id	sid	tid	token	lemma	pos
iajs-4130	1	1	398	398	NUM
iajs-4130	1	2	©	©	ADP
iajs-4130	1	3	2025	2025	NUM
iajs-4130	1	4	the	the	DET
iajs-4130	1	5	author(s	author(s	NOUN
iajs-4130	1	6	)	)	PUNCT
iajs-4130	1	7	.	.	PUNCT
iajs-4130	2	1	published	publish	VERB
iajs-4130	2	2	by	by	ADP
iajs-4130	2	3	college	college	NOUN
iajs-4130	2	4	of	of	ADP
iajs-4130	2	5	education	education	NOUN
iajs-4130	2	6	for	for	ADP
iajs-4130	2	7	pure	pure	ADJ
iajs-4130	2	8	science	science	NOUN
iajs-4130	2	9	(	(	PUNCT
iajs-4130	2	10	ibn	ibn	PROPN
iajs-4130	2	11	al	al	PROPN
iajs-4130	2	12	-	-	PUNCT
iajs-4130	2	13	haitham	haitham	PROPN
iajs-4130	2	14	)	)	PUNCT
iajs-4130	2	15	,	,	PUNCT
iajs-4130	2	16	university	university	NOUN
iajs-4130	2	17	of	of	ADP
iajs-4130	2	18	baghdad	baghdad	PROPN
iajs-4130	2	19	.	.	PUNCT
iajs-4130	3	1	this	this	PRON
iajs-4130	3	2	is	be	AUX
iajs-4130	3	3	an	an	DET
iajs-4130	3	4	open	open	ADJ
iajs-4130	3	5	-	-	PUNCT
iajs-4130	3	6	access	access	NOUN
iajs-4130	3	7	article	article	NOUN
iajs-4130	3	8	distributed	distribute	VERB
iajs-4130	3	9	under	under	ADP
iajs-4130	3	10	the	the	DET
iajs-4130	3	11	terms	term	NOUN
iajs-4130	3	12	of	of	ADP
iajs-4130	3	13	the	the	DET
iajs-4130	3	14	creative	creative	ADJ
iajs-4130	3	15	commons	common	NOUN
iajs-4130	3	16	attribution	attribution	NOUN
iajs-4130	3	17	4.0	4.0	NUM
iajs-4130	3	18	international	international	ADJ
iajs-4130	3	19	license	license	NOUN
iajs-4130	3	20	the	the	DET
iajs-4130	3	21	radical	radical	NOUN
iajs-4130	3	22	of	of	ADP
iajs-4130	3	23	an	an	DET
iajs-4130	3	24	endo	endo	NOUN
iajs-4130	3	25	-	-	PUNCT
iajs-4130	3	26	restricted	restrict	VERB
iajs-4130	3	27	bounded	bounded	ADJ
iajs-4130	3	28	submodule	submodule	NOUN
iajs-4130	3	29	related	relate	VERB
iajs-4130	3	30	to	to	ADP
iajs-4130	3	31	prime	prime	ADJ
iajs-4130	3	32	submodules	submodules	PROPN
iajs-4130	3	33	mohammed	mohammed	PROPN
iajs-4130	3	34	salman	salman	PROPN
iajs-4130	3	35	murad	murad	PROPN
iajs-4130	3	36	1	1	NUM
iajs-4130	3	37	,	,	PUNCT
iajs-4130	3	38	buthyna	buthyna	PROPN
iajs-4130	3	39	najad	najad	PROPN
iajs-4130	3	40	shihab	shihab	PROPN
iajs-4130	3	41	2	2	NUM
iajs-4130	3	42	*	*	SYM
iajs-4130	3	43	1,2	1,2	NUM
iajs-4130	3	44	department	department	NOUN
iajs-4130	3	45	of	of	ADP
iajs-4130	3	46	mathematics	mathematic	NOUN
iajs-4130	3	47	,	,	PUNCT
iajs-4130	3	48	college	college	NOUN
iajs-4130	3	49	of	of	ADP
iajs-4130	3	50	education	education	NOUN
iajs-4130	3	51	for	for	ADP
iajs-4130	3	52	pure	pure	ADJ
iajs-4130	3	53	sciences	science	NOUN
iajs-4130	3	54	,	,	PUNCT
iajs-4130	3	55	ibn	ibn	PROPN
iajs-4130	3	56	-	-	PUNCT
iajs-4130	3	57	al	al	PROPN
iajs-4130	3	58	-	-	PUNCT
iajs-4130	3	59	haitham	haitham	PROPN
iajs-4130	3	60	,	,	PUNCT
iajs-4130	3	61	university	university	PROPN
iajs-4130	3	62	of	of	ADP
iajs-4130	3	63	baghdad	baghdad	PROPN
iajs-4130	3	64	,	,	PUNCT
iajs-4130	3	65	baghdad	baghdad	PROPN
iajs-4130	3	66	,	,	PUNCT
iajs-4130	3	67	iraq	iraq	PROPN
iajs-4130	3	68	.	.	PUNCT
iajs-4130	3	69	.	.	PUNCT
iajs-4130	4	1	*	*	PUNCT
iajs-4130	4	2	corresponding	correspond	VERB
iajs-4130	4	3	author	author	NOUN
iajs-4130	4	4	.	.	PUNCT
iajs-4130	5	1	received:10	received:10	PROPN
iajs-4130	5	2	february	february	PROPN
iajs-4130	5	3	2025	2025	NUM
iajs-4130	5	4	accepted	accept	VERB
iajs-4130	5	5	:	:	PUNCT
iajs-4130	5	6	13	13	NUM
iajs-4130	5	7	may	may	AUX
iajs-4130	5	8	2025	2025	NUM
iajs-4130	5	9	published	publish	VERB
iajs-4130	5	10	:	:	PUNCT
iajs-4130	5	11	20	20	NUM
iajs-4130	5	12	october	october	NOUN
iajs-4130	5	13	2025	2025	NUM
iajs-4130	5	14	doi.org/10.30526/38.4.4130	doi.org/10.30526/38.4.4130	ADJ
iajs-4130	5	15	abstract	abstract	NOUN
iajs-4130	5	16	in	in	ADP
iajs-4130	5	17	this	this	DET
iajs-4130	5	18	paper	paper	NOUN
iajs-4130	5	19	,	,	PUNCT
iajs-4130	5	20	we	we	PRON
iajs-4130	5	21	present	present	VERB
iajs-4130	5	22	the	the	DET
iajs-4130	5	23	concept	concept	NOUN
iajs-4130	5	24	of	of	ADP
iajs-4130	5	25	the	the	DET
iajs-4130	5	26	radical	radical	NOUN
iajs-4130	5	27	of	of	ADP
iajs-4130	5	28	an	an	DET
iajs-4130	5	29	endo	endo	NOUN
iajs-4130	5	30	-	-	PUNCT
iajs-4130	5	31	restricted	restrict	VERB
iajs-4130	5	32	bounded	bounded	ADJ
iajs-4130	5	33	submodule	submodule	NOUN
iajs-4130	5	34	and	and	CCONJ
iajs-4130	5	35	establish	establish	VERB
iajs-4130	5	36	its	its	PRON
iajs-4130	5	37	characterization	characterization	NOUN
iajs-4130	5	38	,	,	PUNCT
iajs-4130	5	39	which	which	PRON
iajs-4130	5	40	is	be	AUX
iajs-4130	5	41	regarded	regard	VERB
iajs-4130	5	42	as	as	ADP
iajs-4130	5	43	a	a	DET
iajs-4130	5	44	new	new	ADJ
iajs-4130	5	45	notion	notion	NOUN
iajs-4130	5	46	.	.	PUNCT
iajs-4130	6	1	in	in	ADP
iajs-4130	6	2	addition	addition	NOUN
iajs-4130	6	3	,	,	PUNCT
iajs-4130	6	4	we	we	PRON
iajs-4130	6	5	study	study	VERB
iajs-4130	6	6	the	the	DET
iajs-4130	6	7	relationship	relationship	NOUN
iajs-4130	6	8	between	between	ADP
iajs-4130	6	9	the	the	DET
iajs-4130	6	10	radical	radical	NOUN
iajs-4130	6	11	of	of	ADP
iajs-4130	6	12	submodules	submodule	NOUN
iajs-4130	6	13	and	and	CCONJ
iajs-4130	6	14	the	the	DET
iajs-4130	6	15	radical	radical	NOUN
iajs-4130	6	16	of	of	ADP
iajs-4130	6	17	an	an	DET
iajs-4130	6	18	endorestricted	endorestricte	VERB
iajs-4130	6	19	bounded	bounded	ADJ
iajs-4130	6	20	submodules	submodule	NOUN
iajs-4130	6	21	and	and	CCONJ
iajs-4130	6	22	this	this	DET
iajs-4130	6	23	connection	connection	NOUN
iajs-4130	6	24	will	will	AUX
iajs-4130	6	25	give	give	VERB
iajs-4130	6	26	us	we	PRON
iajs-4130	6	27	important	important	ADJ
iajs-4130	6	28	results	result	NOUN
iajs-4130	6	29	in	in	ADP
iajs-4130	6	30	terms	term	NOUN
iajs-4130	6	31	of	of	ADP
iajs-4130	6	32	their	their	PRON
iajs-4130	6	33	radical	radical	ADJ
iajs-4130	6	34	.	.	PUNCT
iajs-4130	7	1	furthermore	furthermore	ADV
iajs-4130	7	2	,	,	PUNCT
iajs-4130	7	3	this	this	DET
iajs-4130	7	4	article	article	NOUN
iajs-4130	7	5	will	will	AUX
iajs-4130	7	6	demonstrate	demonstrate	VERB
iajs-4130	7	7	numerous	numerous	ADJ
iajs-4130	7	8	properties	property	NOUN
iajs-4130	7	9	and	and	CCONJ
iajs-4130	7	10	corollaries	corollary	NOUN
iajs-4130	7	11	that	that	PRON
iajs-4130	7	12	elucidate	elucidate	VERB
iajs-4130	7	13	the	the	DET
iajs-4130	7	14	concept	concept	NOUN
iajs-4130	7	15	of	of	ADP
iajs-4130	7	16	the	the	DET
iajs-4130	7	17	endo	endo	NOUN
iajs-4130	7	18	-	-	PUNCT
iajs-4130	7	19	restricted	restrict	VERB
iajs-4130	7	20	bounded	bounded	ADJ
iajs-4130	7	21	submodule	submodule	NOUN
iajs-4130	7	22	's	's	PART
iajs-4130	7	23	radical	radical	ADJ
iajs-4130	7	24	.	.	PUNCT
iajs-4130	8	1	this	this	DET
iajs-4130	8	2	work	work	NOUN
iajs-4130	8	3	includes	include	VERB
iajs-4130	8	4	a	a	DET
iajs-4130	8	5	new	new	ADJ
iajs-4130	8	6	class	class	NOUN
iajs-4130	8	7	of	of	ADP
iajs-4130	8	8	t	t	NOUN
iajs-4130	8	9	-	-	PUNCT
iajs-4130	8	10	module	module	NOUN
iajs-4130	8	11	as	as	ADV
iajs-4130	8	12	well	well	ADV
iajs-4130	8	13	as	as	ADP
iajs-4130	8	14	a	a	DET
iajs-4130	8	15	t	t	PROPN
iajs-4130	8	16	-	-	PUNCT
iajs-4130	8	17	submodule	submodule	NOUN
iajs-4130	8	18	called	call	VERB
iajs-4130	8	19	the	the	DET
iajs-4130	8	20	endo	endo	NOUN
iajs-4130	8	21	-	-	PUNCT
iajs-4130	8	22	restricted	restrict	VERB
iajs-4130	8	23	bounded	bounded	ADJ
iajs-4130	8	24	module	module	NOUN
iajs-4130	8	25	(	(	PUNCT
iajs-4130	8	26	submodule	submodule	NOUN
iajs-4130	8	27	)	)	PUNCT
iajs-4130	8	28	,	,	PUNCT
iajs-4130	8	29	written	write	VERB
iajs-4130	8	30	briefly	briefly	ADV
iajs-4130	8	31	as	as	ADP
iajs-4130	8	32	the	the	DET
iajs-4130	8	33	endo	endo	NOUN
iajs-4130	8	34	-	-	PUNCT
iajs-4130	8	35	r.b	r.b	PROPN
iajs-4130	8	36	.	.	PROPN
iajs-4130	8	37	module	module	NOUN
iajs-4130	8	38	(	(	PUNCT
iajs-4130	8	39	submodule	submodule	NOUN
iajs-4130	8	40	)	)	PUNCT
iajs-4130	8	41	,	,	PUNCT
iajs-4130	8	42	provided	provide	VERB
iajs-4130	8	43	with	with	ADP
iajs-4130	8	44	some	some	DET
iajs-4130	8	45	examples	example	NOUN
iajs-4130	8	46	that	that	PRON
iajs-4130	8	47	illustrate	illustrate	VERB
iajs-4130	8	48	and	and	CCONJ
iajs-4130	8	49	clarify	clarify	VERB
iajs-4130	8	50	in	in	ADP
iajs-4130	8	51	a	a	DET
iajs-4130	8	52	nice	nice	ADJ
iajs-4130	8	53	way	way	NOUN
iajs-4130	8	54	this	this	DET
iajs-4130	8	55	type	type	NOUN
iajs-4130	8	56	of	of	ADP
iajs-4130	8	57	module	module	NOUN
iajs-4130	8	58	(	(	PUNCT
iajs-4130	8	59	submodule	submodule	NOUN
iajs-4130	8	60	)	)	PUNCT
iajs-4130	8	61	.	.	PUNCT
iajs-4130	9	1	however	however	ADV
iajs-4130	9	2	,	,	PUNCT
iajs-4130	9	3	our	our	PRON
iajs-4130	9	4	focus	focus	NOUN
iajs-4130	9	5	will	will	AUX
iajs-4130	9	6	be	be	AUX
iajs-4130	9	7	on	on	ADP
iajs-4130	9	8	the	the	DET
iajs-4130	9	9	radicals	radical	NOUN
iajs-4130	9	10	of	of	ADP
iajs-4130	9	11	the	the	DET
iajs-4130	9	12	endo	endo	NOUN
iajs-4130	9	13	-	-	PUNCT
iajs-4130	9	14	r.b	r.b	PROPN
iajs-4130	9	15	.	.	PROPN
iajs-4130	9	16	submodule	submodule	PROPN
iajs-4130	9	17	.	.	PUNCT
iajs-4130	10	1	prime	prime	PROPN
iajs-4130	10	2	submodule	submodule	PROPN
iajs-4130	10	3	and	and	CCONJ
iajs-4130	10	4	scalar	scalar	ADJ
iajs-4130	10	5	module	module	NOUN
iajs-4130	10	6	both	both	PRON
iajs-4130	10	7	play	play	VERB
iajs-4130	10	8	a	a	DET
iajs-4130	10	9	crucial	crucial	ADJ
iajs-4130	10	10	role	role	NOUN
iajs-4130	10	11	in	in	ADP
iajs-4130	10	12	many	many	ADJ
iajs-4130	10	13	properties	property	NOUN
iajs-4130	10	14	that	that	PRON
iajs-4130	10	15	show	show	VERB
iajs-4130	10	16	the	the	DET
iajs-4130	10	17	relationship	relationship	NOUN
iajs-4130	10	18	between	between	ADP
iajs-4130	10	19	prime	prime	ADJ
iajs-4130	10	20	submodules	submodule	NOUN
iajs-4130	10	21	and	and	CCONJ
iajs-4130	10	22	endo	endo	NOUN
iajs-4130	10	23	-	-	PUNCT
iajs-4130	10	24	r.b	r.b	NOUN
iajs-4130	10	25	submodules	submodule	NOUN
iajs-4130	10	26	.	.	PUNCT
iajs-4130	11	1	furthermore	furthermore	ADV
iajs-4130	11	2	,	,	PUNCT
iajs-4130	11	3	some	some	DET
iajs-4130	11	4	generalizations	generalization	NOUN
iajs-4130	11	5	of	of	ADP
iajs-4130	11	6	prime	prime	ADJ
iajs-4130	11	7	submodules	submodule	NOUN
iajs-4130	11	8	,	,	PUNCT
iajs-4130	11	9	such	such	ADJ
iajs-4130	11	10	as	as	ADP
iajs-4130	11	11	s	s	NOUN
iajs-4130	11	12	-	-	PUNCT
iajs-4130	11	13	prime	prime	ADJ
iajs-4130	11	14	submodules	submodule	NOUN
iajs-4130	11	15	,	,	PUNCT
iajs-4130	11	16	are	be	AUX
iajs-4130	11	17	involved	involve	VERB
iajs-4130	11	18	in	in	ADP
iajs-4130	11	19	this	this	DET
iajs-4130	11	20	research	research	NOUN
iajs-4130	11	21	.	.	PUNCT
iajs-4130	12	1	keywords	keyword	NOUN
iajs-4130	12	2	:	:	PUNCT
iajs-4130	12	3	endo	endo	NOUN
iajs-4130	12	4	-	-	PUNCT
iajs-4130	12	5	r.b	r.b	NOUN
iajs-4130	12	6	radical	radical	ADJ
iajs-4130	12	7	submodule	submodule	NOUN
iajs-4130	12	8	,	,	PUNCT
iajs-4130	12	9	bounded	bound	VERB
iajs-4130	12	10	module	module	NOUN
iajs-4130	12	11	,	,	PUNCT
iajs-4130	12	12	s	s	NOUN
iajs-4130	12	13	-	-	PUNCT
iajs-4130	12	14	prime	prime	ADJ
iajs-4130	12	15	submodule	submodule	NOUN
iajs-4130	12	16	,	,	PUNCT
iajs-4130	12	17	scalar	scalar	ADJ
iajs-4130	12	18	module	module	NOUN
iajs-4130	12	19	.	.	PUNCT
iajs-4130	13	1	1	1	X
iajs-4130	13	2	.	.	X
iajs-4130	13	3	introduction	introduction	NOUN
iajs-4130	13	4	the	the	DET
iajs-4130	13	5	ring	ring	NOUN
iajs-4130	13	6	in	in	ADP
iajs-4130	13	7	this	this	DET
iajs-4130	13	8	paper	paper	NOUN
iajs-4130	13	9	is	be	AUX
iajs-4130	13	10	commutative	commutative	ADJ
iajs-4130	13	11	with	with	ADP
iajs-4130	13	12	identity	identity	NOUN
iajs-4130	13	13	denoted	denote	VERB
iajs-4130	13	14	by	by	ADP
iajs-4130	13	15	t	t	PROPN
iajs-4130	13	16	and	and	CCONJ
iajs-4130	13	17	is	be	AUX
iajs-4130	13	18	a	a	DET
iajs-4130	13	19	unitary	unitary	ADJ
iajs-4130	13	20	left	left	ADJ
iajs-4130	13	21	-	-	PUNCT
iajs-4130	13	22	tmodule	tmodule	NOUN
iajs-4130	13	23	.	.	PUNCT
iajs-4130	14	1	motivated	motivate	VERB
iajs-4130	14	2	by	by	ADP
iajs-4130	14	3	the	the	DET
iajs-4130	14	4	notion	notion	NOUN
iajs-4130	14	5	of	of	ADP
iajs-4130	14	6	bounded	bounded	ADJ
iajs-4130	14	7	module	module	NOUN
iajs-4130	14	8	,	,	PUNCT
iajs-4130	14	9	where	where	SCONJ
iajs-4130	14	10	a	a	DET
iajs-4130	14	11	t	t	NOUN
iajs-4130	14	12	-	-	PUNCT
iajs-4130	14	13	module	module	NOUN
iajs-4130	14	14	is	be	AUX
iajs-4130	14	15	called	call	VERB
iajs-4130	14	16	bounded	bounded	ADJ
iajs-4130	14	17	if	if	SCONJ
iajs-4130	14	18	there	there	PRON
iajs-4130	14	19	exists	exist	VERB
iajs-4130	14	20	such	such	ADJ
iajs-4130	14	21	that	that	SCONJ
iajs-4130	14	22	(	(	PUNCT
iajs-4130	14	23	)	)	PUNCT
iajs-4130	14	24	(	(	PUNCT
iajs-4130	14	25	)	)	PUNCT
iajs-4130	14	26	(	(	PUNCT
iajs-4130	14	27	1–3	1–3	NOUN
iajs-4130	14	28	)	)	PUNCT
iajs-4130	14	29	.	.	PUNCT
iajs-4130	15	1	a	a	DET
iajs-4130	15	2	t	t	PROPN
iajs-4130	15	3	-	-	PUNCT
iajs-4130	15	4	submodule	submodule	NOUN
iajs-4130	15	5	a	a	PRON
iajs-4130	15	6	is	be	AUX
iajs-4130	15	7	called	call	VERB
iajs-4130	15	8	bounded	bound	VERB
iajs-4130	15	9	if	if	SCONJ
iajs-4130	15	10	there	there	PRON
iajs-4130	15	11	exists	exist	VERB
iajs-4130	15	12	an	an	DET
iajs-4130	15	13	element	element	NOUN
iajs-4130	15	14	such	such	ADJ
iajs-4130	15	15	that	that	SCONJ
iajs-4130	15	16	(	(	PUNCT
iajs-4130	15	17	)	)	PUNCT
iajs-4130	15	18	(	(	PUNCT
iajs-4130	15	19	)	)	PUNCT
iajs-4130	15	20	(	(	PUNCT
iajs-4130	15	21	3	3	NUM
iajs-4130	15	22	)	)	PUNCT
iajs-4130	15	23	.	.	PUNCT
iajs-4130	16	1	we	we	PRON
iajs-4130	16	2	introduced	introduce	VERB
iajs-4130	16	3	a	a	DET
iajs-4130	16	4	new	new	ADJ
iajs-4130	16	5	concept	concept	NOUN
iajs-4130	16	6	of	of	ADP
iajs-4130	16	7	module	module	NOUN
iajs-4130	16	8	(	(	PUNCT
iajs-4130	16	9	submodule	submodule	NOUN
iajs-4130	16	10	)	)	PUNCT
iajs-4130	16	11	namely	namely	ADV
iajs-4130	16	12	an	an	DET
iajs-4130	16	13	endo	endo	NOUN
iajs-4130	16	14	-	-	PUNCT
iajs-4130	16	15	restricted	restrict	VERB
iajs-4130	16	16	bounded	bounded	ADJ
iajs-4130	16	17	submodule	submodule	NOUN
iajs-4130	16	18	(	(	PUNCT
iajs-4130	16	19	module	module	NOUN
iajs-4130	16	20	)	)	PUNCT
iajs-4130	16	21	(	(	PUNCT
iajs-4130	16	22	endo	endo	NOUN
iajs-4130	16	23	-	-	PUNCT
iajs-4130	16	24	r.b	r.b	NOUN
iajs-4130	16	25	submodule	submodule	NOUN
iajs-4130	16	26	(	(	PUNCT
iajs-4130	16	27	module	module	NOUN
iajs-4130	16	28	)	)	PUNCT
iajs-4130	16	29	)	)	PUNCT
iajs-4130	16	30	and	and	CCONJ
iajs-4130	16	31	then	then	ADV
iajs-4130	16	32	we	we	PRON
iajs-4130	16	33	turned	turn	VERB
iajs-4130	16	34	to	to	ADP
iajs-4130	16	35	the	the	DET
iajs-4130	16	36	main	main	ADJ
iajs-4130	16	37	purpose	purpose	NOUN
iajs-4130	16	38	,	,	PUNCT
iajs-4130	16	39	which	which	PRON
iajs-4130	16	40	is	be	AUX
iajs-4130	16	41	the	the	DET
iajs-4130	16	42	radical	radical	NOUN
iajs-4130	16	43	of	of	ADP
iajs-4130	16	44	an	an	DET
iajs-4130	16	45	endo	endo	NOUN
iajs-4130	16	46	-	-	PUNCT
iajs-4130	16	47	r.b	r.b	NOUN
iajs-4130	16	48	submodule	submodule	NOUN
iajs-4130	16	49	that	that	PRON
iajs-4130	16	50	will	will	AUX
iajs-4130	16	51	be	be	AUX
iajs-4130	16	52	defined	define	VERB
iajs-4130	16	53	later	later	ADV
iajs-4130	16	54	in	in	ADP
iajs-4130	16	55	this	this	DET
iajs-4130	16	56	work	work	NOUN
iajs-4130	16	57	with	with	ADP
iajs-4130	16	58	some	some	DET
iajs-4130	16	59	important	important	ADJ
iajs-4130	16	60	properties	property	NOUN
iajs-4130	16	61	.	.	PUNCT
iajs-4130	17	1	an	an	DET
iajs-4130	17	2	endo	endo	NOUN
iajs-4130	17	3	-	-	PUNCT
iajs-4130	17	4	r.b	r.b	NOUN
iajs-4130	17	5	submodule	submodule	NOUN
iajs-4130	17	6	is	be	AUX
iajs-4130	17	7	a	a	DET
iajs-4130	17	8	new	new	ADJ
iajs-4130	17	9	type	type	NOUN
iajs-4130	17	10	of	of	ADP
iajs-4130	17	11	t	t	PROPN
iajs-4130	17	12	-	-	PUNCT
iajs-4130	17	13	submodule	submodule	NOUN
iajs-4130	17	14	that	that	PRON
iajs-4130	17	15	has	have	AUX
iajs-4130	17	16	not	not	PART
iajs-4130	17	17	been	be	AUX
iajs-4130	17	18	recognized	recognize	VERB
iajs-4130	17	19	previously	previously	ADV
iajs-4130	17	20	by	by	ADP
iajs-4130	17	21	other	other	ADJ
iajs-4130	17	22	authors	author	NOUN
iajs-4130	17	23	,	,	PUNCT
iajs-4130	17	24	where	where	SCONJ
iajs-4130	17	25	a	a	DET
iajs-4130	17	26	proper	proper	ADJ
iajs-4130	17	27	submodule	submodule	NOUN
iajs-4130	17	28	a	a	PRON
iajs-4130	17	29	of	of	ADP
iajs-4130	17	30	a	a	DET
iajs-4130	17	31	t	t	NOUN
iajs-4130	17	32	-	-	PUNCT
iajs-4130	17	33	module	module	NOUN
iajs-4130	17	34	is	be	AUX
iajs-4130	17	35	called	call	VERB
iajs-4130	17	36	endor.b	endor.b	NOUN
iajs-4130	17	37	whenever	whenever	SCONJ
iajs-4130	17	38	(	(	PUNCT
iajs-4130	17	39	)	)	PUNCT
iajs-4130	17	40	,	,	PUNCT
iajs-4130	17	41	(	(	PUNCT
iajs-4130	17	42	)	)	PUNCT
iajs-4130	17	43	implies	imply	VERB
iajs-4130	17	44	that	that	SCONJ
iajs-4130	17	45	(	(	PUNCT
iajs-4130	17	46	(	(	PUNCT
iajs-4130	17	47	)	)	PUNCT
iajs-4130	17	48	)	)	PUNCT
iajs-4130	17	49	(	(	PUNCT
iajs-4130	17	50	)	)	PUNCT
iajs-4130	17	51	.	.	PUNCT
iajs-4130	18	1	we	we	PRON
iajs-4130	18	2	say	say	VERB
iajs-4130	18	3	that	that	PRON
iajs-4130	18	4	is	be	AUX
iajs-4130	18	5	an	an	DET
iajs-4130	18	6	endo	endo	NOUN
iajs-4130	18	7	-	-	PUNCT
iajs-4130	18	8	r.b	r.b	PROPN
iajs-4130	18	9	t	t	NOUN
iajs-4130	18	10	-	-	PUNCT
iajs-4130	18	11	module	module	NOUN
iajs-4130	18	12	if	if	SCONJ
iajs-4130	18	13	every	every	DET
iajs-4130	18	14	proper	proper	ADJ
iajs-4130	18	15	submodule	submodule	NOUN
iajs-4130	18	16	is	be	AUX
iajs-4130	18	17	an	an	DET
iajs-4130	18	18	endo	endo	NOUN
iajs-4130	18	19	-	-	PUNCT
iajs-4130	18	20	r.b	r.b	NOUN
iajs-4130	18	21	.	.	PROPN
iajs-4130	19	1	also	also	ADV
iajs-4130	19	2	,	,	PUNCT
iajs-4130	19	3	prime	prime	ADJ
iajs-4130	19	4	https://orcid.org/0009-0005-7288-4292	https://orcid.org/0009-0005-7288-4292	NOUN
iajs-4130	19	5	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4130	19	6	https://orcid.org/0000-0002-6553-2089	https://orcid.org/0000-0002-6553-2089	NOUN
iajs-4130	19	7	mailto:buthynashihab@gmail.com	mailto:buthynashihab@gmail.com	X
iajs-4130	19	8	https://orcid.org/0009-0005-7288-4292	https://orcid.org/0009-0005-7288-4292	ADJ
iajs-4130	19	9	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4130	19	10	https://orcid.org/0000-0002-6553-2089	https://orcid.org/0000-0002-6553-2089	NOUN
iajs-4130	19	11	mailto:buthynashihab@gmail.com	mailto:buthynashihab@gmail.com	X
iajs-4130	19	12	https://orcid.org/0009-0005-7288-4292	https://orcid.org/0009-0005-7288-4292	ADJ
iajs-4130	19	13	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4130	19	14	https://orcid.org/0000-0002-6553-2089	https://orcid.org/0000-0002-6553-2089	NOUN
iajs-4130	19	15	mailto:buthynashihab@gmail.com	mailto:buthynashihab@gmail.com	X
iajs-4130	19	16	https://orcid.org/0009-0005-7288-4292	https://orcid.org/0009-0005-7288-4292	ADJ
iajs-4130	19	17	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4130	19	18	https://orcid.org/0000-0002-6553-2089	https://orcid.org/0000-0002-6553-2089	NOUN
iajs-4130	19	19	mailto:buthynashihab@gmail.com	mailto:buthynashihab@gmail.com	X
iajs-4130	19	20	https://orcid.org/0009-0005-7288-4292	https://orcid.org/0009-0005-7288-4292	ADJ
iajs-4130	19	21	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4130	19	22	https://orcid.org/0000-0002-6553-2089	https://orcid.org/0000-0002-6553-2089	NOUN
iajs-4130	19	23	mailto:buthynashihab@gmail.com	mailto:buthynashihab@gmail.com	X
iajs-4130	19	24	https://orcid.org/0009-0005-7288-4292	https://orcid.org/0009-0005-7288-4292	ADJ
iajs-4130	19	25	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4130	19	26	https://orcid.org/0000-0002-6553-2089	https://orcid.org/0000-0002-6553-2089	NOUN
iajs-4130	19	27	mailto:buthynashihab@gmail.com	mailto:buthynashihab@gmail.com	PROPN
iajs-4130	19	28	ihjpas	ihjpas	PROPN
iajs-4130	19	29	.	.	PUNCT
iajs-4130	20	1	2025,38(4	2025,38(4	NOUN
iajs-4130	20	2	)	)	PUNCT
iajs-4130	20	3	399	399	NUM
iajs-4130	20	4	submodules	submodule	NOUN
iajs-4130	20	5	play	play	VERB
iajs-4130	20	6	a	a	DET
iajs-4130	20	7	crucial	crucial	ADJ
iajs-4130	20	8	role	role	NOUN
iajs-4130	20	9	in	in	ADP
iajs-4130	20	10	order	order	NOUN
iajs-4130	20	11	to	to	PART
iajs-4130	20	12	recall	recall	VERB
iajs-4130	20	13	a	a	DET
iajs-4130	20	14	radical	radical	NOUN
iajs-4130	20	15	of	of	ADP
iajs-4130	20	16	a	a	DET
iajs-4130	20	17	submodule	submodule	NOUN
iajs-4130	20	18	and	and	CCONJ
iajs-4130	20	19	its	its	PRON
iajs-4130	20	20	connection	connection	NOUN
iajs-4130	20	21	with	with	ADP
iajs-4130	20	22	the	the	DET
iajs-4130	20	23	radical	radical	NOUN
iajs-4130	20	24	of	of	ADP
iajs-4130	20	25	an	an	DET
iajs-4130	20	26	endo	endo	NOUN
iajs-4130	20	27	-	-	PUNCT
iajs-4130	20	28	r.b	r.b	NOUN
iajs-4130	20	29	submodule	submodule	NOUN
iajs-4130	20	30	,	,	PUNCT
iajs-4130	20	31	where	where	SCONJ
iajs-4130	20	32	a	a	DET
iajs-4130	20	33	submodule	submodule	NOUN
iajs-4130	20	34	a	a	PRON
iajs-4130	20	35	of	of	ADP
iajs-4130	20	36	a	a	DET
iajs-4130	20	37	t	t	NOUN
iajs-4130	20	38	-	-	PUNCT
iajs-4130	20	39	module	module	NOUN
iajs-4130	20	40	is	be	AUX
iajs-4130	20	41	said	say	VERB
iajs-4130	20	42	to	to	PART
iajs-4130	20	43	be	be	AUX
iajs-4130	20	44	prime	prime	ADJ
iajs-4130	20	45	whenever	whenever	SCONJ
iajs-4130	20	46	,	,	PUNCT
iajs-4130	20	47	(	(	PUNCT
iajs-4130	20	48	4–7	4–7	NOUN
iajs-4130	20	49	)	)	PUNCT
iajs-4130	20	50	.	.	PUNCT
iajs-4130	21	1	in	in	ADP
iajs-4130	21	2	addition	addition	NOUN
iajs-4130	21	3	,	,	PUNCT
iajs-4130	21	4	we	we	PRON
iajs-4130	21	5	found	find	VERB
iajs-4130	21	6	that	that	SCONJ
iajs-4130	21	7	if	if	SCONJ
iajs-4130	21	8	a	a	PRON
iajs-4130	21	9	is	be	AUX
iajs-4130	21	10	an	an	DET
iajs-4130	21	11	endo	endo	NOUN
iajs-4130	21	12	-	-	PUNCT
iajs-4130	21	13	r.b	r.b	NOUN
iajs-4130	21	14	,	,	PUNCT
iajs-4130	21	15	then	then	ADV
iajs-4130	21	16	it	it	PRON
iajs-4130	21	17	is	be	AUX
iajs-4130	21	18	not	not	PART
iajs-4130	21	19	necessary	necessary	ADJ
iajs-4130	21	20	that	that	SCONJ
iajs-4130	21	21	a	a	PRON
iajs-4130	21	22	be	be	AUX
iajs-4130	21	23	a	a	DET
iajs-4130	21	24	prime	prime	NOUN
iajs-4130	21	25	,	,	PUNCT
iajs-4130	21	26	and	and	CCONJ
iajs-4130	21	27	the	the	DET
iajs-4130	21	28	following	follow	VERB
iajs-4130	21	29	example	example	NOUN
iajs-4130	21	30	shows	show	VERB
iajs-4130	21	31	that	that	SCONJ
iajs-4130	21	32	:	:	PUNCT
iajs-4130	21	33	let	let	VERB
iajs-4130	22	1	=	=	PRON
iajs-4130	22	2	〈	〈	NOUN
iajs-4130	22	3	̅	̅	NOUN
iajs-4130	22	4	〉	〉	NOUN
iajs-4130	22	5	〈	〈	NOUN
iajs-4130	22	6	̅	̅	NOUN
iajs-4130	22	7	〉	〉	NOUN
iajs-4130	22	8	(	(	PUNCT
iajs-4130	22	9	̅	̅	NOUN
iajs-4130	22	10	̅	̅	NOUN
iajs-4130	22	11	)	)	PUNCT
iajs-4130	22	12	(	(	PUNCT
iajs-4130	22	13	̅	̅	NOUN
iajs-4130	22	14	,	,	PUNCT
iajs-4130	22	15	̅	̅	NOUN
iajs-4130	22	16	)	)	PUNCT
iajs-4130	22	17	(	(	PUNCT
iajs-4130	22	18	̅	̅	NUM
iajs-4130	22	19	̅	̅	NOUN
iajs-4130	22	20	)	)	PUNCT
iajs-4130	22	21	,	,	PUNCT
iajs-4130	22	22	then	then	ADV
iajs-4130	22	23	a	a	PRON
iajs-4130	22	24	is	be	AUX
iajs-4130	22	25	an	an	DET
iajs-4130	22	26	endo	endo	NOUN
iajs-4130	22	27	-	-	PUNCT
iajs-4130	22	28	r.b	r.b	NOUN
iajs-4130	22	29	.	.	PUNCT
iajs-4130	23	1	however	however	ADV
iajs-4130	23	2	,	,	PUNCT
iajs-4130	23	3	a	a	PRON
iajs-4130	23	4	is	be	AUX
iajs-4130	23	5	not	not	PART
iajs-4130	23	6	a	a	DET
iajs-4130	23	7	prime	prime	ADJ
iajs-4130	23	8	submodule	submodule	NOUN
iajs-4130	23	9	since	since	SCONJ
iajs-4130	23	10	if	if	SCONJ
iajs-4130	23	11	(	(	PUNCT
iajs-4130	23	12	̅	̅	NOUN
iajs-4130	23	13	̅	̅	NOUN
iajs-4130	23	14	)	)	PUNCT
iajs-4130	23	15	(	(	PUNCT
iajs-4130	23	16	̅	̅	NUM
iajs-4130	23	17	̅	̅	NOUN
iajs-4130	23	18	)	)	PUNCT
iajs-4130	23	19	but	but	CCONJ
iajs-4130	23	20	(	(	PUNCT
iajs-4130	23	21	̅	̅	NUM
iajs-4130	23	22	̅	̅	NOUN
iajs-4130	23	23	)	)	PUNCT
iajs-4130	23	24	,	,	PUNCT
iajs-4130	23	25	also	also	ADV
iajs-4130	23	26	,	,	PUNCT
iajs-4130	23	27	we	we	PRON
iajs-4130	23	28	found	find	VERB
iajs-4130	23	29	another	another	DET
iajs-4130	23	30	example	example	NOUN
iajs-4130	23	31	that	that	PRON
iajs-4130	23	32	shows	show	VERB
iajs-4130	23	33	if	if	SCONJ
iajs-4130	23	34	a	a	DET
iajs-4130	23	35	submodule	submodule	NOUN
iajs-4130	23	36	a	a	PRON
iajs-4130	23	37	is	be	AUX
iajs-4130	23	38	prime	prime	ADJ
iajs-4130	23	39	,	,	PUNCT
iajs-4130	23	40	then	then	ADV
iajs-4130	23	41	it	it	PRON
iajs-4130	23	42	is	be	AUX
iajs-4130	23	43	not	not	PART
iajs-4130	23	44	necessary	necessary	ADJ
iajs-4130	23	45	to	to	PART
iajs-4130	23	46	be	be	AUX
iajs-4130	23	47	an	an	DET
iajs-4130	23	48	endo	endo	NOUN
iajs-4130	23	49	-	-	PUNCT
iajs-4130	23	50	r.b	r.b	NOUN
iajs-4130	23	51	.	.	PROPN
iajs-4130	23	52	assume	assume	VERB
iajs-4130	23	53	that	that	SCONJ
iajs-4130	23	54	=	=	PUNCT
iajs-4130	23	55	as	as	ADP
iajs-4130	23	56	a	a	DET
iajs-4130	23	57	z	z	NOUN
iajs-4130	23	58	-	-	PUNCT
iajs-4130	23	59	module	module	NOUN
iajs-4130	23	60	and	and	CCONJ
iajs-4130	23	61	〈	〈	NOUN
iajs-4130	23	62	̅	̅	NOUN
iajs-4130	23	63	〉	〉	NOUN
iajs-4130	23	64	.	.	PUNCT
iajs-4130	24	1	define	define	NOUN
iajs-4130	24	2	as	as	ADP
iajs-4130	24	3	the	the	DET
iajs-4130	24	4	same	same	ADJ
iajs-4130	24	5	way	way	NOUN
iajs-4130	24	6	in	in	ADP
iajs-4130	24	7	the	the	DET
iajs-4130	24	8	previous	previous	ADJ
iajs-4130	24	9	example	example	NOUN
iajs-4130	24	10	.	.	PUNCT
iajs-4130	25	1	the	the	DET
iajs-4130	25	2	radical	radical	NOUN
iajs-4130	25	3	of	of	ADP
iajs-4130	25	4	a	a	DET
iajs-4130	25	5	t	t	PROPN
iajs-4130	25	6	-	-	PUNCT
iajs-4130	25	7	submodule	submodule	NOUN
iajs-4130	25	8	denoted	denote	VERB
iajs-4130	25	9	by	by	ADP
iajs-4130	25	10	(	(	PUNCT
iajs-4130	25	11	)	)	PUNCT
iajs-4130	25	12	and	and	CCONJ
iajs-4130	25	13	it	it	PRON
iajs-4130	25	14	is	be	AUX
iajs-4130	25	15	the	the	DET
iajs-4130	25	16	intersection	intersection	NOUN
iajs-4130	25	17	of	of	ADP
iajs-4130	25	18	all	all	DET
iajs-4130	25	19	prime	prime	ADJ
iajs-4130	25	20	submodules	submodule	NOUN
iajs-4130	25	21	of	of	ADP
iajs-4130	25	22	that	that	PRON
iajs-4130	25	23	contain	contain	VERB
iajs-4130	25	24	n.	n.	NOUN
iajs-4130	25	25	in	in	ADP
iajs-4130	25	26	other	other	ADJ
iajs-4130	25	27	word	word	NOUN
iajs-4130	25	28	,	,	PUNCT
iajs-4130	25	29	(	(	PUNCT
iajs-4130	25	30	)	)	PUNCT
iajs-4130	25	31	*	*	PUNCT
iajs-4130	26	1	|	|	ADV
iajs-4130	26	2	+	+	ADJ
iajs-4130	26	3	(	(	PUNCT
iajs-4130	26	4	8–11	8–11	NOUN
iajs-4130	26	5	)	)	PUNCT
iajs-4130	26	6	.	.	PUNCT
iajs-4130	27	1	in	in	ADP
iajs-4130	27	2	this	this	DET
iajs-4130	27	3	paper	paper	NOUN
iajs-4130	27	4	,	,	PUNCT
iajs-4130	27	5	we	we	PRON
iajs-4130	27	6	prove	prove	VERB
iajs-4130	27	7	many	many	ADJ
iajs-4130	27	8	properties	property	NOUN
iajs-4130	27	9	provided	provide	VERB
iajs-4130	27	10	with	with	ADP
iajs-4130	27	11	some	some	DET
iajs-4130	27	12	crucial	crucial	ADJ
iajs-4130	27	13	conditions	condition	NOUN
iajs-4130	27	14	,	,	PUNCT
iajs-4130	27	15	and	and	CCONJ
iajs-4130	27	16	we	we	PRON
iajs-4130	27	17	will	will	AUX
iajs-4130	27	18	see	see	VERB
iajs-4130	27	19	the	the	DET
iajs-4130	27	20	relationship	relationship	NOUN
iajs-4130	27	21	between	between	ADP
iajs-4130	27	22	(	(	PUNCT
iajs-4130	27	23	)	)	PUNCT
iajs-4130	27	24	and	and	CCONJ
iajs-4130	27	25	the	the	DET
iajs-4130	27	26	radical	radical	ADJ
iajs-4130	27	27	of	of	ADP
iajs-4130	27	28	endo	endo	NOUN
iajs-4130	27	29	-	-	PUNCT
iajs-4130	27	30	r.b	r.b	NOUN
iajs-4130	27	31	submodules	submodule	NOUN
iajs-4130	27	32	.	.	PUNCT
iajs-4130	28	1	2	2	X
iajs-4130	28	2	.	.	X
iajs-4130	28	3	endo	endo	NOUN
iajs-4130	28	4	-	-	PUNCT
iajs-4130	28	5	r.b	r.b	NOUN
iajs-4130	28	6	submodules	submodule	NOUN
iajs-4130	28	7	and	and	CCONJ
iajs-4130	28	8	modules	module	NOUN
iajs-4130	28	9	in	in	ADP
iajs-4130	28	10	this	this	DET
iajs-4130	28	11	section	section	NOUN
iajs-4130	28	12	,	,	PUNCT
iajs-4130	28	13	we	we	PRON
iajs-4130	28	14	give	give	VERB
iajs-4130	28	15	a	a	DET
iajs-4130	28	16	brief	brief	ADJ
iajs-4130	28	17	introduction	introduction	NOUN
iajs-4130	28	18	about	about	ADP
iajs-4130	28	19	endo	endo	NOUN
iajs-4130	28	20	-	-	PUNCT
iajs-4130	28	21	r.b	r.b	NOUN
iajs-4130	28	22	submodules	submodule	NOUN
iajs-4130	28	23	as	as	ADV
iajs-4130	28	24	well	well	ADV
iajs-4130	28	25	as	as	ADP
iajs-4130	28	26	its	its	PRON
iajs-4130	28	27	modules	module	NOUN
iajs-4130	28	28	with	with	ADP
iajs-4130	28	29	some	some	DET
iajs-4130	28	30	examples	example	NOUN
iajs-4130	28	31	.	.	PUNCT
iajs-4130	29	1	we	we	PRON
iajs-4130	29	2	refer	refer	VERB
iajs-4130	29	3	to	to	ADP
iajs-4130	29	4	a	a	DET
iajs-4130	29	5	proper	proper	ADJ
iajs-4130	29	6	submodule	submodule	NOUN
iajs-4130	29	7	by	by	ADP
iajs-4130	29	8	the	the	DET
iajs-4130	29	9	symbol	symbol	NOUN
iajs-4130	29	10	and	and	CCONJ
iajs-4130	29	11	(	(	PUNCT
iajs-4130	29	12	)	)	PUNCT
iajs-4130	29	13	is	be	AUX
iajs-4130	29	14	the	the	DET
iajs-4130	29	15	set	set	NOUN
iajs-4130	29	16	of	of	ADP
iajs-4130	29	17	all	all	DET
iajs-4130	29	18	endomorphisms	endomorphism	NOUN
iajs-4130	29	19	of	of	ADP
iajs-4130	29	20	a	a	DET
iajs-4130	29	21	t	t	NOUN
iajs-4130	29	22	-	-	PUNCT
iajs-4130	29	23	module	module	NOUN
iajs-4130	29	24	.	.	PUNCT
iajs-4130	30	1	2.1.definition	2.1.definition	NUM
iajs-4130	30	2	:	:	PUNCT
iajs-4130	30	3	a	a	DET
iajs-4130	30	4	proper	proper	ADJ
iajs-4130	30	5	t	t	PROPN
iajs-4130	30	6	-	-	PUNCT
iajs-4130	30	7	submodule	submodule	NOUN
iajs-4130	30	8	a	a	PRON
iajs-4130	30	9	of	of	ADP
iajs-4130	30	10	is	be	AUX
iajs-4130	30	11	said	say	VERB
iajs-4130	30	12	to	to	PART
iajs-4130	30	13	be	be	AUX
iajs-4130	30	14	endo	endo	NOUN
iajs-4130	30	15	-	-	PUNCT
iajs-4130	30	16	r.b	r.b	NOUN
iajs-4130	30	17	if	if	SCONJ
iajs-4130	30	18	there	there	PRON
iajs-4130	30	19	exists	exist	VERB
iajs-4130	30	20	an	an	DET
iajs-4130	30	21	endomorphism	endomorphism	NOUN
iajs-4130	30	22	of	of	ADP
iajs-4130	30	23	(	(	PUNCT
iajs-4130	30	24	(	(	PUNCT
iajs-4130	30	25	)	)	PUNCT
iajs-4130	30	26	)	)	PUNCT
iajs-4130	30	27	and	and	CCONJ
iajs-4130	30	28	(	(	PUNCT
iajs-4130	30	29	)	)	PUNCT
iajs-4130	30	30	for	for	ADP
iajs-4130	30	31	some	some	DET
iajs-4130	30	32	such	such	ADJ
iajs-4130	30	33	that	that	SCONJ
iajs-4130	30	34	(	(	PUNCT
iajs-4130	30	35	(	(	PUNCT
iajs-4130	30	36	)	)	PUNCT
iajs-4130	30	37	)	)	PUNCT
iajs-4130	31	1	(	(	PUNCT
iajs-4130	31	2	)	)	PUNCT
iajs-4130	31	3	we	we	PRON
iajs-4130	31	4	need	need	VERB
iajs-4130	31	5	just	just	ADV
iajs-4130	31	6	one	one	NUM
iajs-4130	31	7	an	an	DET
iajs-4130	31	8	endomorphism	endomorphism	NOUN
iajs-4130	31	9	of	of	ADP
iajs-4130	31	10	with	with	ADP
iajs-4130	31	11	an	an	DET
iajs-4130	31	12	element	element	NOUN
iajs-4130	31	13	belonging	belong	VERB
iajs-4130	31	14	to	to	ADP
iajs-4130	31	15	a	a	DET
iajs-4130	31	16	t	t	NOUN
iajs-4130	31	17	-	-	PUNCT
iajs-4130	31	18	module	module	NOUN
iajs-4130	31	19	defined	define	VERB
iajs-4130	31	20	in	in	ADP
iajs-4130	31	21	some	some	DET
iajs-4130	31	22	way	way	NOUN
iajs-4130	31	23	that	that	PRON
iajs-4130	31	24	satisfies	satisfy	VERB
iajs-4130	31	25	two	two	NUM
iajs-4130	31	26	conditions	condition	NOUN
iajs-4130	31	27	,	,	PUNCT
iajs-4130	31	28	(	(	PUNCT
iajs-4130	31	29	)	)	PUNCT
iajs-4130	31	30	and	and	CCONJ
iajs-4130	31	31	(	(	PUNCT
iajs-4130	31	32	(	(	PUNCT
iajs-4130	31	33	)	)	PUNCT
iajs-4130	31	34	)	)	PUNCT
iajs-4130	31	35	(	(	PUNCT
iajs-4130	31	36	)	)	PUNCT
iajs-4130	31	37	examples	example	NOUN
iajs-4130	31	38	1	1	NUM
iajs-4130	31	39	)	)	PUNCT
iajs-4130	31	40	suppose	suppose	VERB
iajs-4130	31	41	that	that	SCONJ
iajs-4130	31	42	=	=	SYM
iajs-4130	31	43	z	z	NOUN
iajs-4130	31	44	,	,	PUNCT
iajs-4130	31	45	and	and	CCONJ
iajs-4130	31	46	let	let	VERB
iajs-4130	31	47	〈	〈	PRON
iajs-4130	31	48	̅	̅	NOUN
iajs-4130	31	49	〉	〉	NOUN
iajs-4130	31	50	.	.	PUNCT
iajs-4130	32	1	then	then	ADV
iajs-4130	32	2	we	we	PRON
iajs-4130	32	3	can	can	AUX
iajs-4130	32	4	find	find	VERB
iajs-4130	32	5	z	z	NOUN
iajs-4130	32	6	z	z	AUX
iajs-4130	32	7	defined	define	VERB
iajs-4130	32	8	by	by	ADP
iajs-4130	32	9	(	(	PUNCT
iajs-4130	32	10	̅	̅	NOUN
iajs-4130	32	11	)	)	PUNCT
iajs-4130	32	12	(	(	PUNCT
iajs-4130	32	13	̅	̅	NOUN
iajs-4130	32	14	)	)	PUNCT
iajs-4130	32	15	.	.	PUNCT
iajs-4130	33	1	it	it	PRON
iajs-4130	33	2	is	be	AUX
iajs-4130	33	3	clear	clear	ADJ
iajs-4130	33	4	that	that	PRON
iajs-4130	33	5	is	be	AUX
iajs-4130	33	6	an	an	DET
iajs-4130	33	7	endomorphism	endomorphism	NOUN
iajs-4130	33	8	.	.	PUNCT
iajs-4130	34	1	now	now	ADV
iajs-4130	34	2	,	,	PUNCT
iajs-4130	34	3	let	let	VERB
iajs-4130	34	4	(	(	PUNCT
iajs-4130	34	5	̅	̅	NOUN
iajs-4130	34	6	)	)	PUNCT
iajs-4130	34	7	(	(	PUNCT
iajs-4130	34	8	̅	̅	NOUN
iajs-4130	34	9	)	)	PUNCT
iajs-4130	34	10	(	(	PUNCT
iajs-4130	34	11	̅	̅	NOUN
iajs-4130	34	12	)	)	PUNCT
iajs-4130	34	13	,	,	PUNCT
iajs-4130	34	14	and	and	CCONJ
iajs-4130	34	15	hence	hence	ADV
iajs-4130	34	16	we	we	PRON
iajs-4130	34	17	see	see	VERB
iajs-4130	34	18	that	that	PRON
iajs-4130	34	19	(	(	PUNCT
iajs-4130	34	20	̅	̅	NOUN
iajs-4130	34	21	)	)	PUNCT
iajs-4130	34	22	(	(	PUNCT
iajs-4130	34	23	)	)	PUNCT
iajs-4130	34	24	〈	〈	NOUN
iajs-4130	34	25	̅	̅	NOUN
iajs-4130	34	26	〉	〉	NOUN
iajs-4130	34	27	.	.	PUNCT
iajs-4130	35	1	we	we	PRON
iajs-4130	35	2	conclude	conclude	VERB
iajs-4130	35	3	that	that	SCONJ
iajs-4130	35	4	a	a	DET
iajs-4130	35	5	submodule	submodule	NOUN
iajs-4130	35	6	a	a	PRON
iajs-4130	35	7	is	be	AUX
iajs-4130	35	8	an	an	DET
iajs-4130	35	9	endo	endo	NOUN
iajs-4130	35	10	-	-	PUNCT
iajs-4130	35	11	r.b	r.b	NOUN
iajs-4130	35	12	submodule	submodule	NOUN
iajs-4130	35	13	of	of	ADP
iajs-4130	35	14	.	.	PROPN
iajs-4130	35	15	2	2	X
iajs-4130	35	16	)	)	PUNCT
iajs-4130	35	17	assume	assume	VERB
iajs-4130	35	18	=	=	PUNCT
iajs-4130	35	19	as	as	ADP
iajs-4130	35	20	a	a	DET
iajs-4130	35	21	z	z	NOUN
iajs-4130	35	22	-	-	PUNCT
iajs-4130	35	23	module	module	NOUN
iajs-4130	35	24	and	and	CCONJ
iajs-4130	35	25	a=	a=	ADJ
iajs-4130	35	26	〈	〈	NOUN
iajs-4130	35	27	̅	̅	NOUN
iajs-4130	35	28	〉	〉	NOUN
iajs-4130	35	29	.	.	PUNCT
iajs-4130	36	1	define	define	VERB
iajs-4130	36	2	as	as	ADP
iajs-4130	36	3	(	(	PUNCT
iajs-4130	36	4	̅	̅	NOUN
iajs-4130	36	5	)	)	PUNCT
iajs-4130	36	6	̅	̅	NOUN
iajs-4130	36	7	̅	̅	NOUN
iajs-4130	36	8	and	and	CCONJ
iajs-4130	36	9	is	be	AUX
iajs-4130	36	10	an	an	DET
iajs-4130	36	11	endomorphism	endomorphism	NOUN
iajs-4130	36	12	.	.	PUNCT
iajs-4130	37	1	then	then	ADV
iajs-4130	37	2	(	(	PUNCT
iajs-4130	37	3	̅	̅	NOUN
iajs-4130	37	4	)	)	PUNCT
iajs-4130	37	5	,	,	PUNCT
iajs-4130	37	6	but	but	CCONJ
iajs-4130	37	7	(	(	PUNCT
iajs-4130	37	8	)	)	PUNCT
iajs-4130	37	9	〈	〈	NOUN
iajs-4130	37	10	̅	̅	NOUN
iajs-4130	37	11	〉	〉	NOUN
iajs-4130	37	12	is	be	AUX
iajs-4130	37	13	not	not	PART
iajs-4130	37	14	equal	equal	ADJ
iajs-4130	37	15	to	to	ADP
iajs-4130	37	16	(	(	PUNCT
iajs-4130	37	17	(	(	PUNCT
iajs-4130	37	18	̅	̅	NOUN
iajs-4130	37	19	)	)	PUNCT
iajs-4130	37	20	)	)	PUNCT
iajs-4130	37	21	(	(	PUNCT
iajs-4130	37	22	̅	̅	NOUN
iajs-4130	37	23	)	)	PUNCT
iajs-4130	37	24	.	.	PUNCT
iajs-4130	38	1	therefore	therefore	ADV
iajs-4130	38	2	,	,	PUNCT
iajs-4130	38	3	a	a	PRON
iajs-4130	38	4	is	be	AUX
iajs-4130	38	5	not	not	PART
iajs-4130	38	6	the	the	DET
iajs-4130	38	7	endo	endo	NOUN
iajs-4130	38	8	-	-	PUNCT
iajs-4130	38	9	r.b	r.b	NOUN
iajs-4130	38	10	submodule	submodule	NOUN
iajs-4130	38	11	.	.	PUNCT
iajs-4130	39	1	now	now	ADV
iajs-4130	39	2	,	,	PUNCT
iajs-4130	39	3	we	we	PRON
iajs-4130	39	4	are	be	AUX
iajs-4130	39	5	ready	ready	ADJ
iajs-4130	39	6	to	to	PART
iajs-4130	39	7	give	give	VERB
iajs-4130	39	8	the	the	DET
iajs-4130	39	9	definition	definition	NOUN
iajs-4130	39	10	of	of	ADP
iajs-4130	39	11	the	the	DET
iajs-4130	39	12	endo	endo	NOUN
iajs-4130	39	13	-	-	PUNCT
iajs-4130	39	14	r.b	r.b	NOUN
iajs-4130	39	15	module	module	NOUN
iajs-4130	39	16	with	with	ADP
iajs-4130	39	17	some	some	DET
iajs-4130	39	18	examples	example	NOUN
iajs-4130	39	19	that	that	PRON
iajs-4130	39	20	explain	explain	VERB
iajs-4130	39	21	the	the	DET
iajs-4130	39	22	structure	structure	NOUN
iajs-4130	39	23	of	of	ADP
iajs-4130	39	24	the	the	DET
iajs-4130	39	25	definition	definition	NOUN
iajs-4130	39	26	.	.	PUNCT
iajs-4130	40	1	2.2	2.2	NUM
iajs-4130	40	2	.	.	PUNCT
iajs-4130	40	3	definition	definition	NOUN
iajs-4130	40	4	a	a	DET
iajs-4130	40	5	t	t	NOUN
iajs-4130	40	6	-	-	PUNCT
iajs-4130	40	7	module	module	NOUN
iajs-4130	40	8	is	be	AUX
iajs-4130	40	9	called	call	VERB
iajs-4130	40	10	endo	endo	NOUN
iajs-4130	40	11	-	-	PUNCT
iajs-4130	40	12	r.b	r.b	NOUN
iajs-4130	40	13	module	module	NOUN
iajs-4130	40	14	if	if	SCONJ
iajs-4130	40	15	every	every	DET
iajs-4130	40	16	proper	proper	ADJ
iajs-4130	40	17	submodule	submodule	NOUN
iajs-4130	40	18	of	of	ADP
iajs-4130	40	19	is	be	AUX
iajs-4130	40	20	an	an	DET
iajs-4130	40	21	endo	endo	NOUN
iajs-4130	40	22	-	-	PUNCT
iajs-4130	40	23	r.b	r.b	NOUN
iajs-4130	40	24	submodule	submodule	NOUN
iajs-4130	40	25	.	.	PUNCT
iajs-4130	41	1	examples	example	NOUN
iajs-4130	41	2	(	(	PUNCT
iajs-4130	41	3	1	1	X
iajs-4130	41	4	)	)	PUNCT
iajs-4130	41	5	as	as	ADP
iajs-4130	41	6	a	a	DET
iajs-4130	41	7	z	z	NOUN
iajs-4130	41	8	-	-	PUNCT
iajs-4130	41	9	module	module	NOUN
iajs-4130	41	10	is	be	AUX
iajs-4130	41	11	endo	endo	NOUN
iajs-4130	41	12	-	-	PUNCT
iajs-4130	41	13	r.b	r.b	NOUN
iajs-4130	41	14	where	where	SCONJ
iajs-4130	41	15	p	p	NOUN
iajs-4130	41	16	is	be	AUX
iajs-4130	41	17	a	a	DET
iajs-4130	41	18	prime	prime	ADJ
iajs-4130	41	19	number	number	NOUN
iajs-4130	41	20	since	since	SCONJ
iajs-4130	41	21	(	(	PUNCT
iajs-4130	41	22	̅	̅	NOUN
iajs-4130	41	23	)	)	PUNCT
iajs-4130	41	24	is	be	AUX
iajs-4130	41	25	the	the	DET
iajs-4130	41	26	only	only	ADJ
iajs-4130	41	27	proper	proper	ADJ
iajs-4130	41	28	submodule	submodule	NOUN
iajs-4130	41	29	of	of	ADP
iajs-4130	41	30	.	.	PUNCT
iajs-4130	42	1	to	to	PART
iajs-4130	42	2	show	show	VERB
iajs-4130	42	3	that	that	SCONJ
iajs-4130	42	4	,	,	PUNCT
iajs-4130	42	5	take	take	VERB
iajs-4130	42	6	an	an	DET
iajs-4130	42	7	endomorphism	endomorphism	NOUN
iajs-4130	42	8	(	(	PUNCT
iajs-4130	42	9	)	)	PUNCT
iajs-4130	42	10	as	as	ADP
iajs-4130	42	11	such	such	ADJ
iajs-4130	42	12	that	that	PRON
iajs-4130	42	13	(	(	PUNCT
iajs-4130	42	14	̅	̅	NOUN
iajs-4130	42	15	)	)	PUNCT
iajs-4130	42	16	̅	̅	NOUN
iajs-4130	42	17	̅	̅	NOUN
iajs-4130	42	18	then	then	ADV
iajs-4130	42	19	(	(	PUNCT
iajs-4130	42	20	̅	̅	NOUN
iajs-4130	42	21	)	)	PUNCT
iajs-4130	42	22	(	(	PUNCT
iajs-4130	42	23	̅	̅	NOUN
iajs-4130	42	24	)	)	PUNCT
iajs-4130	42	25	(	(	PUNCT
iajs-4130	42	26	̅	̅	NOUN
iajs-4130	42	27	)	)	PUNCT
iajs-4130	42	28	(	(	PUNCT
iajs-4130	42	29	̅	̅	NOUN
iajs-4130	42	30	)	)	PUNCT
iajs-4130	42	31	(	(	PUNCT
iajs-4130	42	32	̅	̅	NOUN
iajs-4130	42	33	)	)	PUNCT
iajs-4130	42	34	and	and	CCONJ
iajs-4130	42	35	(	(	PUNCT
iajs-4130	42	36	〈	〈	NOUN
iajs-4130	42	37	̅	̅	NOUN
iajs-4130	42	38	〉	〉	NOUN
iajs-4130	42	39	)	)	PUNCT
iajs-4130	42	40	(	(	PUNCT
iajs-4130	42	41	(	(	PUNCT
iajs-4130	42	42	̅	̅	NOUN
iajs-4130	42	43	)	)	PUNCT
iajs-4130	42	44	)	)	PUNCT
iajs-4130	42	45	ihjpas	ihjpas	PROPN
iajs-4130	42	46	.	.	PUNCT
iajs-4130	43	1	2025,38(4	2025,38(4	NOUN
iajs-4130	43	2	)	)	PUNCT
iajs-4130	43	3	400	400	NUM
iajs-4130	43	4	(	(	PUNCT
iajs-4130	43	5	2	2	X
iajs-4130	43	6	)	)	PUNCT
iajs-4130	43	7	let	let	VERB
iajs-4130	43	8	as	as	ADP
iajs-4130	43	9	a	a	DET
iajs-4130	43	10	z	z	NOUN
iajs-4130	43	11	-	-	PUNCT
iajs-4130	43	12	module	module	NOUN
iajs-4130	43	13	.	.	PUNCT
iajs-4130	44	1	define	define	VERB
iajs-4130	44	2	as	as	ADP
iajs-4130	44	3	(	(	PUNCT
iajs-4130	44	4	̅	̅	NOUN
iajs-4130	44	5	̅	̅	NOUN
iajs-4130	44	6	)	)	PUNCT
iajs-4130	44	7	(	(	PUNCT
iajs-4130	44	8	̅	̅	NOUN
iajs-4130	44	9	)	)	PUNCT
iajs-4130	44	10	(	(	PUNCT
iajs-4130	44	11	̅	̅	NUM
iajs-4130	44	12	̅	̅	NOUN
iajs-4130	44	13	)	)	PUNCT
iajs-4130	44	14	.	.	PUNCT
iajs-4130	45	1	then	then	ADV
iajs-4130	45	2	,	,	PUNCT
iajs-4130	45	3	if	if	SCONJ
iajs-4130	45	4	we	we	PRON
iajs-4130	45	5	take	take	VERB
iajs-4130	45	6	(	(	PUNCT
iajs-4130	45	7	̅	̅	NOUN
iajs-4130	45	8	)	)	PUNCT
iajs-4130	45	9	as	as	ADP
iajs-4130	45	10	a	a	DET
iajs-4130	45	11	submodule	submodule	NOUN
iajs-4130	45	12	of	of	ADP
iajs-4130	45	13	we	we	PRON
iajs-4130	45	14	get	get	VERB
iajs-4130	45	15	that	that	DET
iajs-4130	45	16	(	(	PUNCT
iajs-4130	45	17	̅	̅	NUM
iajs-4130	45	18	̅	̅	NOUN
iajs-4130	45	19	)	)	PUNCT
iajs-4130	45	20	(	(	PUNCT
iajs-4130	45	21	̅	̅	NOUN
iajs-4130	45	22	)	)	PUNCT
iajs-4130	45	23	.	.	PUNCT
iajs-4130	46	1	thus	thus	ADV
iajs-4130	46	2	,	,	PUNCT
iajs-4130	46	3	we	we	PRON
iajs-4130	46	4	conclude	conclude	VERB
iajs-4130	46	5	that	that	PRON
iajs-4130	46	6	is	be	AUX
iajs-4130	46	7	not	not	PART
iajs-4130	46	8	an	an	DET
iajs-4130	46	9	endo	endo	NOUN
iajs-4130	46	10	-	-	PUNCT
iajs-4130	46	11	r.b	r.b	PROPN
iajs-4130	46	12	z	z	PROPN
iajs-4130	46	13	-	-	PUNCT
iajs-4130	46	14	submodule	submodule	NOUN
iajs-4130	46	15	because	because	SCONJ
iajs-4130	46	16	if	if	SCONJ
iajs-4130	46	17	(	(	PUNCT
iajs-4130	46	18	̅	̅	NOUN
iajs-4130	46	19	̅	̅	NOUN
iajs-4130	46	20	)	)	PUNCT
iajs-4130	46	21	,	,	PUNCT
iajs-4130	46	22	then	then	ADV
iajs-4130	46	23	(	(	PUNCT
iajs-4130	46	24	)	)	PUNCT
iajs-4130	46	25	(	(	PUNCT
iajs-4130	46	26	̅	̅	NUM
iajs-4130	46	27	̅	̅	NOUN
iajs-4130	46	28	)	)	PUNCT
iajs-4130	46	29	and	and	CCONJ
iajs-4130	46	30	,	,	PUNCT
iajs-4130	46	31	hence	hence	ADV
iajs-4130	46	32	,	,	PUNCT
iajs-4130	46	33	is	be	AUX
iajs-4130	46	34	not	not	PART
iajs-4130	46	35	an	an	DET
iajs-4130	46	36	endo	endo	NOUN
iajs-4130	46	37	-	-	PUNCT
iajs-4130	46	38	r.b	r.b	PROPN
iajs-4130	46	39	z	z	NOUN
iajs-4130	46	40	-	-	PUNCT
iajs-4130	46	41	module	module	NOUN
iajs-4130	46	42	.	.	PUNCT
iajs-4130	47	1	2.3	2.3	NUM
iajs-4130	47	2	.	.	PUNCT
iajs-4130	47	3	remark	remark	NOUN
iajs-4130	47	4	(	(	PUNCT
iajs-4130	47	5	1	1	NUM
iajs-4130	47	6	)	)	PUNCT
iajs-4130	47	7	every	every	DET
iajs-4130	47	8	endo	endo	NOUN
iajs-4130	47	9	-	-	PUNCT
iajs-4130	47	10	r.b.t	r.b.t	NOUN
iajs-4130	47	11	-	-	PUNCT
iajs-4130	47	12	module	module	NOUN
iajs-4130	47	13	is	be	AUX
iajs-4130	47	14	bounded	bound	VERB
iajs-4130	47	15	,	,	PUNCT
iajs-4130	47	16	but	but	CCONJ
iajs-4130	47	17	the	the	DET
iajs-4130	47	18	converse	converse	NOUN
iajs-4130	47	19	is	be	AUX
iajs-4130	47	20	not	not	PART
iajs-4130	47	21	true	true	ADJ
iajs-4130	47	22	.	.	PUNCT
iajs-4130	48	1	to	to	PART
iajs-4130	48	2	show	show	VERB
iajs-4130	48	3	that	that	SCONJ
iajs-4130	48	4	,	,	PUNCT
iajs-4130	48	5	let	let	VERB
iajs-4130	48	6	as	as	ADP
iajs-4130	48	7	a	a	DET
iajs-4130	48	8	z	z	NOUN
iajs-4130	48	9	-	-	PUNCT
iajs-4130	48	10	module	module	NOUN
iajs-4130	48	11	and	and	CCONJ
iajs-4130	48	12	let	let	VERB
iajs-4130	48	13	(	(	PUNCT
iajs-4130	48	14	̅	̅	NOUN
iajs-4130	48	15	)	)	PUNCT
iajs-4130	48	16	be	be	AUX
iajs-4130	48	17	a	a	DET
iajs-4130	48	18	submodule	submodule	NOUN
iajs-4130	48	19	of	of	ADP
iajs-4130	48	20	define	define	NOUN
iajs-4130	48	21	where	where	SCONJ
iajs-4130	48	22	(	(	PUNCT
iajs-4130	48	23	)	)	PUNCT
iajs-4130	48	24	by	by	ADP
iajs-4130	48	25	(	(	PUNCT
iajs-4130	48	26	̅	̅	NOUN
iajs-4130	48	27	̅	̅	NOUN
iajs-4130	48	28	)	)	PUNCT
iajs-4130	48	29	(	(	PUNCT
iajs-4130	48	30	̅	̅	NUM
iajs-4130	48	31	̅	̅	NOUN
iajs-4130	48	32	)	)	PUNCT
iajs-4130	48	33	since	since	SCONJ
iajs-4130	48	34	(	(	PUNCT
iajs-4130	48	35	̅	̅	NUM
iajs-4130	48	36	̅	̅	NOUN
iajs-4130	48	37	)	)	PUNCT
iajs-4130	48	38	(	(	PUNCT
iajs-4130	48	39	̅	̅	NUM
iajs-4130	48	40	̅	̅	NOUN
iajs-4130	48	41	)	)	PUNCT
iajs-4130	48	42	but	but	CCONJ
iajs-4130	48	43	(	(	PUNCT
iajs-4130	48	44	)	)	PUNCT
iajs-4130	48	45	(	(	PUNCT
iajs-4130	48	46	̅	̅	NUM
iajs-4130	48	47	̅	̅	NOUN
iajs-4130	48	48	)	)	PUNCT
iajs-4130	48	49	.	.	PUNCT
iajs-4130	49	1	therefore	therefore	ADV
iajs-4130	49	2	,	,	PUNCT
iajs-4130	49	3	is	be	AUX
iajs-4130	49	4	not	not	PART
iajs-4130	49	5	an	an	DET
iajs-4130	49	6	endo	endo	NOUN
iajs-4130	49	7	-	-	PUNCT
iajs-4130	49	8	r.b	r.b	NOUN
iajs-4130	49	9	,	,	PUNCT
iajs-4130	49	10	while	while	SCONJ
iajs-4130	49	11	is	be	AUX
iajs-4130	49	12	a	a	DET
iajs-4130	49	13	bounded	bounded	ADJ
iajs-4130	49	14	t	t	NOUN
iajs-4130	49	15	-	-	PUNCT
iajs-4130	49	16	module	module	NOUN
iajs-4130	49	17	since	since	SCONJ
iajs-4130	49	18	there	there	PRON
iajs-4130	49	19	exists	exist	VERB
iajs-4130	49	20	an	an	DET
iajs-4130	49	21	element	element	NOUN
iajs-4130	49	22	(	(	PUNCT
iajs-4130	49	23	̅	̅	NOUN
iajs-4130	49	24	̅	̅	NOUN
iajs-4130	49	25	)	)	PUNCT
iajs-4130	49	26	such	such	ADJ
iajs-4130	49	27	that	that	PRON
iajs-4130	49	28	(	(	PUNCT
iajs-4130	49	29	)	)	PUNCT
iajs-4130	49	30	(	(	PUNCT
iajs-4130	49	31	̅	̅	NUM
iajs-4130	49	32	̅	̅	NOUN
iajs-4130	49	33	)	)	PUNCT
iajs-4130	49	34	.	.	PUNCT
iajs-4130	50	1	(	(	PUNCT
iajs-4130	50	2	2	2	X
iajs-4130	50	3	)	)	PUNCT
iajs-4130	50	4	every	every	DET
iajs-4130	50	5	proper	proper	ADJ
iajs-4130	50	6	submodule	submodule	NOUN
iajs-4130	50	7	of	of	ADP
iajs-4130	50	8	the	the	DET
iajs-4130	50	9	endo	endo	NOUN
iajs-4130	50	10	-	-	PUNCT
iajs-4130	50	11	r.b	r.b	NOUN
iajs-4130	50	12	module	module	NOUN
iajs-4130	50	13	is	be	AUX
iajs-4130	50	14	also	also	ADV
iajs-4130	50	15	an	an	DET
iajs-4130	50	16	endo	endo	NOUN
iajs-4130	50	17	-	-	PUNCT
iajs-4130	50	18	r.b	r.b	NOUN
iajs-4130	50	19	.	.	PUNCT
iajs-4130	51	1	(	(	PUNCT
iajs-4130	51	2	3	3	X
iajs-4130	51	3	)	)	PUNCT
iajs-4130	51	4	the	the	DET
iajs-4130	51	5	intersection	intersection	NOUN
iajs-4130	51	6	of	of	ADP
iajs-4130	51	7	two	two	NUM
iajs-4130	51	8	endo	endo	NOUN
iajs-4130	51	9	-	-	PUNCT
iajs-4130	51	10	r	r	NOUN
iajs-4130	51	11	,	,	PUNCT
iajs-4130	51	12	b	b	NOUN
iajs-4130	51	13	submodules	submodule	NOUN
iajs-4130	51	14	of	of	ADP
iajs-4130	51	15	is	be	AUX
iajs-4130	51	16	an	an	DET
iajs-4130	51	17	endo	endo	NOUN
iajs-4130	51	18	-	-	PUNCT
iajs-4130	51	19	r.b	r.b	NOUN
iajs-4130	51	20	since	since	SCONJ
iajs-4130	51	21	if	if	SCONJ
iajs-4130	51	22	and	and	CCONJ
iajs-4130	51	23	are	be	AUX
iajs-4130	51	24	two	two	NUM
iajs-4130	51	25	endo	endo	NOUN
iajs-4130	51	26	-	-	PUNCT
iajs-4130	51	27	r.b	r.b	NOUN
iajs-4130	51	28	submodules	submodule	NOUN
iajs-4130	51	29	.	.	PUNCT
iajs-4130	52	1	then	then	ADV
iajs-4130	52	2	,	,	PUNCT
iajs-4130	52	3	which	which	PRON
iajs-4130	52	4	implies	imply	VERB
iajs-4130	52	5	that	that	PRON
iajs-4130	52	6	is	be	AUX
iajs-4130	52	7	an	an	DET
iajs-4130	52	8	endo	endo	NOUN
iajs-4130	52	9	-	-	PUNCT
iajs-4130	52	10	r.b	r.b	NOUN
iajs-4130	52	11	submodule	submodule	NOUN
iajs-4130	52	12	.	.	PUNCT
iajs-4130	53	1	2.4	2.4	NUM
iajs-4130	53	2	.	.	PUNCT
iajs-4130	53	3	proposition	proposition	NOUN
iajs-4130	53	4	let	let	AUX
iajs-4130	53	5	be	be	AUX
iajs-4130	53	6	a	a	DET
iajs-4130	53	7	t	t	NOUN
iajs-4130	53	8	-	-	PUNCT
iajs-4130	53	9	module	module	NOUN
iajs-4130	53	10	and	and	CCONJ
iajs-4130	53	11	.	.	PUNCT
iajs-4130	54	1	then	then	ADV
iajs-4130	54	2	(	(	PUNCT
iajs-4130	54	3	1	1	X
iajs-4130	54	4	)	)	PUNCT
iajs-4130	54	5	if	if	SCONJ
iajs-4130	54	6	n	n	PRON
iajs-4130	54	7	is	be	AUX
iajs-4130	54	8	an	an	DET
iajs-4130	54	9	endo	endo	NOUN
iajs-4130	54	10	-	-	PUNCT
iajs-4130	54	11	r.b	r.b	NOUN
iajs-4130	54	12	submodule	submodule	NOUN
iajs-4130	54	13	of	of	ADP
iajs-4130	54	14	and	and	CCONJ
iajs-4130	54	15	an	an	DET
iajs-4130	54	16	epimorphism	epimorphism	NOUN
iajs-4130	54	17	(	(	PUNCT
iajs-4130	54	18	)	)	PUNCT
iajs-4130	54	19	,	,	PUNCT
iajs-4130	54	20	then	then	ADV
iajs-4130	54	21	(	(	PUNCT
iajs-4130	54	22	)	)	PUNCT
iajs-4130	54	23	is	be	AUX
iajs-4130	54	24	an	an	DET
iajs-4130	54	25	endo	endo	NOUN
iajs-4130	54	26	-	-	PUNCT
iajs-4130	54	27	r.b	r.b	NOUN
iajs-4130	54	28	submodule	submodule	NOUN
iajs-4130	54	29	of	of	ADP
iajs-4130	54	30	(	(	PUNCT
iajs-4130	54	31	2	2	X
iajs-4130	54	32	)	)	PUNCT
iajs-4130	54	33	if	if	SCONJ
iajs-4130	54	34	n	n	PRON
iajs-4130	54	35	is	be	AUX
iajs-4130	54	36	an	an	DET
iajs-4130	54	37	endo	endo	NOUN
iajs-4130	54	38	-	-	PUNCT
iajs-4130	54	39	r.b	r.b	NOUN
iajs-4130	54	40	submodule	submodule	NOUN
iajs-4130	54	41	of	of	ADP
iajs-4130	54	42	,	,	PUNCT
iajs-4130	54	43	then	then	ADV
iajs-4130	54	44	(	(	PUNCT
iajs-4130	54	45	)	)	PUNCT
iajs-4130	54	46	is	be	AUX
iajs-4130	54	47	an	an	DET
iajs-4130	54	48	endo	endo	NOUN
iajs-4130	54	49	-	-	PUNCT
iajs-4130	54	50	r.b	r.b	NOUN
iajs-4130	54	51	submodule	submodule	NOUN
iajs-4130	54	52	of	of	ADP
iajs-4130	54	53	.	.	PUNCT
iajs-4130	55	1	proof	proof	NOUN
iajs-4130	55	2	.	.	PUNCT
iajs-4130	56	1	(	(	PUNCT
iajs-4130	56	2	1	1	X
iajs-4130	56	3	)	)	PUNCT
iajs-4130	56	4	define	define	VERB
iajs-4130	56	5	as	as	ADP
iajs-4130	56	6	(	(	PUNCT
iajs-4130	56	7	)	)	PUNCT
iajs-4130	56	8	and	and	CCONJ
iajs-4130	56	9	suppose	suppose	VERB
iajs-4130	56	10	,	,	PUNCT
iajs-4130	56	11	then	then	ADV
iajs-4130	56	12	(	(	PUNCT
iajs-4130	56	13	)	)	PUNCT
iajs-4130	56	14	because	because	SCONJ
iajs-4130	56	15	n	n	NUM
iajs-4130	56	16	is	be	AUX
iajs-4130	56	17	an	an	DET
iajs-4130	56	18	endo	endo	NOUN
iajs-4130	56	19	-	-	PUNCT
iajs-4130	56	20	r.b	r.b	NOUN
iajs-4130	56	21	submodule	submodule	NOUN
iajs-4130	56	22	of	of	ADP
iajs-4130	56	23	it	it	PRON
iajs-4130	56	24	is	be	AUX
iajs-4130	56	25	clear	clear	ADJ
iajs-4130	56	26	that	that	SCONJ
iajs-4130	56	27	(	(	PUNCT
iajs-4130	56	28	)	)	PUNCT
iajs-4130	56	29	we	we	PRON
iajs-4130	56	30	can	can	AUX
iajs-4130	56	31	define	define	VERB
iajs-4130	56	32	as	as	ADP
iajs-4130	56	33	(	(	PUNCT
iajs-4130	56	34	(	(	PUNCT
iajs-4130	56	35	)	)	PUNCT
iajs-4130	56	36	)	)	PUNCT
iajs-4130	56	37	.	.	PUNCT
iajs-4130	57	1	then	then	ADV
iajs-4130	57	2	,	,	PUNCT
iajs-4130	57	3	we	we	PRON
iajs-4130	57	4	have	have	VERB
iajs-4130	57	5	the	the	DET
iajs-4130	57	6	following	following	ADJ
iajs-4130	57	7	commutative	commutative	ADJ
iajs-4130	57	8	diagram	diagram	NOUN
iajs-4130	57	9	:	:	PUNCT
iajs-4130	57	10	since	since	SCONJ
iajs-4130	57	11	(	(	PUNCT
iajs-4130	57	12	)	)	PUNCT
iajs-4130	57	13	(	(	PUNCT
iajs-4130	57	14	)	)	PUNCT
iajs-4130	57	15	(	(	PUNCT
iajs-4130	57	16	(	(	PUNCT
iajs-4130	57	17	)	)	PUNCT
iajs-4130	57	18	)	)	PUNCT
iajs-4130	58	1	it	it	PRON
iajs-4130	58	2	is	be	AUX
iajs-4130	58	3	obvious	obvious	ADJ
iajs-4130	58	4	that	that	SCONJ
iajs-4130	58	5	(	(	PUNCT
iajs-4130	58	6	(	(	PUNCT
iajs-4130	58	7	)	)	PUNCT
iajs-4130	58	8	)	)	PUNCT
iajs-4130	58	9	(	(	PUNCT
iajs-4130	58	10	)	)	PUNCT
iajs-4130	58	11	.	.	PUNCT
iajs-4130	59	1	(	(	PUNCT
iajs-4130	59	2	(	(	PUNCT
iajs-4130	59	3	)	)	PUNCT
iajs-4130	59	4	)	)	PUNCT
iajs-4130	59	5	(	(	PUNCT
iajs-4130	59	6	)	)	PUNCT
iajs-4130	59	7	(	(	PUNCT
iajs-4130	59	8	)	)	PUNCT
iajs-4130	59	9	because	because	SCONJ
iajs-4130	59	10	is	be	AUX
iajs-4130	59	11	an	an	DET
iajs-4130	59	12	epimorphism	epimorphism	NOUN
iajs-4130	59	13	.	.	PUNCT
iajs-4130	60	1	also	also	ADV
iajs-4130	60	2	,	,	PUNCT
iajs-4130	60	3	it	it	PRON
iajs-4130	60	4	is	be	AUX
iajs-4130	60	5	clear	clear	ADJ
iajs-4130	60	6	that	that	SCONJ
iajs-4130	60	7	(	(	PUNCT
iajs-4130	60	8	(	(	PUNCT
iajs-4130	60	9	)	)	PUNCT
iajs-4130	60	10	)	)	PUNCT
iajs-4130	61	1	(	(	PUNCT
iajs-4130	61	2	(	(	PUNCT
iajs-4130	61	3	(	(	PUNCT
iajs-4130	61	4	)	)	PUNCT
iajs-4130	61	5	)	)	PUNCT
iajs-4130	61	6	another	another	DET
iajs-4130	61	7	containment	containment	NOUN
iajs-4130	61	8	,	,	PUNCT
iajs-4130	61	9	let	let	VERB
iajs-4130	61	10	(	(	PUNCT
iajs-4130	61	11	(	(	PUNCT
iajs-4130	61	12	(	(	PUNCT
iajs-4130	61	13	(	(	PUNCT
iajs-4130	61	14	)	)	PUNCT
iajs-4130	61	15	)	)	PUNCT
iajs-4130	61	16	,	,	PUNCT
iajs-4130	61	17	then	then	ADV
iajs-4130	61	18	(	(	PUNCT
iajs-4130	61	19	(	(	PUNCT
iajs-4130	61	20	)	)	PUNCT
iajs-4130	61	21	)	)	PUNCT
iajs-4130	61	22	implies	imply	VERB
iajs-4130	61	23	that	that	SCONJ
iajs-4130	61	24	(	(	PUNCT
iajs-4130	61	25	)	)	PUNCT
iajs-4130	61	26	for	for	ADP
iajs-4130	61	27	all	all	PRON
iajs-4130	61	28	,	,	PUNCT
iajs-4130	61	29	then	then	ADV
iajs-4130	61	30	and	and	CCONJ
iajs-4130	61	31	since	since	SCONJ
iajs-4130	61	32	is	be	AUX
iajs-4130	61	33	an	an	DET
iajs-4130	61	34	epimorphism	epimorphism	NOUN
iajs-4130	61	35	(	(	PUNCT
iajs-4130	61	36	(	(	PUNCT
iajs-4130	61	37	)	)	PUNCT
iajs-4130	61	38	)	)	PUNCT
iajs-4130	61	39	we	we	PRON
iajs-4130	61	40	obtain	obtain	VERB
iajs-4130	61	41	(	(	PUNCT
iajs-4130	61	42	)	)	PUNCT
iajs-4130	61	43	and	and	CCONJ
iajs-4130	61	44	hence	hence	ADV
iajs-4130	61	45	,	,	PUNCT
iajs-4130	61	46	(	(	PUNCT
iajs-4130	61	47	(	(	PUNCT
iajs-4130	61	48	)	)	PUNCT
iajs-4130	61	49	)	)	PUNCT
iajs-4130	62	1	(	(	PUNCT
iajs-4130	62	2	2	2	X
iajs-4130	62	3	)	)	PUNCT
iajs-4130	62	4	suppose	suppose	VERB
iajs-4130	62	5	that	that	SCONJ
iajs-4130	62	6	,	,	PUNCT
iajs-4130	62	7	then	then	ADV
iajs-4130	62	8	there	there	PRON
iajs-4130	62	9	exists	exist	VERB
iajs-4130	62	10	defined	define	VERB
iajs-4130	62	11	as	as	ADP
iajs-4130	62	12	:	:	PUNCT
iajs-4130	62	13	(	(	PUNCT
iajs-4130	62	14	)	)	PUNCT
iajs-4130	62	15	and	and	CCONJ
iajs-4130	62	16	(	(	PUNCT
iajs-4130	62	17	)	)	PUNCT
iajs-4130	62	18	.	.	PUNCT
iajs-4130	63	1	assume	assume	VERB
iajs-4130	63	2	(	(	PUNCT
iajs-4130	63	3	)	)	PUNCT
iajs-4130	63	4	and	and	CCONJ
iajs-4130	63	5	define	define	VERB
iajs-4130	63	6	:	:	PUNCT
iajs-4130	63	7	(	(	PUNCT
iajs-4130	63	8	(	(	PUNCT
iajs-4130	63	9	)	)	PUNCT
iajs-4130	63	10	)	)	PUNCT
iajs-4130	63	11	(	(	PUNCT
iajs-4130	63	12	)	)	PUNCT
iajs-4130	63	13	since	since	SCONJ
iajs-4130	63	14	is	be	AUX
iajs-4130	63	15	an	an	DET
iajs-4130	63	16	isomorphism	isomorphism	NOUN
iajs-4130	63	17	,	,	PUNCT
iajs-4130	63	18	then	then	ADV
iajs-4130	63	19	is	be	AUX
iajs-4130	63	20	onto	onto	ADP
iajs-4130	63	21	and	and	CCONJ
iajs-4130	63	22	hence	hence	ADV
iajs-4130	63	23	,	,	PUNCT
iajs-4130	63	24	(	(	PUNCT
iajs-4130	63	25	)	)	PUNCT
iajs-4130	63	26	from	from	ADP
iajs-4130	63	27	this	this	DET
iajs-4130	63	28	fact	fact	NOUN
iajs-4130	63	29	,	,	PUNCT
iajs-4130	63	30	we	we	PRON
iajs-4130	63	31	get	get	VERB
iajs-4130	63	32	(	(	PUNCT
iajs-4130	63	33	(	(	PUNCT
iajs-4130	63	34	)	)	PUNCT
iajs-4130	63	35	)	)	PUNCT
iajs-4130	64	1	(	(	PUNCT
iajs-4130	64	2	)	)	PUNCT
iajs-4130	64	3	it	it	PRON
iajs-4130	64	4	is	be	AUX
iajs-4130	64	5	clear	clear	ADJ
iajs-4130	64	6	that	that	SCONJ
iajs-4130	64	7	:	:	PUNCT
iajs-4130	64	8	(	(	PUNCT
iajs-4130	64	9	(	(	PUNCT
iajs-4130	64	10	)	)	PUNCT
iajs-4130	64	11	)	)	PUNCT
iajs-4130	64	12	(	(	PUNCT
iajs-4130	64	13	(	(	PUNCT
iajs-4130	64	14	(	(	PUNCT
iajs-4130	64	15	)	)	PUNCT
iajs-4130	64	16	)	)	PUNCT
iajs-4130	64	17	now	now	ADV
iajs-4130	64	18	,	,	PUNCT
iajs-4130	64	19	let	let	VERB
iajs-4130	64	20	(	(	PUNCT
iajs-4130	64	21	(	(	PUNCT
iajs-4130	64	22	(	(	PUNCT
iajs-4130	64	23	)	)	PUNCT
iajs-4130	64	24	)	)	PUNCT
iajs-4130	64	25	(	(	PUNCT
iajs-4130	64	26	(	(	PUNCT
iajs-4130	64	27	)	)	PUNCT
iajs-4130	64	28	)	)	PUNCT
iajs-4130	64	29	.	.	PUNCT
iajs-4130	65	1	thus	thus	ADV
iajs-4130	65	2	,	,	PUNCT
iajs-4130	65	3	(	(	PUNCT
iajs-4130	65	4	)	)	PUNCT
iajs-4130	65	5	for	for	ADP
iajs-4130	65	6	all	all	PRON
iajs-4130	65	7	,	,	PUNCT
iajs-4130	65	8	then	then	ADV
iajs-4130	65	9	(	(	PUNCT
iajs-4130	65	10	)	)	PUNCT
iajs-4130	65	11	.	.	PUNCT
iajs-4130	66	1	therefore	therefore	ADV
iajs-4130	66	2	,	,	PUNCT
iajs-4130	66	3	since	since	SCONJ
iajs-4130	66	4	is	be	AUX
iajs-4130	66	5	outomor	outomor	ADJ
iajs-4130	66	6	ω	ω	PROPN
iajs-4130	66	7	ω	ω	NOUN
iajs-4130	67	1	𝐼ω	𝐼ω	PROPN
iajs-4130	67	2	ω	ω	PROPN
iajs-4130	67	3	𝜓	𝜓	PROPN
iajs-4130	67	4	𝜑	𝜑	X
iajs-4130	67	5	𝜓	𝜓	NOUN
iajs-4130	67	6	∘	∘	NOUN
iajs-4130	67	7	𝜑	𝜑	ADP
iajs-4130	67	8	𝐼ω	𝐼ω	PROPN
iajs-4130	67	9	ihjpas	ihjpa	VERB
iajs-4130	67	10	.	.	PUNCT
iajs-4130	68	1	2025,38(4	2025,38(4	X
iajs-4130	68	2	)	)	PUNCT
iajs-4130	69	1	401	401	NUM
iajs-4130	69	2	then	then	ADV
iajs-4130	69	3	(	(	PUNCT
iajs-4130	69	4	(	(	PUNCT
iajs-4130	69	5	(	(	PUNCT
iajs-4130	69	6	)	)	PUNCT
iajs-4130	69	7	)	)	PUNCT
iajs-4130	70	1	(	(	PUNCT
iajs-4130	70	2	(	(	PUNCT
iajs-4130	70	3	)	)	PUNCT
iajs-4130	70	4	)	)	PUNCT
iajs-4130	70	5	the	the	DET
iajs-4130	70	6	next	next	ADJ
iajs-4130	70	7	proposition	proposition	NOUN
iajs-4130	70	8	shows	show	VERB
iajs-4130	70	9	that	that	SCONJ
iajs-4130	70	10	a	a	DET
iajs-4130	70	11	divisible	divisible	ADJ
iajs-4130	70	12	t	t	NOUN
iajs-4130	70	13	-	-	PUNCT
iajs-4130	70	14	module	module	NOUN
iajs-4130	70	15	plays	play	VERB
iajs-4130	70	16	an	an	DET
iajs-4130	70	17	important	important	ADJ
iajs-4130	70	18	role	role	NOUN
iajs-4130	70	19	for	for	ADP
iajs-4130	70	20	a	a	DET
iajs-4130	70	21	cyclic	cyclic	ADJ
iajs-4130	70	22	submodule	submodule	NOUN
iajs-4130	70	23	to	to	PART
iajs-4130	70	24	be	be	AUX
iajs-4130	70	25	an	an	DET
iajs-4130	70	26	endo	endo	NOUN
iajs-4130	70	27	-	-	PUNCT
iajs-4130	70	28	r.b	r.b	NOUN
iajs-4130	70	29	.	.	NOUN
iajs-4130	70	30	3	3	X
iajs-4130	70	31	.	.	PUNCT
iajs-4130	71	1	the	the	DET
iajs-4130	71	2	radical	radical	NOUN
iajs-4130	71	3	of	of	ADP
iajs-4130	71	4	an	an	DET
iajs-4130	71	5	endo	endo	NOUN
iajs-4130	71	6	-	-	PUNCT
iajs-4130	71	7	r.b	r.b	PROPN
iajs-4130	71	8	t	t	PROPN
iajs-4130	71	9	-	-	PUNCT
iajs-4130	71	10	submodule	submodule	NOUN
iajs-4130	71	11	in	in	ADP
iajs-4130	71	12	this	this	DET
iajs-4130	71	13	section	section	NOUN
iajs-4130	71	14	,	,	PUNCT
iajs-4130	71	15	we	we	PRON
iajs-4130	71	16	are	be	AUX
iajs-4130	71	17	ready	ready	ADJ
iajs-4130	71	18	to	to	PART
iajs-4130	71	19	focus	focus	VERB
iajs-4130	71	20	on	on	ADP
iajs-4130	71	21	the	the	DET
iajs-4130	71	22	main	main	ADJ
iajs-4130	71	23	purpose	purpose	NOUN
iajs-4130	71	24	of	of	ADP
iajs-4130	71	25	this	this	DET
iajs-4130	71	26	paper	paper	NOUN
iajs-4130	71	27	and	and	CCONJ
iajs-4130	71	28	give	give	VERB
iajs-4130	71	29	some	some	DET
iajs-4130	71	30	definitions	definition	NOUN
iajs-4130	71	31	and	and	CCONJ
iajs-4130	71	32	properties	property	NOUN
iajs-4130	71	33	that	that	PRON
iajs-4130	71	34	illustrate	illustrate	VERB
iajs-4130	71	35	the	the	DET
iajs-4130	71	36	notion	notion	NOUN
iajs-4130	71	37	of	of	ADP
iajs-4130	71	38	the	the	DET
iajs-4130	71	39	radical	radical	NOUN
iajs-4130	71	40	of	of	ADP
iajs-4130	71	41	an	an	DET
iajs-4130	71	42	endo	endo	NOUN
iajs-4130	71	43	-	-	PUNCT
iajs-4130	71	44	r.b	r.b	NOUN
iajs-4130	71	45	submodule	submodule	NOUN
iajs-4130	71	46	.	.	PUNCT
iajs-4130	72	1	3.1	3.1	NUM
iajs-4130	72	2	.	.	PUNCT
iajs-4130	72	3	definition	definition	NOUN
iajs-4130	72	4	the	the	DET
iajs-4130	72	5	radical	radical	NOUN
iajs-4130	72	6	of	of	ADP
iajs-4130	72	7	an	an	DET
iajs-4130	72	8	endo	endo	NOUN
iajs-4130	72	9	-	-	PUNCT
iajs-4130	72	10	r.b	r.b	NOUN
iajs-4130	72	11	submodule	submodule	PROPN
iajs-4130	72	12	n	n	PROPN
iajs-4130	72	13	of	of	ADP
iajs-4130	72	14	a	a	DET
iajs-4130	72	15	t	t	NOUN
iajs-4130	72	16	-	-	PUNCT
iajs-4130	72	17	module	module	NOUN
iajs-4130	72	18	is	be	AUX
iajs-4130	72	19	denoted	denote	VERB
iajs-4130	72	20	by	by	ADP
iajs-4130	72	21	endo	endo	NOUN
iajs-4130	72	22	(	(	PUNCT
iajs-4130	72	23	)	)	PUNCT
iajs-4130	72	24	and	and	CCONJ
iajs-4130	72	25	defined	define	VERB
iajs-4130	72	26	as	as	ADP
iajs-4130	72	27	the	the	DET
iajs-4130	72	28	intersection	intersection	NOUN
iajs-4130	72	29	of	of	ADP
iajs-4130	72	30	all	all	DET
iajs-4130	72	31	endo	endo	NOUN
iajs-4130	72	32	-	-	PUNCT
iajs-4130	72	33	r.b	r.b	NOUN
iajs-4130	72	34	submodules	submodule	NOUN
iajs-4130	72	35	of	of	ADP
iajs-4130	72	36	that	that	PRON
iajs-4130	72	37	contains	contain	VERB
iajs-4130	72	38	n.	n.	NOUN
iajs-4130	72	39	if	if	SCONJ
iajs-4130	72	40	there	there	PRON
iajs-4130	72	41	exists	exist	VERB
iajs-4130	72	42	no	no	DET
iajs-4130	72	43	endo	endo	NOUN
iajs-4130	72	44	-	-	PUNCT
iajs-4130	72	45	r.b	r.b	NOUN
iajs-4130	72	46	submodule	submodule	NOUN
iajs-4130	72	47	of	of	ADP
iajs-4130	72	48	containing	contain	VERB
iajs-4130	72	49	n	n	CCONJ
iajs-4130	72	50	,	,	PUNCT
iajs-4130	72	51	then	then	ADV
iajs-4130	72	52	we	we	PRON
iajs-4130	72	53	write	write	VERB
iajs-4130	72	54	:	:	PUNCT
iajs-4130	72	55	endo	endo	NOUN
iajs-4130	72	56	(	(	PUNCT
iajs-4130	72	57	)	)	PUNCT
iajs-4130	72	58	.	.	PUNCT
iajs-4130	73	1	if	if	SCONJ
iajs-4130	73	2	=	=	NOUN
iajs-4130	73	3	t	t	PROPN
iajs-4130	73	4	and	and	CCONJ
iajs-4130	73	5	n	n	PROPN
iajs-4130	73	6	is	be	AUX
iajs-4130	73	7	an	an	DET
iajs-4130	73	8	ideal	ideal	NOUN
iajs-4130	73	9	of	of	ADP
iajs-4130	73	10	t	t	PROPN
iajs-4130	73	11	,	,	PUNCT
iajs-4130	73	12	then	then	ADV
iajs-4130	73	13	endo	endo	PROPN
iajs-4130	73	14	(	(	PUNCT
iajs-4130	73	15	)	)	PUNCT
iajs-4130	73	16	is	be	AUX
iajs-4130	73	17	the	the	DET
iajs-4130	73	18	intersection	intersection	NOUN
iajs-4130	73	19	of	of	ADP
iajs-4130	73	20	all	all	DET
iajs-4130	73	21	endo	endo	NOUN
iajs-4130	73	22	-	-	PUNCT
iajs-4130	73	23	r.b	r.b	NOUN
iajs-4130	73	24	ideals	ideal	NOUN
iajs-4130	73	25	of	of	ADP
iajs-4130	73	26	t	t	NOUN
iajs-4130	73	27	containing	contain	VERB
iajs-4130	73	28	n.	n.	PROPN
iajs-4130	73	29	3.2	3.2	NUM
iajs-4130	73	30	.	.	PUNCT
iajs-4130	74	1	definition	definition	NOUN
iajs-4130	74	2	a	a	DET
iajs-4130	74	3	proper	proper	ADJ
iajs-4130	74	4	submodule	submodule	NOUN
iajs-4130	74	5	n	n	PROPN
iajs-4130	74	6	of	of	ADP
iajs-4130	74	7	an	an	DET
iajs-4130	74	8	t	t	NOUN
iajs-4130	74	9	-	-	PUNCT
iajs-4130	74	10	module	module	NOUN
iajs-4130	74	11	is	be	AUX
iajs-4130	74	12	called	call	VERB
iajs-4130	74	13	an	an	DET
iajs-4130	74	14	endo	endo	NOUN
iajs-4130	74	15	-	-	PUNCT
iajs-4130	74	16	r.b	r.b	NOUN
iajs-4130	74	17	radical	radical	ADJ
iajs-4130	74	18	submodule	submodule	NOUN
iajs-4130	74	19	if	if	SCONJ
iajs-4130	74	20	endo	endo	PROPN
iajs-4130	74	21	(	(	PUNCT
iajs-4130	74	22	)	)	PUNCT
iajs-4130	74	23	3.3	3.3	NUM
iajs-4130	74	24	.	.	PUNCT
iajs-4130	75	1	remark	remark	NOUN
iajs-4130	75	2	let	let	VERB
iajs-4130	75	3	n	n	PRON
iajs-4130	75	4	be	be	AUX
iajs-4130	75	5	an	an	DET
iajs-4130	75	6	endo	endo	NOUN
iajs-4130	75	7	-	-	PUNCT
iajs-4130	75	8	r.b	r.b	NOUN
iajs-4130	75	9	submodule	submodule	NOUN
iajs-4130	75	10	of	of	ADP
iajs-4130	75	11	a	a	DET
iajs-4130	75	12	t	t	NOUN
iajs-4130	75	13	-	-	PUNCT
iajs-4130	75	14	module	module	NOUN
iajs-4130	75	15	,	,	PUNCT
iajs-4130	75	16	then	then	ADV
iajs-4130	75	17	endo	endo	PROPN
iajs-4130	75	18	(	(	PUNCT
iajs-4130	75	19	)	)	PUNCT
iajs-4130	75	20	is	be	AUX
iajs-4130	75	21	also	also	ADV
iajs-4130	75	22	an	an	DET
iajs-4130	75	23	endo	endo	NOUN
iajs-4130	75	24	-	-	PUNCT
iajs-4130	75	25	r.b	r.b	NOUN
iajs-4130	75	26	submodule	submodule	NOUN
iajs-4130	75	27	.	.	PUNCT
iajs-4130	76	1	proof	proof	NOUN
iajs-4130	76	2	.	.	PUNCT
iajs-4130	77	1	by	by	ADP
iajs-4130	77	2	the	the	DET
iajs-4130	77	3	definition	definition	NOUN
iajs-4130	77	4	of	of	ADP
iajs-4130	77	5	the	the	DET
iajs-4130	77	6	radical	radical	NOUN
iajs-4130	77	7	of	of	ADP
iajs-4130	77	8	an	an	DET
iajs-4130	77	9	endo	endo	NOUN
iajs-4130	77	10	-	-	PUNCT
iajs-4130	77	11	r.b	r.b	NOUN
iajs-4130	77	12	submodule	submodule	NOUN
iajs-4130	77	13	,	,	PUNCT
iajs-4130	77	14	we	we	PRON
iajs-4130	77	15	have	have	VERB
iajs-4130	77	16	endo	endo	NOUN
iajs-4130	77	17	(	(	PUNCT
iajs-4130	77	18	)	)	PUNCT
iajs-4130	77	19	*	*	PUNCT
iajs-4130	77	20	k|	k|	PROPN
iajs-4130	77	21	k	k	PROPN
iajs-4130	77	22	is	be	AUX
iajs-4130	77	23	an	an	DET
iajs-4130	77	24	endo	endo	NOUN
iajs-4130	77	25	-	-	PUNCT
iajs-4130	77	26	r.b	r.b	NOUN
iajs-4130	77	27	submodule	submodule	NOUN
iajs-4130	77	28	and	and	CCONJ
iajs-4130	77	29	+	+	NOUN
iajs-4130	77	30	.	.	PUNCT
iajs-4130	78	1	by	by	ADP
iajs-4130	78	2	using	use	VERB
iajs-4130	78	3	induction	induction	NOUN
iajs-4130	78	4	and	and	CCONJ
iajs-4130	78	5	remark	remark	NOUN
iajs-4130	78	6	(	(	PUNCT
iajs-4130	78	7	2.5	2.5	NUM
iajs-4130	78	8	)	)	PUNCT
iajs-4130	78	9	,	,	PUNCT
iajs-4130	78	10	we	we	PRON
iajs-4130	78	11	conclude	conclude	VERB
iajs-4130	78	12	that	that	SCONJ
iajs-4130	78	13	endo	endo	NOUN
iajs-4130	78	14	(	(	PUNCT
iajs-4130	78	15	)	)	PUNCT
iajs-4130	78	16	is	be	AUX
iajs-4130	78	17	an	an	DET
iajs-4130	78	18	endo	endo	NOUN
iajs-4130	78	19	-	-	PUNCT
iajs-4130	78	20	r.b	r.b	NOUN
iajs-4130	78	21	submodule	submodule	NOUN
iajs-4130	78	22	.	.	PUNCT
iajs-4130	79	1	3.4	3.4	NUM
iajs-4130	79	2	.	.	PUNCT
iajs-4130	79	3	proposition	proposition	NOUN
iajs-4130	79	4	let	let	AUX
iajs-4130	79	5	be	be	AUX
iajs-4130	79	6	an	an	DET
iajs-4130	79	7	epiomorphism	epiomorphism	NOUN
iajs-4130	79	8	and	and	CCONJ
iajs-4130	79	9	with	with	ADP
iajs-4130	79	10	.	.	PUNCT
iajs-4130	80	1	then	then	ADV
iajs-4130	80	2	,	,	PUNCT
iajs-4130	80	3	(	(	PUNCT
iajs-4130	80	4	1	1	NUM
iajs-4130	80	5	)	)	PUNCT
iajs-4130	80	6	.	.	PUNCT
iajs-4130	81	1	(	(	PUNCT
iajs-4130	81	2	)	)	PUNCT
iajs-4130	81	3	/	/	SYM
iajs-4130	81	4	(	(	PUNCT
iajs-4130	81	5	)	)	PUNCT
iajs-4130	81	6	(	(	PUNCT
iajs-4130	81	7	2	2	NUM
iajs-4130	81	8	)	)	PUNCT
iajs-4130	81	9	.	.	PUNCT
iajs-4130	82	1	(	(	PUNCT
iajs-4130	82	2	)	)	PUNCT
iajs-4130	82	3	/	/	SYM
iajs-4130	82	4	(	(	PUNCT
iajs-4130	82	5	)	)	PUNCT
iajs-4130	82	6	where	where	SCONJ
iajs-4130	82	7	proof	proof	NOUN
iajs-4130	82	8	.	.	PUNCT
iajs-4130	83	1	(	(	PUNCT
iajs-4130	83	2	1	1	X
iajs-4130	83	3	)	)	PUNCT
iajs-4130	83	4	by	by	ADP
iajs-4130	83	5	the	the	DET
iajs-4130	83	6	definition	definition	NOUN
iajs-4130	83	7	of	of	ADP
iajs-4130	83	8	the	the	DET
iajs-4130	83	9	endo	endo	NOUN
iajs-4130	83	10	-	-	PUNCT
iajs-4130	83	11	r.b	r.b	NOUN
iajs-4130	83	12	radical	radical	NOUN
iajs-4130	83	13	of	of	ADP
iajs-4130	83	14	submodule	submodule	NOUN
iajs-4130	83	15	,	,	PUNCT
iajs-4130	83	16	we	we	PRON
iajs-4130	83	17	have	have	VERB
iajs-4130	83	18	.	.	PUNCT
iajs-4130	84	1	(	(	PUNCT
iajs-4130	84	2	)	)	PUNCT
iajs-4130	84	3	/	/	SYM
iajs-4130	84	4	(	(	PUNCT
iajs-4130	84	5	)	)	PUNCT
iajs-4130	84	6	where	where	SCONJ
iajs-4130	84	7	k	k	PROPN
iajs-4130	84	8	is	be	AUX
iajs-4130	84	9	an	an	DET
iajs-4130	84	10	endo	endo	NOUN
iajs-4130	84	11	-	-	PUNCT
iajs-4130	84	12	r.b	r.b	NOUN
iajs-4130	84	13	submodule	submodule	NOUN
iajs-4130	84	14	and	and	CCONJ
iajs-4130	84	15	.	.	PUNCT
iajs-4130	85	1	since	since	SCONJ
iajs-4130	85	2	,	,	PUNCT
iajs-4130	85	3	then	then	ADV
iajs-4130	85	4	(	(	PUNCT
iajs-4130	85	5	(	(	PUNCT
iajs-4130	85	6	)	)	PUNCT
iajs-4130	85	7	)	)	PUNCT
iajs-4130	86	1	(	(	PUNCT
iajs-4130	86	2	)	)	PUNCT
iajs-4130	86	3	,	,	PUNCT
iajs-4130	86	4	where	where	SCONJ
iajs-4130	86	5	(	(	PUNCT
iajs-4130	86	6	)	)	PUNCT
iajs-4130	86	7	(	(	PUNCT
iajs-4130	86	8	)	)	PUNCT
iajs-4130	86	9	and	and	CCONJ
iajs-4130	86	10	the	the	DET
iajs-4130	86	11	intersection	intersection	NOUN
iajs-4130	86	12	works	work	VERB
iajs-4130	86	13	over	over	ADP
iajs-4130	86	14	all	all	DET
iajs-4130	86	15	endo	endo	NOUN
iajs-4130	86	16	-	-	PUNCT
iajs-4130	86	17	r.b	r.b	NOUN
iajs-4130	86	18	submodules	submodule	NOUN
iajs-4130	86	19	(	(	PUNCT
iajs-4130	86	20	)	)	PUNCT
iajs-4130	86	21	therefore	therefore	ADV
iajs-4130	86	22	,	,	PUNCT
iajs-4130	86	23	.	.	PUNCT
iajs-4130	87	1	(	(	PUNCT
iajs-4130	87	2	)	)	PUNCT
iajs-4130	87	3	/	/	SYM
iajs-4130	87	4	(	(	PUNCT
iajs-4130	87	5	)	)	PUNCT
iajs-4130	87	6	(	(	PUNCT
iajs-4130	87	7	2	2	X
iajs-4130	87	8	)	)	PUNCT
iajs-4130	87	9	let	let	VERB
iajs-4130	87	10	,	,	PUNCT
iajs-4130	87	11	then	then	ADV
iajs-4130	87	12	.	.	PUNCT
iajs-4130	88	1	(	(	PUNCT
iajs-4130	88	2	)	)	PUNCT
iajs-4130	88	3	/	/	SYM
iajs-4130	88	4	where	where	SCONJ
iajs-4130	88	5	the	the	DET
iajs-4130	88	6	intersection	intersection	NOUN
iajs-4130	88	7	is	be	AUX
iajs-4130	88	8	over	over	ADP
iajs-4130	88	9	all	all	DET
iajs-4130	88	10	endo	endo	NOUN
iajs-4130	88	11	-	-	PUNCT
iajs-4130	88	12	r.b	r.b	NOUN
iajs-4130	88	13	submodules	submodule	NOUN
iajs-4130	88	14	d	d	PROPN
iajs-4130	88	15	of	of	ADP
iajs-4130	88	16	with	with	ADP
iajs-4130	88	17	.	.	PUNCT
iajs-4130	89	1	then	then	ADV
iajs-4130	89	2	(	(	PUNCT
iajs-4130	89	3	(	(	PUNCT
iajs-4130	89	4	)	)	PUNCT
iajs-4130	89	5	(	(	PUNCT
iajs-4130	89	6	)	)	PUNCT
iajs-4130	89	7	(	(	PUNCT
iajs-4130	89	8	)	)	PUNCT
iajs-4130	89	9	where	where	SCONJ
iajs-4130	89	10	the	the	DET
iajs-4130	89	11	intersection	intersection	NOUN
iajs-4130	89	12	is	be	AUX
iajs-4130	89	13	over	over	ADP
iajs-4130	89	14	all	all	DET
iajs-4130	89	15	endor.b	endor.b	NOUN
iajs-4130	89	16	submodules	submodule	NOUN
iajs-4130	89	17	(	(	PUNCT
iajs-4130	89	18	)	)	PUNCT
iajs-4130	89	19	of	of	ADP
iajs-4130	89	20	with	with	ADP
iajs-4130	89	21	(	(	PUNCT
iajs-4130	89	22	)	)	PUNCT
iajs-4130	89	23	(	(	PUNCT
iajs-4130	89	24	)	)	PUNCT
iajs-4130	89	25	.	.	PUNCT
iajs-4130	90	1	hence	hence	ADV
iajs-4130	90	2	.	.	PUNCT
iajs-4130	91	1	(	(	PUNCT
iajs-4130	91	2	)	)	PUNCT
iajs-4130	91	3	/	/	SYM
iajs-4130	91	4	(	(	PUNCT
iajs-4130	91	5	)	)	PUNCT
iajs-4130	91	6	3.5	3.5	NUM
iajs-4130	91	7	.	.	PUNCT
iajs-4130	92	1	proposition	proposition	NOUN
iajs-4130	92	2	let	let	AUX
iajs-4130	92	3	be	be	AUX
iajs-4130	92	4	an	an	DET
iajs-4130	92	5	t	t	NOUN
iajs-4130	92	6	-	-	PUNCT
iajs-4130	92	7	module	module	NOUN
iajs-4130	92	8	and	and	CCONJ
iajs-4130	92	9	.	.	PUNCT
iajs-4130	93	1	then	then	ADV
iajs-4130	93	2	,	,	PUNCT
iajs-4130	93	3	the	the	DET
iajs-4130	93	4	following	follow	VERB
iajs-4130	93	5	statements	statement	NOUN
iajs-4130	93	6	hold	hold	VERB
iajs-4130	93	7	:	:	PUNCT
iajs-4130	93	8	(	(	PUNCT
iajs-4130	93	9	1	1	X
iajs-4130	93	10	)	)	PUNCT
iajs-4130	93	11	(	(	PUNCT
iajs-4130	93	12	)	)	PUNCT
iajs-4130	93	13	(	(	PUNCT
iajs-4130	93	14	2	2	NUM
iajs-4130	93	15	)	)	PUNCT
iajs-4130	93	16	,	,	PUNCT
iajs-4130	93	17	then	then	ADV
iajs-4130	93	18	endo	endo	PROPN
iajs-4130	93	19	(	(	PUNCT
iajs-4130	93	20	)	)	PUNCT
iajs-4130	93	21	(	(	PUNCT
iajs-4130	93	22	)	)	PUNCT
iajs-4130	93	23	(	(	PUNCT
iajs-4130	93	24	3	3	X
iajs-4130	93	25	)	)	PUNCT
iajs-4130	93	26	endo	endo	NOUN
iajs-4130	93	27	.	.	PUNCT
iajs-4130	94	1	(	(	PUNCT
iajs-4130	94	2	)	)	PUNCT
iajs-4130	94	3	/	/	SYM
iajs-4130	94	4	(	(	PUNCT
iajs-4130	94	5	)	)	PUNCT
iajs-4130	94	6	(	(	PUNCT
iajs-4130	94	7	4	4	NUM
iajs-4130	94	8	)	)	PUNCT
iajs-4130	94	9	(	(	PUNCT
iajs-4130	94	10	)	)	PUNCT
iajs-4130	94	11	(	(	PUNCT
iajs-4130	94	12	)	)	PUNCT
iajs-4130	94	13	(	(	PUNCT
iajs-4130	94	14	)	)	PUNCT
iajs-4130	94	15	(	(	PUNCT
iajs-4130	94	16	5	5	NUM
iajs-4130	94	17	)	)	PUNCT
iajs-4130	94	18	(	(	PUNCT
iajs-4130	94	19	)	)	PUNCT
iajs-4130	94	20	(	(	PUNCT
iajs-4130	94	21	(	(	PUNCT
iajs-4130	94	22	)	)	PUNCT
iajs-4130	94	23	(	(	PUNCT
iajs-4130	94	24	)	)	PUNCT
iajs-4130	94	25	)	)	PUNCT
iajs-4130	94	26	ihjpas	ihjpas	PROPN
iajs-4130	94	27	.	.	PUNCT
iajs-4130	95	1	2025,38(4	2025,38(4	X
iajs-4130	95	2	)	)	PUNCT
iajs-4130	96	1	402	402	NUM
iajs-4130	96	2	proof	proof	NOUN
iajs-4130	96	3	.	.	PUNCT
iajs-4130	97	1	(	(	PUNCT
iajs-4130	97	2	1	1	X
iajs-4130	97	3	)	)	PUNCT
iajs-4130	97	4	by	by	ADP
iajs-4130	97	5	the	the	DET
iajs-4130	97	6	definition	definition	NOUN
iajs-4130	97	7	,	,	PUNCT
iajs-4130	97	8	we	we	PRON
iajs-4130	97	9	have	have	VERB
iajs-4130	97	10	that	that	PRON
iajs-4130	97	11	(	(	PUNCT
iajs-4130	97	12	)	)	PUNCT
iajs-4130	97	13	where	where	SCONJ
iajs-4130	97	14	the	the	DET
iajs-4130	97	15	intersection	intersection	NOUN
iajs-4130	97	16	runs	run	VERB
iajs-4130	97	17	over	over	ADP
iajs-4130	97	18	all	all	DET
iajs-4130	97	19	endo	endo	NOUN
iajs-4130	97	20	-	-	PUNCT
iajs-4130	97	21	r.b	r.b	NOUN
iajs-4130	97	22	submodules	submodule	NOUN
iajs-4130	98	1	k	k	PROPN
iajs-4130	98	2	of	of	ADP
iajs-4130	98	3	with	with	ADP
iajs-4130	98	4	so	so	SCONJ
iajs-4130	98	5	that	that	PRON
iajs-4130	98	6	(	(	PUNCT
iajs-4130	98	7	)	)	PUNCT
iajs-4130	98	8	.	.	PUNCT
iajs-4130	99	1	(	(	PUNCT
iajs-4130	99	2	2	2	X
iajs-4130	99	3	)	)	PUNCT
iajs-4130	99	4	assume	assume	VERB
iajs-4130	99	5	that	that	SCONJ
iajs-4130	99	6	and	and	CCONJ
iajs-4130	99	7	let	let	VERB
iajs-4130	99	8	k	k	PRON
iajs-4130	99	9	be	be	AUX
iajs-4130	99	10	an	an	DET
iajs-4130	99	11	endo	endo	NOUN
iajs-4130	99	12	-	-	PUNCT
iajs-4130	99	13	r.b	r.b	NOUN
iajs-4130	99	14	submodule	submodule	NOUN
iajs-4130	99	15	of	of	ADP
iajs-4130	99	16	with	with	ADP
iajs-4130	99	17	.	.	PUNCT
iajs-4130	100	1	then	then	ADV
iajs-4130	100	2	implies	imply	VERB
iajs-4130	100	3	that	that	PRON
iajs-4130	100	4	.	.	PUNCT
iajs-4130	101	1	(	(	PUNCT
iajs-4130	101	2	)	)	PUNCT
iajs-4130	101	3	(	(	PUNCT
iajs-4130	101	4	)	)	PUNCT
iajs-4130	101	5	.	.	PUNCT
iajs-4130	102	1	(	(	PUNCT
iajs-4130	102	2	3	3	X
iajs-4130	102	3	)	)	PUNCT
iajs-4130	102	4	since	since	ADV
iajs-4130	102	5	.	.	PUNCT
iajs-4130	103	1	(	(	PUNCT
iajs-4130	103	2	)	)	PUNCT
iajs-4130	103	3	/	/	SYM
iajs-4130	103	4	where	where	SCONJ
iajs-4130	103	5	the	the	DET
iajs-4130	103	6	intersection	intersection	NOUN
iajs-4130	103	7	is	be	AUX
iajs-4130	103	8	taken	take	VERB
iajs-4130	103	9	on	on	ADP
iajs-4130	103	10	all	all	ADV
iajs-4130	103	11	over	over	ADP
iajs-4130	103	12	endo	endo	NOUN
iajs-4130	103	13	-	-	PUNCT
iajs-4130	103	14	r.b	r.b	NOUN
iajs-4130	103	15	submodules	submodule	NOUN
iajs-4130	103	16	w	w	NOUN
iajs-4130	103	17	of	of	ADP
iajs-4130	103	18	with	with	ADP
iajs-4130	103	19	(	(	PUNCT
iajs-4130	103	20	)	)	PUNCT
iajs-4130	103	21	and	and	CCONJ
iajs-4130	103	22	from	from	ADP
iajs-4130	103	23	no	no	DET
iajs-4130	103	24	(	(	PUNCT
iajs-4130	103	25	1	1	NUM
iajs-4130	103	26	)	)	PUNCT
iajs-4130	103	27	,	,	PUNCT
iajs-4130	103	28	we	we	PRON
iajs-4130	103	29	get	get	VERB
iajs-4130	103	30	that	that	PRON
iajs-4130	103	31	(	(	PUNCT
iajs-4130	103	32	)	)	PUNCT
iajs-4130	103	33	therefore	therefore	ADV
iajs-4130	103	34	,	,	PUNCT
iajs-4130	103	35	.	.	PUNCT
iajs-4130	104	1	(	(	PUNCT
iajs-4130	104	2	)	)	PUNCT
iajs-4130	104	3	/	/	SYM
iajs-4130	104	4	(	(	PUNCT
iajs-4130	104	5	)	)	PUNCT
iajs-4130	104	6	and	and	CCONJ
iajs-4130	104	7	by	by	ADP
iajs-4130	104	8	no.(1	no.(1	NOUN
iajs-4130	104	9	)	)	PUNCT
iajs-4130	104	10	,	,	PUNCT
iajs-4130	104	11	we	we	PRON
iajs-4130	104	12	obtain	obtain	VERB
iajs-4130	104	13	(	(	PUNCT
iajs-4130	104	14	)	)	PUNCT
iajs-4130	104	15	(	(	PUNCT
iajs-4130	104	16	(	(	PUNCT
iajs-4130	104	17	)	)	PUNCT
iajs-4130	104	18	)	)	PUNCT
iajs-4130	104	19	.	.	PUNCT
iajs-4130	105	1	(	(	PUNCT
iajs-4130	105	2	4	4	X
iajs-4130	105	3	)	)	PUNCT
iajs-4130	105	4	let	let	VERB
iajs-4130	105	5	k	k	X
iajs-4130	105	6	be	be	AUX
iajs-4130	105	7	an	an	DET
iajs-4130	105	8	endo	endo	NOUN
iajs-4130	105	9	-	-	PUNCT
iajs-4130	105	10	r.b	r.b	NOUN
iajs-4130	105	11	submodule	submodule	NOUN
iajs-4130	105	12	of	of	ADP
iajs-4130	105	13	containing	contain	VERB
iajs-4130	105	14	n	n	PROPN
iajs-4130	105	15	and	and	CCONJ
iajs-4130	105	16	l.	l.	NOUN
iajs-4130	105	17	since	since	SCONJ
iajs-4130	105	18	we	we	PRON
iajs-4130	105	19	have	have	VERB
iajs-4130	105	20	that	that	PRON
iajs-4130	105	21	(	(	PUNCT
iajs-4130	105	22	)	)	PUNCT
iajs-4130	105	23	.	.	PUNCT
iajs-4130	106	1	thus	thus	ADV
iajs-4130	106	2	,	,	PUNCT
iajs-4130	106	3	(	(	PUNCT
iajs-4130	106	4	)	)	PUNCT
iajs-4130	106	5	(	(	PUNCT
iajs-4130	106	6	)	)	PUNCT
iajs-4130	106	7	.	.	PUNCT
iajs-4130	107	1	similarly	similarly	ADV
iajs-4130	107	2	,	,	PUNCT
iajs-4130	107	3	we	we	PRON
iajs-4130	107	4	have	have	VERB
iajs-4130	107	5	(	(	PUNCT
iajs-4130	107	6	)	)	PUNCT
iajs-4130	107	7	(	(	PUNCT
iajs-4130	107	8	)	)	PUNCT
iajs-4130	107	9	therefore	therefore	ADV
iajs-4130	107	10	,	,	PUNCT
iajs-4130	107	11	(	(	PUNCT
iajs-4130	107	12	)	)	PUNCT
iajs-4130	107	13	(	(	PUNCT
iajs-4130	107	14	)	)	PUNCT
iajs-4130	107	15	(	(	PUNCT
iajs-4130	107	16	)	)	PUNCT
iajs-4130	107	17	.	.	PUNCT
iajs-4130	108	1	(	(	PUNCT
iajs-4130	108	2	5	5	NUM
iajs-4130	108	3	)	)	PUNCT
iajs-4130	108	4	since	since	SCONJ
iajs-4130	108	5	(	(	PUNCT
iajs-4130	108	6	)	)	PUNCT
iajs-4130	108	7	(	(	PUNCT
iajs-4130	108	8	)	)	PUNCT
iajs-4130	108	9	,	,	PUNCT
iajs-4130	108	10	then	then	ADV
iajs-4130	108	11	by	by	ADP
iajs-4130	108	12	(	(	PUNCT
iajs-4130	108	13	2	2	NUM
iajs-4130	108	14	)	)	PUNCT
iajs-4130	108	15	(	(	PUNCT
iajs-4130	108	16	)	)	PUNCT
iajs-4130	108	17	(	(	PUNCT
iajs-4130	108	18	(	(	PUNCT
iajs-4130	108	19	)	)	PUNCT
iajs-4130	108	20	(	(	PUNCT
iajs-4130	108	21	)	)	PUNCT
iajs-4130	108	22	)	)	PUNCT
iajs-4130	108	23	now	now	ADV
iajs-4130	108	24	,	,	PUNCT
iajs-4130	108	25	let	let	VERB
iajs-4130	108	26	k	k	PRON
iajs-4130	108	27	be	be	AUX
iajs-4130	108	28	an	an	DET
iajs-4130	108	29	endo	endo	NOUN
iajs-4130	108	30	-	-	PUNCT
iajs-4130	108	31	r.b	r.b	NOUN
iajs-4130	108	32	submodule	submodule	NOUN
iajs-4130	108	33	of	of	ADP
iajs-4130	108	34	containing	contain	VERB
iajs-4130	108	35	n+l	n+l	NOUN
iajs-4130	108	36	.	.	PUNCT
iajs-4130	109	1	since	since	SCONJ
iajs-4130	109	2	then	then	ADV
iajs-4130	109	3	and	and	CCONJ
iajs-4130	109	4	.	.	PUNCT
iajs-4130	110	1	thus	thus	ADV
iajs-4130	110	2	,	,	PUNCT
iajs-4130	110	3	(	(	PUNCT
iajs-4130	110	4	)	)	PUNCT
iajs-4130	110	5	(	(	PUNCT
iajs-4130	110	6	)	)	PUNCT
iajs-4130	110	7	.	.	PUNCT
iajs-4130	110	8	therefore	therefore	ADV
iajs-4130	110	9	,	,	PUNCT
iajs-4130	110	10	.	.	PUNCT
iajs-4130	111	1	(	(	PUNCT
iajs-4130	111	2	)	)	PUNCT
iajs-4130	111	3	(	(	PUNCT
iajs-4130	111	4	)	)	PUNCT
iajs-4130	111	5	/	/	SYM
iajs-4130	111	6	(	(	PUNCT
iajs-4130	111	7	)	)	PUNCT
iajs-4130	111	8	.	.	PUNCT
iajs-4130	112	1	hence	hence	ADV
iajs-4130	112	2	,	,	PUNCT
iajs-4130	112	3	(	(	PUNCT
iajs-4130	112	4	)	)	PUNCT
iajs-4130	112	5	(	(	PUNCT
iajs-4130	112	6	(	(	PUNCT
iajs-4130	112	7	)	)	PUNCT
iajs-4130	112	8	(	(	PUNCT
iajs-4130	112	9	)	)	PUNCT
iajs-4130	112	10	)	)	PUNCT
iajs-4130	112	11	.	.	PUNCT
iajs-4130	112	12	recall	recall	VERB
iajs-4130	112	13	a	a	DET
iajs-4130	112	14	t	t	NOUN
iajs-4130	112	15	-	-	PUNCT
iajs-4130	112	16	module	module	NOUN
iajs-4130	112	17	called	call	VERB
iajs-4130	112	18	a	a	DET
iajs-4130	112	19	multiplication	multiplication	NOUN
iajs-4130	112	20	module	module	NOUN
iajs-4130	112	21	if	if	SCONJ
iajs-4130	112	22	for	for	ADP
iajs-4130	112	23	every	every	DET
iajs-4130	112	24	submodule	submodule	NOUN
iajs-4130	112	25	a	a	PRON
iajs-4130	112	26	of	of	ADP
iajs-4130	112	27	there	there	ADV
iajs-4130	112	28	exists	exist	VERB
iajs-4130	112	29	an	an	DET
iajs-4130	112	30	ideal	ideal	NOUN
iajs-4130	112	31	i	i	PRON
iajs-4130	112	32	of	of	ADP
iajs-4130	112	33	t	t	PROPN
iajs-4130	112	34	such	such	ADJ
iajs-4130	112	35	that	that	PRON
iajs-4130	112	36	.(12–14	.(12–14	NUM
iajs-4130	112	37	)	)	PUNCT
iajs-4130	112	38	recall	recall	VERB
iajs-4130	112	39	a	a	DET
iajs-4130	112	40	t	t	NOUN
iajs-4130	112	41	-	-	PUNCT
iajs-4130	112	42	module	module	NOUN
iajs-4130	112	43	said	say	VERB
iajs-4130	112	44	to	to	PART
iajs-4130	112	45	be	be	AUX
iajs-4130	112	46	a	a	DET
iajs-4130	112	47	scalar	scalar	ADJ
iajs-4130	112	48	module	module	NOUN
iajs-4130	112	49	if	if	SCONJ
iajs-4130	112	50	for	for	ADP
iajs-4130	112	51	each	each	PRON
iajs-4130	112	52	(	(	PUNCT
iajs-4130	112	53	)	)	PUNCT
iajs-4130	112	54	there	there	PRON
iajs-4130	112	55	exists	exist	VERB
iajs-4130	112	56	such	such	ADJ
iajs-4130	112	57	that	that	SCONJ
iajs-4130	112	58	(	(	PUNCT
iajs-4130	112	59	)	)	PUNCT
iajs-4130	112	60	(	(	PUNCT
iajs-4130	112	61	15–17	15–17	NUM
iajs-4130	112	62	)	)	PUNCT
iajs-4130	112	63	.	.	PUNCT
iajs-4130	113	1	next	next	ADV
iajs-4130	113	2	,	,	PUNCT
iajs-4130	113	3	the	the	DET
iajs-4130	113	4	scalar	scalar	ADJ
iajs-4130	113	5	module	module	NOUN
iajs-4130	113	6	plays	play	VERB
iajs-4130	113	7	a	a	DET
iajs-4130	113	8	crucial	crucial	ADJ
iajs-4130	113	9	role	role	NOUN
iajs-4130	113	10	to	to	PART
iajs-4130	113	11	connect	connect	VERB
iajs-4130	113	12	prime	prime	ADJ
iajs-4130	113	13	and	and	CCONJ
iajs-4130	113	14	endo	endo	NOUN
iajs-4130	113	15	-	-	PUNCT
iajs-4130	113	16	r.b	r.b	NOUN
iajs-4130	113	17	submodules	submodule	NOUN
iajs-4130	113	18	.	.	PUNCT
iajs-4130	114	1	we	we	PRON
iajs-4130	114	2	need	need	VERB
iajs-4130	114	3	the	the	DET
iajs-4130	114	4	next	next	ADJ
iajs-4130	114	5	proposition	proposition	NOUN
iajs-4130	114	6	to	to	PART
iajs-4130	114	7	see	see	VERB
iajs-4130	114	8	this	this	DET
iajs-4130	114	9	relationship	relationship	NOUN
iajs-4130	114	10	.	.	PUNCT
iajs-4130	115	1	3.6	3.6	NUM
iajs-4130	115	2	.	.	PUNCT
iajs-4130	115	3	proposition	proposition	NOUN
iajs-4130	115	4	let	let	AUX
iajs-4130	115	5	be	be	AUX
iajs-4130	115	6	a	a	DET
iajs-4130	115	7	scalar	scalar	ADJ
iajs-4130	115	8	t	t	NOUN
iajs-4130	115	9	-	-	PUNCT
iajs-4130	115	10	module	module	NOUN
iajs-4130	115	11	,	,	PUNCT
iajs-4130	115	12	and	and	CCONJ
iajs-4130	115	13	a	a	PRON
iajs-4130	115	14	is	be	AUX
iajs-4130	115	15	a	a	DET
iajs-4130	115	16	prime	prime	ADJ
iajs-4130	115	17	submodule	submodule	NOUN
iajs-4130	115	18	.	.	PUNCT
iajs-4130	116	1	then	then	ADV
iajs-4130	116	2	a	a	PRON
iajs-4130	116	3	is	be	AUX
iajs-4130	116	4	an	an	DET
iajs-4130	116	5	endo	endo	NOUN
iajs-4130	116	6	-	-	PUNCT
iajs-4130	116	7	r.b	r.b	NOUN
iajs-4130	116	8	.	.	NOUN
iajs-4130	116	9	proof	proof	NOUN
iajs-4130	116	10	.	.	PUNCT
iajs-4130	117	1	let	let	VERB
iajs-4130	117	2	(	(	PUNCT
iajs-4130	117	3	)	)	PUNCT
iajs-4130	117	4	.	.	PUNCT
iajs-4130	118	1	since	since	SCONJ
iajs-4130	118	2	is	be	AUX
iajs-4130	118	3	a	a	DET
iajs-4130	118	4	scalar	scalar	ADJ
iajs-4130	118	5	module	module	NOUN
iajs-4130	118	6	,	,	PUNCT
iajs-4130	118	7	then	then	ADV
iajs-4130	118	8	for	for	ADP
iajs-4130	118	9	all	all	PRON
iajs-4130	118	10	(	(	PUNCT
iajs-4130	118	11	)	)	PUNCT
iajs-4130	118	12	there	there	PRON
iajs-4130	118	13	exists	exist	VERB
iajs-4130	118	14	such	such	ADJ
iajs-4130	118	15	that	that	SCONJ
iajs-4130	118	16	(	(	PUNCT
iajs-4130	118	17	)	)	PUNCT
iajs-4130	118	18	.	.	PUNCT
iajs-4130	119	1	thus	thus	ADV
iajs-4130	119	2	,	,	PUNCT
iajs-4130	119	3	(	(	PUNCT
iajs-4130	119	4	)	)	PUNCT
iajs-4130	119	5	.	.	PUNCT
iajs-4130	120	1	we	we	PRON
iajs-4130	120	2	claim	claim	VERB
iajs-4130	120	3	that	that	SCONJ
iajs-4130	120	4	(	(	PUNCT
iajs-4130	120	5	)	)	PUNCT
iajs-4130	120	6	(	(	PUNCT
iajs-4130	120	7	)	)	PUNCT
iajs-4130	120	8	let	let	VERB
iajs-4130	120	9	(	(	PUNCT
iajs-4130	120	10	)	)	PUNCT
iajs-4130	120	11	,	,	PUNCT
iajs-4130	120	12	then	then	ADV
iajs-4130	120	13	(	(	PUNCT
iajs-4130	120	14	)	)	PUNCT
iajs-4130	120	15	(	(	PUNCT
iajs-4130	120	16	)	)	PUNCT
iajs-4130	120	17	(	(	PUNCT
iajs-4130	120	18	)	)	PUNCT
iajs-4130	120	19	for	for	ADP
iajs-4130	120	20	all	all	PRON
iajs-4130	120	21	since	since	SCONJ
iajs-4130	120	22	a	a	PRON
iajs-4130	120	23	is	be	AUX
iajs-4130	120	24	a	a	DET
iajs-4130	120	25	prime	prime	ADJ
iajs-4130	120	26	submodule	submodule	NOUN
iajs-4130	120	27	,	,	PUNCT
iajs-4130	120	28	then	then	ADV
iajs-4130	120	29	either	either	ADV
iajs-4130	120	30	.	.	PUNCT
iajs-4130	121	1	we	we	PRON
iajs-4130	121	2	conclude	conclude	VERB
iajs-4130	121	3	that	that	PRON
iajs-4130	121	4	(	(	PUNCT
iajs-4130	121	5	)	)	PUNCT
iajs-4130	121	6	.	.	PUNCT
iajs-4130	122	1	3.7	3.7	NUM
iajs-4130	122	2	.	.	PUNCT
iajs-4130	122	3	proposition	proposition	NOUN
iajs-4130	122	4	let	let	AUX
iajs-4130	122	5	be	be	AUX
iajs-4130	122	6	a	a	DET
iajs-4130	122	7	multiplication	multiplication	NOUN
iajs-4130	122	8	finitely	finitely	ADV
iajs-4130	122	9	generated	generate	VERB
iajs-4130	122	10	t	t	NOUN
iajs-4130	122	11	-	-	PUNCT
iajs-4130	122	12	module	module	NOUN
iajs-4130	122	13	and	and	CCONJ
iajs-4130	122	14	then	then	ADV
iajs-4130	122	15	,	,	PUNCT
iajs-4130	122	16	if	if	SCONJ
iajs-4130	122	17	and	and	CCONJ
iajs-4130	122	18	only	only	ADV
iajs-4130	122	19	if	if	SCONJ
iajs-4130	122	20	(	(	PUNCT
iajs-4130	122	21	)	)	PUNCT
iajs-4130	122	22	(	(	PUNCT
iajs-4130	122	23	)	)	PUNCT
iajs-4130	122	24	.	.	PUNCT
iajs-4130	123	1	proof	proof	NOUN
iajs-4130	123	2	.	.	PUNCT
iajs-4130	124	1	by	by	ADP
iajs-4130	124	2	proposition	proposition	NOUN
iajs-4130	124	3	(	(	PUNCT
iajs-4130	124	4	3.5	3.5	NUM
iajs-4130	124	5	)	)	PUNCT
iajs-4130	124	6	,	,	PUNCT
iajs-4130	124	7	we	we	PRON
iajs-4130	124	8	have	have	VERB
iajs-4130	124	9	that	that	PRON
iajs-4130	124	10	(	(	PUNCT
iajs-4130	124	11	)	)	PUNCT
iajs-4130	124	12	and	and	CCONJ
iajs-4130	124	13	(	(	PUNCT
iajs-4130	124	14	)	)	PUNCT
iajs-4130	124	15	so	so	ADV
iajs-4130	124	16	it	it	PRON
iajs-4130	124	17	ihjpas	ihjpa	VERB
iajs-4130	124	18	.	.	PUNCT
iajs-4130	125	1	2025,38(4	2025,38(4	NOUN
iajs-4130	125	2	)	)	PUNCT
iajs-4130	126	1	403	403	NUM
iajs-4130	126	2	is	be	AUX
iajs-4130	126	3	obvious	obvious	ADJ
iajs-4130	126	4	that	that	SCONJ
iajs-4130	126	5	(	(	PUNCT
iajs-4130	126	6	)	)	PUNCT
iajs-4130	126	7	(	(	PUNCT
iajs-4130	126	8	)	)	PUNCT
iajs-4130	126	9	conversely	conversely	ADV
iajs-4130	126	10	,	,	PUNCT
iajs-4130	126	11	assume	assume	VERB
iajs-4130	126	12	that	that	SCONJ
iajs-4130	126	13	(	(	PUNCT
iajs-4130	126	14	)	)	PUNCT
iajs-4130	126	15	(	(	PUNCT
iajs-4130	126	16	)	)	PUNCT
iajs-4130	126	17	and	and	CCONJ
iajs-4130	126	18	let	let	VERB
iajs-4130	126	19	.	.	PUNCT
iajs-4130	127	1	since	since	SCONJ
iajs-4130	127	2	is	be	AUX
iajs-4130	127	3	finitely	finitely	ADV
iajs-4130	127	4	generated	generate	VERB
iajs-4130	127	5	,	,	PUNCT
iajs-4130	127	6	then	then	ADV
iajs-4130	127	7	there	there	PRON
iajs-4130	127	8	exists	exist	VERB
iajs-4130	127	9	a	a	DET
iajs-4130	127	10	maximal	maximal	ADJ
iajs-4130	127	11	submodule	submodule	NOUN
iajs-4130	127	12	k	k	PROPN
iajs-4130	127	13	of	of	ADP
iajs-4130	127	14	such	such	ADJ
iajs-4130	127	15	that	that	PRON
iajs-4130	127	16	.	.	PUNCT
iajs-4130	128	1	k	k	PROPN
iajs-4130	128	2	is	be	AUX
iajs-4130	128	3	prime	prime	ADJ
iajs-4130	128	4	submodule	submodule	NOUN
iajs-4130	128	5	and	and	CCONJ
iajs-4130	128	6	since	since	SCONJ
iajs-4130	128	7	is	be	AUX
iajs-4130	128	8	a	a	DET
iajs-4130	128	9	multiplication	multiplication	NOUN
iajs-4130	128	10	t	t	NOUN
iajs-4130	128	11	-	-	PUNCT
iajs-4130	128	12	module	module	NOUN
iajs-4130	128	13	,	,	PUNCT
iajs-4130	128	14	then	then	ADV
iajs-4130	128	15	by	by	ADP
iajs-4130	128	16	(	(	PUNCT
iajs-4130	128	17	18	18	NUM
iajs-4130	128	18	)	)	PUNCT
iajs-4130	128	19	,	,	PUNCT
iajs-4130	128	20	is	be	AUX
iajs-4130	128	21	a	a	DET
iajs-4130	128	22	scalar	scalar	ADJ
iajs-4130	128	23	t	t	NOUN
iajs-4130	128	24	-	-	PUNCT
iajs-4130	128	25	module	module	NOUN
iajs-4130	128	26	and	and	CCONJ
iajs-4130	128	27	hence	hence	ADV
iajs-4130	128	28	,	,	PUNCT
iajs-4130	128	29	by	by	ADP
iajs-4130	128	30	proposition	proposition	NOUN
iajs-4130	128	31	(	(	PUNCT
iajs-4130	128	32	3.6	3.6	NUM
iajs-4130	128	33	)	)	PUNCT
iajs-4130	128	34	,	,	PUNCT
iajs-4130	128	35	we	we	PRON
iajs-4130	128	36	get	get	VERB
iajs-4130	128	37	that	that	SCONJ
iajs-4130	128	38	k	k	PROPN
iajs-4130	128	39	is	be	AUX
iajs-4130	128	40	an	an	DET
iajs-4130	128	41	endo	endo	NOUN
iajs-4130	128	42	-	-	PUNCT
iajs-4130	128	43	r.b	r.b	NOUN
iajs-4130	128	44	submodule	submodule	NOUN
iajs-4130	128	45	.	.	PUNCT
iajs-4130	129	1	therefore	therefore	ADV
iajs-4130	129	2	,	,	PUNCT
iajs-4130	129	3	(	(	PUNCT
iajs-4130	129	4	)	)	PUNCT
iajs-4130	129	5	and	and	CCONJ
iajs-4130	129	6	(	(	PUNCT
iajs-4130	129	7	)	)	PUNCT
iajs-4130	129	8	thus	thus	ADV
iajs-4130	129	9	(	(	PUNCT
iajs-4130	129	10	)	)	PUNCT
iajs-4130	129	11	(	(	PUNCT
iajs-4130	129	12	)	)	PUNCT
iajs-4130	129	13	implies	imply	VERB
iajs-4130	129	14	that	that	SCONJ
iajs-4130	129	15	k=	k=	VERB
iajs-4130	129	16	which	which	PRON
iajs-4130	129	17	is	be	AUX
iajs-4130	129	18	a	a	DET
iajs-4130	129	19	contradiction	contradiction	NOUN
iajs-4130	129	20	.	.	PUNCT
iajs-4130	130	1	hence	hence	ADV
iajs-4130	130	2	n+l=	n+l=	NOUN
iajs-4130	130	3	.	.	PUNCT
iajs-4130	131	1	recall	recall	VERB
iajs-4130	131	2	a	a	DET
iajs-4130	131	3	submodule	submodule	NOUN
iajs-4130	131	4	n	n	PROPN
iajs-4130	131	5	of	of	ADP
iajs-4130	131	6	a	a	DET
iajs-4130	131	7	–	–	PUNCT
iajs-4130	131	8	module	module	NOUN
iajs-4130	131	9	is	be	AUX
iajs-4130	131	10	said	say	VERB
iajs-4130	131	11	to	to	PART
iajs-4130	131	12	be	be	AUX
iajs-4130	131	13	completely	completely	ADV
iajs-4130	131	14	irreducible	irreducible	ADJ
iajs-4130	131	15	if	if	SCONJ
iajs-4130	131	16	for	for	ADP
iajs-4130	131	17	any	any	DET
iajs-4130	131	18	two	two	NUM
iajs-4130	131	19	submodules	submodule	NOUN
iajs-4130	131	20	of	of	ADP
iajs-4130	131	21	,	,	PUNCT
iajs-4130	131	22	implies	imply	VERB
iajs-4130	131	23	that	that	SCONJ
iajs-4130	131	24	either	either	CCONJ
iajs-4130	131	25	or	or	CCONJ
iajs-4130	131	26	(	(	PUNCT
iajs-4130	131	27	19,20	19,20	NOUN
iajs-4130	131	28	)	)	PUNCT
iajs-4130	131	29	3.8	3.8	NUM
iajs-4130	131	30	.	.	PUNCT
iajs-4130	132	1	proposition	proposition	NOUN
iajs-4130	132	2	let	let	AUX
iajs-4130	132	3	be	be	AUX
iajs-4130	132	4	a	a	DET
iajs-4130	132	5	t	t	NOUN
iajs-4130	132	6	-	-	PUNCT
iajs-4130	132	7	module	module	NOUN
iajs-4130	132	8	and	and	CCONJ
iajs-4130	132	9	if	if	SCONJ
iajs-4130	132	10	every	every	DET
iajs-4130	132	11	endo	endo	NOUN
iajs-4130	132	12	-	-	PUNCT
iajs-4130	132	13	r.b	r.b	NOUN
iajs-4130	132	14	submodules	submodule	NOUN
iajs-4130	132	15	that	that	PRON
iajs-4130	132	16	contains	contain	VERB
iajs-4130	132	17	is	be	AUX
iajs-4130	132	18	completely	completely	ADV
iajs-4130	132	19	irreducible	irreducible	ADJ
iajs-4130	132	20	submodule	submodule	NOUN
iajs-4130	132	21	,	,	PUNCT
iajs-4130	132	22	then	then	ADV
iajs-4130	132	23	(	(	PUNCT
iajs-4130	132	24	)	)	PUNCT
iajs-4130	132	25	(	(	PUNCT
iajs-4130	132	26	)	)	PUNCT
iajs-4130	132	27	(	(	PUNCT
iajs-4130	132	28	)	)	PUNCT
iajs-4130	132	29	proof	proof	NOUN
iajs-4130	132	30	.	.	PUNCT
iajs-4130	133	1	it	it	PRON
iajs-4130	133	2	is	be	AUX
iajs-4130	133	3	clear	clear	ADJ
iajs-4130	133	4	that	that	SCONJ
iajs-4130	133	5	(	(	PUNCT
iajs-4130	133	6	)	)	PUNCT
iajs-4130	133	7	(	(	PUNCT
iajs-4130	133	8	)	)	PUNCT
iajs-4130	133	9	(	(	PUNCT
iajs-4130	133	10	)	)	PUNCT
iajs-4130	133	11	.	.	PUNCT
iajs-4130	134	1	let	let	VERB
iajs-4130	134	2	k	k	PRON
iajs-4130	134	3	be	be	AUX
iajs-4130	134	4	an	an	DET
iajs-4130	134	5	endo	endo	NOUN
iajs-4130	134	6	-	-	PUNCT
iajs-4130	134	7	r.b	r.b	NOUN
iajs-4130	134	8	submodule	submodule	NOUN
iajs-4130	134	9	such	such	ADJ
iajs-4130	134	10	that	that	SCONJ
iajs-4130	134	11	since	since	SCONJ
iajs-4130	134	12	k	k	PROPN
iajs-4130	134	13	is	be	AUX
iajs-4130	134	14	completely	completely	ADV
iajs-4130	134	15	irreducible	irreducible	ADJ
iajs-4130	134	16	,	,	PUNCT
iajs-4130	134	17	then	then	ADV
iajs-4130	134	18	either	either	PRON
iajs-4130	134	19	implies	imply	VERB
iajs-4130	134	20	that	that	SCONJ
iajs-4130	134	21	(	(	PUNCT
iajs-4130	134	22	)	)	PUNCT
iajs-4130	134	23	or	or	CCONJ
iajs-4130	134	24	(	(	PUNCT
iajs-4130	134	25	)	)	PUNCT
iajs-4130	134	26	.	.	PUNCT
iajs-4130	135	1	hence	hence	ADV
iajs-4130	135	2	(	(	PUNCT
iajs-4130	135	3	)	)	PUNCT
iajs-4130	135	4	(	(	PUNCT
iajs-4130	135	5	)	)	PUNCT
iajs-4130	135	6	and	and	CCONJ
iajs-4130	135	7	this	this	PRON
iajs-4130	135	8	holds	hold	VERB
iajs-4130	135	9	for	for	ADP
iajs-4130	135	10	any	any	DET
iajs-4130	135	11	k	k	NOUN
iajs-4130	135	12	and	and	CCONJ
iajs-4130	135	13	the	the	DET
iajs-4130	135	14	intersection	intersection	NOUN
iajs-4130	135	15	of	of	ADP
iajs-4130	135	16	all	all	PRON
iajs-4130	135	17	k	k	PROPN
iajs-4130	135	18	is	be	AUX
iajs-4130	135	19	an	an	PRON
iajs-4130	135	20	(	(	PUNCT
iajs-4130	135	21	)	)	PUNCT
iajs-4130	135	22	.	.	PUNCT
iajs-4130	136	1	then	then	ADV
iajs-4130	136	2	either	either	CCONJ
iajs-4130	136	3	(	(	PUNCT
iajs-4130	136	4	)	)	PUNCT
iajs-4130	136	5	(	(	PUNCT
iajs-4130	136	6	)	)	PUNCT
iajs-4130	136	7	or	or	CCONJ
iajs-4130	136	8	(	(	PUNCT
iajs-4130	136	9	)	)	PUNCT
iajs-4130	136	10	(	(	PUNCT
iajs-4130	136	11	)	)	PUNCT
iajs-4130	136	12	.	.	PUNCT
iajs-4130	137	1	therefore	therefore	ADV
iajs-4130	137	2	,	,	PUNCT
iajs-4130	137	3	(	(	PUNCT
iajs-4130	137	4	)	)	PUNCT
iajs-4130	137	5	(	(	PUNCT
iajs-4130	137	6	)	)	PUNCT
iajs-4130	137	7	(	(	PUNCT
iajs-4130	137	8	)	)	PUNCT
iajs-4130	137	9	thus	thus	ADV
iajs-4130	137	10	(	(	PUNCT
iajs-4130	137	11	)	)	PUNCT
iajs-4130	137	12	(	(	PUNCT
iajs-4130	137	13	)	)	PUNCT
iajs-4130	137	14	(	(	PUNCT
iajs-4130	137	15	)	)	PUNCT
iajs-4130	137	16	3.9	3.9	NUM
iajs-4130	137	17	.	.	PUNCT
iajs-4130	137	18	proposition	proposition	NOUN
iajs-4130	137	19	let	let	AUX
iajs-4130	137	20	be	be	AUX
iajs-4130	137	21	a	a	DET
iajs-4130	137	22	scalar	scalar	ADJ
iajs-4130	137	23	t	t	NOUN
iajs-4130	137	24	-	-	PUNCT
iajs-4130	137	25	module	module	NOUN
iajs-4130	137	26	and	and	CCONJ
iajs-4130	137	27	.	.	PUNCT
iajs-4130	138	1	then	then	ADV
iajs-4130	138	2	(	(	PUNCT
iajs-4130	138	3	)	)	PUNCT
iajs-4130	138	4	(	(	PUNCT
iajs-4130	138	5	)	)	PUNCT
iajs-4130	138	6	proof	proof	NOUN
iajs-4130	138	7	.	.	PUNCT
iajs-4130	139	1	let	let	VERB
iajs-4130	139	2	k	k	PRON
iajs-4130	139	3	be	be	AUX
iajs-4130	139	4	a	a	DET
iajs-4130	139	5	prime	prime	ADJ
iajs-4130	139	6	submodule	submodule	NOUN
iajs-4130	139	7	that	that	SCONJ
iajs-4130	139	8	containing	contain	VERB
iajs-4130	139	9	n	n	CCONJ
iajs-4130	139	10	,	,	PUNCT
iajs-4130	139	11	then	then	ADV
iajs-4130	139	12	by	by	ADP
iajs-4130	139	13	proposition	proposition	NOUN
iajs-4130	139	14	(	(	PUNCT
iajs-4130	139	15	3.6	3.6	NUM
iajs-4130	139	16	)	)	PUNCT
iajs-4130	139	17	,	,	PUNCT
iajs-4130	139	18	k	k	PROPN
iajs-4130	139	19	is	be	AUX
iajs-4130	139	20	an	an	DET
iajs-4130	139	21	endo	endo	NOUN
iajs-4130	139	22	-	-	PUNCT
iajs-4130	139	23	r.b	r.b	NOUN
iajs-4130	139	24	submodule	submodule	NOUN
iajs-4130	139	25	containing	contain	VERB
iajs-4130	139	26	n	n	PROPN
iajs-4130	139	27	implies	imply	VERB
iajs-4130	139	28	that	that	PRON
iajs-4130	139	29	(	(	PUNCT
iajs-4130	139	30	)	)	PUNCT
iajs-4130	139	31	.	.	PUNCT
iajs-4130	140	1	hence	hence	ADV
iajs-4130	140	2	,	,	PUNCT
iajs-4130	140	3	(	(	PUNCT
iajs-4130	140	4	)	)	PUNCT
iajs-4130	140	5	for	for	ADP
iajs-4130	140	6	all	all	DET
iajs-4130	140	7	endo	endo	NOUN
iajs-4130	140	8	-	-	PUNCT
iajs-4130	140	9	r.b	r.b	NOUN
iajs-4130	140	10	submodules	submodule	NOUN
iajs-4130	140	11	containing	contain	VERB
iajs-4130	140	12	n.	n.	NOUN
iajs-4130	140	13	therefore	therefore	ADV
iajs-4130	140	14	,	,	PUNCT
iajs-4130	140	15	(	(	PUNCT
iajs-4130	140	16	)	)	PUNCT
iajs-4130	140	17	(	(	PUNCT
iajs-4130	140	18	)	)	PUNCT
iajs-4130	140	19	3.10	3.10	NUM
iajs-4130	140	20	.	.	PUNCT
iajs-4130	141	1	proposition	proposition	NOUN
iajs-4130	141	2	let	let	VERB
iajs-4130	141	3	n	n	NOUN
iajs-4130	141	4	and	and	CCONJ
iajs-4130	141	5	l	l	NOUN
iajs-4130	141	6	be	be	AUX
iajs-4130	141	7	two	two	NUM
iajs-4130	141	8	submodules	submodule	NOUN
iajs-4130	141	9	of	of	ADP
iajs-4130	141	10	a	a	DET
iajs-4130	141	11	t	t	NOUN
iajs-4130	141	12	-	-	PUNCT
iajs-4130	141	13	module	module	NOUN
iajs-4130	141	14	,	,	PUNCT
iajs-4130	141	15	then	then	ADV
iajs-4130	141	16	(	(	PUNCT
iajs-4130	141	17	)	)	PUNCT
iajs-4130	141	18	(	(	PUNCT
iajs-4130	141	19	)	)	PUNCT
iajs-4130	141	20	if	if	SCONJ
iajs-4130	141	21	and	and	CCONJ
iajs-4130	141	22	only	only	ADV
iajs-4130	141	23	if	if	SCONJ
iajs-4130	141	24	1	1	NUM
iajs-4130	141	25	.	.	PUNCT
iajs-4130	141	26	is	be	AUX
iajs-4130	141	27	a	a	DET
iajs-4130	141	28	radical	radical	ADJ
iajs-4130	141	29	endo	endo	NOUN
iajs-4130	141	30	-	-	PUNCT
iajs-4130	141	31	r.b	r.b	NOUN
iajs-4130	141	32	submodule	submodule	NOUN
iajs-4130	141	33	.	.	PUNCT
iajs-4130	142	1	2	2	X
iajs-4130	142	2	.	.	X
iajs-4130	142	3	(	(	PUNCT
iajs-4130	142	4	)	)	PUNCT
iajs-4130	142	5	(	(	PUNCT
iajs-4130	142	6	)	)	PUNCT
iajs-4130	142	7	(	(	PUNCT
iajs-4130	142	8	)	)	PUNCT
iajs-4130	142	9	.	.	PUNCT
iajs-4130	143	1	proof	proof	NOUN
iajs-4130	143	2	.	.	PUNCT
iajs-4130	144	1	suppose	suppose	VERB
iajs-4130	144	2	that	that	SCONJ
iajs-4130	144	3	(	(	PUNCT
iajs-4130	144	4	)	)	PUNCT
iajs-4130	144	5	(	(	PUNCT
iajs-4130	144	6	)	)	PUNCT
iajs-4130	144	7	then	then	ADV
iajs-4130	144	8	(	(	PUNCT
iajs-4130	144	9	)	)	PUNCT
iajs-4130	144	10	(	(	PUNCT
iajs-4130	144	11	)	)	PUNCT
iajs-4130	144	12	(	(	PUNCT
iajs-4130	144	13	)	)	PUNCT
iajs-4130	144	14	and	and	CCONJ
iajs-4130	144	15	the	the	DET
iajs-4130	144	16	inequality	inequality	NOUN
iajs-4130	144	17	(	(	PUNCT
iajs-4130	144	18	)	)	PUNCT
iajs-4130	144	19	(	(	PUNCT
iajs-4130	144	20	)	)	PUNCT
iajs-4130	144	21	(	(	PUNCT
iajs-4130	144	22	)	)	PUNCT
iajs-4130	144	23	holds	hold	VERB
iajs-4130	144	24	by	by	ADP
iajs-4130	144	25	proposition	proposition	NOUN
iajs-4130	144	26	(	(	PUNCT
iajs-4130	144	27	3.5	3.5	NUM
iajs-4130	144	28	)	)	PUNCT
iajs-4130	144	29	.	.	PUNCT
iajs-4130	145	1	now	now	ADV
iajs-4130	145	2	,	,	PUNCT
iajs-4130	145	3	by	by	ADP
iajs-4130	145	4	the	the	DET
iajs-4130	145	5	assumption	assumption	NOUN
iajs-4130	145	6	and	and	CCONJ
iajs-4130	145	7	proving	prove	VERB
iajs-4130	145	8	part	part	NOUN
iajs-4130	145	9	(	(	PUNCT
iajs-4130	145	10	2	2	NUM
iajs-4130	145	11	)	)	PUNCT
iajs-4130	145	12	,	,	PUNCT
iajs-4130	145	13	we	we	PRON
iajs-4130	145	14	have	have	AUX
iajs-4130	145	15	is	be	AUX
iajs-4130	145	16	a	a	DET
iajs-4130	145	17	radical	radical	ADJ
iajs-4130	145	18	endo	endo	NOUN
iajs-4130	145	19	-	-	PUNCT
iajs-4130	145	20	r.b	r.b	NOUN
iajs-4130	145	21	submodule	submodule	NOUN
iajs-4130	145	22	.	.	PUNCT
iajs-4130	146	1	conversely	conversely	ADV
iajs-4130	146	2	,	,	PUNCT
iajs-4130	146	3	it	it	PRON
iajs-4130	146	4	is	be	AUX
iajs-4130	146	5	obvious	obvious	ADJ
iajs-4130	146	6	that	that	SCONJ
iajs-4130	146	7	(	(	PUNCT
iajs-4130	146	8	1	1	X
iajs-4130	146	9	)	)	PUNCT
iajs-4130	146	10	and	and	CCONJ
iajs-4130	146	11	(	(	PUNCT
iajs-4130	146	12	2	2	X
iajs-4130	146	13	)	)	PUNCT
iajs-4130	146	14	prove	prove	VERB
iajs-4130	146	15	that	that	SCONJ
iajs-4130	146	16	:	:	PUNCT
iajs-4130	146	17	ihjpas	ihjpas	PROPN
iajs-4130	146	18	.	.	PUNCT
iajs-4130	147	1	2025,38(4	2025,38(4	X
iajs-4130	147	2	)	)	PUNCT
iajs-4130	147	3	404	404	NUM
iajs-4130	147	4	(	(	PUNCT
iajs-4130	147	5	)	)	PUNCT
iajs-4130	147	6	(	(	PUNCT
iajs-4130	147	7	)	)	PUNCT
iajs-4130	147	8	recall	recall	VERB
iajs-4130	147	9	a	a	DET
iajs-4130	147	10	submodule	submodule	NOUN
iajs-4130	147	11	a	a	PRON
iajs-4130	147	12	of	of	ADP
iajs-4130	147	13	a	a	DET
iajs-4130	147	14	t	t	NOUN
iajs-4130	147	15	-	-	PUNCT
iajs-4130	147	16	module	module	NOUN
iajs-4130	147	17	called	call	VERB
iajs-4130	147	18	an	an	DET
iajs-4130	147	19	s	s	NOUN
iajs-4130	147	20	-	-	NOUN
iajs-4130	147	21	prime	prime	NOUN
iajs-4130	147	22	if	if	SCONJ
iajs-4130	147	23	there	there	PRON
iajs-4130	147	24	exists	exist	VERB
iajs-4130	147	25	(	(	PUNCT
iajs-4130	147	26	)	)	PUNCT
iajs-4130	147	27	such	such	ADJ
iajs-4130	147	28	that	that	SCONJ
iajs-4130	147	29	(	(	PUNCT
iajs-4130	147	30	)	)	PUNCT
iajs-4130	147	31	implies	imply	VERB
iajs-4130	147	32	that	that	SCONJ
iajs-4130	147	33	either	either	CCONJ
iajs-4130	147	34	(	(	PUNCT
iajs-4130	147	35	)	)	PUNCT
iajs-4130	147	36	(	(	PUNCT
iajs-4130	147	37	21–23	21–23	NUM
iajs-4130	147	38	)	)	PUNCT
iajs-4130	147	39	.	.	PUNCT
iajs-4130	148	1	3.11	3.11	NUM
iajs-4130	148	2	.	.	PUNCT
iajs-4130	149	1	lemma	lemma	PROPN
iajs-4130	149	2	let	let	AUX
iajs-4130	149	3	be	be	AUX
iajs-4130	149	4	a	a	DET
iajs-4130	149	5	scalar	scalar	ADJ
iajs-4130	149	6	t	t	NOUN
iajs-4130	149	7	-	-	PUNCT
iajs-4130	149	8	module	module	NOUN
iajs-4130	149	9	and	and	CCONJ
iajs-4130	149	10	a	a	PRON
iajs-4130	149	11	is	be	AUX
iajs-4130	149	12	an	an	DET
iajs-4130	149	13	endo	endo	NOUN
iajs-4130	149	14	-	-	PUNCT
iajs-4130	149	15	r.b	r.b	NOUN
iajs-4130	149	16	.	.	PROPN
iajs-4130	149	17	submodule	submodule	PROPN
iajs-4130	149	18	of	of	ADP
iajs-4130	149	19	.	.	PUNCT
iajs-4130	150	1	then	then	ADV
iajs-4130	150	2	a	a	PRON
iajs-4130	150	3	is	be	AUX
iajs-4130	150	4	an	an	DET
iajs-4130	150	5	s	s	ADJ
iajs-4130	150	6	-	-	PUNCT
iajs-4130	150	7	prime	prime	ADJ
iajs-4130	150	8	submodule	submodule	NOUN
iajs-4130	150	9	proof	proof	NOUN
iajs-4130	150	10	let	let	VERB
iajs-4130	150	11	(	(	PUNCT
iajs-4130	150	12	)	)	PUNCT
iajs-4130	150	13	and	and	CCONJ
iajs-4130	150	14	define	define	VERB
iajs-4130	150	15	as	as	ADP
iajs-4130	150	16	(	(	PUNCT
iajs-4130	150	17	)	)	PUNCT
iajs-4130	150	18	.	.	PUNCT
iajs-4130	151	1	suppose	suppose	VERB
iajs-4130	151	2	that	that	SCONJ
iajs-4130	151	3	then	then	ADV
iajs-4130	151	4	,	,	PUNCT
iajs-4130	151	5	we	we	PRON
iajs-4130	151	6	have	have	VERB
iajs-4130	151	7	to	to	PART
iajs-4130	151	8	prove	prove	VERB
iajs-4130	151	9	that	that	PRON
iajs-4130	151	10	(	(	PUNCT
iajs-4130	151	11	)	)	PUNCT
iajs-4130	151	12	since	since	SCONJ
iajs-4130	151	13	is	be	AUX
iajs-4130	151	14	a	a	DET
iajs-4130	151	15	scalar	scalar	ADJ
iajs-4130	151	16	and	and	CCONJ
iajs-4130	151	17	a	a	PRON
iajs-4130	151	18	is	be	AUX
iajs-4130	151	19	an	an	DET
iajs-4130	151	20	endo	endo	NOUN
iajs-4130	151	21	-	-	PUNCT
iajs-4130	151	22	r.b	r.b	NOUN
iajs-4130	151	23	submodule	submodule	NOUN
iajs-4130	151	24	,	,	PUNCT
iajs-4130	151	25	then	then	ADV
iajs-4130	151	26	(	(	PUNCT
iajs-4130	151	27	)	)	PUNCT
iajs-4130	151	28	which	which	PRON
iajs-4130	151	29	means	mean	VERB
iajs-4130	151	30	that	that	SCONJ
iajs-4130	151	31	(	(	PUNCT
iajs-4130	151	32	)	)	PUNCT
iajs-4130	151	33	therefore	therefore	ADV
iajs-4130	151	34	,	,	PUNCT
iajs-4130	151	35	a	a	PRON
iajs-4130	151	36	is	be	AUX
iajs-4130	151	37	an	an	DET
iajs-4130	151	38	s	s	NOUN
iajs-4130	151	39	-	-	PUNCT
iajs-4130	151	40	prime	prime	ADJ
iajs-4130	151	41	submodule	submodule	NOUN
iajs-4130	151	42	.	.	PUNCT
iajs-4130	152	1	note	note	VERB
iajs-4130	152	2	that	that	SCONJ
iajs-4130	152	3	every	every	DET
iajs-4130	152	4	s	s	NOUN
iajs-4130	152	5	-	-	PUNCT
iajs-4130	152	6	prime	prime	ADJ
iajs-4130	152	7	submodule	submodule	NOUN
iajs-4130	152	8	is	be	AUX
iajs-4130	152	9	a	a	DET
iajs-4130	152	10	prime	prime	ADJ
iajs-4130	152	11	submodule	submodule	NOUN
iajs-4130	152	12	.	.	PUNCT
iajs-4130	153	1	3.12	3.12	NUM
iajs-4130	153	2	.	.	PUNCT
iajs-4130	154	1	lemma	lemma	PROPN
iajs-4130	154	2	let	let	AUX
iajs-4130	154	3	be	be	AUX
iajs-4130	154	4	a	a	DET
iajs-4130	154	5	scalar	scalar	ADJ
iajs-4130	154	6	t	t	NOUN
iajs-4130	154	7	-	-	PUNCT
iajs-4130	154	8	module	module	NOUN
iajs-4130	154	9	and	and	CCONJ
iajs-4130	154	10	then	then	ADV
iajs-4130	154	11	,	,	PUNCT
iajs-4130	154	12	(	(	PUNCT
iajs-4130	154	13	)	)	PUNCT
iajs-4130	154	14	(	(	PUNCT
iajs-4130	154	15	)	)	PUNCT
iajs-4130	154	16	.	.	PUNCT
iajs-4130	155	1	proof	proof	NOUN
iajs-4130	155	2	.	.	PUNCT
iajs-4130	156	1	since	since	SCONJ
iajs-4130	156	2	every	every	DET
iajs-4130	156	3	s	s	NOUN
iajs-4130	156	4	-	-	PUNCT
iajs-4130	156	5	prime	prime	ADJ
iajs-4130	156	6	submodule	submodule	NOUN
iajs-4130	156	7	is	be	AUX
iajs-4130	156	8	a	a	DET
iajs-4130	156	9	prime	prime	NOUN
iajs-4130	156	10	,	,	PUNCT
iajs-4130	156	11	then	then	ADV
iajs-4130	156	12	the	the	DET
iajs-4130	156	13	proof	proof	NOUN
iajs-4130	156	14	is	be	AUX
iajs-4130	156	15	by	by	ADP
iajs-4130	156	16	lemma	lemma	PROPN
iajs-4130	156	17	(	(	PUNCT
iajs-4130	156	18	3.11	3.11	NUM
iajs-4130	156	19	)	)	PUNCT
iajs-4130	156	20	and	and	CCONJ
iajs-4130	156	21	proposition	proposition	NOUN
iajs-4130	156	22	(	(	PUNCT
iajs-4130	156	23	3.6	3.6	NUM
iajs-4130	156	24	)	)	PUNCT
iajs-4130	156	25	.	.	PUNCT
iajs-4130	157	1	3.13	3.13	NUM
iajs-4130	157	2	.	.	PUNCT
iajs-4130	158	1	proposition	proposition	NOUN
iajs-4130	158	2	let	let	VERB
iajs-4130	158	3	n	n	PRON
iajs-4130	158	4	and	and	CCONJ
iajs-4130	158	5	l	l	NOUN
iajs-4130	158	6	are	be	AUX
iajs-4130	158	7	two	two	NUM
iajs-4130	158	8	submodules	submodule	NOUN
iajs-4130	158	9	of	of	ADP
iajs-4130	158	10	a	a	DET
iajs-4130	158	11	multiplication	multiplication	NOUN
iajs-4130	158	12	finitely	finitely	ADV
iajs-4130	158	13	generated	generate	VERB
iajs-4130	158	14	t	t	NOUN
iajs-4130	158	15	-	-	PUNCT
iajs-4130	158	16	module	module	NOUN
iajs-4130	158	17	such	such	ADJ
iajs-4130	158	18	that	that	PRON
iajs-4130	158	19	,	,	PUNCT
iajs-4130	158	20	,	,	PUNCT
iajs-4130	158	21	are	be	AUX
iajs-4130	158	22	radical	radical	ADJ
iajs-4130	158	23	ideals	ideal	NOUN
iajs-4130	158	24	,	,	PUNCT
iajs-4130	158	25	then	then	ADV
iajs-4130	158	26	,	,	PUNCT
iajs-4130	158	27	,	,	PUNCT
iajs-4130	158	28	(	(	PUNCT
iajs-4130	158	29	)	)	PUNCT
iajs-4130	158	30	proof	proof	NOUN
iajs-4130	158	31	.	.	PUNCT
iajs-4130	159	1	clearly	clearly	ADV
iajs-4130	159	2	,	,	PUNCT
iajs-4130	159	3	,	,	PUNCT
iajs-4130	159	4	,	,	PUNCT
iajs-4130	159	5	√	√	NUM
iajs-4130	159	6	,	,	PUNCT
iajs-4130	159	7	√	√	NUM
iajs-4130	159	8	,	,	PUNCT
iajs-4130	159	9	√	√	NUM
iajs-4130	159	10	,	,	PUNCT
iajs-4130	159	11	since	since	SCONJ
iajs-4130	159	12	is	be	AUX
iajs-4130	159	13	finitely	finitely	ADV
iajs-4130	159	14	generated	generate	VERB
iajs-4130	159	15	,	,	PUNCT
iajs-4130	159	16	then	then	ADV
iajs-4130	159	17	by	by	ADP
iajs-4130	159	18	theorem	theorem	NOUN
iajs-4130	159	19	4.4	4.4	NUM
iajs-4130	159	20	in	in	ADP
iajs-4130	159	21	(	(	PUNCT
iajs-4130	159	22	24	24	NUM
iajs-4130	159	23	)	)	PUNCT
iajs-4130	159	24	,	,	PUNCT
iajs-4130	159	25	we	we	PRON
iajs-4130	159	26	have	have	VERB
iajs-4130	159	27	√	√	NUM
iajs-4130	159	28	,	,	PUNCT
iajs-4130	159	29	,	,	PUNCT
iajs-4130	159	30	(	(	PUNCT
iajs-4130	159	31	)	)	PUNCT
iajs-4130	159	32	using	use	VERB
iajs-4130	159	33	lemma	lemma	PROPN
iajs-4130	159	34	(	(	PUNCT
iajs-4130	159	35	3.12	3.12	NUM
iajs-4130	159	36	)	)	PUNCT
iajs-4130	159	37	,	,	PUNCT
iajs-4130	159	38	we	we	PRON
iajs-4130	159	39	conclude	conclude	VERB
iajs-4130	159	40	that	that	PRON
iajs-4130	159	41	,	,	PUNCT
iajs-4130	159	42	,	,	PUNCT
iajs-4130	159	43	(	(	PUNCT
iajs-4130	159	44	)	)	PUNCT
iajs-4130	159	45	-	-	PUNCT
iajs-4130	159	46	.	.	PUNCT
iajs-4130	160	1	3.14	3.14	NUM
iajs-4130	160	2	.	.	PUNCT
iajs-4130	161	1	proposition	proposition	NOUN
iajs-4130	161	2	let	let	VERB
iajs-4130	161	3	n	n	PRON
iajs-4130	161	4	and	and	CCONJ
iajs-4130	161	5	l	l	NOUN
iajs-4130	161	6	are	be	AUX
iajs-4130	161	7	two	two	NUM
iajs-4130	161	8	submodules	submodule	NOUN
iajs-4130	161	9	of	of	ADP
iajs-4130	161	10	a	a	DET
iajs-4130	161	11	multiplication	multiplication	NOUN
iajs-4130	161	12	finitely	finitely	ADV
iajs-4130	161	13	generated	generate	VERB
iajs-4130	161	14	t	t	NOUN
iajs-4130	161	15	-	-	PUNCT
iajs-4130	161	16	module	module	NOUN
iajs-4130	161	17	.	.	PUNCT
iajs-4130	162	1	then	then	ADV
iajs-4130	162	2	[	[	X
iajs-4130	162	3	endo	endo	X
iajs-4130	162	4	(	(	PUNCT
iajs-4130	162	5	)	)	PUNCT
iajs-4130	163	1	[	[	X
iajs-4130	163	2	endo	endo	NOUN
iajs-4130	163	3	(	(	PUNCT
iajs-4130	163	4	)	)	PUNCT
iajs-4130	163	5	endo	endo	NOUN
iajs-4130	163	6	(	(	PUNCT
iajs-4130	163	7	)	)	PUNCT
iajs-4130	163	8	proof	proof	NOUN
iajs-4130	163	9	.	.	PUNCT
iajs-4130	164	1	since	since	SCONJ
iajs-4130	164	2	is	be	AUX
iajs-4130	164	3	a	a	DET
iajs-4130	164	4	multiplication	multiplication	NOUN
iajs-4130	164	5	finitely	finitely	ADV
iajs-4130	164	6	generated	generate	VERB
iajs-4130	164	7	module	module	NOUN
iajs-4130	164	8	,	,	PUNCT
iajs-4130	164	9	then	then	ADV
iajs-4130	164	10	is	be	AUX
iajs-4130	164	11	a	a	DET
iajs-4130	164	12	scalar	scalar	ADJ
iajs-4130	164	13	module	module	NOUN
iajs-4130	164	14	and	and	CCONJ
iajs-4130	164	15	hence	hence	ADV
iajs-4130	164	16	by	by	ADP
iajs-4130	164	17	lemma	lemma	PROPN
iajs-4130	164	18	(	(	PUNCT
iajs-4130	164	19	3.12	3.12	NUM
iajs-4130	164	20	)	)	PUNCT
iajs-4130	164	21	,	,	PUNCT
iajs-4130	164	22	we	we	PRON
iajs-4130	164	23	have	have	VERB
iajs-4130	164	24	(	(	PUNCT
iajs-4130	164	25	)	)	PUNCT
iajs-4130	164	26	(	(	PUNCT
iajs-4130	164	27	)	)	PUNCT
iajs-4130	164	28	.	.	PUNCT
iajs-4130	165	1	thus	thus	ADV
iajs-4130	165	2	,	,	PUNCT
iajs-4130	165	3	[	[	X
iajs-4130	165	4	endo	endo	NOUN
iajs-4130	165	5	(	(	PUNCT
iajs-4130	165	6	)	)	PUNCT
iajs-4130	165	7	,	,	PUNCT
iajs-4130	165	8	(	(	PUNCT
iajs-4130	165	9	)	)	PUNCT
iajs-4130	165	10	√	√	NUM
iajs-4130	165	11	,	,	PUNCT
iajs-4130	165	12	√	√	NUM
iajs-4130	165	13	,	,	PUNCT
iajs-4130	165	14	√	√	NUM
iajs-4130	165	15	,	,	PUNCT
iajs-4130	165	16	[	[	X
iajs-4130	165	17	endo	endo	NOUN
iajs-4130	165	18	[	[	X
iajs-4130	165	19	endo	endo	NOUN
iajs-4130	165	20	,	,	PUNCT
iajs-4130	165	21	(	(	PUNCT
iajs-4130	165	22	)	)	PUNCT
iajs-4130	165	23	(	(	PUNCT
iajs-4130	165	24	)	)	PUNCT
iajs-4130	165	25	3.15	3.15	X
iajs-4130	165	26	.	.	PUNCT
iajs-4130	166	1	proposition	proposition	NOUN
iajs-4130	166	2	let	let	AUX
iajs-4130	166	3	be	be	AUX
iajs-4130	166	4	a	a	DET
iajs-4130	166	5	multiplication	multiplication	NOUN
iajs-4130	166	6	finitely	finitely	ADV
iajs-4130	166	7	generated	generate	VERB
iajs-4130	166	8	t	t	NOUN
iajs-4130	166	9	-	-	PUNCT
iajs-4130	166	10	module	module	NOUN
iajs-4130	166	11	and	and	CCONJ
iajs-4130	166	12	.	.	PUNCT
iajs-4130	167	1	then	then	ADV
iajs-4130	167	2	√	√	VERB
iajs-4130	167	3	,	,	PUNCT
iajs-4130	167	4	(	(	PUNCT
iajs-4130	167	5	)	)	PUNCT
iajs-4130	167	6	.	.	PUNCT
iajs-4130	168	1	proof	proof	NOUN
iajs-4130	168	2	.	.	PUNCT
iajs-4130	169	1	let	let	VERB
iajs-4130	169	2	k	k	PRON
iajs-4130	169	3	be	be	AUX
iajs-4130	169	4	an	an	DET
iajs-4130	169	5	endo	endo	NOUN
iajs-4130	169	6	-	-	PUNCT
iajs-4130	169	7	r.b	r.b	NOUN
iajs-4130	169	8	submodule	submodule	NOUN
iajs-4130	169	9	of	of	ADP
iajs-4130	169	10	containing	contain	VERB
iajs-4130	169	11	n.	n.	PROPN
iajs-4130	169	12	also	also	ADV
iajs-4130	169	13	,	,	PUNCT
iajs-4130	169	14	by	by	ADP
iajs-4130	169	15	lemma	lemma	PROPN
iajs-4130	169	16	(	(	PUNCT
iajs-4130	169	17	3.11	3.11	NUM
iajs-4130	169	18	)	)	PUNCT
iajs-4130	169	19	,	,	PUNCT
iajs-4130	169	20	if	if	SCONJ
iajs-4130	169	21	is	be	AUX
iajs-4130	169	22	a	a	DET
iajs-4130	169	23	scalar	scalar	ADJ
iajs-4130	169	24	and	and	CCONJ
iajs-4130	169	25	n	n	PRON
iajs-4130	169	26	is	be	AUX
iajs-4130	169	27	an	an	DET
iajs-4130	169	28	endo	endo	NOUN
iajs-4130	169	29	-	-	PUNCT
iajs-4130	169	30	r.b	r.b	NOUN
iajs-4130	169	31	,	,	PUNCT
iajs-4130	169	32	then	then	ADV
iajs-4130	169	33	n	n	PRON
iajs-4130	169	34	is	be	AUX
iajs-4130	169	35	an	an	DET
iajs-4130	169	36	s	s	NOUN
iajs-4130	169	37	-	-	NOUN
iajs-4130	169	38	prime	prime	NOUN
iajs-4130	169	39	and	and	CCONJ
iajs-4130	169	40	every	every	DET
iajs-4130	169	41	s	s	NOUN
iajs-4130	169	42	-	-	ADJ
iajs-4130	169	43	prime	prime	NOUN
iajs-4130	169	44	is	be	AUX
iajs-4130	169	45	a	a	DET
iajs-4130	169	46	prime	prime	ADJ
iajs-4130	169	47	submodule	submodule	NOUN
iajs-4130	169	48	.	.	PUNCT
iajs-4130	170	1	ihjpas	ihjpas	PROPN
iajs-4130	170	2	.	.	PUNCT
iajs-4130	171	1	2025,38(4	2025,38(4	NOUN
iajs-4130	171	2	)	)	PUNCT
iajs-4130	171	3	405	405	NUM
iajs-4130	171	4	therefore	therefore	ADV
iajs-4130	171	5	,	,	PUNCT
iajs-4130	171	6	k	k	PROPN
iajs-4130	171	7	is	be	AUX
iajs-4130	171	8	a	a	DET
iajs-4130	171	9	prime	prime	ADJ
iajs-4130	171	10	submodule	submodule	NOUN
iajs-4130	171	11	and	and	CCONJ
iajs-4130	171	12	,	,	PUNCT
iajs-4130	171	13	is	be	AUX
iajs-4130	171	14	a	a	DET
iajs-4130	171	15	prime	prime	ADJ
iajs-4130	171	16	ideal	ideal	NOUN
iajs-4130	171	17	implies	imply	VERB
iajs-4130	171	18	that	that	SCONJ
iajs-4130	171	19	,	,	PUNCT
iajs-4130	171	20	,	,	PUNCT
iajs-4130	171	21	and	and	CCONJ
iajs-4130	171	22	√	√	NUM
iajs-4130	171	23	,	,	PUNCT
iajs-4130	171	24	,	,	PUNCT
iajs-4130	171	25	-	-	PUNCT
iajs-4130	171	26	.	.	PUNCT
iajs-4130	171	27	hence	hence	ADV
iajs-4130	171	28	√	√	NUM
iajs-4130	171	29	,	,	PUNCT
iajs-4130	171	30	,	,	PUNCT
iajs-4130	171	31	and	and	CCONJ
iajs-4130	171	32	since	since	SCONJ
iajs-4130	171	33	k	k	PROPN
iajs-4130	171	34	is	be	AUX
iajs-4130	171	35	an	an	DET
iajs-4130	171	36	arbitrary	arbitrary	ADJ
iajs-4130	171	37	endor.b	endor.b	NOUN
iajs-4130	171	38	submodule	submodule	NOUN
iajs-4130	171	39	containing	contain	VERB
iajs-4130	171	40	n	n	PROPN
iajs-4130	171	41	so	so	ADV
iajs-4130	171	42	,	,	PUNCT
iajs-4130	171	43	we	we	PRON
iajs-4130	171	44	have	have	VERB
iajs-4130	171	45	that	that	DET
iajs-4130	171	46	√	√	VERB
iajs-4130	171	47	,	,	PUNCT
iajs-4130	171	48	(	(	PUNCT
iajs-4130	171	49	)	)	PUNCT
iajs-4130	171	50	.	.	PUNCT
iajs-4130	172	1	since	since	SCONJ
iajs-4130	172	2	is	be	AUX
iajs-4130	172	3	a	a	DET
iajs-4130	172	4	multiplication	multiplication	NOUN
iajs-4130	172	5	finitely	finitely	ADV
iajs-4130	172	6	generated	generate	VERB
iajs-4130	172	7	t	t	NOUN
iajs-4130	172	8	-	-	PUNCT
iajs-4130	172	9	module	module	NOUN
iajs-4130	172	10	then	then	ADV
iajs-4130	172	11	is	be	AUX
iajs-4130	172	12	a	a	DET
iajs-4130	172	13	scalar	scalar	ADJ
iajs-4130	172	14	module	module	NOUN
iajs-4130	172	15	and	and	CCONJ
iajs-4130	172	16	using	use	VERB
iajs-4130	172	17	lemma	lemma	PROPN
iajs-4130	172	18	(	(	PUNCT
iajs-4130	172	19	3.12	3.12	NUM
iajs-4130	172	20	)	)	PUNCT
iajs-4130	172	21	,	,	PUNCT
iajs-4130	172	22	we	we	PRON
iajs-4130	172	23	have	have	VERB
iajs-4130	172	24	that	that	PRON
iajs-4130	172	25	(	(	PUNCT
iajs-4130	172	26	)	)	PUNCT
iajs-4130	172	27	(	(	PUNCT
iajs-4130	172	28	)	)	PUNCT
iajs-4130	172	29	.	.	PUNCT
iajs-4130	173	1	by	by	ADP
iajs-4130	173	2	(	(	PUNCT
iajs-4130	173	3	24),we	24),we	PROPN
iajs-4130	173	4	get	get	VERB
iajs-4130	173	5	that	that	PRON
iajs-4130	173	6	(	(	PUNCT
iajs-4130	173	7	)	)	PUNCT
iajs-4130	173	8	√	√	NUM
iajs-4130	173	9	,	,	PUNCT
iajs-4130	173	10	therefore	therefore	ADV
iajs-4130	173	11	,	,	PUNCT
iajs-4130	173	12	√	√	NUM
iajs-4130	173	13	,	,	PUNCT
iajs-4130	173	14	(	(	PUNCT
iajs-4130	173	15	)	)	PUNCT
iajs-4130	173	16	(	(	PUNCT
iajs-4130	173	17	)	)	PUNCT
iajs-4130	173	18	.	.	PUNCT
iajs-4130	174	1	let	let	VERB
iajs-4130	174	2	n	n	PRON
iajs-4130	174	3	be	be	AUX
iajs-4130	174	4	a	a	DET
iajs-4130	174	5	submodule	submodule	NOUN
iajs-4130	174	6	of	of	ADP
iajs-4130	174	7	a	a	DET
iajs-4130	174	8	t	t	NOUN
iajs-4130	174	9	-	-	PUNCT
iajs-4130	174	10	module	module	NOUN
iajs-4130	174	11	and	and	CCONJ
iajs-4130	174	12	q	q	NOUN
iajs-4130	174	13	be	be	AUX
iajs-4130	174	14	a	a	DET
iajs-4130	174	15	multiplicative	multiplicative	ADJ
iajs-4130	174	16	set	set	NOUN
iajs-4130	174	17	of	of	ADP
iajs-4130	174	18	t	t	PROPN
iajs-4130	174	19	,	,	PUNCT
iajs-4130	174	20	then	then	ADV
iajs-4130	174	21	(	(	PUNCT
iajs-4130	174	22	)	)	PUNCT
iajs-4130	174	23	*	*	PUNCT
iajs-4130	175	1	+	+	CCONJ
iajs-4130	175	2	is	be	AUX
iajs-4130	175	3	a	a	DET
iajs-4130	175	4	submodule	submodule	NOUN
iajs-4130	175	5	of	of	ADP
iajs-4130	175	6	contains	contain	VERB
iajs-4130	175	7	n	n	X
iajs-4130	175	8	(	(	PUNCT
iajs-4130	175	9	25,26	25,26	NUM
iajs-4130	175	10	)	)	PUNCT
iajs-4130	175	11	and	and	CCONJ
iajs-4130	175	12	the	the	DET
iajs-4130	175	13	closure	closure	NOUN
iajs-4130	175	14	of	of	ADP
iajs-4130	175	15	a	a	DET
iajs-4130	175	16	submodule	submodule	NOUN
iajs-4130	175	17	a	a	PRON
iajs-4130	175	18	is	be	AUX
iajs-4130	175	19	denoted	denote	VERB
iajs-4130	175	20	by	by	ADP
iajs-4130	175	21	(	(	PUNCT
iajs-4130	175	22	)	)	PUNCT
iajs-4130	175	23	*	*	PUNCT
iajs-4130	175	24	,	,	PUNCT
iajs-4130	175	25	+	+	CCONJ
iajs-4130	175	26	(	(	PUNCT
iajs-4130	175	27	27,28	27,28	NUM
iajs-4130	175	28	)	)	PUNCT
iajs-4130	175	29	3.16	3.16	NUM
iajs-4130	175	30	.	.	PUNCT
iajs-4130	176	1	proposition	proposition	NOUN
iajs-4130	176	2	let	let	AUX
iajs-4130	176	3	be	be	AUX
iajs-4130	176	4	a	a	DET
iajs-4130	176	5	t	t	NOUN
iajs-4130	176	6	-	-	PUNCT
iajs-4130	176	7	module	module	NOUN
iajs-4130	176	8	and	and	CCONJ
iajs-4130	176	9	then	then	ADV
iajs-4130	176	10	1	1	X
iajs-4130	176	11	)	)	PUNCT
iajs-4130	176	12	(	(	PUNCT
iajs-4130	176	13	)	)	PUNCT
iajs-4130	176	14	(	(	PUNCT
iajs-4130	176	15	(	(	PUNCT
iajs-4130	176	16	)	)	PUNCT
iajs-4130	176	17	)	)	PUNCT
iajs-4130	176	18	where	where	SCONJ
iajs-4130	176	19	q	q	NOUN
iajs-4130	176	20	is	be	AUX
iajs-4130	176	21	a	a	DET
iajs-4130	176	22	multiplicative	multiplicative	ADJ
iajs-4130	176	23	set	set	NOUN
iajs-4130	176	24	of	of	ADP
iajs-4130	176	25	t.	t.	PROPN
iajs-4130	176	26	2	2	NUM
iajs-4130	176	27	)	)	PUNCT
iajs-4130	176	28	(	(	PUNCT
iajs-4130	176	29	)	)	PUNCT
iajs-4130	176	30	(	(	PUNCT
iajs-4130	176	31	(	(	PUNCT
iajs-4130	176	32	)	)	PUNCT
iajs-4130	176	33	)	)	PUNCT
iajs-4130	176	34	.	.	PUNCT
iajs-4130	177	1	3	3	X
iajs-4130	177	2	)	)	PUNCT
iajs-4130	177	3	(	(	PUNCT
iajs-4130	177	4	)	)	PUNCT
iajs-4130	177	5	(	(	PUNCT
iajs-4130	177	6	,	,	PUNCT
iajs-4130	177	7	-	-	PUNCT
iajs-4130	177	8	)	)	PUNCT
iajs-4130	177	9	for	for	ADP
iajs-4130	177	10	every	every	DET
iajs-4130	177	11	ideal	ideal	ADJ
iajs-4130	177	12	i	i	PRON
iajs-4130	177	13	of	of	ADP
iajs-4130	177	14	t.	t.	NOUN
iajs-4130	177	15	proof	proof	NOUN
iajs-4130	177	16	.	.	PUNCT
iajs-4130	178	1	1	1	X
iajs-4130	178	2	)	)	PUNCT
iajs-4130	178	3	since	since	SCONJ
iajs-4130	178	4	n(q	n(q	PROPN
iajs-4130	178	5	)	)	PUNCT
iajs-4130	178	6	is	be	AUX
iajs-4130	178	7	a	a	DET
iajs-4130	178	8	submodule	submodule	NOUN
iajs-4130	178	9	of	of	ADP
iajs-4130	178	10	contains	contain	NOUN
iajs-4130	178	11	n	n	CCONJ
iajs-4130	178	12	,	,	PUNCT
iajs-4130	178	13	then	then	ADV
iajs-4130	178	14	by	by	ADP
iajs-4130	178	15	proposition	proposition	NOUN
iajs-4130	178	16	(	(	PUNCT
iajs-4130	178	17	3.5	3.5	NUM
iajs-4130	178	18	)	)	PUNCT
iajs-4130	178	19	,	,	PUNCT
iajs-4130	178	20	we	we	PRON
iajs-4130	178	21	have	have	VERB
iajs-4130	178	22	that	that	PRON
iajs-4130	178	23	(	(	PUNCT
iajs-4130	178	24	)	)	PUNCT
iajs-4130	178	25	(	(	PUNCT
iajs-4130	178	26	(	(	PUNCT
iajs-4130	178	27	)	)	PUNCT
iajs-4130	178	28	)	)	PUNCT
iajs-4130	178	29	2	2	X
iajs-4130	178	30	)	)	PUNCT
iajs-4130	178	31	it	it	PRON
iajs-4130	178	32	is	be	AUX
iajs-4130	178	33	clear	clear	ADJ
iajs-4130	178	34	since	since	SCONJ
iajs-4130	178	35	(	(	PUNCT
iajs-4130	178	36	)	)	PUNCT
iajs-4130	178	37	is	be	AUX
iajs-4130	178	38	a	a	DET
iajs-4130	178	39	submodule	submodule	NOUN
iajs-4130	178	40	of	of	ADP
iajs-4130	178	41	contains	contain	NOUN
iajs-4130	178	42	n.	n.	NOUN
iajs-4130	178	43	3	3	NUM
iajs-4130	178	44	)	)	PUNCT
iajs-4130	178	45	since	since	SCONJ
iajs-4130	178	46	for	for	ADP
iajs-4130	178	47	ideal	ideal	ADJ
iajs-4130	178	48	i	i	PRON
iajs-4130	178	49	of	of	ADP
iajs-4130	178	50	t	t	PROPN
iajs-4130	178	51	,	,	PUNCT
iajs-4130	178	52	we	we	PRON
iajs-4130	178	53	have	have	VERB
iajs-4130	178	54	,	,	PUNCT
iajs-4130	178	55	-	-	PUNCT
iajs-4130	178	56	.	.	PUNCT
iajs-4130	179	1	therefore	therefore	ADV
iajs-4130	179	2	,	,	PUNCT
iajs-4130	179	3	the	the	DET
iajs-4130	179	4	result	result	NOUN
iajs-4130	179	5	follows	follow	VERB
iajs-4130	179	6	directly	directly	ADV
iajs-4130	179	7	from	from	ADP
iajs-4130	179	8	proposition	proposition	NOUN
iajs-4130	179	9	(	(	PUNCT
iajs-4130	179	10	3.5	3.5	NUM
iajs-4130	179	11	)	)	PUNCT
iajs-4130	179	12	recall	recall	VERB
iajs-4130	179	13	a	a	DET
iajs-4130	179	14	submodule	submodule	NOUN
iajs-4130	179	15	h	h	NOUN
iajs-4130	179	16	of	of	ADP
iajs-4130	179	17	a	a	DET
iajs-4130	179	18	t	t	NOUN
iajs-4130	179	19	-	-	PUNCT
iajs-4130	179	20	module	module	NOUN
iajs-4130	179	21	said	say	VERB
iajs-4130	179	22	to	to	PART
iajs-4130	179	23	be	be	AUX
iajs-4130	179	24	fully	fully	ADV
iajs-4130	179	25	invariant	invariant	ADJ
iajs-4130	179	26	if	if	SCONJ
iajs-4130	179	27	(	(	PUNCT
iajs-4130	179	28	)	)	PUNCT
iajs-4130	179	29	for	for	ADP
iajs-4130	179	30	every	every	PRON
iajs-4130	179	31	(	(	PUNCT
iajs-4130	179	32	)	)	PUNCT
iajs-4130	179	33	(	(	PUNCT
iajs-4130	179	34	29	29	NUM
iajs-4130	179	35	)	)	PUNCT
iajs-4130	179	36	.	.	PUNCT
iajs-4130	180	1	3.17	3.17	X
iajs-4130	180	2	.	.	PUNCT
iajs-4130	180	3	proposition	proposition	NOUN
iajs-4130	180	4	let	let	VERB
iajs-4130	180	5	n	n	NOUN
iajs-4130	180	6	and	and	CCONJ
iajs-4130	180	7	l	l	NOUN
iajs-4130	180	8	be	be	AUX
iajs-4130	180	9	two	two	NUM
iajs-4130	180	10	fully	fully	ADV
iajs-4130	180	11	invariant	invariant	ADJ
iajs-4130	180	12	submodules	submodule	NOUN
iajs-4130	180	13	of	of	ADP
iajs-4130	180	14	a	a	DET
iajs-4130	180	15	–	–	PUNCT
iajs-4130	180	16	module	module	NOUN
iajs-4130	180	17	m	m	NOUN
iajs-4130	180	18	and	and	CCONJ
iajs-4130	180	19	consider	consider	VERB
iajs-4130	180	20	,	,	PUNCT
iajs-4130	180	21	then	then	ADV
iajs-4130	180	22	(	(	PUNCT
iajs-4130	180	23	)	)	PUNCT
iajs-4130	180	24	(	(	PUNCT
iajs-4130	180	25	)	)	PUNCT
iajs-4130	180	26	(	(	PUNCT
iajs-4130	180	27	)	)	PUNCT
iajs-4130	180	28	.	.	PUNCT
iajs-4130	181	1	proof	proof	NOUN
iajs-4130	181	2	.	.	PUNCT
iajs-4130	182	1	since	since	SCONJ
iajs-4130	182	2	,	,	PUNCT
iajs-4130	182	3	then	then	ADV
iajs-4130	182	4	(	(	PUNCT
iajs-4130	182	5	)	)	PUNCT
iajs-4130	182	6	(	(	PUNCT
iajs-4130	182	7	)	)	PUNCT
iajs-4130	182	8	(	(	PUNCT
iajs-4130	182	9	)	)	PUNCT
iajs-4130	182	10	.	.	PUNCT
iajs-4130	183	1	thus	thus	ADV
iajs-4130	183	2	,	,	PUNCT
iajs-4130	183	3	(	(	PUNCT
iajs-4130	183	4	)	)	PUNCT
iajs-4130	183	5	(	(	PUNCT
iajs-4130	183	6	)	)	PUNCT
iajs-4130	183	7	(	(	PUNCT
iajs-4130	183	8	)	)	PUNCT
iajs-4130	183	9	.	.	PUNCT
iajs-4130	184	1	the	the	DET
iajs-4130	184	2	set	set	NOUN
iajs-4130	184	3	of	of	ADP
iajs-4130	184	4	all	all	DET
iajs-4130	184	5	endo	endo	NOUN
iajs-4130	184	6	-	-	PUNCT
iajs-4130	184	7	r.b	r.b	NOUN
iajs-4130	184	8	submodules	submodule	NOUN
iajs-4130	184	9	of	of	ADP
iajs-4130	184	10	a	a	DET
iajs-4130	184	11	t	t	NOUN
iajs-4130	184	12	-	-	PUNCT
iajs-4130	184	13	module	module	NOUN
iajs-4130	184	14	is	be	AUX
iajs-4130	184	15	denoted	denote	VERB
iajs-4130	184	16	by	by	ADP
iajs-4130	184	17	(	(	PUNCT
iajs-4130	184	18	)	)	PUNCT
iajs-4130	184	19	.	.	PUNCT
iajs-4130	185	1	consider	consider	VERB
iajs-4130	185	2	the	the	DET
iajs-4130	185	3	notation	notation	NOUN
iajs-4130	185	4	:	:	PUNCT
iajs-4130	185	5	(	(	PUNCT
iajs-4130	185	6	)	)	PUNCT
iajs-4130	186	1	*	*	PUNCT
iajs-4130	186	2	|	|	ADV
iajs-4130	186	3	(	(	PUNCT
iajs-4130	186	4	)	)	PUNCT
iajs-4130	186	5	+	+	CCONJ
iajs-4130	186	6	and	and	CCONJ
iajs-4130	186	7	so	so	ADV
iajs-4130	186	8	(	(	PUNCT
iajs-4130	186	9	)	)	PUNCT
iajs-4130	186	10	⋂	⋂	PROPN
iajs-4130	186	11	(	(	PUNCT
iajs-4130	186	12	)	)	PUNCT
iajs-4130	186	13	.	.	PUNCT
iajs-4130	187	1	3.18	3.18	NUM
iajs-4130	187	2	.	.	PUNCT
iajs-4130	187	3	proposition	proposition	NOUN
iajs-4130	187	4	let	let	AUX
iajs-4130	187	5	be	be	AUX
iajs-4130	187	6	a	a	DET
iajs-4130	187	7	t	t	NOUN
iajs-4130	187	8	-	-	PUNCT
iajs-4130	187	9	module	module	NOUN
iajs-4130	187	10	,	,	PUNCT
iajs-4130	187	11	then	then	ADV
iajs-4130	187	12	the	the	DET
iajs-4130	187	13	following	follow	VERB
iajs-4130	187	14	holds	hold	VERB
iajs-4130	187	15	(	(	PUNCT
iajs-4130	187	16	1	1	NUM
iajs-4130	187	17	)	)	PUNCT
iajs-4130	187	18	(	(	PUNCT
iajs-4130	187	19	)	)	PUNCT
iajs-4130	187	20	(	(	PUNCT
iajs-4130	187	21	)	)	PUNCT
iajs-4130	187	22	and	and	CCONJ
iajs-4130	187	23	(	(	PUNCT
iajs-4130	187	24	)	)	PUNCT
iajs-4130	187	25	.	.	PUNCT
iajs-4130	188	1	(	(	PUNCT
iajs-4130	188	2	2	2	X
iajs-4130	188	3	)	)	PUNCT
iajs-4130	188	4	(	(	PUNCT
iajs-4130	188	5	)	)	PUNCT
iajs-4130	188	6	(	(	PUNCT
iajs-4130	188	7	)	)	PUNCT
iajs-4130	188	8	(	(	PUNCT
iajs-4130	188	9	)	)	PUNCT
iajs-4130	188	10	(	(	PUNCT
iajs-4130	188	11	3	3	X
iajs-4130	188	12	)	)	PUNCT
iajs-4130	188	13	(	(	PUNCT
iajs-4130	188	14	)	)	PUNCT
iajs-4130	188	15	(	(	PUNCT
iajs-4130	188	16	)	)	PUNCT
iajs-4130	188	17	(	(	PUNCT
iajs-4130	188	18	)	)	PUNCT
iajs-4130	188	19	for	for	ADP
iajs-4130	188	20	any	any	DET
iajs-4130	188	21	fully	fully	ADV
iajs-4130	188	22	invariant	invariant	ADJ
iajs-4130	188	23	submodule	submodule	NOUN
iajs-4130	188	24	n	n	CCONJ
iajs-4130	188	25	,	,	PUNCT
iajs-4130	188	26	l	l	NOUN
iajs-4130	188	27	of	of	ADP
iajs-4130	188	28	.	.	PUNCT
iajs-4130	189	1	proof	proof	NOUN
iajs-4130	189	2	.	.	PUNCT
iajs-4130	190	1	(	(	PUNCT
iajs-4130	190	2	1	1	X
iajs-4130	190	3	)	)	PUNCT
iajs-4130	190	4	and	and	CCONJ
iajs-4130	190	5	(	(	PUNCT
iajs-4130	190	6	2	2	X
iajs-4130	190	7	)	)	PUNCT
iajs-4130	190	8	are	be	AUX
iajs-4130	190	9	obvious	obvious	ADJ
iajs-4130	190	10	.	.	PUNCT
iajs-4130	191	1	(	(	PUNCT
iajs-4130	191	2	3	3	X
iajs-4130	191	3	)	)	PUNCT
iajs-4130	191	4	let	let	VERB
iajs-4130	191	5	n	n	PRON
iajs-4130	191	6	and	and	CCONJ
iajs-4130	191	7	l	l	NOUN
iajs-4130	191	8	are	be	AUX
iajs-4130	191	9	two	two	NUM
iajs-4130	191	10	fully	fully	ADV
iajs-4130	191	11	invariant	invariant	ADJ
iajs-4130	191	12	submodules	submodule	NOUN
iajs-4130	191	13	of	of	ADP
iajs-4130	191	14	,	,	PUNCT
iajs-4130	191	15	then	then	ADV
iajs-4130	191	16	by	by	ADP
iajs-4130	191	17	the	the	DET
iajs-4130	191	18	definition	definition	NOUN
iajs-4130	191	19	,	,	PUNCT
iajs-4130	191	20	we	we	PRON
iajs-4130	191	21	have	have	VERB
iajs-4130	191	22	(	(	PUNCT
iajs-4130	191	23	)	)	PUNCT
iajs-4130	192	1	*	*	PUNCT
iajs-4130	192	2	|	|	INTJ
iajs-4130	192	3	(	(	PUNCT
iajs-4130	192	4	)	)	PUNCT
iajs-4130	192	5	+	+	CCONJ
iajs-4130	192	6	(	(	PUNCT
iajs-4130	192	7	)	)	PUNCT
iajs-4130	192	8	*	*	PUNCT
iajs-4130	192	9	|	|	INTJ
iajs-4130	192	10	(	(	PUNCT
iajs-4130	192	11	)	)	PUNCT
iajs-4130	192	12	+	+	CCONJ
iajs-4130	192	13	(	(	PUNCT
iajs-4130	192	14	)	)	PUNCT
iajs-4130	192	15	(	(	PUNCT
iajs-4130	192	16	)	)	PUNCT
iajs-4130	192	17	*	*	PUNCT
iajs-4130	193	1	|	|	ADV
iajs-4130	193	2	(	(	PUNCT
iajs-4130	193	3	)	)	PUNCT
iajs-4130	193	4	+	+	CCONJ
iajs-4130	193	5	(	(	PUNCT
iajs-4130	193	6	)	)	PUNCT
iajs-4130	193	7	*	*	PUNCT
iajs-4130	193	8	|	|	ADV
iajs-4130	193	9	(	(	PUNCT
iajs-4130	193	10	)	)	PUNCT
iajs-4130	193	11	+	+	CCONJ
iajs-4130	193	12	now	now	ADV
iajs-4130	193	13	,	,	PUNCT
iajs-4130	193	14	take	take	VERB
iajs-4130	193	15	(	(	PUNCT
iajs-4130	193	16	)	)	PUNCT
iajs-4130	193	17	(	(	PUNCT
iajs-4130	193	18	)	)	PUNCT
iajs-4130	193	19	,	,	PUNCT
iajs-4130	193	20	then	then	ADV
iajs-4130	193	21	is	be	AUX
iajs-4130	193	22	an	an	DET
iajs-4130	193	23	endo	endo	NOUN
iajs-4130	193	24	-	-	PUNCT
iajs-4130	193	25	r.b	r.b	NOUN
iajs-4130	193	26	submodule	submodule	NOUN
iajs-4130	193	27	such	such	ADJ
iajs-4130	193	28	that	that	DET
iajs-4130	193	29	ihjpas	ihjpas	PROPN
iajs-4130	193	30	.	.	PUNCT
iajs-4130	194	1	2025,38(4	2025,38(4	X
iajs-4130	194	2	)	)	PUNCT
iajs-4130	194	3	406	406	NUM
iajs-4130	194	4	therefore	therefore	ADV
iajs-4130	194	5	,	,	PUNCT
iajs-4130	194	6	and	and	CCONJ
iajs-4130	194	7	and	and	CCONJ
iajs-4130	194	8	hence	hence	ADV
iajs-4130	194	9	(	(	PUNCT
iajs-4130	194	10	)	)	PUNCT
iajs-4130	194	11	.	.	PUNCT
iajs-4130	195	1	recall	recall	VERB
iajs-4130	195	2	the	the	DET
iajs-4130	195	3	radical	radical	NOUN
iajs-4130	195	4	of	of	ADP
iajs-4130	195	5	a	a	DET
iajs-4130	195	6	t	t	NOUN
iajs-4130	195	7	-	-	PUNCT
iajs-4130	195	8	module	module	NOUN
iajs-4130	195	9	denoted	denote	VERB
iajs-4130	195	10	(	(	PUNCT
iajs-4130	195	11	)	)	PUNCT
iajs-4130	195	12	and	and	CCONJ
iajs-4130	195	13	it	it	PRON
iajs-4130	195	14	is	be	AUX
iajs-4130	195	15	the	the	DET
iajs-4130	195	16	intersection	intersection	NOUN
iajs-4130	195	17	of	of	ADP
iajs-4130	195	18	all	all	DET
iajs-4130	195	19	maximal	maximal	ADJ
iajs-4130	195	20	submodules	submodule	NOUN
iajs-4130	195	21	of	of	ADP
iajs-4130	195	22	(	(	PUNCT
iajs-4130	195	23	30	30	NUM
iajs-4130	195	24	)	)	PUNCT
iajs-4130	195	25	.	.	PUNCT
iajs-4130	196	1	3.19	3.19	NUM
iajs-4130	196	2	.	.	PUNCT
iajs-4130	196	3	proposition	proposition	NOUN
iajs-4130	196	4	suppose	suppose	VERB
iajs-4130	196	5	that	that	SCONJ
iajs-4130	196	6	(	(	PUNCT
iajs-4130	196	7	)	)	PUNCT
iajs-4130	196	8	and	and	CCONJ
iajs-4130	196	9	let	let	VERB
iajs-4130	196	10	where	where	SCONJ
iajs-4130	196	11	k	k	PROPN
iajs-4130	196	12	is	be	AUX
iajs-4130	196	13	a	a	DET
iajs-4130	196	14	direct	direct	ADJ
iajs-4130	196	15	summand	summand	NOUN
iajs-4130	196	16	of	of	ADP
iajs-4130	196	17	.	.	PUNCT
iajs-4130	197	1	then	then	ADV
iajs-4130	197	2	(	(	PUNCT
iajs-4130	197	3	)	)	PUNCT
iajs-4130	197	4	(	(	PUNCT
iajs-4130	197	5	)	)	PUNCT
iajs-4130	197	6	if	if	SCONJ
iajs-4130	197	7	and	and	CCONJ
iajs-4130	197	8	only	only	ADV
iajs-4130	197	9	if	if	SCONJ
iajs-4130	197	10	k	k	PROPN
iajs-4130	197	11	=	=	NOUN
iajs-4130	197	12	l.	l.	NOUN
iajs-4130	197	13	proof	proof	NOUN
iajs-4130	197	14	.	.	PUNCT
iajs-4130	198	1	since	since	SCONJ
iajs-4130	198	2	k	k	PROPN
iajs-4130	198	3	is	be	AUX
iajs-4130	198	4	a	a	DET
iajs-4130	198	5	direct	direct	ADJ
iajs-4130	198	6	summand	summand	NOUN
iajs-4130	198	7	of	of	ADP
iajs-4130	198	8	,	,	PUNCT
iajs-4130	198	9	then	then	ADV
iajs-4130	198	10	there	there	PRON
iajs-4130	198	11	exists	exist	VERB
iajs-4130	198	12	a	a	DET
iajs-4130	198	13	submodule	submodule	NOUN
iajs-4130	198	14	of	of	ADP
iajs-4130	198	15	such	such	ADJ
iajs-4130	198	16	that	that	PRON
iajs-4130	198	17	.	.	PUNCT
iajs-4130	199	1	hence	hence	ADV
iajs-4130	199	2	,	,	PUNCT
iajs-4130	199	3	(	(	PUNCT
iajs-4130	199	4	)	)	PUNCT
iajs-4130	199	5	so	so	SCONJ
iajs-4130	199	6	that	that	SCONJ
iajs-4130	199	7	(	(	PUNCT
iajs-4130	199	8	)	)	PUNCT
iajs-4130	199	9	(	(	PUNCT
iajs-4130	199	10	)	)	PUNCT
iajs-4130	199	11	(	(	PUNCT
iajs-4130	199	12	)	)	PUNCT
iajs-4130	199	13	.	.	PUNCT
iajs-4130	200	1	therefore	therefore	ADV
iajs-4130	200	2	,	,	PUNCT
iajs-4130	200	3	(	(	PUNCT
iajs-4130	200	4	)	)	PUNCT
iajs-4130	200	5	and	and	CCONJ
iajs-4130	200	6	this	this	PRON
iajs-4130	200	7	can	can	AUX
iajs-4130	200	8	be	be	AUX
iajs-4130	200	9	written	write	VERB
iajs-4130	200	10	as	as	ADP
iajs-4130	200	11	(	(	PUNCT
iajs-4130	200	12	)	)	PUNCT
iajs-4130	200	13	(	(	PUNCT
iajs-4130	200	14	)	)	PUNCT
iajs-4130	200	15	implies	imply	VERB
iajs-4130	200	16	that	that	SCONJ
iajs-4130	200	17	since	since	SCONJ
iajs-4130	200	18	rad	rad	PROPN
iajs-4130	200	19	(	(	PUNCT
iajs-4130	200	20	)	)	PUNCT
iajs-4130	200	21	is	be	AUX
iajs-4130	200	22	an	an	DET
iajs-4130	200	23	essential	essential	ADJ
iajs-4130	200	24	submodule	submodule	NOUN
iajs-4130	200	25	of	of	ADP
iajs-4130	200	26	.	.	PUNCT
iajs-4130	201	1	thus	thus	ADV
iajs-4130	201	2	,	,	PUNCT
iajs-4130	201	3	k	k	X
iajs-4130	201	4	=	=	NOUN
iajs-4130	201	5	l.	l.	NOUN
iajs-4130	201	6	4	4	NUM
iajs-4130	201	7	.	.	PUNCT
iajs-4130	202	1	conclusion	conclusion	NOUN
iajs-4130	202	2	we	we	PRON
iajs-4130	202	3	discussed	discuss	VERB
iajs-4130	202	4	in	in	ADP
iajs-4130	202	5	this	this	DET
iajs-4130	202	6	paper	paper	NOUN
iajs-4130	202	7	the	the	DET
iajs-4130	202	8	formula	formula	NOUN
iajs-4130	202	9	of	of	ADP
iajs-4130	202	10	the	the	DET
iajs-4130	202	11	radical	radical	NOUN
iajs-4130	202	12	of	of	ADP
iajs-4130	202	13	an	an	DET
iajs-4130	202	14	endo	endo	NOUN
iajs-4130	202	15	-	-	PUNCT
iajs-4130	202	16	r.b	r.b	NOUN
iajs-4130	202	17	submodule	submodule	NOUN
iajs-4130	202	18	as	as	ADP
iajs-4130	202	19	a	a	DET
iajs-4130	202	20	new	new	ADJ
iajs-4130	202	21	type	type	NOUN
iajs-4130	202	22	and	and	CCONJ
iajs-4130	202	23	proved	prove	VERB
iajs-4130	202	24	that	that	SCONJ
iajs-4130	202	25	it	it	PRON
iajs-4130	202	26	is	be	AUX
iajs-4130	202	27	an	an	DET
iajs-4130	202	28	endo	endo	NOUN
iajs-4130	202	29	-	-	PUNCT
iajs-4130	202	30	r.b	r.b	NOUN
iajs-4130	202	31	submodule	submodule	NOUN
iajs-4130	202	32	of	of	ADP
iajs-4130	202	33	a	a	DET
iajs-4130	202	34	t	t	NOUN
iajs-4130	202	35	-	-	PUNCT
iajs-4130	202	36	module	module	NOUN
iajs-4130	202	37	.	.	PUNCT
iajs-4130	203	1	also	also	ADV
iajs-4130	203	2	,	,	PUNCT
iajs-4130	203	3	the	the	DET
iajs-4130	203	4	relationship	relationship	NOUN
iajs-4130	203	5	between	between	ADP
iajs-4130	203	6	prime	prime	ADJ
iajs-4130	203	7	and	and	CCONJ
iajs-4130	203	8	endo	endo	NOUN
iajs-4130	203	9	-	-	PUNCT
iajs-4130	203	10	r.b	r.b	NOUN
iajs-4130	203	11	submodules	submodule	NOUN
iajs-4130	203	12	helps	help	VERB
iajs-4130	203	13	us	we	PRON
iajs-4130	203	14	to	to	PART
iajs-4130	203	15	give	give	VERB
iajs-4130	203	16	many	many	ADJ
iajs-4130	203	17	properties	property	NOUN
iajs-4130	203	18	.	.	PUNCT
iajs-4130	204	1	in	in	ADP
iajs-4130	204	2	this	this	DET
iajs-4130	204	3	paper	paper	NOUN
iajs-4130	204	4	,	,	PUNCT
iajs-4130	204	5	we	we	PRON
iajs-4130	204	6	show	show	VERB
iajs-4130	204	7	that	that	SCONJ
iajs-4130	204	8	the	the	DET
iajs-4130	204	9	radical	radical	NOUN
iajs-4130	204	10	of	of	ADP
iajs-4130	204	11	the	the	DET
iajs-4130	204	12	submodule	submodule	NOUN
iajs-4130	204	13	n	n	PROPN
iajs-4130	204	14	and	and	CCONJ
iajs-4130	204	15	the	the	DET
iajs-4130	204	16	radical	radical	NOUN
iajs-4130	204	17	of	of	ADP
iajs-4130	204	18	an	an	DET
iajs-4130	204	19	endo	endo	NOUN
iajs-4130	204	20	-	-	PUNCT
iajs-4130	204	21	r.b	r.b	NOUN
iajs-4130	204	22	submodule	submodule	NOUN
iajs-4130	204	23	merge	merge	VERB
iajs-4130	204	24	under	under	ADP
iajs-4130	204	25	certain	certain	ADJ
iajs-4130	204	26	conditions	condition	NOUN
iajs-4130	204	27	that	that	PRON
iajs-4130	204	28	will	will	AUX
iajs-4130	204	29	be	be	AUX
iajs-4130	204	30	useful	useful	ADJ
iajs-4130	204	31	to	to	ADP
iajs-4130	204	32	other	other	ADJ
iajs-4130	204	33	researchers	researcher	NOUN
iajs-4130	204	34	in	in	ADP
iajs-4130	204	35	order	order	NOUN
iajs-4130	204	36	to	to	PART
iajs-4130	204	37	present	present	VERB
iajs-4130	204	38	other	other	ADJ
iajs-4130	204	39	results	result	NOUN
iajs-4130	204	40	.	.	PUNCT
iajs-4130	205	1	acknowledgment	acknowledgment	NOUN
iajs-4130	205	2	i	i	PRON
iajs-4130	205	3	would	would	AUX
iajs-4130	205	4	like	like	VERB
iajs-4130	205	5	to	to	PART
iajs-4130	205	6	thank	thank	VERB
iajs-4130	205	7	my	my	PRON
iajs-4130	205	8	supervisor	supervisor	NOUN
iajs-4130	205	9	professor	professor	PROPN
iajs-4130	205	10	dr	dr	PROPN
iajs-4130	205	11	.	.	PROPN
iajs-4130	205	12	buthyna	buthyna	PROPN
iajs-4130	205	13	.	.	PUNCT
iajs-4130	206	1	n.	n.	PROPN
iajs-4130	206	2	shihab	shihab	PROPN
iajs-4130	206	3	for	for	ADP
iajs-4130	206	4	her	her	PRON
iajs-4130	206	5	remarkable	remarkable	ADJ
iajs-4130	206	6	advice	advice	NOUN
iajs-4130	206	7	and	and	CCONJ
iajs-4130	206	8	support	support	NOUN
iajs-4130	206	9	that	that	PRON
iajs-4130	206	10	helps	help	VERB
iajs-4130	206	11	me	i	PRON
iajs-4130	206	12	write	write	VERB
iajs-4130	206	13	this	this	DET
iajs-4130	206	14	paper	paper	NOUN
iajs-4130	206	15	.	.	PUNCT
iajs-4130	207	1	conflict	conflict	NOUN
iajs-4130	207	2	of	of	ADP
iajs-4130	207	3	interest	interest	NOUN
iajs-4130	207	4	the	the	DET
iajs-4130	207	5	authors	author	NOUN
iajs-4130	207	6	declare	declare	VERB
iajs-4130	207	7	no	no	DET
iajs-4130	207	8	conflicts	conflict	NOUN
iajs-4130	207	9	of	of	ADP
iajs-4130	207	10	interest	interest	NOUN
iajs-4130	207	11	.	.	PUNCT
iajs-4130	208	1	funding	fund	VERB
iajs-4130	208	2	no	no	DET
iajs-4130	208	3	funding	funding	NOUN
iajs-4130	208	4	was	be	AUX
iajs-4130	208	5	received	receive	VERB
iajs-4130	208	6	for	for	ADP
iajs-4130	208	7	the	the	DET
iajs-4130	208	8	article	article	NOUN
iajs-4130	208	9	.	.	PUNCT
iajs-4130	209	1	references	reference	NOUN
iajs-4130	209	2	1	1	NUM
iajs-4130	209	3	.	.	PUNCT
iajs-4130	210	1	mahmood	mahmood	PROPN
iajs-4130	210	2	ls	ls	PROPN
iajs-4130	210	3	,	,	PUNCT
iajs-4130	210	4	al	al	PROPN
iajs-4130	210	5	-	-	PUNCT
iajs-4130	210	6	ani	ani	NOUN
iajs-4130	210	7	as	as	ADP
iajs-4130	210	8	.	.	PUNCT
iajs-4130	210	9	bounded	bound	VERB
iajs-4130	210	10	modules	module	NOUN
iajs-4130	210	11	.	.	PUNCT
iajs-4130	211	1	ibn	ibn	PROPN
iajs-4130	211	2	al	al	PROPN
iajs-4130	211	3	-	-	PUNCT
iajs-4130	211	4	haitham	haitham	PROPN
iajs-4130	211	5	j	j	PROPN
iajs-4130	211	6	pure	pure	PROPN
iajs-4130	211	7	appl	appl	PROPN
iajs-4130	211	8	sci	sci	PROPN
iajs-4130	211	9	.	.	PUNCT
iajs-4130	212	1	2017;19(3):75	2017;19(3):75	NUM
iajs-4130	212	2	–	–	PUNCT
iajs-4130	212	3	91	91	NUM
iajs-4130	212	4	.	.	PUNCT
iajs-4130	212	5	https://jih.uobaghdad.edu.iq/index.php/j/article/view/1650	https://jih.uobaghdad.edu.iq/index.php/j/article/view/1650	NOUN
iajs-4130	212	6	.	.	PUNCT
iajs-4130	213	1	2	2	X
iajs-4130	213	2	.	.	X
iajs-4130	213	3	abdul	abdul	PROPN
iajs-4130	213	4	-	-	PUNCT
iajs-4130	213	5	al	al	PROPN
iajs-4130	213	6	-	-	PUNCT
iajs-4130	213	7	kalik	kalik	PROPN
iajs-4130	213	8	aj	aj	PROPN
iajs-4130	213	9	.	.	PROPN
iajs-4130	213	10	semi	semi	ADJ
iajs-4130	213	11	–	–	PUNCT
iajs-4130	213	12	bounded	bounded	ADJ
iajs-4130	213	13	modules	module	NOUN
iajs-4130	213	14	.	.	PUNCT
iajs-4130	214	1	baghdad	baghdad	PROPN
iajs-4130	214	2	sci	sci	PROPN
iajs-4130	214	3	j.	j.	PROPN
iajs-4130	214	4	2012;9(4):720–7	2012;9(4):720–7	PROPN
iajs-4130	214	5	.	.	PUNCT
iajs-4130	215	1	https://doi.org/10.21123/bsj.2012.9.4.720-727	https://doi.org/10.21123/bsj.2012.9.4.720-727	PROPN
iajs-4130	215	2	3	3	NUM
iajs-4130	215	3	.	.	PUNCT
iajs-4130	215	4	shihab	shihab	PROPN
iajs-4130	215	5	bn	bn	PROPN
iajs-4130	215	6	,	,	PUNCT
iajs-4130	215	7	murad	murad	PROPN
iajs-4130	215	8	ms	ms	PROPN
iajs-4130	215	9	.	.	PROPN
iajs-4130	215	10	investigation	investigation	NOUN
iajs-4130	215	11	of	of	ADP
iajs-4130	215	12	bounded	bounded	ADJ
iajs-4130	215	13	modules	module	NOUN
iajs-4130	215	14	and	and	CCONJ
iajs-4130	215	15	some	some	DET
iajs-4130	215	16	related	related	ADJ
iajs-4130	215	17	concepts	concept	NOUN
iajs-4130	215	18	.	.	PUNCT
iajs-4130	216	1	babylonian	babylonian	PROPN
iajs-4130	216	2	j	j	PROPN
iajs-4130	216	3	math	math	NOUN
iajs-4130	216	4	.	.	PUNCT
iajs-4130	217	1	2024;2024:102–11	2024;2024:102–11	NUM
iajs-4130	217	2	.	.	PUNCT
iajs-4130	218	1	https://doi.org/10.58496/bjm/2024/013	https://doi.org/10.58496/bjm/2024/013	PROPN
iajs-4130	218	2	4	4	X
iajs-4130	218	3	.	.	PUNCT
iajs-4130	218	4	jamali	jamali	PROPN
iajs-4130	218	5	m	m	PROPN
iajs-4130	218	6	,	,	PUNCT
iajs-4130	218	7	jahani	jahani	NOUN
iajs-4130	218	8	-	-	PUNCT
iajs-4130	218	9	nezhad	nezhad	VERB
iajs-4130	218	10	r.	r.	PROPN
iajs-4130	218	11	on	on	ADP
iajs-4130	218	12	classical	classical	ADJ
iajs-4130	218	13	weakly	weakly	ADJ
iajs-4130	218	14	prime	prime	ADJ
iajs-4130	218	15	submodules	submodule	NOUN
iajs-4130	218	16	.	.	PUNCT
iajs-4130	219	1	facta	facta	PROPN
iajs-4130	219	2	univ	univ	PROPN
iajs-4130	219	3	ser	ser	PROPN
iajs-4130	219	4	math	math	PROPN
iajs-4130	219	5	informatics	informatics	PROPN
iajs-4130	219	6	.	.	PUNCT
iajs-4130	220	1	2022;37(1):17–30	2022;37(1):17–30	NUM
iajs-4130	220	2	.	.	PUNCT
iajs-4130	221	1	https://doi.org/10.22190/fumi200906003j	https://doi.org/10.22190/fumi200906003j	PRON
iajs-4130	221	2	5	5	PROPN
iajs-4130	221	3	.	.	PROPN
iajs-4130	221	4	ali	ali	PROPN
iajs-4130	221	5	wa	wa	PROPN
iajs-4130	221	6	,	,	PUNCT
iajs-4130	221	7	mothafar	mothafar	ADV
iajs-4130	221	8	ns	ns	ADJ
iajs-4130	221	9	.	.	PROPN
iajs-4130	221	10	on	on	ADP
iajs-4130	221	11	quasi	quasi	ADJ
iajs-4130	221	12	-	-	ADJ
iajs-4130	221	13	small	small	ADJ
iajs-4130	221	14	prime	prime	ADJ
iajs-4130	221	15	submodules	submodule	NOUN
iajs-4130	221	16	.	.	PUNCT
iajs-4130	222	1	iraqi	iraqi	ADJ
iajs-4130	222	2	j	j	PROPN
iajs-4130	222	3	sci.2022;63(4):1692–9	sci.2022;63(4):1692–9	X
iajs-4130	222	4	.	.	PUNCT
iajs-4130	223	1	https://doi.org/10.24996/ijs.2022.63.4.26	https://doi.org/10.24996/ijs.2022.63.4.26	PROPN
iajs-4130	223	2	6	6	NUM
iajs-4130	223	3	.	.	PUNCT
iajs-4130	224	1	jasem	jasem	PROPN
iajs-4130	224	2	fd	fd	PROPN
iajs-4130	224	3	,	,	PUNCT
iajs-4130	224	4	elewi	elewi	VERB
iajs-4130	224	5	aa	aa	NOUN
iajs-4130	224	6	.	.	PUNCT
iajs-4130	225	1	2	2	NUM
iajs-4130	225	2	-	-	PUNCT
iajs-4130	225	3	prime	prime	ADJ
iajs-4130	225	4	submodules	submodule	NOUN
iajs-4130	225	5	of	of	ADP
iajs-4130	225	6	modules	module	NOUN
iajs-4130	225	7	.	.	PUNCT
iajs-4130	226	1	iraqi	iraqi	PROPN
iajs-4130	226	2	j	j	PROPN
iajs-4130	226	3	sci	sci	PROPN
iajs-4130	226	4	.	.	PROPN
iajs-4130	226	5	2022	2022	NUM
iajs-4130	226	6	aug	aug	PROPN
iajs-4130	226	7	31;63(8):3605–11	31;63(8):3605–11	PROPN
iajs-4130	226	8	.	.	PUNCT
iajs-4130	227	1	https://doi.org/10.24996/ijs.2022.63.8.34	https://doi.org/10.24996/ijs.2022.63.8.34	PROPN
iajs-4130	227	2	7	7	NUM
iajs-4130	227	3	.	.	PUNCT
iajs-4130	228	1	sabah	sabah	PROPN
iajs-4130	228	2	sadiaq	sadiaq	PROPN
iajs-4130	228	3	a	a	DET
iajs-4130	228	4	,	,	PUNCT
iajs-4130	228	5	mohammadali	mohammadali	ADJ
iajs-4130	228	6	k	k	NOUN
iajs-4130	228	7	,	,	PUNCT
iajs-4130	228	8	weakly	weakly	ADJ
iajs-4130	228	9	h.	h.	NOUN
iajs-4130	228	10	nearly	nearly	ADV
iajs-4130	228	11	prime	prime	ADJ
iajs-4130	228	12	submodules	submodule	NOUN
iajs-4130	228	13	.	.	PUNCT
iajs-4130	229	1	ibn	ibn	PROPN
iajs-4130	229	2	alhaitham	alhaitham	PROPN
iajs-4130	229	3	j	j	PROPN
iajs-4130	229	4	pure	pure	PROPN
iajs-4130	229	5	appl	appl	PROPN
iajs-4130	229	6	sci	sci	PROPN
iajs-4130	229	7	.	.	PUNCT
iajs-4130	230	1	2021;34(1):38–46	2021;34(1):38–46	NUM
iajs-4130	230	2	.	.	PUNCT
iajs-4130	230	3	https://doi.org/10.30526/34.1.2556	https://doi.org/10.30526/34.1.2556	SYM
iajs-4130	230	4	https://jih.uobaghdad.edu.iq/index.php/j/article/view/1650	https://jih.uobaghdad.edu.iq/index.php/j/article/view/1650	NOUN
iajs-4130	230	5	https://doi.org/10.21123/bsj.2012.9.4.720-727	https://doi.org/10.21123/bsj.2012.9.4.720-727	VERB
iajs-4130	230	6	https://doi.org/10.58496/bjm/2024/013	https://doi.org/10.58496/bjm/2024/013	X
iajs-4130	230	7	https://doi.org/10.22190/fumi200906003j	https://doi.org/10.22190/fumi200906003j	VERB
iajs-4130	230	8	https://doi.org/10.24996/ijs.2022.63.4.26	https://doi.org/10.24996/ijs.2022.63.4.26	ADJ
iajs-4130	230	9	https://doi.org/10.24996/ijs.2022.63.8.34	https://doi.org/10.24996/ijs.2022.63.8.34	NOUN
iajs-4130	230	10	https://doi.org/10.30526/34.1.2556	https://doi.org/10.30526/34.1.2556	NUM
iajs-4130	230	11	ihjpas	ihjpas	PROPN
iajs-4130	230	12	.	.	PUNCT
iajs-4130	231	1	2025,38(4	2025,38(4	NOUN
iajs-4130	231	2	)	)	PUNCT
iajs-4130	231	3	407	407	NUM
iajs-4130	231	4	8	8	NUM
iajs-4130	231	5	.	.	PUNCT
iajs-4130	232	1	majeed	majeed	PROPN
iajs-4130	232	2	rn	rn	PROPN
iajs-4130	232	3	,	,	PUNCT
iajs-4130	232	4	ahmed	ahme	VERB
iajs-4130	232	5	g	g	PROPN
iajs-4130	232	6	,	,	PUNCT
iajs-4130	232	7	fiadh	fiadh	PROPN
iajs-4130	232	8	ms	ms	PROPN
iajs-4130	232	9	,	,	PUNCT
iajs-4130	232	10	hadi	hadi	PROPN
iajs-4130	232	11	a.	a.	PROPN
iajs-4130	232	12	rad	rad	PROPN
iajs-4130	232	13	-	-	PROPN
iajs-4130	232	14	quasi	quasi	ADJ
iajs-4130	232	15	-	-	ADJ
iajs-4130	232	16	prime	prime	ADJ
iajs-4130	232	17	submodules	submodule	NOUN
iajs-4130	232	18	,	,	PUNCT
iajs-4130	232	19	lemya	lemya	PROPN
iajs-4130	232	20	abd	abd	PROPN
iajs-4130	232	21	.	.	PUNCT
iajs-4130	233	1	iraqi	iraqi	PROPN
iajs-4130	233	2	j	j	PROPN
iajs-4130	233	3	comput	comput	PROPN
iajs-4130	233	4	sci	sci	PROPN
iajs-4130	233	5	math	math	PROPN
iajs-4130	233	6	.	.	PUNCT
iajs-4130	234	1	2024;5(2):21–5	2024;5(2):21–5	PROPN
iajs-4130	234	2	.	.	PUNCT
iajs-4130	235	1	https://doi.org/10.52866/2788-7421.1222	https://doi.org/10.52866/2788-7421.1222	PROPN
iajs-4130	235	2	9	9	NUM
iajs-4130	235	3	.	.	PUNCT
iajs-4130	236	1	al	al	PROPN
iajs-4130	236	2	-	-	PUNCT
iajs-4130	236	3	ragab	ragab	PROPN
iajs-4130	236	4	om	om	PROPN
iajs-4130	236	5	,	,	PUNCT
iajs-4130	236	6	al	al	PROPN
iajs-4130	236	7	-	-	PUNCT
iajs-4130	236	8	mothafar	mothafar	ADV
iajs-4130	236	9	ns	ns	ADJ
iajs-4130	236	10	.	.	PUNCT
iajs-4130	236	11	quasi	quasi	ADJ
iajs-4130	236	12	-	-	ADJ
iajs-4130	236	13	radical	radical	ADJ
iajs-4130	236	14	semiprime	semiprime	NOUN
iajs-4130	236	15	submodules	submodule	NOUN
iajs-4130	236	16	.	.	PUNCT
iajs-4130	237	1	iraqi	iraqi	PROPN
iajs-4130	237	2	j	j	PROPN
iajs-4130	237	3	sci	sci	PROPN
iajs-4130	237	4	.	.	PROPN
iajs-4130	237	5	2022;63(5):2148–54	2022;63(5):2148–54	PROPN
iajs-4130	237	6	.	.	PUNCT
iajs-4130	238	1	https://doi.org/10.24996/ijs.2022.63.5.28	https://doi.org/10.24996/ijs.2022.63.5.28	PROPN
iajs-4130	238	2	10	10	NUM
iajs-4130	238	3	.	.	PUNCT
iajs-4130	239	1	ansari	ansari	ADJ
iajs-4130	239	2	-	-	PUNCT
iajs-4130	239	3	toroghy	toroghy	ADJ
iajs-4130	239	4	h	h	NOUN
iajs-4130	239	5	,	,	PUNCT
iajs-4130	239	6	farshadifar	farshadifar	ADJ
iajs-4130	239	7	f	f	NOUN
iajs-4130	239	8	,	,	PUNCT
iajs-4130	239	9	mahboobi	mahboobi	NOUN
iajs-4130	239	10	-	-	NOUN
iajs-4130	239	11	abkenar	abkenar	ADJ
iajs-4130	239	12	f.	f.	NOUN
iajs-4130	240	1	the	the	DET
iajs-4130	240	2	secondary	secondary	ADJ
iajs-4130	240	3	radicals	radical	NOUN
iajs-4130	240	4	of	of	ADP
iajs-4130	240	5	submodules	submodule	NOUN
iajs-4130	240	6	.	.	PUNCT
iajs-4130	241	1	algebr	algebr	PROPN
iajs-4130	241	2	struct	struct	VERB
iajs-4130	241	3	their	their	PRON
iajs-4130	241	4	appl	appl	NOUN
iajs-4130	241	5	.	.	PUNCT
iajs-4130	242	1	2020;7(2):1–13	2020;7(2):1–13	NUM
iajs-4130	242	2	.	.	PUNCT
iajs-4130	243	1	https://doi.org/10.22034/as.2020.1786	https://doi.org/10.22034/as.2020.1786	NOUN
iajs-4130	243	2	11	11	NUM
iajs-4130	243	3	.	.	PUNCT
iajs-4130	244	1	ahmadi	ahmadi	PROPN
iajs-4130	244	2	m	m	PROPN
iajs-4130	244	3	,	,	PUNCT
iajs-4130	244	4	moghaderi	moghaderi	PROPN
iajs-4130	244	5	j.	j.	PROPN
iajs-4130	244	6	n	n	CCONJ
iajs-4130	244	7	-	-	PUNCT
iajs-4130	244	8	submodules	submodules	NOUN
iajs-4130	244	9	.	.	PUNCT
iajs-4130	245	1	iran	iran	PROPN
iajs-4130	245	2	j	j	PROPN
iajs-4130	245	3	math	math	PROPN
iajs-4130	245	4	sci	sci	PROPN
iajs-4130	245	5	informatics	informatics	PROPN
iajs-4130	245	6	.	.	PUNCT
iajs-4130	246	1	2022;17(1):177–90	2022;17(1):177–90	NUM
iajs-4130	246	2	.	.	PUNCT
iajs-4130	247	1	https://doi.org/10.52547/ijmsi.17.1.177	https://doi.org/10.52547/ijmsi.17.1.177	PRON
iajs-4130	247	2	12	12	NUM
iajs-4130	247	3	.	.	PUNCT
iajs-4130	248	1	anderson	anderson	PROPN
iajs-4130	248	2	dd	dd	PROPN
iajs-4130	248	3	,	,	PUNCT
iajs-4130	248	4	arabaci	arabaci	PROPN
iajs-4130	248	5	t	t	PROPN
iajs-4130	248	6	,	,	PUNCT
iajs-4130	248	7	tekir	tekir	PROPN
iajs-4130	248	8	ü	ü	NOUN
iajs-4130	248	9	,	,	PUNCT
iajs-4130	248	10	koç	koç	PROPN
iajs-4130	248	11	s.	s.	PROPN
iajs-4130	248	12	on	on	ADP
iajs-4130	248	13	s	s	NOUN
iajs-4130	248	14	-	-	PUNCT
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iajs-4130	248	16	modules	module	NOUN
iajs-4130	248	17	.	.	PUNCT
iajs-4130	249	1	commun	commun	PROPN
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iajs-4130	249	3	.	.	PUNCT
iajs-4130	250	1	2020;48(8):3398–407	2020;48(8):3398–407	X
iajs-4130	250	2	.	.	PUNCT
iajs-4130	251	1	https://doi.org/10.1080/00927872.2020.1737873	https://doi.org/10.1080/00927872.2020.1737873	PROPN
iajs-4130	251	2	13	13	NUM
iajs-4130	251	3	.	.	PUNCT
iajs-4130	252	1	abdullah	abdullah	PROPN
iajs-4130	252	2	o.	o.	PROPN
iajs-4130	252	3	,	,	PUNCT
iajs-4130	252	4	mohammadali	mohammadali	PROPN
iajs-4130	252	5	hk	hk	PROPN
iajs-4130	252	6	.	.	PUNCT
iajs-4130	252	7	extend	extend	VERB
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iajs-4130	252	10	quasi-2	quasi-2	ADJ
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iajs-4130	252	13	submodules	submodule	NOUN
iajs-4130	252	14	(	(	PUNCT
iajs-4130	252	15	i	i	NOUN
iajs-4130	252	16	)	)	PUNCT
iajs-4130	252	17	.	.	PUNCT
iajs-4130	253	1	ibn	ibn	PROPN
iajs-4130	253	2	al	al	PROPN
iajs-4130	253	3	-	-	PUNCT
iajs-4130	253	4	haitham	haitham	PROPN
iajs-4130	253	5	j	j	PROPN
iajs-4130	253	6	pure	pure	PROPN
iajs-4130	253	7	appl	appl	PROPN
iajs-4130	253	8	sci	sci	PROPN
iajs-4130	253	9	.	.	PUNCT
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iajs-4130	254	2	.	.	PUNCT
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iajs-4130	255	3	.	.	PUNCT
iajs-4130	256	1	tolooei	tolooei	PROPN
iajs-4130	256	2	y.	y.	PROPN
iajs-4130	256	3	multiplication	multiplication	NOUN
iajs-4130	256	4	modules	module	NOUN
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iajs-4130	256	6	are	be	AUX
iajs-4130	256	7	finitely	finitely	ADV
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iajs-4130	256	9	.	.	PUNCT
iajs-4130	257	1	j	j	PROPN
iajs-4130	257	2	algebr	algebr	PROPN
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iajs-4130	257	4	.	.	PUNCT
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iajs-4130	258	2	.	.	PUNCT
iajs-4130	259	1	https://doi.org/10.22044/jas.2019.8699.1421	https://doi.org/10.22044/jas.2019.8699.1421	PROPN
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iajs-4130	259	3	.	.	PUNCT
iajs-4130	260	1	obaid	obaid	PROPN
iajs-4130	260	2	is	be	AUX
iajs-4130	260	3	,	,	PUNCT
iajs-4130	260	4	hussain	hussain	PROPN
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iajs-4130	260	6	,	,	PUNCT
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iajs-4130	260	8	dj	dj	NOUN
iajs-4130	260	9	.	.	PUNCT
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iajs-4130	260	14	quasi	quasi	ADJ
iajs-4130	260	15	-	-	ADJ
iajs-4130	260	16	dedekind	dedekind	ADJ
iajs-4130	260	17	modules	module	NOUN
iajs-4130	260	18	with	with	ADP
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iajs-4130	260	20	and	and	CCONJ
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iajs-4130	260	22	modules	module	NOUN
iajs-4130	260	23	.	.	PUNCT
iajs-4130	261	1	iraqi	iraqi	PROPN
iajs-4130	261	2	j	j	PROPN
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iajs-4130	261	4	.	.	PUNCT
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iajs-4130	261	6	.	.	PUNCT
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iajs-4130	262	2	16	16	NUM
iajs-4130	262	3	.	.	PUNCT
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iajs-4130	263	6	.	.	PUNCT
iajs-4130	264	1	we	we	PRON
iajs-4130	264	2	-	-	PUNCT
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iajs-4130	264	4	submodules	submodule	NOUN
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iajs-4130	264	7	-	-	PUNCT
iajs-4130	264	8	semi	semi	ADJ
iajs-4130	264	9	-	-	ADJ
iajs-4130	264	10	prime	prime	ADJ
iajs-4130	264	11	submodules	submodule	NOUN
iajs-4130	264	12	.	.	PUNCT
iajs-4130	265	1	ibn	ibn	PROPN
iajs-4130	265	2	alhaitham	alhaitham	PROPN
iajs-4130	265	3	j	j	PROPN
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iajs-4130	265	5	appl	appl	PROPN
iajs-4130	265	6	sci	sci	PROPN
iajs-4130	265	7	.	.	PUNCT
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iajs-4130	266	2	.	.	PUNCT
iajs-4130	267	1	https://doi.org/10.30526/31.3.2000	https://doi.org/10.30526/31.3.2000	PROPN
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iajs-4130	267	3	.	.	PUNCT
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iajs-4130	269	4	.	.	PUNCT
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iajs-4130	270	2	.	.	PUNCT
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iajs-4130	271	2	18	18	NUM
iajs-4130	271	3	.	.	PUNCT
iajs-4130	272	1	shihab	shihab	PROPN
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iajs-4130	272	3	.	.	PUNCT
iajs-4130	272	4	scalar	scalar	ADJ
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iajs-4130	272	6	modules	module	NOUN
iajs-4130	272	7	.	.	PUNCT
iajs-4130	273	1	msc	msc	PROPN
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iajs-4130	274	7	al	al	PROPN
iajs-4130	274	8	-	-	PUNCT
iajs-4130	274	9	haitham	haitham	PROPN
iajs-4130	274	10	,	,	PUNCT
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iajs-4130	274	14	;	;	PUNCT
iajs-4130	274	15	2004	2004	NUM
iajs-4130	274	16	.	.	PUNCT
iajs-4130	275	1	19	19	NUM
iajs-4130	275	2	.	.	X
iajs-4130	275	3	ajeel	ajeel	PROPN
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iajs-4130	275	5	,	,	PUNCT
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iajs-4130	275	7	hk	hk	PROPN
iajs-4130	275	8	.	.	PUNCT
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iajs-4130	276	2	prime	prime	ADJ
iajs-4130	276	3	submodules	submodule	NOUN
iajs-4130	276	4	and	and	CCONJ
iajs-4130	276	5	some	some	DET
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iajs-4130	276	8	.	.	PUNCT
iajs-4130	277	1	ibn	ibn	PROPN
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iajs-4130	277	3	-	-	PUNCT
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iajs-4130	277	7	appl	appl	PROPN
iajs-4130	277	8	sci	sci	PROPN
iajs-4130	277	9	.	.	PUNCT
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iajs-4130	278	2	.	.	PUNCT
iajs-4130	279	1	https://doi.org/10.30526/32.2.2148	https://doi.org/10.30526/32.2.2148	NOUN
iajs-4130	279	2	.	.	PROPN
iajs-4130	279	3	20	20	NUM
iajs-4130	279	4	.	.	PUNCT
iajs-4130	279	5	atani	atani	PROPN
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iajs-4130	279	7	,	,	PUNCT
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iajs-4130	279	9	f.	f.	PROPN
iajs-4130	279	10	on	on	ADP
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iajs-4130	279	12	prime	prime	ADJ
iajs-4130	279	13	submodules	submodule	NOUN
iajs-4130	279	14	.	.	PUNCT
iajs-4130	280	1	tamkang	tamkang	PROPN
iajs-4130	281	1	j	j	PROPN
iajs-4130	281	2	math	math	NOUN
iajs-4130	281	3	.	.	PUNCT
iajs-4130	282	1	2007;38(3):247–52	2007;38(3):247–52	X
iajs-4130	282	2	.	.	PUNCT
iajs-4130	283	1	https://doi.org/10.5556/j.tkjm.38.2007.77	https://doi.org/10.5556/j.tkjm.38.2007.77	PROPN
iajs-4130	283	2	.	.	PROPN
iajs-4130	283	3	21	21	NUM
iajs-4130	283	4	.	.	PUNCT
iajs-4130	284	1	hadi	hadi	PROPN
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iajs-4130	284	5	,	,	PUNCT
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iajs-4130	284	7	ag	ag	PROPN
iajs-4130	284	8	.	.	PROPN
iajs-4130	285	1	strongly	strongly	PROPN
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iajs-4130	285	3	-	-	ADJ
iajs-4130	285	4	prime	prime	ADJ
iajs-4130	285	5	submodules	submodule	NOUN
iajs-4130	285	6	.	.	PUNCT
iajs-4130	286	1	al	al	PROPN
iajs-4130	286	2	-	-	PUNCT
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iajs-4130	286	4	j	j	PROPN
iajs-4130	286	5	sci	sci	PROPN
iajs-4130	286	6	.	.	PUNCT
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iajs-4130	287	2	–	–	PUNCT
iajs-4130	287	3	10	10	NUM
iajs-4130	287	4	.	.	PUNCT
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iajs-4130	288	2	.	.	PUNCT
iajs-4130	289	1	22	22	NUM
iajs-4130	289	2	.	.	PUNCT
iajs-4130	290	1	saleh	saleh	PROPN
iajs-4130	290	2	k	k	PROPN
iajs-4130	290	3	,	,	PUNCT
iajs-4130	290	4	astuti	astuti	ADP
iajs-4130	290	5	p	p	X
iajs-4130	290	6	,	,	PUNCT
iajs-4130	290	7	muchtadi	muchtadi	NOUN
iajs-4130	290	8	-	-	PUNCT
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iajs-4130	290	10	i.	i.	NOUN
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iajs-4130	290	18	of	of	ADP
iajs-4130	290	19	a	a	DET
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iajs-4130	290	22	over	over	ADP
iajs-4130	290	23	a	a	DET
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iajs-4130	290	25	ideal	ideal	ADJ
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iajs-4130	290	27	.	.	PUNCT
iajs-4130	291	1	in	in	ADP
iajs-4130	291	2	:	:	PUNCT
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iajs-4130	291	6	.	.	PUNCT
iajs-4130	292	1	aip	aip	PROPN
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iajs-4130	292	4	2017	2017	NUM
iajs-4130	292	5	.	.	PUNCT
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iajs-4130	293	2	.	.	PUNCT
iajs-4130	294	1	23	23	NUM
iajs-4130	294	2	.	.	PUNCT
iajs-4130	295	1	jabbar	jabbar	PROPN
iajs-4130	295	2	ak	ak	PROPN
iajs-4130	295	3	,	,	PUNCT
iajs-4130	295	4	hamaali	hamaali	ADJ
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iajs-4130	295	6	,	,	PUNCT
iajs-4130	295	7	abdul	abdul	PROPN
iajs-4130	295	8	-	-	PUNCT
iajs-4130	295	9	jabbar	jabbar	PROPN
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iajs-4130	295	11	.	.	PUNCT
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iajs-4130	296	4	-	-	NOUN
iajs-4130	296	5	prime	prime	ADJ
iajs-4130	296	6	and	and	CCONJ
iajs-4130	296	7	locally	locally	ADV
iajs-4130	296	8	s	s	NOUN
iajs-4130	296	9	-	-	ADJ
iajs-4130	296	10	primary	primary	ADJ
iajs-4130	296	11	submodules	submodule	NOUN
iajs-4130	296	12	.	.	PUNCT
iajs-4130	297	1	j	j	PROPN
iajs-4130	297	2	univ	univ	PROPN
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iajs-4130	297	5	appl	appl	PROPN
iajs-4130	297	6	sci	sci	PROPN
iajs-4130	297	7	.	.	PUNCT
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iajs-4130	298	2	.	.	PUNCT
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iajs-4130	299	2	.	.	PUNCT
iajs-4130	300	1	24	24	NUM
iajs-4130	300	2	.	.	X
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iajs-4130	301	2	rl	rl	PROPN
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iajs-4130	301	4	,	,	PUNCT
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iajs-4130	301	6	me	i	PRON
iajs-4130	301	7	.	.	PUNCT
iajs-4130	302	1	prime	prime	ADJ
iajs-4130	302	2	submodules	submodule	NOUN
iajs-4130	302	3	.	.	PUNCT
iajs-4130	303	1	commun	commun	PROPN
iajs-4130	303	2	algebr	algebr	PROPN
iajs-4130	303	3	.	.	PUNCT
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iajs-4130	304	2	.	.	PUNCT
iajs-4130	305	1	https://doi.org/10.1080/00927879208824432	https://doi.org/10.1080/00927879208824432	NUM
iajs-4130	306	1	25	25	NUM
iajs-4130	306	2	.	.	PUNCT
iajs-4130	307	1	azizi	azizi	PROPN
iajs-4130	307	2	a.	a.	PROPN
iajs-4130	307	3	radical	radical	ADJ
iajs-4130	307	4	formula	formula	NOUN
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iajs-4130	307	6	prime	prime	ADJ
iajs-4130	307	7	submodules	submodule	NOUN
iajs-4130	307	8	.	.	PUNCT
iajs-4130	308	1	j	j	PROPN
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iajs-4130	308	3	.	.	PUNCT
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iajs-4130	309	2	.	.	PUNCT
iajs-4130	310	1	https://doi.org/10.1016/j.jalgebra.2006.07.006	https://doi.org/10.1016/j.jalgebra.2006.07.006	PROPN
iajs-4130	310	2	.	.	PUNCT
iajs-4130	311	1	26	26	NUM
iajs-4130	311	2	.	.	PUNCT
iajs-4130	312	1	azizi	azizi	PROPN
iajs-4130	312	2	a.	a.	PROPN
iajs-4130	312	3	radical	radical	ADJ
iajs-4130	312	4	formula	formula	NOUN
iajs-4130	312	5	and	and	CCONJ
iajs-4130	312	6	weakly	weakly	ADJ
iajs-4130	312	7	prime	prime	ADJ
iajs-4130	312	8	submodules	submodule	NOUN
iajs-4130	312	9	.	.	PUNCT
iajs-4130	313	1	glas	glas	PROPN
iajs-4130	313	2	math	math	PROPN
iajs-4130	313	3	j.	j.	PROPN
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iajs-4130	313	5	.	.	PUNCT
iajs-4130	314	1	https://doi.org/10.1017/s0017089509005072	https://doi.org/10.1017/s0017089509005072	NUM
iajs-4130	314	2	.	.	PUNCT
iajs-4130	315	1	27	27	NUM
iajs-4130	315	2	.	.	PUNCT
iajs-4130	316	1	kumar	kumar	PROPN
iajs-4130	316	2	s	s	PROPN
iajs-4130	316	3	,	,	PUNCT
iajs-4130	316	4	gupta	gupta	PROPN
iajs-4130	316	5	aj	aj	PROPN
iajs-4130	316	6	.	.	PUNCT
iajs-4130	316	7	purely	purely	ADV
iajs-4130	316	8	extending	extend	VERB
iajs-4130	316	9	modules	module	NOUN
iajs-4130	316	10	and	and	CCONJ
iajs-4130	316	11	their	their	PRON
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iajs-4130	316	13	.	.	PUNCT
iajs-4130	317	1	kyungpook	kyungpook	PROPN
iajs-4130	317	2	math	math	PROPN
iajs-4130	317	3	j.	j.	PROPN
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iajs-4130	317	5	.	.	PUNCT
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iajs-4130	318	2	.	.	PROPN
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iajs-4130	318	4	.	.	PUNCT
iajs-4130	319	1	yahyaa	yahyaa	PROPN
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iajs-4130	319	3	,	,	PUNCT
iajs-4130	319	4	yassen	yassen	PROPN
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iajs-4130	319	6	.	.	PROPN
iajs-4130	319	7	closed	close	VERB
iajs-4130	319	8	-	-	PUNCT
iajs-4130	319	9	small	small	ADJ
iajs-4130	319	10	submodules	submodule	NOUN
iajs-4130	319	11	and	and	CCONJ
iajs-4130	319	12	closed	closed	ADJ
iajs-4130	319	13	-	-	PUNCT
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iajs-4130	319	15	modules	module	NOUN
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iajs-4130	322	8	.	.	PUNCT
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iajs-4130	327	18	.	.	PUNCT
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iajs-4130	328	5	.	.	PUNCT
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iajs-4130	329	2	.	.	PUNCT
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iajs-4130	330	12	https://doi.org/10.30526/32.2.2148	https://doi.org/10.30526/32.2.2148	PUNCT
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