id	sid	tid	token	lemma	pos
iajs-1215	1	1	2009	2009	NUM
iajs-1215	1	2	)	)	PUNCT
iajs-1215	1	3	3	3	NUM
iajs-1215	1	4	(	(	PUNCT
iajs-1215	1	5	22مجلة	22مجلة	NUM
iajs-1215	1	6	ابن	ابن	PROPN
iajs-1215	1	7	الھیثم	الھیثم	PROPN
iajs-1215	1	8	للعلوم	للعلوم	PROPN
iajs-1215	1	9	الصرفة	الصرفة	PROPN
iajs-1215	1	10	والتطبیقیة	والتطبیقیة	PROPN
iajs-1215	1	11	المجلد	المجلد	PROPN
iajs-1215	1	12	حول	حول	PROPN
iajs-1215	1	13	المقاسات	المقاسات	PROPN
iajs-1215	1	14	الجزئیة	الجزئیة	PROPN
iajs-1215	1	15	األولیة	األولیة	PROPN
iajs-1215	1	16	الضعیفة	الضعیفة	VERB
iajs-1215	1	17	أنعام	أنعام	PROPN
iajs-1215	1	18	محمد	محمد	PROPN
iajs-1215	1	19	علي	علي	NOUN
iajs-1215	1	20	هادي	هادي	NOUN
iajs-1215	1	21	ابن	ابن	PROPN
iajs-1215	1	22	الهیثم،جامعة	الهیثم،جامعة	PROPN
iajs-1215	1	23	بغداد	بغداد	PROPN
iajs-1215	1	24	-	-	PUNCT
iajs-1215	1	25	قسم	قسم	PROPN
iajs-1215	1	26	الریاضیات	الریاضیات	PROPN
iajs-1215	1	27	،	،	PROPN
iajs-1215	1	28	كلیةالتربیة	كلیةالتربیة	NOUN
iajs-1215	1	29	الخالصة	الخالصة	VERB
iajs-1215	1	30	یكـون	یكـون	PROPN
iajs-1215	1	31	mفـي	mفـي	PROPN
iajs-1215	1	32	nُنعـرف	nُنعـرف	PROPN
iajs-1215	2	1	ان	ان	ADV
iajs-1215	2	2	مقاسـاً	مقاسـاً	PROPN
iajs-1215	2	3	جزئیـاً	جزئیـاً	NOUN
iajs-1215	2	4	فعلیـاً	فعلیـاً	VERB
iajs-1215	2	5	.	.	PUNCT
iajs-1215	3	1	rمقاساً	rمقاساً	PROPN
iajs-1215	3	2	أیسـر	أیسـر	NOUN
iajs-1215	3	3	علـى	علـى	INTJ
iajs-1215	3	4	mحلقة	mحلقة	PROPN
iajs-1215	3	5	ابدالیة	ابدالیة	PROPN
iajs-1215	3	6	ذا	ذا	PROPN
iajs-1215	3	7	محاید	محاید	PROPN
iajs-1215	3	8	ولیكن	ولیكن	PROPN
iajs-1215	3	9	rلتكن	rلتكن	PROPN
iajs-1215	3	10	فـي	فـي	INTJ
iajs-1215	3	11	.	.	PUNCT
iajs-1215	4	1	r	r	NOUN
iajs-1215	4	2			NOUN
iajs-1215	4	3	(	(	PUNCT
iajs-1215	4	4	n	n	CCONJ
iajs-1215	4	5	:	:	PUNCT
iajs-1215	4	6	m)أو	m)أو	PROPN
iajs-1215	4	7	x	x	SYM
iajs-1215	4	8			PROPN
iajs-1215	4	9	nیـؤدي	nیـؤدي	NOUN
iajs-1215	4	10	الـى	الـى	VERB
iajs-1215	4	11	r	r	PROPN
iajs-1215	4	12	x	x	SYM
iajs-1215	4	13			NOUN
iajs-1215	4	14	n	n	CCONJ
iajs-1215	4	15	≠	≠	PROPN
iajs-1215	4	16	0	0	NUM
iajs-1215	5	1	و	و	NOUN
iajs-1215	5	2	،	،	NOUN
iajs-1215	5	3	x	x	X
iajs-1215	5	4			NOUN
iajs-1215	5	5	mو	mو	ADP
iajs-1215	5	6	،	،	PROPN
iajs-1215	5	7	r	r	NOUN
iajs-1215	5	8			NOUN
iajs-1215	5	9	rأولیـاً	rأولیـاً	ADP
iajs-1215	5	10	ضـعیفاً	ضـعیفاً	NOUN
iajs-1215	5	11	اذا	اذا	PROPN
iajs-1215	5	12	كـان	كـان	PROPN
iajs-1215	5	13	لكـل	لكـل	PROPN
iajs-1215	5	14	ا	ا	PROPN
iajs-1215	5	15	كـان	كـان	PROPN
iajs-1215	5	16	،	،	PROPN
iajs-1215	6	1	یسـمى	یسـمى	PROPN
iajs-1215	6	2	أولیـاً	أولیـاً	PROPN
iajs-1215	6	3	ضـعیفاً	ضـعیفاً	PROPN
iajs-1215	6	4	اذrفـي	اذrفـي	PROPN
iajs-1215	6	5	pان	pان	NOUN
iajs-1215	7	1	مثالیـاً	مثالیـاً	NOUN
iajs-1215	7	2	فعلیـاً	فعلیـاً	NOUN
iajs-1215	7	3	اذ	اذ	NOUN
iajs-1215	7	4	الحقیقة	الحقیقة	ADJ
iajs-1215	7	5	ان	ان	AUX
iajs-1215	7	6	هذا	هذا	ADJ
iajs-1215	7	7	المفهوم	المفهوم	NOUN
iajs-1215	8	1	هو	هو	INTJ
iajs-1215	8	2	تعمیم	تعمیم	NOUN
iajs-1215	8	3	لمفهوم	لمفهوم	PROPN
iajs-1215	8	4	مثالي	مثالي	NOUN
iajs-1215	8	5	أولي	أولي	PROPN
iajs-1215	8	6	ضعیف	ضعیف	PROPN
iajs-1215	8	7	،	،	PROPN
iajs-1215	8	8	.	.	PUNCT
iajs-1215	9	1	a	a	DET
iajs-1215	9	2			NOUN
iajs-1215	9	3	p	p	NOUN
iajs-1215	9	4	أو	أو	PROPN
iajs-1215	9	5	b	b	PROPN
iajs-1215	9	6			PROPN
iajs-1215	9	7	pیؤدي	pیؤدي	NOUN
iajs-1215	9	8	الى	الى	VERB
iajs-1215	9	9	ان	ان	PROPN
iajs-1215	9	10	a	a	DET
iajs-1215	9	11	b	b	NOUN
iajs-1215	9	12			NOUN
iajs-1215	9	13	p	p	PROPN
iajs-1215	9	14	≠	≠	PROPN
iajs-1215	9	15	0و	0و	PROPN
iajs-1215	9	16	،	،	X
iajs-1215	9	17	a	a	PROPN
iajs-1215	9	18	,	,	PUNCT
iajs-1215	9	19	b	b	PROPN
iajs-1215	9	20			PROPN
iajs-1215	9	21	rلكل	rلكل	NOUN
iajs-1215	9	22	.خواص	.خواص	PRON
iajs-1215	9	23	مختلفة	مختلفة	ADJ
iajs-1215	9	24	عن	عن	PROPN
iajs-1215	9	25	المقاسات	المقاسات	PROPN
iajs-1215	9	26	الجزئیة	الجزئیة	PROPN
iajs-1215	9	27	األولیة	األولیة	PROPN
iajs-1215	9	28	الضعیفة	الضعیفة	VERB
iajs-1215	9	29	قد	قد	ADP
iajs-1215	9	30	أعطیت	أعطیت	PROPN
iajs-1215	9	31	ibn	ibn	PROPN
iajs-1215	9	32	alhaitham	alhaitham	NOUN
iajs-1215	10	1	j.	j.	PROPN
iajs-1215	11	1	fo	fo	ADP
iajs-1215	11	2	r	r	NOUN
iajs-1215	11	3	pure	pure	ADJ
iajs-1215	11	4	&	&	CCONJ
iajs-1215	11	5	appl	appl	PROPN
iajs-1215	11	6	.	.	PUNCT
iajs-1215	12	1	sc	sc	PROPN
iajs-1215	13	1	i	i	PRON
iajs-1215	13	2	vo	vo	INTJ
iajs-1215	13	3	l.22	l.22	X
iajs-1215	13	4	(	(	PUNCT
iajs-1215	13	5	3	3	NUM
iajs-1215	13	6	)	)	PUNCT
iajs-1215	13	7	2009	2009	NUM
iajs-1215	13	8	on	on	ADP
iajs-1215	13	9	weakly	weakly	ADJ
iajs-1215	13	10	prime	prime	ADJ
iajs-1215	13	11	submodules	submodules	PROPN
iajs-1215	13	12	i.	i.	PROPN
iajs-1215	13	13	m.a.hadi	m.a.hadi	PROPN
iajs-1215	13	14	department	department	PROPN
iajs-1215	13	15	of	of	ADP
iajs-1215	13	16	mathematics	mathematics	PROPN
iajs-1215	13	17	,	,	PUNCT
iajs-1215	13	18	ibn	ibn	PROPN
iajs-1215	13	19	-	-	PUNCT
iajs-1215	13	20	al	al	PROPN
iajs-1215	13	21	-	-	PUNCT
iajs-1215	13	22	haitham	haitham	PROPN
iajs-1215	13	23	college	college	PROPN
iajs-1215	13	24	of	of	ADP
iajs-1215	13	25	education	education	PROPN
iajs-1215	13	26	university	university	PROPN
iajs-1215	13	27	of	of	ADP
iajs-1215	13	28	baghdad	baghdad	PROPN
iajs-1215	13	29	abstract	abstract	ADV
iajs-1215	13	30	let	let	VERB
iajs-1215	13	31	r	r	PRON
iajs-1215	13	32	be	be	AUX
iajs-1215	13	33	a	a	DET
iajs-1215	13	34	commutative	commutative	ADJ
iajs-1215	13	35	ring	ring	NOUN
iajs-1215	13	36	with	with	ADP
iajs-1215	13	37	unity	unity	NOUN
iajs-1215	13	38	and	and	CCONJ
iajs-1215	13	39	let	let	VERB
iajs-1215	13	40	m	m	PRON
iajs-1215	13	41	be	be	AUX
iajs-1215	13	42	a	a	DET
iajs-1215	13	43	left	left	ADJ
iajs-1215	13	44	r	r	NOUN
iajs-1215	13	45	-	-	PUNCT
iajs-1215	13	46	module	module	NOUN
iajs-1215	13	47	.	.	PUNCT
iajs-1215	14	1	we	we	PRON
iajs-1215	14	2	define	define	VERB
iajs-1215	14	3	a	a	DET
iajs-1215	14	4	proper	proper	ADJ
iajs-1215	14	5	submodule	submodule	NOUN
iajs-1215	14	6	n	n	PROPN
iajs-1215	14	7	of	of	ADP
iajs-1215	14	8	m	m	PRON
iajs-1215	14	9	to	to	PART
iajs-1215	14	10	be	be	AUX
iajs-1215	14	11	a	a	DET
iajs-1215	14	12	weakly	weakly	ADJ
iajs-1215	14	13	prime	prime	NOUN
iajs-1215	14	14	if	if	SCONJ
iajs-1215	14	15	whenever	whenever	SCONJ
iajs-1215	14	16	r	r	NOUN
iajs-1215	14	17			NOUN
iajs-1215	14	18	r	r	NOUN
iajs-1215	14	19	,	,	PUNCT
iajs-1215	14	20	x	x	SYM
iajs-1215	14	21			NOUN
iajs-1215	14	22	m	m	PROPN
iajs-1215	14	23	,	,	PUNCT
iajs-1215	14	24	0	0	NUM
iajs-1215	14	25			NOUN
iajs-1215	14	26	r	r	NOUN
iajs-1215	14	27	x	x	SYM
iajs-1215	14	28			NOUN
iajs-1215	14	29	n	n	PRON
iajs-1215	14	30	implies	imply	VERB
iajs-1215	14	31	x	x	PUNCT
iajs-1215	14	32			NOUN
iajs-1215	14	33	n	n	CCONJ
iajs-1215	14	34	or	or	CCONJ
iajs-1215	14	35	r	r	NOUN
iajs-1215	14	36			NOUN
iajs-1215	14	37	(	(	PUNCT
iajs-1215	14	38	n	n	CCONJ
iajs-1215	14	39	:	:	PUNCT
iajs-1215	14	40	m	m	NUM
iajs-1215	14	41	)	)	PUNCT
iajs-1215	14	42	.	.	PUNCT
iajs-1215	15	1	in	in	ADP
iajs-1215	15	2	fact	fact	NOUN
iajs-1215	15	3	this	this	DET
iajs-1215	15	4	concept	concept	NOUN
iajs-1215	15	5	is	be	AUX
iajs-1215	15	6	a	a	DET
iajs-1215	15	7	generalization	generalization	NOUN
iajs-1215	15	8	of	of	ADP
iajs-1215	15	9	the	the	DET
iajs-1215	15	10	concept	concept	NOUN
iajs-1215	15	11	weakly	weakly	ADJ
iajs-1215	15	12	prime	prime	ADJ
iajs-1215	15	13	ideal	ideal	NOUN
iajs-1215	15	14	,	,	PUNCT
iajs-1215	15	15	where	where	SCONJ
iajs-1215	15	16	a	a	DET
iajs-1215	15	17	proper	proper	ADJ
iajs-1215	15	18	ideal	ideal	NOUN
iajs-1215	15	19	p	p	NOUN
iajs-1215	15	20	of	of	ADP
iajs-1215	15	21	r	r	NOUN
iajs-1215	15	22	is	be	AUX
iajs-1215	15	23	called	call	VERB
iajs-1215	15	24	a	a	DET
iajs-1215	15	25	weakly	weakly	ADJ
iajs-1215	15	26	prime	prime	NOUN
iajs-1215	15	27	,	,	PUNCT
iajs-1215	15	28	if	if	SCONJ
iajs-1215	15	29	for	for	ADP
iajs-1215	15	30	all	all	DET
iajs-1215	15	31	a	a	PRON
iajs-1215	15	32	,	,	PUNCT
iajs-1215	15	33	b	b	NOUN
iajs-1215	15	34			PROPN
iajs-1215	15	35	r	r	NOUN
iajs-1215	15	36	,	,	PUNCT
iajs-1215	15	37	0	0	NUM
iajs-1215	15	38			PROPN
iajs-1215	15	39	a	a	DET
iajs-1215	15	40	b	b	PROPN
iajs-1215	15	41			NOUN
iajs-1215	15	42	p	p	NOUN
iajs-1215	15	43	implies	imply	VERB
iajs-1215	15	44	a	a	DET
iajs-1215	15	45			NOUN
iajs-1215	15	46	p	p	NOUN
iajs-1215	15	47	or	or	CCONJ
iajs-1215	15	48	b	b	NOUN
iajs-1215	15	49			PROPN
iajs-1215	16	1	p.	p.	NOUN
iajs-1215	16	2	various	various	ADJ
iajs-1215	16	3	properties	property	NOUN
iajs-1215	16	4	of	of	ADP
iajs-1215	16	5	weakly	weakly	ADJ
iajs-1215	16	6	prime	prime	ADJ
iajs-1215	16	7	submodules	submodule	NOUN
iajs-1215	16	8	are	be	AUX
iajs-1215	16	9	considered	consider	VERB
iajs-1215	16	10	.	.	PUNCT
iajs-1215	17	1	1.introduction	1.introduction	NUM
iajs-1215	17	2	throughout	throughout	ADP
iajs-1215	17	3	this	this	DET
iajs-1215	17	4	paper	paper	NOUN
iajs-1215	17	5	,	,	PUNCT
iajs-1215	17	6	r	r	NOUN
iajs-1215	17	7	be	be	VERB
iajs-1215	17	8	a	a	DET
iajs-1215	17	9	commutative	commutative	ADJ
iajs-1215	17	10	ring	ring	NOUN
iajs-1215	17	11	with	with	ADP
iajs-1215	17	12	identity	identity	NOUN
iajs-1215	17	13	and	and	CCONJ
iajs-1215	17	14	m	m	AUX
iajs-1215	17	15	be	be	AUX
iajs-1215	17	16	a	a	DET
iajs-1215	17	17	unity	unity	NOUN
iajs-1215	17	18	rmodule	rmodule	NOUN
iajs-1215	17	19	.	.	PUNCT
iajs-1215	18	1	a	a	DET
iajs-1215	18	2	proper	proper	ADJ
iajs-1215	18	3	submodule	submodule	NOUN
iajs-1215	18	4	n	n	PROPN
iajs-1215	18	5	of	of	ADP
iajs-1215	18	6	m	m	PROPN
iajs-1215	18	7	is	be	AUX
iajs-1215	18	8	said	say	VERB
iajs-1215	18	9	to	to	PART
iajs-1215	18	10	be	be	AUX
iajs-1215	18	11	prime	prime	ADJ
iajs-1215	18	12	if	if	SCONJ
iajs-1215	18	13	whenever	whenever	SCONJ
iajs-1215	18	14	r	r	NOUN
iajs-1215	18	15			NOUN
iajs-1215	18	16	r	r	NOUN
iajs-1215	18	17	,	,	PUNCT
iajs-1215	18	18	x	x	SYM
iajs-1215	18	19			NOUN
iajs-1215	18	20	m	m	VERB
iajs-1215	18	21	,	,	PUNCT
iajs-1215	18	22	rx	rx	VERB
iajs-1215	18	23			NOUN
iajs-1215	18	24	n	n	PRON
iajs-1215	18	25	implies	imply	VERB
iajs-1215	18	26	either	either	CCONJ
iajs-1215	18	27	x	x	SYM
iajs-1215	18	28			NOUN
iajs-1215	18	29	n	n	CCONJ
iajs-1215	18	30	or	or	CCONJ
iajs-1215	18	31	r	r	NOUN
iajs-1215	18	32			NOUN
iajs-1215	18	33	(	(	PUNCT
iajs-1215	18	34	n	n	CCONJ
iajs-1215	18	35	:	:	PUNCT
iajs-1215	18	36	m	m	NUM
iajs-1215	18	37	)	)	PUNCT
iajs-1215	18	38	,	,	PUNCT
iajs-1215	18	39	where	where	SCONJ
iajs-1215	18	40	(	(	PUNCT
iajs-1215	18	41	n	n	NUM
iajs-1215	18	42	:	:	PUNCT
iajs-1215	18	43	m	m	X
iajs-1215	18	44	)	)	PUNCT
iajs-1215	19	1	=	=	PRON
iajs-1215	19	2	{	{	PUNCT
iajs-1215	19	3	r	r	NOUN
iajs-1215	19	4			NOUN
iajs-1215	19	5	r	r	NOUN
iajs-1215	19	6	:	:	PUNCT
iajs-1215	19	7	r	r	NOUN
iajs-1215	19	8	m	m	NOUN
iajs-1215	19	9			ADJ
iajs-1215	19	10	n	n	CCONJ
iajs-1215	19	11	}	}	PUNCT
iajs-1215	19	12	,	,	PUNCT
iajs-1215	19	13	see	see	VERB
iajs-1215	19	14	(	(	PUNCT
iajs-1215	19	15	1	1	NUM
iajs-1215	19	16	)	)	PUNCT
iajs-1215	19	17	.	.	PUNCT
iajs-1215	20	1	semiprime	semiprime	NOUN
iajs-1215	20	2	submodules	submodule	NOUN
iajs-1215	20	3	was	be	AUX
iajs-1215	20	4	given	give	VERB
iajs-1215	20	5	by	by	ADP
iajs-1215	20	6	dauns	daun	NOUN
iajs-1215	20	7	in	in	ADP
iajs-1215	20	8	(	(	PUNCT
iajs-1215	20	9	2	2	NUM
iajs-1215	20	10	)	)	PUNCT
iajs-1215	20	11	,	,	PUNCT
iajs-1215	20	12	as	as	ADP
iajs-1215	20	13	a	a	DET
iajs-1215	20	14	generalization	generalization	NOUN
iajs-1215	20	15	of	of	ADP
iajs-1215	20	16	prime	prime	ADJ
iajs-1215	20	17	submodules	submodule	NOUN
iajs-1215	20	18	,	,	PUNCT
iajs-1215	20	19	where	where	SCONJ
iajs-1215	20	20	a	a	DET
iajs-1215	20	21	proper	proper	ADJ
iajs-1215	20	22	submodule	submodule	NOUN
iajs-1215	20	23	n	n	PROPN
iajs-1215	20	24	of	of	ADP
iajs-1215	20	25	m	m	PROPN
iajs-1215	20	26	is	be	AUX
iajs-1215	20	27	semiprime	semiprime	NOUN
iajs-1215	20	28	if	if	SCONJ
iajs-1215	20	29	rk	rk	PROPN
iajs-1215	20	30	x	x	SYM
iajs-1215	20	31			NOUN
iajs-1215	20	32	n	n	CCONJ
iajs-1215	20	33	,	,	PUNCT
iajs-1215	20	34	for	for	ADP
iajs-1215	20	35	r	r	NOUN
iajs-1215	20	36			PROPN
iajs-1215	20	37	r	r	NOUN
iajs-1215	20	38	,	,	PUNCT
iajs-1215	20	39	x	x	SYM
iajs-1215	20	40			PROPN
iajs-1215	20	41	m	m	PROPN
iajs-1215	20	42	,	,	PUNCT
iajs-1215	20	43	k	k	X
iajs-1215	20	44			PROPN
iajs-1215	20	45	z+	z+	NUM
iajs-1215	20	46	(	(	PUNCT
iajs-1215	20	47	set	set	VERB
iajs-1215	20	48	of	of	ADP
iajs-1215	20	49	positive	positive	ADJ
iajs-1215	20	50	integers	integer	NOUN
iajs-1215	20	51	)	)	PUNCT
iajs-1215	20	52	implies	imply	VERB
iajs-1215	20	53	rx	rx	VERB
iajs-1215	20	54			PROPN
iajs-1215	20	55	m.	m.	NOUN
iajs-1215	21	1	also	also	ADV
iajs-1215	21	2	eman	eman	PROPN
iajs-1215	21	3	a.a	a.a	PROPN
iajs-1215	21	4	.	.	PROPN
iajs-1215	21	5	in	in	ADP
iajs-1215	21	6	(	(	PUNCT
iajs-1215	21	7	3	3	X
iajs-1215	21	8	)	)	PUNCT
iajs-1215	21	9	studied	study	VERB
iajs-1215	21	10	these	these	DET
iajs-1215	21	11	notions	notion	NOUN
iajs-1215	21	12	.	.	PUNCT
iajs-1215	22	1	in	in	ADP
iajs-1215	22	2	1999	1999	NUM
iajs-1215	22	3	,	,	PUNCT
iajs-1215	22	4	quasi	quasi	ADJ
iajs-1215	22	5	-	-	ADJ
iajs-1215	22	6	prime	prime	ADJ
iajs-1215	22	7	submodules	submodule	NOUN
iajs-1215	22	8	was	be	AUX
iajs-1215	22	9	introduced	introduce	VERB
iajs-1215	22	10	and	and	CCONJ
iajs-1215	22	11	studied	study	VERB
iajs-1215	22	12	by	by	ADP
iajs-1215	22	13	muntaha	muntaha	NOUN
iajs-1215	22	14	(	(	PUNCT
iajs-1215	22	15	see	see	VERB
iajs-1215	22	16	(	(	PUNCT
iajs-1215	22	17	4	4	NUM
iajs-1215	22	18	)	)	PUNCT
iajs-1215	22	19	)	)	PUNCT
iajs-1215	22	20	,	,	PUNCT
iajs-1215	22	21	as	as	ADP
iajs-1215	22	22	another	another	DET
iajs-1215	22	23	generalization	generalization	NOUN
iajs-1215	22	24	of	of	ADP
iajs-1215	22	25	prime	prime	ADJ
iajs-1215	22	26	submodules	submodule	NOUN
iajs-1215	22	27	,	,	PUNCT
iajs-1215	22	28	where	where	SCONJ
iajs-1215	22	29	a	a	DET
iajs-1215	22	30	proper	proper	ADJ
iajs-1215	22	31	submodule	submodule	NOUN
iajs-1215	22	32	n	n	PROPN
iajs-1215	22	33	of	of	ADP
iajs-1215	22	34	m	m	PROPN
iajs-1215	22	35	is	be	AUX
iajs-1215	22	36	quasiprime	quasiprime	ADJ
iajs-1215	22	37	if	if	SCONJ
iajs-1215	22	38	r1r2	r1r2	NOUN
iajs-1215	22	39	m	m	PROPN
iajs-1215	22	40			NOUN
iajs-1215	22	41	n	n	CCONJ
iajs-1215	22	42	,	,	PUNCT
iajs-1215	22	43	for	for	ADP
iajs-1215	22	44	r1	r1	NOUN
iajs-1215	22	45	,	,	PUNCT
iajs-1215	22	46	r2	r2	PROPN
iajs-1215	22	47			PROPN
iajs-1215	22	48	r	r	NOUN
iajs-1215	22	49	,	,	PUNCT
iajs-1215	22	50	m	m	VERB
iajs-1215	22	51			NOUN
iajs-1215	22	52	m	m	VERB
iajs-1215	22	53	implies	imply	VERB
iajs-1215	22	54	r1	r1	PROPN
iajs-1215	22	55	m	m	PROPN
iajs-1215	22	56			NOUN
iajs-1215	22	57	n	n	CCONJ
iajs-1215	22	58	or	or	CCONJ
iajs-1215	22	59	r2	r2	PROPN
iajs-1215	22	60	m	m	PROPN
iajs-1215	22	61			NOUN
iajs-1215	22	62	n	n	CCONJ
iajs-1215	22	63	;	;	PUNCT
iajs-1215	22	64	equivalently	equivalently	ADV
iajs-1215	22	65	,	,	PUNCT
iajs-1215	22	66	n	n	PRON
iajs-1215	22	67	is	be	AUX
iajs-1215	22	68	quasi	quasi	ADJ
iajs-1215	22	69	-	-	ADJ
iajs-1215	22	70	prime	prime	ADJ
iajs-1215	22	71	if	if	SCONJ
iajs-1215	22	72	the	the	DET
iajs-1215	22	73	ideal	ideal	NOUN
iajs-1215	22	74	(	(	PUNCT
iajs-1215	22	75	n	n	NUM
iajs-1215	22	76	:	:	PUNCT
iajs-1215	22	77	m	m	VERB
iajs-1215	22	78	)	)	PUNCT
iajs-1215	22	79	is	be	AUX
iajs-1215	22	80	prime	prime	ADJ
iajs-1215	22	81	for	for	ADP
iajs-1215	22	82	all	all	DET
iajs-1215	22	83	m	m	PROPN
iajs-1215	22	84			NOUN
iajs-1215	22	85	m.	m.	NOUN
iajs-1215	22	86	in	in	ADP
iajs-1215	22	87	2004	2004	NUM
iajs-1215	22	88	,	,	PUNCT
iajs-1215	22	89	m.behoodi	m.behoodi	NOUN
iajs-1215	22	90	and	and	CCONJ
iajs-1215	22	91	h.koohi	h.koohi	NOUN
iajs-1215	22	92	in	in	ADP
iajs-1215	22	93	(	(	PUNCT
iajs-1215	22	94	5	5	NUM
iajs-1215	22	95	)	)	PUNCT
iajs-1215	22	96	gave	give	VERB
iajs-1215	22	97	the	the	DET
iajs-1215	22	98	notion	notion	NOUN
iajs-1215	22	99	of	of	ADP
iajs-1215	22	100	weakly	weakly	ADJ
iajs-1215	22	101	prime	prime	ADJ
iajs-1215	22	102	submodules	submodule	NOUN
iajs-1215	22	103	,	,	PUNCT
iajs-1215	22	104	where	where	SCONJ
iajs-1215	22	105	a	a	DET
iajs-1215	22	106	proper	proper	ADJ
iajs-1215	22	107	submodule	submodule	NOUN
iajs-1215	22	108	n	n	PROPN
iajs-1215	22	109	of	of	ADP
iajs-1215	22	110	m	m	PROPN
iajs-1215	22	111	weakly	weakly	ADJ
iajs-1215	22	112	prime	prime	ADJ
iajs-1215	22	113	if	if	SCONJ
iajs-1215	22	114	(	(	PUNCT
iajs-1215	22	115	n	n	NUM
iajs-1215	22	116	:	:	PUNCT
iajs-1215	22	117	k	k	X
iajs-1215	22	118	)	)	PUNCT
iajs-1215	22	119	is	be	AUX
iajs-1215	22	120	a	a	DET
iajs-1215	22	121	prime	prime	ADJ
iajs-1215	22	122	ideal	ideal	NOUN
iajs-1215	22	123	,	,	PUNCT
iajs-1215	22	124	for	for	ADP
iajs-1215	22	125	all	all	DET
iajs-1215	22	126	submodules	submodule	NOUN
iajs-1215	22	127	k	k	PROPN
iajs-1215	22	128	of	of	ADP
iajs-1215	22	129	m.	m.	NOUN
iajs-1215	22	130	also	also	ADV
iajs-1215	22	131	this	this	DET
iajs-1215	22	132	notion	notion	NOUN
iajs-1215	22	133	was	be	AUX
iajs-1215	22	134	studied	study	VERB
iajs-1215	22	135	by	by	ADP
iajs-1215	22	136	a.azizi	a.azizi	NOUN
iajs-1215	22	137	in	in	ADP
iajs-1215	22	138	(	(	PUNCT
iajs-1215	22	139	6	6	NUM
iajs-1215	22	140	)	)	PUNCT
iajs-1215	22	141	,	,	PUNCT
iajs-1215	22	142	2006	2006	NUM
iajs-1215	22	143	.	.	PUNCT
iajs-1215	23	1	by	by	ADP
iajs-1215	23	2	th.2.14	th.2.14	PROPN
iajs-1215	23	3	in	in	ADP
iajs-1215	23	4	(	(	PUNCT
iajs-1215	23	5	4	4	NUM
iajs-1215	23	6	)	)	PUNCT
iajs-1215	23	7	,	,	PUNCT
iajs-1215	23	8	we	we	PRON
iajs-1215	23	9	obtain	obtain	VERB
iajs-1215	23	10	that	that	SCONJ
iajs-1215	23	11	the	the	DET
iajs-1215	23	12	two	two	NUM
iajs-1215	23	13	concepts	concept	NOUN
iajs-1215	23	14	weakly	weakly	ADJ
iajs-1215	23	15	prime	prime	ADJ
iajs-1215	23	16	submodules	submodule	NOUN
iajs-1215	23	17	and	and	CCONJ
iajs-1215	23	18	quasiprime	quasiprime	ADJ
iajs-1215	23	19	submodules	submodule	NOUN
iajs-1215	23	20	are	be	AUX
iajs-1215	23	21	equivalent	equivalent	ADJ
iajs-1215	23	22	.	.	PUNCT
iajs-1215	24	1	in	in	ADP
iajs-1215	24	2	this	this	DET
iajs-1215	24	3	paper	paper	NOUN
iajs-1215	24	4	,	,	PUNCT
iajs-1215	24	5	we	we	PRON
iajs-1215	24	6	give	give	VERB
iajs-1215	24	7	another	another	DET
iajs-1215	24	8	generalization	generalization	NOUN
iajs-1215	24	9	of	of	ADP
iajs-1215	24	10	prime	prime	ADJ
iajs-1215	24	11	submodules	submodule	NOUN
iajs-1215	24	12	namely	namely	ADV
iajs-1215	24	13	weakly	weakly	ADJ
iajs-1215	24	14	prime	prime	ADJ
iajs-1215	24	15	submodules	submodule	NOUN
iajs-1215	24	16	,	,	PUNCT
iajs-1215	24	17	however	however	ADV
iajs-1215	24	18	this	this	DET
iajs-1215	24	19	concept	concept	NOUN
iajs-1215	24	20	is	be	AUX
iajs-1215	24	21	different	different	ADJ
iajs-1215	24	22	from	from	ADP
iajs-1215	24	23	the	the	DET
iajs-1215	24	24	concept	concept	NOUN
iajs-1215	24	25	of	of	ADP
iajs-1215	24	26	quasi	quasi	ADJ
iajs-1215	24	27	-	-	ADJ
iajs-1215	24	28	prime	prime	ADJ
iajs-1215	24	29	submodule	submodule	NOUN
iajs-1215	24	30	(	(	PUNCT
iajs-1215	24	31	see	see	VERB
iajs-1215	24	32	remarks	remark	VERB
iajs-1215	24	33	2.1.(5	2.1.(5	NUM
iajs-1215	24	34	)	)	PUNCT
iajs-1215	24	35	)	)	PUNCT
iajs-1215	24	36	.	.	PUNCT
iajs-1215	25	1	in	in	ADP
iajs-1215	25	2	fact	fact	NOUN
iajs-1215	25	3	,	,	PUNCT
iajs-1215	25	4	d.d.anderson	d.d.anderson	NOUN
iajs-1215	25	5	and	and	CCONJ
iajs-1215	25	6	e.smith	e.smith	ADP
iajs-1215	25	7	in	in	ADP
iajs-1215	25	8	(	(	PUNCT
iajs-1215	25	9	7	7	X
iajs-1215	25	10	)	)	PUNCT
iajs-1215	25	11	gave	give	VERB
iajs-1215	25	12	the	the	DET
iajs-1215	25	13	following	following	NOUN
iajs-1215	25	14	:	:	PUNCT
iajs-1215	25	15	a	a	DET
iajs-1215	25	16	proper	proper	ADJ
iajs-1215	25	17	ideal	ideal	NOUN
iajs-1215	25	18	i	i	PRON
iajs-1215	25	19	of	of	ADP
iajs-1215	25	20	r	r	NOUN
iajs-1215	25	21	is	be	AUX
iajs-1215	25	22	said	say	VERB
iajs-1215	25	23	to	to	PART
iajs-1215	25	24	be	be	AUX
iajs-1215	25	25	a	a	DET
iajs-1215	25	26	weakly	weakly	ADJ
iajs-1215	25	27	prime	prime	NOUN
iajs-1215	25	28	if	if	SCONJ
iajs-1215	25	29	0	0	NUM
iajs-1215	25	30			PROPN
iajs-1215	25	31	ab	ab	PROPN
iajs-1215	25	32			PROPN
iajs-1215	25	33	i	i	PRON
iajs-1215	25	34	,	,	PUNCT
iajs-1215	25	35	for	for	ADP
iajs-1215	25	36	a	a	DET
iajs-1215	25	37	,	,	PUNCT
iajs-1215	25	38	b	b	NOUN
iajs-1215	25	39			PROPN
iajs-1215	25	40	r	r	NOUN
iajs-1215	25	41	,	,	PUNCT
iajs-1215	25	42	then	then	ADV
iajs-1215	25	43	a	a	DET
iajs-1215	25	44			NOUN
iajs-1215	25	45	i	i	PRON
iajs-1215	25	46	or	or	CCONJ
iajs-1215	25	47	b	b	PROPN
iajs-1215	25	48			PROPN
iajs-1215	25	49	i.	i.	NOUN
iajs-1215	25	50	we	we	PRON
iajs-1215	25	51	define	define	VERB
iajs-1215	25	52	a	a	DET
iajs-1215	25	53	proper	proper	ADJ
iajs-1215	25	54	submodule	submodule	NOUN
iajs-1215	25	55	n	n	PROPN
iajs-1215	25	56	of	of	ADP
iajs-1215	25	57	m	m	PROPN
iajs-1215	25	58	is	be	AUX
iajs-1215	25	59	weakly	weakly	ADV
iajs-1215	25	60	prime	prime	ADJ
iajs-1215	25	61	if	if	SCONJ
iajs-1215	25	62	whenever	whenever	SCONJ
iajs-1215	25	63	r	r	NOUN
iajs-1215	25	64			NOUN
iajs-1215	25	65	r	r	NOUN
iajs-1215	25	66	,	,	PUNCT
iajs-1215	25	67	x	x	X
iajs-1215	25	68			NOUN
iajs-1215	25	69	m	m	VERB
iajs-1215	25	70	,	,	PUNCT
iajs-1215	25	71	0	0	NUM
iajs-1215	25	72			NOUN
iajs-1215	25	73	rx	rx	VERB
iajs-1215	25	74			PROPN
iajs-1215	25	75	n	n	PRON
iajs-1215	25	76	implies	imply	VERB
iajs-1215	25	77	x	x	PUNCT
iajs-1215	25	78			NOUN
iajs-1215	25	79	n	n	CCONJ
iajs-1215	25	80	or	or	CCONJ
iajs-1215	25	81	r	r	NOUN
iajs-1215	25	82			NOUN
iajs-1215	25	83	(	(	PUNCT
iajs-1215	25	84	n	n	CCONJ
iajs-1215	25	85	:	:	PUNCT
iajs-1215	25	86	m	m	NUM
iajs-1215	25	87	)	)	PUNCT
iajs-1215	25	88	.	.	PUNCT
iajs-1215	26	1	moreover	moreover	ADV
iajs-1215	26	2	s.e.atani	s.e.atani	NOUN
iajs-1215	26	3	and	and	CCONJ
iajs-1215	26	4	f.farzalipour	f.farzalipour	VERB
iajs-1215	26	5	in	in	ADP
iajs-1215	26	6	(	(	PUNCT
iajs-1215	26	7	8)	8)	NUM
iajs-1215	26	8	introduced	introduce	VERB
iajs-1215	26	9	the	the	DET
iajs-1215	26	10	notion	notion	NOUN
iajs-1215	26	11	of	of	ADP
iajs-1215	26	12	weakly	weakly	ADJ
iajs-1215	26	13	primary	primary	ADJ
iajs-1215	26	14	submodules	submodule	NOUN
iajs-1215	26	15	,	,	PUNCT
iajs-1215	26	16	where	where	SCONJ
iajs-1215	26	17	a	a	DET
iajs-1215	26	18	proper	proper	ADJ
iajs-1215	26	19	submodule	submodule	NOUN
iajs-1215	26	20	n	n	PROPN
iajs-1215	26	21	of	of	ADP
iajs-1215	26	22	m	m	PROPN
iajs-1215	26	23	is	be	AUX
iajs-1215	26	24	a	a	DET
iajs-1215	26	25	weakly	weakly	ADJ
iajs-1215	26	26	primary	primary	NOUN
iajs-1215	26	27	if	if	SCONJ
iajs-1215	26	28	whenever	whenever	SCONJ
iajs-1215	26	29	r	r	NOUN
iajs-1215	26	30			NOUN
iajs-1215	26	31	r	r	NOUN
iajs-1215	26	32	,	,	PUNCT
iajs-1215	26	33	x	x	X
iajs-1215	26	34			NOUN
iajs-1215	26	35	m	m	VERB
iajs-1215	26	36	,	,	PUNCT
iajs-1215	26	37	0	0	NUM
iajs-1215	26	38			NOUN
iajs-1215	26	39	rx	rx	VERB
iajs-1215	26	40			PROPN
iajs-1215	26	41	n	n	PRON
iajs-1215	26	42	implies	imply	VERB
iajs-1215	26	43	x	x	PUNCT
iajs-1215	26	44			NOUN
iajs-1215	26	45	n	n	CCONJ
iajs-1215	26	46	or	or	CCONJ
iajs-1215	26	47	r	r	NOUN
iajs-1215	26	48	n	n	PROPN
iajs-1215	26	49			NOUN
iajs-1215	26	50	(	(	PUNCT
iajs-1215	26	51	n	n	CCONJ
iajs-1215	26	52	:	:	PUNCT
iajs-1215	26	53	m	m	VERB
iajs-1215	26	54	)	)	PUNCT
iajs-1215	26	55	for	for	ADP
iajs-1215	26	56	some	some	DET
iajs-1215	26	57	n	n	ADJ
iajs-1215	26	58			NOUN
iajs-1215	26	59	z+	z+	NUM
iajs-1215	26	60	.	.	PUNCT
iajs-1215	27	1	also	also	ADV
iajs-1215	27	2	they	they	PRON
iajs-1215	27	3	gave	give	VERB
iajs-1215	27	4	that	that	PRON
iajs-1215	27	5	:	:	PUNCT
iajs-1215	27	6	a	a	DET
iajs-1215	27	7	proper	proper	ADJ
iajs-1215	27	8	ideal	ideal	NOUN
iajs-1215	27	9	of	of	ADP
iajs-1215	27	10	r	r	NOUN
iajs-1215	27	11	is	be	AUX
iajs-1215	27	12	a	a	DET
iajs-1215	27	13	weakly	weakly	ADJ
iajs-1215	27	14	primary	primary	NOUN
iajs-1215	27	15	if	if	SCONJ
iajs-1215	27	16	it	it	PRON
iajs-1215	27	17	is	be	AUX
iajs-1215	27	18	a	a	DET
iajs-1215	27	19	weakly	weakly	ADJ
iajs-1215	27	20	prime	prime	ADJ
iajs-1215	27	21	submodule	submodule	NOUN
iajs-1215	27	22	of	of	ADP
iajs-1215	27	23	the	the	DET
iajs-1215	27	24	r	r	NOUN
iajs-1215	27	25	-	-	PUNCT
iajs-1215	27	26	module	module	NOUN
iajs-1215	27	27	r	r	NOUN
iajs-1215	27	28	,	,	PUNCT
iajs-1215	27	29	(	(	PUNCT
iajs-1215	27	30	see	see	VERB
iajs-1215	27	31	(	(	PUNCT
iajs-1215	27	32	8)	8)	NUM
iajs-1215	27	33	)	)	PUNCT
iajs-1215	27	34	.	.	PUNCT
iajs-1215	28	1	in	in	ADP
iajs-1215	28	2	this	this	DET
iajs-1215	28	3	paper	paper	NOUN
iajs-1215	28	4	we	we	PRON
iajs-1215	28	5	study	study	VERB
iajs-1215	28	6	weakly	weakly	ADJ
iajs-1215	28	7	prime	prime	ADJ
iajs-1215	28	8	submodules	submodule	NOUN
iajs-1215	28	9	and	and	CCONJ
iajs-1215	28	10	give	give	VERB
iajs-1215	28	11	many	many	ADJ
iajs-1215	28	12	basic	basic	ADJ
iajs-1215	28	13	properties	property	NOUN
iajs-1215	28	14	related	relate	VERB
iajs-1215	28	15	to	to	ADP
iajs-1215	28	16	this	this	DET
iajs-1215	28	17	concept	concept	NOUN
iajs-1215	28	18	.	.	PUNCT
iajs-1215	29	1	ibn	ibn	PROPN
iajs-1215	29	2	alhaitham	alhaitham	PROPN
iajs-1215	30	1	j.	j.	PROPN
iajs-1215	31	1	fo	fo	ADP
iajs-1215	31	2	r	r	NOUN
iajs-1215	31	3	pure	pure	ADJ
iajs-1215	31	4	&	&	CCONJ
iajs-1215	31	5	appl	appl	PROPN
iajs-1215	31	6	.	.	PUNCT
iajs-1215	32	1	sc	sc	PROPN
iajs-1215	33	1	i	i	PRON
iajs-1215	33	2	vo	vo	INTJ
iajs-1215	33	3	l.22	l.22	X
iajs-1215	33	4	(	(	PUNCT
iajs-1215	33	5	3	3	NUM
iajs-1215	33	6	)	)	PUNCT
iajs-1215	33	7	2009	2009	NUM
iajs-1215	33	8	2.basic	2.basic	NUM
iajs-1215	33	9	properties	property	NOUN
iajs-1215	33	10	as	as	SCONJ
iajs-1215	33	11	we	we	PRON
iajs-1215	33	12	mentioned	mention	VERB
iajs-1215	33	13	in	in	ADP
iajs-1215	33	14	the	the	DET
iajs-1215	33	15	introduction	introduction	NOUN
iajs-1215	33	16	,	,	PUNCT
iajs-1215	33	17	we	we	PRON
iajs-1215	33	18	introduce	introduce	VERB
iajs-1215	33	19	the	the	DET
iajs-1215	33	20	following	following	NOUN
iajs-1215	33	21	:	:	PUNCT
iajs-1215	33	22	definition	definition	NOUN
iajs-1215	33	23	2.0	2.0	NUM
iajs-1215	33	24	:	:	PUNCT
iajs-1215	33	25	a	a	DET
iajs-1215	33	26	proper	proper	ADJ
iajs-1215	33	27	submodule	submodule	NOUN
iajs-1215	33	28	n	n	PROPN
iajs-1215	33	29	of	of	ADP
iajs-1215	33	30	an	an	DET
iajs-1215	33	31	r	r	NOUN
iajs-1215	33	32	-	-	PUNCT
iajs-1215	33	33	module	module	NOUN
iajs-1215	33	34	mis	mis	NOUN
iajs-1215	33	35	weakly	weakly	ADJ
iajs-1215	33	36	prime	prime	NOUN
iajs-1215	33	37	if	if	SCONJ
iajs-1215	33	38	whenever	whenever	SCONJ
iajs-1215	33	39	rr	rr	NOUN
iajs-1215	33	40	,	,	PUNCT
iajs-1215	33	41	x	x	X
iajs-1215	33	42	m	m	X
iajs-1215	33	43	,	,	PUNCT
iajs-1215	33	44	0	0	X
iajs-1215	33	45	rxn	rxn	PRON
iajs-1215	33	46	implies	imply	VERB
iajs-1215	33	47	xn	xn	PROPN
iajs-1215	33	48	or	or	CCONJ
iajs-1215	33	49	r(n	r(n	NOUN
iajs-1215	33	50	:	:	PUNCT
iajs-1215	33	51	m	m	NOUN
iajs-1215	33	52	)	)	PUNCT
iajs-1215	33	53	.	.	PUNCT
iajs-1215	34	1	in	in	ADP
iajs-1215	34	2	this	this	DET
iajs-1215	34	3	section	section	NOUN
iajs-1215	34	4	,	,	PUNCT
iajs-1215	34	5	we	we	PRON
iajs-1215	34	6	will	will	AUX
iajs-1215	34	7	give	give	VERB
iajs-1215	34	8	basic	basic	ADJ
iajs-1215	34	9	properties	property	NOUN
iajs-1215	34	10	of	of	ADP
iajs-1215	34	11	weakly	weakly	ADJ
iajs-1215	34	12	prime	prime	ADJ
iajs-1215	34	13	submodules	submodule	NOUN
iajs-1215	34	14	.	.	PUNCT
iajs-1215	35	1	some	some	PRON
iajs-1215	35	2	of	of	ADP
iajs-1215	35	3	these	these	PRON
iajs-1215	35	4	are	be	AUX
iajs-1215	35	5	extension	extension	NOUN
iajs-1215	35	6	of	of	ADP
iajs-1215	35	7	the	the	DET
iajs-1215	35	8	results	result	NOUN
iajs-1215	35	9	about	about	ADP
iajs-1215	35	10	weakly	weakly	ADJ
iajs-1215	35	11	prime	prime	ADJ
iajs-1215	35	12	ideals	ideal	NOUN
iajs-1215	35	13	,	,	PUNCT
iajs-1215	35	14	which	which	PRON
iajs-1215	35	15	are	be	AUX
iajs-1215	35	16	given	give	VERB
iajs-1215	35	17	in	in	ADP
iajs-1215	35	18	(	(	PUNCT
iajs-1215	35	19	7	7	NUM
iajs-1215	35	20	)	)	PUNCT
iajs-1215	35	21	.	.	PUNCT
iajs-1215	36	1	let	let	VERB
iajs-1215	36	2	us	we	PRON
iajs-1215	36	3	start	start	VERB
iajs-1215	36	4	with	with	ADP
iajs-1215	36	5	the	the	DET
iajs-1215	36	6	following	following	NOUN
iajs-1215	36	7	:	:	PUNCT
iajs-1215	36	8	remarks	remark	VERB
iajs-1215	36	9	2.1	2.1	NUM
iajs-1215	36	10	:	:	PUNCT
iajs-1215	36	11	(	(	PUNCT
iajs-1215	36	12	1	1	X
iajs-1215	36	13	)	)	PUNCT
iajs-1215	36	14	it	it	PRON
iajs-1215	36	15	is	be	AUX
iajs-1215	36	16	clear	clear	ADJ
iajs-1215	36	17	that	that	SCONJ
iajs-1215	36	18	every	every	DET
iajs-1215	36	19	prime	prime	ADJ
iajs-1215	36	20	submodule	submodule	NOUN
iajs-1215	36	21	is	be	AUX
iajs-1215	36	22	weakly	weakly	ADV
iajs-1215	36	23	prime	prime	ADJ
iajs-1215	36	24	.	.	PUNCT
iajs-1215	37	1	however	however	ADV
iajs-1215	37	2	,	,	PUNCT
iajs-1215	37	3	since	since	SCONJ
iajs-1215	37	4	(	(	PUNCT
iajs-1215	37	5	0	0	X
iajs-1215	37	6	)	)	PUNCT
iajs-1215	37	7	the	the	DET
iajs-1215	37	8	zero	zero	NUM
iajs-1215	37	9	submodule	submodule	NOUN
iajs-1215	37	10	of	of	ADP
iajs-1215	37	11	any	any	DET
iajs-1215	37	12	module	module	NOUN
iajs-1215	37	13	)	)	PUNCT
iajs-1215	37	14	is	be	AUX
iajs-1215	37	15	always	always	ADV
iajs-1215	37	16	weakly	weakly	ADJ
iajs-1215	37	17	prime	prime	ADJ
iajs-1215	37	18	(	(	PUNCT
iajs-1215	37	19	by	by	ADP
iajs-1215	37	20	definition	definition	NOUN
iajs-1215	37	21	)	)	PUNCT
iajs-1215	37	22	,	,	PUNCT
iajs-1215	37	23	a	a	DET
iajs-1215	37	24	weakly	weakly	ADJ
iajs-1215	37	25	prime	prime	ADJ
iajs-1215	37	26	submodule	submodule	NOUN
iajs-1215	37	27	may	may	AUX
iajs-1215	37	28	not	not	PART
iajs-1215	37	29	be	be	AUX
iajs-1215	37	30	prime	prime	ADJ
iajs-1215	37	31	;	;	PUNCT
iajs-1215	37	32	for	for	ADP
iajs-1215	37	33	example	example	NOUN
iajs-1215	37	34	:	:	PUNCT
iajs-1215	37	35	the	the	DET
iajs-1215	37	36	zero	zero	NUM
iajs-1215	37	37	submodule	submodule	NOUN
iajs-1215	37	38	of	of	ADP
iajs-1215	37	39	the	the	DET
iajs-1215	37	40	z	z	NOUN
iajs-1215	37	41	-	-	PUNCT
iajs-1215	37	42	module	module	NOUN
iajs-1215	37	43	z4	z4	NOUN
iajs-1215	37	44	is	be	AUX
iajs-1215	37	45	weakly	weakly	ADJ
iajs-1215	37	46	prime	prime	ADJ
iajs-1215	37	47	,	,	PUNCT
iajs-1215	37	48	but	but	CCONJ
iajs-1215	37	49	it	it	PRON
iajs-1215	37	50	is	be	AUX
iajs-1215	37	51	not	not	PART
iajs-1215	37	52	prime	prime	ADJ
iajs-1215	37	53	.	.	PUNCT
iajs-1215	38	1	moreover	moreover	ADV
iajs-1215	38	2	it	it	PRON
iajs-1215	38	3	is	be	AUX
iajs-1215	38	4	easy	easy	ADJ
iajs-1215	38	5	to	to	PART
iajs-1215	38	6	check	check	VERB
iajs-1215	38	7	that	that	SCONJ
iajs-1215	38	8	in	in	ADP
iajs-1215	38	9	the	the	DET
iajs-1215	38	10	class	class	NOUN
iajs-1215	38	11	of	of	ADP
iajs-1215	38	12	torsion	torsion	NOUN
iajs-1215	38	13	free	free	ADJ
iajs-1215	38	14	modules	module	NOUN
iajs-1215	38	15	,	,	PUNCT
iajs-1215	38	16	the	the	DET
iajs-1215	38	17	concepts	concept	NOUN
iajs-1215	38	18	of	of	ADP
iajs-1215	38	19	prime	prime	ADJ
iajs-1215	38	20	submodule	submodule	NOUN
iajs-1215	38	21	and	and	CCONJ
iajs-1215	38	22	weakly	weakly	ADJ
iajs-1215	38	23	prime	prime	ADJ
iajs-1215	38	24	submodule	submodule	NOUN
iajs-1215	38	25	are	be	AUX
iajs-1215	38	26	equivalent	equivalent	ADJ
iajs-1215	38	27	.	.	PUNCT
iajs-1215	39	1	(	(	PUNCT
iajs-1215	39	2	2	2	X
iajs-1215	39	3	)	)	PUNCT
iajs-1215	39	4	every	every	DET
iajs-1215	39	5	weakly	weakly	ADJ
iajs-1215	39	6	prime	prime	ADJ
iajs-1215	39	7	ideal	ideal	NOUN
iajs-1215	39	8	p	p	NOUN
iajs-1215	39	9	of	of	ADP
iajs-1215	39	10	a	a	DET
iajs-1215	39	11	ring	ring	NOUN
iajs-1215	39	12	r	r	NOUN
iajs-1215	39	13	is	be	AUX
iajs-1215	39	14	a	a	DET
iajs-1215	39	15	weakly	weakly	ADJ
iajs-1215	39	16	prime	prime	ADJ
iajs-1215	39	17	submodule	submodule	NOUN
iajs-1215	39	18	of	of	ADP
iajs-1215	39	19	the	the	DET
iajs-1215	39	20	r	r	NOUN
iajs-1215	39	21	-	-	PUNCT
iajs-1215	39	22	module	module	NOUN
iajs-1215	39	23	r.	r.	NOUN
iajs-1215	39	24	(	(	PUNCT
iajs-1215	39	25	3	3	NUM
iajs-1215	39	26	)	)	PUNCT
iajs-1215	39	27	every	every	DET
iajs-1215	39	28	weakly	weakly	ADJ
iajs-1215	39	29	prime	prime	ADJ
iajs-1215	39	30	submodule	submodule	NOUN
iajs-1215	39	31	is	be	AUX
iajs-1215	39	32	weakly	weakly	ADV
iajs-1215	39	33	primary	primary	ADJ
iajs-1215	39	34	,	,	PUNCT
iajs-1215	39	35	but	but	CCONJ
iajs-1215	39	36	the	the	DET
iajs-1215	39	37	converse	converse	NOUN
iajs-1215	39	38	is	be	AUX
iajs-1215	39	39	false	false	ADJ
iajs-1215	39	40	as	as	SCONJ
iajs-1215	39	41	the	the	DET
iajs-1215	39	42	following	follow	VERB
iajs-1215	39	43	example	example	NOUN
iajs-1215	39	44	shows	show	VERB
iajs-1215	39	45	.	.	PUNCT
iajs-1215	40	1	the	the	DET
iajs-1215	40	2	submodule	submodule	NOUN
iajs-1215	40	3	n	n	NOUN
iajs-1215	40	4	=	=	SYM
iajs-1215	40	5	)	)	PUNCT
iajs-1215	40	6	4	4	NUM
iajs-1215	40	7	(	(	PUNCT
iajs-1215	40	8	of	of	ADP
iajs-1215	40	9	the	the	DET
iajs-1215	40	10	z	z	NOUN
iajs-1215	40	11	-	-	PUNCT
iajs-1215	40	12	module	module	NOUN
iajs-1215	40	13	z8	z8	NOUN
iajs-1215	40	14	is	be	AUX
iajs-1215	40	15	weakly	weakly	ADV
iajs-1215	40	16	primary	primary	ADJ
iajs-1215	40	17	but	but	CCONJ
iajs-1215	40	18	it	it	PRON
iajs-1215	40	19	is	be	AUX
iajs-1215	40	20	not	not	PART
iajs-1215	40	21	weakly	weakly	ADJ
iajs-1215	40	22	prime	prime	ADJ
iajs-1215	40	23	.	.	PUNCT
iajs-1215	41	1	(	(	PUNCT
iajs-1215	41	2	4	4	X
iajs-1215	41	3	)	)	PUNCT
iajs-1215	41	4	it	it	PRON
iajs-1215	41	5	is	be	AUX
iajs-1215	41	6	easy	easy	ADJ
iajs-1215	41	7	to	to	PART
iajs-1215	41	8	check	check	VERB
iajs-1215	41	9	that	that	PRON
iajs-1215	41	10	:	:	PUNCT
iajs-1215	41	11	if	if	SCONJ
iajs-1215	41	12	p	p	NOUN
iajs-1215	41	13	is	be	AUX
iajs-1215	41	14	a	a	DET
iajs-1215	41	15	weakly	weakly	ADJ
iajs-1215	41	16	primary	primary	ADJ
iajs-1215	41	17	submodule	submodule	NOUN
iajs-1215	41	18	of	of	ADP
iajs-1215	41	19	an	an	DET
iajs-1215	41	20	r	r	NOUN
iajs-1215	41	21	-	-	PUNCT
iajs-1215	41	22	module	module	NOUN
iajs-1215	41	23	m	m	NOUN
iajs-1215	41	24	and	and	CCONJ
iajs-1215	41	25	(	(	PUNCT
iajs-1215	41	26	p	p	X
iajs-1215	41	27	:	:	PUNCT
iajs-1215	41	28	m	m	VERB
iajs-1215	41	29	)	)	PUNCT
iajs-1215	41	30	is	be	AUX
iajs-1215	41	31	a	a	DET
iajs-1215	41	32	semiprime	semiprime	NOUN
iajs-1215	41	33	ideal	ideal	NOUN
iajs-1215	41	34	,	,	PUNCT
iajs-1215	41	35	then	then	ADV
iajs-1215	41	36	p	p	NOUN
iajs-1215	41	37	is	be	AUX
iajs-1215	41	38	weakly	weakly	ADV
iajs-1215	41	39	prime	prime	ADJ
iajs-1215	41	40	.	.	PUNCT
iajs-1215	42	1	(	(	PUNCT
iajs-1215	42	2	5	5	NUM
iajs-1215	42	3	)	)	PUNCT
iajs-1215	42	4	(	(	PUNCT
iajs-1215	42	5	a	a	X
iajs-1215	42	6	)	)	PUNCT
iajs-1215	42	7	weakly	weakly	ADJ
iajs-1215	42	8	prime	prime	ADJ
iajs-1215	42	9	submodule	submodule	NOUN
iajs-1215	42	10	need	need	AUX
iajs-1215	42	11	not	not	PART
iajs-1215	42	12	be	be	AUX
iajs-1215	42	13	quasi	quasi	ADJ
iajs-1215	42	14	-	-	ADJ
iajs-1215	42	15	prime	prime	ADJ
iajs-1215	42	16	as	as	SCONJ
iajs-1215	42	17	the	the	DET
iajs-1215	42	18	following	follow	VERB
iajs-1215	42	19	example	example	NOUN
iajs-1215	42	20	shows	show	VERB
iajs-1215	42	21	:	:	PUNCT
iajs-1215	42	22	the	the	DET
iajs-1215	42	23	zero	zero	NUM
iajs-1215	42	24	submodule	submodule	NOUN
iajs-1215	42	25	of	of	ADP
iajs-1215	42	26	the	the	DET
iajs-1215	42	27	z	z	NOUN
iajs-1215	42	28	-	-	PUNCT
iajs-1215	42	29	module	module	NOUN
iajs-1215	42	30	z12	z12	NOUN
iajs-1215	42	31	is	be	AUX
iajs-1215	42	32	weakly	weakly	ADV
iajs-1215	42	33	prime	prime	ADJ
iajs-1215	42	34	,	,	PUNCT
iajs-1215	42	35	but	but	CCONJ
iajs-1215	42	36	it	it	PRON
iajs-1215	42	37	is	be	AUX
iajs-1215	42	38	not	not	PART
iajs-1215	42	39	quasi	quasi	ADJ
iajs-1215	42	40	-	-	NOUN
iajs-1215	42	41	prime	prime	ADJ
iajs-1215	42	42	since	since	SCONJ
iajs-1215	42	43	(	(	PUNCT
iajs-1215	42	44	0	0	NUM
iajs-1215	42	45	:	:	SYM
iajs-1215	42	46	3	3	X
iajs-1215	42	47	)	)	PUNCT
iajs-1215	42	48	=	=	NOUN
iajs-1215	42	49	4z	4z	NOUN
iajs-1215	42	50	which	which	PRON
iajs-1215	42	51	is	be	AUX
iajs-1215	42	52	not	not	PART
iajs-1215	42	53	a	a	DET
iajs-1215	42	54	prime	prime	ADJ
iajs-1215	42	55	ideal	ideal	NOUN
iajs-1215	42	56	of	of	ADP
iajs-1215	42	57	z.	z.	PROPN
iajs-1215	42	58	(	(	PUNCT
iajs-1215	42	59	b	b	X
iajs-1215	42	60	)	)	PUNCT
iajs-1215	42	61	quasi	quasi	ADJ
iajs-1215	42	62	-	-	ADJ
iajs-1215	42	63	prime	prime	ADJ
iajs-1215	42	64	submodule	submodule	NOUN
iajs-1215	42	65	need	need	AUX
iajs-1215	42	66	not	not	PART
iajs-1215	42	67	be	be	AUX
iajs-1215	42	68	weakly	weakly	ADJ
iajs-1215	42	69	prime	prime	ADJ
iajs-1215	42	70	submodule	submodule	NOUN
iajs-1215	42	71	,	,	PUNCT
iajs-1215	42	72	as	as	SCONJ
iajs-1215	42	73	the	the	DET
iajs-1215	42	74	following	follow	VERB
iajs-1215	42	75	example	example	NOUN
iajs-1215	42	76	shows	show	VERB
iajs-1215	42	77	if	if	SCONJ
iajs-1215	42	78	m	m	NOUN
iajs-1215	42	79	is	be	AUX
iajs-1215	42	80	the	the	DET
iajs-1215	42	81	z	z	NOUN
iajs-1215	42	82	-	-	PUNCT
iajs-1215	42	83	module	module	NOUN
iajs-1215	42	84	zz	zz	NOUN
iajs-1215	42	85	,	,	PUNCT
iajs-1215	42	86	and	and	CCONJ
iajs-1215	42	87	n	n	CCONJ
iajs-1215	42	88	=	=	SYM
iajs-1215	42	89	2z(0	2z(0	NUM
iajs-1215	42	90	)	)	PUNCT
iajs-1215	42	91	,	,	PUNCT
iajs-1215	42	92	then	then	ADV
iajs-1215	42	93	n	n	PRON
iajs-1215	42	94	is	be	AUX
iajs-1215	42	95	a	a	DET
iajs-1215	42	96	quasi	quasi	ADJ
iajs-1215	42	97	-	-	ADJ
iajs-1215	42	98	prime	prime	ADJ
iajs-1215	42	99	submodule	submodule	NOUN
iajs-1215	42	100	of	of	ADP
iajs-1215	42	101	m	m	PROPN
iajs-1215	42	102	(	(	PUNCT
iajs-1215	42	103	see	see	VERB
iajs-1215	42	104	(	(	PUNCT
iajs-1215	42	105	4	4	NUM
iajs-1215	42	106	)	)	PUNCT
iajs-1215	42	107	,	,	PUNCT
iajs-1215	42	108	rem.2.1.2(1	rem.2.1.2(1	NOUN
iajs-1215	42	109	)	)	PUNCT
iajs-1215	42	110	)	)	PUNCT
iajs-1215	42	111	.	.	PUNCT
iajs-1215	43	1	but	but	CCONJ
iajs-1215	43	2	n	n	PRON
iajs-1215	43	3	is	be	AUX
iajs-1215	43	4	not	not	PART
iajs-1215	43	5	weakly	weakly	ADJ
iajs-1215	43	6	prime	prime	ADJ
iajs-1215	43	7	submodule	submodule	NOUN
iajs-1215	43	8	,	,	PUNCT
iajs-1215	43	9	since	since	SCONJ
iajs-1215	43	10	(	(	PUNCT
iajs-1215	43	11	0,0	0,0	NOUN
iajs-1215	43	12	)	)	PUNCT
iajs-1215	43	13			NOUN
iajs-1215	43	14	2	2	NUM
iajs-1215	43	15	(	(	PUNCT
iajs-1215	43	16	3,0	3,0	NUM
iajs-1215	43	17	)	)	PUNCT
iajs-1215	43	18			NOUN
iajs-1215	43	19	n	n	CCONJ
iajs-1215	43	20	,	,	PUNCT
iajs-1215	43	21	(	(	PUNCT
iajs-1215	43	22	3,0	3,0	NUM
iajs-1215	43	23	)	)	PUNCT
iajs-1215	43	24			CCONJ
iajs-1215	43	25	n	n	NOUN
iajs-1215	43	26	and	and	CCONJ
iajs-1215	43	27	2	2	NUM
iajs-1215	43	28			X
iajs-1215	43	29	(	(	PUNCT
iajs-1215	43	30	n	n	CCONJ
iajs-1215	43	31	:	:	PUNCT
iajs-1215	43	32	m	m	X
iajs-1215	43	33	)	)	PUNCT
iajs-1215	43	34	=	=	SYM
iajs-1215	43	35	(	(	PUNCT
iajs-1215	43	36	0	0	NUM
iajs-1215	43	37	)	)	PUNCT
iajs-1215	43	38	.	.	PUNCT
iajs-1215	44	1	(	(	PUNCT
iajs-1215	44	2	6	6	NUM
iajs-1215	44	3	)	)	PUNCT
iajs-1215	44	4	if	if	SCONJ
iajs-1215	44	5	p	p	NOUN
iajs-1215	44	6	is	be	AUX
iajs-1215	44	7	a	a	DET
iajs-1215	44	8	proper	proper	ADJ
iajs-1215	44	9	submodule	submodule	NOUN
iajs-1215	44	10	of	of	ADP
iajs-1215	44	11	an	an	DET
iajs-1215	44	12	r	r	NOUN
iajs-1215	44	13	-	-	PUNCT
iajs-1215	44	14	module	module	NOUN
iajs-1215	44	15	m.	m.	NOUN
iajs-1215	44	16	then	then	ADV
iajs-1215	44	17	p	p	NOUN
iajs-1215	44	18	is	be	AUX
iajs-1215	44	19	a	a	DET
iajs-1215	44	20	weakly	weakly	ADJ
iajs-1215	44	21	prime	prime	ADJ
iajs-1215	44	22	r	r	NOUN
iajs-1215	44	23	-	-	PUNCT
iajs-1215	44	24	submodule	submodule	NOUN
iajs-1215	44	25	of	of	ADP
iajs-1215	44	26	m	m	PROPN
iajs-1215	44	27	iff	iff	PROPN
iajs-1215	44	28	p	p	NOUN
iajs-1215	44	29	is	be	AUX
iajs-1215	44	30	a	a	DET
iajs-1215	44	31	weakly	weakly	ADJ
iajs-1215	44	32	prime	prime	ADJ
iajs-1215	44	33	r	r	NOUN
iajs-1215	44	34	/	/	SYM
iajs-1215	44	35	i	i	NOUN
iajs-1215	44	36	-	-	PUNCT
iajs-1215	44	37	submodule	submodule	NOUN
iajs-1215	44	38	of	of	ADP
iajs-1215	44	39	m	m	PROPN
iajs-1215	44	40	,	,	PUNCT
iajs-1215	44	41	where	where	SCONJ
iajs-1215	44	42	i	i	PRON
iajs-1215	44	43	is	be	AUX
iajs-1215	44	44	an	an	DET
iajs-1215	44	45	ideal	ideal	NOUN
iajs-1215	44	46	of	of	ADP
iajs-1215	44	47	r	r	NOUN
iajs-1215	44	48	with	with	ADP
iajs-1215	44	49	i	i	PRON
iajs-1215	44	50			PROPN
iajs-1215	44	51	ann	ann	PROPN
iajs-1215	44	52	m.	m.	NOUN
iajs-1215	44	53	the	the	DET
iajs-1215	44	54	following	following	ADJ
iajs-1215	44	55	result	result	NOUN
iajs-1215	44	56	gives	give	VERB
iajs-1215	44	57	characterizations	characterization	NOUN
iajs-1215	44	58	of	of	ADP
iajs-1215	44	59	weakly	weakly	ADJ
iajs-1215	44	60	prime	prime	ADJ
iajs-1215	44	61	submodules	submodule	NOUN
iajs-1215	44	62	.	.	PUNCT
iajs-1215	45	1	theorem	theorem	VERB
iajs-1215	45	2	2.2	2.2	NUM
iajs-1215	45	3	:	:	PUNCT
iajs-1215	45	4	let	let	VERB
iajs-1215	45	5	m	m	PRON
iajs-1215	45	6	be	be	AUX
iajs-1215	45	7	an	an	DET
iajs-1215	45	8	r	r	NOUN
iajs-1215	45	9	-	-	PUNCT
iajs-1215	45	10	module	module	NOUN
iajs-1215	45	11	.	.	PUNCT
iajs-1215	46	1	the	the	DET
iajs-1215	46	2	following	follow	VERB
iajs-1215	46	3	asserations	asseration	NOUN
iajs-1215	46	4	are	be	AUX
iajs-1215	46	5	equivalent	equivalent	ADJ
iajs-1215	46	6	:	:	PUNCT
iajs-1215	46	7	1	1	X
iajs-1215	46	8	.	.	X
iajs-1215	46	9	p	p	NOUN
iajs-1215	46	10	is	be	AUX
iajs-1215	46	11	a	a	DET
iajs-1215	46	12	weakly	weakly	ADJ
iajs-1215	46	13	prime	prime	ADJ
iajs-1215	46	14	submodule	submodule	NOUN
iajs-1215	46	15	of	of	ADP
iajs-1215	46	16	m.	m.	NOUN
iajs-1215	46	17	2	2	NUM
iajs-1215	46	18	.	.	PUNCT
iajs-1215	47	1	(	(	PUNCT
iajs-1215	47	2	p	p	X
iajs-1215	47	3	:x	:x	PROPN
iajs-1215	47	4	)	)	PUNCT
iajs-1215	48	1	=	=	PUNCT
iajs-1215	48	2	(	(	PUNCT
iajs-1215	48	3	p	p	X
iajs-1215	48	4	:	:	PUNCT
iajs-1215	48	5	m	m	NOUN
iajs-1215	48	6	)	)	PUNCT
iajs-1215	48	7			NOUN
iajs-1215	48	8	(	(	PUNCT
iajs-1215	48	9	0	0	NUM
iajs-1215	48	10	:x	:x	PROPN
iajs-1215	48	11	)	)	PUNCT
iajs-1215	48	12	for	for	ADP
iajs-1215	48	13	any	any	DET
iajs-1215	48	14	x	x	ADJ
iajs-1215	48	15			NOUN
iajs-1215	48	16	m	m	PROPN
iajs-1215	48	17	,	,	PUNCT
iajs-1215	48	18	x	x	PROPN
iajs-1215	48	19			VERB
iajs-1215	48	20	p.	p.	NOUN
iajs-1215	48	21	3	3	NUM
iajs-1215	48	22	.	.	PUNCT
iajs-1215	48	23	(	(	PUNCT
iajs-1215	48	24	p	p	X
iajs-1215	48	25	:x	:x	PROPN
iajs-1215	48	26	)	)	PUNCT
iajs-1215	48	27	=	=	PUNCT
iajs-1215	49	1	(	(	PUNCT
iajs-1215	49	2	p	p	X
iajs-1215	49	3	:	:	PUNCT
iajs-1215	49	4	m	m	NUM
iajs-1215	49	5	)	)	PUNCT
iajs-1215	49	6	or	or	CCONJ
iajs-1215	49	7	(	(	PUNCT
iajs-1215	49	8	p	p	X
iajs-1215	49	9	:x	:x	PROPN
iajs-1215	49	10	)	)	PUNCT
iajs-1215	49	11	=	=	PUNCT
iajs-1215	50	1	(	(	PUNCT
iajs-1215	50	2	0	0	NUM
iajs-1215	50	3	:x	:x	PROPN
iajs-1215	50	4	)	)	PUNCT
iajs-1215	50	5	for	for	ADP
iajs-1215	50	6	any	any	DET
iajs-1215	50	7	x	x	ADJ
iajs-1215	50	8			NOUN
iajs-1215	50	9	m	m	PROPN
iajs-1215	50	10	,	,	PUNCT
iajs-1215	50	11	x	x	PROPN
iajs-1215	50	12			VERB
iajs-1215	50	13	p.	p.	NOUN
iajs-1215	50	14	4	4	NUM
iajs-1215	50	15	.	.	PUNCT
iajs-1215	51	1	if	if	SCONJ
iajs-1215	51	2	(	(	PUNCT
iajs-1215	51	3	0	0	X
iajs-1215	51	4	)	)	PUNCT
iajs-1215	51	5			NOUN
iajs-1215	51	6	(	(	PUNCT
iajs-1215	51	7	a)n	a)n	ADJ
iajs-1215	51	8			PROPN
iajs-1215	51	9	p	p	PROPN
iajs-1215	51	10	,	,	PUNCT
iajs-1215	51	11	then	then	ADV
iajs-1215	51	12	either	either	CCONJ
iajs-1215	51	13	n	n	CCONJ
iajs-1215	51	14			PROPN
iajs-1215	51	15	p	p	PROPN
iajs-1215	51	16	or	or	CCONJ
iajs-1215	51	17	(	(	PUNCT
iajs-1215	51	18	a	a	NOUN
iajs-1215	51	19	)	)	PUNCT
iajs-1215	51	20			PROPN
iajs-1215	51	21	(	(	PUNCT
iajs-1215	51	22	p	p	X
iajs-1215	51	23	:	:	PUNCT
iajs-1215	51	24	m	m	NUM
iajs-1215	51	25	)	)	PUNCT
iajs-1215	51	26	,	,	PUNCT
iajs-1215	51	27	where	where	SCONJ
iajs-1215	51	28	a	a	DET
iajs-1215	51	29			NOUN
iajs-1215	51	30	r	r	NOUN
iajs-1215	51	31	,	,	PUNCT
iajs-1215	51	32	n	n	X
iajs-1215	51	33	is	be	AUX
iajs-1215	51	34	a	a	DET
iajs-1215	51	35	submodule	submodule	NOUN
iajs-1215	51	36	of	of	ADP
iajs-1215	51	37	m.	m.	NOUN
iajs-1215	51	38	proof	proof	NOUN
iajs-1215	51	39	.	.	PUNCT
iajs-1215	52	1	(	(	PUNCT
iajs-1215	52	2	1	1	X
iajs-1215	52	3	)	)	PUNCT
iajs-1215	52	4			NOUN
iajs-1215	52	5	(	(	PUNCT
iajs-1215	52	6	2	2	X
iajs-1215	52	7	)	)	PUNCT
iajs-1215	52	8	let	let	VERB
iajs-1215	52	9	r	r	NOUN
iajs-1215	52	10			NOUN
iajs-1215	52	11	(	(	PUNCT
iajs-1215	52	12	p	p	X
iajs-1215	52	13	:x	:x	PROPN
iajs-1215	52	14	)	)	PUNCT
iajs-1215	53	1	and	and	CCONJ
iajs-1215	53	2	x	x	X
iajs-1215	53	3			X
iajs-1215	53	4	p.	p.	NOUN
iajs-1215	53	5	then	then	ADV
iajs-1215	54	1	r	r	NOUN
iajs-1215	54	2	x	x	X
iajs-1215	54	3			PROPN
iajs-1215	54	4	p.	p.	NOUN
iajs-1215	54	5	suppose	suppose	VERB
iajs-1215	55	1	r	r	NOUN
iajs-1215	55	2	x	x	SYM
iajs-1215	55	3	≠	≠	PROPN
iajs-1215	55	4	0	0	NUM
iajs-1215	55	5	.	.	PUNCT
iajs-1215	56	1	hence	hence	ADV
iajs-1215	56	2	r	r	VERB
iajs-1215	56	3			NOUN
iajs-1215	56	4	(	(	PUNCT
iajs-1215	56	5	p	p	X
iajs-1215	56	6	:	:	PUNCT
iajs-1215	56	7	m	m	NUM
iajs-1215	56	8	)	)	PUNCT
iajs-1215	56	9	because	because	SCONJ
iajs-1215	56	10	p	p	NOUN
iajs-1215	56	11	is	be	AUX
iajs-1215	56	12	weakly	weakly	ADV
iajs-1215	56	13	prime	prime	ADJ
iajs-1215	56	14	and	and	CCONJ
iajs-1215	56	15	x	x	NOUN
iajs-1215	56	16			NOUN
iajs-1215	56	17	p.	p.	NOUN
iajs-1215	56	18	if	if	SCONJ
iajs-1215	56	19	r	r	NOUN
iajs-1215	56	20	x	x	NOUN
iajs-1215	56	21	=	=	SYM
iajs-1215	56	22	0	0	NUM
iajs-1215	56	23	,	,	PUNCT
iajs-1215	56	24	then	then	ADV
iajs-1215	56	25	r	r	NOUN
iajs-1215	56	26			PROPN
iajs-1215	56	27	(	(	PUNCT
iajs-1215	56	28	0	0	NUM
iajs-1215	56	29	:x	:x	NUM
iajs-1215	56	30	)	)	PUNCT
iajs-1215	56	31	.	.	PUNCT
iajs-1215	57	1	thus	thus	ADV
iajs-1215	57	2	(	(	PUNCT
iajs-1215	57	3	p	p	X
iajs-1215	57	4	:x	:x	PROPN
iajs-1215	57	5	)	)	PUNCT
iajs-1215	57	6			PROPN
iajs-1215	57	7	(	(	PUNCT
iajs-1215	57	8	p	p	X
iajs-1215	57	9	:	:	PUNCT
iajs-1215	57	10	m	m	NOUN
iajs-1215	57	11	)	)	PUNCT
iajs-1215	57	12			NOUN
iajs-1215	57	13	(	(	PUNCT
iajs-1215	57	14	0	0	NUM
iajs-1215	57	15	:x	:x	NUM
iajs-1215	57	16	)	)	PUNCT
iajs-1215	57	17	.	.	PUNCT
iajs-1215	58	1	now	now	ADV
iajs-1215	58	2	if	if	SCONJ
iajs-1215	58	3	r	r	NOUN
iajs-1215	58	4			NOUN
iajs-1215	58	5	(	(	PUNCT
iajs-1215	58	6	p	p	X
iajs-1215	58	7	:	:	PUNCT
iajs-1215	58	8	m	m	NOUN
iajs-1215	58	9	)	)	PUNCT
iajs-1215	58	10			NOUN
iajs-1215	58	11	(	(	PUNCT
iajs-1215	58	12	0	0	NUM
iajs-1215	58	13	:x	:x	NUM
iajs-1215	58	14	)	)	PUNCT
iajs-1215	58	15	,	,	PUNCT
iajs-1215	58	16	then	then	ADV
iajs-1215	58	17	either	either	CCONJ
iajs-1215	58	18	r	r	NOUN
iajs-1215	58	19			NOUN
iajs-1215	58	20	(	(	PUNCT
iajs-1215	58	21	p	p	X
iajs-1215	58	22	:	:	PUNCT
iajs-1215	58	23	m	m	NUM
iajs-1215	58	24	)	)	PUNCT
iajs-1215	58	25	or	or	CCONJ
iajs-1215	58	26	r	r	NOUN
iajs-1215	58	27			NOUN
iajs-1215	58	28	(	(	PUNCT
iajs-1215	58	29	0	0	NUM
iajs-1215	58	30	:x	:x	NUM
iajs-1215	58	31	)	)	PUNCT
iajs-1215	58	32	.	.	PUNCT
iajs-1215	59	1	hence	hence	ADV
iajs-1215	59	2	,	,	PUNCT
iajs-1215	59	3	when	when	SCONJ
iajs-1215	59	4	r	r	NOUN
iajs-1215	59	5			NOUN
iajs-1215	59	6	(	(	PUNCT
iajs-1215	59	7	0	0	NUM
iajs-1215	59	8	:x	:x	NUM
iajs-1215	59	9	)	)	PUNCT
iajs-1215	59	10	,	,	PUNCT
iajs-1215	59	11	r	r	NOUN
iajs-1215	59	12	x	x	SYM
iajs-1215	59	13	=	=	SYM
iajs-1215	59	14	0	0	NUM
iajs-1215	59	15			NOUN
iajs-1215	59	16	p	p	NOUN
iajs-1215	59	17	and	and	CCONJ
iajs-1215	59	18	so	so	ADV
iajs-1215	59	19	r	r	NOUN
iajs-1215	59	20			NOUN
iajs-1215	59	21	(	(	PUNCT
iajs-1215	59	22	p	p	X
iajs-1215	59	23	:x	:x	PROPN
iajs-1215	59	24	)	)	PUNCT
iajs-1215	59	25	.	.	PUNCT
iajs-1215	60	1	if	if	SCONJ
iajs-1215	60	2	r	r	NOUN
iajs-1215	60	3			NOUN
iajs-1215	60	4	(	(	PUNCT
iajs-1215	60	5	p	p	X
iajs-1215	60	6	:	:	PUNCT
iajs-1215	60	7	m	m	VERB
iajs-1215	60	8	)	)	PUNCT
iajs-1215	60	9	then	then	ADV
iajs-1215	60	10	r	r	NOUN
iajs-1215	60	11	m	m	NOUN
iajs-1215	60	12			PROPN
iajs-1215	60	13	p	p	X
iajs-1215	60	14	,	,	PUNCT
iajs-1215	60	15	and	and	CCONJ
iajs-1215	60	16	this	this	PRON
iajs-1215	60	17	implies	imply	VERB
iajs-1215	60	18	r	r	NOUN
iajs-1215	60	19	x	x	X
iajs-1215	60	20			NOUN
iajs-1215	61	1	p.	p.	NOUN
iajs-1215	61	2	hence	hence	ADV
iajs-1215	61	3	r	r	VERB
iajs-1215	61	4			NOUN
iajs-1215	61	5	(	(	PUNCT
iajs-1215	61	6	p	p	X
iajs-1215	61	7	:x	:x	PROPN
iajs-1215	61	8	)	)	PUNCT
iajs-1215	61	9	.	.	PUNCT
iajs-1215	62	1	thus	thus	ADV
iajs-1215	62	2	(	(	PUNCT
iajs-1215	62	3	p	p	X
iajs-1215	62	4	:	:	PUNCT
iajs-1215	62	5	m	m	NOUN
iajs-1215	62	6	)	)	PUNCT
iajs-1215	62	7			NOUN
iajs-1215	62	8	(	(	PUNCT
iajs-1215	62	9	0	0	NUM
iajs-1215	62	10	:x	:x	PROPN
iajs-1215	62	11	)	)	PUNCT
iajs-1215	62	12			PROPN
iajs-1215	62	13	(	(	PUNCT
iajs-1215	62	14	p	p	X
iajs-1215	62	15	:x	:x	PROPN
iajs-1215	62	16	)	)	PUNCT
iajs-1215	62	17	and	and	CCONJ
iajs-1215	62	18	therefore	therefore	ADV
iajs-1215	62	19	(	(	PUNCT
iajs-1215	62	20	p	p	X
iajs-1215	62	21	:	:	PUNCT
iajs-1215	62	22	m	m	NOUN
iajs-1215	62	23	)	)	PUNCT
iajs-1215	62	24			NOUN
iajs-1215	62	25	(	(	PUNCT
iajs-1215	62	26	0	0	NUM
iajs-1215	62	27	:x	:x	PROPN
iajs-1215	62	28	)	)	PUNCT
iajs-1215	62	29	=	=	PUNCT
iajs-1215	62	30	(	(	PUNCT
iajs-1215	62	31	p	p	X
iajs-1215	62	32	:x	:x	PROPN
iajs-1215	62	33	)	)	PUNCT
iajs-1215	62	34	.	.	PUNCT
iajs-1215	63	1	(	(	PUNCT
iajs-1215	63	2	2	2	X
iajs-1215	63	3	)	)	PUNCT
iajs-1215	63	4			NOUN
iajs-1215	63	5	(	(	PUNCT
iajs-1215	63	6	3	3	X
iajs-1215	63	7	)	)	PUNCT
iajs-1215	63	8	it	it	PRON
iajs-1215	63	9	is	be	AUX
iajs-1215	63	10	well	well	ADV
iajs-1215	63	11	-	-	PUNCT
iajs-1215	63	12	known	know	VERB
iajs-1215	63	13	that	that	SCONJ
iajs-1215	63	14	the	the	DET
iajs-1215	63	15	union	union	NOUN
iajs-1215	63	16	of	of	ADP
iajs-1215	63	17	two	two	NUM
iajs-1215	63	18	ideals	ideal	NOUN
iajs-1215	63	19	a	a	PRON
iajs-1215	63	20	,	,	PUNCT
iajs-1215	63	21	b	b	NOUN
iajs-1215	63	22	of	of	ADP
iajs-1215	63	23	r	r	NOUN
iajs-1215	63	24	is	be	AUX
iajs-1215	63	25	an	an	DET
iajs-1215	63	26	ideal	ideal	NOUN
iajs-1215	63	27	if	if	SCONJ
iajs-1215	63	28	a	a	DET
iajs-1215	63	29			PROPN
iajs-1215	63	30	b	b	PROPN
iajs-1215	63	31	or	or	CCONJ
iajs-1215	63	32	b	b	NOUN
iajs-1215	63	33			PROPN
iajs-1215	63	34	a.	a.	NOUN
iajs-1215	63	35	by	by	ADP
iajs-1215	63	36	condition	condition	NOUN
iajs-1215	63	37	,	,	PUNCT
iajs-1215	63	38	the	the	DET
iajs-1215	63	39	ideals	ideal	NOUN
iajs-1215	63	40	(	(	PUNCT
iajs-1215	63	41	p	p	X
iajs-1215	63	42	:	:	PUNCT
iajs-1215	63	43	m	m	VERB
iajs-1215	63	44	)	)	PUNCT
iajs-1215	63	45	is	be	AUX
iajs-1215	63	46	the	the	DET
iajs-1215	63	47	union	union	NOUN
iajs-1215	63	48	of	of	ADP
iajs-1215	63	49	the	the	DET
iajs-1215	63	50	ideals	ideal	NOUN
iajs-1215	63	51	(	(	PUNCT
iajs-1215	63	52	p	p	X
iajs-1215	63	53	:	:	PUNCT
iajs-1215	63	54	m	m	NUM
iajs-1215	63	55	)	)	PUNCT
iajs-1215	63	56	,	,	PUNCT
iajs-1215	63	57	(	(	PUNCT
iajs-1215	63	58	0	0	NUM
iajs-1215	63	59	:x	:x	NUM
iajs-1215	63	60	)	)	PUNCT
iajs-1215	63	61	,	,	PUNCT
iajs-1215	63	62	so	so	ADV
iajs-1215	63	63	either	either	ADV
iajs-1215	63	64	(	(	PUNCT
iajs-1215	63	65	p	p	X
iajs-1215	63	66	:	:	PUNCT
iajs-1215	63	67	m	m	NOUN
iajs-1215	63	68	)	)	PUNCT
iajs-1215	63	69			PROPN
iajs-1215	63	70	(	(	PUNCT
iajs-1215	63	71	0	0	NUM
iajs-1215	63	72	:x	:x	NUM
iajs-1215	63	73	)	)	PUNCT
iajs-1215	63	74	or	or	CCONJ
iajs-1215	63	75	(	(	PUNCT
iajs-1215	63	76	0	0	NUM
iajs-1215	63	77	:x	:x	PROPN
iajs-1215	63	78	)	)	PUNCT
iajs-1215	63	79			PROPN
iajs-1215	63	80	(	(	PUNCT
iajs-1215	63	81	p	p	X
iajs-1215	63	82	:	:	PUNCT
iajs-1215	63	83	m	m	NUM
iajs-1215	63	84	)	)	PUNCT
iajs-1215	63	85	.	.	PUNCT
iajs-1215	64	1	thus	thus	ADV
iajs-1215	64	2	either	either	CCONJ
iajs-1215	64	3	(	(	PUNCT
iajs-1215	64	4	p	p	X
iajs-1215	64	5	:x	:x	PROPN
iajs-1215	64	6	)	)	PUNCT
iajs-1215	64	7	=	=	PUNCT
iajs-1215	64	8	(	(	PUNCT
iajs-1215	64	9	0	0	NUM
iajs-1215	64	10	:x	:x	NUM
iajs-1215	64	11	)	)	PUNCT
iajs-1215	64	12	or	or	CCONJ
iajs-1215	64	13	(	(	PUNCT
iajs-1215	64	14	p	p	X
iajs-1215	64	15	:x	:x	PROPN
iajs-1215	64	16	)	)	PUNCT
iajs-1215	65	1	=	=	PUNCT
iajs-1215	65	2	(	(	PUNCT
iajs-1215	65	3	p	p	X
iajs-1215	65	4	:	:	PUNCT
iajs-1215	65	5	m	m	NUM
iajs-1215	65	6	)	)	PUNCT
iajs-1215	65	7	.	.	PUNCT
iajs-1215	66	1	ibn	ibn	PROPN
iajs-1215	66	2	alhaitham	alhaitham	NOUN
iajs-1215	67	1	j.	j.	PROPN
iajs-1215	68	1	fo	fo	ADP
iajs-1215	68	2	r	r	NOUN
iajs-1215	68	3	pure	pure	ADJ
iajs-1215	68	4	&	&	CCONJ
iajs-1215	68	5	appl	appl	PROPN
iajs-1215	68	6	.	.	PUNCT
iajs-1215	69	1	sc	sc	PROPN
iajs-1215	70	1	i	i	PRON
iajs-1215	70	2	vo	vo	INTJ
iajs-1215	70	3	l.22	l.22	X
iajs-1215	70	4	(	(	PUNCT
iajs-1215	70	5	3	3	NUM
iajs-1215	70	6	)	)	PUNCT
iajs-1215	70	7	2009	2009	NUM
iajs-1215	70	8	(	(	PUNCT
iajs-1215	70	9	3	3	NUM
iajs-1215	70	10	)	)	PUNCT
iajs-1215	70	11			NOUN
iajs-1215	70	12	(	(	PUNCT
iajs-1215	70	13	4	4	X
iajs-1215	70	14	)	)	PUNCT
iajs-1215	70	15	if	if	SCONJ
iajs-1215	70	16	0	0	NUM
iajs-1215	70	17			NOUN
iajs-1215	70	18	(	(	PUNCT
iajs-1215	70	19	a	a	NOUN
iajs-1215	70	20	)	)	PUNCT
iajs-1215	70	21	n	n	NOUN
iajs-1215	70	22			PROPN
iajs-1215	70	23	p.	p.	NOUN
iajs-1215	70	24	suppose	suppose	VERB
iajs-1215	70	25	that	that	SCONJ
iajs-1215	70	26	n	n	PROPN
iajs-1215	70	27			PROPN
iajs-1215	70	28	p	p	X
iajs-1215	70	29	and	and	CCONJ
iajs-1215	70	30	(	(	PUNCT
iajs-1215	70	31	a	a	X
iajs-1215	70	32	)	)	PUNCT
iajs-1215	70	33			PROPN
iajs-1215	70	34	(	(	PUNCT
iajs-1215	70	35	p	p	X
iajs-1215	70	36	:	:	PUNCT
iajs-1215	70	37	m	m	NUM
iajs-1215	70	38	)	)	PUNCT
iajs-1215	70	39	.	.	PUNCT
iajs-1215	71	1	n	n	PROPN
iajs-1215	71	2			PROPN
iajs-1215	71	3	p	p	PROPN
iajs-1215	71	4	implies	imply	VERB
iajs-1215	71	5	that	that	SCONJ
iajs-1215	71	6	there	there	PRON
iajs-1215	71	7	exists	exist	VERB
iajs-1215	71	8	x	x	PUNCT
iajs-1215	71	9			NOUN
iajs-1215	71	10	n	n	CCONJ
iajs-1215	71	11	and	and	CCONJ
iajs-1215	71	12	x	x	ADJ
iajs-1215	71	13			NOUN
iajs-1215	71	14	p	p	X
iajs-1215	71	15	,	,	PUNCT
iajs-1215	71	16	hence	hence	ADV
iajs-1215	71	17	ax	ax	VERB
iajs-1215	71	18			NOUN
iajs-1215	71	19	an	an	DET
iajs-1215	71	20			PROPN
iajs-1215	71	21	p	p	X
iajs-1215	71	22	;	;	PUNCT
iajs-1215	71	23	that	that	PRON
iajs-1215	71	24	is	be	AUX
iajs-1215	71	25	a	a	DET
iajs-1215	71	26			NOUN
iajs-1215	71	27	(	(	PUNCT
iajs-1215	71	28	p	p	X
iajs-1215	71	29	:x	:x	PROPN
iajs-1215	71	30	)	)	PUNCT
iajs-1215	71	31	.	.	PUNCT
iajs-1215	72	1	by	by	ADP
iajs-1215	72	2	condition	condition	NOUN
iajs-1215	72	3	(	(	PUNCT
iajs-1215	72	4	3	3	NUM
iajs-1215	72	5	)	)	PUNCT
iajs-1215	72	6	,	,	PUNCT
iajs-1215	72	7	either	either	CCONJ
iajs-1215	72	8	(	(	PUNCT
iajs-1215	72	9	p	p	X
iajs-1215	72	10	:x	:x	PROPN
iajs-1215	72	11	)	)	PUNCT
iajs-1215	73	1	=	=	PUNCT
iajs-1215	73	2	(	(	PUNCT
iajs-1215	73	3	p	p	X
iajs-1215	73	4	:	:	PUNCT
iajs-1215	73	5	m	m	NUM
iajs-1215	73	6	)	)	PUNCT
iajs-1215	73	7	or	or	CCONJ
iajs-1215	73	8	(	(	PUNCT
iajs-1215	73	9	p	p	X
iajs-1215	73	10	:x	:x	PROPN
iajs-1215	73	11	)	)	PUNCT
iajs-1215	73	12	=	=	PUNCT
iajs-1215	74	1	(	(	PUNCT
iajs-1215	74	2	0	0	NUM
iajs-1215	74	3	:x	:x	NUM
iajs-1215	74	4	)	)	PUNCT
iajs-1215	74	5	.	.	PUNCT
iajs-1215	75	1	if	if	SCONJ
iajs-1215	75	2	(	(	PUNCT
iajs-1215	75	3	p	p	X
iajs-1215	75	4	:x	:x	PROPN
iajs-1215	75	5	)	)	PUNCT
iajs-1215	75	6	=	=	PUNCT
iajs-1215	76	1	(	(	PUNCT
iajs-1215	76	2	p	p	X
iajs-1215	76	3	:	:	PUNCT
iajs-1215	76	4	m	m	NUM
iajs-1215	76	5	)	)	PUNCT
iajs-1215	76	6	,	,	PUNCT
iajs-1215	76	7	we	we	PRON
iajs-1215	76	8	get	get	VERB
iajs-1215	76	9	a	a	DET
iajs-1215	76	10			NOUN
iajs-1215	76	11	(	(	PUNCT
iajs-1215	76	12	p	p	X
iajs-1215	76	13	:	:	PUNCT
iajs-1215	76	14	m	m	VERB
iajs-1215	76	15	)	)	PUNCT
iajs-1215	76	16	which	which	PRON
iajs-1215	76	17	is	be	AUX
iajs-1215	76	18	a	a	DET
iajs-1215	76	19	contradiction	contradiction	NOUN
iajs-1215	76	20	.	.	PUNCT
iajs-1215	77	1	thus	thus	ADV
iajs-1215	77	2	(	(	PUNCT
iajs-1215	77	3	p	p	X
iajs-1215	77	4	:x	:x	PROPN
iajs-1215	77	5	)	)	PUNCT
iajs-1215	77	6	=	=	PUNCT
iajs-1215	78	1	(	(	PUNCT
iajs-1215	78	2	0	0	NUM
iajs-1215	78	3	:x	:x	NUM
iajs-1215	78	4	)	)	PUNCT
iajs-1215	79	1	and	and	CCONJ
iajs-1215	79	2	so	so	ADV
iajs-1215	79	3	ax	ax	NOUN
iajs-1215	79	4	=	=	NOUN
iajs-1215	79	5	0	0	X
iajs-1215	79	6	.	.	PUNCT
iajs-1215	80	1	on	on	ADP
iajs-1215	80	2	the	the	DET
iajs-1215	80	3	other	other	ADJ
iajs-1215	80	4	hand	hand	NOUN
iajs-1215	80	5	,	,	PUNCT
iajs-1215	80	6	0	0	NUM
iajs-1215	80	7			NOUN
iajs-1215	80	8	(	(	PUNCT
iajs-1215	80	9	a	a	NOUN
iajs-1215	80	10	)	)	PUNCT
iajs-1215	80	11	n	n	NOUN
iajs-1215	80	12			PROPN
iajs-1215	80	13	p	p	PROPN
iajs-1215	80	14	implies	imply	VERB
iajs-1215	80	15	that	that	SCONJ
iajs-1215	80	16	there	there	PRON
iajs-1215	80	17	exists	exist	VERB
iajs-1215	80	18	y	y	PROPN
iajs-1215	80	19			PROPN
iajs-1215	80	20	n	n	CCONJ
iajs-1215	80	21	such	such	ADJ
iajs-1215	80	22	that	that	DET
iajs-1215	80	23	0	0	NUM
iajs-1215	80	24			PROPN
iajs-1215	80	25	ay	ay	NOUN
iajs-1215	80	26			PROPN
iajs-1215	80	27	p	p	PROPN
iajs-1215	81	1	and	and	CCONJ
iajs-1215	81	2	so	so	ADV
iajs-1215	81	3	a	a	DET
iajs-1215	81	4			NOUN
iajs-1215	81	5	(	(	PUNCT
iajs-1215	81	6	p	p	X
iajs-1215	81	7	:	:	PUNCT
iajs-1215	81	8	y	y	PROPN
iajs-1215	81	9	)	)	PUNCT
iajs-1215	81	10	.	.	PUNCT
iajs-1215	82	1	moreover	moreover	ADV
iajs-1215	82	2	we	we	PRON
iajs-1215	82	3	can	can	AUX
iajs-1215	82	4	see	see	VERB
iajs-1215	82	5	that	that	SCONJ
iajs-1215	82	6	y	y	PROPN
iajs-1215	82	7			NOUN
iajs-1215	82	8	p	p	X
iajs-1215	82	9	,	,	PUNCT
iajs-1215	82	10	for	for	ADP
iajs-1215	82	11	if	if	SCONJ
iajs-1215	82	12	we	we	PRON
iajs-1215	82	13	assume	assume	VERB
iajs-1215	82	14	that	that	SCONJ
iajs-1215	82	15	y	y	PROPN
iajs-1215	82	16			PUNCT
iajs-1215	82	17	p	p	X
iajs-1215	82	18	,	,	PUNCT
iajs-1215	82	19	then	then	ADV
iajs-1215	82	20	by	by	ADP
iajs-1215	82	21	condition	condition	NOUN
iajs-1215	82	22	3	3	NUM
iajs-1215	82	23	,	,	PUNCT
iajs-1215	82	24	either	either	CCONJ
iajs-1215	82	25	(	(	PUNCT
iajs-1215	82	26	p	p	X
iajs-1215	82	27	:	:	PUNCT
iajs-1215	82	28	y	y	NOUN
iajs-1215	82	29	)	)	PUNCT
iajs-1215	82	30	=	=	PUNCT
iajs-1215	83	1	(	(	PUNCT
iajs-1215	83	2	p	p	X
iajs-1215	83	3	:	:	PUNCT
iajs-1215	83	4	m	m	NUM
iajs-1215	83	5	)	)	PUNCT
iajs-1215	83	6	or	or	CCONJ
iajs-1215	83	7	(	(	PUNCT
iajs-1215	83	8	p	p	X
iajs-1215	83	9	:	:	NOUN
iajs-1215	83	10	y	y	NOUN
iajs-1215	83	11	)	)	PUNCT
iajs-1215	83	12	=	=	SYM
iajs-1215	84	1	(	(	PUNCT
iajs-1215	84	2	0	0	NUM
iajs-1215	84	3	:	:	SYM
iajs-1215	84	4	y	y	NOUN
iajs-1215	84	5	)	)	PUNCT
iajs-1215	84	6	.	.	PUNCT
iajs-1215	85	1	if	if	SCONJ
iajs-1215	85	2	(	(	PUNCT
iajs-1215	85	3	p	p	X
iajs-1215	85	4	:	:	NOUN
iajs-1215	85	5	y	y	NOUN
iajs-1215	85	6	)	)	PUNCT
iajs-1215	85	7	=	=	PUNCT
iajs-1215	86	1	(	(	PUNCT
iajs-1215	86	2	p	p	X
iajs-1215	86	3	:	:	PUNCT
iajs-1215	86	4	m	m	NUM
iajs-1215	86	5	)	)	PUNCT
iajs-1215	86	6	,	,	PUNCT
iajs-1215	86	7	then	then	ADV
iajs-1215	86	8	a	a	DET
iajs-1215	86	9			NOUN
iajs-1215	86	10	(	(	PUNCT
iajs-1215	86	11	p	p	X
iajs-1215	86	12	:	:	PUNCT
iajs-1215	86	13	m	m	VERB
iajs-1215	86	14	)	)	PUNCT
iajs-1215	86	15	which	which	PRON
iajs-1215	86	16	is	be	AUX
iajs-1215	86	17	a	a	DET
iajs-1215	86	18	contradiction	contradiction	NOUN
iajs-1215	86	19	.	.	PUNCT
iajs-1215	87	1	if	if	SCONJ
iajs-1215	87	2	(	(	PUNCT
iajs-1215	87	3	p	p	X
iajs-1215	87	4	:	:	NOUN
iajs-1215	87	5	y	y	NOUN
iajs-1215	87	6	)	)	PUNCT
iajs-1215	87	7	=	=	SYM
iajs-1215	87	8	(	(	PUNCT
iajs-1215	87	9	0,y	0,y	NOUN
iajs-1215	87	10	)	)	PUNCT
iajs-1215	87	11	,	,	PUNCT
iajs-1215	87	12	we	we	PRON
iajs-1215	87	13	get	get	VERB
iajs-1215	87	14	ay	ay	NOUN
iajs-1215	87	15	=	=	SYM
iajs-1215	87	16	0	0	NUM
iajs-1215	87	17	which	which	PRON
iajs-1215	87	18	is	be	AUX
iajs-1215	87	19	a	a	DET
iajs-1215	87	20	contradiction	contradiction	NOUN
iajs-1215	87	21	.	.	PUNCT
iajs-1215	88	1	moreover	moreover	ADV
iajs-1215	88	2	0	0	NUM
iajs-1215	88	3			PROPN
iajs-1215	88	4	ay	ay	NOUN
iajs-1215	88	5	=	=	PUNCT
iajs-1215	88	6	ay	ay	PROPN
iajs-1215	88	7	+	+	NUM
iajs-1215	88	8	ax	ax	NOUN
iajs-1215	88	9	=	=	PUNCT
iajs-1215	88	10	a(y	a(y	PROPN
iajs-1215	88	11	+	+	CCONJ
iajs-1215	88	12	x	x	X
iajs-1215	88	13	)	)	PUNCT
iajs-1215	88	14			NOUN
iajs-1215	88	15	p	p	X
iajs-1215	88	16	;	;	PUNCT
iajs-1215	88	17	that	that	PRON
iajs-1215	88	18	is	be	AUX
iajs-1215	88	19	a	a	DET
iajs-1215	88	20			NOUN
iajs-1215	88	21	(	(	PUNCT
iajs-1215	88	22	p	p	X
iajs-1215	88	23	:	:	PUNCT
iajs-1215	88	24	y	y	PROPN
iajs-1215	88	25	+	+	PROPN
iajs-1215	88	26	x	x	NOUN
iajs-1215	88	27	)	)	PUNCT
iajs-1215	88	28	.	.	PUNCT
iajs-1215	89	1	but	but	CCONJ
iajs-1215	89	2	y	y	PROPN
iajs-1215	89	3	+	+	CCONJ
iajs-1215	89	4	x	x	X
iajs-1215	89	5			X
iajs-1215	89	6	p	p	X
iajs-1215	89	7	because	because	SCONJ
iajs-1215	89	8	x	x	PROPN
iajs-1215	89	9			PUNCT
iajs-1215	89	10	p	p	X
iajs-1215	89	11	,	,	PUNCT
iajs-1215	89	12	y	y	PROPN
iajs-1215	89	13			PROPN
iajs-1215	89	14	p	p	NOUN
iajs-1215	89	15	,	,	PUNCT
iajs-1215	89	16	hence	hence	ADV
iajs-1215	89	17	by	by	ADP
iajs-1215	89	18	condition	condition	NOUN
iajs-1215	89	19	3	3	NUM
iajs-1215	89	20	,	,	PUNCT
iajs-1215	89	21	either	either	CCONJ
iajs-1215	89	22	(	(	PUNCT
iajs-1215	89	23	p	p	X
iajs-1215	89	24	:	:	PUNCT
iajs-1215	89	25	y	y	PROPN
iajs-1215	89	26	+	+	CCONJ
iajs-1215	89	27	x	x	X
iajs-1215	89	28	)	)	PUNCT
iajs-1215	90	1	=	=	SYM
iajs-1215	90	2	(	(	PUNCT
iajs-1215	90	3	p	p	X
iajs-1215	90	4	:	:	PUNCT
iajs-1215	90	5	m	m	NUM
iajs-1215	90	6	)	)	PUNCT
iajs-1215	90	7	or	or	CCONJ
iajs-1215	90	8	(	(	PUNCT
iajs-1215	90	9	p	p	X
iajs-1215	90	10	:	:	PUNCT
iajs-1215	90	11	y	y	PROPN
iajs-1215	90	12	+	+	CCONJ
iajs-1215	90	13	x	x	X
iajs-1215	90	14	)	)	PUNCT
iajs-1215	90	15	=	=	SYM
iajs-1215	90	16	(	(	PUNCT
iajs-1215	90	17	0	0	NUM
iajs-1215	90	18	:	:	PUNCT
iajs-1215	90	19	y	y	PROPN
iajs-1215	90	20	+	+	CCONJ
iajs-1215	90	21	x	x	X
iajs-1215	90	22	)	)	PUNCT
iajs-1215	90	23	.	.	PUNCT
iajs-1215	91	1	if	if	SCONJ
iajs-1215	91	2	(	(	PUNCT
iajs-1215	91	3	p	p	X
iajs-1215	91	4	:	:	PUNCT
iajs-1215	91	5	y	y	PROPN
iajs-1215	91	6	+	+	CCONJ
iajs-1215	91	7	x	x	X
iajs-1215	91	8	)	)	PUNCT
iajs-1215	91	9	=	=	SYM
iajs-1215	91	10	(	(	PUNCT
iajs-1215	91	11	p	p	X
iajs-1215	91	12	:	:	PUNCT
iajs-1215	91	13	m	m	NUM
iajs-1215	91	14	)	)	PUNCT
iajs-1215	91	15	then	then	ADV
iajs-1215	91	16	a	a	DET
iajs-1215	91	17			NOUN
iajs-1215	91	18	(	(	PUNCT
iajs-1215	91	19	p	p	X
iajs-1215	91	20	:	:	PUNCT
iajs-1215	91	21	m	m	VERB
iajs-1215	91	22	)	)	PUNCT
iajs-1215	91	23	which	which	PRON
iajs-1215	91	24	is	be	AUX
iajs-1215	91	25	a	a	DET
iajs-1215	91	26	contradiction	contradiction	NOUN
iajs-1215	91	27	.	.	PUNCT
iajs-1215	92	1	if	if	SCONJ
iajs-1215	92	2	(	(	PUNCT
iajs-1215	92	3	p	p	X
iajs-1215	92	4	:	:	PUNCT
iajs-1215	92	5	y	y	PROPN
iajs-1215	92	6	+	+	CCONJ
iajs-1215	92	7	x	x	X
iajs-1215	92	8	)	)	PUNCT
iajs-1215	92	9	=	=	SYM
iajs-1215	93	1	(	(	PUNCT
iajs-1215	93	2	0	0	NUM
iajs-1215	93	3	:	:	PUNCT
iajs-1215	93	4	y	y	PROPN
iajs-1215	93	5	+	+	CCONJ
iajs-1215	93	6	x	x	X
iajs-1215	93	7	)	)	PUNCT
iajs-1215	93	8	,	,	PUNCT
iajs-1215	93	9	then	then	ADV
iajs-1215	93	10	a(y	a(y	PROPN
iajs-1215	93	11	+	+	PUNCT
iajs-1215	93	12	x	x	X
iajs-1215	93	13	)	)	PUNCT
iajs-1215	94	1	=	=	SYM
iajs-1215	94	2	0	0	PUNCT
iajs-1215	94	3	and	and	CCONJ
iajs-1215	94	4	hence	hence	ADV
iajs-1215	94	5	ay	ay	NOUN
iajs-1215	95	1	+	+	CCONJ
iajs-1215	95	2	ax	ax	NOUN
iajs-1215	95	3	=	=	PUNCT
iajs-1215	95	4	ay	ay	PROPN
iajs-1215	96	1	+	+	NOUN
iajs-1215	96	2	0	0	NUM
iajs-1215	96	3	=	=	SYM
iajs-1215	96	4	0	0	NUM
iajs-1215	96	5	which	which	PRON
iajs-1215	96	6	is	be	AUX
iajs-1215	96	7	a	a	DET
iajs-1215	96	8	contradiction	contradiction	NOUN
iajs-1215	96	9	.	.	PUNCT
iajs-1215	97	1	therefore	therefore	ADV
iajs-1215	97	2	our	our	PRON
iajs-1215	97	3	assumption	assumption	NOUN
iajs-1215	97	4	is	be	AUX
iajs-1215	97	5	false	false	ADJ
iajs-1215	97	6	and	and	CCONJ
iajs-1215	97	7	so	so	ADV
iajs-1215	97	8	either	either	CCONJ
iajs-1215	97	9	n	n	CCONJ
iajs-1215	97	10			PROPN
iajs-1215	97	11	p	p	PROPN
iajs-1215	97	12	or	or	CCONJ
iajs-1215	97	13	(	(	PUNCT
iajs-1215	97	14	a	a	NOUN
iajs-1215	97	15	)	)	PUNCT
iajs-1215	97	16			PROPN
iajs-1215	97	17	(	(	PUNCT
iajs-1215	97	18	p	p	X
iajs-1215	97	19	:	:	PUNCT
iajs-1215	97	20	m	m	NUM
iajs-1215	97	21	)	)	PUNCT
iajs-1215	97	22	.	.	PUNCT
iajs-1215	98	1	(	(	PUNCT
iajs-1215	98	2	4	4	X
iajs-1215	98	3	)	)	PUNCT
iajs-1215	98	4			NOUN
iajs-1215	98	5	(	(	PUNCT
iajs-1215	98	6	1	1	X
iajs-1215	98	7	)	)	PUNCT
iajs-1215	98	8	let	let	VERB
iajs-1215	98	9	r	r	NOUN
iajs-1215	98	10			NOUN
iajs-1215	98	11	r	r	NOUN
iajs-1215	98	12	,	,	PUNCT
iajs-1215	98	13	x	x	SYM
iajs-1215	98	14			PROPN
iajs-1215	98	15	m	m	PROPN
iajs-1215	98	16	,	,	PUNCT
iajs-1215	99	1	such	such	ADJ
iajs-1215	99	2	that	that	SCONJ
iajs-1215	99	3	0	0	NUM
iajs-1215	99	4			NOUN
iajs-1215	99	5	r	r	NOUN
iajs-1215	99	6	x	x	NOUN
iajs-1215	99	7			NOUN
iajs-1215	99	8	p.	p.	NOUN
iajs-1215	99	9	then	then	ADV
iajs-1215	100	1	0	0	NUM
iajs-1215	100	2			NOUN
iajs-1215	100	3	(	(	PUNCT
iajs-1215	100	4	r	r	NOUN
iajs-1215	100	5	)	)	PUNCT
iajs-1215	100	6	(	(	PUNCT
iajs-1215	100	7	x	x	X
iajs-1215	100	8	)	)	PUNCT
iajs-1215	100	9			PROPN
iajs-1215	100	10	p.	p.	NOUN
iajs-1215	100	11	by	by	ADP
iajs-1215	100	12	condition	condition	NOUN
iajs-1215	100	13	(	(	PUNCT
iajs-1215	100	14	4	4	NUM
iajs-1215	100	15	)	)	PUNCT
iajs-1215	100	16	,	,	PUNCT
iajs-1215	100	17	(	(	PUNCT
iajs-1215	100	18	x	x	X
iajs-1215	100	19	)	)	PUNCT
iajs-1215	100	20			PROPN
iajs-1215	100	21	p	p	NOUN
iajs-1215	100	22	or	or	CCONJ
iajs-1215	100	23	(	(	PUNCT
iajs-1215	100	24	r	r	NOUN
iajs-1215	100	25	)	)	PUNCT
iajs-1215	100	26			PROPN
iajs-1215	100	27	(	(	PUNCT
iajs-1215	100	28	p	p	X
iajs-1215	100	29	:	:	PUNCT
iajs-1215	100	30	m	m	NUM
iajs-1215	100	31	)	)	PUNCT
iajs-1215	100	32	and	and	CCONJ
iajs-1215	100	33	hence	hence	ADV
iajs-1215	100	34	either	either	CCONJ
iajs-1215	100	35	x	x	SYM
iajs-1215	100	36			NOUN
iajs-1215	100	37	p	p	NOUN
iajs-1215	100	38	or	or	CCONJ
iajs-1215	100	39	r	r	NOUN
iajs-1215	100	40			NOUN
iajs-1215	100	41	(	(	PUNCT
iajs-1215	100	42	p	p	X
iajs-1215	100	43	:	:	PUNCT
iajs-1215	100	44	m	m	NUM
iajs-1215	100	45	)	)	PUNCT
iajs-1215	100	46	;	;	PUNCT
iajs-1215	100	47	that	that	PRON
iajs-1215	100	48	is	is	ADV
iajs-1215	100	49	,	,	PUNCT
iajs-1215	100	50	p	p	PRON
iajs-1215	100	51	is	be	AUX
iajs-1215	100	52	weakly	weakly	ADV
iajs-1215	100	53	prime	prime	ADJ
iajs-1215	100	54	.	.	PUNCT
iajs-1215	101	1	remark	remark	VERB
iajs-1215	101	2	2.3	2.3	NUM
iajs-1215	101	3	it	it	PRON
iajs-1215	101	4	is	be	AUX
iajs-1215	101	5	known	know	VERB
iajs-1215	101	6	that	that	SCONJ
iajs-1215	101	7	if	if	SCONJ
iajs-1215	101	8	p	p	NOUN
iajs-1215	101	9	is	be	AUX
iajs-1215	101	10	a	a	DET
iajs-1215	101	11	prime	prime	ADJ
iajs-1215	101	12	submodule	submodule	NOUN
iajs-1215	101	13	of	of	ADP
iajs-1215	101	14	an	an	DET
iajs-1215	101	15	r	r	NOUN
iajs-1215	101	16	-	-	PUNCT
iajs-1215	101	17	module	module	NOUN
iajs-1215	101	18	m	m	NOUN
iajs-1215	101	19	,	,	PUNCT
iajs-1215	101	20	then	then	ADV
iajs-1215	101	21	(	(	PUNCT
iajs-1215	101	22	p	p	X
iajs-1215	101	23	:	:	PUNCT
iajs-1215	101	24	m	m	VERB
iajs-1215	101	25	)	)	PUNCT
iajs-1215	101	26	is	be	AUX
iajs-1215	101	27	a	a	DET
iajs-1215	101	28	prime	prime	ADJ
iajs-1215	101	29	ideal	ideal	NOUN
iajs-1215	101	30	of	of	ADP
iajs-1215	101	31	r.	r.	PROPN
iajs-1215	101	32	however	however	ADV
iajs-1215	101	33	the	the	DET
iajs-1215	101	34	"	"	PUNCT
iajs-1215	101	35	weak	weak	ADJ
iajs-1215	101	36	"	"	PUNCT
iajs-1215	101	37	analogs	analog	NOUN
iajs-1215	101	38	of	of	ADP
iajs-1215	101	39	this	this	DET
iajs-1215	101	40	statement	statement	NOUN
iajs-1215	101	41	is	be	AUX
iajs-1215	101	42	not	not	PART
iajs-1215	101	43	true	true	ADJ
iajs-1215	101	44	in	in	ADP
iajs-1215	101	45	general	general	ADJ
iajs-1215	101	46	,	,	PUNCT
iajs-1215	101	47	for	for	ADP
iajs-1215	101	48	example	example	NOUN
iajs-1215	102	1	:	:	PUNCT
iajs-1215	102	2	the	the	DET
iajs-1215	102	3	zero	zero	NUM
iajs-1215	102	4	submodule	submodule	NOUN
iajs-1215	102	5	of	of	ADP
iajs-1215	102	6	the	the	DET
iajs-1215	102	7	z	z	NOUN
iajs-1215	102	8	-	-	PUNCT
iajs-1215	102	9	module	module	NOUN
iajs-1215	102	10	z4	z4	NOUN
iajs-1215	102	11	,	,	PUNCT
iajs-1215	102	12	is	be	AUX
iajs-1215	102	13	weakly	weakly	ADJ
iajs-1215	102	14	prime	prime	ADJ
iajs-1215	102	15	,	,	PUNCT
iajs-1215	102	16	but	but	CCONJ
iajs-1215	102	17	(	(	PUNCT
iajs-1215	102	18	0	0	NUM
iajs-1215	102	19			PROPN
iajs-1215	102	20	:	:	PUNCT
iajs-1215	102	21	z4	z4	X
iajs-1215	102	22	)	)	PUNCT
iajs-1215	102	23	=	=	SYM
iajs-1215	102	24	4	4	NUM
iajs-1215	102	25	z	z	NOUN
iajs-1215	102	26	is	be	AUX
iajs-1215	102	27	not	not	PART
iajs-1215	102	28	a	a	DET
iajs-1215	102	29	weakly	weakly	ADJ
iajs-1215	102	30	prime	prime	ADJ
iajs-1215	102	31	ideal	ideal	NOUN
iajs-1215	102	32	of	of	ADP
iajs-1215	102	33	z.	z.	PROPN
iajs-1215	102	34	we	we	PRON
iajs-1215	102	35	give	give	VERB
iajs-1215	102	36	the	the	DET
iajs-1215	102	37	following	following	NOUN
iajs-1215	102	38	:	:	PUNCT
iajs-1215	102	39	proposition	proposition	NOUN
iajs-1215	102	40	2.4	2.4	NUM
iajs-1215	102	41	:	:	PUNCT
iajs-1215	102	42	if	if	SCONJ
iajs-1215	102	43	p	p	NOUN
iajs-1215	102	44	is	be	AUX
iajs-1215	102	45	a	a	DET
iajs-1215	102	46	weakly	weakly	ADJ
iajs-1215	102	47	prime	prime	ADJ
iajs-1215	102	48	submodule	submodule	NOUN
iajs-1215	102	49	of	of	ADP
iajs-1215	102	50	a	a	DET
iajs-1215	102	51	faithful	faithful	ADJ
iajs-1215	102	52	r	r	NOUN
iajs-1215	102	53	-	-	PUNCT
iajs-1215	102	54	module	module	NOUN
iajs-1215	102	55	m	m	NOUN
iajs-1215	102	56	,	,	PUNCT
iajs-1215	102	57	then	then	ADV
iajs-1215	102	58	(	(	PUNCT
iajs-1215	102	59	p	p	NOUN
iajs-1215	102	60	r	r	NOUN
iajs-1215	102	61	:	:	PUNCT
iajs-1215	102	62	m	m	X
iajs-1215	102	63	)	)	PUNCT
iajs-1215	102	64	is	be	AUX
iajs-1215	102	65	a	a	DET
iajs-1215	102	66	weakly	weakly	ADJ
iajs-1215	102	67	prime	prime	ADJ
iajs-1215	102	68	ideal	ideal	NOUN
iajs-1215	102	69	of	of	ADP
iajs-1215	102	70	r.	r.	PROPN
iajs-1215	102	71	proof	proof	NOUN
iajs-1215	102	72	.	.	PUNCT
iajs-1215	103	1	let	let	VERB
iajs-1215	103	2	a	a	DET
iajs-1215	103	3	,	,	PUNCT
iajs-1215	103	4	b	b	PROPN
iajs-1215	103	5			PROPN
iajs-1215	103	6	r.	r.	VERB
iajs-1215	103	7	if	if	SCONJ
iajs-1215	103	8	0	0	NUM
iajs-1215	103	9			VERB
iajs-1215	103	10	a	a	DET
iajs-1215	103	11	b	b	NOUN
iajs-1215	103	12			NOUN
iajs-1215	103	13	(	(	PUNCT
iajs-1215	103	14	p	p	X
iajs-1215	103	15	:	:	PUNCT
iajs-1215	103	16	m	m	NUM
iajs-1215	103	17	)	)	PUNCT
iajs-1215	103	18	;	;	PUNCT
iajs-1215	103	19	then	then	ADV
iajs-1215	103	20	a	a	DET
iajs-1215	103	21	b	b	NOUN
iajs-1215	103	22	m	m	NOUN
iajs-1215	103	23			PROPN
iajs-1215	103	24	p.	p.	NOUN
iajs-1215	103	25	since	since	SCONJ
iajs-1215	103	26	m	m	PROPN
iajs-1215	103	27	is	be	AUX
iajs-1215	103	28	faithful	faithful	ADJ
iajs-1215	103	29	,	,	PUNCT
iajs-1215	103	30	a	a	DET
iajs-1215	103	31	b	b	NOUN
iajs-1215	103	32	m	m	VERB
iajs-1215	103	33			NOUN
iajs-1215	103	34	(	(	PUNCT
iajs-1215	103	35	0	0	NUM
iajs-1215	103	36	)	)	PUNCT
iajs-1215	103	37	,	,	PUNCT
iajs-1215	103	38	hence	hence	ADV
iajs-1215	103	39	0	0	NUM
iajs-1215	103	40			NOUN
iajs-1215	103	41	(	(	PUNCT
iajs-1215	103	42	a	a	NOUN
iajs-1215	103	43	)	)	PUNCT
iajs-1215	103	44	(	(	PUNCT
iajs-1215	103	45	b	b	NOUN
iajs-1215	103	46	m	m	NOUN
iajs-1215	103	47	)	)	PUNCT
iajs-1215	103	48			PROPN
iajs-1215	103	49	p	p	NOUN
iajs-1215	103	50	and	and	CCONJ
iajs-1215	103	51	so	so	ADV
iajs-1215	103	52	by	by	ADP
iajs-1215	103	53	th.2.2	th.2.2	NOUN
iajs-1215	103	54	either	either	CCONJ
iajs-1215	103	55	(	(	PUNCT
iajs-1215	103	56	a	a	NOUN
iajs-1215	103	57	)	)	PUNCT
iajs-1215	103	58			PROPN
iajs-1215	103	59	(	(	PUNCT
iajs-1215	103	60	p	p	X
iajs-1215	103	61	:	:	PUNCT
iajs-1215	103	62	m	m	NUM
iajs-1215	103	63	)	)	PUNCT
iajs-1215	103	64	or	or	CCONJ
iajs-1215	103	65	b	b	NOUN
iajs-1215	103	66	m	m	NOUN
iajs-1215	103	67			PROPN
iajs-1215	103	68	p	p	X
iajs-1215	103	69	;	;	PUNCT
iajs-1215	103	70	that	that	PRON
iajs-1215	103	71	is	is	ADV
iajs-1215	103	72	,	,	PUNCT
iajs-1215	103	73	either	either	CCONJ
iajs-1215	103	74	a	a	DET
iajs-1215	103	75			NOUN
iajs-1215	103	76	(	(	PUNCT
iajs-1215	103	77	p	p	X
iajs-1215	103	78	:	:	PUNCT
iajs-1215	103	79	m	m	NUM
iajs-1215	103	80	)	)	PUNCT
iajs-1215	103	81	or	or	CCONJ
iajs-1215	103	82	b	b	NOUN
iajs-1215	103	83			NOUN
iajs-1215	103	84	(	(	PUNCT
iajs-1215	103	85	p	p	X
iajs-1215	103	86	:	:	PUNCT
iajs-1215	103	87	m	m	NUM
iajs-1215	103	88	)	)	PUNCT
iajs-1215	103	89	.	.	PUNCT
iajs-1215	104	1	thus	thus	ADV
iajs-1215	104	2	(	(	PUNCT
iajs-1215	104	3	p	p	X
iajs-1215	104	4	:	:	PUNCT
iajs-1215	104	5	m	m	VERB
iajs-1215	104	6	)	)	PUNCT
iajs-1215	104	7	is	be	AUX
iajs-1215	104	8	a	a	DET
iajs-1215	104	9	weakly	weakly	ADJ
iajs-1215	104	10	prime	prime	ADJ
iajs-1215	104	11	ideal	ideal	NOUN
iajs-1215	104	12	of	of	ADP
iajs-1215	104	13	r.	r.	PROPN
iajs-1215	104	14	the	the	DET
iajs-1215	104	15	converse	converse	NOUN
iajs-1215	104	16	of	of	ADP
iajs-1215	104	17	prop.2.4	prop.2.4	PROPN
iajs-1215	104	18	is	be	AUX
iajs-1215	104	19	not	not	PART
iajs-1215	104	20	true	true	ADJ
iajs-1215	104	21	as	as	SCONJ
iajs-1215	104	22	the	the	DET
iajs-1215	104	23	following	follow	VERB
iajs-1215	104	24	example	example	NOUN
iajs-1215	104	25	shows	show	VERB
iajs-1215	104	26	:	:	PUNCT
iajs-1215	104	27	let	let	VERB
iajs-1215	104	28	m	m	PRON
iajs-1215	104	29	be	be	AUX
iajs-1215	104	30	the	the	DET
iajs-1215	104	31	z	z	NOUN
iajs-1215	104	32	-	-	PUNCT
iajs-1215	104	33	module	module	NOUN
iajs-1215	104	34	z	z	PROPN
iajs-1215	104	35			PROPN
iajs-1215	104	36	z	z	PROPN
iajs-1215	104	37	,	,	PUNCT
iajs-1215	104	38	let	let	VERB
iajs-1215	104	39	p	p	NOUN
iajs-1215	104	40	=	=	X
iajs-1215	104	41	(	(	PUNCT
iajs-1215	104	42	0	0	X
iajs-1215	104	43	)	)	PUNCT
iajs-1215	104	44			NOUN
iajs-1215	104	45	2z	2z	NUM
iajs-1215	104	46	.	.	PUNCT
iajs-1215	105	1	then	then	ADV
iajs-1215	105	2	(	(	PUNCT
iajs-1215	105	3	p	p	PROPN
iajs-1215	105	4			PROPN
iajs-1215	105	5	:	:	PUNCT
iajs-1215	105	6	m	m	X
iajs-1215	105	7	)	)	PUNCT
iajs-1215	105	8	=	=	SYM
iajs-1215	105	9	(	(	PUNCT
iajs-1215	105	10	0	0	NUM
iajs-1215	105	11	)	)	PUNCT
iajs-1215	105	12	which	which	PRON
iajs-1215	105	13	is	be	AUX
iajs-1215	105	14	a	a	DET
iajs-1215	105	15	weakly	weakly	ADJ
iajs-1215	105	16	prime	prime	ADJ
iajs-1215	105	17	ideal	ideal	NOUN
iajs-1215	105	18	in	in	ADP
iajs-1215	105	19	z	z	PROPN
iajs-1215	105	20	,	,	PUNCT
iajs-1215	105	21	however	however	ADV
iajs-1215	105	22	p	p	PROPN
iajs-1215	105	23	is	be	AUX
iajs-1215	105	24	not	not	PART
iajs-1215	105	25	a	a	DET
iajs-1215	105	26	weakly	weakly	ADJ
iajs-1215	105	27	prime	prime	ADJ
iajs-1215	105	28	submodule	submodule	NOUN
iajs-1215	105	29	of	of	ADP
iajs-1215	105	30	m	m	PROPN
iajs-1215	105	31	because	because	SCONJ
iajs-1215	105	32	(	(	PUNCT
iajs-1215	105	33	0,0	0,0	NOUN
iajs-1215	105	34	)	)	PUNCT
iajs-1215	105	35			NOUN
iajs-1215	105	36	2(0,1	2(0,1	NUM
iajs-1215	105	37	)	)	PUNCT
iajs-1215	105	38			NOUN
iajs-1215	105	39	p	p	NOUN
iajs-1215	105	40	,	,	PUNCT
iajs-1215	105	41	but	but	CCONJ
iajs-1215	105	42	(	(	PUNCT
iajs-1215	105	43	0,1	0,1	NUM
iajs-1215	105	44	)	)	PUNCT
iajs-1215	105	45			NOUN
iajs-1215	105	46	p	p	NOUN
iajs-1215	105	47	and	and	CCONJ
iajs-1215	105	48	2	2	NUM
iajs-1215	105	49			X
iajs-1215	105	50	(	(	PUNCT
iajs-1215	105	51	p	p	PROPN
iajs-1215	105	52			PROPN
iajs-1215	105	53	:	:	PUNCT
iajs-1215	105	54	m	m	X
iajs-1215	105	55	)	)	PUNCT
iajs-1215	105	56	=	=	SYM
iajs-1215	105	57	(	(	PUNCT
iajs-1215	105	58	0	0	NUM
iajs-1215	105	59	)	)	PUNCT
iajs-1215	105	60	.	.	PUNCT
iajs-1215	106	1	also	also	ADV
iajs-1215	106	2	,	,	PUNCT
iajs-1215	106	3	we	we	PRON
iajs-1215	106	4	have	have	VERB
iajs-1215	106	5	the	the	DET
iajs-1215	106	6	following	following	NOUN
iajs-1215	106	7	:	:	PUNCT
iajs-1215	106	8	proposition	proposition	NOUN
iajs-1215	106	9	2.5	2.5	NUM
iajs-1215	106	10	:	:	PUNCT
iajs-1215	106	11	let	let	VERB
iajs-1215	106	12	p	p	PRON
iajs-1215	106	13	be	be	AUX
iajs-1215	106	14	a	a	DET
iajs-1215	106	15	weakly	weakly	ADJ
iajs-1215	106	16	prime	prime	ADJ
iajs-1215	106	17	submodule	submodule	NOUN
iajs-1215	106	18	of	of	ADP
iajs-1215	106	19	an	an	DET
iajs-1215	106	20	r	r	NOUN
iajs-1215	106	21	-	-	PUNCT
iajs-1215	106	22	module	module	NOUN
iajs-1215	106	23	m.	m.	NOUN
iajs-1215	106	24	then	then	ADV
iajs-1215	106	25	(	(	PUNCT
iajs-1215	106	26	p	p	NOUN
iajs-1215	106	27	r	r	NOUN
iajs-1215	106	28	:	:	PUNCT
iajs-1215	106	29	m	m	X
iajs-1215	106	30	)	)	PUNCT
iajs-1215	106	31	is	be	AUX
iajs-1215	106	32	a	a	DET
iajs-1215	106	33	weakly	weakly	ADJ
iajs-1215	106	34	prime	prime	ADJ
iajs-1215	106	35	ideal	ideal	NOUN
iajs-1215	106	36	of	of	ADP
iajs-1215	106	37	r	r	NOUN
iajs-1215	106	38	,	,	PUNCT
iajs-1215	106	39	where	where	SCONJ
iajs-1215	106	40			ADJ
iajs-1215	106	41	annrr	annrr	X
iajs-1215	106	42	/	/	PUNCT
iajs-1215	106	43	.	.	PUNCT
iajs-1215	107	1	proof	proof	NOUN
iajs-1215	107	2	.	.	PUNCT
iajs-1215	108	1	by	by	ADP
iajs-1215	108	2	remark	remark	NOUN
iajs-1215	108	3	2.1	2.1	NUM
iajs-1215	108	4	(	(	PUNCT
iajs-1215	108	5	6	6	NUM
iajs-1215	108	6	)	)	PUNCT
iajs-1215	108	7	,	,	PUNCT
iajs-1215	108	8	p	p	NOUN
iajs-1215	108	9	is	be	AUX
iajs-1215	108	10	a	a	DET
iajs-1215	108	11	weakly	weakly	ADJ
iajs-1215	108	12	prime	prime	ADJ
iajs-1215	108	13	r	r	NOUN
iajs-1215	108	14	-submodule	-submodule	NOUN
iajs-1215	108	15	of	of	ADP
iajs-1215	108	16	m.	m.	NOUN
iajs-1215	108	17	but	but	CCONJ
iajs-1215	108	18	m	m	NOUN
iajs-1215	108	19	is	be	AUX
iajs-1215	108	20	a	a	DET
iajs-1215	108	21	faithful	faithful	ADJ
iajs-1215	108	22	r	r	NOUN
iajs-1215	108	23	module	module	NOUN
iajs-1215	108	24	,	,	PUNCT
iajs-1215	108	25	so	so	ADV
iajs-1215	108	26	by	by	ADP
iajs-1215	108	27	prop.2.6	prop.2.6	PROPN
iajs-1215	108	28	,	,	PUNCT
iajs-1215	108	29	(	(	PUNCT
iajs-1215	108	30	p	p	NOUN
iajs-1215	108	31	r	r	NOUN
iajs-1215	108	32	:	:	PUNCT
iajs-1215	108	33	m	m	X
iajs-1215	108	34	)	)	PUNCT
iajs-1215	108	35	is	be	AUX
iajs-1215	108	36	a	a	DET
iajs-1215	108	37	weakly	weakly	ADJ
iajs-1215	108	38	prime	prime	ADJ
iajs-1215	108	39	ideal	ideal	NOUN
iajs-1215	108	40	of	of	ADP
iajs-1215	108	41	r	r	NOUN
iajs-1215	108	42	.	.	PUNCT
iajs-1215	109	1	recall	recall	VERB
iajs-1215	109	2	that	that	SCONJ
iajs-1215	109	3	an	an	DET
iajs-1215	109	4	r	r	NOUN
iajs-1215	109	5	-	-	PUNCT
iajs-1215	109	6	module	module	NOUN
iajs-1215	109	7	m	m	NOUN
iajs-1215	109	8	is	be	AUX
iajs-1215	109	9	called	call	VERB
iajs-1215	109	10	multiplication	multiplication	NOUN
iajs-1215	109	11	module	module	NOUN
iajs-1215	109	12	if	if	SCONJ
iajs-1215	109	13	for	for	ADP
iajs-1215	109	14	each	each	DET
iajs-1215	109	15	submodule	submodule	NOUN
iajs-1215	109	16	n	n	PROPN
iajs-1215	109	17	of	of	ADP
iajs-1215	109	18	m	m	PROPN
iajs-1215	109	19	,	,	PUNCT
iajs-1215	109	20	n	n	PROPN
iajs-1215	110	1	=	=	VERB
iajs-1215	111	1	i	i	PRON
iajs-1215	111	2	m	m	VERB
iajs-1215	111	3	for	for	ADP
iajs-1215	111	4	some	some	DET
iajs-1215	111	5	ideal	ideal	ADJ
iajs-1215	111	6	i	i	PRON
iajs-1215	111	7	of	of	ADP
iajs-1215	111	8	r	r	NOUN
iajs-1215	111	9	,	,	PUNCT
iajs-1215	111	10	equivalently	equivalently	ADV
iajs-1215	111	11	n	n	NOUN
iajs-1215	111	12	=	=	SYM
iajs-1215	111	13	(	(	PUNCT
iajs-1215	111	14	n	n	CCONJ
iajs-1215	111	15	:	:	PUNCT
iajs-1215	111	16	m)m	m)m	X
iajs-1215	111	17	(	(	PUNCT
iajs-1215	111	18	see	see	VERB
iajs-1215	111	19	(	(	PUNCT
iajs-1215	111	20	9	9	NUM
iajs-1215	111	21	)	)	PUNCT
iajs-1215	111	22	)	)	PUNCT
iajs-1215	111	23	.	.	PUNCT
iajs-1215	112	1	in	in	ADP
iajs-1215	112	2	the	the	DET
iajs-1215	112	3	class	class	NOUN
iajs-1215	112	4	of	of	ADP
iajs-1215	112	5	finitely	finitely	ADV
iajs-1215	112	6	generated	generate	VERB
iajs-1215	112	7	faithful	faithful	ADJ
iajs-1215	112	8	multiplication	multiplication	NOUN
iajs-1215	112	9	modules	module	NOUN
iajs-1215	112	10	,	,	PUNCT
iajs-1215	112	11	we	we	PRON
iajs-1215	112	12	have	have	VERB
iajs-1215	112	13	the	the	DET
iajs-1215	112	14	following	following	NOUN
iajs-1215	112	15	:	:	PUNCT
iajs-1215	112	16	theorem	theorem	VERB
iajs-1215	112	17	2.6	2.6	NUM
iajs-1215	112	18	:	:	PUNCT
iajs-1215	112	19	let	let	VERB
iajs-1215	112	20	m	m	PRON
iajs-1215	112	21	be	be	AUX
iajs-1215	112	22	a	a	DET
iajs-1215	112	23	faithful	faithful	ADJ
iajs-1215	112	24	finitely	finitely	ADV
iajs-1215	112	25	generated	generate	VERB
iajs-1215	112	26	multiplication	multiplication	NOUN
iajs-1215	112	27	r	r	NOUN
iajs-1215	112	28	-	-	PUNCT
iajs-1215	112	29	module	module	NOUN
iajs-1215	112	30	,	,	PUNCT
iajs-1215	112	31	let	let	VERB
iajs-1215	112	32	n	n	PRON
iajs-1215	112	33	be	be	AUX
iajs-1215	112	34	a	a	DET
iajs-1215	112	35	proper	proper	ADJ
iajs-1215	112	36	submodule	submodule	NOUN
iajs-1215	112	37	of	of	ADP
iajs-1215	112	38	m.	m.	NOUN
iajs-1215	112	39	then	then	ADV
iajs-1215	112	40	the	the	DET
iajs-1215	112	41	following	follow	VERB
iajs-1215	112	42	statements	statement	NOUN
iajs-1215	112	43	are	be	AUX
iajs-1215	112	44	equivalent	equivalent	ADJ
iajs-1215	112	45	1	1	NUM
iajs-1215	112	46	.	.	PUNCT
iajs-1215	113	1	n	n	PRON
iajs-1215	113	2	is	be	AUX
iajs-1215	113	3	a	a	DET
iajs-1215	113	4	weakly	weakly	ADJ
iajs-1215	113	5	prime	prime	ADJ
iajs-1215	113	6	submodule	submodule	NOUN
iajs-1215	113	7	of	of	ADP
iajs-1215	113	8	m.	m.	NOUN
iajs-1215	113	9	2	2	NUM
iajs-1215	113	10	.	.	PUNCT
iajs-1215	114	1	(	(	PUNCT
iajs-1215	114	2	n	n	CCONJ
iajs-1215	114	3	r	r	NOUN
iajs-1215	114	4	:	:	PUNCT
iajs-1215	114	5	m	m	X
iajs-1215	114	6	)	)	PUNCT
iajs-1215	114	7	is	be	AUX
iajs-1215	114	8	a	a	DET
iajs-1215	114	9	weakly	weakly	ADJ
iajs-1215	114	10	prime	prime	ADJ
iajs-1215	114	11	ideal	ideal	NOUN
iajs-1215	114	12	of	of	ADP
iajs-1215	114	13	r.	r.	PROPN
iajs-1215	114	14	3	3	NUM
iajs-1215	114	15	.	.	PUNCT
iajs-1215	115	1	n	n	NOUN
iajs-1215	115	2	=	=	NOUN
iajs-1215	116	1	i	i	PRON
iajs-1215	116	2	m	m	VERB
iajs-1215	116	3	for	for	ADP
iajs-1215	116	4	some	some	DET
iajs-1215	116	5	weakly	weakly	ADJ
iajs-1215	116	6	prime	prime	ADJ
iajs-1215	116	7	ideal	ideal	NOUN
iajs-1215	116	8	i	i	PRON
iajs-1215	116	9	of	of	ADP
iajs-1215	116	10	r.	r.	PROPN
iajs-1215	116	11	proof	proof	NOUN
iajs-1215	116	12	.	.	PUNCT
iajs-1215	117	1	(	(	PUNCT
iajs-1215	117	2	1	1	X
iajs-1215	117	3	)	)	PUNCT
iajs-1215	117	4			NOUN
iajs-1215	117	5	(	(	PUNCT
iajs-1215	117	6	2	2	X
iajs-1215	117	7	)	)	PUNCT
iajs-1215	117	8	it	it	PRON
iajs-1215	117	9	holds	hold	VERB
iajs-1215	117	10	by	by	ADP
iajs-1215	117	11	prop	prop	NOUN
iajs-1215	117	12	.	.	PUNCT
iajs-1215	118	1	2.4	2.4	NUM
iajs-1215	118	2	ibn	ibn	PROPN
iajs-1215	118	3	alhaitham	alhaitham	NOUN
iajs-1215	118	4	j.	j.	PROPN
iajs-1215	119	1	fo	fo	ADP
iajs-1215	119	2	r	r	NOUN
iajs-1215	119	3	pure	pure	ADJ
iajs-1215	119	4	&	&	CCONJ
iajs-1215	119	5	appl	appl	PROPN
iajs-1215	119	6	.	.	PUNCT
iajs-1215	120	1	sc	sc	PROPN
iajs-1215	121	1	i	i	PRON
iajs-1215	121	2	vo	vo	INTJ
iajs-1215	121	3	l.22	l.22	X
iajs-1215	121	4	(	(	PUNCT
iajs-1215	121	5	3	3	NUM
iajs-1215	121	6	)	)	PUNCT
iajs-1215	121	7	2009	2009	NUM
iajs-1215	121	8	(	(	PUNCT
iajs-1215	121	9	2	2	NUM
iajs-1215	121	10	)	)	PUNCT
iajs-1215	121	11			NOUN
iajs-1215	121	12	(	(	PUNCT
iajs-1215	121	13	1	1	X
iajs-1215	121	14	)	)	PUNCT
iajs-1215	121	15	let	let	VERB
iajs-1215	121	16	r	r	NOUN
iajs-1215	121	17			NOUN
iajs-1215	121	18	r	r	NOUN
iajs-1215	121	19	,	,	PUNCT
iajs-1215	121	20	x	x	SYM
iajs-1215	121	21			PROPN
iajs-1215	121	22	m	m	PROPN
iajs-1215	121	23	,	,	PUNCT
iajs-1215	121	24	such	such	ADJ
iajs-1215	121	25	that	that	SCONJ
iajs-1215	121	26	0	0	NUM
iajs-1215	121	27			NOUN
iajs-1215	121	28	r	r	NOUN
iajs-1215	121	29	x	x	SYM
iajs-1215	121	30			PROPN
iajs-1215	121	31	n.	n.	NOUN
iajs-1215	121	32	(	(	PUNCT
iajs-1215	121	33	x	x	X
iajs-1215	121	34	)	)	PUNCT
iajs-1215	121	35	is	be	AUX
iajs-1215	121	36	a	a	DET
iajs-1215	121	37	submodule	submodule	NOUN
iajs-1215	121	38	of	of	ADP
iajs-1215	121	39	m	m	PROPN
iajs-1215	121	40	,	,	PUNCT
iajs-1215	121	41	hence	hence	ADV
iajs-1215	121	42	(	(	PUNCT
iajs-1215	121	43	x	x	X
iajs-1215	121	44	)	)	PUNCT
iajs-1215	122	1	=	=	PUNCT
iajs-1215	122	2	j	j	PROPN
iajs-1215	122	3	m	m	VERB
iajs-1215	122	4	for	for	ADP
iajs-1215	122	5	some	some	DET
iajs-1215	122	6	ideal	ideal	ADJ
iajs-1215	122	7	j	j	PROPN
iajs-1215	122	8	of	of	ADP
iajs-1215	122	9	r.	r.	PROPN
iajs-1215	123	1	thus	thus	ADV
iajs-1215	123	2	0	0	NUM
iajs-1215	123	3			NOUN
iajs-1215	124	1	r	r	X
iajs-1215	124	2	j	j	PROPN
iajs-1215	124	3	m	m	PROPN
iajs-1215	124	4			PROPN
iajs-1215	124	5	n	n	PROPN
iajs-1215	124	6	=	=	SYM
iajs-1215	124	7	(	(	PUNCT
iajs-1215	124	8	n	n	CCONJ
iajs-1215	124	9	:	:	PUNCT
iajs-1215	124	10	m	m	NOUN
iajs-1215	124	11	)	)	PUNCT
iajs-1215	124	12	m.	m.	NOUN
iajs-1215	125	1	but	but	CCONJ
iajs-1215	125	2	m	m	NOUN
iajs-1215	125	3	is	be	AUX
iajs-1215	125	4	a	a	DET
iajs-1215	125	5	faithful	faithful	ADJ
iajs-1215	125	6	finitely	finitely	ADV
iajs-1215	125	7	generated	generate	VERB
iajs-1215	125	8	multiplication	multiplication	NOUN
iajs-1215	125	9	r	r	NOUN
iajs-1215	125	10	-	-	PUNCT
iajs-1215	125	11	module	module	NOUN
iajs-1215	125	12	,	,	PUNCT
iajs-1215	125	13	so	so	ADV
iajs-1215	125	14	by	by	ADP
iajs-1215	125	15	(	(	PUNCT
iajs-1215	125	16	10	10	NUM
iajs-1215	125	17	,	,	PUNCT
iajs-1215	125	18	th.3.1	th.3.1	ADJ
iajs-1215	125	19	)	)	PUNCT
iajs-1215	125	20	r	r	NOUN
iajs-1215	125	21	j	j	PROPN
iajs-1215	125	22			PROPN
iajs-1215	125	23	(	(	PUNCT
iajs-1215	125	24	n	n	NOUN
iajs-1215	125	25	r	r	NOUN
iajs-1215	125	26	:	:	PUNCT
iajs-1215	125	27	m	m	ADJ
iajs-1215	125	28	)	)	PUNCT
iajs-1215	125	29	.	.	PUNCT
iajs-1215	126	1	moreover	moreover	ADV
iajs-1215	126	2	r	r	X
iajs-1215	126	3	j	j	PROPN
iajs-1215	126	4			PROPN
iajs-1215	126	5	(	(	PUNCT
iajs-1215	126	6	0	0	NUM
iajs-1215	126	7	)	)	PUNCT
iajs-1215	126	8	and	and	CCONJ
iajs-1215	126	9	since	since	SCONJ
iajs-1215	126	10	(	(	PUNCT
iajs-1215	126	11	n	n	PRON
iajs-1215	126	12	r	r	NOUN
iajs-1215	126	13	:	:	PUNCT
iajs-1215	126	14	m	m	X
iajs-1215	126	15	)	)	PUNCT
iajs-1215	126	16	is	be	AUX
iajs-1215	126	17	weakly	weakly	ADJ
iajs-1215	126	18	prime	prime	ADJ
iajs-1215	126	19	ideal	ideal	NOUN
iajs-1215	126	20	,	,	PUNCT
iajs-1215	126	21	either	either	CCONJ
iajs-1215	126	22	r	r	NOUN
iajs-1215	126	23			NOUN
iajs-1215	126	24	(	(	PUNCT
iajs-1215	126	25	n	n	NOUN
iajs-1215	126	26	r	r	NOUN
iajs-1215	126	27	:	:	PUNCT
iajs-1215	126	28	m	m	X
iajs-1215	126	29	)	)	PUNCT
iajs-1215	126	30	or	or	CCONJ
iajs-1215	126	31	j	j	PROPN
iajs-1215	126	32			PROPN
iajs-1215	126	33	(	(	PUNCT
iajs-1215	126	34	n	n	NOUN
iajs-1215	126	35	r	r	NOUN
iajs-1215	126	36	:	:	PUNCT
iajs-1215	126	37	m	m	X
iajs-1215	126	38	)	)	PUNCT
iajs-1215	126	39	(	(	PUNCT
iajs-1215	126	40	see	see	VERB
iajs-1215	126	41	th.3	th.3	PROPN
iajs-1215	126	42	in	in	ADP
iajs-1215	126	43	(	(	PUNCT
iajs-1215	126	44	7	7	NUM
iajs-1215	126	45	)	)	PUNCT
iajs-1215	126	46	)	)	PUNCT
iajs-1215	126	47	.	.	PUNCT
iajs-1215	127	1	hence	hence	ADV
iajs-1215	127	2	either	either	CCONJ
iajs-1215	127	3	r	r	NOUN
iajs-1215	127	4			NOUN
iajs-1215	127	5	(	(	PUNCT
iajs-1215	127	6	n	n	NOUN
iajs-1215	127	7	r	r	NOUN
iajs-1215	127	8	:	:	PUNCT
iajs-1215	127	9	m	m	X
iajs-1215	127	10	)	)	PUNCT
iajs-1215	127	11	or	or	CCONJ
iajs-1215	127	12	(	(	PUNCT
iajs-1215	127	13	x	x	X
iajs-1215	127	14	)	)	PUNCT
iajs-1215	127	15	=	=	SYM
iajs-1215	128	1	j	j	PROPN
iajs-1215	128	2	m	m	VERB
iajs-1215	128	3			PROPN
iajs-1215	128	4	(	(	PUNCT
iajs-1215	128	5	n	n	NOUN
iajs-1215	128	6	r	r	NOUN
iajs-1215	128	7	:	:	PUNCT
iajs-1215	128	8	m	m	X
iajs-1215	128	9	)	)	PUNCT
iajs-1215	128	10	m	m	PROPN
iajs-1215	128	11	=	=	SYM
iajs-1215	128	12	n	n	CCONJ
iajs-1215	128	13	,	,	PUNCT
iajs-1215	128	14	that	that	ADV
iajs-1215	128	15	is	is	ADV
iajs-1215	128	16	r	r	NOUN
iajs-1215	128	17			NOUN
iajs-1215	128	18	(	(	PUNCT
iajs-1215	128	19	n	n	NOUN
iajs-1215	128	20	r	r	NOUN
iajs-1215	128	21	:	:	PUNCT
iajs-1215	128	22	m	m	X
iajs-1215	128	23	)	)	PUNCT
iajs-1215	128	24	or	or	CCONJ
iajs-1215	128	25	x	x	ADJ
iajs-1215	128	26			PROPN
iajs-1215	128	27	n.	n.	NOUN
iajs-1215	128	28	thus	thus	ADV
iajs-1215	128	29	n	n	ADV
iajs-1215	128	30	is	be	AUX
iajs-1215	128	31	weakly	weakly	ADV
iajs-1215	128	32	prime	prime	ADJ
iajs-1215	128	33	.	.	PUNCT
iajs-1215	129	1	(	(	PUNCT
iajs-1215	129	2	2	2	X
iajs-1215	129	3	)	)	PUNCT
iajs-1215	129	4			NOUN
iajs-1215	129	5	(	(	PUNCT
iajs-1215	129	6	3	3	NUM
iajs-1215	129	7	)	)	PUNCT
iajs-1215	129	8	since	since	SCONJ
iajs-1215	129	9	(	(	PUNCT
iajs-1215	129	10	n	n	PRON
iajs-1215	129	11	r	r	NOUN
iajs-1215	129	12	:	:	PUNCT
iajs-1215	129	13	m	m	X
iajs-1215	129	14	)	)	PUNCT
iajs-1215	129	15	is	be	AUX
iajs-1215	129	16	weakly	weakly	ADV
iajs-1215	129	17	prime	prime	ADJ
iajs-1215	129	18	and	and	CCONJ
iajs-1215	129	19	n	n	NOUN
iajs-1215	129	20	=	=	SYM
iajs-1215	129	21	(	(	PUNCT
iajs-1215	129	22	n	n	NOUN
iajs-1215	129	23	r	r	NOUN
iajs-1215	129	24	:	:	PUNCT
iajs-1215	129	25	m	m	X
iajs-1215	129	26	)	)	PUNCT
iajs-1215	129	27	m	m	PROPN
iajs-1215	129	28	,	,	PUNCT
iajs-1215	129	29	so	so	ADV
iajs-1215	129	30	condition	condition	NOUN
iajs-1215	129	31	(	(	PUNCT
iajs-1215	129	32	3	3	X
iajs-1215	129	33	)	)	PUNCT
iajs-1215	129	34	hold	hold	NOUN
iajs-1215	129	35	.	.	PUNCT
iajs-1215	130	1	(	(	PUNCT
iajs-1215	130	2	3	3	X
iajs-1215	130	3	)	)	PUNCT
iajs-1215	130	4			NOUN
iajs-1215	130	5	(	(	PUNCT
iajs-1215	130	6	2	2	NUM
iajs-1215	130	7	)	)	PUNCT
iajs-1215	130	8	by	by	ADP
iajs-1215	130	9	(	(	PUNCT
iajs-1215	130	10	3	3	NUM
iajs-1215	130	11	)	)	PUNCT
iajs-1215	130	12	,	,	PUNCT
iajs-1215	130	13	n	n	NOUN
iajs-1215	130	14	=	=	VERB
iajs-1215	131	1	i	i	PRON
iajs-1215	131	2	m	m	VERB
iajs-1215	131	3	for	for	ADP
iajs-1215	131	4	some	some	DET
iajs-1215	131	5	weakly	weakly	ADJ
iajs-1215	131	6	prime	prime	ADJ
iajs-1215	131	7	ideal	ideal	NOUN
iajs-1215	131	8	i	i	PRON
iajs-1215	131	9	of	of	ADP
iajs-1215	131	10	r.	r.	PROPN
iajs-1215	132	1	but	but	CCONJ
iajs-1215	132	2	m	m	PROPN
iajs-1215	132	3	is	be	AUX
iajs-1215	132	4	a	a	DET
iajs-1215	132	5	multiplication	multiplication	NOUN
iajs-1215	132	6	module	module	NOUN
iajs-1215	132	7	,	,	PUNCT
iajs-1215	132	8	so	so	SCONJ
iajs-1215	132	9	n	n	NOUN
iajs-1215	132	10	=	=	SYM
iajs-1215	132	11	(	(	PUNCT
iajs-1215	132	12	n	n	NOUN
iajs-1215	132	13	r	r	NOUN
iajs-1215	132	14	:	:	PUNCT
iajs-1215	132	15	m	m	X
iajs-1215	132	16	)	)	PUNCT
iajs-1215	132	17	m.	m.	NOUN
iajs-1215	133	1	hence	hence	ADV
iajs-1215	133	2	i	i	PRON
iajs-1215	133	3	m	m	VERB
iajs-1215	133	4	=	=	SYM
iajs-1215	133	5	(	(	PUNCT
iajs-1215	133	6	n	n	CCONJ
iajs-1215	133	7	:	:	PUNCT
iajs-1215	133	8	m	m	VERB
iajs-1215	133	9	)	)	PUNCT
iajs-1215	133	10	m	m	PROPN
iajs-1215	133	11	and	and	CCONJ
iajs-1215	133	12	so	so	ADV
iajs-1215	133	13	by	by	ADP
iajs-1215	133	14	(	(	PUNCT
iajs-1215	133	15	10	10	NUM
iajs-1215	133	16	,	,	PUNCT
iajs-1215	133	17	th.3.1	th.3.1	NOUN
iajs-1215	133	18	)	)	PUNCT
iajs-1215	133	19	i	i	NOUN
iajs-1215	133	20	=	=	PUNCT
iajs-1215	133	21	(	(	PUNCT
iajs-1215	133	22	n	n	NOUN
iajs-1215	133	23	r	r	NOUN
iajs-1215	133	24	:	:	PUNCT
iajs-1215	133	25	m	m	NOUN
iajs-1215	133	26	)	)	PUNCT
iajs-1215	133	27	.	.	PUNCT
iajs-1215	134	1	proposition	proposition	NOUN
iajs-1215	134	2	2.7	2.7	NUM
iajs-1215	134	3	:	:	PUNCT
iajs-1215	134	4	let	let	VERB
iajs-1215	134	5	p	p	PRON
iajs-1215	134	6	be	be	AUX
iajs-1215	134	7	a	a	DET
iajs-1215	134	8	weakly	weakly	ADJ
iajs-1215	134	9	prime	prime	ADJ
iajs-1215	134	10	submodule	submodule	NOUN
iajs-1215	134	11	of	of	ADP
iajs-1215	134	12	an	an	DET
iajs-1215	134	13	r	r	NOUN
iajs-1215	134	14	-	-	PUNCT
iajs-1215	134	15	module	module	NOUN
iajs-1215	134	16	m.	m.	NOUN
iajs-1215	134	17	if	if	SCONJ
iajs-1215	134	18	p	p	NOUN
iajs-1215	134	19	is	be	AUX
iajs-1215	134	20	not	not	PART
iajs-1215	134	21	prime	prime	ADJ
iajs-1215	134	22	,	,	PUNCT
iajs-1215	134	23	then	then	ADV
iajs-1215	134	24	(	(	PUNCT
iajs-1215	134	25	p	p	X
iajs-1215	134	26	:	:	PUNCT
iajs-1215	134	27	m	m	NOUN
iajs-1215	134	28	)	)	PUNCT
iajs-1215	135	1	p	p	NOUN
iajs-1215	135	2	=	=	SYM
iajs-1215	135	3	(	(	PUNCT
iajs-1215	135	4	0	0	NUM
iajs-1215	135	5	)	)	PUNCT
iajs-1215	135	6	.	.	PUNCT
iajs-1215	136	1	proof	proof	NOUN
iajs-1215	136	2	.	.	PUNCT
iajs-1215	137	1	suppose	suppose	VERB
iajs-1215	137	2	(	(	PUNCT
iajs-1215	137	3	p	p	X
iajs-1215	137	4	:	:	PUNCT
iajs-1215	137	5	m	m	NOUN
iajs-1215	137	6	)	)	PUNCT
iajs-1215	137	7	p	p	PROPN
iajs-1215	137	8			PROPN
iajs-1215	137	9	0	0	NUM
iajs-1215	137	10	.	.	PUNCT
iajs-1215	138	1	we	we	PRON
iajs-1215	138	2	will	will	AUX
iajs-1215	138	3	show	show	VERB
iajs-1215	138	4	that	that	SCONJ
iajs-1215	138	5	p	p	NOUN
iajs-1215	138	6	is	be	AUX
iajs-1215	138	7	prime	prime	ADJ
iajs-1215	138	8	.	.	PUNCT
iajs-1215	139	1	let	let	VERB
iajs-1215	139	2	r	r	NOUN
iajs-1215	139	3	x	x	PRON
iajs-1215	139	4			NOUN
iajs-1215	140	1	p.	p.	NOUN
iajs-1215	140	2	if	if	SCONJ
iajs-1215	140	3	r	r	NOUN
iajs-1215	140	4	x	x	SYM
iajs-1215	140	5			NOUN
iajs-1215	140	6	0	0	NUM
iajs-1215	140	7	,	,	PUNCT
iajs-1215	140	8	then	then	ADV
iajs-1215	140	9	either	either	CCONJ
iajs-1215	140	10	x	x	SYM
iajs-1215	140	11			NOUN
iajs-1215	140	12	p	p	NOUN
iajs-1215	140	13	or	or	CCONJ
iajs-1215	140	14	r	r	NOUN
iajs-1215	140	15			NOUN
iajs-1215	140	16	(	(	PUNCT
iajs-1215	140	17	p	p	X
iajs-1215	140	18	:	:	PUNCT
iajs-1215	140	19	m	m	NUM
iajs-1215	140	20	)	)	PUNCT
iajs-1215	140	21	,	,	PUNCT
iajs-1215	140	22	since	since	SCONJ
iajs-1215	140	23	p	p	NOUN
iajs-1215	140	24	is	be	AUX
iajs-1215	140	25	weakly	weakly	ADV
iajs-1215	140	26	prime	prime	ADJ
iajs-1215	140	27	.	.	PUNCT
iajs-1215	141	1	now	now	ADV
iajs-1215	141	2	assume	assume	VERB
iajs-1215	141	3	r	r	NOUN
iajs-1215	141	4	x	x	NOUN
iajs-1215	141	5	=	=	SYM
iajs-1215	141	6	0	0	X
iajs-1215	141	7	.	.	PUNCT
iajs-1215	142	1	first	first	ADV
iajs-1215	142	2	suppose	suppose	VERB
iajs-1215	142	3	rp	rp	PROPN
iajs-1215	142	4			PROPN
iajs-1215	142	5	(	(	PUNCT
iajs-1215	142	6	0	0	NUM
iajs-1215	142	7	)	)	PUNCT
iajs-1215	142	8	,	,	PUNCT
iajs-1215	142	9	so	so	CCONJ
iajs-1215	142	10	there	there	PRON
iajs-1215	142	11	exists	exist	VERB
iajs-1215	142	12	y	y	PROPN
iajs-1215	142	13			PROPN
iajs-1215	142	14	p	p	PRON
iajs-1215	142	15	such	such	ADJ
iajs-1215	142	16	that	that	DET
iajs-1215	142	17	0	0	NUM
iajs-1215	142	18			NOUN
iajs-1215	142	19	r	r	NOUN
iajs-1215	142	20	y	y	PROPN
iajs-1215	142	21			PROPN
iajs-1215	143	1	p.	p.	NOUN
iajs-1215	143	2	hence	hence	ADV
iajs-1215	143	3	0	0	NUM
iajs-1215	144	1			NOUN
iajs-1215	144	2	r	r	NOUN
iajs-1215	144	3	y	y	NOUN
iajs-1215	144	4	=	=	SYM
iajs-1215	144	5	r	r	NOUN
iajs-1215	144	6	(	(	PUNCT
iajs-1215	144	7	x	x	PROPN
iajs-1215	144	8	+	+	NUM
iajs-1215	144	9	y	y	NOUN
iajs-1215	144	10	)	)	PUNCT
iajs-1215	144	11			NOUN
iajs-1215	144	12	p	p	X
iajs-1215	144	13	which	which	PRON
iajs-1215	144	14	implies	imply	VERB
iajs-1215	144	15	that	that	SCONJ
iajs-1215	144	16	either	either	ADV
iajs-1215	144	17	x	x	X
iajs-1215	144	18	+	+	NUM
iajs-1215	144	19	y	y	PROPN
iajs-1215	144	20			NOUN
iajs-1215	144	21	p	p	NOUN
iajs-1215	144	22	or	or	CCONJ
iajs-1215	144	23	r	r	NOUN
iajs-1215	144	24			NOUN
iajs-1215	144	25	(	(	PUNCT
iajs-1215	144	26	p	p	X
iajs-1215	144	27	:	:	PUNCT
iajs-1215	144	28	m	m	NUM
iajs-1215	144	29	)	)	PUNCT
iajs-1215	144	30	.	.	PUNCT
iajs-1215	145	1	hence	hence	ADV
iajs-1215	145	2	either	either	CCONJ
iajs-1215	145	3	x	x	SYM
iajs-1215	145	4			NOUN
iajs-1215	145	5	p	p	NOUN
iajs-1215	145	6	or	or	CCONJ
iajs-1215	145	7	r	r	NOUN
iajs-1215	145	8			NOUN
iajs-1215	145	9	(	(	PUNCT
iajs-1215	145	10	p	p	X
iajs-1215	145	11	:	:	PUNCT
iajs-1215	145	12	m	m	NUM
iajs-1215	145	13	)	)	PUNCT
iajs-1215	145	14	.	.	PUNCT
iajs-1215	146	1	now	now	ADV
iajs-1215	146	2	we	we	PRON
iajs-1215	146	3	can	can	AUX
iajs-1215	146	4	assume	assume	VERB
iajs-1215	146	5	that	that	SCONJ
iajs-1215	146	6	r	r	NOUN
iajs-1215	146	7	p	p	X
iajs-1215	146	8	=	=	NOUN
iajs-1215	146	9	0	0	PROPN
iajs-1215	147	1	and	and	CCONJ
iajs-1215	147	2	(	(	PUNCT
iajs-1215	147	3	p	p	X
iajs-1215	147	4	:	:	PUNCT
iajs-1215	147	5	m	m	NOUN
iajs-1215	147	6	)	)	PUNCT
iajs-1215	147	7	x	x	PUNCT
iajs-1215	148	1	=	=	PUNCT
iajs-1215	148	2	0	0	X
iajs-1215	148	3	.	.	PUNCT
iajs-1215	149	1	since	since	SCONJ
iajs-1215	149	2	(	(	PUNCT
iajs-1215	149	3	p	p	X
iajs-1215	149	4	:	:	PUNCT
iajs-1215	149	5	m	m	NOUN
iajs-1215	149	6	)	)	PUNCT
iajs-1215	149	7	p	p	PROPN
iajs-1215	149	8			PROPN
iajs-1215	149	9	(	(	PUNCT
iajs-1215	149	10	0	0	NUM
iajs-1215	149	11	)	)	PUNCT
iajs-1215	149	12	,	,	PUNCT
iajs-1215	149	13	there	there	PRON
iajs-1215	149	14	exists	exist	VERB
iajs-1215	149	15	s	s	PART
iajs-1215	149	16			NOUN
iajs-1215	149	17	(	(	PUNCT
iajs-1215	149	18	p	p	X
iajs-1215	149	19	:	:	PUNCT
iajs-1215	149	20	m	m	NUM
iajs-1215	149	21	)	)	PUNCT
iajs-1215	149	22	,	,	PUNCT
iajs-1215	149	23	y	y	PROPN
iajs-1215	149	24			PROPN
iajs-1215	149	25	p	p	PRON
iajs-1215	149	26	such	such	ADJ
iajs-1215	149	27	that	that	DET
iajs-1215	149	28	0	0	NUM
iajs-1215	149	29			NOUN
iajs-1215	149	30	s	s	X
iajs-1215	149	31	y	y	PROPN
iajs-1215	149	32			PROPN
iajs-1215	149	33	p.	p.	NOUN
iajs-1215	150	1	thus	thus	ADV
iajs-1215	150	2	(	(	PUNCT
iajs-1215	150	3	r	r	NOUN
iajs-1215	150	4	+	+	NUM
iajs-1215	150	5	s	s	NOUN
iajs-1215	150	6	)	)	PUNCT
iajs-1215	150	7	(	(	PUNCT
iajs-1215	150	8	x	x	X
iajs-1215	150	9	+	+	NUM
iajs-1215	150	10	y	y	NOUN
iajs-1215	150	11	)	)	PUNCT
iajs-1215	151	1	=	=	SYM
iajs-1215	151	2	r	r	NOUN
iajs-1215	151	3	x	x	PUNCT
iajs-1215	152	1	+	+	SYM
iajs-1215	152	2	s	s	NOUN
iajs-1215	152	3	x	x	X
iajs-1215	153	1	+	+	CCONJ
iajs-1215	153	2	r	r	NOUN
iajs-1215	153	3	y	y	NOUN
iajs-1215	154	1	+	+	SYM
iajs-1215	154	2	s	s	PART
iajs-1215	154	3	y	y	NOUN
iajs-1215	154	4	=	=	PUNCT
iajs-1215	154	5	0	0	PUNCT
iajs-1215	155	1	+	+	CCONJ
iajs-1215	155	2	0	0	NUM
iajs-1215	156	1	+	+	CCONJ
iajs-1215	156	2	0	0	NUM
iajs-1215	157	1	+	+	NUM
iajs-1215	157	2	s	s	AUX
iajs-1215	157	3	y	y	NOUN
iajs-1215	157	4	=	=	SYM
iajs-1215	157	5	s	s	PROPN
iajs-1215	157	6	y	y	NOUN
iajs-1215	157	7	that	that	PRON
iajs-1215	157	8	is	be	AUX
iajs-1215	157	9	0	0	NUM
iajs-1215	157	10			NOUN
iajs-1215	157	11	(	(	PUNCT
iajs-1215	157	12	r	r	NOUN
iajs-1215	157	13	+	+	NUM
iajs-1215	157	14	s	s	NOUN
iajs-1215	157	15	)	)	PUNCT
iajs-1215	157	16	(	(	PUNCT
iajs-1215	157	17	x	x	X
iajs-1215	157	18	+	+	NUM
iajs-1215	157	19	y	y	NOUN
iajs-1215	157	20	)	)	PUNCT
iajs-1215	157	21			NOUN
iajs-1215	158	1	p.	p.	NOUN
iajs-1215	158	2	then	then	ADV
iajs-1215	158	3	p	p	NOUN
iajs-1215	158	4	is	be	AUX
iajs-1215	158	5	weakly	weakly	ADJ
iajs-1215	158	6	prime	prime	ADJ
iajs-1215	158	7	gives	give	VERB
iajs-1215	158	8	x	x	PUNCT
iajs-1215	158	9	+	+	NUM
iajs-1215	158	10	y	y	PROPN
iajs-1215	158	11			NOUN
iajs-1215	158	12	p	p	NOUN
iajs-1215	158	13	or	or	CCONJ
iajs-1215	158	14	r	r	NOUN
iajs-1215	158	15	+	+	PROPN
iajs-1215	158	16	s	s	NOUN
iajs-1215	158	17			NOUN
iajs-1215	158	18	(	(	PUNCT
iajs-1215	158	19	p	p	X
iajs-1215	158	20	:	:	PUNCT
iajs-1215	158	21	m	m	NUM
iajs-1215	158	22	)	)	PUNCT
iajs-1215	158	23	.	.	PUNCT
iajs-1215	159	1	hence	hence	ADV
iajs-1215	159	2	x	x	PUNCT
iajs-1215	159	3			NOUN
iajs-1215	159	4	p	p	NOUN
iajs-1215	159	5	or	or	CCONJ
iajs-1215	159	6	r	r	NOUN
iajs-1215	159	7			NOUN
iajs-1215	159	8	(	(	PUNCT
iajs-1215	159	9	p	p	X
iajs-1215	159	10	:	:	PUNCT
iajs-1215	159	11	m	m	NUM
iajs-1215	159	12	)	)	PUNCT
iajs-1215	159	13	.	.	PUNCT
iajs-1215	160	1	now	now	ADV
iajs-1215	160	2	we	we	PRON
iajs-1215	160	3	have	have	VERB
iajs-1215	160	4	the	the	DET
iajs-1215	160	5	following	following	NOUN
iajs-1215	160	6	:	:	PUNCT
iajs-1215	160	7	proposition	proposition	NOUN
iajs-1215	160	8	2.8	2.8	NUM
iajs-1215	160	9	:	:	PUNCT
iajs-1215	160	10	let	let	VERB
iajs-1215	160	11	m	m	PRON
iajs-1215	160	12	and	and	CCONJ
iajs-1215	160	13	m	m	VERB
iajs-1215	160	14			ADJ
iajs-1215	160	15	be	be	VERB
iajs-1215	160	16	r	r	NOUN
iajs-1215	160	17	-	-	PUNCT
iajs-1215	160	18	modules	module	NOUN
iajs-1215	160	19	and	and	CCONJ
iajs-1215	160	20	let	let	VERB
iajs-1215	160	21	f	f	X
iajs-1215	160	22	:	:	PUNCT
iajs-1215	160	23	m	m	VERB
iajs-1215	160	24			NOUN
iajs-1215	160	25	m	m	AUX
iajs-1215	160	26			ADJ
iajs-1215	160	27	be	be	AUX
iajs-1215	160	28	an	an	DET
iajs-1215	160	29	r	r	NOUN
iajs-1215	160	30	-	-	PUNCT
iajs-1215	160	31	epimorphism	epimorphism	NOUN
iajs-1215	160	32	.	.	PUNCT
iajs-1215	161	1	if	if	SCONJ
iajs-1215	161	2	n	n	PRON
iajs-1215	161	3	is	be	AUX
iajs-1215	161	4	a	a	DET
iajs-1215	161	5	weakly	weakly	ADJ
iajs-1215	161	6	prime	prime	ADJ
iajs-1215	161	7	submodule	submodule	NOUN
iajs-1215	161	8	of	of	ADP
iajs-1215	161	9	m	m	PRON
iajs-1215	161	10	such	such	ADJ
iajs-1215	161	11	that	that	DET
iajs-1215	161	12	ker	ker	PROPN
iajs-1215	161	13	f	f	PROPN
iajs-1215	161	14			PROPN
iajs-1215	161	15	n	n	CCONJ
iajs-1215	161	16	,	,	PUNCT
iajs-1215	161	17	then	then	ADV
iajs-1215	161	18	f	f	PROPN
iajs-1215	161	19	(	(	PUNCT
iajs-1215	161	20	n	n	CCONJ
iajs-1215	161	21	)	)	PUNCT
iajs-1215	161	22	is	be	AUX
iajs-1215	161	23	a	a	DET
iajs-1215	161	24	weakly	weakly	ADJ
iajs-1215	161	25	prime	prime	ADJ
iajs-1215	161	26	submodule	submodule	NOUN
iajs-1215	161	27	of	of	ADP
iajs-1215	161	28	m	m	PROPN
iajs-1215	161	29	.	.	PROPN
iajs-1215	161	30	proof	proof	NOUN
iajs-1215	161	31	.	.	PUNCT
iajs-1215	162	1	let	let	VERB
iajs-1215	162	2	r	r	NOUN
iajs-1215	162	3			NOUN
iajs-1215	162	4	r	r	NOUN
iajs-1215	162	5	,	,	PUNCT
iajs-1215	162	6	y	y	PROPN
iajs-1215	162	7			PROPN
iajs-1215	162	8	m	m	VERB
iajs-1215	162	9			ADJ
iajs-1215	162	10	,	,	PUNCT
iajs-1215	162	11	such	such	ADJ
iajs-1215	162	12	that	that	SCONJ
iajs-1215	162	13	0	0	NUM
iajs-1215	162	14			NOUN
iajs-1215	162	15	r	r	NOUN
iajs-1215	162	16	y	y	PROPN
iajs-1215	162	17			PROPN
iajs-1215	162	18	f	f	PROPN
iajs-1215	162	19	(	(	PUNCT
iajs-1215	162	20	n	n	CCONJ
iajs-1215	162	21	)	)	PUNCT
iajs-1215	162	22	.	.	PUNCT
iajs-1215	163	1	then	then	ADV
iajs-1215	163	2	there	there	PRON
iajs-1215	163	3	exists	exist	VERB
iajs-1215	163	4	x	x	PUNCT
iajs-1215	163	5			NOUN
iajs-1215	163	6	n	n	CCONJ
iajs-1215	163	7	such	such	ADJ
iajs-1215	163	8	that	that	SCONJ
iajs-1215	163	9	0	0	NUM
iajs-1215	163	10			NOUN
iajs-1215	163	11	r	r	NOUN
iajs-1215	163	12	y	y	NOUN
iajs-1215	163	13	=	=	SYM
iajs-1215	163	14	f	f	PROPN
iajs-1215	163	15	(	(	PUNCT
iajs-1215	163	16	x	x	NOUN
iajs-1215	163	17	)	)	PUNCT
iajs-1215	163	18	,	,	PUNCT
iajs-1215	163	19	and	and	CCONJ
iajs-1215	163	20	since	since	SCONJ
iajs-1215	163	21	f	f	PROPN
iajs-1215	163	22	is	be	AUX
iajs-1215	163	23	an	an	DET
iajs-1215	163	24	epimorphism	epimorphism	NOUN
iajs-1215	163	25	,	,	PUNCT
iajs-1215	163	26	y	y	PROPN
iajs-1215	163	27	=	=	SYM
iajs-1215	163	28	f	f	PROPN
iajs-1215	163	29	(	(	PUNCT
iajs-1215	163	30	x1	x1	PROPN
iajs-1215	163	31	)	)	PUNCT
iajs-1215	163	32	for	for	ADP
iajs-1215	163	33	some	some	DET
iajs-1215	163	34	x1	x1	NOUN
iajs-1215	163	35	m.	m.	NOUN
iajs-1215	163	36	thus	thus	ADV
iajs-1215	163	37	f	f	X
iajs-1215	164	1	(	(	PUNCT
iajs-1215	164	2	r	r	NOUN
iajs-1215	164	3	x1	x1	PROPN
iajs-1215	164	4	–	–	PUNCT
iajs-1215	164	5	x	x	X
iajs-1215	164	6	)	)	PUNCT
iajs-1215	164	7	=	=	SYM
iajs-1215	164	8	0	0	PUNCT
iajs-1215	165	1	and	and	CCONJ
iajs-1215	165	2	so	so	ADV
iajs-1215	165	3	r	r	NOUN
iajs-1215	165	4	x1	x1	PROPN
iajs-1215	165	5	–	–	PUNCT
iajs-1215	165	6	x	x	SYM
iajs-1215	165	7			NOUN
iajs-1215	165	8	ker	ker	X
iajs-1215	166	1	f	f	PROPN
iajs-1215	166	2			PROPN
iajs-1215	166	3	n.	n.	PROPN
iajs-1215	166	4	it	it	PRON
iajs-1215	166	5	follows	follow	VERB
iajs-1215	166	6	that	that	SCONJ
iajs-1215	166	7	0	0	NUM
iajs-1215	166	8			NOUN
iajs-1215	166	9	r	r	NOUN
iajs-1215	166	10	x1	x1	PROPN
iajs-1215	166	11			NOUN
iajs-1215	166	12	n	n	CCONJ
iajs-1215	166	13	and	and	CCONJ
iajs-1215	166	14	since	since	SCONJ
iajs-1215	166	15	n	n	PRON
iajs-1215	166	16	is	be	AUX
iajs-1215	166	17	weakly	weakly	ADV
iajs-1215	166	18	prime	prime	ADJ
iajs-1215	166	19	either	either	CCONJ
iajs-1215	166	20	x1	x1	ADJ
iajs-1215	166	21			NOUN
iajs-1215	166	22	n	n	CCONJ
iajs-1215	166	23	or	or	CCONJ
iajs-1215	166	24	r	r	NOUN
iajs-1215	166	25	(n	(n	NOUN
iajs-1215	166	26	r	r	NOUN
iajs-1215	166	27	:	:	PUNCT
iajs-1215	166	28	m	m	NOUN
iajs-1215	166	29	)	)	PUNCT
iajs-1215	166	30	.	.	PUNCT
iajs-1215	167	1	thus	thus	ADV
iajs-1215	167	2	y	y	PROPN
iajs-1215	167	3	=	=	SYM
iajs-1215	167	4	f	f	PROPN
iajs-1215	167	5	(	(	PUNCT
iajs-1215	167	6	x1	x1	PROPN
iajs-1215	167	7	)	)	PUNCT
iajs-1215	167	8			NOUN
iajs-1215	167	9	f	f	X
iajs-1215	167	10	(	(	PUNCT
iajs-1215	167	11	n	n	CCONJ
iajs-1215	167	12	)	)	PUNCT
iajs-1215	167	13	or	or	CCONJ
iajs-1215	167	14	r	r	NOUN
iajs-1215	167	15			NOUN
iajs-1215	167	16	(	(	PUNCT
iajs-1215	167	17	f	f	PROPN
iajs-1215	167	18	(	(	PUNCT
iajs-1215	167	19	n	n	CCONJ
iajs-1215	167	20	):	):	PUNCT
iajs-1215	167	21	m	m	NOUN
iajs-1215	167	22			ADJ
iajs-1215	167	23	)	)	PUNCT
iajs-1215	167	24	;	;	PUNCT
iajs-1215	167	25	that	that	PRON
iajs-1215	167	26	is	be	AUX
iajs-1215	167	27	f	f	PROPN
iajs-1215	167	28	(	(	PUNCT
iajs-1215	167	29	n	n	CCONJ
iajs-1215	167	30	)	)	PUNCT
iajs-1215	167	31	is	be	AUX
iajs-1215	167	32	weakly	weakly	ADV
iajs-1215	167	33	prime	prime	ADJ
iajs-1215	167	34	.	.	PUNCT
iajs-1215	168	1	as	as	ADP
iajs-1215	168	2	a	a	DET
iajs-1215	168	3	particular	particular	ADJ
iajs-1215	168	4	case	case	NOUN
iajs-1215	168	5	of	of	ADP
iajs-1215	168	6	prop.(2.8	prop.(2.8	NOUN
iajs-1215	168	7	)	)	PUNCT
iajs-1215	168	8	,	,	PUNCT
iajs-1215	168	9	we	we	PRON
iajs-1215	168	10	have	have	VERB
iajs-1215	168	11	the	the	DET
iajs-1215	168	12	following	following	NOUN
iajs-1215	168	13	:	:	PUNCT
iajs-1215	168	14	if	if	SCONJ
iajs-1215	168	15	n	n	CCONJ
iajs-1215	168	16	,	,	PUNCT
iajs-1215	168	17	w	w	PROPN
iajs-1215	168	18	are	be	AUX
iajs-1215	168	19	submodules	submodule	NOUN
iajs-1215	168	20	of	of	ADP
iajs-1215	168	21	an	an	DET
iajs-1215	168	22	r	r	NOUN
iajs-1215	168	23	-	-	PUNCT
iajs-1215	168	24	module	module	NOUN
iajs-1215	168	25	m	m	NOUN
iajs-1215	168	26	such	such	ADJ
iajs-1215	168	27	that	that	SCONJ
iajs-1215	168	28	n	n	PROPN
iajs-1215	168	29			PROPN
iajs-1215	168	30	w	w	PROPN
iajs-1215	168	31	and	and	CCONJ
iajs-1215	168	32	n	n	PROPN
iajs-1215	168	33	is	be	AUX
iajs-1215	168	34	weakly	weakly	ADV
iajs-1215	168	35	prime	prime	ADJ
iajs-1215	168	36	,	,	PUNCT
iajs-1215	168	37	then	then	ADV
iajs-1215	168	38	n	n	CCONJ
iajs-1215	168	39	/	/	SYM
iajs-1215	168	40	w	w	PROPN
iajs-1215	168	41	is	be	AUX
iajs-1215	168	42	a	a	DET
iajs-1215	168	43	weakly	weakly	ADJ
iajs-1215	168	44	prime	prime	ADJ
iajs-1215	168	45	rsubmodule	rsubmodule	NOUN
iajs-1215	168	46	of	of	ADP
iajs-1215	168	47	m	m	PROPN
iajs-1215	168	48	/	/	SYM
iajs-1215	168	49	n.	n.	PROPN
iajs-1215	168	50	the	the	DET
iajs-1215	168	51	following	follow	VERB
iajs-1215	168	52	result	result	NOUN
iajs-1215	168	53	discussos	discusso	VERB
iajs-1215	168	54	the	the	DET
iajs-1215	168	55	localization	localization	NOUN
iajs-1215	168	56	of	of	ADP
iajs-1215	168	57	weakly	weakly	ADJ
iajs-1215	168	58	prime	prime	ADJ
iajs-1215	168	59	submodules	submodule	NOUN
iajs-1215	168	60	.	.	PUNCT
iajs-1215	169	1	proposition	proposition	NOUN
iajs-1215	169	2	2.9	2.9	NUM
iajs-1215	169	3	:	:	PUNCT
iajs-1215	169	4	let	let	VERB
iajs-1215	169	5	p	p	PRON
iajs-1215	169	6	be	be	AUX
iajs-1215	169	7	a	a	DET
iajs-1215	169	8	weakly	weakly	ADJ
iajs-1215	169	9	prime	prime	ADJ
iajs-1215	169	10	r	r	NOUN
iajs-1215	169	11	-	-	PUNCT
iajs-1215	169	12	submodule	submodule	NOUN
iajs-1215	169	13	and	and	CCONJ
iajs-1215	169	14	s	s	AUX
iajs-1215	169	15	be	be	AUX
iajs-1215	169	16	a	a	DET
iajs-1215	169	17	multiplicative	multiplicative	ADJ
iajs-1215	169	18	subset	subset	NOUN
iajs-1215	169	19	of	of	ADP
iajs-1215	169	20	r	r	NOUN
iajs-1215	169	21	with	with	ADP
iajs-1215	169	22	(	(	PUNCT
iajs-1215	169	23	p	p	NOUN
iajs-1215	169	24	r	r	NOUN
iajs-1215	169	25	:	:	PUNCT
iajs-1215	169	26	m	m	X
iajs-1215	169	27	)	)	PUNCT
iajs-1215	170	1			X
iajs-1215	170	2	s	s	NOUN
iajs-1215	170	3	=	=	PUNCT
iajs-1215	170	4			X
iajs-1215	170	5	.	.	PUNCT
iajs-1215	171	1	then	then	ADV
iajs-1215	171	2	ps	ps	PROPN
iajs-1215	171	3	is	be	AUX
iajs-1215	171	4	weakly	weakly	ADJ
iajs-1215	171	5	prime	prime	ADJ
iajs-1215	171	6	rs	rs	NOUN
iajs-1215	171	7	-	-	PUNCT
iajs-1215	171	8	submodule	submodule	NOUN
iajs-1215	171	9	of	of	ADP
iajs-1215	171	10	m	m	PROPN
iajs-1215	171	11	s.	s.	PROPN
iajs-1215	171	12	proof	proof	PROPN
iajs-1215	171	13	.	.	PUNCT
iajs-1215	172	1	let	let	VERB
iajs-1215	172	2	b	b	X
iajs-1215	172	3	a	a	DET
iajs-1215	172	4			NOUN
iajs-1215	172	5	rs	rs	NOUN
iajs-1215	172	6	and	and	CCONJ
iajs-1215	172	7	c	c	NOUN
iajs-1215	172	8	x	x	SYM
iajs-1215	173	1			NOUN
iajs-1215	173	2	m	m	VERB
iajs-1215	173	3	s	s	VERB
iajs-1215	173	4	such	such	ADJ
iajs-1215	173	5	that	that	SCONJ
iajs-1215	173	6	0s	0s	PROPN
iajs-1215	173	7			PROPN
iajs-1215	173	8	b	b	PROPN
iajs-1215	173	9	a	a	DET
iajs-1215	173	10	c	c	NOUN
iajs-1215	173	11	x	x	SYM
iajs-1215	173	12			PROPN
iajs-1215	173	13	ps	ps	PROPN
iajs-1215	173	14	.	.	PROPN
iajs-1215	174	1	hence	hence	ADV
iajs-1215	174	2	0s	0s	PROPN
iajs-1215	174	3			PROPN
iajs-1215	174	4	cb	cb	PROPN
iajs-1215	174	5	xa	xa	PROPN
iajs-1215	174	6			PROPN
iajs-1215	174	7	ps	ps	PROPN
iajs-1215	175	1	and	and	CCONJ
iajs-1215	175	2	so	so	ADV
iajs-1215	175	3	there	there	PRON
iajs-1215	175	4	exists	exist	VERB
iajs-1215	175	5	y	y	PROPN
iajs-1215	175	6			PROPN
iajs-1215	175	7	p	p	PROPN
iajs-1215	175	8	and	and	CCONJ
iajs-1215	175	9	d	d	ADP
iajs-1215	175	10			NOUN
iajs-1215	175	11	s	s	VERB
iajs-1215	175	12	such	such	ADJ
iajs-1215	175	13	that	that	SCONJ
iajs-1215	175	14	d	d	X
iajs-1215	175	15	y	y	PROPN
iajs-1215	175	16	cb	cb	PROPN
iajs-1215	175	17	xa	xa	PROPN
iajs-1215	175	18			PROPN
iajs-1215	175	19	,	,	PUNCT
iajs-1215	175	20	and	and	CCONJ
iajs-1215	175	21	this	this	PRON
iajs-1215	175	22	implies	imply	VERB
iajs-1215	175	23	that	that	SCONJ
iajs-1215	175	24	there	there	PRON
iajs-1215	175	25	exists	exist	VERB
iajs-1215	175	26	t	t	PROPN
iajs-1215	175	27			PROPN
iajs-1215	175	28	s	s	VERB
iajs-1215	175	29	such	such	ADJ
iajs-1215	176	1	that	that	SCONJ
iajs-1215	176	2	t	t	PROPN
iajs-1215	176	3	a	a	PRON
iajs-1215	176	4	d	d	X
iajs-1215	176	5	x	x	X
iajs-1215	176	6	=	=	SYM
iajs-1215	176	7	t	t	PROPN
iajs-1215	176	8	b	b	X
iajs-1215	176	9	c	c	PROPN
iajs-1215	176	10	y.	y.	PROPN
iajs-1215	176	11	on	on	ADP
iajs-1215	176	12	the	the	DET
iajs-1215	176	13	other	other	ADJ
iajs-1215	176	14	hand	hand	NOUN
iajs-1215	176	15	1	1	NUM
iajs-1215	176	16	0	0	NUM
iajs-1215	176	17			PROPN
iajs-1215	176	18	cb	cb	PROPN
iajs-1215	176	19	xa	xa	PROPN
iajs-1215	176	20	=	=	PROPN
iajs-1215	176	21	(	(	PUNCT
iajs-1215	176	22	0s	0s	NOUN
iajs-1215	176	23	)	)	PUNCT
iajs-1215	176	24	which	which	PRON
iajs-1215	176	25	implies	imply	VERB
iajs-1215	176	26	that	that	SCONJ
iajs-1215	176	27	f	f	PROPN
iajs-1215	176	28	a	a	X
iajs-1215	176	29	x	x	X
iajs-1215	176	30			NOUN
iajs-1215	176	31	0	0	NUM
iajs-1215	176	32	for	for	ADP
iajs-1215	176	33	all	all	DET
iajs-1215	176	34	f	f	PROPN
iajs-1215	176	35			PROPN
iajs-1215	176	36	s.	s.	PROPN
iajs-1215	176	37	hence	hence	ADV
iajs-1215	176	38	0	0	NUM
iajs-1215	176	39			PROPN
iajs-1215	176	40	t	t	VERB
iajs-1215	176	41	a	a	DET
iajs-1215	176	42	d	d	X
iajs-1215	176	43	x	x	X
iajs-1215	176	44			NOUN
iajs-1215	177	1	p.	p.	NOUN
iajs-1215	178	1	but	but	CCONJ
iajs-1215	178	2	p	p	NOUN
iajs-1215	178	3	is	be	AUX
iajs-1215	178	4	a	a	DET
iajs-1215	178	5	weakly	weakly	ADJ
iajs-1215	178	6	prime	prime	ADJ
iajs-1215	178	7	r	r	NOUN
iajs-1215	178	8	-	-	PUNCT
iajs-1215	178	9	submodule	submodule	NOUN
iajs-1215	178	10	of	of	ADP
iajs-1215	178	11	m	m	PROPN
iajs-1215	178	12	,	,	PUNCT
iajs-1215	178	13	so	so	ADV
iajs-1215	179	1	either	either	DET
iajs-1215	179	2	t	t	PROPN
iajs-1215	179	3	d	d	X
iajs-1215	179	4	x	x	X
iajs-1215	179	5			NOUN
iajs-1215	179	6	p	p	NOUN
iajs-1215	179	7	or	or	CCONJ
iajs-1215	179	8	a	a	DET
iajs-1215	179	9			NOUN
iajs-1215	179	10	ibn	ibn	PROPN
iajs-1215	179	11	alhaitham	alhaitham	NOUN
iajs-1215	179	12	j.	j.	PROPN
iajs-1215	180	1	fo	fo	ADP
iajs-1215	180	2	r	r	NOUN
iajs-1215	180	3	pure	pure	ADJ
iajs-1215	180	4	&	&	CCONJ
iajs-1215	180	5	appl	appl	PROPN
iajs-1215	180	6	.	.	PUNCT
iajs-1215	181	1	sc	sc	PROPN
iajs-1215	182	1	i	i	PRON
iajs-1215	182	2	vo	vo	INTJ
iajs-1215	182	3	l.22	l.22	X
iajs-1215	182	4	(	(	PUNCT
iajs-1215	182	5	3	3	NUM
iajs-1215	182	6	)	)	PUNCT
iajs-1215	182	7	2009	2009	NUM
iajs-1215	182	8	(	(	PUNCT
iajs-1215	182	9	p	p	X
iajs-1215	182	10	:	:	PUNCT
iajs-1215	182	11	m	m	NUM
iajs-1215	182	12	)	)	PUNCT
iajs-1215	182	13	and	and	CCONJ
iajs-1215	182	14	hence	hence	ADV
iajs-1215	182	15	either	either	CCONJ
iajs-1215	182	16	cdt	cdt	PROPN
iajs-1215	182	17	xdt	xdt	PROPN
iajs-1215	182	18			PROPN
iajs-1215	182	19	ps	ps	PROPN
iajs-1215	182	20	or	or	CCONJ
iajs-1215	182	21	b	b	PROPN
iajs-1215	182	22	a	a	DET
iajs-1215	182	23			NOUN
iajs-1215	182	24	(	(	PUNCT
iajs-1215	182	25	p	p	X
iajs-1215	182	26	:	:	PUNCT
iajs-1215	182	27	m)s	m)s	ADJ
iajs-1215	182	28	.	.	PUNCT
iajs-1215	183	1	because	because	SCONJ
iajs-1215	183	2	(	(	PUNCT
iajs-1215	183	3	p	p	NOUN
iajs-1215	183	4	r	r	NOUN
iajs-1215	183	5	:	:	PUNCT
iajs-1215	183	6	m)s	m)s	ADJ
iajs-1215	183	7			PROPN
iajs-1215	183	8	(	(	PUNCT
iajs-1215	183	9	ps	ps	NOUN
iajs-1215	183	10	sr	sr	PROPN
iajs-1215	183	11	:	:	PUNCT
iajs-1215	183	12	m	m	PROPN
iajs-1215	183	13	s	s	X
iajs-1215	183	14	)	)	PUNCT
iajs-1215	183	15	,	,	PUNCT
iajs-1215	183	16	we	we	PRON
iajs-1215	183	17	have	have	VERB
iajs-1215	183	18	either	either	CCONJ
iajs-1215	183	19	c	c	NOUN
iajs-1215	183	20	x	x	SYM
iajs-1215	183	21			NOUN
iajs-1215	183	22	ps	ps	PROPN
iajs-1215	183	23	or	or	CCONJ
iajs-1215	183	24	b	b	PROPN
iajs-1215	183	25	a	a	DET
iajs-1215	183	26			NOUN
iajs-1215	183	27	(	(	PUNCT
iajs-1215	183	28	ps	ps	PROPN
iajs-1215	183	29	sr	sr	PROPN
iajs-1215	183	30	:	:	PUNCT
iajs-1215	183	31	m	m	PROPN
iajs-1215	183	32	s	s	NOUN
iajs-1215	183	33	)	)	PUNCT
iajs-1215	183	34	.	.	PUNCT
iajs-1215	184	1	as	as	ADP
iajs-1215	184	2	a	a	DET
iajs-1215	184	3	generalization	generalization	NOUN
iajs-1215	184	4	of	of	ADP
iajs-1215	184	5	cohen	cohen	PROPN
iajs-1215	184	6	theorem	theorem	PROPN
iajs-1215	184	7	,	,	PUNCT
iajs-1215	184	8	the	the	DET
iajs-1215	184	9	following	follow	VERB
iajs-1215	184	10	was	be	AUX
iajs-1215	184	11	given	give	VERB
iajs-1215	184	12	in	in	ADP
iajs-1215	184	13	(	(	PUNCT
iajs-1215	184	14	(	(	PUNCT
iajs-1215	184	15	3),prop.4.15,ch.1	3),prop.4.15,ch.1	NUM
iajs-1215	184	16	)	)	PUNCT
iajs-1215	184	17	.	.	PUNCT
iajs-1215	185	1	let	let	VERB
iajs-1215	185	2	m	m	PRON
iajs-1215	185	3	be	be	AUX
iajs-1215	185	4	a	a	DET
iajs-1215	185	5	finitely	finitely	ADV
iajs-1215	185	6	generated	generate	VERB
iajs-1215	185	7	r	r	NOUN
iajs-1215	185	8	-	-	PUNCT
iajs-1215	185	9	module	module	NOUN
iajs-1215	185	10	,	,	PUNCT
iajs-1215	185	11	then	then	ADV
iajs-1215	185	12	m	m	NOUN
iajs-1215	185	13	is	be	AUX
iajs-1215	185	14	noetherian	noetherian	ADJ
iajs-1215	185	15	iff	iff	PROPN
iajs-1215	185	16	every	every	DET
iajs-1215	185	17	prime	prime	ADJ
iajs-1215	185	18	submodule	submodule	NOUN
iajs-1215	185	19	is	be	AUX
iajs-1215	185	20	finitely	finitely	ADV
iajs-1215	185	21	generated	generate	VERB
iajs-1215	185	22	.	.	PUNCT
iajs-1215	186	1	since	since	SCONJ
iajs-1215	186	2	every	every	DET
iajs-1215	186	3	prime	prime	ADJ
iajs-1215	186	4	submodule	submodule	NOUN
iajs-1215	186	5	is	be	AUX
iajs-1215	186	6	weakly	weakly	ADV
iajs-1215	186	7	prime	prime	ADJ
iajs-1215	186	8	(	(	PUNCT
iajs-1215	186	9	by	by	ADP
iajs-1215	186	10	rem.2.1.(11	rem.2.1.(11	NOUN
iajs-1215	186	11	)	)	PUNCT
iajs-1215	186	12	)	)	PUNCT
iajs-1215	186	13	,	,	PUNCT
iajs-1215	186	14	we	we	PRON
iajs-1215	186	15	have	have	VERB
iajs-1215	186	16	the	the	DET
iajs-1215	186	17	following	following	NOUN
iajs-1215	186	18	:	:	PUNCT
iajs-1215	186	19	proposition2.10	proposition2.10	NOUN
iajs-1215	186	20	:	:	PUNCT
iajs-1215	186	21	let	let	VERB
iajs-1215	186	22	m	m	PRON
iajs-1215	186	23	be	be	AUX
iajs-1215	186	24	a	a	DET
iajs-1215	186	25	finitely	finitely	ADV
iajs-1215	186	26	generated	generate	VERB
iajs-1215	186	27	r	r	NOUN
iajs-1215	186	28	-	-	PUNCT
iajs-1215	186	29	module	module	NOUN
iajs-1215	186	30	.	.	PUNCT
iajs-1215	187	1	then	then	ADV
iajs-1215	187	2	m	m	VERB
iajs-1215	187	3	is	be	AUX
iajs-1215	187	4	noetherian	noetherian	ADJ
iajs-1215	187	5	if	if	SCONJ
iajs-1215	187	6	every	every	DET
iajs-1215	187	7	weakly	weakly	ADJ
iajs-1215	187	8	prime	prime	ADJ
iajs-1215	187	9	submodule	submodule	NOUN
iajs-1215	187	10	is	be	AUX
iajs-1215	187	11	finitely	finitely	ADV
iajs-1215	187	12	generated	generate	VERB
iajs-1215	187	13	.	.	PUNCT
iajs-1215	188	1	noteice	noteice	NOUN
iajs-1215	188	2	that	that	SCONJ
iajs-1215	188	3	the	the	DET
iajs-1215	188	4	condition	condition	NOUN
iajs-1215	188	5	m	m	AUX
iajs-1215	188	6	is	be	AUX
iajs-1215	188	7	finitely	finitely	ADV
iajs-1215	188	8	generated	generate	VERB
iajs-1215	188	9	that	that	SCONJ
iajs-1215	188	10	cannt	cannt	NOUN
iajs-1215	188	11	be	be	AUX
iajs-1215	188	12	dropped	drop	VERB
iajs-1215	188	13	from	from	ADP
iajs-1215	188	14	prop	prop	NOUN
iajs-1215	188	15	.2.10	.2.10	PROPN
iajs-1215	188	16	,	,	PUNCT
iajs-1215	188	17	as	as	SCONJ
iajs-1215	188	18	the	the	DET
iajs-1215	188	19	following	follow	VERB
iajs-1215	188	20	example	example	NOUN
iajs-1215	188	21	shows	show	VERB
iajs-1215	188	22	:	:	PUNCT
iajs-1215	188	23	the	the	DET
iajs-1215	188	24	z	z	NOUN
iajs-1215	188	25	-	-	PUNCT
iajs-1215	188	26	module	module	NOUN
iajs-1215	188	27	z	z	NOUN
iajs-1215	188	28	p	p	ADJ
iajs-1215	188	29	is	be	AUX
iajs-1215	188	30	not	not	PART
iajs-1215	188	31	finitely	finitely	ADV
iajs-1215	188	32	generated	generate	VERB
iajs-1215	188	33	,	,	PUNCT
iajs-1215	188	34	also	also	ADV
iajs-1215	188	35	it	it	PRON
iajs-1215	188	36	is	be	AUX
iajs-1215	188	37	not	not	PART
iajs-1215	188	38	noetherian	noetherian	ADJ
iajs-1215	188	39	.	.	PUNCT
iajs-1215	189	1	however	however	ADV
iajs-1215	189	2	if	if	SCONJ
iajs-1215	189	3	g	g	PROPN
iajs-1215	189	4	is	be	AUX
iajs-1215	189	5	a	a	DET
iajs-1215	189	6	nonzero	nonzero	NOUN
iajs-1215	189	7	submodule	submodule	NOUN
iajs-1215	189	8	,	,	PUNCT
iajs-1215	189	9	then	then	ADV
iajs-1215	189	10	1	1	NUM
iajs-1215	189	11	g	g	AUX
iajs-1215	189	12			PROPN
iajs-1215	189	13			X
iajs-1215	190	1	i	i	PRON
iajs-1215	190	2	z	z	PROPN
iajs-1215	190	3	p	p	NOUN
iajs-1215	190	4	for	for	ADP
iajs-1215	190	5	some	some	DET
iajs-1215	190	6	i	i	PRON
iajs-1215	190	7			NOUN
iajs-1215	190	8	z+	z+	NUM
iajs-1215	190	9	,	,	PUNCT
iajs-1215	190	10	and	and	CCONJ
iajs-1215	190	11	0	0	NUM
iajs-1215	190	12			NOUN
iajs-1215	190	13	p	p	X
iajs-1215	190	14	(	(	PUNCT
iajs-1215	190	15	1	1	NUM
iajs-1215	190	16			ADV
iajs-1215	190	17	i	i	PRON
iajs-1215	190	18	+1	+1	NOUN
iajs-1215	190	19	z	z	PROPN
iajs-1215	190	20	p	p	NOUN
iajs-1215	190	21	)	)	PUNCT
iajs-1215	191	1			PROPN
iajs-1215	191	2	g.	g.	PROPN
iajs-1215	192	1	but	but	CCONJ
iajs-1215	192	2	p	p	PROPN
iajs-1215	192	3			PROPN
iajs-1215	192	4	(	(	PUNCT
iajs-1215	192	5	g	g	NOUN
iajs-1215	192	6	:	:	PUNCT
iajs-1215	192	7	z	z	NOUN
iajs-1215	192	8	p	p	ADJ
iajs-1215	192	9	)	)	PUNCT
iajs-1215	193	1	=	=	SYM
iajs-1215	193	2	0	0	NUM
iajs-1215	193	3	and	and	CCONJ
iajs-1215	193	4	1	1	NUM
iajs-1215	193	5			ADV
iajs-1215	194	1	i	i	PRON
iajs-1215	195	1	+1	+1	INTJ
iajs-1215	196	1	z	z	PROPN
iajs-1215	196	2	p	p	X
iajs-1215	196	3			NOUN
iajs-1215	196	4	g	g	NOUN
iajs-1215	196	5	;	;	PUNCT
iajs-1215	196	6	that	that	PRON
iajs-1215	196	7	is	be	AUX
iajs-1215	196	8	g	g	NOUN
iajs-1215	196	9	is	be	AUX
iajs-1215	196	10	not	not	PART
iajs-1215	196	11	weakly	weakly	ADV
iajs-1215	196	12	prime	prime	ADJ
iajs-1215	196	13	.	.	PUNCT
iajs-1215	197	1	thus	thus	ADV
iajs-1215	197	2	(	(	PUNCT
iajs-1215	197	3	0	0	X
iajs-1215	197	4	)	)	PUNCT
iajs-1215	197	5	is	be	AUX
iajs-1215	197	6	the	the	DET
iajs-1215	197	7	only	only	ADJ
iajs-1215	197	8	weakly	weakly	ADJ
iajs-1215	197	9	prime	prime	ADJ
iajs-1215	197	10	submodule	submodule	NOUN
iajs-1215	197	11	of	of	ADP
iajs-1215	197	12	z	z	PROPN
iajs-1215	197	13	p	p	ADJ
iajs-1215	198	1	and	and	CCONJ
iajs-1215	198	2	it	it	PRON
iajs-1215	198	3	is	be	AUX
iajs-1215	198	4	obviously	obviously	ADV
iajs-1215	198	5	finitely	finitely	ADV
iajs-1215	198	6	generated	generate	VERB
iajs-1215	198	7	.	.	PUNCT
iajs-1215	199	1	theorem	theorem	VERB
iajs-1215	199	2	2.11	2.11	NUM
iajs-1215	199	3	:	:	PUNCT
iajs-1215	199	4	let	let	VERB
iajs-1215	199	5	m	m	PROPN
iajs-1215	199	6	1	1	NUM
iajs-1215	199	7	,	,	PUNCT
iajs-1215	199	8	m	m	VERB
iajs-1215	199	9	2	2	NUM
iajs-1215	199	10	be	be	VERB
iajs-1215	199	11	r	r	NOUN
iajs-1215	199	12	-	-	PUNCT
iajs-1215	199	13	modules	module	NOUN
iajs-1215	199	14	and	and	CCONJ
iajs-1215	199	15	let	let	VERB
iajs-1215	199	16	n	n	PRON
iajs-1215	199	17	be	be	AUX
iajs-1215	199	18	a	a	DET
iajs-1215	199	19	proper	proper	ADJ
iajs-1215	199	20	submodule	submodule	NOUN
iajs-1215	199	21	of	of	ADP
iajs-1215	199	22	m1	m1	PROPN
iajs-1215	199	23	.	.	PUNCT
iajs-1215	200	1	then	then	ADV
iajs-1215	200	2	w	w	PROPN
iajs-1215	200	3	=	=	PUNCT
iajs-1215	200	4	n	n	PROPN
iajs-1215	200	5			ADJ
iajs-1215	200	6	m	m	VERB
iajs-1215	200	7	2	2	NUM
iajs-1215	200	8	is	be	AUX
iajs-1215	200	9	a	a	DET
iajs-1215	200	10	weakly	weakly	ADJ
iajs-1215	200	11	prime	prime	ADJ
iajs-1215	200	12	submodule	submodule	NOUN
iajs-1215	200	13	of	of	ADP
iajs-1215	200	14	m	m	PROPN
iajs-1215	200	15	=	=	NOUN
iajs-1215	200	16	m	m	VERB
iajs-1215	200	17	1	1	NUM
iajs-1215	200	18	m	m	NOUN
iajs-1215	200	19	2	2	NUM
iajs-1215	200	20			X
iajs-1215	200	21	n	n	X
iajs-1215	200	22	is	be	AUX
iajs-1215	200	23	a	a	DET
iajs-1215	200	24	weakly	weakly	ADJ
iajs-1215	200	25	prime	prime	ADJ
iajs-1215	200	26	submodule	submodule	NOUN
iajs-1215	200	27	of	of	ADP
iajs-1215	200	28	m	m	PROPN
iajs-1215	200	29	1	1	NUM
iajs-1215	200	30	and	and	CCONJ
iajs-1215	200	31	for	for	ADP
iajs-1215	200	32	r	r	NOUN
iajs-1215	200	33			PROPN
iajs-1215	200	34	r	r	NOUN
iajs-1215	200	35	,	,	PUNCT
iajs-1215	200	36	x	x	SYM
iajs-1215	200	37			NOUN
iajs-1215	200	38	m	m	VERB
iajs-1215	200	39	1	1	NUM
iajs-1215	200	40	with	with	ADP
iajs-1215	200	41	r	r	NOUN
iajs-1215	200	42	x	x	SYM
iajs-1215	200	43	=	=	SYM
iajs-1215	200	44	0	0	NUM
iajs-1215	200	45	,	,	PUNCT
iajs-1215	200	46	but	but	CCONJ
iajs-1215	200	47	x	x	X
iajs-1215	200	48			NOUN
iajs-1215	200	49	n	n	CCONJ
iajs-1215	200	50	,	,	PUNCT
iajs-1215	200	51	r	r	NOUN
iajs-1215	200	52			VERB
iajs-1215	200	53	(	(	PUNCT
iajs-1215	200	54	n	n	CCONJ
iajs-1215	200	55	:	:	PUNCT
iajs-1215	200	56	m	m	NOUN
iajs-1215	200	57	1	1	NUM
iajs-1215	200	58	)	)	PUNCT
iajs-1215	200	59	implies	imply	VERB
iajs-1215	200	60	r	r	NOUN
iajs-1215	200	61			PROPN
iajs-1215	200	62	ann	ann	PROPN
iajs-1215	200	63	m	m	PROPN
iajs-1215	200	64	2	2	NUM
iajs-1215	200	65	.	.	PUNCT
iajs-1215	201	1	proof	proof	NOUN
iajs-1215	201	2	.	.	PUNCT
iajs-1215	202	1	(	(	PUNCT
iajs-1215	202	2			NOUN
iajs-1215	202	3	)	)	PUNCT
iajs-1215	202	4	let	let	VERB
iajs-1215	202	5	r	r	NOUN
iajs-1215	202	6			NOUN
iajs-1215	202	7	r	r	NOUN
iajs-1215	202	8	,	,	PUNCT
iajs-1215	202	9	x	x	SYM
iajs-1215	202	10			NOUN
iajs-1215	202	11	m	m	VERB
iajs-1215	202	12	1	1	NUM
iajs-1215	203	1	such	such	ADJ
iajs-1215	203	2	that	that	SCONJ
iajs-1215	203	3	0	0	NUM
iajs-1215	203	4			NOUN
iajs-1215	203	5	r	r	NOUN
iajs-1215	203	6	x	x	X
iajs-1215	203	7			PROPN
iajs-1215	203	8	n.	n.	NOUN
iajs-1215	203	9	then	then	ADV
iajs-1215	203	10	(	(	PUNCT
iajs-1215	203	11	0,0	0,0	NOUN
iajs-1215	203	12	)	)	PUNCT
iajs-1215	203	13	≠	≠	PROPN
iajs-1215	203	14	r	r	NOUN
iajs-1215	203	15	(	(	PUNCT
iajs-1215	203	16	x,0	x,0	PROPN
iajs-1215	203	17	)	)	PUNCT
iajs-1215	203	18			PROPN
iajs-1215	203	19	w	w	PROPN
iajs-1215	203	20	,	,	PUNCT
iajs-1215	203	21	but	but	CCONJ
iajs-1215	203	22	w	w	NOUN
iajs-1215	203	23	is	be	AUX
iajs-1215	203	24	weakly	weakly	ADV
iajs-1215	203	25	prime	prime	ADJ
iajs-1215	203	26	,	,	PUNCT
iajs-1215	203	27	so	so	ADV
iajs-1215	203	28	either	either	CCONJ
iajs-1215	203	29	(	(	PUNCT
iajs-1215	203	30	x,0	x,0	PROPN
iajs-1215	203	31	)	)	PUNCT
iajs-1215	204	1			NOUN
iajs-1215	204	2	w	w	NOUN
iajs-1215	204	3	or	or	CCONJ
iajs-1215	204	4	r	r	NOUN
iajs-1215	204	5			NOUN
iajs-1215	204	6	(	(	PUNCT
iajs-1215	204	7	w	w	NOUN
iajs-1215	204	8	r	r	NOUN
iajs-1215	204	9	:	:	PUNCT
iajs-1215	204	10	m	m	NOUN
iajs-1215	204	11	)	)	PUNCT
iajs-1215	204	12	.	.	PUNCT
iajs-1215	205	1	thus	thus	ADV
iajs-1215	205	2	either	either	CCONJ
iajs-1215	205	3	x	x	SYM
iajs-1215	205	4			NOUN
iajs-1215	205	5	n	n	CCONJ
iajs-1215	205	6	or	or	CCONJ
iajs-1215	205	7	r	r	NOUN
iajs-1215	205	8			NOUN
iajs-1215	205	9	(	(	PUNCT
iajs-1215	205	10	n	n	NOUN
iajs-1215	205	11	r	r	NOUN
iajs-1215	205	12	:	:	PUNCT
iajs-1215	205	13	m	m	PROPN
iajs-1215	205	14	1	1	NUM
iajs-1215	205	15	)	)	PUNCT
iajs-1215	205	16	,	,	PUNCT
iajs-1215	205	17	so	so	SCONJ
iajs-1215	205	18	that	that	SCONJ
iajs-1215	205	19	n	n	NOUN
iajs-1215	205	20	is	be	AUX
iajs-1215	205	21	weakly	weakly	ADV
iajs-1215	205	22	prime	prime	ADJ
iajs-1215	205	23	.	.	PUNCT
iajs-1215	206	1	now	now	ADV
iajs-1215	206	2	,	,	PUNCT
iajs-1215	206	3	if	if	SCONJ
iajs-1215	206	4	r	r	NOUN
iajs-1215	206	5			NOUN
iajs-1215	206	6	r	r	NOUN
iajs-1215	206	7	,	,	PUNCT
iajs-1215	206	8	x	x	SYM
iajs-1215	206	9			NOUN
iajs-1215	206	10	m	m	VERB
iajs-1215	206	11	1	1	NUM
iajs-1215	206	12	such	such	ADJ
iajs-1215	206	13	that	that	DET
iajs-1215	206	14	r	r	NOUN
iajs-1215	206	15	x	x	NOUN
iajs-1215	206	16	=	=	SYM
iajs-1215	206	17	0	0	NUM
iajs-1215	206	18	,	,	PUNCT
iajs-1215	206	19	x	x	PUNCT
iajs-1215	206	20			NOUN
iajs-1215	206	21	n	n	ADP
iajs-1215	206	22	and	and	CCONJ
iajs-1215	206	23	r	r	PROPN
iajs-1215	206	24			PROPN
iajs-1215	206	25	(	(	PUNCT
iajs-1215	206	26	n	n	CCONJ
iajs-1215	206	27	:	:	PUNCT
iajs-1215	206	28	m	m	PROPN
iajs-1215	206	29	1	1	NUM
iajs-1215	206	30	)	)	PUNCT
iajs-1215	206	31	.	.	PUNCT
iajs-1215	207	1	assume	assume	VERB
iajs-1215	207	2	that	that	SCONJ
iajs-1215	207	3	r	r	NOUN
iajs-1215	207	4			NOUN
iajs-1215	207	5	ann	ann	PROPN
iajs-1215	207	6	m	m	PROPN
iajs-1215	207	7	2	2	NUM
iajs-1215	207	8	,	,	PUNCT
iajs-1215	207	9	so	so	SCONJ
iajs-1215	207	10	there	there	PRON
iajs-1215	207	11	exists	exist	VERB
iajs-1215	207	12	m	m	VERB
iajs-1215	207	13			NOUN
iajs-1215	207	14	m	m	VERB
iajs-1215	207	15	2	2	NUM
iajs-1215	207	16	such	such	ADJ
iajs-1215	207	17	that	that	SCONJ
iajs-1215	207	18	r	r	NOUN
iajs-1215	207	19	m2	m2	PROPN
iajs-1215	207	20			PROPN
iajs-1215	207	21	0	0	NUM
iajs-1215	207	22	.	.	PUNCT
iajs-1215	208	1	thus	thus	ADV
iajs-1215	208	2	r	r	NOUN
iajs-1215	208	3	(	(	PUNCT
iajs-1215	208	4	x	x	NOUN
iajs-1215	208	5	,	,	PUNCT
iajs-1215	208	6	m2	m2	PROPN
iajs-1215	208	7	)	)	PUNCT
iajs-1215	208	8	=	=	PUNCT
iajs-1215	208	9	(	(	PUNCT
iajs-1215	208	10	r	r	NOUN
iajs-1215	208	11	x	x	PROPN
iajs-1215	208	12	,	,	PUNCT
iajs-1215	208	13	r	r	PROPN
iajs-1215	208	14	m2	m2	PROPN
iajs-1215	208	15	)	)	PUNCT
iajs-1215	208	16	=	=	PUNCT
iajs-1215	208	17	(	(	PUNCT
iajs-1215	208	18	0	0	NUM
iajs-1215	208	19	,	,	PUNCT
iajs-1215	208	20	r	r	PROPN
iajs-1215	208	21	m2	m2	PROPN
iajs-1215	208	22	)	)	PUNCT
iajs-1215	208	23			PROPN
iajs-1215	208	24	(	(	PUNCT
iajs-1215	208	25	0,0	0,0	NOUN
iajs-1215	208	26	)	)	PUNCT
iajs-1215	208	27	and	and	CCONJ
iajs-1215	208	28	hence	hence	ADV
iajs-1215	208	29	(	(	PUNCT
iajs-1215	208	30	0,0	0,0	NOUN
iajs-1215	208	31	)	)	PUNCT
iajs-1215	208	32			NOUN
iajs-1215	208	33	r	r	NOUN
iajs-1215	208	34	(	(	PUNCT
iajs-1215	208	35	x	x	NOUN
iajs-1215	208	36	,	,	PUNCT
iajs-1215	208	37	m2	m2	PROPN
iajs-1215	208	38	)	)	PUNCT
iajs-1215	208	39			NOUN
iajs-1215	208	40	n	n	PROPN
iajs-1215	208	41			PROPN
iajs-1215	208	42	m	m	VERB
iajs-1215	208	43	2	2	NUM
iajs-1215	208	44	=	=	SYM
iajs-1215	208	45	w.	w.	NOUN
iajs-1215	208	46	since	since	SCONJ
iajs-1215	208	47	w	w	PROPN
iajs-1215	208	48	is	be	AUX
iajs-1215	208	49	weakly	weakly	ADV
iajs-1215	208	50	prime	prime	ADJ
iajs-1215	208	51	,	,	PUNCT
iajs-1215	208	52	so	so	ADV
iajs-1215	208	53	either	either	CCONJ
iajs-1215	208	54	(	(	PUNCT
iajs-1215	208	55	x	x	X
iajs-1215	208	56	,	,	PUNCT
iajs-1215	208	57	m2	m2	PROPN
iajs-1215	208	58	)	)	PUNCT
iajs-1215	208	59			NOUN
iajs-1215	208	60	n	n	PROPN
iajs-1215	208	61			PROPN
iajs-1215	208	62	m	m	VERB
iajs-1215	208	63	2	2	NUM
iajs-1215	208	64	or	or	CCONJ
iajs-1215	208	65	r	r	NOUN
iajs-1215	208	66			NOUN
iajs-1215	208	67	(	(	PUNCT
iajs-1215	208	68	n	n	CCONJ
iajs-1215	208	69			PROPN
iajs-1215	208	70	m	m	VERB
iajs-1215	208	71	2	2	NUM
iajs-1215	208	72	r	r	NOUN
iajs-1215	208	73	:	:	PUNCT
iajs-1215	208	74	m	m	VERB
iajs-1215	208	75	1	1	NUM
iajs-1215	208	76			ADJ
iajs-1215	208	77	m	m	VERB
iajs-1215	208	78	2	2	NUM
iajs-1215	208	79	)	)	PUNCT
iajs-1215	208	80	.	.	PUNCT
iajs-1215	209	1	thus	thus	ADV
iajs-1215	209	2	either	either	CCONJ
iajs-1215	209	3	x	x	SYM
iajs-1215	209	4			NOUN
iajs-1215	209	5	n	n	CCONJ
iajs-1215	209	6	or	or	CCONJ
iajs-1215	209	7	r	r	NOUN
iajs-1215	209	8			NOUN
iajs-1215	209	9	(	(	PUNCT
iajs-1215	209	10	n	n	NOUN
iajs-1215	209	11	r	r	NOUN
iajs-1215	209	12	:	:	PUNCT
iajs-1215	209	13	m	m	PROPN
iajs-1215	209	14	1	1	NUM
iajs-1215	209	15	)	)	PUNCT
iajs-1215	209	16	which	which	PRON
iajs-1215	209	17	is	be	AUX
iajs-1215	209	18	a	a	DET
iajs-1215	209	19	contradiction	contradiction	NOUN
iajs-1215	209	20	with	with	ADP
iajs-1215	209	21	hypothesis	hypothesis	NOUN
iajs-1215	209	22	.	.	PUNCT
iajs-1215	210	1	(	(	PUNCT
iajs-1215	210	2			AUX
iajs-1215	210	3	)	)	PUNCT
iajs-1215	210	4	let	let	VERB
iajs-1215	210	5	r	r	NOUN
iajs-1215	210	6			NOUN
iajs-1215	210	7	r	r	NOUN
iajs-1215	210	8	,	,	PUNCT
iajs-1215	210	9	(	(	PUNCT
iajs-1215	210	10	x	x	NOUN
iajs-1215	210	11	,	,	PUNCT
iajs-1215	210	12	y	y	PROPN
iajs-1215	210	13	)	)	PUNCT
iajs-1215	210	14			NOUN
iajs-1215	210	15	m.	m.	NOUN
iajs-1215	211	1	assume	assume	VERB
iajs-1215	211	2	(	(	PUNCT
iajs-1215	211	3	0,0	0,0	NOUN
iajs-1215	211	4	)	)	PUNCT
iajs-1215	211	5			NOUN
iajs-1215	211	6	r	r	NOUN
iajs-1215	211	7	(	(	PUNCT
iajs-1215	211	8	x	x	NOUN
iajs-1215	211	9	,	,	PUNCT
iajs-1215	211	10	y	y	PROPN
iajs-1215	211	11	)	)	PUNCT
iajs-1215	211	12			NOUN
iajs-1215	211	13	n	n	PROPN
iajs-1215	211	14			PROPN
iajs-1215	211	15	m	m	VERB
iajs-1215	211	16	2	2	NUM
iajs-1215	211	17	,	,	PUNCT
iajs-1215	211	18	so	so	ADV
iajs-1215	211	19	if	if	SCONJ
iajs-1215	211	20	r	r	NOUN
iajs-1215	211	21	x	x	SYM
iajs-1215	211	22			NOUN
iajs-1215	211	23	0	0	NUM
iajs-1215	211	24	,	,	PUNCT
iajs-1215	211	25	then	then	ADV
iajs-1215	211	26	either	either	CCONJ
iajs-1215	211	27	x	x	SYM
iajs-1215	211	28			NOUN
iajs-1215	211	29	n	n	CCONJ
iajs-1215	211	30	or	or	CCONJ
iajs-1215	211	31	r	r	NOUN
iajs-1215	211	32			NOUN
iajs-1215	211	33	(	(	PUNCT
iajs-1215	211	34	n	n	CCONJ
iajs-1215	211	35	:	:	PUNCT
iajs-1215	211	36	m	m	PROPN
iajs-1215	211	37	1	1	NUM
iajs-1215	211	38	)	)	PUNCT
iajs-1215	211	39	,	,	PUNCT
iajs-1215	211	40	since	since	SCONJ
iajs-1215	211	41	n	n	PRON
iajs-1215	211	42	is	be	AUX
iajs-1215	211	43	weakly	weakly	ADV
iajs-1215	211	44	prime	prime	ADJ
iajs-1215	211	45	.	.	PUNCT
iajs-1215	212	1	thus	thus	ADV
iajs-1215	212	2	either	either	CCONJ
iajs-1215	212	3	(	(	PUNCT
iajs-1215	212	4	x	x	X
iajs-1215	212	5	,	,	PUNCT
iajs-1215	212	6	y	y	PROPN
iajs-1215	212	7	)	)	PUNCT
iajs-1215	212	8			NOUN
iajs-1215	212	9	n	n	PROPN
iajs-1215	213	1			PROPN
iajs-1215	213	2	m	m	VERB
iajs-1215	213	3	2	2	NUM
iajs-1215	213	4	or	or	CCONJ
iajs-1215	213	5	r	r	NOUN
iajs-1215	213	6			NOUN
iajs-1215	213	7	(	(	PUNCT
iajs-1215	213	8	n	n	CCONJ
iajs-1215	213	9			PROPN
iajs-1215	213	10	m	m	VERB
iajs-1215	213	11	2	2	NUM
iajs-1215	213	12	:	:	PUNCT
iajs-1215	213	13	m1	m1	PROPN
iajs-1215	213	14			PROPN
iajs-1215	213	15	m	m	VERB
iajs-1215	213	16	2	2	NUM
iajs-1215	213	17	)	)	PUNCT
iajs-1215	213	18	.	.	PUNCT
iajs-1215	214	1	if	if	SCONJ
iajs-1215	214	2	r	r	NOUN
iajs-1215	214	3	x	x	NOUN
iajs-1215	214	4	=	=	NOUN
iajs-1215	214	5	0	0	X
iajs-1215	214	6	.	.	PUNCT
iajs-1215	214	7	suppose	suppose	VERB
iajs-1215	214	8	x	x	X
iajs-1215	214	9			NOUN
iajs-1215	214	10	n	n	CCONJ
iajs-1215	214	11	and	and	CCONJ
iajs-1215	214	12	r	r	PROPN
iajs-1215	214	13			PROPN
iajs-1215	214	14	(	(	PUNCT
iajs-1215	214	15	n1	n1	NOUN
iajs-1215	214	16	:	:	PUNCT
iajs-1215	214	17	m1	m1	PROPN
iajs-1215	214	18	)	)	PUNCT
iajs-1215	214	19	,	,	PUNCT
iajs-1215	214	20	then	then	ADV
iajs-1215	214	21	by	by	ADP
iajs-1215	214	22	hypothesis	hypothesis	NOUN
iajs-1215	214	23	r	r	NOUN
iajs-1215	214	24			PROPN
iajs-1215	214	25	ann	ann	PROPN
iajs-1215	214	26	m	m	PROPN
iajs-1215	214	27	2	2	NUM
iajs-1215	214	28	and	and	CCONJ
iajs-1215	214	29	so	so	ADV
iajs-1215	214	30	r	r	NOUN
iajs-1215	214	31	(	(	PUNCT
iajs-1215	214	32	x	x	NOUN
iajs-1215	214	33	,	,	PUNCT
iajs-1215	214	34	y	y	NOUN
iajs-1215	214	35	)	)	PUNCT
iajs-1215	214	36	=	=	SYM
iajs-1215	214	37	(	(	PUNCT
iajs-1215	214	38	0,0	0,0	NOUN
iajs-1215	214	39	)	)	PUNCT
iajs-1215	214	40	which	which	PRON
iajs-1215	214	41	is	be	AUX
iajs-1215	214	42	a	a	DET
iajs-1215	214	43	contradiction	contradiction	NOUN
iajs-1215	214	44	.	.	PUNCT
iajs-1215	215	1	thus	thus	ADV
iajs-1215	215	2	either	either	CCONJ
iajs-1215	215	3	x	x	SYM
iajs-1215	215	4			NOUN
iajs-1215	215	5	n	n	CCONJ
iajs-1215	215	6	or	or	CCONJ
iajs-1215	215	7	r	r	NOUN
iajs-1215	215	8			NOUN
iajs-1215	215	9	(	(	PUNCT
iajs-1215	215	10	n1	n1	NOUN
iajs-1215	215	11	r	r	NOUN
iajs-1215	215	12	:	:	PUNCT
iajs-1215	215	13	m	m	NOUN
iajs-1215	215	14	1	1	NUM
iajs-1215	215	15	)	)	PUNCT
iajs-1215	215	16	and	and	CCONJ
iajs-1215	215	17	hence	hence	ADV
iajs-1215	215	18	either	either	CCONJ
iajs-1215	215	19	(	(	PUNCT
iajs-1215	215	20	x	x	X
iajs-1215	215	21	,	,	PUNCT
iajs-1215	215	22	y	y	PROPN
iajs-1215	215	23	)	)	PUNCT
iajs-1215	215	24			NOUN
iajs-1215	216	1	n	n	PROPN
iajs-1215	216	2			PROPN
iajs-1215	216	3	m	m	VERB
iajs-1215	216	4	2	2	NUM
iajs-1215	216	5	or	or	CCONJ
iajs-1215	216	6	r	r	NOUN
iajs-1215	216	7			NOUN
iajs-1215	216	8	(	(	PUNCT
iajs-1215	216	9	n1	n1	PROPN
iajs-1215	216	10			PROPN
iajs-1215	216	11	m	m	VERB
iajs-1215	216	12	2	2	NUM
iajs-1215	216	13	r	r	NOUN
iajs-1215	216	14	:	:	PUNCT
iajs-1215	216	15	m	m	VERB
iajs-1215	216	16	1	1	NUM
iajs-1215	216	17			ADJ
iajs-1215	216	18	m	m	VERB
iajs-1215	216	19	2	2	NUM
iajs-1215	216	20	)	)	PUNCT
iajs-1215	216	21	.	.	PUNCT
iajs-1215	217	1	it	it	PRON
iajs-1215	217	2	is	be	AUX
iajs-1215	217	3	known	know	VERB
iajs-1215	217	4	that	that	SCONJ
iajs-1215	217	5	if	if	SCONJ
iajs-1215	217	6	q	q	NOUN
iajs-1215	217	7	is	be	AUX
iajs-1215	217	8	a	a	DET
iajs-1215	217	9	primary	primary	ADJ
iajs-1215	217	10	submodule	submodule	NOUN
iajs-1215	217	11	then	then	ADV
iajs-1215	217	12	)	)	PUNCT
iajs-1215	217	13	:(	:(	PUNCT
iajs-1215	218	1			NOUN
iajs-1215	218	2	q	q	NOUN
iajs-1215	218	3	is	be	AUX
iajs-1215	218	4	a	a	DET
iajs-1215	218	5	prime	prime	ADJ
iajs-1215	218	6	ideal	ideal	NOUN
iajs-1215	218	7	,	,	PUNCT
iajs-1215	218	8	see	see	VERB
iajs-1215	218	9	(	(	PUNCT
iajs-1215	218	10	11	11	NUM
iajs-1215	218	11	,	,	PUNCT
iajs-1215	218	12	prop	prop	NOUN
iajs-1215	218	13	.	.	PUNCT
iajs-1215	218	14	2.11	2.11	NUM
iajs-1215	218	15	,	,	PUNCT
iajs-1215	218	16	p.41	p.41	NOUN
iajs-1215	218	17	)	)	PUNCT
iajs-1215	218	18	.	.	PUNCT
iajs-1215	219	1	sometimes	sometimes	ADV
iajs-1215	219	2	q	q	X
iajs-1215	219	3	is	be	AUX
iajs-1215	219	4	called	call	VERB
iajs-1215	219	5	p	p	NOUN
iajs-1215	219	6	-	-	PUNCT
iajs-1215	219	7	primary	primary	ADJ
iajs-1215	219	8	,	,	PUNCT
iajs-1215	219	9	see(11	see(11	NOUN
iajs-1215	219	10	,	,	PUNCT
iajs-1215	219	11	p.42	p.42	NOUN
iajs-1215	219	12	)	)	PUNCT
iajs-1215	219	13	.	.	PUNCT
iajs-1215	220	1	now	now	ADV
iajs-1215	220	2	we	we	PRON
iajs-1215	220	3	have	have	VERB
iajs-1215	220	4	the	the	DET
iajs-1215	220	5	following	follow	VERB
iajs-1215	220	6	result	result	NOUN
iajs-1215	220	7	:	:	PUNCT
iajs-1215	220	8	corollary	corollary	ADJ
iajs-1215	220	9	2.12	2.12	NUM
iajs-1215	220	10	:	:	PUNCT
iajs-1215	220	11	let	let	VERB
iajs-1215	220	12	q	q	NOUN
iajs-1215	220	13	be	be	AUX
iajs-1215	220	14	p	p	NOUN
iajs-1215	220	15	-	-	PUNCT
iajs-1215	220	16	primary	primary	ADJ
iajs-1215	220	17	submodules	submodule	NOUN
iajs-1215	220	18	of	of	ADP
iajs-1215	220	19	an	an	DET
iajs-1215	220	20	r	r	NOUN
iajs-1215	220	21	-	-	PUNCT
iajs-1215	220	22	module	module	NOUN
iajs-1215	220	23	m	m	NOUN
iajs-1215	220	24	1	1	NUM
iajs-1215	220	25	with	with	ADP
iajs-1215	220	26			NOUN
iajs-1215	220	27	q	q	NOUN
iajs-1215	220	28	=	=	SYM
iajs-1215	220	29	(	(	PUNCT
iajs-1215	220	30	0	0	NUM
iajs-1215	220	31	)	)	PUNCT
iajs-1215	220	32	.	.	PUNCT
iajs-1215	221	1	if	if	SCONJ
iajs-1215	221	2	n	n	PRON
iajs-1215	221	3	is	be	AUX
iajs-1215	221	4	a	a	DET
iajs-1215	221	5	weakly	weakly	ADJ
iajs-1215	221	6	prime	prime	ADJ
iajs-1215	221	7	submodule	submodule	NOUN
iajs-1215	221	8	of	of	ADP
iajs-1215	221	9	m	m	PROPN
iajs-1215	221	10	1	1	NUM
iajs-1215	221	11	and	and	CCONJ
iajs-1215	221	12	m2	m2	PROPN
iajs-1215	221	13	is	be	AUX
iajs-1215	221	14	an	an	DET
iajs-1215	221	15	r	r	NOUN
iajs-1215	221	16	-	-	PUNCT
iajs-1215	221	17	module	module	NOUN
iajs-1215	221	18	such	such	ADJ
iajs-1215	221	19	that	that	SCONJ
iajs-1215	221	20	p	p	PROPN
iajs-1215	221	21			PROPN
iajs-1215	221	22	annr	annr	NOUN
iajs-1215	221	23	m	m	VERB
iajs-1215	221	24	2	2	NUM
iajs-1215	221	25	,	,	PUNCT
iajs-1215	221	26	then	then	ADV
iajs-1215	221	27	n	n	CCONJ
iajs-1215	221	28			ADJ
iajs-1215	221	29	m	m	VERB
iajs-1215	221	30	2	2	NUM
iajs-1215	221	31	is	be	AUX
iajs-1215	221	32	a	a	DET
iajs-1215	221	33	weakly	weakly	ADJ
iajs-1215	221	34	prime	prime	ADJ
iajs-1215	221	35	submodule	submodule	NOUN
iajs-1215	221	36	in	in	ADP
iajs-1215	221	37	m	m	PROPN
iajs-1215	221	38	1	1	NUM
iajs-1215	221	39			ADJ
iajs-1215	221	40	m	m	VERB
iajs-1215	221	41	2	2	NUM
iajs-1215	221	42	.	.	PUNCT
iajs-1215	221	43	proof	proof	NOUN
iajs-1215	221	44	.	.	PUNCT
iajs-1215	222	1	let	let	VERB
iajs-1215	223	1	r	r	NOUN
iajs-1215	223	2			NOUN
iajs-1215	223	3	r	r	NOUN
iajs-1215	223	4	,	,	PUNCT
iajs-1215	223	5	x	x	SYM
iajs-1215	223	6			NOUN
iajs-1215	223	7	m	m	VERB
iajs-1215	223	8	1	1	NUM
iajs-1215	223	9	with	with	ADP
iajs-1215	223	10	r	r	NOUN
iajs-1215	223	11	x	x	SYM
iajs-1215	224	1	=	=	NOUN
iajs-1215	224	2	0	0	X
iajs-1215	224	3	.	.	PUNCT
iajs-1215	225	1	if	if	SCONJ
iajs-1215	225	2	x	x	NOUN
iajs-1215	225	3			X
iajs-1215	225	4	n1	n1	PROPN
iajs-1215	225	5	(	(	PUNCT
iajs-1215	225	6	so	so	ADV
iajs-1215	225	7	x	x	SYM
iajs-1215	225	8			NOUN
iajs-1215	225	9	0	0	NUM
iajs-1215	225	10	)	)	PUNCT
iajs-1215	225	11	and	and	CCONJ
iajs-1215	225	12	r	r	PROPN
iajs-1215	225	13			PROPN
iajs-1215	225	14	(	(	PUNCT
iajs-1215	225	15	n	n	CCONJ
iajs-1215	225	16	:	:	PUNCT
iajs-1215	225	17	m	m	PROPN
iajs-1215	225	18	1	1	NUM
iajs-1215	225	19	)	)	PUNCT
iajs-1215	225	20	.	.	PUNCT
iajs-1215	226	1	we	we	PRON
iajs-1215	226	2	will	will	AUX
iajs-1215	226	3	prove	prove	VERB
iajs-1215	226	4	that	that	SCONJ
iajs-1215	226	5	r	r	NOUN
iajs-1215	226	6			PROPN
iajs-1215	226	7	ann	ann	PROPN
iajs-1215	226	8	m	m	PROPN
iajs-1215	226	9	2	2	NUM
iajs-1215	226	10	and	and	CCONJ
iajs-1215	226	11	hence	hence	ADV
iajs-1215	226	12	the	the	DET
iajs-1215	226	13	result	result	NOUN
iajs-1215	226	14	is	be	AUX
iajs-1215	226	15	obtained	obtain	VERB
iajs-1215	226	16	by	by	ADP
iajs-1215	226	17	previous	previous	ADJ
iajs-1215	226	18	theorem	theorem	PROPN
iajs-1215	226	19	.	.	PUNCT
iajs-1215	226	20	suppose	suppose	VERB
iajs-1215	226	21	that	that	SCONJ
iajs-1215	226	22	r	r	NOUN
iajs-1215	226	23			NOUN
iajs-1215	226	24	ann	ann	PROPN
iajs-1215	226	25	m	m	PROPN
iajs-1215	226	26	2	2	NUM
iajs-1215	226	27	.	.	PUNCT
iajs-1215	227	1	hence	hence	ADV
iajs-1215	227	2	r	r	NOUN
iajs-1215	227	3			NOUN
iajs-1215	227	4	p.	p.	NOUN
iajs-1215	227	5	on	on	ADP
iajs-1215	227	6	the	the	DET
iajs-1215	227	7	other	other	ADJ
iajs-1215	227	8	hand	hand	NOUN
iajs-1215	227	9	,	,	PUNCT
iajs-1215	227	10	r	r	NOUN
iajs-1215	227	11	x	x	SYM
iajs-1215	227	12	=	=	SYM
iajs-1215	227	13	0	0	PUNCT
iajs-1215	227	14	=	=	NOUN
iajs-1215	227	15			X
iajs-1215	227	16	q	q	NOUN
iajs-1215	227	17	,	,	PUNCT
iajs-1215	227	18	but	but	CCONJ
iajs-1215	227	19			NUM
iajs-1215	227	20	q	q	NOUN
iajs-1215	227	21	is	be	AUX
iajs-1215	227	22	a	a	DET
iajs-1215	227	23	p	p	NOUN
iajs-1215	227	24	-	-	PUNCT
iajs-1215	227	25	primary	primary	ADJ
iajs-1215	227	26	submodule	submodule	NOUN
iajs-1215	227	27	by	by	ADP
iajs-1215	227	28	(	(	PUNCT
iajs-1215	227	29	11,prop.1.1	11,prop.1.1	PROPN
iajs-1215	227	30	,	,	PUNCT
iajs-1215	227	31	p.15	p.15	PROPN
iajs-1215	227	32	)	)	PUNCT
iajs-1215	227	33	,	,	PUNCT
iajs-1215	227	34	so	so	CCONJ
iajs-1215	227	35	either	either	ADV
iajs-1215	227	36	x	x	PUNCT
iajs-1215	227	37			NOUN
iajs-1215	227	38			PUNCT
iajs-1215	227	39	q	q	NOUN
iajs-1215	227	40	=	=	SYM
iajs-1215	227	41	0	0	NUM
iajs-1215	227	42	or	or	CCONJ
iajs-1215	227	43	r	r	NOUN
iajs-1215	227	44			PROPN
iajs-1215	227	45	p	p	NOUN
iajs-1215	227	46	,	,	PUNCT
iajs-1215	227	47	which	which	PRON
iajs-1215	227	48	is	be	AUX
iajs-1215	227	49	a	a	DET
iajs-1215	227	50	contradiction	contradiction	NOUN
iajs-1215	227	51	.	.	PUNCT
iajs-1215	228	1	thus	thus	ADV
iajs-1215	228	2	r	r	NOUN
iajs-1215	228	3			NOUN
iajs-1215	228	4	ann	ann	PROPN
iajs-1215	228	5	m	m	PROPN
iajs-1215	228	6	2	2	NUM
iajs-1215	228	7	.	.	PUNCT
iajs-1215	228	8	ibn	ibn	PROPN
iajs-1215	228	9	alhaitham	alhaitham	NOUN
iajs-1215	228	10	j.	j.	PROPN
iajs-1215	229	1	fo	fo	ADP
iajs-1215	229	2	r	r	NOUN
iajs-1215	229	3	pure	pure	ADJ
iajs-1215	229	4	&	&	CCONJ
iajs-1215	229	5	appl	appl	PROPN
iajs-1215	229	6	.	.	PUNCT
iajs-1215	230	1	sc	sc	PROPN
iajs-1215	231	1	i	i	PRON
iajs-1215	231	2	vo	vo	INTJ
iajs-1215	231	3	l.22	l.22	X
iajs-1215	231	4	(	(	PUNCT
iajs-1215	231	5	3	3	NUM
iajs-1215	231	6	)	)	PUNCT
iajs-1215	231	7	2009	2009	NUM
iajs-1215	231	8	remark	remark	NOUN
iajs-1215	231	9	2.13	2.13	NUM
iajs-1215	231	10	:	:	PUNCT
iajs-1215	231	11	let	let	VERB
iajs-1215	231	12	m	m	PROPN
iajs-1215	231	13	1	1	NUM
iajs-1215	231	14	,	,	PUNCT
iajs-1215	231	15	m	m	VERB
iajs-1215	231	16	2	2	NUM
iajs-1215	231	17	be	be	VERB
iajs-1215	231	18	r	r	NOUN
iajs-1215	231	19	-	-	PUNCT
iajs-1215	231	20	modules	module	NOUN
iajs-1215	231	21	.	.	PUNCT
iajs-1215	232	1	if	if	SCONJ
iajs-1215	232	2	(	(	PUNCT
iajs-1215	232	3	0	0	X
iajs-1215	232	4	)	)	PUNCT
iajs-1215	232	5	is	be	AUX
iajs-1215	232	6	a	a	DET
iajs-1215	232	7	prime	prime	ADJ
iajs-1215	232	8	submodule	submodule	NOUN
iajs-1215	232	9	of	of	ADP
iajs-1215	232	10	m1	m1	PROPN
iajs-1215	232	11	,	,	PUNCT
iajs-1215	232	12	then	then	ADV
iajs-1215	232	13	(	(	PUNCT
iajs-1215	232	14	0	0	X
iajs-1215	232	15	)	)	PUNCT
iajs-1215	232	16			PROPN
iajs-1215	232	17	m2	m2	PROPN
iajs-1215	232	18	is	be	AUX
iajs-1215	232	19	a	a	DET
iajs-1215	232	20	weakly	weakly	ADJ
iajs-1215	232	21	prime	prime	ADJ
iajs-1215	232	22	submodule	submodule	NOUN
iajs-1215	232	23	of	of	ADP
iajs-1215	232	24	m	m	PROPN
iajs-1215	232	25	=	=	ADJ
iajs-1215	232	26	m	m	PROPN
iajs-1215	232	27	1	1	NUM
iajs-1215	232	28			ADJ
iajs-1215	232	29	m	m	VERB
iajs-1215	232	30	2	2	NUM
iajs-1215	232	31	.	.	PUNCT
iajs-1215	232	32	proof	proof	NOUN
iajs-1215	232	33	.	.	PUNCT
iajs-1215	233	1	let	let	VERB
iajs-1215	233	2	r	r	NOUN
iajs-1215	233	3			NOUN
iajs-1215	233	4	r	r	NOUN
iajs-1215	233	5	,	,	PUNCT
iajs-1215	233	6	(	(	PUNCT
iajs-1215	233	7	x	x	NOUN
iajs-1215	233	8	,	,	PUNCT
iajs-1215	233	9	y	y	PROPN
iajs-1215	233	10	)	)	PUNCT
iajs-1215	233	11			NOUN
iajs-1215	233	12	m.	m.	NOUN
iajs-1215	234	1	if	if	SCONJ
iajs-1215	234	2	(	(	PUNCT
iajs-1215	234	3	0,0	0,0	NOUN
iajs-1215	234	4	)	)	PUNCT
iajs-1215	234	5			NOUN
iajs-1215	234	6	r	r	NOUN
iajs-1215	234	7	(	(	PUNCT
iajs-1215	234	8	x	x	NOUN
iajs-1215	234	9	,	,	PUNCT
iajs-1215	234	10	y	y	PROPN
iajs-1215	234	11	)	)	PUNCT
iajs-1215	234	12			NOUN
iajs-1215	234	13	(	(	PUNCT
iajs-1215	234	14	0	0	NUM
iajs-1215	234	15	)	)	PUNCT
iajs-1215	234	16			PROPN
iajs-1215	234	17	m	m	ADJ
iajs-1215	234	18	2	2	NUM
iajs-1215	234	19	,	,	PUNCT
iajs-1215	234	20	then	then	ADV
iajs-1215	234	21	r	r	NOUN
iajs-1215	234	22	x	x	SYM
iajs-1215	234	23	=	=	SYM
iajs-1215	234	24	0	0	NUM
iajs-1215	234	25	and	and	CCONJ
iajs-1215	234	26	r	r	PROPN
iajs-1215	234	27	y	y	PROPN
iajs-1215	234	28			PROPN
iajs-1215	234	29	m	m	VERB
iajs-1215	234	30	2	2	NUM
iajs-1215	234	31	.	.	PUNCT
iajs-1215	235	1	since	since	SCONJ
iajs-1215	235	2	(	(	PUNCT
iajs-1215	235	3	0	0	NUM
iajs-1215	235	4	)	)	PUNCT
iajs-1215	235	5	is	be	AUX
iajs-1215	235	6	prime	prime	ADJ
iajs-1215	235	7	in	in	ADP
iajs-1215	235	8	m	m	PROPN
iajs-1215	235	9	1	1	NUM
iajs-1215	235	10	,	,	PUNCT
iajs-1215	235	11	either	either	CCONJ
iajs-1215	235	12	x	x	PUNCT
iajs-1215	235	13	=	=	SYM
iajs-1215	235	14	0	0	NUM
iajs-1215	235	15	or	or	CCONJ
iajs-1215	235	16	r	r	NOUN
iajs-1215	235	17			NOUN
iajs-1215	235	18	(	(	PUNCT
iajs-1215	235	19	0	0	NUM
iajs-1215	235	20	:	:	PUNCT
iajs-1215	235	21	m1	m1	NOUN
iajs-1215	235	22	)	)	PUNCT
iajs-1215	235	23	.	.	PUNCT
iajs-1215	236	1	hence	hence	ADV
iajs-1215	236	2	either	either	CCONJ
iajs-1215	236	3	(	(	PUNCT
iajs-1215	236	4	x	x	X
iajs-1215	236	5	,	,	PUNCT
iajs-1215	236	6	y	y	NOUN
iajs-1215	236	7	)	)	PUNCT
iajs-1215	236	8	=	=	SYM
iajs-1215	236	9	(	(	PUNCT
iajs-1215	236	10	0,y	0,y	NOUN
iajs-1215	236	11	)	)	PUNCT
iajs-1215	236	12			NOUN
iajs-1215	236	13	(	(	PUNCT
iajs-1215	236	14	0	0	NUM
iajs-1215	236	15	)	)	PUNCT
iajs-1215	236	16			PROPN
iajs-1215	236	17	m	m	VERB
iajs-1215	236	18	2	2	NUM
iajs-1215	236	19	or	or	CCONJ
iajs-1215	236	20	r	r	NOUN
iajs-1215	236	21			NOUN
iajs-1215	236	22	(	(	PUNCT
iajs-1215	236	23	(	(	PUNCT
iajs-1215	236	24	0	0	NUM
iajs-1215	236	25	)	)	PUNCT
iajs-1215	237	1	+	+	NUM
iajs-1215	237	2	m	m	NUM
iajs-1215	237	3	2	2	NUM
iajs-1215	237	4	:	:	PUNCT
iajs-1215	237	5	m1	m1	PROPN
iajs-1215	237	6			PROPN
iajs-1215	237	7	m	m	VERB
iajs-1215	237	8	2	2	NUM
iajs-1215	237	9	)	)	PUNCT
iajs-1215	237	10	;	;	PUNCT
iajs-1215	237	11	that	that	ADV
iajs-1215	237	12	is	is	ADV
iajs-1215	237	13	(	(	PUNCT
iajs-1215	237	14	0	0	NUM
iajs-1215	237	15	)	)	PUNCT
iajs-1215	237	16			PROPN
iajs-1215	237	17	m	m	VERB
iajs-1215	237	18	2	2	NUM
iajs-1215	237	19	is	be	AUX
iajs-1215	237	20	weakly	weakly	ADV
iajs-1215	237	21	prime	prime	ADJ
iajs-1215	237	22	in	in	ADP
iajs-1215	237	23	m.	m.	NOUN
iajs-1215	237	24	thus	thus	ADV
iajs-1215	237	25	we	we	PRON
iajs-1215	237	26	can	can	AUX
iajs-1215	237	27	give	give	VERB
iajs-1215	237	28	the	the	DET
iajs-1215	237	29	following	follow	VERB
iajs-1215	237	30	example	example	NOUN
iajs-1215	237	31	:	:	PUNCT
iajs-1215	237	32	n	n	X
iajs-1215	237	33	=	=	SYM
iajs-1215	237	34	(	(	PUNCT
iajs-1215	237	35	0	0	X
iajs-1215	237	36	)	)	PUNCT
iajs-1215	237	37			PROPN
iajs-1215	237	38	z4	z4	PROPN
iajs-1215	237	39	is	be	AUX
iajs-1215	237	40	a	a	DET
iajs-1215	237	41	weakly	weakly	ADJ
iajs-1215	237	42	prime	prime	ADJ
iajs-1215	237	43	submodule	submodule	NOUN
iajs-1215	237	44	of	of	ADP
iajs-1215	237	45	the	the	DET
iajs-1215	237	46	z	z	NOUN
iajs-1215	237	47	-	-	PUNCT
iajs-1215	237	48	module	module	NOUN
iajs-1215	237	49	z	z	PROPN
iajs-1215	237	50			PROPN
iajs-1215	237	51	z4	z4	PROPN
iajs-1215	237	52	.	.	PUNCT
iajs-1215	238	1	next	next	ADV
iajs-1215	238	2	we	we	PRON
iajs-1215	238	3	have	have	VERB
iajs-1215	238	4	the	the	DET
iajs-1215	238	5	following	following	NOUN
iajs-1215	238	6	:	:	PUNCT
iajs-1215	238	7	proposition	proposition	NOUN
iajs-1215	238	8	2.14	2.14	NUM
iajs-1215	238	9	:	:	PUNCT
iajs-1215	238	10	let	let	VERB
iajs-1215	238	11	m	m	PROPN
iajs-1215	238	12	1	1	NUM
iajs-1215	238	13	,	,	PUNCT
iajs-1215	238	14	m	m	VERB
iajs-1215	238	15	2	2	NUM
iajs-1215	238	16	be	be	VERB
iajs-1215	238	17	r	r	NOUN
iajs-1215	238	18	-	-	PUNCT
iajs-1215	238	19	modules	module	NOUN
iajs-1215	238	20	.	.	PUNCT
iajs-1215	239	1	if	if	SCONJ
iajs-1215	239	2	n	n	PRON
iajs-1215	239	3	=	=	SYM
iajs-1215	239	4	u	u	NOUN
iajs-1215	239	5			PROPN
iajs-1215	239	6	w	w	AUX
iajs-1215	239	7	be	be	AUX
iajs-1215	239	8	a	a	DET
iajs-1215	239	9	weakly	weakly	ADJ
iajs-1215	239	10	prime	prime	ADJ
iajs-1215	239	11	submodule	submodule	NOUN
iajs-1215	239	12	in	in	ADP
iajs-1215	239	13	m	m	PROPN
iajs-1215	239	14	=	=	ADJ
iajs-1215	239	15	m	m	PROPN
iajs-1215	239	16	1	1	NUM
iajs-1215	239	17			ADJ
iajs-1215	239	18	m	m	VERB
iajs-1215	239	19	2	2	NUM
iajs-1215	239	20	,	,	PUNCT
iajs-1215	239	21	then	then	ADV
iajs-1215	239	22	u	u	NOUN
iajs-1215	239	23	,	,	PUNCT
iajs-1215	239	24	w	w	PROPN
iajs-1215	239	25	are	be	AUX
iajs-1215	239	26	weakly	weakly	ADJ
iajs-1215	239	27	prime	prime	ADJ
iajs-1215	239	28	submodules	submodule	NOUN
iajs-1215	239	29	in	in	ADP
iajs-1215	239	30	m1	m1	PROPN
iajs-1215	239	31	,	,	PUNCT
iajs-1215	239	32	m	m	PROPN
iajs-1215	239	33	2	2	NUM
iajs-1215	239	34	respectively	respectively	ADV
iajs-1215	239	35	.	.	PUNCT
iajs-1215	240	1	proof	proof	NOUN
iajs-1215	240	2	.	.	PUNCT
iajs-1215	241	1	the	the	DET
iajs-1215	241	2	proof	proof	NOUN
iajs-1215	241	3	is	be	AUX
iajs-1215	241	4	a	a	DET
iajs-1215	241	5	straight	straight	ADJ
iajs-1215	241	6	forword	forword	NOUN
iajs-1215	241	7	,	,	PUNCT
iajs-1215	241	8	so	so	SCONJ
iajs-1215	241	9	it	it	PRON
iajs-1215	241	10	is	be	AUX
iajs-1215	241	11	omitted	omit	VERB
iajs-1215	241	12	.	.	PUNCT
iajs-1215	242	1	remark	remark	PROPN
iajs-1215	242	2	2.15	2.15	NUM
iajs-1215	242	3	:	:	PUNCT
iajs-1215	243	1	the	the	DET
iajs-1215	243	2	converse	converse	NOUN
iajs-1215	243	3	of	of	ADP
iajs-1215	243	4	proposition	proposition	NOUN
iajs-1215	243	5	2.14	2.14	NUM
iajs-1215	243	6	is	be	AUX
iajs-1215	243	7	not	not	PART
iajs-1215	243	8	true	true	ADJ
iajs-1215	243	9	in	in	ADP
iajs-1215	243	10	general	general	ADJ
iajs-1215	243	11	as	as	ADP
iajs-1215	243	12	the	the	DET
iajs-1215	243	13	following	follow	VERB
iajs-1215	243	14	example	example	NOUN
iajs-1215	243	15	shows	show	VERB
iajs-1215	243	16	.	.	PUNCT
iajs-1215	244	1	example	example	NOUN
iajs-1215	244	2	:	:	PUNCT
iajs-1215	244	3	(	(	PUNCT
iajs-1215	244	4	0	0	X
iajs-1215	244	5	)	)	PUNCT
iajs-1215	244	6	is	be	AUX
iajs-1215	244	7	a	a	DET
iajs-1215	244	8	weakly	weakly	ADJ
iajs-1215	244	9	prime	prime	ADJ
iajs-1215	244	10	submodule	submodule	NOUN
iajs-1215	244	11	of	of	ADP
iajs-1215	244	12	the	the	DET
iajs-1215	244	13	z	z	NOUN
iajs-1215	244	14	-	-	PUNCT
iajs-1215	244	15	module	module	NOUN
iajs-1215	244	16	z	z	NOUN
iajs-1215	244	17	,	,	PUNCT
iajs-1215	244	18	(	(	PUNCT
iajs-1215	244	19	2z	2z	NUM
iajs-1215	244	20	)	)	PUNCT
iajs-1215	244	21	is	be	AUX
iajs-1215	244	22	a	a	DET
iajs-1215	244	23	prime	prime	ADJ
iajs-1215	244	24	submodule	submodule	NOUN
iajs-1215	244	25	of	of	ADP
iajs-1215	244	26	the	the	DET
iajs-1215	244	27	z	z	NOUN
iajs-1215	244	28	-	-	PUNCT
iajs-1215	244	29	module	module	NOUN
iajs-1215	244	30	z	z	NOUN
iajs-1215	245	1	so	so	SCONJ
iajs-1215	245	2	it	it	PRON
iajs-1215	245	3	is	be	AUX
iajs-1215	245	4	weakly	weakly	ADV
iajs-1215	245	5	prime	prime	ADJ
iajs-1215	245	6	.	.	PUNCT
iajs-1215	246	1	but	but	CCONJ
iajs-1215	246	2	n	n	CCONJ
iajs-1215	246	3	=	=	SYM
iajs-1215	246	4	(	(	PUNCT
iajs-1215	246	5	0	0	X
iajs-1215	246	6	)	)	PUNCT
iajs-1215	246	7			ADJ
iajs-1215	246	8	2z	2z	NUM
iajs-1215	246	9	is	be	AUX
iajs-1215	246	10	not	not	PART
iajs-1215	246	11	weakly	weakly	ADJ
iajs-1215	246	12	prime	prime	ADJ
iajs-1215	246	13	in	in	ADP
iajs-1215	246	14	the	the	DET
iajs-1215	246	15	zmodule	zmodule	NOUN
iajs-1215	246	16	z	z	PROPN
iajs-1215	246	17			PROPN
iajs-1215	246	18	z.	z.	PROPN
iajs-1215	247	1	for	for	ADP
iajs-1215	247	2	the	the	DET
iajs-1215	247	3	next	next	ADJ
iajs-1215	247	4	results	result	NOUN
iajs-1215	247	5	we	we	PRON
iajs-1215	247	6	will	will	AUX
iajs-1215	247	7	assume	assume	VERB
iajs-1215	247	8	that	that	SCONJ
iajs-1215	247	9	r	r	NOUN
iajs-1215	247	10	=	=	SYM
iajs-1215	247	11	r1	r1	PROPN
iajs-1215	247	12			PROPN
iajs-1215	247	13	r2	r2	PROPN
iajs-1215	247	14	where	where	SCONJ
iajs-1215	247	15	each	each	DET
iajs-1215	247	16	ri	ri	PROPN
iajs-1215	247	17	is	be	AUX
iajs-1215	247	18	a	a	DET
iajs-1215	247	19	commutative	commutative	ADJ
iajs-1215	247	20	ring	ring	NOUN
iajs-1215	247	21	with	with	ADP
iajs-1215	247	22	identity	identity	NOUN
iajs-1215	247	23	,	,	PUNCT
iajs-1215	247	24	m	m	AUX
iajs-1215	247	25	i	i	PRON
iajs-1215	247	26	be	be	VERB
iajs-1215	247	27	an	an	DET
iajs-1215	247	28	ri	ri	NOUN
iajs-1215	247	29	-	-	PUNCT
iajs-1215	247	30	module	module	NOUN
iajs-1215	247	31	,	,	PUNCT
iajs-1215	247	32	where	where	SCONJ
iajs-1215	247	33	i	i	PRON
iajs-1215	247	34	=	=	SYM
iajs-1215	247	35	1,2	1,2	NUM
iajs-1215	247	36	.	.	PUNCT
iajs-1215	248	1	and	and	CCONJ
iajs-1215	248	2	m	m	PROPN
iajs-1215	248	3	=	=	NOUN
iajs-1215	248	4	m	m	VERB
iajs-1215	248	5	1	1	NUM
iajs-1215	248	6			NOUN
iajs-1215	248	7	m	m	PROPN
iajs-1215	248	8	2	2	NUM
iajs-1215	248	9	be	be	AUX
iajs-1215	248	10	the	the	DET
iajs-1215	248	11	r	r	NOUN
iajs-1215	248	12	-	-	PUNCT
iajs-1215	248	13	module	module	NOUN
iajs-1215	248	14	with	with	ADP
iajs-1215	248	15	action	action	NOUN
iajs-1215	248	16	(	(	PUNCT
iajs-1215	248	17	r1,r2	r1,r2	PROPN
iajs-1215	248	18	)	)	PUNCT
iajs-1215	248	19	(	(	PUNCT
iajs-1215	248	20	m1,m2	m1,m2	PROPN
iajs-1215	248	21	)	)	PUNCT
iajs-1215	249	1	=	=	PUNCT
iajs-1215	249	2	(	(	PUNCT
iajs-1215	249	3	r1	r1	PROPN
iajs-1215	249	4	m1	m1	PROPN
iajs-1215	249	5	,	,	PUNCT
iajs-1215	249	6	r2	r2	PROPN
iajs-1215	249	7	m2	m2	PROPN
iajs-1215	249	8	)	)	PUNCT
iajs-1215	249	9	where	where	SCONJ
iajs-1215	249	10	ri	ri	PROPN
iajs-1215	249	11			PROPN
iajs-1215	249	12	ri	ri	PROPN
iajs-1215	249	13	,	,	PUNCT
iajs-1215	249	14	mi	mi	PROPN
iajs-1215	249	15			PROPN
iajs-1215	249	16	m	m	VERB
iajs-1215	250	1	i	i	PRON
iajs-1215	250	2	,	,	PUNCT
iajs-1215	250	3	i	i	NOUN
iajs-1215	250	4	=	=	SYM
iajs-1215	250	5	1,2	1,2	NUM
iajs-1215	250	6	.	.	PUNCT
iajs-1215	251	1	proposition	proposition	NOUN
iajs-1215	251	2	2.16	2.16	NUM
iajs-1215	251	3	:	:	PUNCT
iajs-1215	251	4	if	if	SCONJ
iajs-1215	251	5	p	p	NOUN
iajs-1215	251	6	is	be	AUX
iajs-1215	251	7	a	a	DET
iajs-1215	251	8	proper	proper	ADJ
iajs-1215	251	9	r1	r1	NOUN
iajs-1215	251	10	-	-	PUNCT
iajs-1215	251	11	submodule	submodule	NOUN
iajs-1215	251	12	of	of	ADP
iajs-1215	251	13	m	m	PROPN
iajs-1215	251	14	1	1	NUM
iajs-1215	251	15	,	,	PUNCT
iajs-1215	251	16	then	then	ADV
iajs-1215	251	17	the	the	DET
iajs-1215	251	18	following	following	ADJ
iajs-1215	251	19	statements	statement	NOUN
iajs-1215	251	20	are	be	AUX
iajs-1215	251	21	equivalent	equivalent	ADJ
iajs-1215	251	22	1	1	NUM
iajs-1215	251	23	.	.	PUNCT
iajs-1215	252	1	p	p	NOUN
iajs-1215	252	2	is	be	AUX
iajs-1215	252	3	a	a	DET
iajs-1215	252	4	prime	prime	ADJ
iajs-1215	252	5	r1	r1	NOUN
iajs-1215	252	6	-	-	PUNCT
iajs-1215	252	7	submodule	submodule	NOUN
iajs-1215	252	8	of	of	ADP
iajs-1215	252	9	m	m	PROPN
iajs-1215	252	10	1	1	NUM
iajs-1215	252	11	.	.	NOUN
iajs-1215	252	12	2	2	NUM
iajs-1215	252	13	.	.	X
iajs-1215	253	1	p	p	NOUN
iajs-1215	253	2			PROPN
iajs-1215	253	3	m	m	PROPN
iajs-1215	253	4	2	2	NUM
iajs-1215	253	5	is	be	AUX
iajs-1215	253	6	a	a	DET
iajs-1215	253	7	prime	prime	ADJ
iajs-1215	253	8	r	r	NOUN
iajs-1215	253	9	-	-	PUNCT
iajs-1215	253	10	submodule	submodule	NOUN
iajs-1215	253	11	of	of	ADP
iajs-1215	253	12	m	m	PROPN
iajs-1215	253	13	=	=	ADJ
iajs-1215	253	14	m	m	VERB
iajs-1215	253	15	1	1	NUM
iajs-1215	253	16			NOUN
iajs-1215	253	17	m	m	PROPN
iajs-1215	253	18	2	2	NUM
iajs-1215	253	19	.	.	NOUN
iajs-1215	253	20	3	3	NUM
iajs-1215	253	21	.	.	X
iajs-1215	254	1	p	p	NOUN
iajs-1215	254	2			PROPN
iajs-1215	254	3	m	m	PROPN
iajs-1215	254	4	2	2	NUM
iajs-1215	254	5	is	be	AUX
iajs-1215	254	6	a	a	DET
iajs-1215	254	7	weakly	weakly	ADJ
iajs-1215	254	8	prime	prime	ADJ
iajs-1215	254	9	r	r	NOUN
iajs-1215	254	10	-	-	PUNCT
iajs-1215	254	11	submodule	submodule	NOUN
iajs-1215	254	12	of	of	ADP
iajs-1215	254	13	m	m	PROPN
iajs-1215	254	14	=	=	ADJ
iajs-1215	254	15	m	m	VERB
iajs-1215	254	16	1	1	NUM
iajs-1215	254	17			NOUN
iajs-1215	254	18	m	m	PROPN
iajs-1215	254	19	2	2	NUM
iajs-1215	254	20	.	.	PUNCT
iajs-1215	254	21	proof	proof	NOUN
iajs-1215	254	22	.	.	PUNCT
iajs-1215	255	1	(	(	PUNCT
iajs-1215	255	2	1	1	X
iajs-1215	255	3	)	)	PUNCT
iajs-1215	255	4			NOUN
iajs-1215	255	5	(	(	PUNCT
iajs-1215	255	6	2	2	X
iajs-1215	255	7	)	)	PUNCT
iajs-1215	255	8	let	let	VERB
iajs-1215	255	9	(	(	PUNCT
iajs-1215	255	10	r1,r2	r1,r2	PROPN
iajs-1215	255	11	)	)	PUNCT
iajs-1215	255	12			NOUN
iajs-1215	255	13	r	r	NOUN
iajs-1215	255	14	,	,	PUNCT
iajs-1215	255	15	(	(	PUNCT
iajs-1215	255	16	x	x	NOUN
iajs-1215	255	17	,	,	PUNCT
iajs-1215	255	18	y	y	PROPN
iajs-1215	255	19	)	)	PUNCT
iajs-1215	255	20			NOUN
iajs-1215	255	21	m	m	VERB
iajs-1215	255	22	such	such	ADJ
iajs-1215	255	23	that	that	SCONJ
iajs-1215	255	24	(	(	PUNCT
iajs-1215	255	25	r1,r2	r1,r2	PROPN
iajs-1215	255	26	)	)	PUNCT
iajs-1215	255	27	(	(	PUNCT
iajs-1215	255	28	x	x	X
iajs-1215	255	29	,	,	PUNCT
iajs-1215	255	30	y	y	PROPN
iajs-1215	255	31	)	)	PUNCT
iajs-1215	255	32			NOUN
iajs-1215	255	33	p	p	PROPN
iajs-1215	255	34			PROPN
iajs-1215	255	35	m	m	PROPN
iajs-1215	255	36	2	2	NUM
iajs-1215	255	37	.	.	PUNCT
iajs-1215	255	38	then	then	ADV
iajs-1215	255	39	r1	r1	PROPN
iajs-1215	255	40	x	x	PUNCT
iajs-1215	255	41			PROPN
iajs-1215	255	42	p	p	NOUN
iajs-1215	255	43	and	and	CCONJ
iajs-1215	255	44	since	since	SCONJ
iajs-1215	255	45	p	p	NOUN
iajs-1215	255	46	is	be	AUX
iajs-1215	255	47	prime	prime	ADJ
iajs-1215	255	48	,	,	PUNCT
iajs-1215	255	49	either	either	CCONJ
iajs-1215	255	50	x	x	PUNCT
iajs-1215	255	51			NOUN
iajs-1215	255	52	p	p	NOUN
iajs-1215	255	53	or	or	CCONJ
iajs-1215	255	54	r1	r1	PROPN
iajs-1215	255	55			NOUN
iajs-1215	255	56	(	(	PUNCT
iajs-1215	255	57	p	p	NOUN
iajs-1215	255	58	1	1	NUM
iajs-1215	255	59	:	:	PUNCT
iajs-1215	255	60	r	r	NOUN
iajs-1215	255	61	m	m	NOUN
iajs-1215	255	62	1	1	NUM
iajs-1215	255	63	)	)	PUNCT
iajs-1215	255	64	.	.	PUNCT
iajs-1215	256	1	if	if	SCONJ
iajs-1215	256	2	x	x	PRON
iajs-1215	256	3			NOUN
iajs-1215	256	4	p	p	NOUN
iajs-1215	256	5	,	,	PUNCT
iajs-1215	256	6	then	then	ADV
iajs-1215	256	7	(	(	PUNCT
iajs-1215	256	8	x	x	X
iajs-1215	256	9	,	,	PUNCT
iajs-1215	256	10	y	y	PROPN
iajs-1215	256	11	)	)	PUNCT
iajs-1215	256	12			NOUN
iajs-1215	256	13	p	p	PROPN
iajs-1215	256	14			PROPN
iajs-1215	256	15	m	m	PROPN
iajs-1215	256	16	2	2	NUM
iajs-1215	256	17	.	.	PUNCT
iajs-1215	257	1	if	if	SCONJ
iajs-1215	257	2	r1	r1	VERB
iajs-1215	257	3			NOUN
iajs-1215	257	4	(	(	PUNCT
iajs-1215	257	5	p	p	NOUN
iajs-1215	257	6	1	1	NUM
iajs-1215	257	7	:	:	PUNCT
iajs-1215	257	8	r	r	NOUN
iajs-1215	257	9	m	m	NOUN
iajs-1215	257	10	1	1	NUM
iajs-1215	257	11	)	)	PUNCT
iajs-1215	257	12	,	,	PUNCT
iajs-1215	257	13	then	then	ADV
iajs-1215	257	14	(	(	PUNCT
iajs-1215	257	15	r1,r2	r1,r2	PROPN
iajs-1215	257	16	)	)	PUNCT
iajs-1215	257	17			NOUN
iajs-1215	257	18	(	(	PUNCT
iajs-1215	257	19	p	p	X
iajs-1215	257	20			NOUN
iajs-1215	257	21	m	m	PROPN
iajs-1215	257	22	2	2	NUM
iajs-1215	257	23	r	r	NOUN
iajs-1215	257	24	:	:	PUNCT
iajs-1215	257	25	m	m	NOUN
iajs-1215	257	26	)	)	PUNCT
iajs-1215	257	27	.	.	PUNCT
iajs-1215	258	1	thus	thus	ADV
iajs-1215	258	2	p	p	X
iajs-1215	258	3			PROPN
iajs-1215	258	4	m	m	PROPN
iajs-1215	258	5	2	2	NUM
iajs-1215	258	6	is	be	AUX
iajs-1215	258	7	a	a	DET
iajs-1215	258	8	prime	prime	ADJ
iajs-1215	258	9	r	r	NOUN
iajs-1215	258	10	-	-	PUNCT
iajs-1215	258	11	submodule	submodule	NOUN
iajs-1215	258	12	of	of	ADP
iajs-1215	258	13	m.	m.	NOUN
iajs-1215	258	14	(	(	PUNCT
iajs-1215	258	15	2	2	NUM
iajs-1215	258	16	)	)	PUNCT
iajs-1215	258	17			NOUN
iajs-1215	258	18	(	(	PUNCT
iajs-1215	258	19	3	3	X
iajs-1215	258	20	)	)	PUNCT
iajs-1215	258	21	it	it	PRON
iajs-1215	258	22	holds	hold	VERB
iajs-1215	258	23	by	by	ADP
iajs-1215	258	24	remark	remark	NOUN
iajs-1215	258	25	2.1	2.1	NUM
iajs-1215	258	26	(	(	PUNCT
iajs-1215	258	27	1	1	NUM
iajs-1215	258	28	)	)	PUNCT
iajs-1215	258	29	.	.	PUNCT
iajs-1215	259	1	(	(	PUNCT
iajs-1215	259	2	3	3	X
iajs-1215	259	3	)	)	PUNCT
iajs-1215	259	4			NOUN
iajs-1215	259	5	(	(	PUNCT
iajs-1215	259	6	1	1	X
iajs-1215	259	7	)	)	PUNCT
iajs-1215	259	8	let	let	VERB
iajs-1215	259	9	r	r	NOUN
iajs-1215	259	10			NOUN
iajs-1215	259	11	r1	r1	NOUN
iajs-1215	259	12	,	,	PUNCT
iajs-1215	259	13	x	x	SYM
iajs-1215	259	14			PROPN
iajs-1215	259	15	m	m	VERB
iajs-1215	259	16	1	1	NUM
iajs-1215	260	1	such	such	ADJ
iajs-1215	260	2	that	that	DET
iajs-1215	260	3	r	r	NOUN
iajs-1215	260	4	x	x	X
iajs-1215	260	5			NOUN
iajs-1215	261	1	p.	p.	NOUN
iajs-1215	261	2	then	then	ADV
iajs-1215	261	3	for	for	ADP
iajs-1215	261	4	each	each	DET
iajs-1215	261	5	y	y	PROPN
iajs-1215	261	6			PROPN
iajs-1215	261	7	m	m	VERB
iajs-1215	261	8	2	2	NUM
iajs-1215	261	9	,	,	PUNCT
iajs-1215	261	10	y	y	PROPN
iajs-1215	261	11			PROPN
iajs-1215	261	12	0	0	NUM
iajs-1215	261	13	,	,	PUNCT
iajs-1215	261	14	(	(	PUNCT
iajs-1215	261	15	0,0	0,0	NOUN
iajs-1215	261	16	)	)	PUNCT
iajs-1215	261	17			NOUN
iajs-1215	261	18	(	(	PUNCT
iajs-1215	261	19	r,1	r,1	PROPN
iajs-1215	261	20	)	)	PUNCT
iajs-1215	261	21	(	(	PUNCT
iajs-1215	261	22	x	x	X
iajs-1215	261	23	,	,	PUNCT
iajs-1215	261	24	y	y	PROPN
iajs-1215	261	25	)	)	PUNCT
iajs-1215	261	26			NOUN
iajs-1215	261	27	p	p	PROPN
iajs-1215	261	28			PROPN
iajs-1215	261	29	m	m	PROPN
iajs-1215	261	30	2	2	NUM
iajs-1215	261	31	.	.	PUNCT
iajs-1215	262	1	but	but	CCONJ
iajs-1215	262	2	p	p	NOUN
iajs-1215	262	3			PROPN
iajs-1215	262	4	m	m	PROPN
iajs-1215	262	5	2	2	NUM
iajs-1215	262	6	is	be	AUX
iajs-1215	262	7	a	a	DET
iajs-1215	262	8	weakly	weakly	ADJ
iajs-1215	262	9	prime	prime	ADJ
iajs-1215	262	10	r	r	NOUN
iajs-1215	262	11	-	-	PUNCT
iajs-1215	262	12	submodule	submodule	NOUN
iajs-1215	262	13	of	of	ADP
iajs-1215	262	14	m	m	PROPN
iajs-1215	262	15	,	,	PUNCT
iajs-1215	262	16	so	so	ADV
iajs-1215	262	17	either	either	CCONJ
iajs-1215	262	18	(	(	PUNCT
iajs-1215	262	19	x	x	NOUN
iajs-1215	262	20	,	,	PUNCT
iajs-1215	262	21	y	y	PROPN
iajs-1215	262	22	)	)	PUNCT
iajs-1215	262	23			NOUN
iajs-1215	262	24	p	p	PROPN
iajs-1215	262	25			PROPN
iajs-1215	262	26	m	m	PROPN
iajs-1215	262	27	2	2	NUM
iajs-1215	262	28	or	or	CCONJ
iajs-1215	262	29	(	(	PUNCT
iajs-1215	262	30	r,1	r,1	PROPN
iajs-1215	262	31	)	)	PUNCT
iajs-1215	262	32			NOUN
iajs-1215	262	33	(	(	PUNCT
iajs-1215	262	34	p	p	X
iajs-1215	262	35			NOUN
iajs-1215	262	36	m	m	PROPN
iajs-1215	262	37	2	2	NUM
iajs-1215	262	38	r	r	NOUN
iajs-1215	262	39	:	:	PUNCT
iajs-1215	262	40	m	m	NOUN
iajs-1215	262	41	)	)	PUNCT
iajs-1215	262	42	.	.	PUNCT
iajs-1215	263	1	thus	thus	ADV
iajs-1215	263	2	either	either	CCONJ
iajs-1215	263	3	x	x	SYM
iajs-1215	263	4			NOUN
iajs-1215	263	5	p	p	NOUN
iajs-1215	263	6	or	or	CCONJ
iajs-1215	263	7	r	r	NOUN
iajs-1215	263	8			NOUN
iajs-1215	263	9	(	(	PUNCT
iajs-1215	263	10	p	p	NOUN
iajs-1215	263	11	1	1	NUM
iajs-1215	263	12	:	:	PUNCT
iajs-1215	263	13	r	r	NOUN
iajs-1215	263	14	m	m	NOUN
iajs-1215	263	15	1	1	NUM
iajs-1215	263	16	)	)	PUNCT
iajs-1215	263	17	;	;	PUNCT
iajs-1215	263	18	that	that	PRON
iajs-1215	263	19	is	be	AUX
iajs-1215	263	20	p	p	NOUN
iajs-1215	263	21	is	be	AUX
iajs-1215	263	22	a	a	DET
iajs-1215	263	23	prime	prime	ADJ
iajs-1215	263	24	r1submodule	r1submodule	NOUN
iajs-1215	263	25	of	of	ADP
iajs-1215	263	26	m	m	PROPN
iajs-1215	263	27	1	1	NUM
iajs-1215	263	28	.	.	PUNCT
iajs-1215	264	1	similarly	similarly	ADV
iajs-1215	264	2	we	we	PRON
iajs-1215	264	3	have	have	VERB
iajs-1215	264	4	proposition	proposition	NOUN
iajs-1215	264	5	2.17	2.17	NUM
iajs-1215	264	6	:	:	PUNCT
iajs-1215	264	7	if	if	SCONJ
iajs-1215	264	8	p	p	NOUN
iajs-1215	264	9	is	be	AUX
iajs-1215	264	10	a	a	DET
iajs-1215	264	11	proper	proper	ADJ
iajs-1215	264	12	r2	r2	NOUN
iajs-1215	264	13	-	-	PUNCT
iajs-1215	264	14	submodule	submodule	NOUN
iajs-1215	264	15	of	of	ADP
iajs-1215	264	16	m	m	PROPN
iajs-1215	264	17	2	2	NUM
iajs-1215	264	18	,	,	PUNCT
iajs-1215	264	19	then	then	ADV
iajs-1215	264	20	the	the	DET
iajs-1215	264	21	following	following	ADJ
iajs-1215	264	22	statements	statement	NOUN
iajs-1215	264	23	are	be	AUX
iajs-1215	264	24	equivalent	equivalent	ADJ
iajs-1215	264	25	1	1	NUM
iajs-1215	264	26	.	.	PUNCT
iajs-1215	265	1	p	p	NOUN
iajs-1215	265	2	is	be	AUX
iajs-1215	265	3	a	a	DET
iajs-1215	265	4	prime	prime	ADJ
iajs-1215	265	5	r2	r2	NOUN
iajs-1215	265	6	-	-	PUNCT
iajs-1215	265	7	submodule	submodule	NOUN
iajs-1215	265	8	of	of	ADP
iajs-1215	265	9	m	m	PROPN
iajs-1215	265	10	2	2	NUM
iajs-1215	265	11	.	.	NOUN
iajs-1215	265	12	2	2	NUM
iajs-1215	265	13	.	.	X
iajs-1215	265	14	m	m	VERB
iajs-1215	265	15	1	1	NUM
iajs-1215	265	16	p	p	PROPN
iajs-1215	265	17	is	be	AUX
iajs-1215	265	18	a	a	DET
iajs-1215	265	19	prime	prime	ADJ
iajs-1215	265	20	r	r	NOUN
iajs-1215	265	21	-	-	PUNCT
iajs-1215	265	22	submodule	submodule	NOUN
iajs-1215	265	23	of	of	ADP
iajs-1215	265	24	m	m	PROPN
iajs-1215	265	25	=	=	ADJ
iajs-1215	265	26	m	m	VERB
iajs-1215	265	27	1	1	NUM
iajs-1215	265	28			NOUN
iajs-1215	265	29	m	m	PROPN
iajs-1215	265	30	2	2	NUM
iajs-1215	265	31	.	.	NOUN
iajs-1215	266	1	3	3	NUM
iajs-1215	266	2	.	.	X
iajs-1215	266	3	m	m	PROPN
iajs-1215	266	4	1	1	NUM
iajs-1215	266	5	p	p	PROPN
iajs-1215	266	6	is	be	AUX
iajs-1215	266	7	a	a	DET
iajs-1215	266	8	weakly	weakly	ADJ
iajs-1215	266	9	prime	prime	ADJ
iajs-1215	266	10	r	r	NOUN
iajs-1215	266	11	-	-	PUNCT
iajs-1215	266	12	submodule	submodule	NOUN
iajs-1215	266	13	of	of	ADP
iajs-1215	266	14	m	m	PROPN
iajs-1215	266	15	=	=	ADJ
iajs-1215	266	16	m	m	VERB
iajs-1215	266	17	1	1	NUM
iajs-1215	266	18			NOUN
iajs-1215	266	19	m	m	PROPN
iajs-1215	266	20	2	2	NUM
iajs-1215	266	21	.	.	PUNCT
iajs-1215	266	22	proposition	proposition	NOUN
iajs-1215	266	23	2.18	2.18	NUM
iajs-1215	266	24	:	:	PUNCT
iajs-1215	266	25	let	let	VERB
iajs-1215	266	26	m	m	PROPN
iajs-1215	266	27	1	1	NUM
iajs-1215	266	28	,	,	PUNCT
iajs-1215	266	29	m	m	VERB
iajs-1215	266	30	2	2	NUM
iajs-1215	266	31	be	be	VERB
iajs-1215	266	32	r1	r1	VERB
iajs-1215	266	33	,	,	PUNCT
iajs-1215	266	34	r2	r2	NOUN
iajs-1215	266	35	-	-	PUNCT
iajs-1215	266	36	modules	module	NOUN
iajs-1215	266	37	respectively	respectively	ADV
iajs-1215	266	38	.	.	PUNCT
iajs-1215	267	1	if	if	SCONJ
iajs-1215	267	2	p	p	PROPN
iajs-1215	267	3	=	=	PROPN
iajs-1215	267	4	p1	p1	PROPN
iajs-1215	267	5			NOUN
iajs-1215	267	6	p2	p2	PROPN
iajs-1215	267	7	is	be	AUX
iajs-1215	267	8	a	a	DET
iajs-1215	267	9	weakly	weakly	ADJ
iajs-1215	267	10	prime	prime	ADJ
iajs-1215	267	11	r	r	NOUN
iajs-1215	267	12	-	-	PUNCT
iajs-1215	267	13	submodule	submodule	NOUN
iajs-1215	267	14	of	of	ADP
iajs-1215	267	15	m	m	PROPN
iajs-1215	267	16	=	=	ADJ
iajs-1215	267	17	m	m	VERB
iajs-1215	267	18	1	1	NUM
iajs-1215	267	19			NOUN
iajs-1215	267	20	m	m	PROPN
iajs-1215	267	21	2	2	NUM
iajs-1215	267	22	,	,	PUNCT
iajs-1215	267	23	then	then	ADV
iajs-1215	267	24	either	either	CCONJ
iajs-1215	267	25	p	p	X
iajs-1215	267	26	=	=	NOUN
iajs-1215	267	27	0	0	NUM
iajs-1215	267	28	or	or	CCONJ
iajs-1215	267	29	p	p	NOUN
iajs-1215	267	30	is	be	AUX
iajs-1215	267	31	a	a	DET
iajs-1215	267	32	prime	prime	ADJ
iajs-1215	267	33	submodule	submodule	NOUN
iajs-1215	267	34	of	of	ADP
iajs-1215	267	35	m.	m.	NOUN
iajs-1215	267	36	proof	proof	NOUN
iajs-1215	267	37	.	.	PUNCT
iajs-1215	268	1	assume	assume	VERB
iajs-1215	268	2	p	p	PROPN
iajs-1215	268	3			NOUN
iajs-1215	268	4	0	0	NUM
iajs-1215	268	5	,	,	PUNCT
iajs-1215	268	6	so	so	SCONJ
iajs-1215	268	7	either	either	CCONJ
iajs-1215	268	8	p1	p1	PROPN
iajs-1215	268	9			PROPN
iajs-1215	268	10	0	0	NUM
iajs-1215	268	11	or	or	CCONJ
iajs-1215	268	12	p2	p2	PROPN
iajs-1215	268	13			NOUN
iajs-1215	268	14	0	0	NUM
iajs-1215	268	15	.	.	PUNCT
iajs-1215	268	16	suppose	suppose	VERB
iajs-1215	268	17	that	that	SCONJ
iajs-1215	268	18	p2	p2	PROPN
iajs-1215	268	19			PROPN
iajs-1215	268	20	0	0	NUM
iajs-1215	268	21	,	,	PUNCT
iajs-1215	268	22	hence	hence	ADV
iajs-1215	268	23	there	there	PRON
iajs-1215	268	24	exists	exist	VERB
iajs-1215	268	25	y	y	PROPN
iajs-1215	268	26			PROPN
iajs-1215	268	27	p2	p2	PROPN
iajs-1215	268	28	,	,	PUNCT
iajs-1215	268	29	y	y	PROPN
iajs-1215	268	30			PROPN
iajs-1215	268	31	0	0	X
iajs-1215	268	32	.	.	PUNCT
iajs-1215	269	1	let	let	VERB
iajs-1215	269	2	r	r	NOUN
iajs-1215	269	3			PROPN
iajs-1215	269	4	(	(	PUNCT
iajs-1215	269	5	p1	p1	PROPN
iajs-1215	269	6	1	1	NUM
iajs-1215	269	7	:	:	PUNCT
iajs-1215	269	8	r	r	NOUN
iajs-1215	269	9	m	m	NOUN
iajs-1215	269	10	1	1	NUM
iajs-1215	269	11	)	)	PUNCT
iajs-1215	269	12	and	and	CCONJ
iajs-1215	269	13	let	let	VERB
iajs-1215	269	14	x	x	PRON
iajs-1215	269	15			PROPN
iajs-1215	269	16	m	m	VERB
iajs-1215	269	17	1	1	NUM
iajs-1215	269	18	,	,	PUNCT
iajs-1215	269	19	then	then	ADV
iajs-1215	269	20	(	(	PUNCT
iajs-1215	269	21	0,0	0,0	NOUN
iajs-1215	269	22	)	)	PUNCT
iajs-1215	269	23			NOUN
iajs-1215	269	24	(	(	PUNCT
iajs-1215	269	25	r,1	r,1	PROPN
iajs-1215	269	26	)	)	PUNCT
iajs-1215	269	27	(	(	PUNCT
iajs-1215	269	28	x	x	X
iajs-1215	269	29	,	,	PUNCT
iajs-1215	269	30	y	y	NOUN
iajs-1215	269	31	)	)	PUNCT
iajs-1215	269	32	=	=	SYM
iajs-1215	270	1	(	(	PUNCT
iajs-1215	270	2	r	r	NOUN
iajs-1215	270	3	x	x	PROPN
iajs-1215	270	4	,	,	PUNCT
iajs-1215	270	5	y	y	PROPN
iajs-1215	270	6	)	)	PUNCT
iajs-1215	270	7			NOUN
iajs-1215	270	8	p1	p1	NOUN
iajs-1215	270	9			NOUN
iajs-1215	270	10	p2	p2	PROPN
iajs-1215	270	11	=	=	PUNCT
iajs-1215	271	1	p.	p.	NOUN
iajs-1215	271	2	since	since	SCONJ
iajs-1215	271	3	p	p	NOUN
iajs-1215	271	4	is	be	AUX
iajs-1215	271	5	weakly	weakly	ADJ
iajs-1215	271	6	prime	prime	ADJ
iajs-1215	271	7	in	in	ADP
iajs-1215	271	8	m	m	PROPN
iajs-1215	271	9	,	,	PUNCT
iajs-1215	271	10	either	either	CCONJ
iajs-1215	271	11	(	(	PUNCT
iajs-1215	271	12	x	x	NOUN
iajs-1215	271	13	,	,	PUNCT
iajs-1215	271	14	y	y	PROPN
iajs-1215	271	15	)	)	PUNCT
iajs-1215	271	16			NOUN
iajs-1215	271	17	p	p	NOUN
iajs-1215	271	18	or	or	CCONJ
iajs-1215	271	19	(	(	PUNCT
iajs-1215	271	20	r,1	r,1	PROPN
iajs-1215	271	21	)	)	PUNCT
iajs-1215	271	22			NOUN
iajs-1215	271	23	(	(	PUNCT
iajs-1215	271	24	p1	p1	PROPN
iajs-1215	271	25			NOUN
iajs-1215	271	26	p2	p2	PROPN
iajs-1215	271	27	:	:	PUNCT
iajs-1215	271	28	m1	m1	PROPN
iajs-1215	271	29			PROPN
iajs-1215	271	30	m	m	PROPN
iajs-1215	271	31	2	2	NUM
iajs-1215	271	32	)	)	PUNCT
iajs-1215	271	33	.	.	PUNCT
iajs-1215	272	1	hence	hence	ADV
iajs-1215	272	2	if	if	SCONJ
iajs-1215	272	3	(	(	PUNCT
iajs-1215	272	4	x	x	NOUN
iajs-1215	272	5	,	,	PUNCT
iajs-1215	272	6	y	y	PROPN
iajs-1215	272	7	)	)	PUNCT
iajs-1215	272	8			NOUN
iajs-1215	272	9	p	p	NOUN
iajs-1215	272	10	,	,	PUNCT
iajs-1215	272	11	then	then	ADV
iajs-1215	272	12	x	x	SYM
iajs-1215	272	13			PROPN
iajs-1215	272	14	p1	p1	NOUN
iajs-1215	272	15	and	and	CCONJ
iajs-1215	272	16	so	so	ADV
iajs-1215	272	17	m	m	VERB
iajs-1215	272	18	1	1	NUM
iajs-1215	272	19	=	=	SYM
iajs-1215	272	20	p1	p1	NOUN
iajs-1215	272	21	which	which	PRON
iajs-1215	272	22	implies	imply	VERB
iajs-1215	272	23	p	p	X
iajs-1215	272	24	=	=	NOUN
iajs-1215	272	25	m	m	PROPN
iajs-1215	272	26	1	1	NUM
iajs-1215	272	27			NOUN
iajs-1215	272	28	p2	p2	NOUN
iajs-1215	272	29	.	.	PUNCT
iajs-1215	273	1	if	if	SCONJ
iajs-1215	273	2	(	(	PUNCT
iajs-1215	273	3	r,1	r,1	PROPN
iajs-1215	273	4	)	)	PUNCT
iajs-1215	273	5			NOUN
iajs-1215	273	6	(	(	PUNCT
iajs-1215	273	7	p1	p1	PROPN
iajs-1215	273	8			NOUN
iajs-1215	273	9	p2	p2	PROPN
iajs-1215	273	10	:	:	PUNCT
iajs-1215	273	11	m1	m1	PROPN
iajs-1215	273	12			PROPN
iajs-1215	273	13	m	m	PROPN
iajs-1215	273	14	2	2	NUM
iajs-1215	273	15	)	)	PUNCT
iajs-1215	273	16	,	,	PUNCT
iajs-1215	273	17	then	then	ADV
iajs-1215	273	18	m2	m2	PROPN
iajs-1215	273	19	=	=	PROPN
iajs-1215	273	20	p2	p2	PROPN
iajs-1215	273	21	which	which	PRON
iajs-1215	273	22	implies	imply	VERB
iajs-1215	273	23	p	p	X
iajs-1215	273	24	=	=	PROPN
iajs-1215	273	25	p1	p1	PROPN
iajs-1215	273	26			NOUN
iajs-1215	273	27	m	m	PROPN
iajs-1215	273	28	2	2	NUM
iajs-1215	273	29	.	.	PUNCT
iajs-1215	274	1	hence	hence	ADV
iajs-1215	274	2	by	by	ADP
iajs-1215	274	3	propositions	proposition	NOUN
iajs-1215	274	4	2.16	2.16	NUM
iajs-1215	274	5	,	,	PUNCT
iajs-1215	274	6	2.17	2.17	NUM
iajs-1215	274	7	,	,	PUNCT
iajs-1215	274	8	p	p	NOUN
iajs-1215	274	9	is	be	AUX
iajs-1215	274	10	a	a	DET
iajs-1215	274	11	prime	prime	ADJ
iajs-1215	274	12	r	r	NOUN
iajs-1215	274	13	-	-	PUNCT
iajs-1215	274	14	submodule	submodule	NOUN
iajs-1215	274	15	of	of	ADP
iajs-1215	274	16	m.	m.	PROPN
iajs-1215	274	17	ibn	ibn	PROPN
iajs-1215	274	18	alhaitham	alhaitham	PROPN
iajs-1215	275	1	j.	j.	PROPN
iajs-1215	276	1	fo	fo	ADP
iajs-1215	276	2	r	r	NOUN
iajs-1215	276	3	pure	pure	ADJ
iajs-1215	276	4	&	&	CCONJ
iajs-1215	276	5	appl	appl	PROPN
iajs-1215	276	6	.	.	PUNCT
iajs-1215	277	1	sc	sc	PROPN
iajs-1215	278	1	i	i	PRON
iajs-1215	278	2	vo	vo	INTJ
iajs-1215	278	3	l.22	l.22	X
iajs-1215	278	4	(	(	PUNCT
iajs-1215	278	5	3	3	NUM
iajs-1215	278	6	)	)	PUNCT
iajs-1215	278	7	2009	2009	NUM
iajs-1215	278	8	references	reference	NOUN
iajs-1215	278	9	1	1	NUM
iajs-1215	278	10	.	.	PUNCT
iajs-1215	278	11	c.p.lu	c.p.lu	PROPN
iajs-1215	278	12	,	,	PUNCT
iajs-1215	278	13	(	(	PUNCT
iajs-1215	278	14	1984	1984	NUM
iajs-1215	278	15	)	)	PUNCT
iajs-1215	278	16	,	,	PUNCT
iajs-1215	278	17	"	"	PUNCT
iajs-1215	278	18	prime	prime	ADJ
iajs-1215	278	19	submodules	submodule	NOUN
iajs-1215	278	20	of	of	ADP
iajs-1215	278	21	modules	module	NOUN
iajs-1215	278	22	"	"	PUNCT
iajs-1215	278	23	,	,	PUNCT
iajs-1215	278	24	comment	comment	NOUN
iajs-1215	278	25	.	.	PUNCT
iajs-1215	279	1	math	math	NOUN
iajs-1215	279	2	.	.	PUNCT
iajs-1215	280	1	univ	univ	PROPN
iajs-1215	280	2	.	.	PUNCT
iajs-1215	281	1	st	st	PROPN
iajs-1215	281	2	,	,	PUNCT
iajs-1215	281	3	paul	paul	PROPN
iajs-1215	281	4	,	,	PUNCT
iajs-1215	281	5	33	33	NUM
iajs-1215	281	6	,	,	PUNCT
iajs-1215	281	7	61	61	NUM
iajs-1215	281	8	-	-	SYM
iajs-1215	281	9	69	69	NUM
iajs-1215	281	10	.	.	PUNCT
iajs-1215	282	1	2	2	X
iajs-1215	282	2	.	.	X
iajs-1215	282	3	dauns	daun	NOUN
iajs-1215	282	4	,	,	PUNCT
iajs-1215	282	5	j.	j.	PROPN
iajs-1215	282	6	(	(	PUNCT
iajs-1215	282	7	1980	1980	NUM
iajs-1215	282	8	)	)	PUNCT
iajs-1215	282	9	,	,	PUNCT
iajs-1215	282	10	"	"	PUNCT
iajs-1215	282	11	prime	prime	ADJ
iajs-1215	282	12	submodules	submodule	NOUN
iajs-1215	282	13	and	and	CCONJ
iajs-1215	282	14	one	one	NUM
iajs-1215	282	15	sided	sided	ADJ
iajs-1215	282	16	ideals	ideal	NOUN
iajs-1215	282	17	in	in	ADP
iajs-1215	282	18	ring	ring	NOUN
iajs-1215	282	19	theory	theory	NOUN
iajs-1215	282	20	and	and	CCONJ
iajs-1215	282	21	algebra	algebra	NOUN
iajs-1215	282	22	iii	iii	PROPN
iajs-1215	282	23	"	"	PUNCT
iajs-1215	282	24	,	,	PUNCT
iajs-1215	282	25	proc	proc	NOUN
iajs-1215	282	26	.	.	PUNCT
iajs-1215	283	1	of	of	ADP
iajs-1215	283	2	3	3	NUM
iajs-1215	283	3	rd	rd	PROPN
iajs-1215	283	4	oklahoma	oklahoma	PROPN
iajs-1215	283	5	conference	conference	PROPN
iajs-1215	283	6	b.r	b.r	PROPN
iajs-1215	283	7	mc	mc	PROPN
iajs-1215	283	8	donald	donald	PROPN
iajs-1215	283	9	(	(	PUNCT
iajs-1215	283	10	editor	editor	NOUN
iajs-1215	283	11	)	)	PUNCT
iajs-1215	283	12	dekker	dekker	PROPN
iajs-1215	283	13	,	,	PUNCT
iajs-1215	283	14	new	new	PROPN
iajs-1215	283	15	york	york	PROPN
iajs-1215	283	16	,	,	PUNCT
iajs-1215	283	17	301	301	NUM
iajs-1215	283	18	-	-	SYM
iajs-1215	283	19	344	344	NUM
iajs-1215	283	20	.	.	PUNCT
iajs-1215	283	21	3	3	NUM
iajs-1215	283	22	.	.	NOUN
iajs-1215	283	23	athab	athab	PROPN
iajs-1215	283	24	,	,	PUNCT
iajs-1215	283	25	e.a	e.a	PROPN
iajs-1215	283	26	.	.	PROPN
iajs-1215	283	27	(	(	PUNCT
iajs-1215	283	28	1996	1996	NUM
iajs-1215	283	29	)	)	PUNCT
iajs-1215	283	30	,	,	PUNCT
iajs-1215	283	31	"	"	PUNCT
iajs-1215	283	32	prime	prime	ADJ
iajs-1215	283	33	submodules	submodule	NOUN
iajs-1215	283	34	and	and	CCONJ
iajs-1215	283	35	weakly	weakly	ADJ
iajs-1215	283	36	prime	prime	ADJ
iajs-1215	283	37	submodules	submodule	NOUN
iajs-1215	283	38	"	"	PUNCT
iajs-1215	283	39	,	,	PUNCT
iajs-1215	283	40	m.sc	m.sc	PROPN
iajs-1215	283	41	thesis	thesis	NOUN
iajs-1215	283	42	,	,	PUNCT
iajs-1215	283	43	univ	univ	PROPN
iajs-1215	283	44	.	.	PROPN
iajs-1215	283	45	of	of	ADP
iajs-1215	283	46	baghdad	baghdad	PROPN
iajs-1215	283	47	.	.	PUNCT
iajs-1215	284	1	4	4	X
iajs-1215	284	2	.	.	X
iajs-1215	284	3	hassin	hassin	PROPN
iajs-1215	284	4	,	,	PUNCT
iajs-1215	284	5	m.a	m.a	PROPN
iajs-1215	284	6	.	.	PROPN
iajs-1215	284	7	(	(	PUNCT
iajs-1215	284	8	1990	1990	NUM
iajs-1215	284	9	)	)	PUNCT
iajs-1215	284	10	,	,	PUNCT
iajs-1215	284	11	"	"	PUNCT
iajs-1215	284	12	quasi	quasi	ADJ
iajs-1215	284	13	-	-	ADJ
iajs-1215	284	14	prime	prime	ADJ
iajs-1215	284	15	modules	module	NOUN
iajs-1215	284	16	and	and	CCONJ
iajs-1215	284	17	quasiprime	quasiprime	ADJ
iajs-1215	284	18	submodules	submodule	NOUN
iajs-1215	284	19	"	"	PUNCT
iajs-1215	284	20	,	,	PUNCT
iajs-1215	284	21	m.sc	m.sc	PROPN
iajs-1215	284	22	thesis	thesis	NOUN
iajs-1215	284	23	,	,	PUNCT
iajs-1215	284	24	univ	univ	PROPN
iajs-1215	284	25	.	.	PROPN
iajs-1215	284	26	of	of	ADP
iajs-1215	284	27	baghdad	baghdad	PROPN
iajs-1215	284	28	.	.	PUNCT
iajs-1215	285	1	5	5	NUM
iajs-1215	285	2	.	.	X
iajs-1215	285	3	m.behoodi	m.behoodi	NOUN
iajs-1215	285	4	and	and	CCONJ
iajs-1215	285	5	h.koohi	h.koohi	NOUN
iajs-1215	285	6	,	,	PUNCT
iajs-1215	285	7	(	(	PUNCT
iajs-1215	285	8	2004	2004	NUM
iajs-1215	285	9	)	)	PUNCT
iajs-1215	285	10	,	,	PUNCT
iajs-1215	285	11	"	"	PUNCT
iajs-1215	285	12	weakly	weakly	ADJ
iajs-1215	285	13	prime	prime	ADJ
iajs-1215	285	14	submodules	submodule	NOUN
iajs-1215	285	15	"	"	PUNCT
iajs-1215	285	16	,	,	PUNCT
iajs-1215	285	17	vietnam	vietnam	PROPN
iajs-1215	285	18	j.	j.	PROPN
iajs-1215	285	19	math	math	PROPN
iajs-1215	285	20	,	,	PUNCT
iajs-1215	285	21	32(2	32(2	NUM
iajs-1215	285	22	)	)	PUNCT
iajs-1215	285	23	,	,	PUNCT
iajs-1215	285	24	185	185	NUM
iajs-1215	285	25	-	-	SYM
iajs-1215	285	26	195	195	NUM
iajs-1215	285	27	.	.	NOUN
iajs-1215	285	28	6	6	NUM
iajs-1215	285	29	.	.	X
iajs-1215	286	1	azizi	azizi	PROPN
iajs-1215	286	2	,	,	PUNCT
iajs-1215	286	3	a.	a.	NOUN
iajs-1215	286	4	(	(	PUNCT
iajs-1215	286	5	2006	2006	NUM
iajs-1215	286	6	)	)	PUNCT
iajs-1215	286	7	,	,	PUNCT
iajs-1215	286	8	glasgow	glasgow	PROPN
iajs-1215	286	9	math	math	NOUN
iajs-1215	286	10	.	.	PUNCT
iajs-1215	287	1	j.	j.	PROPN
iajs-1215	287	2	,	,	PUNCT
iajs-1215	287	3	48	48	NUM
iajs-1215	287	4	:	:	PUNCT
iajs-1215	287	5	343	343	NUM
iajs-1215	287	6	-	-	SYM
iajs-1215	287	7	346	346	NUM
iajs-1215	287	8	.	.	PUNCT
iajs-1215	288	1	7	7	NUM
iajs-1215	288	2	.	.	X
iajs-1215	288	3	anderson	anderson	PROPN
iajs-1215	288	4	,	,	PUNCT
iajs-1215	288	5	d.d	d.d	PROPN
iajs-1215	288	6	.	.	PROPN
iajs-1215	288	7	and	and	CCONJ
iajs-1215	288	8	eric	eric	PROPN
iajs-1215	288	9	smith	smith	PROPN
iajs-1215	288	10	,	,	PUNCT
iajs-1215	288	11	(	(	PUNCT
iajs-1215	288	12	2003	2003	NUM
iajs-1215	288	13	)	)	PUNCT
iajs-1215	288	14	,	,	PUNCT
iajs-1215	288	15	"	"	PUNCT
iajs-1215	288	16	weakly	weakly	ADJ
iajs-1215	288	17	prime	prime	ADJ
iajs-1215	288	18	ideals	ideal	NOUN
iajs-1215	288	19	"	"	PUNCT
iajs-1215	288	20	,	,	PUNCT
iajs-1215	288	21	houston	houston	PROPN
iajs-1215	288	22	j.of	j.of	PROPN
iajs-1215	288	23	math	math	NOUN
iajs-1215	288	24	.	.	PUNCT
iajs-1215	288	25	,	,	PUNCT
iajs-1215	288	26	vol.29	vol.29	NOUN
iajs-1215	288	27	,	,	PUNCT
iajs-1215	288	28	no.4	no.4	PROPN
iajs-1215	288	29	,	,	PUNCT
iajs-1215	288	30	831	831	NUM
iajs-1215	288	31	-	-	SYM
iajs-1215	288	32	840	840	NUM
iajs-1215	288	33	.	.	PUNCT
iajs-1215	288	34	8	8	NUM
iajs-1215	288	35	.	.	X
iajs-1215	289	1	atani	atani	PROPN
iajs-1215	289	2	,	,	PUNCT
iajs-1215	289	3	s.e	s.e	PROPN
iajs-1215	289	4	.	.	PROPN
iajs-1215	289	5	and	and	CCONJ
iajs-1215	289	6	forzalipour	forzalipour	NOUN
iajs-1215	289	7	,	,	PUNCT
iajs-1215	289	8	f.	f.	PROPN
iajs-1215	289	9	(	(	PUNCT
iajs-1215	289	10	2005	2005	NUM
iajs-1215	289	11	)	)	PUNCT
iajs-1215	289	12	,	,	PUNCT
iajs-1215	289	13	georgian	georgian	PROPN
iajs-1215	289	14	mathematical	mathematical	ADJ
iajs-1215	289	15	journal	journal	NOUN
iajs-1215	289	16	,	,	PUNCT
iajs-1215	289	17	12	12	NUM
iajs-1215	289	18	,	,	PUNCT
iajs-1215	289	19	1	1	NUM
iajs-1215	289	20	-	-	SYM
iajs-1215	289	21	7	7	NUM
iajs-1215	289	22	.	.	NOUN
iajs-1215	289	23	9	9	NUM
iajs-1215	289	24	.	.	NUM
iajs-1215	289	25	barned	barne	VERB
iajs-1215	289	26	,	,	PUNCT
iajs-1215	289	27	a.	a.	NOUN
iajs-1215	289	28	(	(	PUNCT
iajs-1215	289	29	1981	1981	NUM
iajs-1215	289	30	)	)	PUNCT
iajs-1215	289	31	,	,	PUNCT
iajs-1215	289	32	j.	j.	PROPN
iajs-1215	289	33	algebra	algebra	PROPN
iajs-1215	289	34	,	,	PUNCT
iajs-1215	289	35	71	71	NUM
iajs-1215	289	36	,	,	PUNCT
iajs-1215	289	37	174	174	NUM
iajs-1215	289	38	-	-	SYM
iajs-1215	289	39	178	178	NUM
iajs-1215	289	40	.	.	PUNCT
iajs-1215	290	1	10	10	NUM
iajs-1215	290	2	.	.	PUNCT
iajs-1215	291	1	el	el	PROPN
iajs-1215	291	2	bast	bast	PROPN
iajs-1215	291	3	,	,	PUNCT
iajs-1215	291	4	z.a	z.a	PROPN
iajs-1215	291	5	.	.	PROPN
iajs-1215	291	6	smith	smith	PROPN
iajs-1215	291	7	,	,	PUNCT
iajs-1215	291	8	p.f	p.f	PROPN
iajs-1215	291	9	.	.	PROPN
iajs-1215	291	10	(	(	PUNCT
iajs-1215	291	11	1988	1988	NUM
iajs-1215	291	12	)	)	PUNCT
iajs-1215	291	13	,	,	PUNCT
iajs-1215	291	14	"	"	PUNCT
iajs-1215	291	15	multiplication	multiplication	NOUN
iajs-1215	291	16	modules	module	NOUN
iajs-1215	291	17	"	"	PUNCT
iajs-1215	291	18	,	,	PUNCT
iajs-1215	291	19	comm	comm	NOUN
iajs-1215	291	20	in	in	ADP
iajs-1215	291	21	algebra	algebra	NOUN
iajs-1215	291	22	,	,	PUNCT
iajs-1215	291	23	16	16	NUM
iajs-1215	291	24	,	,	PUNCT
iajs-1215	291	25	755	755	NUM
iajs-1215	291	26	-	-	SYM
iajs-1215	291	27	779	779	NUM
iajs-1215	291	28	.	.	PROPN
iajs-1215	291	29	11	11	NUM
iajs-1215	291	30	.	.	PUNCT
iajs-1215	292	1	larsen	larsen	PROPN
iajs-1215	292	2	,	,	PUNCT
iajs-1215	292	3	m.d	m.d	PROPN
iajs-1215	292	4	.	.	PROPN
iajs-1215	292	5	and	and	CCONJ
iajs-1215	292	6	mc	mc	PROPN
iajs-1215	292	7	carthy	carthy	PROPN
iajs-1215	292	8	,	,	PUNCT
iajs-1215	292	9	p.j	p.j	PROPN
iajs-1215	292	10	.	.	PROPN
iajs-1215	292	11	(	(	PUNCT
iajs-1215	292	12	1971	1971	NUM
iajs-1215	292	13	)	)	PUNCT
iajs-1215	292	14	,	,	PUNCT
iajs-1215	292	15	"	"	PUNCT
iajs-1215	292	16	multiplicative	multiplicative	ADJ
iajs-1215	292	17	theory	theory	NOUN
iajs-1215	292	18	of	of	ADP
iajs-1215	292	19	ideals	ideal	NOUN
iajs-1215	292	20	"	"	PUNCT
iajs-1215	292	21	,	,	PUNCT
iajs-1215	292	22	academic	academic	ADJ
iajs-1215	292	23	press	press	NOUN
iajs-1215	292	24	,	,	PUNCT
iajs-1215	292	25	newyork	newyork	NOUN
iajs-1215	292	26	and	and	CCONJ
iajs-1215	292	27	london	london	PROPN
iajs-1215	292	28	,	,	PUNCT
iajs-1215	292	29	.	.	PUNCT
iajs-1215	293	1	12	12	NUM
iajs-1215	293	2	.	.	PUNCT
iajs-1215	294	1	al	al	PROPN
iajs-1215	294	2	-	-	PUNCT
iajs-1215	294	3	kalik	kalik	PROPN
iajs-1215	294	4	,	,	PUNCT
iajs-1215	294	5	a.j	a.j	PROPN
iajs-1215	294	6	.	.	PROPN
iajs-1215	294	7	a.	a.	PROPN
iajs-1215	294	8	(	(	PUNCT
iajs-1215	294	9	2005	2005	NUM
iajs-1215	294	10	)	)	PUNCT
iajs-1215	294	11	,	,	PUNCT
iajs-1215	294	12	"	"	PUNCT
iajs-1215	294	13	primary	primary	ADJ
iajs-1215	294	14	modules	module	NOUN
iajs-1215	294	15	"	"	PUNCT
iajs-1215	294	16	,	,	PUNCT
iajs-1215	294	17	m.sc	m.sc	PROPN
iajs-1215	294	18	thesis	thesis	NOUN
iajs-1215	294	19	,	,	PUNCT
iajs-1215	294	20	college	college	NOUN
iajs-1215	294	21	of	of	ADP
iajs-1215	294	22	education	education	NOUN
iajs-1215	294	23	,	,	PUNCT
iajs-1215	294	24	university	university	NOUN
iajs-1215	294	25	of	of	ADP
iajs-1215	294	26	baghdad	baghdad	PROPN
iajs-1215	294	27	,	,	PUNCT
iajs-1215	294	28	.	.	PUNCT