id	sid	tid	token	lemma	pos
iajs-1808	1	1	microsoft	microsoft	PROPN
iajs-1808	1	2	word	word	NOUN
iajs-1808	1	3	353	353	NUM
iajs-1808	1	4	-	-	SYM
iajs-1808	1	5	362	362	NUM
iajs-1808	1	6	ihsciconf	ihsciconf	NOUN
iajs-1808	1	7	2017	2017	NUM
iajs-1808	1	8	special	special	ADJ
iajs-1808	1	9	issue	issue	NOUN
iajs-1808	1	10	ibn	ibn	PROPN
iajs-1808	1	11	al	al	PROPN
iajs-1808	1	12	-	-	PUNCT
iajs-1808	1	13	haitham	haitham	PROPN
iajs-1808	1	14	journal	journal	PROPN
iajs-1808	1	15	for	for	ADP
iajs-1808	1	16	pure	pure	ADJ
iajs-1808	1	17	and	and	CCONJ
iajs-1808	1	18	applied	apply	VERB
iajs-1808	1	19	science	science	NOUN
iajs-1808	1	20	https://doi.org/	https://doi.org/	NOUN
iajs-1808	1	21	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	1	22	for	for	ADP
iajs-1808	1	23	more	more	ADJ
iajs-1808	1	24	information	information	NOUN
iajs-1808	1	25	about	about	ADP
iajs-1808	1	26	the	the	DET
iajs-1808	1	27	conference	conference	NOUN
iajs-1808	1	28	please	please	INTJ
iajs-1808	1	29	visit	visit	VERB
iajs-1808	1	30	the	the	DET
iajs-1808	1	31	websites	website	NOUN
iajs-1808	1	32	:	:	PUNCT
iajs-1808	1	33	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	1	34	  	  	SPACE
iajs-1808	1	35	www.ihsciconf.org	www.ihsciconf.org	NOUN
iajs-1808	1	36	   	   	SPACE
iajs-1808	1	37	mathematics|353	mathematics|353	PROPN
iajs-1808	1	38	    	    	SPACE
iajs-1808	1	39	strongly	strongly	ADV
iajs-1808	1	40	𝑪𝟏𝟏-condition	𝑪𝟏𝟏-condition	NOUN
iajs-1808	1	41	modules	module	NOUN
iajs-1808	1	42	and	and	CCONJ
iajs-1808	1	43	strongly	strongly	ADV
iajs-1808	1	44	𝑻𝟏𝟏-type	𝑻𝟏𝟏-type	VERB
iajs-1808	1	45	modules	module	NOUN
iajs-1808	1	46	inaam	inaam	PROPN
iajs-1808	1	47	m	m	PROPN
iajs-1808	1	48	a	a	DET
iajs-1808	1	49	hadi	hadi	PROPN
iajs-1808	1	50	inaam1976@yahoo.com	inaam1976@yahoo.com	PROPN
iajs-1808	1	51	dept	dept	PROPN
iajs-1808	1	52	.	.	PROPN
iajs-1808	2	1	of	of	ADP
iajs-1808	2	2	mathematics	mathematics	PROPN
iajs-1808	2	3	,	,	PUNCT
iajs-1808	2	4	college	college	NOUN
iajs-1808	2	5	of	of	ADP
iajs-1808	2	6	education	education	NOUN
iajs-1808	2	7	for	for	ADP
iajs-1808	2	8	pure	pure	ADJ
iajs-1808	2	9	science	science	NOUN
iajs-1808	2	10	(	(	PUNCT
iajs-1808	2	11	ibn	ibn	PROPN
iajs-1808	2	12	-	-	PUNCT
iajs-1808	2	13	al	al	PROPN
iajs-1808	2	14	-	-	PUNCT
iajs-1808	2	15	haitham	haitham	PROPN
iajs-1808	2	16	)	)	PUNCT
iajs-1808	2	17	university	university	PROPN
iajs-1808	2	18	of	of	ADP
iajs-1808	2	19	baghdad	baghdad	PROPN
iajs-1808	2	20	farhan	farhan	PROPN
iajs-1808	3	1	d	d	X
iajs-1808	3	2	shyaa	shyaa	PROPN
iajs-1808	3	3	farhan.shyaa@qu.edu.iq	farhan.shyaa@qu.edu.iq	ADJ
iajs-1808	3	4	dept	dept	NOUN
iajs-1808	3	5	.	.	PROPN
iajs-1808	3	6	of	of	ADP
iajs-1808	3	7	mathematics	mathematics	PROPN
iajs-1808	3	8	,	,	PUNCT
iajs-1808	3	9	university	university	PROPN
iajs-1808	3	10	of	of	ADP
iajs-1808	3	11	al	al	PROPN
iajs-1808	3	12	-	-	PUNCT
iajs-1808	3	13	qadsiyah	qadsiyah	PROPN
iajs-1808	3	14	abstract	abstract	NOUN
iajs-1808	3	15	in	in	ADP
iajs-1808	3	16	this	this	DET
iajs-1808	3	17	paper	paper	NOUN
iajs-1808	3	18	,	,	PUNCT
iajs-1808	3	19	we	we	PRON
iajs-1808	3	20	introduced	introduce	VERB
iajs-1808	3	21	module	module	NOUN
iajs-1808	3	22	that	that	PRON
iajs-1808	3	23	satisfying	satisfy	VERB
iajs-1808	3	24	strongly	strongly	ADV
iajs-1808	3	25	𝐶	𝐶	PROPN
iajs-1808	3	26	-condition	-condition	NOUN
iajs-1808	3	27	modules	module	NOUN
iajs-1808	3	28	and	and	CCONJ
iajs-1808	3	29	strongly	strongly	ADV
iajs-1808	3	30	𝑇	𝑇	PROPN
iajs-1808	3	31	-type	-type	NOUN
iajs-1808	3	32	modules	module	NOUN
iajs-1808	3	33	as	as	ADP
iajs-1808	3	34	generalizations	generalization	NOUN
iajs-1808	3	35	of	of	ADP
iajs-1808	3	36	t	t	NOUN
iajs-1808	3	37	-	-	PUNCT
iajs-1808	3	38	extending	extending	NOUN
iajs-1808	3	39	.	.	PUNCT
iajs-1808	4	1	a	a	DET
iajs-1808	4	2	module	module	NOUN
iajs-1808	4	3	𝑀	𝑀	PROPN
iajs-1808	4	4	is	be	AUX
iajs-1808	4	5	said	say	VERB
iajs-1808	4	6	strongly	strongly	ADV
iajs-1808	4	7	𝐶	𝐶	PROPN
iajs-1808	4	8	condition	condition	NOUN
iajs-1808	4	9	if	if	SCONJ
iajs-1808	4	10	for	for	ADP
iajs-1808	4	11	every	every	DET
iajs-1808	4	12	submodule	submodule	NOUN
iajs-1808	4	13	of	of	ADP
iajs-1808	4	14	𝑀	𝑀	PROPN
iajs-1808	4	15	has	have	VERB
iajs-1808	4	16	a	a	DET
iajs-1808	4	17	complement	complement	NOUN
iajs-1808	4	18	which	which	PRON
iajs-1808	4	19	is	be	AUX
iajs-1808	4	20	fully	fully	ADV
iajs-1808	4	21	invariant	invariant	ADJ
iajs-1808	4	22	direct	direct	ADJ
iajs-1808	4	23	summand	summand	NOUN
iajs-1808	4	24	.	.	PUNCT
iajs-1808	5	1	a	a	DET
iajs-1808	5	2	module	module	NOUN
iajs-1808	5	3	𝑀	𝑀	PROPN
iajs-1808	5	4	is	be	AUX
iajs-1808	5	5	said	say	VERB
iajs-1808	5	6	to	to	PART
iajs-1808	5	7	be	be	AUX
iajs-1808	5	8	strongly	strongly	ADV
iajs-1808	5	9	𝑇	𝑇	PROPN
iajs-1808	5	10	-type	-type	NOUN
iajs-1808	5	11	modules	module	NOUN
iajs-1808	5	12	if	if	SCONJ
iajs-1808	5	13	every	every	DET
iajs-1808	5	14	t	t	PROPN
iajs-1808	5	15	-	-	PUNCT
iajs-1808	5	16	closed	close	VERB
iajs-1808	5	17	submodule	submodule	NOUN
iajs-1808	5	18	has	have	VERB
iajs-1808	5	19	a	a	DET
iajs-1808	5	20	complement	complement	NOUN
iajs-1808	5	21	which	which	PRON
iajs-1808	5	22	is	be	AUX
iajs-1808	5	23	a	a	DET
iajs-1808	5	24	fully	fully	ADV
iajs-1808	5	25	invariant	invariant	ADJ
iajs-1808	5	26	direct	direct	ADJ
iajs-1808	5	27	summand	summand	NOUN
iajs-1808	5	28	.	.	PUNCT
iajs-1808	6	1	many	many	ADJ
iajs-1808	6	2	characterizations	characterization	NOUN
iajs-1808	6	3	for	for	ADP
iajs-1808	6	4	modules	module	NOUN
iajs-1808	6	5	with	with	ADP
iajs-1808	6	6	strongly	strongly	ADV
iajs-1808	6	7	𝐶	𝐶	PROPN
iajs-1808	6	8	-condition	-condition	NOUN
iajs-1808	6	9	for	for	ADP
iajs-1808	6	10	strongly	strongly	ADV
iajs-1808	6	11	𝑇	𝑇	PROPN
iajs-1808	6	12	-type	-type	NOUN
iajs-1808	6	13	module	module	NOUN
iajs-1808	6	14	are	be	AUX
iajs-1808	6	15	given	give	VERB
iajs-1808	6	16	.	.	PUNCT
iajs-1808	7	1	also	also	ADV
iajs-1808	7	2	many	many	ADJ
iajs-1808	7	3	connections	connection	NOUN
iajs-1808	7	4	between	between	ADP
iajs-1808	7	5	these	these	DET
iajs-1808	7	6	types	type	NOUN
iajs-1808	7	7	of	of	ADP
iajs-1808	7	8	module	module	NOUN
iajs-1808	7	9	and	and	CCONJ
iajs-1808	7	10	some	some	DET
iajs-1808	7	11	related	related	ADJ
iajs-1808	7	12	types	type	NOUN
iajs-1808	7	13	of	of	ADP
iajs-1808	7	14	modules	module	NOUN
iajs-1808	7	15	are	be	AUX
iajs-1808	7	16	investigated	investigate	VERB
iajs-1808	7	17	.	.	PUNCT
iajs-1808	8	1	keywords	keyword	NOUN
iajs-1808	8	2	.	.	PUNCT
iajs-1808	9	1	𝐶	𝐶	PROPN
iajs-1808	9	2	-condition	-condition	PROPN
iajs-1808	9	3	,	,	PUNCT
iajs-1808	9	4	strongly	strongly	ADV
iajs-1808	9	5	𝐶	𝐶	PROPN
iajs-1808	9	6	-condition	-condition	NOUN
iajs-1808	9	7	modules	module	NOUN
iajs-1808	9	8	,	,	PUNCT
iajs-1808	9	9	𝑇	𝑇	PROPN
iajs-1808	9	10	-type	-type	NOUN
iajs-1808	9	11	modules	module	NOUN
iajs-1808	9	12	strongly	strongly	ADV
iajs-1808	9	13	𝑇	𝑇	PROPN
iajs-1808	9	14	-type	-type	NOUN
iajs-1808	9	15	modules	module	NOUN
iajs-1808	9	16	,	,	PUNCT
iajs-1808	9	17	t	t	NOUN
iajs-1808	9	18	-	-	PUNCT
iajs-1808	9	19	semisimple	semisimple	NOUN
iajs-1808	9	20	modules	module	NOUN
iajs-1808	9	21	,	,	PUNCT
iajs-1808	9	22	strongly	strongly	ADV
iajs-1808	9	23	t	t	NOUN
iajs-1808	9	24	-	-	PUNCT
iajs-1808	9	25	semisimple	semisimple	NOUN
iajs-1808	9	26	modules	module	NOUN
iajs-1808	9	27	,	,	PUNCT
iajs-1808	9	28	strongly	strongly	ADV
iajs-1808	9	29	extending	extend	VERB
iajs-1808	9	30	modules	module	NOUN
iajs-1808	9	31	,	,	PUNCT
iajs-1808	9	32	t	t	NOUN
iajs-1808	9	33	-	-	PUNCT
iajs-1808	9	34	extending	extend	VERB
iajs-1808	9	35	modules	module	NOUN
iajs-1808	9	36	and	and	CCONJ
iajs-1808	9	37	strongly	strongly	ADV
iajs-1808	9	38	t	t	NOUN
iajs-1808	9	39	-	-	PUNCT
iajs-1808	9	40	extending	extend	VERB
iajs-1808	9	41	modules	module	NOUN
iajs-1808	9	42	.	.	PUNCT
iajs-1808	10	1	ihsciconf	ihsciconf	PROPN
iajs-1808	10	2	2017	2017	NUM
iajs-1808	10	3	special	special	ADJ
iajs-1808	10	4	issue	issue	NOUN
iajs-1808	10	5	ibn	ibn	PROPN
iajs-1808	10	6	al	al	PROPN
iajs-1808	10	7	-	-	PUNCT
iajs-1808	10	8	haitham	haitham	PROPN
iajs-1808	10	9	journal	journal	PROPN
iajs-1808	10	10	for	for	ADP
iajs-1808	10	11	pure	pure	ADJ
iajs-1808	10	12	and	and	CCONJ
iajs-1808	10	13	applied	apply	VERB
iajs-1808	10	14	science	science	NOUN
iajs-1808	10	15	https://doi.org/	https://doi.org/	NOUN
iajs-1808	10	16	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	10	17	for	for	ADP
iajs-1808	10	18	more	more	ADJ
iajs-1808	10	19	information	information	NOUN
iajs-1808	10	20	about	about	ADP
iajs-1808	10	21	the	the	DET
iajs-1808	10	22	conference	conference	NOUN
iajs-1808	10	23	please	please	INTJ
iajs-1808	10	24	visit	visit	VERB
iajs-1808	10	25	the	the	DET
iajs-1808	10	26	websites	website	NOUN
iajs-1808	10	27	:	:	PUNCT
iajs-1808	10	28	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	10	29	  	  	SPACE
iajs-1808	10	30	www.ihsciconf.org	www.ihsciconf.org	NOUN
iajs-1808	10	31	   	   	SPACE
iajs-1808	10	32	mathematics|354	mathematics|354	NOUN
iajs-1808	10	33	    	    	SPACE
iajs-1808	10	34	1	1	NUM
iajs-1808	10	35	.	.	PUNCT
iajs-1808	11	1	introduction	introduction	NOUN
iajs-1808	11	2	first	first	ADV
iajs-1808	11	3	,	,	PUNCT
iajs-1808	11	4	a	a	DET
iajs-1808	11	5	few	few	ADJ
iajs-1808	11	6	concepts	concept	NOUN
iajs-1808	11	7	and	and	CCONJ
iajs-1808	11	8	some	some	DET
iajs-1808	11	9	results	result	NOUN
iajs-1808	11	10	which	which	PRON
iajs-1808	11	11	are	be	AUX
iajs-1808	11	12	relevant	relevant	ADJ
iajs-1808	11	13	for	for	ADP
iajs-1808	11	14	our	our	PRON
iajs-1808	11	15	work	work	NOUN
iajs-1808	11	16	are	be	AUX
iajs-1808	11	17	recalled	recall	VERB
iajs-1808	11	18	.	.	PUNCT
iajs-1808	12	1	throughout	throughout	ADV
iajs-1808	12	2	,	,	PUNCT
iajs-1808	12	3	all	all	DET
iajs-1808	12	4	rings	ring	NOUN
iajs-1808	12	5	are	be	AUX
iajs-1808	12	6	associative	associative	ADJ
iajs-1808	12	7	rings	ring	NOUN
iajs-1808	12	8	with	with	ADP
iajs-1808	12	9	unity	unity	NOUN
iajs-1808	12	10	and	and	CCONJ
iajs-1808	12	11	all	all	DET
iajs-1808	12	12	modules	module	NOUN
iajs-1808	12	13	are	be	AUX
iajs-1808	12	14	right	right	ADJ
iajs-1808	12	15	unitary	unitary	ADJ
iajs-1808	12	16	modules	module	NOUN
iajs-1808	12	17	.	.	PUNCT
iajs-1808	13	1	a	a	DET
iajs-1808	13	2	submodule	submodule	NOUN
iajs-1808	13	3	𝑁	𝑁	PROPN
iajs-1808	13	4	of	of	ADP
iajs-1808	13	5	𝑀	𝑀	PROPN
iajs-1808	13	6	is	be	AUX
iajs-1808	13	7	called	call	VERB
iajs-1808	13	8	closed	closed	ADJ
iajs-1808	13	9	in	in	ADP
iajs-1808	13	10	𝑀	𝑀	PROPN
iajs-1808	13	11	if	if	SCONJ
iajs-1808	13	12	has	have	VERB
iajs-1808	13	13	no	no	DET
iajs-1808	13	14	proper	proper	ADJ
iajs-1808	13	15	essential	essential	ADJ
iajs-1808	13	16	extension	extension	NOUN
iajs-1808	13	17	in	in	ADP
iajs-1808	13	18	𝑀[8	𝑀[8	PROPN
iajs-1808	13	19	]	]	PUNCT
iajs-1808	13	20	,	,	PUNCT
iajs-1808	13	21	that	that	PRON
iajs-1808	13	22	means	mean	VERB
iajs-1808	13	23	if	if	SCONJ
iajs-1808	13	24	𝑁	𝑁	PROPN
iajs-1808	13	25	is	be	AUX
iajs-1808	13	26	essential	essential	ADJ
iajs-1808	13	27	in	in	ADP
iajs-1808	13	28	𝑊	𝑊	PROPN
iajs-1808	13	29	,	,	PUNCT
iajs-1808	13	30	where	where	SCONJ
iajs-1808	13	31	𝑊	𝑊	PROPN
iajs-1808	13	32	𝑀,then	𝑀,then	PUNCT
iajs-1808	13	33	𝑁	𝑁	PROPN
iajs-1808	13	34	𝑊	𝑊	PROPN
iajs-1808	13	35	,	,	PUNCT
iajs-1808	13	36	where	where	SCONJ
iajs-1808	13	37	a	a	DET
iajs-1808	13	38	submodule	submodule	NOUN
iajs-1808	13	39	𝑁	𝑁	PROPN
iajs-1808	13	40	is	be	AUX
iajs-1808	13	41	essential	essential	ADJ
iajs-1808	13	42	in	in	ADP
iajs-1808	13	43	an	an	DET
iajs-1808	13	44	𝑅-module	𝑅-module	PROPN
iajs-1808	13	45	𝑀	𝑀	PROPN
iajs-1808	13	46	(	(	PUNCT
iajs-1808	13	47	briefly	briefly	ADV
iajs-1808	13	48	(	(	PUNCT
iajs-1808	13	49	𝑁	𝑁	PROPN
iajs-1808	13	50	𝑀	𝑀	PROPN
iajs-1808	13	51	if	if	SCONJ
iajs-1808	13	52	𝑁	𝑁	PROPN
iajs-1808	13	53	∩	∩	ADJ
iajs-1808	13	54	𝐾	𝐾	NOUN
iajs-1808	13	55	0	0	NUM
iajs-1808	13	56	,	,	PUNCT
iajs-1808	13	57	𝐾	𝐾	PROPN
iajs-1808	13	58	𝑀	𝑀	PROPN
iajs-1808	13	59	implies	imply	VERB
iajs-1808	13	60	𝐾	𝐾	PROPN
iajs-1808	13	61	0	0	PUNCT
iajs-1808	14	1	[	[	X
iajs-1808	14	2	8	8	NUM
iajs-1808	14	3	]	]	PUNCT
iajs-1808	14	4	.	.	PUNCT
iajs-1808	15	1	as	as	ADP
iajs-1808	15	2	a	a	DET
iajs-1808	15	3	generalization	generalization	NOUN
iajs-1808	15	4	of	of	ADP
iajs-1808	15	5	essential	essential	ADJ
iajs-1808	15	6	submodule	submodule	NOUN
iajs-1808	15	7	,	,	PUNCT
iajs-1808	15	8	asgari	asgari	ADJ
iajs-1808	15	9	in	in	ADP
iajs-1808	15	10	[	[	PUNCT
iajs-1808	15	11	2	2	NUM
iajs-1808	15	12	]	]	PUNCT
iajs-1808	15	13	,	,	PUNCT
iajs-1808	15	14	introduced	introduce	VERB
iajs-1808	15	15	the	the	DET
iajs-1808	15	16	notion	notion	NOUN
iajs-1808	15	17	of	of	ADP
iajs-1808	15	18	tessential	tessential	ADJ
iajs-1808	15	19	submodule	submodule	NOUN
iajs-1808	15	20	,	,	PUNCT
iajs-1808	15	21	where	where	SCONJ
iajs-1808	15	22	a	a	DET
iajs-1808	15	23	submodule	submodule	NOUN
iajs-1808	15	24	𝑁	𝑁	PROPN
iajs-1808	15	25	of	of	ADP
iajs-1808	15	26	𝑀	𝑀	PROPN
iajs-1808	15	27	is	be	AUX
iajs-1808	15	28	called	call	VERB
iajs-1808	15	29	t	t	NOUN
iajs-1808	15	30	-	-	PUNCT
iajs-1808	15	31	essential	essential	ADJ
iajs-1808	15	32	(	(	PUNCT
iajs-1808	15	33	denoted	denote	VERB
iajs-1808	15	34	by	by	ADP
iajs-1808	15	35	𝑁	𝑁	PROPN
iajs-1808	15	36	𝑀	𝑀	PROPN
iajs-1808	15	37	if	if	SCONJ
iajs-1808	15	38	whenever	whenever	SCONJ
iajs-1808	15	39	𝑊	𝑊	PROPN
iajs-1808	15	40	𝑀	𝑀	PROPN
iajs-1808	15	41	,	,	PUNCT
iajs-1808	15	42	𝑁	𝑁	PROPN
iajs-1808	15	43	∩	∩	ADJ
iajs-1808	15	44	𝑊	𝑊	NOUN
iajs-1808	15	45	𝑍	𝑍	PROPN
iajs-1808	15	46	𝑀	𝑀	PROPN
iajs-1808	15	47	implies	imply	VERB
iajs-1808	15	48	𝑊	𝑊	PROPN
iajs-1808	15	49	𝑍	𝑍	PROPN
iajs-1808	15	50	𝑀	𝑀	PROPN
iajs-1808	15	51	.	.	PUNCT
iajs-1808	16	1	𝑍	𝑍	PROPN
iajs-1808	16	2	𝑀	𝑀	PROPN
iajs-1808	16	3	is	be	AUX
iajs-1808	16	4	called	call	VERB
iajs-1808	16	5	the	the	DET
iajs-1808	16	6	second	second	ADJ
iajs-1808	16	7	singular	singular	ADJ
iajs-1808	16	8	submodule	submodule	NOUN
iajs-1808	16	9	and	and	CCONJ
iajs-1808	16	10	is	be	AUX
iajs-1808	16	11	defined	define	VERB
iajs-1808	16	12	by	by	ADP
iajs-1808	16	13	𝑍	𝑍	NOUN
iajs-1808	16	14	[	[	NOUN
iajs-1808	16	15	8	8	NUM
iajs-1808	16	16	]	]	PUNCT
iajs-1808	16	17	,	,	PUNCT
iajs-1808	16	18	where	where	SCONJ
iajs-1808	16	19	𝑍	𝑍	VERB
iajs-1808	16	20	𝑀	𝑀	PROPN
iajs-1808	16	21	𝑥	𝑥	NOUN
iajs-1808	16	22	∈	∈	PROPN
iajs-1808	16	23	𝑀	𝑀	PROPN
iajs-1808	16	24	:	:	PUNCT
iajs-1808	16	25	𝑥𝐼	𝑥𝐼	PROPN
iajs-1808	16	26	0	0	NUM
iajs-1808	17	1	for	for	ADP
iajs-1808	17	2	some	some	DET
iajs-1808	17	3	essential	essential	ADJ
iajs-1808	17	4	ideal	ideal	NOUN
iajs-1808	17	5	of	of	ADP
iajs-1808	17	6	𝑅	𝑅	NOUN
iajs-1808	17	7	}	}	PUNCT
iajs-1808	17	8	.	.	PUNCT
iajs-1808	18	1	equivalently	equivalently	ADV
iajs-1808	18	2	𝑍	𝑍	VERB
iajs-1808	18	3	𝑀	𝑀	PROPN
iajs-1808	18	4	𝑥	𝑥	NOUN
iajs-1808	18	5	∈	∈	PROPN
iajs-1808	18	6	𝑀	𝑀	PROPN
iajs-1808	18	7	:	:	PUNCT
iajs-1808	18	8	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
iajs-1808	18	9	𝑥	𝑥	PROPN
iajs-1808	18	10	𝑅	𝑅	PROPN
iajs-1808	18	11	and	and	CCONJ
iajs-1808	18	12	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
iajs-1808	18	13	𝑥	𝑥	PRON
iajs-1808	18	14	𝑟	𝑟	PRON
iajs-1808	18	15	∈	∈	PROPN
iajs-1808	18	16	𝑀	𝑀	PROPN
iajs-1808	18	17	:	:	PUNCT
iajs-1808	18	18	𝑥𝑟	𝑥𝑟	NOUN
iajs-1808	18	19	0	0	NUM
iajs-1808	18	20	.	.	PUNCT
iajs-1808	19	1	𝑀	𝑀	PROPN
iajs-1808	19	2	is	be	AUX
iajs-1808	19	3	called	call	VERB
iajs-1808	19	4	singular	singular	ADJ
iajs-1808	19	5	(	(	PUNCT
iajs-1808	19	6	nonsingular	nonsingular	ADJ
iajs-1808	19	7	)	)	PUNCT
iajs-1808	19	8	if	if	SCONJ
iajs-1808	19	9	𝑍	𝑍	NOUN
iajs-1808	19	10	𝑀	𝑀	PROPN
iajs-1808	19	11	𝑀	𝑀	PROPN
iajs-1808	19	12	𝑍	𝑍	VERB
iajs-1808	19	13	𝑀	𝑀	PROPN
iajs-1808	19	14	0	0	NUM
iajs-1808	19	15	.	.	PUNCT
iajs-1808	20	1	note	note	VERB
iajs-1808	20	2	that	that	SCONJ
iajs-1808	20	3	𝑍	𝑍	VERB
iajs-1808	20	4	𝑀	𝑀	PROPN
iajs-1808	20	5	𝑥	𝑥	NOUN
iajs-1808	20	6	∈	∈	PROPN
iajs-1808	20	7	𝑀	𝑀	PROPN
iajs-1808	20	8	:	:	PUNCT
iajs-1808	20	9	𝑥𝐼	𝑥𝐼	PROPN
iajs-1808	20	10	0	0	NUM
iajs-1808	21	1	for	for	ADP
iajs-1808	21	2	some	some	DET
iajs-1808	21	3	t	t	NOUN
iajs-1808	21	4	-	-	PUNCT
iajs-1808	21	5	essential	essential	ADJ
iajs-1808	21	6	ideal	ideal	NOUN
iajs-1808	21	7	𝐼	𝐼	PROPN
iajs-1808	21	8	of	of	ADP
iajs-1808	21	9	𝑅	𝑅	PROPN
iajs-1808	21	10	.𝑀	.𝑀	PROPN
iajs-1808	21	11	is	be	AUX
iajs-1808	21	12	called	call	VERB
iajs-1808	21	13	𝑍	𝑍	NOUN
iajs-1808	21	14	-torsion	-torsion	NOUN
iajs-1808	21	15	if	if	SCONJ
iajs-1808	21	16	𝑍	𝑍	PROPN
iajs-1808	21	17	𝑀	𝑀	PROPN
iajs-1808	21	18	𝑀[8	𝑀[8	PROPN
iajs-1808	21	19	]	]	X
iajs-1808	21	20	.	.	PUNCT
iajs-1808	22	1	a	a	DET
iajs-1808	22	2	submodule	submodule	NOUN
iajs-1808	22	3	𝑁	𝑁	PROPN
iajs-1808	22	4	is	be	AUX
iajs-1808	22	5	called	call	VERB
iajs-1808	22	6	t	t	NOUN
iajs-1808	22	7	-	-	PUNCT
iajs-1808	22	8	closed	close	VERB
iajs-1808	22	9	(	(	PUNCT
iajs-1808	22	10	denoted	denote	VERB
iajs-1808	22	11	by	by	ADP
iajs-1808	22	12	𝑁	𝑁	PROPN
iajs-1808	22	13	𝑀	𝑀	PROPN
iajs-1808	22	14	if	if	SCONJ
iajs-1808	22	15	𝑁	𝑁	PROPN
iajs-1808	22	16	has	have	VERB
iajs-1808	22	17	no	no	DET
iajs-1808	22	18	proper	proper	ADJ
iajs-1808	22	19	t	t	NOUN
iajs-1808	22	20	-	-	PUNCT
iajs-1808	22	21	essential	essential	ADJ
iajs-1808	22	22	extension	extension	NOUN
iajs-1808	22	23	in	in	ADP
iajs-1808	22	24	𝑀[2	𝑀[2	PROPN
iajs-1808	22	25	]	]	PUNCT
iajs-1808	22	26	.	.	PUNCT
iajs-1808	23	1	it	it	PRON
iajs-1808	23	2	is	be	AUX
iajs-1808	23	3	clear	clear	ADJ
iajs-1808	23	4	that	that	SCONJ
iajs-1808	23	5	every	every	DET
iajs-1808	23	6	t	t	PROPN
iajs-1808	23	7	-	-	PUNCT
iajs-1808	23	8	closed	close	VERB
iajs-1808	23	9	submodule	submodule	NOUN
iajs-1808	23	10	is	be	AUX
iajs-1808	23	11	closed	close	VERB
iajs-1808	23	12	,	,	PUNCT
iajs-1808	23	13	but	but	CCONJ
iajs-1808	23	14	the	the	DET
iajs-1808	23	15	convers	conver	NOUN
iajs-1808	23	16	is	be	AUX
iajs-1808	23	17	not	not	PART
iajs-1808	23	18	true	true	ADJ
iajs-1808	23	19	.	.	PUNCT
iajs-1808	24	1	however	however	ADV
iajs-1808	24	2	,	,	PUNCT
iajs-1808	24	3	under	under	ADP
iajs-1808	24	4	the	the	DET
iajs-1808	24	5	class	class	NOUN
iajs-1808	24	6	of	of	ADP
iajs-1808	24	7	nonsingular	nonsingular	ADJ
iajs-1808	24	8	,	,	PUNCT
iajs-1808	24	9	the	the	DET
iajs-1808	24	10	two	two	NUM
iajs-1808	24	11	concepts	concept	NOUN
iajs-1808	24	12	are	be	AUX
iajs-1808	24	13	equivalent	equivalent	ADJ
iajs-1808	24	14	.	.	PUNCT
iajs-1808	25	1	recall	recall	VERB
iajs-1808	25	2	that	that	SCONJ
iajs-1808	25	3			PROPN
iajs-1808	25	4	a	a	DET
iajs-1808	25	5	module	module	NOUN
iajs-1808	25	6	𝑀	𝑀	PROPN
iajs-1808	25	7	is	be	AUX
iajs-1808	25	8	called	call	VERB
iajs-1808	25	9	extending	extend	VERB
iajs-1808	25	10	if	if	SCONJ
iajs-1808	25	11	for	for	ADP
iajs-1808	25	12	every	every	DET
iajs-1808	25	13	submodule	submodule	NOUN
iajs-1808	25	14	𝑁	𝑁	PROPN
iajs-1808	25	15	of	of	ADP
iajs-1808	25	16	𝑀	𝑀	PROPN
iajs-1808	25	17	then	then	ADV
iajs-1808	25	18	there	there	PRON
iajs-1808	25	19	exists	exist	VERB
iajs-1808	25	20	a	a	DET
iajs-1808	25	21	direct	direct	ADJ
iajs-1808	25	22	summand	summand	NOUN
iajs-1808	25	23	𝑊	𝑊	PROPN
iajs-1808	25	24	𝑊	𝑊	PROPN
iajs-1808	25	25	⨁	⨁	PROPN
iajs-1808	25	26	𝑀	𝑀	PROPN
iajs-1808	25	27	)	)	PUNCT
iajs-1808	26	1	such	such	ADJ
iajs-1808	26	2	that	that	SCONJ
iajs-1808	26	3	𝑁	𝑁	PROPN
iajs-1808	26	4	𝑊	𝑊	PROPN
iajs-1808	26	5			NOUN
iajs-1808	27	1	[	[	X
iajs-1808	27	2	6	6	NUM
iajs-1808	27	3	]	]	PUNCT
iajs-1808	27	4	.	.	PUNCT
iajs-1808	28	1	equivalently	equivalently	ADV
iajs-1808	28	2	𝑀	𝑀	PROPN
iajs-1808	28	3	is	be	AUX
iajs-1808	28	4	extending	extend	VERB
iajs-1808	28	5	module	module	NOUN
iajs-1808	28	6	if	if	SCONJ
iajs-1808	28	7	every	every	DET
iajs-1808	28	8	closed	closed	ADJ
iajs-1808	28	9	submodule	submodule	NOUN
iajs-1808	28	10	of	of	ADP
iajs-1808	28	11	𝑀	𝑀	PROPN
iajs-1808	28	12	is	be	AUX
iajs-1808	28	13	a	a	DET
iajs-1808	28	14	direct	direct	ADJ
iajs-1808	28	15	summand.	summand.	NOUN
iajs-1808	28	16	as	as	ADP
iajs-1808	28	17	a	a	DET
iajs-1808	28	18	generalization	generalization	NOUN
iajs-1808	28	19	of	of	ADP
iajs-1808	28	20	extending	extend	VERB
iajs-1808	28	21	module	module	NOUN
iajs-1808	28	22	,	,	PUNCT
iajs-1808	28	23	asgari	asgari	PROPN
iajs-1808	29	1	[	[	X
iajs-1808	29	2	2	2	NUM
iajs-1808	29	3	]	]	PUNCT
iajs-1808	29	4	introduced	introduce	VERB
iajs-1808	29	5	the	the	DET
iajs-1808	29	6	concept	concept	NOUN
iajs-1808	29	7	t	t	NOUN
iajs-1808	29	8	-	-	PUNCT
iajs-1808	29	9	extending	extend	VERB
iajs-1808	29	10	module	module	NOUN
iajs-1808	29	11	,	,	PUNCT
iajs-1808	29	12	where	where	SCONJ
iajs-1808	29	13	a	a	NUM
iajs-1808	29	14	module	module	NOUN
iajs-1808	29	15	𝑀	𝑀	PROPN
iajs-1808	29	16	is	be	AUX
iajs-1808	29	17	t	t	NOUN
iajs-1808	29	18	-	-	PUNCT
iajs-1808	29	19	extending	extend	VERB
iajs-1808	29	20	if	if	SCONJ
iajs-1808	29	21	every	every	DET
iajs-1808	29	22	t	t	NOUN
iajs-1808	29	23	-	-	PUNCT
iajs-1808	29	24	closed	close	VERB
iajs-1808	29	25	submodule	submodule	NOUN
iajs-1808	29	26	is	be	AUX
iajs-1808	29	27	a	a	DET
iajs-1808	29	28	direct	direct	ADJ
iajs-1808	29	29	summand.	summand.	NOUN
iajs-1808	29	30	equivalently,𝑀	equivalently,𝑀	NOUN
iajs-1808	29	31	is	be	AUX
iajs-1808	29	32	t	t	NOUN
iajs-1808	29	33	-	-	PUNCT
iajs-1808	29	34	extending	extend	VERB
iajs-1808	29	35	if	if	SCONJ
iajs-1808	29	36	every	every	DET
iajs-1808	29	37	submodule	submodule	NOUN
iajs-1808	29	38	of	of	ADP
iajs-1808	29	39	𝑀	𝑀	PROPN
iajs-1808	29	40	is	be	AUX
iajs-1808	29	41	t	t	NOUN
iajs-1808	29	42	-	-	PUNCT
iajs-1808	29	43	essential	essential	ADJ
iajs-1808	29	44	in	in	ADP
iajs-1808	29	45	a	a	DET
iajs-1808	29	46	direct	direct	ADJ
iajs-1808	29	47	summand	summand	NOUN
iajs-1808	29	48	[2	[2	PROPN
iajs-1808	29	49	]	]	PUNCT
iajs-1808	29	50	.	.	PUNCT
iajs-1808	30	1	the	the	DET
iajs-1808	30	2	notion	notion	NOUN
iajs-1808	30	3	of	of	ADP
iajs-1808	30	4	a	a	DET
iajs-1808	30	5	strongly	strongly	ADV
iajs-1808	30	6	extending	extend	VERB
iajs-1808	30	7	module	module	NOUN
iajs-1808	30	8	is	be	AUX
iajs-1808	30	9	introduced	introduce	VERB
iajs-1808	30	10	in	in	ADP
iajs-1808	30	11	[	[	X
iajs-1808	30	12	13	13	NUM
iajs-1808	30	13	]	]	PUNCT
iajs-1808	30	14	,	,	PUNCT
iajs-1808	30	15	which	which	PRON
iajs-1808	30	16	is	be	AUX
iajs-1808	30	17	a	a	DET
iajs-1808	30	18	subclass	subclass	NOUN
iajs-1808	30	19	of	of	ADP
iajs-1808	30	20	the	the	DET
iajs-1808	30	21	class	class	NOUN
iajs-1808	30	22	of	of	ADP
iajs-1808	30	23	extending	extend	VERB
iajs-1808	30	24	module	module	NOUN
iajs-1808	30	25	,	,	PUNCT
iajs-1808	30	26	where	where	SCONJ
iajs-1808	30	27	an	an	NUM
iajs-1808	30	28	r	r	NOUN
iajs-1808	30	29	-	-	PUNCT
iajs-1808	30	30	module	module	NOUN
iajs-1808	30	31	m	m	NOUN
iajs-1808	30	32	is	be	AUX
iajs-1808	30	33	called	call	VERB
iajs-1808	30	34	strongly	strongly	ADV
iajs-1808	30	35	extending	extend	VERB
iajs-1808	30	36	if	if	SCONJ
iajs-1808	30	37	each	each	DET
iajs-1808	30	38	submodule	submodule	NOUN
iajs-1808	30	39	of	of	ADP
iajs-1808	30	40	m	m	PROPN
iajs-1808	30	41	is	be	AUX
iajs-1808	30	42	essential	essential	ADJ
iajs-1808	30	43	in	in	ADP
iajs-1808	30	44	a	a	DET
iajs-1808	30	45	fully	fully	ADV
iajs-1808	30	46	invariant	invariant	ADJ
iajs-1808	30	47	direct	direct	ADJ
iajs-1808	30	48	summand	summand	NOUN
iajs-1808	30	49	of	of	ADP
iajs-1808	30	50	m	m	PROPN
iajs-1808	30	51			PROPN
iajs-1808	30	52	,	,	PUNCT
iajs-1808	30	53	where	where	SCONJ
iajs-1808	30	54	a	a	NUM
iajs-1808	30	55	submodule	submodule	NOUN
iajs-1808	30	56	n	n	NUM
iajs-1808	30	57	is	be	AUX
iajs-1808	30	58	called	call	VERB
iajs-1808	30	59	fully	fully	ADV
iajs-1808	30	60	invariant	invariant	ADJ
iajs-1808	30	61	if	if	SCONJ
iajs-1808	30	62	for	for	ADP
iajs-1808	30	63	each	each	DET
iajs-1808	30	64	𝑓	𝑓	PRON
iajs-1808	30	65	∈	∈	PROPN
iajs-1808	30	66	end	end	NOUN
iajs-1808	30	67	m	m	VERB
iajs-1808	30	68	,	,	PUNCT
iajs-1808	30	69	𝑓	𝑓	DET
iajs-1808	30	70	n	n	DET
iajs-1808	30	71	n.an	n.an	NOUN
iajs-1808	30	72	𝑅-module	𝑅-module	PROPN
iajs-1808	30	73	is	be	AUX
iajs-1808	30	74	called	call	VERB
iajs-1808	30	75	strongly	strongly	ADV
iajs-1808	30	76	t	t	NOUN
iajs-1808	30	77	-	-	PUNCT
iajs-1808	30	78	extending	extend	VERB
iajs-1808	30	79	if	if	SCONJ
iajs-1808	30	80	every	every	DET
iajs-1808	30	81	submodule	submodule	NOUN
iajs-1808	30	82	𝑁	𝑁	PROPN
iajs-1808	30	83	of	of	ADP
iajs-1808	30	84	m	m	PRON
iajs-1808	30	85	,	,	PUNCT
iajs-1808	30	86	there	there	PRON
iajs-1808	30	87	exists	exist	VERB
iajs-1808	30	88	a	a	DET
iajs-1808	30	89	fully	fully	ADV
iajs-1808	30	90	invariant	invariant	ADJ
iajs-1808	30	91	direct	direct	ADJ
iajs-1808	30	92	summand	summand	NOUN
iajs-1808	30	93	𝑊	𝑊	PROPN
iajs-1808	30	94	of	of	ADP
iajs-1808	30	95	m	m	PRON
iajs-1808	30	96	such	such	ADJ
iajs-1808	30	97	that	that	SCONJ
iajs-1808	30	98	𝑁	𝑁	PROPN
iajs-1808	30	99	𝑊[7].equivalently	𝑊[7].equivalently	ADV
iajs-1808	30	100	“	"	PUNCT
iajs-1808	30	101	m	m	NOUN
iajs-1808	30	102	is	be	AUX
iajs-1808	30	103	strongly	strongly	ADV
iajs-1808	30	104	t	t	NOUN
iajs-1808	30	105	-	-	PUNCT
iajs-1808	30	106	extending	extend	VERB
iajs-1808	30	107	if	if	SCONJ
iajs-1808	30	108	every	every	DET
iajs-1808	30	109	t	t	NOUN
iajs-1808	30	110	-	-	PUNCT
iajs-1808	30	111	closed	close	VERB
iajs-1808	30	112	submodule	submodule	NOUN
iajs-1808	30	113	of	of	ADP
iajs-1808	30	114	𝑀	𝑀	PROPN
iajs-1808	30	115	is	be	AUX
iajs-1808	30	116	fully	fully	ADV
iajs-1808	30	117	invariant	invariant	ADJ
iajs-1808	30	118	direct	direct	ADJ
iajs-1808	30	119	summand[7	summand[7	NOUN
iajs-1808	30	120	]	]	X
iajs-1808	30	121	.	.	PUNCT
iajs-1808	31	1	a	a	DET
iajs-1808	31	2	module	module	NOUN
iajs-1808	31	3	m	m	VERB
iajs-1808	31	4	is	be	AUX
iajs-1808	31	5	called	call	VERB
iajs-1808	31	6	duo	duo	NOUN
iajs-1808	31	7	if	if	SCONJ
iajs-1808	31	8	every	every	DET
iajs-1808	31	9	submodule	submodule	NOUN
iajs-1808	31	10	of	of	ADP
iajs-1808	31	11	m	m	PROPN
iajs-1808	31	12	is	be	AUX
iajs-1808	31	13	fully	fully	ADV
iajs-1808	31	14	invariant	invariant	ADJ
iajs-1808	32	1	[	[	X
iajs-1808	32	2	12	12	NUM
iajs-1808	32	3	]	]	PUNCT
iajs-1808	32	4	.	.	PUNCT
iajs-1808	33	1	hence	hence	ADV
iajs-1808	33	2	the	the	DET
iajs-1808	33	3	two	two	NUM
iajs-1808	33	4	concepts	concept	NOUN
iajs-1808	33	5	strongly	strongly	ADV
iajs-1808	33	6	extending	extend	VERB
iajs-1808	33	7	and	and	CCONJ
iajs-1808	33	8	extending	extend	VERB
iajs-1808	33	9	are	be	AUX
iajs-1808	33	10	equivalent	equivalent	ADJ
iajs-1808	33	11	in	in	ADP
iajs-1808	33	12	the	the	DET
iajs-1808	33	13	class	class	NOUN
iajs-1808	33	14	of	of	ADP
iajs-1808	33	15	duo	duo	NOUN
iajs-1808	33	16	modules	module	NOUN
iajs-1808	33	17	.	.	PUNCT
iajs-1808	34	1	asgari	asgari	PROPN
iajs-1808	34	2	and	and	CCONJ
iajs-1808	34	3	haghany	haghany	NOUN
iajs-1808	34	4	introduced	introduce	VERB
iajs-1808	34	5	the	the	DET
iajs-1808	34	6	concept	concept	NOUN
iajs-1808	34	7	of	of	ADP
iajs-1808	34	8	t	t	NOUN
iajs-1808	34	9	-	-	PUNCT
iajs-1808	34	10	semisimple	semisimple	NOUN
iajs-1808	34	11	modules	module	NOUN
iajs-1808	34	12	and	and	CCONJ
iajs-1808	34	13	t	t	NOUN
iajs-1808	34	14	-	-	PUNCT
iajs-1808	34	15	semisimple	semisimple	NOUN
iajs-1808	34	16	rings	ring	NOUN
iajs-1808	34	17	.	.	PUNCT
iajs-1808	35	1	a	a	DET
iajs-1808	35	2	module	module	NOUN
iajs-1808	35	3	m	m	VERB
iajs-1808	35	4	is	be	AUX
iajs-1808	35	5	called	call	VERB
iajs-1808	35	6	t	t	NOUN
iajs-1808	35	7	-	-	PUNCT
iajs-1808	35	8	semisimple	semisimple	NOUN
iajs-1808	35	9	if	if	SCONJ
iajs-1808	35	10	every	every	DET
iajs-1808	35	11	submodule	submodule	NOUN
iajs-1808	35	12	n	n	PRON
iajs-1808	35	13	contains	contain	VERB
iajs-1808	35	14	a	a	DET
iajs-1808	35	15	direct	direct	ADJ
iajs-1808	35	16	summand	summand	NOUN
iajs-1808	35	17	k	k	PROPN
iajs-1808	35	18	of	of	ADP
iajs-1808	35	19	m	m	PROPN
iajs-1808	35	20	such	such	ADJ
iajs-1808	35	21	that	that	SCONJ
iajs-1808	35	22	k	k	PROPN
iajs-1808	35	23	is	be	AUX
iajs-1808	35	24	tessential	tessential	ADJ
iajs-1808	35	25	in	in	ADP
iajs-1808	35	26	n	n	PRON
iajs-1808	35	27	[	[	X
iajs-1808	35	28	3	3	NUM
iajs-1808	35	29	]	]	PUNCT
iajs-1808	35	30	.	.	PUNCT
iajs-1808	36	1	in	in	ADP
iajs-1808	36	2	[	[	X
iajs-1808	36	3	9	9	NUM
iajs-1808	36	4	]	]	SYM
iajs-1808	36	5	inaam	inaam	NOUN
iajs-1808	36	6	and	and	CCONJ
iajs-1808	36	7	farhan	farhan	PROPN
iajs-1808	36	8	introduced	introduce	VERB
iajs-1808	36	9	and	and	CCONJ
iajs-1808	36	10	studied	study	VERB
iajs-1808	36	11	strongly	strongly	ADV
iajs-1808	36	12	t	t	NOUN
iajs-1808	36	13	-	-	PUNCT
iajs-1808	36	14	semisimple	semisimple	NOUN
iajs-1808	36	15	,	,	PUNCT
iajs-1808	36	16	where	where	SCONJ
iajs-1808	36	17	an	an	DET
iajs-1808	36	18	r	r	NOUN
iajs-1808	36	19	-	-	PUNCT
iajs-1808	36	20	module	module	NOUN
iajs-1808	36	21	is	be	AUX
iajs-1808	36	22	called	call	VERB
iajs-1808	36	23	strongly	strongly	ADV
iajs-1808	36	24	t	t	NOUN
iajs-1808	36	25	-	-	PUNCT
iajs-1808	36	26	semisimple	semisimple	NOUN
iajs-1808	36	27	if	if	SCONJ
iajs-1808	36	28	for	for	ADP
iajs-1808	36	29	each	each	DET
iajs-1808	36	30	submodule	submodule	NOUN
iajs-1808	36	31	n	n	PROPN
iajs-1808	36	32	of	of	ADP
iajs-1808	36	33	m	m	VERB
iajs-1808	36	34	there	there	PRON
iajs-1808	36	35	exists	exist	VERB
iajs-1808	36	36	a	a	DET
iajs-1808	36	37	fully	fully	ADV
iajs-1808	36	38	invariant	invariant	ADJ
iajs-1808	36	39	direct	direct	ADJ
iajs-1808	36	40	summand	summand	NOUN
iajs-1808	36	41	k	k	PROPN
iajs-1808	36	42	such	such	ADJ
iajs-1808	36	43	that	that	SCONJ
iajs-1808	37	1	k	k	PROPN
iajs-1808	37	2	n	n	X
iajs-1808	38	1	[	[	X
iajs-1808	38	2	9	9	NUM
iajs-1808	38	3	]	]	PUNCT
iajs-1808	38	4	.	.	PUNCT
iajs-1808	38	5	inaam	inaam	PROPN
iajs-1808	38	6	and	and	CCONJ
iajs-1808	38	7	farhan	farhan	PROPN
iajs-1808	38	8	in	in	ADP
iajs-1808	38	9	[	[	X
iajs-1808	38	10	10	10	NUM
iajs-1808	38	11	]	]	PUNCT
iajs-1808	38	12	introduced	introduce	VERB
iajs-1808	38	13	and	and	CCONJ
iajs-1808	38	14	studied	study	VERB
iajs-1808	38	15	fi	fi	PROPN
iajs-1808	38	16	-	-	ADJ
iajs-1808	38	17	t	t	NOUN
iajs-1808	38	18	-	-	PUNCT
iajs-1808	38	19	semisimple	semisimple	NOUN
iajs-1808	38	20	and	and	CCONJ
iajs-1808	38	21	strongly	strongly	ADV
iajs-1808	38	22	fi	fi	ADJ
iajs-1808	38	23	-	-	ADJ
iajs-1808	38	24	t	t	NOUN
iajs-1808	38	25	-	-	PUNCT
iajs-1808	38	26	semisimple	semisimple	NOUN
iajs-1808	38	27	.	.	PUNCT
iajs-1808	39	1	an	an	NUM
iajs-1808	39	2	r	r	NOUN
iajs-1808	39	3	-	-	PUNCT
iajs-1808	39	4	module	module	NOUN
iajs-1808	39	5	m	m	NOUN
iajs-1808	39	6	is	be	AUX
iajs-1808	39	7	called	call	VERB
iajs-1808	39	8	fi	fi	NOUN
iajs-1808	39	9	-	-	NOUN
iajs-1808	39	10	tsemisimple	tsemisimple	NOUN
iajs-1808	39	11	if	if	SCONJ
iajs-1808	39	12	for	for	ADP
iajs-1808	39	13	each	each	DET
iajs-1808	39	14	fully	fully	ADV
iajs-1808	39	15	invariant	invariant	ADJ
iajs-1808	39	16	submodule	submodule	NOUN
iajs-1808	39	17	𝑁	𝑁	PROPN
iajs-1808	39	18	of	of	ADP
iajs-1808	39	19	𝑀	𝑀	PROPN
iajs-1808	39	20	,	,	PUNCT
iajs-1808	39	21	there	there	PRON
iajs-1808	39	22	exists	exist	VERB
iajs-1808	39	23	k	k	PROPN
iajs-1808	39	24	⨁	⨁	PROPN
iajs-1808	39	25	m	m	VERB
iajs-1808	39	26	such	such	ADJ
iajs-1808	39	27	that	that	SCONJ
iajs-1808	39	28	k	k	PROPN
iajs-1808	39	29	n.	n.	PROPN
iajs-1808	39	30	an	an	DET
iajs-1808	39	31	r	r	NOUN
iajs-1808	39	32	-	-	PUNCT
iajs-1808	39	33	module	module	NOUN
iajs-1808	39	34	𝑀	𝑀	PROPN
iajs-1808	39	35	is	be	AUX
iajs-1808	39	36	called	call	VERB
iajs-1808	39	37	strongly	strongly	ADV
iajs-1808	39	38	fi	fi	NOUN
iajs-1808	39	39	-	-	ADJ
iajs-1808	39	40	t	t	NOUN
iajs-1808	39	41	-	-	PUNCT
iajs-1808	39	42	semisimple	semisimple	NOUN
iajs-1808	39	43	if	if	SCONJ
iajs-1808	39	44	for	for	ADP
iajs-1808	39	45	each	each	DET
iajs-1808	39	46	fully	fully	ADV
iajs-1808	39	47	invariant	invariant	ADJ
iajs-1808	39	48	submodule	submodule	NOUN
iajs-1808	39	49	𝑁	𝑁	PROPN
iajs-1808	39	50	of	of	ADP
iajs-1808	39	51	𝑀	𝑀	PROPN
iajs-1808	39	52	,	,	PUNCT
iajs-1808	39	53	there	there	PRON
iajs-1808	39	54	exists	exist	VERB
iajs-1808	39	55	a	a	DET
iajs-1808	39	56	fully	fully	ADV
iajs-1808	39	57	invariant	invariant	ADJ
iajs-1808	39	58	direct	direct	ADJ
iajs-1808	39	59	summand	summand	NOUN
iajs-1808	39	60	𝐾	𝐾	PROPN
iajs-1808	40	1	such	such	ADJ
iajs-1808	40	2	that	that	SCONJ
iajs-1808	40	3	k	k	PROPN
iajs-1808	40	4	n	n	ADV
iajs-1808	40	5	[10	[10	NOUN
iajs-1808	40	6	]	]	PUNCT
iajs-1808	40	7	.	.	PUNCT
iajs-1808	41	1	recall	recall	VERB
iajs-1808	41	2	that:	that:	DET
iajs-1808	41	3	an	an	DET
iajs-1808	41	4	𝑅-module	𝑅-module	PROPN
iajs-1808	41	5	𝑀	𝑀	PROPN
iajs-1808	41	6	is	be	AUX
iajs-1808	41	7	said	say	VERB
iajs-1808	41	8	to	to	PART
iajs-1808	41	9	be	be	AUX
iajs-1808	41	10	satisfy	satisfy	NOUN
iajs-1808	41	11	𝐶	𝐶	PROPN
iajs-1808	41	12	-condition	-condition	PROPN
iajs-1808	41	13	if	if	SCONJ
iajs-1808	41	14	every	every	DET
iajs-1808	41	15	submodule	submodule	NOUN
iajs-1808	41	16	of	of	ADP
iajs-1808	41	17	𝑀	𝑀	PROPN
iajs-1808	41	18	has	have	VERB
iajs-1808	41	19	a	a	DET
iajs-1808	41	20	complement	complement	NOUN
iajs-1808	41	21	which	which	PRON
iajs-1808	41	22	is	be	AUX
iajs-1808	41	23	a	a	DET
iajs-1808	41	24	direct	direct	ADJ
iajs-1808	41	25	summand[15	summand[15	NOUN
iajs-1808	41	26	]	]	PUNCT
iajs-1808	41	27	.	.	PUNCT
iajs-1808	42	1	asgari	asgari	PROPN
iajs-1808	43	1	[	[	X
iajs-1808	43	2	4	4	NUM
iajs-1808	43	3	]	]	PUNCT
iajs-1808	43	4	,	,	PUNCT
iajs-1808	43	5	restricted	restrict	VERB
iajs-1808	43	6	𝐶	𝐶	PROPN
iajs-1808	43	7	condition	condition	NOUN
iajs-1808	43	8	to	to	ADP
iajs-1808	43	9	tclosed	tclosed	ADJ
iajs-1808	43	10	condition	condition	NOUN
iajs-1808	43	11	of	of	ADP
iajs-1808	43	12	𝑀.	𝑀.	PROPN
iajs-1808	43	13	she	she	PRON
iajs-1808	43	14	defined	define	VERB
iajs-1808	43	15	the	the	DET
iajs-1808	43	16	following	following	NOUN
iajs-1808	43	17	.	.	PUNCT
iajs-1808	44	1	an	an	NUM
iajs-1808	44	2	𝑅-module	𝑅-module	PROPN
iajs-1808	44	3	𝑀	𝑀	PROPN
iajs-1808	44	4	said	say	VERB
iajs-1808	44	5	to	to	PART
iajs-1808	44	6	be	be	AUX
iajs-1808	44	7	𝑇	𝑇	PROPN
iajs-1808	44	8	-type	-type	NOUN
iajs-1808	44	9	module	module	NOUN
iajs-1808	44	10	(	(	PUNCT
iajs-1808	44	11	or	or	CCONJ
iajs-1808	44	12	𝑀	𝑀	PROPN
iajs-1808	44	13	satidfy	satidfy	NOUN
iajs-1808	44	14	𝑇	𝑇	PROPN
iajs-1808	44	15	-condition	-condition	PROPN
iajs-1808	44	16	)	)	PUNCT
iajs-1808	44	17	if	if	SCONJ
iajs-1808	44	18	every	every	DET
iajs-1808	44	19	t	t	PROPN
iajs-1808	44	20	-	-	PUNCT
iajs-1808	44	21	closed	close	VERB
iajs-1808	44	22	submodule	submodule	NOUN
iajs-1808	44	23	has	have	VERB
iajs-1808	44	24	a	a	DET
iajs-1808	44	25	complement	complement	NOUN
iajs-1808	44	26	which	which	PRON
iajs-1808	44	27	is	be	AUX
iajs-1808	44	28	a	a	DET
iajs-1808	44	29	direct	direct	ADJ
iajs-1808	44	30	summand	summand	NOUN
iajs-1808	44	31	.	.	PUNCT
iajs-1808	45	1	a	a	DET
iajs-1808	45	2	ring	ring	NOUN
iajs-1808	45	3	is	be	AUX
iajs-1808	45	4	said	say	VERB
iajs-1808	45	5	to	to	PART
iajs-1808	45	6	be	be	AUX
iajs-1808	45	7	right	right	ADJ
iajs-1808	45	8	𝑇	𝑇	PROPN
iajs-1808	45	9	-type	-type	NOUN
iajs-1808	45	10	ring	ring	NOUN
iajs-1808	45	11	if	if	SCONJ
iajs-1808	45	12	𝑅	𝑅	PROPN
iajs-1808	45	13	is	be	AUX
iajs-1808	45	14	a	a	DET
iajs-1808	45	15	𝑇	𝑇	PROPN
iajs-1808	45	16	-type	-type	NOUN
iajs-1808	45	17	module	module	NOUN
iajs-1808	45	18	[	[	X
iajs-1808	45	19	4	4	NUM
iajs-1808	45	20	]	]	PUNCT
iajs-1808	45	21	.	.	PUNCT
iajs-1808	46	1	ihsciconf	ihsciconf	PROPN
iajs-1808	46	2	2017	2017	NUM
iajs-1808	46	3	special	special	ADJ
iajs-1808	46	4	issue	issue	NOUN
iajs-1808	46	5	ibn	ibn	PROPN
iajs-1808	46	6	al	al	PROPN
iajs-1808	46	7	-	-	PUNCT
iajs-1808	46	8	haitham	haitham	PROPN
iajs-1808	46	9	journal	journal	PROPN
iajs-1808	46	10	for	for	ADP
iajs-1808	46	11	pure	pure	ADJ
iajs-1808	46	12	and	and	CCONJ
iajs-1808	46	13	applied	apply	VERB
iajs-1808	46	14	science	science	NOUN
iajs-1808	46	15	https://doi.org/	https://doi.org/	NOUN
iajs-1808	46	16	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	46	17	for	for	ADP
iajs-1808	46	18	more	more	ADJ
iajs-1808	46	19	information	information	NOUN
iajs-1808	46	20	about	about	ADP
iajs-1808	46	21	the	the	DET
iajs-1808	46	22	conference	conference	NOUN
iajs-1808	46	23	please	please	INTJ
iajs-1808	46	24	visit	visit	VERB
iajs-1808	46	25	the	the	DET
iajs-1808	46	26	websites	website	NOUN
iajs-1808	46	27	:	:	PUNCT
iajs-1808	46	28	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	46	29	  	  	SPACE
iajs-1808	46	30	www.ihsciconf.org	www.ihsciconf.org	NOUN
iajs-1808	46	31	   	   	SPACE
iajs-1808	46	32	mathematics|355	mathematics|355	PROPN
iajs-1808	46	33	    	    	SPACE
iajs-1808	46	34	this	this	DET
iajs-1808	46	35	paper	paper	NOUN
iajs-1808	46	36	consists	consist	VERB
iajs-1808	46	37	of	of	ADP
iajs-1808	46	38	three	three	NUM
iajs-1808	46	39	sections	section	NOUN
iajs-1808	46	40	.	.	PUNCT
iajs-1808	47	1	in	in	ADP
iajs-1808	47	2	section	section	NOUN
iajs-1808	47	3	two	two	NUM
iajs-1808	47	4	we	we	PRON
iajs-1808	47	5	deal	deal	VERB
iajs-1808	47	6	with	with	ADP
iajs-1808	47	7	certain	certain	ADJ
iajs-1808	47	8	known	know	VERB
iajs-1808	47	9	results	result	NOUN
iajs-1808	47	10	which	which	PRON
iajs-1808	47	11	are	be	AUX
iajs-1808	47	12	worthwhile	worthwhile	ADJ
iajs-1808	47	13	throughout	throughout	ADP
iajs-1808	47	14	the	the	DET
iajs-1808	47	15	paper	paper	NOUN
iajs-1808	47	16	.	.	PUNCT
iajs-1808	48	1	in	in	ADP
iajs-1808	48	2	section	section	NOUN
iajs-1808	48	3	three	three	NUM
iajs-1808	48	4	we	we	PRON
iajs-1808	48	5	investigate	investigate	VERB
iajs-1808	48	6	certain	certain	ADJ
iajs-1808	48	7	types	type	NOUN
iajs-1808	48	8	of	of	ADP
iajs-1808	48	9	module	module	NOUN
iajs-1808	48	10	namely	namely	ADV
iajs-1808	48	11	strongly	strongly	ADV
iajs-1808	48	12	𝐶	𝐶	PROPN
iajs-1808	48	13	-condition	-condition	PROPN
iajs-1808	48	14	and	and	CCONJ
iajs-1808	48	15	strongly	strongly	ADV
iajs-1808	48	16	𝑇	𝑇	PROPN
iajs-1808	48	17	-type	-type	NOUN
iajs-1808	48	18	modules	module	NOUN
iajs-1808	48	19	.	.	PUNCT
iajs-1808	49	1	an	an	DET
iajs-1808	49	2	𝑅-module	𝑅-module	PROPN
iajs-1808	49	3	𝑀	𝑀	PROPN
iajs-1808	49	4	is	be	AUX
iajs-1808	49	5	said	say	VERB
iajs-1808	49	6	to	to	PART
iajs-1808	49	7	be	be	AUX
iajs-1808	49	8	satisfy	satisfy	VERB
iajs-1808	49	9	strongly𝐶	strongly𝐶	CCONJ
iajs-1808	49	10	-condition	-condition	NOUN
iajs-1808	49	11	if	if	SCONJ
iajs-1808	49	12	every	every	DET
iajs-1808	49	13	submodule	submodule	NOUN
iajs-1808	49	14	of	of	ADP
iajs-1808	49	15	𝑀	𝑀	PROPN
iajs-1808	49	16	has	have	VERB
iajs-1808	49	17	a	a	DET
iajs-1808	49	18	complement	complement	NOUN
iajs-1808	49	19	which	which	PRON
iajs-1808	49	20	is	be	AUX
iajs-1808	49	21	a	a	DET
iajs-1808	49	22	fully	fully	ADV
iajs-1808	49	23	invariant	invariant	ADJ
iajs-1808	49	24	direct	direct	ADJ
iajs-1808	49	25	summand	summand	NOUN
iajs-1808	49	26	.	.	PUNCT
iajs-1808	50	1	also	also	ADV
iajs-1808	50	2	;	;	PUNCT
iajs-1808	50	3	an	an	DET
iajs-1808	50	4	𝑅-module	𝑅-module	PROPN
iajs-1808	50	5	m	m	NOUN
iajs-1808	50	6	is	be	AUX
iajs-1808	50	7	called	call	VERB
iajs-1808	50	8	strongly	strongly	ADV
iajs-1808	50	9	𝑇	𝑇	PROPN
iajs-1808	50	10	-type	-type	NOUN
iajs-1808	50	11	module	module	NOUN
iajs-1808	50	12	if	if	SCONJ
iajs-1808	50	13	every	every	DET
iajs-1808	50	14	tclosed	tclose	VERB
iajs-1808	50	15	submodule	submodule	NOUN
iajs-1808	50	16	has	have	VERB
iajs-1808	50	17	a	a	DET
iajs-1808	50	18	complement	complement	NOUN
iajs-1808	50	19	which	which	PRON
iajs-1808	50	20	is	be	AUX
iajs-1808	50	21	a	a	DET
iajs-1808	50	22	fully	fully	ADV
iajs-1808	50	23	invariant	invariant	ADJ
iajs-1808	50	24	direct	direct	ADJ
iajs-1808	50	25	summand	summand	NOUN
iajs-1808	50	26	.	.	PUNCT
iajs-1808	51	1	we	we	PRON
iajs-1808	51	2	give	give	VERB
iajs-1808	51	3	several	several	ADJ
iajs-1808	51	4	characterizations	characterization	NOUN
iajs-1808	51	5	of	of	ADP
iajs-1808	51	6	strongly	strongly	ADV
iajs-1808	51	7	𝐶	𝐶	PROPN
iajs-1808	51	8	-condition	-condition	PROPN
iajs-1808	51	9	and	and	CCONJ
iajs-1808	51	10	strongly	strongly	ADV
iajs-1808	51	11	𝑇	𝑇	PROPN
iajs-1808	51	12	-type	-type	NOUN
iajs-1808	51	13	module	module	NOUN
iajs-1808	51	14	.	.	PUNCT
iajs-1808	52	1	in	in	ADP
iajs-1808	52	2	particular	particular	ADJ
iajs-1808	52	3	𝑀	𝑀	PROPN
iajs-1808	52	4	is	be	AUX
iajs-1808	52	5	strongly	strongly	ADV
iajs-1808	52	6	𝑇	𝑇	PROPN
iajs-1808	52	7	-type	-type	NOUN
iajs-1808	52	8	module	module	NOUN
iajs-1808	52	9	if	if	SCONJ
iajs-1808	52	10	and	and	CCONJ
iajs-1808	52	11	only	only	ADV
iajs-1808	52	12	if	if	SCONJ
iajs-1808	52	13	𝑀	𝑀	PROPN
iajs-1808	52	14	𝑍	𝑍	VERB
iajs-1808	52	15	𝑀	𝑀	PROPN
iajs-1808	52	16	⨁𝑀	⨁𝑀	NOUN
iajs-1808	52	17	,	,	PUNCT
iajs-1808	52	18	where	where	SCONJ
iajs-1808	52	19	𝑀	𝑀	PROPN
iajs-1808	52	20	satisfies	satisfy	VERB
iajs-1808	52	21	strongly	strongly	ADV
iajs-1808	52	22	𝐶	𝐶	PROPN
iajs-1808	52	23	-condition	-condition	PROPN
iajs-1808	52	24	.	.	PUNCT
iajs-1808	53	1	every	every	DET
iajs-1808	53	2	module	module	NOUN
iajs-1808	53	3	with	with	ADP
iajs-1808	53	4	strongly	strongly	ADV
iajs-1808	53	5	𝐶	𝐶	PROPN
iajs-1808	53	6	-condition	-condition	NOUN
iajs-1808	53	7	is	be	AUX
iajs-1808	53	8	strongly	strongly	ADV
iajs-1808	53	9	𝑇	𝑇	PROPN
iajs-1808	53	10	-type	-type	NOUN
iajs-1808	53	11	module	module	NOUN
iajs-1808	53	12	,	,	PUNCT
iajs-1808	53	13	but	but	CCONJ
iajs-1808	53	14	not	not	PART
iajs-1808	53	15	conversely	conversely	ADV
iajs-1808	53	16	(	(	PUNCT
iajs-1808	53	17	see	see	VERB
iajs-1808	53	18	remark	remark	NOUN
iajs-1808	53	19	3.6(1	3.6(1	NUM
iajs-1808	53	20	)	)	PUNCT
iajs-1808	53	21	)	)	PUNCT
iajs-1808	53	22	.	.	PUNCT
iajs-1808	54	1	the	the	DET
iajs-1808	54	2	two	two	NUM
iajs-1808	54	3	concepts	concept	NOUN
iajs-1808	54	4	are	be	AUX
iajs-1808	54	5	equivalent	equivalent	ADJ
iajs-1808	54	6	under	under	ADP
iajs-1808	54	7	certain	certain	ADJ
iajs-1808	54	8	classes	class	NOUN
iajs-1808	54	9	of	of	ADP
iajs-1808	54	10	module	module	NOUN
iajs-1808	54	11	is	be	AUX
iajs-1808	54	12	given	give	VERB
iajs-1808	54	13	.	.	PUNCT
iajs-1808	55	1	furthermore	furthermore	ADV
iajs-1808	55	2	,	,	PUNCT
iajs-1808	55	3	many	many	ADJ
iajs-1808	55	4	connections	connection	NOUN
iajs-1808	55	5	between	between	ADP
iajs-1808	55	6	strongly	strongly	ADV
iajs-1808	55	7	𝑇	𝑇	PROPN
iajs-1808	55	8	-type	-type	NOUN
iajs-1808	55	9	modules	module	NOUN
iajs-1808	55	10	strongly	strongly	ADV
iajs-1808	55	11	tsemisimple	tsemisimple	NOUN
iajs-1808	55	12	module	module	NOUN
iajs-1808	55	13	,	,	PUNCT
iajs-1808	55	14	strongly	strongly	ADV
iajs-1808	55	15	fi	fi	ADJ
iajs-1808	55	16	-	-	ADJ
iajs-1808	55	17	t	t	NOUN
iajs-1808	55	18	-	-	PUNCT
iajs-1808	55	19	semisimple	semisimple	NOUN
iajs-1808	55	20	module	module	NOUN
iajs-1808	55	21	,	,	PUNCT
iajs-1808	55	22	fi	fi	NOUN
iajs-1808	55	23	-	-	ADJ
iajs-1808	55	24	t	t	NOUN
iajs-1808	55	25	-	-	PUNCT
iajs-1808	55	26	semisimple	semisimple	NOUN
iajs-1808	55	27	module	module	NOUN
iajs-1808	55	28	,	,	PUNCT
iajs-1808	55	29	strongly	strongly	ADV
iajs-1808	55	30	extending	extend	VERB
iajs-1808	55	31	module	module	NOUN
iajs-1808	55	32	,	,	PUNCT
iajs-1808	55	33	strongly	strongly	ADV
iajs-1808	55	34	t	t	NOUN
iajs-1808	55	35	-	-	PUNCT
iajs-1808	55	36	extending	extend	VERB
iajs-1808	55	37	module	module	NOUN
iajs-1808	55	38	are	be	AUX
iajs-1808	55	39	presented	present	VERB
iajs-1808	55	40	.	.	PUNCT
iajs-1808	56	1	2	2	X
iajs-1808	56	2	.	.	PUNCT
iajs-1808	56	3	preliminaries	preliminary	NOUN
iajs-1808	56	4	proposition	proposition	NOUN
iajs-1808	56	5	(	(	PUNCT
iajs-1808	56	6	1.1)[2	1.1)[2	NOUN
iajs-1808	56	7	]	]	X
iajs-1808	56	8	:	:	PUNCT
iajs-1808	56	9	the	the	DET
iajs-1808	56	10	following	follow	VERB
iajs-1808	56	11	statements	statement	NOUN
iajs-1808	56	12	are	be	AUX
iajs-1808	56	13	equivalent	equivalent	ADJ
iajs-1808	56	14	for	for	ADP
iajs-1808	56	15	a	a	DET
iajs-1808	56	16	submodule	submodule	NOUN
iajs-1808	56	17	𝐴	𝐴	PROPN
iajs-1808	56	18	of	of	ADP
iajs-1808	56	19	an	an	DET
iajs-1808	56	20	𝑅module	𝑅module	PROPN
iajs-1808	56	21	𝑀	𝑀	PROPN
iajs-1808	56	22	𝐴	𝐴	PROPN
iajs-1808	56	23	is	be	AUX
iajs-1808	56	24	t	t	NOUN
iajs-1808	56	25	-	-	PUNCT
iajs-1808	56	26	essential	essential	ADJ
iajs-1808	56	27	in	in	ADP
iajs-1808	56	28	𝑀	𝑀	PROPN
iajs-1808	56	29	;	;	PUNCT
iajs-1808	56	30	(	(	PUNCT
iajs-1808	56	31	𝐴+𝑍	𝐴+𝑍	NOUN
iajs-1808	56	32	𝑀	𝑀	PROPN
iajs-1808	56	33	/𝑍	/𝑍	PUNCT
iajs-1808	56	34	𝑀	𝑀	PROPN
iajs-1808	56	35	is	be	AUX
iajs-1808	56	36	essential	essential	ADJ
iajs-1808	56	37	in	in	ADP
iajs-1808	56	38	𝑀/𝑍	𝑀/𝑍	PROPN
iajs-1808	56	39	𝑀	𝑀	PROPN
iajs-1808	56	40	;	;	PUNCT
iajs-1808	56	41	𝐴+𝑍	𝐴+𝑍	PROPN
iajs-1808	56	42	𝑀	𝑀	PROPN
iajs-1808	56	43	is	be	AUX
iajs-1808	56	44	essential	essential	ADJ
iajs-1808	56	45	in	in	ADP
iajs-1808	56	46	𝑀	𝑀	PROPN
iajs-1808	56	47	;	;	PUNCT
iajs-1808	56	48	𝑀/𝐴	𝑀/𝐴	VERB
iajs-1808	56	49	is	be	AUX
iajs-1808	56	50	𝑍	𝑍	PROPN
iajs-1808	56	51	𝑡𝑜𝑟𝑠𝑖𝑜𝑛..	𝑡𝑜𝑟𝑠𝑖𝑜𝑛..	AUX
iajs-1808	56	52	lemma	lemma	PROPN
iajs-1808	56	53	(	(	PUNCT
iajs-1808	56	54	1.2)[2]:	1.2)[2]:	NUM
iajs-1808	56	55	let	let	VERB
iajs-1808	56	56	m	m	PRON
iajs-1808	56	57	be	be	AUX
iajs-1808	56	58	an	an	DET
iajs-1808	56	59	𝑅module	𝑅module	PROPN
iajs-1808	56	60	.	.	PUNCT
iajs-1808	57	1	then	then	ADV
iajs-1808	57	2	if	if	SCONJ
iajs-1808	57	3	m	m	PROPN
iajs-1808	57	4	,	,	PUNCT
iajs-1808	57	5	then	then	ADV
iajs-1808	57	6	𝑍	𝑍	VERB
iajs-1808	57	7	𝑀	𝑀	PROPN
iajs-1808	57	8	𝐶.	𝐶.	PROPN
iajs-1808	57	9	0	0	NUM
iajs-1808	57	10	𝑀	𝑀	PROPN
iajs-1808	58	1	if	if	SCONJ
iajs-1808	58	2	and	and	CCONJ
iajs-1808	58	3	only	only	ADV
iajs-1808	58	4	if	if	SCONJ
iajs-1808	58	5	m	m	NOUN
iajs-1808	58	6	is	be	AUX
iajs-1808	58	7	nonsingular	nonsingular	ADJ
iajs-1808	58	8	.	.	PUNCT
iajs-1808	59	1	if	if	SCONJ
iajs-1808	59	2	𝐴	𝐴	PROPN
iajs-1808	59	3	𝐶	𝐶	PROPN
iajs-1808	59	4	,	,	PUNCT
iajs-1808	59	5	then	then	ADV
iajs-1808	59	6	𝐶	𝐶	PROPN
iajs-1808	59	7	m	m	VERB
iajs-1808	59	8	if	if	SCONJ
iajs-1808	59	9	and	and	CCONJ
iajs-1808	59	10	only	only	ADV
iajs-1808	59	11	if	if	SCONJ
iajs-1808	59	12	.	.	PROPN
iajs-1808	59	13	theorem	theorem	NOUN
iajs-1808	59	14	(	(	PUNCT
iajs-1808	59	15	1.3)[9	1.3)[9	NOUN
iajs-1808	59	16	]	]	X
iajs-1808	59	17	:	:	PUNCT
iajs-1808	59	18			NOUN
iajs-1808	59	19	the	the	DET
iajs-1808	59	20	following	follow	VERB
iajs-1808	59	21	statements	statement	NOUN
iajs-1808	59	22	are	be	AUX
iajs-1808	59	23	equivalent	equivalent	ADJ
iajs-1808	59	24	for	for	ADP
iajs-1808	59	25	an	an	DET
iajs-1808	59	26	𝑅	𝑅	PROPN
iajs-1808	59	27	-module	-module	PROPN
iajs-1808	59	28	𝑀	𝑀	PROPN
iajs-1808	59	29	:	:	PUNCT
iajs-1808	59	30	𝑀	𝑀	PROPN
iajs-1808	59	31	is	be	AUX
iajs-1808	59	32	strongly	strongly	ADV
iajs-1808	59	33	t	t	NOUN
iajs-1808	59	34	-	-	PUNCT
iajs-1808	59	35	semisimple	semisimple	NOUN
iajs-1808	59	36	,	,	PUNCT
iajs-1808	59	37	is	be	AUX
iajs-1808	59	38	a	a	DET
iajs-1808	59	39	fully	fully	ADV
iajs-1808	59	40	stable	stable	ADJ
iajs-1808	59	41	semisimple	semisimple	NOUN
iajs-1808	59	42	and	and	CCONJ
iajs-1808	59	43	isomorphic	isomorphic	ADJ
iajs-1808	59	44	to	to	ADP
iajs-1808	59	45	a	a	DET
iajs-1808	59	46	stable	stable	ADJ
iajs-1808	59	47	submodule	submodule	NOUN
iajs-1808	59	48	of	of	ADP
iajs-1808	59	49	𝑀	𝑀	PROPN
iajs-1808	59	50	,	,	PUNCT
iajs-1808	59	51	𝑀	𝑀	PROPN
iajs-1808	59	52	=	=	SYM
iajs-1808	59	53	𝑍	𝑍	PROPN
iajs-1808	59	54	(	(	PUNCT
iajs-1808	59	55	𝑀)⨁𝑀	𝑀)⨁𝑀	NOUN
iajs-1808	59	56	where	where	SCONJ
iajs-1808	59	57	𝑀	𝑀	PROPN
iajs-1808	59	58	is	be	AUX
iajs-1808	59	59	a	a	DET
iajs-1808	59	60	nonsingular	nonsingular	ADJ
iajs-1808	59	61	semisimple	semisimple	NOUN
iajs-1808	59	62	fully	fully	ADV
iajs-1808	59	63	stable	stable	ADJ
iajs-1808	59	64	module	module	NOUN
iajs-1808	59	65	and	and	CCONJ
iajs-1808	59	66	𝑀	𝑀	PROPN
iajs-1808	59	67	is	be	AUX
iajs-1808	59	68	stable	stable	ADJ
iajs-1808	59	69	in	in	ADP
iajs-1808	59	70	𝑀	𝑀	PROPN
iajs-1808	59	71	,	,	PUNCT
iajs-1808	59	72	every	every	DET
iajs-1808	59	73	nonsingular	nonsingular	ADJ
iajs-1808	59	74	submodule	submodule	NOUN
iajs-1808	59	75	is	be	AUX
iajs-1808	59	76	stable	stable	ADJ
iajs-1808	59	77	direct	direct	ADJ
iajs-1808	59	78	summand	summand	NOUN
iajs-1808	59	79	,	,	PUNCT
iajs-1808	59	80	every	every	DET
iajs-1808	59	81	submodule	submodule	NOUN
iajs-1808	59	82	of	of	ADP
iajs-1808	59	83	𝑀	𝑀	PROPN
iajs-1808	59	84	which	which	PRON
iajs-1808	59	85	contains	contain	VERB
iajs-1808	59	86	𝑍	𝑍	PROPN
iajs-1808	59	87	(	(	PUNCT
iajs-1808	59	88	𝑀	𝑀	PROPN
iajs-1808	59	89	)	)	PUNCT
iajs-1808	59	90	is	be	AUX
iajs-1808	59	91	a	a	DET
iajs-1808	59	92	direct	direct	ADJ
iajs-1808	59	93	summand	summand	NOUN
iajs-1808	59	94	of	of	ADP
iajs-1808	59	95	𝑀	𝑀	PROPN
iajs-1808	59	96	and	and	CCONJ
iajs-1808	59	97	is	be	AUX
iajs-1808	59	98	fully	fully	ADV
iajs-1808	59	99	stable	stable	ADJ
iajs-1808	59	100	and	and	CCONJ
iajs-1808	59	101	isomorphic	isomorphic	ADJ
iajs-1808	59	102	to	to	ADP
iajs-1808	59	103	a	a	DET
iajs-1808	59	104	stable	stable	ADJ
iajs-1808	59	105	submodule	submodule	NOUN
iajs-1808	59	106	of	of	ADP
iajs-1808	59	107	𝑀.	𝑀.	PROPN
iajs-1808	59	108	proposition	proposition	NOUN
iajs-1808	59	109	(	(	PUNCT
iajs-1808	59	110	1.4)[10]:	1.4)[10]:	NOUN
iajs-1808	59	111	let	let	VERB
iajs-1808	59	112	𝑀	𝑀	PROPN
iajs-1808	59	113	be	be	AUX
iajs-1808	59	114	an	an	DET
iajs-1808	59	115	𝑅-module	𝑅-module	PROPN
iajs-1808	59	116	with	with	ADP
iajs-1808	59	117	the	the	DET
iajs-1808	59	118	property	property	NOUN
iajs-1808	59	119	,	,	PUNCT
iajs-1808	59	120	complement	complement	NOUN
iajs-1808	59	121	of	of	ADP
iajs-1808	59	122	any	any	DET
iajs-1808	59	123	submodule	submodule	NOUN
iajs-1808	59	124	of	of	ADP
iajs-1808	59	125	𝑀	𝑀	PROPN
iajs-1808	59	126	is	be	AUX
iajs-1808	59	127	stable	stable	ADJ
iajs-1808	59	128	.	.	PUNCT
iajs-1808	60	1	the	the	DET
iajs-1808	60	2	following	follow	VERB
iajs-1808	60	3	statements	statement	NOUN
iajs-1808	60	4	are	be	AUX
iajs-1808	60	5	equivalent	equivalent	ADJ
iajs-1808	60	6	.	.	PUNCT
iajs-1808	61	1	𝑀	𝑀	PROPN
iajs-1808	61	2	is	be	AUX
iajs-1808	61	3	strongly	strongly	ADV
iajs-1808	61	4	fi	fi	ADJ
iajs-1808	61	5	-	-	ADJ
iajs-1808	61	6	t	t	NOUN
iajs-1808	61	7	-	-	PUNCT
iajs-1808	61	8	semisimple	semisimple	NOUN
iajs-1808	61	9	;	;	PUNCT
iajs-1808	61	10	𝑀	𝑀	PROPN
iajs-1808	61	11	is	be	AUX
iajs-1808	61	12	fi	fi	ADJ
iajs-1808	61	13	-	-	ADJ
iajs-1808	61	14	t	t	NOUN
iajs-1808	61	15	-	-	PUNCT
iajs-1808	61	16	semisimple;.	semisimple;.	PROPN
iajs-1808	61	17	let	let	NOUN
iajs-1808	61	18	(	(	PUNCT
iajs-1808	61	19	⋇	⋇	NOUN
iajs-1808	61	20	)	)	PUNCT
iajs-1808	61	21	means	mean	VERB
iajs-1808	61	22	the	the	DET
iajs-1808	61	23	following	following	NOUN
iajs-1808	61	24	:	:	PUNCT
iajs-1808	61	25	for	for	ADP
iajs-1808	61	26	an	an	DET
iajs-1808	61	27	𝑹-module	𝑹-module	PROPN
iajs-1808	61	28	𝑴	𝑴	PROPN
iajs-1808	61	29	,	,	PUNCT
iajs-1808	61	30	the	the	DET
iajs-1808	61	31	complement	complement	NOUN
iajs-1808	61	32	of	of	ADP
iajs-1808	61	33	𝒁𝟐	𝒁𝟐	PROPN
iajs-1808	61	34	𝑴	𝑴	PROPN
iajs-1808	61	35	is	be	AUX
iajs-1808	61	36	stable	stable	ADJ
iajs-1808	61	37	in	in	ADP
iajs-1808	61	38	𝑴[10	𝑴[10	PROPN
iajs-1808	61	39	]	]	PUNCT
iajs-1808	61	40	.	.	PUNCT
iajs-1808	62	1	proposition	proposition	NOUN
iajs-1808	62	2	(	(	PUNCT
iajs-1808	62	3	1.5)[10]:	1.5)[10]:	PRON
iajs-1808	62	4	let	let	VERB
iajs-1808	62	5	𝑀	𝑀	PRON
iajs-1808	62	6	be	be	AUX
iajs-1808	62	7	an	an	DET
iajs-1808	62	8	𝑅-module	𝑅-module	PROPN
iajs-1808	62	9	which	which	DET
iajs-1808	62	10	satisfies	satisfie	NOUN
iajs-1808	62	11	(	(	PUNCT
iajs-1808	62	12	⋇	⋇	NOUN
iajs-1808	62	13	.	.	PUNCT
iajs-1808	63	1	if	if	SCONJ
iajs-1808	63	2	𝑀	𝑀	PROPN
iajs-1808	63	3	is	be	AUX
iajs-1808	63	4	strongly	strongly	ADV
iajs-1808	63	5	fi	fi	NOUN
iajs-1808	63	6	-	-	NOUN
iajs-1808	63	7	tsemisimple	tsemisimple	ADJ
iajs-1808	63	8	,	,	PUNCT
iajs-1808	63	9	then	then	ADV
iajs-1808	63	10	is	be	AUX
iajs-1808	63	11	fi	fi	NOUN
iajs-1808	63	12	-	-	NOUN
iajs-1808	63	13	semisimple	semisimple	NOUN
iajs-1808	63	14	,	,	PUNCT
iajs-1808	63	15	and	and	CCONJ
iajs-1808	63	16	hence	hence	ADV
iajs-1808	63	17	it	it	PRON
iajs-1808	63	18	is	be	AUX
iajs-1808	63	19	strongly	strongly	ADV
iajs-1808	63	20	fi	fi	NOUN
iajs-1808	63	21	-	-	ADJ
iajs-1808	63	22	t	t	NOUN
iajs-1808	63	23	-	-	PUNCT
iajs-1808	63	24	semisimple.	semisimple.	NOUN
iajs-1808	63	25	corollary	corollary	ADJ
iajs-1808	63	26	(	(	PUNCT
iajs-1808	63	27	1.6)[10]:	1.6)[10]:	NUM
iajs-1808	63	28	for	for	ADP
iajs-1808	63	29	an	an	DET
iajs-1808	63	30	𝑅-module	𝑅-module	PROPN
iajs-1808	63	31	𝑀	𝑀	PROPN
iajs-1808	63	32	which	which	PRON
iajs-1808	63	33	satisfies	satisfy	VERB
iajs-1808	63	34	⋇	⋇	NOUN
iajs-1808	63	35	𝑀	𝑀	PROPN
iajs-1808	63	36	is	be	AUX
iajs-1808	63	37	strongly	strongly	ADV
iajs-1808	63	38	fi	fi	ADJ
iajs-1808	63	39	-	-	ADJ
iajs-1808	63	40	t	t	NOUN
iajs-1808	63	41	-	-	PUNCT
iajs-1808	63	42	semisimple	semisimple	NOUN
iajs-1808	63	43	if	if	SCONJ
iajs-1808	63	44	and	and	CCONJ
iajs-1808	63	45	only	only	ADV
iajs-1808	63	46	if	if	SCONJ
iajs-1808	63	47	for	for	ADP
iajs-1808	63	48	every	every	DET
iajs-1808	63	49	fully	fully	ADV
iajs-1808	63	50	invariant	invariant	ADJ
iajs-1808	63	51	submodule	submodule	NOUN
iajs-1808	63	52	𝑁	𝑁	PROPN
iajs-1808	63	53	of	of	ADP
iajs-1808	63	54	𝑀	𝑀	PROPN
iajs-1808	63	55	such	such	ADJ
iajs-1808	63	56	that	that	SCONJ
iajs-1808	63	57	𝑁	𝑁	PROPN
iajs-1808	63	58	⊇	⊇	NOUN
iajs-1808	63	59	𝑍	𝑍	PROPN
iajs-1808	63	60	𝑀	𝑀	PROPN
iajs-1808	63	61	,	,	PUNCT
iajs-1808	63	62	is	be	AUX
iajs-1808	63	63	strongly	strongly	ADV
iajs-1808	63	64	fi	fi	ADJ
iajs-1808	63	65	-	-	ADJ
iajs-1808	63	66	tsemisimple,.	tsemisimple,.	VERB
iajs-1808	63	67	ihsciconf	ihsciconf	NOUN
iajs-1808	63	68	2017	2017	NUM
iajs-1808	63	69	special	special	ADJ
iajs-1808	63	70	issue	issue	NOUN
iajs-1808	63	71	ibn	ibn	PROPN
iajs-1808	63	72	al	al	PROPN
iajs-1808	63	73	-	-	PUNCT
iajs-1808	63	74	haitham	haitham	PROPN
iajs-1808	63	75	journal	journal	PROPN
iajs-1808	63	76	for	for	ADP
iajs-1808	63	77	pure	pure	ADJ
iajs-1808	63	78	and	and	CCONJ
iajs-1808	63	79	applied	apply	VERB
iajs-1808	63	80	science	science	NOUN
iajs-1808	63	81	https://doi.org/	https://doi.org/	NOUN
iajs-1808	63	82	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	63	83	for	for	ADP
iajs-1808	63	84	more	more	ADJ
iajs-1808	63	85	information	information	NOUN
iajs-1808	63	86	about	about	ADP
iajs-1808	63	87	the	the	DET
iajs-1808	63	88	conference	conference	NOUN
iajs-1808	63	89	please	please	INTJ
iajs-1808	63	90	visit	visit	VERB
iajs-1808	63	91	the	the	DET
iajs-1808	63	92	websites	website	NOUN
iajs-1808	63	93	:	:	PUNCT
iajs-1808	63	94	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	63	95	  	  	SPACE
iajs-1808	63	96	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1808	63	97	   	   	SPACE
iajs-1808	63	98	mathematics|356	mathematics|356	NOUN
iajs-1808	63	99	    	    	SPACE
iajs-1808	63	100	proposition	proposition	NOUN
iajs-1808	63	101	(	(	PUNCT
iajs-1808	63	102	1.7)[2]:	1.7)[2]:	NUM
iajs-1808	63	103	let	let	VERB
iajs-1808	63	104	𝑀	𝑀	PROPN
iajs-1808	63	105	be	be	AUX
iajs-1808	63	106	a	a	DET
iajs-1808	63	107	nonsingular	nonsingular	ADJ
iajs-1808	63	108	module	module	NOUN
iajs-1808	63	109	.	.	PUNCT
iajs-1808	64	1	then	then	ADV
iajs-1808	64	2	𝑀	𝑀	PROPN
iajs-1808	64	3	is	be	AUX
iajs-1808	64	4	strongly	strongly	ADV
iajs-1808	64	5	extending	extend	VERB
iajs-1808	64	6	if	if	SCONJ
iajs-1808	64	7	and	and	CCONJ
iajs-1808	64	8	only	only	ADV
iajs-1808	64	9	if	if	SCONJ
iajs-1808	64	10	𝑀	𝑀	PROPN
iajs-1808	64	11	is	be	AUX
iajs-1808	64	12	strongly	strongly	ADV
iajs-1808	64	13	t	t	NOUN
iajs-1808	64	14	-	-	PUNCT
iajs-1808	64	15	extending.	extending.	NOUN
iajs-1808	64	16	2	2	NUM
iajs-1808	64	17	.	.	X
iajs-1808	64	18	t	t	NOUN
iajs-1808	64	19	-	-	PUNCT
iajs-1808	64	20	semisimple	semisimple	NOUN
iajs-1808	64	21	modules	module	NOUN
iajs-1808	64	22	and	and	CCONJ
iajs-1808	64	23	𝑻𝟏𝟏-type	𝑻𝟏𝟏-type	NOUN
iajs-1808	64	24	modules	module	NOUN
iajs-1808	64	25	remarks	remark	NOUN
iajs-1808	64	26	and	and	CCONJ
iajs-1808	64	27	examples	example	NOUN
iajs-1808	64	28	(	(	PUNCT
iajs-1808	64	29	2.1	2.1	NUM
iajs-1808	64	30	):	):	PUNCT
iajs-1808	64	31	it	it	PRON
iajs-1808	64	32	is	be	AUX
iajs-1808	64	33	clear	clear	ADJ
iajs-1808	64	34	that	that	SCONJ
iajs-1808	64	35	module	module	PROPN
iajs-1808	64	36	satisfying	satisfy	VERB
iajs-1808	64	37	𝐶	𝐶	PROPN
iajs-1808	64	38	implies	imply	VERB
iajs-1808	64	39	𝑇	𝑇	PROPN
iajs-1808	64	40	-type	-type	NOUN
iajs-1808	64	41	-	-	PUNCT
iajs-1808	64	42	module	module	NOUN
iajs-1808	65	1	[	[	X
iajs-1808	65	2	4	4	NUM
iajs-1808	65	3	]	]	PUNCT
iajs-1808	65	4	.	.	PUNCT
iajs-1808	66	1	every	every	NOUN
iajs-1808	66	2	t	t	NOUN
iajs-1808	66	3	-	-	PUNCT
iajs-1808	66	4	extending	extend	VERB
iajs-1808	66	5	module	module	NOUN
iajs-1808	66	6	(	(	PUNCT
iajs-1808	66	7	hence	hence	ADV
iajs-1808	66	8	every	every	DET
iajs-1808	66	9	extending	extend	VERB
iajs-1808	66	10	module	module	NOUN
iajs-1808	66	11	)	)	PUNCT
iajs-1808	66	12	is	be	AUX
iajs-1808	66	13	a	a	DET
iajs-1808	66	14	𝑇	𝑇	PROPN
iajs-1808	66	15	-type	-type	NOUN
iajs-1808	66	16	module	module	NOUN
iajs-1808	66	17	[	[	X
iajs-1808	66	18	4	4	NUM
iajs-1808	66	19	]	]	PUNCT
iajs-1808	66	20	.	.	PUNCT
iajs-1808	67	1	every	every	NOUN
iajs-1808	67	2	𝑍	𝑍	NOUN
iajs-1808	67	3	-torsion	-torsion	NOUN
iajs-1808	67	4	is	be	AUX
iajs-1808	67	5	𝑇	𝑇	PROPN
iajs-1808	67	6	-type	-type	NOUN
iajs-1808	67	7	module	module	NOUN
iajs-1808	67	8	and	and	CCONJ
iajs-1808	67	9	every	every	DET
iajs-1808	67	10	finite	finite	NOUN
iajs-1808	67	11	generated	generate	VERB
iajs-1808	67	12	abelian	abelian	PROPN
iajs-1808	67	13	group	group	NOUN
iajs-1808	67	14	is	be	AUX
iajs-1808	67	15	a	a	DET
iajs-1808	67	16	𝑇	𝑇	PROPN
iajs-1808	67	17	-type	-type	NOUN
iajs-1808	67	18	module	module	NOUN
iajs-1808	67	19	[	[	X
iajs-1808	67	20	4	4	NUM
iajs-1808	67	21	]	]	PUNCT
iajs-1808	67	22	.	.	PUNCT
iajs-1808	68	1	proposition	proposition	NOUN
iajs-1808	68	2	(	(	PUNCT
iajs-1808	68	3	2.2	2.2	NUM
iajs-1808	68	4	):	):	PUNCT
iajs-1808	68	5	every	every	DET
iajs-1808	68	6	t	t	PROPN
iajs-1808	68	7	-	-	PUNCT
iajs-1808	68	8	semisimple	semisimple	NOUN
iajs-1808	68	9	module	module	NOUN
iajs-1808	68	10	is	be	AUX
iajs-1808	68	11	𝑇	𝑇	PROPN
iajs-1808	68	12	-type	-type	NOUN
iajs-1808	68	13	module	module	NOUN
iajs-1808	68	14	.	.	PUNCT
iajs-1808	69	1	proof	proof	NOUN
iajs-1808	69	2	:	:	PUNCT
iajs-1808	69	3	by	by	ADP
iajs-1808	69	4	[	[	X
iajs-1808	69	5	3	3	NUM
iajs-1808	69	6	,	,	PUNCT
iajs-1808	69	7	proposition	proposition	NOUN
iajs-1808	69	8	2.16	2.16	NUM
iajs-1808	69	9	]	]	PUNCT
iajs-1808	69	10	,	,	PUNCT
iajs-1808	69	11	every	every	DET
iajs-1808	69	12	t	t	NOUN
iajs-1808	69	13	-	-	PUNCT
iajs-1808	69	14	semisimple	semisimple	NOUN
iajs-1808	69	15	is	be	AUX
iajs-1808	69	16	t	t	NOUN
iajs-1808	69	17	-	-	PUNCT
iajs-1808	69	18	extending	extending	NOUN
iajs-1808	69	19	,	,	PUNCT
iajs-1808	69	20	hence	hence	ADV
iajs-1808	69	21	by	by	ADP
iajs-1808	69	22	remarks	remark	NOUN
iajs-1808	69	23	and	and	CCONJ
iajs-1808	69	24	examples	example	NOUN
iajs-1808	69	25	2.1(2	2.1(2	NUM
iajs-1808	69	26	)	)	PUNCT
iajs-1808	69	27	it	it	PRON
iajs-1808	69	28	is	be	AUX
iajs-1808	69	29	𝑇	𝑇	PROPN
iajs-1808	69	30	-type	-type	NOUN
iajs-1808	69	31	module	module	NOUN
iajs-1808	69	32	.	.	PUNCT
iajs-1808	70	1	□	□	PUNCT
iajs-1808	70	2	remark	remark	NOUN
iajs-1808	70	3	(	(	PUNCT
iajs-1808	70	4	2.3)[4]:	2.3)[4]:	NUM
iajs-1808	70	5	the	the	DET
iajs-1808	70	6	class	class	NOUN
iajs-1808	70	7	of	of	ADP
iajs-1808	70	8	𝑇	𝑇	NOUN
iajs-1808	70	9	-type	-type	NOUN
iajs-1808	70	10	modules	module	NOUN
iajs-1808	70	11	properly	properly	ADV
iajs-1808	70	12	contains	contain	VERB
iajs-1808	70	13	the	the	DET
iajs-1808	70	14	modules	module	NOUN
iajs-1808	70	15	satisfying	satisfy	VERB
iajs-1808	70	16	𝐶	𝐶	PROPN
iajs-1808	70	17	condition	condition	NOUN
iajs-1808	70	18	and	and	CCONJ
iajs-1808	70	19	the	the	DET
iajs-1808	70	20	class	class	NOUN
iajs-1808	70	21	of	of	ADP
iajs-1808	70	22	t	t	PROPN
iajs-1808	70	23	-	-	PUNCT
iajs-1808	70	24	extending	extend	VERB
iajs-1808	70	25	modules	modules	NOUN
iajs-1808	70	26	.	.	PUNCT
iajs-1808	71	1	examples	example	NOUN
iajs-1808	71	2	(	(	PUNCT
iajs-1808	71	3	2.4	2.4	NUM
iajs-1808	71	4	):	):	PUNCT
iajs-1808	71	5	let	let	VERB
iajs-1808	71	6	𝑅	𝑅	PROPN
iajs-1808	71	7	𝑍	𝑍	PROPN
iajs-1808	71	8	𝑋	𝑋	NOUN
iajs-1808	71	9	,	,	PUNCT
iajs-1808	71	10	𝑅	𝑅	PROPN
iajs-1808	71	11	is	be	AUX
iajs-1808	71	12	uniform	uniform	ADJ
iajs-1808	71	13	,	,	PUNCT
iajs-1808	71	14	nonsingular	nonsingular	ADJ
iajs-1808	71	15	𝑅-module	𝑅-module	PROPN
iajs-1808	71	16	.	.	PUNCT
iajs-1808	72	1	by	by	ADP
iajs-1808	72	2	[	[	X
iajs-1808	72	3	13	13	NUM
iajs-1808	72	4	,	,	PUNCT
iajs-1808	72	5	theorem	theorem	VERB
iajs-1808	72	6	2.4	2.4	NUM
iajs-1808	72	7	]	]	PUNCT
iajs-1808	72	8	𝑅⨁𝑅	𝑅⨁𝑅	X
iajs-1808	72	9	satisfies	satisfy	VERB
iajs-1808	72	10	𝐶	𝐶	PROPN
iajs-1808	72	11	-condition	-condition	PROPN
iajs-1808	72	12	.	.	PUNCT
iajs-1808	73	1	hence	hence	ADV
iajs-1808	73	2	𝑅⨁𝑅	𝑅⨁𝑅	NOUN
iajs-1808	73	3	is	be	AUX
iajs-1808	73	4	𝑇	𝑇	PROPN
iajs-1808	73	5	-type	-type	NOUN
iajs-1808	73	6	module	module	NOUN
iajs-1808	73	7	.	.	PUNCT
iajs-1808	74	1	but	but	CCONJ
iajs-1808	74	2	𝑅⨁𝑅	𝑅⨁𝑅	NOUN
iajs-1808	74	3	is	be	AUX
iajs-1808	74	4	not	not	PART
iajs-1808	74	5	t	t	NOUN
iajs-1808	74	6	-	-	PUNCT
iajs-1808	74	7	semisimple	semisimple	NOUN
iajs-1808	74	8	,	,	PUNCT
iajs-1808	74	9	because	because	SCONJ
iajs-1808	74	10	it	it	PRON
iajs-1808	74	11	is	be	AUX
iajs-1808	74	12	so	so	ADV
iajs-1808	74	13	,	,	PUNCT
iajs-1808	74	14	then	then	ADV
iajs-1808	74	15	𝑅⨁𝑅	𝑅⨁𝑅	NOUN
iajs-1808	74	16	is	be	AUX
iajs-1808	74	17	t	t	NOUN
iajs-1808	74	18	-	-	PUNCT
iajs-1808	74	19	extending	extending	NOUN
iajs-1808	74	20	,	,	PUNCT
iajs-1808	74	21	which	which	PRON
iajs-1808	74	22	is	be	AUX
iajs-1808	74	23	a	a	DET
iajs-1808	74	24	contradiction	contradiction	NOUN
iajs-1808	74	25	since	since	SCONJ
iajs-1808	74	26	by	by	ADP
iajs-1808	74	27	[	[	X
iajs-1808	74	28	5,example	5,example	NUM
iajs-1808	74	29	2.4	2.4	NUM
iajs-1808	74	30	]	]	PUNCT
iajs-1808	74	31	𝑅⨁𝑅	𝑅⨁𝑅	NOUN
iajs-1808	74	32	is	be	AUX
iajs-1808	74	33	not	not	PART
iajs-1808	74	34	extending	extend	VERB
iajs-1808	74	35	,	,	PUNCT
iajs-1808	74	36	hence	hence	ADV
iajs-1808	74	37	not	not	PART
iajs-1808	74	38	t	t	NOUN
iajs-1808	74	39	-	-	PUNCT
iajs-1808	74	40	extending	extending	NOUN
iajs-1808	74	41	,	,	PUNCT
iajs-1808	74	42	since	since	SCONJ
iajs-1808	74	43	𝑅⨁𝑅	𝑅⨁𝑅	NOUN
iajs-1808	74	44	is	be	AUX
iajs-1808	74	45	nonsingular	nonsingular	ADJ
iajs-1808	74	46	see	see	VERB
iajs-1808	74	47	remarks	remark	NOUN
iajs-1808	74	48	and	and	CCONJ
iajs-1808	74	49	examples	example	NOUN
iajs-1808	74	50	2.1(2	2.1(2	NUM
iajs-1808	74	51	)	)	PUNCT
iajs-1808	74	52	.	.	PUNCT
iajs-1808	75	1	the	the	DET
iajs-1808	75	2	𝑍-module	𝑍-module	PROPN
iajs-1808	75	3	𝑍	𝑍	PROPN
iajs-1808	75	4	is	be	AUX
iajs-1808	75	5	not	not	PART
iajs-1808	75	6	t	t	NOUN
iajs-1808	75	7	-	-	PUNCT
iajs-1808	75	8	semisimple	semisimple	NOUN
iajs-1808	75	9	.	.	PUNCT
iajs-1808	76	1	but	but	CCONJ
iajs-1808	76	2	𝑍	𝑍	PROPN
iajs-1808	76	3	is	be	AUX
iajs-1808	76	4	indecomposable	indecomposable	ADJ
iajs-1808	76	5	and	and	CCONJ
iajs-1808	76	6	nonsingular	nonsingular	ADJ
iajs-1808	76	7	uniform	uniform	NOUN
iajs-1808	76	8	,	,	PUNCT
iajs-1808	76	9	so	so	SCONJ
iajs-1808	76	10	𝑍	𝑍	PROPN
iajs-1808	76	11	is	be	AUX
iajs-1808	76	12	𝑇	𝑇	PROPN
iajs-1808	76	13	-type	-type	NOUN
iajs-1808	76	14	module	module	NOUN
iajs-1808	76	15	see[4	see[4	NOUN
iajs-1808	76	16	,	,	PUNCT
iajs-1808	76	17	corollary	corollary	ADJ
iajs-1808	76	18	2.8	2.8	NUM
iajs-1808	76	19	]	]	PUNCT
iajs-1808	76	20	.	.	PUNCT
iajs-1808	77	1	an	an	DET
iajs-1808	77	2	𝑍-module	𝑍-module	PROPN
iajs-1808	77	3	𝑄	𝑄	PROPN
iajs-1808	77	4	is	be	AUX
iajs-1808	77	5	indecomposable	indecomposable	ADJ
iajs-1808	77	6	,	,	PUNCT
iajs-1808	77	7	nonsingular	nonsingular	ADJ
iajs-1808	77	8	,	,	PUNCT
iajs-1808	77	9	uniform	uniform	NOUN
iajs-1808	77	10	so	so	ADV
iajs-1808	77	11	𝑄	𝑄	PROPN
iajs-1808	77	12	is	be	AUX
iajs-1808	77	13	𝑇	𝑇	PROPN
iajs-1808	77	14	-type	-type	NOUN
iajs-1808	77	15	module[4	module[4	NOUN
iajs-1808	77	16	,	,	PUNCT
iajs-1808	77	17	corollary	corollary	ADJ
iajs-1808	77	18	2.8	2.8	NUM
iajs-1808	77	19	]	]	PUNCT
iajs-1808	77	20	any	any	VERB
iajs-1808	77	21	direct	direct	ADJ
iajs-1808	77	22	summand	summand	NOUN
iajs-1808	77	23	of	of	ADP
iajs-1808	77	24	uniform	uniform	NOUN
iajs-1808	77	25	is	be	AUX
iajs-1808	77	26	𝐶	𝐶	PROPN
iajs-1808	77	27	-type	-type	NOUN
iajs-1808	77	28	module	module	NOUN
iajs-1808	77	29	,	,	PUNCT
iajs-1808	77	30	so	so	ADV
iajs-1808	77	31	is	be	AUX
iajs-1808	77	32	𝑇	𝑇	PROPN
iajs-1808	77	33	-type	-type	NOUN
iajs-1808	77	34	module	module	NOUN
iajs-1808	77	35	[15	[15	NOUN
iajs-1808	77	36	]	]	PUNCT
iajs-1808	77	37	.	.	PUNCT
iajs-1808	78	1	(	(	PUNCT
iajs-1808	78	2	5	5	X
iajs-1808	78	3	)	)	PUNCT
iajs-1808	78	4	it	it	PRON
iajs-1808	78	5	is	be	AUX
iajs-1808	78	6	clear	clear	ADJ
iajs-1808	78	7	the	the	DET
iajs-1808	78	8	𝑍	𝑍	PROPN
iajs-1808	78	9	module	module	NOUN
iajs-1808	78	10	𝑄⨁𝑍	𝑄⨁𝑍	NOUN
iajs-1808	78	11	,	,	PUNCT
iajs-1808	78	12	𝑍	𝑍	ADJ
iajs-1808	78	13	⨁𝑍	⨁𝑍	NOUN
iajs-1808	78	14	,	,	PUNCT
iajs-1808	78	15	𝑍	𝑍	PROPN
iajs-1808	78	16	⨁𝑍	⨁𝑍	NOUN
iajs-1808	78	17	are	be	AUX
iajs-1808	78	18	𝑇	𝑇	PROPN
iajs-1808	78	19	-type	-type	NOUN
iajs-1808	78	20	module	module	NOUN
iajs-1808	78	21	.	.	PUNCT
iajs-1808	79	1	also	also	ADV
iajs-1808	79	2	notice	notice	VERB
iajs-1808	79	3	that	that	SCONJ
iajs-1808	79	4	𝑄⨁𝑍	𝑄⨁𝑍	NOUN
iajs-1808	79	5	is	be	AUX
iajs-1808	79	6	not	not	PART
iajs-1808	79	7	t	t	NOUN
iajs-1808	79	8	-	-	PUNCT
iajs-1808	79	9	semisimple	semisimple	NOUN
iajs-1808	79	10	.	.	PUNCT
iajs-1808	80	1	3	3	X
iajs-1808	80	2	.	.	X
iajs-1808	80	3	strongly	strongly	ADV
iajs-1808	80	4	𝑪𝟏𝟏-modules	𝑪𝟏𝟏-module	NOUN
iajs-1808	80	5	and	and	CCONJ
iajs-1808	80	6	strongly	strongly	ADV
iajs-1808	80	7	𝑻𝟏𝟏-type	𝑻𝟏𝟏-type	VERB
iajs-1808	80	8	modules	module	NOUN
iajs-1808	80	9	.	.	PUNCT
iajs-1808	81	1	in	in	ADP
iajs-1808	81	2	this	this	DET
iajs-1808	81	3	section	section	NOUN
iajs-1808	81	4	,	,	PUNCT
iajs-1808	81	5	we	we	PRON
iajs-1808	81	6	generalize	generalize	VERB
iajs-1808	81	7	modules	module	NOUN
iajs-1808	81	8	with	with	ADP
iajs-1808	81	9	𝐶	𝐶	PROPN
iajs-1808	81	10	-condition	-condition	PROPN
iajs-1808	81	11	and	and	CCONJ
iajs-1808	81	12	𝑇	𝑇	NOUN
iajs-1808	81	13	-type	-type	NOUN
iajs-1808	81	14	modules	module	NOUN
iajs-1808	81	15	into	into	ADP
iajs-1808	81	16	modules	module	NOUN
iajs-1808	81	17	with	with	ADP
iajs-1808	81	18	strongly	strongly	ADV
iajs-1808	81	19	𝐶	𝐶	PROPN
iajs-1808	81	20	conditions	condition	NOUN
iajs-1808	81	21	and	and	CCONJ
iajs-1808	81	22	strongly	strongly	ADV
iajs-1808	81	23	𝑇	𝑇	PROPN
iajs-1808	81	24	-type	-type	NOUN
iajs-1808	81	25	modules	module	NOUN
iajs-1808	81	26	.	.	PUNCT
iajs-1808	82	1	we	we	PRON
iajs-1808	82	2	study	study	VERB
iajs-1808	82	3	these	these	DET
iajs-1808	82	4	concepts	concept	NOUN
iajs-1808	82	5	and	and	CCONJ
iajs-1808	82	6	their	their	PRON
iajs-1808	82	7	connection	connection	NOUN
iajs-1808	82	8	with	with	ADP
iajs-1808	82	9	strongly	strongly	ADV
iajs-1808	82	10	t	t	NOUN
iajs-1808	82	11	-	-	PUNCT
iajs-1808	82	12	semisimple	semisimple	NOUN
iajs-1808	82	13	modules	module	NOUN
iajs-1808	82	14	and	and	CCONJ
iajs-1808	82	15	other	other	ADJ
iajs-1808	82	16	related	related	ADJ
iajs-1808	82	17	classes	class	NOUN
iajs-1808	82	18	of	of	ADP
iajs-1808	82	19	modules	module	NOUN
iajs-1808	82	20	.	.	PUNCT
iajs-1808	83	1	definition	definition	NOUN
iajs-1808	83	2	(	(	PUNCT
iajs-1808	83	3	3.1	3.1	NUM
iajs-1808	83	4	):	):	PUNCT
iajs-1808	83	5	an	an	DET
iajs-1808	83	6	𝑅-module	𝑅-module	PROPN
iajs-1808	83	7	𝑀	𝑀	PROPN
iajs-1808	83	8	said	say	VERB
iajs-1808	83	9	to	to	PART
iajs-1808	83	10	be	be	AUX
iajs-1808	83	11	satisfy	satisfy	VERB
iajs-1808	83	12	strongly	strongly	ADV
iajs-1808	83	13	𝐶	𝐶	PROPN
iajs-1808	83	14	condition	condition	NOUN
iajs-1808	83	15	(	(	PUNCT
iajs-1808	83	16	𝑀	𝑀	PROPN
iajs-1808	83	17	is	be	AUX
iajs-1808	83	18	strongly	strongly	ADV
iajs-1808	83	19	𝐶	𝐶	PROPN
iajs-1808	83	20	)	)	PUNCT
iajs-1808	83	21	if	if	SCONJ
iajs-1808	83	22	every	every	DET
iajs-1808	83	23	submodule	submodule	NOUN
iajs-1808	83	24	has	have	VERB
iajs-1808	83	25	a	a	DET
iajs-1808	83	26	complement	complement	NOUN
iajs-1808	83	27	which	which	PRON
iajs-1808	83	28	is	be	AUX
iajs-1808	83	29	fully	fully	ADV
iajs-1808	83	30	invariant	invariant	ADJ
iajs-1808	83	31	direct	direct	ADJ
iajs-1808	83	32	summand	summand	NOUN
iajs-1808	83	33	.	.	PUNCT
iajs-1808	84	1	lemma	lemma	PROPN
iajs-1808	84	2	(	(	PUNCT
iajs-1808	84	3	3.2)[15]:	3.2)[15]:	NOUN
iajs-1808	84	4	let	let	VERB
iajs-1808	84	5	𝑁	𝑁	PROPN
iajs-1808	84	6	𝑀	𝑀	PROPN
iajs-1808	84	7	,	,	PUNCT
iajs-1808	84	8	let	let	VERB
iajs-1808	84	9	𝐾	𝐾	PRON
iajs-1808	84	10	be	be	AUX
iajs-1808	84	11	a	a	DET
iajs-1808	84	12	direct	direct	ADJ
iajs-1808	84	13	summand	summand	NOUN
iajs-1808	84	14	of	of	ADP
iajs-1808	84	15	𝑀	𝑀	PROPN
iajs-1808	84	16	.	.	PUNCT
iajs-1808	85	1	𝐾	𝐾	PROPN
iajs-1808	85	2	is	be	AUX
iajs-1808	85	3	a	a	DET
iajs-1808	85	4	complement	complement	NOUN
iajs-1808	85	5	of	of	ADP
iajs-1808	85	6	𝑁	𝑁	PROPN
iajs-1808	85	7	if	if	SCONJ
iajs-1808	85	8	and	and	CCONJ
iajs-1808	85	9	only	only	ADV
iajs-1808	85	10	if	if	SCONJ
iajs-1808	85	11	𝐾⋂𝑁	𝐾⋂𝑁	NOUN
iajs-1808	85	12	0	0	PUNCT
iajs-1808	85	13	and	and	CCONJ
iajs-1808	85	14	𝐾⨁𝑁	𝐾⨁𝑁	PROPN
iajs-1808	85	15	𝑀	𝑀	PROPN
iajs-1808	85	16	.	.	VERB
iajs-1808	85	17	the	the	DET
iajs-1808	85	18	following	follow	VERB
iajs-1808	85	19	lemma	lemma	PROPN
iajs-1808	85	20	is	be	AUX
iajs-1808	85	21	clear	clear	ADJ
iajs-1808	85	22	.	.	PUNCT
iajs-1808	86	1	lemma	lemma	PROPN
iajs-1808	86	2	(	(	PUNCT
iajs-1808	86	3	3.3	3.3	NUM
iajs-1808	86	4	):	):	PUNCT
iajs-1808	86	5	if	if	SCONJ
iajs-1808	86	6	𝑁	𝑁	PROPN
iajs-1808	86	7	𝑀	𝑀	PROPN
iajs-1808	86	8	and	and	CCONJ
iajs-1808	86	9	𝐾	𝐾	PROPN
iajs-1808	86	10	is	be	AUX
iajs-1808	86	11	a	a	DET
iajs-1808	86	12	fully	fully	ADV
iajs-1808	86	13	invariant	invariant	ADJ
iajs-1808	86	14	direct	direct	ADJ
iajs-1808	86	15	summand	summand	NOUN
iajs-1808	86	16	then	then	ADV
iajs-1808	86	17	𝐾	𝐾	PROPN
iajs-1808	86	18	is	be	AUX
iajs-1808	86	19	a	a	DET
iajs-1808	86	20	fully	fully	ADV
iajs-1808	86	21	invariant	invariant	ADJ
iajs-1808	86	22	complement	complement	NOUN
iajs-1808	86	23	of	of	ADP
iajs-1808	86	24	𝑁	𝑁	PROPN
iajs-1808	86	25	if	if	SCONJ
iajs-1808	86	26	and	and	CCONJ
iajs-1808	86	27	only	only	ADV
iajs-1808	86	28	if	if	SCONJ
iajs-1808	86	29	𝐾⋂𝑁	𝐾⋂𝑁	NOUN
iajs-1808	86	30	0	0	PUNCT
iajs-1808	86	31	and	and	CCONJ
iajs-1808	86	32	𝐾⨁𝑁	𝐾⨁𝑁	PROPN
iajs-1808	86	33	𝑀.	𝑀.	PROPN
iajs-1808	86	34	the	the	DET
iajs-1808	86	35	following	follow	VERB
iajs-1808	86	36	proposition	proposition	NOUN
iajs-1808	86	37	gives	give	VERB
iajs-1808	86	38	characterizations	characterization	NOUN
iajs-1808	86	39	for	for	ADP
iajs-1808	86	40	module	module	NOUN
iajs-1808	86	41	with	with	ADP
iajs-1808	86	42	strongly	strongly	ADV
iajs-1808	86	43	𝐶	𝐶	PROPN
iajs-1808	86	44	-condition	-condition	PROPN
iajs-1808	86	45	.	.	PUNCT
iajs-1808	87	1	proposition	proposition	NOUN
iajs-1808	87	2	(	(	PUNCT
iajs-1808	87	3	3.4	3.4	NUM
iajs-1808	87	4	):	):	PUNCT
iajs-1808	87	5	the	the	DET
iajs-1808	87	6	following	following	ADJ
iajs-1808	87	7	statements	statement	NOUN
iajs-1808	87	8	are	be	AUX
iajs-1808	87	9	equivalent	equivalent	ADJ
iajs-1808	87	10	for	for	ADP
iajs-1808	87	11	a	a	DET
iajs-1808	87	12	module	module	NOUN
iajs-1808	87	13	𝑀	𝑀	PROPN
iajs-1808	87	14	𝑀	𝑀	PROPN
iajs-1808	87	15	satisfies	satisfy	VERB
iajs-1808	87	16	strongly	strongly	ADV
iajs-1808	87	17	𝐶	𝐶	PROPN
iajs-1808	87	18	-condition	-condition	PROPN
iajs-1808	87	19	;	;	PUNCT
iajs-1808	87	20	for	for	ADP
iajs-1808	87	21	any	any	DET
iajs-1808	87	22	complement	complement	NOUN
iajs-1808	87	23	submodule	submodule	NOUN
iajs-1808	87	24	𝐿	𝐿	PROPN
iajs-1808	87	25	in	in	ADP
iajs-1808	87	26	𝑀	𝑀	PROPN
iajs-1808	87	27	,	,	PUNCT
iajs-1808	87	28	there	there	PRON
iajs-1808	87	29	exists	exist	VERB
iajs-1808	87	30	a	a	DET
iajs-1808	87	31	fully	fully	ADV
iajs-1808	87	32	invariant	invariant	ADJ
iajs-1808	87	33	direct	direct	ADJ
iajs-1808	87	34	summand	summand	NOUN
iajs-1808	87	35	𝐾	𝐾	PROPN
iajs-1808	87	36	of	of	ADP
iajs-1808	87	37	𝑀	𝑀	PROPN
iajs-1808	87	38	such	such	ADJ
iajs-1808	87	39	that	that	SCONJ
iajs-1808	87	40	𝐾	𝐾	PROPN
iajs-1808	87	41	is	be	AUX
iajs-1808	87	42	a	a	DET
iajs-1808	87	43	complement	complement	NOUN
iajs-1808	87	44	of	of	ADP
iajs-1808	87	45	𝐿	𝐿	PROPN
iajs-1808	87	46	in	in	ADP
iajs-1808	87	47	𝑀	𝑀	PROPN
iajs-1808	87	48	;	;	PUNCT
iajs-1808	87	49	for	for	ADP
iajs-1808	87	50	any	any	DET
iajs-1808	87	51	submodule	submodule	NOUN
iajs-1808	87	52	𝑁	𝑁	PROPN
iajs-1808	87	53	of	of	ADP
iajs-1808	87	54	𝑀	𝑀	PROPN
iajs-1808	87	55	,	,	PUNCT
iajs-1808	87	56	there	there	PRON
iajs-1808	87	57	exists	exist	VERB
iajs-1808	87	58	a	a	DET
iajs-1808	87	59	fully	fully	ADV
iajs-1808	87	60	invariant	invariant	ADJ
iajs-1808	87	61	direct	direct	ADJ
iajs-1808	87	62	summand	summand	NOUN
iajs-1808	87	63	𝐾	𝐾	PROPN
iajs-1808	87	64	of	of	ADP
iajs-1808	87	65	𝑀	𝑀	PROPN
iajs-1808	87	66	such	such	ADJ
iajs-1808	87	67	that	that	DET
iajs-1808	87	68	𝑁⋂𝐾	𝑁⋂𝐾	ADP
iajs-1808	87	69	0	0	NUM
iajs-1808	87	70	and	and	CCONJ
iajs-1808	87	71	𝑁⨁𝐾	𝑁⨁𝐾	NOUN
iajs-1808	87	72	is	be	AUX
iajs-1808	87	73	an	an	DET
iajs-1808	87	74	essential	essential	ADJ
iajs-1808	87	75	submodule	submodule	NOUN
iajs-1808	87	76	of	of	ADP
iajs-1808	87	77	𝑀	𝑀	PROPN
iajs-1808	87	78	;	;	PUNCT
iajs-1808	87	79	for	for	ADP
iajs-1808	87	80	any	any	DET
iajs-1808	87	81	complement	complement	NOUN
iajs-1808	87	82	submodule	submodule	NOUN
iajs-1808	87	83	𝐿	𝐿	PROPN
iajs-1808	87	84	in	in	ADP
iajs-1808	87	85	𝑀	𝑀	PROPN
iajs-1808	87	86	,	,	PUNCT
iajs-1808	87	87	there	there	PRON
iajs-1808	87	88	exists	exist	VERB
iajs-1808	87	89	a	a	DET
iajs-1808	87	90	fully	fully	ADV
iajs-1808	87	91	invariant	invariant	ADJ
iajs-1808	87	92	direct	direct	ADJ
iajs-1808	87	93	summand	summand	NOUN
iajs-1808	87	94	𝐾	𝐾	PROPN
iajs-1808	87	95	of	of	ADP
iajs-1808	87	96	𝑀	𝑀	PROPN
iajs-1808	87	97	such	such	ADJ
iajs-1808	87	98	that	that	PRON
iajs-1808	87	99	𝐿⋂𝐾	𝐿⋂𝐾	NOUN
iajs-1808	87	100	0	0	NUM
iajs-1808	87	101	and	and	CCONJ
iajs-1808	87	102	𝐿⨁𝐾	𝐿⨁𝐾	ADJ
iajs-1808	87	103	𝑀.	𝑀.	NOUN
iajs-1808	87	104	proof	proof	NOUN
iajs-1808	87	105	:	:	PUNCT
iajs-1808	87	106	(	(	PUNCT
iajs-1808	87	107	1	1	X
iajs-1808	87	108	)	)	PUNCT
iajs-1808	87	109			NOUN
iajs-1808	87	110	(	(	PUNCT
iajs-1808	87	111	2	2	NUM
iajs-1808	87	112	)	)	PUNCT
iajs-1808	87	113	for	for	ADP
iajs-1808	87	114	any	any	DET
iajs-1808	87	115	complement	complement	NOUN
iajs-1808	87	116	submodule	submodule	NOUN
iajs-1808	87	117	𝐿	𝐿	PROPN
iajs-1808	87	118	in	in	ADP
iajs-1808	87	119	𝑀.	𝑀.	NOUN
iajs-1808	87	120	by	by	ADP
iajs-1808	87	121	strongly	strongly	ADV
iajs-1808	87	122	𝐶	𝐶	PROPN
iajs-1808	87	123	-condition	-condition	PROPN
iajs-1808	87	124	,	,	PUNCT
iajs-1808	87	125	there	there	PRON
iajs-1808	87	126	exists	exist	VERB
iajs-1808	87	127	a	a	DET
iajs-1808	87	128	fully	fully	ADV
iajs-1808	87	129	invariant	invariant	ADJ
iajs-1808	87	130	direct	direct	ADJ
iajs-1808	87	131	summand	summand	NOUN
iajs-1808	87	132	𝐾	𝐾	PROPN
iajs-1808	87	133	of	of	ADP
iajs-1808	87	134	𝑀	𝑀	PROPN
iajs-1808	87	135	which	which	PRON
iajs-1808	87	136	is	be	AUX
iajs-1808	87	137	a	a	DET
iajs-1808	87	138	complement	complement	NOUN
iajs-1808	87	139	of	of	ADP
iajs-1808	87	140	𝐿	𝐿	PROPN
iajs-1808	87	141	in	in	ADP
iajs-1808	87	142	𝑀.	𝑀.	PROPN
iajs-1808	87	143	ihsciconf	ihsciconf	NOUN
iajs-1808	87	144	2017	2017	NUM
iajs-1808	87	145	special	special	ADJ
iajs-1808	87	146	issue	issue	NOUN
iajs-1808	87	147	ibn	ibn	PROPN
iajs-1808	87	148	al	al	PROPN
iajs-1808	87	149	-	-	PUNCT
iajs-1808	87	150	haitham	haitham	PROPN
iajs-1808	87	151	journal	journal	PROPN
iajs-1808	87	152	for	for	ADP
iajs-1808	87	153	pure	pure	ADJ
iajs-1808	87	154	and	and	CCONJ
iajs-1808	87	155	applied	apply	VERB
iajs-1808	87	156	science	science	NOUN
iajs-1808	87	157	https://doi.org/	https://doi.org/	NOUN
iajs-1808	87	158	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	87	159	for	for	ADP
iajs-1808	87	160	more	more	ADJ
iajs-1808	87	161	information	information	NOUN
iajs-1808	87	162	about	about	ADP
iajs-1808	87	163	the	the	DET
iajs-1808	87	164	conference	conference	NOUN
iajs-1808	87	165	please	please	INTJ
iajs-1808	87	166	visit	visit	VERB
iajs-1808	87	167	the	the	DET
iajs-1808	87	168	websites	website	NOUN
iajs-1808	87	169	:	:	PUNCT
iajs-1808	87	170	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	87	171	  	  	SPACE
iajs-1808	87	172	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1808	87	173	   	   	SPACE
iajs-1808	87	174	mathematics|357	mathematics|357	NOUN
iajs-1808	87	175	    	    	SPACE
iajs-1808	87	176	(	(	PUNCT
iajs-1808	87	177	3	3	NUM
iajs-1808	87	178	)	)	PUNCT
iajs-1808	87	179			NOUN
iajs-1808	87	180	(	(	PUNCT
iajs-1808	87	181	4	4	NUM
iajs-1808	87	182	)	)	PUNCT
iajs-1808	87	183	and	and	CCONJ
iajs-1808	87	184	(	(	PUNCT
iajs-1808	87	185	2	2	X
iajs-1808	87	186	)	)	PUNCT
iajs-1808	87	187			NOUN
iajs-1808	87	188	(	(	PUNCT
iajs-1808	87	189	4	4	X
iajs-1808	87	190	)	)	PUNCT
iajs-1808	87	191	are	be	AUX
iajs-1808	87	192	obvious	obvious	ADJ
iajs-1808	87	193	.	.	PUNCT
iajs-1808	88	1	(	(	PUNCT
iajs-1808	88	2	1	1	X
iajs-1808	88	3	)	)	PUNCT
iajs-1808	88	4			NOUN
iajs-1808	88	5	(	(	PUNCT
iajs-1808	88	6	3	3	X
iajs-1808	88	7	)	)	PUNCT
iajs-1808	88	8	it	it	PRON
iajs-1808	88	9	is	be	AUX
iajs-1808	88	10	clear	clear	ADJ
iajs-1808	88	11	by	by	ADP
iajs-1808	88	12	lemma	lemma	PROPN
iajs-1808	88	13	(	(	PUNCT
iajs-1808	88	14	3.3	3.3	NUM
iajs-1808	88	15	)	)	PUNCT
iajs-1808	88	16	.	.	PUNCT
iajs-1808	89	1	(	(	PUNCT
iajs-1808	89	2	4	4	X
iajs-1808	89	3	)	)	PUNCT
iajs-1808	89	4			NOUN
iajs-1808	89	5	(	(	PUNCT
iajs-1808	89	6	1	1	X
iajs-1808	89	7	)	)	PUNCT
iajs-1808	89	8	let	let	VERB
iajs-1808	89	9	𝐴	𝐴	PROPN
iajs-1808	89	10	be	be	AUX
iajs-1808	89	11	any	any	DET
iajs-1808	89	12	submodule	submodule	NOUN
iajs-1808	89	13	of	of	ADP
iajs-1808	89	14	𝑀.	𝑀.	PROPN
iajs-1808	89	15	then	then	ADV
iajs-1808	89	16	there	there	PRON
iajs-1808	89	17	exists	exist	VERB
iajs-1808	89	18	a	a	DET
iajs-1808	89	19	complement	complement	NOUN
iajs-1808	89	20	(	(	PUNCT
iajs-1808	89	21	so	so	ADV
iajs-1808	89	22	closed	closed	ADJ
iajs-1808	89	23	submodule	submodule	NOUN
iajs-1808	89	24	𝐵	𝐵	PROPN
iajs-1808	89	25	in	in	ADP
iajs-1808	89	26	𝑀	𝑀	PROPN
iajs-1808	89	27	)	)	PUNCT
iajs-1808	89	28	such	such	ADJ
iajs-1808	89	29	that	that	SCONJ
iajs-1808	89	30	𝐴	𝐴	PROPN
iajs-1808	89	31	𝐵	𝐵	PROPN
iajs-1808	89	32	,	,	PUNCT
iajs-1808	89	33	by	by	ADP
iajs-1808	89	34	[	[	X
iajs-1808	89	35	8,excerces	8,excerces	NUM
iajs-1808	89	36	13,p.20	13,p.20	NUM
iajs-1808	89	37	]	]	PUNCT
iajs-1808	89	38	.	.	PUNCT
iajs-1808	90	1	by	by	ADP
iajs-1808	90	2	hypothesis	hypothesis	NOUN
iajs-1808	90	3	,	,	PUNCT
iajs-1808	90	4	there	there	PRON
iajs-1808	90	5	exists	exist	VERB
iajs-1808	90	6	a	a	DET
iajs-1808	90	7	fully	fully	ADV
iajs-1808	90	8	invariant	invariant	ADJ
iajs-1808	90	9	direct	direct	ADJ
iajs-1808	90	10	summand	summand	NOUN
iajs-1808	90	11	𝐾	𝐾	PROPN
iajs-1808	90	12	of	of	ADP
iajs-1808	90	13	𝑀	𝑀	PROPN
iajs-1808	91	1	such	such	ADJ
iajs-1808	91	2	that	that	SCONJ
iajs-1808	91	3	𝐵⋂𝐾	𝐵⋂𝐾	PROPN
iajs-1808	91	4	0	0	NUM
iajs-1808	92	1	and	and	CCONJ
iajs-1808	92	2	𝐵⨁𝐾	𝐵⨁𝐾	NOUN
iajs-1808	92	3	𝑀.	𝑀.	PROPN
iajs-1808	92	4	hence	hence	ADV
iajs-1808	92	5	by	by	ADP
iajs-1808	92	6	lemma	lemma	PROPN
iajs-1808	92	7	(	(	PUNCT
iajs-1808	92	8	3.2	3.2	NUM
iajs-1808	92	9	)	)	PUNCT
iajs-1808	92	10	𝐾	𝐾	NOUN
iajs-1808	92	11	is	be	AUX
iajs-1808	92	12	a	a	DET
iajs-1808	92	13	complement	complement	NOUN
iajs-1808	92	14	of	of	ADP
iajs-1808	92	15	𝐵	𝐵	NOUN
iajs-1808	92	16	in	in	ADP
iajs-1808	92	17	𝑀.	𝑀.	PROPN
iajs-1808	92	18	so	so	ADV
iajs-1808	92	19	𝐵⋂𝐾	𝐵⋂𝐾	NOUN
iajs-1808	92	20	0	0	NUM
iajs-1808	92	21	,	,	PUNCT
iajs-1808	92	22	which	which	PRON
iajs-1808	92	23	implies	imply	VERB
iajs-1808	92	24	𝐾⋂𝐴	𝐾⋂𝐴	NOUN
iajs-1808	92	25	0	0	NUM
iajs-1808	92	26	.	.	PUNCT
iajs-1808	93	1	suppose	suppose	VERB
iajs-1808	93	2	that	that	SCONJ
iajs-1808	93	3	𝐾	𝐾	PROPN
iajs-1808	93	4	𝑀	𝑀	PROPN
iajs-1808	93	5	and	and	CCONJ
iajs-1808	93	6	,	,	PUNCT
iajs-1808	93	7	𝐾	𝐾	PROPN
iajs-1808	93	8	𝐾.	𝐾.	PROPN
iajs-1808	93	9	therefore	therefore	ADV
iajs-1808	93	10	𝐾	𝐾	PROPN
iajs-1808	93	11	⋂𝐵	⋂𝐵	PROPN
iajs-1808	93	12	0	0	NUM
iajs-1808	93	13	and	and	CCONJ
iajs-1808	93	14	hence	hence	ADV
iajs-1808	93	15	𝐾	𝐾	PROPN
iajs-1808	93	16	⋂𝐵	⋂𝐵	PROPN
iajs-1808	93	17	⋂𝐴	⋂𝐴	NUM
iajs-1808	93	18	0	0	NUM
iajs-1808	93	19	(	(	PUNCT
iajs-1808	93	20	since	since	SCONJ
iajs-1808	93	21	𝐴	𝐴	PROPN
iajs-1808	93	22	𝐵	𝐵	PROPN
iajs-1808	93	23	,	,	PUNCT
iajs-1808	93	24	so	so	SCONJ
iajs-1808	93	25	that	that	SCONJ
iajs-1808	93	26	𝐾	𝐾	PROPN
iajs-1808	93	27	⋂𝐴	⋂𝐴	PROPN
iajs-1808	93	28	0	0	NUM
iajs-1808	93	29	.	.	PUNCT
iajs-1808	94	1	thus	thus	ADV
iajs-1808	94	2	𝐾	𝐾	PROPN
iajs-1808	94	3	is	be	AUX
iajs-1808	94	4	a	a	DET
iajs-1808	94	5	complement	complement	NOUN
iajs-1808	94	6	of	of	ADP
iajs-1808	94	7	𝐴	𝐴	PROPN
iajs-1808	94	8	in	in	ADP
iajs-1808	94	9	𝑀.	𝑀.	PROPN
iajs-1808	94	10	□	□	PUNCT
iajs-1808	94	11	as	as	SCONJ
iajs-1808	94	12	we	we	PRON
iajs-1808	94	13	have	have	AUX
iajs-1808	94	14	seen	see	VERB
iajs-1808	94	15	t	t	NOUN
iajs-1808	94	16	-	-	PUNCT
iajs-1808	94	17	semisimple	semisimple	NOUN
iajs-1808	94	18	is	be	AUX
iajs-1808	94	19	𝑇	𝑇	PROPN
iajs-1808	94	20	-type	-type	NOUN
iajs-1808	94	21	module	module	NOUN
iajs-1808	94	22	.	.	PUNCT
iajs-1808	95	1	we	we	PRON
iajs-1808	95	2	claim	claim	VERB
iajs-1808	95	3	that	that	SCONJ
iajs-1808	95	4	strongly	strongly	ADV
iajs-1808	95	5	t	t	NOUN
iajs-1808	95	6	-	-	PUNCT
iajs-1808	95	7	semisimple	semisimple	NOUN
iajs-1808	95	8	modules	module	NOUN
iajs-1808	95	9	imply	imply	VERB
iajs-1808	95	10	modules	module	NOUN
iajs-1808	95	11	which	which	PRON
iajs-1808	95	12	are	be	AUX
iajs-1808	95	13	strongly	strongly	ADV
iajs-1808	95	14	than	than	ADP
iajs-1808	95	15	module	module	NOUN
iajs-1808	95	16	with	with	ADP
iajs-1808	95	17	𝑇	𝑇	PROPN
iajs-1808	95	18	-type	-type	NOUN
iajs-1808	95	19	module	module	NOUN
iajs-1808	95	20	.	.	PUNCT
iajs-1808	96	1	hence	hence	ADV
iajs-1808	96	2	this	this	PRON
iajs-1808	96	3	leads	lead	VERB
iajs-1808	96	4	us	we	PRON
iajs-1808	96	5	to	to	PART
iajs-1808	96	6	define	define	VERB
iajs-1808	96	7	the	the	DET
iajs-1808	96	8	following	following	NOUN
iajs-1808	96	9	:	:	PUNCT
iajs-1808	96	10	definition	definition	NOUN
iajs-1808	96	11	(	(	PUNCT
iajs-1808	96	12	3.5	3.5	NUM
iajs-1808	96	13	):	):	PUNCT
iajs-1808	96	14	an	an	DET
iajs-1808	96	15	𝑅-module	𝑅-module	PROPN
iajs-1808	96	16	is	be	AUX
iajs-1808	96	17	said	say	VERB
iajs-1808	96	18	to	to	PART
iajs-1808	96	19	be	be	AUX
iajs-1808	96	20	strongly	strongly	ADV
iajs-1808	96	21	𝑇	𝑇	PROPN
iajs-1808	96	22	(	(	PUNCT
iajs-1808	96	23	or	or	CCONJ
iajs-1808	96	24	strongly	strongly	ADV
iajs-1808	96	25	𝑇	𝑇	PROPN
iajs-1808	96	26	-type	-type	NOUN
iajs-1808	96	27	modules	module	NOUN
iajs-1808	96	28	)	)	PUNCT
iajs-1808	96	29	if	if	SCONJ
iajs-1808	96	30	for	for	ADP
iajs-1808	96	31	each	each	DET
iajs-1808	96	32	t	t	NOUN
iajs-1808	96	33	-	-	PUNCT
iajs-1808	96	34	closed	close	VERB
iajs-1808	96	35	submodule	submodule	NOUN
iajs-1808	96	36	,	,	PUNCT
iajs-1808	96	37	there	there	PRON
iajs-1808	96	38	exists	exist	VERB
iajs-1808	96	39	a	a	DET
iajs-1808	96	40	complement	complement	NOUN
iajs-1808	96	41	which	which	PRON
iajs-1808	96	42	is	be	AUX
iajs-1808	96	43	a	a	DET
iajs-1808	96	44	fully	fully	ADV
iajs-1808	96	45	invariant	invariant	ADJ
iajs-1808	96	46	direct	direct	ADJ
iajs-1808	96	47	summand	summand	NOUN
iajs-1808	96	48	.	.	PUNCT
iajs-1808	97	1	remarks	remark	NOUN
iajs-1808	97	2	(	(	PUNCT
iajs-1808	97	3	3.6	3.6	NUM
iajs-1808	97	4	):	):	PUNCT
iajs-1808	97	5	(	(	PUNCT
iajs-1808	97	6	1	1	X
iajs-1808	97	7	)	)	PUNCT
iajs-1808	97	8	it	it	PRON
iajs-1808	97	9	is	be	AUX
iajs-1808	97	10	clear	clear	ADJ
iajs-1808	97	11	that	that	SCONJ
iajs-1808	97	12	every	every	DET
iajs-1808	97	13	module	module	NOUN
iajs-1808	97	14	,	,	PUNCT
iajs-1808	97	15	which	which	PRON
iajs-1808	97	16	satisfies	satisfy	VERB
iajs-1808	97	17	strongly	strongly	ADV
iajs-1808	97	18	𝐶	𝐶	PROPN
iajs-1808	97	19	-condition	-condition	PROPN
iajs-1808	97	20	,	,	PUNCT
iajs-1808	97	21	is	be	AUX
iajs-1808	97	22	a	a	DET
iajs-1808	97	23	strongly	strongly	ADV
iajs-1808	97	24	𝑇	𝑇	NOUN
iajs-1808	97	25	-type	-type	NOUN
iajs-1808	97	26	module	module	NOUN
iajs-1808	97	27	,	,	PUNCT
iajs-1808	97	28	but	but	CCONJ
iajs-1808	97	29	the	the	DET
iajs-1808	97	30	converse	converse	NOUN
iajs-1808	97	31	is	be	AUX
iajs-1808	97	32	not	not	PART
iajs-1808	97	33	true	true	ADJ
iajs-1808	97	34	in	in	ADP
iajs-1808	97	35	general	general	ADJ
iajs-1808	97	36	,	,	PUNCT
iajs-1808	97	37	as	as	SCONJ
iajs-1808	97	38	the	the	DET
iajs-1808	97	39	following	follow	VERB
iajs-1808	97	40	example	example	NOUN
iajs-1808	97	41	shows	show	VERB
iajs-1808	97	42	:	:	PUNCT
iajs-1808	97	43	let	let	VERB
iajs-1808	97	44	𝑀	𝑀	PROPN
iajs-1808	97	45	𝑍	𝑍	VERB
iajs-1808	97	46	⨁𝑍	⨁𝑍	NOUN
iajs-1808	97	47	as	as	SCONJ
iajs-1808	97	48	𝑍-module	𝑍-module	PROPN
iajs-1808	97	49	𝑀	𝑀	PROPN
iajs-1808	97	50	is	be	AUX
iajs-1808	97	51	𝑇	𝑇	PROPN
iajs-1808	97	52	-type	-type	NOUN
iajs-1808	97	53	module	module	NOUN
iajs-1808	97	54	by	by	ADP
iajs-1808	97	55	[	[	X
iajs-1808	97	56	4	4	NUM
iajs-1808	97	57	,	,	PUNCT
iajs-1808	97	58	corollary	corollary	ADJ
iajs-1808	97	59	2.6	2.6	NUM
iajs-1808	97	60	]	]	PUNCT
iajs-1808	97	61	.	.	PUNCT
iajs-1808	98	1	𝑀	𝑀	PROPN
iajs-1808	98	2	is	be	AUX
iajs-1808	98	3	strongly	strongly	ADV
iajs-1808	98	4	𝑇	𝑇	PROPN
iajs-1808	98	5	-type	-type	NOUN
iajs-1808	98	6	module	module	NOUN
iajs-1808	98	7	,	,	PUNCT
iajs-1808	98	8	but	but	CCONJ
iajs-1808	98	9	it	it	PRON
iajs-1808	98	10	is	be	AUX
iajs-1808	98	11	not	not	PART
iajs-1808	98	12	strongly	strongly	ADV
iajs-1808	98	13	𝐶	𝐶	PROPN
iajs-1808	98	14	-type	-type	NOUN
iajs-1808	98	15	module	module	NOUN
iajs-1808	98	16	.	.	PUNCT
iajs-1808	99	1	if	if	SCONJ
iajs-1808	99	2	𝑁	𝑁	PROPN
iajs-1808	99	3	2	2	NUM
iajs-1808	99	4	⨁	⨁	PROPN
iajs-1808	99	5	0	0	NUM
iajs-1808	99	6	2	2	NUM
iajs-1808	99	7	,	,	PUNCT
iajs-1808	99	8	0	0	NUM
iajs-1808	99	9	,	,	PUNCT
iajs-1808	99	10	4	4	NUM
iajs-1808	99	11	,	,	PUNCT
iajs-1808	99	12	0	0	NUM
iajs-1808	99	13	,	,	PUNCT
iajs-1808	99	14	6	6	NUM
iajs-1808	99	15	,	,	PUNCT
iajs-1808	99	16	0	0	NUM
iajs-1808	99	17	,	,	PUNCT
iajs-1808	99	18	0	0	NUM
iajs-1808	99	19	,	,	PUNCT
iajs-1808	99	20	0	0	NUM
iajs-1808	99	21	,	,	PUNCT
iajs-1808	99	22	𝑁	𝑁	PROPN
iajs-1808	99	23	∩	∩	NOUN
iajs-1808	99	24	0	0	NUM
iajs-1808	99	25	⨁𝑍	⨁𝑍	PROPN
iajs-1808	99	26	0	0	NUM
iajs-1808	99	27	,	,	PUNCT
iajs-1808	99	28	0	0	NUM
iajs-1808	99	29	and	and	CCONJ
iajs-1808	99	30	𝑁⨁𝑊	𝑁⨁𝑊	NOUN
iajs-1808	99	31	2)⨁𝑍	2)⨁𝑍	NUM
iajs-1808	99	32	𝑀	𝑀	PROPN
iajs-1808	99	33	where	where	SCONJ
iajs-1808	99	34	𝑊	𝑊	PROPN
iajs-1808	99	35	0	0	NUM
iajs-1808	99	36	⨁𝑍	⨁𝑍	PROPN
iajs-1808	99	37	and	and	CCONJ
iajs-1808	99	38	𝑊	𝑊	PROPN
iajs-1808	99	39	⨁	⨁	PROPN
iajs-1808	99	40	𝑀	𝑀	PROPN
iajs-1808	99	41	,	,	PUNCT
iajs-1808	99	42	then	then	ADV
iajs-1808	99	43	𝑊	𝑊	PROPN
iajs-1808	99	44	is	be	AUX
iajs-1808	99	45	a	a	DET
iajs-1808	99	46	complement	complement	NOUN
iajs-1808	99	47	of	of	ADP
iajs-1808	99	48	𝑁.	𝑁.	PROPN
iajs-1808	99	49	also	also	ADV
iajs-1808	99	50	𝑁	𝑁	PROPN
iajs-1808	99	51	∩	∩	ADJ
iajs-1808	99	52	𝐾	𝐾	NOUN
iajs-1808	99	53	0	0	NUM
iajs-1808	99	54	,	,	PUNCT
iajs-1808	99	55	where	where	SCONJ
iajs-1808	99	56	𝐾	𝐾	PROPN
iajs-1808	99	57	(	(	PUNCT
iajs-1808	99	58	4	4	NUM
iajs-1808	99	59	,	,	PUNCT
iajs-1808	99	60	1	1	NUM
iajs-1808	99	61	,	,	PUNCT
iajs-1808	99	62	0	0	NUM
iajs-1808	99	63	,	,	PUNCT
iajs-1808	99	64	0	0	NUM
iajs-1808	99	65	)	)	PUNCT
iajs-1808	99	66	}	}	PUNCT
iajs-1808	99	67	.	.	PUNCT
iajs-1808	100	1	but	but	CCONJ
iajs-1808	100	2	⨁𝐾	⨁𝐾	PROPN
iajs-1808	100	3	0	0	NUM
iajs-1808	100	4	,	,	PUNCT
iajs-1808	100	5	0	0	NUM
iajs-1808	100	6	,	,	PUNCT
iajs-1808	100	7	2	2	NUM
iajs-1808	100	8	,	,	PUNCT
iajs-1808	100	9	0	0	NUM
iajs-1808	100	10	)	)	PUNCT
iajs-1808	100	11	,	,	PUNCT
iajs-1808	100	12	(	(	PUNCT
iajs-1808	100	13	4	4	NUM
iajs-1808	100	14	,	,	PUNCT
iajs-1808	100	15	0	0	NUM
iajs-1808	100	16	)	)	PUNCT
iajs-1808	100	17	,	,	PUNCT
iajs-1808	100	18	(	(	PUNCT
iajs-1808	100	19	6	6	NUM
iajs-1808	100	20	,	,	PUNCT
iajs-1808	100	21	0)}⨁	0)}⨁	NUM
iajs-1808	100	22	4	4	NUM
iajs-1808	100	23	,	,	PUNCT
iajs-1808	100	24	1	1	NUM
iajs-1808	100	25	,	,	PUNCT
iajs-1808	100	26	0	0	NUM
iajs-1808	100	27	,	,	PUNCT
iajs-1808	100	28	0	0	NUM
iajs-1808	100	29	=	=	SYM
iajs-1808	100	30	4	4	NUM
iajs-1808	100	31	,	,	PUNCT
iajs-1808	100	32	1	1	NUM
iajs-1808	100	33	,	,	PUNCT
iajs-1808	100	34	0	0	NUM
iajs-1808	100	35	,	,	PUNCT
iajs-1808	100	36	0	0	NUM
iajs-1808	100	37	,	,	PUNCT
iajs-1808	100	38	6	6	NUM
iajs-1808	100	39	,	,	PUNCT
iajs-1808	100	40	1	1	NUM
iajs-1808	100	41	,	,	PUNCT
iajs-1808	100	42	2	2	NUM
iajs-1808	100	43	,	,	PUNCT
iajs-1808	100	44	0	0	NUM
iajs-1808	100	45	,	,	PUNCT
iajs-1808	100	46	0	0	NUM
iajs-1808	100	47	,	,	PUNCT
iajs-1808	100	48	1	1	NUM
iajs-1808	100	49	,	,	PUNCT
iajs-1808	100	50	4	4	NUM
iajs-1808	100	51	,	,	PUNCT
iajs-1808	100	52	0	0	NUM
iajs-1808	100	53	)	)	PUNCT
iajs-1808	100	54	,	,	PUNCT
iajs-1808	100	55	(	(	PUNCT
iajs-1808	100	56	2	2	NUM
iajs-1808	100	57	,	,	PUNCT
iajs-1808	100	58	1	1	NUM
iajs-1808	100	59	,	,	PUNCT
iajs-1808	100	60	6	6	NUM
iajs-1808	100	61	,	,	PUNCT
iajs-1808	100	62	0	0	X
iajs-1808	100	63	𝑈	𝑈	PROPN
iajs-1808	100	64	and	and	CCONJ
iajs-1808	100	65	𝑈	𝑈	PROPN
iajs-1808	100	66	𝑀	𝑀	PROPN
iajs-1808	100	67	,	,	PUNCT
iajs-1808	100	68	hence	hence	ADV
iajs-1808	100	69	is	be	AUX
iajs-1808	100	70	a	a	DET
iajs-1808	100	71	complement	complement	NOUN
iajs-1808	100	72	of	of	ADP
iajs-1808	100	73	𝑁	𝑁	PROPN
iajs-1808	100	74	,	,	PUNCT
iajs-1808	100	75	but	but	CCONJ
iajs-1808	100	76	𝐾	𝐾	NOUN
iajs-1808	100	77	≰⨁	≰⨁	VERB
iajs-1808	100	78	𝑀.	𝑀.	PROPN
iajs-1808	100	79	thus	thus	ADV
iajs-1808	100	80	𝑊	𝑊	PROPN
iajs-1808	100	81	is	be	AUX
iajs-1808	100	82	a	a	DET
iajs-1808	100	83	unique	unique	ADJ
iajs-1808	100	84	complement	complement	NOUN
iajs-1808	100	85	of	of	ADP
iajs-1808	100	86	𝑁	𝑁	PROPN
iajs-1808	100	87	which	which	PRON
iajs-1808	100	88	is	be	AUX
iajs-1808	100	89	a	a	DET
iajs-1808	100	90	direct	direct	ADJ
iajs-1808	100	91	summand	summand	NOUN
iajs-1808	100	92	,	,	PUNCT
iajs-1808	100	93	but	but	CCONJ
iajs-1808	100	94	𝑊	𝑊	NOUN
iajs-1808	100	95	is	be	AUX
iajs-1808	100	96	not	not	PART
iajs-1808	100	97	fully	fully	ADV
iajs-1808	100	98	invariant	invariant	ADJ
iajs-1808	100	99	submodule	submodule	NOUN
iajs-1808	100	100	as	as	SCONJ
iajs-1808	100	101	there	there	PRON
iajs-1808	100	102	exists	exist	VERB
iajs-1808	100	103	𝑓	𝑓	PRON
iajs-1808	100	104	:	:	PUNCT
iajs-1808	100	105	𝑊	𝑊	NOUN
iajs-1808	100	106	0	0	NUM
iajs-1808	100	107	,	,	PUNCT
iajs-1808	100	108	0	0	NUM
iajs-1808	100	109	,	,	PUNCT
iajs-1808	100	110	0	0	NUM
iajs-1808	100	111	,	,	PUNCT
iajs-1808	100	112	1	1	NUM
iajs-1808	100	113	⟼	⟼	PROPN
iajs-1808	100	114	𝑀	𝑀	PROPN
iajs-1808	100	115	defined	define	VERB
iajs-1808	100	116	by	by	ADP
iajs-1808	100	117	𝑓	𝑓	DET
iajs-1808	100	118	0	0	NUM
iajs-1808	100	119	,	,	PUNCT
iajs-1808	100	120	0	0	NUM
iajs-1808	100	121	0	0	NUM
iajs-1808	100	122	,	,	PUNCT
iajs-1808	100	123	0	0	NUM
iajs-1808	100	124	𝑓	𝑓	DET
iajs-1808	100	125	0	0	NUM
iajs-1808	100	126	,	,	PUNCT
iajs-1808	100	127	1	1	NUM
iajs-1808	100	128	4	4	NUM
iajs-1808	100	129	,	,	PUNCT
iajs-1808	100	130	1	1	NUM
iajs-1808	100	131	,	,	PUNCT
iajs-1808	100	132	𝑓	𝑓	PRON
iajs-1808	100	133	is	be	AUX
iajs-1808	100	134	𝑍-homorphism	𝑍-homorphism	PROPN
iajs-1808	100	135	and	and	CCONJ
iajs-1808	100	136	𝑓	𝑓	DET
iajs-1808	100	137	𝑊	𝑊	PROPN
iajs-1808	100	138	≰	≰	PROPN
iajs-1808	100	139	𝑊.	𝑊.	PROPN
iajs-1808	100	140	thus	thus	ADV
iajs-1808	100	141	𝑀	𝑀	PROPN
iajs-1808	100	142	does	do	AUX
iajs-1808	100	143	not	not	PART
iajs-1808	100	144	satisfy	satisfy	VERB
iajs-1808	100	145	strongly	strongly	ADV
iajs-1808	100	146	𝐶	𝐶	PROPN
iajs-1808	100	147	-type	-type	NOUN
iajs-1808	100	148	module	module	NOUN
iajs-1808	100	149	.	.	PUNCT
iajs-1808	101	1	also	also	ADV
iajs-1808	101	2	𝑀	𝑀	PROPN
iajs-1808	101	3	is	be	AUX
iajs-1808	101	4	singular	singular	ADJ
iajs-1808	101	5	,	,	PUNCT
iajs-1808	101	6	so	so	ADV
iajs-1808	101	7	𝑀	𝑀	PROPN
iajs-1808	101	8	is	be	AUX
iajs-1808	101	9	the	the	DET
iajs-1808	101	10	only	only	ADJ
iajs-1808	101	11	t	t	PROPN
iajs-1808	101	12	-	-	PUNCT
iajs-1808	101	13	closed	close	VERB
iajs-1808	101	14	submodule	submodule	NOUN
iajs-1808	101	15	and	and	CCONJ
iajs-1808	101	16	has	have	VERB
iajs-1808	101	17	a	a	DET
iajs-1808	101	18	complement	complement	NOUN
iajs-1808	101	19	which	which	PRON
iajs-1808	101	20	is	be	AUX
iajs-1808	101	21	the	the	DET
iajs-1808	101	22	zero	zero	NUM
iajs-1808	101	23	submodules	submodule	NOUN
iajs-1808	101	24	and	and	CCONJ
iajs-1808	101	25	it	it	PRON
iajs-1808	101	26	is	be	AUX
iajs-1808	101	27	clear	clear	ADJ
iajs-1808	101	28	direct	direct	ADJ
iajs-1808	101	29	summand	summand	NOUN
iajs-1808	101	30	fully	fully	ADV
iajs-1808	101	31	invariant	invariant	ADJ
iajs-1808	101	32	submodule	submodule	NOUN
iajs-1808	101	33	.	.	PUNCT
iajs-1808	102	1	thus	thus	ADV
iajs-1808	102	2	𝑀	𝑀	PROPN
iajs-1808	102	3	is	be	AUX
iajs-1808	102	4	strongly	strongly	ADV
iajs-1808	102	5	𝑇	𝑇	PROPN
iajs-1808	102	6	-type	-type	NOUN
iajs-1808	102	7	.	.	PUNCT
iajs-1808	103	1	(	(	PUNCT
iajs-1808	103	2	2	2	X
iajs-1808	103	3	)	)	PUNCT
iajs-1808	103	4	let	let	VERB
iajs-1808	103	5	𝑀	𝑀	PROPN
iajs-1808	103	6	be	be	AUX
iajs-1808	103	7	an	an	DET
iajs-1808	103	8	𝑅-module	𝑅-module	PROPN
iajs-1808	103	9	which	which	PRON
iajs-1808	103	10	satisfies	satisfy	VERB
iajs-1808	103	11	strongly	strongly	ADV
iajs-1808	103	12	𝐶	𝐶	PROPN
iajs-1808	103	13	-condition	-condition	PROPN
iajs-1808	103	14	.	.	PUNCT
iajs-1808	104	1	then	then	ADV
iajs-1808	104	2	𝑀	𝑀	PROPN
iajs-1808	104	3	is	be	AUX
iajs-1808	104	4	strongly	strongly	ADV
iajs-1808	104	5	fi	fi	NOUN
iajs-1808	104	6	-	-	NOUN
iajs-1808	104	7	tsemisimple	tsemisimple	ADJ
iajs-1808	104	8	if	if	SCONJ
iajs-1808	105	1	and	and	CCONJ
iajs-1808	105	2	only	only	ADV
iajs-1808	105	3	if	if	SCONJ
iajs-1808	105	4	𝑀	𝑀	PROPN
iajs-1808	105	5	is	be	AUX
iajs-1808	105	6	fi	fi	ADJ
iajs-1808	105	7	-	-	ADJ
iajs-1808	105	8	t	t	NOUN
iajs-1808	105	9	-	-	PUNCT
iajs-1808	105	10	semisimple	semisimple	NOUN
iajs-1808	105	11	.	.	PUNCT
iajs-1808	106	1	proof	proof	NOUN
iajs-1808	106	2	:	:	PUNCT
iajs-1808	106	3	it	it	PRON
iajs-1808	106	4	follows	follow	VERB
iajs-1808	106	5	directly	directly	ADV
iajs-1808	106	6	by	by	ADP
iajs-1808	106	7	proposition	proposition	NOUN
iajs-1808	106	8	1.4	1.4	NUM
iajs-1808	106	9	,	,	PUNCT
iajs-1808	106	10	and	and	CCONJ
iajs-1808	106	11	definition	definition	NOUN
iajs-1808	106	12	3.1	3.1	NUM
iajs-1808	106	13	.	.	PUNCT
iajs-1808	107	1	□	□	PUNCT
iajs-1808	107	2	proposition	proposition	NOUN
iajs-1808	107	3	(	(	PUNCT
iajs-1808	107	4	3.7	3.7	NUM
iajs-1808	107	5	):	):	PUNCT
iajs-1808	107	6	let	let	VERB
iajs-1808	107	7	𝑀	𝑀	PRON
iajs-1808	107	8	be	be	AUX
iajs-1808	107	9	a	a	DET
iajs-1808	107	10	nonsingular	nonsingular	ADJ
iajs-1808	107	11	𝑅-module	𝑅-module	PROPN
iajs-1808	107	12	.	.	PUNCT
iajs-1808	108	1	𝑀	𝑀	PROPN
iajs-1808	108	2	is	be	AUX
iajs-1808	108	3	strongly	strongly	ADV
iajs-1808	108	4	𝐶	𝐶	PROPN
iajs-1808	108	5	-condition	-condition	NOUN
iajs-1808	108	6	module	module	NOUN
iajs-1808	108	7	if	if	SCONJ
iajs-1808	108	8	and	and	CCONJ
iajs-1808	108	9	only	only	ADV
iajs-1808	108	10	if	if	SCONJ
iajs-1808	108	11	𝑀	𝑀	PROPN
iajs-1808	108	12	is	be	AUX
iajs-1808	108	13	strongly	strongly	ADV
iajs-1808	108	14	𝑇	𝑇	PROPN
iajs-1808	108	15	-type	-type	NOUN
iajs-1808	108	16	module	module	NOUN
iajs-1808	108	17	.	.	PUNCT
iajs-1808	109	1	proof:	proof:	PROPN
iajs-1808	109	2	it	it	PRON
iajs-1808	109	3	is	be	AUX
iajs-1808	109	4	clear	clear	ADJ
iajs-1808	109	5	.	.	PUNCT
iajs-1808	110	1			NOUN
iajs-1808	110	2	let	let	VERB
iajs-1808	110	3	𝐴	𝐴	PROPN
iajs-1808	110	4	𝑀	𝑀	PROPN
iajs-1808	110	5	,	,	PUNCT
iajs-1808	110	6	by	by	ADP
iajs-1808	110	7	[	[	X
iajs-1808	110	8	8,exercies	8,exercies	NUM
iajs-1808	110	9	13,p.20	13,p.20	NUM
iajs-1808	110	10	]	]	PUNCT
iajs-1808	110	11	,	,	PUNCT
iajs-1808	110	12	there	there	PRON
iajs-1808	110	13	exists	exist	VERB
iajs-1808	110	14	a	a	DET
iajs-1808	110	15	closed	closed	ADJ
iajs-1808	110	16	submodule	submodule	NOUN
iajs-1808	110	17	𝑊	𝑊	PROPN
iajs-1808	110	18	of	of	ADP
iajs-1808	110	19	𝑀	𝑀	PROPN
iajs-1808	110	20	such	such	ADJ
iajs-1808	110	21	that	that	SCONJ
iajs-1808	110	22	𝐴	𝐴	PROPN
iajs-1808	110	23	𝑊.	𝑊.	PROPN
iajs-1808	110	24	since	since	SCONJ
iajs-1808	110	25	𝑀	𝑀	PROPN
iajs-1808	110	26	is	be	AUX
iajs-1808	110	27	nonsingular	nonsingular	ADJ
iajs-1808	110	28	,	,	PUNCT
iajs-1808	110	29	𝑊	𝑊	PROPN
iajs-1808	110	30	is	be	AUX
iajs-1808	110	31	t	t	NOUN
iajs-1808	110	32	-	-	PUNCT
iajs-1808	110	33	closed	close	VERB
iajs-1808	110	34	in	in	ADP
iajs-1808	110	35	𝑀.	𝑀.	PROPN
iajs-1808	110	36	hence	hence	ADV
iajs-1808	110	37	there	there	PRON
iajs-1808	110	38	exists	exist	VERB
iajs-1808	110	39	a	a	DET
iajs-1808	110	40	fully	fully	ADV
iajs-1808	110	41	invariant	invariant	ADJ
iajs-1808	110	42	direct	direct	ADJ
iajs-1808	110	43	summand	summand	NOUN
iajs-1808	110	44	𝐷	𝐷	PROPN
iajs-1808	110	45	of	of	ADP
iajs-1808	110	46	𝑀	𝑀	PROPN
iajs-1808	110	47	such	such	ADJ
iajs-1808	110	48	that	that	SCONJ
iajs-1808	110	49	𝑊⨁𝐷	𝑊⨁𝐷	PROPN
iajs-1808	110	50	𝑀.	𝑀.	PROPN
iajs-1808	110	51	it	it	PRON
iajs-1808	110	52	follows	follow	VERB
iajs-1808	110	53	that	that	SCONJ
iajs-1808	110	54	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	110	55	𝑊⨁𝐷	𝑊⨁𝐷	NOUN
iajs-1808	110	56	𝑀.	𝑀.	NOUN
iajs-1808	110	57	thus	thus	ADV
iajs-1808	110	58	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	110	59	𝑀.	𝑀.	NOUN
iajs-1808	110	60	so	so	SCONJ
iajs-1808	110	61	that	that	SCONJ
iajs-1808	110	62	𝑀	𝑀	PROPN
iajs-1808	110	63	satisfies	satisfy	VERB
iajs-1808	110	64	strongly	strongly	ADV
iajs-1808	110	65	𝐶	𝐶	PROPN
iajs-1808	110	66	-condition	-condition	PROPN
iajs-1808	110	67	.	.	PUNCT
iajs-1808	111	1	□	□	PUNCT
iajs-1808	111	2	recall	recall	VERB
iajs-1808	111	3	that	that	PRON
iajs-1808	111	4	an	an	NUM
iajs-1808	111	5	𝑅-module	𝑅-module	PROPN
iajs-1808	111	6	𝑀	𝑀	PROPN
iajs-1808	111	7	is	be	AUX
iajs-1808	111	8	called	call	VERB
iajs-1808	111	9	multiplication	multiplication	NOUN
iajs-1808	111	10	module	module	NOUN
iajs-1808	111	11	if	if	SCONJ
iajs-1808	111	12	for	for	ADP
iajs-1808	111	13	each	each	DET
iajs-1808	111	14	𝑁	𝑁	PROPN
iajs-1808	111	15	𝑀	𝑀	PROPN
iajs-1808	111	16	,	,	PUNCT
iajs-1808	111	17	there	there	PRON
iajs-1808	111	18	exists	exist	VERB
iajs-1808	111	19	an	an	DET
iajs-1808	111	20	ideal	ideal	ADJ
iajs-1808	111	21	𝐼	𝐼	PROPN
iajs-1808	111	22	of	of	ADP
iajs-1808	111	23	𝑅	𝑅	PROPN
iajs-1808	111	24	with	with	ADP
iajs-1808	111	25	𝑁	𝑁	PROPN
iajs-1808	111	26	𝑀𝐼[17	𝑀𝐼[17	PROPN
iajs-1808	111	27	]	]	PUNCT
iajs-1808	111	28	.	.	PUNCT
iajs-1808	112	1	equivalently	equivalently	ADV
iajs-1808	112	2	an	an	PROPN
iajs-1808	112	3	𝑅-module	𝑅-module	PROPN
iajs-1808	112	4	𝑀	𝑀	PROPN
iajs-1808	112	5	is	be	AUX
iajs-1808	112	6	a	a	DET
iajs-1808	112	7	multiplication	multiplication	NOUN
iajs-1808	112	8	module	module	NOUN
iajs-1808	112	9	if	if	SCONJ
iajs-1808	112	10	for	for	ADP
iajs-1808	112	11	each	each	DET
iajs-1808	112	12	𝑁	𝑁	PROPN
iajs-1808	112	13	𝑀,𝑁	𝑀,𝑁	NOUN
iajs-1808	112	14	𝑁	𝑁	PROPN
iajs-1808	112	15	:	:	PUNCT
iajs-1808	112	16	𝑀	𝑀	PROPN
iajs-1808	112	17	𝑀	𝑀	PROPN
iajs-1808	112	18	,	,	PUNCT
iajs-1808	112	19	where	where	SCONJ
iajs-1808	112	20	𝑁	𝑁	PROPN
iajs-1808	112	21	:	:	PUNCT
iajs-1808	112	22	𝑀	𝑀	PROPN
iajs-1808	112	23	𝑟	𝑟	NOUN
iajs-1808	112	24	∈	∈	PROPN
iajs-1808	112	25	𝑅	𝑅	PROPN
iajs-1808	112	26	:	:	PUNCT
iajs-1808	112	27	𝑀𝑟	𝑀𝑟	PROPN
iajs-1808	112	28	𝑁	𝑁	PROPN
iajs-1808	112	29	"	"	PUNCT
iajs-1808	112	30	.[17	.[17	PUNCT
iajs-1808	112	31	]	]	X
iajs-1808	112	32	proposition	proposition	NOUN
iajs-1808	112	33	(	(	PUNCT
iajs-1808	112	34	3.8	3.8	NUM
iajs-1808	112	35	):	):	PUNCT
iajs-1808	112	36	let	let	VERB
iajs-1808	112	37	𝑀	𝑀	PRON
iajs-1808	112	38	be	be	AUX
iajs-1808	112	39	a	a	DET
iajs-1808	112	40	multiplication	multiplication	NOUN
iajs-1808	112	41	(	(	PUNCT
iajs-1808	112	42	hence	hence	ADV
iajs-1808	112	43	𝑀	𝑀	PROPN
iajs-1808	112	44	is	be	AUX
iajs-1808	112	45	duo	duo	ADJ
iajs-1808	112	46	or	or	CCONJ
iajs-1808	112	47	fully	fully	ADV
iajs-1808	112	48	stable	stable	ADJ
iajs-1808	112	49	)	)	PUNCT
iajs-1808	112	50	.	.	PUNCT
iajs-1808	113	1	then	then	ADV
iajs-1808	113	2	𝑀	𝑀	PROPN
iajs-1808	113	3	is	be	AUX
iajs-1808	113	4	𝑇	𝑇	PROPN
iajs-1808	113	5	-type	-type	NOUN
iajs-1808	113	6	module	module	NOUN
iajs-1808	113	7	if	if	SCONJ
iajs-1808	113	8	and	and	CCONJ
iajs-1808	113	9	only	only	ADV
iajs-1808	113	10	if	if	SCONJ
iajs-1808	113	11	𝑀	𝑀	PROPN
iajs-1808	113	12	is	be	AUX
iajs-1808	113	13	strongly	strongly	ADV
iajs-1808	113	14	𝑇	𝑇	PROPN
iajs-1808	113	15	-type	-type	NOUN
iajs-1808	113	16	module	module	NOUN
iajs-1808	113	17	.	.	PUNCT
iajs-1808	114	1	𝑀	𝑀	PROPN
iajs-1808	114	2	is	be	AUX
iajs-1808	114	3	𝐶	𝐶	PROPN
iajs-1808	114	4	-type	-type	PROPN
iajs-1808	114	5	if	if	SCONJ
iajs-1808	114	6	and	and	CCONJ
iajs-1808	114	7	only	only	ADV
iajs-1808	114	8	if	if	SCONJ
iajs-1808	114	9	𝑀	𝑀	PROPN
iajs-1808	114	10	is	be	AUX
iajs-1808	114	11	strongly	strongly	ADV
iajs-1808	114	12	𝐶	𝐶	PROPN
iajs-1808	114	13	-type	-type	NOUN
iajs-1808	114	14	module	module	NOUN
iajs-1808	114	15	.	.	PUNCT
iajs-1808	115	1	we	we	PRON
iajs-1808	115	2	will	will	AUX
iajs-1808	115	3	give	give	VERB
iajs-1808	115	4	some	some	DET
iajs-1808	115	5	properties	property	NOUN
iajs-1808	115	6	of	of	ADP
iajs-1808	115	7	strongly	strongly	ADV
iajs-1808	115	8	𝑇	𝑇	PROPN
iajs-1808	115	9	-type	-type	NOUN
iajs-1808	115	10	modules	module	NOUN
iajs-1808	115	11	.	.	PUNCT
iajs-1808	116	1	theorem	theorem	NOUN
iajs-1808	116	2	(	(	PUNCT
iajs-1808	116	3	3.9	3.9	NUM
iajs-1808	116	4	):	):	PUNCT
iajs-1808	116	5	consider	consider	VERB
iajs-1808	116	6	the	the	DET
iajs-1808	116	7	following	follow	VERB
iajs-1808	116	8	statements	statement	NOUN
iajs-1808	116	9	for	for	ADP
iajs-1808	116	10	a	a	DET
iajs-1808	116	11	module	module	NOUN
iajs-1808	116	12	𝑀	𝑀	PROPN
iajs-1808	116	13	𝑀	𝑀	PROPN
iajs-1808	116	14	is	be	AUX
iajs-1808	116	15	strongly	strongly	ADV
iajs-1808	116	16	𝑇	𝑇	PROPN
iajs-1808	116	17	-type	-type	NOUN
iajs-1808	116	18	module	module	NOUN
iajs-1808	116	19	;	;	PUNCT
iajs-1808	116	20	ihsciconf	ihsciconf	PROPN
iajs-1808	116	21	2017	2017	NUM
iajs-1808	116	22	special	special	ADJ
iajs-1808	116	23	issue	issue	NOUN
iajs-1808	116	24	ibn	ibn	PROPN
iajs-1808	116	25	al	al	PROPN
iajs-1808	116	26	-	-	PUNCT
iajs-1808	116	27	haitham	haitham	PROPN
iajs-1808	116	28	journal	journal	PROPN
iajs-1808	116	29	for	for	ADP
iajs-1808	116	30	pure	pure	ADJ
iajs-1808	116	31	and	and	CCONJ
iajs-1808	116	32	applied	apply	VERB
iajs-1808	116	33	science	science	NOUN
iajs-1808	116	34	https://doi.org/	https://doi.org/	NOUN
iajs-1808	116	35	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	116	36	for	for	ADP
iajs-1808	116	37	more	more	ADJ
iajs-1808	116	38	information	information	NOUN
iajs-1808	116	39	about	about	ADP
iajs-1808	116	40	the	the	DET
iajs-1808	116	41	conference	conference	NOUN
iajs-1808	116	42	please	please	INTJ
iajs-1808	116	43	visit	visit	VERB
iajs-1808	116	44	the	the	DET
iajs-1808	116	45	websites	website	NOUN
iajs-1808	116	46	:	:	PUNCT
iajs-1808	116	47	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	116	48	  	  	SPACE
iajs-1808	116	49	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1808	116	50	   	   	SPACE
iajs-1808	116	51	mathematics|358	mathematics|358	NOUN
iajs-1808	116	52	    	    	SPACE
iajs-1808	116	53	𝑀	𝑀	PROPN
iajs-1808	116	54	𝑍	𝑍	VERB
iajs-1808	116	55	𝑀	𝑀	PROPN
iajs-1808	116	56	⨁𝑀	⨁𝑀	NOUN
iajs-1808	116	57	,	,	PUNCT
iajs-1808	116	58	where	where	SCONJ
iajs-1808	116	59	𝑀	𝑀	PROPN
iajs-1808	116	60	is	be	AUX
iajs-1808	116	61	a	a	DET
iajs-1808	116	62	fully	fully	ADV
iajs-1808	116	63	invariant	invariant	ADJ
iajs-1808	116	64	submodule	submodule	NOUN
iajs-1808	116	65	in	in	ADP
iajs-1808	116	66	𝑀	𝑀	PROPN
iajs-1808	116	67	and	and	CCONJ
iajs-1808	116	68	satisfies	satisfy	VERB
iajs-1808	116	69	strongly	strongly	ADV
iajs-1808	116	70	𝐶	𝐶	PROPN
iajs-1808	116	71	condition	condition	NOUN
iajs-1808	116	72	;	;	PUNCT
iajs-1808	116	73	for	for	ADP
iajs-1808	116	74	every	every	DET
iajs-1808	116	75	submodule	submodule	NOUN
iajs-1808	116	76	𝐴	𝐴	PROPN
iajs-1808	116	77	of	of	ADP
iajs-1808	116	78	𝑀	𝑀	PROPN
iajs-1808	116	79	,	,	PUNCT
iajs-1808	116	80	there	there	PRON
iajs-1808	116	81	exists	exist	VERB
iajs-1808	116	82	a	a	DET
iajs-1808	116	83	fully	fully	ADV
iajs-1808	116	84	invariant	invariant	ADJ
iajs-1808	116	85	direct	direct	ADJ
iajs-1808	116	86	summand	summand	NOUN
iajs-1808	116	87	𝐷	𝐷	PROPN
iajs-1808	116	88	of	of	ADP
iajs-1808	116	89	𝑀	𝑀	PROPN
iajs-1808	116	90	such	such	ADJ
iajs-1808	116	91	that	that	SCONJ
iajs-1808	116	92	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	116	93	𝑀.	𝑀.	PROPN
iajs-1808	116	94	for	for	ADP
iajs-1808	116	95	every	every	DET
iajs-1808	116	96	t	t	NOUN
iajs-1808	116	97	-	-	PUNCT
iajs-1808	116	98	closed	close	VERB
iajs-1808	116	99	submodule	submodule	NOUN
iajs-1808	116	100	𝐶	𝐶	PROPN
iajs-1808	116	101	of	of	ADP
iajs-1808	116	102	𝑀	𝑀	PROPN
iajs-1808	116	103	,	,	PUNCT
iajs-1808	116	104	there	there	PRON
iajs-1808	116	105	exists	exist	VERB
iajs-1808	116	106	a	a	DET
iajs-1808	116	107	fully	fully	ADV
iajs-1808	116	108	invariant	invariant	ADJ
iajs-1808	116	109	direct	direct	ADJ
iajs-1808	116	110	summand	summand	NOUN
iajs-1808	116	111	𝐷	𝐷	PROPN
iajs-1808	116	112	of	of	ADP
iajs-1808	116	113	𝑀	𝑀	PROPN
iajs-1808	116	114	such	such	ADJ
iajs-1808	116	115	that	that	PRON
iajs-1808	116	116	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	116	117	𝑀.	𝑀.	PROPN
iajs-1808	116	118	for	for	ADP
iajs-1808	116	119	every	every	DET
iajs-1808	116	120	t	t	NOUN
iajs-1808	116	121	-	-	PUNCT
iajs-1808	116	122	closed	close	VERB
iajs-1808	116	123	submodule	submodule	NOUN
iajs-1808	116	124	𝐶	𝐶	PROPN
iajs-1808	116	125	of	of	ADP
iajs-1808	116	126	𝑀	𝑀	PROPN
iajs-1808	116	127	,	,	PUNCT
iajs-1808	116	128	there	there	PRON
iajs-1808	116	129	exists	exist	VERB
iajs-1808	116	130	a	a	DET
iajs-1808	116	131	fully	fully	ADV
iajs-1808	116	132	invariant	invariant	ADJ
iajs-1808	116	133	direct	direct	ADJ
iajs-1808	116	134	summand	summand	NOUN
iajs-1808	116	135	𝐷	𝐷	PROPN
iajs-1808	116	136	of	of	ADP
iajs-1808	116	137	𝑀	𝑀	PROPN
iajs-1808	116	138	such	such	ADJ
iajs-1808	116	139	that	that	PRON
iajs-1808	116	140	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	116	141	𝑀.	𝑀.	PROPN
iajs-1808	116	142	then	then	ADV
iajs-1808	116	143	(	(	PUNCT
iajs-1808	116	144	1	1	NUM
iajs-1808	116	145	)	)	PUNCT
iajs-1808	116	146	,	,	PUNCT
iajs-1808	116	147	(	(	PUNCT
iajs-1808	116	148	3	3	NUM
iajs-1808	116	149	)	)	PUNCT
iajs-1808	116	150	,	,	PUNCT
iajs-1808	116	151	(	(	PUNCT
iajs-1808	116	152	4	4	NUM
iajs-1808	116	153	)	)	PUNCT
iajs-1808	116	154	and	and	CCONJ
iajs-1808	116	155	(	(	PUNCT
iajs-1808	116	156	5	5	X
iajs-1808	116	157	)	)	PUNCT
iajs-1808	116	158	are	be	AUX
iajs-1808	116	159	equivalent	equivalent	ADJ
iajs-1808	116	160	,	,	PUNCT
iajs-1808	116	161	(	(	PUNCT
iajs-1808	116	162	1	1	NUM
iajs-1808	116	163	)	)	PUNCT
iajs-1808	116	164	(2	(2	PUNCT
iajs-1808	116	165	)	)	PUNCT
iajs-1808	116	166	if	if	SCONJ
iajs-1808	116	167	is	be	AUX
iajs-1808	116	168	fully	fully	ADV
iajs-1808	116	169	invariant	invariant	ADJ
iajs-1808	116	170	of	of	ADP
iajs-1808	116	171	for	for	ADP
iajs-1808	116	172	each	each	DET
iajs-1808	116	173	fully	fully	ADV
iajs-1808	116	174	invariant	invariant	ADJ
iajs-1808	116	175	submodule	submodule	NOUN
iajs-1808	116	176	𝐿	𝐿	PROPN
iajs-1808	116	177	of	of	ADP
iajs-1808	116	178	𝑀	𝑀	PROPN
iajs-1808	116	179	containg	contae	VERB
iajs-1808	116	180	𝑍	𝑍	PROPN
iajs-1808	116	181	𝑀	𝑀	PROPN
iajs-1808	116	182	,	,	PUNCT
iajs-1808	116	183	and	and	CCONJ
iajs-1808	116	184	(	(	PUNCT
iajs-1808	116	185	2	2	X
iajs-1808	116	186	)	)	PUNCT
iajs-1808	116	187			NOUN
iajs-1808	116	188	(	(	PUNCT
iajs-1808	116	189	5	5	NUM
iajs-1808	116	190	)	)	PUNCT
iajs-1808	116	191	.	.	PUNCT
iajs-1808	117	1	proof	proof	NOUN
iajs-1808	117	2	:	:	PUNCT
iajs-1808	117	3	(	(	PUNCT
iajs-1808	117	4	1)(5	1)(5	NUM
iajs-1808	117	5	)	)	PUNCT
iajs-1808	117	6	let	let	VERB
iajs-1808	117	7	𝐶	𝐶	PROPN
iajs-1808	117	8	be	be	AUX
iajs-1808	117	9	a	a	DET
iajs-1808	117	10	t	t	NOUN
iajs-1808	117	11	-	-	PUNCT
iajs-1808	117	12	closed	close	VERB
iajs-1808	117	13	submodule	submodule	NOUN
iajs-1808	117	14	of	of	ADP
iajs-1808	117	15	𝑀.	𝑀.	PROPN
iajs-1808	117	16	by	by	ADP
iajs-1808	117	17	condition	condition	NOUN
iajs-1808	117	18	(	(	PUNCT
iajs-1808	117	19	1	1	X
iajs-1808	117	20	)	)	PUNCT
iajs-1808	117	21	there	there	PRON
iajs-1808	117	22	exists	exist	VERB
iajs-1808	117	23	a	a	DET
iajs-1808	117	24	complement	complement	NOUN
iajs-1808	117	25	𝐷	𝐷	NOUN
iajs-1808	117	26	to	to	ADP
iajs-1808	117	27	𝐶	𝐶	PROPN
iajs-1808	117	28	such	such	DET
iajs-1808	117	29	that	that	DET
iajs-1808	117	30	𝐷	𝐷	PROPN
iajs-1808	117	31	⨁	⨁	PROPN
iajs-1808	117	32	𝑀	𝑀	PROPN
iajs-1808	117	33	,	,	PUNCT
iajs-1808	117	34	𝐷	𝐷	PROPN
iajs-1808	117	35	is	be	AUX
iajs-1808	117	36	fully	fully	ADV
iajs-1808	117	37	invariant	invariant	ADJ
iajs-1808	117	38	.	.	PUNCT
iajs-1808	118	1	thus	thus	ADV
iajs-1808	118	2	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	118	3	𝑀.	𝑀.	PROPN
iajs-1808	118	4	(	(	PUNCT
iajs-1808	118	5	3)	3)	NUM
iajs-1808	118	6	(	(	PUNCT
iajs-1808	118	7	1	1	NUM
iajs-1808	118	8	)	)	PUNCT
iajs-1808	118	9	let	let	VERB
iajs-1808	118	10	𝐶	𝐶	PROPN
iajs-1808	118	11	be	be	AUX
iajs-1808	118	12	a	a	DET
iajs-1808	118	13	t	t	NOUN
iajs-1808	118	14	-	-	PUNCT
iajs-1808	118	15	closed	close	VERB
iajs-1808	118	16	submodule	submodule	NOUN
iajs-1808	118	17	of	of	ADP
iajs-1808	118	18	𝑀.	𝑀.	PROPN
iajs-1808	118	19	by	by	ADP
iajs-1808	118	20	hypothesis	hypothesis	NOUN
iajs-1808	118	21	there	there	PRON
iajs-1808	118	22	exists	exist	VERB
iajs-1808	118	23	a	a	DET
iajs-1808	118	24	fully	fully	ADV
iajs-1808	118	25	invariant	invariant	ADJ
iajs-1808	118	26	direct	direct	ADJ
iajs-1808	118	27	summand	summand	NOUN
iajs-1808	118	28	𝐷	𝐷	PROPN
iajs-1808	118	29	of	of	ADP
iajs-1808	118	30	𝑀	𝑀	PROPN
iajs-1808	118	31	such	such	ADJ
iajs-1808	118	32	that	that	SCONJ
iajs-1808	118	33	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	118	34	𝑀.	𝑀.	PROPN
iajs-1808	118	35	let	let	VERB
iajs-1808	118	36	𝐸	𝐸	PRON
iajs-1808	118	37	be	be	AUX
iajs-1808	118	38	a	a	DET
iajs-1808	118	39	complement	complement	NOUN
iajs-1808	118	40	of	of	ADP
iajs-1808	118	41	𝐶	𝐶	PROPN
iajs-1808	118	42	,	,	PUNCT
iajs-1808	118	43	then	then	ADV
iajs-1808	118	44	𝐶⋂𝐸	𝐶⋂𝐸	PROPN
iajs-1808	118	45	0	0	NUM
iajs-1808	118	46	and	and	CCONJ
iajs-1808	118	47	𝐶⨁𝐸	𝐶⨁𝐸	NOUN
iajs-1808	118	48	𝑀.	𝑀.	ADP
iajs-1808	118	49	we	we	PRON
iajs-1808	118	50	claim	claim	VERB
iajs-1808	118	51	that	that	SCONJ
iajs-1808	118	52	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	118	53	𝐶⨁𝐸.	𝐶⨁𝐸.	PROPN
iajs-1808	118	54	let	let	AUX
iajs-1808	118	55	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	118	56	⋂𝑋	⋂𝑋	NUM
iajs-1808	118	57	0	0	NUM
iajs-1808	118	58	,	,	PUNCT
iajs-1808	118	59	where	where	SCONJ
iajs-1808	118	60	𝑋	𝑋	PROPN
iajs-1808	118	61	𝐶⨁𝐸.	𝐶⨁𝐸.	PROPN
iajs-1808	118	62	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	118	63	⋂𝑋	⋂𝑋	PROPN
iajs-1808	118	64	0	0	NUM
iajs-1808	118	65	𝑍	𝑍	PROPN
iajs-1808	118	66	𝑀	𝑀	PROPN
iajs-1808	118	67	.	.	PUNCT
iajs-1808	119	1	thus	thus	ADV
iajs-1808	119	2	implies	imply	VERB
iajs-1808	119	3	𝑋	𝑋	PROPN
iajs-1808	119	4	𝑍	𝑍	PROPN
iajs-1808	119	5	𝑀	𝑀	PROPN
iajs-1808	119	6	since	since	SCONJ
iajs-1808	119	7	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	119	8	𝑀.	𝑀.	PROPN
iajs-1808	119	9	but	but	CCONJ
iajs-1808	119	10	𝑍	𝑍	PROPN
iajs-1808	119	11	𝑀	𝑀	PROPN
iajs-1808	119	12	𝐶	𝐶	PROPN
iajs-1808	119	13	(	(	PUNCT
iajs-1808	119	14	since	since	SCONJ
iajs-1808	119	15	𝐶	𝐶	PROPN
iajs-1808	119	16	is	be	AUX
iajs-1808	119	17	t	t	NOUN
iajs-1808	119	18	-	-	PUNCT
iajs-1808	119	19	closed	close	VERB
iajs-1808	119	20	)	)	PUNCT
iajs-1808	119	21	hence	hence	ADV
iajs-1808	119	22	𝑋	𝑋	PROPN
iajs-1808	119	23	𝐶.	𝐶.	PROPN
iajs-1808	119	24	it	it	PRON
iajs-1808	119	25	follows	follow	VERB
iajs-1808	119	26	that	that	PRON
iajs-1808	119	27	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	119	28	⋂𝑋	⋂𝑋	PROPN
iajs-1808	119	29	𝑋	𝑋	PROPN
iajs-1808	119	30	0	0	NUM
iajs-1808	119	31	.	.	PUNCT
iajs-1808	120	1	thus	thus	ADV
iajs-1808	120	2	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	120	3	𝐶⨁𝐸.	𝐶⨁𝐸.	PROPN
iajs-1808	120	4	it	it	PRON
iajs-1808	120	5	follows	follow	VERB
iajs-1808	120	6	that	that	SCONJ
iajs-1808	120	7	𝐷	𝐷	PROPN
iajs-1808	120	8	𝐸.	𝐸.	PROPN
iajs-1808	120	9	however,𝐷	however,𝐷	NOUN
iajs-1808	120	10	⨁	⨁	PROPN
iajs-1808	120	11	𝑀	𝑀	PROPN
iajs-1808	120	12	and	and	CCONJ
iajs-1808	120	13	so	so	ADV
iajs-1808	120	14	𝐷	𝐷	PROPN
iajs-1808	120	15	is	be	AUX
iajs-1808	120	16	closed	close	VERB
iajs-1808	120	17	in	in	ADP
iajs-1808	120	18	𝑀.	𝑀.	PROPN
iajs-1808	120	19	which	which	PRON
iajs-1808	120	20	implies	imply	VERB
iajs-1808	120	21	𝐷	𝐷	PROPN
iajs-1808	120	22	𝐸,that	𝐸,that	PUNCT
iajs-1808	120	23	is	be	AUX
iajs-1808	120	24	𝐸	𝐸	PRON
iajs-1808	120	25	a	a	DET
iajs-1808	120	26	complement	complement	NOUN
iajs-1808	120	27	of	of	ADP
iajs-1808	120	28	𝐶	𝐶	PROPN
iajs-1808	120	29	,	,	PUNCT
iajs-1808	120	30	which	which	PRON
iajs-1808	120	31	is	be	AUX
iajs-1808	120	32	a	a	DET
iajs-1808	120	33	fully	fully	ADV
iajs-1808	120	34	invariant	invariant	ADJ
iajs-1808	120	35	direct	direct	ADJ
iajs-1808	120	36	summand	summand	NOUN
iajs-1808	120	37	.	.	PUNCT
iajs-1808	121	1	thus	thus	ADV
iajs-1808	121	2	𝑀	𝑀	PROPN
iajs-1808	121	3	is	be	AUX
iajs-1808	121	4	a	a	DET
iajs-1808	121	5	strongly	strongly	ADV
iajs-1808	121	6	𝑇	𝑇	NOUN
iajs-1808	121	7	-type	-type	NOUN
iajs-1808	121	8	module	module	NOUN
iajs-1808	121	9	.	.	PUNCT
iajs-1808	122	1	(	(	PUNCT
iajs-1808	122	2	4	4	NUM
iajs-1808	122	3	)	)	PUNCT
iajs-1808	122	4			NOUN
iajs-1808	122	5	(	(	PUNCT
iajs-1808	122	6	3	3	X
iajs-1808	122	7	)	)	PUNCT
iajs-1808	122	8	let	let	VERB
iajs-1808	122	9	𝐴	𝐴	PROPN
iajs-1808	122	10	𝑀.	𝑀.	PROPN
iajs-1808	122	11	by	by	ADP
iajs-1808	122	12	[	[	X
iajs-1808	122	13	4	4	NUM
iajs-1808	122	14	,	,	PUNCT
iajs-1808	122	15	lemma	lemma	PROPN
iajs-1808	122	16	2.3	2.3	NUM
iajs-1808	122	17	]	]	PUNCT
iajs-1808	122	18	,	,	PUNCT
iajs-1808	122	19	there	there	PRON
iajs-1808	122	20	exists	exist	VERB
iajs-1808	122	21	a	a	DET
iajs-1808	122	22	t	t	NOUN
iajs-1808	122	23	-	-	PUNCT
iajs-1808	122	24	closed	close	VERB
iajs-1808	122	25	𝐶	𝐶	PROPN
iajs-1808	122	26	of	of	ADP
iajs-1808	122	27	𝐴	𝐴	PROPN
iajs-1808	122	28	such	such	ADJ
iajs-1808	122	29	that	that	SCONJ
iajs-1808	122	30	𝐴	𝐴	PROPN
iajs-1808	122	31	𝐶.	𝐶.	VERB
iajs-1808	122	32	by	by	ADP
iajs-1808	122	33	hypothesis	hypothesis	NOUN
iajs-1808	122	34	,	,	PUNCT
iajs-1808	122	35	there	there	PRON
iajs-1808	122	36	exists	exist	VERB
iajs-1808	122	37	a	a	DET
iajs-1808	122	38	fully	fully	ADV
iajs-1808	122	39	invariant	invariant	ADJ
iajs-1808	122	40	direct	direct	ADJ
iajs-1808	122	41	summand	summand	NOUN
iajs-1808	122	42	𝐷	𝐷	PROPN
iajs-1808	122	43	such	such	ADJ
iajs-1808	122	44	that	that	SCONJ
iajs-1808	122	45	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	122	46	𝑀.	𝑀.	PROPN
iajs-1808	122	47	but	but	CCONJ
iajs-1808	122	48	𝐴	𝐴	PROPN
iajs-1808	122	49	𝐶	𝐶	PROPN
iajs-1808	122	50	,	,	PUNCT
iajs-1808	122	51	we	we	PRON
iajs-1808	122	52	conclude	conclude	VERB
iajs-1808	122	53	that	that	SCONJ
iajs-1808	122	54	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	122	55	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	122	56	and	and	CCONJ
iajs-1808	122	57	hence	hence	ADV
iajs-1808	122	58	𝐴⨁𝐷	𝐴⨁𝐷	VERB
iajs-1808	122	59	𝑀.	𝑀.	PROPN
iajs-1808	122	60	(	(	PUNCT
iajs-1808	122	61	5	5	NUM
iajs-1808	122	62	)	)	PUNCT
iajs-1808	122	63	(4	(4	PUNCT
iajs-1808	122	64	)	)	PUNCT
iajs-1808	122	65	the	the	DET
iajs-1808	122	66	implication	implication	NOUN
iajs-1808	122	67	is	be	AUX
iajs-1808	122	68	clear	clear	ADJ
iajs-1808	122	69	since	since	SCONJ
iajs-1808	122	70	every	every	DET
iajs-1808	122	71	essential	essential	ADJ
iajs-1808	122	72	submodule	submodule	NOUN
iajs-1808	122	73	is	be	AUX
iajs-1808	122	74	t	t	NOUN
iajs-1808	122	75	-	-	PUNCT
iajs-1808	122	76	essential	essential	ADJ
iajs-1808	122	77	submodule	submodule	NOUN
iajs-1808	122	78	.	.	PUNCT
iajs-1808	123	1	(	(	PUNCT
iajs-1808	123	2	2	2	NUM
iajs-1808	123	3	)	)	PUNCT
iajs-1808	123	4	(5	(5	PUNCT
iajs-1808	123	5	)	)	PUNCT
iajs-1808	123	6	let	let	VERB
iajs-1808	123	7	𝐶	𝐶	PROPN
iajs-1808	123	8	be	be	AUX
iajs-1808	123	9	a	a	DET
iajs-1808	123	10	t	t	NOUN
iajs-1808	123	11	-	-	PUNCT
iajs-1808	123	12	closed	close	VERB
iajs-1808	123	13	submodule	submodule	NOUN
iajs-1808	123	14	of	of	ADP
iajs-1808	123	15	𝑀.	𝑀.	PROPN
iajs-1808	123	16	hence	hence	ADV
iajs-1808	123	17	by	by	ADP
iajs-1808	123	18	lemma	lemma	PROPN
iajs-1808	123	19	(	(	PUNCT
iajs-1808	123	20	1.2	1.2	NUM
iajs-1808	123	21	)	)	PUNCT
iajs-1808	123	22	,	,	PUNCT
iajs-1808	123	23	𝑍	𝑍	PROPN
iajs-1808	123	24	𝑀	𝑀	PROPN
iajs-1808	123	25	𝐶	𝐶	PROPN
iajs-1808	123	26	and	and	CCONJ
iajs-1808	123	27	so	so	ADV
iajs-1808	123	28	𝐶	𝐶	PROPN
iajs-1808	123	29	𝑍	𝑍	PROPN
iajs-1808	123	30	𝑀	𝑀	PROPN
iajs-1808	123	31	⨁	⨁	PROPN
iajs-1808	123	32	𝐶⋂𝑀	𝐶⋂𝑀	NUM
iajs-1808	123	33	.	.	PUNCT
iajs-1808	124	1	moreover	moreover	ADV
iajs-1808	124	2	,	,	PUNCT
iajs-1808	124	3	𝐶⋂𝑀	𝐶⋂𝑀	PRON
iajs-1808	124	4	is	be	AUX
iajs-1808	124	5	a	a	DET
iajs-1808	124	6	t	t	NOUN
iajs-1808	124	7	-	-	PUNCT
iajs-1808	124	8	closed	close	VERB
iajs-1808	124	9	submodule	submodule	NOUN
iajs-1808	124	10	of	of	ADP
iajs-1808	124	11	𝑀	𝑀	PROPN
iajs-1808	124	12	by	by	ADP
iajs-1808	124	13	[	[	X
iajs-1808	124	14	2	2	NUM
iajs-1808	124	15	,	,	PUNCT
iajs-1808	124	16	proposition	proposition	NOUN
iajs-1808	124	17	2.9	2.9	NUM
iajs-1808	124	18	]	]	PUNCT
iajs-1808	124	19	.	.	PUNCT
iajs-1808	125	1	since	since	SCONJ
iajs-1808	125	2	𝑀	𝑀	PROPN
iajs-1808	125	3	satisfies	satisfy	VERB
iajs-1808	125	4	strongly	strongly	ADV
iajs-1808	125	5	𝐶	𝐶	PROPN
iajs-1808	125	6	condition	condition	NOUN
iajs-1808	125	7	,	,	PUNCT
iajs-1808	125	8	there	there	PRON
iajs-1808	125	9	exists	exist	VERB
iajs-1808	125	10	a	a	DET
iajs-1808	125	11	fully	fully	ADV
iajs-1808	125	12	invariant	invariant	ADJ
iajs-1808	125	13	direct	direct	ADJ
iajs-1808	125	14	summand	summand	NOUN
iajs-1808	125	15	𝐷	𝐷	PROPN
iajs-1808	125	16	of	of	ADP
iajs-1808	125	17	𝑀	𝑀	PROPN
iajs-1808	125	18	such	such	ADJ
iajs-1808	125	19	that	that	SCONJ
iajs-1808	125	20	(	(	PUNCT
iajs-1808	125	21	𝐶⋂𝑀	𝐶⋂𝑀	X
iajs-1808	125	22	⨁𝐷	⨁𝐷	PUNCT
iajs-1808	125	23	𝑀	𝑀	PROPN
iajs-1808	125	24	.	.	PUNCT
iajs-1808	126	1	but	but	CCONJ
iajs-1808	126	2	𝐷	𝐷	PROPN
iajs-1808	126	3	⨁	⨁	PROPN
iajs-1808	126	4	𝑀	𝑀	PROPN
iajs-1808	126	5	and	and	CCONJ
iajs-1808	126	6	𝑀	𝑀	PROPN
iajs-1808	126	7	⨁	⨁	PROPN
iajs-1808	126	8	𝑀	𝑀	PROPN
iajs-1808	126	9	,	,	PUNCT
iajs-1808	126	10	then	then	ADV
iajs-1808	126	11	𝐷	𝐷	PROPN
iajs-1808	126	12	⨁	⨁	PROPN
iajs-1808	126	13	𝑀	𝑀	PROPN
iajs-1808	126	14	and	and	CCONJ
iajs-1808	126	15	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	126	16	𝑍	𝑍	PROPN
iajs-1808	126	17	𝑀	𝑀	PROPN
iajs-1808	126	18	⨁	⨁	PROPN
iajs-1808	126	19	𝐶⋂𝑀	𝐶⋂𝑀	NUM
iajs-1808	126	20	⨁𝐷	⨁𝐷	PUNCT
iajs-1808	126	21	𝑍	𝑍	VERB
iajs-1808	126	22	𝑀	𝑀	PROPN
iajs-1808	126	23	⨁	⨁	PROPN
iajs-1808	126	24	𝐶⋂𝑀	𝐶⋂𝑀	NUM
iajs-1808	126	25	⨁𝐷	⨁𝐷	NOUN
iajs-1808	126	26	]	]	PUNCT
iajs-1808	126	27	𝑍	𝑍	PROPN
iajs-1808	126	28	𝑀	𝑀	PROPN
iajs-1808	126	29	⨁𝑀	⨁𝑀	NOUN
iajs-1808	126	30	𝑀.hence	𝑀.hence	PROPN
iajs-1808	126	31	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	126	32	𝑀	𝑀	PROPN
iajs-1808	126	33	,	,	PUNCT
iajs-1808	126	34	but	but	CCONJ
iajs-1808	126	35	𝐷	𝐷	NOUN
iajs-1808	126	36	is	be	AUX
iajs-1808	126	37	fully	fully	ADV
iajs-1808	126	38	invariant	invariant	ADJ
iajs-1808	126	39	in	in	ADP
iajs-1808	126	40	𝑀	𝑀	PROPN
iajs-1808	126	41	and	and	CCONJ
iajs-1808	126	42	𝑀	𝑀	PROPN
iajs-1808	126	43	is	be	AUX
iajs-1808	126	44	fully	fully	ADV
iajs-1808	126	45	invariant	invariant	ADJ
iajs-1808	126	46	in	in	ADP
iajs-1808	126	47	𝑀.	𝑀.	PROPN
iajs-1808	127	1	hence	hence	ADV
iajs-1808	127	2	𝐷	𝐷	NOUN
iajs-1808	127	3	is	be	AUX
iajs-1808	127	4	fully	fully	ADV
iajs-1808	127	5	invariant	invariant	ADJ
iajs-1808	127	6	in	in	ADP
iajs-1808	127	7	𝑀.	𝑀.	PROPN
iajs-1808	127	8	(	(	PUNCT
iajs-1808	127	9	1	1	NUM
iajs-1808	127	10	)	)	PUNCT
iajs-1808	127	11	(2	(2	NUM
iajs-1808	127	12	)	)	PUNCT
iajs-1808	127	13	since	since	SCONJ
iajs-1808	127	14	𝑀	𝑀	PROPN
iajs-1808	127	15	is	be	AUX
iajs-1808	127	16	strongly	strongly	ADV
iajs-1808	127	17	𝑇	𝑇	PROPN
iajs-1808	127	18	-type	-type	NOUN
iajs-1808	127	19	module	module	NOUN
iajs-1808	127	20	and	and	CCONJ
iajs-1808	127	21	𝑍	𝑍	PROPN
iajs-1808	127	22	𝑀	𝑀	PROPN
iajs-1808	127	23	is	be	AUX
iajs-1808	127	24	a	a	DET
iajs-1808	127	25	t	t	NOUN
iajs-1808	127	26	-	-	PUNCT
iajs-1808	127	27	closed	close	VERB
iajs-1808	127	28	submodule	submodule	NOUN
iajs-1808	127	29	of	of	ADP
iajs-1808	127	30	𝑀	𝑀	PROPN
iajs-1808	127	31	,	,	PUNCT
iajs-1808	127	32	there	there	PRON
iajs-1808	127	33	exists	exist	VERB
iajs-1808	127	34	a	a	DET
iajs-1808	127	35	complement	complement	NOUN
iajs-1808	127	36	𝑀	𝑀	PROPN
iajs-1808	127	37	to	to	ADP
iajs-1808	127	38	𝑍	𝑍	PROPN
iajs-1808	127	39	𝑀	𝑀	PROPN
iajs-1808	127	40	which	which	PRON
iajs-1808	127	41	is	be	AUX
iajs-1808	127	42	a	a	DET
iajs-1808	127	43	fully	fully	ADV
iajs-1808	127	44	invariant	invariant	ADJ
iajs-1808	127	45	direct	direct	ADJ
iajs-1808	127	46	summand	summand	NOUN
iajs-1808	127	47	,	,	PUNCT
iajs-1808	127	48	say	say	VERB
iajs-1808	127	49	𝑀	𝑀	PROPN
iajs-1808	127	50	𝐿⨁𝑀	𝐿⨁𝑀	PROPN
iajs-1808	127	51	.	.	PUNCT
iajs-1808	128	1	since	since	SCONJ
iajs-1808	128	2	𝑀	𝑀	PROPN
iajs-1808	128	3	is	be	AUX
iajs-1808	128	4	nonsingular	nonsingular	ADJ
iajs-1808	128	5	,	,	PUNCT
iajs-1808	128	6	we	we	PRON
iajs-1808	128	7	have	have	VERB
iajs-1808	128	8	𝑍	𝑍	NOUN
iajs-1808	128	9	𝑀	𝑀	PROPN
iajs-1808	128	10	𝑍	𝑍	PROPN
iajs-1808	128	11	𝐿	𝐿	PROPN
iajs-1808	128	12	.	.	PUNCT
iajs-1808	129	1	but	but	CCONJ
iajs-1808	129	2	𝑍	𝑍	VERB
iajs-1808	129	3	𝑀	𝑀	PROPN
iajs-1808	129	4	⨁𝑀	⨁𝑀	PROPN
iajs-1808	129	5	𝑀	𝑀	PROPN
iajs-1808	129	6	since	since	SCONJ
iajs-1808	129	7	𝑀	𝑀	PROPN
iajs-1808	129	8	is	be	AUX
iajs-1808	129	9	complement	complement	NOUN
iajs-1808	129	10	to	to	ADP
iajs-1808	129	11	𝑍	𝑍	PROPN
iajs-1808	129	12	𝑀	𝑀	PROPN
iajs-1808	129	13	,	,	PUNCT
iajs-1808	129	14	so	so	ADV
iajs-1808	129	15	by	by	ADP
iajs-1808	129	16	proposition	proposition	NOUN
iajs-1808	129	17	(	(	PUNCT
iajs-1808	129	18	1.1	1.1	NUM
iajs-1808	129	19	)	)	PUNCT
iajs-1808	129	20	is	be	AUX
iajs-1808	129	21	𝑍	𝑍	NOUN
iajs-1808	129	22	-torsion	-torsion	NOUN
iajs-1808	129	23	,	,	PUNCT
iajs-1808	129	24	thus	thus	ADV
iajs-1808	129	25	𝐿	𝐿	PROPN
iajs-1808	129	26	is	be	AUX
iajs-1808	129	27	𝑍	𝑍	PROPN
iajs-1808	129	28	-torsion	-torsion	NOUN
iajs-1808	129	29	(	(	PUNCT
iajs-1808	129	30	since	since	SCONJ
iajs-1808	129	31	𝐿	𝐿	PROPN
iajs-1808	129	32	≃	≃	NOUN
iajs-1808	129	33	.	.	PUNCT
iajs-1808	130	1	so	so	ADV
iajs-1808	130	2	𝐿	𝐿	PROPN
iajs-1808	130	3	𝑍	𝑍	PROPN
iajs-1808	130	4	𝐿	𝐿	NOUN
iajs-1808	130	5	𝑍	𝑍	PROPN
iajs-1808	130	6	𝑀	𝑀	PROPN
iajs-1808	130	7	and	and	CCONJ
iajs-1808	130	8	hence	hence	ADV
iajs-1808	130	9	𝐿	𝐿	PROPN
iajs-1808	130	10	𝑍	𝑍	PROPN
iajs-1808	130	11	𝑀	𝑀	PROPN
iajs-1808	130	12	.	.	PUNCT
iajs-1808	131	1	therefore	therefore	ADV
iajs-1808	131	2	𝑀	𝑀	PROPN
iajs-1808	131	3	𝑍	𝑍	VERB
iajs-1808	131	4	𝑀	𝑀	PROPN
iajs-1808	131	5	⨁𝑀	⨁𝑀	NOUN
iajs-1808	131	6	.	.	PUNCT
iajs-1808	132	1	now	now	ADV
iajs-1808	132	2	to	to	PART
iajs-1808	132	3	show	show	VERB
iajs-1808	132	4	that	that	SCONJ
iajs-1808	132	5	𝑀	𝑀	PROPN
iajs-1808	132	6	≃	≃	NOUN
iajs-1808	132	7	≃	≃	NOUN
iajs-1808	132	8	𝑀	𝑀	PROPN
iajs-1808	132	9	satisfies	satisfy	VERB
iajs-1808	132	10	strongly	strongly	ADV
iajs-1808	132	11	𝐶	𝐶	PROPN
iajs-1808	132	12	condition	condition	NOUN
iajs-1808	132	13	.	.	PUNCT
iajs-1808	133	1	let	let	VERB
iajs-1808	133	2	�	�	PRON
iajs-1808	133	3	̅	̅	VERB
iajs-1808	133	4	�	�	NOUN
iajs-1808	133	5	=	=	PUNCT
iajs-1808	133	6	be	be	VERB
iajs-1808	133	7	a	a	DET
iajs-1808	133	8	closed	closed	ADJ
iajs-1808	133	9	submodule	submodule	NOUN
iajs-1808	133	10	of	of	ADP
iajs-1808	133	11	𝑀	𝑀	PROPN
iajs-1808	133	12	so	so	SCONJ
iajs-1808	133	13	�	�	PROPN
iajs-1808	133	14	̅	̅	NOUN
iajs-1808	133	15	�	�	PROPN
iajs-1808	133	16	is	be	AUX
iajs-1808	133	17	t	t	NOUN
iajs-1808	133	18	-	-	PUNCT
iajs-1808	133	19	closed	close	VERB
iajs-1808	133	20	in	in	ADP
iajs-1808	133	21	𝑀	𝑀	PROPN
iajs-1808	133	22	and	and	CCONJ
iajs-1808	133	23	𝐶	𝐶	PROPN
iajs-1808	133	24	is	be	AUX
iajs-1808	133	25	t	t	NOUN
iajs-1808	133	26	-	-	PUNCT
iajs-1808	133	27	closed	close	VERB
iajs-1808	133	28	submodule	submodule	NOUN
iajs-1808	133	29	of	of	ADP
iajs-1808	133	30	𝑀	𝑀	PROPN
iajs-1808	133	31	by	by	ADP
iajs-1808	133	32	lemma	lemma	PROPN
iajs-1808	133	33	1.2(3).but	1.2(3).but	PROPN
iajs-1808	133	34	𝑀	𝑀	PROPN
iajs-1808	133	35	is	be	AUX
iajs-1808	133	36	a	a	DET
iajs-1808	133	37	strongly	strongly	ADV
iajs-1808	133	38	𝑇	𝑇	PROPN
iajs-1808	133	39	-type	-type	NOUN
iajs-1808	133	40	,	,	PUNCT
iajs-1808	133	41	so	so	SCONJ
iajs-1808	133	42	there	there	PRON
iajs-1808	133	43	exists	exist	VERB
iajs-1808	133	44	a	a	DET
iajs-1808	133	45	complement	complement	NOUN
iajs-1808	133	46	𝐷	𝐷	NOUN
iajs-1808	133	47	of	of	ADP
iajs-1808	133	48	𝐶	𝐶	PROPN
iajs-1808	133	49	in	in	ADP
iajs-1808	133	50	𝑀	𝑀	PROPN
iajs-1808	133	51	which	which	PRON
iajs-1808	133	52	is	be	AUX
iajs-1808	133	53	a	a	DET
iajs-1808	133	54	fully	fully	ADV
iajs-1808	133	55	invariant	invariant	ADJ
iajs-1808	133	56	direct	direct	ADJ
iajs-1808	133	57	summand	summand	NOUN
iajs-1808	133	58	of	of	ADP
iajs-1808	133	59	𝑀.	𝑀.	PROPN
iajs-1808	133	60	say	say	VERB
iajs-1808	133	61	𝑀	𝑀	PROPN
iajs-1808	133	62	𝐷⨁𝐷	𝐷⨁𝐷	VERB
iajs-1808	133	63	for	for	ADP
iajs-1808	133	64	some𝐷	some𝐷	PROPN
iajs-1808	133	65	𝑀.	𝑀.	PROPN
iajs-1808	133	66	since	since	SCONJ
iajs-1808	133	67	𝑍	𝑍	PROPN
iajs-1808	133	68	𝑀	𝑀	PROPN
iajs-1808	133	69	𝑍	𝑍	PROPN
iajs-1808	133	70	𝐷	𝐷	PROPN
iajs-1808	133	71	⨁𝑍	⨁𝑍	NOUN
iajs-1808	133	72	𝐷	𝐷	NOUN
iajs-1808	133	73	we	we	PRON
iajs-1808	133	74	get𝑀	get𝑀	VERB
iajs-1808	133	75	⨁	⨁	PROPN
iajs-1808	133	76	⨁	⨁	PROPN
iajs-1808	133	77	≃	≃	PROPN
iajs-1808	133	78	⨁	⨁	PROPN
iajs-1808	133	79	𝐷⨁𝐷	𝐷⨁𝐷	PROPN
iajs-1808	133	80	.	.	PUNCT
iajs-1808	134	1	cleary	cleary	PROPN
iajs-1808	134	2	𝐷	𝐷	PROPN
iajs-1808	134	3	∩	∩	PROPN
iajs-1808	134	4	𝐷	𝐷	NOUN
iajs-1808	134	5	=	=	NOUN
iajs-1808	134	6	0	0	NUM
iajs-1808	134	7	and	and	CCONJ
iajs-1808	134	8	�	�	PROPN
iajs-1808	134	9	̅	̅	NOUN
iajs-1808	134	10	�	�	NOUN
iajs-1808	134	11	⨁𝐷	⨁𝐷	PUNCT
iajs-1808	134	12	𝑀.	𝑀.	PROPN
iajs-1808	134	13	but	but	CCONJ
iajs-1808	134	14	𝐷	𝐷	PROPN
iajs-1808	134	15	,	,	PUNCT
iajs-1808	134	16	𝑍	𝑍	PROPN
iajs-1808	134	17	𝑀	𝑀	PROPN
iajs-1808	134	18	are	be	AUX
iajs-1808	134	19	fully	fully	ADV
iajs-1808	134	20	invariant	invariant	ADJ
iajs-1808	134	21	in	in	ADP
iajs-1808	134	22	𝑀	𝑀	PROPN
iajs-1808	134	23	and	and	CCONJ
iajs-1808	134	24	by	by	ADP
iajs-1808	134	25	hypothesis	hypothesis	NOUN
iajs-1808	134	26	𝐷	𝐷	NOUN
iajs-1808	134	27	is	be	AUX
iajs-1808	134	28	fully	fully	ADV
iajs-1808	134	29	invariant	invariant	ADJ
iajs-1808	134	30	in	in	ADP
iajs-1808	134	31	𝑀.	𝑀.	PROPN
iajs-1808	134	32	□	□	PUNCT
iajs-1808	134	33	ihsciconf	ihsciconf	PROPN
iajs-1808	134	34	2017	2017	NUM
iajs-1808	134	35	special	special	ADJ
iajs-1808	134	36	issue	issue	NOUN
iajs-1808	134	37	ibn	ibn	PROPN
iajs-1808	134	38	al	al	PROPN
iajs-1808	134	39	-	-	PUNCT
iajs-1808	134	40	haitham	haitham	PROPN
iajs-1808	134	41	journal	journal	PROPN
iajs-1808	134	42	for	for	ADP
iajs-1808	134	43	pure	pure	ADJ
iajs-1808	134	44	and	and	CCONJ
iajs-1808	134	45	applied	apply	VERB
iajs-1808	134	46	science	science	NOUN
iajs-1808	134	47	https://doi.org/	https://doi.org/	NOUN
iajs-1808	134	48	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	134	49	for	for	ADP
iajs-1808	134	50	more	more	ADJ
iajs-1808	134	51	information	information	NOUN
iajs-1808	134	52	about	about	ADP
iajs-1808	134	53	the	the	DET
iajs-1808	134	54	conference	conference	NOUN
iajs-1808	134	55	please	please	INTJ
iajs-1808	134	56	visit	visit	VERB
iajs-1808	134	57	the	the	DET
iajs-1808	134	58	websites	website	NOUN
iajs-1808	134	59	:	:	PUNCT
iajs-1808	134	60	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	AUX
iajs-1808	134	61	  	  	SPACE
iajs-1808	134	62	www.ihsciconf.org	www.ihsciconf.org	ADV
iajs-1808	134	63	   	   	SPACE
iajs-1808	134	64	mathematics|359	mathematics|359	PROPN
iajs-1808	134	65	    	    	SPACE
iajs-1808	134	66	recall	recall	VERB
iajs-1808	134	67	that	that	PRON
iajs-1808	134	68	.	.	PUNCT
iajs-1808	135	1	a	a	NUM
iajs-1808	135	2	submodule	submodule	NOUN
iajs-1808	135	3	𝑁	𝑁	PROPN
iajs-1808	135	4	of	of	ADP
iajs-1808	135	5	𝑅-module	𝑅-module	PROPN
iajs-1808	135	6	𝑀	𝑀	PROPN
iajs-1808	135	7	is	be	AUX
iajs-1808	135	8	called	call	VERB
iajs-1808	135	9	stable	stable	ADJ
iajs-1808	135	10	,	,	PUNCT
iajs-1808	135	11	if	if	SCONJ
iajs-1808	135	12	𝑓	𝑓	DET
iajs-1808	135	13	𝑁	𝑁	PROPN
iajs-1808	135	14	𝑁	𝑁	PROPN
iajs-1808	135	15	for	for	ADP
iajs-1808	135	16	each	each	DET
iajs-1808	135	17	𝑅homomorphism	𝑅homomorphism	PROPN
iajs-1808	135	18	𝑓	𝑓	X
iajs-1808	135	19	:	:	PUNCT
iajs-1808	135	20	𝑁𝑀	𝑁𝑀	NUM
iajs-1808	135	21	.	.	PUNCT
iajs-1808	136	1	an	an	DET
iajs-1808	136	2	𝑅-module	𝑅-module	PROPN
iajs-1808	136	3	𝑀	𝑀	PROPN
iajs-1808	136	4	is	be	AUX
iajs-1808	136	5	fully	fully	ADV
iajs-1808	136	6	stable	stable	ADJ
iajs-1808	136	7	if	if	SCONJ
iajs-1808	136	8	every	every	DET
iajs-1808	136	9	submodule	submodule	NOUN
iajs-1808	136	10	of	of	ADP
iajs-1808	136	11	𝑀	𝑀	PROPN
iajs-1808	136	12	is	be	AUX
iajs-1808	136	13	stable.[1	stable.[1	PROPN
iajs-1808	136	14	]	]	X
iajs-1808	136	15	remarks	remark	NOUN
iajs-1808	136	16	(	(	PUNCT
iajs-1808	136	17	3.10	3.10	NUM
iajs-1808	136	18	):	):	PUNCT
iajs-1808	136	19	if	if	SCONJ
iajs-1808	136	20	an	an	DET
iajs-1808	136	21	𝑅-module	𝑅-module	PROPN
iajs-1808	136	22	𝑀	𝑀	PROPN
iajs-1808	136	23	is	be	AUX
iajs-1808	136	24	fully	fully	ADV
iajs-1808	136	25	stable	stable	ADJ
iajs-1808	136	26	(	(	PUNCT
iajs-1808	136	27	where	where	SCONJ
iajs-1808	136	28	an	an	DET
iajs-1808	136	29	𝑅-module	𝑅-module	PROPN
iajs-1808	136	30	𝑀	𝑀	PROPN
iajs-1808	136	31	is	be	AUX
iajs-1808	136	32	fully	fully	ADV
iajs-1808	136	33	stable	stable	ADJ
iajs-1808	136	34	if	if	SCONJ
iajs-1808	136	35	every	every	DET
iajs-1808	136	36	submodule	submodule	NOUN
iajs-1808	136	37	of	of	ADP
iajs-1808	136	38	𝑀	𝑀	PROPN
iajs-1808	136	39	is	be	AUX
iajs-1808	136	40	stable	stable	ADJ
iajs-1808	136	41	)	)	PUNCT
iajs-1808	136	42	and	and	CCONJ
iajs-1808	136	43	semisimple	semisimple	NOUN
iajs-1808	136	44	,	,	PUNCT
iajs-1808	136	45	then	then	ADV
iajs-1808	136	46	𝑀	𝑀	PROPN
iajs-1808	136	47	is	be	AUX
iajs-1808	136	48	strongly	strongly	ADV
iajs-1808	136	49	𝐶	𝐶	PROPN
iajs-1808	136	50	-condition	-condition	NOUN
iajs-1808	136	51	module	module	NOUN
iajs-1808	136	52	.	.	PUNCT
iajs-1808	137	1	proof	proof	NOUN
iajs-1808	137	2	:	:	PUNCT
iajs-1808	137	3	let	let	VERB
iajs-1808	137	4	𝑁	𝑁	PROPN
iajs-1808	137	5	𝑀	𝑀	PROPN
iajs-1808	137	6	,	,	PUNCT
iajs-1808	137	7	then	then	ADV
iajs-1808	137	8	𝑁	𝑁	PROPN
iajs-1808	137	9	⨁	⨁	PROPN
iajs-1808	137	10	𝑀	𝑀	PROPN
iajs-1808	137	11	,	,	PUNCT
iajs-1808	137	12	and	and	CCONJ
iajs-1808	137	13	so	so	ADV
iajs-1808	137	14	there	there	PRON
iajs-1808	137	15	exists	exist	VERB
iajs-1808	137	16	𝑊	𝑊	PROPN
iajs-1808	137	17	𝑀	𝑀	PROPN
iajs-1808	137	18	such	such	ADJ
iajs-1808	137	19	that	that	SCONJ
iajs-1808	137	20	𝑁⨁𝑊	𝑁⨁𝑊	PROPN
iajs-1808	137	21	𝑀	𝑀	PROPN
iajs-1808	137	22	,	,	PUNCT
iajs-1808	137	23	hence	hence	ADV
iajs-1808	137	24	𝑊	𝑊	PROPN
iajs-1808	137	25	is	be	AUX
iajs-1808	137	26	a	a	DET
iajs-1808	137	27	complement	complement	NOUN
iajs-1808	137	28	of	of	ADP
iajs-1808	137	29	𝑁.	𝑁.	PROPN
iajs-1808	137	30	but	but	CCONJ
iajs-1808	137	31	𝑀	𝑀	PROPN
iajs-1808	137	32	is	be	AUX
iajs-1808	137	33	fully	fully	ADV
iajs-1808	137	34	stable	stable	ADJ
iajs-1808	137	35	,	,	PUNCT
iajs-1808	137	36	so	so	ADV
iajs-1808	137	37	𝑊	𝑊	PROPN
iajs-1808	137	38	is	be	AUX
iajs-1808	137	39	a	a	DET
iajs-1808	137	40	fully	fully	ADV
iajs-1808	137	41	invariant	invariant	ADJ
iajs-1808	137	42	,	,	PUNCT
iajs-1808	137	43	moreover	moreover	ADV
iajs-1808	137	44	𝑊	𝑊	PROPN
iajs-1808	137	45	⨁	⨁	PROPN
iajs-1808	137	46	𝑀.	𝑀.	PROPN
iajs-1808	137	47	thus	thus	ADV
iajs-1808	137	48	𝑀	𝑀	PROPN
iajs-1808	137	49	is	be	AUX
iajs-1808	137	50	strongly	strongly	ADV
iajs-1808	137	51	𝐶	𝐶	PROPN
iajs-1808	137	52	-condition	-condition	NOUN
iajs-1808	137	53	module	module	NOUN
iajs-1808	137	54	.	.	PUNCT
iajs-1808	138	1	□	□	PUNCT
iajs-1808	138	2	let	let	VERB
iajs-1808	138	3	𝑀	𝑀	PRON
iajs-1808	138	4	be	be	AUX
iajs-1808	138	5	a	a	DET
iajs-1808	138	6	strongly	strongly	ADV
iajs-1808	138	7	𝑇	𝑇	NOUN
iajs-1808	138	8	-type	-type	NOUN
iajs-1808	138	9	module	module	NOUN
iajs-1808	138	10	.	.	PUNCT
iajs-1808	139	1	then	then	ADV
iajs-1808	139	2	𝑀	𝑀	PROPN
iajs-1808	139	3	is	be	AUX
iajs-1808	139	4	strongly	strongly	ADV
iajs-1808	139	5	-	-	PUNCT
iajs-1808	139	6	fi	fi	NOUN
iajs-1808	139	7	-	-	PUNCT
iajs-1808	139	8	t	t	NOUN
iajs-1808	139	9	-	-	PUNCT
iajs-1808	139	10	semisimple	semisimple	NOUN
iajs-1808	139	11	if	if	SCONJ
iajs-1808	140	1	and	and	CCONJ
iajs-1808	140	2	only	only	ADV
iajs-1808	140	3	if	if	SCONJ
iajs-1808	140	4	every	every	DET
iajs-1808	140	5	fully	fully	ADV
iajs-1808	140	6	invariant	invariant	ADJ
iajs-1808	140	7	t	t	PROPN
iajs-1808	140	8	-	-	PUNCT
iajs-1808	140	9	closed	close	VERB
iajs-1808	140	10	submodule	submodule	NOUN
iajs-1808	140	11	of	of	ADP
iajs-1808	140	12	𝑀	𝑀	PROPN
iajs-1808	140	13	is	be	AUX
iajs-1808	140	14	strongly	strongly	ADV
iajs-1808	140	15	-	-	PUNCT
iajs-1808	140	16	fi	fi	NOUN
iajs-1808	140	17	-	-	PUNCT
iajs-1808	140	18	t	t	NOUN
iajs-1808	140	19	-	-	PUNCT
iajs-1808	140	20	semisimple	semisimple	NOUN
iajs-1808	140	21	.	.	PUNCT
iajs-1808	141	1	proof	proof	NOUN
iajs-1808	141	2	:	:	PUNCT
iajs-1808	141	3	since	since	SCONJ
iajs-1808	141	4	𝑀	𝑀	PROPN
iajs-1808	141	5	is	be	AUX
iajs-1808	141	6	strongly	strongly	ADV
iajs-1808	141	7	𝑇	𝑇	PROPN
iajs-1808	141	8	-type	-type	NOUN
iajs-1808	141	9	module	module	NOUN
iajs-1808	141	10	and	and	CCONJ
iajs-1808	141	11	𝑍	𝑍	PROPN
iajs-1808	141	12	𝑀	𝑀	PROPN
iajs-1808	141	13	is	be	AUX
iajs-1808	141	14	t	t	NOUN
iajs-1808	141	15	-	-	PUNCT
iajs-1808	141	16	closed	closed	ADJ
iajs-1808	141	17	,	,	PUNCT
iajs-1808	141	18	then	then	ADV
iajs-1808	141	19	there	there	PRON
iajs-1808	141	20	exists	exist	VERB
iajs-1808	141	21	a	a	DET
iajs-1808	141	22	𝑍	𝑍	NOUN
iajs-1808	141	23	𝑀	𝑀	PROPN
iajs-1808	141	24	has	have	VERB
iajs-1808	141	25	a	a	DET
iajs-1808	141	26	complement	complement	NOUN
iajs-1808	141	27	which	which	PRON
iajs-1808	141	28	is	be	AUX
iajs-1808	141	29	a	a	DET
iajs-1808	141	30	fully	fully	ADV
iajs-1808	141	31	invariant	invariant	ADJ
iajs-1808	141	32	direct	direct	ADJ
iajs-1808	141	33	summand	summand	NOUN
iajs-1808	141	34	,	,	PUNCT
iajs-1808	141	35	that	that	PRON
iajs-1808	141	36	is	be	AUX
iajs-1808	141	37	𝑍	𝑍	PROPN
iajs-1808	141	38	𝑀	𝑀	PROPN
iajs-1808	141	39	has	have	VERB
iajs-1808	141	40	a	a	DET
iajs-1808	141	41	complement	complement	NOUN
iajs-1808	141	42	which	which	PRON
iajs-1808	141	43	is	be	AUX
iajs-1808	141	44	stable	stable	ADJ
iajs-1808	141	45	direct	direct	ADJ
iajs-1808	141	46	summand	summand	NOUN
iajs-1808	141	47	(	(	PUNCT
iajs-1808	141	48	so	so	ADV
iajs-1808	141	49	condition	condition	NOUN
iajs-1808	141	50	∗	∗	NOUN
iajs-1808	141	51	hold	hold	NOUN
iajs-1808	141	52	)	)	PUNCT
iajs-1808	141	53	.	.	PUNCT
iajs-1808	142	1	then	then	ADV
iajs-1808	142	2	by	by	ADP
iajs-1808	142	3	corollary	corollary	ADJ
iajs-1808	142	4	1.6	1.6	NUM
iajs-1808	142	5	every	every	DET
iajs-1808	142	6	fully	fully	ADV
iajs-1808	142	7	invariant	invariant	ADJ
iajs-1808	142	8	submodule	submodule	NOUN
iajs-1808	142	9	𝑁	𝑁	PROPN
iajs-1808	142	10	of	of	ADP
iajs-1808	142	11	𝑀	𝑀	PROPN
iajs-1808	142	12	,	,	PUNCT
iajs-1808	142	13	𝑁	𝑁	PROPN
iajs-1808	142	14	⊇	⊇	NOUN
iajs-1808	142	15	𝑍	𝑍	PROPN
iajs-1808	142	16	𝑀	𝑀	PROPN
iajs-1808	142	17	is	be	AUX
iajs-1808	142	18	strongly	strongly	ADV
iajs-1808	142	19	fi	fi	ADJ
iajs-1808	142	20	-	-	ADJ
iajs-1808	142	21	t	t	NOUN
iajs-1808	142	22	-	-	PUNCT
iajs-1808	142	23	semisimple	semisimple	NOUN
iajs-1808	142	24	.	.	PUNCT
iajs-1808	143	1	thus	thus	ADV
iajs-1808	143	2	every	every	DET
iajs-1808	143	3	fully	fully	ADV
iajs-1808	143	4	invariant	invariant	ADJ
iajs-1808	143	5	t	t	PROPN
iajs-1808	143	6	-	-	PUNCT
iajs-1808	143	7	closed	close	VERB
iajs-1808	143	8	submodule	submodule	NOUN
iajs-1808	143	9	of	of	ADP
iajs-1808	143	10	𝑀	𝑀	PROPN
iajs-1808	143	11	is	be	AUX
iajs-1808	143	12	strongly	strongly	ADV
iajs-1808	143	13	fi	fi	ADJ
iajs-1808	143	14	-	-	ADJ
iajs-1808	143	15	t	t	NOUN
iajs-1808	143	16	-	-	PUNCT
iajs-1808	143	17	semisimple	semisimple	NOUN
iajs-1808	143	18	.	.	PUNCT
iajs-1808	144	1	□	□	PUNCT
iajs-1808	144	2	let	let	VERB
iajs-1808	144	3	𝑀	𝑀	PRON
iajs-1808	144	4	be	be	AUX
iajs-1808	144	5	a	a	DET
iajs-1808	144	6	strongly	strongly	ADV
iajs-1808	144	7	𝑇	𝑇	NOUN
iajs-1808	144	8	-type	-type	NOUN
iajs-1808	144	9	module	module	NOUN
iajs-1808	144	10	.	.	PUNCT
iajs-1808	145	1	if	if	SCONJ
iajs-1808	145	2	𝑀	𝑀	PROPN
iajs-1808	145	3	strongly	strongly	ADV
iajs-1808	145	4	fi	fi	NOUN
iajs-1808	145	5	-	-	ADJ
iajs-1808	145	6	t	t	NOUN
iajs-1808	145	7	-	-	PUNCT
iajs-1808	145	8	semisimple	semisimple	NOUN
iajs-1808	145	9	,	,	PUNCT
iajs-1808	145	10	then	then	ADV
iajs-1808	145	11	is	be	AUX
iajs-1808	145	12	fisemisimple	fisemisimple	ADJ
iajs-1808	145	13	and	and	CCONJ
iajs-1808	145	14	hence	hence	ADV
iajs-1808	145	15	it	it	PRON
iajs-1808	145	16	is	be	AUX
iajs-1808	145	17	strongly	strongly	ADV
iajs-1808	145	18	fi	fi	NOUN
iajs-1808	145	19	-	-	NOUN
iajs-1808	145	20	semisimple	semisimple	NOUN
iajs-1808	145	21	.	.	PUNCT
iajs-1808	146	1	proof	proof	NOUN
iajs-1808	146	2	:	:	PUNCT
iajs-1808	146	3	since	since	SCONJ
iajs-1808	146	4	𝑀	𝑀	PROPN
iajs-1808	146	5	is	be	AUX
iajs-1808	146	6	strongly	strongly	ADV
iajs-1808	146	7	𝑇	𝑇	PROPN
iajs-1808	146	8	-type	-type	NOUN
iajs-1808	146	9	and	and	CCONJ
iajs-1808	146	10	𝑍	𝑍	PROPN
iajs-1808	146	11	𝑀	𝑀	PROPN
iajs-1808	146	12	is	be	AUX
iajs-1808	146	13	t	t	NOUN
iajs-1808	146	14	-	-	PUNCT
iajs-1808	146	15	closed	closed	ADJ
iajs-1808	146	16	,	,	PUNCT
iajs-1808	146	17	then	then	ADV
iajs-1808	146	18	there	there	PRON
iajs-1808	146	19	exists	exist	VERB
iajs-1808	146	20	a	a	DET
iajs-1808	146	21	complement	complement	NOUN
iajs-1808	146	22	of	of	ADP
iajs-1808	146	23	𝑍	𝑍	PROPN
iajs-1808	146	24	𝑀	𝑀	PROPN
iajs-1808	146	25	which	which	PRON
iajs-1808	146	26	is	be	AUX
iajs-1808	146	27	a	a	DET
iajs-1808	146	28	fully	fully	ADV
iajs-1808	146	29	invariant	invariant	ADJ
iajs-1808	146	30	direct	direct	ADJ
iajs-1808	146	31	summand	summand	NOUN
iajs-1808	146	32	.	.	PUNCT
iajs-1808	147	1	thus	thus	ADV
iajs-1808	147	2	(	(	PUNCT
iajs-1808	147	3	condition	condition	NOUN
iajs-1808	147	4	(	(	PUNCT
iajs-1808	147	5	∗	∗	NOUN
iajs-1808	147	6	hold	hold	NOUN
iajs-1808	147	7	)	)	PUNCT
iajs-1808	147	8	.	.	PUNCT
iajs-1808	148	1	hence	hence	ADV
iajs-1808	148	2	the	the	DET
iajs-1808	148	3	result	result	NOUN
iajs-1808	148	4	is	be	AUX
iajs-1808	148	5	followed	follow	VERB
iajs-1808	148	6	by	by	ADP
iajs-1808	148	7	proposition	proposition	NOUN
iajs-1808	148	8	1.5	1.5	NUM
iajs-1808	148	9	.	.	PUNCT
iajs-1808	149	1	□	□	PUNCT
iajs-1808	149	2	proposition	proposition	NOUN
iajs-1808	149	3	(	(	PUNCT
iajs-1808	149	4	3.11	3.11	NUM
iajs-1808	149	5	):	):	PUNCT
iajs-1808	149	6	if	if	SCONJ
iajs-1808	149	7	an	an	DET
iajs-1808	149	8	𝑅-module	𝑅-module	PROPN
iajs-1808	149	9	𝑀	𝑀	PROPN
iajs-1808	149	10	strongly	strongly	ADV
iajs-1808	149	11	t	t	NOUN
iajs-1808	149	12	-	-	PUNCT
iajs-1808	149	13	semisimple	semisimple	NOUN
iajs-1808	149	14	,	,	PUNCT
iajs-1808	149	15	then	then	ADV
iajs-1808	149	16	𝑀	𝑀	PROPN
iajs-1808	149	17	strongly	strongly	ADV
iajs-1808	149	18	𝑇	𝑇	PROPN
iajs-1808	149	19	-type	-type	NOUN
iajs-1808	149	20	module	module	NOUN
iajs-1808	149	21	.	.	PUNCT
iajs-1808	150	1	proof	proof	NOUN
iajs-1808	150	2	:	:	PUNCT
iajs-1808	150	3	by	by	ADP
iajs-1808	150	4	theorem	theorem	NOUN
iajs-1808	150	5	1.3	1.3	NUM
iajs-1808	150	6	,	,	PUNCT
iajs-1808	150	7	𝑀	𝑀	PROPN
iajs-1808	150	8	𝑍	𝑍	VERB
iajs-1808	150	9	𝑀	𝑀	PROPN
iajs-1808	150	10	⨁𝑀	⨁𝑀	NOUN
iajs-1808	150	11	,	,	PUNCT
iajs-1808	150	12	where	where	SCONJ
iajs-1808	150	13	𝑀	𝑀	PROPN
iajs-1808	150	14	is	be	AUX
iajs-1808	150	15	nonsingular	nonsingular	ADJ
iajs-1808	150	16	semisimple	semisimple	NOUN
iajs-1808	150	17	fully	fully	ADV
iajs-1808	150	18	stable	stable	ADJ
iajs-1808	150	19	and	and	CCONJ
iajs-1808	150	20	𝑀	𝑀	PROPN
iajs-1808	150	21	is	be	AUX
iajs-1808	150	22	stable	stable	ADJ
iajs-1808	150	23	in	in	ADP
iajs-1808	150	24	𝑀	𝑀	PROPN
iajs-1808	150	25	.	.	PUNCT
iajs-1808	151	1	but	but	CCONJ
iajs-1808	151	2	𝑀	𝑀	PROPN
iajs-1808	151	3	is	be	AUX
iajs-1808	151	4	fully	fully	ADV
iajs-1808	151	5	stable	stable	ADJ
iajs-1808	151	6	semisimple	semisimple	NOUN
iajs-1808	151	7	then	then	ADV
iajs-1808	151	8	𝑀	𝑀	PROPN
iajs-1808	151	9	is	be	AUX
iajs-1808	151	10	strongly	strongly	ADV
iajs-1808	151	11	𝐶	𝐶	PROPN
iajs-1808	151	12	-condition	-condition	NOUN
iajs-1808	151	13	module	module	NOUN
iajs-1808	151	14	by	by	ADP
iajs-1808	151	15	remarks	remark	NOUN
iajs-1808	151	16	3.10(1	3.10(1	NOUN
iajs-1808	151	17	)	)	PUNCT
iajs-1808	151	18	.	.	PUNCT
iajs-1808	152	1	hence	hence	ADV
iajs-1808	152	2	𝑀	𝑀	PROPN
iajs-1808	152	3	satisfies	satisfy	VERB
iajs-1808	152	4	condition	condition	NOUN
iajs-1808	152	5	(	(	PUNCT
iajs-1808	152	6	2	2	NUM
iajs-1808	152	7	)	)	PUNCT
iajs-1808	152	8	of	of	ADP
iajs-1808	152	9	theorem	theorem	ADJ
iajs-1808	152	10	3.9	3.9	NUM
iajs-1808	152	11	.	.	PUNCT
iajs-1808	153	1	where	where	SCONJ
iajs-1808	153	2	(	(	PUNCT
iajs-1808	153	3	251	251	NUM
iajs-1808	153	4	)	)	PUNCT
iajs-1808	153	5	.	.	PUNCT
iajs-1808	154	1	thus	thus	ADV
iajs-1808	154	2	𝑀	𝑀	PROPN
iajs-1808	154	3	is	be	AUX
iajs-1808	154	4	strongly	strongly	ADV
iajs-1808	154	5	𝑇	𝑇	PROPN
iajs-1808	154	6	-type	-type	NOUN
iajs-1808	154	7	module	module	NOUN
iajs-1808	154	8	.	.	PUNCT
iajs-1808	155	1	□	□	PUNCT
iajs-1808	155	2	theorem	theorem	NOUN
iajs-1808	155	3	(	(	PUNCT
iajs-1808	155	4	3.12	3.12	NUM
iajs-1808	155	5	):	):	PUNCT
iajs-1808	155	6	every	every	DET
iajs-1808	155	7	strongly	strongly	ADV
iajs-1808	155	8	extending	extend	VERB
iajs-1808	155	9	module	module	NOUN
iajs-1808	155	10	is	be	AUX
iajs-1808	155	11	strongly	strongly	ADV
iajs-1808	155	12	𝑇	𝑇	PROPN
iajs-1808	155	13	-type	-type	NOUN
iajs-1808	155	14	module	module	NOUN
iajs-1808	155	15	.	.	PUNCT
iajs-1808	156	1	proof	proof	NOUN
iajs-1808	156	2	:	:	PUNCT
iajs-1808	156	3	let	let	VERB
iajs-1808	156	4	𝑁	𝑁	PROPN
iajs-1808	156	5	be	be	AUX
iajs-1808	156	6	a	a	DET
iajs-1808	156	7	t	t	NOUN
iajs-1808	156	8	-	-	PUNCT
iajs-1808	156	9	closed	close	VERB
iajs-1808	156	10	submodule	submodule	NOUN
iajs-1808	156	11	of	of	ADP
iajs-1808	156	12	𝑀.	𝑀.	PROPN
iajs-1808	156	13	hence	hence	ADV
iajs-1808	156	14	𝑁	𝑁	PROPN
iajs-1808	156	15	is	be	AUX
iajs-1808	156	16	a	a	DET
iajs-1808	156	17	closed	closed	ADJ
iajs-1808	156	18	submodule	submodule	NOUN
iajs-1808	156	19	.	.	PUNCT
iajs-1808	157	1	as	as	SCONJ
iajs-1808	157	2	𝑀	𝑀	PROPN
iajs-1808	157	3	is	be	AUX
iajs-1808	157	4	strongly	strongly	ADV
iajs-1808	157	5	extending	extending	ADJ
iajs-1808	157	6	,	,	PUNCT
iajs-1808	157	7	𝑁	𝑁	PROPN
iajs-1808	157	8	is	be	AUX
iajs-1808	157	9	a	a	DET
iajs-1808	157	10	fully	fully	ADV
iajs-1808	157	11	invariant	invariant	ADJ
iajs-1808	157	12	direct	direct	ADJ
iajs-1808	157	13	summand	summand	NOUN
iajs-1808	157	14	.	.	PUNCT
iajs-1808	158	1	then	then	ADV
iajs-1808	158	2	𝑀	𝑀	PROPN
iajs-1808	158	3	𝑁	𝑁	PROPN
iajs-1808	158	4	⨁𝑊	⨁𝑊	NOUN
iajs-1808	158	5	for	for	ADP
iajs-1808	158	6	some	some	DET
iajs-1808	158	7	𝑊	𝑊	PROPN
iajs-1808	158	8	𝑀	𝑀	PROPN
iajs-1808	159	1	and	and	CCONJ
iajs-1808	159	2	so	so	ADV
iajs-1808	159	3	𝑊	𝑊	PROPN
iajs-1808	159	4	is	be	AUX
iajs-1808	159	5	a	a	DET
iajs-1808	159	6	complement	complement	NOUN
iajs-1808	159	7	of	of	ADP
iajs-1808	159	8	𝑁.	𝑁.	PROPN
iajs-1808	159	9	to	to	PART
iajs-1808	159	10	see	see	VERB
iajs-1808	159	11	this	this	PRON
iajs-1808	159	12	let	let	VERB
iajs-1808	159	13	𝑊	𝑊	PROPN
iajs-1808	159	14	𝑀	𝑀	PROPN
iajs-1808	159	15	and	and	CCONJ
iajs-1808	159	16	𝑊	𝑊	PROPN
iajs-1808	159	17	𝑊	𝑊	PROPN
iajs-1808	159	18	𝑀	𝑀	PROPN
iajs-1808	159	19	and	and	CCONJ
iajs-1808	159	20	𝑁⋂𝑊	𝑁⋂𝑊	X
iajs-1808	159	21	0	0	NUM
iajs-1808	159	22	,	,	PUNCT
iajs-1808	159	23	then	then	ADV
iajs-1808	159	24	𝑀	𝑀	PROPN
iajs-1808	159	25	𝑁⨁𝑊	𝑁⨁𝑊	VERB
iajs-1808	159	26	⊆	⊆	NUM
iajs-1808	159	27	𝑁⨁𝑊	𝑁⨁𝑊	NOUN
iajs-1808	159	28	,	,	PUNCT
iajs-1808	159	29	so	so	ADV
iajs-1808	159	30	𝑀	𝑀	PROPN
iajs-1808	159	31	𝑁⨁𝑊	𝑁⨁𝑊	VERB
iajs-1808	159	32	𝑁⨁𝑊.	𝑁⨁𝑊.	X
iajs-1808	159	33	assume	assume	VERB
iajs-1808	159	34	𝑥	𝑥	X
iajs-1808	159	35	∈	∈	PROPN
iajs-1808	159	36	𝑊	𝑊	NOUN
iajs-1808	159	37	then	then	ADV
iajs-1808	159	38	𝑥	𝑥	PROPN
iajs-1808	159	39	𝑛	𝑛	PROPN
iajs-1808	159	40	𝑦	𝑦	NOUN
iajs-1808	159	41	,	,	PUNCT
iajs-1808	159	42	𝑛	𝑛	PRON
iajs-1808	159	43	∈	∈	PROPN
iajs-1808	159	44	𝑁	𝑁	PROPN
iajs-1808	159	45	,	,	PUNCT
iajs-1808	159	46	𝑦	𝑦	NOUN
iajs-1808	159	47	∈	∈	NOUN
iajs-1808	159	48	𝑊	𝑊	PROPN
iajs-1808	159	49	𝑊	𝑊	PROPN
iajs-1808	159	50	,	,	PUNCT
iajs-1808	159	51	then	then	ADV
iajs-1808	159	52	𝑥	𝑥	PROPN
iajs-1808	159	53	𝑦	𝑦	NOUN
iajs-1808	159	54	𝑛	𝑛	PRON
iajs-1808	159	55	∈	∈	X
iajs-1808	159	56	𝑁⋂𝑊	𝑁⋂𝑊	X
iajs-1808	159	57	0	0	NUM
iajs-1808	159	58	,	,	PUNCT
iajs-1808	159	59	hence	hence	ADV
iajs-1808	159	60	𝑥	𝑥	PRON
iajs-1808	159	61	𝑦	𝑦	NOUN
iajs-1808	159	62	0	0	NUM
iajs-1808	159	63	implies	imply	VERB
iajs-1808	159	64	𝑥	𝑥	PROPN
iajs-1808	159	65	𝑦	𝑦	NUM
iajs-1808	159	66	∈	∈	X
iajs-1808	159	67	𝑊.	𝑊.	NOUN
iajs-1808	159	68	hence	hence	ADV
iajs-1808	159	69	𝑊	𝑊	PROPN
iajs-1808	159	70	𝑊	𝑊	PROPN
iajs-1808	159	71	,	,	PUNCT
iajs-1808	159	72	moreover	moreover	ADV
iajs-1808	159	73	𝑊	𝑊	PROPN
iajs-1808	159	74	⨁	⨁	PROPN
iajs-1808	159	75	𝑀	𝑀	PROPN
iajs-1808	159	76	,	,	PUNCT
iajs-1808	159	77	so	so	ADV
iajs-1808	159	78	𝑊	𝑊	PROPN
iajs-1808	159	79	is	be	AUX
iajs-1808	159	80	closed	close	VERB
iajs-1808	159	81	submodule	submodule	NOUN
iajs-1808	159	82	and	and	CCONJ
iajs-1808	159	83	hence	hence	ADV
iajs-1808	159	84	𝑊	𝑊	PROPN
iajs-1808	159	85	is	be	AUX
iajs-1808	159	86	a	a	DET
iajs-1808	159	87	fully	fully	ADV
iajs-1808	159	88	invariant	invariant	ADJ
iajs-1808	159	89	direct	direct	ADJ
iajs-1808	159	90	summand	summand	NOUN
iajs-1808	159	91	.	.	PUNCT
iajs-1808	160	1	thus	thus	ADV
iajs-1808	160	2	𝑀	𝑀	PROPN
iajs-1808	160	3	is	be	AUX
iajs-1808	160	4	strongly	strongly	ADV
iajs-1808	160	5	𝑇	𝑇	PROPN
iajs-1808	160	6	-type	-type	NOUN
iajs-1808	160	7	module	module	NOUN
iajs-1808	160	8	.	.	PUNCT
iajs-1808	161	1	□	□	PUNCT
iajs-1808	161	2	proposition	proposition	NOUN
iajs-1808	161	3	(	(	PUNCT
iajs-1808	161	4	3.13	3.13	NUM
iajs-1808	161	5	):	):	PUNCT
iajs-1808	161	6	if	if	SCONJ
iajs-1808	161	7	𝑀	𝑀	PROPN
iajs-1808	161	8	is	be	AUX
iajs-1808	161	9	a	a	DET
iajs-1808	161	10	strongly	strongly	ADV
iajs-1808	161	11	t	t	NOUN
iajs-1808	161	12	-	-	PUNCT
iajs-1808	161	13	extending	extend	VERB
iajs-1808	161	14	𝑅-module	𝑅-module	PROPN
iajs-1808	161	15	then	then	ADV
iajs-1808	161	16	is	be	AUX
iajs-1808	161	17	strongly	strongly	ADV
iajs-1808	161	18	𝑇	𝑇	PROPN
iajs-1808	161	19	-type	-type	NOUN
iajs-1808	161	20	module	module	NOUN
iajs-1808	161	21	and	and	CCONJ
iajs-1808	161	22	every	every	DET
iajs-1808	161	23	complement	complement	NOUN
iajs-1808	161	24	to	to	ADP
iajs-1808	161	25	a	a	DET
iajs-1808	161	26	nonsingular	nonsingular	ADJ
iajs-1808	161	27	direct	direct	ADJ
iajs-1808	161	28	summand	summand	NOUN
iajs-1808	161	29	is	be	AUX
iajs-1808	161	30	fully	fully	ADV
iajs-1808	161	31	invariant	invariant	ADJ
iajs-1808	161	32	direct	direct	ADJ
iajs-1808	161	33	summand	summand	NOUN
iajs-1808	161	34	.	.	PUNCT
iajs-1808	162	1	proof	proof	NOUN
iajs-1808	162	2	:	:	PUNCT
iajs-1808	162	3	since	since	SCONJ
iajs-1808	162	4	𝑀	𝑀	PROPN
iajs-1808	162	5	is	be	AUX
iajs-1808	162	6	strongly	strongly	ADV
iajs-1808	162	7	t	t	NOUN
iajs-1808	162	8	-	-	PUNCT
iajs-1808	162	9	extending	extending	NOUN
iajs-1808	162	10	,	,	PUNCT
iajs-1808	162	11	then	then	ADV
iajs-1808	162	12	𝑀	𝑀	PROPN
iajs-1808	162	13	𝑍	𝑍	VERB
iajs-1808	162	14	𝑀	𝑀	PROPN
iajs-1808	162	15	⨁𝑀	⨁𝑀	NOUN
iajs-1808	162	16	,	,	PUNCT
iajs-1808	162	17	𝑀	𝑀	PROPN
iajs-1808	162	18	is	be	AUX
iajs-1808	162	19	strongly	strongly	ADV
iajs-1808	162	20	extending	extend	VERB
iajs-1808	162	21	module[7	module[7	NOUN
iajs-1808	162	22	]	]	PUNCT
iajs-1808	162	23	.	.	PUNCT
iajs-1808	163	1	hence	hence	ADV
iajs-1808	163	2	,	,	PUNCT
iajs-1808	163	3	𝑀	𝑀	PROPN
iajs-1808	163	4	is	be	AUX
iajs-1808	163	5	strongly	strongly	ADV
iajs-1808	163	6	𝑇	𝑇	PROPN
iajs-1808	163	7	-type	-type	NOUN
iajs-1808	163	8	module	module	NOUN
iajs-1808	163	9	by	by	ADP
iajs-1808	163	10	proposition	proposition	NOUN
iajs-1808	163	11	(	(	PUNCT
iajs-1808	163	12	3.12	3.12	NUM
iajs-1808	163	13	)	)	PUNCT
iajs-1808	163	14	.	.	PUNCT
iajs-1808	164	1	but	but	CCONJ
iajs-1808	164	2	𝑀	𝑀	PROPN
iajs-1808	164	3	is	be	AUX
iajs-1808	164	4	nonsingular	nonsingular	ADJ
iajs-1808	164	5	,	,	PUNCT
iajs-1808	164	6	so	so	ADV
iajs-1808	164	7	𝑀	𝑀	PROPN
iajs-1808	164	8	satisfies	satisfy	VERB
iajs-1808	164	9	strongly	strongly	ADV
iajs-1808	164	10	𝐶	𝐶	PROPN
iajs-1808	164	11	-condition	-condition	NOUN
iajs-1808	164	12	module	module	NOUN
iajs-1808	164	13	by	by	ADP
iajs-1808	164	14	proposition	proposition	NOUN
iajs-1808	164	15	(	(	PUNCT
iajs-1808	164	16	3.7	3.7	NUM
iajs-1808	164	17	)	)	PUNCT
iajs-1808	164	18	.	.	PUNCT
iajs-1808	165	1	thus	thus	ADV
iajs-1808	165	2	𝑀	𝑀	PROPN
iajs-1808	165	3	satisfies	satisfy	VERB
iajs-1808	165	4	condition	condition	NOUN
iajs-1808	165	5	(	(	PUNCT
iajs-1808	165	6	2	2	NUM
iajs-1808	165	7	)	)	PUNCT
iajs-1808	165	8	of	of	ADP
iajs-1808	165	9	theorem	theorem	NOUN
iajs-1808	165	10	3.9	3.9	NUM
iajs-1808	165	11	so	so	ADV
iajs-1808	165	12	𝑀	𝑀	PROPN
iajs-1808	165	13	is	be	AUX
iajs-1808	165	14	a	a	DET
iajs-1808	165	15	strongly	strongly	ADV
iajs-1808	165	16	𝑇	𝑇	NOUN
iajs-1808	165	17	-type	-type	NOUN
iajs-1808	165	18	module	module	NOUN
iajs-1808	165	19	.	.	PUNCT
iajs-1808	166	1	now	now	ADV
iajs-1808	166	2	let	let	VERB
iajs-1808	166	3	𝐶	𝐶	PROPN
iajs-1808	166	4	be	be	AUX
iajs-1808	166	5	a	a	DET
iajs-1808	166	6	complement	complement	NOUN
iajs-1808	166	7	of	of	ADP
iajs-1808	166	8	a	a	DET
iajs-1808	166	9	nonsingular	nonsingular	ADJ
iajs-1808	166	10	submodule	submodule	NOUN
iajs-1808	166	11	of	of	ADP
iajs-1808	166	12	𝑀	𝑀	PROPN
iajs-1808	166	13	,	,	PUNCT
iajs-1808	166	14	so	so	ADV
iajs-1808	166	15	by	by	ADP
iajs-1808	166	16	[	[	X
iajs-1808	166	17	2	2	NUM
iajs-1808	166	18	,	,	PUNCT
iajs-1808	166	19	proposition	proposition	NOUN
iajs-1808	166	20	2.6(52	2.6(52	NUM
iajs-1808	166	21	)	)	PUNCT
iajs-1808	166	22	]	]	PUNCT
iajs-1808	166	23	𝐶	𝐶	PROPN
iajs-1808	166	24	is	be	AUX
iajs-1808	166	25	a	a	DET
iajs-1808	166	26	t	t	NOUN
iajs-1808	166	27	-	-	PUNCT
iajs-1808	166	28	closed	close	VERB
iajs-1808	166	29	submodule	submodule	NOUN
iajs-1808	166	30	of	of	ADP
iajs-1808	166	31	𝑀.	𝑀.	PROPN
iajs-1808	166	32	hence	hence	ADV
iajs-1808	166	33	𝐶	𝐶	PROPN
iajs-1808	166	34	is	be	AUX
iajs-1808	166	35	a	a	DET
iajs-1808	166	36	fully	fully	ADV
iajs-1808	166	37	invariant	invariant	ADJ
iajs-1808	166	38	direct	direct	ADJ
iajs-1808	166	39	summand	summand	NOUN
iajs-1808	166	40	of	of	ADP
iajs-1808	166	41	𝑀	𝑀	PROPN
iajs-1808	166	42	by	by	ADP
iajs-1808	166	43	definition	definition	NOUN
iajs-1808	166	44	of	of	ADP
iajs-1808	166	45	strongly	strongly	ADV
iajs-1808	166	46	textending	textende	VERB
iajs-1808	166	47	proposition	proposition	NOUN
iajs-1808	166	48	(	(	PUNCT
iajs-1808	166	49	1.7	1.7	NUM
iajs-1808	166	50	)	)	PUNCT
iajs-1808	166	51	.	.	PUNCT
iajs-1808	167	1	not	not	PART
iajs-1808	167	2	that	that	SCONJ
iajs-1808	167	3	if	if	SCONJ
iajs-1808	167	4	every	every	DET
iajs-1808	167	5	complement	complement	NOUN
iajs-1808	167	6	of	of	ADP
iajs-1808	167	7	nonsingular	nonsingular	ADJ
iajs-1808	167	8	submodule	submodule	NOUN
iajs-1808	167	9	of	of	ADP
iajs-1808	167	10	an	an	DET
iajs-1808	167	11	𝑅-module	𝑅-module	PROPN
iajs-1808	167	12	is	be	AUX
iajs-1808	167	13	fully	fully	ADV
iajs-1808	167	14	invariant	invariant	ADJ
iajs-1808	167	15	direct	direct	ADJ
iajs-1808	167	16	summand	summand	NOUN
iajs-1808	167	17	implies	imply	VERB
iajs-1808	167	18	𝑀	𝑀	PROPN
iajs-1808	167	19	is	be	AUX
iajs-1808	167	20	strongly	strongly	ADV
iajs-1808	167	21	t	t	NOUN
iajs-1808	167	22	-	-	PUNCT
iajs-1808	167	23	extending	extending	NOUN
iajs-1808	167	24	,	,	PUNCT
iajs-1808	167	25	since	since	SCONJ
iajs-1808	167	26	by	by	ADP
iajs-1808	167	27	[	[	X
iajs-1808	167	28	2	2	NUM
iajs-1808	167	29	,	,	PUNCT
iajs-1808	167	30	proposition	proposition	NOUN
iajs-1808	167	31	2.6(52	2.6(52	NUM
iajs-1808	167	32	)	)	PUNCT
iajs-1808	167	33	]	]	PUNCT
iajs-1808	168	1	every	every	DET
iajs-1808	168	2	t	t	PROPN
iajs-1808	168	3	-	-	PUNCT
iajs-1808	168	4	closed	close	VERB
iajs-1808	168	5	ihsciconf	ihsciconf	NOUN
iajs-1808	168	6	2017	2017	NUM
iajs-1808	168	7	special	special	ADJ
iajs-1808	168	8	issue	issue	NOUN
iajs-1808	168	9	ibn	ibn	PROPN
iajs-1808	168	10	al	al	PROPN
iajs-1808	168	11	-	-	PUNCT
iajs-1808	168	12	haitham	haitham	PROPN
iajs-1808	168	13	journal	journal	PROPN
iajs-1808	168	14	for	for	ADP
iajs-1808	168	15	pure	pure	ADJ
iajs-1808	168	16	and	and	CCONJ
iajs-1808	168	17	applied	apply	VERB
iajs-1808	168	18	science	science	NOUN
iajs-1808	168	19	https://doi.org/	https://doi.org/	NOUN
iajs-1808	168	20	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	168	21	for	for	ADP
iajs-1808	168	22	more	more	ADJ
iajs-1808	168	23	information	information	NOUN
iajs-1808	168	24	about	about	ADP
iajs-1808	168	25	the	the	DET
iajs-1808	168	26	conference	conference	NOUN
iajs-1808	168	27	please	please	INTJ
iajs-1808	168	28	visit	visit	VERB
iajs-1808	168	29	the	the	DET
iajs-1808	168	30	websites	website	NOUN
iajs-1808	168	31	:	:	PUNCT
iajs-1808	168	32	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	168	33	  	  	SPACE
iajs-1808	168	34	www.ihsciconf.org	www.ihsciconf.org	NOUN
iajs-1808	168	35	   	   	SPACE
iajs-1808	168	36	mathematics|360	mathematics|360	NOUN
iajs-1808	168	37	    	    	SPACE
iajs-1808	168	38	is	be	AUX
iajs-1808	168	39	a	a	DET
iajs-1808	168	40	complement	complement	NOUN
iajs-1808	168	41	of	of	ADP
iajs-1808	168	42	nonsingular	nonsingular	ADJ
iajs-1808	168	43	submodule	submodule	NOUN
iajs-1808	168	44	and	and	CCONJ
iajs-1808	168	45	so	so	SCONJ
iajs-1808	168	46	that	that	SCONJ
iajs-1808	168	47	every	every	DET
iajs-1808	168	48	t	t	NOUN
iajs-1808	168	49	-	-	PUNCT
iajs-1808	168	50	closed	close	VERB
iajs-1808	168	51	submodule	submodule	NOUN
iajs-1808	168	52	is	be	AUX
iajs-1808	168	53	fully	fully	ADV
iajs-1808	168	54	invariant	invariant	ADJ
iajs-1808	168	55	direct	direct	ADJ
iajs-1808	168	56	summand	summand	NOUN
iajs-1808	168	57	(	(	PUNCT
iajs-1808	168	58	that	that	PRON
iajs-1808	168	59	is	is	ADV
iajs-1808	168	60	𝑀	𝑀	PROPN
iajs-1808	168	61	is	be	AUX
iajs-1808	168	62	strongly	strongly	ADV
iajs-1808	168	63	t	t	NOUN
iajs-1808	168	64	-	-	PUNCT
iajs-1808	168	65	extending	extending	NOUN
iajs-1808	168	66	)	)	PUNCT
iajs-1808	168	67	.	.	PUNCT
iajs-1808	169	1	□	□	PUNCT
iajs-1808	169	2	proposition	proposition	NOUN
iajs-1808	169	3	(	(	PUNCT
iajs-1808	169	4	3.14	3.14	NUM
iajs-1808	169	5	):	):	PUNCT
iajs-1808	169	6	let	let	VERB
iajs-1808	169	7	𝑀	𝑀	PROPN
iajs-1808	169	8	𝑀	𝑀	PROPN
iajs-1808	169	9	⨁𝑀	⨁𝑀	PROPN
iajs-1808	169	10	,	,	PUNCT
iajs-1808	169	11	𝑀	𝑀	PROPN
iajs-1808	169	12	is	be	AUX
iajs-1808	169	13	a	a	DET
iajs-1808	169	14	fully	fully	ADV
iajs-1808	169	15	invariant	invariant	ADJ
iajs-1808	169	16	submodule	submodule	NOUN
iajs-1808	169	17	in	in	ADP
iajs-1808	169	18	𝑀.	𝑀.	PROPN
iajs-1808	169	19	then	then	ADV
iajs-1808	169	20	the	the	DET
iajs-1808	169	21	following	follow	VERB
iajs-1808	169	22	conditions	condition	NOUN
iajs-1808	169	23	are	be	AUX
iajs-1808	169	24	equivalent	equivalent	ADJ
iajs-1808	169	25	:	:	PUNCT
iajs-1808	169	26	𝑀	𝑀	PROPN
iajs-1808	169	27	is	be	AUX
iajs-1808	169	28	strongly	strongly	ADV
iajs-1808	169	29	𝑇	𝑇	PROPN
iajs-1808	169	30	-type	-type	NOUN
iajs-1808	169	31	module	module	NOUN
iajs-1808	169	32	;	;	PUNCT
iajs-1808	169	33	for	for	ADP
iajs-1808	169	34	every	every	DET
iajs-1808	169	35	submodule	submodule	NOUN
iajs-1808	169	36	𝐴	𝐴	PROPN
iajs-1808	169	37	of	of	ADP
iajs-1808	169	38	𝑀	𝑀	PROPN
iajs-1808	169	39	,	,	PUNCT
iajs-1808	169	40	there	there	PRON
iajs-1808	169	41	exists	exist	VERB
iajs-1808	169	42	a	a	DET
iajs-1808	169	43	fully	fully	ADV
iajs-1808	169	44	invariant	invariant	ADJ
iajs-1808	169	45	direct	direct	ADJ
iajs-1808	169	46	summand	summand	NOUN
iajs-1808	169	47	𝐷	𝐷	PROPN
iajs-1808	169	48	of	of	ADP
iajs-1808	169	49	𝑀	𝑀	PROPN
iajs-1808	169	50	such	such	ADJ
iajs-1808	169	51	that	that	SCONJ
iajs-1808	169	52	𝑀	𝑀	PROPN
iajs-1808	169	53	𝐷	𝐷	PROPN
iajs-1808	169	54	and	and	CCONJ
iajs-1808	169	55	𝐴⨁𝐷	𝐴⨁𝐷	VERB
iajs-1808	169	56	𝑀.	𝑀.	PROPN
iajs-1808	169	57	for	for	ADP
iajs-1808	169	58	every	every	DET
iajs-1808	169	59	t	t	NOUN
iajs-1808	169	60	-	-	PUNCT
iajs-1808	169	61	closed	close	VERB
iajs-1808	169	62	submodule	submodule	NOUN
iajs-1808	169	63	𝐶	𝐶	PROPN
iajs-1808	169	64	of	of	ADP
iajs-1808	169	65	𝑀	𝑀	PROPN
iajs-1808	169	66	,	,	PUNCT
iajs-1808	169	67	there	there	PRON
iajs-1808	169	68	exists	exist	VERB
iajs-1808	169	69	a	a	DET
iajs-1808	169	70	fully	fully	ADV
iajs-1808	169	71	invariant	invariant	ADJ
iajs-1808	169	72	direct	direct	ADJ
iajs-1808	169	73	summand	summand	NOUN
iajs-1808	169	74	𝐷	𝐷	PROPN
iajs-1808	169	75	of	of	ADP
iajs-1808	169	76	𝑀	𝑀	PROPN
iajs-1808	169	77	such	such	ADJ
iajs-1808	169	78	that	that	SCONJ
iajs-1808	169	79	𝑀	𝑀	PROPN
iajs-1808	169	80	𝐷	𝐷	PROPN
iajs-1808	169	81	and	and	CCONJ
iajs-1808	169	82	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	169	83	𝑀	𝑀	PROPN
iajs-1808	169	84	;	;	PUNCT
iajs-1808	169	85	for	for	ADP
iajs-1808	169	86	every	every	DET
iajs-1808	169	87	t	t	PROPN
iajs-1808	169	88	-	-	PUNCT
iajs-1808	169	89	closed	close	VERB
iajs-1808	169	90	submodule	submodule	NOUN
iajs-1808	169	91	𝐶	𝐶	PROPN
iajs-1808	169	92	of	of	ADP
iajs-1808	169	93	𝑀	𝑀	PROPN
iajs-1808	169	94	,	,	PUNCT
iajs-1808	169	95	there	there	PRON
iajs-1808	169	96	exists	exist	VERB
iajs-1808	169	97	a	a	DET
iajs-1808	169	98	fully	fully	ADV
iajs-1808	169	99	invariant	invariant	ADJ
iajs-1808	169	100	direct	direct	ADJ
iajs-1808	169	101	summand	summand	NOUN
iajs-1808	169	102	𝐷	𝐷	PROPN
iajs-1808	169	103	of	of	ADP
iajs-1808	169	104	𝑀	𝑀	PROPN
iajs-1808	169	105	such	such	ADJ
iajs-1808	169	106	that	that	SCONJ
iajs-1808	169	107	𝑀	𝑀	PROPN
iajs-1808	169	108	𝐷	𝐷	PROPN
iajs-1808	169	109	and	and	CCONJ
iajs-1808	169	110	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	169	111	𝑀.	𝑀.	NOUN
iajs-1808	169	112	proof	proof	NOUN
iajs-1808	169	113	:	:	PUNCT
iajs-1808	169	114	(	(	PUNCT
iajs-1808	169	115	1	1	X
iajs-1808	169	116	)	)	PUNCT
iajs-1808	169	117			NOUN
iajs-1808	169	118	(	(	PUNCT
iajs-1808	169	119	2	2	NUM
iajs-1808	169	120	)	)	PUNCT
iajs-1808	169	121	since	since	SCONJ
iajs-1808	169	122	𝑀	𝑀	PROPN
iajs-1808	169	123	is	be	AUX
iajs-1808	169	124	strongly	strongly	ADV
iajs-1808	169	125	𝑇	𝑇	PROPN
iajs-1808	169	126	-type	-type	NOUN
iajs-1808	169	127	module	module	NOUN
iajs-1808	169	128	,	,	PUNCT
iajs-1808	169	129	then	then	ADV
iajs-1808	169	130	by	by	ADP
iajs-1808	169	131	condition	condition	NOUN
iajs-1808	169	132	(	(	PUNCT
iajs-1808	169	133	3	3	NUM
iajs-1808	169	134	)	)	PUNCT
iajs-1808	169	135	of	of	ADP
iajs-1808	169	136	theorem	theorem	ADJ
iajs-1808	169	137	3.9	3.9	NUM
iajs-1808	169	138	for	for	ADP
iajs-1808	169	139	each	each	DET
iajs-1808	169	140	𝐴	𝐴	PROPN
iajs-1808	169	141	𝑀	𝑀	PROPN
iajs-1808	169	142	,	,	PUNCT
iajs-1808	169	143	there	there	PRON
iajs-1808	169	144	exists	exist	VERB
iajs-1808	169	145	a	a	DET
iajs-1808	169	146	fully	fully	ADV
iajs-1808	169	147	invariant	invariant	ADJ
iajs-1808	169	148	direct	direct	ADJ
iajs-1808	169	149	summand	summand	NOUN
iajs-1808	169	150	𝐷	𝐷	PROPN
iajs-1808	169	151	of	of	ADP
iajs-1808	169	152	𝑀	𝑀	PROPN
iajs-1808	169	153	such	such	ADJ
iajs-1808	169	154	that	that	SCONJ
iajs-1808	169	155	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	169	156	𝑀	𝑀	PROPN
iajs-1808	169	157	.	.	PUNCT
iajs-1808	170	1	but	but	CCONJ
iajs-1808	170	2	𝐷	𝐷	PROPN
iajs-1808	170	3	⨁	⨁	PROPN
iajs-1808	170	4	𝑀	𝑀	PROPN
iajs-1808	170	5	implies	imply	VERB
iajs-1808	170	6	that	that	SCONJ
iajs-1808	170	7	𝐷⨁𝑀	𝐷⨁𝑀	SCONJ
iajs-1808	170	8	⨁	⨁	PROPN
iajs-1808	170	9	𝑀.	𝑀.	PROPN
iajs-1808	170	10	also	also	ADV
iajs-1808	170	11	,	,	PUNCT
iajs-1808	170	12	we	we	PRON
iajs-1808	170	13	can	can	AUX
iajs-1808	170	14	show	show	VERB
iajs-1808	170	15	that	that	SCONJ
iajs-1808	170	16	𝐷⨁𝑀	𝐷⨁𝑀	NOUN
iajs-1808	170	17	is	be	AUX
iajs-1808	170	18	fully	fully	ADV
iajs-1808	170	19	invariant	invariant	ADJ
iajs-1808	170	20	in	in	ADP
iajs-1808	170	21	𝑀.	𝑀.	PROPN
iajs-1808	170	22	let	let	VERB
iajs-1808	170	23	𝑓	𝑓	DET
iajs-1808	170	24	∈	∈	PROPN
iajs-1808	170	25	𝐸𝑛𝑑	𝐸𝑛𝑑	PROPN
iajs-1808	170	26	𝑀	𝑀	PROPN
iajs-1808	170	27	𝐸𝑛𝑑	𝐸𝑛𝑑	PROPN
iajs-1808	170	28	𝑀	𝑀	PROPN
iajs-1808	170	29	𝐻𝑜𝑚	𝐻𝑜𝑚	PROPN
iajs-1808	170	30	𝑀	𝑀	PROPN
iajs-1808	170	31	,	,	PUNCT
iajs-1808	170	32	𝑀	𝑀	PROPN
iajs-1808	170	33	𝐻𝑜𝑚	𝐻𝑜𝑚	PROPN
iajs-1808	170	34	𝑀	𝑀	PROPN
iajs-1808	170	35	,	,	PUNCT
iajs-1808	170	36	𝑀	𝑀	PROPN
iajs-1808	170	37	𝐸𝑛𝑑	𝐸𝑛𝑑	PROPN
iajs-1808	170	38	𝑀	𝑀	PROPN
iajs-1808	170	39	.	.	PUNCT
iajs-1808	171	1	but	but	CCONJ
iajs-1808	171	2	𝑀	𝑀	PROPN
iajs-1808	171	3	is	be	AUX
iajs-1808	171	4	fully	fully	ADV
iajs-1808	171	5	invariant	invariant	ADJ
iajs-1808	171	6	in	in	ADP
iajs-1808	171	7	𝑀	𝑀	PROPN
iajs-1808	171	8	by	by	ADP
iajs-1808	171	9	hypothesis	hypothesis	NOUN
iajs-1808	171	10	so	so	SCONJ
iajs-1808	171	11	hom(𝑀	hom(𝑀	NUM
iajs-1808	171	12	,	,	PUNCT
iajs-1808	171	13	𝑀	𝑀	PROPN
iajs-1808	171	14	0	0	NUM
iajs-1808	171	15	,	,	PUNCT
iajs-1808	171	16	then	then	ADV
iajs-1808	171	17	𝑓	𝑓	PRON
iajs-1808	171	18	𝑓	𝑓	DET
iajs-1808	171	19	0	0	NUM
iajs-1808	171	20	𝑓	𝑓	PRON
iajs-1808	171	21	𝑓	𝑓	DET
iajs-1808	171	22	for	for	ADP
iajs-1808	171	23	some	some	DET
iajs-1808	171	24	𝑓	𝑓	DET
iajs-1808	171	25	∈	∈	PROPN
iajs-1808	171	26	𝐸𝑛𝑑	𝐸𝑛𝑑	PROPN
iajs-1808	171	27	𝑀	𝑀	PROPN
iajs-1808	171	28	,	,	PUNCT
iajs-1808	171	29	𝑓	𝑓	DET
iajs-1808	171	30	∈	∈	PROPN
iajs-1808	171	31	𝐻𝑜𝑚	𝐻𝑜𝑚	PROPN
iajs-1808	171	32	𝑀	𝑀	PROPN
iajs-1808	171	33	,	,	PUNCT
iajs-1808	171	34	𝑀	𝑀	PROPN
iajs-1808	171	35	,	,	PUNCT
iajs-1808	171	36	𝑓	𝑓	DET
iajs-1808	171	37	∈	∈	PROPN
iajs-1808	171	38	𝐸𝑛𝑑	𝐸𝑛𝑑	PROPN
iajs-1808	171	39	𝑀	𝑀	PROPN
iajs-1808	171	40	.hence	.hence	ADP
iajs-1808	172	1	𝑓	𝑓	DET
iajs-1808	172	2	𝐷⨁𝑀	𝐷⨁𝑀	ADP
iajs-1808	172	3	≃	≃	NOUN
iajs-1808	172	4	𝑓	𝑓	DET
iajs-1808	172	5	0	0	NUM
iajs-1808	172	6	𝑓	𝑓	PRON
iajs-1808	172	7	𝑓	𝑓	DET
iajs-1808	172	8	𝐷	𝐷	PROPN
iajs-1808	172	9	𝑀	𝑀	PROPN
iajs-1808	172	10	𝐷	𝐷	PROPN
iajs-1808	172	11	𝑀	𝑀	PROPN
iajs-1808	172	12	,	,	PUNCT
iajs-1808	172	13	hence	hence	ADV
iajs-1808	172	14	𝐷⨁𝑀	𝐷⨁𝑀	ADP
iajs-1808	172	15	is	be	AUX
iajs-1808	172	16	fully	fully	ADV
iajs-1808	172	17	invariant	invariant	ADJ
iajs-1808	172	18	in	in	ADP
iajs-1808	172	19	𝑀.	𝑀.	PROPN
iajs-1808	172	20	moreover	moreover	ADV
iajs-1808	172	21	,	,	PUNCT
iajs-1808	172	22	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	172	23	𝑀	𝑀	PROPN
iajs-1808	172	24	implies	imply	VERB
iajs-1808	172	25	that	that	SCONJ
iajs-1808	172	26	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	172	27	⨁𝑀	⨁𝑀	NOUN
iajs-1808	172	28	𝑀	𝑀	PROPN
iajs-1808	172	29	⨁𝑀	⨁𝑀	PROPN
iajs-1808	172	30	𝑀.	𝑀.	PROPN
iajs-1808	172	31	(	(	PUNCT
iajs-1808	172	32	2	2	NUM
iajs-1808	172	33	)	)	PUNCT
iajs-1808	172	34			NOUN
iajs-1808	172	35	(	(	PUNCT
iajs-1808	172	36	3	3	X
iajs-1808	172	37	)	)	PUNCT
iajs-1808	172	38	it	it	PRON
iajs-1808	172	39	is	be	AUX
iajs-1808	172	40	obvious	obvious	ADJ
iajs-1808	172	41	(	(	PUNCT
iajs-1808	172	42	3	3	NUM
iajs-1808	172	43	)	)	PUNCT
iajs-1808	172	44			NOUN
iajs-1808	172	45	(	(	PUNCT
iajs-1808	172	46	4	4	NUM
iajs-1808	172	47	)	)	PUNCT
iajs-1808	172	48	for	for	ADP
iajs-1808	172	49	every	every	DET
iajs-1808	172	50	t	t	PROPN
iajs-1808	172	51	-	-	PUNCT
iajs-1808	172	52	closed	close	VERB
iajs-1808	172	53	submodule	submodule	NOUN
iajs-1808	172	54	𝐶	𝐶	PROPN
iajs-1808	172	55	of	of	ADP
iajs-1808	172	56	𝑀	𝑀	PROPN
iajs-1808	172	57	,	,	PUNCT
iajs-1808	172	58	there	there	PRON
iajs-1808	172	59	exists	exist	VERB
iajs-1808	172	60	a	a	DET
iajs-1808	172	61	fully	fully	ADV
iajs-1808	172	62	invariant	invariant	ADJ
iajs-1808	172	63	direct	direct	ADJ
iajs-1808	172	64	summand	summand	NOUN
iajs-1808	172	65	𝐷	𝐷	PROPN
iajs-1808	172	66	of	of	ADP
iajs-1808	172	67	𝑀	𝑀	PROPN
iajs-1808	172	68	such	such	ADJ
iajs-1808	172	69	that	that	SCONJ
iajs-1808	172	70	𝑀	𝑀	PROPN
iajs-1808	172	71	𝐷	𝐷	PROPN
iajs-1808	172	72	and	and	CCONJ
iajs-1808	172	73	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	172	74	𝑀	𝑀	PROPN
iajs-1808	172	75	.	.	PUNCT
iajs-1808	173	1	then	then	ADV
iajs-1808	173	2	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	173	3	𝑍	𝑍	PROPN
iajs-1808	173	4	𝑀	𝑀	PROPN
iajs-1808	173	5	𝑀	𝑀	PROPN
iajs-1808	173	6	by	by	ADP
iajs-1808	173	7	proposition	proposition	NOUN
iajs-1808	173	8	(	(	PUNCT
iajs-1808	173	9	1.1	1.1	NUM
iajs-1808	173	10	)	)	PUNCT
iajs-1808	173	11	.	.	PUNCT
iajs-1808	174	1	but	but	CCONJ
iajs-1808	174	2	𝑍	𝑍	PROPN
iajs-1808	174	3	𝑀	𝑀	PROPN
iajs-1808	174	4	𝑍	𝑍	PROPN
iajs-1808	174	5	𝑀	𝑀	PROPN
iajs-1808	174	6	⨁𝑍	⨁𝑍	PROPN
iajs-1808	174	7	𝑀	𝑀	PROPN
iajs-1808	174	8	.	.	PUNCT
iajs-1808	175	1	as	as	SCONJ
iajs-1808	175	2	𝐶	𝐶	PROPN
iajs-1808	175	3	is	be	AUX
iajs-1808	175	4	t	t	NOUN
iajs-1808	175	5	-	-	PUNCT
iajs-1808	175	6	closed	close	VERB
iajs-1808	175	7	in	in	ADP
iajs-1808	175	8	𝑀	𝑀	PROPN
iajs-1808	175	9	,	,	PUNCT
iajs-1808	175	10	𝐶	𝐶	PROPN
iajs-1808	175	11	⊇	⊇	NOUN
iajs-1808	175	12	𝑍	𝑍	PROPN
iajs-1808	175	13	𝑀	𝑀	PROPN
iajs-1808	175	14	by	by	ADP
iajs-1808	175	15	lemma	lemma	PROPN
iajs-1808	175	16	1.2(1	1.2(1	NUM
iajs-1808	175	17	)	)	PUNCT
iajs-1808	175	18	.	.	PUNCT
iajs-1808	176	1	also	also	ADV
iajs-1808	176	2	as	as	ADP
iajs-1808	176	3	𝑀	𝑀	PROPN
iajs-1808	176	4	𝐷	𝐷	PROPN
iajs-1808	176	5	,	,	PUNCT
iajs-1808	176	6	then	then	ADV
iajs-1808	176	7	𝑍	𝑍	VERB
iajs-1808	176	8	𝑀	𝑀	PROPN
iajs-1808	176	9	𝑍	𝑍	PROPN
iajs-1808	176	10	𝐷	𝐷	NOUN
iajs-1808	176	11	𝐷.	𝐷.	NOUN
iajs-1808	176	12	it	it	PRON
iajs-1808	176	13	follows	follow	VERB
iajs-1808	176	14	that	that	SCONJ
iajs-1808	176	15	⨁𝐷	⨁𝐷	NOUN
iajs-1808	176	16	𝑍	𝑍	PROPN
iajs-1808	176	17	𝑀	𝑀	PROPN
iajs-1808	176	18	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	176	19	𝑍	𝑍	PROPN
iajs-1808	176	20	𝑀	𝑀	PROPN
iajs-1808	176	21	⨁𝑍	⨁𝑍	PROPN
iajs-1808	176	22	𝑀	𝑀	PROPN
iajs-1808	176	23	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	176	24	𝑀.	𝑀.	PROPN
iajs-1808	176	25	(	(	PUNCT
iajs-1808	176	26	4	4	NUM
iajs-1808	176	27	)	)	PUNCT
iajs-1808	176	28			NOUN
iajs-1808	176	29	(	(	PUNCT
iajs-1808	176	30	1	1	X
iajs-1808	176	31	)	)	PUNCT
iajs-1808	176	32	let	let	VERB
iajs-1808	176	33	𝐶	𝐶	PROPN
iajs-1808	176	34	be	be	AUX
iajs-1808	176	35	a	a	DET
iajs-1808	176	36	t	t	NOUN
iajs-1808	176	37	-	-	PUNCT
iajs-1808	176	38	closed	closed	NOUN
iajs-1808	176	39	of	of	ADP
iajs-1808	176	40	𝑀	𝑀	PROPN
iajs-1808	176	41	.	.	PUNCT
iajs-1808	177	1	by	by	ADP
iajs-1808	177	2	condition	condition	NOUN
iajs-1808	177	3	(	(	PUNCT
iajs-1808	177	4	4	4	X
iajs-1808	177	5	)	)	PUNCT
iajs-1808	177	6	there	there	PRON
iajs-1808	177	7	exists	exist	VERB
iajs-1808	177	8	a	a	DET
iajs-1808	177	9	fully	fully	ADV
iajs-1808	177	10	invariant	invariant	ADJ
iajs-1808	177	11	direct	direct	ADJ
iajs-1808	177	12	summand	summand	NOUN
iajs-1808	177	13	𝐷	𝐷	PROPN
iajs-1808	177	14	of	of	ADP
iajs-1808	177	15	𝑀	𝑀	PROPN
iajs-1808	177	16	such	such	ADJ
iajs-1808	177	17	that	that	SCONJ
iajs-1808	177	18	𝑀	𝑀	PROPN
iajs-1808	177	19	𝐷	𝐷	PROPN
iajs-1808	177	20	and	and	CCONJ
iajs-1808	177	21	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	177	22	𝑀.	𝑀.	PROPN
iajs-1808	177	23	but	but	CCONJ
iajs-1808	177	24	𝐷	𝐷	PROPN
iajs-1808	177	25	is	be	AUX
iajs-1808	177	26	a	a	DET
iajs-1808	177	27	fully	fully	ADV
iajs-1808	177	28	invariant	invariant	ADJ
iajs-1808	177	29	submodule	submodule	NOUN
iajs-1808	177	30	in	in	ADP
iajs-1808	177	31	𝑀	𝑀	PROPN
iajs-1808	177	32	implies	imply	VERB
iajs-1808	177	33	,	,	PUNCT
iajs-1808	177	34	𝐷	𝐷	PROPN
iajs-1808	177	35	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	177	36	⨁	⨁	PROPN
iajs-1808	177	37	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	177	38	,	,	PUNCT
iajs-1808	177	39	such	such	ADJ
iajs-1808	177	40	that	that	SCONJ
iajs-1808	177	41	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	177	42	is	be	AUX
iajs-1808	177	43	fully	fully	ADV
iajs-1808	177	44	invariant	invariant	ADJ
iajs-1808	177	45	in	in	ADP
iajs-1808	177	46	𝑀	𝑀	PROPN
iajs-1808	177	47	and	and	CCONJ
iajs-1808	177	48	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	177	49	𝑀	𝑀	PROPN
iajs-1808	177	50	since	since	SCONJ
iajs-1808	177	51	𝑀	𝑀	PROPN
iajs-1808	177	52	𝐷.	𝐷.	PROPN
iajs-1808	177	53	hence	hence	ADV
iajs-1808	177	54	𝐷	𝐷	PROPN
iajs-1808	177	55	𝐷⋂𝑀	𝐷⋂𝑀	NOUN
iajs-1808	177	56	⨁𝑀	⨁𝑀	PROPN
iajs-1808	177	57	and	and	CCONJ
iajs-1808	177	58	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	177	59	⨁	⨁	PROPN
iajs-1808	177	60	𝑀	𝑀	PROPN
iajs-1808	177	61	.	.	PUNCT
iajs-1808	178	1	now	now	ADV
iajs-1808	178	2	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	178	3	𝐶⨁	𝐶⨁	PROPN
iajs-1808	178	4	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	178	5	⨁𝑀	⨁𝑀	PROPN
iajs-1808	178	6	𝑀	𝑀	PROPN
iajs-1808	178	7	𝑀	𝑀	PROPN
iajs-1808	178	8	⨁𝑀	⨁𝑀	PROPN
iajs-1808	178	9	.	.	PUNCT
iajs-1808	179	1	hence	hence	ADV
iajs-1808	179	2	𝐶⨁	𝐶⨁	PROPN
iajs-1808	179	3	𝐷⋂𝑀	𝐷⋂𝑀	PROPN
iajs-1808	179	4	𝑀	𝑀	PROPN
iajs-1808	179	5	.	.	PUNCT
iajs-1808	180	1	thus	thus	ADV
iajs-1808	180	2	𝑀	𝑀	PROPN
iajs-1808	180	3	satisfies	satisfy	VERB
iajs-1808	180	4	condition	condition	NOUN
iajs-1808	180	5	(	(	PUNCT
iajs-1808	180	6	5	5	NUM
iajs-1808	180	7	)	)	PUNCT
iajs-1808	180	8	of	of	ADP
iajs-1808	180	9	theorem	theorem	NOUN
iajs-1808	180	10	(	(	PUNCT
iajs-1808	180	11	3.9	3.9	NUM
iajs-1808	180	12	)	)	PUNCT
iajs-1808	180	13	,	,	PUNCT
iajs-1808	180	14	which	which	PRON
iajs-1808	180	15	implies	imply	VERB
iajs-1808	180	16	𝑀	𝑀	PROPN
iajs-1808	180	17	is	be	AUX
iajs-1808	180	18	strongly	strongly	ADV
iajs-1808	180	19	type-𝑇	type-𝑇	NOUN
iajs-1808	180	20	module	module	NOUN
iajs-1808	180	21	.	.	PUNCT
iajs-1808	181	1	□	□	PUNCT
iajs-1808	181	2	for	for	ADP
iajs-1808	181	3	𝑅-modules	𝑅-modules	PROPN
iajs-1808	181	4	𝑁	𝑁	PROPN
iajs-1808	181	5	and	and	CCONJ
iajs-1808	181	6	𝐴.	𝐴.	PROPN
iajs-1808	181	7	𝑁	𝑁	PROPN
iajs-1808	181	8	is	be	AUX
iajs-1808	181	9	said	say	VERB
iajs-1808	181	10	to	to	PART
iajs-1808	181	11	be	be	AUX
iajs-1808	181	12	𝐴projective	𝐴projective	PROPN
iajs-1808	181	13	,	,	PUNCT
iajs-1808	181	14	if	if	SCONJ
iajs-1808	181	15	every	every	DET
iajs-1808	181	16	submodule	submodule	NOUN
iajs-1808	181	17	𝑋	𝑋	PROPN
iajs-1808	181	18	of	of	ADP
iajs-1808	181	19	𝐴	𝐴	PROPN
iajs-1808	181	20	,	,	PUNCT
iajs-1808	181	21	any	any	DET
iajs-1808	181	22	homomorphism	homomorphism	NOUN
iajs-1808	181	23	∅	∅	NOUN
iajs-1808	181	24	:	:	PUNCT
iajs-1808	181	25	𝑁	𝑁	PROPN
iajs-1808	181	26	⟼	⟼	PROPN
iajs-1808	181	27	can	can	AUX
iajs-1808	181	28	be	be	AUX
iajs-1808	181	29	lifted	lift	VERB
iajs-1808	181	30	to	to	ADP
iajs-1808	181	31	a	a	DET
iajs-1808	181	32	homorphism	homorphism	NOUN
iajs-1808	181	33	,	,	PUNCT
iajs-1808	181	34	𝜓	𝜓	NOUN
iajs-1808	181	35	:	:	PUNCT
iajs-1808	181	36	𝑁	𝑁	PROPN
iajs-1808	181	37	⟼	⟼	PROPN
iajs-1808	181	38	𝐴	𝐴	PROPN
iajs-1808	181	39	,	,	PUNCT
iajs-1808	181	40	that	that	PRON
iajs-1808	181	41	is	be	AUX
iajs-1808	181	42	if	if	SCONJ
iajs-1808	181	43	𝜋	𝜋	NOUN
iajs-1808	181	44	:	:	PUNCT
iajs-1808	181	45	𝐴	𝐴	PROPN
iajs-1808	181	46	⟼	⟼	PROPN
iajs-1808	181	47	,	,	PUNCT
iajs-1808	181	48	be	be	AUX
iajs-1808	181	49	the	the	DET
iajs-1808	181	50	-	-	PUNCT
iajs-1808	181	51	natural	natural	ADJ
iajs-1808	181	52	epiomorphism	epiomorphism	NOUN
iajs-1808	181	53	,	,	PUNCT
iajs-1808	181	54	then	then	ADV
iajs-1808	181	55	there	there	PRON
iajs-1808	181	56	exists	exist	VERB
iajs-1808	181	57	a	a	DET
iajs-1808	181	58	homorphism	homorphism	NOUN
iajs-1808	181	59	𝜓	𝜓	NOUN
iajs-1808	181	60	:	:	PUNCT
iajs-1808	181	61	𝑁	𝑁	PROPN
iajs-1808	181	62	⟼	⟼	PROPN
iajs-1808	181	63	𝐴	𝐴	PROPN
iajs-1808	181	64	such	such	ADJ
iajs-1808	181	65	that	that	SCONJ
iajs-1808	181	66	𝜋	𝜋	NOUN
iajs-1808	181	67	∘	∘	PROPN
iajs-1808	181	68	𝜓	𝜓	AUX
iajs-1808	181	69	∅.	∅.	AUX
iajs-1808	181	70	∅𝜓	∅𝜓	VERB
iajs-1808	181	71	a	a	DET
iajs-1808	181	72	  	  	SPACE
iajs-1808	181	73	a	a	NOUN
iajs-1808	181	74	/	/	SYM
iajs-1808	181	75	x	x	NOUN
iajs-1808	181	76	  	  	SPACE
iajs-1808	181	77	n	n	CCONJ
iajs-1808	181	78	  	  	SPACE
iajs-1808	181	79			NOUN
iajs-1808	181	80	  	  	SPACE
iajs-1808	181	81	ihsciconf	ihsciconf	NOUN
iajs-1808	181	82	2017	2017	NUM
iajs-1808	181	83	special	special	ADJ
iajs-1808	181	84	issue	issue	NOUN
iajs-1808	181	85	ibn	ibn	PROPN
iajs-1808	181	86	al	al	PROPN
iajs-1808	181	87	-	-	PUNCT
iajs-1808	181	88	haitham	haitham	PROPN
iajs-1808	181	89	journal	journal	PROPN
iajs-1808	181	90	for	for	ADP
iajs-1808	181	91	pure	pure	ADJ
iajs-1808	181	92	and	and	CCONJ
iajs-1808	181	93	applied	apply	VERB
iajs-1808	181	94	science	science	NOUN
iajs-1808	181	95	https://doi.org/	https://doi.org/	NOUN
iajs-1808	181	96	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	181	97	for	for	ADP
iajs-1808	181	98	more	more	ADJ
iajs-1808	181	99	information	information	NOUN
iajs-1808	181	100	about	about	ADP
iajs-1808	181	101	the	the	DET
iajs-1808	181	102	conference	conference	NOUN
iajs-1808	181	103	please	please	INTJ
iajs-1808	181	104	visit	visit	VERB
iajs-1808	181	105	the	the	DET
iajs-1808	181	106	websites	website	NOUN
iajs-1808	181	107	:	:	PUNCT
iajs-1808	181	108	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	181	109	  	  	SPACE
iajs-1808	181	110	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1808	181	111	   	   	SPACE
iajs-1808	181	112	mathematics|361	mathematics|361	NOUN
iajs-1808	181	113	    	    	SPACE
iajs-1808	181	114	𝑀	𝑀	PROPN
iajs-1808	181	115	is	be	AUX
iajs-1808	181	116	called	call	VERB
iajs-1808	181	117	projective	projective	ADJ
iajs-1808	181	118	if	if	SCONJ
iajs-1808	181	119	𝑀	𝑀	PROPN
iajs-1808	181	120	is	be	AUX
iajs-1808	181	121	𝑁-projective	𝑁-projective	PROPN
iajs-1808	181	122	for	for	ADP
iajs-1808	181	123	every	every	DET
iajs-1808	181	124	𝑅-module	𝑅-module	PROPN
iajs-1808	181	125	𝑁.	𝑁.	PROPN
iajs-1808	181	126	if	if	SCONJ
iajs-1808	181	127	𝑀	𝑀	PROPN
iajs-1808	181	128	is	be	AUX
iajs-1808	181	129	𝑀-projective	𝑀-projective	PROPN
iajs-1808	181	130	,	,	PUNCT
iajs-1808	181	131	𝑀	𝑀	PROPN
iajs-1808	181	132	is	be	AUX
iajs-1808	181	133	called	call	VERB
iajs-1808	181	134	self	self	NOUN
iajs-1808	181	135	-	-	PUNCT
iajs-1808	181	136	projective.	projective.	VERB
iajs-1808	182	1	[	[	X
iajs-1808	182	2	11	11	NUM
iajs-1808	182	3	]	]	SYM
iajs-1808	182	4	proposition	proposition	NOUN
iajs-1808	182	5	(	(	PUNCT
iajs-1808	182	6	3.15	3.15	NUM
iajs-1808	182	7	):	):	PUNCT
iajs-1808	182	8	if	if	SCONJ
iajs-1808	182	9	an	an	DET
iajs-1808	182	10	𝑅-module	𝑅-module	PROPN
iajs-1808	182	11	𝑀	𝑀	PROPN
iajs-1808	182	12	is	be	AUX
iajs-1808	182	13	strongly	strongly	ADV
iajs-1808	182	14	𝑇	𝑇	PROPN
iajs-1808	182	15	-type	-type	NOUN
iajs-1808	182	16	module	module	NOUN
iajs-1808	182	17	and	and	CCONJ
iajs-1808	182	18	𝐿	𝐿	PROPN
iajs-1808	182	19	is	be	AUX
iajs-1808	182	20	fully	fully	ADV
iajs-1808	182	21	invariant	invariant	ADJ
iajs-1808	182	22	direct	direct	ADJ
iajs-1808	182	23	summand	summand	NOUN
iajs-1808	182	24	of	of	ADP
iajs-1808	182	25	𝑀	𝑀	PROPN
iajs-1808	182	26	,	,	PUNCT
iajs-1808	182	27	then	then	ADV
iajs-1808	182	28	𝐿	𝐿	PROPN
iajs-1808	182	29	is	be	AUX
iajs-1808	182	30	strongly	strongly	ADV
iajs-1808	182	31	𝑇	𝑇	PROPN
iajs-1808	182	32	-type	-type	NOUN
iajs-1808	182	33	module	module	NOUN
iajs-1808	182	34	and	and	CCONJ
iajs-1808	182	35	is	be	AUX
iajs-1808	182	36	strongly	strongly	ADV
iajs-1808	182	37	𝑇	𝑇	PROPN
iajs-1808	182	38	-type	-type	NOUN
iajs-1808	182	39	module	module	NOUN
iajs-1808	182	40	if	if	SCONJ
iajs-1808	182	41	𝑀	𝑀	PROPN
iajs-1808	182	42	is	be	AUX
iajs-1808	182	43	selfprojective	selfprojective	ADJ
iajs-1808	182	44	.	.	PUNCT
iajs-1808	183	1	proof	proof	NOUN
iajs-1808	183	2	:	:	PUNCT
iajs-1808	183	3	to	to	PART
iajs-1808	183	4	prove	prove	VERB
iajs-1808	183	5	𝐿	𝐿	PROPN
iajs-1808	183	6	is	be	AUX
iajs-1808	183	7	strongly	strongly	ADV
iajs-1808	183	8	𝑇	𝑇	PROPN
iajs-1808	183	9	-type	-type	NOUN
iajs-1808	183	10	module	module	NOUN
iajs-1808	183	11	.	.	PUNCT
iajs-1808	184	1	let	let	VERB
iajs-1808	184	2	𝐴	𝐴	PROPN
iajs-1808	184	3	be	be	AUX
iajs-1808	184	4	a	a	DET
iajs-1808	184	5	submodule	submodule	NOUN
iajs-1808	184	6	of	of	ADP
iajs-1808	184	7	𝐿	𝐿	PROPN
iajs-1808	184	8	,	,	PUNCT
iajs-1808	184	9	hence	hence	ADV
iajs-1808	184	10	𝐴	𝐴	PROPN
iajs-1808	184	11	is	be	AUX
iajs-1808	184	12	a	a	DET
iajs-1808	184	13	submodule	submodule	NOUN
iajs-1808	184	14	of	of	ADP
iajs-1808	184	15	𝑀	𝑀	PROPN
iajs-1808	184	16	,	,	PUNCT
iajs-1808	184	17	and	and	CCONJ
iajs-1808	184	18	so	so	ADV
iajs-1808	184	19	by	by	ADP
iajs-1808	184	20	condition	condition	NOUN
iajs-1808	184	21	(	(	PUNCT
iajs-1808	184	22	3	3	NUM
iajs-1808	184	23	)	)	PUNCT
iajs-1808	184	24	of	of	ADP
iajs-1808	184	25	theorem	theorem	NOUN
iajs-1808	184	26	3.9	3.9	NUM
iajs-1808	184	27	,	,	PUNCT
iajs-1808	184	28	there	there	PRON
iajs-1808	184	29	exists	exist	VERB
iajs-1808	184	30	a	a	DET
iajs-1808	184	31	fully	fully	ADV
iajs-1808	184	32	invariant	invariant	ADJ
iajs-1808	184	33	direct	direct	ADJ
iajs-1808	184	34	summand	summand	NOUN
iajs-1808	184	35	𝐷	𝐷	PROPN
iajs-1808	184	36	of	of	ADP
iajs-1808	184	37	𝑀	𝑀	PROPN
iajs-1808	184	38	,	,	PUNCT
iajs-1808	184	39	such	such	ADJ
iajs-1808	184	40	that	that	SCONJ
iajs-1808	184	41	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	184	42	𝑀.	𝑀.	NOUN
iajs-1808	184	43	hence	hence	ADV
iajs-1808	184	44	𝐴⨁𝐷	𝐴⨁𝐷	NOUN
iajs-1808	184	45	∩	∩	X
iajs-1808	184	46	𝐿	𝐿	PROPN
iajs-1808	184	47	𝐿	𝐿	PROPN
iajs-1808	184	48	and	and	CCONJ
iajs-1808	184	49	so	so	ADV
iajs-1808	184	50	𝐴⨁	𝐴⨁	PROPN
iajs-1808	184	51	𝐷	𝐷	PROPN
iajs-1808	184	52	∩	∩	ADJ
iajs-1808	184	53	𝐿	𝐿	PROPN
iajs-1808	184	54	𝐿.	𝐿.	VERB
iajs-1808	184	55	on	on	ADP
iajs-1808	184	56	the	the	DET
iajs-1808	184	57	other	other	ADJ
iajs-1808	184	58	hand	hand	NOUN
iajs-1808	184	59	,	,	PUNCT
iajs-1808	184	60	𝐷	𝐷	PROPN
iajs-1808	184	61	⨁	⨁	PROPN
iajs-1808	184	62	𝑀	𝑀	PROPN
iajs-1808	184	63	implies	imply	VERB
iajs-1808	184	64	𝑀	𝑀	PROPN
iajs-1808	184	65	𝐷⨁𝐷	𝐷⨁𝐷	NOUN
iajs-1808	184	66	for	for	ADP
iajs-1808	184	67	some	some	DET
iajs-1808	184	68	𝐷	𝐷	NOUN
iajs-1808	184	69	𝑀.	𝑀.	PROPN
iajs-1808	184	70	as	as	SCONJ
iajs-1808	184	71	𝐿	𝐿	PROPN
iajs-1808	184	72	is	be	AUX
iajs-1808	184	73	fully	fully	ADV
iajs-1808	184	74	invariant	invariant	ADJ
iajs-1808	184	75	submodule	submodule	NOUN
iajs-1808	184	76	in	in	ADP
iajs-1808	184	77	𝑀	𝑀	PROPN
iajs-1808	184	78	,	,	PUNCT
iajs-1808	184	79	𝐿	𝐿	PROPN
iajs-1808	184	80	𝐷	𝐷	PROPN
iajs-1808	184	81	∩	∩	ADJ
iajs-1808	184	82	𝐿	𝐿	PROPN
iajs-1808	184	83	⨁	⨁	PROPN
iajs-1808	184	84	𝐷	𝐷	PROPN
iajs-1808	184	85	∩	∩	ADJ
iajs-1808	184	86	𝐿	𝐿	PROPN
iajs-1808	184	87	,	,	PUNCT
iajs-1808	184	88	where	where	SCONJ
iajs-1808	184	89	𝐷	𝐷	NOUN
iajs-1808	184	90	∩	∩	ADJ
iajs-1808	184	91	𝐿	𝐿	PROPN
iajs-1808	184	92	is	be	AUX
iajs-1808	184	93	fully	fully	ADV
iajs-1808	184	94	invariant	invariant	ADJ
iajs-1808	184	95	in	in	ADP
iajs-1808	184	96	𝐷	𝐷	PROPN
iajs-1808	184	97	,	,	PUNCT
iajs-1808	184	98	𝐷	𝐷	PROPN
iajs-1808	184	99	⋂𝐿	⋂𝐿	PROPN
iajs-1808	184	100	is	be	AUX
iajs-1808	184	101	fully	fully	ADV
iajs-1808	184	102	invariant	invariant	ADJ
iajs-1808	184	103	submodule	submodule	NOUN
iajs-1808	184	104	in	in	ADP
iajs-1808	184	105	𝐷	𝐷	PROPN
iajs-1808	184	106	.now	.now	NOUN
iajs-1808	185	1	𝐷⋂𝐿	𝐷⋂𝐿	NOUN
iajs-1808	185	2	is	be	AUX
iajs-1808	185	3	fully	fully	ADV
iajs-1808	185	4	invariant	invariant	ADJ
iajs-1808	185	5	in	in	ADP
iajs-1808	185	6	𝐷	𝐷	PROPN
iajs-1808	185	7	and	and	CCONJ
iajs-1808	185	8	𝐷	𝐷	PROPN
iajs-1808	185	9	is	be	AUX
iajs-1808	185	10	fully	fully	ADV
iajs-1808	185	11	invariant	invariant	ADJ
iajs-1808	185	12	in	in	ADP
iajs-1808	185	13	𝑀	𝑀	PROPN
iajs-1808	185	14	,	,	PUNCT
iajs-1808	185	15	so	so	ADV
iajs-1808	185	16	𝐷⋂𝐿	𝐷⋂𝐿	NOUN
iajs-1808	185	17	is	be	AUX
iajs-1808	185	18	fully	fully	ADV
iajs-1808	185	19	invariant	invariant	ADJ
iajs-1808	185	20	in	in	ADP
iajs-1808	185	21	𝑀.	𝑀.	PROPN
iajs-1808	185	22	also	also	ADV
iajs-1808	185	23	𝐷	𝐷	NOUN
iajs-1808	185	24	∩	∩	ADJ
iajs-1808	185	25	𝐿	𝐿	PROPN
iajs-1808	185	26	⨁	⨁	PROPN
iajs-1808	185	27	𝐿	𝐿	PROPN
iajs-1808	185	28	,	,	PUNCT
iajs-1808	185	29	𝐿	𝐿	PROPN
iajs-1808	185	30	⨁	⨁	PROPN
iajs-1808	185	31	𝑀	𝑀	PROPN
iajs-1808	185	32	,	,	PUNCT
iajs-1808	185	33	so	so	SCONJ
iajs-1808	185	34	𝐷	𝐷	PROPN
iajs-1808	185	35	∩	∩	ADJ
iajs-1808	185	36	𝐿	𝐿	PROPN
iajs-1808	185	37	⨁	⨁	PROPN
iajs-1808	185	38	𝑀.	𝑀.	PROPN
iajs-1808	185	39	thus	thus	ADV
iajs-1808	185	40	by	by	ADP
iajs-1808	185	41	[	[	X
iajs-1808	185	42	4	4	NUM
iajs-1808	185	43	,	,	PUNCT
iajs-1808	185	44	lemma	lemma	PROPN
iajs-1808	185	45	2.3	2.3	NUM
iajs-1808	185	46	]	]	PUNCT
iajs-1808	185	47	𝐷	𝐷	NOUN
iajs-1808	185	48	∩	∩	ADJ
iajs-1808	185	49	𝐿	𝐿	PROPN
iajs-1808	185	50	is	be	AUX
iajs-1808	185	51	fully	fully	ADV
iajs-1808	185	52	invariant	invariant	ADJ
iajs-1808	185	53	direct	direct	ADJ
iajs-1808	185	54	summand	summand	NOUN
iajs-1808	185	55	of	of	ADP
iajs-1808	185	56	𝑀.	𝑀.	PROPN
iajs-1808	185	57	let	let	AUX
iajs-1808	185	58	be	be	AUX
iajs-1808	185	59	a	a	DET
iajs-1808	185	60	t	t	NOUN
iajs-1808	185	61	-	-	PUNCT
iajs-1808	185	62	closed	close	VERB
iajs-1808	185	63	submodule	submodule	NOUN
iajs-1808	185	64	in	in	ADP
iajs-1808	185	65	.	.	PUNCT
iajs-1808	186	1	then	then	ADV
iajs-1808	186	2	𝐶	𝐶	PROPN
iajs-1808	186	3	is	be	AUX
iajs-1808	186	4	a	a	DET
iajs-1808	186	5	t	t	NOUN
iajs-1808	186	6	-	-	PUNCT
iajs-1808	186	7	closed	close	VERB
iajs-1808	186	8	in	in	ADP
iajs-1808	186	9	𝑀.	𝑀.	PROPN
iajs-1808	186	10	as	as	SCONJ
iajs-1808	186	11	𝑀	𝑀	PROPN
iajs-1808	186	12	is	be	AUX
iajs-1808	186	13	strongly	strongly	ADV
iajs-1808	186	14	𝑇	𝑇	PROPN
iajs-1808	186	15	-type	-type	NOUN
iajs-1808	186	16	module	module	NOUN
iajs-1808	186	17	there	there	PRON
iajs-1808	186	18	exists	exist	VERB
iajs-1808	186	19	a	a	DET
iajs-1808	186	20	fully	fully	ADV
iajs-1808	186	21	invariant	invariant	ADJ
iajs-1808	186	22	direct	direct	ADJ
iajs-1808	186	23	summand	summand	NOUN
iajs-1808	186	24	𝐷	𝐷	PROPN
iajs-1808	186	25	of	of	ADP
iajs-1808	186	26	𝑀	𝑀	PROPN
iajs-1808	186	27	such	such	ADJ
iajs-1808	186	28	that	that	PRON
iajs-1808	186	29	𝐶⨁𝐷	𝐶⨁𝐷	VERB
iajs-1808	186	30	𝑀	𝑀	PROPN
iajs-1808	186	31	by	by	ADP
iajs-1808	186	32	theorem	theorem	ADJ
iajs-1808	186	33	3.9	3.9	NUM
iajs-1808	186	34	.	.	PUNCT
iajs-1808	187	1	let	let	VERB
iajs-1808	187	2	𝑀	𝑀	PROPN
iajs-1808	187	3	𝐷⨁𝐷	𝐷⨁𝐷	VERB
iajs-1808	187	4	for	for	ADP
iajs-1808	187	5	some	some	DET
iajs-1808	187	6	𝐷	𝐷	NOUN
iajs-1808	187	7	𝑀	𝑀	PROPN
iajs-1808	187	8	and	and	CCONJ
iajs-1808	187	9	since	since	SCONJ
iajs-1808	187	10	𝐿	𝐿	PROPN
iajs-1808	187	11	is	be	AUX
iajs-1808	187	12	fully	fully	ADV
iajs-1808	187	13	invariant	invariant	ADJ
iajs-1808	187	14	in	in	ADP
iajs-1808	187	15	𝑀	𝑀	PROPN
iajs-1808	187	16	,	,	PUNCT
iajs-1808	187	17	𝐿	𝐿	PROPN
iajs-1808	187	18	𝐷⋂𝐿	𝐷⋂𝐿	NOUN
iajs-1808	187	19	⨁	⨁	PROPN
iajs-1808	187	20	𝐷	𝐷	PROPN
iajs-1808	187	21	⋂𝐿	⋂𝐿	NOUN
iajs-1808	187	22	such	such	ADJ
iajs-1808	187	23	that	that	DET
iajs-1808	187	24	𝐷⋂𝐿	𝐷⋂𝐿	NOUN
iajs-1808	187	25	is	be	AUX
iajs-1808	187	26	fully	fully	ADV
iajs-1808	187	27	invariant	invariant	ADJ
iajs-1808	187	28	in	in	ADP
iajs-1808	187	29	𝐷	𝐷	PROPN
iajs-1808	187	30	,	,	PUNCT
iajs-1808	187	31	𝐷	𝐷	PROPN
iajs-1808	187	32	⋂𝐿	⋂𝐿	PROPN
iajs-1808	187	33	is	be	AUX
iajs-1808	187	34	fully	fully	ADV
iajs-1808	187	35	invariant	invariant	ADJ
iajs-1808	187	36	in	in	ADP
iajs-1808	187	37	𝐷	𝐷	PROPN
iajs-1808	187	38	.	.	PUNCT
iajs-1808	188	1	then	then	ADV
iajs-1808	188	2	⨁	⨁	PROPN
iajs-1808	188	3	⋂	⋂	PROPN
iajs-1808	188	4	⨁	⨁	PROPN
iajs-1808	188	5	⋂	⋂	PROPN
iajs-1808	188	6	≃	≃	PROPN
iajs-1808	188	7	⋂	⋂	PROPN
iajs-1808	188	8	⨁	⨁	PROPN
iajs-1808	188	9	⋂	⋂	PROPN
iajs-1808	188	10	≃	≃	PROPN
iajs-1808	188	11	⨁	⨁	PROPN
iajs-1808	188	12	.	.	PUNCT
iajs-1808	189	1	but	but	CCONJ
iajs-1808	189	2	it	it	PRON
iajs-1808	189	3	is	be	AUX
iajs-1808	189	4	easy	easy	ADJ
iajs-1808	189	5	to	to	PART
iajs-1808	189	6	see	see	VERB
iajs-1808	189	7	that	that	PRON
iajs-1808	189	8	⨁	⨁	PROPN
iajs-1808	189	9	.	.	PUNCT
iajs-1808	190	1	as	as	ADP
iajs-1808	190	2	𝐿	𝐿	PROPN
iajs-1808	190	3	⨁	⨁	PROPN
iajs-1808	190	4	𝑀	𝑀	PROPN
iajs-1808	190	5	,	,	PUNCT
iajs-1808	190	6	𝐿	𝐿	PROPN
iajs-1808	190	7	is	be	AUX
iajs-1808	190	8	closed	close	VERB
iajs-1808	190	9	and	and	CCONJ
iajs-1808	190	10	this	this	PRON
iajs-1808	190	11	implies	imply	VERB
iajs-1808	190	12	that	that	SCONJ
iajs-1808	190	13	⨁	⨁	PROPN
iajs-1808	190	14	by	by	ADP
iajs-1808	190	15	[	[	X
iajs-1808	190	16	8,proposition	8,proposition	NUM
iajs-1808	190	17	1.4,p.18	1.4,p.18	NUM
iajs-1808	190	18	]	]	PUNCT
iajs-1808	190	19	.	.	PUNCT
iajs-1808	191	1	thus	thus	ADV
iajs-1808	191	2	⨁	⨁	PROPN
iajs-1808	191	3	.	.	PUNCT
iajs-1808	192	1	on	on	ADP
iajs-1808	192	2	the	the	DET
iajs-1808	192	3	other	other	ADJ
iajs-1808	192	4	hand	hand	NOUN
iajs-1808	192	5	,	,	PUNCT
iajs-1808	192	6	since	since	SCONJ
iajs-1808	192	7	𝐷	𝐷	NOUN
iajs-1808	192	8	is	be	AUX
iajs-1808	192	9	a	a	DET
iajs-1808	192	10	fully	fully	ADV
iajs-1808	192	11	invariant	invariant	ADJ
iajs-1808	192	12	submodule	submodule	NOUN
iajs-1808	192	13	in	in	ADP
iajs-1808	192	14	𝑀	𝑀	PROPN
iajs-1808	192	15	and	and	CCONJ
iajs-1808	192	16	,	,	PUNCT
iajs-1808	192	17	𝐿	𝐿	PROPN
iajs-1808	192	18	is	be	AUX
iajs-1808	192	19	fully	fully	ADV
iajs-1808	192	20	invariant	invariant	ADJ
iajs-1808	192	21	in	in	ADP
iajs-1808	192	22	𝑀	𝑀	PROPN
iajs-1808	192	23	,	,	PUNCT
iajs-1808	192	24	then	then	ADV
iajs-1808	192	25	𝐷⨁𝐿	𝐷⨁𝐿	PROPN
iajs-1808	192	26	is	be	AUX
iajs-1808	192	27	fully	fully	ADV
iajs-1808	192	28	invariant	invariant	ADJ
iajs-1808	192	29	in	in	ADP
iajs-1808	192	30	𝑀.	𝑀.	PROPN
iajs-1808	192	31	hence	hence	ADV
iajs-1808	192	32	is	be	AUX
iajs-1808	192	33	fully	fully	ADV
iajs-1808	192	34	invariant	invariant	ADJ
iajs-1808	192	35	in	in	ADP
iajs-1808	192	36	by	by	ADP
iajs-1808	192	37	[	[	X
iajs-1808	192	38	16	16	NUM
iajs-1808	192	39	,	,	PUNCT
iajs-1808	192	40	lemma	lemma	PROPN
iajs-1808	192	41	1.1.20(2	1.1.20(2	PROPN
iajs-1808	192	42	)	)	PUNCT
iajs-1808	192	43	]	]	PUNCT
iajs-1808	193	1	(	(	PUNCT
iajs-1808	193	2	since	since	SCONJ
iajs-1808	193	3	𝑀	𝑀	PROPN
iajs-1808	193	4	is	be	AUX
iajs-1808	193	5	self	self	NOUN
iajs-1808	193	6	-	-	PUNCT
iajs-1808	193	7	projective	projective	NOUN
iajs-1808	193	8	)	)	PUNCT
iajs-1808	193	9	.	.	PUNCT
iajs-1808	194	1	thus	thus	ADV
iajs-1808	194	2	is	be	AUX
iajs-1808	194	3	a	a	DET
iajs-1808	194	4	fully	fully	ADV
iajs-1808	194	5	invariant	invariant	ADJ
iajs-1808	194	6	direct	direct	ADJ
iajs-1808	194	7	summand	summand	NOUN
iajs-1808	194	8	of	of	ADP
iajs-1808	194	9	and	and	CCONJ
iajs-1808	194	10	⨁	⨁	PROPN
iajs-1808	194	11	.	.	PUNCT
iajs-1808	195	1	therefore	therefore	ADV
iajs-1808	195	2	is	be	AUX
iajs-1808	195	3	strongly	strongly	ADV
iajs-1808	195	4	𝑇	𝑇	PROPN
iajs-1808	195	5	-type	-type	NOUN
iajs-1808	195	6	module	module	NOUN
iajs-1808	195	7	by	by	ADP
iajs-1808	195	8	theorem	theorem	ADJ
iajs-1808	195	9	3.9(13	3.9(13	PROPN
iajs-1808	195	10	)	)	PUNCT
iajs-1808	195	11	.	.	PUNCT
iajs-1808	196	1	□	□	PUNCT
iajs-1808	196	2	corollary	corollary	ADJ
iajs-1808	196	3	(	(	PUNCT
iajs-1808	196	4	3.16	3.16	NUM
iajs-1808	196	5	):	):	PUNCT
iajs-1808	196	6	if	if	SCONJ
iajs-1808	196	7	𝑅	𝑅	PROPN
iajs-1808	196	8	is	be	AUX
iajs-1808	196	9	a	a	DET
iajs-1808	196	10	commutative	commutative	ADJ
iajs-1808	196	11	strongly	strongly	ADV
iajs-1808	196	12	𝑇	𝑇	PROPN
iajs-1808	196	13	-type	-type	NOUN
iajs-1808	196	14	module	module	NOUN
iajs-1808	196	15	and	and	CCONJ
iajs-1808	196	16	𝐿	𝐿	PROPN
iajs-1808	196	17	⨁	⨁	PROPN
iajs-1808	196	18	𝑅	𝑅	PROPN
iajs-1808	196	19	,	,	PUNCT
iajs-1808	196	20	then	then	ADV
iajs-1808	196	21	is	be	AUX
iajs-1808	196	22	a	a	DET
iajs-1808	196	23	strongly	strongly	ADV
iajs-1808	196	24	𝑇	𝑇	NOUN
iajs-1808	196	25	-type	-type	NOUN
iajs-1808	196	26	module	module	NOUN
iajs-1808	196	27	.	.	PUNCT
iajs-1808	197	1	corollary	corollary	NOUN
iajs-1808	197	2	(	(	PUNCT
iajs-1808	197	3	3.17	3.17	NUM
iajs-1808	197	4	):	):	PUNCT
iajs-1808	197	5	let	let	VERB
iajs-1808	197	6	𝑀	𝑀	PRON
iajs-1808	197	7	be	be	AUX
iajs-1808	197	8	a	a	DET
iajs-1808	197	9	multiplication	multiplication	NOUN
iajs-1808	197	10	strongly	strongly	ADV
iajs-1808	197	11	𝑇	𝑇	PROPN
iajs-1808	197	12	-type	-type	NOUN
iajs-1808	197	13	module	module	NOUN
iajs-1808	197	14	and	and	CCONJ
iajs-1808	197	15	𝐿	𝐿	PROPN
iajs-1808	197	16	⨁	⨁	PROPN
iajs-1808	197	17	𝑀.	𝑀.	PROPN
iajs-1808	197	18	then	then	ADV
iajs-1808	197	19	is	be	AUX
iajs-1808	197	20	strongly	strongly	ADV
iajs-1808	197	21	𝑇	𝑇	PROPN
iajs-1808	197	22	-type	-type	NOUN
iajs-1808	197	23	modu	modu	NOUN
iajs-1808	197	24	references	reference	NOUN
iajs-1808	197	25	[	[	X
iajs-1808	197	26	1	1	NUM
iajs-1808	197	27	]	]	PUNCT
iajs-1808	197	28	m.	m.	NOUN
iajs-1808	197	29	s.	s.	PROPN
iajs-1808	197	30	abas	abas	PROPN
iajs-1808	197	31	on	on	ADP
iajs-1808	197	32	fully	fully	ADV
iajs-1808	197	33	stable	stable	ADJ
iajs-1808	197	34	modules	module	NOUN
iajs-1808	197	35	ph.d	ph.d	PROPN
iajs-1808	197	36	thesis	thesis	PROPN
iajs-1808	197	37	college	college	PROPN
iajs-1808	197	38	of	of	ADP
iajs-1808	197	39	science	science	PROPN
iajs-1808	197	40	university	university	PROPN
iajs-1808	197	41	of	of	ADP
iajs-1808	197	42	baghdad	baghdad	PROPN
iajs-1808	197	43	.	.	PUNCT
iajs-1808	198	1	1991	1991	NUM
iajs-1808	199	1	[	[	X
iajs-1808	199	2	2	2	X
iajs-1808	199	3	]	]	X
iajs-1808	199	4	sh	sh	PROPN
iajs-1808	199	5	.	.	PROPN
iajs-1808	199	6	asgari	asgari	PROPN
iajs-1808	199	7	and	and	CCONJ
iajs-1808	199	8	a.	a.	NOUN
iajs-1808	199	9	haghany	haghany	PROPN
iajs-1808	199	10	t	t	PROPN
iajs-1808	199	11	-	-	PUNCT
iajs-1808	199	12	extending	extend	VERB
iajs-1808	199	13	modules	module	NOUN
iajs-1808	199	14	and	and	CCONJ
iajs-1808	199	15	t	t	PROPN
iajs-1808	199	16	-	-	PUNCT
iajs-1808	199	17	baer	baer	PROPN
iajs-1808	199	18	modules	module	NOUN
iajs-1808	199	19	comm.algebra	comm.algebra	NOUN
iajs-1808	199	20	39	39	NUM
iajs-1808	199	21	1605	1605	NUM
iajs-1808	199	22	-	-	SYM
iajs-1808	199	23	1623	1623	NUM
iajs-1808	199	24	.	.	PUNCT
iajs-1808	200	1	2011	2011	NUM
iajs-1808	201	1	[	[	X
iajs-1808	201	2	3	3	X
iajs-1808	201	3	]	]	X
iajs-1808	201	4	sh	sh	PROPN
iajs-1808	201	5	.	.	PROPN
iajs-1808	201	6	asgari	asgari	PROPN
iajs-1808	201	7	,	,	PUNCT
iajs-1808	201	8	a.	a.	NOUN
iajs-1808	201	9	haghany	haghany	PROPN
iajs-1808	201	10	and	and	CCONJ
iajs-1808	201	11	y	y	PROPN
iajs-1808	201	12	tolooei	tolooei	PROPN
iajs-1808	201	13	.	.	PUNCT
iajs-1808	202	1	t	t	NOUN
iajs-1808	202	2	-	-	PUNCT
iajs-1808	202	3	semisimple	semisimple	NOUN
iajs-1808	202	4	modules	module	NOUN
iajs-1808	202	5	and	and	CCONJ
iajs-1808	202	6	t	t	NOUN
iajs-1808	202	7	-	-	PUNCT
iajs-1808	202	8	semisimple	semisimple	NOUN
iajs-1808	202	9	rings	ring	NOUN
iajs-1808	202	10	comm	comm	NOUN
iajs-1808	202	11	algebra	algebra	NOUN
iajs-1808	202	12	41	41	NUM
iajs-1808	202	13	5	5	NUM
iajs-1808	202	14	.1882	.1882	PROPN
iajs-1808	202	15	-	-	PUNCT
iajs-1808	202	16	1902	1902	NUM
iajs-1808	202	17	.	.	PUNCT
iajs-1808	203	1	2013	2013	NUM
iajs-1808	204	1	[	[	X
iajs-1808	204	2	4	4	X
iajs-1808	204	3	]	]	X
iajs-1808	204	4	sh	sh	PROPN
iajs-1808	204	5	.	.	PROPN
iajs-1808	204	6	asgari	asgari	PROPN
iajs-1808	204	7	,	,	PUNCT
iajs-1808	204	8	a.	a.	NOUN
iajs-1808	204	9	haghany	haghany	NOUN
iajs-1808	204	10	and	and	CCONJ
iajs-1808	204	11	a	a	DET
iajs-1808	204	12	.r	.r	NOUN
iajs-1808	204	13	.rezaei	.rezaei	X
iajs-1808	204	14	.	.	PUNCT
iajs-1808	205	1	module	module	NOUN
iajs-1808	205	2	whose	whose	DET
iajs-1808	205	3	t	t	NOUN
iajs-1808	205	4	-	-	PUNCT
iajs-1808	205	5	closed	close	VERB
iajs-1808	205	6	submodules	submodule	NOUN
iajs-1808	205	7	have	have	VERB
iajs-1808	205	8	a	a	DET
iajs-1808	205	9	summand	summand	NOUN
iajs-1808	205	10	as	as	ADP
iajs-1808	205	11	a	a	DET
iajs-1808	205	12	complement	complement	NOUN
iajs-1808	205	13	comm.algebra	comm.algebra	NOUN
iajs-1808	205	14	42	42	NUM
iajs-1808	205	15	pp	pp	NOUN
iajs-1808	205	16	.	.	PUNCT
iajs-1808	206	1	5299–5318	5299–5318	NUM
iajs-1808	206	2	.	.	PUNCT
iajs-1808	206	3	2014	2014	NUM
iajs-1808	207	1	[	[	X
iajs-1808	207	2	5	5	NUM
iajs-1808	207	3	]	]	PUNCT
iajs-1808	207	4	a.	a.	NOUN
iajs-1808	207	5	w.	w.	PROPN
iajs-1808	207	6	chatters	chatter	VERB
iajs-1808	207	7	and	and	CCONJ
iajs-1808	207	8	s.	s.	PROPN
iajs-1808	207	9	m.	m.	PROPN
iajs-1808	207	10	khuri	khuri	PROPN
iajs-1808	207	11	.endomorphism	.endomorphism	PROPN
iajs-1808	207	12	rings	ring	NOUN
iajs-1808	207	13	of	of	ADP
iajs-1808	207	14	modules	module	NOUN
iajs-1808	207	15	over	over	ADP
iajs-1808	207	16	nonsingular	nonsingular	PROPN
iajs-1808	207	17	cs	cs	PROPN
iajs-1808	207	18	rings	rings	PROPN
iajs-1808	207	19	j.	j.	PROPN
iajs-1808	207	20	london	london	PROPN
iajs-1808	207	21	math	math	PROPN
iajs-1808	207	22	.	.	PUNCT
iajs-1808	208	1	soc	soc	PROPN
iajs-1808	208	2	.	.	PUNCT
iajs-1808	209	1	21	21	NUM
iajs-1808	209	2	.	.	PUNCT
iajs-1808	210	1	434	434	NUM
iajs-1808	210	2	-	-	SYM
iajs-1808	210	3	444	444	NUM
iajs-1808	210	4	.	.	PUNCT
iajs-1808	211	1	[	[	X
iajs-1808	211	2	6	6	NUM
iajs-1808	211	3	]	]	X
iajs-1808	211	4	n.	n.	NOUN
iajs-1808	211	5	v.	v.	ADP
iajs-1808	211	6	dung	dung	PROPN
iajs-1808	211	7	,	,	PUNCT
iajs-1808	211	8	d.	d.	PROPN
iajs-1808	211	9	v.	v.	PROPN
iajs-1808	211	10	huynh	huynh	PROPN
iajs-1808	211	11	,	,	PUNCT
iajs-1808	211	12	p.	p.	PROPN
iajs-1808	211	13	f.	f.	PROPN
iajs-1808	211	14	smith	smith	PROPN
iajs-1808	211	15	and	and	CCONJ
iajs-1808	211	16	r.	r.	PROPN
iajs-1808	211	17	wisbauer	wisbauer	PROPN
iajs-1808	211	18	.	.	PUNCT
iajs-1808	212	1	extending	extend	VERB
iajs-1808	212	2	modules	module	NOUN
iajs-1808	212	3	pitman	pitman	NOUN
iajs-1808	212	4	ihsciconf	ihsciconf	PROPN
iajs-1808	212	5	2017	2017	NUM
iajs-1808	212	6	special	special	ADJ
iajs-1808	212	7	issue	issue	NOUN
iajs-1808	212	8	ibn	ibn	PROPN
iajs-1808	212	9	al	al	PROPN
iajs-1808	212	10	-	-	PUNCT
iajs-1808	212	11	haitham	haitham	PROPN
iajs-1808	212	12	journal	journal	PROPN
iajs-1808	212	13	for	for	ADP
iajs-1808	212	14	pure	pure	ADJ
iajs-1808	212	15	and	and	CCONJ
iajs-1808	212	16	applied	apply	VERB
iajs-1808	212	17	science	science	NOUN
iajs-1808	212	18	https://doi.org/	https://doi.org/	NOUN
iajs-1808	212	19	10.30526/2017.ihsciconf.1808	10.30526/2017.ihsciconf.1808	NUM
iajs-1808	212	20	for	for	ADP
iajs-1808	212	21	more	more	ADJ
iajs-1808	212	22	information	information	NOUN
iajs-1808	212	23	about	about	ADP
iajs-1808	212	24	the	the	DET
iajs-1808	212	25	conference	conference	NOUN
iajs-1808	212	26	please	please	INTJ
iajs-1808	212	27	visit	visit	VERB
iajs-1808	212	28	the	the	DET
iajs-1808	212	29	websites	website	NOUN
iajs-1808	212	30	:	:	PUNCT
iajs-1808	212	31	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1808	212	32	  	  	SPACE
iajs-1808	212	33	www.ihsciconf.org	www.ihsciconf.org	ADV
iajs-1808	212	34	   	   	SPACE
iajs-1808	212	35	mathematics|362	mathematics|362	VERB
iajs-1808	212	36	    	    	SPACE
iajs-1808	212	37	research	research	NOUN
iajs-1808	212	38	notes	note	NOUN
iajs-1808	212	39	in	in	ADP
iajs-1808	212	40	mathematics	mathematics	PROPN
iajs-1808	212	41	313	313	NUM
iajs-1808	212	42	longman	longman	PROPN
iajs-1808	212	43	harlow	harlow	NOUN
iajs-1808	212	44	,	,	PUNCT
iajs-1808	212	45	1994	1994	NUM
iajs-1808	213	1	[	[	X
iajs-1808	213	2	7	7	X
iajs-1808	213	3	]	]	PUNCT
iajs-1808	213	4	s.	s.	PROPN
iajs-1808	213	5	ebrahimi	ebrahimi	PROPN
iajs-1808	213	6	,	,	PUNCT
iajs-1808	213	7	dolati	dolati	PROPN
iajs-1808	213	8	pish	pish	PROPN
iajs-1808	213	9	hesari	hesari	PROPN
iajs-1808	213	10	and	and	CCONJ
iajs-1808	213	11	m.	m.	PROPN
iajs-1808	213	12	khoramdel	khoramdel	PROPN
iajs-1808	213	13	.	.	PUNCT
iajs-1808	214	1	strongly	strongly	ADV
iajs-1808	214	2	t	t	NOUN
iajs-1808	214	3	-	-	PUNCT
iajs-1808	214	4	extending	extend	VERB
iajs-1808	214	5	and	and	CCONJ
iajs-1808	214	6	strongly	strongly	ADV
iajs-1808	214	7	tbaer	tbaer	X
iajs-1808	214	8	international	international	ADJ
iajs-1808	214	9	electronic	electronic	ADJ
iajs-1808	214	10	journal	journal	NOUN
iajs-1808	214	11	of	of	ADP
iajs-1808	214	12	algebra	algebra	PROPN
iajs-1808	214	13	20	20	NUM
iajs-1808	214	14	.	.	PUNCT
iajs-1808	215	1	86	86	NUM
iajs-1808	215	2	-	-	SYM
iajs-1808	215	3	98	98	NUM
iajs-1808	215	4	.	.	PUNCT
iajs-1808	216	1	2016	2016	NUM
iajs-1808	217	1	[	[	X
iajs-1808	217	2	8	8	NUM
iajs-1808	217	3	]	]	PUNCT
iajs-1808	217	4	k.	k.	PROPN
iajs-1808	217	5	r.	r.	PROPN
iajs-1808	217	6	good	good	PROPN
iajs-1808	217	7	earl	earl	PROPN
iajs-1808	217	8	.	.	PUNCT
iajs-1808	218	1	ring	ring	NOUN
iajs-1808	218	2	theory	theory	PROPN
iajs-1808	218	3	non	non	ADJ
iajs-1808	218	4	singular	singular	PROPN
iajs-1808	218	5	rings	ring	NOUN
iajs-1808	218	6	and	and	CCONJ
iajs-1808	218	7	modules	module	NOUN
iajs-1808	218	8	marcel	marcel	PROPN
iajs-1808	218	9	dekker	dekker	PROPN
iajs-1808	218	10	inc	inc	PROPN
iajs-1808	218	11	.	.	PROPN
iajs-1808	218	12	new	new	PROPN
iajs-1808	218	13	york	york	PROPN
iajs-1808	218	14	and	and	CCONJ
iajs-1808	218	15	basel	basel	PROPN
iajs-1808	218	16	.	.	PUNCT
iajs-1808	219	1	1976	1976	NUM
iajs-1808	220	1	[	[	X
iajs-1808	220	2	9	9	NUM
iajs-1808	220	3	]	]	PUNCT
iajs-1808	220	4	m.	m.	NOUN
iajs-1808	220	5	a.	a.	NOUN
iajs-1808	220	6	inaam	inaam	PROPN
iajs-1808	220	7	and	and	CCONJ
iajs-1808	220	8	farhan	farhan	PROPN
iajs-1808	220	9	d	d	PROPN
iajs-1808	220	10	shyaa	shyaa	PROPN
iajs-1808	220	11	.	.	PUNCT
iajs-1808	221	1	strongly	strongly	ADV
iajs-1808	221	2	t	t	NOUN
iajs-1808	221	3	-	-	PUNCT
iajs-1808	221	4	semisimple	semisimple	NOUN
iajs-1808	221	5	modules	module	NOUN
iajs-1808	221	6	and	and	CCONJ
iajs-1808	221	7	strongly	strongly	ADV
iajs-1808	221	8	tsemisimple	tsemisimple	PROPN
iajs-1808	221	9	rings	ring	NOUN
iajs-1808	221	10	international	international	PROPN
iajs-1808	221	11	journal	journal	NOUN
iajs-1808	221	12	of	of	ADP
iajs-1808	221	13	pure	pure	ADJ
iajs-1808	221	14	and	and	CCONJ
iajs-1808	221	15	applied	applied	ADJ
iajs-1808	221	16	mathematics	mathematic	NOUN
iajs-1808	221	17	15	15	NUM
iajs-1808	221	18	1	1	NUM
iajs-1808	221	19	.	.	PUNCT
iajs-1808	222	1	27	27	NUM
iajs-1808	222	2	-	-	SYM
iajs-1808	222	3	41	41	NUM
iajs-1808	222	4	.	.	NOUN
iajs-1808	223	1	2017	2017	NUM
iajs-1808	224	1	[	[	SYM
iajs-1808	224	2	10	10	NUM
iajs-1808	224	3	]	]	PUNCT
iajs-1808	224	4	m.	m.	NOUN
iajs-1808	224	5	a.	a.	NOUN
iajs-1808	224	6	inaam	inaam	PROPN
iajs-1808	224	7	and	and	CCONJ
iajs-1808	224	8	farhan	farhan	PROPN
iajs-1808	224	9	d	d	PROPN
iajs-1808	224	10	shyaa	shyaa	NOUN
iajs-1808	224	11	.	.	PUNCT
iajs-1808	225	1	fi	fi	NOUN
iajs-1808	225	2	-	-	NOUN
iajs-1808	225	3	semisimple	semisimple	ADJ
iajs-1808	225	4	fi	fi	PROPN
iajs-1808	225	5	-	-	PUNCT
iajs-1808	225	6	t	t	NOUN
iajs-1808	225	7	-	-	PUNCT
iajs-1808	225	8	semisimple	semisimple	NOUN
iajs-1808	225	9	and	and	CCONJ
iajs-1808	225	10	strongly	strongly	ADV
iajs-1808	225	11	fi	fi	ADJ
iajs-1808	225	12	-	-	ADJ
iajs-1808	225	13	tsemisimple	tsemisimple	ADJ
iajs-1808	225	14	modules	module	NOUN
iajs-1808	225	15	international	international	ADJ
iajs-1808	225	16	journal	journal	NOUN
iajs-1808	225	17	of	of	ADP
iajs-1808	225	18	pure	pure	ADJ
iajs-1808	225	19	and	and	CCONJ
iajs-1808	225	20	applied	applied	ADJ
iajs-1808	225	21	mathematics	mathematic	NOUN
iajs-1808	225	22	15	15	NUM
iajs-1808	225	23	2	2	NUM
iajs-1808	225	24	.	.	PUNCT
iajs-1808	225	25	285300	285300	NUM
iajs-1808	225	26	.	.	PUNCT
iajs-1808	226	1	2017	2017	NUM
iajs-1808	227	1	[	[	X
iajs-1808	227	2	11	11	NUM
iajs-1808	227	3	]	]	PUNCT
iajs-1808	227	4	s.	s.	PROPN
iajs-1808	227	5	h.	h.	PROPN
iajs-1808	227	6	mohamed	mohamed	PROPN
iajs-1808	227	7	and	and	CCONJ
iajs-1808	227	8	b.	b.	PROPN
iajs-1808	227	9	j.	j.	PROPN
iajs-1808	227	10	muller	muller	PROPN
iajs-1808	227	11	.continuous	.continuous	PROPN
iajs-1808	227	12	and	and	CCONJ
iajs-1808	227	13	discrete	discrete	ADJ
iajs-1808	227	14	modules	module	NOUN
iajs-1808	227	15	cambridge	cambridge	PROPN
iajs-1808	227	16	university	university	PROPN
iajs-1808	227	17	press	press	PROPN
iajs-1808	227	18	cambridge	cambridge	PROPN
iajs-1808	227	19	.	.	PUNCT
iajs-1808	228	1	1990	1990	NUM
iajs-1808	229	1	[	[	X
iajs-1808	229	2	12]a.c	12]a.c	NUM
iajs-1808	229	3	.	.	PUNCT
iajs-1808	229	4	özcan	özcan	PROPN
iajs-1808	229	5	,	,	PUNCT
iajs-1808	229	6	a.	a.	NOUN
iajs-1808	229	7	harmanci	harmanci	PROPN
iajs-1808	229	8	and	and	CCONJ
iajs-1808	229	9	f.	f.	PROPN
iajs-1808	229	10	smithp	smithp	PROPN
iajs-1808	229	11	.	.	PUNCT
iajs-1808	230	1	duo	duo	NOUN
iajs-1808	230	2	modules	module	NOUN
iajs-1808	230	3	glasgow	glasgow	VERB
iajs-1808	230	4	math.j	math.j	NOUN
iajs-1808	230	5	.	.	PROPN
iajs-1808	231	1	48	48	NUM
iajs-1808	231	2	.	.	PUNCT
iajs-1808	232	1	533	533	NUM
iajs-1808	232	2	-	-	SYM
iajs-1808	232	3	545	545	NUM
iajs-1808	232	4	.	.	PUNCT
iajs-1808	233	1	2006	2006	NUM
iajs-1808	234	1	[	[	X
iajs-1808	234	2	13	13	NUM
iajs-1808	234	3	]	]	SYM
iajs-1808	234	4	a	a	PRON
iajs-1808	234	5	.saad	.saad	INTJ
iajs-1808	234	6	al	al	PROPN
iajs-1808	234	7	-	-	PUNCT
iajs-1808	234	8	saadi	saadi	NOUN
iajs-1808	234	9	.	.	PUNCT
iajs-1808	235	1	s	s	X
iajs-1808	235	2	-	-	PUNCT
iajs-1808	235	3	extending	extend	VERB
iajs-1808	235	4	modules	module	NOUN
iajs-1808	235	5	and	and	CCONJ
iajs-1808	235	6	related	related	ADJ
iajs-1808	235	7	concepts	concept	NOUN
iajs-1808	235	8	ph.d	ph.d	PROPN
iajs-1808	235	9	thesis	thesis	PROPN
iajs-1808	235	10	college	college	PROPN
iajs-1808	235	11	of	of	ADP
iajs-1808	235	12	science	science	PROPN
iajs-1808	235	13	almustansiriyah	almustansiriyah	PROPN
iajs-1808	235	14	university	university	PROPN
iajs-1808	235	15	.	.	PUNCT
iajs-1808	236	1	2007	2007	NUM
iajs-1808	237	1	[	[	X
iajs-1808	237	2	14	14	NUM
iajs-1808	237	3	]	]	X
iajs-1808	237	4	saad	saad	PROPN
iajs-1808	237	5	abulkadhim	abulkadhim	PROPN
iajs-1808	237	6	al	al	PROPN
iajs-1808	237	7	-	-	PUNCT
iajs-1808	237	8	saadi	saadi	PROPN
iajs-1808	237	9	and	and	CCONJ
iajs-1808	237	10	tamadher	tamadher	PROPN
iajs-1808	237	11	arif	arif	PROPN
iajs-1808	237	12	ibrahim	ibrahim	PROPN
iajs-1808	237	13	2014	2014	NUM
iajs-1808	237	14	strongly	strongly	ADV
iajs-1808	237	15	rickart	rickart	NOUN
iajs-1808	237	16	modules	module	NOUN
iajs-1808	237	17	,	,	PUNCT
iajs-1808	237	18	journal	journal	NOUN
iajs-1808	237	19	of	of	ADP
iajs-1808	237	20	advances	advance	NOUN
iajs-1808	237	21	in	in	ADP
iajs-1808	237	22	mathematics	mathematic	NOUN
iajs-1808	237	23	19	19	NUM
iajs-1808	237	24	4	4	NUM
iajs-1808	237	25	.	.	PUNCT
iajs-1808	237	26	2507	2507	NUM
iajs-1808	237	27	-	-	SYM
iajs-1808	237	28	2514	2514	NUM
iajs-1808	238	1	[	[	X
iajs-1808	238	2	15	15	NUM
iajs-1808	238	3	]	]	PUNCT
iajs-1808	238	4	p.	p.	PROPN
iajs-1808	238	5	f.	f.	PROPN
iajs-1808	238	6	smith	smith	PROPN
iajs-1808	238	7	and	and	CCONJ
iajs-1808	238	8	a.	a.	NOUN
iajs-1808	238	9	tercan	tercan	PROPN
iajs-1808	238	10	.generalizations	.generalizations	PROPN
iajs-1808	238	11	of	of	ADP
iajs-1808	238	12	cs	cs	ADJ
iajs-1808	238	13	-	-	ADJ
iajs-1808	238	14	modules	module	NOUN
iajs-1808	238	15	comm	comm	NOUN
iajs-1808	238	16	algebra	algebra	NOUN
iajs-1808	238	17	21	21	NUM
iajs-1808	238	18	.	.	PUNCT
iajs-1808	238	19	1809	1809	NUM
iajs-1808	238	20	–	–	PUNCT
iajs-1808	238	21	1847	1847	NUM
iajs-1808	238	22	.	.	PUNCT
iajs-1808	239	1	1993	1993	NUM
iajs-1808	240	1	[	[	X
iajs-1808	240	2	16	16	NUM
iajs-1808	240	3	]	]	X
iajs-1808	240	4	i.	i.	PROPN
iajs-1808	240	5	e.	e.	PROPN
iajs-1808	240	6	wijayanti	wijayanti	PROPN
iajs-1808	240	7	.	.	PUNCT
iajs-1808	241	1	coprime	coprime	NOUN
iajs-1808	241	2	modules	module	NOUN
iajs-1808	241	3	and	and	CCONJ
iajs-1808	241	4	comodules	comodule	VERB
iajs-1808	241	5	ph	ph	PROPN
iajs-1808	241	6	d	d	PROPN
iajs-1808	241	7	thesis	thesis	PROPN
iajs-1808	241	8	university	university	PROPN
iajs-1808	241	9	of	of	ADP
iajs-1808	241	10	dusseidorf	dusseidorf	PROPN
iajs-1808	241	11	.	.	PUNCT
iajs-1808	242	1	2006	2006	NUM
iajs-1808	243	1	[	[	X
iajs-1808	243	2	17	17	NUM
iajs-1808	243	3	]	]	PUNCT
iajs-1808	243	4	zeinb	zeinb	PROPN
iajs-1808	243	5	abd	abd	PROPN
iajs-1808	243	6	el	el	PROPN
iajs-1808	243	7	-	-	PUNCT
iajs-1808	243	8	bast	bast	NOUN
iajs-1808	243	9	and	and	CCONJ
iajs-1808	243	10	parrick	parrick	NOUN
iajs-1808	243	11	f	f	PROPN
iajs-1808	243	12	smith	smith	PROPN
iajs-1808	243	13	.	.	PUNCT
iajs-1808	244	1	multiplication	multiplication	NOUN
iajs-1808	244	2	modules	module	NOUN
iajs-1808	244	3	comm	comm	NOUN
iajs-1808	244	4	algebra	algebra	NOUN
iajs-1808	244	5	16	16	NUM
iajs-1808	244	6	4	4	NUM
iajs-1808	244	7	.	.	PUNCT
iajs-1808	245	1	755	755	NUM
iajs-1808	245	2	-	-	SYM
iajs-1808	245	3	779	779	NUM
iajs-1808	245	4	.	.	PUNCT
iajs-1808	245	5	1988	1988	NUM
iajs-1808	245	6	  	  	SPACE