id	sid	tid	token	lemma	pos
iajs-2804	1	1	ibn	ibn	PROPN
iajs-2804	1	2	al	al	PROPN
iajs-2804	1	3	-	-	PUNCT
iajs-2804	1	4	haitham	haitham	PROPN
iajs-2804	1	5	jour	jour	X
iajs-2804	1	6	.	.	PROPN
iajs-2804	1	7	for	for	ADP
iajs-2804	1	8	pure	pure	ADJ
iajs-2804	1	9	&	&	CCONJ
iajs-2804	1	10	appl	appl	PROPN
iajs-2804	1	11	.	.	PUNCT
iajs-2804	2	1	sci	sci	PROPN
iajs-2804	2	2	.	.	PROPN
iajs-2804	3	1	53	53	NUM
iajs-2804	3	2	(	(	PUNCT
iajs-2804	3	3	1)2022	1)2022	PROPN
iajs-2804	3	4	102	102	NUM
iajs-2804	3	5	this	this	DET
iajs-2804	3	6	work	work	NOUN
iajs-2804	3	7	is	be	AUX
iajs-2804	3	8	licensed	license	VERB
iajs-2804	3	9	under	under	ADP
iajs-2804	3	10	a	a	DET
iajs-2804	3	11	creative	creative	ADJ
iajs-2804	3	12	commons	common	NOUN
iajs-2804	3	13	attribution	attribution	NOUN
iajs-2804	3	14	4.0	4.0	NUM
iajs-2804	3	15	international	international	ADJ
iajs-2804	3	16	license	license	NOUN
iajs-2804	3	17	.	.	PUNCT
iajs-2804	4	1	fuzzy	fuzzy	ADJ
iajs-2804	4	2	soc	soc	NOUN
iajs-2804	4	3	-	-	PUNCT
iajs-2804	4	4	semi	semi	ADJ
iajs-2804	4	5	-	-	ADJ
iajs-2804	4	6	prime	prime	ADJ
iajs-2804	4	7	sub	sub	ADJ
iajs-2804	4	8	-	-	NOUN
iajs-2804	4	9	modules	module	NOUN
iajs-2804	4	10	abstract	abstract	NOUN
iajs-2804	4	11	in	in	ADP
iajs-2804	4	12	this	this	DET
iajs-2804	4	13	paper	paper	NOUN
iajs-2804	4	14	,	,	PUNCT
iajs-2804	4	15	we	we	PRON
iajs-2804	4	16	study	study	VERB
iajs-2804	4	17	a	a	DET
iajs-2804	4	18	new	new	ADJ
iajs-2804	4	19	concept	concept	NOUN
iajs-2804	4	20	of	of	ADP
iajs-2804	4	21	fuzzy	fuzzy	ADJ
iajs-2804	4	22	sub	sub	NOUN
iajs-2804	4	23	-	-	NOUN
iajs-2804	4	24	module	module	NOUN
iajs-2804	4	25	,	,	PUNCT
iajs-2804	4	26	called	call	VERB
iajs-2804	4	27	fuzzy	fuzzy	ADJ
iajs-2804	4	28	socle	socle	NOUN
iajs-2804	4	29	semi	semi	ADJ
iajs-2804	4	30	-	-	ADJ
iajs-2804	4	31	prime	prime	ADJ
iajs-2804	4	32	sub	sub	NOUN
iajs-2804	4	33	-	-	NOUN
iajs-2804	4	34	module	module	NOUN
iajs-2804	4	35	that	that	PRON
iajs-2804	4	36	is	be	AUX
iajs-2804	4	37	a	a	DET
iajs-2804	4	38	generalization	generalization	NOUN
iajs-2804	4	39	the	the	DET
iajs-2804	4	40	concept	concept	NOUN
iajs-2804	4	41	of	of	ADP
iajs-2804	4	42	semi	semi	ADJ
iajs-2804	4	43	-	-	ADJ
iajs-2804	4	44	prime	prime	ADJ
iajs-2804	4	45	fuzzy	fuzzy	ADJ
iajs-2804	4	46	sub	sub	NOUN
iajs-2804	4	47	-	-	NOUN
iajs-2804	4	48	module	module	NOUN
iajs-2804	4	49	and	and	CCONJ
iajs-2804	4	50	fuzzy	fuzzy	ADJ
iajs-2804	4	51	of	of	ADP
iajs-2804	4	52	approximately	approximately	ADV
iajs-2804	4	53	semi	semi	ADJ
iajs-2804	4	54	-	-	ADJ
iajs-2804	4	55	prime	prime	ADJ
iajs-2804	4	56	sub	sub	NOUN
iajs-2804	4	57	-	-	NOUN
iajs-2804	4	58	module	module	NOUN
iajs-2804	4	59	in	in	ADP
iajs-2804	4	60	the	the	DET
iajs-2804	4	61	ordinary	ordinary	ADJ
iajs-2804	4	62	sense	sense	NOUN
iajs-2804	4	63	.	.	PUNCT
iajs-2804	5	1	this	this	PRON
iajs-2804	5	2	leads	lead	VERB
iajs-2804	5	3	us	we	PRON
iajs-2804	5	4	to	to	PART
iajs-2804	5	5	introduce	introduce	VERB
iajs-2804	5	6	level	level	NOUN
iajs-2804	5	7	property	property	NOUN
iajs-2804	5	8	which	which	PRON
iajs-2804	5	9	studies	study	VERB
iajs-2804	5	10	the	the	DET
iajs-2804	5	11	relation	relation	NOUN
iajs-2804	5	12	between	between	ADP
iajs-2804	5	13	the	the	DET
iajs-2804	5	14	ordinary	ordinary	ADJ
iajs-2804	5	15	and	and	CCONJ
iajs-2804	5	16	fuzzy	fuzzy	ADJ
iajs-2804	5	17	sense	sense	NOUN
iajs-2804	5	18	of	of	ADP
iajs-2804	5	19	approximately	approximately	ADV
iajs-2804	5	20	semi	semi	ADJ
iajs-2804	5	21	-	-	ADJ
iajs-2804	5	22	prime	prime	ADJ
iajs-2804	5	23	sub	sub	NOUN
iajs-2804	5	24	-	-	NOUN
iajs-2804	5	25	module	module	NOUN
iajs-2804	5	26	.	.	PUNCT
iajs-2804	6	1	also	also	ADV
iajs-2804	6	2	,	,	PUNCT
iajs-2804	6	3	some	some	PRON
iajs-2804	6	4	of	of	ADP
iajs-2804	6	5	its	its	PRON
iajs-2804	6	6	characteristics	characteristic	NOUN
iajs-2804	6	7	and	and	CCONJ
iajs-2804	6	8	notions	notion	NOUN
iajs-2804	6	9	such	such	ADJ
iajs-2804	6	10	as	as	ADP
iajs-2804	6	11	the	the	DET
iajs-2804	6	12	intersection	intersection	NOUN
iajs-2804	6	13	,	,	PUNCT
iajs-2804	6	14	image	image	NOUN
iajs-2804	6	15	and	and	CCONJ
iajs-2804	6	16	external	external	ADJ
iajs-2804	6	17	direct	direct	ADJ
iajs-2804	6	18	sum	sum	NOUN
iajs-2804	6	19	of	of	ADP
iajs-2804	6	20	fuzzy	fuzzy	ADJ
iajs-2804	6	21	socle	socle	NOUN
iajs-2804	6	22	semi	semi	ADJ
iajs-2804	6	23	-	-	ADJ
iajs-2804	6	24	prime	prime	ADJ
iajs-2804	6	25	sub	sub	NOUN
iajs-2804	6	26	-	-	NOUN
iajs-2804	6	27	modules	module	NOUN
iajs-2804	6	28	are	be	AUX
iajs-2804	6	29	introduced	introduce	VERB
iajs-2804	6	30	.	.	PUNCT
iajs-2804	7	1	furthermore	furthermore	ADV
iajs-2804	7	2	,	,	PUNCT
iajs-2804	7	3	the	the	DET
iajs-2804	7	4	relation	relation	NOUN
iajs-2804	7	5	between	between	ADP
iajs-2804	7	6	the	the	DET
iajs-2804	7	7	fuzzy	fuzzy	ADJ
iajs-2804	7	8	socle	socle	NOUN
iajs-2804	7	9	semi	semi	ADJ
iajs-2804	7	10	-	-	ADJ
iajs-2804	7	11	prime	prime	ADJ
iajs-2804	7	12	sub	sub	NOUN
iajs-2804	7	13	-	-	NOUN
iajs-2804	7	14	module	module	ADJ
iajs-2804	7	15	and	and	CCONJ
iajs-2804	7	16	other	other	ADJ
iajs-2804	7	17	types	type	NOUN
iajs-2804	7	18	of	of	ADP
iajs-2804	7	19	fuzzy	fuzzy	ADJ
iajs-2804	7	20	sub	sub	ADJ
iajs-2804	7	21	-	-	ADJ
iajs-2804	7	22	module	module	NOUN
iajs-2804	7	23	presented	present	VERB
iajs-2804	7	24	.	.	PUNCT
iajs-2804	8	1	keyword	keyword	NOUN
iajs-2804	8	2	:	:	PUNCT
iajs-2804	9	1	ℱ-module	ℱ-module	PROPN
iajs-2804	9	2	,	,	PUNCT
iajs-2804	9	3	ℱ-sub	ℱ-sub	NOUN
iajs-2804	9	4	-	-	NOUN
iajs-2804	9	5	module	module	NOUN
iajs-2804	9	6	,	,	PUNCT
iajs-2804	9	7	ℱ-prime	ℱ-prime	PROPN
iajs-2804	9	8	sub	sub	NOUN
iajs-2804	9	9	-	-	NOUN
iajs-2804	9	10	module	module	NOUN
iajs-2804	9	11	,	,	PUNCT
iajs-2804	9	12	socle	socle	NOUN
iajs-2804	9	13	of	of	ADP
iajs-2804	9	14	ℱ-module	ℱ-module	PROPN
iajs-2804	9	15	.	.	PUNCT
iajs-2804	10	1	1.introduction	1.introduction	NUM
iajs-2804	10	2	the	the	DET
iajs-2804	10	3	concept	concept	NOUN
iajs-2804	10	4	of	of	ADP
iajs-2804	10	5	fuzzy	fuzzy	ADJ
iajs-2804	10	6	sets	set	NOUN
iajs-2804	10	7	was	be	AUX
iajs-2804	10	8	introduced	introduce	VERB
iajs-2804	10	9	by	by	ADP
iajs-2804	10	10	zadeh	zadeh	PROPN
iajs-2804	10	11	in1965[1	in1965[1	PROPN
iajs-2804	10	12	]	]	X
iajs-2804	10	13	.	.	PUNCT
iajs-2804	11	1	many	many	ADJ
iajs-2804	11	2	authors	author	NOUN
iajs-2804	11	3	indeed	indeed	ADV
iajs-2804	11	4	presented	present	VERB
iajs-2804	11	5	fuzzy	fuzzy	ADJ
iajs-2804	11	6	subrings	subring	NOUN
iajs-2804	11	7	and	and	CCONJ
iajs-2804	11	8	fuzzy	fuzzy	ADJ
iajs-2804	11	9	ideals	ideal	NOUN
iajs-2804	11	10	.	.	PUNCT
iajs-2804	12	1	the	the	DET
iajs-2804	12	2	concept	concept	NOUN
iajs-2804	12	3	of	of	ADP
iajs-2804	12	4	fuzzy	fuzzy	ADJ
iajs-2804	12	5	module	module	NOUN
iajs-2804	12	6	was	be	AUX
iajs-2804	12	7	introduced	introduce	VERB
iajs-2804	12	8	by	by	ADP
iajs-2804	12	9	negoita	negoita	PROPN
iajs-2804	12	10	and	and	CCONJ
iajs-2804	12	11	relescu	relescu	NOUN
iajs-2804	12	12	in	in	ADP
iajs-2804	12	13	1975	1975	NUM
iajs-2804	12	14	[	[	X
iajs-2804	12	15	2	2	NUM
iajs-2804	12	16	]	]	PUNCT
iajs-2804	12	17	.	.	PUNCT
iajs-2804	13	1	since	since	SCONJ
iajs-2804	13	2	then	then	ADV
iajs-2804	13	3	several	several	ADJ
iajs-2804	13	4	authors	author	NOUN
iajs-2804	13	5	have	have	AUX
iajs-2804	13	6	studied	study	VERB
iajs-2804	13	7	fuzzy	fuzzy	ADJ
iajs-2804	13	8	modules	module	NOUN
iajs-2804	13	9	.	.	PUNCT
iajs-2804	14	1	the	the	DET
iajs-2804	14	2	concept	concept	NOUN
iajs-2804	14	3	of	of	ADP
iajs-2804	14	4	semi	semi	ADJ
iajs-2804	14	5	-	-	ADJ
iajs-2804	14	6	prime	prime	ADJ
iajs-2804	14	7	fuzzy	fuzzy	ADJ
iajs-2804	14	8	sub	sub	NOUN
iajs-2804	14	9	-	-	NOUN
iajs-2804	14	10	module	module	NOUN
iajs-2804	14	11	was	be	AUX
iajs-2804	14	12	introduced	introduce	VERB
iajs-2804	14	13	by	by	ADP
iajs-2804	14	14	rabi	rabi	NOUN
iajs-2804	14	15	2004[3	2004[3	NUM
iajs-2804	14	16	]	]	X
iajs-2804	14	17	.	.	PUNCT
iajs-2804	15	1	the	the	DET
iajs-2804	15	2	concept	concept	NOUN
iajs-2804	15	3	of	of	ADP
iajs-2804	15	4	approximately	approximately	ADV
iajs-2804	15	5	semi	semi	ADJ
iajs-2804	15	6	-	-	ADJ
iajs-2804	15	7	prime	prime	ADJ
iajs-2804	15	8	sub	sub	NOUN
iajs-2804	15	9	-	-	NOUN
iajs-2804	15	10	module	module	NOUN
iajs-2804	15	11	was	be	AUX
iajs-2804	15	12	introduced	introduce	VERB
iajs-2804	15	13	by	by	ADP
iajs-2804	15	14	ali	ali	PROPN
iajs-2804	15	15	2019[4	2019[4	NUM
iajs-2804	15	16	]	]	PUNCT
iajs-2804	15	17	.	.	PUNCT
iajs-2804	16	1	the	the	DET
iajs-2804	16	2	socle	socle	NOUN
iajs-2804	16	3	of	of	ADP
iajs-2804	16	4	m	m	PROPN
iajs-2804	16	5	is	be	AUX
iajs-2804	16	6	a	a	DET
iajs-2804	16	7	summation	summation	NOUN
iajs-2804	16	8	of	of	ADP
iajs-2804	16	9	simple	simple	ADJ
iajs-2804	16	10	sub	sub	NOUN
iajs-2804	16	11	-	-	NOUN
iajs-2804	16	12	modules	module	NOUN
iajs-2804	16	13	of	of	ADP
iajs-2804	16	14	an	an	DET
iajs-2804	16	15	ℛ-module	ℛ-module	PROPN
iajs-2804	16	16	m	m	PROPN
iajs-2804	16	17	and	and	CCONJ
iajs-2804	16	18	denoted	denote	VERB
iajs-2804	16	19	by	by	ADP
iajs-2804	16	20	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	PROPN
iajs-2804	16	21	)	)	PUNCT
iajs-2804	16	22	.	.	PUNCT
iajs-2804	17	1	but	but	CCONJ
iajs-2804	17	2	,	,	PUNCT
iajs-2804	17	3	the	the	DET
iajs-2804	17	4	fuzzy	fuzzy	ADJ
iajs-2804	17	5	socle	socle	NOUN
iajs-2804	17	6	of	of	ADP
iajs-2804	17	7	ℱ-module	ℱ-module	PROPN
iajs-2804	17	8	x	x	PUNCT
iajs-2804	17	9	an	an	DET
iajs-2804	17	10	ℛ-module	ℛ-module	PROPN
iajs-2804	17	11	m	m	NOUN
iajs-2804	17	12	is	be	AUX
iajs-2804	17	13	a	a	DET
iajs-2804	17	14	summation	summation	NOUN
iajs-2804	17	15	of	of	ADP
iajs-2804	17	16	simple	simple	ADJ
iajs-2804	17	17	ℱ-sub	ℱ-sub	NOUN
iajs-2804	17	18	-	-	NOUN
iajs-2804	17	19	modules	module	NOUN
iajs-2804	17	20	of	of	ADP
iajs-2804	17	21	𝑋	𝑋	NOUN
iajs-2804	17	22	and	and	CCONJ
iajs-2804	17	23	denoted	denote	VERB
iajs-2804	17	24	by	by	ADP
iajs-2804	17	25	𝐹	𝐹	PROPN
iajs-2804	17	26	−	−	PROPN
iajs-2804	17	27	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	17	28	)	)	PUNCT
iajs-2804	17	29	.	.	PUNCT
iajs-2804	18	1	ibn	ibn	PROPN
iajs-2804	18	2	al	al	PROPN
iajs-2804	18	3	haitham	haitham	PROPN
iajs-2804	18	4	journal	journal	PROPN
iajs-2804	18	5	for	for	ADP
iajs-2804	18	6	pure	pure	ADJ
iajs-2804	18	7	and	and	CCONJ
iajs-2804	18	8	applied	apply	VERB
iajs-2804	18	9	science	science	NOUN
iajs-2804	18	10	journal	journal	PROPN
iajs-2804	18	11	homepage	homepage	NOUN
iajs-2804	18	12	:	:	PUNCT
iajs-2804	18	13	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2804	18	14	doi	doi	NOUN
iajs-2804	18	15	:	:	PUNCT
iajs-2804	18	16	10.30526/35.1.2804	10.30526/35.1.2804	PROPN
iajs-2804	18	17	article	article	NOUN
iajs-2804	18	18	history	history	NOUN
iajs-2804	18	19	:	:	PUNCT
iajs-2804	18	20	received	receive	VERB
iajs-2804	18	21	1	1	NUM
iajs-2804	18	22	,	,	PUNCT
iajs-2804	18	23	november	november	PROPN
iajs-2804	18	24	,	,	PUNCT
iajs-2804	18	25	2021	2021	NUM
iajs-2804	18	26	,	,	PUNCT
iajs-2804	18	27	accepted,16	accepted,16	NOUN
iajs-2804	18	28	,	,	PUNCT
iajs-2804	18	29	december	december	PROPN
iajs-2804	18	30	,	,	PUNCT
iajs-2804	18	31	2021	2021	NUM
iajs-2804	18	32	,	,	PUNCT
iajs-2804	18	33	published	publish	VERB
iajs-2804	18	34	in	in	ADP
iajs-2804	18	35	january	january	PROPN
iajs-2804	18	36	2022	2022	NUM
iajs-2804	18	37	.	.	PUNCT
iajs-2804	19	1	saad	saad	PROPN
iajs-2804	19	2	s.merie	s.merie	PROPN
iajs-2804	19	3	saadsaleem@uokirkuk.edu.iq	saadsaleem@uokirkuk.edu.iq	PROPN
iajs-2804	19	4	depatment	depatment	NOUN
iajs-2804	19	5	of	of	ADP
iajs-2804	19	6	mthmatics	mthmatic	NOUN
iajs-2804	19	7	,	,	PUNCT
iajs-2804	19	8	college	college	NOUN
iajs-2804	19	9	of	of	ADP
iajs-2804	19	10	education	education	NOUN
iajs-2804	19	11	of	of	ADP
iajs-2804	19	12	pure	pure	ADJ
iajs-2804	19	13	science	science	NOUN
iajs-2804	19	14	,	,	PUNCT
iajs-2804	19	15	ibn	ibn	PROPN
iajs-2804	19	16	alhaitham	alhaitham	NOUN
iajs-2804	19	17	,	,	PUNCT
iajs-2804	19	18	university	university	NOUN
iajs-2804	19	19	of	of	ADP
iajs-2804	19	20	baghdad	baghdad	PROPN
iajs-2804	19	21	,	,	PUNCT
iajs-2804	19	22	baghdad	baghdad	PROPN
iajs-2804	19	23	–	–	PUNCT
iajs-2804	19	24	iraq	iraq	PROPN
iajs-2804	19	25	.	.	PUNCT
iajs-2804	20	1	hatam	hatam	PROPN
iajs-2804	20	2	yahya	yahya	PROPN
iajs-2804	20	3	khalf	khalf	PROPN
iajs-2804	20	4	dr.hatamyahya@yahoo.com	dr.hatamyahya@yahoo.com	VERB
iajs-2804	20	5	depatment	depatment	NOUN
iajs-2804	20	6	of	of	ADP
iajs-2804	20	7	mthmatics	mthmatic	NOUN
iajs-2804	20	8	,	,	PUNCT
iajs-2804	20	9	college	college	NOUN
iajs-2804	20	10	of	of	ADP
iajs-2804	20	11	education	education	NOUN
iajs-2804	20	12	of	of	ADP
iajs-2804	20	13	pure	pure	ADJ
iajs-2804	20	14	science	science	NOUN
iajs-2804	20	15	,	,	PUNCT
iajs-2804	20	16	ibn	ibn	PROPN
iajs-2804	20	17	alhaitham	alhaitham	NOUN
iajs-2804	20	18	,	,	PUNCT
iajs-2804	20	19	university	university	NOUN
iajs-2804	20	20	of	of	ADP
iajs-2804	20	21	baghdad	baghdad	PROPN
iajs-2804	20	22	,	,	PUNCT
iajs-2804	20	23	baghdad	baghdad	PROPN
iajs-2804	20	24	–	–	PUNCT
iajs-2804	20	25	iraq	iraq	PROPN
iajs-2804	20	26	.	.	PUNCT
iajs-2804	21	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2804	21	2	mailto:saadsaleem@uokirkuk.edu.iq	mailto:saadsaleem@uokirkuk.edu.iq	PROPN
iajs-2804	21	3	mailto:dr.hatamyahya@yahoo.com	mailto:dr.hatamyahya@yahoo.com	PROPN
iajs-2804	21	4	ibn	ibn	PROPN
iajs-2804	21	5	al	al	PROPN
iajs-2804	21	6	-	-	PUNCT
iajs-2804	21	7	haitham	haitham	PROPN
iajs-2804	21	8	jour	jour	X
iajs-2804	21	9	.	.	PROPN
iajs-2804	22	1	for	for	ADP
iajs-2804	22	2	pure	pure	ADJ
iajs-2804	22	3	&	&	CCONJ
iajs-2804	22	4	appl	appl	PROPN
iajs-2804	22	5	.	.	PUNCT
iajs-2804	23	1	sci	sci	PROPN
iajs-2804	23	2	.	.	PROPN
iajs-2804	24	1	53	53	NUM
iajs-2804	24	2	(	(	PUNCT
iajs-2804	24	3	1)2022	1)2022	NOUN
iajs-2804	24	4	103	103	NUM
iajs-2804	24	5	preliminaries	preliminary	NOUN
iajs-2804	24	6	"	"	PUNCT
iajs-2804	24	7	there	there	PRON
iajs-2804	24	8	are	be	VERB
iajs-2804	24	9	various	various	ADJ
iajs-2804	24	10	definitions	definition	NOUN
iajs-2804	24	11	and	and	CCONJ
iajs-2804	24	12	characteristics	characteristic	NOUN
iajs-2804	24	13	in	in	ADP
iajs-2804	24	14	this	this	DET
iajs-2804	24	15	section	section	NOUN
iajs-2804	24	16	of	of	ADP
iajs-2804	24	17	ℱ-sets	ℱ-sets	PROPN
iajs-2804	24	18	,	,	PUNCT
iajs-2804	24	19	ℱ-modules	ℱ-modules	PROPN
iajs-2804	24	20	,	,	PUNCT
iajs-2804	24	21	and	and	CCONJ
iajs-2804	24	22	prime	prime	ADJ
iajs-2804	24	23	ℱ-sub	ℱ-sub	NOUN
iajs-2804	24	24	-	-	NOUN
iajs-2804	24	25	modules	module	NOUN
iajs-2804	24	26	.	.	PUNCT
iajs-2804	25	1	definition	definition	NOUN
iajs-2804	25	2	1.1	1.1	NUM
iajs-2804	25	3	[	[	X
iajs-2804	25	4	1	1	NUM
iajs-2804	25	5	]	]	PUNCT
iajs-2804	25	6	let	let	VERB
iajs-2804	25	7	d	d	PRON
iajs-2804	25	8	be	be	AUX
iajs-2804	25	9	a	a	DET
iajs-2804	25	10	nonempty	nonempty	ADV
iajs-2804	25	11	set	set	VERB
iajs-2804	26	1	and	and	CCONJ
iajs-2804	26	2	i	i	PRON
iajs-2804	26	3	is	be	AUX
iajs-2804	26	4	closed	closed	ADJ
iajs-2804	26	5	interval	interval	NOUN
iajs-2804	26	6	[	[	X
iajs-2804	26	7	0	0	NUM
iajs-2804	26	8	,	,	PUNCT
iajs-2804	26	9	1	1	NUM
iajs-2804	26	10	]	]	PUNCT
iajs-2804	26	11	of	of	ADP
iajs-2804	26	12	real	real	ADJ
iajs-2804	26	13	numbers	number	NOUN
iajs-2804	26	14	.	.	PUNCT
iajs-2804	27	1	an	an	DET
iajs-2804	27	2	ℱ-set	ℱ-set	PROPN
iajs-2804	27	3	b	b	PROPN
iajs-2804	27	4	in	in	ADP
iajs-2804	27	5	d	d	PROPN
iajs-2804	27	6	(	(	PUNCT
iajs-2804	27	7	an	an	DET
iajs-2804	27	8	ℱ-subset	ℱ-subset	PROPN
iajs-2804	27	9	of	of	ADP
iajs-2804	27	10	d	d	PROPN
iajs-2804	27	11	)	)	PUNCT
iajs-2804	27	12	is	be	AUX
iajs-2804	27	13	a	a	DET
iajs-2804	27	14	function	function	NOUN
iajs-2804	27	15	from	from	ADP
iajs-2804	27	16	d	d	PROPN
iajs-2804	27	17	into	into	ADP
iajs-2804	27	18	i.	i.	NOUN
iajs-2804	27	19	definition	definition	NOUN
iajs-2804	27	20	1.2	1.2	NUM
iajs-2804	27	21	[	[	X
iajs-2804	27	22	1	1	NUM
iajs-2804	27	23	]	]	PUNCT
iajs-2804	27	24	an	an	DET
iajs-2804	27	25	ℱ-set	ℱ-set	PROPN
iajs-2804	27	26	b	b	PROPN
iajs-2804	27	27	of	of	ADP
iajs-2804	27	28	a	a	DET
iajs-2804	27	29	set	set	NOUN
iajs-2804	27	30	d	d	NOUN
iajs-2804	27	31	is	be	AUX
iajs-2804	27	32	said	say	VERB
iajs-2804	27	33	to	to	PART
iajs-2804	27	34	be	be	AUX
iajs-2804	27	35	ℱ-constant	ℱ-constant	ADJ
iajs-2804	27	36	if	if	SCONJ
iajs-2804	27	37	𝐵(𝑥	𝐵(𝑥	NUM
iajs-2804	27	38	)	)	PUNCT
iajs-2804	27	39	=	=	SYM
iajs-2804	27	40	𝑡	𝑡	NOUN
iajs-2804	27	41	,	,	PUNCT
iajs-2804	27	42	∀	∀	VERB
iajs-2804	27	43	𝑥	𝑥	PRON
iajs-2804	27	44	∈	∈	PROPN
iajs-2804	27	45	𝐷	𝐷	NOUN
iajs-2804	27	46	𝑡	𝑡	PROPN
iajs-2804	27	47	∈	∈	PROPN
iajs-2804	28	1	[	[	X
iajs-2804	28	2	0	0	NUM
iajs-2804	28	3	,	,	PUNCT
iajs-2804	28	4	1	1	NUM
iajs-2804	28	5	]	]	PUNCT
iajs-2804	28	6	definition	definition	NOUN
iajs-2804	28	7	1.3	1.3	NUM
iajs-2804	28	8	[	[	X
iajs-2804	28	9	1	1	NUM
iajs-2804	28	10	]	]	PUNCT
iajs-2804	28	11	let	let	VERB
iajs-2804	28	12	𝑥𝑡	𝑥𝑡	ADP
iajs-2804	28	13	:	:	PUNCT
iajs-2804	28	14	𝐷	𝐷	NOUN
iajs-2804	28	15	→	→	SYM
iajs-2804	28	16	[	[	X
iajs-2804	28	17	0	0	NUM
iajs-2804	28	18	,	,	PUNCT
iajs-2804	28	19	1	1	NUM
iajs-2804	28	20	]	]	PUNCT
iajs-2804	28	21	be	be	AUX
iajs-2804	28	22	an	an	DET
iajs-2804	28	23	ℱ-set	ℱ-set	NOUN
iajs-2804	28	24	in	in	ADP
iajs-2804	28	25	d	d	PROPN
iajs-2804	28	26	,	,	PUNCT
iajs-2804	28	27	where	where	SCONJ
iajs-2804	28	28	x	x	PROPN
iajs-2804	28	29	∈d	∈d	PROPN
iajs-2804	28	30	,	,	PUNCT
iajs-2804	28	31	t	t	PROPN
iajs-2804	28	32	∈	∈	PROPN
iajs-2804	29	1	[	[	X
iajs-2804	29	2	0	0	NUM
iajs-2804	29	3	,	,	PUNCT
iajs-2804	29	4	1	1	NUM
iajs-2804	29	5	]	]	PUNCT
iajs-2804	29	6	defined	define	VERB
iajs-2804	29	7	by	by	ADP
iajs-2804	29	8	:	:	PUNCT
iajs-2804	29	9	𝑥𝑡(𝑦	𝑥𝑡(𝑦	NUM
iajs-2804	29	10	)	)	PUNCT
iajs-2804	29	11	=	=	SYM
iajs-2804	29	12	{	{	PUNCT
iajs-2804	29	13	𝑡	𝑡	X
iajs-2804	29	14	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	29	15	𝑥	𝑥	NOUN
iajs-2804	29	16	=	=	PUNCT
iajs-2804	29	17	𝑦	𝑦	SYM
iajs-2804	29	18	0	0	NUM
iajs-2804	29	19	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	29	20	𝑥	𝑥	PROPN
iajs-2804	29	21	≠	≠	PROPN
iajs-2804	29	22	𝑦	𝑦	NOUN
iajs-2804	29	23	for	for	ADP
iajs-2804	29	24	all	all	DET
iajs-2804	29	25	𝑦	𝑦	PROPN
iajs-2804	29	26	∈	∈	NOUN
iajs-2804	29	27	𝐷.	𝐷.	NOUN
iajs-2804	29	28	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	29	29	is	be	AUX
iajs-2804	29	30	said	say	VERB
iajs-2804	29	31	to	to	PART
iajs-2804	29	32	be	be	AUX
iajs-2804	29	33	an	an	DET
iajs-2804	29	34	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	29	35	or	or	CCONJ
iajs-2804	29	36	ℱ-point	ℱ-point	PROPN
iajs-2804	29	37	in	in	ADP
iajs-2804	29	38	d.	d.	PROPN
iajs-2804	29	39	definition	definition	NOUN
iajs-2804	29	40	1.4	1.4	NUM
iajs-2804	30	1	[	[	X
iajs-2804	30	2	5	5	NUM
iajs-2804	30	3	]	]	PUNCT
iajs-2804	30	4	let	let	VERB
iajs-2804	30	5	𝐵	𝐵	PRON
iajs-2804	30	6	be	be	AUX
iajs-2804	30	7	an	an	DET
iajs-2804	30	8	ℱ-set	ℱ-set	NOUN
iajs-2804	30	9	in	in	ADP
iajs-2804	30	10	d	d	PROPN
iajs-2804	30	11	,	,	PUNCT
iajs-2804	30	12	for	for	ADP
iajs-2804	30	13	all	all	DET
iajs-2804	30	14	t	t	NOUN
iajs-2804	30	15	∈	∈	PROPN
iajs-2804	31	1	[	[	X
iajs-2804	31	2	0	0	NUM
iajs-2804	31	3	,	,	PUNCT
iajs-2804	31	4	1	1	NUM
iajs-2804	31	5	]	]	PUNCT
iajs-2804	31	6	,	,	PUNCT
iajs-2804	31	7	the	the	DET
iajs-2804	31	8	set	set	NOUN
iajs-2804	31	9	b𝑡	b𝑡	NOUN
iajs-2804	31	10	=	=	PUNCT
iajs-2804	31	11	{	{	PUNCT
iajs-2804	31	12	𝑥	𝑥	PROPN
iajs-2804	31	13	∈	∈	PROPN
iajs-2804	31	14	𝐷	𝐷	NOUN
iajs-2804	31	15	;	;	PUNCT
iajs-2804	31	16	b(𝑥	b(𝑥	PROPN
iajs-2804	31	17	)	)	PUNCT
iajs-2804	31	18	≥	≥	PRON
iajs-2804	31	19	𝑡	𝑡	PROPN
iajs-2804	31	20	}	}	PUNCT
iajs-2804	31	21	is	be	AUX
iajs-2804	31	22	said	say	VERB
iajs-2804	31	23	to	to	PART
iajs-2804	31	24	be	be	AUX
iajs-2804	31	25	a	a	DET
iajs-2804	31	26	level	level	NOUN
iajs-2804	31	27	subset	subset	NOUN
iajs-2804	31	28	of	of	ADP
iajs-2804	31	29	𝐵.	𝐵.	PROPN
iajs-2804	31	30	remark	remark	NOUN
iajs-2804	31	31	1.5	1.5	NUM
iajs-2804	31	32	[	[	SYM
iajs-2804	31	33	6	6	NUM
iajs-2804	31	34	]	]	PUNCT
iajs-2804	31	35	let	let	VERB
iajs-2804	31	36	α	α	NOUN
iajs-2804	31	37	and	and	CCONJ
iajs-2804	31	38	β	β	X
iajs-2804	31	39	be	be	AUX
iajs-2804	31	40	two	two	NUM
iajs-2804	31	41	ℱ-sets	ℱ-sets	PROPN
iajs-2804	31	42	in	in	ADP
iajs-2804	31	43	s	s	NOUN
iajs-2804	31	44	,	,	PUNCT
iajs-2804	31	45	then	then	ADV
iajs-2804	31	46	:	:	PUNCT
iajs-2804	31	47	1	1	NUM
iajs-2804	31	48	α	α	X
iajs-2804	31	49	=	=	PUNCT
iajs-2804	31	50	𝛣	𝛣	PROPN
iajs-2804	31	51	if	if	SCONJ
iajs-2804	31	52	and	and	CCONJ
iajs-2804	31	53	only	only	ADV
iajs-2804	31	54	if	if	SCONJ
iajs-2804	31	55	α(𝑥	α(𝑥	NOUN
iajs-2804	31	56	)	)	PUNCT
iajs-2804	31	57	=	=	PUNCT
iajs-2804	32	1	𝛣(𝑥	𝛣(𝑥	VERB
iajs-2804	32	2	)	)	PUNCT
iajs-2804	32	3	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	32	4	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2804	32	5	𝑥	𝑥	X
iajs-2804	32	6	∈	∈	PROPN
iajs-2804	32	7	s.	s.	PROPN
iajs-2804	32	8	2	2	NUM
iajs-2804	32	9	α	α	NOUN
iajs-2804	32	10	⊆	⊆	NUM
iajs-2804	32	11	𝛣	𝛣	PROPN
iajs-2804	32	12	if	if	SCONJ
iajs-2804	32	13	and	and	CCONJ
iajs-2804	32	14	only	only	ADV
iajs-2804	32	15	if	if	SCONJ
iajs-2804	32	16	α(𝑥	α(𝑥	NOUN
iajs-2804	32	17	)	)	PUNCT
iajs-2804	32	18	≤	≤	NOUN
iajs-2804	32	19	𝛣(𝑥	𝛣(𝑥	NUM
iajs-2804	32	20	)	)	PUNCT
iajs-2804	32	21	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	32	22	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2804	32	23	𝑥	𝑥	X
iajs-2804	32	24	∈	∈	PROPN
iajs-2804	32	25	s.	s.	PROPN
iajs-2804	32	26	3	3	NUM
iajs-2804	32	27	α	α	NOUN
iajs-2804	32	28	=	=	PUNCT
iajs-2804	32	29	𝛣	𝛣	PROPN
iajs-2804	32	30	if	if	SCONJ
iajs-2804	32	31	and	and	CCONJ
iajs-2804	32	32	only	only	ADV
iajs-2804	32	33	if	if	SCONJ
iajs-2804	32	34	α𝑡	α𝑡	ADP
iajs-2804	32	35	=	=	SYM
iajs-2804	32	36	𝛣𝑡	𝛣𝑡	PROPN
iajs-2804	32	37	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	32	38	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2804	32	39	𝑡	𝑡	PROPN
iajs-2804	32	40	∈	∈	PROPN
iajs-2804	33	1	[	[	X
iajs-2804	33	2	0,1	0,1	NUM
iajs-2804	33	3	]	]	PUNCT
iajs-2804	33	4	.	.	PUNCT
iajs-2804	34	1	if	if	SCONJ
iajs-2804	34	2	α	α	PRON
iajs-2804	34	3	<	<	X
iajs-2804	34	4	𝐵	𝐵	NOUN
iajs-2804	34	5	and	and	CCONJ
iajs-2804	34	6	there	there	PRON
iajs-2804	34	7	exists	exist	VERB
iajs-2804	34	8	x	x	X
iajs-2804	34	9	∈s	∈s	NOUN
iajs-2804	34	10	such	such	ADJ
iajs-2804	34	11	that	that	SCONJ
iajs-2804	34	12	α(𝑥	α(𝑥	NOUN
iajs-2804	34	13	)	)	PUNCT
iajs-2804	34	14	<	<	X
iajs-2804	34	15	𝛣(𝑥	𝛣(𝑥	NUM
iajs-2804	34	16	)	)	PUNCT
iajs-2804	34	17	,	,	PUNCT
iajs-2804	34	18	then	then	ADV
iajs-2804	34	19	a	a	PRON
iajs-2804	34	20	is	be	AUX
iajs-2804	34	21	a	a	DET
iajs-2804	34	22	proper	proper	ADJ
iajs-2804	34	23	ℱ-subset	ℱ-subset	PROPN
iajs-2804	34	24	of	of	ADP
iajs-2804	34	25	β	β	PROPN
iajs-2804	34	26	and	and	CCONJ
iajs-2804	34	27	written	write	VERB
iajs-2804	34	28	as	as	ADP
iajs-2804	34	29	α	α	X
iajs-2804	34	30	<	<	X
iajs-2804	34	31	𝛣.	𝛣.	NOUN
iajs-2804	34	32	by	by	ADP
iajs-2804	34	33	part	part	NOUN
iajs-2804	34	34	(	(	PUNCT
iajs-2804	34	35	2	2	NUM
iajs-2804	34	36	)	)	PUNCT
iajs-2804	34	37	,	,	PUNCT
iajs-2804	34	38	we	we	PRON
iajs-2804	34	39	can	can	AUX
iajs-2804	34	40	deduce	deduce	VERB
iajs-2804	34	41	that	that	PRON
iajs-2804	34	42	𝑥𝑡	𝑥𝑡	ADP
iajs-2804	34	43	⊆	⊆	NUM
iajs-2804	34	44	α	α	NOUN
iajs-2804	34	45	if	if	SCONJ
iajs-2804	34	46	and	and	CCONJ
iajs-2804	34	47	only	only	ADV
iajs-2804	34	48	if	if	SCONJ
iajs-2804	34	49	α(𝑥	α(𝑥	NOUN
iajs-2804	34	50	)	)	PUNCT
iajs-2804	34	51	≥	≥	NUM
iajs-2804	34	52	𝑡	𝑡	PROPN
iajs-2804	34	53	.	.	PUNCT
iajs-2804	35	1	definition	definition	NOUN
iajs-2804	35	2	1.6	1.6	NUM
iajs-2804	36	1	[	[	X
iajs-2804	36	2	6	6	NUM
iajs-2804	36	3	]	]	X
iajs-2804	36	4	if	if	SCONJ
iajs-2804	36	5	μ	μ	PROPN
iajs-2804	36	6	is	be	AUX
iajs-2804	36	7	an	an	DET
iajs-2804	36	8	ℛ-module	ℛ-module	PROPN
iajs-2804	36	9	.	.	PUNCT
iajs-2804	36	10	an	an	DET
iajs-2804	36	11	ℱ-set	ℱ-set	NOUN
iajs-2804	36	12	x	x	PROPN
iajs-2804	36	13	of	of	ADP
iajs-2804	36	14	μ	μ	PROPN
iajs-2804	36	15	is	be	AUX
iajs-2804	36	16	called	call	VERB
iajs-2804	36	17	ℱ-module	ℱ-module	PROPN
iajs-2804	36	18	of	of	ADP
iajs-2804	36	19	an	an	DET
iajs-2804	36	20	ℛ-module	ℛ-module	PROPN
iajs-2804	36	21	μ	μ	PROPN
iajs-2804	36	22	if	if	SCONJ
iajs-2804	36	23	:	:	PUNCT
iajs-2804	36	24	1	1	NUM
iajs-2804	36	25	𝑋(𝑥	𝑋(𝑥	NUM
iajs-2804	36	26	−	−	PROPN
iajs-2804	36	27	𝑦	𝑦	NUM
iajs-2804	36	28	)	)	PUNCT
iajs-2804	36	29	≥	≥	NOUN
iajs-2804	36	30	min{𝑋(𝑥	min{𝑋(𝑥	PROPN
iajs-2804	36	31	)	)	PUNCT
iajs-2804	36	32	,	,	PUNCT
iajs-2804	36	33	𝑋(𝑦)}𝑓𝑜𝑟	𝑋(𝑦)}𝑓𝑜𝑟	NOUN
iajs-2804	36	34	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-2804	36	35	𝑥	𝑥	NOUN
iajs-2804	36	36	,	,	PUNCT
iajs-2804	36	37	𝑦	𝑦	PROPN
iajs-2804	36	38	∈	∈	NOUN
iajs-2804	36	39	μ	μ	NUM
iajs-2804	36	40	}	}	PUNCT
iajs-2804	36	41	.	.	PUNCT
iajs-2804	37	1	2	2	NUM
iajs-2804	37	2	𝑋(𝑟𝑥	𝑋(𝑟𝑥	NOUN
iajs-2804	37	3	)	)	PUNCT
iajs-2804	37	4	≥	≥	NOUN
iajs-2804	37	5	𝑋(𝑥	𝑋(𝑥	NUM
iajs-2804	37	6	)	)	PUNCT
iajs-2804	37	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	37	8	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2804	37	9	𝑥	𝑥	PRON
iajs-2804	37	10	∈	∈	PROPN
iajs-2804	37	11	μ	μ	PROPN
iajs-2804	37	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2804	37	13	𝑟	𝑟	X
iajs-2804	37	14	∈	∈	PROPN
iajs-2804	37	15	ℛ	ℛ	PROPN
iajs-2804	37	16	.	.	NOUN
iajs-2804	37	17	3	3	NUM
iajs-2804	37	18	𝑋(0	𝑋(0	NOUN
iajs-2804	37	19	)	)	PUNCT
iajs-2804	37	20	=	=	SYM
iajs-2804	37	21	1	1	X
iajs-2804	37	22	.	.	X
iajs-2804	37	23	proposition	proposition	NOUN
iajs-2804	37	24	1.7	1.7	NUM
iajs-2804	38	1	[	[	X
iajs-2804	38	2	7	7	NUM
iajs-2804	38	3	]	]	PUNCT
iajs-2804	38	4	let	let	VERB
iajs-2804	38	5	𝐶	𝐶	PROPN
iajs-2804	38	6	be	be	AUX
iajs-2804	38	7	an	an	DET
iajs-2804	38	8	ℱ-set	ℱ-set	NOUN
iajs-2804	38	9	of	of	ADP
iajs-2804	38	10	an	an	DET
iajs-2804	38	11	ℛ-module	ℛ-module	PROPN
iajs-2804	38	12	μ	μ	PROPN
iajs-2804	38	13	.	.	PUNCT
iajs-2804	39	1	then	then	ADV
iajs-2804	39	2	the	the	DET
iajs-2804	39	3	level	level	NOUN
iajs-2804	39	4	subset	subset	VERB
iajs-2804	39	5	c𝑡	c𝑡	NOUN
iajs-2804	39	6	of	of	ADP
iajs-2804	39	7	μ	μ	PROPN
iajs-2804	39	8	,	,	PUNCT
iajs-2804	39	9	∀	∀	X
iajs-2804	39	10	t	t	NOUN
iajs-2804	39	11	∈	∈	PROPN
iajs-2804	40	1	[	[	X
iajs-2804	40	2	0	0	NUM
iajs-2804	40	3	,	,	PUNCT
iajs-2804	40	4	1	1	NUM
iajs-2804	40	5	]	]	PUNCT
iajs-2804	40	6	is	be	AUX
iajs-2804	40	7	a	a	DET
iajs-2804	40	8	submodule	submodule	NOUN
iajs-2804	40	9	of	of	ADP
iajs-2804	40	10	m	m	PROPN
iajs-2804	40	11	if	if	SCONJ
iajs-2804	40	12	and	and	CCONJ
iajs-2804	40	13	only	only	ADV
iajs-2804	40	14	if	if	SCONJ
iajs-2804	40	15	c	c	PROPN
iajs-2804	40	16	is	be	AUX
iajs-2804	40	17	an	an	DET
iajs-2804	40	18	ℱ-sub	ℱ-sub	NOUN
iajs-2804	40	19	-	-	NOUN
iajs-2804	40	20	module	module	NOUN
iajs-2804	40	21	of	of	ADP
iajs-2804	40	22	ℱ-module	ℱ-module	PROPN
iajs-2804	40	23	of	of	ADP
iajs-2804	40	24	an	an	DET
iajs-2804	40	25	ℛ-module	ℛ-module	PROPN
iajs-2804	40	26	μ	μ	PROPN
iajs-2804	40	27	.	.	PUNCT
iajs-2804	41	1	definition	definition	NOUN
iajs-2804	41	2	1.8	1.8	NUM
iajs-2804	41	3	[	[	SYM
iajs-2804	41	4	8	8	NUM
iajs-2804	41	5	]	]	PUNCT
iajs-2804	41	6	let	let	VERB
iajs-2804	41	7	x	x	PRON
iajs-2804	41	8	and	and	CCONJ
iajs-2804	41	9	a	a	DET
iajs-2804	41	10	be	be	AUX
iajs-2804	41	11	two	two	NUM
iajs-2804	41	12	ℱ-modules	ℱ-modules	PROPN
iajs-2804	41	13	of	of	ADP
iajs-2804	41	14	ℛ-module	ℛ-module	PROPN
iajs-2804	41	15	μ	μ	PROPN
iajs-2804	41	16	.	.	PUNCT
iajs-2804	42	1	a	a	PRON
iajs-2804	42	2	is	be	AUX
iajs-2804	42	3	said	say	VERB
iajs-2804	42	4	to	to	PART
iajs-2804	42	5	be	be	AUX
iajs-2804	42	6	an	an	DET
iajs-2804	42	7	ℱ-sub	ℱ-sub	NOUN
iajs-2804	42	8	-	-	NOUN
iajs-2804	42	9	module	module	NOUN
iajs-2804	42	10	of	of	ADP
iajs-2804	42	11	x	x	PRON
iajs-2804	42	12	if	if	SCONJ
iajs-2804	42	13	α	α	NOUN
iajs-2804	42	14	⊆	⊆	NUM
iajs-2804	42	15	𝑋.	𝑋.	PROPN
iajs-2804	42	16	proposition	proposition	NOUN
iajs-2804	42	17	1.9	1.9	NUM
iajs-2804	42	18	[	[	SYM
iajs-2804	42	19	5	5	NUM
iajs-2804	42	20	]	]	PUNCT
iajs-2804	42	21	ibn	ibn	PROPN
iajs-2804	42	22	al	al	PROPN
iajs-2804	42	23	-	-	PUNCT
iajs-2804	42	24	haitham	haitham	PROPN
iajs-2804	42	25	jour	jour	X
iajs-2804	42	26	.	.	PROPN
iajs-2804	42	27	for	for	ADP
iajs-2804	42	28	pure	pure	ADJ
iajs-2804	42	29	&	&	CCONJ
iajs-2804	42	30	appl	appl	PROPN
iajs-2804	42	31	.	.	PUNCT
iajs-2804	43	1	sci	sci	PROPN
iajs-2804	43	2	.	.	PROPN
iajs-2804	44	1	53	53	NUM
iajs-2804	44	2	(	(	PUNCT
iajs-2804	44	3	1)2022	1)2022	PROPN
iajs-2804	44	4	104	104	NUM
iajs-2804	44	5	let	let	VERB
iajs-2804	44	6	α	α	PRON
iajs-2804	44	7	be	be	AUX
iajs-2804	44	8	an	an	DET
iajs-2804	44	9	ℱ-set	ℱ-set	NOUN
iajs-2804	44	10	of	of	ADP
iajs-2804	44	11	an	an	DET
iajs-2804	44	12	ℛ-module	ℛ-module	PROPN
iajs-2804	44	13	μ	μ	PROPN
iajs-2804	44	14	.	.	PUNCT
iajs-2804	45	1	then	then	ADV
iajs-2804	45	2	the	the	DET
iajs-2804	45	3	level	level	NOUN
iajs-2804	45	4	subset	subset	VERB
iajs-2804	45	5	α𝑡	α𝑡	ADP
iajs-2804	45	6	,	,	PUNCT
iajs-2804	45	7	t	t	PROPN
iajs-2804	45	8	∈	∈	PROPN
iajs-2804	46	1	[	[	X
iajs-2804	46	2	0	0	NUM
iajs-2804	46	3	,	,	PUNCT
iajs-2804	46	4	1	1	NUM
iajs-2804	46	5	]	]	PUNCT
iajs-2804	46	6	is	be	AUX
iajs-2804	46	7	a	a	DET
iajs-2804	46	8	sub	sub	NOUN
iajs-2804	46	9	-	-	NOUN
iajs-2804	46	10	module	module	NOUN
iajs-2804	46	11	of	of	ADP
iajs-2804	46	12	μ	μ	PROPN
iajs-2804	46	13	if	if	SCONJ
iajs-2804	46	14	α	α	PRON
iajs-2804	46	15	is	be	AUX
iajs-2804	46	16	an	an	DET
iajs-2804	46	17	ℱ-sub	ℱ-sub	NOUN
iajs-2804	46	18	-	-	NOUN
iajs-2804	46	19	module	module	NOUN
iajs-2804	46	20	of	of	ADP
iajs-2804	46	21	x	x	SYM
iajs-2804	46	22	where	where	SCONJ
iajs-2804	46	23	x	x	PRON
iajs-2804	46	24	is	be	AUX
iajs-2804	46	25	an	an	DET
iajs-2804	46	26	ℱ-module	ℱ-module	PROPN
iajs-2804	46	27	of	of	ADP
iajs-2804	46	28	an	an	DET
iajs-2804	46	29	ℛ-module	ℛ-module	PROPN
iajs-2804	46	30	μ	μ	PROPN
iajs-2804	46	31	.	.	PUNCT
iajs-2804	47	1	now	now	ADV
iajs-2804	47	2	,	,	PUNCT
iajs-2804	47	3	we	we	PRON
iajs-2804	47	4	go	go	VERB
iajs-2804	47	5	over	over	ADP
iajs-2804	47	6	various	various	ADJ
iajs-2804	47	7	ℱ-sub	ℱ-sub	NOUN
iajs-2804	47	8	-	-	NOUN
iajs-2804	47	9	module	module	NOUN
iajs-2804	47	10	attributes	attribute	NOUN
iajs-2804	47	11	that	that	PRON
iajs-2804	47	12	will	will	AUX
iajs-2804	47	13	be	be	AUX
iajs-2804	47	14	useful	useful	ADJ
iajs-2804	47	15	in	in	ADP
iajs-2804	47	16	the	the	DET
iajs-2804	47	17	next	next	ADJ
iajs-2804	47	18	section	section	NOUN
iajs-2804	47	19	.	.	PUNCT
iajs-2804	48	1	lemma	lemma	PROPN
iajs-2804	48	2	1.10	1.10	NUM
iajs-2804	49	1	[	[	X
iajs-2804	49	2	6	6	NUM
iajs-2804	49	3	]	]	X
iajs-2804	49	4	if	if	SCONJ
iajs-2804	49	5	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	49	6	be	be	AUX
iajs-2804	49	7	an	an	DET
iajs-2804	49	8	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	49	9	of	of	ADP
iajs-2804	49	10	ℛ	ℛ	PROPN
iajs-2804	49	11	and	and	CCONJ
iajs-2804	49	12	α	α	PROPN
iajs-2804	49	13	be	be	VERB
iajs-2804	49	14	an	an	DET
iajs-2804	49	15	ℱ-module	ℱ-module	PROPN
iajs-2804	49	16	of	of	ADP
iajs-2804	49	17	an	an	DET
iajs-2804	49	18	ℛ-module	ℛ-module	PROPN
iajs-2804	49	19	μ.then	μ.then	NOUN
iajs-2804	49	20	for	for	ADP
iajs-2804	49	21	any	any	DET
iajs-2804	49	22	w	w	PROPN
iajs-2804	49	23	∈	∈	PROPN
iajs-2804	49	24	μ	μ	NOUN
iajs-2804	49	25	(	(	PUNCT
iajs-2804	49	26	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	49	27	α)(𝑤	α)(𝑤	NUM
iajs-2804	49	28	)	)	PUNCT
iajs-2804	49	29	=	=	PRON
iajs-2804	49	30	{	{	PUNCT
iajs-2804	49	31	sup{inf	sup{inf	PROPN
iajs-2804	49	32	(	(	PUNCT
iajs-2804	49	33	𝑡	𝑡	NOUN
iajs-2804	49	34	,	,	PUNCT
iajs-2804	49	35	𝐴(𝑥	𝐴(𝑥	NOUN
iajs-2804	49	36	)	)	PUNCT
iajs-2804	49	37	)	)	PUNCT
iajs-2804	49	38	}	}	PUNCT
iajs-2804	49	39	:	:	PUNCT
iajs-2804	49	40	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	49	41	𝑤	𝑤	X
iajs-2804	49	42	=	=	SYM
iajs-2804	49	43	𝑟𝑥	𝑟𝑥	X
iajs-2804	49	44	}	}	PUNCT
iajs-2804	49	45	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	49	46	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	VERB
iajs-2804	49	47	𝑥	𝑥	ADP
iajs-2804	49	48	∈	∈	PROPN
iajs-2804	49	49	μ	μ	NOUN
iajs-2804	49	50	0	0	NUM
iajs-2804	49	51	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2804	49	52	where	where	SCONJ
iajs-2804	49	53	𝑟𝑡	𝑟𝑡	NOUN
iajs-2804	49	54	:	:	PUNCT
iajs-2804	49	55	ℛ	ℛ	PROPN
iajs-2804	49	56	→	→	SYM
iajs-2804	49	57	[	[	X
iajs-2804	49	58	0	0	NUM
iajs-2804	49	59	,	,	PUNCT
iajs-2804	49	60	1	1	NUM
iajs-2804	49	61	]	]	PUNCT
iajs-2804	49	62	,	,	PUNCT
iajs-2804	49	63	defined	define	VERB
iajs-2804	49	64	by	by	ADP
iajs-2804	49	65	𝑟𝑡(𝑧	𝑟𝑡(𝑧	NOUN
iajs-2804	49	66	)	)	PUNCT
iajs-2804	49	67	=	=	PRON
iajs-2804	49	68	{	{	PUNCT
iajs-2804	49	69	𝑡	𝑡	X
iajs-2804	49	70	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	49	71	𝑟	𝑟	NOUN
iajs-2804	49	72	=	=	SYM
iajs-2804	49	73	𝑧	𝑧	X
iajs-2804	49	74	0	0	NUM
iajs-2804	49	75	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	49	76	𝑟	𝑟	NOUN
iajs-2804	49	77	≠	≠	PROPN
iajs-2804	49	78	𝑧	𝑧	PROPN
iajs-2804	49	79	for	for	ADP
iajs-2804	49	80	all	all	PRON
iajs-2804	49	81	𝑧	𝑧	DET
iajs-2804	49	82	∈	∈	NOUN
iajs-2804	49	83	ℛ	ℛ	ADJ
iajs-2804	49	84	definition	definition	NOUN
iajs-2804	49	85	1.11	1.11	NUM
iajs-2804	49	86	[	[	X
iajs-2804	49	87	6	6	NUM
iajs-2804	49	88	]	]	PUNCT
iajs-2804	49	89	let	let	VERB
iajs-2804	49	90	α	α	PRON
iajs-2804	49	91	and	and	CCONJ
iajs-2804	49	92	𝛣	𝛣	PROPN
iajs-2804	49	93	be	be	VERB
iajs-2804	49	94	two	two	NUM
iajs-2804	49	95	ℱ-sub	ℱ-sub	NOUN
iajs-2804	49	96	-	-	NOUN
iajs-2804	49	97	modules	module	NOUN
iajs-2804	49	98	of	of	ADP
iajs-2804	49	99	an	an	DET
iajs-2804	49	100	ℱ-module	ℱ-module	PROPN
iajs-2804	49	101	x	x	PROPN
iajs-2804	49	102	of	of	ADP
iajs-2804	49	103	ℛ-module	ℛ-module	PROPN
iajs-2804	49	104	μ	μ	PROPN
iajs-2804	49	105	.	.	PUNCT
iajs-2804	50	1	the	the	DET
iajs-2804	50	2	residual	residual	ADJ
iajs-2804	50	3	quotient	quotient	NOUN
iajs-2804	50	4	of	of	ADP
iajs-2804	50	5	α	α	NOUN
iajs-2804	50	6	and	and	CCONJ
iajs-2804	50	7	𝛣	𝛣	PROPN
iajs-2804	50	8	denoted	denote	VERB
iajs-2804	50	9	by	by	ADP
iajs-2804	50	10	(	(	PUNCT
iajs-2804	50	11	α	α	PROPN
iajs-2804	50	12	∶	∶	PROPN
iajs-2804	50	13	𝛣	𝛣	PROPN
iajs-2804	50	14	)	)	PUNCT
iajs-2804	50	15	is	be	AUX
iajs-2804	50	16	the	the	DET
iajs-2804	50	17	ℱ-subset	ℱ-subset	PROPN
iajs-2804	50	18	of	of	ADP
iajs-2804	50	19	ℛ	ℛ	PROPN
iajs-2804	50	20	defined	define	VERB
iajs-2804	50	21	by	by	ADP
iajs-2804	50	22	:	:	PUNCT
iajs-2804	50	23	(	(	PUNCT
iajs-2804	50	24	α	α	PROPN
iajs-2804	50	25	∶	∶	NOUN
iajs-2804	50	26	𝛣)(r	𝛣)(r	PROPN
iajs-2804	50	27	)	)	PUNCT
iajs-2804	50	28	=	=	SYM
iajs-2804	50	29	sup	sup	NOUN
iajs-2804	50	30	{	{	PUNCT
iajs-2804	50	31	t	t	NOUN
iajs-2804	50	32	∈	∈	PROPN
iajs-2804	51	1	[	[	X
iajs-2804	51	2	0	0	NUM
iajs-2804	51	3	,	,	PUNCT
iajs-2804	51	4	1	1	NUM
iajs-2804	51	5	]	]	X
iajs-2804	51	6	∶	∶	NOUN
iajs-2804	51	7	𝑟𝑡	𝑟𝑡	NUM
iajs-2804	51	8	𝛣	𝛣	PROPN
iajs-2804	51	9	⊆	⊆	NUM
iajs-2804	51	10	α	α	NOUN
iajs-2804	51	11	}	}	PUNCT
iajs-2804	51	12	,	,	PUNCT
iajs-2804	51	13	for	for	ADP
iajs-2804	51	14	all	all	PRON
iajs-2804	51	15	𝑟	𝑟	DET
iajs-2804	51	16	∈	∈	ADJ
iajs-2804	51	17	ℛ.	ℛ.	NOUN
iajs-2804	51	18	that	that	PRON
iajs-2804	51	19	is	be	AUX
iajs-2804	51	20	(	(	PUNCT
iajs-2804	51	21	α	α	PROPN
iajs-2804	51	22	∶	∶	NOUN
iajs-2804	51	23	𝛣	𝛣	PROPN
iajs-2804	51	24	)	)	PUNCT
iajs-2804	51	25	=	=	PRON
iajs-2804	51	26	{	{	PUNCT
iajs-2804	51	27	𝑟𝑡	𝑟𝑡	NUM
iajs-2804	51	28	∶	∶	VERB
iajs-2804	51	29	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	51	30	b	b	NOUN
iajs-2804	51	31	⊆	⊆	NUM
iajs-2804	51	32	α	α	NOUN
iajs-2804	51	33	;	;	PUNCT
iajs-2804	51	34	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	51	35	is	be	AUX
iajs-2804	51	36	an	an	DET
iajs-2804	51	37	ℱ	ℱ	PROPN
iajs-2804	51	38	−	−	PROPN
iajs-2804	51	39	singleton	singleton	NOUN
iajs-2804	51	40	of	of	ADP
iajs-2804	51	41	ℛ	ℛ	PROPN
iajs-2804	51	42	}	}	PUNCT
iajs-2804	51	43	.	.	PUNCT
iajs-2804	52	1	if	if	SCONJ
iajs-2804	52	2	𝛣	𝛣	PROPN
iajs-2804	52	3	=	=	PUNCT
iajs-2804	52	4	〈	〈	PROPN
iajs-2804	52	5	𝑥𝑘	𝑥𝑘	NOUN
iajs-2804	52	6	〉	〉	NOUN
iajs-2804	52	7	,	,	PUNCT
iajs-2804	52	8	then	then	ADV
iajs-2804	52	9	(	(	PUNCT
iajs-2804	52	10	α	α	PROPN
iajs-2804	52	11	∶	∶	NOUN
iajs-2804	52	12	〈	〈	NOUN
iajs-2804	52	13	𝑥𝑘	𝑥𝑘	X
iajs-2804	52	14	〉	〉	NOUN
iajs-2804	52	15	)	)	PUNCT
iajs-2804	52	16	=	=	PRON
iajs-2804	52	17	{	{	PUNCT
iajs-2804	52	18	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	52	19	∶	∶	NOUN
iajs-2804	52	20	𝑟𝑡	𝑟𝑡	NOUN
iajs-2804	52	21	𝑥𝑘	𝑥𝑘	NOUN
iajs-2804	52	22	⊆	⊆	NUM
iajs-2804	52	23	α	α	NOUN
iajs-2804	52	24	;	;	PUNCT
iajs-2804	52	25	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	52	26	is	be	AUX
iajs-2804	52	27	an	an	DET
iajs-2804	52	28	ℱ	ℱ	PROPN
iajs-2804	52	29	−	−	PROPN
iajs-2804	52	30	singleton	singleton	NOUN
iajs-2804	52	31	of	of	ADP
iajs-2804	52	32	ℛ	ℛ	PROPN
iajs-2804	52	33	}	}	PUNCT
iajs-2804	52	34	.	.	PUNCT
iajs-2804	53	1	lemma	lemma	PROPN
iajs-2804	53	2	1.12	1.12	NUM
iajs-2804	54	1	[	[	X
iajs-2804	54	2	9	9	NUM
iajs-2804	54	3	]	]	PUNCT
iajs-2804	54	4	let	let	VERB
iajs-2804	54	5	α	α	PRON
iajs-2804	54	6	be	be	AUX
iajs-2804	54	7	an	an	DET
iajs-2804	54	8	ℱ-sub	ℱ-sub	NOUN
iajs-2804	54	9	-	-	NOUN
iajs-2804	54	10	module	module	NOUN
iajs-2804	54	11	of	of	ADP
iajs-2804	54	12	ℱ-module	ℱ-module	PROPN
iajs-2804	54	13	x	x	PROPN
iajs-2804	54	14	,	,	PUNCT
iajs-2804	54	15	(	(	PUNCT
iajs-2804	54	16	α	α	NOUN
iajs-2804	54	17	𝑡	𝑡	NOUN
iajs-2804	54	18	:	:	PUNCT
iajs-2804	54	19	𝑋𝑡	𝑋𝑡	PROPN
iajs-2804	54	20	)	)	PUNCT
iajs-2804	54	21	≥	≥	NOUN
iajs-2804	54	22	(	(	PUNCT
iajs-2804	54	23	α	α	NOUN
iajs-2804	54	24	:	:	PUNCT
iajs-2804	54	25	𝑋)𝑡	𝑋)𝑡	NOUN
iajs-2804	54	26	,	,	PUNCT
iajs-2804	54	27	for	for	ADP
iajs-2804	54	28	all	all	DET
iajs-2804	54	29	t	t	NOUN
iajs-2804	54	30	∈	∈	PROPN
iajs-2804	55	1	[	[	X
iajs-2804	55	2	0	0	NUM
iajs-2804	55	3	,	,	PUNCT
iajs-2804	55	4	1	1	NUM
iajs-2804	55	5	]	]	PUNCT
iajs-2804	55	6	.	.	PUNCT
iajs-2804	56	1	also	also	ADV
iajs-2804	56	2	,	,	PUNCT
iajs-2804	56	3	we	we	PRON
iajs-2804	56	4	can	can	AUX
iajs-2804	56	5	prove	prove	VERB
iajs-2804	56	6	that	that	SCONJ
iajs-2804	56	7	by	by	ADP
iajs-2804	56	8	lemma	lemma	PROPN
iajs-2804	56	9	2.3.3.[6	2.3.3.[6	PROPN
iajs-2804	56	10	]	]	X
iajs-2804	56	11	.	.	PUNCT
iajs-2804	57	1	it	it	PRON
iajs-2804	57	2	follows	follow	VERB
iajs-2804	57	3	that	that	SCONJ
iajs-2804	57	4	if	if	SCONJ
iajs-2804	57	5	,	,	PUNCT
iajs-2804	57	6	𝑋	𝑋	PROPN
iajs-2804	57	7	=	=	SYM
iajs-2804	57	8	α	α	PROPN
iajs-2804	57	9	⊕	⊕	PROPN
iajs-2804	57	10	𝛣,where	𝛣,where	X
iajs-2804	57	11	𝐴	𝐴	PROPN
iajs-2804	57	12	,	,	PUNCT
iajs-2804	57	13	𝛣	𝛣	PROPN
iajs-2804	57	14	≤	≤	NOUN
iajs-2804	57	15	𝑋	𝑋	PROPN
iajs-2804	57	16	then	then	ADV
iajs-2804	57	17	𝑋𝑡	𝑋𝑡	PROPN
iajs-2804	57	18	=	=	PUNCT
iajs-2804	57	19	(	(	PUNCT
iajs-2804	57	20	α	α	PROPN
iajs-2804	57	21	⊕	⊕	PROPN
iajs-2804	57	22	𝛣)𝑡	𝛣)𝑡	PUNCT
iajs-2804	57	23	=	=	PUNCT
iajs-2804	57	24	α𝑡	α𝑡	ADP
iajs-2804	57	25	⊕	⊕	PROPN
iajs-2804	57	26	𝛣𝑡	𝛣𝑡	PROPN
iajs-2804	57	27	.	.	PUNCT
iajs-2804	58	1	definition	definition	NOUN
iajs-2804	58	2	1.13	1.13	NUM
iajs-2804	58	3	[	[	X
iajs-2804	58	4	10	10	NUM
iajs-2804	58	5	]	]	PUNCT
iajs-2804	58	6	let	let	VERB
iajs-2804	58	7	f	f	PRON
iajs-2804	58	8	be	be	AUX
iajs-2804	58	9	a	a	DET
iajs-2804	58	10	mapping	mapping	NOUN
iajs-2804	58	11	from	from	ADP
iajs-2804	58	12	a	a	DET
iajs-2804	58	13	set	set	NOUN
iajs-2804	58	14	μ	μ	NOUN
iajs-2804	58	15	into	into	ADP
iajs-2804	58	16	a	a	DET
iajs-2804	58	17	set	set	NOUN
iajs-2804	58	18	ν	ν	NOUN
iajs-2804	58	19	and	and	CCONJ
iajs-2804	58	20	let	let	VERB
iajs-2804	58	21	α	α	PRON
iajs-2804	58	22	be	be	AUX
iajs-2804	58	23	ℱ-set	ℱ-set	NOUN
iajs-2804	58	24	in	in	ADP
iajs-2804	58	25	μ	μ	NUM
iajs-2804	58	26	.	.	PUNCT
iajs-2804	59	1	the	the	DET
iajs-2804	59	2	image	image	NOUN
iajs-2804	59	3	of	of	ADP
iajs-2804	59	4	α	α	PROPN
iajs-2804	59	5	is	be	AUX
iajs-2804	59	6	denoted	denote	VERB
iajs-2804	59	7	by	by	ADP
iajs-2804	59	8	f	f	PROPN
iajs-2804	59	9	(	(	PUNCT
iajs-2804	59	10	α	α	NOUN
iajs-2804	59	11	)	)	PUNCT
iajs-2804	59	12	,	,	PUNCT
iajs-2804	59	13	where	where	SCONJ
iajs-2804	59	14	f	f	PROPN
iajs-2804	59	15	(	(	PUNCT
iajs-2804	59	16	α	α	NOUN
iajs-2804	59	17	)	)	PUNCT
iajs-2804	59	18	is	be	AUX
iajs-2804	59	19	defined	define	VERB
iajs-2804	59	20	by	by	ADP
iajs-2804	59	21	:	:	PUNCT
iajs-2804	59	22	𝑓	𝑓	PROPN
iajs-2804	59	23	(	(	PUNCT
iajs-2804	59	24	α	α	NOUN
iajs-2804	59	25	)	)	PUNCT
iajs-2804	59	26	(	(	PUNCT
iajs-2804	59	27	𝑦	𝑦	X
iajs-2804	59	28	)	)	PUNCT
iajs-2804	59	29	=	=	PRON
iajs-2804	59	30	{	{	PUNCT
iajs-2804	59	31	sup{α(𝑧	sup{α(𝑧	PROPN
iajs-2804	59	32	):	):	PUNCT
iajs-2804	59	33	𝑧	𝑧	PROPN
iajs-2804	59	34	∈	∈	PROPN
iajs-2804	59	35	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
iajs-2804	59	36	)	)	PUNCT
iajs-2804	59	37	≠	≠	PROPN
iajs-2804	59	38	∅	∅	NOUN
iajs-2804	59	39	}	}	PUNCT
iajs-2804	59	40	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	59	41	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2804	59	42	𝑦	𝑦	PROPN
iajs-2804	59	43	∈	∈	PROPN
iajs-2804	59	44	ν	ν	NOUN
iajs-2804	59	45	0	0	NUM
iajs-2804	59	46	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2804	59	47	note	note	VERB
iajs-2804	59	48	that	that	SCONJ
iajs-2804	59	49	,	,	PUNCT
iajs-2804	59	50	if	if	SCONJ
iajs-2804	59	51	𝑓	𝑓	PRON
iajs-2804	59	52	is	be	AUX
iajs-2804	59	53	a	a	DET
iajs-2804	59	54	bijective	bijective	ADJ
iajs-2804	59	55	mapping	mapping	NOUN
iajs-2804	59	56	,	,	PUNCT
iajs-2804	59	57	then	then	ADV
iajs-2804	59	58	𝑓	𝑓	PRON
iajs-2804	59	59	(	(	PUNCT
iajs-2804	59	60	α)(𝑦	α)(𝑦	NOUN
iajs-2804	59	61	)	)	PUNCT
iajs-2804	59	62	=	=	SYM
iajs-2804	59	63	α(𝑓−1(𝑦	α(𝑓−1(𝑦	PROPN
iajs-2804	59	64	)	)	PUNCT
iajs-2804	59	65	)	)	PUNCT
iajs-2804	59	66	proposition	proposition	NOUN
iajs-2804	59	67	1.14	1.14	NUM
iajs-2804	59	68	[	[	X
iajs-2804	59	69	11	11	NUM
iajs-2804	59	70	]	]	PUNCT
iajs-2804	59	71	let	let	VERB
iajs-2804	59	72	f	f	PRON
iajs-2804	59	73	be	be	AUX
iajs-2804	59	74	a	a	DET
iajs-2804	59	75	mapping	mapping	NOUN
iajs-2804	59	76	from	from	ADP
iajs-2804	59	77	a	a	DET
iajs-2804	59	78	set	set	NOUN
iajs-2804	59	79	μ	μ	NOUN
iajs-2804	59	80	into	into	ADP
iajs-2804	59	81	a	a	DET
iajs-2804	59	82	set	set	NOUN
iajs-2804	59	83	ν	ν	NOUN
iajs-2804	59	84	.	.	PUNCT
iajs-2804	59	85	assume	assume	VERB
iajs-2804	59	86	that	that	SCONJ
iajs-2804	59	87	x	x	PROPN
iajs-2804	59	88	and	and	CCONJ
iajs-2804	59	89	y	y	PROPN
iajs-2804	59	90	are	be	AUX
iajs-2804	59	91	ℱ-modules	ℱ-modules	PROPN
iajs-2804	59	92	of	of	ADP
iajs-2804	59	93	m	m	NOUN
iajs-2804	59	94	and	and	CCONJ
iajs-2804	59	95	n	n	PRON
iajs-2804	59	96	respectively	respectively	ADV
iajs-2804	59	97	,	,	PUNCT
iajs-2804	59	98	let	let	VERB
iajs-2804	59	99	α	α	PRON
iajs-2804	59	100	be	be	AUX
iajs-2804	59	101	an	an	DET
iajs-2804	59	102	ℱ-sub	ℱ-sub	NOUN
iajs-2804	59	103	-	-	NOUN
iajs-2804	59	104	module	module	NOUN
iajs-2804	59	105	of	of	ADP
iajs-2804	59	106	x	x	NOUN
iajs-2804	59	107	,	,	PUNCT
iajs-2804	59	108	then	then	ADV
iajs-2804	59	109	f	f	X
iajs-2804	59	110	(	(	PUNCT
iajs-2804	59	111	α	α	NOUN
iajs-2804	59	112	)	)	PUNCT
iajs-2804	59	113	is	be	AUX
iajs-2804	59	114	an	an	DET
iajs-2804	59	115	ℱ-sub	ℱ-sub	NOUN
iajs-2804	59	116	-	-	NOUN
iajs-2804	59	117	module	module	NOUN
iajs-2804	59	118	of	of	ADP
iajs-2804	59	119	y.	y.	PROPN
iajs-2804	59	120	definition	definition	NOUN
iajs-2804	59	121	1.15	1.15	NUM
iajs-2804	59	122	[	[	X
iajs-2804	59	123	12	12	NUM
iajs-2804	59	124	]	]	PUNCT
iajs-2804	59	125	an	an	DET
iajs-2804	59	126	ℱ-subset	ℱ-subset	PROPN
iajs-2804	59	127	k	k	PROPN
iajs-2804	59	128	of	of	ADP
iajs-2804	59	129	a	a	DET
iajs-2804	59	130	ring	ring	NOUN
iajs-2804	59	131	ℛ	ℛ	PROPN
iajs-2804	59	132	is	be	AUX
iajs-2804	59	133	called	call	VERB
iajs-2804	59	134	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	59	135	of	of	ADP
iajs-2804	59	136	ℛ	ℛ	PROPN
iajs-2804	59	137	,	,	PUNCT
iajs-2804	59	138	if	if	SCONJ
iajs-2804	59	139	∀	∀	NOUN
iajs-2804	59	140	𝑥	𝑥	NOUN
iajs-2804	59	141	,	,	PUNCT
iajs-2804	59	142	𝑦	𝑦	NOUN
iajs-2804	59	143	∈	∈	PROPN
iajs-2804	59	144	ℛ	ℛ	NOUN
iajs-2804	59	145	:	:	PUNCT
iajs-2804	59	146	1	1	NUM
iajs-2804	59	147	𝐾(𝑥	𝐾(𝑥	SYM
iajs-2804	59	148	−	−	NUM
iajs-2804	59	149	𝑦	𝑦	NUM
iajs-2804	59	150	)	)	PUNCT
iajs-2804	59	151	≥	≥	PROPN
iajs-2804	59	152	min	min	PROPN
iajs-2804	59	153	{	{	PUNCT
iajs-2804	59	154	𝐾(𝑥	𝐾(𝑥	NOUN
iajs-2804	59	155	)	)	PUNCT
iajs-2804	59	156	,	,	PUNCT
iajs-2804	59	157	𝐾(𝑦	𝐾(𝑦	NOUN
iajs-2804	59	158	)	)	PUNCT
iajs-2804	59	159	}	}	PUNCT
iajs-2804	59	160	.	.	PUNCT
iajs-2804	60	1	2	2	NUM
iajs-2804	60	2	𝐾(𝑥𝑦	𝐾(𝑥𝑦	NOUN
iajs-2804	60	3	)	)	PUNCT
iajs-2804	60	4	≥	≥	PROPN
iajs-2804	60	5	max	max	PROPN
iajs-2804	60	6	{	{	PUNCT
iajs-2804	60	7	𝐾(𝑥	𝐾(𝑥	NOUN
iajs-2804	60	8	)	)	PUNCT
iajs-2804	60	9	,	,	PUNCT
iajs-2804	60	10	𝐾(𝑦	𝐾(𝑦	NOUN
iajs-2804	60	11	)	)	PUNCT
iajs-2804	60	12	}	}	PUNCT
iajs-2804	60	13	.	.	PUNCT
iajs-2804	61	1	definition	definition	NOUN
iajs-2804	61	2	1.16	1.16	NUM
iajs-2804	62	1	[	[	X
iajs-2804	62	2	13	13	NUM
iajs-2804	62	3	]	]	PUNCT
iajs-2804	62	4	let	let	VERB
iajs-2804	62	5	x	x	PRON
iajs-2804	62	6	be	be	AUX
iajs-2804	62	7	an	an	DET
iajs-2804	62	8	ℱ-module	ℱ-module	PROPN
iajs-2804	62	9	of	of	ADP
iajs-2804	62	10	an	an	DET
iajs-2804	62	11	ℛ-module	ℛ-module	PROPN
iajs-2804	62	12	μ	μ	PROPN
iajs-2804	62	13	,	,	PUNCT
iajs-2804	62	14	let	let	VERB
iajs-2804	62	15	a	a	PRON
iajs-2804	62	16	be	be	AUX
iajs-2804	62	17	an	an	DET
iajs-2804	62	18	ℱ-sub	ℱ-sub	NOUN
iajs-2804	62	19	-	-	NOUN
iajs-2804	62	20	module	module	NOUN
iajs-2804	62	21	of	of	ADP
iajs-2804	62	22	x	x	X
iajs-2804	62	23	and	and	CCONJ
iajs-2804	62	24	k	k	PROPN
iajs-2804	62	25	be	be	AUX
iajs-2804	62	26	an	an	DET
iajs-2804	62	27	ℱ	ℱ	PROPN
iajs-2804	62	28	ideal	ideal	NOUN
iajs-2804	62	29	of	of	ADP
iajs-2804	62	30	ℛ	ℛ	PROPN
iajs-2804	62	31	,	,	PUNCT
iajs-2804	62	32	the	the	DET
iajs-2804	62	33	product	product	NOUN
iajs-2804	62	34	ka	ka	PROPN
iajs-2804	62	35	of	of	ADP
iajs-2804	62	36	k	k	PROPN
iajs-2804	62	37	and	and	CCONJ
iajs-2804	62	38	α	α	PROPN
iajs-2804	62	39	is	be	AUX
iajs-2804	62	40	defined	define	VERB
iajs-2804	62	41	by	by	ADP
iajs-2804	62	42	:	:	PUNCT
iajs-2804	62	43	kα(𝑥	kα(𝑥	NOUN
iajs-2804	62	44	)	)	PUNCT
iajs-2804	63	1	=	=	PRON
iajs-2804	63	2	{	{	PUNCT
iajs-2804	63	3	sup	sup	PROPN
iajs-2804	63	4	{	{	PUNCT
iajs-2804	63	5	𝑖𝑛𝑓{𝐾(𝑟1	𝑖𝑛𝑓{𝐾(𝑟1	PROPN
iajs-2804	63	6	)	)	PUNCT
iajs-2804	63	7	,	,	PUNCT
iajs-2804	63	8	…	…	PUNCT
iajs-2804	63	9	.	.	PUNCT
iajs-2804	64	1	,	,	PUNCT
iajs-2804	64	2	𝐾(𝑟𝑛	𝐾(𝑟𝑛	NOUN
iajs-2804	64	3	)	)	PUNCT
iajs-2804	64	4	,	,	PUNCT
iajs-2804	64	5	α(𝑥1	α(𝑥1	ADV
iajs-2804	64	6	)	)	PUNCT
iajs-2804	64	7	,	,	PUNCT
iajs-2804	64	8	…	…	PUNCT
iajs-2804	64	9	,	,	PUNCT
iajs-2804	64	10	α(𝑥𝑛	α(𝑥𝑛	NUM
iajs-2804	64	11	)	)	PUNCT
iajs-2804	64	12	}	}	PUNCT
iajs-2804	64	13	}	}	PUNCT
iajs-2804	64	14	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	64	15	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	VERB
iajs-2804	64	16	𝑟𝑖	𝑟𝑖	PROPN
iajs-2804	64	17	∈	∈	PROPN
iajs-2804	64	18	ℛ	ℛ	PROPN
iajs-2804	64	19	,	,	PUNCT
iajs-2804	64	20	𝑥𝑖	𝑥𝑖	PROPN
iajs-2804	64	21	∈	∈	PROPN
iajs-2804	64	22	μ	μ	PROPN
iajs-2804	64	23	,	,	PUNCT
iajs-2804	64	24	𝑛	𝑛	PRON
iajs-2804	64	25	∈	∈	PROPN
iajs-2804	64	26	ν	ν	NOUN
iajs-2804	64	27	0	0	NUM
iajs-2804	64	28	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
iajs-2804	64	29	ibn	ibn	PROPN
iajs-2804	64	30	al	al	PROPN
iajs-2804	64	31	-	-	PUNCT
iajs-2804	64	32	haitham	haitham	PROPN
iajs-2804	64	33	jour	jour	X
iajs-2804	64	34	.	.	PROPN
iajs-2804	65	1	for	for	ADP
iajs-2804	65	2	pure	pure	ADJ
iajs-2804	65	3	&	&	CCONJ
iajs-2804	65	4	appl	appl	PROPN
iajs-2804	65	5	.	.	PUNCT
iajs-2804	66	1	sci	sci	PROPN
iajs-2804	66	2	.	.	PROPN
iajs-2804	67	1	53	53	NUM
iajs-2804	67	2	(	(	PUNCT
iajs-2804	67	3	1)2022	1)2022	PROPN
iajs-2804	67	4	105	105	NUM
iajs-2804	67	5	note	note	NOUN
iajs-2804	67	6	that	that	SCONJ
iajs-2804	67	7	k	k	PROPN
iajs-2804	67	8	α	α	PROPN
iajs-2804	67	9	is	be	AUX
iajs-2804	67	10	an	an	DET
iajs-2804	67	11	ℱ-sub	ℱ-sub	NOUN
iajs-2804	67	12	-	-	NOUN
iajs-2804	67	13	module	module	NOUN
iajs-2804	67	14	of	of	ADP
iajs-2804	67	15	x	x	PRON
iajs-2804	67	16	,	,	PUNCT
iajs-2804	67	17	and	and	CCONJ
iajs-2804	67	18	(	(	PUNCT
iajs-2804	67	19	kα)𝑡	kα)𝑡	PROPN
iajs-2804	67	20	=	=	SYM
iajs-2804	67	21	k𝑡	k𝑡	VERB
iajs-2804	67	22	α𝑡	α𝑡	ADP
iajs-2804	67	23	,	,	PUNCT
iajs-2804	67	24	∀	∀	X
iajs-2804	67	25	t	t	NOUN
iajs-2804	67	26	∈	∈	PROPN
iajs-2804	68	1	[	[	X
iajs-2804	68	2	0	0	NUM
iajs-2804	68	3	,	,	PUNCT
iajs-2804	68	4	1	1	NUM
iajs-2804	68	5	]	]	PUNCT
iajs-2804	68	6	.	.	PUNCT
iajs-2804	69	1	definition	definition	NOUN
iajs-2804	69	2	1.17	1.17	NUM
iajs-2804	69	3	[	[	X
iajs-2804	69	4	9	9	NUM
iajs-2804	69	5	]	]	PUNCT
iajs-2804	69	6	let	let	VERB
iajs-2804	69	7	x	x	PRON
iajs-2804	69	8	be	be	AUX
iajs-2804	69	9	an	an	DET
iajs-2804	69	10	ℱ-module	ℱ-module	PROPN
iajs-2804	69	11	of	of	ADP
iajs-2804	69	12	an	an	DET
iajs-2804	69	13	ℛ-module	ℛ-module	PROPN
iajs-2804	69	14	μ	μ	PROPN
iajs-2804	69	15	,	,	PUNCT
iajs-2804	69	16	an	an	DET
iajs-2804	69	17	ℱ-sub	ℱ-sub	NOUN
iajs-2804	69	18	-	-	NOUN
iajs-2804	69	19	module	module	NOUN
iajs-2804	69	20	u	u	NOUN
iajs-2804	69	21	of	of	ADP
iajs-2804	69	22	x	x	PRON
iajs-2804	69	23	is	be	AUX
iajs-2804	69	24	called	call	VERB
iajs-2804	69	25	completely	completely	ADV
iajs-2804	69	26	prime	prime	ADJ
iajs-2804	69	27	if	if	SCONJ
iajs-2804	69	28	whenever	whenever	SCONJ
iajs-2804	69	29	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	69	30	⊆	⊆	NUM
iajs-2804	69	31	𝑈,with	𝑈,with	NOUN
iajs-2804	69	32	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	69	33	≠	≠	PROPN
iajs-2804	69	34	01	01	NUM
iajs-2804	69	35	is	be	AUX
iajs-2804	69	36	an	an	DET
iajs-2804	69	37	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	69	38	of	of	ADP
iajs-2804	69	39	ℛ	ℛ	PROPN
iajs-2804	69	40	and	and	CCONJ
iajs-2804	69	41	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	69	42	is	be	AUX
iajs-2804	69	43	an	an	DET
iajs-2804	69	44	ℱsingleton	ℱsingleton	PROPN
iajs-2804	69	45	of	of	ADP
iajs-2804	69	46	ximplies	ximplie	NOUN
iajs-2804	70	1	that	that	SCONJ
iajs-2804	70	2	𝑚𝑡	𝑚𝑡	ADP
iajs-2804	70	3	⊆	⊆	NUM
iajs-2804	70	4	𝑈	𝑈	PROPN
iajs-2804	70	5	for	for	ADP
iajs-2804	70	6	each	each	DET
iajs-2804	70	7	t	t	PROPN
iajs-2804	70	8	,	,	PUNCT
iajs-2804	70	9	b	b	X
iajs-2804	70	10	∈	∈	PROPN
iajs-2804	70	11	[	[	X
iajs-2804	70	12	0,1	0,1	NUM
iajs-2804	70	13	]	]	PUNCT
iajs-2804	71	1	.	.	PUNCT
iajs-2804	72	1	definition	definition	NOUN
iajs-2804	72	2	1.18	1.18	NUM
iajs-2804	73	1	[	[	X
iajs-2804	73	2	6	6	NUM
iajs-2804	73	3	]	]	PUNCT
iajs-2804	73	4	let	let	VERB
iajs-2804	73	5	α	α	NOUN
iajs-2804	73	6	and	and	CCONJ
iajs-2804	73	7	β	β	X
iajs-2804	73	8	be	be	AUX
iajs-2804	73	9	two	two	NUM
iajs-2804	73	10	ℱ-sub	ℱ-sub	NOUN
iajs-2804	73	11	-	-	NOUN
iajs-2804	73	12	modules	module	NOUN
iajs-2804	73	13	of	of	ADP
iajs-2804	73	14	an	an	DET
iajs-2804	73	15	r	r	NOUN
iajs-2804	73	16	-	-	PUNCT
iajs-2804	73	17	module	module	NOUN
iajs-2804	73	18	μ	μ	NOUN
iajs-2804	73	19	.	.	PUNCT
iajs-2804	74	1	the	the	DET
iajs-2804	74	2	addition	addition	NOUN
iajs-2804	74	3	a	a	DET
iajs-2804	74	4	+	+	X
iajs-2804	74	5	β	β	X
iajs-2804	74	6	is	be	AUX
iajs-2804	74	7	defined	define	VERB
iajs-2804	74	8	by	by	ADP
iajs-2804	74	9	:	:	PUNCT
iajs-2804	74	10	(	(	PUNCT
iajs-2804	74	11	α	α	NOUN
iajs-2804	74	12	+	+	X
iajs-2804	74	13	β)(x	β)(x	NUM
iajs-2804	74	14	)	)	PUNCT
iajs-2804	74	15	=	=	SYM
iajs-2804	74	16	sup{𝑚𝑖𝑛{α(𝑦	sup{𝑚𝑖𝑛{α(𝑦	PROPN
iajs-2804	74	17	)	)	PUNCT
iajs-2804	74	18	,	,	PUNCT
iajs-2804	74	19	β(𝑧	β(𝑧	PROPN
iajs-2804	74	20	)	)	PUNCT
iajs-2804	74	21	}	}	PUNCT
iajs-2804	74	22	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	VERB
iajs-2804	74	23	𝑥	𝑥	NOUN
iajs-2804	74	24	=	=	PUNCT
iajs-2804	74	25	𝑦	𝑦	PROPN
iajs-2804	74	26	+	+	CCONJ
iajs-2804	74	27	𝑧	𝑧	X
iajs-2804	74	28	,	,	PUNCT
iajs-2804	74	29	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	74	30	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-2804	74	31	𝑥	𝑥	PROPN
iajs-2804	74	32	,	,	PUNCT
iajs-2804	74	33	𝑦	𝑦	NOUN
iajs-2804	74	34	,	,	PUNCT
iajs-2804	74	35	𝑧	𝑧	PROPN
iajs-2804	74	36	∈	∈	PROPN
iajs-2804	74	37	μ	μ	PROPN
iajs-2804	74	38	}	}	PUNCT
iajs-2804	74	39	.	.	PUNCT
iajs-2804	75	1	furthermore	furthermore	ADV
iajs-2804	75	2	,	,	PUNCT
iajs-2804	75	3	α	α	PROPN
iajs-2804	75	4	+	+	X
iajs-2804	75	5	β	β	X
iajs-2804	75	6	is	be	AUX
iajs-2804	75	7	an	an	DET
iajs-2804	75	8	ℱ-sub	ℱ-sub	NOUN
iajs-2804	75	9	-	-	NOUN
iajs-2804	75	10	module	module	NOUN
iajs-2804	75	11	of	of	ADP
iajs-2804	75	12	an	an	DET
iajs-2804	75	13	ℛ-module	ℛ-module	PROPN
iajs-2804	75	14	μ	μ	PROPN
iajs-2804	75	15	.	.	PUNCT
iajs-2804	75	16	corollary	corollary	ADJ
iajs-2804	75	17	1.19	1.19	NUM
iajs-2804	75	18	[	[	NOUN
iajs-2804	75	19	8	8	NUM
iajs-2804	75	20	]	]	X
iajs-2804	75	21	if	if	SCONJ
iajs-2804	75	22	x	x	PRON
iajs-2804	75	23	is	be	AUX
iajs-2804	75	24	an	an	DET
iajs-2804	75	25	ℱ-module	ℱ-module	PROPN
iajs-2804	75	26	of	of	ADP
iajs-2804	75	27	an	an	DET
iajs-2804	75	28	ℛ-module	ℛ-module	PROPN
iajs-2804	75	29	μ	μ	PROPN
iajs-2804	75	30	and	and	CCONJ
iajs-2804	75	31	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	75	32	⊆	⊆	NUM
iajs-2804	75	33	x	x	NOUN
iajs-2804	75	34	,	,	PUNCT
iajs-2804	75	35	then	then	ADV
iajs-2804	75	36	for	for	ADP
iajs-2804	75	37	all	all	DET
iajs-2804	75	38	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	75	39	𝑟𝑘	𝑟𝑘	NOUN
iajs-2804	75	40	of	of	ADP
iajs-2804	75	41	ℛ	ℛ	PROPN
iajs-2804	75	42	,	,	PUNCT
iajs-2804	75	43	𝑟𝑘	𝑟𝑘	NOUN
iajs-2804	75	44	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	75	45	=	=	X
iajs-2804	75	46	(	(	PUNCT
iajs-2804	75	47	𝑟𝑥)𝜆	𝑟𝑥)𝜆	X
iajs-2804	75	48	,	,	PUNCT
iajs-2804	75	49	where	where	SCONJ
iajs-2804	75	50	𝜆	𝜆	PROPN
iajs-2804	75	51	=	=	SYM
iajs-2804	75	52	min	min	PROPN
iajs-2804	75	53	{	{	PUNCT
iajs-2804	75	54	𝑡	𝑡	PROPN
iajs-2804	75	55	,	,	PUNCT
iajs-2804	75	56	𝑘	𝑘	NOUN
iajs-2804	75	57	}	}	PUNCT
iajs-2804	75	58	.	.	PUNCT
iajs-2804	76	1	proposition	proposition	NOUN
iajs-2804	76	2	1.20	1.20	NUM
iajs-2804	76	3	[	[	X
iajs-2804	76	4	6	6	NUM
iajs-2804	76	5	]	]	PUNCT
iajs-2804	76	6	let	let	VERB
iajs-2804	76	7	α	α	NOUN
iajs-2804	76	8	and	and	CCONJ
iajs-2804	76	9	β	β	X
iajs-2804	76	10	be	be	AUX
iajs-2804	76	11	two	two	NUM
iajs-2804	76	12	ℱ-sub	ℱ-sub	NOUN
iajs-2804	76	13	-	-	NOUN
iajs-2804	76	14	modules	module	NOUN
iajs-2804	76	15	of	of	ADP
iajs-2804	76	16	an	an	DET
iajs-2804	76	17	ℱ-module	ℱ-module	PROPN
iajs-2804	76	18	x	x	PROPN
iajs-2804	76	19	of	of	ADP
iajs-2804	76	20	an	an	DET
iajs-2804	76	21	ℛ-module	ℛ-module	PROPN
iajs-2804	76	22	μ	μ	PROPN
iajs-2804	76	23	.	.	PUNCT
iajs-2804	77	1	then	then	ADV
iajs-2804	77	2	the	the	DET
iajs-2804	77	3	residual	residual	ADJ
iajs-2804	77	4	quotient	quotient	NOUN
iajs-2804	77	5	of	of	ADP
iajs-2804	77	6	α	α	PROPN
iajs-2804	77	7	and	and	CCONJ
iajs-2804	77	8	𝛣	𝛣	PROPN
iajs-2804	77	9	(	(	PUNCT
iajs-2804	77	10	α	α	NOUN
iajs-2804	77	11	∶	∶	NOUN
iajs-2804	77	12	β	β	X
iajs-2804	77	13	)	)	PUNCT
iajs-2804	77	14	is	be	AUX
iajs-2804	77	15	an	an	DET
iajs-2804	77	16	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	77	17	of	of	ADP
iajs-2804	77	18	ℛ.	ℛ.	PROPN
iajs-2804	77	19	proposition	proposition	NOUN
iajs-2804	77	20	1.21	1.21	NUM
iajs-2804	77	21	[	[	X
iajs-2804	77	22	14	14	NUM
iajs-2804	77	23	]	]	PUNCT
iajs-2804	77	24	let	let	VERB
iajs-2804	77	25	𝑓	𝑓	X
iajs-2804	77	26	:	:	PUNCT
iajs-2804	78	1	𝑀	𝑀	PROPN
iajs-2804	78	2	⟶	⟶	NOUN
iajs-2804	78	3	ν	ν	NOUN
iajs-2804	78	4	be	be	AUX
iajs-2804	78	5	an	an	DET
iajs-2804	78	6	ℛ-homomorphisim	ℛ-homomorphisim	PROPN
iajs-2804	78	7	,	,	PUNCT
iajs-2804	78	8	then	then	ADV
iajs-2804	78	9	𝑓(𝑆𝑜𝑐(𝑀	𝑓(𝑆𝑜𝑐(𝑀	NUM
iajs-2804	78	10	)	)	PUNCT
iajs-2804	78	11	)	)	PUNCT
iajs-2804	79	1	⊆	⊆	NUM
iajs-2804	79	2	𝑆𝑜𝑐(ν	𝑆𝑜𝑐(ν	PRON
iajs-2804	79	3	)	)	PUNCT
iajs-2804	79	4	.	.	PUNCT
iajs-2804	80	1	definition	definition	NOUN
iajs-2804	80	2	1.22	1.22	NUM
iajs-2804	80	3	[	[	X
iajs-2804	80	4	15	15	NUM
iajs-2804	80	5	]	]	PUNCT
iajs-2804	80	6	let	let	VERB
iajs-2804	80	7	x	x	PRON
iajs-2804	80	8	be	be	AUX
iajs-2804	80	9	an	an	DET
iajs-2804	80	10	ℱ-module	ℱ-module	PROPN
iajs-2804	80	11	of	of	ADP
iajs-2804	80	12	an	an	DET
iajs-2804	80	13	r	r	NOUN
iajs-2804	80	14	-	-	PUNCT
iajs-2804	80	15	module	module	NOUN
iajs-2804	80	16	μ	μ	NOUN
iajs-2804	80	17	,	,	PUNCT
iajs-2804	80	18	x	x	VERB
iajs-2804	80	19	is	be	AUX
iajs-2804	80	20	called	call	VERB
iajs-2804	80	21	ℱ-simple	ℱ-simple	PROPN
iajs-2804	80	22	if	if	SCONJ
iajs-2804	81	1	and	and	CCONJ
iajs-2804	81	2	only	only	ADV
iajs-2804	81	3	if	if	SCONJ
iajs-2804	81	4	x	x	PRON
iajs-2804	81	5	has	have	AUX
iajs-2804	81	6	no	no	DET
iajs-2804	81	7	proper	proper	ADJ
iajs-2804	81	8	ℱ-sub	ℱ-sub	NOUN
iajs-2804	81	9	-	-	NOUN
iajs-2804	81	10	modules	module	NOUN
iajs-2804	81	11	(	(	PUNCT
iajs-2804	81	12	in	in	ADP
iajs-2804	81	13	fact	fact	NOUN
iajs-2804	81	14	x	x	PUNCT
iajs-2804	81	15	is	be	AUX
iajs-2804	81	16	ℱ-simple	ℱ-simple	PROPN
iajs-2804	81	17	if	if	SCONJ
iajs-2804	81	18	and	and	CCONJ
iajs-2804	81	19	only	only	ADV
iajs-2804	81	20	if	if	SCONJ
iajs-2804	81	21	x	x	PRON
iajs-2804	81	22	has	have	VERB
iajs-2804	81	23	only	only	ADV
iajs-2804	81	24	itself	itself	PRON
iajs-2804	81	25	and	and	CCONJ
iajs-2804	81	26	01	01	NUM
iajs-2804	81	27	)	)	PUNCT
iajs-2804	81	28	.	.	PUNCT
iajs-2804	82	1	definition	definition	NOUN
iajs-2804	82	2	1.23	1.23	NUM
iajs-2804	82	3	[	[	X
iajs-2804	82	4	16	16	NUM
iajs-2804	82	5	]	]	PUNCT
iajs-2804	82	6	𝐴	𝐴	PROPN
iajs-2804	82	7	ℱ-module	ℱ-module	PROPN
iajs-2804	82	8	𝑋	𝑋	PROPN
iajs-2804	82	9	is	be	AUX
iajs-2804	82	10	called	call	VERB
iajs-2804	82	11	semi	semi	ADJ
iajs-2804	82	12	-	-	ADJ
iajs-2804	82	13	simple	simple	ADJ
iajs-2804	82	14	if	if	SCONJ
iajs-2804	82	15	𝑋	𝑋	PROPN
iajs-2804	82	16	is	be	AUX
iajs-2804	82	17	a	a	DET
iajs-2804	82	18	summation	summation	NOUN
iajs-2804	82	19	of	of	ADP
iajs-2804	82	20	simple	simple	ADJ
iajs-2804	82	21	ℱ-sub	ℱ-sub	NOUN
iajs-2804	82	22	-	-	NOUN
iajs-2804	82	23	modules	module	NOUN
iajs-2804	82	24	of	of	ADP
iajs-2804	82	25	𝑋	𝑋	PROPN
iajs-2804	82	26	.	.	PUNCT
iajs-2804	83	1	moreover	moreover	ADV
iajs-2804	83	2	,	,	PUNCT
iajs-2804	83	3	𝑋	𝑋	PROPN
iajs-2804	83	4	is	be	AUX
iajs-2804	83	5	called	call	VERB
iajs-2804	83	6	semi	semi	ADJ
iajs-2804	83	7	-	-	ADJ
iajs-2804	83	8	simple	simple	ADJ
iajs-2804	83	9	if	if	SCONJ
iajs-2804	83	10	𝑋	𝑋	PROPN
iajs-2804	83	11	=	=	SYM
iajs-2804	83	12	𝐹	𝐹	PROPN
iajs-2804	83	13	−	−	PROPN
iajs-2804	83	14	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	83	15	)	)	PUNCT
iajs-2804	83	16	.	.	PUNCT
iajs-2804	84	1	definition	definition	NOUN
iajs-2804	84	2	1.24	1.24	NUM
iajs-2804	84	3	[	[	X
iajs-2804	84	4	9	9	NUM
iajs-2804	84	5	]	]	PUNCT
iajs-2804	84	6	let	let	VERB
iajs-2804	84	7	x	x	PRON
iajs-2804	84	8	be	be	AUX
iajs-2804	84	9	an	an	DET
iajs-2804	84	10	ℱ-module	ℱ-module	PROPN
iajs-2804	84	11	of	of	ADP
iajs-2804	84	12	an	an	DET
iajs-2804	84	13	ℛ-module	ℛ-module	PROPN
iajs-2804	84	14	μ	μ	PROPN
iajs-2804	84	15	,	,	PUNCT
iajs-2804	84	16	x	x	PROPN
iajs-2804	84	17	is	be	AUX
iajs-2804	84	18	said	say	VERB
iajs-2804	84	19	to	to	PART
iajs-2804	84	20	be	be	AUX
iajs-2804	84	21	faithful	faithful	ADJ
iajs-2804	84	22	if	if	SCONJ
iajs-2804	84	23	𝐹	𝐹	PROPN
iajs-2804	85	1	−	−	VERB
iajs-2804	86	1	𝑎𝑛𝑛𝑋	𝑎𝑛𝑛𝑋	INTJ
iajs-2804	87	1	=	=	NOUN
iajs-2804	87	2	01	01	NUM
iajs-2804	87	3	.	.	PUNCT
iajs-2804	88	1	where	where	SCONJ
iajs-2804	88	2	𝐹	𝐹	PROPN
iajs-2804	88	3	−	−	VERB
iajs-2804	89	1	𝑎𝑛𝑛𝑋	𝑎𝑛𝑛𝑋	INTJ
iajs-2804	90	1	=	=	PUNCT
iajs-2804	90	2	{	{	PUNCT
iajs-2804	90	3	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	90	4	∶	∶	NOUN
iajs-2804	90	5	𝑟𝑡	𝑟𝑡	ADP
iajs-2804	90	6	𝑥𝑙	𝑥𝑙	NOUN
iajs-2804	90	7	=	=	SYM
iajs-2804	90	8	01	01	NUM
iajs-2804	90	9	;	;	PUNCT
iajs-2804	90	10	for	for	ADP
iajs-2804	90	11	all	all	DET
iajs-2804	90	12	𝑥𝑙	𝑥𝑙	NUM
iajs-2804	90	13	⊆	⊆	NUM
iajs-2804	90	14	x	x	PUNCT
iajs-2804	90	15	and	and	CCONJ
iajs-2804	90	16	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	90	17	be	be	AUX
iajs-2804	90	18	an	an	DET
iajs-2804	90	19	ℱ	ℱ	PROPN
iajs-2804	90	20	−	−	PROPN
iajs-2804	90	21	singleton	singleton	NOUN
iajs-2804	90	22	of	of	ADP
iajs-2804	90	23	ℛ	ℛ	PROPN
iajs-2804	90	24	}	}	PUNCT
iajs-2804	90	25	.	.	PUNCT
iajs-2804	91	1	definition	definition	NOUN
iajs-2804	91	2	1.25	1.25	NUM
iajs-2804	91	3	[	[	X
iajs-2804	91	4	17	17	NUM
iajs-2804	91	5	]	]	PUNCT
iajs-2804	91	6	let	let	VERB
iajs-2804	91	7	x	x	PRON
iajs-2804	91	8	be	be	AUX
iajs-2804	91	9	an	an	DET
iajs-2804	91	10	ℱ-module	ℱ-module	PROPN
iajs-2804	91	11	of	of	ADP
iajs-2804	91	12	an	an	DET
iajs-2804	91	13	ℛ-module	ℛ-module	PROPN
iajs-2804	91	14	μ	μ	PROPN
iajs-2804	91	15	,	,	PUNCT
iajs-2804	91	16	x	x	PROPN
iajs-2804	91	17	is	be	AUX
iajs-2804	91	18	said	say	VERB
iajs-2804	91	19	to	to	PART
iajs-2804	91	20	be	be	AUX
iajs-2804	91	21	cancellative	cancellative	ADJ
iajs-2804	91	22	if	if	SCONJ
iajs-2804	91	23	whenever	whenever	SCONJ
iajs-2804	91	24	𝑟𝑡	𝑟𝑡	ADP
iajs-2804	91	25	𝑥𝑙	𝑥𝑙	NOUN
iajs-2804	91	26	=	=	SYM
iajs-2804	91	27	𝑟𝑡	𝑟𝑡	NUM
iajs-2804	91	28	𝑦𝑑	𝑦𝑑	VERB
iajs-2804	91	29	for	for	ADP
iajs-2804	91	30	all	all	DET
iajs-2804	91	31	𝑥𝑙	𝑥𝑙	NOUN
iajs-2804	91	32	,	,	PUNCT
iajs-2804	91	33	𝑦𝑑	𝑦𝑑	VERB
iajs-2804	91	34	⊆	⊆	NUM
iajs-2804	91	35	x	x	PUNCT
iajs-2804	91	36	and	and	CCONJ
iajs-2804	91	37	𝑟𝑡	𝑟𝑡	PROPN
iajs-2804	91	38	be	be	AUX
iajs-2804	91	39	an	an	DET
iajs-2804	91	40	ℱ	ℱ	PROPN
iajs-2804	91	41	−	−	PROPN
iajs-2804	91	42	singleton	singleton	NOUN
iajs-2804	91	43	of	of	ADP
iajs-2804	91	44	ℛ	ℛ	PROPN
iajs-2804	91	45	then	then	ADV
iajs-2804	91	46	𝑥𝑙	𝑥𝑙	PRON
iajs-2804	91	47	=	=	SYM
iajs-2804	91	48	𝑦𝑑	𝑦𝑑	X
iajs-2804	91	49	.	.	PUNCT
iajs-2804	92	1	definition	definition	NOUN
iajs-2804	92	2	1.26	1.26	NUM
iajs-2804	92	3	[	[	X
iajs-2804	92	4	3	3	X
iajs-2804	92	5	]	]	X
iajs-2804	92	6	a	a	DET
iajs-2804	92	7	proper	proper	ADJ
iajs-2804	92	8	ℱ-sub	ℱ-sub	ADJ
iajs-2804	92	9	-	-	NOUN
iajs-2804	92	10	module	module	ADJ
iajs-2804	92	11	u	u	NOUN
iajs-2804	92	12	of	of	ADP
iajs-2804	92	13	an	an	DET
iajs-2804	92	14	ℱ-module	ℱ-module	PROPN
iajs-2804	92	15	x	x	PROPN
iajs-2804	92	16	of	of	ADP
iajs-2804	92	17	an	an	DET
iajs-2804	92	18	ℛ-module	ℛ-module	PROPN
iajs-2804	92	19	m	m	NOUN
iajs-2804	92	20	is	be	AUX
iajs-2804	92	21	called	call	VERB
iajs-2804	92	22	semi	semi	ADJ
iajs-2804	92	23	-	-	ADJ
iajs-2804	92	24	prime	prime	ADJ
iajs-2804	92	25	ℱsub	ℱsub	PROPN
iajs-2804	92	26	-	-	PUNCT
iajs-2804	92	27	module	module	NOUN
iajs-2804	92	28	of	of	ADP
iajs-2804	92	29	x	x	SYM
iajs-2804	92	30	if	if	SCONJ
iajs-2804	92	31	whenever	whenever	SCONJ
iajs-2804	92	32	𝑟𝑏	𝑟𝑏	PRON
iajs-2804	92	33	𝑛𝑚𝑡	𝑛𝑚𝑡	VERB
iajs-2804	92	34	⊆	⊆	NUM
iajs-2804	92	35	𝑈,where	𝑈,where	NUM
iajs-2804	92	36	𝑟𝑏	𝑟𝑏	NUM
iajs-2804	92	37	is	be	AUX
iajs-2804	92	38	an	an	DET
iajs-2804	92	39	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	92	40	of	of	ADP
iajs-2804	92	41	ℛ	ℛ	PROPN
iajs-2804	92	42	,	,	PUNCT
iajs-2804	92	43	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	92	44	is	be	AUX
iajs-2804	92	45	an	an	DET
iajs-2804	92	46	ℱsingleton	ℱsingleton	PROPN
iajs-2804	92	47	of	of	ADP
iajs-2804	92	48	x	x	PROPN
iajs-2804	92	49	and	and	CCONJ
iajs-2804	92	50	n	n	PRON
iajs-2804	92	51	∈	∈	NOUN
iajs-2804	92	52	𝑍+implies	𝑍+implie	NOUN
iajs-2804	92	53	that	that	PRON
iajs-2804	92	54	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	92	55	⊆	⊆	NUM
iajs-2804	92	56	𝑈	𝑈	PROPN
iajs-2804	92	57	for	for	ADP
iajs-2804	92	58	each	each	DET
iajs-2804	92	59	t	t	PROPN
iajs-2804	92	60	,	,	PUNCT
iajs-2804	92	61	b	b	X
iajs-2804	92	62	∈	∈	PROPN
iajs-2804	93	1	[	[	X
iajs-2804	93	2	0,1	0,1	NUM
iajs-2804	93	3	]	]	PUNCT
iajs-2804	93	4	.	.	PUNCT
iajs-2804	94	1	definition	definition	NOUN
iajs-2804	94	2	1.27	1.27	NUM
iajs-2804	94	3	[	[	X
iajs-2804	94	4	4	4	NUM
iajs-2804	94	5	]	]	X
iajs-2804	94	6	a	a	DET
iajs-2804	94	7	proper	proper	ADJ
iajs-2804	94	8	sub	sub	NOUN
iajs-2804	94	9	-	-	NOUN
iajs-2804	94	10	module	module	ADJ
iajs-2804	94	11	e	e	NOUN
iajs-2804	94	12	of	of	ADP
iajs-2804	94	13	an	an	DET
iajs-2804	94	14	ℛ-module	ℛ-module	PROPN
iajs-2804	94	15	μ	μ	PROPN
iajs-2804	94	16	is	be	AUX
iajs-2804	94	17	called	call	VERB
iajs-2804	94	18	pproximately	pproximately	ADV
iajs-2804	94	19	semi	semi	ADJ
iajs-2804	94	20	prime	prime	NOUN
iajs-2804	94	21	(	(	PUNCT
iajs-2804	94	22	for	for	ADP
iajs-2804	94	23	a	a	DET
iajs-2804	94	24	short	short	ADJ
iajs-2804	94	25	app	app	ADJ
iajs-2804	94	26	-	-	PUNCT
iajs-2804	94	27	semi	semi	ADJ
iajs-2804	94	28	-	-	ADJ
iajs-2804	94	29	prime	prime	ADJ
iajs-2804	94	30	)	)	PUNCT
iajs-2804	94	31	sub	sub	NOUN
iajs-2804	94	32	-	-	NOUN
iajs-2804	94	33	module	module	NOUN
iajs-2804	94	34	of	of	ADP
iajs-2804	94	35	μ	μ	NUM
iajs-2804	94	36	if	if	SCONJ
iajs-2804	94	37	whenever	whenever	SCONJ
iajs-2804	94	38	𝑎𝑚	𝑎𝑚	PROPN
iajs-2804	94	39	∈	∈	PROPN
iajs-2804	94	40	𝐸	𝐸	PROPN
iajs-2804	94	41	,	,	PUNCT
iajs-2804	94	42	for	for	ADP
iajs-2804	94	43	𝑎	𝑎	PROPN
iajs-2804	94	44	∈	∈	PROPN
iajs-2804	94	45	ℛ	ℛ	PROPN
iajs-2804	94	46	,	,	PUNCT
iajs-2804	94	47	𝑚	𝑚	PROPN
iajs-2804	94	48	∈	∈	PROPN
iajs-2804	94	49	μ𝑖𝑚𝑝𝑙𝑖𝑒𝑠	μ𝑖𝑚𝑝𝑙𝑖𝑒𝑠	NOUN
iajs-2804	94	50	that	that	PRON
iajs-2804	94	51	𝑎𝑚	𝑎𝑚	PROPN
iajs-2804	94	52	∈	∈	PROPN
iajs-2804	94	53	𝐸	𝐸	PROPN
iajs-2804	94	54	+	+	CCONJ
iajs-2804	94	55	𝑆𝑜𝑐(μ	𝑆𝑜𝑐(μ	NOUN
iajs-2804	94	56	)	)	PUNCT
iajs-2804	94	57	.	.	PUNCT
iajs-2804	95	1	ibn	ibn	PROPN
iajs-2804	95	2	al	al	PROPN
iajs-2804	95	3	-	-	PUNCT
iajs-2804	95	4	haitham	haitham	PROPN
iajs-2804	95	5	jour	jour	X
iajs-2804	95	6	.	.	PROPN
iajs-2804	95	7	for	for	ADP
iajs-2804	95	8	pure	pure	ADJ
iajs-2804	95	9	&	&	CCONJ
iajs-2804	95	10	appl	appl	PROPN
iajs-2804	95	11	.	.	PUNCT
iajs-2804	96	1	sci	sci	PROPN
iajs-2804	96	2	.	.	PROPN
iajs-2804	97	1	53	53	NUM
iajs-2804	97	2	(	(	PUNCT
iajs-2804	97	3	1)2022	1)2022	NUM
iajs-2804	97	4	106	106	NUM
iajs-2804	97	5	definition	definition	NOUN
iajs-2804	97	6	1.28	1.28	NUM
iajs-2804	97	7	[	[	X
iajs-2804	97	8	9	9	NUM
iajs-2804	97	9	]	]	PUNCT
iajs-2804	97	10	an	an	DET
iajs-2804	97	11	ℱ-sub	ℱ-sub	NOUN
iajs-2804	97	12	-	-	NOUN
iajs-2804	97	13	module	module	NOUN
iajs-2804	97	14	n	n	NOUN
iajs-2804	97	15	of	of	ADP
iajs-2804	97	16	an	an	DET
iajs-2804	97	17	ℱ-module	ℱ-module	PROPN
iajs-2804	97	18	x	x	PROPN
iajs-2804	97	19	of	of	ADP
iajs-2804	97	20	an	an	DET
iajs-2804	97	21	ℛ-module	ℛ-module	PROPN
iajs-2804	97	22	m	m	NOUN
iajs-2804	97	23	is	be	AUX
iajs-2804	97	24	called	call	VERB
iajs-2804	97	25	weakly	weakly	ADV
iajs-2804	97	26	pure	pure	ADJ
iajs-2804	97	27	ℱ-submodule	ℱ-submodule	PROPN
iajs-2804	97	28	of	of	ADP
iajs-2804	97	29	x	x	PRON
iajs-2804	97	30	if	if	SCONJ
iajs-2804	97	31	for	for	ADP
iajs-2804	97	32	any	any	DET
iajs-2804	97	33	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	97	34	𝑟𝑏	𝑟𝑏	NOUN
iajs-2804	97	35	of	of	ADP
iajs-2804	97	36	ℛ	ℛ	PROPN
iajs-2804	97	37	implies	imply	VERB
iajs-2804	97	38	that	that	SCONJ
iajs-2804	97	39	𝑟𝑏𝑁	𝑟𝑏𝑁	PROPN
iajs-2804	97	40	=	=	SYM
iajs-2804	97	41	𝑟𝑏𝑋	𝑟𝑏𝑋	NOUN
iajs-2804	97	42	∩	∩	NOUN
iajs-2804	97	43	𝑁	𝑁	PROPN
iajs-2804	97	44	with	with	ADP
iajs-2804	97	45	b	b	PROPN
iajs-2804	97	46	∈	∈	PROPN
iajs-2804	97	47	[	[	X
iajs-2804	97	48	0,1	0,1	NUM
iajs-2804	97	49	]	]	PUNCT
iajs-2804	97	50	.	.	PUNCT
iajs-2804	98	1	lemma	lemma	PROPN
iajs-2804	98	2	1.29	1.29	NUM
iajs-2804	99	1	[	[	X
iajs-2804	99	2	18	18	NUM
iajs-2804	99	3	]	]	PUNCT
iajs-2804	99	4	let	let	VERB
iajs-2804	99	5	x	x	PRON
iajs-2804	99	6	be	be	AUX
iajs-2804	99	7	an	an	DET
iajs-2804	99	8	ℱ-module	ℱ-module	PROPN
iajs-2804	99	9	of	of	ADP
iajs-2804	99	10	an	an	DET
iajs-2804	99	11	ℛ-module	ℛ-module	PROPN
iajs-2804	99	12	m	m	NOUN
iajs-2804	99	13	and	and	CCONJ
iajs-2804	99	14	let	let	VERB
iajs-2804	99	15	α	α	PRON
iajs-2804	99	16	,	,	PUNCT
iajs-2804	99	17	β	β	PROPN
iajs-2804	99	18	and	and	CCONJ
iajs-2804	99	19	c	c	PROPN
iajs-2804	99	20	are	be	AUX
iajs-2804	99	21	ℱ-sub	ℱ-sub	NOUN
iajs-2804	99	22	-	-	NOUN
iajs-2804	99	23	modules	module	NOUN
iajs-2804	99	24	of	of	ADP
iajs-2804	99	25	x	x	SYM
iajs-2804	99	26	such	such	ADJ
iajs-2804	99	27	that	that	SCONJ
iajs-2804	99	28	c	c	PROPN
iajs-2804	99	29	⊆	⊆	NUM
iajs-2804	99	30	β	β	X
iajs-2804	99	31	.	.	PUNCT
iajs-2804	100	1	then	then	ADV
iajs-2804	100	2	𝐶	𝐶	PROPN
iajs-2804	100	3	+	+	CCONJ
iajs-2804	100	4	(	(	PUNCT
iajs-2804	100	5	β⋂α	β⋂α	NOUN
iajs-2804	100	6	)	)	PUNCT
iajs-2804	100	7	=	=	PUNCT
iajs-2804	100	8	(	(	PUNCT
iajs-2804	100	9	𝐶	𝐶	PROPN
iajs-2804	100	10	+	+	CCONJ
iajs-2804	100	11	α)⋂β	α)⋂β	NOUN
iajs-2804	100	12	.	.	PUNCT
iajs-2804	101	1	proposition	proposition	NOUN
iajs-2804	101	2	1.30	1.30	NUM
iajs-2804	102	1	[	[	X
iajs-2804	102	2	14	14	NUM
iajs-2804	102	3	]	]	X
iajs-2804	102	4	if	if	SCONJ
iajs-2804	102	5	μ	μ	PROPN
iajs-2804	102	6	be	be	VERB
iajs-2804	102	7	a	a	DET
iajs-2804	102	8	faithful	faithful	ADJ
iajs-2804	102	9	multiplication	multiplication	NOUN
iajs-2804	102	10	ℛ-module	ℛ-module	PROPN
iajs-2804	102	11	,	,	PUNCT
iajs-2804	102	12	then	then	ADV
iajs-2804	102	13	𝑆𝑜𝑐(ℛ)μ	𝑆𝑜𝑐(ℛ)μ	PROPN
iajs-2804	102	14	=	=	SYM
iajs-2804	102	15	𝑆𝑜𝑐(μ	𝑆𝑜𝑐(μ	PROPN
iajs-2804	102	16	)	)	PUNCT
iajs-2804	102	17	definition	definition	NOUN
iajs-2804	102	18	1.31	1.31	NUM
iajs-2804	102	19	[	[	X
iajs-2804	102	20	15	15	NUM
iajs-2804	102	21	]	]	PUNCT
iajs-2804	102	22	let	let	VERB
iajs-2804	102	23	x	x	PRON
iajs-2804	102	24	be	be	AUX
iajs-2804	102	25	an	an	DET
iajs-2804	102	26	ℱ-module	ℱ-module	PROPN
iajs-2804	102	27	of	of	ADP
iajs-2804	102	28	an	an	DET
iajs-2804	102	29	ℛmodule	ℛmodule	PROPN
iajs-2804	102	30	μ	μ	NUM
iajs-2804	102	31	.	.	PUNCT
iajs-2804	103	1	x	x	PUNCT
iajs-2804	103	2	is	be	AUX
iajs-2804	103	3	called	call	VERB
iajs-2804	103	4	multiplication	multiplication	NOUN
iajs-2804	103	5	ℱ-module	ℱ-module	PROPN
iajs-2804	103	6	if	if	SCONJ
iajs-2804	103	7	and	and	CCONJ
iajs-2804	103	8	only	only	ADV
iajs-2804	103	9	if	if	SCONJ
iajs-2804	103	10	for	for	ADP
iajs-2804	103	11	each	each	DET
iajs-2804	103	12	ℱsub	ℱsub	PROPN
iajs-2804	103	13	-	-	PUNCT
iajs-2804	103	14	module	module	NOUN
iajs-2804	103	15	α	α	NOUN
iajs-2804	103	16	of	of	ADP
iajs-2804	103	17	x	x	PRON
iajs-2804	103	18	,	,	PUNCT
iajs-2804	103	19	there	there	PRON
iajs-2804	103	20	exists	exist	VERB
iajs-2804	103	21	an	an	DET
iajs-2804	103	22	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	103	23	k	k	PROPN
iajs-2804	103	24	of	of	ADP
iajs-2804	103	25	ℛ	ℛ	PROPN
iajs-2804	103	26	such	such	ADJ
iajs-2804	103	27	that	that	SCONJ
iajs-2804	103	28	α	α	PROPN
iajs-2804	103	29	=	=	SYM
iajs-2804	103	30	kx	kx	PROPN
iajs-2804	103	31	.	.	PUNCT
iajs-2804	103	32	proposition	proposition	NOUN
iajs-2804	103	33	1.32	1.32	NUM
iajs-2804	104	1	[	[	X
iajs-2804	104	2	15	15	NUM
iajs-2804	104	3	]	]	PUNCT
iajs-2804	104	4	an	an	DET
iajs-2804	104	5	ℱ-module	ℱ-module	PROPN
iajs-2804	104	6	x	x	PROPN
iajs-2804	104	7	of	of	ADP
iajs-2804	104	8	an	an	DET
iajs-2804	104	9	ℛ-module	ℛ-module	PROPN
iajs-2804	104	10	μ	μ	PROPN
iajs-2804	104	11	is	be	AUX
iajs-2804	104	12	a	a	DET
iajs-2804	104	13	multiplication	multiplication	NOUN
iajs-2804	104	14	if	if	SCONJ
iajs-2804	104	15	and	and	CCONJ
iajs-2804	104	16	only	only	ADV
iajs-2804	104	17	if	if	SCONJ
iajs-2804	104	18	every	every	DET
iajs-2804	104	19	non	non	ADJ
iajs-2804	104	20	-	-	ADJ
iajs-2804	104	21	empty	empty	ADJ
iajs-2804	104	22	ℱ	ℱ	PROPN
iajs-2804	104	23	sub	sub	NOUN
iajs-2804	104	24	-	-	NOUN
iajs-2804	104	25	module	module	NOUN
iajs-2804	104	26	a	a	PRON
iajs-2804	104	27	of	of	ADP
iajs-2804	104	28	x	x	SYM
iajs-2804	104	29	such	such	ADJ
iajs-2804	104	30	that	that	SCONJ
iajs-2804	104	31	α	α	NOUN
iajs-2804	104	32	=	=	SYM
iajs-2804	104	33	(	(	PUNCT
iajs-2804	104	34	α:𝑅	α:𝑅	PRON
iajs-2804	104	35	𝑋)𝑋	𝑋)𝑋	VERB
iajs-2804	104	36	.	.	PUNCT
iajs-2804	105	1	definition	definition	NOUN
iajs-2804	105	2	1.33	1.33	NUM
iajs-2804	105	3	[	[	SYM
iajs-2804	105	4	19	19	NUM
iajs-2804	105	5	]	]	PUNCT
iajs-2804	105	6	a	a	DET
iajs-2804	105	7	sub	sub	ADJ
iajs-2804	105	8	-	-	ADJ
iajs-2804	105	9	module	module	ADJ
iajs-2804	105	10	𝑉	𝑉	PROPN
iajs-2804	105	11	of	of	ADP
iajs-2804	105	12	ℛ-module	ℛ-module	PROPN
iajs-2804	105	13	μ	μ	PROPN
iajs-2804	105	14	is	be	AUX
iajs-2804	105	15	called	call	VERB
iajs-2804	105	16	essential	essential	ADJ
iajs-2804	105	17	if	if	SCONJ
iajs-2804	105	18	𝐻	𝐻	PROPN
iajs-2804	105	19	∩	∩	NOUN
iajs-2804	105	20	v	v	ADP
iajs-2804	105	21	≠	≠	PROPN
iajs-2804	105	22	0	0	NUM
iajs-2804	105	23	.	.	PUNCT
iajs-2804	106	1	for	for	ADP
iajs-2804	106	2	non	non	ADJ
iajs-2804	106	3	-	-	ADJ
iajs-2804	106	4	trivial	trivial	ADJ
iajs-2804	106	5	sub	sub	ADJ
iajs-2804	106	6	-	-	ADJ
iajs-2804	106	7	module	module	ADJ
iajs-2804	106	8	h	h	NOUN
iajs-2804	106	9	of	of	ADP
iajs-2804	106	10	μ	μ	PROPN
iajs-2804	106	11	.	.	PUNCT
iajs-2804	107	1	definition	definition	NOUN
iajs-2804	107	2	1.34	1.34	NUM
iajs-2804	107	3	[	[	X
iajs-2804	107	4	9	9	NUM
iajs-2804	107	5	]	]	PUNCT
iajs-2804	107	6	let	let	VERB
iajs-2804	107	7	x	x	PRON
iajs-2804	107	8	be	be	AUX
iajs-2804	107	9	an	an	DET
iajs-2804	107	10	ℱ-module	ℱ-module	PROPN
iajs-2804	107	11	of	of	ADP
iajs-2804	107	12	an	an	DET
iajs-2804	107	13	ℛ-module	ℛ-module	PROPN
iajs-2804	107	14	μ	μ	PROPN
iajs-2804	107	15	.	.	PUNCT
iajs-2804	108	1	an	an	DET
iajs-2804	108	2	ℱ-sub	ℱ-sub	NOUN
iajs-2804	108	3	-	-	NOUN
iajs-2804	108	4	module	module	NOUN
iajs-2804	108	5	a	a	PRON
iajs-2804	108	6	of	of	ADP
iajs-2804	108	7	x	x	PRON
iajs-2804	108	8	is	be	AUX
iajs-2804	108	9	called	call	VERB
iajs-2804	108	10	essential	essential	ADJ
iajs-2804	108	11	if	if	SCONJ
iajs-2804	108	12	𝐴	𝐴	PROPN
iajs-2804	108	13	∩	∩	ADJ
iajs-2804	108	14	𝐵	𝐵	NOUN
iajs-2804	108	15	≠	≠	PROPN
iajs-2804	108	16	01	01	NUM
iajs-2804	108	17	,	,	PUNCT
iajs-2804	108	18	for	for	ADP
iajs-2804	108	19	nontrivial	nontrivial	ADJ
iajs-2804	108	20	ℱ-sub	ℱ-sub	NOUN
iajs-2804	108	21	-	-	NOUN
iajs-2804	108	22	module	module	NOUN
iajs-2804	108	23	b	b	PROPN
iajs-2804	108	24	of	of	ADP
iajs-2804	108	25	x.	x.	NOUN
iajs-2804	108	26	finally	finally	ADV
iajs-2804	108	27	,	,	PUNCT
iajs-2804	108	28	(	(	PUNCT
iajs-2804	108	29	shortly	shortly	ADV
iajs-2804	108	30	fuzzy	fuzzy	ADJ
iajs-2804	108	31	set	set	NOUN
iajs-2804	108	32	,	,	PUNCT
iajs-2804	108	33	fuzzy	fuzzy	ADJ
iajs-2804	108	34	sub	sub	NOUN
iajs-2804	108	35	-	-	NOUN
iajs-2804	108	36	module	module	ADJ
iajs-2804	108	37	,	,	PUNCT
iajs-2804	108	38	fuzzy	fuzzy	ADJ
iajs-2804	108	39	ideal	ideal	ADJ
iajs-2804	108	40	,	,	PUNCT
iajs-2804	108	41	fuzzy	fuzzy	ADJ
iajs-2804	108	42	module	module	NOUN
iajs-2804	108	43	and	and	CCONJ
iajs-2804	108	44	fuzzy	fuzzy	ADJ
iajs-2804	108	45	singleton	singleton	NOUN
iajs-2804	108	46	are	be	AUX
iajs-2804	108	47	ℱ-set	ℱ-set	NOUN
iajs-2804	108	48	,	,	PUNCT
iajs-2804	108	49	ℱ-sub	ℱ-sub	NOUN
iajs-2804	108	50	-	-	NOUN
iajs-2804	108	51	module	module	NOUN
iajs-2804	108	52	,	,	PUNCT
iajs-2804	108	53	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	108	54	,	,	PUNCT
iajs-2804	108	55	ℱ-module	ℱ-module	PROPN
iajs-2804	108	56	and	and	CCONJ
iajs-2804	108	57	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	108	58	)	)	PUNCT
iajs-2804	108	59	.	.	PUNCT
iajs-2804	108	60	"	"	PUNCT
iajs-2804	109	1	𝓕-soc	𝓕-soc	NOUN
iajs-2804	109	2	-	-	PUNCT
iajs-2804	109	3	semi	semi	ADJ
iajs-2804	109	4	-	-	ADJ
iajs-2804	109	5	prime	prime	ADJ
iajs-2804	109	6	sub	sub	NOUN
iajs-2804	109	7	-	-	NOUN
iajs-2804	109	8	modules	module	NOUN
iajs-2804	109	9	in	in	ADP
iajs-2804	109	10	this	this	DET
iajs-2804	109	11	section	section	NOUN
iajs-2804	109	12	,	,	PUNCT
iajs-2804	109	13	we	we	PRON
iajs-2804	109	14	offer	offer	VERB
iajs-2804	109	15	the	the	DET
iajs-2804	109	16	concept	concept	NOUN
iajs-2804	109	17	of	of	ADP
iajs-2804	109	18	an	an	DET
iajs-2804	109	19	ℱ-soc	ℱ-soc	NOUN
iajs-2804	109	20	-	-	PUNCT
iajs-2804	109	21	semi	semi	ADJ
iajs-2804	109	22	-	-	ADJ
iajs-2804	109	23	prime	prime	ADJ
iajs-2804	109	24	sub	sub	NOUN
iajs-2804	109	25	-	-	NOUN
iajs-2804	109	26	module	module	NOUN
iajs-2804	109	27	as	as	ADP
iajs-2804	109	28	a	a	DET
iajs-2804	109	29	generalization	generalization	NOUN
iajs-2804	109	30	of	of	ADP
iajs-2804	109	31	ordinary	ordinary	ADJ
iajs-2804	109	32	concept(approximately	concept(approximately	ADV
iajs-2804	109	33	semi	semi	ADJ
iajs-2804	109	34	-	-	ADJ
iajs-2804	109	35	prime	prime	ADJ
iajs-2804	109	36	sub	sub	NOUN
iajs-2804	109	37	-	-	NOUN
iajs-2804	109	38	module	module	NOUN
iajs-2804	109	39	)	)	PUNCT
iajs-2804	109	40	.	.	PUNCT
iajs-2804	110	1	some	some	DET
iajs-2804	110	2	characterizations	characterization	NOUN
iajs-2804	110	3	of	of	ADP
iajs-2804	110	4	ℱsoc	ℱsoc	PROPN
iajs-2804	110	5	-	-	PUNCT
iajs-2804	110	6	prime	prime	ADJ
iajs-2804	110	7	sub	sub	NOUN
iajs-2804	110	8	-	-	NOUN
iajs-2804	110	9	module	module	NOUN
iajs-2804	110	10	are	be	AUX
iajs-2804	110	11	introduced	introduce	VERB
iajs-2804	110	12	.	.	PUNCT
iajs-2804	111	1	definition	definition	NOUN
iajs-2804	111	2	2.1	2.1	NUM
iajs-2804	111	3	let	let	VERB
iajs-2804	111	4	𝑟𝑏	𝑟𝑏	PART
iajs-2804	111	5	be	be	AUX
iajs-2804	111	6	an	an	DET
iajs-2804	111	7	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	111	8	of	of	ADP
iajs-2804	111	9	ℛ	ℛ	PROPN
iajs-2804	111	10	and	and	CCONJ
iajs-2804	111	11	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	111	12	is	be	AUX
iajs-2804	111	13	an	an	DET
iajs-2804	111	14	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	111	15	of	of	ADP
iajs-2804	111	16	x	x	SYM
iajs-2804	111	17	,	,	PUNCT
iajs-2804	111	18	then	then	ADV
iajs-2804	111	19	a	a	DET
iajs-2804	111	20	proper	proper	ADJ
iajs-2804	111	21	ℱ-sub	ℱ-sub	NOUN
iajs-2804	111	22	-	-	NOUN
iajs-2804	111	23	module	module	ADJ
iajs-2804	111	24	u	u	NOUN
iajs-2804	111	25	of	of	ADP
iajs-2804	111	26	an	an	DET
iajs-2804	111	27	ℱ-module	ℱ-module	PROPN
iajs-2804	111	28	x	x	PROPN
iajs-2804	111	29	of	of	ADP
iajs-2804	111	30	an	an	DET
iajs-2804	111	31	ℛ-module	ℛ-module	PROPN
iajs-2804	111	32	m	m	NOUN
iajs-2804	111	33	is	be	AUX
iajs-2804	111	34	called	call	VERB
iajs-2804	111	35	an	an	DET
iajs-2804	111	36	ℱ-socle	ℱ-socle	PROPN
iajs-2804	111	37	semi	semi	NOUN
iajs-2804	111	38	-	-	ADJ
iajs-2804	111	39	prime	prime	ADJ
iajs-2804	111	40	(	(	PUNCT
iajs-2804	111	41	for	for	ADP
iajs-2804	111	42	short	short	ADJ
iajs-2804	111	43	ℱ-socsemi	ℱ-socsemi	PROPN
iajs-2804	111	44	-	-	PUNCT
iajs-2804	111	45	prime	prime	NOUN
iajs-2804	111	46	)	)	PUNCT
iajs-2804	111	47	sub	sub	NOUN
iajs-2804	111	48	-	-	NOUN
iajs-2804	111	49	module(ideal	module(ideal	ADJ
iajs-2804	111	50	)	)	PUNCT
iajs-2804	111	51	of	of	ADP
iajs-2804	111	52	x	x	PRON
iajs-2804	111	53	if	if	SCONJ
iajs-2804	111	54	whenever	whenever	SCONJ
iajs-2804	111	55	𝑟𝑏	𝑟𝑏	PRON
iajs-2804	111	56	𝑛𝑚𝑡	𝑛𝑚𝑡	VERB
iajs-2804	111	57	⊆	⊆	NUM
iajs-2804	111	58	𝑈	𝑈	NOUN
iajs-2804	111	59	with	with	ADP
iajs-2804	111	60	n	n	PRON
iajs-2804	111	61	∈	∈	NOUN
iajs-2804	112	1	𝑍+	𝑍+	NOUN
iajs-2804	112	2	implies	imply	VERB
iajs-2804	112	3	that	that	SCONJ
iajs-2804	112	4	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	112	5	⊆	⊆	NUM
iajs-2804	112	6	𝑈	𝑈	PROPN
iajs-2804	112	7	+	+	CCONJ
iajs-2804	112	8	ℱ	ℱ	PROPN
iajs-2804	112	9	−	−	PROPN
iajs-2804	112	10	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	112	11	)	)	PUNCT
iajs-2804	112	12	for	for	ADP
iajs-2804	112	13	each	each	DET
iajs-2804	112	14	t	t	PROPN
iajs-2804	112	15	,	,	PUNCT
iajs-2804	112	16	b	b	X
iajs-2804	112	17	∈	∈	PROPN
iajs-2804	113	1	[	[	X
iajs-2804	113	2	0,1	0,1	NUM
iajs-2804	113	3	]	]	PUNCT
iajs-2804	113	4	.	.	PUNCT
iajs-2804	114	1	furthermore	furthermore	ADV
iajs-2804	114	2	,	,	PUNCT
iajs-2804	114	3	if	if	SCONJ
iajs-2804	114	4	𝑟𝑏and	𝑟𝑏and	NOUN
iajs-2804	114	5	𝑠ℎ	𝑠ℎ	NOUN
iajs-2804	114	6	are	be	AUX
iajs-2804	114	7	ℱ-singletons	ℱ-singletons	PROPN
iajs-2804	114	8	of	of	ADP
iajs-2804	114	9	ℛ	ℛ	PROPN
iajs-2804	114	10	,	,	PUNCT
iajs-2804	114	11	then	then	ADV
iajs-2804	114	12	a	a	DET
iajs-2804	114	13	proper	proper	ADJ
iajs-2804	114	14	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	114	15	l	l	NOUN
iajs-2804	114	16	of	of	ADP
iajs-2804	114	17	ℛ	ℛ	PROPN
iajs-2804	114	18	is	be	AUX
iajs-2804	114	19	called	call	VERB
iajs-2804	114	20	an	an	DET
iajs-2804	114	21	ℱ-socle	ℱ-socle	PROPN
iajs-2804	114	22	semi	semi	NOUN
iajs-2804	114	23	-	-	ADJ
iajs-2804	114	24	prime	prime	ADJ
iajs-2804	114	25	(	(	PUNCT
iajs-2804	114	26	for	for	ADP
iajs-2804	114	27	short	short	ADJ
iajs-2804	114	28	ℱ-soc	ℱ-soc	PROPN
iajs-2804	114	29	-	-	PUNCT
iajs-2804	114	30	semi	semi	ADJ
iajs-2804	114	31	-	-	ADJ
iajs-2804	114	32	prime	prime	ADJ
iajs-2804	114	33	)	)	PUNCT
iajs-2804	114	34	ideal	ideal	NOUN
iajs-2804	114	35	of	of	ADP
iajs-2804	114	36	ℛ	ℛ	PROPN
iajs-2804	114	37	if	if	SCONJ
iajs-2804	114	38	whenever	whenever	SCONJ
iajs-2804	114	39	𝑟𝑏	𝑟𝑏	AUX
iajs-2804	114	40	𝑛𝑠ℎ	𝑛𝑠ℎ	VERB
iajs-2804	114	41	⊆	⊆	NUM
iajs-2804	114	42	𝐿	𝐿	PROPN
iajs-2804	114	43	with	with	ADP
iajs-2804	114	44	n	n	PRON
iajs-2804	114	45	∈	∈	PROPN
iajs-2804	114	46	𝑍+implies	𝑍+implie	NOUN
iajs-2804	114	47	that	that	SCONJ
iajs-2804	114	48	𝑟𝑏𝑠ℎ	𝑟𝑏𝑠ℎ	NOUN
iajs-2804	114	49	⊆	⊆	NUM
iajs-2804	114	50	𝐿	𝐿	PROPN
iajs-2804	114	51	+	+	CCONJ
iajs-2804	114	52	ℱ	ℱ	PROPN
iajs-2804	114	53	−	−	PROPN
iajs-2804	114	54	𝑆𝑜𝑐(ℛ	𝑆𝑜𝑐(ℛ	NOUN
iajs-2804	114	55	)	)	PUNCT
iajs-2804	114	56	for	for	ADP
iajs-2804	114	57	each	each	DET
iajs-2804	114	58	h	h	NOUN
iajs-2804	114	59	,	,	PUNCT
iajs-2804	114	60	b	b	PROPN
iajs-2804	114	61	∈	∈	PROPN
iajs-2804	115	1	[	[	X
iajs-2804	115	2	0,1	0,1	NUM
iajs-2804	115	3	]	]	PUNCT
iajs-2804	115	4	.	.	PUNCT
iajs-2804	116	1	we	we	PRON
iajs-2804	116	2	will	will	AUX
iajs-2804	116	3	adopt	adopt	VERB
iajs-2804	116	4	the	the	DET
iajs-2804	116	5	definition	definition	NOUN
iajs-2804	116	6	of	of	ADP
iajs-2804	116	7	an	an	DET
iajs-2804	116	8	ℱ-socle	ℱ-socle	PROPN
iajs-2804	116	9	of	of	ADP
iajs-2804	116	10	x	x	PUNCT
iajs-2804	116	11	in	in	ADP
iajs-2804	116	12	this	this	DET
iajs-2804	116	13	research	research	NOUN
iajs-2804	116	14	as	as	SCONJ
iajs-2804	116	15	follows	follow	VERB
iajs-2804	116	16	:	:	PUNCT
iajs-2804	116	17	ibn	ibn	PROPN
iajs-2804	116	18	al	al	PROPN
iajs-2804	116	19	-	-	PUNCT
iajs-2804	116	20	haitham	haitham	PROPN
iajs-2804	116	21	jour	jour	X
iajs-2804	116	22	.	.	PROPN
iajs-2804	117	1	for	for	ADP
iajs-2804	117	2	pure	pure	ADJ
iajs-2804	117	3	&	&	CCONJ
iajs-2804	117	4	appl	appl	PROPN
iajs-2804	117	5	.	.	PUNCT
iajs-2804	118	1	sci	sci	PROPN
iajs-2804	118	2	.	.	PROPN
iajs-2804	119	1	53	53	NUM
iajs-2804	119	2	(	(	PUNCT
iajs-2804	119	3	1)2022	1)2022	PROPN
iajs-2804	119	4	107	107	NUM
iajs-2804	119	5	ℱ	ℱ	PROPN
iajs-2804	119	6	−	−	PROPN
iajs-2804	119	7	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	119	8	):	):	PUNCT
iajs-2804	119	9	𝑀	𝑀	PROPN
iajs-2804	119	10	→	→	PUNCT
iajs-2804	120	1	[	[	X
iajs-2804	120	2	0,1	0,1	NUM
iajs-2804	120	3	]	]	PUNCT
iajs-2804	120	4	such	such	ADJ
iajs-2804	120	5	that	that	SCONJ
iajs-2804	120	6	:	:	PUNCT
iajs-2804	120	7	ℱ	ℱ	PROPN
iajs-2804	120	8	−	−	NOUN
iajs-2804	120	9	𝑆𝑜𝑐(𝑋)(𝑚	𝑆𝑜𝑐(𝑋)(𝑚	NOUN
iajs-2804	120	10	)	)	PUNCT
iajs-2804	121	1	=	=	PRON
iajs-2804	121	2	{	{	PUNCT
iajs-2804	121	3	1	1	NUM
iajs-2804	121	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	121	5	𝑚	𝑚	ADP
iajs-2804	121	6	∈	∈	PROPN
iajs-2804	121	7	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	121	8	)	)	PUNCT
iajs-2804	121	9	ℎ	ℎ	PROPN
iajs-2804	121	10	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	121	11	𝑚	𝑚	PROPN
iajs-2804	121	12	∉	∉	PROPN
iajs-2804	121	13	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	PROPN
iajs-2804	121	14	)	)	PUNCT
iajs-2804	121	15	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
iajs-2804	121	16	0	0	PUNCT
iajs-2804	121	17	<	<	X
iajs-2804	121	18	ℎ	ℎ	X
iajs-2804	121	19	<	<	X
iajs-2804	121	20	1	1	NUM
iajs-2804	121	21	lemma	lemma	PROPN
iajs-2804	121	22	2.2	2.2	NUM
iajs-2804	121	23	(	(	PUNCT
iajs-2804	121	24	ℱ	ℱ	PROPN
iajs-2804	121	25	−	−	PROPN
iajs-2804	121	26	𝑆𝑜𝑐(𝑋))𝑡	𝑆𝑜𝑐(𝑋))𝑡	PROPN
iajs-2804	121	27	=	=	NOUN
iajs-2804	121	28	𝑆𝑜𝑐(𝑋𝑡	𝑆𝑜𝑐(𝑋𝑡	PROPN
iajs-2804	121	29	)	)	PUNCT
iajs-2804	121	30	for	for	ADP
iajs-2804	121	31	any	any	DET
iajs-2804	121	32	ℱ-module	ℱ-module	PROPN
iajs-2804	121	33	x	x	PUNCT
iajs-2804	121	34	for	for	ADP
iajs-2804	121	35	each	each	DET
iajs-2804	121	36	t	t	NOUN
iajs-2804	121	37	∈	∈	PROPN
iajs-2804	121	38	(	(	PUNCT
iajs-2804	121	39	0,1	0,1	NOUN
iajs-2804	121	40	]	]	PUNCT
iajs-2804	121	41	with	with	ADP
iajs-2804	121	42	(	(	PUNCT
iajs-2804	121	43	ℱ	ℱ	PROPN
iajs-2804	121	44	−	−	PROPN
iajs-2804	121	45	𝑆𝑜𝑐(𝑋))𝑡	𝑆𝑜𝑐(𝑋))𝑡	PROPN
iajs-2804	121	46	≠	≠	PROPN
iajs-2804	122	1	𝑋𝑡	𝑋𝑡	PROPN
iajs-2804	122	2	proof	proof	NOUN
iajs-2804	122	3	:	:	PUNCT
iajs-2804	122	4	ℱ	ℱ	PROPN
iajs-2804	122	5	−	−	PROPN
iajs-2804	122	6	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	122	7	):	):	PUNCT
iajs-2804	122	8	𝑀	𝑀	PROPN
iajs-2804	122	9	→	→	PUNCT
iajs-2804	123	1	[	[	X
iajs-2804	123	2	0,1	0,1	NUM
iajs-2804	123	3	]	]	PUNCT
iajs-2804	123	4	such	such	ADJ
iajs-2804	123	5	that	that	SCONJ
iajs-2804	123	6	:	:	PUNCT
iajs-2804	123	7	ℱ	ℱ	PROPN
iajs-2804	123	8	−	−	NOUN
iajs-2804	123	9	𝑆𝑜𝑐(𝑋)(𝑚	𝑆𝑜𝑐(𝑋)(𝑚	NOUN
iajs-2804	123	10	)	)	PUNCT
iajs-2804	124	1	=	=	PRON
iajs-2804	124	2	{	{	PUNCT
iajs-2804	124	3	1	1	NUM
iajs-2804	124	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	124	5	𝑚	𝑚	ADP
iajs-2804	124	6	∈	∈	PROPN
iajs-2804	124	7	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	124	8	)	)	PUNCT
iajs-2804	124	9	ℎ	ℎ	PROPN
iajs-2804	124	10	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	124	11	𝑚	𝑚	PROPN
iajs-2804	124	12	∉	∉	PROPN
iajs-2804	124	13	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	PROPN
iajs-2804	124	14	)	)	PUNCT
iajs-2804	124	15	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
iajs-2804	124	16	0	0	PUNCT
iajs-2804	124	17	<	<	X
iajs-2804	124	18	ℎ	ℎ	X
iajs-2804	124	19	<	<	X
iajs-2804	124	20	1	1	NUM
iajs-2804	124	21	now	now	ADV
iajs-2804	124	22	,	,	PUNCT
iajs-2804	124	23	(	(	PUNCT
iajs-2804	124	24	ℱ	ℱ	PROPN
iajs-2804	124	25	−	−	PROPN
iajs-2804	124	26	𝑆𝑜𝑐(𝑋))𝑡	𝑆𝑜𝑐(𝑋))𝑡	PROPN
iajs-2804	124	27	=	=	NOUN
iajs-2804	124	28	{	{	PUNCT
iajs-2804	124	29	𝑚	𝑚	PROPN
iajs-2804	124	30	∈	∈	PROPN
iajs-2804	124	31	𝑀	𝑀	PROPN
iajs-2804	124	32	∶	∶	NOUN
iajs-2804	124	33	(	(	PUNCT
iajs-2804	124	34	ℱ	ℱ	PROPN
iajs-2804	124	35	−	−	ADP
iajs-2804	124	36	𝑆𝑜𝑐(𝑋))(𝑚	𝑆𝑜𝑐(𝑋))(𝑚	ADJ
iajs-2804	124	37	)	)	PUNCT
iajs-2804	124	38	≥	≥	NOUN
iajs-2804	124	39	𝑡	𝑡	PROPN
iajs-2804	124	40	}	}	PUNCT
iajs-2804	124	41	so	so	CCONJ
iajs-2804	124	42	,	,	PUNCT
iajs-2804	124	43	if	if	SCONJ
iajs-2804	124	44	𝑡	𝑡	X
iajs-2804	124	45	=	=	VERB
iajs-2804	124	46	1	1	NUM
iajs-2804	124	47	then	then	ADV
iajs-2804	124	48	(	(	PUNCT
iajs-2804	124	49	ℱ	ℱ	PROPN
iajs-2804	124	50	−	−	PROPN
iajs-2804	124	51	𝑆𝑜𝑐(𝑋))𝑡	𝑆𝑜𝑐(𝑋))𝑡	PROPN
iajs-2804	124	52	=	=	SYM
iajs-2804	124	53	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	124	54	)	)	PUNCT
iajs-2804	124	55	=	=	SYM
iajs-2804	124	56	𝑆𝑜𝑐(𝑋𝑡	𝑆𝑜𝑐(𝑋𝑡	PROPN
iajs-2804	124	57	)	)	PUNCT
iajs-2804	125	1	if	if	SCONJ
iajs-2804	125	2	0	0	NUM
iajs-2804	125	3	<	<	X
iajs-2804	125	4	𝑡	𝑡	X
iajs-2804	125	5	≤	≤	ADJ
iajs-2804	125	6	ℎ	ℎ	NOUN
iajs-2804	125	7	then	then	ADV
iajs-2804	125	8	(	(	PUNCT
iajs-2804	125	9	ℱ	ℱ	PROPN
iajs-2804	125	10	−	−	PROPN
iajs-2804	125	11	𝑆𝑜𝑐(𝑋))𝑡	𝑆𝑜𝑐(𝑋))𝑡	PROPN
iajs-2804	125	12	=	=	SYM
iajs-2804	125	13	𝑀	𝑀	PROPN
iajs-2804	125	14	=	=	PUNCT
iajs-2804	126	1	𝑋𝑡	𝑋𝑡	PROPN
iajs-2804	126	2	that	that	PRON
iajs-2804	126	3	is	be	AUX
iajs-2804	126	4	a	a	DET
iajs-2804	126	5	contradiction	contradiction	NOUN
iajs-2804	126	6	if	if	SCONJ
iajs-2804	126	7	ℎ	ℎ	PROPN
iajs-2804	126	8	<	<	X
iajs-2804	126	9	𝑡	𝑡	X
iajs-2804	126	10	<	<	X
iajs-2804	126	11	1	1	NUM
iajs-2804	126	12	then	then	ADV
iajs-2804	126	13	(	(	PUNCT
iajs-2804	126	14	ℱ	ℱ	PROPN
iajs-2804	126	15	−	−	PROPN
iajs-2804	126	16	𝑆𝑜𝑐(𝑋))𝑡	𝑆𝑜𝑐(𝑋))𝑡	PROPN
iajs-2804	126	17	=	=	SYM
iajs-2804	126	18	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	126	19	)	)	PUNCT
iajs-2804	126	20	=	=	SYM
iajs-2804	126	21	𝑆𝑜𝑐(𝑋𝑡	𝑆𝑜𝑐(𝑋𝑡	PROPN
iajs-2804	126	22	)	)	PUNCT
iajs-2804	126	23	lemma	lemma	PROPN
iajs-2804	126	24	2.3	2.3	NUM
iajs-2804	126	25	let	let	VERB
iajs-2804	126	26	x	x	PRON
iajs-2804	126	27	be	be	AUX
iajs-2804	126	28	an	an	DET
iajs-2804	126	29	ℱ-module	ℱ-module	PROPN
iajs-2804	126	30	of	of	ADP
iajs-2804	126	31	an	an	DET
iajs-2804	126	32	ℛ-module	ℛ-module	PROPN
iajs-2804	126	33	m	m	NOUN
iajs-2804	126	34	with	with	ADP
iajs-2804	126	35	x(m)=1	x(m)=1	NUM
iajs-2804	126	36	for	for	ADP
iajs-2804	126	37	each	each	DET
iajs-2804	126	38	𝑚	𝑚	PROPN
iajs-2804	126	39	∈	∈	PROPN
iajs-2804	126	40	𝑀	𝑀	PROPN
iajs-2804	126	41	,	,	PUNCT
iajs-2804	126	42	if	if	SCONJ
iajs-2804	126	43	u	u	NOUN
iajs-2804	126	44	is	be	AUX
iajs-2804	126	45	an	an	DET
iajs-2804	126	46	ℱ-submodule	ℱ-submodule	PROPN
iajs-2804	126	47	of	of	ADP
iajs-2804	126	48	x	x	PRON
iajs-2804	126	49	is	be	AUX
iajs-2804	126	50	defined	define	VERB
iajs-2804	126	51	by	by	ADP
iajs-2804	126	52	𝑈	𝑈	PROPN
iajs-2804	126	53	:	:	PUNCT
iajs-2804	126	54	𝑀	𝑀	PROPN
iajs-2804	126	55	→	→	PUNCT
iajs-2804	127	1	[	[	X
iajs-2804	127	2	0,1	0,1	NUM
iajs-2804	127	3	]	]	PUNCT
iajs-2804	127	4	such	such	ADJ
iajs-2804	127	5	that	that	SCONJ
iajs-2804	127	6	:	:	PUNCT
iajs-2804	127	7	𝑈(𝑚	𝑈(𝑚	NOUN
iajs-2804	127	8	)	)	PUNCT
iajs-2804	127	9	=	=	SYM
iajs-2804	127	10	{	{	PUNCT
iajs-2804	127	11	1	1	NUM
iajs-2804	127	12	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	127	13	𝑚	𝑚	ADP
iajs-2804	127	14	∈	∈	PROPN
iajs-2804	127	15	𝐸	𝐸	PROPN
iajs-2804	127	16	𝑘	𝑘	X
iajs-2804	127	17	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	127	18	𝑚	𝑚	X
iajs-2804	127	19	∉	∉	PROPN
iajs-2804	127	20	𝐸	𝐸	PROPN
iajs-2804	127	21	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
iajs-2804	127	22	0	0	PUNCT
iajs-2804	127	23	<	<	X
iajs-2804	127	24	𝑘	𝑘	X
iajs-2804	127	25	<	<	X
iajs-2804	127	26	1	1	NUM
iajs-2804	127	27	where	where	SCONJ
iajs-2804	127	28	e	e	NOUN
iajs-2804	127	29	is	be	AUX
iajs-2804	127	30	a	a	DET
iajs-2804	127	31	sub	sub	NOUN
iajs-2804	127	32	-	-	NOUN
iajs-2804	127	33	module	module	NOUN
iajs-2804	127	34	of	of	ADP
iajs-2804	127	35	m.	m.	NOUN
iajs-2804	127	36	then	then	ADV
iajs-2804	127	37	u	u	NOUN
iajs-2804	127	38	is	be	AUX
iajs-2804	127	39	an	an	DET
iajs-2804	127	40	ℱ-soc	ℱ-soc	NOUN
iajs-2804	127	41	-	-	PUNCT
iajs-2804	127	42	semi	semi	ADJ
iajs-2804	127	43	-	-	ADJ
iajs-2804	127	44	prime	prime	ADJ
iajs-2804	127	45	sub	sub	NOUN
iajs-2804	127	46	-	-	NOUN
iajs-2804	127	47	module	module	NOUN
iajs-2804	127	48	of	of	ADP
iajs-2804	127	49	x	x	PRON
iajs-2804	127	50	if	if	SCONJ
iajs-2804	127	51	and	and	CCONJ
iajs-2804	127	52	only	only	ADV
iajs-2804	127	53	if	if	SCONJ
iajs-2804	127	54	e	e	NOUN
iajs-2804	127	55	is	be	AUX
iajs-2804	127	56	an	an	DET
iajs-2804	127	57	app	app	ADJ
iajs-2804	127	58	-	-	PUNCT
iajs-2804	127	59	semi	semi	ADJ
iajs-2804	127	60	-	-	ADJ
iajs-2804	127	61	prime	prime	ADJ
iajs-2804	127	62	sub	sub	NOUN
iajs-2804	127	63	-	-	NOUN
iajs-2804	127	64	module	module	NOUN
iajs-2804	127	65	of	of	ADP
iajs-2804	127	66	m.	m.	NOUN
iajs-2804	127	67	proof	proof	NOUN
iajs-2804	127	68	:	:	PUNCT
iajs-2804	127	69	first	first	ADV
iajs-2804	127	70	of	of	ADP
iajs-2804	127	71	all	all	PRON
iajs-2804	127	72	,	,	PUNCT
iajs-2804	127	73	we	we	PRON
iajs-2804	127	74	must	must	AUX
iajs-2804	127	75	define	define	VERB
iajs-2804	127	76	𝑈	𝑈	PROPN
iajs-2804	127	77	+	+	CCONJ
iajs-2804	127	78	ℱ	ℱ	PROPN
iajs-2804	127	79	−	−	PROPN
iajs-2804	127	80	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	127	81	)	)	PUNCT
iajs-2804	127	82	.	.	PUNCT
iajs-2804	128	1	(	(	PUNCT
iajs-2804	128	2	𝑈	𝑈	NOUN
iajs-2804	128	3	+	+	CCONJ
iajs-2804	128	4	ℱ	ℱ	PROPN
iajs-2804	128	5	−	−	ADP
iajs-2804	128	6	𝑆𝑜𝑐(𝑋))(𝑚	𝑆𝑜𝑐(𝑋))(𝑚	ADJ
iajs-2804	128	7	)	)	PUNCT
iajs-2804	129	1	=	=	SYM
iajs-2804	129	2	sup	sup	NOUN
iajs-2804	129	3	{	{	PUNCT
iajs-2804	129	4	min(𝑈(𝑦	min(𝑈(𝑦	NOUN
iajs-2804	129	5	)	)	PUNCT
iajs-2804	129	6	,	,	PUNCT
iajs-2804	129	7	ℱ	ℱ	PROPN
iajs-2804	129	8	−	−	NOUN
iajs-2804	129	9	𝑆𝑜𝑐(𝑋)(𝑧	𝑆𝑜𝑐(𝑋)(𝑧	NOUN
iajs-2804	129	10	)	)	PUNCT
iajs-2804	129	11	)	)	PUNCT
iajs-2804	129	12	,	,	PUNCT
iajs-2804	129	13	𝑦	𝑦	NOUN
iajs-2804	129	14	+	+	CCONJ
iajs-2804	129	15	𝑧	𝑧	X
iajs-2804	129	16	=	=	PUNCT
iajs-2804	129	17	𝑚	𝑚	NOUN
iajs-2804	129	18	}	}	PUNCT
iajs-2804	129	19	so	so	ADV
iajs-2804	129	20	,	,	PUNCT
iajs-2804	129	21	we	we	PRON
iajs-2804	129	22	have	have	VERB
iajs-2804	129	23	(	(	PUNCT
iajs-2804	129	24	𝑈	𝑈	NOUN
iajs-2804	129	25	+	+	CCONJ
iajs-2804	129	26	ℱ	ℱ	PROPN
iajs-2804	129	27	−	−	ADP
iajs-2804	129	28	𝑆𝑜𝑐(𝑋))(𝑚	𝑆𝑜𝑐(𝑋))(𝑚	ADJ
iajs-2804	129	29	)	)	PUNCT
iajs-2804	130	1	=	=	PRON
iajs-2804	130	2	{	{	PUNCT
iajs-2804	130	3	1	1	NUM
iajs-2804	130	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	130	5	𝑚	𝑚	ADP
iajs-2804	130	6	∈	∈	PROPN
iajs-2804	130	7	𝐸	𝐸	PROPN
iajs-2804	130	8	+	+	CCONJ
iajs-2804	130	9	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	130	10	)	)	PUNCT
iajs-2804	130	11	𝑠	𝑠	NOUN
iajs-2804	131	1	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	131	2	𝑚	𝑚	PROPN
iajs-2804	131	3	∉	∉	PROPN
iajs-2804	131	4	𝐸	𝐸	PROPN
iajs-2804	131	5	+	+	CCONJ
iajs-2804	131	6	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	131	7	)	)	PUNCT
iajs-2804	131	8	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
iajs-2804	131	9	𝑠	𝑠	NOUN
iajs-2804	131	10	=	=	SYM
iajs-2804	131	11	max	max	PROPN
iajs-2804	131	12	{	{	PUNCT
iajs-2804	131	13	𝑘	𝑘	PROPN
iajs-2804	131	14	,	,	PUNCT
iajs-2804	131	15	ℎ	ℎ	PROPN
iajs-2804	131	16	}	}	PUNCT
iajs-2804	131	17	where	where	SCONJ
iajs-2804	131	18	ℱ	ℱ	PROPN
iajs-2804	131	19	−	−	PROPN
iajs-2804	131	20	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	131	21	):	):	PUNCT
iajs-2804	131	22	𝑀	𝑀	PROPN
iajs-2804	131	23	→	→	PUNCT
iajs-2804	132	1	[	[	X
iajs-2804	132	2	0,1	0,1	NUM
iajs-2804	132	3	]	]	PUNCT
iajs-2804	132	4	such	such	ADJ
iajs-2804	132	5	that	that	SCONJ
iajs-2804	132	6	:	:	PUNCT
iajs-2804	132	7	ℱ	ℱ	PROPN
iajs-2804	132	8	−	−	NOUN
iajs-2804	132	9	𝑆𝑜𝑐(𝑋)(𝑚	𝑆𝑜𝑐(𝑋)(𝑚	NOUN
iajs-2804	132	10	)	)	PUNCT
iajs-2804	133	1	=	=	PRON
iajs-2804	133	2	{	{	PUNCT
iajs-2804	133	3	1	1	NUM
iajs-2804	133	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	133	5	𝑚	𝑚	ADP
iajs-2804	133	6	∈	∈	PROPN
iajs-2804	133	7	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	133	8	)	)	PUNCT
iajs-2804	133	9	ℎ	ℎ	PROPN
iajs-2804	133	10	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	133	11	𝑚	𝑚	PROPN
iajs-2804	133	12	∉	∉	PROPN
iajs-2804	133	13	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	PROPN
iajs-2804	133	14	)	)	PUNCT
iajs-2804	133	15	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
iajs-2804	133	16	0	0	PUNCT
iajs-2804	133	17	<	<	X
iajs-2804	133	18	ℎ	ℎ	X
iajs-2804	133	19	<	<	X
iajs-2804	133	20	1	1	NUM
iajs-2804	133	21	now	now	ADV
iajs-2804	133	22	,	,	PUNCT
iajs-2804	133	23	suppose	suppose	VERB
iajs-2804	133	24	e	e	NOUN
iajs-2804	133	25	is	be	AUX
iajs-2804	133	26	an	an	DET
iajs-2804	133	27	app	app	ADJ
iajs-2804	133	28	-	-	PUNCT
iajs-2804	133	29	semi	semi	ADJ
iajs-2804	133	30	-	-	ADJ
iajs-2804	133	31	prime	prime	ADJ
iajs-2804	133	32	sub	sub	NOUN
iajs-2804	133	33	-	-	NOUN
iajs-2804	133	34	module	module	NOUN
iajs-2804	133	35	of	of	ADP
iajs-2804	133	36	m	m	PROPN
iajs-2804	133	37	,	,	PUNCT
iajs-2804	133	38	to	to	PART
iajs-2804	133	39	prove	prove	VERB
iajs-2804	133	40	that	that	SCONJ
iajs-2804	133	41	u	u	NOUN
iajs-2804	133	42	is	be	AUX
iajs-2804	133	43	an	an	DET
iajs-2804	133	44	ℱ-soc	ℱ-soc	NOUN
iajs-2804	133	45	-	-	PUNCT
iajs-2804	133	46	semi	semi	ADJ
iajs-2804	133	47	-	-	ADJ
iajs-2804	133	48	prime	prime	ADJ
iajs-2804	133	49	sub	sub	NOUN
iajs-2804	133	50	-	-	NOUN
iajs-2804	133	51	module	module	NOUN
iajs-2804	133	52	of	of	ADP
iajs-2804	133	53	x.	x.	NOUN
iajs-2804	133	54	let	let	VERB
iajs-2804	133	55	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	133	56	⊆	⊆	NUM
iajs-2804	133	57	ℛ	ℛ	PROPN
iajs-2804	133	58	and	and	CCONJ
iajs-2804	133	59	𝑚𝑡	𝑚𝑡	ADP
iajs-2804	133	60	⊆	⊆	NUM
iajs-2804	133	61	𝑋	𝑋	PROPN
iajs-2804	133	62	for	for	ADP
iajs-2804	133	63	each	each	DET
iajs-2804	133	64	t	t	PROPN
iajs-2804	133	65	,	,	PUNCT
iajs-2804	133	66	b	b	PROPN
iajs-2804	133	67	∈	∈	PROPN
iajs-2804	134	1	[	[	X
iajs-2804	134	2	0,1	0,1	NUM
iajs-2804	134	3	]	]	PUNCT
iajs-2804	134	4	such	such	ADJ
iajs-2804	134	5	that	that	SCONJ
iajs-2804	134	6	(	(	PUNCT
iajs-2804	134	7	𝑟𝑏)𝑛𝑚𝑡	𝑟𝑏)𝑛𝑚𝑡	NOUN
iajs-2804	134	8	⊆	⊆	NUM
iajs-2804	134	9	𝑈,thus	𝑈,thus	X
iajs-2804	134	10	(	(	PUNCT
iajs-2804	134	11	𝑟𝑛)𝑏𝑚𝑡	𝑟𝑛)𝑏𝑚𝑡	NUM
iajs-2804	134	12	⊆	⊆	X
iajs-2804	134	13	𝑈	𝑈	PROPN
iajs-2804	134	14	that	that	PRON
iajs-2804	134	15	is	be	AUX
iajs-2804	134	16	either	either	CCONJ
iajs-2804	134	17	𝑟𝑛𝑚	𝑟𝑛𝑚	PROPN
iajs-2804	134	18	∈	∈	PROPN
iajs-2804	134	19	𝐸	𝐸	PROPN
iajs-2804	134	20	or	or	CCONJ
iajs-2804	134	21	𝑟𝑛𝑚	𝑟𝑛𝑚	VERB
iajs-2804	134	22	∉	∉	PROPN
iajs-2804	134	23	𝐸.	𝐸.	PROPN
iajs-2804	134	24	ibn	ibn	PROPN
iajs-2804	134	25	al	al	PROPN
iajs-2804	134	26	-	-	PUNCT
iajs-2804	134	27	haitham	haitham	PROPN
iajs-2804	134	28	jour	jour	X
iajs-2804	134	29	.	.	PROPN
iajs-2804	135	1	for	for	ADP
iajs-2804	135	2	pure	pure	ADJ
iajs-2804	135	3	&	&	CCONJ
iajs-2804	135	4	appl	appl	PROPN
iajs-2804	135	5	.	.	PUNCT
iajs-2804	136	1	sci	sci	PROPN
iajs-2804	136	2	.	.	PROPN
iajs-2804	137	1	53	53	NUM
iajs-2804	137	2	(	(	PUNCT
iajs-2804	137	3	1)2022	1)2022	NUM
iajs-2804	137	4	108	108	NUM
iajs-2804	137	5	1	1	NUM
iajs-2804	137	6	)	)	PUNCT
iajs-2804	137	7	if	if	SCONJ
iajs-2804	137	8	𝑟𝑛𝑚	𝑟𝑛𝑚	NOUN
iajs-2804	137	9	∈	∈	PROPN
iajs-2804	137	10	𝐸	𝐸	PROPN
iajs-2804	137	11	,	,	PUNCT
iajs-2804	137	12	then	then	ADV
iajs-2804	137	13	𝑟𝑚	𝑟𝑚	ADP
iajs-2804	137	14	∈	∈	PROPN
iajs-2804	137	15	𝐸	𝐸	PROPN
iajs-2804	137	16	+	+	CCONJ
iajs-2804	137	17	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	137	18	)	)	PUNCT
iajs-2804	137	19	.	.	PUNCT
iajs-2804	138	1	hence	hence	ADV
iajs-2804	138	2	(	(	PUNCT
iajs-2804	138	3	𝑈	𝑈	PROPN
iajs-2804	138	4	+	+	CCONJ
iajs-2804	138	5	ℱ	ℱ	PROPN
iajs-2804	138	6	−	−	NOUN
iajs-2804	138	7	𝑆𝑜𝑐(𝑋))(𝑟𝑚	𝑆𝑜𝑐(𝑋))(𝑟𝑚	NOUN
iajs-2804	138	8	)	)	PUNCT
iajs-2804	138	9	=	=	PUNCT
iajs-2804	139	1	1	1	NUM
iajs-2804	139	2	this	this	PRON
iajs-2804	139	3	implies	imply	VERB
iajs-2804	139	4	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	X
iajs-2804	139	5	=	=	PUNCT
iajs-2804	139	6	(	(	PUNCT
iajs-2804	139	7	𝑟𝑚)𝑡	𝑟𝑚)𝑡	PROPN
iajs-2804	139	8	⊆	⊆	NUM
iajs-2804	139	9	(	(	PUNCT
iajs-2804	139	10	𝑟𝑚)1	𝑟𝑚)1	PROPN
iajs-2804	139	11	⊆	⊆	NUM
iajs-2804	139	12	𝑈	𝑈	PROPN
iajs-2804	139	13	+	+	CCONJ
iajs-2804	139	14	ℱ	ℱ	PROPN
iajs-2804	139	15	−	−	PROPN
iajs-2804	139	16	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	139	17	)	)	PUNCT
iajs-2804	139	18	.	.	PUNCT
iajs-2804	140	1	2	2	X
iajs-2804	140	2	)	)	PUNCT
iajs-2804	140	3	if	if	SCONJ
iajs-2804	140	4	𝑟𝑛𝑚	𝑟𝑛𝑚	PROPN
iajs-2804	140	5	∉	∉	PROPN
iajs-2804	140	6	𝐸	𝐸	PROPN
iajs-2804	140	7	then	then	ADV
iajs-2804	140	8	𝑈(𝑟𝑛𝑚	𝑈(𝑟𝑛𝑚	VERB
iajs-2804	140	9	)	)	PUNCT
iajs-2804	140	10	=	=	SYM
iajs-2804	140	11	𝑘	𝑘	PROPN
iajs-2804	140	12	with	with	ADP
iajs-2804	140	13	𝑚	𝑚	PROPN
iajs-2804	140	14	∉	∉	PROPN
iajs-2804	140	15	𝐸	𝐸	PROPN
iajs-2804	140	16	thus	thus	ADV
iajs-2804	140	17	𝑈(𝑚	𝑈(𝑚	NOUN
iajs-2804	140	18	)	)	PUNCT
iajs-2804	140	19	=	=	SYM
iajs-2804	140	20	𝑘.	𝑘.	NOUN
iajs-2804	140	21	since	since	SCONJ
iajs-2804	140	22	(	(	PUNCT
iajs-2804	140	23	𝑟𝑏)𝑛𝑚𝑡	𝑟𝑏)𝑛𝑚𝑡	NOUN
iajs-2804	140	24	⊆	⊆	NUM
iajs-2804	140	25	𝑈	𝑈	NOUN
iajs-2804	140	26	then	then	ADV
iajs-2804	140	27	(	(	PUNCT
iajs-2804	140	28	𝑟𝑛𝑚)ℷ	𝑟𝑛𝑚)ℷ	PROPN
iajs-2804	140	29	⊆	⊆	NUM
iajs-2804	140	30	𝑈	𝑈	NOUN
iajs-2804	140	31	where	where	SCONJ
iajs-2804	140	32	ℷ	ℷ	PROPN
iajs-2804	140	33	=	=	SYM
iajs-2804	140	34	min	min	PROPN
iajs-2804	140	35	{	{	PUNCT
iajs-2804	140	36	𝑏	𝑏	NOUN
iajs-2804	140	37	,	,	PUNCT
iajs-2804	140	38	𝑡	𝑡	X
iajs-2804	140	39	}	}	PUNCT
iajs-2804	140	40	,	,	PUNCT
iajs-2804	140	41	that	that	PRON
iajs-2804	140	42	is	be	AUX
iajs-2804	140	43	𝑈(𝑟𝑛𝑚	𝑈(𝑟𝑛𝑚	NOUN
iajs-2804	140	44	)	)	PUNCT
iajs-2804	140	45	≥	≥	NOUN
iajs-2804	141	1	ℷ	ℷ	X
iajs-2804	141	2	thus	thus	ADV
iajs-2804	141	3	𝑘	𝑘	DET
iajs-2804	141	4	≥	≥	NOUN
iajs-2804	141	5	ℷ	ℷ	INTJ
iajs-2804	141	6	.	.	PUNCT
iajs-2804	142	1	now	now	ADV
iajs-2804	142	2	,	,	PUNCT
iajs-2804	142	3	if	if	SCONJ
iajs-2804	142	4	ℷ	ℷ	PROPN
iajs-2804	142	5	=	=	SYM
iajs-2804	142	6	t	t	PROPN
iajs-2804	142	7	this	this	PRON
iajs-2804	142	8	implies	imply	VERB
iajs-2804	142	9	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	142	10	⊆	⊆	NUM
iajs-2804	142	11	𝑚𝑘	𝑚𝑘	ADP
iajs-2804	142	12	⊆	⊆	NUM
iajs-2804	142	13	𝑈	𝑈	PROPN
iajs-2804	142	14	⊆	⊆	NUM
iajs-2804	142	15	𝑈	𝑈	PROPN
iajs-2804	142	16	+	+	CCONJ
iajs-2804	142	17	ℱ	ℱ	PROPN
iajs-2804	142	18	−	−	PROPN
iajs-2804	142	19	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	142	20	)	)	PUNCT
iajs-2804	142	21	.	.	PUNCT
iajs-2804	143	1	that	that	PRON
iajs-2804	143	2	is	be	AUX
iajs-2804	143	3	mean	mean	VERB
iajs-2804	143	4	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	ADJ
iajs-2804	143	5	⊆	⊆	NUM
iajs-2804	143	6	𝑟𝑏𝑚𝑘	𝑟𝑏𝑚𝑘	NOUN
iajs-2804	143	7	⊆	⊆	NUM
iajs-2804	143	8	𝑈	𝑈	PROPN
iajs-2804	143	9	⊆	⊆	NUM
iajs-2804	143	10	𝑈	𝑈	PROPN
iajs-2804	143	11	+	+	CCONJ
iajs-2804	143	12	ℱ	ℱ	PROPN
iajs-2804	143	13	−	−	PROPN
iajs-2804	143	14	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	143	15	)	)	PUNCT
iajs-2804	143	16	if	if	SCONJ
iajs-2804	143	17	ℷ	ℷ	PROPN
iajs-2804	143	18	=	=	SYM
iajs-2804	143	19	b	b	PROPN
iajs-2804	143	20	,	,	PUNCT
iajs-2804	143	21	𝑈(ℎ	𝑈(ℎ	NOUN
iajs-2804	143	22	)	)	PUNCT
iajs-2804	143	23	≥	≥	NOUN
iajs-2804	143	24	𝑘	𝑘	NOUN
iajs-2804	143	25	for	for	ADP
iajs-2804	143	26	any	any	DET
iajs-2804	143	27	ℎ	ℎ	PROPN
iajs-2804	143	28	∈	∈	PROPN
iajs-2804	143	29	𝑀	𝑀	PROPN
iajs-2804	143	30	,	,	PUNCT
iajs-2804	143	31	and	and	CCONJ
iajs-2804	143	32	:	:	PUNCT
iajs-2804	143	33	(	(	PUNCT
iajs-2804	143	34	𝑟𝑛	𝑟𝑛	ADP
iajs-2804	143	35	𝑏	𝑏	DET
iajs-2804	143	36	𝑋𝑀)(ℎ	𝑋𝑀)(ℎ	PROPN
iajs-2804	143	37	)	)	PUNCT
iajs-2804	143	38	=	=	PRON
iajs-2804	143	39	{	{	PUNCT
iajs-2804	144	1	𝑏	𝑏	NOUN
iajs-2804	144	2	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	144	3	ℎ	ℎ	PROPN
iajs-2804	144	4	=	=	SYM
iajs-2804	144	5	𝑟𝑛𝑎	𝑟𝑛𝑎	NOUN
iajs-2804	144	6	0	0	NUM
iajs-2804	144	7	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
iajs-2804	144	8	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2804	144	9	𝑠𝑜𝑚𝑒	𝑠𝑜𝑚𝑒	NOUN
iajs-2804	144	10	𝑎	𝑎	DET
iajs-2804	144	11	∈	∈	PROPN
iajs-2804	144	12	𝑀	𝑀	NOUN
iajs-2804	144	13	then	then	ADV
iajs-2804	144	14	we	we	PRON
iajs-2804	144	15	get	get	VERB
iajs-2804	144	16	(	(	PUNCT
iajs-2804	144	17	𝑟𝑛	𝑟𝑛	ADP
iajs-2804	144	18	𝑏	𝑏	DET
iajs-2804	144	19	𝑋𝑀)(ℎ	𝑋𝑀)(ℎ	PROPN
iajs-2804	144	20	)	)	PUNCT
iajs-2804	144	21	≤	≤	NOUN
iajs-2804	144	22	𝑈(ℎ	𝑈(ℎ	NOUN
iajs-2804	144	23	)	)	PUNCT
iajs-2804	144	24	,	,	PUNCT
iajs-2804	144	25	hence	hence	ADV
iajs-2804	144	26	𝑟𝑛	𝑟𝑛	ADP
iajs-2804	144	27	𝑏𝑋𝑀	𝑏𝑋𝑀	PROPN
iajs-2804	145	1	⊆	⊆	NUM
iajs-2804	145	2	𝑈	𝑈	PROPN
iajs-2804	145	3	⊆	⊆	NUM
iajs-2804	145	4	𝑈	𝑈	PROPN
iajs-2804	145	5	+	+	CCONJ
iajs-2804	145	6	ℱ	ℱ	PROPN
iajs-2804	145	7	−	−	PROPN
iajs-2804	145	8	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	145	9	)	)	PUNCT
iajs-2804	146	1	so	so	ADV
iajs-2804	146	2	,	,	PUNCT
iajs-2804	146	3	each	each	DET
iajs-2804	146	4	case	case	NOUN
iajs-2804	146	5	implies	imply	VERB
iajs-2804	146	6	that	that	SCONJ
iajs-2804	146	7	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	146	8	⊆	⊆	NUM
iajs-2804	146	9	𝑈	𝑈	PROPN
iajs-2804	146	10	+	+	CCONJ
iajs-2804	146	11	ℱ	ℱ	PROPN
iajs-2804	146	12	−	−	PROPN
iajs-2804	146	13	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	146	14	)	)	PUNCT
iajs-2804	146	15	therefore	therefore	ADV
iajs-2804	146	16	u	u	NOUN
iajs-2804	146	17	is	be	AUX
iajs-2804	146	18	an	an	DET
iajs-2804	146	19	ℱ-soc	ℱ-soc	NOUN
iajs-2804	146	20	-	-	PUNCT
iajs-2804	146	21	semi	semi	ADJ
iajs-2804	146	22	-	-	ADJ
iajs-2804	146	23	prime	prime	ADJ
iajs-2804	146	24	sub	sub	NOUN
iajs-2804	146	25	-	-	NOUN
iajs-2804	146	26	module	module	NOUN
iajs-2804	146	27	of	of	ADP
iajs-2804	146	28	x.	x.	NOUN
iajs-2804	146	29	conversely	conversely	ADV
iajs-2804	146	30	suppose	suppose	VERB
iajs-2804	146	31	u	u	NOUN
iajs-2804	146	32	is	be	AUX
iajs-2804	146	33	an	an	DET
iajs-2804	146	34	ℱ-soc	ℱ-soc	NOUN
iajs-2804	146	35	-	-	PUNCT
iajs-2804	146	36	semi	semi	NOUN
iajs-2804	146	37	-	-	ADJ
iajs-2804	146	38	prime	prime	ADJ
iajs-2804	146	39	of	of	ADP
iajs-2804	146	40	x.	x.	NOUN
iajs-2804	146	41	let	let	VERB
iajs-2804	147	1	𝑎𝑛𝑥	𝑎𝑛𝑥	NOUN
iajs-2804	147	2	∈	∈	PROPN
iajs-2804	147	3	𝑈𝑡	𝑈𝑡	PROPN
iajs-2804	147	4	,	,	PUNCT
iajs-2804	147	5	with	with	ADP
iajs-2804	147	6	𝑎	𝑎	PROPN
iajs-2804	147	7	∈	∈	PROPN
iajs-2804	147	8	ℛ	ℛ	PROPN
iajs-2804	147	9	,	,	PUNCT
iajs-2804	147	10	n	n	PRON
iajs-2804	147	11	∈	∈	NOUN
iajs-2804	147	12	𝑍+	𝑍+	NOUN
iajs-2804	147	13	and	and	CCONJ
iajs-2804	147	14	𝑥	𝑥	PRON
iajs-2804	147	15	∈	∈	PROPN
iajs-2804	148	1	𝑋𝑡	𝑋𝑡	PROPN
iajs-2804	148	2	it	it	PRON
iajs-2804	148	3	follows	follow	VERB
iajs-2804	148	4	that	that	SCONJ
iajs-2804	148	5	(	(	PUNCT
iajs-2804	148	6	𝑎𝑛𝑥)𝑡	𝑎𝑛𝑥)𝑡	PROPN
iajs-2804	148	7	⊆	⊆	NUM
iajs-2804	148	8	𝑈	𝑈	PROPN
iajs-2804	148	9	,	,	PUNCT
iajs-2804	148	10	that	that	ADV
iajs-2804	148	11	is	is	ADV
iajs-2804	148	12	(	(	PUNCT
iajs-2804	148	13	𝑎𝑛)𝑡𝑥𝑡	𝑎𝑛)𝑡𝑥𝑡	NOUN
iajs-2804	148	14	=	=	SYM
iajs-2804	148	15	(	(	PUNCT
iajs-2804	148	16	𝑎𝑡)𝑛𝑥𝑡	𝑎𝑡)𝑛𝑥𝑡	NOUN
iajs-2804	148	17	⊆	⊆	NUM
iajs-2804	148	18	𝑈.	𝑈.	NOUN
iajs-2804	148	19	but	but	CCONJ
iajs-2804	148	20	u	u	NOUN
iajs-2804	148	21	is	be	AUX
iajs-2804	148	22	an	an	DET
iajs-2804	148	23	ℱ-soc	ℱ-soc	NOUN
iajs-2804	148	24	-	-	PUNCT
iajs-2804	148	25	semi	semi	NOUN
iajs-2804	148	26	-	-	ADJ
iajs-2804	148	27	prime	prime	ADJ
iajs-2804	148	28	of	of	ADP
iajs-2804	148	29	x	x	PRON
iajs-2804	148	30	,	,	PUNCT
iajs-2804	148	31	then	then	ADV
iajs-2804	148	32	we	we	PRON
iajs-2804	148	33	get	get	VERB
iajs-2804	148	34	𝑎𝑡𝑥𝑡	𝑎𝑡𝑥𝑡	ADJ
iajs-2804	149	1	=	=	PUNCT
iajs-2804	149	2	(	(	PUNCT
iajs-2804	149	3	𝑎𝑥)𝑡	𝑎𝑥)𝑡	PROPN
iajs-2804	149	4	⊆	⊆	NUM
iajs-2804	149	5	𝑈	𝑈	PROPN
iajs-2804	149	6	+	+	CCONJ
iajs-2804	149	7	ℱ	ℱ	PROPN
iajs-2804	149	8	−	−	PROPN
iajs-2804	149	9	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	149	10	)	)	PUNCT
iajs-2804	149	11	.	.	PUNCT
iajs-2804	150	1	thus	thus	ADV
iajs-2804	150	2	we	we	PRON
iajs-2804	150	3	get	get	VERB
iajs-2804	150	4	(	(	PUNCT
iajs-2804	150	5	𝑈	𝑈	NOUN
iajs-2804	150	6	+	+	CCONJ
iajs-2804	150	7	ℱ	ℱ	PROPN
iajs-2804	150	8	−	−	PROPN
iajs-2804	150	9	𝑆𝑜𝑐(𝑋))(𝑎𝑥	𝑆𝑜𝑐(𝑋))(𝑎𝑥	PROPN
iajs-2804	150	10	)	)	PUNCT
iajs-2804	150	11	≥	≥	NUM
iajs-2804	150	12	𝑡	𝑡	PROPN
iajs-2804	150	13	,	,	PUNCT
iajs-2804	150	14	hence	hence	ADV
iajs-2804	150	15	,	,	PUNCT
iajs-2804	150	16	by	by	ADP
iajs-2804	150	17	(	(	PUNCT
iajs-2804	150	18	lemma	lemma	PROPN
iajs-2804	150	19	1.12	1.12	NUM
iajs-2804	150	20	)	)	PUNCT
iajs-2804	150	21	and	and	CCONJ
iajs-2804	150	22	(	(	PUNCT
iajs-2804	150	23	lemma	lemma	PROPN
iajs-2804	150	24	2.2	2.2	NUM
iajs-2804	150	25	)	)	PUNCT
iajs-2804	150	26	,	,	PUNCT
iajs-2804	150	27	we	we	PRON
iajs-2804	150	28	have	have	VERB
iajs-2804	150	29	𝑎𝑥	𝑎𝑥	X
iajs-2804	150	30	∈	∈	NOUN
iajs-2804	150	31	(	(	PUNCT
iajs-2804	150	32	𝑈	𝑈	NOUN
iajs-2804	150	33	+	+	CCONJ
iajs-2804	150	34	ℱ	ℱ	PROPN
iajs-2804	150	35	−	−	PROPN
iajs-2804	150	36	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	150	37	)	)	PUNCT
iajs-2804	150	38	)	)	PUNCT
iajs-2804	151	1	𝑡	𝑡	PROPN
iajs-2804	151	2	=	=	PUNCT
iajs-2804	152	1	𝑈𝑡	𝑈𝑡	PROPN
iajs-2804	152	2	+	+	CCONJ
iajs-2804	152	3	(	(	PUNCT
iajs-2804	152	4	ℱ	ℱ	PROPN
iajs-2804	152	5	−	−	PROPN
iajs-2804	152	6	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	152	7	)	)	PUNCT
iajs-2804	152	8	)	)	PUNCT
iajs-2804	153	1	𝑡	𝑡	PROPN
iajs-2804	153	2	=	=	SYM
iajs-2804	154	1	𝑈𝑡	𝑈𝑡	PROPN
iajs-2804	154	2	+	+	NUM
iajs-2804	154	3	𝑆𝑜𝑐(𝑋𝑡	𝑆𝑜𝑐(𝑋𝑡	NOUN
iajs-2804	154	4	)	)	PUNCT
iajs-2804	154	5	.	.	PUNCT
iajs-2804	155	1	that	that	PRON
iajs-2804	155	2	is	be	AUX
iajs-2804	155	3	mean	mean	VERB
iajs-2804	155	4	𝑈𝑡	𝑈𝑡	PROPN
iajs-2804	155	5	is	be	AUX
iajs-2804	155	6	an	an	DET
iajs-2804	155	7	app	app	ADJ
iajs-2804	155	8	-	-	PUNCT
iajs-2804	155	9	semi	semi	ADJ
iajs-2804	155	10	-	-	ADJ
iajs-2804	155	11	prime	prime	ADJ
iajs-2804	155	12	sub	sub	NOUN
iajs-2804	155	13	-	-	NOUN
iajs-2804	155	14	module	module	NOUN
iajs-2804	155	15	of	of	ADP
iajs-2804	155	16	𝑋𝑡	𝑋𝑡	PROPN
iajs-2804	155	17	.	.	PUNCT
iajs-2804	156	1	hence	hence	ADV
iajs-2804	156	2	𝑈1	𝑈1	X
iajs-2804	156	3	=	=	SYM
iajs-2804	156	4	𝐸	𝐸	PROPN
iajs-2804	156	5	is	be	AUX
iajs-2804	156	6	an	an	DET
iajs-2804	156	7	app	app	ADJ
iajs-2804	156	8	-	-	PUNCT
iajs-2804	156	9	semi	semi	ADJ
iajs-2804	156	10	-	-	ADJ
iajs-2804	156	11	prime	prime	ADJ
iajs-2804	156	12	sub	sub	NOUN
iajs-2804	156	13	-	-	NOUN
iajs-2804	156	14	module	module	NOUN
iajs-2804	156	15	of	of	ADP
iajs-2804	156	16	m.	m.	NOUN
iajs-2804	156	17	the	the	DET
iajs-2804	156	18	following	follow	VERB
iajs-2804	156	19	example	example	NOUN
iajs-2804	156	20	shows	show	VERB
iajs-2804	156	21	that	that	SCONJ
iajs-2804	156	22	the	the	DET
iajs-2804	156	23	definition	definition	NOUN
iajs-2804	156	24	of	of	ADP
iajs-2804	156	25	an	an	DET
iajs-2804	156	26	ℱ-socle	ℱ-socle	PROPN
iajs-2804	156	27	of	of	ADP
iajs-2804	156	28	x	x	PRON
iajs-2804	156	29	that	that	SCONJ
iajs-2804	156	30	we	we	PRON
iajs-2804	156	31	adopt	adopt	VERB
iajs-2804	156	32	in	in	ADP
iajs-2804	156	33	this	this	DET
iajs-2804	156	34	research	research	NOUN
iajs-2804	156	35	is	be	AUX
iajs-2804	156	36	necessary	necessary	ADJ
iajs-2804	156	37	to	to	PART
iajs-2804	156	38	prove	prove	VERB
iajs-2804	156	39	one	one	NUM
iajs-2804	156	40	side	side	NOUN
iajs-2804	156	41	of	of	ADP
iajs-2804	156	42	above	above	ADP
iajs-2804	156	43	lemma	lemma	PROPN
iajs-2804	156	44	.	.	PUNCT
iajs-2804	156	45	example	example	NOUN
iajs-2804	156	46	2.4	2.4	NUM
iajs-2804	156	47	let	let	VERB
iajs-2804	156	48	𝑀	𝑀	PROPN
iajs-2804	156	49	=	=	PUNCT
iajs-2804	157	1	𝑍12	𝑍12	PROPN
iajs-2804	157	2	as	as	ADP
iajs-2804	157	3	a	a	DET
iajs-2804	157	4	z	z	NOUN
iajs-2804	157	5	-	-	PUNCT
iajs-2804	157	6	module	module	NOUN
iajs-2804	157	7	and	and	CCONJ
iajs-2804	157	8	𝑋	𝑋	PROPN
iajs-2804	157	9	:	:	PUNCT
iajs-2804	157	10	𝑀	𝑀	PROPN
iajs-2804	157	11	→	→	PUNCT
iajs-2804	157	12	[	[	X
iajs-2804	157	13	0,1	0,1	NUM
iajs-2804	157	14	]	]	PUNCT
iajs-2804	157	15	,	,	PUNCT
iajs-2804	157	16	𝑈	𝑈	PROPN
iajs-2804	157	17	:	:	PUNCT
iajs-2804	157	18	𝑀	𝑀	PROPN
iajs-2804	157	19	→	→	PUNCT
iajs-2804	158	1	[	[	X
iajs-2804	158	2	0,1	0,1	NUM
iajs-2804	158	3	]	]	PUNCT
iajs-2804	158	4	defined	define	VERB
iajs-2804	158	5	by	by	ADP
iajs-2804	158	6	:	:	PUNCT
iajs-2804	158	7	𝑋(𝑚	𝑋(𝑚	PROPN
iajs-2804	158	8	)	)	PUNCT
iajs-2804	158	9	=	=	SYM
iajs-2804	158	10	1	1	NUM
iajs-2804	158	11	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	158	12	𝑚	𝑚	PRON
iajs-2804	158	13	∈	∈	PROPN
iajs-2804	158	14	𝑍12	𝑍12	PROPN
iajs-2804	158	15	𝑈(𝑚	𝑈(𝑚	NOUN
iajs-2804	158	16	)	)	PUNCT
iajs-2804	158	17	=	=	NOUN
iajs-2804	158	18	{	{	PUNCT
iajs-2804	158	19	1	1	NUM
iajs-2804	158	20	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	158	21	𝑚	𝑚	PROPN
iajs-2804	158	22	∈	∈	PROPN
iajs-2804	158	23	〈	〈	PROPN
iajs-2804	158	24	0̅	0̅	PROPN
iajs-2804	158	25	〉	〉	NOUN
iajs-2804	158	26	1/4	1/4	NUM
iajs-2804	158	27	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2804	158	28	and	and	CCONJ
iajs-2804	158	29	an	an	DET
iajs-2804	158	30	ℱ-socle	ℱ-socle	PROPN
iajs-2804	158	31	of	of	ADP
iajs-2804	158	32	x	x	PUNCT
iajs-2804	158	33	is	be	AUX
iajs-2804	158	34	defined	define	VERB
iajs-2804	158	35	by	by	ADP
iajs-2804	158	36	ℱ	ℱ	PROPN
iajs-2804	158	37	−	−	PROPN
iajs-2804	158	38	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	158	39	):	):	PUNCT
iajs-2804	158	40	𝑀	𝑀	PROPN
iajs-2804	158	41	→	→	PUNCT
iajs-2804	159	1	[	[	X
iajs-2804	159	2	0,1	0,1	NUM
iajs-2804	159	3	]	]	PUNCT
iajs-2804	159	4	such	such	ADJ
iajs-2804	159	5	that	that	SCONJ
iajs-2804	159	6	:	:	PUNCT
iajs-2804	159	7	ℱ	ℱ	PROPN
iajs-2804	159	8	−	−	NOUN
iajs-2804	159	9	𝑆𝑜𝑐(𝑋)(𝑚	𝑆𝑜𝑐(𝑋)(𝑚	NOUN
iajs-2804	159	10	)	)	PUNCT
iajs-2804	160	1	=	=	PRON
iajs-2804	160	2	{	{	PUNCT
iajs-2804	160	3	1	1	NUM
iajs-2804	160	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	160	5	𝑥	𝑥	NOUN
iajs-2804	160	6	=	=	SYM
iajs-2804	160	7	0̅	0̅	NUM
iajs-2804	160	8	2/3	2/3	NUM
iajs-2804	160	9	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	160	10	𝑚	𝑚	X
iajs-2804	160	11	∈	∈	PROPN
iajs-2804	160	12	〈	〈	NOUN
iajs-2804	160	13	2̅	2̅	NOUN
iajs-2804	160	14	〉	〉	NOUN
iajs-2804	160	15	−	−	NOUN
iajs-2804	160	16	{	{	PUNCT
iajs-2804	160	17	0̅	0̅	PROPN
iajs-2804	160	18	}	}	PUNCT
iajs-2804	160	19	1/3	1/3	NUM
iajs-2804	160	20	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2804	160	21	where	where	SCONJ
iajs-2804	160	22	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	160	23	)	)	PUNCT
iajs-2804	160	24	=	=	PUNCT
iajs-2804	160	25	〈	〈	PROPN
iajs-2804	160	26	2̅	2̅	NUM
iajs-2804	160	27	〉	〉	NOUN
iajs-2804	160	28	.	.	PUNCT
iajs-2804	161	1	that	that	PRON
iajs-2804	161	2	’s	’	VERB
iajs-2804	161	3	clear	clear	ADJ
iajs-2804	161	4	x	x	VERB
iajs-2804	161	5	is	be	AUX
iajs-2804	161	6	an	an	DET
iajs-2804	161	7	ℱ-module	ℱ-module	PROPN
iajs-2804	161	8	and	and	CCONJ
iajs-2804	161	9	u	u	NOUN
iajs-2804	161	10	be	be	VERB
iajs-2804	161	11	an	an	DET
iajs-2804	161	12	ℱ-sub	ℱ-sub	NOUN
iajs-2804	161	13	-	-	NOUN
iajs-2804	161	14	module	module	NOUN
iajs-2804	161	15	of	of	ADP
iajs-2804	161	16	x.	x.	NOUN
iajs-2804	161	17	we	we	PRON
iajs-2804	161	18	have	have	AUX
iajs-2804	161	19	𝑈𝑡	𝑈𝑡	PROPN
iajs-2804	161	20	is	be	AUX
iajs-2804	161	21	an	an	DET
iajs-2804	161	22	app	app	ADJ
iajs-2804	161	23	-	-	PUNCT
iajs-2804	161	24	semi	semi	ADJ
iajs-2804	161	25	-	-	ADJ
iajs-2804	161	26	prime	prime	ADJ
iajs-2804	161	27	sub	sub	NOUN
iajs-2804	161	28	-	-	NOUN
iajs-2804	161	29	module	module	NOUN
iajs-2804	161	30	of	of	ADP
iajs-2804	161	31	m	m	PRON
iajs-2804	161	32	for	for	ADP
iajs-2804	161	33	every	every	DET
iajs-2804	161	34	𝑡	𝑡	X
iajs-2804	161	35	>	>	X
iajs-2804	161	36	0	0	PUNCT
iajs-2804	161	37	.	.	PUNCT
iajs-2804	162	1	now	now	ADV
iajs-2804	162	2	,	,	PUNCT
iajs-2804	162	3	ibn	ibn	PROPN
iajs-2804	162	4	al	al	PROPN
iajs-2804	162	5	-	-	PUNCT
iajs-2804	162	6	haitham	haitham	PROPN
iajs-2804	162	7	jour	jour	X
iajs-2804	162	8	.	.	PROPN
iajs-2804	162	9	for	for	ADP
iajs-2804	162	10	pure	pure	ADJ
iajs-2804	162	11	&	&	CCONJ
iajs-2804	162	12	appl	appl	PROPN
iajs-2804	162	13	.	.	PUNCT
iajs-2804	163	1	sci	sci	PROPN
iajs-2804	163	2	.	.	PROPN
iajs-2804	164	1	53	53	NUM
iajs-2804	164	2	(	(	PUNCT
iajs-2804	164	3	1)2022	1)2022	NUM
iajs-2804	164	4	109	109	NUM
iajs-2804	164	5	(	(	PUNCT
iajs-2804	164	6	𝑈	𝑈	PROPN
iajs-2804	164	7	+	+	CCONJ
iajs-2804	164	8	ℱ	ℱ	PROPN
iajs-2804	164	9	−	−	ADP
iajs-2804	164	10	𝑆𝑜𝑐(𝑋))(𝑚	𝑆𝑜𝑐(𝑋))(𝑚	ADJ
iajs-2804	164	11	)	)	PUNCT
iajs-2804	165	1	=	=	PRON
iajs-2804	165	2	{	{	PUNCT
iajs-2804	165	3	1	1	NUM
iajs-2804	165	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	165	5	𝑥	𝑥	NOUN
iajs-2804	165	6	=	=	SYM
iajs-2804	165	7	0̅	0̅	NUM
iajs-2804	165	8	2/3	2/3	NUM
iajs-2804	165	9	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	165	10	𝑚	𝑚	X
iajs-2804	165	11	∈	∈	PROPN
iajs-2804	165	12	〈	〈	NOUN
iajs-2804	165	13	2̅	2̅	NOUN
iajs-2804	165	14	〉	〉	NOUN
iajs-2804	165	15	−	−	NOUN
iajs-2804	165	16	{	{	PUNCT
iajs-2804	165	17	0̅	0̅	PROPN
iajs-2804	165	18	}	}	PUNCT
iajs-2804	165	19	1/3	1/3	NUM
iajs-2804	165	20	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2804	166	1	but	but	CCONJ
iajs-2804	166	2	,	,	PUNCT
iajs-2804	166	3	u	u	NOUN
iajs-2804	166	4	is	be	AUX
iajs-2804	166	5	not	not	PART
iajs-2804	166	6	an	an	DET
iajs-2804	166	7	ℱ-soc	ℱ-soc	NOUN
iajs-2804	166	8	-	-	PUNCT
iajs-2804	166	9	semi	semi	ADJ
iajs-2804	166	10	-	-	ADJ
iajs-2804	166	11	prime	prime	ADJ
iajs-2804	166	12	sub	sub	NOUN
iajs-2804	166	13	-	-	NOUN
iajs-2804	166	14	module	module	NOUN
iajs-2804	166	15	of	of	ADP
iajs-2804	166	16	x	x	PRON
iajs-2804	166	17	,	,	PUNCT
iajs-2804	166	18	since	since	SCONJ
iajs-2804	166	19	for	for	ADP
iajs-2804	166	20	an	an	DET
iajs-2804	166	21	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	166	22	3̅3	3̅3	NUM
iajs-2804	166	23	4	4	NUM
iajs-2804	166	24	⊆	⊆	NUM
iajs-2804	166	25	𝑋	𝑋	NOUN
iajs-2804	166	26	and	and	CCONJ
iajs-2804	166	27	an	an	DET
iajs-2804	166	28	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	166	29	23	23	NUM
iajs-2804	166	30	4	4	NUM
iajs-2804	166	31	of	of	ADP
iajs-2804	166	32	ℛ	ℛ	NOUN
iajs-2804	166	33	such	such	ADJ
iajs-2804	166	34	that	that	SCONJ
iajs-2804	166	35	(	(	PUNCT
iajs-2804	166	36	22)3	22)3	NUM
iajs-2804	166	37	4	4	NUM
iajs-2804	166	38	3̅3	3̅3	NUM
iajs-2804	166	39	4	4	NUM
iajs-2804	166	40	=	=	SYM
iajs-2804	166	41	0̅3	0̅3	NUM
iajs-2804	166	42	4	4	NUM
iajs-2804	166	43	,	,	PUNCT
iajs-2804	166	44	where	where	SCONJ
iajs-2804	166	45	0̅3	0̅3	NUM
iajs-2804	166	46	4	4	NUM
iajs-2804	166	47	⊆	⊆	NUM
iajs-2804	166	48	𝑈	𝑈	PROPN
iajs-2804	166	49	since	since	SCONJ
iajs-2804	166	50	𝑈(0̅	𝑈(0̅	NOUN
iajs-2804	166	51	)	)	PUNCT
iajs-2804	166	52	=	=	SYM
iajs-2804	166	53	1	1	X
iajs-2804	166	54	>	>	SYM
iajs-2804	166	55	3	3	NUM
iajs-2804	166	56	4	4	NUM
iajs-2804	166	57	.	.	PUNCT
iajs-2804	167	1	but	but	CCONJ
iajs-2804	167	2	23	23	NUM
iajs-2804	167	3	4	4	NUM
iajs-2804	167	4	3̅3	3̅3	NUM
iajs-2804	167	5	4	4	NUM
iajs-2804	167	6	=	=	SYM
iajs-2804	167	7	6̅3	6̅3	NUM
iajs-2804	167	8	4	4	NUM
iajs-2804	167	9	⊈	⊈	X
iajs-2804	167	10	𝑈	𝑈	NOUN
iajs-2804	167	11	+	+	CCONJ
iajs-2804	167	12	ℱ	ℱ	PROPN
iajs-2804	167	13	−	−	PROPN
iajs-2804	167	14	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	167	15	)	)	PUNCT
iajs-2804	167	16	since	since	SCONJ
iajs-2804	167	17	(	(	PUNCT
iajs-2804	167	18	𝑈	𝑈	PROPN
iajs-2804	167	19	+	+	CCONJ
iajs-2804	167	20	ℱ	ℱ	PROPN
iajs-2804	167	21	−	−	PROPN
iajs-2804	167	22	𝑆𝑜𝑐(𝑋))(6̅	𝑆𝑜𝑐(𝑋))(6̅	NOUN
iajs-2804	167	23	)	)	PUNCT
iajs-2804	167	24	=	=	SYM
iajs-2804	167	25	2	2	NUM
iajs-2804	167	26	3	3	NUM
iajs-2804	167	27	≱	≱	ADP
iajs-2804	167	28	3	3	NUM
iajs-2804	167	29	4	4	NUM
iajs-2804	167	30	.	.	PUNCT
iajs-2804	168	1	hence	hence	ADV
iajs-2804	168	2	,	,	PUNCT
iajs-2804	168	3	u	u	PROPN
iajs-2804	168	4	is	be	AUX
iajs-2804	168	5	not	not	PART
iajs-2804	168	6	an	an	DET
iajs-2804	168	7	ℱ-soc	ℱ-soc	NOUN
iajs-2804	168	8	-	-	PUNCT
iajs-2804	168	9	semi	semi	NOUN
iajs-2804	168	10	-	-	ADJ
iajs-2804	168	11	prime	prime	ADJ
iajs-2804	168	12	of	of	ADP
iajs-2804	168	13	sub	sub	NOUN
iajs-2804	168	14	-	-	NOUN
iajs-2804	168	15	module	module	NOUN
iajs-2804	168	16	of	of	ADP
iajs-2804	168	17	x.	x.	NOUN
iajs-2804	168	18	proposition	proposition	NOUN
iajs-2804	168	19	2.5	2.5	NUM
iajs-2804	168	20	let	let	VERB
iajs-2804	168	21	u	u	PRON
iajs-2804	168	22	and	and	CCONJ
iajs-2804	168	23	v	v	NOUN
iajs-2804	168	24	are	be	AUX
iajs-2804	168	25	ℱ-sub	ℱ-sub	NOUN
iajs-2804	168	26	-	-	NOUN
iajs-2804	168	27	modules	module	NOUN
iajs-2804	168	28	of	of	ADP
iajs-2804	168	29	an	an	DET
iajs-2804	168	30	ℱ-module	ℱ-module	PROPN
iajs-2804	168	31	x	x	PROPN
iajs-2804	168	32	of	of	ADP
iajs-2804	168	33	an	an	DET
iajs-2804	168	34	ℛ-module	ℛ-module	PROPN
iajs-2804	168	35	m	m	PROPN
iajs-2804	168	36	with	with	ADP
iajs-2804	168	37	v	v	NUM
iajs-2804	168	38	is	be	AUX
iajs-2804	168	39	an	an	DET
iajs-2804	168	40	ℱsemiprime	ℱsemiprime	ADJ
iajs-2804	168	41	sub	sub	NOUN
iajs-2804	168	42	-	-	NOUN
iajs-2804	168	43	module	module	NOUN
iajs-2804	168	44	of	of	ADP
iajs-2804	168	45	x.	x.	NOUN
iajs-2804	169	1	then	then	ADV
iajs-2804	169	2	[	[	X
iajs-2804	169	3	𝑈:ℛ	𝑈:ℛ	PROPN
iajs-2804	169	4	𝑉	𝑉	PROPN
iajs-2804	169	5	]	]	PUNCT
iajs-2804	169	6	is	be	AUX
iajs-2804	169	7	an	an	DET
iajs-2804	169	8	ℱ-soc	ℱ-soc	NOUN
iajs-2804	169	9	-	-	PUNCT
iajs-2804	169	10	semi	semi	ADJ
iajs-2804	169	11	-	-	ADJ
iajs-2804	169	12	prime	prime	ADJ
iajs-2804	169	13	ideal	ideal	NOUN
iajs-2804	169	14	of	of	ADP
iajs-2804	169	15	ℛ.	ℛ.	PROPN
iajs-2804	169	16	proof	proof	NOUN
iajs-2804	169	17	:	:	PUNCT
iajs-2804	169	18	suppose	suppose	VERB
iajs-2804	169	19	that	that	SCONJ
iajs-2804	169	20	𝑟𝑏	𝑟𝑏	AUX
iajs-2804	169	21	𝑛𝑚𝑡	𝑛𝑚𝑡	VERB
iajs-2804	169	22	⊆	⊆	NUM
iajs-2804	169	23	[	[	X
iajs-2804	169	24	𝑈:ℛ	𝑈:ℛ	PROPN
iajs-2804	169	25	𝑉	𝑉	PROPN
iajs-2804	169	26	]	]	PUNCT
iajs-2804	169	27	,	,	PUNCT
iajs-2804	169	28	for	for	ADP
iajs-2804	169	29	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	169	30	⊆	⊆	NUM
iajs-2804	169	31	ℛ	ℛ	PROPN
iajs-2804	169	32	,	,	PUNCT
iajs-2804	169	33	𝑚𝑡	𝑚𝑡	ADP
iajs-2804	169	34	⊆	⊆	X
iajs-2804	169	35	𝑋,thus	𝑋,thus	X
iajs-2804	169	36	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	169	37	𝑛𝑚𝑡𝑉	𝑛𝑚𝑡𝑉	NOUN
iajs-2804	169	38	⊆	⊆	NUM
iajs-2804	169	39	𝑈.so	𝑈.so	NOUN
iajs-2804	169	40	we	we	PRON
iajs-2804	169	41	have	have	VERB
iajs-2804	169	42	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	169	43	𝑛(𝑚𝑡𝑉	𝑛(𝑚𝑡𝑉	NOUN
iajs-2804	169	44	)	)	PUNCT
iajs-2804	169	45	⊆	⊆	NUM
iajs-2804	169	46	𝑈	𝑈	PROPN
iajs-2804	169	47	,	,	PUNCT
iajs-2804	169	48	but	but	CCONJ
iajs-2804	169	49	v	v	NOUN
iajs-2804	169	50	is	be	AUX
iajs-2804	169	51	an	an	DET
iajs-2804	169	52	ℱ-semi	ℱ-semi	PROPN
iajs-2804	169	53	-	-	PUNCT
iajs-2804	169	54	prime	prime	ADJ
iajs-2804	169	55	sub	sub	NOUN
iajs-2804	169	56	-	-	NOUN
iajs-2804	169	57	module	module	NOUN
iajs-2804	169	58	of	of	ADP
iajs-2804	169	59	x.	x.	NOUN
iajs-2804	169	60	that	that	PRON
iajs-2804	169	61	is	be	AUX
iajs-2804	169	62	𝑟𝑏(𝑚𝑡𝑉	𝑟𝑏(𝑚𝑡𝑉	NOUN
iajs-2804	169	63	)	)	PUNCT
iajs-2804	169	64	⊆	⊆	PROPN
iajs-2804	169	65	𝑈	𝑈	PROPN
iajs-2804	169	66	,	,	PUNCT
iajs-2804	169	67	hence	hence	ADV
iajs-2804	169	68	𝑟𝑏𝑚𝑡𝑉	𝑟𝑏𝑚𝑡𝑉	PROPN
iajs-2804	169	69	⊆	⊆	NUM
iajs-2804	169	70	𝑈	𝑈	PROPN
iajs-2804	169	71	that	that	PRON
iajs-2804	169	72	is	be	AUX
iajs-2804	169	73	mean	mean	VERB
iajs-2804	169	74	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	ADJ
iajs-2804	169	75	⊆	⊆	NUM
iajs-2804	169	76	[	[	X
iajs-2804	169	77	𝑈:ℛ	𝑈:ℛ	PROPN
iajs-2804	169	78	𝑉	𝑉	PROPN
iajs-2804	169	79	]	]	PUNCT
iajs-2804	169	80	⊆	⊆	NUM
iajs-2804	169	81	[	[	X
iajs-2804	169	82	𝑈:ℛ	𝑈:ℛ	PROPN
iajs-2804	169	83	𝑉	𝑉	PROPN
iajs-2804	169	84	]	]	X
iajs-2804	169	85	+	+	CCONJ
iajs-2804	169	86	ℱ	ℱ	PROPN
iajs-2804	169	87	−	−	PROPN
iajs-2804	169	88	𝑆𝑜𝑐(ℛ	𝑆𝑜𝑐(ℛ	NOUN
iajs-2804	169	89	)	)	PUNCT
iajs-2804	169	90	.	.	PUNCT
iajs-2804	170	1	proposition	proposition	NOUN
iajs-2804	170	2	2.6	2.6	NUM
iajs-2804	170	3	let	let	VERB
iajs-2804	170	4	u	u	PRON
iajs-2804	170	5	and	and	CCONJ
iajs-2804	170	6	v	v	NOUN
iajs-2804	170	7	are	be	AUX
iajs-2804	170	8	ℱ-soc	ℱ-soc	NOUN
iajs-2804	170	9	-	-	PUNCT
iajs-2804	170	10	semi	semi	ADJ
iajs-2804	170	11	-	-	ADJ
iajs-2804	170	12	prime	prime	ADJ
iajs-2804	170	13	sub	sub	NOUN
iajs-2804	170	14	-	-	NOUN
iajs-2804	170	15	modules	module	NOUN
iajs-2804	170	16	of	of	ADP
iajs-2804	170	17	an	an	DET
iajs-2804	170	18	ℱ-module	ℱ-module	PROPN
iajs-2804	170	19	x	x	PROPN
iajs-2804	170	20	of	of	ADP
iajs-2804	170	21	an	an	DET
iajs-2804	170	22	ℛ-module	ℛ-module	PROPN
iajs-2804	170	23	m	m	NOUN
iajs-2804	170	24	with	with	ADP
iajs-2804	170	25	ℱ	ℱ	PROPN
iajs-2804	170	26	−	−	PROPN
iajs-2804	170	27	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	170	28	)	)	PUNCT
iajs-2804	170	29	⊆	⊆	NUM
iajs-2804	170	30	𝑈	𝑈	PROPN
iajs-2804	170	31	,	,	PUNCT
iajs-2804	170	32	then	then	ADV
iajs-2804	170	33	u	u	NOUN
iajs-2804	170	34	∩v	∩v	NOUN
iajs-2804	170	35	is	be	AUX
iajs-2804	170	36	an	an	DET
iajs-2804	170	37	ℱ-soc	ℱ-soc	NOUN
iajs-2804	170	38	-	-	PUNCT
iajs-2804	170	39	semi	semi	ADJ
iajs-2804	170	40	-	-	ADJ
iajs-2804	170	41	prime	prime	ADJ
iajs-2804	170	42	sub	sub	NOUN
iajs-2804	170	43	-	-	NOUN
iajs-2804	170	44	module	module	NOUN
iajs-2804	170	45	of	of	ADP
iajs-2804	170	46	x.	x.	NOUN
iajs-2804	170	47	proof	proof	NOUN
iajs-2804	170	48	:	:	PUNCT
iajs-2804	170	49	let	let	VERB
iajs-2804	170	50	𝑟𝑏	𝑟𝑏	PART
iajs-2804	170	51	𝑛𝑚𝑡	𝑛𝑚𝑡	VERB
iajs-2804	170	52	⊆	⊆	NUM
iajs-2804	170	53	u	u	NOUN
iajs-2804	170	54	∩v	∩v	NOUN
iajs-2804	170	55	,	,	PUNCT
iajs-2804	170	56	for	for	ADP
iajs-2804	170	57	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	170	58	⊆	⊆	NUM
iajs-2804	170	59	ℛ	ℛ	PROPN
iajs-2804	170	60	,	,	PUNCT
iajs-2804	170	61	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	170	62	⊆	⊆	NUM
iajs-2804	170	63	𝑋	𝑋	PROPN
iajs-2804	170	64	,	,	PUNCT
iajs-2804	170	65	that	that	PRON
iajs-2804	170	66	is	be	AUX
iajs-2804	170	67	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	170	68	𝑛𝑚𝑡	𝑛𝑚𝑡	VERB
iajs-2804	170	69	⊆	⊆	NUM
iajs-2804	170	70	u	u	NOUN
iajs-2804	170	71	and	and	CCONJ
iajs-2804	170	72	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	170	73	𝑛𝑚𝑡	𝑛𝑚𝑡	NOUN
iajs-2804	170	74	⊆	⊆	NUM
iajs-2804	170	75	v.	v.	CCONJ
iajs-2804	170	76	but	but	CCONJ
iajs-2804	170	77	u	u	PROPN
iajs-2804	170	78	and	and	CCONJ
iajs-2804	170	79	v	v	NOUN
iajs-2804	170	80	are	be	AUX
iajs-2804	170	81	ℱ-soc	ℱ-soc	NOUN
iajs-2804	170	82	-	-	PUNCT
iajs-2804	170	83	semi	semi	ADJ
iajs-2804	170	84	-	-	ADJ
iajs-2804	170	85	prime	prime	ADJ
iajs-2804	170	86	sub	sub	NOUN
iajs-2804	170	87	-	-	NOUN
iajs-2804	170	88	modules	module	NOUN
iajs-2804	170	89	of	of	ADP
iajs-2804	170	90	x	x	PRON
iajs-2804	170	91	,	,	PUNCT
iajs-2804	170	92	this	this	PRON
iajs-2804	170	93	implies	imply	VERB
iajs-2804	170	94	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	ADJ
iajs-2804	170	95	⊆	⊆	NUM
iajs-2804	170	96	𝑈	𝑈	PROPN
iajs-2804	170	97	+	+	CCONJ
iajs-2804	170	98	ℱ	ℱ	PROPN
iajs-2804	170	99	−	−	PROPN
iajs-2804	170	100	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	170	101	)	)	PUNCT
iajs-2804	170	102	and	and	CCONJ
iajs-2804	170	103	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	170	104	⊆	⊆	NUM
iajs-2804	170	105	𝑉	𝑉	PROPN
iajs-2804	170	106	+	+	CCONJ
iajs-2804	170	107	ℱ	ℱ	PROPN
iajs-2804	170	108	−	−	PROPN
iajs-2804	170	109	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	170	110	)	)	PUNCT
iajs-2804	170	111	.	.	PUNCT
iajs-2804	171	1	that	that	PRON
iajs-2804	171	2	is	be	AUX
iajs-2804	171	3	mean	mean	VERB
iajs-2804	171	4	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	ADJ
iajs-2804	171	5	⊆	⊆	NUM
iajs-2804	171	6	(	(	PUNCT
iajs-2804	171	7	𝑈	𝑈	NOUN
iajs-2804	171	8	+	+	CCONJ
iajs-2804	171	9	ℱ	ℱ	PROPN
iajs-2804	171	10	−	−	PROPN
iajs-2804	171	11	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	171	12	)	)	PUNCT
iajs-2804	171	13	)	)	PUNCT
iajs-2804	171	14	∩	∩	NOUN
iajs-2804	171	15	(	(	PUNCT
iajs-2804	171	16	𝑉	𝑉	PROPN
iajs-2804	171	17	+	+	NOUN
iajs-2804	171	18	ℱ	ℱ	PROPN
iajs-2804	171	19	−	−	PROPN
iajs-2804	171	20	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	171	21	)	)	PUNCT
iajs-2804	171	22	)	)	PUNCT
iajs-2804	171	23	,	,	PUNCT
iajs-2804	171	24	by	by	ADP
iajs-2804	171	25	using	use	VERB
iajs-2804	171	26	modular	modular	ADJ
iajs-2804	171	27	law	law	NOUN
iajs-2804	171	28	we	we	PRON
iajs-2804	171	29	get	get	VERB
iajs-2804	171	30	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	ADJ
iajs-2804	171	31	⊆	⊆	NUM
iajs-2804	171	32	(	(	PUNCT
iajs-2804	171	33	u	u	NOUN
iajs-2804	171	34	∩	∩	NOUN
iajs-2804	171	35	v	v	NOUN
iajs-2804	171	36	)	)	PUNCT
iajs-2804	171	37	+	+	CCONJ
iajs-2804	171	38	ℱ	ℱ	PROPN
iajs-2804	171	39	−	−	PROPN
iajs-2804	171	40	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	171	41	)	)	PUNCT
iajs-2804	171	42	.	.	PUNCT
iajs-2804	172	1	hence	hence	ADV
iajs-2804	172	2	u	u	NOUN
iajs-2804	172	3	∩v	∩v	NOUN
iajs-2804	172	4	is	be	AUX
iajs-2804	172	5	an	an	DET
iajs-2804	172	6	ℱ-socsemi	ℱ-socsemi	PROPN
iajs-2804	172	7	-	-	PUNCT
iajs-2804	172	8	prime	prime	ADJ
iajs-2804	172	9	sub	sub	NOUN
iajs-2804	172	10	-	-	NOUN
iajs-2804	172	11	module	module	NOUN
iajs-2804	172	12	of	of	ADP
iajs-2804	172	13	x.	x.	NOUN
iajs-2804	172	14	remark	remark	PROPN
iajs-2804	172	15	2.7	2.7	NUM
iajs-2804	172	16	every	every	DET
iajs-2804	172	17	ℱ-semi	ℱ-semi	PROPN
iajs-2804	172	18	-	-	PUNCT
iajs-2804	172	19	prime	prime	ADJ
iajs-2804	172	20	sub	sub	NOUN
iajs-2804	172	21	-	-	NOUN
iajs-2804	172	22	module	module	NOUN
iajs-2804	172	23	is	be	AUX
iajs-2804	172	24	an	an	DET
iajs-2804	172	25	ℱ-soc	ℱ-soc	NOUN
iajs-2804	172	26	-	-	PUNCT
iajs-2804	172	27	semi	semi	ADJ
iajs-2804	172	28	-	-	ADJ
iajs-2804	172	29	prime	prime	ADJ
iajs-2804	172	30	sub	sub	NOUN
iajs-2804	172	31	-	-	NOUN
iajs-2804	172	32	module	module	NOUN
iajs-2804	172	33	,	,	PUNCT
iajs-2804	172	34	but	but	CCONJ
iajs-2804	172	35	the	the	DET
iajs-2804	172	36	converse	converse	NOUN
iajs-2804	172	37	is	be	AUX
iajs-2804	172	38	not	not	PART
iajs-2804	172	39	true	true	ADJ
iajs-2804	172	40	.	.	PUNCT
iajs-2804	173	1	proof	proof	NOUN
iajs-2804	173	2	:	:	PUNCT
iajs-2804	173	3	suppose	suppose	VERB
iajs-2804	173	4	u	u	PRON
iajs-2804	173	5	be	be	VERB
iajs-2804	173	6	an	an	DET
iajs-2804	173	7	ℱ-semi	ℱ-semi	PROPN
iajs-2804	173	8	-	-	PUNCT
iajs-2804	173	9	prime	prime	ADJ
iajs-2804	173	10	sub	sub	NOUN
iajs-2804	173	11	-	-	NOUN
iajs-2804	173	12	module	module	NOUN
iajs-2804	173	13	of	of	ADP
iajs-2804	173	14	an	an	DET
iajs-2804	173	15	ℱ-module	ℱ-module	PROPN
iajs-2804	173	16	x	x	PROPN
iajs-2804	173	17	of	of	ADP
iajs-2804	173	18	an	an	DET
iajs-2804	173	19	ℛ-module	ℛ-module	PROPN
iajs-2804	173	20	m	m	PROPN
iajs-2804	173	21	and	and	CCONJ
iajs-2804	173	22	𝑟𝑏	𝑟𝑏	AUX
iajs-2804	173	23	𝑛𝑚𝑡	𝑛𝑚𝑡	NOUN
iajs-2804	173	24	⊆	⊆	NUM
iajs-2804	173	25	𝑈,for	𝑈,for	NOUN
iajs-2804	173	26	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	173	27	⊆	⊆	PROPN
iajs-2804	173	28	𝑅	𝑅	PROPN
iajs-2804	173	29	,	,	PUNCT
iajs-2804	173	30	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	173	31	⊆	⊆	NUM
iajs-2804	173	32	𝑋.	𝑋.	PROPN
iajs-2804	173	33	since	since	SCONJ
iajs-2804	173	34	u	u	NOUN
iajs-2804	173	35	is	be	AUX
iajs-2804	173	36	an	an	DET
iajs-2804	173	37	ℱ-semi	ℱ-semi	PROPN
iajs-2804	173	38	-	-	PUNCT
iajs-2804	173	39	prime	prime	ADJ
iajs-2804	173	40	sub	sub	NOUN
iajs-2804	173	41	-	-	NOUN
iajs-2804	173	42	module	module	NOUN
iajs-2804	173	43	,	,	PUNCT
iajs-2804	173	44	then	then	ADV
iajs-2804	173	45	we	we	PRON
iajs-2804	173	46	get	get	VERB
iajs-2804	173	47	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	ADJ
iajs-2804	173	48	⊆	⊆	NUM
iajs-2804	173	49	𝑈	𝑈	PROPN
iajs-2804	173	50	⊆	⊆	NUM
iajs-2804	173	51	𝑈	𝑈	PROPN
iajs-2804	173	52	+	+	CCONJ
iajs-2804	173	53	ℱ	ℱ	PROPN
iajs-2804	173	54	−	−	PROPN
iajs-2804	173	55	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	173	56	)	)	PUNCT
iajs-2804	173	57	,	,	PUNCT
iajs-2804	173	58	thus	thus	ADV
iajs-2804	173	59	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	173	60	⊆	⊆	NUM
iajs-2804	173	61	𝑈	𝑈	PROPN
iajs-2804	173	62	+	+	CCONJ
iajs-2804	173	63	ℱ	ℱ	PROPN
iajs-2804	173	64	−	−	PROPN
iajs-2804	173	65	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	173	66	)	)	PUNCT
iajs-2804	173	67	.	.	PUNCT
iajs-2804	174	1	therefore	therefore	ADV
iajs-2804	174	2	u	u	PROPN
iajs-2804	174	3	is	be	AUX
iajs-2804	174	4	an	an	DET
iajs-2804	174	5	ℱ-socprime	ℱ-socprime	PROPN
iajs-2804	174	6	sub	sub	NOUN
iajs-2804	174	7	-	-	NOUN
iajs-2804	174	8	module	module	NOUN
iajs-2804	174	9	.	.	PUNCT
iajs-2804	175	1	the	the	DET
iajs-2804	175	2	following	follow	VERB
iajs-2804	175	3	example	example	NOUN
iajs-2804	175	4	show	show	VERB
iajs-2804	175	5	that	that	SCONJ
iajs-2804	175	6	the	the	DET
iajs-2804	175	7	converse	converse	NOUN
iajs-2804	175	8	is	be	AUX
iajs-2804	175	9	not	not	PART
iajs-2804	175	10	true	true	ADJ
iajs-2804	175	11	example	example	NOUN
iajs-2804	175	12	2.8	2.8	NUM
iajs-2804	175	13	consider	consider	VERB
iajs-2804	175	14	𝑀	𝑀	PROPN
iajs-2804	175	15	=	=	PUNCT
iajs-2804	175	16	𝑍12	𝑍12	PROPN
iajs-2804	175	17	as	as	ADP
iajs-2804	175	18	a	a	DET
iajs-2804	175	19	z	z	NOUN
iajs-2804	175	20	-	-	PUNCT
iajs-2804	175	21	module	module	NOUN
iajs-2804	175	22	and	and	CCONJ
iajs-2804	175	23	𝑋	𝑋	PROPN
iajs-2804	175	24	:	:	PUNCT
iajs-2804	175	25	𝑀	𝑀	PROPN
iajs-2804	175	26	→	→	PUNCT
iajs-2804	176	1	[	[	X
iajs-2804	176	2	0,1	0,1	NUM
iajs-2804	176	3	]	]	PUNCT
iajs-2804	176	4	,	,	PUNCT
iajs-2804	176	5	𝑈	𝑈	PROPN
iajs-2804	176	6	:	:	PUNCT
iajs-2804	176	7	𝑀	𝑀	PROPN
iajs-2804	176	8	→	→	PUNCT
iajs-2804	177	1	[	[	X
iajs-2804	177	2	0,1	0,1	NUM
iajs-2804	177	3	]	]	PUNCT
iajs-2804	177	4	defined	define	VERB
iajs-2804	177	5	by	by	ADP
iajs-2804	177	6	:	:	PUNCT
iajs-2804	177	7	𝑋(𝑚	𝑋(𝑚	PROPN
iajs-2804	177	8	)	)	PUNCT
iajs-2804	177	9	=	=	SYM
iajs-2804	177	10	1	1	NUM
iajs-2804	177	11	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	177	12	𝑚	𝑚	PRON
iajs-2804	177	13	∈	∈	PROPN
iajs-2804	177	14	𝑍12	𝑍12	PROPN
iajs-2804	177	15	ibn	ibn	PROPN
iajs-2804	177	16	al	al	PROPN
iajs-2804	177	17	-	-	PUNCT
iajs-2804	177	18	haitham	haitham	PROPN
iajs-2804	177	19	jour	jour	X
iajs-2804	177	20	.	.	PROPN
iajs-2804	178	1	for	for	ADP
iajs-2804	178	2	pure	pure	ADJ
iajs-2804	178	3	&	&	CCONJ
iajs-2804	178	4	appl	appl	PROPN
iajs-2804	178	5	.	.	PUNCT
iajs-2804	179	1	sci	sci	PROPN
iajs-2804	179	2	.	.	PROPN
iajs-2804	180	1	53	53	NUM
iajs-2804	180	2	(	(	PUNCT
iajs-2804	180	3	1)2022	1)2022	NUM
iajs-2804	180	4	110	110	NUM
iajs-2804	180	5	𝑈(𝑚	𝑈(𝑚	NOUN
iajs-2804	180	6	)	)	PUNCT
iajs-2804	180	7	=	=	SYM
iajs-2804	180	8	{	{	PUNCT
iajs-2804	180	9	1	1	NUM
iajs-2804	180	10	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	180	11	𝑚	𝑚	PROPN
iajs-2804	180	12	∈	∈	PROPN
iajs-2804	180	13	〈	〈	PROPN
iajs-2804	180	14	0̅	0̅	NOUN
iajs-2804	180	15	〉	〉	NOUN
iajs-2804	180	16	1/5	1/5	NUM
iajs-2804	180	17	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	180	18	𝑚	𝑚	PROPN
iajs-2804	180	19	∉	∉	PROPN
iajs-2804	180	20	〈	〈	PROPN
iajs-2804	180	21	0̅	0̅	PROPN
iajs-2804	180	22	〉	〉	NOUN
iajs-2804	180	23	and	and	CCONJ
iajs-2804	180	24	an	an	DET
iajs-2804	180	25	ℱ-socle	ℱ-socle	PROPN
iajs-2804	180	26	of	of	ADP
iajs-2804	180	27	x	x	PUNCT
iajs-2804	180	28	is	be	AUX
iajs-2804	180	29	defined	define	VERB
iajs-2804	180	30	by	by	ADP
iajs-2804	180	31	ℱ	ℱ	PROPN
iajs-2804	180	32	−	−	PROPN
iajs-2804	180	33	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	180	34	):	):	PUNCT
iajs-2804	180	35	𝑀	𝑀	PROPN
iajs-2804	180	36	→	→	PUNCT
iajs-2804	181	1	[	[	X
iajs-2804	181	2	0,1	0,1	NUM
iajs-2804	181	3	]	]	PUNCT
iajs-2804	181	4	such	such	ADJ
iajs-2804	181	5	that	that	SCONJ
iajs-2804	181	6	:	:	PUNCT
iajs-2804	181	7	ℱ	ℱ	PROPN
iajs-2804	181	8	−	−	NOUN
iajs-2804	181	9	𝑆𝑜𝑐(𝑋)(𝑚	𝑆𝑜𝑐(𝑋)(𝑚	NOUN
iajs-2804	181	10	)	)	PUNCT
iajs-2804	182	1	=	=	PRON
iajs-2804	182	2	{	{	PUNCT
iajs-2804	182	3	1	1	NUM
iajs-2804	182	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	182	5	𝑚	𝑚	PROPN
iajs-2804	182	6	∈	∈	PROPN
iajs-2804	182	7	〈	〈	X
iajs-2804	182	8	2̅	2̅	NOUN
iajs-2804	182	9	〉	〉	NOUN
iajs-2804	182	10	1/3	1/3	NUM
iajs-2804	182	11	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	182	12	𝑚	𝑚	PROPN
iajs-2804	182	13	∉	∉	PROPN
iajs-2804	182	14	〈	〈	PROPN
iajs-2804	182	15	2̅	2̅	NOUN
iajs-2804	182	16	〉	〉	NOUN
iajs-2804	182	17	where	where	SCONJ
iajs-2804	182	18	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	182	19	)	)	PUNCT
iajs-2804	182	20	=	=	PUNCT
iajs-2804	182	21	〈	〈	PROPN
iajs-2804	182	22	2̅	2̅	NUM
iajs-2804	182	23	〉	〉	NOUN
iajs-2804	182	24	.	.	PUNCT
iajs-2804	183	1	that	that	PRON
iajs-2804	183	2	’s	’	VERB
iajs-2804	183	3	clear	clear	ADJ
iajs-2804	183	4	x	x	VERB
iajs-2804	183	5	is	be	AUX
iajs-2804	183	6	an	an	DET
iajs-2804	183	7	ℱ-module	ℱ-module	PROPN
iajs-2804	183	8	and	and	CCONJ
iajs-2804	183	9	u	u	NOUN
iajs-2804	183	10	be	be	VERB
iajs-2804	183	11	an	an	DET
iajs-2804	183	12	ℱ-sub	ℱ-sub	NOUN
iajs-2804	183	13	-	-	NOUN
iajs-2804	183	14	module	module	NOUN
iajs-2804	183	15	of	of	ADP
iajs-2804	183	16	x.	x.	NOUN
iajs-2804	183	17	from	from	ADP
iajs-2804	183	18	(	(	PUNCT
iajs-2804	183	19	[	[	X
iajs-2804	183	20	4	4	X
iajs-2804	183	21	]	]	PUNCT
iajs-2804	183	22	remark	remark	NOUN
iajs-2804	183	23	2.3.2	2.3.2	NUM
iajs-2804	183	24	)	)	PUNCT
iajs-2804	183	25	〈	〈	NOUN
iajs-2804	183	26	0̅	0̅	PROPN
iajs-2804	183	27	〉	〉	NOUN
iajs-2804	183	28	is	be	AUX
iajs-2804	183	29	an	an	DET
iajs-2804	183	30	app	app	ADJ
iajs-2804	183	31	-	-	PUNCT
iajs-2804	183	32	semi	semi	ADJ
iajs-2804	183	33	-	-	ADJ
iajs-2804	183	34	prime	prime	ADJ
iajs-2804	183	35	sub	sub	NOUN
iajs-2804	183	36	-	-	NOUN
iajs-2804	183	37	module	module	NOUN
iajs-2804	183	38	of	of	ADP
iajs-2804	183	39	m	m	PRON
iajs-2804	183	40	,	,	PUNCT
iajs-2804	183	41	so	so	ADV
iajs-2804	183	42	by	by	ADP
iajs-2804	183	43	(	(	PUNCT
iajs-2804	183	44	lemma	lemma	PROPN
iajs-2804	183	45	2.3	2.3	NUM
iajs-2804	183	46	)	)	PUNCT
iajs-2804	183	47	we	we	PRON
iajs-2804	183	48	get	get	VERB
iajs-2804	183	49	u	u	NOUN
iajs-2804	183	50	is	be	AUX
iajs-2804	183	51	an	an	DET
iajs-2804	183	52	ℱ-soc	ℱ-soc	NOUN
iajs-2804	183	53	-	-	PUNCT
iajs-2804	183	54	semi	semi	ADJ
iajs-2804	183	55	-	-	ADJ
iajs-2804	183	56	prime	prime	ADJ
iajs-2804	183	57	sub	sub	NOUN
iajs-2804	183	58	-	-	NOUN
iajs-2804	183	59	module	module	NOUN
iajs-2804	183	60	of	of	ADP
iajs-2804	183	61	x.	x.	NOUN
iajs-2804	183	62	but	but	CCONJ
iajs-2804	183	63	,	,	PUNCT
iajs-2804	183	64	u	u	NOUN
iajs-2804	183	65	is	be	AUX
iajs-2804	183	66	not	not	PART
iajs-2804	183	67	an	an	DET
iajs-2804	183	68	ℱ-semi	ℱ-semi	PROPN
iajs-2804	183	69	-	-	PUNCT
iajs-2804	183	70	prime	prime	ADJ
iajs-2804	183	71	sub	sub	NOUN
iajs-2804	183	72	-	-	NOUN
iajs-2804	183	73	module	module	NOUN
iajs-2804	183	74	of	of	ADP
iajs-2804	183	75	x	x	PRON
iajs-2804	183	76	,	,	PUNCT
iajs-2804	183	77	since	since	SCONJ
iajs-2804	183	78	for	for	ADP
iajs-2804	183	79	an	an	DET
iajs-2804	183	80	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	183	81	31	31	NUM
iajs-2804	183	82	3	3	NUM
iajs-2804	183	83	⊆	⊆	NUM
iajs-2804	183	84	𝑋	𝑋	NOUN
iajs-2804	183	85	and	and	CCONJ
iajs-2804	183	86	an	an	DET
iajs-2804	183	87	ℱsingleton	ℱsingleton	PROPN
iajs-2804	183	88	21	21	NUM
iajs-2804	183	89	3	3	NUM
iajs-2804	183	90	of	of	ADP
iajs-2804	183	91	ℛ	ℛ	PROPN
iajs-2804	184	1	such	such	ADJ
iajs-2804	184	2	that	that	SCONJ
iajs-2804	184	3	(	(	PUNCT
iajs-2804	184	4	21	21	NUM
iajs-2804	184	5	3	3	NUM
iajs-2804	184	6	)	)	PUNCT
iajs-2804	184	7	2	2	NUM
iajs-2804	184	8	31	31	NUM
iajs-2804	184	9	3	3	NUM
iajs-2804	184	10	=	=	SYM
iajs-2804	184	11	01	01	NUM
iajs-2804	184	12	3	3	NUM
iajs-2804	184	13	where	where	SCONJ
iajs-2804	184	14	01	01	NUM
iajs-2804	184	15	3	3	NUM
iajs-2804	184	16	⊆	⊆	NUM
iajs-2804	184	17	𝑈	𝑈	PROPN
iajs-2804	184	18	since	since	SCONJ
iajs-2804	184	19	𝑈(0	𝑈(0	NUM
iajs-2804	184	20	)	)	PUNCT
iajs-2804	184	21	=	=	SYM
iajs-2804	184	22	1	1	NUM
iajs-2804	184	23	>	>	SYM
iajs-2804	184	24	1	1	NUM
iajs-2804	184	25	3	3	NUM
iajs-2804	184	26	.	.	PUNCT
iajs-2804	185	1	but	but	CCONJ
iajs-2804	185	2	21	21	NUM
iajs-2804	185	3	3	3	NUM
iajs-2804	185	4	31	31	NUM
iajs-2804	185	5	3	3	NUM
iajs-2804	185	6	=	=	SYM
iajs-2804	185	7	61	61	NUM
iajs-2804	185	8	3	3	NUM
iajs-2804	185	9	⊈	⊈	PROPN
iajs-2804	185	10	𝑈	𝑈	PROPN
iajs-2804	185	11	since	since	SCONJ
iajs-2804	185	12	𝑈(6	𝑈(6	NUM
iajs-2804	185	13	)	)	PUNCT
iajs-2804	185	14	=	=	SYM
iajs-2804	185	15	1	1	NUM
iajs-2804	185	16	5	5	NUM
iajs-2804	185	17	≯	≯	VERB
iajs-2804	185	18	1	1	NUM
iajs-2804	185	19	3	3	NUM
iajs-2804	185	20	.	.	PUNCT
iajs-2804	186	1	hence	hence	ADV
iajs-2804	186	2	,	,	PUNCT
iajs-2804	186	3	u	u	PROPN
iajs-2804	186	4	is	be	AUX
iajs-2804	186	5	not	not	PART
iajs-2804	186	6	an	an	DET
iajs-2804	186	7	ℱ-semi	ℱ-semi	PROPN
iajs-2804	186	8	-	-	PUNCT
iajs-2804	186	9	prime	prime	ADJ
iajs-2804	186	10	sub	sub	NOUN
iajs-2804	186	11	-	-	NOUN
iajs-2804	186	12	module	module	NOUN
iajs-2804	186	13	of	of	ADP
iajs-2804	186	14	x.	x.	NOUN
iajs-2804	186	15	remark	remark	PROPN
iajs-2804	186	16	2.9	2.9	NUM
iajs-2804	186	17	every	every	PRON
iajs-2804	186	18	completely	completely	ADV
iajs-2804	186	19	ℱ-sub	ℱ-sub	NOUN
iajs-2804	186	20	-	-	NOUN
iajs-2804	186	21	module	module	NOUN
iajs-2804	186	22	of	of	ADP
iajs-2804	186	23	an	an	DET
iajs-2804	186	24	ℱ-module	ℱ-module	PROPN
iajs-2804	186	25	x	x	PROPN
iajs-2804	186	26	of	of	ADP
iajs-2804	186	27	an	an	DET
iajs-2804	186	28	ℛ-module	ℛ-module	PROPN
iajs-2804	186	29	m	m	NOUN
iajs-2804	186	30	is	be	AUX
iajs-2804	186	31	an	an	DET
iajs-2804	186	32	ℱ-soc	ℱ-soc	NOUN
iajs-2804	186	33	-	-	PUNCT
iajs-2804	186	34	semiprime	semiprime	NOUN
iajs-2804	186	35	sub	sub	NOUN
iajs-2804	186	36	-	-	NOUN
iajs-2804	186	37	module	module	NOUN
iajs-2804	186	38	of	of	ADP
iajs-2804	186	39	x	x	PRON
iajs-2804	186	40	,	,	PUNCT
iajs-2804	186	41	but	but	CCONJ
iajs-2804	186	42	the	the	DET
iajs-2804	186	43	converse	converse	NOUN
iajs-2804	186	44	is	be	AUX
iajs-2804	186	45	not	not	PART
iajs-2804	186	46	true	true	ADJ
iajs-2804	186	47	.	.	PUNCT
iajs-2804	187	1	proof	proof	NOUN
iajs-2804	187	2	:	:	PUNCT
iajs-2804	187	3	we	we	PRON
iajs-2804	187	4	take	take	VERB
iajs-2804	187	5	u	u	PRON
iajs-2804	187	6	as	as	ADP
iajs-2804	187	7	a	a	DET
iajs-2804	187	8	completely	completely	ADV
iajs-2804	187	9	ℱ-sub	ℱ-sub	NOUN
iajs-2804	187	10	-	-	NOUN
iajs-2804	187	11	module	module	NOUN
iajs-2804	187	12	of	of	ADP
iajs-2804	187	13	x	x	PUNCT
iajs-2804	187	14	with	with	ADP
iajs-2804	187	15	𝑟𝑏	𝑟𝑏	NUM
iajs-2804	187	16	𝑛𝑚𝑡	𝑛𝑚𝑡	NOUN
iajs-2804	187	17	⊆	⊆	NUM
iajs-2804	187	18	𝑈,for	𝑈,for	NOUN
iajs-2804	187	19	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	187	20	⊆	⊆	PROPN
iajs-2804	187	21	𝑅	𝑅	PROPN
iajs-2804	187	22	,	,	PUNCT
iajs-2804	187	23	𝑚𝑡	𝑚𝑡	ADP
iajs-2804	187	24	⊆	⊆	NUM
iajs-2804	187	25	𝑋.now	𝑋.now	NOUN
iajs-2804	187	26	,	,	PUNCT
iajs-2804	187	27	if	if	SCONJ
iajs-2804	187	28	𝑟𝑏	𝑟𝑏	NUM
iajs-2804	187	29	=	=	SYM
iajs-2804	188	1	01	01	NUM
iajs-2804	189	1	then	then	ADV
iajs-2804	189	2	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	189	3	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	189	4	=	=	ADJ
iajs-2804	189	5	0𝑡	0𝑡	PROPN
iajs-2804	190	1	⊆	⊆	NUM
iajs-2804	190	2	01	01	NUM
iajs-2804	190	3	⊆	⊆	NUM
iajs-2804	190	4	𝑈.we	𝑈.we	NOUN
iajs-2804	190	5	get	get	VERB
iajs-2804	190	6	u	u	NOUN
iajs-2804	190	7	is	be	AUX
iajs-2804	190	8	an	an	DET
iajs-2804	190	9	ℱ-socsemi	ℱ-socsemi	PROPN
iajs-2804	190	10	-	-	PUNCT
iajs-2804	190	11	prime	prime	ADJ
iajs-2804	190	12	sub	sub	NOUN
iajs-2804	190	13	-	-	NOUN
iajs-2804	190	14	module	module	NOUN
iajs-2804	190	15	of	of	ADP
iajs-2804	190	16	x.	x.	NOUN
iajs-2804	190	17	if	if	SCONJ
iajs-2804	190	18	𝑟𝑏	𝑟𝑏	PROPN
iajs-2804	190	19	≠	≠	PROPN
iajs-2804	190	20	01	01	NUM
iajs-2804	190	21	,	,	PUNCT
iajs-2804	190	22	thus	thus	ADV
iajs-2804	190	23	(	(	PUNCT
iajs-2804	190	24	𝑟𝑏)𝑛−1	𝑟𝑏)𝑛−1	NOUN
iajs-2804	190	25	(	(	PUNCT
iajs-2804	190	26	𝑟𝑏	𝑟𝑏	PROPN
iajs-2804	190	27	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	190	28	)	)	PUNCT
iajs-2804	190	29	⊆	⊆	PROPN
iajs-2804	190	30	𝑈	𝑈	PROPN
iajs-2804	190	31	,	,	PUNCT
iajs-2804	190	32	we	we	PRON
iajs-2804	190	33	get	get	VERB
iajs-2804	190	34	(	(	PUNCT
iajs-2804	190	35	𝑟	𝑟	X
iajs-2804	190	36	𝑛−1)𝑏	𝑛−1)𝑏	X
iajs-2804	190	37	(	(	PUNCT
iajs-2804	190	38	𝑟𝑚)𝑑	𝑟𝑚)𝑑	PROPN
iajs-2804	190	39	⊆	⊆	NUM
iajs-2804	190	40	𝑈	𝑈	PROPN
iajs-2804	190	41	where	where	SCONJ
iajs-2804	190	42	𝑑	𝑑	PROPN
iajs-2804	190	43	=	=	SYM
iajs-2804	190	44	min	min	PROPN
iajs-2804	190	45	{	{	PUNCT
iajs-2804	190	46	𝑏	𝑏	NOUN
iajs-2804	190	47	,	,	PUNCT
iajs-2804	190	48	𝑡}.now	𝑡}.now	PROPN
iajs-2804	190	49	,	,	PUNCT
iajs-2804	190	50	since	since	SCONJ
iajs-2804	190	51	u	u	NOUN
iajs-2804	190	52	is	be	AUX
iajs-2804	190	53	a	a	DET
iajs-2804	190	54	completely	completely	ADV
iajs-2804	190	55	ℱ-sub	ℱ-sub	NOUN
iajs-2804	190	56	-	-	NOUN
iajs-2804	190	57	module	module	NOUN
iajs-2804	190	58	of	of	ADP
iajs-2804	190	59	x	x	PRON
iajs-2804	190	60	,	,	PUNCT
iajs-2804	190	61	then	then	ADV
iajs-2804	190	62	we	we	PRON
iajs-2804	190	63	have	have	VERB
iajs-2804	190	64	(	(	PUNCT
iajs-2804	191	1	𝑟𝑚)𝑑	𝑟𝑚)𝑑	PROPN
iajs-2804	191	2	⊆	⊆	NUM
iajs-2804	191	3	𝑈	𝑈	PROPN
iajs-2804	191	4	⊆	⊆	NUM
iajs-2804	191	5	𝑈	𝑈	PROPN
iajs-2804	191	6	+	+	CCONJ
iajs-2804	191	7	ℱ	ℱ	PROPN
iajs-2804	191	8	−	−	PROPN
iajs-2804	191	9	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	191	10	)	)	PUNCT
iajs-2804	191	11	,	,	PUNCT
iajs-2804	191	12	thus	thus	ADV
iajs-2804	191	13	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	191	14	⊆	⊆	NUM
iajs-2804	191	15	𝑈	𝑈	PROPN
iajs-2804	191	16	+	+	CCONJ
iajs-2804	191	17	ℱ	ℱ	PROPN
iajs-2804	191	18	−	−	PROPN
iajs-2804	191	19	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	191	20	)	)	PUNCT
iajs-2804	191	21	.	.	PUNCT
iajs-2804	192	1	therefore	therefore	ADV
iajs-2804	192	2	u	u	PROPN
iajs-2804	192	3	is	be	AUX
iajs-2804	192	4	an	an	DET
iajs-2804	192	5	ℱ-soc	ℱ-soc	NOUN
iajs-2804	192	6	-	-	PUNCT
iajs-2804	192	7	semi	semi	ADJ
iajs-2804	192	8	-	-	ADJ
iajs-2804	192	9	prime	prime	ADJ
iajs-2804	192	10	submodule	submodule	NOUN
iajs-2804	192	11	.	.	PUNCT
iajs-2804	193	1	the	the	DET
iajs-2804	193	2	following	follow	VERB
iajs-2804	193	3	example	example	NOUN
iajs-2804	193	4	show	show	VERB
iajs-2804	193	5	that	that	SCONJ
iajs-2804	193	6	the	the	DET
iajs-2804	193	7	converse	converse	NOUN
iajs-2804	193	8	is	be	AUX
iajs-2804	193	9	not	not	PART
iajs-2804	193	10	true	true	ADJ
iajs-2804	193	11	example	example	NOUN
iajs-2804	193	12	2.10	2.10	NUM
iajs-2804	193	13	consider	consider	VERB
iajs-2804	193	14	𝑀	𝑀	NOUN
iajs-2804	193	15	=	=	PUNCT
iajs-2804	193	16	𝑍	𝑍	PROPN
iajs-2804	193	17	as	as	ADP
iajs-2804	193	18	a	a	DET
iajs-2804	193	19	z	z	NOUN
iajs-2804	193	20	-	-	PUNCT
iajs-2804	193	21	module	module	NOUN
iajs-2804	193	22	and	and	CCONJ
iajs-2804	193	23	𝑋	𝑋	PROPN
iajs-2804	193	24	:	:	PUNCT
iajs-2804	193	25	𝑀	𝑀	PROPN
iajs-2804	193	26	→	→	PUNCT
iajs-2804	194	1	[	[	X
iajs-2804	194	2	0,1	0,1	NUM
iajs-2804	194	3	]	]	PUNCT
iajs-2804	194	4	,	,	PUNCT
iajs-2804	194	5	𝑈	𝑈	PROPN
iajs-2804	194	6	:	:	PUNCT
iajs-2804	194	7	𝑀	𝑀	PROPN
iajs-2804	194	8	→	→	PUNCT
iajs-2804	195	1	[	[	X
iajs-2804	195	2	0,1	0,1	NUM
iajs-2804	195	3	]	]	PUNCT
iajs-2804	195	4	defined	define	VERB
iajs-2804	195	5	by	by	ADP
iajs-2804	195	6	:	:	PUNCT
iajs-2804	195	7	𝑋(𝑚	𝑋(𝑚	PROPN
iajs-2804	195	8	)	)	PUNCT
iajs-2804	195	9	=	=	SYM
iajs-2804	195	10	1	1	NUM
iajs-2804	195	11	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	195	12	𝑚	𝑚	PRON
iajs-2804	195	13	∈	∈	PROPN
iajs-2804	195	14	𝑍	𝑍	PROPN
iajs-2804	195	15	𝑈(𝑚	𝑈(𝑚	NOUN
iajs-2804	195	16	)	)	PUNCT
iajs-2804	195	17	=	=	SYM
iajs-2804	195	18	{	{	PUNCT
iajs-2804	195	19	1	1	NUM
iajs-2804	195	20	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	195	21	𝑚	𝑚	NOUN
iajs-2804	195	22	∈	∈	PROPN
iajs-2804	195	23	2𝑍	2𝑍	PROPN
iajs-2804	195	24	1/4	1/4	NUM
iajs-2804	195	25	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	195	26	𝑚	𝑚	PROPN
iajs-2804	195	27	∉	∉	PROPN
iajs-2804	195	28	2𝑍	2𝑍	PROPN
iajs-2804	195	29	and	and	CCONJ
iajs-2804	195	30	an	an	DET
iajs-2804	195	31	ℱ-socle	ℱ-socle	PROPN
iajs-2804	195	32	of	of	ADP
iajs-2804	195	33	x	x	PUNCT
iajs-2804	195	34	is	be	AUX
iajs-2804	195	35	defined	define	VERB
iajs-2804	195	36	by	by	ADP
iajs-2804	195	37	ℱ	ℱ	PROPN
iajs-2804	195	38	−	−	PROPN
iajs-2804	195	39	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	195	40	):	):	PUNCT
iajs-2804	195	41	𝑀	𝑀	PROPN
iajs-2804	195	42	→	→	PUNCT
iajs-2804	196	1	[	[	X
iajs-2804	196	2	0,1	0,1	NUM
iajs-2804	196	3	]	]	PUNCT
iajs-2804	196	4	such	such	ADJ
iajs-2804	196	5	that	that	SCONJ
iajs-2804	196	6	:	:	PUNCT
iajs-2804	196	7	ℱ	ℱ	PROPN
iajs-2804	196	8	−	−	NOUN
iajs-2804	196	9	𝑆𝑜𝑐(𝑋)(𝑚	𝑆𝑜𝑐(𝑋)(𝑚	NOUN
iajs-2804	196	10	)	)	PUNCT
iajs-2804	197	1	=	=	PRON
iajs-2804	197	2	{	{	PUNCT
iajs-2804	197	3	1	1	NUM
iajs-2804	197	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	197	5	𝑚	𝑚	ADP
iajs-2804	197	6	∈	∈	PROPN
iajs-2804	197	7	{	{	PUNCT
iajs-2804	197	8	0	0	NUM
iajs-2804	197	9	}	}	PUNCT
iajs-2804	197	10	1/3	1/3	NUM
iajs-2804	197	11	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	197	12	𝑚	𝑚	ADP
iajs-2804	197	13	∉	∉	X
iajs-2804	197	14	{	{	PUNCT
iajs-2804	197	15	0	0	NUM
iajs-2804	197	16	}	}	PUNCT
iajs-2804	197	17	(	(	PUNCT
iajs-2804	197	18	𝑈	𝑈	PROPN
iajs-2804	197	19	+	+	CCONJ
iajs-2804	197	20	ℱ	ℱ	PROPN
iajs-2804	197	21	−	−	ADP
iajs-2804	197	22	𝑆𝑜𝑐(𝑋))(𝑚	𝑆𝑜𝑐(𝑋))(𝑚	ADJ
iajs-2804	197	23	)	)	PUNCT
iajs-2804	198	1	=	=	PRON
iajs-2804	198	2	{	{	PUNCT
iajs-2804	198	3	1	1	NUM
iajs-2804	198	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	198	5	𝑚	𝑚	NOUN
iajs-2804	198	6	∈	∈	PROPN
iajs-2804	198	7	2𝑍	2𝑍	NOUN
iajs-2804	198	8	1/3	1/3	NUM
iajs-2804	198	9	𝑖𝑓	𝑖𝑓	ADP
iajs-2804	198	10	𝑚	𝑚	PROPN
iajs-2804	198	11	∉	∉	PROPN
iajs-2804	198	12	2𝑍	2𝑍	PROPN
iajs-2804	198	13	where	where	SCONJ
iajs-2804	198	14	𝑆𝑜𝑐(𝑀	𝑆𝑜𝑐(𝑀	NOUN
iajs-2804	198	15	)	)	PUNCT
iajs-2804	198	16	=	=	PRON
iajs-2804	198	17	{	{	PUNCT
iajs-2804	198	18	0	0	NUM
iajs-2804	198	19	}	}	PUNCT
iajs-2804	198	20	.	.	PUNCT
iajs-2804	199	1	that	that	PRON
iajs-2804	199	2	’s	’	VERB
iajs-2804	199	3	clear	clear	ADJ
iajs-2804	199	4	x	x	VERB
iajs-2804	199	5	is	be	AUX
iajs-2804	199	6	an	an	DET
iajs-2804	199	7	ℱ-module	ℱ-module	PROPN
iajs-2804	199	8	and	and	CCONJ
iajs-2804	199	9	u	u	NOUN
iajs-2804	199	10	be	be	VERB
iajs-2804	199	11	an	an	DET
iajs-2804	199	12	ℱ-sub	ℱ-sub	NOUN
iajs-2804	199	13	-	-	NOUN
iajs-2804	199	14	module	module	NOUN
iajs-2804	199	15	of	of	ADP
iajs-2804	199	16	x.	x.	PROPN
iajs-2804	199	17	ibn	ibn	PROPN
iajs-2804	199	18	al	al	PROPN
iajs-2804	199	19	-	-	PUNCT
iajs-2804	199	20	haitham	haitham	PROPN
iajs-2804	199	21	jour	jour	X
iajs-2804	199	22	.	.	PROPN
iajs-2804	199	23	for	for	ADP
iajs-2804	199	24	pure	pure	ADJ
iajs-2804	199	25	&	&	CCONJ
iajs-2804	199	26	appl	appl	PROPN
iajs-2804	199	27	.	.	PUNCT
iajs-2804	200	1	sci	sci	PROPN
iajs-2804	200	2	.	.	PROPN
iajs-2804	201	1	53	53	NUM
iajs-2804	201	2	(	(	PUNCT
iajs-2804	201	3	1)2022	1)2022	NOUN
iajs-2804	201	4	111	111	NUM
iajs-2804	201	5	2𝑍	2𝑍	NOUN
iajs-2804	201	6	is	be	AUX
iajs-2804	201	7	an	an	DET
iajs-2804	201	8	app	app	ADJ
iajs-2804	201	9	-	-	PUNCT
iajs-2804	201	10	semi	semi	ADJ
iajs-2804	201	11	-	-	ADJ
iajs-2804	201	12	prime	prime	ADJ
iajs-2804	201	13	sub	sub	NOUN
iajs-2804	201	14	-	-	NOUN
iajs-2804	201	15	module	module	NOUN
iajs-2804	201	16	of	of	ADP
iajs-2804	201	17	m	m	PRON
iajs-2804	201	18	,	,	PUNCT
iajs-2804	201	19	so	so	ADV
iajs-2804	201	20	by	by	ADP
iajs-2804	201	21	(	(	PUNCT
iajs-2804	201	22	lemma	lemma	PROPN
iajs-2804	201	23	2.3	2.3	NUM
iajs-2804	201	24	)	)	PUNCT
iajs-2804	201	25	we	we	PRON
iajs-2804	201	26	get	get	VERB
iajs-2804	201	27	u	u	NOUN
iajs-2804	201	28	is	be	AUX
iajs-2804	201	29	an	an	DET
iajs-2804	201	30	ℱ-soc	ℱ-soc	NOUN
iajs-2804	201	31	-	-	PUNCT
iajs-2804	201	32	semiprime	semiprime	NOUN
iajs-2804	201	33	sub	sub	NOUN
iajs-2804	201	34	-	-	NOUN
iajs-2804	201	35	module	module	NOUN
iajs-2804	201	36	of	of	ADP
iajs-2804	201	37	x.	x.	NOUN
iajs-2804	201	38	but	but	CCONJ
iajs-2804	201	39	u	u	NOUN
iajs-2804	201	40	is	be	AUX
iajs-2804	201	41	not	not	PART
iajs-2804	201	42	completely	completely	ADV
iajs-2804	201	43	ℱ-sub	ℱ-sub	NOUN
iajs-2804	201	44	-	-	NOUN
iajs-2804	201	45	module	module	NOUN
iajs-2804	201	46	of	of	ADP
iajs-2804	201	47	x	x	PRON
iajs-2804	201	48	,	,	PUNCT
iajs-2804	201	49	since	since	SCONJ
iajs-2804	201	50	for	for	ADP
iajs-2804	201	51	an	an	DET
iajs-2804	201	52	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	201	53	51	51	NUM
iajs-2804	201	54	3	3	NUM
iajs-2804	201	55	⊆	⊆	NUM
iajs-2804	201	56	𝑋	𝑋	NOUN
iajs-2804	201	57	and	and	CCONJ
iajs-2804	201	58	an	an	DET
iajs-2804	201	59	ℱsingleton	ℱsingleton	PROPN
iajs-2804	201	60	21	21	NUM
iajs-2804	201	61	2	2	NUM
iajs-2804	201	62	of	of	ADP
iajs-2804	201	63	ℛ	ℛ	PROPN
iajs-2804	201	64	such	such	ADJ
iajs-2804	201	65	that	that	DET
iajs-2804	201	66	21	21	NUM
iajs-2804	201	67	2	2	NUM
iajs-2804	201	68	51	51	NUM
iajs-2804	201	69	3	3	NUM
iajs-2804	201	70	=	=	SYM
iajs-2804	201	71	101	101	NUM
iajs-2804	201	72	3	3	NUM
iajs-2804	201	73	where	where	SCONJ
iajs-2804	201	74	101	101	NUM
iajs-2804	201	75	3	3	NUM
iajs-2804	201	76	⊆	⊆	NUM
iajs-2804	201	77	𝑈	𝑈	PROPN
iajs-2804	201	78	since	since	SCONJ
iajs-2804	201	79	𝑈(10	𝑈(10	NUM
iajs-2804	201	80	)	)	PUNCT
iajs-2804	201	81	=	=	SYM
iajs-2804	201	82	1	1	NUM
iajs-2804	201	83	>	>	SYM
iajs-2804	201	84	1	1	NUM
iajs-2804	201	85	3	3	NUM
iajs-2804	201	86	.	.	PUNCT
iajs-2804	202	1	but	but	CCONJ
iajs-2804	202	2	51	51	NUM
iajs-2804	202	3	3	3	NUM
iajs-2804	202	4	⊈	⊈	PROPN
iajs-2804	202	5	𝑈	𝑈	NOUN
iajs-2804	202	6	since	since	SCONJ
iajs-2804	202	7	𝑈(5	𝑈(5	NUM
iajs-2804	202	8	)	)	PUNCT
iajs-2804	202	9	=	=	SYM
iajs-2804	202	10	1	1	NUM
iajs-2804	202	11	4	4	NUM
iajs-2804	202	12	≯	≯	VERB
iajs-2804	202	13	1	1	NUM
iajs-2804	202	14	3	3	NUM
iajs-2804	202	15	.	.	PUNCT
iajs-2804	203	1	hence	hence	ADV
iajs-2804	203	2	,	,	PUNCT
iajs-2804	203	3	u	u	PROPN
iajs-2804	203	4	is	be	AUX
iajs-2804	203	5	not	not	PART
iajs-2804	203	6	completely	completely	ADV
iajs-2804	203	7	ℱ-sub	ℱ-sub	NOUN
iajs-2804	203	8	-	-	NOUN
iajs-2804	203	9	module	module	NOUN
iajs-2804	203	10	of	of	ADP
iajs-2804	203	11	x.	x.	NOUN
iajs-2804	203	12	proposition	proposition	NOUN
iajs-2804	203	13	2.11	2.11	NUM
iajs-2804	203	14	let	let	VERB
iajs-2804	203	15	u	u	PRON
iajs-2804	203	16	be	be	AUX
iajs-2804	203	17	an	an	DET
iajs-2804	203	18	ℱ-soc	ℱ-soc	NOUN
iajs-2804	203	19	-	-	PUNCT
iajs-2804	203	20	semi	semi	ADJ
iajs-2804	203	21	-	-	ADJ
iajs-2804	203	22	prime	prime	ADJ
iajs-2804	203	23	sub	sub	NOUN
iajs-2804	203	24	-	-	NOUN
iajs-2804	203	25	module	module	NOUN
iajs-2804	203	26	of	of	ADP
iajs-2804	203	27	an	an	DET
iajs-2804	203	28	ℱ-module	ℱ-module	PROPN
iajs-2804	203	29	x	x	PROPN
iajs-2804	203	30	of	of	ADP
iajs-2804	203	31	an	an	DET
iajs-2804	203	32	ℛ-module	ℛ-module	PROPN
iajs-2804	203	33	m	m	PROPN
iajs-2804	203	34	,	,	PUNCT
iajs-2804	203	35	then	then	ADV
iajs-2804	203	36	u	u	NOUN
iajs-2804	203	37	is	be	AUX
iajs-2804	203	38	an	an	DET
iajs-2804	203	39	ℱ-soc	ℱ-soc	NOUN
iajs-2804	203	40	-	-	PUNCT
iajs-2804	203	41	semi	semi	ADJ
iajs-2804	203	42	-	-	ADJ
iajs-2804	203	43	prime	prime	ADJ
iajs-2804	203	44	sub	sub	NOUN
iajs-2804	203	45	-	-	NOUN
iajs-2804	203	46	module	module	NOUN
iajs-2804	203	47	of	of	ADP
iajs-2804	203	48	x	x	PRON
iajs-2804	203	49	if	if	SCONJ
iajs-2804	203	50	and	and	CCONJ
iajs-2804	203	51	only	only	ADV
iajs-2804	203	52	if	if	SCONJ
iajs-2804	203	53	∀	∀	X
iajs-2804	203	54	ℱ-sub	ℱ-sub	NOUN
iajs-2804	203	55	-	-	NOUN
iajs-2804	203	56	module	module	NOUN
iajs-2804	203	57	s	s	NOUN
iajs-2804	203	58	of	of	ADP
iajs-2804	203	59	x	x	X
iajs-2804	203	60	and	and	CCONJ
iajs-2804	203	61	an	an	DET
iajs-2804	203	62	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	203	63	j	j	PROPN
iajs-2804	203	64	of	of	ADP
iajs-2804	203	65	ℛ	ℛ	PROPN
iajs-2804	203	66	with	with	ADP
iajs-2804	203	67	(	(	PUNCT
iajs-2804	203	68	𝐽)ns	𝐽)ns	PROPN
iajs-2804	203	69	⊆u	⊆u	VERB
iajs-2804	203	70	for	for	ADP
iajs-2804	203	71	𝑛	𝑛	DET
iajs-2804	203	72	∈	∈	PROPN
iajs-2804	203	73	𝑍+implies	𝑍+implie	NOUN
iajs-2804	203	74	that	that	PRON
iajs-2804	203	75	𝐽s	𝐽	VERB
iajs-2804	203	76	⊆	⊆	NUM
iajs-2804	203	77	𝑈	𝑈	PROPN
iajs-2804	203	78	+	+	CCONJ
iajs-2804	203	79	ℱ	ℱ	PROPN
iajs-2804	203	80	−	−	PROPN
iajs-2804	203	81	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	203	82	)	)	PUNCT
iajs-2804	203	83	proof	proof	NOUN
iajs-2804	203	84	:	:	PUNCT
iajs-2804	203	85	(	(	PUNCT
iajs-2804	203	86	)	)	PUNCT
iajs-2804	203	87	assume	assume	VERB
iajs-2804	203	88	that	that	SCONJ
iajs-2804	203	89	(	(	PUNCT
iajs-2804	203	90	𝐽)𝑛s	𝐽)𝑛s	PROPN
iajs-2804	203	91	⊆u	⊆u	VERB
iajs-2804	203	92	,	,	PUNCT
iajs-2804	203	93	for	for	ADP
iajs-2804	203	94	s	s	PROPN
iajs-2804	203	95	is	be	AUX
iajs-2804	203	96	an	an	DET
iajs-2804	203	97	ℱ-sub	ℱ-sub	NOUN
iajs-2804	203	98	-	-	NOUN
iajs-2804	203	99	module	module	NOUN
iajs-2804	203	100	of	of	ADP
iajs-2804	203	101	x	x	PUNCT
iajs-2804	203	102	and	and	CCONJ
iajs-2804	203	103	𝐽	𝐽	PROPN
iajs-2804	203	104	is	be	AUX
iajs-2804	203	105	an	an	DET
iajs-2804	203	106	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	203	107	of	of	ADP
iajs-2804	203	108	ℛ	ℛ	PROPN
iajs-2804	203	109	,	,	PUNCT
iajs-2804	203	110	let	let	VERB
iajs-2804	203	111	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	203	112	⊆	⊆	NUM
iajs-2804	203	113	𝐽𝑆	𝐽𝑆	PROPN
iajs-2804	203	114	with	with	ADP
iajs-2804	203	115	𝑡	𝑡	PROPN
iajs-2804	203	116	∈	∈	PROPN
iajs-2804	204	1	[	[	X
iajs-2804	204	2	0,1	0,1	NUM
iajs-2804	204	3	]	]	PUNCT
iajs-2804	204	4	then	then	ADV
iajs-2804	204	5	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	204	6	=	=	PUNCT
iajs-2804	204	7	(	(	PUNCT
iajs-2804	204	8	𝑐1)ℎ1(𝑦1)𝑡1	𝑐1)ℎ1(𝑦1)𝑡1	PROPN
iajs-2804	204	9	+	+	CCONJ
iajs-2804	204	10	(	(	PUNCT
iajs-2804	204	11	𝑐2)ℎ2(𝑦2)𝑡2	𝑐2)ℎ2(𝑦2)𝑡2	NOUN
iajs-2804	204	12	+	+	CCONJ
iajs-2804	204	13	⋯	⋯	VERB
iajs-2804	204	14	+	+	CCONJ
iajs-2804	204	15	(	(	PUNCT
iajs-2804	204	16	𝑐𝑛)ℎ𝑛(𝑦𝑛)𝑡𝑛	𝑐𝑛)ℎ𝑛(𝑦𝑛)𝑡𝑛	NOUN
iajs-2804	204	17	,	,	PUNCT
iajs-2804	204	18	for	for	ADP
iajs-2804	204	19	every	every	DET
iajs-2804	204	20	(	(	PUNCT
iajs-2804	204	21	𝑐𝑖)ℎ𝑖	𝑐𝑖)ℎ𝑖	SYM
iajs-2804	204	22	⊆	⊆	NUM
iajs-2804	204	23	𝐽	𝐽	PROPN
iajs-2804	204	24	and	and	CCONJ
iajs-2804	204	25	(	(	PUNCT
iajs-2804	204	26	𝑦𝑖)𝑡𝑖	𝑦𝑖)𝑡𝑖	PROPN
iajs-2804	204	27	⊆	⊆	NUM
iajs-2804	204	28	𝑈	𝑈	NOUN
iajs-2804	204	29	where	where	SCONJ
iajs-2804	204	30	ℎ𝑖	ℎ𝑖	NOUN
iajs-2804	204	31	,	,	PUNCT
iajs-2804	204	32	𝑡𝑖	𝑡𝑖	NOUN
iajs-2804	204	33	∈	∈	NOUN
iajs-2804	204	34	[	[	X
iajs-2804	204	35	0,1	0,1	NUM
iajs-2804	204	36	]	]	PUNCT
iajs-2804	204	37	for	for	ADP
iajs-2804	204	38	every	every	DET
iajs-2804	204	39	i=1,2,	i=1,2,	NOUN
iajs-2804	204	40	…	…	PUNCT
iajs-2804	204	41	..	..	PUNCT
iajs-2804	204	42	,n	,n	NOUN
iajs-2804	204	43	.	.	PUNCT
iajs-2804	205	1	now	now	ADV
iajs-2804	205	2	,	,	PUNCT
iajs-2804	205	3	we	we	PRON
iajs-2804	205	4	get	get	AUX
iajs-2804	205	5	(	(	PUNCT
iajs-2804	205	6	(	(	PUNCT
iajs-2804	205	7	𝑐𝑖)ℎ𝑖	𝑐𝑖)ℎ𝑖	NOUN
iajs-2804	205	8	)	)	PUNCT
iajs-2804	205	9	𝑛(𝑦𝑖)𝑡𝑖	𝑛(𝑦𝑖)𝑡𝑖	VERB
iajs-2804	205	10	⊆	⊆	NUM
iajs-2804	205	11	(	(	PUNCT
iajs-2804	205	12	𝐽)𝑛s	𝐽)𝑛s	PROPN
iajs-2804	205	13	⊆u	⊆u	VERB
iajs-2804	205	14	hence	hence	ADV
iajs-2804	205	15	(	(	PUNCT
iajs-2804	205	16	(	(	PUNCT
iajs-2804	205	17	𝑐𝑖)ℎ𝑖	𝑐𝑖)ℎ𝑖	X
iajs-2804	205	18	)	)	PUNCT
iajs-2804	205	19	𝑛(𝑦𝑖)𝑡𝑖	𝑛(𝑦𝑖)𝑡𝑖	VERB
iajs-2804	205	20	⊆	⊆	NUM
iajs-2804	205	21	u.	u.	NOUN
iajs-2804	205	22	but	but	CCONJ
iajs-2804	205	23	u	u	NOUN
iajs-2804	205	24	is	be	AUX
iajs-2804	205	25	an	an	DET
iajs-2804	205	26	ℱ-soc	ℱ-soc	NOUN
iajs-2804	205	27	-	-	PUNCT
iajs-2804	205	28	semi	semi	ADJ
iajs-2804	205	29	-	-	ADJ
iajs-2804	205	30	prime	prime	ADJ
iajs-2804	205	31	submodule	submodule	NOUN
iajs-2804	205	32	of	of	ADP
iajs-2804	205	33	x	x	PROPN
iajs-2804	205	34	implies	imply	VERB
iajs-2804	205	35	that	that	SCONJ
iajs-2804	205	36	(	(	PUNCT
iajs-2804	205	37	𝑐𝑖)ℎ𝑖(𝑦𝑖)𝑡𝑖	𝑐𝑖)ℎ𝑖(𝑦𝑖)𝑡𝑖	VERB
iajs-2804	205	38	⊆	⊆	NUM
iajs-2804	205	39	u+ℱ	u+ℱ	ADJ
iajs-2804	205	40	−	−	PROPN
iajs-2804	205	41	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	205	42	)	)	PUNCT
iajs-2804	205	43	for	for	ADP
iajs-2804	205	44	each	each	DET
iajs-2804	205	45	i=1,2,	i=1,2,	NOUN
iajs-2804	205	46	…	…	PUNCT
iajs-2804	205	47	..	..	PUNCT
iajs-2804	205	48	,n	,n	PUNCT
iajs-2804	205	49	.	.	PUNCT
iajs-2804	206	1	so	so	ADV
iajs-2804	206	2	we	we	PRON
iajs-2804	206	3	have	have	VERB
iajs-2804	206	4	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	206	5	⊆	⊆	NUM
iajs-2804	206	6	u+ℱ	u+ℱ	ADJ
iajs-2804	206	7	−	−	PROPN
iajs-2804	206	8	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	206	9	)	)	PUNCT
iajs-2804	206	10	.	.	PUNCT
iajs-2804	207	1	it	it	PRON
iajs-2804	207	2	follows	follow	VERB
iajs-2804	207	3	that	that	SCONJ
iajs-2804	207	4	𝐽s	𝐽	VERB
iajs-2804	207	5	⊆	⊆	NUM
iajs-2804	207	6	𝑈	𝑈	PROPN
iajs-2804	207	7	+	+	CCONJ
iajs-2804	207	8	ℱ	ℱ	PROPN
iajs-2804	207	9	−	−	PROPN
iajs-2804	207	10	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	207	11	)	)	PUNCT
iajs-2804	207	12	.	.	PUNCT
iajs-2804	208	1	(	(	PUNCT
iajs-2804	208	2	)	)	PUNCT
iajs-2804	208	3	let	let	VERB
iajs-2804	208	4	(	(	PUNCT
iajs-2804	208	5	𝑟𝑏)𝑛𝑥𝑡	𝑟𝑏)𝑛𝑥𝑡	NOUN
iajs-2804	208	6	⊆	⊆	NUM
iajs-2804	208	7	𝑈	𝑈	NOUN
iajs-2804	208	8	for	for	ADP
iajs-2804	208	9	𝑟𝑏	𝑟𝑏	NUM
iajs-2804	208	10	⊆	⊆	NUM
iajs-2804	208	11	ℛ	ℛ	PROPN
iajs-2804	208	12	and	and	CCONJ
iajs-2804	208	13	𝑛	𝑛	PRON
iajs-2804	208	14	∈	∈	NOUN
iajs-2804	208	15	𝑍+	𝑍+	PUNCT
iajs-2804	209	1	then	then	ADV
iajs-2804	209	2	〈	〈	PROPN
iajs-2804	209	3	𝑟𝑏	𝑟𝑏	PROPN
iajs-2804	209	4	𝑛〉〈𝑥𝑡	𝑛〉〈𝑥𝑡	ADV
iajs-2804	209	5	〉	〉	NOUN
iajs-2804	209	6	⊆	⊆	NUM
iajs-2804	209	7	𝑈	𝑈	PROPN
iajs-2804	209	8	,	,	PUNCT
iajs-2804	209	9	that	that	PRON
iajs-2804	209	10	is	be	AUX
iajs-2804	209	11	〈	〈	PROPN
iajs-2804	209	12	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	209	13	〉	〉	NOUN
iajs-2804	209	14	𝑛〈𝑥𝑡	𝑛〈𝑥𝑡	NOUN
iajs-2804	209	15	〉	〉	NOUN
iajs-2804	209	16	⊆	⊆	NUM
iajs-2804	209	17	𝑈	𝑈	PROPN
iajs-2804	209	18	then	then	ADV
iajs-2804	209	19	by	by	ADP
iajs-2804	209	20	hypothesis	hypothesis	NOUN
iajs-2804	209	21	we	we	PRON
iajs-2804	209	22	get	get	VERB
iajs-2804	209	23	〈	〈	NOUN
iajs-2804	209	24	𝑟𝑏〉〈𝑥𝑡	𝑟𝑏〉〈𝑥𝑡	NOUN
iajs-2804	209	25	〉	〉	NOUN
iajs-2804	209	26	⊆	⊆	NUM
iajs-2804	209	27	𝑈	𝑈	PROPN
iajs-2804	209	28	,	,	PUNCT
iajs-2804	209	29	hence	hence	ADV
iajs-2804	209	30	𝑟𝑏𝑥𝑡	𝑟𝑏𝑥𝑡	NOUN
iajs-2804	209	31	⊆	⊆	NUM
iajs-2804	209	32	𝑈.	𝑈.	NOUN
iajs-2804	209	33	that	that	PRON
iajs-2804	209	34	is	be	AUX
iajs-2804	209	35	mean	mean	ADJ
iajs-2804	209	36	u	u	NOUN
iajs-2804	209	37	is	be	AUX
iajs-2804	209	38	an	an	DET
iajs-2804	209	39	ℱ-socsemi	ℱ-socsemi	PROPN
iajs-2804	209	40	-	-	PUNCT
iajs-2804	209	41	prime	prime	ADJ
iajs-2804	209	42	sub	sub	NOUN
iajs-2804	209	43	-	-	NOUN
iajs-2804	209	44	module	module	NOUN
iajs-2804	209	45	of	of	ADP
iajs-2804	209	46	x.	x.	PROPN
iajs-2804	209	47	corollary	corollary	PROPN
iajs-2804	209	48	2.12	2.12	NUM
iajs-2804	209	49	let	let	VERB
iajs-2804	209	50	u	u	PRON
iajs-2804	209	51	be	be	AUX
iajs-2804	209	52	an	an	DET
iajs-2804	209	53	ℱ-sub	ℱ-sub	NOUN
iajs-2804	209	54	-	-	NOUN
iajs-2804	209	55	module	module	NOUN
iajs-2804	209	56	of	of	ADP
iajs-2804	209	57	an	an	DET
iajs-2804	209	58	ℱ-module	ℱ-module	PROPN
iajs-2804	209	59	x	x	PROPN
iajs-2804	209	60	of	of	ADP
iajs-2804	209	61	an	an	DET
iajs-2804	209	62	ℛ-module	ℛ-module	PROPN
iajs-2804	209	63	m	m	PROPN
iajs-2804	209	64	,	,	PUNCT
iajs-2804	209	65	then	then	ADV
iajs-2804	209	66	u	u	NOUN
iajs-2804	209	67	is	be	AUX
iajs-2804	209	68	an	an	DET
iajs-2804	209	69	ℱ-soc	ℱ-soc	NOUN
iajs-2804	209	70	-	-	PUNCT
iajs-2804	209	71	semiprime	semiprime	NOUN
iajs-2804	209	72	sub	sub	NOUN
iajs-2804	209	73	-	-	NOUN
iajs-2804	209	74	module	module	NOUN
iajs-2804	209	75	of	of	ADP
iajs-2804	209	76	x	x	PRON
iajs-2804	209	77	if	if	SCONJ
iajs-2804	209	78	and	and	CCONJ
iajs-2804	209	79	only	only	ADV
iajs-2804	209	80	if	if	SCONJ
iajs-2804	209	81	∀	∀	X
iajs-2804	209	82	ℱ-sub	ℱ-sub	NOUN
iajs-2804	209	83	-	-	NOUN
iajs-2804	209	84	module	module	NOUN
iajs-2804	209	85	s	s	NOUN
iajs-2804	209	86	of	of	ADP
iajs-2804	209	87	x	x	X
iajs-2804	209	88	and	and	CCONJ
iajs-2804	209	89	every	every	DET
iajs-2804	209	90	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	209	91	𝑟𝑏	𝑟𝑏	NOUN
iajs-2804	209	92	of	of	ADP
iajs-2804	209	93	ℛ	ℛ	PROPN
iajs-2804	209	94	with	with	ADP
iajs-2804	209	95	(	(	PUNCT
iajs-2804	209	96	𝑟𝑏)𝑛	𝑟𝑏)𝑛	PROPN
iajs-2804	209	97	s	s	PART
iajs-2804	209	98	⊆u	⊆u	VERB
iajs-2804	209	99	implies	imply	VERB
iajs-2804	209	100	that	that	SCONJ
iajs-2804	209	101	𝑟𝑏s	𝑟𝑏s	VERB
iajs-2804	209	102	⊆	⊆	NUM
iajs-2804	209	103	𝑈	𝑈	PROPN
iajs-2804	209	104	+	+	CCONJ
iajs-2804	209	105	ℱ	ℱ	PROPN
iajs-2804	209	106	−	−	PROPN
iajs-2804	209	107	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	209	108	)	)	PUNCT
iajs-2804	209	109	.	.	PUNCT
iajs-2804	210	1	proof	proof	NOUN
iajs-2804	210	2	:	:	PUNCT
iajs-2804	210	3	it	it	PRON
iajs-2804	210	4	is	be	AUX
iajs-2804	210	5	clear	clear	ADJ
iajs-2804	210	6	from	from	ADP
iajs-2804	210	7	(	(	PUNCT
iajs-2804	210	8	proposition	proposition	NOUN
iajs-2804	210	9	2.11	2.11	NUM
iajs-2804	210	10	)	)	PUNCT
iajs-2804	210	11	.	.	PUNCT
iajs-2804	211	1	corollary	corollary	NOUN
iajs-2804	211	2	2.13	2.13	NUM
iajs-2804	211	3	let	let	VERB
iajs-2804	211	4	l	l	NOUN
iajs-2804	211	5	be	be	AUX
iajs-2804	211	6	an	an	DET
iajs-2804	211	7	ℱideal	ℱideal	PROPN
iajs-2804	211	8	of	of	ADP
iajs-2804	211	9	ℛ	ℛ	PROPN
iajs-2804	211	10	,	,	PUNCT
iajs-2804	211	11	then	then	ADV
iajs-2804	211	12	l	l	NOUN
iajs-2804	211	13	is	be	AUX
iajs-2804	211	14	an	an	DET
iajs-2804	211	15	ℱ-soc	ℱ-soc	NOUN
iajs-2804	211	16	-	-	PUNCT
iajs-2804	211	17	semi	semi	ADJ
iajs-2804	211	18	-	-	ADJ
iajs-2804	211	19	prime	prime	ADJ
iajs-2804	211	20	ideal	ideal	NOUN
iajs-2804	211	21	of	of	ADP
iajs-2804	211	22	ℛ	ℛ	PROPN
iajs-2804	211	23	if	if	NOUN
iajs-2804	211	24	and	and	CCONJ
iajs-2804	211	25	only	only	ADV
iajs-2804	211	26	if	if	SCONJ
iajs-2804	211	27	∀	∀	X
iajs-2804	211	28	ℱ-sub	ℱ-sub	X
iajs-2804	211	29	ideal	ideal	ADJ
iajs-2804	211	30	j	j	PROPN
iajs-2804	211	31	of	of	ADP
iajs-2804	211	32	ℛ	ℛ	PROPN
iajs-2804	211	33	and	and	CCONJ
iajs-2804	211	34	every	every	DET
iajs-2804	211	35	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	211	36	𝑟𝑏	𝑟𝑏	NOUN
iajs-2804	211	37	of	of	ADP
iajs-2804	211	38	ℛ	ℛ	PROPN
iajs-2804	211	39	with	with	ADP
iajs-2804	211	40	(	(	PUNCT
iajs-2804	211	41	𝑟𝑏)𝑛	𝑟𝑏)𝑛	PROPN
iajs-2804	211	42	j	j	PROPN
iajs-2804	211	43	⊆l	⊆l	NOUN
iajs-2804	211	44	implies	imply	VERB
iajs-2804	211	45	that	that	SCONJ
iajs-2804	211	46	𝑟𝑏j	𝑟𝑏j	VERB
iajs-2804	211	47	⊆	⊆	NUM
iajs-2804	211	48	𝐿	𝐿	PROPN
iajs-2804	211	49	+	+	CCONJ
iajs-2804	211	50	ℱ	ℱ	PROPN
iajs-2804	211	51	−	−	PROPN
iajs-2804	211	52	𝑆𝑜𝑐(ℛ	𝑆𝑜𝑐(ℛ	NOUN
iajs-2804	211	53	)	)	PUNCT
iajs-2804	211	54	.	.	PUNCT
iajs-2804	212	1	proof	proof	NOUN
iajs-2804	212	2	:	:	PUNCT
iajs-2804	212	3	clearly	clearly	ADV
iajs-2804	212	4	from	from	ADP
iajs-2804	212	5	(	(	PUNCT
iajs-2804	212	6	proposition	proposition	NOUN
iajs-2804	212	7	2.11	2.11	NUM
iajs-2804	212	8	)	)	PUNCT
iajs-2804	212	9	.	.	PUNCT
iajs-2804	213	1	ibn	ibn	PROPN
iajs-2804	213	2	al	al	PROPN
iajs-2804	213	3	-	-	PUNCT
iajs-2804	213	4	haitham	haitham	PROPN
iajs-2804	213	5	jour	jour	X
iajs-2804	213	6	.	.	PROPN
iajs-2804	213	7	for	for	ADP
iajs-2804	213	8	pure	pure	ADJ
iajs-2804	213	9	&	&	CCONJ
iajs-2804	213	10	appl	appl	PROPN
iajs-2804	213	11	.	.	PUNCT
iajs-2804	214	1	sci	sci	PROPN
iajs-2804	214	2	.	.	PROPN
iajs-2804	215	1	53	53	NUM
iajs-2804	215	2	(	(	PUNCT
iajs-2804	215	3	1)2022	1)2022	NUM
iajs-2804	215	4	112	112	NUM
iajs-2804	215	5	proposition	proposition	NOUN
iajs-2804	215	6	2.14	2.14	NUM
iajs-2804	215	7	:	:	PUNCT
iajs-2804	215	8	if	if	SCONJ
iajs-2804	215	9	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	215	10	𝑛	𝑛	PRON
iajs-2804	215	11	𝑈	𝑈	PROPN
iajs-2804	215	12	ℱ-soc	ℱ-soc	PROPN
iajs-2804	215	13	-	-	PUNCT
iajs-2804	215	14	semi	semi	ADJ
iajs-2804	215	15	-	-	ADJ
iajs-2804	215	16	prime	prime	ADJ
iajs-2804	215	17	sub	sub	NOUN
iajs-2804	216	1	-	-	NOUN
iajs-2804	216	2	module	module	NOUN
iajs-2804	216	3	of	of	ADP
iajs-2804	216	4	cancellative	cancellative	ADJ
iajs-2804	216	5	ℱ-module	ℱ-module	PROPN
iajs-2804	216	6	x.	x.	NOUN
iajs-2804	216	7	where	where	SCONJ
iajs-2804	216	8	u	u	NOUN
iajs-2804	216	9	is	be	AUX
iajs-2804	216	10	an	an	DET
iajs-2804	216	11	ℱ-submodules	ℱ-submodules	PROPN
iajs-2804	216	12	of	of	ADP
iajs-2804	216	13	x	x	PUNCT
iajs-2804	216	14	and	and	CCONJ
iajs-2804	216	15	𝑟𝑏	𝑟𝑏	PROPN
iajs-2804	216	16	is	be	AUX
iajs-2804	216	17	an	an	DET
iajs-2804	216	18	idempotent	idempotent	NOUN
iajs-2804	216	19	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	216	20	of	of	ADP
iajs-2804	216	21	r.	r.	PROPN
iajs-2804	216	22	then	then	ADV
iajs-2804	216	23	𝑈	𝑈	PROPN
iajs-2804	216	24	⊆	⊆	NUM
iajs-2804	216	25	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	216	26	𝑛−1	𝑛−1	PROPN
iajs-2804	216	27	𝑈	𝑈	PROPN
iajs-2804	217	1	+	+	CCONJ
iajs-2804	217	2	𝐹	𝐹	PROPN
iajs-2804	217	3	−	−	PROPN
iajs-2804	217	4	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	217	5	)	)	PUNCT
iajs-2804	217	6	proof	proof	NOUN
iajs-2804	217	7	:	:	PUNCT
iajs-2804	217	8	let	let	VERB
iajs-2804	217	9	𝑎𝑡	𝑎𝑡	PRON
iajs-2804	217	10	⊆	⊆	NUM
iajs-2804	217	11	𝑈	𝑈	PROPN
iajs-2804	217	12	this	this	PRON
iajs-2804	217	13	implies	imply	VERB
iajs-2804	217	14	𝑟𝑏	𝑟𝑏	PART
iajs-2804	217	15	𝑛𝑎𝑡	𝑛𝑎𝑡	VERB
iajs-2804	217	16	⊆	⊆	NUM
iajs-2804	217	17	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	217	18	𝑛𝑈	𝑛𝑈	NOUN
iajs-2804	217	19	,	,	PUNCT
iajs-2804	217	20	for	for	SCONJ
iajs-2804	217	21	𝑟𝑏	𝑟𝑏	NUM
iajs-2804	217	22	is	be	VERB
iajs-2804	217	23	an	an	DET
iajs-2804	217	24	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	217	25	of	of	ADP
iajs-2804	217	26	r.	r.	PROPN
iajs-2804	217	27	but	but	CCONJ
iajs-2804	217	28	,	,	PUNCT
iajs-2804	217	29	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	217	30	𝑛𝑈	𝑛𝑈	NOUN
iajs-2804	217	31	is	be	AUX
iajs-2804	217	32	an	an	DET
iajs-2804	217	33	ℱsoc	ℱsoc	PROPN
iajs-2804	217	34	-	-	PUNCT
iajs-2804	217	35	semi	semi	ADJ
iajs-2804	217	36	-	-	ADJ
iajs-2804	217	37	prime	prime	ADJ
iajs-2804	217	38	sub	sub	NOUN
iajs-2804	217	39	-	-	NOUN
iajs-2804	217	40	module	module	NOUN
iajs-2804	217	41	of	of	ADP
iajs-2804	217	42	x	x	PUNCT
iajs-2804	217	43	with	with	ADP
iajs-2804	217	44	𝑎𝑡	𝑎𝑡	PRON
iajs-2804	217	45	⊆	⊆	NUM
iajs-2804	217	46	𝑋,where	𝑋,where	X
iajs-2804	217	47	𝑡	𝑡	NOUN
iajs-2804	217	48	,	,	PUNCT
iajs-2804	217	49	𝑏	𝑏	PROPN
iajs-2804	217	50	∈	∈	PROPN
iajs-2804	218	1	[	[	X
iajs-2804	218	2	0,1	0,1	NUM
iajs-2804	218	3	]	]	PUNCT
iajs-2804	218	4	.	.	PUNCT
iajs-2804	219	1	therefore	therefore	ADV
iajs-2804	219	2	𝑟𝑏𝑎𝑡	𝑟𝑏𝑎𝑡	NOUN
iajs-2804	220	1	⊆	⊆	NUM
iajs-2804	220	2	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	220	3	𝑛𝑈	𝑛𝑈	NOUN
iajs-2804	220	4	+	+	CCONJ
iajs-2804	220	5	𝐹	𝐹	PROPN
iajs-2804	220	6	−	−	PROPN
iajs-2804	220	7	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	220	8	)	)	PUNCT
iajs-2804	220	9	,	,	PUNCT
iajs-2804	220	10	that	that	PRON
iajs-2804	220	11	is	be	AUX
iajs-2804	220	12	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	220	13	2	2	NUM
iajs-2804	220	14	𝑎𝑡	𝑎𝑡	ADP
iajs-2804	220	15	⊆	⊆	NUM
iajs-2804	220	16	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	220	17	𝑛+1	𝑛+1	PROPN
iajs-2804	220	18	𝑈	𝑈	PROPN
iajs-2804	221	1	+	+	CCONJ
iajs-2804	221	2	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	221	3	𝐹	𝐹	PROPN
iajs-2804	221	4	−	−	PROPN
iajs-2804	221	5	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	221	6	)	)	PUNCT
iajs-2804	221	7	,	,	PUNCT
iajs-2804	221	8	thus	thus	ADV
iajs-2804	221	9	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	221	10	2𝑎𝑡	2𝑎𝑡	NOUN
iajs-2804	221	11	⊆	⊆	NUM
iajs-2804	221	12	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	221	13	2𝑟𝑏	2𝑟𝑏	ADJ
iajs-2804	221	14	𝑛−1𝑈	𝑛−1𝑈	NOUN
iajs-2804	221	15	+	+	CCONJ
iajs-2804	221	16	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	221	17	𝐹	𝐹	PRON
iajs-2804	221	18	−	−	PROPN
iajs-2804	221	19	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	PROPN
iajs-2804	221	20	)	)	PUNCT
iajs-2804	221	21	)	)	PUNCT
iajs-2804	221	22	,	,	PUNCT
iajs-2804	221	23	but	but	CCONJ
iajs-2804	221	24	𝑟𝑏	𝑟𝑏	PRON
iajs-2804	221	25	is	be	AUX
iajs-2804	221	26	an	an	DET
iajs-2804	221	27	idempotent	idempotent	NOUN
iajs-2804	221	28	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	221	29	of	of	ADP
iajs-2804	221	30	r.	r.	PROPN
iajs-2804	222	1	so	so	ADV
iajs-2804	222	2	we	we	PRON
iajs-2804	222	3	get	get	VERB
iajs-2804	222	4	𝑟𝑏𝑎𝑡	𝑟𝑏𝑎𝑡	NOUN
iajs-2804	222	5	⊆	⊆	NUM
iajs-2804	222	6	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	222	7	𝑛𝑈	𝑛𝑈	NOUN
iajs-2804	222	8	+	+	CCONJ
iajs-2804	222	9	𝑟𝑏𝐹	𝑟𝑏𝐹	PROPN
iajs-2804	222	10	−	−	ADP
iajs-2804	222	11	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	222	12	)	)	PUNCT
iajs-2804	222	13	.	.	PUNCT
iajs-2804	223	1	but	but	CCONJ
iajs-2804	223	2	,	,	PUNCT
iajs-2804	223	3	x	x	X
iajs-2804	223	4	is	be	AUX
iajs-2804	223	5	a	a	DET
iajs-2804	223	6	cancellative	cancellative	ADJ
iajs-2804	223	7	ℱ-module	ℱ-module	PROPN
iajs-2804	223	8	,	,	PUNCT
iajs-2804	223	9	we	we	PRON
iajs-2804	223	10	have	have	VERB
iajs-2804	223	11	𝑎𝑡	𝑎𝑡	PRON
iajs-2804	223	12	⊆	⊆	NUM
iajs-2804	223	13	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	223	14	𝑛−1	𝑛−1	PROPN
iajs-2804	223	15	𝑈	𝑈	PROPN
iajs-2804	224	1	+	+	CCONJ
iajs-2804	224	2	𝐹	𝐹	PROPN
iajs-2804	224	3	−	−	PROPN
iajs-2804	224	4	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	224	5	)	)	PUNCT
iajs-2804	224	6	,	,	PUNCT
iajs-2804	224	7	that	that	PRON
iajs-2804	224	8	is	be	AUX
iajs-2804	224	9	mean	mean	VERB
iajs-2804	224	10	𝑈	𝑈	PROPN
iajs-2804	224	11	⊆	⊆	NUM
iajs-2804	224	12	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	224	13	𝑛−1	𝑛−1	PROPN
iajs-2804	224	14	𝑈	𝑈	PROPN
iajs-2804	225	1	+	+	CCONJ
iajs-2804	225	2	𝐹	𝐹	PROPN
iajs-2804	225	3	−	−	PROPN
iajs-2804	225	4	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	225	5	)	)	PUNCT
iajs-2804	225	6	.	.	PUNCT
iajs-2804	226	1	remark	remark	VERB
iajs-2804	226	2	2.15	2.15	NUM
iajs-2804	226	3	every	every	DET
iajs-2804	226	4	ℱ-semi	ℱ-semi	PROPN
iajs-2804	226	5	-	-	PUNCT
iajs-2804	226	6	prime	prime	ADJ
iajs-2804	226	7	sub	sub	NOUN
iajs-2804	226	8	-	-	NOUN
iajs-2804	226	9	module	module	NOUN
iajs-2804	226	10	is	be	AUX
iajs-2804	226	11	an	an	DET
iajs-2804	226	12	ℱ-soc	ℱ-soc	NOUN
iajs-2804	226	13	-	-	PUNCT
iajs-2804	226	14	semi	semi	ADJ
iajs-2804	226	15	-	-	ADJ
iajs-2804	226	16	prime	prime	ADJ
iajs-2804	226	17	sub	sub	NOUN
iajs-2804	226	18	-	-	NOUN
iajs-2804	226	19	module	module	NOUN
iajs-2804	226	20	.	.	PUNCT
iajs-2804	227	1	proof	proof	NOUN
iajs-2804	227	2	:	:	PUNCT
iajs-2804	227	3	it	it	PRON
iajs-2804	227	4	is	be	AUX
iajs-2804	227	5	clear	clear	ADJ
iajs-2804	227	6	by	by	ADP
iajs-2804	227	7	definition	definition	NOUN
iajs-2804	227	8	of	of	ADP
iajs-2804	227	9	ℱ-semi	ℱ-semi	PROPN
iajs-2804	227	10	-	-	ADJ
iajs-2804	227	11	prime	prime	ADJ
iajs-2804	227	12	sub	sub	NOUN
iajs-2804	227	13	-	-	NOUN
iajs-2804	227	14	module	module	NOUN
iajs-2804	227	15	.	.	PUNCT
iajs-2804	228	1	remark	remark	VERB
iajs-2804	228	2	2.16	2.16	NUM
iajs-2804	228	3	if	if	SCONJ
iajs-2804	228	4	u	u	NOUN
iajs-2804	228	5	is	be	AUX
iajs-2804	228	6	an	an	DET
iajs-2804	228	7	ℱ-soc	ℱ-soc	NOUN
iajs-2804	228	8	-	-	PUNCT
iajs-2804	228	9	semi	semi	ADJ
iajs-2804	228	10	-	-	ADJ
iajs-2804	228	11	prime	prime	ADJ
iajs-2804	228	12	sub	sub	NOUN
iajs-2804	228	13	-	-	NOUN
iajs-2804	228	14	module	module	NOUN
iajs-2804	228	15	of	of	ADP
iajs-2804	228	16	ℱ-module	ℱ-module	PROPN
iajs-2804	228	17	x	x	NOUN
iajs-2804	228	18	,	,	PUNCT
iajs-2804	228	19	with	with	ADP
iajs-2804	228	20	ℱ	ℱ	PROPN
iajs-2804	228	21	−	−	ADP
iajs-2804	228	22	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	228	23	)	)	PUNCT
iajs-2804	228	24	⊆	⊆	NUM
iajs-2804	228	25	𝑈.	𝑈.	PROPN
iajs-2804	228	26	then	then	ADV
iajs-2804	228	27	u	u	NOUN
iajs-2804	228	28	is	be	AUX
iajs-2804	228	29	an	an	DET
iajs-2804	228	30	ℱ-semi	ℱ-semi	PROPN
iajs-2804	228	31	-	-	PUNCT
iajs-2804	228	32	prime	prime	ADJ
iajs-2804	228	33	sub	sub	NOUN
iajs-2804	228	34	-	-	NOUN
iajs-2804	228	35	module	module	NOUN
iajs-2804	228	36	.	.	PUNCT
iajs-2804	229	1	proof	proof	NOUN
iajs-2804	229	2	:	:	PUNCT
iajs-2804	229	3	assume	assume	VERB
iajs-2804	229	4	that	that	SCONJ
iajs-2804	229	5	u	u	NOUN
iajs-2804	229	6	is	be	AUX
iajs-2804	229	7	an	an	DET
iajs-2804	229	8	ℱ-soc	ℱ-soc	NOUN
iajs-2804	229	9	-	-	PUNCT
iajs-2804	229	10	semi	semi	ADJ
iajs-2804	229	11	-	-	ADJ
iajs-2804	229	12	prime	prime	ADJ
iajs-2804	229	13	sub	sub	NOUN
iajs-2804	229	14	-	-	NOUN
iajs-2804	229	15	module	module	NOUN
iajs-2804	229	16	of	of	ADP
iajs-2804	229	17	an	an	DET
iajs-2804	229	18	ℱ-module	ℱ-module	PROPN
iajs-2804	229	19	x	x	PROPN
iajs-2804	229	20	of	of	ADP
iajs-2804	229	21	an	an	DET
iajs-2804	229	22	ℛ-module	ℛ-module	PROPN
iajs-2804	229	23	m.	m.	NOUN
iajs-2804	229	24	let	let	VERB
iajs-2804	229	25	(	(	PUNCT
iajs-2804	229	26	𝑟𝑛)𝑏𝑚𝑡	𝑟𝑛)𝑏𝑚𝑡	X
iajs-2804	229	27	=	=	PUNCT
iajs-2804	229	28	(	(	PUNCT
iajs-2804	229	29	𝑟𝑏)𝑛𝑚𝑡	𝑟𝑏)𝑛𝑚𝑡	NOUN
iajs-2804	229	30	⊆	⊆	NUM
iajs-2804	229	31	𝑈	𝑈	PROPN
iajs-2804	229	32	,	,	PUNCT
iajs-2804	229	33	for	for	ADP
iajs-2804	229	34	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	229	35	⊆	⊆	NUM
iajs-2804	229	36	ℛ	ℛ	PROPN
iajs-2804	229	37	,	,	PUNCT
iajs-2804	229	38	𝑚𝑡	𝑚𝑡	ADP
iajs-2804	229	39	⊆	⊆	NUM
iajs-2804	229	40	𝑋,where	𝑋,where	X
iajs-2804	229	41	𝑡	𝑡	NOUN
iajs-2804	229	42	,	,	PUNCT
iajs-2804	229	43	𝑏	𝑏	PROPN
iajs-2804	229	44	∈	∈	PROPN
iajs-2804	230	1	[	[	X
iajs-2804	230	2	0,1	0,1	NUM
iajs-2804	230	3	]	]	PUNCT
iajs-2804	230	4	.	.	PUNCT
iajs-2804	231	1	since	since	SCONJ
iajs-2804	231	2	u	u	NOUN
iajs-2804	231	3	is	be	AUX
iajs-2804	231	4	an	an	DET
iajs-2804	231	5	ℱsoc	ℱsoc	PROPN
iajs-2804	231	6	-	-	PUNCT
iajs-2804	231	7	semi	semi	ADJ
iajs-2804	231	8	-	-	ADJ
iajs-2804	231	9	prime	prime	ADJ
iajs-2804	231	10	sub	sub	NOUN
iajs-2804	231	11	-	-	NOUN
iajs-2804	231	12	module	module	NOUN
iajs-2804	231	13	,	,	PUNCT
iajs-2804	231	14	then	then	ADV
iajs-2804	231	15	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	VERB
iajs-2804	231	16	⊆	⊆	NUM
iajs-2804	231	17	𝑈	𝑈	PROPN
iajs-2804	231	18	+	+	CCONJ
iajs-2804	231	19	ℱ	ℱ	PROPN
iajs-2804	231	20	−	−	PROPN
iajs-2804	231	21	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	231	22	)	)	PUNCT
iajs-2804	231	23	⊆	⊆	NUM
iajs-2804	231	24	𝑈	𝑈	PROPN
iajs-2804	231	25	but	but	CCONJ
iajs-2804	231	26	ℱ	ℱ	PROPN
iajs-2804	231	27	−	−	PROPN
iajs-2804	231	28	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	231	29	)	)	PUNCT
iajs-2804	231	30	⊆	⊆	NUM
iajs-2804	231	31	𝑈.	𝑈.	PROPN
iajs-2804	231	32	hence	hence	ADV
iajs-2804	231	33	u	u	NOUN
iajs-2804	231	34	is	be	AUX
iajs-2804	231	35	an	an	DET
iajs-2804	231	36	ℱ-semi	ℱ-semi	PROPN
iajs-2804	231	37	-	-	PUNCT
iajs-2804	231	38	prime	prime	ADJ
iajs-2804	231	39	sub	sub	NOUN
iajs-2804	231	40	-	-	NOUN
iajs-2804	231	41	module	module	NOUN
iajs-2804	231	42	.	.	PUNCT
iajs-2804	232	1	corollary	corollary	ADJ
iajs-2804	232	2	2.17	2.17	NUM
iajs-2804	232	3	if	if	SCONJ
iajs-2804	232	4	u	u	NOUN
iajs-2804	232	5	is	be	AUX
iajs-2804	232	6	an	an	DET
iajs-2804	232	7	ℱ-soc	ℱ-soc	NOUN
iajs-2804	232	8	-	-	PUNCT
iajs-2804	232	9	semi	semi	ADJ
iajs-2804	232	10	-	-	ADJ
iajs-2804	232	11	prime	prime	ADJ
iajs-2804	232	12	sub	sub	NOUN
iajs-2804	232	13	-	-	NOUN
iajs-2804	232	14	module	module	NOUN
iajs-2804	232	15	of	of	ADP
iajs-2804	232	16	ℱ-module	ℱ-module	PROPN
iajs-2804	232	17	x	x	NOUN
iajs-2804	232	18	,	,	PUNCT
iajs-2804	232	19	with	with	SCONJ
iajs-2804	232	20	u	u	NOUN
iajs-2804	232	21	be	be	VERB
iajs-2804	232	22	an	an	DET
iajs-2804	232	23	ℱ-essential	ℱ-essential	PROPN
iajs-2804	232	24	submodule	submodule	NOUN
iajs-2804	232	25	of	of	ADP
iajs-2804	232	26	x.	x.	NOUN
iajs-2804	232	27	then	then	ADV
iajs-2804	232	28	u	u	NOUN
iajs-2804	232	29	is	be	AUX
iajs-2804	232	30	an	an	DET
iajs-2804	232	31	ℱ-semi	ℱ-semi	PROPN
iajs-2804	232	32	-	-	PUNCT
iajs-2804	232	33	prime	prime	ADJ
iajs-2804	232	34	sub	sub	NOUN
iajs-2804	232	35	-	-	NOUN
iajs-2804	232	36	module	module	NOUN
iajs-2804	232	37	.	.	PUNCT
iajs-2804	233	1	proof	proof	NOUN
iajs-2804	233	2	:	:	PUNCT
iajs-2804	233	3	since	since	SCONJ
iajs-2804	233	4	u	u	PRON
iajs-2804	233	5	be	be	VERB
iajs-2804	233	6	an	an	DET
iajs-2804	233	7	ℱ-essential	ℱ-essential	ADJ
iajs-2804	233	8	sub	sub	NOUN
iajs-2804	233	9	-	-	NOUN
iajs-2804	233	10	module	module	NOUN
iajs-2804	233	11	of	of	ADP
iajs-2804	233	12	x	x	NOUN
iajs-2804	233	13	,	,	PUNCT
iajs-2804	233	14	then	then	ADV
iajs-2804	233	15	by	by	ADP
iajs-2804	233	16	definition	definition	NOUN
iajs-2804	233	17	of	of	ADP
iajs-2804	233	18	ℱ-socle	ℱ-socle	PROPN
iajs-2804	233	19	we	we	PRON
iajs-2804	233	20	have	have	VERB
iajs-2804	233	21	ℱ	ℱ	PROPN
iajs-2804	233	22	−	−	ADP
iajs-2804	233	23	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	233	24	)	)	PUNCT
iajs-2804	233	25	⊆	⊆	NUM
iajs-2804	233	26	𝑈	𝑈	PROPN
iajs-2804	233	27	and	and	CCONJ
iajs-2804	233	28	by	by	ADP
iajs-2804	233	29	(	(	PUNCT
iajs-2804	233	30	remark	remark	NOUN
iajs-2804	233	31	2.16	2.16	NUM
iajs-2804	233	32	)	)	PUNCT
iajs-2804	233	33	that	that	PRON
iajs-2804	233	34	is	be	AUX
iajs-2804	233	35	complete	complete	ADJ
iajs-2804	233	36	the	the	DET
iajs-2804	233	37	proof	proof	NOUN
iajs-2804	233	38	.	.	PUNCT
iajs-2804	234	1	corollary	corollary	ADJ
iajs-2804	234	2	2.18	2.18	NUM
iajs-2804	234	3	if	if	SCONJ
iajs-2804	234	4	u	u	NOUN
iajs-2804	234	5	is	be	AUX
iajs-2804	234	6	an	an	DET
iajs-2804	234	7	ℱ-sub	ℱ-sub	NOUN
iajs-2804	234	8	-	-	NOUN
iajs-2804	234	9	module	module	NOUN
iajs-2804	234	10	of	of	ADP
iajs-2804	234	11	ℱ-module	ℱ-module	PROPN
iajs-2804	234	12	x	x	NOUN
iajs-2804	234	13	,	,	PUNCT
iajs-2804	234	14	with	with	ADP
iajs-2804	234	15	ℱ	ℱ	PROPN
iajs-2804	234	16	−	−	ADP
iajs-2804	234	17	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	234	18	)	)	PUNCT
iajs-2804	234	19	⊆	⊆	NUM
iajs-2804	234	20	𝑈.	𝑈.	PROPN
iajs-2804	234	21	then	then	ADV
iajs-2804	234	22	u	u	NOUN
iajs-2804	234	23	is	be	AUX
iajs-2804	234	24	an	an	DET
iajs-2804	234	25	ℱ-semiprime	ℱ-semiprime	PROPN
iajs-2804	234	26	sub	sub	NOUN
iajs-2804	234	27	-	-	NOUN
iajs-2804	234	28	module	module	NOUN
iajs-2804	234	29	of	of	ADP
iajs-2804	234	30	x	x	PRON
iajs-2804	234	31	if	if	SCONJ
iajs-2804	234	32	and	and	CCONJ
iajs-2804	234	33	only	only	ADV
iajs-2804	234	34	if	if	SCONJ
iajs-2804	234	35	u	u	NOUN
iajs-2804	234	36	is	be	AUX
iajs-2804	234	37	an	an	DET
iajs-2804	234	38	ℱ-soc	ℱ-soc	NOUN
iajs-2804	234	39	-	-	PUNCT
iajs-2804	234	40	prime	prime	ADJ
iajs-2804	234	41	sub	sub	NOUN
iajs-2804	234	42	-	-	NOUN
iajs-2804	234	43	module	module	NOUN
iajs-2804	234	44	of	of	ADP
iajs-2804	234	45	x.	x.	NOUN
iajs-2804	234	46	proof	proof	NOUN
iajs-2804	234	47	:	:	PUNCT
iajs-2804	234	48	consequently	consequently	ADV
iajs-2804	234	49	from	from	ADP
iajs-2804	234	50	(	(	PUNCT
iajs-2804	234	51	remark	remark	NOUN
iajs-2804	234	52	2.7	2.7	NUM
iajs-2804	234	53	)	)	PUNCT
iajs-2804	234	54	and	and	CCONJ
iajs-2804	234	55	(	(	PUNCT
iajs-2804	234	56	remark	remark	NOUN
iajs-2804	234	57	2.16	2.16	NUM
iajs-2804	234	58	)	)	PUNCT
iajs-2804	234	59	.	.	PUNCT
iajs-2804	235	1	ibn	ibn	PROPN
iajs-2804	235	2	al	al	PROPN
iajs-2804	235	3	-	-	PUNCT
iajs-2804	235	4	haitham	haitham	PROPN
iajs-2804	235	5	jour	jour	X
iajs-2804	235	6	.	.	PROPN
iajs-2804	235	7	for	for	ADP
iajs-2804	235	8	pure	pure	ADJ
iajs-2804	235	9	&	&	CCONJ
iajs-2804	235	10	appl	appl	PROPN
iajs-2804	235	11	.	.	PUNCT
iajs-2804	236	1	sci	sci	PROPN
iajs-2804	236	2	.	.	PROPN
iajs-2804	237	1	53	53	NUM
iajs-2804	237	2	(	(	PUNCT
iajs-2804	237	3	1)2022	1)2022	NOUN
iajs-2804	237	4	113	113	NUM
iajs-2804	237	5	remark	remark	NOUN
iajs-2804	237	6	2.19	2.19	NUM
iajs-2804	237	7	let	let	VERB
iajs-2804	237	8	u	u	PRON
iajs-2804	237	9	and	and	CCONJ
iajs-2804	237	10	v	v	NOUN
iajs-2804	237	11	are	be	AUX
iajs-2804	237	12	ℱ-sub	ℱ-sub	NOUN
iajs-2804	237	13	-	-	NOUN
iajs-2804	237	14	modules	module	NOUN
iajs-2804	237	15	of	of	ADP
iajs-2804	237	16	ℱ-module	ℱ-module	PROPN
iajs-2804	237	17	x.	x.	NOUN
iajs-2804	237	18	if	if	SCONJ
iajs-2804	237	19	u+v	u+v	NUM
iajs-2804	237	20	is	be	AUX
iajs-2804	237	21	an	an	DET
iajs-2804	237	22	ℱ-semi	ℱ-semi	PROPN
iajs-2804	237	23	-	-	PUNCT
iajs-2804	237	24	prime	prime	ADJ
iajs-2804	237	25	sub	sub	NOUN
iajs-2804	237	26	-	-	NOUN
iajs-2804	237	27	module	module	NOUN
iajs-2804	237	28	of	of	ADP
iajs-2804	237	29	x	x	PUNCT
iajs-2804	237	30	with	with	ADP
iajs-2804	237	31	v	v	NUM
iajs-2804	237	32	⊆	⊆	NUM
iajs-2804	237	33	ℱ	ℱ	PROPN
iajs-2804	237	34	−	−	PROPN
iajs-2804	237	35	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	237	36	)	)	PUNCT
iajs-2804	237	37	,	,	PUNCT
iajs-2804	237	38	then	then	ADV
iajs-2804	237	39	u	u	NOUN
iajs-2804	237	40	is	be	AUX
iajs-2804	237	41	an	an	DET
iajs-2804	237	42	ℱ-soc	ℱ-soc	NOUN
iajs-2804	237	43	-	-	PUNCT
iajs-2804	237	44	semi	semi	ADJ
iajs-2804	237	45	-	-	ADJ
iajs-2804	237	46	prime	prime	ADJ
iajs-2804	237	47	sub	sub	NOUN
iajs-2804	237	48	-	-	NOUN
iajs-2804	237	49	module	module	NOUN
iajs-2804	237	50	of	of	ADP
iajs-2804	237	51	x.	x.	NOUN
iajs-2804	237	52	proof	proof	NOUN
iajs-2804	237	53	:	:	PUNCT
iajs-2804	237	54	let	let	VERB
iajs-2804	237	55	(	(	PUNCT
iajs-2804	237	56	𝑟𝑛)𝑏𝑥𝑘	𝑟𝑛)𝑏𝑥𝑘	NUM
iajs-2804	237	57	=	=	SYM
iajs-2804	237	58	(	(	PUNCT
iajs-2804	237	59	𝑟𝑏)𝑛𝑥𝑘	𝑟𝑏)𝑛𝑥𝑘	PROPN
iajs-2804	237	60	⊆	⊆	NUM
iajs-2804	237	61	𝑈	𝑈	PROPN
iajs-2804	237	62	,	,	PUNCT
iajs-2804	237	63	for	for	ADP
iajs-2804	237	64	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	237	65	⊆	⊆	NUM
iajs-2804	237	66	ℛ	ℛ	PROPN
iajs-2804	237	67	,	,	PUNCT
iajs-2804	237	68	𝑥𝑘	𝑥𝑘	PRON
iajs-2804	237	69	⊆	⊆	NUM
iajs-2804	237	70	𝑋,where	𝑋,where	X
iajs-2804	237	71	𝑘	𝑘	NOUN
iajs-2804	237	72	,	,	PUNCT
iajs-2804	237	73	𝑏	𝑏	PROPN
iajs-2804	237	74	∈	∈	PROPN
iajs-2804	238	1	[	[	X
iajs-2804	238	2	0,1	0,1	NUM
iajs-2804	238	3	]	]	PUNCT
iajs-2804	238	4	.	.	PUNCT
iajs-2804	239	1	this	this	PRON
iajs-2804	239	2	implies	imply	VERB
iajs-2804	239	3	(	(	PUNCT
iajs-2804	239	4	𝑟𝑛)𝑏𝑥𝑘	𝑟𝑛)𝑏𝑥𝑘	NUM
iajs-2804	239	5	⊆	⊆	NUM
iajs-2804	239	6	u	u	NOUN
iajs-2804	239	7	+	+	X
iajs-2804	240	1	v.	v.	CCONJ
iajs-2804	241	1	but	but	CCONJ
iajs-2804	241	2	u+v	u+v	NUM
iajs-2804	241	3	is	be	AUX
iajs-2804	241	4	an	an	DET
iajs-2804	241	5	ℱ-semi	ℱ-semi	PROPN
iajs-2804	241	6	-	-	PUNCT
iajs-2804	241	7	prime	prime	ADJ
iajs-2804	241	8	sub	sub	NOUN
iajs-2804	241	9	-	-	NOUN
iajs-2804	241	10	module	module	NOUN
iajs-2804	241	11	of	of	ADP
iajs-2804	241	12	x	x	NOUN
iajs-2804	241	13	,	,	PUNCT
iajs-2804	241	14	hence	hence	ADV
iajs-2804	241	15	𝑟𝑏𝑥𝑘	𝑟𝑏𝑥𝑘	NOUN
iajs-2804	241	16	⊆	⊆	PROPN
iajs-2804	241	17	𝑈	𝑈	PROPN
iajs-2804	241	18	+	+	CCONJ
iajs-2804	242	1	𝑉	𝑉	PROPN
iajs-2804	242	2	⊆	⊆	NUM
iajs-2804	242	3	𝑈	𝑈	PROPN
iajs-2804	242	4	+	+	CCONJ
iajs-2804	242	5	ℱ	ℱ	PROPN
iajs-2804	242	6	−	−	PROPN
iajs-2804	242	7	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	242	8	)	)	PUNCT
iajs-2804	242	9	since	since	SCONJ
iajs-2804	242	10	v	v	NUM
iajs-2804	242	11	⊆	⊆	NUM
iajs-2804	242	12	ℱ	ℱ	PROPN
iajs-2804	242	13	−	−	PROPN
iajs-2804	242	14	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	242	15	)	)	PUNCT
iajs-2804	242	16	.	.	PUNCT
iajs-2804	243	1	that	that	PRON
iajs-2804	243	2	is	be	AUX
iajs-2804	243	3	u	u	NOUN
iajs-2804	243	4	is	be	AUX
iajs-2804	243	5	an	an	DET
iajs-2804	243	6	ℱ-soc	ℱ-soc	NOUN
iajs-2804	243	7	-	-	PUNCT
iajs-2804	243	8	semi	semi	ADJ
iajs-2804	243	9	-	-	ADJ
iajs-2804	243	10	prime	prime	ADJ
iajs-2804	243	11	sub	sub	NOUN
iajs-2804	243	12	-	-	NOUN
iajs-2804	243	13	module	module	NOUN
iajs-2804	243	14	of	of	ADP
iajs-2804	243	15	x.	x.	NOUN
iajs-2804	243	16	theorem	theorem	VERB
iajs-2804	243	17	2.20	2.20	NUM
iajs-2804	243	18	any	any	DET
iajs-2804	243	19	ℱ-sub	ℱ-sub	NOUN
iajs-2804	243	20	-	-	NOUN
iajs-2804	243	21	module	module	NOUN
iajs-2804	243	22	of	of	ADP
iajs-2804	243	23	semi	semi	ADJ
iajs-2804	243	24	-	-	ADJ
iajs-2804	243	25	simple	simple	ADJ
iajs-2804	243	26	ℱ-module	ℱ-module	PROPN
iajs-2804	243	27	x	x	PUNCT
iajs-2804	243	28	is	be	AUX
iajs-2804	243	29	an	an	DET
iajs-2804	243	30	ℱ-soc	ℱ-soc	NOUN
iajs-2804	243	31	-	-	PUNCT
iajs-2804	243	32	semi	semi	ADJ
iajs-2804	243	33	-	-	ADJ
iajs-2804	243	34	prime	prime	ADJ
iajs-2804	243	35	sub	sub	NOUN
iajs-2804	243	36	-	-	NOUN
iajs-2804	243	37	module	module	NOUN
iajs-2804	243	38	of	of	ADP
iajs-2804	243	39	x.	x.	NOUN
iajs-2804	243	40	proof	proof	NOUN
iajs-2804	243	41	:	:	PUNCT
iajs-2804	243	42	if	if	SCONJ
iajs-2804	243	43	u	u	NOUN
iajs-2804	243	44	is	be	AUX
iajs-2804	243	45	an	an	DET
iajs-2804	243	46	ℱ-sub	ℱ-sub	NOUN
iajs-2804	243	47	-	-	NOUN
iajs-2804	243	48	module	module	NOUN
iajs-2804	243	49	of	of	ADP
iajs-2804	243	50	an	an	DET
iajs-2804	243	51	ℱ-module	ℱ-module	PROPN
iajs-2804	243	52	x	x	PROPN
iajs-2804	243	53	of	of	ADP
iajs-2804	243	54	an	an	DET
iajs-2804	243	55	ℛ-module	ℛ-module	PROPN
iajs-2804	243	56	m.	m.	NOUN
iajs-2804	243	57	let	let	VERB
iajs-2804	243	58	(	(	PUNCT
iajs-2804	243	59	𝑟𝑛)𝑏𝑥𝑘	𝑟𝑛)𝑏𝑥𝑘	NUM
iajs-2804	243	60	=	=	SYM
iajs-2804	243	61	(	(	PUNCT
iajs-2804	243	62	𝑟𝑏)𝑛𝑥𝑘	𝑟𝑏)𝑛𝑥𝑘	PROPN
iajs-2804	243	63	⊆	⊆	NUM
iajs-2804	243	64	𝑈	𝑈	PROPN
iajs-2804	243	65	,	,	PUNCT
iajs-2804	243	66	for	for	ADP
iajs-2804	243	67	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	243	68	⊆	⊆	NUM
iajs-2804	243	69	ℛ	ℛ	PROPN
iajs-2804	243	70	,	,	PUNCT
iajs-2804	243	71	𝑥𝑘	𝑥𝑘	PRON
iajs-2804	243	72	⊆	⊆	NUM
iajs-2804	243	73	𝑋,where	𝑋,where	X
iajs-2804	243	74	𝑘	𝑘	NOUN
iajs-2804	243	75	,	,	PUNCT
iajs-2804	243	76	𝑏	𝑏	PROPN
iajs-2804	243	77	∈	∈	PROPN
iajs-2804	244	1	[	[	X
iajs-2804	244	2	0,1	0,1	NUM
iajs-2804	244	3	]	]	PUNCT
iajs-2804	244	4	.	.	PUNCT
iajs-2804	245	1	but	but	CCONJ
iajs-2804	245	2	,	,	PUNCT
iajs-2804	245	3	x	x	X
iajs-2804	245	4	is	be	AUX
iajs-2804	245	5	a	a	DET
iajs-2804	245	6	semi	semi	ADJ
iajs-2804	245	7	-	-	ADJ
iajs-2804	245	8	simple	simple	ADJ
iajs-2804	245	9	ℱ-module	ℱ-module	PROPN
iajs-2804	245	10	,	,	PUNCT
iajs-2804	245	11	thus	thus	ADV
iajs-2804	245	12	𝑋	𝑋	NOUN
iajs-2804	245	13	=	=	SYM
iajs-2804	245	14	ℱ	ℱ	PROPN
iajs-2804	245	15	−	−	NOUN
iajs-2804	245	16	𝑆𝑜𝑐(𝑋).we	𝑆𝑜𝑐(𝑋).we	NOUN
iajs-2804	245	17	have	have	VERB
iajs-2804	245	18	𝑥𝑘	𝑥𝑘	NUM
iajs-2804	245	19	⊆	⊆	NUM
iajs-2804	245	20	𝑋	𝑋	PROPN
iajs-2804	245	21	=	=	SYM
iajs-2804	245	22	ℱ	ℱ	PROPN
iajs-2804	245	23	−	−	PROPN
iajs-2804	245	24	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	245	25	)	)	PUNCT
iajs-2804	246	1	⊆	⊆	NUM
iajs-2804	246	2	𝑈	𝑈	PROPN
iajs-2804	246	3	+	+	CCONJ
iajs-2804	246	4	ℱ	ℱ	PROPN
iajs-2804	246	5	−	−	PROPN
iajs-2804	246	6	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	246	7	)	)	PUNCT
iajs-2804	246	8	,	,	PUNCT
iajs-2804	246	9	this	this	PRON
iajs-2804	246	10	implies	imply	VERB
iajs-2804	246	11	𝑟𝑏𝑥𝑘	𝑟𝑏𝑥𝑘	NOUN
iajs-2804	246	12	⊆	⊆	NUM
iajs-2804	246	13	𝑟𝑏𝑋	𝑟𝑏𝑋	NOUN
iajs-2804	246	14	=	=	SYM
iajs-2804	246	15	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	246	16	ℱ	ℱ	PROPN
iajs-2804	246	17	−	−	PROPN
iajs-2804	246	18	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	246	19	)	)	PUNCT
iajs-2804	246	20	⊆	⊆	NUM
iajs-2804	246	21	𝑟𝑏(𝑈	𝑟𝑏(𝑈	NOUN
iajs-2804	246	22	+	+	CCONJ
iajs-2804	246	23	ℱ	ℱ	PROPN
iajs-2804	246	24	−	−	PROPN
iajs-2804	246	25	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	246	26	)	)	PUNCT
iajs-2804	246	27	)	)	PUNCT
iajs-2804	247	1	⊆	⊆	X
iajs-2804	247	2	𝑈	𝑈	NOUN
iajs-2804	247	3	+	+	CCONJ
iajs-2804	247	4	ℱ	ℱ	PROPN
iajs-2804	247	5	−	−	PROPN
iajs-2804	247	6	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	247	7	)	)	PUNCT
iajs-2804	247	8	that	that	PRON
iajs-2804	247	9	is	be	AUX
iajs-2804	247	10	mean	mean	VERB
iajs-2804	247	11	u	u	NOUN
iajs-2804	247	12	is	be	AUX
iajs-2804	247	13	an	an	DET
iajs-2804	247	14	ℱ-socsemi	ℱ-socsemi	PROPN
iajs-2804	247	15	-	-	PUNCT
iajs-2804	247	16	prime	prime	ADJ
iajs-2804	247	17	sub	sub	NOUN
iajs-2804	247	18	-	-	NOUN
iajs-2804	247	19	module	module	NOUN
iajs-2804	247	20	of	of	ADP
iajs-2804	247	21	x.	x.	NOUN
iajs-2804	247	22	proposition	proposition	NOUN
iajs-2804	247	23	2.21	2.21	NUM
iajs-2804	247	24	:	:	PUNCT
iajs-2804	247	25	if	if	SCONJ
iajs-2804	247	26	u	u	NOUN
iajs-2804	247	27	is	be	AUX
iajs-2804	247	28	a	a	DET
iajs-2804	247	29	weakly	weakly	ADJ
iajs-2804	247	30	pure	pure	ADJ
iajs-2804	247	31	ℱ-sub	ℱ-sub	NOUN
iajs-2804	247	32	-	-	NOUN
iajs-2804	247	33	module	module	NOUN
iajs-2804	247	34	of	of	ADP
iajs-2804	247	35	ℱ-module	ℱ-module	PROPN
iajs-2804	247	36	x	x	PUNCT
iajs-2804	247	37	with	with	ADP
iajs-2804	247	38	(	(	PUNCT
iajs-2804	247	39	𝑟𝑛)𝑏u	𝑟𝑛)𝑏u	NOUN
iajs-2804	247	40	is	be	AUX
iajs-2804	247	41	an	an	DET
iajs-2804	247	42	ℱ-soc	ℱ-soc	NOUN
iajs-2804	247	43	-	-	PUNCT
iajs-2804	247	44	semi	semi	ADJ
iajs-2804	247	45	-	-	ADJ
iajs-2804	247	46	prime	prime	ADJ
iajs-2804	247	47	sub	sub	NOUN
iajs-2804	247	48	-	-	NOUN
iajs-2804	247	49	module	module	NOUN
iajs-2804	247	50	of	of	ADP
iajs-2804	247	51	x	x	PUNCT
iajs-2804	247	52	for	for	ADP
iajs-2804	247	53	every	every	DET
iajs-2804	247	54	non	non	ADJ
iajs-2804	247	55	-	-	ADJ
iajs-2804	247	56	empty	empty	ADJ
iajs-2804	247	57	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	247	58	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	247	59	of	of	ADP
iajs-2804	247	60	r	r	NOUN
iajs-2804	247	61	,	,	PUNCT
iajs-2804	247	62	then	then	ADV
iajs-2804	247	63	u	u	NOUN
iajs-2804	247	64	is	be	AUX
iajs-2804	247	65	an	an	DET
iajs-2804	247	66	ℱ-soc	ℱ-soc	NOUN
iajs-2804	247	67	-	-	PUNCT
iajs-2804	247	68	semi	semi	ADJ
iajs-2804	247	69	-	-	ADJ
iajs-2804	247	70	prime	prime	ADJ
iajs-2804	247	71	sub	sub	NOUN
iajs-2804	247	72	-	-	NOUN
iajs-2804	247	73	module	module	NOUN
iajs-2804	247	74	of	of	ADP
iajs-2804	247	75	x.	x.	NOUN
iajs-2804	247	76	proof	proof	NOUN
iajs-2804	247	77	:	:	PUNCT
iajs-2804	247	78	suppose	suppose	VERB
iajs-2804	247	79	that	that	SCONJ
iajs-2804	247	80	(	(	PUNCT
iajs-2804	247	81	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	247	82	⊆	⊆	NUM
iajs-2804	247	83	𝑈,with	𝑈,with	SYM
iajs-2804	247	84	𝑟𝑏	𝑟𝑏	PART
iajs-2804	247	85	is	be	AUX
iajs-2804	247	86	an	an	DET
iajs-2804	247	87	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	247	88	of	of	ADP
iajs-2804	247	89	r	r	NOUN
iajs-2804	247	90	and	and	CCONJ
iajs-2804	247	91	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	247	92	⊆	⊆	NUM
iajs-2804	247	93	𝑋,where	𝑋,where	X
iajs-2804	247	94	𝑡	𝑡	NOUN
iajs-2804	247	95	,	,	PUNCT
iajs-2804	247	96	𝑏	𝑏	PROPN
iajs-2804	247	97	∈	∈	PROPN
iajs-2804	248	1	[	[	X
iajs-2804	248	2	0,1].also	0,1].also	NOUN
iajs-2804	248	3	(	(	PUNCT
iajs-2804	248	4	𝑟𝑛)𝑏	𝑟𝑛)𝑏	NOUN
iajs-2804	248	5	𝑥𝑡	𝑥𝑡	ADP
iajs-2804	248	6	⊆	⊆	NUM
iajs-2804	248	7	(	(	PUNCT
iajs-2804	248	8	𝑟𝑛)𝑏𝑋	𝑟𝑛)𝑏𝑋	X
iajs-2804	248	9	this	this	PRON
iajs-2804	248	10	implies	imply	VERB
iajs-2804	248	11	(	(	PUNCT
iajs-2804	248	12	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	248	13	⊆	⊆	NUM
iajs-2804	248	14	𝑈	𝑈	PROPN
iajs-2804	248	15	∩	∩	NOUN
iajs-2804	248	16	(	(	PUNCT
iajs-2804	248	17	𝑟𝑛)𝑏𝑋	𝑟𝑛)𝑏𝑋	X
iajs-2804	248	18	=	=	SYM
iajs-2804	248	19	(	(	PUNCT
iajs-2804	248	20	𝑟𝑛)𝑏𝑈	𝑟𝑛)𝑏𝑈	NUM
iajs-2804	248	21	since	since	SCONJ
iajs-2804	248	22	u	u	NOUN
iajs-2804	248	23	is	be	AUX
iajs-2804	248	24	a	a	DET
iajs-2804	248	25	weakly	weakly	ADJ
iajs-2804	248	26	pure	pure	ADJ
iajs-2804	248	27	ℱ-sub	ℱ-sub	NOUN
iajs-2804	248	28	-	-	NOUN
iajs-2804	248	29	module	module	NOUN
iajs-2804	248	30	of	of	ADP
iajs-2804	248	31	x	x	PRON
iajs-2804	248	32	,	,	PUNCT
iajs-2804	248	33	but	but	CCONJ
iajs-2804	248	34	(	(	PUNCT
iajs-2804	248	35	𝑟𝑛)𝑏𝑈	𝑟𝑛)𝑏𝑈	NUM
iajs-2804	248	36	is	be	AUX
iajs-2804	248	37	an	an	DET
iajs-2804	248	38	ℱ-soc	ℱ-soc	NOUN
iajs-2804	248	39	-	-	PUNCT
iajs-2804	248	40	semi	semi	ADJ
iajs-2804	248	41	-	-	ADJ
iajs-2804	248	42	prime	prime	ADJ
iajs-2804	248	43	sub	sub	NOUN
iajs-2804	248	44	-	-	NOUN
iajs-2804	248	45	module	module	NOUN
iajs-2804	248	46	of	of	ADP
iajs-2804	248	47	x	x	NOUN
iajs-2804	248	48	,	,	PUNCT
iajs-2804	248	49	hence	hence	ADV
iajs-2804	248	50	𝑟𝑏𝑥𝑡	𝑟𝑏𝑥𝑡	NOUN
iajs-2804	248	51	⊆	⊆	NUM
iajs-2804	248	52	(	(	PUNCT
iajs-2804	248	53	𝑟𝑛)𝑏𝑈	𝑟𝑛)𝑏𝑈	NUM
iajs-2804	248	54	+	+	CCONJ
iajs-2804	248	55	𝐹	𝐹	PROPN
iajs-2804	248	56	−	−	PROPN
iajs-2804	248	57	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	248	58	)	)	PUNCT
iajs-2804	248	59	⊆	⊆	NUM
iajs-2804	248	60	𝑈	𝑈	PROPN
iajs-2804	248	61	+	+	NUM
iajs-2804	248	62	𝐹	𝐹	PROPN
iajs-2804	248	63	−	−	PROPN
iajs-2804	248	64	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	248	65	)	)	PUNCT
iajs-2804	248	66	.	.	PUNCT
iajs-2804	249	1	thus	thus	ADV
iajs-2804	249	2	u	u	PRON
iajs-2804	249	3	is	be	AUX
iajs-2804	249	4	an	an	DET
iajs-2804	249	5	ℱ-soc	ℱ-soc	NOUN
iajs-2804	249	6	-	-	PUNCT
iajs-2804	249	7	semi	semi	ADJ
iajs-2804	249	8	-	-	ADJ
iajs-2804	249	9	prime	prime	ADJ
iajs-2804	249	10	sub	sub	NOUN
iajs-2804	249	11	-	-	NOUN
iajs-2804	249	12	module	module	NOUN
iajs-2804	249	13	of	of	ADP
iajs-2804	249	14	x.	x.	PROPN
iajs-2804	249	15	lemma	lemma	PROPN
iajs-2804	249	16	2.22	2.22	NUM
iajs-2804	249	17	:	:	PUNCT
iajs-2804	249	18	(	(	PUNCT
iajs-2804	249	19	𝐴⨁𝐵	𝐴⨁𝐵	NOUN
iajs-2804	249	20	)	)	PUNCT
iajs-2804	249	21	+	+	CCONJ
iajs-2804	249	22	𝐹	𝐹	PROPN
iajs-2804	249	23	−	−	PROPN
iajs-2804	249	24	𝑆𝑜𝑐(𝑋⨁𝑌	𝑆𝑜𝑐(𝑋⨁𝑌	PROPN
iajs-2804	249	25	)	)	PUNCT
iajs-2804	249	26	=	=	PUNCT
iajs-2804	250	1	(	(	PUNCT
iajs-2804	250	2	𝐴	𝐴	PROPN
iajs-2804	250	3	+	+	CCONJ
iajs-2804	250	4	𝐹	𝐹	PROPN
iajs-2804	250	5	−	−	PROPN
iajs-2804	250	6	𝑆𝑜𝑐(𝑋))⨁(𝐵	𝑆𝑜𝑐(𝑋))⨁(𝐵	PRON
iajs-2804	250	7	+	+	CCONJ
iajs-2804	250	8	𝐹	𝐹	PROPN
iajs-2804	250	9	−	−	PROPN
iajs-2804	250	10	𝑆𝑜𝑐(𝑌	𝑆𝑜𝑐(𝑌	NUM
iajs-2804	250	11	)	)	PUNCT
iajs-2804	250	12	)	)	PUNCT
iajs-2804	251	1	for	for	ADP
iajs-2804	251	2	every	every	DET
iajs-2804	251	3	fuzzy	fuzzy	ADJ
iajs-2804	251	4	submodules	submodule	NOUN
iajs-2804	251	5	a	a	PRON
iajs-2804	251	6	and	and	CCONJ
iajs-2804	251	7	b	b	NOUN
iajs-2804	251	8	of	of	ADP
iajs-2804	251	9	fuzzy	fuzzy	ADJ
iajs-2804	251	10	modules	module	NOUN
iajs-2804	251	11	x	x	PUNCT
iajs-2804	251	12	and	and	CCONJ
iajs-2804	251	13	y	y	PROPN
iajs-2804	251	14	respectively	respectively	ADV
iajs-2804	251	15	.	.	PUNCT
iajs-2804	252	1	proof	proof	NOUN
iajs-2804	252	2	:	:	PUNCT
iajs-2804	252	3	from	from	ADP
iajs-2804	252	4	(	(	PUNCT
iajs-2804	252	5	lemma	lemma	PROPN
iajs-2804	252	6	2.2	2.2	NUM
iajs-2804	252	7	)	)	PUNCT
iajs-2804	252	8	we	we	PRON
iajs-2804	252	9	get	get	VERB
iajs-2804	252	10	(	(	PUNCT
iajs-2804	252	11	𝐹	𝐹	PROPN
iajs-2804	252	12	−	−	PROPN
iajs-2804	252	13	𝑆𝑜𝑐(𝑋⨁𝑌))𝑡	𝑆𝑜𝑐(𝑋⨁𝑌))𝑡	ADV
iajs-2804	252	14	=	=	SYM
iajs-2804	252	15	𝑆𝑜𝑐((𝑋⨁𝑌)𝑡	𝑆𝑜𝑐((𝑋⨁𝑌)𝑡	X
iajs-2804	252	16	)	)	PUNCT
iajs-2804	252	17	for	for	ADP
iajs-2804	252	18	each	each	DET
iajs-2804	252	19	t	t	NOUN
iajs-2804	252	20	∈	∈	PROPN
iajs-2804	252	21	(	(	PUNCT
iajs-2804	252	22	0,1	0,1	NOUN
iajs-2804	252	23	]	]	PUNCT
iajs-2804	252	24	.	.	PUNCT
iajs-2804	253	1	but	but	CCONJ
iajs-2804	253	2	,	,	PUNCT
iajs-2804	253	3	𝑆𝑜𝑐((𝑋⨁𝑌)𝑡	𝑆𝑜𝑐((𝑋⨁𝑌)𝑡	AUX
iajs-2804	253	4	)	)	PUNCT
iajs-2804	253	5	=	=	SYM
iajs-2804	253	6	𝑆𝑜𝑐(𝑋𝑡⨁𝑌𝑡	𝑆𝑜𝑐(𝑋𝑡⨁𝑌𝑡	X
iajs-2804	253	7	)	)	PUNCT
iajs-2804	253	8	and	and	CCONJ
iajs-2804	253	9	we	we	PRON
iajs-2804	253	10	have	have	VERB
iajs-2804	253	11	𝑆𝑜𝑐(𝑋𝑡⨁𝑌𝑡	𝑆𝑜𝑐(𝑋𝑡⨁𝑌𝑡	NOUN
iajs-2804	253	12	)	)	PUNCT
iajs-2804	253	13	=	=	SYM
iajs-2804	253	14	𝑆𝑜𝑐(𝑋𝑡)⨁𝑆𝑜𝑐(𝑌𝑡	𝑆𝑜𝑐(𝑋𝑡)⨁𝑆𝑜𝑐(𝑌𝑡	PROPN
iajs-2804	253	15	)	)	PUNCT
iajs-2804	254	1	that	that	PRON
iajs-2804	254	2	is	be	AUX
iajs-2804	254	3	(	(	PUNCT
iajs-2804	254	4	𝐹	𝐹	PROPN
iajs-2804	254	5	−	−	PROPN
iajs-2804	254	6	𝑆𝑜𝑐(𝑋⨁𝑌))𝑡	𝑆𝑜𝑐(𝑋⨁𝑌))𝑡	ADV
iajs-2804	254	7	=	=	SYM
iajs-2804	254	8	𝑆𝑜𝑐(𝑋𝑡)⨁𝑆𝑜𝑐(𝑌𝑡	𝑆𝑜𝑐(𝑋𝑡)⨁𝑆𝑜𝑐(𝑌𝑡	PROPN
iajs-2804	254	9	)	)	PUNCT
iajs-2804	255	1	=	=	PUNCT
iajs-2804	255	2	(	(	PUNCT
iajs-2804	255	3	𝐹	𝐹	PROPN
iajs-2804	255	4	−	−	PROPN
iajs-2804	255	5	𝑆𝑜𝑐(𝑋))𝑡⨁(𝐹	𝑆𝑜𝑐(𝑋))𝑡⨁(𝐹	NOUN
iajs-2804	255	6	−	−	PROPN
iajs-2804	255	7	𝑆𝑜𝑐(𝑌))𝑡	𝑆𝑜𝑐(𝑌))𝑡	ADV
iajs-2804	256	1	thus	thus	ADV
iajs-2804	256	2	(	(	PUNCT
iajs-2804	256	3	𝐹	𝐹	PROPN
iajs-2804	256	4	−	−	PROPN
iajs-2804	256	5	𝑆𝑜𝑐(𝑋⨁𝑌))𝑡	𝑆𝑜𝑐(𝑋⨁𝑌))𝑡	ADV
iajs-2804	256	6	=	=	PUNCT
iajs-2804	257	1	[	[	X
iajs-2804	257	2	(	(	PUNCT
iajs-2804	257	3	𝐹	𝐹	PROPN
iajs-2804	257	4	−	−	PROPN
iajs-2804	257	5	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	257	6	)	)	PUNCT
iajs-2804	257	7	)	)	PUNCT
iajs-2804	257	8	⨁(𝐹	⨁(𝐹	NOUN
iajs-2804	257	9	−	−	PROPN
iajs-2804	257	10	𝑆𝑜𝑐(𝑌)]𝑡	𝑆𝑜𝑐(𝑌)]𝑡	VERB
iajs-2804	257	11	𝐹	𝐹	PROPN
iajs-2804	257	12	−	−	PROPN
iajs-2804	257	13	𝑆𝑜𝑐(𝑋⨁𝑌	𝑆𝑜𝑐(𝑋⨁𝑌	PROPN
iajs-2804	257	14	)	)	PUNCT
iajs-2804	258	1	=	=	SYM
iajs-2804	258	2	𝐹	𝐹	PROPN
iajs-2804	258	3	−	−	PROPN
iajs-2804	258	4	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	258	5	)	)	PUNCT
iajs-2804	258	6	⨁	⨁	PROPN
iajs-2804	258	7	𝐹	𝐹	PROPN
iajs-2804	258	8	−	−	PROPN
iajs-2804	258	9	𝑆𝑜𝑐(𝑌	𝑆𝑜𝑐(𝑌	PROPN
iajs-2804	258	10	)	)	PUNCT
iajs-2804	258	11	hence	hence	ADV
iajs-2804	258	12	from	from	ADP
iajs-2804	258	13	(	(	PUNCT
iajs-2804	258	14	remark	remark	NOUN
iajs-2804	258	15	1.5	1.5	NUM
iajs-2804	258	16	)	)	PUNCT
iajs-2804	258	17	then	then	ADV
iajs-2804	258	18	ibn	ibn	PROPN
iajs-2804	258	19	al	al	PROPN
iajs-2804	258	20	-	-	PUNCT
iajs-2804	258	21	haitham	haitham	PROPN
iajs-2804	258	22	jour	jour	X
iajs-2804	258	23	.	.	PROPN
iajs-2804	259	1	for	for	ADP
iajs-2804	259	2	pure	pure	ADJ
iajs-2804	259	3	&	&	CCONJ
iajs-2804	259	4	appl	appl	PROPN
iajs-2804	259	5	.	.	PUNCT
iajs-2804	260	1	sci	sci	PROPN
iajs-2804	260	2	.	.	PROPN
iajs-2804	261	1	53	53	NUM
iajs-2804	261	2	(	(	PUNCT
iajs-2804	261	3	1)2022	1)2022	NOUN
iajs-2804	261	4	114	114	NUM
iajs-2804	261	5	proposition	proposition	NOUN
iajs-2804	261	6	2.23	2.23	NUM
iajs-2804	261	7	:	:	PUNCT
iajs-2804	261	8	if	if	SCONJ
iajs-2804	261	9	u	u	NOUN
iajs-2804	261	10	and	and	CCONJ
iajs-2804	261	11	v	v	NOUN
iajs-2804	261	12	are	be	AUX
iajs-2804	261	13	ℱ-sub	ℱ-sub	NOUN
iajs-2804	261	14	-	-	NOUN
iajs-2804	261	15	modules	module	NOUN
iajs-2804	261	16	of	of	ADP
iajs-2804	261	17	ℱ-modules	ℱ-modules	PROPN
iajs-2804	261	18	x	x	X
iajs-2804	261	19	and	and	CCONJ
iajs-2804	261	20	y	y	PROPN
iajs-2804	261	21	respectively	respectively	ADV
iajs-2804	261	22	,	,	PUNCT
iajs-2804	261	23	then	then	ADV
iajs-2804	261	24	1	1	X
iajs-2804	261	25	)	)	PUNCT
iajs-2804	261	26	if	if	SCONJ
iajs-2804	261	27	𝑈⨁𝑌	𝑈⨁𝑌	NOUN
iajs-2804	261	28	is	be	AUX
iajs-2804	261	29	an	an	DET
iajs-2804	261	30	ℱ-soc	ℱ-soc	NOUN
iajs-2804	261	31	-	-	PUNCT
iajs-2804	261	32	semi	semi	ADJ
iajs-2804	261	33	-	-	ADJ
iajs-2804	261	34	prime	prime	ADJ
iajs-2804	261	35	sub	sub	NOUN
iajs-2804	261	36	-	-	NOUN
iajs-2804	261	37	module	module	NOUN
iajs-2804	261	38	of	of	ADP
iajs-2804	261	39	𝑋⨁𝑌	𝑋⨁𝑌	ADJ
iajs-2804	261	40	thus	thus	ADV
iajs-2804	261	41	u	u	NOUN
iajs-2804	261	42	is	be	AUX
iajs-2804	261	43	an	an	DET
iajs-2804	261	44	ℱ-soc	ℱ-soc	NOUN
iajs-2804	261	45	-	-	PUNCT
iajs-2804	261	46	semi	semi	ADJ
iajs-2804	261	47	-	-	ADJ
iajs-2804	261	48	prime	prime	ADJ
iajs-2804	261	49	sub	sub	NOUN
iajs-2804	261	50	-	-	NOUN
iajs-2804	261	51	module	module	NOUN
iajs-2804	261	52	of	of	ADP
iajs-2804	261	53	x.	x.	NOUN
iajs-2804	261	54	2	2	NUM
iajs-2804	261	55	)	)	PUNCT
iajs-2804	261	56	if	if	SCONJ
iajs-2804	261	57	𝑋⨁𝑉	𝑋⨁𝑉	NOUN
iajs-2804	261	58	is	be	AUX
iajs-2804	261	59	an	an	DET
iajs-2804	261	60	ℱ-soc	ℱ-soc	NOUN
iajs-2804	261	61	-	-	PUNCT
iajs-2804	261	62	semi	semi	ADJ
iajs-2804	261	63	-	-	ADJ
iajs-2804	261	64	prime	prime	ADJ
iajs-2804	261	65	sub	sub	NOUN
iajs-2804	261	66	-	-	NOUN
iajs-2804	261	67	module	module	NOUN
iajs-2804	261	68	of	of	ADP
iajs-2804	261	69	𝑋⨁𝑌	𝑋⨁𝑌	ADJ
iajs-2804	261	70	thus	thus	ADV
iajs-2804	261	71	v	v	NOUN
iajs-2804	261	72	is	be	AUX
iajs-2804	261	73	an	an	DET
iajs-2804	261	74	ℱ-soc	ℱ-soc	NOUN
iajs-2804	261	75	-	-	PUNCT
iajs-2804	261	76	semi	semi	ADJ
iajs-2804	261	77	-	-	ADJ
iajs-2804	261	78	prime	prime	ADJ
iajs-2804	261	79	sub	sub	NOUN
iajs-2804	261	80	-	-	NOUN
iajs-2804	261	81	module	module	NOUN
iajs-2804	261	82	of	of	ADP
iajs-2804	261	83	x.	x.	NOUN
iajs-2804	261	84	proof	proof	NOUN
iajs-2804	261	85	:	:	PUNCT
iajs-2804	261	86	1	1	X
iajs-2804	261	87	)	)	PUNCT
iajs-2804	261	88	suppose	suppose	VERB
iajs-2804	261	89	that	that	SCONJ
iajs-2804	261	90	𝑈⨁𝑌	𝑈⨁𝑌	NOUN
iajs-2804	261	91	is	be	AUX
iajs-2804	261	92	an	an	DET
iajs-2804	261	93	ℱ-soc	ℱ-soc	NOUN
iajs-2804	261	94	-	-	PUNCT
iajs-2804	261	95	semi	semi	ADJ
iajs-2804	261	96	-	-	ADJ
iajs-2804	261	97	prime	prime	ADJ
iajs-2804	261	98	sub	sub	NOUN
iajs-2804	261	99	-	-	NOUN
iajs-2804	261	100	module	module	NOUN
iajs-2804	261	101	of	of	ADP
iajs-2804	261	102	𝑋⨁𝑌	𝑋⨁𝑌	NOUN
iajs-2804	261	103	and	and	CCONJ
iajs-2804	261	104	𝑟𝑏	𝑟𝑏	PROPN
iajs-2804	261	105	is	be	AUX
iajs-2804	261	106	an	an	DET
iajs-2804	261	107	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	261	108	of	of	ADP
iajs-2804	261	109	r	r	NOUN
iajs-2804	261	110	and	and	CCONJ
iajs-2804	261	111	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	261	112	⊆	⊆	PROPN
iajs-2804	261	113	𝑋	𝑋	NOUN
iajs-2804	261	114	such	such	ADJ
iajs-2804	261	115	that	that	SCONJ
iajs-2804	261	116	(	(	PUNCT
iajs-2804	261	117	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	261	118	⊆	⊆	NUM
iajs-2804	261	119	𝑈.	𝑈.	NOUN
iajs-2804	261	120	then	then	ADV
iajs-2804	261	121	(	(	PUNCT
iajs-2804	261	122	𝑟𝑛)𝑏(𝑥𝑡	𝑟𝑛)𝑏(𝑥𝑡	PROPN
iajs-2804	261	123	,	,	PUNCT
iajs-2804	261	124	𝑦𝑝	𝑦𝑝	NOUN
iajs-2804	261	125	)	)	PUNCT
iajs-2804	261	126	=	=	SYM
iajs-2804	261	127	(	(	PUNCT
iajs-2804	261	128	(	(	PUNCT
iajs-2804	261	129	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	261	130	,	,	PUNCT
iajs-2804	261	131	(	(	PUNCT
iajs-2804	261	132	𝑟𝑛)𝑏𝑦𝑝	𝑟𝑛)𝑏𝑦𝑝	X
iajs-2804	261	133	)	)	PUNCT
iajs-2804	261	134	⊆	⊆	NUM
iajs-2804	261	135	𝑈⨁𝑌	𝑈⨁𝑌	NOUN
iajs-2804	261	136	,	,	PUNCT
iajs-2804	261	137	for	for	ADP
iajs-2804	261	138	any	any	DET
iajs-2804	261	139	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	261	140	𝑦𝑝	𝑦𝑝	NOUN
iajs-2804	261	141	⊆	⊆	NUM
iajs-2804	261	142	𝑌	𝑌	PROPN
iajs-2804	261	143	,	,	PUNCT
iajs-2804	261	144	but	but	CCONJ
iajs-2804	261	145	𝑈⨁𝑌	𝑈⨁𝑌	NOUN
iajs-2804	261	146	is	be	AUX
iajs-2804	261	147	an	an	DET
iajs-2804	261	148	ℱ-soc	ℱ-soc	NOUN
iajs-2804	261	149	-	-	PUNCT
iajs-2804	261	150	semi	semi	ADJ
iajs-2804	261	151	-	-	ADJ
iajs-2804	261	152	prime	prime	ADJ
iajs-2804	261	153	sub	sub	NOUN
iajs-2804	261	154	-	-	NOUN
iajs-2804	261	155	module	module	NOUN
iajs-2804	261	156	of	of	ADP
iajs-2804	261	157	𝑋⨁𝑌.	𝑋⨁𝑌.	NUM
iajs-2804	261	158	thus	thus	ADV
iajs-2804	261	159	(	(	PUNCT
iajs-2804	261	160	𝑟𝑏𝑥𝑡	𝑟𝑏𝑥𝑡	NOUN
iajs-2804	261	161	,	,	PUNCT
iajs-2804	261	162	𝑟𝑏𝑦𝑝	𝑟𝑏𝑦𝑝	NOUN
iajs-2804	261	163	)	)	PUNCT
iajs-2804	261	164	⊆	⊆	NUM
iajs-2804	261	165	(	(	PUNCT
iajs-2804	261	166	𝑈⨁𝑌	𝑈⨁𝑌	NOUN
iajs-2804	261	167	)	)	PUNCT
iajs-2804	262	1	+	+	CCONJ
iajs-2804	262	2	𝐹	𝐹	PROPN
iajs-2804	262	3	−	−	PROPN
iajs-2804	262	4	𝑆𝑜𝑐(𝑋⨁𝑌	𝑆𝑜𝑐(𝑋⨁𝑌	PROPN
iajs-2804	262	5	)	)	PUNCT
iajs-2804	262	6	,	,	PUNCT
iajs-2804	262	7	by	by	ADP
iajs-2804	262	8	(	(	PUNCT
iajs-2804	262	9	lemma	lemma	PROPN
iajs-2804	262	10	2.22	2.22	NUM
iajs-2804	262	11	)	)	PUNCT
iajs-2804	262	12	we	we	PRON
iajs-2804	262	13	get	get	VERB
iajs-2804	262	14	(	(	PUNCT
iajs-2804	262	15	𝑟𝑏𝑥𝑡	𝑟𝑏𝑥𝑡	X
iajs-2804	262	16	,	,	PUNCT
iajs-2804	262	17	𝑟𝑏𝑦𝑝	𝑟𝑏𝑦𝑝	NOUN
iajs-2804	262	18	)	)	PUNCT
iajs-2804	263	1	⊆	⊆	NUM
iajs-2804	263	2	(	(	PUNCT
iajs-2804	263	3	𝑈	𝑈	NOUN
iajs-2804	263	4	+	+	NOUN
iajs-2804	263	5	𝐹	𝐹	PROPN
iajs-2804	263	6	−	−	NOUN
iajs-2804	264	1	𝑆𝑜𝑐(𝑋))⨁(𝑌	𝑆𝑜𝑐(𝑋))⨁(𝑌	PROPN
iajs-2804	265	1	+	+	CCONJ
iajs-2804	265	2	𝐹	𝐹	PROPN
iajs-2804	265	3	−	−	PROPN
iajs-2804	265	4	𝑆𝑜𝑐(𝑌	𝑆𝑜𝑐(𝑌	PROPN
iajs-2804	265	5	)	)	PUNCT
iajs-2804	265	6	)	)	PUNCT
iajs-2804	265	7	.	.	PUNCT
iajs-2804	266	1	that	that	PRON
iajs-2804	266	2	is	be	AUX
iajs-2804	266	3	𝑟𝑏𝑥𝑡	𝑟𝑏𝑥𝑡	NOUN
iajs-2804	266	4	⊆	⊆	NUM
iajs-2804	266	5	𝑈	𝑈	PROPN
iajs-2804	266	6	+	+	CCONJ
iajs-2804	266	7	𝐹	𝐹	PROPN
iajs-2804	266	8	−	−	PROPN
iajs-2804	266	9	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	266	10	)	)	PUNCT
iajs-2804	266	11	,	,	PUNCT
iajs-2804	266	12	therefore	therefore	ADV
iajs-2804	266	13	u	u	NOUN
iajs-2804	266	14	is	be	AUX
iajs-2804	266	15	an	an	DET
iajs-2804	266	16	ℱ-soc	ℱ-soc	NOUN
iajs-2804	266	17	-	-	PUNCT
iajs-2804	266	18	semi	semi	ADJ
iajs-2804	266	19	-	-	ADJ
iajs-2804	266	20	prime	prime	ADJ
iajs-2804	266	21	sub	sub	NOUN
iajs-2804	266	22	-	-	NOUN
iajs-2804	266	23	module	module	NOUN
iajs-2804	266	24	of	of	ADP
iajs-2804	266	25	x.	x.	NOUN
iajs-2804	266	26	2	2	NUM
iajs-2804	266	27	)	)	PUNCT
iajs-2804	266	28	similarly	similarly	ADV
iajs-2804	266	29	as	as	ADP
iajs-2804	266	30	the	the	DET
iajs-2804	266	31	idea	idea	NOUN
iajs-2804	266	32	in	in	ADP
iajs-2804	266	33	(	(	PUNCT
iajs-2804	266	34	1	1	NUM
iajs-2804	266	35	)	)	PUNCT
iajs-2804	266	36	.	.	PUNCT
iajs-2804	267	1	lemma	lemma	PROPN
iajs-2804	267	2	2.24	2.24	NUM
iajs-2804	267	3	:	:	PUNCT
iajs-2804	267	4	if	if	SCONJ
iajs-2804	267	5	x	x	PRON
iajs-2804	267	6	is	be	AUX
iajs-2804	267	7	an	an	DET
iajs-2804	267	8	ℱ-module	ℱ-module	PROPN
iajs-2804	267	9	of	of	ADP
iajs-2804	267	10	an	an	DET
iajs-2804	267	11	ℛ-module	ℛ-module	PROPN
iajs-2804	267	12	m	m	PROPN
iajs-2804	267	13	,	,	PUNCT
iajs-2804	267	14	and	and	CCONJ
iajs-2804	267	15	m	m	AUX
iajs-2804	267	16	be	be	AUX
iajs-2804	267	17	a	a	DET
iajs-2804	267	18	faithful	faithful	ADJ
iajs-2804	267	19	multiplication	multiplication	NOUN
iajs-2804	267	20	ℛ-module	ℛ-module	PROPN
iajs-2804	267	21	,	,	PUNCT
iajs-2804	267	22	then	then	ADV
iajs-2804	267	23	:	:	PUNCT
iajs-2804	267	24	ℱ	ℱ	PROPN
iajs-2804	267	25	−	−	PROPN
iajs-2804	267	26	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	267	27	)	)	PUNCT
iajs-2804	268	1	=	=	NOUN
iajs-2804	268	2	𝑋	𝑋	PROPN
iajs-2804	268	3	ℱ	ℱ	PROPN
iajs-2804	268	4	−	−	PROPN
iajs-2804	268	5	𝑆𝑜𝑐(ℛ	𝑆𝑜𝑐(ℛ	NOUN
iajs-2804	268	6	)	)	PUNCT
iajs-2804	268	7	.	.	PUNCT
iajs-2804	269	1	proposition	proposition	NOUN
iajs-2804	269	2	2.25	2.25	NUM
iajs-2804	269	3	:	:	PUNCT
iajs-2804	269	4	let	let	VERB
iajs-2804	269	5	x	x	PRON
iajs-2804	269	6	be	be	AUX
iajs-2804	269	7	a	a	DET
iajs-2804	269	8	finitely	finitely	ADV
iajs-2804	269	9	generated	generate	VERB
iajs-2804	269	10	multiplication	multiplication	NOUN
iajs-2804	269	11	and	and	CCONJ
iajs-2804	269	12	faithful	faithful	ADJ
iajs-2804	269	13	ℱ-module	ℱ-module	PROPN
iajs-2804	269	14	of	of	ADP
iajs-2804	269	15	an	an	DET
iajs-2804	269	16	ℛ-module	ℛ-module	PROPN
iajs-2804	269	17	m	m	PROPN
iajs-2804	269	18	,	,	PUNCT
iajs-2804	269	19	if	if	SCONJ
iajs-2804	269	20	j	j	PROPN
iajs-2804	269	21	is	be	AUX
iajs-2804	269	22	an	an	DET
iajs-2804	269	23	ℱ-soc	ℱ-soc	NOUN
iajs-2804	269	24	-	-	PUNCT
iajs-2804	269	25	semi	semi	ADJ
iajs-2804	269	26	-	-	ADJ
iajs-2804	269	27	prime	prime	ADJ
iajs-2804	269	28	ideal	ideal	NOUN
iajs-2804	269	29	of	of	ADP
iajs-2804	269	30	ℛ	ℛ	PROPN
iajs-2804	269	31	then	then	ADV
iajs-2804	269	32	jx	jx	PROPN
iajs-2804	269	33	is	be	AUX
iajs-2804	269	34	an	an	DET
iajs-2804	269	35	ℱ-soc	ℱ-soc	NOUN
iajs-2804	269	36	-	-	PUNCT
iajs-2804	269	37	semi	semi	ADJ
iajs-2804	269	38	-	-	ADJ
iajs-2804	269	39	prime	prime	ADJ
iajs-2804	269	40	sub	sub	NOUN
iajs-2804	269	41	-	-	NOUN
iajs-2804	269	42	module	module	NOUN
iajs-2804	269	43	of	of	ADP
iajs-2804	269	44	x.	x.	NOUN
iajs-2804	269	45	proof	proof	NOUN
iajs-2804	269	46	:	:	PUNCT
iajs-2804	269	47	assume	assume	VERB
iajs-2804	269	48	that	that	SCONJ
iajs-2804	269	49	𝑟𝑏	𝑟𝑏	PRON
iajs-2804	269	50	is	be	VERB
iajs-2804	269	51	an	an	DET
iajs-2804	269	52	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	269	53	of	of	ADP
iajs-2804	269	54	ℛ	ℛ	PROPN
iajs-2804	269	55	and	and	CCONJ
iajs-2804	269	56	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	269	57	⊆	⊆	PROPN
iajs-2804	269	58	𝑋	𝑋	NOUN
iajs-2804	269	59	such	such	ADJ
iajs-2804	269	60	that	that	PRON
iajs-2804	269	61	(	(	PUNCT
iajs-2804	269	62	𝑟𝑛)𝑏𝑥𝑘	𝑟𝑛)𝑏𝑥𝑘	NUM
iajs-2804	269	63	=	=	SYM
iajs-2804	269	64	(	(	PUNCT
iajs-2804	269	65	𝑟𝑏)𝑛𝑥𝑘	𝑟𝑏)𝑛𝑥𝑘	PROPN
iajs-2804	269	66	⊆	⊆	NUM
iajs-2804	269	67	jx	jx	PROPN
iajs-2804	269	68	,	,	PUNCT
iajs-2804	269	69	where	where	SCONJ
iajs-2804	269	70	𝑘	𝑘	X
iajs-2804	269	71	,	,	PUNCT
iajs-2804	269	72	𝑏	𝑏	PROPN
iajs-2804	269	73	∈	∈	PROPN
iajs-2804	270	1	[	[	X
iajs-2804	270	2	0,1].that	0,1].that	X
iajs-2804	270	3	is	be	AUX
iajs-2804	270	4	(	(	PUNCT
iajs-2804	270	5	𝑟𝑛)𝑏〈𝑥𝑡	𝑟𝑛)𝑏〈𝑥𝑡	NOUN
iajs-2804	270	6	〉	〉	NOUN
iajs-2804	270	7	⊆	⊆	NUM
iajs-2804	270	8	jx	jx	PROPN
iajs-2804	270	9	.	.	PUNCT
iajs-2804	271	1	but	but	CCONJ
iajs-2804	271	2	x	x	X
iajs-2804	271	3	is	be	AUX
iajs-2804	271	4	a	a	DET
iajs-2804	271	5	multiplication	multiplication	NOUN
iajs-2804	271	6	ℱ-module	ℱ-module	PROPN
iajs-2804	271	7	,	,	PUNCT
iajs-2804	271	8	thus	thus	ADV
iajs-2804	271	9	there	there	PRON
iajs-2804	271	10	exists	exist	VERB
iajs-2804	271	11	an	an	DET
iajs-2804	271	12	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	271	13	l	l	NOUN
iajs-2804	271	14	of	of	ADP
iajs-2804	271	15	ℛ	ℛ	PROPN
iajs-2804	271	16	with	with	ADP
iajs-2804	271	17	〈	〈	NOUN
iajs-2804	271	18	𝑥𝑡	𝑥𝑡	NOUN
iajs-2804	271	19	〉	〉	NOUN
iajs-2804	271	20	=	=	SYM
iajs-2804	271	21	𝐿𝑋.	𝐿𝑋.	NOUN
iajs-2804	271	22	then	then	ADV
iajs-2804	271	23	we	we	PRON
iajs-2804	271	24	get	get	VERB
iajs-2804	271	25	(	(	PUNCT
iajs-2804	271	26	𝑟𝑛)𝑏𝐿𝑋	𝑟𝑛)𝑏𝐿𝑋	PROPN
iajs-2804	271	27	⊆	⊆	NUM
iajs-2804	271	28	jx	jx	PROPN
iajs-2804	271	29	,	,	PUNCT
iajs-2804	271	30	so	so	ADV
iajs-2804	271	31	(	(	PUNCT
iajs-2804	271	32	𝑟𝑛)𝑏𝐿	𝑟𝑛)𝑏𝐿	PROPN
iajs-2804	271	33	⊆	⊆	NUM
iajs-2804	271	34	j	j	NOUN
iajs-2804	271	35	+	+	CCONJ
iajs-2804	271	36	ℱ	ℱ	PROPN
iajs-2804	271	37	−	−	NOUN
iajs-2804	271	38	ann(x	ann(x	PROPN
iajs-2804	271	39	)	)	PUNCT
iajs-2804	271	40	=	=	SYM
iajs-2804	271	41	j	j	PROPN
iajs-2804	271	42	since	since	SCONJ
iajs-2804	271	43	x	x	PRON
iajs-2804	271	44	is	be	AUX
iajs-2804	271	45	a	a	DET
iajs-2804	271	46	faithful	faithful	ADJ
iajs-2804	271	47	ℱmodule	ℱmodule	PROPN
iajs-2804	271	48	.	.	PUNCT
iajs-2804	272	1	but	but	CCONJ
iajs-2804	272	2	j	j	PROPN
iajs-2804	272	3	is	be	AUX
iajs-2804	272	4	an	an	DET
iajs-2804	272	5	ℱ-soc	ℱ-soc	NOUN
iajs-2804	272	6	-	-	PUNCT
iajs-2804	272	7	semi	semi	ADJ
iajs-2804	272	8	-	-	ADJ
iajs-2804	272	9	prime	prime	ADJ
iajs-2804	272	10	ideal	ideal	NOUN
iajs-2804	272	11	of	of	ADP
iajs-2804	272	12	ℛ	ℛ	PROPN
iajs-2804	272	13	,	,	PUNCT
iajs-2804	272	14	then	then	ADV
iajs-2804	272	15	by	by	ADP
iajs-2804	272	16	(	(	PUNCT
iajs-2804	272	17	corollary	corollary	ADJ
iajs-2804	272	18	2.13	2.13	NUM
iajs-2804	272	19	)	)	PUNCT
iajs-2804	272	20	implies	imply	VERB
iajs-2804	272	21	that	that	SCONJ
iajs-2804	272	22	𝑟𝑏𝐿	𝑟𝑏𝐿	PROPN
iajs-2804	272	23	⊆	⊆	PROPN
iajs-2804	272	24	𝐽	𝐽	PROPN
iajs-2804	272	25	+	+	CCONJ
iajs-2804	272	26	ℱ	ℱ	PROPN
iajs-2804	272	27	−	−	NOUN
iajs-2804	272	28	𝑆𝑜𝑐(ℛ).now	𝑆𝑜𝑐(ℛ).now	NOUN
iajs-2804	272	29	,	,	PUNCT
iajs-2804	272	30	by	by	ADP
iajs-2804	272	31	multiplying	multiply	VERB
iajs-2804	272	32	both	both	DET
iajs-2804	272	33	sides	side	NOUN
iajs-2804	272	34	with	with	ADP
iajs-2804	272	35	x	x	PUNCT
iajs-2804	272	36	and	and	CCONJ
iajs-2804	272	37	using	use	VERB
iajs-2804	272	38	(	(	PUNCT
iajs-2804	272	39	lemma	lemma	PROPN
iajs-2804	272	40	2.24	2.24	NUM
iajs-2804	272	41	)	)	PUNCT
iajs-2804	272	42	we	we	PRON
iajs-2804	272	43	have	have	VERB
iajs-2804	272	44	𝑟𝑏𝐿𝑋	𝑟𝑏𝐿𝑋	NUM
iajs-2804	272	45	⊆	⊆	NUM
iajs-2804	272	46	𝐽𝑋	𝐽𝑋	PROPN
iajs-2804	272	47	+	+	CCONJ
iajs-2804	272	48	ℱ	ℱ	PROPN
iajs-2804	272	49	−	−	NOUN
iajs-2804	272	50	𝑆𝑜𝑐(ℛ)𝑋	𝑆𝑜𝑐(ℛ)𝑋	NOUN
iajs-2804	273	1	=	=	SYM
iajs-2804	273	2	𝐽𝑋	𝐽𝑋	PROPN
iajs-2804	273	3	+	+	CCONJ
iajs-2804	273	4	ℱ	ℱ	PROPN
iajs-2804	273	5	−	−	PROPN
iajs-2804	273	6	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	273	7	)	)	PUNCT
iajs-2804	273	8	.	.	PUNCT
iajs-2804	274	1	therefore	therefore	ADV
iajs-2804	274	2	,	,	PUNCT
iajs-2804	274	3	jx	jx	PROPN
iajs-2804	274	4	is	be	AUX
iajs-2804	274	5	an	an	DET
iajs-2804	274	6	ℱ-soc	ℱ-soc	NOUN
iajs-2804	274	7	-	-	PUNCT
iajs-2804	274	8	semi	semi	ADJ
iajs-2804	274	9	-	-	ADJ
iajs-2804	274	10	prime	prime	ADJ
iajs-2804	274	11	sub	sub	NOUN
iajs-2804	274	12	-	-	NOUN
iajs-2804	274	13	module	module	NOUN
iajs-2804	274	14	of	of	ADP
iajs-2804	274	15	x.	x.	NOUN
iajs-2804	274	16	proposition	proposition	PROPN
iajs-2804	274	17	2.26	2.26	NUM
iajs-2804	274	18	suppose	suppose	VERB
iajs-2804	274	19	that	that	SCONJ
iajs-2804	274	20	u	u	PROPN
iajs-2804	274	21	is	be	AUX
iajs-2804	274	22	an	an	DET
iajs-2804	274	23	ℱ-soc	ℱ-soc	NOUN
iajs-2804	274	24	-	-	PUNCT
iajs-2804	274	25	semi	semi	ADJ
iajs-2804	274	26	-	-	ADJ
iajs-2804	274	27	prime	prime	ADJ
iajs-2804	274	28	sub	sub	NOUN
iajs-2804	274	29	-	-	NOUN
iajs-2804	274	30	module	module	NOUN
iajs-2804	274	31	of	of	ADP
iajs-2804	274	32	an	an	DET
iajs-2804	274	33	ℱ-module	ℱ-module	PROPN
iajs-2804	274	34	x	x	X
iajs-2804	274	35	and	and	CCONJ
iajs-2804	274	36	v	v	NOUN
iajs-2804	274	37	is	be	AUX
iajs-2804	274	38	an	an	DET
iajs-2804	274	39	ℱ-semiprime	ℱ-semiprime	PROPN
iajs-2804	274	40	sub	sub	NOUN
iajs-2804	274	41	-	-	NOUN
iajs-2804	274	42	module	module	NOUN
iajs-2804	274	43	of	of	ADP
iajs-2804	274	44	x	x	PUNCT
iajs-2804	274	45	with	with	ADP
iajs-2804	274	46	ℱ	ℱ	PROPN
iajs-2804	274	47	−	−	ADP
iajs-2804	274	48	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	274	49	)	)	PUNCT
iajs-2804	274	50	⊆	⊆	NUM
iajs-2804	274	51	𝑉.	𝑉.	NOUN
iajs-2804	274	52	then	then	ADV
iajs-2804	274	53	the	the	DET
iajs-2804	274	54	intersection	intersection	NOUN
iajs-2804	274	55	of	of	ADP
iajs-2804	274	56	u	u	NOUN
iajs-2804	274	57	and	and	CCONJ
iajs-2804	274	58	v	v	NOUN
iajs-2804	274	59	is	be	AUX
iajs-2804	274	60	an	an	DET
iajs-2804	274	61	ℱsoc	ℱsoc	PROPN
iajs-2804	274	62	-	-	PUNCT
iajs-2804	274	63	semi	semi	NOUN
iajs-2804	274	64	-	-	ADJ
iajs-2804	274	65	prime	prime	ADJ
iajs-2804	274	66	of	of	ADP
iajs-2804	274	67	x.	x.	NOUN
iajs-2804	274	68	proof	proof	NOUN
iajs-2804	274	69	:	:	PUNCT
iajs-2804	274	70	if	if	SCONJ
iajs-2804	274	71	𝑟𝑏	𝑟𝑏	PRON
iajs-2804	274	72	is	be	AUX
iajs-2804	274	73	an	an	DET
iajs-2804	274	74	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	274	75	of	of	ADP
iajs-2804	274	76	ℛ	ℛ	PROPN
iajs-2804	274	77	and	and	CCONJ
iajs-2804	274	78	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	274	79	⊆	⊆	NUM
iajs-2804	274	80	𝑋	𝑋	NOUN
iajs-2804	274	81	where	where	SCONJ
iajs-2804	274	82	𝑏	𝑏	NOUN
iajs-2804	274	83	,	,	PUNCT
iajs-2804	274	84	𝑡	𝑡	PROPN
iajs-2804	274	85	∈	∈	PROPN
iajs-2804	275	1	[	[	X
iajs-2804	275	2	0,1	0,1	NUM
iajs-2804	275	3	]	]	PUNCT
iajs-2804	275	4	,	,	PUNCT
iajs-2804	275	5	such	such	ADJ
iajs-2804	275	6	that	that	SCONJ
iajs-2804	275	7	(	(	PUNCT
iajs-2804	275	8	𝑟𝑛)𝑏𝑥𝑘	𝑟𝑛)𝑏𝑥𝑘	NUM
iajs-2804	275	9	=	=	SYM
iajs-2804	275	10	(	(	PUNCT
iajs-2804	275	11	𝑟𝑏)𝑛𝑥𝑘	𝑟𝑏)𝑛𝑥𝑘	PROPN
iajs-2804	275	12	⊆	⊆	NUM
iajs-2804	275	13	𝑈	𝑈	PROPN
iajs-2804	275	14	∩	∩	ADJ
iajs-2804	275	15	𝑉.	𝑉.	NOUN
iajs-2804	275	16	this	this	PRON
iajs-2804	275	17	implies	imply	VERB
iajs-2804	275	18	(	(	PUNCT
iajs-2804	275	19	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	275	20	⊆	⊆	NUM
iajs-2804	275	21	𝑈	𝑈	PROPN
iajs-2804	275	22	and	and	CCONJ
iajs-2804	275	23	(	(	PUNCT
iajs-2804	275	24	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	275	25	⊆	⊆	NUM
iajs-2804	275	26	𝑉	𝑉	PROPN
iajs-2804	275	27	,	,	PUNCT
iajs-2804	275	28	but	but	CCONJ
iajs-2804	275	29	u	u	NOUN
iajs-2804	275	30	is	be	AUX
iajs-2804	275	31	an	an	DET
iajs-2804	275	32	ℱ-soc	ℱ-soc	NOUN
iajs-2804	275	33	-	-	PUNCT
iajs-2804	275	34	semi	semi	ADJ
iajs-2804	275	35	-	-	ADJ
iajs-2804	275	36	prime	prime	ADJ
iajs-2804	275	37	submodule	submodule	NOUN
iajs-2804	275	38	of	of	ADP
iajs-2804	275	39	x.	x.	PROPN
iajs-2804	276	1	so	so	ADV
iajs-2804	276	2	,	,	PUNCT
iajs-2804	276	3	we	we	PRON
iajs-2804	276	4	have	have	VERB
iajs-2804	276	5	𝑟𝑏𝑥𝑘	𝑟𝑏𝑥𝑘	NOUN
iajs-2804	277	1	⊆	⊆	NUM
iajs-2804	277	2	𝑈	𝑈	PROPN
iajs-2804	277	3	+	+	CCONJ
iajs-2804	277	4	ℱ	ℱ	PROPN
iajs-2804	277	5	−	−	NOUN
iajs-2804	277	6	𝑆𝑜𝑐(𝑋).now	𝑆𝑜𝑐(𝑋).now	ADJ
iajs-2804	277	7	,	,	PUNCT
iajs-2804	277	8	since	since	SCONJ
iajs-2804	277	9	v	v	NOUN
iajs-2804	277	10	is	be	AUX
iajs-2804	277	11	an	an	DET
iajs-2804	277	12	ℱ-semi	ℱ-semi	PROPN
iajs-2804	277	13	-	-	ADJ
iajs-2804	277	14	prime	prime	ADJ
iajs-2804	277	15	submodule	submodule	NOUN
iajs-2804	277	16	of	of	ADP
iajs-2804	277	17	x	x	SYM
iajs-2804	277	18	then	then	ADV
iajs-2804	277	19	𝑟𝑏𝑥𝑘	𝑟𝑏𝑥𝑘	NOUN
iajs-2804	277	20	⊆	⊆	NUM
iajs-2804	277	21	𝑉.we	𝑉.we	NOUN
iajs-2804	277	22	get	get	VERB
iajs-2804	277	23	𝑟𝑏𝑥𝑘	𝑟𝑏𝑥𝑘	NOUN
iajs-2804	277	24	⊆	⊆	NUM
iajs-2804	277	25	[	[	X
iajs-2804	277	26	𝑈	𝑈	NOUN
iajs-2804	277	27	+	+	CCONJ
iajs-2804	277	28	ℱ	ℱ	PROPN
iajs-2804	277	29	−	−	PROPN
iajs-2804	277	30	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	277	31	)	)	PUNCT
iajs-2804	277	32	]	]	PUNCT
iajs-2804	277	33	∩	∩	X
iajs-2804	277	34	𝑉	𝑉	PROPN
iajs-2804	277	35	,	,	PUNCT
iajs-2804	277	36	but	but	CCONJ
iajs-2804	277	37	ℱ	ℱ	PROPN
iajs-2804	277	38	−	−	PROPN
iajs-2804	277	39	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	277	40	)	)	PUNCT
iajs-2804	277	41	⊆	⊆	NUM
iajs-2804	277	42	𝑉	𝑉	PROPN
iajs-2804	277	43	ibn	ibn	PROPN
iajs-2804	277	44	al	al	PROPN
iajs-2804	277	45	-	-	PUNCT
iajs-2804	277	46	haitham	haitham	PROPN
iajs-2804	277	47	jour	jour	X
iajs-2804	277	48	.	.	PROPN
iajs-2804	278	1	for	for	ADP
iajs-2804	278	2	pure	pure	ADJ
iajs-2804	278	3	&	&	CCONJ
iajs-2804	278	4	appl	appl	PROPN
iajs-2804	278	5	.	.	PUNCT
iajs-2804	279	1	sci	sci	PROPN
iajs-2804	279	2	.	.	PROPN
iajs-2804	280	1	53	53	NUM
iajs-2804	280	2	(	(	PUNCT
iajs-2804	280	3	1)2022	1)2022	NOUN
iajs-2804	280	4	115	115	NUM
iajs-2804	280	5	then	then	ADV
iajs-2804	280	6	by	by	ADP
iajs-2804	280	7	using	use	VERB
iajs-2804	280	8	(	(	PUNCT
iajs-2804	280	9	lemma	lemma	PROPN
iajs-2804	280	10	1.29	1.29	NUM
iajs-2804	280	11	)	)	PUNCT
iajs-2804	280	12	we	we	PRON
iajs-2804	280	13	have	have	VERB
iajs-2804	280	14	𝑟𝑏𝑥𝑘	𝑟𝑏𝑥𝑘	NOUN
iajs-2804	280	15	⊆	⊆	NUM
iajs-2804	280	16	(	(	PUNCT
iajs-2804	280	17	𝑈	𝑈	PROPN
iajs-2804	280	18	∩	∩	ADJ
iajs-2804	280	19	𝑉	𝑉	PROPN
iajs-2804	280	20	)	)	PUNCT
iajs-2804	280	21	+	+	CCONJ
iajs-2804	280	22	ℱ	ℱ	PROPN
iajs-2804	280	23	−	−	PROPN
iajs-2804	280	24	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	280	25	)	)	PUNCT
iajs-2804	280	26	.	.	PUNCT
iajs-2804	281	1	that	that	PRON
iajs-2804	281	2	is	be	AUX
iajs-2804	281	3	mean	mean	VERB
iajs-2804	281	4	𝑈	𝑈	PROPN
iajs-2804	281	5	∩	∩	NOUN
iajs-2804	281	6	𝑉	𝑉	PROPN
iajs-2804	281	7	is	be	AUX
iajs-2804	281	8	an	an	DET
iajs-2804	281	9	ℱ-soc	ℱ-soc	NOUN
iajs-2804	281	10	-	-	PUNCT
iajs-2804	281	11	semi	semi	NOUN
iajs-2804	281	12	-	-	ADJ
iajs-2804	281	13	prime	prime	ADJ
iajs-2804	281	14	of	of	ADP
iajs-2804	281	15	x.	x.	NOUN
iajs-2804	281	16	proposition	proposition	NOUN
iajs-2804	281	17	2.27	2.27	NUM
iajs-2804	281	18	let	let	VERB
iajs-2804	281	19	x	x	PRON
iajs-2804	281	20	be	be	AUX
iajs-2804	281	21	a	a	DET
iajs-2804	281	22	faithful	faithful	ADJ
iajs-2804	281	23	multiplication	multiplication	NOUN
iajs-2804	281	24	ℱ-module	ℱ-module	PROPN
iajs-2804	281	25	of	of	ADP
iajs-2804	281	26	an	an	DET
iajs-2804	281	27	ℛ-module	ℛ-module	PROPN
iajs-2804	281	28	m	m	PROPN
iajs-2804	281	29	,	,	PUNCT
iajs-2804	281	30	then	then	ADV
iajs-2804	281	31	a	a	DET
iajs-2804	281	32	proper	proper	ADJ
iajs-2804	281	33	ℱ-sub	ℱ-sub	NOUN
iajs-2804	281	34	-	-	NOUN
iajs-2804	281	35	module	module	NOUN
iajs-2804	281	36	u	u	NOUN
iajs-2804	281	37	is	be	AUX
iajs-2804	281	38	an	an	DET
iajs-2804	281	39	ℱ-soc	ℱ-soc	NOUN
iajs-2804	281	40	-	-	PUNCT
iajs-2804	281	41	semi	semi	ADJ
iajs-2804	281	42	-	-	ADJ
iajs-2804	281	43	prime	prime	ADJ
iajs-2804	281	44	sub	sub	NOUN
iajs-2804	281	45	-	-	NOUN
iajs-2804	281	46	module	module	NOUN
iajs-2804	281	47	of	of	ADP
iajs-2804	281	48	if	if	SCONJ
iajs-2804	281	49	and	and	CCONJ
iajs-2804	281	50	only	only	ADV
iajs-2804	281	51	if	if	SCONJ
iajs-2804	281	52	[	[	X
iajs-2804	281	53	𝑈:𝑅	𝑈:𝑅	ADJ
iajs-2804	281	54	𝑋	𝑋	NOUN
iajs-2804	281	55	]	]	PUNCT
iajs-2804	281	56	is	be	AUX
iajs-2804	281	57	an	an	DET
iajs-2804	281	58	ℱ-soc	ℱ-soc	NOUN
iajs-2804	281	59	-	-	PUNCT
iajs-2804	281	60	semi	semi	ADJ
iajs-2804	281	61	-	-	ADJ
iajs-2804	281	62	prime	prime	ADJ
iajs-2804	281	63	ideal	ideal	NOUN
iajs-2804	281	64	of	of	ADP
iajs-2804	281	65	ℛ.	ℛ.	PROPN
iajs-2804	281	66	proof	proof	NOUN
iajs-2804	281	67	:	:	PUNCT
iajs-2804	281	68	let	let	VERB
iajs-2804	281	69	(	(	PUNCT
iajs-2804	281	70	𝑟𝑛)𝑏𝑚𝑡	𝑟𝑛)𝑏𝑚𝑡	X
iajs-2804	281	71	=	=	PUNCT
iajs-2804	281	72	(	(	PUNCT
iajs-2804	281	73	𝑟𝑏)𝑛𝑚𝑡	𝑟𝑏)𝑛𝑚𝑡	NOUN
iajs-2804	281	74	⊆	⊆	NUM
iajs-2804	281	75	[	[	X
iajs-2804	281	76	𝑈:𝑅	𝑈:𝑅	ADJ
iajs-2804	281	77	𝑋	𝑋	NOUN
iajs-2804	281	78	]	]	PUNCT
iajs-2804	281	79	with	with	ADP
iajs-2804	281	80	𝑚𝑡	𝑚𝑡	PROPN
iajs-2804	282	1	and	and	CCONJ
iajs-2804	282	2	𝑟𝑏	𝑟𝑏	ADV
iajs-2804	282	3	are	be	AUX
iajs-2804	282	4	ℱ-singletons	ℱ-singletons	PROPN
iajs-2804	282	5	of	of	ADP
iajs-2804	282	6	ℛ	ℛ	PROPN
iajs-2804	282	7	where	where	SCONJ
iajs-2804	282	8	𝑏	𝑏	NOUN
iajs-2804	282	9	,	,	PUNCT
iajs-2804	282	10	𝑡	𝑡	PROPN
iajs-2804	282	11	∈	∈	PROPN
iajs-2804	283	1	[	[	X
iajs-2804	283	2	0,1]implies	0,1]implie	NOUN
iajs-2804	283	3	that	that	PRON
iajs-2804	283	4	(	(	PUNCT
iajs-2804	283	5	𝑟𝑛)𝑏(𝑚𝑡𝑋	𝑟𝑛)𝑏(𝑚𝑡𝑋	NOUN
iajs-2804	283	6	)	)	PUNCT
iajs-2804	283	7	⊆	⊆	NUM
iajs-2804	283	8	𝑈.	𝑈.	PROPN
iajs-2804	283	9	but	but	CCONJ
iajs-2804	283	10	,	,	PUNCT
iajs-2804	283	11	u	u	NOUN
iajs-2804	283	12	is	be	AUX
iajs-2804	283	13	an	an	DET
iajs-2804	283	14	ℱ-soc	ℱ-soc	NOUN
iajs-2804	283	15	-	-	PUNCT
iajs-2804	283	16	semi	semi	ADJ
iajs-2804	283	17	-	-	ADJ
iajs-2804	283	18	prime	prime	ADJ
iajs-2804	283	19	sub	sub	NOUN
iajs-2804	283	20	-	-	NOUN
iajs-2804	283	21	module	module	NOUN
iajs-2804	283	22	,	,	PUNCT
iajs-2804	283	23	so	so	ADV
iajs-2804	283	24	by	by	ADP
iajs-2804	283	25	(	(	PUNCT
iajs-2804	283	26	corollary	corollary	ADJ
iajs-2804	283	27	2.13	2.13	NUM
iajs-2804	283	28	)	)	PUNCT
iajs-2804	283	29	then	then	ADV
iajs-2804	283	30	𝑟𝑏(𝑚𝑡𝑋	𝑟𝑏(𝑚𝑡𝑋	X
iajs-2804	283	31	)	)	PUNCT
iajs-2804	283	32	⊆	⊆	NUM
iajs-2804	283	33	𝑈	𝑈	PROPN
iajs-2804	283	34	+	+	CCONJ
iajs-2804	283	35	ℱ	ℱ	PROPN
iajs-2804	283	36	−	−	PROPN
iajs-2804	283	37	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	283	38	)	)	PUNCT
iajs-2804	283	39	.	.	PUNCT
iajs-2804	284	1	since	since	SCONJ
iajs-2804	284	2	x	x	PRON
iajs-2804	284	3	is	be	AUX
iajs-2804	284	4	a	a	DET
iajs-2804	284	5	multiplication	multiplication	NOUN
iajs-2804	284	6	ℱ-module	ℱ-module	PROPN
iajs-2804	284	7	,	,	PUNCT
iajs-2804	284	8	then	then	ADV
iajs-2804	284	9	by	by	ADP
iajs-2804	284	10	(	(	PUNCT
iajs-2804	284	11	preposition	preposition	NOUN
iajs-2804	284	12	1.32	1.32	NUM
iajs-2804	284	13	)	)	PUNCT
iajs-2804	284	14	𝑈	𝑈	NOUN
iajs-2804	284	15	=	=	PUNCT
iajs-2804	285	1	[	[	X
iajs-2804	285	2	𝑈:𝑅	𝑈:𝑅	X
iajs-2804	285	3	𝑋]𝑋	𝑋]𝑋	NOUN
iajs-2804	285	4	,	,	PUNCT
iajs-2804	285	5	and	and	CCONJ
iajs-2804	285	6	since	since	SCONJ
iajs-2804	285	7	x	x	PRON
iajs-2804	285	8	is	be	AUX
iajs-2804	285	9	a	a	DET
iajs-2804	285	10	faithful	faithful	ADJ
iajs-2804	285	11	multiplication	multiplication	NOUN
iajs-2804	285	12	,	,	PUNCT
iajs-2804	285	13	so	so	ADV
iajs-2804	285	14	by	by	ADP
iajs-2804	285	15	(	(	PUNCT
iajs-2804	285	16	lemma	lemma	PROPN
iajs-2804	285	17	2.24	2.24	NUM
iajs-2804	285	18	)	)	PUNCT
iajs-2804	285	19	𝐹	𝐹	PROPN
iajs-2804	285	20	−	−	NOUN
iajs-2804	285	21	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	285	22	)	)	PUNCT
iajs-2804	286	1	=	=	SYM
iajs-2804	286	2	ℱ	ℱ	PROPN
iajs-2804	286	3	−	−	PROPN
iajs-2804	287	1	𝑆𝑜𝑐(ℛ)𝑋.	𝑆𝑜𝑐(ℛ)𝑋.	CCONJ
iajs-2804	287	2	therefore	therefore	ADV
iajs-2804	287	3	𝑟𝑏𝑚𝑡𝑋	𝑟𝑏𝑚𝑡𝑋	VERB
iajs-2804	287	4	⊆	⊆	NUM
iajs-2804	288	1	[	[	X
iajs-2804	288	2	𝑈:𝑅	𝑈:𝑅	NUM
iajs-2804	288	3	𝑋]𝑋	𝑋]𝑋	X
iajs-2804	288	4	+	+	CCONJ
iajs-2804	288	5	ℱ	ℱ	PROPN
iajs-2804	288	6	−	−	PROPN
iajs-2804	288	7	𝑆𝑜𝑐(ℛ)𝑋	𝑆𝑜𝑐(ℛ)𝑋	PROPN
iajs-2804	288	8	,	,	PUNCT
iajs-2804	288	9	this	this	PRON
iajs-2804	288	10	implies	imply	VERB
iajs-2804	288	11	𝑟𝑏𝑚𝑡	𝑟𝑏𝑚𝑡	PRON
iajs-2804	288	12	⊆	⊆	NUM
iajs-2804	288	13	[	[	X
iajs-2804	288	14	𝑈:𝑅	𝑈:𝑅	ADJ
iajs-2804	288	15	𝑋	𝑋	NOUN
iajs-2804	288	16	]	]	X
iajs-2804	288	17	+	+	CCONJ
iajs-2804	288	18	ℱ	ℱ	PROPN
iajs-2804	288	19	−	−	NOUN
iajs-2804	288	20	𝑆𝑜𝑐(ℛ	𝑆𝑜𝑐(ℛ	NOUN
iajs-2804	288	21	)	)	PUNCT
iajs-2804	288	22	.	.	PUNCT
iajs-2804	289	1	thus	thus	ADV
iajs-2804	289	2	[	[	X
iajs-2804	289	3	𝑈:𝑅	𝑈:𝑅	ADJ
iajs-2804	289	4	𝑋	𝑋	NOUN
iajs-2804	289	5	]	]	PUNCT
iajs-2804	289	6	is	be	AUX
iajs-2804	289	7	an	an	DET
iajs-2804	289	8	ℱ-soc	ℱ-soc	PROPN
iajs-2804	289	9	-	-	PUNCT
iajs-2804	289	10	semiprime	semiprime	NOUN
iajs-2804	289	11	ideal	ideal	NOUN
iajs-2804	289	12	of	of	ADP
iajs-2804	289	13	ℛ	ℛ	PROPN
iajs-2804	289	14	.	.	PUNCT
iajs-2804	290	1	conversely	conversely	ADV
iajs-2804	290	2	let	let	VERB
iajs-2804	290	3	(	(	PUNCT
iajs-2804	290	4	𝑟𝑛)𝑏𝐷	𝑟𝑛)𝑏𝐷	NOUN
iajs-2804	290	5	=	=	SYM
iajs-2804	290	6	(	(	PUNCT
iajs-2804	290	7	𝑟𝑏)𝑛𝐷	𝑟𝑏)𝑛𝐷	NUM
iajs-2804	290	8	⊆	⊆	NUM
iajs-2804	290	9	𝑈	𝑈	NOUN
iajs-2804	290	10	with	with	ADP
iajs-2804	290	11	𝑟𝑏	𝑟𝑏	INTJ
iajs-2804	290	12	be	be	AUX
iajs-2804	290	13	an	an	DET
iajs-2804	290	14	ℱ-singleton	ℱ-singleton	PROPN
iajs-2804	290	15	of	of	ADP
iajs-2804	290	16	ℛ	ℛ	PROPN
iajs-2804	290	17	and	and	CCONJ
iajs-2804	290	18	d	d	PROPN
iajs-2804	290	19	is	be	AUX
iajs-2804	290	20	an	an	DET
iajs-2804	290	21	ℱ-sub	ℱ-sub	NOUN
iajs-2804	290	22	-	-	NOUN
iajs-2804	290	23	module	module	NOUN
iajs-2804	290	24	of	of	ADP
iajs-2804	290	25	x.	x.	NOUN
iajs-2804	290	26	since	since	SCONJ
iajs-2804	290	27	x	x	PRON
iajs-2804	290	28	is	be	AUX
iajs-2804	290	29	a	a	DET
iajs-2804	290	30	multiplication	multiplication	NOUN
iajs-2804	290	31	ℱ-module	ℱ-module	PROPN
iajs-2804	290	32	,	,	PUNCT
iajs-2804	290	33	then	then	ADV
iajs-2804	290	34	𝐷	𝐷	PROPN
iajs-2804	290	35	=	=	SYM
iajs-2804	290	36	𝐽𝑋	𝐽𝑋	PROPN
iajs-2804	290	37	for	for	ADP
iajs-2804	290	38	some	some	DET
iajs-2804	290	39	an	an	DET
iajs-2804	290	40	ℱ-ideal	ℱ-ideal	PROPN
iajs-2804	290	41	of	of	ADP
iajs-2804	290	42	ℛ	ℛ	PROPN
iajs-2804	290	43	,	,	PUNCT
iajs-2804	290	44	we	we	PRON
iajs-2804	290	45	get	get	VERB
iajs-2804	290	46	(	(	PUNCT
iajs-2804	290	47	𝑟𝑛)𝑏𝐽𝑋	𝑟𝑛)𝑏𝐽𝑋	NOUN
iajs-2804	290	48	⊆	⊆	NUM
iajs-2804	290	49	𝑈	𝑈	PROPN
iajs-2804	290	50	that	that	PRON
iajs-2804	290	51	is	be	AUX
iajs-2804	290	52	mean	mean	ADJ
iajs-2804	290	53	(	(	PUNCT
iajs-2804	290	54	𝑟𝑛)𝑏𝐽	𝑟𝑛)𝑏𝐽	PROPN
iajs-2804	290	55	⊆	⊆	NUM
iajs-2804	291	1	[	[	X
iajs-2804	291	2	𝑈:𝑅	𝑈:𝑅	X
iajs-2804	291	3	𝑋],but	𝑋],but	PROPN
iajs-2804	291	4	[	[	X
iajs-2804	291	5	𝑈:𝑅	𝑈:𝑅	PROPN
iajs-2804	291	6	𝑋	𝑋	NOUN
iajs-2804	291	7	]	]	PUNCT
iajs-2804	291	8	is	be	AUX
iajs-2804	291	9	an	an	DET
iajs-2804	291	10	ℱ-soc	ℱ-soc	NOUN
iajs-2804	291	11	-	-	PUNCT
iajs-2804	291	12	semi	semi	ADJ
iajs-2804	291	13	-	-	ADJ
iajs-2804	291	14	prime	prime	ADJ
iajs-2804	291	15	ideal	ideal	NOUN
iajs-2804	291	16	of	of	ADP
iajs-2804	291	17	ℛ	ℛ	PROPN
iajs-2804	291	18	,	,	PUNCT
iajs-2804	291	19	so	so	ADV
iajs-2804	291	20	by	by	ADP
iajs-2804	291	21	(	(	PUNCT
iajs-2804	291	22	corollary	corollary	ADJ
iajs-2804	291	23	2.12	2.12	NUM
iajs-2804	291	24	)	)	PUNCT
iajs-2804	291	25	we	we	PRON
iajs-2804	291	26	have	have	VERB
iajs-2804	291	27	𝑟𝑏𝐽	𝑟𝑏𝐽	NOUN
iajs-2804	291	28	⊆	⊆	NUM
iajs-2804	291	29	[	[	X
iajs-2804	291	30	𝑈:𝑅	𝑈:𝑅	ADJ
iajs-2804	291	31	𝑋	𝑋	NOUN
iajs-2804	291	32	]	]	X
iajs-2804	292	1	+	+	PROPN
iajs-2804	292	2	𝐹	𝐹	PROPN
iajs-2804	292	3	−	−	NOUN
iajs-2804	292	4	𝑆𝑜𝑐(ℛ	𝑆𝑜𝑐(ℛ	NOUN
iajs-2804	292	5	)	)	PUNCT
iajs-2804	292	6	,	,	PUNCT
iajs-2804	292	7	this	this	PRON
iajs-2804	292	8	implies	imply	VERB
iajs-2804	292	9	𝑟𝑏𝐽𝑋	𝑟𝑏𝐽𝑋	NUM
iajs-2804	292	10	⊆	⊆	NUM
iajs-2804	292	11	[	[	X
iajs-2804	292	12	𝑈:𝑅	𝑈:𝑅	NUM
iajs-2804	292	13	𝑋]𝑋	𝑋]𝑋	X
iajs-2804	292	14	+	+	SYM
iajs-2804	292	15	ℱ	ℱ	PROPN
iajs-2804	292	16	−	−	PROPN
iajs-2804	292	17	𝑆𝑜𝑐(ℛ)𝑋	𝑆𝑜𝑐(ℛ)𝑋	NOUN
iajs-2804	292	18	,	,	PUNCT
iajs-2804	292	19	then	then	ADV
iajs-2804	292	20	by	by	ADP
iajs-2804	292	21	(	(	PUNCT
iajs-2804	292	22	lemma	lemma	PROPN
iajs-2804	292	23	2.25	2.25	NUM
iajs-2804	292	24	)	)	PUNCT
iajs-2804	292	25	we	we	PRON
iajs-2804	292	26	get	get	VERB
iajs-2804	292	27	𝑟𝑏𝐽𝑋	𝑟𝑏𝐽𝑋	NUM
iajs-2804	292	28	⊆	⊆	NUM
iajs-2804	292	29	𝑈	𝑈	PROPN
iajs-2804	292	30	+	+	CCONJ
iajs-2804	292	31	ℱ	ℱ	PROPN
iajs-2804	292	32	−	−	PROPN
iajs-2804	292	33	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	292	34	)	)	PUNCT
iajs-2804	292	35	.	.	PUNCT
iajs-2804	293	1	lemma	lemma	PROPN
iajs-2804	293	2	2.28	2.28	NUM
iajs-2804	293	3	let	let	VERB
iajs-2804	293	4	𝑓	𝑓	PRON
iajs-2804	293	5	:	:	PUNCT
iajs-2804	293	6	m	m	PROPN
iajs-2804	293	7	→	→	SYM
iajs-2804	293	8	�	�	NOUN
iajs-2804	293	9	̅	̅	NOUN
iajs-2804	293	10	�	�	NOUN
iajs-2804	293	11	be	be	VERB
iajs-2804	293	12	isomorphism	isomorphism	NOUN
iajs-2804	293	13	mapping	mapping	NOUN
iajs-2804	293	14	from	from	ADP
iajs-2804	293	15	an	an	DET
iajs-2804	293	16	ℛ-module	ℛ-module	PROPN
iajs-2804	293	17	m	m	NOUN
iajs-2804	293	18	into	into	ADP
iajs-2804	293	19	an	an	DET
iajs-2804	293	20	ℛ-module	ℛ-module	PROPN
iajs-2804	293	21	�	�	PROPN
iajs-2804	293	22	̅	̅	NOUN
iajs-2804	293	23	�	�	NOUN
iajs-2804	293	24	.if	.if	PUNCT
iajs-2804	293	25	x	x	SYM
iajs-2804	293	26	and	and	CCONJ
iajs-2804	293	27	�	�	PROPN
iajs-2804	293	28	̅	̅	NOUN
iajs-2804	293	29	�	�	NOUN
iajs-2804	293	30	are	be	AUX
iajs-2804	293	31	ℱ-modules	ℱ-modules	PROPN
iajs-2804	293	32	of	of	ADP
iajs-2804	293	33	an	an	DET
iajs-2804	293	34	ℛ-modules	ℛ-modules	PROPN
iajs-2804	293	35	m	m	NOUN
iajs-2804	293	36	and	and	CCONJ
iajs-2804	293	37	�	�	PROPN
iajs-2804	293	38	̅	̅	NOUN
iajs-2804	293	39	�	�	NOUN
iajs-2804	293	40	respectively	respectively	ADV
iajs-2804	293	41	.	.	PUNCT
iajs-2804	294	1	then	then	ADV
iajs-2804	294	2	f	f	X
iajs-2804	294	3	(	(	PUNCT
iajs-2804	294	4	ℱ	ℱ	PROPN
iajs-2804	294	5	−	−	PROPN
iajs-2804	294	6	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	294	7	)	)	PUNCT
iajs-2804	294	8	)	)	PUNCT
iajs-2804	294	9	⊆	⊆	NUM
iajs-2804	294	10	ℱ	ℱ	PROPN
iajs-2804	294	11	−	−	PROPN
iajs-2804	294	12	𝑆𝑜𝑐(	𝑆𝑜𝑐(	PROPN
iajs-2804	294	13	�	�	PROPN
iajs-2804	294	14	̅	̅	NOUN
iajs-2804	294	15	�	�	NOUN
iajs-2804	294	16	)	)	PUNCT
iajs-2804	294	17	.	.	PUNCT
iajs-2804	295	1	proposition	proposition	NOUN
iajs-2804	295	2	2.29	2.29	NUM
iajs-2804	295	3	let	let	VERB
iajs-2804	295	4	𝑓	𝑓	X
iajs-2804	295	5	:	:	PUNCT
iajs-2804	295	6	𝑋	𝑋	PROPN
iajs-2804	295	7	→	→	SYM
iajs-2804	295	8	�	�	PROPN
iajs-2804	295	9	̅	̅	NOUN
iajs-2804	295	10	�	�	NOUN
iajs-2804	295	11	be	be	VERB
iajs-2804	295	12	an	an	DET
iajs-2804	295	13	ℱ-isomorphism	ℱ-isomorphism	PROPN
iajs-2804	295	14	from	from	ADP
iajs-2804	295	15	ℱ-module	ℱ-module	PROPN
iajs-2804	295	16	𝑋	𝑋	PROPN
iajs-2804	295	17	into	into	ADP
iajs-2804	295	18	ℱ-module	ℱ-module	PROPN
iajs-2804	295	19	�	�	PROPN
iajs-2804	295	20	̅	̅	NOUN
iajs-2804	295	21	�	�	PROPN
iajs-2804	295	22	,	,	PUNCT
iajs-2804	295	23	with	with	SCONJ
iajs-2804	295	24	u	u	NOUN
iajs-2804	295	25	is	be	AUX
iajs-2804	295	26	an	an	DET
iajs-2804	295	27	ℱ-socsemi	ℱ-socsemi	PROPN
iajs-2804	295	28	-	-	PUNCT
iajs-2804	295	29	prime	prime	ADJ
iajs-2804	295	30	sub	sub	NOUN
iajs-2804	295	31	-	-	NOUN
iajs-2804	295	32	module	module	NOUN
iajs-2804	295	33	of	of	ADP
iajs-2804	295	34	𝑋	𝑋	PROPN
iajs-2804	295	35	,	,	PUNCT
iajs-2804	295	36	such	such	ADJ
iajs-2804	295	37	that	that	DET
iajs-2804	295	38	ker	ker	NOUN
iajs-2804	295	39	(	(	PUNCT
iajs-2804	295	40	𝑓	𝑓	X
iajs-2804	295	41	)	)	PUNCT
iajs-2804	295	42	⊆	⊆	NUM
iajs-2804	295	43	𝑈.	𝑈.	PROPN
iajs-2804	295	44	then	then	ADV
iajs-2804	295	45	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	295	46	)	)	PUNCT
iajs-2804	295	47	is	be	AUX
iajs-2804	295	48	an	an	DET
iajs-2804	295	49	ℱ-soc	ℱ-soc	NOUN
iajs-2804	295	50	-	-	PUNCT
iajs-2804	295	51	semi	semi	ADJ
iajs-2804	295	52	-	-	ADJ
iajs-2804	295	53	prime	prime	ADJ
iajs-2804	295	54	sub	sub	NOUN
iajs-2804	295	55	-	-	NOUN
iajs-2804	295	56	module	module	NOUN
iajs-2804	295	57	of	of	ADP
iajs-2804	295	58	�	�	PROPN
iajs-2804	295	59	̅	̅	NOUN
iajs-2804	295	60	�	�	NOUN
iajs-2804	295	61	.	.	PUNCT
iajs-2804	296	1	proof	proof	NOUN
iajs-2804	296	2	:	:	PUNCT
iajs-2804	296	3	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	296	4	)	)	PUNCT
iajs-2804	296	5	is	be	AUX
iajs-2804	296	6	a	a	DET
iajs-2804	296	7	proper	proper	ADJ
iajs-2804	296	8	ℱ-sub	ℱ-sub	NOUN
iajs-2804	296	9	-	-	NOUN
iajs-2804	296	10	module	module	NOUN
iajs-2804	296	11	of	of	ADP
iajs-2804	296	12	�	�	PROPN
iajs-2804	296	13	̅	̅	NOUN
iajs-2804	296	14	�	�	NOUN
iajs-2804	296	15	.	.	PUNCT
iajs-2804	297	1	if	if	SCONJ
iajs-2804	297	2	not	not	PART
iajs-2804	297	3	,	,	PUNCT
iajs-2804	297	4	then	then	ADV
iajs-2804	297	5	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	297	6	)	)	PUNCT
iajs-2804	297	7	=	=	PUNCT
iajs-2804	297	8	�	�	PROPN
iajs-2804	297	9	̅	̅	NOUN
iajs-2804	297	10	�	�	NOUN
iajs-2804	297	11	.	.	PUNCT
iajs-2804	298	1	let	let	VERB
iajs-2804	298	2	𝑥𝑡	𝑥𝑡	ADP
iajs-2804	298	3	⊆	⊆	PROPN
iajs-2804	298	4	𝑋	𝑋	PROPN
iajs-2804	298	5	,	,	PUNCT
iajs-2804	298	6	so	so	ADV
iajs-2804	298	7	𝑓(𝑥𝑡	𝑓(𝑥𝑡	ADJ
iajs-2804	298	8	)	)	PUNCT
iajs-2804	298	9	⊆	⊆	NUM
iajs-2804	298	10	�	�	NOUN
iajs-2804	298	11	̅	̅	NOUN
iajs-2804	298	12	�	�	NOUN
iajs-2804	298	13	=	=	SYM
iajs-2804	298	14	𝑓(𝑈	𝑓(𝑈	PROPN
iajs-2804	298	15	)	)	PUNCT
iajs-2804	298	16	,	,	PUNCT
iajs-2804	298	17	that	that	PRON
iajs-2804	298	18	is	be	AUX
iajs-2804	298	19	there	there	PRON
iajs-2804	298	20	exists	exist	VERB
iajs-2804	298	21	𝑦𝑠	𝑦𝑠	ADP
iajs-2804	298	22	⊆	⊆	NUM
iajs-2804	298	23	𝑈	𝑈	PROPN
iajs-2804	298	24	where	where	SCONJ
iajs-2804	298	25	𝑠	𝑠	NOUN
iajs-2804	298	26	,	,	PUNCT
iajs-2804	298	27	𝑡	𝑡	PROPN
iajs-2804	298	28	∈	∈	PROPN
iajs-2804	299	1	[	[	X
iajs-2804	299	2	0,1	0,1	NUM
iajs-2804	299	3	]	]	PUNCT
iajs-2804	299	4	such	such	ADJ
iajs-2804	299	5	that	that	SCONJ
iajs-2804	299	6	𝑓(𝑥𝑡	𝑓(𝑥𝑡	VERB
iajs-2804	299	7	)	)	PUNCT
iajs-2804	299	8	=	=	SYM
iajs-2804	299	9	𝑓(𝑦𝑠)implies	𝑓(𝑦𝑠)implie	NOUN
iajs-2804	299	10	that	that	SCONJ
iajs-2804	299	11	𝑓(𝑥𝑡	𝑓(𝑥𝑡	VERB
iajs-2804	299	12	)	)	PUNCT
iajs-2804	299	13	−	−	ADP
iajs-2804	299	14	𝑓(𝑦𝑠	𝑓(𝑦𝑠	NUM
iajs-2804	299	15	)	)	PUNCT
iajs-2804	299	16	=	=	SYM
iajs-2804	300	1	01	01	NUM
iajs-2804	300	2	then	then	ADV
iajs-2804	300	3	𝑓(𝑥𝑡	𝑓(𝑥𝑡	PROPN
iajs-2804	300	4	−	−	PROPN
iajs-2804	300	5	𝑦𝑠	𝑦𝑠	NOUN
iajs-2804	300	6	)	)	PUNCT
iajs-2804	300	7	=	=	SYM
iajs-2804	300	8	01	01	NUM
iajs-2804	300	9	,	,	PUNCT
iajs-2804	300	10	thus	thus	ADV
iajs-2804	300	11	𝑥𝑡	𝑥𝑡	ADV
iajs-2804	300	12	−	−	PROPN
iajs-2804	300	13	𝑦𝑠	𝑦𝑠	INTJ
iajs-2804	300	14	⊆	⊆	NUM
iajs-2804	300	15	ker	ker	NOUN
iajs-2804	300	16	(	(	PUNCT
iajs-2804	300	17	𝑓	𝑓	X
iajs-2804	300	18	)	)	PUNCT
iajs-2804	300	19	⊆	⊆	PROPN
iajs-2804	300	20	𝑈	𝑈	PROPN
iajs-2804	300	21	,	,	PUNCT
iajs-2804	300	22	it	it	PRON
iajs-2804	300	23	follows	follow	VERB
iajs-2804	300	24	that	that	SCONJ
iajs-2804	300	25	𝑥𝑡	𝑥𝑡	ADP
iajs-2804	300	26	⊆	⊆	NUM
iajs-2804	300	27	𝑈.thus	𝑈.thus	X
iajs-2804	300	28	𝑈	𝑈	PROPN
iajs-2804	300	29	=	=	SYM
iajs-2804	300	30	𝑋	𝑋	PROPN
iajs-2804	300	31	that	that	PRON
iajs-2804	300	32	is	be	AUX
iajs-2804	300	33	a	a	DET
iajs-2804	300	34	contradiction	contradiction	NOUN
iajs-2804	300	35	.	.	PUNCT
iajs-2804	301	1	now	now	ADV
iajs-2804	301	2	,	,	PUNCT
iajs-2804	301	3	let	let	VERB
iajs-2804	301	4	(	(	PUNCT
iajs-2804	301	5	𝑟𝑏)𝑛𝑧𝑐	𝑟𝑏)𝑛𝑧𝑐	NOUN
iajs-2804	301	6	⊆	⊆	NUM
iajs-2804	301	7	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	301	8	)	)	PUNCT
iajs-2804	301	9	with	with	ADP
iajs-2804	301	10	𝑟𝑏	𝑟𝑏	ADP
iajs-2804	301	11	⊆	⊆	NUM
iajs-2804	301	12	ℛ	ℛ	NOUN
iajs-2804	301	13	and	and	CCONJ
iajs-2804	301	14	𝑧𝑐	𝑧𝑐	NOUN
iajs-2804	301	15	⊆	⊆	NUM
iajs-2804	301	16	�	�	NOUN
iajs-2804	301	17	̅	̅	NOUN
iajs-2804	301	18	�	�	NOUN
iajs-2804	301	19	with	with	ADP
iajs-2804	301	20	𝑏	𝑏	PROPN
iajs-2804	301	21	,	,	PUNCT
iajs-2804	301	22	𝑐	𝑐	PROPN
iajs-2804	301	23	∈	∈	PROPN
iajs-2804	302	1	[	[	X
iajs-2804	302	2	0,1],but	0,1],but	NOUN
iajs-2804	302	3	𝑓	𝑓	PRON
iajs-2804	302	4	is	be	AUX
iajs-2804	302	5	onto	onto	ADP
iajs-2804	302	6	𝑓(𝑥𝑡	𝑓(𝑥𝑡	ADJ
iajs-2804	302	7	)	)	PUNCT
iajs-2804	302	8	=	=	SYM
iajs-2804	302	9	𝑧𝑐	𝑧𝑐	NOUN
iajs-2804	302	10	for	for	ADP
iajs-2804	302	11	some	some	PRON
iajs-2804	302	12	𝑥𝑡	𝑥𝑡	ADP
iajs-2804	302	13	⊆	⊆	PROPN
iajs-2804	302	14	𝑋	𝑋	PROPN
iajs-2804	302	15	,	,	PUNCT
iajs-2804	302	16	therefore	therefore	ADV
iajs-2804	302	17	(	(	PUNCT
iajs-2804	302	18	𝑟𝑛)𝑏𝑧𝑐	𝑟𝑛)𝑏𝑧𝑐	X
iajs-2804	302	19	=	=	X
iajs-2804	302	20	(	(	PUNCT
iajs-2804	302	21	𝑟𝑛)𝑏𝑓(𝑥𝑡	𝑟𝑛)𝑏𝑓(𝑥𝑡	NUM
iajs-2804	302	22	)	)	PUNCT
iajs-2804	302	23	=	=	PUNCT
iajs-2804	302	24	𝑓((𝑟𝑛)𝑏𝑥𝑡	𝑓((𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	302	25	)	)	PUNCT
iajs-2804	302	26	⊆	⊆	NUM
iajs-2804	302	27	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	302	28	)	)	PUNCT
iajs-2804	302	29	,	,	PUNCT
iajs-2804	302	30	this	this	PRON
iajs-2804	302	31	implies	imply	VERB
iajs-2804	302	32	that	that	SCONJ
iajs-2804	302	33	there	there	PRON
iajs-2804	302	34	exists	exist	VERB
iajs-2804	302	35	𝑘ℎ	𝑘ℎ	ADP
iajs-2804	302	36	⊆	⊆	NUM
iajs-2804	302	37	𝑈	𝑈	PROPN
iajs-2804	302	38	with	with	ADP
iajs-2804	302	39	ℎ	ℎ	PROPN
iajs-2804	302	40	∈	∈	PROPN
iajs-2804	303	1	[	[	X
iajs-2804	303	2	0,1	0,1	NUM
iajs-2804	303	3	]	]	PUNCT
iajs-2804	303	4	such	such	ADJ
iajs-2804	303	5	that	that	SCONJ
iajs-2804	303	6	𝑓(𝑘ℎ	𝑓(𝑘ℎ	NOUN
iajs-2804	303	7	)	)	PUNCT
iajs-2804	303	8	=	=	PUNCT
iajs-2804	303	9	𝑓((𝑟𝑛)𝑏𝑥𝑡	𝑓((𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	303	10	)	)	PUNCT
iajs-2804	303	11	,	,	PUNCT
iajs-2804	303	12	that	that	PRON
iajs-2804	303	13	is	be	AUX
iajs-2804	303	14	𝑓(𝑘ℎ	𝑓(𝑘ℎ	PROPN
iajs-2804	303	15	−	−	PROPN
iajs-2804	303	16	(	(	PUNCT
iajs-2804	303	17	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	303	18	)	)	PUNCT
iajs-2804	303	19	=	=	SYM
iajs-2804	303	20	01	01	NUM
iajs-2804	303	21	,	,	PUNCT
iajs-2804	303	22	so	so	ADV
iajs-2804	304	1	𝑘ℎ	𝑘ℎ	INTJ
iajs-2804	304	2	−	−	PROPN
iajs-2804	304	3	(	(	PUNCT
iajs-2804	304	4	𝑟𝑛)𝑏𝑥𝑡	𝑟𝑛)𝑏𝑥𝑡	NOUN
iajs-2804	304	5	⊆	⊆	NUM
iajs-2804	304	6	ker	ker	NOUN
iajs-2804	304	7	(	(	PUNCT
iajs-2804	304	8	𝑓	𝑓	X
iajs-2804	304	9	)	)	PUNCT
iajs-2804	304	10	⊆	⊆	NUM
iajs-2804	304	11	𝑈.	𝑈.	PROPN
iajs-2804	304	12	it	it	PRON
iajs-2804	304	13	follows	follow	VERB
iajs-2804	304	14	that	that	SCONJ
iajs-2804	304	15	(	(	PUNCT
iajs-2804	304	16	𝑟𝑏)𝑛𝑥𝑡	𝑟𝑏)𝑛𝑥𝑡	NOUN
iajs-2804	304	17	⊆	⊆	NUM
iajs-2804	304	18	𝑈.	𝑈.	PROPN
iajs-2804	304	19	but	but	CCONJ
iajs-2804	304	20	,	,	PUNCT
iajs-2804	304	21	u	u	NOUN
iajs-2804	304	22	is	be	AUX
iajs-2804	304	23	an	an	DET
iajs-2804	304	24	ℱ-soc	ℱ-soc	NOUN
iajs-2804	304	25	-	-	PUNCT
iajs-2804	304	26	semi	semi	ADJ
iajs-2804	304	27	-	-	ADJ
iajs-2804	304	28	prime	prime	ADJ
iajs-2804	304	29	sub	sub	NOUN
iajs-2804	304	30	-	-	NOUN
iajs-2804	304	31	module	module	NOUN
iajs-2804	304	32	of	of	ADP
iajs-2804	304	33	𝑋	𝑋	PROPN
iajs-2804	304	34	,	,	PUNCT
iajs-2804	304	35	thus	thus	ADV
iajs-2804	304	36	𝑟𝑏𝑥𝑡	𝑟𝑏𝑥𝑡	NOUN
iajs-2804	304	37	⊆	⊆	NUM
iajs-2804	304	38	𝑈	𝑈	PROPN
iajs-2804	304	39	+	+	CCONJ
iajs-2804	304	40	ℱ	ℱ	PROPN
iajs-2804	304	41	−	−	PROPN
iajs-2804	304	42	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	304	43	)	)	PUNCT
iajs-2804	304	44	.	.	PUNCT
iajs-2804	305	1	then	then	ADV
iajs-2804	305	2	by	by	ADP
iajs-2804	305	3	(	(	PUNCT
iajs-2804	305	4	lemma	lemma	PROPN
iajs-2804	305	5	2.28	2.28	NUM
iajs-2804	305	6	)	)	PUNCT
iajs-2804	305	7	we	we	PRON
iajs-2804	305	8	have	have	VERB
iajs-2804	305	9	𝑟𝑏𝑧𝑐	𝑟𝑏𝑧𝑐	NUM
iajs-2804	305	10	=	=	SYM
iajs-2804	305	11	𝑟𝑏𝑓(𝑥𝑡	𝑟𝑏𝑓(𝑥𝑡	NUM
iajs-2804	305	12	)	)	PUNCT
iajs-2804	305	13	⊆	⊆	NUM
iajs-2804	305	14	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	305	15	)	)	PUNCT
iajs-2804	306	1	+	+	CCONJ
iajs-2804	306	2	𝑓(ℱ	𝑓(ℱ	ADV
iajs-2804	306	3	−	−	PROPN
iajs-2804	306	4	𝑆𝑜𝑐(𝑋	𝑆𝑜𝑐(𝑋	NUM
iajs-2804	306	5	)	)	PUNCT
iajs-2804	306	6	)	)	PUNCT
iajs-2804	306	7	⊆	⊆	NUM
iajs-2804	306	8	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	306	9	)	)	PUNCT
iajs-2804	307	1	+	+	CCONJ
iajs-2804	307	2	ℱ	ℱ	PROPN
iajs-2804	307	3	−	−	PROPN
iajs-2804	307	4	𝑆𝑜𝑐(	𝑆𝑜𝑐(	PROPN
iajs-2804	307	5	�	�	PROPN
iajs-2804	307	6	̅	̅	NOUN
iajs-2804	307	7	�	�	NOUN
iajs-2804	307	8	)	)	PUNCT
iajs-2804	307	9	.	.	PUNCT
iajs-2804	308	1	hence	hence	ADV
iajs-2804	308	2	𝑓(𝑈	𝑓(𝑈	NUM
iajs-2804	308	3	)	)	PUNCT
iajs-2804	308	4	is	be	AUX
iajs-2804	308	5	an	an	DET
iajs-2804	308	6	ℱ-soc	ℱ-soc	NOUN
iajs-2804	308	7	-	-	PUNCT
iajs-2804	308	8	semi	semi	ADJ
iajs-2804	308	9	-	-	ADJ
iajs-2804	308	10	prime	prime	ADJ
iajs-2804	308	11	submodule	submodule	NOUN
iajs-2804	308	12	of	of	ADP
iajs-2804	308	13	�	�	PROPN
iajs-2804	308	14	̅	̅	NOUN
iajs-2804	308	15	�	�	PROPN
iajs-2804	308	16	.	.	PUNCT
iajs-2804	309	1	ibn	ibn	PROPN
iajs-2804	309	2	al	al	PROPN
iajs-2804	309	3	-	-	PUNCT
iajs-2804	309	4	haitham	haitham	PROPN
iajs-2804	309	5	jour	jour	X
iajs-2804	309	6	.	.	PROPN
iajs-2804	309	7	for	for	ADP
iajs-2804	309	8	pure	pure	ADJ
iajs-2804	309	9	&	&	CCONJ
iajs-2804	309	10	appl	appl	PROPN
iajs-2804	309	11	.	.	PUNCT
iajs-2804	310	1	sci	sci	PROPN
iajs-2804	310	2	.	.	PROPN
iajs-2804	311	1	53	53	NUM
iajs-2804	311	2	(	(	PUNCT
iajs-2804	311	3	1)2022	1)2022	PROPN
iajs-2804	311	4	116	116	NUM
iajs-2804	311	5	2.conclusion	2.conclusion	NUM
iajs-2804	311	6	through	through	ADP
iajs-2804	311	7	this	this	DET
iajs-2804	311	8	research	research	NOUN
iajs-2804	311	9	,	,	PUNCT
iajs-2804	311	10	we	we	PRON
iajs-2804	311	11	were	be	AUX
iajs-2804	311	12	able	able	ADJ
iajs-2804	311	13	to	to	PART
iajs-2804	311	14	know	know	VERB
iajs-2804	311	15	some	some	PRON
iajs-2804	311	16	of	of	ADP
iajs-2804	311	17	the	the	DET
iajs-2804	311	18	fuzzy	fuzzy	ADJ
iajs-2804	311	19	algebraic	algebraic	ADJ
iajs-2804	311	20	properties	property	NOUN
iajs-2804	311	21	of	of	ADP
iajs-2804	311	22	fuzzy	fuzzy	ADJ
iajs-2804	311	23	socle	socle	NOUN
iajs-2804	311	24	semi	semi	ADJ
iajs-2804	311	25	-	-	ADJ
iajs-2804	311	26	prime	prime	ADJ
iajs-2804	311	27	sub	sub	NOUN
iajs-2804	311	28	-	-	NOUN
iajs-2804	311	29	modules	module	NOUN
iajs-2804	311	30	and	and	CCONJ
iajs-2804	311	31	the	the	DET
iajs-2804	311	32	relationship	relationship	NOUN
iajs-2804	311	33	with	with	ADP
iajs-2804	311	34	other	other	ADJ
iajs-2804	311	35	concepts	concept	NOUN
iajs-2804	311	36	.	.	PUNCT
iajs-2804	312	1	the	the	DET
iajs-2804	312	2	idea	idea	NOUN
iajs-2804	312	3	of	of	ADP
iajs-2804	312	4	fuzzy	fuzzy	ADJ
iajs-2804	312	5	socle	socle	NOUN
iajs-2804	312	6	semi	semi	ADJ
iajs-2804	312	7	-	-	ADJ
iajs-2804	312	8	prime	prime	ADJ
iajs-2804	312	9	sub	sub	NOUN
iajs-2804	312	10	-	-	NOUN
iajs-2804	312	11	modules	module	NOUN
iajs-2804	312	12	is	be	AUX
iajs-2804	312	13	dualized	dualize	VERB
iajs-2804	312	14	in	in	ADP
iajs-2804	312	15	this	this	DET
iajs-2804	312	16	study	study	NOUN
iajs-2804	312	17	by	by	ADP
iajs-2804	312	18	introducing	introduce	VERB
iajs-2804	312	19	several	several	ADJ
iajs-2804	312	20	characteristics	characteristic	NOUN
iajs-2804	312	21	and	and	CCONJ
iajs-2804	312	22	properties	property	NOUN
iajs-2804	312	23	of	of	ADP
iajs-2804	312	24	semi	semi	ADJ
iajs-2804	312	25	-	-	ADJ
iajs-2804	312	26	prime	prime	ADJ
iajs-2804	312	27	fuzzy	fuzzy	ADJ
iajs-2804	312	28	sub	sub	NOUN
iajs-2804	312	29	-	-	NOUN
iajs-2804	312	30	modules	module	NOUN
iajs-2804	312	31	.	.	PUNCT
iajs-2804	313	1	this	this	DET
iajs-2804	313	2	approach	approach	NOUN
iajs-2804	313	3	has	have	AUX
iajs-2804	313	4	opened	open	VERB
iajs-2804	313	5	up	up	ADP
iajs-2804	313	6	new	new	ADJ
iajs-2804	313	7	possibilities	possibility	NOUN
iajs-2804	313	8	for	for	ADP
iajs-2804	313	9	studying	study	VERB
iajs-2804	313	10	the	the	DET
iajs-2804	313	11	fuzzy	fuzzy	ADJ
iajs-2804	313	12	dimension	dimension	NOUN
iajs-2804	313	13	.	.	PUNCT
iajs-2804	314	1	thus	thus	ADV
iajs-2804	314	2	,	,	PUNCT
iajs-2804	314	3	socle	socle	NOUN
iajs-2804	314	4	semi	semi	ADJ
iajs-2804	314	5	-	-	ADJ
iajs-2804	314	6	prime	prime	ADJ
iajs-2804	314	7	module	module	NOUN
iajs-2804	314	8	and	and	CCONJ
iajs-2804	314	9	completely	completely	ADV
iajs-2804	314	10	socle	socle	NOUN
iajs-2804	314	11	semi	semi	ADJ
iajs-2804	314	12	-	-	ADJ
iajs-2804	314	13	prime	prime	ADJ
iajs-2804	314	14	sub	sub	NOUN
iajs-2804	314	15	-	-	NOUN
iajs-2804	314	16	modules	module	NOUN
iajs-2804	314	17	can	can	AUX
iajs-2804	314	18	be	be	AUX
iajs-2804	314	19	defined	define	VERB
iajs-2804	314	20	utilizing	utilize	VERB
iajs-2804	314	21	the	the	DET
iajs-2804	314	22	concept	concept	NOUN
iajs-2804	314	23	of	of	ADP
iajs-2804	314	24	fuzzy	fuzzy	ADJ
iajs-2804	314	25	socle	socle	NOUN
iajs-2804	314	26	semi	semi	ADJ
iajs-2804	314	27	-	-	ADJ
iajs-2804	314	28	prime	prime	ADJ
iajs-2804	314	29	sub	sub	NOUN
iajs-2804	314	30	-	-	NOUN
iajs-2804	314	31	modules	module	NOUN
iajs-2804	314	32	.	.	PUNCT
iajs-2804	315	1	references	reference	NOUN
iajs-2804	315	2	1	1	NUM
iajs-2804	315	3	.	.	PUNCT
iajs-2804	315	4	zadeh	zadeh	PROPN
iajs-2804	315	5	l.	l.	PROPN
iajs-2804	315	6	a.	a.	PROPN
iajs-2804	315	7	1965	1965	NUM
iajs-2804	315	8	.	.	PUNCT
iajs-2804	316	1	fuzzy	fuzzy	ADJ
iajs-2804	316	2	sets	set	NOUN
iajs-2804	316	3	,	,	PUNCT
iajs-2804	316	4	information	information	NOUN
iajs-2804	316	5	and	and	CCONJ
iajs-2804	316	6	control	control	NOUN
iajs-2804	316	7	,	,	PUNCT
iajs-2804	316	8	8	8	NUM
iajs-2804	316	9	:	:	SYM
iajs-2804	316	10	338	338	NUM
iajs-2804	316	11	-	-	SYM
iajs-2804	316	12	353	353	NUM
iajs-2804	316	13	,	,	PUNCT
iajs-2804	316	14	1965	1965	NUM
iajs-2804	316	15	.	.	PUNCT
iajs-2804	317	1	2	2	X
iajs-2804	317	2	.	.	X
iajs-2804	317	3	naegoita	naegoita	PROPN
iajs-2804	317	4	,	,	PUNCT
iajs-2804	317	5	c.	c.	PROPN
iajs-2804	317	6	v.	v.	PROPN
iajs-2804	317	7	;	;	PUNCT
iajs-2804	317	8	ralescu	ralescu	NOUN
iajs-2804	317	9	,	,	PUNCT
iajs-2804	317	10	d.	d.	PROPN
iajs-2804	317	11	a.	a.	NOUN
iajs-2804	317	12	application	application	NOUN
iajs-2804	317	13	of	of	ADP
iajs-2804	317	14	fuzzy	fuzzy	ADJ
iajs-2804	317	15	sets	set	NOUN
iajs-2804	317	16	in	in	ADP
iajs-2804	317	17	system	system	NOUN
iajs-2804	317	18	analysis	analysis	NOUN
iajs-2804	317	19	,	,	PUNCT
iajs-2804	317	20	birkhauser	birkhauser	NOUN
iajs-2804	317	21	,	,	PUNCT
iajs-2804	317	22	basel	basel	PROPN
iajs-2804	317	23	,	,	PUNCT
iajs-2804	317	24	switzerland	switzerland	PROPN
iajs-2804	317	25	,	,	PUNCT
iajs-2804	317	26	1975	1975	NUM
iajs-2804	317	27	.	.	PUNCT
iajs-2804	318	1	3	3	X
iajs-2804	318	2	.	.	X
iajs-2804	318	3	hadi	hadi	PROPN
iajs-2804	318	4	,	,	PUNCT
iajs-2804	318	5	i.	i.	PROPN
iajs-2804	318	6	m.	m.	PROPN
iajs-2804	318	7	a.	a.	PROPN
iajs-2804	318	8	semiprime	semiprime	PROPN
iajs-2804	318	9	fuzzy	fuzzy	ADJ
iajs-2804	318	10	sub	sub	NOUN
iajs-2804	318	11	-	-	NOUN
iajs-2804	318	12	modules	module	NOUN
iajs-2804	318	13	of	of	ADP
iajs-2804	318	14	fuzzy	fuzzy	ADJ
iajs-2804	318	15	modules	module	NOUN
iajs-2804	318	16	,	,	PUNCT
iajs-2804	318	17	ibn	ibn	PROPN
iajs-2804	318	18	-	-	PUNCT
iajs-2804	318	19	al	al	PROPN
iajs-2804	318	20	-	-	PUNCT
iajs-2804	318	21	haitham	haitham	PROPN
iajs-2804	318	22	j.	j.	PROPN
iajs-2804	318	23	for	for	ADP
iajs-2804	318	24	pure	pure	ADJ
iajs-2804	318	25	and	and	CCONJ
iajs-2804	318	26	appl	appl	NOUN
iajs-2804	318	27	.	.	PUNCT
iajs-2804	319	1	sci	sci	PROPN
iajs-2804	319	2	.	.	PROPN
iajs-2804	319	3	,	,	PUNCT
iajs-2804	319	4	2004,17(3),112	2004,17(3),112	PROPN
iajs-2804	319	5	-	-	SYM
iajs-2804	319	6	123	123	NUM
iajs-2804	319	7	.	.	PUNCT
iajs-2804	320	1	4	4	X
iajs-2804	320	2	.	.	X
iajs-2804	321	1	al	al	PROPN
iajs-2804	321	2	,	,	PUNCT
iajs-2804	321	3	i	i	PROPN
iajs-2804	321	4	s.a	s.a	PROPN
iajs-2804	321	5	.	.	PROPN
iajs-2804	321	6	approximately	approximately	ADV
iajs-2804	321	7	prime	prime	ADJ
iajs-2804	321	8	sub	sub	NOUN
iajs-2804	321	9	-	-	NOUN
iajs-2804	321	10	modules	module	NOUN
iajs-2804	321	11	and	and	CCONJ
iajs-2804	321	12	some	some	PRON
iajs-2804	321	13	of	of	ADP
iajs-2804	321	14	their	their	PRON
iajs-2804	321	15	generalizations	generalization	NOUN
iajs-2804	321	16	m.sc.thesis	m.sc.thesis	NOUN
iajs-2804	321	17	,	,	PUNCT
iajs-2804	321	18	university	university	NOUN
iajs-2804	321	19	of	of	ADP
iajs-2804	321	20	tikrit	tikrit	NOUN
iajs-2804	321	21	.	.	PUNCT
iajs-2804	322	1	2019	2019	NUM
iajs-2804	322	2	5	5	NUM
iajs-2804	322	3	.	.	PUNCT
iajs-2804	323	1	martinez	martinez	PROPN
iajs-2804	323	2	,	,	PUNCT
iajs-2804	323	3	l.	l.	PROPN
iajs-2804	323	4	,	,	PUNCT
iajs-2804	323	5	fuzzy	fuzzy	ADJ
iajs-2804	323	6	modules	module	NOUN
iajs-2804	323	7	over	over	ADP
iajs-2804	323	8	fuzzy	fuzzy	ADJ
iajs-2804	323	9	rings	ring	NOUN
iajs-2804	323	10	in	in	ADP
iajs-2804	323	11	connection	connection	NOUN
iajs-2804	323	12	with	with	ADP
iajs-2804	323	13	fuzzy	fuzzy	ADJ
iajs-2804	323	14	ideals	ideal	NOUN
iajs-2804	323	15	of	of	ADP
iajs-2804	323	16	rings	ring	NOUN
iajs-2804	323	17	,	,	PUNCT
iajs-2804	323	18	j.fuzzy	j.fuzzy	ADJ
iajs-2804	324	1	math.1996	math.1996	PROPN
iajs-2804	324	2	,	,	PUNCT
iajs-2804	324	3	4	4	NUM
iajs-2804	324	4	,	,	PUNCT
iajs-2804	324	5	843	843	NUM
iajs-2804	324	6	-	-	NOUN
iajs-2804	324	7	857	857	NUM
iajs-2804	324	8	.	.	PUNCT
iajs-2804	325	1	6	6	NUM
iajs-2804	325	2	.	.	X
iajs-2804	325	3	zahedi	zahedi	PROPN
iajs-2804	325	4	,	,	PUNCT
iajs-2804	325	5	m.	m.	NOUN
iajs-2804	325	6	m.	m.	NOUN
iajs-2804	325	7	on	on	ADP
iajs-2804	325	8	l	l	ADJ
iajs-2804	325	9	-	-	ADJ
iajs-2804	325	10	fuzzy	fuzzy	ADJ
iajs-2804	325	11	residual	residual	ADJ
iajs-2804	325	12	quotient	quotient	NOUN
iajs-2804	325	13	modules	module	NOUN
iajs-2804	325	14	and	and	CCONJ
iajs-2804	325	15	p.	p.	NOUN
iajs-2804	325	16	primary	primary	ADJ
iajs-2804	325	17	sub	sub	NOUN
iajs-2804	325	18	-	-	NOUN
iajs-2804	325	19	modules	module	NOUN
iajs-2804	325	20	,	,	PUNCT
iajs-2804	325	21	fuzzy	fuzzy	ADJ
iajs-2804	325	22	sets	set	NOUN
iajs-2804	325	23	and	and	CCONJ
iajs-2804	325	24	systems	system	NOUN
iajs-2804	325	25	,	,	PUNCT
iajs-2804	325	26	1992,51	1992,51	NUM
iajs-2804	325	27	:	:	PUNCT
iajs-2804	325	28	333	333	NUM
iajs-2804	325	29	-	-	SYM
iajs-2804	325	30	344	344	NUM
iajs-2804	325	31	.	.	PUNCT
iajs-2804	326	1	7	7	NUM
iajs-2804	326	2	.	.	X
iajs-2804	326	3	mukherjee	mukherjee	NOUN
iajs-2804	326	4	,	,	PUNCT
iajs-2804	326	5	t.	t.	PROPN
iajs-2804	326	6	k.	k.	PROPN
iajs-2804	326	7	;	;	PUNCT
iajs-2804	326	8	sen	sen	PROPN
iajs-2804	326	9	,	,	PUNCT
iajs-2804	326	10	m.	m.	PROPN
iajs-2804	326	11	k.	k.	PROPN
iajs-2804	326	12	;	;	PUNCT
iajs-2804	326	13	roy	roy	PROPN
iajs-2804	326	14	,	,	PUNCT
iajs-2804	326	15	d.	d.	PROPN
iajs-2804	326	16	on	on	ADP
iajs-2804	326	17	fuzzy	fuzzy	ADJ
iajs-2804	326	18	sub	sub	NOUN
iajs-2804	326	19	-	-	NOUN
iajs-2804	326	20	modules	module	NOUN
iajs-2804	326	21	and	and	CCONJ
iajs-2804	326	22	their	their	PRON
iajs-2804	326	23	radicals	radical	NOUN
iajs-2804	326	24	,	,	PUNCT
iajs-2804	326	25	j.	j.	PROPN
iajs-2804	326	26	fuzzy	fuzzy	PROPN
iajs-2804	326	27	math	math	PROPN
iajs-2804	326	28	.	.	PUNCT
iajs-2804	326	29	,	,	PUNCT
iajs-2804	326	30	1996	1996	NUM
iajs-2804	326	31	,	,	PUNCT
iajs-2804	326	32	4	4	NUM
iajs-2804	326	33	,	,	PUNCT
iajs-2804	326	34	549	549	NUM
iajs-2804	326	35	-	-	SYM
iajs-2804	326	36	558	558	NUM
iajs-2804	326	37	.	.	PUNCT
iajs-2804	327	1	8.mashinchi	8.mashinchi	NUM
iajs-2804	327	2	,	,	PUNCT
iajs-2804	327	3	m.	m.	NOUN
iajs-2804	327	4	;	;	PUNCT
iajs-2804	327	5	zahedi	zahedi	PROPN
iajs-2804	327	6	,	,	PUNCT
iajs-2804	327	7	m.m	m.m	PROPN
iajs-2804	327	8	.	.	PROPN
iajs-2804	327	9	,	,	PUNCT
iajs-2804	327	10	2,"on	2,"on	NUM
iajs-2804	327	11	l	l	ADJ
iajs-2804	327	12	-	-	ADJ
iajs-2804	327	13	fuzzy	fuzzy	ADJ
iajs-2804	327	14	primary	primary	ADJ
iajs-2804	327	15	sub	sub	NOUN
iajs-2804	327	16	-	-	NOUN
iajs-2804	327	17	modules	module	NOUN
iajs-2804	327	18	,	,	PUNCT
iajs-2804	327	19	fuzzy	fuzzy	ADJ
iajs-2804	327	20	sets	set	NOUN
iajs-2804	327	21	systems	system	NOUN
iajs-2804	327	22	,	,	PUNCT
iajs-2804	327	23	199,49,231	199,49,231	NUM
iajs-2804	327	24	-	-	SYM
iajs-2804	327	25	236	236	NUM
iajs-2804	327	26	.	.	PUNCT
iajs-2804	328	1	9.rabi	9.rabi	NUM
iajs-2804	329	1	h.	h.	PROPN
iajs-2804	329	2	j.	j.	PROPN
iajs-2804	329	3	.	.	PUNCT
iajs-2804	330	1	prime	prime	PROPN
iajs-2804	330	2	fuzzy	fuzzy	ADJ
iajs-2804	330	3	sub	sub	NOUN
iajs-2804	330	4	-	-	NOUN
iajs-2804	330	5	module	module	ADJ
iajs-2804	330	6	and	and	CCONJ
iajs-2804	330	7	prime	prime	ADJ
iajs-2804	330	8	fuzzy	fuzzy	ADJ
iajs-2804	330	9	modules	module	NOUN
iajs-2804	330	10	,	,	PUNCT
iajs-2804	330	11	m.	m.	PROPN
iajs-2804	330	12	sc	sc	PROPN
iajs-2804	330	13	.	.	PUNCT
iajs-2804	331	1	thesis	thesis	PROPN
iajs-2804	331	2	,	,	PUNCT
iajs-2804	331	3	university	university	NOUN
iajs-2804	331	4	of	of	ADP
iajs-2804	331	5	baghdad	baghdad	PROPN
iajs-2804	331	6	.	.	PUNCT
iajs-2804	332	1	2001	2001	NUM
iajs-2804	332	2	10.zahedi	10.zahedi	NUM
iajs-2804	332	3	,	,	PUNCT
iajs-2804	332	4	m.	m.	NOUN
iajs-2804	332	5	m.	m.	NOUN
iajs-2804	332	6	.	.	PUNCT
iajs-2804	333	1	a	a	DET
iajs-2804	333	2	characterization	characterization	NOUN
iajs-2804	333	3	of	of	ADP
iajs-2804	333	4	l	l	ADJ
iajs-2804	333	5	-	-	ADJ
iajs-2804	333	6	fuzzy	fuzzy	ADJ
iajs-2804	333	7	prime	prime	ADJ
iajs-2804	333	8	ideals	ideal	NOUN
iajs-2804	333	9	,	,	PUNCT
iajs-2804	333	10	fuzzy	fuzzy	ADJ
iajs-2804	333	11	sets	set	NOUN
iajs-2804	333	12	and	and	CCONJ
iajs-2804	333	13	systems	system	NOUN
iajs-2804	333	14	,	,	PUNCT
iajs-2804	333	15	1991,44	1991,44	NUM
iajs-2804	333	16	:	:	PUNCT
iajs-2804	333	17	147	147	NUM
iajs-2804	333	18	-	-	SYM
iajs-2804	333	19	160	160	NUM
iajs-2804	333	20	.	.	PUNCT
iajs-2804	333	21	11	11	NUM
iajs-2804	333	22	.	.	PUNCT
iajs-2804	334	1	al	al	PROPN
iajs-2804	334	2	-	-	PUNCT
iajs-2804	334	3	abege	abege	PROPN
iajs-2804	334	4	a.	a.	NOUN
iajs-2804	334	5	m.	m.	PROPN
iajs-2804	334	6	h	h	PROPN
iajs-2804	334	7	,	,	PUNCT
iajs-2804	334	8	near	near	ADP
iajs-2804	334	9	-	-	PUNCT
iajs-2804	334	10	ring	ring	NOUN
iajs-2804	334	11	,	,	PUNCT
iajs-2804	334	12	near	near	ADP
iajs-2804	334	13	module	module	NOUN
iajs-2804	334	14	and	and	CCONJ
iajs-2804	334	15	their	their	PRON
iajs-2804	334	16	spectrum	spectrum	NOUN
iajs-2804	334	17	,	,	PUNCT
iajs-2804	334	18	m.sc.thesis	m.sc.thesis	NOUN
iajs-2804	334	19	,	,	PUNCT
iajs-2804	334	20	university	university	NOUN
iajs-2804	334	21	of	of	ADP
iajs-2804	334	22	kufa	kufa	PROPN
iajs-2804	334	23	,	,	PUNCT
iajs-2804	334	24	college	college	NOUN
iajs-2804	334	25	of	of	ADP
iajs-2804	334	26	mathematics	mathematic	NOUN
iajs-2804	334	27	and	and	CCONJ
iajs-2804	334	28	computers	computer	NOUN
iajs-2804	334	29	sciences	science	NOUN
iajs-2804	334	30	.	.	PUNCT
iajs-2804	335	1	2010	2010	NUM
iajs-2804	335	2	12.mashinchi	12.mashinchi	NUM
iajs-2804	335	3	,	,	PUNCT
iajs-2804	335	4	m.	m.	NOUN
iajs-2804	335	5	;	;	PUNCT
iajs-2804	335	6	zahedi	zahedi	PROPN
iajs-2804	335	7	,	,	PUNCT
iajs-2804	335	8	m.m	m.m	PROPN
iajs-2804	335	9	.	.	PROPN
iajs-2804	335	10	,	,	PUNCT
iajs-2804	335	11	on	on	ADP
iajs-2804	335	12	l	l	ADJ
iajs-2804	335	13	-	-	ADJ
iajs-2804	335	14	fuzzy	fuzzy	ADJ
iajs-2804	335	15	primary	primary	ADJ
iajs-2804	335	16	sub	sub	NOUN
iajs-2804	335	17	-	-	NOUN
iajs-2804	335	18	modules	module	NOUN
iajs-2804	335	19	,	,	PUNCT
iajs-2804	335	20	fuzzy	fuzzy	ADJ
iajs-2804	335	21	sets	set	NOUN
iajs-2804	335	22	systems	system	NOUN
iajs-2804	335	23	,	,	PUNCT
iajs-2804	335	24	1996	1996	NUM
iajs-2804	335	25	,	,	PUNCT
iajs-2804	335	26	4	4	NUM
iajs-2804	335	27	,	,	PUNCT
iajs-2804	335	28	843	843	NUM
iajs-2804	335	29	-	-	NOUN
iajs-2804	335	30	857	857	NUM
iajs-2804	335	31	.	.	PUNCT
iajs-2804	336	1	13	13	NUM
iajs-2804	336	2	.	.	X
iajs-2804	337	1	gada	gada	PROPN
iajs-2804	337	2	,	,	PUNCT
iajs-2804	337	3	a.a	a.a	PROPN
iajs-2804	337	4	.	.	PROPN
iajs-2804	337	5	,	,	PUNCT
iajs-2804	337	6	fuzzy	fuzzy	ADJ
iajs-2804	337	7	spectrum	spectrum	NOUN
iajs-2804	337	8	of	of	ADP
iajs-2804	337	9	modules	module	NOUN
iajs-2804	337	10	over	over	ADP
iajs-2804	337	11	commutative	commutative	ADJ
iajs-2804	337	12	rings	ring	NOUN
iajs-2804	337	13	,	,	PUNCT
iajs-2804	337	14	m.sc.thesis	m.sc.thesis	NOUN
iajs-2804	337	15	,	,	PUNCT
iajs-2804	337	16	university	university	NOUN
iajs-2804	337	17	of	of	ADP
iajs-2804	337	18	baghdad	baghdad	PROPN
iajs-2804	337	19	.	.	PUNCT
iajs-2804	338	1	2000	2000	NUM
iajs-2804	338	2	14.kasch	14.kasch	NUM
iajs-2804	338	3	,	,	PUNCT
iajs-2804	338	4	f.	f.	PROPN
iajs-2804	338	5	.	.	PUNCT
iajs-2804	339	1	modules	module	NOUN
iajs-2804	339	2	and	and	CCONJ
iajs-2804	339	3	rings	ring	NOUN
iajs-2804	339	4	,	,	PUNCT
iajs-2804	339	5	academic	academic	ADJ
iajs-2804	339	6	press	press	NOUN
iajs-2804	339	7	.	.	PUNCT
iajs-2804	340	1	1982	1982	NUM
iajs-2804	340	2	15.hatam	15.hatam	NUM
iajs-2804	340	3	y.	y.	PROPN
iajs-2804	340	4	k.	k.	PROPN
iajs-2804	340	5	,	,	PUNCT
iajs-2804	340	6	fuzzy	fuzzy	ADJ
iajs-2804	340	7	quasi	quasi	ADJ
iajs-2804	340	8	-	-	ADJ
iajs-2804	340	9	prime	prime	ADJ
iajs-2804	340	10	modules	module	NOUN
iajs-2804	340	11	and	and	CCONJ
iajs-2804	340	12	fuzzy	fuzzy	ADJ
iajs-2804	340	13	quasi	quasi	NOUN
iajs-2804	340	14	-	-	NOUN
iajs-2804	340	15	prime	prime	ADJ
iajs-2804	340	16	.	.	PUNCT
iajs-2804	341	1	sub	sub	NOUN
iajs-2804	341	2	-	-	NOUN
iajs-2804	341	3	modules	module	NOUN
iajs-2804	341	4	,	,	PUNCT
iajs-2804	341	5	m.sc	m.sc	PROPN
iajs-2804	342	1	.	.	PUNCT
iajs-2804	342	2	thesis	thesis	NOUN
iajs-2804	342	3	,	,	PUNCT
iajs-2804	342	4	university	university	NOUN
iajs-2804	342	5	of	of	ADP
iajs-2804	342	6	baghdad	baghdad	PROPN
iajs-2804	342	7	.	.	PUNCT
iajs-2804	343	1	2001	2001	NUM
iajs-2804	343	2	16	16	NUM
iajs-2804	343	3	..	..	PUNCT
iajs-2804	343	4	kalita	kalita	PROPN
iajs-2804	343	5	,	,	PUNCT
iajs-2804	343	6	m.	m.	NOUN
iajs-2804	343	7	c	c	PROPN
iajs-2804	343	8	a	a	DET
iajs-2804	343	9	study	study	NOUN
iajs-2804	343	10	of	of	ADP
iajs-2804	343	11	fuzzy	fuzzy	ADJ
iajs-2804	343	12	algebraic	algebraic	ADJ
iajs-2804	343	13	structures	structure	NOUN
iajs-2804	343	14	:	:	PUNCT
iajs-2804	343	15	some	some	DET
iajs-2804	343	16	special	special	ADJ
iajs-2804	343	17	types	type	NOUN
iajs-2804	343	18	,	,	PUNCT
iajs-2804	343	19	ph.d	ph.d	PROPN
iajs-2804	343	20	thesis	thesis	NOUN
iajs-2804	343	21	,	,	PUNCT
iajs-2804	343	22	gauhati	gauhati	PROPN
iajs-2804	343	23	university	university	PROPN
iajs-2804	343	24	,	,	PUNCT
iajs-2804	343	25	gauhati	gauhati	PROPN
iajs-2804	343	26	,	,	PUNCT
iajs-2804	343	27	india	india	PROPN
iajs-2804	343	28	,	,	PUNCT
iajs-2804	343	29	2007	2007	NUM
iajs-2804	343	30	.	.	PUNCT
iajs-2804	344	1	17.wafaa	17.wafaa	NUM
iajs-2804	344	2	,	,	PUNCT
iajs-2804	344	3	h.	h.	PROPN
iajs-2804	344	4	h.t	h.t	PROPN
iajs-2804	344	5	-	-	PUNCT
iajs-2804	344	6	abso	abso	PROPN
iajs-2804	344	7	fuzzy	fuzzy	ADJ
iajs-2804	344	8	sub	sub	NOUN
iajs-2804	344	9	-	-	NOUN
iajs-2804	344	10	modules	module	NOUN
iajs-2804	344	11	and	and	CCONJ
iajs-2804	344	12	t	t	PROPN
iajs-2804	344	13	-	-	PUNCT
iajs-2804	344	14	abso	abso	PROPN
iajs-2804	344	15	fuzzy	fuzzy	ADJ
iajs-2804	344	16	modules	module	NOUN
iajs-2804	344	17	and	and	CCONJ
iajs-2804	344	18	some	some	DET
iajs-2804	344	19	their	their	PRON
iajs-2804	344	20	generalizations	generalization	NOUN
iajs-2804	344	21	,	,	PUNCT
iajs-2804	344	22	ph.d	ph.d	PROPN
iajs-2804	344	23	.	.	PUNCT
iajs-2804	345	1	thesis	thesis	NOUN
iajs-2804	345	2	,	,	PUNCT
iajs-2804	345	3	university	university	NOUN
iajs-2804	345	4	of	of	ADP
iajs-2804	345	5	baghdad	baghdad	PROPN
iajs-2804	345	6	.	.	PUNCT
iajs-2804	346	1	2018	2018	NUM
iajs-2804	346	2	.	.	PUNCT
iajs-2804	347	1	ibn	ibn	PROPN
iajs-2804	347	2	al	al	PROPN
iajs-2804	347	3	-	-	PUNCT
iajs-2804	347	4	haitham	haitham	PROPN
iajs-2804	347	5	jour	jour	X
iajs-2804	347	6	.	.	PROPN
iajs-2804	347	7	for	for	ADP
iajs-2804	347	8	pure	pure	ADJ
iajs-2804	347	9	&	&	CCONJ
iajs-2804	347	10	appl	appl	PROPN
iajs-2804	347	11	.	.	PUNCT
iajs-2804	348	1	sci	sci	PROPN
iajs-2804	348	2	.	.	PROPN
iajs-2804	349	1	53	53	NUM
iajs-2804	349	2	(	(	PUNCT
iajs-2804	349	3	1)2022	1)2022	PROPN
iajs-2804	349	4	117	117	NUM
iajs-2804	349	5	18	18	NUM
iajs-2804	349	6	.	.	PUNCT
iajs-2804	350	1	hadi	hadi	PROPN
iajs-2804	350	2	,	,	PUNCT
iajs-2804	350	3	g.	g.	PROPN
iajs-2804	350	4	rashed	rashed	PROPN
iajs-2804	350	5	,	,	PUNCT
iajs-2804	350	6	fully	fully	ADV
iajs-2804	350	7	cancellation	cancellation	NOUN
iajs-2804	350	8	fuzzy	fuzzy	ADJ
iajs-2804	350	9	modules	module	NOUN
iajs-2804	350	10	and	and	CCONJ
iajs-2804	350	11	some	some	DET
iajs-2804	350	12	generalizations	generalization	NOUN
iajs-2804	350	13	,	,	PUNCT
iajs-2804	350	14	m.sc	m.sc	PROPN
iajs-2804	350	15	.	.	PUNCT
iajs-2804	351	1	thesis	thesis	NOUN
iajs-2804	351	2	,	,	PUNCT
iajs-2804	351	3	university	university	NOUN
iajs-2804	351	4	of	of	ADP
iajs-2804	351	5	baghdad	baghdad	PROPN
iajs-2804	351	6	.	.	PUNCT
iajs-2804	352	1	2017	2017	NUM
iajs-2804	352	2	19.goodreal	19.goodreal	NUM
iajs-2804	352	3	,	,	PUNCT
iajs-2804	352	4	k.	k.	PROPN
iajs-2804	352	5	r.	r.	PROPN
iajs-2804	352	6	ring	ring	PROPN
iajs-2804	352	7	theory	theory	NOUN
iajs-2804	352	8	–	–	PUNCT
iajs-2804	352	9	non	non	X
iajs-2804	352	10	singular	singular	PROPN
iajs-2804	352	11	rings	ring	NOUN
iajs-2804	352	12	and	and	CCONJ
iajs-2804	352	13	modules	module	NOUN
iajs-2804	352	14	,	,	PUNCT
iajs-2804	352	15	marci	marci	NOUN
iajs-2804	352	16	-	-	NOUN
iajs-2804	352	17	dekker	dekker	PROPN
iajs-2804	352	18	,	,	PUNCT
iajs-2804	352	19	new	new	PROPN
iajs-2804	352	20	york	york	PROPN
iajs-2804	352	21	and	and	CCONJ
iajs-2804	352	22	basel	basel	PROPN
iajs-2804	352	23	.	.	PUNCT
iajs-2804	353	1	1976	1976	NUM