id	sid	tid	token	lemma	pos
iajs-1618	1	1	conseguences	conseguence	NOUN
iajs-1618	1	2	of	of	ADP
iajs-1618	1	3	soil	soil	NOUN
iajs-1618	1	4	crude	crude	ADJ
iajs-1618	1	5	oil	oil	NOUN
iajs-1618	1	6	pollution	pollution	NOUN
iajs-1618	1	7	on	on	ADP
iajs-1618	1	8	some	some	DET
iajs-1618	1	9	wood	wood	NOUN
iajs-1618	1	10	properties	property	NOUN
iajs-1618	1	11	of	of	ADP
iajs-1618	1	12	olive	olive	NOUN
iajs-1618	1	13	trees	tree	NOUN
iajs-1618	1	14	mathematics	mathematic	NOUN
iajs-1618	1	15	|227	|227	PROPN
iajs-1618	1	16	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	X
iajs-1618	1	17	7102	7102	NUM
iajs-1618	1	18	(	(	PUNCT
iajs-1618	1	19	عام	عام	PROPN
iajs-1618	1	20	3	3	NUM
iajs-1618	1	21	(	(	PUNCT
iajs-1618	1	22	العدد	العدد	PROPN
iajs-1618	1	23	)	)	PUNCT
iajs-1618	1	24	30مجلة	30مجلة	PROPN
iajs-1618	1	25	إبن	إبن	VERB
iajs-1618	1	26	الهيثم	الهيثم	ADJ
iajs-1618	1	27	للعلوم	للعلوم	NOUN
iajs-1618	1	28	الصرفة	الصرفة	NOUN
iajs-1618	2	1	و	و	PRON
iajs-1618	2	2	التطبيقية	التطبيقية	ADV
iajs-1618	2	3	المجلد	المجلد	ADV
iajs-1618	2	4	)	)	PUNCT
iajs-1618	3	1	ibn	ibn	PROPN
iajs-1618	3	2	al	al	PROPN
iajs-1618	3	3	-	-	PUNCT
iajs-1618	3	4	haitham	haitham	PROPN
iajs-1618	3	5	j.	j.	PROPN
iajs-1618	3	6	for	for	ADP
iajs-1618	3	7	pure	pure	PROPN
iajs-1618	3	8	&	&	CCONJ
iajs-1618	3	9	appl	appl	PROPN
iajs-1618	3	10	.	.	PUNCT
iajs-1618	4	1	sci	sci	PROPN
iajs-1618	4	2	.	.	PUNCT
iajs-1618	5	1	vol.03	vol.03	PROPN
iajs-1618	5	2	(	(	PUNCT
iajs-1618	5	3	3	3	NUM
iajs-1618	5	4	)	)	PUNCT
iajs-1618	5	5	2017	2017	NUM
iajs-1618	5	6	modules	module	NOUN
iajs-1618	5	7	with	with	ADP
iajs-1618	5	8	chain	chain	NOUN
iajs-1618	5	9	conditions	condition	NOUN
iajs-1618	5	10	on	on	ADP
iajs-1618	5	11	s	s	NOUN
iajs-1618	5	12	-	-	PUNCT
iajs-1618	5	13	closed	close	VERB
iajs-1618	5	14	submodules	submodule	NOUN
iajs-1618	5	15	rana	rana	PROPN
iajs-1618	5	16	noori	noori	PROPN
iajs-1618	5	17	majeed	majeed	PROPN
iajs-1618	5	18	mohammed	mohammed	PROPN
iajs-1618	5	19	dept	dept	PROPN
iajs-1618	5	20	.	.	PROPN
iajs-1618	5	21	of	of	ADP
iajs-1618	5	22	mathematic	mathematic	PROPN
iajs-1618	5	23	/	/	SYM
iajs-1618	5	24	college	college	NOUN
iajs-1618	5	25	of	of	ADP
iajs-1618	5	26	education	education	NOUN
iajs-1618	5	27	for	for	ADP
iajs-1618	5	28	pure	pure	ADJ
iajs-1618	5	29	science-(ibn	science-(ibn	PROPN
iajs-1618	5	30	al	al	PROPN
iajs-1618	5	31	-	-	PUNCT
iajs-1618	5	32	haitham)/	haitham)/	PROPN
iajs-1618	5	33	university	university	NOUN
iajs-1618	5	34	of	of	ADP
iajs-1618	5	35	baghdad	baghdad	PROPN
iajs-1618	5	36	rana.n.m@ihcoedu.uobaghdad.edu.iq	rana.n.m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-1618	5	37	received	receive	VERB
iajs-1618	5	38	in	in	ADP
iajs-1618	5	39	:	:	PUNCT
iajs-1618	5	40	31	31	NUM
iajs-1618	5	41	/may	/may	PUNCT
iajs-1618	5	42	/2017	/2017	NOUN
iajs-1618	5	43	,	,	PUNCT
iajs-1618	5	44	accepted	accept	VERB
iajs-1618	5	45	in	in	ADP
iajs-1618	5	46	:	:	PUNCT
iajs-1618	5	47	27	27	NUM
iajs-1618	5	48	/august/	/august/	SYM
iajs-1618	5	49	2017	2017	NUM
iajs-1618	5	50	abstract	abstract	NOUN
iajs-1618	5	51	let	let	VERB
iajs-1618	5	52	l	l	NOUN
iajs-1618	5	53	be	be	AUX
iajs-1618	5	54	a	a	DET
iajs-1618	5	55	commutative	commutative	ADJ
iajs-1618	5	56	ring	ring	NOUN
iajs-1618	5	57	with	with	ADP
iajs-1618	5	58	identity	identity	NOUN
iajs-1618	5	59	and	and	CCONJ
iajs-1618	5	60	let	let	VERB
iajs-1618	5	61	w	w	NOUN
iajs-1618	5	62	be	be	AUX
iajs-1618	5	63	a	a	DET
iajs-1618	5	64	unitary	unitary	ADJ
iajs-1618	5	65	left	leave	VERB
iajs-1618	5	66	lmodule	lmodule	NOUN
iajs-1618	5	67	.	.	PUNCT
iajs-1618	6	1	a	a	DET
iajs-1618	6	2	submodule	submodule	PROPN
iajs-1618	6	3	d	d	PROPN
iajs-1618	6	4	of	of	ADP
iajs-1618	6	5	an	an	DET
iajs-1618	6	6	lmodule	lmodule	NOUN
iajs-1618	6	7	w	w	NOUN
iajs-1618	6	8	is	be	AUX
iajs-1618	6	9	called	call	VERB
iajs-1618	6	10	sclosed	sclosed	ADJ
iajs-1618	6	11	submodule	submodule	NOUN
iajs-1618	6	12	denoted	denote	VERB
iajs-1618	6	13	by	by	ADP
iajs-1618	6	14	d	d	PROPN
iajs-1618	6	15	≤sc	≤sc	NOUN
iajs-1618	6	16	w	w	NOUN
iajs-1618	6	17	,	,	PUNCT
iajs-1618	6	18	if	if	SCONJ
iajs-1618	6	19	d	d	PROPN
iajs-1618	6	20	has	have	VERB
iajs-1618	6	21	no	no	DET
iajs-1618	6	22	proper	proper	ADJ
iajs-1618	6	23	sessential	sessential	ADJ
iajs-1618	6	24	extension	extension	NOUN
iajs-1618	6	25	in	in	ADP
iajs-1618	6	26	w	w	PROPN
iajs-1618	6	27	,	,	PUNCT
iajs-1618	6	28	that	that	ADV
iajs-1618	6	29	is	is	ADV
iajs-1618	6	30	,	,	PUNCT
iajs-1618	6	31	whenever	whenever	SCONJ
iajs-1618	6	32	d	d	PROPN
iajs-1618	6	33	≤	≤	NUM
iajs-1618	6	34	w	w	ADP
iajs-1618	6	35	such	such	ADJ
iajs-1618	6	36	that	that	SCONJ
iajs-1618	6	37	d	d	PROPN
iajs-1618	6	38	≤se	≤se	NUM
iajs-1618	6	39	h≤	h≤	NUM
iajs-1618	6	40	w	w	PROPN
iajs-1618	6	41	,	,	PUNCT
iajs-1618	6	42	then	then	ADV
iajs-1618	6	43	d	d	PROPN
iajs-1618	6	44	=	=	SYM
iajs-1618	6	45	h.	h.	PROPN
iajs-1618	6	46	in	in	ADP
iajs-1618	6	47	this	this	DET
iajs-1618	6	48	paper	paper	NOUN
iajs-1618	6	49	,	,	PUNCT
iajs-1618	6	50	we	we	PRON
iajs-1618	6	51	study	study	VERB
iajs-1618	6	52	modules	module	NOUN
iajs-1618	6	53	which	which	PRON
iajs-1618	6	54	satisfies	satisfy	VERB
iajs-1618	6	55	the	the	DET
iajs-1618	6	56	ascending	ascend	VERB
iajs-1618	6	57	chain	chain	NOUN
iajs-1618	6	58	conditions	condition	NOUN
iajs-1618	6	59	(	(	PUNCT
iajs-1618	6	60	acc	acc	PROPN
iajs-1618	6	61	)	)	PUNCT
iajs-1618	6	62	and	and	CCONJ
iajs-1618	6	63	descending	descend	VERB
iajs-1618	6	64	chain	chain	NOUN
iajs-1618	6	65	conditions	condition	NOUN
iajs-1618	6	66	(	(	PUNCT
iajs-1618	6	67	dcc	dcc	PROPN
iajs-1618	6	68	)	)	PUNCT
iajs-1618	6	69	on	on	ADP
iajs-1618	6	70	this	this	DET
iajs-1618	6	71	kind	kind	NOUN
iajs-1618	6	72	of	of	ADP
iajs-1618	6	73	submodules	submodule	NOUN
iajs-1618	6	74	.	.	PUNCT
iajs-1618	7	1	keywords	keyword	NOUN
iajs-1618	7	2	:	:	PUNCT
iajs-1618	7	3	s	s	X
iajs-1618	7	4	-	-	ADJ
iajs-1618	7	5	essential	essential	ADJ
iajs-1618	7	6	submodules	submodule	NOUN
iajs-1618	7	7	,	,	PUNCT
iajs-1618	7	8	s	s	NOUN
iajs-1618	7	9	-	-	PUNCT
iajs-1618	7	10	closed	closed	ADJ
iajs-1618	7	11	submodules	submodule	NOUN
iajs-1618	7	12	,	,	PUNCT
iajs-1618	7	13	ascending	ascend	VERB
iajs-1618	7	14	and	and	CCONJ
iajs-1618	7	15	descending	descend	VERB
iajs-1618	7	16	chain	chain	NOUN
iajs-1618	7	17	conditions	condition	NOUN
iajs-1618	7	18	.	.	PUNCT
iajs-1618	8	1	mailto:rana.n.m@ihcoedu.uobaghdad.edu.iq	mailto:rana.n.m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-1618	8	2	mathematics	mathematics	PROPN
iajs-1618	8	3	|228	|228	PROPN
iajs-1618	9	1	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	PROPN
iajs-1618	9	2	7102	7102	NUM
iajs-1618	9	3	(	(	PUNCT
iajs-1618	9	4	عام	عام	PROPN
iajs-1618	9	5	3	3	NUM
iajs-1618	9	6	(	(	PUNCT
iajs-1618	9	7	العدد	العدد	PROPN
iajs-1618	9	8	)	)	PUNCT
iajs-1618	9	9	30مجلة	30مجلة	PROPN
iajs-1618	9	10	إبن	إبن	VERB
iajs-1618	9	11	الهيثم	الهيثم	ADJ
iajs-1618	9	12	للعلوم	للعلوم	NOUN
iajs-1618	9	13	الصرفة	الصرفة	NOUN
iajs-1618	10	1	و	و	PRON
iajs-1618	10	2	التطبيقية	التطبيقية	ADV
iajs-1618	10	3	المجلد	المجلد	ADV
iajs-1618	10	4	)	)	PUNCT
iajs-1618	11	1	ibn	ibn	PROPN
iajs-1618	11	2	al	al	PROPN
iajs-1618	11	3	-	-	PUNCT
iajs-1618	11	4	haitham	haitham	PROPN
iajs-1618	11	5	j.	j.	PROPN
iajs-1618	11	6	for	for	ADP
iajs-1618	11	7	pure	pure	PROPN
iajs-1618	11	8	&	&	CCONJ
iajs-1618	11	9	appl	appl	PROPN
iajs-1618	11	10	.	.	PUNCT
iajs-1618	12	1	sci	sci	PROPN
iajs-1618	12	2	.	.	PUNCT
iajs-1618	13	1	vol.03	vol.03	PROPN
iajs-1618	13	2	(	(	PUNCT
iajs-1618	13	3	3	3	NUM
iajs-1618	13	4	)	)	PUNCT
iajs-1618	13	5	2017	2017	NUM
iajs-1618	13	6	introduction	introduction	NOUN
iajs-1618	13	7	throughout	throughout	ADP
iajs-1618	13	8	this	this	DET
iajs-1618	13	9	paper	paper	NOUN
iajs-1618	13	10	,	,	PUNCT
iajs-1618	13	11	l	l	NOUN
iajs-1618	13	12	represents	represent	VERB
iajs-1618	13	13	a	a	DET
iajs-1618	13	14	commutative	commutative	ADJ
iajs-1618	13	15	ring	ring	NOUN
iajs-1618	13	16	with	with	ADP
iajs-1618	13	17	unity	unity	NOUN
iajs-1618	13	18	and	and	CCONJ
iajs-1618	13	19	w	w	AUX
iajs-1618	13	20	be	be	AUX
iajs-1618	13	21	a	a	DET
iajs-1618	13	22	left	left	ADJ
iajs-1618	13	23	unitary	unitary	ADJ
iajs-1618	13	24	lmodule	lmodule	NOUN
iajs-1618	13	25	.	.	PUNCT
iajs-1618	14	1	its	its	PUNCT
iajs-1618	15	1	well	well	INTJ
iajs-1618	15	2	known	know	VERB
iajs-1618	15	3	that	that	SCONJ
iajs-1618	15	4	“	"	PUNCT
iajs-1618	15	5	a	a	DET
iajs-1618	15	6	submodule	submodule	PROPN
iajs-1618	15	7	d	d	PROPN
iajs-1618	15	8	of	of	ADP
iajs-1618	15	9	w	w	PROPN
iajs-1618	15	10	is	be	AUX
iajs-1618	15	11	called	call	VERB
iajs-1618	15	12	small	small	ADJ
iajs-1618	15	13	denoted	denote	VERB
iajs-1618	15	14	by	by	ADP
iajs-1618	15	15	d	d	PROPN
iajs-1618	15	16	<	<	X
iajs-1618	15	17	<	<	X
iajs-1618	15	18	w	w	NOUN
iajs-1618	15	19	if	if	SCONJ
iajs-1618	16	1	and	and	CCONJ
iajs-1618	16	2	only	only	ADV
iajs-1618	16	3	if	if	SCONJ
iajs-1618	16	4	d	d	PROPN
iajs-1618	16	5	+	+	NOUN
iajs-1618	16	6	u	u	NOUN
iajs-1618	16	7	=	=	SYM
iajs-1618	16	8	w	w	NOUN
iajs-1618	16	9	implies	imply	VERB
iajs-1618	16	10	u	u	NOUN
iajs-1618	16	11	=	=	NOUN
iajs-1618	16	12	w	w	NOUN
iajs-1618	16	13	for	for	ADP
iajs-1618	16	14	each	each	DET
iajs-1618	16	15	u	u	PROPN
iajs-1618	16	16	submodule	submodule	NOUN
iajs-1618	16	17	of	of	ADP
iajs-1618	16	18	w	w	PROPN
iajs-1618	16	19	(	(	PUNCT
iajs-1618	16	20	u≤w	u≤w	PROPN
iajs-1618	16	21	)	)	PUNCT
iajs-1618	16	22	”	"	PUNCT
iajs-1618	17	1	[	[	X
iajs-1618	17	2	2	2	NUM
iajs-1618	17	3	]	]	PUNCT
iajs-1618	17	4	,	,	PUNCT
iajs-1618	17	5	and	and	CCONJ
iajs-1618	17	6	“	"	PUNCT
iajs-1618	17	7	a	a	DET
iajs-1618	17	8	submodule	submodule	PROPN
iajs-1618	17	9	d	d	PROPN
iajs-1618	17	10	of	of	ADP
iajs-1618	17	11	an	an	DET
iajs-1618	17	12	l	l	NOUN
iajs-1618	17	13	module	module	NOUN
iajs-1618	17	14	w	w	NOUN
iajs-1618	17	15	is	be	AUX
iajs-1618	17	16	called	call	VERB
iajs-1618	17	17	an	an	DET
iajs-1618	17	18	essential	essential	ADJ
iajs-1618	17	19	submodule	submodule	NOUN
iajs-1618	17	20	of	of	ADP
iajs-1618	17	21	w	w	PROPN
iajs-1618	17	22	and	and	CCONJ
iajs-1618	17	23	denoted	denote	VERB
iajs-1618	17	24	by	by	ADP
iajs-1618	17	25	d≤ew	d≤ew	PROPN
iajs-1618	17	26	if	if	SCONJ
iajs-1618	17	27	every	every	DET
iajs-1618	17	28	non	non	ADJ
iajs-1618	17	29	-	-	ADJ
iajs-1618	17	30	zero	zero	NUM
iajs-1618	17	31	submodule	submodule	NOUN
iajs-1618	17	32	of	of	ADP
iajs-1618	17	33	w	w	PROPN
iajs-1618	17	34	has	have	VERB
iajs-1618	17	35	non	non	ADJ
iajs-1618	17	36	-	-	ADJ
iajs-1618	17	37	zero	zero	NUM
iajs-1618	17	38	intersection	intersection	NOUN
iajs-1618	17	39	with	with	ADP
iajs-1618	17	40	d	d	NOUN
iajs-1618	17	41	”	"	PUNCT
iajs-1618	17	42	[	[	X
iajs-1618	17	43	3	3	NUM
iajs-1618	17	44	]	]	PUNCT
iajs-1618	17	45	,	,	PUNCT
iajs-1618	17	46	while	while	SCONJ
iajs-1618	17	47	“	"	PUNCT
iajs-1618	17	48	a	a	DET
iajs-1618	17	49	submodule	submodule	PROPN
iajs-1618	17	50	d	d	PROPN
iajs-1618	17	51	of	of	ADP
iajs-1618	17	52	an	an	DET
iajs-1618	17	53	lmodule	lmodule	NOUN
iajs-1618	17	54	w	w	NOUN
iajs-1618	17	55	is	be	AUX
iajs-1618	17	56	said	say	VERB
iajs-1618	17	57	to	to	PART
iajs-1618	17	58	be	be	AUX
iajs-1618	17	59	a	a	DET
iajs-1618	17	60	closed	closed	ADJ
iajs-1618	17	61	submodule	submodule	NOUN
iajs-1618	17	62	of	of	ADP
iajs-1618	17	63	w	w	PROPN
iajs-1618	17	64	if	if	SCONJ
iajs-1618	17	65	d	d	PROPN
iajs-1618	17	66	has	have	VERB
iajs-1618	17	67	no	no	DET
iajs-1618	17	68	proper	proper	ADJ
iajs-1618	17	69	essential	essential	ADJ
iajs-1618	17	70	extension	extension	NOUN
iajs-1618	17	71	inside	inside	ADP
iajs-1618	17	72	w	w	PROPN
iajs-1618	17	73	,	,	PUNCT
iajs-1618	17	74	that	that	PRON
iajs-1618	17	75	is	be	AUX
iajs-1618	17	76	if	if	SCONJ
iajs-1618	17	77	d≤e	d≤e	PROPN
iajs-1618	17	78	h≤	h≤	AUX
iajs-1618	17	79	w	w	ADV
iajs-1618	17	80	then	then	ADV
iajs-1618	18	1	d	d	X
iajs-1618	18	2	=	=	NOUN
iajs-1618	18	3	h	h	NOUN
iajs-1618	18	4	”	"	PUNCT
iajs-1618	18	5	[	[	X
iajs-1618	18	6	3	3	NUM
iajs-1618	18	7	]	]	PUNCT
iajs-1618	18	8	.	.	PUNCT
iajs-1618	19	1	as	as	ADP
iajs-1618	19	2	a	a	DET
iajs-1618	19	3	generalization	generalization	NOUN
iajs-1618	19	4	of	of	ADP
iajs-1618	19	5	essential	essential	ADJ
iajs-1618	19	6	submodules	submodule	NOUN
iajs-1618	19	7	,	,	PUNCT
iajs-1618	19	8	in	in	ADP
iajs-1618	19	9	[	[	X
iajs-1618	19	10	4	4	X
iajs-1618	19	11	]	]	PUNCT
iajs-1618	19	12	“	"	PUNCT
iajs-1618	19	13	zhou	zhou	PROPN
iajs-1618	19	14	and	and	CCONJ
iajs-1618	19	15	zhang	zhang	PROPN
iajs-1618	19	16	”	"	PUNCT
iajs-1618	19	17	introduced	introduce	VERB
iajs-1618	19	18	the	the	DET
iajs-1618	19	19	concept	concept	NOUN
iajs-1618	19	20	of	of	ADP
iajs-1618	19	21	sessential	sessential	ADJ
iajs-1618	19	22	submodule	submodule	NOUN
iajs-1618	19	23	,	,	PUNCT
iajs-1618	19	24	where	where	SCONJ
iajs-1618	19	25	“	"	PUNCT
iajs-1618	19	26	a	a	DET
iajs-1618	19	27	submodule	submodule	PROPN
iajs-1618	19	28	d	d	PROPN
iajs-1618	19	29	of	of	ADP
iajs-1618	19	30	an	an	DET
iajs-1618	19	31	l	l	NOUN
iajs-1618	19	32	-	-	NOUN
iajs-1618	19	33	module	module	NOUN
iajs-1618	19	34	w	w	NOUN
iajs-1618	19	35	is	be	AUX
iajs-1618	19	36	said	say	VERB
iajs-1618	19	37	to	to	PART
iajs-1618	19	38	be	be	AUX
iajs-1618	19	39	an	an	DET
iajs-1618	19	40	s	s	NOUN
iajs-1618	19	41	-essential	-essential	ADJ
iajs-1618	19	42	submodule	submodule	NOUN
iajs-1618	19	43	of	of	ADP
iajs-1618	19	44	w	w	PROPN
iajs-1618	19	45	denoted	denote	VERB
iajs-1618	19	46	by	by	ADP
iajs-1618	19	47	d≤se	d≤se	NOUN
iajs-1618	19	48	w	w	NOUN
iajs-1618	19	49	if	if	SCONJ
iajs-1618	19	50	d∩h=0	d∩h=0	PROPN
iajs-1618	19	51	with	with	ADP
iajs-1618	19	52	h	h	NOUN
iajs-1618	19	53	is	be	AUX
iajs-1618	19	54	a	a	DET
iajs-1618	19	55	small	small	ADJ
iajs-1618	19	56	submodule	submodule	NOUN
iajs-1618	19	57	of	of	ADP
iajs-1618	19	58	w	w	PROPN
iajs-1618	19	59	implies	imply	VERB
iajs-1618	19	60	h=	h=	NOUN
iajs-1618	19	61	0	0	X
iajs-1618	19	62	.	.	PUNCT
iajs-1618	20	1	“	"	PUNCT
iajs-1618	20	2	mehdi	mehdi	ADJ
iajs-1618	20	3	sadiq	sadiq	NOUN
iajs-1618	20	4	and	and	CCONJ
iajs-1618	20	5	faten	faten	NOUN
iajs-1618	20	6	”	"	PUNCT
iajs-1618	20	7	in	in	ADP
iajs-1618	20	8	[	[	X
iajs-1618	20	9	1	1	NUM
iajs-1618	20	10	]	]	PUNCT
iajs-1618	20	11	introduced	introduce	VERB
iajs-1618	20	12	and	and	CCONJ
iajs-1618	20	13	studied	study	VERB
iajs-1618	20	14	the	the	DET
iajs-1618	20	15	notion	notion	NOUN
iajs-1618	20	16	of	of	ADP
iajs-1618	20	17	sclosed	sclose	VERB
iajs-1618	20	18	submodules	submodule	NOUN
iajs-1618	20	19	,	,	PUNCT
iajs-1618	20	20	“	"	PUNCT
iajs-1618	20	21	a	a	DET
iajs-1618	20	22	submodule	submodule	PROPN
iajs-1618	20	23	d	d	PROPN
iajs-1618	20	24	of	of	ADP
iajs-1618	20	25	an	an	DET
iajs-1618	20	26	lmodule	lmodule	NOUN
iajs-1618	20	27	w	w	NOUN
iajs-1618	20	28	is	be	AUX
iajs-1618	20	29	called	call	VERB
iajs-1618	20	30	s	s	PART
iajs-1618	20	31	-	-	PUNCT
iajs-1618	20	32	closed	closed	ADJ
iajs-1618	20	33	submodule	submodule	NOUN
iajs-1618	20	34	denoted	denote	VERB
iajs-1618	20	35	by	by	ADP
iajs-1618	20	36	d≤sc	d≤sc	PROPN
iajs-1618	20	37	w	w	PROPN
iajs-1618	20	38	,	,	PUNCT
iajs-1618	20	39	if	if	SCONJ
iajs-1618	20	40	d	d	PROPN
iajs-1618	20	41	has	have	VERB
iajs-1618	20	42	no	no	DET
iajs-1618	20	43	proper	proper	ADJ
iajs-1618	20	44	sessential	sessential	ADJ
iajs-1618	20	45	extension	extension	NOUN
iajs-1618	20	46	in	in	ADP
iajs-1618	20	47	w	w	PROPN
iajs-1618	20	48	,	,	PUNCT
iajs-1618	20	49	that	that	ADV
iajs-1618	20	50	is	is	ADV
iajs-1618	20	51	,	,	PUNCT
iajs-1618	20	52	whenever	whenever	SCONJ
iajs-1618	20	53	d	d	PROPN
iajs-1618	20	54	≤	≤	NUM
iajs-1618	20	55	w	w	ADP
iajs-1618	20	56	such	such	ADJ
iajs-1618	20	57	that	that	SCONJ
iajs-1618	20	58	d≤se	d≤se	PRON
iajs-1618	20	59	h≤	h≤	NOUN
iajs-1618	20	60	w	w	NOUN
iajs-1618	20	61	,	,	PUNCT
iajs-1618	20	62	then	then	ADV
iajs-1618	20	63	d=	d=	ADJ
iajs-1618	20	64	h.	h.	NOUN
iajs-1618	20	65	this	this	DET
iajs-1618	20	66	paper	paper	NOUN
iajs-1618	20	67	consists	consist	VERB
iajs-1618	20	68	of	of	ADP
iajs-1618	20	69	two	two	NUM
iajs-1618	20	70	sections	section	NOUN
iajs-1618	20	71	.	.	PUNCT
iajs-1618	21	1	in	in	ADP
iajs-1618	21	2	section	section	NOUN
iajs-1618	21	3	one	one	NUM
iajs-1618	21	4	,	,	PUNCT
iajs-1618	21	5	we	we	PRON
iajs-1618	21	6	give	give	VERB
iajs-1618	21	7	some	some	DET
iajs-1618	21	8	other	other	ADJ
iajs-1618	21	9	properties	property	NOUN
iajs-1618	21	10	and	and	CCONJ
iajs-1618	21	11	examples	example	NOUN
iajs-1618	21	12	of	of	ADP
iajs-1618	21	13	s	s	NOUN
iajs-1618	21	14	-	-	ADJ
iajs-1618	21	15	essential	essential	ADJ
iajs-1618	21	16	submodules	submodule	NOUN
iajs-1618	21	17	and	and	CCONJ
iajs-1618	21	18	sclosed	sclose	VERB
iajs-1618	21	19	submodules	submodule	NOUN
iajs-1618	21	20	.	.	PUNCT
iajs-1618	22	1	in	in	ADP
iajs-1618	22	2	section	section	NOUN
iajs-1618	22	3	two	two	NUM
iajs-1618	22	4	,	,	PUNCT
iajs-1618	22	5	we	we	PRON
iajs-1618	22	6	study	study	VERB
iajs-1618	22	7	chain	chain	NOUN
iajs-1618	22	8	conditions	condition	NOUN
iajs-1618	22	9	(	(	PUNCT
iajs-1618	22	10	that	that	PRON
iajs-1618	22	11	is	be	AUX
iajs-1618	22	12	ascending	ascend	VERB
iajs-1618	22	13	and	and	CCONJ
iajs-1618	22	14	descending	descend	VERB
iajs-1618	22	15	chain	chain	NOUN
iajs-1618	22	16	conditions	condition	NOUN
iajs-1618	22	17	)	)	PUNCT
iajs-1618	22	18	on	on	ADP
iajs-1618	22	19	s	s	NOUN
iajs-1618	22	20	-	-	PUNCT
iajs-1618	22	21	closed	closed	ADJ
iajs-1618	22	22	submodules	submodule	NOUN
iajs-1618	22	23	.	.	PUNCT
iajs-1618	23	1	1	1	X
iajs-1618	23	2	.	.	X
iajs-1618	23	3	s	s	X
iajs-1618	23	4	-	-	ADJ
iajs-1618	23	5	essential	essential	ADJ
iajs-1618	23	6	submodules	submodule	NOUN
iajs-1618	23	7	and	and	CCONJ
iajs-1618	23	8	sclosed	sclose	VERB
iajs-1618	23	9	submodules	submodule	NOUN
iajs-1618	23	10	definition	definition	NOUN
iajs-1618	23	11	1	1	NUM
iajs-1618	23	12	.	.	PUNCT
iajs-1618	23	13	1	1	NUM
iajs-1618	23	14	:	:	PUNCT
iajs-1618	24	1	[	[	X
iajs-1618	24	2	4	4	X
iajs-1618	24	3	]	]	PUNCT
iajs-1618	24	4	a	a	DET
iajs-1618	24	5	submodule	submodule	NOUN
iajs-1618	24	6	d	d	PROPN
iajs-1618	24	7	of	of	ADP
iajs-1618	24	8	an	an	DET
iajs-1618	24	9	l	l	NOUN
iajs-1618	24	10	-	-	NOUN
iajs-1618	24	11	module	module	NOUN
iajs-1618	24	12	w	w	NOUN
iajs-1618	24	13	is	be	AUX
iajs-1618	24	14	said	say	VERB
iajs-1618	24	15	to	to	PART
iajs-1618	24	16	be	be	AUX
iajs-1618	24	17	an	an	DET
iajs-1618	24	18	sessential	sessential	ADJ
iajs-1618	24	19	submodule	submodule	NOUN
iajs-1618	24	20	of	of	ADP
iajs-1618	24	21	w	w	PROPN
iajs-1618	24	22	denoted	denote	VERB
iajs-1618	24	23	by	by	ADP
iajs-1618	24	24	d	d	PROPN
iajs-1618	24	25	≤se	≤se	PROPN
iajs-1618	24	26	w	w	ADP
iajs-1618	24	27	if	if	SCONJ
iajs-1618	24	28	d∩h=	d∩h=	PROPN
iajs-1618	24	29	0	0	NUM
iajs-1618	24	30	with	with	SCONJ
iajs-1618	24	31	h	h	NOUN
iajs-1618	24	32	is	be	AUX
iajs-1618	24	33	a	a	DET
iajs-1618	24	34	small	small	ADJ
iajs-1618	24	35	submodule	submodule	NOUN
iajs-1618	24	36	of	of	ADP
iajs-1618	24	37	w	w	PROPN
iajs-1618	24	38	implies	imply	VERB
iajs-1618	24	39	h=	h=	NOUN
iajs-1618	24	40	0	0	X
iajs-1618	24	41	.	.	PUNCT
iajs-1618	25	1	remarks	remark	NOUN
iajs-1618	25	2	and	and	CCONJ
iajs-1618	25	3	examples	example	NOUN
iajs-1618	25	4	1	1	NUM
iajs-1618	25	5	.	.	SYM
iajs-1618	25	6	2	2	NUM
iajs-1618	25	7	:	:	SYM
iajs-1618	25	8	1	1	NUM
iajs-1618	25	9	)	)	PUNCT
iajs-1618	25	10	its	its	PUNCT
iajs-1618	25	11	clear	clear	ADJ
iajs-1618	25	12	that	that	SCONJ
iajs-1618	25	13	every	every	DET
iajs-1618	25	14	essential	essential	ADJ
iajs-1618	25	15	submodule	submodule	NOUN
iajs-1618	25	16	is	be	AUX
iajs-1618	25	17	an	an	DET
iajs-1618	25	18	sessential	sessential	ADJ
iajs-1618	25	19	submodule	submodule	NOUN
iajs-1618	25	20	,	,	PUNCT
iajs-1618	25	21	hence	hence	ADV
iajs-1618	25	22	every	every	DET
iajs-1618	25	23	submodule	submodule	NOUN
iajs-1618	25	24	of	of	ADP
iajs-1618	25	25	z	z	PROPN
iajs-1618	25	26	-module	-module	PROPN
iajs-1618	25	27	z	z	PROPN
iajs-1618	25	28	,	,	PUNCT
iajs-1618	25	29	(	(	PUNCT
iajs-1618	25	30	where	where	SCONJ
iajs-1618	25	31	p	p	NOUN
iajs-1618	25	32	is	be	AUX
iajs-1618	25	33	a	a	DET
iajs-1618	25	34	prime	prime	ADJ
iajs-1618	25	35	number	number	NOUN
iajs-1618	25	36	,	,	PUNCT
iajs-1618	25	37	nz+	nz+	NOUN
iajs-1618	25	38	)	)	PUNCT
iajs-1618	25	39	is	be	AUX
iajs-1618	25	40	sessential	sessential	ADJ
iajs-1618	25	41	.	.	PUNCT
iajs-1618	26	1	2	2	X
iajs-1618	26	2	)	)	PUNCT
iajs-1618	26	3	if	if	SCONJ
iajs-1618	26	4	w	w	NOUN
iajs-1618	26	5	is	be	AUX
iajs-1618	26	6	an	an	DET
iajs-1618	26	7	lmodule	lmodule	NOUN
iajs-1618	26	8	such	such	ADJ
iajs-1618	26	9	that	that	SCONJ
iajs-1618	26	10	(	(	PUNCT
iajs-1618	26	11	0	0	NUM
iajs-1618	26	12	)	)	PUNCT
iajs-1618	26	13	is	be	AUX
iajs-1618	26	14	the	the	DET
iajs-1618	26	15	only	only	ADJ
iajs-1618	26	16	small	small	ADJ
iajs-1618	26	17	submodule	submodule	NOUN
iajs-1618	26	18	then	then	ADV
iajs-1618	26	19	every	every	DET
iajs-1618	26	20	submodule	submodule	NOUN
iajs-1618	26	21	is	be	AUX
iajs-1618	26	22	sessential	sessential	ADJ
iajs-1618	26	23	submodule	submodule	NOUN
iajs-1618	26	24	in	in	ADP
iajs-1618	26	25	w.	w.	PROPN
iajs-1618	26	26	in	in	ADP
iajs-1618	26	27	particular	particular	ADJ
iajs-1618	26	28	,	,	PUNCT
iajs-1618	26	29	for	for	ADP
iajs-1618	26	30	each	each	DET
iajs-1618	26	31	submodule	submodule	NOUN
iajs-1618	26	32	of	of	ADP
iajs-1618	26	33	semisimple	semisimple	NOUN
iajs-1618	26	34	module	module	NOUN
iajs-1618	26	35	(	(	PUNCT
iajs-1618	26	36	or	or	CCONJ
iajs-1618	26	37	free	free	ADJ
iajs-1618	26	38	z	z	NOUN
iajs-1618	26	39	˗module	˗module	NOUN
iajs-1618	26	40	)	)	PUNCT
iajs-1618	26	41	is	be	AUX
iajs-1618	26	42	sessential	sessential	ADJ
iajs-1618	26	43	.	.	PUNCT
iajs-1618	27	1	hence	hence	ADV
iajs-1618	27	2	its	its	PUNCT
iajs-1618	27	3	clear	clear	ADJ
iajs-1618	27	4	that	that	SCONJ
iajs-1618	27	5	every	every	DET
iajs-1618	27	6	submodule	submodule	NOUN
iajs-1618	27	7	of	of	ADP
iajs-1618	27	8	z	z	PROPN
iajs-1618	27	9	˗module	˗module	PROPN
iajs-1618	27	10	z6	z6	PROPN
iajs-1618	27	11	is	be	AUX
iajs-1618	27	12	sessential	sessential	ADJ
iajs-1618	27	13	,	,	PUNCT
iajs-1618	27	14	however	however	ADV
iajs-1618	27	15	they	they	PRON
iajs-1618	27	16	are	be	AUX
iajs-1618	27	17	not	not	PART
iajs-1618	27	18	essential	essential	ADJ
iajs-1618	27	19	.	.	PUNCT
iajs-1618	28	1	also	also	ADV
iajs-1618	28	2	every	every	DET
iajs-1618	28	3	submodule	submodule	NOUN
iajs-1618	28	4	of	of	ADP
iajs-1618	28	5	the	the	DET
iajs-1618	28	6	z˗	z˗	PROPN
iajs-1618	28	7	module	module	NOUN
iajs-1618	28	8	z	z	PROPN
iajs-1618	28	9			PROPN
iajs-1618	28	10	z	z	PROPN
iajs-1618	28	11	is	be	AUX
iajs-1618	28	12	sessential	sessential	ADJ
iajs-1618	28	13	submodule	submodule	NOUN
iajs-1618	28	14	.	.	PUNCT
iajs-1618	29	1	3	3	X
iajs-1618	29	2	)	)	PUNCT
iajs-1618	29	3	let	let	VERB
iajs-1618	29	4	a	a	PRON
iajs-1618	29	5	be	be	AUX
iajs-1618	29	6	a	a	DET
iajs-1618	29	7	submodule	submodule	NOUN
iajs-1618	29	8	of	of	ADP
iajs-1618	29	9	an	an	DET
iajs-1618	29	10	l	l	NOUN
iajs-1618	29	11	-module	-module	PROPN
iajs-1618	29	12	w	w	NOUN
iajs-1618	29	13	,	,	PUNCT
iajs-1618	29	14	then	then	ADV
iajs-1618	29	15	there	there	PRON
iajs-1618	29	16	exists	exist	VERB
iajs-1618	29	17	a	a	DET
iajs-1618	29	18	closed	closed	ADJ
iajs-1618	29	19	submodule	submodule	NOUN
iajs-1618	29	20	h	h	NOUN
iajs-1618	29	21	of	of	ADP
iajs-1618	29	22	w	w	ADP
iajs-1618	29	23	such	such	ADJ
iajs-1618	29	24	that	that	SCONJ
iajs-1618	29	25	a≤e	a≤e	PROPN
iajs-1618	29	26	h	h	NOUN
iajs-1618	29	27	,	,	PUNCT
iajs-1618	29	28	it	it	PRON
iajs-1618	29	29	is	be	AUX
iajs-1618	29	30	clear	clear	ADJ
iajs-1618	29	31	by	by	ADP
iajs-1618	29	32	[	[	X
iajs-1618	29	33	3	3	NUM
iajs-1618	29	34	,	,	PUNCT
iajs-1618	29	35	exc.13	exc.13	NOUN
iajs-1618	29	36	,	,	PUNCT
iajs-1618	29	37	p.20	p.20	X
iajs-1618	29	38	]	]	X
iajs-1618	29	39	,	,	PUNCT
iajs-1618	29	40	hence	hence	ADV
iajs-1618	29	41	a≤se	a≤se	PROPN
iajs-1618	29	42	h.	h.	NOUN
iajs-1618	29	43	4	4	NUM
iajs-1618	29	44	)	)	PUNCT
iajs-1618	29	45	in	in	ADP
iajs-1618	29	46	z24	z24	PROPN
iajs-1618	29	47	as	as	ADP
iajs-1618	29	48	zmodule	zmodule	NOUN
iajs-1618	29	49	,	,	PUNCT
iajs-1618	29	50	we	we	PRON
iajs-1618	29	51	have	have	VERB
iajs-1618	29	52	<	<	X
iajs-1618	29	53	̅	̅	NOUN
iajs-1618	29	54	>	>	X
iajs-1618	29	55	,	,	PUNCT
iajs-1618	29	56	<	<	X
iajs-1618	29	57	̅	̅	NOUN
iajs-1618	29	58	>	>	X
iajs-1618	29	59	,	,	PUNCT
iajs-1618	29	60	<	<	X
iajs-1618	29	61	̅	̅	NOUN
iajs-1618	29	62	>	>	X
iajs-1618	29	63	,	,	PUNCT
iajs-1618	29	64	<	<	X
iajs-1618	29	65	̅	̅	NOUN
iajs-1618	29	66	>	>	X
iajs-1618	29	67	,	,	PUNCT
iajs-1618	29	68	<	<	X
iajs-1618	29	69	̅̅̅̅	̅̅̅̅	X
iajs-1618	29	70	>	>	X
iajs-1618	29	71	,	,	PUNCT
iajs-1618	29	72	and	and	CCONJ
iajs-1618	29	73	z24	z24	PROPN
iajs-1618	29	74	are	be	AUX
iajs-1618	29	75	sessential	sessential	ADJ
iajs-1618	29	76	submodules	submodule	NOUN
iajs-1618	29	77	in	in	ADP
iajs-1618	29	78	z24	z24	PROPN
iajs-1618	29	79	,	,	PUNCT
iajs-1618	29	80	but	but	CCONJ
iajs-1618	29	81	<	<	X
iajs-1618	29	82	̅	̅	X
iajs-1618	29	83	>	>	X
iajs-1618	29	84	is	be	AUX
iajs-1618	29	85	not	not	PART
iajs-1618	29	86	since	since	SCONJ
iajs-1618	29	87	<	<	X
iajs-1618	29	88	̅>∩	̅>∩	X
iajs-1618	29	89	<	<	X
iajs-1618	29	90	̅	̅	NOUN
iajs-1618	29	91	>	>	X
iajs-1618	29	92	=	=	NOUN
iajs-1618	29	93	{	{	PUNCT
iajs-1618	29	94	0	0	NUM
iajs-1618	29	95	}	}	PUNCT
iajs-1618	29	96	while	while	SCONJ
iajs-1618	29	97	<	<	X
iajs-1618	29	98	̅	̅	X
iajs-1618	29	99	>	>	X
iajs-1618	29	100			NOUN
iajs-1618	29	101	0	0	NUM
iajs-1618	29	102	is	be	AUX
iajs-1618	29	103	a	a	DET
iajs-1618	29	104	small	small	ADJ
iajs-1618	29	105	submodule	submodule	NOUN
iajs-1618	29	106	in	in	ADP
iajs-1618	29	107	z24	z24	PROPN
iajs-1618	29	108	.	.	PROPN
iajs-1618	29	109	5	5	NUM
iajs-1618	29	110	)	)	PUNCT
iajs-1618	29	111	for	for	ADP
iajs-1618	29	112	a	a	DET
iajs-1618	29	113	nonzero	nonzero	ADJ
iajs-1618	29	114	r	r	NOUN
iajs-1618	29	115	-	-	PUNCT
iajs-1618	29	116	module	module	NOUN
iajs-1618	29	117	w	w	NOUN
iajs-1618	29	118	,	,	PUNCT
iajs-1618	29	119	w	w	PROPN
iajs-1618	29	120	≤se	≤se	PROPN
iajs-1618	29	121	w.	w.	NOUN
iajs-1618	29	122	6	6	NUM
iajs-1618	29	123	)	)	PUNCT
iajs-1618	29	124	the	the	DET
iajs-1618	29	125	two	two	NUM
iajs-1618	29	126	concepts	concept	NOUN
iajs-1618	29	127	essential	essential	ADJ
iajs-1618	29	128	and	and	CCONJ
iajs-1618	29	129	sessential	sessential	ADJ
iajs-1618	29	130	are	be	AUX
iajs-1618	29	131	coincide	coincide	VERB
iajs-1618	29	132	under	under	ADP
iajs-1618	29	133	the	the	DET
iajs-1618	29	134	class	class	NOUN
iajs-1618	29	135	of	of	ADP
iajs-1618	29	136	hollow	hollow	ADJ
iajs-1618	29	137	modules	module	NOUN
iajs-1618	29	138	,	,	PUNCT
iajs-1618	29	139	by[1	by[1	NOUN
iajs-1618	29	140	,	,	PUNCT
iajs-1618	29	141	remark	remark	NOUN
iajs-1618	29	142	(	(	PUNCT
iajs-1618	29	143	2.3	2.3	NUM
iajs-1618	29	144	)	)	PUNCT
iajs-1618	29	145	]	]	PUNCT
iajs-1618	29	146	,	,	PUNCT
iajs-1618	29	147	where	where	SCONJ
iajs-1618	29	148	“	"	PUNCT
iajs-1618	29	149	an	an	DET
iajs-1618	29	150	l	l	NOUN
iajs-1618	29	151	-module	-module	PROPN
iajs-1618	29	152	w	w	NOUN
iajs-1618	29	153	is	be	AUX
iajs-1618	29	154	called	call	VERB
iajs-1618	29	155	hollow	hollow	ADJ
iajs-1618	29	156	if	if	SCONJ
iajs-1618	29	157	every	every	DET
iajs-1618	29	158	proper	proper	ADJ
iajs-1618	29	159	submodule	submodule	NOUN
iajs-1618	29	160	of	of	ADP
iajs-1618	29	161	w	w	PROPN
iajs-1618	29	162	is	be	AUX
iajs-1618	29	163	small	small	ADJ
iajs-1618	29	164	”	"	PUNCT
iajs-1618	29	165	.	.	PUNCT
iajs-1618	30	1	[	[	X
iajs-1618	30	2	5	5	NUM
iajs-1618	30	3	]	]	PUNCT
iajs-1618	30	4	proposition	proposition	NOUN
iajs-1618	30	5	1	1	NUM
iajs-1618	30	6	.	.	PUNCT
iajs-1618	30	7	3	3	NUM
iajs-1618	30	8	:	:	PUNCT
iajs-1618	30	9	let	let	AUX
iajs-1618	30	10	w	w	PART
iajs-1618	30	11	be	be	AUX
iajs-1618	30	12	an	an	DET
iajs-1618	30	13	l	l	NOUN
iajs-1618	30	14	-	-	NOUN
iajs-1618	30	15	module	module	NOUN
iajs-1618	30	16	and	and	CCONJ
iajs-1618	30	17	let	let	VERB
iajs-1618	30	18	s	s	PRON
iajs-1618	30	19	≤se	≤se	NUM
iajs-1618	30	20	t	t	NOUN
iajs-1618	30	21	≤	≤	NUM
iajs-1618	30	22	m	m	NOUN
iajs-1618	30	23	and	and	CCONJ
iajs-1618	30	24	s	s	PRON
iajs-1618	30	25	≤se	≤se	NUM
iajs-1618	30	26	tʹ	tʹ	NOUN
iajs-1618	30	27	≤	≤	NUM
iajs-1618	30	28	w	w	NOUN
iajs-1618	30	29	,	,	PUNCT
iajs-1618	30	30	then	then	ADV
iajs-1618	30	31	s	s	VERB
iajs-1618	30	32	∩	∩	ADJ
iajs-1618	30	33	sʹ	sʹ	ADJ
iajs-1618	30	34	≤se	≤se	NUM
iajs-1618	30	35	t	t	PROPN
iajs-1618	30	36	∩	∩	X
iajs-1618	30	37	tʹ.	tʹ.	PROPN
iajs-1618	30	38	proof	proof	NOUN
iajs-1618	30	39	:	:	PUNCT
iajs-1618	30	40	let	let	VERB
iajs-1618	30	41	u	u	PRON
iajs-1618	30	42	<	<	X
iajs-1618	30	43	<	<	X
iajs-1618	30	44	t	t	NOUN
iajs-1618	30	45	∩	∩	X
iajs-1618	30	46	tʹ	tʹ	X
iajs-1618	30	47	and	and	CCONJ
iajs-1618	30	48	(	(	PUNCT
iajs-1618	30	49	s	s	X
iajs-1618	30	50	∩	∩	ADJ
iajs-1618	30	51	sʹ	sʹ	ADJ
iajs-1618	30	52	)	)	PUNCT
iajs-1618	30	53	∩	∩	NOUN
iajs-1618	30	54	u	u	NOUN
iajs-1618	30	55	=	=	SYM
iajs-1618	30	56	(	(	PUNCT
iajs-1618	30	57	0	0	NUM
iajs-1618	30	58	)	)	PUNCT
iajs-1618	30	59	,	,	PUNCT
iajs-1618	30	60	hence	hence	ADV
iajs-1618	30	61	s	s	PART
iajs-1618	30	62	∩	∩	NOUN
iajs-1618	30	63	(	(	PUNCT
iajs-1618	30	64	sʹ	sʹ	X
iajs-1618	30	65	∩	∩	ADJ
iajs-1618	30	66	u	u	NOUN
iajs-1618	30	67	)	)	PUNCT
iajs-1618	30	68	=	=	SYM
iajs-1618	31	1	0	0	X
iajs-1618	31	2	.	.	PUNCT
iajs-1618	32	1	but	but	CCONJ
iajs-1618	32	2	u	u	PRON
iajs-1618	32	3	<	<	X
iajs-1618	32	4	<	<	X
iajs-1618	32	5	(	(	PUNCT
iajs-1618	32	6	t	t	PROPN
iajs-1618	32	7	∩	∩	NOUN
iajs-1618	32	8	tʹ	tʹ	CCONJ
iajs-1618	32	9	)	)	PUNCT
iajs-1618	32	10	implies	imply	VERB
iajs-1618	32	11	u	u	NOUN
iajs-1618	32	12	<	<	X
iajs-1618	32	13	<	<	X
iajs-1618	32	14	tʹ	tʹ	X
iajs-1618	32	15	and	and	CCONJ
iajs-1618	32	16	u	u	X
iajs-1618	32	17	<	<	X
iajs-1618	32	18	<	<	X
iajs-1618	32	19	t.	t.	PROPN
iajs-1618	32	20	as	as	ADP
iajs-1618	32	21	sʹ∩u	sʹ∩u	PROPN
iajs-1618	32	22	u	u	NOUN
iajs-1618	32	23	<	<	X
iajs-1618	32	24	<	<	X
iajs-1618	32	25	t	t	PROPN
iajs-1618	32	26	,	,	PUNCT
iajs-1618	32	27	then	then	ADV
iajs-1618	32	28	sʹ∩u	sʹ∩u	VERB
iajs-1618	32	29	<	<	X
iajs-1618	32	30	<	<	X
iajs-1618	32	31	t.	t.	PROPN
iajs-1618	32	32	but	but	CCONJ
iajs-1618	32	33	s	s	PROPN
iajs-1618	32	34	≤se	≤se	NUM
iajs-1618	32	35	t	t	PROPN
iajs-1618	32	36	,	,	PUNCT
iajs-1618	32	37	hence	hence	ADV
iajs-1618	32	38	sʹ∩u=	sʹ∩u=	PROPN
iajs-1618	32	39	0	0	PUNCT
iajs-1618	32	40	.	.	PUNCT
iajs-1618	33	1	it	it	PRON
iajs-1618	33	2	follows	follow	VERB
iajs-1618	33	3	that	that	SCONJ
iajs-1618	33	4	u	u	NOUN
iajs-1618	33	5	=	=	NOUN
iajs-1618	33	6	0	0	PUNCT
iajs-1618	33	7	since	since	SCONJ
iajs-1618	33	8	s	s	PRON
iajs-1618	33	9	≤se	≤se	NUM
iajs-1618	33	10	tʹ	tʹ	X
iajs-1618	33	11	and	and	CCONJ
iajs-1618	33	12	u	u	X
iajs-1618	33	13	<	<	X
iajs-1618	33	14	<	<	X
iajs-1618	33	15	tʹ.	tʹ.	PROPN
iajs-1618	33	16	the	the	DET
iajs-1618	33	17	following	following	ADJ
iajs-1618	33	18	result	result	NOUN
iajs-1618	33	19	follows	follow	VERB
iajs-1618	33	20	by	by	ADP
iajs-1618	33	21	proposition	proposition	NOUN
iajs-1618	33	22	1.3	1.3	NUM
iajs-1618	33	23	directly	directly	ADV
iajs-1618	33	24	.	.	PUNCT
iajs-1618	34	1	mathematics	mathematic	NOUN
iajs-1618	34	2	|229	|229	AUX
iajs-1618	34	3	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	NOUN
iajs-1618	34	4	7102	7102	NUM
iajs-1618	34	5	(	(	PUNCT
iajs-1618	34	6	عام	عام	PROPN
iajs-1618	34	7	3	3	NUM
iajs-1618	34	8	(	(	PUNCT
iajs-1618	34	9	العدد	العدد	PROPN
iajs-1618	34	10	)	)	PUNCT
iajs-1618	34	11	30مجلة	30مجلة	PROPN
iajs-1618	34	12	إبن	إبن	VERB
iajs-1618	34	13	الهيثم	الهيثم	ADJ
iajs-1618	34	14	للعلوم	للعلوم	NOUN
iajs-1618	34	15	الصرفة	الصرفة	NOUN
iajs-1618	35	1	و	و	PRON
iajs-1618	35	2	التطبيقية	التطبيقية	ADV
iajs-1618	35	3	المجلد	المجلد	ADV
iajs-1618	35	4	)	)	PUNCT
iajs-1618	36	1	ibn	ibn	PROPN
iajs-1618	36	2	al	al	PROPN
iajs-1618	36	3	-	-	PUNCT
iajs-1618	36	4	haitham	haitham	PROPN
iajs-1618	36	5	j.	j.	PROPN
iajs-1618	36	6	for	for	ADP
iajs-1618	36	7	pure	pure	PROPN
iajs-1618	36	8	&	&	CCONJ
iajs-1618	36	9	appl	appl	PROPN
iajs-1618	36	10	.	.	PUNCT
iajs-1618	37	1	sci	sci	PROPN
iajs-1618	37	2	.	.	PUNCT
iajs-1618	38	1	vol.03	vol.03	PROPN
iajs-1618	38	2	(	(	PUNCT
iajs-1618	38	3	3	3	NUM
iajs-1618	38	4	)	)	PUNCT
iajs-1618	38	5	2017	2017	NUM
iajs-1618	38	6	corollary	corollary	NOUN
iajs-1618	38	7	1	1	NUM
iajs-1618	38	8	.	.	PUNCT
iajs-1618	38	9	4	4	NUM
iajs-1618	38	10	:	:	PUNCT
iajs-1618	38	11	let	let	VERB
iajs-1618	38	12	c	c	X
iajs-1618	38	13	,	,	PUNCT
iajs-1618	38	14	d	d	X
iajs-1618	38	15	be	be	AUX
iajs-1618	38	16	submodules	submodule	NOUN
iajs-1618	38	17	of	of	ADP
iajs-1618	38	18	w	w	NOUN
iajs-1618	38	19	such	such	ADJ
iajs-1618	38	20	that	that	SCONJ
iajs-1618	38	21	c≤sew	c≤sew	PROPN
iajs-1618	38	22	and	and	CCONJ
iajs-1618	38	23	d≤sew	d≤sew	PROPN
iajs-1618	38	24	.	.	PUNCT
iajs-1618	39	1	then	then	ADV
iajs-1618	39	2	c∩d	c∩d	VERB
iajs-1618	39	3	≤se	≤se	NUM
iajs-1618	39	4	w	w	PROPN
iajs-1618	39	5	,	,	PUNCT
iajs-1618	39	6	[	[	X
iajs-1618	39	7	4	4	NUM
iajs-1618	39	8	,	,	PUNCT
iajs-1618	39	9	proposition	proposition	NOUN
iajs-1618	39	10	2.7(1)(b	2.7(1)(b	NOUN
iajs-1618	39	11	)	)	PUNCT
iajs-1618	39	12	]	]	PUNCT
iajs-1618	39	13	.	.	PUNCT
iajs-1618	40	1	proposition	proposition	NOUN
iajs-1618	40	2	1	1	NUM
iajs-1618	40	3	.	.	X
iajs-1618	40	4	5	5	NUM
iajs-1618	40	5	:	:	PUNCT
iajs-1618	40	6	let	let	VERB
iajs-1618	40	7	w	w	X
iajs-1618	40	8	=	=	VERB
iajs-1618	40	9	w1⊕w2	w1⊕w2	PROPN
iajs-1618	40	10	,	,	PUNCT
iajs-1618	40	11	and	and	CCONJ
iajs-1618	40	12	let	let	VERB
iajs-1618	40	13	a	a	DET
iajs-1618	40	14	=	=	X
iajs-1618	40	15	a1⊕a2	a1⊕a2	PROPN
iajs-1618	40	16	≤se	≤se	NUM
iajs-1618	40	17	b1⊕b2	b1⊕b2	PROPN
iajs-1618	40	18	,	,	PUNCT
iajs-1618	40	19	where	where	SCONJ
iajs-1618	40	20	b1	b1	NOUN
iajs-1618	40	21	≤	≤	NOUN
iajs-1618	40	22	w1	w1	NOUN
iajs-1618	40	23	and	and	CCONJ
iajs-1618	40	24	b2≤	b2≤	NUM
iajs-1618	40	25	w2	w2	NOUN
iajs-1618	40	26	.	.	PUNCT
iajs-1618	41	1	then	then	ADV
iajs-1618	41	2	a1≤se	a1≤se	VERB
iajs-1618	41	3	b1	b1	NOUN
iajs-1618	41	4	and	and	CCONJ
iajs-1618	41	5	a2≤se	a2≤se	PROPN
iajs-1618	41	6	b2	b2	NOUN
iajs-1618	41	7	.	.	PUNCT
iajs-1618	42	1	proof	proof	NOUN
iajs-1618	42	2	:	:	PUNCT
iajs-1618	42	3	suppose	suppose	VERB
iajs-1618	42	4	a1	a1	NOUN
iajs-1618	42	5	is	be	AUX
iajs-1618	42	6	not	not	PART
iajs-1618	42	7	an	an	DET
iajs-1618	42	8	s	s	NOUN
iajs-1618	42	9	-	-	ADJ
iajs-1618	42	10	essential	essential	ADJ
iajs-1618	42	11	submodule	submodule	NOUN
iajs-1618	42	12	in	in	ADP
iajs-1618	42	13	b1	b1	NOUN
iajs-1618	42	14	.	.	PUNCT
iajs-1618	43	1	so	so	ADV
iajs-1618	43	2	there	there	PRON
iajs-1618	43	3	exists	exist	VERB
iajs-1618	43	4	a	a	DET
iajs-1618	43	5	nonzero	nonzero	ADJ
iajs-1618	43	6	small	small	ADJ
iajs-1618	43	7	submodule	submodule	NOUN
iajs-1618	43	8	d1	d1	PROPN
iajs-1618	43	9	in	in	ADP
iajs-1618	43	10	b1	b1	NOUN
iajs-1618	43	11	such	such	DET
iajs-1618	43	12	that	that	DET
iajs-1618	43	13	a1∩d1=(0	a1∩d1=(0	NOUN
iajs-1618	43	14	)	)	PUNCT
iajs-1618	43	15	.	.	PUNCT
iajs-1618	44	1	since	since	SCONJ
iajs-1618	44	2	d1⊕(0	d1⊕(0	NUM
iajs-1618	44	3	)	)	PUNCT
iajs-1618	44	4	is	be	AUX
iajs-1618	44	5	a	a	DET
iajs-1618	44	6	small	small	ADJ
iajs-1618	44	7	submodule	submodule	NOUN
iajs-1618	44	8	in	in	ADP
iajs-1618	44	9	b1⊕b2	b1⊕b2	PROPN
iajs-1618	44	10	and	and	CCONJ
iajs-1618	44	11	(	(	PUNCT
iajs-1618	44	12	a1⊕a2	a1⊕a2	NOUN
iajs-1618	44	13	)	)	PUNCT
iajs-1618	44	14	∩	∩	NOUN
iajs-1618	44	15	(	(	PUNCT
iajs-1618	44	16	d1⊕(0	d1⊕(0	NOUN
iajs-1618	44	17	)	)	PUNCT
iajs-1618	44	18	)	)	PUNCT
iajs-1618	45	1	=	=	SYM
iajs-1618	45	2	(	(	PUNCT
iajs-1618	45	3	a1∩d1	a1∩d1	PROPN
iajs-1618	45	4	)	)	PUNCT
iajs-1618	45	5	⊕	⊕	PROPN
iajs-1618	45	6	(	(	PUNCT
iajs-1618	45	7	a2∩(0	a2∩(0	PROPN
iajs-1618	45	8	)	)	PUNCT
iajs-1618	45	9	)	)	PUNCT
iajs-1618	46	1	=	=	PUNCT
iajs-1618	46	2	(	(	PUNCT
iajs-1618	46	3	0	0	NUM
iajs-1618	46	4	)	)	PUNCT
iajs-1618	46	5	.	.	PUNCT
iajs-1618	47	1	then	then	ADV
iajs-1618	47	2	a1⊕a2	a1⊕a2	PROPN
iajs-1618	47	3	is	be	AUX
iajs-1618	47	4	not	not	PART
iajs-1618	47	5	an	an	DET
iajs-1618	47	6	s	s	NOUN
iajs-1618	47	7	-	-	ADJ
iajs-1618	47	8	essential	essential	ADJ
iajs-1618	47	9	submodule	submodule	NOUN
iajs-1618	47	10	in	in	ADP
iajs-1618	47	11	b1⊕b2	b1⊕b2	PROPN
iajs-1618	47	12	which	which	PRON
iajs-1618	47	13	is	be	AUX
iajs-1618	47	14	a	a	DET
iajs-1618	47	15	contradiction	contradiction	NOUN
iajs-1618	47	16	.	.	PUNCT
iajs-1618	48	1	thus	thus	ADV
iajs-1618	48	2	a1	a1	VERB
iajs-1618	48	3	≤se	≤se	NUM
iajs-1618	48	4	b1	b1	NOUN
iajs-1618	48	5	and	and	CCONJ
iajs-1618	48	6	by	by	ADP
iajs-1618	48	7	the	the	DET
iajs-1618	48	8	same	same	ADJ
iajs-1618	48	9	way	way	NOUN
iajs-1618	48	10	of	of	ADP
iajs-1618	48	11	proof	proof	NOUN
iajs-1618	48	12	that	that	PRON
iajs-1618	48	13	a2	a2	PROPN
iajs-1618	48	14	≤se	≤se	NUM
iajs-1618	48	15	b2	b2	NOUN
iajs-1618	48	16	.	.	PUNCT
iajs-1618	49	1	proposition	proposition	NOUN
iajs-1618	49	2	1	1	NUM
iajs-1618	49	3	.	.	PUNCT
iajs-1618	49	4	6	6	NUM
iajs-1618	49	5	:	:	PUNCT
iajs-1618	49	6	let	let	VERB
iajs-1618	49	7	w	w	PART
iajs-1618	49	8	be	be	AUX
iajs-1618	49	9	a	a	DET
iajs-1618	49	10	faithful	faithful	ADJ
iajs-1618	49	11	multiplication	multiplication	NOUN
iajs-1618	49	12	finitely	finitely	ADV
iajs-1618	49	13	generated	generate	VERB
iajs-1618	49	14	(	(	PUNCT
iajs-1618	49	15	denoted	denote	VERB
iajs-1618	49	16	by	by	ADP
iajs-1618	49	17	fmfg	fmfg	ADJ
iajs-1618	49	18	)	)	PUNCT
iajs-1618	49	19	lmodule	lmodule	NOUN
iajs-1618	49	20	,	,	PUNCT
iajs-1618	49	21	and	and	CCONJ
iajs-1618	49	22	u	u	PRON
iajs-1618	49	23	a	a	DET
iajs-1618	49	24	submodule	submodule	NOUN
iajs-1618	49	25	of	of	ADP
iajs-1618	49	26	w.	w.	PROPN
iajs-1618	49	27	then	then	ADV
iajs-1618	49	28	u	u	PRON
iajs-1618	49	29	≤se	≤se	PRON
iajs-1618	49	30	w	w	NOUN
iajs-1618	49	31	if	if	SCONJ
iajs-1618	50	1	and	and	CCONJ
iajs-1618	50	2	only	only	ADV
iajs-1618	50	3	if	if	SCONJ
iajs-1618	50	4	there	there	PRON
iajs-1618	50	5	exists	exist	VERB
iajs-1618	50	6	an	an	DET
iajs-1618	50	7	s	s	NOUN
iajs-1618	50	8	-essential	-essential	ADJ
iajs-1618	50	9	ideal	ideal	ADJ
iajs-1618	50	10	e	e	NOUN
iajs-1618	50	11	of	of	ADP
iajs-1618	50	12	l	l	NOUN
iajs-1618	51	1	such	such	ADJ
iajs-1618	51	2	that	that	PRON
iajs-1618	51	3	u	u	NOUN
iajs-1618	51	4	=	=	X
iajs-1618	51	5	ew	ew	PROPN
iajs-1618	51	6	.	.	NOUN
iajs-1618	51	7	proof	proof	NOUN
iajs-1618	51	8	:	:	PUNCT
iajs-1618	51	9	(	(	PUNCT
iajs-1618	51	10			NOUN
iajs-1618	51	11	)	)	PUNCT
iajs-1618	51	12	let	let	VERB
iajs-1618	51	13	u≤se	u≤se	PROPN
iajs-1618	51	14	w	w	VERB
iajs-1618	51	15	.	.	PUNCT
iajs-1618	52	1	as	as	SCONJ
iajs-1618	52	2	w	w	PROPN
iajs-1618	52	3	is	be	AUX
iajs-1618	52	4	a	a	DET
iajs-1618	52	5	multiplication	multiplication	NOUN
iajs-1618	52	6	lmodule	lmodule	NOUN
iajs-1618	52	7	,	,	PUNCT
iajs-1618	52	8	so	so	ADV
iajs-1618	52	9	u=	u=	ADV
iajs-1618	52	10	ew	ew	VERB
iajs-1618	52	11	for	for	ADP
iajs-1618	52	12	some	some	DET
iajs-1618	52	13	e	e	NOUN
iajs-1618	52	14	≤	≤	X
iajs-1618	52	15	l.	l.	NOUN
iajs-1618	52	16	to	to	PART
iajs-1618	52	17	prove	prove	VERB
iajs-1618	52	18	that	that	SCONJ
iajs-1618	52	19	e	e	PROPN
iajs-1618	52	20	≤se	≤se	NUM
iajs-1618	52	21	l	l	NOUN
iajs-1618	52	22	,	,	PUNCT
iajs-1618	52	23	assume	assume	VERB
iajs-1618	52	24	j	j	PROPN
iajs-1618	52	25	is	be	AUX
iajs-1618	52	26	a	a	DET
iajs-1618	52	27	small	small	ADJ
iajs-1618	52	28	ideal	ideal	NOUN
iajs-1618	52	29	of	of	ADP
iajs-1618	52	30	l	l	NOUN
iajs-1618	52	31	and	and	CCONJ
iajs-1618	52	32	e	e	PROPN
iajs-1618	52	33	∩	∩	X
iajs-1618	52	34	j	j	PROPN
iajs-1618	52	35	=	=	SYM
iajs-1618	52	36	0	0	PROPN
iajs-1618	52	37	,	,	PUNCT
iajs-1618	52	38	hence	hence	ADV
iajs-1618	52	39	(	(	PUNCT
iajs-1618	52	40	e	e	X
iajs-1618	52	41	∩	∩	X
iajs-1618	52	42	j	j	PROPN
iajs-1618	52	43	)	)	PUNCT
iajs-1618	52	44	w	w	PROPN
iajs-1618	52	45	=	=	NOUN
iajs-1618	52	46	0	0	PROPN
iajs-1618	52	47	.	.	PUNCT
iajs-1618	53	1	then	then	ADV
iajs-1618	53	2	by	by	ADP
iajs-1618	53	3	[	[	X
iajs-1618	53	4	6	6	NUM
iajs-1618	53	5	,	,	PUNCT
iajs-1618	53	6	th	th	X
iajs-1618	53	7	.	.	PUNCT
iajs-1618	53	8	1.6(i	1.6(i	NUM
iajs-1618	53	9	)	)	PUNCT
iajs-1618	53	10	,	,	PUNCT
iajs-1618	53	11	p.	p.	NOUN
iajs-1618	53	12	759	759	NUM
iajs-1618	53	13	]	]	X
iajs-1618	53	14	ew	ew	PROPN
iajs-1618	53	15	∩	∩	PROPN
iajs-1618	53	16	jw	jw	PROPN
iajs-1618	53	17	=	=	SYM
iajs-1618	53	18	0	0	PROPN
iajs-1618	53	19	,	,	PUNCT
iajs-1618	53	20	that	that	PRON
iajs-1618	53	21	is	be	AUX
iajs-1618	53	22	u	u	NOUN
iajs-1618	53	23	∩	∩	NOUN
iajs-1618	53	24	jw	jw	NOUN
iajs-1618	53	25	=	=	NOUN
iajs-1618	53	26	0	0	PROPN
iajs-1618	53	27	.	.	PUNCT
iajs-1618	54	1	but	but	CCONJ
iajs-1618	54	2	by	by	ADP
iajs-1618	54	3	[	[	X
iajs-1618	54	4	8	8	NUM
iajs-1618	54	5	,	,	PUNCT
iajs-1618	54	6	prop.1.1.8	prop.1.1.8	PROPN
iajs-1618	54	7	]	]	X
iajs-1618	54	8	jw	jw	PROPN
iajs-1618	54	9	is	be	AUX
iajs-1618	54	10	a	a	DET
iajs-1618	54	11	small	small	ADJ
iajs-1618	54	12	submodule	submodule	NOUN
iajs-1618	54	13	of	of	ADP
iajs-1618	54	14	w	w	PROPN
iajs-1618	54	15	and	and	CCONJ
iajs-1618	54	16	u	u	PRON
iajs-1618	54	17	≤se	≤se	PROPN
iajs-1618	54	18	w	w	PROPN
iajs-1618	54	19	,	,	PUNCT
iajs-1618	54	20	so	so	ADV
iajs-1618	54	21	jw	jw	PROPN
iajs-1618	54	22	=	=	NOUN
iajs-1618	55	1	0	0	PROPN
iajs-1618	55	2	.	.	PUNCT
iajs-1618	56	1	hence	hence	ADV
iajs-1618	56	2	j=	j=	VERB
iajs-1618	56	3	0	0	PUNCT
iajs-1618	57	1	(	(	PUNCT
iajs-1618	57	2	since	since	SCONJ
iajs-1618	57	3	w	w	PROPN
iajs-1618	57	4	is	be	AUX
iajs-1618	57	5	a	a	DET
iajs-1618	57	6	faithful	faithful	ADJ
iajs-1618	57	7	module	module	NOUN
iajs-1618	57	8	)	)	PUNCT
iajs-1618	57	9	.	.	PUNCT
iajs-1618	58	1	thus	thus	ADV
iajs-1618	58	2	e	e	X
iajs-1618	58	3	≤se	≤se	PROPN
iajs-1618	58	4	l.	l.	PROPN
iajs-1618	58	5	(	(	PUNCT
iajs-1618	58	6			NOUN
iajs-1618	58	7	)	)	PUNCT
iajs-1618	58	8	to	to	PART
iajs-1618	58	9	prove	prove	VERB
iajs-1618	58	10	u	u	PRON
iajs-1618	58	11	≤se	≤se	PROPN
iajs-1618	58	12	w.	w.	PROPN
iajs-1618	58	13	assume	assume	VERB
iajs-1618	58	14	v	v	NUM
iajs-1618	58	15	is	be	AUX
iajs-1618	58	16	a	a	DET
iajs-1618	58	17	small	small	ADJ
iajs-1618	58	18	submodule	submodule	NOUN
iajs-1618	58	19	of	of	ADP
iajs-1618	58	20	w	w	PROPN
iajs-1618	58	21	,	,	PUNCT
iajs-1618	58	22	hence	hence	ADV
iajs-1618	58	23	v	v	NOUN
iajs-1618	58	24	=	=	SYM
iajs-1618	58	25	jw	jw	PROPN
iajs-1618	58	26	for	for	ADP
iajs-1618	58	27	some	some	DET
iajs-1618	58	28	j	j	NOUN
iajs-1618	58	29	<	<	X
iajs-1618	58	30	<	<	X
iajs-1618	58	31	l.	l.	X
iajs-1618	58	32	if	if	SCONJ
iajs-1618	58	33	u	u	PROPN
iajs-1618	58	34	∩	∩	NOUN
iajs-1618	58	35	v	v	NOUN
iajs-1618	58	36	=	=	SYM
iajs-1618	58	37	0	0	NUM
iajs-1618	58	38	,	,	PUNCT
iajs-1618	58	39	then	then	ADV
iajs-1618	58	40	ew	ew	PROPN
iajs-1618	58	41	∩	∩	PROPN
iajs-1618	58	42	jw	jw	PROPN
iajs-1618	58	43	=	=	PUNCT
iajs-1618	58	44	0	0	PROPN
iajs-1618	59	1	and	and	CCONJ
iajs-1618	59	2	so	so	ADV
iajs-1618	59	3	(	(	PUNCT
iajs-1618	59	4	e	e	PROPN
iajs-1618	59	5	∩	∩	PROPN
iajs-1618	59	6	j	j	PROPN
iajs-1618	59	7	)	)	PUNCT
iajs-1618	59	8	w	w	PROPN
iajs-1618	60	1	=	=	NOUN
iajs-1618	60	2	0	0	X
iajs-1618	60	3	.	.	PUNCT
iajs-1618	61	1	hence	hence	ADV
iajs-1618	61	2	e	e	PROPN
iajs-1618	61	3	∩	∩	NOUN
iajs-1618	61	4	j	j	PROPN
iajs-1618	62	1	=	=	SYM
iajs-1618	62	2	0	0	PROPN
iajs-1618	63	1	since	since	SCONJ
iajs-1618	63	2	w	w	PROPN
iajs-1618	63	3	is	be	AUX
iajs-1618	63	4	faithful	faithful	ADJ
iajs-1618	63	5	.	.	PUNCT
iajs-1618	64	1	thus	thus	ADV
iajs-1618	64	2	j=	j=	VERB
iajs-1618	64	3	0	0	PUNCT
iajs-1618	65	1	because	because	SCONJ
iajs-1618	65	2	e	e	PROPN
iajs-1618	65	3	≤se	≤se	PROPN
iajs-1618	65	4	l.	l.	NOUN
iajs-1618	65	5	it	it	PRON
iajs-1618	65	6	follows	follow	VERB
iajs-1618	65	7	that	that	DET
iajs-1618	65	8	v=	v=	NOUN
iajs-1618	65	9	0	0	PUNCT
iajs-1618	65	10	and	and	CCONJ
iajs-1618	65	11	u	u	PROPN
iajs-1618	65	12	≤se	≤se	PROPN
iajs-1618	65	13	w.	w.	PROPN
iajs-1618	65	14	theorem	theorem	VERB
iajs-1618	65	15	1	1	NUM
iajs-1618	65	16	.	.	PUNCT
iajs-1618	65	17	7	7	NUM
iajs-1618	65	18	:	:	PUNCT
iajs-1618	65	19	let	let	VERB
iajs-1618	65	20	w	w	PART
iajs-1618	65	21	be	be	AUX
iajs-1618	65	22	a	a	DET
iajs-1618	65	23	fmfg	fmfg	ADJ
iajs-1618	65	24	lmodule	lmodule	NOUN
iajs-1618	65	25	.	.	PUNCT
iajs-1618	66	1	then	then	ADV
iajs-1618	66	2	i≤se	i≤se	VERB
iajs-1618	66	3	j≤	j≤	NOUN
iajs-1618	66	4	l	l	NOUN
iajs-1618	66	5	if	if	SCONJ
iajs-1618	67	1	and	and	CCONJ
iajs-1618	67	2	only	only	ADV
iajs-1618	67	3	if	if	SCONJ
iajs-1618	67	4	iw≤se	iw≤se	PROPN
iajs-1618	67	5	jw	jw	PROPN
iajs-1618	67	6	.	.	PUNCT
iajs-1618	67	7	proof	proof	PROPN
iajs-1618	67	8	:	:	PUNCT
iajs-1618	67	9	(	(	PUNCT
iajs-1618	67	10			NOUN
iajs-1618	67	11	)	)	PUNCT
iajs-1618	67	12	let	let	VERB
iajs-1618	67	13	u	u	PRON
iajs-1618	67	14	be	be	AUX
iajs-1618	67	15	a	a	DET
iajs-1618	67	16	small	small	ADJ
iajs-1618	67	17	submodule	submodule	NOUN
iajs-1618	67	18	in	in	ADP
iajs-1618	67	19	jw≤	jw≤	NOUN
iajs-1618	67	20	w	w	NOUN
iajs-1618	67	21	,	,	PUNCT
iajs-1618	67	22	so	so	ADV
iajs-1618	67	23	u≤	u≤	PROPN
iajs-1618	67	24	w.	w.	NOUN
iajs-1618	67	25	thus	thus	ADV
iajs-1618	67	26	u=	u=	ADV
iajs-1618	67	27	kw	kw	VERB
iajs-1618	67	28	for	for	ADP
iajs-1618	67	29	some	some	DET
iajs-1618	67	30	k≤	k≤	PROPN
iajs-1618	67	31	l.	l.	PROPN
iajs-1618	67	32	as	as	ADP
iajs-1618	67	33	kw≤jw	kw≤jw	PROPN
iajs-1618	67	34	then	then	ADV
iajs-1618	67	35	k≤	k≤	PROPN
iajs-1618	67	36	j	j	PROPN
iajs-1618	67	37	,	,	PUNCT
iajs-1618	67	38	by	by	ADP
iajs-1618	67	39	[	[	X
iajs-1618	67	40	6	6	NUM
iajs-1618	67	41	,	,	PUNCT
iajs-1618	67	42	th.3.1	th.3.1	NOUN
iajs-1618	67	43	]	]	PUNCT
iajs-1618	67	44	to	to	PART
iajs-1618	67	45	prove	prove	VERB
iajs-1618	67	46	k	k	PROPN
iajs-1618	67	47	is	be	AUX
iajs-1618	67	48	a	a	DET
iajs-1618	67	49	small	small	ADJ
iajs-1618	67	50	submodule	submodule	NOUN
iajs-1618	67	51	in	in	ADP
iajs-1618	67	52	j	j	PROPN
iajs-1618	67	53	,	,	PUNCT
iajs-1618	67	54	let	let	VERB
iajs-1618	67	55	k+h	k+h	X
iajs-1618	67	56	=	=	SYM
iajs-1618	67	57	j	j	PROPN
iajs-1618	67	58	,	,	PUNCT
iajs-1618	67	59	so	so	ADV
iajs-1618	68	1	kw	kw	INTJ
iajs-1618	69	1	+	+	CCONJ
iajs-1618	69	2	hw	hw	PROPN
iajs-1618	69	3	=	=	PROPN
iajs-1618	69	4	jw	jw	PROPN
iajs-1618	69	5	.	.	PUNCT
iajs-1618	70	1	that	that	PRON
iajs-1618	70	2	is	be	AUX
iajs-1618	70	3	hw	hw	ADP
iajs-1618	70	4	=	=	PROPN
iajs-1618	70	5	jw	jw	PROPN
iajs-1618	70	6	(	(	PUNCT
iajs-1618	70	7	since	since	SCONJ
iajs-1618	70	8	kw	kw	PROPN
iajs-1618	70	9	=	=	PROPN
iajs-1618	70	10	u	u	NOUN
iajs-1618	70	11	which	which	PRON
iajs-1618	70	12	is	be	AUX
iajs-1618	70	13	a	a	DET
iajs-1618	70	14	small	small	ADJ
iajs-1618	70	15	submodule	submodule	NOUN
iajs-1618	70	16	in	in	ADP
iajs-1618	70	17	jw	jw	PROPN
iajs-1618	70	18	)	)	PUNCT
iajs-1618	70	19	.	.	PUNCT
iajs-1618	71	1	hence	hence	ADV
iajs-1618	71	2	hw	hw	PROPN
iajs-1618	72	1	=	=	PROPN
iajs-1618	72	2	jw	jw	PROPN
iajs-1618	73	1	and	and	CCONJ
iajs-1618	73	2	so	so	ADV
iajs-1618	73	3	h	h	PROPN
iajs-1618	73	4	=	=	PROPN
iajs-1618	73	5	j	j	PROPN
iajs-1618	73	6	,	,	PUNCT
iajs-1618	73	7	that	that	PRON
iajs-1618	73	8	is	is	ADV
iajs-1618	73	9	k	k	PROPN
iajs-1618	73	10	is	be	AUX
iajs-1618	73	11	a	a	DET
iajs-1618	73	12	small	small	ADJ
iajs-1618	73	13	submodule	submodule	NOUN
iajs-1618	73	14	in	in	ADP
iajs-1618	73	15	j.	j.	PROPN
iajs-1618	74	1	if	if	SCONJ
iajs-1618	74	2	iw	iw	INTJ
iajs-1618	74	3	∩u	∩u	NOUN
iajs-1618	75	1	=	=	PUNCT
iajs-1618	75	2	0	0	NUM
iajs-1618	75	3	,	,	PUNCT
iajs-1618	75	4	then	then	ADV
iajs-1618	75	5	iw	iw	PROPN
iajs-1618	75	6	∩	∩	ADJ
iajs-1618	75	7	kw	kw	NOUN
iajs-1618	75	8	=	=	NOUN
iajs-1618	75	9	0	0	NUM
iajs-1618	75	10	.	.	PUNCT
iajs-1618	76	1	thus	thus	ADV
iajs-1618	76	2	(	(	PUNCT
iajs-1618	76	3	i∩k)w	i∩k)w	PROPN
iajs-1618	76	4	=	=	SYM
iajs-1618	76	5	0	0	NUM
iajs-1618	76	6	,	,	PUNCT
iajs-1618	76	7	so	so	ADV
iajs-1618	76	8	i∩k=0	i∩k=0	PROPN
iajs-1618	77	1	(	(	PUNCT
iajs-1618	77	2	since	since	SCONJ
iajs-1618	77	3	w	w	PROPN
iajs-1618	77	4	is	be	AUX
iajs-1618	77	5	faithful	faithful	ADJ
iajs-1618	77	6	multiplication	multiplication	NOUN
iajs-1618	77	7	)	)	PUNCT
iajs-1618	77	8	.	.	PUNCT
iajs-1618	78	1	but	but	CCONJ
iajs-1618	78	2	i≤se	i≤se	NOUN
iajs-1618	78	3	j	j	PROPN
iajs-1618	78	4	and	and	CCONJ
iajs-1618	78	5	k	k	PROPN
iajs-1618	78	6	is	be	AUX
iajs-1618	78	7	a	a	DET
iajs-1618	78	8	small	small	ADJ
iajs-1618	78	9	submodule	submodule	NOUN
iajs-1618	78	10	in	in	ADP
iajs-1618	78	11	j	j	PROPN
iajs-1618	78	12	,	,	PUNCT
iajs-1618	78	13	hence	hence	ADV
iajs-1618	78	14	k=	k=	NOUN
iajs-1618	78	15	0	0	X
iajs-1618	78	16	.	.	PUNCT
iajs-1618	79	1	it	it	PRON
iajs-1618	79	2	follows	follow	VERB
iajs-1618	79	3	u=	u=	ADV
iajs-1618	79	4	0	0	NUM
iajs-1618	79	5	,	,	PUNCT
iajs-1618	79	6	thus	thus	ADV
iajs-1618	79	7	iw≤se	iw≤se	PROPN
iajs-1618	79	8	jw	jw	PROPN
iajs-1618	79	9	.	.	PUNCT
iajs-1618	79	10	(	(	PUNCT
iajs-1618	79	11			NUM
iajs-1618	79	12	)	)	PUNCT
iajs-1618	79	13	if	if	SCONJ
iajs-1618	79	14	iw≤se	iw≤se	PROPN
iajs-1618	79	15	jw	jw	VERB
iajs-1618	79	16	to	to	PART
iajs-1618	79	17	prove	prove	VERB
iajs-1618	79	18	i≤se	i≤se	NOUN
iajs-1618	79	19	j≤	j≤	PROPN
iajs-1618	79	20	l.	l.	PROPN
iajs-1618	79	21	let	let	VERB
iajs-1618	79	22	k	k	PROPN
iajs-1618	79	23	be	be	AUX
iajs-1618	79	24	a	a	DET
iajs-1618	79	25	small	small	ADJ
iajs-1618	79	26	submodule	submodule	NOUN
iajs-1618	79	27	of	of	ADP
iajs-1618	79	28	j.	j.	PROPN
iajs-1618	79	29	assume	assume	VERB
iajs-1618	79	30	i∩k	i∩k	NOUN
iajs-1618	79	31	=	=	SYM
iajs-1618	79	32	0	0	NUM
iajs-1618	79	33	,	,	PUNCT
iajs-1618	79	34	then	then	ADV
iajs-1618	79	35	(	(	PUNCT
iajs-1618	79	36	i∩k	i∩k	NOUN
iajs-1618	79	37	)	)	PUNCT
iajs-1618	79	38	w	w	NOUN
iajs-1618	80	1	=	=	NOUN
iajs-1618	80	2	0	0	NUM
iajs-1618	80	3	,	,	PUNCT
iajs-1618	80	4	so	so	ADV
iajs-1618	80	5	iw	iw	ADJ
iajs-1618	80	6	∩	∩	ADJ
iajs-1618	80	7	kw	kw	NOUN
iajs-1618	80	8	=	=	NOUN
iajs-1618	80	9	0	0	X
iajs-1618	80	10	.	.	PUNCT
iajs-1618	81	1	let	let	VERB
iajs-1618	82	1	kw	kw	PROPN
iajs-1618	82	2	+	+	PROPN
iajs-1618	82	3	h	h	NOUN
iajs-1618	82	4	=	=	SYM
iajs-1618	82	5	jw	jw	PROPN
iajs-1618	82	6	.	.	PUNCT
iajs-1618	83	1	since	since	SCONJ
iajs-1618	83	2	w	w	PROPN
iajs-1618	83	3	is	be	AUX
iajs-1618	83	4	a	a	DET
iajs-1618	83	5	multiplication	multiplication	NOUN
iajs-1618	83	6	module	module	NOUN
iajs-1618	83	7	,	,	PUNCT
iajs-1618	83	8	thus	thus	ADV
iajs-1618	83	9	h	h	NOUN
iajs-1618	83	10	=	=	SYM
iajs-1618	83	11	cw	cw	NOUN
iajs-1618	83	12	.	.	PROPN
iajs-1618	83	13	hence	hence	ADV
iajs-1618	83	14	kw	kw	PROPN
iajs-1618	83	15	+	+	PROPN
iajs-1618	83	16	cw	cw	NOUN
iajs-1618	83	17	=	=	PROPN
iajs-1618	83	18	jw	jw	PROPN
iajs-1618	83	19	.	.	PROPN
iajs-1618	84	1	since	since	SCONJ
iajs-1618	84	2	kw	kw	PROPN
iajs-1618	84	3	is	be	AUX
iajs-1618	84	4	a	a	DET
iajs-1618	84	5	small	small	ADJ
iajs-1618	84	6	submodule	submodule	NOUN
iajs-1618	84	7	in	in	ADP
iajs-1618	84	8	jw	jw	PROPN
iajs-1618	84	9	,	,	PUNCT
iajs-1618	84	10	then	then	ADV
iajs-1618	84	11	cw	cw	PROPN
iajs-1618	84	12	=	=	PROPN
iajs-1618	84	13	jw	jw	PROPN
iajs-1618	84	14	and	and	CCONJ
iajs-1618	84	15	hence	hence	ADV
iajs-1618	84	16	c	c	PROPN
iajs-1618	85	1	=	=	PUNCT
iajs-1618	85	2	j.	j.	PROPN
iajs-1618	85	3	thus	thus	ADV
iajs-1618	85	4	h	h	PROPN
iajs-1618	85	5	=	=	SYM
iajs-1618	85	6	jw	jw	PROPN
iajs-1618	85	7	and	and	CCONJ
iajs-1618	85	8	kw	kw	PROPN
iajs-1618	85	9	is	be	AUX
iajs-1618	85	10	a	a	DET
iajs-1618	85	11	small	small	ADJ
iajs-1618	85	12	submodule	submodule	NOUN
iajs-1618	85	13	of	of	ADP
iajs-1618	85	14	jw	jw	PROPN
iajs-1618	85	15	.	.	PUNCT
iajs-1618	86	1	now	now	ADV
iajs-1618	86	2	,	,	PUNCT
iajs-1618	86	3	iw	iw	PROPN
iajs-1618	86	4	∩	∩	ADJ
iajs-1618	86	5	kw	kw	NOUN
iajs-1618	86	6	=	=	NOUN
iajs-1618	86	7	0	0	NUM
iajs-1618	86	8	and	and	CCONJ
iajs-1618	86	9	kw	kw	PROPN
iajs-1618	86	10	is	be	AUX
iajs-1618	86	11	a	a	DET
iajs-1618	86	12	small	small	ADJ
iajs-1618	86	13	submodule	submodule	NOUN
iajs-1618	86	14	in	in	ADP
iajs-1618	86	15	jw	jw	PROPN
iajs-1618	86	16	implies	imply	VERB
iajs-1618	86	17	kw	kw	PROPN
iajs-1618	86	18	=	=	SYM
iajs-1618	86	19	0	0	PUNCT
iajs-1618	86	20	(	(	PUNCT
iajs-1618	86	21	since	since	SCONJ
iajs-1618	86	22	iw≤se	iw≤se	PROPN
iajs-1618	86	23	jw	jw	PROPN
iajs-1618	86	24	)	)	PUNCT
iajs-1618	86	25	and	and	CCONJ
iajs-1618	86	26	so	so	ADV
iajs-1618	86	27	k=0	k=0	PROPN
iajs-1618	86	28	.	.	PUNCT
iajs-1618	87	1	it	it	PRON
iajs-1618	87	2	follows	follow	VERB
iajs-1618	87	3	i≤se	i≤se	PROPN
iajs-1618	87	4	j.	j.	PROPN
iajs-1618	87	5	recall	recall	VERB
iajs-1618	87	6	that	that	PRON
iajs-1618	87	7	,	,	PUNCT
iajs-1618	87	8	“	"	PUNCT
iajs-1618	87	9	a	a	DET
iajs-1618	87	10	non	non	ADJ
iajs-1618	87	11	-	-	ADJ
iajs-1618	87	12	zero	zero	ADJ
iajs-1618	87	13	l	l	NOUN
iajs-1618	87	14	-	-	NOUN
iajs-1618	87	15	module	module	NOUN
iajs-1618	87	16	w	w	NOUN
iajs-1618	87	17	is	be	AUX
iajs-1618	87	18	called	call	VERB
iajs-1618	87	19	small	small	ADJ
iajs-1618	87	20	-uniform	-uniform	NOUN
iajs-1618	87	21	(	(	PUNCT
iajs-1618	87	22	shortly	shortly	ADV
iajs-1618	87	23	,	,	PUNCT
iajs-1618	87	24	by	by	ADP
iajs-1618	87	25	s	s	NOUN
iajs-1618	87	26	-uniform	-uniform	NOUN
iajs-1618	87	27	)	)	PUNCT
iajs-1618	87	28	if	if	SCONJ
iajs-1618	87	29	every	every	DET
iajs-1618	87	30	nonzero	nonzero	PROPN
iajs-1618	87	31	submodule	submodule	PROPN
iajs-1618	87	32	of	of	ADP
iajs-1618	87	33	w	w	PROPN
iajs-1618	87	34	is	be	AUX
iajs-1618	87	35	s	s	NOUN
iajs-1618	87	36	-essential	-essential	ADJ
iajs-1618	87	37	.	.	PUNCT
iajs-1618	88	1	a	a	DET
iajs-1618	88	2	ring	ring	NOUN
iajs-1618	88	3	l	l	NOUN
iajs-1618	88	4	is	be	AUX
iajs-1618	88	5	called	call	VERB
iajs-1618	88	6	s	s	NOUN
iajs-1618	88	7	-	-	PUNCT
iajs-1618	88	8	uniform	uniform	ADJ
iajs-1618	88	9	if	if	SCONJ
iajs-1618	88	10	l	l	NOUN
iajs-1618	88	11	is	be	AUX
iajs-1618	88	12	an	an	DET
iajs-1618	88	13	suniform	suniform	NOUN
iajs-1618	88	14	l	l	NOUN
iajs-1618	88	15	-	-	NOUN
iajs-1618	88	16	module	module	NOUN
iajs-1618	88	17	”	"	PUNCT
iajs-1618	88	18	.	.	PUNCT
iajs-1618	89	1	[	[	X
iajs-1618	89	2	9	9	NUM
iajs-1618	89	3	]	]	X
iajs-1618	89	4	mathematics	mathematic	NOUN
iajs-1618	89	5	|230	|230	PROPN
iajs-1618	90	1	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	NOUN
iajs-1618	90	2	7102	7102	NUM
iajs-1618	90	3	(	(	PUNCT
iajs-1618	90	4	عام	عام	PROPN
iajs-1618	90	5	3	3	NUM
iajs-1618	90	6	(	(	PUNCT
iajs-1618	90	7	العدد	العدد	PROPN
iajs-1618	90	8	)	)	PUNCT
iajs-1618	90	9	30مجلة	30مجلة	PROPN
iajs-1618	90	10	إبن	إبن	VERB
iajs-1618	90	11	الهيثم	الهيثم	ADJ
iajs-1618	90	12	للعلوم	للعلوم	NOUN
iajs-1618	90	13	الصرفة	الصرفة	NOUN
iajs-1618	91	1	و	و	PRON
iajs-1618	91	2	التطبيقية	التطبيقية	ADV
iajs-1618	91	3	المجلد	المجلد	ADV
iajs-1618	91	4	)	)	PUNCT
iajs-1618	92	1	ibn	ibn	PROPN
iajs-1618	92	2	al	al	PROPN
iajs-1618	92	3	-	-	PUNCT
iajs-1618	92	4	haitham	haitham	PROPN
iajs-1618	92	5	j.	j.	PROPN
iajs-1618	92	6	for	for	ADP
iajs-1618	92	7	pure	pure	PROPN
iajs-1618	92	8	&	&	CCONJ
iajs-1618	92	9	appl	appl	PROPN
iajs-1618	92	10	.	.	PUNCT
iajs-1618	93	1	sci	sci	PROPN
iajs-1618	93	2	.	.	PUNCT
iajs-1618	94	1	vol.03	vol.03	PROPN
iajs-1618	94	2	(	(	PUNCT
iajs-1618	94	3	3	3	NUM
iajs-1618	94	4	)	)	PUNCT
iajs-1618	94	5	2017	2017	NUM
iajs-1618	94	6	corollary	corollary	NOUN
iajs-1618	94	7	1	1	NUM
iajs-1618	94	8	.	.	NOUN
iajs-1618	94	9	8	8	NUM
iajs-1618	94	10	:	:	PUNCT
iajs-1618	94	11	let	let	VERB
iajs-1618	94	12	w	w	PART
iajs-1618	94	13	be	be	AUX
iajs-1618	94	14	a	a	DET
iajs-1618	94	15	fmfg	fmfg	ADJ
iajs-1618	94	16	l	l	NOUN
iajs-1618	94	17	-	-	NOUN
iajs-1618	94	18	module	module	NOUN
iajs-1618	94	19	.	.	PUNCT
iajs-1618	95	1	then	then	ADV
iajs-1618	95	2	w	w	PROPN
iajs-1618	95	3	is	be	AUX
iajs-1618	95	4	s	s	NOUN
iajs-1618	95	5	-	-	PUNCT
iajs-1618	95	6	uniform	uniform	ADJ
iajs-1618	95	7	module	module	NOUN
iajs-1618	95	8	if	if	SCONJ
iajs-1618	95	9	and	and	CCONJ
iajs-1618	95	10	only	only	ADV
iajs-1618	95	11	if	if	SCONJ
iajs-1618	95	12	l	l	NOUN
iajs-1618	95	13	is	be	AUX
iajs-1618	95	14	s	s	NOUN
iajs-1618	95	15	-	-	PUNCT
iajs-1618	95	16	uniform	uniform	ADJ
iajs-1618	95	17	ring	ring	NOUN
iajs-1618	95	18	.	.	PUNCT
iajs-1618	96	1	definition	definition	NOUN
iajs-1618	96	2	1	1	NUM
iajs-1618	96	3	.	.	PUNCT
iajs-1618	96	4	9	9	NUM
iajs-1618	96	5	:	:	PUNCT
iajs-1618	97	1	[	[	X
iajs-1618	97	2	1	1	X
iajs-1618	97	3	]	]	PUNCT
iajs-1618	97	4	a	a	DET
iajs-1618	97	5	submodule	submodule	NOUN
iajs-1618	97	6	d	d	PROPN
iajs-1618	97	7	of	of	ADP
iajs-1618	97	8	an	an	DET
iajs-1618	97	9	l	l	NOUN
iajs-1618	97	10	-module	-module	PROPN
iajs-1618	97	11	w	w	NOUN
iajs-1618	97	12	is	be	AUX
iajs-1618	97	13	called	call	VERB
iajs-1618	97	14	s	s	PART
iajs-1618	97	15	-	-	PUNCT
iajs-1618	97	16	closed	closed	ADJ
iajs-1618	97	17	submodule	submodule	NOUN
iajs-1618	97	18	denoted	denote	VERB
iajs-1618	97	19	by	by	ADP
iajs-1618	97	20	d	d	PROPN
iajs-1618	97	21	≤sc	≤sc	NOUN
iajs-1618	97	22	w	w	NOUN
iajs-1618	97	23	,	,	PUNCT
iajs-1618	97	24	if	if	SCONJ
iajs-1618	97	25	d	d	PROPN
iajs-1618	97	26	has	have	VERB
iajs-1618	97	27	no	no	DET
iajs-1618	97	28	proper	proper	ADJ
iajs-1618	97	29	s	s	NOUN
iajs-1618	97	30	-essential	-essential	ADJ
iajs-1618	97	31	extension	extension	NOUN
iajs-1618	97	32	in	in	ADP
iajs-1618	97	33	w	w	PROPN
iajs-1618	97	34	,	,	PUNCT
iajs-1618	97	35	that	that	ADV
iajs-1618	97	36	is	is	ADV
iajs-1618	97	37	,	,	PUNCT
iajs-1618	97	38	whenever	whenever	SCONJ
iajs-1618	97	39	d	d	VERB
iajs-1618	97	40	≤w	≤w	NOUN
iajs-1618	97	41	such	such	ADJ
iajs-1618	97	42	that	that	SCONJ
iajs-1618	97	43	d	d	ADP
iajs-1618	97	44	≤se	≤se	NUM
iajs-1618	97	45	k	k	PROPN
iajs-1618	97	46	≤	≤	PROPN
iajs-1618	97	47	w	w	PROPN
iajs-1618	97	48	,	,	PUNCT
iajs-1618	97	49	then	then	ADV
iajs-1618	97	50	d	d	PROPN
iajs-1618	97	51	=	=	PUNCT
iajs-1618	97	52	k.	k.	PROPN
iajs-1618	98	1	an	an	DET
iajs-1618	98	2	ideal	ideal	ADJ
iajs-1618	98	3	e	e	NOUN
iajs-1618	98	4	of	of	ADP
iajs-1618	98	5	l	l	NOUN
iajs-1618	98	6	is	be	AUX
iajs-1618	98	7	called	call	VERB
iajs-1618	98	8	an	an	DET
iajs-1618	98	9	s	s	NOUN
iajs-1618	98	10	-	-	PUNCT
iajs-1618	98	11	closed	closed	ADJ
iajs-1618	98	12	,	,	PUNCT
iajs-1618	98	13	if	if	SCONJ
iajs-1618	98	14	its	its	PROPN
iajs-1618	98	15	an	an	DET
iajs-1618	98	16	sclosed	sclose	VERB
iajs-1618	98	17	submodule	submodule	NOUN
iajs-1618	98	18	in	in	ADP
iajs-1618	98	19	l.	l.	PROPN
iajs-1618	98	20	where	where	SCONJ
iajs-1618	98	21	every	every	DET
iajs-1618	98	22	sclosed	sclose	VERB
iajs-1618	98	23	submodule	submodule	NOUN
iajs-1618	98	24	in	in	ADP
iajs-1618	98	25	w	w	PROPN
iajs-1618	98	26	is	be	AUX
iajs-1618	98	27	closed	close	VERB
iajs-1618	98	28	in	in	ADP
iajs-1618	98	29	w	w	PROPN
iajs-1618	99	1	but	but	CCONJ
iajs-1618	99	2	the	the	DET
iajs-1618	99	3	converse	converse	NOUN
iajs-1618	99	4	is	be	AUX
iajs-1618	99	5	not	not	PART
iajs-1618	99	6	true	true	ADJ
iajs-1618	99	7	.	.	PUNCT
iajs-1618	100	1	examples	example	NOUN
iajs-1618	100	2	1	1	NUM
iajs-1618	100	3	.	.	PUNCT
iajs-1618	100	4	10	10	NUM
iajs-1618	100	5	:	:	SYM
iajs-1618	100	6	1	1	X
iajs-1618	100	7	)	)	PUNCT
iajs-1618	100	8	in	in	ADP
iajs-1618	100	9	z24	z24	PROPN
iajs-1618	100	10	as	as	ADP
iajs-1618	100	11	a	a	DET
iajs-1618	100	12	z	z	NOUN
iajs-1618	100	13	-	-	PUNCT
iajs-1618	100	14	module	module	NOUN
iajs-1618	100	15	.	.	PUNCT
iajs-1618	101	1	z24	z24	PROPN
iajs-1618	101	2	and	and	CCONJ
iajs-1618	101	3	<	<	X
iajs-1618	101	4	̅	̅	X
iajs-1618	101	5	>	>	X
iajs-1618	101	6	are	be	AUX
iajs-1618	101	7	the	the	DET
iajs-1618	101	8	only	only	ADJ
iajs-1618	101	9	s	s	NOUN
iajs-1618	101	10	-	-	PUNCT
iajs-1618	101	11	closed	closed	ADJ
iajs-1618	101	12	submodules	submodule	NOUN
iajs-1618	101	13	while	while	SCONJ
iajs-1618	101	14	<	<	X
iajs-1618	101	15	̅	̅	X
iajs-1618	101	16	>	>	X
iajs-1618	101	17	,	,	PUNCT
iajs-1618	101	18	<	<	X
iajs-1618	101	19	̅	̅	NOUN
iajs-1618	101	20	>	>	X
iajs-1618	101	21	,	,	PUNCT
iajs-1618	101	22	<	<	X
iajs-1618	101	23	̅	̅	NOUN
iajs-1618	101	24	>	>	X
iajs-1618	101	25	,	,	PUNCT
iajs-1618	101	26	<	<	X
iajs-1618	101	27	̅	̅	X
iajs-1618	101	28	>	>	X
iajs-1618	101	29	and	and	CCONJ
iajs-1618	101	30	<	<	X
iajs-1618	101	31	̅̅̅̅	̅̅̅̅	X
iajs-1618	101	32	>	>	X
iajs-1618	101	33	are	be	AUX
iajs-1618	101	34	not	not	PART
iajs-1618	101	35	because	because	SCONJ
iajs-1618	101	36	they	they	PRON
iajs-1618	101	37	have	have	VERB
iajs-1618	101	38	a	a	DET
iajs-1618	101	39	proper	proper	ADJ
iajs-1618	101	40	s	s	NOUN
iajs-1618	101	41	-	-	ADJ
iajs-1618	101	42	essential	essential	ADJ
iajs-1618	101	43	submodule	submodule	NOUN
iajs-1618	101	44	which	which	PRON
iajs-1618	101	45	is	be	AUX
iajs-1618	101	46	z24	z24	PROPN
iajs-1618	101	47	.	.	PUNCT
iajs-1618	102	1	all	all	DET
iajs-1618	102	2	submodules	submodule	NOUN
iajs-1618	102	3	of	of	ADP
iajs-1618	102	4	z24	z24	PROPN
iajs-1618	102	5	have	have	VERB
iajs-1618	102	6	the	the	DET
iajs-1618	102	7	following	follow	VERB
iajs-1618	102	8	properties	property	NOUN
iajs-1618	102	9	.	.	PUNCT
iajs-1618	103	1	a	a	DET
iajs-1618	103	2	≤	≤	PROPN
iajs-1618	103	3	z24	z24	NOUN
iajs-1618	103	4	a	a	PRON
iajs-1618	103	5	<	<	X
iajs-1618	103	6	<	<	X
iajs-1618	103	7	z24	z24	PROPN
iajs-1618	103	8	a	a	DET
iajs-1618	103	9	≤se	≤se	PROPN
iajs-1618	103	10	z24	z24	X
iajs-1618	103	11	a	a	DET
iajs-1618	103	12	≤sc	≤sc	NOUN
iajs-1618	103	13	z24	z24	NOUN
iajs-1618	103	14	<	<	X
iajs-1618	103	15	̅	̅	PROPN
iajs-1618	103	16	>	>	X
iajs-1618	103	17			ADP
iajs-1618	103	18			PROPN
iajs-1618	103	19			VERB
iajs-1618	103	20	<	<	X
iajs-1618	103	21	̅	̅	PROPN
iajs-1618	103	22	>	>	X
iajs-1618	103	23			PROPN
iajs-1618	103	24			ADV
iajs-1618	103	25			PROPN
iajs-1618	103	26	<	<	X
iajs-1618	103	27	̅	̅	PROPN
iajs-1618	103	28	>	>	X
iajs-1618	103	29			PROPN
iajs-1618	103	30			ADV
iajs-1618	103	31			PROPN
iajs-1618	103	32	<	<	X
iajs-1618	103	33	̅	̅	PROPN
iajs-1618	103	34	>	>	X
iajs-1618	103	35			PROPN
iajs-1618	103	36			ADV
iajs-1618	103	37			PROPN
iajs-1618	103	38	<	<	X
iajs-1618	103	39	>	>	X
iajs-1618	103	40			PROPN
iajs-1618	103	41			ADV
iajs-1618	103	42			PROPN
iajs-1618	103	43	<	<	X
iajs-1618	103	44	̅	̅	PROPN
iajs-1618	103	45	>	>	X
iajs-1618	103	46			PROPN
iajs-1618	103	47			PROPN
iajs-1618	103	48			ADV
iajs-1618	103	49	<	<	X
iajs-1618	103	50	̅̅̅̅	̅̅̅̅	X
iajs-1618	103	51	>	>	X
iajs-1618	103	52			PROPN
iajs-1618	103	53			PROPN
iajs-1618	103	54			PROPN
iajs-1618	103	55	z24	z24	PROPN
iajs-1618	103	56			PROPN
iajs-1618	103	57			ADV
iajs-1618	103	58			ADV
iajs-1618	103	59	similarly	similarly	ADV
iajs-1618	103	60	,	,	PUNCT
iajs-1618	103	61	<	<	X
iajs-1618	103	62	̅	̅	NOUN
iajs-1618	103	63	>	>	X
iajs-1618	103	64	,	,	PUNCT
iajs-1618	103	65	<	<	X
iajs-1618	103	66	̅	̅	X
iajs-1618	103	67	>	>	X
iajs-1618	103	68	and	and	CCONJ
iajs-1618	103	69	<	<	X
iajs-1618	103	70	̅	̅	NOUN
iajs-1618	103	71	>	>	X
iajs-1618	103	72	are	be	AUX
iajs-1618	103	73	not	not	PART
iajs-1618	103	74	small	small	ADJ
iajs-1618	103	75	submodules	submodule	NOUN
iajs-1618	103	76	in	in	ADP
iajs-1618	103	77	<	<	X
iajs-1618	103	78	̅	̅	NOUN
iajs-1618	103	79	>	>	X
iajs-1618	103	80	in	in	ADP
iajs-1618	103	81	z24	z24	PROPN
iajs-1618	103	82	but	but	CCONJ
iajs-1618	103	83	<	<	AUX
iajs-1618	103	84	̅̅̅̅	̅̅̅̅	X
iajs-1618	103	85	>	>	X
iajs-1618	103	86	is	be	AUX
iajs-1618	103	87	a	a	DET
iajs-1618	103	88	small	small	ADJ
iajs-1618	103	89	submodule	submodule	NOUN
iajs-1618	103	90	in	in	ADP
iajs-1618	103	91	<	<	X
iajs-1618	103	92	̅	̅	NOUN
iajs-1618	103	93	>	>	X
iajs-1618	103	94	and	and	CCONJ
iajs-1618	103	95	<	<	X
iajs-1618	103	96	̅>∩	̅>∩	X
iajs-1618	103	97	<	<	X
iajs-1618	103	98	̅̅̅̅	̅̅̅̅	X
iajs-1618	103	99	>	>	X
iajs-1618	103	100	{0	{0	NOUN
iajs-1618	103	101	}	}	PUNCT
iajs-1618	103	102	thus	thus	ADV
iajs-1618	103	103	<	<	X
iajs-1618	103	104	̅	̅	X
iajs-1618	103	105	>	>	X
iajs-1618	103	106	is	be	AUX
iajs-1618	103	107	an	an	DET
iajs-1618	103	108	s	s	NOUN
iajs-1618	103	109	-	-	ADJ
iajs-1618	103	110	essential	essential	ADJ
iajs-1618	103	111	submodule	submodule	NOUN
iajs-1618	103	112	in	in	ADP
iajs-1618	103	113	<	<	X
iajs-1618	103	114	̅	̅	NOUN
iajs-1618	103	115	>	>	PUNCT
iajs-1618	103	116	,	,	PUNCT
iajs-1618	103	117	so	so	SCONJ
iajs-1618	103	118	it	it	PRON
iajs-1618	103	119	is	be	AUX
iajs-1618	103	120	not	not	PART
iajs-1618	103	121	an	an	DET
iajs-1618	103	122	s	s	ADV
iajs-1618	103	123	-	-	PUNCT
iajs-1618	103	124	closed	closed	ADJ
iajs-1618	103	125	submodule	submodule	NOUN
iajs-1618	103	126	in	in	ADP
iajs-1618	103	127	<	<	X
iajs-1618	103	128	̅	̅	NOUN
iajs-1618	103	129	>	>	X
iajs-1618	103	130	.	.	PROPN
iajs-1618	104	1	2	2	X
iajs-1618	104	2	)	)	PUNCT
iajs-1618	104	3	if	if	SCONJ
iajs-1618	104	4	w	w	NOUN
iajs-1618	104	5	is	be	AUX
iajs-1618	104	6	a	a	DET
iajs-1618	104	7	simple	simple	ADJ
iajs-1618	104	8	module	module	NOUN
iajs-1618	104	9	,	,	PUNCT
iajs-1618	104	10	then	then	ADV
iajs-1618	104	11	<	<	X
iajs-1618	104	12	̅	̅	X
iajs-1618	104	13	>	>	X
iajs-1618	104	14	and	and	CCONJ
iajs-1618	104	15	w	w	PROPN
iajs-1618	104	16	are	be	AUX
iajs-1618	104	17	sclosed	sclose	VERB
iajs-1618	104	18	submodules	submodule	NOUN
iajs-1618	104	19	.	.	PUNCT
iajs-1618	105	1	3	3	X
iajs-1618	105	2	)	)	PUNCT
iajs-1618	105	3	let	let	VERB
iajs-1618	105	4	w	w	NOUN
iajs-1618	105	5	be	be	AUX
iajs-1618	105	6	an	an	DET
iajs-1618	105	7	l	l	NOUN
iajs-1618	105	8	-	-	NOUN
iajs-1618	105	9	module	module	NOUN
iajs-1618	105	10	.	.	PUNCT
iajs-1618	106	1	if	if	SCONJ
iajs-1618	106	2	every	every	DET
iajs-1618	106	3	submodule	submodule	NOUN
iajs-1618	106	4	of	of	ADP
iajs-1618	106	5	w	w	PROPN
iajs-1618	106	6	is	be	AUX
iajs-1618	106	7	s	s	NOUN
iajs-1618	106	8	-	-	PUNCT
iajs-1618	106	9	closed	closed	ADJ
iajs-1618	106	10	(	(	PUNCT
iajs-1618	106	11	hence	hence	ADV
iajs-1618	106	12	every	every	DET
iajs-1618	106	13	submodule	submodule	NOUN
iajs-1618	106	14	is	be	AUX
iajs-1618	106	15	closed	closed	ADJ
iajs-1618	106	16	)	)	PUNCT
iajs-1618	106	17	,	,	PUNCT
iajs-1618	106	18	then	then	ADV
iajs-1618	106	19	w	w	PROPN
iajs-1618	106	20	is	be	AUX
iajs-1618	106	21	semisimple	semisimple	NOUN
iajs-1618	106	22	module	module	NOUN
iajs-1618	106	23	,	,	PUNCT
iajs-1618	106	24	however	however	ADV
iajs-1618	106	25	the	the	DET
iajs-1618	106	26	converse	converse	NOUN
iajs-1618	106	27	is	be	AUX
iajs-1618	106	28	not	not	PART
iajs-1618	106	29	true	true	ADJ
iajs-1618	106	30	,	,	PUNCT
iajs-1618	106	31	for	for	ADP
iajs-1618	106	32	example	example	NOUN
iajs-1618	106	33	in	in	ADP
iajs-1618	106	34	z6	z6	PROPN
iajs-1618	106	35	,	,	PUNCT
iajs-1618	106	36	z6	z6	PROPN
iajs-1618	106	37	is	be	AUX
iajs-1618	106	38	a	a	DET
iajs-1618	106	39	zmodule	zmodule	NOUN
iajs-1618	106	40	is	be	AUX
iajs-1618	106	41	semisimple	semisimple	ADJ
iajs-1618	106	42	but	but	CCONJ
iajs-1618	106	43	the	the	DET
iajs-1618	106	44	submodules	submodule	NOUN
iajs-1618	106	45	<	<	X
iajs-1618	106	46	̅	̅	NOUN
iajs-1618	106	47	>	>	X
iajs-1618	106	48	,	,	PUNCT
iajs-1618	106	49	<	<	X
iajs-1618	106	50	̅	̅	NOUN
iajs-1618	106	51	>	>	X
iajs-1618	106	52	,	,	PUNCT
iajs-1618	106	53	<	<	X
iajs-1618	106	54	̅	̅	X
iajs-1618	106	55	>	>	X
iajs-1618	106	56	are	be	AUX
iajs-1618	106	57	not	not	PART
iajs-1618	106	58	sclosed	sclose	VERB
iajs-1618	106	59	.	.	PUNCT
iajs-1618	107	1	proposition	proposition	NOUN
iajs-1618	107	2	1	1	NUM
iajs-1618	107	3	.	.	NOUN
iajs-1618	107	4	11	11	NUM
iajs-1618	107	5	:	:	PUNCT
iajs-1618	107	6	let	let	VERB
iajs-1618	107	7	w	w	PART
iajs-1618	107	8	be	be	AUX
iajs-1618	107	9	an	an	DET
iajs-1618	107	10	l	l	NOUN
iajs-1618	107	11	-	-	NOUN
iajs-1618	107	12	module	module	NOUN
iajs-1618	107	13	such	such	ADJ
iajs-1618	107	14	that	that	SCONJ
iajs-1618	107	15	the	the	DET
iajs-1618	107	16	s	s	NOUN
iajs-1618	107	17	-	-	ADJ
iajs-1618	107	18	essential	essential	ADJ
iajs-1618	107	19	submodules	submodule	NOUN
iajs-1618	107	20	satisfy	satisfy	VERB
iajs-1618	107	21	transitive	transitive	ADJ
iajs-1618	107	22	property	property	NOUN
iajs-1618	107	23	.	.	PUNCT
iajs-1618	108	1	then	then	ADV
iajs-1618	108	2	for	for	ADP
iajs-1618	108	3	each	each	PRON
iajs-1618	108	4	a	a	DET
iajs-1618	108	5	≤	≤	NUM
iajs-1618	108	6	w	w	NOUN
iajs-1618	108	7	,	,	PUNCT
iajs-1618	108	8	there	there	PRON
iajs-1618	108	9	exists	exist	VERB
iajs-1618	108	10	an	an	DET
iajs-1618	108	11	s	s	NOUN
iajs-1618	108	12	-	-	PUNCT
iajs-1618	108	13	closed	closed	ADJ
iajs-1618	108	14	submodule	submodule	NOUN
iajs-1618	108	15	such	such	ADJ
iajs-1618	108	16	that	that	SCONJ
iajs-1618	108	17	a	a	DET
iajs-1618	108	18	≤se	≤se	NUM
iajs-1618	108	19	h.	h.	NOUN
iajs-1618	108	20	proof	proof	NOUN
iajs-1618	108	21	:	:	PUNCT
iajs-1618	108	22	let	let	VERB
iajs-1618	108	23	s={k≤w	s={k≤w	PROPN
iajs-1618	108	24	:	:	PUNCT
iajs-1618	108	25	a	a	DET
iajs-1618	108	26	≤se	≤se	NUM
iajs-1618	108	27	k	k	NOUN
iajs-1618	108	28	}	}	PUNCT
iajs-1618	108	29	.	.	PUNCT
iajs-1618	109	1	v	v	NOUN
iajs-1618	109	2	since	since	SCONJ
iajs-1618	109	3	a	a	PROPN
iajs-1618	109	4	v.	v.	CCONJ
iajs-1618	109	5	so	so	ADV
iajs-1618	109	6	by	by	ADP
iajs-1618	109	7	“	"	PUNCT
iajs-1618	109	8	zorn	zorn	PROPN
iajs-1618	109	9	’s	’s	PART
iajs-1618	109	10	lemma	lemma	PROPN
iajs-1618	109	11	”	"	PUNCT
iajs-1618	109	12	s	s	PART
iajs-1618	109	13	has	have	VERB
iajs-1618	109	14	a	a	DET
iajs-1618	109	15	maximal	maximal	ADJ
iajs-1618	109	16	element	element	NOUN
iajs-1618	109	17	say	say	VERB
iajs-1618	109	18	h.	h.	PROPN
iajs-1618	109	19	to	to	PART
iajs-1618	109	20	prove	prove	VERB
iajs-1618	109	21	h	h	NOUN
iajs-1618	109	22	is	be	AUX
iajs-1618	109	23	an	an	DET
iajs-1618	109	24	s	s	NOUN
iajs-1618	109	25	-closed	-close	VERB
iajs-1618	109	26	submodule	submodule	NOUN
iajs-1618	109	27	in	in	ADP
iajs-1618	109	28	w.	w.	PROPN
iajs-1618	109	29	assume	assume	VERB
iajs-1618	109	30	h	h	PROPN
iajs-1618	109	31	≤se	≤se	NUM
iajs-1618	109	32	d	d	PROPN
iajs-1618	109	33	≤	≤	X
iajs-1618	109	34	w.	w.	NOUN
iajs-1618	109	35	since	since	SCONJ
iajs-1618	109	36	a	a	DET
iajs-1618	109	37	≤se	≤se	NUM
iajs-1618	109	38	h	h	NOUN
iajs-1618	109	39	and	and	CCONJ
iajs-1618	109	40	h	h	NOUN
iajs-1618	109	41	≤se	≤se	NUM
iajs-1618	109	42	d	d	PROPN
iajs-1618	109	43	,	,	PUNCT
iajs-1618	109	44	then	then	ADV
iajs-1618	109	45	a	a	DET
iajs-1618	109	46	≤se	≤se	PROPN
iajs-1618	109	47	d	d	NOUN
iajs-1618	109	48	(	(	PUNCT
iajs-1618	109	49	by	by	ADP
iajs-1618	109	50	transitive	transitive	ADJ
iajs-1618	109	51	property	property	NOUN
iajs-1618	109	52	)	)	PUNCT
iajs-1618	109	53	,	,	PUNCT
iajs-1618	109	54	and	and	CCONJ
iajs-1618	109	55	so	so	ADV
iajs-1618	109	56	d	d	ADP
iajs-1618	109	57			PROPN
iajs-1618	109	58	s.	s.	PROPN
iajs-1618	109	59	hence	hence	ADV
iajs-1618	109	60	h	h	PROPN
iajs-1618	110	1	=	=	SYM
iajs-1618	110	2	d	d	PROPN
iajs-1618	110	3	(	(	PUNCT
iajs-1618	110	4	by	by	ADP
iajs-1618	110	5	maximality	maximality	NOUN
iajs-1618	110	6	of	of	ADP
iajs-1618	110	7	l	l	PROPN
iajs-1618	110	8	)	)	PUNCT
iajs-1618	110	9	.	.	PUNCT
iajs-1618	111	1	thus	thus	ADV
iajs-1618	111	2	h	h	NOUN
iajs-1618	111	3	is	be	AUX
iajs-1618	111	4	an	an	DET
iajs-1618	111	5	s	s	NOUN
iajs-1618	111	6	-	-	PUNCT
iajs-1618	111	7	closed	closed	ADJ
iajs-1618	111	8	submodule	submodule	NOUN
iajs-1618	111	9	.	.	PUNCT
iajs-1618	112	1	the	the	DET
iajs-1618	112	2	following	follow	VERB
iajs-1618	112	3	proposition	proposition	NOUN
iajs-1618	112	4	has	have	AUX
iajs-1618	112	5	been	be	AUX
iajs-1618	112	6	given	give	VERB
iajs-1618	112	7	in	in	ADP
iajs-1618	112	8	[	[	X
iajs-1618	112	9	1	1	NUM
iajs-1618	112	10	]	]	PUNCT
iajs-1618	112	11	,	,	PUNCT
iajs-1618	112	12	we	we	PRON
iajs-1618	112	13	will	will	AUX
iajs-1618	112	14	mention	mention	VERB
iajs-1618	112	15	it	it	PRON
iajs-1618	112	16	with	with	ADP
iajs-1618	112	17	its	its	PRON
iajs-1618	112	18	proof	proof	NOUN
iajs-1618	112	19	for	for	ADP
iajs-1618	112	20	the	the	DET
iajs-1618	112	21	sake	sake	NOUN
iajs-1618	112	22	of	of	ADP
iajs-1618	112	23	completness	completness	NOUN
iajs-1618	112	24	.	.	PUNCT
iajs-1618	113	1	proposition	proposition	NOUN
iajs-1618	113	2	1	1	NUM
iajs-1618	113	3	.	.	PUNCT
iajs-1618	113	4	12	12	NUM
iajs-1618	113	5	:	:	PUNCT
iajs-1618	113	6	lea	lea	NOUN
iajs-1618	113	7	a	a	DET
iajs-1618	113	8	be	be	AUX
iajs-1618	113	9	a	a	DET
iajs-1618	113	10	submodule	submodule	NOUN
iajs-1618	113	11	of	of	ADP
iajs-1618	113	12	b	b	NOUN
iajs-1618	113	13	,	,	PUNCT
iajs-1618	113	14	and	and	CCONJ
iajs-1618	113	15	let	let	VERB
iajs-1618	113	16	b	b	PRON
iajs-1618	113	17	an	an	DET
iajs-1618	113	18	s	s	NOUN
iajs-1618	113	19	-	-	PUNCT
iajs-1618	113	20	closed	closed	ADJ
iajs-1618	113	21	submodule	submodule	NOUN
iajs-1618	113	22	of	of	ADP
iajs-1618	113	23	w	w	PROPN
iajs-1618	113	24	,	,	PUNCT
iajs-1618	113	25	then	then	ADV
iajs-1618	113	26	(	(	PUNCT
iajs-1618	113	27	b	b	X
iajs-1618	113	28	/	/	SYM
iajs-1618	113	29	a	a	NOUN
iajs-1618	113	30	)	)	PUNCT
iajs-1618	113	31	is	be	AUX
iajs-1618	113	32	an	an	DET
iajs-1618	113	33	s	s	NOUN
iajs-1618	113	34	-closed	-close	VERB
iajs-1618	113	35	submodule	submodule	NOUN
iajs-1618	113	36	of	of	ADP
iajs-1618	113	37	(	(	PUNCT
iajs-1618	113	38	w	w	PROPN
iajs-1618	113	39	/	/	SYM
iajs-1618	113	40	a	a	NOUN
iajs-1618	113	41	)	)	PUNCT
iajs-1618	113	42	.	.	PUNCT
iajs-1618	114	1	proof	proof	NOUN
iajs-1618	114	2	:	:	PUNCT
iajs-1618	114	3	assume	assume	VERB
iajs-1618	114	4	(	(	PUNCT
iajs-1618	114	5	b	b	X
iajs-1618	114	6	/	/	SYM
iajs-1618	114	7	a	a	NOUN
iajs-1618	114	8	)	)	PUNCT
iajs-1618	114	9	≤se	≤se	NOUN
iajs-1618	114	10	(	(	PUNCT
iajs-1618	114	11	c	c	NOUN
iajs-1618	114	12	/	/	SYM
iajs-1618	114	13	a	a	NOUN
iajs-1618	114	14	)	)	PUNCT
iajs-1618	114	15	where	where	SCONJ
iajs-1618	114	16	(	(	PUNCT
iajs-1618	114	17	c	c	X
iajs-1618	114	18	/	/	SYM
iajs-1618	114	19	a	a	NOUN
iajs-1618	114	20	)	)	PUNCT
iajs-1618	114	21	≤	≤	NOUN
iajs-1618	114	22	(	(	PUNCT
iajs-1618	114	23	w	w	NOUN
iajs-1618	114	24	/	/	SYM
iajs-1618	114	25	a	a	NOUN
iajs-1618	114	26	)	)	PUNCT
iajs-1618	114	27	.	.	PUNCT
iajs-1618	115	1	let	let	VERB
iajs-1618	115	2	π	π	NOUN
iajs-1618	115	3	:	:	PUNCT
iajs-1618	115	4	w	w	X
iajs-1618	115	5			NOUN
iajs-1618	115	6	(	(	PUNCT
iajs-1618	115	7	w	w	NOUN
iajs-1618	115	8	/	/	SYM
iajs-1618	115	9	a	a	PRON
iajs-1618	115	10	)	)	PUNCT
iajs-1618	115	11	be	be	AUX
iajs-1618	115	12	a	a	DET
iajs-1618	115	13	natural	natural	ADJ
iajs-1618	115	14	projection	projection	NOUN
iajs-1618	115	15	map	map	NOUN
iajs-1618	115	16	.	.	PUNCT
iajs-1618	116	1	then	then	ADV
iajs-1618	116	2	b=	b=	VERB
iajs-1618	116	3	(	(	PUNCT
iajs-1618	116	4	b	b	X
iajs-1618	116	5	/	/	SYM
iajs-1618	116	6	a	a	NOUN
iajs-1618	116	7	)	)	PUNCT
iajs-1618	116	8	,	,	PUNCT
iajs-1618	116	9	and	and	CCONJ
iajs-1618	116	10	so	so	ADV
iajs-1618	116	11	by	by	ADP
iajs-1618	116	12	[	[	X
iajs-1618	116	13	4	4	NUM
iajs-1618	116	14	,	,	PUNCT
iajs-1618	116	15	prop.27(2	prop.27(2	NOUN
iajs-1618	116	16	)	)	PUNCT
iajs-1618	116	17	,	,	PUNCT
iajs-1618	116	18	p.1054	p.1054	NOUN
iajs-1618	116	19	]	]	X
iajs-1618	116	20	b	b	X
iajs-1618	116	21	≤se	≤se	PROPN
iajs-1618	116	22	c.	c.	PROPN
iajs-1618	116	23	but	but	CCONJ
iajs-1618	116	24	b	b	PROPN
iajs-1618	116	25	is	be	AUX
iajs-1618	116	26	an	an	DET
iajs-1618	116	27	sclosed	sclose	VERB
iajs-1618	116	28	submodule	submodule	NOUN
iajs-1618	116	29	in	in	ADP
iajs-1618	116	30	w.	w.	PROPN
iajs-1618	116	31	thus	thus	ADV
iajs-1618	116	32	b	b	PROPN
iajs-1618	116	33	=	=	PROPN
iajs-1618	116	34	c.	c.	PROPN
iajs-1618	116	35	mathematics	mathematic	NOUN
iajs-1618	116	36	|231	|231	X
iajs-1618	117	1	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	X
iajs-1618	117	2	7102	7102	NUM
iajs-1618	117	3	(	(	PUNCT
iajs-1618	117	4	عام	عام	PROPN
iajs-1618	117	5	3	3	NUM
iajs-1618	117	6	(	(	PUNCT
iajs-1618	117	7	العدد	العدد	PROPN
iajs-1618	117	8	)	)	PUNCT
iajs-1618	117	9	30مجلة	30مجلة	PROPN
iajs-1618	117	10	إبن	إبن	VERB
iajs-1618	117	11	الهيثم	الهيثم	ADJ
iajs-1618	117	12	للعلوم	للعلوم	NOUN
iajs-1618	117	13	الصرفة	الصرفة	NOUN
iajs-1618	118	1	و	و	PRON
iajs-1618	118	2	التطبيقية	التطبيقية	ADV
iajs-1618	118	3	المجلد	المجلد	ADV
iajs-1618	118	4	)	)	PUNCT
iajs-1618	119	1	ibn	ibn	PROPN
iajs-1618	119	2	al	al	PROPN
iajs-1618	119	3	-	-	PUNCT
iajs-1618	119	4	haitham	haitham	PROPN
iajs-1618	119	5	j.	j.	PROPN
iajs-1618	119	6	for	for	ADP
iajs-1618	119	7	pure	pure	PROPN
iajs-1618	119	8	&	&	CCONJ
iajs-1618	119	9	appl	appl	PROPN
iajs-1618	119	10	.	.	PUNCT
iajs-1618	120	1	sci	sci	PROPN
iajs-1618	120	2	.	.	PUNCT
iajs-1618	121	1	vol.03	vol.03	PROPN
iajs-1618	121	2	(	(	PUNCT
iajs-1618	121	3	3	3	NUM
iajs-1618	121	4	)	)	PUNCT
iajs-1618	121	5	2017	2017	NUM
iajs-1618	121	6	it	it	PRON
iajs-1618	121	7	follows	follow	VERB
iajs-1618	121	8	that	that	SCONJ
iajs-1618	121	9	(	(	PUNCT
iajs-1618	121	10	b	b	X
iajs-1618	121	11	/	/	SYM
iajs-1618	121	12	a	a	NOUN
iajs-1618	121	13	)	)	PUNCT
iajs-1618	121	14	=	=	SYM
iajs-1618	121	15	(	(	PUNCT
iajs-1618	121	16	c/	c/	NOUN
iajs-1618	121	17	a	a	NOUN
iajs-1618	121	18	)	)	PUNCT
iajs-1618	121	19	and	and	CCONJ
iajs-1618	121	20	(	(	PUNCT
iajs-1618	121	21	b/	b/	PRON
iajs-1618	121	22	a	a	PRON
iajs-1618	121	23	)	)	PUNCT
iajs-1618	121	24	is	be	AUX
iajs-1618	121	25	an	an	DET
iajs-1618	121	26	s	s	NOUN
iajs-1618	121	27	-	-	PUNCT
iajs-1618	121	28	closed	closed	ADJ
iajs-1618	121	29	submodule	submodule	NOUN
iajs-1618	121	30	in	in	ADP
iajs-1618	121	31	(	(	PUNCT
iajs-1618	121	32	w	w	NOUN
iajs-1618	121	33	/	/	SYM
iajs-1618	121	34	a	a	NOUN
iajs-1618	121	35	)	)	PUNCT
iajs-1618	121	36	.	.	PUNCT
iajs-1618	122	1	proposition	proposition	NOUN
iajs-1618	122	2	1	1	NUM
iajs-1618	122	3	.	.	PUNCT
iajs-1618	122	4	13	13	NUM
iajs-1618	122	5	:	:	PUNCT
iajs-1618	122	6	let	let	VERB
iajs-1618	122	7	a	a	DET
iajs-1618	122	8	≤	≤	NUM
iajs-1618	122	9	b	b	NOUN
iajs-1618	122	10	≤	≤	NUM
iajs-1618	122	11	w	w	ADP
iajs-1618	122	12	such	such	ADJ
iajs-1618	122	13	that	that	SCONJ
iajs-1618	122	14	a	a	PRON
iajs-1618	122	15	is	be	AUX
iajs-1618	122	16	an	an	DET
iajs-1618	122	17	s	s	NOUN
iajs-1618	122	18	-closed	-close	VERB
iajs-1618	122	19	submodule	submodule	NOUN
iajs-1618	122	20	of	of	ADP
iajs-1618	122	21	an	an	DET
iajs-1618	122	22	l	l	NOUN
iajs-1618	122	23	-	-	NOUN
iajs-1618	122	24	module	module	NOUN
iajs-1618	122	25	w.	w.	NOUN
iajs-1618	123	1	then	then	ADV
iajs-1618	123	2	b	b	X
iajs-1618	123	3	≤sc	≤sc	NOUN
iajs-1618	124	1	w	w	NOUN
iajs-1618	124	2	if	if	SCONJ
iajs-1618	125	1	and	and	CCONJ
iajs-1618	125	2	only	only	ADV
iajs-1618	125	3	if	if	SCONJ
iajs-1618	125	4	≤sc	≤sc	X
iajs-1618	125	5	.	.	PUNCT
iajs-1618	126	1	proof	proof	NOUN
iajs-1618	126	2	:	:	PUNCT
iajs-1618	126	3	(	(	PUNCT
iajs-1618	126	4			NOUN
iajs-1618	126	5	)	)	PUNCT
iajs-1618	126	6	see	see	VERB
iajs-1618	126	7	[	[	X
iajs-1618	126	8	1	1	NUM
iajs-1618	126	9	,	,	PUNCT
iajs-1618	126	10	coro.2.7	coro.2.7	PROPN
iajs-1618	126	11	]	]	X
iajs-1618	126	12	(	(	PUNCT
iajs-1618	126	13			X
iajs-1618	126	14	)	)	PUNCT
iajs-1618	126	15	suppose	suppose	VERB
iajs-1618	126	16	≤sc	≤sc	PRON
iajs-1618	126	17	and	and	CCONJ
iajs-1618	126	18	let	let	VERB
iajs-1618	126	19	b	b	NUM
iajs-1618	126	20	≤se	≤se	NUM
iajs-1618	126	21	h	h	NOUN
iajs-1618	126	22	≤	≤	NOUN
iajs-1618	126	23	w.	w.	NOUN
iajs-1618	126	24	since	since	SCONJ
iajs-1618	126	25	a	a	DET
iajs-1618	126	26	≤sc	≤sc	NOUN
iajs-1618	126	27	w	w	NOUN
iajs-1618	126	28	and	and	CCONJ
iajs-1618	126	29	a	a	DET
iajs-1618	126	30	≤	≤	PROPN
iajs-1618	126	31	b	b	NOUN
iajs-1618	126	32	then	then	ADV
iajs-1618	126	33	≤se	≤se	PROPN
iajs-1618	126	34	implies	imply	VERB
iajs-1618	126	35	b	b	NUM
iajs-1618	126	36	≤se	≤se	NUM
iajs-1618	126	37	w	w	VERB
iajs-1618	126	38	by	by	ADP
iajs-1618	126	39	[	[	X
iajs-1618	126	40	1	1	NUM
iajs-1618	126	41	,	,	PUNCT
iajs-1618	126	42	remarks	remark	NOUN
iajs-1618	126	43	and	and	CCONJ
iajs-1618	126	44	examples	example	NOUN
iajs-1618	126	45	2.2(6	2.2(6	NUM
iajs-1618	126	46	)	)	PUNCT
iajs-1618	126	47	]	]	PUNCT
iajs-1618	126	48	.	.	PUNCT
iajs-1618	127	1	that	that	PRON
iajs-1618	127	2	is	be	AUX
iajs-1618	127	3	a	a	DET
iajs-1618	127	4	≤sc	≤sc	NOUN
iajs-1618	127	5	b	b	NOUN
iajs-1618	127	6	by	by	ADP
iajs-1618	127	7	[	[	X
iajs-1618	127	8	1	1	NUM
iajs-1618	127	9	,	,	PUNCT
iajs-1618	127	10	propo.2.8	propo.2.8	PROPN
iajs-1618	127	11	]	]	PUNCT
iajs-1618	127	12	.	.	PUNCT
iajs-1618	128	1	to	to	PART
iajs-1618	128	2	prove	prove	VERB
iajs-1618	128	3	a	a	DET
iajs-1618	128	4	≤sc	≤sc	NOUN
iajs-1618	128	5	h	h	NOUN
iajs-1618	128	6	,	,	PUNCT
iajs-1618	128	7	suppose	suppose	VERB
iajs-1618	128	8	that	that	SCONJ
iajs-1618	128	9	a≤se	a≤se	PROPN
iajs-1618	128	10	c	c	VERB
iajs-1618	128	11	for	for	ADP
iajs-1618	128	12	some	some	DET
iajs-1618	128	13	submodule	submodule	NOUN
iajs-1618	128	14	c	c	PROPN
iajs-1618	128	15	of	of	ADP
iajs-1618	128	16	h.	h.	PROPN
iajs-1618	128	17	as	as	SCONJ
iajs-1618	128	18	a	a	PRON
iajs-1618	128	19	is	be	AUX
iajs-1618	128	20	an	an	DET
iajs-1618	128	21	sclosed	sclosed	ADJ
iajs-1618	128	22	submodule	submodule	NOUN
iajs-1618	128	23	of	of	ADP
iajs-1618	128	24	w	w	PROPN
iajs-1618	128	25	,	,	PUNCT
iajs-1618	128	26	thus	thus	ADV
iajs-1618	128	27	a	a	DET
iajs-1618	128	28	=	=	SYM
iajs-1618	128	29	c	c	NOUN
iajs-1618	128	30	.	.	PUNCT
iajs-1618	129	1	hence	hence	ADV
iajs-1618	129	2	a	a	PRON
iajs-1618	129	3	is	be	AUX
iajs-1618	129	4	an	an	DET
iajs-1618	129	5	s	s	NOUN
iajs-1618	129	6	-closed	-close	VERB
iajs-1618	129	7	submodule	submodule	NOUN
iajs-1618	129	8	of	of	ADP
iajs-1618	129	9	h	h	NOUN
iajs-1618	129	10	and	and	CCONJ
iajs-1618	129	11	b	b	X
iajs-1618	129	12	≤se	≤se	NUM
iajs-1618	129	13	h	h	NOUN
iajs-1618	129	14	,	,	PUNCT
iajs-1618	129	15	that	that	PRON
iajs-1618	129	16	is	is	ADV
iajs-1618	129	17	≤se	≤se	NUM
iajs-1618	129	18	,	,	PUNCT
iajs-1618	129	19	by	by	ADP
iajs-1618	129	20	[	[	X
iajs-1618	129	21	1	1	NUM
iajs-1618	129	22	,	,	PUNCT
iajs-1618	129	23	remarks	remark	NOUN
iajs-1618	129	24	and	and	CCONJ
iajs-1618	129	25	examples	example	NOUN
iajs-1618	129	26	2.2(6	2.2(6	NUM
iajs-1618	129	27	)	)	PUNCT
iajs-1618	129	28	]	]	PUNCT
iajs-1618	129	29	.	.	PUNCT
iajs-1618	130	1	but	but	CCONJ
iajs-1618	130	2	≤sc	≤sc	VERB
iajs-1618	130	3	,	,	PUNCT
iajs-1618	131	1	so	so	ADV
iajs-1618	131	2	=	=	X
iajs-1618	131	3	.	.	PUNCT
iajs-1618	132	1	then	then	ADV
iajs-1618	132	2	b	b	X
iajs-1618	132	3	=	=	NOUN
iajs-1618	132	4	h	h	NOUN
iajs-1618	132	5	which	which	PRON
iajs-1618	132	6	means	mean	VERB
iajs-1618	132	7	b	b	X
iajs-1618	132	8	≤sc	≤sc	PROPN
iajs-1618	132	9	w.	w.	NOUN
iajs-1618	132	10	proposition	proposition	NOUN
iajs-1618	132	11	1	1	NUM
iajs-1618	132	12	.	.	PUNCT
iajs-1618	132	13	14	14	NUM
iajs-1618	132	14	:	:	PUNCT
iajs-1618	132	15	let	let	VERB
iajs-1618	132	16	w	w	PART
iajs-1618	132	17	be	be	AUX
iajs-1618	132	18	a	a	DET
iajs-1618	132	19	fmfg	fmfg	ADJ
iajs-1618	132	20	l	l	NOUN
iajs-1618	132	21	-	-	NOUN
iajs-1618	132	22	module	module	NOUN
iajs-1618	132	23	,	,	PUNCT
iajs-1618	132	24	and	and	CCONJ
iajs-1618	132	25	c≤	c≤	PROPN
iajs-1618	132	26	w.	w.	PROPN
iajs-1618	132	27	c	c	PROPN
iajs-1618	132	28	is	be	AUX
iajs-1618	132	29	an	an	DET
iajs-1618	132	30	s	s	NOUN
iajs-1618	132	31	-	-	PUNCT
iajs-1618	132	32	closed	closed	ADJ
iajs-1618	132	33	submodule	submodule	NOUN
iajs-1618	132	34	in	in	ADP
iajs-1618	132	35	w	w	PROPN
iajs-1618	132	36	if	if	SCONJ
iajs-1618	133	1	and	and	CCONJ
iajs-1618	133	2	only	only	ADV
iajs-1618	133	3	if	if	SCONJ
iajs-1618	133	4	c	c	NOUN
iajs-1618	133	5	=	=	VERB
iajs-1618	133	6	hw	hw	PROPN
iajs-1618	133	7	for	for	ADP
iajs-1618	133	8	some	some	DET
iajs-1618	133	9	s	s	NOUN
iajs-1618	133	10	-	-	PUNCT
iajs-1618	133	11	closed	closed	ADJ
iajs-1618	133	12	ideal	ideal	ADJ
iajs-1618	133	13	h	h	NOUN
iajs-1618	133	14	in	in	ADP
iajs-1618	133	15	l.	l.	PROPN
iajs-1618	133	16	proof	proof	PROPN
iajs-1618	133	17	:	:	PUNCT
iajs-1618	133	18	(	(	PUNCT
iajs-1618	133	19			NOUN
iajs-1618	133	20	)	)	PUNCT
iajs-1618	133	21	let	let	VERB
iajs-1618	133	22	c≤	c≤	PROPN
iajs-1618	133	23	w	w	INTJ
iajs-1618	133	24	,	,	PUNCT
iajs-1618	133	25	then	then	ADV
iajs-1618	133	26	c	c	X
iajs-1618	133	27	=	=	SYM
iajs-1618	134	1	hw	hw	PROPN
iajs-1618	134	2	.	.	PUNCT
iajs-1618	134	3	to	to	PART
iajs-1618	134	4	prove	prove	VERB
iajs-1618	134	5	h	h	NOUN
iajs-1618	134	6	is	be	AUX
iajs-1618	134	7	an	an	DET
iajs-1618	134	8	s	s	NOUN
iajs-1618	134	9	-	-	PUNCT
iajs-1618	134	10	closed	closed	ADJ
iajs-1618	134	11	ideal	ideal	NOUN
iajs-1618	134	12	in	in	ADP
iajs-1618	134	13	l	l	PROPN
iajs-1618	134	14	.	.	PUNCT
iajs-1618	135	1	assume	assume	VERB
iajs-1618	135	2	h≤se	h≤se	NOUN
iajs-1618	135	3	j.	j.	PROPN
iajs-1618	135	4	hence	hence	ADV
iajs-1618	135	5	hw≤se	hw≤se	PROPN
iajs-1618	135	6	jm	jm	PROPN
iajs-1618	135	7	by	by	ADP
iajs-1618	135	8	(	(	PUNCT
iajs-1618	135	9	th	th	NOUN
iajs-1618	135	10	.	.	NOUN
iajs-1618	135	11	1.7	1.7	NUM
iajs-1618	135	12	)	)	PUNCT
iajs-1618	135	13	,	,	PUNCT
iajs-1618	135	14	thus	thus	ADV
iajs-1618	135	15	c≤se	c≤se	VERB
iajs-1618	135	16	jw	jw	NOUN
iajs-1618	135	17	so	so	ADV
iajs-1618	135	18	c	c	PROPN
iajs-1618	135	19	=	=	SYM
iajs-1618	135	20	jw	jw	PROPN
iajs-1618	136	1	that	that	PRON
iajs-1618	136	2	is	be	AUX
iajs-1618	136	3	hw	hw	ADP
iajs-1618	136	4	=	=	PROPN
iajs-1618	136	5	jw	jw	PROPN
iajs-1618	136	6	.	.	PUNCT
iajs-1618	137	1	since	since	SCONJ
iajs-1618	137	2	w	w	PROPN
iajs-1618	137	3	is	be	AUX
iajs-1618	137	4	fmfg	fmfg	ADJ
iajs-1618	137	5	module	module	NOUN
iajs-1618	137	6	so	so	ADV
iajs-1618	137	7	h	h	PROPN
iajs-1618	137	8	=	=	SYM
iajs-1618	137	9	j	j	PROPN
iajs-1618	137	10	,	,	PUNCT
iajs-1618	137	11	hence	hence	ADV
iajs-1618	137	12	h	h	NOUN
iajs-1618	137	13	is	be	AUX
iajs-1618	137	14	an	an	DET
iajs-1618	137	15	s	s	NOUN
iajs-1618	137	16	-closed	-close	VERB
iajs-1618	137	17	ideal	ideal	NOUN
iajs-1618	137	18	in	in	ADP
iajs-1618	137	19	l.	l.	PROPN
iajs-1618	137	20	(	(	PUNCT
iajs-1618	137	21			X
iajs-1618	137	22	)	)	PUNCT
iajs-1618	137	23	similarly	similarly	ADV
iajs-1618	137	24	.	.	PUNCT
iajs-1618	138	1	2	2	X
iajs-1618	138	2	.	.	X
iajs-1618	138	3	ascending	ascend	VERB
iajs-1618	138	4	(	(	PUNCT
iajs-1618	138	5	descending	descending	NOUN
iajs-1618	138	6	)	)	PUNCT
iajs-1618	138	7	chain	chain	NOUN
iajs-1618	138	8	conditions	condition	NOUN
iajs-1618	138	9	on	on	ADP
iajs-1618	138	10	s	s	NOUN
iajs-1618	138	11	-	-	PUNCT
iajs-1618	138	12	closed	closed	ADJ
iajs-1618	138	13	submodules	submodule	NOUN
iajs-1618	138	14	in	in	ADP
iajs-1618	138	15	this	this	DET
iajs-1618	138	16	section	section	NOUN
iajs-1618	138	17	,	,	PUNCT
iajs-1618	138	18	we	we	PRON
iajs-1618	138	19	study	study	VERB
iajs-1618	138	20	modules	module	NOUN
iajs-1618	138	21	with	with	ADP
iajs-1618	138	22	chain	chain	NOUN
iajs-1618	138	23	conditions	condition	NOUN
iajs-1618	138	24	on	on	ADP
iajs-1618	138	25	s	s	VERB
iajs-1618	138	26	˗closed	˗close	VERB
iajs-1618	138	27	submodules	submodule	NOUN
iajs-1618	138	28	.	.	PUNCT
iajs-1618	139	1	definition	definition	NOUN
iajs-1618	139	2	2	2	NUM
iajs-1618	139	3	.	.	PUNCT
iajs-1618	139	4	1	1	NUM
iajs-1618	139	5	:	:	PUNCT
iajs-1618	139	6	an	an	DET
iajs-1618	139	7	l	l	NOUN
iajs-1618	139	8	-	-	NOUN
iajs-1618	139	9	module	module	NOUN
iajs-1618	139	10	w	w	NOUN
iajs-1618	139	11	is	be	AUX
iajs-1618	139	12	said	say	VERB
iajs-1618	139	13	to	to	PART
iajs-1618	139	14	have	have	VERB
iajs-1618	139	15	the	the	DET
iajs-1618	139	16	ascending	ascend	VERB
iajs-1618	139	17	(	(	PUNCT
iajs-1618	139	18	descending	descending	NOUN
iajs-1618	139	19	)	)	PUNCT
iajs-1618	139	20	chain	chain	NOUN
iajs-1618	139	21	condition	condition	NOUN
iajs-1618	139	22	,	,	PUNCT
iajs-1618	140	1	briefly	briefly	ADV
iajs-1618	140	2	a	a	DET
iajs-1618	140	3	c	c	NOUN
iajs-1618	140	4	c	c	NOUN
iajs-1618	140	5	(	(	PUNCT
iajs-1618	140	6	d	d	NOUN
iajs-1618	140	7	c	c	NOUN
iajs-1618	140	8	c	c	NOUN
iajs-1618	140	9	)	)	PUNCT
iajs-1618	140	10	on	on	ADP
iajs-1618	140	11	s	s	NOUN
iajs-1618	140	12	-	-	PUNCT
iajs-1618	140	13	closed	closed	ADJ
iajs-1618	140	14	submodules	submodule	NOUN
iajs-1618	140	15	if	if	SCONJ
iajs-1618	140	16	every	every	DET
iajs-1618	140	17	ascending	ascend	VERB
iajs-1618	140	18	(	(	PUNCT
iajs-1618	140	19	descending	descending	NOUN
iajs-1618	140	20	)	)	PUNCT
iajs-1618	140	21	chain	chain	NOUN
iajs-1618	140	22	a1	a1	NOUN
iajs-1618	140	23			PROPN
iajs-1618	140	24	a2	a2	PROPN
iajs-1618	140	25			PROPN
iajs-1618	140	26	…	…	PUNCT
iajs-1618	140	27	(	(	PUNCT
iajs-1618	140	28	a1	a1	PROPN
iajs-1618	140	29			PROPN
iajs-1618	140	30	a2	a2	PROPN
iajs-1618	140	31			PROPN
iajs-1618	140	32	…	…	PUNCT
iajs-1618	140	33	)	)	PUNCT
iajs-1618	140	34	of	of	ADP
iajs-1618	140	35	s	s	NOUN
iajs-1618	140	36	-	-	PUNCT
iajs-1618	140	37	closed	closed	ADJ
iajs-1618	140	38	submodules	submodule	NOUN
iajs-1618	140	39	of	of	ADP
iajs-1618	140	40	w	w	PROPN
iajs-1618	140	41	is	be	AUX
iajs-1618	140	42	finite	finite	ADJ
iajs-1618	140	43	.	.	PUNCT
iajs-1618	141	1	that	that	PRON
iajs-1618	141	2	is	be	AUX
iajs-1618	141	3	there	there	PRON
iajs-1618	141	4	exists	exist	VERB
iajs-1618	141	5	k	k	PROPN
iajs-1618	141	6	z+	z+	NUM
iajs-1618	141	7	such	such	ADJ
iajs-1618	141	8	that	that	SCONJ
iajs-1618	141	9	an	an	DET
iajs-1618	141	10	=	=	SYM
iajs-1618	141	11	ak	ak	PROPN
iajs-1618	141	12	for	for	ADP
iajs-1618	141	13	all	all	DET
iajs-1618	141	14	nk	nk	PROPN
iajs-1618	141	15	.	.	PUNCT
iajs-1618	142	1	recall	recall	VERB
iajs-1618	142	2	that	that	PRON
iajs-1618	142	3	,	,	PUNCT
iajs-1618	142	4	“	"	PUNCT
iajs-1618	142	5	a	a	DET
iajs-1618	142	6	noetherian	noetherian	ADJ
iajs-1618	142	7	module	module	NOUN
iajs-1618	142	8	is	be	AUX
iajs-1618	142	9	a	a	DET
iajs-1618	142	10	module	module	NOUN
iajs-1618	142	11	that	that	PRON
iajs-1618	142	12	satisfies	satisfy	VERB
iajs-1618	142	13	the	the	DET
iajs-1618	142	14	ascending	ascend	VERB
iajs-1618	142	15	chain	chain	NOUN
iajs-1618	142	16	condition	condition	NOUN
iajs-1618	142	17	on	on	ADP
iajs-1618	142	18	its	its	PRON
iajs-1618	142	19	submodules	submodule	NOUN
iajs-1618	142	20	.	.	PUNCT
iajs-1618	143	1	also	also	ADV
iajs-1618	143	2	,	,	PUNCT
iajs-1618	143	3	an	an	DET
iajs-1618	143	4	artinian	artinian	ADJ
iajs-1618	143	5	module	module	NOUN
iajs-1618	143	6	is	be	AUX
iajs-1618	143	7	a	a	DET
iajs-1618	143	8	module	module	NOUN
iajs-1618	143	9	that	that	PRON
iajs-1618	143	10	satisfies	satisfy	VERB
iajs-1618	143	11	the	the	DET
iajs-1618	143	12	descending	descend	VERB
iajs-1618	143	13	chain	chain	NOUN
iajs-1618	143	14	condition	condition	NOUN
iajs-1618	143	15	on	on	ADP
iajs-1618	143	16	its	its	PRON
iajs-1618	143	17	submodules	submodule	NOUN
iajs-1618	143	18	”	"	PUNCT
iajs-1618	143	19	.	.	PUNCT
iajs-1618	144	1	[	[	X
iajs-1618	144	2	3	3	NUM
iajs-1618	144	3	]	]	PUNCT
iajs-1618	144	4	remarks	remark	VERB
iajs-1618	144	5	2	2	NUM
iajs-1618	144	6	.	.	SYM
iajs-1618	144	7	2	2	NUM
iajs-1618	144	8	:	:	SYM
iajs-1618	144	9	1	1	X
iajs-1618	144	10	.	.	X
iajs-1618	145	1	every	every	DET
iajs-1618	145	2	noetherian	noetherian	ADJ
iajs-1618	145	3	(	(	PUNCT
iajs-1618	145	4	respectively	respectively	ADV
iajs-1618	145	5	artinian	artinian	ADJ
iajs-1618	145	6	)	)	PUNCT
iajs-1618	145	7	module	module	NOUN
iajs-1618	145	8	satisfies	satisfie	NOUN
iajs-1618	145	9	a	a	DET
iajs-1618	145	10	c	c	NOUN
iajs-1618	145	11	c	c	NOUN
iajs-1618	145	12	(	(	PUNCT
iajs-1618	145	13	respectively	respectively	ADV
iajs-1618	145	14	d	d	X
iajs-1618	145	15	c	c	NOUN
iajs-1618	145	16	c	c	NOUN
iajs-1618	145	17	)	)	PUNCT
iajs-1618	145	18	on	on	ADP
iajs-1618	145	19	s	s	VERB
iajs-1618	145	20	-closed	-close	VERB
iajs-1618	145	21	submodules	submodule	NOUN
iajs-1618	145	22	.	.	PUNCT
iajs-1618	146	1	2	2	X
iajs-1618	146	2	.	.	X
iajs-1618	146	3	if	if	SCONJ
iajs-1618	146	4	w	w	NOUN
iajs-1618	146	5	satisfies	satisfy	VERB
iajs-1618	146	6	a	a	DET
iajs-1618	146	7	c	c	NOUN
iajs-1618	146	8	c	c	NOUN
iajs-1618	146	9	(	(	PUNCT
iajs-1618	146	10	respectively	respectively	ADV
iajs-1618	146	11	d	d	PROPN
iajs-1618	146	12	c	c	NOUN
iajs-1618	146	13	c	c	NOUN
iajs-1618	146	14	)	)	PUNCT
iajs-1618	146	15	on	on	ADP
iajs-1618	146	16	closed	closed	ADJ
iajs-1618	146	17	submodules	submodule	NOUN
iajs-1618	146	18	,	,	PUNCT
iajs-1618	146	19	then	then	ADV
iajs-1618	146	20	w	w	NOUN
iajs-1618	146	21	satisfies	satisfie	NOUN
iajs-1618	146	22	a	a	DET
iajs-1618	146	23	c	c	NOUN
iajs-1618	146	24	c	c	NOUN
iajs-1618	146	25	(	(	PUNCT
iajs-1618	146	26	respectively	respectively	ADV
iajs-1618	146	27	d	d	PROPN
iajs-1618	146	28	c	c	NOUN
iajs-1618	146	29	c	c	NOUN
iajs-1618	146	30	)	)	PUNCT
iajs-1618	146	31	on	on	ADP
iajs-1618	146	32	s	s	VERB
iajs-1618	146	33	-closed	-close	VERB
iajs-1618	146	34	submodules	submodule	NOUN
iajs-1618	146	35	.	.	PUNCT
iajs-1618	147	1	proof	proof	NOUN
iajs-1618	147	2	:	:	PUNCT
iajs-1618	147	3	it	it	PRON
iajs-1618	147	4	is	be	AUX
iajs-1618	147	5	clear	clear	ADJ
iajs-1618	147	6	since	since	SCONJ
iajs-1618	147	7	every	every	DET
iajs-1618	147	8	s	s	NOUN
iajs-1618	147	9	-closed	-close	VERB
iajs-1618	147	10	submodule	submodule	NOUN
iajs-1618	147	11	in	in	ADP
iajs-1618	147	12	w	w	PROPN
iajs-1618	147	13	is	be	AUX
iajs-1618	147	14	closed	close	VERB
iajs-1618	147	15	submodule	submodule	NOUN
iajs-1618	147	16	in	in	ADP
iajs-1618	147	17	w.	w.	PROPN
iajs-1618	147	18	the	the	DET
iajs-1618	147	19	converse	converse	NOUN
iajs-1618	147	20	is	be	AUX
iajs-1618	147	21	true	true	ADJ
iajs-1618	147	22	if	if	SCONJ
iajs-1618	147	23	w	w	NOUN
iajs-1618	147	24	is	be	AUX
iajs-1618	147	25	hollow	hollow	ADJ
iajs-1618	147	26	by	by	ADP
iajs-1618	147	27	remark	remark	NOUN
iajs-1618	147	28	1.2(6	1.2(6	NUM
iajs-1618	147	29	)	)	PUNCT
iajs-1618	147	30	or	or	CCONJ
iajs-1618	147	31	uniform	uniform	NOUN
iajs-1618	147	32	module	module	NOUN
iajs-1618	147	33	,	,	PUNCT
iajs-1618	147	34	where	where	SCONJ
iajs-1618	147	35	“	"	PUNCT
iajs-1618	147	36	a	a	DET
iajs-1618	147	37	uniform	uniform	ADJ
iajs-1618	147	38	module	module	NOUN
iajs-1618	147	39	is	be	AUX
iajs-1618	147	40	a	a	DET
iajs-1618	147	41	nonzero	nonzero	NOUN
iajs-1618	147	42	module	module	NOUN
iajs-1618	147	43	w	w	ADP
iajs-1618	147	44	which	which	PRON
iajs-1618	147	45	is	be	AUX
iajs-1618	147	46	every	every	DET
iajs-1618	147	47	non	non	ADJ
iajs-1618	147	48	-	-	ADJ
iajs-1618	147	49	zero	zero	NUM
iajs-1618	147	50	submodule	submodule	NOUN
iajs-1618	147	51	of	of	ADP
iajs-1618	147	52	w	w	PROPN
iajs-1618	147	53	is	be	AUX
iajs-1618	147	54	essential	essential	ADJ
iajs-1618	147	55	in	in	ADP
iajs-1618	147	56	w	w	NOUN
iajs-1618	147	57	”	"	PUNCT
iajs-1618	147	58	.	.	PUNCT
iajs-1618	148	1	[	[	X
iajs-1618	148	2	3	3	X
iajs-1618	148	3	]	]	PUNCT
iajs-1618	148	4	recall	recall	NOUN
iajs-1618	148	5	that	that	SCONJ
iajs-1618	148	6	,	,	PUNCT
iajs-1618	148	7	“	"	PUNCT
iajs-1618	148	8	an	an	DET
iajs-1618	148	9	l	l	NOUN
iajs-1618	148	10	-	-	NOUN
iajs-1618	148	11	module	module	NOUN
iajs-1618	148	12	w	w	NOUN
iajs-1618	148	13	is	be	AUX
iajs-1618	148	14	called	call	VERB
iajs-1618	148	15	chained	chain	VERB
iajs-1618	148	16	if	if	SCONJ
iajs-1618	148	17	for	for	ADP
iajs-1618	148	18	all	all	DET
iajs-1618	148	19	submodules	submodule	NOUN
iajs-1618	148	20	c	c	PROPN
iajs-1618	148	21	and	and	CCONJ
iajs-1618	148	22	d	d	PROPN
iajs-1618	148	23	of	of	ADP
iajs-1618	148	24	w	w	NOUN
iajs-1618	149	1	either	either	CCONJ
iajs-1618	149	2	c	c	PROPN
iajs-1618	149	3	≤	≤	NUM
iajs-1618	149	4	d	d	NOUN
iajs-1618	149	5	or	or	CCONJ
iajs-1618	149	6	d	d	NOUN
iajs-1618	149	7	≤	≤	PROPN
iajs-1618	149	8	c	c	NOUN
iajs-1618	149	9	”	"	PUNCT
iajs-1618	149	10	.	.	PUNCT
iajs-1618	150	1	[	[	X
iajs-1618	150	2	7	7	X
iajs-1618	150	3	]	]	SYM
iajs-1618	150	4	proposition	proposition	NOUN
iajs-1618	150	5	2	2	NUM
iajs-1618	150	6	.	.	PUNCT
iajs-1618	150	7	3	3	NUM
iajs-1618	150	8	:	:	PUNCT
iajs-1618	150	9	let	let	VERB
iajs-1618	150	10	w	w	PART
iajs-1618	150	11	be	be	AUX
iajs-1618	150	12	a	a	DET
iajs-1618	150	13	chained	chained	ADJ
iajs-1618	150	14	l	l	NOUN
iajs-1618	150	15	-module	-module	NOUN
iajs-1618	150	16	,	,	PUNCT
iajs-1618	150	17	and	and	CCONJ
iajs-1618	150	18	let	let	VERB
iajs-1618	150	19	a	a	PRON
iajs-1618	150	20	be	be	AUX
iajs-1618	150	21	an	an	DET
iajs-1618	150	22	s	s	NOUN
iajs-1618	150	23	-	-	PUNCT
iajs-1618	150	24	closed	closed	ADJ
iajs-1618	150	25	submodule	submodule	NOUN
iajs-1618	150	26	of	of	ADP
iajs-1618	150	27	w.	w.	PROPN
iajs-1618	150	28	if	if	SCONJ
iajs-1618	150	29	w	w	PROPN
iajs-1618	150	30	satisfied	satisfy	VERB
iajs-1618	150	31	a	a	DET
iajs-1618	150	32	c	c	NOUN
iajs-1618	150	33	c	c	NOUN
iajs-1618	150	34	(	(	PUNCT
iajs-1618	150	35	respectively	respectively	ADV
iajs-1618	150	36	d	d	PROPN
iajs-1618	150	37	c	c	NOUN
iajs-1618	150	38	c	c	NOUN
iajs-1618	150	39	)	)	PUNCT
iajs-1618	150	40	on	on	ADP
iajs-1618	150	41	s	s	VERB
iajs-1618	150	42	-closed	-close	VERB
iajs-1618	150	43	submodules	submodule	NOUN
iajs-1618	150	44	,	,	PUNCT
iajs-1618	150	45	then	then	ADV
iajs-1618	150	46	a	a	DET
iajs-1618	150	47	satisfies	satisfie	NOUN
iajs-1618	150	48	the	the	DET
iajs-1618	150	49	a	a	DET
iajs-1618	150	50	c	c	NOUN
iajs-1618	150	51	c	c	NOUN
iajs-1618	150	52	(	(	PUNCT
iajs-1618	150	53	respectively	respectively	ADV
iajs-1618	150	54	d	d	PROPN
iajs-1618	150	55	c	c	NOUN
iajs-1618	150	56	c	c	NOUN
iajs-1618	150	57	)	)	PUNCT
iajs-1618	150	58	on	on	ADP
iajs-1618	150	59	s	s	NOUN
iajs-1618	150	60	-	-	PUNCT
iajs-1618	150	61	closed	closed	ADJ
iajs-1618	150	62	submodules	submodule	NOUN
iajs-1618	150	63	.	.	PUNCT
iajs-1618	151	1	proof	proof	NOUN
iajs-1618	151	2	:	:	PUNCT
iajs-1618	151	3	assume	assume	VERB
iajs-1618	151	4	w	w	NOUN
iajs-1618	151	5	satisfies	satisfie	NOUN
iajs-1618	151	6	a	a	DET
iajs-1618	151	7	c	c	NOUN
iajs-1618	151	8	c	c	NOUN
iajs-1618	151	9	on	on	ADP
iajs-1618	151	10	s	s	NOUN
iajs-1618	151	11	-	-	PUNCT
iajs-1618	151	12	closed	closed	ADJ
iajs-1618	151	13	submodules	submodule	NOUN
iajs-1618	151	14	and	and	CCONJ
iajs-1618	151	15	a1	a1	NOUN
iajs-1618	151	16			PROPN
iajs-1618	151	17	a2	a2	PROPN
iajs-1618	151	18			PROPN
iajs-1618	151	19	…	…	PUNCT
iajs-1618	151	20	be	be	AUX
iajs-1618	151	21	ascending	ascend	VERB
iajs-1618	151	22	chain	chain	NOUN
iajs-1618	151	23	of	of	ADP
iajs-1618	151	24	sclosed	sclose	VERB
iajs-1618	151	25	submodules	submodule	NOUN
iajs-1618	151	26	of	of	ADP
iajs-1618	151	27	a.	a.	NOUN
iajs-1618	151	28	since	since	SCONJ
iajs-1618	151	29	a	a	PRON
iajs-1618	151	30	is	be	AUX
iajs-1618	151	31	an	an	DET
iajs-1618	151	32	s	s	NOUN
iajs-1618	151	33	-	-	PUNCT
iajs-1618	151	34	closed	closed	ADJ
iajs-1618	151	35	submodule	submodule	NOUN
iajs-1618	151	36	of	of	ADP
iajs-1618	151	37	w	w	PROPN
iajs-1618	151	38	and	and	CCONJ
iajs-1618	151	39	w	w	PROPN
iajs-1618	151	40	satisfy	satisfy	NOUN
iajs-1618	151	41	chained	chain	VERB
iajs-1618	151	42	condition	condition	NOUN
iajs-1618	151	43	,	,	PUNCT
iajs-1618	151	44	so	so	CCONJ
iajs-1618	151	45	by	by	ADP
iajs-1618	151	46	[	[	X
iajs-1618	151	47	1	1	NUM
iajs-1618	151	48	,	,	PUNCT
iajs-1618	151	49	prop.2.11	prop.2.11	PROPN
iajs-1618	151	50	,	,	PUNCT
iajs-1618	151	51	p.345	p.345	NOUN
iajs-1618	151	52	]	]	PUNCT
iajs-1618	152	1	ai	ai	VERB
iajs-1618	152	2	is	be	AUX
iajs-1618	152	3	an	an	DET
iajs-1618	152	4	s	s	NOUN
iajs-1618	152	5	-	-	PUNCT
iajs-1618	152	6	closed	closed	ADJ
iajs-1618	152	7	submodule	submodule	NOUN
iajs-1618	152	8	of	of	ADP
iajs-1618	152	9	w	w	PROPN
iajs-1618	152	10	for	for	ADP
iajs-1618	152	11	each	each	DET
iajs-1618	152	12	i	i	NOUN
iajs-1618	152	13	=	=	NOUN
iajs-1618	152	14	1	1	NUM
iajs-1618	152	15	,	,	PUNCT
iajs-1618	152	16	2	2	NUM
iajs-1618	152	17	,	,	PUNCT
iajs-1618	152	18	…	…	PUNCT
iajs-1618	152	19	.	.	PUNCT
iajs-1618	153	1	hence	hence	ADV
iajs-1618	153	2	a1	a1	PROPN
iajs-1618	153	3			PROPN
iajs-1618	153	4	a2	a2	PROPN
iajs-1618	153	5			PROPN
iajs-1618	153	6	…	…	PUNCT
iajs-1618	153	7	be	be	AUX
iajs-1618	153	8	ascending	ascend	VERB
iajs-1618	153	9	chain	chain	NOUN
iajs-1618	153	10	of	of	ADP
iajs-1618	153	11	s	s	NOUN
iajs-1618	153	12	-	-	PUNCT
iajs-1618	153	13	closed	closed	ADJ
iajs-1618	153	14	submodules	submodule	NOUN
iajs-1618	153	15	of	of	ADP
iajs-1618	153	16	w.	w.	PROPN
iajs-1618	153	17	but	but	CCONJ
iajs-1618	153	18	w	w	PROPN
iajs-1618	153	19	satisfies	satisfie	NOUN
iajs-1618	153	20	a	a	DET
iajs-1618	153	21	c	c	NOUN
iajs-1618	153	22	c	c	NOUN
iajs-1618	153	23	on	on	ADP
iajs-1618	153	24	sclosed	sclose	VERB
iajs-1618	153	25	submodules	submodule	NOUN
iajs-1618	153	26	,	,	PUNCT
iajs-1618	153	27	thus	thus	ADV
iajs-1618	153	28	kz+	kz+	X
iajs-1618	153	29	such	such	ADJ
iajs-1618	153	30	that	that	SCONJ
iajs-1618	153	31	an	an	DET
iajs-1618	153	32	=	=	SYM
iajs-1618	153	33	ak	ak	PROPN
iajs-1618	153	34	for	for	ADP
iajs-1618	153	35	all	all	DET
iajs-1618	153	36	nk	nk	NOUN
iajs-1618	153	37	.	.	PUNCT
iajs-1618	154	1	that	that	PRON
iajs-1618	154	2	is	be	AUX
iajs-1618	154	3	a	a	DET
iajs-1618	154	4	satisfies	satisfie	NOUN
iajs-1618	154	5	a	a	DET
iajs-1618	154	6	c	c	NOUN
iajs-1618	154	7	c	c	NOUN
iajs-1618	154	8	on	on	ADP
iajs-1618	154	9	sclosed	sclose	VERB
iajs-1618	154	10	submodules	submodule	NOUN
iajs-1618	154	11	.	.	PUNCT
iajs-1618	155	1	https://en.wikipedia.org/wiki/module_%28mathematics%29	https://en.wikipedia.org/wiki/module_%28mathematics%29	NOUN
iajs-1618	155	2	https://en.wikipedia.org/wiki/ascending_chain_condition	https://en.wikipedia.org/wiki/ascending_chain_condition	NUM
iajs-1618	155	3	https://en.wikipedia.org/wiki/ascending_chain_condition	https://en.wikipedia.org/wiki/ascending_chain_condition	NOUN
iajs-1618	155	4	https://en.wikipedia.org/wiki/submodule	https://en.wikipedia.org/wiki/submodule	ADJ
iajs-1618	155	5	https://en.wikipedia.org/wiki/module_%28mathematics%29	https://en.wikipedia.org/wiki/module_%28mathematics%29	NOUN
iajs-1618	155	6	https://en.wikipedia.org/wiki/descending_chain_condition	https://en.wikipedia.org/wiki/descending_chain_condition	NOUN
iajs-1618	155	7	mathematics	mathematic	NOUN
iajs-1618	155	8	|232	|232	PUNCT
iajs-1618	156	1	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	NOUN
iajs-1618	156	2	7102	7102	NUM
iajs-1618	156	3	(	(	PUNCT
iajs-1618	156	4	عام	عام	PROPN
iajs-1618	156	5	3	3	NUM
iajs-1618	156	6	(	(	PUNCT
iajs-1618	156	7	العدد	العدد	PROPN
iajs-1618	156	8	)	)	PUNCT
iajs-1618	156	9	30مجلة	30مجلة	PROPN
iajs-1618	156	10	إبن	إبن	VERB
iajs-1618	156	11	الهيثم	الهيثم	ADJ
iajs-1618	156	12	للعلوم	للعلوم	NOUN
iajs-1618	156	13	الصرفة	الصرفة	NOUN
iajs-1618	157	1	و	و	PRON
iajs-1618	157	2	التطبيقية	التطبيقية	ADV
iajs-1618	157	3	المجلد	المجلد	ADV
iajs-1618	157	4	)	)	PUNCT
iajs-1618	158	1	ibn	ibn	PROPN
iajs-1618	158	2	al	al	PROPN
iajs-1618	158	3	-	-	PUNCT
iajs-1618	158	4	haitham	haitham	PROPN
iajs-1618	158	5	j.	j.	PROPN
iajs-1618	158	6	for	for	ADP
iajs-1618	158	7	pure	pure	PROPN
iajs-1618	158	8	&	&	CCONJ
iajs-1618	158	9	appl	appl	PROPN
iajs-1618	158	10	.	.	PUNCT
iajs-1618	159	1	sci	sci	PROPN
iajs-1618	159	2	.	.	PUNCT
iajs-1618	160	1	vol.03	vol.03	PROPN
iajs-1618	160	2	(	(	PUNCT
iajs-1618	160	3	3	3	NUM
iajs-1618	160	4	)	)	PUNCT
iajs-1618	160	5	2017	2017	NUM
iajs-1618	160	6	similarly	similarly	ADV
iajs-1618	160	7	,	,	PUNCT
iajs-1618	160	8	if	if	SCONJ
iajs-1618	160	9	w	w	ADJ
iajs-1618	160	10	satisfies	satisfie	NOUN
iajs-1618	160	11	d	d	X
iajs-1618	160	12	c	c	PROPN
iajs-1618	160	13	c	c	NOUN
iajs-1618	160	14	on	on	ADP
iajs-1618	160	15	sclosed	sclose	VERB
iajs-1618	160	16	submodules	submodule	NOUN
iajs-1618	160	17	,	,	PUNCT
iajs-1618	160	18	then	then	ADV
iajs-1618	160	19	a	a	DET
iajs-1618	160	20	satisfies	satisfie	NOUN
iajs-1618	160	21	d	d	X
iajs-1618	160	22	c	c	PROPN
iajs-1618	160	23	c	c	NOUN
iajs-1618	160	24	on	on	ADP
iajs-1618	160	25	sclosed	sclose	VERB
iajs-1618	160	26	submodules	submodule	NOUN
iajs-1618	160	27	of	of	ADP
iajs-1618	160	28	a.	a.	NOUN
iajs-1618	160	29	proposition	proposition	NOUN
iajs-1618	160	30	2	2	NUM
iajs-1618	160	31	.	.	NOUN
iajs-1618	160	32	4	4	NUM
iajs-1618	160	33	:	:	PUNCT
iajs-1618	160	34	let	let	VERB
iajs-1618	160	35	w	w	X
iajs-1618	160	36	=	=	PUNCT
iajs-1618	160	37	w1⊕w2	w1⊕w2	PROPN
iajs-1618	160	38	be	be	VERB
iajs-1618	160	39	an	an	DET
iajs-1618	160	40	l	l	NOUN
iajs-1618	160	41	-	-	PUNCT
iajs-1618	160	42	module	module	NOUN
iajs-1618	160	43	satisfies	satisfie	NOUN
iajs-1618	160	44	a	a	DET
iajs-1618	160	45	c	c	NOUN
iajs-1618	160	46	c	c	NOUN
iajs-1618	160	47	(	(	PUNCT
iajs-1618	160	48	respectively	respectively	ADV
iajs-1618	160	49	d	d	PROPN
iajs-1618	160	50	c	c	NOUN
iajs-1618	160	51	c	c	NOUN
iajs-1618	160	52	)	)	PUNCT
iajs-1618	160	53	on	on	ADP
iajs-1618	160	54	sclosed	sclose	VERB
iajs-1618	160	55	submodules	submodule	NOUN
iajs-1618	160	56	.	.	PUNCT
iajs-1618	161	1	then	then	ADV
iajs-1618	161	2	w1	w1	PROPN
iajs-1618	161	3	and	and	CCONJ
iajs-1618	161	4	w2	w2	NOUN
iajs-1618	161	5	satisfy	satisfy	VERB
iajs-1618	161	6	a	a	DET
iajs-1618	161	7	c	c	NOUN
iajs-1618	161	8	c	c	NOUN
iajs-1618	161	9	(	(	PUNCT
iajs-1618	161	10	respectively	respectively	ADV
iajs-1618	161	11	d	d	PROPN
iajs-1618	161	12	c	c	NOUN
iajs-1618	161	13	c	c	NOUN
iajs-1618	161	14	)	)	PUNCT
iajs-1618	161	15	on	on	ADP
iajs-1618	161	16	sclosed	sclose	VERB
iajs-1618	161	17	submodules	submodule	NOUN
iajs-1618	161	18	.	.	PUNCT
iajs-1618	162	1	proof	proof	NOUN
iajs-1618	162	2	:	:	PUNCT
iajs-1618	162	3	suppose	suppose	VERB
iajs-1618	162	4	w	w	ADP
iajs-1618	162	5	satisfies	satisfie	NOUN
iajs-1618	162	6	a	a	DET
iajs-1618	162	7	c	c	NOUN
iajs-1618	162	8	c	c	NOUN
iajs-1618	162	9	(	(	PUNCT
iajs-1618	162	10	respectively	respectively	ADV
iajs-1618	162	11	d	d	PROPN
iajs-1618	162	12	c	c	NOUN
iajs-1618	162	13	c	c	NOUN
iajs-1618	162	14	)	)	PUNCT
iajs-1618	162	15	on	on	ADP
iajs-1618	162	16	s	s	NOUN
iajs-1618	162	17	-	-	PUNCT
iajs-1618	162	18	closed	closed	ADJ
iajs-1618	162	19	submodule	submodule	NOUN
iajs-1618	162	20	and	and	CCONJ
iajs-1618	162	21	a1	a1	NOUN
iajs-1618	162	22			PROPN
iajs-1618	162	23	a2	a2	PROPN
iajs-1618	162	24			PROPN
iajs-1618	162	25	…	…	PUNCT
iajs-1618	162	26	(	(	PUNCT
iajs-1618	162	27	respectively	respectively	ADV
iajs-1618	162	28	a1	a1	NOUN
iajs-1618	162	29			PROPN
iajs-1618	162	30	a2	a2	PROPN
iajs-1618	162	31			PROPN
iajs-1618	162	32	…	…	PUNCT
iajs-1618	162	33	)	)	PUNCT
iajs-1618	162	34	be	be	AUX
iajs-1618	162	35	ascending	ascend	VERB
iajs-1618	162	36	(	(	PUNCT
iajs-1618	162	37	respectively	respectively	ADV
iajs-1618	162	38	descending	descending	NOUN
iajs-1618	162	39	)	)	PUNCT
iajs-1618	162	40	chain	chain	NOUN
iajs-1618	162	41	of	of	ADP
iajs-1618	162	42	s	s	NOUN
iajs-1618	162	43	-	-	PUNCT
iajs-1618	162	44	closed	closed	ADJ
iajs-1618	162	45	submodules	submodule	NOUN
iajs-1618	162	46	of	of	ADP
iajs-1618	162	47	w1	w1	NOUN
iajs-1618	162	48	.	.	PUNCT
iajs-1618	163	1	thus	thus	ADV
iajs-1618	163	2	a1⊕w2	a1⊕w2	PROPN
iajs-1618	163	3	,	,	PUNCT
iajs-1618	163	4	a2⊕	a2⊕	ADJ
iajs-1618	163	5	w2	w2	NOUN
iajs-1618	163	6	,	,	PUNCT
iajs-1618	163	7	…	…	PUNCT
iajs-1618	163	8	are	be	AUX
iajs-1618	163	9	s	s	NOUN
iajs-1618	163	10	-	-	PUNCT
iajs-1618	163	11	closed	closed	ADJ
iajs-1618	163	12	submodules	submodule	NOUN
iajs-1618	163	13	of	of	ADP
iajs-1618	163	14	w1⊕w2	w1⊕w2	PROPN
iajs-1618	163	15	,	,	PUNCT
iajs-1618	163	16	by	by	ADP
iajs-1618	163	17	[	[	X
iajs-1618	163	18	1	1	NUM
iajs-1618	163	19	,	,	PUNCT
iajs-1618	163	20	prop.2.5	prop.2.5	PROPN
iajs-1618	163	21	]	]	PUNCT
iajs-1618	163	22	.	.	PUNCT
iajs-1618	164	1	that	that	PRON
iajs-1618	164	2	is	be	AUX
iajs-1618	164	3	a1⊕w2	a1⊕w2	PROPN
iajs-1618	164	4			PROPN
iajs-1618	164	5	a2⊕	a2⊕	PROPN
iajs-1618	164	6	w2	w2	NOUN
iajs-1618	164	7			PROPN
iajs-1618	164	8	…	…	PUNCT
iajs-1618	164	9	(	(	PUNCT
iajs-1618	164	10	respectively	respectively	ADV
iajs-1618	164	11	a1⊕	a1⊕	PROPN
iajs-1618	164	12	w2	w2	NOUN
iajs-1618	164	13			PROPN
iajs-1618	164	14	a2⊕	a2⊕	PROPN
iajs-1618	164	15	w2	w2	NOUN
iajs-1618	164	16			PROPN
iajs-1618	164	17	…	…	PUNCT
iajs-1618	164	18	)	)	PUNCT
iajs-1618	165	1	is	be	AUX
iajs-1618	165	2	a	a	DET
iajs-1618	165	3	chain	chain	NOUN
iajs-1618	165	4	of	of	ADP
iajs-1618	165	5	s	s	NOUN
iajs-1618	165	6	-	-	PUNCT
iajs-1618	165	7	closed	closed	ADJ
iajs-1618	165	8	submodules	submodule	NOUN
iajs-1618	165	9	of	of	ADP
iajs-1618	165	10	w	w	NOUN
iajs-1618	165	11	,	,	PUNCT
iajs-1618	165	12	but	but	CCONJ
iajs-1618	165	13	w	w	NOUN
iajs-1618	165	14	satisfies	satisfie	NOUN
iajs-1618	165	15	a	a	DET
iajs-1618	165	16	c	c	NOUN
iajs-1618	165	17	c	c	NOUN
iajs-1618	165	18	(	(	PUNCT
iajs-1618	165	19	respectively	respectively	ADV
iajs-1618	165	20	d	d	PROPN
iajs-1618	165	21	c	c	NOUN
iajs-1618	165	22	c	c	NOUN
iajs-1618	165	23	)	)	PUNCT
iajs-1618	165	24	on	on	ADP
iajs-1618	165	25	s	s	NOUN
iajs-1618	165	26	-	-	PUNCT
iajs-1618	165	27	closed	closed	ADJ
iajs-1618	165	28	submodules	submodule	NOUN
iajs-1618	165	29	.	.	PUNCT
iajs-1618	166	1	so	so	ADV
iajs-1618	166	2	there	there	PRON
iajs-1618	166	3	exists	exist	VERB
iajs-1618	166	4	kz+	kz+	NOUN
iajs-1618	166	5	such	such	ADJ
iajs-1618	166	6	that	that	SCONJ
iajs-1618	166	7	an⊕	an⊕	CCONJ
iajs-1618	166	8	w2	w2	PROPN
iajs-1618	166	9	=	=	PROPN
iajs-1618	166	10	ak	ak	PROPN
iajs-1618	166	11	⊕	⊕	PROPN
iajs-1618	166	12	w2	w2	NOUN
iajs-1618	166	13	for	for	ADP
iajs-1618	166	14	all	all	PRON
iajs-1618	166	15	n	n	PRON
iajs-1618	166	16			NUM
iajs-1618	166	17	k.	k.	PUNCT
iajs-1618	167	1	so	so	ADV
iajs-1618	167	2	an	an	DET
iajs-1618	167	3	=	=	X
iajs-1618	167	4	ak	ak	PROPN
iajs-1618	167	5	for	for	ADP
iajs-1618	167	6	all	all	DET
iajs-1618	167	7	nk	nk	NOUN
iajs-1618	167	8	.	.	PUNCT
iajs-1618	168	1	hence	hence	ADV
iajs-1618	168	2	w1	w1	NOUN
iajs-1618	168	3	satisfies	satisfie	NOUN
iajs-1618	168	4	a	a	DET
iajs-1618	168	5	c	c	NOUN
iajs-1618	168	6	c	c	NOUN
iajs-1618	168	7	(	(	PUNCT
iajs-1618	168	8	respectively	respectively	ADV
iajs-1618	168	9	d	d	PROPN
iajs-1618	168	10	c	c	NOUN
iajs-1618	168	11	c	c	NOUN
iajs-1618	168	12	)	)	PUNCT
iajs-1618	168	13	on	on	ADP
iajs-1618	168	14	s	s	NOUN
iajs-1618	168	15	-	-	PUNCT
iajs-1618	168	16	close	close	ADJ
iajs-1618	168	17	submodules	submodule	NOUN
iajs-1618	168	18	.	.	PUNCT
iajs-1618	169	1	by	by	ADP
iajs-1618	169	2	the	the	DET
iajs-1618	169	3	same	same	ADJ
iajs-1618	169	4	way	way	NOUN
iajs-1618	169	5	of	of	ADP
iajs-1618	169	6	proof	proof	NOUN
iajs-1618	169	7	,	,	PUNCT
iajs-1618	169	8	w2	w2	NOUN
iajs-1618	169	9	satisfies	satisfy	VERB
iajs-1618	169	10	a	a	DET
iajs-1618	169	11	c	c	NOUN
iajs-1618	169	12	c	c	NOUN
iajs-1618	169	13	(	(	PUNCT
iajs-1618	169	14	respectively	respectively	ADV
iajs-1618	169	15	d	d	PROPN
iajs-1618	169	16	c	c	NOUN
iajs-1618	169	17	c	c	NOUN
iajs-1618	169	18	)	)	PUNCT
iajs-1618	169	19	on	on	ADP
iajs-1618	169	20	s	s	NOUN
iajs-1618	169	21	-	-	PUNCT
iajs-1618	169	22	closed	closed	ADJ
iajs-1618	169	23	submodules	submodule	NOUN
iajs-1618	169	24	.	.	PUNCT
iajs-1618	170	1	recall	recall	VERB
iajs-1618	170	2	that	that	PRON
iajs-1618	170	3	,	,	PUNCT
iajs-1618	170	4	“	"	PUNCT
iajs-1618	170	5	a	a	DET
iajs-1618	170	6	submodule	submodule	NOUN
iajs-1618	170	7	c	c	PROPN
iajs-1618	170	8	is	be	AUX
iajs-1618	170	9	fully	fully	ADV
iajs-1618	170	10	invariant	invariant	ADJ
iajs-1618	170	11	in	in	ADP
iajs-1618	170	12	w	w	PROPN
iajs-1618	170	13	if	if	SCONJ
iajs-1618	170	14	f(c	f(c	PROPN
iajs-1618	170	15	)	)	PUNCT
iajs-1618	170	16			PROPN
iajs-1618	170	17	c	c	PROPN
iajs-1618	170	18	for	for	ADP
iajs-1618	170	19	all	all	DET
iajs-1618	170	20	f	f	PROPN
iajs-1618	170	21			PROPN
iajs-1618	170	22	endr(w	endr(w	PROPN
iajs-1618	170	23	)	)	PUNCT
iajs-1618	170	24	”	"	PUNCT
iajs-1618	170	25	.	.	PUNCT
iajs-1618	171	1	[	[	X
iajs-1618	171	2	3	3	X
iajs-1618	171	3	]	]	PUNCT
iajs-1618	171	4	proposition	proposition	NOUN
iajs-1618	171	5	2	2	NUM
iajs-1618	171	6	.	.	NOUN
iajs-1618	171	7	5	5	NUM
iajs-1618	171	8	:	:	PUNCT
iajs-1618	171	9	let	let	VERB
iajs-1618	171	10	w	w	X
iajs-1618	171	11	=	=	PUNCT
iajs-1618	171	12	w1⊕w2	w1⊕w2	PROPN
iajs-1618	171	13	be	be	VERB
iajs-1618	171	14	an	an	DET
iajs-1618	171	15	r	r	NOUN
iajs-1618	171	16	-	-	PUNCT
iajs-1618	171	17	module	module	NOUN
iajs-1618	171	18	where	where	SCONJ
iajs-1618	171	19	w1	w1	NOUN
iajs-1618	171	20	and	and	CCONJ
iajs-1618	171	21	w2	w2	NOUN
iajs-1618	171	22	are	be	AUX
iajs-1618	171	23	s	s	NOUN
iajs-1618	171	24	-	-	PUNCT
iajs-1618	171	25	closed	closed	ADJ
iajs-1618	171	26	submodules	submodule	NOUN
iajs-1618	171	27	of	of	ADP
iajs-1618	171	28	w	w	PROPN
iajs-1618	171	29	.	.	PUNCT
iajs-1618	172	1	then	then	ADV
iajs-1618	172	2	w	w	NOUN
iajs-1618	172	3	satisfies	satisfie	NOUN
iajs-1618	172	4	a	a	DET
iajs-1618	172	5	c	c	NOUN
iajs-1618	172	6	c	c	NOUN
iajs-1618	172	7	(	(	PUNCT
iajs-1618	172	8	respectively	respectively	ADV
iajs-1618	172	9	d	d	PROPN
iajs-1618	172	10	c	c	NOUN
iajs-1618	172	11	c	c	NOUN
iajs-1618	172	12	)	)	PUNCT
iajs-1618	172	13	on	on	ADP
iajs-1618	172	14	nonzero	nonzero	PROPN
iajs-1618	172	15	sclosed	sclose	VERB
iajs-1618	172	16	submodules	submodule	NOUN
iajs-1618	172	17	if	if	SCONJ
iajs-1618	172	18	and	and	CCONJ
iajs-1618	172	19	only	only	ADV
iajs-1618	172	20	if	if	SCONJ
iajs-1618	172	21	w1	w1	NOUN
iajs-1618	172	22	and	and	CCONJ
iajs-1618	172	23	w2	w2	NOUN
iajs-1618	172	24	satisfy	satisfy	VERB
iajs-1618	172	25	a	a	DET
iajs-1618	172	26	c	c	NOUN
iajs-1618	172	27	c	c	NOUN
iajs-1618	172	28	(	(	PUNCT
iajs-1618	172	29	respectively	respectively	ADV
iajs-1618	172	30	d	d	PROPN
iajs-1618	172	31	c	c	NOUN
iajs-1618	172	32	c	c	NOUN
iajs-1618	172	33	)	)	PUNCT
iajs-1618	172	34	on	on	ADP
iajs-1618	172	35	nonzero	nonzero	PROPN
iajs-1618	172	36	s	s	PART
iajs-1618	172	37	-	-	PUNCT
iajs-1618	172	38	closed	closed	ADJ
iajs-1618	172	39	submodules	submodule	NOUN
iajs-1618	172	40	,	,	PUNCT
iajs-1618	172	41	provided	provide	VERB
iajs-1618	172	42	that	that	SCONJ
iajs-1618	172	43	every	every	DET
iajs-1618	172	44	sclosed	sclose	VERB
iajs-1618	172	45	submodule	submodule	NOUN
iajs-1618	172	46	of	of	ADP
iajs-1618	172	47	w	w	PROPN
iajs-1618	172	48	is	be	AUX
iajs-1618	172	49	a	a	DET
iajs-1618	172	50	fully	fully	ADV
iajs-1618	172	51	invariant	invariant	ADJ
iajs-1618	172	52	.	.	PUNCT
iajs-1618	173	1	proof	proof	NOUN
iajs-1618	173	2	:	:	PUNCT
iajs-1618	173	3	(	(	PUNCT
iajs-1618	173	4			NOUN
iajs-1618	173	5	)	)	PUNCT
iajs-1618	173	6	see	see	VERB
iajs-1618	173	7	proposition	proposition	NOUN
iajs-1618	173	8	2.4	2.4	NUM
iajs-1618	173	9	.	.	PUNCT
iajs-1618	174	1	(	(	PUNCT
iajs-1618	174	2			X
iajs-1618	174	3	)	)	PUNCT
iajs-1618	174	4	suppose	suppose	VERB
iajs-1618	174	5	w1	w1	NOUN
iajs-1618	174	6	and	and	CCONJ
iajs-1618	174	7	w2	w2	NOUN
iajs-1618	174	8	satisfy	satisfy	VERB
iajs-1618	174	9	a	a	DET
iajs-1618	174	10	c	c	NOUN
iajs-1618	174	11	c	c	NOUN
iajs-1618	174	12	(	(	PUNCT
iajs-1618	174	13	respectively	respectively	ADV
iajs-1618	174	14	d	d	PROPN
iajs-1618	174	15	c	c	NOUN
iajs-1618	174	16	c	c	NOUN
iajs-1618	174	17	)	)	PUNCT
iajs-1618	174	18	on	on	ADP
iajs-1618	174	19	s	s	NOUN
iajs-1618	174	20	-	-	PUNCT
iajs-1618	174	21	closed	closed	ADJ
iajs-1618	174	22	submodules	submodule	NOUN
iajs-1618	174	23	,	,	PUNCT
iajs-1618	174	24	to	to	PART
iajs-1618	174	25	prove	prove	VERB
iajs-1618	174	26	w	w	NOUN
iajs-1618	174	27	satisfy	satisfy	NOUN
iajs-1618	174	28	a	a	DET
iajs-1618	174	29	c	c	NOUN
iajs-1618	174	30	c	c	NOUN
iajs-1618	174	31	(	(	PUNCT
iajs-1618	174	32	respectively	respectively	ADV
iajs-1618	174	33	d	d	PROPN
iajs-1618	174	34	c	c	NOUN
iajs-1618	174	35	c	c	NOUN
iajs-1618	174	36	)	)	PUNCT
iajs-1618	174	37	on	on	ADP
iajs-1618	174	38	s	s	NOUN
iajs-1618	174	39	-	-	PUNCT
iajs-1618	174	40	closed	closed	ADJ
iajs-1618	174	41	submodules	submodule	NOUN
iajs-1618	174	42	.	.	PUNCT
iajs-1618	175	1	let	let	AUX
iajs-1618	175	2	and	and	CCONJ
iajs-1618	175	3	a1	a1	VERB
iajs-1618	175	4			PROPN
iajs-1618	175	5	a2	a2	PROPN
iajs-1618	175	6			PROPN
iajs-1618	175	7	…	…	PUNCT
iajs-1618	175	8	(	(	PUNCT
iajs-1618	175	9	respectively	respectively	ADV
iajs-1618	175	10	a1	a1	NOUN
iajs-1618	175	11			PROPN
iajs-1618	175	12	a2	a2	PROPN
iajs-1618	175	13			PROPN
iajs-1618	175	14	…	…	PUNCT
iajs-1618	175	15	)	)	PUNCT
iajs-1618	175	16	be	be	AUX
iajs-1618	175	17	ascending	ascend	VERB
iajs-1618	175	18	(	(	PUNCT
iajs-1618	175	19	respectively	respectively	ADV
iajs-1618	175	20	descending	descending	NOUN
iajs-1618	175	21	)	)	PUNCT
iajs-1618	175	22	chain	chain	NOUN
iajs-1618	175	23	of	of	ADP
iajs-1618	175	24	s	s	NOUN
iajs-1618	175	25	-	-	PUNCT
iajs-1618	175	26	closed	closed	ADJ
iajs-1618	175	27	submodules	submodule	NOUN
iajs-1618	175	28	of	of	ADP
iajs-1618	175	29	w.	w.	NOUN
iajs-1618	175	30	let	let	VERB
iajs-1618	175	31	πi	πi	ADV
iajs-1618	175	32	:	:	PUNCT
iajs-1618	175	33	w	w	PROPN
iajs-1618	175	34			PROPN
iajs-1618	175	35	wi	wi	PROPN
iajs-1618	175	36	be	be	AUX
iajs-1618	175	37	a	a	DET
iajs-1618	175	38	projection	projection	NOUN
iajs-1618	175	39	map	map	NOUN
iajs-1618	175	40	for	for	ADP
iajs-1618	175	41	each	each	DET
iajs-1618	175	42	i	i	NOUN
iajs-1618	175	43	=	=	NOUN
iajs-1618	175	44	1	1	NUM
iajs-1618	175	45	,	,	PUNCT
iajs-1618	175	46	2	2	NUM
iajs-1618	175	47	.	.	PUNCT
iajs-1618	175	48	suppose	suppose	VERB
iajs-1618	175	49	that	that	SCONJ
iajs-1618	175	50	ai	ai	VERB
iajs-1618	175	51	=	=	PUNCT
iajs-1618	175	52	(	(	PUNCT
iajs-1618	175	53	ai	ai	PROPN
iajs-1618	175	54	∩	∩	ADJ
iajs-1618	175	55	w1	w1	NOUN
iajs-1618	175	56	)	)	PUNCT
iajs-1618	175	57	⊕	⊕	PROPN
iajs-1618	175	58	(	(	PUNCT
iajs-1618	175	59	ai	ai	VERB
iajs-1618	175	60	∩w2	∩w2	NOUN
iajs-1618	175	61	)	)	PUNCT
iajs-1618	175	62	by	by	ADP
iajs-1618	175	63	[	[	X
iajs-1618	175	64	10	10	NUM
iajs-1618	175	65	,	,	PUNCT
iajs-1618	175	66	lemma.2.1	lemma.2.1	NOUN
iajs-1618	175	67	]	]	X
iajs-1618	175	68	.	.	PUNCT
iajs-1618	176	1	note	note	VERB
iajs-1618	176	2	that	that	SCONJ
iajs-1618	176	3	,	,	PUNCT
iajs-1618	176	4	ai	ai	INTJ
iajs-1618	176	5	,	,	PUNCT
iajs-1618	176	6	w1	w1	NOUN
iajs-1618	176	7	and	and	CCONJ
iajs-1618	176	8	w2	w2	NOUN
iajs-1618	176	9	are	be	AUX
iajs-1618	176	10	s	s	NOUN
iajs-1618	176	11	-	-	PUNCT
iajs-1618	176	12	closed	closed	ADJ
iajs-1618	176	13	submodules	submodule	NOUN
iajs-1618	176	14	of	of	ADP
iajs-1618	176	15	w	w	PROPN
iajs-1618	176	16	,	,	PUNCT
iajs-1618	176	17	for	for	ADP
iajs-1618	176	18	each	each	DET
iajs-1618	176	19	i.	i.	NOUN
iajs-1618	176	20	thus	thus	ADV
iajs-1618	176	21	by	by	ADP
iajs-1618	176	22	[	[	X
iajs-1618	176	23	1	1	NUM
iajs-1618	176	24	,	,	PUNCT
iajs-1618	176	25	remarks	remark	NOUN
iajs-1618	176	26	and	and	CCONJ
iajs-1618	176	27	examples	example	NOUN
iajs-1618	176	28	2.2	2.2	NUM
iajs-1618	176	29	(	(	PUNCT
iajs-1618	176	30	3	3	NUM
iajs-1618	176	31	)	)	PUNCT
iajs-1618	176	32	]	]	PUNCT
iajs-1618	177	1	(	(	PUNCT
iajs-1618	177	2	ai	ai	PROPN
iajs-1618	177	3	∩	∩	ADJ
iajs-1618	177	4	w1	w1	NOUN
iajs-1618	177	5	)	)	PUNCT
iajs-1618	177	6	and	and	CCONJ
iajs-1618	177	7	(	(	PUNCT
iajs-1618	177	8	ai	ai	VERB
iajs-1618	177	9	∩w2	∩w2	NOUN
iajs-1618	177	10	)	)	PUNCT
iajs-1618	177	11	are	be	AUX
iajs-1618	177	12	s	s	NOUN
iajs-1618	177	13	-	-	PUNCT
iajs-1618	177	14	closed	closed	ADJ
iajs-1618	177	15	submodules	submodule	NOUN
iajs-1618	177	16	of	of	ADP
iajs-1618	177	17	w.	w.	PROPN
iajs-1618	177	18	since	since	SCONJ
iajs-1618	177	19	(	(	PUNCT
iajs-1618	177	20	ai	ai	PROPN
iajs-1618	177	21	∩	∩	ADJ
iajs-1618	177	22	w1	w1	NOUN
iajs-1618	177	23	)	)	PUNCT
iajs-1618	177	24			PROPN
iajs-1618	177	25	w1	w1	PROPN
iajs-1618	177	26			PROPN
iajs-1618	177	27	w	w	PROPN
iajs-1618	177	28	,	,	PUNCT
iajs-1618	177	29	so	so	ADV
iajs-1618	177	30	by	by	ADP
iajs-1618	177	31	[	[	X
iajs-1618	177	32	1	1	NUM
iajs-1618	177	33	,	,	PUNCT
iajs-1618	177	34	prop.2.8	prop.2.8	NOUN
iajs-1618	177	35	,	,	PUNCT
iajs-1618	177	36	p.345	p.345	X
iajs-1618	177	37	]	]	PUNCT
iajs-1618	177	38	(	(	PUNCT
iajs-1618	177	39	ai	ai	PROPN
iajs-1618	177	40	∩	∩	ADJ
iajs-1618	177	41	w1	w1	NOUN
iajs-1618	177	42	)	)	PUNCT
iajs-1618	177	43	is	be	AUX
iajs-1618	177	44	an	an	DET
iajs-1618	177	45	s	s	NOUN
iajs-1618	177	46	-	-	PUNCT
iajs-1618	177	47	closed	closed	ADJ
iajs-1618	177	48	submodule	submodule	NOUN
iajs-1618	177	49	of	of	ADP
iajs-1618	177	50	w1	w1	PROPN
iajs-1618	177	51	and	and	CCONJ
iajs-1618	177	52	(	(	PUNCT
iajs-1618	177	53	ai	ai	PROPN
iajs-1618	177	54	∩w2	∩w2	PROPN
iajs-1618	177	55	)	)	PUNCT
iajs-1618	177	56	is	be	AUX
iajs-1618	177	57	an	an	DET
iajs-1618	177	58	s	s	NOUN
iajs-1618	177	59	-	-	PUNCT
iajs-1618	177	60	closed	closed	ADJ
iajs-1618	177	61	submodule	submodule	NOUN
iajs-1618	177	62	in	in	ADP
iajs-1618	177	63	w2	w2	NOUN
iajs-1618	177	64	for	for	ADP
iajs-1618	177	65	each	each	DET
iajs-1618	177	66	i	i	NOUN
iajs-1618	177	67	=	=	SYM
iajs-1618	177	68	1,2	1,2	NUM
iajs-1618	177	69	,	,	PUNCT
iajs-1618	177	70	…	…	PUNCT
iajs-1618	177	71	.	.	PUNCT
iajs-1618	178	1	in	in	ADP
iajs-1618	178	2	fact	fact	NOUN
iajs-1618	178	3	if	if	SCONJ
iajs-1618	178	4	ai	ai	VERB
iajs-1618	178	5	∩	∩	NOUN
iajs-1618	178	6	wj	wj	PROPN
iajs-1618	178	7	=	=	PUNCT
iajs-1618	178	8	0	0	PROPN
iajs-1618	178	9	for	for	ADP
iajs-1618	178	10	all	all	DET
iajs-1618	178	11	i	i	NOUN
iajs-1618	178	12	=	=	SYM
iajs-1618	178	13	1,2	1,2	NUM
iajs-1618	178	14	,	,	PUNCT
iajs-1618	178	15	…	…	PUNCT
iajs-1618	178	16	and	and	CCONJ
iajs-1618	178	17	j	j	X
iajs-1618	178	18	=	=	SYM
iajs-1618	178	19	1,2	1,2	NUM
iajs-1618	178	20	then	then	ADV
iajs-1618	178	21	ai	ai	VERB
iajs-1618	178	22	=	=	PUNCT
iajs-1618	178	23	(	(	PUNCT
iajs-1618	178	24	ai	ai	PROPN
iajs-1618	178	25	∩	∩	ADJ
iajs-1618	178	26	w1	w1	NOUN
iajs-1618	178	27	)	)	PUNCT
iajs-1618	178	28	⊕	⊕	PROPN
iajs-1618	178	29	(	(	PUNCT
iajs-1618	178	30	ai	ai	PROPN
iajs-1618	178	31	∩	∩	ADJ
iajs-1618	178	32	w2	w2	NOUN
iajs-1618	178	33	)	)	PUNCT
iajs-1618	179	1	=	=	SYM
iajs-1618	179	2	0	0	NUM
iajs-1618	179	3	which	which	PRON
iajs-1618	179	4	is	be	AUX
iajs-1618	179	5	a	a	DET
iajs-1618	179	6	contradiction	contradiction	NOUN
iajs-1618	179	7	with	with	ADP
iajs-1618	179	8	our	our	PRON
iajs-1618	179	9	assumption	assumption	NOUN
iajs-1618	179	10	.	.	PUNCT
iajs-1618	180	1	that	that	PRON
iajs-1618	180	2	is	be	AUX
iajs-1618	180	3	ai	ai	ADP
iajs-1618	180	4	∩	∩	NOUN
iajs-1618	180	5	wj	wj	PROPN
iajs-1618	180	6	are	be	AUX
iajs-1618	180	7	nonzero	nonzero	PROPN
iajs-1618	180	8	s	s	PART
iajs-1618	180	9	-	-	PUNCT
iajs-1618	180	10	closed	closed	ADJ
iajs-1618	180	11	submodules	submodule	NOUN
iajs-1618	180	12	in	in	ADP
iajs-1618	180	13	w	w	NOUN
iajs-1618	180	14	for	for	ADP
iajs-1618	180	15	each	each	DET
iajs-1618	180	16	i	i	NOUN
iajs-1618	180	17	=	=	NOUN
iajs-1618	180	18	1	1	NUM
iajs-1618	180	19	,	,	PUNCT
iajs-1618	180	20	2	2	NUM
iajs-1618	180	21	,	,	PUNCT
iajs-1618	180	22	…	…	PUNCT
iajs-1618	180	23	and	and	CCONJ
iajs-1618	180	24	j	j	X
iajs-1618	180	25	=	=	SYM
iajs-1618	180	26	1	1	NUM
iajs-1618	180	27	,	,	PUNCT
iajs-1618	180	28	2	2	NUM
iajs-1618	180	29	.	.	PUNCT
iajs-1618	181	1	so	so	ADV
iajs-1618	181	2	we	we	PRON
iajs-1618	181	3	have	have	VERB
iajs-1618	181	4	the	the	DET
iajs-1618	181	5	following	following	NOUN
iajs-1618	181	6	ascending	ascend	VERB
iajs-1618	181	7	(	(	PUNCT
iajs-1618	181	8	respectively	respectively	ADV
iajs-1618	181	9	descending	descending	NOUN
iajs-1618	181	10	)	)	PUNCT
iajs-1618	181	11	chain	chain	NOUN
iajs-1618	181	12	of	of	ADP
iajs-1618	181	13	nonzero	nonzero	PROPN
iajs-1618	181	14	sclosed	sclose	VERB
iajs-1618	181	15	submodules	submodule	NOUN
iajs-1618	181	16	in	in	ADP
iajs-1618	181	17	wj	wj	PROPN
iajs-1618	181	18	,	,	PUNCT
iajs-1618	181	19	(	(	PUNCT
iajs-1618	181	20	a1	a1	NOUN
iajs-1618	181	21	∩	∩	ADJ
iajs-1618	181	22	wj	wj	NOUN
iajs-1618	181	23	)	)	PUNCT
iajs-1618	181	24			PROPN
iajs-1618	181	25	(	(	PUNCT
iajs-1618	181	26	a2	a2	PROPN
iajs-1618	181	27	∩wj	∩wj	ADJ
iajs-1618	181	28	)	)	PUNCT
iajs-1618	181	29			PROPN
iajs-1618	181	30	…	…	PUNCT
iajs-1618	181	31	(	(	PUNCT
iajs-1618	181	32	respectively	respectively	ADV
iajs-1618	181	33	a1∩wj	a1∩wj	ADJ
iajs-1618	181	34			PROPN
iajs-1618	181	35	a2	a2	PROPN
iajs-1618	181	36	∩wj	∩wj	ADJ
iajs-1618	181	37			PROPN
iajs-1618	181	38	…	…	PUNCT
iajs-1618	181	39	)	)	PUNCT
iajs-1618	181	40	for	for	ADP
iajs-1618	181	41	each	each	PRON
iajs-1618	181	42	j	j	PROPN
iajs-1618	182	1	=	=	SYM
iajs-1618	182	2	1	1	NUM
iajs-1618	182	3	,	,	PUNCT
iajs-1618	182	4	2	2	NUM
iajs-1618	182	5	.	.	PUNCT
iajs-1618	183	1	but	but	CCONJ
iajs-1618	183	2	wj	wj	PROPN
iajs-1618	183	3	satisfies	satisfy	VERB
iajs-1618	183	4	a	a	DET
iajs-1618	183	5	c	c	NOUN
iajs-1618	183	6	c	c	NOUN
iajs-1618	183	7	(	(	PUNCT
iajs-1618	183	8	respectively	respectively	ADV
iajs-1618	183	9	d	d	PROPN
iajs-1618	183	10	c	c	NOUN
iajs-1618	183	11	c	c	NOUN
iajs-1618	183	12	)	)	PUNCT
iajs-1618	183	13	on	on	ADP
iajs-1618	183	14	s	s	NOUN
iajs-1618	183	15	-	-	PUNCT
iajs-1618	183	16	closed	closed	ADJ
iajs-1618	183	17	submodules	submodule	NOUN
iajs-1618	183	18	for	for	ADP
iajs-1618	183	19	each	each	PRON
iajs-1618	183	20	j	j	PROPN
iajs-1618	184	1	=	=	SYM
iajs-1618	184	2	1	1	NUM
iajs-1618	184	3	,	,	PUNCT
iajs-1618	184	4	2	2	NUM
iajs-1618	184	5	.	.	PUNCT
iajs-1618	185	1	thus	thus	ADV
iajs-1618	185	2	there	there	PRON
iajs-1618	185	3	exists	exist	VERB
iajs-1618	185	4	kjz+	kjz+	ADV
iajs-1618	185	5	such	such	ADJ
iajs-1618	185	6	that	that	SCONJ
iajs-1618	185	7	an	an	DET
iajs-1618	185	8	∩	∩	ADJ
iajs-1618	185	9	wj	wj	X
iajs-1618	185	10	=	=	PUNCT
iajs-1618	185	11	akj	akj	ADJ
iajs-1618	185	12	∩	∩	NOUN
iajs-1618	185	13	wj	wj	PROPN
iajs-1618	185	14	,	,	PUNCT
iajs-1618	185	15	for	for	ADP
iajs-1618	185	16	all	all	DET
iajs-1618	185	17	nkj	nkj	PROPN
iajs-1618	185	18	and	and	CCONJ
iajs-1618	185	19	j	j	NOUN
iajs-1618	185	20	=	=	SYM
iajs-1618	185	21	1	1	NUM
iajs-1618	185	22	,	,	PUNCT
iajs-1618	185	23	2	2	NUM
iajs-1618	185	24	.	.	PUNCT
iajs-1618	186	1	let	let	VERB
iajs-1618	186	2	k	k	NOUN
iajs-1618	186	3	=	=	SYM
iajs-1618	186	4	max	max	PROPN
iajs-1618	186	5	{	{	PUNCT
iajs-1618	186	6	k1	k1	PROPN
iajs-1618	186	7	,	,	PUNCT
iajs-1618	186	8	k2	k2	NOUN
iajs-1618	186	9	}	}	PUNCT
iajs-1618	186	10	,	,	PUNCT
iajs-1618	186	11	so	so	ADV
iajs-1618	186	12	an	an	PRON
iajs-1618	186	13	=	=	X
iajs-1618	186	14	(	(	PUNCT
iajs-1618	186	15	an	an	DET
iajs-1618	186	16	∩	∩	ADJ
iajs-1618	186	17	w1	w1	NOUN
iajs-1618	186	18	)	)	PUNCT
iajs-1618	186	19	⊕	⊕	PROPN
iajs-1618	186	20	(	(	PUNCT
iajs-1618	186	21	an	an	DET
iajs-1618	186	22	∩w2	∩w2	NOUN
iajs-1618	186	23	)	)	PUNCT
iajs-1618	186	24	=	=	SYM
iajs-1618	186	25	(	(	PUNCT
iajs-1618	186	26	ak	ak	PROPN
iajs-1618	186	27	∩	∩	ADJ
iajs-1618	186	28	w1	w1	PROPN
iajs-1618	186	29	)	)	PUNCT
iajs-1618	186	30	⊕	⊕	PROPN
iajs-1618	186	31	(	(	PUNCT
iajs-1618	186	32	ak	ak	PROPN
iajs-1618	186	33	∩w2	∩w2	PROPN
iajs-1618	186	34	)	)	PUNCT
iajs-1618	186	35	=	=	SYM
iajs-1618	186	36	ak	ak	PROPN
iajs-1618	186	37	,	,	PUNCT
iajs-1618	186	38	for	for	ADP
iajs-1618	186	39	all	all	DET
iajs-1618	186	40	nk	nk	NOUN
iajs-1618	186	41	.	.	PUNCT
iajs-1618	187	1	hence	hence	ADV
iajs-1618	187	2	w	w	NOUN
iajs-1618	187	3	satisfies	satisfie	NOUN
iajs-1618	187	4	a	a	DET
iajs-1618	187	5	c	c	NOUN
iajs-1618	187	6	c	c	NOUN
iajs-1618	187	7	(	(	PUNCT
iajs-1618	187	8	respectively	respectively	ADV
iajs-1618	187	9	d	d	PROPN
iajs-1618	187	10	c	c	NOUN
iajs-1618	187	11	c	c	NOUN
iajs-1618	187	12	)	)	PUNCT
iajs-1618	187	13	.	.	PUNCT
iajs-1618	188	1	remark	remark	PROPN
iajs-1618	188	2	2	2	NUM
iajs-1618	188	3	.	.	NOUN
iajs-1618	188	4	6	6	NUM
iajs-1618	188	5	:	:	PUNCT
iajs-1618	188	6	we	we	PRON
iajs-1618	188	7	can	can	AUX
iajs-1618	188	8	generalize	generalize	VERB
iajs-1618	188	9	proposition	proposition	NOUN
iajs-1618	188	10	2.5	2.5	NUM
iajs-1618	188	11	for	for	ADP
iajs-1618	188	12	finite	finite	ADJ
iajs-1618	188	13	index	index	NOUN
iajs-1618	188	14	i	i	PRON
iajs-1618	188	15	of	of	ADP
iajs-1618	188	16	the	the	DET
iajs-1618	188	17	direct	direct	ADJ
iajs-1618	188	18	sum	sum	NOUN
iajs-1618	188	19	of	of	ADP
iajs-1618	188	20	l	l	NOUN
iajs-1618	188	21	-	-	PUNCT
iajs-1618	188	22	modules	module	NOUN
iajs-1618	188	23	.	.	PUNCT
iajs-1618	189	1	proposition	proposition	NOUN
iajs-1618	189	2	2	2	NUM
iajs-1618	189	3	.	.	NOUN
iajs-1618	189	4	7	7	NUM
iajs-1618	189	5	:	:	PUNCT
iajs-1618	189	6	let	let	VERB
iajs-1618	189	7	a	a	DET
iajs-1618	189	8	≤	≤	NUM
iajs-1618	189	9	b	b	NOUN
iajs-1618	189	10	≤	≤	NUM
iajs-1618	189	11	w	w	ADP
iajs-1618	189	12	such	such	ADJ
iajs-1618	189	13	that	that	SCONJ
iajs-1618	189	14	a	a	PRON
iajs-1618	189	15	is	be	AUX
iajs-1618	189	16	an	an	DET
iajs-1618	189	17	s	s	NOUN
iajs-1618	189	18	-	-	PUNCT
iajs-1618	189	19	closed	closed	ADJ
iajs-1618	189	20	submodule	submodule	NOUN
iajs-1618	189	21	of	of	ADP
iajs-1618	189	22	an	an	DET
iajs-1618	189	23	lmodule	lmodule	NOUN
iajs-1618	189	24	w.	w.	PROPN
iajs-1618	189	25	w	w	PROPN
iajs-1618	189	26	satisfies	satisfy	VERB
iajs-1618	189	27	a	a	DET
iajs-1618	189	28	c	c	NOUN
iajs-1618	189	29	c	c	NOUN
iajs-1618	189	30	(	(	PUNCT
iajs-1618	189	31	respectively	respectively	ADV
iajs-1618	189	32	d	d	PROPN
iajs-1618	189	33	c	c	NOUN
iajs-1618	189	34	c	c	NOUN
iajs-1618	189	35	)	)	PUNCT
iajs-1618	189	36	on	on	ADP
iajs-1618	189	37	s	s	NOUN
iajs-1618	189	38	-	-	PUNCT
iajs-1618	189	39	closed	closed	ADJ
iajs-1618	189	40	submodules	submodule	NOUN
iajs-1618	189	41	if	if	SCONJ
iajs-1618	189	42	and	and	CCONJ
iajs-1618	189	43	only	only	ADV
iajs-1618	189	44	if	if	SCONJ
iajs-1618	189	45	satisfies	satisfy	VERB
iajs-1618	189	46	a	a	DET
iajs-1618	189	47	c	c	NOUN
iajs-1618	189	48	c	c	NOUN
iajs-1618	189	49	(	(	PUNCT
iajs-1618	189	50	respectivelyd	respectivelyd	NOUN
iajs-1618	189	51	c	c	PROPN
iajs-1618	189	52	c	c	NOUN
iajs-1618	189	53	)	)	PUNCT
iajs-1618	189	54	on	on	ADP
iajs-1618	189	55	s	s	NOUN
iajs-1618	189	56	-	-	PUNCT
iajs-1618	189	57	closed	closed	ADJ
iajs-1618	189	58	submodules	submodule	NOUN
iajs-1618	189	59	.	.	PUNCT
iajs-1618	190	1	proof	proof	NOUN
iajs-1618	190	2	:	:	PUNCT
iajs-1618	190	3	(	(	PUNCT
iajs-1618	190	4			NOUN
iajs-1618	190	5	)	)	PUNCT
iajs-1618	190	6	suppose	suppose	VERB
iajs-1618	190	7	w	w	NOUN
iajs-1618	190	8	satisfied	satisfy	VERB
iajs-1618	190	9	a	a	DET
iajs-1618	190	10	c	c	NOUN
iajs-1618	190	11	c	c	NOUN
iajs-1618	190	12	on	on	ADP
iajs-1618	190	13	sclosed	sclose	VERB
iajs-1618	190	14	submodules	submodule	NOUN
iajs-1618	190	15	,	,	PUNCT
iajs-1618	190	16	and	and	CCONJ
iajs-1618	190	17	let	let	VERB
iajs-1618	190	18			PROPN
iajs-1618	190	19			PROPN
iajs-1618	190	20	…	…	PUNCT
iajs-1618	190	21	,	,	PUNCT
iajs-1618	190	22	be	be	AUX
iajs-1618	190	23	ascending	ascend	VERB
iajs-1618	190	24	chain	chain	NOUN
iajs-1618	190	25	of	of	ADP
iajs-1618	190	26	s	s	NOUN
iajs-1618	190	27	-	-	PUNCT
iajs-1618	190	28	closed	closed	ADJ
iajs-1618	190	29	submodules	submodule	NOUN
iajs-1618	190	30	of	of	ADP
iajs-1618	190	31	,	,	PUNCT
iajs-1618	190	32	then	then	ADV
iajs-1618	190	33	bi	bi	NOUN
iajs-1618	190	34	is	be	AUX
iajs-1618	190	35	an	an	DET
iajs-1618	190	36	s	s	NOUN
iajs-1618	190	37	-	-	PUNCT
iajs-1618	190	38	closed	closed	ADJ
iajs-1618	190	39	submodule	submodule	NOUN
iajs-1618	190	40	of	of	ADP
iajs-1618	190	41	w	w	PROPN
iajs-1618	190	42	by	by	ADP
iajs-1618	190	43	(	(	PUNCT
iajs-1618	190	44	proposition	proposition	NOUN
iajs-1618	190	45	1.12	1.12	NUM
iajs-1618	190	46	)	)	PUNCT
iajs-1618	190	47	.	.	PUNCT
iajs-1618	191	1	thus	thus	ADV
iajs-1618	191	2	there	there	PRON
iajs-1618	191	3	exists	exist	VERB
iajs-1618	191	4	k	k	PROPN
iajs-1618	191	5			PROPN
iajs-1618	191	6	z+	z+	NUM
iajs-1618	191	7	such	such	ADJ
iajs-1618	191	8	that	that	PRON
iajs-1618	191	9	bn	bn	NOUN
iajs-1618	191	10	=	=	PUNCT
iajs-1618	191	11	bk	bk	NOUN
iajs-1618	191	12	for	for	ADP
iajs-1618	191	13	all	all	DET
iajs-1618	191	14	nk	nk	NOUN
iajs-1618	191	15	.	.	PUNCT
iajs-1618	192	1	hence	hence	ADV
iajs-1618	192	2	=	=	PUNCT
iajs-1618	192	3	for	for	ADP
iajs-1618	192	4	all	all	PRON
iajs-1618	192	5	nk	nk	PROPN
iajs-1618	192	6	.that	.that	PRON
iajs-1618	192	7	is	be	AUX
iajs-1618	192	8	satisfies	satisfie	NOUN
iajs-1618	192	9	a	a	DET
iajs-1618	192	10	c	c	NOUN
iajs-1618	192	11	c	c	NOUN
iajs-1618	192	12	on	on	ADP
iajs-1618	192	13	s	s	NOUN
iajs-1618	192	14	-	-	PUNCT
iajs-1618	192	15	closed	closed	ADJ
iajs-1618	192	16	submodules	submodule	NOUN
iajs-1618	192	17	.	.	PUNCT
iajs-1618	193	1	mathematics	mathematic	NOUN
iajs-1618	193	2	|233	|233	PART
iajs-1618	194	1	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	X
iajs-1618	194	2	7102	7102	NUM
iajs-1618	194	3	(	(	PUNCT
iajs-1618	194	4	عام	عام	PROPN
iajs-1618	194	5	3	3	NUM
iajs-1618	194	6	(	(	PUNCT
iajs-1618	194	7	العدد	العدد	PROPN
iajs-1618	194	8	)	)	PUNCT
iajs-1618	194	9	30مجلة	30مجلة	PROPN
iajs-1618	194	10	إبن	إبن	VERB
iajs-1618	194	11	الهيثم	الهيثم	ADJ
iajs-1618	194	12	للعلوم	للعلوم	NOUN
iajs-1618	194	13	الصرفة	الصرفة	NOUN
iajs-1618	195	1	و	و	PRON
iajs-1618	195	2	التطبيقية	التطبيقية	ADV
iajs-1618	195	3	المجلد	المجلد	ADV
iajs-1618	195	4	)	)	PUNCT
iajs-1618	196	1	ibn	ibn	PROPN
iajs-1618	196	2	al	al	PROPN
iajs-1618	196	3	-	-	PUNCT
iajs-1618	196	4	haitham	haitham	PROPN
iajs-1618	196	5	j.	j.	PROPN
iajs-1618	196	6	for	for	ADP
iajs-1618	196	7	pure	pure	PROPN
iajs-1618	196	8	&	&	CCONJ
iajs-1618	196	9	appl	appl	PROPN
iajs-1618	196	10	.	.	PUNCT
iajs-1618	197	1	sci	sci	PROPN
iajs-1618	197	2	.	.	PUNCT
iajs-1618	198	1	vol.03	vol.03	PROPN
iajs-1618	198	2	(	(	PUNCT
iajs-1618	198	3	3	3	NUM
iajs-1618	198	4	)	)	PUNCT
iajs-1618	198	5	2017	2017	NUM
iajs-1618	198	6	(	(	PUNCT
iajs-1618	198	7			NUM
iajs-1618	198	8	)	)	PUNCT
iajs-1618	198	9	suppose	suppose	VERB
iajs-1618	198	10	satisfies	satisfie	NOUN
iajs-1618	198	11	a	a	DET
iajs-1618	198	12	c	c	NOUN
iajs-1618	198	13	c	c	NOUN
iajs-1618	198	14	on	on	ADP
iajs-1618	198	15	sclosed	sclose	VERB
iajs-1618	198	16	submodules	submodule	NOUN
iajs-1618	198	17	.	.	PUNCT
iajs-1618	199	1	let	let	VERB
iajs-1618	199	2	a	a	DET
iajs-1618	199	3			ADJ
iajs-1618	199	4	a1	a1	NOUN
iajs-1618	199	5			PROPN
iajs-1618	199	6	a2	a2	PROPN
iajs-1618	199	7			PROPN
iajs-1618	199	8	…	…	PUNCT
iajs-1618	199	9	be	be	AUX
iajs-1618	199	10	a	a	DET
iajs-1618	199	11	chain	chain	NOUN
iajs-1618	199	12	of	of	ADP
iajs-1618	199	13	s	s	NOUN
iajs-1618	199	14	-	-	PUNCT
iajs-1618	199	15	closed	closed	ADJ
iajs-1618	199	16	submodules	submodule	NOUN
iajs-1618	199	17	of	of	ADP
iajs-1618	199	18	w.	w.	PROPN
iajs-1618	199	19	since	since	SCONJ
iajs-1618	199	20	a	a	DET
iajs-1618	199	21			PROPN
iajs-1618	199	22	a1	a1	NOUN
iajs-1618	199	23	and	and	CCONJ
iajs-1618	199	24	a	a	DET
iajs-1618	199	25			PROPN
iajs-1618	199	26	a2	a2	PROPN
iajs-1618	199	27	,	,	PUNCT
iajs-1618	199	28	…	…	PUNCT
iajs-1618	199	29	and	and	CCONJ
iajs-1618	199	30	a	a	PRON
iajs-1618	199	31	is	be	AUX
iajs-1618	199	32	an	an	DET
iajs-1618	199	33	s	s	NOUN
iajs-1618	199	34	-	-	PUNCT
iajs-1618	199	35	closed	closed	ADJ
iajs-1618	199	36	submodule	submodule	NOUN
iajs-1618	199	37	of	of	ADP
iajs-1618	199	38	w	w	PROPN
iajs-1618	199	39	,	,	PUNCT
iajs-1618	199	40	then	then	ADV
iajs-1618	199	41	by	by	ADP
iajs-1618	199	42	[	[	X
iajs-1618	199	43	1	1	NUM
iajs-1618	199	44	,	,	PUNCT
iajs-1618	199	45	coro.2.7	coro.2.7	PROPN
iajs-1618	199	46	,	,	PUNCT
iajs-1618	199	47	p.345	p.345	NOUN
iajs-1618	199	48	]	]	PUNCT
iajs-1618	199	49	is	be	AUX
iajs-1618	199	50	an	an	DET
iajs-1618	199	51	s	s	NOUN
iajs-1618	199	52	-	-	PUNCT
iajs-1618	199	53	closed	closed	ADJ
iajs-1618	199	54	submodule	submodule	NOUN
iajs-1618	199	55	of	of	ADP
iajs-1618	199	56	w	w	PROPN
iajs-1618	199	57	for	for	ADP
iajs-1618	199	58	each	each	DET
iajs-1618	199	59	i.	i.	NOUN
iajs-1618	199	60	thus	thus	ADV
iajs-1618	199	61	we	we	PRON
iajs-1618	199	62	have	have	VERB
iajs-1618	199	63			PROPN
iajs-1618	199	64			PROPN
iajs-1618	199	65	…	…	PUNCT
iajs-1618	199	66	is	be	AUX
iajs-1618	199	67	an	an	DET
iajs-1618	199	68	ascending	ascend	VERB
iajs-1618	199	69	chain	chain	NOUN
iajs-1618	199	70	of	of	ADP
iajs-1618	199	71	s	s	NOUN
iajs-1618	199	72	-	-	PUNCT
iajs-1618	199	73	closed	closed	ADJ
iajs-1618	199	74	submodules	submodule	NOUN
iajs-1618	199	75	of	of	ADP
iajs-1618	199	76	,	,	PUNCT
iajs-1618	199	77	hence	hence	ADV
iajs-1618	199	78	by	by	ADP
iajs-1618	199	79	our	our	PRON
iajs-1618	199	80	assumption	assumption	NOUN
iajs-1618	199	81	satisfies	satisfy	VERB
iajs-1618	199	82	a	a	DET
iajs-1618	199	83	c	c	NOUN
iajs-1618	199	84	c	c	NOUN
iajs-1618	199	85	on	on	ADP
iajs-1618	199	86	s	s	NOUN
iajs-1618	199	87	-	-	PUNCT
iajs-1618	199	88	closed	closed	ADJ
iajs-1618	199	89	submodules	submodule	NOUN
iajs-1618	199	90	so	so	SCONJ
iajs-1618	199	91	there	there	PRON
iajs-1618	199	92	exists	exist	VERB
iajs-1618	199	93	k	k	PROPN
iajs-1618	199	94			PROPN
iajs-1618	199	95	z+	z+	NUM
iajs-1618	199	96	such	such	ADJ
iajs-1618	199	97	that	that	SCONJ
iajs-1618	199	98	=	=	PUNCT
iajs-1618	199	99	for	for	ADP
iajs-1618	199	100	all	all	DET
iajs-1618	199	101	nk	nk	NOUN
iajs-1618	199	102	.	.	PUNCT
iajs-1618	200	1	that	that	PRON
iajs-1618	200	2	is	be	AUX
iajs-1618	200	3	an	an	DET
iajs-1618	200	4	=	=	SYM
iajs-1618	200	5	ak	ak	PROPN
iajs-1618	200	6	for	for	ADP
iajs-1618	200	7	all	all	DET
iajs-1618	200	8	nk	nk	NOUN
iajs-1618	200	9	which	which	PRON
iajs-1618	200	10	means	mean	VERB
iajs-1618	200	11	w	w	NOUN
iajs-1618	200	12	satisfied	satisfied	ADJ
iajs-1618	200	13	a	a	DET
iajs-1618	200	14	c	c	NOUN
iajs-1618	200	15	c	c	NOUN
iajs-1618	200	16	on	on	ADP
iajs-1618	200	17	s	s	NOUN
iajs-1618	200	18	-	-	PUNCT
iajs-1618	200	19	closed	closed	ADJ
iajs-1618	200	20	submodules	submodule	NOUN
iajs-1618	200	21	.	.	PUNCT
iajs-1618	201	1	by	by	ADP
iajs-1618	201	2	the	the	DET
iajs-1618	201	3	same	same	ADJ
iajs-1618	201	4	way	way	NOUN
iajs-1618	201	5	we	we	PRON
iajs-1618	201	6	can	can	AUX
iajs-1618	201	7	prove	prove	VERB
iajs-1618	201	8	that	that	SCONJ
iajs-1618	201	9	w	w	NOUN
iajs-1618	201	10	satisfies	satisfie	NOUN
iajs-1618	201	11	d	d	X
iajs-1618	201	12	c	c	PROPN
iajs-1618	201	13	c	c	NOUN
iajs-1618	201	14	on	on	ADP
iajs-1618	201	15	s	s	NOUN
iajs-1618	201	16	-	-	PUNCT
iajs-1618	201	17	closed	closed	ADJ
iajs-1618	201	18	submodules	submodule	NOUN
iajs-1618	201	19	if	if	SCONJ
iajs-1618	201	20	and	and	CCONJ
iajs-1618	201	21	only	only	ADV
iajs-1618	201	22	if	if	SCONJ
iajs-1618	201	23	satisfies	satisfie	NOUN
iajs-1618	201	24	d	d	X
iajs-1618	201	25	c	c	NOUN
iajs-1618	201	26	c	c	NOUN
iajs-1618	201	27	on	on	ADP
iajs-1618	201	28	s	s	NOUN
iajs-1618	201	29	-	-	PUNCT
iajs-1618	201	30	closed	closed	ADJ
iajs-1618	201	31	submodules	submodule	NOUN
iajs-1618	201	32	.	.	PUNCT
iajs-1618	202	1	proposition	proposition	NOUN
iajs-1618	202	2	2	2	NUM
iajs-1618	202	3	.	.	NOUN
iajs-1618	202	4	8	8	NUM
iajs-1618	202	5	:	:	PUNCT
iajs-1618	202	6	let	let	VERB
iajs-1618	202	7	w	w	NOUN
iajs-1618	202	8	=	=	VERB
iajs-1618	202	9	w1	w1	PROPN
iajs-1618	202	10	⊕	⊕	PROPN
iajs-1618	202	11	w2	w2	NOUN
iajs-1618	202	12	be	be	AUX
iajs-1618	202	13	an	an	DET
iajs-1618	202	14	l	l	NOUN
iajs-1618	202	15	-	-	NOUN
iajs-1618	202	16	module	module	NOUN
iajs-1618	202	17	and	and	CCONJ
iajs-1618	202	18	l	l	NOUN
iajs-1618	202	19	=	=	PUNCT
iajs-1618	202	20	ann(w1	ann(w1	NOUN
iajs-1618	202	21	)	)	PUNCT
iajs-1618	202	22	+	+	CCONJ
iajs-1618	202	23	ann(w2	ann(w2	NOUN
iajs-1618	202	24	)	)	PUNCT
iajs-1618	202	25	.	.	PUNCT
iajs-1618	203	1	then	then	ADV
iajs-1618	203	2	w	w	NOUN
iajs-1618	203	3	satisfies	satisfie	NOUN
iajs-1618	203	4	a	a	DET
iajs-1618	203	5	c	c	NOUN
iajs-1618	203	6	c	c	NOUN
iajs-1618	203	7	(	(	PUNCT
iajs-1618	203	8	respectively	respectively	ADV
iajs-1618	203	9	d	d	PROPN
iajs-1618	203	10	c	c	NOUN
iajs-1618	203	11	c	c	NOUN
iajs-1618	203	12	)	)	PUNCT
iajs-1618	203	13	on	on	ADP
iajs-1618	203	14	sclosed	sclose	VERB
iajs-1618	203	15	submodules	submodule	NOUN
iajs-1618	203	16	if	if	SCONJ
iajs-1618	203	17	and	and	CCONJ
iajs-1618	203	18	only	only	ADV
iajs-1618	203	19	if	if	SCONJ
iajs-1618	203	20	w1	w1	NOUN
iajs-1618	203	21	and	and	CCONJ
iajs-1618	203	22	w2	w2	NOUN
iajs-1618	203	23	satisfy	satisfy	VERB
iajs-1618	203	24	a	a	DET
iajs-1618	203	25	c	c	NOUN
iajs-1618	203	26	c	c	NOUN
iajs-1618	203	27	(	(	PUNCT
iajs-1618	203	28	respectively	respectively	ADV
iajs-1618	203	29	d	d	PROPN
iajs-1618	203	30	c	c	NOUN
iajs-1618	203	31	c	c	NOUN
iajs-1618	203	32	)	)	PUNCT
iajs-1618	203	33	on	on	ADP
iajs-1618	203	34	sclosed	sclose	VERB
iajs-1618	203	35	submodules	submodule	NOUN
iajs-1618	203	36	.	.	PUNCT
iajs-1618	204	1	proof	proof	NOUN
iajs-1618	204	2	:	:	PUNCT
iajs-1618	204	3	(	(	PUNCT
iajs-1618	204	4			NOUN
iajs-1618	204	5	)	)	PUNCT
iajs-1618	204	6	see	see	VERB
iajs-1618	204	7	proposition	proposition	NOUN
iajs-1618	204	8	2.4	2.4	NUM
iajs-1618	204	9	.	.	PUNCT
iajs-1618	205	1	(	(	PUNCT
iajs-1618	205	2			X
iajs-1618	205	3	)	)	PUNCT
iajs-1618	205	4	let	let	AUX
iajs-1618	205	5	e1	e1	PROPN
iajs-1618	205	6			PROPN
iajs-1618	205	7	e2	e2	PROPN
iajs-1618	205	8			PROPN
iajs-1618	205	9	…	…	PUNCT
iajs-1618	205	10	be	be	AUX
iajs-1618	205	11	an	an	DET
iajs-1618	205	12	ascending	ascend	VERB
iajs-1618	205	13	chain	chain	NOUN
iajs-1618	205	14	of	of	ADP
iajs-1618	205	15	sclosed	sclose	VERB
iajs-1618	205	16	submodules	submodule	NOUN
iajs-1618	205	17	of	of	ADP
iajs-1618	205	18	w	w	PROPN
iajs-1618	205	19	(	(	PUNCT
iajs-1618	205	20	since	since	SCONJ
iajs-1618	205	21	l	l	NOUN
iajs-1618	205	22	=	=	SYM
iajs-1618	205	23	ann(w1	ann(w1	NOUN
iajs-1618	205	24	)	)	PUNCT
iajs-1618	205	25	+	+	CCONJ
iajs-1618	205	26	ann(w2	ann(w2	NOUN
iajs-1618	205	27	)	)	PUNCT
iajs-1618	205	28	,	,	PUNCT
iajs-1618	205	29	every	every	DET
iajs-1618	205	30	submodule	submodule	NOUN
iajs-1618	205	31	ei	ei	PROPN
iajs-1618	205	32	of	of	ADP
iajs-1618	205	33	w	w	PROPN
iajs-1618	205	34	has	have	VERB
iajs-1618	205	35	the	the	DET
iajs-1618	205	36	form	form	NOUN
iajs-1618	205	37	ni⊕ki	ni⊕ki	NUM
iajs-1618	205	38	for	for	ADP
iajs-1618	205	39	some	some	DET
iajs-1618	205	40	ni	ni	PROPN
iajs-1618	205	41	≤	≤	PROPN
iajs-1618	205	42	w1	w1	NOUN
iajs-1618	205	43	and	and	CCONJ
iajs-1618	205	44	ki	ki	PROPN
iajs-1618	205	45	≤	≤	PROPN
iajs-1618	205	46	w2	w2	NOUN
iajs-1618	205	47	)	)	PUNCT
iajs-1618	205	48	.	.	PUNCT
iajs-1618	206	1	hence	hence	ADV
iajs-1618	206	2	by	by	ADP
iajs-1618	206	3	[	[	X
iajs-1618	206	4	1	1	NUM
iajs-1618	206	5	,	,	PUNCT
iajs-1618	206	6	prop.2.5	prop.2.5	PROPN
iajs-1618	206	7	]	]	X
iajs-1618	206	8	ni	ni	PROPN
iajs-1618	206	9	is	be	AUX
iajs-1618	206	10	an	an	DET
iajs-1618	206	11	sclosed	sclose	VERB
iajs-1618	206	12	submodule	submodule	NOUN
iajs-1618	206	13	in	in	ADP
iajs-1618	206	14	w1	w1	NOUN
iajs-1618	206	15	,	,	PUNCT
iajs-1618	206	16	and	and	CCONJ
iajs-1618	206	17	ki	ki	PROPN
iajs-1618	206	18	is	be	AUX
iajs-1618	206	19	an	an	DET
iajs-1618	206	20	sclosed	sclosed	ADJ
iajs-1618	206	21	submodule	submodule	NOUN
iajs-1618	206	22	of	of	ADP
iajs-1618	206	23	w2	w2	NOUN
iajs-1618	206	24	for	for	ADP
iajs-1618	206	25	all	all	DET
iajs-1618	206	26	i=	i=	ADJ
iajs-1618	206	27	1	1	NUM
iajs-1618	206	28	,	,	PUNCT
iajs-1618	206	29	2	2	NUM
iajs-1618	206	30	,	,	PUNCT
iajs-1618	206	31	…	…	PUNCT
iajs-1618	206	32	.so	.so	PUNCT
iajs-1618	207	1	n1	n1	ADJ
iajs-1618	207	2			PROPN
iajs-1618	207	3	n2	n2	ADJ
iajs-1618	207	4			PROPN
iajs-1618	207	5	…	…	PUNCT
iajs-1618	207	6	is	be	AUX
iajs-1618	207	7	an	an	DET
iajs-1618	207	8	ascending	ascend	VERB
iajs-1618	207	9	chain	chain	NOUN
iajs-1618	207	10	of	of	ADP
iajs-1618	207	11	sclosed	sclose	VERB
iajs-1618	207	12	submodules	submodule	NOUN
iajs-1618	207	13	of	of	ADP
iajs-1618	207	14	w1	w1	NOUN
iajs-1618	207	15	and	and	CCONJ
iajs-1618	207	16	k1	k1	PROPN
iajs-1618	207	17			PROPN
iajs-1618	207	18	k2	k2	PROPN
iajs-1618	207	19			PROPN
iajs-1618	207	20	…	…	PUNCT
iajs-1618	207	21	is	be	AUX
iajs-1618	207	22	an	an	DET
iajs-1618	207	23	ascending	ascend	VERB
iajs-1618	207	24	chain	chain	NOUN
iajs-1618	207	25	of	of	ADP
iajs-1618	207	26	sclosed	sclose	VERB
iajs-1618	207	27	submodules	submodule	NOUN
iajs-1618	207	28	of	of	ADP
iajs-1618	207	29	w2	w2	NOUN
iajs-1618	207	30	.	.	PUNCT
iajs-1618	208	1	since	since	SCONJ
iajs-1618	208	2	w1	w1	NOUN
iajs-1618	208	3	and	and	CCONJ
iajs-1618	208	4	w2	w2	NOUN
iajs-1618	208	5	satisfy	satisfy	VERB
iajs-1618	208	6	a	a	DET
iajs-1618	208	7	c	c	NOUN
iajs-1618	208	8	c	c	NOUN
iajs-1618	208	9	on	on	ADP
iajs-1618	208	10	sclosed	sclose	VERB
iajs-1618	208	11	submodules	submodule	NOUN
iajs-1618	208	12	,	,	PUNCT
iajs-1618	208	13	then	then	ADV
iajs-1618	208	14	there	there	PRON
iajs-1618	208	15	exists	exist	VERB
iajs-1618	208	16	t	t	PROPN
iajs-1618	208	17	,	,	PUNCT
iajs-1618	208	18	r	r	NOUN
iajs-1618	208	19			NOUN
iajs-1618	208	20	z+	z+	NUM
iajs-1618	208	21	such	such	ADJ
iajs-1618	208	22	that	that	SCONJ
iajs-1618	208	23	nt	not	PART
iajs-1618	208	24	=	=	PROPN
iajs-1618	208	25	nt+i	nt+i	PROPN
iajs-1618	208	26	and	and	CCONJ
iajs-1618	208	27	kr	kr	PROPN
iajs-1618	208	28	=	=	SYM
iajs-1618	208	29	kr+i	kr+i	PROPN
iajs-1618	208	30	,	,	PUNCT
iajs-1618	208	31	for	for	ADP
iajs-1618	208	32	each	each	DET
iajs-1618	208	33	i	i	NOUN
iajs-1618	208	34	=	=	NOUN
iajs-1618	208	35	1	1	NUM
iajs-1618	208	36	,	,	PUNCT
iajs-1618	208	37	2	2	NUM
iajs-1618	208	38	,	,	PUNCT
iajs-1618	208	39	…	…	PUNCT
iajs-1618	208	40	.	.	PUNCT
iajs-1618	209	1	take	take	VERB
iajs-1618	209	2	s	s	PART
iajs-1618	209	3	=	=	X
iajs-1618	209	4	max	max	PROPN
iajs-1618	209	5	{	{	PUNCT
iajs-1618	209	6	t	t	PROPN
iajs-1618	209	7	,	,	PUNCT
iajs-1618	209	8	r	r	NOUN
iajs-1618	209	9	}	}	PUNCT
iajs-1618	209	10	,	,	PUNCT
iajs-1618	209	11	hence	hence	ADV
iajs-1618	209	12	ns⊕ks	ns⊕ks	PROPN
iajs-1618	209	13			PROPN
iajs-1618	209	14	ns+i⊕ks+i	ns+i⊕ks+i	PROPN
iajs-1618	209	15	,	,	PUNCT
iajs-1618	209	16	for	for	ADP
iajs-1618	209	17	each	each	DET
iajs-1618	209	18	i	i	NOUN
iajs-1618	209	19	=	=	NOUN
iajs-1618	209	20	1	1	NUM
iajs-1618	209	21	,	,	PUNCT
iajs-1618	209	22	2	2	NUM
iajs-1618	209	23	,	,	PUNCT
iajs-1618	209	24	…	…	PUNCT
iajs-1618	209	25	.	.	PUNCT
iajs-1618	210	1	that	that	PRON
iajs-1618	210	2	is	be	AUX
iajs-1618	210	3	w	w	ADJ
iajs-1618	210	4	satisfies	satisfie	NOUN
iajs-1618	210	5	a	a	DET
iajs-1618	210	6	c	c	NOUN
iajs-1618	210	7	c	c	NOUN
iajs-1618	210	8	on	on	ADP
iajs-1618	210	9	sclosed	sclose	VERB
iajs-1618	210	10	submodules	submodule	NOUN
iajs-1618	210	11	.	.	PUNCT
iajs-1618	211	1	by	by	ADP
iajs-1618	211	2	the	the	DET
iajs-1618	211	3	same	same	ADJ
iajs-1618	211	4	way	way	NOUN
iajs-1618	211	5	we	we	PRON
iajs-1618	211	6	can	can	AUX
iajs-1618	211	7	prove	prove	VERB
iajs-1618	211	8	that	that	SCONJ
iajs-1618	211	9	w	w	NOUN
iajs-1618	211	10	satisfies	satisfie	NOUN
iajs-1618	211	11	d	d	X
iajs-1618	211	12	c	c	PROPN
iajs-1618	211	13	c	c	NOUN
iajs-1618	211	14	on	on	ADP
iajs-1618	211	15	sclosed	sclose	VERB
iajs-1618	211	16	submodules	submodule	NOUN
iajs-1618	211	17	if	if	SCONJ
iajs-1618	211	18	and	and	CCONJ
iajs-1618	211	19	only	only	ADV
iajs-1618	211	20	if	if	SCONJ
iajs-1618	211	21	w1	w1	NOUN
iajs-1618	211	22	and	and	CCONJ
iajs-1618	211	23	w2	w2	NOUN
iajs-1618	211	24	satisfy	satisfy	NOUN
iajs-1618	211	25	d	d	PROPN
iajs-1618	211	26	c	c	PROPN
iajs-1618	211	27	c	c	NOUN
iajs-1618	211	28	on	on	ADP
iajs-1618	211	29	sclosed	sclose	VERB
iajs-1618	211	30	submodules	submodule	NOUN
iajs-1618	211	31	.	.	PUNCT
iajs-1618	212	1	proposition	proposition	NOUN
iajs-1618	212	2	2	2	NUM
iajs-1618	212	3	.	.	NOUN
iajs-1618	212	4	9	9	NUM
iajs-1618	212	5	:	:	PUNCT
iajs-1618	212	6	let	let	VERB
iajs-1618	212	7	w	w	PART
iajs-1618	212	8	be	be	AUX
iajs-1618	212	9	an	an	DET
iajs-1618	212	10	l	l	NOUN
iajs-1618	212	11	-	-	NOUN
iajs-1618	212	12	module	module	NOUN
iajs-1618	212	13	such	such	ADJ
iajs-1618	212	14	that	that	SCONJ
iajs-1618	212	15	the	the	DET
iajs-1618	212	16	sum	sum	NOUN
iajs-1618	212	17	of	of	ADP
iajs-1618	212	18	any	any	DET
iajs-1618	212	19	two	two	NUM
iajs-1618	212	20	sclosed	sclose	VERB
iajs-1618	212	21	submodules	submodule	NOUN
iajs-1618	212	22	of	of	ADP
iajs-1618	212	23	w	w	PROPN
iajs-1618	212	24	is	be	AUX
iajs-1618	212	25	again	again	ADV
iajs-1618	212	26	an	an	DET
iajs-1618	212	27	sclosed	sclosed	ADJ
iajs-1618	212	28	submodule	submodule	NOUN
iajs-1618	212	29	.	.	PUNCT
iajs-1618	213	1	if	if	SCONJ
iajs-1618	213	2	a	a	PRON
iajs-1618	213	3	is	be	AUX
iajs-1618	213	4	an	an	DET
iajs-1618	213	5	sclosed	sclose	VERB
iajs-1618	213	6	submodule	submodule	NOUN
iajs-1618	213	7	of	of	ADP
iajs-1618	213	8	w	w	NOUN
iajs-1618	213	9	such	such	ADJ
iajs-1618	213	10	that	that	SCONJ
iajs-1618	213	11	a	a	PRON
iajs-1618	213	12	and	and	CCONJ
iajs-1618	213	13	satisfy	satisfy	VERB
iajs-1618	213	14	a	a	DET
iajs-1618	213	15	c	c	NOUN
iajs-1618	213	16	c	c	NOUN
iajs-1618	213	17	(	(	PUNCT
iajs-1618	213	18	respectively	respectively	ADV
iajs-1618	213	19	d	d	PROPN
iajs-1618	213	20	c	c	NOUN
iajs-1618	213	21	c	c	NOUN
iajs-1618	213	22	)	)	PUNCT
iajs-1618	213	23	on	on	ADP
iajs-1618	213	24	s	s	NOUN
iajs-1618	213	25	-	-	PUNCT
iajs-1618	213	26	closed	closed	ADJ
iajs-1618	213	27	submodules	submodule	NOUN
iajs-1618	213	28	,	,	PUNCT
iajs-1618	213	29	then	then	ADV
iajs-1618	213	30	w	w	NOUN
iajs-1618	213	31	satisfies	satisfie	NOUN
iajs-1618	213	32	a	a	DET
iajs-1618	213	33	c	c	NOUN
iajs-1618	213	34	c	c	NOUN
iajs-1618	213	35	(	(	PUNCT
iajs-1618	213	36	respectively	respectively	ADV
iajs-1618	213	37	d	d	PROPN
iajs-1618	213	38	c	c	NOUN
iajs-1618	213	39	c	c	NOUN
iajs-1618	213	40	)	)	PUNCT
iajs-1618	213	41	on	on	ADP
iajs-1618	213	42	sclosed	sclose	VERB
iajs-1618	213	43	submodules	submodule	NOUN
iajs-1618	213	44	.	.	PUNCT
iajs-1618	214	1	proof	proof	NOUN
iajs-1618	214	2	:	:	PUNCT
iajs-1618	214	3	assume	assume	VERB
iajs-1618	214	4	b1	b1	PROPN
iajs-1618	214	5			PROPN
iajs-1618	214	6	b2	b2	PROPN
iajs-1618	214	7			NOUN
iajs-1618	214	8	…	…	PUNCT
iajs-1618	214	9	be	be	AUX
iajs-1618	214	10	ascending	ascend	VERB
iajs-1618	214	11	chain	chain	NOUN
iajs-1618	214	12	of	of	ADP
iajs-1618	214	13	s	s	NOUN
iajs-1618	214	14	-closed	-close	VERB
iajs-1618	214	15	submodules	submodule	NOUN
iajs-1618	214	16	of	of	ADP
iajs-1618	214	17	an	an	DET
iajs-1618	214	18	l	l	NOUN
iajs-1618	214	19	-	-	NOUN
iajs-1618	214	20	module	module	NOUN
iajs-1618	214	21	w	w	NOUN
iajs-1618	214	22	,	,	PUNCT
iajs-1618	214	23	then	then	ADV
iajs-1618	214	24	by	by	ADP
iajs-1618	214	25	[	[	X
iajs-1618	214	26	1	1	NUM
iajs-1618	214	27	,	,	PUNCT
iajs-1618	214	28	remaks	remak	NOUN
iajs-1618	214	29	and	and	CCONJ
iajs-1618	214	30	examples	example	NOUN
iajs-1618	214	31	2.2(3	2.2(3	NUM
iajs-1618	214	32	)	)	PUNCT
iajs-1618	214	33	,	,	PUNCT
iajs-1618	214	34	p.343	p.343	X
iajs-1618	214	35	]	]	PUNCT
iajs-1618	214	36	bi	bi	NOUN
iajs-1618	214	37	∩	∩	NOUN
iajs-1618	214	38	a	a	PRON
iajs-1618	214	39	is	be	AUX
iajs-1618	214	40	an	an	DET
iajs-1618	214	41	s	s	NOUN
iajs-1618	214	42	-	-	PUNCT
iajs-1618	214	43	closed	closed	ADJ
iajs-1618	214	44	submodule	submodule	NOUN
iajs-1618	214	45	of	of	ADP
iajs-1618	214	46	w	w	PROPN
iajs-1618	214	47	,	,	PUNCT
iajs-1618	214	48	for	for	ADP
iajs-1618	214	49	each	each	DET
iajs-1618	214	50	i	i	NOUN
iajs-1618	214	51	=	=	NOUN
iajs-1618	214	52	1	1	NUM
iajs-1618	214	53	,	,	PUNCT
iajs-1618	214	54	2	2	NUM
iajs-1618	214	55	,	,	PUNCT
iajs-1618	214	56	…	…	PUNCT
iajs-1618	214	57	,	,	PUNCT
iajs-1618	214	58	but	but	CCONJ
iajs-1618	214	59	(	(	PUNCT
iajs-1618	214	60	bi∩a	bi∩a	PROPN
iajs-1618	214	61	)	)	PUNCT
iajs-1618	214	62			PROPN
iajs-1618	214	63	a	a	PRON
iajs-1618	214	64	,	,	PUNCT
iajs-1618	214	65	thus	thus	ADV
iajs-1618	214	66	bi∩a	bi∩a	NOUN
iajs-1618	214	67	is	be	AUX
iajs-1618	214	68	an	an	DET
iajs-1618	214	69	s	s	NOUN
iajs-1618	214	70	-	-	PUNCT
iajs-1618	214	71	closed	closed	ADJ
iajs-1618	214	72	submodule	submodule	NOUN
iajs-1618	214	73	of	of	ADP
iajs-1618	214	74	a	a	PRON
iajs-1618	214	75	,	,	PUNCT
iajs-1618	214	76	for	for	ADP
iajs-1618	214	77	each	each	DET
iajs-1618	214	78	i	i	NOUN
iajs-1618	214	79	=	=	NOUN
iajs-1618	214	80	1	1	NUM
iajs-1618	214	81	,	,	PUNCT
iajs-1618	214	82	2	2	NUM
iajs-1618	214	83	,	,	PUNCT
iajs-1618	214	84	…	…	PUNCT
iajs-1618	214	85	,	,	PUNCT
iajs-1618	214	86	by	by	ADP
iajs-1618	214	87	[	[	X
iajs-1618	214	88	1	1	NUM
iajs-1618	214	89	,	,	PUNCT
iajs-1618	214	90	prop	prop	NOUN
iajs-1618	214	91	.	.	PUNCT
iajs-1618	215	1	2.8	2.8	NUM
iajs-1618	215	2	,	,	PUNCT
iajs-1618	215	3	p.345	p.345	NOUN
iajs-1618	215	4	]	]	PUNCT
iajs-1618	215	5	.	.	PUNCT
iajs-1618	216	1	also	also	ADV
iajs-1618	216	2	,	,	PUNCT
iajs-1618	216	3	bi	bi	PROPN
iajs-1618	216	4	+	+	CCONJ
iajs-1618	216	5	a	a	PRON
iajs-1618	216	6	is	be	AUX
iajs-1618	216	7	an	an	DET
iajs-1618	216	8	s	s	NOUN
iajs-1618	216	9	closed	closed	ADJ
iajs-1618	216	10	submodule	submodule	NOUN
iajs-1618	216	11	of	of	ADP
iajs-1618	216	12	w	w	PROPN
iajs-1618	216	13	(	(	PUNCT
iajs-1618	216	14	by	by	ADP
iajs-1618	216	15	our	our	PRON
iajs-1618	216	16	assumption	assumption	NOUN
iajs-1618	216	17	)	)	PUNCT
iajs-1618	216	18	,	,	PUNCT
iajs-1618	216	19	hence	hence	ADV
iajs-1618	216	20	is	be	AUX
iajs-1618	216	21	an	an	DET
iajs-1618	216	22	s	s	NOUN
iajs-1618	216	23	closed	closed	ADJ
iajs-1618	216	24	submodule	submodule	NOUN
iajs-1618	216	25	of	of	ADP
iajs-1618	216	26	,	,	PUNCT
iajs-1618	216	27	for	for	ADP
iajs-1618	216	28	each	each	DET
iajs-1618	216	29	i	i	NOUN
iajs-1618	216	30	=	=	NOUN
iajs-1618	216	31	1	1	NUM
iajs-1618	216	32	,	,	PUNCT
iajs-1618	216	33	2	2	NUM
iajs-1618	216	34	,	,	PUNCT
iajs-1618	216	35	…	…	PUNCT
iajs-1618	216	36	,	,	PUNCT
iajs-1618	216	37	by	by	ADP
iajs-1618	216	38	proposition	proposition	NOUN
iajs-1618	216	39	1.12	1.12	NUM
iajs-1618	216	40	.	.	PUNCT
iajs-1618	217	1	now	now	ADV
iajs-1618	217	2	consider	consider	VERB
iajs-1618	217	3	the	the	DET
iajs-1618	217	4	two	two	NUM
iajs-1618	217	5	following	follow	VERB
iajs-1618	217	6	two	two	NUM
iajs-1618	217	7	ascending	ascend	VERB
iajs-1618	217	8	chain	chain	NOUN
iajs-1618	217	9	of	of	ADP
iajs-1618	217	10	s	s	NOUN
iajs-1618	217	11	-	-	PUNCT
iajs-1618	217	12	closed	closed	ADJ
iajs-1618	217	13	submodules	submodule	NOUN
iajs-1618	217	14	of	of	ADP
iajs-1618	217	15	a	a	PRON
iajs-1618	217	16	and	and	CCONJ
iajs-1618	217	17	:	:	PUNCT
iajs-1618	217	18	b1	b1	NOUN
iajs-1618	217	19	∩	∩	NOUN
iajs-1618	217	20	a	a	DET
iajs-1618	217	21			PROPN
iajs-1618	217	22	b2	b2	NOUN
iajs-1618	217	23	∩	∩	NOUN
iajs-1618	217	24	a	a	DET
iajs-1618	217	25			NOUN
iajs-1618	217	26	…	…	PUNCT
iajs-1618	217	27	,	,	PUNCT
iajs-1618	217	28	and	and	CCONJ
iajs-1618	217	29			PROPN
iajs-1618	217	30			PROPN
iajs-1618	217	31	…	…	PUNCT
iajs-1618	217	32	,	,	PUNCT
iajs-1618	217	33	but	but	CCONJ
iajs-1618	217	34	a	a	PRON
iajs-1618	217	35	and	and	CCONJ
iajs-1618	217	36	satisfy	satisfy	VERB
iajs-1618	217	37	a	a	DET
iajs-1618	217	38	c	c	NOUN
iajs-1618	217	39	c	c	NOUN
iajs-1618	217	40	on	on	ADP
iajs-1618	217	41	s	s	NOUN
iajs-1618	217	42	-	-	PUNCT
iajs-1618	217	43	closed	closed	ADJ
iajs-1618	217	44	submodules	submodule	NOUN
iajs-1618	217	45	.	.	PUNCT
iajs-1618	218	1	therefore	therefore	ADV
iajs-1618	218	2	,	,	PUNCT
iajs-1618	218	3	there	there	PRON
iajs-1618	218	4	exists	exist	VERB
iajs-1618	218	5	k1	k1	PROPN
iajs-1618	218	6	,	,	PUNCT
iajs-1618	218	7	k2	k2	ADJ
iajs-1618	218	8			PROPN
iajs-1618	218	9	z+	z+	NUM
iajs-1618	218	10	such	such	ADJ
iajs-1618	218	11	that	that	DET
iajs-1618	218	12	bn	bn	ADP
iajs-1618	218	13	∩	∩	NOUN
iajs-1618	218	14	a	a	DET
iajs-1618	218	15	=	=	PUNCT
iajs-1618	218	16	bk1	bk1	NOUN
iajs-1618	218	17	∩	∩	PROPN
iajs-1618	218	18	a	a	X
iajs-1618	218	19	,	,	PUNCT
iajs-1618	218	20	for	for	ADP
iajs-1618	218	21	each	each	DET
iajs-1618	218	22	n	n	CCONJ
iajs-1618	218	23			NUM
iajs-1618	218	24	k1	k1	NOUN
iajs-1618	218	25	,	,	PUNCT
iajs-1618	218	26	and	and	CCONJ
iajs-1618	218	27	=	=	SYM
iajs-1618	218	28	,	,	PUNCT
iajs-1618	218	29	for	for	ADP
iajs-1618	218	30	each	each	DET
iajs-1618	218	31	n	n	CCONJ
iajs-1618	218	32			NUM
iajs-1618	218	33	k2	k2	NOUN
iajs-1618	218	34	.	.	PUNCT
iajs-1618	219	1	by	by	ADP
iajs-1618	219	2	isomorphism	isomorphism	PROPN
iajs-1618	219	3	theorem	theorem	VERB
iajs-1618	219	4			PROPN
iajs-1618	220	1	[	[	X
iajs-1618	220	2	2	2	NUM
iajs-1618	220	3	,	,	PUNCT
iajs-1618	220	4	th	th	X
iajs-1618	220	5	.	.	PUNCT
iajs-1618	221	1	3.4.3	3.4.3	NUM
iajs-1618	221	2	,	,	PUNCT
iajs-1618	221	3	p.	p.	NOUN
iajs-1618	221	4	56	56	NUM
iajs-1618	221	5	]	]	PUNCT
iajs-1618	221	6	,	,	PUNCT
iajs-1618	222	1	so	so	CCONJ
iajs-1618	222	2			PROPN
iajs-1618	222	3	.	.	PUNCT
iajs-1618	223	1	hence	hence	ADV
iajs-1618	223	2	,	,	PUNCT
iajs-1618	223	3	=	=	PRON
iajs-1618	223	4	,	,	PUNCT
iajs-1618	223	5	which	which	PRON
iajs-1618	223	6	means	mean	VERB
iajs-1618	223	7	bn	bn	INTJ
iajs-1618	223	8	∩	∩	NOUN
iajs-1618	223	9	a	a	DET
iajs-1618	223	10	=	=	SYM
iajs-1618	223	11	bk2	bk2	NOUN
iajs-1618	223	12	∩	∩	NOUN
iajs-1618	223	13	a	a	X
iajs-1618	223	14	,	,	PUNCT
iajs-1618	223	15	for	for	ADP
iajs-1618	223	16	each	each	DET
iajs-1618	223	17	n	n	CCONJ
iajs-1618	223	18			NUM
iajs-1618	223	19	k2	k2	NOUN
iajs-1618	223	20	.	.	PUNCT
iajs-1618	224	1	let	let	VERB
iajs-1618	224	2	k	k	PROPN
iajs-1618	224	3	=	=	SYM
iajs-1618	224	4	max	max	PROPN
iajs-1618	224	5	{	{	PUNCT
iajs-1618	224	6	k1	k1	PROPN
iajs-1618	224	7	,	,	PUNCT
iajs-1618	224	8	k2	k2	NOUN
iajs-1618	224	9	}	}	PUNCT
iajs-1618	224	10	,	,	PUNCT
iajs-1618	224	11	thus	thus	ADV
iajs-1618	224	12	bn	bn	ADP
iajs-1618	224	13	∩	∩	NOUN
iajs-1618	224	14	a	a	DET
iajs-1618	224	15	=	=	X
iajs-1618	224	16	bk	bk	PROPN
iajs-1618	224	17	∩	∩	NOUN
iajs-1618	224	18	a	a	PRON
iajs-1618	224	19	for	for	ADP
iajs-1618	224	20	each	each	DET
iajs-1618	224	21	n	n	NUM
iajs-1618	224	22			NUM
iajs-1618	224	23	k	k	PROPN
iajs-1618	224	24	and	and	CCONJ
iajs-1618	224	25	bn	bn	ADP
iajs-1618	224	26	∩	∩	NOUN
iajs-1618	224	27	a	a	DET
iajs-1618	224	28	=	=	X
iajs-1618	224	29	bk	bk	NOUN
iajs-1618	224	30	∩	∩	PROPN
iajs-1618	224	31	bn	bn	VERB
iajs-1618	224	32	for	for	ADP
iajs-1618	224	33	each	each	DET
iajs-1618	224	34	n	n	NOUN
iajs-1618	224	35			NUM
iajs-1618	224	36	k.	k.	PROPN
iajs-1618	225	1	now	now	ADV
iajs-1618	225	2	,	,	PUNCT
iajs-1618	225	3	for	for	ADP
iajs-1618	225	4	each	each	DET
iajs-1618	225	5	n	n	PRON
iajs-1618	225	6			NUM
iajs-1618	225	7	k	k	PROPN
iajs-1618	225	8	,	,	PUNCT
iajs-1618	225	9	bn=	bn=	ADJ
iajs-1618	225	10	bn	bn	ADJ
iajs-1618	225	11	∩	∩	NOUN
iajs-1618	225	12	(	(	PUNCT
iajs-1618	225	13	bn	bn	X
iajs-1618	225	14	+	+	NOUN
iajs-1618	225	15	a	a	X
iajs-1618	225	16	)	)	PUNCT
iajs-1618	225	17	=	=	PUNCT
iajs-1618	225	18	bn∩	bn∩	NOUN
iajs-1618	225	19	(	(	PUNCT
iajs-1618	225	20	bk	bk	VERB
iajs-1618	225	21	+	+	NOUN
iajs-1618	225	22	a	a	X
iajs-1618	225	23	)	)	PUNCT
iajs-1618	225	24	=	=	SYM
iajs-1618	225	25	bk	bk	NOUN
iajs-1618	225	26	∩	∩	NOUN
iajs-1618	225	27	(	(	PUNCT
iajs-1618	225	28	bk	bk	VERB
iajs-1618	225	29	+	+	NOUN
iajs-1618	225	30	a	a	X
iajs-1618	225	31	)	)	PUNCT
iajs-1618	225	32	=	=	SYM
iajs-1618	225	33	bk	bk	NOUN
iajs-1618	225	34	.	.	PUNCT
iajs-1618	226	1	thus	thus	ADV
iajs-1618	226	2	,	,	PUNCT
iajs-1618	226	3	m	m	VERB
iajs-1618	226	4	satisfies	satisfy	VERB
iajs-1618	226	5	a	a	DET
iajs-1618	226	6	c	c	NOUN
iajs-1618	226	7	c	c	NOUN
iajs-1618	226	8	on	on	ADP
iajs-1618	226	9	s	s	NOUN
iajs-1618	226	10	-	-	PUNCT
iajs-1618	226	11	closed	closed	ADJ
iajs-1618	226	12	submodules	submodule	NOUN
iajs-1618	226	13	.	.	PUNCT
iajs-1618	227	1	by	by	ADP
iajs-1618	227	2	a	a	DET
iajs-1618	227	3	similarly	similarly	ADV
iajs-1618	227	4	proof	proof	NOUN
iajs-1618	227	5	w	w	NOUN
iajs-1618	227	6	satisfies	satisfie	NOUN
iajs-1618	227	7	d	d	X
iajs-1618	227	8	c	c	NOUN
iajs-1618	227	9	c	c	NOUN
iajs-1618	227	10	on	on	ADP
iajs-1618	227	11	s	s	NOUN
iajs-1618	227	12	-	-	PUNCT
iajs-1618	227	13	closed	closed	ADJ
iajs-1618	227	14	submodules	submodule	NOUN
iajs-1618	227	15	.	.	PUNCT
iajs-1618	228	1	mathematics	mathematic	NOUN
iajs-1618	228	2	|234	|234	PRON
iajs-1618	229	1	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	X
iajs-1618	229	2	7102	7102	NUM
iajs-1618	229	3	(	(	PUNCT
iajs-1618	229	4	عام	عام	PROPN
iajs-1618	229	5	3	3	NUM
iajs-1618	229	6	(	(	PUNCT
iajs-1618	229	7	العدد	العدد	PROPN
iajs-1618	229	8	)	)	PUNCT
iajs-1618	229	9	30مجلة	30مجلة	PROPN
iajs-1618	229	10	إبن	إبن	VERB
iajs-1618	229	11	الهيثم	الهيثم	ADJ
iajs-1618	229	12	للعلوم	للعلوم	NOUN
iajs-1618	229	13	الصرفة	الصرفة	NOUN
iajs-1618	230	1	و	و	PRON
iajs-1618	230	2	التطبيقية	التطبيقية	ADV
iajs-1618	230	3	المجلد	المجلد	ADV
iajs-1618	230	4	)	)	PUNCT
iajs-1618	231	1	ibn	ibn	PROPN
iajs-1618	231	2	al	al	PROPN
iajs-1618	231	3	-	-	PUNCT
iajs-1618	231	4	haitham	haitham	PROPN
iajs-1618	231	5	j.	j.	PROPN
iajs-1618	231	6	for	for	ADP
iajs-1618	231	7	pure	pure	PROPN
iajs-1618	231	8	&	&	CCONJ
iajs-1618	231	9	appl	appl	PROPN
iajs-1618	231	10	.	.	PUNCT
iajs-1618	232	1	sci	sci	PROPN
iajs-1618	232	2	.	.	PUNCT
iajs-1618	233	1	vol.03	vol.03	PROPN
iajs-1618	233	2	(	(	PUNCT
iajs-1618	233	3	3	3	NUM
iajs-1618	233	4	)	)	PUNCT
iajs-1618	233	5	2017	2017	NUM
iajs-1618	233	6	proposition	proposition	NOUN
iajs-1618	233	7	2	2	NUM
iajs-1618	233	8	.	.	NOUN
iajs-1618	233	9	10	10	NUM
iajs-1618	233	10	:	:	PUNCT
iajs-1618	233	11	let	let	VERB
iajs-1618	233	12	w	w	PART
iajs-1618	233	13	be	be	AUX
iajs-1618	233	14	a	a	DET
iajs-1618	233	15	fmfg	fmfg	ADJ
iajs-1618	233	16	l	l	NOUN
iajs-1618	233	17	-	-	NOUN
iajs-1618	233	18	module	module	NOUN
iajs-1618	233	19	.	.	PUNCT
iajs-1618	234	1	then	then	ADV
iajs-1618	234	2	w	w	NOUN
iajs-1618	234	3	satisfies	satisfie	NOUN
iajs-1618	234	4	a	a	DET
iajs-1618	234	5	c	c	NOUN
iajs-1618	234	6	c	c	NOUN
iajs-1618	234	7	(	(	PUNCT
iajs-1618	234	8	respectively	respectively	ADV
iajs-1618	234	9	d	d	PROPN
iajs-1618	234	10	c	c	NOUN
iajs-1618	234	11	c	c	NOUN
iajs-1618	234	12	)	)	PUNCT
iajs-1618	234	13	on	on	ADP
iajs-1618	234	14	sclosed	sclose	VERB
iajs-1618	234	15	submodules	submodule	NOUN
iajs-1618	234	16	if	if	SCONJ
iajs-1618	234	17	and	and	CCONJ
iajs-1618	234	18	only	only	ADV
iajs-1618	234	19	if	if	SCONJ
iajs-1618	234	20	l	l	NOUN
iajs-1618	234	21	satisfies	satisfy	VERB
iajs-1618	234	22	a	a	DET
iajs-1618	234	23	c	c	NOUN
iajs-1618	234	24	c	c	NOUN
iajs-1618	234	25	(	(	PUNCT
iajs-1618	234	26	respectively	respectively	ADV
iajs-1618	234	27	d	d	PROPN
iajs-1618	234	28	c	c	NOUN
iajs-1618	234	29	c	c	NOUN
iajs-1618	234	30	)	)	PUNCT
iajs-1618	234	31	on	on	ADP
iajs-1618	234	32	sclosed	sclose	VERB
iajs-1618	234	33	ideals	ideal	NOUN
iajs-1618	234	34	.	.	PUNCT
iajs-1618	235	1	proof	proof	NOUN
iajs-1618	235	2	:	:	PUNCT
iajs-1618	235	3	(	(	PUNCT
iajs-1618	235	4			NOUN
iajs-1618	235	5	)	)	PUNCT
iajs-1618	235	6	suppose	suppose	VERB
iajs-1618	235	7	w	w	NOUN
iajs-1618	235	8	satisfies	satisfie	NOUN
iajs-1618	235	9	a	a	DET
iajs-1618	235	10	c	c	NOUN
iajs-1618	235	11	c	c	NOUN
iajs-1618	235	12	(	(	PUNCT
iajs-1618	235	13	respectively	respectively	ADV
iajs-1618	235	14	d	d	PROPN
iajs-1618	235	15	c	c	NOUN
iajs-1618	235	16	c	c	NOUN
iajs-1618	235	17	)	)	PUNCT
iajs-1618	235	18	on	on	ADP
iajs-1618	235	19	sclosed	sclose	VERB
iajs-1618	235	20	submodules	submodule	NOUN
iajs-1618	235	21	.	.	PUNCT
iajs-1618	236	1	to	to	PART
iajs-1618	236	2	prove	prove	VERB
iajs-1618	236	3	l	l	NOUN
iajs-1618	236	4	satisfies	satisfie	NOUN
iajs-1618	236	5	a	a	DET
iajs-1618	236	6	c	c	NOUN
iajs-1618	236	7	c	c	NOUN
iajs-1618	236	8	(	(	PUNCT
iajs-1618	236	9	respectively	respectively	ADV
iajs-1618	236	10	d	d	PROPN
iajs-1618	236	11	c	c	NOUN
iajs-1618	236	12	c	c	NOUN
iajs-1618	236	13	)	)	PUNCT
iajs-1618	236	14	on	on	ADP
iajs-1618	236	15	sclosed	sclose	VERB
iajs-1618	236	16	ideals	ideal	NOUN
iajs-1618	236	17	.	.	PUNCT
iajs-1618	237	1	let	let	VERB
iajs-1618	237	2	i1	i1	PROPN
iajs-1618	237	3			PROPN
iajs-1618	237	4	i2	i2	PROPN
iajs-1618	237	5			PROPN
iajs-1618	237	6	…	…	PUNCT
iajs-1618	237	7	(	(	PUNCT
iajs-1618	237	8	i1	i1	PROPN
iajs-1618	237	9			PROPN
iajs-1618	237	10	i2	i2	PROPN
iajs-1618	237	11			PROPN
iajs-1618	237	12	…	…	PUNCT
iajs-1618	237	13	)	)	PUNCT
iajs-1618	237	14	be	be	AUX
iajs-1618	237	15	an	an	DET
iajs-1618	237	16	ascending	ascend	VERB
iajs-1618	237	17	(	(	PUNCT
iajs-1618	237	18	respectively	respectively	ADV
iajs-1618	237	19	descending	descending	NOUN
iajs-1618	237	20	)	)	PUNCT
iajs-1618	237	21	chain	chain	NOUN
iajs-1618	237	22	of	of	ADP
iajs-1618	237	23	s	s	NOUN
iajs-1618	237	24	-	-	PUNCT
iajs-1618	237	25	closed	closed	ADJ
iajs-1618	237	26	ideals	ideal	NOUN
iajs-1618	237	27	of	of	ADP
iajs-1618	237	28	l.	l.	NOUN
iajs-1618	237	29	thus	thus	ADV
iajs-1618	237	30	by	by	ADP
iajs-1618	237	31	(	(	PUNCT
iajs-1618	237	32	proposition	proposition	NOUN
iajs-1618	237	33	1.14	1.14	NUM
iajs-1618	237	34	)	)	PUNCT
iajs-1618	237	35	a1	a1	NOUN
iajs-1618	237	36	=	=	SYM
iajs-1618	237	37	i1w	i1w	PROPN
iajs-1618	237	38			PROPN
iajs-1618	237	39	a2	a2	PROPN
iajs-1618	237	40	=	=	PUNCT
iajs-1618	237	41	i2w	i2w	PROPN
iajs-1618	237	42			PROPN
iajs-1618	237	43	…	…	PUNCT
iajs-1618	237	44	(	(	PUNCT
iajs-1618	237	45	respectively	respectively	ADV
iajs-1618	237	46	a1	a1	NOUN
iajs-1618	237	47	=	=	SYM
iajs-1618	237	48	i1w	i1w	PROPN
iajs-1618	237	49			PROPN
iajs-1618	237	50	a2	a2	PROPN
iajs-1618	237	51	=	=	PUNCT
iajs-1618	237	52	i2w	i2w	PROPN
iajs-1618	237	53			PROPN
iajs-1618	237	54	…	…	PUNCT
iajs-1618	237	55	)	)	PUNCT
iajs-1618	237	56	is	be	AUX
iajs-1618	237	57	an	an	DET
iajs-1618	237	58	ascending	ascend	VERB
iajs-1618	237	59	(	(	PUNCT
iajs-1618	237	60	respectively	respectively	ADV
iajs-1618	237	61	descending	descending	NOUN
iajs-1618	237	62	)	)	PUNCT
iajs-1618	237	63	chain	chain	NOUN
iajs-1618	237	64	of	of	ADP
iajs-1618	237	65	s	s	NOUN
iajs-1618	237	66	-	-	PUNCT
iajs-1618	237	67	closed	closed	ADJ
iajs-1618	237	68	submodules	submodule	NOUN
iajs-1618	237	69	of	of	ADP
iajs-1618	237	70	w.	w.	PROPN
iajs-1618	237	71	but	but	CCONJ
iajs-1618	237	72	w	w	PROPN
iajs-1618	237	73	satisfies	satisfie	NOUN
iajs-1618	237	74	a	a	DET
iajs-1618	237	75	c	c	NOUN
iajs-1618	237	76	c	c	NOUN
iajs-1618	237	77	(	(	PUNCT
iajs-1618	237	78	respectively	respectively	ADV
iajs-1618	237	79	d	d	PROPN
iajs-1618	237	80	c	c	NOUN
iajs-1618	237	81	c	c	NOUN
iajs-1618	237	82	)	)	PUNCT
iajs-1618	237	83	on	on	ADP
iajs-1618	237	84	s	s	NOUN
iajs-1618	237	85	-	-	PUNCT
iajs-1618	237	86	closed	closed	ADJ
iajs-1618	237	87	submodules	submodule	NOUN
iajs-1618	237	88	,	,	PUNCT
iajs-1618	237	89	so	so	SCONJ
iajs-1618	237	90	there	there	PRON
iajs-1618	237	91	exists	exist	VERB
iajs-1618	237	92	k	k	PROPN
iajs-1618	237	93			PROPN
iajs-1618	237	94	z+	z+	NUM
iajs-1618	237	95	such	such	ADJ
iajs-1618	237	96	that	that	SCONJ
iajs-1618	237	97	an	an	DET
iajs-1618	237	98	=	=	SYM
iajs-1618	237	99	ak	ak	PROPN
iajs-1618	237	100	for	for	ADP
iajs-1618	237	101	all	all	DET
iajs-1618	237	102	nk	nk	PROPN
iajs-1618	237	103	,	,	PUNCT
iajs-1618	237	104	hence	hence	ADV
iajs-1618	237	105	inw	inw	PROPN
iajs-1618	237	106	=	=	NOUN
iajs-1618	237	107	ikw	ikw	NOUN
iajs-1618	237	108	for	for	ADP
iajs-1618	237	109	all	all	DET
iajs-1618	237	110	nk	nk	PROPN
iajs-1618	237	111	,	,	PUNCT
iajs-1618	237	112	that	that	PRON
iajs-1618	237	113	is	be	AUX
iajs-1618	237	114	in	in	ADP
iajs-1618	237	115	=	=	PUNCT
iajs-1618	237	116	ik	ik	PROPN
iajs-1618	237	117	for	for	ADP
iajs-1618	237	118	all	all	DET
iajs-1618	237	119	nk	nk	PROPN
iajs-1618	237	120	.	.	PUNCT
iajs-1618	238	1	so	so	ADV
iajs-1618	238	2	l	l	NOUN
iajs-1618	238	3	satisfies	satisfy	VERB
iajs-1618	238	4	a	a	DET
iajs-1618	238	5	c	c	NOUN
iajs-1618	238	6	c	c	NOUN
iajs-1618	238	7	(	(	PUNCT
iajs-1618	238	8	respectively	respectively	ADV
iajs-1618	238	9	d	d	PROPN
iajs-1618	238	10	c	c	NOUN
iajs-1618	238	11	c	c	NOUN
iajs-1618	238	12	)	)	PUNCT
iajs-1618	238	13	on	on	ADP
iajs-1618	238	14	sclosed	sclose	VERB
iajs-1618	238	15	ideals	ideal	NOUN
iajs-1618	238	16	.	.	PUNCT
iajs-1618	239	1	(	(	PUNCT
iajs-1618	239	2			NOUN
iajs-1618	239	3	)	)	PUNCT
iajs-1618	239	4	similarly	similarly	ADV
iajs-1618	239	5	.	.	PUNCT
iajs-1618	240	1	recall	recall	VERB
iajs-1618	240	2	that	that	PRON
iajs-1618	240	3	,	,	PUNCT
iajs-1618	240	4	“	"	PUNCT
iajs-1618	240	5	an	an	DET
iajs-1618	240	6	lmodule	lmodule	NOUN
iajs-1618	240	7	w	w	NOUN
iajs-1618	240	8	is	be	AUX
iajs-1618	240	9	called	call	VERB
iajs-1618	240	10	a	a	DET
iajs-1618	240	11	scalar	scalar	ADJ
iajs-1618	240	12	module	module	NOUN
iajs-1618	240	13	if	if	SCONJ
iajs-1618	240	14	every	every	DET
iajs-1618	240	15	lendomorphism	lendomorphism	NOUN
iajs-1618	240	16	of	of	ADP
iajs-1618	240	17	w	w	PROPN
iajs-1618	240	18	is	be	AUX
iajs-1618	240	19	a	a	DET
iajs-1618	240	20	scalar	scalar	ADJ
iajs-1618	240	21	homomorphism	homomorphism	NOUN
iajs-1618	240	22	,	,	PUNCT
iajs-1618	240	23	that	that	PRON
iajs-1618	240	24	is	be	AUX
iajs-1618	240	25	for	for	ADP
iajs-1618	240	26	each	each	DET
iajs-1618	240	27	0	0	NUM
iajs-1618	240	28	≠	≠	PROPN
iajs-1618	240	29	f	f	NOUN
iajs-1618	240	30	end(w	end(w	PROPN
iajs-1618	240	31	)	)	PUNCT
iajs-1618	240	32	,	,	PUNCT
iajs-1618	240	33	there	there	PRON
iajs-1618	240	34	exists	exist	VERB
iajs-1618	240	35	0	0	NUM
iajs-1618	240	36	≠	≠	PROPN
iajs-1618	240	37	s	s	PROPN
iajs-1618	240	38	l	l	NOUN
iajs-1618	240	39	such	such	ADJ
iajs-1618	240	40	that	that	SCONJ
iajs-1618	240	41	f(a)=	f(a)=	PROPN
iajs-1618	240	42	sa	sa	NOUN
iajs-1618	240	43	for	for	ADP
iajs-1618	240	44	all	all	DET
iajs-1618	240	45	aw	aw	ADJ
iajs-1618	240	46	”	"	PUNCT
iajs-1618	240	47	.	.	PUNCT
iajs-1618	241	1	[	[	X
iajs-1618	241	2	11	11	NUM
iajs-1618	241	3	]	]	X
iajs-1618	241	4	corollary	corollary	ADJ
iajs-1618	241	5	2	2	NUM
iajs-1618	241	6	.	.	NOUN
iajs-1618	241	7	11	11	NUM
iajs-1618	241	8	:	:	PUNCT
iajs-1618	241	9	let	let	VERB
iajs-1618	241	10	w	w	PART
iajs-1618	241	11	be	be	AUX
iajs-1618	241	12	a	a	DET
iajs-1618	241	13	fmfg	fmfg	ADJ
iajs-1618	241	14	l	l	NOUN
iajs-1618	241	15	-	-	NOUN
iajs-1618	241	16	module	module	NOUN
iajs-1618	241	17	.	.	PUNCT
iajs-1618	242	1	then	then	ADV
iajs-1618	242	2	w	w	NOUN
iajs-1618	242	3	satisfies	satisfie	NOUN
iajs-1618	242	4	a	a	DET
iajs-1618	242	5	c	c	NOUN
iajs-1618	242	6	c	c	NOUN
iajs-1618	242	7	(	(	PUNCT
iajs-1618	242	8	respectively	respectively	ADV
iajs-1618	242	9	d	d	PROPN
iajs-1618	242	10	c	c	NOUN
iajs-1618	242	11	c	c	NOUN
iajs-1618	242	12	)	)	PUNCT
iajs-1618	242	13	on	on	ADP
iajs-1618	242	14	sclosed	sclose	VERB
iajs-1618	242	15	submodules	submodule	NOUN
iajs-1618	242	16	if	if	SCONJ
iajs-1618	242	17	and	and	CCONJ
iajs-1618	242	18	only	only	ADV
iajs-1618	242	19	if	if	SCONJ
iajs-1618	242	20	end(w	end(w	PROPN
iajs-1618	242	21	)	)	PUNCT
iajs-1618	242	22	satisfies	satisfy	VERB
iajs-1618	242	23	a	a	DET
iajs-1618	242	24	c	c	NOUN
iajs-1618	242	25	c	c	NOUN
iajs-1618	242	26	(	(	PUNCT
iajs-1618	242	27	respectively	respectively	ADV
iajs-1618	242	28	d	d	PROPN
iajs-1618	242	29	c	c	NOUN
iajs-1618	242	30	c	c	NOUN
iajs-1618	242	31	)	)	PUNCT
iajs-1618	242	32	on	on	ADP
iajs-1618	242	33	sclosed	sclose	VERB
iajs-1618	242	34	ideals	ideal	NOUN
iajs-1618	242	35	.	.	PUNCT
iajs-1618	243	1	proof	proof	NOUN
iajs-1618	243	2	:	:	PUNCT
iajs-1618	243	3	(	(	PUNCT
iajs-1618	243	4			NOUN
iajs-1618	243	5	)	)	PUNCT
iajs-1618	243	6	since	since	SCONJ
iajs-1618	243	7	w	w	PROPN
iajs-1618	243	8	be	be	AUX
iajs-1618	243	9	a	a	DET
iajs-1618	243	10	fmfg	fmfg	ADJ
iajs-1618	243	11	l	l	NOUN
iajs-1618	243	12	-	-	NOUN
iajs-1618	243	13	module	module	NOUN
iajs-1618	243	14	,	,	PUNCT
iajs-1618	243	15	then	then	ADV
iajs-1618	243	16	w	w	PROPN
iajs-1618	243	17	is	be	AUX
iajs-1618	243	18	a	a	DET
iajs-1618	243	19	scalar	scalar	ADJ
iajs-1618	243	20	module	module	NOUN
iajs-1618	243	21	by	by	ADP
iajs-1618	243	22	[	[	X
iajs-1618	243	23	11	11	NUM
iajs-1618	243	24	,	,	PUNCT
iajs-1618	243	25	coro.1.1.11	coro.1.1.11	VERB
iajs-1618	243	26	]	]	PUNCT
iajs-1618	243	27	,	,	PUNCT
iajs-1618	243	28	end(w	end(w	PROPN
iajs-1618	243	29	)	)	PUNCT
iajs-1618	243	30			PROPN
iajs-1618	243	31	by	by	ADP
iajs-1618	243	32	[	[	X
iajs-1618	243	33	12	12	NUM
iajs-1618	243	34	,	,	PUNCT
iajs-1618	243	35	lemma	lemma	PROPN
iajs-1618	243	36	6.2	6.2	NUM
iajs-1618	243	37	]	]	PUNCT
iajs-1618	243	38	.	.	PUNCT
iajs-1618	244	1	but	but	CCONJ
iajs-1618	244	2	ann(w	ann(w	PROPN
iajs-1618	244	3	)	)	PUNCT
iajs-1618	244	4	=	=	SYM
iajs-1618	245	1	0	0	NUM
iajs-1618	245	2	,	,	PUNCT
iajs-1618	245	3	so	so	ADV
iajs-1618	245	4	end(w	end(w	PROPN
iajs-1618	245	5	)	)	PUNCT
iajs-1618	245	6			PROPN
iajs-1618	245	7	l.	l.	PROPN
iajs-1618	245	8	hence	hence	ADV
iajs-1618	245	9	the	the	DET
iajs-1618	245	10	result	result	NOUN
iajs-1618	245	11	follows	follow	VERB
iajs-1618	245	12	by	by	ADP
iajs-1618	245	13	proposition	proposition	NOUN
iajs-1618	245	14	2.10	2.10	NUM
iajs-1618	245	15	.	.	PUNCT
iajs-1618	246	1	(	(	PUNCT
iajs-1618	246	2			NOUN
iajs-1618	246	3	)	)	PUNCT
iajs-1618	246	4	similarly	similarly	ADV
iajs-1618	246	5	.	.	PUNCT
iajs-1618	247	1	future	future	ADJ
iajs-1618	247	2	works	work	NOUN
iajs-1618	247	3	:	:	PUNCT
iajs-1618	247	4	1	1	X
iajs-1618	247	5	.	.	X
iajs-1618	247	6	give	give	VERB
iajs-1618	247	7	an	an	DET
iajs-1618	247	8	example	example	NOUN
iajs-1618	247	9	shows	show	VERB
iajs-1618	247	10	that	that	SCONJ
iajs-1618	247	11	every	every	DET
iajs-1618	247	12	noetherian	noetherian	NOUN
iajs-1618	247	13	(	(	PUNCT
iajs-1618	247	14	respectively	respectively	ADV
iajs-1618	247	15	artinian	artinian	ADJ
iajs-1618	247	16	)	)	PUNCT
iajs-1618	247	17	module	module	NOUN
iajs-1618	247	18	satisfies	satisfie	NOUN
iajs-1618	247	19	acc	acc	PROPN
iajs-1618	247	20	(	(	PUNCT
iajs-1618	247	21	respectively	respectively	ADV
iajs-1618	247	22	d	d	X
iajs-1618	247	23	c	c	NOUN
iajs-1618	247	24	c	c	NOUN
iajs-1618	247	25	)	)	PUNCT
iajs-1618	247	26	on	on	ADP
iajs-1618	247	27	s	s	VERB
iajs-1618	247	28	-closed	-close	VERB
iajs-1618	247	29	submodules	submodule	NOUN
iajs-1618	247	30	.	.	PUNCT
iajs-1618	248	1	2	2	X
iajs-1618	248	2	.	.	X
iajs-1618	248	3	give	give	VERB
iajs-1618	248	4	an	an	DET
iajs-1618	248	5	example	example	NOUN
iajs-1618	248	6	shows	show	VERB
iajs-1618	248	7	that	that	SCONJ
iajs-1618	248	8	the	the	DET
iajs-1618	248	9	converse	converse	NOUN
iajs-1618	248	10	of	of	ADP
iajs-1618	248	11	(	(	PUNCT
iajs-1618	248	12	remark	remark	NOUN
iajs-1618	248	13	2.2(2	2.2(2	NUM
iajs-1618	248	14	)	)	PUNCT
iajs-1618	248	15	)	)	PUNCT
iajs-1618	248	16	is	be	AUX
iajs-1618	248	17	not	not	PART
iajs-1618	248	18	true	true	ADJ
iajs-1618	248	19	in	in	ADP
iajs-1618	248	20	general	general	ADJ
iajs-1618	248	21	.	.	PUNCT
iajs-1618	249	1	3	3	X
iajs-1618	249	2	.	.	X
iajs-1618	249	3	give	give	VERB
iajs-1618	249	4	an	an	DET
iajs-1618	249	5	example	example	NOUN
iajs-1618	249	6	shows	show	VERB
iajs-1618	249	7	that	that	SCONJ
iajs-1618	249	8	the	the	DET
iajs-1618	249	9	converse	converse	NOUN
iajs-1618	249	10	of	of	ADP
iajs-1618	249	11	(	(	PUNCT
iajs-1618	249	12	proposition	proposition	NOUN
iajs-1618	249	13	2.4	2.4	NUM
iajs-1618	249	14	)	)	PUNCT
iajs-1618	249	15	is	be	AUX
iajs-1618	249	16	not	not	PART
iajs-1618	249	17	true	true	ADJ
iajs-1618	249	18	in	in	ADP
iajs-1618	249	19	general	general	ADJ
iajs-1618	249	20	.	.	PUNCT
iajs-1618	250	1	references	reference	NOUN
iajs-1618	250	2	1	1	X
iajs-1618	250	3	.	.	X
iajs-1618	251	1	mehdi	mehdi	PROPN
iajs-1618	251	2	sadiq	sadiq	PROPN
iajs-1618	251	3	abbas	abbas	PROPN
iajs-1618	251	4	and	and	CCONJ
iajs-1618	251	5	faten	faten	VERB
iajs-1618	251	6	hashim	hashim	PROPN
iajs-1618	251	7	mohammed	mohammed	PROPN
iajs-1618	251	8	,	,	PUNCT
iajs-1618	251	9	(	(	PUNCT
iajs-1618	251	10	2016	2016	NUM
iajs-1618	251	11	)	)	PUNCT
iajs-1618	251	12	,	,	PUNCT
iajs-1618	251	13	“	"	PUNCT
iajs-1618	251	14	small	small	ADJ
iajs-1618	251	15	-	-	PUNCT
iajs-1618	251	16	closed	close	VERB
iajs-1618	251	17	submodules	submodule	NOUN
iajs-1618	251	18	”	"	PUNCT
iajs-1618	251	19	,	,	PUNCT
iajs-1618	251	20	international	international	ADJ
iajs-1618	251	21	journal	journal	NOUN
iajs-1618	251	22	of	of	ADP
iajs-1618	251	23	scientific	scientific	ADJ
iajs-1618	251	24	and	and	CCONJ
iajs-1618	251	25	technical	technical	ADJ
iajs-1618	251	26	research	research	NOUN
iajs-1618	251	27	,	,	PUNCT
iajs-1618	251	28	6	6	NUM
iajs-1618	251	29	,	,	PUNCT
iajs-1618	251	30	1	1	NUM
iajs-1618	251	31	.	.	NOUN
iajs-1618	251	32	2	2	NUM
iajs-1618	251	33	.	.	X
iajs-1618	251	34	kasch	kasch	PROPN
iajs-1618	251	35	,	,	PUNCT
iajs-1618	251	36	f.	f.	PROPN
iajs-1618	251	37	(	(	PUNCT
iajs-1618	251	38	1982	1982	NUM
iajs-1618	251	39	)	)	PUNCT
iajs-1618	251	40	,	,	PUNCT
iajs-1618	251	41	“	"	PUNCT
iajs-1618	251	42	modules	module	NOUN
iajs-1618	251	43	and	and	CCONJ
iajs-1618	251	44	rings	ring	NOUN
iajs-1618	251	45	”	"	PUNCT
iajs-1618	251	46	,	,	PUNCT
iajs-1618	251	47	academic	academic	ADJ
iajs-1618	251	48	press	press	PROPN
iajs-1618	251	49	,	,	PUNCT
iajs-1618	251	50	inc	inc	PROPN
iajs-1618	251	51	.	.	PROPN
iajs-1618	251	52	london	london	PROPN
iajs-1618	251	53	.	.	PUNCT
iajs-1618	252	1	3	3	X
iajs-1618	252	2	.	.	X
iajs-1618	252	3	goodearlو	goodearlو	PROPN
iajs-1618	252	4	k.r	k.r	PROPN
iajs-1618	252	5	.	.	PUNCT
iajs-1618	252	6	(	(	PUNCT
iajs-1618	252	7	1976	1976	NUM
iajs-1618	252	8	)	)	PUNCT
iajs-1618	252	9	,	,	PUNCT
iajs-1618	252	10	“	"	PUNCT
iajs-1618	252	11	ring	ring	NOUN
iajs-1618	252	12	theory	theory	NOUN
iajs-1618	252	13	,	,	PUNCT
iajs-1618	252	14	nonsingular	nonsingular	ADJ
iajs-1618	252	15	rings	ring	NOUN
iajs-1618	252	16	and	and	CCONJ
iajs-1618	252	17	modules	module	NOUN
iajs-1618	252	18	”	"	PUNCT
iajs-1618	252	19	,	,	PUNCT
iajs-1618	252	20	marcel	marcel	PROPN
iajs-1618	252	21	dekker	dekker	PROPN
iajs-1618	252	22	,	,	PUNCT
iajs-1618	252	23	inc	inc	PROPN
iajs-1618	252	24	.	.	PROPN
iajs-1618	252	25	new	new	PROPN
iajs-1618	252	26	york	york	PROPN
iajs-1618	252	27	and	and	CCONJ
iajs-1618	252	28	basel	basel	PROPN
iajs-1618	252	29	.	.	PUNCT
iajs-1618	253	1	4	4	X
iajs-1618	253	2	.	.	X
iajs-1618	253	3	zhou	zhou	PROPN
iajs-1618	253	4	,	,	PUNCT
iajs-1618	253	5	d.x	d.x	PROPN
iajs-1618	253	6	.	.	PROPN
iajs-1618	253	7	and	and	CCONJ
iajs-1618	253	8	zhang	zhang	PROPN
iajs-1618	253	9	,	,	PUNCT
iajs-1618	253	10	x.r	x.r	PROPN
iajs-1618	253	11	.	.	PUNCT
iajs-1618	253	12	(	(	PUNCT
iajs-1618	253	13	2011	2011	NUM
iajs-1618	253	14	)	)	PUNCT
iajs-1618	253	15	,	,	PUNCT
iajs-1618	253	16	“	"	PUNCT
iajs-1618	253	17	small	small	ADJ
iajs-1618	253	18	-	-	PUNCT
iajs-1618	253	19	essential	essential	ADJ
iajs-1618	253	20	submodules	submodule	NOUN
iajs-1618	253	21	and	and	CCONJ
iajs-1618	253	22	morita	morita	PROPN
iajs-1618	253	23	duality	duality	PROPN
iajs-1618	253	24	”	"	PUNCT
iajs-1618	253	25	,	,	PUNCT
iajs-1618	253	26	southeast	southeast	ADJ
iajs-1618	253	27	asian	asian	ADJ
iajs-1618	253	28	bulletin	bulletin	NOUN
iajs-1618	253	29	of	of	ADP
iajs-1618	253	30	mathematics	mathematic	NOUN
iajs-1618	253	31	,	,	PUNCT
iajs-1618	253	32	35	35	NUM
iajs-1618	253	33	,	,	PUNCT
iajs-1618	253	34	(	(	PUNCT
iajs-1618	253	35	1051	1051	NUM
iajs-1618	253	36	-	-	SYM
iajs-1618	253	37	1062	1062	NUM
iajs-1618	253	38	)	)	PUNCT
iajs-1618	253	39	.	.	PUNCT
iajs-1618	254	1	5	5	X
iajs-1618	254	2	.	.	X
iajs-1618	254	3	fleury	fleury	PROPN
iajs-1618	254	4	,	,	PUNCT
iajs-1618	254	5	p.	p.	NOUN
iajs-1618	254	6	(	(	PUNCT
iajs-1618	254	7	1974	1974	NUM
iajs-1618	254	8	)	)	PUNCT
iajs-1618	254	9	,	,	PUNCT
iajs-1618	254	10	“	"	PUNCT
iajs-1618	254	11	hollow	hollow	ADJ
iajs-1618	254	12	modules	module	NOUN
iajs-1618	254	13	and	and	CCONJ
iajs-1618	254	14	local	local	ADJ
iajs-1618	254	15	endomorphism	endomorphism	PROPN
iajs-1618	254	16	rings	ring	NOUN
iajs-1618	254	17	”	"	PUNCT
iajs-1618	254	18	,	,	PUNCT
iajs-1618	254	19	pacific	pacific	PROPN
iajs-1618	254	20	j.	j.	PROPN
iajs-1618	254	21	math	math	PROPN
iajs-1618	254	22	.	.	PUNCT
iajs-1618	255	1	53	53	NUM
iajs-1618	255	2	,	,	PUNCT
iajs-1618	255	3	2	2	NUM
iajs-1618	255	4	,	,	PUNCT
iajs-1618	255	5	(	(	PUNCT
iajs-1618	255	6	379	379	NUM
iajs-1618	255	7	-385	-385	NUM
iajs-1618	255	8	)	)	PUNCT
iajs-1618	255	9	.	.	PUNCT
iajs-1618	256	1	6	6	X
iajs-1618	256	2	.	.	X
iajs-1618	256	3	el	el	NOUN
iajs-1618	256	4	-	-	PUNCT
iajs-1618	256	5	bast	bast	NOUN
iajs-1618	256	6	,	,	PUNCT
iajs-1618	256	7	z.	z.	PROPN
iajs-1618	256	8	a.	a.	PROPN
iajs-1618	256	9	and	and	CCONJ
iajs-1618	256	10	smith	smith	PROPN
iajs-1618	256	11	,	,	PUNCT
iajs-1618	256	12	p.	p.	PROPN
iajs-1618	256	13	f.	f.	PROPN
iajs-1618	256	14	(	(	PUNCT
iajs-1618	256	15	1988	1988	NUM
iajs-1618	256	16	)	)	PUNCT
iajs-1618	256	17	,	,	PUNCT
iajs-1618	256	18	“	"	PUNCT
iajs-1618	256	19	multiplication	multiplication	NOUN
iajs-1618	256	20	modules	module	NOUN
iajs-1618	256	21	”	"	PUNCT
iajs-1618	256	22	,	,	PUNCT
iajs-1618	256	23	comm	comm	NOUN
iajs-1618	256	24	.	.	PUNCT
iajs-1618	257	1	in	in	ADP
iajs-1618	257	2	algebra	algebra	NOUN
iajs-1618	257	3	,	,	PUNCT
iajs-1618	257	4	16	16	NUM
iajs-1618	257	5	,	,	PUNCT
iajs-1618	257	6	4	4	NUM
iajs-1618	257	7	,	,	PUNCT
iajs-1618	257	8	(	(	PUNCT
iajs-1618	257	9	755	755	NUM
iajs-1618	257	10	-	-	SYM
iajs-1618	257	11	779	779	NUM
iajs-1618	257	12	)	)	PUNCT
iajs-1618	257	13	.	.	PUNCT
iajs-1618	258	1	7	7	X
iajs-1618	258	2	.	.	X
iajs-1618	258	3	osofsky	osofsky	ADJ
iajs-1618	258	4	,	,	PUNCT
iajs-1618	258	5	b.	b.	PROPN
iajs-1618	258	6	l.	l.	PROPN
iajs-1618	258	7	(	(	PUNCT
iajs-1618	258	8	1991	1991	NUM
iajs-1618	258	9	)	)	PUNCT
iajs-1618	258	10	,	,	PUNCT
iajs-1618	258	11	“	"	PUNCT
iajs-1618	258	12	a	a	DET
iajs-1618	258	13	construction	construction	NOUN
iajs-1618	258	14	of	of	ADP
iajs-1618	258	15	nonstandard	nonstandard	ADJ
iajs-1618	258	16	uniserial	uniserial	ADJ
iajs-1618	258	17	modules	module	NOUN
iajs-1618	258	18	over	over	ADP
iajs-1618	258	19	valuation	valuation	NOUN
iajs-1618	258	20	domain	domain	NOUN
iajs-1618	258	21	”	"	PUNCT
iajs-1618	258	22	,	,	PUNCT
iajs-1618	258	23	bull	bull	NOUN
iajs-1618	258	24	.	.	PUNCT
iajs-1618	259	1	amer	amer	PROPN
iajs-1618	259	2	.	.	PUNCT
iajs-1618	259	3	math	math	PROPN
iajs-1618	259	4	.	.	PUNCT
iajs-1618	260	1	soc	soc	PROPN
iajs-1618	260	2	.	.	PUNCT
iajs-1618	261	1	(	(	PUNCT
iajs-1618	261	2	n.s	n.s	PROPN
iajs-1618	261	3	.	.	PROPN
iajs-1618	261	4	)	)	PUNCT
iajs-1618	261	5	,	,	PUNCT
iajs-1618	261	6	25	25	NUM
iajs-1618	261	7	,	,	PUNCT
iajs-1618	261	8	1	1	NUM
iajs-1618	261	9	,	,	PUNCT
iajs-1618	261	10	(	(	PUNCT
iajs-1618	261	11	89	89	NUM
iajs-1618	261	12	-	-	SYM
iajs-1618	261	13	97	97	NUM
iajs-1618	261	14	)	)	PUNCT
iajs-1618	261	15	.	.	PUNCT
iajs-1618	262	1	8	8	X
iajs-1618	262	2	.	.	X
iajs-1618	262	3	iman	iman	PROPN
iajs-1618	262	4	,	,	PUNCT
iajs-1618	262	5	ali	ali	PROPN
iajs-1618	262	6	athab	athab	PROPN
iajs-1618	262	7	,	,	PUNCT
iajs-1618	262	8	(	(	PUNCT
iajs-1618	262	9	2004	2004	NUM
iajs-1618	262	10	)	)	PUNCT
iajs-1618	262	11	,	,	PUNCT
iajs-1618	262	12	“	"	PUNCT
iajs-1618	262	13	some	some	DET
iajs-1618	262	14	generalizations	generalization	NOUN
iajs-1618	262	15	of	of	ADP
iajs-1618	262	16	projective	projective	ADJ
iajs-1618	262	17	module	module	NOUN
iajs-1618	262	18	”	"	PUNCT
iajs-1618	262	19	,	,	PUNCT
iajs-1618	262	20	ph.d	ph.d	PROPN
iajs-1618	262	21	.	.	PUNCT
iajs-1618	263	1	thesis	thesis	NOUN
iajs-1618	263	2	,	,	PUNCT
iajs-1618	263	3	university	university	NOUN
iajs-1618	263	4	of	of	ADP
iajs-1618	263	5	baghdad	baghdad	PROPN
iajs-1618	263	6	.	.	PUNCT
iajs-1618	264	1	9	9	X
iajs-1618	264	2	.	.	X
iajs-1618	264	3	mehdi	mehdi	PROPN
iajs-1618	264	4	sadiq	sadiq	PROPN
iajs-1618	264	5	abbas	abbas	PROPN
iajs-1618	264	6	and	and	CCONJ
iajs-1618	264	7	faten	faten	VERB
iajs-1618	264	8	hashim	hashim	PROPN
iajs-1618	264	9	mohammed	mohammed	PROPN
iajs-1618	264	10	,	,	PUNCT
iajs-1618	264	11	(	(	PUNCT
iajs-1618	264	12	2016	2016	NUM
iajs-1618	264	13	)	)	PUNCT
iajs-1618	264	14	,	,	PUNCT
iajs-1618	264	15	“	"	PUNCT
iajs-1618	264	16	small	small	ADJ
iajs-1618	264	17	-	-	PUNCT
iajs-1618	264	18	extending	extend	VERB
iajs-1618	264	19	modules	module	NOUN
iajs-1618	264	20	”	"	PUNCT
iajs-1618	264	21	,	,	PUNCT
iajs-1618	264	22	mathematics	mathematic	NOUN
iajs-1618	264	23	and	and	CCONJ
iajs-1618	264	24	statics	statics	PROPN
iajs-1618	264	25	journal	journal	PROPN
iajs-1618	264	26	,	,	PUNCT
iajs-1618	264	27	2	2	NUM
iajs-1618	264	28	,	,	PUNCT
iajs-1618	264	29	1	1	NUM
iajs-1618	264	30	,	,	PUNCT
iajs-1618	264	31	(	(	PUNCT
iajs-1618	264	32	33	33	NUM
iajs-1618	264	33	-	-	SYM
iajs-1618	264	34	39	39	NUM
iajs-1618	264	35	)	)	PUNCT
iajs-1618	264	36	.	.	PUNCT
iajs-1618	265	1	10	10	NUM
iajs-1618	265	2	.	.	X
iajs-1618	265	3	ozcan	ozcan	PROPN
iajs-1618	265	4	,	,	PUNCT
iajs-1618	265	5	a.c	a.c	PROPN
iajs-1618	265	6	.	.	PROPN
iajs-1618	265	7	;	;	PUNCT
iajs-1618	266	1	harmanci	harmanci	PROPN
iajs-1618	266	2	,	,	PUNCT
iajs-1618	266	3	a.	a.	NOUN
iajs-1618	266	4	and	and	CCONJ
iajs-1618	266	5	smith	smith	PROPN
iajs-1618	266	6	,	,	PUNCT
iajs-1618	266	7	p.f	p.f	PROPN
iajs-1618	266	8	.	.	PROPN
iajs-1618	266	9	(	(	PUNCT
iajs-1618	266	10	2006	2006	NUM
iajs-1618	266	11	)	)	PUNCT
iajs-1618	266	12	,	,	PUNCT
iajs-1618	266	13	“	"	PUNCT
iajs-1618	266	14	duo	duo	NOUN
iajs-1618	266	15	modules	module	NOUN
iajs-1618	266	16	”	"	PUNCT
iajs-1618	266	17	,	,	PUNCT
iajs-1618	266	18	glasgow	glasgow	PROPN
iajs-1618	266	19	math.j.48	math.j.48	PROPN
iajs-1618	266	20	,	,	PUNCT
iajs-1618	266	21	(	(	PUNCT
iajs-1618	266	22	533	533	NUM
iajs-1618	266	23	-	-	SYM
iajs-1618	266	24	545	545	NUM
iajs-1618	266	25	)	)	PUNCT
iajs-1618	266	26	.	.	PUNCT
iajs-1618	267	1	http://www.ams.org/bull/	http://www.ams.org/bull/	PROPN
iajs-1618	267	2	mathematics	mathematics	PROPN
iajs-1618	267	3	|235	|235	VERB
iajs-1618	267	4	https://doi.org/10.30526/30.3.1618	https://doi.org/10.30526/30.3.1618	NOUN
iajs-1618	267	5	7102	7102	NUM
iajs-1618	267	6	(	(	PUNCT
iajs-1618	267	7	عام	عام	PROPN
iajs-1618	267	8	3	3	NUM
iajs-1618	267	9	(	(	PUNCT
iajs-1618	267	10	العدد	العدد	PROPN
iajs-1618	267	11	)	)	PUNCT
iajs-1618	267	12	30مجلة	30مجلة	PROPN
iajs-1618	268	1	إبن	إبن	VERB
iajs-1618	268	2	الهيثم	الهيثم	ADJ
iajs-1618	268	3	للعلوم	للعلوم	NOUN
iajs-1618	268	4	الصرفة	الصرفة	NOUN
iajs-1618	269	1	و	و	PRON
iajs-1618	269	2	التطبيقية	التطبيقية	ADV
iajs-1618	269	3	المجلد	المجلد	ADV
iajs-1618	269	4	)	)	PUNCT
iajs-1618	270	1	ibn	ibn	PROPN
iajs-1618	270	2	al	al	PROPN
iajs-1618	270	3	-	-	PUNCT
iajs-1618	270	4	haitham	haitham	PROPN
iajs-1618	270	5	j.	j.	PROPN
iajs-1618	270	6	for	for	ADP
iajs-1618	270	7	pure	pure	PROPN
iajs-1618	270	8	&	&	CCONJ
iajs-1618	270	9	appl	appl	PROPN
iajs-1618	270	10	.	.	PUNCT
iajs-1618	271	1	sci	sci	PROPN
iajs-1618	271	2	.	.	PUNCT
iajs-1618	272	1	vol.03	vol.03	PROPN
iajs-1618	272	2	(	(	PUNCT
iajs-1618	272	3	3	3	NUM
iajs-1618	272	4	)	)	PUNCT
iajs-1618	272	5	2017	2017	NUM
iajs-1618	272	6	11	11	NUM
iajs-1618	272	7	.	.	PUNCT
iajs-1618	273	1	bothaynah	bothaynah	PROPN
iajs-1618	273	2	najad	najad	PROPN
iajs-1618	273	3	shihab	shihab	PROPN
iajs-1618	273	4	,	,	PUNCT
iajs-1618	273	5	(	(	PUNCT
iajs-1618	273	6	2004	2004	NUM
iajs-1618	273	7	)	)	PUNCT
iajs-1618	273	8	,	,	PUNCT
iajs-1618	273	9	“	"	PUNCT
iajs-1618	273	10	scalar	scalar	ADJ
iajs-1618	273	11	reflexive	reflexive	ADJ
iajs-1618	273	12	modules	module	NOUN
iajs-1618	273	13	”	"	PUNCT
iajs-1618	273	14	,	,	PUNCT
iajs-1618	273	15	ph.d	ph.d	PROPN
iajs-1618	273	16	thesis	thesis	NOUN
iajs-1618	273	17	.	.	PUNCT
iajs-1618	274	1	university	university	NOUN
iajs-1618	274	2	of	of	ADP
iajs-1618	274	3	baghdad	baghdad	PROPN
iajs-1618	274	4	.	.	PUNCT
iajs-1618	275	1	12	12	NUM
iajs-1618	275	2	.	.	PUNCT
iajs-1618	276	1	eman	eman	PROPN
iajs-1618	276	2	abd	abd	PROPN
iajs-1618	276	3	al	al	PROPN
iajs-1618	276	4	-	-	PROPN
iajs-1618	276	5	amer	amer	PROPN
iajs-1618	276	6	mohammad	mohammad	PROPN
iajs-1618	276	7	ali	ali	PROPN
iajs-1618	276	8	,	,	PUNCT
iajs-1618	276	9	(	(	PUNCT
iajs-1618	276	10	2006	2006	NUM
iajs-1618	276	11	)	)	PUNCT
iajs-1618	276	12	,	,	PUNCT
iajs-1618	276	13	“	"	PUNCT
iajs-1618	276	14	on	on	ADP
iajs-1618	276	15	ikeda	ikeda	NOUN
iajs-1618	276	16	-	-	PUNCT
iajs-1618	276	17	nakayama	nakayama	PROPN
iajs-1618	276	18	modules	module	NOUN
iajs-1618	276	19	”	"	PUNCT
iajs-1618	276	20	,	,	PUNCT
iajs-1618	276	21	ph.d	ph.d	PROPN
iajs-1618	276	22	.	.	PUNCT
iajs-1618	277	1	thesis	thesis	NOUN
iajs-1618	277	2	,	,	PUNCT
iajs-1618	277	3	university	university	NOUN
iajs-1618	277	4	of	of	ADP
iajs-1618	277	5	baghdad	baghdad	PROPN
iajs-1618	277	6	.	.	PUNCT