id	sid	tid	token	lemma	pos
iajs-3092	1	1	ihjpas	ihjpas	PROPN
iajs-3092	1	2	.	.	PUNCT
iajs-3092	2	1	36	36	NUM
iajs-3092	2	2	(	(	PUNCT
iajs-3092	2	3	3	3	NUM
iajs-3092	2	4	)	)	PUNCT
iajs-3092	2	5	2023	2023	NUM
iajs-3092	2	6	372	372	NUM
iajs-3092	2	7	this	this	DET
iajs-3092	2	8	work	work	NOUN
iajs-3092	2	9	is	be	AUX
iajs-3092	2	10	licensed	license	VERB
iajs-3092	2	11	under	under	ADP
iajs-3092	2	12	a	a	DET
iajs-3092	2	13	creative	creative	ADJ
iajs-3092	2	14	commons	common	NOUN
iajs-3092	2	15	attribution	attribution	NOUN
iajs-3092	2	16	4.0	4.0	NUM
iajs-3092	2	17	international	international	ADJ
iajs-3092	2	18	license	license	NOUN
iajs-3092	2	19	*	*	PUNCT
iajs-3092	2	20	corresponding	correspond	VERB
iajs-3092	2	21	author	author	NOUN
iajs-3092	2	22	:	:	PUNCT
iajs-3092	2	23	sajidak.mohammed@uokufa.edu.iq	sajidak.mohammed@uokufa.edu.iq	ADJ
iajs-3092	2	24	abstract	abstract	NOUN
iajs-3092	2	25	in	in	ADP
iajs-3092	2	26	previous	previous	ADJ
iajs-3092	2	27	our	our	PRON
iajs-3092	2	28	research	research	NOUN
iajs-3092	2	29	,	,	PUNCT
iajs-3092	2	30	the	the	DET
iajs-3092	2	31	concepts	concept	NOUN
iajs-3092	2	32	of	of	ADP
iajs-3092	2	33	visible	visible	ADJ
iajs-3092	2	34	submodules	submodule	NOUN
iajs-3092	2	35	and	and	CCONJ
iajs-3092	2	36	fully	fully	ADV
iajs-3092	2	37	visible	visible	ADJ
iajs-3092	2	38	modules	module	NOUN
iajs-3092	2	39	were	be	AUX
iajs-3092	2	40	introduced	introduce	VERB
iajs-3092	2	41	,	,	PUNCT
iajs-3092	2	42	and	and	CCONJ
iajs-3092	2	43	then	then	ADV
iajs-3092	2	44	these	these	DET
iajs-3092	2	45	two	two	NUM
iajs-3092	2	46	concepts	concept	NOUN
iajs-3092	2	47	were	be	AUX
iajs-3092	2	48	fuzzified	fuzzifie	VERB
iajs-3092	2	49	to	to	PART
iajs-3092	2	50	fuzzy	fuzzy	ADJ
iajs-3092	2	51	visible	visible	ADJ
iajs-3092	2	52	submodules	submodule	NOUN
iajs-3092	2	53	and	and	CCONJ
iajs-3092	2	54	fully	fully	ADV
iajs-3092	2	55	fuzzy	fuzzy	ADJ
iajs-3092	2	56	.	.	PUNCT
iajs-3092	3	1	the	the	DET
iajs-3092	3	2	main	main	ADJ
iajs-3092	3	3	goal	goal	NOUN
iajs-3092	3	4	of	of	ADP
iajs-3092	3	5	this	this	DET
iajs-3092	3	6	paper	paper	NOUN
iajs-3092	3	7	is	be	AUX
iajs-3092	3	8	to	to	PART
iajs-3092	3	9	study	study	VERB
iajs-3092	3	10	the	the	DET
iajs-3092	3	11	relationships	relationship	NOUN
iajs-3092	3	12	between	between	ADP
iajs-3092	3	13	fully	fully	ADV
iajs-3092	3	14	fuzzy	fuzzy	ADJ
iajs-3092	3	15	visible	visible	ADJ
iajs-3092	3	16	modules	module	NOUN
iajs-3092	3	17	and	and	CCONJ
iajs-3092	3	18	some	some	DET
iajs-3092	3	19	types	type	NOUN
iajs-3092	3	20	of	of	ADP
iajs-3092	3	21	fuzzy	fuzzy	ADJ
iajs-3092	3	22	modules	module	NOUN
iajs-3092	3	23	such	such	ADJ
iajs-3092	3	24	as	as	ADP
iajs-3092	3	25	semiprime	semiprime	NOUN
iajs-3092	3	26	,	,	PUNCT
iajs-3092	3	27	prime	prime	ADJ
iajs-3092	3	28	,	,	PUNCT
iajs-3092	3	29	quasi	quasi	ADJ
iajs-3092	3	30	,	,	PUNCT
iajs-3092	3	31	divisible	divisible	ADJ
iajs-3092	3	32	,	,	PUNCT
iajs-3092	3	33	f	f	NOUN
iajs-3092	3	34	-	-	PUNCT
iajs-3092	3	35	regular	regular	ADJ
iajs-3092	3	36	,	,	PUNCT
iajs-3092	3	37	quasi	quasi	NOUN
iajs-3092	3	38	injective	injective	NOUN
iajs-3092	3	39	,	,	PUNCT
iajs-3092	3	40	and	and	CCONJ
iajs-3092	3	41	duo	duo	ADJ
iajs-3092	3	42	fuzzy	fuzzy	ADJ
iajs-3092	3	43	modules	module	NOUN
iajs-3092	3	44	,	,	PUNCT
iajs-3092	3	45	where	where	SCONJ
iajs-3092	3	46	under	under	ADP
iajs-3092	3	47	certain	certain	ADJ
iajs-3092	3	48	conditions	condition	NOUN
iajs-3092	3	49	it	it	PRON
iajs-3092	3	50	has	have	AUX
iajs-3092	3	51	been	be	AUX
iajs-3092	3	52	proven	prove	VERB
iajs-3092	3	53	that	that	SCONJ
iajs-3092	3	54	each	each	DET
iajs-3092	3	55	fully	fully	ADV
iajs-3092	3	56	fuzzy	fuzzy	ADJ
iajs-3092	3	57	visible	visible	ADJ
iajs-3092	3	58	module	module	NOUN
iajs-3092	3	59	is	be	AUX
iajs-3092	3	60	fuzzy	fuzzy	ADJ
iajs-3092	3	61	duo	duo	NOUN
iajs-3092	3	62	.	.	PUNCT
iajs-3092	4	1	in	in	ADP
iajs-3092	4	2	addition	addition	NOUN
iajs-3092	4	3	,	,	PUNCT
iajs-3092	4	4	there	there	PRON
iajs-3092	4	5	are	be	VERB
iajs-3092	4	6	many	many	ADJ
iajs-3092	4	7	various	various	ADJ
iajs-3092	4	8	properties	property	NOUN
iajs-3092	4	9	and	and	CCONJ
iajs-3092	4	10	important	important	ADJ
iajs-3092	4	11	results	result	NOUN
iajs-3092	4	12	obtained	obtain	VERB
iajs-3092	4	13	through	through	ADP
iajs-3092	4	14	this	this	DET
iajs-3092	4	15	research	research	NOUN
iajs-3092	4	16	,	,	PUNCT
iajs-3092	4	17	which	which	PRON
iajs-3092	4	18	have	have	AUX
iajs-3092	4	19	been	be	AUX
iajs-3092	4	20	illustrated	illustrate	VERB
iajs-3092	4	21	.	.	PUNCT
iajs-3092	5	1	also	also	ADV
iajs-3092	5	2	,	,	PUNCT
iajs-3092	5	3	fuzzy	fuzzy	ADJ
iajs-3092	5	4	artinian	artinian	ADJ
iajs-3092	5	5	modules	module	NOUN
iajs-3092	5	6	and	and	CCONJ
iajs-3092	5	7	fuzzy	fuzzy	ADJ
iajs-3092	5	8	fully	fully	ADV
iajs-3092	5	9	stable	stable	ADJ
iajs-3092	5	10	modules	module	NOUN
iajs-3092	5	11	have	have	AUX
iajs-3092	5	12	been	be	AUX
iajs-3092	5	13	introduced	introduce	VERB
iajs-3092	5	14	,	,	PUNCT
iajs-3092	5	15	and	and	CCONJ
iajs-3092	5	16	we	we	PRON
iajs-3092	5	17	study	study	VERB
iajs-3092	5	18	the	the	DET
iajs-3092	5	19	relationships	relationship	NOUN
iajs-3092	5	20	between	between	ADP
iajs-3092	5	21	these	these	DET
iajs-3092	5	22	kinds	kind	NOUN
iajs-3092	5	23	of	of	ADP
iajs-3092	5	24	modules	module	NOUN
iajs-3092	5	25	and	and	CCONJ
iajs-3092	5	26	fully	fully	ADV
iajs-3092	5	27	fuzzy	fuzzy	ADJ
iajs-3092	5	28	visible	visible	ADJ
iajs-3092	5	29	modules	module	NOUN
iajs-3092	5	30	.	.	PUNCT
iajs-3092	6	1	many	many	ADJ
iajs-3092	6	2	other	other	ADJ
iajs-3092	6	3	intersecting	intersecting	NOUN
iajs-3092	6	4	results	result	NOUN
iajs-3092	6	5	we	we	PRON
iajs-3092	6	6	found	find	VERB
iajs-3092	6	7	.	.	PUNCT
iajs-3092	7	1	keywords	keyword	NOUN
iajs-3092	7	2	:	:	PUNCT
iajs-3092	7	3	fully	fully	ADV
iajs-3092	7	4	fuzzy	fuzzy	ADJ
iajs-3092	7	5	visible	visible	ADJ
iajs-3092	7	6	modules	module	NOUN
iajs-3092	7	7	,	,	PUNCT
iajs-3092	7	8	fully	fully	ADV
iajs-3092	7	9	fuzzy	fuzzy	ADJ
iajs-3092	7	10	stable	stable	ADJ
iajs-3092	7	11	modules	module	NOUN
iajs-3092	7	12	,	,	PUNCT
iajs-3092	7	13	fuzzy	fuzzy	ADJ
iajs-3092	7	14	artiain	artiain	NOUN
iajs-3092	7	15	modules	module	NOUN
iajs-3092	7	16	,	,	PUNCT
iajs-3092	7	17	fuzzy	fuzzy	ADJ
iajs-3092	7	18	regular	regular	ADJ
iajs-3092	7	19	rings	ring	NOUN
iajs-3092	7	20	,	,	PUNCT
iajs-3092	7	21	fuzzy	fuzzy	ADJ
iajs-3092	7	22	quasi	quasi	ADJ
iajs-3092	7	23	injective	injective	ADJ
iajs-3092	7	24	modules	module	NOUN
iajs-3092	7	25	.	.	PUNCT
iajs-3092	8	1	1	1	X
iajs-3092	8	2	.	.	X
iajs-3092	8	3	introduction	introduction	NOUN
iajs-3092	8	4	.	.	PUNCT
iajs-3092	9	1	the	the	DET
iajs-3092	9	2	notion	notion	NOUN
iajs-3092	9	3	of	of	ADP
iajs-3092	9	4	fuzzy	fuzzy	ADJ
iajs-3092	9	5	set	set	NOUN
iajs-3092	9	6	has	have	AUX
iajs-3092	9	7	been	be	AUX
iajs-3092	9	8	presented	present	VERB
iajs-3092	9	9	by	by	ADP
iajs-3092	9	10	zadeh	zadeh	PROPN
iajs-3092	9	11	in	in	ADP
iajs-3092	9	12	1965[1].the	1965[1].the	NUM
iajs-3092	9	13	fuzzy	fuzzy	ADJ
iajs-3092	9	14	groups	group	NOUN
iajs-3092	9	15	have	have	AUX
iajs-3092	9	16	been	be	AUX
iajs-3092	9	17	introduced	introduce	VERB
iajs-3092	9	18	by	by	ADP
iajs-3092	9	19	rosenfeld	rosenfeld	PROPN
iajs-3092	9	20	in	in	ADP
iajs-3092	9	21	1971	1971	NUM
iajs-3092	10	1	[	[	X
iajs-3092	10	2	2	2	NUM
iajs-3092	10	3	]	]	PUNCT
iajs-3092	10	4	.	.	PUNCT
iajs-3092	11	1	then	then	ADV
iajs-3092	11	2	,	,	PUNCT
iajs-3092	11	3	many	many	ADJ
iajs-3092	11	4	other	other	ADJ
iajs-3092	11	5	researchers	researcher	NOUN
iajs-3092	11	6	studied	study	VERB
iajs-3092	11	7	different	different	ADJ
iajs-3092	11	8	applications	application	NOUN
iajs-3092	11	9	of	of	ADP
iajs-3092	11	10	fuzzy	fuzzy	ADJ
iajs-3092	11	11	sets	set	NOUN
iajs-3092	11	12	of	of	ADP
iajs-3092	11	13	algebra	algebra	NOUN
iajs-3092	11	14	.	.	PUNCT
iajs-3092	12	1	the	the	DET
iajs-3092	12	2	concept	concept	NOUN
iajs-3092	12	3	of	of	ADP
iajs-3092	12	4	fuzzy	fuzzy	ADJ
iajs-3092	12	5	modules	module	NOUN
iajs-3092	12	6	was	be	AUX
iajs-3092	12	7	presented	present	VERB
iajs-3092	12	8	by	by	ADP
iajs-3092	12	9	negoita	negoita	PROPN
iajs-3092	12	10	and	and	CCONJ
iajs-3092	12	11	relescn	relescn	NOUN
iajs-3092	12	12	in	in	ADP
iajs-3092	12	13	1975	1975	NUM
iajs-3092	12	14	[	[	X
iajs-3092	12	15	3	3	NUM
iajs-3092	12	16	]	]	PUNCT
iajs-3092	12	17	.	.	PUNCT
iajs-3092	13	1	the	the	DET
iajs-3092	13	2	notion	notion	NOUN
iajs-3092	13	3	of	of	ADP
iajs-3092	13	4	fuzzy	fuzzy	ADJ
iajs-3092	13	5	visible	visible	ADJ
iajs-3092	13	6	submodules	submodule	NOUN
iajs-3092	13	7	was	be	AUX
iajs-3092	13	8	introduced	introduce	VERB
iajs-3092	13	9	by	by	ADP
iajs-3092	13	10	sajda	sajda	NOUN
iajs-3092	13	11	k.m	k.m	PROPN
iajs-3092	13	12	.	.	PROPN
iajs-3092	13	13	and	and	CCONJ
iajs-3092	13	14	buthyna	buthyna	PROPN
iajs-3092	13	15	n.	n.	PROPN
iajs-3092	13	16	s.	s.	PROPN
iajs-3092	13	17	2021	2021	NUM
iajs-3092	14	1	[	[	X
iajs-3092	14	2	4	4	NUM
iajs-3092	14	3	]	]	PUNCT
iajs-3092	14	4	.	.	PUNCT
iajs-3092	15	1	the	the	DET
iajs-3092	15	2	concept	concept	NOUN
iajs-3092	15	3	of	of	ADP
iajs-3092	15	4	fully	fully	ADV
iajs-3092	15	5	fuzzy	fuzzy	ADJ
iajs-3092	15	6	visible	visible	ADJ
iajs-3092	15	7	modules	module	NOUN
iajs-3092	15	8	has	have	AUX
iajs-3092	15	9	been	be	AUX
iajs-3092	15	10	introduced	introduce	VERB
iajs-3092	15	11	by	by	ADP
iajs-3092	15	12	sajda	sajda	NOUN
iajs-3092	15	13	k.m	k.m	PROPN
iajs-3092	15	14	.	.	PROPN
iajs-3092	15	15	and	and	CCONJ
iajs-3092	15	16	buthyna	buthyna	PROPN
iajs-3092	15	17	n.s	n.s	PROPN
iajs-3092	15	18	.	.	PROPN
iajs-3092	16	1	in	in	ADP
iajs-3092	16	2	2022	2022	NUM
iajs-3092	16	3	[	[	X
iajs-3092	16	4	5	5	NUM
iajs-3092	16	5	]	]	PUNCT
iajs-3092	16	6	.	.	PUNCT
iajs-3092	17	1	in	in	ADP
iajs-3092	17	2	this	this	DET
iajs-3092	17	3	paper	paper	NOUN
iajs-3092	17	4	,	,	PUNCT
iajs-3092	17	5	the	the	DET
iajs-3092	17	6	relationships	relationship	NOUN
iajs-3092	17	7	between	between	ADP
iajs-3092	17	8	fully	fully	ADV
iajs-3092	17	9	fuzzy	fuzzy	ADJ
iajs-3092	17	10	visible	visible	ADJ
iajs-3092	17	11	modules	module	NOUN
iajs-3092	17	12	and	and	CCONJ
iajs-3092	17	13	other	other	ADJ
iajs-3092	17	14	different	different	ADJ
iajs-3092	17	15	modules	module	NOUN
iajs-3092	17	16	have	have	AUX
iajs-3092	17	17	been	be	AUX
iajs-3092	17	18	explained	explain	VERB
iajs-3092	17	19	,	,	PUNCT
iajs-3092	17	20	like	like	ADP
iajs-3092	17	21	fuzzy	fuzzy	ADJ
iajs-3092	17	22	(	(	PUNCT
iajs-3092	17	23	prime	prime	ADJ
iajs-3092	17	24	,	,	PUNCT
iajs-3092	17	25	semiprime	semiprime	NOUN
iajs-3092	17	26	,	,	PUNCT
iajs-3092	17	27	semisimple	semisimple	NOUN
iajs-3092	17	28	,	,	PUNCT
iajs-3092	17	29	and	and	CCONJ
iajs-3092	17	30	fegular	fegular	ADJ
iajs-3092	17	31	)	)	PUNCT
iajs-3092	17	32	.	.	PUNCT
iajs-3092	18	1	also	also	ADV
iajs-3092	18	2	,	,	PUNCT
iajs-3092	18	3	the	the	DET
iajs-3092	18	4	notions	notion	NOUN
iajs-3092	18	5	of	of	ADP
iajs-3092	18	6	stable	stable	ADJ
iajs-3092	18	7	submodules	submodule	NOUN
iajs-3092	18	8	,	,	PUNCT
iajs-3092	18	9	fully	fully	ADV
iajs-3092	18	10	stable	stable	ADJ
iajs-3092	18	11	modules	module	NOUN
iajs-3092	18	12	,	,	PUNCT
iajs-3092	18	13	and	and	CCONJ
iajs-3092	18	14	quasi	quasi	ADJ
iajs-3092	18	15	-	-	ADJ
iajs-3092	18	16	injective	injective	ADJ
iajs-3092	18	17	modules	module	NOUN
iajs-3092	18	18	doi.org/10.30526/36.3.3092	doi.org/10.30526/36.3.3092	VERB
iajs-3092	18	19	article	article	NOUN
iajs-3092	18	20	history	history	NOUN
iajs-3092	18	21	:	:	PUNCT
iajs-3092	18	22	received	receive	VERB
iajs-3092	18	23	30	30	NUM
iajs-3092	18	24	october	october	NOUN
iajs-3092	18	25	2022	2022	NUM
iajs-3092	18	26	,	,	PUNCT
iajs-3092	18	27	accepted	accept	VERB
iajs-3092	18	28	19	19	NUM
iajs-3092	18	29	december	december	PROPN
iajs-3092	18	30	2022	2022	NUM
iajs-3092	18	31	,	,	PUNCT
iajs-3092	18	32	published	publish	VERB
iajs-3092	18	33	in	in	ADP
iajs-3092	18	34	july	july	PROPN
iajs-3092	18	35	2023	2023	NUM
iajs-3092	18	36	.	.	PUNCT
iajs-3092	19	1	ibn	ibn	PROPN
iajs-3092	19	2	al	al	PROPN
iajs-3092	19	3	-	-	PUNCT
iajs-3092	19	4	haitham	haitham	PROPN
iajs-3092	19	5	journal	journal	PROPN
iajs-3092	19	6	for	for	ADP
iajs-3092	19	7	pure	pure	ADJ
iajs-3092	19	8	and	and	CCONJ
iajs-3092	19	9	applied	applied	ADJ
iajs-3092	19	10	sciences	sciences	PROPN
iajs-3092	19	11	journal	journal	PROPN
iajs-3092	19	12	homepage	homepage	NOUN
iajs-3092	19	13	:	:	PUNCT
iajs-3092	19	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	VERB
iajs-3092	19	15	fully	fully	ADV
iajs-3092	19	16	fuzzy	fuzzy	ADJ
iajs-3092	19	17	visible	visible	ADJ
iajs-3092	19	18	modules	module	NOUN
iajs-3092	19	19	with	with	ADP
iajs-3092	19	20	other	other	ADJ
iajs-3092	19	21	related	relate	VERB
iajs-3092	19	22	concepts	concept	NOUN
iajs-3092	19	23	*	*	PUNCT
iajs-3092	19	24	sajda	sajda	NOUN
iajs-3092	19	25	k.mohammed	k.mohammed	ADJ
iajs-3092	19	26	department	department	NOUN
iajs-3092	19	27	of	of	ADP
iajs-3092	19	28	mathematics	mathematics	PROPN
iajs-3092	19	29	faculty	faculty	NOUN
iajs-3092	19	30	of	of	ADP
iajs-3092	19	31	education	education	NOUN
iajs-3092	19	32	for	for	ADP
iajs-3092	19	33	girls	girl	NOUN
iajs-3092	19	34	,	,	PUNCT
iajs-3092	19	35	university	university	PROPN
iajs-3092	19	36	of	of	ADP
iajs-3092	19	37	kufa	kufa	PROPN
iajs-3092	19	38	,	,	PUNCT
iajs-3092	19	39	iraq	iraq	PROPN
iajs-3092	19	40	department	department	PROPN
iajs-3092	19	41	of	of	ADP
iajs-3092	19	42	mathematics	mathematics	PROPN
iajs-3092	19	43	,	,	PUNCT
iajs-3092	19	44	college	college	NOUN
iajs-3092	19	45	of	of	ADP
iajs-3092	19	46	education	education	NOUN
iajs-3092	19	47	for	for	ADP
iajs-3092	19	48	pure	pure	ADJ
iajs-3092	19	49	science	science	NOUN
iajs-3092	19	50	ibn	ibn	PROPN
iajs-3092	19	51	al	al	PROPN
iajs-3092	19	52	-	-	PUNCT
iajs-3092	19	53	haitham	haitham	PROPN
iajs-3092	19	54	,	,	PUNCT
iajs-3092	19	55	university	university	PROPN
iajs-3092	19	56	of	of	ADP
iajs-3092	19	57	baghdad	baghdad	PROPN
iajs-3092	19	58	,	,	PUNCT
iajs-3092	19	59	baghdad	baghdad	PROPN
iajs-3092	19	60	,	,	PUNCT
iajs-3092	19	61	iraq	iraq	PROPN
iajs-3092	19	62	.	.	PUNCT
iajs-3092	20	1	sajidak.mohammed@uokufa.edu.iq	sajidak.mohammed@uokufa.edu.iq	PROPN
iajs-3092	20	2	buthyna	buthyna	PROPN
iajs-3092	20	3	n.	n.	PROPN
iajs-3092	20	4	shihab	shihab	PROPN
iajs-3092	20	5	department	department	PROPN
iajs-3092	20	6	of	of	ADP
iajs-3092	20	7	mathematics	mathematics	PROPN
iajs-3092	20	8	,	,	PUNCT
iajs-3092	20	9	college	college	NOUN
iajs-3092	20	10	of	of	ADP
iajs-3092	20	11	education	education	NOUN
iajs-3092	20	12	for	for	ADP
iajs-3092	20	13	pure	pure	ADJ
iajs-3092	20	14	science	science	NOUN
iajs-3092	20	15	ibn	ibn	PROPN
iajs-3092	20	16	al	al	PROPN
iajs-3092	20	17	-	-	PUNCT
iajs-3092	20	18	haitham	haitham	PROPN
iajs-3092	20	19	,	,	PUNCT
iajs-3092	20	20	university	university	PROPN
iajs-3092	20	21	of	of	ADP
iajs-3092	20	22	baghdad	baghdad	PROPN
iajs-3092	20	23	,	,	PUNCT
iajs-3092	20	24	baghdad	baghdad	PROPN
iajs-3092	20	25	,	,	PUNCT
iajs-3092	20	26	iraq	iraq	PROPN
iajs-3092	20	27	.	.	PUNCT
iajs-3092	21	1	bothaina.n.s@ihcoedu.uobaghda.edu.iq	bothaina.n.s@ihcoedu.uobaghda.edu.iq	PROPN
iajs-3092	21	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3092	21	3	mailto:sajidak.mohammed@uokufa.edu.iq	mailto:sajidak.mohammed@uokufa.edu.iq	PROPN
iajs-3092	21	4	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3092	21	5	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3092	21	6	mailto:sajidak.mohammed@uokufa.edu.iq	mailto:sajidak.mohammed@uokufa.edu.iq	PROPN
iajs-3092	21	7	mailto:sajidak.mohammed@uokufa.edu.iq	mailto:sajidak.mohammed@uokufa.edu.iq	PROPN
iajs-3092	21	8	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3092	21	9	http://en.uobaghdad.edu.iq/?page_id=15060	http://en.uobaghdad.edu.iq/?page_id=15060	NOUN
iajs-3092	21	10	mailto:bothaina.n.s@ihcoedu.uobaghda.edu.iq	mailto:bothaina.n.s@ihcoedu.uobaghda.edu.iq	PROPN
iajs-3092	21	11	ihjpas	ihjpas	PROPN
iajs-3092	21	12	.	.	PUNCT
iajs-3092	22	1	36	36	NUM
iajs-3092	22	2	(	(	PUNCT
iajs-3092	22	3	3	3	NUM
iajs-3092	22	4	)	)	PUNCT
iajs-3092	22	5	2023	2023	NUM
iajs-3092	22	6	373	373	NUM
iajs-3092	22	7	are	be	AUX
iajs-3092	22	8	fuzzyified	fuzzyifie	VERB
iajs-3092	22	9	.	.	PUNCT
iajs-3092	23	1	many	many	ADJ
iajs-3092	23	2	important	important	ADJ
iajs-3092	23	3	properties	property	NOUN
iajs-3092	23	4	and	and	CCONJ
iajs-3092	23	5	results	result	NOUN
iajs-3092	23	6	between	between	ADP
iajs-3092	23	7	these	these	DET
iajs-3092	23	8	above	above	ADJ
iajs-3092	23	9	concepts	concept	NOUN
iajs-3092	23	10	and	and	CCONJ
iajs-3092	23	11	fully	fully	ADV
iajs-3092	23	12	fuzzy	fuzzy	ADJ
iajs-3092	23	13	visible	visible	ADJ
iajs-3092	23	14	modules	module	NOUN
iajs-3092	23	15	have	have	AUX
iajs-3092	23	16	been	be	AUX
iajs-3092	23	17	proven	prove	VERB
iajs-3092	23	18	.	.	PUNCT
iajs-3092	24	1	through	through	ADP
iajs-3092	24	2	this	this	DET
iajs-3092	24	3	paper	paper	NOUN
iajs-3092	24	4	will	will	AUX
iajs-3092	24	5	be	be	AUX
iajs-3092	24	6	a	a	DET
iajs-3092	24	7	unitary	unitary	ADJ
iajs-3092	24	8	module	module	NOUN
iajs-3092	24	9	over	over	ADP
iajs-3092	24	10	a	a	DET
iajs-3092	24	11	commutative	commutative	ADJ
iajs-3092	24	12	ring	ring	NOUN
iajs-3092	24	13	with	with	ADP
iajs-3092	24	14	identity	identity	NOUN
iajs-3092	24	15	and	and	CCONJ
iajs-3092	24	16	for	for	ADP
iajs-3092	24	17	each	each	DET
iajs-3092	24	18	fuzzy	fuzzy	ADJ
iajs-3092	24	19	ideal	ideal	ADJ
iajs-3092	24	20	ʉ	ʉ	PROPN
iajs-3092	24	21	of	of	ADP
iajs-3092	24	22	𝔽	𝔽	PROPN
iajs-3092	24	23	,	,	PUNCT
iajs-3092	24	24	ʉ	ʉ	PROPN
iajs-3092	24	25	(	(	PUNCT
iajs-3092	24	26	0)=1	0)=1	NUM
iajs-3092	24	27	.	.	NOUN
iajs-3092	25	1	2	2	NUM
iajs-3092	25	2	.	.	NUM
iajs-3092	25	3	preliminaries	preliminary	NOUN
iajs-3092	25	4	.	.	PUNCT
iajs-3092	26	1	this	this	DET
iajs-3092	26	2	section	section	NOUN
iajs-3092	26	3	contains	contain	VERB
iajs-3092	26	4	some	some	DET
iajs-3092	26	5	definitions	definition	NOUN
iajs-3092	26	6	and	and	CCONJ
iajs-3092	26	7	properties	property	NOUN
iajs-3092	26	8	that	that	PRON
iajs-3092	26	9	will	will	AUX
iajs-3092	26	10	be	be	AUX
iajs-3092	26	11	needed	need	VERB
iajs-3092	26	12	in	in	ADP
iajs-3092	26	13	our	our	PRON
iajs-3092	26	14	work	work	NOUN
iajs-3092	26	15	.	.	PUNCT
iajs-3092	27	1	2.1	2.1	NUM
iajs-3092	27	2	definition	definition	NOUN
iajs-3092	27	3	:	:	PUNCT
iajs-3092	27	4	let	let	VERB
iajs-3092	27	5	𝔽	𝔽	PROPN
iajs-3092	27	6	≠	≠	PROPN
iajs-3092	27	7	∅	∅	NOUN
iajs-3092	27	8	be	be	AUX
iajs-3092	27	9	a	a	DET
iajs-3092	27	10	set	set	NOUN
iajs-3092	27	11	and	and	CCONJ
iajs-3092	27	12	ɨ	ɨ	NOUN
iajs-3092	27	13	=	=	PUNCT
iajs-3092	28	1	[	[	X
iajs-3092	28	2	0	0	NUM
iajs-3092	28	3	,	,	PUNCT
iajs-3092	28	4	1	1	NUM
iajs-3092	28	5	]	]	PUNCT
iajs-3092	28	6	of	of	ADP
iajs-3092	28	7	the	the	DET
iajs-3092	28	8	real	real	ADJ
iajs-3092	28	9	line	line	NOUN
iajs-3092	28	10	(	(	PUNCT
iajs-3092	28	11	real	real	ADJ
iajs-3092	28	12	numbers	number	NOUN
iajs-3092	28	13	)	)	PUNCT
iajs-3092	28	14	.	.	PUNCT
iajs-3092	29	1	let	let	VERB
iajs-3092	29	2	ɦ	ɦ	PRON
iajs-3092	29	3	:	:	PUNCT
iajs-3092	29	4	𝔽	𝔽	PROPN
iajs-3092	29	5	→	→	SYM
iajs-3092	29	6	ɨ	ɨ	NOUN
iajs-3092	29	7	is	be	AUX
iajs-3092	29	8	a	a	DET
iajs-3092	29	9	function	function	NOUN
iajs-3092	29	10	.	.	PUNCT
iajs-3092	30	1	then	then	ADV
iajs-3092	30	2	,	,	PUNCT
iajs-3092	30	3	ɦ	ɦ	PRON
iajs-3092	30	4	is	be	AUX
iajs-3092	30	5	known	know	VERB
iajs-3092	30	6	as	as	ADP
iajs-3092	30	7	a	a	DET
iajs-3092	30	8	fuzzy	fuzzy	ADJ
iajs-3092	30	9	set	set	NOUN
iajs-3092	30	10	in	in	ADP
iajs-3092	30	11	𝔽	𝔽	PROPN
iajs-3092	30	12	(	(	PUNCT
iajs-3092	30	13	a	a	DET
iajs-3092	30	14	fuzzy	fuzzy	ADJ
iajs-3092	30	15	subset	subset	NOUN
iajs-3092	30	16	of	of	ADP
iajs-3092	30	17	,	,	PUNCT
iajs-3092	30	18	[	[	X
iajs-3092	30	19	1	1	NUM
iajs-3092	30	20	]	]	PUNCT
iajs-3092	30	21	.	.	PUNCT
iajs-3092	31	1	let	let	VERB
iajs-3092	31	2	fsf(𝔽	fsf(𝔽	NUM
iajs-3092	31	3	)	)	PUNCT
iajs-3092	31	4	=	=	PRON
iajs-3092	31	5	{	{	PUNCT
iajs-3092	31	6	ɦ	ɦ	X
iajs-3092	31	7	:	:	PUNCT
iajs-3092	31	8	ɦ	ɦ	NUM
iajs-3092	31	9	:	:	PUNCT
iajs-3092	31	10	𝔽	𝔽	PROPN
iajs-3092	31	11	→	→	SYM
iajs-3092	31	12	ɨ	ɨ	NOUN
iajs-3092	31	13	is	be	AUX
iajs-3092	31	14	a	a	DET
iajs-3092	31	15	function	function	NOUN
iajs-3092	31	16	}	}	PUNCT
iajs-3092	31	17	.	.	PUNCT
iajs-3092	32	1	2.2	2.2	NUM
iajs-3092	32	2	definition	definition	NOUN
iajs-3092	32	3	.	.	PUNCT
iajs-3092	33	1	let	let	VERB
iajs-3092	33	2	𝕐	𝕐	PRON
iajs-3092	33	3	≠	≠	PROPN
iajs-3092	33	4	∅	∅	NOUN
iajs-3092	33	5	and	and	CCONJ
iajs-3092	33	6	ℭ	ℭ	PROPN
iajs-3092	33	7	∈	∈	PROPN
iajs-3092	33	8	𝐹𝑈𝑆(𝕐	𝐹𝑈𝑆(𝕐	NUM
iajs-3092	33	9	)	)	PUNCT
iajs-3092	33	10	.	.	PUNCT
iajs-3092	34	1	a	a	DET
iajs-3092	34	2	level	level	NOUN
iajs-3092	34	3	set	set	NOUN
iajs-3092	34	4	of	of	ADP
iajs-3092	34	5	𝕐	𝕐	PROPN
iajs-3092	34	6	with	with	ADP
iajs-3092	34	7	respect	respect	NOUN
iajs-3092	34	8	to	to	ADP
iajs-3092	34	9	ℭ	ℭ	PROPN
iajs-3092	34	10	denoted	denote	VERB
iajs-3092	34	11	by	by	ADP
iajs-3092	34	12	the	the	DET
iajs-3092	34	13	set	set	PROPN
iajs-3092	34	14	ℭ𝑡	ℭ𝑡	PROPN
iajs-3092	34	15	,	,	PUNCT
iajs-3092	34	16	where	where	SCONJ
iajs-3092	34	17	ℭ𝑡={x∈	ℭ𝑡={x∈	PROPN
iajs-3092	34	18	𝕐	𝕐	PROPN
iajs-3092	34	19	∶	∶	NOUN
iajs-3092	34	20	ℭ	ℭ	PROPN
iajs-3092	34	21	(	(	PUNCT
iajs-3092	34	22	x)≥	x)≥	PROPN
iajs-3092	34	23	t	t	PROPN
iajs-3092	34	24	}	}	PUNCT
iajs-3092	34	25	∀𝑡	∀𝑡	PROPN
iajs-3092	34	26	∈	∈	PROPN
iajs-3092	35	1	[	[	X
iajs-3092	35	2	0,1	0,1	NUM
iajs-3092	35	3	]	]	PUNCT
iajs-3092	35	4	,	,	PUNCT
iajs-3092	35	5	[	[	PUNCT
iajs-3092	35	6	68	68	NUM
iajs-3092	35	7	]	]	PUNCT
iajs-3092	35	8	.	.	PUNCT
iajs-3092	36	1	2.3	2.3	NUM
iajs-3092	36	2	definition	definition	NOUN
iajs-3092	36	3	:	:	PUNCT
iajs-3092	36	4	suppose	suppose	VERB
iajs-3092	36	5	that	that	SCONJ
iajs-3092	36	6	𝕄	𝕄	PROPN
iajs-3092	36	7	be	be	VERB
iajs-3092	36	8	an	an	DET
iajs-3092	36	9	𝔽	𝔽	PROPN
iajs-3092	36	10	–	–	PUNCT
iajs-3092	36	11	module	module	NOUN
iajs-3092	36	12	and	and	CCONJ
iajs-3092	36	13	₮	₮	ADP
iajs-3092	36	14	∈	∈	PROPN
iajs-3092	36	15	𝐹𝑆𝐸(𝕄	𝐹𝑆𝐸(𝕄	NOUN
iajs-3092	36	16	)	)	PUNCT
iajs-3092	36	17	.	.	PUNCT
iajs-3092	37	1	ℙ	ℙ	NOUN
iajs-3092	37	2	is	be	AUX
iajs-3092	37	3	known	know	VERB
iajs-3092	37	4	as	as	ADP
iajs-3092	37	5	a	a	DET
iajs-3092	37	6	fuzzy	fuzzy	ADJ
iajs-3092	37	7	module	module	NOUN
iajs-3092	37	8	of	of	ADP
iajs-3092	37	9	an	an	DET
iajs-3092	37	10	𝔽	𝔽	PROPN
iajs-3092	37	11	-module	-module	NOUN
iajs-3092	37	12	if	if	SCONJ
iajs-3092	37	13	,	,	PUNCT
iajs-3092	37	14	1	1	NUM
iajs-3092	37	15	₮	₮	PROPN
iajs-3092	37	16	(	(	PUNCT
iajs-3092	37	17	ʑ	ʑ	PROPN
iajs-3092	37	18	ƫ)≥min	ƫ)≥min	X
iajs-3092	37	19	{	{	PUNCT
iajs-3092	37	20	₮	₮	PROPN
iajs-3092	37	21	(	(	PUNCT
iajs-3092	37	22	ʑ	ʑ	NOUN
iajs-3092	37	23	)	)	PUNCT
iajs-3092	37	24	,	,	PUNCT
iajs-3092	37	25	₮	₮	PROPN
iajs-3092	37	26	(	(	PUNCT
iajs-3092	37	27	ƫ	ƫ	NOUN
iajs-3092	37	28	)	)	PUNCT
iajs-3092	37	29	}	}	PUNCT
iajs-3092	37	30	,	,	PUNCT
iajs-3092	37	31	for	for	ADP
iajs-3092	37	32	all	all	DET
iajs-3092	37	33	ʑ	ʑ	PROPN
iajs-3092	37	34	,	,	PUNCT
iajs-3092	37	35	ƫ∈	ƫ∈	NOUN
iajs-3092	37	36	𝕄	𝕄	PROPN
iajs-3092	37	37	,	,	PUNCT
iajs-3092	37	38	2	2	NUM
iajs-3092	37	39	₮	₮	PROPN
iajs-3092	37	40	(	(	PUNCT
iajs-3092	37	41	rʑ)≥	rʑ)≥	NOUN
iajs-3092	37	42	₮	₮	PROPN
iajs-3092	37	43	(	(	PUNCT
iajs-3092	37	44	ʑ	ʑ	PROPN
iajs-3092	37	45	)	)	PUNCT
iajs-3092	37	46	,	,	PUNCT
iajs-3092	37	47	for	for	ADP
iajs-3092	37	48	all	all	PRON
iajs-3092	37	49	ʑ∈	ʑ∈	ADP
iajs-3092	37	50	𝕄	𝕄	PROPN
iajs-3092	37	51	,	,	PUNCT
iajs-3092	37	52	r∈	r∈	PROPN
iajs-3092	37	53	𝔽	𝔽	PROPN
iajs-3092	37	54	,	,	PUNCT
iajs-3092	37	55	3	3	NUM
iajs-3092	37	56	₮	₮	PROPN
iajs-3092	37	57	(	(	PUNCT
iajs-3092	37	58	0)=1	0)=1	NUM
iajs-3092	38	1	[	[	PUNCT
iajs-3092	38	2	9	9	NUM
iajs-3092	38	3	-	-	SYM
iajs-3092	38	4	10	10	NUM
iajs-3092	38	5	]	]	PUNCT
iajs-3092	38	6	.	.	PUNCT
iajs-3092	39	1	let	let	VERB
iajs-3092	39	2	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	X
iajs-3092	39	3	)	)	PUNCT
iajs-3092	39	4	=	=	SYM
iajs-3092	39	5	{	{	PUNCT
iajs-3092	39	6	₮	₮	ADP
iajs-3092	39	7	:	:	PUNCT
iajs-3092	39	8	₮	₮	PROPN
iajs-3092	39	9	is	be	AUX
iajs-3092	39	10	𝑎	𝑎	DET
iajs-3092	39	11	fuzzy	fuzzy	ADJ
iajs-3092	39	12	modules	module	NOUN
iajs-3092	39	13	of	of	ADP
iajs-3092	39	14	an	an	DET
iajs-3092	39	15	𝔽	𝔽	PROPN
iajs-3092	39	16	−	−	PROPN
iajs-3092	39	17	module	module	NOUN
iajs-3092	39	18	𝕄	𝕄	PROPN
iajs-3092	39	19	}	}	PUNCT
iajs-3092	39	20	.	.	PUNCT
iajs-3092	40	1	2.4	2.4	NUM
iajs-3092	40	2	definition	definition	NOUN
iajs-3092	40	3	:	:	PUNCT
iajs-3092	40	4	let	let	VERB
iajs-3092	40	5	ℭ	ℭ	PROPN
iajs-3092	40	6	,	,	PUNCT
iajs-3092	40	7	ℙ	ℙ	PROPN
iajs-3092	40	8	∈	∈	PROPN
iajs-3092	40	9	fum(𝕄	fum(𝕄	PROPN
iajs-3092	40	10	)	)	PUNCT
iajs-3092	40	11	.	.	PUNCT
iajs-3092	41	1	ℭ	ℭ	PROPN
iajs-3092	41	2	is	be	AUX
iajs-3092	41	3	known	know	VERB
iajs-3092	41	4	as	as	ADP
iajs-3092	41	5	a	a	DET
iajs-3092	41	6	fuzzy	fuzzy	ADJ
iajs-3092	41	7	submodule	submodule	NOUN
iajs-3092	41	8	of	of	ADP
iajs-3092	41	9	ℙ	ℙ	PRON
iajs-3092	41	10	if	if	SCONJ
iajs-3092	41	11	ℭ	ℭ	PROPN
iajs-3092	41	12	⊆	⊆	NUM
iajs-3092	41	13	ℙ	ℙ	NOUN
iajs-3092	42	1	[	[	NOUN
iajs-3092	42	2	1113	1113	NUM
iajs-3092	42	3	]	]	PUNCT
iajs-3092	42	4	.	.	PUNCT
iajs-3092	43	1	let	let	VERB
iajs-3092	43	2	fus(ℙ	fus(ℙ	PRON
iajs-3092	43	3	)	)	PUNCT
iajs-3092	44	1	=	=	PRON
iajs-3092	44	2	{	{	PUNCT
iajs-3092	44	3	ℭ	ℭ	NOUN
iajs-3092	44	4	:	:	PUNCT
iajs-3092	44	5	ℭ	ℭ	PROPN
iajs-3092	44	6	∈	∈	PROPN
iajs-3092	44	7	fum(𝕄	fum(𝕄	X
iajs-3092	44	8	)	)	PUNCT
iajs-3092	44	9	and	and	CCONJ
iajs-3092	44	10	ℭ	ℭ	PROPN
iajs-3092	44	11	⊆	⊆	NUM
iajs-3092	44	12	ℙ	ℙ	PROPN
iajs-3092	44	13	}	}	PUNCT
iajs-3092	44	14	.	.	PUNCT
iajs-3092	45	1	2.5	2.5	NUM
iajs-3092	45	2	definition	definition	NOUN
iajs-3092	45	3	:	:	PUNCT
iajs-3092	45	4	let	let	VERB
iajs-3092	45	5	𝔽	𝔽	PRON
iajs-3092	45	6	be	be	AUX
iajs-3092	45	7	a	a	DET
iajs-3092	45	8	ring	ring	NOUN
iajs-3092	45	9	and	and	CCONJ
iajs-3092	45	10	ί	ί	NOUN
iajs-3092	45	11	∈	∈	PROPN
iajs-3092	45	12	𝐹𝑆𝐸(𝔽	𝐹𝑆𝐸(𝔽	NOUN
iajs-3092	45	13	)	)	PUNCT
iajs-3092	45	14	.	.	PUNCT
iajs-3092	46	1	ί	ί	PROPN
iajs-3092	46	2	is	be	AUX
iajs-3092	46	3	referred	refer	VERB
iajs-3092	46	4	to	to	ADP
iajs-3092	46	5	as	as	ADP
iajs-3092	46	6	a	a	DET
iajs-3092	46	7	fuzzy	fuzzy	ADJ
iajs-3092	46	8	ideal	ideal	NOUN
iajs-3092	46	9	of	of	ADP
iajs-3092	46	10	𝔽	𝔽	PROPN
iajs-3092	46	11	,	,	PUNCT
iajs-3092	46	12	if	if	SCONJ
iajs-3092	46	13	∀	∀	X
iajs-3092	46	14	ʑ	ʑ	NOUN
iajs-3092	46	15	,	,	PUNCT
iajs-3092	46	16	ƫ∈	ƫ∈	NOUN
iajs-3092	46	17	𝔽	𝔽	PROPN
iajs-3092	46	18	:	:	PUNCT
iajs-3092	46	19	1-ί	1-ί	NUM
iajs-3092	46	20	(	(	PUNCT
iajs-3092	46	21	ʑ	ʑ	PROPN
iajs-3092	46	22	-y)≥	-y)≥	PROPN
iajs-3092	46	23	min	min	PROPN
iajs-3092	46	24	{	{	PUNCT
iajs-3092	46	25	ί	ί	PROPN
iajs-3092	46	26	(	(	PUNCT
iajs-3092	46	27	ʑ	ʑ	PROPN
iajs-3092	46	28	)	)	PUNCT
iajs-3092	46	29	,	,	PUNCT
iajs-3092	46	30	ί	ί	PROPN
iajs-3092	46	31	(	(	PUNCT
iajs-3092	46	32	ƫ	ƫ	NOUN
iajs-3092	46	33	)	)	PUNCT
iajs-3092	46	34	}	}	PUNCT
iajs-3092	46	35	2-ί	2-ί	NUM
iajs-3092	46	36	(	(	PUNCT
iajs-3092	46	37	ʑ.y)≥	ʑ.y)≥	PROPN
iajs-3092	46	38	max	max	PROPN
iajs-3092	46	39	{	{	PUNCT
iajs-3092	46	40	ί	ί	PROPN
iajs-3092	46	41	(	(	PUNCT
iajs-3092	46	42	ʑ	ʑ	PROPN
iajs-3092	46	43	)	)	PUNCT
iajs-3092	46	44	,	,	PUNCT
iajs-3092	46	45	ί	ί	PROPN
iajs-3092	46	46	(	(	PUNCT
iajs-3092	46	47	ƫ	ƫ	NOUN
iajs-3092	46	48	)	)	PUNCT
iajs-3092	46	49	}	}	PUNCT
iajs-3092	46	50	,	,	PUNCT
iajs-3092	46	51	[	[	X
iajs-3092	46	52	14	14	NUM
iajs-3092	46	53	]	]	PUNCT
iajs-3092	46	54	.	.	PUNCT
iajs-3092	47	1	fui(ℝ	fui(ℝ	PUNCT
iajs-3092	47	2	)	)	PUNCT
iajs-3092	47	3	will	will	AUX
iajs-3092	47	4	be	be	AUX
iajs-3092	47	5	used	use	VERB
iajs-3092	47	6	to	to	PART
iajs-3092	47	7	represent	represent	VERB
iajs-3092	47	8	the	the	DET
iajs-3092	47	9	set	set	NOUN
iajs-3092	47	10	of	of	ADP
iajs-3092	47	11	all	all	DET
iajs-3092	47	12	fuzzy	fuzzy	ADJ
iajs-3092	47	13	ideals	ideal	NOUN
iajs-3092	47	14	of	of	ADP
iajs-3092	47	15	𝔽	𝔽	PROPN
iajs-3092	47	16	.	.	PUNCT
iajs-3092	48	1	2.6	2.6	NUM
iajs-3092	48	2	definition	definition	NOUN
iajs-3092	48	3	:	:	PUNCT
iajs-3092	48	4	let	let	VERB
iajs-3092	48	5	ℙ	ℙ	PRON
iajs-3092	48	6	∈	∈	NOUN
iajs-3092	48	7	𝐹𝑈𝑀(𝕄),ℭ	𝐹𝑈𝑀(𝕄),ℭ	NUM
iajs-3092	48	8	∈	∈	NOUN
iajs-3092	48	9	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	48	10	)	)	PUNCT
iajs-3092	48	11	and	and	CCONJ
iajs-3092	48	12	ί	ί	PRON
iajs-3092	48	13	∈	∈	NOUN
iajs-3092	48	14	𝐹𝑈𝐼(𝔽).the	𝐹𝑈𝐼(𝔽).the	DET
iajs-3092	48	15	product	product	NOUN
iajs-3092	48	16	(	(	PUNCT
iajs-3092	48	17	ί	ί	PROPN
iajs-3092	48	18	ℭ)(x	ℭ)(x	NOUN
iajs-3092	48	19	)	)	PUNCT
iajs-3092	48	20	=	=	PRON
iajs-3092	48	21	{	{	PUNCT
iajs-3092	48	22	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
iajs-3092	48	23	{	{	PUNCT
iajs-3092	48	24	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
iajs-3092	48	25	{	{	PUNCT
iajs-3092	48	26	ί	ί	PROPN
iajs-3092	48	27	(	(	PUNCT
iajs-3092	48	28	𝑟1	𝑟1	NOUN
iajs-3092	48	29	)	)	PUNCT
iajs-3092	48	30	,	,	PUNCT
iajs-3092	48	31	…	…	PUNCT
iajs-3092	48	32	,	,	PUNCT
iajs-3092	48	33	ί	ί	PROPN
iajs-3092	48	34	(	(	PUNCT
iajs-3092	48	35	𝑟𝑛	𝑟𝑛	ADJ
iajs-3092	48	36	)	)	PUNCT
iajs-3092	48	37	,	,	PUNCT
iajs-3092	48	38	ℭ(𝑥1	ℭ(𝑥1	PROPN
iajs-3092	48	39	)	)	PUNCT
iajs-3092	48	40	,	,	PUNCT
iajs-3092	48	41	…	…	PUNCT
iajs-3092	48	42	,	,	PUNCT
iajs-3092	48	43	ℭ(𝑥𝑛)𝑓𝑜𝑟	ℭ(𝑥𝑛)𝑓𝑜𝑟	PROPN
iajs-3092	48	44	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	PROPN
iajs-3092	48	45	𝑟𝑖	𝑟𝑖	ADP
iajs-3092	48	46	∈	∈	PROPN
iajs-3092	48	47	𝔽	𝔽	PROPN
iajs-3092	48	48	,	,	PUNCT
iajs-3092	48	49	𝑥𝑖	𝑥𝑖	PROPN
iajs-3092	48	50	∈	∈	PROPN
iajs-3092	48	51	𝕄	𝕄	PROPN
iajs-3092	48	52	,	,	PUNCT
iajs-3092	48	53	𝑛	𝑛	PRON
iajs-3092	48	54	∈	∈	PROPN
iajs-3092	48	55	𝑁	𝑁	PROPN
iajs-3092	48	56	,	,	PUNCT
iajs-3092	48	57	0	0	NUM
iajs-3092	48	58	𝑜.	𝑜.	NOUN
iajs-3092	48	59	𝑤.	𝑤.	NOUN
iajs-3092	49	1	𝑥	𝑥	NOUN
iajs-3092	49	2	=	=	PUNCT
iajs-3092	49	3	∑	∑	PUNCT
iajs-3092	49	4	𝑟𝑖𝑥𝑖	𝑟𝑖𝑥𝑖	VERB
iajs-3092	49	5	𝑛	𝑛	PRON
iajs-3092	49	6	𝑖=1	𝑖=1	PUNCT
iajs-3092	49	7	,	,	PUNCT
iajs-3092	50	1	[	[	X
iajs-3092	50	2	9	9	NUM
iajs-3092	50	3	]	]	PUNCT
iajs-3092	50	4	.	.	PUNCT
iajs-3092	51	1	note	note	VERB
iajs-3092	51	2	that	that	SCONJ
iajs-3092	51	3	ί	ί	VERB
iajs-3092	51	4	ℭ	ℭ	PROPN
iajs-3092	51	5	∈	∈	PROPN
iajs-3092	51	6	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	51	7	)	)	PUNCT
iajs-3092	51	8	if	if	SCONJ
iajs-3092	51	9	ί	ί	PROPN
iajs-3092	51	10	(	(	PUNCT
iajs-3092	51	11	0)=1	0)=1	NUM
iajs-3092	51	12	,	,	PUNCT
iajs-3092	51	13	[	[	X
iajs-3092	51	14	9	9	NUM
iajs-3092	51	15	]	]	PUNCT
iajs-3092	51	16	.	.	PUNCT
iajs-3092	52	1	and	and	CCONJ
iajs-3092	52	2	(	(	PUNCT
iajs-3092	52	3	ί	ί	NOUN
iajs-3092	52	4	ℭ)𝑡	ℭ)𝑡	NOUN
iajs-3092	52	5	=	=	SYM
iajs-3092	52	6	ί	ί	PROPN
iajs-3092	52	7	𝑡ℭ𝑡for	𝑡ℭ𝑡for	PROPN
iajs-3092	52	8	each	each	DET
iajs-3092	52	9	𝑡	𝑡	X
iajs-3092	52	10	∈	∈	PROPN
iajs-3092	53	1	[	[	X
iajs-3092	53	2	0,1	0,1	NUM
iajs-3092	53	3	]	]	PUNCT
iajs-3092	53	4	,	,	PUNCT
iajs-3092	53	5	[	[	X
iajs-3092	53	6	15	15	NUM
iajs-3092	53	7	]	]	PUNCT
iajs-3092	53	8	.	.	PUNCT
iajs-3092	54	1	2.7	2.7	NUM
iajs-3092	54	2	definition	definition	NOUN
iajs-3092	54	3	let	let	VERB
iajs-3092	54	4	ᵮ	ᵮ	PRON
iajs-3092	54	5	be	be	AUX
iajs-3092	54	6	a	a	DET
iajs-3092	54	7	mapping	mapping	NOUN
iajs-3092	54	8	from	from	ADP
iajs-3092	54	9	a	a	DET
iajs-3092	54	10	set	set	VERB
iajs-3092	54	11	𝕄	𝕄	PROPN
iajs-3092	54	12	into	into	ADP
iajs-3092	54	13	a	a	DET
iajs-3092	54	14	set	set	NOUN
iajs-3092	54	15	n	n	CCONJ
iajs-3092	54	16	,	,	PUNCT
iajs-3092	54	17	let	let	VERB
iajs-3092	54	18	𝔸	𝔸	PROPN
iajs-3092	54	19	∈	∈	NOUN
iajs-3092	54	20	fse(𝕄	fse(𝕄	NUM
iajs-3092	54	21	)	)	PUNCT
iajs-3092	54	22	and	and	CCONJ
iajs-3092	54	23	𝔹	𝔹	VERB
iajs-3092	54	24	∈	∈	NOUN
iajs-3092	54	25	𝐹𝑆𝐸(𝑁	𝐹𝑆𝐸(𝑁	NOUN
iajs-3092	54	26	)	)	PUNCT
iajs-3092	54	27	.	.	PUNCT
iajs-3092	55	1	the	the	DET
iajs-3092	55	2	image	image	NOUN
iajs-3092	55	3	of	of	ADP
iajs-3092	55	4	𝔸	𝔸	PROPN
iajs-3092	55	5	denoted	denote	VERB
iajs-3092	55	6	by	by	ADP
iajs-3092	55	7	ᵮ	ᵮ	PROPN
iajs-3092	55	8	(	(	PUNCT
iajs-3092	55	9	𝔸	𝔸	PROPN
iajs-3092	55	10	)	)	PUNCT
iajs-3092	55	11	∈	∈	PROPN
iajs-3092	55	12	𝐹𝑆𝐸(𝑁	𝐹𝑆𝐸(𝑁	NOUN
iajs-3092	55	13	)	)	PUNCT
iajs-3092	55	14	defined	define	VERB
iajs-3092	55	15	by	by	ADP
iajs-3092	55	16	:	:	PUNCT
iajs-3092	55	17	ᵮ(𝔸)(ℯ	ᵮ(𝔸)(ℯ	PROPN
iajs-3092	55	18	)	)	PUNCT
iajs-3092	55	19	=	=	PRON
iajs-3092	55	20	{	{	PUNCT
iajs-3092	55	21	{	{	PUNCT
iajs-3092	55	22	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
iajs-3092	55	23	{	{	PUNCT
iajs-3092	55	24	𝔸(𝑧	𝔸(𝑧	NOUN
iajs-3092	55	25	):	):	PUNCT
iajs-3092	55	26	𝑧	𝑧	PROPN
iajs-3092	55	27	∈	∈	PROPN
iajs-3092	55	28	ᵮ−1(ℯ	ᵮ−1(ℯ	PROPN
iajs-3092	55	29	)	)	PUNCT
iajs-3092	55	30	,	,	PUNCT
iajs-3092	55	31	𝑖𝑓	𝑖𝑓	ADP
iajs-3092	55	32	ᵮ−1(ℯ	ᵮ−1(ℯ	PROPN
iajs-3092	55	33	)	)	PUNCT
iajs-3092	55	34	≠	≠	PROPN
iajs-3092	55	35	∅	∅	NOUN
iajs-3092	55	36	,	,	PUNCT
iajs-3092	55	37	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3092	55	38	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3092	55	39	ℯ	ℯ	PART
iajs-3092	55	40	∈	∈	PROPN
iajs-3092	56	1	𝑁	𝑁	NOUN
iajs-3092	56	2	0	0	NUM
iajs-3092	56	3	𝑜.	𝑜.	NOUN
iajs-3092	56	4	𝑤.	𝑤.	NOUN
iajs-3092	56	5	,	,	PUNCT
iajs-3092	56	6	and	and	CCONJ
iajs-3092	56	7	the	the	DET
iajs-3092	56	8	inverse	inverse	ADJ
iajs-3092	56	9	image	image	NOUN
iajs-3092	56	10	of	of	ADP
iajs-3092	56	11	𝔹	𝔹	PROPN
iajs-3092	56	12	denoted	denote	VERB
iajs-3092	56	13	by	by	ADP
iajs-3092	56	14	𝑓−1(𝔹	𝑓−1(𝔹	NOUN
iajs-3092	56	15	)	)	PUNCT
iajs-3092	56	16	∈	∈	NOUN
iajs-3092	56	17	𝐹𝑆𝐸(𝕄)𝑖𝑠	𝐹𝑆𝐸(𝕄)𝑖𝑠	NUM
iajs-3092	56	18	defined	define	VERB
iajs-3092	56	19	by	by	ADP
iajs-3092	56	20	:	:	PUNCT
iajs-3092	56	21	ᵮ−1(𝔹)(𝑥	ᵮ−1(𝔹)(𝑥	X
iajs-3092	56	22	)	)	PUNCT
iajs-3092	56	23	=	=	SYM
iajs-3092	56	24	𝔹(ᵮ(𝑥	𝔹(ᵮ(𝑥	NUM
iajs-3092	56	25	)	)	PUNCT
iajs-3092	56	26	)	)	PUNCT
iajs-3092	56	27	,	,	PUNCT
iajs-3092	56	28	for	for	ADP
iajs-3092	56	29	all	all	DET
iajs-3092	56	30	∈	∈	PROPN
iajs-3092	56	31	𝕄	𝕄	PROPN
iajs-3092	57	1	[	[	X
iajs-3092	57	2	15	15	NUM
iajs-3092	57	3	-	-	SYM
iajs-3092	57	4	16	16	NUM
iajs-3092	57	5	]	]	PUNCT
iajs-3092	57	6	.	.	PUNCT
iajs-3092	58	1	2.8	2.8	NUM
iajs-3092	58	2	definition	definition	NOUN
iajs-3092	58	3	:	:	PUNCT
iajs-3092	58	4	let	let	VERB
iajs-3092	58	5	𝕏	𝕏	NOUN
iajs-3092	58	6	∈fum(𝕄1	∈fum(𝕄1	VERB
iajs-3092	58	7	)	)	PUNCT
iajs-3092	58	8	and	and	CCONJ
iajs-3092	58	9	𝕐	𝕐	PROPN
iajs-3092	58	10	∈fm(𝕄2	∈fm(𝕄2	NUM
iajs-3092	58	11	)	)	PUNCT
iajs-3092	58	12	.	.	PUNCT
iajs-3092	59	1	ℊ	ℊ	PROPN
iajs-3092	60	1	∶	∶	NOUN
iajs-3092	60	2	𝕏	𝕏	NOUN
iajs-3092	60	3	→	→	PUNCT
iajs-3092	60	4	𝕐	𝕐	PROPN
iajs-3092	60	5	is	be	AUX
iajs-3092	60	6	called	call	VERB
iajs-3092	60	7	a	a	DET
iajs-3092	60	8	fuzzy	fuzzy	ADJ
iajs-3092	60	9	homomorphism	homomorphism	NOUN
iajs-3092	60	10	if	if	SCONJ
iajs-3092	60	11	ℊ	ℊ	NOUN
iajs-3092	60	12	:	:	PUNCT
iajs-3092	60	13	𝕄1	𝕄1	PROPN
iajs-3092	60	14	→	→	SYM
iajs-3092	60	15	𝕄2	𝕄2	PROPN
iajs-3092	60	16	is	be	AUX
iajs-3092	60	17	𝔽-homomorphism	𝔽-homomorphism	PROPN
iajs-3092	60	18	and	and	CCONJ
iajs-3092	60	19	𝕐(ℊ(𝒽	𝕐(ℊ(𝒽	NOUN
iajs-3092	60	20	)	)	PUNCT
iajs-3092	60	21	=	=	SYM
iajs-3092	60	22	𝕏(𝒽	𝕏(𝒽	PROPN
iajs-3092	60	23	)	)	PUNCT
iajs-3092	60	24	for	for	ADP
iajs-3092	60	25	each	each	DET
iajs-3092	60	26	𝒽	𝒽	DET
iajs-3092	60	27	∈	∈	NOUN
iajs-3092	60	28	𝕄1	𝕄1	NOUN
iajs-3092	60	29	[	[	X
iajs-3092	60	30	15	15	NUM
iajs-3092	60	31	]	]	PUNCT
iajs-3092	60	32	.	.	PUNCT
iajs-3092	61	1	ihjpas	ihjpas	PROPN
iajs-3092	61	2	.	.	PUNCT
iajs-3092	62	1	36	36	NUM
iajs-3092	62	2	(	(	PUNCT
iajs-3092	62	3	3	3	NUM
iajs-3092	62	4	)	)	PUNCT
iajs-3092	62	5	2023	2023	NUM
iajs-3092	62	6	374	374	NUM
iajs-3092	62	7	2.9	2.9	NUM
iajs-3092	62	8	definition	definition	NOUN
iajs-3092	62	9	:	:	PUNCT
iajs-3092	62	10	let	let	VERB
iajs-3092	62	11	∅	∅	NOUN
iajs-3092	62	12	≠	≠	NOUN
iajs-3092	62	13	ℭ	ℭ	PROPN
iajs-3092	62	14	∈	∈	PROPN
iajs-3092	62	15	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	62	16	)	)	PUNCT
iajs-3092	62	17	.	.	PUNCT
iajs-3092	63	1	the	the	DET
iajs-3092	63	2	fuzzy	fuzzy	ADJ
iajs-3092	63	3	annihilator	annihilator	NOUN
iajs-3092	63	4	of	of	ADP
iajs-3092	63	5	a	a	DET
iajs-3092	63	6	denoted	denote	VERB
iajs-3092	63	7	by	by	ADP
iajs-3092	63	8	f	f	PROPN
iajs-3092	63	9	-	-	PUNCT
iajs-3092	63	10	ann	ann	PROPN
iajs-3092	63	11	ℭ	ℭ	PROPN
iajs-3092	63	12	is	be	AUX
iajs-3092	63	13	defined	define	VERB
iajs-3092	63	14	by	by	ADP
iajs-3092	63	15	:	:	PUNCT
iajs-3092	63	16	(	(	PUNCT
iajs-3092	63	17	f	f	X
iajs-3092	63	18	-	-	PUNCT
iajs-3092	63	19	ann	ann	PROPN
iajs-3092	63	20	ℭ)(r)=sup{t∶	ℭ)(r)=sup{t∶	NOUN
iajs-3092	63	21	t	t	PROPN
iajs-3092	63	22	∈[0,1],𝑟𝑡	∈[0,1],𝑟𝑡	NOUN
iajs-3092	63	23	ℭ	ℭ	PROPN
iajs-3092	63	24	⊆	⊆	NUM
iajs-3092	63	25	01	01	NUM
iajs-3092	63	26	}	}	PUNCT
iajs-3092	63	27	,	,	PUNCT
iajs-3092	63	28	for	for	ADP
iajs-3092	63	29	all	all	DET
iajs-3092	63	30	r∈	r∈	PROPN
iajs-3092	63	31	𝔽	𝔽	PROPN
iajs-3092	64	1	[	[	X
iajs-3092	64	2	17	17	NUM
iajs-3092	64	3	-	-	SYM
iajs-3092	64	4	18	18	NUM
iajs-3092	64	5	]	]	PUNCT
iajs-3092	64	6	.	.	PUNCT
iajs-3092	65	1	note	note	VERB
iajs-3092	65	2	that	that	SCONJ
iajs-3092	65	3	that	that	SCONJ
iajs-3092	65	4	f	f	X
iajs-3092	65	5	-	-	PUNCT
iajs-3092	65	6	ann	ann	PROPN
iajs-3092	65	7	ℭ	ℭ	PROPN
iajs-3092	65	8	=(	=(	NOUN
iajs-3092	65	9	01∶	01∶	PROPN
iajs-3092	65	10	ℭ	ℭ	PROPN
iajs-3092	65	11	)	)	PUNCT
iajs-3092	65	12	,	,	PUNCT
iajs-3092	65	13	"	"	PUNCT
iajs-3092	66	1	[	[	X
iajs-3092	66	2	16	16	NUM
iajs-3092	66	3	]	]	PUNCT
iajs-3092	66	4	.	.	PUNCT
iajs-3092	67	1	"	"	PUNCT
iajs-3092	67	2	hence	hence	ADV
iajs-3092	67	3	(	(	PUNCT
iajs-3092	67	4	(	(	PUNCT
iajs-3092	67	5	𝐹	𝐹	PROPN
iajs-3092	67	6	−	−	PROPN
iajs-3092	67	7	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
iajs-3092	67	8	ℭ))𝑡	ℭ))𝑡	PROPN
iajs-3092	67	9	⊆	⊆	NUM
iajs-3092	67	10	ann	ann	NOUN
iajs-3092	67	11	ℭ𝑡	ℭ𝑡	PROPN
iajs-3092	68	1	[	[	X
iajs-3092	68	2	19	19	NUM
iajs-3092	68	3	]	]	PUNCT
iajs-3092	68	4	.	.	PUNCT
iajs-3092	69	1	2.10	2.10	NUM
iajs-3092	69	2	remark	remark	NOUN
iajs-3092	69	3	:	:	PUNCT
iajs-3092	69	4	for	for	ADP
iajs-3092	69	5	a	a	DET
iajs-3092	69	6	∈	∈	PROPN
iajs-3092	69	7	fum(𝕄).we	fum(𝕄).we	NOUN
iajs-3092	69	8	let	let	VERB
iajs-3092	69	9	ℭ∗	ℭ∗	NOUN
iajs-3092	69	10	=	=	SYM
iajs-3092	69	11	{	{	PUNCT
iajs-3092	69	12	𝑥	𝑥	PUNCT
iajs-3092	69	13	∈	∈	PROPN
iajs-3092	69	14	𝕄	𝕄	PROPN
iajs-3092	69	15	:	:	PUNCT
iajs-3092	69	16	ℭ(𝑥	ℭ(𝑥	PROPN
iajs-3092	69	17	)	)	PUNCT
iajs-3092	69	18	=	=	SYM
iajs-3092	70	1	1}[20	1}[20	NUM
iajs-3092	70	2	−	−	NOUN
iajs-3092	70	3	21	21	NUM
iajs-3092	70	4	]	]	PUNCT
iajs-3092	70	5	.	.	PUNCT
iajs-3092	71	1	2.11	2.11	NUM
iajs-3092	71	2	proposition	proposition	NOUN
iajs-3092	71	3	:	:	PUNCT
iajs-3092	71	4	let	let	VERB
iajs-3092	71	5	ℙ	ℙ	PRON
iajs-3092	71	6	∈	∈	PROPN
iajs-3092	71	7	fum(m	fum(m	PROPN
iajs-3092	71	8	)	)	PUNCT
iajs-3092	71	9	.then	.then	AUX
iajs-3092	72	1	𝐹	𝐹	PROPN
iajs-3092	72	2	−	−	PROPN
iajs-3092	72	3	𝑎𝑛𝑛	𝑎𝑛𝑛	VERB
iajs-3092	72	4	ℙ	ℙ	PROPN
iajs-3092	72	5	∈	∈	NOUN
iajs-3092	72	6	𝐹𝑈𝐼(𝔽)[9	𝐹𝑈𝐼(𝔽)[9	NOUN
iajs-3092	72	7	]	]	PUNCT
iajs-3092	72	8	.	.	PUNCT
iajs-3092	73	1	2.12	2.12	NUM
iajs-3092	73	2	definition	definition	NOUN
iajs-3092	73	3	:	:	PUNCT
iajs-3092	73	4	let	let	VERB
iajs-3092	73	5	ҡ	ҡ	PRON
iajs-3092	73	6	∈fui(𝔽	∈fui(𝔽	VERB
iajs-3092	73	7	)	)	PUNCT
iajs-3092	73	8	.	.	PUNCT
iajs-3092	74	1	ҡ	ҡ	PRON
iajs-3092	74	2	is	be	AUX
iajs-3092	74	3	called	call	VERB
iajs-3092	74	4	a	a	DET
iajs-3092	74	5	principle	principle	ADJ
iajs-3092	74	6	fuzzy	fuzzy	ADJ
iajs-3092	74	7	ideal	ideal	NOUN
iajs-3092	74	8	if	if	SCONJ
iajs-3092	74	9	∃𝓀𝑡	∃𝓀𝑡	NOUN
iajs-3092	74	10	⊆	⊆	NUM
iajs-3092	74	11	ҡ	ҡ	X
iajs-3092	74	12	such	such	ADJ
iajs-3092	74	13	that	that	SCONJ
iajs-3092	74	14	ҡ	ҡ	PROPN
iajs-3092	74	15	=	=	SYM
iajs-3092	74	16	(	(	PUNCT
iajs-3092	74	17	𝓀𝑡	𝓀𝑡	NOUN
iajs-3092	74	18	)	)	PUNCT
iajs-3092	74	19	for	for	ADP
iajs-3092	74	20	each	each	PRON
iajs-3092	74	21	𝒾𝑠	𝒾𝑠	ADP
iajs-3092	74	22	⊆	⊆	NUM
iajs-3092	74	23	ҡ	ҡ	NOUN
iajs-3092	74	24	,	,	PUNCT
iajs-3092	74	25	∃	∃	PROPN
iajs-3092	74	26	a	a	DET
iajs-3092	74	27	fuzzy	fuzzy	ADJ
iajs-3092	74	28	singleton	singleton	NOUN
iajs-3092	74	29	ℯ𝑙	ℯ𝑙	NUM
iajs-3092	74	30	of	of	ADP
iajs-3092	74	31	𝔽	𝔽	PROPN
iajs-3092	74	32	such	such	ADJ
iajs-3092	74	33	that	that	PRON
iajs-3092	74	34	𝒾𝑠	𝒾𝑠	ADP
iajs-3092	74	35	=	=	SYM
iajs-3092	74	36	ℯ𝑙𝒽𝑡	ℯ𝑙𝒽𝑡	PROPN
iajs-3092	74	37	,	,	PUNCT
iajs-3092	74	38	where	where	SCONJ
iajs-3092	74	39	𝑠	𝑠	PROPN
iajs-3092	74	40	,	,	PUNCT
iajs-3092	74	41	𝑙	𝑙	PROPN
iajs-3092	74	42	,	,	PUNCT
iajs-3092	74	43	𝑡	𝑡	PROPN
iajs-3092	74	44	∈	∈	PROPN
iajs-3092	75	1	[	[	X
iajs-3092	75	2	0,1	0,1	NUM
iajs-3092	75	3	]	]	PUNCT
iajs-3092	75	4	,	,	PUNCT
iajs-3092	75	5	which	which	PRON
iajs-3092	75	6	is	be	AUX
iajs-3092	75	7	ҡ	ҡ	X
iajs-3092	75	8	=	=	PUNCT
iajs-3092	75	9	(	(	PUNCT
iajs-3092	75	10	𝓀𝑡	𝓀𝑡	X
iajs-3092	75	11	)	)	PUNCT
iajs-3092	75	12	=	=	NOUN
iajs-3092	75	13	{	{	PUNCT
iajs-3092	75	14	𝒾𝑠	𝒾𝑠	ADP
iajs-3092	75	15	⊆	⊆	NUM
iajs-3092	75	16	𝐻	𝐻	NOUN
iajs-3092	75	17	:	:	PUNCT
iajs-3092	75	18	𝒾𝑠	𝒾𝑠	ADP
iajs-3092	75	19	=	=	PUNCT
iajs-3092	75	20	ℯ𝑙	ℯ𝑙	NOUN
iajs-3092	75	21	𝓀𝑡	𝓀𝑡	ADP
iajs-3092	75	22	for	for	ADP
iajs-3092	75	23	some	some	DET
iajs-3092	75	24	fuzzy	fuzzy	ADJ
iajs-3092	75	25	singleton	singleton	PROPN
iajs-3092	75	26	𝑎𝑙	𝑎𝑙	NOUN
iajs-3092	75	27	of	of	ADP
iajs-3092	75	28	𝔽	𝔽	PROPN
iajs-3092	75	29	}	}	PUNCT
iajs-3092	76	1	[	[	X
iajs-3092	76	2	22	22	NUM
iajs-3092	76	3	]	]	PUNCT
iajs-3092	76	4	.	.	PUNCT
iajs-3092	77	1	2.13	2.13	NUM
iajs-3092	77	2	definition	definition	NOUN
iajs-3092	77	3	:	:	PUNCT
iajs-3092	77	4	let	let	VERB
iajs-3092	77	5	ϯ	ϯ	X
iajs-3092	77	6	∈	∈	PROPN
iajs-3092	77	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	VERB
iajs-3092	77	8	)	)	PUNCT
iajs-3092	77	9	and	and	CCONJ
iajs-3092	77	10	ϯ≠	ϯ≠	VERB
iajs-3092	77	11	đ∈fus	đ∈fus	PROPN
iajs-3092	77	12	(	(	PUNCT
iajs-3092	77	13	ϯ).then	ϯ).then	PROPN
iajs-3092	77	14	đ	đ	PROPN
iajs-3092	77	15	is	be	AUX
iajs-3092	77	16	known	know	VERB
iajs-3092	77	17	to	to	ADP
iajs-3092	77	18	as	as	ADP
iajs-3092	77	19	a	a	DET
iajs-3092	77	20	fuzzy	fuzzy	ADJ
iajs-3092	77	21	visible	visible	ADJ
iajs-3092	77	22	submodule	submodule	NOUN
iajs-3092	77	23	if	if	SCONJ
iajs-3092	77	24	đ	đ	PROPN
iajs-3092	77	25	=	=	NOUN
iajs-3092	77	26	ɨđ	ɨđ	ADP
iajs-3092	77	27	∀	∀	NOUN
iajs-3092	77	28	non	non	ADJ
iajs-3092	77	29	-	-	ADJ
iajs-3092	77	30	empty	empty	ADJ
iajs-3092	77	31	ɨ	ɨ	NOUN
iajs-3092	77	32	∈	∈	ADJ
iajs-3092	77	33	𝐹𝑈𝐼(𝔽	𝐹𝑈𝐼(𝔽	NOUN
iajs-3092	77	34	)	)	PUNCT
iajs-3092	77	35	.	.	PUNCT
iajs-3092	78	1	ɬ∈	ɬ∈	PROPN
iajs-3092	78	2	𝐹𝑈𝐼(𝔽	𝐹𝑈𝐼(𝔽	PROPN
iajs-3092	78	3	)	)	PUNCT
iajs-3092	78	4	is	be	AUX
iajs-3092	78	5	defined	define	VERB
iajs-3092	78	6	visible	visible	ADJ
iajs-3092	78	7	if	if	SCONJ
iajs-3092	78	8	it	it	PRON
iajs-3092	78	9	is	be	AUX
iajs-3092	78	10	visible	visible	ADJ
iajs-3092	78	11	of	of	ADP
iajs-3092	78	12	a	a	DET
iajs-3092	78	13	fuzzy	fuzzy	ADJ
iajs-3092	78	14	𝔽	𝔽	PROPN
iajs-3092	78	15	-module	-module	NOUN
iajs-3092	78	16	𝔽	𝔽	PROPN
iajs-3092	79	1	[	[	NOUN
iajs-3092	79	2	4	4	NUM
iajs-3092	79	3	]	]	PUNCT
iajs-3092	79	4	.	.	PUNCT
iajs-3092	80	1	2.14	2.14	NUM
iajs-3092	80	2	definition	definition	NOUN
iajs-3092	80	3	:	:	PUNCT
iajs-3092	80	4	let	let	VERB
iajs-3092	80	5	𝕐	𝕐	PRON
iajs-3092	80	6	∈	∈	PROPN
iajs-3092	80	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	VERB
iajs-3092	80	8	)	)	PUNCT
iajs-3092	80	9	.	.	PUNCT
iajs-3092	81	1	then	then	ADV
iajs-3092	81	2	𝕐	𝕐	PROPN
iajs-3092	81	3	is	be	AUX
iajs-3092	81	4	said	say	VERB
iajs-3092	81	5	to	to	PART
iajs-3092	81	6	be	be	AUX
iajs-3092	81	7	fully	fully	ADV
iajs-3092	81	8	fuzzy	fuzzy	ADJ
iajs-3092	81	9	visible	visible	ADJ
iajs-3092	81	10	module	module	NOUN
iajs-3092	81	11	if	if	SCONJ
iajs-3092	81	12	for	for	ADP
iajs-3092	81	13	any	any	DET
iajs-3092	81	14	𝕐	𝕐	PROPN
iajs-3092	81	15	≠	≠	PROPN
iajs-3092	81	16	ℨ	ℨ	NOUN
iajs-3092	81	17	∈	∈	NOUN
iajs-3092	81	18	𝐹𝑈𝑆	𝐹𝑈𝑆	X
iajs-3092	81	19	(	(	PUNCT
iajs-3092	81	20	𝕐	𝕐	NOUN
iajs-3092	81	21	)	)	PUNCT
iajs-3092	81	22	is	be	AUX
iajs-3092	81	23	a	a	DET
iajs-3092	81	24	fuzzy	fuzzy	ADJ
iajs-3092	81	25	visible	visible	ADJ
iajs-3092	81	26	[	[	X
iajs-3092	81	27	5	5	NUM
iajs-3092	81	28	]	]	PUNCT
iajs-3092	81	29	.	.	PUNCT
iajs-3092	82	1	2.15	2.15	NUM
iajs-3092	82	2	proposition	proposition	NOUN
iajs-3092	82	3	:	:	PUNCT
iajs-3092	82	4	let	let	VERB
iajs-3092	82	5	𝕐	𝕐	PRON
iajs-3092	82	6	∈	∈	PROPN
iajs-3092	82	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	ADV
iajs-3092	82	8	)	)	PUNCT
iajs-3092	82	9	.	.	PUNCT
iajs-3092	83	1	then	then	ADV
iajs-3092	83	2	ℙ	ℙ	PROPN
iajs-3092	83	3	is	be	AUX
iajs-3092	83	4	fully	fully	ADV
iajs-3092	83	5	fuzzy	fuzzy	ADJ
iajs-3092	83	6	visible	visible	ADJ
iajs-3092	83	7	module	module	NOUN
iajs-3092	83	8	if	if	SCONJ
iajs-3092	83	9	and	and	CCONJ
iajs-3092	83	10	only	only	ADV
iajs-3092	83	11	if	if	SCONJ
iajs-3092	83	12	𝕐t	𝕐t	PROPN
iajs-3092	83	13	is	be	AUX
iajs-3092	83	14	fully	fully	ADV
iajs-3092	83	15	visible	visible	ADJ
iajs-3092	83	16	module,∀	module,∀	PROPN
iajs-3092	83	17	𝑡	𝑡	PROPN
iajs-3092	83	18	∈	∈	PROPN
iajs-3092	83	19	(	(	PUNCT
iajs-3092	83	20	0,1	0,1	NOUN
iajs-3092	83	21	]	]	PUNCT
iajs-3092	84	1	[	[	X
iajs-3092	84	2	5	5	NUM
iajs-3092	84	3	]	]	PUNCT
iajs-3092	84	4	.	.	PUNCT
iajs-3092	85	1	2.16	2.16	NUM
iajs-3092	85	2	definition	definition	NOUN
iajs-3092	85	3	:	:	PUNCT
iajs-3092	85	4	let	let	VERB
iajs-3092	85	5	𝕏	𝕏	PRON
iajs-3092	85	6	∈	∈	PRON
iajs-3092	85	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	PRON
iajs-3092	85	8	)	)	PUNCT
iajs-3092	85	9	.	.	PUNCT
iajs-3092	86	1	𝑋≠	𝑋≠	VERB
iajs-3092	86	2	ℭ∈fus(𝕏	ℭ∈fus(𝕏	PROPN
iajs-3092	86	3	)	)	PUNCT
iajs-3092	86	4	,	,	PUNCT
iajs-3092	86	5	ℭ	ℭ	PROPN
iajs-3092	86	6	is	be	AUX
iajs-3092	86	7	termed	term	VERB
iajs-3092	86	8	as	as	ADP
iajs-3092	86	9	a	a	DET
iajs-3092	86	10	semiprime	semiprime	NOUN
iajs-3092	86	11	fuzzy	fuzzy	ADJ
iajs-3092	86	12	submodule	submodule	NOUN
iajs-3092	86	13	if	if	SCONJ
iajs-3092	86	14	for	for	ADP
iajs-3092	86	15	each	each	DET
iajs-3092	86	16	fuzzy	fuzzy	ADJ
iajs-3092	86	17	singleton	singleton	NOUN
iajs-3092	86	18	𝑟𝑙	𝑟𝑙	NOUN
iajs-3092	86	19	of	of	ADP
iajs-3092	86	20	𝔽	𝔽	PROPN
iajs-3092	86	21	,	,	PUNCT
iajs-3092	86	22	𝑥𝑠	𝑥𝑠	VERB
iajs-3092	86	23	⊆	⊆	NUM
iajs-3092	86	24	𝕏	𝕏	NOUN
iajs-3092	86	25	,	,	PUNCT
iajs-3092	86	26	𝑟𝑙	𝑟𝑙	AUX
iajs-3092	86	27	2𝑥𝑠	2𝑥𝑠	ADJ
iajs-3092	86	28	⊆	⊆	NUM
iajs-3092	86	29	ℭ	ℭ	PROPN
iajs-3092	86	30	implies	imply	VERB
iajs-3092	86	31	𝑟𝑙𝑥𝑠	𝑟𝑙𝑥𝑠	NOUN
iajs-3092	86	32	⊆	⊆	NUM
iajs-3092	86	33	ℭ	ℭ	PROPN
iajs-3092	87	1	[	[	X
iajs-3092	87	2	23	23	NUM
iajs-3092	87	3	]	]	PUNCT
iajs-3092	87	4	.	.	PUNCT
iajs-3092	88	1	2.17	2.17	NUM
iajs-3092	88	2	remark	remark	NOUN
iajs-3092	88	3	:	:	PUNCT
iajs-3092	88	4	we	we	PRON
iajs-3092	88	5	assume	assume	VERB
iajs-3092	88	6	that	that	SCONJ
iajs-3092	88	7	if	if	SCONJ
iajs-3092	88	8	ℭ∗	ℭ∗	NOUN
iajs-3092	88	9	=	=	SYM
iajs-3092	88	10	𝔔∗	𝔔∗	NOUN
iajs-3092	88	11	.	.	PUNCT
iajs-3092	89	1	then	then	ADV
iajs-3092	89	2	,	,	PUNCT
iajs-3092	89	3	ℭ	ℭ	PROPN
iajs-3092	89	4	=	=	SYM
iajs-3092	89	5	𝔔	𝔔	PROPN
iajs-3092	89	6	is	be	AUX
iajs-3092	89	7	called	call	VERB
iajs-3092	89	8	condition	condition	NOUN
iajs-3092	89	9	(	(	PUNCT
iajs-3092	89	10	*	*	PUNCT
iajs-3092	89	11	)	)	PUNCT
iajs-3092	90	1	[	[	X
iajs-3092	90	2	20	20	NUM
iajs-3092	90	3	]	]	PUNCT
iajs-3092	90	4	.	.	PUNCT
iajs-3092	91	1	3	3	X
iajs-3092	91	2	.	.	X
iajs-3092	91	3	fully	fully	ADV
iajs-3092	91	4	fuzzy	fuzzy	ADJ
iajs-3092	91	5	visible	visible	ADJ
iajs-3092	91	6	modules	module	NOUN
iajs-3092	91	7	with	with	ADP
iajs-3092	91	8	different	different	ADJ
iajs-3092	91	9	modules	module	NOUN
iajs-3092	91	10	:	:	PUNCT
iajs-3092	91	11	in	in	ADP
iajs-3092	91	12	this	this	DET
iajs-3092	91	13	section	section	NOUN
iajs-3092	91	14	,	,	PUNCT
iajs-3092	91	15	the	the	DET
iajs-3092	91	16	relationships	relationship	NOUN
iajs-3092	91	17	between	between	ADP
iajs-3092	91	18	fully	fully	ADV
iajs-3092	91	19	fuzzy	fuzzy	ADJ
iajs-3092	91	20	visible	visible	ADJ
iajs-3092	91	21	modules	module	NOUN
iajs-3092	91	22	and	and	CCONJ
iajs-3092	91	23	other	other	ADJ
iajs-3092	91	24	fuzzy	fuzzy	ADJ
iajs-3092	91	25	modules	module	NOUN
iajs-3092	91	26	such	such	ADJ
iajs-3092	91	27	as	as	ADP
iajs-3092	91	28	(	(	PUNCT
iajs-3092	91	29	semiprime	semiprime	NOUN
iajs-3092	91	30	,	,	PUNCT
iajs-3092	91	31	prime	prime	ADJ
iajs-3092	91	32	quasi	quasi	ADJ
iajs-3092	91	33	prime	prime	NOUN
iajs-3092	91	34	,	,	PUNCT
iajs-3092	91	35	fully	fully	ADV
iajs-3092	91	36	stable	stable	ADJ
iajs-3092	91	37	,	,	PUNCT
iajs-3092	91	38	artiain	artiain	NOUN
iajs-3092	91	39	,	,	PUNCT
iajs-3092	91	40	essential	essential	ADJ
iajs-3092	91	41	and	and	CCONJ
iajs-3092	91	42	quasi	quasi	ADJ
iajs-3092	91	43	injective	injective	PROPN
iajs-3092	91	44	)	)	PUNCT
iajs-3092	91	45	have	have	AUX
iajs-3092	91	46	been	be	AUX
iajs-3092	91	47	investigated	investigate	VERB
iajs-3092	91	48	.	.	PUNCT
iajs-3092	92	1	many	many	ADJ
iajs-3092	92	2	interesting	interesting	ADJ
iajs-3092	92	3	outcomes	outcome	NOUN
iajs-3092	92	4	were	be	AUX
iajs-3092	92	5	identified	identify	VERB
iajs-3092	92	6	.	.	PUNCT
iajs-3092	93	1	3.1	3.1	NUM
iajs-3092	93	2	proposition	proposition	NOUN
iajs-3092	93	3	:	:	PUNCT
iajs-3092	93	4	let	let	VERB
iajs-3092	93	5	ℙ	ℙ	PRON
iajs-3092	93	6	∈	∈	NOUN
iajs-3092	93	7	fum(𝕄	fum(𝕄	NOUN
iajs-3092	93	8	)	)	PUNCT
iajs-3092	93	9	over	over	ADP
iajs-3092	93	10	principle	principle	ADJ
iajs-3092	93	11	fuzzy	fuzzy	ADJ
iajs-3092	93	12	ideal	ideal	ADJ
iajs-3092	93	13	ring	ring	NOUN
iajs-3092	93	14	.	.	PUNCT
iajs-3092	94	1	then	then	ADV
iajs-3092	94	2	ℙ	ℙ	PROPN
iajs-3092	94	3	is	be	AUX
iajs-3092	94	4	fully	fully	ADV
iajs-3092	94	5	fuzzy	fuzzy	ADJ
iajs-3092	94	6	visible	visible	ADJ
iajs-3092	94	7	if	if	SCONJ
iajs-3092	94	8	and	and	CCONJ
iajs-3092	94	9	only	only	ADV
iajs-3092	94	10	if	if	SCONJ
iajs-3092	94	11	every	every	DET
iajs-3092	94	12	proper	proper	ADJ
iajs-3092	94	13	fuzzy	fuzzy	ADJ
iajs-3092	94	14	submodule	submodule	NOUN
iajs-3092	94	15	of	of	ADP
iajs-3092	94	16	ℙ	ℙ	PROPN
iajs-3092	94	17	is	be	AUX
iajs-3092	94	18	fuzzy	fuzzy	ADJ
iajs-3092	94	19	semiprime	semiprime	NOUN
iajs-3092	94	20	.	.	PUNCT
iajs-3092	95	1	proof	proof	NOUN
iajs-3092	95	2	:	:	PUNCT
iajs-3092	95	3	let	let	VERB
iajs-3092	95	4	ℙ	ℙ	PRON
iajs-3092	95	5	be	be	AUX
iajs-3092	95	6	a	a	DET
iajs-3092	95	7	fully	fully	ADV
iajs-3092	95	8	fuzzy	fuzzy	ADJ
iajs-3092	95	9	visible	visible	ADJ
iajs-3092	95	10	,	,	PUNCT
iajs-3092	95	11	then	then	ADV
iajs-3092	95	12	ℙt	ℙt	PROPN
iajs-3092	95	13	is	be	AUX
iajs-3092	95	14	a	a	DET
iajs-3092	95	15	fully	fully	ADV
iajs-3092	95	16	visible	visible	ADJ
iajs-3092	95	17	,	,	PUNCT
iajs-3092	95	18	where	where	SCONJ
iajs-3092	95	19	𝑡	𝑡	PROPN
iajs-3092	95	20	∈	∈	PROPN
iajs-3092	95	21	(	(	PUNCT
iajs-3092	95	22	0,1	0,1	NOUN
iajs-3092	95	23	]	]	PUNCT
iajs-3092	95	24	by	by	ADP
iajs-3092	95	25	[	[	X
iajs-3092	95	26	5,proposition	5,proposition	NUM
iajs-3092	95	27	3.2	3.2	NUM
iajs-3092	95	28	]	]	PUNCT
iajs-3092	95	29	and	and	CCONJ
iajs-3092	95	30	let	let	VERB
iajs-3092	95	31	ℙ	ℙ	PRON
iajs-3092	95	32	≠	≠	PROPN
iajs-3092	95	33	𝔚	𝔚	NOUN
iajs-3092	95	34	∈	∈	NOUN
iajs-3092	95	35	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	95	36	)	)	PUNCT
iajs-3092	95	37	.	.	PUNCT
iajs-3092	96	1	then	then	ADV
iajs-3092	96	2	𝔚t	𝔚t	PROPN
iajs-3092	96	3	is	be	AUX
iajs-3092	96	4	a	a	DET
iajs-3092	96	5	proper	proper	ADJ
iajs-3092	96	6	submodule	submodule	NOUN
iajs-3092	96	7	of	of	ADP
iajs-3092	96	8	ℙt	ℙt	PROPN
iajs-3092	96	9	and	and	CCONJ
iajs-3092	96	10	hence	hence	ADV
iajs-3092	96	11	𝔚t	𝔚t	PROPN
iajs-3092	96	12	is	be	AUX
iajs-3092	96	13	a	a	DET
iajs-3092	96	14	semprime	semprime	NOUN
iajs-3092	96	15	submodule	submodule	NOUN
iajs-3092	96	16	by	by	ADP
iajs-3092	96	17	[	[	PUNCT
iajs-3092	96	18	24	24	NUM
iajs-3092	96	19	,	,	PUNCT
iajs-3092	96	20	proposition	proposition	NOUN
iajs-3092	96	21	2.6.1	2.6.1	NUM
iajs-3092	96	22	]	]	PUNCT
iajs-3092	96	23	.	.	PUNCT
iajs-3092	97	1	therefore	therefore	ADV
iajs-3092	97	2	𝔚	𝔚	PROPN
iajs-3092	97	3	is	be	AUX
iajs-3092	97	4	a	a	DET
iajs-3092	97	5	fuzzy	fuzzy	ADJ
iajs-3092	97	6	semiprime	semiprime	NOUN
iajs-3092	97	7	by	by	ADP
iajs-3092	97	8	[	[	X
iajs-3092	97	9	23	23	NUM
iajs-3092	97	10	,	,	PUNCT
iajs-3092	97	11	proposition	proposition	NOUN
iajs-3092	97	12	3.2.6	3.2.6	NUM
iajs-3092	97	13	]	]	PUNCT
iajs-3092	97	14	.	.	PUNCT
iajs-3092	98	1	conversely	conversely	ADV
iajs-3092	98	2	,	,	PUNCT
iajs-3092	98	3	let	let	VERB
iajs-3092	98	4	ℙ	ℙ	PRON
iajs-3092	98	5	≠	≠	PROPN
iajs-3092	98	6	𝔚	𝔚	NOUN
iajs-3092	98	7	∈	∈	PROPN
iajs-3092	98	8	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	98	9	)	)	PUNCT
iajs-3092	98	10	,	,	PUNCT
iajs-3092	98	11	then	then	ADV
iajs-3092	98	12	𝔚	𝔚	PROPN
iajs-3092	98	13	is	be	AUX
iajs-3092	98	14	a	a	DET
iajs-3092	98	15	fuzzy	fuzzy	ADJ
iajs-3092	98	16	semprime	semprime	NOUN
iajs-3092	98	17	and	and	CCONJ
iajs-3092	98	18	hence	hence	ADV
iajs-3092	98	19	𝔚t	𝔚t	PROPN
iajs-3092	98	20	is	be	AUX
iajs-3092	98	21	semiprime	semiprime	NOUN
iajs-3092	98	22	by	by	ADP
iajs-3092	98	23	[	[	X
iajs-3092	98	24	23	23	NUM
iajs-3092	98	25	,	,	PUNCT
iajs-3092	98	26	proposition	proposition	NOUN
iajs-3092	98	27	3.2.6	3.2.6	NUM
iajs-3092	98	28	]	]	PUNCT
iajs-3092	98	29	,	,	PUNCT
iajs-3092	98	30	therefore	therefore	ADV
iajs-3092	98	31	ℙt	ℙt	PROPN
iajs-3092	98	32	is	be	AUX
iajs-3092	98	33	a	a	DET
iajs-3092	98	34	fully	fully	ADV
iajs-3092	98	35	visible	visible	ADJ
iajs-3092	98	36	by	by	ADP
iajs-3092	98	37	[	[	PUNCT
iajs-3092	98	38	24	24	NUM
iajs-3092	98	39	,	,	PUNCT
iajs-3092	98	40	proposition	proposition	NOUN
iajs-3092	98	41	2.6.1	2.6.1	NUM
iajs-3092	98	42	]	]	PUNCT
iajs-3092	98	43	and	and	CCONJ
iajs-3092	98	44	hence	hence	ADV
iajs-3092	98	45	ℙ	ℙ	PROPN
iajs-3092	98	46	is	be	AUX
iajs-3092	98	47	fuzzy	fuzzy	ADJ
iajs-3092	98	48	fully	fully	ADV
iajs-3092	98	49	visible	visible	ADJ
iajs-3092	98	50	.	.	PUNCT
iajs-3092	99	1	3.2	3.2	NUM
iajs-3092	99	2	proposition	proposition	NOUN
iajs-3092	99	3	:	:	PUNCT
iajs-3092	99	4	let	let	VERB
iajs-3092	99	5	ℙ	ℙ	PRON
iajs-3092	99	6	∈	∈	NOUN
iajs-3092	99	7	fum(𝕄	fum(𝕄	NOUN
iajs-3092	99	8	)	)	PUNCT
iajs-3092	99	9	over	over	ADP
iajs-3092	99	10	principle	principle	ADJ
iajs-3092	99	11	fuzzy	fuzzy	ADJ
iajs-3092	99	12	ideal	ideal	ADJ
iajs-3092	99	13	ring	ring	NOUN
iajs-3092	99	14	and	and	CCONJ
iajs-3092	99	15	ℙ	ℙ	PROPN
iajs-3092	99	16	≠	≠	PROPN
iajs-3092	99	17	𝔚	𝔚	NOUN
iajs-3092	99	18	∈	∈	PROPN
iajs-3092	99	19	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	99	20	)	)	PUNCT
iajs-3092	99	21	.	.	PUNCT
iajs-3092	100	1	then	then	ADV
iajs-3092	100	2	,	,	PUNCT
iajs-3092	100	3	𝔚	𝔚	PROPN
iajs-3092	100	4	visible	visible	ADJ
iajs-3092	100	5	if	if	SCONJ
iajs-3092	100	6	and	and	CCONJ
iajs-3092	100	7	only	only	ADV
iajs-3092	100	8	if	if	SCONJ
iajs-3092	100	9	𝔚	𝔚	PROPN
iajs-3092	100	10	divisible	divisible	VERB
iajs-3092	100	11	.	.	PUNCT
iajs-3092	101	1	proof	proof	NOUN
iajs-3092	101	2	:	:	PUNCT
iajs-3092	101	3	assume	assume	VERB
iajs-3092	101	4	that	that	SCONJ
iajs-3092	101	5	𝔚	𝔚	PROPN
iajs-3092	101	6	is	be	AUX
iajs-3092	101	7	visible	visible	ADJ
iajs-3092	101	8	then	then	ADV
iajs-3092	101	9	∀	∀	NOUN
iajs-3092	101	10	𝑟𝑙	𝑟𝑙	ADP
iajs-3092	101	11	≠	≠	PROPN
iajs-3092	101	12	01	01	NUM
iajs-3092	101	13	,	,	PUNCT
iajs-3092	101	14	we	we	PRON
iajs-3092	101	15	have	have	VERB
iajs-3092	101	16	<	<	X
iajs-3092	101	17	𝑟𝑙	𝑟𝑙	X
iajs-3092	101	18	>	>	X
iajs-3092	101	19	𝔚	𝔚	PROPN
iajs-3092	101	20	=	=	PUNCT
iajs-3092	101	21	𝔚	𝔚	PROPN
iajs-3092	101	22	,	,	PUNCT
iajs-3092	101	23	then	then	ADV
iajs-3092	101	24	𝑟𝑙𝔚	𝑟𝑙𝔚	PROPN
iajs-3092	101	25	=	=	SYM
iajs-3092	102	1	𝔚.therefore	𝔚.therefore	PUNCT
iajs-3092	102	2	𝔚	𝔚	PROPN
iajs-3092	102	3	is	be	AUX
iajs-3092	102	4	divisible	divisible	ADJ
iajs-3092	102	5	.	.	PUNCT
iajs-3092	103	1	conversely	conversely	ADV
iajs-3092	103	2	,	,	PUNCT
iajs-3092	103	3	clear	clear	ADJ
iajs-3092	103	4	.	.	PUNCT
iajs-3092	104	1	3.3	3.3	NUM
iajs-3092	104	2	proposition	proposition	NOUN
iajs-3092	104	3	:	:	PUNCT
iajs-3092	104	4	let	let	VERB
iajs-3092	104	5	𝕐	𝕐	PRON
iajs-3092	104	6	be	be	AUX
iajs-3092	104	7	a	a	DET
iajs-3092	104	8	fully	fully	ADV
iajs-3092	104	9	fuzzy	fuzzy	ADJ
iajs-3092	104	10	visible	visible	ADJ
iajs-3092	104	11	module	module	NOUN
iajs-3092	104	12	over	over	ADP
iajs-3092	104	13	𝔽.	𝔽.	PROPN
iajs-3092	104	14	then	then	ADV
iajs-3092	104	15	every	every	DET
iajs-3092	104	16	proper	proper	ADJ
iajs-3092	104	17	fuzzy	fuzzy	ADJ
iajs-3092	104	18	submodule	submodule	NOUN
iajs-3092	104	19	of	of	ADP
iajs-3092	104	20	𝕐	𝕐	PROPN
iajs-3092	104	21	is	be	AUX
iajs-3092	104	22	divisible	divisible	ADJ
iajs-3092	104	23	.	.	PUNCT
iajs-3092	105	1	ihjpas	ihjpas	PROPN
iajs-3092	105	2	.	.	PUNCT
iajs-3092	106	1	36	36	NUM
iajs-3092	106	2	(	(	PUNCT
iajs-3092	106	3	3	3	NUM
iajs-3092	106	4	)	)	PUNCT
iajs-3092	106	5	2023	2023	NUM
iajs-3092	106	6	375	375	NUM
iajs-3092	106	7	proof	proof	NOUN
iajs-3092	106	8	:	:	PUNCT
iajs-3092	106	9	let	let	VERB
iajs-3092	106	10	𝕐	𝕐	PRON
iajs-3092	106	11	be	be	AUX
iajs-3092	106	12	a	a	DET
iajs-3092	106	13	fully	fully	ADV
iajs-3092	106	14	fuzzy	fuzzy	ADJ
iajs-3092	106	15	visible	visible	ADJ
iajs-3092	106	16	module	module	NOUN
iajs-3092	106	17	,	,	PUNCT
iajs-3092	106	18	then	then	ADV
iajs-3092	106	19	every	every	DET
iajs-3092	106	20	proper	proper	ADJ
iajs-3092	106	21	fuzzy	fuzzy	ADJ
iajs-3092	106	22	submodule	submodule	NOUN
iajs-3092	106	23	of	of	ADP
iajs-3092	106	24	𝕐	𝕐	PROPN
iajs-3092	106	25	is	be	AUX
iajs-3092	106	26	visible	visible	ADJ
iajs-3092	106	27	and	and	CCONJ
iajs-3092	106	28	hence	hence	ADV
iajs-3092	106	29	is	be	AUX
iajs-3092	106	30	divisible	divisible	ADJ
iajs-3092	106	31	by	by	ADP
iajs-3092	106	32	above	above	ADP
iajs-3092	106	33	proposition	proposition	NOUN
iajs-3092	106	34	.	.	PUNCT
iajs-3092	107	1	3.4	3.4	NUM
iajs-3092	107	2	proposition	proposition	NOUN
iajs-3092	107	3	:	:	PUNCT
iajs-3092	107	4	if	if	SCONJ
iajs-3092	107	5	ℙ	ℙ	PROPN
iajs-3092	107	6	is	be	AUX
iajs-3092	107	7	a	a	DET
iajs-3092	107	8	fully	fully	ADV
iajs-3092	107	9	fuzzy	fuzzy	ADJ
iajs-3092	107	10	visible	visible	ADJ
iajs-3092	107	11	module	module	NOUN
iajs-3092	107	12	,	,	PUNCT
iajs-3092	107	13	then	then	ADV
iajs-3092	107	14	ℙ	ℙ	PROPN
iajs-3092	107	15	is	be	AUX
iajs-3092	107	16	f	f	PROPN
iajs-3092	107	17	–	–	PUNCT
iajs-3092	107	18	regular	regular	ADJ
iajs-3092	107	19	fuzzy	fuzzy	ADJ
iajs-3092	107	20	module	module	NOUN
iajs-3092	107	21	.	.	PUNCT
iajs-3092	108	1	proof	proof	NOUN
iajs-3092	108	2	:	:	PUNCT
iajs-3092	108	3	by	by	ADP
iajs-3092	108	4	using	use	VERB
iajs-3092	108	5	definition	definition	NOUN
iajs-3092	108	6	(	(	PUNCT
iajs-3092	108	7	2.14	2.14	NUM
iajs-3092	108	8	)	)	PUNCT
iajs-3092	108	9	and	and	CCONJ
iajs-3092	108	10	[	[	X
iajs-3092	108	11	4	4	NUM
iajs-3092	108	12	,	,	PUNCT
iajs-3092	108	13	proposition	proposition	NOUN
iajs-3092	108	14	3.17	3.17	NUM
iajs-3092	108	15	]	]	PUNCT
iajs-3092	108	16	,	,	PUNCT
iajs-3092	108	17	the	the	DET
iajs-3092	108	18	result	result	NOUN
iajs-3092	108	19	is	be	AUX
iajs-3092	108	20	true	true	ADJ
iajs-3092	108	21	.	.	PUNCT
iajs-3092	109	1	the	the	DET
iajs-3092	109	2	converse	converse	NOUN
iajs-3092	109	3	of	of	ADP
iajs-3092	109	4	the	the	DET
iajs-3092	109	5	above	above	ADJ
iajs-3092	109	6	proposition	proposition	NOUN
iajs-3092	109	7	is	be	AUX
iajs-3092	109	8	not	not	PART
iajs-3092	109	9	correct	correct	ADJ
iajs-3092	109	10	as	as	SCONJ
iajs-3092	109	11	we	we	PRON
iajs-3092	109	12	see	see	VERB
iajs-3092	109	13	in	in	ADP
iajs-3092	109	14	the	the	DET
iajs-3092	109	15	following	follow	VERB
iajs-3092	109	16	example	example	NOUN
iajs-3092	109	17	.	.	PUNCT
iajs-3092	110	1	consider	consider	VERB
iajs-3092	110	2	𝕄	𝕄	PROPN
iajs-3092	110	3	=	=	SYM
iajs-3092	110	4	𝑍6	𝑍6	PROPN
iajs-3092	110	5	and	and	CCONJ
iajs-3092	110	6	𝔽	𝔽	PROPN
iajs-3092	110	7	=	=	SYM
iajs-3092	110	8	𝑍.	𝑍.	PROPN
iajs-3092	110	9	let	let	VERB
iajs-3092	110	10	ℙ	ℙ	NOUN
iajs-3092	110	11	:	:	PUNCT
iajs-3092	110	12	𝑍6	𝑍6	PROPN
iajs-3092	110	13	→	→	PUNCT
iajs-3092	110	14	[	[	X
iajs-3092	110	15	0,1	0,1	NUM
iajs-3092	110	16	]	]	PUNCT
iajs-3092	110	17	,	,	PUNCT
iajs-3092	110	18	as	as	ADP
iajs-3092	110	19	ℙ(𝑥	ℙ(𝑥	NOUN
iajs-3092	110	20	)	)	PUNCT
iajs-3092	110	21	=	=	SYM
iajs-3092	110	22	1	1	NUM
iajs-3092	110	23	∀𝑥	∀𝑥	PROPN
iajs-3092	110	24	∈	∈	PROPN
iajs-3092	110	25	𝑍6	𝑍6	PROPN
iajs-3092	110	26	.	.	PUNCT
iajs-3092	111	1	𝑍6	𝑍6	PROPN
iajs-3092	112	1	𝑎𝑠	𝑎𝑠	PROPN
iajs-3092	112	2	𝑍	𝑍	PROPN
iajs-3092	112	3	−	−	NOUN
iajs-3092	112	4	module	module	NOUN
iajs-3092	112	5	is	be	AUX
iajs-3092	112	6	fregular	fregular	ADJ
iajs-3092	112	7	by	by	ADP
iajs-3092	112	8	[	[	X
iajs-3092	112	9	24	24	NUM
iajs-3092	112	10	,	,	PUNCT
iajs-3092	112	11	p97	p97	NOUN
iajs-3092	112	12	]	]	PUNCT
iajs-3092	112	13	and	and	CCONJ
iajs-3092	112	14	hence	hence	ADV
iajs-3092	112	15	ℙ	ℙ	PROPN
iajs-3092	112	16	is	be	AUX
iajs-3092	112	17	f	f	X
iajs-3092	112	18	-	-	PUNCT
iajs-3092	112	19	regular	regular	ADJ
iajs-3092	112	20	by	by	ADP
iajs-3092	112	21	[	[	X
iajs-3092	112	22	25	25	NUM
iajs-3092	112	23	,	,	PUNCT
iajs-3092	112	24	proposition	proposition	NOUN
iajs-3092	112	25	3.1.3	3.1.3	NUM
iajs-3092	112	26	]	]	PUNCT
iajs-3092	112	27	.	.	PUNCT
iajs-3092	113	1	but	but	CCONJ
iajs-3092	113	2	𝑍6	𝑍6	PROPN
iajs-3092	113	3	is	be	AUX
iajs-3092	113	4	not	not	PART
iajs-3092	113	5	fully	fully	ADV
iajs-3092	113	6	visible	visible	ADJ
iajs-3092	113	7	by	by	ADP
iajs-3092	113	8	[	[	X
iajs-3092	113	9	24	24	NUM
iajs-3092	113	10	,	,	PUNCT
iajs-3092	113	11	p97	p97	NOUN
iajs-3092	113	12	]	]	PUNCT
iajs-3092	113	13	and	and	CCONJ
iajs-3092	113	14	hence	hence	ADV
iajs-3092	113	15	ℙ	ℙ	PROPN
iajs-3092	113	16	is	be	AUX
iajs-3092	113	17	not	not	PART
iajs-3092	113	18	fully	fully	ADV
iajs-3092	113	19	visible	visible	ADJ
iajs-3092	113	20	.	.	PUNCT
iajs-3092	114	1	3.5	3.5	NUM
iajs-3092	114	2	proposition	proposition	NOUN
iajs-3092	114	3	:	:	PUNCT
iajs-3092	114	4	let	let	VERB
iajs-3092	114	5	ℙ	ℙ	PRON
iajs-3092	114	6	be	be	AUX
iajs-3092	114	7	a	a	DET
iajs-3092	114	8	fuzzy	fuzzy	ADJ
iajs-3092	114	9	divisible	divisible	ADJ
iajs-3092	114	10	module	module	NOUN
iajs-3092	114	11	over	over	ADP
iajs-3092	114	12	a	a	DET
iajs-3092	114	13	field	field	NOUN
iajs-3092	114	14	,	,	PUNCT
iajs-3092	114	15	then	then	ADV
iajs-3092	114	16	ℙ	ℙ	PROPN
iajs-3092	114	17	is	be	AUX
iajs-3092	114	18	a	a	DET
iajs-3092	114	19	fully	fully	ADV
iajs-3092	114	20	fuzzy	fuzzy	ADJ
iajs-3092	114	21	visible	visible	ADJ
iajs-3092	114	22	module	module	NOUN
iajs-3092	114	23	.	.	PUNCT
iajs-3092	115	1	proof	proof	NOUN
iajs-3092	115	2	:	:	PUNCT
iajs-3092	115	3	from	from	ADP
iajs-3092	115	4	[	[	X
iajs-3092	115	5	25	25	NUM
iajs-3092	115	6	,	,	PUNCT
iajs-3092	115	7	proposition	proposition	NOUN
iajs-3092	115	8	3.1.8	3.1.8	NUM
iajs-3092	115	9	]	]	PUNCT
iajs-3092	115	10	,	,	PUNCT
iajs-3092	115	11	we	we	PRON
iajs-3092	115	12	have	have	VERB
iajs-3092	115	13	ℙ	ℙ	PROPN
iajs-3092	115	14	is	be	AUX
iajs-3092	115	15	an	an	DET
iajs-3092	115	16	fregular	fregular	ADJ
iajs-3092	115	17	fuzzy	fuzzy	ADJ
iajs-3092	115	18	module	module	NOUN
iajs-3092	115	19	,	,	PUNCT
iajs-3092	115	20	then	then	ADV
iajs-3092	115	21	ℙt	ℙt	PROPN
iajs-3092	115	22	is	be	AUX
iajs-3092	115	23	a	a	DET
iajs-3092	115	24	f	f	PROPN
iajs-3092	115	25	regular	regular	ADJ
iajs-3092	115	26	module	module	NOUN
iajs-3092	115	27	,	,	PUNCT
iajs-3092	115	28	for	for	ADP
iajs-3092	115	29	every	every	DET
iajs-3092	115	30	𝑡	𝑡	PROPN
iajs-3092	115	31	∈	∈	PROPN
iajs-3092	115	32	(	(	PUNCT
iajs-3092	115	33	0,1	0,1	NOUN
iajs-3092	115	34	]	]	PUNCT
iajs-3092	115	35	by	by	ADP
iajs-3092	115	36	[	[	X
iajs-3092	115	37	25	25	NUM
iajs-3092	115	38	,	,	PUNCT
iajs-3092	115	39	proposition	proposition	NOUN
iajs-3092	115	40	3.1.2	3.1.2	NOUN
iajs-3092	115	41	]	]	PUNCT
iajs-3092	115	42	.	.	PUNCT
iajs-3092	116	1	ℙt	ℙt	PROPN
iajs-3092	116	2	is	be	AUX
iajs-3092	116	3	divisible	divisible	ADJ
iajs-3092	116	4	because	because	SCONJ
iajs-3092	116	5	ℙ	ℙ	PROPN
iajs-3092	116	6	is	be	AUX
iajs-3092	116	7	a	a	DET
iajs-3092	116	8	fuzzy	fuzzy	ADJ
iajs-3092	116	9	divisible	divisible	ADJ
iajs-3092	116	10	module	module	NOUN
iajs-3092	116	11	by	by	ADP
iajs-3092	116	12	[	[	PUNCT
iajs-3092	116	13	23	23	NUM
iajs-3092	116	14	,	,	PUNCT
iajs-3092	116	15	proposition	proposition	NOUN
iajs-3092	116	16	2.2.12	2.2.12	NUM
iajs-3092	116	17	]	]	PUNCT
iajs-3092	116	18	and	and	CCONJ
iajs-3092	116	19	hence	hence	ADV
iajs-3092	116	20	every	every	DET
iajs-3092	116	21	proper	proper	ADJ
iajs-3092	116	22	pure	pure	ADJ
iajs-3092	116	23	submodule	submodule	NOUN
iajs-3092	116	24	of	of	ADP
iajs-3092	116	25	ℙt	ℙt	PROPN
iajs-3092	116	26	is	be	AUX
iajs-3092	116	27	visible	visible	ADJ
iajs-3092	116	28	by	by	ADP
iajs-3092	116	29	[	[	PUNCT
iajs-3092	116	30	26	26	NUM
iajs-3092	116	31	,	,	PUNCT
iajs-3092	116	32	proposition	proposition	NOUN
iajs-3092	116	33	1.11	1.11	NUM
iajs-3092	116	34	]	]	PUNCT
iajs-3092	116	35	.	.	PUNCT
iajs-3092	117	1	therefore	therefore	ADV
iajs-3092	117	2	ℙt	ℙt	PROPN
iajs-3092	117	3	is	be	AUX
iajs-3092	117	4	fully	fully	ADV
iajs-3092	117	5	visible	visible	ADJ
iajs-3092	117	6	,	,	PUNCT
iajs-3092	117	7	then	then	ADV
iajs-3092	117	8	ℙ	ℙ	PROPN
iajs-3092	117	9	is	be	AUX
iajs-3092	117	10	fuzzy	fuzzy	ADJ
iajs-3092	117	11	fully	fully	ADV
iajs-3092	117	12	visible	visible	ADJ
iajs-3092	117	13	.	.	PUNCT
iajs-3092	118	1	3.6	3.6	NUM
iajs-3092	118	2	corollary	corollary	NOUN
iajs-3092	118	3	:	:	PUNCT
iajs-3092	118	4	suppose	suppose	VERB
iajs-3092	118	5	that	that	SCONJ
iajs-3092	118	6	ℙ	ℙ	PROPN
iajs-3092	118	7	is	be	AUX
iajs-3092	118	8	a	a	DET
iajs-3092	118	9	non	non	ADJ
iajs-3092	118	10	-	-	ADJ
iajs-3092	118	11	constant	constant	ADJ
iajs-3092	118	12	fully	fully	ADV
iajs-3092	118	13	fuzzy	fuzzy	ADJ
iajs-3092	118	14	visible	visible	ADJ
iajs-3092	118	15	module	module	NOUN
iajs-3092	118	16	over	over	ADP
iajs-3092	118	17	a	a	DET
iajs-3092	118	18	fuzzy	fuzzy	ADJ
iajs-3092	118	19	integral	integral	ADJ
iajs-3092	118	20	domain	domain	NOUN
iajs-3092	118	21	𝔽.	𝔽.	PROPN
iajs-3092	118	22	then	then	ADV
iajs-3092	118	23	,	,	PUNCT
iajs-3092	118	24	ℝ	ℝ	PROPN
iajs-3092	118	25	is	be	AUX
iajs-3092	118	26	a	a	DET
iajs-3092	118	27	fuzzy	fuzzy	ADJ
iajs-3092	118	28	field	field	NOUN
iajs-3092	118	29	.	.	PUNCT
iajs-3092	119	1	proof	proof	NOUN
iajs-3092	119	2	:	:	PUNCT
iajs-3092	119	3	depending	depend	VERB
iajs-3092	119	4	on	on	ADP
iajs-3092	119	5	proposition	proposition	NOUN
iajs-3092	119	6	(	(	PUNCT
iajs-3092	119	7	3.4	3.4	NUM
iajs-3092	119	8	)	)	PUNCT
iajs-3092	119	9	and	and	CCONJ
iajs-3092	119	10	[	[	PUNCT
iajs-3092	119	11	22	22	NUM
iajs-3092	119	12	,	,	PUNCT
iajs-3092	119	13	remark	remark	NOUN
iajs-3092	119	14	and	and	CCONJ
iajs-3092	119	15	examples	example	NOUN
iajs-3092	119	16	1.3.4(4	1.3.4(4	NOUN
iajs-3092	119	17	)	)	PUNCT
iajs-3092	119	18	]	]	PUNCT
iajs-3092	119	19	,	,	PUNCT
iajs-3092	119	20	we	we	PRON
iajs-3092	119	21	get	get	VERB
iajs-3092	119	22	the	the	DET
iajs-3092	119	23	result	result	NOUN
iajs-3092	119	24	.	.	PUNCT
iajs-3092	120	1	3.7	3.7	NUM
iajs-3092	120	2	corollary	corollary	NOUN
iajs-3092	120	3	:	:	PUNCT
iajs-3092	120	4	let	let	VERB
iajs-3092	120	5	ℙ	ℙ	PRON
iajs-3092	120	6	∈	∈	NOUN
iajs-3092	120	7	fum(𝕄	fum(𝕄	PROPN
iajs-3092	120	8	)	)	PUNCT
iajs-3092	120	9	.	.	PUNCT
iajs-3092	121	1	if	if	SCONJ
iajs-3092	121	2	ℙ	ℙ	PROPN
iajs-3092	121	3	is	be	AUX
iajs-3092	121	4	fully	fully	ADV
iajs-3092	121	5	visible	visible	ADJ
iajs-3092	121	6	,	,	PUNCT
iajs-3092	121	7	then	then	ADV
iajs-3092	121	8	𝐹	𝐹	PROPN
iajs-3092	121	9	−	−	PROPN
iajs-3092	121	10	𝐽(ℙ	𝐽(ℙ	NOUN
iajs-3092	121	11	)	)	PUNCT
iajs-3092	121	12	=	=	SYM
iajs-3092	121	13	01	01	NUM
iajs-3092	121	14	,	,	PUNCT
iajs-3092	121	15	where	where	SCONJ
iajs-3092	121	16	ℙ	ℙ	NOUN
iajs-3092	121	17	satisfies	satisfy	VERB
iajs-3092	121	18	condition(∗	condition(∗	NUM
iajs-3092	121	19	)	)	PUNCT
iajs-3092	121	20	.	.	PUNCT
iajs-3092	122	1	proof	proof	NOUN
iajs-3092	122	2	:	:	PUNCT
iajs-3092	122	3	ℙ∗	ℙ∗	X
iajs-3092	122	4	is	be	AUX
iajs-3092	122	5	a	a	DET
iajs-3092	122	6	fully	fully	ADV
iajs-3092	122	7	visible	visible	ADJ
iajs-3092	122	8	because	because	SCONJ
iajs-3092	122	9	ℙ	ℙ	PROPN
iajs-3092	122	10	is	be	AUX
iajs-3092	122	11	a	a	DET
iajs-3092	122	12	fully	fully	ADV
iajs-3092	122	13	fuzzy	fuzzy	ADJ
iajs-3092	122	14	visible	visible	ADJ
iajs-3092	122	15	by	by	ADP
iajs-3092	122	16	[	[	X
iajs-3092	122	17	5	5	NUM
iajs-3092	122	18	,	,	PUNCT
iajs-3092	122	19	proposition3.2	proposition3.2	NOUN
iajs-3092	122	20	]	]	PUNCT
iajs-3092	122	21	and	and	CCONJ
iajs-3092	122	22	hence𝐽(ℙ∗	hence𝐽(ℙ∗	X
iajs-3092	122	23	)	)	PUNCT
iajs-3092	122	24	=	=	PRON
iajs-3092	123	1	{	{	PUNCT
iajs-3092	123	2	0	0	NUM
iajs-3092	123	3	}	}	PUNCT
iajs-3092	123	4	=	=	SYM
iajs-3092	123	5	(	(	PUNCT
iajs-3092	123	6	01)∗	01)∗	NOUN
iajs-3092	123	7	,	,	PUNCT
iajs-3092	123	8	by	by	ADP
iajs-3092	123	9	[	[	PUNCT
iajs-3092	123	10	24	24	NUM
iajs-3092	123	11	,	,	PUNCT
iajs-3092	123	12	corollary	corollary	ADJ
iajs-3092	123	13	2.6.6	2.6.6	NOUN
iajs-3092	123	14	]	]	PUNCT
iajs-3092	123	15	.	.	PUNCT
iajs-3092	124	1	but	but	CCONJ
iajs-3092	124	2	𝐽(ℙ∗	𝐽(ℙ∗	ADJ
iajs-3092	124	3	)	)	PUNCT
iajs-3092	124	4	=	=	PUNCT
iajs-3092	125	1	(	(	PUNCT
iajs-3092	125	2	𝐹	𝐹	PROPN
iajs-3092	125	3	−	−	PROPN
iajs-3092	125	4	𝐽(ℙ	𝐽(ℙ	NOUN
iajs-3092	125	5	)	)	PUNCT
iajs-3092	125	6	)	)	PUNCT
iajs-3092	126	1	∗	∗	NOUN
iajs-3092	126	2	=	=	PUNCT
iajs-3092	126	3	(	(	PUNCT
iajs-3092	126	4	01)∗	01)∗	NOUN
iajs-3092	126	5	,	,	PUNCT
iajs-3092	126	6	then	then	ADV
iajs-3092	126	7	𝐹	𝐹	PROPN
iajs-3092	126	8	−	−	PROPN
iajs-3092	126	9	𝐽(ℙ	𝐽(ℙ	NOUN
iajs-3092	126	10	)	)	PUNCT
iajs-3092	126	11	=	=	SYM
iajs-3092	127	1	01	01	NUM
iajs-3092	127	2	.	.	NOUN
iajs-3092	127	3	3.8	3.8	NUM
iajs-3092	127	4	definition	definition	NOUN
iajs-3092	127	5	:	:	PUNCT
iajs-3092	127	6	let	let	VERB
iajs-3092	127	7	ℙ	ℙ	PRON
iajs-3092	127	8	∈	∈	NOUN
iajs-3092	127	9	fum(𝕄	fum(𝕄	PROPN
iajs-3092	127	10	)	)	PUNCT
iajs-3092	127	11	.	.	PUNCT
iajs-3092	128	1	then	then	ADV
iajs-3092	128	2	ℙ	ℙ	PROPN
iajs-3092	128	3	is	be	AUX
iajs-3092	128	4	called	call	VERB
iajs-3092	128	5	fuzzy	fuzzy	ADJ
iajs-3092	128	6	artiain	artiain	NOUN
iajs-3092	128	7	module	module	NOUN
iajs-3092	128	8	if	if	SCONJ
iajs-3092	128	9	every	every	DET
iajs-3092	128	10	descending	descend	VERB
iajs-3092	128	11	chain	chain	NOUN
iajs-3092	128	12	fuzzy	fuzzy	ADJ
iajs-3092	128	13	submodules	submodule	NOUN
iajs-3092	128	14	of	of	ADP
iajs-3092	128	15	ℙ	ℙ	PROPN
iajs-3092	128	16	is	be	AUX
iajs-3092	128	17	finite	finite	ADJ
iajs-3092	128	18	.	.	PUNCT
iajs-3092	129	1	3.9	3.9	NUM
iajs-3092	129	2	theorem	theorem	VERB
iajs-3092	129	3	:	:	PUNCT
iajs-3092	129	4	if	if	SCONJ
iajs-3092	129	5	ℙ	ℙ	PROPN
iajs-3092	129	6	∈	∈	NOUN
iajs-3092	129	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	NOUN
iajs-3092	129	8	)	)	PUNCT
iajs-3092	129	9	artiain	artiain	NOUN
iajs-3092	129	10	𝔽	𝔽	PROPN
iajs-3092	129	11	-module	-module	PROPN
iajs-3092	129	12	,	,	PUNCT
iajs-3092	129	13	then	then	ADV
iajs-3092	129	14	ℙ∗	ℙ∗	X
iajs-3092	129	15	is	be	AUX
iajs-3092	129	16	artiain	artiain	NOUN
iajs-3092	129	17	modules	module	NOUN
iajs-3092	129	18	.	.	PUNCT
iajs-3092	130	1	proof	proof	NOUN
iajs-3092	130	2	:	:	PUNCT
iajs-3092	130	3	let	let	VERB
iajs-3092	130	4	ℙ	ℙ	PRON
iajs-3092	130	5	∈	∈	PROPN
iajs-3092	130	6	𝐹𝑀(𝕄	𝐹𝑀(𝕄	NOUN
iajs-3092	130	7	)	)	PUNCT
iajs-3092	130	8	artiain	artiain	NOUN
iajs-3092	130	9	ℝ	ℝ	NOUN
iajs-3092	130	10	-module	-module	NOUN
iajs-3092	130	11	and	and	CCONJ
iajs-3092	130	12	𝐴1	𝐴1	PROPN
iajs-3092	130	13	⊇	⊇	PROPN
iajs-3092	130	14	𝐴2	𝐴2	PROPN
iajs-3092	130	15	⊇	⊇	PROPN
iajs-3092	130	16	⋯	⋯	PROPN
iajs-3092	130	17	be	be	AUX
iajs-3092	130	18	descending	descend	VERB
iajs-3092	130	19	chain	chain	NOUN
iajs-3092	130	20	of	of	ADP
iajs-3092	130	21	submodules	submodule	NOUN
iajs-3092	130	22	of	of	ADP
iajs-3092	130	23	ℙ∗.	ℙ∗.	NOUN
iajs-3092	130	24	for	for	ADP
iajs-3092	130	25	each	each	DET
iajs-3092	130	26	positive	positive	ADJ
iajs-3092	130	27	integer	integer	NOUN
iajs-3092	131	1	i	i	PRON
iajs-3092	131	2	,	,	PUNCT
iajs-3092	131	3	we	we	PRON
iajs-3092	131	4	define	define	VERB
iajs-3092	131	5	the	the	DET
iajs-3092	131	6	mapping	mapping	NOUN
iajs-3092	131	7	ℙ𝑖	ℙ𝑖	PROPN
iajs-3092	131	8	:	:	PUNCT
iajs-3092	131	9	𝕄	𝕄	PROPN
iajs-3092	131	10	→	→	SYM
iajs-3092	131	11	[	[	X
iajs-3092	131	12	0,1	0,1	NUM
iajs-3092	131	13	]	]	PUNCT
iajs-3092	131	14	by	by	ADP
iajs-3092	131	15	ℙ𝑖(𝑥	ℙ𝑖(𝑥	PROPN
iajs-3092	131	16	)	)	PUNCT
iajs-3092	132	1	=	=	PRON
iajs-3092	132	2	{	{	PUNCT
iajs-3092	132	3	1	1	NUM
iajs-3092	132	4	𝑖𝑓	𝑖𝑓	ADP
iajs-3092	132	5	𝑥	𝑥	PRON
iajs-3092	132	6	∈	∈	PROPN
iajs-3092	133	1	𝐴𝑖	𝐴𝑖	PROPN
iajs-3092	133	2	0	0	PUNCT
iajs-3092	133	3	𝑜.	𝑜.	NOUN
iajs-3092	133	4	𝑤	𝑤	ADP
iajs-3092	133	5	,	,	PUNCT
iajs-3092	133	6	clearly	clearly	ADV
iajs-3092	133	7	,	,	PUNCT
iajs-3092	133	8	ℙ𝑖	ℙ𝑖	PROPN
iajs-3092	133	9	∈	∈	PROPN
iajs-3092	133	10	fs(ℙ	fs(ℙ	NOUN
iajs-3092	133	11	)	)	PUNCT
iajs-3092	133	12	.	.	PUNCT
iajs-3092	134	1	then	then	ADV
iajs-3092	134	2	ℙ1	ℙ1	PROPN
iajs-3092	134	3	⊇	⊇	PROPN
iajs-3092	134	4	ℙ2	ℙ2	PROPN
iajs-3092	134	5	⊇	⊇	PROPN
iajs-3092	134	6	⋯	⋯	PROPN
iajs-3092	134	7	is	be	AUX
iajs-3092	134	8	descending	descend	VERB
iajs-3092	134	9	chain	chain	NOUN
iajs-3092	134	10	of	of	ADP
iajs-3092	134	11	fuzzy	fuzzy	ADJ
iajs-3092	134	12	submodules	submodule	NOUN
iajs-3092	134	13	of	of	ADP
iajs-3092	134	14	ℙ	ℙ	NOUN
iajs-3092	134	15	,	,	PUNCT
iajs-3092	134	16	so	so	SCONJ
iajs-3092	134	17	there	there	PRON
iajs-3092	134	18	exist	exist	VERB
iajs-3092	134	19	a	a	DET
iajs-3092	134	20	positive	positive	ADJ
iajs-3092	134	21	integer	integer	NOUN
iajs-3092	134	22	m	m	VERB
iajs-3092	134	23	such	such	ADJ
iajs-3092	134	24	that	that	SCONJ
iajs-3092	134	25	ℙ𝑚	ℙ𝑚	PROPN
iajs-3092	134	26	=	=	SYM
iajs-3092	134	27	ℙ𝑚+ℎ	ℙ𝑚+ℎ	PROPN
iajs-3092	134	28	for	for	ADP
iajs-3092	134	29	every	every	DET
iajs-3092	134	30	positive	positive	ADJ
iajs-3092	134	31	integer	integer	NOUN
iajs-3092	134	32	h.	h.	PROPN
iajs-3092	135	1	it	it	PRON
iajs-3092	135	2	is	be	AUX
iajs-3092	135	3	clear	clear	ADJ
iajs-3092	135	4	that	that	SCONJ
iajs-3092	135	5	(	(	PUNCT
iajs-3092	135	6	ℙ𝑖)∗	ℙ𝑖)∗	X
iajs-3092	135	7	=	=	SYM
iajs-3092	135	8	𝐴𝑖	𝐴𝑖	PROPN
iajs-3092	135	9	.	.	PUNCT
iajs-3092	136	1	now	now	ADV
iajs-3092	136	2	,	,	PUNCT
iajs-3092	136	3	let	let	VERB
iajs-3092	136	4	h	h	PRON
iajs-3092	136	5	be	be	AUX
iajs-3092	136	6	a	a	DET
iajs-3092	136	7	positive	positive	ADJ
iajs-3092	136	8	integer	integer	NOUN
iajs-3092	136	9	and	and	CCONJ
iajs-3092	136	10	𝑎	𝑎	NOUN
iajs-3092	136	11	∈	∈	PROPN
iajs-3092	136	12	𝐴𝑚+ℎ	𝐴𝑚+ℎ	ADP
iajs-3092	136	13	so	so	ADV
iajs-3092	136	14	,	,	PUNCT
iajs-3092	136	15	ℙ𝑚(𝑎	ℙ𝑚(𝑎	ADJ
iajs-3092	136	16	)	)	PUNCT
iajs-3092	136	17	=	=	SYM
iajs-3092	136	18	ℙ𝑚+ℎ(𝑎	ℙ𝑚+ℎ(𝑎	NOUN
iajs-3092	136	19	)	)	PUNCT
iajs-3092	136	20	=	=	SYM
iajs-3092	137	1	1	1	NUM
iajs-3092	137	2	,	,	PUNCT
iajs-3092	137	3	hence	hence	ADV
iajs-3092	137	4	𝑎	𝑎	PROPN
iajs-3092	137	5	∈	∈	ADJ
iajs-3092	137	6	𝐴𝑚.	𝐴𝑚.	NOUN
iajs-3092	137	7	thus	thus	ADV
iajs-3092	137	8	𝐴𝑚	𝐴𝑚	PROPN
iajs-3092	137	9	⊇	⊇	NOUN
iajs-3092	137	10	𝐴𝑚+ℎ	𝐴𝑚+ℎ	PROPN
iajs-3092	137	11	⊇	⊇	PROPN
iajs-3092	137	12	𝐴𝑚.	𝐴𝑚.	PROPN
iajs-3092	137	13	therefore	therefore	ADV
iajs-3092	137	14	ℙ∗	ℙ∗	PROPN
iajs-3092	137	15	is	be	AUX
iajs-3092	137	16	artiain	artiain	NOUN
iajs-3092	137	17	.	.	PUNCT
iajs-3092	138	1	the	the	DET
iajs-3092	138	2	converse	converse	NOUN
iajs-3092	138	3	of	of	ADP
iajs-3092	138	4	above	above	ADJ
iajs-3092	138	5	theorem	theorem	NOUN
iajs-3092	138	6	is	be	AUX
iajs-3092	138	7	not	not	PART
iajs-3092	138	8	true	true	ADJ
iajs-3092	138	9	as	as	ADP
iajs-3092	138	10	in	in	ADP
iajs-3092	138	11	the	the	DET
iajs-3092	138	12	following	follow	VERB
iajs-3092	138	13	example	example	NOUN
iajs-3092	138	14	.	.	PUNCT
iajs-3092	139	1	let	let	VERB
iajs-3092	139	2	𝕄	𝕄	PROPN
iajs-3092	139	3	=	=	SYM
iajs-3092	139	4	𝔽	𝔽	PROPN
iajs-3092	139	5	and	and	CCONJ
iajs-3092	139	6	𝔽	𝔽	PROPN
iajs-3092	139	7	=	=	SYM
iajs-3092	139	8	ℝ	ℝ	PROPN
iajs-3092	139	9	.	.	PUNCT
iajs-3092	140	1	express	express	VERB
iajs-3092	140	2	ℙ	ℙ	PROPN
iajs-3092	140	3	:	:	PUNCT
iajs-3092	140	4	𝕄	𝕄	PROPN
iajs-3092	140	5	→	→	PUNCT
iajs-3092	141	1	[	[	X
iajs-3092	141	2	0,1	0,1	NUM
iajs-3092	141	3	]	]	PUNCT
iajs-3092	141	4	by	by	ADP
iajs-3092	141	5	ℙ(x	ℙ(x	PROPN
iajs-3092	141	6	)	)	PUNCT
iajs-3092	141	7	=	=	SYM
iajs-3092	141	8	1	1	NUM
iajs-3092	141	9	∀x	∀x	NUM
iajs-3092	141	10	∈	∈	PROPN
iajs-3092	141	11	𝕄.	𝕄.	PROPN
iajs-3092	141	12	clearly	clearly	ADV
iajs-3092	141	13	,	,	PUNCT
iajs-3092	141	14	ℙ	ℙ	PROPN
iajs-3092	141	15	∈	∈	PROPN
iajs-3092	141	16	fum(𝕄	fum(𝕄	PROPN
iajs-3092	141	17	)	)	PUNCT
iajs-3092	141	18	.	.	PUNCT
iajs-3092	142	1	describe	describe	VERB
iajs-3092	142	2	ℙ	ℙ	NUM
iajs-3092	142	3	:	:	PUNCT
iajs-3092	142	4	𝕄	𝕄	PROPN
iajs-3092	142	5	→	→	PUNCT
iajs-3092	142	6	[	[	X
iajs-3092	142	7	0,1	0,1	NUM
iajs-3092	142	8	]	]	PUNCT
iajs-3092	142	9	by	by	ADP
iajs-3092	142	10	ℙ𝑚	ℙ𝑚	PROPN
iajs-3092	142	11	(	(	PUNCT
iajs-3092	142	12	𝑥	𝑥	NOUN
iajs-3092	142	13	)	)	PUNCT
iajs-3092	142	14	=	=	NOUN
iajs-3092	142	15	{	{	PUNCT
iajs-3092	143	1	1	1	NUM
iajs-3092	143	2	𝑖𝑓	𝑖𝑓	ADP
iajs-3092	143	3	𝑥	𝑥	NOUN
iajs-3092	143	4	=	=	SYM
iajs-3092	143	5	0	0	NUM
iajs-3092	143	6	1	1	NUM
iajs-3092	143	7	2𝑚	2𝑚	NOUN
iajs-3092	143	8	𝑖𝑓	𝑖𝑓	ADP
iajs-3092	143	9	𝑥	𝑥	NOUN
iajs-3092	143	10	≠	≠	PROPN
iajs-3092	143	11	0	0	NUM
iajs-3092	143	12	,	,	PUNCT
iajs-3092	143	13	𝑚	𝑚	PROPN
iajs-3092	143	14	∈	∈	PROPN
iajs-3092	143	15	𝑍+	𝑍+	NOUN
iajs-3092	143	16	.	.	PUNCT
iajs-3092	144	1	clear	clear	ADJ
iajs-3092	144	2	that	that	SCONJ
iajs-3092	144	3	ℙ𝑚	ℙ𝑚	PROPN
iajs-3092	144	4	∈	∈	PROPN
iajs-3092	144	5	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	144	6	)	)	PUNCT
iajs-3092	144	7	for	for	ADP
iajs-3092	144	8	each	each	DET
iajs-3092	144	9	m.	m.	NOUN
iajs-3092	144	10	then	then	ADV
iajs-3092	144	11	ℙ1	ℙ1	PROPN
iajs-3092	144	12	⊇	⊇	PROPN
iajs-3092	144	13	ℙ2	ℙ2	PROPN
iajs-3092	144	14	⊇	⊇	PROPN
iajs-3092	144	15	⋯	⋯	PROPN
iajs-3092	144	16	is	be	AUX
iajs-3092	144	17	an	an	DET
iajs-3092	144	18	infinite	infinite	ADJ
iajs-3092	144	19	strictily	strictily	NOUN
iajs-3092	144	20	descending	descend	VERB
iajs-3092	144	21	chain	chain	NOUN
iajs-3092	144	22	of	of	ADP
iajs-3092	144	23	fuzzy	fuzzy	ADJ
iajs-3092	144	24	submodules	submodule	NOUN
iajs-3092	144	25	of	of	ADP
iajs-3092	144	26	ℙ.	ℙ.	PROPN
iajs-3092	144	27	thus	thus	ADV
iajs-3092	144	28	ℙ	ℙ	NOUN
iajs-3092	144	29	is	be	AUX
iajs-3092	144	30	not	not	PART
iajs-3092	144	31	fuzzy	fuzzy	ADJ
iajs-3092	144	32	artiain	artiain	NOUN
iajs-3092	144	33	.	.	PUNCT
iajs-3092	145	1	but	but	CCONJ
iajs-3092	145	2	,	,	PUNCT
iajs-3092	145	3	ℙ∗	ℙ∗	X
iajs-3092	145	4	=	=	SYM
iajs-3092	145	5	𝕄	𝕄	PROPN
iajs-3092	145	6	,	,	PUNCT
iajs-3092	145	7	is	be	AUX
iajs-3092	145	8	artiain	artiain	VERB
iajs-3092	145	9	𝔽	𝔽	PROPN
iajs-3092	146	1	−module	−module	NOUN
iajs-3092	146	2	.	.	PUNCT
iajs-3092	147	1	3.10	3.10	NUM
iajs-3092	147	2	corollary	corollary	NOUN
iajs-3092	147	3	:	:	PUNCT
iajs-3092	147	4	let	let	VERB
iajs-3092	147	5	ℙ	ℙ	PRON
iajs-3092	147	6	be	be	AUX
iajs-3092	147	7	a	a	DET
iajs-3092	147	8	fully	fully	ADV
iajs-3092	147	9	fuzzy	fuzzy	ADJ
iajs-3092	147	10	visible	visible	ADJ
iajs-3092	147	11	artiain	artiain	NOUN
iajs-3092	147	12	module	module	NOUN
iajs-3092	147	13	on	on	ADP
iajs-3092	147	14	𝔽.	𝔽.	PROPN
iajs-3092	147	15	then	then	ADV
iajs-3092	147	16	ℙ	ℙ	PROPN
iajs-3092	147	17	is	be	AUX
iajs-3092	147	18	fuzzy	fuzzy	ADJ
iajs-3092	147	19	semisimple	semisimple	NOUN
iajs-3092	147	20	module	module	NOUN
iajs-3092	147	21	provided	provide	VERB
iajs-3092	147	22	that	that	SCONJ
iajs-3092	147	23	ℙ	ℙ	NOUN
iajs-3092	147	24	satisfies	satisfy	VERB
iajs-3092	147	25	condition	condition	NOUN
iajs-3092	147	26	(	(	PUNCT
iajs-3092	147	27	∗	∗	NOUN
iajs-3092	147	28	)	)	PUNCT
iajs-3092	147	29	.	.	PUNCT
iajs-3092	148	1	ihjpas	ihjpas	PROPN
iajs-3092	148	2	.	.	PUNCT
iajs-3092	149	1	36	36	NUM
iajs-3092	149	2	(	(	PUNCT
iajs-3092	149	3	3	3	NUM
iajs-3092	149	4	)	)	PUNCT
iajs-3092	149	5	2023	2023	NUM
iajs-3092	149	6	376	376	NUM
iajs-3092	149	7	proof	proof	NOUN
iajs-3092	149	8	:	:	PUNCT
iajs-3092	149	9	ℙ∗	ℙ∗	X
iajs-3092	149	10	is	be	AUX
iajs-3092	149	11	a	a	DET
iajs-3092	149	12	fully	fully	ADV
iajs-3092	149	13	visible	visible	ADJ
iajs-3092	149	14	module	module	NOUN
iajs-3092	149	15	since	since	SCONJ
iajs-3092	149	16	ℙ	ℙ	PROPN
iajs-3092	149	17	is	be	AUX
iajs-3092	149	18	fully	fully	ADV
iajs-3092	149	19	fuzzy	fuzzy	ADJ
iajs-3092	149	20	visible	visible	ADJ
iajs-3092	149	21	and	and	CCONJ
iajs-3092	149	22	ℙ∗	ℙ∗	ADJ
iajs-3092	149	23	is	be	AUX
iajs-3092	149	24	artiain	artiain	VERB
iajs-3092	149	25	by	by	ADP
iajs-3092	149	26	theorem	theorem	ADJ
iajs-3092	149	27	3.9	3.9	NUM
iajs-3092	149	28	.	.	PUNCT
iajs-3092	150	1	therefore	therefore	ADV
iajs-3092	150	2	ℙ∗	ℙ∗	PROPN
iajs-3092	150	3	is	be	AUX
iajs-3092	150	4	semisimple	semisimple	NOUN
iajs-3092	150	5	by	by	ADP
iajs-3092	150	6	[	[	X
iajs-3092	150	7	24	24	NUM
iajs-3092	150	8	,	,	PUNCT
iajs-3092	150	9	proposition	proposition	NOUN
iajs-3092	150	10	2.6.7	2.6.7	NUM
iajs-3092	150	11	]	]	PUNCT
iajs-3092	150	12	.	.	PUNCT
iajs-3092	151	1	then	then	ADV
iajs-3092	151	2	,	,	PUNCT
iajs-3092	151	3	ℙ	ℙ	PROPN
iajs-3092	151	4	is	be	AUX
iajs-3092	151	5	fuzzy	fuzzy	ADJ
iajs-3092	151	6	semisimple	semisimple	NOUN
iajs-3092	151	7	by	by	ADP
iajs-3092	151	8	[	[	X
iajs-3092	151	9	20	20	NUM
iajs-3092	151	10	,	,	PUNCT
iajs-3092	151	11	proposition	proposition	NOUN
iajs-3092	151	12	2.8(1	2.8(1	NOUN
iajs-3092	151	13	)	)	PUNCT
iajs-3092	151	14	]	]	PUNCT
iajs-3092	151	15	.	.	PUNCT
iajs-3092	152	1	3.11	3.11	NUM
iajs-3092	152	2	definition	definition	NOUN
iajs-3092	152	3	:	:	PUNCT
iajs-3092	152	4	a	a	DET
iajs-3092	152	5	fuzzy	fuzzy	ADJ
iajs-3092	152	6	ring	ring	NOUN
iajs-3092	152	7	𝔽	𝔽	PROPN
iajs-3092	152	8	is	be	AUX
iajs-3092	152	9	called	call	VERB
iajs-3092	152	10	regular	regular	ADJ
iajs-3092	152	11	if	if	SCONJ
iajs-3092	152	12	and	and	CCONJ
iajs-3092	152	13	only	only	ADV
iajs-3092	152	14	if	if	SCONJ
iajs-3092	152	15	every	every	DET
iajs-3092	152	16	fuzzy	fuzzy	ADJ
iajs-3092	152	17	singleton	singleton	NOUN
iajs-3092	152	18	in	in	ADP
iajs-3092	152	19	𝔽	𝔽	PROPN
iajs-3092	152	20	is	be	AUX
iajs-3092	152	21	fuzzy	fuzzy	ADJ
iajs-3092	152	22	regular	regular	ADJ
iajs-3092	152	23	,	,	PUNCT
iajs-3092	152	24	i.e.	i.e.	X
iajs-3092	152	25	∀	∀	X
iajs-3092	152	26	fuzzy	fuzzy	ADJ
iajs-3092	152	27	singleton	singleton	NOUN
iajs-3092	152	28	𝑥𝑡	𝑥𝑡	ADP
iajs-3092	152	29	of	of	ADP
iajs-3092	152	30	𝔽	𝔽	PROPN
iajs-3092	152	31	,	,	PUNCT
iajs-3092	152	32	𝑡	𝑡	PROPN
iajs-3092	152	33	∈	∈	PROPN
iajs-3092	152	34	(	(	PUNCT
iajs-3092	152	35	0,1	0,1	NOUN
iajs-3092	152	36	]	]	PUNCT
iajs-3092	152	37	,	,	PUNCT
iajs-3092	152	38	there	there	PRON
iajs-3092	152	39	exist	exist	VERB
iajs-3092	152	40	a	a	DET
iajs-3092	152	41	fuzzy	fuzzy	ADJ
iajs-3092	152	42	singleton	singleton	NOUN
iajs-3092	152	43	𝑦𝑘	𝑦𝑘	NOUN
iajs-3092	152	44	of	of	ADP
iajs-3092	152	45	𝔽	𝔽	PROPN
iajs-3092	152	46	such	such	ADJ
iajs-3092	152	47	that	that	PRON
iajs-3092	152	48	𝑥𝑡	𝑥𝑡	ADP
iajs-3092	152	49	=	=	NOUN
iajs-3092	152	50	𝑥𝑡𝑦𝑘𝑥𝑡.	𝑥𝑡𝑦𝑘𝑥𝑡.	ADJ
iajs-3092	152	51	3.12	3.12	NUM
iajs-3092	152	52	proposition	proposition	NOUN
iajs-3092	152	53	:	:	PUNCT
iajs-3092	152	54	a	a	DET
iajs-3092	152	55	fuzzy	fuzzy	ADJ
iajs-3092	152	56	ring	ring	NOUN
iajs-3092	152	57	𝔽	𝔽	PROPN
iajs-3092	152	58	is	be	AUX
iajs-3092	152	59	regular	regular	ADJ
iajs-3092	152	60	if	if	SCONJ
iajs-3092	152	61	and	and	CCONJ
iajs-3092	152	62	only	only	ADV
iajs-3092	152	63	if	if	SCONJ
iajs-3092	152	64	𝔽	𝔽	PROPN
iajs-3092	152	65	is	be	AUX
iajs-3092	152	66	regular	regular	ADJ
iajs-3092	152	67	ring	ring	NOUN
iajs-3092	152	68	.	.	PUNCT
iajs-3092	153	1	proof	proof	NOUN
iajs-3092	153	2	:	:	PUNCT
iajs-3092	153	3	suppose	suppose	VERB
iajs-3092	153	4	that	that	SCONJ
iajs-3092	153	5	𝔽	𝔽	PROPN
iajs-3092	153	6	is	be	AUX
iajs-3092	153	7	regular	regular	ADJ
iajs-3092	153	8	,	,	PUNCT
iajs-3092	153	9	then	then	ADV
iajs-3092	153	10	∀𝑥	∀𝑥	PROPN
iajs-3092	153	11	∈	∈	PROPN
iajs-3092	153	12	𝔽	𝔽	PROPN
iajs-3092	153	13	,	,	PUNCT
iajs-3092	153	14	∃𝑦	∃𝑦	PROPN
iajs-3092	153	15	∈	∈	PROPN
iajs-3092	153	16	𝔽	𝔽	PROPN
iajs-3092	153	17	,	,	PUNCT
iajs-3092	153	18	s.t	s.t	PROPN
iajs-3092	153	19	.	.	PUNCT
iajs-3092	153	20	𝑥	𝑥	X
iajs-3092	153	21	=	=	PUNCT
iajs-3092	153	22	𝑥𝑦𝑥	𝑥𝑦𝑥	NOUN
iajs-3092	153	23	,	,	PUNCT
iajs-3092	153	24	hence	hence	ADV
iajs-3092	153	25	𝑥𝑡	𝑥𝑡	ADV
iajs-3092	153	26	=	=	PUNCT
iajs-3092	153	27	(	(	PUNCT
iajs-3092	153	28	𝑥𝑦𝑥)𝑡	𝑥𝑦𝑥)𝑡	NOUN
iajs-3092	153	29	=	=	SYM
iajs-3092	153	30	𝑥𝑡𝑦𝑡𝑥𝑡	𝑥𝑡𝑦𝑡𝑥𝑡	NOUN
iajs-3092	153	31	,	,	PUNCT
iajs-3092	153	32	where	where	SCONJ
iajs-3092	153	33	𝑥𝑡	𝑥𝑡	ADV
iajs-3092	153	34	,	,	PUNCT
iajs-3092	153	35	𝑦𝑡	𝑦𝑡	VERB
iajs-3092	153	36	fuzzy	fuzzy	ADJ
iajs-3092	153	37	singleton	singleton	NOUN
iajs-3092	153	38	of	of	ADP
iajs-3092	153	39	𝔽	𝔽	PROPN
iajs-3092	153	40	,	,	PUNCT
iajs-3092	153	41	𝑡	𝑡	PROPN
iajs-3092	153	42	∈	∈	PROPN
iajs-3092	153	43	(	(	PUNCT
iajs-3092	153	44	0,1	0,1	NOUN
iajs-3092	153	45	]	]	PUNCT
iajs-3092	153	46	.	.	PUNCT
iajs-3092	154	1	therefore	therefore	ADV
iajs-3092	154	2	,	,	PUNCT
iajs-3092	154	3	a	a	DET
iajs-3092	154	4	fuzzy	fuzzy	ADJ
iajs-3092	154	5	ring	ring	NOUN
iajs-3092	154	6	𝔽	𝔽	PROPN
iajs-3092	154	7	is	be	AUX
iajs-3092	154	8	regular	regular	ADJ
iajs-3092	154	9	.	.	PUNCT
iajs-3092	155	1	conversely	conversely	ADV
iajs-3092	155	2	,	,	PUNCT
iajs-3092	155	3	let	let	VERB
iajs-3092	155	4	𝑀𝔽	𝑀𝔽	PROPN
iajs-3092	155	5	be	be	AUX
iajs-3092	155	6	a	a	DET
iajs-3092	155	7	fuzzy	fuzzy	ADJ
iajs-3092	155	8	ring	ring	NOUN
iajs-3092	155	9	over	over	ADP
iajs-3092	155	10	𝔽	𝔽	PROPN
iajs-3092	155	11	,	,	PUNCT
iajs-3092	155	12	then	then	ADV
iajs-3092	155	13	∀	∀	X
iajs-3092	155	14	fuzzy	fuzzy	ADJ
iajs-3092	155	15	singleton	singleton	PROPN
iajs-3092	155	16	xt	xt	PROPN
iajs-3092	155	17	of	of	ADP
iajs-3092	155	18	𝔽	𝔽	PROPN
iajs-3092	155	19	,	,	PUNCT
iajs-3092	155	20	∃𝑦𝑘	∃𝑦𝑘	VERB
iajs-3092	155	21	fuzzy	fuzzy	ADJ
iajs-3092	155	22	singleton	singleton	NOUN
iajs-3092	155	23	of	of	ADP
iajs-3092	155	24	𝔽	𝔽	PROPN
iajs-3092	155	25	such	such	ADJ
iajs-3092	155	26	that	that	PRON
iajs-3092	155	27	𝑥𝑡	𝑥𝑡	ADP
iajs-3092	155	28	=	=	PUNCT
iajs-3092	155	29	𝑥𝑡𝑦𝑘𝑥𝑡	𝑥𝑡𝑦𝑘𝑥𝑡	PROPN
iajs-3092	155	30	=	=	SYM
iajs-3092	155	31	(	(	PUNCT
iajs-3092	155	32	𝑥𝑦𝑥)𝑚𝑖𝑛{𝑡,𝑘	𝑥𝑦𝑥)𝑚𝑖𝑛{𝑡,𝑘	X
iajs-3092	155	33	}	}	PUNCT
iajs-3092	155	34	,	,	PUNCT
iajs-3092	155	35	hence	hence	ADV
iajs-3092	155	36	𝑥	𝑥	PROPN
iajs-3092	155	37	=	=	PUNCT
iajs-3092	155	38	𝑥𝑦𝑥	𝑥𝑦𝑥	NOUN
iajs-3092	155	39	and	and	CCONJ
iajs-3092	155	40	𝑡	𝑡	X
iajs-3092	155	41	=	=	SYM
iajs-3092	155	42	𝑚𝑖𝑛{𝑡	𝑚𝑖𝑛{𝑡	PROPN
iajs-3092	155	43	,	,	PUNCT
iajs-3092	155	44	𝑘	𝑘	NOUN
iajs-3092	155	45	}	}	PUNCT
iajs-3092	155	46	.	.	PUNCT
iajs-3092	156	1	therefore	therefore	ADV
iajs-3092	156	2	𝔽	𝔽	PROPN
iajs-3092	156	3	is	be	AUX
iajs-3092	156	4	a	a	DET
iajs-3092	156	5	regular	regular	ADJ
iajs-3092	156	6	.	.	PUNCT
iajs-3092	157	1	3.13	3.13	NUM
iajs-3092	157	2	corollary	corollary	NOUN
iajs-3092	157	3	:	:	PUNCT
iajs-3092	157	4	let	let	VERB
iajs-3092	157	5	ℙ	ℙ	PRON
iajs-3092	157	6	be	be	AUX
iajs-3092	157	7	a	a	DET
iajs-3092	157	8	fully	fully	ADV
iajs-3092	157	9	fuzzy	fuzzy	ADJ
iajs-3092	157	10	visible	visible	ADJ
iajs-3092	157	11	and	and	CCONJ
iajs-3092	157	12	artiain	artiain	NOUN
iajs-3092	157	13	module	module	NOUN
iajs-3092	157	14	on	on	ADP
iajs-3092	157	15	𝔽	𝔽	PROPN
iajs-3092	157	16	,	,	PUNCT
iajs-3092	157	17	satisfied	satisfied	ADJ
iajs-3092	157	18	condition	condition	NOUN
iajs-3092	157	19	(	(	PUNCT
iajs-3092	157	20	*	*	NOUN
iajs-3092	157	21	)	)	PUNCT
iajs-3092	157	22	,	,	PUNCT
iajs-3092	157	23	then	then	ADV
iajs-3092	157	24	𝑀𝔽/(𝐹	𝑀𝔽/(𝐹	AUX
iajs-3092	157	25	−	−	PROPN
iajs-3092	157	26	𝑎𝑛𝑛𝑥𝑡)∗	𝑎𝑛𝑛𝑥𝑡)∗	PROPN
iajs-3092	157	27	is	be	AUX
iajs-3092	157	28	fuzzy	fuzzy	ADJ
iajs-3092	157	29	regular	regular	ADJ
iajs-3092	157	30	∀𝑥𝑡	∀𝑥𝑡	NOUN
iajs-3092	157	31	⊆	⊆	NUM
iajs-3092	157	32	ℙ.	ℙ.	NOUN
iajs-3092	157	33	proof	proof	NOUN
iajs-3092	157	34	:	:	PUNCT
iajs-3092	157	35	ℙ∗	ℙ∗	X
iajs-3092	157	36	is	be	AUX
iajs-3092	157	37	fully	fully	ADV
iajs-3092	157	38	visible	visible	ADJ
iajs-3092	157	39	and	and	CCONJ
iajs-3092	157	40	artiain	artiain	NOUN
iajs-3092	157	41	module	module	NOUN
iajs-3092	157	42	as	as	ADP
iajs-3092	157	43	in	in	ADP
iajs-3092	157	44	corollary	corollary	ADJ
iajs-3092	157	45	(	(	PUNCT
iajs-3092	157	46	3.10	3.10	NUM
iajs-3092	157	47	)	)	PUNCT
iajs-3092	157	48	.	.	PUNCT
iajs-3092	158	1	then	then	ADV
iajs-3092	158	2	𝔽/(𝐹	𝔽/(𝐹	PROPN
iajs-3092	158	3	−	−	PROPN
iajs-3092	159	1	𝑎𝑛𝑛𝑥𝑡)∗	𝑎𝑛𝑛𝑥𝑡)∗	PROPN
iajs-3092	159	2	is	be	AUX
iajs-3092	159	3	regular	regular	ADJ
iajs-3092	159	4	by	by	ADP
iajs-3092	159	5	[	[	X
iajs-3092	159	6	24	24	NUM
iajs-3092	159	7	,	,	PUNCT
iajs-3092	159	8	corollary	corollary	ADJ
iajs-3092	159	9	2.6.10	2.6.10	NUM
iajs-3092	159	10	]	]	X
iajs-3092	159	11	,	,	PUNCT
iajs-3092	159	12	then	then	ADV
iajs-3092	159	13	m𝔽/(f	m𝔽/(f	VERB
iajs-3092	159	14	−	−	PROPN
iajs-3092	159	15	𝑎𝑛𝑛𝑥𝑡)∗	𝑎𝑛𝑛𝑥𝑡)∗	PROPN
iajs-3092	159	16	is	be	AUX
iajs-3092	159	17	fuzzy	fuzzy	ADJ
iajs-3092	159	18	regular	regular	ADV
iajs-3092	159	19	by	by	ADP
iajs-3092	159	20	[	[	X
iajs-3092	159	21	27	27	NUM
iajs-3092	159	22	,	,	PUNCT
iajs-3092	159	23	theorem	theorem	VERB
iajs-3092	159	24	3.2.10	3.2.10	NUM
iajs-3092	159	25	]	]	PUNCT
iajs-3092	159	26	and	and	CCONJ
iajs-3092	159	27	proposition	proposition	NOUN
iajs-3092	159	28	(	(	PUNCT
iajs-3092	159	29	3.12	3.12	NUM
iajs-3092	159	30	)	)	PUNCT
iajs-3092	159	31	.	.	PUNCT
iajs-3092	160	1	3.14	3.14	NUM
iajs-3092	160	2	proposition	proposition	NOUN
iajs-3092	160	3	:	:	PUNCT
iajs-3092	160	4	let	let	VERB
iajs-3092	160	5	ℙ	ℙ	PRON
iajs-3092	160	6	∈	∈	NOUN
iajs-3092	160	7	𝐹𝑈𝑀(𝑀	𝐹𝑈𝑀(𝑀	NOUN
iajs-3092	160	8	)	)	PUNCT
iajs-3092	160	9	over	over	ADP
iajs-3092	160	10	principle	principle	ADJ
iajs-3092	160	11	fuzzy	fuzzy	ADJ
iajs-3092	160	12	ideal	ideal	ADJ
iajs-3092	160	13	ring	ring	NOUN
iajs-3092	160	14	.	.	PUNCT
iajs-3092	161	1	if	if	SCONJ
iajs-3092	161	2	every	every	DET
iajs-3092	161	3	ℙ	ℙ	PROPN
iajs-3092	161	4	≠	≠	PROPN
iajs-3092	161	5	𝔚	𝔚	NOUN
iajs-3092	161	6	∈	∈	PROPN
iajs-3092	161	7	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	161	8	)	)	PUNCT
iajs-3092	161	9	over	over	ADP
iajs-3092	161	10	𝔽	𝔽	PROPN
iajs-3092	161	11	is	be	AUX
iajs-3092	161	12	divisible	divisible	ADJ
iajs-3092	161	13	,	,	PUNCT
iajs-3092	161	14	then	then	ADV
iajs-3092	161	15	ℙ	ℙ	PROPN
iajs-3092	161	16	is	be	AUX
iajs-3092	161	17	fuzzy	fuzzy	ADJ
iajs-3092	161	18	fully	fully	ADV
iajs-3092	161	19	visible	visible	ADJ
iajs-3092	161	20	and	and	CCONJ
iajs-3092	161	21	hence	hence	ADV
iajs-3092	161	22	ℙ	ℙ	PROPN
iajs-3092	161	23	is	be	AUX
iajs-3092	161	24	fregular	fregular	ADJ
iajs-3092	161	25	fuzzy	fuzzy	ADJ
iajs-3092	161	26	.	.	PUNCT
iajs-3092	162	1	proof	proof	NOUN
iajs-3092	162	2	:	:	PUNCT
iajs-3092	162	3	by	by	ADP
iajs-3092	162	4	using	use	VERB
iajs-3092	162	5	proposition	proposition	NOUN
iajs-3092	162	6	3.2	3.2	NUM
iajs-3092	162	7	,	,	PUNCT
iajs-3092	162	8	we	we	PRON
iajs-3092	162	9	have	have	VERB
iajs-3092	162	10	ℙ	ℙ	PRON
iajs-3092	162	11	a	a	PRON
iajs-3092	162	12	is	be	AUX
iajs-3092	162	13	fuzzy	fuzzy	ADJ
iajs-3092	162	14	fully	fully	ADV
iajs-3092	162	15	visible	visible	ADJ
iajs-3092	162	16	,	,	PUNCT
iajs-3092	162	17	then	then	ADV
iajs-3092	162	18	ℙ	ℙ	PROPN
iajs-3092	162	19	is	be	AUX
iajs-3092	162	20	f	f	X
iajs-3092	162	21	-	-	PUNCT
iajs-3092	162	22	regular	regular	ADJ
iajs-3092	162	23	by	by	ADP
iajs-3092	162	24	proposition	proposition	NOUN
iajs-3092	162	25	(	(	PUNCT
iajs-3092	162	26	3	3	NUM
iajs-3092	162	27	.	.	NOUN
iajs-3092	162	28	4	4	NUM
iajs-3092	162	29	)	)	PUNCT
iajs-3092	162	30	.	.	PUNCT
iajs-3092	163	1	3.15	3.15	NUM
iajs-3092	163	2	proposition	proposition	NOUN
iajs-3092	163	3	:	:	PUNCT
iajs-3092	163	4	let	let	VERB
iajs-3092	163	5	𝔽	𝔽	PRON
iajs-3092	163	6	be	be	AUX
iajs-3092	163	7	a	a	DET
iajs-3092	163	8	principle	principle	ADJ
iajs-3092	163	9	fuzzy	fuzzy	ADJ
iajs-3092	163	10	ideal	ideal	ADJ
iajs-3092	163	11	ring	ring	NOUN
iajs-3092	163	12	and	and	CCONJ
iajs-3092	163	13	ℙ	ℙ	NOUN
iajs-3092	163	14	is	be	AUX
iajs-3092	163	15	a	a	DET
iajs-3092	163	16	fuzzy	fuzzy	ADJ
iajs-3092	163	17	divisible	divisible	NOUN
iajs-3092	163	18	,	,	PUNCT
iajs-3092	163	19	then	then	ADV
iajs-3092	163	20	the	the	DET
iajs-3092	163	21	following	follow	VERB
iajs-3092	163	22	is	be	AUX
iajs-3092	163	23	equivalent	equivalent	ADJ
iajs-3092	163	24	:	:	PUNCT
iajs-3092	163	25	1	1	X
iajs-3092	163	26	.	.	X
iajs-3092	163	27	ℙ	ℙ	NOUN
iajs-3092	163	28	is	be	AUX
iajs-3092	163	29	a	a	DET
iajs-3092	163	30	fully	fully	ADV
iajs-3092	163	31	fuzzy	fuzzy	ADJ
iajs-3092	163	32	visible	visible	ADJ
iajs-3092	163	33	module	module	NOUN
iajs-3092	163	34	.	.	PUNCT
iajs-3092	164	1	2	2	X
iajs-3092	164	2	.	.	X
iajs-3092	164	3	each	each	DET
iajs-3092	164	4	ℙ	ℙ	PROPN
iajs-3092	164	5	≠	≠	PROPN
iajs-3092	164	6	ᵹ	ᵹ	PROPN
iajs-3092	164	7	∈	∈	PROPN
iajs-3092	164	8	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	164	9	)	)	PUNCT
iajs-3092	164	10	is	be	AUX
iajs-3092	164	11	pure	pure	ADJ
iajs-3092	164	12	.	.	PUNCT
iajs-3092	165	1	3	3	X
iajs-3092	165	2	.	.	X
iajs-3092	165	3	each	each	DET
iajs-3092	165	4	ℙ	ℙ	PROPN
iajs-3092	165	5	≠	≠	PROPN
iajs-3092	165	6	ᵹ	ᵹ	PROPN
iajs-3092	165	7	∈	∈	PROPN
iajs-3092	165	8	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	165	9	)	)	PUNCT
iajs-3092	165	10	is	be	AUX
iajs-3092	165	11	divisible	divisible	ADJ
iajs-3092	165	12	.	.	PUNCT
iajs-3092	166	1	4	4	X
iajs-3092	166	2	.	.	X
iajs-3092	166	3	each	each	DET
iajs-3092	166	4	ℙ	ℙ	PROPN
iajs-3092	166	5	≠	≠	PROPN
iajs-3092	166	6	ᵹ	ᵹ	PROPN
iajs-3092	166	7	∈	∈	PROPN
iajs-3092	166	8	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	166	9	)	)	PUNCT
iajs-3092	166	10	is	be	AUX
iajs-3092	166	11	visible	visible	ADJ
iajs-3092	166	12	.	.	PUNCT
iajs-3092	167	1	proof:1	proof:1	NOUN
iajs-3092	167	2	⟹	⟹	NUM
iajs-3092	167	3	2	2	NUM
iajs-3092	167	4	directly	directly	ADV
iajs-3092	167	5	from	from	ADP
iajs-3092	167	6	[	[	X
iajs-3092	167	7	4	4	NUM
iajs-3092	167	8	,	,	PUNCT
iajs-3092	167	9	proposition	proposition	NOUN
iajs-3092	167	10	3.17	3.17	NUM
iajs-3092	167	11	]	]	PUNCT
iajs-3092	167	12	.	.	PUNCT
iajs-3092	168	1	2	2	NUM
iajs-3092	168	2	⟹	⟹	NUM
iajs-3092	168	3	3	3	NUM
iajs-3092	168	4	since	since	SCONJ
iajs-3092	168	5	𝔽	𝔽	PROPN
iajs-3092	168	6	be	be	VERB
iajs-3092	168	7	a	a	DET
iajs-3092	168	8	principle	principle	ADJ
iajs-3092	168	9	fuzzy	fuzzy	ADJ
iajs-3092	168	10	ideal	ideal	ADJ
iajs-3092	168	11	ring	ring	NOUN
iajs-3092	168	12	and	and	CCONJ
iajs-3092	168	13	ℙ	ℙ	NOUN
iajs-3092	168	14	is	be	AUX
iajs-3092	168	15	a	a	DET
iajs-3092	168	16	fuzzy	fuzzy	ADJ
iajs-3092	168	17	divisible	divisible	NOUN
iajs-3092	168	18	,	,	PUNCT
iajs-3092	168	19	then	then	ADV
iajs-3092	168	20	by[4	by[4	ADJ
iajs-3092	168	21	,	,	PUNCT
iajs-3092	168	22	proposition	proposition	NOUN
iajs-3092	168	23	3.19	3.19	NUM
iajs-3092	168	24	]	]	PUNCT
iajs-3092	168	25	we	we	PRON
iajs-3092	168	26	have	have	VERB
iajs-3092	168	27	each	each	DET
iajs-3092	168	28	proper	proper	ADJ
iajs-3092	168	29	submodule	submodule	NOUN
iajs-3092	168	30	of	of	ADP
iajs-3092	168	31	ℙ	ℙ	PROPN
iajs-3092	168	32	is	be	AUX
iajs-3092	168	33	visible	visible	ADJ
iajs-3092	168	34	and	and	CCONJ
iajs-3092	168	35	hence	hence	ADV
iajs-3092	168	36	divisible	divisible	ADJ
iajs-3092	168	37	.	.	PUNCT
iajs-3092	169	1	3	3	NUM
iajs-3092	169	2	⟹	⟹	NUM
iajs-3092	169	3	4	4	NUM
iajs-3092	169	4	directly	directly	ADV
iajs-3092	169	5	from	from	ADP
iajs-3092	169	6	proposition	proposition	NOUN
iajs-3092	169	7	3.2	3.2	NUM
iajs-3092	169	8	.	.	NOUN
iajs-3092	169	9	4	4	NUM
iajs-3092	169	10	⟹	⟹	NUM
iajs-3092	169	11	1	1	NUM
iajs-3092	169	12	clear	clear	ADJ
iajs-3092	169	13	.	.	PUNCT
iajs-3092	170	1	3.16	3.16	NUM
iajs-3092	170	2	proposition	proposition	NOUN
iajs-3092	170	3	:	:	PUNCT
iajs-3092	170	4	let	let	VERB
iajs-3092	170	5	ℙ	ℙ	PRON
iajs-3092	170	6	be	be	AUX
iajs-3092	170	7	a	a	DET
iajs-3092	170	8	fully	fully	ADV
iajs-3092	170	9	fuzzy	fuzzy	ADJ
iajs-3092	170	10	visible	visible	ADJ
iajs-3092	170	11	module	module	NOUN
iajs-3092	170	12	over	over	ADP
iajs-3092	170	13	a	a	DET
iajs-3092	170	14	local	local	ADJ
iajs-3092	170	15	ring	ring	NOUN
iajs-3092	170	16	,	,	PUNCT
iajs-3092	170	17	then	then	ADV
iajs-3092	170	18	ℙ	ℙ	PROPN
iajs-3092	170	19	is	be	AUX
iajs-3092	170	20	a	a	DET
iajs-3092	170	21	fuzzy	fuzzy	ADJ
iajs-3092	170	22	semisimple	semisimple	NOUN
iajs-3092	170	23	module	module	NOUN
iajs-3092	170	24	such	such	ADJ
iajs-3092	170	25	that	that	DET
iajs-3092	170	26	ℙ	ℙ	NOUN
iajs-3092	170	27	satisfied	satisfied	ADJ
iajs-3092	170	28	condition	condition	NOUN
iajs-3092	170	29	(	(	PUNCT
iajs-3092	170	30	∗	∗	NOUN
iajs-3092	170	31	)	)	PUNCT
iajs-3092	170	32	.	.	PUNCT
iajs-3092	171	1	proof	proof	NOUN
iajs-3092	171	2	:	:	PUNCT
iajs-3092	171	3	ℙ∗	ℙ∗	X
iajs-3092	171	4	is	be	AUX
iajs-3092	171	5	a	a	DET
iajs-3092	171	6	fully	fully	ADV
iajs-3092	171	7	visible	visible	ADJ
iajs-3092	171	8	module	module	NOUN
iajs-3092	171	9	by	by	ADP
iajs-3092	171	10	[	[	X
iajs-3092	171	11	5	5	NUM
iajs-3092	171	12	,	,	PUNCT
iajs-3092	171	13	proposition	proposition	NOUN
iajs-3092	171	14	3.2	3.2	NUM
iajs-3092	171	15	]	]	PUNCT
iajs-3092	171	16	,	,	PUNCT
iajs-3092	171	17	then	then	ADV
iajs-3092	171	18	ℙ∗	ℙ∗	X
iajs-3092	171	19	is	be	AUX
iajs-3092	171	20	semisimple	semisimple	NOUN
iajs-3092	171	21	by	by	ADP
iajs-3092	171	22	[	[	X
iajs-3092	171	23	24	24	NUM
iajs-3092	171	24	,	,	PUNCT
iajs-3092	171	25	proposition	proposition	NOUN
iajs-3092	171	26	2.6.13	2.6.13	NUM
iajs-3092	171	27	]	]	PUNCT
iajs-3092	171	28	,	,	PUNCT
iajs-3092	171	29	hence	hence	ADV
iajs-3092	171	30	ℙ	ℙ	PROPN
iajs-3092	171	31	is	be	AUX
iajs-3092	171	32	a	a	DET
iajs-3092	171	33	semisimple	semisimple	NOUN
iajs-3092	171	34	by	by	ADP
iajs-3092	171	35	[	[	X
iajs-3092	171	36	20	20	NUM
iajs-3092	171	37	,	,	PUNCT
iajs-3092	171	38	proposition	proposition	NOUN
iajs-3092	171	39	.	.	PUNCT
iajs-3092	172	1	2.8(1	2.8(1	NOUN
iajs-3092	172	2	)	)	PUNCT
iajs-3092	172	3	]	]	PUNCT
iajs-3092	172	4	.	.	PUNCT
iajs-3092	173	1	3.17	3.17	NUM
iajs-3092	173	2	proposition	proposition	NOUN
iajs-3092	173	3	:	:	PUNCT
iajs-3092	173	4	let	let	VERB
iajs-3092	173	5	ℙ	ℙ	PRON
iajs-3092	173	6	be	be	AUX
iajs-3092	173	7	a	a	DET
iajs-3092	173	8	fuzzy	fuzzy	ADJ
iajs-3092	173	9	divisible	divisible	ADJ
iajs-3092	173	10	over	over	ADP
iajs-3092	173	11	principle	principle	ADJ
iajs-3092	173	12	fuzzy	fuzzy	ADJ
iajs-3092	173	13	ideal	ideal	ADJ
iajs-3092	173	14	ring	ring	NOUN
iajs-3092	173	15	,	,	PUNCT
iajs-3092	173	16	then	then	ADV
iajs-3092	173	17	ℙ	ℙ	PROPN
iajs-3092	173	18	is	be	AUX
iajs-3092	173	19	fully	fully	ADV
iajs-3092	173	20	visible	visible	ADJ
iajs-3092	173	21	.	.	PUNCT
iajs-3092	174	1	proof	proof	NOUN
iajs-3092	174	2	:	:	PUNCT
iajs-3092	174	3	depending	depend	VERB
iajs-3092	174	4	on	on	ADP
iajs-3092	174	5	[	[	X
iajs-3092	174	6	4	4	NUM
iajs-3092	174	7	,	,	PUNCT
iajs-3092	174	8	proposition	proposition	NOUN
iajs-3092	174	9	3.19	3.19	NUM
iajs-3092	174	10	]	]	PUNCT
iajs-3092	174	11	,	,	PUNCT
iajs-3092	174	12	every	every	DET
iajs-3092	174	13	proper	proper	ADJ
iajs-3092	174	14	fuzzy	fuzzy	ADJ
iajs-3092	174	15	submodule	submodule	NOUN
iajs-3092	174	16	of	of	ADP
iajs-3092	174	17	ℙ	ℙ	PROPN
iajs-3092	174	18	is	be	AUX
iajs-3092	174	19	visible	visible	ADJ
iajs-3092	174	20	and	and	CCONJ
iajs-3092	174	21	hence	hence	ADV
iajs-3092	174	22	ℙ	ℙ	PROPN
iajs-3092	174	23	is	be	AUX
iajs-3092	174	24	fully	fully	ADV
iajs-3092	174	25	visible	visible	ADJ
iajs-3092	174	26	.	.	PUNCT
iajs-3092	175	1	3.18	3.18	NUM
iajs-3092	175	2	proposition	proposition	NOUN
iajs-3092	175	3	:	:	PUNCT
iajs-3092	175	4	let	let	VERB
iajs-3092	175	5	ℙ	ℙ	PRON
iajs-3092	175	6	be	be	AUX
iajs-3092	175	7	a	a	DET
iajs-3092	175	8	fully	fully	ADV
iajs-3092	175	9	fuzzy	fuzzy	ADJ
iajs-3092	175	10	visible	visible	ADJ
iajs-3092	175	11	module	module	NOUN
iajs-3092	175	12	over	over	ADP
iajs-3092	175	13	𝔽	𝔽	PROPN
iajs-3092	175	14	,	,	PUNCT
iajs-3092	175	15	then	then	ADV
iajs-3092	175	16	ℙ	ℙ	PROPN
iajs-3092	175	17	is	be	AUX
iajs-3092	175	18	a	a	DET
iajs-3092	175	19	fuzzy	fuzzy	ADJ
iajs-3092	175	20	torsion	torsion	NOUN
iajs-3092	175	21	free	free	ADJ
iajs-3092	175	22	.	.	PUNCT
iajs-3092	176	1	proof	proof	NOUN
iajs-3092	176	2	:	:	PUNCT
iajs-3092	176	3	ℙt	ℙt	PROPN
iajs-3092	176	4	is	be	AUX
iajs-3092	176	5	fully	fully	ADV
iajs-3092	176	6	visible	visible	ADJ
iajs-3092	176	7	∀	∀	X
iajs-3092	176	8	𝑡	𝑡	X
iajs-3092	176	9	∈	∈	PROPN
iajs-3092	176	10	(	(	PUNCT
iajs-3092	176	11	0,1	0,1	NOUN
iajs-3092	176	12	]	]	PUNCT
iajs-3092	176	13	by	by	ADP
iajs-3092	176	14	[	[	X
iajs-3092	176	15	5	5	NUM
iajs-3092	176	16	,	,	PUNCT
iajs-3092	176	17	proposition	proposition	NOUN
iajs-3092	176	18	3.2	3.2	NUM
iajs-3092	176	19	]	]	PUNCT
iajs-3092	176	20	,	,	PUNCT
iajs-3092	176	21	then	then	ADV
iajs-3092	176	22	ℙt	ℙt	PROPN
iajs-3092	176	23	torsion	torsion	NOUN
iajs-3092	176	24	free	free	ADJ
iajs-3092	176	25	by	by	ADP
iajs-3092	176	26	[	[	PUNCT
iajs-3092	176	27	24	24	NUM
iajs-3092	176	28	,	,	PUNCT
iajs-3092	176	29	proposition	proposition	NOUN
iajs-3092	176	30	2.6.16	2.6.16	NUM
iajs-3092	176	31	]	]	PUNCT
iajs-3092	176	32	.	.	PUNCT
iajs-3092	177	1	therefore	therefore	ADV
iajs-3092	177	2	ℙ	ℙ	PROPN
iajs-3092	177	3	is	be	AUX
iajs-3092	177	4	fuzzy	fuzzy	ADJ
iajs-3092	177	5	torsion	torsion	NOUN
iajs-3092	177	6	free	free	ADJ
iajs-3092	177	7	by	by	ADP
iajs-3092	177	8	{	{	PUNCT
iajs-3092	177	9	17	17	NUM
iajs-3092	177	10	,	,	PUNCT
iajs-3092	177	11	proposition	proposition	NOUN
iajs-3092	177	12	2.2.23	2.2.23	NUM
iajs-3092	177	13	]	]	PUNCT
iajs-3092	177	14	.	.	PUNCT
iajs-3092	178	1	ihjpas	ihjpas	PROPN
iajs-3092	178	2	.	.	PUNCT
iajs-3092	179	1	36	36	NUM
iajs-3092	179	2	(	(	PUNCT
iajs-3092	179	3	3	3	NUM
iajs-3092	179	4	)	)	PUNCT
iajs-3092	179	5	2023	2023	NUM
iajs-3092	179	6	377	377	NUM
iajs-3092	179	7	3.19	3.19	NUM
iajs-3092	179	8	proposition	proposition	NOUN
iajs-3092	179	9	:	:	PUNCT
iajs-3092	179	10	let	let	VERB
iajs-3092	179	11	ℙ	ℙ	PRON
iajs-3092	179	12	be	be	AUX
iajs-3092	179	13	a	a	DET
iajs-3092	179	14	fully	fully	ADV
iajs-3092	179	15	fuzzy	fuzzy	ADJ
iajs-3092	179	16	visible	visible	ADJ
iajs-3092	179	17	module	module	NOUN
iajs-3092	179	18	,	,	PUNCT
iajs-3092	179	19	ℙ	ℙ	PROPN
iajs-3092	179	20	≠	≠	PROPN
iajs-3092	179	21	𝔚	𝔚	NOUN
iajs-3092	179	22	∈	∈	PROPN
iajs-3092	179	23	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	179	24	)	)	PUNCT
iajs-3092	179	25	.	.	PUNCT
iajs-3092	180	1	then	then	ADV
iajs-3092	180	2	,	,	PUNCT
iajs-3092	180	3	𝔚	𝔚	PROPN
iajs-3092	180	4	is	be	AUX
iajs-3092	180	5	a	a	DET
iajs-3092	180	6	prime	prime	ADJ
iajs-3092	180	7	submodule	submodule	NOUN
iajs-3092	180	8	.	.	PUNCT
iajs-3092	181	1	proof	proof	NOUN
iajs-3092	181	2	:	:	PUNCT
iajs-3092	181	3	because	because	SCONJ
iajs-3092	181	4	ℙ	ℙ	PROPN
iajs-3092	181	5	is	be	AUX
iajs-3092	181	6	a	a	DET
iajs-3092	181	7	fully	fully	ADV
iajs-3092	181	8	fuzzy	fuzzy	ADJ
iajs-3092	181	9	visible	visible	ADJ
iajs-3092	181	10	module	module	NOUN
iajs-3092	181	11	and	and	CCONJ
iajs-3092	181	12	ℙ	ℙ	PROPN
iajs-3092	181	13	≠	≠	PROPN
iajs-3092	181	14	𝔚	𝔚	NOUN
iajs-3092	181	15	∈	∈	PROPN
iajs-3092	181	16	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	181	17	)	)	PUNCT
iajs-3092	181	18	,	,	PUNCT
iajs-3092	181	19	then	then	ADV
iajs-3092	181	20	𝔚	𝔚	PROPN
iajs-3092	181	21	is	be	AUX
iajs-3092	181	22	pure	pure	ADJ
iajs-3092	181	23	,	,	PUNCT
iajs-3092	181	24	hence	hence	ADV
iajs-3092	181	25	𝔚	𝔚	PROPN
iajs-3092	181	26	is	be	AUX
iajs-3092	181	27	weakly	weakly	ADV
iajs-3092	181	28	pure	pure	ADJ
iajs-3092	181	29	fuzzy	fuzzy	ADJ
iajs-3092	181	30	submodule	submodule	NOUN
iajs-3092	181	31	of	of	ADP
iajs-3092	181	32	ℙ.	ℙ.	PROPN
iajs-3092	181	33	therefore	therefore	ADV
iajs-3092	181	34	𝔚	𝔚	PROPN
iajs-3092	181	35	is	be	AUX
iajs-3092	181	36	a	a	DET
iajs-3092	181	37	prime	prime	ADJ
iajs-3092	181	38	fuzzy	fuzzy	NOUN
iajs-3092	181	39	by	by	ADP
iajs-3092	181	40	[	[	X
iajs-3092	181	41	17	17	NUM
iajs-3092	181	42	,	,	PUNCT
iajs-3092	181	43	proposition	proposition	NOUN
iajs-3092	181	44	2.2.22	2.2.22	NUM
iajs-3092	181	45	]	]	X
iajs-3092	181	46	.	.	PUNCT
iajs-3092	182	1	3.20	3.20	NUM
iajs-3092	182	2	corollary	corollary	NOUN
iajs-3092	182	3	:	:	PUNCT
iajs-3092	182	4	if	if	SCONJ
iajs-3092	182	5	ℙ	ℙ	PROPN
iajs-3092	182	6	is	be	AUX
iajs-3092	182	7	a	a	DET
iajs-3092	182	8	fully	fully	ADV
iajs-3092	182	9	fuzzy	fuzzy	ADJ
iajs-3092	182	10	visible	visible	ADJ
iajs-3092	182	11	module	module	NOUN
iajs-3092	182	12	,	,	PUNCT
iajs-3092	182	13	then	then	ADV
iajs-3092	182	14	every	every	DET
iajs-3092	182	15	proper	proper	ADJ
iajs-3092	182	16	fuzzy	fuzzy	ADJ
iajs-3092	182	17	submodule	submodule	NOUN
iajs-3092	182	18	𝔚	𝔚	PROPN
iajs-3092	182	19	of	of	ADP
iajs-3092	182	20	ℙ	ℙ	PROPN
iajs-3092	182	21	is	be	AUX
iajs-3092	182	22	quasi	quasi	ADJ
iajs-3092	182	23	prime	prime	NOUN
iajs-3092	182	24	.	.	PUNCT
iajs-3092	183	1	proof	proof	NOUN
iajs-3092	183	2	:	:	PUNCT
iajs-3092	183	3	depending	depend	VERB
iajs-3092	183	4	proposition	proposition	NOUN
iajs-3092	183	5	3.19	3.19	NUM
iajs-3092	183	6	and	and	CCONJ
iajs-3092	183	7	[	[	X
iajs-3092	183	8	23	23	NUM
iajs-3092	183	9	,	,	PUNCT
iajs-3092	183	10	proposition	proposition	NOUN
iajs-3092	183	11	3.2.2	3.2.2	NUM
iajs-3092	183	12	]	]	PUNCT
iajs-3092	183	13	,	,	PUNCT
iajs-3092	183	14	the	the	DET
iajs-3092	183	15	result	result	NOUN
iajs-3092	183	16	hold	hold	VERB
iajs-3092	183	17	.	.	PUNCT
iajs-3092	184	1	3.21	3.21	NUM
iajs-3092	184	2	corollary	corollary	NOUN
iajs-3092	184	3	:	:	PUNCT
iajs-3092	184	4	every	every	DET
iajs-3092	184	5	fully	fully	ADV
iajs-3092	184	6	fuzzy	fuzzy	ADJ
iajs-3092	184	7	visible	visible	ADJ
iajs-3092	184	8	module	module	NOUN
iajs-3092	184	9	is	be	AUX
iajs-3092	184	10	fuzzy	fuzzy	ADJ
iajs-3092	184	11	t	t	NOUN
iajs-3092	184	12	-	-	PUNCT
iajs-3092	184	13	regular	regular	NOUN
iajs-3092	184	14	.	.	PUNCT
iajs-3092	185	1	proof	proof	NOUN
iajs-3092	185	2	:	:	PUNCT
iajs-3092	185	3	depending	depend	VERB
iajs-3092	185	4	on	on	ADP
iajs-3092	185	5	proposition	proposition	NOUN
iajs-3092	185	6	(	(	PUNCT
iajs-3092	185	7	3.4	3.4	NUM
iajs-3092	185	8	)	)	PUNCT
iajs-3092	185	9	and	and	CCONJ
iajs-3092	185	10	[	[	X
iajs-3092	185	11	22	22	NUM
iajs-3092	185	12	,	,	PUNCT
iajs-3092	185	13	remark	remark	NOUN
iajs-3092	185	14	and	and	CCONJ
iajs-3092	185	15	examples	example	NOUN
iajs-3092	185	16	2.1.5(1	2.1.5(1	NUM
iajs-3092	185	17	)	)	PUNCT
iajs-3092	185	18	]	]	PUNCT
iajs-3092	185	19	.	.	PUNCT
iajs-3092	186	1	the	the	DET
iajs-3092	186	2	converse	converse	NOUN
iajs-3092	186	3	of	of	ADP
iajs-3092	186	4	the	the	DET
iajs-3092	186	5	corollary	corollary	ADJ
iajs-3092	186	6	(	(	PUNCT
iajs-3092	186	7	3.21	3.21	NUM
iajs-3092	186	8	)	)	PUNCT
iajs-3092	186	9	is	be	AUX
iajs-3092	186	10	not	not	PART
iajs-3092	186	11	true	true	ADJ
iajs-3092	186	12	as	as	SCONJ
iajs-3092	186	13	we	we	PRON
iajs-3092	186	14	see	see	VERB
iajs-3092	186	15	in	in	ADP
iajs-3092	186	16	the	the	DET
iajs-3092	186	17	following	follow	VERB
iajs-3092	186	18	example	example	NOUN
iajs-3092	186	19	:	:	PUNCT
iajs-3092	186	20	let	let	VERB
iajs-3092	186	21	𝕄	𝕄	PROPN
iajs-3092	186	22	=	=	PUNCT
iajs-3092	186	23	𝑍4	𝑍4	PROPN
iajs-3092	186	24	,	,	PUNCT
iajs-3092	186	25	𝔽	𝔽	PROPN
iajs-3092	186	26	=	=	SYM
iajs-3092	186	27	𝑍.	𝑍.	PROPN
iajs-3092	186	28	define	define	VERB
iajs-3092	186	29	𝑃	𝑃	NOUN
iajs-3092	186	30	:	:	PUNCT
iajs-3092	186	31	𝕄	𝕄	PROPN
iajs-3092	186	32	→	→	SYM
iajs-3092	186	33	𝑍4	𝑍4	NOUN
iajs-3092	186	34	as	as	ADP
iajs-3092	186	35	ℙ(𝑥	ℙ(𝑥	NOUN
iajs-3092	186	36	)	)	PUNCT
iajs-3092	186	37	=	=	PRON
iajs-3092	186	38	{	{	PUNCT
iajs-3092	186	39	1	1	NUM
iajs-3092	186	40	if	if	SCONJ
iajs-3092	186	41	𝑥	𝑥	PRON
iajs-3092	186	42	∈	∈	PROPN
iajs-3092	186	43	𝑍4	𝑍4	NOUN
iajs-3092	186	44	0	0	PUNCT
iajs-3092	186	45	𝑜.	𝑜.	NOUN
iajs-3092	186	46	𝑤	𝑤	ADP
iajs-3092	186	47	,	,	PUNCT
iajs-3092	186	48	clearly	clearly	ADV
iajs-3092	186	49	,	,	PUNCT
iajs-3092	186	50	ℙ	ℙ	PROPN
iajs-3092	186	51	∈	∈	PROPN
iajs-3092	186	52	fum(𝕄	fum(𝕄	NUM
iajs-3092	186	53	)	)	PUNCT
iajs-3092	186	54	,	,	PUNCT
iajs-3092	186	55	then	then	ADV
iajs-3092	186	56	ℙt	ℙt	PROPN
iajs-3092	186	57	=	=	SYM
iajs-3092	186	58	z4	z4	PROPN
iajs-3092	186	59	,	,	PUNCT
iajs-3092	186	60	∀	∀	PUNCT
iajs-3092	186	61	𝑡	𝑡	NOUN
iajs-3092	186	62	∈	∈	PROPN
iajs-3092	186	63	(	(	PUNCT
iajs-3092	186	64	0,1	0,1	NOUN
iajs-3092	186	65	]	]	PUNCT
iajs-3092	186	66	as	as	SCONJ
iajs-3092	186	67	z	z	NOUN
iajs-3092	186	68	-	-	PUNCT
iajs-3092	186	69	module	module	NOUN
iajs-3092	186	70	is	be	AUX
iajs-3092	186	71	t	t	NOUN
iajs-3092	186	72	-	-	PUNCT
iajs-3092	186	73	regular	regular	NOUN
iajs-3092	186	74	by	by	ADP
iajs-3092	186	75	[	[	PUNCT
iajs-3092	186	76	24	24	NUM
iajs-3092	186	77	,	,	PUNCT
iajs-3092	186	78	p99	p99	PROPN
iajs-3092	186	79	]	]	PUNCT
iajs-3092	186	80	,	,	PUNCT
iajs-3092	186	81	then	then	ADV
iajs-3092	186	82	ℙ	ℙ	PROPN
iajs-3092	186	83	is	be	AUX
iajs-3092	186	84	fuzzy	fuzzy	ADJ
iajs-3092	186	85	t	t	NOUN
iajs-3092	186	86	-	-	PUNCT
iajs-3092	186	87	regular	regular	NOUN
iajs-3092	186	88	by	by	ADP
iajs-3092	186	89	[	[	X
iajs-3092	186	90	22	22	NUM
iajs-3092	186	91	,	,	PUNCT
iajs-3092	186	92	proposition	proposition	NOUN
iajs-3092	186	93	2.1.2	2.1.2	NUM
iajs-3092	186	94	]	]	PUNCT
iajs-3092	186	95	.	.	PUNCT
iajs-3092	187	1	but	but	CCONJ
iajs-3092	187	2	z4	z4	PROPN
iajs-3092	187	3	as	as	ADP
iajs-3092	187	4	z	z	NOUN
iajs-3092	187	5	-	-	PUNCT
iajs-3092	187	6	module	module	NOUN
iajs-3092	187	7	is	be	AUX
iajs-3092	187	8	not	not	PART
iajs-3092	187	9	fully	fully	ADV
iajs-3092	187	10	visible	visible	ADJ
iajs-3092	187	11	module	module	NOUN
iajs-3092	187	12	by	by	ADP
iajs-3092	187	13	[	[	X
iajs-3092	187	14	24	24	NUM
iajs-3092	187	15	,	,	PUNCT
iajs-3092	187	16	remark	remark	NOUN
iajs-3092	187	17	and	and	CCONJ
iajs-3092	187	18	examples	example	NOUN
iajs-3092	187	19	2.1.2	2.1.2	NUM
iajs-3092	187	20	]	]	PUNCT
iajs-3092	187	21	,	,	PUNCT
iajs-3092	187	22	then	then	ADV
iajs-3092	187	23	ℙ	ℙ	PROPN
iajs-3092	187	24	is	be	AUX
iajs-3092	187	25	not	not	PART
iajs-3092	187	26	fully	fully	ADV
iajs-3092	187	27	fuzzy	fuzzy	ADJ
iajs-3092	187	28	visible	visible	ADJ
iajs-3092	187	29	.	.	PUNCT
iajs-3092	188	1	3.22	3.22	NUM
iajs-3092	188	2	definition	definition	NOUN
iajs-3092	188	3	:	:	PUNCT
iajs-3092	188	4	let	let	VERB
iajs-3092	188	5	ℙ	ℙ	PRON
iajs-3092	188	6	∈	∈	NOUN
iajs-3092	188	7	fum(𝕄	fum(𝕄	PROPN
iajs-3092	188	8	)	)	PUNCT
iajs-3092	188	9	,	,	PUNCT
iajs-3092	188	10	𝔚	𝔚	PROPN
iajs-3092	188	11	∈fus(ℙ	∈fus(ℙ	NOUN
iajs-3092	188	12	)	)	PUNCT
iajs-3092	188	13	.	.	PUNCT
iajs-3092	189	1	we	we	PRON
iajs-3092	189	2	say	say	VERB
iajs-3092	189	3	that	that	SCONJ
iajs-3092	189	4	𝔚	𝔚	PROPN
iajs-3092	189	5	is	be	AUX
iajs-3092	189	6	stable	stable	ADJ
iajs-3092	189	7	if	if	SCONJ
iajs-3092	189	8	𝑓(𝔚	𝑓(𝔚	NUM
iajs-3092	189	9	)	)	PUNCT
iajs-3092	189	10	⊆	⊆	NUM
iajs-3092	189	11	𝔚	𝔚	NOUN
iajs-3092	189	12	for	for	ADP
iajs-3092	189	13	each	each	DET
iajs-3092	189	14	fuzzy	fuzzy	ADJ
iajs-3092	189	15	𝔽	𝔽	PROPN
iajs-3092	189	16	-homomorphism	-homomorphism	PROPN
iajs-3092	189	17	𝑓	𝑓	PRON
iajs-3092	189	18	:	:	PUNCT
iajs-3092	189	19	𝔚	𝔚	PROPN
iajs-3092	189	20	→	→	SYM
iajs-3092	189	21	ℙ.	ℙ.	PROPN
iajs-3092	189	22	3.23	3.23	NUM
iajs-3092	189	23	proposition	proposition	NOUN
iajs-3092	189	24	:	:	PUNCT
iajs-3092	189	25	let	let	VERB
iajs-3092	189	26	ℙ	ℙ	PRON
iajs-3092	189	27	∈	∈	NOUN
iajs-3092	189	28	fum(𝕄	fum(𝕄	NOUN
iajs-3092	189	29	)	)	PUNCT
iajs-3092	189	30	and	and	CCONJ
iajs-3092	189	31	𝔚	𝔚	PROPN
iajs-3092	189	32	∈	∈	PROPN
iajs-3092	189	33	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	189	34	)	)	PUNCT
iajs-3092	189	35	constant	constant	ADJ
iajs-3092	189	36	on	on	ADP
iajs-3092	189	37	𝑘𝑒𝑟	𝑘𝑒𝑟	PROPN
iajs-3092	189	38	𝑓.	𝑓.	NOUN
iajs-3092	189	39	then	then	ADV
iajs-3092	189	40	𝔚	𝔚	PROPN
iajs-3092	189	41	is	be	AUX
iajs-3092	189	42	fuzzy	fuzzy	ADJ
iajs-3092	189	43	stable	stable	ADJ
iajs-3092	189	44	if	if	SCONJ
iajs-3092	189	45	and	and	CCONJ
iajs-3092	189	46	only	only	ADV
iajs-3092	189	47	if	if	SCONJ
iajs-3092	189	48	𝔚∗	𝔚∗	PRON
iajs-3092	189	49	is	be	AUX
iajs-3092	189	50	stable	stable	ADJ
iajs-3092	189	51	and	and	CCONJ
iajs-3092	189	52	ℙ	ℙ	NOUN
iajs-3092	189	53	satisfies	satisfie	NOUN
iajs-3092	189	54	condition	condition	NOUN
iajs-3092	189	55	(	(	PUNCT
iajs-3092	189	56	∗	∗	NOUN
iajs-3092	189	57	)	)	PUNCT
iajs-3092	189	58	.	.	PUNCT
iajs-3092	190	1	proof	proof	NOUN
iajs-3092	190	2	:	:	PUNCT
iajs-3092	190	3	let	let	VERB
iajs-3092	190	4	𝔚	𝔚	PRON
iajs-3092	190	5	be	be	AUX
iajs-3092	190	6	a	a	DET
iajs-3092	190	7	fuzzy	fuzzy	ADJ
iajs-3092	190	8	stable	stable	NOUN
iajs-3092	190	9	.	.	PUNCT
iajs-3092	191	1	then	then	ADV
iajs-3092	191	2	,	,	PUNCT
iajs-3092	191	3	𝑓(𝔚	𝑓(𝔚	PROPN
iajs-3092	191	4	)	)	PUNCT
iajs-3092	191	5	⊆	⊆	NUM
iajs-3092	191	6	𝔚	𝔚	NOUN
iajs-3092	191	7	by	by	ADP
iajs-3092	191	8	definition	definition	NOUN
iajs-3092	191	9	(	(	PUNCT
iajs-3092	191	10	3.21	3.21	NUM
iajs-3092	191	11	)	)	PUNCT
iajs-3092	191	12	.	.	PUNCT
iajs-3092	192	1	since	since	SCONJ
iajs-3092	192	2	𝔚	𝔚	PROPN
iajs-3092	192	3	is	be	AUX
iajs-3092	192	4	constant	constant	ADJ
iajs-3092	192	5	on	on	ADP
iajs-3092	192	6	ker	ker	PROPN
iajs-3092	192	7	𝑓	𝑓	PRON
iajs-3092	192	8	,	,	PUNCT
iajs-3092	192	9	then	then	ADV
iajs-3092	192	10	𝑓(𝔚∗	𝑓(𝔚∗	PROPN
iajs-3092	192	11	)	)	PUNCT
iajs-3092	192	12	⊆	⊆	NUM
iajs-3092	192	13	𝔚∗	𝔚∗	NUM
iajs-3092	192	14	by	by	ADP
iajs-3092	192	15	[	[	X
iajs-3092	192	16	27	27	NUM
iajs-3092	192	17	,	,	PUNCT
iajs-3092	192	18	lemma	lemma	PROPN
iajs-3092	192	19	3.2.5	3.2.5	NUM
iajs-3092	192	20	]	]	PUNCT
iajs-3092	192	21	.	.	PUNCT
iajs-3092	193	1	therefore	therefore	ADV
iajs-3092	193	2	𝔚∗	𝔚∗	PROPN
iajs-3092	193	3	is	be	AUX
iajs-3092	193	4	stable	stable	ADJ
iajs-3092	193	5	.	.	PUNCT
iajs-3092	194	1	conversely	conversely	ADV
iajs-3092	194	2	,	,	PUNCT
iajs-3092	194	3	let	let	VERB
iajs-3092	194	4	𝔚∗	𝔚∗	PRON
iajs-3092	194	5	is	be	AUX
iajs-3092	194	6	stable	stable	ADJ
iajs-3092	194	7	.	.	PUNCT
iajs-3092	195	1	then	then	ADV
iajs-3092	195	2	𝑓(𝔚∗	𝑓(𝔚∗	PROPN
iajs-3092	195	3	)	)	PUNCT
iajs-3092	195	4	⊆	⊆	NUM
iajs-3092	195	5	𝔚∗	𝔚∗	NUM
iajs-3092	195	6	and	and	CCONJ
iajs-3092	195	7	hence	hence	ADV
iajs-3092	195	8	(	(	PUNCT
iajs-3092	195	9	𝑓(𝔚	𝑓(𝔚	PROPN
iajs-3092	195	10	)	)	PUNCT
iajs-3092	195	11	)	)	PUNCT
iajs-3092	196	1	∗	∗	VERB
iajs-3092	196	2	⊆	⊆	NUM
iajs-3092	196	3	𝔚∗	𝔚∗	NUM
iajs-3092	196	4	by	by	ADP
iajs-3092	196	5	[	[	X
iajs-3092	196	6	27	27	NUM
iajs-3092	196	7	,	,	PUNCT
iajs-3092	196	8	lemma	lemma	PROPN
iajs-3092	196	9	3.2.5	3.2.5	NUM
iajs-3092	196	10	]	]	PUNCT
iajs-3092	196	11	,	,	PUNCT
iajs-3092	196	12	since	since	SCONJ
iajs-3092	196	13	𝔚	𝔚	PROPN
iajs-3092	196	14	satisfies	satisfy	VERB
iajs-3092	196	15	condition	condition	NOUN
iajs-3092	196	16	(	(	PUNCT
iajs-3092	196	17	∗	∗	PROPN
iajs-3092	196	18	)	)	PUNCT
iajs-3092	196	19	,	,	PUNCT
iajs-3092	196	20	then	then	ADV
iajs-3092	196	21	𝑓(𝔚	𝑓(𝔚	NUM
iajs-3092	196	22	)	)	PUNCT
iajs-3092	196	23	⊆	⊆	NUM
iajs-3092	196	24	𝔚.	𝔚.	NOUN
iajs-3092	196	25	therefore	therefore	ADV
iajs-3092	196	26	𝔚	𝔚	PROPN
iajs-3092	196	27	is	be	AUX
iajs-3092	196	28	fuzzy	fuzzy	ADJ
iajs-3092	196	29	stable	stable	ADJ
iajs-3092	196	30	.	.	PUNCT
iajs-3092	197	1	3.24example	3.24example	NUM
iajs-3092	197	2	:	:	PUNCT
iajs-3092	197	3	let	let	VERB
iajs-3092	197	4	𝕄	𝕄	PROPN
iajs-3092	197	5	=	=	SYM
iajs-3092	197	6	𝔽	𝔽	PROPN
iajs-3092	197	7	and	and	CCONJ
iajs-3092	197	8	𝔽	𝔽	PROPN
iajs-3092	197	9	=	=	PUNCT
iajs-3092	197	10	ℝ.	ℝ.	AUX
iajs-3092	197	11	define	define	VERB
iajs-3092	197	12	ҡ	ҡ	X
iajs-3092	197	13	:	:	PUNCT
iajs-3092	197	14	𝔽	𝔽	PROPN
iajs-3092	197	15	→	→	SYM
iajs-3092	198	1	[	[	X
iajs-3092	198	2	0,1	0,1	NUM
iajs-3092	198	3	]	]	PUNCT
iajs-3092	198	4	by	by	ADP
iajs-3092	198	5	ҡ(𝑥	ҡ(𝑥	NOUN
iajs-3092	198	6	)	)	PUNCT
iajs-3092	198	7	=	=	SYM
iajs-3092	198	8	1	1	NUM
iajs-3092	198	9	∀𝑥	∀𝑥	NOUN
iajs-3092	198	10	∈	∈	PROPN
iajs-3092	198	11	𝔽	𝔽	PROPN
iajs-3092	198	12	.	.	PUNCT
iajs-3092	199	1	consider	consider	VERB
iajs-3092	199	2	𝔛	𝔛	NOUN
iajs-3092	199	3	:	:	PUNCT
iajs-3092	199	4	𝔽	𝔽	PROPN
iajs-3092	199	5	→	→	SYM
iajs-3092	199	6	[	[	X
iajs-3092	199	7	0,1	0,1	NUM
iajs-3092	199	8	]	]	PUNCT
iajs-3092	199	9	as	as	ADP
iajs-3092	199	10	𝔛(𝑥	𝔛(𝑥	X
iajs-3092	199	11	)	)	PUNCT
iajs-3092	199	12	=	=	PRON
iajs-3092	199	13	{	{	PUNCT
iajs-3092	200	1	1	1	NUM
iajs-3092	200	2	𝑖𝑓	𝑖𝑓	ADP
iajs-3092	200	3	𝑥	𝑥	PRON
iajs-3092	200	4	∈	∈	PROPN
iajs-3092	200	5	q	q	NOUN
iajs-3092	200	6	0	0	NUM
iajs-3092	200	7	𝑜.	𝑜.	NOUN
iajs-3092	200	8	𝑤	𝑤	X
iajs-3092	200	9	.	.	PUNCT
iajs-3092	201	1	clear	clear	ADJ
iajs-3092	201	2	that	that	SCONJ
iajs-3092	201	3	𝔛	𝔛	PROPN
iajs-3092	201	4	∈	∈	PROPN
iajs-3092	201	5	𝐹𝑈𝑆(ҡ	𝐹𝑈𝑆(ҡ	NUM
iajs-3092	201	6	)	)	PUNCT
iajs-3092	201	7	.	.	PUNCT
iajs-3092	202	1	ҡ∗	ҡ∗	NOUN
iajs-3092	202	2	=	=	SYM
iajs-3092	202	3	ℝ	ℝ	PROPN
iajs-3092	202	4	,	,	PUNCT
iajs-3092	202	5	𝔛∗	𝔛∗	NOUN
iajs-3092	202	6	=	=	SYM
iajs-3092	203	1	q	q	NOUN
iajs-3092	203	2	.	.	PUNCT
iajs-3092	204	1	we	we	PRON
iajs-3092	204	2	know	know	VERB
iajs-3092	204	3	that	that	SCONJ
iajs-3092	204	4	𝑄	𝑄	PRON
iajs-3092	204	5	is	be	AUX
iajs-3092	204	6	stable	stable	ADJ
iajs-3092	204	7	of	of	ADP
iajs-3092	204	8	𝔽	𝔽	PROPN
iajs-3092	204	9	,	,	PUNCT
iajs-3092	204	10	then	then	ADV
iajs-3092	204	11	𝔛	𝔛	PROPN
iajs-3092	204	12	is	be	AUX
iajs-3092	204	13	a	a	DET
iajs-3092	204	14	fuzzy	fuzzy	ADJ
iajs-3092	204	15	stable	stable	NOUN
iajs-3092	204	16	of	of	ADP
iajs-3092	204	17	ҡ	ҡ	PRON
iajs-3092	204	18	by	by	ADP
iajs-3092	204	19	proposition	proposition	NOUN
iajs-3092	204	20	(	(	PUNCT
iajs-3092	204	21	3.22	3.22	NUM
iajs-3092	204	22	)	)	PUNCT
iajs-3092	204	23	,	,	PUNCT
iajs-3092	204	24	where	where	SCONJ
iajs-3092	204	25	𝔛is	𝔛is	PROPN
iajs-3092	204	26	coustant	coustant	NOUN
iajs-3092	204	27	on	on	ADP
iajs-3092	204	28	𝑘𝑒𝑟	𝑘𝑒𝑟	PROPN
iajs-3092	204	29	𝑓.	𝑓.	NOUN
iajs-3092	204	30	3.25	3.25	NUM
iajs-3092	204	31	example	example	NOUN
iajs-3092	204	32	:	:	PUNCT
iajs-3092	204	33	let	let	VERB
iajs-3092	204	34	𝕄	𝕄	PROPN
iajs-3092	204	35	=	=	SYM
iajs-3092	204	36	𝑍	𝑍	PROPN
iajs-3092	204	37	and	and	CCONJ
iajs-3092	204	38	𝔽	𝔽	PROPN
iajs-3092	204	39	=	=	SYM
iajs-3092	204	40	𝑍.	𝑍.	PROPN
iajs-3092	204	41	define	define	VERB
iajs-3092	204	42	ℙ	ℙ	NOUN
iajs-3092	204	43	:	:	PUNCT
iajs-3092	204	44	𝑍	𝑍	PROPN
iajs-3092	204	45	→	→	SYM
iajs-3092	204	46	[	[	X
iajs-3092	204	47	0,1	0,1	NUM
iajs-3092	204	48	]	]	PUNCT
iajs-3092	204	49	by	by	ADP
iajs-3092	204	50	ℙ(𝑥	ℙ(𝑥	PRON
iajs-3092	204	51	)	)	PUNCT
iajs-3092	204	52	=	=	SYM
iajs-3092	205	1	1	1	NUM
iajs-3092	205	2	∀𝑥	∀𝑥	NOUN
iajs-3092	205	3	∈	∈	PROPN
iajs-3092	205	4	𝑍.	𝑍.	PROPN
iajs-3092	205	5	suppose	suppose	VERB
iajs-3092	205	6	that	that	SCONJ
iajs-3092	205	7	𝔚	𝔚	NOUN
iajs-3092	205	8	:	:	PUNCT
iajs-3092	205	9	𝑍	𝑍	PROPN
iajs-3092	205	10	→	→	SYM
iajs-3092	205	11	[	[	X
iajs-3092	205	12	0,1	0,1	NUM
iajs-3092	205	13	]	]	PUNCT
iajs-3092	205	14	defined	define	VERB
iajs-3092	205	15	as	as	ADP
iajs-3092	205	16	𝔚(𝑥	𝔚(𝑥	NUM
iajs-3092	205	17	)	)	PUNCT
iajs-3092	205	18	=	=	PRON
iajs-3092	205	19	{	{	PUNCT
iajs-3092	205	20	1	1	NUM
iajs-3092	205	21	𝑖𝑓	𝑖𝑓	ADP
iajs-3092	205	22	𝑥	𝑥	PRON
iajs-3092	205	23	∈	∈	PROPN
iajs-3092	205	24	2𝑍	2𝑍	NOUN
iajs-3092	205	25	0	0	NUM
iajs-3092	205	26	𝑜.	𝑜.	NOUN
iajs-3092	205	27	𝑤	𝑤	X
iajs-3092	205	28	.	.	PUNCT
iajs-3092	206	1	it	it	PRON
iajs-3092	206	2	is	be	AUX
iajs-3092	206	3	clear	clear	ADJ
iajs-3092	206	4	that	that	SCONJ
iajs-3092	206	5	𝔚	𝔚	PROPN
iajs-3092	206	6	∈	∈	PROPN
iajs-3092	206	7	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	206	8	)	)	PUNCT
iajs-3092	206	9	,	,	PUNCT
iajs-3092	206	10	ℙ∗	ℙ∗	PUNCT
iajs-3092	207	1	=	=	SYM
iajs-3092	207	2	𝑍	𝑍	PROPN
iajs-3092	207	3	and	and	CCONJ
iajs-3092	207	4	𝔚∗	𝔚∗	NUM
iajs-3092	208	1	=	=	NOUN
iajs-3092	208	2	2𝑍.	2𝑍.	NUM
iajs-3092	208	3	then	then	ADV
iajs-3092	208	4	𝔚∗	𝔚∗	PRON
iajs-3092	208	5	is	be	AUX
iajs-3092	208	6	not	not	PART
iajs-3092	208	7	stable	stable	ADJ
iajs-3092	208	8	by	by	ADP
iajs-3092	208	9	[	[	X
iajs-3092	208	10	28	28	NUM
iajs-3092	208	11	,	,	PUNCT
iajs-3092	208	12	example	example	NOUN
iajs-3092	208	13	and	and	CCONJ
iajs-3092	208	14	remarks	remark	VERB
iajs-3092	208	15	1.2(a	1.2(a	NUM
iajs-3092	208	16	)	)	PUNCT
iajs-3092	208	17	]	]	PUNCT
iajs-3092	208	18	,	,	PUNCT
iajs-3092	208	19	and	and	CCONJ
iajs-3092	208	20	,	,	PUNCT
iajs-3092	208	21	hence	hence	ADV
iajs-3092	208	22	,	,	PUNCT
iajs-3092	208	23	𝔚	𝔚	PROPN
iajs-3092	208	24	is	be	AUX
iajs-3092	208	25	not	not	PART
iajs-3092	208	26	fuzzy	fuzzy	ADJ
iajs-3092	208	27	stable	stable	ADJ
iajs-3092	208	28	by	by	ADP
iajs-3092	208	29	proposition	proposition	NOUN
iajs-3092	208	30	3.23	3.23	NUM
iajs-3092	208	31	.	.	PUNCT
iajs-3092	209	1	3.26	3.26	NUM
iajs-3092	209	2	definition	definition	NOUN
iajs-3092	209	3	:	:	PUNCT
iajs-3092	209	4	let	let	VERB
iajs-3092	209	5	ℙ	ℙ	PRON
iajs-3092	209	6	∈	∈	PROPN
iajs-3092	209	7	fum(m	fum(m	PROPN
iajs-3092	209	8	)	)	PUNCT
iajs-3092	209	9	.	.	PUNCT
iajs-3092	210	1	ℙ	ℙ	NOUN
iajs-3092	210	2	is	be	AUX
iajs-3092	210	3	termed	term	VERB
iajs-3092	210	4	to	to	PART
iajs-3092	210	5	be	be	AUX
iajs-3092	210	6	fuzzy	fuzzy	ADJ
iajs-3092	210	7	fully	fully	ADV
iajs-3092	210	8	stable	stable	ADJ
iajs-3092	210	9	if	if	SCONJ
iajs-3092	210	10	and	and	CCONJ
iajs-3092	210	11	only	only	ADV
iajs-3092	210	12	if	if	SCONJ
iajs-3092	210	13	every	every	DET
iajs-3092	210	14	ɮ	ɮ	PROPN
iajs-3092	210	15	∈	∈	NOUN
iajs-3092	210	16	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	210	17	)	)	PUNCT
iajs-3092	210	18	is	be	AUX
iajs-3092	210	19	fuzzy	fuzzy	ADJ
iajs-3092	210	20	stable	stable	ADJ
iajs-3092	210	21	.	.	PUNCT
iajs-3092	211	1	3.27	3.27	NUM
iajs-3092	211	2	proposition	proposition	NOUN
iajs-3092	211	3	:	:	PUNCT
iajs-3092	211	4	let	let	VERB
iajs-3092	211	5	ℙ	ℙ	PRON
iajs-3092	211	6	be	be	AUX
iajs-3092	211	7	a	a	DET
iajs-3092	211	8	fuzzy	fuzzy	ADJ
iajs-3092	211	9	𝔽	𝔽	PROPN
iajs-3092	211	10	–	–	PUNCT
iajs-3092	211	11	module	module	NOUN
iajs-3092	211	12	,	,	PUNCT
iajs-3092	211	13	such	such	ADJ
iajs-3092	211	14	that	that	SCONJ
iajs-3092	211	15	every	every	DET
iajs-3092	211	16	fuzzy	fuzzy	ADJ
iajs-3092	211	17	submodule	submodule	NOUN
iajs-3092	211	18	of	of	ADP
iajs-3092	211	19	ℙ	ℙ	PROPN
iajs-3092	211	20	constant	constant	ADJ
iajs-3092	211	21	on	on	ADP
iajs-3092	211	22	𝑘𝑒𝑟	𝑘𝑒𝑟	PROPN
iajs-3092	211	23	𝑓	𝑓	NOUN
iajs-3092	211	24	and	and	CCONJ
iajs-3092	211	25	satisfies	satisfy	VERB
iajs-3092	211	26	condition	condition	NOUN
iajs-3092	211	27	(	(	PUNCT
iajs-3092	211	28	*	*	NOUN
iajs-3092	211	29	)	)	PUNCT
iajs-3092	211	30	.	.	PUNCT
iajs-3092	212	1	then	then	ADV
iajs-3092	212	2	ℙ	ℙ	PROPN
iajs-3092	212	3	is	be	AUX
iajs-3092	212	4	a	a	DET
iajs-3092	212	5	fuzzy	fuzzy	ADJ
iajs-3092	212	6	fully	fully	ADV
iajs-3092	212	7	stable	stable	ADJ
iajs-3092	212	8	if	if	SCONJ
iajs-3092	212	9	and	and	CCONJ
iajs-3092	212	10	only	only	ADV
iajs-3092	212	11	if	if	SCONJ
iajs-3092	212	12	ℙ∗	ℙ∗	ADJ
iajs-3092	212	13	is	be	AUX
iajs-3092	212	14	fully	fully	ADV
iajs-3092	212	15	stable	stable	ADJ
iajs-3092	212	16	.	.	PUNCT
iajs-3092	213	1	proof	proof	NOUN
iajs-3092	213	2	:	:	PUNCT
iajs-3092	213	3	in	in	ADP
iajs-3092	213	4	a	a	DET
iajs-3092	213	5	similar	similar	ADJ
iajs-3092	213	6	way	way	NOUN
iajs-3092	213	7	of	of	ADP
iajs-3092	213	8	proposition	proposition	NOUN
iajs-3092	213	9	(	(	PUNCT
iajs-3092	213	10	3.23	3.23	NUM
iajs-3092	213	11	)	)	PUNCT
iajs-3092	213	12	.	.	PUNCT
iajs-3092	214	1	3.28	3.28	NUM
iajs-3092	214	2	proposition	proposition	NOUN
iajs-3092	214	3	:	:	PUNCT
iajs-3092	214	4	let	let	VERB
iajs-3092	214	5	ℙ	ℙ	PRON
iajs-3092	214	6	be	be	AUX
iajs-3092	214	7	a	a	DET
iajs-3092	214	8	fully	fully	ADV
iajs-3092	214	9	fuzzy	fuzzy	ADJ
iajs-3092	214	10	visible	visible	ADJ
iajs-3092	214	11	𝔽	𝔽	PROPN
iajs-3092	214	12	-module	-module	NOUN
iajs-3092	214	13	and	and	CCONJ
iajs-3092	214	14	endℝ(ℙ∗	endℝ(ℙ∗	NOUN
iajs-3092	214	15	)	)	PUNCT
iajs-3092	214	16	is	be	AUX
iajs-3092	214	17	commutative	commutative	ADJ
iajs-3092	214	18	.	.	PUNCT
iajs-3092	215	1	then	then	ADV
iajs-3092	215	2	ℙ	ℙ	PROPN
iajs-3092	215	3	is	be	AUX
iajs-3092	215	4	fuzzy	fuzzy	ADJ
iajs-3092	215	5	fully	fully	ADV
iajs-3092	215	6	stable	stable	ADJ
iajs-3092	215	7	such	such	ADJ
iajs-3092	215	8	that	that	SCONJ
iajs-3092	215	9	∀	∀	NOUN
iajs-3092	215	10	𝔚	𝔚	NOUN
iajs-3092	215	11	⊆	⊆	NUM
iajs-3092	215	12	ℙ	ℙ	NOUN
iajs-3092	215	13	constant	constant	ADJ
iajs-3092	215	14	on	on	ADP
iajs-3092	215	15	𝑘𝑒𝑟	𝑘𝑒𝑟	PROPN
iajs-3092	215	16	𝑓	𝑓	NOUN
iajs-3092	215	17	and	and	CCONJ
iajs-3092	215	18	satisfies	satisfy	VERB
iajs-3092	215	19	condition	condition	NOUN
iajs-3092	215	20	(	(	PUNCT
iajs-3092	215	21	*	*	NOUN
iajs-3092	215	22	)	)	PUNCT
iajs-3092	215	23	.	.	PUNCT
iajs-3092	216	1	proof	proof	NOUN
iajs-3092	216	2	:	:	PUNCT
iajs-3092	216	3	ℙ∗	ℙ∗	X
iajs-3092	216	4	is	be	AUX
iajs-3092	216	5	a	a	DET
iajs-3092	216	6	fully	fully	ADV
iajs-3092	216	7	visible	visible	ADJ
iajs-3092	216	8	by[3	by[3	NOUN
iajs-3092	216	9	,	,	PUNCT
iajs-3092	216	10	proposition	proposition	NOUN
iajs-3092	216	11	3.2	3.2	NUM
iajs-3092	216	12	]	]	PUNCT
iajs-3092	216	13	,	,	PUNCT
iajs-3092	216	14	then	then	ADV
iajs-3092	216	15	ℙ∗	ℙ∗	X
iajs-3092	216	16	is	be	AUX
iajs-3092	216	17	fully	fully	ADV
iajs-3092	216	18	stable	stable	ADJ
iajs-3092	216	19	by	by	ADP
iajs-3092	216	20	[	[	X
iajs-3092	216	21	24	24	NUM
iajs-3092	216	22	,	,	PUNCT
iajs-3092	216	23	proposition	proposition	NOUN
iajs-3092	216	24	(	(	PUNCT
iajs-3092	216	25	2.6.22	2.6.22	NUM
iajs-3092	216	26	)	)	PUNCT
iajs-3092	216	27	]	]	PUNCT
iajs-3092	216	28	,	,	PUNCT
iajs-3092	216	29	and	and	CCONJ
iajs-3092	216	30	hence	hence	ADV
iajs-3092	216	31	each	each	DET
iajs-3092	216	32	submodule	submodule	NOUN
iajs-3092	216	33	of	of	ADP
iajs-3092	216	34	ℙ∗	ℙ∗	PROPN
iajs-3092	216	35	will	will	AUX
iajs-3092	216	36	be	be	AUX
iajs-3092	216	37	stable	stable	ADJ
iajs-3092	216	38	,	,	PUNCT
iajs-3092	216	39	then	then	ADV
iajs-3092	216	40	∀	∀	NUM
iajs-3092	216	41	𝔚	𝔚	NOUN
iajs-3092	216	42	∈	∈	PROPN
iajs-3092	216	43	𝐹𝑆(ℙ	𝐹𝑆(ℙ	PROPN
iajs-3092	216	44	)	)	PUNCT
iajs-3092	216	45	,	,	PUNCT
iajs-3092	216	46	𝔚∗	𝔚∗	PRON
iajs-3092	216	47	will	will	AUX
iajs-3092	216	48	be	be	AUX
iajs-3092	216	49	stable	stable	ADJ
iajs-3092	216	50	,	,	PUNCT
iajs-3092	216	51	then	then	ADV
iajs-3092	216	52	by	by	ADP
iajs-3092	216	53	proposition	proposition	NOUN
iajs-3092	216	54	3.22	3.22	NUM
iajs-3092	216	55	,	,	PUNCT
iajs-3092	216	56	𝔚	𝔚	PROPN
iajs-3092	216	57	is	be	AUX
iajs-3092	216	58	a	a	DET
iajs-3092	216	59	stable	stable	ADJ
iajs-3092	216	60	fuzzy	fuzzy	ADJ
iajs-3092	216	61	submodule	submodule	NOUN
iajs-3092	216	62	.	.	PUNCT
iajs-3092	217	1	therefore	therefore	ADV
iajs-3092	217	2	,	,	PUNCT
iajs-3092	217	3	ℙ	ℙ	PROPN
iajs-3092	217	4	is	be	AUX
iajs-3092	217	5	fuzzy	fuzzy	ADJ
iajs-3092	217	6	fully	fully	ADV
iajs-3092	217	7	stable	stable	ADJ
iajs-3092	217	8	.	.	PUNCT
iajs-3092	218	1	ihjpas	ihjpas	PROPN
iajs-3092	218	2	.	.	PUNCT
iajs-3092	219	1	36	36	NUM
iajs-3092	219	2	(	(	PUNCT
iajs-3092	219	3	3	3	NUM
iajs-3092	219	4	)	)	PUNCT
iajs-3092	219	5	2023	2023	NUM
iajs-3092	219	6	378	378	NUM
iajs-3092	219	7	3.29	3.29	NUM
iajs-3092	219	8	proposition	proposition	NOUN
iajs-3092	219	9	:	:	PUNCT
iajs-3092	219	10	let	let	VERB
iajs-3092	219	11	ℙ	ℙ	PRON
iajs-3092	219	12	be	be	AUX
iajs-3092	219	13	a	a	DET
iajs-3092	219	14	fully	fully	ADV
iajs-3092	219	15	fuzzy	fuzzy	ADJ
iajs-3092	219	16	visible	visible	ADJ
iajs-3092	219	17	module	module	NOUN
iajs-3092	219	18	over	over	ADP
iajs-3092	219	19	𝔽.	𝔽.	PROPN
iajs-3092	219	20	then	then	ADV
iajs-3092	219	21	each	each	DET
iajs-3092	219	22	jℙ	jℙ	NOUN
iajs-3092	219	23	∈	∈	PROPN
iajs-3092	219	24	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	219	25	)	)	PUNCT
iajs-3092	219	26	is	be	AUX
iajs-3092	219	27	an	an	DET
iajs-3092	219	28	essential	essential	ADJ
iajs-3092	219	29	submodule	submodule	NOUN
iajs-3092	219	30	of	of	ADP
iajs-3092	219	31	ℙ	ℙ	PROPN
iajs-3092	219	32	for	for	ADP
iajs-3092	219	33	each	each	DET
iajs-3092	219	34	non	non	ADJ
iajs-3092	219	35	-	-	ADJ
iajs-3092	219	36	emptyj	emptyj	ADJ
iajs-3092	219	37	∈	∈	PROPN
iajs-3092	219	38	𝐹𝑈𝐼(𝔽	𝐹𝑈𝐼(𝔽	NOUN
iajs-3092	219	39	)	)	PUNCT
iajs-3092	219	40	.	.	PUNCT
iajs-3092	220	1	proof	proof	NOUN
iajs-3092	220	2	:	:	PUNCT
iajs-3092	220	3	let	let	VERB
iajs-3092	220	4	ξ	ξ	X
iajs-3092	220	5	be	be	AUX
iajs-3092	220	6	a	a	DET
iajs-3092	220	7	non	non	ADJ
iajs-3092	220	8	-	-	ADJ
iajs-3092	220	9	trivial	trivial	ADJ
iajs-3092	220	10	fuzzy	fuzzy	ADJ
iajs-3092	220	11	submodule	submodule	NOUN
iajs-3092	220	12	of	of	ADP
iajs-3092	220	13	ℙ	ℙ	PRON
iajs-3092	220	14	such	such	ADJ
iajs-3092	220	15	that	that	SCONJ
iajs-3092	220	16	jℙ	jℙ	NOUN
iajs-3092	220	17	∩	∩	PROPN
iajs-3092	220	18	𝔛	𝔛	PROPN
iajs-3092	220	19	=	=	NOUN
iajs-3092	220	20	01	01	PROPN
iajs-3092	220	21	.	.	PUNCT
iajs-3092	221	1	but	but	CCONJ
iajs-3092	221	2	𝔛	𝔛	PROPN
iajs-3092	221	3	is	be	AUX
iajs-3092	221	4	visible	visible	ADJ
iajs-3092	221	5	and	and	CCONJ
iajs-3092	221	6	hence	hence	ADV
iajs-3092	221	7	ξ	ξ	PROPN
iajs-3092	221	8	is	be	AUX
iajs-3092	221	9	pure	pure	ADJ
iajs-3092	221	10	.	.	PUNCT
iajs-3092	222	1	then	then	ADV
iajs-3092	222	2	jξ	jξ	ADP
iajs-3092	222	3	=	=	SYM
iajs-3092	222	4	jℙ	jℙ	PROPN
iajs-3092	222	5	∩	∩	X
iajs-3092	222	6	ξ	ξ	X
iajs-3092	222	7	=	=	SYM
iajs-3092	222	8	01	01	NUM
iajs-3092	222	9	,	,	PUNCT
iajs-3092	222	10	so	so	ADV
iajs-3092	222	11	jξ	jξ	PROPN
iajs-3092	222	12	=	=	SYM
iajs-3092	222	13	01	01	NUM
iajs-3092	222	14	and	and	CCONJ
iajs-3092	222	15	hence	hence	ADV
iajs-3092	222	16	ξ	ξ	X
iajs-3092	222	17	=	=	SYM
iajs-3092	222	18	01	01	PROPN
iajs-3092	222	19	.	.	PUNCT
iajs-3092	223	1	therefore	therefore	ADV
iajs-3092	223	2	jℙ	jℙ	PROPN
iajs-3092	223	3	is	be	AUX
iajs-3092	223	4	a	a	DET
iajs-3092	223	5	fuzzy	fuzzy	ADJ
iajs-3092	223	6	essential	essential	ADJ
iajs-3092	223	7	.	.	PUNCT
iajs-3092	224	1	3.30	3.30	NUM
iajs-3092	224	2	proposition	proposition	NOUN
iajs-3092	224	3	:	:	PUNCT
iajs-3092	224	4	let	let	VERB
iajs-3092	224	5	ℙ	ℙ	PRON
iajs-3092	224	6	be	be	AUX
iajs-3092	224	7	a	a	DET
iajs-3092	224	8	fully	fully	ADV
iajs-3092	224	9	fuzzy	fuzzy	ADJ
iajs-3092	224	10	visible	visible	ADJ
iajs-3092	224	11	multiplication	multiplication	NOUN
iajs-3092	224	12	module	module	NOUN
iajs-3092	224	13	,	,	PUNCT
iajs-3092	224	14	then	then	ADV
iajs-3092	224	15	every	every	DET
iajs-3092	224	16	non	non	ADJ
iajs-3092	224	17	-	-	ADJ
iajs-3092	224	18	empty	empty	ADJ
iajs-3092	224	19	fuzzy	fuzzy	ADJ
iajs-3092	224	20	submodule	submodule	NOUN
iajs-3092	224	21	of	of	ADP
iajs-3092	224	22	ℙ	ℙ	PROPN
iajs-3092	224	23	is	be	AUX
iajs-3092	224	24	an	an	DET
iajs-3092	224	25	essential	essential	ADJ
iajs-3092	224	26	of	of	ADP
iajs-3092	224	27	ℙ.	ℙ.	NOUN
iajs-3092	224	28	proof	proof	NOUN
iajs-3092	224	29	:	:	PUNCT
iajs-3092	224	30	because	because	SCONJ
iajs-3092	224	31	ℙ	ℙ	PROPN
iajs-3092	224	32	is	be	AUX
iajs-3092	224	33	fully	fully	ADV
iajs-3092	224	34	fuzzy	fuzzy	ADJ
iajs-3092	224	35	visible	visible	ADJ
iajs-3092	224	36	,	,	PUNCT
iajs-3092	224	37	then	then	ADV
iajs-3092	224	38	every	every	DET
iajs-3092	224	39	fuzzy	fuzzy	ADJ
iajs-3092	224	40	submodule	submodule	PROPN
iajs-3092	224	41	jℙ	jℙ	PROPN
iajs-3092	224	42	is	be	AUX
iajs-3092	224	43	an	an	DET
iajs-3092	224	44	essential	essential	ADJ
iajs-3092	224	45	for	for	ADP
iajs-3092	224	46	each	each	DET
iajs-3092	224	47	non	non	ADJ
iajs-3092	224	48	-	-	ADJ
iajs-3092	224	49	empty	empty	ADJ
iajs-3092	224	50	j	j	PROPN
iajs-3092	224	51	∈	∈	PROPN
iajs-3092	224	52	𝐹𝑈𝐼(𝔽	𝐹𝑈𝐼(𝔽	NOUN
iajs-3092	224	53	)	)	PUNCT
iajs-3092	224	54	by	by	ADP
iajs-3092	224	55	proposition	proposition	NOUN
iajs-3092	224	56	(	(	PUNCT
iajs-3092	224	57	3.29	3.29	NUM
iajs-3092	224	58	)	)	PUNCT
iajs-3092	224	59	.	.	PUNCT
iajs-3092	225	1	but	but	CCONJ
iajs-3092	225	2	ℙ	ℙ	PROPN
iajs-3092	225	3	is	be	AUX
iajs-3092	225	4	fuzzy	fuzzy	ADJ
iajs-3092	225	5	multiplication	multiplication	NOUN
iajs-3092	225	6	,	,	PUNCT
iajs-3092	225	7	then	then	ADV
iajs-3092	225	8	every	every	DET
iajs-3092	225	9	non	non	ADJ
iajs-3092	225	10	–	–	PUNCT
iajs-3092	225	11	empty	empty	ADJ
iajs-3092	225	12	fuzzy	fuzzy	ADJ
iajs-3092	225	13	submodule	submodule	NOUN
iajs-3092	225	14	of	of	ADP
iajs-3092	225	15	ℙ	ℙ	PROPN
iajs-3092	225	16	is	be	AUX
iajs-3092	225	17	essential	essential	ADJ
iajs-3092	225	18	.	.	PUNCT
iajs-3092	226	1	3.31	3.31	NUM
iajs-3092	226	2	proposition	proposition	NOUN
iajs-3092	226	3	:	:	PUNCT
iajs-3092	226	4	let	let	VERB
iajs-3092	226	5	ℙ	ℙ	PRON
iajs-3092	226	6	be	be	AUX
iajs-3092	226	7	a	a	DET
iajs-3092	226	8	fully	fully	ADV
iajs-3092	226	9	fuzzy	fuzzy	ADJ
iajs-3092	226	10	visible	visible	ADJ
iajs-3092	226	11	module	module	NOUN
iajs-3092	226	12	over	over	ADP
iajs-3092	226	13	a	a	DET
iajs-3092	226	14	domain	domain	NOUN
iajs-3092	226	15	𝔽	𝔽	PROPN
iajs-3092	226	16	and	and	CCONJ
iajs-3092	226	17	endr(ℙt	endr(ℙt	PROPN
iajs-3092	226	18	)	)	PUNCT
iajs-3092	226	19	is	be	AUX
iajs-3092	226	20	commutative	commutative	ADJ
iajs-3092	226	21	ring	ring	NOUN
iajs-3092	226	22	.	.	PUNCT
iajs-3092	227	1	then	then	ADV
iajs-3092	227	2	ℙ	ℙ	PROPN
iajs-3092	227	3	is	be	AUX
iajs-3092	227	4	fuzzy	fuzzy	ADJ
iajs-3092	227	5	divisible	divisible	ADJ
iajs-3092	227	6	.	.	PUNCT
iajs-3092	228	1	proof	proof	NOUN
iajs-3092	228	2	:	:	PUNCT
iajs-3092	228	3	since	since	SCONJ
iajs-3092	228	4	ℙt	ℙt	PROPN
iajs-3092	228	5	is	be	AUX
iajs-3092	228	6	fully	fully	ADV
iajs-3092	228	7	visible	visible	ADJ
iajs-3092	228	8	,	,	PUNCT
iajs-3092	228	9	then	then	ADV
iajs-3092	228	10	ℙt	ℙt	PROPN
iajs-3092	228	11	is	be	AUX
iajs-3092	228	12	divisible	divisible	ADJ
iajs-3092	228	13	by	by	ADP
iajs-3092	228	14	[	[	PUNCT
iajs-3092	228	15	24	24	NUM
iajs-3092	228	16	,	,	PUNCT
iajs-3092	228	17	proposition	proposition	NOUN
iajs-3092	228	18	2.6.42(2	2.6.42(2	NUM
iajs-3092	228	19	)	)	PUNCT
iajs-3092	228	20	]	]	PUNCT
iajs-3092	228	21	.	.	PUNCT
iajs-3092	229	1	then	then	ADV
iajs-3092	229	2	ℙ	ℙ	PROPN
iajs-3092	229	3	is	be	AUX
iajs-3092	229	4	divisible	divisible	ADJ
iajs-3092	229	5	by	by	ADP
iajs-3092	229	6	[	[	X
iajs-3092	229	7	17	17	NUM
iajs-3092	229	8	]	]	PUNCT
iajs-3092	229	9	.	.	PUNCT
iajs-3092	230	1	let	let	VERB
iajs-3092	230	2	𝔓	𝔓	PROPN
iajs-3092	230	3	∈	∈	NOUN
iajs-3092	230	4	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	NOUN
iajs-3092	230	5	)	)	PUNCT
iajs-3092	230	6	and	and	CCONJ
iajs-3092	230	7	𝔊	𝔊	PROPN
iajs-3092	230	8	∈	∈	PROPN
iajs-3092	230	9	𝐹𝑈𝑀(𝑁	𝐹𝑈𝑀(𝑁	NOUN
iajs-3092	230	10	)	)	PUNCT
iajs-3092	230	11	.	.	PUNCT
iajs-3092	231	1	if	if	SCONJ
iajs-3092	231	2	ᵮ	ᵮ	DET
iajs-3092	231	3	∶	∶	NOUN
iajs-3092	231	4	𝕄	𝕄	PROPN
iajs-3092	231	5	→	→	PUNCT
iajs-3092	231	6	𝑁	𝑁	PROPN
iajs-3092	231	7	is	be	AUX
iajs-3092	231	8	a	a	DET
iajs-3092	231	9	𝔽	𝔽	PROPN
iajs-3092	231	10	–	–	PUNCT
iajs-3092	231	11	module	module	NOUN
iajs-3092	231	12	homomorphism	homomorphism	NOUN
iajs-3092	231	13	and	and	CCONJ
iajs-3092	231	14	𝔊(ᵮ(𝑥	𝔊(ᵮ(𝑥	NOUN
iajs-3092	231	15	)	)	PUNCT
iajs-3092	231	16	)	)	PUNCT
iajs-3092	231	17	≥	≥	NOUN
iajs-3092	231	18	𝔓(𝑥	𝔓(𝑥	NUM
iajs-3092	231	19	)	)	PUNCT
iajs-3092	231	20	,	,	PUNCT
iajs-3092	231	21	∀	∀	PUNCT
iajs-3092	231	22	𝑥	𝑥	PRON
iajs-3092	231	23	∈	∈	PROPN
iajs-3092	231	24	𝕄	𝕄	PROPN
iajs-3092	231	25	,	,	PUNCT
iajs-3092	231	26	then	then	ADV
iajs-3092	231	27	ᵮ	ᵮ	NOUN
iajs-3092	231	28	is	be	AUX
iajs-3092	231	29	called	call	VERB
iajs-3092	231	30	a	a	DET
iajs-3092	231	31	fuzzy	fuzzy	ADJ
iajs-3092	231	32	homomorphism	homomorphism	NOUN
iajs-3092	231	33	and	and	CCONJ
iajs-3092	231	34	denoted	denote	VERB
iajs-3092	231	35	by	by	ADP
iajs-3092	231	36	ᵮ	ᵮ	NOUN
iajs-3092	231	37	:	:	PUNCT
iajs-3092	231	38	𝔓	𝔓	PROPN
iajs-3092	231	39	→	→	SYM
iajs-3092	231	40	𝔊	𝔊	PROPN
iajs-3092	231	41	[	[	PUNCT
iajs-3092	231	42	29].this	29].this	PROPN
iajs-3092	231	43	definition	definition	NOUN
iajs-3092	231	44	can	can	AUX
iajs-3092	231	45	be	be	AUX
iajs-3092	231	46	used	use	VERB
iajs-3092	231	47	to	to	PART
iajs-3092	231	48	introduce	introduce	VERB
iajs-3092	231	49	the	the	DET
iajs-3092	231	50	following	following	ADJ
iajs-3092	231	51	concept	concept	NOUN
iajs-3092	231	52	.	.	PUNCT
iajs-3092	232	1	3.32	3.32	NUM
iajs-3092	232	2	definition	definition	NOUN
iajs-3092	232	3	:	:	PUNCT
iajs-3092	232	4	let	let	VERB
iajs-3092	232	5	ℙ	ℙ	PRON
iajs-3092	232	6	∈	∈	NOUN
iajs-3092	232	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	VERB
iajs-3092	232	8	)	)	PUNCT
iajs-3092	232	9	.	.	PUNCT
iajs-3092	233	1	ℙ	ℙ	NOUN
iajs-3092	233	2	is	be	AUX
iajs-3092	233	3	termed	term	VERB
iajs-3092	233	4	to	to	PART
iajs-3092	233	5	be	be	AUX
iajs-3092	233	6	fuzzy	fuzzy	ADJ
iajs-3092	233	7	quasi	quasi	ADJ
iajs-3092	233	8	injective	injective	ADJ
iajs-3092	233	9	if	if	SCONJ
iajs-3092	233	10	for	for	ADP
iajs-3092	233	11	each	each	DET
iajs-3092	233	12	submodule	submodule	NOUN
iajs-3092	233	13	n	n	PROPN
iajs-3092	233	14	of	of	ADP
iajs-3092	233	15	𝕄	𝕄	PROPN
iajs-3092	233	16	and	and	CCONJ
iajs-3092	233	17	each	each	DET
iajs-3092	233	18	fuzzy	fuzzy	ADJ
iajs-3092	233	19	homomorphism	homomorphism	NOUN
iajs-3092	233	20	ᵮ	ᵮ	X
iajs-3092	233	21	∶	∶	NOUN
iajs-3092	233	22	𝔚	𝔚	X
iajs-3092	233	23	→	→	SYM
iajs-3092	233	24	ℙ	ℙ	PROPN
iajs-3092	233	25	,	,	PUNCT
iajs-3092	233	26	where	where	SCONJ
iajs-3092	233	27	𝔚	𝔚	PROPN
iajs-3092	233	28	∈	∈	PROPN
iajs-3092	233	29	𝐹𝑈𝑀(𝑁	𝐹𝑈𝑀(𝑁	PART
iajs-3092	233	30	)	)	PUNCT
iajs-3092	233	31	can	can	AUX
iajs-3092	233	32	be	be	AUX
iajs-3092	233	33	extended	extend	VERB
iajs-3092	233	34	to	to	ADP
iajs-3092	233	35	fuzzy	fuzzy	ADJ
iajs-3092	233	36	endomorphism	endomorphism	X
iajs-3092	233	37	ᶃ	ᶃ	NUM
iajs-3092	233	38	such	such	ADJ
iajs-3092	233	39	that	that	DET
iajs-3092	233	40	ᵮ	ᵮ	NOUN
iajs-3092	233	41	=	=	X
iajs-3092	233	42	ᶃ	ᶃ	NUM
iajs-3092	233	43	∘	∘	X
iajs-3092	233	44	i	i	PRON
iajs-3092	233	45	,	,	PUNCT
iajs-3092	233	46	where	where	SCONJ
iajs-3092	233	47	i	i	PRON
iajs-3092	233	48	:	:	PUNCT
iajs-3092	233	49	𝔚	𝔚	PROPN
iajs-3092	233	50	→	→	SYM
iajs-3092	233	51	𝕄	𝕄	PROPN
iajs-3092	233	52	inclusion	inclusion	NOUN
iajs-3092	233	53	homomorphis	homomorphis	ADP
iajs-3092	233	54	.	.	PUNCT
iajs-3092	234	1	3.33	3.33	NUM
iajs-3092	234	2	proposition	proposition	NOUN
iajs-3092	234	3	:	:	PUNCT
iajs-3092	234	4	let	let	VERB
iajs-3092	234	5	𝕍	𝕍	PROPN
iajs-3092	234	6	∈	∈	PROPN
iajs-3092	234	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	VERB
iajs-3092	234	8	)	)	PUNCT
iajs-3092	234	9	.	.	PUNCT
iajs-3092	235	1	ℙ	ℙ	NOUN
iajs-3092	235	2	is	be	AUX
iajs-3092	235	3	fuzzy	fuzzy	ADJ
iajs-3092	235	4	quasi	quasi	ADJ
iajs-3092	235	5	injective	injective	PROPN
iajs-3092	235	6	iff	iff	PROPN
iajs-3092	235	7	𝕄	𝕄	PROPN
iajs-3092	235	8	is	be	AUX
iajs-3092	235	9	quasi	quasi	NOUN
iajs-3092	235	10	injective	injective	ADJ
iajs-3092	235	11	and	and	CCONJ
iajs-3092	235	12	𝕍(ℯ	𝕍(ℯ	NUM
iajs-3092	235	13	)	)	PUNCT
iajs-3092	235	14	=	=	SYM
iajs-3092	235	15	1	1	NUM
iajs-3092	235	16	∀ℯ	∀ℯ	NOUN
iajs-3092	235	17	∈	∈	NOUN
iajs-3092	235	18	𝕄.	𝕄.	NOUN
iajs-3092	235	19	proof	proof	NOUN
iajs-3092	235	20	:	:	PUNCT
iajs-3092	235	21	let	let	VERB
iajs-3092	235	22	𝕍	𝕍	PRON
iajs-3092	235	23	be	be	AUX
iajs-3092	235	24	a	a	DET
iajs-3092	235	25	fuzzy	fuzzy	ADJ
iajs-3092	235	26	quasi	quasi	ADJ
iajs-3092	235	27	injective	injective	ADJ
iajs-3092	235	28	𝔽	𝔽	PROPN
iajs-3092	235	29	-module	-module	PROPN
iajs-3092	235	30	and	and	CCONJ
iajs-3092	235	31	n	n	PRON
iajs-3092	235	32	be	be	VERB
iajs-3092	235	33	a	a	DET
iajs-3092	235	34	submodule	submodule	NOUN
iajs-3092	235	35	of	of	ADP
iajs-3092	235	36	𝕄.	𝕄.	PROPN
iajs-3092	235	37	define	define	VERB
iajs-3092	235	38	𝔚	𝔚	NOUN
iajs-3092	235	39	:	:	PUNCT
iajs-3092	235	40	n	n	PROPN
iajs-3092	235	41	→	→	SYM
iajs-3092	235	42	[	[	X
iajs-3092	235	43	0,1	0,1	NUM
iajs-3092	235	44	]	]	PUNCT
iajs-3092	235	45	by	by	ADP
iajs-3092	235	46	𝔚(𝓆	𝔚(𝓆	X
iajs-3092	235	47	)	)	PUNCT
iajs-3092	235	48	=	=	PRON
iajs-3092	235	49	{	{	PUNCT
iajs-3092	235	50	1	1	NUM
iajs-3092	235	51	if	if	SCONJ
iajs-3092	235	52	𝓆	𝓆	X
iajs-3092	235	53	∈	∈	PROPN
iajs-3092	235	54	n	n	NOUN
iajs-3092	235	55	0	0	NUM
iajs-3092	236	1	o.	o.	PROPN
iajs-3092	236	2	w	w	PROPN
iajs-3092	236	3	,	,	PUNCT
iajs-3092	236	4	then	then	ADV
iajs-3092	236	5	∀	∀	X
iajs-3092	236	6	ᵮ	ᵮ	NOUN
iajs-3092	236	7	:	:	PUNCT
iajs-3092	236	8	𝔚	𝔚	NOUN
iajs-3092	236	9	→	→	SYM
iajs-3092	236	10	𝕍	𝕍	PROPN
iajs-3092	236	11	be	be	AUX
iajs-3092	236	12	a	a	DET
iajs-3092	236	13	fuzzy	fuzzy	ADJ
iajs-3092	236	14	homomorphism	homomorphism	NOUN
iajs-3092	236	15	∃	∃	PROPN
iajs-3092	236	16	ᶃ	ᶃ	NUM
iajs-3092	236	17	:	:	PUNCT
iajs-3092	236	18	𝕍	𝕍	NOUN
iajs-3092	236	19	→	→	SYM
iajs-3092	236	20	𝕍	𝕍	PROPN
iajs-3092	236	21	s.t	s.t	PROPN
iajs-3092	236	22	.	.	PROPN
iajs-3092	236	23	ᵮ	ᵮ	PROPN
iajs-3092	237	1	=	=	NOUN
iajs-3092	237	2	ᶃ	ᶃ	NUM
iajs-3092	237	3	∘	∘	X
iajs-3092	237	4	i	i	PRON
iajs-3092	237	5	and	and	CCONJ
iajs-3092	237	6	hence	hence	ADV
iajs-3092	237	7	∀	∀	X
iajs-3092	237	8	submodule	submodule	PROPN
iajs-3092	237	9	n	n	PROPN
iajs-3092	237	10	of	of	ADP
iajs-3092	237	11	𝕄	𝕄	PROPN
iajs-3092	237	12	and	and	CCONJ
iajs-3092	237	13	ᵮ	ᵮ	NOUN
iajs-3092	237	14	:	:	PUNCT
iajs-3092	237	15	n	n	PROPN
iajs-3092	237	16	→	→	SYM
iajs-3092	237	17	𝕄	𝕄	PROPN
iajs-3092	237	18	,	,	PUNCT
iajs-3092	237	19	there	there	PRON
iajs-3092	237	20	exist	exist	VERB
iajs-3092	237	21	g	g	NOUN
iajs-3092	237	22	:	:	PUNCT
iajs-3092	237	23	𝕄	𝕄	PROPN
iajs-3092	237	24	→	→	SYM
iajs-3092	237	25	𝕄	𝕄	PROPN
iajs-3092	237	26	such	such	ADJ
iajs-3092	237	27	that	that	DET
iajs-3092	237	28	ᵮ	ᵮ	NOUN
iajs-3092	237	29	=	=	X
iajs-3092	237	30	ᶃ	ᶃ	NUM
iajs-3092	237	31	∘	∘	NUM
iajs-3092	237	32	i.now	i.now	PROPN
iajs-3092	237	33	,	,	PUNCT
iajs-3092	237	34	if	if	SCONJ
iajs-3092	237	35	𝕍(𝓆	𝕍(𝓆	NUM
iajs-3092	237	36	)	)	PUNCT
iajs-3092	237	37	≠	≠	PROPN
iajs-3092	237	38	1	1	NUM
iajs-3092	237	39	,	,	PUNCT
iajs-3092	237	40	since	since	SCONJ
iajs-3092	237	41	𝕍(ᶃ(𝓆	𝕍(ᶃ(𝓆	NUM
iajs-3092	237	42	)	)	PUNCT
iajs-3092	237	43	)	)	PUNCT
iajs-3092	237	44	≥	≥	X
iajs-3092	237	45	𝕍(𝓆	𝕍(𝓆	NUM
iajs-3092	237	46	)	)	PUNCT
iajs-3092	237	47	by	by	ADP
iajs-3092	237	48	definition	definition	NOUN
iajs-3092	237	49	of	of	ADP
iajs-3092	237	50	fuzzy	fuzzy	ADJ
iajs-3092	237	51	homomorphism	homomorphism	NOUN
iajs-3092	237	52	,	,	PUNCT
iajs-3092	237	53	then	then	ADV
iajs-3092	237	54	𝕍(ᶃ(𝓆	𝕍(ᶃ(𝓆	NOUN
iajs-3092	237	55	)	)	PUNCT
iajs-3092	237	56	)	)	PUNCT
iajs-3092	237	57	≥	≥	X
iajs-3092	237	58	𝕍(𝓆	𝕍(𝓆	NUM
iajs-3092	237	59	)	)	PUNCT
iajs-3092	237	60	=	=	SYM
iajs-3092	237	61	𝕍(i(𝓆	𝕍(i(𝓆	NUM
iajs-3092	237	62	)	)	PUNCT
iajs-3092	237	63	)	)	PUNCT
iajs-3092	237	64	≥	≥	NOUN
iajs-3092	237	65	𝔚(𝓆	𝔚(𝓆	NUM
iajs-3092	237	66	)	)	PUNCT
iajs-3092	237	67	=	=	SYM
iajs-3092	237	68	1	1	NUM
iajs-3092	237	69	,	,	PUNCT
iajs-3092	237	70	hence	hence	ADV
iajs-3092	237	71	𝕍(𝓆	𝕍(𝓆	NUM
iajs-3092	237	72	)	)	PUNCT
iajs-3092	237	73	=	=	SYM
iajs-3092	237	74	1	1	NUM
iajs-3092	237	75	∀𝓆	∀𝓆	SYM
iajs-3092	237	76	∈	∈	PROPN
iajs-3092	237	77	𝕄.	𝕄.	PROPN
iajs-3092	237	78	conversely	conversely	ADV
iajs-3092	237	79	,	,	PUNCT
iajs-3092	237	80	let	let	VERB
iajs-3092	237	81	𝕄	𝕄	PROPN
iajs-3092	237	82	be	be	AUX
iajs-3092	237	83	a	a	DET
iajs-3092	237	84	quasi	quasi	ADJ
iajs-3092	237	85	-	-	ADJ
iajs-3092	237	86	injective	injective	ADJ
iajs-3092	237	87	,	,	PUNCT
iajs-3092	237	88	n	n	X
iajs-3092	237	89	be	be	VERB
iajs-3092	237	90	a	a	DET
iajs-3092	237	91	submodule	submodule	NOUN
iajs-3092	237	92	of	of	ADP
iajs-3092	237	93	𝕄	𝕄	PROPN
iajs-3092	237	94	and	and	CCONJ
iajs-3092	237	95	𝕍(𝓆	𝕍(𝓆	NUM
iajs-3092	237	96	)	)	PUNCT
iajs-3092	237	97	=	=	SYM
iajs-3092	237	98	1	1	NUM
iajs-3092	237	99	∀𝓆	∀𝓆	NOUN
iajs-3092	237	100	∈	∈	PROPN
iajs-3092	237	101	𝕄	𝕄	PROPN
iajs-3092	237	102	,	,	PUNCT
iajs-3092	237	103	let	let	VERB
iajs-3092	237	104	𝔚	𝔚	PROPN
iajs-3092	237	105	∈	∈	PROPN
iajs-3092	237	106	𝐹𝑈𝑀(𝑁	𝐹𝑈𝑀(𝑁	PRON
iajs-3092	237	107	)	)	PUNCT
iajs-3092	237	108	.	.	PUNCT
iajs-3092	238	1	then	then	ADV
iajs-3092	238	2	∀	∀	PUNCT
iajs-3092	238	3	ᵮ	ᵮ	NOUN
iajs-3092	238	4	:	:	PUNCT
iajs-3092	238	5	n	n	PROPN
iajs-3092	238	6	→	→	SYM
iajs-3092	238	7	m	m	PROPN
iajs-3092	238	8	∃	∃	PROPN
iajs-3092	238	9	ᶃ	ᶃ	NUM
iajs-3092	238	10	:	:	PUNCT
iajs-3092	238	11	𝕄	𝕄	PROPN
iajs-3092	238	12	→	→	SYM
iajs-3092	238	13	𝕄s.t	𝕄s.t	PROPN
iajs-3092	238	14	.	.	PUNCT
iajs-3092	238	15	ᵮ	ᵮ	NOUN
iajs-3092	239	1	=	=	NOUN
iajs-3092	239	2	ᶃ	ᶃ	NUM
iajs-3092	239	3	∘	∘	X
iajs-3092	239	4	i	i	PRON
iajs-3092	239	5	,	,	PUNCT
iajs-3092	239	6	i	i	PRON
iajs-3092	239	7	:	:	PUNCT
iajs-3092	239	8	n	n	PROPN
iajs-3092	239	9	→	→	SYM
iajs-3092	239	10	𝕄	𝕄	PROPN
iajs-3092	239	11	inclusion	inclusion	NOUN
iajs-3092	239	12	homomorphism	homomorphism	NOUN
iajs-3092	239	13	,	,	PUNCT
iajs-3092	239	14	hence	hence	ADV
iajs-3092	239	15	∀	∀	NOUN
iajs-3092	239	16	ᵮ	ᵮ	NOUN
iajs-3092	239	17	∶	∶	NOUN
iajs-3092	239	18	𝔚	𝔚	NOUN
iajs-3092	239	19	→	→	PUNCT
iajs-3092	239	20	𝕍	𝕍	PROPN
iajs-3092	239	21	∃	∃	PROPN
iajs-3092	239	22	ᶃ	ᶃ	NUM
iajs-3092	239	23	:	:	PUNCT
iajs-3092	239	24	𝕍	𝕍	NOUN
iajs-3092	239	25	→	→	SYM
iajs-3092	239	26	𝕍	𝕍	PROPN
iajs-3092	239	27	s.t	s.t	PROPN
iajs-3092	239	28	.	.	PROPN
iajs-3092	239	29	ᵮ	ᵮ	PROPN
iajs-3092	240	1	=	=	NOUN
iajs-3092	240	2	ᶃ	ᶃ	NUM
iajs-3092	240	3	∘	∘	X
iajs-3092	240	4	i	i	PRON
iajs-3092	240	5	,	,	PUNCT
iajs-3092	241	1	where	where	SCONJ
iajs-3092	241	2	(	(	PUNCT
iajs-3092	241	3	ᶃ	ᶃ	NOUN
iajs-3092	241	4	∘	∘	X
iajs-3092	241	5	i)(𝓆	i)(𝓆	NOUN
iajs-3092	241	6	)	)	PUNCT
iajs-3092	241	7	=	=	PUNCT
iajs-3092	242	1	𝕍(ᶃ	𝕍(ᶃ	SYM
iajs-3092	242	2	∘	∘	X
iajs-3092	242	3	i)(𝓆	i)(𝓆	NOUN
iajs-3092	242	4	)	)	PUNCT
iajs-3092	242	5	=	=	SYM
iajs-3092	242	6	𝕍(ᶃ(𝓆	𝕍(ᶃ(𝓆	NUM
iajs-3092	242	7	)	)	PUNCT
iajs-3092	242	8	)	)	PUNCT
iajs-3092	243	1	=	=	PUNCT
iajs-3092	244	1	𝕍(𝓆	𝕍(𝓆	X
iajs-3092	244	2	)	)	PUNCT
iajs-3092	244	3	=	=	SYM
iajs-3092	244	4	1	1	NUM
iajs-3092	244	5	≥	≥	NOUN
iajs-3092	244	6	𝕍(𝓆	𝕍(𝓆	NUM
iajs-3092	244	7	)	)	PUNCT
iajs-3092	244	8	=	=	SYM
iajs-3092	244	9	1	1	NUM
iajs-3092	244	10	≥	≥	NOUN
iajs-3092	244	11	𝔚(𝓆	𝔚(𝓆	NUM
iajs-3092	244	12	)	)	PUNCT
iajs-3092	244	13	.	.	PUNCT
iajs-3092	245	1	3.34	3.34	NUM
iajs-3092	245	2	definition	definition	NOUN
iajs-3092	245	3	:	:	PUNCT
iajs-3092	245	4	a	a	DET
iajs-3092	245	5	fuzzy	fuzzy	ADJ
iajs-3092	245	6	ring	ring	NOUN
iajs-3092	245	7	𝔽	𝔽	PROPN
iajs-3092	245	8	is	be	AUX
iajs-3092	245	9	self	self	NOUN
iajs-3092	245	10	injective	injective	ADJ
iajs-3092	245	11	if	if	SCONJ
iajs-3092	245	12	it	it	PRON
iajs-3092	245	13	is	be	AUX
iajs-3092	245	14	fuzzy	fuzzy	ADJ
iajs-3092	245	15	injective	injective	ADJ
iajs-3092	245	16	𝔽	𝔽	PROPN
iajs-3092	245	17	-module	-module	PROPN
iajs-3092	245	18	.	.	PUNCT
iajs-3092	246	1	3.35	3.35	NUM
iajs-3092	246	2	corollary	corollary	NOUN
iajs-3092	246	3	:	:	PUNCT
iajs-3092	246	4	suppose	suppose	VERB
iajs-3092	246	5	that	that	SCONJ
iajs-3092	246	6	ℙ	ℙ	PROPN
iajs-3092	246	7	is	be	AUX
iajs-3092	246	8	a	a	DET
iajs-3092	246	9	fuzzy	fuzzy	ADJ
iajs-3092	246	10	fully	fully	ADV
iajs-3092	246	11	visible	visible	ADJ
iajs-3092	246	12	𝔽	𝔽	PROPN
iajs-3092	246	13	module	module	NOUN
iajs-3092	246	14	𝕄	𝕄	PROPN
iajs-3092	246	15	and	and	CCONJ
iajs-3092	246	16	endr(p∗	endr(p∗	NOUN
iajs-3092	246	17	)	)	PUNCT
iajs-3092	246	18	is	be	AUX
iajs-3092	246	19	commutative	commutative	ADJ
iajs-3092	246	20	,	,	PUNCT
iajs-3092	246	21	where	where	SCONJ
iajs-3092	246	22	ℙ	ℙ	NOUN
iajs-3092	246	23	satisfies	satisfy	VERB
iajs-3092	246	24	condition	condition	NOUN
iajs-3092	246	25	(	(	PUNCT
iajs-3092	246	26	∗	∗	NOUN
iajs-3092	246	27	)	)	PUNCT
iajs-3092	246	28	.	.	PUNCT
iajs-3092	247	1	if	if	SCONJ
iajs-3092	247	2	ℙ	ℙ	PROPN
iajs-3092	247	3	is	be	AUX
iajs-3092	247	4	a	a	DET
iajs-3092	247	5	fuzzy	fuzzy	ADJ
iajs-3092	247	6	quasi	quasi	ADJ
iajs-3092	247	7	injective	injective	ADJ
iajs-3092	247	8	module	module	NOUN
iajs-3092	247	9	and	and	CCONJ
iajs-3092	247	10	ℙ	ℙ	PROPN
iajs-3092	247	11	(	(	PUNCT
iajs-3092	247	12	x)=1	x)=1	PROPN
iajs-3092	247	13	∀	∀	X
iajs-3092	247	14	x	x	SYM
iajs-3092	247	15	∈	∈	PROPN
iajs-3092	247	16	𝕄	𝕄	PROPN
iajs-3092	247	17	,	,	PUNCT
iajs-3092	247	18	then	then	ADV
iajs-3092	247	19	1	1	NUM
iajs-3092	247	20	.	.	PUNCT
iajs-3092	248	1	every	every	DET
iajs-3092	248	2	fuzzy	fuzzy	ADJ
iajs-3092	248	3	submodule	submodule	NOUN
iajs-3092	248	4	𝔚	𝔚	PROPN
iajs-3092	248	5	of	of	ADP
iajs-3092	248	6	ℙ	ℙ	PROPN
iajs-3092	248	7	constant	constant	ADJ
iajs-3092	248	8	on	on	ADP
iajs-3092	248	9	ker	ker	PROPN
iajs-3092	248	10	ᵮ	ᵮ	NOUN
iajs-3092	248	11	is	be	AUX
iajs-3092	248	12	fuzzy	fuzzy	ADJ
iajs-3092	248	13	duo	duo	NOUN
iajs-3092	248	14	module	module	NOUN
iajs-3092	248	15	.	.	PUNCT
iajs-3092	249	1	2	2	X
iajs-3092	249	2	.	.	X
iajs-3092	249	3	every	every	DET
iajs-3092	249	4	homomorphism	homomorphism	PROPN
iajs-3092	249	5	image	image	NOUN
iajs-3092	249	6	constant	constant	ADJ
iajs-3092	249	7	on	on	ADP
iajs-3092	249	8	ker	ker	PROPN
iajs-3092	249	9	ᵮ	ᵮ	NOUN
iajs-3092	249	10	is	be	AUX
iajs-3092	249	11	duo	duo	NOUN
iajs-3092	249	12	module	module	NOUN
iajs-3092	249	13	.	.	PUNCT
iajs-3092	250	1	3	3	X
iajs-3092	250	2	.	.	X
iajs-3092	250	3	a	a	DET
iajs-3092	250	4	fuzzy	fuzzy	ADJ
iajs-3092	250	5	ring	ring	NOUN
iajs-3092	250	6	γ	γ	X
iajs-3092	250	7	on	on	ADP
iajs-3092	250	8	end𝔽(ℙ∗	end𝔽(ℙ∗	NOUN
iajs-3092	250	9	)	)	PUNCT
iajs-3092	250	10	is	be	AUX
iajs-3092	250	11	self	self	NOUN
iajs-3092	250	12	injective	injective	ADJ
iajs-3092	250	13	.	.	PUNCT
iajs-3092	251	1	proof	proof	NOUN
iajs-3092	251	2	:	:	PUNCT
iajs-3092	251	3	1	1	X
iajs-3092	251	4	.	.	X
iajs-3092	251	5	since	since	SCONJ
iajs-3092	251	6	ℙ	ℙ	PROPN
iajs-3092	251	7	is	be	AUX
iajs-3092	251	8	fully	fully	ADV
iajs-3092	251	9	fuzzy	fuzzy	ADJ
iajs-3092	251	10	visible	visible	ADJ
iajs-3092	251	11	and	and	CCONJ
iajs-3092	251	12	endr(ℙ∗	endr(ℙ∗	NOUN
iajs-3092	251	13	)	)	PUNCT
iajs-3092	251	14	is	be	AUX
iajs-3092	251	15	commutative	commutative	ADJ
iajs-3092	251	16	,	,	PUNCT
iajs-3092	251	17	then	then	ADV
iajs-3092	251	18	ℙ∗	ℙ∗	X
iajs-3092	251	19	is	be	AUX
iajs-3092	251	20	fully	fully	ADV
iajs-3092	251	21	visible	visible	ADJ
iajs-3092	251	22	.	.	PUNCT
iajs-3092	252	1	but	but	CCONJ
iajs-3092	252	2	ℙ	ℙ	PROPN
iajs-3092	252	3	is	be	AUX
iajs-3092	252	4	fuzzy	fuzzy	ADJ
iajs-3092	252	5	quasi	quasi	ADJ
iajs-3092	252	6	injective	injective	NOUN
iajs-3092	252	7	,	,	PUNCT
iajs-3092	252	8	then	then	ADV
iajs-3092	252	9	𝕄	𝕄	PROPN
iajs-3092	252	10	is	be	AUX
iajs-3092	252	11	a	a	DET
iajs-3092	252	12	quasi	quasi	NOUN
iajs-3092	252	13	injective	injective	ADJ
iajs-3092	252	14	and	and	CCONJ
iajs-3092	252	15	ℙ(x	ℙ(x	NUM
iajs-3092	252	16	)	)	PUNCT
iajs-3092	252	17	=	=	SYM
iajs-3092	252	18	1	1	NUM
iajs-3092	252	19	∀x	∀x	NUM
iajs-3092	252	20	∈	∈	PROPN
iajs-3092	252	21	𝕄.	𝕄.	PROPN
iajs-3092	252	22	therefore	therefore	ADV
iajs-3092	252	23	ihjpas	ihjpa	VERB
iajs-3092	252	24	.	.	PUNCT
iajs-3092	253	1	36	36	NUM
iajs-3092	253	2	(	(	PUNCT
iajs-3092	253	3	3	3	NUM
iajs-3092	253	4	)	)	PUNCT
iajs-3092	253	5	2023	2023	NUM
iajs-3092	253	6	379	379	NUM
iajs-3092	253	7	ℙ∗is	ℙ∗is	ADJ
iajs-3092	253	8	duo	duo	NOUN
iajs-3092	253	9	by	by	ADP
iajs-3092	253	10	[	[	X
iajs-3092	253	11	5	5	NUM
iajs-3092	253	12	,	,	PUNCT
iajs-3092	253	13	proposition	proposition	NOUN
iajs-3092	253	14	3.8	3.8	NUM
iajs-3092	253	15	]	]	PUNCT
iajs-3092	253	16	.	.	PUNCT
iajs-3092	254	1	let	let	VERB
iajs-3092	254	2	𝔚	𝔚	PRON
iajs-3092	254	3	∈	∈	PROPN
iajs-3092	254	4	𝐹𝑈𝑆(ℙ	𝐹𝑈𝑆(ℙ	NOUN
iajs-3092	254	5	)	)	PUNCT
iajs-3092	254	6	,	,	PUNCT
iajs-3092	254	7	then	then	ADV
iajs-3092	254	8	𝔚∗	𝔚∗	NUM
iajs-3092	254	9	submodule	submodule	NOUN
iajs-3092	254	10	of	of	ADP
iajs-3092	254	11	ℙ∗	ℙ∗	PROPN
iajs-3092	254	12	,	,	PUNCT
iajs-3092	254	13	then	then	ADV
iajs-3092	254	14	𝔚∗	𝔚∗	PROPN
iajs-3092	254	15	is	be	AUX
iajs-3092	254	16	duo	duo	NOUN
iajs-3092	254	17	which	which	PRON
iajs-3092	254	18	implies	imply	VERB
iajs-3092	254	19	that	that	SCONJ
iajs-3092	254	20	𝔚	𝔚	PROPN
iajs-3092	254	21	is	be	AUX
iajs-3092	254	22	duo	duo	ADJ
iajs-3092	254	23	by	by	ADP
iajs-3092	254	24	[	[	X
iajs-3092	254	25	5	5	NUM
iajs-3092	254	26	,	,	PUNCT
iajs-3092	254	27	proposition	proposition	NOUN
iajs-3092	254	28	3.8	3.8	NUM
iajs-3092	254	29	]	]	PUNCT
iajs-3092	254	30	.	.	PUNCT
iajs-3092	255	1	2	2	X
iajs-3092	255	2	.	.	X
iajs-3092	255	3	since	since	SCONJ
iajs-3092	255	4	every	every	DET
iajs-3092	255	5	fuzzy	fuzzy	ADJ
iajs-3092	255	6	homomorphic	homomorphic	ADJ
iajs-3092	255	7	image	image	NOUN
iajs-3092	255	8	is	be	AUX
iajs-3092	255	9	a	a	DET
iajs-3092	255	10	fuzzy	fuzzy	ADJ
iajs-3092	255	11	submodule	submodule	NOUN
iajs-3092	255	12	of	of	ADP
iajs-3092	255	13	ℙ	ℙ	PROPN
iajs-3092	255	14	,	,	PUNCT
iajs-3092	255	15	then	then	ADV
iajs-3092	255	16	by	by	ADP
iajs-3092	255	17	(	(	PUNCT
iajs-3092	255	18	1	1	NUM
iajs-3092	255	19	)	)	PUNCT
iajs-3092	255	20	,	,	PUNCT
iajs-3092	255	21	we	we	PRON
iajs-3092	255	22	get	get	VERB
iajs-3092	255	23	the	the	DET
iajs-3092	255	24	result	result	NOUN
iajs-3092	255	25	.	.	PUNCT
iajs-3092	256	1	3	3	X
iajs-3092	256	2	.	.	X
iajs-3092	256	3	since	since	SCONJ
iajs-3092	256	4	end𝔽(ℙ∗	end𝔽(ℙ∗	NOUN
iajs-3092	256	5	)	)	PUNCT
iajs-3092	256	6	is	be	AUX
iajs-3092	256	7	end𝔽(ℙ∗	end𝔽(ℙ∗	NOUN
iajs-3092	256	8	)	)	PUNCT
iajs-3092	256	9	−injective	−injective	VERB
iajs-3092	256	10	by	by	ADP
iajs-3092	256	11	[	[	X
iajs-3092	256	12	24	24	NUM
iajs-3092	256	13	,	,	PUNCT
iajs-3092	256	14	proposition	proposition	NOUN
iajs-3092	256	15	2.6.25	2.6.25	NUM
iajs-3092	256	16	]	]	PUNCT
iajs-3092	256	17	,	,	PUNCT
iajs-3092	256	18	then	then	ADV
iajs-3092	256	19	γend𝔽	γend𝔽	VERB
iajs-3092	256	20	(	(	PUNCT
iajs-3092	256	21	ℙ∗	ℙ∗	X
iajs-3092	256	22	)	)	PUNCT
iajs-3092	256	23	is	be	AUX
iajs-3092	256	24	a	a	DET
iajs-3092	256	25	fuzzy	fuzzy	ADJ
iajs-3092	256	26	self	self	NOUN
iajs-3092	256	27	injective	injective	ADJ
iajs-3092	256	28	[	[	X
iajs-3092	256	29	29	29	NUM
iajs-3092	256	30	,	,	PUNCT
iajs-3092	256	31	proposition	proposition	NOUN
iajs-3092	256	32	4.3(i	4.3(i	NUM
iajs-3092	256	33	)	)	PUNCT
iajs-3092	256	34	]	]	PUNCT
iajs-3092	256	35	.	.	PUNCT
iajs-3092	257	1	3.36	3.36	NUM
iajs-3092	257	2	corollary	corollary	NOUN
iajs-3092	257	3	:	:	PUNCT
iajs-3092	257	4	let	let	VERB
iajs-3092	257	5	ℙ	ℙ	PRON
iajs-3092	257	6	be	be	AUX
iajs-3092	257	7	a	a	DET
iajs-3092	257	8	finitely	finitely	ADV
iajs-3092	257	9	generated	generate	VERB
iajs-3092	257	10	fully	fully	ADV
iajs-3092	257	11	visible	visible	ADJ
iajs-3092	257	12	module	module	NOUN
iajs-3092	257	13	over	over	ADP
iajs-3092	257	14	dedkined	dedkine	VERB
iajs-3092	257	15	domain	domain	NOUN
iajs-3092	257	16	,	,	PUNCT
iajs-3092	257	17	then	then	ADV
iajs-3092	257	18	ℙ	ℙ	PROPN
iajs-3092	257	19	is	be	AUX
iajs-3092	257	20	fuzzy	fuzzy	ADJ
iajs-3092	257	21	duo	duo	NOUN
iajs-3092	257	22	multiplication	multiplication	NOUN
iajs-3092	257	23	module	module	NOUN
iajs-3092	257	24	,	,	PUNCT
iajs-3092	257	25	where	where	SCONJ
iajs-3092	257	26	ℙ	ℙ	NOUN
iajs-3092	257	27	satisfies	satisfy	VERB
iajs-3092	257	28	condition	condition	NOUN
iajs-3092	257	29	(	(	PUNCT
iajs-3092	257	30	∗	∗	NOUN
iajs-3092	257	31	)	)	PUNCT
iajs-3092	257	32	and	and	CCONJ
iajs-3092	257	33	every	every	DET
iajs-3092	257	34	fuzzy	fuzzy	ADJ
iajs-3092	257	35	submodule	submodule	NOUN
iajs-3092	257	36	of	of	ADP
iajs-3092	257	37	ℙ	ℙ	PROPN
iajs-3092	257	38	constant	constant	ADJ
iajs-3092	257	39	on	on	ADP
iajs-3092	257	40	ker	ker	PROPN
iajs-3092	257	41	f.	f.	PROPN
iajs-3092	257	42	proof	proof	PROPN
iajs-3092	257	43	:	:	PUNCT
iajs-3092	257	44	ℙ∗	ℙ∗	X
iajs-3092	257	45	is	be	AUX
iajs-3092	257	46	finitely	finitely	ADV
iajs-3092	257	47	generated	generate	VERB
iajs-3092	257	48	module	module	NOUN
iajs-3092	257	49	,	,	PUNCT
iajs-3092	257	50	then	then	ADV
iajs-3092	257	51	ℙ∗	ℙ∗	X
iajs-3092	257	52	is	be	AUX
iajs-3092	257	53	duo	duo	NOUN
iajs-3092	257	54	multiplication	multiplication	NOUN
iajs-3092	257	55	module	module	NOUN
iajs-3092	257	56	by	by	ADP
iajs-3092	257	57	[	[	X
iajs-3092	257	58	24	24	NUM
iajs-3092	257	59	,	,	PUNCT
iajs-3092	257	60	corollary	corollary	ADJ
iajs-3092	257	61	2.6.26	2.6.26	NUM
iajs-3092	257	62	]	]	PUNCT
iajs-3092	257	63	,	,	PUNCT
iajs-3092	257	64	but	but	CCONJ
iajs-3092	257	65	ℙ∗	ℙ∗	X
iajs-3092	257	66	is	be	AUX
iajs-3092	257	67	fully	fully	ADV
iajs-3092	257	68	visible	visible	ADJ
iajs-3092	257	69	,	,	PUNCT
iajs-3092	257	70	then	then	ADV
iajs-3092	257	71	ℙ	ℙ	PROPN
iajs-3092	257	72	is	be	AUX
iajs-3092	257	73	duo	duo	ADJ
iajs-3092	257	74	multiplication	multiplication	NOUN
iajs-3092	257	75	by	by	ADP
iajs-3092	257	76	[	[	X
iajs-3092	257	77	25	25	NUM
iajs-3092	257	78	]	]	PUNCT
iajs-3092	257	79	and	and	CCONJ
iajs-3092	257	80	[	[	PUNCT
iajs-3092	257	81	5,proposition	5,proposition	NUM
iajs-3092	257	82	3.8	3.8	NUM
iajs-3092	257	83	]	]	PUNCT
iajs-3092	257	84	.	.	PUNCT
iajs-3092	258	1	3.37	3.37	NUM
iajs-3092	258	2	proposition	proposition	NOUN
iajs-3092	258	3	:	:	PUNCT
iajs-3092	258	4	if	if	SCONJ
iajs-3092	258	5	ℙ	ℙ	PROPN
iajs-3092	258	6	∈	∈	NOUN
iajs-3092	258	7	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	VERB
iajs-3092	258	8	)	)	PUNCT
iajs-3092	258	9	over	over	ADP
iajs-3092	258	10	a	a	DET
iajs-3092	258	11	fuzzy	fuzzy	ADJ
iajs-3092	258	12	principle	principle	ADJ
iajs-3092	258	13	ideal	ideal	NOUN
iajs-3092	258	14	ring	ring	NOUN
iajs-3092	258	15	and	and	CCONJ
iajs-3092	258	16	𝔚	𝔚	NOUN
iajs-3092	258	17	∈	∈	PROPN
iajs-3092	258	18	𝐹𝑈𝑀(ℙ	𝐹𝑈𝑀(ℙ	PROPN
iajs-3092	258	19	)	)	PUNCT
iajs-3092	258	20	be	be	VERB
iajs-3092	258	21	a	a	DET
iajs-3092	258	22	such	such	ADJ
iajs-3092	258	23	that	that	SCONJ
iajs-3092	258	24	∀𝑥𝑡	∀𝑥𝑡	PROPN
iajs-3092	258	25	⊆	⊆	NUM
iajs-3092	258	26	𝔚	𝔚	PROPN
iajs-3092	258	27	and	and	CCONJ
iajs-3092	258	28	a	a	DET
iajs-3092	258	29	fuzzy	fuzzy	ADJ
iajs-3092	258	30	singleton	singleton	NOUN
iajs-3092	258	31	𝑟ℓ	𝑟ℓ	NOUN
iajs-3092	258	32	of	of	ADP
iajs-3092	258	33	𝔽	𝔽	PROPN
iajs-3092	258	34	,	,	PUNCT
iajs-3092	258	35	𝑥𝑡	𝑥𝑡	ADV
iajs-3092	258	36	=	=	PUNCT
iajs-3092	258	37	𝑟ℓ𝑠𝑘𝑥𝑡	𝑟ℓ𝑠𝑘𝑥𝑡	PROPN
iajs-3092	258	38	for	for	ADP
iajs-3092	258	39	some	some	DET
iajs-3092	258	40	fuzzy	fuzzy	ADJ
iajs-3092	258	41	singleton	singleton	PROPN
iajs-3092	258	42	𝑠𝑘	𝑠𝑘	NOUN
iajs-3092	258	43	of	of	ADP
iajs-3092	258	44	ℝ	ℝ	PROPN
iajs-3092	258	45	,	,	PUNCT
iajs-3092	258	46	then	then	ADV
iajs-3092	258	47	ℙ	ℙ	PROPN
iajs-3092	258	48	is	be	AUX
iajs-3092	258	49	a	a	DET
iajs-3092	258	50	fully	fully	ADV
iajs-3092	258	51	visible	visible	ADJ
iajs-3092	258	52	fuzzy	fuzzy	ADJ
iajs-3092	258	53	module	module	NOUN
iajs-3092	258	54	.	.	PUNCT
iajs-3092	259	1	proof	proof	NOUN
iajs-3092	259	2	:	:	PUNCT
iajs-3092	259	3	let	let	VERB
iajs-3092	259	4	𝑥𝑡	𝑥𝑡	ADP
iajs-3092	259	5	⊆	⊆	NUM
iajs-3092	259	6	𝔚	𝔚	PROPN
iajs-3092	259	7	and	and	CCONJ
iajs-3092	259	8	fuzzy	fuzzy	ADJ
iajs-3092	259	9	singleton	singleton	PROPN
iajs-3092	259	10	𝑟ℓ	𝑟ℓ	PROPN
iajs-3092	259	11	of	of	ADP
iajs-3092	259	12	𝔽.	𝔽.	PROPN
iajs-3092	259	13	then	then	ADV
iajs-3092	259	14	𝑥𝑡	𝑥𝑡	ADV
iajs-3092	259	15	⊆	⊆	X
iajs-3092	259	16	<	<	X
iajs-3092	259	17	𝑥𝑡	𝑥𝑡	X
iajs-3092	259	18	>	>	X
iajs-3092	259	19	⊆	⊆	NUM
iajs-3092	259	20	𝔚	𝔚	NOUN
iajs-3092	259	21	,	,	PUNCT
iajs-3092	259	22	then	then	ADV
iajs-3092	259	23	𝑥𝑡	𝑥𝑡	ADP
iajs-3092	259	24	=	=	VERB
iajs-3092	259	25	𝑠𝑘𝑥𝑡	𝑠𝑘𝑥𝑡	NOUN
iajs-3092	259	26	for	for	ADP
iajs-3092	259	27	some	some	DET
iajs-3092	259	28	𝑠𝑘	𝑠𝑘	NOUN
iajs-3092	259	29	⊆	⊆	NUM
iajs-3092	259	30	𝑅.	𝑅.	NOUN
iajs-3092	259	31	then	then	ADV
iajs-3092	259	32	for	for	ADP
iajs-3092	259	33	each	each	DET
iajs-3092	259	34	𝑟ℓ	𝑟ℓ	PROPN
iajs-3092	259	35	⊆	⊆	NUM
iajs-3092	259	36	𝔽	𝔽	PROPN
iajs-3092	259	37	,	,	PUNCT
iajs-3092	259	38	we	we	PRON
iajs-3092	259	39	get	get	VERB
iajs-3092	259	40	𝑠𝑘𝑥𝑡	𝑠𝑘𝑥𝑡	NOUN
iajs-3092	259	41	=	=	NOUN
iajs-3092	259	42	𝑟ℓ𝑠𝑘	𝑟ℓ𝑠𝑘	NOUN
iajs-3092	260	1	=	=	PUNCT
iajs-3092	260	2	𝑟ℓ𝑠𝑘𝑥𝑡	𝑟ℓ𝑠𝑘𝑥𝑡	NOUN
iajs-3092	260	3	⊆	⊆	NUM
iajs-3092	260	4	𝑟ℓ𝔚.	𝑟ℓ𝔚.	PROPN
iajs-3092	260	5	therefore	therefore	ADV
iajs-3092	260	6	,	,	PUNCT
iajs-3092	260	7	𝑥𝑡	𝑥𝑡	ADV
iajs-3092	260	8	⊆	⊆	NUM
iajs-3092	260	9	𝑟ℓ𝔚	𝑟ℓ𝔚	NOUN
iajs-3092	260	10	,	,	PUNCT
iajs-3092	260	11	but	but	CCONJ
iajs-3092	260	12	𝔽	𝔽	PROPN
iajs-3092	260	13	is	be	AUX
iajs-3092	260	14	a	a	DET
iajs-3092	260	15	fuzzy	fuzzy	ADJ
iajs-3092	260	16	principle	principle	ADJ
iajs-3092	260	17	ideal	ideal	NOUN
iajs-3092	260	18	ring	ring	NOUN
iajs-3092	260	19	,	,	PUNCT
iajs-3092	260	20	then	then	ADV
iajs-3092	260	21	𝑥𝑡	𝑥𝑡	ADV
iajs-3092	260	22	⊆	⊆	NUM
iajs-3092	260	23	𝐽𝔚	𝐽𝔚	PROPN
iajs-3092	260	24	and	and	CCONJ
iajs-3092	260	25	hence	hence	ADV
iajs-3092	260	26	𝔚	𝔚	PROPN
iajs-3092	260	27	⊆	⊆	NUM
iajs-3092	260	28	𝐽𝔚	𝐽𝔚	PROPN
iajs-3092	260	29	,	,	PUNCT
iajs-3092	260	30	but	but	CCONJ
iajs-3092	260	31	𝐽𝔚	𝐽𝔚	PROPN
iajs-3092	260	32	⊆	⊆	NUM
iajs-3092	260	33	𝔚.	𝔚.	NOUN
iajs-3092	260	34	therefore	therefore	ADV
iajs-3092	260	35	𝔚	𝔚	PROPN
iajs-3092	260	36	=	=	PUNCT
iajs-3092	260	37	𝐽𝔚	𝐽𝔚	PROPN
iajs-3092	260	38	for	for	ADP
iajs-3092	260	39	each	each	DET
iajs-3092	260	40	nonempty	nonempty	ADJ
iajs-3092	260	41	fuzzy	fuzzy	ADJ
iajs-3092	260	42	ideal	ideal	ADJ
iajs-3092	260	43	j	j	PROPN
iajs-3092	260	44	of	of	ADP
iajs-3092	260	45	𝔽and	𝔽and	PROPN
iajs-3092	260	46	hence	hence	ADV
iajs-3092	260	47	𝔚	𝔚	PROPN
iajs-3092	260	48	is	be	AUX
iajs-3092	260	49	a	a	DET
iajs-3092	260	50	fuzzy	fuzzy	ADJ
iajs-3092	260	51	visible	visible	ADJ
iajs-3092	260	52	submodule	submodule	NOUN
iajs-3092	260	53	,	,	PUNCT
iajs-3092	260	54	thus	thus	ADV
iajs-3092	260	55	the	the	DET
iajs-3092	260	56	result	result	NOUN
iajs-3092	260	57	holds	hold	VERB
iajs-3092	260	58	.	.	PUNCT
iajs-3092	261	1	3.38	3.38	NUM
iajs-3092	261	2	proposition	proposition	NOUN
iajs-3092	261	3	:	:	PUNCT
iajs-3092	261	4	let	let	VERB
iajs-3092	261	5	𝔽	𝔽	PRON
iajs-3092	261	6	be	be	AUX
iajs-3092	261	7	a	a	DET
iajs-3092	261	8	principle	principle	ADJ
iajs-3092	261	9	ideal	ideal	ADJ
iajs-3092	261	10	field	field	NOUN
iajs-3092	261	11	and	and	CCONJ
iajs-3092	261	12	𝔚	𝔚	PROPN
iajs-3092	261	13	be	be	AUX
iajs-3092	261	14	a	a	DET
iajs-3092	261	15	fuzzy	fuzzy	ADJ
iajs-3092	261	16	submodule	submodule	NOUN
iajs-3092	261	17	of	of	ADP
iajs-3092	261	18	a	a	DET
iajs-3092	261	19	fuzzy	fuzzy	ADJ
iajs-3092	261	20	module	module	NOUN
iajs-3092	261	21	of	of	ADP
iajs-3092	261	22	an	an	DET
iajs-3092	261	23	𝔽	𝔽	PROPN
iajs-3092	261	24	-module	-module	PROPN
iajs-3092	261	25	𝕄.	𝕄.	PROPN
iajs-3092	261	26	then	then	ADV
iajs-3092	261	27	1	1	X
iajs-3092	261	28	.	.	PUNCT
iajs-3092	262	1	ℙ	ℙ	NOUN
iajs-3092	262	2	is	be	AUX
iajs-3092	262	3	a	a	DET
iajs-3092	262	4	fully	fully	ADV
iajs-3092	262	5	fuzzy	fuzzy	ADJ
iajs-3092	262	6	visible	visible	ADJ
iajs-3092	262	7	.	.	PUNCT
iajs-3092	263	1	2.γ𝔽/(f	2.γ𝔽/(f	NUM
iajs-3092	263	2	−	−	NOUN
iajs-3092	264	1	annxt)∗	annxt)∗	ADV
iajs-3092	264	2	is	be	AUX
iajs-3092	264	3	fuzzy	fuzzy	ADJ
iajs-3092	264	4	negular	negular	ADJ
iajs-3092	264	5	ring	ring	NOUN
iajs-3092	264	6	∀xt	∀xt	NOUN
iajs-3092	264	7	⊆	⊆	NUM
iajs-3092	264	8	ℙ	ℙ	NOUN
iajs-3092	264	9	,	,	PUNCT
iajs-3092	264	10	where	where	SCONJ
iajs-3092	264	11	γ𝔽	γ𝔽	NOUN
iajs-3092	264	12	is	be	AUX
iajs-3092	264	13	a	a	DET
iajs-3092	264	14	fuzzy	fuzzy	ADJ
iajs-3092	264	15	ring	ring	NOUN
iajs-3092	264	16	of	of	ADP
iajs-3092	264	17	𝔽.	𝔽.	PROPN
iajs-3092	264	18	3.∀	3.∀	PROPN
iajs-3092	264	19	xt	xt	ADP
iajs-3092	264	20	⊆	⊆	NUM
iajs-3092	264	21	ℙ	ℙ	PROPN
iajs-3092	264	22	∃rℓ	∃rℓ	ADJ
iajs-3092	264	23	fuzzy	fuzzy	ADJ
iajs-3092	264	24	singleton	singleton	NOUN
iajs-3092	264	25	of	of	ADP
iajs-3092	264	26	𝔽	𝔽	PROPN
iajs-3092	264	27	,	,	PUNCT
iajs-3092	264	28	xt	xt	PROPN
iajs-3092	264	29	=	=	SYM
iajs-3092	264	30	rℓhkxt	rℓhkxt	PROPN
iajs-3092	264	31	,	,	PUNCT
iajs-3092	264	32	k	k	PROPN
iajs-3092	264	33	∈	∈	PROPN
iajs-3092	264	34	(	(	PUNCT
iajs-3092	264	35	0,1	0,1	NOUN
iajs-3092	264	36	]	]	PUNCT
iajs-3092	264	37	.	.	PUNCT
iajs-3092	265	1	proof	proof	NOUN
iajs-3092	265	2	:	:	PUNCT
iajs-3092	265	3	1	1	NUM
iajs-3092	265	4	⇒	⇒	NOUN
iajs-3092	265	5	2	2	NUM
iajs-3092	265	6	p∗	p∗	NOUN
iajs-3092	265	7	is	be	AUX
iajs-3092	265	8	fully	fully	ADV
iajs-3092	265	9	visible	visible	ADJ
iajs-3092	265	10	by	by	ADP
iajs-3092	265	11	propositon	propositon	NOUN
iajs-3092	265	12	(	(	PUNCT
iajs-3092	265	13	2.15	2.15	NUM
iajs-3092	265	14	)	)	PUNCT
iajs-3092	265	15	and	and	CCONJ
iajs-3092	265	16	hence	hence	ADV
iajs-3092	265	17	𝔽/(f	𝔽/(f	NOUN
iajs-3092	265	18	−	−	PROPN
iajs-3092	266	1	annxt)∗	annxt)∗	ADV
iajs-3092	266	2	is	be	AUX
iajs-3092	266	3	regular	regular	ADJ
iajs-3092	266	4	ring	ring	NOUN
iajs-3092	266	5	by	by	ADP
iajs-3092	266	6	[	[	X
iajs-3092	266	7	24	24	NUM
iajs-3092	266	8	,	,	PUNCT
iajs-3092	266	9	proposition	proposition	NOUN
iajs-3092	266	10	2.6.27].therefore	2.6.27].therefore	NOUN
iajs-3092	266	11	γ𝔽/(f	γ𝔽/(f	NOUN
iajs-3092	266	12	−	−	PROPN
iajs-3092	267	1	annxt)∗	annxt)∗	ADV
iajs-3092	267	2	is	be	AUX
iajs-3092	267	3	fuzzy	fuzzy	ADJ
iajs-3092	267	4	negular	negular	ADJ
iajs-3092	267	5	ring	ring	NOUN
iajs-3092	267	6	by	by	ADP
iajs-3092	267	7	[	[	X
iajs-3092	267	8	27	27	NUM
iajs-3092	267	9	,	,	PUNCT
iajs-3092	267	10	theorem	theorem	VERB
iajs-3092	267	11	3.2.10	3.2.10	NUM
iajs-3092	267	12	]	]	PUNCT
iajs-3092	267	13	and	and	CCONJ
iajs-3092	267	14	proposition	proposition	NOUN
iajs-3092	267	15	3.12	3.12	NUM
iajs-3092	267	16	.	.	NOUN
iajs-3092	267	17	2	2	NUM
iajs-3092	267	18	⟹	⟹	NUM
iajs-3092	267	19	3	3	NUM
iajs-3092	267	20	since	since	SCONJ
iajs-3092	267	21	γ𝔽/(f	γ𝔽/(f	NOUN
iajs-3092	267	22	−	−	PROPN
iajs-3092	267	23	annxt)∗	annxt)∗	ADV
iajs-3092	267	24	is	be	AUX
iajs-3092	267	25	fuzzy	fuzzy	ADJ
iajs-3092	267	26	regular	regular	ADJ
iajs-3092	267	27	,	,	PUNCT
iajs-3092	267	28	then	then	ADV
iajs-3092	267	29	𝔽/(f	𝔽/(f	NOUN
iajs-3092	267	30	−	−	PROPN
iajs-3092	268	1	annxt)∗	annxt)∗	ADV
iajs-3092	268	2	is	be	AUX
iajs-3092	268	3	regular	regular	ADJ
iajs-3092	268	4	ring	ring	NOUN
iajs-3092	268	5	by	by	ADP
iajs-3092	268	6	[	[	X
iajs-3092	268	7	24,theorem	24,theorem	NUM
iajs-3092	268	8	3.2.10	3.2.10	NUM
iajs-3092	268	9	]	]	PUNCT
iajs-3092	268	10	,	,	PUNCT
iajs-3092	268	11	hence	hence	ADV
iajs-3092	268	12	∀x	∀x	X
iajs-3092	268	13	∈	∈	PROPN
iajs-3092	268	14	ℙ𝑡	ℙ𝑡	X
iajs-3092	268	15	,	,	PUNCT
iajs-3092	268	16	r	r	NOUN
iajs-3092	268	17	∈	∈	PROPN
iajs-3092	268	18	𝔽	𝔽	PROPN
iajs-3092	268	19	,	,	PUNCT
iajs-3092	268	20	x	x	X
iajs-3092	268	21	=	=	PUNCT
iajs-3092	268	22	rhx	rhx	NOUN
iajs-3092	268	23	for	for	ADP
iajs-3092	268	24	some	some	DET
iajs-3092	268	25	h	h	NOUN
iajs-3092	268	26	∈	∈	NOUN
iajs-3092	268	27	𝔽	𝔽	PROPN
iajs-3092	268	28	by	by	ADP
iajs-3092	268	29	[	[	PUNCT
iajs-3092	268	30	24	24	NUM
iajs-3092	268	31	,	,	PUNCT
iajs-3092	268	32	proposition	proposition	NOUN
iajs-3092	268	33	2.6.27	2.6.27	NUM
iajs-3092	268	34	]	]	PUNCT
iajs-3092	268	35	.	.	PUNCT
iajs-3092	269	1	therefore	therefore	ADV
iajs-3092	269	2	,	,	PUNCT
iajs-3092	269	3	∀	∀	X
iajs-3092	269	4	xt	xt	ADP
iajs-3092	269	5	⊆	⊆	NUM
iajs-3092	269	6	ℙ	ℙ	PROPN
iajs-3092	269	7	,	,	PUNCT
iajs-3092	269	8	∃γℓ	∃γℓ	ADJ
iajs-3092	269	9	fuzzy	fuzzy	ADJ
iajs-3092	269	10	singlet	singlet	NOUN
iajs-3092	269	11	of	of	ADP
iajs-3092	269	12	𝔽	𝔽	PROPN
iajs-3092	269	13	s.t	s.t	PROPN
iajs-3092	269	14	.	.	PROPN
iajs-3092	269	15	xt	xt	PROPN
iajs-3092	270	1	=	=	PUNCT
iajs-3092	271	1	rℓhkxt	rℓhkxt	ADV
iajs-3092	271	2	,	,	PUNCT
iajs-3092	271	3	t	t	PROPN
iajs-3092	271	4	=	=	SYM
iajs-3092	271	5	min{ℓ	min{ℓ	PROPN
iajs-3092	271	6	,	,	PUNCT
iajs-3092	271	7	k	k	PROPN
iajs-3092	271	8	,	,	PUNCT
iajs-3092	271	9	t	t	PROPN
iajs-3092	271	10	}	}	PUNCT
iajs-3092	271	11	.	.	PUNCT
iajs-3092	272	1	2	2	NUM
iajs-3092	272	2	⟹	⟹	NUM
iajs-3092	272	3	3	3	NUM
iajs-3092	272	4	directly	directly	ADV
iajs-3092	272	5	from	from	ADP
iajs-3092	272	6	proposition	proposition	NOUN
iajs-3092	272	7	3.37	3.37	NUM
iajs-3092	272	8	3.39	3.39	NUM
iajs-3092	272	9	corollary	corollary	NOUN
iajs-3092	272	10	:	:	PUNCT
iajs-3092	272	11	if	if	SCONJ
iajs-3092	272	12	𝔽	𝔽	PROPN
iajs-3092	272	13	is	be	AUX
iajs-3092	272	14	a	a	DET
iajs-3092	272	15	principle	principle	ADJ
iajs-3092	272	16	ideal	ideal	ADJ
iajs-3092	272	17	field	field	NOUN
iajs-3092	272	18	,	,	PUNCT
iajs-3092	272	19	then	then	ADV
iajs-3092	272	20	𝕄𝔽	𝕄𝔽	PROPN
iajs-3092	272	21	is	be	AUX
iajs-3092	272	22	fuzzy	fuzzy	ADJ
iajs-3092	272	23	regular	regular	ADV
iajs-3092	272	24	,	,	PUNCT
iajs-3092	272	25	if	if	SCONJ
iajs-3092	272	26	and	and	CCONJ
iajs-3092	272	27	only	only	ADV
iajs-3092	272	28	if	if	SCONJ
iajs-3092	272	29	𝕄𝔽is	𝕄𝔽is	PROPN
iajs-3092	272	30	a	a	DET
iajs-3092	272	31	fuzzy	fuzzy	ADJ
iajs-3092	272	32	𝔽	𝔽	PROPN
iajs-3092	272	33	-module	-module	NOUN
iajs-3092	272	34	is	be	AUX
iajs-3092	272	35	fully	fully	ADV
iajs-3092	272	36	visible	visible	ADJ
iajs-3092	272	37	.	.	PUNCT
iajs-3092	273	1	proof	proof	NOUN
iajs-3092	273	2	:	:	PUNCT
iajs-3092	273	3	let	let	VERB
iajs-3092	273	4	𝕄𝔽	𝕄𝔽	PROPN
iajs-3092	273	5	be	be	AUX
iajs-3092	273	6	a	a	DET
iajs-3092	273	7	fuzzy	fuzzy	ADJ
iajs-3092	273	8	regular	regular	ADJ
iajs-3092	273	9	ring	ring	NOUN
iajs-3092	273	10	,	,	PUNCT
iajs-3092	273	11	then	then	ADV
iajs-3092	273	12	𝔽	𝔽	PROPN
iajs-3092	273	13	is	be	AUX
iajs-3092	273	14	regular	regular	ADJ
iajs-3092	273	15	ring	ring	NOUN
iajs-3092	273	16	by	by	ADP
iajs-3092	273	17	proposition	proposition	NOUN
iajs-3092	273	18	3.12	3.12	NUM
iajs-3092	273	19	,	,	PUNCT
iajs-3092	273	20	then	then	ADV
iajs-3092	273	21	𝔽	𝔽	PROPN
iajs-3092	273	22	is	be	AUX
iajs-3092	273	23	fully	fully	ADV
iajs-3092	273	24	visible	visible	ADJ
iajs-3092	273	25	by	by	ADP
iajs-3092	273	26	[	[	PUNCT
iajs-3092	273	27	24	24	NUM
iajs-3092	273	28	,	,	PUNCT
iajs-3092	273	29	corollary	corollary	NOUN
iajs-3092	273	30	2.6.28	2.6.28	NUM
iajs-3092	273	31	]	]	X
iajs-3092	273	32	,	,	PUNCT
iajs-3092	273	33	and	and	CCONJ
iajs-3092	273	34	hence	hence	ADV
iajs-3092	273	35	m𝔽	m𝔽	NOUN
iajs-3092	273	36	is	be	AUX
iajs-3092	273	37	fully	fully	ADV
iajs-3092	273	38	visible	visible	ADJ
iajs-3092	273	39	.	.	PUNCT
iajs-3092	274	1	3.40	3.40	NUM
iajs-3092	274	2	corollary	corollary	NOUN
iajs-3092	274	3	:	:	PUNCT
iajs-3092	274	4	if	if	SCONJ
iajs-3092	274	5	𝔽	𝔽	PROPN
iajs-3092	274	6	is	be	AUX
iajs-3092	274	7	a	a	DET
iajs-3092	274	8	field	field	NOUN
iajs-3092	274	9	and	and	CCONJ
iajs-3092	274	10	ℙ	ℙ	NOUN
iajs-3092	274	11	∈	∈	NOUN
iajs-3092	274	12	𝐹𝑈𝑀(𝕄	𝐹𝑈𝑀(𝕄	NOUN
iajs-3092	274	13	)	)	PUNCT
iajs-3092	274	14	such	such	ADJ
iajs-3092	274	15	that	that	SCONJ
iajs-3092	274	16	𝕄𝔽	𝕄𝔽	PROPN
iajs-3092	274	17	is	be	AUX
iajs-3092	274	18	a	a	DET
iajs-3092	274	19	fuzzy	fuzzy	ADJ
iajs-3092	274	20	ring	ring	NOUN
iajs-3092	274	21	on	on	ADP
iajs-3092	274	22	𝔽	𝔽	PROPN
iajs-3092	274	23	,	,	PUNCT
iajs-3092	274	24	then	then	ADV
iajs-3092	274	25	ℙ	ℙ	PROPN
iajs-3092	274	26	is	be	AUX
iajs-3092	274	27	fully	fully	ADV
iajs-3092	274	28	fuzzy	fuzzy	ADJ
iajs-3092	274	29	visible	visible	ADJ
iajs-3092	274	30	module	module	NOUN
iajs-3092	274	31	,	,	PUNCT
iajs-3092	274	32	if	if	SCONJ
iajs-3092	274	33	𝕄𝔽/(f	𝕄𝔽/(f	VERB
iajs-3092	274	34	−	−	PROPN
iajs-3092	274	35	ann	ann	PROPN
iajs-3092	274	36	ℙ	ℙ	PROPN
iajs-3092	274	37	)	)	PUNCT
iajs-3092	274	38	𝑡	𝑡	PROPN
iajs-3092	274	39	is	be	AUX
iajs-3092	274	40	a	a	DET
iajs-3092	274	41	fuzzy	fuzzy	ADJ
iajs-3092	274	42	regular	regular	ADJ
iajs-3092	274	43	ring	ring	NOUN
iajs-3092	274	44	,	,	PUNCT
iajs-3092	274	45	where	where	SCONJ
iajs-3092	274	46	(	(	PUNCT
iajs-3092	274	47	f	f	X
iajs-3092	274	48	−	−	PROPN
iajs-3092	274	49	annℙ)t	annℙ)t	PROPN
iajs-3092	274	50	=	=	SYM
iajs-3092	274	51	annℙt	annℙt	PROPN
iajs-3092	274	52	,	,	PUNCT
iajs-3092	274	53	∀	∀	PUNCT
iajs-3092	274	54	𝑡	𝑡	X
iajs-3092	274	55	∈	∈	PROPN
iajs-3092	274	56	(	(	PUNCT
iajs-3092	274	57	0,1	0,1	NOUN
iajs-3092	274	58	]	]	PUNCT
iajs-3092	274	59	.	.	PUNCT
iajs-3092	275	1	proof	proof	NOUN
iajs-3092	275	2	:	:	PUNCT
iajs-3092	275	3	since	since	SCONJ
iajs-3092	275	4	𝕄𝔽/f	𝕄𝔽/f	ADP
iajs-3092	275	5	−	−	PROPN
iajs-3092	275	6	annℙ	annℙ	PROPN
iajs-3092	275	7	is	be	AUX
iajs-3092	275	8	a	a	DET
iajs-3092	275	9	fuzzy	fuzzy	ADJ
iajs-3092	275	10	regular	regular	NOUN
iajs-3092	275	11	,	,	PUNCT
iajs-3092	275	12	then	then	ADV
iajs-3092	275	13	𝔽/(f	𝔽/(f	NOUN
iajs-3092	275	14	−	−	PROPN
iajs-3092	275	15	annℙ)t	annℙ)t	PROPN
iajs-3092	275	16	is	be	AUX
iajs-3092	275	17	a	a	DET
iajs-3092	275	18	regular	regular	ADJ
iajs-3092	275	19	ring	ring	NOUN
iajs-3092	275	20	by	by	ADP
iajs-3092	275	21	[	[	X
iajs-3092	275	22	27,theorem	27,theorem	NUM
iajs-3092	275	23	3.2.10	3.2.10	NUM
iajs-3092	275	24	]	]	PUNCT
iajs-3092	275	25	and	and	CCONJ
iajs-3092	275	26	proposition	proposition	NOUN
iajs-3092	275	27	3.15	3.15	NUM
iajs-3092	275	28	,	,	PUNCT
iajs-3092	275	29	but	but	CCONJ
iajs-3092	275	30	(	(	PUNCT
iajs-3092	275	31	f	f	X
iajs-3092	275	32	−	−	PROPN
iajs-3092	275	33	annℙ)t	annℙ)t	PROPN
iajs-3092	275	34	=	=	SYM
iajs-3092	275	35	annℙ𝑡	annℙ𝑡	PROPN
iajs-3092	275	36	,	,	PUNCT
iajs-3092	275	37	∀	∀	X
iajs-3092	275	38	𝑡	𝑡	NOUN
iajs-3092	275	39	∈	∈	PROPN
iajs-3092	275	40	(	(	PUNCT
iajs-3092	275	41	0,1	0,1	NUM
iajs-3092	275	42	]	]	PUNCT
iajs-3092	275	43	,	,	PUNCT
iajs-3092	275	44	then	then	ADV
iajs-3092	275	45	ℙ𝑡	ℙ𝑡	PROPN
iajs-3092	275	46	is	be	AUX
iajs-3092	275	47	fully	fully	ADV
iajs-3092	275	48	visible	visible	ADJ
iajs-3092	275	49	by	by	ADP
iajs-3092	275	50	[	[	X
iajs-3092	275	51	24	24	NUM
iajs-3092	275	52	]	]	PUNCT
iajs-3092	275	53	.	.	PUNCT
iajs-3092	276	1	therefore	therefore	ADV
iajs-3092	276	2	ℙ	ℙ	PROPN
iajs-3092	276	3	is	be	AUX
iajs-3092	276	4	fully	fully	ADV
iajs-3092	276	5	fuzzy	fuzzy	ADJ
iajs-3092	276	6	visible	visible	ADJ
iajs-3092	276	7	.	.	PUNCT
iajs-3092	277	1	3.41	3.41	NUM
iajs-3092	277	2	proposition	proposition	NOUN
iajs-3092	277	3	:	:	PUNCT
iajs-3092	277	4	if	if	SCONJ
iajs-3092	277	5	ℙ	ℙ	PROPN
iajs-3092	277	6	is	be	AUX
iajs-3092	277	7	divisible	divisible	ADJ
iajs-3092	277	8	over	over	ADP
iajs-3092	277	9	p.i.d	p.i.d	PROPN
iajs-3092	277	10	.	.	PUNCT
iajs-3092	278	1	and	and	CCONJ
iajs-3092	278	2	d	d	X
iajs-3092	278	3	f	f	X
iajs-3092	278	4	-	-	PUNCT
iajs-3092	278	5	ann	ann	PROPN
iajs-3092	278	6	ℙ	ℙ	PROPN
iajs-3092	278	7	is	be	AUX
iajs-3092	278	8	semimaximal	semimaximal	ADJ
iajs-3092	278	9	ideal	ideal	NOUN
iajs-3092	278	10	of	of	ADP
iajs-3092	278	11	𝔽.	𝔽.	PROPN
iajs-3092	278	12	then	then	ADV
iajs-3092	278	13	ℙ	ℙ	PROPN
iajs-3092	278	14	is	be	AUX
iajs-3092	278	15	a	a	DET
iajs-3092	278	16	fully	fully	ADV
iajs-3092	278	17	fuzzy	fuzzy	ADJ
iajs-3092	278	18	visible	visible	ADJ
iajs-3092	278	19	,	,	PUNCT
iajs-3092	279	1	where	where	SCONJ
iajs-3092	279	2	(	(	PUNCT
iajs-3092	279	3	f	f	X
iajs-3092	279	4	−	−	X
iajs-3092	279	5	annℙ)∗	annℙ)∗	VERB
iajs-3092	279	6	=	=	NOUN
iajs-3092	279	7	annℙ∗.	annℙ∗.	NOUN
iajs-3092	279	8	and	and	CCONJ
iajs-3092	279	9	ℙ(x	ℙ(x	NUM
iajs-3092	279	10	)	)	PUNCT
iajs-3092	279	11	=	=	SYM
iajs-3092	279	12	1	1	NUM
iajs-3092	279	13	∀x	∀x	NUM
iajs-3092	279	14	∈	∈	PROPN
iajs-3092	279	15	𝕄.	𝕄.	PROPN
iajs-3092	279	16	ihjpas	ihjpa	NOUN
iajs-3092	279	17	.	.	PUNCT
iajs-3092	280	1	36	36	NUM
iajs-3092	280	2	(	(	PUNCT
iajs-3092	280	3	3	3	NUM
iajs-3092	280	4	)	)	PUNCT
iajs-3092	280	5	2023	2023	NUM
iajs-3092	280	6	380	380	NUM
iajs-3092	280	7	proof	proof	NOUN
iajs-3092	280	8	:	:	PUNCT
iajs-3092	280	9	p∗	p∗	NOUN
iajs-3092	280	10	is	be	AUX
iajs-3092	280	11	divisible	divisible	ADJ
iajs-3092	280	12	module	module	NOUN
iajs-3092	280	13	by	by	ADP
iajs-3092	280	14	[	[	X
iajs-3092	280	15	23	23	NUM
iajs-3092	280	16	]	]	PUNCT
iajs-3092	280	17	,	,	PUNCT
iajs-3092	280	18	since	since	SCONJ
iajs-3092	280	19	(	(	PUNCT
iajs-3092	280	20	f	f	X
iajs-3092	280	21	−	−	PROPN
iajs-3092	280	22	annℙ)∗	annℙ)∗	NUM
iajs-3092	280	23	=	=	SYM
iajs-3092	280	24	annℙ∗	annℙ∗	PROPN
iajs-3092	280	25	and	and	CCONJ
iajs-3092	280	26	f	f	PROPN
iajs-3092	280	27	−	−	PROPN
iajs-3092	280	28	annℙ	annℙ	PROPN
iajs-3092	280	29	is	be	AUX
iajs-3092	280	30	semimaximal	semimaximal	ADJ
iajs-3092	280	31	,	,	PUNCT
iajs-3092	280	32	then	then	ADV
iajs-3092	280	33	annp∗	annp∗	NOUN
iajs-3092	280	34	is	be	AUX
iajs-3092	280	35	semimaximal	semimaximal	ADJ
iajs-3092	280	36	ideal	ideal	NOUN
iajs-3092	280	37	by	by	ADP
iajs-3092	280	38	[	[	X
iajs-3092	280	39	20	20	NUM
iajs-3092	280	40	,	,	PUNCT
iajs-3092	280	41	proposition	proposition	NOUN
iajs-3092	280	42	(	(	PUNCT
iajs-3092	280	43	2.4	2.4	NUM
iajs-3092	280	44	)	)	PUNCT
iajs-3092	280	45	]	]	PUNCT
iajs-3092	280	46	.	.	PUNCT
iajs-3092	281	1	then	then	ADV
iajs-3092	281	2	ℙ∗	ℙ∗	PROPN
iajs-3092	281	3	is	be	AUX
iajs-3092	281	4	fully	fully	ADV
iajs-3092	281	5	visible	visible	ADJ
iajs-3092	281	6	[	[	PUNCT
iajs-3092	281	7	24	24	NUM
iajs-3092	281	8	,	,	PUNCT
iajs-3092	281	9	proposition	proposition	NOUN
iajs-3092	281	10	2.6.38	2.6.38	NUM
iajs-3092	281	11	]	]	PUNCT
iajs-3092	281	12	.	.	PUNCT
iajs-3092	282	1	therefore	therefore	ADV
iajs-3092	282	2	ℙ	ℙ	PROPN
iajs-3092	282	3	is	be	AUX
iajs-3092	282	4	fuzzy	fuzzy	ADJ
iajs-3092	282	5	fully	fully	ADV
iajs-3092	282	6	visible	visible	ADJ
iajs-3092	282	7	.	.	PUNCT
iajs-3092	283	1	3.42	3.42	NUM
iajs-3092	283	2	proposition	proposition	NOUN
iajs-3092	283	3	:	:	PUNCT
iajs-3092	283	4	let	let	VERB
iajs-3092	283	5	γ𝔽	γ𝔽	ADV
iajs-3092	283	6	be	be	AUX
iajs-3092	283	7	a	a	DET
iajs-3092	283	8	fuzzy	fuzzy	ADJ
iajs-3092	283	9	visible	visible	ADJ
iajs-3092	283	10	ring	ring	NOUN
iajs-3092	283	11	defined	define	VERB
iajs-3092	283	12	as	as	ADP
iajs-3092	283	13	γ𝔽(r	γ𝔽(r	PROPN
iajs-3092	283	14	)	)	PUNCT
iajs-3092	283	15	=	=	SYM
iajs-3092	283	16	1	1	NUM
iajs-3092	283	17	∀r	∀r	NOUN
iajs-3092	283	18	∈	∈	PROPN
iajs-3092	283	19	𝔽	𝔽	NOUN
iajs-3092	283	20	,	,	PUNCT
iajs-3092	283	21	if	if	SCONJ
iajs-3092	283	22	ℙ	ℙ	PROPN
iajs-3092	283	23	is	be	AUX
iajs-3092	283	24	a	a	DET
iajs-3092	283	25	fuzzy	fuzzy	ADJ
iajs-3092	283	26	𝔽	𝔽	PROPN
iajs-3092	283	27	module	module	NOUN
iajs-3092	283	28	m	m	NOUN
iajs-3092	283	29	,	,	PUNCT
iajs-3092	283	30	then	then	ADV
iajs-3092	283	31	ℙ	ℙ	PROPN
iajs-3092	283	32	is	be	AUX
iajs-3092	283	33	flat	flat	ADJ
iajs-3092	283	34	.	.	PUNCT
iajs-3092	284	1	proof	proof	NOUN
iajs-3092	284	2	:	:	PUNCT
iajs-3092	284	3	since	since	SCONJ
iajs-3092	284	4	(	(	PUNCT
iajs-3092	284	5	γ𝔽)t	γ𝔽)t	PROPN
iajs-3092	284	6	=	=	SYM
iajs-3092	284	7	𝔽	𝔽	PROPN
iajs-3092	284	8	,	,	PUNCT
iajs-3092	284	9	then	then	ADV
iajs-3092	284	10	𝔽	𝔽	PROPN
iajs-3092	284	11	is	be	AUX
iajs-3092	284	12	visible	visible	ADJ
iajs-3092	284	13	,	,	PUNCT
iajs-3092	284	14	then	then	ADV
iajs-3092	284	15	m	m	VERB
iajs-3092	284	16	is	be	AUX
iajs-3092	284	17	flat	flat	ADJ
iajs-3092	284	18	by	by	ADP
iajs-3092	284	19	[	[	PUNCT
iajs-3092	284	20	24	24	NUM
iajs-3092	284	21	,	,	PUNCT
iajs-3092	284	22	proposition	proposition	NOUN
iajs-3092	284	23	2.6.39	2.6.39	NOUN
iajs-3092	284	24	]	]	PUNCT
iajs-3092	284	25	.	.	PUNCT
iajs-3092	285	1	therefore	therefore	ADV
iajs-3092	285	2	ℙ	ℙ	PROPN
iajs-3092	285	3	is	be	AUX
iajs-3092	285	4	fuzzy	fuzzy	ADJ
iajs-3092	285	5	flat	flat	ADJ
iajs-3092	285	6	by	by	ADP
iajs-3092	285	7	[	[	PUNCT
iajs-3092	285	8	29	29	NUM
iajs-3092	285	9	,	,	PUNCT
iajs-3092	285	10	theorem	theorem	VERB
iajs-3092	285	11	3.15	3.15	NUM
iajs-3092	285	12	]	]	SYM
iajs-3092	285	13	4	4	NUM
iajs-3092	285	14	.	.	X
iajs-3092	285	15	conclusion	conclusion	NOUN
iajs-3092	285	16	:	:	PUNCT
iajs-3092	285	17	in	in	ADP
iajs-3092	285	18	this	this	DET
iajs-3092	285	19	paper	paper	NOUN
iajs-3092	285	20	,	,	PUNCT
iajs-3092	285	21	a	a	DET
iajs-3092	285	22	number	number	NOUN
iajs-3092	285	23	of	of	ADP
iajs-3092	285	24	characteristics	characteristic	NOUN
iajs-3092	285	25	and	and	CCONJ
iajs-3092	285	26	properties	property	NOUN
iajs-3092	285	27	that	that	PRON
iajs-3092	285	28	explain	explain	VERB
iajs-3092	285	29	the	the	DET
iajs-3092	285	30	relationship	relationship	NOUN
iajs-3092	285	31	of	of	ADP
iajs-3092	285	32	fully	fully	ADV
iajs-3092	285	33	fuzzy	fuzzy	ADJ
iajs-3092	285	34	visible	visible	ADJ
iajs-3092	285	35	modules	module	NOUN
iajs-3092	285	36	with	with	ADP
iajs-3092	285	37	other	other	ADJ
iajs-3092	285	38	modules	module	NOUN
iajs-3092	285	39	are	be	AUX
iajs-3092	285	40	proved	prove	VERB
iajs-3092	285	41	.	.	PUNCT
iajs-3092	286	1	in	in	ADP
iajs-3092	286	2	addition	addition	NOUN
iajs-3092	286	3	many	many	ADJ
iajs-3092	286	4	useful	useful	ADJ
iajs-3092	286	5	examples	example	NOUN
iajs-3092	286	6	with	with	ADP
iajs-3092	286	7	a	a	DET
iajs-3092	286	8	collection	collection	NOUN
iajs-3092	286	9	of	of	ADP
iajs-3092	286	10	important	important	ADJ
iajs-3092	286	11	results	result	NOUN
iajs-3092	286	12	are	be	AUX
iajs-3092	286	13	obtained	obtain	VERB
iajs-3092	286	14	.	.	PUNCT
iajs-3092	287	1	references	reference	NOUN
iajs-3092	287	2	1	1	NUM
iajs-3092	287	3	.	.	PUNCT
iajs-3092	288	1	zadeh	zadeh	PROPN
iajs-3092	288	2	,	,	PUNCT
iajs-3092	288	3	l.	l.	PROPN
iajs-3092	288	4	a	a	X
iajs-3092	288	5	..	..	PUNCT
iajs-3092	288	6	"	"	PUNCT
iajs-3092	288	7	fuzzy	fuzzy	ADJ
iajs-3092	288	8	sets	set	NOUN
iajs-3092	288	9	"	"	PUNCT
iajs-3092	288	10	,	,	PUNCT
iajs-3092	288	11	information	information	NOUN
iajs-3092	288	12	and	and	CCONJ
iajs-3092	288	13	control	control	NOUN
iajs-3092	288	14	,	,	PUNCT
iajs-3092	288	15	1965	1965	NUM
iajs-3092	288	16	,	,	PUNCT
iajs-3092	288	17	8	8	NUM
iajs-3092	288	18	,	,	PUNCT
iajs-3092	288	19	338	338	NUM
iajs-3092	288	20	-	-	SYM
iajs-3092	288	21	353	353	NUM
iajs-3092	288	22	.	.	NOUN
iajs-3092	289	1	2	2	NUM
iajs-3092	289	2	.	.	X
iajs-3092	289	3	rosenfeld	rosenfeld	PROPN
iajs-3092	289	4	,	,	PUNCT
iajs-3092	289	5	a.	a.	NOUN
iajs-3092	289	6	"	"	PUNCT
iajs-3092	289	7	fuzzy	fuzzy	ADJ
iajs-3092	289	8	groups	group	NOUN
iajs-3092	289	9	"	"	PUNCT
iajs-3092	289	10	,	,	PUNCT
iajs-3092	289	11	journal	journal	NOUN
iajs-3092	289	12	of	of	ADP
iajs-3092	289	13	mathematical	mathematical	ADJ
iajs-3092	289	14	analysis	analysis	NOUN
iajs-3092	289	15	and	and	CCONJ
iajs-3092	289	16	applications	application	NOUN
iajs-3092	289	17	.	.	PUNCT
iajs-3092	290	1	1971,35	1971,35	NUM
iajs-3092	290	2	,	,	PUNCT
iajs-3092	290	3	512	512	NUM
iajs-3092	290	4	-	-	SYM
iajs-3092	290	5	517	517	NUM
iajs-3092	290	6	.	.	NOUN
iajs-3092	291	1	3	3	X
iajs-3092	291	2	.	.	X
iajs-3092	291	3	negoita	negoita	PROPN
iajs-3092	291	4	c.v	c.v	PROPN
iajs-3092	291	5	.	.	PROPN
iajs-3092	291	6	;	;	PUNCT
iajs-3092	292	1	ralescu	ralescu	NOUN
iajs-3092	292	2	,	,	PUNCT
iajs-3092	292	3	d.a	d.a	PROPN
iajs-3092	292	4	.	.	PUNCT
iajs-3092	293	1	"	"	PUNCT
iajs-3092	293	2	applications	application	NOUN
iajs-3092	293	3	of	of	ADP
iajs-3092	293	4	fuzzy	fuzzy	ADJ
iajs-3092	293	5	sets	set	NOUN
iajs-3092	293	6	and	and	CCONJ
iajs-3092	293	7	system	system	NOUN
iajs-3092	293	8	analysis"(birkhauser	analysis"(birkhauser	PROPN
iajs-3092	293	9	,	,	PUNCT
iajs-3092	293	10	basel	basel	PROPN
iajs-3092	293	11	)	)	PUNCT
iajs-3092	293	12	.	.	PUNCT
iajs-3092	294	1	1975	1975	NUM
iajs-3092	294	2	.	.	PUNCT
iajs-3092	295	1	4	4	X
iajs-3092	295	2	.	.	X
iajs-3092	295	3	sajda	sajda	NOUN
iajs-3092	295	4	,	,	PUNCT
iajs-3092	295	5	k.m	k.m	PROPN
iajs-3092	295	6	.	.	PROPN
iajs-3092	295	7	;	;	PUNCT
iajs-3092	295	8	buthyna	buthyna	PROPN
iajs-3092	295	9	,	,	PUNCT
iajs-3092	295	10	n.s	n.s	PROPN
iajs-3092	295	11	.	.	PUNCT
iajs-3092	295	12	"	"	PUNCT
iajs-3092	295	13	fuzzy	fuzzy	ADJ
iajs-3092	295	14	visible	visible	ADJ
iajs-3092	295	15	submodules	submodule	NOUN
iajs-3092	295	16	with	with	ADP
iajs-3092	295	17	some	some	DET
iajs-3092	295	18	results	result	NOUN
iajs-3092	295	19	"	"	PUNCT
iajs-3092	295	20	,	,	PUNCT
iajs-3092	295	21	int.j.nonlinear	int.j.nonlinear	ADJ
iajs-3092	295	22	anal	anal	PROPN
iajs-3092	295	23	.	.	PUNCT
iajs-3092	296	1	appl	appl	PROPN
iajs-3092	296	2	.	.	PROPN
iajs-3092	296	3	2021	2021	NUM
iajs-3092	296	4	,	,	PUNCT
iajs-3092	296	5	12,special	12,special	NUM
iajs-3092	296	6	issue	issue	NOUN
iajs-3092	296	7	,	,	PUNCT
iajs-3092	296	8	2231	2231	NUM
iajs-3092	296	9	-	-	SYM
iajs-3092	296	10	2241	2241	NUM
iajs-3092	296	11	.	.	PUNCT
iajs-3092	297	1	5	5	NUM
iajs-3092	297	2	.	.	X
iajs-3092	297	3	sajda	sajda	NOUN
iajs-3092	297	4	,	,	PUNCT
iajs-3092	297	5	.k.m	.k.m	PROPN
iajs-3092	297	6	.	.	PROPN
iajs-3092	297	7	;	;	PUNCT
iajs-3092	297	8	buthyna	buthyna	PROPN
iajs-3092	297	9	,	,	PUNCT
iajs-3092	297	10	n.	n.	PROPN
iajs-3092	297	11	s.	s.	PROPN
iajs-3092	298	1	"	"	PUNCT
iajs-3092	298	2	fully	fully	ADV
iajs-3092	298	3	fuzzy	fuzzy	ADJ
iajs-3092	298	4	visible	visible	ADJ
iajs-3092	298	5	modules	module	NOUN
iajs-3092	298	6	with	with	ADP
iajs-3092	298	7	distinguishing	distinguish	VERB
iajs-3092	298	8	result	result	NOUN
iajs-3092	298	9	"	"	PUNCT
iajs-3092	298	10	,	,	PUNCT
iajs-3092	298	11	accepted	accept	VERB
iajs-3092	298	12	for	for	ADP
iajs-3092	298	13	publication	publication	NOUN
iajs-3092	298	14	in	in	ADP
iajs-3092	298	15	journal	journal	NOUN
iajs-3092	298	16	of	of	ADP
iajs-3092	298	17	discrete	discrete	ADJ
iajs-3092	298	18	mathematical	mathematical	ADJ
iajs-3092	298	19	sciences	science	NOUN
iajs-3092	298	20	and	and	CCONJ
iajs-3092	298	21	cryptography,2022	cryptography,2022	NOUN
iajs-3092	298	22	.	.	PUNCT
iajs-3092	299	1	6	6	NUM
iajs-3092	299	2	.	.	X
iajs-3092	299	3	john	john	PROPN
iajs-3092	299	4	.	.	PUNCT
iajs-3092	299	5	n.	n.	PROPN
iajs-3092	299	6	m.	m.	PROPN
iajs-3092	299	7	;	;	PUNCT
iajs-3092	299	8	malik	malik	PROPN
iajs-3092	299	9	,	,	PUNCT
iajs-3092	299	10	d.	d.	PROPN
iajs-3092	299	11	s.	s.	PROPN
iajs-3092	300	1	"	"	PUNCT
iajs-3092	300	2	fuzzy	fuzzy	ADJ
iajs-3092	300	3	maximal	maximal	ADJ
iajs-3092	300	4	,	,	PUNCT
iajs-3092	300	5	radical	radical	ADJ
iajs-3092	300	6	,	,	PUNCT
iajs-3092	300	7	and	and	CCONJ
iajs-3092	300	8	primary	primary	ADJ
iajs-3092	300	9	ideals	ideal	NOUN
iajs-3092	300	10	of	of	ADP
iajs-3092	300	11	a	a	DET
iajs-3092	300	12	ring	ring	NOUN
iajs-3092	300	13	"	"	PUNCT
iajs-3092	300	14	,	,	PUNCT
iajs-3092	300	15	information	information	NOUN
iajs-3092	300	16	sciences	science	NOUN
iajs-3092	300	17	.	.	PUNCT
iajs-3092	300	18	1991	1991	NUM
iajs-3092	300	19	,	,	PUNCT
iajs-3092	300	20	53	53	NUM
iajs-3092	300	21	,	,	PUNCT
iajs-3092	300	22	237	237	NUM
iajs-3092	300	23	-	-	SYM
iajs-3092	300	24	250	250	NUM
iajs-3092	300	25	.	.	PUNCT
iajs-3092	301	1	7	7	X
iajs-3092	301	2	.	.	X
iajs-3092	301	3	hassan	hassan	PROPN
iajs-3092	301	4	.	.	PUNCT
iajs-3092	302	1	k.	k.	PROPN
iajs-3092	302	2	m.	m.	PROPN
iajs-3092	302	3	;	;	PUNCT
iajs-3092	302	4	hatam	hatam	NOUN
iajs-3092	302	5	.	.	PUNCT
iajs-3092	303	1	y.	y.	PROPN
iajs-3092	303	2	k.	k.	PUNCT
iajs-3092	304	1	"	"	PUNCT
iajs-3092	304	2	some	some	DET
iajs-3092	304	3	properties	property	NOUN
iajs-3092	304	4	of	of	ADP
iajs-3092	304	5	the	the	DET
iajs-3092	304	6	essential	essential	ADJ
iajs-3092	304	7	fuzzy	fuzzy	ADJ
iajs-3092	304	8	and	and	CCONJ
iajs-3092	304	9	closed	close	VERB
iajs-3092	304	10	fuzzy	fuzzy	ADJ
iajs-3092	304	11	submodules	submodule	NOUN
iajs-3092	304	12	"	"	PUNCT
iajs-3092	304	13	,	,	PUNCT
iajs-3092	304	14	iraqi	iraqi	ADJ
iajs-3092	304	15	journal	journal	NOUN
iajs-3092	304	16	of	of	ADP
iajs-3092	304	17	science	science	NOUN
iajs-3092	304	18	.	.	PUNCT
iajs-3092	305	1	2020	2020	NUM
iajs-3092	305	2	,	,	PUNCT
iajs-3092	305	3	61	61	NUM
iajs-3092	305	4	,	,	PUNCT
iajs-3092	305	5	890	890	NUM
iajs-3092	305	6	-	-	SYM
iajs-3092	305	7	897	897	NUM
iajs-3092	305	8	.	.	PUNCT
iajs-3092	306	1	8	8	NUM
iajs-3092	306	2	.	.	PUNCT
iajs-3092	307	1	asmaa	asmaa	PROPN
iajs-3092	307	2	z.m	z.m	PROPN
iajs-3092	307	3	.	.	PROPN
iajs-3092	307	4	;	;	PUNCT
iajs-3092	307	5	mahmood	mahmood	PROPN
iajs-3092	307	6	,	,	PUNCT
iajs-3092	307	7	r.d	r.d	PROPN
iajs-3092	307	8	.	.	PROPN
iajs-3092	307	9	and	and	CCONJ
iajs-3092	307	10	marwa	marwa	PROPN
iajs-3092	307	11	m.f	m.f	PROPN
iajs-3092	307	12	.	.	PUNCT
iajs-3092	308	1	"	"	PUNCT
iajs-3092	308	2	on	on	ADP
iajs-3092	308	3	fuzzy	fuzzy	ADJ
iajs-3092	308	4	essential	essential	ADJ
iajs-3092	308	5	ideals	ideal	NOUN
iajs-3092	308	6	of	of	ADP
iajs-3092	308	7	rings	ring	NOUN
iajs-3092	308	8	"	"	PUNCT
iajs-3092	308	9	,	,	PUNCT
iajs-3092	308	10	international	international	ADJ
iajs-3092	308	11	mathematical	mathematical	ADJ
iajs-3092	308	12	form	form	NOUN
iajs-3092	308	13	.	.	PUNCT
iajs-3092	309	1	2020	2020	NUM
iajs-3092	309	2	,	,	PUNCT
iajs-3092	309	3	15	15	NUM
iajs-3092	309	4	,	,	PUNCT
iajs-3092	309	5	383	383	NUM
iajs-3092	309	6	-	-	SYM
iajs-3092	309	7	391	391	NUM
iajs-3092	309	8	.	.	PUNCT
iajs-3092	310	1	9	9	NUM
iajs-3092	310	2	.	.	X
iajs-3092	310	3	zahedi	zahedi	PROPN
iajs-3092	310	4	,	,	PUNCT
iajs-3092	310	5	m.	m.	NOUN
iajs-3092	310	6	m.	m.	NOUN
iajs-3092	310	7	"on	"on	NUM
iajs-3092	310	8	l	l	NOUN
iajs-3092	310	9	-	-	ADJ
iajs-3092	310	10	fuzzy	fuzzy	ADJ
iajs-3092	310	11	residual	residual	ADJ
iajs-3092	310	12	quotient	quotient	NOUN
iajs-3092	310	13	modules	module	NOUN
iajs-3092	310	14	and	and	CCONJ
iajs-3092	310	15	pprimary	pprimary	ADJ
iajs-3092	310	16	submodules	submodule	NOUN
iajs-3092	310	17	"	"	PUNCT
iajs-3092	310	18	,	,	PUNCT
iajs-3092	310	19	fuzzy	fuzzy	ADJ
iajs-3092	310	20	sets	set	NOUN
iajs-3092	310	21	and	and	CCONJ
iajs-3092	310	22	systems	system	NOUN
iajs-3092	310	23	.	.	PUNCT
iajs-3092	311	1	1992	1992	NUM
iajs-3092	311	2	,	,	PUNCT
iajs-3092	311	3	51	51	NUM
iajs-3092	311	4	,	,	PUNCT
iajs-3092	311	5	333	333	NUM
iajs-3092	311	6	-	-	SYM
iajs-3092	311	7	344	344	NUM
iajs-3092	311	8	,	,	PUNCT
iajs-3092	311	9	10	10	NUM
iajs-3092	311	10	.	.	PUNCT
iajs-3092	312	1	saad	saad	PROPN
iajs-3092	312	2	s.m	s.m	PROPN
iajs-3092	312	3	.	.	PROPN
iajs-3092	312	4	;	;	PUNCT
iajs-3092	312	5	hatam.y.k	hatam.y.k	PROPN
iajs-3092	312	6	,	,	PUNCT
iajs-3092	312	7	"	"	PUNCT
iajs-3092	312	8	fuzzy	fuzzy	ADJ
iajs-3092	312	9	soc	soc	NOUN
iajs-3092	312	10	-	-	PUNCT
iajs-3092	312	11	t	t	PROPN
iajs-3092	312	12	-	-	PUNCT
iajs-3092	312	13	abso	abso	PROPN
iajs-3092	312	14	sub	sub	NOUN
iajs-3092	312	15	-	-	NOUN
iajs-3092	312	16	modules	module	NOUN
iajs-3092	312	17	"	"	PUNCT
iajs-3092	312	18	,	,	PUNCT
iajs-3092	312	19	iraqi	iraqi	ADJ
iajs-3092	312	20	journal	journal	NOUN
iajs-3092	312	21	for	for	ADP
iajs-3092	312	22	computer	computer	NOUN
iajs-3092	312	23	science	science	NOUN
iajs-3092	312	24	and	and	CCONJ
iajs-3092	312	25	mathematics	mathematic	NOUN
iajs-3092	312	26	.	.	PUNCT
iajs-3092	313	1	2022	2022	NUM
iajs-3092	313	2	,	,	PUNCT
iajs-3092	313	3	3	3	NUM
iajs-3092	313	4	,	,	PUNCT
iajs-3092	313	5	124	124	NUM
iajs-3092	313	6	-	-	SYM
iajs-3092	313	7	134	134	NUM
iajs-3092	313	8	.	.	PUNCT
iajs-3092	314	1	11	11	NUM
iajs-3092	314	2	.	.	PUNCT
iajs-3092	315	1	mashinchi	mashinchi	PROPN
iajs-3092	315	2	,	,	PUNCT
iajs-3092	315	3	m.	m.	NOUN
iajs-3092	315	4	;	;	PUNCT
iajs-3092	315	5	zahedi	zahedi	PROPN
iajs-3092	315	6	,	,	PUNCT
iajs-3092	315	7	m.m	m.m	PROPN
iajs-3092	315	8	.	.	PUNCT
iajs-3092	315	9	"	"	PUNCT
iajs-3092	315	10	on	on	ADP
iajs-3092	315	11	l	l	ADJ
iajs-3092	315	12	-	-	ADJ
iajs-3092	315	13	fuzzy	fuzzy	ADJ
iajs-3092	315	14	primary	primary	ADJ
iajs-3092	315	15	submodules	submodules	NOUN
iajs-3092	315	16	"	"	PUNCT
iajs-3092	315	17	,	,	PUNCT
iajs-3092	315	18	fuzzy	fuzzy	ADJ
iajs-3092	315	19	sets	set	NOUN
iajs-3092	315	20	and	and	CCONJ
iajs-3092	315	21	systems	system	NOUN
iajs-3092	315	22	,	,	PUNCT
iajs-3092	315	23	1992,49	1992,49	NUM
iajs-3092	315	24	,	,	PUNCT
iajs-3092	315	25	843	843	NUM
iajs-3092	315	26	-	-	NOUN
iajs-3092	315	27	857	857	NUM
iajs-3092	315	28	,	,	PUNCT
iajs-3092	315	29	.	.	PUNCT
iajs-3092	316	1	12	12	NUM
iajs-3092	316	2	.	.	PUNCT
iajs-3092	317	1	maysoun	maysoun	PROPN
iajs-3092	317	2	a.	a.	PROPN
iajs-3092	317	3	h.	h.	PROPN
iajs-3092	317	4	;	;	PUNCT
iajs-3092	317	5	hatam	hatam	PROPN
iajs-3092	317	6	,	,	PUNCT
iajs-3092	317	7	h.y	h.y	PROPN
iajs-3092	317	8	.	.	PUNCT
iajs-3092	317	9	"	"	PUNCT
iajs-3092	317	10	fuzzy	fuzzy	ADJ
iajs-3092	317	11	maximal	maximal	ADJ
iajs-3092	317	12	sub	sub	NOUN
iajs-3092	317	13	-	-	NOUN
iajs-3092	317	14	modules	module	NOUN
iajs-3092	317	15	"	"	PUNCT
iajs-3092	317	16	,	,	PUNCT
iajs-3092	317	17	iraqi	iraqi	ADJ
iajs-3092	317	18	journal	journal	NOUN
iajs-3092	317	19	of	of	ADP
iajs-3092	317	20	science.2020	science.2020	PROPN
iajs-3092	317	21	,	,	PUNCT
iajs-3092	317	22	61	61	NUM
iajs-3092	317	23	,	,	PUNCT
iajs-3092	317	24	1164	1164	NUM
iajs-3092	317	25	-	-	SYM
iajs-3092	317	26	1172	1172	NUM
iajs-3092	317	27	.	.	PUNCT
iajs-3092	318	1	13	13	NUM
iajs-3092	318	2	.	.	X
iajs-3092	318	3	shaheed	shaheed	PROPN
iajs-3092	318	4	,	,	PUNCT
iajs-3092	318	5	j.k	j.k	PROPN
iajs-3092	318	6	;	;	PUNCT
iajs-3092	318	7	hatam	hatam	PROPN
iajs-3092	318	8	,	,	PUNCT
iajs-3092	318	9	h.y	h.y	PROPN
iajs-3092	318	10	.	.	PROPN
iajs-3092	318	11	,	,	PUNCT
iajs-3092	318	12	"	"	PUNCT
iajs-3092	318	13	loc	loc	ADJ
iajs-3092	318	14	-	-	ADJ
iajs-3092	318	15	hollow	hollow	ADJ
iajs-3092	318	16	fuzzy	fuzzy	ADJ
iajs-3092	318	17	modules	module	NOUN
iajs-3092	318	18	with	with	ADP
iajs-3092	318	19	related	related	ADJ
iajs-3092	318	20	modules	module	NOUN
iajs-3092	318	21	"	"	PUNCT
iajs-3092	318	22	,	,	PUNCT
iajs-3092	318	23	ibn	ibn	PROPN
iajs-3092	318	24	al	al	PROPN
iajs-3092	318	25	haitham	haitham	PROPN
iajs-3092	318	26	journal	journal	PROPN
iajs-3092	318	27	for	for	ADP
iajs-3092	318	28	pure	pure	ADJ
iajs-3092	318	29	and	and	CCONJ
iajs-3092	318	30	applied	applied	ADJ
iajs-3092	318	31	science	science	NOUN
iajs-3092	318	32	,	,	PUNCT
iajs-3092	318	33	2022	2022	NUM
iajs-3092	318	34	,	,	PUNCT
iajs-3092	318	35	35	35	NUM
iajs-3092	318	36	,	,	PUNCT
iajs-3092	318	37	84	84	NUM
iajs-3092	318	38	-	-	SYM
iajs-3092	318	39	96	96	NUM
iajs-3092	318	40	.	.	PUNCT
iajs-3092	319	1	14	14	NUM
iajs-3092	319	2	.	.	PUNCT
iajs-3092	320	1	kumar	kumar	PROPN
iajs-3092	320	2	,	,	PUNCT
iajs-3092	320	3	r.	r.	PROPN
iajs-3092	320	4	"	"	PUNCT
iajs-3092	320	5	fuzzy	fuzzy	ADJ
iajs-3092	320	6	semiprimary	semiprimary	ADJ
iajs-3092	320	7	ideal	ideal	NOUN
iajs-3092	320	8	of	of	ADP
iajs-3092	320	9	rings	ring	NOUN
iajs-3092	320	10	"	"	PUNCT
iajs-3092	320	11	,	,	PUNCT
iajs-3092	320	12	fuzzy	fuzzy	ADJ
iajs-3092	320	13	sets	set	NOUN
iajs-3092	320	14	and	and	CCONJ
iajs-3092	320	15	systems	system	NOUN
iajs-3092	320	16	.	.	PUNCT
iajs-3092	320	17	1991	1991	NUM
iajs-3092	320	18	,	,	PUNCT
iajs-3092	320	19	42	42	NUM
iajs-3092	320	20	,	,	PUNCT
iajs-3092	320	21	263	263	NUM
iajs-3092	320	22	-	-	SYM
iajs-3092	320	23	272	272	NUM
iajs-3092	320	24	.	.	PUNCT
iajs-3092	320	25	15	15	NUM
iajs-3092	320	26	.	.	PUNCT
iajs-3092	321	1	kumar	kumar	PROPN
iajs-3092	321	2	,	,	PUNCT
iajs-3092	321	3	r.	r.	PROPN
iajs-3092	321	4	s.	s.	PROPN
iajs-3092	321	5	;	;	PUNCT
iajs-3092	321	6	bhambri	bhambri	PROPN
iajs-3092	321	7	k.	k.	PROPN
iajs-3092	321	8	;	;	PUNCT
iajs-3092	321	9	kumar	kumar	PROPN
iajs-3092	321	10	,	,	PUNCT
iajs-3092	321	11	p.	p.	NOUN
iajs-3092	321	12	"	"	PUNCT
iajs-3092	321	13	fuzzy	fuzzy	ADJ
iajs-3092	321	14	submodules	submodule	NOUN
iajs-3092	321	15	:	:	PUNCT
iajs-3092	321	16	some	some	DET
iajs-3092	321	17	analogues	analogue	NOUN
iajs-3092	321	18	and	and	CCONJ
iajs-3092	321	19	deviations	deviation	NOUN
iajs-3092	321	20	"	"	PUNCT
iajs-3092	321	21	,	,	PUNCT
iajs-3092	321	22	fuzzy	fuzzy	ADJ
iajs-3092	321	23	sets	set	NOUN
iajs-3092	321	24	and	and	CCONJ
iajs-3092	321	25	systems	system	NOUN
iajs-3092	321	26	.	.	PUNCT
iajs-3092	322	1	1995	1995	NUM
iajs-3092	322	2	,	,	PUNCT
iajs-3092	322	3	70	70	NUM
iajs-3092	322	4	,	,	PUNCT
iajs-3092	322	5	125	125	NUM
iajs-3092	322	6	-	-	SYM
iajs-3092	322	7	130	130	NUM
iajs-3092	322	8	.	.	PUNCT
iajs-3092	323	1	16	16	NUM
iajs-3092	323	2	.	.	PUNCT
iajs-3092	324	1	nimbhorkar	nimbhorkar	PROPN
iajs-3092	324	2	,	,	PUNCT
iajs-3092	324	3	s.	s.	PROPN
iajs-3092	324	4	;	;	PUNCT
iajs-3092	324	5	khubchandani	khubchandani	PROPN
iajs-3092	324	6	,	,	PUNCT
iajs-3092	324	7	j.	j.	PROPN
iajs-3092	324	8	"	"	PUNCT
iajs-3092	324	9	fuzzy	fuzzy	ADJ
iajs-3092	324	10	essential	essential	ADJ
iajs-3092	324	11	submodules	submodule	NOUN
iajs-3092	324	12	with	with	ADP
iajs-3092	324	13	respect	respect	NOUN
iajs-3092	324	14	to	to	ADP
iajs-3092	324	15	an	an	DET
iajs-3092	324	16	arbitrary	arbitrary	ADJ
iajs-3092	324	17	fuzzy	fuzzy	ADJ
iajs-3092	324	18	submodules	submodule	NOUN
iajs-3092	324	19	"	"	PUNCT
iajs-3092	324	20	,	,	PUNCT
iajs-3092	324	21	twms	twms	PROPN
iajs-3092	324	22	j.	j.	PROPN
iajs-3092	324	23	app	app	PROPN
iajs-3092	324	24	.	.	PROPN
iajs-3092	325	1	and	and	CCONJ
iajs-3092	325	2	eng	eng	PROPN
iajs-3092	325	3	.	.	PROPN
iajs-3092	325	4	math	math	PROPN
iajs-3092	325	5	.	.	PUNCT
iajs-3092	326	1	2022	2022	NUM
iajs-3092	326	2	,	,	PUNCT
iajs-3092	326	3	12	12	NUM
iajs-3092	326	4	,	,	PUNCT
iajs-3092	326	5	435	435	NUM
iajs-3092	326	6	-	-	SYM
iajs-3092	326	7	444	444	NUM
iajs-3092	326	8	.	.	NOUN
iajs-3092	327	1	17	17	NUM
iajs-3092	327	2	.	.	PUNCT
iajs-3092	328	1	rabi	rabi	NOUN
iajs-3092	328	2	,	,	PUNCT
iajs-3092	328	3	h.	h.	NOUN
iajs-3092	328	4	"	"	PUNCT
iajs-3092	328	5	prime	prime	ADJ
iajs-3092	328	6	fuzzy	fuzzy	ADJ
iajs-3092	328	7	submodules	submodule	NOUN
iajs-3092	328	8	and	and	CCONJ
iajs-3092	328	9	prime	prime	ADJ
iajs-3092	328	10	fuzzy	fuzzy	ADJ
iajs-3092	328	11	modules	module	NOUN
iajs-3092	328	12	"	"	PUNCT
iajs-3092	328	13	,	,	PUNCT
iajs-3092	328	14	msc	msc	PROPN
iajs-3092	328	15	thesis	thesis	NOUN
iajs-3092	328	16	,	,	PUNCT
iajs-3092	328	17	university	university	NOUN
iajs-3092	328	18	of	of	ADP
iajs-3092	328	19	baghdad	baghdad	PROPN
iajs-3092	328	20	.	.	PUNCT
iajs-3092	329	1	2001	2001	NUM
iajs-3092	329	2	.	.	PUNCT
iajs-3092	330	1	ihjpas	ihjpas	PROPN
iajs-3092	330	2	.	.	PUNCT
iajs-3092	331	1	36	36	NUM
iajs-3092	331	2	(	(	PUNCT
iajs-3092	331	3	3	3	NUM
iajs-3092	331	4	)	)	PUNCT
iajs-3092	331	5	2023	2023	NUM
iajs-3092	331	6	381	381	NUM
iajs-3092	331	7	18	18	NUM
iajs-3092	331	8	.	.	PUNCT
iajs-3092	332	1	hadi	hadi	PROPN
iajs-3092	332	2	,	,	PUNCT
iajs-3092	332	3	g.r	g.r	PROPN
iajs-3092	332	4	;	;	PUNCT
iajs-3092	332	5	hatem	hatem	PROPN
iajs-3092	332	6	,	,	PUNCT
iajs-3092	332	7	y.k	y.k	PROPN
iajs-3092	332	8	.	.	PROPN
iajs-3092	332	9	,	,	PUNCT
iajs-3092	332	10	"	"	PUNCT
iajs-3092	332	11	quasi	quasi	ADJ
iajs-3092	332	12	-	-	ADJ
iajs-3092	332	13	fully	fully	ADV
iajs-3092	332	14	cancellation	cancellation	NOUN
iajs-3092	332	15	fuzzy	fuzzy	ADJ
iajs-3092	332	16	modules	module	NOUN
iajs-3092	332	17	"	"	PUNCT
iajs-3092	332	18	,	,	PUNCT
iajs-3092	332	19	ibn	ibn	PROPN
iajs-3092	332	20	al	al	PROPN
iajs-3092	332	21	haitham	haitham	PROPN
iajs-3092	332	22	journal	journal	PROPN
iajs-3092	332	23	for	for	ADP
iajs-3092	332	24	pure	pure	ADJ
iajs-3092	332	25	and	and	CCONJ
iajs-3092	332	26	applied	applied	ADJ
iajs-3092	332	27	science	science	NOUN
iajs-3092	332	28	,	,	PUNCT
iajs-3092	332	29	2017	2017	NUM
iajs-3092	332	30	,	,	PUNCT
iajs-3092	332	31	30	30	NUM
iajs-3092	332	32	,	,	PUNCT
iajs-3092	332	33	192	192	NUM
iajs-3092	332	34	-	-	SYM
iajs-3092	332	35	207	207	NUM
iajs-3092	332	36	.	.	PUNCT
iajs-3092	333	1	19	19	NUM
iajs-3092	333	2	.	.	X
iajs-3092	333	3	inaam	inaam	PROPN
iajs-3092	333	4	m.	m.	PROPN
iajs-3092	333	5	h.	h.	PROPN
iajs-3092	333	6	;	;	PUNCT
iajs-3092	333	7	maysoun	maysoun	ADV
iajs-3092	333	8	,	,	PUNCT
iajs-3092	333	9	a.	a.	NOUN
iajs-3092	333	10	h.	h.	PROPN
iajs-3092	334	1	"	"	PUNCT
iajs-3092	334	2	cancellation	cancellation	NOUN
iajs-3092	334	3	and	and	CCONJ
iajs-3092	334	4	weakly	weakly	ADJ
iajs-3092	334	5	cancellation	cancellation	NOUN
iajs-3092	334	6	fuzzy	fuzzy	ADJ
iajs-3092	334	7	modules	module	NOUN
iajs-3092	334	8	"	"	PUNCT
iajs-3092	334	9	,	,	PUNCT
iajs-3092	334	10	journal	journal	NOUN
iajs-3092	334	11	of	of	ADP
iajs-3092	334	12	basrah	basrah	PROPN
iajs-3092	334	13	reserches	reserche	NOUN
iajs-3092	334	14	(	(	PUNCT
iajs-3092	334	15	(	(	PUNCT
iajs-3092	334	16	sciences	science	NOUN
iajs-3092	334	17	)	)	PUNCT
iajs-3092	334	18	)	)	PUNCT
iajs-3092	334	19	.	.	PUNCT
iajs-3092	335	1	2011,37	2011,37	NUM
iajs-3092	335	2	4d	4d	NUM
iajs-3092	335	3	,	,	PUNCT
iajs-3092	335	4	20	20	NUM
iajs-3092	335	5	.	.	PUNCT
iajs-3092	336	1	maysoun	maysoun	PROPN
iajs-3092	336	2	,	,	PUNCT
iajs-3092	336	3	a.	a.	NOUN
iajs-3092	336	4	h	h	NOUN
iajs-3092	336	5	;	;	PUNCT
iajs-3092	336	6	hatam	hatam	PROPN
iajs-3092	336	7	h.y	h.y	PROPN
iajs-3092	336	8	.	.	PUNCT
iajs-3092	336	9	"	"	PUNCT
iajs-3092	336	10	fuzzy	fuzzy	ADJ
iajs-3092	336	11	semimaximal	semimaximal	ADJ
iajs-3092	336	12	submodules	submodule	NOUN
iajs-3092	336	13	"	"	PUNCT
iajs-3092	336	14	,	,	PUNCT
iajs-3092	336	15	ibn	ibn	PROPN
iajs-3092	336	16	al	al	PROPN
iajs-3092	336	17	haitham	haitham	PROPN
iajs-3092	336	18	journal	journal	PROPN
iajs-3092	336	19	for	for	ADP
iajs-3092	336	20	pure	pure	ADJ
iajs-3092	336	21	and	and	CCONJ
iajs-3092	336	22	applied	apply	VERB
iajs-3092	336	23	science	science	NOUN
iajs-3092	336	24	.	.	PUNCT
iajs-3092	337	1	2020,33(4	2020,33(4	NOUN
iajs-3092	337	2	)	)	PUNCT
iajs-3092	337	3	,	,	PUNCT
iajs-3092	337	4	137	137	NUM
iajs-3092	337	5	-	-	SYM
iajs-3092	337	6	147	147	NUM
iajs-3092	337	7	.	.	PUNCT
iajs-3092	338	1	21	21	NUM
iajs-3092	338	2	.	.	PUNCT
iajs-3092	339	1	hasan	hasan	PROPN
iajs-3092	339	2	,	,	PUNCT
iajs-3092	339	3	k.m	k.m	PROPN
iajs-3092	339	4	.	.	PROPN
iajs-3092	339	5	;	;	PUNCT
iajs-3092	339	6	hatem	hatem	PROPN
iajs-3092	339	7	,	,	PUNCT
iajs-3092	339	8	y.k	y.k	PROPN
iajs-3092	339	9	.	.	PROPN
iajs-3092	339	10	,	,	PUNCT
iajs-3092	339	11	"	"	PUNCT
iajs-3092	339	12	weak	weak	ADJ
iajs-3092	339	13	essential	essential	ADJ
iajs-3092	339	14	fuzzy	fuzzy	ADJ
iajs-3092	339	15	submodules	submodule	NOUN
iajs-3092	339	16	of	of	ADP
iajs-3092	339	17	fuzzy	fuzzy	ADJ
iajs-3092	339	18	modules	module	NOUN
iajs-3092	339	19	"	"	PUNCT
iajs-3092	339	20	,	,	PUNCT
iajs-3092	339	21	ibn	ibn	PROPN
iajs-3092	339	22	al	al	PROPN
iajs-3092	339	23	haitham	haitham	PROPN
iajs-3092	339	24	journal	journal	PROPN
iajs-3092	339	25	for	for	ADP
iajs-3092	339	26	pure	pure	ADJ
iajs-3092	339	27	and	and	CCONJ
iajs-3092	339	28	applied	apply	VERB
iajs-3092	339	29	science,30	science,30	NOUN
iajs-3092	339	30	,	,	PUNCT
iajs-3092	339	31	2020	2020	NUM
iajs-3092	339	32	,	,	PUNCT
iajs-3092	339	33	65	65	NUM
iajs-3092	339	34	-	-	SYM
iajs-3092	339	35	72	72	NUM
iajs-3092	339	36	.	.	PUNCT
iajs-3092	339	37	22	22	NUM
iajs-3092	339	38	.	.	PUNCT
iajs-3092	340	1	shymaa	shymaa	PROPN
iajs-3092	340	2	,	,	PUNCT
iajs-3092	340	3	a.	a.	NOUN
iajs-3092	340	4	m.	m.	NOUN
iajs-3092	340	5	"	"	PUNCT
iajs-3092	340	6	new	new	ADJ
iajs-3092	340	7	types	type	NOUN
iajs-3092	340	8	of	of	ADP
iajs-3092	340	9	regular	regular	ADJ
iajs-3092	340	10	fuzzy	fuzzy	ADJ
iajs-3092	340	11	modules	module	NOUN
iajs-3092	340	12	and	and	CCONJ
iajs-3092	340	13	pure	pure	ADJ
iajs-3092	340	14	fuzzy	fuzzy	ADJ
iajs-3092	340	15	submodules	submodule	NOUN
iajs-3092	340	16	"	"	PUNCT
iajs-3092	340	17	,	,	PUNCT
iajs-3092	340	18	msc	msc	PROPN
iajs-3092	340	19	thesis	thesis	NOUN
iajs-3092	340	20	,	,	PUNCT
iajs-3092	340	21	college	college	NOUN
iajs-3092	340	22	of	of	ADP
iajs-3092	340	23	education	education	NOUN
iajs-3092	340	24	for	for	ADP
iajs-3092	340	25	pure	pure	ADJ
iajs-3092	340	26	science(ibn	science(ibn	PROPN
iajs-3092	340	27	-	-	PUNCT
iajs-3092	340	28	al	al	PROPN
iajs-3092	340	29	-	-	PUNCT
iajs-3092	340	30	haitham	haitham	PROPN
iajs-3092	340	31	)	)	PUNCT
iajs-3092	340	32	,	,	PUNCT
iajs-3092	340	33	university	university	NOUN
iajs-3092	340	34	of	of	ADP
iajs-3092	340	35	baghdad	baghdad	PROPN
iajs-3092	340	36	.	.	PUNCT
iajs-3092	341	1	2018	2018	NUM
iajs-3092	341	2	.	.	PUNCT
iajs-3092	342	1	23	23	NUM
iajs-3092	342	2	.	.	PUNCT
iajs-3092	343	1	hatam	hatam	NOUN
iajs-3092	343	2	,	,	PUNCT
iajs-3092	343	3	y.k	y.k	PROPN
iajs-3092	343	4	.	.	PUNCT
iajs-3092	343	5	"	"	PUNCT
iajs-3092	343	6	fuzzy	fuzzy	ADJ
iajs-3092	343	7	quasi	quasi	ADJ
iajs-3092	343	8	prime	prime	NOUN
iajs-3092	343	9	modules	module	NOUN
iajs-3092	343	10	and	and	CCONJ
iajs-3092	343	11	fuzzy	fuzzy	ADJ
iajs-3092	343	12	quasi	quasi	NOUN
iajs-3092	343	13	–	–	PUNCT
iajs-3092	343	14	prime	prime	ADJ
iajs-3092	343	15	submodules	submodule	NOUN
iajs-3092	343	16	"	"	PUNCT
iajs-3092	343	17	,	,	PUNCT
iajs-3092	343	18	ms.c	ms.c	ADJ
iajs-3092	343	19	.	.	PUNCT
iajs-3092	344	1	thesis	thesis	NOUN
iajs-3092	344	2	,	,	PUNCT
iajs-3092	344	3	college	college	NOUN
iajs-3092	344	4	of	of	ADP
iajs-3092	344	5	education	education	NOUN
iajs-3092	344	6	for	for	ADP
iajs-3092	344	7	pure	pure	ADJ
iajs-3092	344	8	science(ibn	science(ibn	PROPN
iajs-3092	344	9	-	-	PUNCT
iajs-3092	344	10	al	al	PROPN
iajs-3092	344	11	-	-	PUNCT
iajs-3092	344	12	haitham	haitham	PROPN
iajs-3092	344	13	)	)	PUNCT
iajs-3092	344	14	,	,	PUNCT
iajs-3092	344	15	university	university	NOUN
iajs-3092	344	16	of	of	ADP
iajs-3092	344	17	baghdad	baghdad	PROPN
iajs-3092	344	18	,	,	PUNCT
iajs-3092	344	19	2001	2001	NUM
iajs-3092	344	20	.	.	PUNCT
iajs-3092	345	1	24	24	NUM
iajs-3092	345	2	.	.	PUNCT
iajs-3092	345	3	mahmood	mahmood	PROPN
iajs-3092	345	4	,	,	PUNCT
iajs-3092	345	5	s.	s.	PROPN
iajs-3092	345	6	f.	f.	PROPN
iajs-3092	346	1	"	"	PUNCT
iajs-3092	346	2	visible(w	visible(w	ADV
iajs-3092	346	3	-	-	PUNCT
iajs-3092	346	4	visible	visible	ADJ
iajs-3092	346	5	)	)	PUNCT
iajs-3092	346	6	submodules	submodule	NOUN
iajs-3092	346	7	and	and	CCONJ
iajs-3092	346	8	fully	fully	ADV
iajs-3092	346	9	visible(w	visible(w	ADV
iajs-3092	346	10	-	-	PUNCT
iajs-3092	346	11	visible	visible	ADJ
iajs-3092	346	12	)	)	PUNCT
iajs-3092	346	13	modules	module	NOUN
iajs-3092	346	14	with	with	ADP
iajs-3092	346	15	some	some	PRON
iajs-3092	346	16	of	of	ADP
iajs-3092	346	17	their	their	PRON
iajs-3092	346	18	generalizations	generalization	NOUN
iajs-3092	346	19	"	"	PUNCT
iajs-3092	346	20	,	,	PUNCT
iajs-3092	346	21	ph.d	ph.d	PROPN
iajs-3092	346	22	thesis	thesis	NOUN
iajs-3092	346	23	,	,	PUNCT
iajs-3092	346	24	college	college	NOUN
iajs-3092	346	25	of	of	ADP
iajs-3092	346	26	education	education	NOUN
iajs-3092	346	27	for	for	ADP
iajs-3092	346	28	pure	pure	ADJ
iajs-3092	346	29	science(ibn	science(ibn	NOUN
iajs-3092	346	30	-	-	PUNCT
iajs-3092	346	31	alhaitham	alhaitham	NOUN
iajs-3092	346	32	)	)	PUNCT
iajs-3092	346	33	,	,	PUNCT
iajs-3092	346	34	university	university	NOUN
iajs-3092	346	35	of	of	ADP
iajs-3092	346	36	baghdad	baghdad	PROPN
iajs-3092	346	37	,	,	PUNCT
iajs-3092	346	38	2019	2019	NUM
iajs-3092	346	39	.	.	PUNCT
iajs-3092	347	1	25	25	NUM
iajs-3092	347	2	.	.	PUNCT
iajs-3092	348	1	maysoun	maysoun	PROPN
iajs-3092	348	2	,	,	PUNCT
iajs-3092	348	3	a.	a.	PROPN
iajs-3092	348	4	h.	h.	PROPN
iajs-3092	349	1	"	"	PUNCT
iajs-3092	349	2	f	f	X
iajs-3092	349	3	-	-	PUNCT
iajs-3092	349	4	regular	regular	ADJ
iajs-3092	349	5	fuzzy	fuzzy	ADJ
iajs-3092	349	6	modules	module	NOUN
iajs-3092	349	7	"	"	PUNCT
iajs-3092	349	8	,	,	PUNCT
iajs-3092	349	9	m.sc.thesis	m.sc.thesis	NOUN
iajs-3092	349	10	,	,	PUNCT
iajs-3092	349	11	college	college	NOUN
iajs-3092	349	12	of	of	ADP
iajs-3092	349	13	education	education	NOUN
iajs-3092	349	14	for	for	ADP
iajs-3092	349	15	pure	pure	ADJ
iajs-3092	349	16	science(ibn	science(ibn	PROPN
iajs-3092	349	17	-	-	PUNCT
iajs-3092	349	18	al	al	PROPN
iajs-3092	349	19	-	-	PUNCT
iajs-3092	349	20	haitham	haitham	PROPN
iajs-3092	349	21	)	)	PUNCT
iajs-3092	349	22	,	,	PUNCT
iajs-3092	349	23	university	university	NOUN
iajs-3092	349	24	of	of	ADP
iajs-3092	349	25	baghdad	baghdad	PROPN
iajs-3092	349	26	,	,	PUNCT
iajs-3092	349	27	2002	2002	NUM
iajs-3092	349	28	.	.	PUNCT
iajs-3092	350	1	26	26	NUM
iajs-3092	350	2	.	.	PUNCT
iajs-3092	351	1	s.	s.	PROPN
iajs-3092	351	2	f.	f.	PROPN
iajs-3092	351	3	,	,	PUNCT
iajs-3092	351	4	mahmood	mahmood	PROPN
iajs-3092	351	5	;	;	PUNCT
iajs-3092	351	6	n.s	n.s	PROPN
iajs-3092	351	7	.	.	PROPN
iajs-3092	351	8	buthyna	buthyna	PROPN
iajs-3092	351	9	,	,	PUNCT
iajs-3092	351	10	"	"	PUNCT
iajs-3092	351	11	behavior	behavior	NOUN
iajs-3092	351	12	of	of	ADP
iajs-3092	351	13	visible	visible	ADJ
iajs-3092	351	14	submodules	submodule	NOUN
iajs-3092	351	15	in	in	ADP
iajs-3092	351	16	the	the	DET
iajs-3092	351	17	class	class	NOUN
iajs-3092	351	18	of	of	ADP
iajs-3092	351	19	multiplication	multiplication	NOUN
iajs-3092	351	20	modules	module	NOUN
iajs-3092	351	21	"	"	PUNCT
iajs-3092	351	22	.	.	PUNCT
iajs-3092	352	1	aviable	aviable	ADJ
iajs-3092	352	2	online	online	ADV
iajs-3092	352	3	:	:	PUNCT
iajs-3092	352	4	https://www.researchgate.net/publication/3327113	https://www.researchgate.net/publication/3327113	NOUN
iajs-3092	352	5	92	92	NUM
iajs-3092	352	6	,	,	PUNCT
iajs-3092	352	7	(	(	PUNCT
iajs-3092	352	8	2019	2019	NUM
iajs-3092	352	9	)	)	PUNCT
iajs-3092	352	10	.	.	PUNCT
iajs-3092	353	1	27	27	NUM
iajs-3092	353	2	.	.	X
iajs-3092	353	3	mordeson	mordeson	NOUN
iajs-3092	353	4	,	,	PUNCT
iajs-3092	353	5	j.	j.	PROPN
iajs-3092	353	6	n.	n.	PROPN
iajs-3092	353	7	;	;	PUNCT
iajs-3092	353	8	malik	malik	PROPN
iajs-3092	353	9	,	,	PUNCT
iajs-3092	353	10	d.	d.	PROPN
iajs-3092	353	11	s.	s.	PROPN
iajs-3092	353	12	,	,	PUNCT
iajs-3092	353	13	"	"	PUNCT
iajs-3092	353	14	fuzzy	fuzzy	ADJ
iajs-3092	353	15	commutative	commutative	ADJ
iajs-3092	353	16	algebra	algebra	NOUN
iajs-3092	353	17	"	"	PUNCT
iajs-3092	353	18	;	;	PUNCT
iajs-3092	353	19	world	world	NOUN
iajs-3092	353	20	scientific	scientific	ADJ
iajs-3092	353	21	publishing	publishing	NOUN
iajs-3092	353	22	co.pre.ltd	co.pre.ltd	PROPN
iajs-3092	353	23	:	:	PUNCT
iajs-3092	353	24	usa	usa	PROPN
iajs-3092	353	25	,	,	PUNCT
iajs-3092	353	26	1998	1998	NUM
iajs-3092	353	27	;	;	PUNCT
iajs-3092	353	28	isbn	isbn	ADJ
iajs-3092	353	29	981	981	NUM
iajs-3092	353	30	-	-	PUNCT
iajs-3092	353	31	02	02	NUM
iajs-3092	353	32	-	-	PUNCT
iajs-3092	353	33	3628	3628	NUM
iajs-3092	353	34	-	-	PUNCT
iajs-3092	353	35	x.	x.	NOUN
iajs-3092	353	36	28	28	NUM
iajs-3092	353	37	.	.	PUNCT
iajs-3092	354	1	mahdi	mahdi	PROPN
iajs-3092	354	2	,	,	PUNCT
iajs-3092	354	3	s.	s.	PROPN
iajs-3092	354	4	a.	a.	PROPN
iajs-3092	355	1	"	"	PUNCT
iajs-3092	355	2	on	on	ADP
iajs-3092	355	3	fully	fully	ADV
iajs-3092	355	4	stable	stable	ADJ
iajs-3092	355	5	module	module	NOUN
iajs-3092	355	6	"	"	PUNCT
iajs-3092	355	7	,	,	PUNCT
iajs-3092	355	8	ph.d	ph.d	PROPN
iajs-3092	355	9	.	.	PUNCT
iajs-3092	356	1	thesis	thesis	NOUN
iajs-3092	356	2	,	,	PUNCT
iajs-3092	356	3	university	university	NOUN
iajs-3092	356	4	of	of	ADP
iajs-3092	356	5	baghdad	baghdad	PROPN
iajs-3092	356	6	,	,	PUNCT
iajs-3092	356	7	1991	1991	NUM
iajs-3092	356	8	.	.	PUNCT
iajs-3092	357	1	29	29	NUM
iajs-3092	357	2	.	.	X
iajs-3092	358	1	zahedi	zahedi	PROPN
iajs-3092	358	2	,	,	PUNCT
iajs-3092	358	3	m.m	m.m	PROPN
iajs-3092	358	4	.	.	PROPN
iajs-3092	358	5	;	;	PUNCT
iajs-3092	358	6	ameri	ameri	PROPN
iajs-3092	358	7	,	,	PUNCT
iajs-3092	358	8	r.a	r.a	PROPN
iajs-3092	358	9	.	.	PUNCT
iajs-3092	358	10	"	"	PUNCT
iajs-3092	358	11	fuzzy	fuzzy	ADJ
iajs-3092	358	12	flat	flat	ADJ
iajs-3092	358	13	and	and	CCONJ
iajs-3092	358	14	faithfully	faithfully	ADV
iajs-3092	358	15	flat	flat	ADJ
iajs-3092	358	16	r	r	NOUN
iajs-3092	358	17	-	-	PUNCT
iajs-3092	358	18	modules	module	NOUN
iajs-3092	358	19	"	"	PUNCT
iajs-3092	358	20	,	,	PUNCT
iajs-3092	358	21	the	the	DET
iajs-3092	358	22	journal	journal	NOUN
iajs-3092	358	23	of	of	ADP
iajs-3092	358	24	fuzzy	fuzzy	ADJ
iajs-3092	358	25	mathematics	mathematic	NOUN
iajs-3092	358	26	.	.	PUNCT
iajs-3092	359	1	1997	1997	NUM
iajs-3092	359	2	,	,	PUNCT
iajs-3092	359	3	5	5	NUM
iajs-3092	359	4	,	,	PUNCT
iajs-3092	359	5	855	855	NUM
iajs-3092	359	6	-	-	SYM
iajs-3092	359	7	864	864	NUM
iajs-3092	359	8	.	.	PUNCT