id	sid	tid	token	lemma	pos
iajs-2802	1	1	ibn	ibn	PROPN
iajs-2802	1	2	al	al	PROPN
iajs-2802	1	3	-	-	PUNCT
iajs-2802	1	4	haitham	haitham	PROPN
iajs-2802	1	5	jour	jour	X
iajs-2802	1	6	.	.	PROPN
iajs-2802	1	7	for	for	ADP
iajs-2802	1	8	pure	pure	ADJ
iajs-2802	1	9	&	&	CCONJ
iajs-2802	1	10	appl	appl	PROPN
iajs-2802	1	11	.	.	PUNCT
iajs-2802	2	1	sci	sci	PROPN
iajs-2802	2	2	.	.	PROPN
iajs-2802	3	1	53	53	NUM
iajs-2802	3	2	(	(	PUNCT
iajs-2802	3	3	1)2022	1)2022	PROPN
iajs-2802	3	4	84	84	NUM
iajs-2802	3	5	this	this	DET
iajs-2802	3	6	work	work	NOUN
iajs-2802	3	7	is	be	AUX
iajs-2802	3	8	licensed	license	VERB
iajs-2802	3	9	under	under	ADP
iajs-2802	3	10	a	a	DET
iajs-2802	3	11	creative	creative	ADJ
iajs-2802	3	12	commons	common	NOUN
iajs-2802	3	13	attribution	attribution	NOUN
iajs-2802	3	14	4.0	4.0	NUM
iajs-2802	3	15	international	international	ADJ
iajs-2802	3	16	license	license	NOUN
iajs-2802	3	17	.	.	PUNCT
iajs-2802	4	1	strongly	strongly	ADV
iajs-2802	4	2	maximal	maximal	ADJ
iajs-2802	4	3	submodules	submodule	NOUN
iajs-2802	4	4	with	with	ADP
iajs-2802	4	5	a	a	DET
iajs-2802	4	6	study	study	NOUN
iajs-2802	4	7	of	of	ADP
iajs-2802	4	8	their	their	PRON
iajs-2802	4	9	influence	influence	NOUN
iajs-2802	4	10	on	on	ADP
iajs-2802	4	11	types	type	NOUN
iajs-2802	4	12	of	of	ADP
iajs-2802	4	13	modules	module	NOUN
iajs-2802	4	14	abstract	abstract	ADV
iajs-2802	4	15	let	let	VERB
iajs-2802	4	16	s	s	PRON
iajs-2802	4	17	be	be	AUX
iajs-2802	4	18	a	a	DET
iajs-2802	4	19	commutative	commutative	ADJ
iajs-2802	4	20	ring	ring	NOUN
iajs-2802	4	21	with	with	ADP
iajs-2802	4	22	identity	identity	NOUN
iajs-2802	4	23	,	,	PUNCT
iajs-2802	4	24	and	and	CCONJ
iajs-2802	4	25	a	a	PRON
iajs-2802	4	26	is	be	AUX
iajs-2802	4	27	an	an	DET
iajs-2802	4	28	s	s	NOUN
iajs-2802	4	29	-	-	NOUN
iajs-2802	4	30	module	module	NOUN
iajs-2802	4	31	.	.	PUNCT
iajs-2802	5	1	this	this	DET
iajs-2802	5	2	paper	paper	NOUN
iajs-2802	5	3	introduced	introduce	VERB
iajs-2802	5	4	an	an	DET
iajs-2802	5	5	important	important	ADJ
iajs-2802	5	6	concept	concept	NOUN
iajs-2802	5	7	,	,	PUNCT
iajs-2802	5	8	namely	namely	ADV
iajs-2802	5	9	strongly	strongly	ADV
iajs-2802	5	10	maximal	maximal	ADJ
iajs-2802	5	11	submodule	submodule	NOUN
iajs-2802	5	12	.	.	PUNCT
iajs-2802	6	1	some	some	DET
iajs-2802	6	2	properties	property	NOUN
iajs-2802	6	3	and	and	CCONJ
iajs-2802	6	4	many	many	ADJ
iajs-2802	6	5	results	result	NOUN
iajs-2802	6	6	were	be	AUX
iajs-2802	6	7	proved	prove	VERB
iajs-2802	6	8	as	as	ADV
iajs-2802	6	9	well	well	ADV
iajs-2802	6	10	as	as	ADP
iajs-2802	6	11	the	the	DET
iajs-2802	6	12	behavior	behavior	NOUN
iajs-2802	6	13	of	of	ADP
iajs-2802	6	14	that	that	DET
iajs-2802	6	15	concept	concept	NOUN
iajs-2802	6	16	with	with	ADP
iajs-2802	6	17	its	its	PRON
iajs-2802	6	18	localization	localization	NOUN
iajs-2802	6	19	was	be	AUX
iajs-2802	6	20	studied	study	VERB
iajs-2802	6	21	and	and	CCONJ
iajs-2802	6	22	shown	show	VERB
iajs-2802	6	23	.	.	PUNCT
iajs-2802	7	1	keywords	keyword	NOUN
iajs-2802	7	2	:	:	PUNCT
iajs-2802	7	3	maximal	maximal	ADJ
iajs-2802	7	4	submodule	submodule	NOUN
iajs-2802	7	5	,	,	PUNCT
iajs-2802	7	6	regular	regular	ADJ
iajs-2802	7	7	module	module	NOUN
iajs-2802	7	8	,	,	PUNCT
iajs-2802	7	9	regular	regular	ADJ
iajs-2802	7	10	ring	ring	NOUN
iajs-2802	7	11	,	,	PUNCT
iajs-2802	7	12	semi	semi	ADJ
iajs-2802	7	13	-	-	ADJ
iajs-2802	7	14	simple	simple	ADJ
iajs-2802	7	15	module	module	NOUN
iajs-2802	7	16	,	,	PUNCT
iajs-2802	7	17	prime	prime	ADJ
iajs-2802	7	18	module	module	NOUN
iajs-2802	7	19	.	.	PUNCT
iajs-2802	8	1	1.introduction	1.introduction	NUM
iajs-2802	8	2	along	along	ADP
iajs-2802	8	3	with	with	ADP
iajs-2802	8	4	this	this	DET
iajs-2802	8	5	paper	paper	NOUN
iajs-2802	8	6	,	,	PUNCT
iajs-2802	8	7	s	s	VERB
iajs-2802	8	8	is	be	AUX
iajs-2802	8	9	a	a	DET
iajs-2802	8	10	commutative	commutative	ADJ
iajs-2802	8	11	ring	ring	NOUN
iajs-2802	8	12	with	with	ADP
iajs-2802	8	13	identity	identity	NOUN
iajs-2802	8	14	,	,	PUNCT
iajs-2802	8	15	and	and	CCONJ
iajs-2802	8	16	a	a	PRON
iajs-2802	8	17	is	be	AUX
iajs-2802	8	18	an	an	DET
iajs-2802	8	19	s	s	NOUN
iajs-2802	8	20	-	-	NOUN
iajs-2802	8	21	module	module	NOUN
iajs-2802	8	22	.	.	PUNCT
iajs-2802	9	1	a	a	DET
iajs-2802	9	2	proper	proper	ADJ
iajs-2802	9	3	submodule	submodule	NOUN
iajs-2802	9	4	n	n	PROPN
iajs-2802	9	5	of	of	ADP
iajs-2802	9	6	an	an	DET
iajs-2802	9	7	s	s	NOUN
iajs-2802	9	8	-	-	NOUN
iajs-2802	9	9	module	module	NOUN
iajs-2802	9	10	a	a	PRON
iajs-2802	9	11	is	be	AUX
iajs-2802	9	12	named	name	VERB
iajs-2802	9	13	maximal	maximal	ADJ
iajs-2802	9	14	if	if	SCONJ
iajs-2802	9	15	there	there	PRON
iajs-2802	9	16	exists	exist	VERB
iajs-2802	9	17	a	a	DET
iajs-2802	9	18	submodule	submodule	NOUN
iajs-2802	9	19	d	d	PROPN
iajs-2802	9	20	of	of	ADP
iajs-2802	9	21	a	a	DET
iajs-2802	9	22	such	such	ADJ
iajs-2802	9	23	that	that	DET
iajs-2802	9	24	n⊊d⊆a	n⊊d⊆a	NOUN
iajs-2802	9	25	,	,	PUNCT
iajs-2802	9	26	then	then	ADV
iajs-2802	9	27	d	d	X
iajs-2802	9	28	=	=	PUNCT
iajs-2802	9	29	a[1][11	a[1][11	PROPN
iajs-2802	9	30	]	]	PUNCT
iajs-2802	9	31	.	.	PUNCT
iajs-2802	10	1	equivalently	equivalently	ADV
iajs-2802	10	2	,	,	PUNCT
iajs-2802	10	3	n	n	PRON
iajs-2802	10	4	is	be	AUX
iajs-2802	10	5	the	the	DET
iajs-2802	10	6	maximal	maximal	ADJ
iajs-2802	10	7	submodule	submodule	NOUN
iajs-2802	10	8	in	in	ADP
iajs-2802	10	9	a	a	DET
iajs-2802	10	10	if	if	NOUN
iajs-2802	10	11	and	and	CCONJ
iajs-2802	10	12	only	only	ADV
iajs-2802	10	13	if	if	SCONJ
iajs-2802	10	14	a	a	PRON
iajs-2802	10	15	/	/	SYM
iajs-2802	10	16	n	n	NOUN
iajs-2802	10	17	is	be	AUX
iajs-2802	10	18	a	a	DET
iajs-2802	10	19	simple	simple	ADJ
iajs-2802	10	20	s	s	NOUN
iajs-2802	10	21	-	-	NOUN
iajs-2802	10	22	module	module	NOUN
iajs-2802	10	23	[	[	X
iajs-2802	10	24	1][12	1][12	NUM
iajs-2802	10	25	]	]	X
iajs-2802	10	26	.	.	PUNCT
iajs-2802	11	1	maximal	maximal	ADJ
iajs-2802	11	2	submodules	submodule	NOUN
iajs-2802	11	3	may	may	AUX
iajs-2802	11	4	not	not	PART
iajs-2802	11	5	exist	exist	VERB
iajs-2802	11	6	;	;	PUNCT
iajs-2802	11	7	for	for	ADP
iajs-2802	11	8	instance	instance	NOUN
iajs-2802	11	9	,	,	PUNCT
iajs-2802	11	10	the	the	DET
iajs-2802	11	11	z	z	NOUN
iajs-2802	11	12	-	-	PUNCT
iajs-2802	11	13	module	module	NOUN
iajs-2802	11	14	ℂ	ℂ	PROPN
iajs-2802	11	15	has	have	VERB
iajs-2802	11	16	no	no	DET
iajs-2802	11	17	maximal	maximal	ADJ
iajs-2802	11	18	submodules	submodule	NOUN
iajs-2802	11	19	.	.	PUNCT
iajs-2802	12	1	the	the	DET
iajs-2802	12	2	main	main	ADJ
iajs-2802	12	3	goal	goal	NOUN
iajs-2802	12	4	of	of	ADP
iajs-2802	12	5	this	this	DET
iajs-2802	12	6	paper	paper	NOUN
iajs-2802	12	7	is	be	AUX
iajs-2802	12	8	to	to	PART
iajs-2802	12	9	introduce	introduce	VERB
iajs-2802	12	10	a	a	DET
iajs-2802	12	11	new	new	ADJ
iajs-2802	12	12	concept	concept	NOUN
iajs-2802	12	13	called	call	VERB
iajs-2802	12	14	strongly	strongly	ADV
iajs-2802	12	15	maximal	maximal	ADJ
iajs-2802	12	16	submodule	submodule	NOUN
iajs-2802	12	17	(	(	PUNCT
iajs-2802	12	18	for	for	ADP
iajs-2802	12	19	short	short	ADJ
iajs-2802	12	20	,	,	PUNCT
iajs-2802	12	21	sm	sm	NOUN
iajs-2802	12	22	-	-	PUNCT
iajs-2802	12	23	submodule	submodule	NOUN
iajs-2802	12	24	)	)	PUNCT
iajs-2802	12	25	where	where	SCONJ
iajs-2802	12	26	a	a	DET
iajs-2802	12	27	proper	proper	ADJ
iajs-2802	12	28	non	non	ADJ
iajs-2802	12	29	-	-	ADJ
iajs-2802	12	30	zero	zero	NUM
iajs-2802	12	31	submodule	submodule	NOUN
iajs-2802	12	32	b	b	PROPN
iajs-2802	12	33	of	of	ADP
iajs-2802	12	34	an	an	DET
iajs-2802	12	35	s	s	NOUN
iajs-2802	12	36	-	-	NOUN
iajs-2802	12	37	module	module	NOUN
iajs-2802	12	38	a	a	PRON
iajs-2802	12	39	is	be	AUX
iajs-2802	12	40	said	say	VERB
iajs-2802	12	41	to	to	PART
iajs-2802	12	42	be	be	AUX
iajs-2802	12	43	strongly	strongly	ADV
iajs-2802	12	44	maximal	maximal	ADJ
iajs-2802	12	45	submodule	submodule	NOUN
iajs-2802	12	46	if	if	SCONJ
iajs-2802	12	47	and	and	CCONJ
iajs-2802	12	48	only	only	ADV
iajs-2802	12	49	if	if	SCONJ
iajs-2802	12	50	,	,	PUNCT
iajs-2802	12	51	for	for	SCONJ
iajs-2802	12	52	every	every	DET
iajs-2802	12	53	non	non	ADJ
iajs-2802	12	54	-	-	ADJ
iajs-2802	12	55	zero	zero	NUM
iajs-2802	12	56	ideal	ideal	ADJ
iajs-2802	12	57	e	e	NOUN
iajs-2802	12	58	of	of	ADP
iajs-2802	12	59	s	s	PRON
iajs-2802	12	60	implies	imply	VERB
iajs-2802	12	61	a	a	X
iajs-2802	12	62	/	/	PRON
iajs-2802	12	63	e2b	e2b	PROPN
iajs-2802	12	64	is	be	AUX
iajs-2802	12	65	a	a	DET
iajs-2802	12	66	regular	regular	ADJ
iajs-2802	12	67	module	module	NOUN
iajs-2802	12	68	.	.	PUNCT
iajs-2802	13	1	field	field	NOUN
iajs-2802	13	2	house	house	NOUN
iajs-2802	13	3	is	be	AUX
iajs-2802	13	4	defined	define	VERB
iajs-2802	13	5	in	in	ADP
iajs-2802	13	6	[	[	X
iajs-2802	13	7	9	9	NUM
iajs-2802	13	8	]	]	PUNCT
iajs-2802	13	9	,	,	PUNCT
iajs-2802	13	10	a	a	DET
iajs-2802	13	11	pure	pure	ADJ
iajs-2802	13	12	submodule	submodule	NOUN
iajs-2802	13	13	of	of	ADP
iajs-2802	13	14	the	the	DET
iajs-2802	13	15	form	form	NOUN
iajs-2802	13	16	:	:	PUNCT
iajs-2802	13	17	a	a	DET
iajs-2802	13	18	submodule	submodule	PROPN
iajs-2802	13	19	d	d	NOUN
iajs-2802	13	20	of	of	ADP
iajs-2802	13	21	an	an	DET
iajs-2802	13	22	s	s	NOUN
iajs-2802	13	23	-	-	NOUN
iajs-2802	13	24	module	module	NOUN
iajs-2802	13	25	a	a	PRON
iajs-2802	13	26	is	be	AUX
iajs-2802	13	27	pure	pure	ADJ
iajs-2802	13	28	if	if	SCONJ
iajs-2802	13	29	ia⋂d	ia⋂d	PROPN
iajs-2802	13	30	=	=	SYM
iajs-2802	13	31	i	i	PROPN
iajs-2802	13	32	d	d	PROPN
iajs-2802	13	33	for	for	ADP
iajs-2802	13	34	every	every	DET
iajs-2802	13	35	ideal	ideal	NOUN
iajs-2802	13	36	i	i	PRON
iajs-2802	13	37	of	of	ADP
iajs-2802	13	38	s	s	PRON
iajs-2802	13	39	and	and	CCONJ
iajs-2802	13	40	sahera	sahera	NOUN
iajs-2802	13	41	introduced	introduce	VERB
iajs-2802	13	42	in	in	ADP
iajs-2802	13	43	[	[	X
iajs-2802	13	44	2	2	NUM
iajs-2802	13	45	]	]	PUNCT
iajs-2802	13	46	,	,	PUNCT
iajs-2802	13	47	the	the	DET
iajs-2802	13	48	definition	definition	NOUN
iajs-2802	13	49	of	of	ADP
iajs-2802	13	50	f	f	NOUN
iajs-2802	13	51	-	-	PUNCT
iajs-2802	13	52	regular	regular	ADJ
iajs-2802	13	53	module	module	NOUN
iajs-2802	13	54	,	,	PUNCT
iajs-2802	13	55	where	where	SCONJ
iajs-2802	13	56	module	module	NOUN
iajs-2802	13	57	a	a	PRON
iajs-2802	13	58	is	be	AUX
iajs-2802	13	59	said	say	VERB
iajs-2802	13	60	to	to	PART
iajs-2802	13	61	be	be	AUX
iajs-2802	13	62	f	f	NOUN
iajs-2802	13	63	-	-	NOUN
iajs-2802	13	64	regular	regular	ADJ
iajs-2802	13	65	if	if	SCONJ
iajs-2802	13	66	and	and	CCONJ
iajs-2802	13	67	only	only	ADV
iajs-2802	13	68	if	if	SCONJ
iajs-2802	13	69	every	every	DET
iajs-2802	13	70	submodule	submodule	NOUN
iajs-2802	13	71	of	of	ADP
iajs-2802	13	72	a	a	PRON
iajs-2802	13	73	is	be	AUX
iajs-2802	13	74	pure	pure	ADJ
iajs-2802	13	75	every	every	DET
iajs-2802	13	76	strongly	strongly	ADV
iajs-2802	13	77	maximal	maximal	ADJ
iajs-2802	13	78	submodule	submodule	NOUN
iajs-2802	13	79	is	be	AUX
iajs-2802	13	80	a	a	DET
iajs-2802	13	81	maximal	maximal	ADJ
iajs-2802	13	82	submodule	submodule	NOUN
iajs-2802	13	83	,	,	PUNCT
iajs-2802	13	84	but	but	CCONJ
iajs-2802	13	85	the	the	DET
iajs-2802	13	86	opposite	opposite	NOUN
iajs-2802	13	87	does	do	AUX
iajs-2802	13	88	not	not	PART
iajs-2802	13	89	true	true	ADJ
iajs-2802	13	90	.	.	PUNCT
iajs-2802	14	1	this	this	DET
iajs-2802	14	2	paper	paper	NOUN
iajs-2802	14	3	is	be	AUX
iajs-2802	14	4	divided	divide	VERB
iajs-2802	14	5	into	into	ADP
iajs-2802	14	6	three	three	NUM
iajs-2802	14	7	sections	section	NOUN
iajs-2802	14	8	.	.	PUNCT
iajs-2802	15	1	we	we	PRON
iajs-2802	15	2	reviewed	review	VERB
iajs-2802	15	3	some	some	DET
iajs-2802	15	4	basic	basic	ADJ
iajs-2802	15	5	definitions	definition	NOUN
iajs-2802	15	6	and	and	CCONJ
iajs-2802	15	7	properties	property	NOUN
iajs-2802	15	8	needed	need	VERB
iajs-2802	15	9	in	in	ADP
iajs-2802	15	10	our	our	PRON
iajs-2802	15	11	next	next	ADJ
iajs-2802	15	12	work	work	NOUN
iajs-2802	15	13	.	.	PUNCT
iajs-2802	16	1	section	section	NOUN
iajs-2802	16	2	three	three	NUM
iajs-2802	16	3	introduced	introduce	VERB
iajs-2802	16	4	the	the	DET
iajs-2802	16	5	definition	definition	NOUN
iajs-2802	16	6	of	of	ADP
iajs-2802	16	7	strongly	strongly	ADV
iajs-2802	16	8	maximal	maximal	ADJ
iajs-2802	16	9	submodule	submodule	NOUN
iajs-2802	16	10	.	.	PUNCT
iajs-2802	17	1	lots	lot	NOUN
iajs-2802	17	2	of	of	ADP
iajs-2802	17	3	properties	property	NOUN
iajs-2802	17	4	and	and	CCONJ
iajs-2802	17	5	examples	example	NOUN
iajs-2802	17	6	of	of	ADP
iajs-2802	17	7	this	this	DET
iajs-2802	17	8	concept	concept	NOUN
iajs-2802	17	9	were	be	AUX
iajs-2802	17	10	shown	show	VERB
iajs-2802	17	11	.	.	PUNCT
iajs-2802	18	1	in	in	ADP
iajs-2802	18	2	section	section	NOUN
iajs-2802	18	3	four	four	NUM
iajs-2802	18	4	,	,	PUNCT
iajs-2802	18	5	the	the	DET
iajs-2802	18	6	ibn	ibn	PROPN
iajs-2802	18	7	al	al	PROPN
iajs-2802	18	8	haitham	haitham	PROPN
iajs-2802	18	9	journal	journal	PROPN
iajs-2802	18	10	for	for	ADP
iajs-2802	18	11	pure	pure	ADJ
iajs-2802	18	12	and	and	CCONJ
iajs-2802	18	13	applied	apply	VERB
iajs-2802	18	14	science	science	NOUN
iajs-2802	18	15	journal	journal	PROPN
iajs-2802	18	16	homepage	homepage	NOUN
iajs-2802	18	17	:	:	PUNCT
iajs-2802	18	18	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2802	18	19	doi	doi	NOUN
iajs-2802	18	20	:	:	PUNCT
iajs-2802	18	21	10.30526/35.1.2802	10.30526/35.1.2802	PROPN
iajs-2802	18	22	article	article	NOUN
iajs-2802	18	23	history	history	NOUN
iajs-2802	18	24	:	:	PUNCT
iajs-2802	18	25	received	receive	VERB
iajs-2802	18	26	24	24	NUM
iajs-2802	18	27	,	,	PUNCT
iajs-2802	18	28	august	august	PROPN
iajs-2802	18	29	,	,	PUNCT
iajs-2802	18	30	2021	2021	NUM
iajs-2802	18	31	,	,	PUNCT
iajs-2802	18	32	accepted	accept	VERB
iajs-2802	18	33	26	26	NUM
iajs-2802	18	34	,	,	PUNCT
iajs-2802	18	35	september	september	PROPN
iajs-2802	18	36	,	,	PUNCT
iajs-2802	18	37	2021	2021	NUM
iajs-2802	18	38	,	,	PUNCT
iajs-2802	18	39	published	publish	VERB
iajs-2802	18	40	in	in	ADP
iajs-2802	18	41	january	january	PROPN
iajs-2802	18	42	2022	2022	NUM
iajs-2802	18	43	.	.	PUNCT
iajs-2802	19	1	fatima	fatima	PROPN
iajs-2802	19	2	m.	m.	PROPN
iajs-2802	19	3	mohialdeen	mohialdeen	PROPN
iajs-2802	19	4	mathematics	mathematics	PROPN
iajs-2802	19	5	department	department	PROPN
iajs-2802	19	6	,	,	PUNCT
iajs-2802	19	7	college	college	NOUN
iajs-2802	19	8	of	of	ADP
iajs-2802	19	9	education	education	PROPN
iajs-2802	19	10	ibn	ibn	PROPN
iajs-2802	19	11	al	al	PROPN
iajs-2802	19	12	-	-	PUNCT
iajs-2802	19	13	haitham	haitham	PROPN
iajs-2802	19	14	for	for	ADP
iajs-2802	19	15	pure	pure	ADJ
iajs-2802	19	16	sciences	science	NOUN
iajs-2802	19	17	,	,	PUNCT
iajs-2802	19	18	university	university	NOUN
iajs-2802	19	19	of	of	ADP
iajs-2802	19	20	baghdad	baghdad	PROPN
iajs-2802	19	21	,	,	PUNCT
iajs-2802	19	22	iraq	iraq	PROPN
iajs-2802	19	23	fatima.mohy.fm@gmail.com	fatima.mohy.fm@gmail.com	PROPN
iajs-2802	19	24	buthyna	buthyna	PROPN
iajs-2802	19	25	n.	n.	PROPN
iajs-2802	19	26	shihab	shihab	PROPN
iajs-2802	19	27	mathematics	mathematics	PROPN
iajs-2802	19	28	department	department	PROPN
iajs-2802	19	29	,	,	PUNCT
iajs-2802	19	30	college	college	NOUN
iajs-2802	19	31	of	of	ADP
iajs-2802	19	32	education	education	PROPN
iajs-2802	19	33	ibn	ibn	PROPN
iajs-2802	19	34	al	al	PROPN
iajs-2802	19	35	-	-	PUNCT
iajs-2802	19	36	haitham	haitham	PROPN
iajs-2802	19	37	for	for	ADP
iajs-2802	19	38	pure	pure	ADJ
iajs-2802	19	39	science	science	NOUN
iajs-2802	19	40	,	,	PUNCT
iajs-2802	19	41	university	university	NOUN
iajs-2802	19	42	of	of	ADP
iajs-2802	19	43	baghdad	baghdad	PROPN
iajs-2802	19	44	,	,	PUNCT
iajs-2802	19	45	iraq	iraq	PROPN
iajs-2802	19	46	dr.buthyna@yahoo.com	dr.buthyna@yahoo.com	PROPN
iajs-2802	19	47	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2802	19	48	mailto:fatima.mohy.fm@gmail.com	mailto:fatima.mohy.fm@gmail.com	PROPN
iajs-2802	19	49	mailto:dr.buthyna@yahoo.com	mailto:dr.buthyna@yahoo.com	PROPN
iajs-2802	19	50	ibn	ibn	PROPN
iajs-2802	19	51	al	al	PROPN
iajs-2802	19	52	-	-	PUNCT
iajs-2802	19	53	haitham	haitham	PROPN
iajs-2802	19	54	jour	jour	X
iajs-2802	19	55	.	.	PROPN
iajs-2802	20	1	for	for	ADP
iajs-2802	20	2	pure	pure	ADJ
iajs-2802	20	3	&	&	CCONJ
iajs-2802	20	4	appl	appl	PROPN
iajs-2802	20	5	.	.	PUNCT
iajs-2802	21	1	sci	sci	PROPN
iajs-2802	21	2	.	.	PROPN
iajs-2802	22	1	53	53	NUM
iajs-2802	22	2	(	(	PUNCT
iajs-2802	22	3	1)2022	1)2022	NOUN
iajs-2802	22	4	85	85	NUM
iajs-2802	22	5	behavior	behavior	NOUN
iajs-2802	22	6	of	of	ADP
iajs-2802	22	7	strongly	strongly	ADV
iajs-2802	22	8	maximal	maximal	ADJ
iajs-2802	22	9	submodules	submodule	NOUN
iajs-2802	22	10	under	under	ADP
iajs-2802	22	11	localization	localization	NOUN
iajs-2802	22	12	was	be	AUX
iajs-2802	22	13	some	some	DET
iajs-2802	22	14	characterized	characterize	VERB
iajs-2802	22	15	and	and	CCONJ
iajs-2802	22	16	results	result	NOUN
iajs-2802	22	17	were	be	AUX
iajs-2802	22	18	proved	prove	VERB
iajs-2802	22	19	.	.	PUNCT
iajs-2802	23	1	2.basic	2.basic	NUM
iajs-2802	23	2	concepts	concept	NOUN
iajs-2802	23	3	and	and	CCONJ
iajs-2802	23	4	results	result	NOUN
iajs-2802	23	5	this	this	DET
iajs-2802	23	6	part	part	NOUN
iajs-2802	23	7	includes	include	VERB
iajs-2802	23	8	some	some	DET
iajs-2802	23	9	well	well	ADV
iajs-2802	23	10	-	-	PUNCT
iajs-2802	23	11	known	know	VERB
iajs-2802	23	12	definitions	definition	NOUN
iajs-2802	23	13	,	,	PUNCT
iajs-2802	23	14	concepts	concept	NOUN
iajs-2802	23	15	,	,	PUNCT
iajs-2802	23	16	and	and	CCONJ
iajs-2802	23	17	results	result	NOUN
iajs-2802	23	18	that	that	PRON
iajs-2802	23	19	are	be	AUX
iajs-2802	23	20	useful	useful	ADJ
iajs-2802	23	21	for	for	ADP
iajs-2802	23	22	us	we	PRON
iajs-2802	23	23	in	in	ADP
iajs-2802	23	24	our	our	PRON
iajs-2802	23	25	study	study	NOUN
iajs-2802	23	26	of	of	ADP
iajs-2802	23	27	the	the	DET
iajs-2802	23	28	next	next	ADJ
iajs-2802	23	29	section	section	NOUN
iajs-2802	23	30	.	.	PUNCT
iajs-2802	24	1	proposition	proposition	NOUN
iajs-2802	24	2	(	(	PUNCT
iajs-2802	24	3	2.1	2.1	NUM
iajs-2802	24	4	)	)	PUNCT
iajs-2802	24	5	every	every	DET
iajs-2802	24	6	submodule	submodule	NOUN
iajs-2802	24	7	of	of	ADP
iajs-2802	24	8	the	the	DET
iajs-2802	24	9	regular	regular	ADJ
iajs-2802	24	10	module	module	NOUN
iajs-2802	24	11	is	be	AUX
iajs-2802	24	12	regular	regular	ADJ
iajs-2802	24	13	[	[	X
iajs-2802	24	14	2	2	NUM
iajs-2802	24	15	]	]	PUNCT
iajs-2802	24	16	.	.	PUNCT
iajs-2802	25	1	proposition	proposition	NOUN
iajs-2802	25	2	(	(	PUNCT
iajs-2802	25	3	2.2	2.2	NUM
iajs-2802	25	4	)	)	PUNCT
iajs-2802	25	5	an	an	DET
iajs-2802	25	6	s	s	NOUN
iajs-2802	25	7	-	-	NOUN
iajs-2802	25	8	module	module	NOUN
iajs-2802	25	9	a	a	PRON
iajs-2802	25	10	is	be	AUX
iajs-2802	25	11	cyclic	cyclic	ADJ
iajs-2802	25	12	if	if	SCONJ
iajs-2802	25	13	and	and	CCONJ
iajs-2802	25	14	only	only	ADV
iajs-2802	25	15	if	if	SCONJ
iajs-2802	25	16	it	it	PRON
iajs-2802	25	17	is	be	AUX
iajs-2802	25	18	isomorphic	isomorphic	ADJ
iajs-2802	25	19	to	to	ADP
iajs-2802	25	20	a	a	DET
iajs-2802	25	21	factor	factor	NOUN
iajs-2802	25	22	module	module	NOUN
iajs-2802	25	23	of	of	ADP
iajs-2802	25	24	s	s	NOUN
iajs-2802	25	25	[	[	X
iajs-2802	25	26	4	4	NUM
iajs-2802	25	27	]	]	PUNCT
iajs-2802	25	28	.	.	PUNCT
iajs-2802	26	1	proposition	proposition	NOUN
iajs-2802	26	2	(	(	PUNCT
iajs-2802	26	3	2.3	2.3	NUM
iajs-2802	26	4	)	)	PUNCT
iajs-2802	26	5	an	an	DET
iajs-2802	26	6	s	s	NOUN
iajs-2802	26	7	-	-	NOUN
iajs-2802	26	8	module	module	NOUN
iajs-2802	26	9	a	a	PRON
iajs-2802	26	10	is	be	AUX
iajs-2802	26	11	simple	simple	ADJ
iajs-2802	26	12	if	if	SCONJ
iajs-2802	27	1	and	and	CCONJ
iajs-2802	27	2	only	only	ADV
iajs-2802	27	3	if	if	SCONJ
iajs-2802	27	4	a	a	DET
iajs-2802	27	5	≅	≅	PROPN
iajs-2802	27	6	s	s	NOUN
iajs-2802	27	7	/	/	SYM
iajs-2802	27	8	e	e	NOUN
iajs-2802	27	9	for	for	ADP
iajs-2802	27	10	some	some	DET
iajs-2802	27	11	maximal	maximal	ADJ
iajs-2802	27	12	ideal	ideal	ADJ
iajs-2802	27	13	e	e	NOUN
iajs-2802	27	14	of	of	ADP
iajs-2802	27	15	s	s	PRON
iajs-2802	27	16	[	[	X
iajs-2802	27	17	4	4	NUM
iajs-2802	27	18	]	]	X
iajs-2802	28	1	[	[	X
iajs-2802	28	2	1	1	NUM
iajs-2802	28	3	]	]	PUNCT
iajs-2802	28	4	.	.	PUNCT
iajs-2802	29	1	definition	definition	NOUN
iajs-2802	29	2	(	(	PUNCT
iajs-2802	29	3	2.4	2.4	NUM
iajs-2802	29	4	)	)	PUNCT
iajs-2802	29	5	a	a	DET
iajs-2802	29	6	submodule	submodule	NOUN
iajs-2802	29	7	b	b	PROPN
iajs-2802	29	8	of	of	ADP
iajs-2802	29	9	an	an	DET
iajs-2802	29	10	s	s	NOUN
iajs-2802	29	11	-	-	NOUN
iajs-2802	29	12	module	module	NOUN
iajs-2802	29	13	a	a	PRON
iajs-2802	29	14	is	be	AUX
iajs-2802	29	15	called	call	VERB
iajs-2802	29	16	prime	prime	NOUN
iajs-2802	29	17	if	if	SCONJ
iajs-2802	29	18	b	b	PROPN
iajs-2802	29	19	≠	≠	PROPN
iajs-2802	29	20	a	a	X
iajs-2802	29	21	,	,	PUNCT
iajs-2802	29	22	and	and	CCONJ
iajs-2802	29	23	whenever	whenever	SCONJ
iajs-2802	29	24	tx	tx	PROPN
iajs-2802	29	25	∈	∈	PROPN
iajs-2802	29	26	b	b	PROPN
iajs-2802	29	27	for	for	ADP
iajs-2802	29	28	t	t	PROPN
iajs-2802	29	29	∈	∈	PROPN
iajs-2802	29	30	s	s	PART
iajs-2802	29	31	and	and	CCONJ
iajs-2802	29	32	x	x	PROPN
iajs-2802	29	33	∈	∈	PROPN
iajs-2802	29	34	a	a	PRON
iajs-2802	29	35	we	we	PRON
iajs-2802	29	36	have	have	AUX
iajs-2802	30	1	either	either	CCONJ
iajs-2802	30	2	t	t	PROPN
iajs-2802	30	3	∈	∈	PROPN
iajs-2802	31	1	[	[	X
iajs-2802	31	2	b	b	X
iajs-2802	31	3	:	:	PUNCT
iajs-2802	31	4	a	a	PRON
iajs-2802	31	5	]	]	X
iajs-2802	31	6	or	or	CCONJ
iajs-2802	31	7	x	x	SYM
iajs-2802	31	8	∈	∈	PROPN
iajs-2802	31	9	b	b	PROPN
iajs-2802	32	1	[	[	X
iajs-2802	32	2	5	5	NUM
iajs-2802	32	3	]	]	PUNCT
iajs-2802	32	4	.	.	PUNCT
iajs-2802	33	1	definition	definition	NOUN
iajs-2802	33	2	(	(	PUNCT
iajs-2802	33	3	2.5	2.5	NUM
iajs-2802	33	4	)	)	PUNCT
iajs-2802	33	5	a	a	DET
iajs-2802	33	6	submodule	submodule	PROPN
iajs-2802	33	7	b	b	PROPN
iajs-2802	33	8	of	of	ADP
iajs-2802	33	9	an	an	DET
iajs-2802	33	10	s	s	NOUN
iajs-2802	33	11	-	-	NOUN
iajs-2802	33	12	module	module	NOUN
iajs-2802	33	13	a	a	PRON
iajs-2802	33	14	is	be	AUX
iajs-2802	33	15	called	call	VERB
iajs-2802	33	16	a	a	DET
iajs-2802	33	17	semimaximal	semimaximal	ADJ
iajs-2802	33	18	submodule	submodule	NOUN
iajs-2802	33	19	if	if	SCONJ
iajs-2802	33	20	and	and	CCONJ
iajs-2802	33	21	only	only	ADV
iajs-2802	33	22	if	if	SCONJ
iajs-2802	33	23	a	a	DET
iajs-2802	33	24	/	/	SYM
iajs-2802	33	25	b	b	NOUN
iajs-2802	33	26	is	be	AUX
iajs-2802	33	27	a	a	DET
iajs-2802	33	28	semi	semi	ADJ
iajs-2802	33	29	-	-	ADJ
iajs-2802	33	30	simple	simple	ADJ
iajs-2802	33	31	s	s	NOUN
iajs-2802	33	32	-	-	NOUN
iajs-2802	33	33	module	module	NOUN
iajs-2802	33	34	[	[	NOUN
iajs-2802	33	35	3	3	NUM
iajs-2802	33	36	,	,	PUNCT
iajs-2802	33	37	definition	definition	NOUN
iajs-2802	33	38	(	(	PUNCT
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iajs-2802	33	41	.	.	PUNCT
iajs-2802	34	1	definition	definition	NOUN
iajs-2802	34	2	(	(	PUNCT
iajs-2802	34	3	2.6	2.6	NUM
iajs-2802	34	4	)	)	PUNCT
iajs-2802	34	5	let	let	VERB
iajs-2802	34	6	b	b	X
iajs-2802	34	7	be	be	AUX
iajs-2802	34	8	a	a	DET
iajs-2802	34	9	submodule	submodule	NOUN
iajs-2802	34	10	of	of	ADP
iajs-2802	34	11	an	an	DET
iajs-2802	34	12	s	s	NOUN
iajs-2802	34	13	-	-	NOUN
iajs-2802	34	14	module	module	NOUN
iajs-2802	34	15	a	a	PRON
iajs-2802	34	16	,	,	PUNCT
iajs-2802	34	17	the	the	DET
iajs-2802	34	18	closure	closure	NOUN
iajs-2802	34	19	of	of	ADP
iajs-2802	34	20	b	b	NOUN
iajs-2802	34	21	is	be	AUX
iajs-2802	34	22	denoted	denote	VERB
iajs-2802	34	23	by	by	ADP
iajs-2802	34	24	cl(b	cl(b	NOUN
iajs-2802	34	25	)	)	PUNCT
iajs-2802	34	26	=	=	PRON
iajs-2802	35	1	{	{	PUNCT
iajs-2802	35	2	x	x	PUNCT
iajs-2802	35	3	∈	∈	PROPN
iajs-2802	35	4	a	a	PRON
iajs-2802	35	5	:	:	PUNCT
iajs-2802	35	6	[	[	X
iajs-2802	35	7	b:(x	b:(x	NOUN
iajs-2802	35	8	)	)	PUNCT
iajs-2802	35	9	]	]	PUNCT
iajs-2802	36	1	essential	essential	ADJ
iajs-2802	36	2	in	in	ADP
iajs-2802	36	3	s	s	PRON
iajs-2802	36	4	}	}	PUNCT
iajs-2802	36	5	.	.	PUNCT
iajs-2802	37	1	it	it	PRON
iajs-2802	37	2	is	be	AUX
iajs-2802	37	3	clear	clear	ADJ
iajs-2802	37	4	that	that	SCONJ
iajs-2802	37	5	cl(b	cl(b	NOUN
iajs-2802	37	6	)	)	PUNCT
iajs-2802	37	7	is	be	AUX
iajs-2802	37	8	a	a	DET
iajs-2802	37	9	submodule	submodule	NOUN
iajs-2802	37	10	of	of	ADP
iajs-2802	37	11	a	a	DET
iajs-2802	37	12	containing	contain	VERB
iajs-2802	37	13	b.	b.	NOUN
iajs-2802	37	14	that	that	PRON
iajs-2802	37	15	is	be	AUX
iajs-2802	37	16	b	b	NUM
iajs-2802	37	17	⊆	⊆	NUM
iajs-2802	37	18	cl(b	cl(b	NOUN
iajs-2802	37	19	)	)	PUNCT
iajs-2802	38	1	[	[	X
iajs-2802	38	2	7	7	NUM
iajs-2802	38	3	]	]	PUNCT
iajs-2802	38	4	.	.	PUNCT
iajs-2802	39	1	s3	s3	NOUN
iajs-2802	39	2	:	:	PUNCT
iajs-2802	39	3	strongly	strongly	ADV
iajs-2802	39	4	maximal	maximal	ADJ
iajs-2802	39	5	submodules	submodule	NOUN
iajs-2802	39	6	with	with	ADP
iajs-2802	39	7	its	its	PRON
iajs-2802	39	8	advantages	advantage	NOUN
iajs-2802	39	9	in	in	ADP
iajs-2802	39	10	this	this	DET
iajs-2802	39	11	section	section	NOUN
iajs-2802	39	12	,	,	PUNCT
iajs-2802	39	13	the	the	DET
iajs-2802	39	14	concept	concept	NOUN
iajs-2802	39	15	of	of	ADP
iajs-2802	39	16	strongly	strongly	ADV
iajs-2802	39	17	maximal	maximal	ADJ
iajs-2802	39	18	submodule	submodule	NOUN
iajs-2802	39	19	(	(	PUNCT
iajs-2802	39	20	for	for	ADP
iajs-2802	39	21	short	short	ADJ
iajs-2802	39	22	,	,	PUNCT
iajs-2802	39	23	sm	sm	NOUN
iajs-2802	39	24	-	-	PUNCT
iajs-2802	39	25	submodule	submodule	NOUN
iajs-2802	39	26	)	)	PUNCT
iajs-2802	39	27	was	be	AUX
iajs-2802	39	28	introduced	introduce	VERB
iajs-2802	39	29	,	,	PUNCT
iajs-2802	39	30	which	which	PRON
iajs-2802	39	31	was	be	AUX
iajs-2802	39	32	a	a	DET
iajs-2802	39	33	generalization	generalization	NOUN
iajs-2802	39	34	of	of	ADP
iajs-2802	39	35	the	the	DET
iajs-2802	39	36	concept	concept	NOUN
iajs-2802	39	37	the	the	DET
iajs-2802	39	38	strongly	strongly	ADV
iajs-2802	39	39	maximal	maximal	ADJ
iajs-2802	39	40	ideal	ideal	NOUN
iajs-2802	39	41	in	in	ADP
iajs-2802	39	42	a	a	DET
iajs-2802	39	43	ring	ring	NOUN
iajs-2802	39	44	s.	s.	PROPN
iajs-2802	39	45	several	several	ADJ
iajs-2802	39	46	examples	example	NOUN
iajs-2802	39	47	and	and	CCONJ
iajs-2802	39	48	properties	property	NOUN
iajs-2802	39	49	were	be	AUX
iajs-2802	39	50	proved	prove	VERB
iajs-2802	39	51	also	also	ADV
iajs-2802	39	52	a	a	DET
iajs-2802	39	53	lot	lot	NOUN
iajs-2802	39	54	of	of	ADP
iajs-2802	39	55	characterizations	characterization	NOUN
iajs-2802	39	56	,	,	PUNCT
iajs-2802	39	57	and	and	CCONJ
iajs-2802	39	58	different	different	ADJ
iajs-2802	39	59	results	result	NOUN
iajs-2802	39	60	were	be	AUX
iajs-2802	39	61	presented	present	VERB
iajs-2802	39	62	.	.	PUNCT
iajs-2802	40	1	let	let	VERB
iajs-2802	40	2	us	we	PRON
iajs-2802	40	3	start	start	VERB
iajs-2802	40	4	with	with	ADP
iajs-2802	40	5	our	our	PRON
iajs-2802	40	6	basic	basic	ADJ
iajs-2802	40	7	definition	definition	NOUN
iajs-2802	40	8	.	.	PUNCT
iajs-2802	41	1	definition	definition	NOUN
iajs-2802	41	2	(	(	PUNCT
iajs-2802	41	3	3.1	3.1	NUM
iajs-2802	41	4	)	)	PUNCT
iajs-2802	41	5	let	let	VERB
iajs-2802	41	6	a	a	PRON
iajs-2802	41	7	be	be	AUX
iajs-2802	41	8	an	an	DET
iajs-2802	41	9	s	s	NOUN
iajs-2802	41	10	-	-	NOUN
iajs-2802	41	11	module	module	NOUN
iajs-2802	41	12	,	,	PUNCT
iajs-2802	41	13	and	and	CCONJ
iajs-2802	41	14	b	b	X
iajs-2802	41	15	be	be	AUX
iajs-2802	41	16	a	a	DET
iajs-2802	41	17	non	non	ADJ
iajs-2802	41	18	-	-	ADJ
iajs-2802	41	19	zero	zero	ADJ
iajs-2802	41	20	proper	proper	ADJ
iajs-2802	41	21	submodule	submodule	NOUN
iajs-2802	41	22	of	of	ADP
iajs-2802	41	23	a.	a.	PROPN
iajs-2802	41	24	then	then	ADV
iajs-2802	41	25	b	b	PROPN
iajs-2802	41	26	is	be	AUX
iajs-2802	41	27	named	name	VERB
iajs-2802	41	28	strongly	strongly	ADV
iajs-2802	41	29	maximal	maximal	ADJ
iajs-2802	41	30	submodule	submodule	NOUN
iajs-2802	41	31	(	(	PUNCT
iajs-2802	41	32	for	for	ADP
iajs-2802	41	33	short	short	ADJ
iajs-2802	41	34	sm	sm	NOUN
iajs-2802	41	35	-	-	PUNCT
iajs-2802	41	36	submodule	submodule	NOUN
iajs-2802	41	37	)	)	PUNCT
iajs-2802	42	1	if	if	SCONJ
iajs-2802	42	2	and	and	CCONJ
iajs-2802	42	3	only	only	ADV
iajs-2802	42	4	if	if	SCONJ
iajs-2802	42	5	,	,	PUNCT
iajs-2802	42	6	for	for	SCONJ
iajs-2802	42	7	every	every	DET
iajs-2802	42	8	non	non	ADJ
iajs-2802	42	9	-	-	ADJ
iajs-2802	42	10	zero	zero	NUM
iajs-2802	42	11	ideal	ideal	ADJ
iajs-2802	42	12	e	e	NOUN
iajs-2802	42	13	of	of	ADP
iajs-2802	42	14	s	s	PRON
iajs-2802	42	15	implies	imply	VERB
iajs-2802	42	16	a	a	X
iajs-2802	42	17	/	/	PRON
iajs-2802	42	18	e2b	e2b	PROPN
iajs-2802	42	19	is	be	AUX
iajs-2802	42	20	a	a	DET
iajs-2802	42	21	regular	regular	ADJ
iajs-2802	42	22	module	module	NOUN
iajs-2802	42	23	.	.	PUNCT
iajs-2802	43	1	examples	example	NOUN
iajs-2802	43	2	and	and	CCONJ
iajs-2802	43	3	remarks	remark	NOUN
iajs-2802	43	4	(	(	PUNCT
iajs-2802	43	5	3.2	3.2	NUM
iajs-2802	43	6	)	)	PUNCT
iajs-2802	43	7	ibn	ibn	PROPN
iajs-2802	43	8	al	al	PROPN
iajs-2802	43	9	-	-	PUNCT
iajs-2802	43	10	haitham	haitham	PROPN
iajs-2802	43	11	jour	jour	X
iajs-2802	43	12	.	.	PROPN
iajs-2802	44	1	for	for	ADP
iajs-2802	44	2	pure	pure	ADJ
iajs-2802	44	3	&	&	CCONJ
iajs-2802	44	4	appl	appl	PROPN
iajs-2802	44	5	.	.	PUNCT
iajs-2802	45	1	sci	sci	PROPN
iajs-2802	45	2	.	.	PROPN
iajs-2802	46	1	53	53	NUM
iajs-2802	46	2	(	(	PUNCT
iajs-2802	46	3	1)2022	1)2022	PROPN
iajs-2802	46	4	86	86	NUM
iajs-2802	46	5	1	1	NUM
iajs-2802	46	6	.	.	PUNCT
iajs-2802	47	1	all	all	DET
iajs-2802	47	2	the	the	DET
iajs-2802	47	3	following	follow	VERB
iajs-2802	47	4	modules	module	NOUN
iajs-2802	47	5	have	have	VERB
iajs-2802	47	6	no	no	DET
iajs-2802	47	7	sm	sm	NOUN
iajs-2802	47	8	-	-	PUNCT
iajs-2802	47	9	submodules	submodules	NOUN
iajs-2802	47	10	.	.	PUNCT
iajs-2802	48	1	(	(	PUNCT
iajs-2802	48	2	i	i	NOUN
iajs-2802	48	3	)	)	PUNCT
iajs-2802	48	4	z	z	PROPN
iajs-2802	48	5	as	as	ADP
iajs-2802	48	6	a	a	DET
iajs-2802	48	7	z	z	NOUN
iajs-2802	48	8	-	-	PUNCT
iajs-2802	48	9	module	module	NOUN
iajs-2802	48	10	.	.	PUNCT
iajs-2802	49	1	(	(	PUNCT
iajs-2802	49	2	ii	ii	NOUN
iajs-2802	49	3	)	)	PUNCT
iajs-2802	49	4	zp	zp	PROPN
iajs-2802	49	5	as	as	ADP
iajs-2802	49	6	a	a	DET
iajs-2802	49	7	z	z	NOUN
iajs-2802	49	8	-	-	PUNCT
iajs-2802	49	9	module	module	NOUN
iajs-2802	49	10	,	,	PUNCT
iajs-2802	49	11	p	p	PRON
iajs-2802	49	12	is	be	AUX
iajs-2802	49	13	a	a	DET
iajs-2802	49	14	prime	prime	ADJ
iajs-2802	49	15	number	number	NOUN
iajs-2802	49	16	.	.	PUNCT
iajs-2802	50	1	(	(	PUNCT
iajs-2802	50	2	iii	iii	X
iajs-2802	50	3	)	)	PUNCT
iajs-2802	50	4	zp	zp	NOUN
iajs-2802	50	5	as	as	ADP
iajs-2802	50	6	a	a	DET
iajs-2802	50	7	zp	zp	NOUN
iajs-2802	50	8	-	-	NOUN
iajs-2802	50	9	module	module	NOUN
iajs-2802	50	10	,	,	PUNCT
iajs-2802	50	11	p	p	PRON
iajs-2802	50	12	is	be	AUX
iajs-2802	50	13	a	a	DET
iajs-2802	50	14	prime	prime	ADJ
iajs-2802	50	15	number	number	NOUN
iajs-2802	50	16	.	.	PUNCT
iajs-2802	51	1	2	2	X
iajs-2802	51	2	.	.	X
iajs-2802	51	3	every	every	DET
iajs-2802	51	4	simple	simple	ADJ
iajs-2802	51	5	s	s	NOUN
iajs-2802	51	6	-	-	PUNCT
iajs-2802	51	7	module	module	NOUN
iajs-2802	51	8	has	have	VERB
iajs-2802	51	9	no	no	DET
iajs-2802	51	10	sm	sm	NOUN
iajs-2802	51	11	-	-	PUNCT
iajs-2802	51	12	submodule	submodule	NOUN
iajs-2802	51	13	.	.	PUNCT
iajs-2802	52	1	but	but	CCONJ
iajs-2802	52	2	the	the	DET
iajs-2802	52	3	opposite	opposite	NOUN
iajs-2802	52	4	is	be	AUX
iajs-2802	52	5	not	not	PART
iajs-2802	52	6	true	true	ADJ
iajs-2802	52	7	and	and	CCONJ
iajs-2802	52	8	the	the	DET
iajs-2802	52	9	following	follow	VERB
iajs-2802	52	10	example	example	NOUN
iajs-2802	52	11	shows	show	VERB
iajs-2802	52	12	that	that	SCONJ
iajs-2802	52	13	:	:	PUNCT
iajs-2802	52	14	the	the	DET
iajs-2802	52	15	module	module	NOUN
iajs-2802	52	16	a	a	DET
iajs-2802	52	17	=	=	NOUN
iajs-2802	52	18	z4⊕z	z4⊕z	NOUN
iajs-2802	52	19	as	as	ADP
iajs-2802	52	20	a	a	DET
iajs-2802	52	21	z	z	NOUN
iajs-2802	52	22	-	-	PUNCT
iajs-2802	52	23	module	module	NOUN
iajs-2802	52	24	.	.	PUNCT
iajs-2802	53	1	since	since	SCONJ
iajs-2802	53	2	a	a	PRON
iajs-2802	53	3	has	have	VERB
iajs-2802	53	4	no	no	DET
iajs-2802	53	5	smsubmodules	smsubmodule	NOUN
iajs-2802	53	6	,	,	PUNCT
iajs-2802	53	7	a	a	PRON
iajs-2802	53	8	is	be	AUX
iajs-2802	53	9	not	not	PART
iajs-2802	53	10	a	a	DET
iajs-2802	53	11	simple	simple	ADJ
iajs-2802	53	12	module	module	NOUN
iajs-2802	53	13	.	.	PUNCT
iajs-2802	54	1	also	also	ADV
iajs-2802	54	2	notice	notice	VERB
iajs-2802	54	3	examples	example	NOUN
iajs-2802	54	4	(	(	PUNCT
iajs-2802	54	5	ii	ii	NOUN
iajs-2802	54	6	)	)	PUNCT
iajs-2802	54	7	and	and	CCONJ
iajs-2802	54	8	(	(	PUNCT
iajs-2802	54	9	iii	iii	NOUN
iajs-2802	54	10	)	)	PUNCT
iajs-2802	54	11	in	in	ADP
iajs-2802	54	12	no.(1	no.(1	NOUN
iajs-2802	54	13	)	)	PUNCT
iajs-2802	54	14	.	.	PUNCT
iajs-2802	55	1	3	3	X
iajs-2802	55	2	.	.	X
iajs-2802	55	3	it	it	PRON
iajs-2802	55	4	is	be	AUX
iajs-2802	55	5	important	important	ADJ
iajs-2802	55	6	to	to	PART
iajs-2802	55	7	note	note	VERB
iajs-2802	55	8	that	that	SCONJ
iajs-2802	55	9	it	it	PRON
iajs-2802	55	10	is	be	AUX
iajs-2802	55	11	not	not	PART
iajs-2802	55	12	necessary	necessary	ADJ
iajs-2802	55	13	that	that	SCONJ
iajs-2802	55	14	all	all	DET
iajs-2802	55	15	modules	module	NOUN
iajs-2802	55	16	contain	contain	VERB
iajs-2802	55	17	sm	sm	NOUN
iajs-2802	55	18	-	-	PUNCT
iajs-2802	55	19	submodules	submodule	NOUN
iajs-2802	55	20	;	;	PUNCT
iajs-2802	55	21	for	for	ADP
iajs-2802	55	22	example	example	NOUN
iajs-2802	55	23	zp∞	zp∞	PROPN
iajs-2802	55	24	asz	asz	NOUN
iajs-2802	55	25	-	-	PUNCT
iajs-2802	55	26	module	module	NOUN
iajs-2802	55	27	.	.	PUNCT
iajs-2802	56	1	since	since	SCONJ
iajs-2802	56	2	,	,	PUNCT
iajs-2802	56	3	all	all	DET
iajs-2802	56	4	the	the	DET
iajs-2802	56	5	submodules	submodule	NOUN
iajs-2802	56	6	of	of	ADP
iajs-2802	56	7	zp∞	zp∞	PROPN
iajs-2802	56	8	are	be	AUX
iajs-2802	56	9	of	of	ADP
iajs-2802	56	10	the	the	DET
iajs-2802	56	11	form	form	NOUN
iajs-2802	56	12	<	<	X
iajs-2802	56	13	1	1	NUM
iajs-2802	56	14	/	/	SYM
iajs-2802	56	15	pi	pi	NOUN
iajs-2802	57	1	+	+	CCONJ
iajs-2802	57	2	z	z	X
iajs-2802	57	3	>	>	X
iajs-2802	57	4	,	,	PUNCT
iajs-2802	57	5	where	where	SCONJ
iajs-2802	57	6	p	p	NOUN
iajs-2802	57	7	is	be	AUX
iajs-2802	57	8	a	a	DET
iajs-2802	57	9	prime	prime	ADJ
iajs-2802	57	10	number	number	NOUN
iajs-2802	57	11	and	and	CCONJ
iajs-2802	57	12	i=	i=	PROPN
iajs-2802	57	13	0	0	NUM
iajs-2802	57	14	,	,	PUNCT
iajs-2802	57	15	1	1	NUM
iajs-2802	57	16	,	,	PUNCT
iajs-2802	57	17	2	2	NUM
iajs-2802	57	18	…	…	PUNCT
iajs-2802	57	19	now	now	ADV
iajs-2802	57	20	,	,	PUNCT
iajs-2802	57	21	we	we	PRON
iajs-2802	57	22	write	write	VERB
iajs-2802	57	23	n	n	NOUN
iajs-2802	57	24	=	=	SYM
iajs-2802	57	25	<	<	X
iajs-2802	57	26	1	1	NUM
iajs-2802	57	27	/	/	SYM
iajs-2802	57	28	pi	pi	NOUN
iajs-2802	58	1	+	+	PROPN
iajs-2802	58	2	z	z	X
iajs-2802	58	3	>	>	X
iajs-2802	58	4	and	and	CCONJ
iajs-2802	58	5	let	let	VERB
iajs-2802	58	6	e	e	PRON
iajs-2802	58	7	be	be	AUX
iajs-2802	58	8	an	an	DET
iajs-2802	58	9	ideal	ideal	NOUN
iajs-2802	58	10	of	of	ADP
iajs-2802	58	11	z.	z.	PROPN
iajs-2802	58	12	if	if	SCONJ
iajs-2802	58	13	we	we	PRON
iajs-2802	58	14	take	take	VERB
iajs-2802	58	15	e=	e=	NOUN
iajs-2802	58	16	<	<	X
iajs-2802	58	17	1	1	NUM
iajs-2802	58	18	>	>	PUNCT
iajs-2802	58	19	,	,	PUNCT
iajs-2802	58	20	then	then	ADV
iajs-2802	58	21	,	,	PUNCT
iajs-2802	58	22	zp∞/e2b	zp∞/e2b	NOUN
iajs-2802	58	23	=	=	SYM
iajs-2802	58	24	zp∞/<1>2b	zp∞/<1>2b	PROPN
iajs-2802	58	25	=	=	SYM
iajs-2802	58	26	zp∞/b	zp∞/b	PROPN
iajs-2802	58	27	≅	≅	PROPN
iajs-2802	58	28	zp∞	zp∞	PROPN
iajs-2802	58	29	is	be	AUX
iajs-2802	58	30	not	not	PART
iajs-2802	58	31	a	a	DET
iajs-2802	58	32	regular	regular	ADJ
iajs-2802	58	33	module	module	NOUN
iajs-2802	58	34	and	and	CCONJ
iajs-2802	58	35	hence	hence	ADV
iajs-2802	58	36	b	b	X
iajs-2802	59	1	=	=	PUNCT
iajs-2802	59	2	<	<	X
iajs-2802	59	3	1	1	NUM
iajs-2802	59	4	/	/	SYM
iajs-2802	59	5	pi	pi	NOUN
iajs-2802	60	1	+	+	CCONJ
iajs-2802	60	2	z	z	X
iajs-2802	60	3	>	>	X
iajs-2802	60	4	is	be	AUX
iajs-2802	60	5	not	not	PART
iajs-2802	60	6	smsubmodule	smsubmodule	ADJ
iajs-2802	60	7	of	of	ADP
iajs-2802	60	8	zp∞.	zp∞.	NUM
iajs-2802	60	9	that	that	PRON
iajs-2802	60	10	is	be	AUX
iajs-2802	60	11	,	,	PUNCT
iajs-2802	60	12	zp∞	zp∞	PROPN
iajs-2802	60	13	has	have	VERB
iajs-2802	60	14	no	no	DET
iajs-2802	60	15	sm	sm	NOUN
iajs-2802	60	16	-	-	PUNCT
iajs-2802	60	17	submodules	submodules	NOUN
iajs-2802	60	18	.	.	PUNCT
iajs-2802	61	1	also	also	ADV
iajs-2802	61	2	,	,	PUNCT
iajs-2802	61	3	we	we	PRON
iajs-2802	61	4	can	can	AUX
iajs-2802	61	5	give	give	VERB
iajs-2802	61	6	another	another	DET
iajs-2802	61	7	example	example	NOUN
iajs-2802	61	8	z8	z8	NOUN
iajs-2802	61	9	as	as	ADP
iajs-2802	61	10	a	a	DET
iajs-2802	61	11	z8	z8	NOUN
iajs-2802	61	12	-	-	PUNCT
iajs-2802	61	13	module	module	NOUN
iajs-2802	61	14	that	that	PRON
iajs-2802	61	15	has	have	VERB
iajs-2802	61	16	no	no	DET
iajs-2802	61	17	sm	sm	NOUN
iajs-2802	61	18	-	-	PUNCT
iajs-2802	61	19	submodules	submodules	NOUN
iajs-2802	61	20	.since	.since	NOUN
iajs-2802	61	21	<	<	X
iajs-2802	61	22	2	2	NUM
iajs-2802	61	23	>	>	X
iajs-2802	61	24	and	and	CCONJ
iajs-2802	61	25	<	<	X
iajs-2802	61	26	4	4	NUM
iajs-2802	61	27	>	>	X
iajs-2802	61	28	are	be	AUX
iajs-2802	61	29	not	not	PART
iajs-2802	61	30	smsubmodules	smsubmodule	NOUN
iajs-2802	61	31	in	in	ADP
iajs-2802	61	32	z8	z8	NOUN
iajs-2802	61	33	.	.	PUNCT
iajs-2802	62	1	4	4	X
iajs-2802	62	2	.	.	X
iajs-2802	62	3	the	the	DET
iajs-2802	62	4	submodule	submodule	NOUN
iajs-2802	63	1	<3	<3	X
iajs-2802	63	2	>	>	X
iajs-2802	63	3	of	of	ADP
iajs-2802	63	4	a	a	DET
iajs-2802	63	5	z6	z6	NOUN
iajs-2802	63	6	-	-	PUNCT
iajs-2802	63	7	module	module	NOUN
iajs-2802	63	8	z6	z6	NOUN
iajs-2802	63	9	is	be	AUX
iajs-2802	63	10	an	an	DET
iajs-2802	63	11	sm	sm	NOUN
iajs-2802	63	12	-	-	PUNCT
iajs-2802	63	13	submodule	submodule	NOUN
iajs-2802	63	14	.	.	PUNCT
iajs-2802	64	1	to	to	PART
iajs-2802	64	2	clarify	clarify	VERB
iajs-2802	64	3	,	,	PUNCT
iajs-2802	64	4	let	let	VERB
iajs-2802	64	5	e	e	PRON
iajs-2802	64	6	be	be	AUX
iajs-2802	64	7	an	an	DET
iajs-2802	64	8	ideal	ideal	NOUN
iajs-2802	64	9	of	of	ADP
iajs-2802	64	10	a	a	DET
iajs-2802	64	11	ring	ring	NOUN
iajs-2802	64	12	z6	z6	NOUN
iajs-2802	64	13	.	.	PUNCT
iajs-2802	65	1	if	if	SCONJ
iajs-2802	65	2	e	e	X
iajs-2802	65	3	=	=	PUNCT
iajs-2802	65	4	<	<	X
iajs-2802	65	5	1	1	NUM
iajs-2802	65	6	>	>	X
iajs-2802	65	7	,	,	PUNCT
iajs-2802	65	8	then	then	ADV
iajs-2802	65	9	z6/<1>2	z6/<1>2	PUNCT
iajs-2802	66	1	<3	<3	X
iajs-2802	66	2	>	>	X
iajs-2802	66	3	=	=	PUNCT
iajs-2802	66	4	z6/	z6/	NUM
iajs-2802	66	5	<3	<3	X
iajs-2802	66	6	>	>	X
iajs-2802	66	7	≅z3	≅z3	X
iajs-2802	66	8	is	be	AUX
iajs-2802	66	9	a	a	DET
iajs-2802	66	10	regular	regular	ADJ
iajs-2802	66	11	module	module	NOUN
iajs-2802	66	12	.	.	PUNCT
iajs-2802	67	1	if	if	SCONJ
iajs-2802	67	2	e	e	X
iajs-2802	67	3	=	=	PUNCT
iajs-2802	67	4	<	<	X
iajs-2802	67	5	2	2	NUM
iajs-2802	67	6	>	>	X
iajs-2802	67	7	,	,	PUNCT
iajs-2802	67	8	then	then	ADV
iajs-2802	67	9	z6/<2>2	z6/<2>2	X
iajs-2802	68	1	<3	<3	X
iajs-2802	68	2	>	>	X
iajs-2802	68	3	=	=	SYM
iajs-2802	68	4	z6/<0	z6/<0	PROPN
iajs-2802	68	5	>	>	X
iajs-2802	68	6	≅z6	≅z6	PROPN
iajs-2802	68	7	is	be	AUX
iajs-2802	68	8	a	a	DET
iajs-2802	68	9	regular	regular	ADJ
iajs-2802	68	10	module	module	NOUN
iajs-2802	68	11	.	.	PUNCT
iajs-2802	69	1	if	if	SCONJ
iajs-2802	69	2	e	e	X
iajs-2802	69	3	=	=	PUNCT
iajs-2802	69	4	<3	<3	X
iajs-2802	69	5	>	>	X
iajs-2802	69	6	,	,	PUNCT
iajs-2802	69	7	then	then	ADV
iajs-2802	69	8	z6/	z6/	X
iajs-2802	69	9	<3	<3	X
iajs-2802	69	10	>	>	X
iajs-2802	69	11	2	2	NUM
iajs-2802	69	12	<3	<3	X
iajs-2802	69	13	>	>	X
iajs-2802	69	14	=	=	PUNCT
iajs-2802	70	1	z6/	z6/	NUM
iajs-2802	70	2	<3	<3	X
iajs-2802	70	3	>	>	X
iajs-2802	70	4	≅	≅	PROPN
iajs-2802	70	5	z3	z3	PROPN
iajs-2802	70	6	is	be	AUX
iajs-2802	70	7	a	a	DET
iajs-2802	70	8	regular	regular	ADJ
iajs-2802	70	9	module	module	NOUN
iajs-2802	70	10	.	.	PUNCT
iajs-2802	71	1	therefore	therefore	ADV
iajs-2802	71	2	<3	<3	X
iajs-2802	71	3	>	>	X
iajs-2802	71	4	is	be	AUX
iajs-2802	71	5	an	an	DET
iajs-2802	71	6	sm	sm	NOUN
iajs-2802	71	7	-	-	PUNCT
iajs-2802	71	8	submodule	submodule	NOUN
iajs-2802	71	9	of	of	ADP
iajs-2802	71	10	z6	z6	PROPN
iajs-2802	71	11	.	.	PUNCT
iajs-2802	72	1	in	in	ADP
iajs-2802	72	2	general	general	ADJ
iajs-2802	72	3	,	,	PUNCT
iajs-2802	72	4	all	all	DET
iajs-2802	72	5	submodules	submodule	NOUN
iajs-2802	72	6	of	of	ADP
iajs-2802	72	7	z6	z6	PROPN
iajs-2802	72	8	as	as	ADP
iajs-2802	72	9	a	a	DET
iajs-2802	72	10	z6	z6	NOUN
iajs-2802	72	11	-	-	PUNCT
iajs-2802	72	12	module	module	NOUN
iajs-2802	72	13	are	be	AUX
iajs-2802	72	14	sm	sm	NOUN
iajs-2802	72	15	-	-	PUNCT
iajs-2802	72	16	submodules	submodules	NOUN
iajs-2802	72	17	.	.	PUNCT
iajs-2802	73	1	5	5	X
iajs-2802	73	2	.	.	X
iajs-2802	73	3	in	in	ADP
iajs-2802	73	4	z10	z10	NOUN
iajs-2802	73	5	as	as	ADP
iajs-2802	73	6	a	a	DET
iajs-2802	73	7	z20	z20	NOUN
iajs-2802	73	8	-	-	PUNCT
iajs-2802	73	9	module	module	NOUN
iajs-2802	73	10	,	,	PUNCT
iajs-2802	73	11	the	the	DET
iajs-2802	73	12	submodule	submodule	PROPN
iajs-2802	73	13	b	b	PROPN
iajs-2802	74	1	=	=	PUNCT
iajs-2802	74	2	<	<	X
iajs-2802	74	3	5	5	NUM
iajs-2802	74	4	>	>	X
iajs-2802	74	5	is	be	AUX
iajs-2802	74	6	an	an	DET
iajs-2802	74	7	sm	sm	NOUN
iajs-2802	74	8	-	-	PUNCT
iajs-2802	74	9	submodule	submodule	NOUN
iajs-2802	74	10	.	.	PUNCT
iajs-2802	75	1	since	since	SCONJ
iajs-2802	75	2	if	if	SCONJ
iajs-2802	75	3	we	we	PRON
iajs-2802	75	4	take	take	VERB
iajs-2802	75	5	e	e	NOUN
iajs-2802	75	6	=	=	PUNCT
iajs-2802	75	7	<	<	X
iajs-2802	75	8	1	1	NUM
iajs-2802	75	9	>	>	X
iajs-2802	75	10	,	,	PUNCT
iajs-2802	75	11	<	<	X
iajs-2802	75	12	2	2	NUM
iajs-2802	75	13	>	>	X
iajs-2802	75	14	,	,	PUNCT
iajs-2802	75	15	<	<	X
iajs-2802	75	16	10	10	NUM
iajs-2802	75	17	>	>	X
iajs-2802	75	18	,	,	PUNCT
iajs-2802	75	19	<	<	X
iajs-2802	75	20	4	4	NUM
iajs-2802	75	21	>	>	X
iajs-2802	75	22	,	,	PUNCT
iajs-2802	75	23	<	<	X
iajs-2802	75	24	5	5	NUM
iajs-2802	75	25	>	>	X
iajs-2802	75	26	,	,	PUNCT
iajs-2802	75	27	where	where	SCONJ
iajs-2802	75	28	e	e	NOUN
iajs-2802	75	29	is	be	AUX
iajs-2802	75	30	ideal	ideal	ADJ
iajs-2802	75	31	of	of	ADP
iajs-2802	75	32	z20	z20	PROPN
iajs-2802	75	33	,	,	PUNCT
iajs-2802	75	34	then	then	ADV
iajs-2802	75	35	z20	z20	PROPN
iajs-2802	75	36	/	/	SYM
iajs-2802	75	37	e	e	NOUN
iajs-2802	75	38	2<5	2<5	NOUN
iajs-2802	75	39	>	>	X
iajs-2802	75	40	is	be	AUX
iajs-2802	75	41	a	a	DET
iajs-2802	75	42	regular	regular	ADJ
iajs-2802	75	43	module	module	NOUN
iajs-2802	75	44	for	for	ADP
iajs-2802	75	45	all	all	DET
iajs-2802	75	46	ideal	ideal	ADJ
iajs-2802	75	47	e	e	NOUN
iajs-2802	75	48	of	of	ADP
iajs-2802	75	49	z20	z20	PROPN
iajs-2802	75	50	.	.	PUNCT
iajs-2802	76	1	this	this	PRON
iajs-2802	76	2	ends	end	VERB
iajs-2802	76	3	the	the	DET
iajs-2802	76	4	proof	proof	NOUN
iajs-2802	76	5	of	of	ADP
iajs-2802	76	6	example	example	NOUN
iajs-2802	76	7	.	.	PUNCT
iajs-2802	77	1	6	6	X
iajs-2802	77	2	.	.	X
iajs-2802	77	3	consider	consider	VERB
iajs-2802	77	4	z4	z4	PROPN
iajs-2802	77	5	as	as	ADP
iajs-2802	77	6	a	a	DET
iajs-2802	77	7	z	z	NOUN
iajs-2802	77	8	-	-	PUNCT
iajs-2802	77	9	module	module	NOUN
iajs-2802	77	10	.	.	PUNCT
iajs-2802	78	1	the	the	DET
iajs-2802	78	2	submodule	submodule	NOUN
iajs-2802	78	3	<	<	X
iajs-2802	78	4	2	2	NUM
iajs-2802	78	5	>	>	PRON
iajs-2802	78	6	is	be	AUX
iajs-2802	78	7	not	not	PART
iajs-2802	78	8	sm	sm	NOUN
iajs-2802	78	9	-	-	PUNCT
iajs-2802	78	10	submodule	submodule	NOUN
iajs-2802	78	11	of	of	ADP
iajs-2802	78	12	z4	z4	PROPN
iajs-2802	78	13	.	.	PUNCT
iajs-2802	79	1	since	since	SCONJ
iajs-2802	79	2	z4	z4	PROPN
iajs-2802	79	3	/	/	SYM
iajs-2802	79	4	e	e	PROPN
iajs-2802	79	5	2<2	2<2	NOUN
iajs-2802	79	6	>	>	X
iajs-2802	79	7	is	be	AUX
iajs-2802	79	8	not	not	PART
iajs-2802	79	9	regular	regular	ADJ
iajs-2802	79	10	,	,	PUNCT
iajs-2802	79	11	e	e	X
iajs-2802	79	12	is	be	AUX
iajs-2802	79	13	an	an	DET
iajs-2802	79	14	ideal	ideal	NOUN
iajs-2802	79	15	of	of	ADP
iajs-2802	79	16	z.	z.	PROPN
iajs-2802	79	17	to	to	PART
iajs-2802	79	18	prove	prove	VERB
iajs-2802	79	19	that	that	SCONJ
iajs-2802	79	20	,	,	PUNCT
iajs-2802	79	21	take	take	VERB
iajs-2802	79	22	e	e	NOUN
iajs-2802	79	23	=	=	PUNCT
iajs-2802	79	24	<	<	X
iajs-2802	79	25	4	4	NUM
iajs-2802	79	26	>	>	PUNCT
iajs-2802	79	27	.	.	PUNCT
iajs-2802	80	1	then	then	ADV
iajs-2802	80	2	,	,	PUNCT
iajs-2802	80	3	z4/<4>2	z4/<4>2	X
iajs-2802	80	4	<	<	X
iajs-2802	80	5	2	2	NUM
iajs-2802	80	6	>	>	X
iajs-2802	80	7	≅	≅	PROPN
iajs-2802	80	8	z4	z4	PROPN
iajs-2802	80	9	is	be	AUX
iajs-2802	80	10	not	not	PART
iajs-2802	80	11	regular	regular	ADJ
iajs-2802	80	12	module	module	NOUN
iajs-2802	80	13	.	.	PUNCT
iajs-2802	81	1	7	7	X
iajs-2802	81	2	.	.	X
iajs-2802	81	3	let	let	VERB
iajs-2802	81	4	z4	z4	PROPN
iajs-2802	81	5	as	as	ADP
iajs-2802	81	6	a	a	DET
iajs-2802	81	7	z4	z4	NOUN
iajs-2802	81	8	-	-	PUNCT
iajs-2802	81	9	module	module	NOUN
iajs-2802	81	10	.	.	PUNCT
iajs-2802	82	1	then	then	ADV
iajs-2802	82	2	the	the	DET
iajs-2802	82	3	submodule	submodule	PROPN
iajs-2802	82	4	b	b	PROPN
iajs-2802	82	5	=	=	PUNCT
iajs-2802	82	6	<	<	X
iajs-2802	82	7	2	2	NUM
iajs-2802	82	8	>	>	PRON
iajs-2802	82	9	is	be	AUX
iajs-2802	82	10	not	not	PART
iajs-2802	82	11	an	an	DET
iajs-2802	82	12	sm	sm	NOUN
iajs-2802	82	13	-	-	PUNCT
iajs-2802	82	14	submodule	submodule	NOUN
iajs-2802	82	15	.	.	PUNCT
iajs-2802	83	1	notice	notice	VERB
iajs-2802	83	2	if	if	SCONJ
iajs-2802	83	3	e	e	PROPN
iajs-2802	83	4	=	=	PUNCT
iajs-2802	83	5	<	<	X
iajs-2802	83	6	2	2	NUM
iajs-2802	83	7	>	>	PUNCT
iajs-2802	83	8	then	then	ADV
iajs-2802	83	9	z4	z4	PROPN
iajs-2802	83	10	/	/	SYM
iajs-2802	83	11	e	e	PROPN
iajs-2802	83	12	2	2	NUM
iajs-2802	83	13	<	<	X
iajs-2802	83	14	2	2	NUM
iajs-2802	83	15	>	>	X
iajs-2802	83	16	is	be	AUX
iajs-2802	83	17	not	not	PART
iajs-2802	83	18	regular	regular	ADJ
iajs-2802	83	19	module	module	NOUN
iajs-2802	83	20	.	.	PUNCT
iajs-2802	84	1	ibn	ibn	PROPN
iajs-2802	84	2	al	al	PROPN
iajs-2802	84	3	-	-	PUNCT
iajs-2802	84	4	haitham	haitham	PROPN
iajs-2802	84	5	jour	jour	X
iajs-2802	84	6	.	.	PROPN
iajs-2802	84	7	for	for	ADP
iajs-2802	84	8	pure	pure	ADJ
iajs-2802	84	9	&	&	CCONJ
iajs-2802	84	10	appl	appl	PROPN
iajs-2802	84	11	.	.	PUNCT
iajs-2802	85	1	sci	sci	PROPN
iajs-2802	85	2	.	.	PROPN
iajs-2802	86	1	53	53	NUM
iajs-2802	86	2	(	(	PUNCT
iajs-2802	86	3	1)2022	1)2022	PROPN
iajs-2802	86	4	87	87	NUM
iajs-2802	86	5	8	8	NUM
iajs-2802	86	6	.	.	PUNCT
iajs-2802	87	1	let	let	VERB
iajs-2802	87	2	b	b	NOUN
iajs-2802	87	3	=	=	PUNCT
iajs-2802	87	4	<	<	X
iajs-2802	87	5	2	2	NUM
iajs-2802	87	6	>	>	PUNCT
iajs-2802	87	7	be	be	AUX
iajs-2802	87	8	a	a	DET
iajs-2802	87	9	submodule	submodule	NOUN
iajs-2802	87	10	of	of	ADP
iajs-2802	87	11	a	a	DET
iajs-2802	87	12	z20	z20	NOUN
iajs-2802	87	13	-	-	PUNCT
iajs-2802	87	14	module	module	NOUN
iajs-2802	87	15	z10	z10	NOUN
iajs-2802	87	16	.	.	PUNCT
iajs-2802	88	1	since	since	SCONJ
iajs-2802	88	2	z10/	z10/	NOUN
iajs-2802	88	3	e	e	NOUN
iajs-2802	88	4	2b	2b	NUM
iajs-2802	88	5	=	=	SYM
iajs-2802	88	6	z10/<2>2<2	z10/<2>2<2	PROPN
iajs-2802	88	7	>	>	X
iajs-2802	88	8	≅	≅	PROPN
iajs-2802	88	9	z8	z8	PROPN
iajs-2802	88	10	is	be	AUX
iajs-2802	88	11	not	not	PART
iajs-2802	88	12	a	a	DET
iajs-2802	88	13	regular	regular	ADJ
iajs-2802	88	14	module	module	NOUN
iajs-2802	88	15	,	,	PUNCT
iajs-2802	88	16	where	where	SCONJ
iajs-2802	88	17	e	e	NOUN
iajs-2802	88	18	=	=	PUNCT
iajs-2802	88	19	<	<	X
iajs-2802	88	20	2	2	NUM
iajs-2802	88	21	>	>	X
iajs-2802	88	22	is	be	AUX
iajs-2802	88	23	an	an	DET
iajs-2802	88	24	ideal	ideal	NOUN
iajs-2802	88	25	of	of	ADP
iajs-2802	88	26	z20	z20	PROPN
iajs-2802	88	27	.	.	PUNCT
iajs-2802	89	1	9	9	X
iajs-2802	89	2	.	.	X
iajs-2802	89	3	let	let	VERB
iajs-2802	89	4	a	a	DET
iajs-2802	89	5	=	=	X
iajs-2802	89	6	z6⊕z3	z6⊕z3	PROPN
iajs-2802	89	7	be	be	AUX
iajs-2802	89	8	an	an	DET
iajs-2802	89	9	z12	z12	NUM
iajs-2802	89	10	-	-	PUNCT
iajs-2802	89	11	module	module	NOUN
iajs-2802	89	12	and	and	CCONJ
iajs-2802	89	13	b	b	NOUN
iajs-2802	89	14	=	=	PUNCT
iajs-2802	89	15	<	<	X
iajs-2802	89	16	2>⊕<0	2>⊕<0	X
iajs-2802	89	17	>	>	X
iajs-2802	89	18	be	be	AUX
iajs-2802	89	19	a	a	DET
iajs-2802	89	20	submodule	submodule	NOUN
iajs-2802	89	21	of	of	ADP
iajs-2802	89	22	a.	a.	NOUN
iajs-2802	89	23	then	then	ADV
iajs-2802	89	24	a	a	X
iajs-2802	89	25	/	/	SYM
iajs-2802	89	26	e2b	e2b	PROPN
iajs-2802	89	27	=	=	SYM
iajs-2802	89	28	z6⊕z3	z6⊕z3	NUM
iajs-2802	89	29	/	/	SYM
iajs-2802	89	30	e	e	NOUN
iajs-2802	89	31	2(<2>⊕<0	2(<2>⊕<0	X
iajs-2802	89	32	>	>	PUNCT
iajs-2802	89	33	)	)	PUNCT
iajs-2802	89	34	is	be	AUX
iajs-2802	89	35	regular	regular	ADJ
iajs-2802	89	36	module	module	NOUN
iajs-2802	89	37	where	where	SCONJ
iajs-2802	89	38	e	e	NOUN
iajs-2802	89	39	=	=	PUNCT
iajs-2802	89	40	<	<	X
iajs-2802	89	41	1	1	NUM
iajs-2802	89	42	>	>	PUNCT
iajs-2802	89	43	,	,	PUNCT
iajs-2802	89	44	<	<	X
iajs-2802	89	45	2	2	NUM
iajs-2802	89	46	>	>	X
iajs-2802	89	47	,	,	PUNCT
iajs-2802	90	1	<3	<3	X
iajs-2802	90	2	>	>	X
iajs-2802	90	3	,	,	PUNCT
iajs-2802	90	4	<	<	X
iajs-2802	90	5	4	4	NUM
iajs-2802	90	6	>	>	X
iajs-2802	90	7	and	and	CCONJ
iajs-2802	90	8	<	<	X
iajs-2802	90	9	6	6	NUM
iajs-2802	90	10	>	>	PUNCT
iajs-2802	90	11	be	be	AUX
iajs-2802	90	12	ideals	ideal	NOUN
iajs-2802	90	13	of	of	ADP
iajs-2802	90	14	z12	z12	PROPN
iajs-2802	90	15	.	.	PUNCT
iajs-2802	91	1	therefor	therefor	ADP
iajs-2802	91	2	b	b	PROPN
iajs-2802	91	3	=	=	PUNCT
iajs-2802	91	4	<	<	X
iajs-2802	91	5	2>⊕<0	2>⊕<0	X
iajs-2802	91	6	>	>	X
iajs-2802	91	7	is	be	AUX
iajs-2802	91	8	an	an	DET
iajs-2802	91	9	sm	sm	NOUN
iajs-2802	91	10	-	-	PUNCT
iajs-2802	91	11	submodule	submodule	NOUN
iajs-2802	91	12	of	of	ADP
iajs-2802	91	13	a	a	DET
iajs-2802	91	14	=	=	X
iajs-2802	91	15	z6⊕z3	z6⊕z3	PROPN
iajs-2802	91	16	.	.	PUNCT
iajs-2802	92	1	10	10	NUM
iajs-2802	92	2	.	.	PUNCT
iajs-2802	93	1	consider	consider	VERB
iajs-2802	93	2	a	a	DET
iajs-2802	93	3	=	=	NOUN
iajs-2802	93	4	z6⊕z	z6⊕z	NOUN
iajs-2802	93	5	as	as	ADP
iajs-2802	93	6	a	a	DET
iajs-2802	93	7	z	z	NOUN
iajs-2802	93	8	-	-	PUNCT
iajs-2802	93	9	module	module	NOUN
iajs-2802	93	10	.	.	PUNCT
iajs-2802	94	1	then	then	ADV
iajs-2802	94	2	the	the	DET
iajs-2802	94	3	submodule	submodule	PROPN
iajs-2802	94	4	b	b	PROPN
iajs-2802	94	5	=	=	PUNCT
iajs-2802	94	6	<	<	X
iajs-2802	94	7	3>⊕<2	3>⊕<2	NUM
iajs-2802	94	8	>	>	X
iajs-2802	94	9	of	of	ADP
iajs-2802	94	10	a	a	PRON
iajs-2802	94	11	is	be	AUX
iajs-2802	94	12	not	not	PART
iajs-2802	94	13	smsubmodule	smsubmodule	ADJ
iajs-2802	94	14	.	.	PUNCT
iajs-2802	95	1	since	since	SCONJ
iajs-2802	95	2	a	a	PRON
iajs-2802	95	3	/	/	SYM
iajs-2802	95	4	e2b	e2b	NOUN
iajs-2802	95	5	=	=	NOUN
iajs-2802	95	6	z6⊕z/<2>2	z6⊕z/<2>2	NUM
iajs-2802	95	7	(	(	PUNCT
iajs-2802	95	8	<	<	X
iajs-2802	95	9	3>⊕<2	3>⊕<2	NUM
iajs-2802	95	10	>	>	X
iajs-2802	95	11	)	)	PUNCT
iajs-2802	95	12	=	=	SYM
iajs-2802	95	13	z6⊕z/<0>⊕<8	z6⊕z/<0>⊕<8	NOUN
iajs-2802	95	14	>	>	X
iajs-2802	95	15	is	be	AUX
iajs-2802	95	16	not	not	PART
iajs-2802	95	17	a	a	DET
iajs-2802	95	18	regular	regular	ADJ
iajs-2802	95	19	module	module	NOUN
iajs-2802	95	20	,	,	PUNCT
iajs-2802	95	21	where	where	SCONJ
iajs-2802	95	22	e	e	NOUN
iajs-2802	95	23	=	=	PUNCT
iajs-2802	95	24	<	<	X
iajs-2802	95	25	2	2	NUM
iajs-2802	95	26	>	>	X
iajs-2802	95	27	is	be	AUX
iajs-2802	95	28	an	an	DET
iajs-2802	95	29	ideal	ideal	NOUN
iajs-2802	95	30	of	of	ADP
iajs-2802	95	31	z.	z.	PROPN
iajs-2802	95	32	11	11	NUM
iajs-2802	95	33	.	.	PUNCT
iajs-2802	96	1	every	every	DET
iajs-2802	96	2	sm	sm	PROPN
iajs-2802	96	3	-	-	PUNCT
iajs-2802	96	4	submodule	submodule	NOUN
iajs-2802	96	5	is	be	AUX
iajs-2802	96	6	maximal	maximal	ADJ
iajs-2802	96	7	but	but	CCONJ
iajs-2802	96	8	the	the	DET
iajs-2802	96	9	opposite	opposite	NOUN
iajs-2802	96	10	is	be	AUX
iajs-2802	96	11	not	not	PART
iajs-2802	96	12	true	true	ADJ
iajs-2802	96	13	and	and	CCONJ
iajs-2802	96	14	the	the	DET
iajs-2802	96	15	following	follow	VERB
iajs-2802	96	16	example	example	NOUN
iajs-2802	96	17	shows	show	VERB
iajs-2802	96	18	that	that	SCONJ
iajs-2802	96	19	:	:	PUNCT
iajs-2802	96	20	the	the	DET
iajs-2802	96	21	submodule	submodule	NOUN
iajs-2802	96	22	<	<	X
iajs-2802	96	23	2	2	NUM
iajs-2802	96	24	>	>	X
iajs-2802	96	25	of	of	ADP
iajs-2802	96	26	a	a	DET
iajs-2802	96	27	z	z	NOUN
iajs-2802	96	28	-	-	PUNCT
iajs-2802	96	29	module	module	NOUN
iajs-2802	96	30	z4	z4	NOUN
iajs-2802	96	31	is	be	AUX
iajs-2802	96	32	a	a	DET
iajs-2802	96	33	maximal	maximal	ADJ
iajs-2802	96	34	submodule	submodule	NOUN
iajs-2802	96	35	in	in	ADP
iajs-2802	96	36	z4	z4	PROPN
iajs-2802	96	37	but	but	CCONJ
iajs-2802	96	38	it	it	PRON
iajs-2802	96	39	is	be	AUX
iajs-2802	96	40	not	not	PART
iajs-2802	96	41	a	a	DET
iajs-2802	96	42	sm	sm	NOUN
iajs-2802	96	43	-	-	PUNCT
iajs-2802	96	44	submodule	submodule	NOUN
iajs-2802	96	45	,	,	PUNCT
iajs-2802	96	46	see	see	VERB
iajs-2802	96	47	no.(6	no.(6	NOUN
iajs-2802	96	48	)	)	PUNCT
iajs-2802	96	49	.	.	PUNCT
iajs-2802	97	1	12	12	NUM
iajs-2802	97	2	.	.	PUNCT
iajs-2802	98	1	a	a	DET
iajs-2802	98	2	submodule	submodule	NOUN
iajs-2802	98	3	of	of	ADP
iajs-2802	98	4	an	an	DET
iajs-2802	98	5	sm	sm	NOUN
iajs-2802	98	6	-	-	PUNCT
iajs-2802	98	7	submodule	submodule	NOUN
iajs-2802	98	8	is	be	AUX
iajs-2802	98	9	not	not	PART
iajs-2802	98	10	necessary	necessary	ADJ
iajs-2802	98	11	to	to	PART
iajs-2802	98	12	be	be	AUX
iajs-2802	98	13	an	an	DET
iajs-2802	98	14	sm	sm	NOUN
iajs-2802	98	15	-	-	PUNCT
iajs-2802	98	16	submodule	submodule	NOUN
iajs-2802	98	17	,	,	PUNCT
iajs-2802	98	18	for	for	ADP
iajs-2802	98	19	example	example	NOUN
iajs-2802	98	20	:	:	PUNCT
iajs-2802	98	21	a	a	DET
iajs-2802	98	22	submodule	submodule	NOUN
iajs-2802	98	23	<	<	X
iajs-2802	98	24	2	2	NUM
iajs-2802	98	25	>	>	X
iajs-2802	98	26	in	in	ADP
iajs-2802	98	27	a	a	DET
iajs-2802	98	28	z6	z6	NOUN
iajs-2802	98	29	-	-	PUNCT
iajs-2802	98	30	module	module	NOUN
iajs-2802	98	31	z6	z6	NOUN
iajs-2802	98	32	is	be	AUX
iajs-2802	98	33	sm	sm	NOUN
iajs-2802	98	34	-	-	PUNCT
iajs-2802	98	35	submodule	submodule	NOUN
iajs-2802	98	36	.	.	PUNCT
iajs-2802	99	1	see	see	VERB
iajs-2802	99	2	no.(4	no.(4	PRON
iajs-2802	99	3	)	)	PUNCT
iajs-2802	99	4	,	,	PUNCT
iajs-2802	99	5	while	while	SCONJ
iajs-2802	99	6	<	<	X
iajs-2802	99	7	0	0	NUM
iajs-2802	99	8	>	>	X
iajs-2802	99	9	is	be	AUX
iajs-2802	99	10	a	a	DET
iajs-2802	99	11	submodule	submodule	NOUN
iajs-2802	99	12	of	of	ADP
iajs-2802	99	13	<	<	X
iajs-2802	99	14	2	2	NUM
iajs-2802	99	15	>	>	PUNCT
iajs-2802	99	16	and	and	CCONJ
iajs-2802	99	17	it	it	PRON
iajs-2802	99	18	is	be	AUX
iajs-2802	99	19	not	not	PART
iajs-2802	99	20	sm	sm	NOUN
iajs-2802	99	21	-	-	NOUN
iajs-2802	99	22	submodule	submodule	NOUN
iajs-2802	99	23	.	.	PUNCT
iajs-2802	100	1	13	13	NUM
iajs-2802	100	2	.	.	PUNCT
iajs-2802	101	1	the	the	DET
iajs-2802	101	2	intersection	intersection	NOUN
iajs-2802	101	3	of	of	ADP
iajs-2802	101	4	two	two	NUM
iajs-2802	101	5	sm	sm	NOUN
iajs-2802	101	6	-	-	PUNCT
iajs-2802	101	7	submodules	submodules	NOUN
iajs-2802	101	8	is	be	AUX
iajs-2802	101	9	not	not	PART
iajs-2802	101	10	condition	condition	NOUN
iajs-2802	101	11	to	to	PART
iajs-2802	101	12	be	be	AUX
iajs-2802	101	13	sm	sm	NOUN
iajs-2802	101	14	-	-	PUNCT
iajs-2802	101	15	submodule	submodule	NOUN
iajs-2802	101	16	,	,	PUNCT
iajs-2802	101	17	for	for	ADP
iajs-2802	101	18	example	example	NOUN
iajs-2802	101	19	:	:	PUNCT
iajs-2802	101	20	the	the	DET
iajs-2802	101	21	two	two	NUM
iajs-2802	101	22	submodules	submodule	NOUN
iajs-2802	101	23	<	<	X
iajs-2802	101	24	2	2	NUM
iajs-2802	101	25	>	>	X
iajs-2802	101	26	,	,	PUNCT
iajs-2802	101	27	<3	<3	X
iajs-2802	101	28	>	>	X
iajs-2802	101	29	in	in	ADP
iajs-2802	101	30	z6	z6	NOUN
iajs-2802	101	31	-	-	PUNCT
iajs-2802	101	32	module	module	NOUN
iajs-2802	101	33	z6	z6	NOUN
iajs-2802	101	34	are	be	AUX
iajs-2802	101	35	smsubmodules	smsubmodule	NOUN
iajs-2802	101	36	but	but	CCONJ
iajs-2802	101	37	<	<	X
iajs-2802	101	38	2	2	NUM
iajs-2802	101	39	>	>	X
iajs-2802	101	40	∩	∩	ADJ
iajs-2802	101	41	<3	<3	X
iajs-2802	101	42	>	>	X
iajs-2802	101	43	=	=	PUNCT
iajs-2802	102	1	<	<	X
iajs-2802	102	2	0	0	NUM
iajs-2802	102	3	>	>	X
iajs-2802	102	4	is	be	AUX
iajs-2802	102	5	not	not	PART
iajs-2802	102	6	an	an	DET
iajs-2802	102	7	sm	sm	NOUN
iajs-2802	102	8	-	-	PUNCT
iajs-2802	102	9	submodule	submodule	NOUN
iajs-2802	102	10	14	14	NUM
iajs-2802	102	11	.	.	PUNCT
iajs-2802	103	1	more	more	ADV
iajs-2802	103	2	generally	generally	ADV
iajs-2802	103	3	,	,	PUNCT
iajs-2802	103	4	let	let	VERB
iajs-2802	103	5	{	{	PUNCT
iajs-2802	103	6	bi}i=1	bi}i=1	ADJ
iajs-2802	103	7	n	n	PRON
iajs-2802	103	8	be	be	AUX
iajs-2802	103	9	a	a	DET
iajs-2802	103	10	finite	finite	ADJ
iajs-2802	103	11	collection	collection	NOUN
iajs-2802	103	12	of	of	ADP
iajs-2802	103	13	sm	sm	NOUN
iajs-2802	103	14	-	-	PUNCT
iajs-2802	103	15	submodules	submodule	NOUN
iajs-2802	103	16	of	of	ADP
iajs-2802	103	17	an	an	DET
iajs-2802	103	18	s	s	NOUN
iajs-2802	103	19	-	-	PUNCT
iajs-2802	103	20	module	module	NOUN
iajs-2802	103	21	a.	a.	NOUN
iajs-2802	104	1	then	then	ADV
iajs-2802	104	2	∩i=1	∩i=1	ADV
iajs-2802	104	3	n	n	CCONJ
iajs-2802	104	4	bi	bi	NOUN
iajs-2802	104	5	is	be	AUX
iajs-2802	104	6	not	not	PART
iajs-2802	104	7	always	always	ADV
iajs-2802	104	8	sm	sm	NOUN
iajs-2802	104	9	-	-	PUNCT
iajs-2802	104	10	submodule	submodule	NOUN
iajs-2802	104	11	.	.	PUNCT
iajs-2802	105	1	15	15	NUM
iajs-2802	105	2	.	.	PUNCT
iajs-2802	106	1	the	the	DET
iajs-2802	106	2	direct	direct	ADJ
iajs-2802	106	3	sum	sum	NOUN
iajs-2802	106	4	of	of	ADP
iajs-2802	106	5	two	two	NUM
iajs-2802	106	6	sm	sm	NOUN
iajs-2802	106	7	-	-	PUNCT
iajs-2802	106	8	submodules	submodules	NOUN
iajs-2802	106	9	of	of	ADP
iajs-2802	106	10	an	an	DET
iajs-2802	106	11	s	s	NOUN
iajs-2802	106	12	-	-	NOUN
iajs-2802	106	13	module	module	NOUN
iajs-2802	106	14	a	a	PRON
iajs-2802	106	15	is	be	AUX
iajs-2802	106	16	not	not	PART
iajs-2802	106	17	necessary	necessary	ADJ
iajs-2802	106	18	to	to	PART
iajs-2802	106	19	be	be	AUX
iajs-2802	106	20	an	an	DET
iajs-2802	106	21	smsubmodule	smsubmodule	NOUN
iajs-2802	106	22	,	,	PUNCT
iajs-2802	106	23	for	for	ADP
iajs-2802	106	24	example	example	NOUN
iajs-2802	106	25	:	:	PUNCT
iajs-2802	106	26	let	let	VERB
iajs-2802	106	27	<	<	X
iajs-2802	106	28	2	2	NUM
iajs-2802	106	29	>	>	PUNCT
iajs-2802	106	30	,	,	PUNCT
iajs-2802	106	31	<3	<3	X
iajs-2802	106	32	>	>	X
iajs-2802	106	33	be	be	VERB
iajs-2802	106	34	two	two	NUM
iajs-2802	106	35	sm	sm	NOUN
iajs-2802	106	36	-	-	PUNCT
iajs-2802	106	37	submodules	submodules	NOUN
iajs-2802	106	38	of	of	ADP
iajs-2802	106	39	a	a	DET
iajs-2802	106	40	z6	z6	NOUN
iajs-2802	106	41	-	-	PUNCT
iajs-2802	106	42	module	module	NOUN
iajs-2802	106	43	z6	z6	NOUN
iajs-2802	106	44	,	,	PUNCT
iajs-2802	106	45	but	but	CCONJ
iajs-2802	106	46	<	<	X
iajs-2802	106	47	2>⊕<3>=z6	2>⊕<3>=z6	NUM
iajs-2802	106	48	is	be	AUX
iajs-2802	106	49	not	not	PART
iajs-2802	106	50	an	an	DET
iajs-2802	106	51	sm	sm	NOUN
iajs-2802	106	52	-	-	PUNCT
iajs-2802	106	53	submodule	submodule	NOUN
iajs-2802	106	54	.	.	PUNCT
iajs-2802	107	1	16	16	NUM
iajs-2802	107	2	.	.	PUNCT
iajs-2802	108	1	from	from	ADP
iajs-2802	108	2	the	the	DET
iajs-2802	108	3	fact	fact	NOUN
iajs-2802	108	4	that	that	SCONJ
iajs-2802	108	5	every	every	DET
iajs-2802	108	6	maximal	maximal	ADJ
iajs-2802	108	7	submodule	submodule	NOUN
iajs-2802	108	8	is	be	AUX
iajs-2802	108	9	a	a	DET
iajs-2802	108	10	semimaximal	semimaximal	NOUN
iajs-2802	108	11	by	by	ADP
iajs-2802	108	12	[	[	X
iajs-2802	108	13	3	3	NUM
iajs-2802	108	14	,	,	PUNCT
iajs-2802	108	15	remarks	remark	NOUN
iajs-2802	108	16	and	and	CCONJ
iajs-2802	108	17	examples	example	NOUN
iajs-2802	108	18	(	(	PUNCT
iajs-2802	108	19	2.1.2	2.1.2	NUM
iajs-2802	108	20	)	)	PUNCT
iajs-2802	108	21	,	,	PUNCT
iajs-2802	108	22	p32	p32	NOUN
iajs-2802	108	23	]	]	PUNCT
iajs-2802	108	24	,	,	PUNCT
iajs-2802	108	25	and	and	CCONJ
iajs-2802	108	26	the	the	DET
iajs-2802	108	27	fact	fact	NOUN
iajs-2802	108	28	no.(11	no.(11	PROPN
iajs-2802	108	29	)	)	PUNCT
iajs-2802	108	30	,	,	PUNCT
iajs-2802	108	31	we	we	PRON
iajs-2802	108	32	obtain	obtain	VERB
iajs-2802	108	33	that	that	SCONJ
iajs-2802	108	34	every	every	DET
iajs-2802	108	35	sm	sm	NOUN
iajs-2802	108	36	-	-	PUNCT
iajs-2802	108	37	submodule	submodule	NOUN
iajs-2802	108	38	of	of	ADP
iajs-2802	108	39	an	an	DET
iajs-2802	108	40	smodule	smodule	NOUN
iajs-2802	108	41	a	a	PRON
iajs-2802	108	42	is	be	AUX
iajs-2802	108	43	a	a	DET
iajs-2802	108	44	semimaximal	semimaximal	NOUN
iajs-2802	108	45	while	while	SCONJ
iajs-2802	108	46	the	the	DET
iajs-2802	108	47	converse	converse	NOUN
iajs-2802	108	48	is	be	AUX
iajs-2802	108	49	not	not	PART
iajs-2802	108	50	true	true	ADJ
iajs-2802	108	51	in	in	ADP
iajs-2802	108	52	general	general	ADJ
iajs-2802	108	53	and	and	CCONJ
iajs-2802	108	54	the	the	DET
iajs-2802	108	55	following	following	NOUN
iajs-2802	108	56	shows	show	VERB
iajs-2802	108	57	that	that	SCONJ
iajs-2802	108	58	:	:	PUNCT
iajs-2802	108	59	let	let	VERB
iajs-2802	108	60	6z	6z	NOUN
iajs-2802	108	61	be	be	AUX
iajs-2802	108	62	a	a	DET
iajs-2802	108	63	submodule	submodule	NOUN
iajs-2802	108	64	of	of	ADP
iajs-2802	108	65	a	a	DET
iajs-2802	108	66	z	z	NOUN
iajs-2802	108	67	-	-	PUNCT
iajs-2802	108	68	module	module	NOUN
iajs-2802	108	69	z	z	NOUN
iajs-2802	108	70	.	.	PUNCT
iajs-2802	109	1	then	then	ADV
iajs-2802	109	2	,	,	PUNCT
iajs-2802	109	3	6z	6z	NOUN
iajs-2802	109	4	is	be	AUX
iajs-2802	109	5	a	a	DET
iajs-2802	109	6	semimaximal	semimaximal	ADJ
iajs-2802	109	7	submodule	submodule	NOUN
iajs-2802	109	8	of	of	ADP
iajs-2802	109	9	z.	z.	PROPN
iajs-2802	109	10	since	since	SCONJ
iajs-2802	109	11	6z	6z	NOUN
iajs-2802	109	12	=	=	NOUN
iajs-2802	109	13	2z∩5z	2z∩5z	NUM
iajs-2802	109	14	where	where	SCONJ
iajs-2802	109	15	2z	2z	NUM
iajs-2802	109	16	,	,	PUNCT
iajs-2802	109	17	5z	5z	NOUN
iajs-2802	109	18	are	be	AUX
iajs-2802	109	19	maximal	maximal	ADJ
iajs-2802	109	20	submodules	submodule	NOUN
iajs-2802	109	21	of	of	ADP
iajs-2802	109	22	z.	z.	PROPN
iajs-2802	109	23	but	but	CCONJ
iajs-2802	109	24	6z	6z	NOUN
iajs-2802	109	25	is	be	AUX
iajs-2802	109	26	not	not	PART
iajs-2802	109	27	an	an	DET
iajs-2802	109	28	smsubmodule	smsubmodule	NOUN
iajs-2802	109	29	of	of	ADP
iajs-2802	109	30	z.	z.	PROPN
iajs-2802	109	31	since	since	SCONJ
iajs-2802	109	32	z/(2z)2(6z	z/(2z)2(6z	NUM
iajs-2802	109	33	)	)	PUNCT
iajs-2802	109	34	=	=	SYM
iajs-2802	109	35	z/24z	z/24z	PROPN
iajs-2802	109	36	≅	≅	PROPN
iajs-2802	109	37	z24	z24	PROPN
iajs-2802	109	38	is	be	AUX
iajs-2802	109	39	not	not	PART
iajs-2802	109	40	a	a	DET
iajs-2802	109	41	regular	regular	ADJ
iajs-2802	109	42	module	module	NOUN
iajs-2802	109	43	.	.	PUNCT
iajs-2802	110	1	proposition	proposition	NOUN
iajs-2802	110	2	(	(	PUNCT
iajs-2802	110	3	3.3	3.3	NUM
iajs-2802	110	4	)	)	PUNCT
iajs-2802	110	5	let	let	VERB
iajs-2802	110	6	b	b	NOUN
iajs-2802	110	7	,	,	PUNCT
iajs-2802	110	8	d	d	X
iajs-2802	110	9	be	be	AUX
iajs-2802	110	10	two	two	NUM
iajs-2802	110	11	submodules	submodule	NOUN
iajs-2802	110	12	of	of	ADP
iajs-2802	110	13	an	an	DET
iajs-2802	110	14	s	s	NOUN
iajs-2802	110	15	-	-	NOUN
iajs-2802	110	16	module	module	NOUN
iajs-2802	110	17	a	a	PRON
iajs-2802	110	18	with	with	ADP
iajs-2802	110	19	b⊆d	b⊆d	PROPN
iajs-2802	110	20	.	.	PUNCT
iajs-2802	111	1	then	then	ADV
iajs-2802	111	2	b	b	PROPN
iajs-2802	111	3	is	be	AUX
iajs-2802	111	4	sm	sm	NOUN
iajs-2802	111	5	-	-	PUNCT
iajs-2802	111	6	submodule	submodule	NOUN
iajs-2802	111	7	in	in	ADP
iajs-2802	111	8	d	d	PROPN
iajs-2802	111	9	when	when	SCONJ
iajs-2802	111	10	b	b	NOUN
iajs-2802	111	11	is	be	AUX
iajs-2802	111	12	sm	sm	NOUN
iajs-2802	111	13	-	-	PUNCT
iajs-2802	111	14	submodule	submodule	NOUN
iajs-2802	111	15	in	in	ADP
iajs-2802	111	16	a.	a.	NOUN
iajs-2802	111	17	proof	proof	NOUN
iajs-2802	111	18	:	:	PUNCT
iajs-2802	111	19	suppose	suppose	VERB
iajs-2802	111	20	b	b	X
iajs-2802	111	21	is	be	AUX
iajs-2802	111	22	sm	sm	NOUN
iajs-2802	111	23	-	-	PUNCT
iajs-2802	111	24	submodule	submodule	NOUN
iajs-2802	111	25	in	in	ADP
iajs-2802	111	26	a	a	PRON
iajs-2802	111	27	,	,	PUNCT
iajs-2802	111	28	then	then	ADV
iajs-2802	111	29	a	a	X
iajs-2802	111	30	/	/	PRON
iajs-2802	111	31	e2b	e2b	PROPN
iajs-2802	111	32	is	be	AUX
iajs-2802	111	33	regular	regular	ADJ
iajs-2802	111	34	module	module	NOUN
iajs-2802	111	35	for	for	ADP
iajs-2802	111	36	every	every	DET
iajs-2802	111	37	non	non	ADJ
iajs-2802	111	38	-	-	ADJ
iajs-2802	111	39	zero	zero	NUM
iajs-2802	111	40	ideal	ideal	ADJ
iajs-2802	111	41	e	e	PROPN
iajs-2802	111	42	of	of	ADP
iajs-2802	111	43	s.	s.	PROPN
iajs-2802	111	44	since	since	SCONJ
iajs-2802	111	45	d	d	PROPN
iajs-2802	111	46	/	/	SYM
iajs-2802	111	47	e2b	e2b	PROPN
iajs-2802	111	48	is	be	AUX
iajs-2802	111	49	a	a	DET
iajs-2802	111	50	submodule	submodule	NOUN
iajs-2802	111	51	of	of	ADP
iajs-2802	111	52	a	a	PRON
iajs-2802	111	53	/	/	SYM
iajs-2802	111	54	e2b	e2b	PROPN
iajs-2802	111	55	(	(	PUNCT
iajs-2802	111	56	notice	notice	NOUN
iajs-2802	111	57	,	,	PUNCT
iajs-2802	111	58	e2b	e2b	PROPN
iajs-2802	112	1	⊆	⊆	NUM
iajs-2802	112	2	b	b	X
iajs-2802	112	3	⊆	⊆	NUM
iajs-2802	112	4	d	d	NOUN
iajs-2802	112	5	)	)	PUNCT
iajs-2802	112	6	,	,	PUNCT
iajs-2802	112	7	and	and	CCONJ
iajs-2802	112	8	hence	hence	ADV
iajs-2802	112	9	by	by	ADP
iajs-2802	112	10	proposition	proposition	NOUN
iajs-2802	112	11	(	(	PUNCT
iajs-2802	112	12	2.1	2.1	NUM
iajs-2802	112	13	)	)	PUNCT
iajs-2802	112	14	,	,	PUNCT
iajs-2802	112	15	d	d	X
iajs-2802	112	16	/	/	SYM
iajs-2802	112	17	e2b	e2b	PROPN
iajs-2802	112	18	is	be	AUX
iajs-2802	112	19	a	a	DET
iajs-2802	112	20	regular	regular	ADJ
iajs-2802	112	21	submodule	submodule	NOUN
iajs-2802	112	22	of	of	ADP
iajs-2802	112	23	a	a	PRON
iajs-2802	112	24	/	/	SYM
iajs-2802	112	25	e2b	e2b	NOUN
iajs-2802	112	26	.	.	PUNCT
iajs-2802	113	1	thus	thus	ADV
iajs-2802	113	2	b	b	X
iajs-2802	113	3	is	be	AUX
iajs-2802	113	4	sm	sm	NOUN
iajs-2802	113	5	-	-	PUNCT
iajs-2802	113	6	submodule	submodule	NOUN
iajs-2802	113	7	in	in	ADP
iajs-2802	113	8	d.	d.	PROPN
iajs-2802	113	9	ibn	ibn	PROPN
iajs-2802	113	10	al	al	PROPN
iajs-2802	113	11	-	-	PUNCT
iajs-2802	113	12	haitham	haitham	PROPN
iajs-2802	113	13	jour	jour	X
iajs-2802	113	14	.	.	PROPN
iajs-2802	114	1	for	for	ADP
iajs-2802	114	2	pure	pure	ADJ
iajs-2802	114	3	&	&	CCONJ
iajs-2802	114	4	appl	appl	PROPN
iajs-2802	114	5	.	.	PUNCT
iajs-2802	115	1	sci	sci	PROPN
iajs-2802	115	2	.	.	PROPN
iajs-2802	116	1	53	53	NUM
iajs-2802	116	2	(	(	PUNCT
iajs-2802	116	3	1)2022	1)2022	NUM
iajs-2802	116	4	88	88	NUM
iajs-2802	116	5	next	next	ADV
iajs-2802	116	6	,	,	PUNCT
iajs-2802	116	7	we	we	PRON
iajs-2802	116	8	will	will	AUX
iajs-2802	116	9	give	give	VERB
iajs-2802	116	10	an	an	DET
iajs-2802	116	11	application	application	NOUN
iajs-2802	116	12	to	to	ADP
iajs-2802	116	13	a	a	DET
iajs-2802	116	14	proposition	proposition	NOUN
iajs-2802	116	15	(	(	PUNCT
iajs-2802	116	16	3.3	3.3	NUM
iajs-2802	116	17	)	)	PUNCT
iajs-2802	116	18	corollary	corollary	NOUN
iajs-2802	116	19	(	(	PUNCT
iajs-2802	116	20	3.4	3.4	NUM
iajs-2802	116	21	)	)	PUNCT
iajs-2802	116	22	let	let	VERB
iajs-2802	116	23	a	a	PRON
iajs-2802	116	24	be	be	AUX
iajs-2802	116	25	an	an	DET
iajs-2802	116	26	s	s	NOUN
iajs-2802	116	27	-	-	PUNCT
iajs-2802	116	28	module	module	NOUN
iajs-2802	116	29	and	and	CCONJ
iajs-2802	116	30	b	b	NOUN
iajs-2802	116	31	be	be	AUX
iajs-2802	116	32	a	a	DET
iajs-2802	116	33	proper	proper	ADJ
iajs-2802	116	34	submodule	submodule	NOUN
iajs-2802	116	35	of	of	ADP
iajs-2802	116	36	a.	a.	NOUN
iajs-2802	116	37	if	if	SCONJ
iajs-2802	116	38	b	b	PROPN
iajs-2802	116	39	is	be	AUX
iajs-2802	116	40	an	an	DET
iajs-2802	116	41	sm	sm	NOUN
iajs-2802	116	42	-	-	PUNCT
iajs-2802	116	43	submodule	submodule	NOUN
iajs-2802	116	44	of	of	ADP
iajs-2802	116	45	[	[	X
iajs-2802	116	46	b	b	X
iajs-2802	116	47	aa	aa	NOUN
iajs-2802	116	48	:	:	PUNCT
iajs-2802	116	49	]	]	PUNCT
iajs-2802	117	1	and	and	CCONJ
iajs-2802	117	2	[	[	X
iajs-2802	117	3	b	b	X
iajs-2802	117	4	aa	aa	NOUN
iajs-2802	117	5	:	:	PUNCT
iajs-2802	117	6	]	]	PUNCT
iajs-2802	117	7	is	be	AUX
iajs-2802	117	8	an	an	DET
iajs-2802	117	9	sm	sm	NOUN
iajs-2802	117	10	-	-	PUNCT
iajs-2802	117	11	submodule	submodule	NOUN
iajs-2802	117	12	in	in	ADP
iajs-2802	117	13	a	a	PRON
iajs-2802	117	14	,	,	PUNCT
iajs-2802	117	15	then	then	ADV
iajs-2802	117	16	,	,	PUNCT
iajs-2802	117	17	b	b	PROPN
iajs-2802	117	18	is	be	AUX
iajs-2802	117	19	an	an	DET
iajs-2802	117	20	sm	sm	NOUN
iajs-2802	117	21	-	-	PUNCT
iajs-2802	117	22	submodule	submodule	NOUN
iajs-2802	117	23	in	in	ADP
iajs-2802	117	24	a.	a.	NOUN
iajs-2802	117	25	proof	proof	NOUN
iajs-2802	117	26	:	:	PUNCT
iajs-2802	117	27	it	it	PRON
iajs-2802	117	28	clear	clear	ADJ
iajs-2802	117	29	that	that	SCONJ
iajs-2802	117	30	b⊆	b⊆	PROPN
iajs-2802	118	1	[	[	X
iajs-2802	118	2	b	b	X
iajs-2802	118	3	aa	aa	NOUN
iajs-2802	118	4	:	:	PUNCT
iajs-2802	118	5	]	]	PUNCT
iajs-2802	118	6	⊆	⊆	NUM
iajs-2802	118	7	a.	a.	NOUN
iajs-2802	118	8	therefore	therefore	ADV
iajs-2802	118	9	,	,	PUNCT
iajs-2802	118	10	by	by	ADP
iajs-2802	118	11	using	use	VERB
iajs-2802	118	12	proposition	proposition	NOUN
iajs-2802	118	13	(	(	PUNCT
iajs-2802	118	14	3.3	3.3	NUM
iajs-2802	118	15	)	)	PUNCT
iajs-2802	118	16	,	,	PUNCT
iajs-2802	118	17	we	we	PRON
iajs-2802	118	18	conclude	conclude	VERB
iajs-2802	118	19	that	that	SCONJ
iajs-2802	118	20	b	b	PROPN
iajs-2802	118	21	is	be	AUX
iajs-2802	118	22	sm	sm	NOUN
iajs-2802	118	23	-	-	PUNCT
iajs-2802	118	24	submodule	submodule	NOUN
iajs-2802	118	25	in	in	ADP
iajs-2802	118	26	a.	a.	NOUN
iajs-2802	118	27	now	now	ADV
iajs-2802	118	28	,	,	PUNCT
iajs-2802	118	29	we	we	PRON
iajs-2802	118	30	will	will	AUX
iajs-2802	118	31	give	give	VERB
iajs-2802	118	32	the	the	DET
iajs-2802	118	33	sufficient	sufficient	ADJ
iajs-2802	118	34	condition	condition	NOUN
iajs-2802	118	35	for	for	ADP
iajs-2802	118	36	a	a	DET
iajs-2802	118	37	submodule	submodule	NOUN
iajs-2802	118	38	to	to	PART
iajs-2802	118	39	not	not	PART
iajs-2802	118	40	be	be	AUX
iajs-2802	118	41	sm	sm	NOUN
iajs-2802	118	42	-	-	PUNCT
iajs-2802	118	43	submodule	submodule	NOUN
iajs-2802	118	44	.	.	PUNCT
iajs-2802	119	1	proposition	proposition	NOUN
iajs-2802	119	2	(	(	PUNCT
iajs-2802	119	3	3.5	3.5	NUM
iajs-2802	119	4	)	)	PUNCT
iajs-2802	119	5	let	let	VERB
iajs-2802	119	6	a	a	PRON
iajs-2802	119	7	be	be	AUX
iajs-2802	119	8	an	an	DET
iajs-2802	119	9	s	s	NOUN
iajs-2802	119	10	-	-	NOUN
iajs-2802	119	11	module	module	NOUN
iajs-2802	119	12	.	.	PUNCT
iajs-2802	120	1	if	if	SCONJ
iajs-2802	120	2	a	a	PRON
iajs-2802	120	3	is	be	AUX
iajs-2802	120	4	cyclic	cyclic	ADJ
iajs-2802	120	5	module	module	NOUN
iajs-2802	120	6	(	(	PUNCT
iajs-2802	120	7	if	if	SCONJ
iajs-2802	120	8	a	a	DET
iajs-2802	120	9	=	=	NOUN
iajs-2802	120	10	sx	sx	NOUN
iajs-2802	120	11	for	for	ADP
iajs-2802	120	12	some	some	DET
iajs-2802	120	13	x∈a	x∈a	NOUN
iajs-2802	120	14	)	)	PUNCT
iajs-2802	120	15	and	and	CCONJ
iajs-2802	120	16	anns(x	anns(x	PROPN
iajs-2802	120	17	)	)	PUNCT
iajs-2802	120	18	is	be	AUX
iajs-2802	120	19	maximal	maximal	ADJ
iajs-2802	120	20	ideal	ideal	NOUN
iajs-2802	120	21	of	of	ADP
iajs-2802	120	22	s	s	PROPN
iajs-2802	120	23	,	,	PUNCT
iajs-2802	120	24	then	then	ADV
iajs-2802	120	25	a	a	PRON
iajs-2802	120	26	has	have	VERB
iajs-2802	120	27	no	no	DET
iajs-2802	120	28	sm	sm	NOUN
iajs-2802	120	29	-	-	PUNCT
iajs-2802	120	30	submodule	submodule	NOUN
iajs-2802	120	31	.	.	PUNCT
iajs-2802	121	1	proof	proof	NOUN
iajs-2802	121	2	:	:	PUNCT
iajs-2802	121	3	since	since	SCONJ
iajs-2802	121	4	a=	a=	ADV
iajs-2802	121	5	sx	sx	VERB
iajs-2802	121	6	for	for	ADP
iajs-2802	121	7	some	some	DET
iajs-2802	121	8	x∈a	x∈a	NOUN
iajs-2802	121	9	,	,	PUNCT
iajs-2802	121	10	then	then	ADV
iajs-2802	121	11	a	a	PRON
iajs-2802	121	12	is	be	AUX
iajs-2802	121	13	isomorphic	isomorphic	ADJ
iajs-2802	121	14	to	to	ADP
iajs-2802	121	15	a	a	DET
iajs-2802	121	16	factor	factor	NOUN
iajs-2802	121	17	module	module	NOUN
iajs-2802	121	18	of	of	ADP
iajs-2802	121	19	s	s	NOUN
iajs-2802	121	20	by	by	ADP
iajs-2802	121	21	proposition	proposition	NOUN
iajs-2802	121	22	(	(	PUNCT
iajs-2802	121	23	2.2	2.2	NUM
iajs-2802	121	24	)	)	PUNCT
iajs-2802	121	25	.	.	PUNCT
iajs-2802	122	1	we	we	PRON
iajs-2802	122	2	can	can	AUX
iajs-2802	122	3	define	define	VERB
iajs-2802	122	4	f	f	X
iajs-2802	122	5	:	:	PUNCT
iajs-2802	122	6	s	s	X
iajs-2802	122	7	→	→	PUNCT
iajs-2802	122	8	a	a	DET
iajs-2802	122	9	such	such	ADJ
iajs-2802	122	10	that	that	SCONJ
iajs-2802	122	11	f(r	f(r	NOUN
iajs-2802	122	12	)	)	PUNCT
iajs-2802	123	1	=	=	VERB
iajs-2802	123	2	rx	rx	VERB
iajs-2802	123	3	.	.	PUNCT
iajs-2802	124	1	it	it	PRON
iajs-2802	124	2	is	be	AUX
iajs-2802	124	3	easily	easily	ADV
iajs-2802	124	4	to	to	PART
iajs-2802	124	5	show	show	VERB
iajs-2802	124	6	that	that	SCONJ
iajs-2802	124	7	f	f	PROPN
iajs-2802	124	8	is	be	AUX
iajs-2802	124	9	welldefine	welldefine	ADJ
iajs-2802	124	10	and	and	CCONJ
iajs-2802	124	11	epimorphisim	epimorphisim	ADJ
iajs-2802	124	12	,	,	PUNCT
iajs-2802	124	13	by	by	ADP
iajs-2802	124	14	the	the	DET
iajs-2802	124	15	first	first	ADJ
iajs-2802	124	16	fundamental	fundamental	ADJ
iajs-2802	124	17	theorem	theorem	NOUN
iajs-2802	124	18	of	of	ADP
iajs-2802	124	19	isomorphism	isomorphism	PROPN
iajs-2802	124	20	s	s	PROPN
iajs-2802	124	21	/	/	SYM
iajs-2802	124	22	ker	ker	NOUN
iajs-2802	124	23	f	f	PROPN
iajs-2802	124	24	≅	≅	PROPN
iajs-2802	124	25	a.	a.	PROPN
iajs-2802	124	26	next	next	ADV
iajs-2802	124	27	,	,	PUNCT
iajs-2802	124	28	ker	ker	PROPN
iajs-2802	124	29	f	f	PROPN
iajs-2802	125	1	=	=	NOUN
iajs-2802	125	2	{	{	PUNCT
iajs-2802	125	3	r∈s	r∈s	NOUN
iajs-2802	125	4	:	:	PUNCT
iajs-2802	125	5	f(r	f(r	X
iajs-2802	125	6	)	)	PUNCT
iajs-2802	126	1	=	=	X
iajs-2802	126	2	0a	0a	X
iajs-2802	126	3	}	}	PUNCT
iajs-2802	126	4	=	=	SYM
iajs-2802	126	5	{	{	PUNCT
iajs-2802	126	6	r∈s	r∈s	NOUN
iajs-2802	126	7	:	:	PUNCT
iajs-2802	126	8	rx=0a	rx=0a	AUX
iajs-2802	126	9	}	}	PUNCT
iajs-2802	126	10	=	=	SYM
iajs-2802	126	11	anns(x	anns(x	NOUN
iajs-2802	126	12	)	)	PUNCT
iajs-2802	126	13	that	that	PRON
iajs-2802	126	14	is	be	AUX
iajs-2802	126	15	s	s	PROPN
iajs-2802	126	16	/	/	SYM
iajs-2802	126	17	anns(x	anns(x	NOUN
iajs-2802	126	18	)	)	PUNCT
iajs-2802	126	19	≅a	≅a	NOUN
iajs-2802	126	20	.	.	PUNCT
iajs-2802	127	1	also	also	ADV
iajs-2802	127	2	,	,	PUNCT
iajs-2802	127	3	we	we	PRON
iajs-2802	127	4	have	have	AUX
iajs-2802	127	5	anns(x	anns(x	PROPN
iajs-2802	127	6	)	)	PUNCT
iajs-2802	127	7	is	be	AUX
iajs-2802	127	8	maximal	maximal	ADJ
iajs-2802	127	9	ideal	ideal	NOUN
iajs-2802	127	10	of	of	ADP
iajs-2802	127	11	s	s	PRON
iajs-2802	127	12	which	which	PRON
iajs-2802	127	13	implies	imply	VERB
iajs-2802	127	14	s	s	NOUN
iajs-2802	127	15	/	/	SYM
iajs-2802	127	16	anns(x	anns(x	NOUN
iajs-2802	127	17	)	)	PUNCT
iajs-2802	127	18	is	be	AUX
iajs-2802	127	19	simple	simple	ADJ
iajs-2802	127	20	and	and	CCONJ
iajs-2802	127	21	hence	hence	ADV
iajs-2802	127	22	a	a	PRON
iajs-2802	127	23	is	be	AUX
iajs-2802	127	24	simple	simple	ADJ
iajs-2802	127	25	.	.	PUNCT
iajs-2802	128	1	therefore	therefore	ADV
iajs-2802	128	2	,	,	PUNCT
iajs-2802	128	3	by	by	ADP
iajs-2802	128	4	examples	example	NOUN
iajs-2802	128	5	and	and	CCONJ
iajs-2802	128	6	remarks	remark	NOUN
iajs-2802	128	7	(	(	PUNCT
iajs-2802	128	8	(	(	PUNCT
iajs-2802	128	9	3.2	3.2	NUM
iajs-2802	128	10	)	)	PUNCT
iajs-2802	128	11	no	no	NOUN
iajs-2802	128	12	.	.	PUNCT
iajs-2802	129	1	(	(	PUNCT
iajs-2802	129	2	2	2	NUM
iajs-2802	129	3	)	)	PUNCT
iajs-2802	129	4	)	)	PUNCT
iajs-2802	129	5	,	,	PUNCT
iajs-2802	129	6	we	we	PRON
iajs-2802	129	7	get	get	VERB
iajs-2802	129	8	the	the	DET
iajs-2802	129	9	result	result	NOUN
iajs-2802	129	10	.	.	PUNCT
iajs-2802	130	1	next	next	ADJ
iajs-2802	130	2	is	be	AUX
iajs-2802	130	3	the	the	DET
iajs-2802	130	4	application	application	NOUN
iajs-2802	130	5	of	of	ADP
iajs-2802	130	6	proposition	proposition	NOUN
iajs-2802	130	7	(	(	PUNCT
iajs-2802	130	8	3.5	3.5	NUM
iajs-2802	130	9	)	)	PUNCT
iajs-2802	130	10	corollary	corollary	NOUN
iajs-2802	130	11	(	(	PUNCT
iajs-2802	130	12	3.6	3.6	NUM
iajs-2802	130	13	)	)	PUNCT
iajs-2802	130	14	if	if	SCONJ
iajs-2802	130	15	a	a	DET
iajs-2802	130	16	non	non	ADJ
iajs-2802	130	17	-	-	ADJ
iajs-2802	130	18	zero	zero	ADJ
iajs-2802	130	19	prime	prime	ADJ
iajs-2802	130	20	and	and	CCONJ
iajs-2802	130	21	semi	semi	ADJ
iajs-2802	130	22	-	-	ADJ
iajs-2802	130	23	simple	simple	ADJ
iajs-2802	130	24	s	s	NOUN
iajs-2802	130	25	-	-	NOUN
iajs-2802	130	26	module	module	NOUN
iajs-2802	130	27	a	a	PRON
iajs-2802	130	28	,	,	PUNCT
iajs-2802	130	29	then	then	ADV
iajs-2802	130	30	a	a	PRON
iajs-2802	130	31	has	have	VERB
iajs-2802	130	32	no	no	DET
iajs-2802	130	33	sm	sm	NOUN
iajs-2802	130	34	-	-	PUNCT
iajs-2802	130	35	submodule	submodule	NOUN
iajs-2802	130	36	.	.	PUNCT
iajs-2802	131	1	proof	proof	NOUN
iajs-2802	131	2	:	:	PUNCT
iajs-2802	131	3	suppose	suppose	VERB
iajs-2802	131	4	that	that	SCONJ
iajs-2802	131	5	a	a	PRON
iajs-2802	131	6	is	be	AUX
iajs-2802	131	7	a	a	DET
iajs-2802	131	8	prime	prime	ADJ
iajs-2802	131	9	and	and	CCONJ
iajs-2802	131	10	semi	semi	ADJ
iajs-2802	131	11	-	-	ADJ
iajs-2802	131	12	simple	simple	ADJ
iajs-2802	131	13	module	module	NOUN
iajs-2802	131	14	,	,	PUNCT
iajs-2802	131	15	then	then	ADV
iajs-2802	131	16	,	,	PUNCT
iajs-2802	131	17	we	we	PRON
iajs-2802	131	18	obtain	obtain	VERB
iajs-2802	131	19	a	a	PRON
iajs-2802	131	20	is	be	AUX
iajs-2802	131	21	simple	simple	ADJ
iajs-2802	131	22	module	module	NOUN
iajs-2802	131	23	.	.	PUNCT
iajs-2802	132	1	to	to	PART
iajs-2802	132	2	prove	prove	VERB
iajs-2802	132	3	this	this	PRON
iajs-2802	132	4	,	,	PUNCT
iajs-2802	132	5	assume	assume	VERB
iajs-2802	132	6	that	that	SCONJ
iajs-2802	132	7	a	a	PRON
iajs-2802	132	8	is	be	AUX
iajs-2802	132	9	not	not	PART
iajs-2802	132	10	simple	simple	ADJ
iajs-2802	132	11	which	which	PRON
iajs-2802	132	12	implies	imply	VERB
iajs-2802	132	13	a	a	DET
iajs-2802	132	14	is	be	AUX
iajs-2802	132	15	a	a	DET
iajs-2802	132	16	direct	direct	ADJ
iajs-2802	132	17	sum	sum	NOUN
iajs-2802	132	18	of	of	ADP
iajs-2802	132	19	simple	simple	ADJ
iajs-2802	132	20	smodules	smodule	NOUN
iajs-2802	132	21	.	.	PUNCT
iajs-2802	133	1	then	then	ADV
iajs-2802	133	2	,	,	PUNCT
iajs-2802	133	3	there	there	PRON
iajs-2802	133	4	exists	exist	VERB
iajs-2802	133	5	a	a	DET
iajs-2802	133	6	simple	simple	ADJ
iajs-2802	133	7	module	module	NOUN
iajs-2802	133	8	m1	m1	NOUN
iajs-2802	133	9	and	and	CCONJ
iajs-2802	133	10	m2	m2	PROPN
iajs-2802	133	11	which	which	PRON
iajs-2802	133	12	are	be	AUX
iajs-2802	133	13	a	a	DET
iajs-2802	133	14	direct	direct	ADJ
iajs-2802	133	15	summand	summand	NOUN
iajs-2802	133	16	of	of	ADP
iajs-2802	133	17	a	a	PRON
iajs-2802	133	18	with	with	ADP
iajs-2802	133	19	m1	m1	PROPN
iajs-2802	133	20	≠	≠	PROPN
iajs-2802	133	21	m2	m2	PROPN
iajs-2802	133	22	.	.	PUNCT
iajs-2802	133	23	m1≅	m1≅	PROPN
iajs-2802	133	24	s	s	PART
iajs-2802	133	25	/	/	SYM
iajs-2802	133	26	e	e	NOUN
iajs-2802	133	27	,	,	PUNCT
iajs-2802	133	28	m2≅	m2≅	VERB
iajs-2802	133	29	s	s	PROPN
iajs-2802	133	30	/	/	SYM
iajs-2802	133	31	d	d	NOUN
iajs-2802	133	32	where	where	SCONJ
iajs-2802	133	33	e	e	NOUN
iajs-2802	133	34	and	and	CCONJ
iajs-2802	133	35	d	d	NOUN
iajs-2802	133	36	are	be	AUX
iajs-2802	133	37	maximal	maximal	ADJ
iajs-2802	133	38	ideals	ideal	NOUN
iajs-2802	133	39	of	of	ADP
iajs-2802	133	40	s	s	NOUN
iajs-2802	133	41	,	,	PUNCT
iajs-2802	133	42	by	by	ADP
iajs-2802	133	43	proposition	proposition	NOUN
iajs-2802	133	44	(	(	PUNCT
iajs-2802	133	45	2.3	2.3	NUM
iajs-2802	133	46	)	)	PUNCT
iajs-2802	133	47	.	.	PUNCT
iajs-2802	134	1	but	but	CCONJ
iajs-2802	134	2	a	a	PRON
iajs-2802	134	3	is	be	AUX
iajs-2802	134	4	prime	prime	ADJ
iajs-2802	134	5	module	module	NOUN
iajs-2802	134	6	,	,	PUNCT
iajs-2802	134	7	then	then	ADV
iajs-2802	134	8	anns(m1	anns(m1	VERB
iajs-2802	134	9	)	)	PUNCT
iajs-2802	134	10	=	=	SYM
iajs-2802	134	11	e=	e=	NOUN
iajs-2802	134	12	anns(a	anns(a	NOUN
iajs-2802	134	13	)	)	PUNCT
iajs-2802	134	14	and	and	CCONJ
iajs-2802	134	15	anns(m2	anns(m2	ADJ
iajs-2802	134	16	)	)	PUNCT
iajs-2802	135	1	=	=	PUNCT
iajs-2802	136	1	d	d	NOUN
iajs-2802	136	2	=	=	NOUN
iajs-2802	136	3	anns(a).thus	anns(a).thus	X
iajs-2802	136	4	e	e	NOUN
iajs-2802	136	5	=	=	SYM
iajs-2802	136	6	d	d	X
iajs-2802	136	7	which	which	PRON
iajs-2802	136	8	implies	imply	VERB
iajs-2802	136	9	that	that	DET
iajs-2802	136	10	m1	m1	PROPN
iajs-2802	136	11	=	=	SYM
iajs-2802	136	12	m2	m2	PROPN
iajs-2802	136	13	,	,	PUNCT
iajs-2802	136	14	and	and	CCONJ
iajs-2802	136	15	this	this	PRON
iajs-2802	136	16	is	be	AUX
iajs-2802	136	17	a	a	DET
iajs-2802	136	18	contradiction	contradiction	NOUN
iajs-2802	136	19	.	.	PUNCT
iajs-2802	137	1	hence	hence	ADV
iajs-2802	137	2	,	,	PUNCT
iajs-2802	137	3	a	a	PRON
iajs-2802	137	4	is	be	AUX
iajs-2802	137	5	a	a	DET
iajs-2802	137	6	simple	simple	ADJ
iajs-2802	137	7	module	module	NOUN
iajs-2802	137	8	and	and	CCONJ
iajs-2802	137	9	by	by	ADP
iajs-2802	137	10	using	use	VERB
iajs-2802	137	11	proposition	proposition	NOUN
iajs-2802	137	12	(	(	PUNCT
iajs-2802	137	13	3.5	3.5	NUM
iajs-2802	137	14	)	)	PUNCT
iajs-2802	137	15	,	,	PUNCT
iajs-2802	137	16	we	we	PRON
iajs-2802	137	17	have	have	VERB
iajs-2802	137	18	a	a	DET
iajs-2802	137	19	has	have	AUX
iajs-2802	137	20	no	no	DET
iajs-2802	137	21	sm	sm	NOUN
iajs-2802	137	22	-	-	PUNCT
iajs-2802	137	23	submodule	submodule	NOUN
iajs-2802	137	24	.	.	PUNCT
iajs-2802	138	1	as	as	ADP
iajs-2802	138	2	a	a	DET
iajs-2802	138	3	direct	direct	NOUN
iajs-2802	138	4	of	of	ADP
iajs-2802	138	5	corollary	corollary	ADJ
iajs-2802	138	6	(	(	PUNCT
iajs-2802	138	7	3.6	3.6	NUM
iajs-2802	138	8	)	)	PUNCT
iajs-2802	138	9	,	,	PUNCT
iajs-2802	138	10	we	we	PRON
iajs-2802	138	11	have	have	VERB
iajs-2802	138	12	the	the	DET
iajs-2802	138	13	following	following	NOUN
iajs-2802	138	14	.	.	PUNCT
iajs-2802	139	1	corollary	corollary	ADJ
iajs-2802	139	2	(	(	PUNCT
iajs-2802	139	3	3.7	3.7	NUM
iajs-2802	139	4	)	)	PUNCT
iajs-2802	139	5	ibn	ibn	PROPN
iajs-2802	139	6	al	al	PROPN
iajs-2802	139	7	-	-	PUNCT
iajs-2802	139	8	haitham	haitham	PROPN
iajs-2802	139	9	jour	jour	X
iajs-2802	139	10	.	.	PROPN
iajs-2802	140	1	for	for	ADP
iajs-2802	140	2	pure	pure	ADJ
iajs-2802	140	3	&	&	CCONJ
iajs-2802	140	4	appl	appl	PROPN
iajs-2802	140	5	.	.	PUNCT
iajs-2802	141	1	sci	sci	PROPN
iajs-2802	141	2	.	.	PROPN
iajs-2802	142	1	53	53	NUM
iajs-2802	142	2	(	(	PUNCT
iajs-2802	142	3	1)2022	1)2022	PROPN
iajs-2802	142	4	89	89	NUM
iajs-2802	142	5	let	let	VERB
iajs-2802	142	6	p	p	PRON
iajs-2802	142	7	be	be	AUX
iajs-2802	142	8	a	a	DET
iajs-2802	142	9	prime	prime	ADJ
iajs-2802	142	10	and	and	CCONJ
iajs-2802	142	11	semimaximal	semimaximal	ADJ
iajs-2802	142	12	submodule	submodule	NOUN
iajs-2802	142	13	of	of	ADP
iajs-2802	142	14	an	an	DET
iajs-2802	142	15	s	s	NOUN
iajs-2802	142	16	-	-	PUNCT
iajs-2802	142	17	module	module	NOUN
iajs-2802	142	18	a.	a.	NOUN
iajs-2802	142	19	then	then	ADV
iajs-2802	142	20	the	the	DET
iajs-2802	142	21	quotient	quotient	NOUN
iajs-2802	142	22	module	module	NOUN
iajs-2802	142	23	by	by	ADP
iajs-2802	142	24	p	p	PROPN
iajs-2802	142	25	has	have	VERB
iajs-2802	142	26	no	no	DET
iajs-2802	142	27	sm	sm	NOUN
iajs-2802	142	28	-	-	PUNCT
iajs-2802	142	29	submodule	submodule	NOUN
iajs-2802	142	30	.	.	PUNCT
iajs-2802	143	1	proof	proof	NOUN
iajs-2802	143	2	:	:	PUNCT
iajs-2802	143	3	since	since	SCONJ
iajs-2802	143	4	p	p	NOUN
iajs-2802	143	5	is	be	AUX
iajs-2802	143	6	a	a	DET
iajs-2802	143	7	semimaximal	semimaximal	ADJ
iajs-2802	143	8	submodule	submodule	NOUN
iajs-2802	143	9	of	of	ADP
iajs-2802	143	10	a	a	PRON
iajs-2802	143	11	,	,	PUNCT
iajs-2802	143	12	then	then	ADV
iajs-2802	143	13	by	by	ADP
iajs-2802	143	14	definition	definition	NOUN
iajs-2802	143	15	(	(	PUNCT
iajs-2802	143	16	2.5	2.5	NUM
iajs-2802	143	17	)	)	PUNCT
iajs-2802	143	18	,	,	PUNCT
iajs-2802	143	19	we	we	PRON
iajs-2802	143	20	have	have	VERB
iajs-2802	143	21	a	a	DET
iajs-2802	143	22	/	/	SYM
iajs-2802	143	23	p	p	NOUN
iajs-2802	143	24	is	be	AUX
iajs-2802	143	25	a	a	DET
iajs-2802	143	26	semisimple	semisimple	ADJ
iajs-2802	143	27	s	s	NOUN
iajs-2802	143	28	-	-	NOUN
iajs-2802	143	29	module	module	NOUN
iajs-2802	143	30	.	.	PUNCT
iajs-2802	144	1	on	on	ADP
iajs-2802	144	2	the	the	DET
iajs-2802	144	3	other	other	ADJ
iajs-2802	144	4	hand	hand	NOUN
iajs-2802	144	5	,	,	PUNCT
iajs-2802	144	6	p	p	NOUN
iajs-2802	144	7	is	be	AUX
iajs-2802	144	8	a	a	DET
iajs-2802	144	9	prime	prime	ADJ
iajs-2802	144	10	submodule	submodule	NOUN
iajs-2802	144	11	of	of	ADP
iajs-2802	144	12	a	a	PRON
iajs-2802	144	13	,	,	PUNCT
iajs-2802	144	14	then	then	ADV
iajs-2802	144	15	,	,	PUNCT
iajs-2802	144	16	by	by	ADP
iajs-2802	144	17	[	[	X
iajs-2802	144	18	3	3	NUM
iajs-2802	144	19	,	,	PUNCT
iajs-2802	144	20	proposition	proposition	NOUN
iajs-2802	144	21	(	(	PUNCT
iajs-2802	144	22	1.1.51	1.1.51	NUM
iajs-2802	144	23	)	)	PUNCT
iajs-2802	144	24	]	]	PUNCT
iajs-2802	144	25	we	we	PRON
iajs-2802	144	26	obtain	obtain	VERB
iajs-2802	144	27	that	that	PRON
iajs-2802	144	28	a	a	PRON
iajs-2802	144	29	/	/	SYM
iajs-2802	144	30	p	p	NOUN
iajs-2802	144	31	is	be	AUX
iajs-2802	144	32	a	a	DET
iajs-2802	144	33	prime	prime	ADJ
iajs-2802	144	34	s	s	NOUN
iajs-2802	144	35	-	-	NOUN
iajs-2802	144	36	module	module	NOUN
iajs-2802	144	37	,	,	PUNCT
iajs-2802	144	38	and	and	CCONJ
iajs-2802	144	39	hence	hence	ADV
iajs-2802	144	40	by	by	ADP
iajs-2802	144	41	corollary	corollary	ADJ
iajs-2802	144	42	(	(	PUNCT
iajs-2802	144	43	3.6	3.6	NUM
iajs-2802	144	44	)	)	PUNCT
iajs-2802	144	45	,	,	PUNCT
iajs-2802	144	46	then	then	ADV
iajs-2802	144	47	a	a	X
iajs-2802	144	48	/	/	SYM
iajs-2802	144	49	p	p	NOUN
iajs-2802	144	50	is	be	AUX
iajs-2802	144	51	a	a	DET
iajs-2802	144	52	simple	simple	ADJ
iajs-2802	144	53	s	s	NOUN
iajs-2802	144	54	-	-	NOUN
iajs-2802	144	55	module	module	NOUN
iajs-2802	144	56	and	and	CCONJ
iajs-2802	144	57	hence	hence	ADV
iajs-2802	144	58	a	a	X
iajs-2802	144	59	/	/	SYM
iajs-2802	144	60	p	p	NOUN
iajs-2802	144	61	has	have	VERB
iajs-2802	144	62	no	no	DET
iajs-2802	144	63	sm	sm	NOUN
iajs-2802	144	64	-	-	PUNCT
iajs-2802	144	65	submodule	submodule	NOUN
iajs-2802	144	66	.	.	PUNCT
iajs-2802	145	1	s4	s4	PROPN
iajs-2802	145	2	:	:	PUNCT
iajs-2802	145	3	the	the	DET
iajs-2802	145	4	behavior	behavior	NOUN
iajs-2802	145	5	of	of	ADP
iajs-2802	145	6	sm	sm	NOUN
iajs-2802	145	7	-	-	PUNCT
iajs-2802	145	8	submodules	submodule	NOUN
iajs-2802	145	9	under	under	ADP
iajs-2802	145	10	localization	localization	NOUN
iajs-2802	145	11	.	.	PUNCT
iajs-2802	146	1	let	let	VERB
iajs-2802	146	2	k	k	PRON
iajs-2802	146	3	be	be	AUX
iajs-2802	146	4	a	a	DET
iajs-2802	146	5	subset	subset	NOUN
iajs-2802	146	6	of	of	ADP
iajs-2802	146	7	a	a	DET
iajs-2802	146	8	ring	ring	NOUN
iajs-2802	146	9	s	s	PROPN
iajs-2802	146	10	,	,	PUNCT
iajs-2802	146	11	w	w	PROPN
iajs-2802	146	12	is	be	AUX
iajs-2802	146	13	multiplication	multiplication	NOUN
iajs-2802	146	14	closed	close	VERB
iajs-2802	146	15	if	if	SCONJ
iajs-2802	146	16	the	the	DET
iajs-2802	146	17	two	two	NUM
iajs-2802	146	18	condition	condition	NOUN
iajs-2802	146	19	hold	hold	VERB
iajs-2802	146	20	:	:	PUNCT
iajs-2802	147	1	1	1	X
iajs-2802	147	2	.	.	X
iajs-2802	148	1	i	i	PRON
iajs-2802	148	2	∈	∈	PROPN
iajs-2802	148	3	w	w	NOUN
iajs-2802	148	4	.	.	PUNCT
iajs-2802	149	1	2	2	X
iajs-2802	149	2	.	.	PUNCT
iajs-2802	149	3	xy	xy	PROPN
iajs-2802	150	1	∈	∈	PROPN
iajs-2802	151	1	w	w	NOUN
iajs-2802	151	2	for	for	ADP
iajs-2802	151	3	every	every	DET
iajs-2802	151	4	x	x	X
iajs-2802	151	5	,	,	PUNCT
iajs-2802	151	6	y	y	PROPN
iajs-2802	151	7	∈	∈	PROPN
iajs-2802	151	8	w	w	NOUN
iajs-2802	151	9	.	.	PUNCT
iajs-2802	152	1	we	we	PRON
iajs-2802	152	2	know	know	VERB
iajs-2802	152	3	that	that	SCONJ
iajs-2802	152	4	every	every	DET
iajs-2802	152	5	proper	proper	ADJ
iajs-2802	152	6	ideal	ideal	ADJ
iajs-2802	152	7	e	e	NOUN
iajs-2802	152	8	in	in	ADP
iajs-2802	152	9	s	s	PROPN
iajs-2802	152	10	is	be	AUX
iajs-2802	152	11	prime	prime	ADJ
iajs-2802	152	12	if	if	SCONJ
iajs-2802	152	13	and	and	CCONJ
iajs-2802	152	14	only	only	ADV
iajs-2802	152	15	if	if	SCONJ
iajs-2802	152	16	s	s	NOUN
iajs-2802	152	17	-	-	PUNCT
iajs-2802	152	18	e	e	NOUN
iajs-2802	152	19	is	be	AUX
iajs-2802	152	20	multiplicatively	multiplicatively	ADV
iajs-2802	152	21	closed	closed	ADJ
iajs-2802	152	22	,	,	PUNCT
iajs-2802	152	23	see	see	VERB
iajs-2802	152	24	[	[	X
iajs-2802	152	25	8].if	8].if	NUM
iajs-2802	152	26	a	a	PRON
iajs-2802	152	27	is	be	AUX
iajs-2802	152	28	an	an	DET
iajs-2802	152	29	s	s	NOUN
iajs-2802	152	30	-	-	PUNCT
iajs-2802	152	31	module	module	NOUN
iajs-2802	152	32	and	and	CCONJ
iajs-2802	152	33	w	w	NOUN
iajs-2802	152	34	be	be	AUX
iajs-2802	152	35	a	a	DET
iajs-2802	152	36	multiplicatively	multiplicatively	ADV
iajs-2802	152	37	closed	close	VERB
iajs-2802	152	38	on	on	ADP
iajs-2802	152	39	s	s	PRON
iajs-2802	152	40	such	such	ADJ
iajs-2802	152	41	that	that	SCONJ
iajs-2802	152	42	w≠<0	w≠<0	PROPN
iajs-2802	152	43	>	>	X
iajs-2802	152	44	,	,	PUNCT
iajs-2802	152	45	then	then	ADV
iajs-2802	152	46	sw	sw	PROPN
iajs-2802	152	47	be	be	AUX
iajs-2802	152	48	the	the	DET
iajs-2802	152	49	set	set	NOUN
iajs-2802	152	50	for	for	ADP
iajs-2802	152	51	all	all	DET
iajs-2802	152	52	fractional	fractional	ADJ
iajs-2802	152	53	r	r	NOUN
iajs-2802	152	54	/	/	SYM
iajs-2802	152	55	w	w	NOUN
iajs-2802	152	56	where	where	SCONJ
iajs-2802	152	57	r	r	NOUN
iajs-2802	152	58	∈s	∈s	NUM
iajs-2802	152	59	and	and	CCONJ
iajs-2802	152	60	w	w	NOUN
iajs-2802	152	61	∈w	∈w	NOUN
iajs-2802	152	62	and	and	CCONJ
iajs-2802	152	63	aw	aw	INTJ
iajs-2802	152	64	be	be	AUX
iajs-2802	152	65	the	the	DET
iajs-2802	152	66	set	set	NOUN
iajs-2802	152	67	of	of	ADP
iajs-2802	152	68	all	all	DET
iajs-2802	152	69	fractional	fractional	PROPN
iajs-2802	152	70	m	m	PROPN
iajs-2802	152	71	/	/	SYM
iajs-2802	152	72	w	w	PROPN
iajs-2802	152	73	where	where	SCONJ
iajs-2802	152	74	m	m	VERB
iajs-2802	152	75	∈a	∈a	ADJ
iajs-2802	152	76	and	and	CCONJ
iajs-2802	152	77	w	w	NOUN
iajs-2802	152	78	∈w	∈w	NOUN
iajs-2802	152	79	.	.	PUNCT
iajs-2802	153	1	for	for	ADP
iajs-2802	153	2	m1	m1	PROPN
iajs-2802	153	3	,	,	PUNCT
iajs-2802	153	4	m2∈	m2∈	PROPN
iajs-2802	153	5	a	a	NOUN
iajs-2802	153	6	and	and	CCONJ
iajs-2802	153	7	w1,w2∈	w1,w2∈	PROPN
iajs-2802	153	8	w	w	NOUN
iajs-2802	153	9	,	,	PUNCT
iajs-2802	153	10	m1	m1	NOUN
iajs-2802	153	11	/	/	SYM
iajs-2802	153	12	w1	w1	NOUN
iajs-2802	153	13	=	=	SYM
iajs-2802	153	14	m2	m2	PROPN
iajs-2802	153	15	/	/	SYM
iajs-2802	153	16	w2	w2	NOUN
iajs-2802	153	17	if	if	SCONJ
iajs-2802	153	18	and	and	CCONJ
iajs-2802	153	19	only	only	ADV
iajs-2802	153	20	if	if	SCONJ
iajs-2802	153	21	∃	∃	PROPN
iajs-2802	153	22	t	t	PROPN
iajs-2802	153	23	∈	∈	PROPN
iajs-2802	153	24	w	w	ADP
iajs-2802	153	25	such	such	ADJ
iajs-2802	153	26	that	that	SCONJ
iajs-2802	153	27	t(w1m1	t(w1m1	NOUN
iajs-2802	153	28	-	-	PUNCT
iajs-2802	153	29	w2m2)=	w2m2)=	NOUN
iajs-2802	153	30	0	0	NUM
iajs-2802	153	31	.	.	PUNCT
iajs-2802	154	1	also	also	ADV
iajs-2802	154	2	,	,	PUNCT
iajs-2802	154	3	we	we	PRON
iajs-2802	154	4	can	can	AUX
iajs-2802	154	5	make	make	VERB
iajs-2802	154	6	aw	aw	INTJ
iajs-2802	154	7	in	in	ADP
iajs-2802	154	8	to	to	ADP
iajs-2802	154	9	sw	sw	NOUN
iajs-2802	154	10	-	-	PUNCT
iajs-2802	154	11	module	module	NOUN
iajs-2802	154	12	by	by	ADP
iajs-2802	154	13	setting	set	VERB
iajs-2802	154	14	m1	m1	PROPN
iajs-2802	154	15	/	/	SYM
iajs-2802	154	16	w1+m2	w1+m2	PROPN
iajs-2802	154	17	/	/	SYM
iajs-2802	154	18	w2	w2	NOUN
iajs-2802	154	19	=(	=(	NOUN
iajs-2802	154	20	w2m1+w1m2)/w1w2	w2m1+w1m2)/w1w2	PROPN
iajs-2802	154	21	and	and	CCONJ
iajs-2802	154	22	(	(	PUNCT
iajs-2802	154	23	r	r	NOUN
iajs-2802	154	24	/	/	SYM
iajs-2802	154	25	w1	w1	NOUN
iajs-2802	154	26	)	)	PUNCT
iajs-2802	154	27	(	(	PUNCT
iajs-2802	154	28	m1	m1	NOUN
iajs-2802	154	29	/	/	SYM
iajs-2802	154	30	w2	w2	NOUN
iajs-2802	154	31	)	)	PUNCT
iajs-2802	154	32	=	=	PUNCT
iajs-2802	154	33	rm1	rm1	PROPN
iajs-2802	154	34	/	/	SYM
iajs-2802	154	35	w1w2	w1w2	NOUN
iajs-2802	154	36	for	for	ADP
iajs-2802	154	37	every	every	DET
iajs-2802	154	38	m1,m2	m1,m2	PROPN
iajs-2802	154	39	∈a	∈a	ADJ
iajs-2802	154	40	and	and	CCONJ
iajs-2802	154	41	every	every	DET
iajs-2802	154	42	r	r	NOUN
iajs-2802	154	43	∈	∈	NOUN
iajs-2802	154	44	s	s	PART
iajs-2802	154	45	,	,	PUNCT
iajs-2802	154	46	w1,w2	w1,w2	PROPN
iajs-2802	154	47	∈	∈	PROPN
iajs-2802	154	48	w.	w.	NOUN
iajs-2802	154	49	if	if	SCONJ
iajs-2802	154	50	w	w	PROPN
iajs-2802	154	51	=	=	NOUN
iajs-2802	154	52	s	s	NOUN
iajs-2802	154	53	-	-	PUNCT
iajs-2802	154	54	e	e	X
iajs-2802	154	55	where	where	SCONJ
iajs-2802	154	56	e	e	NOUN
iajs-2802	154	57	is	be	AUX
iajs-2802	154	58	a	a	DET
iajs-2802	154	59	prime	prime	ADJ
iajs-2802	154	60	ideal	ideal	NOUN
iajs-2802	154	61	,	,	PUNCT
iajs-2802	154	62	we	we	PRON
iajs-2802	154	63	used	use	VERB
iajs-2802	154	64	ae	ae	PROPN
iajs-2802	154	65	instead	instead	ADV
iajs-2802	154	66	of	of	ADP
iajs-2802	154	67	aw	aw	INTJ
iajs-2802	154	68	and	and	CCONJ
iajs-2802	154	69	se	se	X
iajs-2802	154	70	instead	instead	ADV
iajs-2802	154	71	of	of	ADP
iajs-2802	154	72	sw	sw	PROPN
iajs-2802	154	73	.	.	PUNCT
iajs-2802	155	1	if	if	SCONJ
iajs-2802	155	2	a	a	DET
iajs-2802	155	3	ring	ring	NOUN
iajs-2802	155	4	has	have	VERB
iajs-2802	155	5	only	only	ADV
iajs-2802	155	6	one	one	NUM
iajs-2802	155	7	maximal	maximal	ADJ
iajs-2802	155	8	ideal	ideal	NOUN
iajs-2802	155	9	,	,	PUNCT
iajs-2802	155	10	then	then	ADV
iajs-2802	155	11	it	it	PRON
iajs-2802	155	12	is	be	AUX
iajs-2802	155	13	called	call	VERB
iajs-2802	155	14	a	a	DET
iajs-2802	155	15	local	local	ADJ
iajs-2802	155	16	ring	ring	NOUN
iajs-2802	155	17	.	.	PUNCT
iajs-2802	156	1	hence	hence	ADV
iajs-2802	156	2	se	se	PROPN
iajs-2802	156	3	is	be	AUX
iajs-2802	156	4	often	often	ADV
iajs-2802	156	5	called	call	VERB
iajs-2802	156	6	the	the	DET
iajs-2802	156	7	localization	localization	NOUN
iajs-2802	156	8	of	of	ADP
iajs-2802	156	9	s	s	PRON
iajs-2802	156	10	at	at	ADP
iajs-2802	156	11	e	e	NOUN
iajs-2802	156	12	,	,	PUNCT
iajs-2802	156	13	similar	similar	ADJ
iajs-2802	156	14	ae	ae	PROPN
iajs-2802	156	15	is	be	AUX
iajs-2802	156	16	the	the	DET
iajs-2802	156	17	r/1,ɐ	r/1,ɐ	NOUN
iajs-2802	156	18	r	r	NOUN
iajs-2802	156	19	∈	∈	NOUN
iajs-2802	156	20	s	s	PART
iajs-2802	156	21	and	and	CCONJ
iajs-2802	156	22	𝜱	𝜱	NOUN
iajs-2802	156	23	:	:	PUNCT
iajs-2802	157	1	a	a	DET
iajs-2802	157	2	→aw	→aw	X
iajs-2802	157	3	such	such	ADJ
iajs-2802	157	4	that	that	SCONJ
iajs-2802	157	5	𝜱(m)=m/1	𝜱(m)=m/1	NOUN
iajs-2802	157	6	,	,	PUNCT
iajs-2802	157	7	ɐm∈a	ɐm∈a	NOUN
iajs-2802	157	8	.	.	NOUN
iajs-2802	157	9	furthermore	furthermore	ADV
iajs-2802	157	10	,	,	PUNCT
iajs-2802	157	11	if	if	SCONJ
iajs-2802	157	12	b	b	PROPN
iajs-2802	157	13	is	be	AUX
iajs-2802	157	14	a	a	DET
iajs-2802	157	15	submodule	submodule	NOUN
iajs-2802	157	16	of	of	ADP
iajs-2802	157	17	an	an	DET
iajs-2802	157	18	s	s	NOUN
iajs-2802	157	19	-	-	NOUN
iajs-2802	157	20	module	module	NOUN
iajs-2802	157	21	a	a	PRON
iajs-2802	157	22	and	and	CCONJ
iajs-2802	157	23	w	w	NOUN
iajs-2802	157	24	be	be	AUX
iajs-2802	157	25	a	a	DET
iajs-2802	157	26	multiplicatively	multiplicatively	ADV
iajs-2802	157	27	closed	close	VERB
iajs-2802	157	28	in	in	ADP
iajs-2802	157	29	s	s	PROPN
iajs-2802	157	30	,	,	PUNCT
iajs-2802	157	31	then	then	ADV
iajs-2802	157	32	bw	bw	PROPN
iajs-2802	157	33	=	=	PUNCT
iajs-2802	157	34	{	{	PUNCT
iajs-2802	157	35	n	n	CCONJ
iajs-2802	157	36	/	/	SYM
iajs-2802	157	37	w	w	NOUN
iajs-2802	157	38	:	:	PUNCT
iajs-2802	157	39	n	n	PRON
iajs-2802	157	40	∈b	∈b	PROPN
iajs-2802	157	41	,	,	PUNCT
iajs-2802	157	42	w∈w	w∈w	VERB
iajs-2802	157	43	}	}	PUNCT
iajs-2802	157	44	be	be	AUX
iajs-2802	157	45	a	a	DET
iajs-2802	157	46	submodule	submodule	NOUN
iajs-2802	157	47	on	on	ADP
iajs-2802	157	48	swmodule	swmodule	NOUN
iajs-2802	157	49	,	,	PUNCT
iajs-2802	157	50	see	see	VERB
iajs-2802	157	51	[	[	X
iajs-2802	157	52	8	8	NUM
iajs-2802	157	53	]	]	PUNCT
iajs-2802	157	54	.	.	PUNCT
iajs-2802	158	1	in	in	ADP
iajs-2802	158	2	this	this	DET
iajs-2802	158	3	section	section	NOUN
iajs-2802	158	4	we	we	PRON
iajs-2802	158	5	study	study	VERB
iajs-2802	158	6	the	the	DET
iajs-2802	158	7	behavior	behavior	NOUN
iajs-2802	158	8	of	of	ADP
iajs-2802	158	9	an	an	DET
iajs-2802	158	10	sm	sm	NOUN
iajs-2802	158	11	-	-	PUNCT
iajs-2802	158	12	submodule	submodule	NOUN
iajs-2802	158	13	under	under	ADP
iajs-2802	158	14	localization	localization	NOUN
iajs-2802	158	15	and	and	CCONJ
iajs-2802	158	16	several	several	ADJ
iajs-2802	158	17	of	of	ADP
iajs-2802	158	18	results	result	NOUN
iajs-2802	158	19	have	have	AUX
iajs-2802	158	20	been	be	AUX
iajs-2802	158	21	proved	prove	VERB
iajs-2802	158	22	.	.	PUNCT
iajs-2802	159	1	the	the	DET
iajs-2802	159	2	following	follow	VERB
iajs-2802	159	3	lemma	lemma	PROPN
iajs-2802	159	4	is	be	AUX
iajs-2802	159	5	needed	need	VERB
iajs-2802	159	6	in	in	ADP
iajs-2802	159	7	our	our	PRON
iajs-2802	159	8	next	next	ADJ
iajs-2802	159	9	result	result	NOUN
iajs-2802	159	10	.	.	PUNCT
iajs-2802	160	1	lemma	lemma	PROPN
iajs-2802	160	2	(	(	PUNCT
iajs-2802	160	3	4.1	4.1	NUM
iajs-2802	160	4	)	)	PUNCT
iajs-2802	161	1	[	[	X
iajs-2802	161	2	10	10	NUM
iajs-2802	161	3	]	]	PUNCT
iajs-2802	161	4	let	let	VERB
iajs-2802	161	5	a	a	PRON
iajs-2802	161	6	be	be	AUX
iajs-2802	161	7	an	an	DET
iajs-2802	161	8	s	s	NOUN
iajs-2802	161	9	-	-	PUNCT
iajs-2802	161	10	module	module	NOUN
iajs-2802	161	11	and	and	CCONJ
iajs-2802	161	12	b	b	NOUN
iajs-2802	161	13	,	,	PUNCT
iajs-2802	161	14	l	l	NOUN
iajs-2802	161	15	are	be	AUX
iajs-2802	161	16	two	two	NUM
iajs-2802	161	17	submodules	submodule	NOUN
iajs-2802	161	18	of	of	ADP
iajs-2802	161	19	a.	a.	NOUN
iajs-2802	161	20	then	then	ADV
iajs-2802	161	21	,	,	PUNCT
iajs-2802	161	22	b	b	X
iajs-2802	161	23	=	=	NOUN
iajs-2802	161	24	l	l	NOUN
iajs-2802	161	25	if	if	SCONJ
iajs-2802	161	26	and	and	CCONJ
iajs-2802	161	27	only	only	ADV
iajs-2802	161	28	if	if	SCONJ
iajs-2802	161	29	bp	bp	PROPN
iajs-2802	161	30	=	=	NOUN
iajs-2802	161	31	lp	lp	NOUN
iajs-2802	161	32	for	for	ADP
iajs-2802	161	33	every	every	DET
iajs-2802	161	34	maximal	maximal	ADJ
iajs-2802	161	35	ideal	ideal	NOUN
iajs-2802	161	36	p	p	NOUN
iajs-2802	161	37	of	of	ADP
iajs-2802	161	38	s.	s.	PROPN
iajs-2802	161	39	the	the	DET
iajs-2802	161	40	following	follow	VERB
iajs-2802	161	41	proposition	proposition	NOUN
iajs-2802	161	42	study	study	VERB
iajs-2802	161	43	the	the	DET
iajs-2802	161	44	relationship	relationship	NOUN
iajs-2802	161	45	between	between	ADP
iajs-2802	161	46	a	a	DET
iajs-2802	161	47	module	module	NOUN
iajs-2802	161	48	a	a	PRON
iajs-2802	161	49	and	and	CCONJ
iajs-2802	161	50	its	its	PRON
iajs-2802	161	51	locally	locally	ADV
iajs-2802	161	52	and	and	CCONJ
iajs-2802	161	53	prove	prove	VERB
iajs-2802	161	54	that	that	SCONJ
iajs-2802	161	55	they	they	PRON
iajs-2802	161	56	are	be	AUX
iajs-2802	161	57	equivalent	equivalent	ADJ
iajs-2802	161	58	.	.	PUNCT
iajs-2802	162	1	proposition	proposition	NOUN
iajs-2802	162	2	(	(	PUNCT
iajs-2802	162	3	4.2	4.2	NUM
iajs-2802	162	4	):	):	PUNCT
iajs-2802	162	5	let	let	VERB
iajs-2802	162	6	a	a	PRON
iajs-2802	162	7	be	be	AUX
iajs-2802	162	8	an	an	DET
iajs-2802	162	9	s	s	NOUN
iajs-2802	162	10	-	-	PUNCT
iajs-2802	162	11	module	module	NOUN
iajs-2802	162	12	and	and	CCONJ
iajs-2802	162	13	b	b	NOUN
iajs-2802	162	14	is	be	AUX
iajs-2802	162	15	nonzero	nonzero	X
iajs-2802	162	16	proper	proper	ADJ
iajs-2802	162	17	submodule	submodule	NOUN
iajs-2802	162	18	of	of	ADP
iajs-2802	162	19	a.	a.	NOUN
iajs-2802	162	20	then	then	ADV
iajs-2802	162	21	,	,	PUNCT
iajs-2802	162	22	bp	bp	PROPN
iajs-2802	162	23	is	be	AUX
iajs-2802	162	24	smsubmodule	smsubmodule	ADJ
iajs-2802	162	25	of	of	ADP
iajs-2802	162	26	an	an	DET
iajs-2802	162	27	sp	sp	NOUN
iajs-2802	162	28	-	-	PUNCT
iajs-2802	162	29	submodule	submodule	NOUN
iajs-2802	162	30	ap	ap	PROPN
iajs-2802	163	1	if	if	SCONJ
iajs-2802	163	2	and	and	CCONJ
iajs-2802	163	3	only	only	ADV
iajs-2802	163	4	if	if	SCONJ
iajs-2802	163	5	b	b	PROPN
iajs-2802	163	6	is	be	AUX
iajs-2802	163	7	sm	sm	NOUN
iajs-2802	163	8	-	-	PUNCT
iajs-2802	163	9	submodule	submodule	NOUN
iajs-2802	163	10	of	of	ADP
iajs-2802	163	11	an	an	DET
iajs-2802	163	12	s	s	NOUN
iajs-2802	163	13	-	-	PUNCT
iajs-2802	163	14	module	module	NOUN
iajs-2802	163	15	a.	a.	NOUN
iajs-2802	163	16	proof	proof	NOUN
iajs-2802	163	17	:	:	PUNCT
iajs-2802	163	18	ibn	ibn	PROPN
iajs-2802	163	19	al	al	PROPN
iajs-2802	163	20	-	-	PUNCT
iajs-2802	163	21	haitham	haitham	PROPN
iajs-2802	163	22	jour	jour	X
iajs-2802	163	23	.	.	PROPN
iajs-2802	164	1	for	for	ADP
iajs-2802	164	2	pure	pure	ADJ
iajs-2802	164	3	&	&	CCONJ
iajs-2802	164	4	appl	appl	PROPN
iajs-2802	164	5	.	.	PUNCT
iajs-2802	165	1	sci	sci	PROPN
iajs-2802	165	2	.	.	PROPN
iajs-2802	166	1	53	53	NUM
iajs-2802	166	2	(	(	PUNCT
iajs-2802	166	3	1)2022	1)2022	NUM
iajs-2802	166	4	90	90	NUM
iajs-2802	166	5	suppose	suppose	VERB
iajs-2802	166	6	that	that	SCONJ
iajs-2802	166	7	b	b	PROPN
iajs-2802	166	8	is	be	AUX
iajs-2802	166	9	nonzero	nonzero	X
iajs-2802	166	10	proper	proper	ADJ
iajs-2802	166	11	submodule	submodule	NOUN
iajs-2802	166	12	of	of	ADP
iajs-2802	166	13	a.	a.	NOUN
iajs-2802	166	14	we	we	PRON
iajs-2802	166	15	must	must	AUX
iajs-2802	166	16	prove	prove	VERB
iajs-2802	166	17	that	that	SCONJ
iajs-2802	166	18	a	a	X
iajs-2802	166	19	/	/	PRON
iajs-2802	166	20	e2b	e2b	PROPN
iajs-2802	166	21	is	be	AUX
iajs-2802	166	22	a	a	DET
iajs-2802	166	23	regular	regular	ADJ
iajs-2802	166	24	s	s	NOUN
iajs-2802	166	25	-	-	NOUN
iajs-2802	166	26	module	module	NOUN
iajs-2802	166	27	for	for	ADP
iajs-2802	166	28	every	every	DET
iajs-2802	166	29	nonzero	nonzero	NOUN
iajs-2802	166	30	ideal	ideal	ADJ
iajs-2802	166	31	e	e	PROPN
iajs-2802	166	32	of	of	ADP
iajs-2802	166	33	s	s	PROPN
iajs-2802	166	34	;	;	PUNCT
iajs-2802	166	35	that	that	PRON
iajs-2802	166	36	is	is	ADV
iajs-2802	166	37	,	,	PUNCT
iajs-2802	166	38	every	every	DET
iajs-2802	166	39	submodule	submodule	NOUN
iajs-2802	166	40	of	of	ADP
iajs-2802	166	41	a	a	PRON
iajs-2802	166	42	/	/	PRON
iajs-2802	166	43	e2b	e2b	PROPN
iajs-2802	166	44	is	be	AUX
iajs-2802	166	45	pure	pure	ADJ
iajs-2802	166	46	.	.	PUNCT
iajs-2802	167	1	let	let	VERB
iajs-2802	167	2	l	l	NOUN
iajs-2802	167	3	/	/	SYM
iajs-2802	167	4	e2b	e2b	PROPN
iajs-2802	167	5	be	be	AUX
iajs-2802	167	6	a	a	DET
iajs-2802	167	7	submodule	submodule	NOUN
iajs-2802	167	8	of	of	ADP
iajs-2802	167	9	a	a	PRON
iajs-2802	167	10	/	/	SYM
iajs-2802	167	11	e2b	e2b	PROPN
iajs-2802	167	12	.	.	PUNCT
iajs-2802	168	1	it	it	PRON
iajs-2802	168	2	is	be	AUX
iajs-2802	168	3	clear	clear	ADJ
iajs-2802	168	4	that	that	SCONJ
iajs-2802	168	5	i(l	i(l	PROPN
iajs-2802	168	6	/	/	SYM
iajs-2802	168	7	e2b	e2b	PROPN
iajs-2802	168	8	)	)	PUNCT
iajs-2802	168	9	⊇	⊇	PROPN
iajs-2802	168	10	i(a	i(a	PROPN
iajs-2802	168	11	/	/	SYM
iajs-2802	168	12	e2b)⋂(l	e2b)⋂(l	PROPN
iajs-2802	168	13	/	/	SYM
iajs-2802	168	14	e2b	e2b	PROPN
iajs-2802	168	15	)	)	PUNCT
iajs-2802	168	16	where	where	SCONJ
iajs-2802	168	17	i	i	PRON
iajs-2802	168	18	is	be	AUX
iajs-2802	168	19	an	an	DET
iajs-2802	168	20	ideal	ideal	NOUN
iajs-2802	168	21	of	of	ADP
iajs-2802	168	22	s.	s.	PROPN
iajs-2802	168	23	now	now	ADV
iajs-2802	168	24	,	,	PUNCT
iajs-2802	168	25	to	to	PART
iajs-2802	168	26	prove	prove	VERB
iajs-2802	168	27	i(l	i(l	PROPN
iajs-2802	168	28	/	/	SYM
iajs-2802	168	29	e2b	e2b	PROPN
iajs-2802	168	30	)	)	PUNCT
iajs-2802	168	31	⊆	⊆	NUM
iajs-2802	168	32	i(a	i(a	PROPN
iajs-2802	168	33	/	/	SYM
iajs-2802	168	34	e2b)⋂(l	e2b)⋂(l	NOUN
iajs-2802	168	35	/	/	SYM
iajs-2802	168	36	e2b	e2b	PROPN
iajs-2802	168	37	)	)	PUNCT
iajs-2802	168	38	.	.	PUNCT
iajs-2802	169	1	let	let	VERB
iajs-2802	169	2	x	x	PUNCT
iajs-2802	169	3	∈	∈	PROPN
iajs-2802	169	4	i(l	i(l	PROPN
iajs-2802	169	5	/	/	SYM
iajs-2802	169	6	e2b	e2b	PROPN
iajs-2802	169	7	)	)	PUNCT
iajs-2802	169	8	.	.	PUNCT
iajs-2802	170	1	then	then	ADV
iajs-2802	170	2	x	x	X
iajs-2802	170	3	=	=	PUNCT
iajs-2802	170	4	∑	∑	PUNCT
iajs-2802	170	5	an	an	DET
iajs-2802	170	6	i=1	i=1	PROPN
iajs-2802	170	7	i(li+e2b	i(li+e2b	PROPN
iajs-2802	170	8	)	)	PUNCT
iajs-2802	170	9	.	.	PUNCT
iajs-2802	171	1	therefore	therefore	ADV
iajs-2802	171	2	xs	xs	PROPN
iajs-2802	171	3	/	/	SYM
iajs-2802	171	4	s	s	PROPN
iajs-2802	171	5	=	=	PUNCT
iajs-2802	171	6	(	(	PUNCT
iajs-2802	171	7	∑	∑	ADP
iajs-2802	171	8	an	an	DET
iajs-2802	171	9	i=1	i=1	PROPN
iajs-2802	171	10	i(li+e2b))s	i(li+e2b))s	PROPN
iajs-2802	171	11	/	/	SYM
iajs-2802	171	12	s	s	PART
iajs-2802	171	13	∈	∈	PROPN
iajs-2802	171	14	ip(lp	ip(lp	PROPN
iajs-2802	171	15	/	/	SYM
iajs-2802	171	16	ep	ep	PROPN
iajs-2802	171	17	2bp	2bp	NOUN
iajs-2802	171	18	)	)	PUNCT
iajs-2802	171	19	but	but	CCONJ
iajs-2802	171	20	lp	lp	NOUN
iajs-2802	171	21	is	be	AUX
iajs-2802	171	22	sm	sm	NOUN
iajs-2802	171	23	-	-	PUNCT
iajs-2802	171	24	submodule	submodule	NOUN
iajs-2802	171	25	in	in	ADP
iajs-2802	171	26	ap	ap	PROPN
iajs-2802	171	27	,	,	PUNCT
iajs-2802	171	28	then	then	ADV
iajs-2802	171	29	ip(lp	ip(lp	PROPN
iajs-2802	171	30	/	/	SYM
iajs-2802	171	31	ep	ep	PROPN
iajs-2802	171	32	2bp	2bp	NOUN
iajs-2802	171	33	)	)	PUNCT
iajs-2802	172	1	=	=	PRON
iajs-2802	172	2	(	(	PUNCT
iajs-2802	172	3	ip(ap	ip(ap	PROPN
iajs-2802	172	4	/	/	SYM
iajs-2802	172	5	ep	ep	PROPN
iajs-2802	172	6	2bp))⋂(lp	2bp))⋂(lp	NUM
iajs-2802	172	7	/	/	SYM
iajs-2802	172	8	ep	ep	PROPN
iajs-2802	172	9	2bp	2bp	PROPN
iajs-2802	172	10	)	)	PUNCT
iajs-2802	172	11	which	which	PRON
iajs-2802	172	12	implies	imply	VERB
iajs-2802	172	13	xs	xs	PROPN
iajs-2802	172	14	/	/	SYM
iajs-2802	172	15	s	s	PART
iajs-2802	172	16	∈	∈	PROPN
iajs-2802	172	17	ip(ap	ip(ap	PROPN
iajs-2802	172	18	/	/	SYM
iajs-2802	172	19	ep	ep	PROPN
iajs-2802	172	20	2bp)⋂(lp	2bp)⋂(lp	NUM
iajs-2802	172	21	/	/	SYM
iajs-2802	172	22	ep	ep	PROPN
iajs-2802	172	23	2bp	2bp	NOUN
iajs-2802	172	24	)	)	PUNCT
iajs-2802	173	1	=	=	PUNCT
iajs-2802	173	2	(	(	PUNCT
iajs-2802	173	3	(	(	PUNCT
iajs-2802	173	4	ipap+ep	ipap+ep	PROPN
iajs-2802	173	5	2bp)/	2bp)/	NUM
iajs-2802	173	6	ep	ep	PROPN
iajs-2802	173	7	2bp)⋂(lp	2bp)⋂(lp	NUM
iajs-2802	173	8	/	/	SYM
iajs-2802	173	9	ep	ep	PROPN
iajs-2802	173	10	2bp	2bp	NOUN
iajs-2802	173	11	)	)	PUNCT
iajs-2802	173	12	=(	=(	NOUN
iajs-2802	173	13	(	(	PUNCT
iajs-2802	173	14	(	(	PUNCT
iajs-2802	173	15	ia)p+(e2b)p)/(e2b)p)⋂(lp/(e2b)p	ia)p+(e2b)p)/(e2b)p)⋂(lp/(e2b)p	PROPN
iajs-2802	173	16	)	)	PUNCT
iajs-2802	173	17	by	by	ADP
iajs-2802	173	18	[	[	PUNCT
iajs-2802	173	19	10	10	NUM
iajs-2802	173	20	]	]	PUNCT
iajs-2802	173	21	.	.	PUNCT
iajs-2802	174	1	and	and	CCONJ
iajs-2802	174	2	hence	hence	ADV
iajs-2802	174	3	as	as	ADP
iajs-2802	174	4	/	/	SYM
iajs-2802	174	5	s	s	PART
iajs-2802	174	6	∈((ia+e2b)p)/(e2b)p)⋂(lp/(e2b)p)=	∈((ia+e2b)p)/(e2b)p)⋂(lp/(e2b)p)=	NOUN
iajs-2802	174	7	(	(	PUNCT
iajs-2802	174	8	(	(	PUNCT
iajs-2802	174	9	ia+e2b)/(e2b))p⋂(l/(e2b))p	ia+e2b)/(e2b))p⋂(l/(e2b))p	NOUN
iajs-2802	174	10	=	=	SYM
iajs-2802	174	11	(	(	PUNCT
iajs-2802	174	12	(	(	PUNCT
iajs-2802	174	13	(	(	PUNCT
iajs-2802	174	14	ia+e2b)/(e2b))⋂(l/(e2b)))p	ia+e2b)/(e2b))⋂(l/(e2b)))p	NOUN
iajs-2802	174	15	.	.	PUNCT
iajs-2802	175	1	therefore	therefore	ADV
iajs-2802	175	2	x	x	X
iajs-2802	175	3	∈	∈	PROPN
iajs-2802	175	4	(	(	PUNCT
iajs-2802	175	5	(	(	PUNCT
iajs-2802	175	6	ia+e2b)/(e2b))⋂(l/(e2b	ia+e2b)/(e2b))⋂(l/(e2b	PROPN
iajs-2802	175	7	)	)	PUNCT
iajs-2802	175	8	)	)	PUNCT
iajs-2802	175	9	which	which	PRON
iajs-2802	175	10	implies	imply	VERB
iajs-2802	175	11	x	x	X
iajs-2802	175	12	∈	∈	PROPN
iajs-2802	175	13	(	(	PUNCT
iajs-2802	175	14	i(a/(e2b))⋂(l/(e2b	i(a/(e2b))⋂(l/(e2b	NOUN
iajs-2802	175	15	)	)	PUNCT
iajs-2802	175	16	)	)	PUNCT
iajs-2802	175	17	and	and	CCONJ
iajs-2802	175	18	hence	hence	ADV
iajs-2802	175	19	i(l	i(l	PROPN
iajs-2802	175	20	/	/	SYM
iajs-2802	175	21	e2b	e2b	PROPN
iajs-2802	175	22	)	)	PUNCT
iajs-2802	175	23	⊆	⊆	NUM
iajs-2802	175	24	(	(	PUNCT
iajs-2802	175	25	i(a/(e2b	i(a/(e2b	PROPN
iajs-2802	175	26	)	)	PUNCT
iajs-2802	175	27	)	)	PUNCT
iajs-2802	175	28	⋂	⋂	PROPN
iajs-2802	175	29	(	(	PUNCT
iajs-2802	175	30	l/(e2b	l/(e2b	PROPN
iajs-2802	175	31	)	)	PUNCT
iajs-2802	175	32	)	)	PUNCT
iajs-2802	175	33	.	.	PUNCT
iajs-2802	176	1	therefore	therefore	ADV
iajs-2802	176	2	i(l	i(l	PROPN
iajs-2802	176	3	/	/	SYM
iajs-2802	176	4	e2b	e2b	PROPN
iajs-2802	176	5	)	)	PUNCT
iajs-2802	176	6	=	=	PUNCT
iajs-2802	176	7	(	(	PUNCT
iajs-2802	176	8	i(a/(e2b))⋂(l/(e2b	i(a/(e2b))⋂(l/(e2b	NOUN
iajs-2802	176	9	)	)	PUNCT
iajs-2802	176	10	)	)	PUNCT
iajs-2802	176	11	.	.	PUNCT
iajs-2802	177	1	thus	thus	ADV
iajs-2802	177	2	l	l	X
iajs-2802	177	3	/	/	SYM
iajs-2802	177	4	e2b	e2b	PROPN
iajs-2802	177	5	is	be	AUX
iajs-2802	177	6	pure	pure	ADJ
iajs-2802	177	7	submodule	submodule	NOUN
iajs-2802	177	8	of	of	ADP
iajs-2802	177	9	a	a	PRON
iajs-2802	177	10	/	/	SYM
iajs-2802	177	11	e2b	e2b	PROPN
iajs-2802	177	12	.	.	PUNCT
iajs-2802	178	1	this	this	PRON
iajs-2802	178	2	proves	prove	VERB
iajs-2802	178	3	that	that	SCONJ
iajs-2802	178	4	a	a	X
iajs-2802	178	5	/	/	PRON
iajs-2802	178	6	e2b	e2b	PROPN
iajs-2802	178	7	is	be	AUX
iajs-2802	178	8	regular	regular	ADJ
iajs-2802	178	9	and	and	CCONJ
iajs-2802	178	10	finally	finally	ADV
iajs-2802	178	11	b	b	PROPN
iajs-2802	178	12	is	be	AUX
iajs-2802	178	13	an	an	DET
iajs-2802	178	14	sm	sm	NOUN
iajs-2802	178	15	-	-	PUNCT
iajs-2802	178	16	submodule	submodule	NOUN
iajs-2802	178	17	of	of	ADP
iajs-2802	178	18	a.	a.	NOUN
iajs-2802	178	19	conversely	conversely	ADV
iajs-2802	178	20	:	:	PUNCT
iajs-2802	178	21	suppose	suppose	VERB
iajs-2802	178	22	that	that	SCONJ
iajs-2802	178	23	b	b	PROPN
iajs-2802	178	24	is	be	AUX
iajs-2802	178	25	an	an	DET
iajs-2802	178	26	sm	sm	NOUN
iajs-2802	178	27	-	-	PUNCT
iajs-2802	178	28	submodule	submodule	NOUN
iajs-2802	178	29	of	of	ADP
iajs-2802	178	30	a.	a.	NOUN
iajs-2802	178	31	to	to	PART
iajs-2802	178	32	prove	prove	VERB
iajs-2802	178	33	bp	bp	PROPN
iajs-2802	178	34	is	be	AUX
iajs-2802	178	35	sm	sm	NOUN
iajs-2802	178	36	-	-	PUNCT
iajs-2802	178	37	submodule	submodule	NOUN
iajs-2802	178	38	of	of	ADP
iajs-2802	178	39	an	an	DET
iajs-2802	178	40	sp	sp	NOUN
iajs-2802	178	41	-	-	PUNCT
iajs-2802	178	42	module	module	NOUN
iajs-2802	178	43	ap	ap	NOUN
iajs-2802	178	44	we	we	PRON
iajs-2802	178	45	must	must	AUX
iajs-2802	178	46	show	show	VERB
iajs-2802	178	47	that	that	SCONJ
iajs-2802	178	48	ap	ap	PROPN
iajs-2802	178	49	/	/	SYM
iajs-2802	178	50	ep	ep	PROPN
iajs-2802	178	51	2bp	2bp	PROPN
iajs-2802	178	52	is	be	AUX
iajs-2802	178	53	regular	regular	ADJ
iajs-2802	178	54	sp	sp	NOUN
iajs-2802	178	55	-	-	PUNCT
iajs-2802	178	56	module	module	NOUN
iajs-2802	178	57	.	.	PUNCT
iajs-2802	179	1	it	it	PRON
iajs-2802	179	2	is	be	AUX
iajs-2802	179	3	clear	clear	ADJ
iajs-2802	179	4	that	that	SCONJ
iajs-2802	179	5	(	(	PUNCT
iajs-2802	179	6	ip(ap	ip(ap	NOUN
iajs-2802	179	7	/	/	SYM
iajs-2802	179	8	ep	ep	PROPN
iajs-2802	179	9	2bp))⋂(lp	2bp))⋂(lp	NUM
iajs-2802	179	10	/	/	SYM
iajs-2802	179	11	ep	ep	PROPN
iajs-2802	179	12	2bp	2bp	NOUN
iajs-2802	179	13	)	)	PUNCT
iajs-2802	179	14	⊆	⊆	NUM
iajs-2802	179	15	ip(lp	ip(lp	PROPN
iajs-2802	179	16	/	/	SYM
iajs-2802	179	17	ep	ep	PROPN
iajs-2802	179	18	2bp	2bp	NOUN
iajs-2802	179	19	)	)	PUNCT
iajs-2802	179	20	.	.	PUNCT
iajs-2802	180	1	to	to	PART
iajs-2802	180	2	prove	prove	VERB
iajs-2802	180	3	ip(lp	ip(lp	PROPN
iajs-2802	180	4	/	/	SYM
iajs-2802	180	5	ep	ep	PROPN
iajs-2802	180	6	2bp	2bp	NOUN
iajs-2802	180	7	)	)	PUNCT
iajs-2802	181	1	⊆	⊆	NUM
iajs-2802	181	2	(	(	PUNCT
iajs-2802	181	3	ip(ap	ip(ap	PROPN
iajs-2802	181	4	/	/	SYM
iajs-2802	181	5	ep	ep	PROPN
iajs-2802	181	6	2bp))⋂(lp	2bp))⋂(lp	NUM
iajs-2802	181	7	/	/	SYM
iajs-2802	181	8	ep	ep	PROPN
iajs-2802	181	9	2bp	2bp	NOUN
iajs-2802	181	10	)	)	PUNCT
iajs-2802	181	11	.	.	PUNCT
iajs-2802	182	1	let	let	VERB
iajs-2802	182	2	x/1	x/1	PROPN
iajs-2802	182	3	∈	∈	PROPN
iajs-2802	182	4	i	i	PRON
iajs-2802	182	5	and	and	CCONJ
iajs-2802	182	6	a	a	PRON
iajs-2802	182	7	/	/	SYM
iajs-2802	182	8	s	s	NOUN
iajs-2802	182	9	∈	∈	PROPN
iajs-2802	182	10	ip(lp	ip(lp	PROPN
iajs-2802	182	11	/	/	SYM
iajs-2802	182	12	ep	ep	PROPN
iajs-2802	182	13	2bp	2bp	NOUN
iajs-2802	182	14	)	)	PUNCT
iajs-2802	182	15	(	(	PUNCT
iajs-2802	182	16	xa	xa	PROPN
iajs-2802	182	17	/	/	SYM
iajs-2802	182	18	s	s	PART
iajs-2802	182	19	+	+	NUM
iajs-2802	182	20	ep	ep	PROPN
iajs-2802	182	21	2bp	2bp	NOUN
iajs-2802	182	22	)	)	PUNCT
iajs-2802	183	1	=	=	PUNCT
iajs-2802	183	2	∑	∑	PUNCT
iajs-2802	183	3	(	(	PUNCT
iajs-2802	183	4	bn	bn	INTJ
iajs-2802	183	5	i=0	i=0	PROPN
iajs-2802	183	6	i	i	NOUN
iajs-2802	183	7	/	/	SYM
iajs-2802	183	8	si)(li	si)(li	PROPN
iajs-2802	183	9	/	/	SYM
iajs-2802	183	10	ti+ep	ti+ep	PROPN
iajs-2802	183	11	2bp	2bp	NOUN
iajs-2802	183	12	2	2	NUM
iajs-2802	183	13	)	)	PUNCT
iajs-2802	183	14	where	where	SCONJ
iajs-2802	183	15	si	si	NOUN
iajs-2802	183	16	,	,	PUNCT
iajs-2802	183	17	ti	ti	PROPN
iajs-2802	183	18	∉p	∉p	PROPN
iajs-2802	183	19	and	and	CCONJ
iajs-2802	183	20	bi	bi	ADJ
iajs-2802	183	21	∈i	∈i	NOUN
iajs-2802	183	22	,	,	PUNCT
iajs-2802	183	23	li	li	PROPN
iajs-2802	183	24	∈l	∈l	PROPN
iajs-2802	183	25	.	.	PUNCT
iajs-2802	184	1	put	put	VERB
iajs-2802	184	2	ci	ci	NOUN
iajs-2802	184	3	=	=	NOUN
iajs-2802	184	4	siti	siti	NOUN
iajs-2802	184	5	.	.	PUNCT
iajs-2802	185	1	therefore	therefore	ADV
iajs-2802	185	2	(	(	PUNCT
iajs-2802	185	3	xa	xa	PROPN
iajs-2802	185	4	/	/	SYM
iajs-2802	185	5	s	s	PART
iajs-2802	185	6	+	+	NUM
iajs-2802	185	7	ep	ep	PROPN
iajs-2802	185	8	2bp	2bp	NOUN
iajs-2802	185	9	)	)	PUNCT
iajs-2802	186	1	=	=	PUNCT
iajs-2802	186	2	(	(	PUNCT
iajs-2802	186	3	(	(	PUNCT
iajs-2802	186	4	b1l1v1+b2l2v2+	b1l1v1+b2l2v2+	X
iajs-2802	186	5	…	…	PUNCT
iajs-2802	186	6	+bnlnvn)/u)+	+bnlnvn)/u)+	ADJ
iajs-2802	186	7	ep	ep	ADJ
iajs-2802	186	8	2bp	2bp	NOUN
iajs-2802	186	9	2	2	NUM
iajs-2802	186	10	where	where	SCONJ
iajs-2802	186	11	u	u	NOUN
iajs-2802	186	12	=	=	X
iajs-2802	186	13	c1c2c3….cn	c1c2c3….cn	PROPN
iajs-2802	186	14	and	and	CCONJ
iajs-2802	186	15	v1	v1	PROPN
iajs-2802	186	16	=	=	SYM
iajs-2802	186	17	c2c3….cn	c2c3….cn	X
iajs-2802	186	18	,	,	PUNCT
iajs-2802	186	19	v2	v2	NOUN
iajs-2802	186	20	=	=	SYM
iajs-2802	186	21	c1c3c4….cn	c1c3c4….cn	PROPN
iajs-2802	186	22	,	,	PUNCT
iajs-2802	186	23	vn	vn	NOUN
iajs-2802	186	24	=	=	SYM
iajs-2802	186	25	c1c2c3	c1c2c3	PROPN
iajs-2802	186	26	…	…	SYM
iajs-2802	186	27	.cn-1	.cn-1	PROPN
iajs-2802	186	28	.	.	PUNCT
iajs-2802	187	1	thus	thus	ADV
iajs-2802	187	2	there	there	PRON
iajs-2802	187	3	exist	exist	VERB
iajs-2802	187	4	k∉	k∉	PROPN
iajs-2802	187	5	p	p	NOUN
iajs-2802	187	6	such	such	ADJ
iajs-2802	187	7	that	that	SCONJ
iajs-2802	187	8	kxau+e2b	kxau+e2b	PROPN
iajs-2802	187	9	=	=	SYM
iajs-2802	187	10	k(b1l1v1+b2l2v2+	k(b1l1v1+b2l2v2+	PROPN
iajs-2802	187	11	…	…	PUNCT
iajs-2802	187	12	+bnlnvn)∈	+bnlnvn)∈	ADP
iajs-2802	187	13	i(l	i(l	PROPN
iajs-2802	187	14	/	/	SYM
iajs-2802	187	15	e2b	e2b	PROPN
iajs-2802	187	16	)	)	PUNCT
iajs-2802	187	17	but	but	CCONJ
iajs-2802	187	18	l	l	NOUN
iajs-2802	187	19	is	be	AUX
iajs-2802	187	20	pure	pure	ADJ
iajs-2802	187	21	submodule	submodule	NOUN
iajs-2802	187	22	in	in	ADP
iajs-2802	187	23	a	a	PRON
iajs-2802	187	24	,	,	PUNCT
iajs-2802	187	25	that	that	PRON
iajs-2802	187	26	is	be	AUX
iajs-2802	187	27	i(l	i(l	PROPN
iajs-2802	187	28	/	/	SYM
iajs-2802	187	29	e2b	e2b	PROPN
iajs-2802	187	30	)	)	PUNCT
iajs-2802	187	31	=	=	SYM
iajs-2802	187	32	i(a/	i(a/	NOUN
iajs-2802	187	33	e2b)⋂	e2b)⋂	PROPN
iajs-2802	187	34	(	(	PUNCT
iajs-2802	187	35	l/	l/	X
iajs-2802	187	36	e2b	e2b	NOUN
iajs-2802	187	37	)	)	PUNCT
iajs-2802	187	38	(	(	PUNCT
iajs-2802	187	39	since	since	SCONJ
iajs-2802	187	40	a/	a/	NOUN
iajs-2802	187	41	e2b	e2b	PROPN
iajs-2802	187	42	is	be	AUX
iajs-2802	187	43	regular	regular	ADJ
iajs-2802	187	44	s	s	NOUN
iajs-2802	187	45	-	-	NOUN
iajs-2802	187	46	module	module	NOUN
iajs-2802	187	47	)	)	PUNCT
iajs-2802	187	48	and	and	CCONJ
iajs-2802	187	49	hence	hence	ADV
iajs-2802	187	50	by	by	ADP
iajs-2802	187	51	(	(	PUNCT
iajs-2802	187	52	13,theorem	13,theorem	NUM
iajs-2802	187	53	(	(	PUNCT
iajs-2802	187	54	2.6	2.6	NUM
iajs-2802	187	55	)	)	PUNCT
iajs-2802	187	56	]	]	PUNCT
iajs-2802	188	1	we	we	PRON
iajs-2802	188	2	have	have	VERB
iajs-2802	188	3	(	(	PUNCT
iajs-2802	188	4	xa+	xa+	PROPN
iajs-2802	188	5	e2b	e2b	PROPN
iajs-2802	188	6	)	)	PUNCT
iajs-2802	188	7	∈	∈	PROPN
iajs-2802	188	8	i(a/	i(a/	VERB
iajs-2802	188	9	e2b	e2b	PROPN
iajs-2802	188	10	⋂	⋂	PROPN
iajs-2802	188	11	l/	l/	NOUN
iajs-2802	188	12	e2b	e2b	NOUN
iajs-2802	188	13	)	)	PUNCT
iajs-2802	188	14	.	.	PUNCT
iajs-2802	189	1	this	this	PRON
iajs-2802	189	2	leads	lead	VERB
iajs-2802	189	3	us	we	PRON
iajs-2802	189	4	to	to	PART
iajs-2802	189	5	write	write	VERB
iajs-2802	189	6	(	(	PUNCT
iajs-2802	189	7	xa	xa	PROPN
iajs-2802	189	8	/	/	PROPN
iajs-2802	189	9	s+ep	s+ep	PROPN
iajs-2802	189	10	2bp)∈(ip(ap	2bp)∈(ip(ap	NUM
iajs-2802	189	11	/	/	SYM
iajs-2802	189	12	ep	ep	PROPN
iajs-2802	189	13	2bp)⋂(lp	2bp)⋂(lp	NUM
iajs-2802	189	14	/	/	SYM
iajs-2802	189	15	ep	ep	PROPN
iajs-2802	189	16	2bp	2bp	NOUN
iajs-2802	189	17	)	)	PUNCT
iajs-2802	189	18	.	.	PUNCT
iajs-2802	190	1	this	this	PRON
iajs-2802	190	2	gives	give	VERB
iajs-2802	190	3	ip(lp	ip(lp	PROPN
iajs-2802	190	4	/	/	SYM
iajs-2802	190	5	ep	ep	PROPN
iajs-2802	190	6	2bp	2bp	NOUN
iajs-2802	190	7	)	)	PUNCT
iajs-2802	191	1	⊆	⊆	NUM
iajs-2802	191	2	(	(	PUNCT
iajs-2802	191	3	ip(ap	ip(ap	PROPN
iajs-2802	191	4	/	/	SYM
iajs-2802	191	5	ep	ep	PROPN
iajs-2802	191	6	2bp))⋂(lp	2bp))⋂(lp	NUM
iajs-2802	191	7	/	/	SYM
iajs-2802	191	8	ep	ep	PROPN
iajs-2802	191	9	2bp	2bp	NOUN
iajs-2802	191	10	)	)	PUNCT
iajs-2802	191	11	and	and	CCONJ
iajs-2802	191	12	ap	ap	PROPN
iajs-2802	191	13	/	/	SYM
iajs-2802	191	14	ep	ep	PROPN
iajs-2802	191	15	2bp	2bp	PROPN
iajs-2802	191	16	is	be	AUX
iajs-2802	191	17	a	a	DET
iajs-2802	191	18	regular	regular	ADJ
iajs-2802	191	19	sp	sp	NOUN
iajs-2802	191	20	-	-	PUNCT
iajs-2802	191	21	module	module	NOUN
iajs-2802	191	22	and	and	CCONJ
iajs-2802	191	23	finally	finally	ADV
iajs-2802	191	24	,	,	PUNCT
iajs-2802	191	25	we	we	PRON
iajs-2802	191	26	obtain	obtain	VERB
iajs-2802	191	27	that	that	PRON
iajs-2802	191	28	bp	bp	PROPN
iajs-2802	191	29	is	be	AUX
iajs-2802	191	30	an	an	DET
iajs-2802	191	31	sm	sm	NOUN
iajs-2802	191	32	-	-	PUNCT
iajs-2802	191	33	submodule	submodule	NOUN
iajs-2802	191	34	of	of	ADP
iajs-2802	191	35	ap	ap	PROPN
iajs-2802	191	36	.	.	PUNCT
iajs-2802	192	1	proposition	proposition	NOUN
iajs-2802	192	2	(	(	PUNCT
iajs-2802	192	3	4.3	4.3	NUM
iajs-2802	192	4	):	):	PUNCT
iajs-2802	192	5	let	let	VERB
iajs-2802	192	6	l	l	NOUN
iajs-2802	192	7	,	,	PUNCT
iajs-2802	192	8	b	b	X
iajs-2802	192	9	be	be	AUX
iajs-2802	192	10	two	two	NUM
iajs-2802	192	11	finitely	finitely	ADV
iajs-2802	192	12	generated	generate	VERB
iajs-2802	192	13	submodules	submodule	NOUN
iajs-2802	192	14	of	of	ADP
iajs-2802	192	15	an	an	DET
iajs-2802	192	16	s	s	NOUN
iajs-2802	192	17	-	-	PUNCT
iajs-2802	192	18	module	module	NOUN
iajs-2802	192	19	a.	a.	NOUN
iajs-2802	192	20	if	if	SCONJ
iajs-2802	192	21	lp	lp	PROPN
iajs-2802	192	22	,	,	PUNCT
iajs-2802	192	23	bp	bp	PROPN
iajs-2802	192	24	are	be	AUX
iajs-2802	192	25	smsubmodules	smsubmodule	NOUN
iajs-2802	192	26	of	of	ADP
iajs-2802	192	27	ap	ap	PROPN
iajs-2802	192	28	,	,	PUNCT
iajs-2802	192	29	then	then	ADV
iajs-2802	192	30	l⋂b	l⋂b	PROPN
iajs-2802	192	31	is	be	AUX
iajs-2802	192	32	an	an	DET
iajs-2802	192	33	sm	sm	NOUN
iajs-2802	192	34	-	-	PUNCT
iajs-2802	192	35	submodule	submodule	NOUN
iajs-2802	192	36	of	of	ADP
iajs-2802	192	37	a.	a.	NOUN
iajs-2802	192	38	proof	proof	NOUN
iajs-2802	192	39	:	:	PUNCT
iajs-2802	192	40	since	since	SCONJ
iajs-2802	192	41	l	l	NOUN
iajs-2802	192	42	,	,	PUNCT
iajs-2802	192	43	b	b	PROPN
iajs-2802	192	44	are	be	AUX
iajs-2802	192	45	two	two	NUM
iajs-2802	192	46	finitely	finitely	ADV
iajs-2802	192	47	generated	generate	VERB
iajs-2802	192	48	submodules	submodule	NOUN
iajs-2802	192	49	of	of	ADP
iajs-2802	192	50	a	a	PRON
iajs-2802	192	51	,	,	PUNCT
iajs-2802	192	52	then	then	ADV
iajs-2802	192	53	by	by	ADP
iajs-2802	192	54	{	{	PUNCT
iajs-2802	192	55	10,p24	10,p24	NUM
iajs-2802	192	56	}	}	PUNCT
iajs-2802	192	57	,	,	PUNCT
iajs-2802	192	58	[	[	X
iajs-2802	192	59	lp	lp	NOUN
iajs-2802	192	60	:	:	PUNCT
iajs-2802	192	61	bp	bp	X
iajs-2802	192	62	]	]	X
iajs-2802	193	1	+	+	CCONJ
iajs-2802	193	2	[	[	X
iajs-2802	193	3	bp	bp	X
iajs-2802	193	4	:	:	PUNCT
iajs-2802	193	5	lp	lp	NOUN
iajs-2802	193	6	]	]	X
iajs-2802	193	7	=	=	SYM
iajs-2802	193	8	sp	sp	NOUN
iajs-2802	193	9	for	for	ADP
iajs-2802	193	10	every	every	DET
iajs-2802	193	11	maximal	maximal	ADJ
iajs-2802	193	12	ideals	ideal	NOUN
iajs-2802	193	13	p	p	NOUN
iajs-2802	193	14	of	of	ADP
iajs-2802	193	15	s.	s.	PROPN
iajs-2802	193	16	thus	thus	ADV
iajs-2802	193	17	,	,	PUNCT
iajs-2802	193	18	lp⋂bp	lp⋂bp	PROPN
iajs-2802	193	19	=	=	PUNCT
iajs-2802	193	20	lp	lp	PROPN
iajs-2802	193	21	or	or	CCONJ
iajs-2802	193	22	lp⋂bp	lp⋂bp	PROPN
iajs-2802	193	23	=	=	SYM
iajs-2802	193	24	bp	bp	PROPN
iajs-2802	193	25	,	,	PUNCT
iajs-2802	193	26	but	but	CCONJ
iajs-2802	193	27	lp	lp	NOUN
iajs-2802	193	28	and	and	CCONJ
iajs-2802	193	29	bp	bp	PROPN
iajs-2802	193	30	are	be	AUX
iajs-2802	193	31	sm	sm	NOUN
iajs-2802	193	32	-	-	PUNCT
iajs-2802	193	33	submodules	submodules	NOUN
iajs-2802	193	34	,	,	PUNCT
iajs-2802	193	35	then	then	ADV
iajs-2802	193	36	lp⋂bp	lp⋂bp	PROPN
iajs-2802	193	37	is	be	AUX
iajs-2802	193	38	an	an	DET
iajs-2802	193	39	sm	sm	NOUN
iajs-2802	193	40	-	-	PUNCT
iajs-2802	193	41	submodule	submodule	NOUN
iajs-2802	193	42	,	,	PUNCT
iajs-2802	193	43	and	and	CCONJ
iajs-2802	193	44	we	we	PRON
iajs-2802	193	45	have	have	VERB
iajs-2802	193	46	lp⋂bp	lp⋂bp	PROPN
iajs-2802	193	47	=	=	SYM
iajs-2802	193	48	(	(	PUNCT
iajs-2802	193	49	l⋂b)p	l⋂b)p	PROPN
iajs-2802	193	50	.	.	PUNCT
iajs-2802	194	1	therefore	therefore	ADV
iajs-2802	194	2	(	(	PUNCT
iajs-2802	194	3	l⋂b)p	l⋂b)p	NOUN
iajs-2802	194	4	is	be	AUX
iajs-2802	194	5	an	an	DET
iajs-2802	194	6	sm	sm	NOUN
iajs-2802	194	7	-	-	PUNCT
iajs-2802	194	8	submodule	submodule	NOUN
iajs-2802	194	9	and	and	CCONJ
iajs-2802	194	10	by	by	ADP
iajs-2802	194	11	proposition(4.2	proposition(4.2	NOUN
iajs-2802	194	12	)	)	PUNCT
iajs-2802	194	13	,	,	PUNCT
iajs-2802	194	14	l⋂b	l⋂b	PROPN
iajs-2802	194	15	is	be	AUX
iajs-2802	194	16	an	an	DET
iajs-2802	194	17	sm	sm	NOUN
iajs-2802	194	18	-	-	PUNCT
iajs-2802	194	19	submodule	submodule	NOUN
iajs-2802	194	20	of	of	ADP
iajs-2802	194	21	a.	a.	NOUN
iajs-2802	194	22	proposition	proposition	NOUN
iajs-2802	194	23	(	(	PUNCT
iajs-2802	194	24	4.4	4.4	NUM
iajs-2802	194	25	):	):	PUNCT
iajs-2802	194	26	let	let	VERB
iajs-2802	194	27	l	l	NOUN
iajs-2802	194	28	,	,	PUNCT
iajs-2802	194	29	b	b	X
iajs-2802	194	30	be	be	AUX
iajs-2802	194	31	two	two	NUM
iajs-2802	194	32	finitely	finitely	ADV
iajs-2802	194	33	generated	generate	VERB
iajs-2802	194	34	submodules	submodule	NOUN
iajs-2802	194	35	of	of	ADP
iajs-2802	194	36	an	an	DET
iajs-2802	194	37	s	s	NOUN
iajs-2802	194	38	-	-	PUNCT
iajs-2802	194	39	module	module	NOUN
iajs-2802	194	40	a.	a.	NOUN
iajs-2802	194	41	then	then	ADV
iajs-2802	194	42	,	,	PUNCT
iajs-2802	194	43	l+b	l+b	NUM
iajs-2802	194	44	is	be	AUX
iajs-2802	194	45	an	an	DET
iajs-2802	194	46	smsubmodules	smsubmodule	NOUN
iajs-2802	194	47	of	of	ADP
iajs-2802	194	48	a	a	PRON
iajs-2802	194	49	,	,	PUNCT
iajs-2802	194	50	if	if	SCONJ
iajs-2802	194	51	lp	lp	PROPN
iajs-2802	194	52	,	,	PUNCT
iajs-2802	194	53	bp	bp	PROPN
iajs-2802	194	54	are	be	AUX
iajs-2802	194	55	sm	sm	NOUN
iajs-2802	194	56	-	-	PUNCT
iajs-2802	194	57	submodules	submodule	NOUN
iajs-2802	194	58	of	of	ADP
iajs-2802	194	59	an	an	DET
iajs-2802	194	60	sp	sp	NOUN
iajs-2802	194	61	-	-	PUNCT
iajs-2802	194	62	module	module	NOUN
iajs-2802	194	63	ap	ap	PROPN
iajs-2802	194	64	.	.	PUNCT
iajs-2802	195	1	ibn	ibn	PROPN
iajs-2802	195	2	al	al	PROPN
iajs-2802	195	3	-	-	PUNCT
iajs-2802	195	4	haitham	haitham	PROPN
iajs-2802	195	5	jour	jour	X
iajs-2802	195	6	.	.	PROPN
iajs-2802	195	7	for	for	ADP
iajs-2802	195	8	pure	pure	ADJ
iajs-2802	195	9	&	&	CCONJ
iajs-2802	195	10	appl	appl	PROPN
iajs-2802	195	11	.	.	PUNCT
iajs-2802	196	1	sci	sci	PROPN
iajs-2802	196	2	.	.	PROPN
iajs-2802	197	1	53	53	NUM
iajs-2802	197	2	(	(	PUNCT
iajs-2802	197	3	1)2022	1)2022	NOUN
iajs-2802	197	4	91	91	NUM
iajs-2802	197	5	proof	proof	NOUN
iajs-2802	197	6	:	:	PUNCT
iajs-2802	197	7	let	let	VERB
iajs-2802	197	8	l	l	NOUN
iajs-2802	197	9	,	,	PUNCT
iajs-2802	197	10	b	b	X
iajs-2802	197	11	be	be	AUX
iajs-2802	197	12	two	two	NUM
iajs-2802	197	13	finitely	finitely	ADV
iajs-2802	197	14	generated	generate	VERB
iajs-2802	197	15	submodules	submodule	NOUN
iajs-2802	197	16	of	of	ADP
iajs-2802	197	17	a.	a.	NOUN
iajs-2802	197	18	then	then	ADV
iajs-2802	197	19	by	by	ADP
iajs-2802	197	20	{	{	PUNCT
iajs-2802	197	21	10	10	NUM
iajs-2802	197	22	,	,	PUNCT
iajs-2802	197	23	p24	p24	NOUN
iajs-2802	197	24	}	}	PUNCT
iajs-2802	197	25	,	,	PUNCT
iajs-2802	197	26	we	we	PRON
iajs-2802	197	27	have	have	VERB
iajs-2802	197	28	[	[	X
iajs-2802	197	29	lp	lp	NOUN
iajs-2802	197	30	:	:	PUNCT
iajs-2802	197	31	bp]+[bp	bp]+[bp	NOUN
iajs-2802	197	32	:	:	PUNCT
iajs-2802	197	33	lp	lp	NOUN
iajs-2802	197	34	]	]	X
iajs-2802	197	35	=	=	SYM
iajs-2802	197	36	sp	sp	NOUN
iajs-2802	197	37	for	for	ADP
iajs-2802	197	38	every	every	DET
iajs-2802	197	39	maximal	maximal	ADJ
iajs-2802	197	40	ideal	ideal	NOUN
iajs-2802	197	41	p	p	PROPN
iajs-2802	197	42	of	of	ADP
iajs-2802	197	43	s.	s.	PROPN
iajs-2802	197	44	let	let	VERB
iajs-2802	197	45	y1	y1	PROPN
iajs-2802	197	46	∈	∈	PROPN
iajs-2802	197	47	[	[	X
iajs-2802	197	48	lp	lp	NOUN
iajs-2802	197	49	:	:	PUNCT
iajs-2802	197	50	bp	bp	X
iajs-2802	197	51	]	]	PUNCT
iajs-2802	197	52	and	and	CCONJ
iajs-2802	197	53	y2	y2	PROPN
iajs-2802	197	54	∈	∈	PROPN
iajs-2802	198	1	[	[	X
iajs-2802	198	2	bp	bp	X
iajs-2802	198	3	:	:	PUNCT
iajs-2802	198	4	lp	lp	NOUN
iajs-2802	198	5	]	]	PUNCT
iajs-2802	198	6	such	such	ADJ
iajs-2802	198	7	that	that	SCONJ
iajs-2802	198	8	y1+y2	y1+y2	PROPN
iajs-2802	198	9	=	=	SYM
iajs-2802	198	10	1	1	NUM
iajs-2802	198	11	=	=	NOUN
iajs-2802	198	12	unity	unity	NOUN
iajs-2802	198	13	of	of	ADP
iajs-2802	198	14	sp	sp	NOUN
iajs-2802	198	15	.	.	PUNCT
iajs-2802	199	1	then	then	ADV
iajs-2802	199	2	,	,	PUNCT
iajs-2802	199	3	either	either	CCONJ
iajs-2802	199	4	y1	y1	NOUN
iajs-2802	199	5	is	be	AUX
iajs-2802	199	6	a	a	DET
iajs-2802	199	7	unit	unit	NOUN
iajs-2802	199	8	element	element	NOUN
iajs-2802	199	9	or	or	CCONJ
iajs-2802	199	10	y2	y2	PROPN
iajs-2802	199	11	is	be	AUX
iajs-2802	199	12	a	a	DET
iajs-2802	199	13	unit	unit	NOUN
iajs-2802	199	14	element	element	NOUN
iajs-2802	199	15	(	(	PUNCT
iajs-2802	199	16	since	since	SCONJ
iajs-2802	199	17	sp	sp	ADP
iajs-2802	199	18	is	be	AUX
iajs-2802	199	19	local	local	ADJ
iajs-2802	199	20	ring	ring	NOUN
iajs-2802	199	21	)	)	PUNCT
iajs-2802	199	22	.	.	PUNCT
iajs-2802	200	1	therefore	therefore	ADV
iajs-2802	200	2	[	[	X
iajs-2802	200	3	lp	lp	NOUN
iajs-2802	200	4	:	:	PUNCT
iajs-2802	200	5	bp	bp	X
iajs-2802	200	6	]	]	X
iajs-2802	200	7	=	=	PUNCT
iajs-2802	200	8	sp	sp	ADP
iajs-2802	200	9	or	or	CCONJ
iajs-2802	200	10	[	[	X
iajs-2802	200	11	bp	bp	X
iajs-2802	200	12	:	:	PUNCT
iajs-2802	200	13	lp	lp	NOUN
iajs-2802	200	14	]	]	X
iajs-2802	200	15	=	=	SYM
iajs-2802	200	16	sp	sp	NOUN
iajs-2802	200	17	and	and	CCONJ
iajs-2802	200	18	hence	hence	ADV
iajs-2802	200	19	either	either	CCONJ
iajs-2802	200	20	lp	lp	ADJ
iajs-2802	200	21	⊆	⊆	NUM
iajs-2802	200	22	bp	bp	PROPN
iajs-2802	200	23	or	or	CCONJ
iajs-2802	200	24	bp	bp	PROPN
iajs-2802	200	25	⊆	⊆	NUM
iajs-2802	200	26	lp	lp	NOUN
iajs-2802	200	27	which	which	PRON
iajs-2802	200	28	implies	imply	VERB
iajs-2802	200	29	lp	lp	NOUN
iajs-2802	200	30	+	+	CCONJ
iajs-2802	200	31	bp	bp	PROPN
iajs-2802	200	32	=	=	SYM
iajs-2802	200	33	lp	lp	PROPN
iajs-2802	200	34	or	or	CCONJ
iajs-2802	200	35	lp	lp	NOUN
iajs-2802	200	36	+	+	CCONJ
iajs-2802	200	37	bp	bp	PROPN
iajs-2802	200	38	=	=	SYM
iajs-2802	200	39	bp	bp	PROPN
iajs-2802	200	40	,	,	PUNCT
iajs-2802	200	41	but	but	CCONJ
iajs-2802	200	42	lp	lp	ADV
iajs-2802	200	43	,	,	PUNCT
iajs-2802	200	44	bp	bp	PROPN
iajs-2802	200	45	are	be	AUX
iajs-2802	200	46	sm	sm	NOUN
iajs-2802	200	47	-	-	PUNCT
iajs-2802	200	48	submodules	submodule	NOUN
iajs-2802	200	49	of	of	ADP
iajs-2802	200	50	ap	ap	PROPN
iajs-2802	200	51	.	.	PUNCT
iajs-2802	201	1	thus	thus	ADV
iajs-2802	201	2	lp	lp	PROPN
iajs-2802	201	3	+	+	NUM
iajs-2802	201	4	bp	bp	PROPN
iajs-2802	201	5	is	be	AUX
iajs-2802	201	6	an	an	DET
iajs-2802	201	7	sm	sm	NOUN
iajs-2802	201	8	-	-	PUNCT
iajs-2802	201	9	submodule	submodule	NOUN
iajs-2802	201	10	and	and	CCONJ
iajs-2802	201	11	(	(	PUNCT
iajs-2802	201	12	l+b)p	l+b)p	NOUN
iajs-2802	201	13	is	be	AUX
iajs-2802	201	14	an	an	DET
iajs-2802	201	15	sm	sm	NOUN
iajs-2802	201	16	-	-	PUNCT
iajs-2802	201	17	submodule	submodule	NOUN
iajs-2802	201	18	and	and	CCONJ
iajs-2802	201	19	by	by	ADP
iajs-2802	201	20	proposition(4.2	proposition(4.2	NOUN
iajs-2802	201	21	)	)	PUNCT
iajs-2802	201	22	,	,	PUNCT
iajs-2802	201	23	l+b	l+b	NUM
iajs-2802	201	24	is	be	AUX
iajs-2802	201	25	an	an	DET
iajs-2802	201	26	sm	sm	NOUN
iajs-2802	201	27	-	-	PUNCT
iajs-2802	201	28	submodule	submodule	NOUN
iajs-2802	201	29	of	of	ADP
iajs-2802	201	30	a.	a.	NOUN
iajs-2802	201	31	5.conclusion	5.conclusion	NUM
iajs-2802	201	32	the	the	DET
iajs-2802	201	33	conclusion	conclusion	NOUN
iajs-2802	201	34	of	of	ADP
iajs-2802	201	35	this	this	DET
iajs-2802	201	36	work	work	NOUN
iajs-2802	201	37	is	be	AUX
iajs-2802	201	38	to	to	PART
iajs-2802	201	39	study	study	VERB
iajs-2802	201	40	an	an	DET
iajs-2802	201	41	important	important	ADJ
iajs-2802	201	42	concept	concept	NOUN
iajs-2802	201	43	,	,	PUNCT
iajs-2802	201	44	namely	namely	ADV
iajs-2802	201	45	strongly	strongly	ADV
iajs-2802	201	46	maximal	maximal	ADJ
iajs-2802	201	47	submodule	submodule	NOUN
iajs-2802	201	48	.	.	PUNCT
iajs-2802	202	1	some	some	DET
iajs-2802	202	2	properties	property	NOUN
iajs-2802	202	3	and	and	CCONJ
iajs-2802	202	4	many	many	ADJ
iajs-2802	202	5	results	result	NOUN
iajs-2802	202	6	were	be	AUX
iajs-2802	202	7	proved	prove	VERB
iajs-2802	202	8	and	and	CCONJ
iajs-2802	202	9	the	the	DET
iajs-2802	202	10	behavior	behavior	NOUN
iajs-2802	202	11	of	of	ADP
iajs-2802	202	12	that	that	DET
iajs-2802	202	13	concept	concept	NOUN
iajs-2802	202	14	with	with	ADP
iajs-2802	202	15	its	its	PRON
iajs-2802	202	16	localization	localization	NOUN
iajs-2802	202	17	were	be	AUX
iajs-2802	202	18	studied	study	VERB
iajs-2802	202	19	and	and	CCONJ
iajs-2802	202	20	shown	show	VERB
iajs-2802	202	21	.	.	PUNCT
iajs-2802	203	1	references	reference	NOUN
iajs-2802	203	2	1	1	NUM
iajs-2802	203	3	.	.	PUNCT
iajs-2802	203	4	burton	burton	PROPN
iajs-2802	203	5	,	,	PUNCT
iajs-2802	203	6	d.	d.	PROPN
iajs-2802	203	7	m.	m.	PROPN
iajs-2802	203	8	1970	1970	NUM
iajs-2802	203	9	.	.	PUNCT
iajs-2802	204	1	a	a	DET
iajs-2802	204	2	first	first	ADJ
iajs-2802	204	3	course	course	NOUN
iajs-2802	204	4	in	in	ADP
iajs-2802	204	5	rings	ring	NOUN
iajs-2802	204	6	and	and	CCONJ
iajs-2802	204	7	ideals	ideal	NOUN
iajs-2802	204	8	.	.	PUNCT
iajs-2802	205	1	addison	addison	PROPN
iajs-2802	205	2	-	-	PUNCT
iajs-2802	205	3	wesley	wesley	PROPN
iajs-2802	205	4	.	.	PUNCT
iajs-2802	206	1	2	2	X
iajs-2802	206	2	.	.	X
iajs-2802	206	3	sahera	sahera	NOUN
iajs-2802	206	4	mahmod	mahmod	NOUN
iajs-2802	206	5	yasin	yasin	NOUN
iajs-2802	206	6	about	about	ADP
iajs-2802	206	7	regular	regular	ADJ
iajs-2802	206	8	module	module	NOUN
iajs-2802	206	9	type	type	NOUN
iajs-2802	206	10	-	-	PUNCT
iajs-2802	206	11	f	f	NOUN
iajs-2802	206	12	,	,	PUNCT
iajs-2802	206	13	master	master	NOUN
iajs-2802	206	14	letter	letter	NOUN
iajs-2802	206	15	,	,	PUNCT
iajs-2802	206	16	scince	scince	NOUN
iajs-2802	206	17	college	college	NOUN
iajs-2802	206	18	,	,	PUNCT
iajs-2802	206	19	univ.of	univ.of	PROPN
iajs-2802	206	20	baghdad	baghdad	PROPN
iajs-2802	206	21	,	,	PUNCT
iajs-2802	206	22	1993	1993	NUM
iajs-2802	206	23	.	.	PUNCT
iajs-2802	207	1	3	3	X
iajs-2802	207	2	.	.	X
iajs-2802	208	1	khalaf	khalaf	PROPN
iajs-2802	208	2	,	,	PUNCT
iajs-2802	208	3	h.y	h.y	PROPN
iajs-2802	208	4	.	.	PROPN
iajs-2802	208	5	semimaximal	semimaximal	ADJ
iajs-2802	208	6	submodules	submodule	NOUN
iajs-2802	208	7	ph.d.thesis	ph.d.thesis	PROPN
iajs-2802	208	8	,	,	PUNCT
iajs-2802	208	9	univ	univ	PROPN
iajs-2802	208	10	.	.	PROPN
iajs-2802	208	11	of	of	ADP
iajs-2802	208	12	baghdad	baghdad	PROPN
iajs-2802	208	13	,	,	PUNCT
iajs-2802	208	14	2007	2007	NUM
iajs-2802	208	15	.	.	PUNCT
iajs-2802	209	1	4	4	X
iajs-2802	209	2	.	.	X
iajs-2802	209	3	frank	frank	PROPN
iajs-2802	209	4	w.	w.	PROPN
iajs-2802	209	5	,	,	PUNCT
iajs-2802	209	6	anderson	anderson	PROPN
iajs-2802	209	7	k.	k.	PROPN
iajs-2802	209	8	and	and	CCONJ
iajs-2802	209	9	.	.	PUNCT
iajs-2802	210	1	fuller	full	ADJ
iajs-2802	210	2	r.	r.	PROPN
iajs-2802	210	3	,	,	PUNCT
iajs-2802	210	4	rings	ring	NOUN
iajs-2802	210	5	and	and	CCONJ
iajs-2802	210	6	categories	category	NOUN
iajs-2802	210	7	of	of	ADP
iajs-2802	210	8	modules	module	NOUN
iajs-2802	210	9	,	,	PUNCT
iajs-2802	210	10	springerverlag	springerverlag	NOUN
iajs-2802	210	11	,	,	PUNCT
iajs-2802	210	12	berlin	berlin	PROPN
iajs-2802	210	13	,	,	PUNCT
iajs-2802	210	14	heidelberg	heidelberg	PROPN
iajs-2802	210	15	,	,	PUNCT
iajs-2802	210	16	new	new	PROPN
iajs-2802	210	17	york	york	PROPN
iajs-2802	210	18	,	,	PUNCT
iajs-2802	210	19	1974	1974	NUM
iajs-2802	210	20	.	.	PUNCT
iajs-2802	211	1	5	5	X
iajs-2802	211	2	.	.	X
iajs-2802	211	3	lu.c.p	lu.c.p	NOUN
iajs-2802	211	4	.	.	PUNCT
iajs-2802	212	1	,	,	PUNCT
iajs-2802	212	2	m	m	NOUN
iajs-2802	212	3	-	-	NOUN
iajs-2802	212	4	radicals	radical	NOUN
iajs-2802	212	5	of	of	ADP
iajs-2802	212	6	submodules	submodule	NOUN
iajs-2802	212	7	in	in	ADP
iajs-2802	212	8	modules	module	NOUN
iajs-2802	212	9	,	,	PUNCT
iajs-2802	212	10	math	math	NOUN
iajs-2802	212	11	.	.	PUNCT
iajs-2802	213	1	japon	japon	PROPN
iajs-2802	213	2	.	.	PROPN
iajs-2802	213	3	,	,	PUNCT
iajs-2802	213	4	34	34	NUM
iajs-2802	213	5	1989	1989	NUM
iajs-2802	213	6	,	,	PUNCT
iajs-2802	213	7	211	211	NUM
iajs-2802	213	8	-	-	SYM
iajs-2802	213	9	219	219	NUM
iajs-2802	213	10	.	.	PUNCT
iajs-2802	213	11	6	6	NUM
iajs-2802	213	12	.	.	X
iajs-2802	213	13	kazem	kazem	PROPN
iajs-2802	213	14	,	,	PUNCT
iajs-2802	213	15	a.d	a.d	PROPN
iajs-2802	213	16	.	.	PUNCT
iajs-2802	214	1	some	some	DET
iajs-2802	214	2	types	type	NOUN
iajs-2802	214	3	of	of	ADP
iajs-2802	214	4	visible	visible	ADJ
iajs-2802	214	5	submodules	submodule	NOUN
iajs-2802	214	6	and	and	CCONJ
iajs-2802	214	7	fully	fully	ADV
iajs-2802	214	8	visible	visible	ADJ
iajs-2802	214	9	modules	module	NOUN
iajs-2802	214	10	m.sc.thesis	m.sc.thesis	NOUN
iajs-2802	214	11	,	,	PUNCT
iajs-2802	214	12	univ	univ	PROPN
iajs-2802	214	13	.	.	PROPN
iajs-2802	214	14	of	of	ADP
iajs-2802	214	15	baghdad,2020	baghdad,2020	NOUN
iajs-2802	214	16	.	.	PUNCT
iajs-2802	215	1	7	7	X
iajs-2802	215	2	.	.	X
iajs-2802	215	3	goldie	goldie	PROPN
iajs-2802	215	4	,	,	PUNCT
iajs-2802	215	5	k.r	k.r	PROPN
iajs-2802	215	6	.	.	PROPN
iajs-2802	215	7	torsion	torsion	NOUN
iajs-2802	215	8	free	free	ADJ
iajs-2802	215	9	modules	module	NOUN
iajs-2802	215	10	and	and	CCONJ
iajs-2802	215	11	rings	ring	NOUN
iajs-2802	215	12	,	,	PUNCT
iajs-2802	215	13	j.algebra,1,268	j.algebra,1,268	PROPN
iajs-2802	215	14	-	-	PUNCT
iajs-2802	215	15	287	287	NUM
iajs-2802	215	16	,	,	PUNCT
iajs-2802	215	17	1964	1964	NUM
iajs-2802	215	18	.	.	PUNCT
iajs-2802	216	1	8	8	NUM
iajs-2802	216	2	.	.	PUNCT
iajs-2802	217	1	larson	larson	PROPN
iajs-2802	217	2	m.d	m.d	PROPN
iajs-2802	217	3	.	.	PROPN
iajs-2802	217	4	,	,	PUNCT
iajs-2802	217	5	mc	mc	PROPN
iajs-2802	217	6	carthy	carthy	PROPN
iajs-2802	217	7	p.j	p.j	PROPN
iajs-2802	217	8	.	.	PROPN
iajs-2802	217	9	,	,	PUNCT
iajs-2802	217	10	1971	1971	NUM
iajs-2802	217	11	,	,	PUNCT
iajs-2802	217	12	multiplication	multiplication	NOUN
iajs-2802	217	13	theory	theory	NOUN
iajs-2802	217	14	of	of	ADP
iajs-2802	217	15	ideals	ideal	NOUN
iajs-2802	217	16	,	,	PUNCT
iajs-2802	217	17	academic	academic	ADJ
iajs-2802	217	18	press	press	NOUN
iajs-2802	217	19	,	,	PUNCT
iajs-2802	217	20	new	new	PROPN
iajs-2802	217	21	york	york	PROPN
iajs-2802	217	22	and	and	CCONJ
iajs-2802	217	23	london	london	PROPN
iajs-2802	217	24	.	.	PUNCT
iajs-2802	218	1	9	9	X
iajs-2802	218	2	.	.	X
iajs-2802	218	3	fieldhouse	fieldhouse	PROPN
iajs-2802	218	4	,	,	PUNCT
iajs-2802	218	5	d.j	d.j	PROPN
iajs-2802	218	6	.	.	PROPN
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iajs-2802	218	8	theories	theory	NOUN
iajs-2802	218	9	,	,	PUNCT
iajs-2802	218	10	math.ann	math.ann	NOUN
iajs-2802	218	11	.	.	NOUN
iajs-2802	218	12	,	,	PUNCT
iajs-2802	218	13	184,1	184,1	NOUN
iajs-2802	218	14	-	-	PUNCT
iajs-2802	218	15	18,1969	18,1969	NUM
iajs-2802	218	16	.	.	PUNCT
iajs-2802	219	1	10	10	NUM
iajs-2802	219	2	.	.	PUNCT
iajs-2802	220	1	mijbass	mijbass	PROPN
iajs-2802	220	2	a.s	a.s	PROPN
iajs-2802	220	3	.	.	PROPN
iajs-2802	220	4	,	,	PUNCT
iajs-2802	220	5	on	on	ADP
iajs-2802	220	6	cancelation	cancelation	NOUN
iajs-2802	220	7	modules	module	NOUN
iajs-2802	220	8	,	,	PUNCT
iajs-2802	220	9	m.sc	m.sc	PROPN
iajs-2802	220	10	.	.	PUNCT
iajs-2802	221	1	thesis	thesis	NOUN
iajs-2802	221	2	,	,	PUNCT
iajs-2802	221	3	baghdad	baghdad	PROPN
iajs-2802	221	4	university	university	PROPN
iajs-2802	221	5	,	,	PUNCT
iajs-2802	221	6	1992	1992	NUM
iajs-2802	221	7	.	.	PUNCT
iajs-2802	222	1	11	11	NUM
iajs-2802	222	2	.	.	X
iajs-2802	222	3	faris	faris	PROPN
iajs-2802	222	4	,	,	PUNCT
iajs-2802	222	5	h.	h.	PROPN
iajs-2802	222	6	i.	i.	PROPN
iajs-2802	222	7	,	,	PUNCT
iajs-2802	222	8	jasim	jasim	PROPN
iajs-2802	222	9	,	,	PUNCT
iajs-2802	222	10	r.	r.	PROPN
iajs-2802	222	11	h.	h.	PROPN
iajs-2802	222	12	,	,	PUNCT
iajs-2802	222	13	&	&	CCONJ
iajs-2802	222	14	mohammed	mohammed	PROPN
iajs-2802	222	15	,	,	PUNCT
iajs-2802	222	16	n.	n.	PROPN
iajs-2802	222	17	j.	j.	PROPN
iajs-2802	222	18	2021	2021	PROPN
iajs-2802	222	19	,	,	PUNCT
iajs-2802	222	20	march	march	PROPN
iajs-2802	222	21	.	.	PUNCT
iajs-2802	223	1	pseudo	pseudo	NOUN
iajs-2802	223	2	maximal	maximal	ADJ
iajs-2802	223	3	submodules	submodule	NOUN
iajs-2802	223	4	.	.	PUNCT
iajs-2802	224	1	in	in	ADP
iajs-2802	224	2	journal	journal	PROPN
iajs-2802	224	3	of	of	ADP
iajs-2802	224	4	physics	physics	PROPN
iajs-2802	224	5	:	:	PUNCT
iajs-2802	224	6	conference	conference	NOUN
iajs-2802	224	7	series	series	NOUN
iajs-2802	224	8	(	(	PUNCT
iajs-2802	224	9	vol	vol	NOUN
iajs-2802	224	10	.	.	PUNCT
iajs-2802	224	11	1818	1818	NUM
iajs-2802	224	12	,	,	PUNCT
iajs-2802	224	13	no	no	INTJ
iajs-2802	224	14	.	.	NOUN
iajs-2802	224	15	1	1	NUM
iajs-2802	224	16	,	,	PUNCT
iajs-2802	224	17	p.	p.	NOUN
iajs-2802	224	18	012055	012055	NUM
iajs-2802	224	19	)	)	PUNCT
iajs-2802	224	20	.	.	PUNCT
iajs-2802	225	1	iop	iop	PROPN
iajs-2802	225	2	publishing	publishing	NOUN
iajs-2802	225	3	.	.	PUNCT
iajs-2802	226	1	12	12	NUM
iajs-2802	226	2	.	.	PUNCT
iajs-2802	227	1	mohammed	mohammed	PROPN
iajs-2802	227	2	,	,	PUNCT
iajs-2802	227	3	a.	a.	PROPN
iajs-2802	227	4	s.	s.	PROPN
iajs-2802	227	5	,	,	PUNCT
iajs-2802	227	6	&	&	CCONJ
iajs-2802	227	7	sallman	sallman	PROPN
iajs-2802	227	8	,	,	PUNCT
iajs-2802	227	9	m.	m.	NOUN
iajs-2802	227	10	d.	d.	PROPN
iajs-2802	227	11	2017	2017	NUM
iajs-2802	227	12	.	.	PUNCT
iajs-2802	228	1	2	2	NUM
iajs-2802	228	2	-	-	PUNCT
iajs-2802	228	3	maximal	maximal	ADJ
iajs-2802	228	4	submodules	submodule	NOUN
iajs-2802	228	5	and	and	CCONJ
iajs-2802	228	6	related	related	ADJ
iajs-2802	228	7	concepts	concept	NOUN
iajs-2802	228	8	.	.	PUNCT
iajs-2802	229	1	journal	journal	PROPN
iajs-2802	229	2	of	of	ADP
iajs-2802	229	3	university	university	PROPN
iajs-2802	229	4	of	of	ADP
iajs-2802	229	5	anbar	anbar	NOUN
iajs-2802	229	6	for	for	ADP
iajs-2802	229	7	pure	pure	ADJ
iajs-2802	229	8	science	science	NOUN
iajs-2802	229	9	,	,	PUNCT
iajs-2802	229	10	11(3	11(3	NUM
iajs-2802	229	11	)	)	PUNCT
iajs-2802	229	12	.	.	PUNCT
iajs-2802	230	1	13	13	NUM
iajs-2802	230	2	.	.	X
iajs-2802	231	1	jud	jud	PROPN
iajs-2802	231	2	h.m	h.m	PROPN
iajs-2802	231	3	.	.	PROPN
iajs-2802	231	4	,	,	PUNCT
iajs-2802	231	5	some	some	DET
iajs-2802	231	6	types	type	NOUN
iajs-2802	231	7	of	of	ADP
iajs-2802	231	8	fully	fully	ADV
iajs-2802	231	9	cancelation	cancelation	NOUN
iajs-2802	231	10	modules	module	NOUN
iajs-2802	231	11	,	,	PUNCT
iajs-2802	231	12	m.sc.thesis	m.sc.thesis	NOUN
iajs-2802	231	13	,	,	PUNCT
iajs-2802	231	14	baghdad	baghdad	PROPN
iajs-2802	231	15	university,2016	university,2016	PROPN
iajs-2802	231	16	.	.	PUNCT