id	sid	tid	token	lemma	pos
iajs-2613	1	1	56	56	NUM
iajs-2613	1	2	ibn	ibn	PROPN
iajs-2613	1	3	al	al	PROPN
iajs-2613	1	4	-	-	PUNCT
iajs-2613	1	5	haitham	haitham	PROPN
iajs-2613	1	6	jour	jour	X
iajs-2613	1	7	.	.	PROPN
iajs-2613	1	8	for	for	ADP
iajs-2613	1	9	pure	pure	ADJ
iajs-2613	1	10	&	&	CCONJ
iajs-2613	1	11	appl	appl	PROPN
iajs-2613	1	12	.	.	PUNCT
iajs-2613	2	1	sci	sci	PROPN
iajs-2613	2	2	.	.	PROPN
iajs-2613	3	1	34	34	NUM
iajs-2613	3	2	(	(	PUNCT
iajs-2613	3	3	1	1	NUM
iajs-2613	3	4	)	)	PUNCT
iajs-2613	3	5	2021	2021	NUM
iajs-2613	3	6	weakly	weakly	ADV
iajs-2613	3	7	approximaitly	approximaitly	ADV
iajs-2613	3	8	quasi	quasi	ADJ
iajs-2613	3	9	-	-	ADJ
iajs-2613	3	10	prime	prime	ADJ
iajs-2613	3	11	submodules	submodule	NOUN
iajs-2613	3	12	and	and	CCONJ
iajs-2613	3	13	related	related	ADJ
iajs-2613	3	14	concepts	concept	NOUN
iajs-2613	3	15	haibat	haibat	PROPN
iajs-2613	3	16	k.	k.	PROPN
iajs-2613	3	17	mohammadali	mohammadali	PROPN
iajs-2613	3	18	shahad	shahad	PROPN
iajs-2613	3	19	j.	j.	PROPN
iajs-2613	3	20	mahmood	mahmood	PROPN
iajs-2613	3	21	dr	dr	PROPN
iajs-2613	3	22	,	,	PUNCT
iajs-2613	3	23	mohammadali	mohammadali	PROPN
iajs-2613	3	24	2013@gmail.com	2013@gmail.com	X
iajs-2613	3	25	hasanenjassm@gmail.com	hasanenjassm@gmail.com	X
iajs-2613	3	26	department	department	PROPN
iajs-2613	3	27	of	of	ADP
iajs-2613	3	28	mathematics	mathematics	PROPN
iajs-2613	3	29	,	,	PUNCT
iajs-2613	3	30	college	college	NOUN
iajs-2613	3	31	of	of	ADP
iajs-2613	3	32	computer	computer	NOUN
iajs-2613	3	33	sciences	sciences	PROPN
iajs-2613	3	34	and	and	CCONJ
iajs-2613	3	35	mathematics	mathematic	NOUN
iajs-2613	3	36	,	,	PUNCT
iajs-2613	3	37	university	university	NOUN
iajs-2613	3	38	of	of	ADP
iajs-2613	3	39	tikrit	tikrit	NOUN
iajs-2613	3	40	,	,	PUNCT
iajs-2613	3	41	tikrit	tikrit	NOUN
iajs-2613	3	42	,	,	PUNCT
iajs-2613	3	43	iraq	iraq	PROPN
iajs-2613	3	44	abstract	abstract	ADV
iajs-2613	3	45	let	let	VERB
iajs-2613	3	46	r	r	PRON
iajs-2613	3	47	be	be	AUX
iajs-2613	3	48	commutative	commutative	ADJ
iajs-2613	3	49	ring	ring	NOUN
iajs-2613	3	50	,	,	PUNCT
iajs-2613	3	51	and	and	CCONJ
iajs-2613	3	52	let	let	VERB
iajs-2613	3	53	t	t	PROPN
iajs-2613	3	54	be	be	AUX
iajs-2613	3	55	unitary	unitary	ADJ
iajs-2613	3	56	left	leave	VERB
iajs-2613	3	57	r	r	NOUN
iajs-2613	3	58	−	−	NOUN
iajs-2613	3	59	module	module	NOUN
iajs-2613	3	60	.in	.in	PUNCT
iajs-2613	3	61	this	this	DET
iajs-2613	3	62	paper	paper	NOUN
iajs-2613	3	63	,	,	PUNCT
iajs-2613	3	64	wapp	wapp	NOUN
iajs-2613	3	65	-	-	PUNCT
iajs-2613	3	66	quasi	quasi	ADJ
iajs-2613	3	67	prime	prime	ADJ
iajs-2613	3	68	submodules	submodule	NOUN
iajs-2613	3	69	are	be	AUX
iajs-2613	3	70	introduced	introduce	VERB
iajs-2613	3	71	as	as	ADP
iajs-2613	3	72	new	new	ADJ
iajs-2613	3	73	generalization	generalization	NOUN
iajs-2613	3	74	of	of	ADP
iajs-2613	3	75	weakly	weakly	ADJ
iajs-2613	3	76	quasi	quasi	ADJ
iajs-2613	3	77	prime	prime	ADJ
iajs-2613	3	78	submodules	submodule	NOUN
iajs-2613	3	79	,	,	PUNCT
iajs-2613	3	80	where	where	SCONJ
iajs-2613	3	81	proper	proper	ADJ
iajs-2613	3	82	submodule	submodule	NOUN
iajs-2613	3	83	c	c	PROPN
iajs-2613	3	84	of	of	ADP
iajs-2613	3	85	an	an	DET
iajs-2613	3	86	r	r	NOUN
iajs-2613	3	87	-	-	PUNCT
iajs-2613	3	88	module	module	NOUN
iajs-2613	3	89	t	t	NOUN
iajs-2613	3	90	is	be	AUX
iajs-2613	3	91	called	call	VERB
iajs-2613	3	92	wapp	wapp	NOUN
iajs-2613	3	93	–	–	PUNCT
iajs-2613	3	94	quasi	quasi	ADJ
iajs-2613	3	95	prime	prime	PROPN
iajs-2613	3	96	submodule	submodule	NOUN
iajs-2613	3	97	of	of	ADP
iajs-2613	3	98	t	t	PROPN
iajs-2613	3	99	,	,	PUNCT
iajs-2613	3	100	if	if	SCONJ
iajs-2613	3	101	whenever	whenever	SCONJ
iajs-2613	3	102	0≠rstϵc	0≠rstϵc	NOUN
iajs-2613	3	103	,	,	PUNCT
iajs-2613	3	104	for	for	ADP
iajs-2613	3	105	r	r	NOUN
iajs-2613	3	106	,	,	PUNCT
iajs-2613	3	107	s	s	PART
iajs-2613	3	108	ϵr	ϵr	X
iajs-2613	3	109	,	,	PUNCT
iajs-2613	3	110	t	t	PROPN
iajs-2613	3	111	ϵt	ϵt	NOUN
iajs-2613	3	112	,	,	PUNCT
iajs-2613	3	113	implies	imply	VERB
iajs-2613	3	114	that	that	SCONJ
iajs-2613	3	115	either	either	CCONJ
iajs-2613	3	116	r	r	NOUN
iajs-2613	3	117	tϵ	tϵ	NOUN
iajs-2613	3	118	c	c	NOUN
iajs-2613	3	119	+	+	NOUN
iajs-2613	3	120	soc(t	soc(t	PROPN
iajs-2613	3	121	)	)	PUNCT
iajs-2613	3	122	or	or	CCONJ
iajs-2613	3	123	s	s	AUX
iajs-2613	3	124	tϵc	tϵc	ADJ
iajs-2613	3	125	+	+	NOUN
iajs-2613	3	126	soc(t	soc(t	NOUN
iajs-2613	3	127	)	)	PUNCT
iajs-2613	3	128	.many	.many	NOUN
iajs-2613	3	129	examples	example	NOUN
iajs-2613	3	130	of	of	ADP
iajs-2613	3	131	characterizations	characterization	NOUN
iajs-2613	3	132	and	and	CCONJ
iajs-2613	3	133	basic	basic	ADJ
iajs-2613	3	134	properties	property	NOUN
iajs-2613	3	135	are	be	AUX
iajs-2613	3	136	given	give	VERB
iajs-2613	3	137	.	.	PUNCT
iajs-2613	4	1	furthermore	furthermore	ADV
iajs-2613	4	2	several	several	ADJ
iajs-2613	4	3	characterizations	characterization	NOUN
iajs-2613	4	4	of	of	ADP
iajs-2613	4	5	wapp	wapp	NOUN
iajs-2613	4	6	-	-	PUNCT
iajs-2613	4	7	quasi	quasi	ADJ
iajs-2613	4	8	prime	prime	ADJ
iajs-2613	4	9	submodules	submodule	NOUN
iajs-2613	4	10	in	in	ADP
iajs-2613	4	11	the	the	DET
iajs-2613	4	12	class	class	NOUN
iajs-2613	4	13	of	of	ADP
iajs-2613	4	14	multiplication	multiplication	NOUN
iajs-2613	4	15	modules	module	NOUN
iajs-2613	4	16	are	be	AUX
iajs-2613	4	17	established	establish	VERB
iajs-2613	4	18	.	.	PUNCT
iajs-2613	5	1	keywords	keyword	NOUN
iajs-2613	5	2	:	:	PUNCT
iajs-2613	5	3	weakly	weakly	ADJ
iajs-2613	5	4	quasi	quasi	ADJ
iajs-2613	5	5	prime	prime	ADJ
iajs-2613	5	6	submodules	submodule	NOUN
iajs-2613	5	7	,	,	PUNCT
iajs-2613	5	8	wapp	wapp	NOUN
iajs-2613	5	9	-	-	PUNCT
iajs-2613	5	10	quasi	quasi	ADJ
iajs-2613	5	11	prime	prime	ADJ
iajs-2613	5	12	submodules	submodule	NOUN
iajs-2613	5	13	,	,	PUNCT
iajs-2613	5	14	socle	socle	NOUN
iajs-2613	5	15	of	of	ADP
iajs-2613	5	16	modules	module	NOUN
iajs-2613	5	17	,	,	PUNCT
iajs-2613	5	18	z	z	NOUN
iajs-2613	5	19	-	-	PUNCT
iajs-2613	5	20	regular	regular	ADJ
iajs-2613	5	21	modules	module	NOUN
iajs-2613	5	22	,	,	PUNCT
iajs-2613	5	23	projective	projective	ADJ
iajs-2613	5	24	modules	module	NOUN
iajs-2613	5	25	.	.	PUNCT
iajs-2613	6	1	1	1	X
iajs-2613	6	2	.	.	X
iajs-2613	6	3	introduction	introduction	NOUN
iajs-2613	6	4	throughout	throughout	ADP
iajs-2613	6	5	this	this	DET
iajs-2613	6	6	paper	paper	NOUN
iajs-2613	6	7	,	,	PUNCT
iajs-2613	6	8	all	all	DET
iajs-2613	6	9	rings	ring	NOUN
iajs-2613	6	10	are	be	AUX
iajs-2613	6	11	commutative	commutative	ADJ
iajs-2613	6	12	with	with	ADP
iajs-2613	6	13	identity	identity	NOUN
iajs-2613	6	14	,	,	PUNCT
iajs-2613	6	15	and	and	CCONJ
iajs-2613	6	16	all	all	DET
iajs-2613	6	17	modules	module	NOUN
iajs-2613	6	18	are	be	AUX
iajs-2613	6	19	left	leave	VERB
iajs-2613	6	20	unitary	unitary	ADJ
iajs-2613	6	21	r	r	NOUN
iajs-2613	6	22	-	-	PUNCT
iajs-2613	6	23	modules	module	NOUN
iajs-2613	6	24	.	.	PUNCT
iajs-2613	7	1	weakly	weakly	ADJ
iajs-2613	7	2	quasi	quasi	ADJ
iajs-2613	7	3	prim	prim	ADJ
iajs-2613	7	4	submodules	submodule	NOUN
iajs-2613	7	5	was	be	AUX
iajs-2613	7	6	first	first	ADV
iajs-2613	7	7	introduced	introduce	VERB
iajs-2613	7	8	and	and	CCONJ
iajs-2613	7	9	studied	study	VERB
iajs-2613	7	10	in	in	ADP
iajs-2613	7	11	2013	2013	NUM
iajs-2613	7	12	by	by	ADP
iajs-2613	7	13	[	[	X
iajs-2613	7	14	1	1	NUM
iajs-2613	7	15	]	]	PUNCT
iajs-2613	7	16	as	as	ADP
iajs-2613	7	17	a	a	DET
iajs-2613	7	18	generalization	generalization	NOUN
iajs-2613	7	19	of	of	ADP
iajs-2613	7	20	a	a	DET
iajs-2613	7	21	weakly	weakly	ADJ
iajs-2613	7	22	prime	prime	ADJ
iajs-2613	7	23	submodule	submodule	NOUN
iajs-2613	7	24	,	,	PUNCT
iajs-2613	7	25	where	where	SCONJ
iajs-2613	7	26	proper	proper	ADJ
iajs-2613	7	27	submodule	submodule	NOUN
iajs-2613	7	28	c	c	PROPN
iajs-2613	7	29	of	of	ADP
iajs-2613	7	30	rmodule	rmodule	PROPN
iajs-2613	7	31	t	t	PROPN
iajs-2613	7	32	was	be	AUX
iajs-2613	7	33	called	call	VERB
iajs-2613	7	34	weakly	weakly	ADJ
iajs-2613	7	35	prime	prime	ADJ
iajs-2613	7	36	submodule	submodule	NOUN
iajs-2613	7	37	of	of	ADP
iajs-2613	7	38	c	c	PROPN
iajs-2613	7	39	,	,	PUNCT
iajs-2613	7	40	if	if	SCONJ
iajs-2613	7	41	whenever	whenever	SCONJ
iajs-2613	7	42	0≠atϵc	0≠atϵc	NOUN
iajs-2613	7	43	,	,	PUNCT
iajs-2613	7	44	for	for	ADP
iajs-2613	7	45	a	a	DET
iajs-2613	7	46	ϵr	ϵr	X
iajs-2613	7	47	,	,	PUNCT
iajs-2613	7	48	t	t	PROPN
iajs-2613	7	49	ϵt	ϵt	NOUN
iajs-2613	7	50	,	,	PUNCT
iajs-2613	7	51	implies	imply	VERB
iajs-2613	7	52	that	that	SCONJ
iajs-2613	7	53	either	either	CCONJ
iajs-2613	7	54	tϵc	tϵc	ADJ
iajs-2613	7	55	or	or	CCONJ
iajs-2613	7	56	at	at	PUNCT
iajs-2613	7	57	c	c	NOUN
iajs-2613	8	1	[	[	X
iajs-2613	8	2	2	2	NUM
iajs-2613	8	3	]	]	PUNCT
iajs-2613	8	4	,	,	PUNCT
iajs-2613	8	5	and	and	CCONJ
iajs-2613	8	6	a	a	DET
iajs-2613	8	7	proper	proper	ADJ
iajs-2613	8	8	submodule	submodule	NOUN
iajs-2613	8	9	c	c	PROPN
iajs-2613	8	10	of	of	ADP
iajs-2613	8	11	r	r	NOUN
iajs-2613	8	12	-	-	PUNCT
iajs-2613	8	13	module	module	NOUN
iajs-2613	8	14	t	t	NOUN
iajs-2613	8	15	is	be	AUX
iajs-2613	8	16	called	call	VERB
iajs-2613	8	17	weakly	weakly	ADJ
iajs-2613	8	18	quasi	quasi	ADJ
iajs-2613	8	19	prime	prime	NOUN
iajs-2613	8	20	submodule	submodule	NOUN
iajs-2613	8	21	of	of	ADP
iajs-2613	8	22	t	t	PROPN
iajs-2613	8	23	,	,	PUNCT
iajs-2613	8	24	if	if	SCONJ
iajs-2613	8	25	whenever	whenever	SCONJ
iajs-2613	8	26	0≠abtϵc	0≠abtϵc	NUM
iajs-2613	8	27	,	,	PUNCT
iajs-2613	8	28	for	for	ADP
iajs-2613	8	29	a	a	DET
iajs-2613	8	30	,	,	PUNCT
iajs-2613	8	31	b	b	PROPN
iajs-2613	8	32	ϵr	ϵr	X
iajs-2613	8	33	t	t	PROPN
iajs-2613	8	34	ϵt	ϵt	ADP
iajs-2613	8	35	,	,	PUNCT
iajs-2613	8	36	implies	imply	VERB
iajs-2613	8	37	that	that	SCONJ
iajs-2613	8	38	either	either	CCONJ
iajs-2613	8	39	at	at	ADP
iajs-2613	8	40	ϵc	ϵc	ADV
iajs-2613	8	41	or	or	CCONJ
iajs-2613	8	42	bt	bt	INTJ
iajs-2613	8	43	ϵc	ϵc	INTJ
iajs-2613	8	44	.	.	PUNCT
iajs-2613	9	1	recently	recently	ADV
iajs-2613	9	2	many	many	ADJ
iajs-2613	9	3	generalization	generalization	NOUN
iajs-2613	9	4	of	of	ADP
iajs-2613	9	5	weakly	weakly	ADJ
iajs-2613	9	6	quasi	quasi	ADJ
iajs-2613	9	7	prime	prime	ADJ
iajs-2613	9	8	submodules	submodule	NOUN
iajs-2613	9	9	were	be	AUX
iajs-2613	9	10	introduced	introduce	VERB
iajs-2613	9	11	see	see	VERB
iajs-2613	9	12	[	[	X
iajs-2613	9	13	3	3	NUM
iajs-2613	9	14	,	,	PUNCT
iajs-2613	9	15	4	4	NUM
iajs-2613	9	16	,	,	PUNCT
iajs-2613	9	17	5	5	NUM
iajs-2613	9	18	]	]	PUNCT
iajs-2613	9	19	.	.	PUNCT
iajs-2613	10	1	in	in	ADP
iajs-2613	10	2	this	this	DET
iajs-2613	10	3	research	research	NOUN
iajs-2613	10	4	we	we	PRON
iajs-2613	10	5	introduced	introduce	VERB
iajs-2613	10	6	another	another	DET
iajs-2613	10	7	generalization	generalization	NOUN
iajs-2613	10	8	of	of	ADP
iajs-2613	10	9	weakly	weakly	ADJ
iajs-2613	10	10	quasi	quasi	ADJ
iajs-2613	10	11	prime	prime	PROPN
iajs-2613	10	12	submodule	submodule	NOUN
iajs-2613	10	13	,	,	PUNCT
iajs-2613	10	14	where	where	SCONJ
iajs-2613	10	15	proper	proper	ADJ
iajs-2613	10	16	submodule	submodule	NOUN
iajs-2613	10	17	c	c	PROPN
iajs-2613	10	18	of	of	ADP
iajs-2613	10	19	r	r	NOUN
iajs-2613	10	20	-	-	PUNCT
iajs-2613	10	21	module	module	NOUN
iajs-2613	10	22	t	t	NOUN
iajs-2613	10	23	is	be	AUX
iajs-2613	10	24	called	call	VERB
iajs-2613	10	25	wapp	wapp	NOUN
iajs-2613	10	26	-	-	PUNCT
iajs-2613	10	27	quasi	quasi	ADJ
iajs-2613	10	28	prime	prime	PROPN
iajs-2613	10	29	submodule	submodule	NOUN
iajs-2613	10	30	of	of	ADP
iajs-2613	10	31	t	t	PROPN
iajs-2613	10	32	,	,	PUNCT
iajs-2613	10	33	if	if	SCONJ
iajs-2613	10	34	whenever	whenever	SCONJ
iajs-2613	10	35	0≠abtϵc	0≠abtϵc	NUM
iajs-2613	10	36	for	for	ADP
iajs-2613	10	37	a	a	DET
iajs-2613	10	38	,	,	PUNCT
iajs-2613	10	39	b	b	NOUN
iajs-2613	10	40	ϵr	ϵr	PROPN
iajs-2613	10	41	,	,	PUNCT
iajs-2613	10	42	tϵt	tϵt	NOUN
iajs-2613	10	43	implies	imply	VERB
iajs-2613	10	44	that	that	SCONJ
iajs-2613	10	45	either	either	PRON
iajs-2613	10	46	atϵc+soc(t	atϵc+soc(t	NOUN
iajs-2613	10	47	)	)	PUNCT
iajs-2613	10	48	or	or	CCONJ
iajs-2613	10	49	bt	bt	X
iajs-2613	10	50	ϵc	ϵc	ADP
iajs-2613	10	51	+	+	NOUN
iajs-2613	10	52	soc(t	soc(t	PROPN
iajs-2613	10	53	)	)	PUNCT
iajs-2613	10	54	.	.	PUNCT
iajs-2613	11	1	soc(t	soc(t	PROPN
iajs-2613	11	2	)	)	PUNCT
iajs-2613	11	3	is	be	AUX
iajs-2613	11	4	the	the	DET
iajs-2613	11	5	socle	socle	NOUN
iajs-2613	11	6	of	of	ADP
iajs-2613	11	7	a	a	DET
iajs-2613	11	8	module	module	NOUN
iajs-2613	11	9	t	t	NOUN
iajs-2613	11	10	,	,	PUNCT
iajs-2613	11	11	defined	define	VERB
iajs-2613	11	12	by	by	ADP
iajs-2613	11	13	the	the	DET
iajs-2613	11	14	intersection	intersection	NOUN
iajs-2613	11	15	of	of	ADP
iajs-2613	11	16	all	all	DET
iajs-2613	11	17	essential	essential	ADJ
iajs-2613	11	18	submodule	submodule	NOUN
iajs-2613	11	19	of	of	ADP
iajs-2613	11	20	t	t	PROPN
iajs-2613	12	1	[	[	X
iajs-2613	12	2	6	6	NUM
iajs-2613	12	3	]	]	PUNCT
iajs-2613	12	4	,	,	PUNCT
iajs-2613	12	5	where	where	SCONJ
iajs-2613	12	6	a	a	DET
iajs-2613	12	7	nonzero	nonzero	PROPN
iajs-2613	12	8	submodule	submodule	NOUN
iajs-2613	12	9	a	a	PRON
iajs-2613	12	10	of	of	ADP
iajs-2613	12	11	an	an	DET
iajs-2613	12	12	r	r	NOUN
iajs-2613	12	13	-	-	PUNCT
iajs-2613	12	14	module	module	NOUN
iajs-2613	12	15	t	t	NOUN
iajs-2613	12	16	is	be	AUX
iajs-2613	12	17	called	call	VERB
iajs-2613	12	18	essential	essential	ADJ
iajs-2613	12	19	if	if	SCONJ
iajs-2613	12	20	a∩	a∩	PROPN
iajs-2613	12	21	b	b	PROPN
iajs-2613	12	22	≠	≠	PROPN
iajs-2613	12	23	(	(	PUNCT
iajs-2613	12	24	0)for	0)for	ADP
iajs-2613	12	25	each	each	DET
iajs-2613	12	26	nonzero	nonzero	PROPN
iajs-2613	12	27	submodule	submodule	PROPN
iajs-2613	12	28	b	b	PROPN
iajs-2613	12	29	of	of	ADP
iajs-2613	12	30	t	t	PROPN
iajs-2613	13	1	[	[	X
iajs-2613	13	2	6	6	NUM
iajs-2613	13	3	]	]	PUNCT
iajs-2613	13	4	.	.	PUNCT
iajs-2613	14	1	recall	recall	VERB
iajs-2613	14	2	that	that	SCONJ
iajs-2613	14	3	r	r	NOUN
iajs-2613	14	4	-	-	PUNCT
iajs-2613	14	5	module	module	NOUN
iajs-2613	14	6	t	t	NOUN
iajs-2613	14	7	is	be	AUX
iajs-2613	14	8	ibn	ibn	PROPN
iajs-2613	14	9	al	al	PROPN
iajs-2613	14	10	haitham	haitham	PROPN
iajs-2613	14	11	journal	journal	PROPN
iajs-2613	14	12	for	for	ADP
iajs-2613	14	13	pure	pure	ADJ
iajs-2613	14	14	and	and	CCONJ
iajs-2613	14	15	applied	apply	VERB
iajs-2613	14	16	science	science	NOUN
iajs-2613	14	17	journal	journal	PROPN
iajs-2613	14	18	homepage	homepage	NOUN
iajs-2613	14	19	:	:	PUNCT
iajs-2613	14	20	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2613	14	21	doi	doi	NOUN
iajs-2613	14	22	:	:	PUNCT
iajs-2613	14	23	10.30526/34.2.2613	10.30526/34.2.2613	ADJ
iajs-2613	14	24	article	article	NOUN
iajs-2613	14	25	history	history	NOUN
iajs-2613	14	26	:	:	PUNCT
iajs-2613	14	27	received	receive	VERB
iajs-2613	14	28	,	,	PUNCT
iajs-2613	14	29	21,april	21,april	NUM
iajs-2613	14	30	,	,	PUNCT
iajs-2613	14	31	2020	2020	NUM
iajs-2613	14	32	,	,	PUNCT
iajs-2613	14	33	accepted	accept	VERB
iajs-2613	14	34	21,june,2020	21,june,2020	NOUN
iajs-2613	14	35	,	,	PUNCT
iajs-2613	14	36	published	publish	VERB
iajs-2613	14	37	in	in	ADP
iajs-2613	14	38	april	april	PROPN
iajs-2613	14	39	2021	2021	NUM
iajs-2613	14	40	file:///c:/users	file:///c:/user	NOUN
iajs-2613	14	41	/	/	SYM
iajs-2613	14	42	المجلة	المجلة	NOUN
iajs-2613	14	43	/	/	SYM
iajs-2613	14	44	desktop	desktop	NOUN
iajs-2613	14	45	/	/	SYM
iajs-2613	14	46	العدد%20الثاني%20هدر	العدد%20الثاني%20هدر	NOUN
iajs-2613	14	47	/	/	SYM
iajs-2613	14	48	شهد.docx	شهد.docx	NUM
iajs-2613	14	49	mailto:2013@gmail.com	mailto:2013@gmail.com	X
iajs-2613	14	50	mailto:hasanenjassm@gmail.com	mailto:hasanenjassm@gmail.com	PROPN
iajs-2613	15	1	57	57	NUM
iajs-2613	15	2	ibn	ibn	PROPN
iajs-2613	15	3	al	al	PROPN
iajs-2613	15	4	-	-	PUNCT
iajs-2613	15	5	haitham	haitham	PROPN
iajs-2613	15	6	jour	jour	X
iajs-2613	15	7	.	.	PROPN
iajs-2613	15	8	for	for	ADP
iajs-2613	15	9	pure	pure	ADJ
iajs-2613	15	10	&	&	CCONJ
iajs-2613	15	11	appl	appl	PROPN
iajs-2613	15	12	.	.	PUNCT
iajs-2613	16	1	sci	sci	PROPN
iajs-2613	16	2	.	.	PROPN
iajs-2613	17	1	34	34	NUM
iajs-2613	17	2	(	(	PUNCT
iajs-2613	17	3	1	1	NUM
iajs-2613	17	4	)	)	PUNCT
iajs-2613	17	5	2021	2021	NUM
iajs-2613	17	6	multiplication	multiplication	NOUN
iajs-2613	17	7	if	if	SCONJ
iajs-2613	17	8	every	every	DET
iajs-2613	17	9	submodule	submodule	NOUN
iajs-2613	17	10	c	c	PROPN
iajs-2613	17	11	of	of	ADP
iajs-2613	17	12	t	t	PROPN
iajs-2613	17	13	is	be	AUX
iajs-2613	17	14	of	of	ADP
iajs-2613	17	15	the	the	DET
iajs-2613	17	16	form	form	NOUN
iajs-2613	17	17	it	it	PRON
iajs-2613	17	18	for	for	ADP
iajs-2613	17	19	some	some	DET
iajs-2613	17	20	ideal	ideal	ADJ
iajs-2613	17	21	i	i	PRON
iajs-2613	17	22	of	of	ADP
iajs-2613	17	23	r	r	NOUN
iajs-2613	17	24	,	,	PUNCT
iajs-2613	17	25	in	in	ADP
iajs-2613	17	26	particular	particular	ADJ
iajs-2613	17	27	c=[c	c=[c	NOUN
iajs-2613	17	28	:	:	PUNCT
iajs-2613	17	29	r	r	NOUN
iajs-2613	17	30	t	t	PROPN
iajs-2613	17	31	]	]	X
iajs-2613	17	32	t	t	NOUN
iajs-2613	18	1	[	[	X
iajs-2613	18	2	7	7	NUM
iajs-2613	18	3	]	]	PUNCT
iajs-2613	18	4	.	.	PUNCT
iajs-2613	19	1	let	let	VERB
iajs-2613	19	2	a	a	PRON
iajs-2613	19	3	and	and	CCONJ
iajs-2613	19	4	b	b	NOUN
iajs-2613	19	5	be	be	AUX
iajs-2613	19	6	a	a	DET
iajs-2613	19	7	submodule	submodule	NOUN
iajs-2613	19	8	of	of	ADP
iajs-2613	19	9	multiplication	multiplication	NOUN
iajs-2613	19	10	module	module	NOUN
iajs-2613	19	11	t	t	PROPN
iajs-2613	19	12	with	with	ADP
iajs-2613	19	13	a	a	DET
iajs-2613	19	14	=	=	NOUN
iajs-2613	19	15	im	im	NOUN
iajs-2613	19	16	and	and	CCONJ
iajs-2613	19	17	b	b	X
iajs-2613	19	18	=	=	PROPN
iajs-2613	19	19	jt	jt	PROPN
iajs-2613	19	20	for	for	ADP
iajs-2613	19	21	some	some	DET
iajs-2613	19	22	ideals	ideal	NOUN
iajs-2613	20	1	i	i	PRON
iajs-2613	20	2	,	,	PUNCT
iajs-2613	20	3	j	j	PROPN
iajs-2613	20	4	of	of	ADP
iajs-2613	20	5	r	r	NOUN
iajs-2613	20	6	,	,	PUNCT
iajs-2613	20	7	then	then	ADV
iajs-2613	20	8	ab	ab	PROPN
iajs-2613	20	9	=	=	NOUN
iajs-2613	20	10	ijt	ijt	NOUN
iajs-2613	20	11	=	=	NOUN
iajs-2613	20	12	ib	ib	NOUN
iajs-2613	20	13	.	.	PUNCT
iajs-2613	21	1	in	in	ADP
iajs-2613	21	2	particular	particular	ADJ
iajs-2613	21	3	at	at	ADP
iajs-2613	21	4	=	=	NOUN
iajs-2613	21	5	itt	itt	NOUN
iajs-2613	21	6	=	=	NOUN
iajs-2613	21	7	it	it	PRON
iajs-2613	21	8	=	=	NOUN
iajs-2613	21	9	a.	a.	NOUN
iajs-2613	21	10	also	also	ADV
iajs-2613	21	11	for	for	ADP
iajs-2613	21	12	any	any	DET
iajs-2613	21	13	t	t	NOUN
iajs-2613	21	14	ϵt	ϵt	ADV
iajs-2613	21	15	,	,	PUNCT
iajs-2613	21	16	at	at	ADP
iajs-2613	21	17	=	=	NOUN
iajs-2613	21	18	a	a	PRON
iajs-2613	21	19	<	<	X
iajs-2613	21	20	t	t	X
iajs-2613	21	21	>	>	PUNCT
iajs-2613	21	22	=	=	PUNCT
iajs-2613	21	23	it	it	PRON
iajs-2613	22	1	[	[	X
iajs-2613	22	2	8	8	NUM
iajs-2613	22	3	]	]	PUNCT
iajs-2613	22	4	.	.	PUNCT
iajs-2613	23	1	recall	recall	VERB
iajs-2613	23	2	that	that	SCONJ
iajs-2613	23	3	an	an	DET
iajs-2613	23	4	r	r	NOUN
iajs-2613	23	5	-	-	PUNCT
iajs-2613	23	6	module	module	NOUN
iajs-2613	23	7	t	t	NOUN
iajs-2613	23	8	is	be	AUX
iajs-2613	23	9	faithful	faithful	ADJ
iajs-2613	23	10	,	,	PUNCT
iajs-2613	23	11	if	if	SCONJ
iajs-2613	23	12	ann(t)=	ann(t)=	VERB
iajs-2613	23	13	(	(	PUNCT
iajs-2613	23	14	0	0	NUM
iajs-2613	23	15	)	)	PUNCT
iajs-2613	24	1	[	[	X
iajs-2613	24	2	7	7	X
iajs-2613	24	3	]	]	PUNCT
iajs-2613	24	4	.a	.a	NOUN
iajs-2613	24	5	rmodule	rmodule	PROPN
iajs-2613	24	6	t	t	PROPN
iajs-2613	24	7	is	be	AUX
iajs-2613	24	8	a	a	DET
iajs-2613	24	9	projective	projective	NOUN
iajs-2613	24	10	if	if	SCONJ
iajs-2613	24	11	for	for	ADP
iajs-2613	24	12	any	any	DET
iajs-2613	24	13	epimorphism	epimorphism	NOUN
iajs-2613	24	14	f	f	NOUN
iajs-2613	24	15	from	from	ADP
iajs-2613	24	16	r	r	NOUN
iajs-2613	24	17	-	-	PUNCT
iajs-2613	24	18	module	module	NOUN
iajs-2613	24	19	x	x	PUNCT
iajs-2613	24	20	into	into	ADP
iajs-2613	24	21	x	x	X
iajs-2613	24	22	’	'	PUNCT
iajs-2613	24	23	and	and	CCONJ
iajs-2613	24	24	for	for	ADP
iajs-2613	24	25	any	any	DET
iajs-2613	24	26	homomorphism	homomorphism	NOUN
iajs-2613	24	27	g	g	NOUN
iajs-2613	24	28	from	from	ADP
iajs-2613	24	29	tin	tin	NOUN
iajs-2613	24	30	to	to	ADP
iajs-2613	24	31	x	x	NOUN
iajs-2613	24	32	’	'	PUNCT
iajs-2613	24	33	there	there	PRON
iajs-2613	24	34	exists	exist	VERB
iajs-2613	24	35	a	a	DET
iajs-2613	24	36	homomorphism	homomorphism	PROPN
iajs-2613	24	37	h	h	NOUN
iajs-2613	24	38	from	from	ADP
iajs-2613	24	39	t	t	PROPN
iajs-2613	24	40	in	in	ADP
iajs-2613	24	41	to	to	ADP
iajs-2613	24	42	x	x	SYM
iajs-2613	24	43	such	such	ADJ
iajs-2613	24	44	that	that	SCONJ
iajs-2613	24	45	f	f	PROPN
iajs-2613	25	1	o	o	NOUN
iajs-2613	25	2	h	h	NOUN
iajs-2613	25	3	=	=	NOUN
iajs-2613	25	4	g	g	X
iajs-2613	26	1	[	[	X
iajs-2613	26	2	7].recall	7].recall	NUM
iajs-2613	26	3	that	that	SCONJ
iajs-2613	26	4	an	an	DET
iajs-2613	26	5	r	r	NOUN
iajs-2613	26	6	-	-	PUNCT
iajs-2613	26	7	module	module	NOUN
iajs-2613	26	8	t	t	NOUN
iajs-2613	26	9	is	be	AUX
iajs-2613	26	10	a	a	DET
iajs-2613	26	11	z	z	NOUN
iajs-2613	26	12	-	-	NOUN
iajs-2613	26	13	regular	regular	ADJ
iajs-2613	26	14	,	,	PUNCT
iajs-2613	26	15	if	if	SCONJ
iajs-2613	26	16	for	for	ADP
iajs-2613	26	17	each	each	DET
iajs-2613	26	18	tϵt	tϵt	NOUN
iajs-2613	26	19	there	there	PRON
iajs-2613	26	20	exists	exist	VERB
iajs-2613	26	21	fϵt*=hom(t	fϵt*=hom(t	NOUN
iajs-2613	26	22	,	,	PUNCT
iajs-2613	26	23	r	r	NOUN
iajs-2613	26	24	)	)	PUNCT
iajs-2613	26	25	such	such	ADJ
iajs-2613	26	26	that	that	SCONJ
iajs-2613	26	27	t	t	PROPN
iajs-2613	26	28	=	=	SYM
iajs-2613	26	29	f(t)t	f(t)t	PROPN
iajs-2613	27	1	[	[	X
iajs-2613	27	2	10	10	NUM
iajs-2613	27	3	]	]	SYM
iajs-2613	27	4	2.basic	2.basic	NUM
iajs-2613	27	5	properties	property	NOUN
iajs-2613	27	6	of	of	ADP
iajs-2613	27	7	wapp	wapp	NOUN
iajs-2613	27	8	-	-	PUNCT
iajs-2613	27	9	quasi	quasi	ADJ
iajs-2613	27	10	prime	prime	NOUN
iajs-2613	27	11	submodule	submodule	NOUN
iajs-2613	27	12	in	in	ADP
iajs-2613	27	13	this	this	DET
iajs-2613	27	14	section	section	NOUN
iajs-2613	27	15	,	,	PUNCT
iajs-2613	27	16	we	we	PRON
iajs-2613	27	17	introduced	introduce	VERB
iajs-2613	27	18	the	the	DET
iajs-2613	27	19	definition	definition	NOUN
iajs-2613	27	20	of	of	ADP
iajs-2613	27	21	wapp	wapp	NOUN
iajs-2613	27	22	-	-	PUNCT
iajs-2613	27	23	quasi	quasi	ADJ
iajs-2613	27	24	prime	prime	ADJ
iajs-2613	27	25	submodules	submodule	NOUN
iajs-2613	27	26	and	and	CCONJ
iajs-2613	27	27	established	establish	VERB
iajs-2613	27	28	some	some	PRON
iajs-2613	27	29	of	of	ADP
iajs-2613	27	30	its	its	PRON
iajs-2613	27	31	basic	basic	ADJ
iajs-2613	27	32	properties	property	NOUN
iajs-2613	27	33	,	,	PUNCT
iajs-2613	27	34	characterization	characterization	NOUN
iajs-2613	27	35	and	and	CCONJ
iajs-2613	27	36	examples	example	NOUN
iajs-2613	27	37	.	.	PUNCT
iajs-2613	28	1	definition(1	definition(1	NOUN
iajs-2613	28	2	)	)	PUNCT
iajs-2613	28	3	a	a	DET
iajs-2613	28	4	proper	proper	ADJ
iajs-2613	28	5	submodule	submodule	NOUN
iajs-2613	28	6	c	c	PROPN
iajs-2613	28	7	of	of	ADP
iajs-2613	28	8	an	an	DET
iajs-2613	28	9	r	r	NOUN
iajs-2613	28	10	−	−	NOUN
iajs-2613	28	11	module	module	NOUN
iajs-2613	28	12	t	t	NOUN
iajs-2613	28	13	is	be	AUX
iajs-2613	28	14	called	call	VERB
iajs-2613	28	15	weakly	weakly	ADV
iajs-2613	28	16	approximaitly	approximaitly	ADV
iajs-2613	28	17	quasi	quasi	ADJ
iajs-2613	28	18	prime	prime	PROPN
iajs-2613	28	19	submodule	submodule	NOUN
iajs-2613	28	20	of	of	ADP
iajs-2613	28	21	t	t	PROPN
iajs-2613	28	22	(	(	PUNCT
iajs-2613	28	23	for	for	ADP
iajs-2613	28	24	short	short	ADJ
iajs-2613	28	25	wapp	wapp	NOUN
iajs-2613	28	26	-	-	PUNCT
iajs-2613	28	27	quasi	quasi	ADJ
iajs-2613	28	28	prime	prime	ADJ
iajs-2613	28	29	submodule	submodule	NOUN
iajs-2613	28	30	)	)	PUNCT
iajs-2613	28	31	,	,	PUNCT
iajs-2613	28	32	if	if	SCONJ
iajs-2613	28	33	whenever	whenever	SCONJ
iajs-2613	28	34	0≠abt	0≠abt	NOUN
iajs-2613	28	35	ϵc	ϵc	VERB
iajs-2613	28	36	,	,	PUNCT
iajs-2613	28	37	for	for	ADP
iajs-2613	28	38	a	a	DET
iajs-2613	28	39	,	,	PUNCT
iajs-2613	28	40	bϵr	bϵr	PROPN
iajs-2613	28	41	,	,	PUNCT
iajs-2613	28	42	tϵt	tϵt	PROPN
iajs-2613	28	43	,	,	PUNCT
iajs-2613	28	44	implies	imply	VERB
iajs-2613	28	45	that	that	SCONJ
iajs-2613	28	46	either	either	CCONJ
iajs-2613	28	47	atϵ	atϵ	PROPN
iajs-2613	28	48	c+soc(t	c+soc(t	PROPN
iajs-2613	28	49	)	)	PUNCT
iajs-2613	28	50	or	or	CCONJ
iajs-2613	28	51	btϵc+soc(t	btϵc+soc(t	NOUN
iajs-2613	28	52	)	)	PUNCT
iajs-2613	28	53	.	.	PUNCT
iajs-2613	29	1	and	and	CCONJ
iajs-2613	29	2	an	an	DET
iajs-2613	29	3	ideal	ideal	ADJ
iajs-2613	29	4	j	j	PROPN
iajs-2613	29	5	of	of	ADP
iajs-2613	29	6	ring	ring	NOUN
iajs-2613	29	7	r	r	NOUN
iajs-2613	29	8	is	be	AUX
iajs-2613	29	9	called	call	VERB
iajs-2613	29	10	wapp	wapp	NOUN
iajs-2613	29	11	-	-	PUNCT
iajs-2613	29	12	quasi	quasi	ADJ
iajs-2613	29	13	prime	prime	ADJ
iajs-2613	29	14	ideal	ideal	NOUN
iajs-2613	29	15	of	of	ADP
iajs-2613	29	16	r	r	NOUN
iajs-2613	29	17	if	if	SCONJ
iajs-2613	29	18	j	j	PROPN
iajs-2613	29	19	is	be	AUX
iajs-2613	29	20	wappquasi	wappquasi	PROPN
iajs-2613	29	21	prime	prime	ADJ
iajs-2613	29	22	submodule	submodule	NOUN
iajs-2613	29	23	of	of	ADP
iajs-2613	29	24	r	r	NOUN
iajs-2613	29	25	-	-	PUNCT
iajs-2613	29	26	module	module	NOUN
iajs-2613	29	27	t.	t.	NOUN
iajs-2613	29	28	examples	example	NOUN
iajs-2613	29	29	and	and	CCONJ
iajs-2613	29	30	remarks(2	remarks(2	NOUN
iajs-2613	29	31	)	)	PUNCT
iajs-2613	29	32	1	1	NUM
iajs-2613	29	33	.	.	PUNCT
iajs-2613	30	1	the	the	DET
iajs-2613	30	2	submodule	submodule	PROPN
iajs-2613	30	3	c=	c=	NOUN
iajs-2613	30	4	<	<	X
iajs-2613	30	5	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	30	6	>	>	X
iajs-2613	30	7	of	of	ADP
iajs-2613	30	8	the	the	DET
iajs-2613	30	9	z	z	NOUN
iajs-2613	30	10	-	-	PUNCT
iajs-2613	30	11	module	module	NOUN
iajs-2613	30	12	z24	z24	NOUN
iajs-2613	30	13	is	be	AUX
iajs-2613	30	14	a	a	DET
iajs-2613	30	15	wapp	wapp	NOUN
iajs-2613	30	16	-	-	PUNCT
iajs-2613	30	17	quasi	quasi	ADJ
iajs-2613	30	18	prime	prime	PROPN
iajs-2613	30	19	submodule	submodule	NOUN
iajs-2613	30	20	of	of	ADP
iajs-2613	30	21	z24	z24	PROPN
iajs-2613	30	22	,	,	PUNCT
iajs-2613	30	23	,	,	PUNCT
iajs-2613	30	24	since	since	SCONJ
iajs-2613	30	25	soc(z24)=	soc(z24)=	PROPN
iajs-2613	30	26	<	<	X
iajs-2613	30	27	4̅	4̅	PROPN
iajs-2613	30	28	>	>	X
iajs-2613	30	29	,	,	PUNCT
iajs-2613	30	30	and	and	CCONJ
iajs-2613	30	31	for	for	ADP
iajs-2613	30	32	0≠abtϵ	0≠abtϵ	NOUN
iajs-2613	31	1	<	<	X
iajs-2613	31	2	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	31	3	>	>	X
iajs-2613	32	1	=	=	X
iajs-2613	32	2	{	{	PUNCT
iajs-2613	32	3	0̅	0̅	NOUN
iajs-2613	32	4	,	,	PUNCT
iajs-2613	32	5	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	32	6	}	}	PUNCT
iajs-2613	32	7	for	for	ADP
iajs-2613	32	8	a	a	DET
iajs-2613	32	9	,	,	PUNCT
iajs-2613	32	10	b	b	X
iajs-2613	32	11	ϵz	ϵz	PROPN
iajs-2613	32	12	,	,	PUNCT
iajs-2613	32	13	tϵz24	tϵz24	PROPN
iajs-2613	32	14	,	,	PUNCT
iajs-2613	32	15	implies	imply	VERB
iajs-2613	32	16	that	that	SCONJ
iajs-2613	32	17	either	either	DET
iajs-2613	32	18	atϵ	atϵ	NOUN
iajs-2613	32	19	<	<	X
iajs-2613	32	20	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	32	21	>	>	X
iajs-2613	32	22	+	+	NOUN
iajs-2613	32	23	soc(z24	soc(z24	NOUN
iajs-2613	32	24	)	)	PUNCT
iajs-2613	32	25	or	or	CCONJ
iajs-2613	32	26	bt	bt	X
iajs-2613	32	27	ϵ	ϵ	X
iajs-2613	32	28	<	<	X
iajs-2613	32	29	12̅̅̅̅	12̅̅̅̅	NUM
iajs-2613	32	30	>	>	X
iajs-2613	32	31	+	+	NOUN
iajs-2613	32	32	soc(z24	soc(z24	NOUN
iajs-2613	32	33	)	)	PUNCT
iajs-2613	32	34	.	.	PUNCT
iajs-2613	33	1	that	that	PRON
iajs-2613	33	2	is	be	AUX
iajs-2613	33	3	either	either	PRON
iajs-2613	33	4	atϵ	atϵ	NOUN
iajs-2613	33	5	<	<	X
iajs-2613	33	6	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	33	7	>	>	X
iajs-2613	34	1	+	+	ADJ
iajs-2613	34	2	soc(z24)=	soc(z24)=	ADJ
iajs-2613	34	3	<	<	X
iajs-2613	34	4	4̅	4̅	PROPN
iajs-2613	34	5	>	>	X
iajs-2613	34	6	or	or	CCONJ
iajs-2613	34	7	btϵ	btϵ	NOUN
iajs-2613	34	8	<	<	X
iajs-2613	34	9	12	12	NUM
iajs-2613	34	10	>	>	PUNCT
iajs-2613	34	11	+	+	ADJ
iajs-2613	34	12	soc(z24)=	soc(z24)=	ADJ
iajs-2613	34	13	<	<	X
iajs-2613	34	14	4̅	4̅	PROPN
iajs-2613	34	15	>	>	X
iajs-2613	34	16	thus	thus	ADV
iajs-2613	34	17	0≠2.3	0≠2.3	NUM
iajs-2613	34	18	.	.	PUNCT
iajs-2613	35	1	2̅ϵ	2̅ϵ	NUM
iajs-2613	35	2	<	<	X
iajs-2613	35	3	12̅̅̅̅	12̅̅̅̅	NUM
iajs-2613	35	4	>	>	X
iajs-2613	35	5	for	for	ADP
iajs-2613	35	6	2,3,ϵz	2,3,ϵz	NUM
iajs-2613	35	7	,	,	PUNCT
iajs-2613	35	8	2̅ϵz24	2̅ϵz24	NUM
iajs-2613	35	9	implies	imply	VERB
iajs-2613	35	10	that	that	SCONJ
iajs-2613	35	11	2	2	X
iajs-2613	35	12	.	.	X
iajs-2613	35	13	2̅=4̅ϵ	2̅=4̅ϵ	NUM
iajs-2613	35	14	<	<	X
iajs-2613	36	1	12̅̅̅̅	12̅̅̅̅	NUM
iajs-2613	36	2	>	>	PUNCT
iajs-2613	37	1	+	+	PROPN
iajs-2613	37	2	<	<	X
iajs-2613	37	3	4̅	4̅	ADJ
iajs-2613	37	4	>	>	PUNCT
iajs-2613	37	5	=	=	X
iajs-2613	37	6	<	<	X
iajs-2613	37	7	4̅	4̅	PROPN
iajs-2613	37	8	>	>	PUNCT
iajs-2613	37	9	=	=	X
iajs-2613	37	10	{	{	PUNCT
iajs-2613	37	11	0̅	0̅	PROPN
iajs-2613	37	12	,	,	PUNCT
iajs-2613	37	13	4̅	4̅	PROPN
iajs-2613	37	14	,	,	PUNCT
iajs-2613	37	15	8̅	8̅	NUM
iajs-2613	37	16	,	,	PUNCT
iajs-2613	37	17	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	37	18	,	,	PUNCT
iajs-2613	37	19	16̅̅̅̅	16̅̅̅̅	NUM
iajs-2613	37	20	,	,	PUNCT
iajs-2613	37	21	20̅̅̅̅	20̅̅̅̅	PROPN
iajs-2613	37	22	}	}	PUNCT
iajs-2613	37	23	.	.	PUNCT
iajs-2613	38	1	2	2	X
iajs-2613	38	2	.	.	X
iajs-2613	38	3	the	the	DET
iajs-2613	38	4	submodule	submodule	PROPN
iajs-2613	38	5	12z	12z	PROPN
iajs-2613	38	6	of	of	ADP
iajs-2613	38	7	the	the	DET
iajs-2613	38	8	z	z	NOUN
iajs-2613	38	9	-	-	PUNCT
iajs-2613	38	10	module	module	NOUN
iajs-2613	38	11	z	z	NOUN
iajs-2613	38	12	is	be	AUX
iajs-2613	38	13	not	not	PART
iajs-2613	38	14	wapp	wapp	NOUN
iajs-2613	38	15	-	-	PUNCT
iajs-2613	38	16	quasi	quasi	ADJ
iajs-2613	38	17	prime	prime	NOUN
iajs-2613	38	18	submodule	submodule	NOUN
iajs-2613	38	19	,	,	PUNCT
iajs-2613	38	20	since	since	SCONJ
iajs-2613	38	21	soc(z)=(0	soc(z)=(0	NUM
iajs-2613	38	22	)	)	PUNCT
iajs-2613	38	23	and	and	CCONJ
iajs-2613	38	24	whenever	whenever	SCONJ
iajs-2613	38	25	0≠3.4.1ϵ	0≠3.4.1ϵ	X
iajs-2613	38	26	12z	12z	X
iajs-2613	38	27	,	,	PUNCT
iajs-2613	38	28	for	for	ADP
iajs-2613	38	29	3,4,1ϵz	3,4,1ϵz	NUM
iajs-2613	38	30	,	,	PUNCT
iajs-2613	38	31	implies	imply	VERB
iajs-2613	38	32	that	that	SCONJ
iajs-2613	38	33	3.112z+soc(z	3.112z+soc(z	NUM
iajs-2613	38	34	)	)	PUNCT
iajs-2613	38	35	and	and	CCONJ
iajs-2613	38	36	4.112z+soc(z	4.112z+soc(z	NOUN
iajs-2613	38	37	)	)	PUNCT
iajs-2613	38	38	3	3	NUM
iajs-2613	38	39	.	.	PUNCT
iajs-2613	39	1	it	it	PRON
iajs-2613	39	2	is	be	AUX
iajs-2613	39	3	clear	clear	ADJ
iajs-2613	39	4	that	that	SCONJ
iajs-2613	39	5	every	every	DET
iajs-2613	39	6	weakly	weakly	ADJ
iajs-2613	39	7	quasi	quasi	ADJ
iajs-2613	39	8	prime	prime	ADJ
iajs-2613	39	9	submodule	submodule	NOUN
iajs-2613	39	10	of	of	ADP
iajs-2613	39	11	an	an	DET
iajs-2613	39	12	r	r	NOUN
iajs-2613	39	13	-	-	PUNCT
iajs-2613	39	14	module	module	NOUN
iajs-2613	39	15	t	t	NOUN
iajs-2613	39	16	is	be	AUX
iajs-2613	39	17	wapp	wapp	NOUN
iajs-2613	39	18	-	-	PUNCT
iajs-2613	39	19	quasi	quasi	ADJ
iajs-2613	39	20	prime	prime	NOUN
iajs-2613	39	21	but	but	CCONJ
iajs-2613	39	22	not	not	PART
iajs-2613	39	23	conversely	conversely	ADV
iajs-2613	39	24	.	.	PUNCT
iajs-2613	40	1	the	the	DET
iajs-2613	40	2	following	following	ADJ
iajs-2613	40	3	example	example	NOUN
iajs-2613	40	4	explains	explain	VERB
iajs-2613	40	5	that	that	SCONJ
iajs-2613	40	6	:	:	PUNCT
iajs-2613	40	7	consider	consider	VERB
iajs-2613	40	8	the	the	DET
iajs-2613	40	9	z	z	NOUN
iajs-2613	40	10	-	-	PUNCT
iajs-2613	40	11	module	module	NOUN
iajs-2613	40	12	z24	z24	NOUN
iajs-2613	40	13	,	,	PUNCT
iajs-2613	40	14	and	and	CCONJ
iajs-2613	40	15	the	the	DET
iajs-2613	40	16	submodule	submodule	NOUN
iajs-2613	40	17	c=	c=	NOUN
iajs-2613	40	18	<	<	X
iajs-2613	40	19	6̅	6̅	X
iajs-2613	40	20	>	>	X
iajs-2613	40	21	=	=	X
iajs-2613	40	22	{	{	PUNCT
iajs-2613	40	23	0̅	0̅	PROPN
iajs-2613	40	24	,	,	PUNCT
iajs-2613	40	25	6̅	6̅	PROPN
iajs-2613	40	26	,	,	PUNCT
iajs-2613	40	27	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	40	28	,	,	PUNCT
iajs-2613	40	29	18̅̅̅̅	18̅̅̅̅	PROPN
iajs-2613	40	30	}	}	PUNCT
iajs-2613	40	31	,	,	PUNCT
iajs-2613	40	32	c	c	NOUN
iajs-2613	40	33	is	be	AUX
iajs-2613	40	34	not	not	PART
iajs-2613	40	35	weakly	weakly	ADJ
iajs-2613	40	36	quasi	quasi	ADJ
iajs-2613	40	37	prime	prime	ADJ
iajs-2613	40	38	submodule	submodule	NOUN
iajs-2613	40	39	of	of	ADP
iajs-2613	40	40	z24	z24	PROPN
iajs-2613	40	41	since	since	SCONJ
iajs-2613	40	42	2.3.1̅ϵc	2.3.1̅ϵc	NUM
iajs-2613	40	43	=	=	NOUN
iajs-2613	40	44	<	<	X
iajs-2613	40	45	6̅	6̅	X
iajs-2613	40	46	>	>	X
iajs-2613	40	47	,	,	PUNCT
iajs-2613	40	48	for	for	ADP
iajs-2613	40	49	2,3	2,3	NUM
iajs-2613	40	50	ϵz	ϵz	NOUN
iajs-2613	40	51	,	,	PUNCT
iajs-2613	40	52	1̅ϵz24	1̅ϵz24	NUM
iajs-2613	40	53	,	,	PUNCT
iajs-2613	40	54	implies	imply	VERB
iajs-2613	40	55	that	that	SCONJ
iajs-2613	40	56	2	2	X
iajs-2613	40	57	.	.	PUNCT
iajs-2613	40	58	1̅=2̅≠	1̅=2̅≠	PROPN
iajs-2613	40	59	<	<	X
iajs-2613	40	60	6̅	6̅	PROPN
iajs-2613	40	61	>	>	X
iajs-2613	40	62	and	and	CCONJ
iajs-2613	40	63	3	3	X
iajs-2613	40	64	.	.	PUNCT
iajs-2613	41	1	1̅=3̅	1̅=3̅	X
iajs-2613	41	2	<	<	X
iajs-2613	42	1	6̅	6̅	PROPN
iajs-2613	42	2	>	>	X
iajs-2613	42	3	.	.	PUNCT
iajs-2613	43	1	bat	bat	NOUN
iajs-2613	43	2	c	c	PROPN
iajs-2613	43	3	is	be	AUX
iajs-2613	43	4	a	a	DET
iajs-2613	43	5	wapp	wapp	NOUN
iajs-2613	43	6	-	-	PUNCT
iajs-2613	43	7	quasi	quasi	ADJ
iajs-2613	43	8	prime	prime	PROPN
iajs-2613	43	9	submodule	submodule	NOUN
iajs-2613	43	10	of	of	ADP
iajs-2613	43	11	z24	z24	PROPN
iajs-2613	43	12	,	,	PUNCT
iajs-2613	43	13	since	since	SCONJ
iajs-2613	43	14	soc(z24)=	soc(z24)=	PROPN
iajs-2613	43	15	<	<	X
iajs-2613	43	16	4̅	4̅	PROPN
iajs-2613	43	17	>	>	X
iajs-2613	43	18	,	,	PUNCT
iajs-2613	43	19	and	and	CCONJ
iajs-2613	43	20	whenever	whenever	SCONJ
iajs-2613	43	21	0≠	0≠	NUM
iajs-2613	43	22	abt	abt	VERB
iajs-2613	43	23	∈	∈	NOUN
iajs-2613	43	24	c	c	NOUN
iajs-2613	44	1	=	=	PUNCT
iajs-2613	44	2	<	<	X
iajs-2613	45	1	6̅	6̅	X
iajs-2613	45	2	>	>	X
iajs-2613	45	3	=	=	PUNCT
iajs-2613	45	4	{	{	PUNCT
iajs-2613	45	5	0̅	0̅	PROPN
iajs-2613	45	6	,	,	PUNCT
iajs-2613	45	7	6̅	6̅	PROPN
iajs-2613	45	8	,	,	PUNCT
iajs-2613	45	9	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	45	10	,	,	PUNCT
iajs-2613	45	11	18̅̅̅̅	18̅̅̅̅	PROPN
iajs-2613	45	12	}	}	PUNCT
iajs-2613	45	13	for	for	ADP
iajs-2613	45	14	a	a	DET
iajs-2613	45	15	,	,	PUNCT
iajs-2613	45	16	b	b	PROPN
iajs-2613	45	17	∈	∈	PROPN
iajs-2613	45	18	z	z	NOUN
iajs-2613	45	19	,	,	PUNCT
iajs-2613	45	20	t	t	PROPN
iajs-2613	45	21	∈z24	∈z24	PROPN
iajs-2613	45	22	implies	imply	VERB
iajs-2613	45	23	that	that	SCONJ
iajs-2613	45	24	either	either	CCONJ
iajs-2613	45	25	at∈	at∈	PROPN
iajs-2613	45	26	c	c	PROPN
iajs-2613	45	27	+	+	CCONJ
iajs-2613	45	28	soc(z24)=	soc(z24)=	ADJ
iajs-2613	45	29	<	<	X
iajs-2613	46	1	6̅	6̅	X
iajs-2613	46	2	>	>	X
iajs-2613	47	1	+	+	ADJ
iajs-2613	47	2	<	<	X
iajs-2613	47	3	4̅	4̅	ADJ
iajs-2613	47	4	>	>	PUNCT
iajs-2613	47	5	=	=	X
iajs-2613	47	6	<	<	X
iajs-2613	47	7	2̅	2̅	NUM
iajs-2613	47	8	>	>	X
iajs-2613	47	9	or	or	CCONJ
iajs-2613	47	10	bt∈	bt∈	ADP
iajs-2613	47	11	c	c	PROPN
iajs-2613	47	12	+	+	CCONJ
iajs-2613	47	13	soc(z24)=	soc(z24)=	ADJ
iajs-2613	47	14	<	<	X
iajs-2613	47	15	6̅	6̅	X
iajs-2613	47	16	>	>	X
iajs-2613	48	1	+	+	ADJ
iajs-2613	48	2	<	<	X
iajs-2613	48	3	4̅	4̅	ADJ
iajs-2613	48	4	>	>	PUNCT
iajs-2613	49	1	=	=	X
iajs-2613	49	2	<	<	X
iajs-2613	49	3	2̅	2̅	NUM
iajs-2613	49	4	>	>	PUNCT
iajs-2613	49	5	.	.	PUNCT
iajs-2613	50	1	that	that	PRON
iajs-2613	50	2	is	be	AUX
iajs-2613	50	3	0≠	0≠	NUM
iajs-2613	50	4	2.3	2.3	NUM
iajs-2613	50	5	.	.	PUNCT
iajs-2613	51	1	1̅	1̅	NUM
iajs-2613	51	2	∈	∈	NOUN
iajs-2613	51	3	c	c	NOUN
iajs-2613	51	4	,	,	PUNCT
iajs-2613	51	5	for	for	ADP
iajs-2613	51	6	2,3	2,3	NUM
iajs-2613	51	7	∈	∈	PROPN
iajs-2613	51	8	z	z	NOUN
iajs-2613	51	9	,	,	PUNCT
iajs-2613	51	10	1̅	1̅	NUM
iajs-2613	51	11	∈z24	∈z24	NOUN
iajs-2613	51	12	,	,	PUNCT
iajs-2613	51	13	implies	imply	VERB
iajs-2613	51	14	that	that	SCONJ
iajs-2613	51	15	2.1̅	2.1̅	NUM
iajs-2613	51	16	∈	∈	PROPN
iajs-2613	51	17	c	c	NOUN
iajs-2613	51	18	+	+	CCONJ
iajs-2613	51	19	soc(z24)=	soc(z24)=	X
iajs-2613	51	20	<	<	X
iajs-2613	51	21	2̅	2̅	NOUN
iajs-2613	51	22	>	>	PUNCT
iajs-2613	51	23	.	.	PUNCT
iajs-2613	52	1	4	4	X
iajs-2613	52	2	.	.	X
iajs-2613	53	1	it	it	PRON
iajs-2613	53	2	is	be	AUX
iajs-2613	53	3	clear	clear	ADJ
iajs-2613	53	4	that	that	SCONJ
iajs-2613	53	5	ever	ever	ADV
iajs-2613	53	6	weakly	weakly	ADJ
iajs-2613	53	7	prime	prime	ADJ
iajs-2613	53	8	submodule	submodule	NOUN
iajs-2613	53	9	of	of	ADP
iajs-2613	53	10	an	an	DET
iajs-2613	53	11	r	r	NOUN
iajs-2613	53	12	-	-	PUNCT
iajs-2613	53	13	module	module	NOUN
iajs-2613	53	14	t	t	NOUN
iajs-2613	53	15	is	be	AUX
iajs-2613	53	16	a	a	DET
iajs-2613	53	17	waap	waap	ADJ
iajs-2613	53	18	-	-	PUNCT
iajs-2613	53	19	quasi	quasi	ADJ
iajs-2613	53	20	prime	prime	NOUN
iajs-2613	53	21	but	but	CCONJ
iajs-2613	53	22	not	not	PART
iajs-2613	53	23	conversely	conversely	ADV
iajs-2613	53	24	.	.	PUNCT
iajs-2613	54	1	the	the	DET
iajs-2613	54	2	following	following	ADJ
iajs-2613	54	3	example	example	NOUN
iajs-2613	54	4	explains	explain	VERB
iajs-2613	54	5	that	that	SCONJ
iajs-2613	54	6	:	:	PUNCT
iajs-2613	54	7	58	58	NUM
iajs-2613	54	8	ibn	ibn	PROPN
iajs-2613	54	9	al	al	PROPN
iajs-2613	54	10	-	-	PUNCT
iajs-2613	54	11	haitham	haitham	PROPN
iajs-2613	54	12	jour	jour	X
iajs-2613	54	13	.	.	PROPN
iajs-2613	55	1	for	for	ADP
iajs-2613	55	2	pure	pure	ADJ
iajs-2613	55	3	&	&	CCONJ
iajs-2613	55	4	appl	appl	PROPN
iajs-2613	55	5	.	.	PUNCT
iajs-2613	56	1	sci	sci	PROPN
iajs-2613	56	2	.	.	PROPN
iajs-2613	57	1	34	34	NUM
iajs-2613	57	2	(	(	PUNCT
iajs-2613	57	3	1	1	NUM
iajs-2613	57	4	)	)	PUNCT
iajs-2613	57	5	2021	2021	NUM
iajs-2613	57	6	consider	consider	VERB
iajs-2613	57	7	the	the	DET
iajs-2613	57	8	z	z	NOUN
iajs-2613	57	9	-	-	PUNCT
iajs-2613	57	10	module	module	NOUN
iajs-2613	57	11	z24	z24	NOUN
iajs-2613	57	12	and	and	CCONJ
iajs-2613	57	13	the	the	DET
iajs-2613	57	14	submodule	submodule	NOUN
iajs-2613	57	15	c=	c=	NOUN
iajs-2613	57	16	<	<	X
iajs-2613	57	17	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	57	18	>	>	PUNCT
iajs-2613	57	19	=	=	NOUN
iajs-2613	57	20	{	{	PUNCT
iajs-2613	57	21	0̅	0̅	PROPN
iajs-2613	57	22	,	,	PUNCT
iajs-2613	57	23	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	57	24	}	}	PUNCT
iajs-2613	57	25	.	.	PUNCT
iajs-2613	58	1	from	from	ADP
iajs-2613	58	2	(	(	PUNCT
iajs-2613	58	3	1	1	NUM
iajs-2613	58	4	)	)	PUNCT
iajs-2613	58	5	,	,	PUNCT
iajs-2613	58	6	c	c	PROPN
iajs-2613	58	7	is	be	AUX
iajs-2613	58	8	wappquasi	wappquasi	PROPN
iajs-2613	58	9	prime	prime	ADJ
iajs-2613	58	10	submodule	submodule	NOUN
iajs-2613	58	11	of	of	ADP
iajs-2613	58	12	z24	z24	PROPN
iajs-2613	58	13	.	.	PUNCT
iajs-2613	59	1	but	but	CCONJ
iajs-2613	59	2	c	c	NOUN
iajs-2613	59	3	is	be	AUX
iajs-2613	59	4	not	not	PART
iajs-2613	59	5	weakly	weakly	ADJ
iajs-2613	59	6	prime	prime	ADJ
iajs-2613	59	7	submodule	submodule	NOUN
iajs-2613	59	8	of	of	ADP
iajs-2613	59	9	z24	z24	PROPN
iajs-2613	59	10	.	.	PUNCT
iajs-2613	60	1	since	since	SCONJ
iajs-2613	60	2	if	if	SCONJ
iajs-2613	60	3	0≠	0≠	NUM
iajs-2613	60	4	3	3	NUM
iajs-2613	60	5	.	.	PUNCT
iajs-2613	61	1	4̅	4̅	PROPN
iajs-2613	61	2	∈	∈	PROPN
iajs-2613	61	3	c	c	NOUN
iajs-2613	61	4	,	,	PUNCT
iajs-2613	61	5	for	for	ADP
iajs-2613	61	6	3ϵz	3ϵz	NOUN
iajs-2613	61	7	,	,	PUNCT
iajs-2613	61	8	4̅ϵz24	4̅ϵz24	ADJ
iajs-2613	61	9	,	,	PUNCT
iajs-2613	61	10	but	but	CCONJ
iajs-2613	61	11	4̅	4̅	PROPN
iajs-2613	61	12	c	c	PROPN
iajs-2613	61	13	and	and	CCONJ
iajs-2613	61	14	3[c	3[c	NUM
iajs-2613	61	15	:	:	PUNCT
iajs-2613	61	16	z24]=6z	z24]=6z	NUM
iajs-2613	61	17	5	5	NUM
iajs-2613	61	18	.	.	PUNCT
iajs-2613	62	1	the	the	DET
iajs-2613	62	2	residual	residual	NOUN
iajs-2613	62	3	of	of	ADP
iajs-2613	62	4	wapp	wapp	NOUN
iajs-2613	62	5	-	-	PUNCT
iajs-2613	62	6	quasi	quasi	ADJ
iajs-2613	62	7	prime	prime	PROPN
iajs-2613	62	8	submodule	submodule	PROPN
iajs-2613	62	9	c	c	PROPN
iajs-2613	62	10	of	of	ADP
iajs-2613	62	11	an	an	DET
iajs-2613	62	12	r	r	NOUN
iajs-2613	62	13	-	-	PUNCT
iajs-2613	62	14	module	module	NOUN
iajs-2613	62	15	t	t	NOUN
iajs-2613	62	16	needs	need	VERB
iajs-2613	62	17	not	not	PART
iajs-2613	62	18	to	to	PART
iajs-2613	62	19	be	be	AUX
iajs-2613	62	20	wapp	wapp	NOUN
iajs-2613	62	21	-	-	PUNCT
iajs-2613	62	22	quasi	quasi	ADJ
iajs-2613	62	23	prime	prime	ADJ
iajs-2613	62	24	ideal	ideal	NOUN
iajs-2613	62	25	of	of	ADP
iajs-2613	62	26	r	r	NOUN
iajs-2613	62	27	.	.	PUNCT
iajs-2613	63	1	the	the	DET
iajs-2613	63	2	following	following	ADJ
iajs-2613	63	3	example	example	NOUN
iajs-2613	63	4	explains	explain	VERB
iajs-2613	63	5	that	that	SCONJ
iajs-2613	63	6	:	:	PUNCT
iajs-2613	63	7	we	we	PRON
iajs-2613	63	8	have	have	AUX
iajs-2613	63	9	seen	see	VERB
iajs-2613	63	10	in(1	in(1	NOUN
iajs-2613	63	11	)	)	PUNCT
iajs-2613	63	12	that	that	SCONJ
iajs-2613	63	13	the	the	DET
iajs-2613	63	14	submodule	submodule	NOUN
iajs-2613	63	15	c=	c=	NOUN
iajs-2613	63	16	<	<	X
iajs-2613	63	17	12̅̅̅̅	12̅̅̅̅	PROPN
iajs-2613	63	18	>	>	X
iajs-2613	63	19	of	of	ADP
iajs-2613	63	20	the	the	DET
iajs-2613	63	21	z	z	NOUN
iajs-2613	63	22	−	−	PROPN
iajs-2613	63	23	module	module	NOUN
iajs-2613	63	24	z24	z24	NOUN
iajs-2613	63	25	is	be	AUX
iajs-2613	63	26	a	a	DET
iajs-2613	63	27	wapp	wapp	NOUN
iajs-2613	63	28	-	-	PUNCT
iajs-2613	63	29	quasi	quasi	ADJ
iajs-2613	63	30	prime	prime	NOUN
iajs-2613	63	31	but	but	CCONJ
iajs-2613	64	1	[	[	X
iajs-2613	64	2	c	c	X
iajs-2613	64	3	:	:	PUNCT
iajs-2613	64	4	z	z	NOUN
iajs-2613	64	5	z24]=	z24]=	NUM
iajs-2613	64	6	[	[	X
iajs-2613	64	7	<	<	X
iajs-2613	64	8	12̅̅̅̅	12̅̅̅̅	NUM
iajs-2613	64	9	>	>	X
iajs-2613	64	10	:	:	PUNCT
iajs-2613	64	11	z	z	NOUN
iajs-2613	64	12	z24]=12z	z24]=12z	NOUN
iajs-2613	64	13	is	be	AUX
iajs-2613	64	14	not	not	PART
iajs-2613	64	15	wapp	wapp	NOUN
iajs-2613	64	16	-	-	PUNCT
iajs-2613	64	17	quasi	quasi	ADJ
iajs-2613	64	18	prime	prime	ADJ
iajs-2613	64	19	ideal	ideal	NOUN
iajs-2613	64	20	by	by	ADP
iajs-2613	64	21	(	(	PUNCT
iajs-2613	64	22	2	2	NUM
iajs-2613	64	23	)	)	PUNCT
iajs-2613	64	24	.	.	PUNCT
iajs-2613	65	1	6	6	X
iajs-2613	65	2	.	.	X
iajs-2613	66	1	the	the	DET
iajs-2613	66	2	submodules	submodules	PROPN
iajs-2613	66	3	pz	pz	PROPN
iajs-2613	66	4	of	of	ADP
iajs-2613	66	5	a	a	DET
iajs-2613	66	6	z	z	NOUN
iajs-2613	66	7	-	-	PUNCT
iajs-2613	66	8	module	module	NOUN
iajs-2613	66	9	z	z	NOUN
iajs-2613	66	10	is	be	AUX
iajs-2613	66	11	a	a	DET
iajs-2613	66	12	wapp	wapp	NOUN
iajs-2613	66	13	-	-	PUNCT
iajs-2613	66	14	quasi	quasi	ADJ
iajs-2613	66	15	prime	prime	NOUN
iajs-2613	66	16	if	if	SCONJ
iajs-2613	67	1	and	and	CCONJ
iajs-2613	67	2	only	only	ADV
iajs-2613	67	3	if	if	SCONJ
iajs-2613	67	4	p	p	NOUN
iajs-2613	67	5	is	be	AUX
iajs-2613	67	6	prime	prime	ADJ
iajs-2613	67	7	number	number	NOUN
iajs-2613	67	8	7	7	NUM
iajs-2613	67	9	.	.	PUNCT
iajs-2613	68	1	the	the	DET
iajs-2613	68	2	intersection	intersection	NOUN
iajs-2613	68	3	of	of	ADP
iajs-2613	68	4	two	two	NUM
iajs-2613	68	5	wapp	wapp	NOUN
iajs-2613	68	6	-	-	PUNCT
iajs-2613	68	7	quasi	quasi	ADJ
iajs-2613	68	8	prime	prime	ADJ
iajs-2613	68	9	submodule	submodule	NOUN
iajs-2613	68	10	of	of	ADP
iajs-2613	68	11	r	r	NOUN
iajs-2613	68	12	-	-	PUNCT
iajs-2613	68	13	module	module	NOUN
iajs-2613	68	14	,	,	PUNCT
iajs-2613	68	15	t	t	NOUN
iajs-2613	68	16	need	need	NOUN
iajs-2613	68	17	,	,	PUNCT
iajs-2613	68	18	not	not	PART
iajs-2613	68	19	to	to	PART
iajs-2613	68	20	be	be	AUX
iajs-2613	68	21	wapp	wapp	NOUN
iajs-2613	68	22	-	-	PUNCT
iajs-2613	68	23	quasi	quasi	ADJ
iajs-2613	68	24	prime	prime	PROPN
iajs-2613	68	25	submodule	submodule	NOUN
iajs-2613	68	26	of	of	ADP
iajs-2613	68	27	t	t	PROPN
iajs-2613	68	28	for	for	ADP
iajs-2613	68	29	example	example	NOUN
iajs-2613	68	30	:	:	PUNCT
iajs-2613	68	31	the	the	DET
iajs-2613	68	32	submodule	submodule	NOUN
iajs-2613	68	33	2z	2z	NUM
iajs-2613	68	34	and	and	CCONJ
iajs-2613	68	35	5z	5z	NOUN
iajs-2613	68	36	of	of	ADP
iajs-2613	68	37	the	the	DET
iajs-2613	68	38	z	z	NOUN
iajs-2613	68	39	-	-	PUNCT
iajs-2613	68	40	module	module	NOUN
iajs-2613	68	41	z	z	NOUN
iajs-2613	68	42	are	be	AUX
iajs-2613	68	43	wapp	wapp	NOUN
iajs-2613	68	44	-	-	PUNCT
iajs-2613	68	45	quasi	quasi	ADJ
iajs-2613	68	46	prime	prime	NOUN
iajs-2613	68	47	submodule	submodule	NOUN
iajs-2613	68	48	by	by	ADP
iajs-2613	68	49	(	(	PUNCT
iajs-2613	68	50	6	6	NUM
iajs-2613	68	51	)	)	PUNCT
iajs-2613	68	52	.	.	PUNCT
iajs-2613	69	1	but	but	CCONJ
iajs-2613	69	2	2z∩5z=10z	2z∩5z=10z	PROPN
iajs-2613	69	3	is	be	AUX
iajs-2613	69	4	not	not	PART
iajs-2613	69	5	wapp	wapp	NOUN
iajs-2613	69	6	-	-	PUNCT
iajs-2613	69	7	quasi	quasi	ADJ
iajs-2613	69	8	prime	prime	PROPN
iajs-2613	69	9	submodule	submodule	NOUN
iajs-2613	69	10	of	of	ADP
iajs-2613	69	11	thez	thez	PROPN
iajs-2613	69	12	−	−	PROPN
iajs-2613	69	13	module	module	NOUN
iajs-2613	69	14	z	z	NOUN
iajs-2613	69	15	,	,	PUNCT
iajs-2613	69	16	since	since	SCONJ
iajs-2613	69	17	0≠2.5.1ϵ10z	0≠2.5.1ϵ10z	PROPN
iajs-2613	69	18	,	,	PUNCT
iajs-2613	69	19	for	for	ADP
iajs-2613	69	20	2,5,1ϵz	2,5,1ϵz	NUM
iajs-2613	69	21	but	but	CCONJ
iajs-2613	69	22	2.1=210z+soc(z	2.1=210z+soc(z	NUM
iajs-2613	69	23	)	)	PUNCT
iajs-2613	69	24	and	and	CCONJ
iajs-2613	69	25	5.1=510z+soc(z	5.1=510z+soc(z	NUM
iajs-2613	69	26	)	)	PUNCT
iajs-2613	69	27	the	the	DET
iajs-2613	69	28	following	follow	VERB
iajs-2613	69	29	proposition	proposition	NOUN
iajs-2613	69	30	are	be	AUX
iajs-2613	69	31	characterizations	characterization	NOUN
iajs-2613	69	32	of	of	ADP
iajs-2613	69	33	wapp	wapp	NOUN
iajs-2613	69	34	-	-	PUNCT
iajs-2613	69	35	quasi	quasi	ADJ
iajs-2613	69	36	prime	prime	ADJ
iajs-2613	69	37	submodules	submodule	NOUN
iajs-2613	69	38	.	.	PUNCT
iajs-2613	70	1	proposition(3	proposition(3	PROPN
iajs-2613	70	2	)	)	PUNCT
iajs-2613	70	3	let	let	VERB
iajs-2613	70	4	𝑇	𝑇	PROPN
iajs-2613	70	5	be	be	AUX
iajs-2613	70	6	an	an	DET
iajs-2613	70	7	r	r	NOUN
iajs-2613	70	8	−	−	NOUN
iajs-2613	70	9	modul	modul	NOUN
iajs-2613	70	10	and	and	CCONJ
iajs-2613	70	11	c	c	PROPN
iajs-2613	70	12	be	be	AUX
iajs-2613	70	13	proper	proper	ADJ
iajs-2613	70	14	submodul	submodul	NOUN
iajs-2613	70	15	of	of	ADP
iajs-2613	70	16	t	t	PROPN
iajs-2613	70	17	,	,	PUNCT
iajs-2613	70	18	then	then	ADV
iajs-2613	70	19	c	c	PROPN
iajs-2613	70	20	is	be	AUX
iajs-2613	70	21	wapp	wapp	NOUN
iajs-2613	70	22	−	−	PROPN
iajs-2613	70	23	quasi	quasi	ADJ
iajs-2613	70	24	prime	prime	PROPN
iajs-2613	70	25	sub	sub	PROPN
iajs-2613	70	26	modul	modul	PROPN
iajs-2613	70	27	of	of	ADP
iajs-2613	70	28	t	t	PROPN
iajs-2613	71	1	if	if	SCONJ
iajs-2613	71	2	and	and	CCONJ
iajs-2613	71	3	only	only	ADV
iajs-2613	71	4	if	if	SCONJ
iajs-2613	71	5	,	,	PUNCT
iajs-2613	71	6	whenever	whenever	SCONJ
iajs-2613	71	7	0≠rsbc	0≠rsbc	NOUN
iajs-2613	71	8	,	,	PUNCT
iajs-2613	71	9	for	for	ADP
iajs-2613	71	10	r	r	NOUN
iajs-2613	71	11	,	,	PUNCT
iajs-2613	71	12	sϵ	sϵ	NOUN
iajs-2613	71	13	r	r	NOUN
iajs-2613	71	14	,	,	PUNCT
iajs-2613	71	15	b	b	PROPN
iajs-2613	71	16	is	be	AUX
iajs-2613	71	17	submodul	submodul	NOUN
iajs-2613	71	18	of	of	ADP
iajs-2613	71	19	t	t	PROPN
iajs-2613	71	20	,	,	PUNCT
iajs-2613	71	21	implies	imply	VERB
iajs-2613	71	22	that	that	SCONJ
iajs-2613	71	23	either	either	CCONJ
iajs-2613	71	24	r	r	NOUN
iajs-2613	71	25	bc	bc	NOUN
iajs-2613	72	1	+	+	VERB
iajs-2613	72	2	soc(t	soc(t	PROPN
iajs-2613	72	3	)	)	PUNCT
iajs-2613	72	4	or	or	CCONJ
iajs-2613	72	5	s	s	NOUN
iajs-2613	72	6	bc	bc	PROPN
iajs-2613	73	1	+	+	ADJ
iajs-2613	73	2	soc(t	soc(t	PROPN
iajs-2613	73	3	)	)	PUNCT
iajs-2613	73	4	.	.	PUNCT
iajs-2613	74	1	proof	proof	NOUN
iajs-2613	74	2	:	:	PUNCT
iajs-2613	74	3	(	(	PUNCT
iajs-2613	74	4			NOUN
iajs-2613	74	5	)	)	PUNCT
iajs-2613	74	6	assum	assum	VERB
iajs-2613	74	7	that	that	SCONJ
iajs-2613	74	8	c	c	PROPN
iajs-2613	74	9	is	be	AUX
iajs-2613	74	10	awpp	awpp	ADJ
iajs-2613	74	11	-	-	PUNCT
iajs-2613	74	12	quasi	quasi	ADJ
iajs-2613	74	13	prime	prime	PROPN
iajs-2613	74	14	submodule	submodule	NOUN
iajs-2613	74	15	of	of	ADP
iajs-2613	74	16	t	t	PROPN
iajs-2613	74	17	and	and	CCONJ
iajs-2613	74	18	0≠rsbc	0≠rsbc	NOUN
iajs-2613	74	19	.	.	PUNCT
iajs-2613	75	1	for	for	ADP
iajs-2613	75	2	r	r	NOUN
iajs-2613	75	3	,	,	PUNCT
iajs-2613	75	4	s	s	PART
iajs-2613	75	5	ϵr	ϵr	X
iajs-2613	75	6	,	,	PUNCT
iajs-2613	75	7	b	b	PROPN
iajs-2613	75	8	is	be	AUX
iajs-2613	75	9	a	a	DET
iajs-2613	75	10	submodule	submodule	NOUN
iajs-2613	75	11	of	of	ADP
iajs-2613	75	12	t	t	PROPN
iajs-2613	75	13	,	,	PUNCT
iajs-2613	75	14	with	with	SCONJ
iajs-2613	75	15	r	r	NOUN
iajs-2613	75	16	b	b	NOUN
iajs-2613	75	17	c+	c+	VERB
iajs-2613	75	18	soc(t	soc(t	NUM
iajs-2613	75	19	)	)	PUNCT
iajs-2613	75	20	and	and	CCONJ
iajs-2613	75	21	s	s	AUX
iajs-2613	75	22	b	b	NOUN
iajs-2613	75	23	c	c	NOUN
iajs-2613	75	24	+	+	PROPN
iajs-2613	75	25	soc(t	soc(t	PROPN
iajs-2613	75	26	)	)	PUNCT
iajs-2613	75	27	,	,	PUNCT
iajs-2613	75	28	that	that	PRON
iajs-2613	75	29	is	be	AUX
iajs-2613	75	30	there	there	PRON
iajs-2613	75	31	exists	exist	VERB
iajs-2613	75	32	a	a	DET
iajs-2613	75	33	nonzero	nonzero	ADJ
iajs-2613	75	34	elements	element	NOUN
iajs-2613	75	35	b1	b1	NOUN
iajs-2613	75	36	,	,	PUNCT
iajs-2613	75	37	b2	b2	NOUN
iajs-2613	75	38	ϵb	ϵb	ADP
iajs-2613	75	39	such	such	ADJ
iajs-2613	75	40	that	that	DET
iajs-2613	75	41	rb1	rb1	NOUN
iajs-2613	75	42	c	c	ADP
iajs-2613	75	43	+	+	NOUN
iajs-2613	75	44	soc(t	soc(t	PROPN
iajs-2613	75	45	)	)	PUNCT
iajs-2613	75	46	and	and	CCONJ
iajs-2613	75	47	s	s	NOUN
iajs-2613	75	48	b2	b2	NOUN
iajs-2613	75	49			NOUN
iajs-2613	75	50	c	c	PROPN
iajs-2613	76	1	+	+	NOUN
iajs-2613	76	2	soc(t).now	soc(t).now	NOUN
iajs-2613	76	3	0≠rsb1	0≠rsb1	NUM
iajs-2613	76	4	ϵc	ϵc	INTJ
iajs-2613	76	5	,	,	PUNCT
iajs-2613	76	6	and	and	CCONJ
iajs-2613	76	7	c	c	PROPN
iajs-2613	76	8	is	be	AUX
iajs-2613	76	9	wapp	wapp	NOUN
iajs-2613	76	10	-	-	PUNCT
iajs-2613	76	11	quasi	quasi	ADJ
iajs-2613	76	12	prime	prime	NOUN
iajs-2613	76	13	submodule	submodule	NOUN
iajs-2613	76	14	and	and	CCONJ
iajs-2613	76	15	r	r	NOUN
iajs-2613	76	16	b1	b1	NOUN
iajs-2613	76	17	c	c	PROPN
iajs-2613	76	18	+	+	PROPN
iajs-2613	76	19	soc(t	soc(t	PROPN
iajs-2613	76	20	)	)	PUNCT
iajs-2613	77	1	,	,	PUNCT
iajs-2613	77	2	implies	imply	VERB
iajs-2613	77	3	that	that	SCONJ
iajs-2613	77	4	s	s	VERB
iajs-2613	77	5	b1	b1	NOUN
iajs-2613	77	6	ϵc	ϵc	ADP
iajs-2613	77	7	+	+	NOUN
iajs-2613	77	8	soc(t	soc(t	PROPN
iajs-2613	77	9	)	)	PUNCT
iajs-2613	77	10	.	.	PUNCT
iajs-2613	78	1	also	also	ADV
iajs-2613	78	2	0≠rsb2	0≠rsb2	NUM
iajs-2613	79	1	ϵc	ϵc	INTJ
iajs-2613	79	2	,	,	PUNCT
iajs-2613	79	3	and	and	CCONJ
iajs-2613	79	4	c	c	NOUN
iajs-2613	79	5	is	be	AUX
iajs-2613	79	6	a	a	DET
iajs-2613	79	7	wapp	wapp	NOUN
iajs-2613	79	8	-	-	PUNCT
iajs-2613	79	9	quasi	quasi	ADJ
iajs-2613	79	10	prime	prime	PROPN
iajs-2613	79	11	submodule	submodule	NOUN
iajs-2613	79	12	of	of	ADP
iajs-2613	79	13	t	t	PROPN
iajs-2613	79	14	,	,	PUNCT
iajs-2613	79	15	and	and	CCONJ
iajs-2613	79	16	sb2	sb2	PROPN
iajs-2613	79	17	c	c	PROPN
iajs-2613	79	18	+	+	PROPN
iajs-2613	79	19	soc(t	soc(t	PROPN
iajs-2613	79	20	)	)	PUNCT
iajs-2613	79	21	.,implies	.,implie	NOUN
iajs-2613	80	1	that	that	DET
iajs-2613	80	2	rb2ϵc+soc(t	rb2ϵc+soc(t	NOUN
iajs-2613	80	3	)	)	PUNCT
iajs-2613	80	4	.	.	PUNCT
iajs-2613	81	1	again	again	ADV
iajs-2613	81	2	since	since	SCONJ
iajs-2613	81	3	0≠rs(b1+b2)ϵc	0≠rs(b1+b2)ϵc	NUM
iajs-2613	81	4	and	and	CCONJ
iajs-2613	81	5	c	c	PROPN
iajs-2613	81	6	is	be	AUX
iajs-2613	81	7	wapp	wapp	NOUN
iajs-2613	81	8	-	-	PUNCT
iajs-2613	81	9	quasi	quasi	ADJ
iajs-2613	81	10	prime	prime	PROPN
iajs-2613	81	11	submodule	submodule	NOUN
iajs-2613	81	12	of	of	ADP
iajs-2613	81	13	t	t	PROPN
iajs-2613	81	14	,	,	PUNCT
iajs-2613	81	15	implies	imply	VERB
iajs-2613	81	16	that	that	SCONJ
iajs-2613	81	17	either	either	CCONJ
iajs-2613	81	18	r(b1+b2)ϵc	r(b1+b2)ϵc	VERB
iajs-2613	81	19	+	+	ADJ
iajs-2613	81	20	soc(t	soc(t	PROPN
iajs-2613	81	21	)	)	PUNCT
iajs-2613	81	22	or	or	CCONJ
iajs-2613	81	23	s(b1+b2)ϵc	s(b1+b2)ϵc	ADJ
iajs-2613	81	24	+	+	ADJ
iajs-2613	81	25	soc(t	soc(t	PROPN
iajs-2613	81	26	)	)	PUNCT
iajs-2613	81	27	.	.	PUNCT
iajs-2613	82	1	if	if	SCONJ
iajs-2613	82	2	r(b1+b2)ϵc	r(b1+b2)ϵc	VERB
iajs-2613	82	3	+	+	ADJ
iajs-2613	82	4	soc(t	soc(t	PROPN
iajs-2613	82	5	)	)	PUNCT
iajs-2613	82	6	,	,	PUNCT
iajs-2613	82	7	that	that	PRON
iajs-2613	82	8	is	be	AUX
iajs-2613	82	9	rb1+rb2ϵc+soc(t	rb1+rb2ϵc+soc(t	PROPN
iajs-2613	82	10	)	)	PUNCT
iajs-2613	82	11	,	,	PUNCT
iajs-2613	82	12	and	and	CCONJ
iajs-2613	82	13	since	since	SCONJ
iajs-2613	82	14	rb2ϵc+soc(t	rb2ϵc+soc(t	NOUN
iajs-2613	82	15	)	)	PUNCT
iajs-2613	82	16	,	,	PUNCT
iajs-2613	82	17	it	it	PRON
iajs-2613	82	18	follows	follow	VERB
iajs-2613	82	19	that	that	SCONJ
iajs-2613	82	20	rb1ϵc+soc(t	rb1ϵc+soc(t	NOUN
iajs-2613	82	21	)	)	PUNCT
iajs-2613	82	22	which	which	PRON
iajs-2613	82	23	is	be	AUX
iajs-2613	82	24	contradiction	contradiction	NOUN
iajs-2613	82	25	.	.	PUNCT
iajs-2613	83	1	if	if	SCONJ
iajs-2613	83	2	s(b1+b2)ϵc	s(b1+b2)ϵc	PRON
iajs-2613	83	3	+	+	ADJ
iajs-2613	83	4	soc(t	soc(t	NOUN
iajs-2613	83	5	)	)	PUNCT
iajs-2613	83	6	,	,	PUNCT
iajs-2613	83	7	that	that	PRON
iajs-2613	83	8	is	be	AUX
iajs-2613	83	9	sb1+sb2ϵc+soc(t	sb1+sb2ϵc+soc(t	PROPN
iajs-2613	83	10	)	)	PUNCT
iajs-2613	83	11	and	and	CCONJ
iajs-2613	83	12	since	since	SCONJ
iajs-2613	83	13	sb1ϵc+soc(t	sb1ϵc+soc(t	PROPN
iajs-2613	83	14	)	)	PUNCT
iajs-2613	83	15	,	,	PUNCT
iajs-2613	83	16	it	it	PRON
iajs-2613	83	17	follows	follow	VERB
iajs-2613	83	18	that	that	SCONJ
iajs-2613	83	19	sb2ϵc+soc(t	sb2ϵc+soc(t	NOUN
iajs-2613	83	20	)	)	PUNCT
iajs-2613	83	21	which	which	PRON
iajs-2613	83	22	is	be	AUX
iajs-2613	83	23	contradiction	contradiction	NOUN
iajs-2613	83	24	.	.	PUNCT
iajs-2613	84	1	hence	hence	ADV
iajs-2613	84	2	r	r	NOUN
iajs-2613	84	3	bc	bc	NOUN
iajs-2613	85	1	+	+	VERB
iajs-2613	85	2	soc(t	soc(t	PROPN
iajs-2613	85	3	)	)	PUNCT
iajs-2613	85	4	or	or	CCONJ
iajs-2613	85	5	s	s	NOUN
iajs-2613	85	6	b	b	PROPN
iajs-2613	85	7	c	c	X
iajs-2613	85	8	+	+	ADJ
iajs-2613	85	9	soc(t	soc(t	PROPN
iajs-2613	85	10	)	)	PUNCT
iajs-2613	85	11	.	.	PUNCT
iajs-2613	86	1	(	(	PUNCT
iajs-2613	86	2			NOUN
iajs-2613	86	3	)	)	PUNCT
iajs-2613	86	4	let	let	VERB
iajs-2613	86	5	0≠rstϵc	0≠rstϵc	NOUN
iajs-2613	86	6	,	,	PUNCT
iajs-2613	86	7	for	for	ADP
iajs-2613	86	8	r	r	NOUN
iajs-2613	86	9	,	,	PUNCT
iajs-2613	86	10	s	s	PART
iajs-2613	86	11	ϵr	ϵr	X
iajs-2613	86	12	,	,	PUNCT
iajs-2613	86	13	t	t	X
iajs-2613	86	14	ϵt	ϵt	ADV
iajs-2613	86	15	,	,	PUNCT
iajs-2613	86	16	it	it	PRON
iajs-2613	86	17	follows	follow	VERB
iajs-2613	86	18	that	that	SCONJ
iajs-2613	86	19	0≠rs	0≠rs	NOUN
iajs-2613	86	20	<	<	X
iajs-2613	86	21	t>c	t>c	PROPN
iajs-2613	86	22	,	,	PUNCT
iajs-2613	86	23	so	so	ADV
iajs-2613	86	24	by	by	ADP
iajs-2613	86	25	hypothesis	hypothesis	NOUN
iajs-2613	86	26	either	either	CCONJ
iajs-2613	86	27	r	r	NOUN
iajs-2613	86	28	<	<	X
iajs-2613	86	29	t>c+	t>c+	NOUN
iajs-2613	86	30	soc(t	soc(t	NOUN
iajs-2613	86	31	)	)	PUNCT
iajs-2613	86	32	or	or	CCONJ
iajs-2613	86	33	s	s	AUX
iajs-2613	86	34	<	<	X
iajs-2613	86	35	t>c+	t>c+	NOUN
iajs-2613	86	36	soc(t	soc(t	NOUN
iajs-2613	86	37	)	)	PUNCT
iajs-2613	86	38	.	.	PUNCT
iajs-2613	87	1	that	that	PRON
iajs-2613	87	2	is	be	AUX
iajs-2613	87	3	either	either	DET
iajs-2613	87	4	r	r	NOUN
iajs-2613	87	5	tϵ	tϵ	NOUN
iajs-2613	87	6	c+	c+	VERB
iajs-2613	87	7	soc(t	soc(t	NUM
iajs-2613	87	8	)	)	PUNCT
iajs-2613	87	9	or	or	CCONJ
iajs-2613	87	10	s	s	AUX
iajs-2613	87	11	tϵ	tϵ	NOUN
iajs-2613	87	12	c+	c+	VERB
iajs-2613	87	13	soc(t	soc(t	NUM
iajs-2613	87	14	)	)	PUNCT
iajs-2613	87	15	.hence	.hence	PUNCT
iajs-2613	88	1	c	c	NOUN
iajs-2613	88	2	is	be	AUX
iajs-2613	88	3	wapp	wapp	NOUN
iajs-2613	88	4	−	−	PROPN
iajs-2613	88	5	quasi	quasi	ADJ
iajs-2613	88	6	prime	prime	PROPN
iajs-2613	88	7	submodul	submodul	NOUN
iajs-2613	88	8	of	of	ADP
iajs-2613	88	9	t.	t.	PROPN
iajs-2613	88	10	59	59	NUM
iajs-2613	88	11	ibn	ibn	PROPN
iajs-2613	88	12	al	al	PROPN
iajs-2613	88	13	-	-	PUNCT
iajs-2613	88	14	haitham	haitham	PROPN
iajs-2613	88	15	jour	jour	X
iajs-2613	88	16	.	.	PROPN
iajs-2613	89	1	for	for	ADP
iajs-2613	89	2	pure	pure	ADJ
iajs-2613	89	3	&	&	CCONJ
iajs-2613	89	4	appl	appl	PROPN
iajs-2613	89	5	.	.	PUNCT
iajs-2613	90	1	sci	sci	PROPN
iajs-2613	90	2	.	.	PROPN
iajs-2613	91	1	34	34	NUM
iajs-2613	91	2	(	(	PUNCT
iajs-2613	91	3	1	1	NUM
iajs-2613	91	4	)	)	PUNCT
iajs-2613	91	5	2021	2021	NUM
iajs-2613	91	6	proposition(4	proposition(4	PROPN
iajs-2613	91	7	)	)	PUNCT
iajs-2613	91	8	let𝑇	let𝑇	ADJ
iajs-2613	91	9	be	be	VERB
iajs-2613	91	10	r	r	NOUN
iajs-2613	91	11	−	−	NOUN
iajs-2613	91	12	module	module	NOUN
iajs-2613	91	13	and	and	CCONJ
iajs-2613	91	14	c	c	NOUN
iajs-2613	91	15	be	be	AUX
iajs-2613	91	16	proper	proper	ADJ
iajs-2613	91	17	submodule	submodule	NOUN
iajs-2613	91	18	of	of	ADP
iajs-2613	91	19	t	t	PROPN
iajs-2613	91	20	.	.	PUNCT
iajs-2613	92	1	then	then	ADV
iajs-2613	92	2	c	c	PROPN
iajs-2613	92	3	is	be	AUX
iajs-2613	92	4	wapp	wapp	NOUN
iajs-2613	92	5	−	−	PROPN
iajs-2613	92	6	quasi	quasi	ADJ
iajs-2613	92	7	prime	prime	PROPN
iajs-2613	92	8	submodul	submodul	NOUN
iajs-2613	92	9	of	of	ADP
iajs-2613	92	10	t	t	PROPN
iajs-2613	92	11	if	if	SCONJ
iajs-2613	93	1	and	and	CCONJ
iajs-2613	93	2	only	only	ADV
iajs-2613	93	3	if	if	SCONJ
iajs-2613	93	4	whenever	whenever	SCONJ
iajs-2613	93	5	0≠ijb	0≠ijb	PROPN
iajs-2613	93	6	c	c	NUM
iajs-2613	93	7	,	,	PUNCT
iajs-2613	93	8	for	for	ADP
iajs-2613	93	9	i	i	PRON
iajs-2613	93	10	,	,	PUNCT
iajs-2613	93	11	j	j	PROPN
iajs-2613	93	12	are	be	AUX
iajs-2613	93	13	ideals	ideal	NOUN
iajs-2613	93	14	of	of	ADP
iajs-2613	93	15	r	r	NOUN
iajs-2613	93	16	and	and	CCONJ
iajs-2613	93	17	b	b	NOUN
iajs-2613	93	18	is	be	AUX
iajs-2613	93	19	a	a	DET
iajs-2613	93	20	submodule	submodule	NOUN
iajs-2613	93	21	of	of	ADP
iajs-2613	93	22	t	t	PROPN
iajs-2613	93	23	,	,	PUNCT
iajs-2613	93	24	implies	imply	VERB
iajs-2613	93	25	that	that	SCONJ
iajs-2613	93	26	either	either	CCONJ
iajs-2613	93	27	ibc	ibc	PROPN
iajs-2613	93	28	+	+	PROPN
iajs-2613	93	29	soc(t	soc(t	PROPN
iajs-2613	93	30	)	)	PUNCT
iajs-2613	93	31	or	or	CCONJ
iajs-2613	93	32	jbc	jbc	NOUN
iajs-2613	93	33	+	+	ADV
iajs-2613	93	34	soc(t	soc(t	PROPN
iajs-2613	93	35	)	)	PUNCT
iajs-2613	93	36	.	.	PUNCT
iajs-2613	94	1	proof	proof	NOUN
iajs-2613	94	2	:	:	PUNCT
iajs-2613	94	3	(	(	PUNCT
iajs-2613	94	4	)assume	)assume	NOUN
iajs-2613	94	5	that	that	DET
iajs-2613	94	6	0≠ijbc	0≠ijbc	NOUN
iajs-2613	94	7	.	.	PUNCT
iajs-2613	95	1	for	for	ADP
iajs-2613	95	2	i	i	PRON
iajs-2613	95	3	,	,	PUNCT
iajs-2613	95	4	j	j	PROPN
iajs-2613	95	5	are	be	AUX
iajs-2613	95	6	ideal	ideal	ADJ
iajs-2613	95	7	of	of	ADP
iajs-2613	95	8	r	r	NOUN
iajs-2613	95	9	,	,	PUNCT
iajs-2613	95	10	b	b	PROPN
iajs-2613	95	11	is	be	AUX
iajs-2613	95	12	a	a	DET
iajs-2613	95	13	submodule	submodule	NOUN
iajs-2613	95	14	of	of	ADP
iajs-2613	95	15	t	t	PROPN
iajs-2613	95	16	,	,	PUNCT
iajs-2613	95	17	with	with	ADP
iajs-2613	95	18	ibc+soc(t	ibc+soc(t	PROPN
iajs-2613	95	19	)	)	PUNCT
iajs-2613	95	20	and	and	CCONJ
iajs-2613	95	21	j	j	PROPN
iajs-2613	95	22	bc	bc	PROPN
iajs-2613	95	23	+	+	PROPN
iajs-2613	95	24	soc(t	soc(t	PROPN
iajs-2613	95	25	)	)	PUNCT
iajs-2613	95	26	,	,	PUNCT
iajs-2613	95	27	so	so	CCONJ
iajs-2613	95	28	there	there	PRON
iajs-2613	95	29	exists	exist	VERB
iajs-2613	95	30	a	a	DET
iajs-2613	95	31	nonzero	nonzero	ADJ
iajs-2613	95	32	elements	element	NOUN
iajs-2613	95	33	b1	b1	NOUN
iajs-2613	95	34	,	,	PUNCT
iajs-2613	95	35	b2	b2	NOUN
iajs-2613	95	36	ϵ	ϵ	PROPN
iajs-2613	95	37	b	b	PROPN
iajs-2613	95	38	and	and	CCONJ
iajs-2613	95	39	a	a	DET
iajs-2613	95	40	nonzero	nonzero	NOUN
iajs-2613	95	41	elements	element	NOUN
iajs-2613	95	42	r	r	NOUN
iajs-2613	95	43	ϵi	ϵi	NOUN
iajs-2613	95	44	,	,	PUNCT
iajs-2613	95	45	s	s	VERB
iajs-2613	95	46	ϵj	ϵj	NOUN
iajs-2613	95	47	such	such	ADJ
iajs-2613	95	48	that	that	DET
iajs-2613	95	49	rb1	rb1	NOUN
iajs-2613	95	50	c	c	ADP
iajs-2613	95	51	+	+	NOUN
iajs-2613	95	52	soc(t	soc(t	PROPN
iajs-2613	95	53	)	)	PUNCT
iajs-2613	95	54	and	and	CCONJ
iajs-2613	96	1	sb2	sb2	PROPN
iajs-2613	96	2	c	c	PROPN
iajs-2613	96	3	+	+	PROPN
iajs-2613	96	4	soc(t).now	soc(t).now	PROPN
iajs-2613	96	5	0≠rsb1	0≠rsb1	NUM
iajs-2613	96	6	ϵc	ϵc	INTJ
iajs-2613	96	7	,	,	PUNCT
iajs-2613	96	8	and	and	CCONJ
iajs-2613	96	9	c	c	NOUN
iajs-2613	96	10	is	be	AUX
iajs-2613	96	11	a	a	DET
iajs-2613	96	12	wapp	wapp	NOUN
iajs-2613	96	13	-	-	PUNCT
iajs-2613	96	14	quasi	quasi	ADJ
iajs-2613	96	15	prime	prime	NOUN
iajs-2613	96	16	submodule	submodule	NOUN
iajs-2613	96	17	and	and	CCONJ
iajs-2613	96	18	rb1	rb1	NOUN
iajs-2613	96	19	c+soc(t	c+soc(t	PROPN
iajs-2613	96	20	)	)	PUNCT
iajs-2613	96	21	,	,	PUNCT
iajs-2613	96	22	implies	imply	VERB
iajs-2613	96	23	that	that	SCONJ
iajs-2613	96	24	sb1ϵ	sb1ϵ	PROPN
iajs-2613	96	25	c+soc(t	c+soc(t	NOUN
iajs-2613	96	26	)	)	PUNCT
iajs-2613	96	27	.	.	PUNCT
iajs-2613	97	1	also	also	ADV
iajs-2613	97	2	0≠rsb2ϵc	0≠rsb2ϵc	NOUN
iajs-2613	97	3	,	,	PUNCT
iajs-2613	97	4	and	and	CCONJ
iajs-2613	97	5	c	c	NOUN
iajs-2613	97	6	is	be	AUX
iajs-2613	97	7	a	a	DET
iajs-2613	97	8	wapp	wapp	NOUN
iajs-2613	97	9	-	-	PUNCT
iajs-2613	97	10	quasi	quasi	ADJ
iajs-2613	97	11	prime	prime	PROPN
iajs-2613	97	12	submodule	submodule	NOUN
iajs-2613	97	13	of	of	ADP
iajs-2613	97	14	t	t	PROPN
iajs-2613	97	15	,	,	PUNCT
iajs-2613	97	16	and	and	CCONJ
iajs-2613	97	17	sb2	sb2	PROPN
iajs-2613	97	18	c+soc(t	c+soc(t	PROPN
iajs-2613	97	19	)	)	PUNCT
iajs-2613	97	20	.,implies	.,implies	PROPN
iajs-2613	97	21	that	that	DET
iajs-2613	97	22	rb2ϵc+soc(t	rb2ϵc+soc(t	NOUN
iajs-2613	97	23	)	)	PUNCT
iajs-2613	97	24	.	.	PUNCT
iajs-2613	98	1	again	again	ADV
iajs-2613	98	2	0≠rs(b1+b2)ϵc	0≠rs(b1+b2)ϵc	NUM
iajs-2613	98	3	and	and	CCONJ
iajs-2613	98	4	c	c	PROPN
iajs-2613	98	5	is	be	AUX
iajs-2613	98	6	wapp	wapp	NOUN
iajs-2613	98	7	-	-	PUNCT
iajs-2613	98	8	quasi	quasi	ADJ
iajs-2613	98	9	prime	prime	PROPN
iajs-2613	98	10	submodule	submodule	NOUN
iajs-2613	98	11	of	of	ADP
iajs-2613	98	12	t	t	PROPN
iajs-2613	98	13	,	,	PUNCT
iajs-2613	98	14	implies	imply	VERB
iajs-2613	98	15	that	that	SCONJ
iajs-2613	98	16	either	either	CCONJ
iajs-2613	98	17	r(b1+b2)ϵc	r(b1+b2)ϵc	VERB
iajs-2613	98	18	+	+	ADJ
iajs-2613	98	19	soc(t	soc(t	PROPN
iajs-2613	98	20	)	)	PUNCT
iajs-2613	98	21	or	or	CCONJ
iajs-2613	98	22	s(b1+b2)ϵc	s(b1+b2)ϵc	ADJ
iajs-2613	98	23	+	+	ADJ
iajs-2613	98	24	soc(t	soc(t	PROPN
iajs-2613	98	25	)	)	PUNCT
iajs-2613	98	26	.	.	PUNCT
iajs-2613	99	1	if	if	SCONJ
iajs-2613	99	2	r(b1+b2)ϵc	r(b1+b2)ϵc	VERB
iajs-2613	99	3	+	+	ADJ
iajs-2613	99	4	soc(t	soc(t	PROPN
iajs-2613	99	5	)	)	PUNCT
iajs-2613	99	6	,	,	PUNCT
iajs-2613	99	7	that	that	PRON
iajs-2613	99	8	is	be	AUX
iajs-2613	99	9	rb1+rb2ϵc+soc(t	rb1+rb2ϵc+soc(t	PROPN
iajs-2613	99	10	)	)	PUNCT
iajs-2613	99	11	,	,	PUNCT
iajs-2613	99	12	and	and	CCONJ
iajs-2613	99	13	rb2ϵc+soc(t	rb2ϵc+soc(t	NOUN
iajs-2613	99	14	)	)	PUNCT
iajs-2613	99	15	,	,	PUNCT
iajs-2613	99	16	implies	imply	VERB
iajs-2613	99	17	that	that	SCONJ
iajs-2613	99	18	rb1ϵc+soc(t	rb1ϵc+soc(t	NOUN
iajs-2613	99	19	)	)	PUNCT
iajs-2613	99	20	contradiction	contradiction	NOUN
iajs-2613	99	21	.	.	PUNCT
iajs-2613	100	1	if	if	SCONJ
iajs-2613	100	2	s(b1+b2)ϵc	s(b1+b2)ϵc	PRON
iajs-2613	100	3	+	+	ADJ
iajs-2613	100	4	soc(t	soc(t	NOUN
iajs-2613	100	5	)	)	PUNCT
iajs-2613	100	6	,	,	PUNCT
iajs-2613	100	7	that	that	PRON
iajs-2613	100	8	is	be	AUX
iajs-2613	100	9	sb1+sb2ϵc+soc(t	sb1+sb2ϵc+soc(t	PROPN
iajs-2613	100	10	)	)	PUNCT
iajs-2613	100	11	and	and	CCONJ
iajs-2613	100	12	sb1ϵc+soc(t	sb1ϵc+soc(t	PROPN
iajs-2613	100	13	)	)	PUNCT
iajs-2613	100	14	,	,	PUNCT
iajs-2613	100	15	implies	imply	VERB
iajs-2613	100	16	that	that	SCONJ
iajs-2613	100	17	sb2ϵc+soc(t	sb2ϵc+soc(t	NOUN
iajs-2613	100	18	)	)	PUNCT
iajs-2613	100	19	which	which	PRON
iajs-2613	100	20	is	be	AUX
iajs-2613	100	21	contradiction	contradiction	NOUN
iajs-2613	100	22	.	.	PUNCT
iajs-2613	101	1	hence	hence	ADV
iajs-2613	101	2	ibc+soc(t	ibc+soc(t	PROPN
iajs-2613	101	3	)	)	PUNCT
iajs-2613	101	4	or	or	CCONJ
iajs-2613	101	5	jbc+soc(t	jbc+soc(t	PROPN
iajs-2613	101	6	)	)	PUNCT
iajs-2613	101	7	.	.	PUNCT
iajs-2613	102	1	(	(	PUNCT
iajs-2613	102	2			NOUN
iajs-2613	102	3	)	)	PUNCT
iajs-2613	102	4	suppose	suppose	VERB
iajs-2613	102	5	that	that	SCONJ
iajs-2613	102	6	0≠rstϵ	0≠rstϵ	PROPN
iajs-2613	102	7	c	c	NOUN
iajs-2613	102	8	,	,	PUNCT
iajs-2613	102	9	for	for	ADP
iajs-2613	102	10	r	r	NOUN
iajs-2613	102	11	,	,	PUNCT
iajs-2613	102	12	s	s	PART
iajs-2613	102	13	ϵr	ϵr	X
iajs-2613	102	14	,	,	PUNCT
iajs-2613	102	15	tϵt	tϵt	NOUN
iajs-2613	102	16	,	,	PUNCT
iajs-2613	102	17	that	that	PRON
iajs-2613	102	18	is	be	AUX
iajs-2613	102	19	0≠<r><s><t>c	0≠<r><s><t>c	NUM
iajs-2613	102	20	,	,	PUNCT
iajs-2613	102	21	so	so	ADV
iajs-2613	102	22	by	by	ADP
iajs-2613	102	23	our	our	PRON
iajs-2613	102	24	assumption	assumption	NOUN
iajs-2613	102	25	either	either	CCONJ
iajs-2613	102	26	(	(	PUNCT
iajs-2613	102	27	r)(t)	r)(t)	NOUN
iajs-2613	102	28	c+	c+	X
iajs-2613	102	29	soc(t	soc(t	NUM
iajs-2613	102	30	)	)	PUNCT
iajs-2613	102	31	or	or	CCONJ
iajs-2613	102	32	(	(	PUNCT
iajs-2613	102	33	r)(t	r)(t	ADJ
iajs-2613	102	34	)	)	PUNCT
iajs-2613	102	35			PROPN
iajs-2613	102	36	c+	c+	VERB
iajs-2613	102	37	soc(t	soc(t	NUM
iajs-2613	102	38	)	)	PUNCT
iajs-2613	102	39	.	.	PUNCT
iajs-2613	103	1	that	that	PRON
iajs-2613	103	2	is	be	AUX
iajs-2613	103	3	either	either	DET
iajs-2613	103	4	rt∈	rt∈	NOUN
iajs-2613	103	5	c	c	PROPN
iajs-2613	103	6	+	+	CCONJ
iajs-2613	103	7	soc(t	soc(t	PROPN
iajs-2613	103	8	)	)	PUNCT
iajs-2613	103	9	or	or	CCONJ
iajs-2613	103	10	st∈	st∈	VERB
iajs-2613	103	11	c	c	NOUN
iajs-2613	103	12	+	+	CCONJ
iajs-2613	103	13	soc(t	soc(t	PROPN
iajs-2613	103	14	)	)	PUNCT
iajs-2613	103	15	.hence	.hence	PUNCT
iajs-2613	104	1	c	c	NOUN
iajs-2613	104	2	is	be	AUX
iajs-2613	104	3	wapp	wapp	NOUN
iajs-2613	104	4	-	-	PUNCT
iajs-2613	104	5	quasi	quasi	ADJ
iajs-2613	104	6	prime	prime	ADJ
iajs-2613	104	7	submodule	submodule	NOUN
iajs-2613	104	8	of	of	ADP
iajs-2613	104	9	t.	t.	PROPN
iajs-2613	104	10	as	as	ADP
iajs-2613	104	11	a	a	DET
iajs-2613	104	12	direct	direct	ADJ
iajs-2613	104	13	consequence	consequence	NOUN
iajs-2613	104	14	of	of	ADP
iajs-2613	104	15	the	the	DET
iajs-2613	104	16	above	above	ADJ
iajs-2613	104	17	propositions	proposition	NOUN
iajs-2613	104	18	,	,	PUNCT
iajs-2613	104	19	we	we	PRON
iajs-2613	104	20	get	get	VERB
iajs-2613	104	21	the	the	DET
iajs-2613	104	22	following	follow	VERB
iajs-2613	104	23	corollaries	corollary	NOUN
iajs-2613	104	24	.	.	PUNCT
iajs-2613	105	1	corollary(5	corollary(5	PROPN
iajs-2613	105	2	)	)	PUNCT
iajs-2613	105	3	let	let	VERB
iajs-2613	105	4	𝑇	𝑇	PROPN
iajs-2613	105	5	be	be	AUX
iajs-2613	105	6	r	r	NOUN
iajs-2613	105	7	−	−	NOUN
iajs-2613	105	8	module	module	NOUN
iajs-2613	105	9	and	and	CCONJ
iajs-2613	105	10	c	c	NOUN
iajs-2613	105	11	be	be	AUX
iajs-2613	105	12	proper	proper	ADJ
iajs-2613	105	13	submodule	submodule	NOUN
iajs-2613	105	14	of	of	ADP
iajs-2613	105	15	t	t	PROPN
iajs-2613	105	16	.	.	PUNCT
iajs-2613	106	1	then	then	ADV
iajs-2613	106	2	c	c	PROPN
iajs-2613	106	3	is	be	AUX
iajs-2613	106	4	wapp	wapp	NOUN
iajs-2613	106	5	−	−	PROPN
iajs-2613	106	6	quasi	quasi	ADJ
iajs-2613	106	7	prime	prime	PROPN
iajs-2613	106	8	submodule	submodule	PROPN
iajs-2613	106	9	of	of	ADP
iajs-2613	106	10	t	t	PROPN
iajs-2613	106	11	iff	iff	PROPN
iajs-2613	106	12	whenever	whenever	SCONJ
iajs-2613	106	13	0≠rit	0≠rit	PROPN
iajs-2613	106	14	c	c	NUM
iajs-2613	106	15	,	,	PUNCT
iajs-2613	106	16	for	for	ADP
iajs-2613	106	17	r	r	NOUN
iajs-2613	106	18	ϵr	ϵr	NOUN
iajs-2613	106	19	,	,	PUNCT
iajs-2613	106	20	i	i	PRON
iajs-2613	106	21	is	be	AUX
iajs-2613	106	22	an	an	DET
iajs-2613	106	23	ideals	ideal	NOUN
iajs-2613	106	24	of	of	ADP
iajs-2613	106	25	r	r	NOUN
iajs-2613	106	26	and	and	CCONJ
iajs-2613	106	27	t∈	t∈	PROPN
iajs-2613	106	28	t	t	PROPN
iajs-2613	106	29	,	,	PUNCT
iajs-2613	106	30	implies	imply	VERB
iajs-2613	106	31	that	that	SCONJ
iajs-2613	106	32	either	either	CCONJ
iajs-2613	106	33	r	r	NOUN
iajs-2613	106	34	t	t	X
iajs-2613	106	35	ϵc	ϵc	NOUN
iajs-2613	106	36	+	+	NOUN
iajs-2613	106	37	soc(t	soc(t	PROPN
iajs-2613	106	38	)	)	PUNCT
iajs-2613	106	39	or	or	CCONJ
iajs-2613	106	40	i	i	PRON
iajs-2613	106	41	t	t	PROPN
iajs-2613	106	42	c	c	PUNCT
iajs-2613	106	43	+	+	PRON
iajs-2613	106	44	soc(t	soc(t	PROPN
iajs-2613	106	45	)	)	PUNCT
iajs-2613	106	46	.	.	PUNCT
iajs-2613	107	1	corollary(6	corollary(6	NOUN
iajs-2613	107	2	)	)	PUNCT
iajs-2613	107	3	let	let	VERB
iajs-2613	107	4	𝑇	𝑇	PROPN
iajs-2613	107	5	be	be	AUX
iajs-2613	107	6	r	r	NOUN
iajs-2613	107	7	−	−	NOUN
iajs-2613	107	8	module	module	NOUN
iajs-2613	107	9	and	and	CCONJ
iajs-2613	107	10	c	c	NOUN
iajs-2613	107	11	be	be	AUX
iajs-2613	107	12	proper	proper	ADJ
iajs-2613	107	13	submodule	submodule	NOUN
iajs-2613	107	14	of	of	ADP
iajs-2613	107	15	t	t	PROPN
iajs-2613	107	16	.	.	PUNCT
iajs-2613	108	1	then	then	ADV
iajs-2613	108	2	c	c	PROPN
iajs-2613	108	3	is	be	AUX
iajs-2613	108	4	wapp	wapp	NOUN
iajs-2613	108	5	−	−	PROPN
iajs-2613	108	6	quasi	quasi	ADJ
iajs-2613	108	7	prime	prime	PROPN
iajs-2613	108	8	submodule	submodule	PROPN
iajs-2613	108	9	of	of	ADP
iajs-2613	108	10	t	t	PROPN
iajs-2613	108	11	iff	iff	PROPN
iajs-2613	108	12	whenever	whenever	SCONJ
iajs-2613	108	13	0≠ijt	0≠ijt	PROPN
iajs-2613	108	14	c	c	NUM
iajs-2613	108	15	,	,	PUNCT
iajs-2613	108	16	for	for	ADP
iajs-2613	108	17	j	j	PROPN
iajs-2613	108	18	,	,	PUNCT
iajs-2613	108	19	,	,	PUNCT
iajs-2613	108	20	i	i	PRON
iajs-2613	108	21	is	be	AUX
iajs-2613	108	22	an	an	DET
iajs-2613	108	23	ideals	ideal	NOUN
iajs-2613	108	24	of	of	ADP
iajs-2613	108	25	r	r	NOUN
iajs-2613	109	1	and	and	CCONJ
iajs-2613	109	2	tϵ	tϵ	NOUN
iajs-2613	109	3	t	t	PROPN
iajs-2613	109	4	,	,	PUNCT
iajs-2613	109	5	implies	imply	VERB
iajs-2613	109	6	that	that	SCONJ
iajs-2613	109	7	either	either	CCONJ
iajs-2613	109	8	j	j	PROPN
iajs-2613	109	9	tc	tc	PROPN
iajs-2613	109	10	+	+	NOUN
iajs-2613	109	11	soc(t	soc(t	PROPN
iajs-2613	109	12	)	)	PUNCT
iajs-2613	109	13	or	or	CCONJ
iajs-2613	109	14	itc	itc	NOUN
iajs-2613	109	15	+	+	NOUN
iajs-2613	109	16	soc(t	soc(t	PROPN
iajs-2613	109	17	)	)	PUNCT
iajs-2613	109	18	.	.	PUNCT
iajs-2613	110	1	corollary(7	corollary(7	PROPN
iajs-2613	110	2	)	)	PUNCT
iajs-2613	110	3	let	let	VERB
iajs-2613	110	4	𝑇	𝑇	PROPN
iajs-2613	110	5	be	be	AUX
iajs-2613	110	6	r	r	NOUN
iajs-2613	110	7	−	−	NOUN
iajs-2613	110	8	module	module	NOUN
iajs-2613	110	9	and	and	CCONJ
iajs-2613	110	10	c	c	NOUN
iajs-2613	110	11	be	be	AUX
iajs-2613	110	12	proper	proper	ADJ
iajs-2613	110	13	submodule	submodule	NOUN
iajs-2613	110	14	of	of	ADP
iajs-2613	110	15	t	t	PROPN
iajs-2613	110	16	.	.	PUNCT
iajs-2613	111	1	then	then	ADV
iajs-2613	111	2	c	c	PROPN
iajs-2613	111	3	is	be	AUX
iajs-2613	111	4	wapp	wapp	NOUN
iajs-2613	111	5	−	−	PROPN
iajs-2613	111	6	quasi	quasi	ADJ
iajs-2613	111	7	prime	prime	PROPN
iajs-2613	111	8	submodul	submodul	NOUN
iajs-2613	111	9	of	of	ADP
iajs-2613	111	10	t	t	PROPN
iajs-2613	111	11	if	if	SCONJ
iajs-2613	111	12	and	and	CCONJ
iajs-2613	111	13	only	only	ADV
iajs-2613	111	14	if	if	SCONJ
iajs-2613	111	15	,	,	PUNCT
iajs-2613	111	16	for	for	ADP
iajs-2613	111	17	each	each	DET
iajs-2613	111	18	r	r	NOUN
iajs-2613	111	19	ϵr	ϵr	NOUN
iajs-2613	111	20	and	and	CCONJ
iajs-2613	111	21	every	every	DET
iajs-2613	111	22	ideal	ideal	NOUN
iajs-2613	111	23	i	i	PRON
iajs-2613	111	24	of	of	ADP
iajs-2613	111	25	r	r	NOUN
iajs-2613	111	26	and	and	CCONJ
iajs-2613	111	27	every	every	DET
iajs-2613	111	28	submodule	submodule	PROPN
iajs-2613	111	29	b	b	PROPN
iajs-2613	111	30	of	of	ADP
iajs-2613	111	31	t	t	PROPN
iajs-2613	111	32	,	,	PUNCT
iajs-2613	111	33	with	with	ADP
iajs-2613	111	34	0≠rib	0≠rib	PROPN
iajs-2613	111	35	c	c	NUM
iajs-2613	111	36	,	,	PUNCT
iajs-2613	111	37	implies	imply	VERB
iajs-2613	111	38	that	that	SCONJ
iajs-2613	111	39	either	either	CCONJ
iajs-2613	111	40	rbc	rbc	ADV
iajs-2613	111	41	+	+	ADJ
iajs-2613	111	42	soc(t	soc(t	PROPN
iajs-2613	111	43	)	)	PUNCT
iajs-2613	111	44	or	or	CCONJ
iajs-2613	111	45	ibc	ibc	NOUN
iajs-2613	112	1	+	+	PROPN
iajs-2613	112	2	soc(t	soc(t	PROPN
iajs-2613	112	3	)	)	PUNCT
iajs-2613	112	4	.	.	PUNCT
iajs-2613	113	1	proposition(8	proposition(8	VERB
iajs-2613	113	2	)	)	PUNCT
iajs-2613	114	1	60	60	NUM
iajs-2613	114	2	ibn	ibn	PROPN
iajs-2613	114	3	al	al	PROPN
iajs-2613	114	4	-	-	PUNCT
iajs-2613	114	5	haitham	haitham	PROPN
iajs-2613	114	6	jour	jour	X
iajs-2613	114	7	.	.	PROPN
iajs-2613	114	8	for	for	ADP
iajs-2613	114	9	pure	pure	ADJ
iajs-2613	114	10	&	&	CCONJ
iajs-2613	114	11	appl	appl	PROPN
iajs-2613	114	12	.	.	PUNCT
iajs-2613	115	1	sci	sci	PROPN
iajs-2613	115	2	.	.	PROPN
iajs-2613	116	1	34	34	NUM
iajs-2613	116	2	(	(	PUNCT
iajs-2613	116	3	1	1	NUM
iajs-2613	116	4	)	)	PUNCT
iajs-2613	116	5	2021	2021	NUM
iajs-2613	116	6	let	let	VERB
iajs-2613	116	7	𝑇	𝑇	PROPN
iajs-2613	116	8	be	be	AUX
iajs-2613	116	9	r	r	NOUN
iajs-2613	116	10	−	−	NOUN
iajs-2613	116	11	module	module	NOUN
iajs-2613	116	12	and	and	CCONJ
iajs-2613	116	13	c	c	NOUN
iajs-2613	116	14	be	be	AUX
iajs-2613	116	15	proper	proper	ADJ
iajs-2613	116	16	submodule	submodule	NOUN
iajs-2613	116	17	of	of	ADP
iajs-2613	116	18	t	t	PROPN
iajs-2613	116	19	.	.	PUNCT
iajs-2613	117	1	then	then	ADV
iajs-2613	117	2	c	c	PROPN
iajs-2613	117	3	is	be	AUX
iajs-2613	117	4	wapp	wapp	NOUN
iajs-2613	117	5	−	−	PROPN
iajs-2613	117	6	quasi	quasi	ADJ
iajs-2613	117	7	prime	prime	PROPN
iajs-2613	117	8	submodul	submodul	NOUN
iajs-2613	117	9	of	of	ADP
iajs-2613	117	10	t	t	PROPN
iajs-2613	117	11	if	if	SCONJ
iajs-2613	117	12	and	and	CCONJ
iajs-2613	117	13	only	only	ADV
iajs-2613	117	14	if	if	SCONJ
iajs-2613	117	15	for	for	ADP
iajs-2613	117	16	each	each	DET
iajs-2613	117	17	r	r	NOUN
iajs-2613	117	18	,	,	PUNCT
iajs-2613	117	19	s	s	PART
iajs-2613	117	20	ϵr	ϵr	X
iajs-2613	117	21	,	,	PUNCT
iajs-2613	117	22	[	[	X
iajs-2613	117	23	c	c	X
iajs-2613	117	24	:	:	PUNCT
iajs-2613	117	25	rs][0	rs][0	NOUN
iajs-2613	117	26	:	:	PUNCT
iajs-2613	117	27	t	t	PROPN
iajs-2613	117	28	rs]∪[c+	rs]∪[c+	PROPN
iajs-2613	117	29	soc(t)::t	soc(t)::t	VERB
iajs-2613	117	30	r]∪[c+soc(t):t	r]∪[c+soc(t):t	PROPN
iajs-2613	117	31	s	s	NOUN
iajs-2613	117	32	]	]	PUNCT
iajs-2613	117	33	.	.	PUNCT
iajs-2613	118	1	proof	proof	NOUN
iajs-2613	118	2	:	:	PUNCT
iajs-2613	118	3	(	(	PUNCT
iajs-2613	118	4			NOUN
iajs-2613	118	5	)	)	PUNCT
iajs-2613	118	6	let	let	VERB
iajs-2613	118	7	tϵ[c	tϵ[c	PROPN
iajs-2613	118	8	:	:	PUNCT
iajs-2613	118	9	t	t	NOUN
iajs-2613	118	10	rs	rs	PROPN
iajs-2613	118	11	]	]	PUNCT
iajs-2613	118	12	,	,	PUNCT
iajs-2613	118	13	implies	imply	VERB
iajs-2613	118	14	that	that	SCONJ
iajs-2613	118	15	rstϵc	rstϵc	ADJ
iajs-2613	118	16	.if	.if	PUNCT
iajs-2613	118	17	rst=0	rst=0	ADJ
iajs-2613	118	18	,	,	PUNCT
iajs-2613	118	19	then	then	ADV
iajs-2613	118	20	tϵ[0	tϵ[0	NOUN
iajs-2613	118	21	:	:	PUNCT
iajs-2613	118	22	t	t	NOUN
iajs-2613	118	23	rs	rs	NOUN
iajs-2613	118	24	]	]	PUNCT
iajs-2613	118	25	,	,	PUNCT
iajs-2613	118	26	and	and	CCONJ
iajs-2613	118	27	hence	hence	ADV
iajs-2613	118	28	tϵ[0	tϵ[0	PROPN
iajs-2613	118	29	:	:	PUNCT
iajs-2613	118	30	t	t	PROPN
iajs-2613	118	31	rs]∪[c+	rs]∪[c+	PROPN
iajs-2613	118	32	soc(t)::t	soc(t)::t	NOUN
iajs-2613	118	33	r]∪[c+	r]∪[c+	VERB
iajs-2613	118	34	soc(t)::t	soc(t)::t	NOUN
iajs-2613	118	35	s	s	X
iajs-2613	118	36	]	]	X
iajs-2613	118	37	.suppose	.suppose	PUNCT
iajs-2613	118	38	that	that	DET
iajs-2613	118	39	0≠rstϵc	0≠rstϵc	NOUN
iajs-2613	118	40	and	and	CCONJ
iajs-2613	118	41	since	since	SCONJ
iajs-2613	118	42	c	c	PROPN
iajs-2613	118	43	is	be	AUX
iajs-2613	118	44	wapp	wapp	NOUN
iajs-2613	118	45	-	-	PUNCT
iajs-2613	118	46	quasi	quasi	ADJ
iajs-2613	118	47	prime	prime	PROPN
iajs-2613	118	48	submodule	submodule	NOUN
iajs-2613	118	49	of	of	ADP
iajs-2613	118	50	t	t	PROPN
iajs-2613	118	51	,	,	PUNCT
iajs-2613	118	52	it	it	PRON
iajs-2613	118	53	follows	follow	VERB
iajs-2613	118	54	that	that	SCONJ
iajs-2613	118	55	either	either	CCONJ
iajs-2613	118	56	rt	rt	PROPN
iajs-2613	118	57	ϵc	ϵc	X
iajs-2613	118	58	+	+	NOUN
iajs-2613	118	59	soc(t	soc(t	PROPN
iajs-2613	118	60	)	)	PUNCT
iajs-2613	118	61	or	or	CCONJ
iajs-2613	118	62	st	st	PROPN
iajs-2613	118	63	ϵc	ϵc	INTJ
iajs-2613	118	64	+	+	PROPN
iajs-2613	118	65	soc(t	soc(t	PROPN
iajs-2613	118	66	)	)	PUNCT
iajs-2613	118	67	,	,	PUNCT
iajs-2613	118	68	implies	imply	VERB
iajs-2613	118	69	that	that	SCONJ
iajs-2613	118	70	either	either	CCONJ
iajs-2613	118	71	tϵ[c+soc(t):t	tϵ[c+soc(t):t	PRON
iajs-2613	118	72	r	r	X
iajs-2613	118	73	]	]	PUNCT
iajs-2613	118	74	or	or	CCONJ
iajs-2613	118	75	tϵ[c+soc(t):t	tϵ[c+soc(t):t	PRON
iajs-2613	118	76	s	s	VERB
iajs-2613	118	77	]	]	X
iajs-2613	118	78	.that	.that	PRON
iajs-2613	118	79	is	be	AUX
iajs-2613	118	80	tϵ[0	tϵ[0	NOUN
iajs-2613	118	81	:	:	PUNCT
iajs-2613	118	82	t	t	PROPN
iajs-2613	118	83	rs]∪[c+	rs]∪[c+	PROPN
iajs-2613	118	84	soc(t)::t	soc(t)::t	NOUN
iajs-2613	118	85	r]∪[c+	r]∪[c+	VERB
iajs-2613	118	86	soc(t)::t	soc(t)::t	NOUN
iajs-2613	118	87	s	s	X
iajs-2613	118	88	]	]	X
iajs-2613	118	89	.hence	.hence	NOUN
iajs-2613	118	90	,	,	PUNCT
iajs-2613	119	1	[	[	X
iajs-2613	119	2	c	c	X
iajs-2613	119	3	:	:	PUNCT
iajs-2613	119	4	t	t	NOUN
iajs-2613	119	5	rs][0	rs][0	PROPN
iajs-2613	119	6	:	:	PUNCT
iajs-2613	119	7	t	t	PROPN
iajs-2613	119	8	rs]∪[c+	rs]∪[c+	PROPN
iajs-2613	119	9	soc(t)::t	soc(t)::t	NOUN
iajs-2613	119	10	r]∪[c+	r]∪[c+	VERB
iajs-2613	119	11	soc(t):t	soc(t):t	PROPN
iajs-2613	119	12	s	s	PART
iajs-2613	119	13	]	]	PUNCT
iajs-2613	119	14	.	.	PUNCT
iajs-2613	120	1	(	(	PUNCT
iajs-2613	120	2	)assume	)assume	NOUN
iajs-2613	120	3	that	that	SCONJ
iajs-2613	120	4	0≠rstϵc	0≠rstϵc	NOUN
iajs-2613	120	5	,	,	PUNCT
iajs-2613	120	6	for	for	ADP
iajs-2613	120	7	r	r	NOUN
iajs-2613	120	8	,	,	PUNCT
iajs-2613	120	9	sϵr	sϵr	NOUN
iajs-2613	120	10	,	,	PUNCT
iajs-2613	120	11	tϵt	tϵt	NOUN
iajs-2613	120	12	,	,	PUNCT
iajs-2613	120	13	implies	imply	VERB
iajs-2613	120	14	that	that	SCONJ
iajs-2613	120	15	tϵ,[c	tϵ,[c	ADP
iajs-2613	121	1	:	:	PUNCT
iajs-2613	121	2	t	t	PROPN
iajs-2613	121	3	rs]	rs]	PROPN
iajs-2613	122	1	[	[	PUNCT
iajs-2613	123	1	[	[	X
iajs-2613	123	2	0	0	NUM
iajs-2613	123	3	:	:	PUNCT
iajs-2613	123	4	t	t	PROPN
iajs-2613	123	5	rs]∪	rs]∪	VERB
iajs-2613	124	1	[	[	X
iajs-2613	124	2	c+	c+	NOUN
iajs-2613	124	3	soc(t):+:t	soc(t):+:t	NOUN
iajs-2613	124	4	r]∪[c+	r]∪[c+	NOUN
iajs-2613	124	5	soc(t):t	soc(t):t	PROPN
iajs-2613	124	6	s	s	X
iajs-2613	124	7	]	]	PUNCT
iajs-2613	124	8	.	.	PUNCT
iajs-2613	125	1	but	but	CCONJ
iajs-2613	125	2	0≠rst	0≠rst	PROPN
iajs-2613	125	3	,	,	PUNCT
iajs-2613	125	4	then	then	ADV
iajs-2613	125	5	t	t	PROPN
iajs-2613	125	6	[0	[0	PROPN
iajs-2613	125	7	:	:	PUNCT
iajs-2613	125	8	t	t	NOUN
iajs-2613	125	9	rs	rs	NOUN
iajs-2613	125	10	]	]	PUNCT
iajs-2613	125	11	,	,	PUNCT
iajs-2613	125	12	hence	hence	ADV
iajs-2613	125	13	tϵ	tϵ	ADP
iajs-2613	126	1	[	[	X
iajs-2613	126	2	c+	c+	VERB
iajs-2613	126	3	soc(t):t	soc(t):t	NOUN
iajs-2613	126	4	r]∪[c+	r]∪[c+	VERB
iajs-2613	126	5	soc(t):t	soc(t):t	PROPN
iajs-2613	126	6	s	s	PART
iajs-2613	126	7	]	]	PUNCT
iajs-2613	126	8	,	,	PUNCT
iajs-2613	126	9	it	it	PRON
iajs-2613	126	10	follows	follow	VERB
iajs-2613	126	11	that	that	SCONJ
iajs-2613	126	12	rtϵc+soc(t	rtϵc+soc(t	NOUN
iajs-2613	126	13	)	)	PUNCT
iajs-2613	126	14	or	or	CCONJ
iajs-2613	126	15	stϵc+soc(t).hence	stϵc+soc(t).hence	NOUN
iajs-2613	126	16	,	,	PUNCT
iajs-2613	126	17	c	c	PROPN
iajs-2613	126	18	is	be	AUX
iajs-2613	126	19	wapp	wapp	NOUN
iajs-2613	126	20	-	-	PUNCT
iajs-2613	126	21	quasi	quasi	ADJ
iajs-2613	126	22	prime	prime	ADJ
iajs-2613	126	23	submodule	submodule	NOUN
iajs-2613	126	24	of	of	ADP
iajs-2613	126	25	t.	t.	PROPN
iajs-2613	126	26	proposition(9	proposition(9	PROPN
iajs-2613	126	27	)	)	PUNCT
iajs-2613	126	28	let	let	VERB
iajs-2613	126	29	𝑇	𝑇	PROPN
iajs-2613	126	30	be	be	AUX
iajs-2613	126	31	r	r	NOUN
iajs-2613	126	32	−	−	NOUN
iajs-2613	126	33	modul	modul	NOUN
iajs-2613	126	34	and	and	CCONJ
iajs-2613	126	35	c	c	PROPN
iajs-2613	126	36	be	be	AUX
iajs-2613	126	37	proper	proper	ADJ
iajs-2613	126	38	submodul	submodul	NOUN
iajs-2613	126	39	of	of	ADP
iajs-2613	126	40	t	t	PROPN
iajs-2613	126	41	.	.	PUNCT
iajs-2613	127	1	then	then	ADV
iajs-2613	127	2	c	c	PROPN
iajs-2613	127	3	is	be	AUX
iajs-2613	127	4	wapp	wapp	NOUN
iajs-2613	127	5	−	−	PROPN
iajs-2613	127	6	quase	quase	PROPN
iajs-2613	127	7	priem	priem	PROPN
iajs-2613	127	8	submodul	submodul	NOUN
iajs-2613	127	9	of	of	ADP
iajs-2613	127	10	t	t	PROPN
iajs-2613	127	11	iff	iff	PROPN
iajs-2613	127	12	for	for	ADP
iajs-2613	127	13	every	every	DET
iajs-2613	127	14	rϵr	rϵr	NOUN
iajs-2613	127	15	,	,	PUNCT
iajs-2613	127	16	and	and	CCONJ
iajs-2613	127	17	tϵt	tϵt	NOUN
iajs-2613	127	18	with	with	ADP
iajs-2613	127	19	rtc	rtc	PROPN
iajs-2613	127	20	+	+	PROPN
iajs-2613	127	21	soc(t	soc(t	PROPN
iajs-2613	127	22	)	)	PUNCT
iajs-2613	127	23	,	,	PUNCT
iajs-2613	128	1	[	[	X
iajs-2613	128	2	c	c	X
iajs-2613	128	3	:	:	PUNCT
iajs-2613	128	4	r	r	NOUN
iajs-2613	128	5	rt][0	rt][0	NOUN
iajs-2613	128	6	:	:	PUNCT
iajs-2613	128	7	r	r	NOUN
iajs-2613	128	8	rt]∪[c+soc(t):r	rt]∪[c+soc(t):r	PROPN
iajs-2613	128	9	t	t	PROPN
iajs-2613	128	10	]	]	PUNCT
iajs-2613	128	11	proof	proof	NOUN
iajs-2613	128	12	:	:	PUNCT
iajs-2613	128	13	(	(	PUNCT
iajs-2613	128	14	)suppose	)suppose	NOUN
iajs-2613	128	15	that	that	SCONJ
iajs-2613	128	16	c	c	PROPN
iajs-2613	128	17	is	be	AUX
iajs-2613	128	18	wapp	wapp	NOUN
iajs-2613	128	19	-	-	PUNCT
iajs-2613	128	20	quase	quase	NOUN
iajs-2613	128	21	,	,	PUNCT
iajs-2613	128	22	and	and	CCONJ
iajs-2613	128	23	let	let	VERB
iajs-2613	128	24	sϵ[c	sϵ[c	PRON
iajs-2613	128	25	:	:	PUNCT
iajs-2613	128	26	r	r	NOUN
iajs-2613	128	27	rt	rt	PROPN
iajs-2613	128	28	]	]	PUNCT
iajs-2613	128	29	,	,	PUNCT
iajs-2613	128	30	implies	imply	VERB
iajs-2613	129	1	that	that	SCONJ
iajs-2613	129	2	rstϵc	rstϵc	ADJ
iajs-2613	129	3	.if	.if	PUNCT
iajs-2613	129	4	rst=0	rst=0	VERB
iajs-2613	129	5	then	then	ADV
iajs-2613	129	6	sϵ[0	sϵ[0	NOUN
iajs-2613	129	7	:	:	PUNCT
iajs-2613	129	8	r	r	NOUN
iajs-2613	129	9	r	r	NOUN
iajs-2613	129	10	]	]	X
iajs-2613	129	11	,	,	PUNCT
iajs-2613	129	12	hence	hence	ADV
iajs-2613	129	13	sϵ[0	sϵ[0	PROPN
iajs-2613	129	14	:	:	PUNCT
iajs-2613	129	15	r	r	NOUN
iajs-2613	129	16	rt]∪[c+soc(t):r	rt]∪[c+soc(t):r	PROPN
iajs-2613	129	17	t	t	PROPN
iajs-2613	129	18	]	]	PUNCT
iajs-2613	129	19	.	.	PUNCT
iajs-2613	130	1	if	if	SCONJ
iajs-2613	130	2	0≠rstϵc	0≠rstϵc	NUM
iajs-2613	130	3	and	and	CCONJ
iajs-2613	130	4	c	c	NOUN
iajs-2613	130	5	is	be	AUX
iajs-2613	130	6	a	a	DET
iajs-2613	130	7	wapp	wapp	NOUN
iajs-2613	130	8	-	-	PUNCT
iajs-2613	130	9	quasi	quasi	ADJ
iajs-2613	130	10	prime	prime	PROPN
iajs-2613	130	11	submodule	submodule	NOUN
iajs-2613	130	12	of	of	ADP
iajs-2613	130	13	t	t	PROPN
iajs-2613	130	14	and	and	CCONJ
iajs-2613	130	15	rtc	rtc	PROPN
iajs-2613	130	16	+	+	PROPN
iajs-2613	130	17	soc(t	soc(t	PROPN
iajs-2613	130	18	)	)	PUNCT
iajs-2613	130	19	,	,	PUNCT
iajs-2613	130	20	then	then	ADV
iajs-2613	130	21	stϵc+	stϵc+	PROPN
iajs-2613	130	22	soc(t	soc(t	PROPN
iajs-2613	130	23	)	)	PUNCT
iajs-2613	130	24	that	that	PRON
iajs-2613	130	25	is	be	AUX
iajs-2613	130	26	sϵ[c+	sϵ[c+	PROPN
iajs-2613	130	27	soc(t):r	soc(t):r	PROPN
iajs-2613	130	28	t	t	PROPN
iajs-2613	130	29	]	]	PUNCT
iajs-2613	130	30	.	.	PUNCT
iajs-2613	131	1	hence	hence	ADV
iajs-2613	131	2	sϵ[0	sϵ[0	PROPN
iajs-2613	131	3	:	:	PUNCT
iajs-2613	131	4	r	r	NOUN
iajs-2613	131	5	rt]∪[c+	rt]∪[c+	PROPN
iajs-2613	131	6	soc(t):r	soc(t):r	PROPN
iajs-2613	131	7	t	t	PROPN
iajs-2613	131	8	]	]	PUNCT
iajs-2613	131	9	.	.	PUNCT
iajs-2613	132	1	thus	thus	ADV
iajs-2613	132	2	,	,	PUNCT
iajs-2613	132	3	[	[	X
iajs-2613	132	4	c	c	X
iajs-2613	132	5	:	:	PUNCT
iajs-2613	132	6	r	r	NOUN
iajs-2613	132	7	rt][0	rt][0	NOUN
iajs-2613	132	8	:	:	PUNCT
iajs-2613	132	9	r	r	NOUN
iajs-2613	132	10	rt]∪[c+	rt]∪[c+	PROPN
iajs-2613	132	11	soc(t):r	soc(t):r	PROPN
iajs-2613	132	12	t	t	PROPN
iajs-2613	132	13	]	]	PUNCT
iajs-2613	132	14	.	.	PUNCT
iajs-2613	133	1	as	as	ADP
iajs-2613	133	2	a	a	DET
iajs-2613	133	3	direct	direct	ADJ
iajs-2613	133	4	consequence	consequence	NOUN
iajs-2613	133	5	of	of	ADP
iajs-2613	133	6	proposition	proposition	NOUN
iajs-2613	133	7	(	(	PUNCT
iajs-2613	133	8	9	9	NUM
iajs-2613	133	9	)	)	PUNCT
iajs-2613	133	10	and	and	CCONJ
iajs-2613	133	11	proposition	proposition	NOUN
iajs-2613	133	12	(	(	PUNCT
iajs-2613	133	13	3),we	3),we	NUM
iajs-2613	133	14	get	get	VERB
iajs-2613	133	15	the	the	DET
iajs-2613	133	16	following	follow	VERB
iajs-2613	133	17	corollary	corollary	NOUN
iajs-2613	133	18	:	:	PUNCT
iajs-2613	133	19	corollary(10	corollary(10	PUNCT
iajs-2613	133	20	)	)	PUNCT
iajs-2613	133	21	let	let	VERB
iajs-2613	133	22	𝑇	𝑇	PROPN
iajs-2613	133	23	be	be	AUX
iajs-2613	133	24	r	r	NOUN
iajs-2613	133	25	−	−	NOUN
iajs-2613	133	26	modul	modul	NOUN
iajs-2613	133	27	and	and	CCONJ
iajs-2613	133	28	c	c	PROPN
iajs-2613	133	29	be	be	AUX
iajs-2613	133	30	proper	proper	ADJ
iajs-2613	133	31	submodul	submodul	NOUN
iajs-2613	133	32	of	of	ADP
iajs-2613	133	33	t	t	PROPN
iajs-2613	133	34	.	.	PUNCT
iajs-2613	134	1	then	then	ADV
iajs-2613	134	2	c	c	PROPN
iajs-2613	134	3	is	be	AUX
iajs-2613	134	4	wapp	wapp	NOUN
iajs-2613	134	5	−	−	PROPN
iajs-2613	134	6	quase	quase	PROPN
iajs-2613	134	7	prim	prim	PROPN
iajs-2613	134	8	submodul	submodul	NOUN
iajs-2613	134	9	of	of	ADP
iajs-2613	134	10	t	t	PROPN
iajs-2613	134	11	iff	iff	PROPN
iajs-2613	134	12	for	for	ADP
iajs-2613	134	13	every	every	DET
iajs-2613	134	14	rϵr	rϵr	NOUN
iajs-2613	134	15	,	,	PUNCT
iajs-2613	134	16	and	and	CCONJ
iajs-2613	134	17	any	any	DET
iajs-2613	134	18	submodule	submodule	PROPN
iajs-2613	134	19	b	b	PROPN
iajs-2613	134	20	of	of	ADP
iajs-2613	134	21	t	t	PROPN
iajs-2613	134	22	with	with	ADP
iajs-2613	134	23	rbc	rbc	PROPN
iajs-2613	134	24	+	+	PROPN
iajs-2613	134	25	soc(t	soc(t	PROPN
iajs-2613	134	26	)	)	PUNCT
iajs-2613	134	27	,	,	PUNCT
iajs-2613	135	1	[	[	X
iajs-2613	135	2	c	c	X
iajs-2613	135	3	:	:	PUNCT
iajs-2613	135	4	r	r	NOUN
iajs-2613	135	5	rb][0	rb][0	NOUN
iajs-2613	135	6	:	:	PUNCT
iajs-2613	135	7	r	r	NOUN
iajs-2613	135	8	rb]∪[c+soc(t):r	rb]∪[c+soc(t):r	NOUN
iajs-2613	135	9	b	b	NOUN
iajs-2613	135	10	]	]	X
iajs-2613	135	11	as	as	ADP
iajs-2613	135	12	a	a	DET
iajs-2613	135	13	direct	direct	ADJ
iajs-2613	135	14	consequence	consequence	NOUN
iajs-2613	135	15	of	of	ADP
iajs-2613	135	16	proposition	proposition	NOUN
iajs-2613	135	17	(	(	PUNCT
iajs-2613	135	18	9	9	NUM
iajs-2613	135	19	)	)	PUNCT
iajs-2613	135	20	and	and	CCONJ
iajs-2613	135	21	proposition	proposition	NOUN
iajs-2613	135	22	(	(	PUNCT
iajs-2613	135	23	4)we	4)we	PRON
iajs-2613	135	24	get	get	VERB
iajs-2613	135	25	the	the	DET
iajs-2613	135	26	following	follow	VERB
iajs-2613	135	27	corollary	corollary	NOUN
iajs-2613	135	28	.	.	PUNCT
iajs-2613	136	1	corollary(11	corollary(11	NOUN
iajs-2613	136	2	)	)	PUNCT
iajs-2613	136	3	let	let	VERB
iajs-2613	136	4	𝑇	𝑇	PROPN
iajs-2613	136	5	be	be	AUX
iajs-2613	136	6	r	r	NOUN
iajs-2613	136	7	−	−	NOUN
iajs-2613	136	8	module	module	NOUN
iajs-2613	136	9	and	and	CCONJ
iajs-2613	136	10	c	c	NOUN
iajs-2613	136	11	be	be	AUX
iajs-2613	136	12	proper	proper	ADJ
iajs-2613	136	13	submodule	submodule	NOUN
iajs-2613	136	14	of	of	ADP
iajs-2613	136	15	t	t	PROPN
iajs-2613	136	16	.	.	PUNCT
iajs-2613	137	1	then	then	ADV
iajs-2613	137	2	c	c	PROPN
iajs-2613	137	3	is	be	AUX
iajs-2613	137	4	wapp	wapp	NOUN
iajs-2613	137	5	−	−	PROPN
iajs-2613	137	6	quasi	quasi	ADJ
iajs-2613	137	7	prime	prime	PROPN
iajs-2613	137	8	submodule	submodule	PROPN
iajs-2613	137	9	of	of	ADP
iajs-2613	137	10	t	t	PROPN
iajs-2613	137	11	iff	iff	PROPN
iajs-2613	137	12	for	for	ADP
iajs-2613	137	13	every	every	DET
iajs-2613	137	14	ideal	ideal	NOUN
iajs-2613	137	15	i	i	PRON
iajs-2613	137	16	of	of	ADP
iajs-2613	137	17	r	r	NOUN
iajs-2613	137	18	,	,	PUNCT
iajs-2613	137	19	and	and	CCONJ
iajs-2613	137	20	every	every	DET
iajs-2613	137	21	submodule	submodule	PROPN
iajs-2613	137	22	b	b	PROPN
iajs-2613	137	23	of	of	ADP
iajs-2613	137	24	t	t	PROPN
iajs-2613	137	25	with	with	ADP
iajs-2613	137	26	ibc	ibc	PROPN
iajs-2613	137	27	+	+	PROPN
iajs-2613	137	28	soc(t	soc(t	PROPN
iajs-2613	137	29	)	)	PUNCT
iajs-2613	137	30	,	,	PUNCT
iajs-2613	137	31	,	,	PUNCT
iajs-2613	137	32	[	[	X
iajs-2613	137	33	c	c	X
iajs-2613	137	34	:	:	PUNCT
iajs-2613	137	35	r	r	NOUN
iajs-2613	137	36	ib][0	ib][0	NOUN
iajs-2613	137	37	:	:	PUNCT
iajs-2613	137	38	r	r	NOUN
iajs-2613	137	39	ib]∪[c+soc(t):r	ib]∪[c+soc(t):r	PROPN
iajs-2613	137	40	b	b	NOUN
iajs-2613	137	41	]	]	PUNCT
iajs-2613	137	42	.	.	PUNCT
iajs-2613	138	1	proposition(12	proposition(12	NOUN
iajs-2613	138	2	)	)	PUNCT
iajs-2613	138	3	61	61	NUM
iajs-2613	138	4	ibn	ibn	PROPN
iajs-2613	138	5	al	al	PROPN
iajs-2613	138	6	-	-	PUNCT
iajs-2613	138	7	haitham	haitham	PROPN
iajs-2613	138	8	jour	jour	X
iajs-2613	138	9	.	.	PROPN
iajs-2613	139	1	for	for	ADP
iajs-2613	139	2	pure	pure	ADJ
iajs-2613	139	3	&	&	CCONJ
iajs-2613	139	4	appl	appl	PROPN
iajs-2613	139	5	.	.	PUNCT
iajs-2613	140	1	sci	sci	PROPN
iajs-2613	140	2	.	.	PROPN
iajs-2613	141	1	34	34	NUM
iajs-2613	141	2	(	(	PUNCT
iajs-2613	141	3	1	1	NUM
iajs-2613	141	4	)	)	PUNCT
iajs-2613	141	5	2021	2021	NUM
iajs-2613	141	6	let	let	VERB
iajs-2613	141	7	𝑇	𝑇	PROPN
iajs-2613	141	8	be	be	AUX
iajs-2613	141	9	r	r	NOUN
iajs-2613	141	10	−	−	NOUN
iajs-2613	141	11	module	module	NOUN
iajs-2613	141	12	and	and	CCONJ
iajs-2613	141	13	c	c	NOUN
iajs-2613	141	14	be	be	AUX
iajs-2613	141	15	proper	proper	ADJ
iajs-2613	141	16	submodule	submodule	NOUN
iajs-2613	141	17	of	of	ADP
iajs-2613	141	18	t	t	PROPN
iajs-2613	141	19	.	.	PUNCT
iajs-2613	142	1	then	then	ADV
iajs-2613	142	2	for	for	ADP
iajs-2613	142	3	every	every	DET
iajs-2613	142	4	s	s	NOUN
iajs-2613	142	5	,	,	PUNCT
iajs-2613	142	6	rϵr	rϵr	NOUN
iajs-2613	142	7	,	,	PUNCT
iajs-2613	142	8	and	and	CCONJ
iajs-2613	142	9	tϵt	tϵt	NOUN
iajs-2613	142	10	,	,	PUNCT
iajs-2613	142	11	[	[	X
iajs-2613	142	12	c	c	X
iajs-2613	142	13	:	:	PUNCT
iajs-2613	142	14	r	r	NOUN
iajs-2613	142	15	rst][0	rst][0	PROPN
iajs-2613	142	16	:	:	PUNCT
iajs-2613	142	17	r	r	NOUN
iajs-2613	142	18	rst]∪[c+soc(t):r	rst]∪[c+soc(t):r	PUNCT
iajs-2613	142	19	r	r	NOUN
iajs-2613	142	20	t	t	PROPN
iajs-2613	142	21	]	]	X
iajs-2613	142	22	]	]	X
iajs-2613	142	23	∪[c+soc(t):r	∪[c+soc(t):r	X
iajs-2613	142	24	s	s	PROPN
iajs-2613	142	25	t	t	X
iajs-2613	142	26	]	]	PUNCT
iajs-2613	142	27	.	.	PUNCT
iajs-2613	143	1	proof	proof	NOUN
iajs-2613	143	2	:	:	PUNCT
iajs-2613	143	3	suppose	suppose	VERB
iajs-2613	143	4	that	that	SCONJ
iajs-2613	143	5	eϵ[c	eϵ[c	NOUN
iajs-2613	143	6	:	:	PUNCT
iajs-2613	143	7	r	r	X
iajs-2613	143	8	rst	rst	PROPN
iajs-2613	143	9	]	]	PUNCT
iajs-2613	143	10	,	,	PUNCT
iajs-2613	143	11	implies	imply	VERB
iajs-2613	143	12	that	that	DET
iajs-2613	143	13	rs(et)ϵc	rs(et)ϵc	NOUN
iajs-2613	143	14	.if	.if	PUNCT
iajs-2613	143	15	rs(et)=0	rs(et)=0	ADJ
iajs-2613	143	16	,	,	PUNCT
iajs-2613	143	17	implies	imply	VERB
iajs-2613	143	18	that	that	SCONJ
iajs-2613	143	19	eϵ[0	eϵ[0	PROPN
iajs-2613	143	20	:	:	PUNCT
iajs-2613	143	21	r	r	X
iajs-2613	143	22	rst	rst	X
iajs-2613	143	23	]	]	PUNCT
iajs-2613	143	24	and	and	CCONJ
iajs-2613	143	25	hence	hence	ADV
iajs-2613	143	26	eϵ[0	eϵ[0	PROPN
iajs-2613	143	27	:	:	PUNCT
iajs-2613	143	28	r	r	NOUN
iajs-2613	143	29	rst]∪[c+soc(t):r	rst]∪[c+soc(t):r	NUM
iajs-2613	143	30	r	r	NOUN
iajs-2613	143	31	t	t	PROPN
iajs-2613	143	32	]	]	X
iajs-2613	143	33	]	]	X
iajs-2613	143	34	∪[c+soc(t):r	∪[c+soc(t):r	PROPN
iajs-2613	143	35	s	s	X
iajs-2613	143	36	t].if	t].if	NOUN
iajs-2613	143	37	rs(et)≠0	rs(et)≠0	NOUN
iajs-2613	143	38	,	,	PUNCT
iajs-2613	143	39	and	and	CCONJ
iajs-2613	143	40	c	c	PROPN
iajs-2613	143	41	is	be	AUX
iajs-2613	143	42	a	a	DET
iajs-2613	143	43	wapp	wapp	NOUN
iajs-2613	143	44	-	-	PUNCT
iajs-2613	143	45	quasi	quasi	ADJ
iajs-2613	143	46	prime	prime	PROPN
iajs-2613	143	47	submodule	submodule	NOUN
iajs-2613	143	48	of	of	ADP
iajs-2613	143	49	t	t	PROPN
iajs-2613	143	50	,	,	PUNCT
iajs-2613	143	51	then	then	ADV
iajs-2613	143	52	either	either	CCONJ
iajs-2613	143	53	r(et)ϵc+soc(t	r(et)ϵc+soc(t	PROPN
iajs-2613	143	54	)	)	PUNCT
iajs-2613	143	55	or	or	CCONJ
iajs-2613	143	56	s(et)ϵc+soc(t	s(et)ϵc+soc(t	NOUN
iajs-2613	143	57	)	)	PUNCT
iajs-2613	143	58	.	.	PUNCT
iajs-2613	144	1	that	that	PRON
iajs-2613	144	2	is	be	AUX
iajs-2613	144	3	either	either	CCONJ
iajs-2613	144	4	eϵ[c+soc(t):rrt	eϵ[c+soc(t):rrt	NOUN
iajs-2613	144	5	]	]	X
iajs-2613	144	6	or	or	CCONJ
iajs-2613	144	7	eϵ[c+soc(t):rst	eϵ[c+soc(t):rst	ADJ
iajs-2613	144	8	]	]	X
iajs-2613	144	9	thus	thus	ADV
iajs-2613	144	10	eϵ[0	eϵ[0	NOUN
iajs-2613	144	11	:	:	PUNCT
iajs-2613	144	12	r	r	NOUN
iajs-2613	144	13	rst]∪[c	rst]∪[c	NOUN
iajs-2613	144	14	+	+	CCONJ
iajs-2613	144	15	soc(t):r	soc(t):r	PROPN
iajs-2613	144	16	r	r	NOUN
iajs-2613	144	17	t	t	PROPN
iajs-2613	144	18	]	]	PUNCT
iajs-2613	144	19	]	]	X
iajs-2613	144	20	∪	∪	ADP
iajs-2613	144	21	[	[	PUNCT
iajs-2613	144	22	c	c	NOUN
iajs-2613	144	23	+	+	CCONJ
iajs-2613	144	24	soc(t):r	soc(t):r	PROPN
iajs-2613	144	25	s	s	X
iajs-2613	144	26	t].therefore	t].therefore	NOUN
iajs-2613	144	27	,	,	PUNCT
iajs-2613	144	28	[	[	X
iajs-2613	144	29	c	c	X
iajs-2613	144	30	:	:	PUNCT
iajs-2613	144	31	r	r	NOUN
iajs-2613	144	32	rst][0	rst][0	NOUN
iajs-2613	144	33	:	:	PUNCT
iajs-2613	144	34	r	r	NOUN
iajs-2613	144	35	rst]∪	rst]∪	PROPN
iajs-2613	144	36	[	[	PUNCT
iajs-2613	144	37	c	c	X
iajs-2613	144	38	+	+	CCONJ
iajs-2613	144	39	soc(t):r	soc(t):r	PROPN
iajs-2613	144	40	r	r	NOUN
iajs-2613	144	41	t	t	PROPN
iajs-2613	144	42	]	]	PUNCT
iajs-2613	144	43	]	]	X
iajs-2613	144	44	∪	∪	ADP
iajs-2613	144	45	[	[	PUNCT
iajs-2613	144	46	c	c	NOUN
iajs-2613	144	47	+	+	CCONJ
iajs-2613	144	48	soc(t):r	soc(t):r	PROPN
iajs-2613	144	49	s	s	PROPN
iajs-2613	144	50	t	t	X
iajs-2613	144	51	]	]	PUNCT
iajs-2613	144	52	.	.	PUNCT
iajs-2613	145	1	the	the	DET
iajs-2613	145	2	following	follow	VERB
iajs-2613	145	3	are	be	AUX
iajs-2613	145	4	characterizations	characterization	NOUN
iajs-2613	145	5	in	in	ADP
iajs-2613	145	6	the	the	DET
iajs-2613	145	7	multiplication	multiplication	NOUN
iajs-2613	145	8	module	module	NOUN
iajs-2613	145	9	.	.	PUNCT
iajs-2613	146	1	proposition(13	proposition(13	PROPN
iajs-2613	146	2	)	)	PUNCT
iajs-2613	146	3	let	let	VERB
iajs-2613	146	4	t	t	NOUN
iajs-2613	146	5	be	be	AUX
iajs-2613	146	6	multiplcation	multiplcation	NOUN
iajs-2613	146	7	r_module	r_module	NOUN
iajs-2613	146	8	and	and	CCONJ
iajs-2613	146	9	c	c	NOUN
iajs-2613	146	10	be	be	AUX
iajs-2613	146	11	proper	proper	ADJ
iajs-2613	146	12	submodule	submodule	NOUN
iajs-2613	146	13	of	of	ADP
iajs-2613	146	14	t	t	PROPN
iajs-2613	146	15	.	.	PUNCT
iajs-2613	147	1	then	then	ADV
iajs-2613	147	2	c	c	PROPN
iajs-2613	147	3	is	be	AUX
iajs-2613	147	4	a	a	DET
iajs-2613	147	5	wapp	wapp	NOUN
iajs-2613	147	6	−	−	PROPN
iajs-2613	147	7	quasi	quasi	ADJ
iajs-2613	147	8	prime	prime	PROPN
iajs-2613	147	9	submodule	submodule	PROPN
iajs-2613	147	10	of	of	ADP
iajs-2613	147	11	t	t	PROPN
iajs-2613	147	12	iff	iff	PROPN
iajs-2613	147	13	0≠k1k2	0≠k1k2	PROPN
iajs-2613	147	14	t	t	PROPN
iajs-2613	147	15	c	c	PUNCT
iajs-2613	147	16	,	,	PUNCT
iajs-2613	147	17	for	for	ADP
iajs-2613	147	18	some	some	DET
iajs-2613	147	19	submodules	submodule	NOUN
iajs-2613	147	20	k1	k1	NOUN
iajs-2613	147	21	,	,	PUNCT
iajs-2613	147	22	k2	k2	NOUN
iajs-2613	147	23	of	of	ADP
iajs-2613	147	24	t	t	PROPN
iajs-2613	147	25	,	,	PUNCT
iajs-2613	147	26	and	and	CCONJ
iajs-2613	147	27	tϵt	tϵt	NOUN
iajs-2613	147	28	implies	imply	VERB
iajs-2613	147	29	that	that	SCONJ
iajs-2613	147	30	either	either	CCONJ
iajs-2613	147	31	k1tc	k1tc	PROPN
iajs-2613	147	32	+	+	ADJ
iajs-2613	147	33	soc(t	soc(t	PROPN
iajs-2613	147	34	)	)	PUNCT
iajs-2613	147	35	or	or	CCONJ
iajs-2613	147	36	k2tc	k2tc	ADP
iajs-2613	147	37	+	+	ADJ
iajs-2613	147	38	soc(t	soc(t	PROPN
iajs-2613	147	39	)	)	PUNCT
iajs-2613	147	40	.	.	PUNCT
iajs-2613	148	1	proof	proof	NOUN
iajs-2613	148	2	:	:	PUNCT
iajs-2613	148	3	(	(	PUNCT
iajs-2613	148	4			NOUN
iajs-2613	148	5	)	)	PUNCT
iajs-2613	148	6	suppos	suppos	NOUN
iajs-2613	148	7	that	that	PRON
iajs-2613	148	8	c	c	PROPN
iajs-2613	148	9	is	be	AUX
iajs-2613	148	10	wapp	wapp	NOUN
iajs-2613	148	11	−	−	PROPN
iajs-2613	148	12	quasi	quasi	ADJ
iajs-2613	148	13	prime	prime	PROPN
iajs-2613	148	14	submodul	submodul	NOUN
iajs-2613	148	15	of	of	ADP
iajs-2613	148	16	t	t	PROPN
iajs-2613	148	17	,	,	PUNCT
iajs-2613	148	18	and	and	CCONJ
iajs-2613	148	19	0≠k1k2	0≠k1k2	PROPN
iajs-2613	148	20	t	t	NOUN
iajs-2613	148	21	c	c	NUM
iajs-2613	148	22	for	for	ADP
iajs-2613	148	23	some	some	DET
iajs-2613	148	24	submodules	submodule	NOUN
iajs-2613	148	25	k1	k1	NOUN
iajs-2613	148	26	,	,	PUNCT
iajs-2613	148	27	k2	k2	NOUN
iajs-2613	148	28	of	of	ADP
iajs-2613	148	29	t	t	PROPN
iajs-2613	148	30	,	,	PUNCT
iajs-2613	148	31	and	and	CCONJ
iajs-2613	148	32	tϵt	tϵt	NOUN
iajs-2613	148	33	.	.	PUNCT
iajs-2613	149	1	since	since	SCONJ
iajs-2613	149	2	t	t	PROPN
iajs-2613	149	3	is	be	AUX
iajs-2613	149	4	a	a	DET
iajs-2613	149	5	multiplication	multiplication	NOUN
iajs-2613	149	6	,	,	PUNCT
iajs-2613	149	7	then	then	ADV
iajs-2613	149	8	k1	k1	NOUN
iajs-2613	149	9	=	=	VERB
iajs-2613	149	10	it	it	PRON
iajs-2613	149	11	and	and	CCONJ
iajs-2613	149	12	k2	k2	PROPN
iajs-2613	149	13	=	=	PROPN
iajs-2613	149	14	jt	jt	PROPN
iajs-2613	149	15	for	for	ADP
iajs-2613	149	16	some	some	DET
iajs-2613	149	17	ideals	ideal	NOUN
iajs-2613	149	18	i	i	PRON
iajs-2613	149	19	,	,	PUNCT
iajs-2613	149	20	j	j	PROPN
iajs-2613	149	21	of	of	ADP
iajs-2613	149	22	r	r	PROPN
iajs-2613	149	23	.thus	.thus	ADV
iajs-2613	149	24	0≠k1k2t	0≠k1k2t	NUM
iajs-2613	149	25	=	=	NOUN
iajs-2613	149	26	ijt	ijt	NOUN
iajs-2613	149	27	c	c	NUM
iajs-2613	149	28	.	.	PUNCT
iajs-2613	150	1	since	since	SCONJ
iajs-2613	150	2	c	c	PROPN
iajs-2613	150	3	is	be	AUX
iajs-2613	150	4	a	a	DET
iajs-2613	150	5	wapp	wapp	NOUN
iajs-2613	150	6	-	-	PUNCT
iajs-2613	150	7	quasi	quasi	ADJ
iajs-2613	150	8	prime	prime	PROPN
iajs-2613	150	9	submodule	submodule	NOUN
iajs-2613	150	10	of	of	ADP
iajs-2613	150	11	t	t	PROPN
iajs-2613	150	12	then	then	ADV
iajs-2613	150	13	by	by	ADP
iajs-2613	150	14	corollary	corollary	ADJ
iajs-2613	150	15	(	(	PUNCT
iajs-2613	150	16	6	6	NUM
iajs-2613	150	17	)	)	PUNCT
iajs-2613	150	18	either	either	CCONJ
iajs-2613	150	19	itc	itc	NOUN
iajs-2613	150	20	+	+	CCONJ
iajs-2613	150	21	soc(t	soc(t	PROPN
iajs-2613	150	22	)	)	PUNCT
iajs-2613	150	23	or	or	CCONJ
iajs-2613	150	24	jt	jt	NOUN
iajs-2613	150	25	c	c	NOUN
iajs-2613	150	26	+	+	CCONJ
iajs-2613	150	27	soc(t	soc(t	PROPN
iajs-2613	150	28	)	)	PUNCT
iajs-2613	150	29	.	.	PUNCT
iajs-2613	151	1	hence	hence	ADV
iajs-2613	151	2	either	either	CCONJ
iajs-2613	151	3	k1t	k1t	PROPN
iajs-2613	151	4	c	c	PROPN
iajs-2613	151	5	+	+	PUNCT
iajs-2613	151	6	soc(t	soc(t	PROPN
iajs-2613	151	7	)	)	PUNCT
iajs-2613	151	8	or	or	CCONJ
iajs-2613	151	9	k2t	k2t	PROPN
iajs-2613	151	10	c	c	PROPN
iajs-2613	151	11	+	+	NUM
iajs-2613	151	12	soc(t	soc(t	PROPN
iajs-2613	151	13	)	)	PUNCT
iajs-2613	151	14	.	.	PUNCT
iajs-2613	152	1	(	(	PUNCT
iajs-2613	152	2			NOUN
iajs-2613	152	3	)	)	PUNCT
iajs-2613	152	4	assume	assume	VERB
iajs-2613	152	5	that	that	SCONJ
iajs-2613	152	6	0≠ijtc	0≠ijtc	NOUN
iajs-2613	152	7	,	,	PUNCT
iajs-2613	152	8	for	for	ADP
iajs-2613	152	9	some	some	DET
iajs-2613	152	10	ideals	ideal	NOUN
iajs-2613	152	11	i	i	PRON
iajs-2613	152	12	,	,	PUNCT
iajs-2613	152	13	j	j	PROPN
iajs-2613	152	14	of	of	ADP
iajs-2613	152	15	r	r	NOUN
iajs-2613	152	16	,	,	PUNCT
iajs-2613	152	17	tϵt	tϵt	NOUN
iajs-2613	152	18	.that	.that	PRON
iajs-2613	152	19	is	be	AUX
iajs-2613	152	20	0≠k1k2	0≠k1k2	PROPN
iajs-2613	152	21	t	t	PROPN
iajs-2613	152	22	c	c	NUM
iajs-2613	152	23	for	for	ADP
iajs-2613	152	24	k1	k1	NOUN
iajs-2613	152	25	=	=	NOUN
iajs-2613	152	26	it	it	PRON
iajs-2613	152	27	and	and	CCONJ
iajs-2613	152	28	k2	k2	PROPN
iajs-2613	152	29	=	=	PROPN
iajs-2613	152	30	jt	jt	PROPN
iajs-2613	152	31	.	.	PUNCT
iajs-2613	153	1	it	it	PRON
iajs-2613	153	2	follows	follow	VERB
iajs-2613	153	3	that	that	SCONJ
iajs-2613	153	4	either	either	CCONJ
iajs-2613	153	5	k1tc+soc(t	k1tc+soc(t	PROPN
iajs-2613	153	6	)	)	PUNCT
iajs-2613	153	7	or	or	CCONJ
iajs-2613	153	8	k2tc+soc(t	k2tc+soc(t	PROPN
iajs-2613	153	9	)	)	PUNCT
iajs-2613	153	10	;	;	PUNCT
iajs-2613	153	11	that	that	PRON
iajs-2613	153	12	is	be	AUX
iajs-2613	153	13	it	it	ADP
iajs-2613	153	14	c	c	NOUN
iajs-2613	153	15	+	+	CCONJ
iajs-2613	153	16	soc(t	soc(t	PROPN
iajs-2613	153	17	)	)	PUNCT
iajs-2613	153	18	or	or	CCONJ
iajs-2613	153	19	jt	jt	NOUN
iajs-2613	153	20	c	c	NOUN
iajs-2613	154	1	+	+	CCONJ
iajs-2613	154	2	soc(t).hence	soc(t).hence	NOUN
iajs-2613	154	3	c	c	PROPN
iajs-2613	154	4	is	be	AUX
iajs-2613	154	5	a	a	DET
iajs-2613	154	6	wapp	wapp	NOUN
iajs-2613	154	7	-	-	PUNCT
iajs-2613	154	8	quasi	quasi	ADJ
iajs-2613	154	9	prime	prime	PROPN
iajs-2613	154	10	submodule	submodule	NOUN
iajs-2613	154	11	of	of	ADP
iajs-2613	154	12	t	t	PROPN
iajs-2613	154	13	by	by	ADP
iajs-2613	154	14	corollary(6	corollary(6	NOUN
iajs-2613	154	15	)	)	PUNCT
iajs-2613	154	16	.	.	PUNCT
iajs-2613	155	1	proposition(14	proposition(14	NOUN
iajs-2613	155	2	)	)	PUNCT
iajs-2613	155	3	let	let	VERB
iajs-2613	155	4	t	t	NOUN
iajs-2613	155	5	be	be	AUX
iajs-2613	155	6	multiplcation	multiplcation	NOUN
iajs-2613	155	7	r_module	r_module	NOUN
iajs-2613	155	8	and	and	CCONJ
iajs-2613	155	9	c	c	NOUN
iajs-2613	155	10	be	be	AUX
iajs-2613	155	11	proper	proper	ADJ
iajs-2613	155	12	submodule	submodule	NOUN
iajs-2613	155	13	of	of	ADP
iajs-2613	155	14	t	t	PROPN
iajs-2613	155	15	.	.	PUNCT
iajs-2613	156	1	then	then	ADV
iajs-2613	156	2	c	c	PROPN
iajs-2613	156	3	is	be	AUX
iajs-2613	156	4	wapp	wapp	NOUN
iajs-2613	156	5	−	−	PROPN
iajs-2613	156	6	quasi	quasi	ADJ
iajs-2613	156	7	prime	prime	PROPN
iajs-2613	156	8	submodule	submodule	PROPN
iajs-2613	156	9	of	of	ADP
iajs-2613	156	10	t	t	PROPN
iajs-2613	156	11	iff	iff	PROPN
iajs-2613	156	12	0≠k1k2h	0≠k1k2h	NUM
iajs-2613	156	13	c	c	X
iajs-2613	156	14	,	,	PUNCT
iajs-2613	156	15	for	for	ADP
iajs-2613	156	16	some	some	DET
iajs-2613	156	17	submodules	submodule	NOUN
iajs-2613	156	18	k1	k1	NOUN
iajs-2613	156	19	,	,	PUNCT
iajs-2613	156	20	k2	k2	NOUN
iajs-2613	156	21	and	and	CCONJ
iajs-2613	156	22	h	h	NOUN
iajs-2613	156	23	of	of	ADP
iajs-2613	156	24	t	t	PROPN
iajs-2613	156	25	,	,	PUNCT
iajs-2613	156	26	implies	imply	VERB
iajs-2613	156	27	that	that	SCONJ
iajs-2613	156	28	either	either	CCONJ
iajs-2613	156	29	k1hc	k1hc	NOUN
iajs-2613	156	30	+	+	SYM
iajs-2613	156	31	soc(t	soc(t	PROPN
iajs-2613	156	32	)	)	PUNCT
iajs-2613	156	33	or	or	CCONJ
iajs-2613	156	34	k2hc	k2hc	NOUN
iajs-2613	156	35	+	+	PROPN
iajs-2613	156	36	soc(t	soc(t	PROPN
iajs-2613	156	37	)	)	PUNCT
iajs-2613	156	38	.	.	PUNCT
iajs-2613	157	1	proof	proof	NOUN
iajs-2613	157	2	:	:	PUNCT
iajs-2613	157	3	(	(	PUNCT
iajs-2613	157	4			NOUN
iajs-2613	157	5	)	)	PUNCT
iajs-2613	157	6	assume	assume	VERB
iajs-2613	157	7	that	that	SCONJ
iajs-2613	157	8	0≠k1k2h	0≠k1k2h	PRON
iajs-2613	157	9	c	c	X
iajs-2613	157	10	for	for	ADP
iajs-2613	157	11	some	some	DET
iajs-2613	157	12	submodules	submodule	NOUN
iajs-2613	157	13	k1	k1	NOUN
iajs-2613	157	14	,	,	PUNCT
iajs-2613	157	15	k2	k2	NOUN
iajs-2613	157	16	and	and	CCONJ
iajs-2613	157	17	h	h	NOUN
iajs-2613	157	18	of	of	ADP
iajs-2613	157	19	t	t	PROPN
iajs-2613	157	20	.	.	PUNCT
iajs-2613	158	1	since	since	SCONJ
iajs-2613	158	2	t	t	PROPN
iajs-2613	158	3	is	be	AUX
iajs-2613	158	4	a	a	DET
iajs-2613	158	5	multiplication	multiplication	NOUN
iajs-2613	158	6	,	,	PUNCT
iajs-2613	158	7	then	then	ADV
iajs-2613	158	8	k1	k1	NOUN
iajs-2613	158	9	=	=	PROPN
iajs-2613	158	10	it	it	PRON
iajs-2613	158	11	,	,	PUNCT
iajs-2613	158	12	k2	k2	PROPN
iajs-2613	158	13	=	=	PROPN
iajs-2613	158	14	jt	jt	PROPN
iajs-2613	158	15	for	for	ADP
iajs-2613	158	16	some	some	DET
iajs-2613	158	17	ideals	ideal	NOUN
iajs-2613	158	18	i	i	PRON
iajs-2613	158	19	,	,	PUNCT
iajs-2613	158	20	j	j	PROPN
iajs-2613	158	21	of	of	ADP
iajs-2613	158	22	r	r	NOUN
iajs-2613	158	23	hence	hence	ADV
iajs-2613	158	24	0≠k1k2h	0≠k1k2h	X
iajs-2613	158	25	=	=	NOUN
iajs-2613	158	26	ijh	ijh	NOUN
iajs-2613	158	27	c	c	NOUN
iajs-2613	158	28	.	.	PUNCT
iajs-2613	159	1	but	but	CCONJ
iajs-2613	159	2	c	c	NOUN
iajs-2613	159	3	is	be	AUX
iajs-2613	159	4	wapp	wapp	NOUN
iajs-2613	159	5	-	-	PUNCT
iajs-2613	159	6	quasi	quasi	ADJ
iajs-2613	159	7	prime	prime	PROPN
iajs-2613	159	8	submodule	submodule	NOUN
iajs-2613	159	9	of	of	ADP
iajs-2613	159	10	t	t	PROPN
iajs-2613	159	11	then	then	ADV
iajs-2613	159	12	by	by	ADP
iajs-2613	159	13	proposition	proposition	NOUN
iajs-2613	159	14	(	(	PUNCT
iajs-2613	159	15	4	4	NUM
iajs-2613	159	16	)	)	PUNCT
iajs-2613	159	17	either	either	CCONJ
iajs-2613	159	18	ih	ih	PUNCT
iajs-2613	159	19	c	c	PROPN
iajs-2613	159	20	+	+	CCONJ
iajs-2613	159	21	soc(t	soc(t	PROPN
iajs-2613	159	22	)	)	PUNCT
iajs-2613	159	23	.	.	PUNCT
iajs-2613	160	1	or	or	CCONJ
iajs-2613	160	2	jh	jh	PROPN
iajs-2613	160	3	c	c	PROPN
iajs-2613	160	4	+	+	CCONJ
iajs-2613	160	5	soc(t	soc(t	PROPN
iajs-2613	160	6	)	)	PUNCT
iajs-2613	160	7	..	..	PUNCT
iajs-2613	161	1	hence	hence	ADV
iajs-2613	161	2	either	either	CCONJ
iajs-2613	161	3	k1h	k1h	PROPN
iajs-2613	161	4	c	c	PROPN
iajs-2613	161	5	+	+	CCONJ
iajs-2613	161	6	soc(t	soc(t	PROPN
iajs-2613	161	7	)	)	PUNCT
iajs-2613	161	8	.	.	PUNCT
iajs-2613	162	1	or	or	CCONJ
iajs-2613	162	2	k2h	k2h	PROPN
iajs-2613	162	3	c	c	PROPN
iajs-2613	162	4	+	+	CCONJ
iajs-2613	162	5	soc(t	soc(t	PROPN
iajs-2613	162	6	)	)	PUNCT
iajs-2613	162	7	..	..	PUNCT
iajs-2613	162	8	(	(	PUNCT
iajs-2613	162	9			NOUN
iajs-2613	162	10	)	)	PUNCT
iajs-2613	162	11	let	let	VERB
iajs-2613	162	12	0≠ijhc	0≠ijhc	INTJ
iajs-2613	162	13	,	,	PUNCT
iajs-2613	162	14	where	where	SCONJ
iajs-2613	162	15	i	i	PRON
iajs-2613	162	16	,	,	PUNCT
iajs-2613	162	17	j	j	PROPN
iajs-2613	162	18	are	be	AUX
iajs-2613	162	19	ideals	ideal	NOUN
iajs-2613	162	20	of	of	ADP
iajs-2613	162	21	r	r	NOUN
iajs-2613	162	22	,	,	PUNCT
iajs-2613	162	23	and	and	CCONJ
iajs-2613	162	24	h	h	NOUN
iajs-2613	162	25	is	be	AUX
iajs-2613	162	26	a	a	DET
iajs-2613	162	27	submodule	submodule	NOUN
iajs-2613	162	28	of	of	ADP
iajs-2613	162	29	t	t	PROPN
iajs-2613	162	30	.since	.since	PROPN
iajs-2613	162	31	t	t	PROPN
iajs-2613	162	32	is	be	AUX
iajs-2613	162	33	multiplication	multiplication	NOUN
iajs-2613	162	34	,	,	PUNCT
iajs-2613	162	35	then	then	ADV
iajs-2613	162	36	0≠ijh	0≠ijh	PROPN
iajs-2613	162	37	=	=	SYM
iajs-2613	162	38	k1k2hc	k1k2hc	NOUN
iajs-2613	162	39	,	,	PUNCT
iajs-2613	162	40	hence	hence	ADV
iajs-2613	162	41	by	by	ADP
iajs-2613	162	42	assumption	assumption	NOUN
iajs-2613	162	43	either	either	CCONJ
iajs-2613	162	44	k1h	k1h	PROPN
iajs-2613	162	45	c	c	PROPN
iajs-2613	162	46	+	+	CCONJ
iajs-2613	162	47	soc(t	soc(t	PROPN
iajs-2613	162	48	)	)	PUNCT
iajs-2613	162	49	or	or	CCONJ
iajs-2613	162	50	k2h	k2h	PROPN
iajs-2613	162	51	c	c	PROPN
iajs-2613	163	1	+	+	CCONJ
iajs-2613	163	2	soc(t).that	soc(t).that	PROPN
iajs-2613	163	3	is	be	AUX
iajs-2613	163	4	either	either	PRON
iajs-2613	163	5	ih	ih	PUNCT
iajs-2613	163	6	c	c	PROPN
iajs-2613	163	7	+	+	CCONJ
iajs-2613	163	8	soc(t	soc(t	PROPN
iajs-2613	163	9	)	)	PUNCT
iajs-2613	163	10	or	or	CCONJ
iajs-2613	163	11	jh	jh	PROPN
iajs-2613	163	12	c	c	PROPN
iajs-2613	163	13	+	+	CCONJ
iajs-2613	163	14	soc(t	soc(t	PROPN
iajs-2613	163	15	)	)	PUNCT
iajs-2613	163	16	.	.	PUNCT
iajs-2613	164	1	thus	thus	ADV
iajs-2613	164	2	by	by	ADP
iajs-2613	164	3	proposition	proposition	NOUN
iajs-2613	164	4	(	(	PUNCT
iajs-2613	164	5	4)c	4)c	PROPN
iajs-2613	164	6	is	be	AUX
iajs-2613	164	7	wapp	wapp	NOUN
iajs-2613	164	8	-	-	PUNCT
iajs-2613	164	9	quasi	quasi	ADJ
iajs-2613	164	10	prime	prime	ADJ
iajs-2613	164	11	submodul	submodul	NOUN
iajs-2613	164	12	of	of	ADP
iajs-2613	164	13	t	t	PROPN
iajs-2613	164	14	.	.	PUNCT
iajs-2613	165	1	62	62	NUM
iajs-2613	165	2	ibn	ibn	PROPN
iajs-2613	165	3	al	al	PROPN
iajs-2613	165	4	-	-	PUNCT
iajs-2613	165	5	haitham	haitham	PROPN
iajs-2613	165	6	jour	jour	X
iajs-2613	165	7	.	.	PROPN
iajs-2613	165	8	for	for	ADP
iajs-2613	165	9	pure	pure	ADJ
iajs-2613	165	10	&	&	CCONJ
iajs-2613	165	11	appl	appl	PROPN
iajs-2613	165	12	.	.	PUNCT
iajs-2613	166	1	sci	sci	PROPN
iajs-2613	166	2	.	.	PROPN
iajs-2613	167	1	34	34	NUM
iajs-2613	167	2	(	(	PUNCT
iajs-2613	167	3	1	1	NUM
iajs-2613	167	4	)	)	PUNCT
iajs-2613	167	5	2021	2021	NUM
iajs-2613	167	6	it	it	PRON
iajs-2613	167	7	is	be	AUX
iajs-2613	167	8	well	well	ADV
iajs-2613	167	9	−	−	PROPN
iajs-2613	167	10	knwon	knwon	VERB
iajs-2613	167	11	that	that	SCONJ
iajs-2613	167	12	if	if	SCONJ
iajs-2613	167	13	t	t	PROPN
iajs-2613	167	14	is	be	AUX
iajs-2613	167	15	z	z	NOUN
iajs-2613	167	16	−	−	NOUN
iajs-2613	167	17	regular	regular	ADJ
iajs-2613	167	18	r	r	NOUN
iajs-2613	167	19	−	−	NOUN
iajs-2613	167	20	module	module	NOUN
iajs-2613	167	21	,	,	PUNCT
iajs-2613	167	22	then	then	ADV
iajs-2613	167	23	soc(t)=soc(r)t	soc(t)=soc(r)t	PROPN
iajs-2613	168	1	[	[	X
iajs-2613	168	2	11;prop.(325	11;prop.(325	NUM
iajs-2613	168	3	)	)	PUNCT
iajs-2613	168	4	]	]	PUNCT
iajs-2613	168	5	.	.	PUNCT
iajs-2613	169	1	proposition(15	proposition(15	NOUN
iajs-2613	169	2	)	)	PUNCT
iajs-2613	169	3	let	let	VERB
iajs-2613	169	4	t	t	NOUN
iajs-2613	169	5	be	be	AUX
iajs-2613	169	6	z_regular	z_regular	NUM
iajs-2613	169	7	multiplcation	multiplcation	NOUN
iajs-2613	169	8	r_module	r_module	NOUN
iajs-2613	169	9	and	and	CCONJ
iajs-2613	169	10	c	c	PROPN
iajs-2613	169	11	be	be	AUX
iajs-2613	169	12	proper	proper	ADJ
iajs-2613	169	13	submodule	submodule	NOUN
iajs-2613	169	14	of	of	ADP
iajs-2613	169	15	t	t	PROPN
iajs-2613	169	16	.	.	PUNCT
iajs-2613	170	1	then	then	ADV
iajs-2613	170	2	c	c	PROPN
iajs-2613	170	3	is	be	AUX
iajs-2613	170	4	wapp	wapp	NOUN
iajs-2613	170	5	−	−	PROPN
iajs-2613	170	6	quasi	quasi	ADJ
iajs-2613	170	7	prime	prime	PROPN
iajs-2613	170	8	submodul	submodul	PROPN
iajs-2613	170	9	of	of	ADP
iajs-2613	170	10	t	t	PROPN
iajs-2613	170	11	iff	iff	PROPN
iajs-2613	171	1	[	[	X
iajs-2613	171	2	c	c	X
iajs-2613	171	3	:	:	PUNCT
iajs-2613	171	4	r	r	NOUN
iajs-2613	171	5	t	t	PROPN
iajs-2613	171	6	]	]	PUNCT
iajs-2613	171	7	is	be	AUX
iajs-2613	171	8	wappquasi	wappquasi	NOUN
iajs-2613	171	9	prime	prime	ADJ
iajs-2613	171	10	ideal	ideal	NOUN
iajs-2613	171	11	of	of	ADP
iajs-2613	171	12	r.	r.	PROPN
iajs-2613	171	13	proof	proof	NOUN
iajs-2613	171	14	:	:	PUNCT
iajs-2613	171	15	(	(	PUNCT
iajs-2613	171	16	)suppose	)suppose	NOUN
iajs-2613	171	17	that	that	SCONJ
iajs-2613	171	18	c	c	PROPN
iajs-2613	171	19	is	be	AUX
iajs-2613	171	20	wapp	wapp	NOUN
iajs-2613	171	21	−	−	PROPN
iajs-2613	171	22	quasi	quasi	ADJ
iajs-2613	171	23	prime	prime	PROPN
iajs-2613	171	24	submodule	submodule	NOUN
iajs-2613	171	25	of	of	ADP
iajs-2613	171	26	t	t	PROPN
iajs-2613	171	27	and	and	CCONJ
iajs-2613	171	28	let	let	VERB
iajs-2613	171	29	0≠abi[c	0≠abi[c	PROPN
iajs-2613	171	30	:	:	PUNCT
iajs-2613	171	31	r	r	NOUN
iajs-2613	171	32	t	t	PROPN
iajs-2613	171	33	]	]	PUNCT
iajs-2613	171	34	,	,	PUNCT
iajs-2613	171	35	for	for	ADP
iajs-2613	171	36	a	a	PRON
iajs-2613	171	37	,	,	PUNCT
iajs-2613	171	38	bϵr	bϵr	PROPN
iajs-2613	171	39	,	,	PUNCT
iajs-2613	171	40	i	i	PRON
iajs-2613	171	41	is	be	AUX
iajs-2613	171	42	an	an	DET
iajs-2613	171	43	ideal	ideal	NOUN
iajs-2613	171	44	of	of	ADP
iajs-2613	171	45	r	r	PROPN
iajs-2613	171	46	.it	.it	PUNCT
iajs-2613	171	47	follows	follow	VERB
iajs-2613	171	48	that	that	PRON
iajs-2613	171	49	0≠ab(it)c	0≠ab(it)c	PRON
iajs-2613	171	50	.	.	PUNCT
iajs-2613	172	1	since	since	SCONJ
iajs-2613	172	2	c	c	PROPN
iajs-2613	172	3	is	be	AUX
iajs-2613	172	4	wappquasi	wappquasi	PROPN
iajs-2613	172	5	prime	prime	ADJ
iajs-2613	172	6	submodule	submodule	NOUN
iajs-2613	172	7	of	of	ADP
iajs-2613	172	8	t	t	PROPN
iajs-2613	172	9	,	,	PUNCT
iajs-2613	172	10	then	then	ADV
iajs-2613	172	11	by	by	ADP
iajs-2613	172	12	proposition(3	proposition(3	PROPN
iajs-2613	172	13	)	)	PUNCT
iajs-2613	172	14	either	either	CCONJ
iajs-2613	172	15	aitc+soc(t	aitc+soc(t	PROPN
iajs-2613	172	16	)	)	PUNCT
iajs-2613	172	17	or	or	CCONJ
iajs-2613	172	18	bitc+soc(t).but	bitc+soc(t).but	PROPN
iajs-2613	172	19	t	t	PROPN
iajs-2613	172	20	is	be	AUX
iajs-2613	172	21	a	a	DET
iajs-2613	172	22	z	z	NOUN
iajs-2613	172	23	–	–	PUNCT
iajs-2613	172	24	regular	regular	ADJ
iajs-2613	172	25	module	module	NOUN
iajs-2613	172	26	,	,	PUNCT
iajs-2613	172	27	then	then	ADV
iajs-2613	172	28	soc(t)=soc(r)t	soc(t)=soc(r)t	ADJ
iajs-2613	172	29	,	,	PUNCT
iajs-2613	172	30	and	and	CCONJ
iajs-2613	172	31	since	since	SCONJ
iajs-2613	172	32	t	t	PROPN
iajs-2613	172	33	is	be	AUX
iajs-2613	172	34	multiplication	multiplication	NOUN
iajs-2613	172	35	,	,	PUNCT
iajs-2613	172	36	then	then	ADV
iajs-2613	172	37	c=[c	c=[c	PROPN
iajs-2613	172	38	:	:	PUNCT
iajs-2613	172	39	r	r	NOUN
iajs-2613	172	40	t]t	t]t	NOUN
iajs-2613	172	41	.	.	PUNCT
iajs-2613	173	1	hence	hence	ADV
iajs-2613	173	2	either	either	CCONJ
iajs-2613	173	3	ait[c	ait[c	PROPN
iajs-2613	173	4	:	:	PUNCT
iajs-2613	173	5	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	173	6	or	or	CCONJ
iajs-2613	173	7	bit[c	bit[c	PROPN
iajs-2613	173	8	:	:	PUNCT
iajs-2613	173	9	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	173	10	.	.	PUNCT
iajs-2613	174	1	thus	thus	ADV
iajs-2613	174	2	either	either	CCONJ
iajs-2613	174	3	ai[c	ai[c	NOUN
iajs-2613	174	4	:	:	PUNCT
iajs-2613	174	5	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	174	6	)	)	PUNCT
iajs-2613	174	7	or	or	CCONJ
iajs-2613	174	8	bi[c	bi[c	NOUN
iajs-2613	174	9	:	:	PUNCT
iajs-2613	174	10	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	174	11	)	)	PUNCT
iajs-2613	174	12	.	.	PUNCT
iajs-2613	175	1	hence	hence	ADV
iajs-2613	175	2	by	by	ADP
iajs-2613	175	3	proposition	proposition	NOUN
iajs-2613	175	4	[	[	X
iajs-2613	175	5	c	c	X
iajs-2613	175	6	:	:	PUNCT
iajs-2613	175	7	rt	rt	X
iajs-2613	175	8	]	]	X
iajs-2613	175	9	is	be	AUX
iajs-2613	175	10	a	a	DET
iajs-2613	175	11	wappquase	wappquase	NOUN
iajs-2613	175	12	priem	priem	NOUN
iajs-2613	175	13	ideal	ideal	NOUN
iajs-2613	175	14	of	of	ADP
iajs-2613	175	15	r.	r.	PROPN
iajs-2613	175	16	(	(	PUNCT
iajs-2613	175	17	)suppose	)suppose	VERB
iajs-2613	175	18	that	that	PRON
iajs-2613	176	1	[	[	X
iajs-2613	176	2	c	c	X
iajs-2613	176	3	:	:	PUNCT
iajs-2613	176	4	rt	rt	X
iajs-2613	176	5	]	]	X
iajs-2613	176	6	is	be	AUX
iajs-2613	176	7	a	a	DET
iajs-2613	176	8	wapp	wapp	NOUN
iajs-2613	176	9	-	-	PUNCT
iajs-2613	176	10	quasi	quasi	ADJ
iajs-2613	176	11	prime	prime	ADJ
iajs-2613	176	12	ideal	ideal	NOUN
iajs-2613	176	13	of	of	ADP
iajs-2613	176	14	r	r	NOUN
iajs-2613	176	15	,	,	PUNCT
iajs-2613	176	16	and	and	CCONJ
iajs-2613	176	17	0≠rsb	0≠rsb	PROPN
iajs-2613	176	18	c	c	PROPN
iajs-2613	176	19	,	,	PUNCT
iajs-2613	176	20	for	for	ADP
iajs-2613	176	21	r	r	NOUN
iajs-2613	176	22	,	,	PUNCT
iajs-2613	176	23	sϵr	sϵr	NOUN
iajs-2613	176	24	,	,	PUNCT
iajs-2613	176	25	and	and	CCONJ
iajs-2613	176	26	b	b	X
iajs-2613	176	27	is	be	AUX
iajs-2613	176	28	a	a	DET
iajs-2613	176	29	submodule	submodule	NOUN
iajs-2613	176	30	of	of	ADP
iajs-2613	176	31	t	t	PROPN
iajs-2613	176	32	.	.	PUNCT
iajs-2613	177	1	since	since	SCONJ
iajs-2613	177	2	t	t	PROPN
iajs-2613	177	3	is	be	AUX
iajs-2613	177	4	a	a	DET
iajs-2613	177	5	multiplication	multiplication	NOUN
iajs-2613	177	6	,	,	PUNCT
iajs-2613	177	7	then	then	ADV
iajs-2613	177	8	b	b	X
iajs-2613	177	9	=	=	PRON
iajs-2613	177	10	it	it	PRON
iajs-2613	177	11	,	,	PUNCT
iajs-2613	177	12	for	for	ADP
iajs-2613	177	13	some	some	DET
iajs-2613	177	14	ideal	ideal	ADJ
iajs-2613	177	15	i	i	PRON
iajs-2613	177	16	of	of	ADP
iajs-2613	177	17	r	r	NOUN
iajs-2613	177	18	,	,	PUNCT
iajs-2613	177	19	that	that	PRON
iajs-2613	177	20	is	be	AUX
iajs-2613	177	21	0≠rsi	0≠rsi	PROPN
iajs-2613	177	22	tc	tc	NOUN
iajs-2613	177	23	,	,	PUNCT
iajs-2613	177	24	it	it	PRON
iajs-2613	177	25	follows	follow	VERB
iajs-2613	177	26	that0≠rsi[c	that0≠rsi[c	NOUN
iajs-2613	177	27	:	:	PUNCT
iajs-2613	177	28	r	r	NOUN
iajs-2613	177	29	t	t	PROPN
iajs-2613	177	30	]	]	PUNCT
iajs-2613	177	31	.	.	PUNCT
iajs-2613	178	1	for	for	ADP
iajs-2613	178	2	[	[	X
iajs-2613	178	3	c	c	X
iajs-2613	178	4	:	:	PUNCT
iajs-2613	178	5	rt	rt	X
iajs-2613	178	6	]	]	X
iajs-2613	178	7	is	be	AUX
iajs-2613	178	8	a	a	DET
iajs-2613	178	9	wapp	wapp	NOUN
iajs-2613	178	10	-	-	PUNCT
iajs-2613	178	11	quasi	quasi	ADJ
iajs-2613	178	12	prime	prime	ADJ
iajs-2613	178	13	ideal	ideal	NOUN
iajs-2613	178	14	,	,	PUNCT
iajs-2613	178	15	then	then	ADV
iajs-2613	178	16	by	by	ADP
iajs-2613	178	17	proposition(3	proposition(3	PROPN
iajs-2613	178	18	)	)	PUNCT
iajs-2613	178	19	either	either	CCONJ
iajs-2613	178	20	ri[c	ri[c	NOUN
iajs-2613	178	21	:	:	PUNCT
iajs-2613	178	22	rt]+soc(r	rt]+soc(r	NOUN
iajs-2613	178	23	)	)	PUNCT
iajs-2613	178	24	or	or	CCONJ
iajs-2613	178	25	si[c	si[c	NOUN
iajs-2613	178	26	:	:	PUNCT
iajs-2613	178	27	rt]+soc(r	rt]+soc(r	PROPN
iajs-2613	178	28	)	)	PUNCT
iajs-2613	178	29	,	,	PUNCT
iajs-2613	178	30	it	it	PRON
iajs-2613	178	31	follows	follow	VERB
iajs-2613	178	32	that	that	SCONJ
iajs-2613	178	33	either	either	CCONJ
iajs-2613	178	34	rit[c	rit[c	PROPN
iajs-2613	178	35	:	:	PUNCT
iajs-2613	178	36	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	178	37	or	or	CCONJ
iajs-2613	178	38	sit[c	sit[c	NUM
iajs-2613	178	39	:	:	PUNCT
iajs-2613	178	40	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	178	41	.	.	PUNCT
iajs-2613	179	1	but	but	CCONJ
iajs-2613	179	2	t	t	PROPN
iajs-2613	179	3	is	be	AUX
iajs-2613	179	4	a	a	DET
iajs-2613	179	5	z	z	NOUN
iajs-2613	179	6	-	-	ADJ
iajs-2613	179	7	regular	regular	ADJ
iajs-2613	179	8	soc(t)=soc(r)t	soc(t)=soc(r)t	ADJ
iajs-2613	179	9	and	and	CCONJ
iajs-2613	179	10	since	since	SCONJ
iajs-2613	179	11	t	t	PROPN
iajs-2613	179	12	is	be	AUX
iajs-2613	179	13	a	a	DET
iajs-2613	179	14	multiplication	multiplication	NOUN
iajs-2613	179	15	,	,	PUNCT
iajs-2613	179	16	then	then	ADV
iajs-2613	179	17	[	[	X
iajs-2613	179	18	c	c	X
iajs-2613	179	19	:	:	PUNCT
iajs-2613	179	20	rt]t	rt]t	PROPN
iajs-2613	179	21	=	=	SYM
iajs-2613	179	22	c	c	X
iajs-2613	179	23	.thus	.thus	ADV
iajs-2613	179	24	either	either	CCONJ
iajs-2613	179	25	rbc+soc(t	rbc+soc(t	PROPN
iajs-2613	179	26	)	)	PUNCT
iajs-2613	179	27	or	or	CCONJ
iajs-2613	179	28	sbc+soc(t).hence	sbc+soc(t).hence	NOUN
iajs-2613	179	29	by	by	ADP
iajs-2613	179	30	proposition	proposition	NOUN
iajs-2613	179	31	(	(	PUNCT
iajs-2613	179	32	3	3	X
iajs-2613	179	33	)	)	PUNCT
iajs-2613	179	34	c	c	NOUN
iajs-2613	179	35	is	be	AUX
iajs-2613	179	36	a	a	DET
iajs-2613	179	37	wapp	wapp	NOUN
iajs-2613	179	38	-	-	PUNCT
iajs-2613	179	39	quasi	quasi	ADJ
iajs-2613	179	40	prime	prime	ADJ
iajs-2613	179	41	submodule	submodule	NOUN
iajs-2613	179	42	of	of	ADP
iajs-2613	179	43	t.	t.	PROPN
iajs-2613	179	44	it	it	PRON
iajs-2613	179	45	is	be	AUX
iajs-2613	179	46	well	well	ADV
iajs-2613	179	47	-	-	PUNCT
iajs-2613	179	48	known	know	VERB
iajs-2613	179	49	that	that	SCONJ
iajs-2613	179	50	if	if	SCONJ
iajs-2613	179	51	an	an	DET
iajs-2613	179	52	r	r	NOUN
iajs-2613	179	53	-	-	PUNCT
iajs-2613	179	54	module	module	NOUN
iajs-2613	179	55	t	t	NOUN
iajs-2613	179	56	is	be	AUX
iajs-2613	179	57	projective	projective	ADJ
iajs-2613	179	58	,	,	PUNCT
iajs-2613	179	59	then	then	ADV
iajs-2613	179	60	soc(t)=soc(r)t	soc(t)=soc(r)t	PROPN
iajs-2613	179	61	[	[	X
iajs-2613	179	62	11;prop.(3	11;prop.(3	NUM
iajs-2613	179	63	-	-	SYM
iajs-2613	179	64	24	24	NUM
iajs-2613	179	65	)	)	PUNCT
iajs-2613	179	66	]	]	PUNCT
iajs-2613	179	67	proposition(16	proposition(16	NOUN
iajs-2613	179	68	)	)	PUNCT
iajs-2613	179	69	let	let	VERB
iajs-2613	179	70	t	t	NOUN
iajs-2613	179	71	be	be	AUX
iajs-2613	179	72	a	a	DET
iajs-2613	179	73	projective	projective	ADJ
iajs-2613	179	74	multiplication	multiplication	NOUN
iajs-2613	179	75	r	r	NOUN
iajs-2613	179	76	-	-	PUNCT
iajs-2613	179	77	module	module	NOUN
iajs-2613	179	78	and	and	CCONJ
iajs-2613	179	79	c	c	AUX
iajs-2613	179	80	be	be	AUX
iajs-2613	179	81	a	a	DET
iajs-2613	179	82	proper	proper	ADJ
iajs-2613	179	83	submodule	submodule	NOUN
iajs-2613	179	84	of	of	ADP
iajs-2613	179	85	t	t	PROPN
iajs-2613	179	86	.	.	PUNCT
iajs-2613	180	1	then	then	ADV
iajs-2613	180	2	c	c	PROPN
iajs-2613	180	3	is	be	AUX
iajs-2613	180	4	wapp	wapp	NOUN
iajs-2613	180	5	-	-	PUNCT
iajs-2613	180	6	quasi	quasi	ADJ
iajs-2613	180	7	prime	prime	PROPN
iajs-2613	180	8	submodule	submodule	NOUN
iajs-2613	180	9	of	of	ADP
iajs-2613	180	10	t	t	PROPN
iajs-2613	180	11	if	if	SCONJ
iajs-2613	181	1	and	and	CCONJ
iajs-2613	181	2	only	only	ADV
iajs-2613	181	3	if	if	SCONJ
iajs-2613	181	4	[	[	X
iajs-2613	181	5	c	c	X
iajs-2613	181	6	:	:	PUNCT
iajs-2613	181	7	r	r	NOUN
iajs-2613	181	8	t	t	PROPN
iajs-2613	181	9	]	]	PUNCT
iajs-2613	181	10	is	be	AUX
iajs-2613	181	11	a	a	DET
iajs-2613	181	12	wappquasi	wappquasi	NOUN
iajs-2613	181	13	prime	prime	ADJ
iajs-2613	181	14	ideal	ideal	NOUN
iajs-2613	181	15	of	of	ADP
iajs-2613	181	16	r.	r.	PROPN
iajs-2613	181	17	proof	proof	NOUN
iajs-2613	181	18	:	:	PUNCT
iajs-2613	181	19	(	(	PUNCT
iajs-2613	181	20	)let	)let	PUNCT
iajs-2613	181	21	0≠rij[c	0≠rij[c	NOUN
iajs-2613	181	22	:	:	PUNCT
iajs-2613	181	23	r	r	NOUN
iajs-2613	181	24	t	t	PROPN
iajs-2613	181	25	]	]	PUNCT
iajs-2613	181	26	,	,	PUNCT
iajs-2613	181	27	for	for	ADP
iajs-2613	181	28	rϵr	rϵr	NOUN
iajs-2613	181	29	,	,	PUNCT
iajs-2613	181	30	i	i	PRON
iajs-2613	181	31	,	,	PUNCT
iajs-2613	181	32	j	j	PROPN
iajs-2613	181	33	are	be	AUX
iajs-2613	181	34	ideal	ideal	ADJ
iajs-2613	181	35	of	of	ADP
iajs-2613	181	36	r	r	NOUN
iajs-2613	181	37	.then	.then	X
iajs-2613	181	38	0≠r	0≠r	PROPN
iajs-2613	181	39	i(jt)c	i(jt)c	PROPN
iajs-2613	181	40	.	.	PUNCT
iajs-2613	182	1	since	since	SCONJ
iajs-2613	182	2	c	c	PROPN
iajs-2613	182	3	is	be	AUX
iajs-2613	182	4	wapp	wapp	NOUN
iajs-2613	182	5	quasi	quasi	ADJ
iajs-2613	182	6	prime	prime	PROPN
iajs-2613	182	7	submodule	submodule	NOUN
iajs-2613	182	8	of	of	ADP
iajs-2613	182	9	t	t	PROPN
iajs-2613	182	10	,	,	PUNCT
iajs-2613	182	11	then	then	ADV
iajs-2613	182	12	by	by	ADP
iajs-2613	182	13	corollary(7	corollary(7	PROPN
iajs-2613	182	14	)	)	PUNCT
iajs-2613	182	15	either	either	CCONJ
iajs-2613	182	16	r(jt)c+soc(t	r(jt)c+soc(t	NOUN
iajs-2613	182	17	)	)	PUNCT
iajs-2613	182	18	or	or	CCONJ
iajs-2613	182	19	i(jt)c+soc(t).now	i(jt)c+soc(t).now	PROPN
iajs-2613	182	20	since	since	SCONJ
iajs-2613	182	21	t	t	PROPN
iajs-2613	182	22	is	be	AUX
iajs-2613	182	23	a	a	DET
iajs-2613	182	24	projective	projective	ADJ
iajs-2613	182	25	module	module	NOUN
iajs-2613	182	26	,	,	PUNCT
iajs-2613	182	27	then	then	ADV
iajs-2613	182	28	soc(t)=soc(r)t	soc(t)=soc(r)t	ADJ
iajs-2613	182	29	,	,	PUNCT
iajs-2613	182	30	and	and	CCONJ
iajs-2613	182	31	since	since	SCONJ
iajs-2613	182	32	t	t	PROPN
iajs-2613	182	33	is	be	AUX
iajs-2613	182	34	multiplication	multiplication	NOUN
iajs-2613	182	35	,	,	PUNCT
iajs-2613	182	36	then	then	ADV
iajs-2613	182	37	c=[c	c=[c	PROPN
iajs-2613	182	38	:	:	PUNCT
iajs-2613	182	39	r	r	NOUN
iajs-2613	182	40	t]t	t]t	NOUN
iajs-2613	182	41	.	.	PUNCT
iajs-2613	183	1	hence	hence	ADV
iajs-2613	183	2	either	either	CCONJ
iajs-2613	183	3	r(jt)[c	r(jt)[c	NOUN
iajs-2613	183	4	:	:	PUNCT
iajs-2613	183	5	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	183	6	or	or	CCONJ
iajs-2613	183	7	i(jt)[c	i(jt)[c	ADJ
iajs-2613	183	8	:	:	PUNCT
iajs-2613	183	9	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	183	10	.	.	PUNCT
iajs-2613	184	1	it	it	PRON
iajs-2613	184	2	follows	follow	VERB
iajs-2613	184	3	that	that	SCONJ
iajs-2613	184	4	,	,	PUNCT
iajs-2613	184	5	either	either	CCONJ
iajs-2613	184	6	rj[c	rj[c	NOUN
iajs-2613	184	7	:	:	PUNCT
iajs-2613	184	8	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	184	9	)	)	PUNCT
iajs-2613	184	10	or	or	CCONJ
iajs-2613	184	11	ij[c	ij[c	NOUN
iajs-2613	184	12	:	:	PUNCT
iajs-2613	184	13	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	184	14	)	)	PUNCT
iajs-2613	184	15	.	.	PUNCT
iajs-2613	185	1	hence	hence	ADV
iajs-2613	185	2	by	by	ADP
iajs-2613	185	3	corollary(7	corollary(7	PROPN
iajs-2613	185	4	)	)	PUNCT
iajs-2613	186	1	[	[	X
iajs-2613	186	2	c	c	X
iajs-2613	186	3	:	:	PUNCT
iajs-2613	186	4	rt	rt	X
iajs-2613	186	5	]	]	X
iajs-2613	186	6	is	be	AUX
iajs-2613	186	7	a	a	DET
iajs-2613	186	8	wapp	wapp	NOUN
iajs-2613	186	9	-	-	PUNCT
iajs-2613	186	10	quasi	quasi	ADJ
iajs-2613	186	11	prime	prime	ADJ
iajs-2613	186	12	ideal	ideal	NOUN
iajs-2613	186	13	of	of	ADP
iajs-2613	186	14	r.	r.	PROPN
iajs-2613	187	1	(	(	PUNCT
iajs-2613	187	2	)let	)let	PROPN
iajs-2613	187	3	t	t	NOUN
iajs-2613	187	4	0≠rib	0≠rib	NOUN
iajs-2613	187	5	c	c	PROPN
iajs-2613	187	6	,	,	PUNCT
iajs-2613	187	7	for	for	ADP
iajs-2613	187	8	r	r	NOUN
iajs-2613	187	9	ϵr	ϵr	NOUN
iajs-2613	187	10	,	,	PUNCT
iajs-2613	187	11	i	i	PRON
iajs-2613	187	12	is	be	AUX
iajs-2613	187	13	an	an	DET
iajs-2613	187	14	ideal	ideal	NOUN
iajs-2613	187	15	in	in	ADP
iajs-2613	187	16	r	r	NOUN
iajs-2613	187	17	,	,	PUNCT
iajs-2613	187	18	and	and	CCONJ
iajs-2613	187	19	b	b	NOUN
iajs-2613	187	20	is	be	AUX
iajs-2613	187	21	submodule	submodule	NOUN
iajs-2613	187	22	of	of	ADP
iajs-2613	187	23	.	.	PUNCT
iajs-2613	188	1	since	since	SCONJ
iajs-2613	188	2	t	t	PROPN
iajs-2613	188	3	is	be	AUX
iajs-2613	188	4	a	a	DET
iajs-2613	188	5	multiplication	multiplication	NOUN
iajs-2613	188	6	,	,	PUNCT
iajs-2613	188	7	then	then	ADV
iajs-2613	188	8	b	b	X
iajs-2613	188	9	=	=	PROPN
iajs-2613	188	10	jt	jt	PROPN
iajs-2613	188	11	,	,	PUNCT
iajs-2613	188	12	for	for	ADP
iajs-2613	188	13	some	some	DET
iajs-2613	188	14	ideal	ideal	ADJ
iajs-2613	188	15	j	j	PROPN
iajs-2613	188	16	of	of	ADP
iajs-2613	188	17	r	r	NOUN
iajs-2613	188	18	.thus	.thus	PRON
iajs-2613	188	19	0≠rij	0≠rij	PROPN
iajs-2613	188	20	tc	tc	NOUN
iajs-2613	188	21	,	,	PUNCT
iajs-2613	188	22	implies	imply	VERB
iajs-2613	188	23	that	that	SCONJ
iajs-2613	188	24	0≠rij[c	0≠rij[c	PROPN
iajs-2613	188	25	:	:	PUNCT
iajs-2613	188	26	r	r	NOUN
iajs-2613	188	27	t	t	PROPN
iajs-2613	188	28	]	]	PUNCT
iajs-2613	188	29	.	.	PUNCT
iajs-2613	189	1	but	but	CCONJ
iajs-2613	189	2	[	[	X
iajs-2613	189	3	c	c	X
iajs-2613	189	4	:	:	PUNCT
iajs-2613	189	5	rt	rt	X
iajs-2613	189	6	]	]	X
iajs-2613	189	7	is	be	AUX
iajs-2613	189	8	a	a	DET
iajs-2613	189	9	wapp	wapp	NOUN
iajs-2613	189	10	-	-	PUNCT
iajs-2613	189	11	quasi	quasi	ADJ
iajs-2613	189	12	prime	prime	ADJ
iajs-2613	189	13	ideal	ideal	NOUN
iajs-2613	189	14	,	,	PUNCT
iajs-2613	189	15	then	then	ADV
iajs-2613	189	16	by	by	ADP
iajs-2613	189	17	corollary(7	corollary(7	NOUN
iajs-2613	189	18	)	)	PUNCT
iajs-2613	189	19	either	either	CCONJ
iajs-2613	189	20	rj[c	rj[c	NOUN
iajs-2613	189	21	:	:	PUNCT
iajs-2613	189	22	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	189	23	)	)	PUNCT
iajs-2613	189	24	or	or	CCONJ
iajs-2613	189	25	ij[c	ij[c	NOUN
iajs-2613	189	26	:	:	PUNCT
iajs-2613	189	27	rt]+soc(r	rt]+soc(r	PROPN
iajs-2613	189	28	)	)	PUNCT
iajs-2613	189	29	,	,	PUNCT
iajs-2613	189	30	that	that	PRON
iajs-2613	189	31	is	be	AUX
iajs-2613	189	32	either	either	CCONJ
iajs-2613	189	33	rjt[c	rjt[c	PROPN
iajs-2613	189	34	:	:	PUNCT
iajs-2613	189	35	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	189	36	or	or	CCONJ
iajs-2613	189	37	ijt[c	ijt[c	PROPN
iajs-2613	189	38	:	:	PUNCT
iajs-2613	189	39	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	189	40	.	.	PUNCT
iajs-2613	190	1	since	since	SCONJ
iajs-2613	190	2	t	t	PROPN
iajs-2613	190	3	is	be	AUX
iajs-2613	190	4	a	a	DET
iajs-2613	190	5	projective	projective	NOUN
iajs-2613	190	6	then	then	ADV
iajs-2613	190	7	soc(t)=soc(r)t	soc(t)=soc(r)t	PROPN
iajs-2613	190	8	and	and	CCONJ
iajs-2613	190	9	since	since	SCONJ
iajs-2613	190	10	t	t	PROPN
iajs-2613	190	11	is	be	AUX
iajs-2613	190	12	a	a	DET
iajs-2613	190	13	multiplication	multiplication	NOUN
iajs-2613	190	14	,	,	PUNCT
iajs-2613	190	15	then	then	ADV
iajs-2613	190	16	[	[	X
iajs-2613	190	17	c	c	X
iajs-2613	190	18	:	:	PUNCT
iajs-2613	190	19	rt]t	rt]t	PROPN
iajs-2613	190	20	=	=	SYM
iajs-2613	190	21	c	c	X
iajs-2613	190	22	.thus	.thus	PRON
iajs-2613	190	23	63	63	NUM
iajs-2613	190	24	ibn	ibn	PROPN
iajs-2613	190	25	al	al	PROPN
iajs-2613	190	26	-	-	PUNCT
iajs-2613	190	27	haitham	haitham	PROPN
iajs-2613	190	28	jour	jour	X
iajs-2613	190	29	.	.	PROPN
iajs-2613	191	1	for	for	ADP
iajs-2613	191	2	pure	pure	ADJ
iajs-2613	191	3	&	&	CCONJ
iajs-2613	191	4	appl	appl	PROPN
iajs-2613	191	5	.	.	PUNCT
iajs-2613	192	1	sci	sci	PROPN
iajs-2613	192	2	.	.	PROPN
iajs-2613	193	1	34	34	NUM
iajs-2613	193	2	(	(	PUNCT
iajs-2613	193	3	1	1	NUM
iajs-2613	193	4	)	)	PUNCT
iajs-2613	193	5	2021	2021	NUM
iajs-2613	193	6	either	either	CCONJ
iajs-2613	193	7	rbc+soc(t	rbc+soc(t	PROPN
iajs-2613	193	8	)	)	PUNCT
iajs-2613	193	9	or	or	CCONJ
iajs-2613	193	10	ibc+soc(t).hence	ibc+soc(t).hence	NOUN
iajs-2613	193	11	by	by	ADP
iajs-2613	193	12	corollary	corollary	ADJ
iajs-2613	193	13	(	(	PUNCT
iajs-2613	193	14	7	7	NUM
iajs-2613	193	15	)	)	PUNCT
iajs-2613	193	16	c	c	NOUN
iajs-2613	193	17	is	be	AUX
iajs-2613	193	18	a	a	DET
iajs-2613	193	19	wapp	wapp	NOUN
iajs-2613	193	20	-	-	PUNCT
iajs-2613	193	21	quasi	quasi	ADJ
iajs-2613	193	22	prime	prime	ADJ
iajs-2613	193	23	submodule	submodule	NOUN
iajs-2613	193	24	of	of	ADP
iajs-2613	193	25	t.	t.	PROPN
iajs-2613	193	26	we	we	PRON
iajs-2613	193	27	need	need	VERB
iajs-2613	193	28	to	to	PART
iajs-2613	193	29	recall	recall	VERB
iajs-2613	193	30	the	the	DET
iajs-2613	193	31	following	follow	VERB
iajs-2613	193	32	lemma	lemma	PROPN
iajs-2613	193	33	before	before	SCONJ
iajs-2613	193	34	we	we	PRON
iajs-2613	193	35	introduce	introduce	VERB
iajs-2613	193	36	the	the	DET
iajs-2613	193	37	next	next	ADJ
iajs-2613	193	38	proposition	proposition	NOUN
iajs-2613	193	39	.	.	PUNCT
iajs-2613	194	1	lemma(17)[12	lemma(17)[12	NOUN
iajs-2613	194	2	,	,	PUNCT
iajs-2613	194	3	coro	coro	NOUN
iajs-2613	194	4	,	,	PUNCT
iajs-2613	194	5	of	of	ADP
iajs-2613	194	6	theo	theo	PROPN
iajs-2613	194	7	,	,	PUNCT
iajs-2613	194	8	(	(	PUNCT
iajs-2613	194	9	9	9	NUM
iajs-2613	194	10	)	)	PUNCT
iajs-2613	194	11	]	]	PUNCT
iajs-2613	194	12	let	let	VERB
iajs-2613	194	13	t	t	NOUN
iajs-2613	194	14	be	be	AUX
iajs-2613	194	15	a	a	DET
iajs-2613	194	16	finitely	finitely	ADV
iajs-2613	194	17	generated	generate	VERB
iajs-2613	194	18	multiplication	multiplication	NOUN
iajs-2613	194	19	r	r	NOUN
iajs-2613	194	20	-	-	PUNCT
iajs-2613	194	21	module	module	NOUN
iajs-2613	194	22	and	and	CCONJ
iajs-2613	194	23	i	i	PRON
iajs-2613	194	24	,	,	PUNCT
iajs-2613	194	25	j	j	PROPN
iajs-2613	194	26	are	be	AUX
iajs-2613	194	27	ideals	ideal	NOUN
iajs-2613	194	28	of	of	ADP
iajs-2613	194	29	r	r	NOUN
iajs-2613	194	30	.	.	PUNCT
iajs-2613	195	1	then	then	ADV
iajs-2613	195	2	itjt	itjt	VERB
iajs-2613	195	3	if	if	SCONJ
iajs-2613	195	4	and	and	CCONJ
iajs-2613	195	5	only	only	ADV
iajs-2613	195	6	if	if	SCONJ
iajs-2613	195	7	ij+annr	ij+annr	NOUN
iajs-2613	195	8	(	(	PUNCT
iajs-2613	195	9	t	t	NOUN
iajs-2613	195	10	)	)	PUNCT
iajs-2613	195	11	.	.	PUNCT
iajs-2613	196	1	proposition(18	proposition(18	NOUN
iajs-2613	196	2	)	)	PUNCT
iajs-2613	196	3	let	let	VERB
iajs-2613	196	4	t	t	NOUN
iajs-2613	196	5	be	be	AUX
iajs-2613	196	6	a	a	DET
iajs-2613	196	7	finitely	finitely	ADV
iajs-2613	196	8	generated	generate	VERB
iajs-2613	196	9	multiplcation	multiplcation	NOUN
iajs-2613	196	10	z_regular	z_regular	NOUN
iajs-2613	196	11	r_module	r_module	NOUN
iajs-2613	197	1	and	and	CCONJ
iajs-2613	197	2	i	i	PRON
iajs-2613	197	3	is	be	AUX
iajs-2613	197	4	wapp	wapp	NOUN
iajs-2613	197	5	−	−	PROPN
iajs-2613	197	6	quasi	quasi	ADJ
iajs-2613	197	7	prime	prime	PROPN
iajs-2613	197	8	ideal	ideal	NOUN
iajs-2613	197	9	of	of	ADP
iajs-2613	197	10	r	r	NOUN
iajs-2613	197	11	with	with	ADP
iajs-2613	197	12	annr	annr	NOUN
iajs-2613	197	13	(	(	PUNCT
iajs-2613	197	14	t)	t)	NOUN
iajs-2613	198	1	i	i	PRON
iajs-2613	198	2	.	.	PUNCT
iajs-2613	199	1	then	then	ADV
iajs-2613	199	2	it	it	PRON
iajs-2613	199	3	is	be	AUX
iajs-2613	199	4	an	an	DET
iajs-2613	199	5	wapp	wapp	NOUN
iajs-2613	199	6	-	-	PUNCT
iajs-2613	199	7	quasi	quasi	ADJ
iajs-2613	199	8	prime	prime	ADJ
iajs-2613	199	9	submodule	submodule	NOUN
iajs-2613	199	10	of	of	ADP
iajs-2613	199	11	t.	t.	PROPN
iajs-2613	199	12	proof	proof	NOUN
iajs-2613	199	13	:	:	PUNCT
iajs-2613	199	14	let	let	VERB
iajs-2613	199	15	0≠i1	0≠i1	PROPN
iajs-2613	199	16	i2	i2	PROPN
iajs-2613	199	17	b	b	VERB
iajs-2613	199	18	it	it	PRON
iajs-2613	199	19	,	,	PUNCT
iajs-2613	199	20	for	for	ADP
iajs-2613	199	21	i1,i2	i1,i2	PROPN
iajs-2613	199	22	are	be	AUX
iajs-2613	199	23	is	be	AUX
iajs-2613	199	24	ideals	ideal	NOUN
iajs-2613	199	25	of	of	ADP
iajs-2613	199	26	r	r	NOUN
iajs-2613	199	27	,	,	PUNCT
iajs-2613	199	28	and	and	CCONJ
iajs-2613	199	29	b	b	NOUN
iajs-2613	199	30	is	be	AUX
iajs-2613	199	31	submodul	submodul	NOUN
iajs-2613	199	32	of	of	ADP
iajs-2613	199	33	t.	t.	PROPN
iajs-2613	199	34	since	since	SCONJ
iajs-2613	199	35	t	t	PROPN
iajs-2613	199	36	is	be	AUX
iajs-2613	199	37	a	a	DET
iajs-2613	199	38	multiplication	multiplication	NOUN
iajs-2613	199	39	then	then	ADV
iajs-2613	199	40	b	b	PROPN
iajs-2613	199	41	=	=	SYM
iajs-2613	199	42	j	j	PROPN
iajs-2613	199	43	t	t	NOUN
iajs-2613	199	44	for	for	ADP
iajs-2613	199	45	some	some	DET
iajs-2613	199	46	ideal	ideal	ADJ
iajs-2613	199	47	j	j	PROPN
iajs-2613	199	48	of	of	ADP
iajs-2613	199	49	r.	r.	PROPN
iajs-2613	199	50	that	that	PRON
iajs-2613	199	51	is	be	AUX
iajs-2613	199	52	let	let	VERB
iajs-2613	199	53	0≠i1	0≠i1	ADJ
iajs-2613	199	54	i2	i2	PROPN
iajs-2613	199	55	(	(	PUNCT
iajs-2613	199	56	j	j	PROPN
iajs-2613	199	57	t)	t)	VERB
iajs-2613	199	58	it	it	PRON
iajs-2613	199	59	,	,	PUNCT
iajs-2613	199	60	it	it	PRON
iajs-2613	199	61	follows	follow	VERB
iajs-2613	199	62	by	by	ADP
iajs-2613	199	63	lemma	lemma	PROPN
iajs-2613	199	64	(	(	PUNCT
iajs-2613	199	65	17	17	NUM
iajs-2613	199	66	)	)	PUNCT
iajs-2613	199	67	0≠i1	0≠i1	NOUN
iajs-2613	199	68	i2	i2	PROPN
iajs-2613	199	69	j	j	NOUN
iajs-2613	199	70	i+annr(t	i+annr(t	NOUN
iajs-2613	199	71	)	)	PUNCT
iajs-2613	199	72	.	.	PUNCT
iajs-2613	200	1	but	but	CCONJ
iajs-2613	200	2	annr(t)i	annr(t)i	PROPN
iajs-2613	200	3	,	,	PUNCT
iajs-2613	200	4	implies	imply	VERB
iajs-2613	200	5	that	that	SCONJ
iajs-2613	200	6	i+annr(t)=i	i+annr(t)=i	NOUN
iajs-2613	200	7	.	.	PUNCT
iajs-2613	201	1	that	that	PRON
iajs-2613	201	2	is	be	AUX
iajs-2613	201	3	0≠i1	0≠i1	ADJ
iajs-2613	201	4	i2	i2	PROPN
iajs-2613	201	5	j	j	PROPN
iajs-2613	201	6	i.	i.	PROPN
iajs-2613	202	1	but	but	CCONJ
iajs-2613	202	2	i	i	PRON
iajs-2613	202	3	is	be	AUX
iajs-2613	202	4	a	a	DET
iajs-2613	202	5	wapp	wapp	NOUN
iajs-2613	202	6	-	-	PUNCT
iajs-2613	202	7	quasi	quasi	ADJ
iajs-2613	202	8	prime	prime	ADJ
iajs-2613	202	9	ideal	ideal	NOUN
iajs-2613	202	10	of	of	ADP
iajs-2613	202	11	r	r	NOUN
iajs-2613	202	12	,	,	PUNCT
iajs-2613	202	13	then	then	ADV
iajs-2613	202	14	by	by	ADP
iajs-2613	202	15	proposition	proposition	NOUN
iajs-2613	202	16	(	(	PUNCT
iajs-2613	202	17	4	4	NUM
iajs-2613	202	18	)	)	PUNCT
iajs-2613	202	19	either0≠i1	either0≠i1	NOUN
iajs-2613	202	20	j	j	NOUN
iajs-2613	202	21	i+soc(r	i+soc(r	NOUN
iajs-2613	202	22	)	)	PUNCT
iajs-2613	202	23	or	or	CCONJ
iajs-2613	202	24	0≠	0≠	NUM
iajs-2613	202	25	i2	i2	PROPN
iajs-2613	202	26	j	j	NOUN
iajs-2613	202	27	i+soc(r	i+soc(r	PROPN
iajs-2613	202	28	)	)	PUNCT
iajs-2613	202	29	.	.	PUNCT
iajs-2613	203	1	it	it	PRON
iajs-2613	203	2	follows	follow	VERB
iajs-2613	203	3	that	that	SCONJ
iajs-2613	203	4	either0≠i1	either0≠i1	PROPN
iajs-2613	203	5	j	j	PROPN
iajs-2613	203	6	t	t	X
iajs-2613	203	7	it+soc(r)t	it+soc(r)t	PROPN
iajs-2613	203	8	or0≠	or0≠	PROPN
iajs-2613	203	9	i2	i2	PROPN
iajs-2613	203	10	j	j	PROPN
iajs-2613	203	11	t	t	X
iajs-2613	203	12	it+soc(r)t	it+soc(r)t	PROPN
iajs-2613	203	13	.	.	PUNCT
iajs-2613	204	1	but	but	CCONJ
iajs-2613	204	2	t	t	PROPN
iajs-2613	204	3	is	be	AUX
iajs-2613	204	4	a	a	DET
iajs-2613	204	5	z	z	NOUN
iajs-2613	204	6	-	-	ADJ
iajs-2613	204	7	regular	regular	ADJ
iajs-2613	204	8	then	then	ADV
iajs-2613	204	9	soc(r)t	soc(r)t	NOUN
iajs-2613	204	10	=	=	PROPN
iajs-2613	204	11	soc(t	soc(t	PROPN
iajs-2613	204	12	)	)	PUNCT
iajs-2613	204	13	.	.	PUNCT
iajs-2613	205	1	hence	hence	ADV
iajs-2613	205	2	either	either	CCONJ
iajs-2613	205	3	0≠i1	0≠i1	ADJ
iajs-2613	205	4	b	b	NUM
iajs-2613	205	5	it+soc(t	it+soc(t	NUM
iajs-2613	205	6	)	)	PUNCT
iajs-2613	205	7	or0≠	or0≠	NOUN
iajs-2613	205	8	i2	i2	NOUN
iajs-2613	205	9	b	b	VERB
iajs-2613	205	10	it+soc(t	it+soc(t	PROPN
iajs-2613	205	11	)	)	PUNCT
iajs-2613	205	12	.	.	PUNCT
iajs-2613	206	1	thus	thus	ADV
iajs-2613	206	2	by	by	ADP
iajs-2613	206	3	proposition	proposition	NOUN
iajs-2613	206	4	(	(	PUNCT
iajs-2613	206	5	4	4	X
iajs-2613	206	6	)	)	PUNCT
iajs-2613	206	7	it	it	PRON
iajs-2613	206	8	is	be	AUX
iajs-2613	206	9	wapp	wapp	NOUN
iajs-2613	206	10	-	-	PUNCT
iajs-2613	206	11	quasi	quasi	ADJ
iajs-2613	206	12	prime	prime	NOUN
iajs-2613	206	13	submodule	submodule	NOUN
iajs-2613	206	14	of	of	ADP
iajs-2613	206	15	t	t	PROPN
iajs-2613	206	16	.	.	PUNCT
iajs-2613	207	1	proposition(19	proposition(19	NOUN
iajs-2613	207	2	)	)	PUNCT
iajs-2613	208	1	let	let	VERB
iajs-2613	208	2	t	t	NOUN
iajs-2613	208	3	be	be	AUX
iajs-2613	208	4	a	a	DET
iajs-2613	208	5	finitely	finitely	ADV
iajs-2613	208	6	generated	generate	VERB
iajs-2613	208	7	multiplication	multiplication	NOUN
iajs-2613	208	8	projective	projective	ADJ
iajs-2613	208	9	r	r	NOUN
iajs-2613	208	10	-	-	PUNCT
iajs-2613	208	11	module	module	NOUN
iajs-2613	208	12	and	and	CCONJ
iajs-2613	208	13	i	i	PRON
iajs-2613	208	14	is	be	AUX
iajs-2613	208	15	a	a	DET
iajs-2613	208	16	wappquasi	wappquasi	NOUN
iajs-2613	208	17	prime	prime	ADJ
iajs-2613	208	18	ideal	ideal	NOUN
iajs-2613	208	19	of	of	ADP
iajs-2613	208	20	r	r	NOUN
iajs-2613	208	21	with	with	ADP
iajs-2613	208	22	annr	annr	NOUN
iajs-2613	208	23	(	(	PUNCT
iajs-2613	208	24	t)	t)	NOUN
iajs-2613	209	1	i	i	PRON
iajs-2613	209	2	.	.	PUNCT
iajs-2613	210	1	then	then	ADV
iajs-2613	210	2	it	it	PRON
iajs-2613	210	3	is	be	AUX
iajs-2613	210	4	wapp	wapp	NOUN
iajs-2613	210	5	-	-	PUNCT
iajs-2613	210	6	quasi	quasi	ADJ
iajs-2613	210	7	prime	prime	ADJ
iajs-2613	210	8	submodule	submodule	NOUN
iajs-2613	210	9	of	of	ADP
iajs-2613	210	10	t.	t.	PROPN
iajs-2613	210	11	proof	proof	NOUN
iajs-2613	210	12	:	:	PUNCT
iajs-2613	210	13	let	let	VERB
iajs-2613	210	14	0≠ri1	0≠ri1	NOUN
iajs-2613	210	15	b	b	VERB
iajs-2613	210	16	it	it	PRON
iajs-2613	210	17	,	,	PUNCT
iajs-2613	210	18	for	for	ADP
iajs-2613	210	19	rϵr	rϵr	NOUN
iajs-2613	210	20	,	,	PUNCT
iajs-2613	210	21	i1	i1	PROPN
iajs-2613	210	22	is	be	AUX
iajs-2613	210	23	an	an	DET
iajs-2613	210	24	ideal	ideal	NOUN
iajs-2613	210	25	of	of	ADP
iajs-2613	210	26	r	r	NOUN
iajs-2613	210	27	,	,	PUNCT
iajs-2613	210	28	and	and	CCONJ
iajs-2613	210	29	b	b	NOUN
iajs-2613	210	30	is	be	AUX
iajs-2613	210	31	submodule	submodule	NOUN
iajs-2613	210	32	of	of	ADP
iajs-2613	210	33	t.	t.	PROPN
iajs-2613	210	34	since	since	SCONJ
iajs-2613	210	35	t	t	PROPN
iajs-2613	210	36	is	be	AUX
iajs-2613	210	37	multiplication	multiplication	NOUN
iajs-2613	210	38	then	then	ADV
iajs-2613	210	39	b	b	PROPN
iajs-2613	210	40	=	=	PROPN
iajs-2613	210	41	jt	jt	PROPN
iajs-2613	210	42	for	for	ADP
iajs-2613	210	43	some	some	DET
iajs-2613	210	44	ideal	ideal	ADJ
iajs-2613	210	45	j	j	PROPN
iajs-2613	210	46	of	of	ADP
iajs-2613	210	47	r.	r.	PROPN
iajs-2613	210	48	that	that	PRON
iajs-2613	210	49	is	be	AUX
iajs-2613	210	50	let	let	VERB
iajs-2613	210	51	0≠ri1	0≠ri1	NUM
iajs-2613	210	52	(	(	PUNCT
iajs-2613	210	53	j	j	PROPN
iajs-2613	210	54	t)	t)	VERB
iajs-2613	210	55	it	it	PRON
iajs-2613	210	56	,	,	PUNCT
iajs-2613	210	57	it	it	PRON
iajs-2613	210	58	follows	follow	VERB
iajs-2613	210	59	by	by	ADP
iajs-2613	210	60	lemma	lemma	PROPN
iajs-2613	210	61	(	(	PUNCT
iajs-2613	210	62	17	17	NUM
iajs-2613	210	63	)	)	PUNCT
iajs-2613	210	64	0≠ri1	0≠ri1	NUM
iajs-2613	210	65	j	j	NOUN
iajs-2613	210	66	i+annr(t	i+annr(t	NOUN
iajs-2613	210	67	)	)	PUNCT
iajs-2613	210	68	.	.	PUNCT
iajs-2613	211	1	but	but	CCONJ
iajs-2613	211	2	annr(t)i	annr(t)i	PROPN
iajs-2613	211	3	,	,	PUNCT
iajs-2613	211	4	implies	imply	VERB
iajs-2613	211	5	that	that	SCONJ
iajs-2613	211	6	i+	i+	PROPN
iajs-2613	212	1	annr(t)=i	annr(t)=i	ADJ
iajs-2613	212	2	.	.	PUNCT
iajs-2613	213	1	hence	hence	ADV
iajs-2613	213	2	0≠ri1	0≠ri1	NUM
iajs-2613	213	3	j	j	NOUN
iajs-2613	213	4	i	i	PRON
iajs-2613	213	5	,	,	PUNCT
iajs-2613	213	6	and	and	CCONJ
iajs-2613	213	7	since	since	SCONJ
iajs-2613	213	8	i	i	PRON
iajs-2613	213	9	is	be	AUX
iajs-2613	213	10	wapp	wapp	NOUN
iajs-2613	213	11	-	-	PUNCT
iajs-2613	213	12	quasi	quasi	ADJ
iajs-2613	213	13	prime	prime	ADJ
iajs-2613	213	14	ideal	ideal	NOUN
iajs-2613	213	15	of	of	ADP
iajs-2613	213	16	r	r	NOUN
iajs-2613	213	17	,	,	PUNCT
iajs-2613	213	18	then	then	ADV
iajs-2613	213	19	by	by	ADP
iajs-2613	213	20	corollary	corollary	ADJ
iajs-2613	213	21	(	(	PUNCT
iajs-2613	213	22	7	7	NUM
iajs-2613	213	23	)	)	PUNCT
iajs-2613	213	24	either0≠i1	either0≠i1	NOUN
iajs-2613	213	25	j	j	NOUN
iajs-2613	213	26	i+soc(r	i+soc(r	NOUN
iajs-2613	213	27	)	)	PUNCT
iajs-2613	213	28	or0≠	or0≠	NOUN
iajs-2613	213	29	r	r	NOUN
iajs-2613	213	30	j	j	NOUN
iajs-2613	213	31	i+soc(r).that	i+soc(r).that	PRON
iajs-2613	213	32	is	be	AUX
iajs-2613	213	33	either	either	CCONJ
iajs-2613	213	34	0≠i1	0≠i1	ADJ
iajs-2613	213	35	j	j	PROPN
iajs-2613	214	1	t	t	X
iajs-2613	214	2	it+soc(r)t	it+soc(r)t	PROPN
iajs-2613	214	3	or0≠	or0≠	PROPN
iajs-2613	214	4	r	r	NOUN
iajs-2613	214	5	j	j	PROPN
iajs-2613	214	6	t	t	X
iajs-2613	214	7	it+soc(r)t	it+soc(r)t	PROPN
iajs-2613	214	8	.	.	PUNCT
iajs-2613	215	1	but	but	CCONJ
iajs-2613	215	2	t	t	PROPN
iajs-2613	215	3	is	be	AUX
iajs-2613	215	4	a	a	DET
iajs-2613	215	5	projective	projective	NOUN
iajs-2613	215	6	then	then	ADV
iajs-2613	215	7	soc(r)t	soc(r)t	NOUN
iajs-2613	215	8	=	=	PROPN
iajs-2613	215	9	soc(t	soc(t	PROPN
iajs-2613	215	10	)	)	PUNCT
iajs-2613	215	11	.	.	PUNCT
iajs-2613	216	1	thus	thus	ADV
iajs-2613	216	2	either	either	CCONJ
iajs-2613	216	3	0≠i1	0≠i1	ADJ
iajs-2613	216	4	b	b	VERB
iajs-2613	216	5	it+soc(t	it+soc(t	NUM
iajs-2613	216	6	)	)	PUNCT
iajs-2613	216	7	or0≠	or0≠	NOUN
iajs-2613	216	8	r	r	NOUN
iajs-2613	216	9	b	b	VERB
iajs-2613	216	10	it+soc(t	it+soc(t	NUM
iajs-2613	216	11	)	)	PUNCT
iajs-2613	216	12	.	.	PUNCT
iajs-2613	217	1	hence	hence	ADV
iajs-2613	217	2	by	by	ADP
iajs-2613	217	3	corollary	corollary	ADJ
iajs-2613	217	4	(	(	PUNCT
iajs-2613	217	5	7	7	X
iajs-2613	217	6	)	)	PUNCT
iajs-2613	217	7	it	it	PRON
iajs-2613	217	8	is	be	AUX
iajs-2613	217	9	wapp	wapp	NOUN
iajs-2613	217	10	-	-	PUNCT
iajs-2613	217	11	quasi	quasi	ADJ
iajs-2613	217	12	prime	prime	NOUN
iajs-2613	217	13	submodule	submodule	NOUN
iajs-2613	217	14	of	of	ADP
iajs-2613	217	15	t	t	PROPN
iajs-2613	217	16	.	.	PUNCT
iajs-2613	218	1	it	it	PRON
iajs-2613	218	2	is	be	AUX
iajs-2613	218	3	well	well	ADV
iajs-2613	218	4	-	-	PUNCT
iajs-2613	218	5	known	know	VERB
iajs-2613	218	6	that	that	SCONJ
iajs-2613	218	7	cyclic	cyclic	ADJ
iajs-2613	218	8	r	r	NOUN
iajs-2613	218	9	-	-	PUNCT
iajs-2613	218	10	module	module	NOUN
iajs-2613	218	11	is	be	AUX
iajs-2613	218	12	multiplication	multiplication	NOUN
iajs-2613	218	13	[	[	X
iajs-2613	218	14	13	13	NUM
iajs-2613	218	15	]	]	PUNCT
iajs-2613	218	16	,	,	PUNCT
iajs-2613	218	17	and	and	CCONJ
iajs-2613	218	18	since	since	SCONJ
iajs-2613	218	19	cyclic	cyclic	ADJ
iajs-2613	218	20	r	r	NOUN
iajs-2613	218	21	-	-	PUNCT
iajs-2613	218	22	module	module	NOUN
iajs-2613	218	23	is	be	AUX
iajs-2613	218	24	a	a	DET
iajs-2613	218	25	finitely	finitely	ADV
iajs-2613	218	26	generated	generate	VERB
iajs-2613	218	27	,	,	PUNCT
iajs-2613	218	28	we	we	PRON
iajs-2613	218	29	get	get	VERB
iajs-2613	218	30	the	the	DET
iajs-2613	218	31	following	follow	VERB
iajs-2613	218	32	corollaries	corollary	NOUN
iajs-2613	218	33	:	:	PUNCT
iajs-2613	218	34	corollary(20	corollary(20	NOUN
iajs-2613	218	35	)	)	PUNCT
iajs-2613	218	36	let	let	VERB
iajs-2613	218	37	t	t	NOUN
iajs-2613	218	38	be	be	AUX
iajs-2613	218	39	a	a	DET
iajs-2613	218	40	cyclic	cyclic	ADJ
iajs-2613	218	41	z	z	NOUN
iajs-2613	218	42	-	-	ADJ
iajs-2613	218	43	regular	regular	ADJ
iajs-2613	218	44	r	r	NOUN
iajs-2613	218	45	-	-	PUNCT
iajs-2613	218	46	module	module	NOUN
iajs-2613	219	1	and	and	CCONJ
iajs-2613	219	2	i	i	PRON
iajs-2613	219	3	is	be	AUX
iajs-2613	219	4	wapp	wapp	NOUN
iajs-2613	219	5	-	-	PUNCT
iajs-2613	219	6	quasi	quasi	ADJ
iajs-2613	219	7	prime	prime	ADJ
iajs-2613	219	8	ideal	ideal	NOUN
iajs-2613	219	9	of	of	ADP
iajs-2613	219	10	r	r	NOUN
iajs-2613	219	11	with	with	ADP
iajs-2613	219	12	annr	annr	NOUN
iajs-2613	219	13	(	(	PUNCT
iajs-2613	219	14	t)	t)	NOUN
iajs-2613	220	1	i	i	PRON
iajs-2613	220	2	.	.	PUNCT
iajs-2613	221	1	then	then	ADV
iajs-2613	221	2	it	it	PRON
iajs-2613	221	3	is	be	AUX
iajs-2613	221	4	an	an	DET
iajs-2613	221	5	wapp	wapp	NOUN
iajs-2613	221	6	-	-	PUNCT
iajs-2613	221	7	quasi	quasi	ADJ
iajs-2613	221	8	prime	prime	ADJ
iajs-2613	221	9	submodule	submodule	NOUN
iajs-2613	221	10	of	of	ADP
iajs-2613	221	11	t.	t.	PROPN
iajs-2613	221	12	corollary(21	corollary(21	NOUN
iajs-2613	221	13	)	)	PUNCT
iajs-2613	221	14	64	64	NUM
iajs-2613	221	15	ibn	ibn	PROPN
iajs-2613	221	16	al	al	PROPN
iajs-2613	221	17	-	-	PUNCT
iajs-2613	221	18	haitham	haitham	PROPN
iajs-2613	221	19	jour	jour	X
iajs-2613	221	20	.	.	PROPN
iajs-2613	222	1	for	for	ADP
iajs-2613	222	2	pure	pure	ADJ
iajs-2613	222	3	&	&	CCONJ
iajs-2613	222	4	appl	appl	PROPN
iajs-2613	222	5	.	.	PUNCT
iajs-2613	223	1	sci	sci	PROPN
iajs-2613	223	2	.	.	PROPN
iajs-2613	224	1	34	34	NUM
iajs-2613	224	2	(	(	PUNCT
iajs-2613	224	3	1	1	NUM
iajs-2613	224	4	)	)	PUNCT
iajs-2613	224	5	2021	2021	NUM
iajs-2613	224	6	let	let	VERB
iajs-2613	224	7	t	t	PROPN
iajs-2613	224	8	be	be	AUX
iajs-2613	224	9	a	a	DET
iajs-2613	224	10	cyclic	cyclic	ADJ
iajs-2613	224	11	projective	projective	ADJ
iajs-2613	224	12	r	r	NOUN
iajs-2613	224	13	-	-	PUNCT
iajs-2613	224	14	module	module	NOUN
iajs-2613	225	1	and	and	CCONJ
iajs-2613	225	2	i	i	PRON
iajs-2613	225	3	is	be	AUX
iajs-2613	225	4	an	an	DET
iajs-2613	225	5	wapp	wapp	NOUN
iajs-2613	225	6	-	-	PUNCT
iajs-2613	225	7	quasi	quasi	ADJ
iajs-2613	225	8	prime	prime	ADJ
iajs-2613	225	9	ideal	ideal	NOUN
iajs-2613	225	10	of	of	ADP
iajs-2613	225	11	r	r	NOUN
iajs-2613	225	12	with	with	ADP
iajs-2613	225	13	annr	annr	NOUN
iajs-2613	225	14	(	(	PUNCT
iajs-2613	225	15	t)	t)	NOUN
iajs-2613	225	16	i	i	PRON
iajs-2613	225	17	.	.	PUNCT
iajs-2613	226	1	then	then	ADV
iajs-2613	226	2	it	it	PRON
iajs-2613	226	3	is	be	AUX
iajs-2613	226	4	an	an	DET
iajs-2613	226	5	wapp	wapp	NOUN
iajs-2613	226	6	-	-	PUNCT
iajs-2613	226	7	quasi	quasi	ADJ
iajs-2613	226	8	prim	prim	PROPN
iajs-2613	226	9	submodule	submodule	NOUN
iajs-2613	226	10	of	of	ADP
iajs-2613	226	11	t.	t.	PROPN
iajs-2613	226	12	it	it	PRON
iajs-2613	226	13	is	be	AUX
iajs-2613	226	14	well	well	ADV
iajs-2613	226	15	-	-	PUNCT
iajs-2613	226	16	known	know	VERB
iajs-2613	226	17	that	that	SCONJ
iajs-2613	226	18	if	if	SCONJ
iajs-2613	226	19	a	a	DET
iajs-2613	226	20	submodule	submodule	NOUN
iajs-2613	226	21	c	c	NOUN
iajs-2613	226	22	of	of	ADP
iajs-2613	226	23	an	an	DET
iajs-2613	226	24	r	r	NOUN
iajs-2613	226	25	-	-	PUNCT
iajs-2613	226	26	module	module	NOUN
iajs-2613	226	27	t	t	NOUN
iajs-2613	226	28	is	be	AUX
iajs-2613	226	29	essential	essential	ADJ
iajs-2613	226	30	in	in	ADP
iajs-2613	226	31	t	t	PROPN
iajs-2613	226	32	,	,	PUNCT
iajs-2613	226	33	then	then	ADV
iajs-2613	226	34	soc(c)=soc(t	soc(c)=soc(t	NOUN
iajs-2613	226	35	)	)	PUNCT
iajs-2613	227	1	[	[	X
iajs-2613	227	2	6	6	NUM
iajs-2613	227	3	,	,	PUNCT
iajs-2613	227	4	p.29	p.29	NOUN
iajs-2613	227	5	]	]	PUNCT
iajs-2613	227	6	.	.	PUNCT
iajs-2613	228	1	proposition(22	proposition(22	NOUN
iajs-2613	228	2	)	)	PUNCT
iajs-2613	228	3	let	let	VERB
iajs-2613	228	4	t	t	PROPN
iajs-2613	228	5	be	be	AUX
iajs-2613	228	6	r	r	NOUN
iajs-2613	228	7	-	-	PUNCT
iajs-2613	228	8	module	module	NOUN
iajs-2613	228	9	,	,	PUNCT
iajs-2613	228	10	and	and	CCONJ
iajs-2613	228	11	a	a	DET
iajs-2613	228	12	,	,	PUNCT
iajs-2613	228	13	b	b	NOUN
iajs-2613	228	14	are	be	AUX
iajs-2613	228	15	submodules	submodule	NOUN
iajs-2613	228	16	of	of	ADP
iajs-2613	228	17	t	t	PROPN
iajs-2613	228	18	with	with	ADP
iajs-2613	228	19	a	a	NOUN
iajs-2613	228	20	b	b	PROPN
iajs-2613	228	21	and	and	CCONJ
iajs-2613	228	22	b	b	NOUN
iajs-2613	228	23	is	be	AUX
iajs-2613	228	24	an	an	DET
iajs-2613	228	25	essential	essential	ADJ
iajs-2613	228	26	in	in	ADP
iajs-2613	228	27	t.	t.	PROPN
iajs-2613	228	28	if	if	SCONJ
iajs-2613	228	29	a	a	PRON
iajs-2613	228	30	is	be	AUX
iajs-2613	228	31	an	an	DET
iajs-2613	228	32	wapp	wapp	NOUN
iajs-2613	228	33	-	-	PUNCT
iajs-2613	228	34	quasi	quasi	ADJ
iajs-2613	228	35	prime	prime	PROPN
iajs-2613	228	36	submodule	submodule	NOUN
iajs-2613	228	37	of	of	ADP
iajs-2613	228	38	t	t	PROPN
iajs-2613	228	39	,	,	PUNCT
iajs-2613	228	40	then	then	ADV
iajs-2613	228	41	a	a	PRON
iajs-2613	228	42	is	be	AUX
iajs-2613	228	43	a	a	DET
iajs-2613	228	44	wapp	wapp	NOUN
iajs-2613	228	45	-	-	PUNCT
iajs-2613	228	46	quasi	quasi	ADJ
iajs-2613	228	47	prime	prime	PROPN
iajs-2613	228	48	submodule	submodule	PROPN
iajs-2613	228	49	of	of	ADP
iajs-2613	228	50	b.	b.	PROPN
iajs-2613	228	51	proof	proof	NOUN
iajs-2613	228	52	:	:	PUNCT
iajs-2613	228	53	let	let	VERB
iajs-2613	228	54	0≠rstϵa	0≠rstϵa	PRON
iajs-2613	228	55	,	,	PUNCT
iajs-2613	228	56	for	for	ADP
iajs-2613	228	57	r	r	NOUN
iajs-2613	228	58	,	,	PUNCT
iajs-2613	228	59	sϵr	sϵr	NOUN
iajs-2613	228	60	,	,	PUNCT
iajs-2613	228	61	tϵb	tϵb	PROPN
iajs-2613	228	62	,	,	PUNCT
iajs-2613	228	63	that	that	PRON
iajs-2613	228	64	is	be	AUX
iajs-2613	228	65	tϵt	tϵt	NOUN
iajs-2613	228	66	.	.	PUNCT
iajs-2613	229	1	since	since	SCONJ
iajs-2613	229	2	a	a	PRON
iajs-2613	229	3	is	be	AUX
iajs-2613	229	4	a	a	DET
iajs-2613	229	5	wapp	wapp	NOUN
iajs-2613	229	6	-	-	PUNCT
iajs-2613	229	7	quasi	quasi	ADJ
iajs-2613	229	8	prime	prime	PROPN
iajs-2613	229	9	submodule	submodule	NOUN
iajs-2613	229	10	of	of	ADP
iajs-2613	229	11	t	t	PROPN
iajs-2613	229	12	,	,	PUNCT
iajs-2613	229	13	then	then	ADV
iajs-2613	229	14	either	either	CCONJ
iajs-2613	229	15	rtϵa	rtϵa	NOUN
iajs-2613	229	16	+	+	ADV
iajs-2613	229	17	soc(t	soc(t	PROPN
iajs-2613	229	18	)	)	PUNCT
iajs-2613	229	19	or	or	CCONJ
iajs-2613	229	20	stϵ	stϵ	ADP
iajs-2613	229	21	a	a	DET
iajs-2613	229	22	+	+	NOUN
iajs-2613	229	23	soc(t	soc(t	NOUN
iajs-2613	229	24	)	)	PUNCT
iajs-2613	229	25	.	.	PUNCT
iajs-2613	230	1	but	but	CCONJ
iajs-2613	230	2	b	b	NOUN
iajs-2613	230	3	is	be	AUX
iajs-2613	230	4	essential	essential	ADJ
iajs-2613	230	5	in	in	ADP
iajs-2613	230	6	t	t	PROPN
iajs-2613	230	7	,	,	PUNCT
iajs-2613	230	8	then	then	ADV
iajs-2613	230	9	soc(b)=soc(t	soc(b)=soc(t	PROPN
iajs-2613	230	10	)	)	PUNCT
iajs-2613	230	11	.	.	PUNCT
iajs-2613	231	1	that	that	PRON
iajs-2613	231	2	is	be	AUX
iajs-2613	231	3	either	either	CCONJ
iajs-2613	231	4	rtϵa+soc(b	rtϵa+soc(b	NOUN
iajs-2613	231	5	)	)	PUNCT
iajs-2613	231	6	or	or	CCONJ
iajs-2613	231	7	stϵa+soc(b).hence	stϵa+soc(b).hence	NOUN
iajs-2613	231	8	a	a	PRON
iajs-2613	231	9	is	be	AUX
iajs-2613	231	10	an	an	DET
iajs-2613	231	11	wapp	wapp	NOUN
iajs-2613	231	12	-	-	PUNCT
iajs-2613	231	13	quasi	quasi	ADJ
iajs-2613	231	14	prime	prime	PROPN
iajs-2613	231	15	submodule	submodule	PROPN
iajs-2613	231	16	of	of	ADP
iajs-2613	231	17	b.	b.	PROPN
iajs-2613	231	18	corollary(23	corollary(23	NOUN
iajs-2613	231	19	)	)	PUNCT
iajs-2613	231	20	let	let	VERB
iajs-2613	231	21	t	t	PROPN
iajs-2613	231	22	be	be	AUX
iajs-2613	231	23	r	r	NOUN
iajs-2613	231	24	-	-	PUNCT
iajs-2613	231	25	module	module	NOUN
iajs-2613	231	26	,	,	PUNCT
iajs-2613	231	27	and	and	CCONJ
iajs-2613	231	28	a	a	DET
iajs-2613	231	29	,	,	PUNCT
iajs-2613	231	30	b	b	NOUN
iajs-2613	231	31	are	be	AUX
iajs-2613	231	32	submodules	submodule	NOUN
iajs-2613	231	33	of	of	ADP
iajs-2613	231	34	t	t	PROPN
iajs-2613	231	35	with	with	ADP
iajs-2613	231	36	a	a	NOUN
iajs-2613	231	37	b	b	NOUN
iajs-2613	231	38	and	and	CCONJ
iajs-2613	231	39	soc(t)	soc(t)	NOUN
iajs-2613	231	40	soc(b	soc(b	PROPN
iajs-2613	231	41	)	)	PUNCT
iajs-2613	231	42	.	.	PUNCT
iajs-2613	232	1	then	then	ADV
iajs-2613	232	2	a	a	PRON
iajs-2613	232	3	is	be	AUX
iajs-2613	232	4	a	a	DET
iajs-2613	232	5	wapp	wapp	NOUN
iajs-2613	232	6	-	-	PUNCT
iajs-2613	232	7	quasi	quasi	ADJ
iajs-2613	232	8	prime	prime	PROPN
iajs-2613	232	9	submodule	submodule	PROPN
iajs-2613	232	10	of	of	ADP
iajs-2613	232	11	b.	b.	PROPN
iajs-2613	233	1	it	it	PRON
iajs-2613	233	2	well	well	ADV
iajs-2613	233	3	-	-	PUNCT
iajs-2613	233	4	known	know	VERB
iajs-2613	233	5	that	that	SCONJ
iajs-2613	233	6	if	if	SCONJ
iajs-2613	233	7	a	a	PRON
iajs-2613	233	8	is	be	AUX
iajs-2613	233	9	a	a	DET
iajs-2613	233	10	submodule	submodule	NOUN
iajs-2613	233	11	of	of	ADP
iajs-2613	233	12	an	an	DET
iajs-2613	233	13	r	r	NOUN
iajs-2613	233	14	-	-	PUNCT
iajs-2613	233	15	module	module	NOUN
iajs-2613	233	16	t	t	NOUN
iajs-2613	233	17	,	,	PUNCT
iajs-2613	233	18	then	then	ADV
iajs-2613	233	19	soc(a)=a∩soc(t	soc(a)=a∩soc(t	ADV
iajs-2613	233	20	)	)	PUNCT
iajs-2613	234	1	[	[	X
iajs-2613	234	2	9,lema	9,lema	NUM
iajs-2613	234	3	2.3.15	2.3.15	NUM
iajs-2613	234	4	]	]	PUNCT
iajs-2613	234	5	proposition(24	proposition(24	NOUN
iajs-2613	234	6	)	)	PUNCT
iajs-2613	234	7	let	let	VERB
iajs-2613	234	8	t	t	NOUN
iajs-2613	234	9	be	be	AUX
iajs-2613	234	10	r	r	NOUN
iajs-2613	234	11	−	−	NOUN
iajs-2613	234	12	module	module	NOUN
iajs-2613	234	13	,	,	PUNCT
iajs-2613	234	14	and	and	CCONJ
iajs-2613	234	15	a	a	DET
iajs-2613	234	16	,	,	PUNCT
iajs-2613	234	17	b	b	NOUN
iajs-2613	234	18	are	be	AUX
iajs-2613	234	19	submodules	submodule	NOUN
iajs-2613	234	20	of	of	ADP
iajs-2613	234	21	t	t	PROPN
iajs-2613	234	22	with	with	ADP
iajs-2613	234	23	b	b	PROPN
iajs-2613	234	24	not	not	PART
iajs-2613	234	25	contain	contain	VERB
iajs-2613	234	26	in	in	ADP
iajs-2613	234	27	a	a	DET
iajs-2613	234	28	,	,	PUNCT
iajs-2613	234	29	and	and	CCONJ
iajs-2613	234	30	soc(t)	soc(t)	PROPN
iajs-2613	234	31	b.	b.	PROPN
iajs-2613	234	32	if	if	SCONJ
iajs-2613	234	33	a	a	PRON
iajs-2613	234	34	is	be	AUX
iajs-2613	234	35	a	a	DET
iajs-2613	234	36	wapp	wapp	NOUN
iajs-2613	234	37	-	-	PUNCT
iajs-2613	234	38	quasi	quasi	ADJ
iajs-2613	234	39	prime	prime	PROPN
iajs-2613	234	40	submodule	submodule	NOUN
iajs-2613	234	41	of	of	ADP
iajs-2613	234	42	t	t	PROPN
iajs-2613	234	43	,	,	PUNCT
iajs-2613	234	44	then	then	ADV
iajs-2613	234	45	a∩	a∩	PROPN
iajs-2613	234	46	b	b	PROPN
iajs-2613	234	47	is	be	AUX
iajs-2613	234	48	a	a	DET
iajs-2613	234	49	wapp	wapp	NOUN
iajs-2613	234	50	-	-	PUNCT
iajs-2613	234	51	quasi	quasi	ADJ
iajs-2613	234	52	prime	prime	PROPN
iajs-2613	234	53	submodule	submodule	PROPN
iajs-2613	234	54	of	of	ADP
iajs-2613	234	55	b.	b.	PROPN
iajs-2613	234	56	proof	proof	NOUN
iajs-2613	234	57	:	:	PUNCT
iajs-2613	234	58	it	it	PRON
iajs-2613	234	59	is	be	AUX
iajs-2613	234	60	clear	clear	ADJ
iajs-2613	234	61	that	that	SCONJ
iajs-2613	234	62	a∩	a∩	PROPN
iajs-2613	234	63	b	b	PROPN
iajs-2613	234	64	is	be	AUX
iajs-2613	234	65	an	an	DET
iajs-2613	234	66	proper	proper	ADJ
iajs-2613	234	67	submodule	submodule	NOUN
iajs-2613	234	68	of	of	ADP
iajs-2613	234	69	b	b	PROPN
iajs-2613	234	70	.now	.now	PUNCT
iajs-2613	234	71	,	,	PUNCT
iajs-2613	234	72	let	let	VERB
iajs-2613	234	73	0≠rstϵ	0≠rstϵ	NOUN
iajs-2613	234	74	a∩	a∩	PROPN
iajs-2613	234	75	b	b	X
iajs-2613	234	76	,	,	PUNCT
iajs-2613	234	77	for	for	ADP
iajs-2613	234	78	r	r	NOUN
iajs-2613	234	79	,	,	PUNCT
iajs-2613	234	80	sϵr	sϵr	NOUN
iajs-2613	234	81	,	,	PUNCT
iajs-2613	234	82	tϵb	tϵb	PROPN
iajs-2613	234	83	,	,	PUNCT
iajs-2613	234	84	implies	imply	VERB
iajs-2613	234	85	that	that	SCONJ
iajs-2613	234	86	0≠rstϵ	0≠rstϵ	NOUN
iajs-2613	234	87	a	a	X
iajs-2613	234	88	,	,	PUNCT
iajs-2613	234	89	since	since	SCONJ
iajs-2613	234	90	a	a	PRON
iajs-2613	234	91	is	be	AUX
iajs-2613	234	92	a	a	DET
iajs-2613	234	93	wapp	wapp	NOUN
iajs-2613	234	94	-	-	PUNCT
iajs-2613	234	95	quasi	quasi	ADJ
iajs-2613	234	96	prime	prime	PROPN
iajs-2613	234	97	submodule	submodule	NOUN
iajs-2613	234	98	of	of	ADP
iajs-2613	234	99	t	t	PROPN
iajs-2613	234	100	,	,	PUNCT
iajs-2613	234	101	then	then	ADV
iajs-2613	234	102	either	either	CCONJ
iajs-2613	234	103	rtϵa+soc(t	rtϵa+soc(t	NOUN
iajs-2613	234	104	)	)	PUNCT
iajs-2613	234	105	or	or	CCONJ
iajs-2613	234	106	stϵa+soc(t	stϵa+soc(t	NOUN
iajs-2613	234	107	)	)	PUNCT
iajs-2613	234	108	,	,	PUNCT
iajs-2613	234	109	hence	hence	ADV
iajs-2613	234	110	either	either	DET
iajs-2613	234	111	rtϵ(a+soc(t	rtϵ(a+soc(t	PROPN
iajs-2613	234	112	)	)	PUNCT
iajs-2613	234	113	)	)	PUNCT
iajs-2613	235	1	∩	∩	PROPN
iajs-2613	235	2	bor	bor	PROPN
iajs-2613	235	3	stϵ(a+soc(t))∩	stϵ(a+soc(t))∩	PROPN
iajs-2613	235	4	b.	b.	PROPN
iajs-2613	235	5	since	since	SCONJ
iajs-2613	235	6	soc(t)b	soc(t)b	NOUN
iajs-2613	235	7	,	,	PUNCT
iajs-2613	235	8	then	then	ADV
iajs-2613	235	9	by	by	ADP
iajs-2613	235	10	module	module	NOUN
iajs-2613	235	11	law	law	NOUN
iajs-2613	235	12	either	either	CCONJ
iajs-2613	235	13	rtϵ(a∩	rtϵ(a∩	PROPN
iajs-2613	235	14	b)+	b)+	PROPN
iajs-2613	235	15	(	(	PUNCT
iajs-2613	235	16	b∩	b∩	PROPN
iajs-2613	235	17	soc(t	soc(t	PROPN
iajs-2613	235	18	)	)	PUNCT
iajs-2613	235	19	)	)	PUNCT
iajs-2613	235	20	or	or	CCONJ
iajs-2613	235	21	stϵ(a∩	stϵ(a∩	NUM
iajs-2613	235	22	b)+(b∩	b)+(b∩	NOUN
iajs-2613	235	23	soc(t	soc(t	NUM
iajs-2613	235	24	)	)	PUNCT
iajs-2613	235	25	)	)	PUNCT
iajs-2613	235	26	.	.	PUNCT
iajs-2613	236	1	that	that	PRON
iajs-2613	236	2	is	be	AUX
iajs-2613	236	3	either	either	CCONJ
iajs-2613	236	4	rtϵ(a∩	rtϵ(a∩	PROPN
iajs-2613	236	5	b)+	b)+	PROPN
iajs-2613	236	6	soc(b	soc(b	PROPN
iajs-2613	236	7	)	)	PUNCT
iajs-2613	236	8	or	or	CCONJ
iajs-2613	236	9	stϵ(a∩	stϵ(a∩	NUM
iajs-2613	236	10	b)+soc(b).thus	b)+soc(b).thus	PUNCT
iajs-2613	236	11	a∩	a∩	PROPN
iajs-2613	236	12	b	b	PROPN
iajs-2613	236	13	is	be	AUX
iajs-2613	236	14	a	a	DET
iajs-2613	236	15	wapp	wapp	NOUN
iajs-2613	236	16	-	-	PUNCT
iajs-2613	236	17	quasi	quasi	ADJ
iajs-2613	236	18	prime	prime	PROPN
iajs-2613	236	19	submodule	submodule	PROPN
iajs-2613	236	20	of	of	ADP
iajs-2613	236	21	b.	b.	PROPN
iajs-2613	237	1	it	it	PRON
iajs-2613	237	2	well	well	ADV
iajs-2613	237	3	-	-	PUNCT
iajs-2613	237	4	known	know	VERB
iajs-2613	237	5	that	that	SCONJ
iajs-2613	237	6	for	for	ADP
iajs-2613	237	7	each	each	DET
iajs-2613	237	8	submodule	submodule	NOUN
iajs-2613	237	9	a	a	PRON
iajs-2613	237	10	of	of	ADP
iajs-2613	237	11	an	an	DET
iajs-2613	237	12	r	r	NOUN
iajs-2613	237	13	-	-	PUNCT
iajs-2613	237	14	module	module	NOUN
iajs-2613	237	15	t	t	NOUN
iajs-2613	237	16	,	,	PUNCT
iajs-2613	237	17	then	then	ADV
iajs-2613	237	18	soc(a)=a	soc(a)=a	PROPN
iajs-2613	237	19	,	,	PUNCT
iajs-2613	237	20	then	then	ADV
iajs-2613	237	21	asoc(t)[9,theo.(9.1.4)(c	asoc(t)[9,theo.(9.1.4)(c	PROPN
iajs-2613	237	22	)	)	PUNCT
iajs-2613	237	23	]	]	PUNCT
iajs-2613	237	24	.	.	PUNCT
iajs-2613	238	1	proposition(25	proposition(25	NOUN
iajs-2613	238	2	)	)	PUNCT
iajs-2613	238	3	let	let	VERB
iajs-2613	238	4	t	t	NOUN
iajs-2613	238	5	be	be	AUX
iajs-2613	238	6	an	an	DET
iajs-2613	238	7	r	r	NOUN
iajs-2613	238	8	−	−	NOUN
iajs-2613	238	9	module	module	NOUN
iajs-2613	238	10	,	,	PUNCT
iajs-2613	238	11	and	and	CCONJ
iajs-2613	238	12	a	a	DET
iajs-2613	238	13	,	,	PUNCT
iajs-2613	238	14	b	b	NOUN
iajs-2613	238	15	are	be	AUX
iajs-2613	238	16	submodules	submodule	NOUN
iajs-2613	238	17	of	of	ADP
iajs-2613	238	18	t	t	PROPN
iajs-2613	238	19	with	with	ADP
iajs-2613	238	20	b	b	PROPN
iajs-2613	238	21	not	not	PART
iajs-2613	238	22	contain	contain	VERB
iajs-2613	238	23	in	in	ADP
iajs-2613	238	24	a	a	PRON
iajs-2613	238	25	,	,	PUNCT
iajs-2613	238	26	with	with	ADP
iajs-2613	238	27	soc(a)=a	soc(a)=a	PROPN
iajs-2613	238	28	and	and	CCONJ
iajs-2613	238	29	soc(b)=b	soc(b)=b	PROPN
iajs-2613	238	30	.	.	PUNCT
iajs-2613	239	1	then	then	ADV
iajs-2613	239	2	a∩	a∩	PROPN
iajs-2613	239	3	b	b	PROPN
iajs-2613	239	4	is	be	AUX
iajs-2613	239	5	a	a	DET
iajs-2613	239	6	wapp	wapp	NOUN
iajs-2613	239	7	-	-	PUNCT
iajs-2613	239	8	quasi	quasi	ADJ
iajs-2613	239	9	prime	prime	ADJ
iajs-2613	239	10	sub	sub	NOUN
iajs-2613	239	11	module	module	NOUN
iajs-2613	239	12	of	of	ADP
iajs-2613	239	13	t.	t.	NOUN
iajs-2613	239	14	proof	proof	NOUN
iajs-2613	239	15	:	:	PUNCT
iajs-2613	239	16	65	65	NUM
iajs-2613	239	17	ibn	ibn	PROPN
iajs-2613	239	18	al	al	PROPN
iajs-2613	239	19	-	-	PUNCT
iajs-2613	239	20	haitham	haitham	PROPN
iajs-2613	239	21	jour	jour	X
iajs-2613	239	22	.	.	PROPN
iajs-2613	240	1	for	for	ADP
iajs-2613	240	2	pure	pure	ADJ
iajs-2613	240	3	&	&	CCONJ
iajs-2613	240	4	appl	appl	PROPN
iajs-2613	240	5	.	.	PUNCT
iajs-2613	241	1	sci	sci	PROPN
iajs-2613	241	2	.	.	PROPN
iajs-2613	242	1	34	34	NUM
iajs-2613	242	2	(	(	PUNCT
iajs-2613	242	3	1	1	NUM
iajs-2613	242	4	)	)	PUNCT
iajs-2613	242	5	2021	2021	NUM
iajs-2613	242	6	let	let	VERB
iajs-2613	242	7	0≠rsl	0≠rsl	NOUN
iajs-2613	242	8	a	a	DET
iajs-2613	242	9	∩	∩	ADJ
iajs-2613	242	10	b	b	NOUN
iajs-2613	242	11	,	,	PUNCT
iajs-2613	242	12	for	for	ADP
iajs-2613	242	13	r	r	NOUN
iajs-2613	242	14	,	,	PUNCT
iajs-2613	242	15	sϵr	sϵr	NOUN
iajs-2613	242	16	,	,	PUNCT
iajs-2613	242	17	l	l	NOUN
iajs-2613	242	18	is	be	AUX
iajs-2613	242	19	submodule	submodule	NOUN
iajs-2613	242	20	of	of	ADP
iajs-2613	242	21	t	t	PROPN
iajs-2613	242	22	,	,	PUNCT
iajs-2613	242	23	then	then	ADV
iajs-2613	242	24	0≠rs	0≠rs	ADJ
iajs-2613	242	25	l	l	NOUN
iajs-2613	242	26			PROPN
iajs-2613	242	27	a	a	PRON
iajs-2613	242	28	,	,	PUNCT
iajs-2613	242	29	and	and	CCONJ
iajs-2613	242	30	0≠rs	0≠rs	NUM
iajs-2613	242	31	l	l	PROPN
iajs-2613	242	32	b.	b.	PROPN
iajs-2613	243	1	but	but	CCONJ
iajs-2613	243	2	both	both	DET
iajs-2613	243	3	a	a	DET
iajs-2613	243	4	,	,	PUNCT
iajs-2613	243	5	b	b	NOUN
iajs-2613	243	6	are	be	AUX
iajs-2613	243	7	wapp	wapp	NOUN
iajs-2613	243	8	-	-	PUNCT
iajs-2613	243	9	quasi	quasi	ADJ
iajs-2613	243	10	prime	prime	PROPN
iajs-2613	243	11	submodule	submodule	NOUN
iajs-2613	243	12	of	of	ADP
iajs-2613	243	13	t	t	PROPN
iajs-2613	243	14	,	,	PUNCT
iajs-2613	243	15	then	then	ADV
iajs-2613	243	16	either	either	CCONJ
iajs-2613	243	17	rla+soc(t	rla+soc(t	PROPN
iajs-2613	243	18	)	)	PUNCT
iajs-2613	243	19	or	or	CCONJ
iajs-2613	243	20	sla+soc(t	sla+soc(t	PROPN
iajs-2613	243	21	)	)	PUNCT
iajs-2613	243	22	,	,	PUNCT
iajs-2613	243	23	and	and	CCONJ
iajs-2613	243	24	rl	rl	PROPN
iajs-2613	243	25	b+soc(t	b+soc(t	PROPN
iajs-2613	243	26	)	)	PUNCT
iajs-2613	243	27	or	or	CCONJ
iajs-2613	243	28	sl	sl	ADJ
iajs-2613	243	29	b+soc(t	b+soc(t	NOUN
iajs-2613	243	30	)	)	PUNCT
iajs-2613	243	31	.	.	PUNCT
iajs-2613	244	1	but	but	CCONJ
iajs-2613	244	2	soc(a)=a	soc(a)=a	PROPN
iajs-2613	244	3	and	and	CCONJ
iajs-2613	244	4	soc(b)=b	soc(b)=b	PROPN
iajs-2613	244	5	,	,	PUNCT
iajs-2613	244	6	then	then	ADV
iajs-2613	244	7	asoc(t	asoc(t	PROPN
iajs-2613	244	8	)	)	PUNCT
iajs-2613	244	9	and	and	CCONJ
iajs-2613	244	10	bsoc(t	bsoc(t	NUM
iajs-2613	244	11	)	)	PUNCT
iajs-2613	244	12	,	,	PUNCT
iajs-2613	244	13	hence	hence	ADV
iajs-2613	244	14	a+soc(t)=soc(t	a+soc(t)=soc(t	ADJ
iajs-2613	244	15	)	)	PUNCT
iajs-2613	244	16	and	and	CCONJ
iajs-2613	244	17	b+soc(t)=soc(t	b+soc(t)=soc(t	PROPN
iajs-2613	244	18	)	)	PUNCT
iajs-2613	244	19	,	,	PUNCT
iajs-2613	244	20	a∩	a∩	PROPN
iajs-2613	244	21	b	b	VERB
iajs-2613	244	22	soc(t	soc(t	PROPN
iajs-2613	244	23	)	)	PUNCT
iajs-2613	244	24	,	,	PUNCT
iajs-2613	244	25	implies	imply	VERB
iajs-2613	244	26	that	that	SCONJ
iajs-2613	244	27	a∩	a∩	PROPN
iajs-2613	244	28	b	b	PROPN
iajs-2613	244	29	+	+	CCONJ
iajs-2613	244	30	soc(t)=soc(t	soc(t)=soc(t	ADJ
iajs-2613	244	31	)	)	PUNCT
iajs-2613	244	32	,	,	PUNCT
iajs-2613	244	33	so	so	CCONJ
iajs-2613	244	34	either	either	CCONJ
iajs-2613	244	35	rlsoc(t)=	rlsoc(t)=	VERB
iajs-2613	244	36	a∩	a∩	PROPN
iajs-2613	244	37	b	b	PROPN
iajs-2613	244	38	+	+	CCONJ
iajs-2613	244	39	soc(t	soc(t	PROPN
iajs-2613	244	40	)	)	PUNCT
iajs-2613	244	41	or	or	CCONJ
iajs-2613	244	42	sl	sl	PRON
iajs-2613	244	43	soc(t)=	soc(t)=	NOUN
iajs-2613	244	44	a∩	a∩	PROPN
iajs-2613	244	45	b	b	PROPN
iajs-2613	244	46	+	+	NUM
iajs-2613	244	47	soc(t	soc(t	PROPN
iajs-2613	244	48	)	)	PUNCT
iajs-2613	244	49	.	.	PUNCT
iajs-2613	245	1	hence	hence	ADV
iajs-2613	245	2	a∩	a∩	PROPN
iajs-2613	245	3	b	b	PROPN
iajs-2613	245	4	is	be	AUX
iajs-2613	245	5	wapp	wapp	NOUN
iajs-2613	245	6	-	-	PUNCT
iajs-2613	245	7	quasi	quasi	ADJ
iajs-2613	245	8	prime	prime	PROPN
iajs-2613	245	9	submodule	submodule	NOUN
iajs-2613	245	10	of	of	ADP
iajs-2613	245	11	t	t	PROPN
iajs-2613	245	12	proposition(26	proposition(26	NOUN
iajs-2613	245	13	)	)	PUNCT
iajs-2613	245	14	let	let	VERB
iajs-2613	245	15	f	f	X
iajs-2613	245	16	:	:	PUNCT
iajs-2613	245	17	t→t′	t→t′	VERB
iajs-2613	245	18	be	be	AUX
iajs-2613	245	19	an	an	DET
iajs-2613	245	20	r	r	NOUN
iajs-2613	245	21	-	-	PUNCT
iajs-2613	245	22	epimorphism	epimorphism	NOUN
iajs-2613	245	23	,	,	PUNCT
iajs-2613	245	24	and	and	CCONJ
iajs-2613	245	25	c	c	PROPN
iajs-2613	245	26	be	be	AUX
iajs-2613	245	27	an	an	DET
iajs-2613	245	28	wapp	wapp	NOUN
iajs-2613	245	29	-	-	PUNCT
iajs-2613	245	30	quasi	quasi	ADJ
iajs-2613	245	31	prime	prime	PROPN
iajs-2613	245	32	submodule	submodule	NOUN
iajs-2613	245	33	of	of	ADP
iajs-2613	245	34	t	t	PROPN
iajs-2613	245	35	with	with	ADP
iajs-2613	245	36	kerfc	kerfc	PROPN
iajs-2613	245	37	.	.	PUNCT
iajs-2613	246	1	then	then	ADV
iajs-2613	246	2	f(c	f(c	PROPN
iajs-2613	246	3	)	)	PUNCT
iajs-2613	246	4	is	be	AUX
iajs-2613	246	5	wapp	wapp	NOUN
iajs-2613	246	6	-	-	PUNCT
iajs-2613	246	7	quasi	quasi	ADJ
iajs-2613	246	8	prime	prime	NOUN
iajs-2613	246	9	submodule	submodule	NOUN
iajs-2613	246	10	of	of	ADP
iajs-2613	246	11	t′.	t′.	NOUN
iajs-2613	246	12	proof	proof	NOUN
iajs-2613	246	13	:	:	PUNCT
iajs-2613	246	14	let	let	VERB
iajs-2613	246	15	f	f	X
iajs-2613	246	16	:	:	PUNCT
iajs-2613	246	17	t→t′	t→t′	VERB
iajs-2613	246	18	be	be	AUX
iajs-2613	246	19	an	an	DET
iajs-2613	246	20	r	r	NOUN
iajs-2613	246	21	-	-	PUNCT
iajs-2613	246	22	epimorphism	epimorphism	NOUN
iajs-2613	246	23	,	,	PUNCT
iajs-2613	246	24	and	and	CCONJ
iajs-2613	246	25	c	c	PROPN
iajs-2613	246	26	be	be	AUX
iajs-2613	246	27	an	an	DET
iajs-2613	246	28	wapp	wapp	NOUN
iajs-2613	246	29	-	-	PUNCT
iajs-2613	246	30	quasi	quasi	ADJ
iajs-2613	246	31	prime	prime	PROPN
iajs-2613	246	32	submodule	submodule	NOUN
iajs-2613	246	33	of	of	ADP
iajs-2613	246	34	t	t	PROPN
iajs-2613	246	35	with	with	ADP
iajs-2613	246	36	kerfc	kerfc	PROPN
iajs-2613	246	37	,	,	PUNCT
iajs-2613	246	38	let	let	VERB
iajs-2613	246	39	0≠rst’ϵf(c	0≠rst’ϵf(c	ADV
iajs-2613	246	40	)	)	PUNCT
iajs-2613	246	41	,	,	PUNCT
iajs-2613	246	42	for	for	ADP
iajs-2613	246	43	r	r	NOUN
iajs-2613	246	44	,	,	PUNCT
iajs-2613	246	45	sϵr	sϵr	NOUN
iajs-2613	246	46	,	,	PUNCT
iajs-2613	246	47	tϵt′	tϵt′	ADJ
iajs-2613	246	48	.since	.since	PUNCT
iajs-2613	247	1	f	f	PROPN
iajs-2613	247	2	is	be	AUX
iajs-2613	247	3	onto	onto	ADP
iajs-2613	247	4	,	,	PUNCT
iajs-2613	247	5	then	then	ADV
iajs-2613	247	6	f(t)=	f(t)=	VERB
iajs-2613	247	7	t	t	X
iajs-2613	247	8	,	,	PUNCT
iajs-2613	247	9	for	for	ADP
iajs-2613	247	10	some	some	DET
iajs-2613	247	11	tϵt	tϵt	NOUN
iajs-2613	247	12	,	,	PUNCT
iajs-2613	247	13	it	it	PRON
iajs-2613	247	14	follows	follow	VERB
iajs-2613	247	15	that	that	PRON
iajs-2613	247	16	0≠rsf(t)ϵf(c	0≠rsf(t)ϵf(c	PROPN
iajs-2613	247	17	)	)	PUNCT
iajs-2613	247	18	,	,	PUNCT
iajs-2613	247	19	0≠f(rst)ϵf(c	0≠f(rst)ϵf(c	NUM
iajs-2613	247	20	)	)	PUNCT
iajs-2613	247	21	,	,	PUNCT
iajs-2613	247	22	so	so	CCONJ
iajs-2613	247	23	there	there	PRON
iajs-2613	247	24	exists	exist	VERB
iajs-2613	247	25	a	a	DET
iajs-2613	247	26	nonzero	nonzero	NOUN
iajs-2613	247	27	xϵc	xϵc	NOUN
iajs-2613	247	28	such	such	ADJ
iajs-2613	247	29	that	that	SCONJ
iajs-2613	247	30	,	,	PUNCT
iajs-2613	247	31	0≠f(rst)=f(x	0≠f(rst)=f(x	PROPN
iajs-2613	247	32	)	)	PUNCT
iajs-2613	247	33	.	.	PUNCT
iajs-2613	248	1	that	that	PRON
iajs-2613	248	2	is	be	AUX
iajs-2613	248	3	f(rst	f(rst	PROPN
iajs-2613	248	4	-	-	PUNCT
iajs-2613	248	5	x	x	NOUN
iajs-2613	248	6	)	)	PUNCT
iajs-2613	248	7	=	=	SYM
iajs-2613	248	8	0	0	NUM
iajs-2613	248	9	,	,	PUNCT
iajs-2613	248	10	implies	imply	VERB
iajs-2613	248	11	that	that	SCONJ
iajs-2613	248	12	rst	rst	PROPN
iajs-2613	248	13	-	-	PUNCT
iajs-2613	248	14	xϵ	xϵ	INTJ
iajs-2613	248	15	kerfc	kerfc	PROPN
iajs-2613	248	16	,	,	PUNCT
iajs-2613	248	17	implies	imply	VERB
iajs-2613	248	18	that	that	SCONJ
iajs-2613	248	19	0≠rstϵc	0≠rstϵc	PROPN
iajs-2613	248	20	.	.	PUNCT
iajs-2613	249	1	but	but	CCONJ
iajs-2613	249	2	c	c	NOUN
iajs-2613	249	3	is	be	AUX
iajs-2613	249	4	a	a	DET
iajs-2613	249	5	wapp	wapp	NOUN
iajs-2613	249	6	-	-	PUNCT
iajs-2613	249	7	quasi	quasi	ADJ
iajs-2613	249	8	prime	prime	PROPN
iajs-2613	249	9	submodule	submodule	NOUN
iajs-2613	249	10	of	of	ADP
iajs-2613	249	11	t	t	PROPN
iajs-2613	249	12	,	,	PUNCT
iajs-2613	249	13	then	then	ADV
iajs-2613	249	14	either	either	DET
iajs-2613	249	15	rtϵc+soc(t	rtϵc+soc(t	NOUN
iajs-2613	249	16	)	)	PUNCT
iajs-2613	249	17	or	or	CCONJ
iajs-2613	249	18	s	s	AUX
iajs-2613	249	19	tϵ	tϵ	NOUN
iajs-2613	249	20	c	c	PROPN
iajs-2613	249	21	+	+	NOUN
iajs-2613	249	22	soc(t	soc(t	PROPN
iajs-2613	249	23	)	)	PUNCT
iajs-2613	249	24	.	.	PUNCT
iajs-2613	250	1	that	that	PRON
iajs-2613	250	2	is	be	AUX
iajs-2613	250	3	either	either	CCONJ
iajs-2613	250	4	r	r	NOUN
iajs-2613	250	5	f(t	f(t	NOUN
iajs-2613	250	6	)	)	PUNCT
iajs-2613	250	7	ϵf(c)+f(soc(t))	ϵf(c)+f(soc(t))	PROPN
iajs-2613	250	8	f(c)+soc(t′	f(c)+soc(t′	PROPN
iajs-2613	250	9	)	)	PUNCT
iajs-2613	250	10	or	or	CCONJ
iajs-2613	250	11	sf(t)ϵf(c	sf(t)ϵf(c	NOUN
iajs-2613	250	12	)	)	PUNCT
iajs-2613	250	13	+	+	NOUN
iajs-2613	250	14	f(soc	f(soc	X
iajs-2613	250	15	(	(	PUNCT
iajs-2613	250	16	t	t	NOUN
iajs-2613	250	17	)	)	PUNCT
iajs-2613	250	18	)	)	PUNCT
iajs-2613	250	19			PROPN
iajs-2613	250	20	f(c	f(c	PROPN
iajs-2613	250	21	)	)	PUNCT
iajs-2613	250	22	+	+	NOUN
iajs-2613	250	23	soc(t′	soc(t′	NOUN
iajs-2613	250	24	)	)	PUNCT
iajs-2613	250	25	.	.	PUNCT
iajs-2613	251	1	thus	thus	ADV
iajs-2613	251	2	either	either	CCONJ
iajs-2613	251	3	r𝑡′	r𝑡′	PROPN
iajs-2613	251	4	ϵ	ϵ	X
iajs-2613	251	5	f(c)+soc(t′	f(c)+soc(t′	PROPN
iajs-2613	251	6	)	)	PUNCT
iajs-2613	251	7	or	or	CCONJ
iajs-2613	251	8	s𝑡′ϵ	s𝑡′ϵ	PROPN
iajs-2613	251	9	f(c	f(c	PROPN
iajs-2613	251	10	)	)	PUNCT
iajs-2613	251	11	+	+	NOUN
iajs-2613	251	12	soc(t′	soc(t′	NOUN
iajs-2613	251	13	)	)	PUNCT
iajs-2613	251	14	.	.	PUNCT
iajs-2613	252	1	hence	hence	ADV
iajs-2613	252	2	f(c	f(c	PROPN
iajs-2613	252	3	)	)	PUNCT
iajs-2613	252	4	is	be	AUX
iajs-2613	252	5	an	an	DET
iajs-2613	252	6	wapp	wapp	NOUN
iajs-2613	252	7	-	-	PUNCT
iajs-2613	252	8	quasi	quasi	ADJ
iajs-2613	252	9	prime	prime	NOUN
iajs-2613	252	10	submodule	submodule	NOUN
iajs-2613	252	11	of	of	ADP
iajs-2613	252	12	t′	t′	NUM
iajs-2613	252	13	.	.	PUNCT
iajs-2613	253	1	proposition(27	proposition(27	NOUN
iajs-2613	253	2	)	)	PUNCT
iajs-2613	253	3	let	let	VERB
iajs-2613	253	4	f	f	X
iajs-2613	253	5	:	:	PUNCT
iajs-2613	253	6	t→t′	t→t′	VERB
iajs-2613	253	7	be	be	AUX
iajs-2613	253	8	an	an	DET
iajs-2613	253	9	r	r	NOUN
iajs-2613	253	10	-	-	PUNCT
iajs-2613	253	11	epimorphism	epimorphism	NOUN
iajs-2613	253	12	,	,	PUNCT
iajs-2613	253	13	and	and	CCONJ
iajs-2613	253	14	c	c	PROPN
iajs-2613	253	15	be	be	VERB
iajs-2613	253	16	wapp	wapp	NOUN
iajs-2613	253	17	-	-	PUNCT
iajs-2613	253	18	quasi	quasi	ADJ
iajs-2613	253	19	prime	prime	ADJ
iajs-2613	253	20	submodule	submodule	NOUN
iajs-2613	253	21	of	of	ADP
iajs-2613	253	22	t′	t′	NUM
iajs-2613	253	23	.	.	PUNCT
iajs-2613	254	1	then	then	ADV
iajs-2613	254	2	f	f	PROPN
iajs-2613	254	3	−1(c	−1(c	PROPN
iajs-2613	254	4	)	)	PUNCT
iajs-2613	254	5	is	be	AUX
iajs-2613	254	6	an	an	DET
iajs-2613	254	7	wapp	wapp	NOUN
iajs-2613	254	8	-	-	PUNCT
iajs-2613	254	9	quasi	quasi	ADJ
iajs-2613	254	10	prime	prime	NOUN
iajs-2613	254	11	submodule	submodule	NOUN
iajs-2613	254	12	of	of	ADP
iajs-2613	254	13	t	t	PROPN
iajs-2613	254	14	.	.	PUNCT
iajs-2613	255	1	prove	prove	VERB
iajs-2613	255	2	:	:	PUNCT
iajs-2613	255	3	it	it	PRON
iajs-2613	255	4	is	be	AUX
iajs-2613	255	5	clearly	clearly	ADV
iajs-2613	255	6	that	that	SCONJ
iajs-2613	255	7	f	f	PROPN
iajs-2613	255	8	−1(c	−1(c	ADV
iajs-2613	255	9	)	)	PUNCT
iajs-2613	255	10	is	be	AUX
iajs-2613	255	11	proper	proper	ADJ
iajs-2613	255	12	submodule	submodule	NOUN
iajs-2613	255	13	of	of	ADP
iajs-2613	255	14	t.	t.	PROPN
iajs-2613	255	15	let	let	VERB
iajs-2613	255	16	0≠rstϵ	0≠rstϵ	NOUN
iajs-2613	255	17	f	f	PROPN
iajs-2613	255	18	−1(c	−1(c	PROPN
iajs-2613	255	19	)	)	PUNCT
iajs-2613	255	20	,	,	PUNCT
iajs-2613	255	21	for	for	ADP
iajs-2613	255	22	r	r	NOUN
iajs-2613	255	23	,	,	PUNCT
iajs-2613	255	24	s	s	PART
iajs-2613	255	25	ϵr	ϵr	X
iajs-2613	255	26	,	,	PUNCT
iajs-2613	255	27	tϵt	tϵt	NOUN
iajs-2613	255	28	,	,	PUNCT
iajs-2613	255	29	it	it	PRON
iajs-2613	255	30	follows	follow	VERB
iajs-2613	255	31	that	that	SCONJ
iajs-2613	255	32	then	then	ADV
iajs-2613	255	33	0≠rsf(t)ϵc	0≠rsf(t)ϵc	NUM
iajs-2613	255	34	,	,	PUNCT
iajs-2613	255	35	but	but	CCONJ
iajs-2613	255	36	c	c	NOUN
iajs-2613	255	37	is	be	AUX
iajs-2613	255	38	a	a	DET
iajs-2613	255	39	wapp	wapp	NOUN
iajs-2613	255	40	-	-	PUNCT
iajs-2613	255	41	quasi	quasi	ADJ
iajs-2613	255	42	prime	prime	NOUN
iajs-2613	255	43	submodule	submodule	NOUN
iajs-2613	255	44	of	of	ADP
iajs-2613	255	45	t	t	PROPN
iajs-2613	255	46	,	,	PUNCT
iajs-2613	255	47	then	then	ADV
iajs-2613	255	48	either	either	CCONJ
iajs-2613	255	49	r	r	NOUN
iajs-2613	255	50	f(t	f(t	NOUN
iajs-2613	255	51	)	)	PUNCT
iajs-2613	255	52	ϵc	ϵc	ADP
iajs-2613	256	1	+	+	ADJ
iajs-2613	256	2	soc(t	soc(t	PROPN
iajs-2613	256	3	)	)	PUNCT
iajs-2613	256	4	or	or	CCONJ
iajs-2613	256	5	sf(t)ϵ	sf(t)ϵ	PROPN
iajs-2613	256	6	c	c	PROPN
iajs-2613	256	7	+	+	PROPN
iajs-2613	256	8	soc(t	soc(t	PROPN
iajs-2613	256	9	)	)	PUNCT
iajs-2613	256	10	.	.	PUNCT
iajs-2613	257	1	thus	thus	ADV
iajs-2613	257	2	either	either	CCONJ
iajs-2613	257	3	r	r	NOUN
iajs-2613	257	4	tϵ	tϵ	NOUN
iajs-2613	257	5	f	f	PROPN
iajs-2613	257	6	-1(c)+f	-1(c)+f	INTJ
iajs-2613	257	7	-1	-1	PUNCT
iajs-2613	257	8	(	(	PUNCT
iajs-2613	257	9	soc(t′))	soc(t′))	PROPN
iajs-2613	257	10	f	f	PROPN
iajs-2613	257	11	-1(c)+soc(t	-1(c)+soc(t	NOUN
iajs-2613	257	12	)	)	PUNCT
iajs-2613	257	13	or	or	CCONJ
iajs-2613	257	14	s	s	VERB
iajs-2613	257	15	tϵ	tϵ	NOUN
iajs-2613	257	16	f	f	PROPN
iajs-2613	257	17	-1(c	-1(c	PROPN
iajs-2613	257	18	)	)	PUNCT
iajs-2613	258	1	+	+	NUM
iajs-2613	258	2	f	f	X
iajs-2613	258	3	−1	−1	NOUN
iajs-2613	258	4	(	(	PUNCT
iajs-2613	258	5	soc	soc	NOUN
iajs-2613	258	6	(	(	PUNCT
iajs-2613	258	7	t′	t′	NUM
iajs-2613	258	8	)	)	PUNCT
iajs-2613	258	9	)	)	PUNCT
iajs-2613	259	1			PROPN
iajs-2613	259	2	f	f	PROPN
iajs-2613	259	3	−1(c	−1(c	PROPN
iajs-2613	259	4	)	)	PUNCT
iajs-2613	260	1	+	+	NOUN
iajs-2613	260	2	soc(t	soc(t	NOUN
iajs-2613	260	3	)	)	PUNCT
iajs-2613	260	4	.hence	.hence	PUNCT
iajs-2613	260	5	f	f	PROPN
iajs-2613	260	6	−1(c	−1(c	PROPN
iajs-2613	260	7	)	)	PUNCT
iajs-2613	260	8	is	be	AUX
iajs-2613	260	9	wapp	wapp	NOUN
iajs-2613	260	10	-	-	PUNCT
iajs-2613	260	11	quasi	quasi	ADJ
iajs-2613	260	12	prime	prime	NOUN
iajs-2613	260	13	submodule	submodule	NOUN
iajs-2613	260	14	of	of	ADP
iajs-2613	260	15	t	t	PROPN
iajs-2613	260	16	.	.	PUNCT
iajs-2613	261	1	proposition(28	proposition(28	PROPN
iajs-2613	261	2	)	)	PUNCT
iajs-2613	261	3	let	let	VERB
iajs-2613	261	4	t	t	NOUN
iajs-2613	261	5	be	be	AUX
iajs-2613	261	6	a	a	DET
iajs-2613	261	7	z	z	NOUN
iajs-2613	261	8	-	-	PUNCT
iajs-2613	261	9	regular	regular	ADJ
iajs-2613	261	10	finitely	finitely	ADV
iajs-2613	261	11	generated	generate	VERB
iajs-2613	261	12	multiplication	multiplication	NOUN
iajs-2613	261	13	r	r	NOUN
iajs-2613	261	14	−	−	NOUN
iajs-2613	261	15	module	module	NOUN
iajs-2613	261	16	,	,	PUNCT
iajs-2613	261	17	and	and	CCONJ
iajs-2613	261	18	c	c	PROPN
iajs-2613	261	19	be	be	AUX
iajs-2613	261	20	a	a	DET
iajs-2613	261	21	proper	proper	ADJ
iajs-2613	261	22	submodule	submodule	NOUN
iajs-2613	261	23	of	of	ADP
iajs-2613	261	24	t	t	PROPN
iajs-2613	261	25	.	.	PUNCT
iajs-2613	262	1	then	then	ADV
iajs-2613	262	2	the	the	DET
iajs-2613	262	3	following	follow	VERB
iajs-2613	262	4	statements	statement	NOUN
iajs-2613	262	5	are	be	AUX
iajs-2613	262	6	equivalent	equivalent	ADJ
iajs-2613	262	7	:	:	PUNCT
iajs-2613	262	8	1	1	X
iajs-2613	262	9	.	.	X
iajs-2613	262	10	c	c	PROPN
iajs-2613	262	11	is	be	AUX
iajs-2613	262	12	wapp	wapp	NOUN
iajs-2613	262	13	-	-	PUNCT
iajs-2613	262	14	quasi	quasi	ADJ
iajs-2613	262	15	prime	prime	NOUN
iajs-2613	262	16	submodule	submodule	NOUN
iajs-2613	262	17	of	of	ADP
iajs-2613	262	18	t	t	PROPN
iajs-2613	262	19	.	.	PUNCT
iajs-2613	263	1	2	2	X
iajs-2613	263	2	.	.	PUNCT
iajs-2613	264	1	[	[	X
iajs-2613	264	2	c	c	X
iajs-2613	264	3	:	:	PUNCT
iajs-2613	264	4	rt	rt	X
iajs-2613	264	5	]	]	X
iajs-2613	264	6	is	be	AUX
iajs-2613	264	7	wapp	wapp	NOUN
iajs-2613	264	8	-	-	PUNCT
iajs-2613	264	9	quasi	quasi	ADJ
iajs-2613	264	10	prime	prime	ADJ
iajs-2613	264	11	ideal	ideal	NOUN
iajs-2613	264	12	of	of	ADP
iajs-2613	264	13	r	r	NOUN
iajs-2613	264	14	.	.	PUNCT
iajs-2613	265	1	3	3	X
iajs-2613	265	2	.	.	X
iajs-2613	266	1	c	c	X
iajs-2613	266	2	=	=	PRON
iajs-2613	266	3	it	it	PRON
iajs-2613	266	4	for	for	ADP
iajs-2613	266	5	some	some	DET
iajs-2613	266	6	wapp	wapp	NOUN
iajs-2613	266	7	-	-	PUNCT
iajs-2613	266	8	quasi	quasi	ADJ
iajs-2613	266	9	prime	prime	PROPN
iajs-2613	266	10	ideal	ideal	NOUN
iajs-2613	266	11	i	i	PRON
iajs-2613	266	12	of	of	ADP
iajs-2613	266	13	r	r	NOUN
iajs-2613	266	14	with	with	ADP
iajs-2613	266	15	annr(t)≤i	annr(t)≤i	NOUN
iajs-2613	266	16	.	.	PUNCT
iajs-2613	267	1	poof	poof	NOUN
iajs-2613	267	2	:	:	PUNCT
iajs-2613	267	3	(	(	PUNCT
iajs-2613	267	4	1	1	X
iajs-2613	267	5	)	)	PUNCT
iajs-2613	267	6			NOUN
iajs-2613	267	7	(	(	PUNCT
iajs-2613	267	8	2	2	X
iajs-2613	267	9	)	)	PUNCT
iajs-2613	267	10	follows	follow	VERB
iajs-2613	267	11	by	by	ADP
iajs-2613	267	12	proposition	proposition	NOUN
iajs-2613	267	13	[	[	X
iajs-2613	267	14	15	15	NUM
iajs-2613	267	15	]	]	SYM
iajs-2613	267	16	66	66	NUM
iajs-2613	267	17	ibn	ibn	PROPN
iajs-2613	267	18	al	al	PROPN
iajs-2613	267	19	-	-	PUNCT
iajs-2613	267	20	haitham	haitham	PROPN
iajs-2613	267	21	jour	jour	X
iajs-2613	267	22	.	.	PROPN
iajs-2613	268	1	for	for	ADP
iajs-2613	268	2	pure	pure	ADJ
iajs-2613	268	3	&	&	CCONJ
iajs-2613	268	4	appl	appl	PROPN
iajs-2613	268	5	.	.	PUNCT
iajs-2613	269	1	sci	sci	PROPN
iajs-2613	269	2	.	.	PROPN
iajs-2613	270	1	34	34	NUM
iajs-2613	270	2	(	(	PUNCT
iajs-2613	270	3	1	1	NUM
iajs-2613	270	4	)	)	PUNCT
iajs-2613	270	5	2021	2021	NUM
iajs-2613	270	6	(	(	PUNCT
iajs-2613	270	7	2	2	NUM
iajs-2613	270	8	)	)	PUNCT
iajs-2613	270	9			NOUN
iajs-2613	270	10	(	(	PUNCT
iajs-2613	270	11	3	3	X
iajs-2613	270	12	)	)	PUNCT
iajs-2613	270	13	follows	follow	VERB
iajs-2613	270	14	directly	directly	ADV
iajs-2613	270	15	.	.	PUNCT
iajs-2613	271	1	(	(	PUNCT
iajs-2613	271	2	3	3	X
iajs-2613	271	3	)	)	PUNCT
iajs-2613	271	4			NOUN
iajs-2613	271	5	(	(	PUNCT
iajs-2613	271	6	2	2	X
iajs-2613	271	7	)	)	PUNCT
iajs-2613	271	8	suppose	suppose	VERB
iajs-2613	271	9	that	that	SCONJ
iajs-2613	271	10	c	c	X
iajs-2613	271	11	=	=	PRON
iajs-2613	271	12	it	it	PRON
iajs-2613	271	13	for	for	ADP
iajs-2613	271	14	some	some	DET
iajs-2613	271	15	a	a	DET
iajs-2613	271	16	some	some	DET
iajs-2613	271	17	wapp	wapp	NOUN
iajs-2613	271	18	-	-	PUNCT
iajs-2613	271	19	quasi	quasi	ADJ
iajs-2613	271	20	prime	prime	ADJ
iajs-2613	271	21	ideal	ideal	NOUN
iajs-2613	271	22	of	of	ADP
iajs-2613	271	23	r.	r.	PROPN
iajs-2613	271	24	since	since	SCONJ
iajs-2613	271	25	t	t	PROPN
iajs-2613	271	26	is	be	AUX
iajs-2613	271	27	multiplication	multiplication	NOUN
iajs-2613	271	28	,	,	PUNCT
iajs-2613	271	29	then	then	ADV
iajs-2613	271	30	c=[c	c=[c	PROPN
iajs-2613	271	31	:	:	PUNCT
iajs-2613	271	32	rt]t	rt]t	PROPN
iajs-2613	271	33	=	=	PRON
iajs-2613	271	34	it	it	PRON
iajs-2613	271	35	and	and	CCONJ
iajs-2613	271	36	since	since	SCONJ
iajs-2613	271	37	m	m	PROPN
iajs-2613	271	38	is	be	AUX
iajs-2613	271	39	finitely	finitely	ADV
iajs-2613	271	40	generated	generate	VERB
iajs-2613	271	41	multiplication	multiplication	NOUN
iajs-2613	271	42	,	,	PUNCT
iajs-2613	271	43	then	then	ADV
iajs-2613	271	44	.[c	.[c	PROPN
iajs-2613	271	45	:	:	PUNCT
iajs-2613	271	46	rt]=	rt]=	VERB
iajs-2613	271	47	i+annr(t	i+annr(t	NOUN
iajs-2613	271	48	)	)	PUNCT
iajs-2613	271	49	.	.	PUNCT
iajs-2613	272	1	but	but	CCONJ
iajs-2613	272	2	annr(t)i	annr(t)i	PROPN
iajs-2613	272	3	it	it	PRON
iajs-2613	272	4	follows	follow	VERB
iajs-2613	272	5	that	that	PRON
iajs-2613	272	6	i+annr(t)=i	i+annr(t)=i	ADV
iajs-2613	272	7	.	.	PUNCT
iajs-2613	273	1	thus	thus	ADV
iajs-2613	273	2	[	[	X
iajs-2613	273	3	c	c	X
iajs-2613	273	4	:	:	PUNCT
iajs-2613	273	5	rt]=i	rt]=i	NOUN
iajs-2613	273	6	is	be	AUX
iajs-2613	273	7	a	a	DET
iajs-2613	273	8	wappquasi	wappquasi	NOUN
iajs-2613	273	9	prime	prime	ADJ
iajs-2613	273	10	ideal	ideal	NOUN
iajs-2613	273	11	of	of	ADP
iajs-2613	273	12	r.	r.	PROPN
iajs-2613	273	13	hence	hence	ADV
iajs-2613	274	1	[	[	X
iajs-2613	274	2	c	c	X
iajs-2613	274	3	:	:	PUNCT
iajs-2613	274	4	rt	rt	X
iajs-2613	274	5	]	]	X
iajs-2613	274	6	is	be	AUX
iajs-2613	274	7	wapp	wapp	NOUN
iajs-2613	274	8	-	-	PUNCT
iajs-2613	274	9	quasi	quasi	ADJ
iajs-2613	274	10	prime	prime	ADJ
iajs-2613	274	11	ideal	ideal	NOUN
iajs-2613	274	12	of	of	ADP
iajs-2613	274	13	r.	r.	PROPN
iajs-2613	274	14	the	the	DET
iajs-2613	274	15	following	follow	VERB
iajs-2613	274	16	corollary	corollary	NOUN
iajs-2613	274	17	is	be	AUX
iajs-2613	274	18	a	a	DET
iajs-2613	274	19	direct	direct	ADJ
iajs-2613	274	20	consequence	consequence	NOUN
iajs-2613	274	21	of	of	ADP
iajs-2613	274	22	proposition	proposition	NOUN
iajs-2613	274	23	(	(	PUNCT
iajs-2613	274	24	28	28	NUM
iajs-2613	274	25	)	)	PUNCT
iajs-2613	274	26	corollary(29	corollary(29	NOUN
iajs-2613	274	27	)	)	PUNCT
iajs-2613	274	28	let	let	VERB
iajs-2613	274	29	t	t	NOUN
iajs-2613	274	30	be	be	AUX
iajs-2613	274	31	a	a	DET
iajs-2613	274	32	cyclic	cyclic	ADJ
iajs-2613	274	33	z	z	NOUN
iajs-2613	274	34	-	-	ADJ
iajs-2613	274	35	regular	regular	ADJ
iajs-2613	274	36	r	r	NOUN
iajs-2613	274	37	-	-	PUNCT
iajs-2613	274	38	module	module	NOUN
iajs-2613	274	39	,	,	PUNCT
iajs-2613	274	40	and	and	CCONJ
iajs-2613	274	41	c	c	PROPN
iajs-2613	274	42	be	be	AUX
iajs-2613	274	43	proper	proper	ADJ
iajs-2613	274	44	submodule	submodule	NOUN
iajs-2613	274	45	of	of	ADP
iajs-2613	274	46	t	t	PROPN
iajs-2613	274	47	.	.	PUNCT
iajs-2613	275	1	then	then	ADV
iajs-2613	275	2	the	the	DET
iajs-2613	275	3	following	follow	VERB
iajs-2613	275	4	statements	statement	NOUN
iajs-2613	275	5	are	be	AUX
iajs-2613	275	6	equipollent	equipollent	NOUN
iajs-2613	275	7	:	:	PUNCT
iajs-2613	276	1	1	1	X
iajs-2613	276	2	.	.	X
iajs-2613	276	3	c	c	PROPN
iajs-2613	276	4	is	be	AUX
iajs-2613	276	5	wapp	wapp	NOUN
iajs-2613	276	6	-	-	PUNCT
iajs-2613	276	7	quasi	quasi	ADJ
iajs-2613	276	8	prime	prime	NOUN
iajs-2613	276	9	submodule	submodule	NOUN
iajs-2613	276	10	of	of	ADP
iajs-2613	276	11	t	t	PROPN
iajs-2613	276	12	.	.	PUNCT
iajs-2613	277	1	2	2	X
iajs-2613	277	2	.	.	PUNCT
iajs-2613	278	1	[	[	X
iajs-2613	278	2	c	c	X
iajs-2613	278	3	:	:	PUNCT
iajs-2613	278	4	rt	rt	X
iajs-2613	278	5	]	]	X
iajs-2613	278	6	is	be	AUX
iajs-2613	278	7	wapp	wapp	NOUN
iajs-2613	278	8	-	-	PUNCT
iajs-2613	278	9	quasi	quasi	ADJ
iajs-2613	278	10	prime	prime	ADJ
iajs-2613	278	11	ideal	ideal	NOUN
iajs-2613	278	12	of	of	ADP
iajs-2613	278	13	r	r	NOUN
iajs-2613	278	14	.	.	PUNCT
iajs-2613	279	1	3	3	X
iajs-2613	279	2	.	.	X
iajs-2613	280	1	c	c	X
iajs-2613	280	2	=	=	PRON
iajs-2613	280	3	it	it	PRON
iajs-2613	280	4	for	for	ADP
iajs-2613	280	5	some	some	DET
iajs-2613	280	6	wapp	wapp	NOUN
iajs-2613	280	7	-	-	PUNCT
iajs-2613	280	8	quasi	quasi	ADJ
iajs-2613	280	9	prime	prime	PROPN
iajs-2613	280	10	ideal	ideal	NOUN
iajs-2613	280	11	i	i	PRON
iajs-2613	280	12	of	of	ADP
iajs-2613	280	13	r	r	NOUN
iajs-2613	280	14	with	with	ADP
iajs-2613	280	15	annr(t)i	annr(t)i	PROPN
iajs-2613	280	16	.	.	PUNCT
iajs-2613	281	1	proposition(30	proposition(30	NOUN
iajs-2613	281	2	)	)	PUNCT
iajs-2613	281	3	let	let	VERB
iajs-2613	281	4	t	t	NOUN
iajs-2613	281	5	be	be	AUX
iajs-2613	281	6	a	a	DET
iajs-2613	281	7	finitely	finitely	ADV
iajs-2613	281	8	generated	generate	VERB
iajs-2613	281	9	multiplication	multiplication	NOUN
iajs-2613	281	10	projective	projective	ADJ
iajs-2613	281	11	r	r	NOUN
iajs-2613	281	12	-	-	PUNCT
iajs-2613	281	13	module	module	NOUN
iajs-2613	281	14	,	,	PUNCT
iajs-2613	281	15	and	and	CCONJ
iajs-2613	281	16	c	c	PROPN
iajs-2613	281	17	be	be	AUX
iajs-2613	281	18	a	a	DET
iajs-2613	281	19	proper	proper	ADJ
iajs-2613	281	20	submodule	submodule	NOUN
iajs-2613	281	21	of	of	ADP
iajs-2613	281	22	t	t	PROPN
iajs-2613	281	23	.	.	PUNCT
iajs-2613	282	1	then	then	ADV
iajs-2613	282	2	the	the	DET
iajs-2613	282	3	following	follow	VERB
iajs-2613	282	4	statements	statement	NOUN
iajs-2613	282	5	are	be	AUX
iajs-2613	282	6	equipollent	equipollent	NOUN
iajs-2613	282	7	:	:	PUNCT
iajs-2613	283	1	1	1	X
iajs-2613	283	2	.	.	X
iajs-2613	283	3	c	c	NOUN
iajs-2613	283	4	is	be	AUX
iajs-2613	283	5	a	a	DET
iajs-2613	283	6	wapp	wapp	NOUN
iajs-2613	283	7	-	-	PUNCT
iajs-2613	283	8	quasi	quasi	ADJ
iajs-2613	283	9	prime	prime	NOUN
iajs-2613	283	10	submodule	submodule	NOUN
iajs-2613	283	11	of	of	ADP
iajs-2613	283	12	t	t	PROPN
iajs-2613	283	13	.	.	PUNCT
iajs-2613	284	1	2	2	X
iajs-2613	284	2	.	.	PUNCT
iajs-2613	285	1	[	[	X
iajs-2613	285	2	c	c	X
iajs-2613	285	3	:	:	PUNCT
iajs-2613	285	4	rt	rt	X
iajs-2613	285	5	]	]	X
iajs-2613	285	6	is	be	AUX
iajs-2613	285	7	wapp	wapp	NOUN
iajs-2613	285	8	-	-	PUNCT
iajs-2613	285	9	quasi	quasi	ADJ
iajs-2613	285	10	prime	prime	ADJ
iajs-2613	285	11	ideal	ideal	NOUN
iajs-2613	285	12	of	of	ADP
iajs-2613	285	13	r	r	NOUN
iajs-2613	285	14	.	.	PUNCT
iajs-2613	286	1	3	3	X
iajs-2613	286	2	.	.	X
iajs-2613	287	1	c	c	X
iajs-2613	287	2	=	=	PRON
iajs-2613	287	3	it	it	PRON
iajs-2613	287	4	for	for	ADP
iajs-2613	287	5	some	some	DET
iajs-2613	287	6	wapp	wapp	NOUN
iajs-2613	287	7	-	-	PUNCT
iajs-2613	287	8	quasi	quasi	ADJ
iajs-2613	287	9	prime	prime	PROPN
iajs-2613	287	10	ideal	ideal	NOUN
iajs-2613	287	11	i	i	PRON
iajs-2613	287	12	of	of	ADP
iajs-2613	287	13	r	r	NOUN
iajs-2613	287	14	with	with	ADP
iajs-2613	287	15	annr(t)i	annr(t)i	PROPN
iajs-2613	287	16	.	.	PUNCT
iajs-2613	288	1	proof	proof	NOUN
iajs-2613	288	2	:	:	PUNCT
iajs-2613	288	3	(	(	PUNCT
iajs-2613	288	4	1	1	X
iajs-2613	288	5	)	)	PUNCT
iajs-2613	288	6			NOUN
iajs-2613	288	7	(	(	PUNCT
iajs-2613	288	8	2	2	X
iajs-2613	288	9	)	)	PUNCT
iajs-2613	288	10	follows	follow	VERB
iajs-2613	288	11	by	by	ADP
iajs-2613	288	12	proposition	proposition	NOUN
iajs-2613	288	13	(	(	PUNCT
iajs-2613	288	14	16	16	NUM
iajs-2613	288	15	)	)	PUNCT
iajs-2613	288	16	(	(	PUNCT
iajs-2613	288	17	2	2	X
iajs-2613	288	18	)	)	PUNCT
iajs-2613	288	19			NOUN
iajs-2613	288	20	(	(	PUNCT
iajs-2613	288	21	3	3	X
iajs-2613	288	22	)	)	PUNCT
iajs-2613	288	23	follows	follow	VERB
iajs-2613	288	24	directly	directly	ADV
iajs-2613	288	25	.	.	PUNCT
iajs-2613	289	1	(	(	PUNCT
iajs-2613	289	2	3	3	X
iajs-2613	289	3	)	)	PUNCT
iajs-2613	289	4			NOUN
iajs-2613	289	5	(	(	PUNCT
iajs-2613	289	6	2	2	X
iajs-2613	289	7	)	)	PUNCT
iajs-2613	289	8	follows	follow	VERB
iajs-2613	289	9	as	as	ADP
iajs-2613	289	10	in	in	ADP
iajs-2613	289	11	proposition(28	proposition(28	NOUN
iajs-2613	289	12	)	)	PUNCT
iajs-2613	289	13	.	.	PUNCT
iajs-2613	290	1	as	as	ADP
iajs-2613	290	2	a	a	DET
iajs-2613	290	3	direct	direct	ADJ
iajs-2613	290	4	consequence	consequence	NOUN
iajs-2613	290	5	of	of	ADP
iajs-2613	290	6	proposition	proposition	NOUN
iajs-2613	290	7	(	(	PUNCT
iajs-2613	290	8	30	30	NUM
iajs-2613	290	9	)	)	PUNCT
iajs-2613	290	10	,	,	PUNCT
iajs-2613	290	11	we	we	PRON
iajs-2613	290	12	get	get	VERB
iajs-2613	290	13	the	the	DET
iajs-2613	290	14	following	follow	VERB
iajs-2613	290	15	corollary	corollary	NOUN
iajs-2613	290	16	:	:	PUNCT
iajs-2613	290	17	corollary(31	corollary(31	NOUN
iajs-2613	290	18	)	)	PUNCT
iajs-2613	290	19	let	let	VERB
iajs-2613	290	20	t	t	NOUN
iajs-2613	290	21	be	be	AUX
iajs-2613	290	22	cyclic	cyclic	ADJ
iajs-2613	290	23	projctive	projctive	ADJ
iajs-2613	290	24	r	r	NOUN
iajs-2613	290	25	−	−	NOUN
iajs-2613	290	26	module	module	NOUN
iajs-2613	290	27	,	,	PUNCT
iajs-2613	290	28	and	and	CCONJ
iajs-2613	290	29	c	c	PROPN
iajs-2613	290	30	be	be	AUX
iajs-2613	290	31	proper	proper	ADJ
iajs-2613	290	32	submodule	submodule	NOUN
iajs-2613	290	33	of	of	ADP
iajs-2613	290	34	t	t	PROPN
iajs-2613	290	35	,	,	PUNCT
iajs-2613	290	36	and	and	CCONJ
iajs-2613	290	37	c	c	PROPN
iajs-2613	290	38	be	be	AUX
iajs-2613	290	39	a	a	DET
iajs-2613	290	40	proper	proper	ADJ
iajs-2613	290	41	submodule	submodule	NOUN
iajs-2613	290	42	of	of	ADP
iajs-2613	290	43	t	t	PROPN
iajs-2613	290	44	.	.	PUNCT
iajs-2613	291	1	then	then	ADV
iajs-2613	291	2	the	the	DET
iajs-2613	291	3	following	follow	VERB
iajs-2613	291	4	statements	statement	NOUN
iajs-2613	291	5	are	be	AUX
iajs-2613	291	6	equipollent	equipollent	NOUN
iajs-2613	291	7	:	:	PUNCT
iajs-2613	292	1	1	1	X
iajs-2613	292	2	.	.	X
iajs-2613	292	3	c	c	PROPN
iajs-2613	292	4	is	be	AUX
iajs-2613	292	5	wapp	wapp	NOUN
iajs-2613	292	6	-	-	PUNCT
iajs-2613	292	7	quasi	quasi	ADJ
iajs-2613	292	8	prime	prime	NOUN
iajs-2613	292	9	submodule	submodule	NOUN
iajs-2613	292	10	of	of	ADP
iajs-2613	292	11	t	t	PROPN
iajs-2613	292	12	.	.	PUNCT
iajs-2613	293	1	2	2	X
iajs-2613	293	2	.	.	PUNCT
iajs-2613	294	1	[	[	X
iajs-2613	294	2	c	c	X
iajs-2613	294	3	:	:	PUNCT
iajs-2613	294	4	rt	rt	X
iajs-2613	294	5	]	]	X
iajs-2613	294	6	is	be	AUX
iajs-2613	294	7	wapp	wapp	NOUN
iajs-2613	294	8	-	-	PUNCT
iajs-2613	294	9	quasi	quasi	ADJ
iajs-2613	294	10	prime	prime	ADJ
iajs-2613	294	11	ideal	ideal	NOUN
iajs-2613	294	12	of	of	ADP
iajs-2613	294	13	r	r	NOUN
iajs-2613	294	14	.	.	PUNCT
iajs-2613	295	1	3	3	X
iajs-2613	295	2	.	.	X
iajs-2613	296	1	c	c	X
iajs-2613	296	2	=	=	PRON
iajs-2613	296	3	it	it	PRON
iajs-2613	296	4	for	for	ADP
iajs-2613	296	5	some	some	DET
iajs-2613	296	6	wapp	wapp	NOUN
iajs-2613	296	7	-	-	PUNCT
iajs-2613	296	8	quasi	quasi	ADJ
iajs-2613	296	9	prime	prime	PROPN
iajs-2613	296	10	ideal	ideal	NOUN
iajs-2613	296	11	i	i	PRON
iajs-2613	296	12	of	of	ADP
iajs-2613	296	13	r	r	NOUN
iajs-2613	296	14	with	with	ADP
iajs-2613	296	15	annr(t)i	annr(t)i	PROPN
iajs-2613	296	16	.	.	PUNCT
iajs-2613	297	1	it	it	PRON
iajs-2613	297	2	is	be	AUX
iajs-2613	297	3	well	well	ADV
iajs-2613	297	4	-	-	PUNCT
iajs-2613	297	5	known	know	VERB
iajs-2613	297	6	that	that	SCONJ
iajs-2613	297	7	if	if	SCONJ
iajs-2613	297	8	t	t	PROPN
iajs-2613	297	9	is	be	AUX
iajs-2613	297	10	faithful	faithful	ADJ
iajs-2613	297	11	multiplicationr	multiplicationr	ADJ
iajs-2613	297	12	−	−	NOUN
iajs-2613	297	13	module	module	NOUN
iajs-2613	297	14	,	,	PUNCT
iajs-2613	297	15	then	then	ADV
iajs-2613	297	16	soc(t)=soc(r)t	soc(t)=soc(r)t	PROPN
iajs-2613	298	1	[	[	X
iajs-2613	298	2	7,coro.(2.14)(1	7,coro.(2.14)(1	NUM
iajs-2613	298	3	)	)	PUNCT
iajs-2613	298	4	]	]	PUNCT
iajs-2613	298	5	.	.	PUNCT
iajs-2613	299	1	proposition(32	proposition(32	NOUN
iajs-2613	299	2	)	)	PUNCT
iajs-2613	299	3	let	let	VERB
iajs-2613	299	4	t	t	NOUN
iajs-2613	299	5	be	be	AUX
iajs-2613	299	6	a	a	DET
iajs-2613	299	7	faithful	faithful	ADJ
iajs-2613	299	8	multiplication	multiplication	NOUN
iajs-2613	299	9	r	r	NOUN
iajs-2613	299	10	−	−	NOUN
iajs-2613	299	11	module	module	NOUN
iajs-2613	299	12	and	and	CCONJ
iajs-2613	299	13	c	c	AUX
iajs-2613	299	14	be	be	AUX
iajs-2613	299	15	a	a	DET
iajs-2613	299	16	proper	proper	ADJ
iajs-2613	299	17	submodule	submodule	NOUN
iajs-2613	299	18	of	of	ADP
iajs-2613	299	19	t	t	PROPN
iajs-2613	299	20	.	.	PUNCT
iajs-2613	300	1	then	then	ADV
iajs-2613	300	2	c	c	PROPN
iajs-2613	300	3	is	be	AUX
iajs-2613	300	4	a	a	DET
iajs-2613	300	5	wapp	wapp	NOUN
iajs-2613	300	6	-	-	PUNCT
iajs-2613	300	7	quasi	quasi	ADJ
iajs-2613	300	8	prime	prime	PROPN
iajs-2613	300	9	submodule	submodule	NOUN
iajs-2613	300	10	of	of	ADP
iajs-2613	300	11	t	t	PROPN
iajs-2613	300	12	iff	iff	PROPN
iajs-2613	301	1	[	[	X
iajs-2613	301	2	c	c	X
iajs-2613	301	3	:	:	PUNCT
iajs-2613	301	4	r	r	NOUN
iajs-2613	301	5	t	t	PROPN
iajs-2613	301	6	]	]	PUNCT
iajs-2613	301	7	is	be	AUX
iajs-2613	301	8	a	a	DET
iajs-2613	301	9	wappquasi	wappquasi	NOUN
iajs-2613	301	10	prime	prime	ADJ
iajs-2613	301	11	ideal	ideal	NOUN
iajs-2613	301	12	of	of	ADP
iajs-2613	301	13	r.	r.	PROPN
iajs-2613	301	14	67	67	NUM
iajs-2613	301	15	ibn	ibn	PROPN
iajs-2613	301	16	al	al	PROPN
iajs-2613	301	17	-	-	PUNCT
iajs-2613	301	18	haitham	haitham	PROPN
iajs-2613	301	19	jour	jour	X
iajs-2613	301	20	.	.	PROPN
iajs-2613	301	21	for	for	ADP
iajs-2613	301	22	pure	pure	ADJ
iajs-2613	301	23	&	&	CCONJ
iajs-2613	301	24	appl	appl	PROPN
iajs-2613	301	25	.	.	PUNCT
iajs-2613	302	1	sci	sci	PROPN
iajs-2613	302	2	.	.	PROPN
iajs-2613	303	1	34	34	NUM
iajs-2613	303	2	(	(	PUNCT
iajs-2613	303	3	1	1	NUM
iajs-2613	303	4	)	)	PUNCT
iajs-2613	303	5	2021	2021	NUM
iajs-2613	303	6	proof	proof	NOUN
iajs-2613	303	7	:	:	PUNCT
iajs-2613	303	8	(	(	PUNCT
iajs-2613	303	9	)let	)let	PUNCT
iajs-2613	303	10	0≠ijk[c	0≠ijk[c	PROPN
iajs-2613	303	11	:	:	PUNCT
iajs-2613	303	12	r	r	NOUN
iajs-2613	303	13	t	t	PROPN
iajs-2613	303	14	]	]	PUNCT
iajs-2613	303	15	,	,	PUNCT
iajs-2613	303	16	where	where	SCONJ
iajs-2613	303	17	i	i	PRON
iajs-2613	303	18	,	,	PUNCT
iajs-2613	303	19	j	j	PROPN
iajs-2613	303	20	and	and	CCONJ
iajs-2613	303	21	k	k	PROPN
iajs-2613	303	22	are	be	AUX
iajs-2613	303	23	ideals	ideal	NOUN
iajs-2613	303	24	of	of	ADP
iajs-2613	303	25	r	r	NOUN
iajs-2613	303	26	.then	.then	X
iajs-2613	303	27	0≠	0≠	NUM
iajs-2613	303	28	ij(kt)c	ij(kt)c	INTJ
iajs-2613	303	29	.	.	PUNCT
iajs-2613	304	1	since	since	SCONJ
iajs-2613	304	2	c	c	PROPN
iajs-2613	304	3	is	be	AUX
iajs-2613	304	4	wappquasi	wappquasi	PROPN
iajs-2613	304	5	prime	prime	ADJ
iajs-2613	304	6	submodule	submodule	NOUN
iajs-2613	304	7	of	of	ADP
iajs-2613	304	8	t	t	PROPN
iajs-2613	304	9	,	,	PUNCT
iajs-2613	304	10	then	then	ADV
iajs-2613	304	11	by	by	ADP
iajs-2613	304	12	proposition(4	proposition(4	PROPN
iajs-2613	304	13	)	)	PUNCT
iajs-2613	304	14	either	either	CCONJ
iajs-2613	304	15	j(kt)c+soc(t	j(kt)c+soc(t	NOUN
iajs-2613	304	16	)	)	PUNCT
iajs-2613	304	17	or	or	CCONJ
iajs-2613	304	18	j(kt)c+soc(t).but	j(kt)c+soc(t).but	PROPN
iajs-2613	304	19	t	t	PROPN
iajs-2613	304	20	is	be	AUX
iajs-2613	304	21	a	a	DET
iajs-2613	304	22	faithful	faithful	ADJ
iajs-2613	304	23	multiplication	multiplication	NOUN
iajs-2613	304	24	,	,	PUNCT
iajs-2613	304	25	it	it	PRON
iajs-2613	304	26	follows	follow	VERB
iajs-2613	304	27	that	that	DET
iajs-2613	304	28	c=[c	c=[c	NOUN
iajs-2613	304	29	:	:	PUNCT
iajs-2613	304	30	r	r	NOUN
iajs-2613	304	31	t]t	t]t	NOUN
iajs-2613	304	32	and	and	CCONJ
iajs-2613	304	33	soc(t)=soc(r)t	soc(t)=soc(r)t	ADJ
iajs-2613	304	34	.	.	PUNCT
iajs-2613	305	1	thus	thus	ADV
iajs-2613	305	2	either	either	DET
iajs-2613	305	3	i(kt)[c	i(kt)[c	NOUN
iajs-2613	305	4	:	:	PUNCT
iajs-2613	305	5	rt]t	rt]t	PROPN
iajs-2613	305	6	+	+	PROPN
iajs-2613	305	7	soc(r)t	soc(r)t	ADJ
iajs-2613	305	8	or	or	CCONJ
iajs-2613	305	9	j(kt)[c	j(kt)[c	PROPN
iajs-2613	305	10	:	:	PUNCT
iajs-2613	305	11	rt]t	rt]t	PROPN
iajs-2613	305	12	+	+	NOUN
iajs-2613	305	13	soc(r)t	soc(r)t	ADJ
iajs-2613	305	14	.	.	PUNCT
iajs-2613	306	1	hence	hence	ADV
iajs-2613	306	2	either	either	CCONJ
iajs-2613	306	3	i	i	PROPN
iajs-2613	306	4	k[c	k[c	PROPN
iajs-2613	306	5	:	:	PUNCT
iajs-2613	306	6	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	306	7	)	)	PUNCT
iajs-2613	306	8	or	or	CCONJ
iajs-2613	306	9	jk[c	jk[c	NOUN
iajs-2613	306	10	:	:	PUNCT
iajs-2613	306	11	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	306	12	)	)	PUNCT
iajs-2613	306	13	.	.	PUNCT
iajs-2613	307	1	thus	thus	ADV
iajs-2613	307	2	by	by	ADP
iajs-2613	307	3	proposition(4	proposition(4	PROPN
iajs-2613	307	4	)	)	PUNCT
iajs-2613	308	1	[	[	X
iajs-2613	308	2	c	c	X
iajs-2613	308	3	:	:	PUNCT
iajs-2613	308	4	rt	rt	X
iajs-2613	308	5	]	]	X
iajs-2613	308	6	is	be	AUX
iajs-2613	308	7	wapp	wapp	NOUN
iajs-2613	308	8	-	-	PUNCT
iajs-2613	308	9	quasi	quasi	ADJ
iajs-2613	308	10	prime	prime	ADJ
iajs-2613	308	11	ideal	ideal	NOUN
iajs-2613	308	12	of	of	ADP
iajs-2613	308	13	r.	r.	PROPN
iajs-2613	308	14	(	(	PUNCT
iajs-2613	309	1	)let	)let	PROPN
iajs-2613	309	2	t	t	NOUN
iajs-2613	309	3	0≠abb	0≠abb	NOUN
iajs-2613	310	1	c	c	NOUN
iajs-2613	310	2	,	,	PUNCT
iajs-2613	310	3	for	for	ADP
iajs-2613	310	4	a	a	DET
iajs-2613	310	5	,	,	PUNCT
iajs-2613	310	6	b	b	NOUN
iajs-2613	310	7	ϵr	ϵr	NOUN
iajs-2613	310	8	,	,	PUNCT
iajs-2613	310	9	and	and	CCONJ
iajs-2613	310	10	b	b	NOUN
iajs-2613	310	11	is	be	AUX
iajs-2613	310	12	submodule	submodule	NOUN
iajs-2613	310	13	of	of	ADP
iajs-2613	310	14	t.	t.	PROPN
iajs-2613	310	15	since	since	SCONJ
iajs-2613	310	16	t	t	PROPN
iajs-2613	310	17	is	be	AUX
iajs-2613	310	18	multiplication	multiplication	NOUN
iajs-2613	310	19	,	,	PUNCT
iajs-2613	310	20	then	then	ADV
iajs-2613	310	21	b	b	X
iajs-2613	310	22	=	=	PROPN
iajs-2613	310	23	jt	jt	PROPN
iajs-2613	310	24	,	,	PUNCT
iajs-2613	310	25	for	for	ADP
iajs-2613	310	26	some	some	DET
iajs-2613	310	27	ideal	ideal	ADJ
iajs-2613	310	28	j	j	PROPN
iajs-2613	310	29	of	of	ADP
iajs-2613	310	30	r	r	NOUN
iajs-2613	310	31	.	.	PUNCT
iajs-2613	311	1	thus	thus	ADV
iajs-2613	311	2	0≠abjtc	0≠abjtc	NUM
iajs-2613	311	3	,	,	PUNCT
iajs-2613	311	4	it	it	PRON
iajs-2613	311	5	follows	follow	VERB
iajs-2613	311	6	that	that	SCONJ
iajs-2613	311	7	0≠abj[c	0≠abj[c	NOUN
iajs-2613	311	8	:	:	PUNCT
iajs-2613	311	9	r	r	NOUN
iajs-2613	311	10	t	t	PROPN
iajs-2613	311	11	]	]	PUNCT
iajs-2613	311	12	.	.	PUNCT
iajs-2613	312	1	but	but	CCONJ
iajs-2613	312	2	[	[	X
iajs-2613	312	3	c	c	X
iajs-2613	312	4	:	:	PUNCT
iajs-2613	312	5	rt	rt	X
iajs-2613	312	6	]	]	X
iajs-2613	312	7	is	be	AUX
iajs-2613	312	8	wapp	wapp	NOUN
iajs-2613	312	9	-	-	PUNCT
iajs-2613	312	10	quasi	quasi	ADJ
iajs-2613	312	11	prime	prime	ADJ
iajs-2613	312	12	ideal	ideal	NOUN
iajs-2613	312	13	of	of	ADP
iajs-2613	312	14	r	r	NOUN
iajs-2613	312	15	,	,	PUNCT
iajs-2613	312	16	then	then	ADV
iajs-2613	312	17	by	by	ADP
iajs-2613	312	18	proposition(3	proposition(3	PROPN
iajs-2613	312	19	)	)	PUNCT
iajs-2613	312	20	either	either	CCONJ
iajs-2613	312	21	aj[c	aj[c	PROPN
iajs-2613	312	22	:	:	PUNCT
iajs-2613	312	23	rt]+soc(r	rt]+soc(r	ADJ
iajs-2613	312	24	)	)	PUNCT
iajs-2613	312	25	or	or	CCONJ
iajs-2613	312	26	bj[c	bj[c	PROPN
iajs-2613	312	27	:	:	PUNCT
iajs-2613	312	28	rt]+soc(r	rt]+soc(r	PROPN
iajs-2613	312	29	)	)	PUNCT
iajs-2613	312	30	,	,	PUNCT
iajs-2613	312	31	it	it	PRON
iajs-2613	312	32	follows	follow	VERB
iajs-2613	312	33	that	that	SCONJ
iajs-2613	312	34	either	either	CCONJ
iajs-2613	312	35	ajt[c	ajt[c	NOUN
iajs-2613	312	36	:	:	PUNCT
iajs-2613	312	37	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	312	38	or	or	CCONJ
iajs-2613	312	39	bjt[c	bjt[c	NUM
iajs-2613	312	40	:	:	PUNCT
iajs-2613	312	41	rt]t+soc(r)t	rt]t+soc(r)t	ADJ
iajs-2613	312	42	.	.	PUNCT
iajs-2613	313	1	but	but	CCONJ
iajs-2613	313	2	t	t	PROPN
iajs-2613	313	3	is	be	AUX
iajs-2613	313	4	a	a	DET
iajs-2613	313	5	faithful	faithful	ADJ
iajs-2613	313	6	multiplication	multiplication	NOUN
iajs-2613	313	7	r	r	NOUN
iajs-2613	313	8	-	-	PUNCT
iajs-2613	313	9	module	module	NOUN
iajs-2613	313	10	then	then	ADV
iajs-2613	313	11	either	either	CCONJ
iajs-2613	313	12	abc+soc(t	abc+soc(t	PROPN
iajs-2613	313	13	)	)	PUNCT
iajs-2613	313	14	or	or	CCONJ
iajs-2613	313	15	bb	bb	NOUN
iajs-2613	313	16	c+soc(t).thus	c+soc(t).thu	NOUN
iajs-2613	313	17	by	by	ADP
iajs-2613	313	18	proposition	proposition	NOUN
iajs-2613	313	19	(	(	PUNCT
iajs-2613	313	20	3	3	X
iajs-2613	313	21	)	)	PUNCT
iajs-2613	313	22	c	c	NOUN
iajs-2613	313	23	is	be	AUX
iajs-2613	313	24	a	a	DET
iajs-2613	313	25	wapp	wapp	NOUN
iajs-2613	313	26	-	-	PUNCT
iajs-2613	313	27	quasi	quasi	ADJ
iajs-2613	313	28	prime	prime	ADJ
iajs-2613	313	29	submodule	submodule	NOUN
iajs-2613	313	30	of	of	ADP
iajs-2613	313	31	t.	t.	PROPN
iajs-2613	313	32	the	the	DET
iajs-2613	313	33	following	follow	VERB
iajs-2613	313	34	corollary	corollary	NOUN
iajs-2613	313	35	is	be	AUX
iajs-2613	313	36	a	a	DET
iajs-2613	313	37	direct	direct	ADJ
iajs-2613	313	38	consequence	consequence	NOUN
iajs-2613	313	39	of	of	ADP
iajs-2613	313	40	proposition(32	proposition(32	NOUN
iajs-2613	313	41	)	)	PUNCT
iajs-2613	313	42	corollary(33	corollary(33	NOUN
iajs-2613	313	43	)	)	PUNCT
iajs-2613	313	44	let	let	VERB
iajs-2613	313	45	t	t	NOUN
iajs-2613	313	46	be	be	AUX
iajs-2613	313	47	a	a	DET
iajs-2613	313	48	faithful	faithful	ADJ
iajs-2613	313	49	cyclic	cyclic	ADJ
iajs-2613	313	50	r	r	NOUN
iajs-2613	313	51	−	−	NOUN
iajs-2613	313	52	module	module	NOUN
iajs-2613	313	53	and	and	CCONJ
iajs-2613	313	54	c	c	AUX
iajs-2613	313	55	be	be	AUX
iajs-2613	313	56	a	a	DET
iajs-2613	313	57	proper	proper	ADJ
iajs-2613	313	58	submodule	submodule	NOUN
iajs-2613	313	59	of	of	ADP
iajs-2613	313	60	t	t	PROPN
iajs-2613	313	61	.	.	PUNCT
iajs-2613	314	1	then	then	ADV
iajs-2613	314	2	c	c	PROPN
iajs-2613	314	3	is	be	AUX
iajs-2613	314	4	wapp	wapp	NOUN
iajs-2613	314	5	-	-	PUNCT
iajs-2613	314	6	quasi	quasi	ADJ
iajs-2613	314	7	prime	prime	PROPN
iajs-2613	314	8	submodule	submodule	NOUN
iajs-2613	314	9	of	of	ADP
iajs-2613	314	10	t	t	PROPN
iajs-2613	314	11	if	if	SCONJ
iajs-2613	315	1	and	and	CCONJ
iajs-2613	315	2	only	only	ADV
iajs-2613	315	3	if	if	SCONJ
iajs-2613	315	4	[	[	X
iajs-2613	315	5	c	c	X
iajs-2613	315	6	:	:	PUNCT
iajs-2613	315	7	r	r	NOUN
iajs-2613	315	8	t	t	PROPN
iajs-2613	315	9	]	]	PUNCT
iajs-2613	315	10	is	be	AUX
iajs-2613	315	11	a	a	DET
iajs-2613	315	12	wappquasi	wappquasi	NOUN
iajs-2613	315	13	prime	prime	ADJ
iajs-2613	315	14	ideal	ideal	NOUN
iajs-2613	315	15	of	of	ADP
iajs-2613	315	16	r.	r.	PROPN
iajs-2613	315	17	3.conclusion	3.conclusion	NUM
iajs-2613	315	18	in	in	ADP
iajs-2613	315	19	this	this	DET
iajs-2613	315	20	proper	proper	ADJ
iajs-2613	315	21	,	,	PUNCT
iajs-2613	315	22	we	we	PRON
iajs-2613	315	23	introduced	introduce	VERB
iajs-2613	315	24	and	and	CCONJ
iajs-2613	315	25	studied	study	VERB
iajs-2613	315	26	the	the	DET
iajs-2613	315	27	concept	concept	NOUN
iajs-2613	315	28	wapp	wapp	NOUN
iajs-2613	315	29	-	-	PUNCT
iajs-2613	315	30	quasi	quasi	ADJ
iajs-2613	315	31	prime	prime	NOUN
iajs-2613	315	32	submodule	submodule	NOUN
iajs-2613	315	33	,	,	PUNCT
iajs-2613	315	34	and	and	CCONJ
iajs-2613	315	35	we	we	PRON
iajs-2613	315	36	established	establish	VERB
iajs-2613	315	37	several	several	ADJ
iajs-2613	315	38	examples	example	NOUN
iajs-2613	315	39	,	,	PUNCT
iajs-2613	315	40	characterizations	characterization	NOUN
iajs-2613	315	41	and	and	CCONJ
iajs-2613	315	42	basic	basic	ADJ
iajs-2613	315	43	properties	property	NOUN
iajs-2613	315	44	of	of	ADP
iajs-2613	315	45	this	this	DET
iajs-2613	315	46	concept	concept	NOUN
iajs-2613	315	47	.	.	PUNCT
iajs-2613	316	1	wapp	wapp	NOUN
iajs-2613	316	2	-	-	PUNCT
iajs-2613	316	3	quasi	quasi	ADJ
iajs-2613	316	4	prime	prime	PROPN
iajs-2613	316	5	submodule	submodule	PROPN
iajs-2613	316	6	is	be	AUX
iajs-2613	316	7	generalization	generalization	NOUN
iajs-2613	316	8	of	of	ADP
iajs-2613	316	9	a	a	DET
iajs-2613	316	10	weakly	weakly	ADJ
iajs-2613	316	11	quasi	quasi	ADJ
iajs-2613	316	12	prime	prime	NOUN
iajs-2613	316	13	submodule	submodule	NOUN
iajs-2613	316	14	so	so	SCONJ
iajs-2613	316	15	we	we	PRON
iajs-2613	316	16	give	give	VERB
iajs-2613	316	17	example	example	NOUN
iajs-2613	316	18	for	for	ADP
iajs-2613	316	19	converse	converse	NOUN
iajs-2613	316	20	.	.	PUNCT
iajs-2613	317	1	among	among	ADP
iajs-2613	317	2	c	c	PROPN
iajs-2613	317	3	,	,	PUNCT
iajs-2613	317	4	the	the	DET
iajs-2613	317	5	main	main	ADJ
iajs-2613	317	6	results	result	NOUN
iajs-2613	317	7	of	of	ADP
iajs-2613	317	8	this	this	DET
iajs-2613	317	9	paper	paper	NOUN
iajs-2613	317	10	are	be	AUX
iajs-2613	317	11	the	the	DET
iajs-2613	317	12	following	following	NOUN
iajs-2613	317	13	:	:	PUNCT
iajs-2613	317	14	1	1	X
iajs-2613	317	15	.	.	X
iajs-2613	317	16	proper	proper	ADJ
iajs-2613	317	17	submoduel	submoduel	ADJ
iajs-2613	317	18	c	c	NOUN
iajs-2613	317	19	of	of	ADP
iajs-2613	317	20	r	r	NOUN
iajs-2613	317	21	-	-	PUNCT
iajs-2613	317	22	module	module	NOUN
iajs-2613	317	23	t	t	NOUN
iajs-2613	317	24	is	be	AUX
iajs-2613	317	25	wapp	wapp	NOUN
iajs-2613	317	26	-	-	PUNCT
iajs-2613	317	27	quasi	quasi	ADJ
iajs-2613	317	28	prime	prime	PROPN
iajs-2613	317	29	submodule	submodule	NOUN
iajs-2613	317	30	of	of	ADP
iajs-2613	317	31	t	t	PROPN
iajs-2613	317	32	iff	iff	PROPN
iajs-2613	317	33	whenever	whenever	SCONJ
iajs-2613	317	34	(	(	PUNCT
iajs-2613	317	35	0)≠rsbc	0)≠rsbc	ADV
iajs-2613	317	36	,	,	PUNCT
iajs-2613	317	37	for	for	ADP
iajs-2613	317	38	r	r	NOUN
iajs-2613	317	39	,	,	PUNCT
iajs-2613	317	40	sϵr	sϵr	NOUN
iajs-2613	317	41	,	,	PUNCT
iajs-2613	317	42	b	b	PROPN
iajs-2613	317	43	is	be	AUX
iajs-2613	317	44	a	a	DET
iajs-2613	317	45	submodule	submodule	NOUN
iajs-2613	317	46	of	of	ADP
iajs-2613	317	47	t	t	PROPN
iajs-2613	317	48	,	,	PUNCT
iajs-2613	317	49	implies	imply	VERB
iajs-2613	317	50	that	that	SCONJ
iajs-2613	317	51	either	either	CCONJ
iajs-2613	317	52	rbc+soc(t	rbc+soc(t	PROPN
iajs-2613	317	53	)	)	PUNCT
iajs-2613	317	54	or	or	CCONJ
iajs-2613	317	55	sbc+soc(t	sbc+soc(t	PROPN
iajs-2613	317	56	)	)	PUNCT
iajs-2613	317	57	2	2	NUM
iajs-2613	317	58	.	.	PUNCT
iajs-2613	317	59	proper	proper	ADJ
iajs-2613	317	60	submodule	submodule	PROPN
iajs-2613	317	61	c	c	PROPN
iajs-2613	317	62	of	of	ADP
iajs-2613	317	63	r	r	NOUN
iajs-2613	317	64	-	-	PUNCT
iajs-2613	317	65	module	module	NOUN
iajs-2613	317	66	t	t	NOUN
iajs-2613	317	67	is	be	AUX
iajs-2613	317	68	wapp	wapp	NOUN
iajs-2613	317	69	-	-	PUNCT
iajs-2613	317	70	quasi	quasi	ADJ
iajs-2613	317	71	prime	prime	PROPN
iajs-2613	317	72	submodule	submodule	NOUN
iajs-2613	317	73	of	of	ADP
iajs-2613	317	74	t	t	PROPN
iajs-2613	317	75	iff	iff	PROPN
iajs-2613	317	76	whenever	whenever	SCONJ
iajs-2613	317	77	(	(	PUNCT
iajs-2613	317	78	0)≠ijbc	0)≠ijbc	NUM
iajs-2613	317	79	,	,	PUNCT
iajs-2613	317	80	for	for	ADP
iajs-2613	317	81	i	i	PRON
iajs-2613	317	82	,	,	PUNCT
iajs-2613	317	83	j	j	PROPN
iajs-2613	317	84	are	be	AUX
iajs-2613	317	85	ideals	ideal	NOUN
iajs-2613	317	86	of	of	ADP
iajs-2613	317	87	r	r	NOUN
iajs-2613	317	88	,	,	PUNCT
iajs-2613	317	89	and	and	CCONJ
iajs-2613	317	90	b	b	NOUN
iajs-2613	317	91	is	be	AUX
iajs-2613	317	92	submodule	submodule	NOUN
iajs-2613	317	93	of	of	ADP
iajs-2613	317	94	t	t	PROPN
iajs-2613	317	95	,	,	PUNCT
iajs-2613	317	96	implies	imply	VERB
iajs-2613	317	97	that	that	SCONJ
iajs-2613	317	98	either	either	CCONJ
iajs-2613	317	99	ibc	ibc	PROPN
iajs-2613	317	100	+	+	PROPN
iajs-2613	317	101	soc(t	soc(t	PROPN
iajs-2613	317	102	)	)	PUNCT
iajs-2613	317	103	.	.	PUNCT
iajs-2613	318	1	or	or	CCONJ
iajs-2613	318	2	jbc	jbc	NOUN
iajs-2613	318	3	+	+	CCONJ
iajs-2613	318	4	soc(t	soc(t	PROPN
iajs-2613	318	5	)	)	PUNCT
iajs-2613	318	6	.	.	PUNCT
iajs-2613	319	1	3	3	X
iajs-2613	319	2	.	.	X
iajs-2613	319	3	proper	proper	ADJ
iajs-2613	319	4	submodule	submodule	PROPN
iajs-2613	319	5	c	c	PROPN
iajs-2613	319	6	of	of	ADP
iajs-2613	319	7	r	r	NOUN
iajs-2613	319	8	-	-	PUNCT
iajs-2613	319	9	module	module	NOUN
iajs-2613	319	10	t	t	NOUN
iajs-2613	319	11	is	be	AUX
iajs-2613	319	12	wapp	wapp	NOUN
iajs-2613	319	13	-	-	PUNCT
iajs-2613	319	14	quasi	quasi	ADJ
iajs-2613	319	15	prime	prime	PROPN
iajs-2613	319	16	submodule	submodule	NOUN
iajs-2613	319	17	of	of	ADP
iajs-2613	319	18	t	t	PROPN
iajs-2613	319	19	iff	iff	PROPN
iajs-2613	319	20	for	for	ADP
iajs-2613	319	21	all	all	DET
iajs-2613	319	22	r	r	NOUN
iajs-2613	319	23	,	,	PUNCT
iajs-2613	319	24	sϵr	sϵr	NOUN
iajs-2613	319	25	,	,	PUNCT
iajs-2613	320	1	[	[	X
iajs-2613	320	2	c	c	X
iajs-2613	320	3	:	:	PUNCT
iajs-2613	320	4	t	t	NOUN
iajs-2613	320	5	rs][0	rs][0	PROPN
iajs-2613	320	6	:	:	PUNCT
iajs-2613	320	7	t	t	PROPN
iajs-2613	320	8	rs]∪[c	rs]∪[c	PROPN
iajs-2613	320	9	:	:	PUNCT
iajs-2613	320	10	dt	dt	PROPN
iajs-2613	320	11	r]∪[c	r]∪[c	ADJ
iajs-2613	320	12	:	:	PUNCT
iajs-2613	320	13	t	t	PROPN
iajs-2613	320	14	s	s	PROPN
iajs-2613	320	15	]	]	X
iajs-2613	320	16	68	68	NUM
iajs-2613	320	17	ibn	ibn	PROPN
iajs-2613	320	18	al	al	PROPN
iajs-2613	320	19	-	-	PUNCT
iajs-2613	320	20	haitham	haitham	PROPN
iajs-2613	320	21	jour	jour	X
iajs-2613	320	22	.	.	PROPN
iajs-2613	321	1	for	for	ADP
iajs-2613	321	2	pure	pure	ADJ
iajs-2613	321	3	&	&	CCONJ
iajs-2613	321	4	appl	appl	PROPN
iajs-2613	321	5	.	.	PUNCT
iajs-2613	322	1	sci	sci	PROPN
iajs-2613	322	2	.	.	PROPN
iajs-2613	323	1	34	34	NUM
iajs-2613	323	2	(	(	PUNCT
iajs-2613	323	3	1	1	NUM
iajs-2613	323	4	)	)	PUNCT
iajs-2613	323	5	2021	2021	NUM
iajs-2613	323	6	4	4	NUM
iajs-2613	323	7	.	.	PUNCT
iajs-2613	324	1	proper	proper	ADJ
iajs-2613	324	2	submoduel	submoduel	ADJ
iajs-2613	324	3	c	c	NOUN
iajs-2613	324	4	of	of	ADP
iajs-2613	324	5	r	r	NOUN
iajs-2613	324	6	-	-	PUNCT
iajs-2613	324	7	module	module	NOUN
iajs-2613	324	8	t	t	NOUN
iajs-2613	324	9	is	be	AUX
iajs-2613	324	10	wapp	wapp	NOUN
iajs-2613	324	11	-	-	PUNCT
iajs-2613	324	12	quasi	quasi	ADJ
iajs-2613	324	13	prime	prime	PROPN
iajs-2613	324	14	submodule	submodule	NOUN
iajs-2613	324	15	of	of	ADP
iajs-2613	324	16	t	t	PROPN
iajs-2613	324	17	iff	iff	PROPN
iajs-2613	324	18	for	for	ADP
iajs-2613	324	19	all	all	DET
iajs-2613	324	20	rϵr	rϵr	NOUN
iajs-2613	324	21	,	,	PUNCT
iajs-2613	324	22	tϵt	tϵt	NOUN
iajs-2613	324	23	with	with	ADP
iajs-2613	324	24	rtc	rtc	PROPN
iajs-2613	324	25	+	+	CCONJ
iajs-2613	324	26	soc(t	soc(t	PROPN
iajs-2613	324	27	)	)	PUNCT
iajs-2613	324	28	,	,	PUNCT
iajs-2613	325	1	[	[	X
iajs-2613	325	2	c	c	X
iajs-2613	325	3	:	:	PUNCT
iajs-2613	325	4	t	t	PROPN
iajs-2613	325	5	rt][0	rt][0	PROPN
iajs-2613	325	6	:	:	PUNCT
iajs-2613	325	7	t	t	PROPN
iajs-2613	325	8	rt]∪	rt]∪	NOUN
iajs-2613	325	9	[	[	PUNCT
iajs-2613	325	10	c	c	NOUN
iajs-2613	325	11	+	+	CCONJ
iajs-2613	325	12	soc(t):t	soc(t):t	PROPN
iajs-2613	325	13	t	t	PROPN
iajs-2613	325	14	]	]	PUNCT
iajs-2613	325	15	.	.	PUNCT
iajs-2613	326	1	5	5	NUM
iajs-2613	326	2	.	.	X
iajs-2613	326	3	proper	proper	ADJ
iajs-2613	326	4	submodule	submodule	PROPN
iajs-2613	326	5	c	c	PROPN
iajs-2613	326	6	of	of	ADP
iajs-2613	326	7	multiplication	multiplication	NOUN
iajs-2613	326	8	r	r	NOUN
iajs-2613	326	9	-	-	PUNCT
iajs-2613	326	10	module	module	NOUN
iajs-2613	326	11	t	t	NOUN
iajs-2613	326	12	is	be	AUX
iajs-2613	326	13	wapp	wapp	NOUN
iajs-2613	326	14	-	-	PUNCT
iajs-2613	326	15	quasi	quasi	ADJ
iajs-2613	326	16	prime	prime	PROPN
iajs-2613	326	17	submodule	submodule	NOUN
iajs-2613	326	18	of	of	ADP
iajs-2613	326	19	t	t	PROPN
iajs-2613	326	20	iff	iff	PROPN
iajs-2613	326	21	whenever	whenever	SCONJ
iajs-2613	326	22	(	(	PUNCT
iajs-2613	326	23	0)≠k1k2tc	0)≠k1k2tc	NUM
iajs-2613	326	24	,	,	PUNCT
iajs-2613	326	25	for	for	ADP
iajs-2613	326	26	some	some	DET
iajs-2613	326	27	submodules	submodule	NOUN
iajs-2613	326	28	k1,k2	k1,k2	PROPN
iajs-2613	326	29	of	of	ADP
iajs-2613	326	30	t	t	PROPN
iajs-2613	326	31	and	and	CCONJ
iajs-2613	326	32	tϵt	tϵt	NOUN
iajs-2613	326	33	,	,	PUNCT
iajs-2613	326	34	implies	imply	VERB
iajs-2613	326	35	that	that	SCONJ
iajs-2613	326	36	either	either	CCONJ
iajs-2613	326	37	k1tc+soc(t	k1tc+soc(t	PROPN
iajs-2613	326	38	)	)	PUNCT
iajs-2613	326	39	or	or	CCONJ
iajs-2613	326	40	k2tc+soc(t	k2tc+soc(t	PROPN
iajs-2613	326	41	)	)	PUNCT
iajs-2613	326	42	6	6	NUM
iajs-2613	326	43	.	.	PUNCT
iajs-2613	326	44	proper	proper	ADJ
iajs-2613	326	45	submodule	submodule	PROPN
iajs-2613	326	46	c	c	PROPN
iajs-2613	326	47	of	of	ADP
iajs-2613	326	48	z	z	NOUN
iajs-2613	326	49	-	-	ADJ
iajs-2613	326	50	regular	regular	ADJ
iajs-2613	326	51	multiplication	multiplication	NOUN
iajs-2613	326	52	r	r	NOUN
iajs-2613	326	53	-	-	PUNCT
iajs-2613	326	54	module	module	NOUN
iajs-2613	326	55	t	t	NOUN
iajs-2613	326	56	is	be	AUX
iajs-2613	326	57	a	a	DET
iajs-2613	326	58	wapp	wapp	NOUN
iajs-2613	326	59	-	-	PUNCT
iajs-2613	326	60	quasi	quasi	ADJ
iajs-2613	326	61	prime	prime	PROPN
iajs-2613	326	62	submodule	submodule	NOUN
iajs-2613	326	63	of	of	ADP
iajs-2613	326	64	t	t	PROPN
iajs-2613	326	65	iff	iff	PROPN
iajs-2613	327	1	[	[	X
iajs-2613	327	2	c	c	X
iajs-2613	327	3	:	:	PUNCT
iajs-2613	327	4	r	r	NOUN
iajs-2613	327	5	t	t	PROPN
iajs-2613	327	6	]	]	PUNCT
iajs-2613	327	7	is	be	AUX
iajs-2613	327	8	wapp	wapp	NOUN
iajs-2613	327	9	-	-	PUNCT
iajs-2613	327	10	quasi	quasi	ADJ
iajs-2613	327	11	prime	prime	ADJ
iajs-2613	327	12	ideal	ideal	NOUN
iajs-2613	327	13	of	of	ADP
iajs-2613	327	14	r.	r.	PROPN
iajs-2613	327	15	7	7	NUM
iajs-2613	327	16	.	.	PUNCT
iajs-2613	328	1	proper	proper	ADJ
iajs-2613	328	2	submodule	submodule	PROPN
iajs-2613	328	3	c	c	PROPN
iajs-2613	328	4	of	of	ADP
iajs-2613	328	5	projective	projective	ADJ
iajs-2613	328	6	multiplication	multiplication	NOUN
iajs-2613	328	7	r	r	NOUN
iajs-2613	328	8	-	-	PUNCT
iajs-2613	328	9	module	module	NOUN
iajs-2613	328	10	t	t	NOUN
iajs-2613	328	11	is	be	AUX
iajs-2613	328	12	wapp	wapp	NOUN
iajs-2613	328	13	-	-	PUNCT
iajs-2613	328	14	quasi	quasi	ADJ
iajs-2613	328	15	prim	prim	PROPN
iajs-2613	328	16	submodule	submodule	NOUN
iajs-2613	328	17	of	of	ADP
iajs-2613	328	18	t	t	PROPN
iajs-2613	328	19	iff	iff	PROPN
iajs-2613	329	1	[	[	X
iajs-2613	329	2	c	c	X
iajs-2613	329	3	:	:	PUNCT
iajs-2613	329	4	r	r	NOUN
iajs-2613	329	5	t	t	PROPN
iajs-2613	329	6	]	]	PUNCT
iajs-2613	329	7	is	be	AUX
iajs-2613	329	8	wapp	wapp	NOUN
iajs-2613	329	9	-	-	PUNCT
iajs-2613	329	10	quasi	quasi	ADJ
iajs-2613	329	11	prime	prime	ADJ
iajs-2613	329	12	ideal	ideal	NOUN
iajs-2613	329	13	of	of	ADP
iajs-2613	329	14	r.	r.	PROPN
iajs-2613	329	15	8	8	NUM
iajs-2613	329	16	.	.	PUNCT
iajs-2613	330	1	if	if	SCONJ
iajs-2613	330	2	t	t	PROPN
iajs-2613	330	3	is	be	AUX
iajs-2613	330	4	a	a	DET
iajs-2613	330	5	cyclic	cyclic	NOUN
iajs-2613	330	6	a	a	DET
iajs-2613	330	7	z	z	NOUN
iajs-2613	330	8	-	-	ADJ
iajs-2613	330	9	regular	regular	ADJ
iajs-2613	330	10	r	r	NOUN
iajs-2613	330	11	-	-	PUNCT
iajs-2613	330	12	module	module	NOUN
iajs-2613	331	1	and	and	CCONJ
iajs-2613	331	2	i	i	PRON
iajs-2613	331	3	is	be	AUX
iajs-2613	331	4	wapp	wapp	NOUN
iajs-2613	331	5	-	-	PUNCT
iajs-2613	331	6	quasi	quasi	ADJ
iajs-2613	331	7	prime	prime	ADJ
iajs-2613	331	8	ideal	ideal	NOUN
iajs-2613	331	9	of	of	ADP
iajs-2613	331	10	r	r	NOUN
iajs-2613	331	11	with	with	ADP
iajs-2613	331	12	annr(t)i	annr(t)i	NOUN
iajs-2613	331	13	.	.	PUNCT
iajs-2613	332	1	then	then	ADV
iajs-2613	332	2	it	it	PRON
iajs-2613	332	3	is	be	AUX
iajs-2613	332	4	wapp	wapp	NOUN
iajs-2613	332	5	-	-	PUNCT
iajs-2613	332	6	quasi	quasi	ADJ
iajs-2613	332	7	submodule	submodule	NOUN
iajs-2613	332	8	of	of	ADP
iajs-2613	332	9	t.	t.	PROPN
iajs-2613	332	10	references	reference	NOUN
iajs-2613	332	11	1	1	NUM
iajs-2613	332	12	.	.	PUNCT
iajs-2613	332	13	waad	waad	PROPN
iajs-2613	332	14	,	,	PUNCT
iajs-2613	332	15	k	k	PROPN
iajs-2613	332	16	.	.	PUNCT
iajs-2613	333	1	h.	h.	PROPN
iajs-2613	333	2	;	;	PUNCT
iajs-2613	333	3	weakly	weakly	ADJ
iajs-2613	333	4	quasi	quasi	ADJ
iajs-2613	333	5	prime	prime	ADJ
iajs-2613	333	6	modules	module	NOUN
iajs-2613	333	7	and	and	CCONJ
iajs-2613	333	8	weakly	weakly	ADJ
iajs-2613	333	9	quasi	quasi	ADJ
iajs-2613	333	10	prime	prime	ADJ
iajs-2613	333	11	submodules	submodule	NOUN
iajs-2613	333	12	;	;	PUNCT
iajs-2613	333	13	m.	m.	PROPN
iajs-2613	333	14	sc	sc	PROPN
iajs-2613	333	15	.	.	PUNCT
iajs-2613	334	1	thesis	thesis	NOUN
iajs-2613	334	2	,	,	PUNCT
iajs-2613	334	3	university	university	NOUN
iajs-2613	334	4	of	of	ADP
iajs-2613	334	5	tikrit	tikrit	NOUN
iajs-2613	334	6	2013	2013	NUM
iajs-2613	334	7	.	.	PUNCT
iajs-2613	335	1	2	2	X
iajs-2613	335	2	.	.	NUM
iajs-2613	335	3	behboodi	behboodi	NOUN
iajs-2613	335	4	,	,	PUNCT
iajs-2613	335	5	m.	m.	NOUN
iajs-2613	335	6	;	;	PUNCT
iajs-2613	335	7	koohy	koohy	PROPN
iajs-2613	335	8	h	h	PROPN
iajs-2613	335	9	.;weakly	.;weakly	ADV
iajs-2613	335	10	prime	prime	ADJ
iajs-2613	335	11	modules	module	NOUN
iajs-2613	335	12	;	;	PUNCT
iajs-2613	335	13	vietnam	vietnam	PROPN
iajs-2613	335	14	j.	j.	PROPN
iajs-2613	335	15	math	math	PROPN
iajs-2613	335	16	.	.	PUNCT
iajs-2613	335	17	,	,	PUNCT
iajs-2613	335	18	2004	2004	NUM
iajs-2613	335	19	,	,	PUNCT
iajs-2613	335	20	32	32	NUM
iajs-2613	335	21	,	,	PUNCT
iajs-2613	335	22	2	2	NUM
iajs-2613	335	23	,	,	PUNCT
iajs-2613	335	24	185	185	NUM
iajs-2613	335	25	-	-	SYM
iajs-2613	335	26	195	195	NUM
iajs-2613	335	27	.	.	NOUN
iajs-2613	335	28	3	3	NUM
iajs-2613	335	29	.	.	X
iajs-2613	335	30	haibat	haibat	PROPN
iajs-2613	335	31	,	,	PUNCT
iajs-2613	335	32	k.	k.	PROPN
iajs-2613	335	33	m.	m.	PROPN
iajs-2613	335	34	;	;	PUNCT
iajs-2613	336	1	khalaf	khalaf	PROPN
iajs-2613	336	2	h.a	h.a	PROPN
iajs-2613	336	3	.	.	PROPN
iajs-2613	336	4	;	;	PUNCT
iajs-2613	336	5	weakly	weakly	ADJ
iajs-2613	336	6	quasi	quasi	ADJ
iajs-2613	336	7	-2	-2	ADV
iajs-2613	336	8	-	-	PUNCT
iajs-2613	336	9	absobing	absobe	VERB
iajs-2613	336	10	submodules	submodule	NOUN
iajs-2613	336	11	;	;	PUNCT
iajs-2613	336	12	tikrit	tikrit	NOUN
iajs-2613	336	13	j.	j.	PROPN
iajs-2613	336	14	of	of	ADP
iajs-2613	336	15	pure	pure	ADJ
iajs-2613	336	16	sci	sci	PROPN
iajs-2613	336	17	.2018,23	.2018,23	PROPN
iajs-2613	336	18	,	,	PUNCT
iajs-2613	336	19	7,101	7,101	NUM
iajs-2613	336	20	-	-	SYM
iajs-2613	336	21	104	104	NUM
iajs-2613	336	22	.	.	PUNCT
iajs-2613	337	1	4	4	X
iajs-2613	337	2	.	.	X
iajs-2613	337	3	haibat	haibat	NOUN
iajs-2613	337	4	,	,	PUNCT
iajs-2613	337	5	k	k	PROPN
iajs-2613	337	6	.	.	PUNCT
iajs-2613	338	1	m.	m.	NOUN
iajs-2613	338	2	;	;	PUNCT
iajs-2613	338	3	saif	saif	PROPN
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iajs-2613	338	5	a.h	a.h	PROPN
iajs-2613	338	6	.	.	PROPN
iajs-2613	338	7	;	;	PUNCT
iajs-2613	338	8	we	we	PRON
iajs-2613	338	9	-	-	PUNCT
iajs-2613	338	10	primary	primary	ADJ
iajs-2613	338	11	submodule	submodule	NOUN
iajs-2613	338	12	and	and	CCONJ
iajs-2613	338	13	we	we	PRON
iajs-2613	338	14	-	-	PUNCT
iajs-2613	338	15	quasi	quasi	ADJ
iajs-2613	338	16	prime	prime	ADJ
iajs-2613	338	17	submodules	submodule	NOUN
iajs-2613	338	18	;	;	PUNCT
iajs-2613	338	19	tikrit	tikrit	NOUN
iajs-2613	338	20	j.	j.	PROPN
iajs-2613	338	21	of	of	ADP
iajs-2613	338	22	pure	pure	ADJ
iajs-2613	338	23	swwci	swwci	NOUN
iajs-2613	338	24	,	,	PUNCT
iajs-2613	338	25	2019	2019	NUM
iajs-2613	338	26	,	,	PUNCT
iajs-2613	338	27	24	24	NUM
iajs-2613	338	28	,	,	PUNCT
iajs-2613	338	29	1	1	NUM
iajs-2613	338	30	,	,	PUNCT
iajs-2613	338	31	68	68	NUM
iajs-2613	338	32	-	-	SYM
iajs-2613	338	33	102	102	NUM
iajs-2613	338	34	.	.	PUNCT
iajs-2613	339	1	5	5	X
iajs-2613	339	2	.	.	X
iajs-2613	339	3	haibat	haibat	PROPN
iajs-2613	339	4	,	,	PUNCT
iajs-2613	339	5	k.	k.	PROPN
iajs-2613	339	6	m.	m.	PROPN
iajs-2613	339	7	;	;	PUNCT
iajs-2613	340	1	omer	omer	PROPN
iajs-2613	340	2	,	,	PUNCT
iajs-2613	340	3	a.a	a.a	PROPN
iajs-2613	340	4	.	.	PROPN
iajs-2613	340	5	;	;	PUNCT
iajs-2613	340	6	pseudo	pseudo	NOUN
iajs-2613	340	7	quasi	quasi	NOUN
iajs-2613	340	8	2	2	NUM
iajs-2613	340	9	-	-	PUNCT
iajs-2613	340	10	absorbing	absorb	VERB
iajs-2613	340	11	submodules	submodule	NOUN
iajs-2613	340	12	and	and	CCONJ
iajs-2613	340	13	some	some	DET
iajs-2613	340	14	related	relate	VERB
iajs-2613	340	15	concepts	concept	NOUN
iajs-2613	340	16	;	;	PUNCT
iajs-2613	340	17	ibn	ibn	PROPN
iajs-2613	340	18	al	al	PROPN
iajs-2613	340	19	-	-	PUNCT
iajs-2613	340	20	hatham	hatham	PROPN
iajs-2613	340	21	j	j	PROPN
iajs-2613	340	22	.	.	PUNCT
iajs-2613	341	1	for	for	ADP
iajs-2613	341	2	pure	pure	ADJ
iajs-2613	341	3	and	and	CCONJ
iajs-2613	341	4	applied	apply	VERB
iajs-2613	341	5	sci	sci	PROPN
iajs-2613	341	6	.,2019,32	.,2019,32	PROPN
iajs-2613	341	7	,	,	PUNCT
iajs-2613	341	8	2	2	NUM
iajs-2613	341	9	,	,	PUNCT
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iajs-2613	341	11	-	-	SYM
iajs-2613	341	12	122	122	NUM
iajs-2613	341	13	.	.	NOUN
iajs-2613	341	14	6	6	NUM
iajs-2613	341	15	.	.	X
iajs-2613	341	16	gooderal	gooderal	ADJ
iajs-2613	341	17	,	,	PUNCT
iajs-2613	341	18	k.r	k.r	PROPN
iajs-2613	341	19	.	.	PROPN
iajs-2613	341	20	;	;	PUNCT
iajs-2613	341	21	ring	ring	NOUN
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iajs-2613	341	23	,	,	PUNCT
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iajs-2613	341	25	singular	singular	NOUN
iajs-2613	341	26	ring	ring	NOUN
iajs-2613	341	27	and	and	CCONJ
iajs-2613	341	28	modules	module	NOUN
iajs-2613	341	29	;	;	PUNCT
iajs-2613	341	30	marcel	marcel	PROPN
iajs-2613	341	31	.dekker	.dekker	X
iajs-2613	341	32	,	,	PUNCT
iajs-2613	341	33	new	new	PROPN
iajs-2613	341	34	york	york	PROPN
iajs-2613	341	35	,	,	PUNCT
iajs-2613	341	36	1976	1976	NUM
iajs-2613	341	37	.	.	PUNCT
iajs-2613	342	1	7	7	X
iajs-2613	342	2	.	.	X
iajs-2613	342	3	el	el	NOUN
iajs-2613	342	4	-	-	PUNCT
iajs-2613	342	5	bast	bast	NOUN
iajs-2613	342	6	,	,	PUNCT
iajs-2613	342	7	z.	z.	PROPN
iajs-2613	342	8	a.	a.	PROPN
iajs-2613	342	9	;	;	PUNCT
iajs-2613	342	10	smith	smith	PROPN
iajs-2613	342	11	p.f	p.f	PROPN
iajs-2613	342	12	.	.	PROPN
iajs-2613	342	13	;	;	PUNCT
iajs-2613	342	14	multiplication	multiplication	NOUN
iajs-2613	342	15	modules	module	NOUN
iajs-2613	342	16	;	;	PUNCT
iajs-2613	342	17	comm	comm	NOUN
iajs-2613	342	18	.	.	PUNCT
iajs-2613	343	1	in	in	ADP
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iajs-2613	343	3	,	,	PUNCT
iajs-2613	343	4	1988	1988	NUM
iajs-2613	343	5	,	,	PUNCT
iajs-2613	343	6	16	16	NUM
iajs-2613	343	7	,	,	PUNCT
iajs-2613	343	8	4	4	NUM
iajs-2613	343	9	,	,	PUNCT
iajs-2613	343	10	755	755	NUM
iajs-2613	343	11	-	-	SYM
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iajs-2613	343	13	.	.	NOUN
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iajs-2613	343	15	.	.	X
iajs-2613	343	16	darani	darani	PROPN
iajs-2613	343	17	,	,	PUNCT
iajs-2613	343	18	a.y	a.y	PROPN
iajs-2613	343	19	.	.	PROPN
iajs-2613	343	20	;	;	PUNCT
iajs-2613	343	21	sohelnai	sohelnai	ADJ
iajs-2613	343	22	,	,	PUNCT
iajs-2613	343	23	f.	f.	PROPN
iajs-2613	343	24	;	;	PUNCT
iajs-2613	343	25	2	2	NUM
iajs-2613	343	26	-	-	ADJ
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iajs-2613	343	30	2	2	NUM
iajs-2613	343	31	-	-	PUNCT
iajs-2613	343	32	absorbing	absorb	VERB
iajs-2613	343	33	submodules	submodule	NOUN
iajs-2613	343	34	;	;	PUNCT
iajs-2613	343	35	tahi	tahi	NOUN
iajs-2613	343	36	journal	journal	NOUN
iajs-2613	343	37	math.2011	math.2011	PROPN
iajs-2613	343	38	,	,	PUNCT
iajs-2613	343	39	9	9	NUM
iajs-2613	343	40	,	,	PUNCT
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iajs-2613	343	42	-	-	SYM
iajs-2613	343	43	584	584	NUM
iajs-2613	343	44	.	.	PUNCT
iajs-2613	344	1	9	9	X
iajs-2613	344	2	.	.	X
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iajs-2613	344	4	,	,	PUNCT
iajs-2613	344	5	f.	f.	PROPN
iajs-2613	344	6	;	;	PUNCT
iajs-2613	344	7	modules	module	NOUN
iajs-2613	344	8	and	and	CCONJ
iajs-2613	344	9	rings	ring	NOUN
iajs-2613	344	10	;	;	PUNCT
iajs-2613	344	11	london	london	PROPN
iajs-2613	344	12	math	math	NOUN
iajs-2613	344	13	.	.	PUNCT
iajs-2613	345	1	soc	soc	PROPN
iajs-2613	345	2	.	.	PUNCT
iajs-2613	346	1	monographs	monograph	NOUN
iajs-2613	346	2	,	,	PUNCT
iajs-2613	346	3	new	new	ADJ
iajs-2613	346	4	-	-	PUNCT
iajs-2613	346	5	york	york	NOUN
iajs-2613	346	6	,	,	PUNCT
iajs-2613	346	7	academic	academic	ADJ
iajs-2613	346	8	press	press	NOUN
iajs-2613	346	9	,	,	PUNCT
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iajs-2613	346	11	.	.	PUNCT
iajs-2613	347	1	10	10	NUM
iajs-2613	347	2	.	.	PUNCT
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iajs-2613	347	4	,	,	PUNCT
iajs-2613	347	5	j.	j.	PROPN
iajs-2613	347	6	m	m	PROPN
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iajs-2613	347	8	modules	module	NOUN
iajs-2613	347	9	;	;	PUNCT
iajs-2613	347	10	trans	trans	PROPN
iajs-2613	347	11	.amer	.amer	PROPN
iajs-2613	347	12	.	.	PUNCT
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iajs-2613	347	16	,	,	PUNCT
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iajs-2613	347	18	-	-	SYM
iajs-2613	347	19	355	355	NUM
iajs-2613	347	20	.	.	PUNCT
iajs-2613	348	1	11	11	NUM
iajs-2613	348	2	.	.	X
iajs-2613	349	1	nuha	nuha	PROPN
iajs-2613	349	2	,	,	PUNCT
iajs-2613	349	3	h.	h.	PROPN
iajs-2613	349	4	h.	h.	PROPN
iajs-2613	349	5	;	;	PUNCT
iajs-2613	349	6	the	the	DET
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iajs-2613	349	8	of	of	ADP
iajs-2613	349	9	modules	module	NOUN
iajs-2613	349	10	;	;	PUNCT
iajs-2613	349	11	m	m	PROPN
iajs-2613	349	12	.sc	.sc	PUNCT
iajs-2613	349	13	.	.	PUNCT
iajs-2613	350	1	thesis	thesis	NOUN
iajs-2613	350	2	,	,	PUNCT
iajs-2613	350	3	university	university	NOUN
iajs-2613	350	4	of	of	ADP
iajs-2613	350	5	bagdad	bagdad	PROPN
iajs-2613	350	6	,	,	PUNCT
iajs-2613	350	7	1996	1996	NUM
iajs-2613	350	8	.	.	PUNCT
iajs-2613	351	1	69	69	NUM
iajs-2613	351	2	ibn	ibn	PROPN
iajs-2613	351	3	al	al	PROPN
iajs-2613	351	4	-	-	PUNCT
iajs-2613	351	5	haitham	haitham	PROPN
iajs-2613	351	6	jour	jour	X
iajs-2613	351	7	.	.	PROPN
iajs-2613	351	8	for	for	ADP
iajs-2613	351	9	pure	pure	ADJ
iajs-2613	351	10	&	&	CCONJ
iajs-2613	351	11	appl	appl	PROPN
iajs-2613	351	12	.	.	PUNCT
iajs-2613	352	1	sci	sci	PROPN
iajs-2613	352	2	.	.	PROPN
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iajs-2613	353	2	(	(	PUNCT
iajs-2613	353	3	1	1	NUM
iajs-2613	353	4	)	)	PUNCT
iajs-2613	353	5	2021	2021	NUM
iajs-2613	353	6	12	12	NUM
iajs-2613	353	7	.	.	PUNCT
iajs-2613	354	1	smith	smith	PROPN
iajs-2613	354	2	,	,	PUNCT
iajs-2613	354	3	p.f	p.f	PROPN
iajs-2613	354	4	.	.	PROPN
iajs-2613	354	5	;	;	PUNCT
iajs-2613	354	6	some	some	DET
iajs-2613	354	7	remarks	remark	NOUN
iajs-2613	354	8	on	on	ADP
iajs-2613	354	9	multiplication	multiplication	NOUN
iajs-2613	354	10	modules	module	NOUN
iajs-2613	354	11	;	;	PUNCT
iajs-2613	354	12	arch	arch	NOUN
iajs-2613	354	13	.	.	PUNCT
iajs-2613	355	1	math	math	NOUN
iajs-2613	355	2	.	.	PUNCT
iajs-2613	356	1	1988	1988	NUM
iajs-2613	356	2	,	,	PUNCT
iajs-2613	356	3	50	50	NUM
iajs-2613	356	4	,	,	PUNCT
iajs-2613	356	5	223	223	NUM
iajs-2613	356	6	-	-	SYM
iajs-2613	356	7	223	223	NUM
iajs-2613	356	8	.	.	PUNCT