id	sid	tid	token	lemma	pos
iajs-2518	1	1	microsoft	microsoft	PROPN
iajs-2518	1	2	word	word	NOUN
iajs-2518	1	3	137	137	NUM
iajs-2518	1	4	-	-	SYM
iajs-2518	1	5	147	147	NUM
iajs-2518	1	6	ibn	ibn	PROPN
iajs-2518	1	7	al	al	PROPN
iajs-2518	1	8	-	-	PUNCT
iajs-2518	1	9	haitham	haitham	PROPN
iajs-2518	1	10	jour	jour	X
iajs-2518	1	11	.	.	PROPN
iajs-2518	2	1	for	for	ADP
iajs-2518	2	2	pure	pure	ADJ
iajs-2518	2	3	&	&	CCONJ
iajs-2518	2	4	appl	appl	PROPN
iajs-2518	2	5	.	.	PUNCT
iajs-2518	3	1	sci	sci	PROPN
iajs-2518	3	2	.	.	PROPN
iajs-2518	4	1	33	33	NUM
iajs-2518	4	2	(	(	PUNCT
iajs-2518	4	3	4	4	NUM
iajs-2518	4	4	)	)	PUNCT
iajs-2518	4	5	2020	2020	NUM
iajs-2518	4	6	137	137	NUM
iajs-2518	4	7	          	          	SPACE
iajs-2518	4	8	fuzzy	fuzzy	ADJ
iajs-2518	4	9	semimaximal	semimaximal	ADJ
iajs-2518	4	10	submodules	submodule	NOUN
iajs-2518	4	11	abstract	abstract	ADV
iajs-2518	4	12	let	let	VERB
iajs-2518	4	13	r	r	PRON
iajs-2518	4	14	be	be	AUX
iajs-2518	4	15	a	a	DET
iajs-2518	4	16	commutative	commutative	ADJ
iajs-2518	4	17	ring	ring	NOUN
iajs-2518	4	18	with	with	ADP
iajs-2518	4	19	unity	unity	NOUN
iajs-2518	4	20	and	and	CCONJ
iajs-2518	4	21	an	an	DET
iajs-2518	4	22	r	r	NOUN
iajs-2518	4	23	-	-	PUNCT
iajs-2518	4	24	submodule	submodule	NOUN
iajs-2518	4	25	n	n	NOUN
iajs-2518	4	26	is	be	AUX
iajs-2518	4	27	called	call	VERB
iajs-2518	4	28	semimaximal	semimaximal	PROPN
iajs-2518	4	29	if	if	SCONJ
iajs-2518	5	1	and	and	CCONJ
iajs-2518	5	2	only	only	ADV
iajs-2518	5	3	if	if	SCONJ
iajs-2518	5	4	m	m	NOUN
iajs-2518	5	5	/	/	SYM
iajs-2518	5	6	n	n	PROPN
iajs-2518	5	7	is	be	AUX
iajs-2518	5	8	a	a	DET
iajs-2518	5	9	semisimple	semisimple	ADJ
iajs-2518	5	10	r	r	NOUN
iajs-2518	5	11	-	-	PUNCT
iajs-2518	5	12	module	module	NOUN
iajs-2518	5	13	.	.	PUNCT
iajs-2518	6	1	the	the	DET
iajs-2518	6	2	main	main	ADJ
iajs-2518	6	3	object	object	NOUN
iajs-2518	6	4	of	of	ADP
iajs-2518	6	5	this	this	DET
iajs-2518	6	6	work	work	NOUN
iajs-2518	6	7	is	be	AUX
iajs-2518	6	8	to	to	PART
iajs-2518	6	9	fuzzy	fuzzy	VERB
iajs-2518	6	10	this	this	DET
iajs-2518	6	11	concept	concept	NOUN
iajs-2518	6	12	,	,	PUNCT
iajs-2518	6	13	study	study	VERB
iajs-2518	6	14	the	the	DET
iajs-2518	6	15	basic	basic	ADJ
iajs-2518	6	16	properties	property	NOUN
iajs-2518	6	17	and	and	CCONJ
iajs-2518	6	18	we	we	PRON
iajs-2518	6	19	investigate	investigate	VERB
iajs-2518	6	20	the	the	DET
iajs-2518	6	21	sufficient	sufficient	ADJ
iajs-2518	6	22	conditions	condition	NOUN
iajs-2518	6	23	of	of	ADP
iajs-2518	6	24	fsubmodules	fsubmodule	NOUN
iajs-2518	6	25	to	to	PART
iajs-2518	6	26	be	be	AUX
iajs-2518	6	27	semimaximal	semimaximal	ADJ
iajs-2518	6	28	.	.	PUNCT
iajs-2518	7	1	also	also	ADV
iajs-2518	7	2	,	,	PUNCT
iajs-2518	7	3	the	the	DET
iajs-2518	7	4	concepts	concept	NOUN
iajs-2518	7	5	of	of	ADP
iajs-2518	7	6	(	(	PUNCT
iajs-2518	7	7	simple	simple	ADJ
iajs-2518	7	8	,	,	PUNCT
iajs-2518	7	9	semisimple	semisimple	NOUN
iajs-2518	7	10	)	)	PUNCT
iajs-2518	7	11	fsubmodules	fsubmodule	NOUN
iajs-2518	7	12	and	and	CCONJ
iajs-2518	7	13	quotient	quotient	NOUN
iajs-2518	7	14	fmodules	fmodule	NOUN
iajs-2518	7	15	are	be	AUX
iajs-2518	7	16	introduced	introduce	VERB
iajs-2518	7	17	and	and	CCONJ
iajs-2518	7	18	given	give	VERB
iajs-2518	7	19	some	some	DET
iajs-2518	7	20	properties	property	NOUN
iajs-2518	7	21	.	.	PUNCT
iajs-2518	8	1	keywords	keyword	NOUN
iajs-2518	8	2	:	:	PUNCT
iajs-2518	8	3	fuzzy	fuzzy	ADJ
iajs-2518	8	4	simple	simple	ADJ
iajs-2518	8	5	(	(	PUNCT
iajs-2518	8	6	semisimple	semisimple	NOUN
iajs-2518	8	7	)	)	PUNCT
iajs-2518	8	8	modules	module	NOUN
iajs-2518	8	9	,	,	PUNCT
iajs-2518	8	10	fuzzy	fuzzy	ADJ
iajs-2518	8	11	quotient	quotient	NOUN
iajs-2518	8	12	modules	module	NOUN
iajs-2518	8	13	,	,	PUNCT
iajs-2518	8	14	fuzzy	fuzzy	ADJ
iajs-2518	8	15	semimaximal	semimaximal	NOUN
iajs-2518	8	16	submodule	submodule	NOUN
iajs-2518	8	17	.	.	PUNCT
iajs-2518	9	1	1.introduction	1.introduction	NUM
iajs-2518	9	2	hatem	hatem	NOUN
iajs-2518	9	3	in	in	ADP
iajs-2518	9	4	[	[	X
iajs-2518	9	5	1	1	NUM
iajs-2518	9	6	]	]	PUNCT
iajs-2518	9	7	introduced	introduce	VERB
iajs-2518	9	8	and	and	CCONJ
iajs-2518	9	9	studies	study	NOUN
iajs-2518	9	10	semimaximal	semimaximal	ADJ
iajs-2518	9	11	ideals	ideal	NOUN
iajs-2518	9	12	of	of	ADP
iajs-2518	9	13	a	a	DET
iajs-2518	9	14	ring	ring	NOUN
iajs-2518	9	15	and	and	CCONJ
iajs-2518	9	16	semimaximal	semimaximal	ADJ
iajs-2518	9	17	submodules	submodule	NOUN
iajs-2518	9	18	,	,	PUNCT
iajs-2518	9	19	where	where	SCONJ
iajs-2518	9	20	an	an	DET
iajs-2518	9	21	ideal	ideal	ADJ
iajs-2518	9	22	j	j	PROPN
iajs-2518	9	23	of	of	ADP
iajs-2518	9	24	a	a	DET
iajs-2518	9	25	ring	ring	NOUN
iajs-2518	9	26	r	r	NOUN
iajs-2518	9	27	is	be	AUX
iajs-2518	9	28	called	call	VERB
iajs-2518	9	29	semimaximal	semimaximal	PROPN
iajs-2518	9	30	if	if	SCONJ
iajs-2518	9	31	j	j	PROPN
iajs-2518	9	32	is	be	AUX
iajs-2518	9	33	a	a	DET
iajs-2518	9	34	finite	finite	ADJ
iajs-2518	9	35	intersection	intersection	NOUN
iajs-2518	9	36	of	of	ADP
iajs-2518	9	37	maximal	maximal	ADJ
iajs-2518	9	38	ideals	ideal	NOUN
iajs-2518	9	39	[	[	X
iajs-2518	9	40	2	2	NUM
iajs-2518	9	41	]	]	PUNCT
iajs-2518	9	42	.	.	PUNCT
iajs-2518	10	1	we	we	PRON
iajs-2518	10	2	add	add	VERB
iajs-2518	10	3	many	many	ADJ
iajs-2518	10	4	other	other	ADJ
iajs-2518	10	5	results	result	NOUN
iajs-2518	10	6	.	.	PUNCT
iajs-2518	11	1	maysoun	maysoun	NOUN
iajs-2518	11	2	in	in	ADP
iajs-2518	11	3	[	[	X
iajs-2518	11	4	3	3	NUM
iajs-2518	11	5	]	]	PUNCT
iajs-2518	11	6	introduced	introduce	VERB
iajs-2518	11	7	the	the	DET
iajs-2518	11	8	definition	definition	NOUN
iajs-2518	11	9	of	of	ADP
iajs-2518	11	10	fuzzy	fuzzy	ADJ
iajs-2518	11	11	simple	simple	ADJ
iajs-2518	11	12	modules	module	NOUN
iajs-2518	11	13	and	and	CCONJ
iajs-2518	11	14	fuzzy	fuzzy	ADJ
iajs-2518	11	15	semisimple	semisimple	NOUN
iajs-2518	11	16	modules	module	NOUN
iajs-2518	11	17	.	.	PUNCT
iajs-2518	12	1	some	some	DET
iajs-2518	12	2	properties	property	NOUN
iajs-2518	12	3	of	of	ADP
iajs-2518	12	4	these	these	DET
iajs-2518	12	5	concepts	concept	NOUN
iajs-2518	12	6	which	which	PRON
iajs-2518	12	7	are	be	AUX
iajs-2518	12	8	useful	useful	ADJ
iajs-2518	12	9	in	in	ADP
iajs-2518	12	10	next	next	ADJ
iajs-2518	12	11	sections	section	NOUN
iajs-2518	12	12	are	be	AUX
iajs-2518	12	13	given	give	VERB
iajs-2518	12	14	.	.	PUNCT
iajs-2518	13	1	moreover	moreover	ADV
iajs-2518	13	2	,	,	PUNCT
iajs-2518	13	3	a	a	DET
iajs-2518	13	4	submodule	submodule	NOUN
iajs-2518	13	5	n	n	PROPN
iajs-2518	13	6	of	of	ADP
iajs-2518	13	7	an	an	DET
iajs-2518	13	8	ℛ	ℛ	NOUN
iajs-2518	13	9	module	module	NOUN
iajs-2518	13	10	ℳ	ℳ	NOUN
iajs-2518	13	11	is	be	AUX
iajs-2518	13	12	said	say	VERB
iajs-2518	13	13	semimaximal	semimaximal	ADJ
iajs-2518	13	14	if	if	SCONJ
iajs-2518	13	15	ℳ	ℳ	PROPN
iajs-2518	13	16	𝑁	𝑁	PROPN
iajs-2518	13	17	⁄	⁄	PROPN
iajs-2518	13	18	is	be	AUX
iajs-2518	13	19	semismiple	semismiple	NOUN
iajs-2518	13	20	rmodule	rmodule	NOUN
iajs-2518	13	21	.	.	PUNCT
iajs-2518	14	1	it	it	PRON
iajs-2518	14	2	is	be	AUX
iajs-2518	14	3	clear	clear	ADJ
iajs-2518	14	4	that	that	SCONJ
iajs-2518	14	5	every	every	DET
iajs-2518	14	6	maximal	maximal	ADJ
iajs-2518	14	7	ideal	ideal	NOUN
iajs-2518	14	8	(	(	PUNCT
iajs-2518	14	9	submodule	submodule	NOUN
iajs-2518	14	10	)	)	PUNCT
iajs-2518	14	11	is	be	AUX
iajs-2518	14	12	a	a	DET
iajs-2518	14	13	semimaximal	semimaximal	NOUN
iajs-2518	14	14	.	.	PUNCT
iajs-2518	15	1	in	in	ADP
iajs-2518	15	2	[	[	X
iajs-2518	15	3	4	4	NUM
iajs-2518	15	4	]	]	PUNCT
iajs-2518	15	5	,	,	PUNCT
iajs-2518	15	6	we	we	PRON
iajs-2518	15	7	fuzzify	fuzzify	VERB
iajs-2518	15	8	the	the	DET
iajs-2518	15	9	concept	concept	NOUN
iajs-2518	15	10	semimaximal	semimaximal	NOUN
iajs-2518	15	11	ideal	ideal	NOUN
iajs-2518	15	12	where	where	SCONJ
iajs-2518	15	13	a	a	DET
iajs-2518	15	14	fuzzy	fuzzy	ADJ
iajs-2518	15	15	ideal	ideal	NOUN
iajs-2518	15	16	ℋ	ℋ	PROPN
iajs-2518	15	17	is	be	AUX
iajs-2518	15	18	called	call	VERB
iajs-2518	15	19	a	a	DET
iajs-2518	15	20	fuzzy	fuzzy	ADJ
iajs-2518	15	21	semimaximal	semimaximal	NOUN
iajs-2518	15	22	ideal	ideal	NOUN
iajs-2518	15	23	if	if	SCONJ
iajs-2518	15	24	𝐻	𝐻	PROPN
iajs-2518	15	25	𝑖𝑠	𝑖𝑠	AUX
iajs-2518	15	26	s	s	VERB
iajs-2518	15	27	a	a	DET
iajs-2518	15	28	finite	finite	ADJ
iajs-2518	15	29	intersection	intersection	NOUN
iajs-2518	15	30	of	of	ADP
iajs-2518	15	31	fuzzy	fuzzy	ADJ
iajs-2518	15	32	maximal	maximal	ADJ
iajs-2518	15	33	ideals	ideal	NOUN
iajs-2518	15	34	also	also	ADV
iajs-2518	15	35	we	we	PRON
iajs-2518	15	36	study	study	VERB
iajs-2518	15	37	many	many	ADJ
iajs-2518	15	38	properties	property	NOUN
iajs-2518	15	39	this	this	DET
iajs-2518	15	40	concept	concept	NOUN
iajs-2518	15	41	.	.	PUNCT
iajs-2518	16	1	in	in	ADP
iajs-2518	16	2	this	this	DET
iajs-2518	16	3	paper	paper	NOUN
iajs-2518	16	4	,	,	PUNCT
iajs-2518	16	5	we	we	PRON
iajs-2518	16	6	fuzzify	fuzzify	VERB
iajs-2518	16	7	the	the	DET
iajs-2518	16	8	concept	concept	NOUN
iajs-2518	16	9	semimaximal	semimaximal	NOUN
iajs-2518	16	10	submodules	submodule	NOUN
iajs-2518	16	11	in	in	ADP
iajs-2518	16	12	to	to	ADP
iajs-2518	16	13	fuzzy	fuzzy	ADJ
iajs-2518	16	14	semimaximal	semimaximal	ADJ
iajs-2518	16	15	submodules	submodule	NOUN
iajs-2518	16	16	.	.	PUNCT
iajs-2518	17	1	also	also	ADV
iajs-2518	17	2	,	,	PUNCT
iajs-2518	17	3	we	we	PRON
iajs-2518	17	4	give	give	VERB
iajs-2518	17	5	many	many	ADJ
iajs-2518	17	6	basic	basic	ADJ
iajs-2518	17	7	properties	property	NOUN
iajs-2518	17	8	of	of	ADP
iajs-2518	17	9	this	this	DET
iajs-2518	17	10	notion	notion	NOUN
iajs-2518	17	11	.	.	PUNCT
iajs-2518	18	1	finally	finally	ADV
iajs-2518	18	2	,	,	PUNCT
iajs-2518	18	3	(	(	PUNCT
iajs-2518	18	4	shortly	shortly	ADV
iajs-2518	18	5	fuzzy	fuzzy	ADJ
iajs-2518	18	6	set	set	NOUN
iajs-2518	18	7	,	,	PUNCT
iajs-2518	18	8	fuzzy	fuzzy	ADJ
iajs-2518	18	9	submodule	submodule	NOUN
iajs-2518	18	10	,	,	PUNCT
iajs-2518	18	11	fuzzy	fuzzy	ADJ
iajs-2518	18	12	ideal	ideal	ADJ
iajs-2518	18	13	and	and	CCONJ
iajs-2518	18	14	fuzzy	fuzzy	ADJ
iajs-2518	18	15	module	module	NOUN
iajs-2518	18	16	is	be	AUX
iajs-2518	18	17	f	f	NOUN
iajs-2518	18	18	-	-	PUNCT
iajs-2518	18	19	set	set	VERB
iajs-2518	18	20	,	,	PUNCT
iajs-2518	18	21	fsubmodule	fsubmodule	NOUN
iajs-2518	18	22	,	,	PUNCT
iajs-2518	18	23	f	f	NOUN
iajs-2518	18	24	-	-	PUNCT
iajs-2518	18	25	ideal	ideal	NOUN
iajs-2518	18	26	,	,	PUNCT
iajs-2518	18	27	and	and	CCONJ
iajs-2518	18	28	f	f	X
iajs-2518	18	29	-	-	PUNCT
iajs-2518	18	30	module	module	NOUN
iajs-2518	18	31	)	)	PUNCT
iajs-2518	18	32	.	.	PUNCT
iajs-2518	19	1	ibn	ibn	PROPN
iajs-2518	19	2	al	al	PROPN
iajs-2518	19	3	haitham	haitham	PROPN
iajs-2518	19	4	journal	journal	PROPN
iajs-2518	19	5	for	for	ADP
iajs-2518	19	6	pure	pure	ADJ
iajs-2518	19	7	and	and	CCONJ
iajs-2518	19	8	applied	apply	VERB
iajs-2518	19	9	science	science	NOUN
iajs-2518	19	10	journal	journal	PROPN
iajs-2518	19	11	homepage	homepage	NOUN
iajs-2518	19	12	:	:	PUNCT
iajs-2518	19	13	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	PROPN
iajs-2518	19	14	maysoun	maysoun	PROPN
iajs-2518	19	15	a.	a.	PROPN
iajs-2518	19	16	hamel	hamel	PROPN
iajs-2518	19	17	department	department	PROPN
iajs-2518	19	18	of	of	ADP
iajs-2518	19	19	mathematics	mathematics	PROPN
iajs-2518	19	20	,	,	PUNCT
iajs-2518	19	21	ibn	ibn	NOUN
iajs-2518	19	22	-	-	PUNCT
iajs-2518	19	23	alhaitham	alhaitham	NOUN
iajs-2518	19	24	college	college	NOUN
iajs-2518	19	25	of	of	ADP
iajs-2518	19	26	education	education	NOUN
iajs-2518	19	27	,	,	PUNCT
iajs-2518	19	28	baghdad	baghdad	PROPN
iajs-2518	19	29	university	university	PROPN
iajs-2518	19	30	,	,	PUNCT
iajs-2518	19	31	baghdad	baghdad	PROPN
iajs-2518	19	32	-	-	PUNCT
iajs-2518	19	33	iraq	iraq	PROPN
iajs-2518	20	1	mathmaysoon@gmail.com	mathmaysoon@gmail.com	PROPN
iajs-2518	20	2	  	  	SPACE
iajs-2518	20	3	hatam	hatam	PROPN
iajs-2518	20	4	y.	y.	PROPN
iajs-2518	20	5	khalaf	khalaf	PROPN
iajs-2518	20	6	department	department	PROPN
iajs-2518	20	7	of	of	ADP
iajs-2518	20	8	mathematics	mathematics	PROPN
iajs-2518	20	9	,	,	PUNCT
iajs-2518	20	10	ibn	ibn	NOUN
iajs-2518	20	11	-	-	PUNCT
iajs-2518	20	12	alhaitham	alhaitham	NOUN
iajs-2518	20	13	college	college	NOUN
iajs-2518	20	14	of	of	ADP
iajs-2518	20	15	education	education	NOUN
iajs-2518	20	16	,	,	PUNCT
iajs-2518	20	17	baghdad	baghdad	PROPN
iajs-2518	20	18	university	university	PROPN
iajs-2518	20	19	,	,	PUNCT
iajs-2518	20	20	baghdad	baghdad	PROPN
iajs-2518	20	21	-	-	PUNCT
iajs-2518	20	22	iraq	iraq	PROPN
iajs-2518	20	23	dr.hatamyahya@yahoo.com	dr.hatamyahya@yahoo.com	PROPN
iajs-2518	20	24	doi	doi	PROPN
iajs-2518	20	25	:	:	PUNCT
iajs-2518	20	26	10.30526/33.4.2518	10.30526/33.4.2518	NUM
iajs-2518	20	27	article	article	NOUN
iajs-2518	20	28	history	history	NOUN
iajs-2518	20	29	:	:	PUNCT
iajs-2518	20	30	received	receive	VERB
iajs-2518	20	31	5	5	NUM
iajs-2518	20	32	december	december	PROPN
iajs-2518	20	33	2019	2019	NUM
iajs-2518	20	34	,	,	PUNCT
iajs-2518	20	35	accepted	accept	VERB
iajs-2518	20	36	12	12	NUM
iajs-2518	20	37	january	january	NOUN
iajs-2518	20	38	2020	2020	NUM
iajs-2518	20	39	,	,	PUNCT
iajs-2518	20	40	published	publish	VERB
iajs-2518	20	41	in	in	ADP
iajs-2518	20	42	october	october	PROPN
iajs-2518	20	43	2020	2020	NUM
iajs-2518	20	44	  	  	SPACE
iajs-2518	20	45	138	138	NUM
iajs-2518	20	46	  	  	SPACE
iajs-2518	20	47	ibn	ibn	PROPN
iajs-2518	20	48	al	al	PROPN
iajs-2518	20	49	-	-	PUNCT
iajs-2518	20	50	haitham	haitham	PROPN
iajs-2518	20	51	jour	jour	X
iajs-2518	20	52	.	.	PROPN
iajs-2518	20	53	for	for	ADP
iajs-2518	20	54	pure	pure	ADJ
iajs-2518	20	55	&	&	CCONJ
iajs-2518	20	56	appl	appl	PROPN
iajs-2518	20	57	.	.	PUNCT
iajs-2518	21	1	sci	sci	PROPN
iajs-2518	21	2	.	.	PROPN
iajs-2518	22	1	33	33	NUM
iajs-2518	22	2	(	(	PUNCT
iajs-2518	22	3	4	4	NUM
iajs-2518	22	4	)	)	PUNCT
iajs-2518	22	5	2020	2020	NUM
iajs-2518	22	6	1	1	NUM
iajs-2518	22	7	.	.	PUNCT
iajs-2518	22	8	preliminaries	preliminary	NOUN
iajs-2518	22	9	this	this	DET
iajs-2518	22	10	section	section	NOUN
iajs-2518	22	11	contains	contain	VERB
iajs-2518	22	12	some	some	DET
iajs-2518	22	13	definitions	definition	NOUN
iajs-2518	22	14	and	and	CCONJ
iajs-2518	22	15	properties	property	NOUN
iajs-2518	22	16	of	of	ADP
iajs-2518	22	17	fuzzy	fuzzy	ADJ
iajs-2518	22	18	set	set	VERB
iajs-2518	22	19	and	and	CCONJ
iajs-2518	22	20	fuzzy	fuzzy	ADJ
iajs-2518	22	21	module	module	NOUN
iajs-2518	22	22	.	.	PUNCT
iajs-2518	23	1	definition	definition	NOUN
iajs-2518	23	2	1.1	1.1	NUM
iajs-2518	24	1	[	[	X
iajs-2518	24	2	5	5	NUM
iajs-2518	24	3	]	]	PUNCT
iajs-2518	24	4	let	let	VERB
iajs-2518	24	5	s	s	PRON
iajs-2518	24	6	be	be	AUX
iajs-2518	24	7	a	a	DET
iajs-2518	24	8	non	non	ADJ
iajs-2518	24	9	-	-	ADJ
iajs-2518	24	10	empty	empty	ADJ
iajs-2518	24	11	set	set	NOUN
iajs-2518	25	1	and	and	CCONJ
iajs-2518	25	2	i	i	PRON
iajs-2518	25	3	be	be	VERB
iajs-2518	25	4	the	the	DET
iajs-2518	25	5	closed	closed	ADJ
iajs-2518	25	6	interval	interval	NOUN
iajs-2518	25	7	0,1	0,1	NUM
iajs-2518	25	8	of	of	ADP
iajs-2518	25	9	the	the	DET
iajs-2518	25	10	real	real	ADJ
iajs-2518	25	11	line	line	NOUN
iajs-2518	25	12	(	(	PUNCT
iajs-2518	25	13	real	real	ADJ
iajs-2518	25	14	numbers	number	NOUN
iajs-2518	25	15	)	)	PUNCT
iajs-2518	25	16	.	.	PUNCT
iajs-2518	26	1	a	a	DET
iajs-2518	26	2	fset	fset	NOUN
iajs-2518	26	3	a	a	PRON
iajs-2518	26	4	in	in	ADP
iajs-2518	26	5	s	s	PROPN
iajs-2518	26	6	(	(	PUNCT
iajs-2518	26	7	a	a	DET
iajs-2518	26	8	fsubset	fsubset	NOUN
iajs-2518	26	9	of	of	ADP
iajs-2518	26	10	s	s	NOUN
iajs-2518	26	11	)	)	PUNCT
iajs-2518	26	12	is	be	AUX
iajs-2518	26	13	a	a	DET
iajs-2518	26	14	function	function	NOUN
iajs-2518	26	15	from	from	ADP
iajs-2518	26	16	s	s	PRON
iajs-2518	26	17	in	in	ADP
iajs-2518	26	18	to	to	ADP
iajs-2518	26	19	i	i	PRON
iajs-2518	26	20	"	"	PUNCT
iajs-2518	26	21	.	.	PUNCT
iajs-2518	27	1	definition	definition	NOUN
iajs-2518	27	2	1.2[6	1.2[6	NUM
iajs-2518	27	3	]	]	X
iajs-2518	27	4	let	let	VERB
iajs-2518	27	5	𝑥	𝑥	PRON
iajs-2518	27	6	:	:	PUNCT
iajs-2518	27	7	𝑆	𝑆	PROPN
iajs-2518	27	8	→	→	SYM
iajs-2518	27	9	0,1	0,1	NUM
iajs-2518	27	10	be	be	VERB
iajs-2518	27	11	are	be	AUX
iajs-2518	27	12	two	two	NUM
iajs-2518	27	13	fset	fset	VERB
iajs-2518	27	14	in	in	ADP
iajs-2518	27	15	s	s	PROPN
iajs-2518	27	16	,	,	PUNCT
iajs-2518	27	17	where	where	SCONJ
iajs-2518	27	18	𝑥	𝑥	DET
iajs-2518	27	19	∈	∈	PROPN
iajs-2518	27	20	𝑆	𝑆	PROPN
iajs-2518	27	21	,	,	PUNCT
iajs-2518	27	22	𝑡	𝑡	PROPN
iajs-2518	27	23	∈	∈	NOUN
iajs-2518	27	24	0,1	0,1	NUM
iajs-2518	27	25	,	,	PUNCT
iajs-2518	27	26	defined	define	VERB
iajs-2518	27	27	by	by	ADP
iajs-2518	27	28	:	:	PUNCT
iajs-2518	27	29	𝑥	𝑥	PROPN
iajs-2518	27	30	𝑦	𝑦	NUM
iajs-2518	27	31	𝑡	𝑡	NOUN
iajs-2518	27	32	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	27	33	𝑥	𝑥	PRON
iajs-2518	27	34	𝑦	𝑦	NOUN
iajs-2518	27	35	0	0	NUM
iajs-2518	27	36	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	27	37	𝑥	𝑥	PROPN
iajs-2518	27	38	𝑦	𝑦	NOUN
iajs-2518	27	39	for	for	ADP
iajs-2518	27	40	all	all	DET
iajs-2518	27	41	𝑦	𝑦	DET
iajs-2518	27	42	∈	∈	NOUN
iajs-2518	27	43	𝑆.	𝑆.	NOUN
iajs-2518	27	44	,	,	PUNCT
iajs-2518	27	45	then	then	ADV
iajs-2518	27	46	𝑥	𝑥	PROPN
iajs-2518	27	47	is	be	AUX
iajs-2518	27	48	called	call	VERB
iajs-2518	27	49	fsingleton	fsingleton	NOUN
iajs-2518	27	50	.	.	PUNCT
iajs-2518	28	1	definition	definition	NOUN
iajs-2518	28	2	1.3[6	1.3[6	NUM
iajs-2518	28	3	]	]	X
iajs-2518	28	4	if	if	SCONJ
iajs-2518	28	5	𝐴	𝐴	PROPN
iajs-2518	28	6	an𝑑	an𝑑	PROPN
iajs-2518	28	7	𝐴	𝐴	PROPN
iajs-2518	28	8	f	f	PROPN
iajs-2518	28	9	sets	set	VERB
iajs-2518	28	10	in	in	ADP
iajs-2518	28	11	s	s	PRON
iajs-2518	28	12	,	,	PUNCT
iajs-2518	28	13	then	then	ADV
iajs-2518	28	14	:	:	PUNCT
iajs-2518	28	15	1	1	NUM
iajs-2518	28	16	𝐴	𝐴	PROPN
iajs-2518	28	17	𝐴	𝐴	PROPN
iajs-2518	28	18	if	if	SCONJ
iajs-2518	28	19	and	and	CCONJ
iajs-2518	28	20	only	only	ADV
iajs-2518	28	21	if	if	SCONJ
iajs-2518	28	22	𝐴	𝐴	PROPN
iajs-2518	28	23	𝑥	𝑥	PROPN
iajs-2518	28	24	𝐴	𝐴	PROPN
iajs-2518	28	25	𝑥	𝑥	PROPN
iajs-2518	28	26	,	,	PUNCT
iajs-2518	28	27	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2518	28	28	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2518	28	29	𝑥	𝑥	DET
iajs-2518	28	30	∈	∈	PROPN
iajs-2518	28	31	𝑆	𝑆	PROPN
iajs-2518	28	32	.	.	PUNCT
iajs-2518	29	1	2𝐴	2𝐴	PROPN
iajs-2518	29	2	𝐴	𝐴	PROPN
iajs-2518	29	3	𝑖𝑓	𝑖𝑓	VERB
iajs-2518	29	4	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	29	5	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
iajs-2518	29	6	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	29	7	𝐴	𝐴	PROPN
iajs-2518	29	8	𝑥	𝑥	PROPN
iajs-2518	29	9	𝐴	𝐴	PROPN
iajs-2518	29	10	𝑥	𝑥	PROPN
iajs-2518	29	11	,	,	PUNCT
iajs-2518	29	12	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2518	29	13	𝑎𝑙𝑙𝑥	𝑎𝑙𝑙𝑥	ADP
iajs-2518	29	14	∈	∈	PROPN
iajs-2518	29	15	𝑆	𝑆	PROPN
iajs-2518	29	16	..	..	PUNCT
iajs-2518	30	1	𝐼𝑓	𝐼𝑓	PROPN
iajs-2518	30	2	𝐴	𝐴	NOUN
iajs-2518	30	3	𝐴	𝐴	PROPN
iajs-2518	30	4	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	30	5	𝑡ℎ𝑒𝑟𝑒	𝑡ℎ𝑒𝑟𝑒	VERB
iajs-2518	30	6	𝑒𝑥𝑖𝑠𝑡𝑠	𝑒𝑥𝑖𝑠𝑡𝑠	NOUN
iajs-2518	30	7	𝑥	𝑥	PRON
iajs-2518	30	8	∈	∈	PROPN
iajs-2518	30	9	𝑆	𝑆	PROPN
iajs-2518	30	10	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2518	30	11	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
iajs-2518	30	12	𝐴	𝐴	PROPN
iajs-2518	30	13	𝑥	𝑥	PROPN
iajs-2518	30	14	𝐴	𝐴	PROPN
iajs-2518	30	15	𝑥	𝑥	PROPN
iajs-2518	30	16	,	,	PUNCT
iajs-2518	30	17	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
iajs-2518	30	18	𝐴	𝐴	PROPN
iajs-2518	30	19	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	31	1	𝑎	𝑎	PRON
iajs-2518	31	2	𝑝𝑟𝑜𝑝𝑒𝑟	𝑝𝑟𝑜𝑝𝑒𝑟	NOUN
iajs-2518	31	3	𝐹	𝐹	PROPN
iajs-2518	31	4	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
iajs-2518	31	5	𝑜𝑓	𝑜𝑓	ADP
iajs-2518	31	6	𝐵	𝐵	PROPN
iajs-2518	31	7	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	31	8	𝑤𝑟𝑖𝑡𝑡𝑒𝑛	𝑤𝑟𝑖𝑡𝑡𝑒𝑛	PROPN
iajs-2518	31	9	𝐴	𝐴	PROPN
iajs-2518	31	10	𝐵	𝐵	PROPN
iajs-2518	31	11	.	.	PUNCT
iajs-2518	32	1	by	by	ADP
iajs-2518	32	2	part	part	NOUN
iajs-2518	32	3	(	(	PUNCT
iajs-2518	32	4	2	2	NUM
iajs-2518	32	5	)	)	PUNCT
iajs-2518	32	6	,	,	PUNCT
iajs-2518	32	7	we	we	PRON
iajs-2518	32	8	can	can	AUX
iajs-2518	32	9	deduce	deduce	VERB
iajs-2518	32	10	that	that	SCONJ
iajs-2518	32	11	𝑥	𝑥	PROPN
iajs-2518	32	12	𝐴	𝐴	PROPN
iajs-2518	32	13	𝑖𝑓	𝑖𝑓	VERB
iajs-2518	32	14	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	32	15	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
iajs-2518	32	16	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	32	17	𝐴	𝐴	PROPN
iajs-2518	32	18	𝑥	𝑥	PART
iajs-2518	32	19	𝑡	𝑡	NOUN
iajs-2518	32	20	"	"	PUNCT
iajs-2518	32	21	.	.	PUNCT
iajs-2518	33	1	definition	definition	NOUN
iajs-2518	33	2	1.4	1.4	NUM
iajs-2518	34	1	[	[	X
iajs-2518	34	2	6],[7	6],[7	NUM
iajs-2518	34	3	]	]	X
iajs-2518	34	4	let	let	VERB
iajs-2518	34	5	ℳ	ℳ	PRON
iajs-2518	34	6	be	be	AUX
iajs-2518	34	7	an	an	DET
iajs-2518	34	8	r	r	NOUN
iajs-2518	34	9	-	-	PUNCT
iajs-2518	34	10	modul𝑒	modul𝑒	ADJ
iajs-2518	34	11	.	.	PUNCT
iajs-2518	35	1	a	a	DET
iajs-2518	35	2	fset	fset	VERB
iajs-2518	35	3	ℋ	ℋ	PROPN
iajs-2518	35	4	of	of	ADP
iajs-2518	35	5	ℳ	ℳ	PROPN
iajs-2518	35	6	is	be	AUX
iajs-2518	35	7	fmodule	fmodule	ADJ
iajs-2518	35	8	of	of	ADP
iajs-2518	35	9	an	an	DET
iajs-2518	35	10	r	r	NOUN
iajs-2518	35	11	-	-	PUNCT
iajs-2518	35	12	module	module	NOUN
iajs-2518	35	13	m	m	NOUN
iajs-2518	35	14	if	if	SCONJ
iajs-2518	35	15	:	:	PUNCT
iajs-2518	35	16	1ℋ	1ℋ	INTJ
iajs-2518	35	17	𝑥	𝑥	PUNCT
iajs-2518	35	18	𝑦	𝑦	NUM
iajs-2518	35	19	min	min	NOUN
iajs-2518	35	20	ℋ	ℋ	NOUN
iajs-2518	35	21	𝑥	𝑥	PROPN
iajs-2518	35	22	,	,	PUNCT
iajs-2518	35	23	ℋ	ℋ	PROPN
iajs-2518	35	24	𝑦	𝑦	NOUN
iajs-2518	35	25	,	,	PUNCT
iajs-2518	35	26	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2518	35	27	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2518	35	28	𝑥	𝑥	PROPN
iajs-2518	35	29	,	,	PUNCT
iajs-2518	35	30	𝑦	𝑦	NOUN
iajs-2518	35	31	∈	∈	PROPN
iajs-2518	35	32	ℳ	ℳ	NOUN
iajs-2518	35	33	2-ℋ	2-ℋ	NUM
iajs-2518	35	34	𝑟𝑥	𝑟𝑥	ADP
iajs-2518	36	1	ℋ	ℋ	PROPN
iajs-2518	36	2	𝑥	𝑥	NOUN
iajs-2518	36	3	𝑓𝑜𝑟𝑎𝑙𝑙	𝑓𝑜𝑟𝑎𝑙𝑙	ADV
iajs-2518	36	4	𝑥	𝑥	X
iajs-2518	36	5	∈	∈	PROPN
iajs-2518	36	6	ℳ	ℳ	PROPN
iajs-2518	36	7	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	36	8	𝑟	𝑟	X
iajs-2518	36	9	∈	∈	NOUN
iajs-2518	36	10	𝑅.	𝑅.	SYM
iajs-2518	36	11	3-ℋ	3-ℋ	NUM
iajs-2518	36	12	0	0	SYM
iajs-2518	37	1	1	1	NUM
iajs-2518	37	2	"	"	PUNCT
iajs-2518	37	3	definition	definition	NOUN
iajs-2518	37	4	1.5[6	1.5[6	NUM
iajs-2518	37	5	]	]	PUNCT
iajs-2518	37	6	let	let	VERB
iajs-2518	37	7	𝑋	𝑋	PROPN
iajs-2518	37	8	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	37	9	𝑋	𝑋	NOUN
iajs-2518	37	10	,	,	PUNCT
iajs-2518	37	11	be	be	AUX
iajs-2518	37	12	two	two	NUM
iajs-2518	37	13	fmodules	fmodule	NOUN
iajs-2518	37	14	of	of	ADP
iajs-2518	37	15	r	r	NOUN
iajs-2518	37	16	-	-	PUNCT
iajs-2518	37	17	module	module	NOUN
iajs-2518	37	18	ℳ	ℳ	NOUN
iajs-2518	37	19	.	.	PUNCT
iajs-2518	38	1	𝑋	𝑋	PROPN
iajs-2518	38	2	is	be	AUX
iajs-2518	38	3	said	say	VERB
iajs-2518	38	4	a	a	DET
iajs-2518	38	5	fsubmodule	fsubmodule	NOUN
iajs-2518	38	6	of	of	ADP
iajs-2518	38	7	𝑋	𝑋	NOUN
iajs-2518	38	8	,	,	PUNCT
iajs-2518	38	9	if	if	SCONJ
iajs-2518	38	10	𝑋	𝑋	PROPN
iajs-2518	38	11	𝑋	𝑋	PROPN
iajs-2518	38	12	.	.	PUNCT
iajs-2518	39	1	proposition	proposition	NOUN
iajs-2518	39	2	1.6	1.6	NUM
iajs-2518	40	1	[	[	X
iajs-2518	40	2	6	6	NUM
iajs-2518	40	3	]	]	PUNCT
iajs-2518	40	4	let	let	VERB
iajs-2518	40	5	a	a	PRON
iajs-2518	40	6	be	be	AUX
iajs-2518	40	7	a	a	DET
iajs-2518	40	8	fset	fset	NOUN
iajs-2518	40	9	of	of	ADP
iajs-2518	40	10	an	an	DET
iajs-2518	40	11	r	r	NOUN
iajs-2518	40	12	-	-	PUNCT
iajs-2518	40	13	module	module	NOUN
iajs-2518	40	14	ℳ	ℳ	NOUN
iajs-2518	40	15	.	.	PUNCT
iajs-2518	41	1	then	then	ADV
iajs-2518	41	2	the	the	DET
iajs-2518	41	3	level	level	NOUN
iajs-2518	41	4	subset	subset	VERB
iajs-2518	41	5	a	a	DET
iajs-2518	41	6	x	x	X
iajs-2518	41	7	∈	∈	PROPN
iajs-2518	41	8	m	m	PROPN
iajs-2518	41	9	,	,	PUNCT
iajs-2518	41	10	a	a	DET
iajs-2518	41	11	x	x	X
iajs-2518	41	12	t	t	NOUN
iajs-2518	41	13	,	,	PUNCT
iajs-2518	41	14	t	t	PROPN
iajs-2518	41	15	∈	∈	PROPN
iajs-2518	41	16	0,1	0,1	NUM
iajs-2518	41	17	is	be	AUX
iajs-2518	41	18	a	a	DET
iajs-2518	41	19	submodule	submodule	NOUN
iajs-2518	41	20	of	of	ADP
iajs-2518	41	21	ℳ	ℳ	PROPN
iajs-2518	41	22	if	if	SCONJ
iajs-2518	42	1	and	and	CCONJ
iajs-2518	42	2	only	only	ADV
iajs-2518	42	3	if	if	SCONJ
iajs-2518	42	4	a	a	PRON
iajs-2518	42	5	is	be	AUX
iajs-2518	42	6	a	a	DET
iajs-2518	42	7	fsubmodule	fsubmodule	NOUN
iajs-2518	42	8	of	of	ADP
iajs-2518	42	9	ℋ	ℋ	PROPN
iajs-2518	42	10	where	where	SCONJ
iajs-2518	42	11	ℋ	ℋ	PROPN
iajs-2518	42	12	is	be	AUX
iajs-2518	42	13	a	a	DET
iajs-2518	42	14	f	f	NOUN
iajs-2518	42	15	module	module	NOUN
iajs-2518	42	16	of	of	ADP
iajs-2518	42	17	an	an	DET
iajs-2518	42	18	r	r	NOUN
iajs-2518	42	19	-	-	PUNCT
iajs-2518	42	20	module	module	NOUN
iajs-2518	42	21	ℳ	ℳ	PROPN
iajs-2518	42	22	now	now	ADV
iajs-2518	42	23	,	,	PUNCT
iajs-2518	42	24	we	we	PRON
iajs-2518	42	25	shall	shall	AUX
iajs-2518	42	26	give	give	VERB
iajs-2518	42	27	some	some	DET
iajs-2518	42	28	properties	property	NOUN
iajs-2518	42	29	of	of	ADP
iajs-2518	42	30	fsubmodules	fsubmodule	NOUN
iajs-2518	42	31	,	,	PUNCT
iajs-2518	42	32	which	which	PRON
iajs-2518	42	33	are	be	AUX
iajs-2518	42	34	used	use	VERB
iajs-2518	42	35	in	in	ADP
iajs-2518	42	36	the	the	DET
iajs-2518	42	37	next	next	ADJ
iajs-2518	42	38	sections	section	NOUN
iajs-2518	42	39	.	.	PUNCT
iajs-2518	43	1	definition	definition	NOUN
iajs-2518	43	2	1.7	1.7	NUM
iajs-2518	43	3	[	[	PUNCT
iajs-2518	43	4	6	6	NUM
iajs-2518	43	5	]	]	PUNCT
iajs-2518	43	6	let	let	VERB
iajs-2518	43	7	a	a	PRON
iajs-2518	43	8	be	be	AUX
iajs-2518	43	9	a	a	DET
iajs-2518	43	10	f	f	NOUN
iajs-2518	43	11	-	-	PUNCT
iajs-2518	43	12	set	set	NOUN
iajs-2518	43	13	of	of	ADP
iajs-2518	43	14	an	an	DET
iajs-2518	43	15	r	r	NOUN
iajs-2518	43	16	-	-	PUNCT
iajs-2518	43	17	module	module	NOUN
iajs-2518	43	18	ℳ	ℳ	NOUN
iajs-2518	43	19	,	,	PUNCT
iajs-2518	43	20	then	then	ADV
iajs-2518	43	21	the	the	DET
iajs-2518	43	22	submodule	submodule	NOUN
iajs-2518	43	23	a	a	DET
iajs-2518	43	24	𝑜𝑓	𝑜𝑓	X
iajs-2518	43	25	ℳ	ℳ	PROPN
iajs-2518	43	26	is	be	AUX
iajs-2518	43	27	called	call	VERB
iajs-2518	43	28	the	the	DET
iajs-2518	43	29	level	level	NOUN
iajs-2518	43	30	submodule	submodule	NOUN
iajs-2518	43	31	of	of	ADP
iajs-2518	43	32	ℳ	ℳ	PROPN
iajs-2518	43	33	,	,	PUNCT
iajs-2518	43	34	where	where	SCONJ
iajs-2518	43	35	𝑡	𝑡	PROPN
iajs-2518	43	36	∈	∈	PROPN
iajs-2518	43	37	0,1	0,1	NUM
iajs-2518	43	38	.	.	PUNCT
iajs-2518	43	39	"	"	PUNCT
iajs-2518	44	1	proposition	proposition	NOUN
iajs-2518	44	2	1.8[7],[8	1.8[7],[8	NUM
iajs-2518	44	3	]	]	PUNCT
iajs-2518	44	4	let	let	VERB
iajs-2518	44	5	a	a	PRON
iajs-2518	44	6	be	be	AUX
iajs-2518	44	7	a	a	DET
iajs-2518	44	8	fmodule	fmodule	NOUN
iajs-2518	44	9	in	in	ADP
iajs-2518	44	10	ℳ	ℳ	PROPN
iajs-2518	44	11	,	,	PUNCT
iajs-2518	44	12	then	then	ADV
iajs-2518	44	13	we	we	PRON
iajs-2518	44	14	define	define	VERB
iajs-2518	44	15	𝐴∗	𝐴∗	NUM
iajs-2518	44	16	𝐴	𝐴	PROPN
iajs-2518	44	17	𝑥	𝑥	X
iajs-2518	44	18	∈	∈	PROPN
iajs-2518	44	19	ℳ	ℳ	PROPN
iajs-2518	44	20	,	,	PUNCT
iajs-2518	44	21	𝐴	𝐴	PROPN
iajs-2518	44	22	𝑥	𝑥	PROPN
iajs-2518	44	23	1	1	NUM
iajs-2518	44	24	𝐴	𝐴	NOUN
iajs-2518	44	25	0	0	NUM
iajs-2518	45	1	=	=	SYM
iajs-2518	45	2	1	1	NUM
iajs-2518	45	3	"	"	PUNCT
iajs-2518	45	4	proposition	proposition	NOUN
iajs-2518	45	5	1.9[11	1.9[11	NUM
iajs-2518	45	6	]	]	PUNCT
iajs-2518	45	7	let	let	VERB
iajs-2518	45	8	a	a	PRON
iajs-2518	45	9	be	be	AUX
iajs-2518	45	10	a	a	DET
iajs-2518	45	11	fuzzy	fuzzy	ADJ
iajs-2518	45	12	module	module	NOUN
iajs-2518	45	13	of	of	ADP
iajs-2518	45	14	an	an	DET
iajs-2518	45	15	rmodule	rmodule	NOUN
iajs-2518	45	16	ℳ	ℳ	NOUN
iajs-2518	45	17	,	,	PUNCT
iajs-2518	45	18	then	then	ADV
iajs-2518	45	19	𝐴∗	𝐴∗	PROPN
iajs-2518	45	20	is	be	AUX
iajs-2518	45	21	a	a	DET
iajs-2518	45	22	submodule	submodule	NOUN
iajs-2518	45	23	of	of	ADP
iajs-2518	45	24	ℳ	ℳ	PROPN
iajs-2518	45	25	.	.	PUNCT
iajs-2518	45	26	"	"	PUNCT
iajs-2518	45	27	.	.	PUNCT
iajs-2518	46	1	we	we	PRON
iajs-2518	46	2	add	add	VERB
iajs-2518	46	3	the	the	DET
iajs-2518	46	4	following	follow	VERB
iajs-2518	46	5	results	result	NOUN
iajs-2518	46	6	:	:	PUNCT
iajs-2518	46	7	proposition	proposition	NOUN
iajs-2518	46	8	1.10	1.10	NUM
iajs-2518	46	9	if	if	SCONJ
iajs-2518	46	10	ℋ	ℋ	PROPN
iajs-2518	46	11	is	be	AUX
iajs-2518	46	12	a	a	DET
iajs-2518	46	13	fmodule	fmodule	NOUN
iajs-2518	46	14	of	of	ADP
iajs-2518	46	15	an	an	DET
iajs-2518	46	16	r	r	NOUN
iajs-2518	46	17	-	-	PUNCT
iajs-2518	46	18	module	module	NOUN
iajs-2518	46	19	ℳ	ℳ	NOUN
iajs-2518	46	20	and	and	CCONJ
iajs-2518	46	21	ℕ	ℕ	PROPN
iajs-2518	46	22	ℋ	ℋ	PROPN
iajs-2518	46	23	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2518	46	24	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
iajs-2518	46	25	ℋ	ℋ	PROPN
iajs-2518	46	26	0	0	NUM
iajs-2518	46	27	1	1	NUM
iajs-2518	46	28	,	,	PUNCT
iajs-2518	46	29	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-2518	46	30	ℕ	ℕ	PROPN
iajs-2518	46	31	0	0	NUM
iajs-2518	46	32	1	1	NUM
iajs-2518	46	33	.	.	PUNCT
iajs-2518	47	1	proof	proof	NOUN
iajs-2518	47	2	:	:	PUNCT
iajs-2518	47	3	it	it	PRON
iajs-2518	47	4	is	be	AUX
iajs-2518	47	5	clear	clear	ADJ
iajs-2518	47	6	by	by	ADP
iajs-2518	47	7	the	the	DET
iajs-2518	47	8	definition	definition	NOUN
iajs-2518	47	9	of	of	ADP
iajs-2518	47	10	fmodule	fmodule	NOUN
iajs-2518	47	11	.	.	PUNCT
iajs-2518	47	12	  	  	SPACE
iajs-2518	48	1	139	139	NUM
iajs-2518	48	2	  	  	SPACE
iajs-2518	48	3	ibn	ibn	PROPN
iajs-2518	48	4	al	al	PROPN
iajs-2518	48	5	-	-	PUNCT
iajs-2518	48	6	haitham	haitham	PROPN
iajs-2518	48	7	jour	jour	X
iajs-2518	48	8	.	.	PROPN
iajs-2518	48	9	for	for	ADP
iajs-2518	48	10	pure	pure	ADJ
iajs-2518	48	11	&	&	CCONJ
iajs-2518	48	12	appl	appl	PROPN
iajs-2518	48	13	.	.	PUNCT
iajs-2518	49	1	sci	sci	PROPN
iajs-2518	49	2	.	.	PROPN
iajs-2518	50	1	33	33	NUM
iajs-2518	50	2	(	(	PUNCT
iajs-2518	50	3	4	4	NUM
iajs-2518	50	4	)	)	PUNCT
iajs-2518	50	5	2020	2020	NUM
iajs-2518	50	6	remark	remark	VERB
iajs-2518	50	7	1.11[11	1.11[11	NUM
iajs-2518	50	8	]	]	X
iajs-2518	50	9	if	if	SCONJ
iajs-2518	50	10	a	a	PRON
iajs-2518	50	11	and	and	CCONJ
iajs-2518	50	12	b	b	NOUN
iajs-2518	50	13	are	be	AUX
iajs-2518	50	14	fsubmodules	fsubmodule	NOUN
iajs-2518	50	15	of	of	ADP
iajs-2518	50	16	fmodule	fmodule	ADJ
iajs-2518	50	17	x	x	NOUN
iajs-2518	50	18	such	such	ADJ
iajs-2518	50	19	that	that	SCONJ
iajs-2518	50	20	𝐴	𝐴	PROPN
iajs-2518	50	21	𝐵	𝐵	PROPN
iajs-2518	50	22	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-2518	50	23	𝐴∗	𝐴∗	NOUN
iajs-2518	50	24	𝐵∗	𝐵∗	NUM
iajs-2518	50	25	proof	proof	NOUN
iajs-2518	50	26	:	:	PUNCT
iajs-2518	50	27	let	let	VERB
iajs-2518	50	28	𝑥	𝑥	X
iajs-2518	50	29	∈	∈	PROPN
iajs-2518	50	30	𝐴∗	𝐴∗	PROPN
iajs-2518	50	31	,	,	PUNCT
iajs-2518	50	32	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-2518	50	33	𝐴	𝐴	PROPN
iajs-2518	50	34	𝑥	𝑥	PROPN
iajs-2518	50	35	𝐴	𝐴	PROPN
iajs-2518	50	36	0	0	NUM
iajs-2518	50	37	.	.	PUNCT
iajs-2518	51	1	but	but	CCONJ
iajs-2518	51	2	b(x	b(x	ADJ
iajs-2518	51	3	)	)	PUNCT
iajs-2518	51	4	𝐴	𝐴	PROPN
iajs-2518	51	5	𝑥	𝑥	PROPN
iajs-2518	51	6	,	,	PUNCT
iajs-2518	51	7	∀𝑥	∀𝑥	PROPN
iajs-2518	51	8	∈	∈	PROPN
iajs-2518	51	9	𝑀	𝑀	PROPN
iajs-2518	51	10	,	,	PUNCT
iajs-2518	51	11	ℎ𝑒𝑛𝑐𝑒	ℎ𝑒𝑛𝑐𝑒	NOUN
iajs-2518	51	12	b	b	PROPN
iajs-2518	51	13	x	x	SYM
iajs-2518	51	14	𝐴	𝐴	PROPN
iajs-2518	51	15	𝑥	𝑥	PROPN
iajs-2518	51	16	𝐴	𝐴	PROPN
iajs-2518	51	17	0	0	NUM
iajs-2518	51	18	𝐵	𝐵	NOUN
iajs-2518	51	19	𝑂	𝑂	NOUN
iajs-2518	51	20	.	.	PUNCT
iajs-2518	52	1	𝑇ℎ𝑢𝑠	𝑇ℎ𝑢𝑠	ADP
iajs-2518	52	2	𝑥	𝑥	PRON
iajs-2518	52	3	∈	∈	PROPN
iajs-2518	52	4	𝐵	𝐵	NOUN
iajs-2518	52	5	𝐵∗	𝐵∗	NOUN
iajs-2518	52	6	remark	remark	NOUN
iajs-2518	52	7	1.12[11	1.12[11	NUM
iajs-2518	52	8	]	]	X
iajs-2518	52	9	the	the	DET
iajs-2518	52	10	convers	conver	NOUN
iajs-2518	52	11	of	of	ADP
iajs-2518	52	12	the	the	DET
iajs-2518	52	13	above	above	ADJ
iajs-2518	52	14	remark	remark	NOUN
iajs-2518	52	15	is	be	AUX
iajs-2518	52	16	not	not	PART
iajs-2518	52	17	true	true	ADJ
iajs-2518	52	18	in	in	ADP
iajs-2518	52	19	general	general	ADJ
iajs-2518	52	20	as	as	SCONJ
iajs-2518	52	21	the	the	DET
iajs-2518	52	22	following	follow	VERB
iajs-2518	52	23	example	example	NOUN
iajs-2518	52	24	shows	show	VERB
iajs-2518	52	25	:	:	PUNCT
iajs-2518	52	26	let	let	VERB
iajs-2518	52	27	𝑋	𝑋	NOUN
iajs-2518	52	28	:	:	PUNCT
iajs-2518	52	29	𝑍	𝑍	PROPN
iajs-2518	52	30	→	→	SYM
iajs-2518	52	31	0,1	0,1	NUM
iajs-2518	52	32	,	,	PUNCT
iajs-2518	52	33	𝑑𝑒𝑓𝑖𝑛𝑒	𝑑𝑒𝑓𝑖𝑛𝑒	ADJ
iajs-2518	52	34	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	52	35	∶	∶	PROPN
iajs-2518	52	36	𝑋	𝑋	NOUN
iajs-2518	52	37	𝑥	𝑥	PROPN
iajs-2518	52	38	1	1	NUM
iajs-2518	52	39	,	,	PUNCT
iajs-2518	52	40	∀	∀	VERB
iajs-2518	52	41	𝑥	𝑥	DET
iajs-2518	52	42	∈	∈	PROPN
iajs-2518	52	43	𝑍	𝑍	NOUN
iajs-2518	52	44	,	,	PUNCT
iajs-2518	52	45	let	let	VERB
iajs-2518	52	46	𝐴	𝐴	PROPN
iajs-2518	52	47	𝑥	𝑥	VERB
iajs-2518	52	48	1	1	NUM
iajs-2518	52	49	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	52	50	𝑥	𝑥	PRON
iajs-2518	52	51	∈	∈	PROPN
iajs-2518	52	52	4𝑍	4𝑍	NOUN
iajs-2518	52	53	,	,	PUNCT
iajs-2518	52	54	0.9	0.9	NUM
iajs-2518	52	55	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2518	52	56	𝐵	𝐵	NOUN
iajs-2518	52	57	𝑥	𝑥	ADP
iajs-2518	52	58	1	1	NUM
iajs-2518	52	59	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	52	60	𝑥	𝑥	DET
iajs-2518	52	61	∈	∈	PROPN
iajs-2518	52	62	2𝑍	2𝑍	NOUN
iajs-2518	52	63	,	,	PUNCT
iajs-2518	52	64	1	1	NUM
iajs-2518	52	65	2	2	NUM
iajs-2518	52	66	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	52	67	a	a	PRON
iajs-2518	52	68	and	and	CCONJ
iajs-2518	52	69	b	b	NOUN
iajs-2518	52	70	are	be	AUX
iajs-2518	52	71	fsubmodules	fsubmodule	NOUN
iajs-2518	52	72	x	x	SYM
iajs-2518	52	73	and	and	CCONJ
iajs-2518	52	74	𝐴∗	𝐴∗	NUM
iajs-2518	52	75	4𝑍	4𝑍	NOUN
iajs-2518	52	76	,	,	PUNCT
iajs-2518	52	77	𝐵∗	𝐵∗	PRON
iajs-2518	52	78	2z	2z	NUM
iajs-2518	52	79	hence	hence	ADV
iajs-2518	52	80	,	,	PUNCT
iajs-2518	52	81	𝐴∗	𝐴∗	PROPN
iajs-2518	52	82	𝐵∗	𝐵∗	NUM
iajs-2518	52	83	.	.	PUNCT
iajs-2518	53	1	𝐵𝑢𝑡	𝐵𝑢𝑡	PROPN
iajs-2518	53	2	𝐴	𝐴	PROPN
iajs-2518	53	3	𝐵	𝐵	PROPN
iajs-2518	53	4	,	,	PUNCT
iajs-2518	53	5	𝑠𝑖𝑛𝑐𝑒	𝑠𝑖𝑛𝑐𝑒	PROPN
iajs-2518	53	6	𝐴	𝐴	PROPN
iajs-2518	53	7	3	3	NUM
iajs-2518	53	8	0.9	0.9	NUM
iajs-2518	53	9	,	,	PUNCT
iajs-2518	53	10	𝐵	𝐵	NOUN
iajs-2518	53	11	3	3	NUM
iajs-2518	53	12	1/2	1/2	NUM
iajs-2518	53	13	0.5.x	0.5.x	NUM
iajs-2518	53	14	remark	remark	NOUN
iajs-2518	53	15	we	we	PRON
iajs-2518	53	16	assume	assume	VERB
iajs-2518	53	17	that	that	SCONJ
iajs-2518	53	18	if	if	SCONJ
iajs-2518	53	19	𝐴∗	𝐴∗	NUM
iajs-2518	53	20	=	=	SYM
iajs-2518	53	21	𝐵∗	𝐵∗	PROPN
iajs-2518	53	22	.	.	PUNCT
iajs-2518	54	1	then	then	ADV
iajs-2518	54	2	,	,	PUNCT
iajs-2518	54	3	𝐴	𝐴	PROPN
iajs-2518	54	4	𝐵	𝐵	PROPN
iajs-2518	54	5	is	be	AUX
iajs-2518	54	6	called	call	VERB
iajs-2518	54	7	condition	condition	NOUN
iajs-2518	54	8	(	(	PUNCT
iajs-2518	54	9	*	*	NOUN
iajs-2518	54	10	)	)	PUNCT
iajs-2518	54	11	maysoon	maysoon	NOUN
iajs-2518	54	12	in[11	in[11	PROPN
iajs-2518	54	13	]	]	PUNCT
iajs-2518	54	14	introduced	introduce	VERB
iajs-2518	54	15	the	the	DET
iajs-2518	54	16	following	following	ADJ
iajs-2518	54	17	definition	definition	NOUN
iajs-2518	54	18	:	:	PUNCT
iajs-2518	54	19	definition	definition	NOUN
iajs-2518	54	20	1.13[11	1.13[11	NUM
iajs-2518	54	21	]	]	PUNCT
iajs-2518	54	22	let	let	VERB
iajs-2518	54	23	x	x	PRON
iajs-2518	54	24	be	be	AUX
iajs-2518	54	25	a	a	DET
iajs-2518	54	26	f	f	NOUN
iajs-2518	54	27	module	module	NOUN
iajs-2518	54	28	of	of	ADP
iajs-2518	54	29	an	an	DET
iajs-2518	54	30	r	r	NOUN
iajs-2518	54	31	–	–	PUNCT
iajs-2518	54	32	module	module	NOUN
iajs-2518	54	33	m	m	NOUN
iajs-2518	54	34	,	,	PUNCT
iajs-2518	54	35	let	let	VERB
iajs-2518	54	36	a	a	PRON
iajs-2518	54	37	be	be	AUX
iajs-2518	54	38	f	f	PROPN
iajs-2518	54	39	submodule	submodule	NOUN
iajs-2518	54	40	of	of	ADP
iajs-2518	54	41	x	x	PROPN
iajs-2518	54	42	𝐷𝑒𝑓𝑖𝑛𝑒	𝐷𝑒𝑓𝑖𝑛𝑒	PROPN
iajs-2518	54	43	𝑋	𝑋	PROPN
iajs-2518	54	44	𝐴:⁄	𝐴:⁄	PROPN
iajs-2518	54	45	𝑀	𝑀	PROPN
iajs-2518	54	46	/𝐴∗	/𝐴∗	PUNCT
iajs-2518	55	1	→	→	SYM
iajs-2518	55	2	0,1	0,1	NUM
iajs-2518	55	3	𝑏𝑦|	𝑏𝑦|	ADJ
iajs-2518	55	4	:	:	PUNCT
iajs-2518	55	5	𝑋	𝑋	PROPN
iajs-2518	55	6	𝐴	𝐴	PROPN
iajs-2518	55	7	𝑎	𝑎	X
iajs-2518	55	8	𝐴∗⁄	𝐴∗⁄	NOUN
iajs-2518	55	9	1	1	NUM
iajs-2518	55	10	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	55	11	𝑎	𝑎	PROPN
iajs-2518	55	12	∈	∈	PROPN
iajs-2518	55	13	𝐴∗	𝐴∗	NUM
iajs-2518	55	14	sup	sup	NOUN
iajs-2518	55	15	𝑋	𝑋	PROPN
iajs-2518	55	16	𝑎	𝑎	PROPN
iajs-2518	55	17	𝑏	𝑏	NOUN
iajs-2518	55	18	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	55	19	𝑏	𝑏	PROPN
iajs-2518	55	20	∈	∈	PROPN
iajs-2518	55	21	𝐴∗	𝐴∗	PROPN
iajs-2518	55	22	,	,	PUNCT
iajs-2518	55	23	𝑎	𝑎	PROPN
iajs-2518	55	24			NOUN
iajs-2518	55	25	𝐴∗	𝐴∗	NUM
iajs-2518	55	26	for	for	SCONJ
iajs-2518	55	27	all	all	DET
iajs-2518	55	28	coset	coset	NOUN
iajs-2518	55	29	𝑎	𝑎	PRON
iajs-2518	55	30	𝐴∗	𝐴∗	NUM
iajs-2518	55	31	∈	∈	PROPN
iajs-2518	55	32	𝑀	𝑀	PROPN
iajs-2518	55	33	𝐴∗	𝐴∗	PROPN
iajs-2518	55	34	"	"	PUNCT
iajs-2518	55	35	.	.	PUNCT
iajs-2518	56	1	p𝐫𝐨𝐩𝐨𝐬𝐢𝐭𝐢𝐨𝐧	p𝐫𝐨𝐩𝐨𝐬𝐢𝐭𝐢𝐨𝐧	PROPN
iajs-2518	56	2	𝟏	𝟏	PROPN
iajs-2518	56	3	.	.	PUNCT
iajs-2518	56	4	𝟏𝟒	𝟏𝟒	NUM
iajs-2518	57	1	𝟏𝟏	𝟏𝟏	NUM
iajs-2518	57	2	𝐼𝑓	𝐼𝑓	PROPN
iajs-2518	57	3	𝑋	𝑋	NOUN
iajs-2518	57	4	𝑖𝑠	𝑖𝑠	ADP
iajs-2518	57	5	𝑎	𝑎	PRON
iajs-2518	57	6	𝐹	𝐹	PROPN
iajs-2518	57	7	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	VERB
iajs-2518	57	8	𝑜𝑓	𝑜𝑓	ADP
iajs-2518	57	9	𝑎𝑛	𝑎𝑛	PROPN
iajs-2518	57	10	𝑅𝑚𝑜𝑑𝑢𝑙𝑒	𝑅𝑚𝑜𝑑𝑢𝑙𝑒	PROPN
iajs-2518	57	11	𝑀	𝑀	PROPN
iajs-2518	57	12	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	57	13	𝐴	𝐴	PROPN
iajs-2518	57	14	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	57	15	𝑎	𝑎	PRON
iajs-2518	57	16	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	57	17	𝑜𝑓	𝑜𝑓	ADP
iajs-2518	57	18	𝑋	𝑋	PROPN
iajs-2518	57	19	,	,	PUNCT
iajs-2518	57	20	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-2518	57	21	𝑋	𝑋	PROPN
iajs-2518	57	22	𝐴⁄	𝐴⁄	PROPN
iajs-2518	57	23	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	57	24	𝑎	𝑎	PRON
iajs-2518	57	25	𝐹	𝐹	PROPN
iajs-2518	57	26	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	VERB
iajs-2518	57	27	𝑜𝑓𝑀	𝑜𝑓𝑀	PROPN
iajs-2518	57	28	/𝐴∗	/𝐴∗	PROPN
iajs-2518	57	29	.	.	PUNCT
iajs-2518	58	1	however	however	ADV
iajs-2518	58	2	,	,	PUNCT
iajs-2518	58	3	in	in	ADP
iajs-2518	58	4	[	[	X
iajs-2518	58	5	12	12	NUM
iajs-2518	58	6	]	]	PUNCT
iajs-2518	58	7	there	there	PRON
iajs-2518	58	8	exists	exist	VERB
iajs-2518	58	9	a	a	DET
iajs-2518	58	10	definition	definition	NOUN
iajs-2518	58	11	of	of	ADP
iajs-2518	58	12	quotient	quotient	NOUN
iajs-2518	58	13	fuzzy	fuzzy	ADJ
iajs-2518	58	14	module	module	NOUN
iajs-2518	58	15	which	which	PRON
iajs-2518	58	16	is	be	AUX
iajs-2518	58	17	an	an	DET
iajs-2518	58	18	equivalent	equivalent	NOUN
iajs-2518	58	19	to	to	ADP
iajs-2518	58	20	definition	definition	NOUN
iajs-2518	58	21	3.1	3.1	NUM
iajs-2518	58	22	where	where	SCONJ
iajs-2518	58	23	𝑋	𝑋	PROPN
iajs-2518	58	24	0	0	NUM
iajs-2518	58	25	1	1	NUM
iajs-2518	58	26	,	,	PUNCT
iajs-2518	58	27	as	as	SCONJ
iajs-2518	58	28	follows	follow	VERB
iajs-2518	58	29	:	:	PUNCT
iajs-2518	58	30	proposition	proposition	NOUN
iajs-2518	58	31	1.15[12	1.15[12	NUM
iajs-2518	58	32	]	]	X
iajs-2518	58	33	let	let	VERB
iajs-2518	58	34	x	x	PRON
iajs-2518	58	35	be	be	AUX
iajs-2518	58	36	a	a	DET
iajs-2518	58	37	f	f	NOUN
iajs-2518	58	38	module	module	NOUN
iajs-2518	58	39	of	of	ADP
iajs-2518	58	40	an	an	DET
iajs-2518	58	41	rmodule	rmodule	NOUN
iajs-2518	58	42	ℳ	ℳ	NOUN
iajs-2518	58	43	and	and	CCONJ
iajs-2518	58	44	a	a	DET
iajs-2518	58	45	be	be	AUX
iajs-2518	58	46	a	a	DET
iajs-2518	58	47	f	f	PROPN
iajs-2518	58	48	submodule	submodule	NOUN
iajs-2518	58	49	of	of	ADP
iajs-2518	58	50	x.	x.	NOUN
iajs-2518	58	51	define	define	VERB
iajs-2518	58	52	𝑋	𝑋	PROPN
iajs-2518	58	53	𝐴	𝐴	PROPN
iajs-2518	58	54	:	:	PUNCT
iajs-2518	58	55	𝑀	𝑀	PROPN
iajs-2518	58	56	𝐴∗	𝐴∗	PROPN
iajs-2518	59	1	⁄	⁄	PROPN
iajs-2518	59	2	→	→	SYM
iajs-2518	59	3	0,1	0,1	NUM
iajs-2518	59	4	⁄	⁄	PROPN
iajs-2518	59	5	,	,	PUNCT
iajs-2518	59	6	such	such	ADJ
iajs-2518	59	7	that	that	SCONJ
iajs-2518	59	8	𝑋	𝑋	PROPN
iajs-2518	59	9	𝐴	𝐴	PROPN
iajs-2518	59	10	a	a	DET
iajs-2518	59	11	𝐴∗	𝐴∗	NUM
iajs-2518	59	12	sup	sup	NOUN
iajs-2518	59	13	𝑋	𝑋	PROPN
iajs-2518	59	14	𝑎	𝑎	PROPN
iajs-2518	59	15	𝑏	𝑏	NOUN
iajs-2518	59	16	,	,	PUNCT
iajs-2518	59	17	𝑎	𝑎	PROPN
iajs-2518	59	18	∈	∈	PROPN
iajs-2518	59	19	ℳ	ℳ	PROPN
iajs-2518	59	20	,	,	PUNCT
iajs-2518	59	21	𝑏	𝑏	PROPN
iajs-2518	59	22	∈	∈	PROPN
iajs-2518	59	23	𝐴∗	𝐴∗	PROPN
iajs-2518	60	1	⁄	⁄	PROPN
iajs-2518	60	2	lemma	lemma	PROPN
iajs-2518	61	1	1.16[11	1.16[11	INTJ
iajs-2518	61	2	]	]	X
iajs-2518	61	3	if	if	SCONJ
iajs-2518	61	4	a	a	PRON
iajs-2518	61	5	be	be	AUX
iajs-2518	61	6	fsubmodule	fsubmodule	NOUN
iajs-2518	61	7	of	of	ADP
iajs-2518	61	8	fmodule	fmodule	ADJ
iajs-2518	61	9	x	x	INTJ
iajs-2518	61	10	,	,	PUNCT
iajs-2518	61	11	then	then	ADV
iajs-2518	61	12	𝑋∗	𝑋∗	ADJ
iajs-2518	62	1	⁄	⁄	ADJ
iajs-2518	62	2	𝐴∗	𝐴∗	NUM
iajs-2518	62	3	x	x	SYM
iajs-2518	62	4	/	/	SYM
iajs-2518	62	5	a	a	DET
iajs-2518	62	6	∗	∗	NOUN
iajs-2518	62	7	.	.	PUNCT
iajs-2518	63	1	proposition	proposition	NOUN
iajs-2518	63	2	1.17[11	1.17[11	NUM
iajs-2518	63	3	]	]	PUNCT
iajs-2518	63	4	let	let	VERB
iajs-2518	63	5	x	x	PRON
iajs-2518	63	6	be	be	AUX
iajs-2518	63	7	a	a	DET
iajs-2518	63	8	fmodule	fmodule	NOUN
iajs-2518	63	9	of	of	ADP
iajs-2518	63	10	an	an	DET
iajs-2518	63	11	rmodule	rmodule	NOUN
iajs-2518	63	12	ℳ	ℳ	NOUN
iajs-2518	63	13	such	such	ADJ
iajs-2518	63	14	that	that	SCONJ
iajs-2518	63	15	𝑋	𝑋	PROPN
iajs-2518	63	16	𝑥	𝑥	PROPN
iajs-2518	63	17	1	1	NUM
iajs-2518	63	18	,	,	PUNCT
iajs-2518	63	19	∀𝑥	∀𝑥	PROPN
iajs-2518	63	20	∈	∈	NOUN
iajs-2518	63	21	ℳ.	ℳ.	NOUN
iajs-2518	63	22	𝑇𝑡ℎ𝑒𝑛	𝑇𝑡ℎ𝑒𝑛	VERB
iajs-2518	63	23	𝑋	𝑋	NOUN
iajs-2518	63	24	𝐴⁄	𝐴⁄	PROPN
iajs-2518	63	25	∗	∗	NOUN
iajs-2518	63	26	𝑋∗|𝐴	𝑋∗|𝐴	PROPN
iajs-2518	63	27	∗	∗	NOUN
iajs-2518	63	28	,	,	PUNCT
iajs-2518	63	29	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-2518	63	30	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
iajs-2518	63	31	𝐴	𝐴	PROPN
iajs-2518	63	32	𝑋.	𝑋.	PROPN
iajs-2518	63	33	proposition	proposition	NOUN
iajs-2518	63	34	1.18[11	1.18[11	NUM
iajs-2518	63	35	]	]	X
iajs-2518	63	36	if	if	SCONJ
iajs-2518	63	37	a	a	DET
iajs-2518	63	38	,	,	PUNCT
iajs-2518	63	39	b	b	NOUN
iajs-2518	63	40	are	be	AUX
iajs-2518	63	41	fsubmodules	fsubmodule	NOUN
iajs-2518	63	42	of	of	ADP
iajs-2518	63	43	fmodule	fmodule	ADJ
iajs-2518	63	44	x	x	X
iajs-2518	63	45	.	.	PUNCT
iajs-2518	64	1	then	then	ADV
iajs-2518	64	2	𝑋	𝑋	PROPN
iajs-2518	64	3	𝐴	𝐴	PROPN
iajs-2518	64	4	∩	∩	NOUN
iajs-2518	64	5	𝐵⁄	𝐵⁄	PROPN
iajs-2518	64	6	𝐹	𝐹	PROPN
iajs-2518	64	7	submodules	submodule	NOUN
iajs-2518	64	8	in	in	ADP
iajs-2518	64	9	𝑋	𝑋	PROPN
iajs-2518	64	10	𝐴	𝐴	PROPN
iajs-2518	64	11	⨁	⨁	PROPN
iajs-2518	64	12	𝑋	𝑋	PROPN
iajs-2518	64	13	𝐵⁄⁄	𝐵⁄⁄	NOUN
iajs-2518	64	14	.	.	PUNCT
iajs-2518	64	15	  	  	SPACE
iajs-2518	65	1	140	140	NUM
iajs-2518	65	2	  	  	SPACE
iajs-2518	65	3	ibn	ibn	PROPN
iajs-2518	65	4	al	al	PROPN
iajs-2518	65	5	-	-	PUNCT
iajs-2518	65	6	haitham	haitham	PROPN
iajs-2518	65	7	jour	jour	X
iajs-2518	65	8	.	.	PROPN
iajs-2518	65	9	for	for	ADP
iajs-2518	65	10	pure	pure	ADJ
iajs-2518	65	11	&	&	CCONJ
iajs-2518	65	12	appl	appl	PROPN
iajs-2518	65	13	.	.	PUNCT
iajs-2518	66	1	sci	sci	PROPN
iajs-2518	66	2	.	.	PROPN
iajs-2518	67	1	33	33	NUM
iajs-2518	67	2	(	(	PUNCT
iajs-2518	67	3	4	4	NUM
iajs-2518	67	4	)	)	PUNCT
iajs-2518	67	5	2020	2020	NUM
iajs-2518	67	6	definition	definition	NOUN
iajs-2518	67	7	1.19	1.19	NUM
iajs-2518	67	8	[	[	X
iajs-2518	67	9	3	3	NUM
iajs-2518	67	10	]	]	PUNCT
iajs-2518	67	11	a	a	DET
iajs-2518	67	12	fsubmodule	fsubmodule	NOUN
iajs-2518	67	13	of	of	ADP
iajs-2518	67	14	fmodule	fmodule	NOUN
iajs-2518	67	15	x	x	PUNCT
iajs-2518	67	16	is	be	AUX
iajs-2518	67	17	called	call	VERB
iajs-2518	67	18	pure	pure	ADJ
iajs-2518	67	19	if	if	SCONJ
iajs-2518	67	20	for	for	ADP
iajs-2518	67	21	each	each	DET
iajs-2518	67	22	f	f	NOUN
iajs-2518	67	23	-	-	PUNCT
iajs-2518	67	24	ideal	ideal	NOUN
iajs-2518	67	25	k	k	PROPN
iajs-2518	67	26	of	of	ADP
iajs-2518	67	27	r	r	NOUN
iajs-2518	67	28	,	,	PUNCT
iajs-2518	67	29	𝐾𝑋	𝐾𝑋	PROPN
iajs-2518	67	30	∩	∩	ADJ
iajs-2518	67	31	𝐴	𝐴	PROPN
iajs-2518	67	32	𝐾𝐴.	𝐾𝐴.	NOUN
iajs-2518	67	33	defintion1.20	defintion1.20	PROPN
iajs-2518	68	1	[	[	X
iajs-2518	68	2	3	3	X
iajs-2518	68	3	]	]	PUNCT
iajs-2518	68	4	a	a	DET
iajs-2518	68	5	f	f	NOUN
iajs-2518	68	6	-	-	PUNCT
iajs-2518	68	7	module	module	NOUN
iajs-2518	68	8	x	x	NOUN
iajs-2518	68	9	of	of	ADP
iajs-2518	68	10	an	an	DET
iajs-2518	68	11	r	r	NOUN
iajs-2518	68	12	-	-	PUNCT
iajs-2518	68	13	module	module	NOUN
iajs-2518	68	14	m	m	NOUN
iajs-2518	68	15	is	be	AUX
iajs-2518	68	16	called	call	VERB
iajs-2518	68	17	f	f	X
iajs-2518	68	18	-	-	PUNCT
iajs-2518	68	19	regular	regular	ADJ
iajs-2518	68	20	if	if	SCONJ
iajs-2518	68	21	every	every	DET
iajs-2518	68	22	f	f	NOUN
iajs-2518	68	23	-	-	PUNCT
iajs-2518	68	24	submodule	submodule	NOUN
iajs-2518	68	25	of	of	ADP
iajs-2518	68	26	x	x	SYM
iajs-2518	68	27	is	be	AUX
iajs-2518	68	28	pure	pure	ADJ
iajs-2518	68	29	.	.	PUNCT
iajs-2518	69	1	definition	definition	NOUN
iajs-2518	69	2	1.21[13	1.21[13	NOUN
iajs-2518	69	3	]	]	PUNCT
iajs-2518	69	4	let	let	VERB
iajs-2518	69	5	a	a	PRON
iajs-2518	69	6	be	be	AUX
iajs-2518	69	7	a	a	DET
iajs-2518	69	8	fsubmodule	fsubmodule	NOUN
iajs-2518	69	9	of	of	ADP
iajs-2518	69	10	fuzzy	fuzzy	ADJ
iajs-2518	69	11	module	module	NOUN
iajs-2518	69	12	x	x	PRON
iajs-2518	69	13	is	be	AUX
iajs-2518	69	14	called	call	VERB
iajs-2518	69	15	an	an	DET
iajs-2518	69	16	essential	essential	ADJ
iajs-2518	69	17	if	if	SCONJ
iajs-2518	69	18	𝐴	𝐴	PROPN
iajs-2518	69	19	∩	∩	ADJ
iajs-2518	69	20	𝐵	𝐵	NOUN
iajs-2518	69	21	0	0	NUM
iajs-2518	69	22	,	,	PUNCT
iajs-2518	69	23	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2518	69	24	𝑛𝑜𝑛	𝑛𝑜𝑛	AUX
iajs-2518	69	25	𝑡𝑟𝑖𝑣𝑖𝑎𝑙	𝑡𝑟𝑖𝑣𝑖𝑎𝑙	VERB
iajs-2518	69	26	𝐹	𝐹	PROPN
iajs-2518	69	27	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	69	28	𝐵	𝐵	NOUN
iajs-2518	69	29	𝑜𝑓	𝑜𝑓	ADP
iajs-2518	69	30	𝑋	𝑋	PROPN
iajs-2518	69	31	.	.	PUNCT
iajs-2518	70	1	definition	definition	NOUN
iajs-2518	70	2	1.22	1.22	NUM
iajs-2518	70	3	[	[	X
iajs-2518	70	4	3	3	X
iajs-2518	70	5	]	]	PUNCT
iajs-2518	70	6	let	let	VERB
iajs-2518	70	7	x	x	PRON
iajs-2518	70	8	be	be	AUX
iajs-2518	70	9	a	a	DET
iajs-2518	70	10	f	f	NOUN
iajs-2518	70	11	-	-	PUNCT
iajs-2518	70	12	module	module	NOUN
iajs-2518	70	13	of	of	ADP
iajs-2518	70	14	an	an	DET
iajs-2518	70	15	rmodule	rmodule	NOUN
iajs-2518	70	16	m	m	NOUN
iajs-2518	70	17	.	.	PUNCT
iajs-2518	71	1	x	x	PRON
iajs-2518	71	2	is	be	AUX
iajs-2518	71	3	called	call	VERB
iajs-2518	71	4	a	a	DET
iajs-2518	71	5	multiplication	multiplication	NOUN
iajs-2518	71	6	fmodule	fmodule	ADV
iajs-2518	71	7	if	if	SCONJ
iajs-2518	72	1	and	and	CCONJ
iajs-2518	72	2	only	only	ADV
iajs-2518	72	3	if	if	SCONJ
iajs-2518	72	4	for	for	ADP
iajs-2518	72	5	each	each	DET
iajs-2518	72	6	f	f	PROPN
iajs-2518	72	7	submodule	submodule	PROPN
iajs-2518	72	8	a	a	PRON
iajs-2518	72	9	of	of	ADP
iajs-2518	72	10	x	x	PUNCT
iajs-2518	72	11	,	,	PUNCT
iajs-2518	72	12	there	there	PRON
iajs-2518	72	13	exists	exist	VERB
iajs-2518	72	14	a	a	DET
iajs-2518	72	15	f	f	PROPN
iajs-2518	72	16	ideal	ideal	NOUN
iajs-2518	72	17	k	k	PROPN
iajs-2518	72	18	of	of	ADP
iajs-2518	72	19	r	r	NOUN
iajs-2518	72	20	such	such	ADJ
iajs-2518	73	1	that	that	DET
iajs-2518	73	2	a=	a=	PROPN
iajs-2518	73	3	kx	kx	X
iajs-2518	73	4	.	.	PUNCT
iajs-2518	73	5	2.fuzzy	2.fuzzy	NUM
iajs-2518	73	6	simple	simple	ADJ
iajs-2518	73	7	(	(	PUNCT
iajs-2518	73	8	semisimple	semisimple	NOUN
iajs-2518	73	9	)	)	PUNCT
iajs-2518	73	10	modules	module	NOUN
iajs-2518	73	11	recall	recall	VERB
iajs-2518	73	12	that	that	SCONJ
iajs-2518	73	13	an	an	DET
iajs-2518	73	14	r	r	NOUN
iajs-2518	73	15	-	-	PUNCT
iajs-2518	73	16	module	module	NOUN
iajs-2518	73	17	m	m	NOUN
iajs-2518	73	18	is	be	AUX
iajs-2518	73	19	called	call	VERB
iajs-2518	73	20	simple	simple	ADJ
iajs-2518	73	21	if	if	SCONJ
iajs-2518	73	22	and	and	CCONJ
iajs-2518	73	23	only	only	ADV
iajs-2518	73	24	if	if	SCONJ
iajs-2518	73	25	has	have	VERB
iajs-2518	73	26	no	no	DET
iajs-2518	73	27	proper	proper	ADJ
iajs-2518	73	28	non	non	ADJ
iajs-2518	73	29	trivial	trivial	ADJ
iajs-2518	73	30	submodules	submodule	NOUN
iajs-2518	74	1	[	[	X
iajs-2518	74	2	2	2	NUM
iajs-2518	74	3	]	]	PUNCT
iajs-2518	74	4	and	and	CCONJ
iajs-2518	74	5	"	"	PUNCT
iajs-2518	74	6	m	m	VERB
iajs-2518	74	7	is	be	AUX
iajs-2518	74	8	called	call	VERB
iajs-2518	74	9	semisimple	semisimple	NOUN
iajs-2518	74	10	if	if	SCONJ
iajs-2518	74	11	and	and	CCONJ
iajs-2518	74	12	only	only	ADV
iajs-2518	74	13	if	if	SCONJ
iajs-2518	74	14	m	m	NOUN
iajs-2518	74	15	is	be	AUX
iajs-2518	74	16	sum	sum	NOUN
iajs-2518	74	17	of	of	ADP
iajs-2518	74	18	simple	simple	ADJ
iajs-2518	74	19	submodules	submodule	NOUN
iajs-2518	74	20	of	of	ADP
iajs-2518	74	21	m	m	PROPN
iajs-2518	75	1	[	[	X
iajs-2518	75	2	2	2	NUM
iajs-2518	75	3	]	]	PUNCT
iajs-2518	75	4	.	.	PUNCT
iajs-2518	76	1	maysoon	maysoon	PROPN
iajs-2518	76	2	in[3	in[3	PROPN
iajs-2518	76	3	]	]	PUNCT
iajs-2518	76	4	introduced	introduce	VERB
iajs-2518	76	5	the	the	DET
iajs-2518	76	6	definition	definition	NOUN
iajs-2518	76	7	of	of	ADP
iajs-2518	76	8	fuzzy	fuzzy	ADJ
iajs-2518	76	9	simple	simple	ADJ
iajs-2518	76	10	modules	module	NOUN
iajs-2518	76	11	and	and	CCONJ
iajs-2518	76	12	fuzzy	fuzzy	ADJ
iajs-2518	76	13	semisimple	semisimple	NOUN
iajs-2518	76	14	modules	module	NOUN
iajs-2518	76	15	.	.	PUNCT
iajs-2518	77	1	some	some	DET
iajs-2518	77	2	properties	property	NOUN
iajs-2518	77	3	of	of	ADP
iajs-2518	77	4	these	these	DET
iajs-2518	77	5	concepts	concept	NOUN
iajs-2518	77	6	which	which	PRON
iajs-2518	77	7	are	be	AUX
iajs-2518	77	8	useful	useful	ADJ
iajs-2518	77	9	in	in	ADP
iajs-2518	77	10	next	next	ADJ
iajs-2518	77	11	sections	section	NOUN
iajs-2518	77	12	are	be	AUX
iajs-2518	77	13	given	give	VERB
iajs-2518	77	14	.	.	PUNCT
iajs-2518	78	1	moreover	moreover	ADV
iajs-2518	78	2	we	we	PRON
iajs-2518	78	3	add	add	VERB
iajs-2518	78	4	many	many	ADJ
iajs-2518	78	5	other	other	ADJ
iajs-2518	78	6	results	result	NOUN
iajs-2518	78	7	.	.	PUNCT
iajs-2518	79	1	definition	definition	NOUN
iajs-2518	79	2	2.1[3	2.1[3	NUM
iajs-2518	79	3	]	]	X
iajs-2518	79	4	a	a	DET
iajs-2518	79	5	fmodule	fmodule	NOUN
iajs-2518	79	6	x	x	PUNCT
iajs-2518	79	7	is	be	AUX
iajs-2518	79	8	called	call	VERB
iajs-2518	79	9	simple	simple	ADJ
iajs-2518	79	10	if	if	SCONJ
iajs-2518	79	11	x	x	PRON
iajs-2518	79	12	has	have	VERB
iajs-2518	79	13	no	no	DET
iajs-2518	79	14	nontrivial	nontrivial	ADJ
iajs-2518	79	15	fsubmodules	fsubmodule	NOUN
iajs-2518	79	16	.	.	PUNCT
iajs-2518	80	1	in	in	ADP
iajs-2518	80	2	other	other	ADJ
iajs-2518	80	3	words	word	NOUN
iajs-2518	80	4	,	,	PUNCT
iajs-2518	80	5	x	x	X
iajs-2518	80	6	is	be	AUX
iajs-2518	80	7	simple	simple	ADJ
iajs-2518	80	8	if	if	SCONJ
iajs-2518	80	9	whenever	whenever	SCONJ
iajs-2518	80	10	𝐴	𝐴	PROPN
iajs-2518	80	11	𝑋	𝑋	PROPN
iajs-2518	80	12	,	,	PUNCT
iajs-2518	80	13	either	either	CCONJ
iajs-2518	80	14	𝐴	𝐴	PROPN
iajs-2518	80	15	𝑋	𝑋	PROPN
iajs-2518	80	16	𝑜𝑟	𝑜𝑟	PRON
iajs-2518	80	17	𝐴	𝐴	PROPN
iajs-2518	80	18	0	0	PUNCT
iajs-2518	81	1	moreover	moreover	ADV
iajs-2518	81	2	,	,	PUNCT
iajs-2518	81	3	let	let	VERB
iajs-2518	81	4	𝐴	𝐴	PROPN
iajs-2518	81	5	𝑋	𝑋	PROPN
iajs-2518	81	6	,	,	PUNCT
iajs-2518	81	7	a	a	PRON
iajs-2518	81	8	is	be	AUX
iajs-2518	81	9	a	a	DET
iajs-2518	81	10	fsimple	fsimple	ADJ
iajs-2518	81	11	submodule	submodule	NOUN
iajs-2518	81	12	of	of	ADP
iajs-2518	81	13	x	x	PRON
iajs-2518	81	14	if	if	SCONJ
iajs-2518	81	15	a	a	PRON
iajs-2518	81	16	is	be	AUX
iajs-2518	81	17	a	a	DET
iajs-2518	81	18	f	f	NOUN
iajs-2518	81	19	simply	simply	ADV
iajs-2518	81	20	module	module	NOUN
iajs-2518	81	21	.	.	PUNCT
iajs-2518	82	1	remarks	remark	VERB
iajs-2518	82	2	2.2	2.2	NUM
iajs-2518	82	3	[	[	X
iajs-2518	82	4	3	3	NUM
iajs-2518	82	5	]	]	PUNCT
iajs-2518	82	6	if	if	SCONJ
iajs-2518	82	7	x	x	PRON
iajs-2518	82	8	is	be	AUX
iajs-2518	82	9	a	a	DET
iajs-2518	82	10	fmodule	fmodule	NOUN
iajs-2518	82	11	,	,	PUNCT
iajs-2518	82	12	then	then	ADV
iajs-2518	82	13	the	the	DET
iajs-2518	82	14	following	follow	VERB
iajs-2518	82	15	are	be	AUX
iajs-2518	82	16	held	hold	VERB
iajs-2518	82	17	:	:	PUNCT
iajs-2518	82	18	1	1	X
iajs-2518	82	19	)	)	PUNCT
iajs-2518	82	20	every	every	DET
iajs-2518	82	21	simple	simple	ADJ
iajs-2518	82	22	fmodule	fmodule	NOUN
iajs-2518	82	23	is	be	AUX
iajs-2518	82	24	f	f	NOUN
iajs-2518	82	25	_	_	NOUN
iajs-2518	82	26	regular	regular	ADJ
iajs-2518	82	27	fmodule	fmodule	NOUN
iajs-2518	82	28	,	,	PUNCT
iajs-2518	82	29	where	where	SCONJ
iajs-2518	82	30	is	be	AUX
iajs-2518	82	31	f	f	NOUN
iajs-2518	82	32	_	_	NOUN
iajs-2518	82	33	regular	regular	ADJ
iajs-2518	82	34	is	be	AUX
iajs-2518	82	35	every	every	DET
iajs-2518	82	36	f	f	PROPN
iajs-2518	82	37	submodule	submodule	NOUN
iajs-2518	82	38	of	of	ADP
iajs-2518	82	39	x	x	PUNCT
iajs-2518	82	40	is	be	AUX
iajs-2518	82	41	pure	pure	ADJ
iajs-2518	82	42	.	.	PUNCT
iajs-2518	83	1	2	2	X
iajs-2518	83	2	)	)	PUNCT
iajs-2518	83	3	if	if	SCONJ
iajs-2518	83	4	x	x	PRON
iajs-2518	83	5	is	be	AUX
iajs-2518	83	6	a	a	DET
iajs-2518	83	7	simple	simple	ADJ
iajs-2518	83	8	fmodule	fmodule	NOUN
iajs-2518	83	9	,	,	PUNCT
iajs-2518	83	10	then	then	ADV
iajs-2518	83	11	𝑋	𝑋	PROPN
iajs-2518	83	12	is	be	AUX
iajs-2518	83	13	a	a	DET
iajs-2518	83	14	simple	simple	ADJ
iajs-2518	83	15	module	module	NOUN
iajs-2518	83	16	,	,	PUNCT
iajs-2518	83	17	∀	∀	X
iajs-2518	83	18	t	t	NOUN
iajs-2518	83	19	0,1	0,1	NUM
iajs-2518	83	20	.	.	PUNCT
iajs-2518	84	1	3)if	3)if	NUM
iajs-2518	84	2	x	x	X
iajs-2518	84	3	is	be	AUX
iajs-2518	84	4	a	a	DET
iajs-2518	84	5	simple	simple	ADJ
iajs-2518	84	6	module	module	NOUN
iajs-2518	84	7	,	,	PUNCT
iajs-2518	84	8	∀	∀	X
iajs-2518	84	9	t	t	NOUN
iajs-2518	84	10	∈	∈	NOUN
iajs-2518	84	11	0,1	0,1	NUM
iajs-2518	84	12	,	,	PUNCT
iajs-2518	84	13	then	then	ADV
iajs-2518	84	14	is	be	AUX
iajs-2518	84	15	not	not	PART
iajs-2518	84	16	necessarily	necessarily	ADV
iajs-2518	84	17	that	that	PRON
iajs-2518	84	18	x	x	PRON
iajs-2518	84	19	is	be	AUX
iajs-2518	84	20	a	a	DET
iajs-2518	84	21	simple	simple	ADJ
iajs-2518	84	22	f	f	NOUN
iajs-2518	84	23	module	module	NOUN
iajs-2518	84	24	.	.	PUNCT
iajs-2518	84	25	"	"	PUNCT
iajs-2518	85	1	proposition	proposition	NOUN
iajs-2518	85	2	2.3	2.3	NUM
iajs-2518	86	1	[	[	X
iajs-2518	86	2	11	11	NUM
iajs-2518	86	3	]	]	PUNCT
iajs-2518	86	4	let	let	VERB
iajs-2518	86	5	x	x	PRON
iajs-2518	86	6	be	be	AUX
iajs-2518	86	7	a	a	DET
iajs-2518	86	8	fmodule	fmodule	NOUN
iajs-2518	86	9	of	of	ADP
iajs-2518	86	10	an	an	DET
iajs-2518	86	11	r	r	NOUN
iajs-2518	86	12	-	-	PUNCT
iajs-2518	86	13	module	module	NOUN
iajs-2518	86	14	m	m	NOUN
iajs-2518	86	15	and	and	CCONJ
iajs-2518	86	16	a	a	PRON
iajs-2518	86	17	be	be	AUX
iajs-2518	86	18	a	a	DET
iajs-2518	86	19	fsubmodule	fsubmodule	NOUN
iajs-2518	86	20	of	of	ADP
iajs-2518	86	21	x	x	PRON
iajs-2518	86	22	if	if	SCONJ
iajs-2518	86	23	a	a	PRON
iajs-2518	86	24	is	be	AUX
iajs-2518	86	25	simple	simple	ADJ
iajs-2518	86	26	,	,	PUNCT
iajs-2518	86	27	then	then	ADV
iajs-2518	86	28	a∗	a∗	PROPN
iajs-2518	86	29	is	be	AUX
iajs-2518	86	30	simple	simple	ADJ
iajs-2518	86	31	submodule	submodule	NOUN
iajs-2518	86	32	in	in	ADP
iajs-2518	86	33	x∗.	x∗.	PUNCT
iajs-2518	86	34	"	"	PUNCT
iajs-2518	86	35	remark	remark	NOUN
iajs-2518	86	36	2.4[11	2.4[11	NUM
iajs-2518	86	37	]	]	PUNCT
iajs-2518	86	38	if	if	SCONJ
iajs-2518	86	39	a∗	a∗	PROPN
iajs-2518	86	40	is	be	AUX
iajs-2518	86	41	a	a	DET
iajs-2518	86	42	simple	simple	ADJ
iajs-2518	86	43	submodule	submodule	NOUN
iajs-2518	86	44	in	in	ADP
iajs-2518	86	45	x∗	x∗	PROPN
iajs-2518	86	46	,	,	PUNCT
iajs-2518	86	47	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-2518	86	48	𝐴	𝐴	PROPN
iajs-2518	86	49	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	86	50	𝑛𝑜𝑡	𝑛𝑜𝑡	NOUN
iajs-2518	86	51	𝐹	𝐹	PROPN
iajs-2518	86	52	𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑖𝑚𝑝𝑙𝑒	ADJ
iajs-2518	86	53	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	86	54	"	"	PUNCT
iajs-2518	86	55	.	.	PUNCT
iajs-2518	87	1	𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧	𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧	NOUN
iajs-2518	87	2	𝟐.	𝟐.	X
iajs-2518	87	3	𝟓	𝟓	NUM
iajs-2518	87	4	[	[	X
iajs-2518	87	5	2	2	NUM
iajs-2518	87	6	]	]	PUNCT
iajs-2518	87	7	𝐴	𝐴	PROPN
iajs-2518	87	8	f	f	PROPN
iajs-2518	87	9	module	module	NOUN
iajs-2518	87	10	𝑋	𝑋	PROPN
iajs-2518	87	11	is	be	AUX
iajs-2518	87	12	called	call	VERB
iajs-2518	87	13	semisimple	semisimple	NOUN
iajs-2518	87	14	if	if	SCONJ
iajs-2518	87	15	𝑋	𝑋	PROPN
iajs-2518	87	16	is	be	AUX
iajs-2518	87	17	sum	sum	NOUN
iajs-2518	87	18	of	of	ADP
iajs-2518	87	19	simple	simple	ADJ
iajs-2518	87	20	f	f	PROPN
iajs-2518	87	21	submodule	submodule	NOUN
iajs-2518	87	22	of	of	ADP
iajs-2518	87	23	𝑋	𝑋	PROPN
iajs-2518	87	24	.	.	PUNCT
iajs-2518	88	1	next	next	ADV
iajs-2518	88	2	,	,	PUNCT
iajs-2518	88	3	we	we	PRON
iajs-2518	88	4	need	need	VERB
iajs-2518	88	5	the	the	DET
iajs-2518	88	6	following	follow	VERB
iajs-2518	88	7	lemma	lemma	PROPN
iajs-2518	88	8	:	:	PUNCT
iajs-2518	88	9	lemma	lemma	PROPN
iajs-2518	88	10	2.6[11	2.6[11	NUM
iajs-2518	88	11	]	]	PUNCT
iajs-2518	88	12	if	if	SCONJ
iajs-2518	88	13	x	x	PRON
iajs-2518	88	14	is	be	AUX
iajs-2518	88	15	a	a	DET
iajs-2518	88	16	fmodule	fmodule	NOUN
iajs-2518	88	17	of	of	ADP
iajs-2518	88	18	an	an	DET
iajs-2518	88	19	rmodule	rmodule	NOUN
iajs-2518	88	20	m	m	PROPN
iajs-2518	88	21	and	and	CCONJ
iajs-2518	88	22	a	a	PRON
iajs-2518	88	23	is	be	AUX
iajs-2518	88	24	a	a	DET
iajs-2518	88	25	f	f	NOUN
iajs-2518	88	26	-direct	-direct	NOUN
iajs-2518	88	27	summand	summand	NOUN
iajs-2518	88	28	in	in	ADP
iajs-2518	88	29	x	x	SYM
iajs-2518	88	30	,	,	PUNCT
iajs-2518	88	31	then	then	ADV
iajs-2518	88	32	𝐴∗	𝐴∗	NUM
iajs-2518	88	33	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	88	34	𝑎	𝑎	DET
iajs-2518	88	35	direct	direct	ADJ
iajs-2518	88	36	summand	summand	NOUN
iajs-2518	88	37	in	in	ADP
iajs-2518	88	38	𝑋∗.	𝑋∗.	PROPN
iajs-2518	88	39	proposition	proposition	NOUN
iajs-2518	88	40	2.7[11	2.7[11	NUM
iajs-2518	88	41	]	]	PUNCT
iajs-2518	88	42	if	if	SCONJ
iajs-2518	88	43	𝑋	𝑋	PROPN
iajs-2518	88	44	is	be	AUX
iajs-2518	88	45	a	a	DET
iajs-2518	88	46	f	f	PROPN
iajs-2518	88	47	semisimple	semisimple	NOUN
iajs-2518	88	48	module	module	NOUN
iajs-2518	88	49	,	,	PUNCT
iajs-2518	88	50	then	then	ADV
iajs-2518	88	51	x∗	x∗	PROPN
iajs-2518	88	52	is	be	AUX
iajs-2518	88	53	a	a	DET
iajs-2518	88	54	semisimple	semisimple	NOUN
iajs-2518	88	55	module	module	NOUN
iajs-2518	88	56	.	.	PUNCT
iajs-2518	89	1	proposition	proposition	NOUN
iajs-2518	89	2	2.8[11	2.8[11	NUM
iajs-2518	89	3	]	]	X
iajs-2518	89	4	if	if	SCONJ
iajs-2518	89	5	x∗	x∗	PROPN
iajs-2518	89	6	is	be	AUX
iajs-2518	89	7	semisimple	semisimple	NOUN
iajs-2518	89	8	,	,	PUNCT
iajs-2518	89	9	then	then	ADV
iajs-2518	89	10	x	x	PUNCT
iajs-2518	89	11	semisimple	semisimple	NOUN
iajs-2518	89	12	when	when	SCONJ
iajs-2518	89	13	condition	condition	NOUN
iajs-2518	89	14	∗	∗	NOUN
iajs-2518	89	15	hold	hold	VERB
iajs-2518	89	16	.	.	PUNCT
iajs-2518	89	17	"	"	PUNCT
iajs-2518	89	18	  	  	SPACE
iajs-2518	90	1	141	141	NUM
iajs-2518	90	2	  	  	SPACE
iajs-2518	90	3	ibn	ibn	PROPN
iajs-2518	90	4	al	al	PROPN
iajs-2518	90	5	-	-	PUNCT
iajs-2518	90	6	haitham	haitham	PROPN
iajs-2518	90	7	jour	jour	X
iajs-2518	90	8	.	.	PROPN
iajs-2518	90	9	for	for	ADP
iajs-2518	90	10	pure	pure	ADJ
iajs-2518	90	11	&	&	CCONJ
iajs-2518	90	12	appl	appl	PROPN
iajs-2518	90	13	.	.	PUNCT
iajs-2518	91	1	sci	sci	PROPN
iajs-2518	91	2	.	.	PROPN
iajs-2518	92	1	33	33	NUM
iajs-2518	92	2	(	(	PUNCT
iajs-2518	92	3	4	4	NUM
iajs-2518	92	4	)	)	PUNCT
iajs-2518	92	5	2020	2020	NUM
iajs-2518	93	1	corollary	corollary	NOUN
iajs-2518	93	2	2.9[11	2.9[11	NUM
iajs-2518	93	3	]	]	X
iajs-2518	93	4	any	any	DET
iajs-2518	93	5	f	f	PROPN
iajs-2518	93	6	submodule	submodule	NOUN
iajs-2518	93	7	of	of	ADP
iajs-2518	93	8	semisimple	semisimple	PROPN
iajs-2518	93	9	fuzzy	fuzzy	ADJ
iajs-2518	93	10	submodule	submodule	NOUN
iajs-2518	93	11	is	be	AUX
iajs-2518	93	12	semisimple	semisimple	NOUN
iajs-2518	93	13	.	.	PUNCT
iajs-2518	94	1	proposition	proposition	NOUN
iajs-2518	94	2	2.10[11	2.10[11	NUM
iajs-2518	94	3	]	]	X
iajs-2518	94	4	ift	ift	NOUN
iajs-2518	94	5	x	x	PUNCT
iajs-2518	94	6	is	be	AUX
iajs-2518	94	7	a	a	DET
iajs-2518	94	8	fmodule	fmodule	NOUN
iajs-2518	94	9	of	of	ADP
iajs-2518	94	10	an	an	DET
iajs-2518	94	11	r	r	NOUN
iajs-2518	94	12	-	-	PUNCT
iajs-2518	94	13	module	module	NOUN
iajs-2518	94	14	m	m	NOUN
iajs-2518	94	15	,	,	PUNCT
iajs-2518	94	16	where	where	SCONJ
iajs-2518	94	17	condition	condition	NOUN
iajs-2518	94	18	(	(	PUNCT
iajs-2518	94	19	*	*	NOUN
iajs-2518	94	20	)	)	PUNCT
iajs-2518	94	21	holds	hold	VERB
iajs-2518	94	22	then	then	ADV
iajs-2518	94	23	,	,	PUNCT
iajs-2518	94	24	the	the	DET
iajs-2518	94	25	following	follow	VERB
iajs-2518	94	26	statements	statement	NOUN
iajs-2518	94	27	are	be	AUX
iajs-2518	94	28	equivalent	equivalent	ADJ
iajs-2518	94	29	:	:	PUNCT
iajs-2518	94	30	1	1	NUM
iajs-2518	94	31	x	x	NOUN
iajs-2518	94	32	is	be	AUX
iajs-2518	94	33	semisimple	semisimple	ADJ
iajs-2518	94	34	2	2	NUM
iajs-2518	94	35	x	x	PUNCT
iajs-2518	94	36	has	have	AUX
iajs-2518	94	37	no	no	DET
iajs-2518	94	38	proper	proper	ADJ
iajs-2518	94	39	essential	essential	ADJ
iajs-2518	94	40	f	f	PROPN
iajs-2518	94	41	submodule	submodule	NOUN
iajs-2518	94	42	.	.	PUNCT
iajs-2518	95	1	3	3	NUM
iajs-2518	95	2	every	every	DET
iajs-2518	95	3	f	f	PROPN
iajs-2518	95	4	submodule	submodule	NOUN
iajs-2518	95	5	of	of	ADP
iajs-2518	95	6	x	x	SYM
iajs-2518	95	7	is	be	AUX
iajs-2518	95	8	a	a	DET
iajs-2518	95	9	direct	direct	ADJ
iajs-2518	95	10	summand	summand	NOUN
iajs-2518	95	11	of	of	ADP
iajs-2518	95	12	x	x	PUNCT
iajs-2518	95	13	"	"	PUNCT
iajs-2518	95	14	.	.	PUNCT
iajs-2518	96	1	proposition	proposition	NOUN
iajs-2518	96	2	2.11	2.11	NUM
iajs-2518	96	3	[	[	X
iajs-2518	96	4	11	11	NUM
iajs-2518	96	5	]	]	PUNCT
iajs-2518	96	6	if	if	SCONJ
iajs-2518	96	7	x	x	PRON
iajs-2518	96	8	is	be	AUX
iajs-2518	96	9	a	a	DET
iajs-2518	96	10	fmodule	fmodule	NOUN
iajs-2518	96	11	of	of	ADP
iajs-2518	96	12	an	an	DET
iajs-2518	96	13	r	r	NOUN
iajs-2518	96	14	–	–	PUNCT
iajs-2518	96	15	module	module	NOUN
iajs-2518	96	16	m	m	NOUN
iajs-2518	96	17	.	.	PUNCT
iajs-2518	97	1	then	then	ADV
iajs-2518	97	2	,	,	PUNCT
iajs-2518	97	3	the	the	DET
iajs-2518	97	4	following	follow	VERB
iajs-2518	97	5	are	be	AUX
iajs-2518	97	6	equivalent	equivalent	ADJ
iajs-2518	97	7	:	:	PUNCT
iajs-2518	97	8	1	1	X
iajs-2518	97	9	.	.	X
iajs-2518	98	1	every	every	DET
iajs-2518	98	2	fuzzy	fuzzy	ADJ
iajs-2518	98	3	submodule	submodule	NOUN
iajs-2518	98	4	of	of	ADP
iajs-2518	98	5	x	x	PUNCT
iajs-2518	98	6	is	be	AUX
iajs-2518	98	7	sum	sum	NOUN
iajs-2518	98	8	of	of	ADP
iajs-2518	98	9	fuzzy	fuzzy	ADJ
iajs-2518	98	10	simple	simple	ADJ
iajs-2518	98	11	submodules	submodule	NOUN
iajs-2518	98	12	.	.	PUNCT
iajs-2518	99	1	2	2	X
iajs-2518	99	2	.	.	X
iajs-2518	99	3	x	x	PRON
iajs-2518	99	4	is	be	AUX
iajs-2518	99	5	a	a	DET
iajs-2518	99	6	direct	direct	ADJ
iajs-2518	99	7	sum	sum	NOUN
iajs-2518	99	8	of	of	ADP
iajs-2518	99	9	fuzzy	fuzzy	ADJ
iajs-2518	99	10	simple	simple	ADJ
iajs-2518	99	11	submodules	submodule	NOUN
iajs-2518	99	12	of	of	ADP
iajs-2518	99	13	x	x	PROPN
iajs-2518	99	14	.	.	PUNCT
iajs-2518	100	1	3	3	X
iajs-2518	100	2	.	.	X
iajs-2518	100	3	every	every	DET
iajs-2518	100	4	fuzzy	fuzzy	ADJ
iajs-2518	100	5	submodule	submodule	NOUN
iajs-2518	100	6	of	of	ADP
iajs-2518	100	7	x	x	SYM
iajs-2518	100	8	is	be	AUX
iajs-2518	100	9	a	a	DET
iajs-2518	100	10	direct	direct	ADJ
iajs-2518	100	11	summand	summand	NOUN
iajs-2518	100	12	of	of	ADP
iajs-2518	100	13	x	x	SYM
iajs-2518	100	14	proof	proof	NOUN
iajs-2518	100	15	:	:	PUNCT
iajs-2518	100	16	it	it	PRON
iajs-2518	100	17	is	be	AUX
iajs-2518	100	18	easy	easy	ADJ
iajs-2518	100	19	.	.	PUNCT
iajs-2518	101	1	proposition	proposition	NOUN
iajs-2518	101	2	2.12	2.12	NUM
iajs-2518	102	1	[	[	X
iajs-2518	102	2	11	11	NUM
iajs-2518	102	3	]	]	PUNCT
iajs-2518	102	4	the	the	DET
iajs-2518	102	5	following	follow	VERB
iajs-2518	102	6	are	be	AUX
iajs-2518	102	7	equivalent	equivalent	ADJ
iajs-2518	102	8	:	:	PUNCT
iajs-2518	102	9	1	1	X
iajs-2518	102	10	.	.	X
iajs-2518	103	1	every	every	DET
iajs-2518	103	2	fsubmodle	fsubmodle	NOUN
iajs-2518	103	3	of	of	ADP
iajs-2518	103	4	semisimple	semisimple	PROPN
iajs-2518	103	5	f	f	PROPN
iajs-2518	103	6	module	module	NOUN
iajs-2518	103	7	is	be	AUX
iajs-2518	103	8	semisimple	semisimple	ADJ
iajs-2518	103	9	.	.	PUNCT
iajs-2518	104	1	2	2	X
iajs-2518	104	2	.	.	X
iajs-2518	104	3	every	every	DET
iajs-2518	104	4	epimorphic	epimorphic	ADJ
iajs-2518	104	5	image	image	NOUN
iajs-2518	104	6	of	of	ADP
iajs-2518	104	7	semisimple	semisimple	NOUN
iajs-2518	104	8	f	f	PROPN
iajs-2518	104	9	module	module	NOUN
iajs-2518	104	10	is	be	AUX
iajs-2518	104	11	semisimple	semisimple	NOUN
iajs-2518	104	12	.	.	PUNCT
iajs-2518	105	1	3	3	X
iajs-2518	105	2	.	.	X
iajs-2518	105	3	every	every	DET
iajs-2518	105	4	sum	sum	NOUN
iajs-2518	105	5	of	of	ADP
iajs-2518	105	6	semisimple	semisimple	NOUN
iajs-2518	105	7	f	f	PROPN
iajs-2518	105	8	modules	module	NOUN
iajs-2518	105	9	is	be	AUX
iajs-2518	105	10	semisimple	semisimple	ADJ
iajs-2518	105	11	.	.	PUNCT
iajs-2518	106	1	proof	proof	NOUN
iajs-2518	106	2	:	:	PUNCT
iajs-2518	106	3	it	it	PRON
iajs-2518	106	4	is	be	AUX
iajs-2518	106	5	easy	easy	ADJ
iajs-2518	106	6	.	.	PUNCT
iajs-2518	107	1	3	3	X
iajs-2518	107	2	.	.	X
iajs-2518	107	3	fuzzy	fuzzy	ADJ
iajs-2518	107	4	semimaximal	semimaximal	ADJ
iajs-2518	107	5	submodules	submodule	NOUN
iajs-2518	107	6	definition	definition	NOUN
iajs-2518	107	7	3	3	NUM
iajs-2518	107	8	.1	.1	NUM
iajs-2518	107	9	if	if	SCONJ
iajs-2518	107	10	a	a	PRON
iajs-2518	107	11	is	be	AUX
iajs-2518	107	12	a	a	DET
iajs-2518	107	13	f	f	PROPN
iajs-2518	107	14	submodule	submodule	NOUN
iajs-2518	107	15	of	of	ADP
iajs-2518	107	16	f	f	PROPN
iajs-2518	107	17	module	module	NOUN
iajs-2518	107	18	x	x	SYM
iajs-2518	107	19	,	,	PUNCT
iajs-2518	107	20	then	then	ADV
iajs-2518	107	21	a	a	PRON
iajs-2518	107	22	is	be	AUX
iajs-2518	107	23	called	call	VERB
iajs-2518	107	24	semimaximal	semimaximal	ADJ
iajs-2518	107	25	if	if	SCONJ
iajs-2518	108	1	and	and	CCONJ
iajs-2518	108	2	only	only	ADV
iajs-2518	108	3	if	if	SCONJ
iajs-2518	108	4	𝑋	𝑋	PROPN
iajs-2518	108	5	𝐴⁄	𝐴⁄	PROPN
iajs-2518	108	6	is	be	AUX
iajs-2518	108	7	a	a	DET
iajs-2518	108	8	semisimple	semisimple	NOUN
iajs-2518	108	9	fmodule	fmodule	ADJ
iajs-2518	108	10	.	.	PUNCT
iajs-2518	109	1	proposition	proposition	NOUN
iajs-2518	109	2	3	3	NUM
iajs-2518	109	3	.	.	NOUN
iajs-2518	109	4	2	2	NUM
iajs-2518	109	5	if	if	SCONJ
iajs-2518	109	6	a	a	PRON
iajs-2518	109	7	is	be	AUX
iajs-2518	109	8	a	a	DET
iajs-2518	109	9	semimaximal	semimaximal	ADJ
iajs-2518	109	10	fuzzy	fuzzy	ADJ
iajs-2518	109	11	submodule	submodule	NOUN
iajs-2518	109	12	of	of	ADP
iajs-2518	109	13	fuzzy	fuzzy	ADJ
iajs-2518	109	14	module	module	NOUN
iajs-2518	109	15	x	x	SYM
iajs-2518	109	16	,	,	PUNCT
iajs-2518	109	17	then	then	ADV
iajs-2518	109	18	𝐴∗	𝐴∗	PROPN
iajs-2518	109	19	is	be	AUX
iajs-2518	109	20	a	a	DET
iajs-2518	109	21	semimaximal	semimaximal	ADJ
iajs-2518	109	22	submodule	submodule	NOUN
iajs-2518	109	23	of	of	ADP
iajs-2518	109	24	𝑋∗	𝑋∗	ADJ
iajs-2518	109	25	.	.	PUNCT
iajs-2518	110	1	proof	proof	NOUN
iajs-2518	110	2	:	:	PUNCT
iajs-2518	110	3	since	since	SCONJ
iajs-2518	110	4	a	a	PRON
iajs-2518	110	5	is	be	AUX
iajs-2518	110	6	semimaxmal	semimaxmal	PROPN
iajs-2518	110	7	f	f	PROPN
iajs-2518	110	8	submodule	submodule	NOUN
iajs-2518	110	9	,	,	PUNCT
iajs-2518	110	10	so	so	CCONJ
iajs-2518	110	11	x	x	X
iajs-2518	110	12	/a	/a	PUNCT
iajs-2518	110	13	is	be	AUX
iajs-2518	110	14	semisimple	semisimple	ADJ
iajs-2518	110	15	.	.	PUNCT
iajs-2518	111	1	hence	hence	ADV
iajs-2518	111	2	,	,	PUNCT
iajs-2518	111	3			ADJ
iajs-2518	111	4	∈^	∈^	PROPN
iajs-2518	111	5	𝐶	𝐶	PROPN
iajs-2518	111	6	|	|	NOUN
iajs-2518	111	7	𝐴	𝐴	PROPN
iajs-2518	111	8	,	,	PUNCT
iajs-2518	111	9	where	where	SCONJ
iajs-2518	111	10	𝐶	𝐶	PROPN
iajs-2518	111	11	|	|	NOUN
iajs-2518	111	12	𝐴	𝐴	PROPN
iajs-2518	111	13	is	be	AUX
iajs-2518	111	14	simple	simple	ADJ
iajs-2518	111	15	f	f	NOUN
iajs-2518	111	16	submodules	submodule	NOUN
iajs-2518	111	17	∀	∀	X
iajs-2518	111	18	𝑖	𝑖	SYM
iajs-2518	111	19	∈	∈	PROPN
iajs-2518	111	20	^	^	PUNCT
iajs-2518	111	21	which	which	PRON
iajs-2518	111	22	implies	imply	VERB
iajs-2518	111	23	(	(	PUNCT
iajs-2518	111	24	xla	xla	PROPN
iajs-2518	111	25	∗	∗	X
iajs-2518	111	26			ADJ
iajs-2518	111	27	∈^	∈^	PROPN
iajs-2518	111	28	𝐶𝑖|𝐴	𝐶𝑖|𝐴	PRON
iajs-2518	111	29	∗	∗	NOUN
iajs-2518	111	30			ADJ
iajs-2518	111	31	∈^	∈^	PROPN
iajs-2518	111	32	𝐶𝑖	𝐶𝑖	PROPN
iajs-2518	111	33	|𝐴	|𝐴	ADV
iajs-2518	111	34	∗	∗	NOUN
iajs-2518	111	35	𝐵𝑢𝑡	𝐵𝑢𝑡	PROPN
iajs-2518	111	36	𝐶	𝐶	PROPN
iajs-2518	111	37	|	|	NOUN
iajs-2518	111	38	𝐴	𝐴	PROPN
iajs-2518	111	39	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	111	40	𝐹	𝐹	PROPN
iajs-2518	111	41	𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	111	42	,	,	PUNCT
iajs-2518	111	43	𝑖𝑚𝑝𝑙𝑖𝑒𝑠	𝑖𝑚𝑝𝑙𝑖𝑒𝑠	NOUN
iajs-2518	111	44	𝐶𝑖|𝐴	𝐶𝑖|𝐴	PRON
iajs-2518	112	1	∗	∗	NOUN
iajs-2518	112	2	𝑖𝑠	𝑖𝑠	INTJ
iajs-2518	112	3	𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	112	4	,	,	PUNCT
iajs-2518	112	5	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	112	6	𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜	𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜	PROPN
iajs-2518	112	7	2	2	NUM
iajs-2518	112	8	.3	.3	NUM
iajs-2518	112	9	∀	∀	NOUN
iajs-2518	112	10	𝑖	𝑖	SYM
iajs-2518	112	11	∈	∈	PROPN
iajs-2518	112	12	^	^	PUNCT
iajs-2518	112	13	.hence	.hence	NOUN
iajs-2518	112	14	,	,	PUNCT
iajs-2518	112	15	(	(	PUNCT
iajs-2518	112	16	𝑋𝑙𝐴	𝑋𝑙𝐴	NOUN
iajs-2518	112	17	∗	∗	NOUN
iajs-2518	112	18	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	112	19	𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	112	20	.	.	PUNCT
iajs-2518	113	1	𝐴𝑠	𝐴𝑠	PROPN
iajs-2518	113	2	𝑋∗|𝐴∗	𝑋∗|𝐴∗	NOUN
iajs-2518	113	3	(	(	PUNCT
iajs-2518	113	4	𝑋𝑙𝐴	𝑋𝑙𝐴	NOUN
iajs-2518	113	5	∗	∗	NOUN
iajs-2518	113	6	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	113	7	𝐿𝑒𝑚𝑚𝑎	𝐿𝑒𝑚𝑚𝑎	PROPN
iajs-2518	113	8	1.16	1.16	NUM
iajs-2518	113	9	.	.	PUNCT
iajs-2518	114	1	𝑇ℎ𝑒𝑛	𝑇ℎ𝑒𝑛	NOUN
iajs-2518	114	2	,	,	PUNCT
iajs-2518	114	3	𝑋∗|𝐴∗	𝑋∗|𝐴∗	PRON
iajs-2518	114	4	𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒.	𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒.	VERB
iajs-2518	114	5	𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒	𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒	PROPN
iajs-2518	114	6	𝐴∗	𝐴∗	PROPN
iajs-2518	114	7	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	114	8	semimaxmal	semimaxmal	PROPN
iajs-2518	114	9	submodule	submodule	NOUN
iajs-2518	114	10	of	of	ADP
iajs-2518	114	11	𝑋∗	𝑋∗	ADJ
iajs-2518	114	12	proposition	proposition	NOUN
iajs-2518	114	13	3	3	NUM
iajs-2518	114	14	.	.	NOUN
iajs-2518	114	15	3	3	NUM
iajs-2518	114	16	let	let	VERB
iajs-2518	114	17	a	a	PRON
iajs-2518	114	18	be	be	AUX
iajs-2518	114	19	f	f	PROPN
iajs-2518	114	20	submodule	submodule	NOUN
iajs-2518	114	21	of	of	ADP
iajs-2518	114	22	f	f	PROPN
iajs-2518	114	23	module	module	NOUN
iajs-2518	114	24	x	x	PUNCT
iajs-2518	114	25	which	which	PRON
iajs-2518	114	26	satisfies	satisfy	VERB
iajs-2518	114	27	x	x	X
iajs-2518	114	28	/	/	SYM
iajs-2518	114	29	a	a	DET
iajs-2518	114	30	∗	∗	NOUN
iajs-2518	114	31	𝑋∗	𝑋∗	NOUN
iajs-2518	114	32	𝐴∗⁄	𝐴∗⁄	NOUN
iajs-2518	114	33	.	.	PUNCT
iajs-2518	115	1	if	if	SCONJ
iajs-2518	115	2	𝐴∗	𝐴∗	PROPN
iajs-2518	115	3	is	be	AUX
iajs-2518	115	4	a	a	DET
iajs-2518	115	5	semimaximal	semimaximal	ADJ
iajs-2518	115	6	submodule	submodule	NOUN
iajs-2518	115	7	of	of	ADP
iajs-2518	115	8	𝑋∗	𝑋∗	ADJ
iajs-2518	115	9	,	,	PUNCT
iajs-2518	115	10	then	then	ADV
iajs-2518	115	11	a	a	PRON
iajs-2518	115	12	is	be	AUX
iajs-2518	115	13	a	a	DET
iajs-2518	115	14	f	f	PROPN
iajs-2518	115	15	semimaximal	semimaximal	NOUN
iajs-2518	115	16	submodule	submodule	NOUN
iajs-2518	115	17	of	of	ADP
iajs-2518	115	18	x	x	PROPN
iajs-2518	115	19	.	.	PUNCT
iajs-2518	116	1	proof	proof	NOUN
iajs-2518	116	2	:	:	PUNCT
iajs-2518	116	3	since	since	SCONJ
iajs-2518	116	4	𝐴∗	𝐴∗	PROPN
iajs-2518	116	5	is	be	AUX
iajs-2518	116	6	a	a	DET
iajs-2518	116	7	semimaxmal	semimaxmal	ADJ
iajs-2518	116	8	submodule	submodule	NOUN
iajs-2518	116	9	of	of	ADP
iajs-2518	116	10	𝑋∗	𝑋∗	ADJ
iajs-2518	116	11	ℎ𝑒𝑛𝑐𝑒	ℎ𝑒𝑛𝑐𝑒	NOUN
iajs-2518	116	12	,	,	PUNCT
iajs-2518	116	13	𝑋∗	𝑋∗	ADJ
iajs-2518	116	14	𝐴∗⁄	𝐴∗⁄	ADV
iajs-2518	116	15	𝑖𝑠	𝑖𝑠	INTJ
iajs-2518	116	16	semisimple	semisimple	NOUN
iajs-2518	116	17	hence	hence	ADV
iajs-2518	116	18	,	,	PUNCT
iajs-2518	116	19	𝑋	𝑋	PROPN
iajs-2518	116	20	𝐴⁄	𝐴⁄	PROPN
iajs-2518	116	21	is	be	AUX
iajs-2518	116	22	semisimple	semisimple	ADJ
iajs-2518	116	23	f	f	PROPN
iajs-2518	116	24	module	module	NOUN
iajs-2518	116	25	.	.	PUNCT
iajs-2518	117	1	therefore	therefore	ADV
iajs-2518	117	2	𝐴	𝐴	PROPN
iajs-2518	117	3	is	be	AUX
iajs-2518	117	4	a	a	DET
iajs-2518	117	5	f	f	PROPN
iajs-2518	117	6	semimaximal	semimaximal	NOUN
iajs-2518	117	7	submodule	submodule	NOUN
iajs-2518	117	8	of	of	ADP
iajs-2518	117	9	x	x	PROPN
iajs-2518	117	10	.	.	PUNCT
iajs-2518	118	1	remarks	remark	NOUN
iajs-2518	118	2	and	and	CCONJ
iajs-2518	118	3	examples	example	NOUN
iajs-2518	118	4	3.4	3.4	NUM
iajs-2518	118	5	1	1	NUM
iajs-2518	118	6	)	)	PUNCT
iajs-2518	118	7	every	every	DET
iajs-2518	118	8	f	f	PROPN
iajs-2518	118	9	maximal	maximal	ADJ
iajs-2518	118	10	submodule	submodule	NOUN
iajs-2518	118	11	of	of	ADP
iajs-2518	118	12	fuzzy	fuzzy	ADJ
iajs-2518	118	13	module	module	NOUN
iajs-2518	118	14	is	be	AUX
iajs-2518	118	15	a	a	DET
iajs-2518	118	16	f	f	X
iajs-2518	118	17	semimaxmal	semimaxmal	PROPN
iajs-2518	118	18	submodule	submodule	NOUN
iajs-2518	118	19	.	.	PUNCT
iajs-2518	118	20	  	  	SPACE
iajs-2518	119	1	142	142	NUM
iajs-2518	119	2	  	  	SPACE
iajs-2518	119	3	ibn	ibn	PROPN
iajs-2518	119	4	al	al	PROPN
iajs-2518	119	5	-	-	PUNCT
iajs-2518	119	6	haitham	haitham	PROPN
iajs-2518	119	7	jour	jour	X
iajs-2518	119	8	.	.	PROPN
iajs-2518	119	9	for	for	ADP
iajs-2518	119	10	pure	pure	ADJ
iajs-2518	119	11	&	&	CCONJ
iajs-2518	119	12	appl	appl	PROPN
iajs-2518	119	13	.	.	PUNCT
iajs-2518	120	1	sci	sci	PROPN
iajs-2518	120	2	.	.	PROPN
iajs-2518	121	1	33	33	NUM
iajs-2518	121	2	(	(	PUNCT
iajs-2518	121	3	4	4	NUM
iajs-2518	121	4	)	)	PUNCT
iajs-2518	121	5	2020	2020	NUM
iajs-2518	122	1	proof	proof	NOUN
iajs-2518	122	2	:	:	PUNCT
iajs-2518	122	3	a	a	PRON
iajs-2518	122	4	is	be	AUX
iajs-2518	122	5	a	a	DET
iajs-2518	122	6	fmaximal	fmaximal	ADJ
iajs-2518	122	7	submodule	submodule	NOUN
iajs-2518	122	8	of	of	ADP
iajs-2518	122	9	fmodule	fmodule	ADJ
iajs-2518	122	10	x	x	NOUN
iajs-2518	122	11	.then	.then	X
iajs-2518	122	12	x	x	X
iajs-2518	122	13	/a	/a	PUNCT
iajs-2518	122	14	is	be	AUX
iajs-2518	122	15	simple	simple	ADJ
iajs-2518	122	16	,	,	PUNCT
iajs-2518	122	17	and	and	CCONJ
iajs-2518	122	18	so	so	ADV
iajs-2518	122	19	x	x	X
iajs-2518	122	20	/a	/a	PUNCT
iajs-2518	122	21	is	be	AUX
iajs-2518	122	22	a	a	DET
iajs-2518	122	23	semisimple	semisimple	NOUN
iajs-2518	122	24	.hence	.hence	NOUN
iajs-2518	122	25	,	,	PUNCT
iajs-2518	122	26	a	a	PRON
iajs-2518	122	27	is	be	AUX
iajs-2518	122	28	semimaximal	semimaximal	ADJ
iajs-2518	122	29	.	.	PUNCT
iajs-2518	123	1	the	the	DET
iajs-2518	123	2	converse	converse	NOUN
iajs-2518	123	3	is	be	AUX
iajs-2518	123	4	not	not	PART
iajs-2518	123	5	true	true	ADJ
iajs-2518	123	6	in	in	ADP
iajs-2518	123	7	general	general	ADJ
iajs-2518	123	8	see	see	VERB
iajs-2518	123	9	the	the	DET
iajs-2518	123	10	following	following	ADJ
iajs-2518	123	11	example	example	NOUN
iajs-2518	123	12	:	:	PUNCT
iajs-2518	123	13	example	example	NOUN
iajs-2518	123	14	:	:	PUNCT
iajs-2518	123	15	let	let	VERB
iajs-2518	123	16	𝑀	𝑀	PROPN
iajs-2518	123	17	𝑍	𝑍	VERB
iajs-2518	123	18	𝑎𝑠	𝑎𝑠	NOUN
iajs-2518	123	19	𝑍	𝑍	NOUN
iajs-2518	123	20	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	123	21	.	.	PUNCT
iajs-2518	124	1	𝐷𝑒𝑓𝑖𝑛𝑒	𝐷𝑒𝑓𝑖𝑛𝑒	PROPN
iajs-2518	124	2	𝑋	𝑋	PROPN
iajs-2518	124	3	𝑥	𝑥	PROPN
iajs-2518	124	4	1	1	NUM
iajs-2518	124	5	,	,	PUNCT
iajs-2518	124	6	∀	∀	VERB
iajs-2518	124	7	𝑥	𝑥	PRON
iajs-2518	124	8	∈	∈	NOUN
iajs-2518	124	9	𝑀	𝑀	PROPN
iajs-2518	124	10	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
iajs-2518	124	11	𝐴	𝐴	PROPN
iajs-2518	124	12	𝑋	𝑋	PROPN
iajs-2518	124	13	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2518	124	14	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
iajs-2518	124	15	𝐴	𝐴	PROPN
iajs-2518	125	1	𝑥	𝑥	NOUN
iajs-2518	126	1	1	1	NUM
iajs-2518	126	2	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	126	3	𝑥	𝑥	PRON
iajs-2518	126	4	∈	∈	PROPN
iajs-2518	126	5	6𝑍	6𝑍	NOUN
iajs-2518	126	6	,	,	PUNCT
iajs-2518	126	7	0	0	NUM
iajs-2518	126	8	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2518	126	9	thus	thus	ADV
iajs-2518	126	10	𝑋∗	𝑋∗	ADV
iajs-2518	127	1	𝐴∗⁄	𝐴∗⁄	ADV
iajs-2518	127	2	𝑀	𝑀	NOUN
iajs-2518	127	3	𝐴∗⁄	𝐴∗⁄	PRON
iajs-2518	127	4	𝑍	𝑍	VERB
iajs-2518	127	5	6𝑍⁄	6𝑍⁄	NUM
iajs-2518	127	6	𝑍	𝑍	NOUN
iajs-2518	127	7	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	127	8	semisemple	semisemple	NOUN
iajs-2518	127	9	and	and	CCONJ
iajs-2518	127	10	a	a	PRON
iajs-2518	127	11	is	be	AUX
iajs-2518	127	12	semimaximal	semimaximal	ADJ
iajs-2518	127	13	.	.	PUNCT
iajs-2518	128	1	but	but	CCONJ
iajs-2518	128	2	𝐴∗	𝐴∗	NUM
iajs-2518	128	3	is	be	AUX
iajs-2518	128	4	not	not	PART
iajs-2518	128	5	a	a	DET
iajs-2518	128	6	maximal	maximal	ADJ
iajs-2518	128	7	𝑖𝑛	𝑖𝑛	NOUN
iajs-2518	128	8	𝑋∗	𝑋∗	ADJ
iajs-2518	128	9	.	.	PUNCT
iajs-2518	129	1	thus	thus	ADV
iajs-2518	129	2	a	a	PRON
iajs-2518	129	3	is	be	AUX
iajs-2518	129	4	not	not	PART
iajs-2518	129	5	maximal	maximal	ADJ
iajs-2518	129	6	f	f	X
iajs-2518	129	7	submodule	submodule	NOUN
iajs-2518	129	8	by	by	ADP
iajs-2518	129	9	prop	prop	PROPN
iajs-2518	129	10	3.3.hence	3.3.hence	PROPN
iajs-2518	129	11	x/𝐴	x/𝐴	PROPN
iajs-2518	129	12	∗	∗	PROPN
iajs-2518	129	13	m	m	PROPN
iajs-2518	129	14	/	/	SYM
iajs-2518	129	15	𝐴∗	𝐴∗	NUM
iajs-2518	129	16	𝑍	𝑍	NOUN
iajs-2518	129	17	is	be	AUX
iajs-2518	129	18	semisemple	semisemple	NOUN
iajs-2518	129	19	,	,	PUNCT
iajs-2518	129	20	and	and	CCONJ
iajs-2518	129	21	s𝑜	s𝑜	NOUN
iajs-2518	129	22	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	129	23	𝑃𝑟𝑜𝑝	𝑃𝑟𝑜𝑝	PROPN
iajs-2518	129	24	2.8	2.8	NUM
iajs-2518	129	25	,	,	PUNCT
iajs-2518	129	26	x/𝐴	x/𝐴	PROPN
iajs-2518	129	27	is	be	AUX
iajs-2518	129	28	semisemple	semisemple	ADJ
iajs-2518	129	29	;	;	PUNCT
iajs-2518	129	30	that	that	PRON
iajs-2518	129	31	is	be	AUX
iajs-2518	129	32	𝐴	𝐴	PROPN
iajs-2518	129	33	is	be	AUX
iajs-2518	129	34	a	a	DET
iajs-2518	129	35	f	f	X
iajs-2518	129	36	semimaxmal	semimaxmal	PROPN
iajs-2518	129	37	submodule	submodule	NOUN
iajs-2518	129	38	2	2	NUM
iajs-2518	129	39	)	)	PUNCT
iajs-2518	129	40	a	a	DET
iajs-2518	129	41	f	f	PROPN
iajs-2518	129	42	submodule	submodule	NOUN
iajs-2518	129	43	of	of	ADP
iajs-2518	129	44	semimaximal	semimaximal	ADJ
iajs-2518	129	45	f	f	NOUN
iajs-2518	129	46	-	-	PUNCT
iajs-2518	129	47	module	module	NOUN
iajs-2518	129	48	need	need	AUX
iajs-2518	129	49	not	not	PART
iajs-2518	129	50	to	to	PART
iajs-2518	129	51	be	be	AUX
iajs-2518	129	52	semimaximal	semimaximal	ADJ
iajs-2518	129	53	.	.	PUNCT
iajs-2518	130	1	for	for	ADP
iajs-2518	130	2	example	example	NOUN
iajs-2518	130	3	:	:	PUNCT
iajs-2518	130	4	let	let	VERB
iajs-2518	130	5	a	a	DET
iajs-2518	130	6	,	,	PUNCT
iajs-2518	130	7	b	b	NOUN
iajs-2518	130	8	≤	≤	NUM
iajs-2518	130	9	x	x	PUNCT
iajs-2518	130	10	,	,	PUNCT
iajs-2518	130	11	such	such	ADJ
iajs-2518	130	12	that	that	SCONJ
iajs-2518	130	13	𝐴	𝐴	PROPN
iajs-2518	130	14	𝑥	𝑥	NOUN
iajs-2518	130	15	1	1	NUM
iajs-2518	130	16	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	130	17	𝑥	𝑥	PRON
iajs-2518	130	18	∈	∈	NOUN
iajs-2518	130	19	2	2	NUM
iajs-2518	130	20	1/2	1/2	NUM
iajs-2518	130	21	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	130	22	𝐵	𝐵	NOUN
iajs-2518	130	23	𝑥	𝑥	ADP
iajs-2518	130	24	1	1	NUM
iajs-2518	130	25	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	130	26	𝑥	𝑥	PRON
iajs-2518	130	27	∈	∈	NOUN
iajs-2518	130	28	8	8	NUM
iajs-2518	130	29	1	1	NUM
iajs-2518	130	30	3⁄	3⁄	NUM
iajs-2518	130	31	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	130	32	,	,	PUNCT
iajs-2518	130	33	so	so	ADV
iajs-2518	130	34	𝑋∗	𝑋∗	ADJ
iajs-2518	130	35	/	/	SYM
iajs-2518	130	36	𝐴∗	𝐴∗	NUM
iajs-2518	130	37	𝑍	𝑍	PROPN
iajs-2518	130	38	/	/	SYM
iajs-2518	130	39	2	2	NUM
iajs-2518	130	40	𝑍	𝑍	NOUN
iajs-2518	130	41	∗	∗	NOUN
iajs-2518	130	42	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	130	43	semisimple	semisimple	NOUN
iajs-2518	130	44	.	.	PUNCT
iajs-2518	131	1	so	so	ADV
iajs-2518	131	2	𝐴∗	𝐴∗	PROPN
iajs-2518	131	3	is	be	AUX
iajs-2518	131	4	semimaximal	semimaximal	ADJ
iajs-2518	131	5	.but	.but	PUNCT
iajs-2518	132	1	𝑋∗	𝑋∗	ADJ
iajs-2518	132	2	𝐵∗	𝐵∗	NUM
iajs-2518	132	3	𝑍	𝑍	PROPN
iajs-2518	132	4	/	/	SYM
iajs-2518	132	5	8	8	NUM
iajs-2518	132	6	𝑍	𝑍	PROPN
iajs-2518	132	7	⁄	⁄	PROPN
iajs-2518	132	8	,	,	PUNCT
iajs-2518	132	9	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	132	10	𝑛𝑜𝑡	𝑛𝑜𝑡	NOUN
iajs-2518	132	11	semisimple	semisimple	NOUN
iajs-2518	132	12	.	.	PUNCT
iajs-2518	133	1	hence	hence	ADV
iajs-2518	133	2	,	,	PUNCT
iajs-2518	133	3	𝐵∗	𝐵∗	PROPN
iajs-2518	133	4	is	be	AUX
iajs-2518	133	5	not	not	PART
iajs-2518	133	6	semimaximal	semimaximal	ADJ
iajs-2518	133	7	submodule	submodule	NOUN
iajs-2518	133	8	.	.	PUNCT
iajs-2518	134	1	thus	thus	ADV
iajs-2518	134	2	b	b	X
iajs-2518	134	3	is	be	AUX
iajs-2518	134	4	not	not	PART
iajs-2518	134	5	semimaximal	semimaximal	ADJ
iajs-2518	134	6	f	f	PROPN
iajs-2518	134	7	submodule	submodule	NOUN
iajs-2518	134	8	by	by	ADP
iajs-2518	134	9	proposition	proposition	NOUN
iajs-2518	134	10	3.3	3.3	NUM
iajs-2518	134	11	.	.	PUNCT
iajs-2518	135	1	3	3	X
iajs-2518	135	2	)	)	PUNCT
iajs-2518	135	3	if	if	SCONJ
iajs-2518	135	4	a	a	PRON
iajs-2518	135	5	and	and	CCONJ
iajs-2518	135	6	b	b	NOUN
iajs-2518	135	7	are	be	AUX
iajs-2518	135	8	fsubmodules	fsubmodule	NOUN
iajs-2518	135	9	of	of	ADP
iajs-2518	135	10	f	f	PROPN
iajs-2518	135	11	module	module	NOUN
iajs-2518	135	12	x	x	PUNCT
iajs-2518	135	13	such	such	ADJ
iajs-2518	135	14	that	that	SCONJ
iajs-2518	135	15	a	a	DET
iajs-2518	135	16	𝐵	𝐵	NOUN
iajs-2518	135	17	𝑋	𝑋	PROPN
iajs-2518	135	18	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	135	19	𝐴	𝐴	PROPN
iajs-2518	135	20	is	be	AUX
iajs-2518	135	21	semimaximal	semimaximal	ADJ
iajs-2518	135	22	in	in	ADP
iajs-2518	135	23	x	x	PROPN
iajs-2518	135	24	.then	.then	PROPN
iajs-2518	135	25	b	b	NOUN
iajs-2518	135	26	is	be	AUX
iajs-2518	135	27	semimaximal	semimaximal	ADJ
iajs-2518	135	28	in	in	ADP
iajs-2518	135	29	x	x	X
iajs-2518	135	30	.	.	PUNCT
iajs-2518	136	1	proof	proof	NOUN
iajs-2518	136	2	:	:	PUNCT
iajs-2518	136	3	since	since	SCONJ
iajs-2518	136	4	a	a	PRON
iajs-2518	136	5	is	be	AUX
iajs-2518	136	6	a	a	DET
iajs-2518	136	7	fuzzy	fuzzy	ADJ
iajs-2518	136	8	semimaximal	semimaximal	ADJ
iajs-2518	136	9	submodule	submodule	NOUN
iajs-2518	136	10	in	in	ADP
iajs-2518	136	11	x	x	X
iajs-2518	136	12	,	,	PUNCT
iajs-2518	136	13	𝑋	𝑋	PROPN
iajs-2518	136	14	𝐴⁄	𝐴⁄	PROPN
iajs-2518	136	15	is	be	AUX
iajs-2518	136	16	semisimple	semisimple	NOUN
iajs-2518	136	17	.	.	PUNCT
iajs-2518	137	1	hence	hence	ADV
iajs-2518	137	2	,	,	PUNCT
iajs-2518	137	3	𝑋	𝑋	PROPN
iajs-2518	137	4	𝐴⁄	𝐴⁄	ADJ
iajs-2518	137	5	𝑋	𝑋	NOUN
iajs-2518	137	6	𝐵⁄⁄	𝐵⁄⁄	NOUN
iajs-2518	137	7	is	be	AUX
iajs-2518	137	8	semisimple	semisimple	ADJ
iajs-2518	137	9	image	image	NOUN
iajs-2518	137	10	of	of	ADP
iajs-2518	137	11	semisimple	semisimple	ADJ
iajs-2518	137	12	proposition	proposition	NOUN
iajs-2518	137	13	2.12	2.12	NUM
iajs-2518	137	14	by	by	ADP
iajs-2518	137	15	the	the	DET
iajs-2518	137	16	second	second	ADJ
iajs-2518	137	17	isomorphism	isomorphism	NOUN
iajs-2518	137	18	theorem[14	theorem[14	PROPN
iajs-2518	137	19	]	]	PUNCT
iajs-2518	137	20	,	,	PUNCT
iajs-2518	137	21	𝑋	𝑋	PROPN
iajs-2518	137	22	𝐴⁄	𝐴⁄	ADJ
iajs-2518	137	23	𝑋	𝑋	NOUN
iajs-2518	137	24	𝐵⁄⁄	𝐵⁄⁄	VERB
iajs-2518	137	25	𝑋	𝑋	PROPN
iajs-2518	137	26	𝐵⁄	𝐵⁄	PROPN
iajs-2518	137	27	is	be	AUX
iajs-2518	137	28	semisimple	semisimple	NOUN
iajs-2518	137	29	.	.	PUNCT
iajs-2518	138	1	thus	thus	ADV
iajs-2518	138	2	,	,	PUNCT
iajs-2518	138	3	b	b	X
iajs-2518	138	4	is	be	AUX
iajs-2518	138	5	a	a	DET
iajs-2518	138	6	fuzzy	fuzzy	ADJ
iajs-2518	138	7	semimaximal	semimaximal	NOUN
iajs-2518	138	8	submodule	submodule	NOUN
iajs-2518	138	9	.	.	PUNCT
iajs-2518	139	1	4let	4let	PROPN
iajs-2518	139	2	a	a	PROPN
iajs-2518	139	3	and	and	CCONJ
iajs-2518	139	4	b	b	NOUN
iajs-2518	139	5	be	be	AUX
iajs-2518	139	6	two	two	NUM
iajs-2518	139	7	f	f	PROPN
iajs-2518	139	8	submodules	submodule	NOUN
iajs-2518	139	9	of	of	ADP
iajs-2518	139	10	fmodule	fmodule	ADJ
iajs-2518	139	11	x	x	X
iajs-2518	139	12	.	.	PUNCT
iajs-2518	140	1	if	if	SCONJ
iajs-2518	140	2	is	be	AUX
iajs-2518	140	3	a	a	DET
iajs-2518	140	4	a	a	DET
iajs-2518	140	5	semimaximal	semimaximal	ADJ
iajs-2518	140	6	f	f	PROPN
iajs-2518	140	7	-submodule	-submodule	NOUN
iajs-2518	140	8	of	of	ADP
iajs-2518	140	9	x	x	PRON
iajs-2518	140	10	,	,	PUNCT
iajs-2518	140	11	then	then	ADV
iajs-2518	140	12	𝐴	𝐴	PROPN
iajs-2518	140	13	𝐵	𝐵	PROPN
iajs-2518	140	14	is	be	AUX
iajs-2518	140	15	also	also	ADV
iajs-2518	140	16	semimaximal	semimaximal	ADJ
iajs-2518	140	17	f	f	PROPN
iajs-2518	140	18	submodule	submodule	PROPN
iajs-2518	140	19	of	of	ADP
iajs-2518	140	20	x.	x.	NOUN
iajs-2518	140	21	proof	proof	NOUN
iajs-2518	140	22	:	:	PUNCT
iajs-2518	140	23	clearly	clearly	ADV
iajs-2518	140	24	𝐴	𝐴	PROPN
iajs-2518	140	25	𝐴	𝐴	PROPN
iajs-2518	140	26	𝐵	𝐵	PROPN
iajs-2518	140	27	and	and	CCONJ
iajs-2518	140	28	,	,	PUNCT
iajs-2518	140	29	hence	hence	ADV
iajs-2518	140	30	the	the	DET
iajs-2518	140	31	result	result	NOUN
iajs-2518	140	32	follows	follow	VERB
iajs-2518	140	33	directly	directly	ADV
iajs-2518	140	34	.	.	PUNCT
iajs-2518	141	1	5	5	X
iajs-2518	141	2	)	)	PUNCT
iajs-2518	141	3	let	let	VERB
iajs-2518	141	4	{	{	PUNCT
iajs-2518	141	5	𝐴	𝐴	PROPN
iajs-2518	141	6	1,2,3	1,2,3	NUM
iajs-2518	141	7	,	,	PUNCT
iajs-2518	141	8	…	…	PUNCT
iajs-2518	141	9	𝑛	𝑛	X
iajs-2518	141	10	}	}	PUNCT
iajs-2518	141	11	be	be	AUX
iajs-2518	141	12	a	a	DET
iajs-2518	141	13	finite	finite	ADJ
iajs-2518	141	14	collection	collection	NOUN
iajs-2518	141	15	of	of	ADP
iajs-2518	141	16	semimaximal	semimaximal	PROPN
iajs-2518	141	17	fsubmodules	fsubmodule	NOUN
iajs-2518	141	18	of	of	ADP
iajs-2518	141	19	f	f	PROPN
iajs-2518	141	20	module	module	NOUN
iajs-2518	141	21	x.	x.	NOUN
iajs-2518	141	22	then	then	ADV
iajs-2518	141	23	𝐴	𝐴	PROPN
iajs-2518	141	24	,	,	PUNCT
iajs-2518	141	25	𝑖	𝑖	PROPN
iajs-2518	141	26	1,2,3	1,2,3	NUM
iajs-2518	141	27	…	…	PUNCT
iajs-2518	141	28	𝑛	𝑛	PROPN
iajs-2518	141	29	is	be	AUX
iajs-2518	141	30	a	a	DET
iajs-2518	141	31	semimaximal	semimaximal	ADJ
iajs-2518	141	32	fsubmodule	fsubmodule	NOUN
iajs-2518	141	33	.	.	PUNCT
iajs-2518	142	1	proof	proof	NOUN
iajs-2518	142	2	:	:	PUNCT
iajs-2518	142	3	it	it	PRON
iajs-2518	142	4	is	be	AUX
iajs-2518	142	5	clear	clear	ADJ
iajs-2518	142	6	by	by	ADP
iajs-2518	142	7	4	4	NUM
iajs-2518	142	8	6	6	NUM
iajs-2518	142	9	)	)	PUNCT
iajs-2518	142	10	let	let	VERB
iajs-2518	142	11	a	a	PRON
iajs-2518	142	12	and	and	CCONJ
iajs-2518	142	13	b	b	NOUN
iajs-2518	142	14	be	be	AUX
iajs-2518	142	15	two	two	NUM
iajs-2518	142	16	f	f	PROPN
iajs-2518	142	17	submodules	submodule	NOUN
iajs-2518	142	18	of	of	ADP
iajs-2518	142	19	f	f	PROPN
iajs-2518	142	20	module	module	NOUN
iajs-2518	142	21	x	x	PUNCT
iajs-2518	142	22	such	such	ADJ
iajs-2518	142	23	that	that	SCONJ
iajs-2518	142	24	a	a	DET
iajs-2518	142	25	𝐵	𝐵	NOUN
iajs-2518	142	26	.	.	PUNCT
iajs-2518	143	1	if	if	SCONJ
iajs-2518	143	2	a	a	PRON
iajs-2518	143	3	is	be	AUX
iajs-2518	143	4	semimaximal	semimaximal	ADJ
iajs-2518	143	5	in	in	ADP
iajs-2518	143	6	b	b	PROPN
iajs-2518	143	7	and	and	CCONJ
iajs-2518	143	8	b	b	NOUN
iajs-2518	143	9	is	be	AUX
iajs-2518	143	10	semimaximal	semimaximal	ADJ
iajs-2518	143	11	in	in	ADP
iajs-2518	143	12	x	x	SYM
iajs-2518	143	13	,	,	PUNCT
iajs-2518	143	14	then	then	ADV
iajs-2518	143	15	a	a	PRON
iajs-2518	143	16	is	be	AUX
iajs-2518	143	17	not	not	PART
iajs-2518	143	18	necessary	necessary	ADJ
iajs-2518	143	19	semimaximal	semimaximal	NOUN
iajs-2518	143	20	of	of	ADP
iajs-2518	143	21	x	x	SYM
iajs-2518	143	22	,	,	PUNCT
iajs-2518	143	23	as	as	SCONJ
iajs-2518	143	24	the	the	DET
iajs-2518	143	25	following	follow	VERB
iajs-2518	143	26	example	example	NOUN
iajs-2518	143	27	shows	show	VERB
iajs-2518	143	28	:	:	PUNCT
iajs-2518	143	29	example	example	NOUN
iajs-2518	143	30	:	:	PUNCT
iajs-2518	143	31	take	take	VERB
iajs-2518	143	32	𝑀	𝑀	NOUN
iajs-2518	143	33	𝑍	𝑍	VERB
iajs-2518	143	34	𝑎𝑠	𝑎𝑠	NOUN
iajs-2518	143	35	𝑍	𝑍	PROPN
iajs-2518	143	36	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	143	37	,	,	PUNCT
iajs-2518	143	38	let	let	VERB
iajs-2518	143	39	𝑋	𝑋	NOUN
iajs-2518	143	40	∶	∶	NOUN
iajs-2518	143	41	𝑀	𝑀	PROPN
iajs-2518	143	42	→	→	SYM
iajs-2518	143	43	0,1	0,1	NUM
iajs-2518	143	44	,	,	PUNCT
iajs-2518	143	45	𝑑𝑒𝑓𝑖𝑛𝑒	𝑑𝑒𝑓𝑖𝑛𝑒	ADJ
iajs-2518	143	46	𝑋	𝑋	NOUN
iajs-2518	143	47	𝑥	𝑥	PROPN
iajs-2518	143	48	1	1	NUM
iajs-2518	143	49	,	,	PUNCT
iajs-2518	143	50	∀𝑥	∀𝑥	PROPN
iajs-2518	143	51	∈	∈	PROPN
iajs-2518	143	52	𝑀	𝑀	PROPN
iajs-2518	143	53	,	,	PUNCT
iajs-2518	143	54	𝐴	𝐴	PROPN
iajs-2518	143	55	𝑥	𝑥	PROPN
iajs-2518	143	56	1	1	NUM
iajs-2518	143	57	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	143	58	𝑥	𝑥	PRON
iajs-2518	143	59	∈	∈	NOUN
iajs-2518	143	60	9	9	NUM
iajs-2518	143	61	0	0	NUM
iajs-2518	143	62	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	143	63	  	  	SPACE
iajs-2518	143	64	143	143	NUM
iajs-2518	143	65	  	  	SPACE
iajs-2518	143	66	ibn	ibn	PROPN
iajs-2518	143	67	al	al	PROPN
iajs-2518	143	68	-	-	PUNCT
iajs-2518	143	69	haitham	haitham	PROPN
iajs-2518	143	70	jour	jour	X
iajs-2518	143	71	.	.	PROPN
iajs-2518	144	1	for	for	ADP
iajs-2518	144	2	pure	pure	ADJ
iajs-2518	144	3	&	&	CCONJ
iajs-2518	144	4	appl	appl	PROPN
iajs-2518	144	5	.	.	PUNCT
iajs-2518	145	1	sci	sci	PROPN
iajs-2518	145	2	.	.	PROPN
iajs-2518	146	1	33	33	NUM
iajs-2518	146	2	(	(	PUNCT
iajs-2518	146	3	4	4	NUM
iajs-2518	146	4	)	)	PUNCT
iajs-2518	146	5	2020	2020	NUM
iajs-2518	147	1	𝐵	𝐵	NOUN
iajs-2518	147	2	𝑥	𝑥	NOUN
iajs-2518	147	3	1	1	NUM
iajs-2518	147	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	147	5	𝑥	𝑥	PRON
iajs-2518	147	6	∈	∈	PROPN
iajs-2518	147	7	3	3	NUM
iajs-2518	147	8	0	0	NUM
iajs-2518	147	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	147	10	𝑋	𝑋	PROPN
iajs-2518	147	11	𝐵⁄	𝐵⁄	PROPN
iajs-2518	147	12	∶	∶	PROPN
iajs-2518	147	13	𝑍	𝑍	VERB
iajs-2518	147	14	3⁄	3⁄	NUM
iajs-2518	147	15	≅	≅	NOUN
iajs-2518	147	16	𝑍	𝑍	PROPN
iajs-2518	147	17	→	→	SYM
iajs-2518	147	18	0,1	0,1	NUM
iajs-2518	147	19	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2518	147	20	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2518	147	21	𝑋	𝑋	PROPN
iajs-2518	147	22	𝐵⁄	𝐵⁄	PROPN
iajs-2518	147	23	𝑥	𝑥	PROPN
iajs-2518	147	24	1	1	NUM
iajs-2518	147	25	,	,	PUNCT
iajs-2518	147	26	∀𝑥	∀𝑥	PROPN
iajs-2518	147	27	∈	∈	NOUN
iajs-2518	147	28	𝑍	𝑍	VERB
iajs-2518	147	29	𝑋	𝑋	NOUN
iajs-2518	147	30	𝐴⁄	𝐴⁄	ADJ
iajs-2518	147	31	∶	∶	NOUN
iajs-2518	147	32	𝑍	𝑍	VERB
iajs-2518	147	33	9	9	NUM
iajs-2518	147	34	⁄	⁄	PROPN
iajs-2518	147	35	≅	≅	PROPN
iajs-2518	147	36	𝑍	𝑍	PROPN
iajs-2518	147	37	→	→	SYM
iajs-2518	147	38	0,1	0,1	NUM
iajs-2518	147	39	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2518	147	40	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2518	147	41	𝑋	𝑋	NOUN
iajs-2518	148	1	𝐴⁄	𝐴⁄	PROPN
iajs-2518	149	1	𝑥	𝑥	NOUN
iajs-2518	149	2	1	1	NUM
iajs-2518	149	3	,	,	PUNCT
iajs-2518	149	4	∀𝑥	∀𝑥	PROPN
iajs-2518	149	5	∈	∈	PROPN
iajs-2518	149	6	𝑍	𝑍	PROPN
iajs-2518	149	7	𝑁𝑜𝑤	𝑁𝑜𝑤	PROPN
iajs-2518	149	8	,	,	PUNCT
iajs-2518	149	9	𝑋	𝑋	NOUN
iajs-2518	149	10	/𝐵	/𝐵	PUNCT
iajs-2518	149	11	∗	∗	NOUN
iajs-2518	149	12	𝑍	𝑍	NOUN
iajs-2518	149	13	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	149	14	𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	149	15	,	,	PUNCT
iajs-2518	149	16	𝑠𝑜	𝑠𝑜	ADP
iajs-2518	149	17	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2518	149	18	𝑋∗	𝑋∗	ADJ
iajs-2518	149	19	𝐵∗⁄	𝐵∗⁄	PUNCT
iajs-2518	149	20	𝑍	𝑍	VERB
iajs-2518	149	21	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	149	22	ℎ𝑒𝑛𝑐𝑒	ℎ𝑒𝑛𝑐𝑒	ADJ
iajs-2518	149	23	𝐵	𝐵	NOUN
iajs-2518	149	24	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	149	25	𝑎	𝑎	PRON
iajs-2518	149	26	𝐹	𝐹	PROPN
iajs-2518	149	27	𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙	𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙	NOUN
iajs-2518	149	28	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	149	29	,	,	PUNCT
iajs-2518	149	30	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	149	31	𝑃𝑟𝑜𝑝	𝑃𝑟𝑜𝑝	PROPN
iajs-2518	149	32	3.3	3.3	NUM
iajs-2518	149	33	𝐵𝑢𝑡	𝐵𝑢𝑡	PROPN
iajs-2518	149	34	𝑋∗	𝑋∗	ADJ
iajs-2518	149	35	𝐴∗	𝐴∗	NUM
iajs-2518	149	36	𝑍	𝑍	PROPN
iajs-2518	149	37	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	149	38	𝑛𝑜𝑡	𝑛𝑜𝑡	NOUN
iajs-2518	149	39	𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	149	40	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	149	41	.	.	PUNCT
iajs-2518	150	1	𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒	𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒	PROPN
iajs-2518	150	2	𝐴∗	𝐴∗	PROPN
iajs-2518	150	3	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	150	4	𝑛𝑜𝑡	𝑛𝑜𝑡	NOUN
iajs-2518	150	5	𝑎	𝑎	PRON
iajs-2518	150	6	𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙	𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙	ADJ
iajs-2518	150	7	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	150	8	𝑖𝑛	𝑖𝑛	PRON
iajs-2518	150	9	𝑋∗	𝑋∗	ADJ
iajs-2518	150	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	150	11	ℎ𝑒𝑛𝑐𝑒	ℎ𝑒𝑛𝑐𝑒	PROPN
iajs-2518	150	12	𝐴	𝐴	PROPN
iajs-2518	150	13	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	150	14	𝑛𝑜𝑡	𝑛𝑜𝑡	PROPN
iajs-2518	150	15	𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙	𝑠𝑒𝑚𝑖𝑚𝑎𝑥𝑖𝑚𝑎𝑙	PROPN
iajs-2518	150	16	𝑖𝑛	𝑖𝑛	PROPN
iajs-2518	150	17	𝑋.	𝑋.	PROPN
iajs-2518	150	18	𝑆𝑖𝑛𝑐𝑒	𝑆𝑖𝑛𝑐𝑒	PROPN
iajs-2518	150	19	𝐴	𝐴	PROPN
iajs-2518	150	20	𝐵	𝐵	NOUN
iajs-2518	150	21	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	150	22	𝐵	𝐵	PROPN
iajs-2518	150	23	𝐴	𝐴	PROPN
iajs-2518	150	24	:	:	PUNCT
iajs-2518	150	25	𝑍	𝑍	PROPN
iajs-2518	150	26	/	/	SYM
iajs-2518	150	27	9	9	NUM
iajs-2518	150	28	→	→	SYM
iajs-2518	150	29	0,1	0,1	NUM
iajs-2518	150	30	⁄	⁄	PROPN
iajs-2518	150	31	∶	∶	NOUN
iajs-2518	150	32	𝑑𝑒𝑓𝑖𝑛𝑒𝑑	𝑑𝑒𝑓𝑖𝑛𝑒𝑑	NOUN
iajs-2518	150	33	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	150	34	∶	∶	PROPN
iajs-2518	150	35	b	b	PROPN
iajs-2518	150	36	a	a	PRON
iajs-2518	150	37	x⁄	x⁄	PROPN
iajs-2518	150	38	1	1	NUM
iajs-2518	150	39	if	if	SCONJ
iajs-2518	150	40	x	x	PROPN
iajs-2518	150	41	∈	∈	NOUN
iajs-2518	150	42	3	3	NUM
iajs-2518	150	43	/	/	SYM
iajs-2518	150	44	9	9	NUM
iajs-2518	150	45	≅	≅	PROPN
iajs-2518	150	46	z	z	PROPN
iajs-2518	150	47	0	0	NUM
iajs-2518	151	1	otherwise	otherwise	ADV
iajs-2518	151	2	now	now	ADV
iajs-2518	151	3	,	,	PUNCT
iajs-2518	151	4	𝐵	𝐵	NOUN
iajs-2518	151	5	/a	/a	PUNCT
iajs-2518	151	6	∗	∗	NOUN
iajs-2518	151	7	3	3	NUM
iajs-2518	151	8	/	/	SYM
iajs-2518	151	9	9	9	NUM
iajs-2518	151	10	≅	≅	NUM
iajs-2518	151	11	𝑍	𝑍	PROPN
iajs-2518	151	12	.but	.but	NOUN
iajs-2518	151	13	𝐵∗	𝐵∗	NOUN
iajs-2518	151	14	𝐴∗	𝐴∗	NOUN
iajs-2518	151	15	3	3	NUM
iajs-2518	151	16	/	/	SYM
iajs-2518	151	17	9	9	NUM
iajs-2518	151	18	≅	≅	NUM
iajs-2518	151	19	𝑍⁄	𝑍⁄	PROPN
iajs-2518	151	20	𝑖𝑠	𝑖𝑠	PROPN
iajs-2518	151	21	𝑠𝑖𝑚𝑝𝑙𝑒	𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	151	22	.	.	PUNCT
iajs-2518	152	1	then	then	ADV
iajs-2518	152	2	𝐴∗	𝐴∗	PROPN
iajs-2518	152	3	is	be	AUX
iajs-2518	152	4	semimaximl	semimaximl	NOUN
iajs-2518	152	5	in	in	ADP
iajs-2518	152	6	𝐵∗	𝐵∗	PROPN
iajs-2518	152	7	.	.	PUNCT
iajs-2518	153	1	proposition	proposition	NOUN
iajs-2518	153	2	3.5	3.5	NUM
iajs-2518	153	3	let	let	VERB
iajs-2518	153	4	𝐴	𝐴	PROPN
iajs-2518	153	5	be	be	AUX
iajs-2518	153	6	a	a	DET
iajs-2518	153	7	semimaximal	semimaximal	ADJ
iajs-2518	153	8	fsubmodule	fsubmodule	NOUN
iajs-2518	153	9	of	of	ADP
iajs-2518	153	10	fmodule	fmodule	ADJ
iajs-2518	153	11	𝑋	𝑋	PROPN
iajs-2518	153	12	,	,	PUNCT
iajs-2518	153	13	𝑖	𝑖	NOUN
iajs-2518	153	14	1,2,3	1,2,3	NUM
iajs-2518	153	15	,	,	PUNCT
iajs-2518	153	16	…	…	PUNCT
iajs-2518	153	17	…	…	PUNCT
iajs-2518	153	18	𝑛	𝑛	VERB
iajs-2518	153	19	then	then	ADV
iajs-2518	153	20	𝐴	𝐴	PROPN
iajs-2518	153	21	is	be	AUX
iajs-2518	153	22	a	a	DET
iajs-2518	153	23	semimaximal	semimaximal	ADJ
iajs-2518	153	24	f	f	PROPN
iajs-2518	153	25	submodule	submodule	NOUN
iajs-2518	153	26	of	of	ADP
iajs-2518	153	27	f	f	PROPN
iajs-2518	153	28	module	module	NOUN
iajs-2518	153	29			ADJ
iajs-2518	153	30	𝑋	𝑋	PROPN
iajs-2518	153	31	,	,	PUNCT
iajs-2518	153	32	𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑	𝑃𝑟𝑜𝑣𝑖𝑑𝑒𝑑	PROPN
iajs-2518	153	33	x	x	PROPN
iajs-2518	153	34	/	/	SYM
iajs-2518	153	35	a	a	DET
iajs-2518	153	36	∗	∗	NOUN
iajs-2518	153	37	𝑋∗	𝑋∗	NOUN
iajs-2518	154	1	𝐴∗⁄	𝐴∗⁄	NOUN
iajs-2518	154	2	.	.	PUNCT
iajs-2518	155	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2518	155	2	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
iajs-2518	155	3	𝐹	𝐹	PROPN
iajs-2518	155	4	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	VERB
iajs-2518	155	5	𝑋	𝑋	PROPN
iajs-2518	155	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	155	7	𝐴	𝐴	PROPN
iajs-2518	155	8	𝑋	𝑋	PROPN
iajs-2518	155	9	,	,	PUNCT
iajs-2518	155	10	∀	∀	NOUN
iajs-2518	155	11	𝑖	𝑖	SYM
iajs-2518	155	12	1,2,3	1,2,3	NUM
iajs-2518	155	13	…	…	PUNCT
iajs-2518	155	14	.	.	PUNCT
iajs-2518	156	1	𝑛.	𝑛.	NOUN
iajs-2518	156	2	proof	proof	NOUN
iajs-2518	156	3	:	:	PUNCT
iajs-2518	156	4	since	since	SCONJ
iajs-2518	156	5	𝐴	𝐴	PROPN
iajs-2518	156	6	is	be	AUX
iajs-2518	156	7	a	a	DET
iajs-2518	156	8	semimaximal	semimaximal	ADJ
iajs-2518	156	9	fsubmodule	fsubmodule	NOUN
iajs-2518	156	10	of	of	ADP
iajs-2518	156	11	fmodule	fmodule	ADJ
iajs-2518	156	12	𝑋	𝑋	PROPN
iajs-2518	156	13	,	,	PUNCT
iajs-2518	156	14	∀𝑖	∀𝑖	PROPN
iajs-2518	156	15	1,2,3	1,2,3	NUM
iajs-2518	156	16	,	,	PUNCT
iajs-2518	156	17	…	…	PUNCT
iajs-2518	156	18	.	.	PUNCT
iajs-2518	157	1	𝑛.	𝑛.	NOUN
iajs-2518	157	2	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-2518	157	3	𝑋	𝑋	PROPN
iajs-2518	157	4	𝐴⁄	𝐴⁄	PROPN
iajs-2518	157	5	,	,	PUNCT
iajs-2518	157	6	is	be	AUX
iajs-2518	157	7	f𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒	f𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒	NOUN
iajs-2518	157	8	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	157	9	,	,	PUNCT
iajs-2518	157	10	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2518	157	11	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	NOUN
iajs-2518	157	12	𝑖	𝑖	SYM
iajs-2518	157	13	1,2,3	1,2,3	NUM
iajs-2518	157	14	…	…	PUNCT
iajs-2518	157	15	.	.	PUNCT
iajs-2518	158	1	𝑛.	𝑛.	NOUN
iajs-2518	158	2	hence	hence	ADV
iajs-2518	158	3			ADJ
iajs-2518	158	4	𝑋	𝑋	PROPN
iajs-2518	158	5	/𝐴	/𝐴	PUNCT
iajs-2518	158	6	isf	isf	ADJ
iajs-2518	158	7	-	-	PUNCT
iajs-2518	158	8	semisimple	semisimple	NOUN
iajs-2518	158	9	module	module	NOUN
iajs-2518	158	10	and	and	CCONJ
iajs-2518	158	11	so	so	ADV
iajs-2518	158	12	by	by	ADP
iajs-2518	158	13	remarks	remark	NOUN
iajs-2518	158	14	and	and	CCONJ
iajs-2518	158	15	examples	example	NOUN
iajs-2518	158	16	3.4(5	3.4(5	NUM
iajs-2518	158	17	)	)	PUNCT
iajs-2518	158	18			ADJ
iajs-2518	158	19	∗	∗	NOUN
iajs-2518	158	20	is	be	AUX
iajs-2518	158	21	a	a	DET
iajs-2518	158	22	semisimple	semisimple	NOUN
iajs-2518	158	23	.	.	PUNCT
iajs-2518	159	1	b	b	X
iajs-2518	159	2	ut	ut	PROPN
iajs-2518	159	3			PROPN
iajs-2518	159	4	𝑋	𝑋	PROPN
iajs-2518	159	5	𝐴	𝐴	PROPN
iajs-2518	159	6	∗	∗	VERB
iajs-2518	159	7			ADJ
iajs-2518	159	8	𝑋	𝑋	PROPN
iajs-2518	159	9	𝐴	𝐴	PROPN
iajs-2518	159	10	∗	∗	NOUN
iajs-2518	159	11	by	by	ADP
iajs-2518	159	12	2	2	NUM
iajs-2518	159	13			ADJ
iajs-2518	159	14	𝑋	𝑋	PROPN
iajs-2518	159	15	∗	∗	NOUN
iajs-2518	159	16	𝐴	𝐴	PROPN
iajs-2518	159	17	∗	∗	VERB
iajs-2518	160	1			ADJ
iajs-2518	160	2	𝑋	𝑋	PROPN
iajs-2518	160	3	∗	∗	NOUN
iajs-2518	160	4			PROPN
iajs-2518	160	5	𝐴	𝐴	PROPN
iajs-2518	160	6	∗	∗	VERB
iajs-2518	160	7			ADJ
iajs-2518	160	8	𝑋	𝑋	PROPN
iajs-2518	160	9	∗	∗	NOUN
iajs-2518	160	10	𝐴	𝐴	PROPN
iajs-2518	160	11	∗	∗	NOUN
iajs-2518	160	12	hence	hence	ADV
iajs-2518	160	13			PROPN
iajs-2518	160	14	𝐴	𝐴	PROPN
iajs-2518	160	15	∗	∗	VERB
iajs-2518	160	16	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	161	1	𝑎	𝑎	DET
iajs-2518	161	2	semimaximal	semimaximal	ADJ
iajs-2518	161	3	submodule	submodule	NOUN
iajs-2518	161	4	in	in	ADP
iajs-2518	161	5			ADJ
iajs-2518	161	6	𝑋	𝑋	PROPN
iajs-2518	161	7	∗	∗	NOUN
iajs-2518	161	8	.	.	PUNCT
iajs-2518	162	1	by	by	ADP
iajs-2518	162	2	hypothesis	hypothesis	NOUN
iajs-2518	162	3			PROPN
iajs-2518	162	4			PROPN
iajs-2518	162	5	∗	∗	PROPN
iajs-2518	162	6			PROPN
iajs-2518	162	7	∗	∗	PROPN
iajs-2518	162	8			PROPN
iajs-2518	162	9	∗	∗	NOUN
iajs-2518	162	10	hence	hence	ADV
iajs-2518	162	11			PROPN
iajs-2518	162	12	𝐴	𝐴	PROPN
iajs-2518	162	13	is	be	AUX
iajs-2518	162	14	a	a	DET
iajs-2518	162	15	f	f	PROPN
iajs-2518	162	16	semimaximal	semimaximal	PROPN
iajs-2518	162	17	submodule	submodule	NOUN
iajs-2518	162	18	in	in	ADP
iajs-2518	162	19			ADJ
iajs-2518	162	20	𝑋	𝑋	PROPN
iajs-2518	162	21	.	.	PUNCT
iajs-2518	163	1	remark	remark	VERB
iajs-2518	163	2	3.6	3.6	NUM
iajs-2518	163	3	if	if	SCONJ
iajs-2518	163	4	x	x	PRON
iajs-2518	163	5	is	be	AUX
iajs-2518	163	6	a	a	DET
iajs-2518	163	7	f	f	NOUN
iajs-2518	163	8	module	module	NOUN
iajs-2518	163	9	of	of	ADP
iajs-2518	163	10	an	an	DET
iajs-2518	163	11	rmodule	rmodule	NOUN
iajs-2518	163	12	m	m	VERB
iajs-2518	163	13	,	,	PUNCT
iajs-2518	163	14	then	then	ADV
iajs-2518	163	15	is	be	AUX
iajs-2518	163	16	not	not	PART
iajs-2518	163	17	necessary	necessary	ADJ
iajs-2518	163	18	that	that	SCONJ
iajs-2518	163	19	x	x	PRON
iajs-2518	163	20	has	have	VERB
iajs-2518	163	21	semimaximal	semimaximal	PROPN
iajs-2518	163	22	f	f	PROPN
iajs-2518	163	23	submodule	submodule	NOUN
iajs-2518	163	24	,	,	PUNCT
iajs-2518	163	25	for	for	ADP
iajs-2518	163	26	example	example	NOUN
iajs-2518	163	27	.	.	PUNCT
iajs-2518	164	1	example	example	NOUN
iajs-2518	164	2	:	:	PUNCT
iajs-2518	164	3	let	let	VERB
iajs-2518	164	4	𝑀	𝑀	PROPN
iajs-2518	164	5	𝑍	𝑍	VERB
iajs-2518	164	6	𝑝	𝑝	PROPN
iajs-2518	164	7	𝑖𝑠	𝑖𝑠	ADP
iajs-2518	164	8	𝑎	𝑎	DET
iajs-2518	164	9	𝑝𝑟𝑖𝑚𝑒	𝑝𝑟𝑖𝑚𝑒	NOUN
iajs-2518	164	10	𝑛𝑢𝑚𝑏𝑒𝑟	𝑛𝑢𝑚𝑏𝑒𝑟	NOUN
iajs-2518	164	11	𝑎𝑠	𝑎𝑠	ADP
iajs-2518	164	12	𝑍	𝑍	PROPN
iajs-2518	164	13	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	164	14	,	,	PUNCT
iajs-2518	164	15	let	let	VERB
iajs-2518	164	16	𝑋	𝑋	NOUN
iajs-2518	164	17	∶	∶	NOUN
iajs-2518	164	18	𝑀	𝑀	PROPN
iajs-2518	164	19	→	→	SYM
iajs-2518	164	20	0,1	0,1	NUM
iajs-2518	164	21	,	,	PUNCT
iajs-2518	164	22	𝑑𝑒𝑓𝑖𝑛𝑒	𝑑𝑒𝑓𝑖𝑛𝑒	ADJ
iajs-2518	164	23	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	164	24	𝑋	𝑋	PROPN
iajs-2518	164	25	𝑥	𝑥	PROPN
iajs-2518	164	26	1	1	NUM
iajs-2518	164	27	,	,	PUNCT
iajs-2518	164	28	∀	∀	VERB
iajs-2518	164	29	𝑥	𝑥	DET
iajs-2518	164	30	∈	∈	PROPN
iajs-2518	164	31	𝑀	𝑀	PROPN
iajs-2518	164	32	.	.	PUNCT
iajs-2518	165	1	assume	assume	VERB
iajs-2518	165	2	x	x	PUNCT
iajs-2518	165	3	has	have	VERB
iajs-2518	165	4	a	a	DET
iajs-2518	165	5	semimaximal	semimaximal	ADJ
iajs-2518	165	6	submodule	submodule	NOUN
iajs-2518	165	7	say	say	VERB
iajs-2518	165	8	a	a	PRON
iajs-2518	165	9	.	.	PUNCT
iajs-2518	166	1	let	let	VERB
iajs-2518	166	2	𝐴∗	𝐴∗	NUM
iajs-2518	166	3	n	n	PART
iajs-2518	166	4	is	be	AUX
iajs-2518	166	5	a	a	DET
iajs-2518	166	6	a	a	DET
iajs-2518	166	7	semimaximal	semimaximal	ADJ
iajs-2518	166	8	submodule	submodule	NOUN
iajs-2518	166	9	in	in	ADP
iajs-2518	166	10	𝑋∗	𝑋∗	ADJ
iajs-2518	166	11	𝑍	𝑍	NOUN
iajs-2518	166	12	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	166	13	𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛	𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛	PROPN
iajs-2518	166	14	3.2	3.2	NUM
iajs-2518	166	15	;	;	PUNCT
iajs-2518	166	16	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
iajs-2518	166	17	𝑖𝑠	𝑖𝑠	NUM
iajs-2518	166	18	𝑋∗/𝑁	𝑋∗/𝑁	PROPN
iajs-2518	166	19	𝑍	𝑍	PROPN
iajs-2518	166	20	/𝑁	/𝑁	PUNCT
iajs-2518	166	21	is	be	AUX
iajs-2518	166	22	semisimple	semisimple	ADJ
iajs-2518	166	23	.	.	PUNCT
iajs-2518	167	1	but	but	CCONJ
iajs-2518	167	2	≅	≅	NUM
iajs-2518	167	3	𝑍	𝑍	PROPN
iajs-2518	167	4	,	,	PUNCT
iajs-2518	167	5	𝑠𝑜	𝑠𝑜	ADP
iajs-2518	167	6	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2518	167	7	𝑍	𝑍	PROPN
iajs-2518	167	8	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	167	9	semisimple	semisimple	NOUN
iajs-2518	167	10	,	,	PUNCT
iajs-2518	167	11	which	which	PRON
iajs-2518	167	12	is	be	AUX
iajs-2518	167	13	a	a	DET
iajs-2518	167	14	contradiction	contradiction	NOUN
iajs-2518	167	15	.	.	PUNCT
iajs-2518	167	16	  	  	SPACE
iajs-2518	168	1	144	144	NUM
iajs-2518	168	2	  	  	SPACE
iajs-2518	168	3	ibn	ibn	PROPN
iajs-2518	168	4	al	al	PROPN
iajs-2518	168	5	-	-	PUNCT
iajs-2518	168	6	haitham	haitham	PROPN
iajs-2518	168	7	jour	jour	X
iajs-2518	168	8	.	.	PROPN
iajs-2518	168	9	for	for	ADP
iajs-2518	168	10	pure	pure	ADJ
iajs-2518	168	11	&	&	CCONJ
iajs-2518	168	12	appl	appl	PROPN
iajs-2518	168	13	.	.	PUNCT
iajs-2518	169	1	sci	sci	PROPN
iajs-2518	169	2	.	.	PROPN
iajs-2518	170	1	33	33	NUM
iajs-2518	170	2	(	(	PUNCT
iajs-2518	170	3	4	4	NUM
iajs-2518	170	4	)	)	PUNCT
iajs-2518	170	5	2020	2020	NUM
iajs-2518	171	1	thus	thus	ADV
iajs-2518	171	2	a	a	PRON
iajs-2518	171	3	is	be	AUX
iajs-2518	171	4	not	not	PART
iajs-2518	171	5	semimaximal	semimaximal	ADJ
iajs-2518	171	6	fsubmodule	fsubmodule	NOUN
iajs-2518	171	7	of	of	ADP
iajs-2518	171	8	x	x	X
iajs-2518	171	9	.	.	PUNCT
iajs-2518	172	1	the	the	DET
iajs-2518	172	2	following	follow	VERB
iajs-2518	172	3	proposition	proposition	NOUN
iajs-2518	172	4	give	give	VERB
iajs-2518	172	5	sufficient	sufficient	ADJ
iajs-2518	172	6	condition	condition	NOUN
iajs-2518	172	7	but	but	CCONJ
iajs-2518	172	8	not	not	PART
iajs-2518	172	9	necessary	necessary	ADJ
iajs-2518	172	10	condition	condition	NOUN
iajs-2518	172	11	for	for	SCONJ
iajs-2518	172	12	f	f	PROPN
iajs-2518	172	13	submodule	submodule	PROPN
iajs-2518	172	14	to	to	PART
iajs-2518	172	15	be	be	AUX
iajs-2518	172	16	semimaximal	semimaximal	ADJ
iajs-2518	172	17	.	.	PUNCT
iajs-2518	173	1	proposition	proposition	NOUN
iajs-2518	173	2	3.7	3.7	NUM
iajs-2518	173	3	let	let	VERB
iajs-2518	173	4	a	a	DET
iajs-2518	173	5	be	be	AUX
iajs-2518	173	6	f	f	PROPN
iajs-2518	173	7	submodule	submodule	NOUN
iajs-2518	173	8	of	of	ADP
iajs-2518	173	9	fuzzy	fuzzy	ADJ
iajs-2518	173	10	module	module	NOUN
iajs-2518	173	11	x	x	PUNCT
iajs-2518	173	12	such	such	ADJ
iajs-2518	173	13	that	that	SCONJ
iajs-2518	173	14	a	a	PRON
iajs-2518	173	15	is	be	AUX
iajs-2518	173	16	intersection	intersection	NOUN
iajs-2518	173	17	of	of	ADP
iajs-2518	173	18	a	a	DET
iajs-2518	173	19	finite	finite	ADJ
iajs-2518	173	20	number	number	NOUN
iajs-2518	173	21	of	of	ADP
iajs-2518	173	22	fmaximal	fmaximal	ADJ
iajs-2518	173	23	submodules	submodule	NOUN
iajs-2518	173	24	of	of	ADP
iajs-2518	173	25	x	x	X
iajs-2518	173	26	.	.	PUNCT
iajs-2518	174	1	then	then	ADV
iajs-2518	174	2	,	,	PUNCT
iajs-2518	174	3	a	a	PRON
iajs-2518	174	4	is	be	AUX
iajs-2518	174	5	a	a	DET
iajs-2518	174	6	f	f	PROPN
iajs-2518	174	7	semimaximal	semimaximal	PROPN
iajs-2518	174	8	submodule	submodule	NOUN
iajs-2518	174	9	.	.	PUNCT
iajs-2518	175	1	proof	proof	NOUN
iajs-2518	175	2	:	:	PUNCT
iajs-2518	175	3	let	let	VERB
iajs-2518	175	4	𝐴	𝐴	PROPN
iajs-2518	175	5	𝐴	𝐴	PROPN
iajs-2518	175	6	∩	∩	NOUN
iajs-2518	175	7	𝐴	𝐴	PROPN
iajs-2518	175	8	∩	∩	NOUN
iajs-2518	175	9	…	…	PUNCT
iajs-2518	175	10	…	…	PUNCT
iajs-2518	175	11	…	…	PUNCT
iajs-2518	175	12	∩	∩	X
iajs-2518	175	13	𝐴	𝐴	PROPN
iajs-2518	175	14	,	,	PUNCT
iajs-2518	175	15	𝐴	𝐴	PROPN
iajs-2518	175	16	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	175	17	𝑎	𝑎	DET
iajs-2518	175	18	𝐹	𝐹	PROPN
iajs-2518	175	19	submodule	submodule	NOUN
iajs-2518	175	20	of	of	ADP
iajs-2518	175	21	x	x	PRON
iajs-2518	175	22	,	,	PUNCT
iajs-2518	175	23	∀i	∀i	X
iajs-2518	175	24	1,2	1,2	NUM
iajs-2518	175	25	,	,	PUNCT
iajs-2518	175	26	…	…	PUNCT
iajs-2518	175	27	…	…	PUNCT
iajs-2518	175	28	…	…	PUNCT
iajs-2518	175	29	.	.	PUNCT
iajs-2518	176	1	n.	n.	PROPN
iajs-2518	176	2	hence	hence	ADV
iajs-2518	176	3	,	,	PUNCT
iajs-2518	176	4	byproposition1.18	byproposition1.18	PROPN
iajs-2518	176	5	,	,	PUNCT
iajs-2518	176	6	𝑋	𝑋	PROPN
iajs-2518	176	7	𝐴⁄	𝐴⁄	PROPN
iajs-2518	176	8	≅	≅	NOUN
iajs-2518	176	9	𝐹	𝐹	PROPN
iajs-2518	176	10	𝑠𝑢𝑏𝑚𝑑𝑢𝑙𝑒	𝑠𝑢𝑏𝑚𝑑𝑢𝑙𝑒	ADP
iajs-2518	176	11	𝑜𝑓	𝑜𝑓	ADP
iajs-2518	176	12	𝑋|𝐴	𝑋|𝐴	NOUN
iajs-2518	176	13			ADJ
iajs-2518	176	14	𝑋|𝐴	𝑋|𝐴	NOUN
iajs-2518	176	15	…	…	PUNCT
iajs-2518	176	16	…	…	PUNCT
iajs-2518	176	17	…	…	PUNCT
iajs-2518	176	18	…	…	PUNCT
iajs-2518	176	19	.	.	PUNCT
iajs-2518	177	1	.	.	PROPN
iajs-2518	177	2	𝑋|𝐴	𝑋|𝐴	ADV
iajs-2518	178	1	but	but	CCONJ
iajs-2518	178	2	is	be	AUX
iajs-2518	178	3	simple	simple	ADJ
iajs-2518	178	4	for	for	ADP
iajs-2518	178	5	each	each	DET
iajs-2518	178	6	𝑋/	𝑋/	PROPN
iajs-2518	178	7	𝐴	𝐴	PROPN
iajs-2518	178	8	,	,	PUNCT
iajs-2518	179	1	so	so	ADV
iajs-2518	179	2	𝑋/	𝑋/	ADJ
iajs-2518	179	3	𝐴	𝐴	PROPN
iajs-2518	179	4			ADJ
iajs-2518	179	5	𝑋	𝑋	PROPN
iajs-2518	179	6	/	/	SYM
iajs-2518	179	7	𝐴	𝐴	PROPN
iajs-2518	179	8	…	…	PUNCT
iajs-2518	179	9	…	…	PUNCT
iajs-2518	179	10	…	…	PUNCT
iajs-2518	179	11	…	…	PUNCT
iajs-2518	179	12	..	..	NOUN
iajs-2518	179	13	𝑋|	𝑋|	NOUN
iajs-2518	179	14	/𝐴	/𝐴	PUNCT
iajs-2518	179	15	is	be	AUX
iajs-2518	179	16	semisimple	semisimple	ADJ
iajs-2518	179	17	and	and	CCONJ
iajs-2518	179	18	since	since	SCONJ
iajs-2518	179	19	a	a	DET
iajs-2518	179	20	fsubmodule	fsubmodule	NOUN
iajs-2518	179	21	of	of	ADP
iajs-2518	179	22	fuzzy	fuzzy	ADJ
iajs-2518	179	23	semisimple	semisimple	NOUN
iajs-2518	179	24	is	be	AUX
iajs-2518	179	25	f	f	PROPN
iajs-2518	179	26	semisimple	semisimple	NOUN
iajs-2518	179	27	,	,	PUNCT
iajs-2518	179	28	therefore	therefore	ADV
iajs-2518	179	29	𝑋	𝑋	PROPN
iajs-2518	179	30	𝐴⁄	𝐴⁄	PROPN
iajs-2518	179	31	is	be	AUX
iajs-2518	179	32	a	a	DET
iajs-2518	179	33	f	f	PROPN
iajs-2518	179	34	semisimple	semisimple	NOUN
iajs-2518	179	35	module	module	NOUN
iajs-2518	179	36	thus	thus	ADV
iajs-2518	179	37	a	a	PRON
iajs-2518	179	38	is	be	AUX
iajs-2518	179	39	fsemimaxmal	fsemimaxmal	ADJ
iajs-2518	179	40	submodule	submodule	NOUN
iajs-2518	179	41	.	.	PUNCT
iajs-2518	180	1	remark	remark	VERB
iajs-2518	180	2	3.8	3.8	NUM
iajs-2518	180	3	the	the	DET
iajs-2518	180	4	converse	converse	NOUN
iajs-2518	180	5	of	of	ADP
iajs-2518	180	6	proposition	proposition	NOUN
iajs-2518	180	7	3.7	3.7	NUM
iajs-2518	180	8	is	be	AUX
iajs-2518	180	9	not	not	PART
iajs-2518	180	10	true	true	ADJ
iajs-2518	180	11	in	in	ADP
iajs-2518	180	12	general	general	ADJ
iajs-2518	180	13	.	.	PUNCT
iajs-2518	181	1	we	we	PRON
iajs-2518	181	2	can	can	AUX
iajs-2518	181	3	give	give	VERB
iajs-2518	181	4	the	the	DET
iajs-2518	181	5	following	follow	VERB
iajs-2518	181	6	example	example	NOUN
iajs-2518	181	7	:	:	PUNCT
iajs-2518	181	8	let	let	VERB
iajs-2518	181	9	𝑀	𝑀	PROPN
iajs-2518	181	10			ADJ
iajs-2518	181	11	𝑍	𝑍	NOUN
iajs-2518	181	12	𝑎𝑠	𝑎𝑠	NOUN
iajs-2518	181	13	𝑍	𝑍	PROPN
iajs-2518	181	14	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	181	15	,	,	PUNCT
iajs-2518	181	16	𝑝	𝑝	NOUN
iajs-2518	181	17	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	181	18	𝑎	𝑎	DET
iajs-2518	181	19	𝑝𝑟𝑖𝑚𝑒	𝑝𝑟𝑖𝑚𝑒	NOUN
iajs-2518	181	20	𝑛𝑢𝑚𝑏𝑒𝑟	𝑛𝑢𝑚𝑏𝑒𝑟	NOUN
iajs-2518	181	21	define	define	VERB
iajs-2518	181	22	𝑋	𝑋	PROPN
iajs-2518	181	23	∶	∶	NOUN
iajs-2518	181	24	𝑀	𝑀	PROPN
iajs-2518	181	25	→	→	SYM
iajs-2518	181	26	0,1	0,1	NUM
iajs-2518	181	27	and	and	CCONJ
iajs-2518	181	28	𝐴	𝐴	PROPN
iajs-2518	181	29	:	:	PUNCT
iajs-2518	181	30	𝑀	𝑀	PROPN
iajs-2518	181	31	→	→	SYM
iajs-2518	181	32	0,1	0,1	NUM
iajs-2518	181	33	by	by	ADP
iajs-2518	181	34	𝑋	𝑋	PROPN
iajs-2518	181	35	𝑥	𝑥	PROPN
iajs-2518	181	36	1	1	NUM
iajs-2518	181	37	,	,	PUNCT
iajs-2518	181	38	∀𝑥	∀𝑥	PROPN
iajs-2518	181	39	∈	∈	PROPN
iajs-2518	181	40	𝑀	𝑀	PROPN
iajs-2518	181	41	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	181	42	𝐴	𝐴	PROPN
iajs-2518	181	43	𝑥	𝑥	PROPN
iajs-2518	181	44	1	1	NUM
iajs-2518	181	45	,	,	PUNCT
iajs-2518	181	46	∀x	∀x	VERB
iajs-2518	181	47	∈	∈	NOUN
iajs-2518	181	48	𝑍	𝑍	NOUN
iajs-2518	181	49	but	but	CCONJ
iajs-2518	181	50	𝑋	𝑋	PROPN
iajs-2518	181	51	𝐴⁄	𝐴⁄	PROPN
iajs-2518	181	52	𝑥	𝑥	PRON
iajs-2518	181	53	𝐴∗	𝐴∗	NUM
iajs-2518	181	54	1	1	NUM
iajs-2518	181	55	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	181	56	𝑥	𝑥	PRON
iajs-2518	181	57	∈	∈	PROPN
iajs-2518	181	58	𝐴∗	𝐴∗	NUM
iajs-2518	181	59	sup	sup	NOUN
iajs-2518	181	60	𝑋	𝑋	PROPN
iajs-2518	181	61	𝑥	𝑥	PROPN
iajs-2518	181	62	𝑏	𝑏	NOUN
iajs-2518	181	63	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	181	64	𝑏	𝑏	PROPN
iajs-2518	181	65	∈	∈	PROPN
iajs-2518	181	66	𝐴∗	𝐴∗	PROPN
iajs-2518	181	67	,	,	PUNCT
iajs-2518	181	68	𝑥	𝑥	DET
iajs-2518	181	69	𝐴∗	𝐴∗	NOUN
iajs-2518	181	70	thus	thus	ADV
iajs-2518	181	71	𝑋	𝑋	PROPN
iajs-2518	181	72	𝐴⁄	𝐴⁄	PROPN
iajs-2518	181	73	𝑥	𝑥	PRON
iajs-2518	181	74	𝐴∗	𝐴∗	NUM
iajs-2518	181	75	1	1	NUM
iajs-2518	181	76	,	,	PUNCT
iajs-2518	181	77	∀	∀	X
iajs-2518	181	78	x	x	SYM
iajs-2518	181	79	𝐴∗	𝐴∗	NUM
iajs-2518	181	80	∈	∈	PROPN
iajs-2518	181	81	𝑀|𝐴∗	𝑀|𝐴∗	VERB
iajs-2518	181	82			ADJ
iajs-2518	181	83	𝑍	𝑍	PROPN
iajs-2518	181	84	,	,	PUNCT
iajs-2518	181	85	𝑝	𝑝	PROPN
iajs-2518	181	86	2	2	NUM
iajs-2518	181	87	,	,	PUNCT
iajs-2518	181	88	is	be	AUX
iajs-2518	181	89	semisimple	semisimple	NOUN
iajs-2518	181	90	.	.	PUNCT
iajs-2518	182	1	thus	thus	ADV
iajs-2518	182	2	𝐴	𝐴	PROPN
iajs-2518	182	3	is	be	AUX
iajs-2518	182	4	𝑎	𝑎	PROPN
iajs-2518	182	5	f	f	X
iajs-2518	182	6	semimaxmal	semimaxmal	PROPN
iajs-2518	182	7	submodule	submodule	NOUN
iajs-2518	182	8	.	.	PUNCT
iajs-2518	183	1	but	but	CCONJ
iajs-2518	183	2	a	a	PRON
iajs-2518	183	3	is	be	AUX
iajs-2518	183	4	not	not	PART
iajs-2518	183	5	a	a	DET
iajs-2518	183	6	finite	finite	ADJ
iajs-2518	183	7	intersection	intersection	NOUN
iajs-2518	183	8	of	of	ADP
iajs-2518	183	9	f	f	PROPN
iajs-2518	183	10	maximal	maximal	ADJ
iajs-2518	183	11	submodule	submodule	NOUN
iajs-2518	183	12	.	.	PUNCT
iajs-2518	184	1	proposition	proposition	NOUN
iajs-2518	184	2	3.9	3.9	NUM
iajs-2518	184	3	if	if	SCONJ
iajs-2518	184	4	x	x	PRON
iajs-2518	184	5	is	be	AUX
iajs-2518	184	6	a	a	DET
iajs-2518	184	7	f	f	NOUN
iajs-2518	184	8	module	module	NOUN
iajs-2518	184	9	of	of	ADP
iajs-2518	184	10	an	an	DET
iajs-2518	184	11	rmodule	rmodule	NOUN
iajs-2518	184	12	m	m	PROPN
iajs-2518	184	13	and	and	CCONJ
iajs-2518	184	14	r	r	NOUN
iajs-2518	184	15	is	be	AUX
iajs-2518	184	16	a	a	DET
iajs-2518	184	17	semisimple	semisimple	NOUN
iajs-2518	184	18	ring	ring	NOUN
iajs-2518	184	19	,	,	PUNCT
iajs-2518	184	20	then	then	ADV
iajs-2518	184	21	every	every	DET
iajs-2518	184	22	f	f	PROPN
iajs-2518	184	23	submodule	submodule	PROPN
iajs-2518	184	24	a	a	PRON
iajs-2518	184	25	of	of	ADP
iajs-2518	184	26	x	x	NOUN
iajs-2518	184	27	is	be	AUX
iajs-2518	184	28	semimaximal	semimaximal	ADJ
iajs-2518	184	29	.provided	.provide	VERB
iajs-2518	184	30	𝑋𝑙𝐴	𝑋𝑙𝐴	PROPN
iajs-2518	184	31	∗	∗	NOUN
iajs-2518	184	32	𝑋∗|𝐴∗	𝑋∗|𝐴∗	NOUN
iajs-2518	184	33	proof	proof	NOUN
iajs-2518	184	34	:	:	PUNCT
iajs-2518	184	35	let	let	VERB
iajs-2518	184	36	r	r	NOUN
iajs-2518	184	37	be	be	AUX
iajs-2518	184	38	semisimple	semisimple	NOUN
iajs-2518	184	39	ring	ring	NOUN
iajs-2518	184	40	.	.	PUNCT
iajs-2518	185	1	then	then	ADV
iajs-2518	185	2	m	m	PROPN
iajs-2518	185	3	is	be	AUX
iajs-2518	185	4	a	a	DET
iajs-2518	185	5	semisimple	semisimple	ADJ
iajs-2518	185	6	r	r	NOUN
iajs-2518	185	7	-	-	PUNCT
iajs-2518	185	8	module	module	NOUN
iajs-2518	185	9	.since	.since	NOUN
iajs-2518	185	10	x	x	PRON
iajs-2518	185	11	is	be	AUX
iajs-2518	185	12	a	a	DET
iajs-2518	185	13	fmodule	fmodule	NOUN
iajs-2518	185	14	over	over	ADP
iajs-2518	185	15	m	m	PROPN
iajs-2518	185	16	,	,	PUNCT
iajs-2518	185	17	then	then	ADV
iajs-2518	185	18	𝑋∗	𝑋∗	ADJ
iajs-2518	185	19	𝑀	𝑀	PROPN
iajs-2518	185	20	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	185	21	ℎ𝑒𝑛𝑐𝑒	ℎ𝑒𝑛𝑐𝑒	PROPN
iajs-2518	185	22	𝑋∗	𝑋∗	ADJ
iajs-2518	185	23	is	be	AUX
iajs-2518	185	24	semisimple	semisimple	NOUN
iajs-2518	185	25	which	which	PRON
iajs-2518	185	26	implies	imply	VERB
iajs-2518	185	27	that	that	SCONJ
iajs-2518	185	28	𝑋∗	𝑋∗	ADJ
iajs-2518	185	29	/	/	SYM
iajs-2518	185	30	𝐴∗	𝐴∗	NUM
iajs-2518	185	31	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	185	32	semisimple	semisimple	NOUN
iajs-2518	185	33	.	.	PUNCT
iajs-2518	186	1	𝑇ℎ𝑢𝑠	𝑇ℎ𝑢𝑠	PROPN
iajs-2518	186	2	𝐴∗	𝐴∗	PROPN
iajs-2518	186	3	𝑖𝑠	𝑖𝑠	CCONJ
iajs-2518	186	4	𝑎	𝑎	DET
iajs-2518	186	5	semimaximal	semimaximal	ADJ
iajs-2518	186	6	submodule	submodule	NOUN
iajs-2518	186	7	of	of	ADP
iajs-2518	186	8	𝑋∗.	𝑋∗.	PROPN
iajs-2518	186	9	then	then	ADV
iajs-2518	186	10	by	by	ADP
iajs-2518	186	11	proposition	proposition	NOUN
iajs-2518	186	12	3.4	3.4	NUM
iajs-2518	186	13	𝐴	𝐴	PROPN
iajs-2518	186	14	is	be	AUX
iajs-2518	186	15	a	a	DET
iajs-2518	186	16	f	f	PROPN
iajs-2518	186	17	semimaximal	semimaximal	NOUN
iajs-2518	186	18	submodule	submodule	NOUN
iajs-2518	186	19	of	of	ADP
iajs-2518	186	20	x	x	PROPN
iajs-2518	186	21	.	.	PUNCT
iajs-2518	187	1	proposition	proposition	NOUN
iajs-2518	187	2	3.10	3.10	NUM
iajs-2518	187	3	the	the	DET
iajs-2518	187	4	intersection	intersection	NOUN
iajs-2518	187	5	of	of	ADP
iajs-2518	187	6	two	two	NUM
iajs-2518	187	7	semimaximal	semimaximal	ADJ
iajs-2518	187	8	fsubmodules	fsubmodule	NOUN
iajs-2518	187	9	is	be	AUX
iajs-2518	187	10	aslo	aslo	PROPN
iajs-2518	187	11	semimaximal	semimaximal	ADJ
iajs-2518	187	12	.	.	PUNCT
iajs-2518	188	1	proof	proof	NOUN
iajs-2518	188	2	:	:	PUNCT
iajs-2518	188	3	if	if	SCONJ
iajs-2518	188	4	a	a	DET
iajs-2518	188	5	,	,	PUNCT
iajs-2518	188	6	b	b	NOUN
iajs-2518	188	7	are	be	AUX
iajs-2518	188	8	two	two	NUM
iajs-2518	188	9	semimaximal	semimaximal	ADJ
iajs-2518	188	10	f	f	PROPN
iajs-2518	188	11	submodules	submodule	NOUN
iajs-2518	188	12	of	of	ADP
iajs-2518	188	13	module	module	NOUN
iajs-2518	188	14	x	x	SYM
iajs-2518	188	15	.then	.then	PROPN
iajs-2518	188	16	𝑋	𝑋	PROPN
iajs-2518	188	17	𝐴	𝐴	PROPN
iajs-2518	188	18	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2518	188	19	𝑋	𝑋	PROPN
iajs-2518	188	20	𝐵⁄⁄	𝐵⁄⁄	NOUN
iajs-2518	188	21	are	be	AUX
iajs-2518	188	22	semisimple	semisimple	ADJ
iajs-2518	188	23	.	.	PUNCT
iajs-2518	189	1	therefore	therefore	ADV
iajs-2518	189	2	,	,	PUNCT
iajs-2518	189	3	𝑋	𝑋	PROPN
iajs-2518	189	4	𝐴	𝐴	PROPN
iajs-2518	189	5	⨁	⨁	PROPN
iajs-2518	189	6	𝑋	𝑋	PROPN
iajs-2518	189	7	𝐵⁄⁄	𝐵⁄⁄	VERB
iajs-2518	189	8	is	be	AUX
iajs-2518	189	9	semisimple	semisimple	NOUN
iajs-2518	189	10	.	.	PUNCT
iajs-2518	190	1	as	as	SCONJ
iajs-2518	190	2	𝑋	𝑋	PROPN
iajs-2518	190	3	𝐴	𝐴	PROPN
iajs-2518	190	4	∩	∩	NOUN
iajs-2518	190	5	𝐵⁄	𝐵⁄	PROPN
iajs-2518	190	6	≅	≅	PROPN
iajs-2518	190	7	f	f	PROPN
iajs-2518	190	8	subodule	subodule	NOUN
iajs-2518	190	9	of	of	ADP
iajs-2518	190	10	𝑋	𝑋	PROPN
iajs-2518	190	11	𝐴	𝐴	PROPN
iajs-2518	190	12	⨁	⨁	PROPN
iajs-2518	190	13	𝑋	𝑋	PROPN
iajs-2518	190	14	𝐵⁄⁄	𝐵⁄⁄	VERB
iajs-2518	190	15	.	.	PUNCT
iajs-2518	191	1	but	but	CCONJ
iajs-2518	191	2	any	any	DET
iajs-2518	191	3	f	f	PROPN
iajs-2518	191	4	submodule	submodule	NOUN
iajs-2518	191	5	of	of	ADP
iajs-2518	191	6	𝑋	𝑋	PROPN
iajs-2518	191	7	𝐴	𝐴	PROPN
iajs-2518	191	8	⨁	⨁	PROPN
iajs-2518	191	9	𝑋	𝑋	PROPN
iajs-2518	191	10	𝐵⁄⁄	𝐵⁄⁄	VERB
iajs-2518	191	11	is	be	AUX
iajs-2518	191	12	semisimple	semisimple	NOUN
iajs-2518	191	13	,	,	PUNCT
iajs-2518	191	14	hence	hence	ADV
iajs-2518	191	15	𝑋	𝑋	PROPN
iajs-2518	191	16	𝐴⋂𝐵⁄	𝐴⋂𝐵⁄	PROPN
iajs-2518	191	17	is	be	AUX
iajs-2518	191	18	semisimple	semisimple	NOUN
iajs-2518	191	19	therefore	therefore	ADV
iajs-2518	191	20	𝐴	𝐴	PROPN
iajs-2518	191	21	∩	∩	NOUN
iajs-2518	191	22	𝐵	𝐵	NOUN
iajs-2518	191	23	is	be	AUX
iajs-2518	191	24	a	a	DET
iajs-2518	191	25	f	f	NOUN
iajs-2518	191	26	-semimaximal	-semimaximal	NOUN
iajs-2518	191	27	submodule	submodule	NOUN
iajs-2518	191	28	of	of	ADP
iajs-2518	191	29	x	x	PROPN
iajs-2518	191	30	.	.	PUNCT
iajs-2518	191	31	  	  	SPACE
iajs-2518	192	1	145	145	NUM
iajs-2518	192	2	  	  	SPACE
iajs-2518	192	3	ibn	ibn	PROPN
iajs-2518	192	4	al	al	PROPN
iajs-2518	192	5	-	-	PUNCT
iajs-2518	192	6	haitham	haitham	PROPN
iajs-2518	192	7	jour	jour	X
iajs-2518	192	8	.	.	PROPN
iajs-2518	192	9	for	for	ADP
iajs-2518	192	10	pure	pure	ADJ
iajs-2518	192	11	&	&	CCONJ
iajs-2518	192	12	appl	appl	PROPN
iajs-2518	192	13	.	.	PUNCT
iajs-2518	193	1	sci	sci	PROPN
iajs-2518	193	2	.	.	PROPN
iajs-2518	194	1	33	33	NUM
iajs-2518	194	2	(	(	PUNCT
iajs-2518	194	3	4	4	NUM
iajs-2518	194	4	)	)	PUNCT
iajs-2518	194	5	2020	2020	NUM
iajs-2518	194	6	proposition	proposition	NOUN
iajs-2518	194	7	3.11	3.11	NUM
iajs-2518	194	8	if	if	SCONJ
iajs-2518	194	9	a	a	DET
iajs-2518	194	10	,	,	PUNCT
iajs-2518	194	11	b	b	NOUN
iajs-2518	194	12	are	be	AUX
iajs-2518	194	13	two	two	NUM
iajs-2518	194	14	f	f	PROPN
iajs-2518	194	15	submodules	submodule	NOUN
iajs-2518	194	16	of	of	ADP
iajs-2518	194	17	f	f	PROPN
iajs-2518	194	18	module	module	NOUN
iajs-2518	194	19	x	x	PUNCT
iajs-2518	194	20	such	such	ADJ
iajs-2518	194	21	that	that	SCONJ
iajs-2518	194	22	a	a	PRON
iajs-2518	194	23	is	be	AUX
iajs-2518	194	24	semimaximal	semimaximal	ADJ
iajs-2518	194	25	in	in	ADP
iajs-2518	194	26	x	x	PUNCT
iajs-2518	194	27	and	and	CCONJ
iajs-2518	194	28	b	b	PROPN
iajs-2518	194	29	contains	contain	VERB
iajs-2518	194	30	a	a	PRON
iajs-2518	194	31	.	.	PUNCT
iajs-2518	195	1	then	then	ADV
iajs-2518	195	2	b	b	PROPN
iajs-2518	195	3	is	be	AUX
iajs-2518	195	4	semimaximal	semimaximal	ADJ
iajs-2518	195	5	in	in	ADP
iajs-2518	195	6	x	x	X
iajs-2518	195	7	.	.	PUNCT
iajs-2518	196	1	proof	proof	NOUN
iajs-2518	196	2	:	:	PUNCT
iajs-2518	196	3	𝐵	𝐵	NOUN
iajs-2518	196	4	𝐴⁄	𝐴⁄	PROPN
iajs-2518	196	5	is	be	AUX
iajs-2518	196	6	fsubmodule	fsubmodule	NOUN
iajs-2518	196	7	of	of	ADP
iajs-2518	196	8	𝑋	𝑋	PROPN
iajs-2518	196	9	𝐴⁄	𝐴⁄	PROPN
iajs-2518	196	10	since	since	SCONJ
iajs-2518	196	11	a	a	DET
iajs-2518	196	12	b	b	NOUN
iajs-2518	196	13	and	and	CCONJ
iajs-2518	196	14	𝑋	𝑋	PROPN
iajs-2518	196	15	𝐴⁄	𝐴⁄	PROPN
iajs-2518	196	16	≅	≅	NOUN
iajs-2518	196	17	𝑋	𝑋	PROPN
iajs-2518	196	18	𝐵⁄	𝐵⁄	PROPN
iajs-2518	196	19	,	,	PUNCT
iajs-2518	196	20	by	by	ADP
iajs-2518	196	21	third	third	ADJ
iajs-2518	196	22	isomorphism	isomorphism	NOUN
iajs-2518	196	23	theorem	theorem	VERB
iajs-2518	196	24	.	.	PUNCT
iajs-2518	197	1	but	but	CCONJ
iajs-2518	197	2	x	x	X
iajs-2518	197	3	/a	/a	PUNCT
iajs-2518	197	4	is	be	AUX
iajs-2518	197	5	semisimple	semisimple	ADJ
iajs-2518	197	6	,	,	PUNCT
iajs-2518	197	7	imples	imple	VERB
iajs-2518	197	8	𝑋	𝑋	NOUN
iajs-2518	197	9	/𝐴	/𝐴	PUNCT
iajs-2518	197	10	/	/	SYM
iajs-2518	197	11	𝐵	𝐵	NOUN
iajs-2518	197	12	/𝐴	/𝐴	PUNCT
iajs-2518	197	13	is	be	AUX
iajs-2518	197	14	semisimple	semisimple	ADJ
iajs-2518	197	15	.	.	PUNCT
iajs-2518	198	1	that	that	PRON
iajs-2518	198	2	is	be	AUX
iajs-2518	198	3	𝑋	𝑋	PROPN
iajs-2518	198	4	/	/	SYM
iajs-2518	198	5	𝐵	𝐵	NOUN
iajs-2518	198	6	is	be	AUX
iajs-2518	198	7	semisimple	semisimple	NOUN
iajs-2518	198	8	.	.	PUNCT
iajs-2518	199	1	therefore	therefore	ADV
iajs-2518	199	2	b	b	PROPN
iajs-2518	199	3	is	be	AUX
iajs-2518	199	4	semimaximal	semimaximal	ADJ
iajs-2518	199	5	in	in	ADP
iajs-2518	199	6	x	x	X
iajs-2518	199	7	.	.	PUNCT
iajs-2518	200	1	the	the	DET
iajs-2518	200	2	following	follow	VERB
iajs-2518	200	3	corollary	corollary	NOUN
iajs-2518	200	4	immediately	immediately	ADV
iajs-2518	200	5	consequence	consequence	NOUN
iajs-2518	200	6	of	of	ADP
iajs-2518	200	7	proposition	proposition	NOUN
iajs-2518	200	8	3.11	3.11	NUM
iajs-2518	200	9	corollary	corollary	NOUN
iajs-2518	200	10	3.12	3.12	NUM
iajs-2518	200	11	if	if	SCONJ
iajs-2518	200	12	å	å	PROPN
iajs-2518	200	13	is	be	AUX
iajs-2518	200	14	a	a	DET
iajs-2518	200	15	semimaximal	semimaximal	ADJ
iajs-2518	200	16	fsubmodule	fsubmodule	NOUN
iajs-2518	200	17	of	of	ADP
iajs-2518	200	18	f	f	PROPN
iajs-2518	200	19	module	module	NOUN
iajs-2518	200	20	x	x	PUNCT
iajs-2518	200	21	and	and	CCONJ
iajs-2518	200	22	k	k	PROPN
iajs-2518	200	23	be	be	AUX
iajs-2518	200	24	a	a	DET
iajs-2518	200	25	f	f	PROPN
iajs-2518	200	26	ideal	ideal	NOUN
iajs-2518	200	27	of	of	ADP
iajs-2518	200	28	a	a	DET
iajs-2518	200	29	ring	ring	NOUN
iajs-2518	200	30	r	r	NOUN
iajs-2518	200	31	.	.	PUNCT
iajs-2518	201	1	then	then	ADV
iajs-2518	201	2	å	å	X
iajs-2518	201	3	:	:	PUNCT
iajs-2518	201	4	𝐾	𝐾	NOUN
iajs-2518	201	5	is	be	AUX
iajs-2518	201	6	a	a	DET
iajs-2518	201	7	fsubmodule	fsubmodule	ADJ
iajs-2518	201	8	semimaximal	semimaximal	NOUN
iajs-2518	201	9	.	.	PUNCT
iajs-2518	202	1	proof	proof	NOUN
iajs-2518	202	2	:	:	PUNCT
iajs-2518	202	3	since	since	SCONJ
iajs-2518	202	4	å	å	PROPN
iajs-2518	202	5	:	:	PUNCT
iajs-2518	202	6	𝐾	𝐾	NOUN
iajs-2518	202	7	is	be	AUX
iajs-2518	202	8	a	a	DET
iajs-2518	202	9	fuzzy	fuzzy	ADJ
iajs-2518	202	10	submodule	submodule	NOUN
iajs-2518	202	11	of	of	ADP
iajs-2518	202	12	x	x	PUNCT
iajs-2518	202	13	containing	contain	VERB
iajs-2518	202	14	å	å	PROPN
iajs-2518	202	15	,	,	PUNCT
iajs-2518	202	16	then	then	ADV
iajs-2518	202	17	result	result	NOUN
iajs-2518	202	18	follows	follow	VERB
iajs-2518	202	19	by	by	ADP
iajs-2518	202	20	proposition	proposition	NOUN
iajs-2518	202	21	3.11	3.11	NUM
iajs-2518	202	22	however	however	ADV
iajs-2518	202	23	the	the	DET
iajs-2518	202	24	converse	converse	NOUN
iajs-2518	202	25	of	of	ADP
iajs-2518	202	26	corollary	corollary	ADJ
iajs-2518	202	27	3.12	3.12	NUM
iajs-2518	202	28	.	.	PUNCT
iajs-2518	202	29	is	be	AUX
iajs-2518	202	30	not	not	PART
iajs-2518	202	31	true	true	ADJ
iajs-2518	202	32	in	in	ADP
iajs-2518	202	33	general	general	ADJ
iajs-2518	202	34	,	,	PUNCT
iajs-2518	202	35	for	for	ADP
iajs-2518	202	36	example	example	NOUN
iajs-2518	202	37	example	example	NOUN
iajs-2518	202	38	3.13	3.13	NUM
iajs-2518	202	39	consider	consider	VERB
iajs-2518	202	40	𝑀	𝑀	PRON
iajs-2518	202	41	𝑍	𝑍	VERB
iajs-2518	202	42	𝑎𝑠	𝑎𝑠	NOUN
iajs-2518	202	43	𝑍	𝑍	PROPN
iajs-2518	202	44	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	202	45	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	202	46	𝑙𝑒𝑡	𝑙𝑒𝑡	PROPN
iajs-2518	202	47	𝑋	𝑋	PROPN
iajs-2518	202	48	∶	∶	PROPN
iajs-2518	202	49	𝑀	𝑀	PROPN
iajs-2518	202	50	→	→	SYM
iajs-2518	202	51	0,1	0,1	NUM
iajs-2518	202	52	,	,	PUNCT
iajs-2518	202	53	𝐴	𝐴	PROPN
iajs-2518	202	54	∶	∶	NOUN
iajs-2518	202	55	𝑀	𝑀	PROPN
iajs-2518	202	56	→	→	SYM
iajs-2518	202	57	0,1	0,1	NUM
iajs-2518	202	58	,	,	PUNCT
iajs-2518	202	59	𝑑𝑒𝑓𝑖𝑛𝑒𝑑	𝑑𝑒𝑓𝑖𝑛𝑒𝑑	NOUN
iajs-2518	202	60	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	202	61	:	:	PUNCT
iajs-2518	202	62	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	PROPN
iajs-2518	202	63	𝑋	𝑋	PROPN
iajs-2518	202	64	𝑥	𝑥	PROPN
iajs-2518	202	65	1	1	NUM
iajs-2518	202	66	,	,	PUNCT
iajs-2518	202	67	∀𝑥	∀𝑥	PROPN
iajs-2518	202	68	∈	∈	PROPN
iajs-2518	202	69	𝑀	𝑀	PROPN
iajs-2518	202	70	,	,	PUNCT
iajs-2518	202	71	𝐴	𝐴	PROPN
iajs-2518	202	72	𝑥	𝑥	PROPN
iajs-2518	202	73	1	1	NUM
iajs-2518	202	74	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	202	75	𝑥	𝑥	PRON
iajs-2518	202	76	∈	∈	NOUN
iajs-2518	202	77	0	0	NUM
iajs-2518	202	78	,	,	PUNCT
iajs-2518	202	79	,	,	PUNCT
iajs-2518	202	80	4	4	NUM
iajs-2518	202	81	,	,	PUNCT
iajs-2518	202	82	8	8	NUM
iajs-2518	202	83	,	,	PUNCT
iajs-2518	202	84	1	1	NUM
iajs-2518	202	85	2⁄	2⁄	NUM
iajs-2518	202	86	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	202	87	a	a	PRON
iajs-2518	202	88	is	be	AUX
iajs-2518	202	89	a	a	DET
iajs-2518	202	90	f	f	PROPN
iajs-2518	202	91	submodule	submodule	NOUN
iajs-2518	202	92	of	of	ADP
iajs-2518	202	93	x	x	X
iajs-2518	202	94	.	.	PUNCT
iajs-2518	203	1	let	let	VERB
iajs-2518	203	2	𝐾	𝐾	NOUN
iajs-2518	203	3	:	:	PUNCT
iajs-2518	203	4	𝑍	𝑍	PROPN
iajs-2518	203	5	→	→	SYM
iajs-2518	203	6	0,1	0,1	NUM
iajs-2518	203	7	defined	define	VERB
iajs-2518	203	8	by	by	ADP
iajs-2518	203	9	:	:	PUNCT
iajs-2518	203	10	𝐾	𝐾	NOUN
iajs-2518	203	11	𝑥	𝑥	PROPN
iajs-2518	203	12	1	1	NUM
iajs-2518	203	13	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	203	14	𝑥	𝑥	PRON
iajs-2518	203	15	∈	∈	PROPN
iajs-2518	203	16	2𝑍	2𝑍	PROPN
iajs-2518	203	17	1|3	1|3	PROPN
iajs-2518	203	18	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
iajs-2518	203	19	k	k	PROPN
iajs-2518	203	20	is	be	AUX
iajs-2518	203	21	a	a	DET
iajs-2518	203	22	fideal	fideal	NOUN
iajs-2518	203	23	of	of	ADP
iajs-2518	203	24	z	z	NOUN
iajs-2518	203	25	,	,	PUNCT
iajs-2518	203	26	then	then	ADV
iajs-2518	203	27	𝑋	𝑋	PROPN
iajs-2518	203	28	𝐴	𝐴	PROPN
iajs-2518	203	29	𝑥⁄	𝑥⁄	PROPN
iajs-2518	203	30	1	1	NUM
iajs-2518	203	31	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	203	32	𝑥	𝑥	PRON
iajs-2518	203	33	∈	∈	PROPN
iajs-2518	203	34	𝑍	𝑍	PROPN
iajs-2518	203	35	0	0	NUM
iajs-2518	203	36	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	203	37	note	note	NOUN
iajs-2518	203	38	that	that	SCONJ
iajs-2518	203	39	∗	∗	NOUN
iajs-2518	203	40	∗	∗	NOUN
iajs-2518	203	41	𝑍	𝑍	PROPN
iajs-2518	203	42	/	/	SYM
iajs-2518	203	43	4	4	NUM
iajs-2518	203	44	𝑍	𝑍	NOUN
iajs-2518	203	45	is	be	AUX
iajs-2518	203	46	not	not	PART
iajs-2518	203	47	semisimple	semisimple	NOUN
iajs-2518	203	48	.	.	PUNCT
iajs-2518	204	1	hence	hence	ADV
iajs-2518	204	2	𝐴∗	𝐴∗	NUM
iajs-2518	204	3	is	be	AUX
iajs-2518	204	4	not	not	PART
iajs-2518	204	5	semimaximal	semimaximal	ADJ
iajs-2518	204	6	.	.	PUNCT
iajs-2518	205	1	thus	thus	ADV
iajs-2518	205	2	a	a	DET
iajs-2518	205	3	is	be	AUX
iajs-2518	205	4	is	be	AUX
iajs-2518	205	5	not	not	PART
iajs-2518	205	6	semimaximal	semimaximal	ADJ
iajs-2518	205	7	proposirion	proposirion	NOUN
iajs-2518	205	8	3.4	3.4	NUM
iajs-2518	205	9	.	.	PUNCT
iajs-2518	206	1	however	however	ADV
iajs-2518	206	2	,	,	PUNCT
iajs-2518	206	3	[	[	PUNCT
iajs-2518	206	4	𝐴∗	𝐴∗	NUM
iajs-2518	206	5	:	:	PUNCT
iajs-2518	206	6	𝐾∗	𝐾∗	NUM
iajs-2518	206	7	4	4	NUM
iajs-2518	206	8	:	:	PUNCT
iajs-2518	206	9	2𝑍	2𝑍	NOUN
iajs-2518	206	10	2	2	NUM
iajs-2518	206	11	is	be	AUX
iajs-2518	206	12	a	a	DET
iajs-2518	206	13	semimaximal	semimaximal	ADJ
iajs-2518	206	14	submodule	submodule	NOUN
iajs-2518	206	15	of	of	ADP
iajs-2518	206	16	𝑍	𝑍	PROPN
iajs-2518	206	17	.	.	PUNCT
iajs-2518	207	1	but	but	CCONJ
iajs-2518	207	2	a	a	DET
iajs-2518	207	3	∶	∶	NOUN
iajs-2518	207	4	k	k	PROPN
iajs-2518	207	5	∗	∗	NOUN
iajs-2518	207	6	𝐴∗	𝐴∗	NUM
iajs-2518	208	1	∶	∶	NOUN
iajs-2518	208	2	𝐾∗	𝐾∗	PUNCT
iajs-2518	209	1	𝑏𝑦	𝑏𝑦	PROPN
iajs-2518	209	2	13	13	NUM
iajs-2518	209	3	.	.	PUNCT
iajs-2518	210	1	therefore	therefore	ADV
iajs-2518	210	2	𝐴	𝐴	PROPN
iajs-2518	210	3	:	:	PUNCT
iajs-2518	210	4	𝐾	𝐾	PROPN
iajs-2518	210	5	is	be	AUX
iajs-2518	210	6	a	a	DET
iajs-2518	210	7	semimaximal	semimaximal	ADJ
iajs-2518	210	8	f	f	PROPN
iajs-2518	210	9	submodule	submodule	NOUN
iajs-2518	210	10	of	of	ADP
iajs-2518	210	11	x	x	PROPN
iajs-2518	210	12	.	.	PUNCT
iajs-2518	211	1	proposition	proposition	NOUN
iajs-2518	211	2	3.14	3.14	NUM
iajs-2518	211	3	if	if	SCONJ
iajs-2518	211	4	a	a	PRON
iajs-2518	211	5	and	and	CCONJ
iajs-2518	211	6	b	b	NOUN
iajs-2518	211	7	are	be	AUX
iajs-2518	211	8	two	two	NUM
iajs-2518	211	9	f	f	PROPN
iajs-2518	211	10	submodules	submodule	NOUN
iajs-2518	211	11	of	of	ADP
iajs-2518	211	12	f	f	PROPN
iajs-2518	211	13	module	module	NOUN
iajs-2518	211	14	x	x	PUNCT
iajs-2518	211	15	such	such	ADJ
iajs-2518	211	16	that	that	SCONJ
iajs-2518	211	17	𝐵	𝐵	PROPN
iajs-2518	211	18	𝐴	𝐴	PROPN
iajs-2518	211	19	then	then	ADV
iajs-2518	211	20	,	,	PUNCT
iajs-2518	211	21	𝐴	𝐴	PROPN
iajs-2518	211	22	is	be	AUX
iajs-2518	211	23	semimaximal	semimaximal	ADJ
iajs-2518	211	24	in	in	ADP
iajs-2518	211	25	𝑋	𝑋	PROPN
iajs-2518	211	26	if	if	SCONJ
iajs-2518	212	1	and	and	CCONJ
iajs-2518	212	2	only	only	ADV
iajs-2518	212	3	if	if	SCONJ
iajs-2518	212	4	𝐴	𝐴	PROPN
iajs-2518	212	5	𝐵⁄	𝐵⁄	PROPN
iajs-2518	212	6	is	be	AUX
iajs-2518	212	7	a	a	DET
iajs-2518	212	8	semimaximal	semimaximal	ADJ
iajs-2518	212	9	fsubmodule	fsubmodule	NOUN
iajs-2518	212	10	of	of	ADP
iajs-2518	212	11	𝑋	𝑋	PROPN
iajs-2518	212	12	𝐵⁄	𝐵⁄	PROPN
iajs-2518	212	13	.	.	PUNCT
iajs-2518	213	1	proof	proof	NOUN
iajs-2518	213	2	:	:	PUNCT
iajs-2518	213	3	if	if	SCONJ
iajs-2518	213	4	a	a	PRON
iajs-2518	213	5	is	be	AUX
iajs-2518	213	6	semimaximal	semimaximal	ADJ
iajs-2518	213	7	in	in	ADP
iajs-2518	213	8	𝑋	𝑋	PROPN
iajs-2518	213	9	.then	.then	PUNCT
iajs-2518	213	10	𝑋	𝑋	PROPN
iajs-2518	213	11	𝐴⁄	𝐴⁄	PROPN
iajs-2518	213	12	is	be	AUX
iajs-2518	213	13	semisimple	semisimple	NOUN
iajs-2518	213	14	,	,	PUNCT
iajs-2518	213	15	which	which	PRON
iajs-2518	213	16	implies	imply	VERB
iajs-2518	213	17	is	be	AUX
iajs-2518	213	18	semisimple	semisimple	NOUN
iajs-2518	213	19	,	,	PUNCT
iajs-2518	213	20	since	since	SCONJ
iajs-2518	213	21	𝑋	𝑋	PROPN
iajs-2518	213	22	𝐴⁄	𝐴⁄	PROPN
iajs-2518	213	23	𝑋	𝑋	NOUN
iajs-2518	213	24	/𝐵	/𝐵	PUNCT
iajs-2518	213	25	/	/	SYM
iajs-2518	213	26	𝐴	𝐴	PROPN
iajs-2518	213	27	/	/	SYM
iajs-2518	213	28	𝐵	𝐵	NOUN
iajs-2518	213	29	.by	.by	PUNCT
iajs-2518	214	1	[	[	X
iajs-2518	214	2	second	second	ADJ
iajs-2518	214	3	isomorphism	isomorphism	NOUN
iajs-2518	214	4	theorem	theorem	VERB
iajs-2518	214	5	[	[	X
iajs-2518	214	6	14	14	NUM
iajs-2518	214	7	]	]	PUNCT
iajs-2518	214	8	hence	hence	ADV
iajs-2518	214	9	𝐴	𝐴	PROPN
iajs-2518	214	10	/	/	SYM
iajs-2518	214	11	𝐵	𝐵	PROPN
iajs-2518	214	12	is	be	AUX
iajs-2518	214	13	a	a	DET
iajs-2518	214	14	semimaximal	semimaximal	ADJ
iajs-2518	214	15	fsubmodule	fsubmodule	NOUN
iajs-2518	214	16	of	of	ADP
iajs-2518	214	17	𝑋	𝑋	PROPN
iajs-2518	214	18	/𝐵	/𝐵	PUNCT
iajs-2518	215	1	.the	.the	PRON
iajs-2518	215	2	converse	converse	NOUN
iajs-2518	215	3	is	be	AUX
iajs-2518	215	4	smilarly	smilarly	ADV
iajs-2518	215	5	.	.	PUNCT
iajs-2518	215	6	  	  	SPACE
iajs-2518	216	1	146	146	NUM
iajs-2518	216	2	  	  	SPACE
iajs-2518	216	3	ibn	ibn	PROPN
iajs-2518	216	4	al	al	PROPN
iajs-2518	216	5	-	-	PUNCT
iajs-2518	216	6	haitham	haitham	PROPN
iajs-2518	216	7	jour	jour	X
iajs-2518	216	8	.	.	PROPN
iajs-2518	216	9	for	for	ADP
iajs-2518	216	10	pure	pure	ADJ
iajs-2518	216	11	&	&	CCONJ
iajs-2518	216	12	appl	appl	PROPN
iajs-2518	216	13	.	.	PUNCT
iajs-2518	217	1	sci	sci	PROPN
iajs-2518	217	2	.	.	PROPN
iajs-2518	218	1	33	33	NUM
iajs-2518	218	2	(	(	PUNCT
iajs-2518	218	3	4	4	NUM
iajs-2518	218	4	)	)	PUNCT
iajs-2518	218	5	2020	2020	NUM
iajs-2518	218	6	proposition	proposition	NOUN
iajs-2518	218	7	3.15	3.15	NUM
iajs-2518	218	8	if	if	SCONJ
iajs-2518	218	9	k	k	PROPN
iajs-2518	218	10	is	be	AUX
iajs-2518	218	11	semimaximal	semimaximal	ADJ
iajs-2518	218	12	f	f	PROPN
iajs-2518	218	13	ideal	ideal	NOUN
iajs-2518	218	14	of	of	ADP
iajs-2518	218	15	a	a	DET
iajs-2518	218	16	ring	ring	NOUN
iajs-2518	218	17	r	r	NOUN
iajs-2518	218	18	and	and	CCONJ
iajs-2518	218	19	x	x	NOUN
iajs-2518	218	20	is	be	AUX
iajs-2518	218	21	be	be	AUX
iajs-2518	218	22	a	a	DET
iajs-2518	218	23	f	f	NOUN
iajs-2518	218	24	module	module	NOUN
iajs-2518	218	25	of	of	ADP
iajs-2518	218	26	rmodule	rmodule	NOUN
iajs-2518	218	27	m	m	PROPN
iajs-2518	218	28	,	,	PUNCT
iajs-2518	218	29	then	then	ADV
iajs-2518	218	30	kx	kx	PROPN
iajs-2518	218	31	is	be	AUX
iajs-2518	218	32	a	a	DET
iajs-2518	218	33	semimaximal	semimaximal	ADJ
iajs-2518	218	34	f	f	PROPN
iajs-2518	218	35	submodule	submodule	NOUN
iajs-2518	218	36	,	,	PUNCT
iajs-2518	218	37	provided	provide	VERB
iajs-2518	218	38	∗	∗	NOUN
iajs-2518	218	39	x∗	x∗	PROPN
iajs-2518	218	40	/	/	SYM
iajs-2518	219	1	k∗	k∗	PROPN
iajs-2518	219	2	x∗	x∗	PROPN
iajs-2518	219	3	poorf	poorf	PROPN
iajs-2518	219	4	:	:	PUNCT
iajs-2518	219	5	suppose	suppose	VERB
iajs-2518	219	6	k	k	PROPN
iajs-2518	219	7	is	be	AUX
iajs-2518	219	8	semimaximal	semimaximal	ADJ
iajs-2518	219	9	f	f	AUX
iajs-2518	219	10	ideal	ideal	VERB
iajs-2518	219	11	a	a	DET
iajs-2518	219	12	ring	ring	NOUN
iajs-2518	219	13	r	r	NOUN
iajs-2518	219	14	,	,	PUNCT
iajs-2518	219	15	then	then	ADV
iajs-2518	219	16	𝐾∗	𝐾∗	PRON
iajs-2518	219	17	is	be	AUX
iajs-2518	219	18	a	a	DET
iajs-2518	219	19	semimaximal	semimaximal	ADJ
iajs-2518	219	20	ideal	ideal	NOUN
iajs-2518	219	21	by	by	ADP
iajs-2518	219	22	[	[	X
iajs-2518	219	23	4	4	NUM
iajs-2518	219	24	]	]	PUNCT
iajs-2518	219	25	.	.	PUNCT
iajs-2518	220	1	hence	hence	ADV
iajs-2518	220	2	by[1	by[1	NUM
iajs-2518	220	3	]	]	X
iajs-2518	220	4	k∗	k∗	PROPN
iajs-2518	220	5	x∗	x∗	PROPN
iajs-2518	220	6	is	be	AUX
iajs-2518	220	7	a	a	DET
iajs-2518	220	8	semimaximal	semimaximal	ADJ
iajs-2518	220	9	submodule	submodule	NOUN
iajs-2518	220	10	and	and	CCONJ
iajs-2518	221	1	so	so	ADV
iajs-2518	221	2	kx	kx	PROPN
iajs-2518	221	3	∗	∗	NOUN
iajs-2518	221	4	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	221	5	𝑎	𝑎	DET
iajs-2518	221	6	semimaximal	semimaximal	ADJ
iajs-2518	221	7	submodule	submodule	NOUN
iajs-2518	221	8	of	of	ADP
iajs-2518	221	9	𝑋∗.	𝑋∗.	PROPN
iajs-2518	221	10	thus	thus	ADV
iajs-2518	221	11	kx	kx	PROPN
iajs-2518	221	12	is	be	AUX
iajs-2518	221	13	a	a	DET
iajs-2518	221	14	f	f	ADJ
iajs-2518	221	15	-	-	PUNCT
iajs-2518	221	16	semimaximal	semimaximal	ADJ
iajs-2518	221	17	submodule	submodule	NOUN
iajs-2518	221	18	.	.	PUNCT
iajs-2518	222	1	remark	remark	PROPN
iajs-2518	222	2	3.16	3.16	NUM
iajs-2518	222	3	the	the	DET
iajs-2518	222	4	following	follow	VERB
iajs-2518	222	5	example	example	NOUN
iajs-2518	222	6	shows	show	VERB
iajs-2518	222	7	that	that	SCONJ
iajs-2518	222	8	the	the	DET
iajs-2518	222	9	converse	converse	NOUN
iajs-2518	222	10	of	of	ADP
iajs-2518	222	11	proposition	proposition	NOUN
iajs-2518	222	12	3.15	3.15	NUM
iajs-2518	222	13	is	be	AUX
iajs-2518	222	14	not	not	PART
iajs-2518	222	15	true	true	ADJ
iajs-2518	222	16	in	in	ADP
iajs-2518	222	17	general	general	ADJ
iajs-2518	222	18	,	,	PUNCT
iajs-2518	222	19	consider	consider	VERB
iajs-2518	222	20	𝑀	𝑀	PRON
iajs-2518	222	21	𝑍	𝑍	VERB
iajs-2518	222	22	𝑎𝑠	𝑎𝑠	NOUN
iajs-2518	222	23	𝑍	𝑍	NOUN
iajs-2518	222	24	𝑚𝑜𝑑𝑢𝑙𝑒	𝑚𝑜𝑑𝑢𝑙𝑒	NOUN
iajs-2518	222	25	,	,	PUNCT
iajs-2518	222	26	𝑑𝑒𝑓𝑖𝑛𝑒	𝑑𝑒𝑓𝑖𝑛𝑒	ADJ
iajs-2518	222	27	𝑋	𝑋	PROPN
iajs-2518	222	28	:	:	PUNCT
iajs-2518	223	1	𝑀	𝑀	PROPN
iajs-2518	223	2	→	→	SYM
iajs-2518	223	3	0,1	0,1	NUM
iajs-2518	223	4	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	223	5	𝑋	𝑋	NOUN
iajs-2518	223	6	𝑥	𝑥	PROPN
iajs-2518	223	7	1	1	NUM
iajs-2518	223	8	,	,	PUNCT
iajs-2518	223	9	∀𝑥	∀𝑥	PROPN
iajs-2518	223	10	∈	∈	PROPN
iajs-2518	223	11	𝑀	𝑀	PROPN
iajs-2518	223	12	𝐷𝑒𝑓𝑖𝑛𝑒	𝐷𝑒𝑓𝑖𝑛𝑒	PROPN
iajs-2518	223	13	𝐾	𝐾	PROPN
iajs-2518	223	14	:	:	PUNCT
iajs-2518	223	15	𝑍	𝑍	PROPN
iajs-2518	223	16	→	→	SYM
iajs-2518	223	17	0,1	0,1	NUM
iajs-2518	223	18	,	,	PUNCT
iajs-2518	223	19	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	223	20	:	:	PUNCT
iajs-2518	223	21	𝐾	𝐾	PROPN
iajs-2518	223	22	𝑥	𝑥	PROPN
iajs-2518	223	23	1	1	NUM
iajs-2518	223	24	𝑖𝑓	𝑖𝑓	ADP
iajs-2518	223	25	𝑥	𝑥	DET
iajs-2518	223	26	∈	∈	PROPN
iajs-2518	223	27	4𝑍	4𝑍	NOUN
iajs-2518	223	28	,	,	PUNCT
iajs-2518	223	29	0	0	NUM
iajs-2518	223	30	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2518	223	31	x	x	PUNCT
iajs-2518	223	32	is	be	AUX
iajs-2518	223	33	a	a	DET
iajs-2518	223	34	f	f	NOUN
iajs-2518	223	35	module	module	NOUN
iajs-2518	223	36	and	and	CCONJ
iajs-2518	223	37	k	k	PROPN
iajs-2518	223	38	is	be	AUX
iajs-2518	223	39	f	f	PROPN
iajs-2518	223	40	ideal	ideal	NOUN
iajs-2518	223	41	of	of	ADP
iajs-2518	223	42	z	z	PROPN
iajs-2518	223	43	(	(	PUNCT
iajs-2518	223	44	it	it	PRON
iajs-2518	223	45	is	be	AUX
iajs-2518	223	46	easy	easy	ADJ
iajs-2518	223	47	to	to	PART
iajs-2518	223	48	prove	prove	VERB
iajs-2518	223	49	)	)	PUNCT
iajs-2518	223	50	.	.	PUNCT
iajs-2518	224	1	note	note	VERB
iajs-2518	224	2	that	that	SCONJ
iajs-2518	224	3	x∗	x∗	PROPN
iajs-2518	224	4	𝑍	𝑍	VERB
iajs-2518	224	5	and	and	CCONJ
iajs-2518	224	6	k∗	k∗	PROPN
iajs-2518	224	7	4𝑍	4𝑍	PROPN
iajs-2518	224	8	which	which	PRON
iajs-2518	224	9	is	be	AUX
iajs-2518	224	10	not	not	PART
iajs-2518	224	11	is	be	AUX
iajs-2518	224	12	semimaximal	semimaximal	ADJ
iajs-2518	224	13	ideal	ideal	NOUN
iajs-2518	224	14	.	.	PUNCT
iajs-2518	225	1	since	since	ADV
iajs-2518	225	2	,	,	PUNCT
iajs-2518	225	3	𝐾𝑋	𝐾𝑋	PROPN
iajs-2518	225	4	∗	∗	NOUN
iajs-2518	225	5	𝐾∗	𝐾∗	PUNCT
iajs-2518	226	1	𝑋∗	𝑋∗	ADJ
iajs-2518	226	2	.then	.then	PUNCT
iajs-2518	226	3	𝐾𝑋	𝐾𝑋	PROPN
iajs-2518	226	4	∗	∗	VERB
iajs-2518	226	5	4𝑍	4𝑍	PROPN
iajs-2518	226	6	𝑍	𝑍	PROPN
iajs-2518	226	7	0′	0′	NUM
iajs-2518	226	8	,	,	PUNCT
iajs-2518	226	9	2′	2′	NUM
iajs-2518	226	10	,	,	PUNCT
iajs-2518	226	11	4′	4′	NUM
iajs-2518	226	12	2	2	NUM
iajs-2518	226	13	and	and	CCONJ
iajs-2518	226	14	so	so	ADV
iajs-2518	226	15	.	.	PUNCT
iajs-2518	227	1	𝑋/𝐾𝑋	𝑋/𝐾𝑋	PROPN
iajs-2518	227	2	∗	∗	NOUN
iajs-2518	227	3	𝑍	𝑍	PROPN
iajs-2518	227	4	/	/	SYM
iajs-2518	227	5	2	2	NUM
iajs-2518	227	6	is	be	AUX
iajs-2518	227	7	simple	simple	ADJ
iajs-2518	227	8	thus	thus	ADV
iajs-2518	227	9	,	,	PUNCT
iajs-2518	227	10	kx	kx	PROPN
iajs-2518	227	11	∗	∗	NOUN
iajs-2518	227	12	is	be	AUX
iajs-2518	227	13	a	a	DET
iajs-2518	227	14	maximal	maximal	ADJ
iajs-2518	227	15	submodule	submodule	NOUN
iajs-2518	227	16	in	in	ADP
iajs-2518	227	17	x∗.	x∗.	PROPN
iajs-2518	227	18	on	on	ADP
iajs-2518	227	19	the	the	DET
iajs-2518	227	20	other	other	ADJ
iajs-2518	227	21	hand	hand	NOUN
iajs-2518	227	22	,	,	PUNCT
iajs-2518	227	23	by	by	ADP
iajs-2518	227	24	proposition	proposition	NOUN
iajs-2518	227	25	1.17	1.17	NUM
iajs-2518	227	26	𝑋/𝐾𝑋	𝑋/𝐾𝑋	PROPN
iajs-2518	227	27	∗	∗	VERB
iajs-2518	227	28	𝑋∗	𝑋∗	NOUN
iajs-2518	228	1	𝐾𝑋	𝐾𝑋	PROPN
iajs-2518	228	2	∗⁄	∗⁄	PROPN
iajs-2518	228	3	thus	thus	ADV
iajs-2518	228	4	𝐾𝑋	𝐾𝑋	PROPN
iajs-2518	228	5	is	be	AUX
iajs-2518	228	6	a	a	DET
iajs-2518	228	7	semimaximal	semimaximal	ADJ
iajs-2518	228	8	f	f	PROPN
iajs-2518	228	9	submodule	submodule	NOUN
iajs-2518	228	10	of	of	ADP
iajs-2518	228	11	x	x	PROPN
iajs-2518	228	12	.	.	PUNCT
iajs-2518	229	1	corollary	corollary	NOUN
iajs-2518	229	2	3.17	3.17	NUM
iajs-2518	229	3	x	x	NOUN
iajs-2518	229	4	is	be	AUX
iajs-2518	229	5	be	be	AUX
iajs-2518	229	6	a	a	DET
iajs-2518	229	7	multiplication	multiplication	NOUN
iajs-2518	229	8	fmodule	fmodule	ADV
iajs-2518	229	9	of	of	ADP
iajs-2518	229	10	an	an	DET
iajs-2518	229	11	r	r	NOUN
iajs-2518	229	12	module	module	NOUN
iajs-2518	229	13	m	m	NOUN
iajs-2518	229	14	.	.	PUNCT
iajs-2518	230	1	if	if	SCONJ
iajs-2518	230	2	every	every	DET
iajs-2518	230	3	f	f	PROPN
iajs-2518	230	4	ideal	ideal	NOUN
iajs-2518	230	5	k	k	PROPN
iajs-2518	230	6	in	in	ADP
iajs-2518	230	7	a	a	DET
iajs-2518	230	8	ring	ring	NOUN
iajs-2518	230	9	r	r	NOUN
iajs-2518	230	10	is	be	AUX
iajs-2518	230	11	semimaximal	semimaximal	ADJ
iajs-2518	230	12	,	,	PUNCT
iajs-2518	230	13	then	then	ADV
iajs-2518	230	14	every	every	DET
iajs-2518	230	15	f	f	PROPN
iajs-2518	230	16	submodule	submodule	NOUN
iajs-2518	230	17	of	of	ADP
iajs-2518	230	18	x	x	PROPN
iajs-2518	230	19	is	be	AUX
iajs-2518	230	20	semimaximal	semimaximal	ADJ
iajs-2518	230	21	.	.	PUNCT
iajs-2518	231	1	provided	provide	VERB
iajs-2518	231	2	that	that	SCONJ
iajs-2518	231	3	𝑋	𝑋	NOUN
iajs-2518	231	4	/𝐴	/𝐴	PUNCT
iajs-2518	231	5	∗	∗	NOUN
iajs-2518	231	6	𝑋∗	𝑋∗	ADJ
iajs-2518	231	7	/𝐴∗	/𝐴∗	PUNCT
iajs-2518	231	8	,	,	PUNCT
iajs-2518	231	9	∀	∀	NOUN
iajs-2518	231	10	𝐴	𝐴	PROPN
iajs-2518	231	11	𝑋	𝑋	PROPN
iajs-2518	231	12	poorf	poorf	PROPN
iajs-2518	231	13	:	:	PUNCT
iajs-2518	231	14	it	it	PRON
iajs-2518	231	15	is	be	AUX
iajs-2518	231	16	directly	directly	ADV
iajs-2518	231	17	from	from	ADP
iajs-2518	231	18	proposition	proposition	NOUN
iajs-2518	231	19	3.15	3.15	NUM
iajs-2518	231	20	.	.	PUNCT
iajs-2518	232	1	proposition	proposition	NOUN
iajs-2518	232	2	3.18	3.18	NUM
iajs-2518	232	3	every	every	DET
iajs-2518	232	4	epimorphic	epimorphic	ADJ
iajs-2518	232	5	image	image	NOUN
iajs-2518	232	6	of	of	ADP
iajs-2518	232	7	semimaximal	semimaximal	PROPN
iajs-2518	232	8	f	f	PROPN
iajs-2518	232	9	submodule	submodule	PROPN
iajs-2518	232	10	is	be	AUX
iajs-2518	232	11	a	a	DET
iajs-2518	232	12	semimaximal	semimaximal	ADJ
iajs-2518	232	13	f	f	PROPN
iajs-2518	232	14	submodule	submodule	NOUN
iajs-2518	232	15	.	.	PUNCT
iajs-2518	233	1	proof	proof	NOUN
iajs-2518	233	2	:	:	PUNCT
iajs-2518	233	3	let	let	VERB
iajs-2518	233	4	𝑔	𝑔	PROPN
iajs-2518	233	5	∶	∶	VERB
iajs-2518	233	6	𝑋	𝑋	ADJ
iajs-2518	233	7	𝐴⁄	𝐴⁄	PROPN
iajs-2518	233	8	→	→	SYM
iajs-2518	233	9	𝑌	𝑌	PROPN
iajs-2518	233	10	/	/	SYM
iajs-2518	233	11	𝑓	𝑓	PROPN
iajs-2518	233	12	𝐴	𝐴	PROPN
iajs-2518	233	13	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2518	233	14	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2518	233	15	𝑔	𝑔	PROPN
iajs-2518	233	16	𝑥	𝑥	PROPN
iajs-2518	233	17	𝐴	𝐴	PROPN
iajs-2518	233	18	𝑦	𝑦	PROPN
iajs-2518	233	19	𝑓	𝑓	PROPN
iajs-2518	233	20	𝐴	𝐴	PROPN
iajs-2518	233	21	,	,	PUNCT
iajs-2518	233	22	∀𝑦	∀𝑦	ADP
iajs-2518	233	23	∈	∈	PROPN
iajs-2518	233	24	𝑌.	𝑌.	PROPN
iajs-2518	233	25	𝑔	𝑔	PROPN
iajs-2518	233	26	is	be	AUX
iajs-2518	233	27	well	well	ADV
iajs-2518	233	28	-	-	PUNCT
iajs-2518	233	29	defined	define	VERB
iajs-2518	233	30	and	and	CCONJ
iajs-2518	233	31	𝑔	𝑔	PROPN
iajs-2518	233	32	is	be	AUX
iajs-2518	233	33	an	an	DET
iajs-2518	233	34	epimorphism	epimorphism	NOUN
iajs-2518	233	35	.	.	PUNCT
iajs-2518	234	1	since	since	SCONJ
iajs-2518	234	2	𝑋	𝑋	PROPN
iajs-2518	234	3	𝐴⁄	𝐴⁄	PROPN
iajs-2518	234	4	is	be	AUX
iajs-2518	234	5	semisimple	semisimple	NOUN
iajs-2518	234	6	.	.	PUNCT
iajs-2518	235	1	hence	hence	ADV
iajs-2518	235	2	,	,	PUNCT
iajs-2518	235	3	𝑔	𝑔	PROPN
iajs-2518	235	4	𝑋	𝑋	PROPN
iajs-2518	235	5	𝐴	𝐴	PROPN
iajs-2518	235	6	𝑌	𝑌	PROPN
iajs-2518	235	7	𝑋	𝑋	PROPN
iajs-2518	235	8	𝐴⁄	𝐴⁄	PROPN
iajs-2518	235	9	𝑌⁄	𝑌⁄	PROPN
iajs-2518	235	10	is	be	AUX
iajs-2518	235	11	semisimple	semisimple	NOUN
iajs-2518	235	12	.	.	PUNCT
iajs-2518	236	1	thus	thus	ADV
iajs-2518	236	2	,	,	PUNCT
iajs-2518	236	3	𝑓	𝑓	DET
iajs-2518	236	4	𝐴	𝐴	PROPN
iajs-2518	236	5	is	be	AUX
iajs-2518	236	6	a	a	DET
iajs-2518	236	7	semimaximal	semimaximal	ADJ
iajs-2518	236	8	fuzzy	fuzzy	ADJ
iajs-2518	236	9	submodule	submodule	NOUN
iajs-2518	236	10	.	.	PUNCT
iajs-2518	237	1	proposition	proposition	NOUN
iajs-2518	237	2	3.19	3.19	NUM
iajs-2518	237	3	let	let	VERB
iajs-2518	237	4	x	x	PRON
iajs-2518	237	5	be	be	AUX
iajs-2518	237	6	a	a	DET
iajs-2518	237	7	finitely	finitely	ADV
iajs-2518	237	8	generated	generate	VERB
iajs-2518	237	9	f	f	PROPN
iajs-2518	237	10	rmodule	rmodule	PROPN
iajs-2518	237	11	m	m	VERB
iajs-2518	237	12	such	such	ADJ
iajs-2518	237	13	that	that	SCONJ
iajs-2518	237	14	𝐹	𝐹	PROPN
iajs-2518	237	15	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
iajs-2518	237	16	𝐴	𝐴	PROPN
iajs-2518	237	17	is	be	AUX
iajs-2518	237	18	a	a	DET
iajs-2518	237	19	semimaximal	semimaximal	ADJ
iajs-2518	237	20	f	f	PROPN
iajs-2518	237	21	ideal	ideal	NOUN
iajs-2518	237	22	,	,	PUNCT
iajs-2518	237	23	.	.	PUNCT
iajs-2518	238	1	∀	∀	PUNCT
iajs-2518	239	1	𝐴	𝐴	NOUN
iajs-2518	239	2	𝑋	𝑋	PROPN
iajs-2518	239	3	.	.	PUNCT
iajs-2518	240	1	if	if	SCONJ
iajs-2518	240	2	x	x	PRON
iajs-2518	240	3	/a	/a	PUNCT
iajs-2518	240	4	∗	∗	NOUN
iajs-2518	240	5	∗	∗	NOUN
iajs-2518	240	6	∗	∗	NOUN
iajs-2518	240	7	,	,	PUNCT
iajs-2518	240	8	∀	∀	X
iajs-2518	240	9	a	a	DET
iajs-2518	240	10	𝑋	𝑋	NOUN
iajs-2518	240	11	,	,	PUNCT
iajs-2518	240	12	then	then	ADV
iajs-2518	240	13	every	every	DET
iajs-2518	240	14	f	f	PROPN
iajs-2518	240	15	submodule	submodule	NOUN
iajs-2518	240	16	of	of	ADP
iajs-2518	240	17	x	x	PROPN
iajs-2518	240	18	is	be	AUX
iajs-2518	240	19	semimaximal	semimaximal	ADJ
iajs-2518	240	20	.	.	PUNCT
iajs-2518	241	1	proof	proof	NOUN
iajs-2518	241	2	:	:	PUNCT
iajs-2518	241	3	since	since	SCONJ
iajs-2518	241	4	x	x	PRON
iajs-2518	241	5	is	be	AUX
iajs-2518	241	6	a	a	DET
iajs-2518	241	7	ffinitely	ffinitely	ADV
iajs-2518	241	8	generated	generate	VERB
iajs-2518	241	9	rmodule	rmodule	NOUN
iajs-2518	241	10	,	,	PUNCT
iajs-2518	241	11	then	then	ADV
iajs-2518	241	12	𝑋∗	𝑋∗	ADV
iajs-2518	241	13	is	be	AUX
iajs-2518	241	14	finitely	finitely	ADV
iajs-2518	241	15	generated	generate	VERB
iajs-2518	241	16	rmodule	rmodule	NOUN
iajs-2518	241	17	[	[	X
iajs-2518	241	18	15	15	NUM
iajs-2518	241	19	]	]	PUNCT
iajs-2518	241	20	.	.	PUNCT
iajs-2518	242	1	since	since	SCONJ
iajs-2518	242	2	f	f	PROPN
iajs-2518	242	3	anna	anna	PROPN
iajs-2518	242	4	∗	∗	PROPN
iajs-2518	242	5	𝑎𝑛𝑛	𝑎𝑛𝑛	ADV
iajs-2518	242	6	𝐴∗	𝐴∗	PROPN
iajs-2518	242	7	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2518	242	8	f	f	PROPN
iajs-2518	242	9	𝑎𝑛𝑛𝐴	𝑎𝑛𝑛𝐴	PROPN
iajs-2518	242	10	𝑖𝑠	𝑖𝑠	PROPN
iajs-2518	242	11	semimaximal	semimaximal	ADJ
iajs-2518	242	12	f	f	NOUN
iajs-2518	242	13	-	-	PUNCT
iajs-2518	242	14	ideal	ideal	NOUN
iajs-2518	242	15	,	,	PUNCT
iajs-2518	242	16	implies	imply	VERB
iajs-2518	242	17	f	f	PROPN
iajs-2518	242	18	𝑎𝑛𝑛𝐴	𝑎𝑛𝑛𝐴	PROPN
iajs-2518	242	19	∗	∗	NOUN
iajs-2518	242	20	is	be	AUX
iajs-2518	242	21	semimaximal	semimaximal	ADJ
iajs-2518	242	22	𝑏𝑦	𝑏𝑦	NOUN
iajs-2518	242	23	4	4	NUM
iajs-2518	242	24	,	,	PUNCT
iajs-2518	242	25	𝑠𝑜	𝑠𝑜	INTJ
iajs-2518	242	26	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
iajs-2518	242	27	𝐴∗	𝐴∗	PROPN
iajs-2518	242	28	𝑖𝑠	𝑖𝑠	PROPN
iajs-2518	242	29	semimaximal	semimaximal	ADJ
iajs-2518	242	30	ideal	ideal	NOUN
iajs-2518	242	31	.	.	PUNCT
iajs-2518	243	1	then	then	ADV
iajs-2518	243	2	by	by	ADP
iajs-2518	243	3	1	1	NUM
iajs-2518	243	4	,	,	PUNCT
iajs-2518	243	5	every	every	DET
iajs-2518	243	6	submodule	submodule	NOUN
iajs-2518	243	7	of	of	ADP
iajs-2518	243	8	𝑋∗	𝑋∗	ADJ
iajs-2518	243	9	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	243	10	semimaximal	semimaximal	NOUN
iajs-2518	243	11	.	.	PUNCT
iajs-2518	243	12	  	  	SPACE
iajs-2518	244	1	147	147	NUM
iajs-2518	244	2	  	  	SPACE
iajs-2518	244	3	ibn	ibn	PROPN
iajs-2518	244	4	al	al	PROPN
iajs-2518	244	5	-	-	PUNCT
iajs-2518	244	6	haitham	haitham	PROPN
iajs-2518	244	7	jour	jour	X
iajs-2518	244	8	.	.	PROPN
iajs-2518	244	9	for	for	ADP
iajs-2518	244	10	pure	pure	ADJ
iajs-2518	244	11	&	&	CCONJ
iajs-2518	244	12	appl	appl	PROPN
iajs-2518	244	13	.	.	PUNCT
iajs-2518	245	1	sci	sci	PROPN
iajs-2518	245	2	.	.	PROPN
iajs-2518	246	1	33	33	NUM
iajs-2518	246	2	(	(	PUNCT
iajs-2518	246	3	4	4	NUM
iajs-2518	246	4	)	)	PUNCT
iajs-2518	246	5	2020	2020	NUM
iajs-2518	247	1	hence	hence	ADV
iajs-2518	247	2	,	,	PUNCT
iajs-2518	247	3	𝐴∗	𝐴∗	NUM
iajs-2518	247	4	𝑖𝑠	𝑖𝑠	NOUN
iajs-2518	247	5	semimaximal	semimaximal	NOUN
iajs-2518	247	6	.	.	PUNCT
iajs-2518	248	1	but	but	CCONJ
iajs-2518	248	2	this	this	PRON
iajs-2518	248	3	implies	imply	VERB
iajs-2518	248	4	a	a	DET
iajs-2518	248	5	is	be	AUX
iajs-2518	248	6	semimaximal	semimaximal	ADJ
iajs-2518	248	7	,	,	PUNCT
iajs-2518	248	8	since	since	SCONJ
iajs-2518	248	9	x	x	PRON
iajs-2518	248	10	/a	/a	PUNCT
iajs-2518	248	11	∗	∗	NOUN
iajs-2518	248	12	∗	∗	NOUN
iajs-2518	248	13	∗	∗	NOUN
iajs-2518	248	14	references	reference	NOUN
iajs-2518	248	15	1	1	NUM
iajs-2518	248	16	.	.	PUNCT
iajs-2518	249	1	hatem	hatem	PROPN
iajs-2518	249	2	,	,	PUNCT
iajs-2518	249	3	y.	y.	PROPN
iajs-2518	249	4	k.	k.	PROPN
iajs-2518	249	5	,	,	PUNCT
iajs-2518	249	6	semimaximal	semimaximal	NOUN
iajs-2518	249	7	modules	module	NOUN
iajs-2518	249	8	,	,	PUNCT
iajs-2518	249	9	ph	ph	PROPN
iajs-2518	249	10	.	.	PROPN
iajs-2518	249	11	d.	d.	PROPN
iajs-2518	249	12	thesis	thesis	PROPN
iajs-2518	249	13	,	,	PUNCT
iajs-2518	249	14	university	university	NOUN
iajs-2518	249	15	of	of	ADP
iajs-2518	249	16	baghdad,2005	baghdad,2005	NOUN
iajs-2518	249	17	.	.	PUNCT
iajs-2518	250	1	2	2	X
iajs-2518	250	2	.	.	X
iajs-2518	250	3	kasch	kasch	PROPN
iajs-2518	250	4	,	,	PUNCT
iajs-2518	250	5	f.	f.	PROPN
iajs-2518	250	6	modules	modules	PROPN
iajs-2518	250	7	and	and	CCONJ
iajs-2518	250	8	rings	ring	NOUN
iajs-2518	250	9	,	,	PUNCT
iajs-2518	250	10	academic	academic	ADJ
iajs-2518	250	11	press	press	NOUN
iajs-2518	250	12	,	,	PUNCT
iajs-2518	250	13	1982	1982	NUM
iajs-2518	250	14	.	.	PUNCT
iajs-2518	251	1	3	3	X
iajs-2518	251	2	.	.	X
iajs-2518	251	3	maysoun	maysoun	PROPN
iajs-2518	251	4	a.	a.	PROPN
iajs-2518	251	5	hamel	hamel	PROPN
iajs-2518	251	6	,	,	PUNCT
iajs-2518	251	7	f	f	X
iajs-2518	251	8	-	-	PUNCT
iajs-2518	251	9	regular	regular	ADJ
iajs-2518	251	10	fuzzy	fuzzy	ADJ
iajs-2518	251	11	modules	module	NOUN
iajs-2518	251	12	,	,	PUNCT
iajs-2518	251	13	m.	m.	PROPN
iajs-2518	251	14	sc	sc	PROPN
iajs-2518	251	15	.	.	PUNCT
iajs-2518	252	1	thesis	thesis	PROPN
iajs-2518	252	2	,	,	PUNCT
iajs-2518	252	3	university	university	NOUN
iajs-2518	252	4	of	of	ADP
iajs-2518	252	5	baghdad	baghdad	PROPN
iajs-2518	252	6	,	,	PUNCT
iajs-2518	252	7	2002	2002	NUM
iajs-2518	252	8	.	.	PUNCT
iajs-2518	253	1	4	4	X
iajs-2518	253	2	.	.	X
iajs-2518	253	3	inaam	inaam	NOUN
iajs-2518	253	4	,	,	PUNCT
iajs-2518	253	5	m.a.hadi	m.a.hadi	NOUN
iajs-2518	253	6	;	;	PUNCT
iajs-2518	253	7	maysoun	maysoun	NOUN
iajs-2518	253	8	,	,	PUNCT
iajs-2518	253	9	a.	a.	PROPN
iajs-2518	253	10	hamil	hamil	PROPN
iajs-2518	253	11	,	,	PUNCT
iajs-2518	253	12	fuzzy	fuzzy	ADJ
iajs-2518	253	13	semimaximal	semimaximal	ADJ
iajs-2518	253	14	ideals	ideal	NOUN
iajs-2518	253	15	,	,	PUNCT
iajs-2518	253	16	ibn	ibn	PROPN
iajs-2518	253	17	,	,	PUNCT
iajs-2518	253	18	al	al	PROPN
iajs-2518	253	19	-	-	PUNCT
iajs-2518	253	20	hathaim	hathaim	NOUN
iajs-2518	253	21	for	for	ADP
iajs-2518	253	22	pure	pure	ADJ
iajs-2518	253	23	and	and	CCONJ
iajs-2518	253	24	application	application	NOUN
iajs-2518	253	25	science,2009,22	science,2009,22	NOUN
iajs-2518	253	26	,	,	PUNCT
iajs-2518	253	27	2,166	2,166	NUM
iajs-2518	253	28	-	-	SYM
iajs-2518	253	29	177	177	NUM
iajs-2518	253	30	5	5	NUM
iajs-2518	253	31	.	.	PUNCT
iajs-2518	254	1	zadeh	zadeh	PROPN
iajs-2518	254	2	,	,	PUNCT
iajs-2518	254	3	l.	l.	PROPN
iajs-2518	254	4	a.	a.	PROPN
iajs-2518	254	5	,	,	PUNCT
iajs-2518	254	6	fuzzy	fuzzy	ADJ
iajs-2518	254	7	sets	set	NOUN
iajs-2518	254	8	,	,	PUNCT
iajs-2518	254	9	information	information	NOUN
iajs-2518	254	10	and	and	CCONJ
iajs-2518	254	11	control	control	NOUN
iajs-2518	254	12	.	.	PUNCT
iajs-2518	255	1	1965	1965	NUM
iajs-2518	255	2	,	,	PUNCT
iajs-2518	255	3	8	8	NUM
iajs-2518	255	4	,	,	PUNCT
iajs-2518	255	5	338	338	NUM
iajs-2518	255	6	-	-	SYM
iajs-2518	255	7	353	353	NUM
iajs-2518	255	8	.	.	NOUN
iajs-2518	256	1	6	6	NUM
iajs-2518	256	2	.	.	X
iajs-2518	256	3	zahedi	zahedi	PROPN
iajs-2518	256	4	,	,	PUNCT
iajs-2518	256	5	m.	m.	NOUN
iajs-2518	256	6	m.	m.	NOUN
iajs-2518	256	7	on	on	ADP
iajs-2518	256	8	l	l	ADJ
iajs-2518	256	9	-	-	ADJ
iajs-2518	256	10	fuzzy	fuzzy	ADJ
iajs-2518	256	11	residual	residual	ADJ
iajs-2518	256	12	quotient	quotient	NOUN
iajs-2518	256	13	modules	module	NOUN
iajs-2518	256	14	and	and	CCONJ
iajs-2518	256	15	p.	p.	NOUN
iajs-2518	256	16	primary	primary	ADJ
iajs-2518	256	17	ubmodules	ubmodule	NOUN
iajs-2518	256	18	,	,	PUNCT
iajs-2518	256	19	fuzzy	fuzzy	ADJ
iajs-2518	256	20	sets	set	NOUN
iajs-2518	256	21	and	and	CCONJ
iajs-2518	256	22	systems	system	NOUN
iajs-2518	256	23	,	,	PUNCT
iajs-2518	256	24	1992,51	1992,51	NUM
iajs-2518	256	25	,	,	PUNCT
iajs-2518	256	26	33	33	NUM
iajs-2518	256	27	-	-	SYM
iajs-2518	256	28	344	344	NUM
iajs-2518	256	29	,	,	PUNCT
iajs-2518	256	30	.	.	PUNCT
iajs-2518	257	1	7	7	X
iajs-2518	257	2	.	.	X
iajs-2518	257	3	zahedi	zahedi	PROPN
iajs-2518	257	4	,	,	PUNCT
iajs-2518	257	5	m.	m.	NOUN
iajs-2518	257	6	m.	m.	NOUN
iajs-2518	257	7	,	,	PUNCT
iajs-2518	257	8	a	a	DET
iajs-2518	257	9	characterization	characterization	NOUN
iajs-2518	257	10	of	of	ADP
iajs-2518	257	11	l	l	ADJ
iajs-2518	257	12	-	-	ADJ
iajs-2518	257	13	fuzzy	fuzzy	ADJ
iajs-2518	257	14	prime	prime	ADJ
iajs-2518	257	15	ideals	ideal	NOUN
iajs-2518	257	16	,	,	PUNCT
iajs-2518	257	17	fuzzy	fuzzy	ADJ
iajs-2518	257	18	sets	set	NOUN
iajs-2518	257	19	and	and	CCONJ
iajs-2518	257	20	systems	system	NOUN
iajs-2518	257	21	,	,	PUNCT
iajs-2518	257	22	1991	1991	NUM
iajs-2518	257	23	,	,	PUNCT
iajs-2518	257	24	44	44	NUM
iajs-2518	257	25	,	,	PUNCT
iajs-2518	257	26	147	147	NUM
iajs-2518	257	27	-	-	SYM
iajs-2518	257	28	160	160	NUM
iajs-2518	257	29	.	.	PUNCT
iajs-2518	258	1	8	8	X
iajs-2518	258	2	.	.	X
iajs-2518	258	3	mashinch	mashinch	NOUN
iajs-2518	258	4	,	,	PUNCT
iajs-2518	258	5	m.	m.	NOUN
iajs-2518	258	6	;	;	PUNCT
iajs-2518	258	7	zahedi	zahedi	PROPN
iajs-2518	258	8	,	,	PUNCT
iajs-2518	258	9	m.	m.	NOUN
iajs-2518	258	10	m.	m.	NOUN
iajs-2518	258	11	on	on	ADP
iajs-2518	258	12	l	l	ADJ
iajs-2518	258	13	-	-	ADJ
iajs-2518	258	14	fuzzy	fuzzy	ADJ
iajs-2518	258	15	primary	primary	ADJ
iajs-2518	258	16	submodule	submodule	NOUN
iajs-2518	258	17	,	,	PUNCT
iajs-2518	258	18	fuzzy	fuzzy	ADJ
iajs-2518	258	19	sets	set	NOUN
iajs-2518	258	20	and	and	CCONJ
iajs-2518	258	21	system.s	system.s	PROPN
iajs-2518	258	22	1992	1992	NUM
iajs-2518	258	23	,	,	PUNCT
iajs-2518	258	24	49,231	49,231	NUM
iajs-2518	258	25	-	-	SYM
iajs-2518	258	26	236	236	NUM
iajs-2518	258	27	.	.	PUNCT
iajs-2518	259	1	9	9	X
iajs-2518	259	2	.	.	X
iajs-2518	259	3	mukhejee	mukhejee	NOUN
iajs-2518	259	4	,	,	PUNCT
iajs-2518	259	5	t.k	t.k	PROPN
iajs-2518	259	6	.	.	PROPN
iajs-2518	259	7	;	;	PUNCT
iajs-2518	259	8	sen	sen	PROPN
iajs-2518	259	9	,	,	PUNCT
iajs-2518	259	10	m.	m.	PROPN
iajs-2518	259	11	k.	k.	PROPN
iajs-2518	259	12	;	;	PUNCT
iajs-2518	259	13	roy	roy	PROPN
iajs-2518	259	14	,	,	PUNCT
iajs-2518	259	15	d.	d.	PROPN
iajs-2518	259	16	,	,	PUNCT
iajs-2518	259	17	on	on	ADP
iajs-2518	259	18	submodules	submodule	NOUN
iajs-2518	259	19	and	and	CCONJ
iajs-2518	259	20	their	their	PRON
iajs-2518	259	21	radicals	radical	NOUN
iajs-2518	259	22	,	,	PUNCT
iajs-2518	259	23	j.	j.	PROPN
iajs-2518	259	24	fuzzy	fuzzy	PROPN
iajs-2518	259	25	math	math	PROPN
iajs-2518	259	26	.	.	PUNCT
iajs-2518	259	27	1996	1996	NUM
iajs-2518	259	28	,	,	PUNCT
iajs-2518	259	29	4	4	NUM
iajs-2518	259	30	,	,	PUNCT
iajs-2518	259	31	549	549	NUM
iajs-2518	259	32	-	-	SYM
iajs-2518	259	33	558	558	NUM
iajs-2518	259	34	.	.	PUNCT
iajs-2518	260	1	10	10	NUM
iajs-2518	260	2	.	.	X
iajs-2518	261	1	abd	abd	PROPN
iajs-2518	261	2	alguad	alguad	PROPN
iajs-2518	261	3	q.	q.	PROPN
iajs-2518	261	4	some	some	DET
iajs-2518	261	5	results	result	NOUN
iajs-2518	261	6	on	on	ADP
iajs-2518	261	7	fuzzy	fuzzy	ADJ
iajs-2518	261	8	modules.m	modules.m	NOUN
iajs-2518	261	9	.	.	PUNCT
iajs-2518	262	1	sc	sc	PROPN
iajs-2518	262	2	.thesis	.thesis	PROPN
iajs-2518	262	3	university	university	PROPN
iajs-2518	262	4	of	of	ADP
iajs-2518	262	5	baghdad	baghdad	PROPN
iajs-2518	262	6	,	,	PUNCT
iajs-2518	262	7	1999	1999	NUM
iajs-2518	262	8	.	.	PUNCT
iajs-2518	263	1	11	11	NUM
iajs-2518	263	2	.	.	PUNCT
iajs-2518	264	1	maysoun	maysoun	PROPN
iajs-2518	264	2	,	,	PUNCT
iajs-2518	264	3	a.	a.	PROPN
iajs-2518	264	4	hamel	hamel	PROPN
iajs-2518	264	5	;	;	PUNCT
iajs-2518	264	6	hateem	hateem	PROPN
iajs-2518	264	7	,	,	PUNCT
iajs-2518	264	8	h.k	h.k	PROPN
iajs-2518	264	9	,	,	PUNCT
iajs-2518	264	10	fuzzy	fuzzy	ADJ
iajs-2518	264	11	maximal	maximal	ADJ
iajs-2518	264	12	submodules	submodule	NOUN
iajs-2518	264	13	,	,	PUNCT
iajs-2518	264	14	iraq	iraq	PROPN
iajs-2518	264	15	journal	journal	PROPN
iajs-2518	264	16	of	of	ADP
iajs-2518	264	17	science	science	NOUN
iajs-2518	264	18	,	,	PUNCT
iajs-2518	264	19	2020	2020	NUM
iajs-2518	264	20	,	,	PUNCT
iajs-2518	264	21	61	61	NUM
iajs-2518	264	22	,	,	PUNCT
iajs-2518	264	23	5	5	NUM
iajs-2518	264	24	,	,	PUNCT
iajs-2518	264	25	1164	1164	NUM
iajs-2518	264	26	-	-	SYM
iajs-2518	264	27	1172	1172	NUM
iajs-2518	264	28	.	.	PUNCT
iajs-2518	265	1	12	12	NUM
iajs-2518	265	2	.	.	PUNCT
iajs-2518	265	3	pan	pan	PROPN
iajs-2518	265	4	,	,	PUNCT
iajs-2518	265	5	f	f	PROPN
iajs-2518	265	6	.	.	PUNCT
iajs-2518	266	1	the	the	DET
iajs-2518	266	2	various	various	ADJ
iajs-2518	266	3	structures	structure	NOUN
iajs-2518	266	4	of	of	ADP
iajs-2518	266	5	fuzzy	fuzzy	ADJ
iajs-2518	266	6	quotient	quotient	NOUN
iajs-2518	266	7	modules	module	NOUN
iajs-2518	266	8	,	,	PUNCT
iajs-2518	266	9	fuzzy	fuzzy	ADJ
iajs-2518	266	10	sets	set	NOUN
iajs-2518	266	11	and	and	CCONJ
iajs-2518	266	12	systems,1992,50,187	systems,1992,50,187	PROPN
iajs-2518	266	13	-	-	PUNCT
iajs-2518	266	14	192	192	NUM
iajs-2518	266	15	.	.	PUNCT
iajs-2518	267	1	13	13	NUM
iajs-2518	267	2	.	.	PUNCT
iajs-2518	268	1	rabi	rabi	NOUN
iajs-2518	268	2	,	,	PUNCT
iajs-2518	268	3	h.	h.	PROPN
iajs-2518	268	4	j	j	PROPN
iajs-2518	268	5	.prime	.prime	X
iajs-2518	268	6	fuzzy	fuzzy	ADJ
iajs-2518	268	7	submodule	submodule	NOUN
iajs-2518	268	8	and	and	CCONJ
iajs-2518	268	9	prime	prime	ADJ
iajs-2518	268	10	fuzzy	fuzzy	ADJ
iajs-2518	268	11	modules	module	NOUN
iajs-2518	268	12	,	,	PUNCT
iajs-2518	268	13	m.sc	m.sc	PROPN
iajs-2518	268	14	.	.	PUNCT
iajs-2518	269	1	thesis	thesis	PROPN
iajs-2518	269	2	university	university	PROPN
iajs-2518	269	3	of	of	ADP
iajs-2518	269	4	baghdad	baghdad	PROPN
iajs-2518	269	5	,	,	PUNCT
iajs-2518	269	6	2001	2001	NUM
iajs-2518	269	7	.	.	PUNCT
iajs-2518	270	1	14	14	NUM
iajs-2518	270	2	.	.	X
iajs-2518	271	1	luis	luis	PROPN
iajs-2518	271	2	,	,	PUNCT
iajs-2518	271	3	m.	m.	NOUN
iajs-2518	271	4	,	,	PUNCT
iajs-2518	271	5	fuzzy	fuzzy	ADJ
iajs-2518	271	6	moodules	moodule	NOUN
iajs-2518	271	7	over	over	ADP
iajs-2518	271	8	fuzzy	fuzzy	ADJ
iajs-2518	271	9	rings	ring	NOUN
iajs-2518	271	10	in	in	ADP
iajs-2518	271	11	connection	connection	NOUN
iajs-2518	271	12	with	with	ADP
iajs-2518	271	13	fuzzy	fuzzy	ADJ
iajs-2518	271	14	ideals	ideal	NOUN
iajs-2518	271	15	of	of	ADP
iajs-2518	271	16	fuzzy	fuzzy	ADJ
iajs-2518	271	17	rins	rin	NOUN
iajs-2518	271	18	,	,	PUNCT
iajs-2518	271	19	the	the	DET
iajs-2518	271	20	journal	journal	NOUN
iajs-2518	271	21	of	of	ADP
iajs-2518	271	22	fuzzy	fuzzy	ADJ
iajs-2518	271	23	mathemtics	mathemtic	NOUN
iajs-2518	271	24	1996	1996	NUM
iajs-2518	271	25	,	,	PUNCT
iajs-2518	271	26	4,4	4,4	NUM
iajs-2518	271	27	.	.	NOUN
iajs-2518	271	28	15	15	NUM
iajs-2518	271	29	.	.	PUNCT
iajs-2518	272	1	inaam	inaam	NOUN
iajs-2518	272	2	,	,	PUNCT
iajs-2518	272	3	m.a.hadi	m.a.hadi	PROPN
iajs-2518	272	4	;	;	PUNCT
iajs-2518	272	5	maysoun	maysoun	NOUN
iajs-2518	272	6	,	,	PUNCT
iajs-2518	272	7	a.	a.	PROPN
iajs-2518	272	8	hamil	hamil	PROPN
iajs-2518	272	9	,	,	PUNCT
iajs-2518	272	10	cancellation	cancellation	NOUN
iajs-2518	272	11	and	and	CCONJ
iajs-2518	272	12	weakly	weakly	ADJ
iajs-2518	272	13	cancellation	cancellation	NOUN
iajs-2518	272	14	fuzzy	fuzzy	ADJ
iajs-2518	272	15	modules	module	NOUN
iajs-2518	272	16	,	,	PUNCT
iajs-2518	272	17	journal	journal	NOUN
iajs-2518	272	18	of	of	ADP
iajs-2518	272	19	basrah	basrah	PROPN
iajs-2518	272	20	reserchs((sciences	reserchs((science	NOUN
iajs-2518	272	21	)	)	PUNCT
iajs-2518	272	22	)	)	PUNCT
iajs-2518	272	23	,	,	PUNCT
iajs-2518	272	24	2011	2011	NUM
iajs-2518	272	25	,	,	PUNCT
iajs-2518	272	26	37	37	NUM
iajs-2518	272	27	,	,	PUNCT
iajs-2518	272	28	4.d	4.d	NUM
iajs-2518	272	29	.	.	PUNCT
iajs-2518	272	30	    	    	SPACE