and	and	CCONJ
cana-3972	622	25	its	its	PRON
cana-3972	622	26	applications	application	NOUN
cana-3972	622	27	,	,	PUNCT
cana-3972	622	28	page	page	NOUN
cana-3972	622	29	1750166	1750166	NUM
cana-3972	622	30	(	(	PUNCT
cana-3972	622	31	25	25	NUM
cana-3972	622	32	pages	page	NOUN
cana-3972	622	33	)	)	PUNCT
cana-3972	622	34	,	,	PUNCT
cana-3972	622	35	2017	2017	NUM
cana-3972	622	36	.	.	PUNCT
cana-3972	623	1	[	[	X
cana-3972	623	2	5	5	NUM
cana-3972	623	3	]	]	X
cana-3972	623	4	n.	n.	NOUN
cana-3972	623	5	ding	ding	PROPN
cana-3972	623	6	,	,	PUNCT
cana-3972	623	7	y.	y.	PROPN
cana-3972	623	8	ibrahim	ibrahim	PROPN
cana-3972	623	9	,	,	PUNCT
cana-3972	623	10	m.	m.	NOUN
cana-3972	623	11	yousif	yousif	PROPN
cana-3972	623	12	,	,	PUNCT
cana-3972	623	13	and	and	CCONJ
cana-3972	623	14	y.	y.	PROPN
cana-3972	623	15	zhou	zhou	PROPN
cana-3972	623	16	.	.	PUNCT
cana-3972	624	1	d4	d4	NOUN
cana-3972	624	2	-	-	PUNCT
cana-3972	624	3	modules	module	NOUN
cana-3972	624	4	.	.	PUNCT
cana-3972	625	1	journal	journal	NOUN
cana-3972	625	2	of	of	ADP
cana-3972	625	3	algebra	algebra	PROPN
cana-3972	625	4	and	and	CCONJ
cana-3972	625	5	its	its	PRON
cana-3972	625	6	applications	application	NOUN
cana-3972	625	7	,	,	PUNCT
cana-3972	625	8	page	page	NOUN
cana-3972	625	9	25	25	NUM
cana-3972	625	10	,	,	PUNCT
cana-3972	625	11	2017	2017	NUM
cana-3972	625	12	.	.	PUNCT
cana-3972	626	1	[	[	X
cana-3972	626	2	6	6	NUM
cana-3972	626	3	]	]	PUNCT
cana-3972	626	4	nanqing	nanqe	VERB
cana-3972	626	5	ding	ding	NOUN
cana-3972	626	6	,	,	PUNCT
cana-3972	626	7	yasser	yasser	PROPN
cana-3972	626	8	ibrahim	ibrahim	PROPN
cana-3972	626	9	,	,	PUNCT
cana-3972	626	10	mohamed	mohamed	PROPN
cana-3972	626	11	yousif	yousif	PROPN
cana-3972	626	12	,	,	PUNCT
cana-3972	626	13	and	and	CCONJ
cana-3972	626	14	yiqiang	yiqiang	PROPN
cana-3972	626	15	zhou	zhou	PROPN
cana-3972	626	16	.	.	PUNCT
cana-3972	627	1	c4	c4	NOUN
cana-3972	627	2	-	-	PUNCT
cana-3972	627	3	modules	module	NOUN
cana-3972	627	4	.	.	PUNCT
cana-3972	628	1	communications	communication	NOUN
cana-3972	628	2	in	in	ADP
cana-3972	628	3	algebra	algebra	NOUN
cana-3972	628	4	,	,	PUNCT
cana-3972	628	5	45	45	NUM
cana-3972	628	6	:	:	SYM
cana-3972	628	7	1727–1740	1727–1740	NUM
cana-3972	628	8	,	,	PUNCT
cana-3972	628	9	2016	2016	NUM
cana-3972	628	10	.	.	PUNCT
cana-3972	629	1	[	[	X
cana-3972	629	2	7	7	X
cana-3972	629	3	]	]	PUNCT
cana-3972	629	4	l.	l.	PROPN
cana-3972	629	5	fuchs	fuchs	PROPN
cana-3972	629	6	.	.	PUNCT
cana-3972	630	1	infinite	infinite	ADJ
cana-3972	630	2	abelian	abelian	ADJ
cana-3972	630	3	groups	group	NOUN
cana-3972	630	4	,	,	PUNCT
cana-3972	630	5	vol	vol	NOUN
cana-3972	630	6	.	.	NOUN
cana-3972	630	7	1	1	NUM
cana-3972	630	8	.	.	PUNCT
cana-3972	631	1	pure	pure	ADJ
cana-3972	631	2	appl	appl	PROPN
cana-3972	631	3	.	.	PUNCT
cana-3972	632	1	math	math	PROPN
cana-3972	632	2	.	.	PUNCT
cana-3972	633	1	ser	ser	PROPN
cana-3972	633	2	.	.	PUNCT
cana-3972	634	1	monogr	monogr	PROPN
cana-3972	634	2	.	.	PROPN
cana-3972	634	3	,	,	PUNCT
cana-3972	634	4	new	new	PROPN
cana-3972	634	5	york	york	PROPN
cana-3972	634	6	,	,	PUNCT
cana-3972	634	7	san	san	PROPN
cana-3972	634	8	francisco	francisco	PROPN
cana-3972	634	9	,	,	PUNCT
cana-3972	634	10	london	london	PROPN
cana-3972	634	11	,	,	PUNCT
cana-3972	634	12	1970	1970	NUM
cana-3972	634	13	.	.	PUNCT
cana-3972	635	1	[	[	X
cana-3972	635	2	8	8	NUM
cana-3972	635	3	]	]	PUNCT
cana-3972	635	4	a.	a.	NOUN
cana-3972	635	5	ghorbani	ghorbani	NOUN
cana-3972	635	6	and	and	CCONJ
cana-3972	635	7	a.	a.	NOUN
cana-3972	635	8	haghany	haghany	NOUN
cana-3972	635	9	.	.	PUNCT
cana-3972	636	1	generalized	generalize	VERB
cana-3972	636	2	hopfian	hopfian	ADJ
cana-3972	636	3	modules	module	NOUN
cana-3972	636	4	.	.	PUNCT
cana-3972	637	1	journal	journal	NOUN
cana-3972	637	2	of	of	ADP
cana-3972	637	3	algebra	algebra	PROPN
cana-3972	637	4	,	,	PUNCT
cana-3972	637	5	pages	page	NOUN
cana-3972	637	6	324–341	324–341	NUM
cana-3972	637	7	,	,	PUNCT
cana-3972	637	8	2002	2002	NUM
cana-3972	637	9	.	.	PUNCT
cana-3972	638	1	[	[	X
cana-3972	638	2	9	9	NUM
cana-3972	638	3	]	]	PUNCT
cana-3972	638	4	c.	c.	PROPN
cana-3972	638	5	w.	w.	PROPN
cana-3972	638	6	han	han	PROPN
cana-3972	638	7	,	,	PUNCT
cana-3972	638	8	b.	b.	PROPN
cana-3972	638	9	y.	y.	PROPN
cana-3972	638	10	lee	lee	PROPN
cana-3972	638	11	,	,	PUNCT
cana-3972	638	12	and	and	CCONJ
cana-3972	638	13	s.	s.	PROPN
cana-3972	638	14	j.	j.	PROPN
cana-3972	638	15	choi	choi	PROPN
cana-3972	638	16	.	.	PUNCT
cana-3972	639	1	direct	direct	ADJ
cana-3972	639	2	projective	projective	ADJ
cana-3972	639	3	modules	module	NOUN
cana-3972	639	4	with	with	ADP
cana-3972	639	5	the	the	DET
cana-3972	639	6	summand	summand	NOUN
cana-3972	639	7	intersection	intersection	NOUN
cana-3972	639	8	property	property	NOUN
cana-3972	639	9	.	.	PUNCT
cana-3972	640	1	pusan	pusan	PROPN
cana-3972	640	2	kyongnam	kyongnam	PROPN
cana-3972	640	3	math	math	PROPN
cana-3972	640	4	j	j	PROPN
cana-3972	640	5	,	,	PUNCT
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cana-3972	640	7	3	3	NUM
cana-3972	640	8	,	,	PUNCT
cana-3972	640	9	1994	1994	NUM
cana-3972	640	10	.	.	PUNCT
cana-3972	641	1	[	[	X
cana-3972	641	2	10	10	NUM
cana-3972	641	3	]	]	X
cana-3972	641	4	t.y	t.y	PROPN
cana-3972	641	5	lam	lam	PROPN
cana-3972	641	6	.	.	PUNCT
cana-3972	642	1	a	a	DET
cana-3972	642	2	first	first	ADJ
cana-3972	642	3	course	course	NOUN
cana-3972	642	4	in	in	ADP
cana-3972	642	5	noncommutative	noncommutative	ADJ
cana-3972	642	6	rings	ring	NOUN
cana-3972	642	7	.	.	PUNCT
cana-3972	643	1	springer	springer	NOUN
cana-3972	643	2	-	-	PUNCT
cana-3972	643	3	verlag	verlag	PROPN
cana-3972	643	4	.	.	PUNCT
cana-3972	643	5	usa	usa	PROPN
cana-3972	643	6	,	,	PUNCT
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cana-3972	643	8	.	.	PUNCT
cana-3972	644	1	[	[	X
cana-3972	644	2	11	11	NUM
cana-3972	644	3	]	]	X
cana-3972	644	4	gangyong	gangyong	PROPN
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cana-3972	644	6	,	,	PUNCT
cana-3972	644	7	s.	s.	PROPN
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cana-3972	644	9	rizvi	rizvi	PROPN
cana-3972	644	10	,	,	PUNCT
cana-3972	644	11	and	and	CCONJ
cana-3972	644	12	cosmin	cosmin	PROPN
cana-3972	644	13	s	s	PROPN
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cana-3972	644	15	.	.	PUNCT
cana-3972	645	1	dual	dual	ADJ
cana-3972	645	2	rickart	rickart	NOUN
cana-3972	645	3	modules	module	NOUN
cana-3972	645	4	.	.	PUNCT
cana-3972	646	1	communications	communication	NOUN
cana-3972	646	2	in	in	ADP
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cana-3972	646	4	,	,	PUNCT
cana-3972	646	5	pages	page	NOUN
cana-3972	646	6	4036	4036	NUM
cana-3972	646	7	–	–	PUNCT
cana-3972	646	8	4058	4058	NUM
cana-3972	646	9	,	,	PUNCT
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cana-3972	646	11	.	.	PUNCT
cana-3972	647	1	[	[	X
cana-3972	647	2	12	12	NUM
cana-3972	647	3	]	]	PUNCT
cana-3972	647	4	gangyong	gangyong	PROPN
cana-3972	647	5	lee	lee	PROPN
cana-3972	647	6	,	,	PUNCT
cana-3972	647	7	s.	s.	PROPN
cana-3972	647	8	tariq	tariq	PROPN
cana-3972	647	9	rizvi	rizvi	PROPN
cana-3972	647	10	,	,	PUNCT
cana-3972	647	11	and	and	CCONJ
cana-3972	647	12	cosmin	cosmin	PROPN
cana-3972	647	13	s	s	PROPN
cana-3972	647	14	roman	roman	NOUN
cana-3972	647	15	.	.	PUNCT
cana-3972	648	1	rickart	rickart	NOUN
cana-3972	648	2	modules	module	NOUN
cana-3972	648	3	.	.	PUNCT
cana-3972	649	1	communications	communication	NOUN
cana-3972	649	2	in	in	ADP
cana-3972	649	3	algebra	algebra	NOUN
cana-3972	649	4	,	,	PUNCT
cana-3972	649	5	page	page	NOUN
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cana-3972	649	7	,	,	PUNCT
cana-3972	649	8	2011	2011	NUM
cana-3972	649	9	.	.	PUNCT
cana-3972	650	1	[	[	X
cana-3972	650	2	13	13	NUM
cana-3972	650	3	]	]	PUNCT
cana-3972	650	4	a.	a.	NOUN
cana-3972	650	5	ö.	ö.	PROPN
cana-3972	650	6	meltem	meltem	PROPN
cana-3972	650	7	,	,	PUNCT
cana-3972	650	8	i.	i.	PROPN
cana-3972	650	9	yasser	yasser	PROPN
cana-3972	650	10	,	,	PUNCT
cana-3972	650	11	a.	a.	PROPN
cana-3972	650	12	ç	ç	PROPN
cana-3972	650	13	.	.	PROPN
cana-3972	650	14	ö.	ö.	PROPN
cana-3972	650	15	,	,	PUNCT
cana-3972	650	16	and	and	CCONJ
cana-3972	650	17	m.	m.	PROPN
cana-3972	650	18	yousif	yousif	PROPN
cana-3972	650	19	.	.	PUNCT
cana-3972	650	20	c4and	c4and	NUM
cana-3972	650	21	d4	d4	NOUN
cana-3972	650	22	-	-	PUNCT
cana-3972	650	23	modules	module	NOUN
cana-3972	650	24	via	via	ADP
cana-3972	650	25	perspective	perspective	ADJ
cana-3972	650	26	direct	direct	ADJ
cana-3972	650	27	summands	summand	NOUN
cana-3972	650	28	.	.	PUNCT
cana-3972	651	1	communications	communication	NOUN
cana-3972	651	2	in	in	ADP
cana-3972	651	3	algebra	algebra	NOUN
cana-3972	651	4	,	,	PUNCT
cana-3972	651	5	page	page	NOUN
cana-3972	651	6	22	22	NUM
cana-3972	651	7	,	,	PUNCT
cana-3972	651	8	2010	2010	NUM
cana-3972	651	9	.	.	PUNCT
cana-3972	652	1	[	[	X
cana-3972	652	2	14	14	NUM
cana-3972	652	3	]	]	PUNCT
cana-3972	652	4	a.	a.	NOUN
cana-3972	652	5	a.	a.	NOUN
cana-3972	652	6	nailevich	nailevich	PROPN
cana-3972	652	7	,	,	PUNCT
cana-3972	652	8	t.	t.	PROPN
cana-3972	652	9	c.	c.	PROPN
cana-3972	652	10	quynh	quynh	PROPN
cana-3972	652	11	,	,	PUNCT
cana-3972	652	12	and	and	CCONJ
cana-3972	652	13	t.	t.	PROPN
cana-3972	652	14	h.	h.	PROPN
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cana-3972	652	17	.	.	PUNCT
cana-3972	653	1	on	on	ADP
cana-3972	653	2	classes	class	NOUN
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cana-3972	653	4	c3	c3	PROPN
cana-3972	653	5	and	and	CCONJ
cana-3972	653	6	d3	d3	PROPN
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cana-3972	653	8	.	.	PUNCT
cana-3972	654	1	hacettepe	hacettepe	PROPN
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cana-3972	654	7	,	,	PUNCT
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cana-3972	654	9	47	47	NUM
cana-3972	654	10	(	(	PUNCT
cana-3972	654	11	2	2	NUM
cana-3972	654	12	)	)	PUNCT
cana-3972	654	13	:	:	PUNCT
cana-3972	654	14	317	317	NUM
cana-3972	654	15	–	–	PUNCT
cana-3972	654	16	329	329	NUM
cana-3972	654	17	,	,	PUNCT
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cana-3972	654	19	.	.	PUNCT
cana-3972	655	1	[	[	X
cana-3972	655	2	15	15	NUM
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cana-3972	655	7	.	.	PUNCT
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cana-3972	656	2	modules	module	NOUN
cana-3972	656	3	and	and	CCONJ
cana-3972	656	4	rings	ring	NOUN
cana-3972	656	5	.	.	PUNCT
cana-3972	657	1	can	can	AUX
cana-3972	657	2	.	.	PUNCT
cana-3972	658	1	j.	j.	PROPN
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cana-3972	658	3	,	,	PUNCT
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cana-3972	658	5	16	16	NUM
cana-3972	658	6	,	,	PUNCT
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cana-3972	658	8	.	.	PUNCT
cana-3972	659	1	[	[	X
cana-3972	659	2	16	16	NUM
cana-3972	659	3	]	]	PUNCT
cana-3972	659	4	k.	k.	PROPN
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cana-3972	659	6	.	.	PUNCT
cana-3972	660	1	lifting	lift	VERB
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cana-3972	660	3	,	,	PUNCT
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cana-3972	660	5	modules	module	NOUN
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cana-3972	660	9	.	.	PUNCT
cana-3972	661	1	pages	page	NOUN
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cana-3972	661	4	,	,	PUNCT
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cana-3972	661	6	.	.	PUNCT
cana-3972	662	1	[	[	X
cana-3972	662	2	17	17	NUM
cana-3972	662	3	]	]	X
cana-3972	662	4	y.	y.	PROPN
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cana-3972	662	7	a.	a.	PROPN
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cana-3972	662	11	.	.	PUNCT
cana-3972	663	1	t	t	NOUN
cana-3972	663	2	-	-	PUNCT
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cana-3972	663	4	and	and	CCONJ
cana-3972	663	5	non	non	ADJ
cana-3972	663	6	-	-	ADJ
cana-3972	663	7	t	t	NOUN
cana-3972	663	8	-	-	PUNCT
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cana-3972	663	11	.	.	PUNCT
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cana-3972	664	3	,	,	PUNCT
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cana-3972	664	5	.	.	PUNCT
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cana-3972	667	10	(	(	PUNCT
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cana-3972	667	12	)	)	PUNCT
cana-3972	667	13	:	:	PUNCT
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cana-3972	668	10	.	.	PUNCT
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cana-3972	669	4	.	.	PUNCT
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cana-3972	670	9	,	,	PUNCT
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cana-3972	670	13	:	:	SYM
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cana-3972	671	2	20	20	NUM
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cana-3972	671	4	y.	y.	PROPN
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cana-3972	672	12	.	.	PUNCT
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cana-3972	673	9	.	.	PUNCT
cana-3972	674	1	[	[	X
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cana-3972	674	3	]	]	PUNCT
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cana-3972	675	2	close	close	ADV
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cana-3972	675	5	.	.	PUNCT
cana-3972	676	1	springer	springer	NOUN
cana-3972	676	2	-	-	PUNCT
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cana-3972	676	5	,	,	PUNCT
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cana-3972	676	7	.	.	PROPN
cana-3972	676	8	,	,	PUNCT
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cana-3972	676	21	,	,	PUNCT
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cana-3972	676	23	netherlands	netherlands	PROPN
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cana-3972	677	2	22	22	NUM
cana-3972	677	3	]	]	PUNCT
cana-3972	677	4	r.	r.	PROPN
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cana-3972	677	6	.	.	PUNCT
cana-3972	678	1	foundations	foundation	NOUN
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cana-3972	678	3	module	module	NOUN
cana-3972	678	4	and	and	CCONJ
cana-3972	678	5	ring	ring	NOUN
cana-3972	678	6	theory	theory	NOUN
cana-3972	678	7	.	.	PUNCT
cana-3972	679	1	gordon	gordon	PROPN
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cana-3972	681	6	-	-	PUNCT
cana-3972	681	7	projective	projective	ADJ
cana-3972	681	8	modules	module	NOUN
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cana-3972	681	10	direct	direct	ADJ
cana-3972	681	11	-	-	PUNCT
cana-3972	681	12	injective	injective	ADJ
cana-3972	681	13	modules	module	NOUN
cana-3972	681	14	.	.	PUNCT
cana-3972	682	1	journal	journal	NOUN
cana-3972	682	2	of	of	ADP
cana-3972	682	3	pure	pure	ADJ
cana-3972	682	4	and	and	CCONJ
cana-3972	682	5	applied	applied	ADJ
cana-3972	682	6	algebra	algebra	NOUN
cana-3972	682	7	,	,	PUNCT
cana-3972	682	8	pages	page	NOUN
cana-3972	682	9	99–104	99–104	PROPN
cana-3972	682	10	,	,	PUNCT
cana-3972	682	11	1993	1993	NUM
cana-3972	682	12	.	.	PUNCT
cana-3972	683	1	[	[	X
cana-3972	683	2	24	24	NUM
cana-3972	683	3	]	]	PUNCT
cana-3972	683	4	m.	m.	NOUN
cana-3972	683	5	yousif	yousif	PROPN
cana-3972	683	6	,	,	PUNCT
cana-3972	683	7	i.	i.	PROPN
cana-3972	683	8	amin	amin	PROPN
cana-3972	683	9	,	,	PUNCT
cana-3972	683	10	and	and	CCONJ
cana-3972	683	11	y.	y.	PROPN
cana-3972	683	12	ibrahim	ibrahim	PROPN
cana-3972	683	13	.	.	PUNCT
cana-3972	684	1	d3	d3	PROPN
cana-3972	684	2	-	-	PUNCT
cana-3972	684	3	modules	module	NOUN
cana-3972	684	4	.	.	PUNCT
cana-3972	685	1	communications	communication	NOUN
cana-3972	685	2	in	in	ADP
cana-3972	685	3	algebra	algebra	NOUN
cana-3972	685	4	,	,	PUNCT
cana-3972	685	5	pages	page	NOUN
cana-3972	685	6	578–592	578–592	NUM
cana-3972	685	7	,	,	PUNCT
cana-3972	685	8	2013	2013	NUM
cana-3972	685	9	.	.	PUNCT