id	sid	tid	token	lemma	pos
iajs-3139	1	1	ihjpas	ihjpas	PROPN
iajs-3139	1	2	.	.	PUNCT
iajs-3139	2	1	36	36	NUM
iajs-3139	2	2	(	(	PUNCT
iajs-3139	2	3	4	4	NUM
iajs-3139	2	4	)	)	PUNCT
iajs-3139	2	5	2023	2023	NUM
iajs-3139	2	6	359	359	NUM
iajs-3139	2	7	this	this	DET
iajs-3139	2	8	work	work	NOUN
iajs-3139	2	9	is	be	AUX
iajs-3139	2	10	licensed	license	VERB
iajs-3139	2	11	under	under	ADP
iajs-3139	2	12	a	a	DET
iajs-3139	2	13	creative	creative	ADJ
iajs-3139	2	14	commons	common	NOUN
iajs-3139	2	15	attribution	attribution	NOUN
iajs-3139	2	16	4.0	4.0	NUM
iajs-3139	2	17	international	international	ADJ
iajs-3139	2	18	license	license	NOUN
iajs-3139	2	19	*	*	PUNCT
iajs-3139	2	20	corresponding	correspond	VERB
iajs-3139	2	21	author	author	NOUN
iajs-3139	2	22	:	:	PUNCT
iajs-3139	2	23	khawlahahmmed@gmail.com	khawlahahmmed@gmail.com	X
iajs-3139	2	24	abstract	abstract	ADJ
iajs-3139	2	25	in	in	ADP
iajs-3139	2	26	this	this	DET
iajs-3139	2	27	work	work	NOUN
iajs-3139	2	28	we	we	PRON
iajs-3139	2	29	discuss	discuss	VERB
iajs-3139	2	30	the	the	DET
iajs-3139	2	31	concept	concept	NOUN
iajs-3139	2	32	of	of	ADP
iajs-3139	2	33	pure	pure	ADJ
iajs-3139	2	34	-	-	PUNCT
iajs-3139	2	35	maximal	maximal	ADJ
iajs-3139	2	36	denoted	denote	VERB
iajs-3139	2	37	by	by	ADP
iajs-3139	2	38	(	(	PUNCT
iajs-3139	2	39	pr	pr	ADV
iajs-3139	2	40	-	-	PUNCT
iajs-3139	2	41	maximal	maximal	ADJ
iajs-3139	2	42	)	)	PUNCT
iajs-3139	2	43	submodules	submodule	NOUN
iajs-3139	2	44	as	as	ADP
iajs-3139	2	45	a	a	DET
iajs-3139	2	46	generalization	generalization	NOUN
iajs-3139	2	47	to	to	ADP
iajs-3139	2	48	the	the	DET
iajs-3139	2	49	type	type	NOUN
iajs-3139	2	50	of	of	ADP
iajs-3139	2	51	rmaximal	rmaximal	ADJ
iajs-3139	2	52	submodule	submodule	NOUN
iajs-3139	2	53	,	,	PUNCT
iajs-3139	2	54	where	where	SCONJ
iajs-3139	2	55	a	a	DET
iajs-3139	2	56	proper	proper	ADJ
iajs-3139	2	57	submodule	submodule	NOUN
iajs-3139	2	58	of	of	ADP
iajs-3139	2	59	an	an	DET
iajs-3139	2	60	rmodule	rmodule	NOUN
iajs-3139	2	61	is	be	AUX
iajs-3139	2	62	called	call	VERB
iajs-3139	2	63	prmaximal	prmaximal	ADJ
iajs-3139	2	64	if	if	SCONJ
iajs-3139	2	65	<	<	X
iajs-3139	2	66	𝐻	𝐻	PROPN
iajs-3139	2	67	≤	≤	NOUN
iajs-3139	2	68	𝑊	𝑊	PROPN
iajs-3139	2	69	,	,	PUNCT
iajs-3139	2	70	for	for	ADP
iajs-3139	2	71	any	any	DET
iajs-3139	2	72	submodule	submodule	NOUN
iajs-3139	2	73	of	of	ADP
iajs-3139	2	74	w	w	PROPN
iajs-3139	2	75	is	be	AUX
iajs-3139	2	76	a	a	DET
iajs-3139	2	77	pure	pure	ADJ
iajs-3139	2	78	submodule	submodule	NOUN
iajs-3139	2	79	of	of	ADP
iajs-3139	2	80	w	w	PROPN
iajs-3139	2	81	,	,	PUNCT
iajs-3139	2	82	we	we	PRON
iajs-3139	2	83	offer	offer	VERB
iajs-3139	2	84	some	some	DET
iajs-3139	2	85	properties	property	NOUN
iajs-3139	2	86	of	of	ADP
iajs-3139	2	87	a	a	DET
iajs-3139	2	88	pr	pr	NOUN
iajs-3139	2	89	-	-	PUNCT
iajs-3139	2	90	maximal	maximal	ADJ
iajs-3139	2	91	submodules	submodule	NOUN
iajs-3139	2	92	,	,	PUNCT
iajs-3139	2	93	and	and	CCONJ
iajs-3139	2	94	we	we	PRON
iajs-3139	2	95	give	give	VERB
iajs-3139	2	96	definition	definition	NOUN
iajs-3139	2	97	of	of	ADP
iajs-3139	2	98	the	the	DET
iajs-3139	2	99	concept	concept	NOUN
iajs-3139	2	100	,	,	PUNCT
iajs-3139	2	101	near	near	ADV
iajs-3139	2	102	-	-	PUNCT
iajs-3139	2	103	maximal	maximal	ADJ
iajs-3139	2	104	,	,	PUNCT
iajs-3139	2	105	a	a	DET
iajs-3139	2	106	proper	proper	ADJ
iajs-3139	2	107	submodule	submodule	NOUN
iajs-3139	2	108	of	of	ADP
iajs-3139	2	109	an	an	DET
iajs-3139	2	110	r	r	NOUN
iajs-3139	2	111	-	-	PUNCT
iajs-3139	2	112	module	module	NOUN
iajs-3139	2	113	is	be	AUX
iajs-3139	2	114	named	name	VERB
iajs-3139	2	115	near	near	ADP
iajs-3139	2	116	(	(	PUNCT
iajs-3139	2	117	n	n	CCONJ
iajs-3139	2	118	-	-	PUNCT
iajs-3139	2	119	maximal	maximal	ADJ
iajs-3139	2	120	)	)	PUNCT
iajs-3139	2	121	whensoever	whensoever	NOUN
iajs-3139	2	122	is	be	AUX
iajs-3139	2	123	pure	pure	ADJ
iajs-3139	2	124	submodule	submodule	NOUN
iajs-3139	2	125	of	of	ADP
iajs-3139	2	126	such	such	ADJ
iajs-3139	2	127	that	that	PRON
iajs-3139	3	1	then	then	ADV
iajs-3139	3	2	k=	k=	INTJ
iajs-3139	3	3	.al	.al	PUNCT
iajs-3139	4	1	so	so	ADV
iajs-3139	4	2	we	we	PRON
iajs-3139	4	3	offer	offer	VERB
iajs-3139	4	4	the	the	DET
iajs-3139	4	5	concept	concept	NOUN
iajs-3139	4	6	pr	pr	NOUN
iajs-3139	4	7	-	-	PUNCT
iajs-3139	4	8	module	module	NOUN
iajs-3139	4	9	,	,	PUNCT
iajs-3139	4	10	an	an	DET
iajs-3139	4	11	r	r	NOUN
iajs-3139	4	12	-	-	PUNCT
iajs-3139	4	13	module	module	NOUN
iajs-3139	4	14	w	w	NOUN
iajs-3139	4	15	is	be	AUX
iajs-3139	4	16	named	name	VERB
iajs-3139	4	17	pr	pr	NOUN
iajs-3139	4	18	-	-	PUNCT
iajs-3139	4	19	module	module	NOUN
iajs-3139	4	20	,	,	PUNCT
iajs-3139	4	21	if	if	SCONJ
iajs-3139	4	22	every	every	DET
iajs-3139	4	23	proper	proper	ADJ
iajs-3139	4	24	submodule	submodule	NOUN
iajs-3139	4	25	of	of	ADP
iajs-3139	4	26	is	be	AUX
iajs-3139	4	27	pr	pr	ADV
iajs-3139	4	28	-	-	PUNCT
iajs-3139	4	29	maximal	maximal	ADJ
iajs-3139	4	30	.	.	PUNCT
iajs-3139	5	1	a	a	DET
iajs-3139	5	2	ring	ring	NOUN
iajs-3139	5	3	is	be	AUX
iajs-3139	5	4	named	name	VERB
iajs-3139	5	5	pr	pr	NOUN
iajs-3139	5	6	-	-	PUNCT
iajs-3139	5	7	ring	ring	NOUN
iajs-3139	5	8	if	if	SCONJ
iajs-3139	5	9	whole	whole	ADJ
iajs-3139	5	10	proper	proper	ADJ
iajs-3139	5	11	ideal	ideal	NOUN
iajs-3139	5	12	of	of	ADP
iajs-3139	5	13	is	be	AUX
iajs-3139	5	14	a	a	DET
iajs-3139	5	15	pr	pr	NOUN
iajs-3139	5	16	-	-	PUNCT
iajs-3139	5	17	maximal	maximal	ADJ
iajs-3139	5	18	ideal	ideal	NOUN
iajs-3139	5	19	,	,	PUNCT
iajs-3139	5	20	we	we	PRON
iajs-3139	5	21	offer	offer	VERB
iajs-3139	5	22	the	the	DET
iajs-3139	5	23	concept	concept	NOUN
iajs-3139	5	24	pure	pure	ADJ
iajs-3139	5	25	local	local	ADJ
iajs-3139	5	26	(	(	PUNCT
iajs-3139	5	27	pr	pr	NOUN
iajs-3139	5	28	-	-	ADJ
iajs-3139	5	29	local	local	ADJ
iajs-3139	5	30	)	)	PUNCT
iajs-3139	5	31	module	module	NOUN
iajs-3139	5	32	an	an	DET
iajs-3139	5	33	rmodule	rmodule	NOUN
iajs-3139	5	34	is	be	AUX
iajs-3139	5	35	named	name	VERB
iajs-3139	5	36	pure	pure	ADJ
iajs-3139	5	37	local	local	ADJ
iajs-3139	5	38	(	(	PUNCT
iajs-3139	5	39	pr	pr	NOUN
iajs-3139	5	40	-	-	ADJ
iajs-3139	5	41	local	local	ADJ
iajs-3139	5	42	)	)	PUNCT
iajs-3139	5	43	module	module	NOUN
iajs-3139	5	44	.	.	PUNCT
iajs-3139	6	1	if	if	SCONJ
iajs-3139	6	2	it	it	PRON
iajs-3139	6	3	has	have	VERB
iajs-3139	6	4	only	only	ADV
iajs-3139	6	5	a	a	DET
iajs-3139	6	6	pr	pr	NOUN
iajs-3139	6	7	-	-	PUNCT
iajs-3139	6	8	maximal	maximal	ADJ
iajs-3139	6	9	submodule	submodule	NOUN
iajs-3139	6	10	which	which	PRON
iajs-3139	6	11	includes	include	VERB
iajs-3139	6	12	all	all	DET
iajs-3139	6	13	proper	proper	ADJ
iajs-3139	6	14	submodule	submodule	NOUN
iajs-3139	6	15	of	of	ADP
iajs-3139	6	16	.	.	PUNCT
iajs-3139	7	1	a	a	DET
iajs-3139	7	2	ring	ring	NOUN
iajs-3139	7	3	is	be	AUX
iajs-3139	7	4	named	name	VERB
iajs-3139	7	5	pure	pure	ADJ
iajs-3139	7	6	local	local	ADJ
iajs-3139	7	7	(	(	PUNCT
iajs-3139	7	8	pr	pr	NOUN
iajs-3139	7	9	-	-	ADJ
iajs-3139	7	10	local	local	ADJ
iajs-3139	7	11	)	)	PUNCT
iajs-3139	7	12	ring	ring	NOUN
iajs-3139	7	13	,	,	PUNCT
iajs-3139	7	14	if	if	SCONJ
iajs-3139	7	15	is	be	AUX
iajs-3139	7	16	a	a	DET
iajs-3139	7	17	prlocal	prlocal	ADJ
iajs-3139	7	18	r	r	NOUN
iajs-3139	7	19	-	-	PUNCT
iajs-3139	7	20	module	module	NOUN
iajs-3139	7	21	.	.	PUNCT
iajs-3139	8	1	we	we	PRON
iajs-3139	8	2	give	give	VERB
iajs-3139	8	3	some	some	DET
iajs-3139	8	4	relatio	relatio	NOUN
iajs-3139	8	5	among	among	ADP
iajs-3139	8	6	pr	pr	NOUN
iajs-3139	8	7	-	-	PUNCT
iajs-3139	8	8	maximal	maximal	ADJ
iajs-3139	8	9	submodules	submodule	NOUN
iajs-3139	8	10	and	and	CCONJ
iajs-3139	8	11	others	other	NOUN
iajs-3139	8	12	related	relate	VERB
iajs-3139	8	13	concept	concept	NOUN
iajs-3139	8	14	.	.	PUNCT
iajs-3139	9	1	keywords	keyword	NOUN
iajs-3139	9	2	:	:	PUNCT
iajs-3139	9	3	,	,	PUNCT
iajs-3139	9	4	r	r	NOUN
iajs-3139	9	5	-	-	PUNCT
iajs-3139	9	6	submodule	submodule	NOUN
iajs-3139	9	7	,	,	PUNCT
iajs-3139	9	8	pr	pr	NOUN
iajs-3139	9	9	-	-	PUNCT
iajs-3139	9	10	module	module	NOUN
iajs-3139	9	11	,	,	PUNCT
iajs-3139	9	12	pr	pr	NOUN
iajs-3139	9	13	-	-	PUNCT
iajs-3139	9	14	maximal	maximal	ADJ
iajs-3139	9	15	,	,	PUNCT
iajs-3139	9	16	pr	pr	NOUN
iajs-3139	9	17	-	-	ADJ
iajs-3139	9	18	local	local	ADJ
iajs-3139	9	19	,	,	PUNCT
iajs-3139	9	20	n	n	CCONJ
iajs-3139	9	21	-	-	PUNCT
iajs-3139	9	22	maximal	maximal	ADJ
iajs-3139	9	23	.	.	PUNCT
iajs-3139	10	1	introduction	introduction	NOUN
iajs-3139	10	2	in	in	ADP
iajs-3139	10	3	this	this	DET
iajs-3139	10	4	work	work	NOUN
iajs-3139	10	5	is	be	AUX
iajs-3139	10	6	commutative	commutative	ADJ
iajs-3139	10	7	ring	ring	NOUN
iajs-3139	10	8	with	with	ADP
iajs-3139	10	9	identity	identity	NOUN
iajs-3139	10	10	,	,	PUNCT
iajs-3139	10	11	and	and	CCONJ
iajs-3139	10	12	all	all	DET
iajs-3139	10	13	r	r	NOUN
iajs-3139	10	14	-	-	PUNCT
iajs-3139	10	15	modules	module	NOUN
iajs-3139	10	16	are	be	AUX
iajs-3139	10	17	left	leave	VERB
iajs-3139	10	18	until	until	ADP
iajs-3139	10	19	.	.	PUNCT
iajs-3139	11	1	a	a	DET
iajs-3139	11	2	proper	proper	ADJ
iajs-3139	11	3	submodule	submodule	NOUN
iajs-3139	11	4	of	of	ADP
iajs-3139	11	5	an	an	DET
iajs-3139	11	6	r	r	NOUN
iajs-3139	11	7	-	-	PUNCT
iajs-3139	11	8	module	module	NOUN
iajs-3139	11	9	is	be	AUX
iajs-3139	11	10	named	name	VERB
iajs-3139	11	11	a	a	DET
iajs-3139	11	12	pure	pure	ADJ
iajs-3139	11	13	submodule	submodule	NOUN
iajs-3139	11	14	“	"	PUNCT
iajs-3139	11	15	if	if	SCONJ
iajs-3139	11	16	for	for	ADP
iajs-3139	11	17	every	every	DET
iajs-3139	11	18	ideal	ideal	NOUN
iajs-3139	11	19	of	of	ADP
iajs-3139	11	20	,	,	PUNCT
iajs-3139	11	21	[	[	X
iajs-3139	11	22	1	1	NUM
iajs-3139	11	23	]	]	PUNCT
iajs-3139	11	24	.	.	PUNCT
iajs-3139	12	1	a	a	DET
iajs-3139	12	2	proper	proper	ADJ
iajs-3139	12	3	submodule	submodule	NOUN
iajs-3139	12	4	of	of	ADP
iajs-3139	12	5	an	an	DET
iajs-3139	12	6	r	r	NOUN
iajs-3139	12	7	-	-	PUNCT
iajs-3139	12	8	module	module	NOUN
iajs-3139	12	9	is	be	AUX
iajs-3139	12	10	named	name	VERB
iajs-3139	12	11	maximal	maximal	ADJ
iajs-3139	12	12	in	in	ADP
iajs-3139	12	13	[	[	X
iajs-3139	12	14	3	3	X
iajs-3139	12	15	]	]	PUNCT
iajs-3139	12	16	“	"	PUNCT
iajs-3139	12	17	if	if	SCONJ
iajs-3139	12	18	whenever	whenever	SCONJ
iajs-3139	12	19	is	be	AUX
iajs-3139	12	20	a	a	DET
iajs-3139	12	21	submodule	submodule	NOUN
iajs-3139	12	22	of	of	ADP
iajs-3139	12	23	r	r	NOUN
iajs-3139	12	24	-	-	PUNCT
iajs-3139	12	25	module	module	NOUN
iajs-3139	12	26	with	with	ADP
iajs-3139	12	27	implies	implie	NOUN
iajs-3139	12	28	.	.	PUNCT
iajs-3139	13	1	abduljaleel	abduljaleel	NOUN
iajs-3139	13	2	and	and	CCONJ
iajs-3139	13	3	yaseen	yaseen	PROPN
iajs-3139	13	4	in	in	ADP
iajs-3139	13	5	[	[	X
iajs-3139	13	6	2	2	NUM
iajs-3139	13	7	]	]	PUNCT
iajs-3139	13	8	offer	offer	VERB
iajs-3139	13	9	the	the	DET
iajs-3139	13	10	concept	concept	NOUN
iajs-3139	13	11	of	of	ADP
iajs-3139	13	12	large	large	ADJ
iajs-3139	13	13	maximal	maximal	ADJ
iajs-3139	13	14	submodules	submodule	NOUN
iajs-3139	13	15	as	as	ADP
iajs-3139	13	16	a	a	DET
iajs-3139	13	17	generalization	generalization	NOUN
iajs-3139	13	18	of	of	ADP
iajs-3139	13	19	the	the	DET
iajs-3139	13	20	concept	concept	NOUN
iajs-3139	13	21	maximal	maximal	ADJ
iajs-3139	13	22	submodules	submodule	NOUN
iajs-3139	13	23	,	,	PUNCT
iajs-3139	13	24	“	"	PUNCT
iajs-3139	13	25	where	where	SCONJ
iajs-3139	13	26	a	a	DET
iajs-3139	13	27	proper	proper	ADJ
iajs-3139	13	28	submodule	submodule	NOUN
iajs-3139	13	29	of	of	ADP
iajs-3139	13	30	an	an	DET
iajs-3139	13	31	rmodule	rmodule	NOUN
iajs-3139	13	32	is	be	AUX
iajs-3139	13	33	named	name	VERB
iajs-3139	13	34	large	large	ADJ
iajs-3139	13	35	-	-	PUNCT
iajs-3139	13	36	maximal(l	maximal(l	NOUN
iajs-3139	13	37	-	-	PUNCT
iajs-3139	13	38	maximal	maximal	ADJ
iajs-3139	13	39	)	)	PUNCT
iajs-3139	13	40	if	if	SCONJ
iajs-3139	13	41	implies	imply	VERB
iajs-3139	13	42	is	be	AUX
iajs-3139	13	43	an	an	DET
iajs-3139	13	44	essential	essential	ADJ
iajs-3139	13	45	submodule	submodule	NOUN
iajs-3139	13	46	of	of	ADP
iajs-3139	13	47	,	,	PUNCT
iajs-3139	13	48	where	where	SCONJ
iajs-3139	13	49	a	a	DET
iajs-3139	13	50	submodule	submodule	NOUN
iajs-3139	13	51	of	of	ADP
iajs-3139	13	52	r	r	NOUN
iajs-3139	13	53	-	-	PUNCT
iajs-3139	13	54	module	module	NOUN
iajs-3139	13	55	is	be	AUX
iajs-3139	13	56	named	name	VERB
iajs-3139	13	57	essential	essential	ADJ
iajs-3139	13	58	,	,	PUNCT
iajs-3139	13	59	if	if	SCONJ
iajs-3139	13	60	for	for	ADP
iajs-3139	13	61	every	every	DET
iajs-3139	13	62	non	non	ADJ
iajs-3139	13	63	-	-	ADJ
iajs-3139	13	64	zero	zero	ADJ
iajs-3139	13	65	doi.org/10.30526/36.4.3139	doi.org/10.30526/36.4.3139	NOUN
iajs-3139	13	66	article	article	NOUN
iajs-3139	13	67	history	history	NOUN
iajs-3139	13	68	:	:	PUNCT
iajs-3139	13	69	received	receive	VERB
iajs-3139	13	70	11	11	NUM
iajs-3139	13	71	december	december	PROPN
iajs-3139	13	72	2022	2022	NUM
iajs-3139	13	73	,	,	PUNCT
iajs-3139	13	74	accepted	accept	VERB
iajs-3139	13	75	28	28	NUM
iajs-3139	13	76	februray	februray	ADJ
iajs-3139	13	77	2023	2023	NUM
iajs-3139	13	78	,	,	PUNCT
iajs-3139	13	79	published	publish	VERB
iajs-3139	13	80	in	in	ADP
iajs-3139	13	81	october	october	PROPN
iajs-3139	13	82	2023	2023	NUM
iajs-3139	13	83	ibn	ibn	PROPN
iajs-3139	13	84	al	al	PROPN
iajs-3139	13	85	-	-	PUNCT
iajs-3139	13	86	haitham	haitham	PROPN
iajs-3139	13	87	journal	journal	PROPN
iajs-3139	13	88	for	for	ADP
iajs-3139	13	89	pure	pure	ADJ
iajs-3139	13	90	and	and	CCONJ
iajs-3139	13	91	applied	applied	ADJ
iajs-3139	13	92	sciences	sciences	PROPN
iajs-3139	13	93	journal	journal	PROPN
iajs-3139	13	94	homepage	homepage	NOUN
iajs-3139	13	95	:	:	PUNCT
iajs-3139	13	96	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	VERB
iajs-3139	13	97	pure	pure	ADJ
iajs-3139	13	98	maximal	maximal	ADJ
iajs-3139	13	99	submodules	submodule	NOUN
iajs-3139	13	100	and	and	CCONJ
iajs-3139	13	101	related	related	ADJ
iajs-3139	13	102	concepts	concept	NOUN
iajs-3139	13	103	khawla	khawla	PROPN
iajs-3139	13	104	ahmed	ahmed	PROPN
iajs-3139	13	105	*	*	PROPN
iajs-3139	13	106	department	department	PROPN
iajs-3139	13	107	of	of	ADP
iajs-3139	13	108	mathematics	mathematics	PROPN
iajs-3139	13	109	,	,	PUNCT
iajs-3139	13	110	college	college	NOUN
iajs-3139	13	111	of	of	ADP
iajs-3139	13	112	science	science	NOUN
iajs-3139	13	113	,	,	PUNCT
iajs-3139	13	114	university	university	NOUN
iajs-3139	13	115	of	of	ADP
iajs-3139	13	116	baghdad	baghdad	PROPN
iajs-3139	13	117	,	,	PUNCT
iajs-3139	13	118	baghdad	baghdad	PROPN
iajs-3139	13	119	,	,	PUNCT
iajs-3139	13	120	iraq	iraq	PROPN
iajs-3139	13	121	.	.	PUNCT
iajs-3139	14	1	nuhad	nuhad	PROPN
iajs-3139	14	2	s.	s.	PROPN
iajs-3139	14	3	al	al	PROPN
iajs-3139	14	4	.	.	PROPN
iajs-3139	15	1	mothafar	mothafar	PROPN
iajs-3139	15	2	department	department	PROPN
iajs-3139	15	3	of	of	ADP
iajs-3139	15	4	mathematics	mathematics	PROPN
iajs-3139	15	5	,	,	PUNCT
iajs-3139	15	6	college	college	NOUN
iajs-3139	15	7	of	of	ADP
iajs-3139	15	8	science	science	NOUN
iajs-3139	15	9	,	,	PUNCT
iajs-3139	15	10	university	university	NOUN
iajs-3139	15	11	of	of	ADP
iajs-3139	15	12	baghdad	baghdad	PROPN
iajs-3139	15	13	,	,	PUNCT
iajs-3139	15	14	baghdad	baghdad	PROPN
iajs-3139	15	15	,	,	PUNCT
iajs-3139	15	16	iraq	iraq	PROPN
iajs-3139	15	17	.	.	PUNCT
iajs-3139	16	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3139	16	2	mailto:khawlahahmmed@gmail.com	mailto:khawlahahmmed@gmail.com	PROPN
iajs-3139	16	3	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-3139	16	4	ihjpas	ihjpa	VERB
iajs-3139	16	5	.	.	PUNCT
iajs-3139	17	1	36	36	NUM
iajs-3139	17	2	(	(	PUNCT
iajs-3139	17	3	4	4	NUM
iajs-3139	17	4	)	)	PUNCT
iajs-3139	17	5	2023	2023	NUM
iajs-3139	17	6	360	360	NUM
iajs-3139	17	7	submodule	submodule	NOUN
iajs-3139	17	8	of	of	ADP
iajs-3139	17	9	,	,	PUNCT
iajs-3139	17	10	[	[	X
iajs-3139	17	11	3	3	NUM
iajs-3139	17	12	]	]	PUNCT
iajs-3139	17	13	.	.	PUNCT
iajs-3139	18	1	many	many	ADJ
iajs-3139	18	2	authors	author	NOUN
iajs-3139	18	3	studies	study	VERB
iajs-3139	18	4	module	module	NOUN
iajs-3139	18	5	and	and	CCONJ
iajs-3139	18	6	submodule	submodule	NOUN
iajs-3139	18	7	for	for	ADP
iajs-3139	18	8	example	example	NOUN
iajs-3139	18	9	see	see	VERB
iajs-3139	18	10	[	[	X
iajs-3139	18	11	4	4	X
iajs-3139	18	12	]	]	PUNCT
iajs-3139	18	13	and	and	CCONJ
iajs-3139	18	14	[	[	X
iajs-3139	18	15	5	5	NUM
iajs-3139	18	16	]	]	PUNCT
iajs-3139	18	17	.	.	PUNCT
iajs-3139	19	1	in	in	ADP
iajs-3139	19	2	[	[	X
iajs-3139	19	3	11	11	NUM
iajs-3139	19	4	]	]	SYM
iajs-3139	19	5	b.h.al	b.h.al	ADJ
iajs-3139	19	6	-	-	PUNCT
iajs-3139	19	7	bahrani	bahrani	NOUN
iajs-3139	19	8	generalization	generalization	NOUN
iajs-3139	19	9	of	of	ADP
iajs-3139	19	10	the	the	DET
iajs-3139	19	11	type	type	NOUN
iajs-3139	19	12	of	of	ADP
iajs-3139	19	13	a	a	DET
iajs-3139	19	14	purely	purely	ADV
iajs-3139	19	15	extending	extend	VERB
iajs-3139	19	16	modules	module	NOUN
iajs-3139	19	17	,	,	PUNCT
iajs-3139	19	18	defined	define	VERB
iajs-3139	19	19	using	use	VERB
iajs-3139	19	20	y	y	NOUN
iajs-3139	19	21	-	-	PUNCT
iajs-3139	19	22	closed	close	VERB
iajs-3139	19	23	submodules	submodule	NOUN
iajs-3139	19	24	,	,	PUNCT
iajs-3139	19	25	in	in	ADP
iajs-3139	19	26	,	,	PUNCT
iajs-3139	19	27	this	this	DET
iajs-3139	19	28	discuss	discuss	NOUN
iajs-3139	19	29	,	,	PUNCT
iajs-3139	19	30	we	we	PRON
iajs-3139	19	31	introduce	introduce	VERB
iajs-3139	19	32	,	,	PUNCT
iajs-3139	19	33	the	the	DET
iajs-3139	19	34	concept	concept	NOUN
iajs-3139	19	35	of	of	ADP
iajs-3139	19	36	pure	pure	ADJ
iajs-3139	19	37	-	-	PUNCT
iajs-3139	19	38	maximal	maximal	ADJ
iajs-3139	19	39	(	(	PUNCT
iajs-3139	19	40	pr	pr	NOUN
iajs-3139	19	41	-	-	PUNCT
iajs-3139	19	42	maximal	maximal	ADJ
iajs-3139	19	43	)	)	PUNCT
iajs-3139	19	44	submodules	submodule	NOUN
iajs-3139	19	45	as	as	ADP
iajs-3139	19	46	a	a	DET
iajs-3139	19	47	generalization	generalization	NOUN
iajs-3139	19	48	of	of	ADP
iajs-3139	19	49	maximal	maximal	ADJ
iajs-3139	19	50	submodule	submodule	NOUN
iajs-3139	19	51	,	,	PUNCT
iajs-3139	19	52	where	where	SCONJ
iajs-3139	19	53	a	a	DET
iajs-3139	19	54	proper	proper	ADJ
iajs-3139	19	55	submodule	submodule	NOUN
iajs-3139	19	56	of	of	ADP
iajs-3139	19	57	an	an	DET
iajs-3139	19	58	r	r	NOUN
iajs-3139	19	59	-	-	PUNCT
iajs-3139	19	60	module	module	NOUN
iajs-3139	19	61	is	be	AUX
iajs-3139	19	62	named	name	VERB
iajs-3139	19	63	pr	pr	ADV
iajs-3139	19	64	-	-	PUNCT
iajs-3139	19	65	maximal	maximal	ADJ
iajs-3139	19	66	,	,	PUNCT
iajs-3139	19	67	if	if	SCONJ
iajs-3139	19	68	,	,	PUNCT
iajs-3139	19	69	for	for	ADP
iajs-3139	19	70	any	any	DET
iajs-3139	19	71	submodule	submodule	NOUN
iajs-3139	19	72	of	of	ADP
iajs-3139	19	73	,	,	PUNCT
iajs-3139	19	74	implies	imply	VERB
iajs-3139	19	75	is	be	AUX
iajs-3139	19	76	a	a	DET
iajs-3139	19	77	pure	pure	ADJ
iajs-3139	19	78	a	a	DET
iajs-3139	19	79	submodule	submodule	NOUN
iajs-3139	19	80	of	of	ADP
iajs-3139	19	81	,	,	PUNCT
iajs-3139	19	82	in	in	ADP
iajs-3139	19	83	section	section	NOUN
iajs-3139	19	84	two	two	NUM
iajs-3139	19	85	we	we	PRON
iajs-3139	19	86	give	give	VERB
iajs-3139	19	87	several	several	ADJ
iajs-3139	19	88	properties	property	NOUN
iajs-3139	19	89	of	of	ADP
iajs-3139	19	90	this	this	DET
iajs-3139	19	91	type	type	NOUN
iajs-3139	19	92	of	of	ADP
iajs-3139	19	93	submodules	submodule	NOUN
iajs-3139	19	94	as	as	SCONJ
iajs-3139	19	95	every	every	DET
iajs-3139	19	96	multiplication	multiplication	NOUN
iajs-3139	19	97	module	module	NOUN
iajs-3139	19	98	contains	contain	VERB
iajs-3139	19	99	a	a	DET
iajs-3139	19	100	pr	pr	NOUN
iajs-3139	19	101	-	-	PUNCT
iajs-3139	19	102	maximal	maximal	ADJ
iajs-3139	19	103	submodule	submodule	NOUN
iajs-3139	19	104	.	.	PUNCT
iajs-3139	20	1	also	also	ADV
iajs-3139	20	2	,	,	PUNCT
iajs-3139	20	3	if	if	SCONJ
iajs-3139	20	4	n	n	CCONJ
iajs-3139	20	5	,	,	PUNCT
iajs-3139	20	6	k	k	PROPN
iajs-3139	20	7	are	be	AUX
iajs-3139	20	8	non	non	ADJ
iajs-3139	20	9	-	-	ADJ
iajs-3139	20	10	zero	zero	NUM
iajs-3139	20	11	submodule	submodule	NOUN
iajs-3139	20	12	of	of	ADP
iajs-3139	20	13	such	such	ADJ
iajs-3139	20	14	that	that	SCONJ
iajs-3139	20	15	𝑁	𝑁	PROPN
iajs-3139	20	16	≤	≤	ADJ
iajs-3139	20	17	𝐾	𝐾	PROPN
iajs-3139	20	18	if	if	SCONJ
iajs-3139	20	19	n	n	NOUN
iajs-3139	20	20	is	be	AUX
iajs-3139	20	21	pr	pr	NOUN
iajs-3139	20	22	-	-	ADJ
iajs-3139	20	23	maximal	maximal	ADJ
iajs-3139	20	24	in	in	ADP
iajs-3139	20	25	w	w	PROPN
iajs-3139	20	26	then	then	ADV
iajs-3139	20	27	k	k	PROPN
iajs-3139	20	28	is	be	AUX
iajs-3139	20	29	pr	pr	ADV
iajs-3139	20	30	-	-	ADJ
iajs-3139	20	31	maximal	maximal	ADJ
iajs-3139	20	32	in	in	ADP
iajs-3139	20	33	w	w	PROPN
iajs-3139	20	34	and	and	CCONJ
iajs-3139	20	35	if	if	SCONJ
iajs-3139	20	36	n	n	PRON
iajs-3139	20	37	is	be	AUX
iajs-3139	20	38	pr	pr	ADV
iajs-3139	20	39	-	-	PUNCT
iajs-3139	20	40	maximal	maximal	ADJ
iajs-3139	20	41	submodule	submodule	NOUN
iajs-3139	20	42	of	of	ADP
iajs-3139	20	43	an	an	DET
iajs-3139	20	44	rmodule	rmodule	NOUN
iajs-3139	20	45	w	w	NOUN
iajs-3139	21	1	and	and	CCONJ
iajs-3139	21	2	i	i	PRON
iajs-3139	21	3	be	be	VERB
iajs-3139	21	4	ideal	ideal	ADJ
iajs-3139	21	5	of	of	ADP
iajs-3139	21	6	r	r	NOUN
iajs-3139	21	7	,	,	PUNCT
iajs-3139	21	8	if	if	SCONJ
iajs-3139	21	9	[	[	X
iajs-3139	21	10	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	21	11	:	:	PUNCT
iajs-3139	21	12	𝐼	𝐼	PROPN
iajs-3139	21	13	]	]	PUNCT
iajs-3139	21	14	is	be	AUX
iajs-3139	21	15	a	a	DET
iajs-3139	21	16	proper	proper	ADJ
iajs-3139	21	17	submodule	submodule	NOUN
iajs-3139	21	18	of	of	ADP
iajs-3139	21	19	w	w	PROPN
iajs-3139	21	20	then	then	ADV
iajs-3139	22	1	[	[	X
iajs-3139	22	2	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	22	3	:	:	PUNCT
iajs-3139	23	1	i	i	PRON
iajs-3139	23	2	]	]	PUNCT
iajs-3139	23	3	is	be	AUX
iajs-3139	23	4	pr	pr	NOUN
iajs-3139	23	5	-	-	PUNCT
iajs-3139	23	6	maximal	maximal	ADJ
iajs-3139	23	7	submodule	submodule	NOUN
iajs-3139	23	8	.	.	PUNCT
iajs-3139	24	1	we	we	PRON
iajs-3139	24	2	study	study	VERB
iajs-3139	24	3	the	the	DET
iajs-3139	24	4	relation	relation	NOUN
iajs-3139	24	5	,	,	PUNCT
iajs-3139	24	6	among	among	ADP
iajs-3139	24	7	pr	pr	NOUN
iajs-3139	24	8	-	-	PUNCT
iajs-3139	24	9	maximal	maximal	ADJ
iajs-3139	24	10	(	(	PUNCT
iajs-3139	24	11	submodules	submodule	NOUN
iajs-3139	24	12	and	and	CCONJ
iajs-3139	24	13	other	other	ADJ
iajs-3139	24	14	related	relate	VERB
iajs-3139	24	15	module	module	NOUN
iajs-3139	24	16	)	)	PUNCT
iajs-3139	24	17	,	,	PUNCT
iajs-3139	24	18	in	in	ADP
iajs-3139	24	19	section	section	NOUN
iajs-3139	24	20	three	three	NUM
iajs-3139	24	21	we	we	PRON
iajs-3139	24	22	study	study	VERB
iajs-3139	24	23	pr	pr	NOUN
iajs-3139	24	24	-	-	PUNCT
iajs-3139	24	25	maximal	maximal	ADJ
iajs-3139	24	26	submodule	submodule	NOUN
iajs-3139	24	27	,	,	PUNCT
iajs-3139	24	28	under	under	ADP
iajs-3139	24	29	the	the	DET
iajs-3139	24	30	multiplication	multiplication	NOUN
iajs-3139	24	31	module	module	NOUN
iajs-3139	24	32	and	and	CCONJ
iajs-3139	24	33	we	we	PRON
iajs-3139	24	34	check	check	VERB
iajs-3139	24	35	some	some	DET
iajs-3139	24	36	condition	condition	NOUN
iajs-3139	24	37	under	under	ADP
iajs-3139	24	38	which	which	PRON
iajs-3139	24	39	pr	pr	NOUN
iajs-3139	24	40	-	-	PUNCT
iajs-3139	24	41	maximal	maximal	ADJ
iajs-3139	24	42	submodules	submodule	NOUN
iajs-3139	24	43	and	and	CCONJ
iajs-3139	24	44	maximal	maximal	ADJ
iajs-3139	24	45	submodules	submodule	NOUN
iajs-3139	24	46	are	be	AUX
iajs-3139	24	47	equivalent	equivalent	ADJ
iajs-3139	24	48	.	.	PUNCT
iajs-3139	25	1	every	every	DET
iajs-3139	25	2	multiplication	multiplication	NOUN
iajs-3139	25	3	module	module	NOUN
iajs-3139	25	4	contains	contain	VERB
iajs-3139	25	5	a	a	DET
iajs-3139	25	6	pr	pr	NOUN
iajs-3139	25	7	-	-	PUNCT
iajs-3139	25	8	maximal	maximal	ADJ
iajs-3139	25	9	submodule	submodule	NOUN
iajs-3139	25	10	.	.	PUNCT
iajs-3139	26	1	also	also	ADV
iajs-3139	26	2	,	,	PUNCT
iajs-3139	26	3	we	we	PRON
iajs-3139	26	4	have	have	VERB
iajs-3139	26	5	every	every	DET
iajs-3139	26	6	cyclic	cyclic	ADJ
iajs-3139	26	7	rmodule	rmodule	NOUN
iajs-3139	26	8	has	have	VERB
iajs-3139	26	9	pr	pr	NOUN
iajs-3139	26	10	-	-	PUNCT
iajs-3139	26	11	maximal	maximal	ADJ
iajs-3139	26	12	submodule	submodule	NOUN
iajs-3139	26	13	.	.	PUNCT
iajs-3139	27	1	we	we	PRON
iajs-3139	27	2	found	find	VERB
iajs-3139	27	3	if	if	SCONJ
iajs-3139	27	4	w	w	NOUN
iajs-3139	27	5	is	be	AUX
iajs-3139	27	6	a	a	DET
iajs-3139	27	7	f	f	ADJ
iajs-3139	27	8	-	-	PUNCT
iajs-3139	27	9	regular	regular	ADJ
iajs-3139	27	10	module	module	NOUN
iajs-3139	27	11	then	then	ADV
iajs-3139	27	12	every	every	DET
iajs-3139	27	13	submodule	submodule	NOUN
iajs-3139	27	14	of	of	ADP
iajs-3139	27	15	w	w	PROPN
iajs-3139	27	16	is	be	AUX
iajs-3139	27	17	pr	pr	ADV
iajs-3139	27	18	-	-	ADV
iajs-3139	27	19	maximal	maximal	ADJ
iajs-3139	27	20	.	.	PUNCT
iajs-3139	28	1	2	2	X
iajs-3139	28	2	.	.	X
iajs-3139	28	3	preliminaries	preliminary	NOUN
iajs-3139	28	4	this	this	DET
iajs-3139	28	5	section	section	NOUN
iajs-3139	28	6	is	be	AUX
iajs-3139	28	7	going	go	VERB
iajs-3139	28	8	to	to	PART
iajs-3139	28	9	review	review	VERB
iajs-3139	28	10	some	some	DET
iajs-3139	28	11	well	well	ADV
iajs-3139	28	12	-	-	PUNCT
iajs-3139	28	13	known	know	VERB
iajs-3139	28	14	definitions	definition	NOUN
iajs-3139	28	15	in	in	ADP
iajs-3139	28	16	a	a	DET
iajs-3139	28	17	algebraic	algebraic	ADJ
iajs-3139	28	18	theory	theory	NOUN
iajs-3139	28	19	.	.	PUNCT
iajs-3139	29	1	definition	definition	NOUN
iajs-3139	29	2	2.1	2.1	NUM
iajs-3139	30	1	[	[	X
iajs-3139	30	2	3	3	NUM
iajs-3139	30	3	]	]	X
iajs-3139	30	4	a	a	DET
iajs-3139	30	5	proper	proper	ADJ
iajs-3139	30	6	submodule	submodule	NOUN
iajs-3139	30	7	of	of	ADP
iajs-3139	30	8	an	an	DET
iajs-3139	30	9	r	r	NOUN
iajs-3139	30	10	-	-	PUNCT
iajs-3139	30	11	module	module	NOUN
iajs-3139	30	12	i	i	PRON
iajs-3139	30	13	s	s	AUX
iajs-3139	30	14	named	name	VERB
iajs-3139	30	15	maximal	maximal	ADJ
iajs-3139	30	16	if	if	SCONJ
iajs-3139	30	17	such	such	ADJ
iajs-3139	30	18	that	that	SCONJ
iajs-3139	30	19	namely	namely	ADV
iajs-3139	30	20	=	=	SYM
iajs-3139	30	21	definition	definition	NOUN
iajs-3139	30	22	2.2	2.2	NUM
iajs-3139	30	23	[	[	X
iajs-3139	30	24	1	1	NUM
iajs-3139	30	25	]	]	PUNCT
iajs-3139	30	26	a	a	DET
iajs-3139	30	27	submodule	submodule	NOUN
iajs-3139	30	28	of	of	ADP
iajs-3139	30	29	an	an	DET
iajs-3139	30	30	-module	-module	NOUN
iajs-3139	30	31	is	be	AUX
iajs-3139	30	32	named	name	VERB
iajs-3139	30	33	pure	pure	ADJ
iajs-3139	30	34	-submodule	-submodule	NOUN
iajs-3139	30	35	if	if	SCONJ
iajs-3139	30	36	for	for	ADP
iajs-3139	30	37	each	each	DET
iajs-3139	30	38	ideal	ideal	NOUN
iajs-3139	30	39	of	of	ADP
iajs-3139	30	40	definition	definition	NOUN
iajs-3139	30	41	2.3	2.3	NUM
iajs-3139	30	42	[	[	X
iajs-3139	30	43	6	6	NUM
iajs-3139	30	44	]	]	PUNCT
iajs-3139	30	45	“	"	PUNCT
iajs-3139	30	46	a	a	DET
iajs-3139	30	47	submodule	submodule	NOUN
iajs-3139	30	48	of	of	ADP
iajs-3139	30	49	an	an	DET
iajs-3139	30	50	r	r	NOUN
iajs-3139	30	51	-	-	PUNCT
iajs-3139	30	52	module	module	NOUN
iajs-3139	30	53	is	be	AUX
iajs-3139	30	54	called	call	VERB
iajs-3139	30	55	weak	weak	ADJ
iajs-3139	30	56	maximal	maximal	ADJ
iajs-3139	30	57	if	if	SCONJ
iajs-3139	30	58	𝑊	𝑊	PROPN
iajs-3139	30	59	is	be	AUX
iajs-3139	30	60	f	f	NOUN
iajs-3139	30	61	-	-	ADJ
iajs-3139	30	62	regular	regular	ADJ
iajs-3139	30	63	r	r	NOUN
iajs-3139	30	64	-	-	PUNCT
iajs-3139	30	65	module	module	NOUN
iajs-3139	30	66	”	"	PUNCT
iajs-3139	30	67	.	.	PUNCT
iajs-3139	31	1	lemma	lemma	PROPN
iajs-3139	31	2	(	(	PUNCT
iajs-3139	31	3	2.4	2.4	NUM
iajs-3139	31	4	)	)	PUNCT
iajs-3139	32	1	[	[	X
iajs-3139	32	2	7	7	X
iajs-3139	32	3	]	]	PUNCT
iajs-3139	32	4	“	"	PUNCT
iajs-3139	32	5	if	if	SCONJ
iajs-3139	32	6	:	:	PUNCT
iajs-3139	32	7	is	be	AUX
iajs-3139	32	8	an	an	DET
iajs-3139	32	9	epimorphism	epimorphism	NOUN
iajs-3139	32	10	and	and	CCONJ
iajs-3139	32	11	is	be	AUX
iajs-3139	32	12	pure	pure	ADJ
iajs-3139	32	13	submodule	submodule	NOUN
iajs-3139	32	14	of	of	ADP
iajs-3139	32	15	,	,	PUNCT
iajs-3139	32	16	then	then	ADV
iajs-3139	32	17	(	(	PUNCT
iajs-3139	32	18	)	)	PUNCT
iajs-3139	32	19	is	be	AUX
iajs-3139	32	20	pure	pure	ADJ
iajs-3139	32	21	in	in	ADP
iajs-3139	32	22	.	.	PUNCT
iajs-3139	32	23	”	"	PUNCT
iajs-3139	33	1	definition	definition	NOUN
iajs-3139	33	2	2.5	2.5	NUM
iajs-3139	33	3	[	[	SYM
iajs-3139	33	4	8	8	NUM
iajs-3139	33	5	]	]	PUNCT
iajs-3139	33	6	“	"	PUNCT
iajs-3139	33	7	an	an	DET
iajs-3139	33	8	r	r	NOUN
iajs-3139	33	9	-	-	PUNCT
iajs-3139	33	10	module	module	NOUN
iajs-3139	33	11	is	be	AUX
iajs-3139	33	12	named	name	VERB
iajs-3139	33	13	pure	pure	ADJ
iajs-3139	33	14	simple	simple	ADJ
iajs-3139	33	15	if	if	SCONJ
iajs-3139	34	1	and	and	CCONJ
iajs-3139	34	2	it	it	PRON
iajs-3139	34	3	has	have	VERB
iajs-3139	34	4	no	no	DET
iajs-3139	34	5	pure	pure	ADJ
iajs-3139	34	6	,	,	PUNCT
iajs-3139	34	7	submodule	submodule	NOUN
iajs-3139	34	8	except	except	SCONJ
iajs-3139	34	9	and	and	CCONJ
iajs-3139	34	10	definition	definition	NOUN
iajs-3139	34	11	2.6	2.6	NUM
iajs-3139	35	1	[	[	X
iajs-3139	35	2	4	4	NUM
iajs-3139	35	3	]	]	PUNCT
iajs-3139	35	4	an	an	DET
iajs-3139	35	5	r	r	NOUN
iajs-3139	35	6	-	-	PUNCT
iajs-3139	35	7	module	module	NOUN
iajs-3139	35	8	is	be	AUX
iajs-3139	35	9	called	call	VERB
iajs-3139	35	10	faithful	faithful	ADJ
iajs-3139	35	11	if	if	SCONJ
iajs-3139	35	12	=	=	SYM
iajs-3139	35	13	definition	definition	NOUN
iajs-3139	35	14	2.7	2.7	NUM
iajs-3139	35	15	[	[	X
iajs-3139	35	16	9,10	9,10	NUM
iajs-3139	35	17	]	]	X
iajs-3139	35	18	“	"	PUNCT
iajs-3139	35	19	an	an	DET
iajs-3139	35	20	r	r	NOUN
iajs-3139	35	21	-	-	PUNCT
iajs-3139	35	22	module	module	NOUN
iajs-3139	35	23	is	be	AUX
iajs-3139	35	24	said	say	VERB
iajs-3139	35	25	to	to	PART
iajs-3139	35	26	be	be	AUX
iajs-3139	35	27	multiplication	multiplication	NOUN
iajs-3139	35	28	if	if	SCONJ
iajs-3139	35	29	for	for	ADP
iajs-3139	35	30	each	each	DET
iajs-3139	35	31	submodule	submodule	NOUN
iajs-3139	35	32	of	of	ADP
iajs-3139	35	33	there	there	ADV
iajs-3139	35	34	exists	exist	VERB
iajs-3139	35	35	an	an	DET
iajs-3139	35	36	ideal	ideal	NOUN
iajs-3139	35	37	of	of	ADP
iajs-3139	35	38	such	such	ADJ
iajs-3139	35	39	that	that	PRON
iajs-3139	35	40	=	=	PRON
iajs-3139	35	41	”	"	PUNCT
iajs-3139	35	42	.	.	PUNCT
iajs-3139	36	1	equivalently	equivalently	ADV
iajs-3139	36	2	is	be	AUX
iajs-3139	36	3	a	a	DET
iajs-3139	36	4	multiplication	multiplication	NOUN
iajs-3139	36	5	r	r	NOUN
iajs-3139	36	6	-	-	PUNCT
iajs-3139	36	7	module	module	NOUN
iajs-3139	36	8	if	if	SCONJ
iajs-3139	36	9	and	and	CCONJ
iajs-3139	36	10	only	only	ADV
iajs-3139	36	11	if	if	SCONJ
iajs-3139	36	12	for	for	ADP
iajs-3139	36	13	each	each	DET
iajs-3139	36	14	submodule	submodule	NOUN
iajs-3139	36	15	of	of	ADP
iajs-3139	36	16	,	,	PUNCT
iajs-3139	36	17	=	=	PUNCT
iajs-3139	36	18	[	[	PUNCT
iajs-3139	36	19	]	]	X
iajs-3139	36	20	.	.	PUNCT
iajs-3139	37	1	proposition	proposition	NOUN
iajs-3139	37	2	2.8	2.8	NUM
iajs-3139	38	1	[	[	SYM
iajs-3139	38	2	10	10	NUM
iajs-3139	38	3	]	]	X
iajs-3139	38	4	if	if	SCONJ
iajs-3139	38	5	is	be	AUX
iajs-3139	38	6	an	an	DET
iajs-3139	38	7	r	r	NOUN
iajs-3139	38	8	-	-	PUNCT
iajs-3139	38	9	module	module	NOUN
iajs-3139	38	10	and	and	CCONJ
iajs-3139	38	11	has	have	VERB
iajs-3139	38	12	an	an	DET
iajs-3139	38	13	,	,	PUNCT
iajs-3139	38	14	unique	unique	ADJ
iajs-3139	38	15	maximal	maximal	ADJ
iajs-3139	38	16	submodule	submodule	NOUN
iajs-3139	38	17	,	,	PUNCT
iajs-3139	38	18	then	then	ADV
iajs-3139	38	19	is	be	AUX
iajs-3139	38	20	called	call	VERB
iajs-3139	38	21	local	local	ADJ
iajs-3139	38	22	module	module	NOUN
iajs-3139	38	23	.	.	PUNCT
iajs-3139	39	1	3	3	X
iajs-3139	39	2	.	.	X
iajs-3139	39	3	pr	pr	NOUN
iajs-3139	39	4	-	-	PUNCT
iajs-3139	39	5	maximal	maximal	ADJ
iajs-3139	39	6	submodules	submodule	NOUN
iajs-3139	39	7	:	:	PUNCT
iajs-3139	39	8	in	in	ADP
iajs-3139	39	9	this	this	DET
iajs-3139	39	10	section	section	NOUN
iajs-3139	39	11	,	,	PUNCT
iajs-3139	39	12	the	the	DET
iajs-3139	39	13	basic	basic	ADJ
iajs-3139	39	14	definitions	definition	NOUN
iajs-3139	39	15	and	and	CCONJ
iajs-3139	39	16	facts	fact	NOUN
iajs-3139	39	17	related	relate	VERB
iajs-3139	39	18	to	to	ADP
iajs-3139	39	19	this	this	DET
iajs-3139	39	20	work	work	NOUN
iajs-3139	39	21	are	be	AUX
iajs-3139	39	22	recalled	recall	VERB
iajs-3139	39	23	,	,	PUNCT
iajs-3139	39	24	which	which	PRON
iajs-3139	39	25	starts	start	VERB
iajs-3139	39	26	with	with	ADP
iajs-3139	39	27	the	the	DET
iajs-3139	39	28	following	follow	VERB
iajs-3139	39	29	definition	definition	NOUN
iajs-3139	39	30	.	.	PUNCT
iajs-3139	40	1	ihjpas	ihjpas	PROPN
iajs-3139	40	2	.	.	PUNCT
iajs-3139	41	1	36	36	NUM
iajs-3139	41	2	(	(	PUNCT
iajs-3139	41	3	4	4	NUM
iajs-3139	41	4	)	)	PUNCT
iajs-3139	41	5	2023	2023	NUM
iajs-3139	41	6	361	361	NUM
iajs-3139	41	7	definition	definition	NOUN
iajs-3139	41	8	(	(	PUNCT
iajs-3139	41	9	3.1	3.1	NUM
iajs-3139	41	10	)	)	PUNCT
iajs-3139	41	11	a	a	DET
iajs-3139	41	12	sound	sound	ADJ
iajs-3139	41	13	r	r	NOUN
iajs-3139	41	14	-	-	PUNCT
iajs-3139	41	15	submodule	submodule	NOUN
iajs-3139	41	16	of	of	ADP
iajs-3139	41	17	an	an	DET
iajs-3139	41	18	r	r	NOUN
iajs-3139	41	19	-	-	PUNCT
iajs-3139	41	20	module	module	NOUN
iajs-3139	41	21	is	be	AUX
iajs-3139	41	22	named	name	VERB
iajs-3139	41	23	pure	pure	ADJ
iajs-3139	41	24	-	-	PUNCT
iajs-3139	41	25	maximal	maximal	ADJ
iajs-3139	41	26	(	(	PUNCT
iajs-3139	41	27	-maximal	-maximal	ADJ
iajs-3139	41	28	)	)	PUNCT
iajs-3139	41	29	submodule	submodule	NOUN
iajs-3139	41	30	of	of	ADP
iajs-3139	41	31	if	if	SCONJ
iajs-3139	41	32	there	there	PRON
iajs-3139	41	33	exists	exist	VERB
iajs-3139	41	34	with	with	ADP
iajs-3139	41	35	then	then	ADV
iajs-3139	41	36	is	be	AUX
iajs-3139	41	37	pure	pure	ADJ
iajs-3139	41	38	submodule	submodule	NOUN
iajs-3139	41	39	of	of	ADP
iajs-3139	41	40	w	w	PROPN
iajs-3139	41	41	(	(	PUNCT
iajs-3139	41	42	𝐻	𝐻	PROPN
iajs-3139	41	43	≤𝑃	≤𝑃	ADP
iajs-3139	41	44	𝑊	𝑊	NOUN
iajs-3139	41	45	)	)	PUNCT
iajs-3139	41	46	.	.	PUNCT
iajs-3139	42	1	remark	remark	NOUN
iajs-3139	42	2	and	and	CCONJ
iajs-3139	42	3	example	example	NOUN
iajs-3139	42	4	(	(	PUNCT
iajs-3139	42	5	3.2	3.2	NUM
iajs-3139	42	6	)	)	PUNCT
iajs-3139	42	7	1	1	NUM
iajs-3139	42	8	.	.	PUNCT
iajs-3139	43	1	every	every	DET
iajs-3139	43	2	maximal	maximal	ADJ
iajs-3139	43	3	submodule	submodule	NOUN
iajs-3139	43	4	of	of	ADP
iajs-3139	43	5	r	r	NOUN
iajs-3139	43	6	-	-	PUNCT
iajs-3139	43	7	module	module	NOUN
iajs-3139	43	8	is	be	AUX
iajs-3139	43	9	pr	pr	NOUN
iajs-3139	43	10	-	-	PUNCT
iajs-3139	43	11	maximal	maximal	ADJ
iajs-3139	43	12	.	.	PUNCT
iajs-3139	44	1	proof	proof	NOUN
iajs-3139	44	2	:	:	PUNCT
iajs-3139	44	3	impose	impose	VERB
iajs-3139	44	4	maximal	maximal	ADJ
iajs-3139	44	5	rsubmodule	rsubmodule	NOUN
iajs-3139	44	6	of	of	ADP
iajs-3139	44	7	r	r	NOUN
iajs-3139	44	8	-	-	PUNCT
iajs-3139	44	9	module	module	NOUN
iajs-3139	44	10	there	there	ADV
iajs-3139	44	11	exist	exist	VERB
iajs-3139	44	12	0	0	NUM
iajs-3139	44	13	h	h	NOUN
iajs-3139	44	14	submodule	submodule	NOUN
iajs-3139	44	15	of	of	ADP
iajs-3139	44	16	such	such	DET
iajs-3139	44	17	that	that	SCONJ
iajs-3139	44	18	since	since	SCONJ
iajs-3139	44	19	is	be	AUX
iajs-3139	44	20	maximal	maximal	ADJ
iajs-3139	44	21	then	then	ADV
iajs-3139	44	22	but	but	CCONJ
iajs-3139	44	23	is	be	AUX
iajs-3139	44	24	pure	pure	ADJ
iajs-3139	44	25	of	of	ADP
iajs-3139	44	26	therefore	therefore	ADV
iajs-3139	44	27	is	be	AUX
iajs-3139	44	28	pr	pr	ADV
iajs-3139	44	29	-	-	ADJ
iajs-3139	44	30	maximal	maximal	ADJ
iajs-3139	44	31	the	the	DET
iajs-3139	44	32	convers	conver	NOUN
iajs-3139	44	33	is	be	AUX
iajs-3139	44	34	not	not	PART
iajs-3139	44	35	true	true	ADJ
iajs-3139	44	36	as	as	ADP
iajs-3139	44	37	the	the	DET
iajs-3139	44	38	following	following	ADJ
iajs-3139	44	39	example	example	NOUN
iajs-3139	44	40	in	in	ADP
iajs-3139	44	41	w=	w=	NOUN
iajs-3139	44	42	+	+	CCONJ
iajs-3139	44	43	as	as	ADP
iajs-3139	44	44	a	a	DET
iajs-3139	44	45	z	z	NOUN
iajs-3139	44	46	-	-	PUNCT
iajs-3139	44	47	module	module	NOUN
iajs-3139	44	48	let	let	NOUN
iajs-3139	44	49	,	,	PUNCT
iajs-3139	44	50	=	=	NOUN
iajs-3139	44	51	2𝑍4	2𝑍4	NUM
iajs-3139	44	52	and	and	CCONJ
iajs-3139	44	53	h=	h=	PRON
iajs-3139	44	54	such	such	ADJ
iajs-3139	44	55	that	that	SCONJ
iajs-3139	44	56	2	2	NUM
iajs-3139	44	57	since	since	SCONJ
iajs-3139	44	58	h=	h=	NOUN
iajs-3139	44	59	is	be	AUX
iajs-3139	44	60	a	a	DET
iajs-3139	44	61	summand	summand	NOUN
iajs-3139	44	62	of	of	ADP
iajs-3139	44	63	=	=	PUNCT
iajs-3139	44	64	,	,	PUNCT
iajs-3139	44	65	hence	hence	ADV
iajs-3139	44	66	is	be	AUX
iajs-3139	44	67	pure	pure	ADJ
iajs-3139	44	68	in	in	ADP
iajs-3139	44	69	,	,	PUNCT
iajs-3139	44	70	then	then	ADV
iajs-3139	44	71	is	be	AUX
iajs-3139	44	72	pr	pr	NOUN
iajs-3139	44	73	-	-	PUNCT
iajs-3139	44	74	maximal	maximal	ADJ
iajs-3139	44	75	submodule	submodule	NOUN
iajs-3139	44	76	of	of	ADP
iajs-3139	44	77	but	but	CCONJ
iajs-3139	44	78	is	be	AUX
iajs-3139	44	79	not	not	PART
iajs-3139	44	80	maximal	maximal	ADJ
iajs-3139	44	81	since	since	ADV
iajs-3139	44	82	.	.	PUNCT
iajs-3139	45	1	2	2	X
iajs-3139	45	2	.	.	X
iajs-3139	45	3	a	a	DET
iajs-3139	45	4	subset	subset	NOUN
iajs-3139	45	5	of	of	ADP
iajs-3139	45	6	pr	pr	NOUN
iajs-3139	45	7	-	-	PUNCT
iajs-3139	45	8	maximal	maximal	ADJ
iajs-3139	45	9	-submodule	-submodule	NOUN
iajs-3139	45	10	need	need	AUX
iajs-3139	45	11	not	not	PART
iajs-3139	45	12	be	be	AUX
iajs-3139	45	13	pr	pr	NOUN
iajs-3139	45	14	-	-	PUNCT
iajs-3139	45	15	maximal	maximal	ADJ
iajs-3139	45	16	-submodule	-submodule	NOUN
iajs-3139	45	17	as	as	ADP
iajs-3139	45	18	the	the	DET
iajs-3139	45	19	breech	breech	NOUN
iajs-3139	45	20	example	example	NOUN
iajs-3139	45	21	in	in	ADP
iajs-3139	45	22	as	as	SCONJ
iajs-3139	45	23	z	z	NOUN
iajs-3139	45	24	-	-	PUNCT
iajs-3139	45	25	module	module	NOUN
iajs-3139	45	26	impose	impose	VERB
iajs-3139	46	1	3𝑍12	3𝑍12	PROPN
iajs-3139	46	2	<	<	X
iajs-3139	47	1	𝑍12	𝑍12	PROPN
iajs-3139	47	2	≤	≤	PUNCT
iajs-3139	48	1	𝑍12	𝑍12	PROPN
iajs-3139	48	2	implies	imply	VERB
iajs-3139	48	3	3	3	NUM
iajs-3139	48	4	is	be	AUX
iajs-3139	48	5	pr	pr	NOUN
iajs-3139	48	6	-	-	PUNCT
iajs-3139	48	7	maximal	maximal	ADJ
iajs-3139	48	8	submodule	submodule	NOUN
iajs-3139	48	9	of	of	ADP
iajs-3139	48	10	since	since	ADV
iajs-3139	48	11	,	,	PUNCT
iajs-3139	48	12	but	but	CCONJ
iajs-3139	48	13	6	6	NUM
iajs-3139	48	14	is	be	AUX
iajs-3139	48	15	not	not	PART
iajs-3139	48	16	pr	pr	ADV
iajs-3139	48	17	-	-	PUNCT
iajs-3139	48	18	maximal	maximal	ADJ
iajs-3139	48	19	submodule	submodule	NOUN
iajs-3139	48	20	of	of	ADP
iajs-3139	48	21	since	since	SCONJ
iajs-3139	48	22	6	6	NUM
iajs-3139	48	23	and	and	CCONJ
iajs-3139	48	24	is	be	AUX
iajs-3139	48	25	not	not	PART
iajs-3139	48	26	pure	pure	ADJ
iajs-3139	48	27	in	in	ADP
iajs-3139	48	28	.	.	PUNCT
iajs-3139	49	1	3	3	X
iajs-3139	49	2	.	.	X
iajs-3139	49	3	𝑍4	𝑍4	NOUN
iajs-3139	49	4	as	as	ADP
iajs-3139	49	5	z	z	NOUN
iajs-3139	49	6	-	-	PUNCT
iajs-3139	49	7	module	module	NOUN
iajs-3139	49	8	we	we	PRON
iajs-3139	49	9	have	have	VERB
iajs-3139	49	10	{	{	PUNCT
iajs-3139	49	11	0̅	0̅	NOUN
iajs-3139	49	12	,	,	PUNCT
iajs-3139	49	13	2̅	2̅	PROPN
iajs-3139	49	14	}	}	PUNCT
iajs-3139	49	15	is	be	AUX
iajs-3139	49	16	pr	pr	NOUN
iajs-3139	49	17	-	-	ADJ
iajs-3139	49	18	maximal	maximal	ADJ
iajs-3139	49	19	since	since	SCONJ
iajs-3139	49	20	𝑍4	𝑍4	NOUN
iajs-3139	49	21	is	be	AUX
iajs-3139	49	22	pure	pure	ADJ
iajs-3139	49	23	of	of	ADP
iajs-3139	49	24	𝑍4	𝑍4	NOUN
iajs-3139	49	25	and	and	CCONJ
iajs-3139	49	26	{	{	PUNCT
iajs-3139	49	27	0̅,2̅	0̅,2̅	PROPN
iajs-3139	49	28	}	}	PUNCT
iajs-3139	49	29	<	<	X
iajs-3139	49	30	𝑍4≤	𝑍4≤	NOUN
iajs-3139	49	31	𝑍4	𝑍4	NOUN
iajs-3139	49	32	.	.	PUNCT
iajs-3139	50	1	4	4	X
iajs-3139	50	2	.	.	X
iajs-3139	50	3	z6=	z6=	PROPN
iajs-3139	50	4	{	{	PUNCT
iajs-3139	50	5	0̅,3̅	0̅,3̅	PROPN
iajs-3139	50	6	}	}	PUNCT
iajs-3139	50	7	⨁	⨁	PROPN
iajs-3139	50	8	{	{	PUNCT
iajs-3139	50	9	0,̅	0,̅	NUM
iajs-3139	50	10	2̅,4̅	2̅,4̅	PROPN
iajs-3139	50	11	}	}	PUNCT
iajs-3139	50	12	,	,	PUNCT
iajs-3139	50	13	3𝑍6is	3𝑍6is	NUM
iajs-3139	50	14	pr	pr	NOUN
iajs-3139	50	15	-	-	PUNCT
iajs-3139	50	16	maximal	maximal	ADJ
iajs-3139	50	17	of	of	ADP
iajs-3139	50	18	z6	z6	PROPN
iajs-3139	50	19	since	since	SCONJ
iajs-3139	50	20	{	{	PUNCT
iajs-3139	50	21	0̅	0̅	PROPN
iajs-3139	50	22	,	,	PUNCT
iajs-3139	50	23	3̅}<z6	3̅}<z6	NUM
iajs-3139	50	24	≤z6	≤z6	NOUN
iajs-3139	50	25	and	and	CCONJ
iajs-3139	50	26	2𝑍6	2𝑍6	NUM
iajs-3139	50	27	is	be	AUX
iajs-3139	50	28	prmaximal	prmaximal	ADJ
iajs-3139	50	29	of	of	ADP
iajs-3139	50	30	𝑍6	𝑍6	PROPN
iajs-3139	50	31	,	,	PUNCT
iajs-3139	50	32	also	also	ADV
iajs-3139	50	33	since	since	SCONJ
iajs-3139	50	34	{	{	PUNCT
iajs-3139	50	35	0̅,2̅	0̅,2̅	PROPN
iajs-3139	50	36	,	,	PUNCT
iajs-3139	50	37	4̅	4̅	PROPN
iajs-3139	50	38	}	}	PUNCT
iajs-3139	50	39	<	<	X
iajs-3139	50	40	z6	z6	PROPN
iajs-3139	50	41	≤z6	≤z6	NOUN
iajs-3139	51	1	5	5	X
iajs-3139	51	2	.	.	PUNCT
iajs-3139	52	1	if	if	SCONJ
iajs-3139	52	2	𝑊	𝑊	PROPN
iajs-3139	52	3	𝐻	𝐻	PROPN
iajs-3139	52	4	is	be	AUX
iajs-3139	52	5	simple	simple	ADJ
iajs-3139	52	6	,	,	PUNCT
iajs-3139	52	7	then	then	ADV
iajs-3139	52	8	is	be	AUX
iajs-3139	52	9	pr	pr	ADV
iajs-3139	52	10	-	-	PUNCT
iajs-3139	52	11	maximal	maximal	ADJ
iajs-3139	52	12	.	.	PUNCT
iajs-3139	53	1	proof	proof	NOUN
iajs-3139	53	2	:	:	PUNCT
iajs-3139	53	3	clearly	clearly	ADV
iajs-3139	53	4	since	since	SCONJ
iajs-3139	53	5	is	be	AUX
iajs-3139	53	6	simple	simple	ADJ
iajs-3139	53	7	implies	implie	NOUN
iajs-3139	53	8	is	be	AUX
iajs-3139	53	9	maximal	maximal	ADJ
iajs-3139	53	10	then	then	ADV
iajs-3139	53	11	assist	assist	VERB
iajs-3139	53	12	remark	remark	NOUN
iajs-3139	53	13	(	(	PUNCT
iajs-3139	53	14	2.2	2.2	NUM
iajs-3139	53	15	)	)	PUNCT
iajs-3139	53	16	every	every	PRON
iajs-3139	53	17	,	,	PUNCT
iajs-3139	53	18	is	be	AUX
iajs-3139	53	19	prmaximal	prmaximal	ADJ
iajs-3139	53	20	.	.	PUNCT
iajs-3139	53	21	.	.	PUNCT
iajs-3139	54	1	let	let	VERB
iajs-3139	54	2	and	and	CCONJ
iajs-3139	54	3	are	be	AUX
iajs-3139	54	4	nonzero	nonzero	PROPN
iajs-3139	54	5	submodule	submodule	NOUN
iajs-3139	54	6	of	of	ADP
iajs-3139	54	7	such	such	DET
iajs-3139	54	8	that	that	SCONJ
iajs-3139	54	9	if	if	SCONJ
iajs-3139	54	10	is	be	AUX
iajs-3139	54	11	pr	pr	NOUN
iajs-3139	54	12	-	-	ADJ
iajs-3139	54	13	maximal	maximal	ADJ
iajs-3139	54	14	of	of	ADP
iajs-3139	54	15	and	and	CCONJ
iajs-3139	54	16	is	be	AUX
iajs-3139	54	17	pr	pr	ADV
iajs-3139	54	18	-	-	ADV
iajs-3139	54	19	maximal	maximal	ADJ
iajs-3139	54	20	of	of	ADP
iajs-3139	54	21	then	then	ADV
iajs-3139	54	22	is	be	AUX
iajs-3139	54	23	not	not	PART
iajs-3139	54	24	pr	pr	NOUN
iajs-3139	54	25	-	-	ADV
iajs-3139	54	26	maximal	maximal	ADJ
iajs-3139	54	27	of	of	ADP
iajs-3139	54	28	for	for	ADP
iajs-3139	54	29	example	example	NOUN
iajs-3139	54	30	let	let	VERB
iajs-3139	54	31	=	=	PRON
iajs-3139	54	32	2𝑍24and	2𝑍24and	VERB
iajs-3139	54	33	=	=	NOUN
iajs-3139	54	34	6	6	NUM
iajs-3139	54	35	,	,	PUNCT
iajs-3139	54	36	=	=	NOUN
iajs-3139	54	37	2	2	NUM
iajs-3139	54	38	,	,	PUNCT
iajs-3139	54	39	is	be	AUX
iajs-3139	54	40	pr	pr	NOUN
iajs-3139	54	41	-	-	ADJ
iajs-3139	54	42	maximal	maximal	ADJ
iajs-3139	54	43	in	in	ADV
iajs-3139	54	44	since	since	SCONJ
iajs-3139	54	45	62	62	NUM
iajs-3139	54	46	and	and	CCONJ
iajs-3139	54	47	2	2	NUM
iajs-3139	54	48	2	2	NUM
iajs-3139	54	49	and	and	CCONJ
iajs-3139	54	50	is	be	AUX
iajs-3139	54	51	pr	pr	NOUN
iajs-3139	54	52	-	-	ADJ
iajs-3139	54	53	maximal	maximal	ADJ
iajs-3139	54	54	in	in	ADV
iajs-3139	54	55	since	since	SCONJ
iajs-3139	54	56	is	be	AUX
iajs-3139	54	57	maximal	maximal	ADJ
iajs-3139	54	58	but	but	CCONJ
iajs-3139	54	59	is	be	AUX
iajs-3139	54	60	not	not	PART
iajs-3139	54	61	pr	pr	NOUN
iajs-3139	54	62	-	-	ADJ
iajs-3139	54	63	maximal	maximal	ADJ
iajs-3139	54	64	in	in	ADV
iajs-3139	54	65	since	since	SCONJ
iajs-3139	54	66	6	6	NUM
iajs-3139	54	67	2	2	NUM
iajs-3139	54	68	and2	and2	NOUN
iajs-3139	54	69	is	be	AUX
iajs-3139	54	70	not	not	PART
iajs-3139	54	71	pure	pure	ADJ
iajs-3139	54	72	in	in	ADP
iajs-3139	54	73	.	.	PUNCT
iajs-3139	55	1	7	7	X
iajs-3139	55	2	.	.	X
iajs-3139	56	1	if	if	SCONJ
iajs-3139	56	2	=	=	PRON
iajs-3139	56	3	as	as	ADP
iajs-3139	56	4	,	,	PUNCT
iajs-3139	56	5	z	z	NOUN
iajs-3139	56	6	-	-	PUNCT
iajs-3139	56	7	module	module	NOUN
iajs-3139	56	8	and	and	CCONJ
iajs-3139	56	9	=	=	NOUN
iajs-3139	56	10	4	4	NUM
iajs-3139	56	11	a	a	DET
iajs-3139	56	12	-submodule	-submodule	NOUN
iajs-3139	56	13	of	of	ADP
iajs-3139	56	14	,	,	PUNCT
iajs-3139	56	15	then	then	ADV
iajs-3139	56	16	is	be	AUX
iajs-3139	56	17	not	not	PART
iajs-3139	56	18	pr	pr	NOUN
iajs-3139	56	19	-	-	ADJ
iajs-3139	56	20	maximal	maximal	ADJ
iajs-3139	56	21	in	in	ADV
iajs-3139	56	22	since	since	SCONJ
iajs-3139	56	23	4	4	NUM
iajs-3139	56	24	2	2	NUM
iajs-3139	56	25	and	and	CCONJ
iajs-3139	56	26	2	2	NUM
iajs-3139	56	27	is	be	AUX
iajs-3139	56	28	not	not	PART
iajs-3139	56	29	pure	pure	ADJ
iajs-3139	56	30	in	in	ADP
iajs-3139	56	31	.	.	PUNCT
iajs-3139	57	1	8	8	X
iajs-3139	57	2	.	.	PUNCT
iajs-3139	58	1	if	if	SCONJ
iajs-3139	58	2	is	be	AUX
iajs-3139	58	3	a	a	DET
iajs-3139	58	4	semi	semi	ADJ
iajs-3139	58	5	simple	simple	ADJ
iajs-3139	58	6	r	r	NOUN
iajs-3139	58	7	-	-	PUNCT
iajs-3139	58	8	module	module	NOUN
iajs-3139	58	9	,	,	PUNCT
iajs-3139	58	10	thence	thence	NOUN
iajs-3139	58	11	every	every	DET
iajs-3139	58	12	sound	sound	ADJ
iajs-3139	58	13	-submodule	-submodule	NOUN
iajs-3139	58	14	of	of	ADP
iajs-3139	58	15	is	be	AUX
iajs-3139	58	16	maximal	maximal	ADJ
iajs-3139	58	17	if	if	SCONJ
iajs-3139	58	18	and	and	CCONJ
iajs-3139	58	19	only	only	ADV
iajs-3139	58	20	if	if	SCONJ
iajs-3139	58	21	it	it	PRON
iajs-3139	58	22	is	be	AUX
iajs-3139	58	23	pr	pr	NOUN
iajs-3139	58	24	-	-	ADJ
iajs-3139	58	25	maximal	maximal	ADJ
iajs-3139	58	26	for	for	ADP
iajs-3139	58	27	example	example	NOUN
iajs-3139	58	28	2	2	NUM
iajs-3139	58	29	and	and	CCONJ
iajs-3139	58	30	3	3	NUM
iajs-3139	58	31	are	be	AUX
iajs-3139	58	32	pr	pr	NOUN
iajs-3139	58	33	-	-	PUNCT
iajs-3139	58	34	maximal	maximal	ADJ
iajs-3139	58	35	submodule	submodule	NOUN
iajs-3139	58	36	of	of	ADP
iajs-3139	58	37	as	as	ADP
iajs-3139	58	38	z	z	NOUN
iajs-3139	58	39	-	-	NOUN
iajs-3139	58	40	module	module	NOUN
iajs-3139	58	41	.	.	PUNCT
iajs-3139	58	42	.	.	PUNCT
iajs-3139	59	1	every	every	DET
iajs-3139	59	2	proper	proper	ADJ
iajs-3139	59	3	submodule	submodule	NOUN
iajs-3139	59	4	contain	contain	VERB
iajs-3139	59	5	in	in	ADP
iajs-3139	59	6	a	a	DET
iajs-3139	59	7	pure	pure	ADJ
iajs-3139	59	8	submodule	submodule	NOUN
iajs-3139	59	9	is	be	AUX
iajs-3139	59	10	,	,	PUNCT
iajs-3139	59	11	pr	pr	NOUN
iajs-3139	59	12	-	-	PUNCT
iajs-3139	59	13	maximal	maximal	ADJ
iajs-3139	59	14	.	.	PUNCT
iajs-3139	60	1	definition	definition	NOUN
iajs-3139	60	2	(	(	PUNCT
iajs-3139	60	3	3.3	3.3	NUM
iajs-3139	60	4	)	)	PUNCT
iajs-3139	60	5	a	a	DET
iajs-3139	60	6	proper	proper	ADJ
iajs-3139	60	7	submodule	submodule	NOUN
iajs-3139	60	8	of	of	ADP
iajs-3139	60	9	an	an	DET
iajs-3139	60	10	r	r	NOUN
iajs-3139	60	11	-	-	PUNCT
iajs-3139	60	12	module	module	NOUN
iajs-3139	60	13	is	be	AUX
iajs-3139	60	14	named	name	VERB
iajs-3139	60	15	near	near	ADP
iajs-3139	60	16	-	-	PUNCT
iajs-3139	60	17	maximal(n	maximal(n	NOUN
iajs-3139	60	18	-	-	ADJ
iajs-3139	60	19	maximal	maximal	ADJ
iajs-3139	60	20	)	)	PUNCT
iajs-3139	60	21	whenever	whenever	SCONJ
iajs-3139	60	22	is	be	AUX
iajs-3139	60	23	pure	pure	ADJ
iajs-3139	60	24	submodule	submodule	NOUN
iajs-3139	60	25	of	of	ADP
iajs-3139	60	26	such	such	ADJ
iajs-3139	60	27	that	that	DET
iajs-3139	60	28	then	then	ADV
iajs-3139	60	29	remark	remark	NOUN
iajs-3139	60	30	(	(	PUNCT
iajs-3139	60	31	3.4	3.4	NUM
iajs-3139	60	32	)	)	PUNCT
iajs-3139	60	33	pr	pr	NOUN
iajs-3139	60	34	-	-	PUNCT
iajs-3139	60	35	maximal	maximal	ADJ
iajs-3139	60	36	submodule	submodule	NOUN
iajs-3139	60	37	need	need	AUX
iajs-3139	60	38	not	not	PART
iajs-3139	60	39	be	be	AUX
iajs-3139	60	40	n	n	ADV
iajs-3139	60	41	-	-	PUNCT
iajs-3139	60	42	maximal	maximal	ADJ
iajs-3139	60	43	as	as	ADP
iajs-3139	60	44	the	the	DET
iajs-3139	60	45	following	follow	VERB
iajs-3139	60	46	example	example	NOUN
iajs-3139	60	47	show	show	NOUN
iajs-3139	60	48	:	:	PUNCT
iajs-3139	60	49	in	in	ADP
iajs-3139	60	50	𝑍4⨁	𝑍4⨁	PROPN
iajs-3139	60	51	𝑍2	𝑍2	PROPN
iajs-3139	60	52	as	as	ADP
iajs-3139	60	53	a	a	DET
iajs-3139	60	54	z	z	NOUN
iajs-3139	60	55	-	-	PUNCT
iajs-3139	60	56	module	module	NOUN
iajs-3139	60	57	2𝑍4⨁	2𝑍4⨁	PROPN
iajs-3139	60	58	(	(	PUNCT
iajs-3139	60	59	0̅	0̅	PROPN
iajs-3139	60	60	)	)	PUNCT
iajs-3139	60	61	is	be	AUX
iajs-3139	60	62	pr	pr	NOUN
iajs-3139	60	63	-	-	ADJ
iajs-3139	60	64	maximal	maximal	ADJ
iajs-3139	60	65	since	since	ADV
iajs-3139	60	66	,	,	PUNCT
iajs-3139	60	67	2𝑍4⨁	2𝑍4⨁	PROPN
iajs-3139	60	68	(	(	PUNCT
iajs-3139	60	69	0̅	0̅	PROPN
iajs-3139	60	70	)	)	PUNCT
iajs-3139	60	71	<	<	X
iajs-3139	60	72	𝑍4⨁	𝑍4⨁	X
iajs-3139	60	73	(	(	PUNCT
iajs-3139	60	74	0̅	0̅	PROPN
iajs-3139	60	75	)	)	PUNCT
iajs-3139	60	76	≤	≤	NOUN
iajs-3139	60	77	𝑍4⨁	𝑍4⨁	PROPN
iajs-3139	61	1	𝑍2and	𝑍2and	PROPN
iajs-3139	61	2	𝑍4⨁	𝑍4⨁	PROPN
iajs-3139	61	3	(	(	PUNCT
iajs-3139	61	4	0̅	0̅	PROPN
iajs-3139	61	5	)	)	PUNCT
iajs-3139	61	6	≤𝑃	≤𝑃	VERB
iajs-3139	61	7	𝑍4⨁	𝑍4⨁	PROPN
iajs-3139	61	8	𝑍2	𝑍2	PROPN
iajs-3139	62	1	but	but	CCONJ
iajs-3139	62	2	𝑍4⨁	𝑍4⨁	PROPN
iajs-3139	62	3	(	(	PUNCT
iajs-3139	62	4	0̅	0̅	PROPN
iajs-3139	62	5	)	)	PUNCT
iajs-3139	63	1	≠	≠	PROPN
iajs-3139	63	2	𝑍4⨁	𝑍4⨁	X
iajs-3139	63	3	𝑍2	𝑍2	VERB
iajs-3139	63	4	so	so	ADV
iajs-3139	63	5	not	not	PART
iajs-3139	63	6	n	n	CCONJ
iajs-3139	63	7	-	-	PUNCT
iajs-3139	63	8	maximal	maximal	ADJ
iajs-3139	63	9	ihjpas	ihjpa	NOUN
iajs-3139	63	10	.	.	PUNCT
iajs-3139	64	1	36	36	NUM
iajs-3139	64	2	(	(	PUNCT
iajs-3139	64	3	4	4	NUM
iajs-3139	64	4	)	)	PUNCT
iajs-3139	64	5	2023	2023	NUM
iajs-3139	64	6	362	362	NUM
iajs-3139	64	7	4𝑍12=	4𝑍12=	NUM
iajs-3139	64	8	{	{	PUNCT
iajs-3139	64	9	0̅,4̅,8̅	0̅,4̅,8̅	NOUN
iajs-3139	64	10	}	}	PUNCT
iajs-3139	64	11	is	be	AUX
iajs-3139	64	12	n	n	ADV
iajs-3139	64	13	-	-	PUNCT
iajs-3139	64	14	maximal	maximal	ADJ
iajs-3139	64	15	in	in	ADP
iajs-3139	64	16	𝑍12	𝑍12	PROPN
iajs-3139	64	17	since	since	SCONJ
iajs-3139	64	18	4𝑍12	4𝑍12	PROPN
iajs-3139	64	19	<	<	X
iajs-3139	65	1	𝑍12	𝑍12	PROPN
iajs-3139	65	2	≤	≤	PUNCT
iajs-3139	66	1	𝑍12	𝑍12	PROPN
iajs-3139	66	2	and	and	CCONJ
iajs-3139	66	3	𝑍12=𝑍12	𝑍12=𝑍12	INTJ
iajs-3139	66	4	,	,	PUNCT
iajs-3139	66	5	but	but	CCONJ
iajs-3139	66	6	not	not	PART
iajs-3139	66	7	pr	pr	ADV
iajs-3139	66	8	-	-	PUNCT
iajs-3139	66	9	maximal	maximal	ADJ
iajs-3139	66	10	since	since	SCONJ
iajs-3139	66	11	4𝑍12	4𝑍12	PROPN
iajs-3139	66	12	<	<	X
iajs-3139	66	13	2𝑍12	2𝑍12	PROPN
iajs-3139	66	14	≤	≤	NOUN
iajs-3139	67	1	𝑍12	𝑍12	VERB
iajs-3139	67	2	and	and	CCONJ
iajs-3139	67	3	2𝑍12	2𝑍12	PROPN
iajs-3139	67	4	is	be	AUX
iajs-3139	67	5	not	not	PART
iajs-3139	67	6	pure	pure	ADJ
iajs-3139	67	7	in	in	ADP
iajs-3139	67	8	𝑍12	𝑍12	PROPN
iajs-3139	67	9	proposition	proposition	NOUN
iajs-3139	67	10	(	(	PUNCT
iajs-3139	67	11	3.5	3.5	NUM
iajs-3139	67	12	)	)	PUNCT
iajs-3139	67	13	if	if	SCONJ
iajs-3139	67	14	is	be	AUX
iajs-3139	67	15	an	an	DET
iajs-3139	67	16	f	f	NOUN
iajs-3139	67	17	-	-	PUNCT
iajs-3139	67	18	regular	regular	ADJ
iajs-3139	67	19	module	module	NOUN
iajs-3139	67	20	then	then	ADV
iajs-3139	67	21	every	every	DET
iajs-3139	67	22	submodule	submodule	NOUN
iajs-3139	67	23	of	of	ADP
iajs-3139	67	24	is	be	AUX
iajs-3139	67	25	,	,	PUNCT
iajs-3139	67	26	pr	pr	NOUN
iajs-3139	67	27	-	-	PUNCT
iajs-3139	67	28	maximal	maximal	ADJ
iajs-3139	67	29	.	.	PUNCT
iajs-3139	68	1	proof	proof	NOUN
iajs-3139	68	2	let	let	AUX
iajs-3139	68	3	be	be	AUX
iajs-3139	68	4	submodule	submodule	NOUN
iajs-3139	68	5	of	of	ADP
iajs-3139	68	6	such	such	DET
iajs-3139	68	7	that	that	SCONJ
iajs-3139	68	8	since	since	SCONJ
iajs-3139	68	9	is	be	AUX
iajs-3139	68	10	f	f	NOUN
iajs-3139	68	11	-	-	ADJ
iajs-3139	68	12	regular	regular	ADJ
iajs-3139	68	13	then	then	ADV
iajs-3139	68	14	respective	respective	ADJ
iajs-3139	68	15	submodule	submodule	NOUN
iajs-3139	68	16	of	of	ADP
iajs-3139	68	17	is	be	AUX
iajs-3139	68	18	pure	pure	ADJ
iajs-3139	68	19	hence	hence	ADV
iajs-3139	68	20	is	be	AUX
iajs-3139	68	21	pure	pure	ADJ
iajs-3139	68	22	submodule	submodule	NOUN
iajs-3139	68	23	of	of	ADP
iajs-3139	68	24	then	then	ADV
iajs-3139	68	25	is	be	AUX
iajs-3139	68	26	pr	pr	ADV
iajs-3139	68	27	-	-	PUNCT
iajs-3139	68	28	maximal	maximal	ADJ
iajs-3139	68	29	.	.	PUNCT
iajs-3139	69	1	remark	remark	NOUN
iajs-3139	69	2	(	(	PUNCT
iajs-3139	69	3	3.6	3.6	NUM
iajs-3139	69	4	)	)	PUNCT
iajs-3139	69	5	if	if	SCONJ
iajs-3139	69	6	is	be	AUX
iajs-3139	69	7	simple	simple	ADJ
iajs-3139	69	8	-module	-module	NOUN
iajs-3139	69	9	thence	thence	NOUN
iajs-3139	69	10	every	every	DET
iajs-3139	69	11	-submodule	-submodule	NOUN
iajs-3139	69	12	of	of	ADP
iajs-3139	69	13	w	w	PROPN
iajs-3139	69	14	is	be	AUX
iajs-3139	69	15	pr	pr	ADV
iajs-3139	69	16	-	-	PUNCT
iajs-3139	69	17	maximal	maximal	ADJ
iajs-3139	69	18	.	.	PUNCT
iajs-3139	70	1	proof	proof	NOUN
iajs-3139	70	2	:	:	PUNCT
iajs-3139	70	3	since	since	SCONJ
iajs-3139	70	4	every	every	DET
iajs-3139	70	5	simple	simple	ADJ
iajs-3139	70	6	module	module	NOUN
iajs-3139	70	7	is	be	AUX
iajs-3139	70	8	regular	regular	ADJ
iajs-3139	70	9	,	,	PUNCT
iajs-3139	70	10	thence	thence	NOUN
iajs-3139	70	11	every	every	DET
iajs-3139	70	12	maximal	maximal	ADJ
iajs-3139	70	13	is	be	AUX
iajs-3139	70	14	pr	pr	ADV
iajs-3139	70	15	-	-	PUNCT
iajs-3139	70	16	maximal	maximal	ADJ
iajs-3139	70	17	.	.	PUNCT
iajs-3139	71	1	proposition	proposition	NOUN
iajs-3139	71	2	(	(	PUNCT
iajs-3139	71	3	3.7	3.7	NUM
iajs-3139	71	4	)	)	PUNCT
iajs-3139	71	5	if	if	SCONJ
iajs-3139	71	6	𝑊	𝑊	PROPN
iajs-3139	71	7	𝐻	𝐻	PROPN
iajs-3139	71	8	is	be	AUX
iajs-3139	71	9	f	f	ADJ
iajs-3139	71	10	-	-	PUNCT
iajs-3139	71	11	regular	regular	ADJ
iajs-3139	71	12	module	module	NOUN
iajs-3139	71	13	then	then	ADV
iajs-3139	71	14	any	any	DET
iajs-3139	71	15	pure	pure	ADJ
iajs-3139	71	16	submodule	submodule	NOUN
iajs-3139	71	17	of	of	ADP
iajs-3139	71	18	is	be	AUX
iajs-3139	71	19	pr	pr	ADV
iajs-3139	71	20	-	-	PUNCT
iajs-3139	71	21	maximal	maximal	ADJ
iajs-3139	71	22	.	.	PUNCT
iajs-3139	72	1	proof	proof	NOUN
iajs-3139	72	2	let	let	VERB
iajs-3139	72	3	h	h	NOUN
iajs-3139	72	4	is	be	AUX
iajs-3139	72	5	a	a	DET
iajs-3139	72	6	pure	pure	ADJ
iajs-3139	72	7	r	r	NOUN
iajs-3139	72	8	-	-	PUNCT
iajs-3139	72	9	submodule	submodule	NOUN
iajs-3139	72	10	of	of	ADP
iajs-3139	72	11	w	w	NOUN
iajs-3139	72	12	such	such	ADJ
iajs-3139	72	13	that	that	DET
iajs-3139	72	14	h	h	NOUN
iajs-3139	72	15	and	and	CCONJ
iajs-3139	72	16	since	since	SCONJ
iajs-3139	72	17	𝑊	𝑊	PROPN
iajs-3139	72	18	𝐻	𝐻	PROPN
iajs-3139	72	19	is	be	AUX
iajs-3139	72	20	f	f	X
iajs-3139	72	21	-	-	PUNCT
iajs-3139	72	22	regular	regular	ADJ
iajs-3139	72	23	then	then	ADV
iajs-3139	72	24	is	be	AUX
iajs-3139	72	25	pure	pure	ADJ
iajs-3139	72	26	submodule	submodule	NOUN
iajs-3139	72	27	of	of	ADP
iajs-3139	72	28	𝑊	𝑊	PROPN
iajs-3139	72	29	𝐻	𝐻	PROPN
iajs-3139	72	30	since	since	SCONJ
iajs-3139	72	31	is	be	AUX
iajs-3139	72	32	pure	pure	ADJ
iajs-3139	72	33	of	of	ADP
iajs-3139	72	34	w	w	NOUN
iajs-3139	72	35	then	then	ADV
iajs-3139	72	36	is	be	AUX
iajs-3139	72	37	pure	pure	ADJ
iajs-3139	72	38	in	in	ADP
iajs-3139	72	39	[	[	PUNCT
iajs-3139	72	40	4	4	NUM
iajs-3139	72	41	]	]	PUNCT
iajs-3139	72	42	implies	imply	VERB
iajs-3139	72	43	is	be	AUX
iajs-3139	72	44	pr	pr	NOUN
iajs-3139	72	45	-	-	PUNCT
iajs-3139	72	46	maximal	maximal	ADJ
iajs-3139	72	47	submodule	submodule	NOUN
iajs-3139	72	48	of	of	ADP
iajs-3139	72	49	.	.	PUNCT
iajs-3139	73	1	corollary	corollary	ADJ
iajs-3139	73	2	(	(	PUNCT
iajs-3139	73	3	3.8	3.8	NUM
iajs-3139	73	4	)	)	PUNCT
iajs-3139	73	5	if	if	SCONJ
iajs-3139	73	6	is	be	AUX
iajs-3139	73	7	a	a	DET
iajs-3139	73	8	weak	weak	ADJ
iajs-3139	73	9	r	r	NOUN
iajs-3139	73	10	-	-	PUNCT
iajs-3139	73	11	maximal	maximal	ADJ
iajs-3139	73	12	submodule	submodule	NOUN
iajs-3139	73	13	of	of	ADP
iajs-3139	73	14	an	an	DET
iajs-3139	73	15	r	r	NOUN
iajs-3139	73	16	-	-	PUNCT
iajs-3139	73	17	module	module	NOUN
iajs-3139	73	18	thence	thence	NOUN
iajs-3139	73	19	every	every	DET
iajs-3139	73	20	pure	pure	ADJ
iajs-3139	73	21	submodule	submodule	NOUN
iajs-3139	73	22	of	of	ADP
iajs-3139	73	23	is	be	AUX
iajs-3139	73	24	r	r	NOUN
iajs-3139	73	25	-	-	PUNCT
iajs-3139	73	26	maximal	maximal	ADJ
iajs-3139	73	27	.	.	PUNCT
iajs-3139	74	1	proposition	proposition	NOUN
iajs-3139	74	2	(	(	PUNCT
iajs-3139	74	3	3.9	3.9	NUM
iajs-3139	74	4	)	)	PUNCT
iajs-3139	74	5	if	if	SCONJ
iajs-3139	74	6	are	be	AUX
iajs-3139	74	7	nonzero	nonzero	PROPN
iajs-3139	74	8	submodule	submodule	NOUN
iajs-3139	74	9	of	of	ADP
iajs-3139	74	10	such	such	DET
iajs-3139	74	11	that	that	SCONJ
iajs-3139	74	12	if	if	SCONJ
iajs-3139	74	13	is	be	AUX
iajs-3139	74	14	pr	pr	NOUN
iajs-3139	74	15	-	-	ADJ
iajs-3139	74	16	maximal	maximal	ADJ
iajs-3139	74	17	in	in	ADV
iajs-3139	74	18	then	then	ADV
iajs-3139	74	19	is	be	AUX
iajs-3139	74	20	prmaximal	prmaximal	ADJ
iajs-3139	74	21	in	in	ADP
iajs-3139	74	22	.	.	PUNCT
iajs-3139	75	1	proof	proof	NOUN
iajs-3139	75	2	let	let	VERB
iajs-3139	75	3	submodule	submodule	NOUN
iajs-3139	75	4	of	of	ADP
iajs-3139	75	5	such	such	ADJ
iajs-3139	75	6	that	that	SCONJ
iajs-3139	75	7	since	since	SCONJ
iajs-3139	75	8	𝐾	𝐾	PROPN
iajs-3139	75	9	and	and	CCONJ
iajs-3139	75	10	n	n	NOUN
iajs-3139	75	11	is	be	AUX
iajs-3139	75	12	pr	pr	ADV
iajs-3139	75	13	-	-	ADJ
iajs-3139	75	14	maximal	maximal	ADJ
iajs-3139	75	15	of	of	ADP
iajs-3139	75	16	w	w	NOUN
iajs-3139	75	17	then	then	ADV
iajs-3139	75	18	is	be	AUX
iajs-3139	75	19	pure	pure	ADJ
iajs-3139	75	20	namely	namely	ADV
iajs-3139	75	21	is	be	AUX
iajs-3139	75	22	pr	pr	NOUN
iajs-3139	75	23	-	-	PUNCT
iajs-3139	75	24	maximal	maximal	ADJ
iajs-3139	75	25	assist	assist	NOUN
iajs-3139	75	26	definition	definition	NOUN
iajs-3139	75	27	(	(	PUNCT
iajs-3139	75	28	3.1	3.1	NUM
iajs-3139	75	29	)	)	PUNCT
iajs-3139	75	30	.	.	PUNCT
iajs-3139	76	1	corollary	corollary	ADJ
iajs-3139	76	2	(	(	PUNCT
iajs-3139	76	3	3.10	3.10	NUM
iajs-3139	76	4	)	)	PUNCT
iajs-3139	76	5	if	if	SCONJ
iajs-3139	76	6	are	be	AUX
iajs-3139	76	7	sound	sound	ADJ
iajs-3139	76	8	submodules	submodule	NOUN
iajs-3139	76	9	of	of	ADP
iajs-3139	76	10	r	r	NOUN
iajs-3139	76	11	-	-	PUNCT
iajs-3139	76	12	module	module	NOUN
iajs-3139	76	13	and	and	CCONJ
iajs-3139	76	14	is	be	AUX
iajs-3139	76	15	pr	pr	ADV
iajs-3139	76	16	-	-	PUNCT
iajs-3139	76	17	maximal	maximal	ADJ
iajs-3139	76	18	of	of	ADP
iajs-3139	76	19	,	,	PUNCT
iajs-3139	76	20	then	then	ADV
iajs-3139	76	21	both	both	DET
iajs-3139	76	22	n	n	ADV
iajs-3139	76	23	and	and	CCONJ
iajs-3139	76	24	are	be	AUX
iajs-3139	76	25	pr	pr	NOUN
iajs-3139	76	26	-	-	PUNCT
iajs-3139	76	27	maximal	maximal	ADJ
iajs-3139	76	28	submodule	submodule	NOUN
iajs-3139	76	29	of	of	ADP
iajs-3139	76	30	.	.	PUNCT
iajs-3139	77	1	proof	proof	NOUN
iajs-3139	77	2	since	since	SCONJ
iajs-3139	77	3	and	and	CCONJ
iajs-3139	77	4	is	be	AUX
iajs-3139	77	5	pr	pr	ADV
iajs-3139	77	6	-	-	PUNCT
iajs-3139	77	7	maximal	maximal	ADJ
iajs-3139	77	8	of	of	ADP
iajs-3139	77	9	,	,	PUNCT
iajs-3139	77	10	thence	thence	NOUN
iajs-3139	77	11	assist	assist	NOUN
iajs-3139	77	12	proposition	proposition	NOUN
iajs-3139	77	13	(	(	PUNCT
iajs-3139	77	14	3.9	3.9	NUM
iajs-3139	77	15	)	)	PUNCT
iajs-3139	77	16	is	be	AUX
iajs-3139	77	17	pr	pr	NOUN
iajs-3139	77	18	-	-	PUNCT
iajs-3139	77	19	maximal	maximal	ADJ
iajs-3139	77	20	similarity	similarity	NOUN
iajs-3139	77	21	we	we	PRON
iajs-3139	77	22	prove	prove	VERB
iajs-3139	77	23	is	be	AUX
iajs-3139	77	24	pr	pr	ADV
iajs-3139	77	25	-	-	PUNCT
iajs-3139	77	26	maximal	maximal	ADJ
iajs-3139	77	27	.	.	PUNCT
iajs-3139	78	1	corollary	corollary	ADJ
iajs-3139	78	2	(	(	PUNCT
iajs-3139	78	3	3.11	3.11	NUM
iajs-3139	78	4	)	)	PUNCT
iajs-3139	78	5	if	if	SCONJ
iajs-3139	78	6	𝑁	𝑁	PROPN
iajs-3139	78	7	,	,	PUNCT
iajs-3139	78	8	𝐾	𝐾	PROPN
iajs-3139	78	9	are	be	AUX
iajs-3139	78	10	nonzero	nonzero	ADJ
iajs-3139	78	11	submodules	submodule	NOUN
iajs-3139	78	12	of	of	ADP
iajs-3139	78	13	r	r	NOUN
iajs-3139	78	14	-	-	PUNCT
iajs-3139	78	15	module	module	NOUN
iajs-3139	78	16	if	if	SCONJ
iajs-3139	78	17	or	or	CCONJ
iajs-3139	78	18	are	be	AUX
iajs-3139	78	19	pr	pr	NOUN
iajs-3139	78	20	-	-	PUNCT
iajs-3139	78	21	maximal	maximal	ADJ
iajs-3139	78	22	,	,	PUNCT
iajs-3139	78	23	then	then	ADV
iajs-3139	78	24	is	be	AUX
iajs-3139	78	25	prmaximal	prmaximal	ADJ
iajs-3139	78	26	.	.	PUNCT
iajs-3139	79	1	proof	proof	NOUN
iajs-3139	79	2	:	:	PUNCT
iajs-3139	79	3	since	since	SCONJ
iajs-3139	79	4	is	be	AUX
iajs-3139	79	5	,	,	PUNCT
iajs-3139	79	6	pr	pr	NOUN
iajs-3139	79	7	-	-	ADJ
iajs-3139	79	8	maximal	maximal	ADJ
iajs-3139	79	9	and	and	CCONJ
iajs-3139	79	10	k	k	PROPN
iajs-3139	79	11	≤w	≤w	NOUN
iajs-3139	79	12	assist	assist	VERB
iajs-3139	79	13	proposition	proposition	NOUN
iajs-3139	79	14	(	(	PUNCT
iajs-3139	79	15	3.9	3.9	NUM
iajs-3139	79	16	)	)	PUNCT
iajs-3139	79	17	implies	imply	VERB
iajs-3139	79	18	is	be	AUX
iajs-3139	79	19	prmaximal	prmaximal	ADJ
iajs-3139	79	20	.	.	PUNCT
iajs-3139	80	1	corollary	corollary	ADJ
iajs-3139	80	2	(	(	PUNCT
iajs-3139	80	3	3.12	3.12	NUM
iajs-3139	80	4	)	)	PUNCT
iajs-3139	80	5	if	if	SCONJ
iajs-3139	80	6	is	be	AUX
iajs-3139	80	7	,	,	PUNCT
iajs-3139	80	8	pr	pr	NOUN
iajs-3139	80	9	-	-	PUNCT
iajs-3139	80	10	maximal	maximal	ADJ
iajs-3139	80	11	submodule	submodule	NOUN
iajs-3139	80	12	of	of	ADP
iajs-3139	80	13	an	an	DET
iajs-3139	80	14	r	r	NOUN
iajs-3139	80	15	-	-	PUNCT
iajs-3139	80	16	module	module	NOUN
iajs-3139	81	1	and	and	CCONJ
iajs-3139	81	2	i	i	PRON
iajs-3139	81	3	is	be	AUX
iajs-3139	81	4	an	an	DET
iajs-3139	81	5	ideal	ideal	NOUN
iajs-3139	81	6	of	of	ADP
iajs-3139	81	7	r	r	NOUN
iajs-3139	81	8	,	,	PUNCT
iajs-3139	81	9	if	if	SCONJ
iajs-3139	81	10	[	[	X
iajs-3139	81	11	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	81	12	:	:	PUNCT
iajs-3139	81	13	𝐼	𝐼	PROPN
iajs-3139	81	14	]	]	PUNCT
iajs-3139	81	15	is	be	AUX
iajs-3139	81	16	a	a	DET
iajs-3139	81	17	proper	proper	ADJ
iajs-3139	81	18	submodule	submodule	NOUN
iajs-3139	81	19	of	of	ADP
iajs-3139	81	20	then	then	ADV
iajs-3139	81	21	,	,	PUNCT
iajs-3139	81	22	[	[	X
iajs-3139	81	23	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	81	24	:	:	PUNCT
iajs-3139	81	25	𝐼	𝐼	PROPN
iajs-3139	81	26	]	]	PUNCT
iajs-3139	81	27	is	be	AUX
iajs-3139	81	28	pr	pr	NOUN
iajs-3139	81	29	-	-	PUNCT
iajs-3139	81	30	maximal	maximal	ADJ
iajs-3139	81	31	-submodule	-submodule	NOUN
iajs-3139	81	32	of	of	ADP
iajs-3139	81	33	.	.	PUNCT
iajs-3139	82	1	then	then	ADV
iajs-3139	82	2	[	[	X
iajs-3139	82	3	[	[	X
iajs-3139	82	4	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	82	5	:	:	PUNCT
iajs-3139	82	6	𝐼	𝐼	PROPN
iajs-3139	82	7	]	]	PUNCT
iajs-3139	82	8	is	be	AUX
iajs-3139	82	9	pr	pr	NOUN
iajs-3139	82	10	-	-	PUNCT
iajs-3139	82	11	maximal	maximal	ADJ
iajs-3139	82	12	submodule	submodule	NOUN
iajs-3139	82	13	of	of	ADP
iajs-3139	82	14	w.	w.	PROPN
iajs-3139	82	15	ihjpas	ihjpas	PROPN
iajs-3139	82	16	.	.	PUNCT
iajs-3139	83	1	36	36	NUM
iajs-3139	83	2	(	(	PUNCT
iajs-3139	83	3	4	4	NUM
iajs-3139	83	4	)	)	PUNCT
iajs-3139	83	5	2023	2023	NUM
iajs-3139	83	6	363	363	NUM
iajs-3139	83	7	proof	proof	NOUN
iajs-3139	83	8	since	since	SCONJ
iajs-3139	83	9	[	[	X
iajs-3139	83	10	𝑁𝑊	𝑁𝑊	PROPN
iajs-3139	83	11	:	:	PUNCT
iajs-3139	83	12	𝐼	𝐼	PROPN
iajs-3139	83	13	]	]	PUNCT
iajs-3139	83	14	assist	assist	NOUN
iajs-3139	83	15	[	[	X
iajs-3139	83	16	4	4	NUM
iajs-3139	83	17	]	]	PUNCT
iajs-3139	83	18	and	and	CCONJ
iajs-3139	83	19	is	be	AUX
iajs-3139	83	20	pr	pr	NOUN
iajs-3139	83	21	-	-	PUNCT
iajs-3139	83	22	maximal	maximal	ADJ
iajs-3139	83	23	submodule	submodule	NOUN
iajs-3139	83	24	of	of	ADP
iajs-3139	83	25	then	then	ADV
iajs-3139	83	26	,	,	PUNCT
iajs-3139	83	27	[	[	X
iajs-3139	83	28	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	83	29	:	:	PUNCT
iajs-3139	83	30	𝐼	𝐼	PROPN
iajs-3139	83	31	]	]	PUNCT
iajs-3139	83	32	is	be	AUX
iajs-3139	83	33	pr	pr	NOUN
iajs-3139	83	34	-	-	PUNCT
iajs-3139	83	35	maximal	maximal	ADJ
iajs-3139	83	36	assist	assist	NOUN
iajs-3139	83	37	proposition	proposition	NOUN
iajs-3139	83	38	(	(	PUNCT
iajs-3139	83	39	2.10	2.10	NUM
iajs-3139	83	40	)	)	PUNCT
iajs-3139	83	41	.	.	PUNCT
iajs-3139	84	1	remark	remark	NOUN
iajs-3139	84	2	(	(	PUNCT
iajs-3139	84	3	3.13	3.13	NUM
iajs-3139	84	4	)	)	PUNCT
iajs-3139	84	5	the	the	DET
iajs-3139	84	6	convers	conver	NOUN
iajs-3139	84	7	of	of	ADP
iajs-3139	84	8	corollary	corollary	ADJ
iajs-3139	84	9	(	(	PUNCT
iajs-3139	84	10	2.12	2.12	NUM
iajs-3139	84	11	)	)	PUNCT
iajs-3139	84	12	is	be	AUX
iajs-3139	84	13	not	not	PART
iajs-3139	84	14	true	true	ADJ
iajs-3139	84	15	since	since	SCONJ
iajs-3139	84	16	submodule	submodule	NOUN
iajs-3139	84	17	of	of	ADP
iajs-3139	84	18	pr	pr	NOUN
iajs-3139	84	19	-	-	ADJ
iajs-3139	84	20	maximal	maximal	ADJ
iajs-3139	84	21	not	not	PART
iajs-3139	84	22	pr	pr	ADV
iajs-3139	84	23	-	-	ADV
iajs-3139	84	24	maximal	maximal	ADJ
iajs-3139	84	25	as	as	ADP
iajs-3139	84	26	the	the	DET
iajs-3139	84	27	following	following	ADJ
iajs-3139	84	28	example	example	NOUN
iajs-3139	84	29	let	let	VERB
iajs-3139	84	30	=	=	PRON
iajs-3139	84	31	𝑍12as	𝑍12a	NOUN
iajs-3139	84	32	z	z	NOUN
iajs-3139	84	33	-	-	PUNCT
iajs-3139	84	34	module	module	NOUN
iajs-3139	84	35	the	the	DET
iajs-3139	84	36	ideal	ideal	NOUN
iajs-3139	84	37	of	of	ADP
iajs-3139	84	38	z	z	NOUN
iajs-3139	84	39	-	-	PUNCT
iajs-3139	84	40	module	module	NOUN
iajs-3139	84	41	the	the	DET
iajs-3139	84	42	ideal	ideal	ADJ
iajs-3139	84	43	i=2z	i=2z	NOUN
iajs-3139	84	44	of	of	ADP
iajs-3139	84	45	z	z	NOUN
iajs-3139	84	46	,	,	PUNCT
iajs-3139	84	47	and	and	CCONJ
iajs-3139	85	1	n=6𝑍12	n=6𝑍12	VERB
iajs-3139	86	1	so	so	ADV
iajs-3139	86	2	[	[	X
iajs-3139	86	3	𝑁𝑊	𝑁𝑊	NOUN
iajs-3139	86	4	:	:	PUNCT
iajs-3139	86	5	𝐼]={0̅,2̅	𝐼]={0̅,2̅	ADV
iajs-3139	86	6	,	,	PUNCT
iajs-3139	86	7	4̅	4̅	ADJ
iajs-3139	86	8	,	,	PUNCT
iajs-3139	86	9	6̅	6̅	PROPN
iajs-3139	86	10	,	,	PUNCT
iajs-3139	86	11	8̅	8̅	NUM
iajs-3139	86	12	,	,	PUNCT
iajs-3139	86	13	10̅̅̅̅	10̅̅̅̅	NUM
iajs-3139	86	14	}	}	PUNCT
iajs-3139	86	15	is	be	AUX
iajs-3139	86	16	pr	pr	NOUN
iajs-3139	86	17	-	-	ADJ
iajs-3139	86	18	maximal	maximal	ADJ
iajs-3139	86	19	since	since	SCONJ
iajs-3139	86	20	{	{	PUNCT
iajs-3139	86	21	0̅,2̅	0̅,2̅	PROPN
iajs-3139	86	22	,	,	PUNCT
iajs-3139	86	23	4̅	4̅	PROPN
iajs-3139	86	24	,	,	PUNCT
iajs-3139	86	25	6̅	6̅	PROPN
iajs-3139	86	26	,	,	PUNCT
iajs-3139	86	27	8̅	8̅	NUM
iajs-3139	86	28	,	,	PUNCT
iajs-3139	86	29	10̅̅̅̅	10̅̅̅̅	NUM
iajs-3139	86	30	}	}	PUNCT
iajs-3139	86	31	<	<	X
iajs-3139	87	1	𝑍12	𝑍12	PROPN
iajs-3139	87	2	≤	≤	PUNCT
iajs-3139	88	1	𝑍12	𝑍12	PROPN
iajs-3139	88	2	but	but	CCONJ
iajs-3139	88	3	𝑁	𝑁	PROPN
iajs-3139	88	4	=	=	PUNCT
iajs-3139	88	5	6𝑍12	6𝑍12	PROPN
iajs-3139	88	6	is	be	AUX
iajs-3139	88	7	not	not	PART
iajs-3139	88	8	pr	pr	NOUN
iajs-3139	88	9	-	-	ADV
iajs-3139	88	10	maximal	maximal	ADJ
iajs-3139	88	11	of	of	ADP
iajs-3139	88	12	𝑍12	𝑍12	PROPN
iajs-3139	88	13	,	,	PUNCT
iajs-3139	89	1	since	since	SCONJ
iajs-3139	89	2	6𝑍12	6𝑍12	VERB
iajs-3139	89	3	<	<	X
iajs-3139	89	4	2𝑍12	2𝑍12	PROPN
iajs-3139	89	5	≤	≤	NOUN
iajs-3139	89	6	𝑍12	𝑍12	VERB
iajs-3139	89	7	and	and	CCONJ
iajs-3139	89	8	2𝑍12	2𝑍12	NUM
iajs-3139	89	9	not	not	PART
iajs-3139	89	10	pure	pure	ADJ
iajs-3139	89	11	in	in	ADP
iajs-3139	89	12	𝑍12	𝑍12	PROPN
iajs-3139	89	13	,	,	PUNCT
iajs-3139	89	14	proposition	proposition	NOUN
iajs-3139	89	15	(	(	PUNCT
iajs-3139	89	16	3.15	3.15	NUM
iajs-3139	89	17	)	)	PUNCT
iajs-3139	89	18	let	let	VERB
iajs-3139	89	19	:	:	PUNCT
iajs-3139	89	20	be	be	AUX
iajs-3139	89	21	an	an	DET
iajs-3139	89	22	epimorphism	epimorphism	NOUN
iajs-3139	89	23	,	,	PUNCT
iajs-3139	89	24	where	where	SCONJ
iajs-3139	89	25	be	be	AUX
iajs-3139	89	26	r	r	NOUN
iajs-3139	89	27	-	-	PUNCT
iajs-3139	89	28	modules	module	NOUN
iajs-3139	89	29	.	.	PUNCT
iajs-3139	90	1	if	if	SCONJ
iajs-3139	90	2	is	be	AUX
iajs-3139	90	3	pr	pr	NOUN
iajs-3139	90	4	-	-	PUNCT
iajs-3139	90	5	maximal	maximal	ADJ
iajs-3139	90	6	submodule	submodule	NOUN
iajs-3139	90	7	of	of	ADP
iajs-3139	90	8	,	,	PUNCT
iajs-3139	90	9	then	then	ADV
iajs-3139	90	10	(	(	PUNCT
iajs-3139	90	11	)	)	PUNCT
iajs-3139	90	12	is	be	AUX
iajs-3139	90	13	pr	pr	NOUN
iajs-3139	90	14	-	-	PUNCT
iajs-3139	90	15	maximal	maximal	ADJ
iajs-3139	90	16	r	r	NOUN
iajs-3139	90	17	-	-	PUNCT
iajs-3139	90	18	submodule	submodule	NOUN
iajs-3139	90	19	of	of	ADP
iajs-3139	90	20	.	.	PUNCT
iajs-3139	91	1	proof	proof	NOUN
iajs-3139	91	2	suppose	suppose	VERB
iajs-3139	91	3	(	(	PUNCT
iajs-3139	91	4	)	)	PUNCT
iajs-3139	91	5	,	,	PUNCT
iajs-3139	91	6	so	so	CCONJ
iajs-3139	91	7	(	(	PUNCT
iajs-3139	91	8	(	(	PUNCT
iajs-3139	91	9	)	)	PUNCT
iajs-3139	91	10	)	)	PUNCT
iajs-3139	91	11	then	then	ADV
iajs-3139	91	12	,	,	PUNCT
iajs-3139	91	13	since	since	SCONJ
iajs-3139	91	14	is	be	AUX
iajs-3139	91	15	pr	pr	NOUN
iajs-3139	91	16	-	-	PUNCT
iajs-3139	91	17	maximal	maximal	ADJ
iajs-3139	91	18	r	r	NOUN
iajs-3139	91	19	-	-	PUNCT
iajs-3139	91	20	submodule	submodule	NOUN
iajs-3139	91	21	of	of	ADP
iajs-3139	91	22	,	,	PUNCT
iajs-3139	91	23	thence	thence	NOUN
iajs-3139	91	24	,	,	PUNCT
iajs-3139	91	25	implies	imply	VERB
iajs-3139	91	26	=	=	PUNCT
iajs-3139	91	27	(	(	PUNCT
iajs-3139	91	28	(	(	PUNCT
iajs-3139	91	29	)	)	PUNCT
iajs-3139	91	30	)	)	PUNCT
iajs-3139	91	31	,	,	PUNCT
iajs-3139	91	32	assist	assist	VERB
iajs-3139	91	33	lemma	lemma	PROPN
iajs-3139	91	34	(	(	PUNCT
iajs-3139	91	35	2.14	2.14	NUM
iajs-3139	91	36	)	)	PUNCT
iajs-3139	91	37	,	,	PUNCT
iajs-3139	91	38	so	so	CCONJ
iajs-3139	91	39	(	(	PUNCT
iajs-3139	91	40	)	)	PUNCT
iajs-3139	91	41	is	be	AUX
iajs-3139	91	42	a	a	DET
iajs-3139	91	43	pr	pr	NOUN
iajs-3139	91	44	-	-	PUNCT
iajs-3139	91	45	maximal	maximal	ADJ
iajs-3139	91	46	r	r	NOUN
iajs-3139	91	47	-	-	PUNCT
iajs-3139	91	48	submodule	submodule	NOUN
iajs-3139	91	49	of	of	ADP
iajs-3139	91	50	theorem	theorem	NOUN
iajs-3139	91	51	(	(	PUNCT
iajs-3139	91	52	3.16	3.16	NUM
iajs-3139	91	53	)	)	PUNCT
iajs-3139	91	54	if	if	SCONJ
iajs-3139	91	55	is	be	AUX
iajs-3139	91	56	a	a	DET
iajs-3139	91	57	pure	pure	ADJ
iajs-3139	91	58	simple	simple	ADJ
iajs-3139	91	59	r	r	NOUN
iajs-3139	91	60	-	-	PUNCT
iajs-3139	91	61	module	module	NOUN
iajs-3139	91	62	and	and	CCONJ
iajs-3139	91	63	be	be	AUX
iajs-3139	91	64	a	a	DET
iajs-3139	91	65	proper	proper	ADJ
iajs-3139	91	66	submodule	submodule	NOUN
iajs-3139	91	67	of	of	ADP
iajs-3139	91	68	.	.	PUNCT
iajs-3139	92	1	then	then	ADV
iajs-3139	92	2	let	let	VERB
iajs-3139	92	3	the	the	DET
iajs-3139	92	4	following	following	NOUN
iajs-3139	92	5	:	:	PUNCT
iajs-3139	93	1	1	1	X
iajs-3139	93	2	.	.	X
iajs-3139	93	3	submodule	submodule	NOUN
iajs-3139	93	4	is	be	AUX
iajs-3139	93	5	maximal	maximal	ADJ
iajs-3139	93	6	.	.	PUNCT
iajs-3139	94	1	2	2	X
iajs-3139	94	2	.	.	X
iajs-3139	94	3	submodule	submodule	NOUN
iajs-3139	94	4	is	be	AUX
iajs-3139	94	5	pr	pr	ADV
iajs-3139	94	6	-	-	PUNCT
iajs-3139	94	7	maximal	maximal	ADJ
iajs-3139	94	8	.	.	PUNCT
iajs-3139	95	1	3	3	X
iajs-3139	95	2	.	.	X
iajs-3139	95	3	submodule	submodule	NOUN
iajs-3139	95	4	is	be	AUX
iajs-3139	95	5	n	n	ADV
iajs-3139	95	6	-	-	PUNCT
iajs-3139	95	7	maximal	maximal	ADJ
iajs-3139	95	8	.	.	PUNCT
iajs-3139	96	1	proof	proof	NOUN
iajs-3139	96	2	:	:	PUNCT
iajs-3139	96	3	(	(	PUNCT
iajs-3139	96	4	1)↔	1)↔	NUM
iajs-3139	96	5	(	(	PUNCT
iajs-3139	96	6	2	2	NUM
iajs-3139	96	7	)	)	PUNCT
iajs-3139	96	8	clearly	clearly	ADV
iajs-3139	96	9	(	(	PUNCT
iajs-3139	96	10	2	2	NUM
iajs-3139	96	11	)	)	PUNCT
iajs-3139	96	12	(	(	PUNCT
iajs-3139	96	13	3	3	X
iajs-3139	96	14	)	)	PUNCT
iajs-3139	96	15	let	let	VERB
iajs-3139	96	16	,	,	PUNCT
iajs-3139	96	17	since	since	SCONJ
iajs-3139	96	18	is	be	AUX
iajs-3139	96	19	,	,	PUNCT
iajs-3139	96	20	pr	pr	NOUN
iajs-3139	96	21	-	-	PUNCT
iajs-3139	96	22	maximal	maximal	ADJ
iajs-3139	96	23	then𝑆	then𝑆	NOUN
iajs-3139	96	24	<	<	X
iajs-3139	96	25	𝑃	𝑃	NOUN
iajs-3139	96	26	𝑊	𝑊	PROPN
iajs-3139	96	27	,	,	PUNCT
iajs-3139	96	28	but	but	CCONJ
iajs-3139	96	29	is	be	AUX
iajs-3139	96	30	semi	semi	ADV
iajs-3139	96	31	simple	simple	ADJ
iajs-3139	96	32	,	,	PUNCT
iajs-3139	96	33	hence	hence	ADV
iajs-3139	96	34	=	=	PUNCT
iajs-3139	96	35	,	,	PUNCT
iajs-3139	96	36	so	so	SCONJ
iajs-3139	96	37	that	that	PRON
iajs-3139	96	38	is	be	AUX
iajs-3139	96	39	n	n	ADV
iajs-3139	96	40	-	-	PUNCT
iajs-3139	96	41	maximal	maximal	ADJ
iajs-3139	96	42	submodule	submodule	NOUN
iajs-3139	96	43	of	of	ADP
iajs-3139	96	44	.	.	PUNCT
iajs-3139	97	1	(	(	PUNCT
iajs-3139	97	2	1	1	X
iajs-3139	97	3	)	)	PUNCT
iajs-3139	97	4	(	(	PUNCT
iajs-3139	97	5	3	3	X
iajs-3139	97	6	)	)	PUNCT
iajs-3139	97	7	clear	clear	ADJ
iajs-3139	97	8	.	.	PUNCT
iajs-3139	98	1	if	if	SCONJ
iajs-3139	98	2	is	be	AUX
iajs-3139	98	3	regular	regular	ADJ
iajs-3139	98	4	,	,	PUNCT
iajs-3139	98	5	then	then	ADV
iajs-3139	98	6	(	(	PUNCT
iajs-3139	98	7	3	3	X
iajs-3139	98	8	)	)	PUNCT
iajs-3139	98	9	(	(	PUNCT
iajs-3139	98	10	1	1	NUM
iajs-3139	98	11	)	)	PUNCT
iajs-3139	98	12	.	.	PUNCT
iajs-3139	99	1	proposition	proposition	NOUN
iajs-3139	99	2	(	(	PUNCT
iajs-3139	99	3	3.17	3.17	NUM
iajs-3139	99	4	)	)	PUNCT
iajs-3139	99	5	if	if	SCONJ
iajs-3139	99	6	is	be	AUX
iajs-3139	99	7	a	a	DET
iajs-3139	99	8	semi	semi	ADJ
iajs-3139	99	9	simple	simple	ADJ
iajs-3139	99	10	r	r	NOUN
iajs-3139	99	11	-	-	PUNCT
iajs-3139	99	12	module	module	NOUN
iajs-3139	99	13	and	and	CCONJ
iajs-3139	99	14	be	be	AUX
iajs-3139	99	15	a	a	DET
iajs-3139	99	16	proper	proper	ADJ
iajs-3139	99	17	submodule	submodule	NOUN
iajs-3139	99	18	of	of	ADP
iajs-3139	99	19	w.	w.	PROPN
iajs-3139	99	20	thence	thence	PROPN
iajs-3139	99	21	consider	consider	VERB
iajs-3139	99	22	the	the	DET
iajs-3139	99	23	following	following	NOUN
iajs-3139	99	24	of	of	ADP
iajs-3139	99	25	1	1	NUM
iajs-3139	99	26	.	.	PUNCT
iajs-3139	100	1	submodule	submodule	NOUN
iajs-3139	100	2	is	be	AUX
iajs-3139	100	3	a	a	DET
iajs-3139	100	4	maximal	maximal	ADJ
iajs-3139	100	5	.	.	PUNCT
iajs-3139	101	1	2	2	X
iajs-3139	101	2	.	.	X
iajs-3139	101	3	submodule	submodule	NOUN
iajs-3139	101	4	is	be	AUX
iajs-3139	101	5	a	a	DET
iajs-3139	101	6	pr	pr	NOUN
iajs-3139	101	7	-	-	PUNCT
iajs-3139	101	8	maximal	maximal	ADJ
iajs-3139	101	9	.	.	PUNCT
iajs-3139	102	1	3	3	X
iajs-3139	102	2	.	.	X
iajs-3139	102	3	is	be	AUX
iajs-3139	102	4	a	a	DET
iajs-3139	102	5	n	n	ADV
iajs-3139	102	6	-	-	PUNCT
iajs-3139	102	7	maximal	maximal	ADJ
iajs-3139	102	8	.	.	PUNCT
iajs-3139	103	1	proof	proof	NOUN
iajs-3139	103	2	:	:	PUNCT
iajs-3139	103	3	(	(	PUNCT
iajs-3139	103	4	1	1	X
iajs-3139	103	5	)	)	PUNCT
iajs-3139	103	6	(	(	PUNCT
iajs-3139	103	7	2	2	X
iajs-3139	103	8	)	)	PUNCT
iajs-3139	103	9	clear	clear	ADJ
iajs-3139	103	10	(	(	PUNCT
iajs-3139	103	11	2	2	NUM
iajs-3139	103	12	)	)	PUNCT
iajs-3139	103	13	(	(	PUNCT
iajs-3139	103	14	3	3	X
iajs-3139	103	15	)	)	PUNCT
iajs-3139	103	16	let	let	VERB
iajs-3139	103	17	,	,	PUNCT
iajs-3139	103	18	since	since	SCONJ
iajs-3139	103	19	is	be	AUX
iajs-3139	103	20	,	,	PUNCT
iajs-3139	103	21	pr	pr	NOUN
iajs-3139	103	22	-	-	PUNCT
iajs-3139	103	23	maximal	maximal	ADJ
iajs-3139	103	24	then𝐾	then𝐾	NOUN
iajs-3139	103	25	<	<	X
iajs-3139	103	26	𝑃	𝑃	NOUN
iajs-3139	103	27	𝑊	𝑊	PROPN
iajs-3139	103	28	,	,	PUNCT
iajs-3139	103	29	but	but	CCONJ
iajs-3139	103	30	is	be	AUX
iajs-3139	103	31	semi	semi	ADV
iajs-3139	103	32	simple	simple	ADJ
iajs-3139	103	33	,	,	PUNCT
iajs-3139	103	34	hence	hence	ADV
iajs-3139	103	35	,	,	PUNCT
iajs-3139	103	36	so	so	SCONJ
iajs-3139	103	37	that	that	PRON
iajs-3139	103	38	is	be	AUX
iajs-3139	103	39	n	n	ADV
iajs-3139	103	40	-	-	PUNCT
iajs-3139	103	41	maximal	maximal	ADJ
iajs-3139	103	42	r	r	NOUN
iajs-3139	103	43	-	-	PUNCT
iajs-3139	103	44	submodule	submodule	NOUN
iajs-3139	103	45	of	of	ADP
iajs-3139	103	46	(	(	PUNCT
iajs-3139	103	47	1	1	NUM
iajs-3139	103	48	)	)	PUNCT
iajs-3139	103	49	(	(	PUNCT
iajs-3139	103	50	3	3	X
iajs-3139	103	51	)	)	PUNCT
iajs-3139	103	52	clear	clear	ADJ
iajs-3139	103	53	.	.	PUNCT
iajs-3139	104	1	if	if	SCONJ
iajs-3139	104	2	w	w	NOUN
iajs-3139	104	3	is	be	AUX
iajs-3139	104	4	regular	regular	ADJ
iajs-3139	104	5	,	,	PUNCT
iajs-3139	104	6	then	then	ADV
iajs-3139	104	7	(	(	PUNCT
iajs-3139	104	8	3	3	X
iajs-3139	104	9	)	)	PUNCT
iajs-3139	104	10	(	(	PUNCT
iajs-3139	104	11	1	1	X
iajs-3139	104	12	)	)	PUNCT
iajs-3139	104	13	remark	remark	NOUN
iajs-3139	104	14	(	(	PUNCT
iajs-3139	104	15	3.18	3.18	NUM
iajs-3139	104	16	):	):	PUNCT
iajs-3139	104	17	in	in	ADV
iajs-3139	104	18	as	as	SCONJ
iajs-3139	104	19	z	z	NOUN
iajs-3139	104	20	-	-	PUNCT
iajs-3139	104	21	module	module	NOUN
iajs-3139	104	22	consider	consider	VERB
iajs-3139	104	23	the	the	DET
iajs-3139	104	24	following	following	NOUN
iajs-3139	104	25	.	.	PUNCT
iajs-3139	105	1	𝐾	𝐾	NOUN
iajs-3139	105	2	<	<	X
iajs-3139	105	3	𝑍12	𝑍12	PROPN
iajs-3139	105	4	maximal	maximal	ADJ
iajs-3139	106	1	𝑁-maximal	𝑁-maximal	ADJ
iajs-3139	106	2	pr	pr	NOUN
iajs-3139	106	3	-	-	PUNCT
iajs-3139	106	4	maximal	maximal	ADJ
iajs-3139	106	5	2𝑍12	2𝑍12	NOUN
iajs-3139	107	1			PROPN
iajs-3139	107	2			PROPN
iajs-3139	107	3			PROPN
iajs-3139	107	4	3𝑍12	3𝑍12	PROPN
iajs-3139	107	5			PROPN
iajs-3139	107	6			PROPN
iajs-3139	107	7			PROPN
iajs-3139	107	8	4𝑍12	4𝑍12	PROPN
iajs-3139	107	9			PROPN
iajs-3139	107	10			ADV
iajs-3139	107	11			PROPN
iajs-3139	107	12	6𝑍12	6𝑍12	VERB
iajs-3139	107	13			PROPN
iajs-3139	108	1			PROPN
iajs-3139	108	2			PROPN
iajs-3139	108	3	ihjpas	ihjpa	VERB
iajs-3139	108	4	.	.	PUNCT
iajs-3139	109	1	36	36	NUM
iajs-3139	109	2	(	(	PUNCT
iajs-3139	109	3	4	4	NUM
iajs-3139	109	4	)	)	PUNCT
iajs-3139	109	5	2023	2023	NUM
iajs-3139	109	6	364	364	NUM
iajs-3139	109	7	4	4	NUM
iajs-3139	109	8	.	.	PUNCT
iajs-3139	110	1	pure	pure	ADJ
iajs-3139	110	2	maximal	maximal	ADJ
iajs-3139	110	3	submodules	submodule	NOUN
iajs-3139	110	4	and	and	CCONJ
iajs-3139	110	5	related	related	ADJ
iajs-3139	110	6	concepts	concept	NOUN
iajs-3139	110	7	proposition	proposition	NOUN
iajs-3139	110	8	(	(	PUNCT
iajs-3139	110	9	4.1	4.1	NUM
iajs-3139	110	10	)	)	PUNCT
iajs-3139	110	11	whole	whole	ADJ
iajs-3139	110	12	multiplication	multiplication	NOUN
iajs-3139	110	13	rmodule	rmodule	NOUN
iajs-3139	110	14	deemed	deem	VERB
iajs-3139	110	15	a	a	DET
iajs-3139	110	16	pr	pr	NOUN
iajs-3139	110	17	-	-	PUNCT
iajs-3139	110	18	maximal	maximal	ADJ
iajs-3139	110	19	r	r	NOUN
iajs-3139	110	20	-	-	PUNCT
iajs-3139	110	21	submodule	submodule	NOUN
iajs-3139	110	22	.	.	PUNCT
iajs-3139	111	1	proof	proof	NOUN
iajs-3139	111	2	:	:	PUNCT
iajs-3139	111	3	since	since	SCONJ
iajs-3139	111	4	whole	whole	ADJ
iajs-3139	111	5	multiplication	multiplication	NOUN
iajs-3139	111	6	module	module	NOUN
iajs-3139	111	7	breakpoint	breakpoint	NOUN
iajs-3139	111	8	maximal	maximal	ADJ
iajs-3139	111	9	r	r	NOUN
iajs-3139	111	10	-	-	PUNCT
iajs-3139	111	11	submodule	submodule	NOUN
iajs-3139	111	12	thence	thence	NOUN
iajs-3139	111	13	assist	assist	NOUN
iajs-3139	111	14	remark	remark	NOUN
iajs-3139	111	15	(	(	PUNCT
iajs-3139	111	16	3.2	3.2	NUM
iajs-3139	111	17	)	)	PUNCT
iajs-3139	111	18	we	we	PRON
iajs-3139	111	19	have	have	VERB
iajs-3139	111	20	respective	respective	ADJ
iajs-3139	111	21	multiplication	multiplication	NOUN
iajs-3139	111	22	module	module	NOUN
iajs-3139	111	23	contains	contain	VERB
iajs-3139	111	24	a	a	DET
iajs-3139	111	25	pr	pr	NOUN
iajs-3139	111	26	-	-	PUNCT
iajs-3139	111	27	maximal	maximal	ADJ
iajs-3139	111	28	submodule	submodule	NOUN
iajs-3139	111	29	.	.	PUNCT
iajs-3139	112	1	corollary	corollary	NOUN
iajs-3139	112	2	(	(	PUNCT
iajs-3139	112	3	4.2	4.2	NUM
iajs-3139	112	4	)	)	PUNCT
iajs-3139	112	5	every	every	DET
iajs-3139	112	6	cyclic	cyclic	ADJ
iajs-3139	112	7	r	r	NOUN
iajs-3139	112	8	-	-	PUNCT
iajs-3139	112	9	module	module	NOUN
iajs-3139	112	10	breakpoint	breakpoint	NOUN
iajs-3139	112	11	pr	pr	NOUN
iajs-3139	112	12	-	-	PUNCT
iajs-3139	112	13	maximal	maximal	ADJ
iajs-3139	112	14	submodule	submodule	NOUN
iajs-3139	112	15	.	.	PUNCT
iajs-3139	113	1	proof	proof	NOUN
iajs-3139	113	2	since	since	SCONJ
iajs-3139	113	3	respective	respective	ADJ
iajs-3139	113	4	cyclic	cyclic	ADJ
iajs-3139	113	5	r	r	NOUN
iajs-3139	113	6	-	-	PUNCT
iajs-3139	113	7	module	module	NOUN
iajs-3139	113	8	is	be	AUX
iajs-3139	113	9	a	a	DET
iajs-3139	113	10	multiplication	multiplication	NOUN
iajs-3139	113	11	r	r	NOUN
iajs-3139	113	12	-	-	PUNCT
iajs-3139	113	13	module	module	NOUN
iajs-3139	113	14	so	so	ADV
iajs-3139	113	15	assist	assist	VERB
iajs-3139	113	16	proposition	proposition	NOUN
iajs-3139	113	17	(	(	PUNCT
iajs-3139	113	18	4.1	4.1	NUM
iajs-3139	113	19	)	)	PUNCT
iajs-3139	113	20	,	,	PUNCT
iajs-3139	113	21	it	it	PRON
iajs-3139	113	22	breakpoint	breakpoint	VERB
iajs-3139	113	23	pr	pr	NOUN
iajs-3139	113	24	-	-	PUNCT
iajs-3139	113	25	maximal	maximal	ADJ
iajs-3139	113	26	.	.	PUNCT
iajs-3139	114	1	theorem	theorem	NOUN
iajs-3139	114	2	(	(	PUNCT
iajs-3139	114	3	4.3	4.3	NUM
iajs-3139	114	4	)	)	PUNCT
iajs-3139	114	5	if	if	SCONJ
iajs-3139	114	6	is	be	AUX
iajs-3139	114	7	faithful	faithful	ADJ
iajs-3139	114	8	finitely	finitely	ADV
iajs-3139	114	9	generated	generate	VERB
iajs-3139	114	10	and	and	CCONJ
iajs-3139	114	11	multiplication	multiplication	NOUN
iajs-3139	114	12	r	r	NOUN
iajs-3139	114	13	-	-	PUNCT
iajs-3139	114	14	module	module	NOUN
iajs-3139	114	15	and	and	CCONJ
iajs-3139	114	16	be	be	AUX
iajs-3139	114	17	a	a	DET
iajs-3139	114	18	submodule	submodule	NOUN
iajs-3139	114	19	of	of	ADP
iajs-3139	114	20	thence	thence	NOUN
iajs-3139	114	21	,	,	PUNCT
iajs-3139	114	22	the	the	DET
iajs-3139	114	23	following	following	NOUN
iajs-3139	114	24	,	,	PUNCT
iajs-3139	114	25	are	be	AUX
iajs-3139	114	26	equivalent	equivalent	ADJ
iajs-3139	114	27	1	1	NUM
iajs-3139	114	28	.	.	PUNCT
iajs-3139	115	1	h	h	PROPN
iajs-3139	115	2	submodule	submodule	PROPN
iajs-3139	115	3	is	be	AUX
iajs-3139	115	4	,	,	PUNCT
iajs-3139	115	5	pr	pr	NOUN
iajs-3139	115	6	-	-	ADJ
iajs-3139	115	7	maximal	maximal	ADJ
iajs-3139	115	8	of	of	ADP
iajs-3139	115	9	w	w	PROPN
iajs-3139	115	10	2	2	NUM
iajs-3139	115	11	.	.	PUNCT
iajs-3139	116	1	[	[	PUNCT
iajs-3139	116	2	:	:	PUNCT
iajs-3139	116	3	]	]	X
iajs-3139	116	4	ideal	ideal	NOUN
iajs-3139	116	5	is	be	AUX
iajs-3139	116	6	pr	pr	NOUN
iajs-3139	116	7	-	-	ADJ
iajs-3139	116	8	maximal	maximal	ADJ
iajs-3139	116	9	of	of	ADP
iajs-3139	116	10	.	.	PUNCT
iajs-3139	117	1	3	3	X
iajs-3139	117	2	.	.	X
iajs-3139	117	3	=	=	PRON
iajs-3139	117	4	pr	pr	NOUN
iajs-3139	117	5	-	-	NOUN
iajs-3139	117	6	maxima	maxima	NOUN
iajs-3139	117	7	for	for	ADP
iajs-3139	117	8	some	some	DET
iajs-3139	117	9	ideal	ideal	NOUN
iajs-3139	117	10	of	of	ADP
iajs-3139	117	11	.	.	PUNCT
iajs-3139	118	1	proof	proof	NOUN
iajs-3139	118	2	:	:	PUNCT
iajs-3139	118	3	(	(	PUNCT
iajs-3139	118	4	1	1	X
iajs-3139	118	5	)	)	PUNCT
iajs-3139	118	6	→	→	X
iajs-3139	118	7	(	(	PUNCT
iajs-3139	118	8	2	2	X
iajs-3139	118	9	)	)	PUNCT
iajs-3139	118	10	let	let	VERB
iajs-3139	118	11	𝐻𝑅	𝐻𝑅	NOUN
iajs-3139	118	12	:	:	PUNCT
iajs-3139	118	13	𝑊	𝑊	NOUN
iajs-3139	118	14	]	]	PUNCT
iajs-3139	118	15	<	<	X
iajs-3139	118	16	𝐽	𝐽	PROPN
iajs-3139	118	17	≤	≤	PROPN
iajs-3139	118	18	𝑅	𝑅	PROPN
iajs-3139	118	19	and	and	CCONJ
iajs-3139	118	20	j	j	PROPN
iajs-3139	118	21	is	be	AUX
iajs-3139	118	22	an	an	DET
iajs-3139	118	23	ideal	ideal	NOUN
iajs-3139	118	24	of	of	ADP
iajs-3139	118	25	,	,	PUNCT
iajs-3139	118	26	since	since	SCONJ
iajs-3139	118	27	is	be	AUX
iajs-3139	118	28	r	r	NOUN
iajs-3139	118	29	-	-	PUNCT
iajs-3139	118	30	module	module	NOUN
iajs-3139	118	31	multiplication	multiplication	NOUN
iajs-3139	118	32	,	,	PUNCT
iajs-3139	118	33	then	then	ADV
iajs-3139	118	34	𝐻	𝐻	PROPN
iajs-3139	118	35	=	=	PUNCT
iajs-3139	119	1	[	[	X
iajs-3139	119	2	𝐻𝑅	𝐻𝑅	NOUN
iajs-3139	119	3	:	:	PUNCT
iajs-3139	119	4	𝑊]𝑊	𝑊]𝑊	PROPN
iajs-3139	119	5	≤	≤	ADJ
iajs-3139	119	6	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	119	7	≤	≤	NOUN
iajs-3139	119	8	𝑅𝑊	𝑅𝑊	PROPN
iajs-3139	119	9	=	=	SYM
iajs-3139	119	10	𝑊	𝑊	PROPN
iajs-3139	119	11	,	,	PUNCT
iajs-3139	119	12	so	so	ADV
iajs-3139	119	13	𝐻	𝐻	PROPN
iajs-3139	119	14	<	<	X
iajs-3139	119	15	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	119	16	≤	≤	PROPN
iajs-3139	119	17	𝑅𝑊	𝑅𝑊	PROPN
iajs-3139	119	18	,	,	PUNCT
iajs-3139	119	19	since	since	SCONJ
iajs-3139	119	20	is	be	AUX
iajs-3139	119	21	,	,	PUNCT
iajs-3139	119	22	prmaximal	prmaximal	ADJ
iajs-3139	119	23	then	then	ADV
iajs-3139	119	24	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	119	25	≤𝑃	≤𝑃	PART
iajs-3139	119	26	𝑅𝑊	𝑅𝑊	PROPN
iajs-3139	119	27	=	=	SYM
iajs-3139	119	28	𝑊	𝑊	PROPN
iajs-3139	119	29	,	,	PUNCT
iajs-3139	119	30	so	so	ADV
iajs-3139	119	31	𝐽	𝐽	PROPN
iajs-3139	119	32	≤𝑃	≤𝑃	PROPN
iajs-3139	119	33	𝑅	𝑅	PROPN
iajs-3139	120	1	[	[	X
iajs-3139	120	2	9	9	NUM
iajs-3139	120	3	]	]	PUNCT
iajs-3139	120	4	and	and	CCONJ
iajs-3139	120	5	hance	hance	PROPN
iajs-3139	120	6	[	[	X
iajs-3139	120	7	𝐻𝑅	𝐻𝑅	PROPN
iajs-3139	120	8	:	:	PUNCT
iajs-3139	120	9	𝑊	𝑊	NOUN
iajs-3139	120	10	]	]	PUNCT
iajs-3139	120	11	is	be	AUX
iajs-3139	120	12	pr	pr	NOUN
iajs-3139	120	13	-	-	PUNCT
iajs-3139	120	14	maximal	maximal	ADJ
iajs-3139	120	15	ideal	ideal	NOUN
iajs-3139	120	16	of	of	ADP
iajs-3139	120	17	r.	r.	PROPN
iajs-3139	120	18	(	(	PUNCT
iajs-3139	120	19	2)→(3	2)→(3	NUM
iajs-3139	120	20	)	)	PUNCT
iajs-3139	120	21	since	since	SCONJ
iajs-3139	120	22	r	r	NOUN
iajs-3139	120	23	-	-	PUNCT
iajs-3139	120	24	module	module	NOUN
iajs-3139	120	25	is	be	AUX
iajs-3139	120	26	multiplication	multiplication	NOUN
iajs-3139	120	27	thence	thence	NOUN
iajs-3139	120	28	for	for	ADP
iajs-3139	120	29	a	a	DET
iajs-3139	120	30	r	r	NOUN
iajs-3139	120	31	-	-	PUNCT
iajs-3139	120	32	submodule	submodule	NOUN
iajs-3139	120	33	of	of	ADP
iajs-3139	120	34	thence	thence	NOUN
iajs-3139	121	1	[	[	X
iajs-3139	121	2	𝐻𝑅	𝐻𝑅	NOUN
iajs-3139	121	3	:	:	PUNCT
iajs-3139	121	4	𝑊	𝑊	PROPN
iajs-3139	121	5	]	]	PUNCT
iajs-3139	121	6	assist	assist	NOUN
iajs-3139	121	7	(	(	PUNCT
iajs-3139	121	8	2	2	NUM
iajs-3139	121	9	)	)	PUNCT
iajs-3139	121	10	each	each	PRON
iajs-3139	121	11	[	[	X
iajs-3139	121	12	𝐻𝑅	𝐻𝑅	NOUN
iajs-3139	121	13	:	:	PUNCT
iajs-3139	121	14	𝑊	𝑊	NOUN
iajs-3139	121	15	]	]	PUNCT
iajs-3139	121	16	is	be	AUX
iajs-3139	121	17	ideal	ideal	ADJ
iajs-3139	121	18	pr	pr	NOUN
iajs-3139	121	19	-	-	ADJ
iajs-3139	121	20	maximal	maximal	ADJ
iajs-3139	121	21	of	of	ADP
iajs-3139	121	22	then	then	ADV
iajs-3139	121	23	,	,	PUNCT
iajs-3139	121	24	=	=	PUNCT
iajs-3139	121	25	for	for	ADP
iajs-3139	121	26	some	some	DET
iajs-3139	121	27	ideal	ideal	ADJ
iajs-3139	121	28	pr	pr	NOUN
iajs-3139	121	29	-	-	ADJ
iajs-3139	121	30	maximal	maximal	ADJ
iajs-3139	121	31	of	of	ADP
iajs-3139	121	32	.	.	PUNCT
iajs-3139	122	1	(	(	PUNCT
iajs-3139	122	2	3)→(1	3)→(1	X
iajs-3139	122	3	)	)	PUNCT
iajs-3139	122	4	let	let	VERB
iajs-3139	122	5	since	since	SCONJ
iajs-3139	122	6	=	=	PUNCT
iajs-3139	122	7	for	for	ADP
iajs-3139	122	8	some	some	DET
iajs-3139	122	9	ideal	ideal	ADJ
iajs-3139	122	10	pr	pr	NOUN
iajs-3139	122	11	-	-	ADJ
iajs-3139	122	12	maximal	maximal	ADJ
iajs-3139	122	13	of	of	ADP
iajs-3139	122	14	,	,	PUNCT
iajs-3139	122	15	since	since	SCONJ
iajs-3139	122	16	w	w	PROPN
iajs-3139	122	17	is	be	AUX
iajs-3139	122	18	,	,	PUNCT
iajs-3139	122	19	multiplication	multiplication	NOUN
iajs-3139	122	20	then,𝐾	then,𝐾	NOUN
iajs-3139	122	21	=	=	SYM
iajs-3139	122	22	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	122	23	for	for	ADP
iajs-3139	122	24	some	some	DET
iajs-3139	122	25	pr	pr	NOUN
iajs-3139	122	26	-	-	PUNCT
iajs-3139	122	27	maximal	maximal	ADJ
iajs-3139	122	28	ideal	ideal	NOUN
iajs-3139	123	1	𝐽of	𝐽of	PROPN
iajs-3139	123	2	since	since	SCONJ
iajs-3139	123	3	w	w	PROPN
iajs-3139	123	4	is	be	AUX
iajs-3139	123	5	multiplication	multiplication	NOUN
iajs-3139	123	6	,	,	PUNCT
iajs-3139	123	7	then	then	ADV
iajs-3139	123	8	,	,	PUNCT
iajs-3139	123	9	𝐻	𝐻	PROPN
iajs-3139	123	10	=	=	SYM
iajs-3139	123	11	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	123	12	<	<	X
iajs-3139	123	13	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	123	14	≤	≤	PROPN
iajs-3139	123	15	𝑅𝑊	𝑅𝑊	PROPN
iajs-3139	124	1	=	=	SYM
iajs-3139	124	2	𝑊but	𝑊but	PROPN
iajs-3139	124	3	w	w	ADP
iajs-3139	124	4	faithful	faithful	ADJ
iajs-3139	124	5	finitely	finitely	ADV
iajs-3139	124	6	generated	generate	VERB
iajs-3139	124	7	multiplication	multiplication	NOUN
iajs-3139	124	8	then	then	ADV
iajs-3139	124	9	𝐼	𝐼	PROPN
iajs-3139	124	10	<	<	X
iajs-3139	124	11	𝐽	𝐽	PROPN
iajs-3139	124	12	≤	≤	PROPN
iajs-3139	124	13	𝑅	𝑅	PROPN
iajs-3139	124	14	,	,	PUNCT
iajs-3139	124	15	by	by	ADP
iajs-3139	124	16	(	(	PUNCT
iajs-3139	124	17	3	3	X
iajs-3139	124	18	)	)	PUNCT
iajs-3139	124	19	𝐼	𝐼	NOUN
iajs-3139	124	20	is	be	AUX
iajs-3139	124	21	pr	pr	NOUN
iajs-3139	124	22	-	-	PUNCT
iajs-3139	124	23	maximal	maximal	ADJ
iajs-3139	124	24	ideal	ideal	NOUN
iajs-3139	124	25	of	of	ADP
iajs-3139	124	26	,	,	PUNCT
iajs-3139	124	27	hance	hance	PROPN
iajs-3139	124	28	𝐽	𝐽	PROPN
iajs-3139	124	29	≤𝑃	≤𝑃	PROPN
iajs-3139	124	30	𝑅	𝑅	PROPN
iajs-3139	124	31	then	then	ADV
iajs-3139	124	32	𝐽𝑊	𝐽𝑊	PROPN
iajs-3139	124	33	≤𝑃	≤𝑃	ADP
iajs-3139	124	34	𝑊	𝑊	NOUN
iajs-3139	124	35	by	by	ADP
iajs-3139	124	36	[	[	PUNCT
iajs-3139	124	37	8	8	NUM
iajs-3139	124	38	]	]	PUNCT
iajs-3139	124	39	implies	imply	VERB
iajs-3139	124	40	is	be	AUX
iajs-3139	124	41	pr	pr	NOUN
iajs-3139	124	42	-	-	ADJ
iajs-3139	124	43	maximal	maximal	ADJ
iajs-3139	124	44	of	of	ADP
iajs-3139	124	45	.	.	PUNCT
iajs-3139	125	1	now	now	ADV
iajs-3139	125	2	,	,	PUNCT
iajs-3139	125	3	we	we	PRON
iajs-3139	125	4	introduce	introduce	VERB
iajs-3139	125	5	the	the	DET
iajs-3139	125	6	following	follow	VERB
iajs-3139	125	7	concept	concept	NOUN
iajs-3139	125	8	.	.	PUNCT
iajs-3139	126	1	definition	definition	NOUN
iajs-3139	126	2	(	(	PUNCT
iajs-3139	126	3	4.4	4.4	NUM
iajs-3139	126	4	)	)	PUNCT
iajs-3139	126	5	an	an	DET
iajs-3139	126	6	r	r	NOUN
iajs-3139	126	7	-	-	PUNCT
iajs-3139	126	8	module	module	NOUN
iajs-3139	126	9	is	be	AUX
iajs-3139	126	10	named	name	VERB
iajs-3139	126	11	pr	pr	NOUN
iajs-3139	126	12	-	-	PUNCT
iajs-3139	126	13	module	module	NOUN
iajs-3139	126	14	,	,	PUNCT
iajs-3139	126	15	if	if	SCONJ
iajs-3139	126	16	every	every	DET
iajs-3139	126	17	proper	proper	ADJ
iajs-3139	126	18	submodule	submodule	NOUN
iajs-3139	126	19	of	of	ADP
iajs-3139	126	20	is	be	AUX
iajs-3139	126	21	pr	pr	ADV
iajs-3139	126	22	-	-	PUNCT
iajs-3139	126	23	maximal	maximal	ADJ
iajs-3139	126	24	.	.	PUNCT
iajs-3139	127	1	a	a	DET
iajs-3139	127	2	ring	ring	NOUN
iajs-3139	127	3	is	be	AUX
iajs-3139	127	4	named	name	VERB
iajs-3139	127	5	pr	pr	NOUN
iajs-3139	127	6	-	-	PUNCT
iajs-3139	127	7	ring	ring	NOUN
iajs-3139	127	8	if	if	SCONJ
iajs-3139	127	9	every	every	DET
iajs-3139	127	10	proper	proper	ADJ
iajs-3139	127	11	ideal	ideal	NOUN
iajs-3139	127	12	of	of	ADP
iajs-3139	127	13	is	be	AUX
iajs-3139	127	14	an	an	DET
iajs-3139	127	15	pr	pr	NOUN
iajs-3139	127	16	-	-	PUNCT
iajs-3139	127	17	maximal	maximal	ADJ
iajs-3139	127	18	ideal	ideal	NOUN
iajs-3139	127	19	.	.	PUNCT
iajs-3139	128	1	examples	example	NOUN
iajs-3139	128	2	(	(	PUNCT
iajs-3139	128	3	4.5	4.5	NUM
iajs-3139	128	4	):	):	PUNCT
iajs-3139	128	5	1	1	NUM
iajs-3139	128	6	.	.	PUNCT
iajs-3139	128	7	is	be	AUX
iajs-3139	128	8	pr	pr	NOUN
iajs-3139	128	9	-	-	PUNCT
iajs-3139	128	10	module	module	NOUN
iajs-3139	128	11	as	as	ADP
iajs-3139	128	12	z	z	NOUN
iajs-3139	128	13	-	-	PUNCT
iajs-3139	128	14	module	module	NOUN
iajs-3139	128	15	2	2	NUM
iajs-3139	128	16	.	.	PUNCT
iajs-3139	128	17	whole	whole	ADJ
iajs-3139	128	18	semisimpl	semisimpl	NOUN
iajs-3139	128	19	r	r	NOUN
iajs-3139	128	20	-	-	PUNCT
iajs-3139	128	21	module	module	NOUN
iajs-3139	128	22	is	be	AUX
iajs-3139	128	23	pr	pr	NOUN
iajs-3139	128	24	-	-	PUNCT
iajs-3139	128	25	module	module	NOUN
iajs-3139	128	26	.	.	PUNCT
iajs-3139	129	1	3	3	X
iajs-3139	129	2	.	.	X
iajs-3139	130	1	as	as	SCONJ
iajs-3139	130	2	z	z	NOUN
iajs-3139	130	3	-	-	PUNCT
iajs-3139	130	4	module	module	NOUN
iajs-3139	130	5	is	be	AUX
iajs-3139	130	6	not	not	PART
iajs-3139	130	7	pr	pr	NOUN
iajs-3139	130	8	-	-	PUNCT
iajs-3139	130	9	module	module	NOUN
iajs-3139	130	10	,	,	PUNCT
iajs-3139	130	11	since	since	SCONJ
iajs-3139	130	12	6	6	NUM
iajs-3139	130	13	is	be	AUX
iajs-3139	130	14	not	not	PART
iajs-3139	130	15	pr	pr	ADV
iajs-3139	130	16	-	-	PUNCT
iajs-3139	130	17	maximal	maximal	ADJ
iajs-3139	130	18	submodule	submodule	NOUN
iajs-3139	130	19	of	of	ADP
iajs-3139	130	20	.	.	PUNCT
iajs-3139	131	1	theorem	theorem	NOUN
iajs-3139	131	2	(	(	PUNCT
iajs-3139	131	3	4.6	4.6	NUM
iajs-3139	131	4	):	):	PUNCT
iajs-3139	131	5	if	if	SCONJ
iajs-3139	131	6	is	be	AUX
iajs-3139	131	7	a	a	DET
iajs-3139	131	8	r	r	NOUN
iajs-3139	131	9	-	-	PUNCT
iajs-3139	131	10	module	module	NOUN
iajs-3139	131	11	finitely	finitely	ADV
iajs-3139	131	12	generated	generate	VERB
iajs-3139	131	13	faithful	faithful	ADJ
iajs-3139	131	14	and	and	CCONJ
iajs-3139	131	15	multiplication	multiplication	NOUN
iajs-3139	131	16	.	.	PUNCT
iajs-3139	132	1	then	then	ADV
iajs-3139	132	2	,	,	PUNCT
iajs-3139	132	3	is	be	AUX
iajs-3139	132	4	a	a	DET
iajs-3139	132	5	pr	pr	NOUN
iajs-3139	132	6	-	-	PUNCT
iajs-3139	132	7	module	module	NOUN
iajs-3139	132	8	iff	iff	NOUN
iajs-3139	132	9	is	be	AUX
iajs-3139	132	10	pr	pr	NOUN
iajs-3139	132	11	-	-	PUNCT
iajs-3139	132	12	ring	ring	NOUN
iajs-3139	132	13	.	.	PUNCT
iajs-3139	133	1	ihjpas	ihjpas	PROPN
iajs-3139	133	2	.	.	PUNCT
iajs-3139	134	1	36	36	NUM
iajs-3139	134	2	(	(	PUNCT
iajs-3139	134	3	4	4	NUM
iajs-3139	134	4	)	)	PUNCT
iajs-3139	134	5	2023	2023	NUM
iajs-3139	134	6	365	365	NUM
iajs-3139	134	7	proof	proof	NOUN
iajs-3139	134	8	:	:	PUNCT
iajs-3139	134	9	)	)	PUNCT
iajs-3139	134	10	suppose	suppose	VERB
iajs-3139	134	11	,	,	PUNCT
iajs-3139	134	12	is	be	AUX
iajs-3139	134	13	a	a	DET
iajs-3139	134	14	pr	pr	NOUN
iajs-3139	134	15	-	-	PUNCT
iajs-3139	134	16	module	module	NOUN
iajs-3139	134	17	and	and	CCONJ
iajs-3139	134	18	impose	impose	VERB
iajs-3139	134	19	is	be	AUX
iajs-3139	134	20	a	a	DET
iajs-3139	134	21	proper	proper	ADJ
iajs-3139	134	22	ideal	ideal	NOUN
iajs-3139	134	23	of	of	ADP
iajs-3139	134	24	.	.	PUNCT
iajs-3139	135	1	since	since	SCONJ
iajs-3139	135	2	is	be	AUX
iajs-3139	135	3	a	a	DET
iajs-3139	135	4	multiplication	multiplication	NOUN
iajs-3139	135	5	r	r	NOUN
iajs-3139	135	6	-	-	PUNCT
iajs-3139	135	7	module	module	NOUN
iajs-3139	135	8	,	,	PUNCT
iajs-3139	135	9	then	then	ADV
iajs-3139	135	10	,	,	PUNCT
iajs-3139	135	11	.	.	PUNCT
iajs-3139	136	1	but	but	CCONJ
iajs-3139	136	2	is	be	AUX
iajs-3139	136	3	pr	pr	NOUN
iajs-3139	136	4	-	-	PUNCT
iajs-3139	136	5	module	module	NOUN
iajs-3139	136	6	,	,	PUNCT
iajs-3139	136	7	hence	hence	ADV
iajs-3139	136	8	is	be	AUX
iajs-3139	136	9	a	a	DET
iajs-3139	136	10	pr	pr	NOUN
iajs-3139	136	11	maximal	maximal	ADJ
iajs-3139	136	12	submodule	submodule	NOUN
iajs-3139	136	13	of	of	ADP
iajs-3139	136	14	.	.	PUNCT
iajs-3139	137	1	assists	assist	NOUN
iajs-3139	137	2	theorem	theorem	VERB
iajs-3139	137	3	(	(	PUNCT
iajs-3139	137	4	3.3	3.3	NUM
iajs-3139	137	5	)	)	PUNCT
iajs-3139	137	6	,	,	PUNCT
iajs-3139	137	7	is	be	AUX
iajs-3139	137	8	an	an	DET
iajs-3139	137	9	ideal	ideal	ADJ
iajs-3139	137	10	,	,	PUNCT
iajs-3139	137	11	pr	pr	NOUN
iajs-3139	137	12	-	-	PUNCT
iajs-3139	137	13	maximal	maximal	ADJ
iajs-3139	137	14	of	of	ADP
iajs-3139	137	15	)	)	PUNCT
iajs-3139	137	16	assume	assume	VERB
iajs-3139	137	17	that	that	SCONJ
iajs-3139	137	18	r	r	NOUN
iajs-3139	137	19	is	be	AUX
iajs-3139	137	20	a	a	DET
iajs-3139	137	21	pr	pr	NOUN
iajs-3139	137	22	-	-	PUNCT
iajs-3139	137	23	ring	ring	NOUN
iajs-3139	137	24	and	and	CCONJ
iajs-3139	137	25	let	let	VERB
iajs-3139	137	26	be	be	AUX
iajs-3139	137	27	a	a	DET
iajs-3139	137	28	proper	proper	ADJ
iajs-3139	137	29	submodule	submodule	NOUN
iajs-3139	137	30	of	of	ADP
iajs-3139	137	31	m.	m.	NOUN
iajs-3139	137	32	since	since	SCONJ
iajs-3139	137	33	m	m	PROPN
iajs-3139	137	34	is	be	AUX
iajs-3139	137	35	r	r	NOUN
iajs-3139	137	36	-	-	PUNCT
iajs-3139	137	37	module	module	NOUN
iajs-3139	137	38	a	a	DET
iajs-3139	137	39	multiplication	multiplication	NOUN
iajs-3139	137	40	,	,	PUNCT
iajs-3139	137	41	thence	thence	NOUN
iajs-3139	137	42	,	,	PUNCT
iajs-3139	137	43	for	for	ADP
iajs-3139	137	44	some	some	DET
iajs-3139	137	45	ideal	ideal	NOUN
iajs-3139	137	46	of	of	ADP
iajs-3139	137	47	.	.	PUNCT
iajs-3139	138	1	since	since	SCONJ
iajs-3139	138	2	is	be	AUX
iajs-3139	138	3	a	a	DET
iajs-3139	138	4	pr	pr	NOUN
iajs-3139	138	5	-	-	PUNCT
iajs-3139	138	6	maximal	maximal	ADJ
iajs-3139	138	7	ideal	ideal	NOUN
iajs-3139	138	8	,	,	PUNCT
iajs-3139	138	9	then	then	ADV
iajs-3139	138	10	assists	assist	NOUN
iajs-3139	138	11	theorem	theorem	PROPN
iajs-3139	138	12	(	(	PUNCT
iajs-3139	138	13	4.3	4.3	NUM
iajs-3139	138	14	)	)	PUNCT
iajs-3139	138	15	is	be	AUX
iajs-3139	138	16	pr	pr	NOUN
iajs-3139	138	17	-	-	PUNCT
iajs-3139	138	18	maximal	maximal	ADJ
iajs-3139	138	19	submodule	submodule	NOUN
iajs-3139	138	20	of	of	ADP
iajs-3139	138	21	.	.	PUNCT
iajs-3139	139	1	thus	thus	ADV
iajs-3139	139	2	is	be	AUX
iajs-3139	139	3	pr	pr	NOUN
iajs-3139	139	4	-	-	PUNCT
iajs-3139	139	5	module	module	NOUN
iajs-3139	139	6	.	.	PUNCT
iajs-3139	140	1	remark	remark	NOUN
iajs-3139	140	2	(	(	PUNCT
iajs-3139	140	3	4.7	4.7	NUM
iajs-3139	140	4	)	)	PUNCT
iajs-3139	140	5	not	not	PART
iajs-3139	140	6	all	all	PRON
iajs-3139	140	7	,	,	PUNCT
iajs-3139	140	8	finite	finite	VERB
iajs-3139	140	9	r	r	NOUN
iajs-3139	140	10	-	-	PUNCT
iajs-3139	140	11	module	module	NOUN
iajs-3139	140	12	is	be	AUX
iajs-3139	140	13	local	local	ADJ
iajs-3139	140	14	for	for	ADP
iajs-3139	140	15	example	example	NOUN
iajs-3139	140	16	=	=	PUNCT
iajs-3139	140	17	{	{	PUNCT
iajs-3139	140	18	,	,	PUNCT
iajs-3139	140	19	,	,	PUNCT
iajs-3139	140	20	,	,	PUNCT
iajs-3139	140	21	,	,	PUNCT
iajs-3139	140	22	}	}	PUNCT
iajs-3139	140	23	is	be	AUX
iajs-3139	140	24	not	not	PART
iajs-3139	140	25	local	local	ADJ
iajs-3139	140	26	since	since	SCONJ
iajs-3139	140	27	2𝑍6	2𝑍6	NUM
iajs-3139	140	28	and	and	CCONJ
iajs-3139	140	29	3𝑍6	3𝑍6	NUM
iajs-3139	140	30	are	be	AUX
iajs-3139	140	31	maximal	maximal	ADJ
iajs-3139	140	32	submodules	submodule	NOUN
iajs-3139	140	33	this	this	PRON
iajs-3139	140	34	not	not	PART
iajs-3139	140	35	local	local	ADJ
iajs-3139	140	36	but	but	CCONJ
iajs-3139	140	37	𝑍4	𝑍4	ADJ
iajs-3139	140	38	r	r	NOUN
iajs-3139	140	39	-	-	PUNCT
iajs-3139	140	40	module	module	NOUN
iajs-3139	140	41	is	be	AUX
iajs-3139	140	42	local	local	ADJ
iajs-3139	140	43	since	since	SCONJ
iajs-3139	140	44	2𝑍4is	2𝑍4is	NUM
iajs-3139	140	45	the	the	DET
iajs-3139	140	46	only	only	ADJ
iajs-3139	140	47	proper	proper	ADJ
iajs-3139	140	48	maximal	maximal	ADJ
iajs-3139	140	49	submodule	submodule	NOUN
iajs-3139	140	50	of	of	ADP
iajs-3139	140	51	𝑍4	𝑍4	PROPN
iajs-3139	140	52	proposition	proposition	NOUN
iajs-3139	140	53	(	(	PUNCT
iajs-3139	140	54	4.8	4.8	NUM
iajs-3139	140	55	)	)	PUNCT
iajs-3139	140	56	every	every	DET
iajs-3139	140	57	local	local	ADJ
iajs-3139	140	58	r	r	NOUN
iajs-3139	140	59	-	-	PUNCT
iajs-3139	140	60	module	module	NOUN
iajs-3139	140	61	has	have	VERB
iajs-3139	140	62	pr	pr	NOUN
iajs-3139	140	63	-	-	PUNCT
iajs-3139	140	64	maximal	maximal	ADJ
iajs-3139	140	65	submodule	submodule	NOUN
iajs-3139	140	66	.	.	PUNCT
iajs-3139	141	1	proof	proof	NOUN
iajs-3139	141	2	:	:	PUNCT
iajs-3139	141	3	let	let	VERB
iajs-3139	141	4	is	be	AUX
iajs-3139	141	5	local	local	ADJ
iajs-3139	141	6	r	r	NOUN
iajs-3139	141	7	-	-	PUNCT
iajs-3139	141	8	module	module	NOUN
iajs-3139	141	9	,	,	PUNCT
iajs-3139	141	10	then	then	ADV
iajs-3139	141	11	has	have	VERB
iajs-3139	141	12	only	only	ADV
iajs-3139	141	13	one	one	NUM
iajs-3139	141	14	maximal	maximal	ADJ
iajs-3139	141	15	r	r	NOUN
iajs-3139	141	16	-	-	PUNCT
iajs-3139	141	17	submodule	submodule	NOUN
iajs-3139	141	18	implies	implie	NOUN
iajs-3139	141	19	has	have	VERB
iajs-3139	141	20	prmaximal	prmaximal	ADJ
iajs-3139	141	21	submodule	submodule	NOUN
iajs-3139	141	22	assist	assist	NOUN
iajs-3139	141	23	remark	remark	NOUN
iajs-3139	141	24	(	(	PUNCT
iajs-3139	141	25	2.2	2.2	NUM
iajs-3139	141	26	)	)	PUNCT
iajs-3139	141	27	.	.	PUNCT
iajs-3139	142	1	remark	remark	NOUN
iajs-3139	142	2	(	(	PUNCT
iajs-3139	142	3	4.9	4.9	NUM
iajs-3139	142	4	)	)	PUNCT
iajs-3139	142	5	the	the	DET
iajs-3139	142	6	convers	conver	NOUN
iajs-3139	142	7	of	of	ADP
iajs-3139	142	8	proposition	proposition	NOUN
iajs-3139	142	9	(	(	PUNCT
iajs-3139	142	10	4.8	4.8	NUM
iajs-3139	142	11	)	)	PUNCT
iajs-3139	142	12	is	be	AUX
iajs-3139	142	13	not	not	PART
iajs-3139	142	14	true	true	ADJ
iajs-3139	142	15	for	for	ADP
iajs-3139	142	16	example	example	NOUN
iajs-3139	142	17	has	have	VERB
iajs-3139	142	18	pr	pr	NOUN
iajs-3139	142	19	-	-	PUNCT
iajs-3139	142	20	maximal	maximal	ADJ
iajs-3139	142	21	submodules	submodule	NOUN
iajs-3139	142	22	but	but	CCONJ
iajs-3139	142	23	not	not	PART
iajs-3139	142	24	local	local	ADJ
iajs-3139	142	25	.	.	PUNCT
iajs-3139	143	1	we	we	PRON
iajs-3139	143	2	need	need	VERB
iajs-3139	143	3	to	to	PART
iajs-3139	143	4	give	give	VERB
iajs-3139	143	5	the	the	DET
iajs-3139	143	6	following	follow	VERB
iajs-3139	143	7	concept	concept	NOUN
iajs-3139	143	8	definition	definition	NOUN
iajs-3139	143	9	(	(	PUNCT
iajs-3139	143	10	4.10	4.10	NUM
iajs-3139	143	11	)	)	PUNCT
iajs-3139	143	12	an	an	DET
iajs-3139	143	13	r	r	NOUN
iajs-3139	143	14	-	-	PUNCT
iajs-3139	143	15	module	module	NOUN
iajs-3139	143	16	is	be	AUX
iajs-3139	143	17	named	name	VERB
iajs-3139	143	18	pure	pure	ADJ
iajs-3139	143	19	local	local	ADJ
iajs-3139	143	20	(	(	PUNCT
iajs-3139	143	21	pr	pr	NOUN
iajs-3139	143	22	-	-	ADJ
iajs-3139	143	23	local	local	ADJ
iajs-3139	143	24	)	)	PUNCT
iajs-3139	143	25	module	module	NOUN
iajs-3139	143	26	.	.	PUNCT
iajs-3139	144	1	if	if	SCONJ
iajs-3139	144	2	it	it	PRON
iajs-3139	144	3	has	have	VERB
iajs-3139	144	4	only	only	ADV
iajs-3139	144	5	one	one	NUM
iajs-3139	144	6	pr	pr	NOUN
iajs-3139	144	7	-	-	PUNCT
iajs-3139	144	8	maximal	maximal	ADJ
iajs-3139	144	9	submodule	submodule	NOUN
iajs-3139	144	10	which	which	PRON
iajs-3139	144	11	contains	contain	VERB
iajs-3139	144	12	all	all	DET
iajs-3139	144	13	proper	proper	ADJ
iajs-3139	144	14	submodule	submodule	NOUN
iajs-3139	144	15	of	of	ADP
iajs-3139	144	16	𝑀.	𝑀.	PROPN
iajs-3139	144	17	a	a	DET
iajs-3139	144	18	ring	ring	NOUN
iajs-3139	144	19	is	be	AUX
iajs-3139	144	20	called	call	VERB
iajs-3139	144	21	pure	pure	ADJ
iajs-3139	144	22	local	local	ADJ
iajs-3139	144	23	(	(	PUNCT
iajs-3139	144	24	pr	pr	NOUN
iajs-3139	144	25	-	-	ADJ
iajs-3139	144	26	local	local	ADJ
iajs-3139	144	27	)	)	PUNCT
iajs-3139	144	28	ring	ring	NOUN
iajs-3139	144	29	,	,	PUNCT
iajs-3139	144	30	if	if	SCONJ
iajs-3139	144	31	is	be	AUX
iajs-3139	144	32	a	a	DET
iajs-3139	144	33	pr	pr	NOUN
iajs-3139	144	34	-	-	ADJ
iajs-3139	144	35	local	local	ADJ
iajs-3139	144	36	r	r	NOUN
iajs-3139	144	37	-	-	PUNCT
iajs-3139	144	38	module	module	NOUN
iajs-3139	144	39	remark	remark	NOUN
iajs-3139	144	40	and	and	CCONJ
iajs-3139	144	41	examples	example	NOUN
iajs-3139	144	42	(	(	PUNCT
iajs-3139	144	43	4.11	4.11	NUM
iajs-3139	144	44	)	)	PUNCT
iajs-3139	144	45	1	1	NUM
iajs-3139	144	46	.	.	PUNCT
iajs-3139	145	1	the	the	DET
iajs-3139	145	2	z	z	NOUN
iajs-3139	145	3	-	-	PUNCT
iajs-3139	145	4	module	module	NOUN
iajs-3139	145	5	3𝑍24	3𝑍24	PROPN
iajs-3139	145	6	is	be	AUX
iajs-3139	145	7	a	a	DET
iajs-3139	145	8	pr	pr	NOUN
iajs-3139	145	9	-	-	ADJ
iajs-3139	145	10	local	local	ADJ
iajs-3139	145	11	module	module	NOUN
iajs-3139	145	12	,	,	PUNCT
iajs-3139	145	13	since	since	SCONJ
iajs-3139	145	14	it	it	PRON
iajs-3139	145	15	has	have	VERB
iajs-3139	145	16	only	only	ADV
iajs-3139	145	17	one	one	NUM
iajs-3139	145	18	r	r	NOUN
iajs-3139	145	19	-	-	PUNCT
iajs-3139	145	20	submodule	submodule	NOUN
iajs-3139	145	21	pr	pr	NOUN
iajs-3139	145	22	-	-	ADV
iajs-3139	145	23	maximal	maximal	ADJ
iajs-3139	145	24	that	that	PRON
iajs-3139	145	25	is	be	AUX
iajs-3139	145	26	6𝑍24	6𝑍24	PROPN
iajs-3139	145	27	.	.	PUNCT
iajs-3139	146	1	2	2	X
iajs-3139	146	2	.	.	X
iajs-3139	146	3	every	every	DET
iajs-3139	146	4	local	local	ADJ
iajs-3139	146	5	is	be	AUX
iajs-3139	146	6	,	,	PUNCT
iajs-3139	146	7	pr	pr	NOUN
iajs-3139	146	8	-	-	ADJ
iajs-3139	146	9	local	local	ADJ
iajs-3139	146	10	but	but	CCONJ
iajs-3139	146	11	not	not	PART
iajs-3139	146	12	conversely	conversely	ADV
iajs-3139	146	13	for	for	ADP
iajs-3139	146	14	example	example	NOUN
iajs-3139	146	15	:	:	PUNCT
iajs-3139	146	16	the	the	DET
iajs-3139	146	17	z	z	NOUN
iajs-3139	146	18	-	-	PUNCT
iajs-3139	146	19	module	module	NOUN
iajs-3139	146	20	2𝑍12	2𝑍12	PROPN
iajs-3139	146	21	in	in	ADP
iajs-3139	146	22	𝑍12	𝑍12	PROPN
iajs-3139	146	23	as	as	ADP
iajs-3139	146	24	z	z	NOUN
iajs-3139	146	25	-	-	PUNCT
iajs-3139	146	26	module	module	NOUN
iajs-3139	146	27	is	be	AUX
iajs-3139	146	28	pr	pr	NOUN
iajs-3139	146	29	-	-	ADJ
iajs-3139	146	30	local	local	ADJ
iajs-3139	146	31	since	since	SCONJ
iajs-3139	146	32	4𝑍12	4𝑍12	PROPN
iajs-3139	146	33	is	be	AUX
iajs-3139	146	34	the	the	DET
iajs-3139	146	35	only	only	ADJ
iajs-3139	146	36	pr	pr	NOUN
iajs-3139	146	37	-	-	PUNCT
iajs-3139	146	38	maximal	maximal	ADJ
iajs-3139	146	39	submodule	submodule	NOUN
iajs-3139	146	40	of	of	ADP
iajs-3139	146	41	2𝑍12.but	2𝑍12.but	NUM
iajs-3139	146	42	not	not	PART
iajs-3139	146	43	local	local	ADJ
iajs-3139	146	44	since	since	SCONJ
iajs-3139	146	45	4𝑍12	4𝑍12	PROPN
iajs-3139	146	46	and	and	CCONJ
iajs-3139	146	47	6𝑍12	6𝑍12	NUM
iajs-3139	146	48	are	be	AUX
iajs-3139	146	49	maximal	maximal	ADJ
iajs-3139	146	50	submodules	submodule	NOUN
iajs-3139	146	51	of	of	ADP
iajs-3139	146	52	2𝑍12	2𝑍12	PROPN
iajs-3139	146	53	.	.	PUNCT
iajs-3139	147	1	3.if	3.if	NUM
iajs-3139	147	2	𝑊	𝑊	PROPN
iajs-3139	147	3	is	be	AUX
iajs-3139	147	4	a	a	DET
iajs-3139	147	5	non	non	ADJ
iajs-3139	147	6	-	-	ADJ
iajs-3139	147	7	zero	zero	NUM
iajs-3139	147	8	multiplication	multiplication	NOUN
iajs-3139	147	9	and	and	CCONJ
iajs-3139	147	10	pr	pr	NOUN
iajs-3139	147	11	-	-	ADJ
iajs-3139	147	12	local	local	ADJ
iajs-3139	147	13	r	r	NOUN
iajs-3139	147	14	-	-	PUNCT
iajs-3139	147	15	module	module	NOUN
iajs-3139	147	16	and	and	CCONJ
iajs-3139	147	17	𝐻	𝐻	PROPN
iajs-3139	147	18	is	be	AUX
iajs-3139	147	19	a	a	DET
iajs-3139	147	20	non	non	ADJ
iajs-3139	147	21	-	-	ADJ
iajs-3139	147	22	zero	zero	ADJ
iajs-3139	147	23	pr	pr	NOUN
iajs-3139	147	24	-	-	PUNCT
iajs-3139	147	25	maximal	maximal	ADJ
iajs-3139	147	26	𝑅submodule	𝑅submodule	PROPN
iajs-3139	147	27	of	of	ADP
iajs-3139	147	28	𝑊	𝑊	PROPN
iajs-3139	147	29	,	,	PUNCT
iajs-3139	147	30	thence	thence	NOUN
iajs-3139	147	31	𝐻	𝐻	NOUN
iajs-3139	147	32	not	not	PART
iajs-3139	147	33	necessary	necessary	ADJ
iajs-3139	147	34	,	,	PUNCT
iajs-3139	147	35	pure	pure	ADJ
iajs-3139	147	36	submodule	submodule	NOUN
iajs-3139	147	37	for	for	ADP
iajs-3139	147	38	example	example	NOUN
iajs-3139	147	39	:	:	PUNCT
iajs-3139	147	40	if	if	SCONJ
iajs-3139	147	41	𝑊=𝑍4	𝑊=𝑍4	VERB
iajs-3139	147	42	as	as	ADP
iajs-3139	147	43	z	z	NOUN
iajs-3139	147	44	-	-	PUNCT
iajs-3139	147	45	module	module	NOUN
iajs-3139	147	46	,	,	PUNCT
iajs-3139	147	47	𝑊	𝑊	PROPN
iajs-3139	147	48	is	be	AUX
iajs-3139	147	49	non	non	ADJ
iajs-3139	147	50	-	-	ADJ
iajs-3139	147	51	zero	zero	NUM
iajs-3139	147	52	multiplication	multiplication	NOUN
iajs-3139	147	53	and	and	CCONJ
iajs-3139	147	54	pr	pr	NOUN
iajs-3139	147	55	-	-	ADJ
iajs-3139	147	56	local	local	ADJ
iajs-3139	147	57	z	z	NOUN
iajs-3139	147	58	-	-	PUNCT
iajs-3139	147	59	module	module	NOUN
iajs-3139	147	60	and	and	CCONJ
iajs-3139	147	61	2𝑍4	2𝑍4	NUM
iajs-3139	147	62	is	be	AUX
iajs-3139	147	63	non	non	ADJ
iajs-3139	147	64	-	-	ADJ
iajs-3139	147	65	zero	zero	NUM
iajs-3139	147	66	,	,	PUNCT
iajs-3139	147	67	submodule	submodule	NOUN
iajs-3139	147	68	,	,	PUNCT
iajs-3139	147	69	pr	pr	NOUN
iajs-3139	147	70	-	-	PUNCT
iajs-3139	147	71	maximal	maximal	ADJ
iajs-3139	147	72	of	of	ADP
iajs-3139	147	73	𝑍4	𝑍4	NOUN
iajs-3139	147	74	but	but	CCONJ
iajs-3139	147	75	not	not	PART
iajs-3139	147	76	pure	pure	ADJ
iajs-3139	147	77	.	.	PUNCT
iajs-3139	148	1	remark	remark	NOUN
iajs-3139	148	2	(	(	PUNCT
iajs-3139	148	3	4.12	4.12	NUM
iajs-3139	148	4	):	):	PUNCT
iajs-3139	148	5	description	description	NOUN
iajs-3139	148	6	of	of	ADP
iajs-3139	148	7	r	r	NOUN
iajs-3139	148	8	-	-	PUNCT
iajs-3139	148	9	module	module	NOUN
iajs-3139	148	10	.	.	PUNCT
iajs-3139	149	1	r	r	NOUN
iajs-3139	149	2	-	-	PUNCT
iajs-3139	149	3	module	module	NOUN
iajs-3139	149	4	local	local	ADJ
iajs-3139	149	5	pr	pr	NOUN
iajs-3139	149	6	-	-	ADJ
iajs-3139	149	7	local	local	ADJ
iajs-3139	149	8	𝑍4	𝑍4	NOUN
iajs-3139	149	9			PROPN
iajs-3139	149	10			PROPN
iajs-3139	149	11	𝑍6	𝑍6	PROPN
iajs-3139	149	12			PROPN
iajs-3139	149	13			PROPN
iajs-3139	149	14	3𝑍12	3𝑍12	PROPN
iajs-3139	149	15			PROPN
iajs-3139	150	1			ADP
iajs-3139	150	2	5	5	X
iajs-3139	150	3	.	.	PUNCT
iajs-3139	150	4	conclusions	conclusion	NOUN
iajs-3139	150	5	we	we	PRON
iajs-3139	150	6	will	will	AUX
iajs-3139	150	7	try	try	VERB
iajs-3139	150	8	to	to	PART
iajs-3139	150	9	generalize	generalize	VERB
iajs-3139	150	10	the	the	DET
iajs-3139	150	11	concept	concept	NOUN
iajs-3139	150	12	of	of	ADP
iajs-3139	150	13	pr	pr	NOUN
iajs-3139	150	14	-	-	PUNCT
iajs-3139	150	15	maximal	maximal	ADJ
iajs-3139	150	16	r	r	NOUN
iajs-3139	150	17	-	-	PUNCT
iajs-3139	150	18	submodules	submodules	NOUN
iajs-3139	150	19	to	to	ADP
iajs-3139	150	20	some	some	DET
iajs-3139	150	21	other	other	ADJ
iajs-3139	150	22	concepts	concept	NOUN
iajs-3139	150	23	in	in	ADP
iajs-3139	150	24	future	future	ADJ
iajs-3139	150	25	works	work	NOUN
iajs-3139	150	26	.	.	PUNCT
iajs-3139	151	1	in	in	ADP
iajs-3139	151	2	this	this	DET
iajs-3139	151	3	study	study	NOUN
iajs-3139	151	4	,	,	PUNCT
iajs-3139	151	5	the	the	DET
iajs-3139	151	6	concepts	concept	NOUN
iajs-3139	151	7	of	of	ADP
iajs-3139	151	8	pr	pr	NOUN
iajs-3139	151	9	-	-	PUNCT
iajs-3139	151	10	maximal	maximal	ADJ
iajs-3139	151	11	r	r	NOUN
iajs-3139	151	12	-	-	PUNCT
iajs-3139	151	13	submodules	submodule	NOUN
iajs-3139	151	14	is	be	AUX
iajs-3139	151	15	studied	study	VERB
iajs-3139	151	16	as	as	ADP
iajs-3139	151	17	a	a	DET
iajs-3139	151	18	generalization	generalization	NOUN
iajs-3139	151	19	of	of	ADP
iajs-3139	151	20	maximal	maximal	ADJ
iajs-3139	151	21	submodule	submodule	NOUN
iajs-3139	151	22	and	and	CCONJ
iajs-3139	151	23	some	some	DET
iajs-3139	151	24	properties	property	NOUN
iajs-3139	151	25	of	of	ADP
iajs-3139	151	26	this	this	DET
iajs-3139	151	27	concept	concept	NOUN
iajs-3139	151	28	are	be	AUX
iajs-3139	151	29	investigated	investigate	VERB
iajs-3139	151	30	such	such	ADJ
iajs-3139	151	31	as	as	ADP
iajs-3139	151	32	:	:	PUNCT
iajs-3139	151	33	ihjpas	ihjpas	PROPN
iajs-3139	151	34	.	.	PUNCT
iajs-3139	152	1	36	36	NUM
iajs-3139	152	2	(	(	PUNCT
iajs-3139	152	3	4	4	NUM
iajs-3139	152	4	)	)	PUNCT
iajs-3139	152	5	2023	2023	NUM
iajs-3139	152	6	366	366	NUM
iajs-3139	152	7	1	1	NUM
iajs-3139	152	8	.	.	PUNCT
iajs-3139	153	1	every	every	DET
iajs-3139	153	2	maximal	maximal	ADJ
iajs-3139	153	3	submodule	submodule	NOUN
iajs-3139	153	4	of	of	ADP
iajs-3139	153	5	r	r	NOUN
iajs-3139	153	6	-	-	PUNCT
iajs-3139	153	7	module	module	NOUN
iajs-3139	153	8	is	be	AUX
iajs-3139	153	9	pr	pr	ADV
iajs-3139	153	10	-	-	PUNCT
iajs-3139	153	11	maximal	maximal	ADJ
iajs-3139	153	12	.	.	PUNCT
iajs-3139	154	1	2	2	X
iajs-3139	154	2	.	.	X
iajs-3139	154	3	a	a	DET
iajs-3139	154	4	subset	subset	NOUN
iajs-3139	154	5	of	of	ADP
iajs-3139	154	6	pr	pr	NOUN
iajs-3139	154	7	-	-	PUNCT
iajs-3139	154	8	maximal	maximal	ADJ
iajs-3139	154	9	-submodule	-submodule	NOUN
iajs-3139	154	10	need	need	AUX
iajs-3139	154	11	not	not	PART
iajs-3139	154	12	be	be	AUX
iajs-3139	154	13	pr	pr	NOUN
iajs-3139	154	14	-	-	PUNCT
iajs-3139	154	15	maximal	maximal	ADJ
iajs-3139	154	16	-submodule	-submodule	NOUN
iajs-3139	154	17	.	.	PUNCT
iajs-3139	155	1	3	3	X
iajs-3139	155	2	.	.	X
iajs-3139	156	1	if	if	SCONJ
iajs-3139	156	2	is	be	AUX
iajs-3139	156	3	an	an	DET
iajs-3139	156	4	f	f	NOUN
iajs-3139	156	5	-	-	PUNCT
iajs-3139	156	6	regular	regular	ADJ
iajs-3139	156	7	module	module	NOUN
iajs-3139	156	8	then	then	ADV
iajs-3139	156	9	every	every	DET
iajs-3139	156	10	submodule	submodule	NOUN
iajs-3139	156	11	of	of	ADP
iajs-3139	156	12	is	be	AUX
iajs-3139	156	13	pr	pr	ADV
iajs-3139	156	14	-	-	PUNCT
iajs-3139	156	15	maximal	maximal	ADJ
iajs-3139	156	16	.	.	PUNCT
iajs-3139	157	1	4	4	X
iajs-3139	157	2	.	.	X
iajs-3139	157	3	if	if	SCONJ
iajs-3139	157	4	are	be	AUX
iajs-3139	157	5	non	non	ADJ
iajs-3139	157	6	-	-	ADJ
iajs-3139	157	7	zero	zero	NUM
iajs-3139	157	8	submodule	submodule	NOUN
iajs-3139	157	9	of	of	ADP
iajs-3139	157	10	such	such	ADJ
iajs-3139	157	11	that	that	SCONJ
iajs-3139	157	12	if	if	SCONJ
iajs-3139	157	13	is	be	AUX
iajs-3139	157	14	pr	pr	NOUN
iajs-3139	157	15	-	-	ADJ
iajs-3139	157	16	maximal	maximal	ADJ
iajs-3139	157	17	in	in	ADV
iajs-3139	157	18	then	then	ADV
iajs-3139	157	19	is	be	AUX
iajs-3139	157	20	pr	pr	NOUN
iajs-3139	157	21	-	-	ADJ
iajs-3139	157	22	maximal	maximal	ADJ
iajs-3139	157	23	in	in	ADV
iajs-3139	157	24	.	.	PUNCT
iajs-3139	158	1	5	5	X
iajs-3139	158	2	.	.	X
iajs-3139	159	1	let	let	AUX
iajs-3139	159	2	:	:	PUNCT
iajs-3139	159	3	be	be	AUX
iajs-3139	159	4	an	an	DET
iajs-3139	159	5	epimorphism	epimorphism	NOUN
iajs-3139	159	6	,	,	PUNCT
iajs-3139	159	7	where	where	SCONJ
iajs-3139	159	8	be	be	AUX
iajs-3139	159	9	r	r	NOUN
iajs-3139	159	10	-	-	PUNCT
iajs-3139	159	11	modules	module	NOUN
iajs-3139	159	12	.	.	PUNCT
iajs-3139	160	1	if	if	SCONJ
iajs-3139	160	2	is	be	AUX
iajs-3139	160	3	pr	pr	NOUN
iajs-3139	160	4	-	-	PUNCT
iajs-3139	160	5	maximal	maximal	ADJ
iajs-3139	160	6	submodule	submodule	NOUN
iajs-3139	160	7	of	of	ADP
iajs-3139	160	8	,	,	PUNCT
iajs-3139	160	9	then	then	ADV
iajs-3139	160	10	(	(	PUNCT
iajs-3139	160	11	)	)	PUNCT
iajs-3139	160	12	is	be	AUX
iajs-3139	160	13	pr	pr	NOUN
iajs-3139	160	14	-	-	PUNCT
iajs-3139	160	15	maximal	maximal	ADJ
iajs-3139	160	16	r	r	NOUN
iajs-3139	160	17	-	-	PUNCT
iajs-3139	160	18	submodule	submodule	NOUN
iajs-3139	160	19	of	of	ADP
iajs-3139	160	20	.	.	PROPN
iajs-3139	161	1	6	6	X
iajs-3139	161	2	.	.	X
iajs-3139	161	3	every	every	DET
iajs-3139	161	4	cyclic	cyclic	ADJ
iajs-3139	161	5	r	r	NOUN
iajs-3139	161	6	-	-	PUNCT
iajs-3139	161	7	module	module	NOUN
iajs-3139	161	8	breakpoint	breakpoint	NOUN
iajs-3139	161	9	pr	pr	NOUN
iajs-3139	161	10	-	-	PUNCT
iajs-3139	161	11	maximal	maximal	ADJ
iajs-3139	161	12	submodule	submodule	NOUN
iajs-3139	161	13	.	.	PUNCT
iajs-3139	162	1	7	7	X
iajs-3139	162	2	.	.	X
iajs-3139	162	3	every	every	DET
iajs-3139	162	4	local	local	ADJ
iajs-3139	162	5	r	r	NOUN
iajs-3139	162	6	-	-	PUNCT
iajs-3139	162	7	module	module	NOUN
iajs-3139	162	8	breakpoint	breakpoint	NOUN
iajs-3139	162	9	pr	pr	NOUN
iajs-3139	162	10	-	-	PUNCT
iajs-3139	162	11	maximal	maximal	ADJ
iajs-3139	162	12	r	r	NOUN
iajs-3139	162	13	-	-	PUNCT
iajs-3139	162	14	submodule	submodule	NOUN
iajs-3139	162	15	.	.	PUNCT
iajs-3139	163	1	references	reference	NOUN
iajs-3139	163	2	1	1	NUM
iajs-3139	163	3	.	.	PUNCT
iajs-3139	163	4	nuhad	nuhad	PROPN
iajs-3139	163	5	s	s	PROPN
iajs-3139	163	6	..	..	PUNCT
iajs-3139	163	7	al	al	PROPN
iajs-3139	163	8	-	-	PUNCT
iajs-3139	163	9	mothafar;mohammed	mothafar;mohammed	PROPN
iajs-3139	163	10	b.h	b.h	PROPN
iajs-3139	163	11	.	.	PROPN
iajs-3139	163	12	alhakeem	alhakeem	PROPN
iajs-3139	163	13	,	,	PUNCT
iajs-3139	163	14	nearly	nearly	ADV
iajs-3139	163	15	semiprime	semiprime	NOUN
iajs-3139	163	16	submodule	submodule	PROPN
iajs-3139	163	17	,	,	PUNCT
iajs-3139	163	18	iraq	iraq	PROPN
iajs-3139	163	19	,	,	PUNCT
iajs-3139	163	20	journal	journal	NOUN
iajs-3139	163	21	of	of	ADP
iajs-3139	163	22	science,2015	science,2015	PROPN
iajs-3139	163	23	,	,	PUNCT
iajs-3139	163	24	56,4b	56,4b	NOUN
iajs-3139	163	25	,	,	PUNCT
iajs-3139	163	26	pp,3210	pp,3210	NOUN
iajs-3139	163	27	-	-	SYM
iajs-3139	163	28	3214	3214	NUM
iajs-3139	163	29	.	.	PUNCT
iajs-3139	164	1	2	2	X
iajs-3139	164	2	.	.	X
iajs-3139	164	3	amira	amira	PROPN
iajs-3139	164	4	,	,	PUNCT
iajs-3139	164	5	a.	a.	NOUN
iajs-3139	164	6	a.	a.	NOUN
iajs-3139	164	7	;	;	PUNCT
iajs-3139	164	8	sahira	sahira	PROPN
iajs-3139	164	9	,	,	PUNCT
iajs-3139	164	10	m.	m.	NOUN
iajs-3139	164	11	y.	y.	PROPN
iajs-3139	164	12	large	large	ADJ
iajs-3139	164	13	-	-	PUNCT
iajs-3139	164	14	maximal	maximal	ADJ
iajs-3139	164	15	submodules	submodule	NOUN
iajs-3139	164	16	,	,	PUNCT
iajs-3139	164	17	j.	j.	PROPN
iajs-3139	164	18	phys	phys	PROPN
iajs-3139	164	19	.	.	PUNCT
iajs-3139	164	20	:	:	PUNCT
iajs-3139	165	1	conf	conf	PROPN
iajs-3139	165	2	.	.	PUNCT
iajs-3139	165	3	ser	ser	PROPN
iajs-3139	165	4	.	.	PROPN
iajs-3139	165	5	2019	2019	NUM
iajs-3139	165	6	,	,	PUNCT
iajs-3139	165	7	1294	1294	NUM
iajs-3139	165	8	,	,	PUNCT
iajs-3139	165	9	15	15	NUM
iajs-3139	165	10	.	.	NOUN
iajs-3139	166	1	3	3	NUM
iajs-3139	166	2	.	.	X
iajs-3139	166	3	shahad	shahad	PROPN
iajs-3139	166	4	,	,	PUNCT
iajs-3139	166	5	h.	h.	PROPN
iajs-3139	166	6	a.	a.	PROPN
iajs-3139	166	7	;	;	PUNCT
iajs-3139	166	8	al	al	PROPN
iajs-3139	166	9	-	-	PUNCT
iajs-3139	166	10	mothafar	mothafar	PROPN
iajs-3139	166	11	,	,	PUNCT
iajs-3139	166	12	n.	n.	PROPN
iajs-3139	166	13	s.	s.	PROPN
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iajs-3139	166	15	p	p	ADJ
iajs-3139	166	16	-	-	PUNCT
iajs-3139	166	17	essential	essential	ADJ
iajs-3139	166	18	submodules	submodule	NOUN
iajs-3139	166	19	,	,	PUNCT
iajs-3139	166	20	iraqi	iraqi	ADJ
iajs-3139	166	21	journal	journal	NOUN
iajs-3139	166	22	of	of	ADP
iajs-3139	166	23	science	science	NOUN
iajs-3139	166	24	,	,	PUNCT
iajs-3139	166	25	2021	2021	NUM
iajs-3139	166	26	;	;	PUNCT
iajs-3139	166	27	62	62	NUM
iajs-3139	166	28	,	,	PUNCT
iajs-3139	166	29	4916–4922	4916–4922	NUM
iajs-3139	166	30	.	.	NOUN
iajs-3139	166	31	4	4	NUM
iajs-3139	166	32	.	.	X
iajs-3139	166	33	ajeel	ajeel	PROPN
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iajs-3139	166	35	a.	a.	PROPN
iajs-3139	166	36	s.	s.	PROPN
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iajs-3139	166	40	,	,	PUNCT
iajs-3139	166	41	h.	h.	PROPN
iajs-3139	166	42	k.	k.	PROPN
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iajs-3139	167	2	prime	prime	ADJ
iajs-3139	167	3	submodules	submodule	NOUN
iajs-3139	167	4	and	and	CCONJ
iajs-3139	167	5	some	some	DET
iajs-3139	167	6	related	related	ADJ
iajs-3139	167	7	concepts	concept	NOUN
iajs-3139	167	8	.	.	PUNCT
iajs-3139	168	1	ibn	ibn	PROPN
iajs-3139	168	2	al	al	PROPN
iajs-3139	168	3	-	-	PUNCT
iajs-3139	168	4	haitham	haitham	PROPN
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iajs-3139	168	6	for	for	ADP
iajs-3139	168	7	pure	pure	ADJ
iajs-3139	168	8	and	and	CCONJ
iajs-3139	168	9	applied	applied	ADJ
iajs-3139	168	10	sciences	science	NOUN
iajs-3139	168	11	,	,	PUNCT
iajs-3139	168	12	2019	2019	NUM
iajs-3139	168	13	,	,	PUNCT
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iajs-3139	168	15	)	)	PUNCT
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iajs-3139	169	2	.	.	X
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iajs-3139	169	5	o.	o.	NOUN
iajs-3139	169	6	a.	a.	NOUN
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iajs-3139	170	1	h.	h.	PROPN
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iajs-3139	170	5	-	-	PUNCT
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iajs-3139	170	8	and	and	CCONJ
iajs-3139	170	9	some	some	DET
iajs-3139	170	10	related	related	ADJ
iajs-3139	170	11	concepts	concept	NOUN
iajs-3139	170	12	.	.	PUNCT
iajs-3139	171	1	ibn	ibn	PROPN
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iajs-3139	171	6	for	for	ADP
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iajs-3139	171	8	and	and	CCONJ
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iajs-3139	171	11	,	,	PUNCT
iajs-3139	171	12	2019,32(2	2019,32(2	PROPN
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iajs-3139	171	14	,	,	PUNCT
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iajs-3139	171	20	.	.	X
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iajs-3139	172	2	,	,	PUNCT
iajs-3139	172	3	m.	m.	NOUN
iajs-3139	172	4	r.	r.	PROPN
iajs-3139	172	5	;	;	PUNCT
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iajs-3139	172	13	,	,	PUNCT
iajs-3139	172	14	m.	m.	PROPN
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iajs-3139	173	8	,	,	PUNCT
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iajs-3139	173	10	of	of	ADP
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iajs-3139	173	12	.	.	PUNCT
iajs-3139	174	1	7	7	X
iajs-3139	174	2	.	.	X
iajs-3139	174	3	thaar	thaar	NOUN
iajs-3139	174	4	;	;	PUNCT
iajs-3139	174	5	y.	y.	PROPN
iajs-3139	174	6	c.	c.	PROPN
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iajs-3139	174	8	quasi	quasi	ADJ
iajs-3139	174	9	-	-	ADJ
iajs-3139	174	10	dedekind	dedekind	ADJ
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iajs-3139	174	12	and	and	CCONJ
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iajs-3139	174	14	prime	prime	ADJ
iajs-3139	174	15	modules	module	NOUN
iajs-3139	174	16	,	,	PUNCT
iajs-3139	174	17	2017	2017	NUM
iajs-3139	174	18	,	,	PUNCT
iajs-3139	174	19	university	university	NOUN
iajs-3139	174	20	of	of	ADP
iajs-3139	174	21	al	al	PROPN
iajs-3139	174	22	-	-	PUNCT
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iajs-3139	174	24	.	.	PUNCT
iajs-3139	175	1	8	8	X
iajs-3139	175	2	.	.	X
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iajs-3139	175	4	;	;	PUNCT
iajs-3139	175	5	r.	r.	PROPN
iajs-3139	175	6	foundotion	foundotion	NOUN
iajs-3139	175	7	of	of	ADP
iajs-3139	175	8	module	module	NOUN
iajs-3139	175	9	and	and	CCONJ
iajs-3139	175	10	ring	ring	NOUN
iajs-3139	175	11	theory	theory	NOUN
iajs-3139	175	12	;	;	PUNCT
iajs-3139	175	13	gordon	gordon	PROPN
iajs-3139	175	14	and	and	CCONJ
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iajs-3139	175	16	,	,	PUNCT
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iajs-3139	175	18	.	.	PUNCT
iajs-3139	176	1	9	9	NUM
iajs-3139	176	2	.	.	X
iajs-3139	176	3	abd	abd	PROPN
iajs-3139	176	4	el	el	PROPN
iajs-3139	176	5	-	-	PUNCT
iajs-3139	176	6	bast	bast	NOUN
iajs-3139	176	7	,	,	PUNCT
iajs-3139	176	8	z.	z.	PROPN
iajs-3139	176	9	;	;	PUNCT
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iajs-3139	176	12	p.	p.	PROPN
iajs-3139	176	13	f.	f.	PROPN
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iajs-3139	176	15	modules	module	NOUN
iajs-3139	176	16	;	;	PUNCT
iajs-3139	176	17	comm	comm	NOUN
iajs-3139	176	18	.	.	PUNCT
iajs-3139	177	1	algebra,1988,16	algebra,1988,16	ADJ
iajs-3139	177	2	,	,	PUNCT
iajs-3139	177	3	755	755	PROPN
iajs-3139	177	4	-	-	SYM
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iajs-3139	177	6	.	.	PROPN
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iajs-3139	178	2	.	.	PUNCT
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iajs-3139	179	2	;	;	PUNCT
iajs-3139	179	3	m.	m.	NOUN
iajs-3139	179	4	a.	a.	PROPN
iajs-3139	179	5	;	;	PUNCT
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iajs-3139	179	13	multiplication	multiplication	NOUN
iajs-3139	179	14	module	module	NOUN
iajs-3139	179	15	;	;	PUNCT
iajs-3139	179	16	contributions	contribution	NOUN
iajs-3139	179	17	to	to	ADP
iajs-3139	179	18	algebra	algebra	NOUN
iajs-3139	179	19	and	and	CCONJ
iajs-3139	179	20	geometry	geometry	NOUN
iajs-3139	179	21	,	,	PUNCT
iajs-3139	179	22	2004	2004	NUM
iajs-3139	179	23	,	,	PUNCT
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iajs-3139	180	2	.	.	PUNCT
iajs-3139	181	1	b.h.al	b.h.al	NOUN
iajs-3139	181	2	-	-	PUNCT
iajs-3139	181	3	bahrani	bahrani	PROPN
iajs-3139	181	4	,	,	PUNCT
iajs-3139	181	5	on	on	ADP
iajs-3139	181	6	purely	purely	ADV
iajs-3139	181	7	y	y	NOUN
iajs-3139	181	8	-	-	PUNCT
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iajs-3139	181	11	,	,	PUNCT
iajs-3139	181	12	iraqi	iraqi	ADJ
iajs-3139	181	13	journal	journal	NOUN
iajs-3139	181	14	of	of	ADP
iajs-3139	181	15	scince	scince	NOUN
iajs-3139	181	16	,	,	PUNCT
iajs-3139	181	17	2013	2013	NUM
iajs-3139	181	18	,	,	PUNCT
iajs-3139	181	19	54,pp.672675	54,pp.672675	NUM
iajs-3139	181	20	,	,	PUNCT