id	sid	tid	token	lemma	pos
iajs-2376	1	1	48	48	NUM
iajs-2376	1	2	ibn	ibn	PROPN
iajs-2376	1	3	al	al	PROPN
iajs-2376	1	4	-	-	PUNCT
iajs-2376	1	5	haitham	haitham	PROPN
iajs-2376	1	6	jour	jour	X
iajs-2376	1	7	.	.	PROPN
iajs-2376	1	8	for	for	ADP
iajs-2376	1	9	pure	pure	ADJ
iajs-2376	1	10	&	&	CCONJ
iajs-2376	1	11	appl	appl	PROPN
iajs-2376	1	12	.	.	PUNCT
iajs-2376	2	1	sci	sci	PROPN
iajs-2376	2	2	.	.	PROPN
iajs-2376	3	1	33	33	NUM
iajs-2376	3	2	(	(	PUNCT
iajs-2376	3	3	1	1	NUM
iajs-2376	3	4	)	)	PUNCT
iajs-2376	3	5	2020	2020	NUM
iajs-2376	3	6	pseudo	pseudo	NOUN
iajs-2376	3	7	weakly	weakly	ADJ
iajs-2376	3	8	closed	closed	ADJ
iajs-2376	3	9	submodules	submodule	NOUN
iajs-2376	3	10	and	and	CCONJ
iajs-2376	3	11	related	related	ADJ
iajs-2376	3	12	concepts	concept	NOUN
iajs-2376	4	1	haibat	haibat	PROPN
iajs-2376	4	2	k.	k.	PROPN
iajs-2376	4	3	mohammadali	mohammadali	PROPN
iajs-2376	4	4	mohammad	mohammad	PROPN
iajs-2376	4	5	e.	e.	PROPN
iajs-2376	4	6	dahash	dahash	PROPN
iajs-2376	4	7	department	department	PROPN
iajs-2376	4	8	of	of	ADP
iajs-2376	4	9	mathematics	mathematics	PROPN
iajs-2376	4	10	college	college	PROPN
iajs-2376	4	11	of	of	ADP
iajs-2376	4	12	computer	computer	NOUN
iajs-2376	4	13	science	science	NOUN
iajs-2376	4	14	and	and	CCONJ
iajs-2376	4	15	mathematics	mathematics	PROPN
iajs-2376	4	16	tikrit	tikrit	NOUN
iajs-2376	4	17	university	university	NOUN
iajs-2376	4	18	abstract	abstract	NOUN
iajs-2376	4	19	let	let	AUX
iajs-2376	4	20	be	be	AUX
iajs-2376	4	21	a	a	DET
iajs-2376	4	22	commutative	commutative	ADJ
iajs-2376	4	23	ring	ring	NOUN
iajs-2376	4	24	with	with	ADP
iajs-2376	4	25	identity	identity	NOUN
iajs-2376	4	26	,	,	PUNCT
iajs-2376	4	27	and	and	CCONJ
iajs-2376	4	28	be	be	AUX
iajs-2376	4	29	a	a	DET
iajs-2376	4	30	unitary	unitary	ADJ
iajs-2376	4	31	left	left	ADJ
iajs-2376	4	32	-module	-module	NOUN
iajs-2376	4	33	.	.	PUNCT
iajs-2376	5	1	in	in	ADP
iajs-2376	5	2	this	this	DET
iajs-2376	5	3	paper	paper	NOUN
iajs-2376	5	4	we	we	PRON
iajs-2376	5	5	introduce	introduce	VERB
iajs-2376	5	6	the	the	DET
iajs-2376	5	7	concept	concept	NOUN
iajs-2376	5	8	pseudo	pseudo	NOUN
iajs-2376	5	9	weakly	weakly	ADV
iajs-2376	5	10	closed	closed	ADJ
iajs-2376	5	11	submodule	submodule	NOUN
iajs-2376	5	12	as	as	ADP
iajs-2376	5	13	a	a	DET
iajs-2376	5	14	generalization	generalization	NOUN
iajs-2376	5	15	of	of	ADP
iajs-2376	5	16	-closed	-closed	ADJ
iajs-2376	5	17	submodules	submodule	NOUN
iajs-2376	5	18	,	,	PUNCT
iajs-2376	5	19	where	where	SCONJ
iajs-2376	5	20	a	a	DET
iajs-2376	5	21	submodule	submodule	NOUN
iajs-2376	5	22	of	of	ADP
iajs-2376	5	23	an	an	DET
iajs-2376	5	24	-module	-module	NOUN
iajs-2376	5	25	is	be	AUX
iajs-2376	5	26	called	call	VERB
iajs-2376	5	27	a	a	DET
iajs-2376	5	28	pseudo	pseudo	NOUN
iajs-2376	5	29	weakly	weakly	ADV
iajs-2376	5	30	closed	closed	ADJ
iajs-2376	5	31	submodule	submodule	NOUN
iajs-2376	5	32	,	,	PUNCT
iajs-2376	5	33	if	if	SCONJ
iajs-2376	5	34	for	for	ADP
iajs-2376	5	35	all	all	PRON
iajs-2376	5	36	,	,	PUNCT
iajs-2376	5	37	there	there	PRON
iajs-2376	5	38	exists	exist	VERB
iajs-2376	5	39	a	a	DET
iajs-2376	5	40	-closed	-closed	ADJ
iajs-2376	5	41	submodule	submodule	NOUN
iajs-2376	5	42	of	of	ADP
iajs-2376	5	43	with	with	ADP
iajs-2376	5	44	is	be	AUX
iajs-2376	5	45	a	a	DET
iajs-2376	5	46	submodule	submodule	NOUN
iajs-2376	5	47	of	of	ADP
iajs-2376	5	48	such	such	ADJ
iajs-2376	5	49	that	that	PRON
iajs-2376	5	50	.	.	PUNCT
iajs-2376	6	1	several	several	ADJ
iajs-2376	6	2	basic	basic	ADJ
iajs-2376	6	3	properties	property	NOUN
iajs-2376	6	4	,	,	PUNCT
iajs-2376	6	5	examples	example	NOUN
iajs-2376	6	6	and	and	CCONJ
iajs-2376	6	7	results	result	NOUN
iajs-2376	6	8	of	of	ADP
iajs-2376	6	9	pseudo	pseudo	NOUN
iajs-2376	6	10	weakly	weakly	ADV
iajs-2376	6	11	closed	closed	ADJ
iajs-2376	6	12	submodules	submodule	NOUN
iajs-2376	6	13	are	be	AUX
iajs-2376	6	14	given	give	VERB
iajs-2376	6	15	.	.	PUNCT
iajs-2376	7	1	furthermore	furthermore	ADV
iajs-2376	7	2	the	the	DET
iajs-2376	7	3	behavior	behavior	NOUN
iajs-2376	7	4	of	of	ADP
iajs-2376	7	5	pseudo	pseudo	NOUN
iajs-2376	7	6	weakly	weakly	ADV
iajs-2376	7	7	closed	closed	ADJ
iajs-2376	7	8	submodules	submodule	NOUN
iajs-2376	7	9	in	in	ADP
iajs-2376	7	10	class	class	NOUN
iajs-2376	7	11	of	of	ADP
iajs-2376	7	12	multiplication	multiplication	NOUN
iajs-2376	7	13	modules	module	NOUN
iajs-2376	7	14	are	be	AUX
iajs-2376	7	15	studied	study	VERB
iajs-2376	7	16	.	.	PUNCT
iajs-2376	8	1	on	on	ADP
iajs-2376	8	2	the	the	DET
iajs-2376	8	3	other	other	ADJ
iajs-2376	8	4	hand	hand	NOUN
iajs-2376	8	5	modules	module	NOUN
iajs-2376	8	6	with	with	ADP
iajs-2376	8	7	chain	chain	NOUN
iajs-2376	8	8	conditions	condition	NOUN
iajs-2376	8	9	on	on	ADP
iajs-2376	8	10	pseudo	pseudo	NOUN
iajs-2376	8	11	weakly	weakly	ADV
iajs-2376	8	12	closed	closed	ADJ
iajs-2376	8	13	submodules	submodule	NOUN
iajs-2376	8	14	are	be	AUX
iajs-2376	8	15	established	establish	VERB
iajs-2376	8	16	.	.	PUNCT
iajs-2376	9	1	also	also	ADV
iajs-2376	9	2	,	,	PUNCT
iajs-2376	9	3	the	the	DET
iajs-2376	9	4	relationships	relationship	NOUN
iajs-2376	9	5	of	of	ADP
iajs-2376	9	6	pseudo	pseudo	NOUN
iajs-2376	9	7	weakly	weakly	ADV
iajs-2376	9	8	closed	closed	ADJ
iajs-2376	9	9	submodules	submodule	NOUN
iajs-2376	9	10	with	with	ADP
iajs-2376	9	11	other	other	ADJ
iajs-2376	9	12	classes	class	NOUN
iajs-2376	9	13	of	of	ADP
iajs-2376	9	14	modules	module	NOUN
iajs-2376	9	15	are	be	AUX
iajs-2376	9	16	discussed	discuss	VERB
iajs-2376	9	17	.	.	PUNCT
iajs-2376	10	1	keywords	keyword	NOUN
iajs-2376	10	2	:	:	PUNCT
iajs-2376	10	3	closed	closed	ADJ
iajs-2376	10	4	submodules	submodule	NOUN
iajs-2376	10	5	,	,	PUNCT
iajs-2376	10	6	-closed	-closed	ADJ
iajs-2376	10	7	submodules	submodule	NOUN
iajs-2376	10	8	,	,	PUNCT
iajs-2376	10	9	pseudo	pseudo	NOUN
iajs-2376	10	10	weakly	weakly	ADJ
iajs-2376	10	11	closed	closed	ADJ
iajs-2376	10	12	submodules	submodule	NOUN
iajs-2376	10	13	,	,	PUNCT
iajs-2376	10	14	semi	semi	ADJ
iajs-2376	10	15	-	-	ADJ
iajs-2376	10	16	prime	prime	ADJ
iajs-2376	10	17	submodules	submodule	NOUN
iajs-2376	10	18	,	,	PUNCT
iajs-2376	10	19	fully	fully	ADV
iajs-2376	10	20	semi	semi	ADJ
iajs-2376	10	21	-	-	ADJ
iajs-2376	10	22	prime	prime	ADJ
iajs-2376	10	23	submodules	submodule	NOUN
iajs-2376	10	24	,	,	PUNCT
iajs-2376	10	25	weakly	weakly	ADJ
iajs-2376	10	26	essential	essential	ADJ
iajs-2376	10	27	submodules	submodule	NOUN
iajs-2376	10	28	.	.	PUNCT
iajs-2376	11	1	1	1	X
iajs-2376	11	2	.	.	X
iajs-2376	11	3	introduction	introduction	NOUN
iajs-2376	11	4	a	a	DET
iajs-2376	11	5	proper	proper	ADJ
iajs-2376	11	6	submodule	submodule	NOUN
iajs-2376	11	7	of	of	ADP
iajs-2376	11	8	an	an	DET
iajs-2376	11	9	-module	-module	NOUN
iajs-2376	11	10	is	be	AUX
iajs-2376	11	11	called	call	VERB
iajs-2376	11	12	closed	close	VERB
iajs-2376	11	13	in	in	ADP
iajs-2376	11	14	,	,	PUNCT
iajs-2376	11	15	provided	provide	VERB
iajs-2376	11	16	that	that	PRON
iajs-2376	11	17	has	have	AUX
iajs-2376	11	18	no	no	DET
iajs-2376	11	19	proper	proper	ADJ
iajs-2376	11	20	essential	essential	ADJ
iajs-2376	11	21	extensions	extension	NOUN
iajs-2376	11	22	in	in	ADP
iajs-2376	11	23	[	[	PUNCT
iajs-2376	11	24	]	]	X
iajs-2376	11	25	.	.	PUNCT
iajs-2376	12	1	where	where	SCONJ
iajs-2376	12	2	a	a	DET
iajs-2376	12	3	non	non	ADJ
iajs-2376	12	4	-	-	ADJ
iajs-2376	12	5	zero	zero	NUM
iajs-2376	12	6	submodule	submodule	NOUN
iajs-2376	12	7	of	of	ADP
iajs-2376	12	8	an	an	DET
iajs-2376	12	9	-module	-module	NOUN
iajs-2376	12	10	is	be	AUX
iajs-2376	12	11	called	call	VERB
iajs-2376	12	12	essential	essential	ADJ
iajs-2376	12	13	in	in	ADP
iajs-2376	12	14	if	if	SCONJ
iajs-2376	12	15	for	for	ADP
iajs-2376	12	16	each	each	DET
iajs-2376	12	17	non	non	ADJ
iajs-2376	12	18	-	-	ADJ
iajs-2376	12	19	zero	zero	NUM
iajs-2376	12	20	submodule	submodule	NOUN
iajs-2376	12	21	of	of	ADP
iajs-2376	12	22	[	[	PUNCT
iajs-2376	12	23	]	]	X
iajs-2376	12	24	.	.	PUNCT
iajs-2376	13	1	and	and	CCONJ
iajs-2376	13	2	a	a	DET
iajs-2376	13	3	nonzero	nonzero	NOUN
iajs-2376	13	4	submodule	submodule	NOUN
iajs-2376	13	5	of	of	ADP
iajs-2376	13	6	is	be	AUX
iajs-2376	13	7	called	call	VERB
iajs-2376	13	8	weakly	weakly	ADJ
iajs-2376	13	9	essential	essential	ADJ
iajs-2376	13	10	submodule	submodule	NOUN
iajs-2376	13	11	of	of	ADP
iajs-2376	13	12	if	if	SCONJ
iajs-2376	13	13	for	for	ADP
iajs-2376	13	14	each	each	DET
iajs-2376	13	15	non	non	ADJ
iajs-2376	13	16	-	-	ADJ
iajs-2376	13	17	zero	zero	NUM
iajs-2376	13	18	semi	semi	ADJ
iajs-2376	13	19	-	-	ADJ
iajs-2376	13	20	prime	prime	ADJ
iajs-2376	13	21	submodule	submodule	NOUN
iajs-2376	13	22	of	of	ADP
iajs-2376	13	23	[	[	PUNCT
iajs-2376	13	24	]	]	X
iajs-2376	13	25	.	.	PUNCT
iajs-2376	14	1	equivalently	equivalently	ADV
iajs-2376	14	2	is	be	AUX
iajs-2376	14	3	weak	weak	ADJ
iajs-2376	14	4	essential	essential	ADJ
iajs-2376	14	5	,	,	PUNCT
iajs-2376	14	6	if	if	SCONJ
iajs-2376	14	7	whenever	whenever	ADV
iajs-2376	14	8	,	,	PUNCT
iajs-2376	14	9	then	then	ADV
iajs-2376	14	10	for	for	ADP
iajs-2376	14	11	every	every	DET
iajs-2376	14	12	semi	semi	ADJ
iajs-2376	14	13	-	-	ADJ
iajs-2376	14	14	prime	prime	ADJ
iajs-2376	14	15	submodule	submodule	NOUN
iajs-2376	14	16	of	of	ADP
iajs-2376	14	17	[	[	PUNCT
iajs-2376	14	18	]	]	X
iajs-2376	14	19	.	.	PUNCT
iajs-2376	15	1	where	where	SCONJ
iajs-2376	15	2	a	a	DET
iajs-2376	15	3	submodule	submodule	NOUN
iajs-2376	15	4	of	of	ADP
iajs-2376	15	5	is	be	AUX
iajs-2376	15	6	called	call	VERB
iajs-2376	15	7	semi	semi	ADJ
iajs-2376	15	8	-	-	ADJ
iajs-2376	15	9	prime	prime	ADJ
iajs-2376	15	10	if	if	SCONJ
iajs-2376	15	11	whenever	whenever	ADV
iajs-2376	15	12	,	,	PUNCT
iajs-2376	15	13	for	for	ADP
iajs-2376	15	14	,	,	PUNCT
iajs-2376	15	15	,	,	PUNCT
iajs-2376	15	16	implies	imply	VERB
iajs-2376	15	17	that	that	SCONJ
iajs-2376	15	18	[	[	X
iajs-2376	15	19	]	]	X
iajs-2376	15	20	.	.	PUNCT
iajs-2376	16	1	the	the	DET
iajs-2376	16	2	concept	concept	NOUN
iajs-2376	16	3	of	of	ADP
iajs-2376	16	4	closed	closed	ADJ
iajs-2376	16	5	submodule	submodule	NOUN
iajs-2376	16	6	recently	recently	ADV
iajs-2376	16	7	extended	extend	VERB
iajs-2376	16	8	by	by	ADP
iajs-2376	16	9	[	[	PUNCT
iajs-2376	16	10	]	]	X
iajs-2376	16	11	to	to	ADP
iajs-2376	16	12	-closed	-close	VERB
iajs-2376	16	13	submodule	submodule	NOUN
iajs-2376	16	14	,	,	PUNCT
iajs-2376	16	15	where	where	SCONJ
iajs-2376	16	16	a	a	DET
iajs-2376	16	17	submodule	submodule	NOUN
iajs-2376	16	18	of	of	ADP
iajs-2376	16	19	is	be	AUX
iajs-2376	16	20	called	call	VERB
iajs-2376	16	21	-closed	-closed	ADJ
iajs-2376	16	22	submodule	submodule	NOUN
iajs-2376	16	23	of	of	ADP
iajs-2376	16	24	if	if	SCONJ
iajs-2376	16	25	has	have	AUX
iajs-2376	16	26	no	no	DET
iajs-2376	16	27	proper	proper	ADJ
iajs-2376	16	28	weak	weak	ADJ
iajs-2376	16	29	-	-	PUNCT
iajs-2376	16	30	essential	essential	ADJ
iajs-2376	16	31	extensions	extension	NOUN
iajs-2376	16	32	in	in	ADP
iajs-2376	16	33	,	,	PUNCT
iajs-2376	16	34	that	that	PRON
iajs-2376	16	35	is	be	AUX
iajs-2376	16	36	if	if	SCONJ
iajs-2376	16	37	is	be	AUX
iajs-2376	16	38	a	a	DET
iajs-2376	16	39	weak	weak	ADJ
iajs-2376	16	40	essential	essential	ADJ
iajs-2376	16	41	submodule	submodule	NOUN
iajs-2376	16	42	of	of	ADP
iajs-2376	16	43	,	,	PUNCT
iajs-2376	16	44	where	where	SCONJ
iajs-2376	16	45	is	be	AUX
iajs-2376	16	46	a	a	DET
iajs-2376	16	47	submodule	submodule	NOUN
iajs-2376	16	48	of	of	ADP
iajs-2376	16	49	,	,	PUNCT
iajs-2376	16	50	then	then	ADV
iajs-2376	16	51	[	[	X
iajs-2376	16	52	6,7	6,7	NUM
iajs-2376	16	53	]	]	PUNCT
iajs-2376	16	54	.	.	PUNCT
iajs-2376	17	1	this	this	DET
iajs-2376	17	2	concept	concept	NOUN
iajs-2376	17	3	is	be	AUX
iajs-2376	17	4	generalized	generalize	VERB
iajs-2376	17	5	in	in	ADP
iajs-2376	17	6	this	this	DET
iajs-2376	17	7	article	article	NOUN
iajs-2376	17	8	to	to	ADP
iajs-2376	17	9	a	a	DET
iajs-2376	17	10	pseudo	pseudo	NOUN
iajs-2376	17	11	weakly	weakly	ADJ
iajs-2376	17	12	closed	closed	ADJ
iajs-2376	17	13	submodule	submodule	NOUN
iajs-2376	17	14	.	.	PUNCT
iajs-2376	18	1	many	many	ADJ
iajs-2376	18	2	basic	basic	ADJ
iajs-2376	18	3	properties	property	NOUN
iajs-2376	18	4	of	of	ADP
iajs-2376	18	5	this	this	DET
iajs-2376	18	6	concept	concept	NOUN
iajs-2376	18	7	are	be	AUX
iajs-2376	18	8	discussed	discuss	VERB
iajs-2376	18	9	.	.	PUNCT
iajs-2376	19	1	finally	finally	ADV
iajs-2376	19	2	,	,	PUNCT
iajs-2376	19	3	we	we	PRON
iajs-2376	19	4	notes	note	VERB
iajs-2376	19	5	that	that	SCONJ
iajs-2376	19	6	throughout	throughout	ADP
iajs-2376	19	7	this	this	DET
iajs-2376	19	8	paper	paper	NOUN
iajs-2376	19	9	all	all	DET
iajs-2376	19	10	rings	ring	NOUN
iajs-2376	19	11	are	be	AUX
iajs-2376	19	12	commutative	commutative	ADJ
iajs-2376	19	13	with	with	ADP
iajs-2376	19	14	identity	identity	NOUN
iajs-2376	19	15	and	and	CCONJ
iajs-2376	19	16	all	all	DET
iajs-2376	19	17	modules	module	NOUN
iajs-2376	19	18	ibn	ibn	PROPN
iajs-2376	19	19	al	al	PROPN
iajs-2376	19	20	haitham	haitham	PROPN
iajs-2376	19	21	journal	journal	PROPN
iajs-2376	19	22	for	for	ADP
iajs-2376	19	23	pure	pure	ADJ
iajs-2376	19	24	and	and	CCONJ
iajs-2376	19	25	applied	apply	VERB
iajs-2376	19	26	science	science	NOUN
iajs-2376	19	27	journal	journal	PROPN
iajs-2376	19	28	homepage	homepage	NOUN
iajs-2376	19	29	:	:	PUNCT
iajs-2376	19	30	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2376	19	31	doi	doi	NOUN
iajs-2376	19	32	:	:	PUNCT
iajs-2376	19	33	10.30526/33.1.2376	10.30526/33.1.2376	PROPN
iajs-2376	19	34	dr.mohammadali2013@gmail.com	dr.mohammadali2013@gmail.com	NOUN
iajs-2376	19	35	article	article	NOUN
iajs-2376	19	36	history	history	NOUN
iajs-2376	19	37	:	:	PUNCT
iajs-2376	19	38	received	receive	VERB
iajs-2376	19	39	11	11	NUM
iajs-2376	19	40	april	april	PROPN
iajs-2376	19	41	2019	2019	NUM
iajs-2376	19	42	,	,	PUNCT
iajs-2376	19	43	accepted	accept	VERB
iajs-2376	19	44	3	3	NUM
iajs-2376	19	45	july	july	PROPN
iajs-2376	19	46	2019	2019	NUM
iajs-2376	19	47	,	,	PUNCT
iajs-2376	19	48	publish	publish	VERB
iajs-2376	19	49	january	january	PROPN
iajs-2376	19	50	2020	2020	NUM
iajs-2376	19	51	.	.	PUNCT
iajs-2376	20	1	mailto:dr.mohammadali2013@gmail.com	mailto:dr.mohammadali2013@gmail.com	X
iajs-2376	20	2	48	48	NUM
iajs-2376	21	1	ibn	ibn	PROPN
iajs-2376	21	2	al	al	PROPN
iajs-2376	21	3	-	-	PUNCT
iajs-2376	21	4	haitham	haitham	PROPN
iajs-2376	21	5	jour	jour	X
iajs-2376	21	6	.	.	PROPN
iajs-2376	21	7	for	for	ADP
iajs-2376	21	8	pure	pure	ADJ
iajs-2376	21	9	&	&	CCONJ
iajs-2376	21	10	appl	appl	PROPN
iajs-2376	21	11	.	.	PUNCT
iajs-2376	22	1	sci	sci	PROPN
iajs-2376	22	2	.	.	PROPN
iajs-2376	23	1	33	33	NUM
iajs-2376	23	2	(	(	PUNCT
iajs-2376	23	3	1	1	NUM
iajs-2376	23	4	)	)	PUNCT
iajs-2376	23	5	2020	2020	NUM
iajs-2376	23	6	are	be	AUX
iajs-2376	23	7	unitary	unitary	ADJ
iajs-2376	23	8	left	leave	VERB
iajs-2376	23	9	-modules	-module	NOUN
iajs-2376	23	10	,	,	PUNCT
iajs-2376	23	11	unless	unless	SCONJ
iajs-2376	23	12	otherwise	otherwise	ADV
iajs-2376	23	13	.	.	PUNCT
iajs-2376	24	1	also	also	ADV
iajs-2376	24	2	,	,	PUNCT
iajs-2376	24	3	in	in	ADP
iajs-2376	24	4	this	this	DET
iajs-2376	24	5	paper	paper	NOUN
iajs-2376	24	6	all	all	DET
iajs-2376	24	7	-module	-module	NOUN
iajs-2376	24	8	under	under	ADP
iajs-2376	24	9	study	study	NOUN
iajs-2376	24	10	contains	contain	VERB
iajs-2376	24	11	semi	semi	ADJ
iajs-2376	24	12	-	-	ADJ
iajs-2376	24	13	prime	prime	ADJ
iajs-2376	24	14	submodules	submodule	NOUN
iajs-2376	24	15	.	.	PUNCT
iajs-2376	25	1	2	2	X
iajs-2376	25	2	.	.	X
iajs-2376	25	3	pseudo	pseudo	NOUN
iajs-2376	25	4	weakly	weakly	ADJ
iajs-2376	25	5	closed	closed	ADJ
iajs-2376	25	6	submodules	submodule	NOUN
iajs-2376	25	7	in	in	ADP
iajs-2376	25	8	this	this	DET
iajs-2376	25	9	section	section	NOUN
iajs-2376	25	10	we	we	PRON
iajs-2376	25	11	introduce	introduce	VERB
iajs-2376	25	12	the	the	DET
iajs-2376	25	13	notion	notion	NOUN
iajs-2376	25	14	of	of	ADP
iajs-2376	25	15	pseudo	pseudo	NOUN
iajs-2376	25	16	weakly	weakly	ADV
iajs-2376	25	17	closed	closed	ADJ
iajs-2376	25	18	submodule	submodule	NOUN
iajs-2376	25	19	as	as	ADP
iajs-2376	25	20	a	a	DET
iajs-2376	25	21	generalization	generalization	NOUN
iajs-2376	25	22	of	of	ADP
iajs-2376	25	23	-closed	-closed	ADJ
iajs-2376	25	24	submodule	submodule	NOUN
iajs-2376	25	25	and	and	CCONJ
iajs-2376	25	26	give	give	VERB
iajs-2376	25	27	some	some	DET
iajs-2376	25	28	basic	basic	ADJ
iajs-2376	25	29	properties	property	NOUN
iajs-2376	25	30	and	and	CCONJ
iajs-2376	25	31	examples	example	NOUN
iajs-2376	25	32	of	of	ADP
iajs-2376	25	33	this	this	DET
iajs-2376	25	34	class	class	NOUN
iajs-2376	25	35	.	.	PUNCT
iajs-2376	26	1	definition	definition	NOUN
iajs-2376	26	2	2.1	2.1	NUM
iajs-2376	26	3	a	a	DET
iajs-2376	26	4	submodule	submodule	NOUN
iajs-2376	26	5	of	of	ADP
iajs-2376	26	6	an	an	DET
iajs-2376	26	7	-module	-module	NOUN
iajs-2376	26	8	is	be	AUX
iajs-2376	26	9	called	call	VERB
iajs-2376	26	10	pseudo	pseudo	NOUN
iajs-2376	26	11	weakly	weakly	ADV
iajs-2376	26	12	closed	closed	ADJ
iajs-2376	26	13	submodule	submodule	NOUN
iajs-2376	26	14	(	(	PUNCT
iajs-2376	26	15	for	for	ADP
iajs-2376	26	16	a	a	DET
iajs-2376	26	17	short	short	ADJ
iajs-2376	26	18	-closed	-close	VERB
iajs-2376	26	19	)	)	PUNCT
iajs-2376	26	20	,	,	PUNCT
iajs-2376	26	21	if	if	SCONJ
iajs-2376	26	22	for	for	ADP
iajs-2376	26	23	each	each	PRON
iajs-2376	26	24	,	,	PUNCT
iajs-2376	26	25	there	there	PRON
iajs-2376	26	26	exists	exist	VERB
iajs-2376	26	27	a	a	DET
iajs-2376	26	28	-closed	-close	VERB
iajs-2376	26	29	of	of	ADP
iajs-2376	26	30	with	with	ADP
iajs-2376	26	31	such	such	ADJ
iajs-2376	26	32	that	that	PRON
iajs-2376	26	33	.	.	PUNCT
iajs-2376	27	1	an	an	DET
iajs-2376	27	2	ideal	ideal	NOUN
iajs-2376	27	3	of	of	ADP
iajs-2376	27	4	a	a	DET
iajs-2376	27	5	ring	ring	NOUN
iajs-2376	27	6	is	be	AUX
iajs-2376	27	7	called	call	VERB
iajs-2376	27	8	-closed	-close	VERB
iajs-2376	27	9	if	if	SCONJ
iajs-2376	27	10	it	it	PRON
iajs-2376	27	11	is	be	AUX
iajs-2376	27	12	-closed	-close	VERB
iajs-2376	27	13	submodule	submodule	NOUN
iajs-2376	27	14	of	of	ADP
iajs-2376	27	15	an	an	DET
iajs-2376	27	16	module	module	NOUN
iajs-2376	27	17	.	.	PUNCT
iajs-2376	28	1	remarks	remark	NOUN
iajs-2376	28	2	and	and	CCONJ
iajs-2376	28	3	examples	example	NOUN
iajs-2376	28	4	2.2	2.2	NUM
iajs-2376	28	5	1	1	NUM
iajs-2376	28	6	.	.	PUNCT
iajs-2376	29	1	it	it	PRON
iajs-2376	29	2	is	be	AUX
iajs-2376	29	3	clear	clear	ADJ
iajs-2376	29	4	that	that	SCONJ
iajs-2376	29	5	every	every	DET
iajs-2376	29	6	-closed	-closed	ADJ
iajs-2376	29	7	submodule	submodule	NOUN
iajs-2376	29	8	of	of	ADP
iajs-2376	29	9	an	an	DET
iajs-2376	29	10	-module	-module	NOUN
iajs-2376	29	11	is	be	AUX
iajs-2376	29	12	-closed	-close	VERB
iajs-2376	29	13	submodule	submodule	NOUN
iajs-2376	29	14	of	of	ADP
iajs-2376	29	15	,	,	PUNCT
iajs-2376	29	16	but	but	CCONJ
iajs-2376	29	17	the	the	DET
iajs-2376	29	18	converse	converse	NOUN
iajs-2376	29	19	is	be	AUX
iajs-2376	29	20	not	not	PART
iajs-2376	29	21	true	true	ADJ
iajs-2376	29	22	in	in	ADP
iajs-2376	29	23	general	general	ADJ
iajs-2376	29	24	as	as	SCONJ
iajs-2376	29	25	the	the	DET
iajs-2376	29	26	following	follow	VERB
iajs-2376	29	27	example	example	NOUN
iajs-2376	29	28	explain	explain	VERB
iajs-2376	29	29	that	that	SCONJ
iajs-2376	29	30	:	:	PUNCT
iajs-2376	29	31	consider	consider	VERB
iajs-2376	29	32	the	the	DET
iajs-2376	29	33	-module	-module	NOUN
iajs-2376	29	34	.	.	PUNCT
iajs-2376	30	1	the	the	DET
iajs-2376	30	2	proper	proper	ADJ
iajs-2376	30	3	submodules	submodule	NOUN
iajs-2376	30	4	of	of	ADP
iajs-2376	30	5	are	be	AUX
iajs-2376	30	6	:	:	PUNCT
iajs-2376	30	7	〈	〈	NOUN
iajs-2376	30	8	̅	̅	NOUN
iajs-2376	30	9	〉	〉	NOUN
iajs-2376	30	10	,	,	PUNCT
iajs-2376	30	11	〈	〈	PROPN
iajs-2376	30	12	̅	̅	NOUN
iajs-2376	30	13	〉	〉	NOUN
iajs-2376	30	14	,	,	PUNCT
iajs-2376	30	15	〈	〈	NOUN
iajs-2376	30	16	̅	̅	NOUN
iajs-2376	30	17	〉	〉	NOUN
iajs-2376	30	18	and	and	CCONJ
iajs-2376	30	19	〈	〈	PROPN
iajs-2376	30	20	̅	̅	NOUN
iajs-2376	30	21	〉	〉	NOUN
iajs-2376	30	22	.	.	PUNCT
iajs-2376	31	1	the	the	DET
iajs-2376	31	2	submodule	submodule	NOUN
iajs-2376	31	3	{	{	PUNCT
iajs-2376	31	4	̅	̅	NOUN
iajs-2376	31	5	̅	̅	NOUN
iajs-2376	31	6	̅	̅	NOUN
iajs-2376	31	7	}	}	PUNCT
iajs-2376	31	8	is	be	AUX
iajs-2376	31	9	-closed	-close	VERB
iajs-2376	31	10	but	but	CCONJ
iajs-2376	31	11	not	not	PART
iajs-2376	31	12	closed	close	VERB
iajs-2376	31	13	because	because	SCONJ
iajs-2376	31	14	̅	̅	NOUN
iajs-2376	31	15	,	,	PUNCT
iajs-2376	31	16	̅	̅	NOUN
iajs-2376	31	17	,	,	PUNCT
iajs-2376	31	18	̅	̅	NOUN
iajs-2376	31	19	,	,	PUNCT
iajs-2376	31	20	̅	̅	NOUN
iajs-2376	31	21	,	,	PUNCT
iajs-2376	31	22	̅	̅	NOUN
iajs-2376	31	23	̅̅̅̅	̅̅̅̅	NOUN
iajs-2376	31	24	are	be	AUX
iajs-2376	31	25	in	in	ADP
iajs-2376	31	26	but	but	CCONJ
iajs-2376	31	27	not	not	PART
iajs-2376	31	28	in	in	ADV
iajs-2376	31	29	,	,	PUNCT
iajs-2376	31	30	then	then	ADV
iajs-2376	31	31	there	there	PRON
iajs-2376	31	32	exists	exist	VERB
iajs-2376	31	33	a	a	DET
iajs-2376	31	34	-closed	-close	VERB
iajs-2376	31	35	submodule	submodule	NOUN
iajs-2376	31	36	〈	〈	NOUN
iajs-2376	31	37	̅	̅	NOUN
iajs-2376	31	38	〉	〉	NOUN
iajs-2376	31	39	in	in	ADP
iajs-2376	31	40	such	such	ADJ
iajs-2376	31	41	that	that	PRON
iajs-2376	31	42	and	and	CCONJ
iajs-2376	31	43	̅	̅	NOUN
iajs-2376	31	44	,	,	PUNCT
iajs-2376	31	45	̅	̅	NOUN
iajs-2376	31	46	,	,	PUNCT
iajs-2376	31	47	̅	̅	NOUN
iajs-2376	31	48	,	,	PUNCT
iajs-2376	31	49	̅	̅	NOUN
iajs-2376	31	50	,	,	PUNCT
iajs-2376	31	51	̅	̅	NOUN
iajs-2376	31	52	̅̅̅̅	̅̅̅̅	NOUN
iajs-2376	31	53	are	be	AUX
iajs-2376	31	54	not	not	PART
iajs-2376	31	55	in	in	ADV
iajs-2376	31	56	.	.	PUNCT
iajs-2376	32	1	now	now	ADV
iajs-2376	32	2	is	be	AUX
iajs-2376	32	3	not	not	PART
iajs-2376	32	4	a	a	DET
iajs-2376	32	5	-closed	-close	VERB
iajs-2376	32	6	since	since	SCONJ
iajs-2376	32	7	is	be	AUX
iajs-2376	32	8	a	a	DET
iajs-2376	32	9	weak	weak	ADJ
iajs-2376	32	10	essential	essential	ADJ
iajs-2376	32	11	submodule	submodule	NOUN
iajs-2376	32	12	of	of	ADP
iajs-2376	32	13	.	.	PUNCT
iajs-2376	33	1	2	2	X
iajs-2376	33	2	.	.	X
iajs-2376	33	3	a	a	DET
iajs-2376	33	4	direct	direct	ADJ
iajs-2376	33	5	summand	summand	NOUN
iajs-2376	33	6	of	of	ADP
iajs-2376	33	7	an	an	DET
iajs-2376	33	8	-module	-module	NOUN
iajs-2376	33	9	is	be	AUX
iajs-2376	33	10	not	not	PART
iajs-2376	33	11	necessary	necessary	ADJ
iajs-2376	33	12	-closed	-close	VERB
iajs-2376	33	13	submodule	submodule	NOUN
iajs-2376	33	14	in	in	ADV
iajs-2376	33	15	,	,	PUNCT
iajs-2376	33	16	for	for	ADP
iajs-2376	33	17	example	example	NOUN
iajs-2376	33	18	:	:	PUNCT
iajs-2376	33	19	consider	consider	VERB
iajs-2376	33	20	the	the	DET
iajs-2376	33	21	-module	-module	NOUN
iajs-2376	33	22	.	.	PUNCT
iajs-2376	34	1	in	in	ADP
iajs-2376	34	2	this	this	DET
iajs-2376	34	3	module	module	NOUN
iajs-2376	34	4	there	there	PRON
iajs-2376	34	5	are	be	VERB
iajs-2376	34	6	ten	ten	NUM
iajs-2376	34	7	proper	proper	ADJ
iajs-2376	34	8	submodules	submodule	NOUN
iajs-2376	34	9	〈	〈	PROPN
iajs-2376	34	10	̅	̅	NOUN
iajs-2376	34	11	〉	〉	NOUN
iajs-2376	34	12	,	,	PUNCT
iajs-2376	34	13	〈	〈	PROPN
iajs-2376	34	14	̅	̅	NOUN
iajs-2376	34	15	〉	〉	NOUN
iajs-2376	34	16	,	,	PUNCT
iajs-2376	34	17	〈	〈	PROPN
iajs-2376	34	18	̅	̅	NOUN
iajs-2376	34	19	〉	〉	NOUN
iajs-2376	34	20	,	,	PUNCT
iajs-2376	34	21	〈	〈	PROPN
iajs-2376	34	22	̅	̅	NOUN
iajs-2376	34	23	〉	〉	NOUN
iajs-2376	34	24	,	,	PUNCT
iajs-2376	34	25	〈	〈	PROPN
iajs-2376	34	26	̅	̅	NOUN
iajs-2376	34	27	〉	〉	NOUN
iajs-2376	34	28	,	,	PUNCT
iajs-2376	34	29	〈	〈	PROPN
iajs-2376	34	30	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	34	31	〉	〉	NOUN
iajs-2376	34	32	,	,	PUNCT
iajs-2376	34	33	〈	〈	PROPN
iajs-2376	34	34	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	34	35	〉	〉	NOUN
iajs-2376	34	36	,	,	PUNCT
iajs-2376	34	37	〈	〈	PROPN
iajs-2376	34	38	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	34	39	〉	〉	NOUN
iajs-2376	34	40	,	,	PUNCT
iajs-2376	34	41	〈	〈	PROPN
iajs-2376	34	42	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	34	43	〉	〉	NOUN
iajs-2376	34	44	,	,	PUNCT
iajs-2376	34	45	〈	〈	PROPN
iajs-2376	34	46	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	34	47	〉	〉	NOUN
iajs-2376	34	48	where	where	SCONJ
iajs-2376	34	49	〈	〈	PROPN
iajs-2376	34	50	̅	̅	NOUN
iajs-2376	34	51	〉	〉	NOUN
iajs-2376	34	52	〈	〈	NOUN
iajs-2376	34	53	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	34	54	〉	〉	NOUN
iajs-2376	34	55	.	.	PUNCT
iajs-2376	35	1	that	that	PRON
iajs-2376	35	2	is	be	AUX
iajs-2376	35	3	both	both	PRON
iajs-2376	35	4	and	and	CCONJ
iajs-2376	35	5	are	be	AUX
iajs-2376	35	6	a	a	DET
iajs-2376	35	7	direct	direct	ADJ
iajs-2376	35	8	summands	summand	NOUN
iajs-2376	35	9	in	in	ADP
iajs-2376	35	10	.	.	PUNCT
iajs-2376	36	1	but	but	CCONJ
iajs-2376	36	2	〈	〈	PROPN
iajs-2376	36	3	̅	̅	NOUN
iajs-2376	36	4	〉	〉	NOUN
iajs-2376	36	5	is	be	AUX
iajs-2376	36	6	not	not	PART
iajs-2376	36	7	-closed	-close	VERB
iajs-2376	36	8	submodule	submodule	NOUN
iajs-2376	36	9	in	in	ADP
iajs-2376	36	10	,	,	PUNCT
iajs-2376	36	11	since	since	SCONJ
iajs-2376	36	12	̅	̅	NOUN
iajs-2376	36	13	,	,	PUNCT
iajs-2376	36	14	̅	̅	NOUN
iajs-2376	36	15	〈	〈	PRON
iajs-2376	36	16	̅	̅	NOUN
iajs-2376	36	17	〉	〉	NOUN
iajs-2376	36	18	,	,	PUNCT
iajs-2376	36	19	then	then	ADV
iajs-2376	36	20	there	there	PRON
iajs-2376	36	21	exists	exist	VERB
iajs-2376	36	22	a	a	DET
iajs-2376	36	23	-closed	-close	VERB
iajs-2376	36	24	submodule	submodule	NOUN
iajs-2376	36	25	〈	〈	NOUN
iajs-2376	36	26	̅	̅	NOUN
iajs-2376	36	27	〉	〉	NOUN
iajs-2376	36	28	in	in	ADP
iajs-2376	36	29	with	with	ADP
iajs-2376	36	30	〈	〈	PROPN
iajs-2376	36	31	̅	̅	NOUN
iajs-2376	36	32	〉	〉	NOUN
iajs-2376	36	33	〈	〈	NOUN
iajs-2376	36	34	̅	̅	NOUN
iajs-2376	36	35	〉	〉	NOUN
iajs-2376	36	36	,	,	PUNCT
iajs-2376	36	37	but	but	CCONJ
iajs-2376	36	38	̅	̅	NOUN
iajs-2376	36	39	〈	〈	PRON
iajs-2376	36	40	̅	̅	NOUN
iajs-2376	36	41	〉	〉	NOUN
iajs-2376	36	42	.	.	PUNCT
iajs-2376	37	1	also	also	ADV
iajs-2376	37	2	〈	〈	PROPN
iajs-2376	37	3	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	37	4	〉	〉	NOUN
iajs-2376	37	5	is	be	AUX
iajs-2376	37	6	not	not	PART
iajs-2376	37	7	-closed	-close	VERB
iajs-2376	37	8	submodule	submodule	NOUN
iajs-2376	37	9	in	in	ADP
iajs-2376	37	10	,	,	PUNCT
iajs-2376	37	11	since	since	SCONJ
iajs-2376	37	12	̅	̅	NOUN
iajs-2376	37	13	,	,	PUNCT
iajs-2376	37	14	̅	̅	NOUN
iajs-2376	37	15	〈	〈	PROPN
iajs-2376	37	16	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	37	17	〉	〉	NOUN
iajs-2376	37	18	,	,	PUNCT
iajs-2376	37	19	then	then	ADV
iajs-2376	37	20	there	there	PRON
iajs-2376	37	21	exists	exist	VERB
iajs-2376	37	22	a	a	DET
iajs-2376	37	23	-closed	-close	VERB
iajs-2376	37	24	submodule	submodule	NOUN
iajs-2376	37	25	〈	〈	NOUN
iajs-2376	37	26	̅	̅	NOUN
iajs-2376	37	27	〉	〉	NOUN
iajs-2376	37	28	in	in	ADP
iajs-2376	37	29	such	such	ADJ
iajs-2376	37	30	that	that	SCONJ
iajs-2376	37	31	〈	〈	PROPN
iajs-2376	37	32	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	37	33	〉	〉	NOUN
iajs-2376	37	34	〈	〈	NOUN
iajs-2376	37	35	̅	̅	NOUN
iajs-2376	37	36	〉	〉	NOUN
iajs-2376	37	37	,	,	PUNCT
iajs-2376	37	38	but	but	CCONJ
iajs-2376	37	39	̅	̅	NOUN
iajs-2376	37	40	〈	〈	PRON
iajs-2376	37	41	̅	̅	NOUN
iajs-2376	37	42	〉	〉	NOUN
iajs-2376	37	43	.	.	PUNCT
iajs-2376	38	1	3	3	NUM
iajs-2376	38	2	.	.	X
iajs-2376	39	1	if	if	SCONJ
iajs-2376	39	2	is	be	AUX
iajs-2376	39	3	a	a	DET
iajs-2376	39	4	-closed	-closed	ADJ
iajs-2376	39	5	submodule	submodule	NOUN
iajs-2376	39	6	in	in	ADP
iajs-2376	39	7	an	an	DET
iajs-2376	39	8	-module	-module	NOUN
iajs-2376	39	9	,	,	PUNCT
iajs-2376	39	10	then	then	ADV
iajs-2376	39	11	[	[	PUNCT
iajs-2376	39	12	]	]	X
iajs-2376	39	13	need	need	AUX
iajs-2376	39	14	not	not	PART
iajs-2376	39	15	be	be	AUX
iajs-2376	39	16	a	a	DET
iajs-2376	39	17	-closed	-close	VERB
iajs-2376	39	18	ideal	ideal	NOUN
iajs-2376	39	19	in	in	ADP
iajs-2376	39	20	.	.	PUNCT
iajs-2376	40	1	for	for	ADP
iajs-2376	40	2	example	example	NOUN
iajs-2376	40	3	:	:	PUNCT
iajs-2376	40	4	consider	consider	VERB
iajs-2376	40	5	the	the	DET
iajs-2376	40	6	-module	-module	NOUN
iajs-2376	40	7	.	.	PUNCT
iajs-2376	41	1	in	in	ADP
iajs-2376	41	2	this	this	DET
iajs-2376	41	3	module	module	NOUN
iajs-2376	41	4	there	there	PRON
iajs-2376	41	5	are	be	VERB
iajs-2376	41	6	four	four	NUM
iajs-2376	41	7	proper	proper	ADJ
iajs-2376	41	8	submodules	submodule	NOUN
iajs-2376	41	9	,	,	PUNCT
iajs-2376	41	10	which	which	PRON
iajs-2376	41	11	are	be	AUX
iajs-2376	41	12	:	:	PUNCT
iajs-2376	41	13	〈	〈	NOUN
iajs-2376	41	14	̅	̅	NOUN
iajs-2376	41	15	〉	〉	NOUN
iajs-2376	41	16	,	,	PUNCT
iajs-2376	41	17	〈	〈	PROPN
iajs-2376	41	18	̅	̅	NOUN
iajs-2376	41	19	〉	〉	NOUN
iajs-2376	41	20	,	,	PUNCT
iajs-2376	41	21	〈	〈	PROPN
iajs-2376	41	22	̅	̅	NOUN
iajs-2376	41	23	〉	〉	NOUN
iajs-2376	41	24	,	,	PUNCT
iajs-2376	41	25	〈	〈	PROPN
iajs-2376	41	26	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	41	27	〉	〉	NOUN
iajs-2376	41	28	.	.	PUNCT
iajs-2376	42	1	not	not	PART
iajs-2376	42	2	that	that	SCONJ
iajs-2376	42	3	the	the	DET
iajs-2376	42	4	submodule	submodule	NOUN
iajs-2376	42	5	is	be	AUX
iajs-2376	42	6	a	a	DET
iajs-2376	42	7	-closed	-closed	ADJ
iajs-2376	42	8	submodule	submodule	NOUN
iajs-2376	42	9	in	in	ADP
iajs-2376	42	10	(	(	PUNCT
iajs-2376	42	11	since	since	SCONJ
iajs-2376	42	12	is	be	AUX
iajs-2376	42	13	a	a	DET
iajs-2376	42	14	-closed	-closed	ADJ
iajs-2376	42	15	submodule	submodule	NOUN
iajs-2376	42	16	in	in	ADP
iajs-2376	42	17	and	and	CCONJ
iajs-2376	42	18	hence	hence	ADV
iajs-2376	42	19	by	by	ADP
iajs-2376	42	20	(	(	PUNCT
iajs-2376	42	21	1	1	X
iajs-2376	42	22	)	)	PUNCT
iajs-2376	42	23	is	be	AUX
iajs-2376	42	24	a	a	DET
iajs-2376	42	25	closed	closed	ADJ
iajs-2376	42	26	)	)	PUNCT
iajs-2376	42	27	.	.	PUNCT
iajs-2376	43	1	while	while	SCONJ
iajs-2376	43	2	[	[	PUNCT
iajs-2376	43	3	]	]	X
iajs-2376	43	4	is	be	AUX
iajs-2376	43	5	not	not	PART
iajs-2376	43	6	a	a	DET
iajs-2376	43	7	-closed	-close	VERB
iajs-2376	43	8	ideal	ideal	NOUN
iajs-2376	43	9	in	in	ADP
iajs-2376	43	10	.	.	PUNCT
iajs-2376	44	1	since	since	SCONJ
iajs-2376	44	2	there	there	PRON
iajs-2376	44	3	is	be	VERB
iajs-2376	44	4	no	no	DET
iajs-2376	44	5	a	a	DET
iajs-2376	44	6	-closed	-closed	ADJ
iajs-2376	44	7	ideal	ideal	NOUN
iajs-2376	44	8	in	in	ADP
iajs-2376	44	9	containing	contain	VERB
iajs-2376	44	10	.	.	PUNCT
iajs-2376	45	1	4	4	X
iajs-2376	45	2	.	.	X
iajs-2376	46	1	if	if	SCONJ
iajs-2376	46	2	is	be	AUX
iajs-2376	46	3	a	a	DET
iajs-2376	46	4	module	module	NOUN
iajs-2376	46	5	and	and	CCONJ
iajs-2376	46	6	is	be	AUX
iajs-2376	46	7	a	a	DET
iajs-2376	46	8	-closed	-closed	ADJ
iajs-2376	46	9	submodule	submodule	NOUN
iajs-2376	46	10	in	in	ADP
iajs-2376	46	11	with	with	ADP
iajs-2376	46	12	is	be	AUX
iajs-2376	46	13	a	a	DET
iajs-2376	46	14	submodule	submodule	NOUN
iajs-2376	46	15	of	of	ADP
iajs-2376	46	16	such	such	ADJ
iajs-2376	46	17	that	that	PRON
iajs-2376	46	18	,	,	PUNCT
iajs-2376	46	19	then	then	ADV
iajs-2376	46	20	it	it	PRON
iajs-2376	46	21	is	be	AUX
iajs-2376	46	22	not	not	PART
iajs-2376	46	23	necessary	necessary	ADJ
iajs-2376	46	24	that	that	PRON
iajs-2376	46	25	is	be	AUX
iajs-2376	46	26	a	a	DET
iajs-2376	46	27	-closed	-closed	ADJ
iajs-2376	46	28	submodule	submodule	NOUN
iajs-2376	46	29	in	in	ADP
iajs-2376	46	30	.	.	PUNCT
iajs-2376	47	1	for	for	ADP
iajs-2376	47	2	example	example	NOUN
iajs-2376	47	3	:	:	PUNCT
iajs-2376	47	4	the	the	DET
iajs-2376	47	5	-module	-module	NOUN
iajs-2376	47	6	is	be	AUX
iajs-2376	47	7	a	a	DET
iajs-2376	47	8	-closed	-closed	ADJ
iajs-2376	47	9	submodule	submodule	NOUN
iajs-2376	47	10	in	in	ADP
iajs-2376	47	11	(	(	PUNCT
iajs-2376	47	12	since	since	SCONJ
iajs-2376	47	13	as	as	SCONJ
iajs-2376	47	14	a	a	DET
iajs-2376	47	15	-module	-module	NOUN
iajs-2376	47	16	is	be	AUX
iajs-2376	47	17	a	a	DET
iajs-2376	47	18	-closed	-close	VERB
iajs-2376	47	19	in	in	ADP
iajs-2376	47	20	)	)	PUNCT
iajs-2376	47	21	and	and	CCONJ
iajs-2376	47	22	,	,	PUNCT
iajs-2376	47	23	but	but	CCONJ
iajs-2376	47	24	is	be	AUX
iajs-2376	47	25	not	not	PART
iajs-2376	47	26	a	a	DET
iajs-2376	47	27	-closed	-close	VERB
iajs-2376	47	28	submodule	submodule	NOUN
iajs-2376	47	29	in	in	ADP
iajs-2376	47	30	.	.	PUNCT
iajs-2376	48	1	5	5	X
iajs-2376	48	2	.	.	X
iajs-2376	49	1	if	if	SCONJ
iajs-2376	49	2	is	be	AUX
iajs-2376	49	3	an	an	DET
iajs-2376	49	4	-module	-module	NOUN
iajs-2376	49	5	,	,	PUNCT
iajs-2376	49	6	and	and	CCONJ
iajs-2376	49	7	are	be	AUX
iajs-2376	49	8	submodules	submodule	NOUN
iajs-2376	49	9	of	of	ADP
iajs-2376	49	10	an	an	DET
iajs-2376	49	11	-module	-module	NOUN
iajs-2376	49	12	such	such	ADJ
iajs-2376	49	13	that	that	PRON
iajs-2376	49	14	and	and	CCONJ
iajs-2376	49	15	is	be	AUX
iajs-2376	49	16	a	a	DET
iajs-2376	49	17	-closed	-closed	ADJ
iajs-2376	49	18	submodule	submodule	NOUN
iajs-2376	49	19	in	in	ADP
iajs-2376	49	20	,	,	PUNCT
iajs-2376	49	21	then	then	ADV
iajs-2376	49	22	need	need	VERB
iajs-2376	49	23	not	not	PART
iajs-2376	49	24	to	to	PART
iajs-2376	49	25	be	be	AUX
iajs-2376	49	26	a	a	DET
iajs-2376	49	27	-closed	-closed	ADJ
iajs-2376	49	28	submodule	submodule	NOUN
iajs-2376	49	29	in	in	ADP
iajs-2376	49	30	.	.	PUNCT
iajs-2376	50	1	for	for	ADP
iajs-2376	50	2	example	example	NOUN
iajs-2376	50	3	:	:	PUNCT
iajs-2376	50	4	48	48	NUM
iajs-2376	50	5	ibn	ibn	PROPN
iajs-2376	50	6	al	al	PROPN
iajs-2376	50	7	-	-	PUNCT
iajs-2376	50	8	haitham	haitham	PROPN
iajs-2376	50	9	jour	jour	X
iajs-2376	50	10	.	.	PROPN
iajs-2376	51	1	for	for	ADP
iajs-2376	51	2	pure	pure	ADJ
iajs-2376	51	3	&	&	CCONJ
iajs-2376	51	4	appl	appl	PROPN
iajs-2376	51	5	.	.	PUNCT
iajs-2376	52	1	sci	sci	PROPN
iajs-2376	52	2	.	.	PROPN
iajs-2376	53	1	33	33	NUM
iajs-2376	53	2	(	(	PUNCT
iajs-2376	53	3	1	1	NUM
iajs-2376	53	4	)	)	PUNCT
iajs-2376	53	5	2020	2020	NUM
iajs-2376	53	6	consider	consider	VERB
iajs-2376	53	7	the	the	DET
iajs-2376	53	8	-module	-module	NOUN
iajs-2376	53	9	is	be	AUX
iajs-2376	53	10	a	a	DET
iajs-2376	53	11	-closed	-closed	ADJ
iajs-2376	53	12	submodule	submodule	NOUN
iajs-2376	53	13	in	in	ADP
iajs-2376	53	14	and	and	CCONJ
iajs-2376	53	15	,	,	PUNCT
iajs-2376	53	16	while	while	SCONJ
iajs-2376	53	17	is	be	AUX
iajs-2376	53	18	not	not	PART
iajs-2376	53	19	closed	close	VERB
iajs-2376	53	20	submodule	submodule	NOUN
iajs-2376	53	21	in	in	ADP
iajs-2376	53	22	,	,	PUNCT
iajs-2376	53	23	,	,	PUNCT
iajs-2376	53	24	since	since	SCONJ
iajs-2376	53	25	there	there	PRON
iajs-2376	53	26	is	be	VERB
iajs-2376	53	27	no	no	DET
iajs-2376	53	28	a	a	DET
iajs-2376	53	29	-closed	-closed	ADJ
iajs-2376	53	30	submodule	submodule	NOUN
iajs-2376	53	31	in	in	ADP
iajs-2376	53	32	containing	contain	VERB
iajs-2376	53	33	.	.	PUNCT
iajs-2376	54	1	6	6	X
iajs-2376	54	2	.	.	X
iajs-2376	55	1	if	if	SCONJ
iajs-2376	55	2	is	be	AUX
iajs-2376	55	3	an	an	DET
iajs-2376	55	4	-module	-module	NOUN
iajs-2376	55	5	,	,	PUNCT
iajs-2376	55	6	and	and	CCONJ
iajs-2376	55	7	are	be	AUX
iajs-2376	55	8	submodules	submodule	NOUN
iajs-2376	55	9	of	of	ADP
iajs-2376	55	10	an	an	DET
iajs-2376	55	11	-module	-module	NOUN
iajs-2376	55	12	such	such	ADJ
iajs-2376	55	13	that	that	PRON
iajs-2376	55	14	and	and	CCONJ
iajs-2376	55	15	is	be	AUX
iajs-2376	55	16	a	a	DET
iajs-2376	55	17	-closed	-closed	ADJ
iajs-2376	55	18	submodule	submodule	NOUN
iajs-2376	55	19	in	in	ADP
iajs-2376	55	20	,	,	PUNCT
iajs-2376	55	21	then	then	ADV
iajs-2376	55	22	need	need	VERB
iajs-2376	55	23	not	not	PART
iajs-2376	55	24	to	to	PART
iajs-2376	55	25	be	be	AUX
iajs-2376	55	26	a	a	DET
iajs-2376	55	27	-closed	-closed	ADJ
iajs-2376	55	28	submodule	submodule	NOUN
iajs-2376	55	29	in	in	ADP
iajs-2376	55	30	.	.	PUNCT
iajs-2376	56	1	for	for	ADP
iajs-2376	56	2	example	example	NOUN
iajs-2376	56	3	:	:	PUNCT
iajs-2376	56	4	consider	consider	VERB
iajs-2376	56	5	the	the	DET
iajs-2376	56	6	-module	-module	NOUN
iajs-2376	56	7	and	and	CCONJ
iajs-2376	56	8	the	the	DET
iajs-2376	56	9	submodules	submodule	NOUN
iajs-2376	56	10	and	and	CCONJ
iajs-2376	56	11	.	.	PUNCT
iajs-2376	57	1	notes	note	NOUN
iajs-2376	57	2	that	that	PRON
iajs-2376	57	3	is	be	AUX
iajs-2376	57	4	a	a	DET
iajs-2376	57	5	closed	closed	ADJ
iajs-2376	57	6	submodule	submodule	NOUN
iajs-2376	57	7	in	in	ADP
iajs-2376	57	8	(	(	PUNCT
iajs-2376	57	9	since	since	SCONJ
iajs-2376	57	10	is	be	AUX
iajs-2376	57	11	a	a	DET
iajs-2376	57	12	-closed	-close	VERB
iajs-2376	57	13	in	in	ADP
iajs-2376	57	14	)	)	PUNCT
iajs-2376	57	15	,	,	PUNCT
iajs-2376	57	16	but	but	CCONJ
iajs-2376	57	17	is	be	AUX
iajs-2376	57	18	not	not	PART
iajs-2376	57	19	-closed	-close	VERB
iajs-2376	57	20	submodule	submodule	NOUN
iajs-2376	57	21	in	in	ADP
iajs-2376	57	22	,	,	PUNCT
iajs-2376	57	23	since	since	SCONJ
iajs-2376	57	24	there	there	PRON
iajs-2376	57	25	is	be	VERB
iajs-2376	57	26	no	no	DET
iajs-2376	57	27	a	a	DET
iajs-2376	57	28	-closed	-closed	ADJ
iajs-2376	57	29	submodule	submodule	NOUN
iajs-2376	57	30	in	in	ADP
iajs-2376	57	31	containing	contain	VERB
iajs-2376	57	32	.	.	PUNCT
iajs-2376	58	1	7	7	X
iajs-2376	58	2	.	.	PUNCT
iajs-2376	58	3	[	[	PUNCT
iajs-2376	58	4	]	]	X
iajs-2376	58	5	shows	show	VERB
iajs-2376	58	6	that	that	SCONJ
iajs-2376	58	7	the	the	DET
iajs-2376	58	8	intersection	intersection	NOUN
iajs-2376	58	9	of	of	ADP
iajs-2376	58	10	two	two	NUM
iajs-2376	58	11	closed	closed	ADJ
iajs-2376	58	12	submodules	submodule	NOUN
iajs-2376	58	13	need	need	VERB
iajs-2376	58	14	not	not	PART
iajs-2376	58	15	to	to	PART
iajs-2376	58	16	be	be	AUX
iajs-2376	58	17	closed	close	VERB
iajs-2376	58	18	submodule	submodule	NOUN
iajs-2376	58	19	”	"	PUNCT
iajs-2376	58	20	.	.	PUNCT
iajs-2376	59	1	also	also	ADV
iajs-2376	59	2	,	,	PUNCT
iajs-2376	59	3	the	the	DET
iajs-2376	59	4	intersection	intersection	NOUN
iajs-2376	59	5	of	of	ADP
iajs-2376	59	6	two	two	NUM
iajs-2376	59	7	-closed	-close	VERB
iajs-2376	59	8	submodules	submodule	NOUN
iajs-2376	59	9	need	need	VERB
iajs-2376	59	10	not	not	PART
iajs-2376	59	11	to	to	PART
iajs-2376	59	12	be	be	AUX
iajs-2376	59	13	closed	close	VERB
iajs-2376	59	14	submodule	submodule	NOUN
iajs-2376	59	15	as	as	SCONJ
iajs-2376	59	16	the	the	DET
iajs-2376	59	17	following	follow	VERB
iajs-2376	59	18	example	example	NOUN
iajs-2376	59	19	shows	show	VERB
iajs-2376	59	20	:	:	PUNCT
iajs-2376	59	21	the	the	DET
iajs-2376	59	22	-module	-module	NOUN
iajs-2376	59	23	the	the	DET
iajs-2376	59	24	submodules	submodule	NOUN
iajs-2376	59	25	〈	〈	NOUN
iajs-2376	59	26	̅	̅	NOUN
iajs-2376	59	27	̅	̅	NOUN
iajs-2376	59	28	〉	〉	NOUN
iajs-2376	59	29	and	and	CCONJ
iajs-2376	59	30	〈	〈	NOUN
iajs-2376	59	31	̅	̅	NOUN
iajs-2376	59	32	̅	̅	NOUN
iajs-2376	59	33	〉	〉	NOUN
iajs-2376	59	34	are	be	AUX
iajs-2376	59	35	-closed	-close	VERB
iajs-2376	59	36	submodules	submodule	NOUN
iajs-2376	59	37	in	in	ADP
iajs-2376	59	38	(	(	PUNCT
iajs-2376	59	39	since	since	SCONJ
iajs-2376	59	40	they	they	PRON
iajs-2376	59	41	are	be	AUX
iajs-2376	59	42	-closed	-close	VERB
iajs-2376	59	43	submodules	submodule	NOUN
iajs-2376	59	44	in	in	ADP
iajs-2376	59	45	)	)	PUNCT
iajs-2376	59	46	,	,	PUNCT
iajs-2376	59	47	but	but	CCONJ
iajs-2376	59	48	〈	〈	PROPN
iajs-2376	59	49	̅	̅	NOUN
iajs-2376	59	50	̅	̅	X
iajs-2376	59	51	〉	〉	NOUN
iajs-2376	59	52	〈	〈	NOUN
iajs-2376	59	53	̅	̅	NOUN
iajs-2376	59	54	̅	̅	X
iajs-2376	59	55	〉	〉	NOUN
iajs-2376	59	56	〈	〈	NOUN
iajs-2376	59	57	̅	̅	NOUN
iajs-2376	59	58	̅	̅	NOUN
iajs-2376	59	59	〉	〉	NOUN
iajs-2376	59	60	is	be	AUX
iajs-2376	59	61	not	not	PART
iajs-2376	59	62	closed	close	VERB
iajs-2376	59	63	submodule	submodule	NOUN
iajs-2376	59	64	in	in	ADP
iajs-2376	59	65	since	since	SCONJ
iajs-2376	59	66	〈	〈	NOUN
iajs-2376	59	67	̅	̅	NOUN
iajs-2376	59	68	̅	̅	NOUN
iajs-2376	59	69	〉	〉	NOUN
iajs-2376	59	70	is	be	AUX
iajs-2376	59	71	closed	close	VERB
iajs-2376	59	72	submodule	submodule	NOUN
iajs-2376	59	73	in	in	ADP
iajs-2376	59	74	.	.	PUNCT
iajs-2376	60	1	8	8	X
iajs-2376	60	2	.	.	NUM
iajs-2376	60	3	closed	close	VERB
iajs-2376	60	4	submodules	submodule	NOUN
iajs-2376	60	5	and	and	CCONJ
iajs-2376	60	6	-closed	-close	VERB
iajs-2376	60	7	submodules	submodule	NOUN
iajs-2376	60	8	are	be	AUX
iajs-2376	60	9	independent	independent	ADJ
iajs-2376	60	10	concepts	concept	NOUN
iajs-2376	60	11	,	,	PUNCT
iajs-2376	60	12	as	as	SCONJ
iajs-2376	60	13	the	the	DET
iajs-2376	60	14	following	follow	VERB
iajs-2376	60	15	examples	example	NOUN
iajs-2376	60	16	show	show	VERB
iajs-2376	60	17	that	that	SCONJ
iajs-2376	60	18	:	:	PUNCT
iajs-2376	60	19	consider	consider	VERB
iajs-2376	60	20	the	the	DET
iajs-2376	60	21	-module	-module	NOUN
iajs-2376	60	22	.	.	PUNCT
iajs-2376	61	1	in	in	ADP
iajs-2376	61	2	this	this	DET
iajs-2376	61	3	module	module	NOUN
iajs-2376	61	4	the	the	DET
iajs-2376	61	5	proper	proper	ADJ
iajs-2376	61	6	submodules	submodule	NOUN
iajs-2376	61	7	of	of	ADP
iajs-2376	61	8	are	be	AUX
iajs-2376	61	9	:	:	PUNCT
iajs-2376	61	10	〈	〈	NOUN
iajs-2376	61	11	̅	̅	NOUN
iajs-2376	61	12	〉	〉	NOUN
iajs-2376	61	13	,	,	PUNCT
iajs-2376	61	14	〈	〈	PROPN
iajs-2376	61	15	̅	̅	NOUN
iajs-2376	61	16	〉	〉	NOUN
iajs-2376	61	17	,	,	PUNCT
iajs-2376	61	18	〈	〈	PROPN
iajs-2376	61	19	̅	̅	NOUN
iajs-2376	61	20	〉	〉	NOUN
iajs-2376	61	21	,	,	PUNCT
iajs-2376	61	22	〈	〈	PROPN
iajs-2376	61	23	̅	̅	NOUN
iajs-2376	61	24	〉	〉	NOUN
iajs-2376	61	25	,	,	PUNCT
iajs-2376	61	26	〈	〈	PROPN
iajs-2376	61	27	̅	̅	NOUN
iajs-2376	61	28	〉	〉	NOUN
iajs-2376	61	29	,	,	PUNCT
iajs-2376	61	30	〈	〈	PROPN
iajs-2376	61	31	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	61	32	〉	〉	NOUN
iajs-2376	61	33	,	,	PUNCT
iajs-2376	61	34	〈	〈	PROPN
iajs-2376	61	35	̅̅̅̅	̅̅̅̅	ADJ
iajs-2376	61	36	〉	〉	NOUN
iajs-2376	61	37	.	.	PUNCT
iajs-2376	62	1	we	we	PRON
iajs-2376	62	2	notes	note	VERB
iajs-2376	62	3	that	that	SCONJ
iajs-2376	62	4	the	the	DET
iajs-2376	62	5	submodule	submodule	NOUN
iajs-2376	62	6	〈	〈	PROPN
iajs-2376	62	7	̅	̅	NOUN
iajs-2376	62	8	〉	〉	NOUN
iajs-2376	62	9	is	be	AUX
iajs-2376	62	10	closed	close	VERB
iajs-2376	62	11	in	in	ADP
iajs-2376	62	12	,	,	PUNCT
iajs-2376	62	13	since	since	SCONJ
iajs-2376	62	14	has	have	VERB
iajs-2376	62	15	no	no	DET
iajs-2376	62	16	proper	proper	ADJ
iajs-2376	62	17	essential	essential	ADJ
iajs-2376	62	18	extension	extension	NOUN
iajs-2376	62	19	in	in	ADP
iajs-2376	62	20	,	,	PUNCT
iajs-2376	62	21	while	while	SCONJ
iajs-2376	62	22	〈	〈	NOUN
iajs-2376	62	23	̅	̅	NOUN
iajs-2376	62	24	〉	〉	NOUN
iajs-2376	62	25	is	be	AUX
iajs-2376	62	26	not	not	PART
iajs-2376	62	27	-closed	-close	VERB
iajs-2376	62	28	submodule	submodule	NOUN
iajs-2376	62	29	in	in	ADP
iajs-2376	62	30	since	since	SCONJ
iajs-2376	62	31	̅	̅	NOUN
iajs-2376	62	32	,	,	PUNCT
iajs-2376	62	33	̅	̅	NOUN
iajs-2376	62	34	,	,	PUNCT
iajs-2376	62	35	then	then	ADV
iajs-2376	62	36	there	there	PRON
iajs-2376	62	37	exists	exist	VERB
iajs-2376	62	38	a	a	DET
iajs-2376	62	39	-closed	-close	VERB
iajs-2376	62	40	submodule	submodule	NOUN
iajs-2376	62	41	〈	〈	NOUN
iajs-2376	62	42	̅	̅	NOUN
iajs-2376	62	43	〉	〉	NOUN
iajs-2376	62	44	in	in	ADP
iajs-2376	62	45	with	with	ADP
iajs-2376	62	46	,	,	PUNCT
iajs-2376	62	47	but	but	CCONJ
iajs-2376	62	48	̅	̅	VERB
iajs-2376	62	49	.	.	PUNCT
iajs-2376	63	1	that	that	PRON
iajs-2376	63	2	is	be	AUX
iajs-2376	63	3	〈	〈	PROPN
iajs-2376	63	4	̅	̅	NOUN
iajs-2376	63	5	〉	〉	NOUN
iajs-2376	63	6	closed	close	VERB
iajs-2376	63	7	submodule	submodule	NOUN
iajs-2376	63	8	in	in	ADP
iajs-2376	63	9	,	,	PUNCT
iajs-2376	63	10	but	but	CCONJ
iajs-2376	63	11	not	not	PART
iajs-2376	63	12	-closed	-close	VERB
iajs-2376	63	13	submodule	submodule	NOUN
iajs-2376	63	14	in	in	ADP
iajs-2376	63	15	.	.	PUNCT
iajs-2376	64	1	in	in	ADP
iajs-2376	64	2	the	the	DET
iajs-2376	64	3	-module	-module	NOUN
iajs-2376	64	4	,	,	PUNCT
iajs-2376	64	5	the	the	DET
iajs-2376	64	6	submodule	submodule	NOUN
iajs-2376	64	7	〈	〈	PROPN
iajs-2376	64	8	̅	̅	NOUN
iajs-2376	64	9	〉	〉	NOUN
iajs-2376	64	10	is	be	AUX
iajs-2376	64	11	a	a	DET
iajs-2376	64	12	closed	closed	ADJ
iajs-2376	64	13	submodule	submodule	NOUN
iajs-2376	64	14	in	in	ADP
iajs-2376	64	15	since	since	SCONJ
iajs-2376	64	16	〈	〈	NOUN
iajs-2376	64	17	̅	̅	NOUN
iajs-2376	64	18	〉	〉	NOUN
iajs-2376	64	19	is	be	AUX
iajs-2376	64	20	a	a	DET
iajs-2376	64	21	-closed	-close	VERB
iajs-2376	64	22	in	in	ADV
iajs-2376	64	23	,	,	PUNCT
iajs-2376	64	24	while	while	SCONJ
iajs-2376	64	25	〈	〈	NOUN
iajs-2376	64	26	̅	̅	NOUN
iajs-2376	64	27	〉	〉	NOUN
iajs-2376	64	28	is	be	AUX
iajs-2376	64	29	not	not	PART
iajs-2376	64	30	closed	close	VERB
iajs-2376	64	31	in	in	ADP
iajs-2376	64	32	,	,	PUNCT
iajs-2376	64	33	since	since	SCONJ
iajs-2376	64	34	〈	〈	PROPN
iajs-2376	64	35	̅	̅	NOUN
iajs-2376	64	36	〉	〉	NOUN
iajs-2376	64	37	is	be	AUX
iajs-2376	64	38	an	an	DET
iajs-2376	64	39	essential	essential	ADJ
iajs-2376	64	40	submodule	submodule	NOUN
iajs-2376	64	41	of	of	ADP
iajs-2376	64	42	.	.	PUNCT
iajs-2376	65	1	we	we	PRON
iajs-2376	65	2	start	start	VERB
iajs-2376	65	3	this	this	DET
iajs-2376	65	4	section	section	NOUN
iajs-2376	65	5	by	by	ADP
iajs-2376	65	6	the	the	DET
iajs-2376	65	7	following	follow	VERB
iajs-2376	65	8	proposition	proposition	NOUN
iajs-2376	65	9	.	.	PUNCT
iajs-2376	66	1	proposition	proposition	NOUN
iajs-2376	66	2	2.3	2.3	NUM
iajs-2376	66	3	if	if	SCONJ
iajs-2376	66	4	is	be	AUX
iajs-2376	66	5	an	an	DET
iajs-2376	66	6	-module	-module	NOUN
iajs-2376	66	7	,	,	PUNCT
iajs-2376	66	8	and	and	CCONJ
iajs-2376	66	9	are	be	AUX
iajs-2376	66	10	submodules	submodule	NOUN
iajs-2376	66	11	of	of	ADP
iajs-2376	66	12	with	with	ADP
iajs-2376	66	13	and	and	CCONJ
iajs-2376	66	14	is	be	AUX
iajs-2376	66	15	a	a	DET
iajs-2376	66	16	-closed	-closed	ADJ
iajs-2376	66	17	submodule	submodule	NOUN
iajs-2376	66	18	in	in	ADP
iajs-2376	66	19	and	and	CCONJ
iajs-2376	66	20	is	be	AUX
iajs-2376	66	21	a	a	DET
iajs-2376	66	22	-closed	-closed	ADJ
iajs-2376	66	23	submodule	submodule	NOUN
iajs-2376	66	24	in	in	ADP
iajs-2376	66	25	,	,	PUNCT
iajs-2376	66	26	then	then	ADV
iajs-2376	66	27	is	be	AUX
iajs-2376	66	28	a	a	DET
iajs-2376	66	29	-closed	-closed	ADJ
iajs-2376	66	30	submodule	submodule	NOUN
iajs-2376	66	31	in	in	ADP
iajs-2376	66	32	,	,	PUNCT
iajs-2376	66	33	provided	provide	VERB
iajs-2376	66	34	that	that	SCONJ
iajs-2376	66	35	for	for	ADP
iajs-2376	66	36	any	any	DET
iajs-2376	66	37	weak	weak	ADJ
iajs-2376	66	38	-	-	PUNCT
iajs-2376	66	39	essential	essential	ADJ
iajs-2376	66	40	extensions	extension	NOUN
iajs-2376	66	41	of	of	ADP
iajs-2376	66	42	.	.	PUNCT
iajs-2376	67	1	proof	proof	NOUN
iajs-2376	67	2	:	:	PUNCT
iajs-2376	67	3	to	to	PART
iajs-2376	67	4	prove	prove	VERB
iajs-2376	67	5	that	that	PRON
iajs-2376	67	6	is	be	AUX
iajs-2376	67	7	a	a	DET
iajs-2376	67	8	-closed	-closed	ADJ
iajs-2376	67	9	submodule	submodule	NOUN
iajs-2376	67	10	in	in	ADV
iajs-2376	67	11	,	,	PUNCT
iajs-2376	67	12	suppose	suppose	VERB
iajs-2376	67	13	that	that	SCONJ
iajs-2376	67	14	with	with	ADP
iajs-2376	67	15	,	,	PUNCT
iajs-2376	67	16	then	then	ADV
iajs-2376	67	17	either	either	ADV
iajs-2376	67	18	or	or	CCONJ
iajs-2376	67	19	.	.	PUNCT
iajs-2376	68	1	if	if	SCONJ
iajs-2376	68	2	and	and	CCONJ
iajs-2376	68	3	since	since	SCONJ
iajs-2376	68	4	is	be	AUX
iajs-2376	68	5	a	a	DET
iajs-2376	68	6	-closed	-closed	ADJ
iajs-2376	68	7	submodule	submodule	NOUN
iajs-2376	68	8	in	in	ADV
iajs-2376	68	9	,	,	PUNCT
iajs-2376	68	10	so	so	CCONJ
iajs-2376	68	11	there	there	PRON
iajs-2376	68	12	exists	exist	VERB
iajs-2376	68	13	a	a	DET
iajs-2376	68	14	-closed	-closed	ADJ
iajs-2376	68	15	submodule	submodule	NOUN
iajs-2376	68	16	in	in	ADP
iajs-2376	68	17	such	such	ADJ
iajs-2376	68	18	that	that	PRON
iajs-2376	68	19	and	and	CCONJ
iajs-2376	68	20	.	.	PUNCT
iajs-2376	69	1	since	since	SCONJ
iajs-2376	69	2	is	be	AUX
iajs-2376	69	3	a	a	DET
iajs-2376	69	4	-closed	-close	VERB
iajs-2376	69	5	in	in	ADP
iajs-2376	69	6	and	and	CCONJ
iajs-2376	69	7	is	be	AUX
iajs-2376	69	8	a	a	DET
iajs-2376	69	9	-closed	-close	VERB
iajs-2376	69	10	in	in	ADP
iajs-2376	69	11	,	,	PUNCT
iajs-2376	69	12	then	then	ADV
iajs-2376	69	13	by	by	SCONJ
iajs-2376	69	14	[	[	PUNCT
iajs-2376	69	15	]	]	X
iajs-2376	69	16	is	be	AUX
iajs-2376	69	17	a	a	DET
iajs-2376	69	18	-closed	-close	VERB
iajs-2376	69	19	in	in	ADV
iajs-2376	69	20	.	.	PUNCT
iajs-2376	70	1	thus	thus	ADV
iajs-2376	70	2	we	we	PRON
iajs-2376	70	3	have	have	VERB
iajs-2376	70	4	a	a	DET
iajs-2376	70	5	-closed	-closed	ADJ
iajs-2376	70	6	submodule	submodule	NOUN
iajs-2376	70	7	in	in	ADP
iajs-2376	70	8	such	such	ADJ
iajs-2376	70	9	that	that	PRON
iajs-2376	70	10	and	and	CCONJ
iajs-2376	70	11	.	.	PUNCT
iajs-2376	71	1	i.e.	i.e.	X
iajs-2376	71	2	is	be	AUX
iajs-2376	71	3	a	a	DET
iajs-2376	71	4	-closed	-closed	ADJ
iajs-2376	71	5	submodule	submodule	NOUN
iajs-2376	71	6	in	in	ADP
iajs-2376	71	7	.	.	PUNCT
iajs-2376	72	1	if	if	SCONJ
iajs-2376	72	2	,	,	PUNCT
iajs-2376	72	3	then	then	ADV
iajs-2376	72	4	nothing	nothing	PRON
iajs-2376	72	5	to	to	PART
iajs-2376	72	6	prove	prove	VERB
iajs-2376	72	7	since	since	SCONJ
iajs-2376	72	8	is	be	AUX
iajs-2376	72	9	a	a	DET
iajs-2376	72	10	-closed	-closed	ADJ
iajs-2376	72	11	submodule	submodule	NOUN
iajs-2376	72	12	in	in	ADP
iajs-2376	72	13	such	such	ADJ
iajs-2376	72	14	that	that	PRON
iajs-2376	72	15	and	and	CCONJ
iajs-2376	72	16	.	.	PUNCT
iajs-2376	73	1	therefore	therefore	ADV
iajs-2376	73	2	is	be	AUX
iajs-2376	73	3	a	a	DET
iajs-2376	73	4	-closed	-closed	ADJ
iajs-2376	73	5	submodule	submodule	NOUN
iajs-2376	73	6	in	in	ADP
iajs-2376	73	7	.	.	PUNCT
iajs-2376	74	1	proposition	proposition	NOUN
iajs-2376	74	2	2.4	2.4	NUM
iajs-2376	74	3	let	let	VERB
iajs-2376	74	4	and	and	CCONJ
iajs-2376	74	5	be	be	AUX
iajs-2376	74	6	submodules	submodule	NOUN
iajs-2376	74	7	of	of	ADP
iajs-2376	74	8	a	a	DET
iajs-2376	74	9	module	module	NOUN
iajs-2376	74	10	with	with	ADP
iajs-2376	74	11	.	.	PUNCT
iajs-2376	75	1	if	if	SCONJ
iajs-2376	75	2	is	be	AUX
iajs-2376	75	3	a	a	DET
iajs-2376	75	4	-closed	-closed	ADJ
iajs-2376	75	5	submodule	submodule	NOUN
iajs-2376	75	6	in	in	ADV
iajs-2376	75	7	,	,	PUNCT
iajs-2376	75	8	and	and	CCONJ
iajs-2376	75	9	is	be	AUX
iajs-2376	75	10	a	a	DET
iajs-2376	75	11	-closed	-close	VERB
iajs-2376	75	12	in	in	ADP
iajs-2376	75	13	,	,	PUNCT
iajs-2376	75	14	then	then	ADV
iajs-2376	75	15	is	be	AUX
iajs-2376	75	16	a	a	DET
iajs-2376	75	17	-closed	-closed	ADJ
iajs-2376	75	18	submodule	submodule	NOUN
iajs-2376	75	19	in	in	ADP
iajs-2376	75	20	,	,	PUNCT
iajs-2376	75	21	provided	provide	VERB
iajs-2376	75	22	that	that	PRON
iajs-2376	75	23	is	be	AUX
iajs-2376	75	24	for	for	ADP
iajs-2376	75	25	any	any	DET
iajs-2376	75	26	weak	weak	ADJ
iajs-2376	75	27	essential	essential	ADJ
iajs-2376	75	28	extensions	extension	NOUN
iajs-2376	75	29	of	of	ADP
iajs-2376	75	30	.	.	PUNCT
iajs-2376	76	1	48	48	NUM
iajs-2376	76	2	ibn	ibn	PROPN
iajs-2376	76	3	al	al	PROPN
iajs-2376	76	4	-	-	PUNCT
iajs-2376	76	5	haitham	haitham	PROPN
iajs-2376	76	6	jour	jour	X
iajs-2376	76	7	.	.	PROPN
iajs-2376	76	8	for	for	ADP
iajs-2376	76	9	pure	pure	ADJ
iajs-2376	76	10	&	&	CCONJ
iajs-2376	76	11	appl	appl	PROPN
iajs-2376	76	12	.	.	PUNCT
iajs-2376	77	1	sci	sci	PROPN
iajs-2376	77	2	.	.	PROPN
iajs-2376	78	1	33	33	NUM
iajs-2376	78	2	(	(	PUNCT
iajs-2376	78	3	1	1	NUM
iajs-2376	78	4	)	)	PUNCT
iajs-2376	78	5	2020	2020	NUM
iajs-2376	79	1	proof	proof	NOUN
iajs-2376	79	2	:	:	PUNCT
iajs-2376	79	3	similar	similar	ADJ
iajs-2376	79	4	as	as	ADP
iajs-2376	79	5	in	in	ADP
iajs-2376	79	6	proposition	proposition	NOUN
iajs-2376	79	7	2.3	2.3	NUM
iajs-2376	79	8	.	.	PUNCT
iajs-2376	80	1	proposition	proposition	NOUN
iajs-2376	80	2	2.5	2.5	NUM
iajs-2376	80	3	if	if	SCONJ
iajs-2376	80	4	is	be	AUX
iajs-2376	80	5	a	a	DET
iajs-2376	80	6	uniserial	uniserial	ADJ
iajs-2376	80	7	-module	-module	NOUN
iajs-2376	80	8	,	,	PUNCT
iajs-2376	80	9	and	and	CCONJ
iajs-2376	80	10	are	be	AUX
iajs-2376	80	11	submodules	submodule	NOUN
iajs-2376	80	12	of	of	ADP
iajs-2376	80	13	such	such	ADJ
iajs-2376	80	14	that	that	PRON
iajs-2376	80	15	and	and	CCONJ
iajs-2376	80	16	is	be	AUX
iajs-2376	80	17	a	a	DET
iajs-2376	80	18	-closed	-closed	ADJ
iajs-2376	80	19	submodule	submodule	NOUN
iajs-2376	80	20	in	in	ADP
iajs-2376	80	21	and	and	CCONJ
iajs-2376	80	22	is	be	AUX
iajs-2376	80	23	a	a	DET
iajs-2376	80	24	-closed	-closed	ADJ
iajs-2376	80	25	submodule	submodule	NOUN
iajs-2376	80	26	in	in	ADP
iajs-2376	80	27	,	,	PUNCT
iajs-2376	80	28	then	then	ADV
iajs-2376	80	29	is	be	AUX
iajs-2376	80	30	a	a	DET
iajs-2376	80	31	-closed	-closed	ADJ
iajs-2376	80	32	submodule	submodule	NOUN
iajs-2376	80	33	in	in	ADV
iajs-2376	80	34	.	.	PUNCT
iajs-2376	81	1	proof	proof	NOUN
iajs-2376	81	2	:	:	PUNCT
iajs-2376	81	3	prove	prove	VERB
iajs-2376	81	4	is	be	AUX
iajs-2376	81	5	direct	direct	ADJ
iajs-2376	81	6	.	.	PUNCT
iajs-2376	82	1	recall	recall	VERB
iajs-2376	82	2	that	that	SCONJ
iajs-2376	82	3	an	an	DET
iajs-2376	82	4	-module	-module	NOUN
iajs-2376	82	5	is	be	AUX
iajs-2376	82	6	completely	completely	ADV
iajs-2376	82	7	essential	essential	ADJ
iajs-2376	82	8	if	if	SCONJ
iajs-2376	82	9	every	every	DET
iajs-2376	82	10	non	non	ADJ
iajs-2376	82	11	-	-	ADJ
iajs-2376	82	12	zero	zero	ADJ
iajs-2376	82	13	weak	weak	ADJ
iajs-2376	82	14	essential	essential	ADJ
iajs-2376	82	15	submodule	submodule	NOUN
iajs-2376	82	16	of	of	ADP
iajs-2376	82	17	is	be	AUX
iajs-2376	82	18	essential	essential	ADJ
iajs-2376	82	19	[	[	X
iajs-2376	82	20	5	5	NUM
iajs-2376	82	21	,	,	PUNCT
iajs-2376	82	22	3	3	NUM
iajs-2376	82	23	]	]	PUNCT
iajs-2376	82	24	.	.	PUNCT
iajs-2376	83	1	as	as	SCONJ
iajs-2376	83	2	we	we	PRON
iajs-2376	83	3	mention	mention	VERB
iajs-2376	83	4	in	in	ADP
iajs-2376	83	5	example	example	NOUN
iajs-2376	83	6	and	and	CCONJ
iajs-2376	83	7	remarks	remark	NOUN
iajs-2376	83	8	(	(	PUNCT
iajs-2376	83	9	2.2)(8	2.2)(8	NUM
iajs-2376	83	10	)	)	PUNCT
iajs-2376	83	11	closed	closed	ADJ
iajs-2376	83	12	submodules	submodule	NOUN
iajs-2376	83	13	and	and	CCONJ
iajs-2376	83	14	-closed	-close	VERB
iajs-2376	83	15	submodules	submodule	NOUN
iajs-2376	83	16	are	be	AUX
iajs-2376	83	17	independent	independent	ADJ
iajs-2376	83	18	,	,	PUNCT
iajs-2376	83	19	then	then	ADV
iajs-2376	83	20	the	the	DET
iajs-2376	83	21	following	follow	VERB
iajs-2376	83	22	propositions	proposition	NOUN
iajs-2376	83	23	show	show	VERB
iajs-2376	83	24	that	that	SCONJ
iajs-2376	83	25	,	,	PUNCT
iajs-2376	83	26	the	the	DET
iajs-2376	83	27	class	class	NOUN
iajs-2376	83	28	of	of	ADP
iajs-2376	83	29	closed	closed	ADJ
iajs-2376	83	30	submodules	submodule	NOUN
iajs-2376	83	31	is	be	AUX
iajs-2376	83	32	contained	contain	VERB
iajs-2376	83	33	in	in	ADP
iajs-2376	83	34	the	the	DET
iajs-2376	83	35	class	class	NOUN
iajs-2376	83	36	of	of	ADP
iajs-2376	83	37	a	a	DET
iajs-2376	83	38	-closed	-close	VERB
iajs-2376	83	39	submodules	submodule	NOUN
iajs-2376	83	40	under	under	ADP
iajs-2376	83	41	certain	certain	ADJ
iajs-2376	83	42	condition	condition	NOUN
iajs-2376	83	43	.	.	PUNCT
iajs-2376	84	1	proposition	proposition	NOUN
iajs-2376	84	2	2.6	2.6	NUM
iajs-2376	84	3	let	let	AUX
iajs-2376	84	4	be	be	AUX
iajs-2376	84	5	a	a	DET
iajs-2376	84	6	non	non	ADJ
iajs-2376	84	7	zero	zero	NUM
iajs-2376	84	8	closed	closed	ADJ
iajs-2376	84	9	submodule	submodule	NOUN
iajs-2376	84	10	of	of	ADP
iajs-2376	84	11	a	a	DET
iajs-2376	84	12	module	module	NOUN
iajs-2376	84	13	such	such	ADJ
iajs-2376	84	14	that	that	SCONJ
iajs-2376	84	15	every	every	DET
iajs-2376	84	16	weak	weak	ADJ
iajs-2376	84	17	essential	essential	ADJ
iajs-2376	84	18	extension	extension	NOUN
iajs-2376	84	19	of	of	ADP
iajs-2376	84	20	is	be	AUX
iajs-2376	84	21	a	a	DET
iajs-2376	84	22	completely	completely	ADV
iajs-2376	84	23	essential	essential	ADJ
iajs-2376	84	24	submodule	submodule	NOUN
iajs-2376	84	25	of	of	ADP
iajs-2376	84	26	,	,	PUNCT
iajs-2376	84	27	then	then	ADV
iajs-2376	84	28	is	be	AUX
iajs-2376	84	29	a	a	DET
iajs-2376	84	30	-closed	-closed	ADJ
iajs-2376	84	31	submodule	submodule	NOUN
iajs-2376	84	32	in	in	ADV
iajs-2376	84	33	.	.	PUNCT
iajs-2376	85	1	proof	proof	NOUN
iajs-2376	85	2	assume	assume	VERB
iajs-2376	85	3	that	that	PRON
iajs-2376	85	4	be	be	VERB
iajs-2376	85	5	a	a	DET
iajs-2376	85	6	non	non	ADJ
iajs-2376	85	7	zero	zero	NUM
iajs-2376	85	8	closed	closed	ADJ
iajs-2376	85	9	submodule	submodule	NOUN
iajs-2376	85	10	of	of	ADP
iajs-2376	85	11	,	,	PUNCT
iajs-2376	85	12	then	then	ADV
iajs-2376	85	13	by	by	ADP
iajs-2376	85	14	[	[	PUNCT
iajs-2376	85	15	]	]	X
iajs-2376	85	16	we	we	PRON
iajs-2376	85	17	get	get	VERB
iajs-2376	85	18	is	be	AUX
iajs-2376	85	19	a	a	DET
iajs-2376	85	20	-closed	-closed	ADJ
iajs-2376	85	21	submodule	submodule	NOUN
iajs-2376	85	22	in	in	ADV
iajs-2376	85	23	,	,	PUNCT
iajs-2376	85	24	so	so	CCONJ
iajs-2376	85	25	by	by	ADP
iajs-2376	85	26	remarks	remark	NOUN
iajs-2376	85	27	and	and	CCONJ
iajs-2376	85	28	examples	example	NOUN
iajs-2376	85	29	(	(	PUNCT
iajs-2376	85	30	2.2	2.2	NUM
iajs-2376	85	31	)	)	PUNCT
iajs-2376	85	32	(	(	PUNCT
iajs-2376	85	33	1	1	X
iajs-2376	85	34	)	)	PUNCT
iajs-2376	85	35	we	we	PRON
iajs-2376	85	36	get	get	VERB
iajs-2376	85	37	is	be	AUX
iajs-2376	85	38	a	a	DET
iajs-2376	85	39	-closed	-closed	ADJ
iajs-2376	85	40	submodule	submodule	NOUN
iajs-2376	85	41	of	of	ADP
iajs-2376	85	42	.	.	PUNCT
iajs-2376	86	1	recall	recall	VERB
iajs-2376	86	2	that	that	SCONJ
iajs-2376	86	3	an	an	DET
iajs-2376	86	4	-module	-module	NOUN
iajs-2376	86	5	is	be	AUX
iajs-2376	86	6	called	call	VERB
iajs-2376	86	7	fully	fully	ADV
iajs-2376	86	8	semi	semi	ADJ
iajs-2376	86	9	-	-	ADJ
iajs-2376	86	10	prime	prime	ADJ
iajs-2376	86	11	,	,	PUNCT
iajs-2376	86	12	if	if	SCONJ
iajs-2376	86	13	every	every	DET
iajs-2376	86	14	proper	proper	ADJ
iajs-2376	86	15	submodule	submodule	NOUN
iajs-2376	86	16	of	of	ADP
iajs-2376	86	17	is	be	AUX
iajs-2376	86	18	a	a	DET
iajs-2376	86	19	semi	semi	ADJ
iajs-2376	86	20	-	-	ADJ
iajs-2376	86	21	prime	prime	ADJ
iajs-2376	86	22	submodule	submodule	NOUN
iajs-2376	86	23	[	[	PUNCT
iajs-2376	86	24	]	]	X
iajs-2376	86	25	.	.	PUNCT
iajs-2376	87	1	proposition	proposition	NOUN
iajs-2376	87	2	2.7	2.7	NUM
iajs-2376	87	3	let	let	VERB
iajs-2376	87	4	be	be	AUX
iajs-2376	87	5	a	a	DET
iajs-2376	87	6	non	non	ADJ
iajs-2376	87	7	zero	zero	NUM
iajs-2376	87	8	closed	closed	ADJ
iajs-2376	87	9	submodule	submodule	NOUN
iajs-2376	87	10	of	of	ADP
iajs-2376	87	11	a	a	DET
iajs-2376	87	12	fully	fully	ADV
iajs-2376	87	13	semi	semi	ADJ
iajs-2376	87	14	-	-	ADJ
iajs-2376	87	15	prime	prime	ADJ
iajs-2376	87	16	-module	-module	NOUN
iajs-2376	87	17	.	.	PUNCT
iajs-2376	88	1	then	then	ADV
iajs-2376	88	2	is	be	AUX
iajs-2376	88	3	a	a	DET
iajs-2376	88	4	-closed	-closed	ADJ
iajs-2376	88	5	submodule	submodule	NOUN
iajs-2376	88	6	in	in	ADV
iajs-2376	88	7	.	.	PUNCT
iajs-2376	89	1	proof	proof	NOUN
iajs-2376	89	2	suppose	suppose	VERB
iajs-2376	89	3	that	that	PRON
iajs-2376	89	4	is	be	AUX
iajs-2376	89	5	a	a	DET
iajs-2376	89	6	non	non	ADJ
iajs-2376	89	7	zero	zero	NUM
iajs-2376	89	8	closed	close	VERB
iajs-2376	89	9	submodule	submodule	NOUN
iajs-2376	89	10	of	of	ADP
iajs-2376	89	11	,	,	PUNCT
iajs-2376	89	12	then	then	ADV
iajs-2376	89	13	by	by	ADP
iajs-2376	89	14	[	[	PUNCT
iajs-2376	89	15	]	]	X
iajs-2376	89	16	we	we	PRON
iajs-2376	89	17	get	get	VERB
iajs-2376	89	18	is	be	AUX
iajs-2376	89	19	a	a	DET
iajs-2376	89	20	-closed	-closed	ADJ
iajs-2376	89	21	submodule	submodule	NOUN
iajs-2376	89	22	in	in	ADP
iajs-2376	89	23	.	.	PUNCT
iajs-2376	90	1	hence	hence	ADV
iajs-2376	90	2	by	by	ADP
iajs-2376	90	3	remarks	remark	NOUN
iajs-2376	90	4	and	and	CCONJ
iajs-2376	90	5	examples	example	NOUN
iajs-2376	90	6	(	(	PUNCT
iajs-2376	90	7	2.2	2.2	NUM
iajs-2376	90	8	)	)	PUNCT
iajs-2376	90	9	(	(	PUNCT
iajs-2376	90	10	1	1	X
iajs-2376	90	11	)	)	PUNCT
iajs-2376	90	12	is	be	AUX
iajs-2376	90	13	a	a	DET
iajs-2376	90	14	-closed	-closed	ADJ
iajs-2376	90	15	submodule	submodule	NOUN
iajs-2376	90	16	in	in	ADP
iajs-2376	90	17	.	.	PUNCT
iajs-2376	91	1	proposition	proposition	NOUN
iajs-2376	91	2	2.8	2.8	NUM
iajs-2376	91	3	if	if	SCONJ
iajs-2376	91	4	and	and	CCONJ
iajs-2376	91	5	are	be	AUX
iajs-2376	91	6	submodules	submodule	NOUN
iajs-2376	91	7	of	of	ADP
iajs-2376	91	8	an	an	DET
iajs-2376	91	9	-module	-module	NOUN
iajs-2376	91	10	,	,	PUNCT
iajs-2376	91	11	with	with	ADP
iajs-2376	91	12	,	,	PUNCT
iajs-2376	91	13	containing	contain	VERB
iajs-2376	91	14	every	every	DET
iajs-2376	91	15	-closed	-close	VERB
iajs-2376	91	16	submodule	submodule	NOUN
iajs-2376	91	17	of	of	ADP
iajs-2376	91	18	and	and	CCONJ
iajs-2376	91	19	is	be	AUX
iajs-2376	91	20	a	a	DET
iajs-2376	91	21	-closed	-closed	ADJ
iajs-2376	91	22	submodule	submodule	NOUN
iajs-2376	91	23	in	in	ADP
iajs-2376	91	24	,	,	PUNCT
iajs-2376	91	25	then	then	ADV
iajs-2376	91	26	is	be	AUX
iajs-2376	91	27	a	a	DET
iajs-2376	91	28	a	a	DET
iajs-2376	91	29	-closed	-closed	ADJ
iajs-2376	91	30	submodule	submodule	NOUN
iajs-2376	91	31	in	in	ADV
iajs-2376	91	32	.	.	PUNCT
iajs-2376	92	1	proof	proof	NOUN
iajs-2376	92	2	let	let	VERB
iajs-2376	92	3	and	and	CCONJ
iajs-2376	92	4	,	,	PUNCT
iajs-2376	92	5	then	then	ADV
iajs-2376	92	6	.	.	PUNCT
iajs-2376	93	1	since	since	SCONJ
iajs-2376	93	2	is	be	AUX
iajs-2376	93	3	a	a	DET
iajs-2376	93	4	-closed	-closed	ADJ
iajs-2376	93	5	submodule	submodule	NOUN
iajs-2376	93	6	in	in	ADP
iajs-2376	93	7	,	,	PUNCT
iajs-2376	93	8	then	then	ADV
iajs-2376	93	9	a	a	DET
iajs-2376	93	10	-closed	-closed	ADJ
iajs-2376	93	11	submodule	submodule	NOUN
iajs-2376	93	12	in	in	ADP
iajs-2376	93	13	with	with	ADP
iajs-2376	93	14	and	and	CCONJ
iajs-2376	93	15	.	.	PUNCT
iajs-2376	94	1	hence	hence	ADV
iajs-2376	94	2	is	be	AUX
iajs-2376	94	3	a	a	DET
iajs-2376	94	4	-closed	-closed	ADJ
iajs-2376	94	5	submodule	submodule	NOUN
iajs-2376	94	6	in	in	ADP
iajs-2376	94	7	with	with	ADP
iajs-2376	94	8	and	and	CCONJ
iajs-2376	94	9	.	.	PUNCT
iajs-2376	95	1	thus	thus	ADV
iajs-2376	95	2	is	be	AUX
iajs-2376	95	3	a	a	DET
iajs-2376	95	4	a	a	DET
iajs-2376	95	5	-closed	-closed	ADJ
iajs-2376	95	6	submodule	submodule	NOUN
iajs-2376	95	7	in	in	ADP
iajs-2376	95	8	.	.	PUNCT
iajs-2376	96	1	44	44	NUM
iajs-2376	96	2	ibn	ibn	PROPN
iajs-2376	96	3	al	al	PROPN
iajs-2376	96	4	-	-	PUNCT
iajs-2376	96	5	haitham	haitham	PROPN
iajs-2376	96	6	jour	jour	X
iajs-2376	96	7	.	.	PROPN
iajs-2376	96	8	for	for	ADP
iajs-2376	96	9	pure	pure	ADJ
iajs-2376	96	10	&	&	CCONJ
iajs-2376	96	11	appl	appl	PROPN
iajs-2376	96	12	.	.	PUNCT
iajs-2376	97	1	sci	sci	PROPN
iajs-2376	97	2	.	.	PROPN
iajs-2376	98	1	33	33	NUM
iajs-2376	98	2	(	(	PUNCT
iajs-2376	98	3	1	1	NUM
iajs-2376	98	4	)	)	PUNCT
iajs-2376	98	5	2020	2020	NUM
iajs-2376	98	6	proposition	proposition	NOUN
iajs-2376	98	7	2.9	2.9	NUM
iajs-2376	98	8	if	if	SCONJ
iajs-2376	98	9	is	be	AUX
iajs-2376	98	10	a	a	DET
iajs-2376	98	11	module	module	NOUN
iajs-2376	98	12	where	where	SCONJ
iajs-2376	98	13	and	and	CCONJ
iajs-2376	98	14	are	be	AUX
iajs-2376	98	15	submodules	submodule	NOUN
iajs-2376	98	16	of	of	ADP
iajs-2376	98	17	,	,	PUNCT
iajs-2376	98	18	provided	provide	VERB
iajs-2376	98	19	that	that	PRON
iajs-2376	98	20	and	and	CCONJ
iajs-2376	98	21	each	each	DET
iajs-2376	98	22	weak	weak	ADJ
iajs-2376	98	23	essential	essential	ADJ
iajs-2376	98	24	extension	extension	NOUN
iajs-2376	98	25	of	of	ADP
iajs-2376	98	26	is	be	AUX
iajs-2376	98	27	completely	completely	ADV
iajs-2376	98	28	essential	essential	ADJ
iajs-2376	98	29	module	module	NOUN
iajs-2376	98	30	,	,	PUNCT
iajs-2376	98	31	where	where	SCONJ
iajs-2376	98	32	is	be	AUX
iajs-2376	98	33	a	a	DET
iajs-2376	98	34	non	non	ADJ
iajs-2376	98	35	-	-	ADJ
iajs-2376	98	36	zero	zero	NUM
iajs-2376	98	37	-closed	-close	VERB
iajs-2376	98	38	submodule	submodule	NOUN
iajs-2376	98	39	in	in	ADP
iajs-2376	98	40	and	and	CCONJ
iajs-2376	98	41	is	be	AUX
iajs-2376	98	42	a	a	DET
iajs-2376	98	43	non	non	ADJ
iajs-2376	98	44	-	-	ADJ
iajs-2376	98	45	zero	zero	NUM
iajs-2376	98	46	-closed	-close	VERB
iajs-2376	98	47	submodule	submodule	NOUN
iajs-2376	98	48	in	in	ADP
iajs-2376	98	49	,	,	PUNCT
iajs-2376	98	50	then	then	ADV
iajs-2376	98	51	is	be	AUX
iajs-2376	98	52	a	a	DET
iajs-2376	98	53	-closed	-closed	ADJ
iajs-2376	98	54	submodule	submodule	NOUN
iajs-2376	98	55	in	in	ADV
iajs-2376	98	56	.	.	PUNCT
iajs-2376	99	1	proof	proof	NOUN
iajs-2376	99	2	let	let	VERB
iajs-2376	99	3	with	with	ADP
iajs-2376	99	4	,	,	PUNCT
iajs-2376	99	5	then	then	ADV
iajs-2376	99	6	either	either	ADV
iajs-2376	99	7	or	or	CCONJ
iajs-2376	99	8	.	.	PUNCT
iajs-2376	100	1	if	if	SCONJ
iajs-2376	100	2	and	and	CCONJ
iajs-2376	100	3	since	since	SCONJ
iajs-2376	100	4	is	be	AUX
iajs-2376	100	5	a	a	DET
iajs-2376	100	6	-closed	-closed	ADJ
iajs-2376	100	7	submodule	submodule	NOUN
iajs-2376	100	8	in	in	ADV
iajs-2376	100	9	,	,	PUNCT
iajs-2376	100	10	so	so	CCONJ
iajs-2376	100	11	there	there	PRON
iajs-2376	100	12	exists	exist	VERB
iajs-2376	100	13	a	a	DET
iajs-2376	100	14	-closed	-closed	ADJ
iajs-2376	100	15	submodule	submodule	NOUN
iajs-2376	100	16	in	in	ADP
iajs-2376	100	17	with	with	ADP
iajs-2376	100	18	and	and	CCONJ
iajs-2376	100	19	.	.	PUNCT
iajs-2376	101	1	by	by	ADP
iajs-2376	101	2	[	[	PUNCT
iajs-2376	101	3	]	]	X
iajs-2376	101	4	is	be	AUX
iajs-2376	101	5	a	a	DET
iajs-2376	101	6	-closed	-close	VERB
iajs-2376	101	7	in	in	ADP
iajs-2376	101	8	,	,	PUNCT
iajs-2376	101	9	then	then	ADV
iajs-2376	101	10	by	by	ADP
iajs-2376	101	11	[	[	PUNCT
iajs-2376	101	12	]	]	X
iajs-2376	101	13	,	,	PUNCT
iajs-2376	101	14	we	we	PRON
iajs-2376	101	15	have	have	AUX
iajs-2376	101	16	is	be	AUX
iajs-2376	101	17	a	a	DET
iajs-2376	101	18	-closed	-close	VERB
iajs-2376	101	19	in	in	ADP
iajs-2376	101	20	such	such	ADJ
iajs-2376	101	21	that	that	PRON
iajs-2376	101	22	and	and	CCONJ
iajs-2376	101	23	.	.	PUNCT
iajs-2376	102	1	similarly	similarly	ADV
iajs-2376	102	2	if	if	SCONJ
iajs-2376	102	3	,	,	PUNCT
iajs-2376	102	4	then	then	ADV
iajs-2376	102	5	a	a	DET
iajs-2376	102	6	-closed	-closed	ADJ
iajs-2376	102	7	submodule	submodule	NOUN
iajs-2376	102	8	in	in	ADP
iajs-2376	102	9	containing	contain	VERB
iajs-2376	102	10	and	and	CCONJ
iajs-2376	102	11	does	do	AUX
iajs-2376	102	12	not	not	PART
iajs-2376	102	13	contain	contain	VERB
iajs-2376	102	14	.	.	PUNCT
iajs-2376	103	1	thus	thus	ADV
iajs-2376	103	2	is	be	AUX
iajs-2376	103	3	a	a	DET
iajs-2376	103	4	-closed	-closed	ADJ
iajs-2376	103	5	submodule	submodule	NOUN
iajs-2376	103	6	in	in	ADP
iajs-2376	103	7	.	.	PUNCT
iajs-2376	104	1	proposition	proposition	NOUN
iajs-2376	104	2	2.10	2.10	NUM
iajs-2376	104	3	if	if	SCONJ
iajs-2376	104	4	is	be	AUX
iajs-2376	104	5	an	an	DET
iajs-2376	104	6	-module	-module	NOUN
iajs-2376	104	7	where	where	SCONJ
iajs-2376	104	8	and	and	CCONJ
iajs-2376	104	9	are	be	AUX
iajs-2376	104	10	submodules	submodule	NOUN
iajs-2376	104	11	of	of	ADP
iajs-2376	104	12	,	,	PUNCT
iajs-2376	104	13	provided	provide	VERB
iajs-2376	104	14	that	that	SCONJ
iajs-2376	104	15	and	and	CCONJ
iajs-2376	104	16	all	all	DET
iajs-2376	104	17	submodules	submodule	NOUN
iajs-2376	104	18	of	of	ADP
iajs-2376	104	19	are	be	AUX
iajs-2376	104	20	completely	completely	ADV
iajs-2376	104	21	essential	essential	ADJ
iajs-2376	104	22	submodules	submodule	NOUN
iajs-2376	104	23	.	.	PUNCT
iajs-2376	105	1	if	if	SCONJ
iajs-2376	105	2	is	be	AUX
iajs-2376	105	3	a	a	DET
iajs-2376	105	4	non	non	ADJ
iajs-2376	105	5	-	-	ADJ
iajs-2376	105	6	zero	zero	NUM
iajs-2376	105	7	-closed	-close	VERB
iajs-2376	105	8	submodule	submodule	NOUN
iajs-2376	105	9	in	in	ADP
iajs-2376	105	10	and	and	CCONJ
iajs-2376	105	11	is	be	AUX
iajs-2376	105	12	a	a	DET
iajs-2376	105	13	non	non	ADJ
iajs-2376	105	14	-	-	ADJ
iajs-2376	105	15	zero	zero	NUM
iajs-2376	105	16	-closed	-close	VERB
iajs-2376	105	17	submodule	submodule	NOUN
iajs-2376	105	18	in	in	ADP
iajs-2376	105	19	,	,	PUNCT
iajs-2376	105	20	then	then	ADV
iajs-2376	105	21	is	be	AUX
iajs-2376	105	22	a	a	DET
iajs-2376	105	23	-closed	-closed	ADJ
iajs-2376	105	24	submodule	submodule	NOUN
iajs-2376	105	25	in	in	ADP
iajs-2376	105	26	if	if	SCONJ
iajs-2376	106	1	and	and	CCONJ
iajs-2376	106	2	only	only	ADV
iajs-2376	106	3	if	if	SCONJ
iajs-2376	106	4	is	be	AUX
iajs-2376	106	5	a	a	DET
iajs-2376	106	6	-closed	-closed	ADJ
iajs-2376	106	7	submodule	submodule	NOUN
iajs-2376	106	8	in	in	ADP
iajs-2376	106	9	and	and	CCONJ
iajs-2376	106	10	is	be	AUX
iajs-2376	106	11	a	a	DET
iajs-2376	106	12	-closed	-closed	ADJ
iajs-2376	106	13	submodule	submodule	NOUN
iajs-2376	106	14	in	in	ADV
iajs-2376	106	15	.	.	PUNCT
iajs-2376	107	1	proof	proof	NOUN
iajs-2376	107	2	to	to	PART
iajs-2376	107	3	prove	prove	VERB
iajs-2376	107	4	is	be	AUX
iajs-2376	107	5	a	a	DET
iajs-2376	107	6	-closed	-closed	ADJ
iajs-2376	107	7	submodule	submodule	NOUN
iajs-2376	107	8	in	in	ADP
iajs-2376	107	9	.	.	PUNCT
iajs-2376	108	1	if	if	SCONJ
iajs-2376	108	2	with	with	ADP
iajs-2376	108	3	.	.	PUNCT
iajs-2376	109	1	then	then	ADV
iajs-2376	109	2	.	.	PUNCT
iajs-2376	110	1	but	but	CCONJ
iajs-2376	110	2	is	be	AUX
iajs-2376	110	3	a	a	DET
iajs-2376	110	4	-closed	-closed	ADJ
iajs-2376	110	5	submodule	submodule	NOUN
iajs-2376	110	6	in	in	ADV
iajs-2376	110	7	,	,	PUNCT
iajs-2376	110	8	so	so	CCONJ
iajs-2376	110	9	there	there	PRON
iajs-2376	110	10	exists	exist	VERB
iajs-2376	110	11	a	a	DET
iajs-2376	110	12	-closed	-closed	ADJ
iajs-2376	110	13	submodule	submodule	NOUN
iajs-2376	110	14	in	in	ADP
iajs-2376	110	15	with	with	ADP
iajs-2376	110	16	and	and	CCONJ
iajs-2376	110	17	.	.	PUNCT
iajs-2376	111	1	since	since	ADV
iajs-2376	111	2	,	,	PUNCT
iajs-2376	111	3	then	then	ADV
iajs-2376	111	4	by	by	ADP
iajs-2376	111	5	[	[	PUNCT
iajs-2376	111	6	]	]	X
iajs-2376	111	7	any	any	DET
iajs-2376	111	8	submodule	submodule	NOUN
iajs-2376	111	9	of	of	ADP
iajs-2376	111	10	can	can	AUX
iajs-2376	111	11	be	be	AUX
iajs-2376	111	12	written	write	VERB
iajs-2376	111	13	as	as	ADP
iajs-2376	111	14	,	,	PUNCT
iajs-2376	111	15	where	where	SCONJ
iajs-2376	111	16	and	and	CCONJ
iajs-2376	111	17	.	.	PUNCT
iajs-2376	112	1	hence	hence	ADV
iajs-2376	112	2	by	by	ADP
iajs-2376	112	3	[	[	PUNCT
iajs-2376	112	4	]	]	X
iajs-2376	112	5	it	it	PRON
iajs-2376	112	6	follows	follow	VERB
iajs-2376	112	7	that	that	PRON
iajs-2376	112	8	is	be	AUX
iajs-2376	112	9	a	a	DET
iajs-2376	112	10	-closed	-closed	ADJ
iajs-2376	112	11	submodule	submodule	NOUN
iajs-2376	112	12	in	in	ADP
iajs-2376	112	13	and	and	CCONJ
iajs-2376	112	14	is	be	AUX
iajs-2376	112	15	a	a	DET
iajs-2376	112	16	closed	closed	ADJ
iajs-2376	112	17	submodule	submodule	NOUN
iajs-2376	112	18	in	in	ADP
iajs-2376	112	19	.	.	PUNCT
iajs-2376	113	1	since	since	SCONJ
iajs-2376	113	2	and	and	CCONJ
iajs-2376	113	3	,	,	PUNCT
iajs-2376	113	4	then	then	ADV
iajs-2376	113	5	.	.	PUNCT
iajs-2376	114	1	therefore	therefore	ADV
iajs-2376	114	2	is	be	AUX
iajs-2376	114	3	a	a	DET
iajs-2376	114	4	closed	closed	ADJ
iajs-2376	114	5	submodule	submodule	NOUN
iajs-2376	114	6	in	in	ADP
iajs-2376	114	7	.	.	PUNCT
iajs-2376	115	1	similarly	similarly	ADV
iajs-2376	115	2	is	be	AUX
iajs-2376	115	3	a	a	DET
iajs-2376	115	4	-closed	-closed	ADJ
iajs-2376	115	5	submodule	submodule	NOUN
iajs-2376	115	6	in	in	ADP
iajs-2376	115	7	.	.	PUNCT
iajs-2376	116	1	suppose	suppose	VERB
iajs-2376	116	2	that	that	PRON
iajs-2376	116	3	is	be	AUX
iajs-2376	116	4	a	a	DET
iajs-2376	116	5	-closed	-closed	ADJ
iajs-2376	116	6	submodule	submodule	NOUN
iajs-2376	116	7	in	in	ADP
iajs-2376	116	8	and	and	CCONJ
iajs-2376	116	9	is	be	AUX
iajs-2376	116	10	a	a	DET
iajs-2376	116	11	-closed	-closed	ADJ
iajs-2376	116	12	submodule	submodule	NOUN
iajs-2376	116	13	in	in	ADV
iajs-2376	116	14	.	.	PUNCT
iajs-2376	117	1	let	let	VERB
iajs-2376	117	2	with	with	ADP
iajs-2376	117	3	,	,	PUNCT
iajs-2376	117	4	then	then	ADV
iajs-2376	117	5	either	either	ADV
iajs-2376	117	6	or	or	CCONJ
iajs-2376	117	7	.	.	PUNCT
iajs-2376	118	1	if	if	SCONJ
iajs-2376	118	2	and	and	CCONJ
iajs-2376	118	3	is	be	AUX
iajs-2376	118	4	a	a	DET
iajs-2376	118	5	-closed	-closed	ADJ
iajs-2376	118	6	submodule	submodule	NOUN
iajs-2376	118	7	in	in	ADP
iajs-2376	118	8	,	,	PUNCT
iajs-2376	118	9	so	so	CCONJ
iajs-2376	118	10	a	a	DET
iajs-2376	118	11	-closed	-closed	ADJ
iajs-2376	118	12	submodule	submodule	NOUN
iajs-2376	118	13	in	in	ADP
iajs-2376	118	14	such	such	ADJ
iajs-2376	118	15	that	that	PRON
iajs-2376	118	16	and	and	CCONJ
iajs-2376	118	17	.	.	PUNCT
iajs-2376	119	1	but	but	CCONJ
iajs-2376	119	2	is	be	AUX
iajs-2376	119	3	a	a	DET
iajs-2376	119	4	-closed	-closed	ADJ
iajs-2376	119	5	submodule	submodule	NOUN
iajs-2376	119	6	in	in	ADP
iajs-2376	119	7	and	and	CCONJ
iajs-2376	119	8	by	by	ADP
iajs-2376	119	9	[	[	PUNCT
iajs-2376	119	10	]	]	X
iajs-2376	119	11	is	be	AUX
iajs-2376	119	12	a	a	DET
iajs-2376	119	13	-closed	-close	VERB
iajs-2376	119	14	in	in	ADV
iajs-2376	119	15	,	,	PUNCT
iajs-2376	119	16	hence	hence	ADV
iajs-2376	119	17	by	by	ADP
iajs-2376	119	18	[	[	PUNCT
iajs-2376	119	19	]	]	X
iajs-2376	119	20	,	,	PUNCT
iajs-2376	119	21	we	we	PRON
iajs-2376	119	22	have	have	VERB
iajs-2376	119	23	is	be	AUX
iajs-2376	119	24	a	a	DET
iajs-2376	119	25	-closed	-closed	ADJ
iajs-2376	119	26	submodule	submodule	NOUN
iajs-2376	119	27	in	in	ADP
iajs-2376	119	28	.	.	PUNCT
iajs-2376	120	1	also	also	ADV
iajs-2376	120	2	and	and	CCONJ
iajs-2376	120	3	.	.	PUNCT
iajs-2376	121	1	similarly	similarly	ADV
iajs-2376	121	2	if	if	SCONJ
iajs-2376	121	3	,	,	PUNCT
iajs-2376	121	4	then	then	ADV
iajs-2376	121	5	there	there	PRON
iajs-2376	121	6	exists	exist	VERB
iajs-2376	121	7	a	a	DET
iajs-2376	121	8	-closed	-closed	ADJ
iajs-2376	121	9	submodule	submodule	NOUN
iajs-2376	121	10	in	in	ADP
iajs-2376	121	11	containing	contain	VERB
iajs-2376	121	12	and	and	CCONJ
iajs-2376	121	13	does	do	AUX
iajs-2376	121	14	not	not	PART
iajs-2376	121	15	containing	contain	VERB
iajs-2376	121	16	.	.	PUNCT
iajs-2376	122	1	thus	thus	ADV
iajs-2376	122	2	is	be	AUX
iajs-2376	122	3	a	a	DET
iajs-2376	122	4	-closed	-closed	ADJ
iajs-2376	122	5	submodule	submodule	NOUN
iajs-2376	122	6	in	in	ADP
iajs-2376	122	7	.	.	PUNCT
iajs-2376	123	1	it	it	PRON
iajs-2376	123	2	is	be	AUX
iajs-2376	123	3	well	well	ADV
iajs-2376	123	4	known	know	VERB
iajs-2376	123	5	that	that	SCONJ
iajs-2376	123	6	a	a	DET
iajs-2376	123	7	fully	fully	ADV
iajs-2376	123	8	semi	semi	ADJ
iajs-2376	123	9	-	-	ADJ
iajs-2376	123	10	prime	prime	ADJ
iajs-2376	123	11	module	module	NOUN
iajs-2376	123	12	is	be	AUX
iajs-2376	123	13	a	a	DET
iajs-2376	123	14	completely	completely	ADV
iajs-2376	123	15	essential	essential	ADJ
iajs-2376	123	16	[	[	PUNCT
iajs-2376	123	17	]	]	X
iajs-2376	123	18	so	so	CCONJ
iajs-2376	123	19	we	we	PRON
iajs-2376	123	20	get	get	VERB
iajs-2376	123	21	the	the	DET
iajs-2376	123	22	following	follow	VERB
iajs-2376	123	23	corollary	corollary	NOUN
iajs-2376	123	24	.	.	PUNCT
iajs-2376	124	1	corollary	corollary	ADJ
iajs-2376	124	2	2.11	2.11	NUM
iajs-2376	124	3	if	if	SCONJ
iajs-2376	124	4	is	be	AUX
iajs-2376	124	5	an	an	DET
iajs-2376	124	6	-module	-module	NOUN
iajs-2376	124	7	where	where	SCONJ
iajs-2376	124	8	and	and	CCONJ
iajs-2376	124	9	are	be	AUX
iajs-2376	124	10	submodules	submodule	NOUN
iajs-2376	124	11	of	of	ADP
iajs-2376	124	12	,	,	PUNCT
iajs-2376	124	13	with	with	ADP
iajs-2376	124	14	and	and	CCONJ
iajs-2376	124	15	all	all	DET
iajs-2376	124	16	submodules	submodule	NOUN
iajs-2376	124	17	of	of	ADP
iajs-2376	124	18	are	be	AUX
iajs-2376	124	19	fully	fully	ADV
iajs-2376	124	20	semi	semi	ADJ
iajs-2376	124	21	-	-	ADJ
iajs-2376	124	22	prime	prime	ADJ
iajs-2376	124	23	and	and	CCONJ
iajs-2376	124	24	is	be	AUX
iajs-2376	124	25	a	a	DET
iajs-2376	124	26	non	non	ADJ
iajs-2376	124	27	-	-	ADJ
iajs-2376	124	28	zero	zero	NUM
iajs-2376	124	29	-closed	-close	VERB
iajs-2376	124	30	submodule	submodule	NOUN
iajs-2376	124	31	in	in	ADP
iajs-2376	124	32	and	and	CCONJ
iajs-2376	124	33	is	be	AUX
iajs-2376	124	34	a	a	DET
iajs-2376	124	35	non	non	ADJ
iajs-2376	124	36	-	-	ADJ
iajs-2376	124	37	zero	zero	NUM
iajs-2376	124	38	-closed	-close	VERB
iajs-2376	124	39	submodule	submodule	NOUN
iajs-2376	124	40	in	in	ADP
iajs-2376	124	41	such	such	ADJ
iajs-2376	124	42	that	that	PRON
iajs-2376	124	43	is	be	AUX
iajs-2376	124	44	a	a	DET
iajs-2376	124	45	48	48	NUM
iajs-2376	124	46	ibn	ibn	PROPN
iajs-2376	124	47	al	al	PROPN
iajs-2376	124	48	-	-	PUNCT
iajs-2376	124	49	haitham	haitham	PROPN
iajs-2376	124	50	jour	jour	X
iajs-2376	124	51	.	.	PROPN
iajs-2376	125	1	for	for	ADP
iajs-2376	125	2	pure	pure	ADJ
iajs-2376	125	3	&	&	CCONJ
iajs-2376	125	4	appl	appl	PROPN
iajs-2376	125	5	.	.	PUNCT
iajs-2376	126	1	sci	sci	PROPN
iajs-2376	126	2	.	.	PROPN
iajs-2376	127	1	33	33	NUM
iajs-2376	127	2	(	(	PUNCT
iajs-2376	127	3	1	1	NUM
iajs-2376	127	4	)	)	PUNCT
iajs-2376	127	5	2020	2020	NUM
iajs-2376	127	6	closed	close	VERB
iajs-2376	127	7	submodule	submodule	NOUN
iajs-2376	127	8	in	in	ADP
iajs-2376	127	9	,	,	PUNCT
iajs-2376	127	10	then	then	ADV
iajs-2376	127	11	is	be	AUX
iajs-2376	127	12	a	a	DET
iajs-2376	127	13	-closed	-closed	ADJ
iajs-2376	127	14	submodule	submodule	NOUN
iajs-2376	127	15	in	in	ADP
iajs-2376	127	16	and	and	CCONJ
iajs-2376	127	17	is	be	AUX
iajs-2376	127	18	a	a	DET
iajs-2376	127	19	-closed	-closed	ADJ
iajs-2376	127	20	submodule	submodule	NOUN
iajs-2376	127	21	in	in	ADP
iajs-2376	127	22	.	.	PUNCT
iajs-2376	128	1	proposition	proposition	NOUN
iajs-2376	128	2	2.12	2.12	NUM
iajs-2376	128	3	if	if	SCONJ
iajs-2376	128	4	is	be	AUX
iajs-2376	128	5	an	an	DET
iajs-2376	128	6	epimorphism	epimorphism	NOUN
iajs-2376	128	7	and	and	CCONJ
iajs-2376	128	8	let	let	VERB
iajs-2376	128	9	be	be	AUX
iajs-2376	128	10	a	a	DET
iajs-2376	128	11	submodule	submodule	NOUN
iajs-2376	128	12	of	of	ADP
iajs-2376	128	13	such	such	ADJ
iajs-2376	128	14	that	that	PRON
iajs-2376	128	15	and	and	CCONJ
iajs-2376	128	16	is	be	AUX
iajs-2376	128	17	a	a	DET
iajs-2376	128	18	-closed	-closed	ADJ
iajs-2376	128	19	submodule	submodule	NOUN
iajs-2376	128	20	in	in	ADP
iajs-2376	128	21	,	,	PUNCT
iajs-2376	128	22	then	then	ADV
iajs-2376	128	23	is	be	AUX
iajs-2376	128	24	a	a	DET
iajs-2376	128	25	-closed	-closed	ADJ
iajs-2376	128	26	submodule	submodule	NOUN
iajs-2376	128	27	in	in	ADP
iajs-2376	128	28	,	,	PUNCT
iajs-2376	128	29	where	where	SCONJ
iajs-2376	128	30	is	be	AUX
iajs-2376	128	31	the	the	DET
iajs-2376	128	32	intersection	intersection	NOUN
iajs-2376	128	33	of	of	ADP
iajs-2376	128	34	all	all	DET
iajs-2376	128	35	semi	semi	ADJ
iajs-2376	128	36	-	-	ADJ
iajs-2376	128	37	prime	prime	ADJ
iajs-2376	128	38	submodule	submodule	NOUN
iajs-2376	128	39	of	of	ADP
iajs-2376	128	40	.	.	PUNCT
iajs-2376	129	1	proof	proof	NOUN
iajs-2376	129	2	suppose	suppose	VERB
iajs-2376	129	3	that	that	PRON
iajs-2376	129	4	is	be	AUX
iajs-2376	129	5	a	a	DET
iajs-2376	129	6	-closed	-closed	ADJ
iajs-2376	129	7	submodule	submodule	NOUN
iajs-2376	129	8	in	in	ADP
iajs-2376	129	9	and	and	CCONJ
iajs-2376	129	10	with	with	ADP
iajs-2376	129	11	.	.	PUNCT
iajs-2376	130	1	since	since	SCONJ
iajs-2376	130	2	is	be	AUX
iajs-2376	130	3	an	an	DET
iajs-2376	130	4	epi	epi	NOUN
iajs-2376	130	5	-	-	NOUN
iajs-2376	130	6	morphism	morphism	NOUN
iajs-2376	130	7	then	then	ADV
iajs-2376	130	8	there	there	PRON
iajs-2376	130	9	exists	exist	VERB
iajs-2376	130	10	with	with	ADP
iajs-2376	130	11	and	and	CCONJ
iajs-2376	130	12	.	.	PUNCT
iajs-2376	131	1	since	since	SCONJ
iajs-2376	131	2	is	be	AUX
iajs-2376	131	3	a	a	DET
iajs-2376	131	4	closed	closed	ADJ
iajs-2376	131	5	submodule	submodule	NOUN
iajs-2376	131	6	in	in	ADP
iajs-2376	131	7	,	,	PUNCT
iajs-2376	131	8	then	then	ADV
iajs-2376	131	9	there	there	PRON
iajs-2376	131	10	exists	exist	VERB
iajs-2376	131	11	a	a	DET
iajs-2376	131	12	-closed	-closed	ADJ
iajs-2376	131	13	submodule	submodule	NOUN
iajs-2376	131	14	in	in	ADP
iajs-2376	131	15	with	with	ADP
iajs-2376	131	16	and	and	CCONJ
iajs-2376	131	17	.	.	PUNCT
iajs-2376	132	1	thus	thus	ADV
iajs-2376	132	2	by	by	ADP
iajs-2376	132	3	[	[	PUNCT
iajs-2376	132	4	]	]	X
iajs-2376	132	5	is	be	AUX
iajs-2376	132	6	a	a	DET
iajs-2376	132	7	-closed	-closed	ADJ
iajs-2376	132	8	submodule	submodule	NOUN
iajs-2376	132	9	in	in	ADP
iajs-2376	132	10	.	.	PUNCT
iajs-2376	133	1	since	since	SCONJ
iajs-2376	133	2	is	be	AUX
iajs-2376	133	3	an	an	DET
iajs-2376	133	4	epimorphism	epimorphism	NOUN
iajs-2376	133	5	,	,	PUNCT
iajs-2376	133	6	then	then	ADV
iajs-2376	133	7	and	and	CCONJ
iajs-2376	133	8	.	.	PUNCT
iajs-2376	134	1	thus	thus	ADV
iajs-2376	134	2	is	be	AUX
iajs-2376	134	3	a	a	DET
iajs-2376	134	4	-closed	-closed	ADJ
iajs-2376	134	5	submodule	submodule	NOUN
iajs-2376	134	6	in	in	ADP
iajs-2376	134	7	.	.	PUNCT
iajs-2376	135	1	corollary	corollary	ADJ
iajs-2376	135	2	2.13	2.13	NUM
iajs-2376	135	3	if	if	SCONJ
iajs-2376	135	4	and	and	CCONJ
iajs-2376	135	5	are	be	AUX
iajs-2376	135	6	submodules	submodule	NOUN
iajs-2376	135	7	of	of	ADP
iajs-2376	135	8	a	a	DET
iajs-2376	135	9	module	module	NOUN
iajs-2376	135	10	with	with	ADP
iajs-2376	135	11	and	and	CCONJ
iajs-2376	135	12	is	be	AUX
iajs-2376	135	13	a	a	DET
iajs-2376	135	14	-closed	-closed	ADJ
iajs-2376	135	15	submodule	submodule	NOUN
iajs-2376	135	16	in	in	ADP
iajs-2376	135	17	,	,	PUNCT
iajs-2376	135	18	then	then	ADV
iajs-2376	135	19	is	be	AUX
iajs-2376	135	20	a	a	DET
iajs-2376	135	21	-closed	-closed	ADJ
iajs-2376	135	22	submodule	submodule	NOUN
iajs-2376	135	23	in	in	ADV
iajs-2376	135	24	.	.	PUNCT
iajs-2376	136	1	proof	proof	NOUN
iajs-2376	136	2	it	it	PRON
iajs-2376	136	3	is	be	AUX
iajs-2376	136	4	clear	clear	ADJ
iajs-2376	136	5	.	.	PUNCT
iajs-2376	137	1	recall	recall	VERB
iajs-2376	137	2	that	that	SCONJ
iajs-2376	137	3	a	a	DET
iajs-2376	137	4	submodule	submodule	NOUN
iajs-2376	137	5	of	of	ADP
iajs-2376	137	6	an	an	DET
iajs-2376	137	7	-module	-module	NOUN
iajs-2376	137	8	is	be	AUX
iajs-2376	137	9	called	call	VERB
iajs-2376	137	10	-closed	-closed	ADJ
iajs-2376	137	11	submodule	submodule	NOUN
iajs-2376	137	12	of	of	ADP
iajs-2376	137	13	,	,	PUNCT
iajs-2376	137	14	if	if	SCONJ
iajs-2376	137	15	is	be	AUX
iajs-2376	137	16	a	a	DET
iajs-2376	137	17	non	non	ADJ
iajs-2376	137	18	-	-	ADJ
iajs-2376	137	19	singular	singular	ADJ
iajs-2376	137	20	module	module	NOUN
iajs-2376	137	21	[	[	PUNCT
iajs-2376	137	22	]	]	X
iajs-2376	137	23	.	.	PUNCT
iajs-2376	138	1	the	the	DET
iajs-2376	138	2	following	follow	VERB
iajs-2376	138	3	proposition	proposition	NOUN
iajs-2376	138	4	gives	give	VERB
iajs-2376	138	5	a	a	DET
iajs-2376	138	6	relationships	relationship	NOUN
iajs-2376	138	7	between	between	ADP
iajs-2376	138	8	-closed	-closed	ADJ
iajs-2376	138	9	submodule	submodule	NOUN
iajs-2376	138	10	of	of	ADP
iajs-2376	138	11	and	and	CCONJ
iajs-2376	138	12	-closed	-closed	ADJ
iajs-2376	138	13	submodule	submodule	NOUN
iajs-2376	138	14	of	of	ADP
iajs-2376	138	15	.	.	PUNCT
iajs-2376	139	1	proposition	proposition	NOUN
iajs-2376	139	2	2.14	2.14	NUM
iajs-2376	139	3	if	if	SCONJ
iajs-2376	139	4	is	be	AUX
iajs-2376	139	5	a	a	DET
iajs-2376	139	6	fully	fully	ADV
iajs-2376	139	7	semi	semi	ADJ
iajs-2376	139	8	-	-	ADJ
iajs-2376	139	9	prime	prime	ADJ
iajs-2376	139	10	module	module	NOUN
iajs-2376	139	11	,	,	PUNCT
iajs-2376	139	12	then	then	ADV
iajs-2376	139	13	every	every	DET
iajs-2376	139	14	nonzero	nonzero	NOUN
iajs-2376	139	15	-closed	-close	VERB
iajs-2376	139	16	submodule	submodule	NOUN
iajs-2376	139	17	is	be	AUX
iajs-2376	139	18	a	a	DET
iajs-2376	139	19	closed	closed	ADJ
iajs-2376	139	20	submodule	submodule	NOUN
iajs-2376	139	21	in	in	ADP
iajs-2376	139	22	.	.	PUNCT
iajs-2376	140	1	proof	proof	NOUN
iajs-2376	140	2	suppose	suppose	VERB
iajs-2376	140	3	that	that	PRON
iajs-2376	140	4	is	be	AUX
iajs-2376	140	5	a	a	DET
iajs-2376	140	6	non	non	ADJ
iajs-2376	140	7	-	-	ADJ
iajs-2376	140	8	zero	zero	NUM
iajs-2376	140	9	-closed	-close	VERB
iajs-2376	140	10	submodule	submodule	NOUN
iajs-2376	140	11	of	of	ADP
iajs-2376	140	12	,	,	PUNCT
iajs-2376	140	13	then	then	ADV
iajs-2376	140	14	by	by	SCONJ
iajs-2376	140	15	[	[	PUNCT
iajs-2376	140	16	]	]	X
iajs-2376	140	17	is	be	AUX
iajs-2376	140	18	a	a	DET
iajs-2376	140	19	closed	closed	ADJ
iajs-2376	140	20	submodule	submodule	NOUN
iajs-2376	140	21	in	in	ADP
iajs-2376	140	22	.	.	PUNCT
iajs-2376	141	1	hence	hence	ADV
iajs-2376	141	2	by	by	ADP
iajs-2376	141	3	remarks	remark	NOUN
iajs-2376	141	4	and	and	CCONJ
iajs-2376	141	5	examples	example	NOUN
iajs-2376	141	6	(	(	PUNCT
iajs-2376	141	7	2.2	2.2	NUM
iajs-2376	141	8	)	)	PUNCT
iajs-2376	141	9	(	(	PUNCT
iajs-2376	141	10	1	1	X
iajs-2376	141	11	)	)	PUNCT
iajs-2376	141	12	is	be	AUX
iajs-2376	141	13	a	a	DET
iajs-2376	141	14	-closed	-closed	ADJ
iajs-2376	141	15	submodule	submodule	NOUN
iajs-2376	141	16	in	in	ADP
iajs-2376	141	17	.	.	PUNCT
iajs-2376	142	1	proposition	proposition	NOUN
iajs-2376	142	2	2.15	2.15	NUM
iajs-2376	142	3	if	if	SCONJ
iajs-2376	142	4	is	be	AUX
iajs-2376	142	5	an	an	DET
iajs-2376	142	6	-module	-module	NOUN
iajs-2376	142	7	,	,	PUNCT
iajs-2376	142	8	and	and	CCONJ
iajs-2376	142	9	are	be	AUX
iajs-2376	142	10	non	non	ADJ
iajs-2376	142	11	-	-	ADJ
iajs-2376	142	12	zero	zero	NUM
iajs-2376	142	13	submodules	submodule	NOUN
iajs-2376	142	14	of	of	ADP
iajs-2376	142	15	with	with	ADP
iajs-2376	142	16	and	and	CCONJ
iajs-2376	142	17	every	every	DET
iajs-2376	142	18	weak	weak	ADJ
iajs-2376	142	19	essential	essential	ADJ
iajs-2376	142	20	extension	extension	NOUN
iajs-2376	142	21	of	of	ADP
iajs-2376	142	22	is	be	AUX
iajs-2376	142	23	a	a	DET
iajs-2376	142	24	completely	completely	ADV
iajs-2376	142	25	essential	essential	ADJ
iajs-2376	142	26	submodule	submodule	NOUN
iajs-2376	142	27	of	of	ADP
iajs-2376	142	28	such	such	ADJ
iajs-2376	142	29	that	that	PRON
iajs-2376	142	30	is	be	AUX
iajs-2376	142	31	a	a	DET
iajs-2376	142	32	-closed	-closed	ADJ
iajs-2376	142	33	submodule	submodule	NOUN
iajs-2376	142	34	in	in	ADP
iajs-2376	142	35	and	and	CCONJ
iajs-2376	142	36	is	be	AUX
iajs-2376	142	37	a	a	DET
iajs-2376	142	38	-closed	-closed	ADJ
iajs-2376	142	39	submodule	submodule	NOUN
iajs-2376	142	40	in	in	ADP
iajs-2376	142	41	,	,	PUNCT
iajs-2376	142	42	then	then	ADV
iajs-2376	142	43	is	be	AUX
iajs-2376	142	44	a	a	DET
iajs-2376	142	45	-closed	-closed	ADJ
iajs-2376	142	46	submodule	submodule	NOUN
iajs-2376	142	47	in	in	ADV
iajs-2376	142	48	.	.	PUNCT
iajs-2376	143	1	proof	proof	NOUN
iajs-2376	143	2	assume	assume	VERB
iajs-2376	143	3	that	that	SCONJ
iajs-2376	143	4	with	with	ADP
iajs-2376	143	5	,	,	PUNCT
iajs-2376	143	6	then	then	ADV
iajs-2376	143	7	either	either	CCONJ
iajs-2376	143	8	with	with	ADP
iajs-2376	143	9	.	.	PUNCT
iajs-2376	144	1	if	if	SCONJ
iajs-2376	144	2	and	and	CCONJ
iajs-2376	144	3	since	since	SCONJ
iajs-2376	144	4	is	be	AUX
iajs-2376	144	5	a	a	DET
iajs-2376	144	6	-closed	-closed	ADJ
iajs-2376	144	7	submodule	submodule	NOUN
iajs-2376	144	8	in	in	ADP
iajs-2376	144	9	,	,	PUNCT
iajs-2376	144	10	then	then	ADV
iajs-2376	144	11	a	a	DET
iajs-2376	144	12	-closed	-closed	ADJ
iajs-2376	144	13	submodule	submodule	NOUN
iajs-2376	144	14	in	in	ADP
iajs-2376	144	15	with	with	ADP
iajs-2376	144	16	and	and	CCONJ
iajs-2376	144	17	.	.	PUNCT
iajs-2376	145	1	since	since	SCONJ
iajs-2376	145	2	is	be	AUX
iajs-2376	145	3	a	a	DET
iajs-2376	145	4	-closed	-closed	ADJ
iajs-2376	145	5	submodule	submodule	NOUN
iajs-2376	145	6	in	in	ADP
iajs-2376	145	7	and	and	CCONJ
iajs-2376	145	8	is	be	AUX
iajs-2376	145	9	a	a	DET
iajs-2376	145	10	-closed	-closed	ADJ
iajs-2376	145	11	submodule	submodule	NOUN
iajs-2376	145	12	in	in	ADP
iajs-2376	145	13	,	,	PUNCT
iajs-2376	145	14	then	then	ADV
iajs-2376	145	15	by	by	ADP
iajs-2376	145	16	[	[	PUNCT
iajs-2376	145	17	]	]	X
iajs-2376	145	18	,	,	PUNCT
iajs-2376	145	19	we	we	PRON
iajs-2376	145	20	get	get	VERB
iajs-2376	145	21	is	be	AUX
iajs-2376	145	22	a	a	DET
iajs-2376	145	23	-closed	-closed	ADJ
iajs-2376	145	24	submodule	submodule	NOUN
iajs-2376	145	25	in	in	ADP
iajs-2376	145	26	.	.	PUNCT
iajs-2376	146	1	thus	thus	ADV
iajs-2376	146	2	is	be	AUX
iajs-2376	146	3	a	a	DET
iajs-2376	146	4	-closed	-closed	ADJ
iajs-2376	146	5	submodule	submodule	NOUN
iajs-2376	146	6	in	in	ADP
iajs-2376	146	7	.	.	PUNCT
iajs-2376	147	1	if	if	SCONJ
iajs-2376	147	2	and	and	CCONJ
iajs-2376	147	3	is	be	AUX
iajs-2376	147	4	a	a	DET
iajs-2376	147	5	-closed	-closed	ADJ
iajs-2376	147	6	submodule	submodule	NOUN
iajs-2376	147	7	in	in	ADP
iajs-2376	147	8	such	such	ADJ
iajs-2376	147	9	that	that	PRON
iajs-2376	147	10	and	and	CCONJ
iajs-2376	147	11	.	.	PUNCT
iajs-2376	148	1	thus	thus	ADV
iajs-2376	148	2	is	be	AUX
iajs-2376	148	3	a	a	DET
iajs-2376	148	4	-closed	-closed	ADJ
iajs-2376	148	5	submodule	submodule	NOUN
iajs-2376	148	6	in	in	ADP
iajs-2376	148	7	.	.	PUNCT
iajs-2376	148	8	89	89	NUM
iajs-2376	149	1	ibn	ibn	PROPN
iajs-2376	149	2	al	al	PROPN
iajs-2376	149	3	-	-	PUNCT
iajs-2376	149	4	haitham	haitham	PROPN
iajs-2376	149	5	jour	jour	X
iajs-2376	149	6	.	.	PROPN
iajs-2376	149	7	for	for	ADP
iajs-2376	149	8	pure	pure	ADJ
iajs-2376	149	9	&	&	CCONJ
iajs-2376	149	10	appl	appl	PROPN
iajs-2376	149	11	.	.	PUNCT
iajs-2376	150	1	sci	sci	PROPN
iajs-2376	150	2	.	.	PROPN
iajs-2376	151	1	33	33	NUM
iajs-2376	151	2	(	(	PUNCT
iajs-2376	151	3	1	1	NUM
iajs-2376	151	4	)	)	PUNCT
iajs-2376	151	5	2020	2020	NUM
iajs-2376	151	6	proposition	proposition	NOUN
iajs-2376	151	7	2.16	2.16	NUM
iajs-2376	151	8	if	if	SCONJ
iajs-2376	151	9	is	be	AUX
iajs-2376	151	10	a	a	DET
iajs-2376	151	11	fully	fully	ADV
iajs-2376	151	12	semi	semi	ADJ
iajs-2376	151	13	-	-	ADJ
iajs-2376	151	14	prime	prime	ADJ
iajs-2376	151	15	-module	-module	NOUN
iajs-2376	151	16	and	and	CCONJ
iajs-2376	151	17	be	be	AUX
iajs-2376	151	18	a	a	DET
iajs-2376	151	19	non	non	ADJ
iajs-2376	151	20	-	-	ADJ
iajs-2376	151	21	zero	zero	NUM
iajs-2376	151	22	-closed	-close	VERB
iajs-2376	151	23	submodule	submodule	NOUN
iajs-2376	151	24	in	in	ADP
iajs-2376	151	25	and	and	CCONJ
iajs-2376	151	26	is	be	AUX
iajs-2376	151	27	a	a	DET
iajs-2376	151	28	-closed	-closed	ADJ
iajs-2376	151	29	submodule	submodule	NOUN
iajs-2376	151	30	in	in	ADP
iajs-2376	151	31	,	,	PUNCT
iajs-2376	151	32	then	then	ADV
iajs-2376	151	33	is	be	AUX
iajs-2376	151	34	a	a	DET
iajs-2376	151	35	-closed	-closed	ADJ
iajs-2376	151	36	submodule	submodule	NOUN
iajs-2376	151	37	in	in	ADV
iajs-2376	151	38	.	.	PUNCT
iajs-2376	152	1	proof	proof	NOUN
iajs-2376	152	2	by	by	ADP
iajs-2376	152	3	using	use	VERB
iajs-2376	152	4	[	[	PUNCT
iajs-2376	152	5	]	]	PUNCT
iajs-2376	152	6	and	and	CCONJ
iajs-2376	152	7	similarly	similarly	ADV
iajs-2376	152	8	as	as	ADP
iajs-2376	152	9	in	in	ADP
iajs-2376	152	10	proposition	proposition	NOUN
iajs-2376	152	11	(	(	PUNCT
iajs-2376	152	12	2.15	2.15	NUM
iajs-2376	152	13	)	)	PUNCT
iajs-2376	152	14	,	,	PUNCT
iajs-2376	152	15	we	we	PRON
iajs-2376	152	16	get	get	VERB
iajs-2376	152	17	the	the	DET
iajs-2376	152	18	result	result	NOUN
iajs-2376	152	19	.	.	PUNCT
iajs-2376	153	1	proposition	proposition	NOUN
iajs-2376	153	2	2.17	2.17	NUM
iajs-2376	153	3	let	let	VERB
iajs-2376	153	4	is	be	AUX
iajs-2376	153	5	a	a	DET
iajs-2376	153	6	fully	fully	ADV
iajs-2376	153	7	semi	semi	ADJ
iajs-2376	153	8	-	-	ADJ
iajs-2376	153	9	prime	prime	ADJ
iajs-2376	153	10	module	module	NOUN
iajs-2376	153	11	,	,	PUNCT
iajs-2376	153	12	and	and	CCONJ
iajs-2376	153	13	are	be	AUX
iajs-2376	153	14	submodules	submodule	NOUN
iajs-2376	153	15	of	of	ADP
iajs-2376	153	16	,	,	PUNCT
iajs-2376	153	17	if	if	SCONJ
iajs-2376	153	18	is	be	AUX
iajs-2376	153	19	a	a	DET
iajs-2376	153	20	-closed	-closed	ADJ
iajs-2376	153	21	submodule	submodule	NOUN
iajs-2376	153	22	in	in	ADP
iajs-2376	153	23	and	and	CCONJ
iajs-2376	153	24	is	be	AUX
iajs-2376	153	25	a	a	DET
iajs-2376	153	26	weak	weak	ADJ
iajs-2376	153	27	essential	essential	ADJ
iajs-2376	153	28	submodule	submodule	NOUN
iajs-2376	153	29	in	in	ADP
iajs-2376	153	30	.	.	PUNCT
iajs-2376	154	1	then	then	ADV
iajs-2376	154	2	is	be	AUX
iajs-2376	154	3	a	a	DET
iajs-2376	154	4	-closed	-closed	ADJ
iajs-2376	154	5	submodule	submodule	NOUN
iajs-2376	154	6	in	in	ADV
iajs-2376	154	7	.	.	PUNCT
iajs-2376	155	1	proof	proof	NOUN
iajs-2376	155	2	suppose	suppose	VERB
iajs-2376	155	3	that	that	SCONJ
iajs-2376	155	4	with	with	ADP
iajs-2376	155	5	implies	implie	NOUN
iajs-2376	155	6	that	that	PRON
iajs-2376	155	7	.	.	PUNCT
iajs-2376	156	1	since	since	SCONJ
iajs-2376	156	2	,	,	PUNCT
iajs-2376	156	3	then	then	ADV
iajs-2376	156	4	.	.	PUNCT
iajs-2376	157	1	since	since	SCONJ
iajs-2376	157	2	is	be	AUX
iajs-2376	157	3	a	a	DET
iajs-2376	157	4	-closed	-closed	ADJ
iajs-2376	157	5	submodule	submodule	NOUN
iajs-2376	157	6	in	in	ADV
iajs-2376	157	7	,	,	PUNCT
iajs-2376	157	8	so	so	CCONJ
iajs-2376	157	9	there	there	PRON
iajs-2376	157	10	exists	exist	VERB
iajs-2376	157	11	a	a	DET
iajs-2376	157	12	-closed	-closed	ADJ
iajs-2376	157	13	submodule	submodule	NOUN
iajs-2376	157	14	in	in	ADP
iajs-2376	157	15	with	with	ADP
iajs-2376	157	16	and	and	CCONJ
iajs-2376	157	17	.	.	PUNCT
iajs-2376	158	1	hence	hence	ADV
iajs-2376	158	2	is	be	AUX
iajs-2376	158	3	a	a	DET
iajs-2376	158	4	closed	closed	ADJ
iajs-2376	158	5	submodule	submodule	NOUN
iajs-2376	158	6	in	in	ADP
iajs-2376	158	7	.	.	PUNCT
iajs-2376	159	1	and	and	CCONJ
iajs-2376	159	2	by	by	ADP
iajs-2376	159	3	[	[	PUNCT
iajs-2376	159	4	]	]	X
iajs-2376	159	5	,	,	PUNCT
iajs-2376	159	6	is	be	AUX
iajs-2376	159	7	essential	essential	ADJ
iajs-2376	159	8	in	in	ADP
iajs-2376	159	9	.	.	PUNCT
iajs-2376	160	1	therefore	therefore	ADV
iajs-2376	160	2	by	by	ADP
iajs-2376	160	3	[	[	PUNCT
iajs-2376	160	4	]	]	X
iajs-2376	160	5	we	we	PRON
iajs-2376	160	6	get	get	VERB
iajs-2376	160	7	is	be	AUX
iajs-2376	160	8	a	a	DET
iajs-2376	160	9	closed	closed	ADJ
iajs-2376	160	10	submodule	submodule	NOUN
iajs-2376	160	11	in	in	ADP
iajs-2376	160	12	.	.	PUNCT
iajs-2376	161	1	hence	hence	ADV
iajs-2376	161	2	by	by	ADP
iajs-2376	161	3	[	[	PUNCT
iajs-2376	161	4	]	]	X
iajs-2376	161	5	we	we	PRON
iajs-2376	161	6	have	have	AUX
iajs-2376	161	7	is	be	AUX
iajs-2376	161	8	a	a	DET
iajs-2376	161	9	-closed	-closed	ADJ
iajs-2376	161	10	submodule	submodule	NOUN
iajs-2376	161	11	in	in	ADP
iajs-2376	161	12	with	with	ADP
iajs-2376	161	13	and	and	CCONJ
iajs-2376	161	14	.	.	PUNCT
iajs-2376	162	1	hence	hence	ADV
iajs-2376	162	2	is	be	AUX
iajs-2376	162	3	a	a	DET
iajs-2376	162	4	-closed	-closed	ADJ
iajs-2376	162	5	submodule	submodule	NOUN
iajs-2376	162	6	in	in	ADP
iajs-2376	162	7	.	.	PUNCT
iajs-2376	163	1	recall	recall	VERB
iajs-2376	163	2	that	that	SCONJ
iajs-2376	163	3	an	an	DET
iajs-2376	163	4	-module	-module	NOUN
iajs-2376	163	5	is	be	AUX
iajs-2376	163	6	called	call	VERB
iajs-2376	163	7	fully	fully	ADV
iajs-2376	163	8	prime	prime	ADJ
iajs-2376	163	9	,	,	PUNCT
iajs-2376	163	10	if	if	SCONJ
iajs-2376	163	11	every	every	DET
iajs-2376	163	12	proper	proper	ADJ
iajs-2376	163	13	submodule	submodule	NOUN
iajs-2376	163	14	of	of	ADP
iajs-2376	163	15	is	be	AUX
iajs-2376	163	16	a	a	DET
iajs-2376	163	17	prime	prime	ADJ
iajs-2376	163	18	submodule	submodule	NOUN
iajs-2376	163	19	[	[	PUNCT
iajs-2376	163	20	]	]	X
iajs-2376	163	21	.	.	PUNCT
iajs-2376	164	1	it	it	PRON
iajs-2376	164	2	is	be	AUX
iajs-2376	164	3	wellknown	wellknown	ADJ
iajs-2376	164	4	every	every	DET
iajs-2376	164	5	fully	fully	ADV
iajs-2376	164	6	prime	prime	ADJ
iajs-2376	164	7	-module	-module	NOUN
iajs-2376	164	8	is	be	AUX
iajs-2376	164	9	a	a	DET
iajs-2376	164	10	fully	fully	ADV
iajs-2376	164	11	semi	semi	ADJ
iajs-2376	164	12	-	-	ADJ
iajs-2376	164	13	prime	prime	ADJ
iajs-2376	164	14	we	we	PRON
iajs-2376	164	15	get	get	VERB
iajs-2376	164	16	the	the	DET
iajs-2376	164	17	following	follow	VERB
iajs-2376	164	18	result	result	NOUN
iajs-2376	164	19	.	.	PUNCT
iajs-2376	165	1	corollary	corollary	ADJ
iajs-2376	165	2	2.18	2.18	NUM
iajs-2376	165	3	let	let	AUX
iajs-2376	165	4	be	be	AUX
iajs-2376	165	5	a	a	DET
iajs-2376	165	6	fully	fully	ADV
iajs-2376	165	7	prime	prime	ADJ
iajs-2376	165	8	module	module	NOUN
iajs-2376	165	9	,	,	PUNCT
iajs-2376	165	10	and	and	CCONJ
iajs-2376	165	11	are	be	AUX
iajs-2376	165	12	submodules	submodule	NOUN
iajs-2376	165	13	of	of	ADP
iajs-2376	165	14	,	,	PUNCT
iajs-2376	165	15	if	if	SCONJ
iajs-2376	165	16	is	be	AUX
iajs-2376	165	17	a	a	DET
iajs-2376	165	18	-closed	-closed	ADJ
iajs-2376	165	19	submodule	submodule	NOUN
iajs-2376	165	20	in	in	ADP
iajs-2376	165	21	and	and	CCONJ
iajs-2376	165	22	is	be	AUX
iajs-2376	165	23	a	a	DET
iajs-2376	165	24	weak	weak	ADJ
iajs-2376	165	25	essential	essential	ADJ
iajs-2376	165	26	submodule	submodule	NOUN
iajs-2376	165	27	in	in	ADP
iajs-2376	165	28	.	.	PUNCT
iajs-2376	166	1	then	then	ADV
iajs-2376	166	2	is	be	AUX
iajs-2376	166	3	a	a	DET
iajs-2376	166	4	-closed	-closed	ADJ
iajs-2376	166	5	submodule	submodule	NOUN
iajs-2376	166	6	in	in	ADP
iajs-2376	166	7	.	.	PUNCT
iajs-2376	167	1	since	since	SCONJ
iajs-2376	167	2	-closed	-closed	ADJ
iajs-2376	167	3	submodule	submodule	NOUN
iajs-2376	167	4	is	be	AUX
iajs-2376	167	5	a	a	DET
iajs-2376	167	6	-closed	-close	VERB
iajs-2376	167	7	submodule	submodule	NOUN
iajs-2376	167	8	,	,	PUNCT
iajs-2376	167	9	then	then	ADV
iajs-2376	167	10	we	we	PRON
iajs-2376	167	11	get	get	VERB
iajs-2376	167	12	the	the	DET
iajs-2376	167	13	following	follow	VERB
iajs-2376	167	14	result	result	NOUN
iajs-2376	167	15	.	.	PUNCT
iajs-2376	168	1	corollary	corollary	ADJ
iajs-2376	168	2	2.19	2.19	NUM
iajs-2376	168	3	let	let	AUX
iajs-2376	168	4	be	be	AUX
iajs-2376	168	5	a	a	DET
iajs-2376	168	6	fully	fully	ADV
iajs-2376	168	7	semi	semi	ADJ
iajs-2376	168	8	-	-	ADJ
iajs-2376	168	9	prime	prime	ADJ
iajs-2376	168	10	module	module	NOUN
iajs-2376	168	11	,	,	PUNCT
iajs-2376	168	12	and	and	CCONJ
iajs-2376	168	13	are	be	AUX
iajs-2376	168	14	submodules	submodule	NOUN
iajs-2376	168	15	of	of	ADP
iajs-2376	168	16	,	,	PUNCT
iajs-2376	168	17	if	if	SCONJ
iajs-2376	168	18	is	be	AUX
iajs-2376	168	19	a	a	DET
iajs-2376	168	20	-closed	-closed	ADJ
iajs-2376	168	21	submodule	submodule	NOUN
iajs-2376	168	22	in	in	ADP
iajs-2376	168	23	and	and	CCONJ
iajs-2376	168	24	is	be	AUX
iajs-2376	168	25	a	a	DET
iajs-2376	168	26	weak	weak	ADJ
iajs-2376	168	27	essential	essential	ADJ
iajs-2376	168	28	submodule	submodule	NOUN
iajs-2376	168	29	in	in	ADP
iajs-2376	168	30	.	.	PUNCT
iajs-2376	169	1	then	then	ADV
iajs-2376	169	2	is	be	AUX
iajs-2376	169	3	a	a	DET
iajs-2376	169	4	-closed	-closed	ADJ
iajs-2376	169	5	submodule	submodule	NOUN
iajs-2376	169	6	in	in	ADP
iajs-2376	169	7	.	.	PUNCT
iajs-2376	170	1	since	since	SCONJ
iajs-2376	170	2	in	in	ADP
iajs-2376	170	3	the	the	DET
iajs-2376	170	4	class	class	NOUN
iajs-2376	170	5	of	of	ADP
iajs-2376	170	6	a	a	DET
iajs-2376	170	7	fully	fully	ADV
iajs-2376	170	8	semi	semi	ADJ
iajs-2376	170	9	-	-	ADJ
iajs-2376	170	10	prime	prime	ADJ
iajs-2376	170	11	modules	module	NOUN
iajs-2376	170	12	,	,	PUNCT
iajs-2376	170	13	-closed	-close	VERB
iajs-2376	170	14	submodule	submodule	NOUN
iajs-2376	170	15	is	be	AUX
iajs-2376	170	16	a	a	DET
iajs-2376	170	17	-closed	-close	VERB
iajs-2376	170	18	submodule	submodule	NOUN
iajs-2376	170	19	we	we	PRON
iajs-2376	170	20	get	get	VERB
iajs-2376	170	21	the	the	DET
iajs-2376	170	22	following	follow	VERB
iajs-2376	170	23	result	result	NOUN
iajs-2376	170	24	.	.	PUNCT
iajs-2376	171	1	corollary	corollary	ADJ
iajs-2376	171	2	2.20	2.20	NUM
iajs-2376	171	3	if	if	SCONJ
iajs-2376	171	4	is	be	AUX
iajs-2376	171	5	a	a	DET
iajs-2376	171	6	fully	fully	ADV
iajs-2376	171	7	semi	semi	ADJ
iajs-2376	171	8	-	-	ADJ
iajs-2376	171	9	prime	prime	ADJ
iajs-2376	171	10	module	module	NOUN
iajs-2376	171	11	,	,	PUNCT
iajs-2376	171	12	and	and	CCONJ
iajs-2376	171	13	are	be	AUX
iajs-2376	171	14	submodules	submodule	NOUN
iajs-2376	171	15	of	of	ADP
iajs-2376	171	16	,	,	PUNCT
iajs-2376	171	17	if	if	SCONJ
iajs-2376	171	18	is	be	AUX
iajs-2376	171	19	a	a	DET
iajs-2376	171	20	-closed	-closed	ADJ
iajs-2376	171	21	submodule	submodule	NOUN
iajs-2376	171	22	in	in	ADP
iajs-2376	171	23	and	and	CCONJ
iajs-2376	171	24	is	be	AUX
iajs-2376	171	25	a	a	DET
iajs-2376	171	26	weak	weak	ADJ
iajs-2376	171	27	essential	essential	ADJ
iajs-2376	171	28	submodule	submodule	NOUN
iajs-2376	171	29	in	in	ADP
iajs-2376	171	30	.	.	PUNCT
iajs-2376	172	1	then	then	ADV
iajs-2376	172	2	is	be	AUX
iajs-2376	172	3	a	a	DET
iajs-2376	172	4	-closed	-closed	ADJ
iajs-2376	172	5	submodule	submodule	NOUN
iajs-2376	172	6	in	in	ADP
iajs-2376	172	7	.	.	PUNCT
iajs-2376	173	1	3	3	X
iajs-2376	173	2	.	.	X
iajs-2376	173	3	-closed	-close	VERB
iajs-2376	173	4	submodules	submodule	NOUN
iajs-2376	173	5	in	in	ADP
iajs-2376	173	6	multiplication	multiplication	NOUN
iajs-2376	173	7	modules	module	NOUN
iajs-2376	173	8	this	this	DET
iajs-2376	173	9	section	section	NOUN
iajs-2376	173	10	is	be	AUX
iajs-2376	173	11	devoted	devote	VERB
iajs-2376	173	12	to	to	PART
iajs-2376	173	13	study	study	VERB
iajs-2376	173	14	the	the	DET
iajs-2376	173	15	behavior	behavior	NOUN
iajs-2376	173	16	of	of	ADP
iajs-2376	173	17	-closed	-close	VERB
iajs-2376	173	18	submodules	submodule	NOUN
iajs-2376	173	19	in	in	ADP
iajs-2376	173	20	the	the	DET
iajs-2376	173	21	class	class	NOUN
iajs-2376	173	22	of	of	ADP
iajs-2376	173	23	multiplication	multiplication	NOUN
iajs-2376	173	24	modules	module	NOUN
iajs-2376	173	25	.	.	PUNCT
iajs-2376	174	1	recall	recall	VERB
iajs-2376	174	2	that	that	SCONJ
iajs-2376	174	3	an	an	DET
iajs-2376	174	4	-module	-module	NOUN
iajs-2376	174	5	is	be	AUX
iajs-2376	174	6	a	a	DET
iajs-2376	174	7	multiplication	multiplication	NOUN
iajs-2376	174	8	if	if	SCONJ
iajs-2376	174	9	every	every	DET
iajs-2376	174	10	submodule	submodule	NOUN
iajs-2376	174	11	of	of	ADP
iajs-2376	174	12	is	be	AUX
iajs-2376	174	13	of	of	ADP
iajs-2376	174	14	the	the	DET
iajs-2376	174	15	form	form	NOUN
iajs-2376	174	16	for	for	ADP
iajs-2376	174	17	some	some	DET
iajs-2376	174	18	ideal	ideal	NOUN
iajs-2376	174	19	of	of	ADP
iajs-2376	174	20	[	[	PUNCT
iajs-2376	174	21	]	]	X
iajs-2376	174	22	”	"	PUNCT
iajs-2376	174	23	.	.	PUNCT
iajs-2376	175	1	and	and	CCONJ
iajs-2376	175	2	an	an	DET
iajs-2376	175	3	-module	-module	NOUN
iajs-2376	175	4	is	be	AUX
iajs-2376	175	5	called	call	VERB
iajs-2376	175	6	faithful	faithful	ADJ
iajs-2376	175	7	if	if	SCONJ
iajs-2376	175	8	for	for	ADP
iajs-2376	175	9	any	any	DET
iajs-2376	175	10	non	non	ADJ
iajs-2376	175	11	-	-	NOUN
iajs-2376	175	12	zero	zero	NUM
iajs-2376	175	13	,	,	PUNCT
iajs-2376	175	14	there	there	PRON
iajs-2376	175	15	is	be	VERB
iajs-2376	175	16	an	an	DET
iajs-2376	175	17	element	element	NOUN
iajs-2376	176	1	such	such	ADJ
iajs-2376	176	2	that	that	SCONJ
iajs-2376	176	3	[	[	X
iajs-2376	176	4	]	]	X
iajs-2376	176	5	.	.	X
iajs-2376	176	6	89	89	NUM
iajs-2376	176	7	ibn	ibn	PROPN
iajs-2376	176	8	al	al	PROPN
iajs-2376	176	9	-	-	PUNCT
iajs-2376	176	10	haitham	haitham	PROPN
iajs-2376	176	11	jour	jour	X
iajs-2376	176	12	.	.	PROPN
iajs-2376	176	13	for	for	ADP
iajs-2376	176	14	pure	pure	ADJ
iajs-2376	176	15	&	&	CCONJ
iajs-2376	176	16	appl	appl	PROPN
iajs-2376	176	17	.	.	PUNCT
iajs-2376	177	1	sci	sci	PROPN
iajs-2376	177	2	.	.	PROPN
iajs-2376	178	1	33	33	NUM
iajs-2376	178	2	(	(	PUNCT
iajs-2376	178	3	1	1	NUM
iajs-2376	178	4	)	)	PUNCT
iajs-2376	178	5	2020	2020	NUM
iajs-2376	178	6	recall	recall	VERB
iajs-2376	178	7	that	that	PRON
iajs-2376	178	8	for	for	ADP
iajs-2376	178	9	any	any	DET
iajs-2376	178	10	-module	-module	NOUN
iajs-2376	178	11	and	and	CCONJ
iajs-2376	178	12	any	any	DET
iajs-2376	178	13	ideals	ideal	NOUN
iajs-2376	178	14	and	and	CCONJ
iajs-2376	178	15	of	of	ADP
iajs-2376	178	16	,	,	PUNCT
iajs-2376	178	17	if	if	SCONJ
iajs-2376	178	18	is	be	AUX
iajs-2376	178	19	a	a	DET
iajs-2376	178	20	semi	semi	ADJ
iajs-2376	178	21	-	-	ADJ
iajs-2376	178	22	prime	prime	ADJ
iajs-2376	178	23	ideal	ideal	NOUN
iajs-2376	178	24	of	of	ADP
iajs-2376	178	25	,	,	PUNCT
iajs-2376	178	26	then	then	ADV
iajs-2376	178	27	is	be	AUX
iajs-2376	178	28	a	a	DET
iajs-2376	178	29	semi	semi	ADJ
iajs-2376	178	30	-	-	ADJ
iajs-2376	178	31	prime	prime	ADJ
iajs-2376	178	32	submodule	submodule	NOUN
iajs-2376	178	33	of	of	ADP
iajs-2376	178	34	.	.	PUNCT
iajs-2376	179	1	this	this	PRON
iajs-2376	179	2	is	be	AUX
iajs-2376	179	3	called	call	VERB
iajs-2376	179	4	condition	condition	NOUN
iajs-2376	179	5	(	(	PUNCT
iajs-2376	179	6	*	*	NOUN
iajs-2376	179	7	)	)	PUNCT
iajs-2376	180	1	[	[	PUNCT
iajs-2376	180	2	]	]	X
iajs-2376	180	3	.	.	PUNCT
iajs-2376	181	1	proposition	proposition	NOUN
iajs-2376	181	2	3.1	3.1	NUM
iajs-2376	181	3	if	if	SCONJ
iajs-2376	181	4	is	be	AUX
iajs-2376	181	5	a	a	DET
iajs-2376	181	6	faithful	faithful	ADJ
iajs-2376	181	7	multiplication	multiplication	NOUN
iajs-2376	181	8	module	module	NOUN
iajs-2376	181	9	satisfies	satisfie	NOUN
iajs-2376	181	10	condition	condition	NOUN
iajs-2376	181	11	(	(	PUNCT
iajs-2376	181	12	*	*	NOUN
iajs-2376	181	13	)	)	PUNCT
iajs-2376	181	14	,	,	PUNCT
iajs-2376	181	15	and	and	CCONJ
iajs-2376	181	16	are	be	AUX
iajs-2376	181	17	ideals	ideal	NOUN
iajs-2376	181	18	of	of	ADP
iajs-2376	181	19	a	a	DET
iajs-2376	181	20	ring	ring	NOUN
iajs-2376	181	21	such	such	ADJ
iajs-2376	181	22	that	that	PRON
iajs-2376	181	23	is	be	AUX
iajs-2376	181	24	a	a	DET
iajs-2376	181	25	-closed	-close	VERB
iajs-2376	181	26	ideal	ideal	NOUN
iajs-2376	181	27	in	in	ADP
iajs-2376	181	28	,	,	PUNCT
iajs-2376	181	29	then	then	ADV
iajs-2376	181	30	is	be	AUX
iajs-2376	181	31	a	a	DET
iajs-2376	181	32	-closed	-closed	ADJ
iajs-2376	181	33	submodule	submodule	NOUN
iajs-2376	181	34	in	in	ADV
iajs-2376	181	35	.	.	PUNCT
iajs-2376	182	1	proof	proof	NOUN
iajs-2376	182	2	let	let	AUX
iajs-2376	182	3	be	be	AUX
iajs-2376	182	4	a	a	DET
iajs-2376	182	5	-closed	-close	VERB
iajs-2376	182	6	ideal	ideal	NOUN
iajs-2376	182	7	in	in	ADV
iajs-2376	182	8	,	,	PUNCT
iajs-2376	182	9	and	and	CCONJ
iajs-2376	182	10	let	let	VERB
iajs-2376	182	11	where	where	SCONJ
iajs-2376	182	12	,	,	PUNCT
iajs-2376	182	13	and	and	CCONJ
iajs-2376	182	14	with	with	ADP
iajs-2376	182	15	i.e.	i.e.	X
iajs-2376	182	16	,	,	PUNCT
iajs-2376	182	17	but	but	CCONJ
iajs-2376	182	18	.	.	PUNCT
iajs-2376	183	1	but	but	CCONJ
iajs-2376	183	2	is	be	AUX
iajs-2376	183	3	a	a	DET
iajs-2376	183	4	-closed	-close	VERB
iajs-2376	183	5	ideal	ideal	NOUN
iajs-2376	183	6	in	in	ADP
iajs-2376	183	7	and	and	CCONJ
iajs-2376	183	8	,	,	PUNCT
iajs-2376	183	9	then	then	ADV
iajs-2376	183	10	there	there	PRON
iajs-2376	183	11	exists	exist	VERB
iajs-2376	183	12	a	a	DET
iajs-2376	183	13	-closed	-closed	ADJ
iajs-2376	183	14	ideal	ideal	NOUN
iajs-2376	183	15	in	in	ADP
iajs-2376	183	16	such	such	ADJ
iajs-2376	183	17	that	that	PRON
iajs-2376	183	18	and	and	CCONJ
iajs-2376	183	19	.	.	PUNCT
iajs-2376	184	1	but	but	CCONJ
iajs-2376	184	2	is	be	AUX
iajs-2376	184	3	a	a	DET
iajs-2376	184	4	faithful	faithful	ADJ
iajs-2376	184	5	and	and	CCONJ
iajs-2376	184	6	multiplication	multiplication	NOUN
iajs-2376	184	7	then	then	ADV
iajs-2376	184	8	and	and	CCONJ
iajs-2376	184	9	.	.	PUNCT
iajs-2376	185	1	but	but	CCONJ
iajs-2376	185	2	by	by	ADP
iajs-2376	185	3	[	[	PUNCT
iajs-2376	185	4	]	]	X
iajs-2376	185	5	,	,	PUNCT
iajs-2376	185	6	is	be	AUX
iajs-2376	185	7	a	a	DET
iajs-2376	185	8	-closed	-closed	ADJ
iajs-2376	185	9	submodule	submodule	NOUN
iajs-2376	185	10	in	in	ADP
iajs-2376	185	11	.	.	PUNCT
iajs-2376	186	1	hence	hence	ADV
iajs-2376	186	2	is	be	AUX
iajs-2376	186	3	a	a	DET
iajs-2376	186	4	-closed	-closed	ADJ
iajs-2376	186	5	submodule	submodule	NOUN
iajs-2376	186	6	in	in	ADP
iajs-2376	186	7	.	.	PUNCT
iajs-2376	187	1	proposition	proposition	NOUN
iajs-2376	187	2	3.2	3.2	NUM
iajs-2376	187	3	if	if	SCONJ
iajs-2376	187	4	is	be	AUX
iajs-2376	187	5	a	a	DET
iajs-2376	187	6	finitely	finitely	ADV
iajs-2376	187	7	generated	generate	VERB
iajs-2376	187	8	,	,	PUNCT
iajs-2376	187	9	faithful	faithful	ADJ
iajs-2376	187	10	and	and	CCONJ
iajs-2376	187	11	multiplication	multiplication	NOUN
iajs-2376	187	12	module	module	NOUN
iajs-2376	187	13	,	,	PUNCT
iajs-2376	187	14	and	and	CCONJ
iajs-2376	187	15	are	be	AUX
iajs-2376	187	16	ideals	ideal	NOUN
iajs-2376	187	17	of	of	ADP
iajs-2376	187	18	a	a	DET
iajs-2376	187	19	ring	ring	NOUN
iajs-2376	187	20	such	such	ADJ
iajs-2376	187	21	that	that	PRON
iajs-2376	187	22	is	be	AUX
iajs-2376	187	23	a	a	DET
iajs-2376	187	24	-closed	-closed	ADJ
iajs-2376	187	25	submodule	submodule	NOUN
iajs-2376	187	26	in	in	ADP
iajs-2376	187	27	,	,	PUNCT
iajs-2376	187	28	then	then	ADV
iajs-2376	187	29	is	be	AUX
iajs-2376	187	30	a	a	DET
iajs-2376	187	31	-closed	-close	VERB
iajs-2376	187	32	ideal	ideal	NOUN
iajs-2376	187	33	in	in	ADV
iajs-2376	187	34	.	.	PUNCT
iajs-2376	188	1	proof	proof	NOUN
iajs-2376	188	2	suppose	suppose	VERB
iajs-2376	188	3	that	that	PRON
iajs-2376	188	4	is	be	AUX
iajs-2376	188	5	a	a	DET
iajs-2376	188	6	-closed	-closed	ADJ
iajs-2376	188	7	submodule	submodule	NOUN
iajs-2376	188	8	in	in	ADV
iajs-2376	188	9	.	.	PUNCT
iajs-2376	189	1	let	let	VERB
iajs-2376	189	2	with	with	ADP
iajs-2376	189	3	,	,	PUNCT
iajs-2376	189	4	there	there	ADV
iajs-2376	189	5	for	for	ADP
iajs-2376	189	6	each	each	PRON
iajs-2376	189	7	,	,	PUNCT
iajs-2376	189	8	and	and	CCONJ
iajs-2376	189	9	.	.	PUNCT
iajs-2376	190	1	but	but	CCONJ
iajs-2376	190	2	is	be	AUX
iajs-2376	190	3	a	a	DET
iajs-2376	190	4	-closed	-closed	ADJ
iajs-2376	190	5	submodule	submodule	NOUN
iajs-2376	190	6	in	in	ADP
iajs-2376	190	7	then	then	ADV
iajs-2376	190	8	there	there	PRON
iajs-2376	190	9	exists	exist	VERB
iajs-2376	190	10	a	a	DET
iajs-2376	190	11	-closed	-closed	ADJ
iajs-2376	190	12	submodule	submodule	NOUN
iajs-2376	190	13	in	in	ADP
iajs-2376	190	14	,	,	PUNCT
iajs-2376	190	15	where	where	SCONJ
iajs-2376	190	16	is	be	AUX
iajs-2376	190	17	an	an	DET
iajs-2376	190	18	ideal	ideal	NOUN
iajs-2376	190	19	in	in	ADP
iajs-2376	190	20	such	such	ADJ
iajs-2376	190	21	that	that	PRON
iajs-2376	190	22	and	and	CCONJ
iajs-2376	190	23	.	.	PUNCT
iajs-2376	191	1	hence	hence	ADV
iajs-2376	191	2	by	by	ADP
iajs-2376	191	3	[	[	PUNCT
iajs-2376	191	4	]	]	X
iajs-2376	191	5	we	we	PRON
iajs-2376	191	6	get	get	VERB
iajs-2376	191	7	is	be	AUX
iajs-2376	191	8	a	a	DET
iajs-2376	191	9	-closed	-close	VERB
iajs-2376	191	10	ideal	ideal	NOUN
iajs-2376	191	11	in	in	ADP
iajs-2376	191	12	,	,	PUNCT
iajs-2376	191	13	and	and	CCONJ
iajs-2376	191	14	with	with	ADP
iajs-2376	191	15	and	and	CCONJ
iajs-2376	191	16	.	.	PUNCT
iajs-2376	192	1	hence	hence	ADV
iajs-2376	192	2	is	be	AUX
iajs-2376	192	3	a	a	DET
iajs-2376	192	4	-closed	-close	VERB
iajs-2376	192	5	ideal	ideal	NOUN
iajs-2376	192	6	in	in	ADP
iajs-2376	192	7	.	.	PUNCT
iajs-2376	193	1	from	from	ADP
iajs-2376	193	2	proposition	proposition	NOUN
iajs-2376	193	3	(	(	PUNCT
iajs-2376	193	4	3.1	3.1	NUM
iajs-2376	193	5	)	)	PUNCT
iajs-2376	193	6	and	and	CCONJ
iajs-2376	193	7	proposition	proposition	NOUN
iajs-2376	193	8	(	(	PUNCT
iajs-2376	193	9	3.2	3.2	NUM
iajs-2376	193	10	)	)	PUNCT
iajs-2376	193	11	,	,	PUNCT
iajs-2376	193	12	we	we	PRON
iajs-2376	193	13	get	get	VERB
iajs-2376	193	14	the	the	DET
iajs-2376	193	15	following	follow	VERB
iajs-2376	193	16	corollaries	corollary	NOUN
iajs-2376	193	17	corollary	corollary	NOUN
iajs-2376	193	18	3.3	3.3	NUM
iajs-2376	193	19	let	let	AUX
iajs-2376	193	20	be	be	AUX
iajs-2376	193	21	a	a	DET
iajs-2376	193	22	finitely	finitely	ADV
iajs-2376	193	23	generated	generate	VERB
iajs-2376	193	24	,	,	PUNCT
iajs-2376	193	25	faithful	faithful	ADJ
iajs-2376	193	26	and	and	CCONJ
iajs-2376	193	27	multiplication	multiplication	NOUN
iajs-2376	193	28	module	module	NOUN
iajs-2376	193	29	which	which	PRON
iajs-2376	193	30	satisfies	satisfy	VERB
iajs-2376	193	31	condition	condition	NOUN
iajs-2376	193	32	(	(	PUNCT
iajs-2376	193	33	*	*	NOUN
iajs-2376	193	34	)	)	PUNCT
iajs-2376	193	35	.	.	PUNCT
iajs-2376	194	1	then	then	ADV
iajs-2376	194	2	is	be	AUX
iajs-2376	194	3	a	a	DET
iajs-2376	194	4	-closed	-close	VERB
iajs-2376	194	5	ideal	ideal	NOUN
iajs-2376	194	6	in	in	ADP
iajs-2376	194	7	if	if	SCONJ
iajs-2376	195	1	and	and	CCONJ
iajs-2376	195	2	only	only	ADV
iajs-2376	195	3	if	if	SCONJ
iajs-2376	195	4	is	be	AUX
iajs-2376	195	5	a	a	DET
iajs-2376	195	6	-closed	-closed	ADJ
iajs-2376	195	7	submodule	submodule	NOUN
iajs-2376	195	8	in	in	ADP
iajs-2376	195	9	.	.	PUNCT
iajs-2376	196	1	corollary	corollary	NOUN
iajs-2376	196	2	3.4	3.4	NUM
iajs-2376	196	3	let	let	AUX
iajs-2376	196	4	be	be	AUX
iajs-2376	196	5	a	a	DET
iajs-2376	196	6	finitely	finitely	ADV
iajs-2376	196	7	generated	generate	VERB
iajs-2376	196	8	multiplication	multiplication	NOUN
iajs-2376	196	9	module	module	NOUN
iajs-2376	196	10	and	and	CCONJ
iajs-2376	196	11	let	let	VERB
iajs-2376	196	12	be	be	AUX
iajs-2376	196	13	a	a	DET
iajs-2376	196	14	submodule	submodule	NOUN
iajs-2376	196	15	of	of	ADP
iajs-2376	196	16	such	such	ADJ
iajs-2376	196	17	that	that	SCONJ
iajs-2376	196	18	satisfies	satisfy	VERB
iajs-2376	196	19	condition	condition	NOUN
iajs-2376	196	20	(	(	PUNCT
iajs-2376	196	21	*	*	NOUN
iajs-2376	196	22	)	)	PUNCT
iajs-2376	196	23	.	.	PUNCT
iajs-2376	197	1	then	then	ADV
iajs-2376	197	2	the	the	DET
iajs-2376	197	3	next	next	ADJ
iajs-2376	197	4	statements	statement	NOUN
iajs-2376	197	5	are	be	AUX
iajs-2376	197	6	equivalents	equivalent	NOUN
iajs-2376	197	7	.	.	PUNCT
iajs-2376	198	1	1	1	X
iajs-2376	198	2	.	.	X
iajs-2376	198	3	is	be	AUX
iajs-2376	198	4	a	a	DET
iajs-2376	198	5	-closed	-closed	ADJ
iajs-2376	198	6	submodule	submodule	NOUN
iajs-2376	198	7	in	in	ADP
iajs-2376	198	8	.	.	PUNCT
iajs-2376	199	1	2	2	X
iajs-2376	199	2	.	.	PUNCT
iajs-2376	199	3	[	[	PUNCT
iajs-2376	199	4	]	]	X
iajs-2376	199	5	is	be	AUX
iajs-2376	199	6	a	a	DET
iajs-2376	199	7	-closed	-close	VERB
iajs-2376	199	8	ideal	ideal	NOUN
iajs-2376	199	9	in	in	ADP
iajs-2376	199	10	.	.	PUNCT
iajs-2376	200	1	3	3	X
iajs-2376	200	2	.	.	X
iajs-2376	200	3	for	for	ADP
iajs-2376	200	4	some	some	DET
iajs-2376	200	5	-closed	-close	VERB
iajs-2376	200	6	ideal	ideal	NOUN
iajs-2376	200	7	in	in	ADV
iajs-2376	200	8	.	.	PUNCT
iajs-2376	201	1	proof	proof	NOUN
iajs-2376	201	2	(	(	PUNCT
iajs-2376	201	3	1	1	NUM
iajs-2376	201	4	)	)	PUNCT
iajs-2376	201	5	(	(	PUNCT
iajs-2376	201	6	2	2	X
iajs-2376	201	7	)	)	PUNCT
iajs-2376	201	8	since	since	SCONJ
iajs-2376	201	9	is	be	AUX
iajs-2376	201	10	a	a	DET
iajs-2376	201	11	multiplication	multiplication	NOUN
iajs-2376	201	12	module	module	NOUN
iajs-2376	201	13	and	and	CCONJ
iajs-2376	201	14	is	be	AUX
iajs-2376	201	15	a	a	DET
iajs-2376	201	16	-closed	-closed	ADJ
iajs-2376	201	17	submodule	submodule	NOUN
iajs-2376	201	18	in	in	ADP
iajs-2376	201	19	,	,	PUNCT
iajs-2376	201	20	then	then	ADV
iajs-2376	201	21	by	by	ADP
iajs-2376	201	22	[	[	PUNCT
iajs-2376	201	23	]	]	X
iajs-2376	201	24	[	[	PUNCT
iajs-2376	201	25	]	]	X
iajs-2376	201	26	,	,	PUNCT
iajs-2376	201	27	hence	hence	ADV
iajs-2376	201	28	by	by	ADP
iajs-2376	201	29	proposition	proposition	NOUN
iajs-2376	201	30	(	(	PUNCT
iajs-2376	201	31	3.1	3.1	NUM
iajs-2376	201	32	)	)	PUNCT
iajs-2376	201	33	[	[	PUNCT
iajs-2376	201	34	]	]	X
iajs-2376	201	35	is	be	AUX
iajs-2376	201	36	a	a	DET
iajs-2376	201	37	-closed	-close	VERB
iajs-2376	201	38	ideal	ideal	NOUN
iajs-2376	201	39	in	in	ADP
iajs-2376	201	40	.	.	PUNCT
iajs-2376	202	1	(	(	PUNCT
iajs-2376	202	2	2	2	X
iajs-2376	202	3	)	)	PUNCT
iajs-2376	202	4	(	(	PUNCT
iajs-2376	202	5	3	3	X
iajs-2376	202	6	)	)	PUNCT
iajs-2376	202	7	following	follow	VERB
iajs-2376	202	8	by	by	ADP
iajs-2376	202	9	[	[	PUNCT
iajs-2376	202	10	]	]	X
iajs-2376	202	11	and	and	CCONJ
iajs-2376	202	12	by	by	ADP
iajs-2376	202	13	proposition	proposition	NOUN
iajs-2376	202	14	(	(	PUNCT
iajs-2376	202	15	3.2	3.2	NUM
iajs-2376	202	16	)	)	PUNCT
iajs-2376	202	17	.	.	PUNCT
iajs-2376	203	1	(	(	PUNCT
iajs-2376	203	2	3	3	X
iajs-2376	203	3	)	)	PUNCT
iajs-2376	203	4	(	(	PUNCT
iajs-2376	203	5	1	1	X
iajs-2376	203	6	)	)	PUNCT
iajs-2376	203	7	since	since	SCONJ
iajs-2376	203	8	is	be	AUX
iajs-2376	203	9	a	a	DET
iajs-2376	203	10	-closed	-close	VERB
iajs-2376	203	11	ideal	ideal	NOUN
iajs-2376	203	12	in	in	ADP
iajs-2376	203	13	,	,	PUNCT
iajs-2376	203	14	then	then	ADV
iajs-2376	203	15	by	by	ADP
iajs-2376	203	16	proposition	proposition	NOUN
iajs-2376	203	17	(	(	PUNCT
iajs-2376	203	18	3.2	3.2	NUM
iajs-2376	203	19	)	)	PUNCT
iajs-2376	203	20	is	be	AUX
iajs-2376	203	21	a	a	DET
iajs-2376	203	22	-closed	-closed	ADJ
iajs-2376	203	23	submodule	submodule	NOUN
iajs-2376	203	24	in	in	ADP
iajs-2376	203	25	,	,	PUNCT
iajs-2376	203	26	but	but	CCONJ
iajs-2376	203	27	and	and	CCONJ
iajs-2376	203	28	.	.	PUNCT
iajs-2376	204	1	hence	hence	ADV
iajs-2376	204	2	is	be	AUX
iajs-2376	204	3	a	a	DET
iajs-2376	204	4	-closed	-closed	ADJ
iajs-2376	204	5	submodule	submodule	NOUN
iajs-2376	204	6	in	in	ADP
iajs-2376	204	7	.	.	PUNCT
iajs-2376	205	1	proposition	proposition	NOUN
iajs-2376	205	2	3.5	3.5	NUM
iajs-2376	205	3	let	let	VERB
iajs-2376	205	4	be	be	AUX
iajs-2376	205	5	a	a	DET
iajs-2376	205	6	non	non	ADJ
iajs-2376	205	7	-	-	ADJ
iajs-2376	205	8	zero	zero	NUM
iajs-2376	205	9	multiplication	multiplication	NOUN
iajs-2376	205	10	module	module	NOUN
iajs-2376	205	11	,	,	PUNCT
iajs-2376	205	12	with	with	ADP
iajs-2376	205	13	only	only	ADV
iajs-2376	205	14	one	one	NUM
iajs-2376	205	15	non	non	ADJ
iajs-2376	205	16	-	-	ADJ
iajs-2376	205	17	zero	zero	ADJ
iajs-2376	205	18	maximal	maximal	ADJ
iajs-2376	205	19	submodule	submodule	NOUN
iajs-2376	205	20	,	,	PUNCT
iajs-2376	205	21	then	then	ADV
iajs-2376	205	22	can	can	AUX
iajs-2376	205	23	not	not	PART
iajs-2376	205	24	be	be	AUX
iajs-2376	205	25	a	a	DET
iajs-2376	205	26	-closed	-closed	ADJ
iajs-2376	205	27	submodule	submodule	NOUN
iajs-2376	205	28	in	in	ADP
iajs-2376	205	29	.	.	PUNCT
iajs-2376	206	1	89	89	NUM
iajs-2376	207	1	ibn	ibn	PROPN
iajs-2376	207	2	al	al	PROPN
iajs-2376	207	3	-	-	PUNCT
iajs-2376	207	4	haitham	haitham	PROPN
iajs-2376	207	5	jour	jour	X
iajs-2376	207	6	.	.	PROPN
iajs-2376	207	7	for	for	ADP
iajs-2376	207	8	pure	pure	ADJ
iajs-2376	207	9	&	&	CCONJ
iajs-2376	207	10	appl	appl	PROPN
iajs-2376	207	11	.	.	PUNCT
iajs-2376	208	1	sci	sci	PROPN
iajs-2376	208	2	.	.	PROPN
iajs-2376	209	1	33	33	NUM
iajs-2376	209	2	(	(	PUNCT
iajs-2376	209	3	1	1	NUM
iajs-2376	209	4	)	)	PUNCT
iajs-2376	209	5	2020	2020	NUM
iajs-2376	210	1	proof	proof	NOUN
iajs-2376	210	2	let	let	AUX
iajs-2376	210	3	be	be	AUX
iajs-2376	210	4	a	a	DET
iajs-2376	210	5	-closed	-close	VERB
iajs-2376	210	6	submodule	submodule	NOUN
iajs-2376	210	7	in	in	ADP
iajs-2376	210	8	,	,	PUNCT
iajs-2376	210	9	and	and	CCONJ
iajs-2376	210	10	,	,	PUNCT
iajs-2376	210	11	,	,	PUNCT
iajs-2376	210	12	then	then	ADV
iajs-2376	210	13	there	there	PRON
iajs-2376	210	14	exists	exist	VERB
iajs-2376	210	15	a	a	DET
iajs-2376	210	16	-closed	-closed	ADJ
iajs-2376	210	17	submodule	submodule	NOUN
iajs-2376	210	18	in	in	ADP
iajs-2376	210	19	such	such	ADJ
iajs-2376	210	20	that	that	PRON
iajs-2376	210	21	and	and	CCONJ
iajs-2376	210	22	.	.	PUNCT
iajs-2376	211	1	thus	thus	ADV
iajs-2376	211	2	by	by	ADP
iajs-2376	211	3	[	[	PUNCT
iajs-2376	211	4	]	]	X
iajs-2376	211	5	,	,	PUNCT
iajs-2376	211	6	is	be	AUX
iajs-2376	211	7	a	a	DET
iajs-2376	211	8	weak	weak	ADJ
iajs-2376	211	9	essential	essential	ADJ
iajs-2376	211	10	submodule	submodule	NOUN
iajs-2376	211	11	in	in	ADP
iajs-2376	211	12	.	.	PUNCT
iajs-2376	212	1	but	but	CCONJ
iajs-2376	212	2	,	,	PUNCT
iajs-2376	212	3	then	then	ADV
iajs-2376	212	4	by	by	ADP
iajs-2376	212	5	[	[	PUNCT
iajs-2376	212	6	]	]	X
iajs-2376	212	7	is	be	AUX
iajs-2376	212	8	a	a	DET
iajs-2376	212	9	weak	weak	ADJ
iajs-2376	212	10	essential	essential	ADJ
iajs-2376	212	11	submodule	submodule	NOUN
iajs-2376	212	12	in	in	ADP
iajs-2376	212	13	.	.	PUNCT
iajs-2376	213	1	hence	hence	ADV
iajs-2376	213	2	by	by	ADP
iajs-2376	213	3	[	[	PUNCT
iajs-2376	213	4	]	]	X
iajs-2376	213	5	,	,	PUNCT
iajs-2376	213	6	thus	thus	ADV
iajs-2376	213	7	,	,	PUNCT
iajs-2376	213	8	implies	imply	VERB
iajs-2376	213	9	that	that	DET
iajs-2376	213	10	contradiction	contradiction	NOUN
iajs-2376	213	11	.	.	PUNCT
iajs-2376	214	1	hence	hence	ADV
iajs-2376	214	2	dose	dose	AUX
iajs-2376	214	3	not	not	PART
iajs-2376	214	4	be	be	AUX
iajs-2376	214	5	a	a	DET
iajs-2376	214	6	-closed	-closed	ADJ
iajs-2376	214	7	submodule	submodule	NOUN
iajs-2376	214	8	in	in	ADP
iajs-2376	214	9	.	.	PUNCT
iajs-2376	215	1	proposition	proposition	NOUN
iajs-2376	215	2	3.6	3.6	NUM
iajs-2376	215	3	if	if	SCONJ
iajs-2376	215	4	is	be	AUX
iajs-2376	215	5	a	a	DET
iajs-2376	215	6	multiplication	multiplication	NOUN
iajs-2376	215	7	module	module	NOUN
iajs-2376	215	8	over	over	ADP
iajs-2376	215	9	regular	regular	ADJ
iajs-2376	215	10	ring	ring	NOUN
iajs-2376	215	11	,	,	PUNCT
iajs-2376	215	12	and	and	CCONJ
iajs-2376	215	13	is	be	AUX
iajs-2376	215	14	a	a	DET
iajs-2376	215	15	non	non	ADJ
iajs-2376	215	16	-	-	ADJ
iajs-2376	215	17	zero	zero	NUM
iajs-2376	215	18	closed	closed	ADJ
iajs-2376	215	19	submodule	submodule	NOUN
iajs-2376	215	20	in	in	ADP
iajs-2376	215	21	,	,	PUNCT
iajs-2376	215	22	then	then	ADV
iajs-2376	215	23	is	be	AUX
iajs-2376	215	24	a	a	DET
iajs-2376	215	25	-closed	-closed	ADJ
iajs-2376	215	26	submodule	submodule	NOUN
iajs-2376	215	27	in	in	ADV
iajs-2376	215	28	.	.	PUNCT
iajs-2376	216	1	proof	proof	NOUN
iajs-2376	216	2	since	since	SCONJ
iajs-2376	216	3	is	be	AUX
iajs-2376	216	4	a	a	DET
iajs-2376	216	5	multiplication	multiplication	NOUN
iajs-2376	216	6	module	module	NOUN
iajs-2376	216	7	over	over	ADP
iajs-2376	216	8	regular	regular	ADJ
iajs-2376	216	9	ring	ring	NOUN
iajs-2376	216	10	,	,	PUNCT
iajs-2376	216	11	then	then	ADV
iajs-2376	216	12	by	by	ADP
iajs-2376	216	13	[	[	PUNCT
iajs-2376	216	14	]	]	X
iajs-2376	216	15	is	be	AUX
iajs-2376	216	16	fully	fully	ADV
iajs-2376	216	17	semi	semi	ADJ
iajs-2376	216	18	-	-	ADJ
iajs-2376	216	19	prime	prime	ADJ
iajs-2376	216	20	.	.	PUNCT
iajs-2376	217	1	hence	hence	ADV
iajs-2376	217	2	is	be	AUX
iajs-2376	217	3	a	a	DET
iajs-2376	217	4	-closed	-closed	ADJ
iajs-2376	217	5	submodule	submodule	NOUN
iajs-2376	217	6	in	in	ADP
iajs-2376	217	7	.	.	PUNCT
iajs-2376	218	1	proposition	proposition	NOUN
iajs-2376	218	2	3.7	3.7	NUM
iajs-2376	218	3	if	if	SCONJ
iajs-2376	218	4	is	be	AUX
iajs-2376	218	5	a	a	DET
iajs-2376	218	6	multiplication	multiplication	NOUN
iajs-2376	218	7	module	module	NOUN
iajs-2376	218	8	over	over	ADP
iajs-2376	218	9	regular	regular	ADJ
iajs-2376	218	10	ring	ring	NOUN
iajs-2376	218	11	,	,	PUNCT
iajs-2376	218	12	and	and	CCONJ
iajs-2376	218	13	is	be	AUX
iajs-2376	218	14	a	a	DET
iajs-2376	218	15	non	non	ADJ
iajs-2376	218	16	-	-	ADJ
iajs-2376	218	17	zero	zero	NUM
iajs-2376	218	18	-closed	-close	VERB
iajs-2376	218	19	submodule	submodule	NOUN
iajs-2376	218	20	in	in	ADP
iajs-2376	218	21	,	,	PUNCT
iajs-2376	218	22	then	then	ADV
iajs-2376	218	23	is	be	AUX
iajs-2376	218	24	a	a	DET
iajs-2376	218	25	-closed	-closed	ADJ
iajs-2376	218	26	submodule	submodule	NOUN
iajs-2376	218	27	in	in	ADV
iajs-2376	218	28	.	.	PUNCT
iajs-2376	219	1	proof	proof	NOUN
iajs-2376	219	2	follows	follow	VERB
iajs-2376	219	3	by	by	ADP
iajs-2376	219	4	[	[	PUNCT
iajs-2376	219	5	]	]	X
iajs-2376	219	6	and	and	CCONJ
iajs-2376	219	7	proposition	proposition	NOUN
iajs-2376	219	8	(	(	PUNCT
iajs-2376	219	9	2.14	2.14	NUM
iajs-2376	219	10	)	)	PUNCT
iajs-2376	219	11	.	.	PUNCT
iajs-2376	220	1	proposition	proposition	NOUN
iajs-2376	220	2	3.8	3.8	NUM
iajs-2376	220	3	if	if	SCONJ
iajs-2376	220	4	is	be	AUX
iajs-2376	220	5	a	a	DET
iajs-2376	220	6	multiplication	multiplication	NOUN
iajs-2376	220	7	modulr	modulr	ADJ
iajs-2376	220	8	over	over	ADP
iajs-2376	220	9	regular	regular	ADJ
iajs-2376	220	10	ring	ring	NOUN
iajs-2376	220	11	,	,	PUNCT
iajs-2376	220	12	and	and	CCONJ
iajs-2376	220	13	be	be	AUX
iajs-2376	220	14	a	a	DET
iajs-2376	220	15	non	non	ADJ
iajs-2376	220	16	-	-	ADJ
iajs-2376	220	17	zero	zero	NUM
iajs-2376	220	18	-closed	-close	VERB
iajs-2376	220	19	submodule	submodule	NOUN
iajs-2376	220	20	in	in	ADP
iajs-2376	220	21	and	and	CCONJ
iajs-2376	220	22	is	be	AUX
iajs-2376	220	23	a	a	DET
iajs-2376	220	24	-closed	-closed	ADJ
iajs-2376	220	25	submodule	submodule	NOUN
iajs-2376	220	26	in	in	ADP
iajs-2376	220	27	,	,	PUNCT
iajs-2376	220	28	then	then	ADV
iajs-2376	220	29	is	be	AUX
iajs-2376	220	30	a	a	DET
iajs-2376	220	31	-closed	-closed	ADJ
iajs-2376	220	32	submodule	submodule	NOUN
iajs-2376	220	33	in	in	ADV
iajs-2376	220	34	.	.	PUNCT
iajs-2376	221	1	proof	proof	NOUN
iajs-2376	221	2	follows	follow	VERB
iajs-2376	221	3	by	by	ADP
iajs-2376	221	4	[	[	PUNCT
iajs-2376	221	5	]	]	X
iajs-2376	221	6	and	and	CCONJ
iajs-2376	221	7	proposition	proposition	NOUN
iajs-2376	221	8	(	(	PUNCT
iajs-2376	221	9	2.16	2.16	NUM
iajs-2376	221	10	)	)	PUNCT
iajs-2376	221	11	.	.	PUNCT
iajs-2376	222	1	proposition	proposition	NOUN
iajs-2376	222	2	3.9	3.9	NUM
iajs-2376	222	3	if	if	SCONJ
iajs-2376	222	4	is	be	AUX
iajs-2376	222	5	a	a	DET
iajs-2376	222	6	multiplication	multiplication	NOUN
iajs-2376	222	7	modulr	modulr	ADJ
iajs-2376	222	8	over	over	ADP
iajs-2376	222	9	regular	regular	ADJ
iajs-2376	222	10	ring	ring	NOUN
iajs-2376	222	11	,	,	PUNCT
iajs-2376	222	12	and	and	CCONJ
iajs-2376	222	13	are	be	AUX
iajs-2376	222	14	submodules	submodule	NOUN
iajs-2376	222	15	of	of	ADP
iajs-2376	222	16	,	,	PUNCT
iajs-2376	222	17	with	with	SCONJ
iajs-2376	222	18	is	be	AUX
iajs-2376	222	19	a	a	DET
iajs-2376	222	20	-closed	-closed	ADJ
iajs-2376	222	21	submodule	submodule	NOUN
iajs-2376	222	22	in	in	ADP
iajs-2376	222	23	and	and	CCONJ
iajs-2376	222	24	is	be	AUX
iajs-2376	222	25	a	a	DET
iajs-2376	222	26	weak	weak	ADJ
iajs-2376	222	27	essential	essential	ADJ
iajs-2376	222	28	submodule	submodule	NOUN
iajs-2376	222	29	in	in	ADP
iajs-2376	222	30	,	,	PUNCT
iajs-2376	222	31	then	then	ADV
iajs-2376	222	32	is	be	AUX
iajs-2376	222	33	a	a	DET
iajs-2376	222	34	-closed	-closed	ADJ
iajs-2376	222	35	submodule	submodule	NOUN
iajs-2376	222	36	in	in	ADV
iajs-2376	222	37	.	.	PUNCT
iajs-2376	223	1	proof	proof	NOUN
iajs-2376	223	2	follows	follow	VERB
iajs-2376	223	3	by	by	ADP
iajs-2376	223	4	[	[	PUNCT
iajs-2376	223	5	]	]	X
iajs-2376	223	6	and	and	CCONJ
iajs-2376	223	7	proposition	proposition	NOUN
iajs-2376	223	8	(	(	PUNCT
iajs-2376	223	9	2.17	2.17	NUM
iajs-2376	223	10	)	)	PUNCT
iajs-2376	223	11	.	.	PUNCT
iajs-2376	224	1	since	since	SCONJ
iajs-2376	224	2	-closed	-closed	ADJ
iajs-2376	224	3	submodule	submodule	NOUN
iajs-2376	224	4	is	be	AUX
iajs-2376	224	5	-closed	-close	VERB
iajs-2376	224	6	,	,	PUNCT
iajs-2376	224	7	we	we	PRON
iajs-2376	224	8	get	get	VERB
iajs-2376	224	9	the	the	DET
iajs-2376	224	10	following	following	NOUN
iajs-2376	224	11	:	:	PUNCT
iajs-2376	224	12	corollary	corollary	ADJ
iajs-2376	224	13	3.10	3.10	NUM
iajs-2376	224	14	if	if	SCONJ
iajs-2376	224	15	is	be	AUX
iajs-2376	224	16	a	a	DET
iajs-2376	224	17	multiplication	multiplication	NOUN
iajs-2376	224	18	modulr	modulr	ADJ
iajs-2376	224	19	over	over	ADP
iajs-2376	224	20	regular	regular	ADJ
iajs-2376	224	21	ring	ring	NOUN
iajs-2376	224	22	,	,	PUNCT
iajs-2376	224	23	and	and	CCONJ
iajs-2376	224	24	are	be	AUX
iajs-2376	224	25	submodules	submodule	NOUN
iajs-2376	224	26	of	of	ADP
iajs-2376	224	27	,	,	PUNCT
iajs-2376	224	28	with	with	SCONJ
iajs-2376	224	29	is	be	AUX
iajs-2376	224	30	a	a	DET
iajs-2376	224	31	-closed	-closed	ADJ
iajs-2376	224	32	submodule	submodule	NOUN
iajs-2376	224	33	in	in	ADP
iajs-2376	224	34	and	and	CCONJ
iajs-2376	224	35	is	be	AUX
iajs-2376	224	36	a	a	DET
iajs-2376	224	37	weak	weak	ADJ
iajs-2376	224	38	essential	essential	ADJ
iajs-2376	224	39	submodule	submodule	NOUN
iajs-2376	224	40	in	in	ADP
iajs-2376	224	41	,	,	PUNCT
iajs-2376	224	42	then	then	ADV
iajs-2376	224	43	is	be	AUX
iajs-2376	224	44	a	a	DET
iajs-2376	224	45	-closed	-closed	ADJ
iajs-2376	224	46	submodule	submodule	NOUN
iajs-2376	224	47	in	in	ADP
iajs-2376	224	48	.	.	PUNCT
iajs-2376	225	1	proposition	proposition	NOUN
iajs-2376	225	2	3.11	3.11	NUM
iajs-2376	225	3	if	if	SCONJ
iajs-2376	225	4	is	be	AUX
iajs-2376	225	5	a	a	DET
iajs-2376	225	6	multiplication	multiplication	NOUN
iajs-2376	225	7	modulr	modulr	ADJ
iajs-2376	225	8	over	over	ADP
iajs-2376	225	9	regular	regular	ADJ
iajs-2376	225	10	ring	ring	NOUN
iajs-2376	225	11	,	,	PUNCT
iajs-2376	225	12	and	and	CCONJ
iajs-2376	225	13	are	be	AUX
iajs-2376	225	14	submodules	submodule	NOUN
iajs-2376	225	15	of	of	ADP
iajs-2376	225	16	,	,	PUNCT
iajs-2376	225	17	with	with	SCONJ
iajs-2376	225	18	is	be	AUX
iajs-2376	225	19	a	a	DET
iajs-2376	225	20	-closed	-closed	ADJ
iajs-2376	225	21	submodule	submodule	NOUN
iajs-2376	225	22	in	in	ADP
iajs-2376	225	23	and	and	CCONJ
iajs-2376	225	24	is	be	AUX
iajs-2376	225	25	a	a	DET
iajs-2376	225	26	weak	weak	ADJ
iajs-2376	225	27	essential	essential	ADJ
iajs-2376	225	28	submodule	submodule	NOUN
iajs-2376	225	29	in	in	ADP
iajs-2376	225	30	,	,	PUNCT
iajs-2376	225	31	then	then	ADV
iajs-2376	225	32	is	be	AUX
iajs-2376	225	33	a	a	DET
iajs-2376	225	34	-closed	-closed	ADJ
iajs-2376	225	35	submodule	submodule	NOUN
iajs-2376	225	36	in	in	ADP
iajs-2376	225	37	.	.	PUNCT
iajs-2376	226	1	89	89	NUM
iajs-2376	227	1	ibn	ibn	PROPN
iajs-2376	227	2	al	al	PROPN
iajs-2376	227	3	-	-	PUNCT
iajs-2376	227	4	haitham	haitham	PROPN
iajs-2376	227	5	jour	jour	X
iajs-2376	227	6	.	.	PROPN
iajs-2376	227	7	for	for	ADP
iajs-2376	227	8	pure	pure	ADJ
iajs-2376	227	9	&	&	CCONJ
iajs-2376	227	10	appl	appl	PROPN
iajs-2376	227	11	.	.	PUNCT
iajs-2376	228	1	sci	sci	PROPN
iajs-2376	228	2	.	.	PROPN
iajs-2376	229	1	33	33	NUM
iajs-2376	229	2	(	(	PUNCT
iajs-2376	229	3	1	1	NUM
iajs-2376	229	4	)	)	PUNCT
iajs-2376	229	5	2020	2020	NUM
iajs-2376	229	6	proof	proof	NOUN
iajs-2376	229	7	follows	follow	VERB
iajs-2376	229	8	by	by	ADP
iajs-2376	229	9	[	[	PUNCT
iajs-2376	229	10	]	]	X
iajs-2376	229	11	and	and	CCONJ
iajs-2376	229	12	corollary	corollary	ADJ
iajs-2376	229	13	(	(	PUNCT
iajs-2376	229	14	2.20	2.20	NUM
iajs-2376	229	15	)	)	PUNCT
iajs-2376	229	16	.	.	PUNCT
iajs-2376	230	1	4	4	X
iajs-2376	230	2	.	.	X
iajs-2376	230	3	chain	chain	NOUN
iajs-2376	230	4	conditions	condition	NOUN
iajs-2376	230	5	on	on	ADP
iajs-2376	230	6	-closed	-close	VERB
iajs-2376	230	7	submodules	submodule	NOUN
iajs-2376	230	8	module	module	NOUN
iajs-2376	230	9	with	with	ADP
iajs-2376	230	10	chain	chain	NOUN
iajs-2376	230	11	condition	condition	NOUN
iajs-2376	230	12	on	on	ADP
iajs-2376	230	13	-closed	-closed	ADJ
iajs-2376	230	14	submodules	submodule	NOUN
iajs-2376	230	15	are	be	AUX
iajs-2376	230	16	studied	study	VERB
iajs-2376	230	17	in	in	ADP
iajs-2376	230	18	this	this	DET
iajs-2376	230	19	section	section	NOUN
iajs-2376	230	20	.	.	PUNCT
iajs-2376	231	1	definition	definition	NOUN
iajs-2376	231	2	4.1	4.1	NUM
iajs-2376	231	3	a	a	DET
iajs-2376	231	4	module	module	NOUN
iajs-2376	231	5	is	be	AUX
iajs-2376	231	6	called	call	VERB
iajs-2376	231	7	a	a	DET
iajs-2376	231	8	module	module	NOUN
iajs-2376	231	9	with	with	ADP
iajs-2376	231	10	ascending	ascend	VERB
iajs-2376	231	11	(	(	PUNCT
iajs-2376	231	12	respectively	respectively	ADV
iajs-2376	231	13	,	,	PUNCT
iajs-2376	231	14	descending	descend	VERB
iajs-2376	231	15	)	)	PUNCT
iajs-2376	231	16	chain	chain	NOUN
iajs-2376	231	17	condition	condition	NOUN
iajs-2376	231	18	(	(	PUNCT
iajs-2376	231	19	briefly	briefly	ADV
iajs-2376	231	20	acc	acc	PROPN
iajs-2376	231	21	respectively	respectively	ADV
iajs-2376	231	22	dcc	dcc	PROPN
iajs-2376	231	23	)	)	PUNCT
iajs-2376	231	24	on	on	ADP
iajs-2376	231	25	-closed	-close	VERB
iajs-2376	231	26	submodules	submodule	NOUN
iajs-2376	231	27	,	,	PUNCT
iajs-2376	231	28	if	if	SCONJ
iajs-2376	231	29	every	every	DET
iajs-2376	231	30	ascending	ascend	VERB
iajs-2376	231	31	(	(	PUNCT
iajs-2376	231	32	respectively	respectively	ADV
iajs-2376	231	33	,	,	PUNCT
iajs-2376	231	34	descending	descending	NOUN
iajs-2376	231	35	)	)	PUNCT
iajs-2376	231	36	chain	chain	NOUN
iajs-2376	231	37	of	of	ADP
iajs-2376	231	38	-closed	-closed	ADJ
iajs-2376	231	39	submodules	submodule	NOUN
iajs-2376	231	40	of	of	ADP
iajs-2376	231	41	is	be	AUX
iajs-2376	231	42	finite	finite	ADJ
iajs-2376	231	43	,	,	PUNCT
iajs-2376	231	44	i.e.	i.e.	X
iajs-2376	231	45	such	such	ADJ
iajs-2376	231	46	that	that	PRON
iajs-2376	231	47	.	.	PUNCT
iajs-2376	232	1	remarks	remark	NOUN
iajs-2376	232	2	and	and	CCONJ
iajs-2376	232	3	examples	example	NOUN
iajs-2376	232	4	4.2	4.2	NUM
iajs-2376	232	5	1	1	NUM
iajs-2376	232	6	.	.	PUNCT
iajs-2376	233	1	every	every	DET
iajs-2376	233	2	noetherian	noetherian	ADJ
iajs-2376	233	3	module	module	NOUN
iajs-2376	233	4	has	have	VERB
iajs-2376	233	5	acc	acc	PROPN
iajs-2376	233	6	on	on	ADP
iajs-2376	233	7	-closed	-close	VERB
iajs-2376	233	8	submodules	submodule	NOUN
iajs-2376	233	9	.	.	PUNCT
iajs-2376	234	1	2	2	X
iajs-2376	234	2	.	.	X
iajs-2376	234	3	every	every	DET
iajs-2376	234	4	artirian	artirian	ADJ
iajs-2376	234	5	module	module	NOUN
iajs-2376	234	6	has	have	VERB
iajs-2376	234	7	dcc	dcc	PROPN
iajs-2376	234	8	on	on	ADP
iajs-2376	234	9	-closed	-close	VERB
iajs-2376	234	10	submodules	submodule	NOUN
iajs-2376	234	11	.	.	PUNCT
iajs-2376	235	1	proposition	proposition	NOUN
iajs-2376	235	2	4.3	4.3	NUM
iajs-2376	235	3	if	if	SCONJ
iajs-2376	235	4	is	be	AUX
iajs-2376	235	5	a	a	DET
iajs-2376	235	6	module	module	NOUN
iajs-2376	235	7	satisfies	satisfie	NOUN
iajs-2376	235	8	acc	acc	PROPN
iajs-2376	235	9	(	(	PUNCT
iajs-2376	235	10	dcc	dcc	PROPN
iajs-2376	235	11	)	)	PUNCT
iajs-2376	235	12	on	on	ADP
iajs-2376	235	13	-closed	-closed	ADJ
iajs-2376	235	14	submodules	submodule	NOUN
iajs-2376	235	15	of	of	ADP
iajs-2376	235	16	,	,	PUNCT
iajs-2376	235	17	then	then	ADV
iajs-2376	235	18	satisfiesacc	satisfiesacc	VERB
iajs-2376	235	19	(	(	PUNCT
iajs-2376	235	20	dcc	dcc	PROPN
iajs-2376	235	21	)	)	PUNCT
iajs-2376	235	22	on	on	ADP
iajs-2376	235	23	-closed	-closed	ADJ
iajs-2376	235	24	submodules	submodule	NOUN
iajs-2376	235	25	of	of	ADP
iajs-2376	235	26	.	.	PUNCT
iajs-2376	236	1	proof	proof	NOUN
iajs-2376	236	2	let	let	VERB
iajs-2376	236	3	…	…	PUNCT
iajs-2376	236	4	…	…	PUNCT
iajs-2376	236	5	.	.	PUNCT
iajs-2376	236	6	,	,	PUNCT
iajs-2376	236	7	be	be	AUX
iajs-2376	236	8	an	an	DET
iajs-2376	236	9	ascending	ascend	VERB
iajs-2376	236	10	chain	chain	NOUN
iajs-2376	236	11	of	of	ADP
iajs-2376	236	12	-closed	-closed	ADJ
iajs-2376	236	13	submodule	submodule	NOUN
iajs-2376	236	14	of	of	ADP
iajs-2376	236	15	for	for	ADP
iajs-2376	236	16	each	each	PRON
iajs-2376	236	17	.	.	PUNCT
iajs-2376	237	1	but	but	CCONJ
iajs-2376	237	2	every	every	DET
iajs-2376	237	3	-closed	-close	VERB
iajs-2376	237	4	submodule	submodule	NOUN
iajs-2376	237	5	is	be	AUX
iajs-2376	237	6	-closed	-close	VERB
iajs-2376	237	7	,	,	PUNCT
iajs-2376	237	8	then	then	ADV
iajs-2376	237	9	is	be	AUX
iajs-2376	237	10	a	a	DET
iajs-2376	237	11	-closed	-close	VERB
iajs-2376	237	12	submodule	submodule	NOUN
iajs-2376	237	13	for	for	ADP
iajs-2376	237	14	each	each	PRON
iajs-2376	237	15	…	…	PUNCT
iajs-2376	237	16	.	.	PUNCT
iajs-2376	238	1	since	since	SCONJ
iajs-2376	238	2	has	have	VERB
iajs-2376	238	3	acc	acc	PROPN
iajs-2376	238	4	on	on	ADP
iajs-2376	238	5	-closed	-close	VERB
iajs-2376	238	6	submodules	submodule	NOUN
iajs-2376	238	7	,	,	PUNCT
iajs-2376	238	8	then	then	ADV
iajs-2376	238	9	such	such	ADJ
iajs-2376	238	10	that	that	PRON
iajs-2376	238	11	.	.	PUNCT
iajs-2376	239	1	thus	thus	ADV
iajs-2376	239	2	has	have	VERB
iajs-2376	239	3	acc	acc	PROPN
iajs-2376	239	4	on	on	ADP
iajs-2376	239	5	-closed	-close	VERB
iajs-2376	239	6	submodules	submodule	NOUN
iajs-2376	239	7	.	.	PUNCT
iajs-2376	240	1	for	for	ADP
iajs-2376	240	2	dcc	dcc	PROPN
iajs-2376	240	3	in	in	ADP
iajs-2376	240	4	similarly	similarly	ADV
iajs-2376	240	5	way	way	NOUN
iajs-2376	240	6	.	.	PUNCT
iajs-2376	241	1	proposition	proposition	NOUN
iajs-2376	241	2	4.4	4.4	NUM
iajs-2376	241	3	if	if	SCONJ
iajs-2376	241	4	is	be	AUX
iajs-2376	241	5	a	a	DET
iajs-2376	241	6	fully	fully	ADV
iajs-2376	241	7	semi	semi	ADJ
iajs-2376	241	8	-	-	ADJ
iajs-2376	241	9	prime	prime	ADJ
iajs-2376	241	10	-module	-module	NOUN
iajs-2376	241	11	satisfies	satisfy	VERB
iajs-2376	241	12	acc	acc	PROPN
iajs-2376	241	13	(	(	PUNCT
iajs-2376	241	14	dcc	dcc	PROPN
iajs-2376	241	15	)	)	PUNCT
iajs-2376	241	16	on	on	ADP
iajs-2376	241	17	-closed	-closed	ADJ
iajs-2376	241	18	submodules	submodule	NOUN
iajs-2376	241	19	of	of	ADP
iajs-2376	241	20	,	,	PUNCT
iajs-2376	241	21	then	then	ADV
iajs-2376	241	22	satisfies	satisfy	VERB
iajs-2376	241	23	acc	acc	PROPN
iajs-2376	241	24	(	(	PUNCT
iajs-2376	241	25	dcc	dcc	PROPN
iajs-2376	241	26	)	)	PUNCT
iajs-2376	241	27	on	on	ADP
iajs-2376	241	28	closed	closed	ADJ
iajs-2376	241	29	submodules	submodule	NOUN
iajs-2376	241	30	of	of	ADP
iajs-2376	241	31	.	.	PUNCT
iajs-2376	242	1	proof	proof	NOUN
iajs-2376	242	2	let	let	VERB
iajs-2376	242	3	…	…	PUNCT
iajs-2376	242	4	…	…	PUNCT
iajs-2376	242	5	.,be	.,be	NOUN
iajs-2376	242	6	an	an	DET
iajs-2376	242	7	ascending	ascend	VERB
iajs-2376	242	8	chain	chain	NOUN
iajs-2376	242	9	of	of	ADP
iajs-2376	242	10	closed	closed	ADJ
iajs-2376	242	11	submodule	submodule	NOUN
iajs-2376	242	12	.	.	PUNCT
iajs-2376	243	1	where	where	SCONJ
iajs-2376	243	2	is	be	AUX
iajs-2376	243	3	a	a	DET
iajs-2376	243	4	closed	closed	ADJ
iajs-2376	243	5	submodule	submodule	NOUN
iajs-2376	243	6	in	in	ADP
iajs-2376	243	7	for	for	ADP
iajs-2376	243	8	each	each	PRON
iajs-2376	243	9	…	…	PUNCT
iajs-2376	243	10	.	.	PUNCT
iajs-2376	244	1	but	but	CCONJ
iajs-2376	244	2	is	be	AUX
iajs-2376	244	3	a	a	DET
iajs-2376	244	4	fully	fully	ADV
iajs-2376	244	5	semi	semi	ADJ
iajs-2376	244	6	-	-	ADJ
iajs-2376	244	7	prime	prime	ADJ
iajs-2376	244	8	module	module	NOUN
iajs-2376	244	9	,	,	PUNCT
iajs-2376	244	10	then	then	ADV
iajs-2376	244	11	by	by	ADP
iajs-2376	244	12	proposition	proposition	NOUN
iajs-2376	244	13	(	(	PUNCT
iajs-2376	244	14	2.7	2.7	NUM
iajs-2376	244	15	)	)	PUNCT
iajs-2376	244	16	is	be	AUX
iajs-2376	244	17	a	a	DET
iajs-2376	244	18	-closed	-closed	ADJ
iajs-2376	244	19	submodule	submodule	NOUN
iajs-2376	244	20	in	in	ADP
iajs-2376	244	21	for	for	ADP
iajs-2376	244	22	each	each	PRON
iajs-2376	244	23	…	…	PUNCT
iajs-2376	244	24	.	.	PUNCT
iajs-2376	245	1	but	but	CCONJ
iajs-2376	245	2	has	have	VERB
iajs-2376	245	3	acc	acc	PROPN
iajs-2376	245	4	on	on	ADP
iajs-2376	245	5	-closed	-close	VERB
iajs-2376	245	6	submodules	submodule	NOUN
iajs-2376	245	7	.	.	PUNCT
iajs-2376	246	1	hence	hence	ADV
iajs-2376	246	2	such	such	ADJ
iajs-2376	246	3	that	that	PRON
iajs-2376	246	4	.	.	PUNCT
iajs-2376	247	1	thus	thus	ADV
iajs-2376	247	2	has	have	VERB
iajs-2376	247	3	acc	acc	PROPN
iajs-2376	247	4	on	on	ADP
iajs-2376	247	5	closed	closed	ADJ
iajs-2376	247	6	submodules	submodule	NOUN
iajs-2376	247	7	.	.	PUNCT
iajs-2376	248	1	similarly	similarly	ADV
iajs-2376	248	2	for	for	ADP
iajs-2376	248	3	dcc	dcc	PROPN
iajs-2376	248	4	.	.	PUNCT
iajs-2376	249	1	proposition	proposition	NOUN
iajs-2376	249	2	4.5	4.5	NUM
iajs-2376	249	3	if	if	SCONJ
iajs-2376	249	4	is	be	AUX
iajs-2376	249	5	a	a	DET
iajs-2376	249	6	module	module	NOUN
iajs-2376	249	7	satisfies	satisfie	NOUN
iajs-2376	249	8	acc	acc	PROPN
iajs-2376	249	9	(	(	PUNCT
iajs-2376	249	10	dcc	dcc	PROPN
iajs-2376	249	11	)	)	PUNCT
iajs-2376	249	12	on	on	ADP
iajs-2376	249	13	-closed	-close	VERB
iajs-2376	249	14	submodules	submodule	NOUN
iajs-2376	249	15	for	for	ADP
iajs-2376	249	16	each	each	PRON
iajs-2376	249	17	…	…	NUM
iajs-2376	249	18	.	.	PROPN
iajs-2376	249	19	,	,	PUNCT
iajs-2376	249	20	with	with	ADP
iajs-2376	249	21	each	each	DET
iajs-2376	249	22	weak	weak	ADJ
iajs-2376	249	23	essential	essential	ADJ
iajs-2376	249	24	extension	extension	NOUN
iajs-2376	249	25	of	of	ADP
iajs-2376	249	26	is	be	AUX
iajs-2376	249	27	completely	completely	ADV
iajs-2376	249	28	essential	essential	ADJ
iajs-2376	249	29	for	for	ADP
iajs-2376	249	30	each	each	PRON
iajs-2376	249	31	…	…	NUM
iajs-2376	249	32	.	.	PUNCT
iajs-2376	249	33	,	,	PUNCT
iajs-2376	249	34	then	then	ADV
iajs-2376	249	35	satisfies	satisfy	VERB
iajs-2376	249	36	acc	acc	PROPN
iajs-2376	249	37	(	(	PUNCT
iajs-2376	249	38	dcc	dcc	PROPN
iajs-2376	249	39	)	)	PUNCT
iajs-2376	249	40	on	on	ADP
iajs-2376	249	41	closed	closed	ADJ
iajs-2376	249	42	submodules	submodule	NOUN
iajs-2376	249	43	for	for	ADP
iajs-2376	249	44	each	each	PRON
iajs-2376	249	45	…	…	PUNCT
iajs-2376	249	46	..	..	PUNCT
iajs-2376	249	47	proof	proof	NOUN
iajs-2376	249	48	let	let	VERB
iajs-2376	249	49	…	…	PUNCT
iajs-2376	249	50	…	…	PUNCT
iajs-2376	249	51	.	.	PUNCT
iajs-2376	249	52	,	,	PUNCT
iajs-2376	249	53	be	be	AUX
iajs-2376	249	54	an	an	DET
iajs-2376	249	55	ascending	ascend	VERB
iajs-2376	249	56	chain	chain	NOUN
iajs-2376	249	57	of	of	ADP
iajs-2376	249	58	closed	closed	ADJ
iajs-2376	249	59	submodule	submodule	NOUN
iajs-2376	249	60	for	for	ADP
iajs-2376	249	61	each	each	PRON
iajs-2376	249	62	…	…	PUNCT
iajs-2376	249	63	.	.	PUNCT
iajs-2376	250	1	then	then	ADV
iajs-2376	250	2	by	by	ADP
iajs-2376	250	3	proposition	proposition	NOUN
iajs-2376	250	4	(	(	PUNCT
iajs-2376	250	5	2.6	2.6	NUM
iajs-2376	250	6	)	)	PUNCT
iajs-2376	250	7	is	be	AUX
iajs-2376	250	8	a	a	DET
iajs-2376	250	9	-closed	-close	VERB
iajs-2376	250	10	submodule	submodule	NOUN
iajs-2376	250	11	for	for	ADP
iajs-2376	250	12	each	each	PRON
iajs-2376	250	13	…	…	PUNCT
iajs-2376	250	14	.	.	PUNCT
iajs-2376	251	1	but	but	CCONJ
iajs-2376	251	2	has	have	VERB
iajs-2376	251	3	acc	acc	PROPN
iajs-2376	251	4	88	88	NUM
iajs-2376	251	5	ibn	ibn	PROPN
iajs-2376	251	6	al	al	PROPN
iajs-2376	251	7	-	-	PUNCT
iajs-2376	251	8	haitham	haitham	PROPN
iajs-2376	251	9	jour	jour	X
iajs-2376	251	10	.	.	PROPN
iajs-2376	251	11	for	for	ADP
iajs-2376	251	12	pure	pure	ADJ
iajs-2376	251	13	&	&	CCONJ
iajs-2376	251	14	appl	appl	PROPN
iajs-2376	251	15	.	.	PUNCT
iajs-2376	252	1	sci	sci	PROPN
iajs-2376	252	2	.	.	PROPN
iajs-2376	253	1	33	33	NUM
iajs-2376	253	2	(	(	PUNCT
iajs-2376	253	3	1	1	NUM
iajs-2376	253	4	)	)	PUNCT
iajs-2376	253	5	2020	2020	NUM
iajs-2376	254	1	on	on	ADP
iajs-2376	254	2	-closed	-close	VERB
iajs-2376	254	3	submodules	submodule	NOUN
iajs-2376	254	4	,	,	PUNCT
iajs-2376	254	5	then	then	ADV
iajs-2376	254	6	such	such	ADJ
iajs-2376	254	7	that	that	PRON
iajs-2376	254	8	.	.	PUNCT
iajs-2376	255	1	hence	hence	ADV
iajs-2376	255	2	has	have	VERB
iajs-2376	255	3	acc	acc	PROPN
iajs-2376	255	4	on	on	ADP
iajs-2376	255	5	closed	close	VERB
iajs-2376	255	6	submodules	submodule	NOUN
iajs-2376	255	7	for	for	ADP
iajs-2376	255	8	each	each	PRON
iajs-2376	255	9	…	…	PUNCT
iajs-2376	255	10	...	...	PUNCT
iajs-2376	255	11	similarly	similarly	ADV
iajs-2376	255	12	for	for	ADP
iajs-2376	255	13	dcc	dcc	PROPN
iajs-2376	255	14	.	.	PUNCT
iajs-2376	256	1	proposition	proposition	NOUN
iajs-2376	256	2	4.6	4.6	NUM
iajs-2376	256	3	let	let	VERB
iajs-2376	256	4	be	be	AUX
iajs-2376	256	5	an	an	DET
iajs-2376	256	6	-module	-module	NOUN
iajs-2376	256	7	and	and	CCONJ
iajs-2376	256	8	be	be	AUX
iajs-2376	256	9	a	a	DET
iajs-2376	256	10	submodule	submodule	NOUN
iajs-2376	256	11	of	of	ADP
iajs-2376	256	12	such	such	ADJ
iajs-2376	256	13	that	that	PRON
iajs-2376	256	14	,	,	PUNCT
iajs-2376	256	15	where	where	SCONJ
iajs-2376	256	16	is	be	AUX
iajs-2376	256	17	any	any	DET
iajs-2376	256	18	-closed	-close	VERB
iajs-2376	256	19	submodule	submodule	NOUN
iajs-2376	256	20	in	in	ADP
iajs-2376	256	21	.	.	PUNCT
iajs-2376	257	1	if	if	SCONJ
iajs-2376	257	2	satisfies	satisfie	NOUN
iajs-2376	257	3	dcc	dcc	PROPN
iajs-2376	257	4	(	(	PUNCT
iajs-2376	257	5	acc	acc	PROPN
iajs-2376	257	6	)	)	PUNCT
iajs-2376	257	7	on	on	ADP
iajs-2376	257	8	-closed	-closed	ADJ
iajs-2376	257	9	submodules	submodule	NOUN
iajs-2376	257	10	of	of	ADP
iajs-2376	257	11	.	.	PUNCT
iajs-2376	258	1	then	then	ADV
iajs-2376	258	2	satisfies	satisfy	VERB
iajs-2376	258	3	acc	acc	PROPN
iajs-2376	258	4	(	(	PUNCT
iajs-2376	258	5	dcc	dcc	PROPN
iajs-2376	258	6	)	)	PUNCT
iajs-2376	258	7	on	on	ADP
iajs-2376	258	8	-closed	-closed	ADJ
iajs-2376	258	9	submodules	submodule	NOUN
iajs-2376	258	10	of	of	ADP
iajs-2376	258	11	.	.	PUNCT
iajs-2376	259	1	proof	proof	NOUN
iajs-2376	259	2	let	let	VERB
iajs-2376	259	3	…	…	PUNCT
iajs-2376	259	4	…	…	PUNCT
iajs-2376	259	5	.	.	PUNCT
iajs-2376	259	6	,	,	PUNCT
iajs-2376	259	7	be	be	AUX
iajs-2376	259	8	a	a	DET
iajs-2376	259	9	descending	descend	VERB
iajs-2376	259	10	chain	chain	NOUN
iajs-2376	259	11	of	of	ADP
iajs-2376	259	12	-closed	-closed	ADJ
iajs-2376	259	13	submodules	submodule	NOUN
iajs-2376	259	14	in	in	ADV
iajs-2376	259	15	for	for	ADP
iajs-2376	259	16	each	each	PRON
iajs-2376	259	17	…	…	NUM
iajs-2376	259	18	.	.	NUM
iajs-2376	259	19	,	,	PUNCT
iajs-2376	259	20	and	and	CCONJ
iajs-2376	259	21	for	for	ADP
iajs-2376	259	22	each	each	PRON
iajs-2376	259	23	…	…	PUNCT
iajs-2376	259	24	.	.	PUNCT
iajs-2376	260	1	then	then	ADV
iajs-2376	260	2	by	by	ADP
iajs-2376	260	3	corollary	corollary	ADJ
iajs-2376	260	4	(	(	PUNCT
iajs-2376	260	5	2.13	2.13	NUM
iajs-2376	260	6	)	)	PUNCT
iajs-2376	260	7	we	we	PRON
iajs-2376	260	8	have	have	AUX
iajs-2376	260	9	is	be	AUX
iajs-2376	260	10	a	a	DET
iajs-2376	260	11	-closed	-close	VERB
iajs-2376	260	12	submodules	submodule	NOUN
iajs-2376	260	13	in	in	ADV
iajs-2376	260	14	for	for	ADP
iajs-2376	260	15	each	each	PRON
iajs-2376	260	16	…	…	PUNCT
iajs-2376	260	17	.	.	PUNCT
iajs-2376	261	1	hence	hence	ADV
iajs-2376	261	2	…	…	PUNCT
iajs-2376	261	3	…	…	PUNCT
iajs-2376	261	4	.	.	PUNCT
iajs-2376	261	5	,	,	PUNCT
iajs-2376	261	6	is	be	AUX
iajs-2376	261	7	a	a	DET
iajs-2376	261	8	descending	descend	VERB
iajs-2376	261	9	chain	chain	NOUN
iajs-2376	261	10	of	of	ADP
iajs-2376	261	11	-closed	-closed	ADJ
iajs-2376	261	12	submodules	submodule	NOUN
iajs-2376	261	13	in	in	ADV
iajs-2376	261	14	.	.	PUNCT
iajs-2376	262	1	but	but	CCONJ
iajs-2376	262	2	has	have	VERB
iajs-2376	262	3	(	(	PUNCT
iajs-2376	262	4	dcc	dcc	PROPN
iajs-2376	262	5	)	)	PUNCT
iajs-2376	262	6	on	on	ADP
iajs-2376	262	7	-closed	-close	VERB
iajs-2376	262	8	submodules	submodule	NOUN
iajs-2376	262	9	,	,	PUNCT
iajs-2376	262	10	so	so	SCONJ
iajs-2376	262	11	such	such	ADJ
iajs-2376	262	12	that	that	PRON
iajs-2376	262	13	.	.	PUNCT
iajs-2376	263	1	hence	hence	ADV
iajs-2376	263	2	satisfies	satisfie	NOUN
iajs-2376	263	3	(	(	PUNCT
iajs-2376	263	4	dcc	dcc	PROPN
iajs-2376	263	5	)	)	PUNCT
iajs-2376	263	6	on	on	ADP
iajs-2376	263	7	closed	closed	ADJ
iajs-2376	263	8	submodules	submodule	NOUN
iajs-2376	263	9	.	.	PUNCT
iajs-2376	264	1	similarly	similarly	ADV
iajs-2376	264	2	for	for	ADP
iajs-2376	264	3	acc	acc	PROPN
iajs-2376	264	4	.	.	PROPN
iajs-2376	264	5	proposition	proposition	NOUN
iajs-2376	264	6	4.7	4.7	NUM
iajs-2376	265	1	if	if	SCONJ
iajs-2376	265	2	is	be	AUX
iajs-2376	265	3	a	a	DET
iajs-2376	265	4	fully	fully	ADV
iajs-2376	265	5	semi	semi	ADJ
iajs-2376	265	6	-	-	ADJ
iajs-2376	265	7	prime	prime	ADJ
iajs-2376	265	8	-module	-module	NOUN
iajs-2376	265	9	such	such	ADJ
iajs-2376	265	10	that	that	SCONJ
iajs-2376	265	11	satisfies	satisfy	VERB
iajs-2376	265	12	acc	acc	PROPN
iajs-2376	265	13	(	(	PUNCT
iajs-2376	265	14	dcc	dcc	PROPN
iajs-2376	265	15	)	)	PUNCT
iajs-2376	265	16	on	on	ADP
iajs-2376	265	17	a	a	DET
iajs-2376	265	18	non	non	ADJ
iajs-2376	265	19	-	-	ADJ
iajs-2376	265	20	zero	zero	ADJ
iajs-2376	265	21	-closed	-close	VERB
iajs-2376	265	22	submodules	submodule	NOUN
iajs-2376	265	23	of	of	ADP
iajs-2376	265	24	,	,	PUNCT
iajs-2376	265	25	then	then	ADV
iajs-2376	265	26	satisfies	satisfy	VERB
iajs-2376	265	27	acc	acc	PROPN
iajs-2376	265	28	(	(	PUNCT
iajs-2376	265	29	dcc	dcc	PROPN
iajs-2376	265	30	)	)	PUNCT
iajs-2376	265	31	on	on	ADP
iajs-2376	265	32	non	non	ADJ
iajs-2376	265	33	-	-	ADJ
iajs-2376	265	34	zero	zero	ADJ
iajs-2376	265	35	-closed	-close	VERB
iajs-2376	265	36	submodules	submodule	NOUN
iajs-2376	265	37	of	of	ADP
iajs-2376	265	38	.	.	PUNCT
iajs-2376	266	1	proof	proof	NOUN
iajs-2376	266	2	let	let	VERB
iajs-2376	266	3	…	…	PUNCT
iajs-2376	266	4	…	…	PUNCT
iajs-2376	266	5	,	,	PUNCT
iajs-2376	266	6	be	be	AUX
iajs-2376	266	7	an	an	DET
iajs-2376	266	8	ascending	ascend	VERB
iajs-2376	266	9	chain	chain	NOUN
iajs-2376	266	10	of	of	ADP
iajs-2376	266	11	a	a	DET
iajs-2376	266	12	non	non	ADJ
iajs-2376	266	13	-	-	ADJ
iajs-2376	266	14	zero	zero	ADJ
iajs-2376	266	15	-closed	-close	VERB
iajs-2376	266	16	submodules	submodule	NOUN
iajs-2376	266	17	of	of	ADP
iajs-2376	266	18	for	for	ADP
iajs-2376	266	19	each	each	PRON
iajs-2376	266	20	…	…	PUNCT
iajs-2376	266	21	..	..	PUNCT
iajs-2376	266	22	then	then	ADV
iajs-2376	266	23	by	by	ADP
iajs-2376	266	24	proposition	proposition	NOUN
iajs-2376	266	25	(	(	PUNCT
iajs-2376	266	26	2.14	2.14	NUM
iajs-2376	266	27	)	)	PUNCT
iajs-2376	266	28	is	be	AUX
iajs-2376	266	29	a	a	DET
iajs-2376	266	30	-closed	-close	VERB
iajs-2376	266	31	submodules	submodule	NOUN
iajs-2376	266	32	in	in	ADV
iajs-2376	266	33	for	for	ADP
iajs-2376	266	34	each	each	PRON
iajs-2376	266	35	…	…	PUNCT
iajs-2376	266	36	.	.	PUNCT
iajs-2376	267	1	hence	hence	ADV
iajs-2376	267	2	…	…	PUNCT
iajs-2376	267	3	,	,	PUNCT
iajs-2376	267	4	be	be	AUX
iajs-2376	267	5	an	an	DET
iajs-2376	267	6	ascending	ascend	VERB
iajs-2376	267	7	chain	chain	NOUN
iajs-2376	267	8	of	of	ADP
iajs-2376	267	9	a	a	DET
iajs-2376	267	10	-closed	-close	VERB
iajs-2376	267	11	submodules	submodule	NOUN
iajs-2376	267	12	in	in	ADV
iajs-2376	267	13	.	.	PUNCT
iajs-2376	268	1	but	but	CCONJ
iajs-2376	268	2	has	have	VERB
iajs-2376	268	3	acc	acc	PROPN
iajs-2376	268	4	on	on	ADP
iajs-2376	268	5	-closed	-close	VERB
iajs-2376	268	6	submodules	submodule	NOUN
iajs-2376	268	7	,	,	PUNCT
iajs-2376	268	8	then	then	ADV
iajs-2376	268	9	such	such	ADJ
iajs-2376	268	10	that	that	PRON
iajs-2376	268	11	.	.	PUNCT
iajs-2376	269	1	hence	hence	ADV
iajs-2376	269	2	has	have	VERB
iajs-2376	269	3	acc	acc	PROPN
iajs-2376	269	4	on	on	ADP
iajs-2376	269	5	-closed	-close	VERB
iajs-2376	269	6	submodules	submodule	NOUN
iajs-2376	269	7	.	.	PUNCT
iajs-2376	270	1	similarly	similarly	ADV
iajs-2376	270	2	for	for	ADP
iajs-2376	270	3	dcc	dcc	PROPN
iajs-2376	270	4	.	.	PUNCT
iajs-2376	271	1	5	5	NUM
iajs-2376	271	2	.	.	PUNCT
iajs-2376	271	3	conclusions	conclusion	NOUN
iajs-2376	271	4	in	in	ADP
iajs-2376	271	5	this	this	DET
iajs-2376	271	6	article	article	NOUN
iajs-2376	271	7	we	we	PRON
iajs-2376	271	8	introduce	introduce	VERB
iajs-2376	271	9	and	and	CCONJ
iajs-2376	271	10	study	study	VERB
iajs-2376	271	11	the	the	DET
iajs-2376	271	12	notion	notion	NOUN
iajs-2376	271	13	of	of	ADP
iajs-2376	271	14	a	a	DET
iajs-2376	271	15	pseudo	pseudo	NOUN
iajs-2376	271	16	weakly	weakly	ADV
iajs-2376	271	17	closed	closed	ADJ
iajs-2376	271	18	submodules	submodule	NOUN
iajs-2376	271	19	as	as	ADP
iajs-2376	271	20	a	a	DET
iajs-2376	271	21	generalization	generalization	NOUN
iajs-2376	271	22	of	of	ADP
iajs-2376	271	23	a	a	DET
iajs-2376	271	24	-closed	-close	VERB
iajs-2376	271	25	submodules	submodule	NOUN
iajs-2376	271	26	.	.	PUNCT
iajs-2376	272	1	among	among	ADP
iajs-2376	272	2	the	the	DET
iajs-2376	272	3	main	main	ADJ
iajs-2376	272	4	results	result	NOUN
iajs-2376	272	5	we	we	PRON
iajs-2376	272	6	get	get	VERB
iajs-2376	272	7	are	be	AUX
iajs-2376	272	8	the	the	DET
iajs-2376	272	9	following	following	NOUN
iajs-2376	272	10	.	.	PUNCT
iajs-2376	273	1	1	1	X
iajs-2376	273	2	.	.	X
iajs-2376	274	1	if	if	SCONJ
iajs-2376	274	2	is	be	AUX
iajs-2376	274	3	an	an	DET
iajs-2376	274	4	-module	-module	NOUN
iajs-2376	274	5	,	,	PUNCT
iajs-2376	274	6	and	and	CCONJ
iajs-2376	274	7	are	be	AUX
iajs-2376	274	8	submodules	submodule	NOUN
iajs-2376	274	9	of	of	ADP
iajs-2376	274	10	with	with	ADP
iajs-2376	274	11	and	and	CCONJ
iajs-2376	274	12	is	be	AUX
iajs-2376	274	13	a	a	DET
iajs-2376	274	14	-closed	-closed	ADJ
iajs-2376	274	15	submodule	submodule	NOUN
iajs-2376	274	16	in	in	ADP
iajs-2376	274	17	and	and	CCONJ
iajs-2376	274	18	is	be	AUX
iajs-2376	274	19	a	a	DET
iajs-2376	274	20	-closed	-closed	ADJ
iajs-2376	274	21	submodule	submodule	NOUN
iajs-2376	274	22	in	in	ADP
iajs-2376	274	23	,	,	PUNCT
iajs-2376	274	24	then	then	ADV
iajs-2376	274	25	is	be	AUX
iajs-2376	274	26	a	a	DET
iajs-2376	274	27	-closed	-closed	ADJ
iajs-2376	274	28	submodule	submodule	NOUN
iajs-2376	274	29	in	in	ADP
iajs-2376	274	30	,	,	PUNCT
iajs-2376	274	31	provided	provide	VERB
iajs-2376	274	32	that	that	PRON
iajs-2376	274	33	contained	contain	VERB
iajs-2376	274	34	in	in	ADP
iajs-2376	274	35	any	any	DET
iajs-2376	274	36	weak	weak	ADJ
iajs-2376	274	37	-	-	PUNCT
iajs-2376	274	38	essential	essential	ADJ
iajs-2376	274	39	extensions	extension	NOUN
iajs-2376	274	40	of	of	ADP
iajs-2376	274	41	.	.	PUNCT
iajs-2376	274	42	2	2	X
iajs-2376	274	43	.	.	X
iajs-2376	274	44	if	if	SCONJ
iajs-2376	274	45	is	be	AUX
iajs-2376	274	46	a	a	DET
iajs-2376	274	47	uniserial	uniserial	ADJ
iajs-2376	274	48	-module	-module	NOUN
iajs-2376	274	49	,	,	PUNCT
iajs-2376	274	50	and	and	CCONJ
iajs-2376	274	51	are	be	AUX
iajs-2376	274	52	submodules	submodule	NOUN
iajs-2376	274	53	of	of	ADP
iajs-2376	274	54	such	such	ADJ
iajs-2376	274	55	that	that	PRON
iajs-2376	274	56	and	and	CCONJ
iajs-2376	274	57	is	be	AUX
iajs-2376	274	58	a	a	DET
iajs-2376	274	59	-closed	-closed	ADJ
iajs-2376	274	60	submodule	submodule	NOUN
iajs-2376	274	61	in	in	ADP
iajs-2376	274	62	and	and	CCONJ
iajs-2376	274	63	is	be	AUX
iajs-2376	274	64	a	a	DET
iajs-2376	274	65	-closed	-closed	ADJ
iajs-2376	274	66	submodule	submodule	NOUN
iajs-2376	274	67	in	in	ADP
iajs-2376	274	68	,	,	PUNCT
iajs-2376	274	69	then	then	ADV
iajs-2376	274	70	is	be	AUX
iajs-2376	274	71	a	a	DET
iajs-2376	274	72	closed	closed	ADJ
iajs-2376	274	73	submodule	submodule	NOUN
iajs-2376	274	74	in	in	ADP
iajs-2376	274	75	.	.	PUNCT
iajs-2376	275	1	3	3	X
iajs-2376	275	2	.	.	X
iajs-2376	276	1	if	if	SCONJ
iajs-2376	276	2	is	be	AUX
iajs-2376	276	3	an	an	DET
iajs-2376	276	4	-module	-module	NOUN
iajs-2376	276	5	where	where	SCONJ
iajs-2376	276	6	and	and	CCONJ
iajs-2376	276	7	are	be	AUX
iajs-2376	276	8	submodules	submodule	NOUN
iajs-2376	276	9	of	of	ADP
iajs-2376	276	10	,	,	PUNCT
iajs-2376	276	11	provided	provide	VERB
iajs-2376	276	12	that	that	SCONJ
iajs-2376	276	13	and	and	CCONJ
iajs-2376	276	14	all	all	DET
iajs-2376	276	15	submodules	submodule	NOUN
iajs-2376	276	16	of	of	ADP
iajs-2376	276	17	are	be	AUX
iajs-2376	276	18	completely	completely	ADV
iajs-2376	276	19	essential	essential	ADJ
iajs-2376	276	20	submodules	submodule	NOUN
iajs-2376	276	21	.	.	PUNCT
iajs-2376	277	1	if	if	SCONJ
iajs-2376	277	2	is	be	AUX
iajs-2376	277	3	a	a	DET
iajs-2376	277	4	non	non	ADJ
iajs-2376	277	5	-	-	ADJ
iajs-2376	277	6	zero	zero	NUM
iajs-2376	277	7	-closed	-close	VERB
iajs-2376	277	8	submodule	submodule	NOUN
iajs-2376	277	9	in	in	ADP
iajs-2376	277	10	and	and	CCONJ
iajs-2376	277	11	is	be	AUX
iajs-2376	277	12	a	a	DET
iajs-2376	277	13	non	non	ADJ
iajs-2376	277	14	-	-	ADJ
iajs-2376	277	15	zero	zero	NUM
iajs-2376	277	16	-closed	-close	VERB
iajs-2376	277	17	submodule	submodule	NOUN
iajs-2376	277	18	in	in	ADP
iajs-2376	277	19	,	,	PUNCT
iajs-2376	277	20	then	then	ADV
iajs-2376	277	21	is	be	AUX
iajs-2376	277	22	a	a	DET
iajs-2376	277	23	-closed	-closed	ADJ
iajs-2376	277	24	submodule	submodule	NOUN
iajs-2376	277	25	in	in	ADP
iajs-2376	277	26	if	if	SCONJ
iajs-2376	278	1	and	and	CCONJ
iajs-2376	278	2	only	only	ADV
iajs-2376	278	3	if	if	SCONJ
iajs-2376	278	4	is	be	AUX
iajs-2376	278	5	a	a	DET
iajs-2376	278	6	closed	closed	ADJ
iajs-2376	278	7	submodule	submodule	NOUN
iajs-2376	278	8	in	in	ADP
iajs-2376	278	9	and	and	CCONJ
iajs-2376	278	10	is	be	AUX
iajs-2376	278	11	a	a	DET
iajs-2376	278	12	-closed	-closed	ADJ
iajs-2376	278	13	submodule	submodule	NOUN
iajs-2376	278	14	in	in	ADP
iajs-2376	278	15	.	.	PUNCT
iajs-2376	279	1	88	88	NUM
iajs-2376	279	2	ibn	ibn	PROPN
iajs-2376	279	3	al	al	PROPN
iajs-2376	279	4	-	-	PUNCT
iajs-2376	279	5	haitham	haitham	PROPN
iajs-2376	279	6	jour	jour	X
iajs-2376	279	7	.	.	PROPN
iajs-2376	279	8	for	for	ADP
iajs-2376	279	9	pure	pure	ADJ
iajs-2376	279	10	&	&	CCONJ
iajs-2376	279	11	appl	appl	PROPN
iajs-2376	279	12	.	.	PUNCT
iajs-2376	280	1	sci	sci	PROPN
iajs-2376	280	2	.	.	PROPN
iajs-2376	281	1	33	33	NUM
iajs-2376	281	2	(	(	PUNCT
iajs-2376	281	3	1	1	NUM
iajs-2376	281	4	)	)	PUNCT
iajs-2376	281	5	2020	2020	NUM
iajs-2376	281	6	4	4	NUM
iajs-2376	281	7	.	.	PUNCT
iajs-2376	282	1	let	let	AUX
iajs-2376	282	2	be	be	AUX
iajs-2376	282	3	a	a	DET
iajs-2376	282	4	finitely	finitely	ADV
iajs-2376	282	5	generated	generate	VERB
iajs-2376	282	6	,	,	PUNCT
iajs-2376	282	7	faithful	faithful	ADJ
iajs-2376	282	8	and	and	CCONJ
iajs-2376	282	9	multiplication	multiplication	NOUN
iajs-2376	282	10	module	module	NOUN
iajs-2376	282	11	which	which	PRON
iajs-2376	282	12	satisfies	satisfy	VERB
iajs-2376	282	13	condition	condition	NOUN
iajs-2376	282	14	(	(	PUNCT
iajs-2376	282	15	*	*	NOUN
iajs-2376	282	16	)	)	PUNCT
iajs-2376	282	17	.	.	PUNCT
iajs-2376	283	1	then	then	ADV
iajs-2376	283	2	is	be	AUX
iajs-2376	283	3	a	a	DET
iajs-2376	283	4	-closed	-close	VERB
iajs-2376	283	5	ideal	ideal	NOUN
iajs-2376	283	6	in	in	ADP
iajs-2376	283	7	if	if	SCONJ
iajs-2376	284	1	and	and	CCONJ
iajs-2376	284	2	only	only	ADV
iajs-2376	284	3	if	if	SCONJ
iajs-2376	284	4	is	be	AUX
iajs-2376	284	5	a	a	DET
iajs-2376	284	6	-closed	-closed	ADJ
iajs-2376	284	7	submodule	submodule	NOUN
iajs-2376	284	8	in	in	ADP
iajs-2376	284	9	.	.	PUNCT
iajs-2376	285	1	5	5	X
iajs-2376	285	2	.	.	X
iajs-2376	286	1	if	if	SCONJ
iajs-2376	286	2	is	be	AUX
iajs-2376	286	3	a	a	DET
iajs-2376	286	4	module	module	NOUN
iajs-2376	286	5	satisfies	satisfie	NOUN
iajs-2376	286	6	acc	acc	PROPN
iajs-2376	286	7	(	(	PUNCT
iajs-2376	286	8	dcc	dcc	PROPN
iajs-2376	286	9	)	)	PUNCT
iajs-2376	286	10	on	on	ADP
iajs-2376	286	11	-closed	-closed	ADJ
iajs-2376	286	12	submodules	submodule	NOUN
iajs-2376	286	13	of	of	ADP
iajs-2376	286	14	,	,	PUNCT
iajs-2376	286	15	then	then	ADV
iajs-2376	286	16	satisfiesacc	satisfiesacc	VERB
iajs-2376	286	17	(	(	PUNCT
iajs-2376	286	18	dcc	dcc	PROPN
iajs-2376	286	19	)	)	PUNCT
iajs-2376	286	20	on	on	ADP
iajs-2376	286	21	-closed	-closed	ADJ
iajs-2376	286	22	submodules	submodule	NOUN
iajs-2376	286	23	of	of	ADP
iajs-2376	286	24	.	.	PUNCT
iajs-2376	287	1	6	6	X
iajs-2376	287	2	.	.	X
iajs-2376	288	1	if	if	SCONJ
iajs-2376	288	2	is	be	AUX
iajs-2376	288	3	a	a	DET
iajs-2376	288	4	fully	fully	ADV
iajs-2376	288	5	semi	semi	ADJ
iajs-2376	288	6	-	-	ADJ
iajs-2376	288	7	prime	prime	ADJ
iajs-2376	288	8	-module	-module	NOUN
iajs-2376	288	9	and	and	CCONJ
iajs-2376	288	10	satisfies	satisfie	NOUN
iajs-2376	288	11	acc	acc	PROPN
iajs-2376	288	12	(	(	PUNCT
iajs-2376	288	13	dcc	dcc	PROPN
iajs-2376	288	14	)	)	PUNCT
iajs-2376	288	15	on	on	ADP
iajs-2376	288	16	-closed	-closed	ADJ
iajs-2376	288	17	submodules	submodule	NOUN
iajs-2376	288	18	of	of	ADP
iajs-2376	288	19	,	,	PUNCT
iajs-2376	288	20	then	then	ADV
iajs-2376	288	21	satisfies	satisfy	VERB
iajs-2376	288	22	acc	acc	PROPN
iajs-2376	288	23	(	(	PUNCT
iajs-2376	288	24	dcc	dcc	PROPN
iajs-2376	288	25	)	)	PUNCT
iajs-2376	288	26	on	on	ADP
iajs-2376	288	27	closed	closed	ADJ
iajs-2376	288	28	submodules	submodule	NOUN
iajs-2376	288	29	of	of	ADP
iajs-2376	288	30	.	.	PUNCT
iajs-2376	289	1	references	reference	NOUN
iajs-2376	289	2	1	1	NUM
iajs-2376	289	3	.	.	PUNCT
iajs-2376	289	4	gooderal	gooderal	ADJ
iajs-2376	289	5	,	,	PUNCT
iajs-2376	289	6	k.	k.	PROPN
iajs-2376	289	7	r.	r.	PROPN
iajs-2376	289	8	ring	ring	PROPN
iajs-2376	289	9	theory	theory	PROPN
iajs-2376	289	10	,	,	PUNCT
iajs-2376	289	11	nonsingular	nonsingular	ADJ
iajs-2376	289	12	ring	ring	NOUN
iajs-2376	289	13	and	and	CCONJ
iajs-2376	289	14	modules	module	NOUN
iajs-2376	289	15	.	.	PUNCT
iajs-2376	290	1	marcel	marcel	PROPN
iajs-2376	290	2	.	.	PUNCT
iajs-2376	290	3	dekker	dekker	PROPN
iajs-2376	290	4	,	,	PUNCT
iajs-2376	290	5	new	new	PROPN
iajs-2376	290	6	york	york	PROPN
iajs-2376	290	7	.	.	PUNCT
iajs-2376	290	8	1976	1976	NUM
iajs-2376	290	9	.	.	PUNCT
iajs-2376	291	1	2	2	NUM
iajs-2376	291	2	.	.	X
iajs-2376	291	3	muna	muna	PROPN
iajs-2376	291	4	,	,	PUNCT
iajs-2376	291	5	a.	a.	NOUN
iajs-2376	291	6	a.	a.	NOUN
iajs-2376	291	7	weak	weak	ADJ
iajs-2376	291	8	essential	essential	ADJ
iajs-2376	291	9	submodules	submodule	NOUN
iajs-2376	291	10	.	.	PUNCT
iajs-2376	292	1	um	um	INTJ
iajs-2376	292	2	-	-	PUNCT
iajs-2376	292	3	science	science	NOUN
iajs-2376	292	4	journal.2009	journal.2009	PROPN
iajs-2376	292	5	,	,	PUNCT
iajs-2376	292	6	6	6	NUM
iajs-2376	292	7	,	,	PUNCT
iajs-2376	292	8	1	1	NUM
iajs-2376	292	9	,	,	PUNCT
iajs-2376	292	10	214	214	NUM
iajs-2376	292	11	-	-	SYM
iajs-2376	292	12	222	222	NUM
iajs-2376	292	13	.	.	PUNCT
iajs-2376	293	1	3	3	NUM
iajs-2376	293	2	.	.	X
iajs-2376	293	3	muna	muna	PROPN
iajs-2376	293	4	,	,	PUNCT
iajs-2376	293	5	a.	a.	NOUN
iajs-2376	293	6	a.	a.	NOUN
iajs-2376	293	7	some	some	PRON
iajs-2376	293	8	results	result	VERB
iajs-2376	293	9	on	on	ADP
iajs-2376	293	10	weak	weak	ADJ
iajs-2376	293	11	essential	essential	ADJ
iajs-2376	293	12	submodules	submodule	NOUN
iajs-2376	293	13	.	.	PUNCT
iajs-2376	294	1	baghdad	baghdad	PROPN
iajs-2376	294	2	science	science	PROPN
iajs-2376	294	3	journal	journal	PROPN
iajs-2376	294	4	.	.	PUNCT
iajs-2376	295	1	2016	2016	NUM
iajs-2376	295	2	,	,	PUNCT
iajs-2376	295	3	13	13	NUM
iajs-2376	295	4	,	,	PUNCT
iajs-2376	295	5	3	3	NUM
iajs-2376	295	6	,	,	PUNCT
iajs-2376	295	7	599	599	NUM
iajs-2376	295	8	-	-	SYM
iajs-2376	295	9	606	606	NUM
iajs-2376	295	10	.	.	PUNCT
iajs-2376	296	1	4	4	NUM
iajs-2376	296	2	.	.	NOUN
iajs-2376	296	3	athab	athab	PROPN
iajs-2376	296	4	,	,	PUNCT
iajs-2376	296	5	e.	e.	PROPN
iajs-2376	296	6	a.	a.	PROPN
iajs-2376	296	7	;	;	PUNCT
iajs-2376	296	8	al	al	PROPN
iajs-2376	296	9	-	-	PUNCT
iajs-2376	296	10	hashimi	hashimi	PROPN
iajs-2376	296	11	,	,	PUNCT
iajs-2376	296	12	b.	b.	PROPN
iajs-2376	296	13	a.	a.	PROPN
iajs-2376	296	14	anote	anote	PROPN
iajs-2376	296	15	on	on	ADP
iajs-2376	296	16	seme	seme	ADJ
iajs-2376	296	17	-	-	PUNCT
iajs-2376	296	18	prime	prime	ADJ
iajs-2376	296	19	submodules	submodule	NOUN
iajs-2376	296	20	in	in	ADP
iajs-2376	296	21	multiplication	multiplication	NOUN
iajs-2376	296	22	modules	module	NOUN
iajs-2376	296	23	.	.	PUNCT
iajs-2376	297	1	iraq	iraq	PROPN
iajs-2376	297	2	j.sc.2000	j.sc.2000	PROPN
iajs-2376	297	3	,	,	PUNCT
iajs-2376	297	4	41	41	NUM
iajs-2376	297	5	,	,	PUNCT
iajs-2376	297	6	1	1	NUM
iajs-2376	297	7	,	,	PUNCT
iajs-2376	297	8	88	88	NUM
iajs-2376	297	9	-	-	SYM
iajs-2376	297	10	93	93	NUM
iajs-2376	297	11	.	.	PUNCT
iajs-2376	298	1	5	5	X
iajs-2376	298	2	.	.	X
iajs-2376	298	3	haibat	haibat	NOUN
iajs-2376	298	4	.	.	PUNCT
iajs-2376	298	5	,	,	PUNCT
iajs-2376	299	1	k.	k.	PROPN
iajs-2376	299	2	m.	m.	PROPN
iajs-2376	299	3	;	;	PUNCT
iajs-2376	299	4	mohammad	mohammad	PROPN
iajs-2376	299	5	,	,	PUNCT
iajs-2376	299	6	e.	e.	PROPN
iajs-2376	299	7	d.	d.	PROPN
iajs-2376	299	8	w	w	PROPN
iajs-2376	299	9	-	-	PUNCT
iajs-2376	299	10	closed	close	VERB
iajs-2376	299	11	submodules	submodule	NOUN
iajs-2376	299	12	and	and	CCONJ
iajs-2376	299	13	related	related	ADJ
iajs-2376	299	14	concepts	concept	NOUN
iajs-2376	299	15	.	.	PUNCT
iajs-2376	300	1	ibnal	ibnal	ADJ
iajs-2376	300	2	-	-	PUNCT
iajs-2376	300	3	haitham	haitham	PROPN
iajs-2376	300	4	journal	journal	NOUN
iajs-2376	300	5	,	,	PUNCT
iajs-2376	300	6	for	for	ADP
iajs-2376	300	7	pure	pure	ADJ
iajs-2376	300	8	and	and	CCONJ
iajs-2376	300	9	apple	apple	NOUN
iajs-2376	300	10	.	.	PUNCT
iajs-2376	301	1	sci.2018	sci.2018	NOUN
iajs-2376	301	2	,	,	PUNCT
iajs-2376	301	3	31	31	NUM
iajs-2376	301	4	,	,	PUNCT
iajs-2376	301	5	2	2	NUM
iajs-2376	301	6	,	,	PUNCT
iajs-2376	301	7	164	164	NUM
iajs-2376	301	8	-	-	SYM
iajs-2376	301	9	178	178	NUM
iajs-2376	301	10	.	.	PUNCT
iajs-2376	302	1	6	6	NUM
iajs-2376	302	2	.	.	X
iajs-2376	302	3	osfsky	osfsky	ADJ
iajs-2376	302	4	,	,	PUNCT
iajs-2376	302	5	b.l	b.l	PROPN
iajs-2376	302	6	.	.	PROPN
iajs-2376	302	7	aconstruction	aconstruction	NOUN
iajs-2376	302	8	of	of	ADP
iajs-2376	302	9	nonstandares	nonstandare	NOUN
iajs-2376	302	10	uniserial	uniserial	ADJ
iajs-2376	302	11	modules	module	NOUN
iajs-2376	302	12	over	over	ADP
iajs-2376	302	13	valuation	valuation	NOUN
iajs-2376	302	14	domain	domain	NOUN
iajs-2376	302	15	.	.	PUNCT
iajs-2376	303	1	bull.amer	bull.amer	NOUN
iajs-2376	303	2	,	,	PUNCT
iajs-2376	303	3	math.soc.1991	math.soc.1991	ADJ
iajs-2376	303	4	,	,	PUNCT
iajs-2376	303	5	25	25	NUM
iajs-2376	303	6	,	,	PUNCT
iajs-2376	303	7	89	89	NUM
iajs-2376	303	8	-	-	SYM
iajs-2376	303	9	97	97	NUM
iajs-2376	303	10	.	.	PUNCT
iajs-2376	304	1	7	7	NUM
iajs-2376	304	2	.	.	X
iajs-2376	304	3	abbas	abbas	PROPN
iajs-2376	304	4	,	,	PUNCT
iajs-2376	304	5	m.	m.	NOUN
iajs-2376	304	6	s.	s.	PROPN
iajs-2376	304	7	on	on	ADP
iajs-2376	304	8	fully	fully	ADV
iajs-2376	304	9	stable	stable	ADJ
iajs-2376	304	10	modules	module	NOUN
iajs-2376	304	11	.	.	PUNCT
iajs-2376	305	1	thesis	thesis	NOUN
iajs-2376	305	2	,	,	PUNCT
iajs-2376	305	3	university	university	NOUN
iajs-2376	305	4	of	of	ADP
iajs-2376	305	5	baghdad	baghdad	PROPN
iajs-2376	305	6	,	,	PUNCT
iajs-2376	305	7	1991	1991	NUM
iajs-2376	305	8	.	.	PUNCT
iajs-2376	306	1	8	8	X
iajs-2376	306	2	.	.	X
iajs-2376	306	3	behoodi	behoodi	PROPN
iajs-2376	306	4	,	,	PUNCT
iajs-2376	306	5	m	m	PROPN
iajs-2376	306	6	;	;	PUNCT
iajs-2376	306	7	karamzadeh	karamzadeh	PROPN
iajs-2376	306	8	,	,	PUNCT
iajs-2376	306	9	o.	o.	INTJ
iajs-2376	306	10	;	;	PUNCT
iajs-2376	306	11	koohy	koohy	ADV
iajs-2376	306	12	,	,	PUNCT
iajs-2376	306	13	h.	h.	PROPN
iajs-2376	306	14	modules	module	NOUN
iajs-2376	306	15	whose	whose	DET
iajs-2376	306	16	certain	certain	ADJ
iajs-2376	306	17	submodules	submodule	NOUN
iajs-2376	306	18	are	be	AUX
iajs-2376	306	19	primes	prime	NOUN
iajs-2376	306	20	.	.	PUNCT
iajs-2376	307	1	vietnam	vietnam	PROPN
iajs-2376	307	2	journal	journal	PROPN
iajs-2376	307	3	of	of	ADP
iajs-2376	307	4	mathematics.2004	mathematics.2004	PROPN
iajs-2376	307	5	,	,	PUNCT
iajs-2376	307	6	32	32	NUM
iajs-2376	307	7	,	,	PUNCT
iajs-2376	307	8	3	3	NUM
iajs-2376	307	9	,	,	PUNCT
iajs-2376	307	10	303	303	NUM
iajs-2376	307	11	-	-	SYM
iajs-2376	307	12	317	317	NUM
iajs-2376	307	13	.	.	PUNCT
iajs-2376	308	1	9	9	NUM
iajs-2376	308	2	.	.	X
iajs-2376	308	3	barnard	barnard	PROPN
iajs-2376	308	4	,	,	PUNCT
iajs-2376	308	5	a.	a.	NOUN
iajs-2376	308	6	multiplication	multiplication	NOUN
iajs-2376	308	7	modules	module	NOUN
iajs-2376	308	8	.	.	PUNCT
iajs-2376	309	1	j.algebra.1981	j.algebra.1981	NOUN
iajs-2376	309	2	,	,	PUNCT
iajs-2376	309	3	71	71	NUM
iajs-2376	309	4	,	,	PUNCT
iajs-2376	309	5	174	174	NUM
iajs-2376	309	6	-	-	SYM
iajs-2376	309	7	178	178	NUM
iajs-2376	309	8	.	.	PUNCT
iajs-2376	310	1	10	10	NUM
iajs-2376	310	2	.	.	PUNCT
iajs-2376	311	1	jabber	jabber	PROPN
iajs-2376	311	2	,	,	PUNCT
iajs-2376	311	3	a.k	a.k	PROPN
iajs-2376	311	4	.	.	PROPN
iajs-2376	311	5	;	;	PUNCT
iajs-2376	311	6	tawfiq	tawfiq	PROPN
iajs-2376	311	7	,	,	PUNCT
iajs-2376	311	8	l.n.m	l.n.m	NOUN
iajs-2376	311	9	.	.	PUNCT
iajs-2376	312	1	new	new	ADJ
iajs-2376	312	2	transform	transform	VERB
iajs-2376	312	3	fundamental	fundamental	ADJ
iajs-2376	312	4	properties	property	NOUN
iajs-2376	312	5	and	and	CCONJ
iajs-2376	312	6	its	its	PRON
iajs-2376	312	7	applications	application	NOUN
iajs-2376	312	8	.	.	PUNCT
iajs-2376	313	1	ibn	ibn	PROPN
iajs-2376	313	2	alhaitham	alhaitham	PROPN
iajs-2376	313	3	journal	journal	NOUN
iajs-2376	313	4	for	for	ADP
iajs-2376	313	5	pure	pure	ADJ
iajs-2376	313	6	and	and	CCONJ
iajs-2376	313	7	applied	apply	VERB
iajs-2376	313	8	science.2018	science.2018	PROPN
iajs-2376	313	9	,	,	PUNCT
iajs-2376	313	10	31	31	NUM
iajs-2376	313	11	,	,	PUNCT
iajs-2376	313	12	1	1	NUM
iajs-2376	313	13	,	,	PUNCT
iajs-2376	313	14	151163	151163	NUM
iajs-2376	313	15	.	.	PUNCT