id	sid	tid	token	lemma	pos
iajs-778	1	1	2011	2011	NUM
iajs-778	1	2	)	)	PUNCT
iajs-778	1	3	2	2	NUM
iajs-778	1	4	(	(	PUNCT
iajs-778	1	5	24مجلة	24مجلة	NUM
iajs-778	1	6	ابن	ابن	VERB
iajs-778	1	7	الهیثم	الهیثم	ADJ
iajs-778	1	8	للعلوم	للعلوم	PROPN
iajs-778	1	9	الصرفة	الصرفة	PROPN
iajs-778	1	10	والتطبیقیة	والتطبیقیة	PROPN
iajs-778	1	11	المجلد	المجلد	PROPN
iajs-778	1	12	المقاسات	المقاسات	PROPN
iajs-778	1	13	الجزئیة	الجزئیة	PROPN
iajs-778	1	14	األولیة	األولیة	PROPN
iajs-778	1	15	المضادة	المضادة	PROPN
iajs-778	1	16	هادي	هادي	NOUN
iajs-778	1	17	انعام	انعام	ADJ
iajs-778	1	18	محمد	محمد	PROPN
iajs-778	1	19	علي	علي	NOUN
iajs-778	1	20	جامعة	جامعة	NOUN
iajs-778	1	21	بغداد،ابن	بغداد،ابن	PROPN
iajs-778	1	22	الهیثم	الهیثم	VERB
iajs-778	1	23	-كلیة	-كلیة	PROPN
iajs-778	1	24	التربیة،قسم	التربیة،قسم	PROPN
iajs-778	1	25	الریاضیات	الریاضیات	NOUN
iajs-778	1	26	2010	2010	NUM
iajs-778	1	27	،	،	NOUN
iajs-778	1	28	شباط	شباط	X
iajs-778	1	29	،	،	NOUN
iajs-778	1	30	28	28	NUM
iajs-778	1	31	:	:	PUNCT
iajs-778	1	32	استلم	استلم	PROPN
iajs-778	1	33	البحث	البحث	VERB
iajs-778	1	34	في	في	ADP
iajs-778	1	35	2010	2010	NUM
iajs-778	1	36	،	،	NOUN
iajs-778	1	37	نیسان	نیسان	NOUN
iajs-778	1	38	،	،	NOUN
iajs-778	1	39	25	25	NUM
iajs-778	1	40	:	:	PUNCT
iajs-778	1	41	قبل	قبل	NOUN
iajs-778	1	42	البحث	البحث	VERB
iajs-778	1	43	في	في	SCONJ
iajs-778	1	44	الخالصة	الخالصة	PROPN
iajs-778	1	45	یقـال	یقـال	PROPN
iajs-778	1	46	عـن	عـن	NOUN
iajs-778	1	47	.	.	PUNCT
iajs-778	2	1	mمقاس	mمقاس	PROPN
iajs-778	2	2	جزئـي	جزئـي	PROPN
iajs-778	2	3	فعلـي	فعلـي	PROPN
iajs-778	2	4	مـن	مـن	PROPN
iajs-778	2	5	nلیكن	nلیكن	PROPN
iajs-778	2	6	.	.	PUNCT
iajs-778	3	1	rمقاساً	rمقاساً	PROPN
iajs-778	3	2	احادیاً	احادیاً	NOUN
iajs-778	3	3	على	على	PROPN
iajs-778	3	4	mحلقة	mحلقة	PROPN
iajs-778	3	5	ابدالیة	ابدالیة	NOUN
iajs-778	3	6	ذو	ذو	PROPN
iajs-778	3	7	محاید	محاید	PROPN
iajs-778	3	8	ولیكن	ولیكن	VERB
iajs-778	3	9	rلتكن	rلتكن	PROPN
iajs-778	3	10	n	n	CCONJ
iajs-778	3	11	مقاساً	مقاساً	PROPN
iajs-778	3	12	جزئیاً	جزئیاً	VERB
iajs-778	3	13	اولي	اولي	NOUN
iajs-778	3	14	مضاد	مضاد	NOUN
iajs-778	3	15	اذا	اذا	NOUN
iajs-778	3	16	كان	كان	PROPN
iajs-778	3	17	المقاس	المقاس	VERB
iajs-778	3	18			PROPN
iajs-778	3	19			PROPN
iajs-778	3	20	اولي	اولي	NOUN
iajs-778	3	21	مضاد	مضاد	NOUN
iajs-778	3	22	،	،	X
iajs-778	3	23	حیث	حیث	NOUN
iajs-778	4	1	ان	ان	PROPN
iajs-778	4	2	المقاس	المقاس	PROPN
iajs-778	4	3			PROPN
iajs-778	4	4			PROPN
iajs-778	4	5	rیسمى	rیسمى	ADJ
iajs-778	4	6	اولي	اولي	NOUN
iajs-778	4	7	مضاد	مضاد	NOUN
iajs-778	4	8	اذا	اذا	PROPN
iajs-778	4	9	كـان	كـان	PROPN
iajs-778	4	10	لكـل	لكـل	PROPN
iajs-778	5	1			PROPN
iajs-778	6	1	r	r	X
iajs-778	6	2	اما	اما	X
iajs-778	6	3	،	،	X
iajs-778	6	4	o	o	PUNCT
iajs-778	6	5			VERB
iajs-778	6	6			PROPN
iajs-778	6	7			VERB
iajs-778	6	8			NOUN
iajs-778	6	9	r	r	NOUN
iajs-778	6	10	أو	أو	NOUN
iajs-778	6	11			PROPN
iajs-778	6	12			PROPN
iajs-778	6	13			PROPN
iajs-778	6	14			PROPN
iajs-778	6	15	r.	r.	PROPN
iajs-778	6	16	.في	.في	PUNCT
iajs-778	7	1	هذا	هذا	NOUN
iajs-778	7	2	البحث	البحث	PROPN
iajs-778	7	3	درسنا	درسنا	PROPN
iajs-778	7	4	المقاسات	المقاسات	PROPN
iajs-778	7	5	الجزئیة	الجزئیة	PROPN
iajs-778	7	6	األولیة	األولیة	PROPN
iajs-778	7	7	المضادة	المضادة	PROPN
iajs-778	7	8	واعطینا	واعطینا	PROPN
iajs-778	7	9	العدید	العدید	NOUN
iajs-778	7	10	من	من	ADP
iajs-778	7	11	الخواص	الخواص	NOUN
iajs-778	7	12	المتعلقة	المتعلقة	PROPN
iajs-778	7	13	بهذا	بهذا	VERB
iajs-778	7	14	المفهوم	المفهوم	PROPN
iajs-778	7	15	ـات	ـات	PROPN
iajs-778	7	16	الجزئیـة	الجزئیـة	PROPN
iajs-778	7	17	الثانیـة	الثانیـة	PROPN
iajs-778	7	18	المقاسـات	المقاسـات	PROPN
iajs-778	7	19	الثانیــة	الثانیــة	PROPN
iajs-778	7	20	الم	الم	PROPN
iajs-778	7	21	-المقاسـات	-المقاسـات	PROPN
iajs-778	7	22	الجزئیــة	الجزئیــة	ADJ
iajs-778	7	23	االولیـة	االولیـة	NOUN
iajs-778	7	24	المضـادة	المضـادة	NOUN
iajs-778	7	25	:	:	PUNCT
iajs-778	7	26	الكلمـات	الكلمـات	PROPN
iajs-778	7	27	المفتاحیـة	المفتاحیـة	X
iajs-778	7	28	المقاســات	المقاســات	PROPN
iajs-778	8	1	-)المضـادة	-)المضـادة	INTJ
iajs-778	9	1	االولیـة(قاسـ	االولیـة(قاسـ	PROPN
iajs-778	9	2	.الثانویة	.الثانویة	VERB
iajs-778	9	3	ibn	ibn	PROPN
iajs-778	9	4	alhaitham	alhaitham	PROPN
iajs-778	9	5	j.	j.	PROPN
iajs-778	9	6	for	for	ADP
iajs-778	9	7	pure	pure	ADJ
iajs-778	9	8	&	&	CCONJ
iajs-778	9	9	appl	appl	PROPN
iajs-778	9	10	.	.	PUNCT
iajs-778	10	1	sci	sci	PROPN
iajs-778	10	2	.	.	PUNCT
iajs-778	10	3	vol.24	vol.24	NOUN
iajs-778	10	4	(	(	PUNCT
iajs-778	10	5	2	2	NUM
iajs-778	10	6	)	)	PUNCT
iajs-778	10	7	2011	2011	NUM
iajs-778	10	8	coprime	coprime	NOUN
iajs-778	10	9	submodules	submodules	PROPN
iajs-778	10	10	i.	i.	PROPN
iajs-778	10	11	m.	m.	PROPN
iajs-778	10	12	a.	a.	PROPN
iajs-778	10	13	hadi	hadi	PROPN
iajs-778	10	14	department	department	PROPN
iajs-778	10	15	of	of	ADP
iajs-778	10	16	mathematics	mathematics	PROPN
iajs-778	10	17	,	,	PUNCT
iajs-778	10	18	ibn	ibn	PROPN
iajs-778	10	19	-	-	PUNCT
iajs-778	10	20	al	al	PROPN
iajs-778	10	21	-	-	PUNCT
iajs-778	10	22	haitham	haitham	PROPN
iajs-778	10	23	,	,	PUNCT
iajs-778	10	24	college	college	NOUN
iajs-778	10	25	of	of	ADP
iajs-778	10	26	education	education	NOUN
iajs-778	10	27	,	,	PUNCT
iajs-778	10	28	university	university	PROPN
iajs-778	10	29	of	of	ADP
iajs-778	10	30	baghdad	baghdad	PROPN
iajs-778	10	31	received	receive	VERB
iajs-778	10	32	in	in	ADP
iajs-778	10	33	:	:	PUNCT
iajs-778	10	34	28	28	NUM
iajs-778	10	35	,	,	PUNCT
iajs-778	10	36	february	february	PROPN
iajs-778	10	37	,	,	PUNCT
iajs-778	10	38	2010	2010	NUM
iajs-778	10	39	accepted	accept	VERB
iajs-778	10	40	in	in	ADP
iajs-778	10	41	:	:	PUNCT
iajs-778	10	42	25	25	NUM
iajs-778	10	43	,	,	PUNCT
iajs-778	10	44	april	april	PROPN
iajs-778	10	45	,	,	PUNCT
iajs-778	10	46	2010	2010	NUM
iajs-778	10	47	abstract	abstract	ADV
iajs-778	10	48	let	let	VERB
iajs-778	10	49	r	r	PRON
iajs-778	10	50	be	be	AUX
iajs-778	10	51	a	a	DET
iajs-778	10	52	commutative	commutative	ADJ
iajs-778	10	53	ring	ring	NOUN
iajs-778	10	54	with	with	ADP
iajs-778	10	55	unity	unity	NOUN
iajs-778	10	56	and	and	CCONJ
iajs-778	10	57	let	let	VERB
iajs-778	10	58	m	m	PRON
iajs-778	10	59	be	be	AUX
iajs-778	10	60	a	a	DET
iajs-778	10	61	unitary	unitary	ADJ
iajs-778	10	62	r	r	NOUN
iajs-778	10	63	-	-	PUNCT
iajs-778	10	64	module	module	NOUN
iajs-778	10	65	.	.	PUNCT
iajs-778	11	1	let	let	VERB
iajs-778	11	2	n	n	PRON
iajs-778	11	3	be	be	AUX
iajs-778	11	4	a	a	DET
iajs-778	11	5	proper	proper	ADJ
iajs-778	11	6	submodule	submodule	NOUN
iajs-778	11	7	of	of	ADP
iajs-778	11	8	m	m	PROPN
iajs-778	11	9	,	,	PUNCT
iajs-778	11	10	n	n	PROPN
iajs-778	11	11	is	be	AUX
iajs-778	11	12	called	call	VERB
iajs-778	11	13	a	a	DET
iajs-778	11	14	coprime	coprime	NOUN
iajs-778	11	15	submodule	submodule	NOUN
iajs-778	11	16	if	if	SCONJ
iajs-778	11	17			PROPN
iajs-778	11	18			NOUN
iajs-778	11	19	is	be	AUX
iajs-778	11	20	a	a	DET
iajs-778	11	21	coprime	coprime	ADJ
iajs-778	11	22	r	r	NOUN
iajs-778	11	23	-	-	PUNCT
iajs-778	11	24	module	module	NOUN
iajs-778	11	25	,	,	PUNCT
iajs-778	11	26	where	where	SCONJ
iajs-778	11	27			PROPN
iajs-778	11	28			NOUN
iajs-778	11	29	is	be	AUX
iajs-778	11	30	a	a	DET
iajs-778	11	31	coprime	coprime	ADJ
iajs-778	11	32	r	r	NOUN
iajs-778	11	33	-	-	PUNCT
iajs-778	11	34	module	module	NOUN
iajs-778	11	35	if	if	SCONJ
iajs-778	11	36	for	for	ADP
iajs-778	11	37	any	any	DET
iajs-778	11	38	r	r	NOUN
iajs-778	11	39			NOUN
iajs-778	11	40	r	r	NOUN
iajs-778	11	41	,	,	PUNCT
iajs-778	11	42	either	either	CCONJ
iajs-778	11	43	o	o	PRON
iajs-778	11	44			VERB
iajs-778	11	45			PROPN
iajs-778	11	46			VERB
iajs-778	11	47			NOUN
iajs-778	11	48	r	r	NOUN
iajs-778	11	49	or	or	CCONJ
iajs-778	11	50			NUM
iajs-778	11	51			ADJ
iajs-778	11	52			PROPN
iajs-778	11	53			VERB
iajs-778	11	54			NOUN
iajs-778	11	55	r	r	NOUN
iajs-778	11	56	.	.	PUNCT
iajs-778	12	1	in	in	ADP
iajs-778	12	2	this	this	DET
iajs-778	12	3	paper	paper	NOUN
iajs-778	12	4	we	we	PRON
iajs-778	12	5	study	study	VERB
iajs-778	12	6	coprime	coprime	NOUN
iajs-778	12	7	submodules	submodule	NOUN
iajs-778	12	8	and	and	CCONJ
iajs-778	12	9	give	give	VERB
iajs-778	12	10	many	many	ADJ
iajs-778	12	11	properties	property	NOUN
iajs-778	12	12	related	relate	VERB
iajs-778	12	13	with	with	ADP
iajs-778	12	14	this	this	DET
iajs-778	12	15	concept	concept	NOUN
iajs-778	12	16	.	.	PUNCT
iajs-778	13	1	key	key	ADJ
iajs-778	13	2	words	word	NOUN
iajs-778	13	3	:	:	PUNCT
iajs-778	13	4	coprime	coprime	NOUN
iajs-778	13	5	submodules	submodule	NOUN
iajs-778	13	6	,	,	PUNCT
iajs-778	13	7	second	second	ADJ
iajs-778	13	8	submodule	submodule	NOUN
iajs-778	13	9	,	,	PUNCT
iajs-778	13	10	second	second	ADJ
iajs-778	13	11	(	(	PUNCT
iajs-778	13	12	coprime	coprime	NOUN
iajs-778	13	13	)	)	PUNCT
iajs-778	13	14	module	module	NOUN
iajs-778	13	15	,	,	PUNCT
iajs-778	13	16	secondary	secondary	ADJ
iajs-778	13	17	module	module	NOUN
iajs-778	13	18	.	.	PUNCT
iajs-778	14	1	introduction	introduction	NOUN
iajs-778	14	2	let	let	VERB
iajs-778	14	3	r	r	PRON
iajs-778	14	4	be	be	AUX
iajs-778	14	5	a	a	DET
iajs-778	14	6	commutative	commutative	ADJ
iajs-778	14	7	ring	ring	NOUN
iajs-778	14	8	with	with	ADP
iajs-778	14	9	unity	unity	NOUN
iajs-778	14	10	and	and	CCONJ
iajs-778	14	11	let	let	VERB
iajs-778	14	12	m	m	PRON
iajs-778	14	13	be	be	AUX
iajs-778	14	14	a	a	DET
iajs-778	14	15	unitary	unitary	ADJ
iajs-778	14	16	r	r	NOUN
iajs-778	14	17	-	-	PUNCT
iajs-778	14	18	module	module	NOUN
iajs-778	14	19	.	.	PUNCT
iajs-778	15	1	it	it	PRON
iajs-778	15	2	is	be	AUX
iajs-778	15	3	wellknown	wellknown	ADJ
iajs-778	15	4	that	that	SCONJ
iajs-778	15	5	a	a	DET
iajs-778	15	6	proper	proper	ADJ
iajs-778	15	7	submodule	submodule	NOUN
iajs-778	15	8	n	n	PROPN
iajs-778	15	9	of	of	ADP
iajs-778	15	10	an	an	DET
iajs-778	15	11	r	r	NOUN
iajs-778	15	12	-	-	PUNCT
iajs-778	15	13	module	module	NOUN
iajs-778	15	14	m	m	NOUN
iajs-778	15	15	is	be	AUX
iajs-778	15	16	called	call	VERB
iajs-778	15	17	prime	prime	ADJ
iajs-778	15	18	if	if	SCONJ
iajs-778	15	19	whenever	whenever	ADV
iajs-778	15	20	rr	rr	NOUN
iajs-778	15	21	,	,	PUNCT
iajs-778	15	22	xm	xm	NOUN
iajs-778	15	23	,	,	PUNCT
iajs-778	15	24	rxn	rxn	NOUN
iajs-778	15	25	implies	imply	VERB
iajs-778	15	26	xn	xn	PROPN
iajs-778	15	27	or	or	CCONJ
iajs-778	15	28	r	r	VERB
iajs-778	15	29			NOUN
iajs-778	16	1	[	[	X
iajs-778	16	2	n	n	CCONJ
iajs-778	16	3	:	:	PUNCT
iajs-778	16	4	m	m	VERB
iajs-778	16	5	]	]	X
iajs-778	16	6	,	,	PUNCT
iajs-778	16	7	where	where	SCONJ
iajs-778	16	8	[	[	X
iajs-778	16	9	n	n	CCONJ
iajs-778	16	10	:	:	PUNCT
iajs-778	16	11	m]={rr	m]={rr	NUM
iajs-778	16	12	:	:	PUNCT
iajs-778	16	13	rmn	rmn	NOUN
iajs-778	16	14	}	}	PUNCT
iajs-778	16	15	.	.	PUNCT
iajs-778	17	1	m	m	PROPN
iajs-778	17	2	is	be	AUX
iajs-778	17	3	called	call	VERB
iajs-778	17	4	a	a	DET
iajs-778	17	5	prime	prime	ADJ
iajs-778	17	6	module	module	NOUN
iajs-778	17	7	if	if	SCONJ
iajs-778	17	8	r	r	NOUN
iajs-778	17	9	ann	ann	PROPN
iajs-778	17	10	m	m	NOUN
iajs-778	17	11	=	=	SYM
iajs-778	17	12	r	r	NOUN
iajs-778	17	13	ann	ann	PROPN
iajs-778	17	14	n	n	PROPN
iajs-778	17	15	for	for	ADP
iajs-778	17	16	all	all	DET
iajs-778	17	17	nonzero	nonzero	PROPN
iajs-778	17	18	submodule	submodule	PROPN
iajs-778	17	19	n	n	PROPN
iajs-778	17	20	of	of	ADP
iajs-778	17	21	m	m	PROPN
iajs-778	17	22	,	,	PUNCT
iajs-778	17	23	equivalently	equivalently	ADV
iajs-778	17	24	m	m	VERB
iajs-778	17	25	is	be	AUX
iajs-778	17	26	a	a	DET
iajs-778	17	27	prime	prime	ADJ
iajs-778	17	28	module	module	NOUN
iajs-778	17	29	iff	iff	PROPN
iajs-778	17	30	(	(	PUNCT
iajs-778	17	31	0	0	NUM
iajs-778	17	32	)	)	PUNCT
iajs-778	17	33	is	be	AUX
iajs-778	17	34	a	a	DET
iajs-778	17	35	prime	prime	ADJ
iajs-778	17	36	submodule	submodule	NOUN
iajs-778	17	37	.	.	PUNCT
iajs-778	18	1	s.yassem	s.yassem	PROPN
iajs-778	18	2	in	in	ADP
iajs-778	18	3	[	[	X
iajs-778	18	4	7	7	NUM
iajs-778	18	5	]	]	PUNCT
iajs-778	18	6	,	,	PUNCT
iajs-778	18	7	introduced	introduce	VERB
iajs-778	18	8	the	the	DET
iajs-778	18	9	notions	notion	NOUN
iajs-778	18	10	of	of	ADP
iajs-778	18	11	second	second	ADJ
iajs-778	18	12	submodules	submodule	NOUN
iajs-778	18	13	and	and	CCONJ
iajs-778	18	14	second	second	ADJ
iajs-778	18	15	modules	module	NOUN
iajs-778	18	16	,	,	PUNCT
iajs-778	18	17	where	where	SCONJ
iajs-778	18	18	a	a	DET
iajs-778	18	19	submodule	submodule	NOUN
iajs-778	18	20	n	n	PROPN
iajs-778	18	21	of	of	ADP
iajs-778	18	22	m	m	PROPN
iajs-778	18	23	is	be	AUX
iajs-778	18	24	called	call	VERB
iajs-778	18	25	second	second	ADJ
iajs-778	18	26	if	if	SCONJ
iajs-778	18	27	for	for	ADP
iajs-778	18	28	any	any	DET
iajs-778	18	29	r	r	NOUN
iajs-778	18	30	r	r	NOUN
iajs-778	18	31	,	,	PUNCT
iajs-778	18	32	the	the	DET
iajs-778	18	33	homothety	homothety	NOUN
iajs-778	18	34	r*end	r*end	PROPN
iajs-778	18	35	m	m	PROPN
iajs-778	18	36	,	,	PUNCT
iajs-778	18	37	is	be	AUX
iajs-778	18	38	either	either	DET
iajs-778	18	39	zero	zero	NUM
iajs-778	18	40	or	or	CCONJ
iajs-778	18	41	surjective	surjective	ADJ
iajs-778	18	42	,	,	PUNCT
iajs-778	18	43	where	where	SCONJ
iajs-778	18	44	r*(m	r*(m	VERB
iajs-778	18	45	)	)	PUNCT
iajs-778	19	1	=	=	SYM
iajs-778	19	2	r	r	NOUN
iajs-778	19	3	m	m	PROPN
iajs-778	19	4	,	,	PUNCT
iajs-778	19	5			PROPN
iajs-778	19	6	m	m	PROPN
iajs-778	19	7			NOUN
iajs-778	19	8	m.	m.	NOUN
iajs-778	19	9	it	it	PRON
iajs-778	19	10	follows	follow	VERB
iajs-778	19	11	that	that	SCONJ
iajs-778	19	12	n	n	PRON
iajs-778	19	13	is	be	AUX
iajs-778	19	14	a	a	DET
iajs-778	19	15	second	second	ADJ
iajs-778	19	16	submodule	submodule	NOUN
iajs-778	19	17	iff	iff	NOUN
iajs-778	19	18	for	for	ADP
iajs-778	19	19	each	each	DET
iajs-778	19	20	r	r	NOUN
iajs-778	19	21	r	r	NOUN
iajs-778	19	22	,	,	PUNCT
iajs-778	19	23	either	either	CCONJ
iajs-778	19	24	rn	rn	PROPN
iajs-778	19	25	=	=	PROPN
iajs-778	19	26	0	0	PROPN
iajs-778	19	27	or	or	CCONJ
iajs-778	19	28	rn	rn	PROPN
iajs-778	19	29	=	=	PROPN
iajs-778	20	1	n.	n.	PROPN
iajs-778	20	2	m	m	PROPN
iajs-778	20	3	is	be	AUX
iajs-778	20	4	called	call	VERB
iajs-778	20	5	a	a	DET
iajs-778	20	6	second	second	ADJ
iajs-778	20	7	module	module	NOUN
iajs-778	20	8	if	if	SCONJ
iajs-778	20	9	m	m	NOUN
iajs-778	20	10	is	be	AUX
iajs-778	20	11	a	a	DET
iajs-778	20	12	second	second	ADJ
iajs-778	20	13	submodule	submodule	NOUN
iajs-778	20	14	of	of	ADP
iajs-778	20	15	itself	itself	PRON
iajs-778	20	16	.	.	PUNCT
iajs-778	21	1	for	for	ADP
iajs-778	21	2	an	an	DET
iajs-778	21	3	r	r	NOUN
iajs-778	21	4	-	-	PUNCT
iajs-778	21	5	module	module	NOUN
iajs-778	21	6	m	m	NOUN
iajs-778	21	7	,	,	PUNCT
iajs-778	21	8	the	the	DET
iajs-778	21	9	following	follow	VERB
iajs-778	21	10	statements	statement	NOUN
iajs-778	21	11	are	be	AUX
iajs-778	21	12	equivalent	equivalent	ADJ
iajs-778	21	13	:	:	PUNCT
iajs-778	21	14	(	(	PUNCT
iajs-778	21	15	1	1	X
iajs-778	21	16	)	)	PUNCT
iajs-778	21	17	m	m	VERB
iajs-778	21	18	is	be	AUX
iajs-778	21	19	a	a	DET
iajs-778	21	20	second	second	ADJ
iajs-778	21	21	module	module	NOUN
iajs-778	21	22	.	.	PUNCT
iajs-778	22	1	(	(	PUNCT
iajs-778	22	2	2	2	X
iajs-778	22	3	)	)	PUNCT
iajs-778	22	4	for	for	ADP
iajs-778	22	5	each	each	DET
iajs-778	22	6	r	r	NOUN
iajs-778	22	7	r	r	NOUN
iajs-778	22	8	,	,	PUNCT
iajs-778	22	9	either	either	CCONJ
iajs-778	22	10	rm	rm	PROPN
iajs-778	22	11	=	=	SYM
iajs-778	22	12	0	0	PROPN
iajs-778	22	13	or	or	CCONJ
iajs-778	22	14	rm	rm	NOUN
iajs-778	22	15	=	=	NOUN
iajs-778	22	16	m	m	PROPN
iajs-778	22	17	.	.	PUNCT
iajs-778	23	1	(	(	PUNCT
iajs-778	23	2	3	3	X
iajs-778	23	3	)	)	PUNCT
iajs-778	23	4	ann	ann	PROPN
iajs-778	23	5	m	m	PROPN
iajs-778	23	6	=	=	PROPN
iajs-778	23	7	ann	ann	PROPN
iajs-778	23	8			PROPN
iajs-778	23	9			PROPN
iajs-778	23	10	for	for	ADP
iajs-778	23	11	all	all	DET
iajs-778	23	12	proper	proper	ADJ
iajs-778	23	13	submodules	submodule	NOUN
iajs-778	23	14	n	n	PROPN
iajs-778	23	15	of	of	ADP
iajs-778	23	16	m	m	PROPN
iajs-778	23	17	.	.	PUNCT
iajs-778	24	1	(	(	PUNCT
iajs-778	24	2	4	4	X
iajs-778	24	3	)	)	PUNCT
iajs-778	24	4	ann	ann	PROPN
iajs-778	24	5	m	m	PROPN
iajs-778	24	6	=	=	PROPN
iajs-778	24	7	ann	ann	PROPN
iajs-778	24	8			PROPN
iajs-778	24	9			PROPN
iajs-778	24	10	for	for	ADP
iajs-778	24	11	all	all	DET
iajs-778	24	12	fully	fully	ADV
iajs-778	24	13	invariant	invariant	ADJ
iajs-778	24	14	sub3	sub3	NOUN
iajs-778	24	15	(	(	PUNCT
iajs-778	24	16	5	5	NUM
iajs-778	24	17	)	)	PUNCT
iajs-778	24	18	modules	module	NOUN
iajs-778	24	19	n	n	PRON
iajs-778	24	20	of	of	ADP
iajs-778	24	21	m	m	PROPN
iajs-778	24	22	.	.	PUNCT
iajs-778	25	1	(	(	PUNCT
iajs-778	25	2	6	6	NUM
iajs-778	25	3	)	)	PUNCT
iajs-778	25	4	ann	ann	PROPN
iajs-778	25	5	m	m	PROPN
iajs-778	25	6	=	=	SYM
iajs-778	25	7	w(m	w(m	PROPN
iajs-778	25	8	)	)	PUNCT
iajs-778	25	9	,	,	PUNCT
iajs-778	25	10	where	where	SCONJ
iajs-778	25	11	w(m)={r	w(m)={r	AUX
iajs-778	25	12	r	r	NOUN
iajs-778	25	13	:	:	PUNCT
iajs-778	25	14	r*end	r*end	NOUN
iajs-778	25	15	m	m	PROPN
iajs-778	25	16	,	,	PUNCT
iajs-778	25	17	r	r	X
iajs-778	25	18	*	*	PUNCT
iajs-778	25	19	is	be	AUX
iajs-778	25	20	not	not	PART
iajs-778	25	21	surjective	surjective	ADJ
iajs-778	25	22	}	}	PUNCT
iajs-778	25	23	.	.	PUNCT
iajs-778	26	1	notice	notice	NOUN
iajs-778	26	2	(	(	PUNCT
iajs-778	26	3	1	1	NUM
iajs-778	26	4	)	)	PUNCT
iajs-778	26	5			NOUN
iajs-778	26	6	(	(	PUNCT
iajs-778	26	7	2	2	X
iajs-778	26	8	)	)	PUNCT
iajs-778	26	9	is	be	AUX
iajs-778	26	10	clear	clear	ADJ
iajs-778	26	11	,	,	PUNCT
iajs-778	26	12	(	(	PUNCT
iajs-778	26	13	1	1	X
iajs-778	26	14	)	)	PUNCT
iajs-778	26	15			NOUN
iajs-778	26	16	(	(	PUNCT
iajs-778	26	17	5	5	X
iajs-778	26	18	)	)	PUNCT
iajs-778	27	1	[	[	X
iajs-778	27	2	7,lemma	7,lemma	PROPN
iajs-778	27	3	1.2	1.2	NUM
iajs-778	27	4	]	]	PUNCT
iajs-778	27	5	,	,	PUNCT
iajs-778	27	6	(	(	PUNCT
iajs-778	27	7	1	1	X
iajs-778	27	8	)	)	PUNCT
iajs-778	27	9			NOUN
iajs-778	27	10	(	(	PUNCT
iajs-778	27	11	3	3	X
iajs-778	27	12	)	)	PUNCT
iajs-778	27	13	[	[	X
iajs-778	27	14	3	3	NUM
iajs-778	27	15	,	,	PUNCT
iajs-778	27	16	theorem	theorem	VERB
iajs-778	27	17	2.1.6	2.1.6	NOUN
iajs-778	27	18	]	]	X
iajs-778	27	19	,	,	PUNCT
iajs-778	27	20	(	(	PUNCT
iajs-778	27	21	3	3	X
iajs-778	27	22	)	)	PUNCT
iajs-778	27	23			NOUN
iajs-778	27	24	(	(	PUNCT
iajs-778	27	25	4	4	X
iajs-778	27	26	)	)	PUNCT
iajs-778	28	1	[	[	X
iajs-778	28	2	6	6	NUM
iajs-778	28	3	,	,	PUNCT
iajs-778	28	4	theorem	theorem	VERB
iajs-778	28	5	1.3.2	1.3.2	NUM
iajs-778	28	6	]	]	PUNCT
iajs-778	28	7	.	.	PUNCT
iajs-778	29	1	notice	notice	VERB
iajs-778	29	2	that	that	SCONJ
iajs-778	29	3	statement	statement	NOUN
iajs-778	29	4	(	(	PUNCT
iajs-778	29	5	3	3	NUM
iajs-778	29	6	)	)	PUNCT
iajs-778	29	7	and	and	CCONJ
iajs-778	29	8	statement	statement	NOUN
iajs-778	29	9	(	(	PUNCT
iajs-778	29	10	4	4	NUM
iajs-778	29	11	)	)	PUNCT
iajs-778	29	12	are	be	AUX
iajs-778	29	13	used	use	VERB
iajs-778	29	14	to	to	PART
iajs-778	29	15	define	define	VERB
iajs-778	29	16	coprime	coprime	NOUN
iajs-778	29	17	module	module	NOUN
iajs-778	29	18	by	by	ADP
iajs-778	29	19	s.	s.	PROPN
iajs-778	29	20	annin	annin	PROPN
iajs-778	29	21	in	in	ADP
iajs-778	29	22	[	[	X
iajs-778	29	23	2	2	NUM
iajs-778	29	24	]	]	PUNCT
iajs-778	29	25	and	and	CCONJ
iajs-778	29	26	i.e	i.e	PRON
iajs-778	29	27	wijayart	wijayart	ADJ
iajs-778	29	28	in	in	ADP
iajs-778	29	29	[	[	X
iajs-778	29	30	6	6	NUM
iajs-778	29	31	]	]	PUNCT
iajs-778	29	32	,	,	PUNCT
iajs-778	29	33	respectively	respectively	ADV
iajs-778	29	34	.	.	PUNCT
iajs-778	30	1	ibn	ibn	PROPN
iajs-778	30	2	alhaitham	alhaitham	PROPN
iajs-778	30	3	j.	j.	PROPN
iajs-778	30	4	for	for	ADP
iajs-778	30	5	pure	pure	ADJ
iajs-778	30	6	&	&	CCONJ
iajs-778	30	7	appl	appl	PROPN
iajs-778	30	8	.	.	PUNCT
iajs-778	31	1	sci	sci	PROPN
iajs-778	31	2	.	.	PUNCT
iajs-778	31	3	vol.24	vol.24	NOUN
iajs-778	31	4	(	(	PUNCT
iajs-778	31	5	2	2	NUM
iajs-778	31	6	)	)	PUNCT
iajs-778	31	7	2011	2011	NUM
iajs-778	31	8	moreover	moreover	ADV
iajs-778	31	9	rasha	rasha	VERB
iajs-778	31	10	in	in	ADP
iajs-778	31	11	[	[	X
iajs-778	31	12	3	3	NUM
iajs-778	31	13	]	]	PUNCT
iajs-778	31	14	studied	study	VERB
iajs-778	31	15	coprime	coprime	NOUN
iajs-778	31	16	modules	module	NOUN
iajs-778	31	17	and	and	CCONJ
iajs-778	31	18	give	give	VERB
iajs-778	31	19	some	some	DET
iajs-778	31	20	generalizations	generalization	NOUN
iajs-778	31	21	of	of	ADP
iajs-778	31	22	these	these	DET
iajs-778	31	23	modules	module	NOUN
iajs-778	31	24	,	,	PUNCT
iajs-778	31	25	(	(	PUNCT
iajs-778	31	26	see	see	VERB
iajs-778	31	27	[	[	X
iajs-778	31	28	3	3	NUM
iajs-778	31	29	]	]	NUM
iajs-778	31	30	)	)	PUNCT
iajs-778	31	31	.	.	PUNCT
iajs-778	32	1	j.abuhilail	j.abuhilail	NOUN
iajs-778	32	2	in	in	ADP
iajs-778	32	3	[	[	X
iajs-778	32	4	1	1	NUM
iajs-778	32	5	]	]	PUNCT
iajs-778	32	6	,	,	PUNCT
iajs-778	32	7	introduced	introduce	VERB
iajs-778	32	8	the	the	DET
iajs-778	32	9	notion	notion	NOUN
iajs-778	32	10	of	of	ADP
iajs-778	32	11	coprime	coprime	NOUN
iajs-778	32	12	submodule	submodule	NOUN
iajs-778	32	13	,	,	PUNCT
iajs-778	32	14	where	where	SCONJ
iajs-778	32	15	a	a	DET
iajs-778	32	16	proper	proper	ADJ
iajs-778	32	17	submodule	submodule	NOUN
iajs-778	32	18	n	n	PROPN
iajs-778	32	19	of	of	ADP
iajs-778	32	20	m	m	PROPN
iajs-778	32	21	is	be	AUX
iajs-778	32	22	called	call	VERB
iajs-778	32	23	coprime	coprime	ADV
iajs-778	32	24	if	if	SCONJ
iajs-778	32	25	ann	ann	PROPN
iajs-778	32	26			PROPN
iajs-778	32	27			PROPN
iajs-778	32	28	=	=	SYM
iajs-778	32	29	w	w	PROPN
iajs-778	32	30	(	(	PUNCT
iajs-778	32	31			PROPN
iajs-778	32	32			NOUN
iajs-778	32	33	)	)	PUNCT
iajs-778	32	34	;	;	PUNCT
iajs-778	32	35	that	that	PRON
iajs-778	32	36	is	be	AUX
iajs-778	32	37	n	n	PRON
iajs-778	32	38	is	be	AUX
iajs-778	32	39	a	a	DET
iajs-778	32	40	coprime	coprime	ADJ
iajs-778	32	41	submodule	submodule	NOUN
iajs-778	32	42	if	if	SCONJ
iajs-778	32	43			PROPN
iajs-778	32	44			NOUN
iajs-778	32	45	is	be	AUX
iajs-778	32	46	a	a	DET
iajs-778	32	47	coprime	coprime	ADJ
iajs-778	32	48	r	r	NOUN
iajs-778	32	49	-	-	NOUN
iajs-778	32	50	module	module	NOUN
iajs-778	32	51	.	.	PUNCT
iajs-778	33	1	our	our	PRON
iajs-778	33	2	aim	aim	NOUN
iajs-778	33	3	in	in	ADP
iajs-778	33	4	this	this	DET
iajs-778	33	5	paper	paper	NOUN
iajs-778	33	6	is	be	AUX
iajs-778	33	7	to	to	PART
iajs-778	33	8	study	study	VERB
iajs-778	33	9	coprime	coprime	NOUN
iajs-778	33	10	submodules	submodule	NOUN
iajs-778	33	11	,	,	PUNCT
iajs-778	33	12	we	we	PRON
iajs-778	33	13	give	give	VERB
iajs-778	33	14	the	the	DET
iajs-778	33	15	basic	basic	ADJ
iajs-778	33	16	properties	property	NOUN
iajs-778	33	17	about	about	ADP
iajs-778	33	18	this	this	DET
iajs-778	33	19	concept	concept	NOUN
iajs-778	33	20	.	.	PUNCT
iajs-778	34	1	also	also	ADV
iajs-778	34	2	,	,	PUNCT
iajs-778	34	3	we	we	PRON
iajs-778	34	4	study	study	VERB
iajs-778	34	5	coprime	coprime	NOUN
iajs-778	34	6	submodules	submodule	NOUN
iajs-778	34	7	in	in	ADP
iajs-778	34	8	certain	certain	ADJ
iajs-778	34	9	classes	class	NOUN
iajs-778	34	10	of	of	ADP
iajs-778	34	11	modules	module	NOUN
iajs-778	34	12	.	.	PUNCT
iajs-778	35	1	1coprime	1coprime	NUM
iajs-778	35	2	submodules	submodule	NOUN
iajs-778	35	3	we	we	PRON
iajs-778	35	4	give	give	VERB
iajs-778	35	5	the	the	DET
iajs-778	35	6	basic	basic	ADJ
iajs-778	35	7	properties	property	NOUN
iajs-778	35	8	related	relate	VERB
iajs-778	35	9	with	with	ADP
iajs-778	35	10	coprime	coprime	NOUN
iajs-778	35	11	submodules	submodule	NOUN
iajs-778	35	12	.	.	PUNCT
iajs-778	36	1	also	also	ADV
iajs-778	36	2	,	,	PUNCT
iajs-778	36	3	we	we	PRON
iajs-778	36	4	study	study	VERB
iajs-778	36	5	their	their	PRON
iajs-778	36	6	behaviour	behaviour	NOUN
iajs-778	36	7	in	in	ADP
iajs-778	36	8	certain	certain	ADJ
iajs-778	36	9	classes	class	NOUN
iajs-778	36	10	of	of	ADP
iajs-778	36	11	modules	module	NOUN
iajs-778	36	12	.	.	PUNCT
iajs-778	37	1	following	follow	VERB
iajs-778	37	2	j.abuhilail	j.abuhilail	NOUN
iajs-778	37	3	in	in	ADP
iajs-778	37	4	[	[	X
iajs-778	37	5	1	1	NUM
iajs-778	37	6	]	]	PUNCT
iajs-778	37	7	,	,	PUNCT
iajs-778	37	8	a	a	DET
iajs-778	37	9	proper	proper	ADJ
iajs-778	37	10	submodule	submodule	NOUN
iajs-778	37	11	n	n	PROPN
iajs-778	37	12	of	of	ADP
iajs-778	37	13	an	an	DET
iajs-778	37	14	r	r	NOUN
iajs-778	37	15	-	-	PUNCT
iajs-778	37	16	module	module	NOUN
iajs-778	37	17	m	m	NOUN
iajs-778	37	18	is	be	AUX
iajs-778	37	19	called	call	VERB
iajs-778	37	20	coprime	coprime	ADV
iajs-778	37	21	if	if	SCONJ
iajs-778	37	22			PROPN
iajs-778	37	23			NOUN
iajs-778	37	24	is	be	AUX
iajs-778	37	25	a	a	DET
iajs-778	37	26	coprime	coprime	ADJ
iajs-778	37	27	r	r	NOUN
iajs-778	37	28	-	-	NOUN
iajs-778	37	29	module	module	NOUN
iajs-778	37	30	.	.	PUNCT
iajs-778	38	1	an	an	DET
iajs-778	38	2	ideal	ideal	ADJ
iajs-778	38	3	i	i	PRON
iajs-778	38	4	of	of	ADP
iajs-778	38	5	a	a	DET
iajs-778	38	6	ring	ring	NOUN
iajs-778	38	7	r	r	NOUN
iajs-778	38	8	is	be	AUX
iajs-778	38	9	called	call	VERB
iajs-778	38	10	coprime	coprime	ADJ
iajs-778	38	11	ideal	ideal	NOUN
iajs-778	38	12	iff	iff	PROPN
iajs-778	38	13	r	r	PROPN
iajs-778	38	14			PROPN
iajs-778	38	15	is	be	AUX
iajs-778	38	16	a	a	DET
iajs-778	38	17	coprime	coprime	ADJ
iajs-778	38	18	r	r	NOUN
iajs-778	38	19	-	-	NOUN
iajs-778	38	20	module	module	NOUN
iajs-778	38	21	.	.	PUNCT
iajs-778	39	1	1.1	1.1	NUM
iajs-778	39	2	remarks	remark	NOUN
iajs-778	39	3	and	and	CCONJ
iajs-778	39	4	examples	example	NOUN
iajs-778	39	5	:	:	PUNCT
iajs-778	39	6	(	(	PUNCT
iajs-778	39	7	1	1	X
iajs-778	39	8	)	)	PUNCT
iajs-778	39	9	n	n	VERB
iajs-778	39	10	is	be	AUX
iajs-778	39	11	coprime	coprime	ADJ
iajs-778	39	12	submodule	submodule	NOUN
iajs-778	39	13	iff	iff	PROPN
iajs-778	39	14	for	for	ADP
iajs-778	39	15	each	each	DET
iajs-778	39	16	r	r	NOUN
iajs-778	39	17			NOUN
iajs-778	39	18	r	r	NOUN
iajs-778	39	19	either	either	CCONJ
iajs-778	39	20	o	o	PROPN
iajs-778	39	21			PROPN
iajs-778	39	22			PROPN
iajs-778	39	23			VERB
iajs-778	39	24			NOUN
iajs-778	39	25	r	r	NOUN
iajs-778	39	26	=	=	PUNCT
iajs-778	39	27	n	n	NOUN
iajs-778	39	28	or	or	CCONJ
iajs-778	39	29			NUM
iajs-778	39	30			ADJ
iajs-778	39	31			PROPN
iajs-778	39	32			VERB
iajs-778	39	33			NOUN
iajs-778	39	34	r	r	NOUN
iajs-778	39	35	,	,	PUNCT
iajs-778	39	36	that	that	PRON
iajs-778	39	37	is	be	AUX
iajs-778	39	38	n	n	PRON
iajs-778	39	39	is	be	AUX
iajs-778	39	40	a	a	DET
iajs-778	39	41	coprime	coprime	ADJ
iajs-778	39	42	submodule	submodule	NOUN
iajs-778	39	43	if	if	SCONJ
iajs-778	39	44	for	for	ADP
iajs-778	39	45	each	each	DET
iajs-778	39	46	r	r	NOUN
iajs-778	39	47	r	r	NOUN
iajs-778	39	48	,	,	PUNCT
iajs-778	39	49	either	either	CCONJ
iajs-778	39	50	r	r	NOUN
iajs-778	39	51			NOUN
iajs-778	39	52	[	[	X
iajs-778	39	53	n	n	CCONJ
iajs-778	39	54	:	:	PUNCT
iajs-778	39	55	m	m	VERB
iajs-778	39	56	]	]	X
iajs-778	39	57	or	or	CCONJ
iajs-778	39	58	for	for	ADP
iajs-778	39	59	any	any	DET
iajs-778	39	60	m	m	NOUN
iajs-778	39	61			NOUN
iajs-778	39	62	m	m	ADP
iajs-778	39	63	,	,	PUNCT
iajs-778	39	64	there	there	PRON
iajs-778	39	65	exists	exist	VERB
iajs-778	39	66	m	m	NOUN
iajs-778	39	67	'	'	PART
iajs-778	39	68			NOUN
iajs-778	39	69	m	m	VERB
iajs-778	39	70	such	such	ADJ
iajs-778	39	71	that	that	SCONJ
iajs-778	39	72	m	m	VERB
iajs-778	39	73	–	–	PUNCT
iajs-778	39	74	r	r	NOUN
iajs-778	39	75	m	m	NOUN
iajs-778	39	76	'	'	PART
iajs-778	39	77			PROPN
iajs-778	39	78	n.	n.	NOUN
iajs-778	39	79	(	(	PUNCT
iajs-778	39	80	2	2	NUM
iajs-778	39	81	)	)	PUNCT
iajs-778	40	1	z	z	NOUN
iajs-778	40	2	is	be	AUX
iajs-778	40	3	a	a	DET
iajs-778	40	4	coprime	coprime	ADJ
iajs-778	40	5	submodule	submodule	NOUN
iajs-778	40	6	of	of	ADP
iajs-778	40	7	the	the	DET
iajs-778	40	8	z	z	NOUN
iajs-778	40	9	-	-	PUNCT
iajs-778	40	10	module	module	NOUN
iajs-778	40	11	q	q	NOUN
iajs-778	40	12	,	,	PUNCT
iajs-778	40	13	since	since	SCONJ
iajs-778	40	14	q	q	PROPN
iajs-778	40	15	z	z	NOUN
iajs-778	40	16	is	be	AUX
iajs-778	40	17	a	a	DET
iajs-778	40	18	coprime	coprime	ADJ
iajs-778	40	19	z	z	NOUN
iajs-778	40	20	-	-	PUNCT
iajs-778	40	21	module	module	NOUN
iajs-778	40	22	[	[	X
iajs-778	40	23	4	4	NUM
iajs-778	40	24	]	]	PUNCT
iajs-778	40	25	,	,	PUNCT
iajs-778	40	26	[	[	X
iajs-778	40	27	6	6	NUM
iajs-778	40	28	]	]	PUNCT
iajs-778	40	29	.	.	PUNCT
iajs-778	41	1	note	note	VERB
iajs-778	41	2	that	that	SCONJ
iajs-778	41	3	z	z	NOUN
iajs-778	41	4	is	be	AUX
iajs-778	41	5	not	not	PART
iajs-778	41	6	coprime	coprime	ADJ
iajs-778	41	7	z	z	NOUN
iajs-778	41	8	-	-	PUNCT
iajs-778	41	9	module	module	NOUN
iajs-778	41	10	,	,	PUNCT
iajs-778	41	11	since	since	SCONJ
iajs-778	41	12	when	when	SCONJ
iajs-778	41	13	r	r	NOUN
iajs-778	41	14	=	=	SYM
iajs-778	41	15	2	2	NUM
iajs-778	41	16			NOUN
iajs-778	41	17	0	0	NUM
iajs-778	41	18	,	,	PUNCT
iajs-778	41	19	2z	2z	NUM
iajs-778	42	1			NOUN
iajs-778	42	2	z.	z.	X
iajs-778	42	3	(	(	PUNCT
iajs-778	42	4	3	3	X
iajs-778	42	5	)	)	PUNCT
iajs-778	42	6	every	every	DET
iajs-778	42	7	submodule	submodule	NOUN
iajs-778	42	8	n	n	PROPN
iajs-778	42	9	of	of	ADP
iajs-778	42	10	the	the	DET
iajs-778	42	11	z	z	NOUN
iajs-778	42	12	-	-	PUNCT
iajs-778	42	13	module	module	NOUN
iajs-778	42	14			PROPN
iajs-778	42	15	p	p	PROPN
iajs-778	42	16	is	be	AUX
iajs-778	42	17	a	a	DET
iajs-778	42	18	coprime	coprime	NOUN
iajs-778	42	19	submodule	submodule	NOUN
iajs-778	42	20	,	,	PUNCT
iajs-778	42	21	since	since	SCONJ
iajs-778	42	22			PROPN
iajs-778	42	23	p	p	X
iajs-778	42	24	/n	/n	PUNCT
iajs-778	42	25			PROPN
iajs-778	42	26			PROPN
iajs-778	42	27	p	p	NOUN
iajs-778	42	28	and	and	CCONJ
iajs-778	42	29			PROPN
iajs-778	42	30	p	p	PROPN
iajs-778	42	31	is	be	AUX
iajs-778	42	32	a	a	DET
iajs-778	42	33	coprime	coprime	ADJ
iajs-778	42	34	z	z	NOUN
iajs-778	42	35	-	-	PUNCT
iajs-778	42	36	module	module	NOUN
iajs-778	42	37	,	,	PUNCT
iajs-778	42	38	hence	hence	ADV
iajs-778	42	39			PROPN
iajs-778	43	1	p	p	X
iajs-778	43	2	/n	/n	PUNCT
iajs-778	43	3	is	be	AUX
iajs-778	43	4	a	a	DET
iajs-778	43	5	coprime	coprime	ADJ
iajs-778	43	6	z	z	NOUN
iajs-778	43	7	-	-	PUNCT
iajs-778	43	8	module	module	NOUN
iajs-778	43	9	.	.	PUNCT
iajs-778	44	1	(	(	PUNCT
iajs-778	44	2	4	4	X
iajs-778	44	3	)	)	PUNCT
iajs-778	44	4	let	let	VERB
iajs-778	44	5	m	m	PRON
iajs-778	44	6	be	be	AUX
iajs-778	44	7	a	a	DET
iajs-778	44	8	coprime	coprime	ADJ
iajs-778	44	9	r	r	NOUN
iajs-778	44	10	-	-	NOUN
iajs-778	44	11	module	module	NOUN
iajs-778	44	12	,	,	PUNCT
iajs-778	44	13	then	then	ADV
iajs-778	44	14	every	every	DET
iajs-778	44	15	proper	proper	ADJ
iajs-778	44	16	submodule	submodule	NOUN
iajs-778	44	17	n	n	PROPN
iajs-778	44	18	of	of	ADP
iajs-778	44	19	m	m	PROPN
iajs-778	44	20	is	be	AUX
iajs-778	44	21	a	a	DET
iajs-778	44	22	coprime	coprime	NOUN
iajs-778	44	23	submodule	submodule	NOUN
iajs-778	44	24	.	.	PUNCT
iajs-778	45	1	proof	proof	NOUN
iajs-778	45	2	:	:	PUNCT
iajs-778	45	3	since	since	SCONJ
iajs-778	45	4	m	m	PROPN
iajs-778	45	5	is	be	AUX
iajs-778	45	6	a	a	DET
iajs-778	45	7	coprime	coprime	ADJ
iajs-778	45	8	r	r	NOUN
iajs-778	45	9	-	-	NOUN
iajs-778	45	10	module	module	NOUN
iajs-778	45	11	,	,	PUNCT
iajs-778	45	12	then	then	ADV
iajs-778	45	13	by	by	ADP
iajs-778	45	14	[	[	X
iajs-778	45	15	3,cor	3,cor	NUM
iajs-778	45	16	.	.	NOUN
iajs-778	45	17	2.1.12	2.1.12	NUM
iajs-778	45	18	]	]	X
iajs-778	45	19	,	,	PUNCT
iajs-778	45	20			PROPN
iajs-778	45	21			NOUN
iajs-778	45	22	is	be	AUX
iajs-778	45	23	a	a	DET
iajs-778	45	24	coprime	coprime	NOUN
iajs-778	45	25	rmodule	rmodule	NOUN
iajs-778	45	26	,	,	PUNCT
iajs-778	45	27	for	for	ADP
iajs-778	45	28	all	all	DET
iajs-778	45	29	n	n	NOUN
iajs-778	45	30	<	<	X
iajs-778	45	31	m.	m.	NOUN
iajs-778	45	32	hence	hence	ADV
iajs-778	45	33	n	n	ADV
iajs-778	45	34	is	be	AUX
iajs-778	45	35	a	a	DET
iajs-778	45	36	coprime	coprime	NOUN
iajs-778	45	37	submodule	submodule	NOUN
iajs-778	45	38	.	.	PUNCT
iajs-778	46	1	(	(	PUNCT
iajs-778	46	2	5	5	X
iajs-778	46	3	)	)	PUNCT
iajs-778	46	4	if	if	SCONJ
iajs-778	46	5	n	n	PRON
iajs-778	46	6	is	be	AUX
iajs-778	46	7	a	a	DET
iajs-778	46	8	maximal	maximal	ADJ
iajs-778	46	9	submodule	submodule	NOUN
iajs-778	46	10	of	of	ADP
iajs-778	46	11	an	an	DET
iajs-778	46	12	r	r	NOUN
iajs-778	46	13	-	-	PUNCT
iajs-778	46	14	module	module	NOUN
iajs-778	46	15	m	m	NOUN
iajs-778	46	16	,	,	PUNCT
iajs-778	46	17	then	then	ADV
iajs-778	46	18	n	n	PRON
iajs-778	46	19	is	be	AUX
iajs-778	46	20	a	a	DET
iajs-778	46	21	coprime	coprime	NOUN
iajs-778	46	22	submodule	submodule	NOUN
iajs-778	46	23	.	.	PUNCT
iajs-778	47	1	proof	proof	NOUN
iajs-778	47	2	:	:	PUNCT
iajs-778	47	3	since	since	SCONJ
iajs-778	47	4	n	n	PRON
iajs-778	47	5	is	be	AUX
iajs-778	47	6	maximal	maximal	ADJ
iajs-778	47	7	,	,	PUNCT
iajs-778	47	8			PROPN
iajs-778	47	9			NOUN
iajs-778	47	10	is	be	AUX
iajs-778	47	11	a	a	DET
iajs-778	47	12	simple	simple	ADJ
iajs-778	47	13	r	r	NOUN
iajs-778	47	14	-	-	PUNCT
iajs-778	47	15	module	module	NOUN
iajs-778	47	16	,	,	PUNCT
iajs-778	47	17	hence	hence	ADV
iajs-778	47	18			PROPN
iajs-778	47	19			PROPN
iajs-778	47	20	is	be	AUX
iajs-778	47	21	a	a	DET
iajs-778	47	22	coprime	coprime	ADJ
iajs-778	47	23	rmodule	rmodule	NOUN
iajs-778	47	24	.	.	PUNCT
iajs-778	48	1	thus	thus	ADV
iajs-778	48	2	n	n	PRON
iajs-778	48	3	is	be	AUX
iajs-778	48	4	a	a	DET
iajs-778	48	5	coprime	coprime	NOUN
iajs-778	48	6	submodule	submodule	NOUN
iajs-778	48	7	.	.	PUNCT
iajs-778	49	1	(	(	PUNCT
iajs-778	49	2	6	6	NUM
iajs-778	49	3	)	)	PUNCT
iajs-778	49	4	the	the	DET
iajs-778	49	5	converse	converse	NOUN
iajs-778	49	6	of	of	ADP
iajs-778	49	7	(	(	PUNCT
iajs-778	49	8	4	4	NUM
iajs-778	49	9	)	)	PUNCT
iajs-778	49	10	is	be	AUX
iajs-778	49	11	not	not	PART
iajs-778	49	12	true	true	ADJ
iajs-778	49	13	in	in	ADP
iajs-778	49	14	general	general	ADJ
iajs-778	49	15	for	for	ADP
iajs-778	49	16	example	example	NOUN
iajs-778	49	17	,	,	PUNCT
iajs-778	49	18	z	z	PROPN
iajs-778	49	19	is	be	AUX
iajs-778	49	20	a	a	DET
iajs-778	49	21	coprime	coprime	ADJ
iajs-778	49	22	submodule	submodule	NOUN
iajs-778	49	23	of	of	ADP
iajs-778	49	24	the	the	DET
iajs-778	49	25	zmodule	zmodule	NOUN
iajs-778	49	26	q	q	PROPN
iajs-778	49	27	(	(	PUNCT
iajs-778	49	28	see	see	VERB
iajs-778	49	29	1.1	1.1	NUM
iajs-778	49	30	(	(	PUNCT
iajs-778	49	31	2	2	NUM
iajs-778	49	32	)	)	PUNCT
iajs-778	49	33	)	)	PUNCT
iajs-778	50	1	but	but	CCONJ
iajs-778	50	2	z	z	NOUN
iajs-778	50	3	is	be	AUX
iajs-778	50	4	not	not	PART
iajs-778	50	5	a	a	DET
iajs-778	50	6	maximal	maximal	ADJ
iajs-778	50	7	submodule	submodule	NOUN
iajs-778	50	8	of	of	ADP
iajs-778	50	9	q.	q.	PROPN
iajs-778	50	10	(	(	PUNCT
iajs-778	50	11	7	7	X
iajs-778	50	12	)	)	PUNCT
iajs-778	50	13	let	let	VERB
iajs-778	50	14	m	m	PRON
iajs-778	50	15	be	be	AUX
iajs-778	50	16	an	an	DET
iajs-778	50	17	r	r	NOUN
iajs-778	50	18	-	-	PUNCT
iajs-778	50	19	module	module	NOUN
iajs-778	50	20	,	,	PUNCT
iajs-778	50	21	let	let	VERB
iajs-778	50	22	i	i	PRON
iajs-778	50	23	be	be	AUX
iajs-778	50	24	an	an	DET
iajs-778	50	25	ideal	ideal	NOUN
iajs-778	50	26	of	of	ADP
iajs-778	50	27	r	r	NOUN
iajs-778	50	28	such	such	ADJ
iajs-778	50	29	that	that	SCONJ
iajs-778	50	30	i	i	PRON
iajs-778	50	31			PROPN
iajs-778	50	32	ann	ann	PROPN
iajs-778	50	33	m	m	PROPN
iajs-778	50	34	,	,	PUNCT
iajs-778	50	35	let	let	VERB
iajs-778	50	36	n	n	PRON
iajs-778	50	37	<	<	X
iajs-778	50	38	m.	m.	NOUN
iajs-778	50	39	then	then	ADV
iajs-778	50	40	n	n	PRON
iajs-778	50	41	is	be	AUX
iajs-778	50	42	a	a	DET
iajs-778	50	43	coprime	coprime	ADJ
iajs-778	50	44	r	r	NOUN
iajs-778	50	45	-	-	PUNCT
iajs-778	50	46	submodule	submodule	NOUN
iajs-778	50	47	of	of	ADP
iajs-778	50	48	m	m	PROPN
iajs-778	50	49			PROPN
iajs-778	50	50	n	n	PUNCT
iajs-778	50	51	is	be	AUX
iajs-778	50	52	a	a	DET
iajs-778	50	53	coprime	coprime	NOUN
iajs-778	50	54	r	r	NOUN
iajs-778	50	55	-submodule	-submodule	NOUN
iajs-778	50	56	of	of	ADP
iajs-778	50	57	m	m	PRON
iajs-778	50	58	,	,	PUNCT
iajs-778	50	59	where	where	SCONJ
iajs-778	50	60	r	r	NOUN
iajs-778	50	61	=	=	NOUN
iajs-778	50	62	r	r	NOUN
iajs-778	50	63	/	/	SYM
iajs-778	50	64	i.	i.	NOUN
iajs-778	50	65			PROPN
iajs-778	50	66			PROPN
iajs-778	50	67	ibn	ibn	PROPN
iajs-778	50	68	alhaitham	alhaitham	NOUN
iajs-778	50	69	j.	j.	PROPN
iajs-778	50	70	for	for	ADP
iajs-778	50	71	pure	pure	ADJ
iajs-778	50	72	&	&	CCONJ
iajs-778	50	73	appl	appl	PROPN
iajs-778	50	74	.	.	PUNCT
iajs-778	51	1	sci	sci	PROPN
iajs-778	51	2	.	.	PUNCT
iajs-778	51	3	vol.24	vol.24	NOUN
iajs-778	51	4	(	(	PUNCT
iajs-778	51	5	2	2	NUM
iajs-778	51	6	)	)	PUNCT
iajs-778	51	7	2011	2011	NUM
iajs-778	51	8	proof	proof	NOUN
iajs-778	51	9	:	:	PUNCT
iajs-778	51	10	(	(	PUNCT
iajs-778	51	11			NOUN
iajs-778	51	12	)	)	PUNCT
iajs-778	51	13	let	let	VERB
iajs-778	51	14	n	n	PRON
iajs-778	51	15	be	be	AUX
iajs-778	51	16	a	a	DET
iajs-778	51	17	coprime	coprime	ADJ
iajs-778	51	18	r	r	NOUN
iajs-778	51	19	-	-	PUNCT
iajs-778	51	20	submodule	submodule	NOUN
iajs-778	51	21	.	.	PUNCT
iajs-778	52	1	then	then	ADV
iajs-778	52	2			PROPN
iajs-778	52	3			PROPN
iajs-778	52	4	is	be	AUX
iajs-778	52	5	a	a	DET
iajs-778	52	6	coprime	coprime	ADJ
iajs-778	52	7	r	r	NOUN
iajs-778	52	8	-	-	PUNCT
iajs-778	52	9	module	module	NOUN
iajs-778	52	10	and	and	CCONJ
iajs-778	52	11	hence	hence	ADV
iajs-778	52	12	by	by	ADP
iajs-778	52	13	[	[	X
iajs-778	52	14	3	3	NUM
iajs-778	52	15	,	,	PUNCT
iajs-778	52	16	cor	cor	NOUN
iajs-778	52	17	.	.	PROPN
iajs-778	53	1	2.1.9	2.1.9	NUM
iajs-778	53	2	]	]	PUNCT
iajs-778	53	3	,	,	PUNCT
iajs-778	53	4			PROPN
iajs-778	53	5			NOUN
iajs-778	53	6	is	be	AUX
iajs-778	53	7	coprime	coprime	ADJ
iajs-778	53	8	r	r	NOUN
iajs-778	53	9	-module	-module	NOUN
iajs-778	53	10	.	.	PUNCT
iajs-778	54	1	thus	thus	ADV
iajs-778	54	2	n	n	PRON
iajs-778	54	3	is	be	AUX
iajs-778	54	4	a	a	DET
iajs-778	54	5	coprime	coprime	ADJ
iajs-778	54	6	r	r	NOUN
iajs-778	54	7	-module	-module	NOUN
iajs-778	54	8	.	.	PUNCT
iajs-778	55	1	(	(	PUNCT
iajs-778	55	2			NOUN
iajs-778	55	3	)	)	PUNCT
iajs-778	55	4	the	the	DET
iajs-778	55	5	proof	proof	NOUN
iajs-778	55	6	is	be	AUX
iajs-778	55	7	similarly	similarly	ADV
iajs-778	55	8	.	.	PUNCT
iajs-778	56	1	1.2	1.2	NUM
iajs-778	56	2	proposition	proposition	NOUN
iajs-778	56	3	:	:	PUNCT
iajs-778	56	4	if	if	SCONJ
iajs-778	56	5	n	n	PRON
iajs-778	56	6	is	be	AUX
iajs-778	56	7	a	a	DET
iajs-778	56	8	coprime	coprime	NOUN
iajs-778	56	9	submodule	submodule	NOUN
iajs-778	56	10	,	,	PUNCT
iajs-778	56	11	then	then	ADV
iajs-778	56	12	[	[	X
iajs-778	56	13	n	n	X
iajs-778	56	14	:	:	PUNCT
iajs-778	56	15	m	m	VERB
iajs-778	56	16	]	]	X
iajs-778	56	17	is	be	AUX
iajs-778	56	18	a	a	DET
iajs-778	56	19	prime	prime	ADJ
iajs-778	56	20	ideal	ideal	NOUN
iajs-778	56	21	.	.	PUNCT
iajs-778	57	1	proof	proof	NOUN
iajs-778	57	2	:	:	PUNCT
iajs-778	57	3	since	since	SCONJ
iajs-778	57	4	n	n	PRON
iajs-778	57	5	is	be	AUX
iajs-778	57	6	a	a	DET
iajs-778	57	7	coprime	coprime	NOUN
iajs-778	57	8	submodule	submodule	NOUN
iajs-778	57	9	,	,	PUNCT
iajs-778	57	10			PROPN
iajs-778	57	11			NOUN
iajs-778	57	12	is	be	AUX
iajs-778	57	13	coprime	coprime	ADJ
iajs-778	57	14	r	r	NOUN
iajs-778	57	15	-	-	PUNCT
iajs-778	57	16	module	module	NOUN
iajs-778	57	17	.	.	PUNCT
iajs-778	58	1	hence	hence	ADV
iajs-778	58	2	ann	ann	PROPN
iajs-778	58	3			PROPN
iajs-778	58	4			PROPN
iajs-778	58	5	is	be	AUX
iajs-778	58	6	a	a	DET
iajs-778	58	7	prime	prime	ADJ
iajs-778	58	8	ideal	ideal	NOUN
iajs-778	58	9	of	of	ADP
iajs-778	58	10	r	r	NOUN
iajs-778	58	11	[	[	X
iajs-778	58	12	3	3	NUM
iajs-778	58	13	,	,	PUNCT
iajs-778	58	14	note	note	VERB
iajs-778	58	15	2.1	2.1	NUM
iajs-778	58	16	]	]	PUNCT
iajs-778	58	17	.	.	PUNCT
iajs-778	59	1	but	but	CCONJ
iajs-778	59	2	ann	ann	PROPN
iajs-778	59	3			PROPN
iajs-778	59	4			PROPN
iajs-778	59	5	=	=	PUNCT
iajs-778	60	1	[	[	X
iajs-778	60	2	n	n	NUM
iajs-778	60	3	:	:	PUNCT
iajs-778	60	4	m	m	VERB
iajs-778	60	5	]	]	X
iajs-778	60	6	,	,	PUNCT
iajs-778	60	7	so	so	SCONJ
iajs-778	60	8	[	[	X
iajs-778	60	9	n	n	NUM
iajs-778	60	10	:	:	PUNCT
iajs-778	60	11	m	m	VERB
iajs-778	60	12	]	]	X
iajs-778	60	13	is	be	AUX
iajs-778	60	14	a	a	DET
iajs-778	60	15	prime	prime	ADJ
iajs-778	60	16	ideal	ideal	NOUN
iajs-778	60	17	.	.	PUNCT
iajs-778	61	1	recall	recall	VERB
iajs-778	61	2	that	that	SCONJ
iajs-778	61	3	an	an	DET
iajs-778	61	4	r	r	NOUN
iajs-778	61	5	-	-	PUNCT
iajs-778	61	6	module	module	NOUN
iajs-778	61	7	m	m	NOUN
iajs-778	61	8	is	be	AUX
iajs-778	61	9	called	call	VERB
iajs-778	61	10	secondary	secondary	ADJ
iajs-778	61	11	if	if	SCONJ
iajs-778	61	12	for	for	ADP
iajs-778	61	13	each	each	DET
iajs-778	61	14	r	r	NOUN
iajs-778	61	15			NOUN
iajs-778	61	16	r	r	NOUN
iajs-778	61	17	,	,	PUNCT
iajs-778	62	1	either	either	CCONJ
iajs-778	62	2	r	r	NOUN
iajs-778	62	3	m	m	NOUN
iajs-778	62	4	=	=	SYM
iajs-778	62	5	0	0	NUM
iajs-778	62	6	or	or	CCONJ
iajs-778	62	7	rnm	rnm	NOUN
iajs-778	62	8	=	=	NOUN
iajs-778	62	9	m	m	PROPN
iajs-778	62	10	,	,	PUNCT
iajs-778	62	11	for	for	ADP
iajs-778	62	12	some	some	DET
iajs-778	62	13	n	n	ADJ
iajs-778	62	14			NOUN
iajs-778	62	15	z+	z+	NUM
iajs-778	62	16	.	.	PUNCT
iajs-778	63	1	[	[	X
iajs-778	63	2	7	7	NUM
iajs-778	63	3	]	]	PUNCT
iajs-778	63	4	.	.	PUNCT
iajs-778	64	1	we	we	PRON
iajs-778	64	2	have	have	VERB
iajs-778	64	3	the	the	DET
iajs-778	64	4	following	following	NOUN
iajs-778	64	5	:	:	PUNCT
iajs-778	64	6	1.3	1.3	NUM
iajs-778	64	7	proposition	proposition	NOUN
iajs-778	64	8	:	:	PUNCT
iajs-778	64	9	let	let	VERB
iajs-778	64	10	m	m	PRON
iajs-778	64	11	be	be	AUX
iajs-778	64	12	a	a	DET
iajs-778	64	13	secondary	secondary	ADJ
iajs-778	64	14	r	r	NOUN
iajs-778	64	15	-	-	PUNCT
iajs-778	64	16	module	module	NOUN
iajs-778	64	17	,	,	PUNCT
iajs-778	64	18	let	let	VERB
iajs-778	64	19	n	n	PRON
iajs-778	64	20	<	<	X
iajs-778	64	21	m	m	PROPN
iajs-778	64	22	.	.	PUNCT
iajs-778	65	1	then	then	ADV
iajs-778	65	2	n	n	PRON
iajs-778	65	3	is	be	AUX
iajs-778	65	4	a	a	DET
iajs-778	65	5	coprime	coprime	ADJ
iajs-778	65	6	submodule	submodule	NOUN
iajs-778	65	7	iff	iff	PROPN
iajs-778	66	1	[	[	X
iajs-778	66	2	n	n	NUM
iajs-778	66	3	:	:	PUNCT
iajs-778	66	4	m	m	VERB
iajs-778	66	5	]	]	X
iajs-778	66	6	is	be	AUX
iajs-778	66	7	a	a	DET
iajs-778	66	8	prime	prime	ADJ
iajs-778	66	9	ideal	ideal	NOUN
iajs-778	66	10	of	of	ADP
iajs-778	66	11	r.	r.	PROPN
iajs-778	66	12	proof	proof	NOUN
iajs-778	66	13	:	:	PUNCT
iajs-778	66	14	(	(	PUNCT
iajs-778	66	15			NOUN
iajs-778	66	16	)	)	PUNCT
iajs-778	66	17	it	it	PRON
iajs-778	66	18	follows	follow	VERB
iajs-778	66	19	by	by	ADP
iajs-778	66	20	prop	prop	NOUN
iajs-778	66	21	.	.	PUNCT
iajs-778	67	1	1.2	1.2	NUM
iajs-778	67	2	.	.	PUNCT
iajs-778	67	3	(	(	PUNCT
iajs-778	67	4			NOUN
iajs-778	67	5	)	)	PUNCT
iajs-778	67	6	since	since	SCONJ
iajs-778	67	7	m	m	PROPN
iajs-778	67	8	is	be	AUX
iajs-778	67	9	a	a	DET
iajs-778	67	10	secondary	secondary	ADJ
iajs-778	67	11	r	r	NOUN
iajs-778	67	12	-	-	PUNCT
iajs-778	67	13	module	module	NOUN
iajs-778	67	14	,	,	PUNCT
iajs-778	67	15	then	then	ADV
iajs-778	67	16			PROPN
iajs-778	68	1			PROPN
iajs-778	68	2	is	be	AUX
iajs-778	68	3	a	a	DET
iajs-778	68	4	secondary	secondary	ADJ
iajs-778	68	5	r	r	NOUN
iajs-778	68	6	-	-	PUNCT
iajs-778	68	7	module	module	NOUN
iajs-778	68	8	.	.	PUNCT
iajs-778	69	1	but	but	CCONJ
iajs-778	69	2	[	[	X
iajs-778	69	3	n	n	CCONJ
iajs-778	69	4	:	:	PUNCT
iajs-778	69	5	m	m	VERB
iajs-778	69	6	]	]	X
iajs-778	69	7	=	=	SYM
iajs-778	69	8	ann	ann	PROPN
iajs-778	69	9			PROPN
iajs-778	69	10			PROPN
iajs-778	69	11	is	be	AUX
iajs-778	69	12	a	a	DET
iajs-778	69	13	prime	prime	ADJ
iajs-778	69	14	ideal	ideal	NOUN
iajs-778	69	15	,	,	PUNCT
iajs-778	69	16	so	so	ADV
iajs-778	69	17	by	by	ADP
iajs-778	69	18	[	[	X
iajs-778	69	19	3,prop.1.2.6	3,prop.1.2.6	X
iajs-778	69	20	]	]	PUNCT
iajs-778	69	21	,	,	PUNCT
iajs-778	69	22			PROPN
iajs-778	69	23			NOUN
iajs-778	69	24	is	be	AUX
iajs-778	69	25	a	a	DET
iajs-778	69	26	coprime	coprime	ADJ
iajs-778	69	27	r	r	NOUN
iajs-778	69	28	-	-	NOUN
iajs-778	69	29	module	module	NOUN
iajs-778	69	30	,	,	PUNCT
iajs-778	69	31	hence	hence	ADV
iajs-778	69	32	n	n	PRON
iajs-778	69	33	is	be	AUX
iajs-778	69	34	a	a	DET
iajs-778	69	35	coprime	coprime	NOUN
iajs-778	69	36	submodule	submodule	NOUN
iajs-778	69	37	.	.	PUNCT
iajs-778	70	1	1.4	1.4	NUM
iajs-778	70	2	proposition	proposition	NOUN
iajs-778	70	3	:	:	PUNCT
iajs-778	70	4	let	let	VERB
iajs-778	70	5	n	n	PRON
iajs-778	70	6	be	be	AUX
iajs-778	70	7	a	a	DET
iajs-778	70	8	proper	proper	ADJ
iajs-778	70	9	submodule	submodule	NOUN
iajs-778	70	10	of	of	ADP
iajs-778	70	11	an	an	DET
iajs-778	70	12	r	r	NOUN
iajs-778	70	13	-	-	PUNCT
iajs-778	70	14	module	module	NOUN
iajs-778	70	15	m.	m.	NOUN
iajs-778	70	16	then	then	ADV
iajs-778	70	17	n	n	PRON
iajs-778	70	18	is	be	AUX
iajs-778	70	19	a	a	DET
iajs-778	70	20	coprime	coprime	ADJ
iajs-778	70	21	submodule	submodule	NOUN
iajs-778	70	22	iff	iff	PROPN
iajs-778	71	1	[	[	X
iajs-778	71	2	n	n	CCONJ
iajs-778	71	3	:	:	PUNCT
iajs-778	71	4	m]=[w	m]=[w	NUM
iajs-778	71	5	:	:	PUNCT
iajs-778	71	6	m	m	VERB
iajs-778	71	7	]	]	X
iajs-778	71	8	for	for	ADP
iajs-778	71	9	all	all	DET
iajs-778	71	10	w	w	PROPN
iajs-778	71	11			PROPN
iajs-778	71	12	n.	n.	NOUN
iajs-778	71	13	proof	proof	NOUN
iajs-778	71	14	:	:	PUNCT
iajs-778	71	15	if	if	SCONJ
iajs-778	71	16	n	n	PRON
iajs-778	71	17	is	be	AUX
iajs-778	71	18	a	a	DET
iajs-778	71	19	coprime	coprime	NOUN
iajs-778	71	20	submodule	submodule	NOUN
iajs-778	71	21	,	,	PUNCT
iajs-778	71	22	then	then	ADV
iajs-778	71	23			PROPN
iajs-778	71	24			PROPN
iajs-778	71	25	is	be	AUX
iajs-778	71	26	a	a	DET
iajs-778	71	27	coprime	coprime	ADJ
iajs-778	71	28	r	r	NOUN
iajs-778	71	29	-	-	NOUN
iajs-778	71	30	module	module	NOUN
iajs-778	71	31	.	.	PUNCT
iajs-778	72	1	hence	hence	ADV
iajs-778	72	2	ann	ann	PROPN
iajs-778	72	3			PROPN
iajs-778	72	4			PROPN
iajs-778	72	5	=	=	PROPN
iajs-778	72	6	ann	ann	PROPN
iajs-778	72	7	w	w	PROPN
iajs-778	72	8			PROPN
iajs-778	72	9			PROPN
iajs-778	72	10			NOUN
iajs-778	72	11	for	for	ADP
iajs-778	72	12	all	all	DET
iajs-778	72	13	wn	wn	PROPN
iajs-778	72	14	.	.	PUNCT
iajs-778	73	1	it	it	PRON
iajs-778	73	2	follows	follow	VERB
iajs-778	73	3	that	that	SCONJ
iajs-778	73	4	ann	ann	PROPN
iajs-778	73	5			PROPN
iajs-778	73	6			PROPN
iajs-778	73	7	=	=	PROPN
iajs-778	73	8	ann	ann	PROPN
iajs-778	73	9	w	w	PROPN
iajs-778	73	10			PROPN
iajs-778	73	11	;	;	PUNCT
iajs-778	73	12	that	that	PRON
iajs-778	73	13	is	be	AUX
iajs-778	73	14	[	[	X
iajs-778	73	15	n	n	NUM
iajs-778	73	16	:	:	PUNCT
iajs-778	74	1	m]=	m]=	NOUN
iajs-778	74	2	[	[	X
iajs-778	74	3	w	w	NOUN
iajs-778	74	4	:	:	PUNCT
iajs-778	74	5	m	m	VERB
iajs-778	74	6	]	]	X
iajs-778	74	7	.	.	PUNCT
iajs-778	75	1	if	if	SCONJ
iajs-778	75	2	[	[	X
iajs-778	75	3	n	n	NUM
iajs-778	75	4	:	:	PUNCT
iajs-778	75	5	m	m	VERB
iajs-778	75	6	]	]	X
iajs-778	76	1	=	=	PUNCT
iajs-778	77	1	[	[	X
iajs-778	77	2	w	w	X
iajs-778	77	3	:	:	PUNCT
iajs-778	77	4	m	m	VERB
iajs-778	77	5	]	]	X
iajs-778	77	6	,	,	PUNCT
iajs-778	77	7	for	for	ADP
iajs-778	77	8	all	all	DET
iajs-778	77	9	w	w	PROPN
iajs-778	77	10			PROPN
iajs-778	77	11	n	n	CCONJ
iajs-778	77	12	,	,	PUNCT
iajs-778	77	13	then	then	ADV
iajs-778	77	14	ann	ann	PROPN
iajs-778	77	15			PROPN
iajs-778	77	16			PROPN
iajs-778	77	17	=	=	PROPN
iajs-778	77	18	ann	ann	PROPN
iajs-778	77	19	w	w	PROPN
iajs-778	77	20			PROPN
iajs-778	77	21	.	.	PUNCT
iajs-778	78	1	but	but	CCONJ
iajs-778	78	2	w	w	PROPN
iajs-778	78	3			PROPN
iajs-778	78	4			PROPN
iajs-778	78	5	w	w	ADP
iajs-778	78	6			VERB
iajs-778	78	7			PROPN
iajs-778	78	8			NOUN
iajs-778	78	9	,	,	PUNCT
iajs-778	78	10	so	so	ADV
iajs-778	78	11	ann	ann	PROPN
iajs-778	78	12			PROPN
iajs-778	78	13			PROPN
iajs-778	78	14	=	=	PROPN
iajs-778	78	15	ann	ann	PROPN
iajs-778	78	16	w	w	PROPN
iajs-778	78	17			PROPN
iajs-778	78	18			PROPN
iajs-778	78	19			NOUN
iajs-778	78	20	and	and	CCONJ
iajs-778	78	21			PROPN
iajs-778	78	22			NOUN
iajs-778	78	23	is	be	AUX
iajs-778	78	24	a	a	DET
iajs-778	78	25	coprime	coprime	ADJ
iajs-778	78	26	r	r	NOUN
iajs-778	78	27	-	-	NOUN
iajs-778	78	28	module	module	NOUN
iajs-778	78	29	.	.	PUNCT
iajs-778	79	1	thus	thus	ADV
iajs-778	79	2	n	n	PRON
iajs-778	79	3	is	be	AUX
iajs-778	79	4	a	a	DET
iajs-778	79	5	coprime	coprime	NOUN
iajs-778	79	6	submodule	submodule	NOUN
iajs-778	79	7	.	.	PUNCT
iajs-778	80	1	1.5	1.5	NUM
iajs-778	80	2	proposition	proposition	NOUN
iajs-778	80	3	:	:	PUNCT
iajs-778	80	4	let	let	VERB
iajs-778	80	5	w	w	PART
iajs-778	80	6	be	be	AUX
iajs-778	80	7	a	a	DET
iajs-778	80	8	coprime	coprime	ADJ
iajs-778	80	9	submodule	submodule	NOUN
iajs-778	80	10	of	of	ADP
iajs-778	80	11	m	m	PRON
iajs-778	80	12	and	and	CCONJ
iajs-778	80	13	let	let	VERB
iajs-778	80	14	n	n	PRON
iajs-778	80	15	<	<	X
iajs-778	80	16	m	m	VERB
iajs-778	80	17	such	such	ADJ
iajs-778	80	18	that	that	SCONJ
iajs-778	80	19	n	n	PROPN
iajs-778	80	20			PROPN
iajs-778	80	21	w.	w.	PROPN
iajs-778	80	22	then	then	ADV
iajs-778	80	23	n	n	PROPN
iajs-778	80	24	is	be	AUX
iajs-778	80	25	a	a	DET
iajs-778	80	26	coprime	coprime	ADJ
iajs-778	80	27	submodule	submodule	NOUN
iajs-778	80	28	of	of	ADP
iajs-778	80	29	m	m	PROPN
iajs-778	80	30	and	and	CCONJ
iajs-778	80	31	w	w	PROPN
iajs-778	80	32			NOUN
iajs-778	80	33	is	be	AUX
iajs-778	80	34	a	a	DET
iajs-778	80	35	coprime	coprime	ADJ
iajs-778	80	36	submodule	submodule	NOUN
iajs-778	80	37	of	of	ADP
iajs-778	80	38	m	m	PROPN
iajs-778	80	39	w	w	NOUN
iajs-778	80	40	.	.	PUNCT
iajs-778	81	1			NOUN
iajs-778	81	2			VERB
iajs-778	81	3	ibn	ibn	PROPN
iajs-778	81	4	alhaitham	alhaitham	NOUN
iajs-778	81	5	j.	j.	PROPN
iajs-778	81	6	for	for	ADP
iajs-778	81	7	pure	pure	ADJ
iajs-778	81	8	&	&	CCONJ
iajs-778	81	9	appl	appl	PROPN
iajs-778	81	10	.	.	PUNCT
iajs-778	82	1	sci	sci	PROPN
iajs-778	82	2	.	.	PUNCT
iajs-778	82	3	vol.24	vol.24	NOUN
iajs-778	82	4	(	(	PUNCT
iajs-778	82	5	2	2	NUM
iajs-778	82	6	)	)	PUNCT
iajs-778	82	7	2011	2011	NUM
iajs-778	82	8	proof	proof	NOUN
iajs-778	82	9	:	:	PUNCT
iajs-778	82	10	since	since	SCONJ
iajs-778	82	11	w	w	NOUN
iajs-778	82	12	is	be	AUX
iajs-778	82	13	a	a	DET
iajs-778	82	14	coprime	coprime	NOUN
iajs-778	82	15	submodule	submodule	NOUN
iajs-778	82	16	,	,	PUNCT
iajs-778	82	17	then	then	ADV
iajs-778	82	18	m	m	VERB
iajs-778	82	19	w	w	PROPN
iajs-778	82	20	is	be	AUX
iajs-778	82	21	a	a	DET
iajs-778	82	22	coprime	coprime	ADJ
iajs-778	82	23	r	r	NOUN
iajs-778	82	24	-	-	NOUN
iajs-778	82	25	module	module	NOUN
iajs-778	82	26	.	.	PUNCT
iajs-778	83	1	hence	hence	ADV
iajs-778	83	2	by	by	ADP
iajs-778	83	3	[	[	X
iajs-778	83	4	rem	rem	X
iajs-778	83	5	and	and	CCONJ
iajs-778	83	6	ex	ex	NOUN
iajs-778	83	7	.	.	PROPN
iajs-778	83	8	1.1	1.1	NUM
iajs-778	83	9	(	(	PUNCT
iajs-778	83	10	4	4	NUM
iajs-778	83	11	)	)	PUNCT
iajs-778	83	12	]	]	PUNCT
iajs-778	83	13	,	,	PUNCT
iajs-778	83	14	w	w	PROPN
iajs-778	83	15			PROPN
iajs-778	83	16	is	be	AUX
iajs-778	83	17	a	a	DET
iajs-778	83	18	coprime	coprime	ADJ
iajs-778	83	19	submodule	submodule	NOUN
iajs-778	83	20	of	of	ADP
iajs-778	83	21	m	m	PROPN
iajs-778	83	22	w	w	NOUN
iajs-778	83	23	.	.	PUNCT
iajs-778	84	1	also	also	ADV
iajs-778	84	2	m	m	PROPN
iajs-778	84	3	w	w	PROPN
iajs-778	84	4	is	be	AUX
iajs-778	84	5	a	a	DET
iajs-778	84	6	coprime	coprime	ADJ
iajs-778	84	7	r	r	NOUN
iajs-778	84	8	-	-	PUNCT
iajs-778	84	9	module	module	NOUN
iajs-778	84	10	implies	imply	VERB
iajs-778	84	11	(	(	PUNCT
iajs-778	84	12	m	m	NOUN
iajs-778	84	13	/	/	SYM
iajs-778	84	14	w	w	NOUN
iajs-778	84	15	)	)	PUNCT
iajs-778	84	16	/	/	SYM
iajs-778	84	17	(	(	PUNCT
iajs-778	84	18	n	n	CCONJ
iajs-778	84	19	/	/	SYM
iajs-778	84	20	w	w	NOUN
iajs-778	84	21	)	)	PUNCT
iajs-778	84	22	is	be	AUX
iajs-778	84	23	a	a	DET
iajs-778	84	24	coprime	coprime	ADJ
iajs-778	84	25	r	r	NOUN
iajs-778	84	26	-	-	PUNCT
iajs-778	84	27	module	module	NOUN
iajs-778	84	28	[	[	X
iajs-778	84	29	3,cor	3,cor	NUM
iajs-778	84	30	.	.	NOUN
iajs-778	84	31	2.1.12	2.1.12	NUM
iajs-778	84	32	]	]	PUNCT
iajs-778	84	33	.	.	PUNCT
iajs-778	85	1	but	but	CCONJ
iajs-778	85	2	(	(	PUNCT
iajs-778	85	3	m	m	AUX
iajs-778	85	4	/w	/w	PRON
iajs-778	85	5	)	)	PUNCT
iajs-778	85	6	/	/	SYM
iajs-778	85	7	(	(	PUNCT
iajs-778	85	8	n	n	CCONJ
iajs-778	85	9	/	/	SYM
iajs-778	85	10	w	w	NOUN
iajs-778	85	11	)	)	PUNCT
iajs-778	85	12			PROPN
iajs-778	85	13	m	m	PROPN
iajs-778	85	14	/	/	SYM
iajs-778	85	15	n	n	CCONJ
iajs-778	85	16	,	,	PUNCT
iajs-778	85	17	hence	hence	ADV
iajs-778	85	18	m	m	PROPN
iajs-778	85	19	/	/	SYM
iajs-778	85	20	n	n	PROPN
iajs-778	85	21	is	be	AUX
iajs-778	85	22	a	a	DET
iajs-778	85	23	coprime	coprime	NOUN
iajs-778	85	24	module	module	NOUN
iajs-778	85	25	by	by	ADP
iajs-778	85	26	[	[	X
iajs-778	85	27	3	3	NUM
iajs-778	85	28	,	,	PUNCT
iajs-778	85	29	cor	cor	NOUN
iajs-778	85	30	.	.	PROPN
iajs-778	85	31	2.1.14	2.1.14	NUM
iajs-778	85	32	]	]	PUNCT
iajs-778	85	33	.	.	PUNCT
iajs-778	86	1	thus	thus	ADV
iajs-778	86	2	n	n	PRON
iajs-778	86	3	is	be	AUX
iajs-778	86	4	a	a	DET
iajs-778	86	5	coprime	coprime	ADJ
iajs-778	86	6	submodule	submodule	NOUN
iajs-778	86	7	of	of	ADP
iajs-778	86	8	m.	m.	NOUN
iajs-778	86	9	1.6	1.6	NUM
iajs-778	86	10	proposition	proposition	NOUN
iajs-778	86	11	:	:	PUNCT
iajs-778	86	12	let	let	VERB
iajs-778	86	13	m	m	PRON
iajs-778	86	14	be	be	AUX
iajs-778	86	15	an	an	DET
iajs-778	86	16	r	r	NOUN
iajs-778	86	17	-	-	PUNCT
iajs-778	86	18	module	module	NOUN
iajs-778	86	19	,	,	PUNCT
iajs-778	86	20	let	let	VERB
iajs-778	86	21	n	n	PRON
iajs-778	86	22	,	,	PUNCT
iajs-778	86	23	w	w	PROPN
iajs-778	86	24	be	be	VERB
iajs-778	86	25	proper	proper	ADJ
iajs-778	86	26	submodules	submodule	NOUN
iajs-778	86	27	of	of	ADP
iajs-778	86	28	m	m	PROPN
iajs-778	86	29	,	,	PUNCT
iajs-778	86	30	n	n	PROPN
iajs-778	86	31			PROPN
iajs-778	86	32	w	w	PROPN
iajs-778	86	33	such	such	ADJ
iajs-778	86	34	that	that	SCONJ
iajs-778	86	35	w	w	PROPN
iajs-778	86	36			PROPN
iajs-778	86	37	is	be	AUX
iajs-778	86	38	a	a	DET
iajs-778	86	39	coprime	coprime	ADJ
iajs-778	86	40	submodule	submodule	NOUN
iajs-778	86	41	of	of	ADP
iajs-778	86	42	m	m	PROPN
iajs-778	86	43	w	w	NOUN
iajs-778	86	44	.then	.then	ADP
iajs-778	86	45	n	n	PRON
iajs-778	86	46	is	be	AUX
iajs-778	86	47	a	a	DET
iajs-778	86	48	coprime	coprime	ADJ
iajs-778	86	49	submodule	submodule	NOUN
iajs-778	86	50	of	of	ADP
iajs-778	86	51	m.	m.	NOUN
iajs-778	86	52	proof	proof	NOUN
iajs-778	86	53	:	:	PUNCT
iajs-778	86	54	since	since	SCONJ
iajs-778	86	55	w	w	PROPN
iajs-778	86	56			PROPN
iajs-778	86	57	is	be	AUX
iajs-778	86	58	a	a	DET
iajs-778	86	59	coprime	coprime	ADJ
iajs-778	86	60	submodule	submodule	NOUN
iajs-778	86	61	of	of	ADP
iajs-778	86	62	m	m	PROPN
iajs-778	86	63	w	w	PROPN
iajs-778	86	64	,	,	PUNCT
iajs-778	86	65	we	we	PRON
iajs-778	86	66	have	have	VERB
iajs-778	86	67	(	(	PUNCT
iajs-778	86	68	m	m	NOUN
iajs-778	86	69	/	/	SYM
iajs-778	86	70	w	w	NOUN
iajs-778	86	71	)	)	PUNCT
iajs-778	86	72	/	/	SYM
iajs-778	86	73	(	(	PUNCT
iajs-778	86	74	n	n	CCONJ
iajs-778	86	75	/	/	SYM
iajs-778	86	76	w	w	NOUN
iajs-778	86	77	)	)	PUNCT
iajs-778	86	78	is	be	AUX
iajs-778	86	79	a	a	DET
iajs-778	86	80	coprime	coprime	NOUN
iajs-778	86	81	module	module	NOUN
iajs-778	86	82	.	.	PUNCT
iajs-778	87	1	thus	thus	ADV
iajs-778	87	2	m	m	PROPN
iajs-778	87	3	/	/	SYM
iajs-778	87	4	n	n	PROPN
iajs-778	87	5	is	be	AUX
iajs-778	87	6	a	a	DET
iajs-778	87	7	coprime	coprime	NOUN
iajs-778	87	8	module	module	NOUN
iajs-778	87	9	and	and	CCONJ
iajs-778	87	10	so	so	ADV
iajs-778	87	11	n	n	PRON
iajs-778	87	12	is	be	AUX
iajs-778	87	13	a	a	DET
iajs-778	87	14	coprime	coprime	ADJ
iajs-778	87	15	submodule	submodule	NOUN
iajs-778	87	16	of	of	ADP
iajs-778	87	17	m.	m.	NOUN
iajs-778	87	18	the	the	DET
iajs-778	87	19	following	follow	VERB
iajs-778	87	20	results	result	NOUN
iajs-778	87	21	follow	follow	VERB
iajs-778	87	22	directly	directly	ADV
iajs-778	87	23	by	by	ADP
iajs-778	87	24	proposition	proposition	NOUN
iajs-778	87	25	1.5	1.5	NUM
iajs-778	87	26	.	.	PUNCT
iajs-778	88	1	1.7	1.7	NUM
iajs-778	88	2	corollary	corollary	NOUN
iajs-778	88	3	:	:	PUNCT
iajs-778	88	4	if	if	SCONJ
iajs-778	88	5	n	n	PRON
iajs-778	88	6	is	be	AUX
iajs-778	88	7	a	a	DET
iajs-778	88	8	coprime	coprime	ADJ
iajs-778	88	9	submodule	submodule	NOUN
iajs-778	88	10	of	of	ADP
iajs-778	88	11	an	an	DET
iajs-778	88	12	r	r	NOUN
iajs-778	88	13	-	-	PUNCT
iajs-778	88	14	module	module	NOUN
iajs-778	88	15	m	m	NOUN
iajs-778	88	16	,	,	PUNCT
iajs-778	88	17	i	i	PRON
iajs-778	88	18	an	an	DET
iajs-778	88	19	ideal	ideal	NOUN
iajs-778	88	20	of	of	ADP
iajs-778	88	21	r.	r.	PROPN
iajs-778	88	22	then	then	ADV
iajs-778	89	1	[	[	X
iajs-778	89	2	n	n	X
iajs-778	89	3	:	:	PUNCT
iajs-778	89	4			PROPN
iajs-778	89	5	i	i	X
iajs-778	89	6	]	]	X
iajs-778	89	7	is	be	AUX
iajs-778	89	8	a	a	DET
iajs-778	89	9	coprime	coprime	ADJ
iajs-778	89	10	submodule	submodule	NOUN
iajs-778	89	11	of	of	ADP
iajs-778	89	12	m.	m.	NOUN
iajs-778	89	13	1.8	1.8	NUM
iajs-778	89	14	corollary	corollary	NOUN
iajs-778	89	15	:	:	PUNCT
iajs-778	89	16	let	let	VERB
iajs-778	89	17	a	a	DET
iajs-778	89	18	,	,	PUNCT
iajs-778	89	19	b	b	NOUN
iajs-778	89	20	be	be	AUX
iajs-778	89	21	proper	proper	ADJ
iajs-778	89	22	submodules	submodule	NOUN
iajs-778	89	23	of	of	ADP
iajs-778	89	24	an	an	DET
iajs-778	89	25	r	r	NOUN
iajs-778	89	26	-	-	PUNCT
iajs-778	89	27	module	module	NOUN
iajs-778	89	28	m.	m.	NOUN
iajs-778	89	29	if	if	SCONJ
iajs-778	89	30	a	a	PRON
iajs-778	89	31	or	or	CCONJ
iajs-778	89	32	b	b	NOUN
iajs-778	89	33	is	be	AUX
iajs-778	89	34	a	a	DET
iajs-778	89	35	coprime	coprime	ADJ
iajs-778	89	36	submodule	submodule	NOUN
iajs-778	89	37	and	and	CCONJ
iajs-778	89	38	a	a	DET
iajs-778	89	39	+	+	NOUN
iajs-778	89	40	b	b	PROPN
iajs-778	89	41			PROPN
iajs-778	89	42	m.	m.	NOUN
iajs-778	89	43	then	then	ADV
iajs-778	89	44	a	a	PRON
iajs-778	89	45	+	+	NOUN
iajs-778	89	46	b	b	NOUN
iajs-778	89	47	is	be	AUX
iajs-778	89	48	a	a	DET
iajs-778	89	49	coprime	coprime	ADJ
iajs-778	89	50	submodule	submodule	NOUN
iajs-778	89	51	of	of	ADP
iajs-778	89	52	m.	m.	NOUN
iajs-778	89	53	1.9	1.9	NUM
iajs-778	89	54	proposition	proposition	NOUN
iajs-778	89	55	:	:	PUNCT
iajs-778	89	56	let	let	VERB
iajs-778	89	57	i	i	PRON
iajs-778	89	58	be	be	AUX
iajs-778	89	59	a	a	DET
iajs-778	89	60	proper	proper	ADJ
iajs-778	89	61	ideal	ideal	NOUN
iajs-778	89	62	of	of	ADP
iajs-778	89	63	a	a	DET
iajs-778	89	64	ring	ring	NOUN
iajs-778	89	65	r.	r.	PROPN
iajs-778	89	66	then	then	ADV
iajs-778	89	67	i	i	PRON
iajs-778	89	68	is	be	AUX
iajs-778	89	69	a	a	DET
iajs-778	89	70	coprime	coprime	ADJ
iajs-778	89	71	ideal	ideal	NOUN
iajs-778	89	72	iff	iff	PROPN
iajs-778	89	73	i	i	PRON
iajs-778	89	74	is	be	AUX
iajs-778	89	75	a	a	DET
iajs-778	89	76	maximal	maximal	ADJ
iajs-778	89	77	ideal	ideal	NOUN
iajs-778	89	78	of	of	ADP
iajs-778	89	79	r.	r.	PROPN
iajs-778	89	80	proof	proof	NOUN
iajs-778	89	81	:	:	PUNCT
iajs-778	89	82	if	if	SCONJ
iajs-778	89	83	i	i	PRON
iajs-778	89	84	is	be	AUX
iajs-778	89	85	a	a	DET
iajs-778	89	86	coprime	coprime	ADJ
iajs-778	89	87	ideal	ideal	NOUN
iajs-778	89	88	of	of	ADP
iajs-778	89	89	r	r	NOUN
iajs-778	89	90	,	,	PUNCT
iajs-778	89	91	then	then	ADV
iajs-778	89	92	r	r	X
iajs-778	89	93	/	/	SYM
iajs-778	89	94	i	i	PRON
iajs-778	89	95	is	be	AUX
iajs-778	89	96	a	a	DET
iajs-778	89	97	coprime	coprime	ADJ
iajs-778	89	98	r	r	NOUN
iajs-778	89	99	-	-	NOUN
iajs-778	89	100	module	module	NOUN
iajs-778	89	101	.	.	PUNCT
iajs-778	90	1	but	but	CCONJ
iajs-778	90	2	r	r	X
iajs-778	90	3	/	/	SYM
iajs-778	90	4	i	i	PRON
iajs-778	90	5	is	be	AUX
iajs-778	90	6	a	a	DET
iajs-778	90	7	multiplication	multiplication	NOUN
iajs-778	90	8	r	r	NOUN
iajs-778	90	9	-	-	PUNCT
iajs-778	90	10	module	module	NOUN
iajs-778	90	11	,	,	PUNCT
iajs-778	90	12	so	so	ADV
iajs-778	90	13	by	by	ADP
iajs-778	90	14	[	[	X
iajs-778	90	15	3,rem	3,rem	NUM
iajs-778	90	16	.	.	PROPN
iajs-778	90	17	and	and	CCONJ
iajs-778	90	18	ex	ex	NOUN
iajs-778	90	19	.	.	NOUN
iajs-778	90	20	2.1.3(5	2.1.3(5	NUM
iajs-778	90	21	)	)	PUNCT
iajs-778	90	22	]	]	X
iajs-778	91	1	r	r	X
iajs-778	91	2	/	/	SYM
iajs-778	91	3	i	i	PRON
iajs-778	91	4	is	be	AUX
iajs-778	91	5	simple	simple	ADJ
iajs-778	91	6	r	r	NOUN
iajs-778	91	7	-	-	PUNCT
iajs-778	91	8	module	module	NOUN
iajs-778	91	9	.	.	PUNCT
iajs-778	92	1	thus	thus	ADV
iajs-778	92	2	i	i	PRON
iajs-778	92	3	is	be	AUX
iajs-778	92	4	a	a	DET
iajs-778	92	5	maximal	maximal	ADJ
iajs-778	92	6	ideal	ideal	NOUN
iajs-778	92	7	of	of	ADP
iajs-778	92	8	r.	r.	PROPN
iajs-778	92	9	the	the	DET
iajs-778	92	10	converse	converse	NOUN
iajs-778	92	11	follows	follow	VERB
iajs-778	92	12	by	by	ADP
iajs-778	92	13	(	(	PUNCT
iajs-778	92	14	rem	rem	PROPN
iajs-778	92	15	.	.	PROPN
iajs-778	92	16	and	and	CCONJ
iajs-778	92	17	ex	ex	PROPN
iajs-778	92	18	.	.	NOUN
iajs-778	92	19	1.1.(5	1.1.(5	NUM
iajs-778	92	20	)	)	PUNCT
iajs-778	92	21	)	)	PUNCT
iajs-778	92	22	.	.	PUNCT
iajs-778	93	1	1.10	1.10	NUM
iajs-778	93	2	corollary	corollary	NOUN
iajs-778	93	3	:	:	PUNCT
iajs-778	93	4	let	let	VERB
iajs-778	93	5	r	r	PRON
iajs-778	93	6	be	be	AUX
iajs-778	93	7	a	a	DET
iajs-778	93	8	ring	ring	NOUN
iajs-778	93	9	.	.	PUNCT
iajs-778	94	1	the	the	DET
iajs-778	94	2	following	follow	VERB
iajs-778	94	3	are	be	AUX
iajs-778	94	4	equivalent	equivalent	ADJ
iajs-778	94	5	:	:	PUNCT
iajs-778	94	6	(	(	PUNCT
iajs-778	94	7	1	1	X
iajs-778	94	8	)	)	PUNCT
iajs-778	94	9	(	(	PUNCT
iajs-778	94	10	0	0	NUM
iajs-778	94	11	)	)	PUNCT
iajs-778	94	12	is	be	AUX
iajs-778	94	13	a	a	DET
iajs-778	94	14	coprime	coprime	ADJ
iajs-778	94	15	submodule	submodule	NOUN
iajs-778	94	16	of	of	ADP
iajs-778	94	17	r.	r.	PROPN
iajs-778	94	18	(	(	PUNCT
iajs-778	94	19	2	2	NUM
iajs-778	94	20	)	)	PUNCT
iajs-778	94	21	r/(0	r/(0	NOUN
iajs-778	94	22	)	)	PUNCT
iajs-778	94	23	�	�	PROPN
iajs-778	94	24	r	r	NOUN
iajs-778	94	25	is	be	AUX
iajs-778	94	26	a	a	DET
iajs-778	94	27	coprime	coprime	NOUN
iajs-778	94	28	ring	ring	NOUN
iajs-778	94	29	(	(	PUNCT
iajs-778	94	30	that	that	PRON
iajs-778	94	31	is	is	ADV
iajs-778	94	32	r	r	NOUN
iajs-778	94	33	is	be	AUX
iajs-778	94	34	a	a	DET
iajs-778	94	35	field	field	NOUN
iajs-778	94	36	)	)	PUNCT
iajs-778	94	37	.	.	PUNCT
iajs-778	95	1	(	(	PUNCT
iajs-778	95	2	3	3	X
iajs-778	95	3	)	)	PUNCT
iajs-778	95	4	(	(	PUNCT
iajs-778	95	5	0	0	NUM
iajs-778	95	6	)	)	PUNCT
iajs-778	95	7	is	be	AUX
iajs-778	95	8	a	a	DET
iajs-778	95	9	maximal	maximal	ADJ
iajs-778	95	10	ideal	ideal	NOUN
iajs-778	95	11	of	of	ADP
iajs-778	95	12	r.	r.	PROPN
iajs-778	95	13	1.11	1.11	NUM
iajs-778	95	14	corollary	corollary	NOUN
iajs-778	95	15	:	:	PUNCT
iajs-778	95	16	let	let	VERB
iajs-778	95	17	r	r	PRON
iajs-778	95	18	be	be	AUX
iajs-778	95	19	a	a	DET
iajs-778	95	20	pid	pid	NOUN
iajs-778	95	21	,	,	PUNCT
iajs-778	95	22	let	let	VERB
iajs-778	95	23	i	i	PRON
iajs-778	95	24	be	be	AUX
iajs-778	95	25	a	a	DET
iajs-778	95	26	nonzero	nonzero	ADJ
iajs-778	95	27	proper	proper	ADJ
iajs-778	95	28	ideal	ideal	NOUN
iajs-778	95	29	of	of	ADP
iajs-778	95	30	r.	r.	PROPN
iajs-778	95	31	then	then	ADV
iajs-778	95	32	the	the	DET
iajs-778	95	33	following	following	NOUN
iajs-778	95	34	are	be	AUX
iajs-778	95	35	equivalent	equivalent	ADJ
iajs-778	95	36	:	:	PUNCT
iajs-778	95	37	(	(	PUNCT
iajs-778	95	38	1	1	X
iajs-778	95	39	)	)	PUNCT
iajs-778	95	40	i	i	PRON
iajs-778	95	41	is	be	AUX
iajs-778	95	42	a	a	DET
iajs-778	95	43	coprime	coprime	ADJ
iajs-778	95	44	ideal	ideal	NOUN
iajs-778	95	45	of	of	ADP
iajs-778	95	46	r.	r.	PROPN
iajs-778	95	47	(	(	PUNCT
iajs-778	95	48	2	2	X
iajs-778	95	49	)	)	PUNCT
iajs-778	95	50	i	i	PRON
iajs-778	95	51	is	be	AUX
iajs-778	95	52	a	a	DET
iajs-778	95	53	maximal	maximal	ADJ
iajs-778	95	54	ideal	ideal	NOUN
iajs-778	95	55	of	of	ADP
iajs-778	95	56	r.	r.	PROPN
iajs-778	95	57	(	(	PUNCT
iajs-778	95	58	3	3	X
iajs-778	95	59	)	)	PUNCT
iajs-778	95	60	i	i	PRON
iajs-778	95	61	is	be	AUX
iajs-778	95	62	a	a	DET
iajs-778	95	63	prime	prime	ADJ
iajs-778	95	64	ideal	ideal	NOUN
iajs-778	95	65	of	of	ADP
iajs-778	95	66	r.	r.	PROPN
iajs-778	95	67	1.12	1.12	NUM
iajs-778	95	68	note	note	NOUN
iajs-778	95	69	:	:	PUNCT
iajs-778	95	70	if	if	SCONJ
iajs-778	95	71	n	n	PRON
iajs-778	95	72	is	be	AUX
iajs-778	95	73	a	a	DET
iajs-778	95	74	coprime	coprime	ADJ
iajs-778	95	75	submodule	submodule	NOUN
iajs-778	95	76	of	of	ADP
iajs-778	95	77	an	an	DET
iajs-778	95	78	r	r	NOUN
iajs-778	95	79	-	-	PUNCT
iajs-778	95	80	module	module	NOUN
iajs-778	95	81	m.	m.	NOUN
iajs-778	95	82	then	then	ADV
iajs-778	95	83	it	it	PRON
iajs-778	95	84	is	be	AUX
iajs-778	95	85	not	not	PART
iajs-778	95	86	necessary	necessary	ADJ
iajs-778	95	87	that	that	SCONJ
iajs-778	95	88	[	[	X
iajs-778	95	89	n	n	NUM
iajs-778	95	90	:	:	PUNCT
iajs-778	95	91	m	m	VERB
iajs-778	95	92	]	]	X
iajs-778	95	93	is	be	AUX
iajs-778	95	94	a	a	DET
iajs-778	95	95	coprime	coprime	ADJ
iajs-778	95	96	ideal	ideal	NOUN
iajs-778	95	97	of	of	ADP
iajs-778	95	98	r	r	NOUN
iajs-778	95	99	,	,	PUNCT
iajs-778	95	100	as	as	SCONJ
iajs-778	95	101	the	the	DET
iajs-778	95	102	following	follow	VERB
iajs-778	95	103	example	example	NOUN
iajs-778	95	104	shows	show	VERB
iajs-778	95	105	:	:	PUNCT
iajs-778	95	106	z	z	NOUN
iajs-778	95	107	is	be	AUX
iajs-778	95	108	a	a	DET
iajs-778	95	109	coprime	coprime	ADJ
iajs-778	95	110	submodule	submodule	NOUN
iajs-778	95	111	of	of	ADP
iajs-778	95	112	the	the	DET
iajs-778	95	113	z	z	NOUN
iajs-778	95	114	-	-	PUNCT
iajs-778	95	115	module	module	NOUN
iajs-778	95	116	q	q	NOUN
iajs-778	95	117	but	but	CCONJ
iajs-778	96	1	[	[	X
iajs-778	96	2	z	z	X
iajs-778	96	3	:	:	PUNCT
iajs-778	96	4	q	q	X
iajs-778	96	5	]	]	X
iajs-778	96	6	=	=	SYM
iajs-778	96	7	(	(	PUNCT
iajs-778	96	8	0	0	NUM
iajs-778	96	9	)	)	PUNCT
iajs-778	96	10	is	be	AUX
iajs-778	96	11	not	not	PART
iajs-778	96	12	a	a	DET
iajs-778	96	13	maximal	maximal	ADJ
iajs-778	96	14	ideal	ideal	NOUN
iajs-778	96	15	of	of	ADP
iajs-778	96	16	z	z	NOUN
iajs-778	96	17	,	,	PUNCT
iajs-778	96	18	that	that	ADV
iajs-778	96	19	is	is	ADV
iajs-778	96	20	(	(	PUNCT
iajs-778	96	21	0	0	NUM
iajs-778	96	22	)	)	PUNCT
iajs-778	96	23	is	be	AUX
iajs-778	96	24	not	not	PART
iajs-778	96	25	coprime	coprime	ADJ
iajs-778	96	26	ideal	ideal	NOUN
iajs-778	96	27	of	of	ADP
iajs-778	96	28	z.	z.	PROPN
iajs-778	96	29	1.13	1.13	NUM
iajs-778	96	30	proposition	proposition	NOUN
iajs-778	96	31	:	:	PUNCT
iajs-778	96	32	let	let	VERB
iajs-778	96	33	m	m	PRON
iajs-778	96	34	be	be	AUX
iajs-778	96	35	a	a	DET
iajs-778	96	36	multiplication	multiplication	NOUN
iajs-778	96	37	r	r	NOUN
iajs-778	96	38	-	-	PUNCT
iajs-778	96	39	module	module	NOUN
iajs-778	96	40	,	,	PUNCT
iajs-778	96	41	let	let	VERB
iajs-778	96	42	n	n	PRON
iajs-778	96	43	be	be	AUX
iajs-778	96	44	a	a	DET
iajs-778	96	45	proper	proper	ADJ
iajs-778	96	46	submodule	submodule	NOUN
iajs-778	96	47	of	of	ADP
iajs-778	96	48	m.	m.	NOUN
iajs-778	96	49	then	then	ADV
iajs-778	96	50	n	n	PRON
iajs-778	96	51	is	be	AUX
iajs-778	96	52	a	a	DET
iajs-778	96	53	coprime	coprime	ADJ
iajs-778	96	54	submodule	submodule	NOUN
iajs-778	96	55	iff	iff	PROPN
iajs-778	97	1	[	[	X
iajs-778	97	2	n	n	NUM
iajs-778	97	3	:	:	PUNCT
iajs-778	97	4	m	m	AUX
iajs-778	97	5	]	]	X
iajs-778	97	6	is	be	AUX
iajs-778	97	7	a	a	DET
iajs-778	97	8	coprime	coprime	ADJ
iajs-778	97	9	ideal	ideal	NOUN
iajs-778	97	10	of	of	ADP
iajs-778	97	11	r.	r.	PROPN
iajs-778	97	12	ibn	ibn	PROPN
iajs-778	97	13	alhaitham	alhaitham	PROPN
iajs-778	97	14	j.	j.	PROPN
iajs-778	97	15	for	for	ADP
iajs-778	97	16	pure	pure	ADJ
iajs-778	97	17	&	&	CCONJ
iajs-778	97	18	appl	appl	PROPN
iajs-778	97	19	.	.	PUNCT
iajs-778	98	1	sci	sci	PROPN
iajs-778	98	2	.	.	PUNCT
iajs-778	98	3	vol.24	vol.24	NOUN
iajs-778	98	4	(	(	PUNCT
iajs-778	98	5	2	2	NUM
iajs-778	98	6	)	)	PUNCT
iajs-778	98	7	2011	2011	NUM
iajs-778	98	8	proof	proof	NOUN
iajs-778	98	9	:	:	PUNCT
iajs-778	98	10	if	if	SCONJ
iajs-778	98	11	n	n	PRON
iajs-778	98	12	is	be	AUX
iajs-778	98	13	a	a	DET
iajs-778	98	14	coprime	coprime	ADJ
iajs-778	98	15	submodule	submodule	NOUN
iajs-778	98	16	of	of	ADP
iajs-778	98	17	m	m	PROPN
iajs-778	98	18	,	,	PUNCT
iajs-778	98	19	then	then	ADV
iajs-778	98	20			PROPN
iajs-778	98	21			PROPN
iajs-778	98	22	is	be	AUX
iajs-778	98	23	a	a	DET
iajs-778	98	24	coprime	coprime	ADJ
iajs-778	98	25	r	r	NOUN
iajs-778	98	26	-	-	NOUN
iajs-778	98	27	module	module	NOUN
iajs-778	98	28	.	.	PUNCT
iajs-778	99	1	but	but	CCONJ
iajs-778	99	2	m	m	PROPN
iajs-778	99	3	is	be	AUX
iajs-778	99	4	a	a	DET
iajs-778	99	5	multiplication	multiplication	NOUN
iajs-778	99	6	r	r	NOUN
iajs-778	99	7	-	-	PUNCT
iajs-778	99	8	module	module	NOUN
iajs-778	99	9	implies	imply	VERB
iajs-778	99	10			PROPN
iajs-778	99	11			NOUN
iajs-778	99	12	is	be	AUX
iajs-778	99	13	a	a	DET
iajs-778	99	14	multiplication	multiplication	NOUN
iajs-778	99	15	r	r	NOUN
iajs-778	99	16	-	-	PUNCT
iajs-778	99	17	module	module	NOUN
iajs-778	99	18	.	.	PUNCT
iajs-778	100	1	hence	hence	ADV
iajs-778	100	2	by	by	ADP
iajs-778	100	3	[	[	X
iajs-778	100	4	3,rem	3,rem	NUM
iajs-778	100	5	.	.	PROPN
iajs-778	100	6	and	and	CCONJ
iajs-778	100	7	ex	ex	NOUN
iajs-778	100	8	.	.	NOUN
iajs-778	100	9	2.1.3(5	2.1.3(5	NUM
iajs-778	100	10	)	)	PUNCT
iajs-778	100	11	]	]	PUNCT
iajs-778	101	1			PROPN
iajs-778	101	2			PROPN
iajs-778	101	3	is	be	AUX
iajs-778	101	4	a	a	DET
iajs-778	101	5	simple	simple	ADJ
iajs-778	101	6	r	r	NOUN
iajs-778	101	7	-	-	PUNCT
iajs-778	101	8	module	module	NOUN
iajs-778	101	9	.	.	PUNCT
iajs-778	102	1	thus	thus	ADV
iajs-778	102	2	n	n	PRON
iajs-778	102	3	is	be	AUX
iajs-778	102	4	a	a	DET
iajs-778	102	5	maximal	maximal	ADJ
iajs-778	102	6	submodule	submodule	NOUN
iajs-778	102	7	of	of	ADP
iajs-778	102	8	m	m	PRON
iajs-778	102	9	which	which	PRON
iajs-778	102	10	implies	imply	VERB
iajs-778	102	11	that	that	SCONJ
iajs-778	102	12	[	[	X
iajs-778	102	13	n	n	CCONJ
iajs-778	102	14	:	:	PUNCT
iajs-778	102	15	m	m	VERB
iajs-778	102	16	]	]	X
iajs-778	102	17	is	be	AUX
iajs-778	102	18	a	a	DET
iajs-778	102	19	maximal	maximal	ADJ
iajs-778	102	20	ideal	ideal	NOUN
iajs-778	102	21	.	.	PUNCT
iajs-778	103	1	then	then	ADV
iajs-778	103	2	by	by	ADP
iajs-778	103	3	prop	prop	NOUN
iajs-778	103	4	.	.	PUNCT
iajs-778	104	1	1.9	1.9	NUM
iajs-778	104	2	,	,	PUNCT
iajs-778	104	3	[	[	X
iajs-778	104	4	n;m	n;m	X
iajs-778	104	5	]	]	X
iajs-778	104	6	is	be	AUX
iajs-778	104	7	a	a	DET
iajs-778	104	8	coprime	coprime	ADJ
iajs-778	104	9	ideal	ideal	NOUN
iajs-778	104	10	.	.	PUNCT
iajs-778	105	1	conversely	conversely	ADV
iajs-778	105	2	,	,	PUNCT
iajs-778	105	3	if	if	SCONJ
iajs-778	105	4	[	[	X
iajs-778	105	5	n	n	NUM
iajs-778	105	6	:	:	PUNCT
iajs-778	105	7	m	m	VERB
iajs-778	105	8	]	]	X
iajs-778	105	9	is	be	AUX
iajs-778	105	10	a	a	DET
iajs-778	105	11	coprime	coprime	ADJ
iajs-778	105	12	ideal	ideal	NOUN
iajs-778	105	13	of	of	ADP
iajs-778	105	14	r	r	NOUN
iajs-778	105	15	,	,	PUNCT
iajs-778	105	16	then	then	ADV
iajs-778	105	17	by	by	ADP
iajs-778	105	18	prop	prop	NOUN
iajs-778	105	19	.	.	PUNCT
iajs-778	106	1	1.9	1.9	NUM
iajs-778	106	2	,	,	PUNCT
iajs-778	106	3	[	[	X
iajs-778	106	4	n	n	CCONJ
iajs-778	106	5	:	:	PUNCT
iajs-778	106	6	m	m	VERB
iajs-778	106	7	]	]	X
iajs-778	106	8	is	be	AUX
iajs-778	106	9	a	a	DET
iajs-778	106	10	maximal	maximal	ADJ
iajs-778	106	11	ideal	ideal	NOUN
iajs-778	106	12	of	of	ADP
iajs-778	106	13	r.	r.	PROPN
iajs-778	106	14	now	now	ADV
iajs-778	106	15	m	m	VERB
iajs-778	106	16	is	be	AUX
iajs-778	106	17	a	a	DET
iajs-778	106	18	multiplication	multiplication	NOUN
iajs-778	106	19	module	module	NOUN
iajs-778	106	20	and	and	CCONJ
iajs-778	106	21	[	[	X
iajs-778	106	22	n;m	n;m	NUM
iajs-778	106	23	]	]	X
iajs-778	106	24	is	be	AUX
iajs-778	106	25	a	a	DET
iajs-778	106	26	maximal	maximal	ADJ
iajs-778	106	27	ideal	ideal	NOUN
iajs-778	106	28	of	of	ADP
iajs-778	106	29	r	r	NOUN
iajs-778	106	30	implies	imply	VERB
iajs-778	106	31	that	that	SCONJ
iajs-778	106	32	n=[n;m]m	n=[n;m]m	NOUN
iajs-778	106	33	is	be	AUX
iajs-778	106	34	a	a	DET
iajs-778	106	35	maximal	maximal	ADJ
iajs-778	106	36	submodule	submodule	NOUN
iajs-778	106	37	of	of	ADP
iajs-778	106	38	m.	m.	NOUN
iajs-778	106	39	thus	thus	ADV
iajs-778	106	40	by	by	ADP
iajs-778	106	41	rem	rem	PROPN
iajs-778	106	42	.	.	PROPN
iajs-778	106	43	and	and	CCONJ
iajs-778	106	44	ex	ex	PROPN
iajs-778	106	45	.	.	PROPN
iajs-778	106	46	1.1	1.1	NUM
iajs-778	106	47	(	(	PUNCT
iajs-778	106	48	5	5	NUM
iajs-778	106	49	)	)	PUNCT
iajs-778	106	50	,	,	PUNCT
iajs-778	106	51	n	n	PRON
iajs-778	106	52	is	be	AUX
iajs-778	106	53	a	a	DET
iajs-778	106	54	coprime	coprime	ADJ
iajs-778	106	55	submodule	submodule	NOUN
iajs-778	106	56	of	of	ADP
iajs-778	106	57	m.	m.	NOUN
iajs-778	106	58	1.14	1.14	NUM
iajs-778	106	59	corollary	corollary	NOUN
iajs-778	106	60	:	:	PUNCT
iajs-778	106	61	let	let	VERB
iajs-778	106	62	m	m	PRON
iajs-778	106	63	be	be	AUX
iajs-778	106	64	a	a	DET
iajs-778	106	65	multiplication	multiplication	NOUN
iajs-778	106	66	r	r	NOUN
iajs-778	106	67	-	-	PUNCT
iajs-778	106	68	module	module	NOUN
iajs-778	106	69	and	and	CCONJ
iajs-778	106	70	let	let	VERB
iajs-778	106	71	n	n	PRON
iajs-778	106	72	<	<	X
iajs-778	106	73	m	m	NOUN
iajs-778	106	74	.	.	PUNCT
iajs-778	107	1	the	the	DET
iajs-778	107	2	following	follow	VERB
iajs-778	107	3	are	be	AUX
iajs-778	107	4	equivalent	equivalent	ADJ
iajs-778	107	5	:	:	PUNCT
iajs-778	107	6	(	(	PUNCT
iajs-778	107	7	1	1	X
iajs-778	107	8	)	)	PUNCT
iajs-778	107	9	n	n	PRON
iajs-778	107	10	is	be	AUX
iajs-778	107	11	a	a	DET
iajs-778	107	12	coprime	coprime	ADJ
iajs-778	107	13	submodule	submodule	NOUN
iajs-778	107	14	of	of	ADP
iajs-778	107	15	m.	m.	NOUN
iajs-778	107	16	(	(	PUNCT
iajs-778	107	17	2	2	NUM
iajs-778	107	18	)	)	PUNCT
iajs-778	107	19	[	[	X
iajs-778	107	20	n	n	NUM
iajs-778	107	21	:	:	PUNCT
iajs-778	107	22	m	m	VERB
iajs-778	107	23	]	]	X
iajs-778	107	24	is	be	AUX
iajs-778	107	25	a	a	DET
iajs-778	107	26	coprime	coprime	ADJ
iajs-778	107	27	ideal	ideal	NOUN
iajs-778	107	28	of	of	ADP
iajs-778	107	29	r.	r.	PROPN
iajs-778	107	30	(	(	PUNCT
iajs-778	107	31	3	3	NUM
iajs-778	107	32	)	)	PUNCT
iajs-778	108	1	[	[	X
iajs-778	108	2	n	n	NUM
iajs-778	108	3	:	:	PUNCT
iajs-778	108	4	m	m	VERB
iajs-778	108	5	]	]	X
iajs-778	108	6	is	be	AUX
iajs-778	108	7	a	a	DET
iajs-778	108	8	maximal	maximal	ADJ
iajs-778	108	9	ideal	ideal	NOUN
iajs-778	108	10	of	of	ADP
iajs-778	108	11	r.	r.	PROPN
iajs-778	108	12	(	(	PUNCT
iajs-778	108	13	4	4	NUM
iajs-778	108	14	)	)	PUNCT
iajs-778	108	15	n	n	PRON
iajs-778	108	16	is	be	AUX
iajs-778	108	17	a	a	DET
iajs-778	108	18	maximal	maximal	ADJ
iajs-778	108	19	submodule	submodule	NOUN
iajs-778	108	20	of	of	ADP
iajs-778	108	21	m.	m.	NOUN
iajs-778	108	22	proof	proof	NOUN
iajs-778	108	23	:	:	PUNCT
iajs-778	108	24	(	(	PUNCT
iajs-778	108	25	1	1	X
iajs-778	108	26	)	)	PUNCT
iajs-778	108	27			NOUN
iajs-778	108	28	(	(	PUNCT
iajs-778	108	29	2	2	X
iajs-778	108	30	)	)	PUNCT
iajs-778	108	31	it	it	PRON
iajs-778	108	32	follows	follow	VERB
iajs-778	108	33	by	by	ADP
iajs-778	108	34	prop	prop	NOUN
iajs-778	108	35	.	.	PUNCT
iajs-778	109	1	1.13	1.13	NUM
iajs-778	109	2	.	.	PUNCT
iajs-778	110	1	(	(	PUNCT
iajs-778	110	2	2	2	X
iajs-778	110	3	)	)	PUNCT
iajs-778	110	4			NOUN
iajs-778	110	5	(	(	PUNCT
iajs-778	110	6	3	3	X
iajs-778	110	7	)	)	PUNCT
iajs-778	110	8	it	it	PRON
iajs-778	110	9	follows	follow	VERB
iajs-778	110	10	by	by	ADP
iajs-778	110	11	prop	prop	NOUN
iajs-778	110	12	.	.	PUNCT
iajs-778	111	1	1.9	1.9	NUM
iajs-778	111	2	.	.	PUNCT
iajs-778	112	1	(	(	PUNCT
iajs-778	112	2	4	4	NUM
iajs-778	112	3	)	)	PUNCT
iajs-778	112	4			NOUN
iajs-778	112	5	(	(	PUNCT
iajs-778	112	6	1	1	X
iajs-778	112	7	)	)	PUNCT
iajs-778	112	8	by	by	ADP
iajs-778	112	9	rem	rem	PROPN
iajs-778	112	10	.	.	PROPN
iajs-778	112	11	and	and	CCONJ
iajs-778	112	12	ex	ex	PROPN
iajs-778	112	13	.	.	PROPN
iajs-778	112	14	1.1	1.1	NUM
iajs-778	112	15	(	(	PUNCT
iajs-778	112	16	5	5	NUM
iajs-778	112	17	)	)	PUNCT
iajs-778	112	18	.	.	PUNCT
iajs-778	113	1	(	(	PUNCT
iajs-778	113	2	3	3	X
iajs-778	113	3	)	)	PUNCT
iajs-778	113	4			NOUN
iajs-778	113	5	(	(	PUNCT
iajs-778	113	6	4	4	NUM
iajs-778	113	7	)	)	PUNCT
iajs-778	113	8	since	since	SCONJ
iajs-778	113	9	m	m	PROPN
iajs-778	113	10	is	be	AUX
iajs-778	113	11	multiplication	multiplication	NOUN
iajs-778	113	12	,	,	PUNCT
iajs-778	113	13	and	and	CCONJ
iajs-778	113	14	[	[	X
iajs-778	113	15	n	n	CCONJ
iajs-778	113	16	:	:	PUNCT
iajs-778	113	17	m	m	VERB
iajs-778	113	18	]	]	X
iajs-778	113	19	is	be	AUX
iajs-778	113	20	a	a	DET
iajs-778	113	21	maximal	maximal	ADJ
iajs-778	113	22	ideal	ideal	NOUN
iajs-778	113	23	,	,	PUNCT
iajs-778	113	24	then	then	ADV
iajs-778	113	25	n	n	PRON
iajs-778	113	26	is	be	AUX
iajs-778	113	27	a	a	DET
iajs-778	113	28	maximal	maximal	ADJ
iajs-778	113	29	submodule	submodule	NOUN
iajs-778	113	30	of	of	ADP
iajs-778	113	31	m.	m.	NOUN
iajs-778	113	32	the	the	DET
iajs-778	113	33	following	follow	VERB
iajs-778	113	34	result	result	NOUN
iajs-778	113	35	shows	show	VERB
iajs-778	113	36	that	that	SCONJ
iajs-778	113	37	a	a	DET
iajs-778	113	38	homomorphic	homomorphic	ADJ
iajs-778	113	39	image	image	NOUN
iajs-778	113	40	of	of	ADP
iajs-778	113	41	a	a	DET
iajs-778	113	42	coprime	coprime	ADJ
iajs-778	113	43	submodule	submodule	NOUN
iajs-778	113	44	is	be	AUX
iajs-778	113	45	a	a	DET
iajs-778	113	46	coprime	coprime	NOUN
iajs-778	113	47	submodule	submodule	NOUN
iajs-778	113	48	.	.	PUNCT
iajs-778	114	1	1.15	1.15	NUM
iajs-778	114	2	theorem	theorem	VERB
iajs-778	114	3	:	:	PUNCT
iajs-778	114	4	let	let	VERB
iajs-778	114	5	:m	:m	PROPN
iajs-778	114	6			ADV
iajs-778	114	7	be	be	AUX
iajs-778	114	8	an	an	DET
iajs-778	114	9	r	r	NOUN
iajs-778	114	10	-	-	PUNCT
iajs-778	114	11	epimorphism	epimorphism	NOUN
iajs-778	114	12	,	,	PUNCT
iajs-778	114	13	let	let	VERB
iajs-778	114	14	n	n	PRON
iajs-778	114	15	<	<	X
iajs-778	114	16	m.	m.	NOUN
iajs-778	114	17	if	if	SCONJ
iajs-778	114	18	n	n	PRON
iajs-778	114	19	is	be	AUX
iajs-778	114	20	a	a	DET
iajs-778	114	21	coprime	coprime	ADJ
iajs-778	114	22	submodule	submodule	NOUN
iajs-778	114	23	of	of	ADP
iajs-778	114	24	m	m	PROPN
iajs-778	114	25	,	,	PUNCT
iajs-778	114	26	then	then	ADV
iajs-778	114	27	(n	(n	X
iajs-778	114	28	)	)	PUNCT
iajs-778	114	29	is	be	AUX
iajs-778	114	30	a	a	DET
iajs-778	114	31	coprime	coprime	ADJ
iajs-778	114	32	submodule	submodule	NOUN
iajs-778	114	33	of	of	ADP
iajs-778	114	34			PROPN
iajs-778	114	35	.	.	PUNCT
iajs-778	115	1	proof	proof	NOUN
iajs-778	115	2	:	:	PUNCT
iajs-778	115	3	to	to	PART
iajs-778	115	4	prove	prove	VERB
iajs-778	115	5	(n	(n	NOUN
iajs-778	115	6	)	)	PUNCT
iajs-778	115	7	is	be	AUX
iajs-778	115	8	a	a	DET
iajs-778	115	9	coprime	coprime	ADJ
iajs-778	115	10	submodule	submodule	NOUN
iajs-778	115	11	of	of	ADP
iajs-778	115	12			PROPN
iajs-778	115	13	,	,	PUNCT
iajs-778	115	14	we	we	PRON
iajs-778	115	15	must	must	AUX
iajs-778	115	16	prove	prove	VERB
iajs-778	115	17	(	(	PUNCT
iajs-778	115	18	)	)	PUNCT
iajs-778	115	19			ADP
iajs-778	115	20			PROPN
iajs-778	115	21			NOUN
iajs-778	115	22	is	be	AUX
iajs-778	115	23	a	a	DET
iajs-778	115	24	coprime	coprime	NOUN
iajs-778	115	25	rmodule	rmodule	NOUN
iajs-778	115	26	,	,	PUNCT
iajs-778	115	27	so	so	SCONJ
iajs-778	115	28	we	we	PRON
iajs-778	115	29	must	must	AUX
iajs-778	115	30	show	show	VERB
iajs-778	115	31	that	that	SCONJ
iajs-778	115	32	(	(	PUNCT
iajs-778	115	33	)	)	PUNCT
iajs-778	115	34	(	(	PUNCT
iajs-778	115	35	)	)	PUNCT
iajs-778	116	1			ADV
iajs-778	116	2			ADP
iajs-778	116	3			ADJ
iajs-778	116	4			NUM
iajs-778	116	5			NOUN
iajs-778	116	6			NOUN
iajs-778	116	7			NOUN
iajs-778	116	8			NOUN
iajs-778	116	9	r	r	NOUN
iajs-778	116	10	for	for	ADP
iajs-778	116	11	all	all	DET
iajs-778	116	12	ann	ann	PROPN
iajs-778	116	13	(	(	PUNCT
iajs-778	116	14	)	)	PUNCT
iajs-778	116	15			ADP
iajs-778	116	16			NOUN
iajs-778	116	17			NOUN
iajs-778	116	18			NOUN
iajs-778	116	19	r	r	NOUN
iajs-778	116	20	.	.	PUNCT
iajs-778	117	1	first	first	ADV
iajs-778	117	2	ann	ann	PROPN
iajs-778	117	3	(	(	PUNCT
iajs-778	117	4	)	)	PUNCT
iajs-778	117	5			ADP
iajs-778	117	6			NOUN
iajs-778	117	7			NOUN
iajs-778	117	8			NOUN
iajs-778	117	9	r	r	NOUN
iajs-778	117	10	,	,	PUNCT
iajs-778	117	11	means	mean	VERB
iajs-778	117	12	that	that	SCONJ
iajs-778	117	13	[	[	PUNCT
iajs-778	117	14	(	(	PUNCT
iajs-778	117	15	)	)	PUNCT
iajs-778	117	16	:	:	PUNCT
iajs-778	117	17	]	]	X
iajs-778	117	18			X
iajs-778	117	19			NOUN
iajs-778	117	20			NOUN
iajs-778	117	21	r	r	NOUN
iajs-778	117	22	.	.	PUNCT
iajs-778	118	1	it	it	PRON
iajs-778	118	2	is	be	AUX
iajs-778	118	3	easy	easy	ADJ
iajs-778	118	4	to	to	PART
iajs-778	118	5	check	check	VERB
iajs-778	118	6	that	that	PRON
iajs-778	118	7	[	[	X
iajs-778	118	8	n	n	NUM
iajs-778	118	9	:	:	PUNCT
iajs-778	118	10	m	m	ADJ
iajs-778	118	11	]	]	X
iajs-778	118	12			PROPN
iajs-778	118	13	[	[	PUNCT
iajs-778	118	14	(	(	PUNCT
iajs-778	118	15	)	)	PUNCT
iajs-778	118	16	:	:	PUNCT
iajs-778	118	17	]	]	PUNCT
iajs-778	118	18			NOUN
iajs-778	118	19			VERB
iajs-778	118	20			VERB
iajs-778	118	21	.	.	PUNCT
iajs-778	119	1	hence	hence	ADV
iajs-778	119	2	[	[	PUNCT
iajs-778	119	3	:	:	PUNCT
iajs-778	119	4	]	]	PUNCT
iajs-778	119	5	ann	ann	PROPN
iajs-778	119	6			PROPN
iajs-778	119	7			INTJ
iajs-778	119	8			NOUN
iajs-778	119	9			VERB
iajs-778	119	10			NUM
iajs-778	119	11			NOUN
iajs-778	119	12	r	r	NOUN
iajs-778	119	13	.	.	PUNCT
iajs-778	120	1	on	on	ADP
iajs-778	120	2	the	the	DET
iajs-778	120	3	other	other	ADJ
iajs-778	120	4	hand	hand	NOUN
iajs-778	120	5	n	n	X
iajs-778	120	6	is	be	AUX
iajs-778	120	7	a	a	DET
iajs-778	120	8	coprime	coprime	NOUN
iajs-778	120	9	submodule	submodule	NOUN
iajs-778	120	10	,	,	PUNCT
iajs-778	120	11	implies	imply	VERB
iajs-778	120	12			PROPN
iajs-778	120	13			PROPN
iajs-778	120	14	is	be	AUX
iajs-778	120	15	a	a	DET
iajs-778	120	16	coprime	coprime	ADJ
iajs-778	120	17	r	r	NOUN
iajs-778	120	18	-	-	NOUN
iajs-778	120	19	module	module	NOUN
iajs-778	120	20	.	.	PUNCT
iajs-778	121	1	hence	hence	ADV
iajs-778	121	2			NUM
iajs-778	121	3			ADJ
iajs-778	121	4			PROPN
iajs-778	121	5			NOUN
iajs-778	121	6			NOUN
iajs-778	121	7	r	r	NOUN
iajs-778	121	8	since	since	SCONJ
iajs-778	121	9	ann	ann	PROPN
iajs-778	121	10	[	[	PUNCT
iajs-778	121	11	:	:	PUNCT
iajs-778	121	12	]	]	X
iajs-778	121	13			NUM
iajs-778	121	14			NUM
iajs-778	121	15			PROPN
iajs-778	121	16			PROPN
iajs-778	121	17			VERB
iajs-778	121	18			NOUN
iajs-778	121	19	r	r	NOUN
iajs-778	121	20	.	.	PUNCT
iajs-778	122	1	now	now	ADV
iajs-778	122	2	,	,	PUNCT
iajs-778	122	3	let	let	VERB
iajs-778	122	4	y	y	PROPN
iajs-778	122	5	+	+	CCONJ
iajs-778	122	6	(n	(n	X
iajs-778	122	7	)	)	PUNCT
iajs-778	122	8			NOUN
iajs-778	122	9	(	(	PUNCT
iajs-778	122	10	)	)	PUNCT
iajs-778	122	11			PROPN
iajs-778	122	12			PROPN
iajs-778	122	13			NOUN
iajs-778	122	14	,	,	PUNCT
iajs-778	122	15	so	so	CCONJ
iajs-778	122	16	y	y	PROPN
iajs-778	122	17	=	=	PUNCT
iajs-778	122	18	(m	(m	PROPN
iajs-778	122	19	)	)	PUNCT
iajs-778	122	20	for	for	ADP
iajs-778	122	21	some	some	DET
iajs-778	122	22	m	m	PROPN
iajs-778	122	23			NOUN
iajs-778	122	24	n	n	CCONJ
iajs-778	122	25	,	,	PUNCT
iajs-778	122	26	since	since	SCONJ
iajs-778	122	27			NOUN
iajs-778	122	28	is	be	AUX
iajs-778	122	29	an	an	DET
iajs-778	122	30	epimorphism	epimorphism	NOUN
iajs-778	122	31	.	.	PUNCT
iajs-778	123	1	thus	thus	ADV
iajs-778	123	2	y	y	PROPN
iajs-778	123	3	+	+	CCONJ
iajs-778	123	4	(n	(n	X
iajs-778	123	5	)	)	PUNCT
iajs-778	123	6	=	=	SYM
iajs-778	123	7	(m	(m	PRON
iajs-778	123	8	)	)	PUNCT
iajs-778	123	9	+	+	NUM
iajs-778	123	10	(n	(n	X
iajs-778	123	11	)	)	PUNCT
iajs-778	123	12	=	=	SYM
iajs-778	124	1			PROPN
iajs-778	124	2	(	(	PUNCT
iajs-778	124	3	m	m	PROPN
iajs-778	124	4	+	+	NOUN
iajs-778	124	5	n	n	CCONJ
iajs-778	124	6	)	)	PUNCT
iajs-778	124	7	.	.	PUNCT
iajs-778	125	1	hence	hence	ADV
iajs-778	125	2	there	there	PRON
iajs-778	125	3	exists	exist	VERB
iajs-778	125	4	m	m	NOUN
iajs-778	125	5	'	'	PART
iajs-778	125	6			NOUN
iajs-778	125	7	m	m	VERB
iajs-778	125	8	such	such	ADJ
iajs-778	125	9	that	that	PRON
iajs-778	125	10	.	.	PUNCT
iajs-778	126	1	m	m	VERB
iajs-778	126	2	+	+	NUM
iajs-778	127	1	n	n	CCONJ
iajs-778	127	2	=	=	SYM
iajs-778	127	3	r	r	NOUN
iajs-778	127	4	m	m	NOUN
iajs-778	127	5	+	+	NOUN
iajs-778	127	6	n	n	CCONJ
iajs-778	127	7	,	,	PUNCT
iajs-778	127	8	so	so	ADV
iajs-778	127	9	y	y	PROPN
iajs-778	127	10	+	+	CCONJ
iajs-778	127	11	(n	(n	X
iajs-778	127	12	)	)	PUNCT
iajs-778	127	13	=	=	SYM
iajs-778	128	1			PROPN
iajs-778	128	2	(	(	PUNCT
iajs-778	128	3	r	r	NOUN
iajs-778	128	4	m	m	VERB
iajs-778	128	5	'	'	PART
iajs-778	128	6	+	+	NOUN
iajs-778	128	7	n	n	CCONJ
iajs-778	128	8	)	)	PUNCT
iajs-778	128	9	=	=	SYM
iajs-778	129	1	r	r	NOUN
iajs-778	129	2	(m	(m	NOUN
iajs-778	129	3	'	'	PUNCT
iajs-778	129	4	)	)	PUNCT
iajs-778	130	1	+	+	CCONJ
iajs-778	130	2	n	n	NOUN
iajs-778	130	3	=	=	SYM
iajs-778	130	4	r	r	NOUN
iajs-778	130	5	(	(	PUNCT
iajs-778	130	6	(m	(m	NOUN
iajs-778	130	7	'	'	PUNCT
iajs-778	130	8	)	)	PUNCT
iajs-778	131	1	+	+	NUM
iajs-778	131	2	n	n	CCONJ
iajs-778	131	3	)	)	PUNCT
iajs-778	131	4			NOUN
iajs-778	131	5			ADP
iajs-778	131	6			NOUN
iajs-778	131	7	r	r	NOUN
iajs-778	131	8	.	.	PUNCT
iajs-778	132	1	thus	thus	ADV
iajs-778	132	2	(	(	PUNCT
iajs-778	132	3	)	)	PUNCT
iajs-778	132	4	(	(	PUNCT
iajs-778	132	5	)	)	PUNCT
iajs-778	133	1			ADV
iajs-778	133	2			ADP
iajs-778	133	3			ADJ
iajs-778	133	4			NUM
iajs-778	133	5			NOUN
iajs-778	133	6			NOUN
iajs-778	133	7			NOUN
iajs-778	133	8			NOUN
iajs-778	133	9	r	r	NOUN
iajs-778	133	10	and	and	CCONJ
iajs-778	133	11	so	so	ADV
iajs-778	133	12	(	(	PUNCT
iajs-778	133	13	)	)	PUNCT
iajs-778	133	14			ADP
iajs-778	133	15			PROPN
iajs-778	133	16			NOUN
iajs-778	133	17	is	be	AUX
iajs-778	133	18	a	a	DET
iajs-778	133	19	coprime	coprime	ADJ
iajs-778	133	20	rmodule	rmodule	NOUN
iajs-778	133	21	.	.	PUNCT
iajs-778	134	1	hence	hence	ADV
iajs-778	134	2	(n	(n	X
iajs-778	134	3	)	)	PUNCT
iajs-778	134	4	is	be	AUX
iajs-778	134	5	a	a	DET
iajs-778	134	6	coprime	coprime	ADJ
iajs-778	134	7	submodule	submodule	NOUN
iajs-778	134	8	of	of	ADP
iajs-778	134	9			PROPN
iajs-778	134	10	.	.	PUNCT
iajs-778	135	1	now	now	ADV
iajs-778	135	2	,	,	PUNCT
iajs-778	135	3	we	we	PRON
iajs-778	135	4	turn	turn	VERB
iajs-778	135	5	our	our	PRON
iajs-778	135	6	attention	attention	NOUN
iajs-778	135	7	to	to	ADP
iajs-778	135	8	direct	direct	ADJ
iajs-778	135	9	sum	sum	NOUN
iajs-778	135	10	of	of	ADP
iajs-778	135	11	coprime	coprime	NOUN
iajs-778	135	12	submodules	submodule	NOUN
iajs-778	135	13	.	.	PUNCT
iajs-778	136	1	ibn	ibn	PROPN
iajs-778	136	2	alhaitham	alhaitham	PROPN
iajs-778	136	3	j.	j.	PROPN
iajs-778	136	4	for	for	ADP
iajs-778	136	5	pure	pure	ADJ
iajs-778	136	6	&	&	CCONJ
iajs-778	136	7	appl	appl	PROPN
iajs-778	136	8	.	.	PUNCT
iajs-778	137	1	sci	sci	PROPN
iajs-778	137	2	.	.	PUNCT
iajs-778	137	3	vol.24	vol.24	NOUN
iajs-778	137	4	(	(	PUNCT
iajs-778	137	5	2	2	NUM
iajs-778	137	6	)	)	PUNCT
iajs-778	137	7	2011	2011	NUM
iajs-778	137	8	1.16	1.16	NUM
iajs-778	137	9	theorem	theorem	VERB
iajs-778	137	10	:	:	PUNCT
iajs-778	137	11	let	let	VERB
iajs-778	137	12	m	m	PROPN
iajs-778	137	13	1	1	NUM
iajs-778	137	14	,	,	PUNCT
iajs-778	137	15	m	m	VERB
iajs-778	137	16	2	2	NUM
iajs-778	137	17	be	be	VERB
iajs-778	137	18	r	r	NOUN
iajs-778	137	19	-	-	PUNCT
iajs-778	137	20	modules	module	NOUN
iajs-778	137	21	,	,	PUNCT
iajs-778	137	22	let	let	VERB
iajs-778	137	23	n1	n1	PROPN
iajs-778	137	24	<	<	X
iajs-778	137	25	m	m	PROPN
iajs-778	137	26	1	1	NUM
iajs-778	137	27	,	,	PUNCT
iajs-778	137	28	n2	n2	ADJ
iajs-778	137	29	<	<	X
iajs-778	137	30	m	m	PROPN
iajs-778	137	31	2	2	NUM
iajs-778	137	32	such	such	ADJ
iajs-778	137	33	that	that	SCONJ
iajs-778	137	34	1	1	NUM
iajs-778	137	35	2ann	2ann	NUM
iajs-778	137	36	ann	ann	NOUN
iajs-778	137	37	1	1	NUM
iajs-778	137	38	2	2	NUM
iajs-778	137	39			NUM
iajs-778	137	40			ADJ
iajs-778	137	41			PROPN
iajs-778	137	42			NOUN
iajs-778	137	43			NOUN
iajs-778	137	44	.	.	PUNCT
iajs-778	138	1	then	then	ADV
iajs-778	138	2	n	n	PROPN
iajs-778	138	3	=	=	SYM
iajs-778	138	4	n1n2	n1n2	PROPN
iajs-778	138	5	is	be	AUX
iajs-778	138	6	a	a	DET
iajs-778	138	7	coprime	coprime	ADJ
iajs-778	138	8	submodule	submodule	NOUN
iajs-778	138	9	of	of	ADP
iajs-778	138	10	m	m	PROPN
iajs-778	138	11	iff	iff	PROPN
iajs-778	138	12	n1	n1	PROPN
iajs-778	138	13	is	be	AUX
iajs-778	138	14	a	a	DET
iajs-778	138	15	coprime	coprime	ADJ
iajs-778	138	16	submodule	submodule	NOUN
iajs-778	138	17	of	of	ADP
iajs-778	138	18	m	m	PROPN
iajs-778	138	19	1	1	NUM
iajs-778	138	20	,	,	PUNCT
iajs-778	138	21	n2	n2	NOUN
iajs-778	138	22	is	be	AUX
iajs-778	138	23	a	a	DET
iajs-778	138	24	coprime	coprime	ADJ
iajs-778	138	25	submodule	submodule	NOUN
iajs-778	138	26	of	of	ADP
iajs-778	138	27	m	m	PROPN
iajs-778	138	28	2	2	NUM
iajs-778	138	29	.	.	PUNCT
iajs-778	139	1	proof	proof	NOUN
iajs-778	139	2	:	:	PUNCT
iajs-778	139	3	(	(	PUNCT
iajs-778	139	4			NOUN
iajs-778	139	5	)	)	PUNCT
iajs-778	139	6	let	let	VERB
iajs-778	139	7	p1	p1	PROPN
iajs-778	139	8	:	:	PUNCT
iajs-778	139	9	m1m	m1m	PROPN
iajs-778	139	10	2	2	PROPN
iajs-778	139	11			PROPN
iajs-778	139	12	m	m	VERB
iajs-778	139	13	1	1	NUM
iajs-778	139	14	,	,	PUNCT
iajs-778	139	15	p2	p2	NOUN
iajs-778	139	16	:	:	PUNCT
iajs-778	139	17	m1m	m1m	PROPN
iajs-778	139	18	2	2	PROPN
iajs-778	139	19			PROPN
iajs-778	139	20	m	m	VERB
iajs-778	139	21	2	2	NUM
iajs-778	139	22	be	be	VERB
iajs-778	139	23	the	the	DET
iajs-778	139	24	natural	natural	ADJ
iajs-778	139	25	projection	projection	NOUN
iajs-778	139	26	.	.	PUNCT
iajs-778	140	1	hence	hence	ADV
iajs-778	140	2	p1(n1n2	p1(n1n2	NOUN
iajs-778	140	3	)	)	PUNCT
iajs-778	140	4	=	=	SYM
iajs-778	140	5	n1	n1	NOUN
iajs-778	140	6	,	,	PUNCT
iajs-778	140	7	p2(n1n2	p2(n1n2	PROPN
iajs-778	140	8	)	)	PUNCT
iajs-778	140	9	=	=	SYM
iajs-778	140	10	n2	n2	NOUN
iajs-778	140	11	and	and	CCONJ
iajs-778	140	12	so	so	ADV
iajs-778	140	13	by	by	ADP
iajs-778	140	14	theorem	theorem	NOUN
iajs-778	140	15	1.15	1.15	NUM
iajs-778	140	16	,	,	PUNCT
iajs-778	140	17	n1	n1	PROPN
iajs-778	140	18	is	be	AUX
iajs-778	140	19	a	a	DET
iajs-778	140	20	coprime	coprime	ADJ
iajs-778	140	21	submodule	submodule	NOUN
iajs-778	140	22	of	of	ADP
iajs-778	140	23	m	m	PROPN
iajs-778	140	24	1	1	NUM
iajs-778	140	25	,	,	PUNCT
iajs-778	140	26	n2	n2	NOUN
iajs-778	140	27	is	be	AUX
iajs-778	140	28	a	a	DET
iajs-778	140	29	coprime	coprime	ADJ
iajs-778	140	30	submodule	submodule	NOUN
iajs-778	140	31	of	of	ADP
iajs-778	140	32	m2	m2	PROPN
iajs-778	140	33	.	.	PUNCT
iajs-778	141	1	conversely	conversely	ADV
iajs-778	141	2	,	,	PUNCT
iajs-778	141	3	to	to	PART
iajs-778	141	4	prove	prove	VERB
iajs-778	141	5	n1n2	n1n2	NOUN
iajs-778	141	6	is	be	AUX
iajs-778	141	7	a	a	DET
iajs-778	141	8	coprime	coprime	ADJ
iajs-778	141	9	submodule	submodule	NOUN
iajs-778	141	10	of	of	ADP
iajs-778	141	11	m	m	PROPN
iajs-778	141	12	1m	1m	PROPN
iajs-778	141	13	2	2	NUM
iajs-778	141	14	.	.	PUNCT
iajs-778	142	1	since	since	SCONJ
iajs-778	142	2	n1	n1	PROPN
iajs-778	142	3	,	,	PUNCT
iajs-778	142	4	n2	n2	NOUN
iajs-778	142	5	are	be	AUX
iajs-778	142	6	coprime	coprime	NOUN
iajs-778	142	7	submodules	submodule	NOUN
iajs-778	142	8	of	of	ADP
iajs-778	142	9	m	m	PROPN
iajs-778	142	10	1	1	NUM
iajs-778	142	11	,	,	PUNCT
iajs-778	142	12	m	m	VERB
iajs-778	142	13	2	2	NUM
iajs-778	142	14	respectively	respectively	ADV
iajs-778	142	15	,	,	PUNCT
iajs-778	142	16	then	then	ADV
iajs-778	142	17	1	1	NUM
iajs-778	142	18	1	1	NUM
iajs-778	142	19			NUM
iajs-778	142	20			NOUN
iajs-778	142	21	and	and	CCONJ
iajs-778	142	22	2	2	NUM
iajs-778	142	23	2	2	NUM
iajs-778	142	24			PROPN
iajs-778	142	25			NOUN
iajs-778	142	26	are	be	AUX
iajs-778	142	27	coprime	coprime	ADJ
iajs-778	142	28	r	r	NOUN
iajs-778	142	29	-	-	PUNCT
iajs-778	142	30	module	module	NOUN
iajs-778	142	31	and	and	CCONJ
iajs-778	142	32	since	since	SCONJ
iajs-778	142	33	1	1	NUM
iajs-778	142	34	2ann	2ann	PROPN
iajs-778	142	35	ann	ann	NOUN
iajs-778	142	36	1	1	NUM
iajs-778	142	37	2	2	NUM
iajs-778	142	38			NUM
iajs-778	142	39			ADJ
iajs-778	142	40			PROPN
iajs-778	142	41			NOUN
iajs-778	142	42			NOUN
iajs-778	142	43	it	it	PRON
iajs-778	142	44	follows	follow	VERB
iajs-778	142	45	that	that	SCONJ
iajs-778	142	46	1	1	NUM
iajs-778	142	47	2	2	NUM
iajs-778	142	48	1	1	NUM
iajs-778	142	49	2	2	NUM
iajs-778	142	50			NOUN
iajs-778	142	51			VERB
iajs-778	142	52			PROPN
iajs-778	142	53			NOUN
iajs-778	142	54			NOUN
iajs-778	142	55	is	be	AUX
iajs-778	142	56	a	a	DET
iajs-778	142	57	coprime	coprime	ADJ
iajs-778	142	58	r	r	NOUN
iajs-778	142	59	-	-	PUNCT
iajs-778	142	60	module	module	NOUN
iajs-778	142	61	(	(	PUNCT
iajs-778	142	62	see	see	VERB
iajs-778	142	63	[	[	X
iajs-778	142	64	7	7	NUM
iajs-778	142	65	]	]	PUNCT
iajs-778	142	66	,	,	PUNCT
iajs-778	143	1	[	[	X
iajs-778	143	2	3,prop	3,prop	NUM
iajs-778	143	3	.	.	NOUN
iajs-778	143	4	2.3.3	2.3.3	NUM
iajs-778	143	5	)	)	PUNCT
iajs-778	143	6	.	.	PUNCT
iajs-778	144	1	but	but	CCONJ
iajs-778	144	2	it	it	PRON
iajs-778	144	3	is	be	AUX
iajs-778	144	4	easy	easy	ADJ
iajs-778	144	5	to	to	PART
iajs-778	144	6	check	check	VERB
iajs-778	144	7	that	that	DET
iajs-778	144	8	1	1	NUM
iajs-778	144	9	2	2	NUM
iajs-778	144	10	1	1	NUM
iajs-778	144	11	2	2	NUM
iajs-778	144	12	1	1	NUM
iajs-778	144	13	2	2	NUM
iajs-778	144	14	1	1	NUM
iajs-778	144	15	2	2	NUM
iajs-778	144	16			NOUN
iajs-778	144	17			PROPN
iajs-778	144	18			PROPN
iajs-778	144	19			PROPN
iajs-778	144	20			NUM
iajs-778	144	21			ADJ
iajs-778	144	22			PROPN
iajs-778	144	23			PROPN
iajs-778	144	24			NOUN
iajs-778	144	25			PROPN
iajs-778	144	26	�	�	PROPN
iajs-778	144	27	.	.	PUNCT
iajs-778	145	1	hence	hence	ADV
iajs-778	145	2	by	by	ADP
iajs-778	145	3	[	[	X
iajs-778	145	4	3,cor	3,cor	NUM
iajs-778	145	5	.	.	PUNCT
iajs-778	145	6	2.1.14	2.1.14	NUM
iajs-778	145	7	]	]	PUNCT
iajs-778	145	8	,	,	PUNCT
iajs-778	145	9	1	1	NUM
iajs-778	145	10	2	2	NUM
iajs-778	145	11	1	1	NUM
iajs-778	145	12	2	2	NUM
iajs-778	145	13			NOUN
iajs-778	145	14			PROPN
iajs-778	145	15			NUM
iajs-778	145	16			PROPN
iajs-778	145	17			NOUN
iajs-778	145	18			NOUN
iajs-778	145	19	is	be	AUX
iajs-778	145	20	a	a	DET
iajs-778	145	21	coprime	coprime	ADJ
iajs-778	145	22	r	r	NOUN
iajs-778	145	23	-	-	NOUN
iajs-778	145	24	module	module	NOUN
iajs-778	145	25	.	.	PUNCT
iajs-778	146	1	thus	thus	ADV
iajs-778	146	2	n1n2	n1n2	PROPN
iajs-778	146	3	is	be	AUX
iajs-778	146	4	a	a	DET
iajs-778	146	5	coprime	coprime	ADJ
iajs-778	146	6	submodule	submodule	NOUN
iajs-778	146	7	of	of	ADP
iajs-778	146	8	m	m	PROPN
iajs-778	146	9	1m	1m	PROPN
iajs-778	146	10	2	2	NUM
iajs-778	146	11	.	.	X
iajs-778	146	12	1.17	1.17	NUM
iajs-778	146	13	remark	remark	NOUN
iajs-778	146	14	:	:	PUNCT
iajs-778	146	15	the	the	DET
iajs-778	146	16	condition	condition	NOUN
iajs-778	146	17	1	1	NUM
iajs-778	146	18	2ann	2ann	PROPN
iajs-778	146	19	ann	ann	NOUN
iajs-778	146	20	1	1	NUM
iajs-778	146	21	2	2	NUM
iajs-778	146	22			NUM
iajs-778	146	23			ADJ
iajs-778	146	24			PROPN
iajs-778	146	25			NOUN
iajs-778	146	26			NOUN
iajs-778	146	27	is	be	AUX
iajs-778	146	28	necessary	necessary	ADJ
iajs-778	146	29	condition	condition	NOUN
iajs-778	146	30	in	in	ADP
iajs-778	146	31	th	th	NOUN
iajs-778	146	32	.	.	PUNCT
iajs-778	146	33	14	14	NUM
iajs-778	146	34	,	,	PUNCT
iajs-778	146	35	as	as	SCONJ
iajs-778	146	36	the	the	DET
iajs-778	146	37	following	follow	VERB
iajs-778	146	38	example	example	NOUN
iajs-778	146	39	shows	show	VERB
iajs-778	146	40	:	:	PUNCT
iajs-778	146	41	consider	consider	VERB
iajs-778	146	42	the	the	DET
iajs-778	146	43	z	z	NOUN
iajs-778	146	44	-	-	PUNCT
iajs-778	146	45	module	module	NOUN
iajs-778	146	46	z.	z.	NOUN
iajs-778	146	47	let	let	VERB
iajs-778	146	48	n1	n1	NOUN
iajs-778	146	49	=	=	SYM
iajs-778	146	50	2z	2z	NUM
iajs-778	146	51	,	,	PUNCT
iajs-778	146	52	n2	n2	NOUN
iajs-778	146	53	=	=	SYM
iajs-778	146	54	3z	3z	NUM
iajs-778	146	55	,	,	PUNCT
iajs-778	146	56	n1	n1	NOUN
iajs-778	146	57	,	,	PUNCT
iajs-778	146	58	n2	n2	NOUN
iajs-778	146	59	are	be	AUX
iajs-778	146	60	maximal	maximal	ADJ
iajs-778	146	61	submodules	submodule	NOUN
iajs-778	146	62	of	of	ADP
iajs-778	146	63	z	z	PROPN
iajs-778	146	64	,	,	PUNCT
iajs-778	146	65	so	so	ADV
iajs-778	146	66	n1	n1	ADJ
iajs-778	146	67	,	,	PUNCT
iajs-778	146	68	n2	n2	NOUN
iajs-778	146	69	are	be	AUX
iajs-778	146	70	coprime	coprime	ADJ
iajs-778	146	71	submodules	submodule	NOUN
iajs-778	146	72	of	of	ADP
iajs-778	146	73	z	z	PROPN
iajs-778	146	74	(	(	PUNCT
iajs-778	146	75	see	see	VERB
iajs-778	146	76	rem	rem	PROPN
iajs-778	146	77	.	.	NOUN
iajs-778	146	78	1.1(5	1.1(5	NUM
iajs-778	146	79	)	)	PUNCT
iajs-778	146	80	)	)	PUNCT
iajs-778	146	81	.	.	PUNCT
iajs-778	147	1	let	let	VERB
iajs-778	147	2	n	n	NOUN
iajs-778	147	3	=	=	SYM
iajs-778	147	4	n1	n1	PROPN
iajs-778	147	5			ADJ
iajs-778	147	6	n2	n2	NOUN
iajs-778	147	7	=	=	PROPN
iajs-778	147	8	2z	2z	NUM
iajs-778	147	9			ADJ
iajs-778	147	10	3z	3z	ADJ
iajs-778	147	11	<	<	X
iajs-778	147	12	zz	zz	NOUN
iajs-778	147	13	.	.	PUNCT
iajs-778	148	1	it	it	PRON
iajs-778	148	2	is	be	AUX
iajs-778	148	3	clear	clear	ADJ
iajs-778	148	4	that	that	SCONJ
iajs-778	148	5	ann	ann	PROPN
iajs-778	148	6	ann	ann	PROPN
iajs-778	148	7	1	1	NUM
iajs-778	148	8	2	2	NUM
iajs-778	148	9			NOUN
iajs-778	148	10			NOUN
iajs-778	148	11			NOUN
iajs-778	148	12			PROPN
iajs-778	148	13			PROPN
iajs-778	148	14	.	.	PUNCT
iajs-778	149	1	now	now	ADV
iajs-778	149	2	2	2	NUM
iajs-778	149	3	3	3	NUM
iajs-778	149	4	6	6	NUM
iajs-778	149	5	1	1	NUM
iajs-778	149	6	2	2	NUM
iajs-778	149	7	1	1	NUM
iajs-778	149	8	2	2	NUM
iajs-778	149	9			PROPN
iajs-778	149	10			NOUN
iajs-778	149	11			NOUN
iajs-778	149	12			NOUN
iajs-778	149	13			NOUN
iajs-778	149	14			NUM
iajs-778	149	15			ADJ
iajs-778	149	16			ADJ
iajs-778	149	17			PROPN
iajs-778	149	18			PROPN
iajs-778	149	19			PROPN
iajs-778	149	20			PROPN
iajs-778	149	21			PROPN
iajs-778	149	22			PROPN
iajs-778	149	23			PROPN
iajs-778	149	24	�	�	PROPN
iajs-778	149	25	�	�	PROPN
iajs-778	149	26	.	.	PUNCT
iajs-778	150	1	but	but	CCONJ
iajs-778	150	2	z6	z6	PROPN
iajs-778	150	3	is	be	AUX
iajs-778	150	4	not	not	PART
iajs-778	150	5	a	a	DET
iajs-778	150	6	coprime	coprime	ADJ
iajs-778	150	7	z	z	NOUN
iajs-778	150	8	-	-	PUNCT
iajs-778	150	9	module	module	NOUN
iajs-778	150	10	,	,	PUNCT
iajs-778	150	11	so	so	CCONJ
iajs-778	150	12	1	1	NUM
iajs-778	150	13	2	2	NUM
iajs-778	150	14			VERB
iajs-778	150	15			NUM
iajs-778	150	16			NOUN
iajs-778	150	17			PROPN
iajs-778	150	18	is	be	AUX
iajs-778	150	19	not	not	PART
iajs-778	150	20	a	a	DET
iajs-778	150	21	coprime	coprime	ADJ
iajs-778	150	22	z	z	NOUN
iajs-778	150	23	-	-	PUNCT
iajs-778	150	24	module	module	NOUN
iajs-778	150	25	.	.	PUNCT
iajs-778	151	1	thus	thus	ADV
iajs-778	151	2	n1	n1	ADJ
iajs-778	151	3			ADJ
iajs-778	151	4	n2	n2	NOUN
iajs-778	151	5	is	be	AUX
iajs-778	151	6	not	not	PART
iajs-778	151	7	a	a	DET
iajs-778	151	8	coprime	coprime	ADJ
iajs-778	151	9	submodule	submodule	NOUN
iajs-778	151	10	of	of	ADP
iajs-778	151	11	zz	zz	NOUN
iajs-778	151	12	.	.	PUNCT
iajs-778	152	1	the	the	DET
iajs-778	152	2	following	follow	VERB
iajs-778	152	3	property	property	NOUN
iajs-778	152	4	explains	explain	VERB
iajs-778	152	5	the	the	DET
iajs-778	152	6	behaviour	behaviour	NOUN
iajs-778	152	7	of	of	ADP
iajs-778	152	8	coprime	coprime	NOUN
iajs-778	152	9	submodules	submodule	NOUN
iajs-778	152	10	under	under	ADP
iajs-778	152	11	localization	localization	NOUN
iajs-778	152	12	.	.	PUNCT
iajs-778	153	1	1.18	1.18	NUM
iajs-778	153	2	proposition	proposition	NOUN
iajs-778	153	3	:	:	PUNCT
iajs-778	153	4	let	let	VERB
iajs-778	153	5	s	s	PRON
iajs-778	153	6	be	be	AUX
iajs-778	153	7	a	a	DET
iajs-778	153	8	multiplicative	multiplicative	ADJ
iajs-778	153	9	subset	subset	NOUN
iajs-778	153	10	of	of	ADP
iajs-778	153	11	a	a	DET
iajs-778	153	12	ring	ring	NOUN
iajs-778	153	13	r.	r.	PROPN
iajs-778	153	14	let	let	VERB
iajs-778	153	15	n	n	PRON
iajs-778	153	16	be	be	AUX
iajs-778	153	17	a	a	DET
iajs-778	153	18	proper	proper	ADJ
iajs-778	153	19	submodule	submodule	NOUN
iajs-778	153	20	of	of	ADP
iajs-778	153	21	an	an	DET
iajs-778	153	22	rmodule	rmodule	NOUN
iajs-778	153	23	m	m	VERB
iajs-778	153	24	such	such	ADJ
iajs-778	153	25	that	that	SCONJ
iajs-778	153	26	s	s	NOUN
iajs-778	153	27	–	–	PUNCT
iajs-778	153	28	1	1	NUM
iajs-778	153	29	n	n	NUM
iajs-778	153	30			NOUN
iajs-778	153	31	s	s	PART
iajs-778	153	32	–	–	PUNCT
iajs-778	153	33	1	1	NUM
iajs-778	153	34	m.	m.	NOUN
iajs-778	153	35	if	if	SCONJ
iajs-778	153	36	n	n	PRON
iajs-778	153	37	is	be	AUX
iajs-778	153	38	a	a	DET
iajs-778	153	39	coprime	coprime	ADJ
iajs-778	153	40	submodule	submodule	NOUN
iajs-778	153	41	of	of	ADP
iajs-778	153	42	m	m	PROPN
iajs-778	153	43	,	,	PUNCT
iajs-778	153	44	then	then	ADV
iajs-778	153	45	s	s	PART
iajs-778	153	46	–	–	PUNCT
iajs-778	153	47	1n	1n	NUM
iajs-778	153	48	is	be	AUX
iajs-778	153	49	coprime	coprime	NOUN
iajs-778	153	50	sbmodule	sbmodule	NOUN
iajs-778	153	51	of	of	ADP
iajs-778	153	52	s	s	NOUN
iajs-778	153	53	–	–	PUNCT
iajs-778	153	54	1	1	NUM
iajs-778	153	55	m.	m.	NOUN
iajs-778	153	56	proof	proof	NOUN
iajs-778	153	57	:	:	PUNCT
iajs-778	153	58	n	n	PRON
iajs-778	153	59	is	be	AUX
iajs-778	153	60	a	a	DET
iajs-778	153	61	coprime	coprime	ADJ
iajs-778	153	62	submodule	submodule	NOUN
iajs-778	153	63	of	of	ADP
iajs-778	153	64	m	m	PROPN
iajs-778	153	65	implies	imply	VERB
iajs-778	153	66			PROPN
iajs-778	153	67			PROPN
iajs-778	153	68	is	be	AUX
iajs-778	153	69	a	a	DET
iajs-778	153	70	coprime	coprime	ADJ
iajs-778	153	71	r	r	NOUN
iajs-778	153	72	-	-	NOUN
iajs-778	153	73	module	module	NOUN
iajs-778	153	74	,	,	PUNCT
iajs-778	153	75	then	then	ADV
iajs-778	153	76	by	by	ADP
iajs-778	153	77	[	[	X
iajs-778	153	78	3,prop.2.1.38	3,prop.2.1.38	NUM
iajs-778	153	79	]	]	X
iajs-778	153	80	,	,	PUNCT
iajs-778	153	81	s	s	PART
iajs-778	153	82	–	–	PUNCT
iajs-778	153	83	1	1	NUM
iajs-778	153	84			PROPN
iajs-778	153	85			PROPN
iajs-778	153	86			NOUN
iajs-778	153	87			PROPN
iajs-778	153	88			PROPN
iajs-778	153	89			PROPN
iajs-778	154	1			ADJ
iajs-778	154	2			PROPN
iajs-778	154	3	is	be	AUX
iajs-778	154	4	a	a	DET
iajs-778	154	5	coprime	coprime	NOUN
iajs-778	154	6	s	s	NOUN
iajs-778	154	7	–	–	PUNCT
iajs-778	154	8	1	1	NUM
iajs-778	154	9	r	r	NOUN
iajs-778	154	10	-	-	PUNCT
iajs-778	154	11	module	module	NOUN
iajs-778	154	12	.	.	PUNCT
iajs-778	155	1	but	but	CCONJ
iajs-778	156	1	[	[	X
iajs-778	156	2	5,lemma	5,lemma	NUM
iajs-778	156	3	9.12,p.173	9.12,p.173	NUM
iajs-778	156	4	]	]	PUNCT
iajs-778	156	5	,	,	PUNCT
iajs-778	156	6	ibn	ibn	PROPN
iajs-778	156	7	alhaitham	alhaitham	NOUN
iajs-778	156	8	j.	j.	PROPN
iajs-778	156	9	for	for	ADP
iajs-778	156	10	pure	pure	ADJ
iajs-778	156	11	&	&	CCONJ
iajs-778	156	12	appl	appl	PROPN
iajs-778	156	13	.	.	PUNCT
iajs-778	157	1	sci	sci	PROPN
iajs-778	157	2	.	.	PUNCT
iajs-778	157	3	vol.24	vol.24	NOUN
iajs-778	157	4	(	(	PUNCT
iajs-778	157	5	2	2	NUM
iajs-778	157	6	)	)	PUNCT
iajs-778	157	7	2011	2011	NUM
iajs-778	157	8	1	1	NUM
iajs-778	157	9	1	1	NUM
iajs-778	157	10	1	1	NUM
iajs-778	157	11	s	s	PART
iajs-778	157	12	s	s	NOUN
iajs-778	157	13	s	s	NOUN
iajs-778	157	14			PROPN
iajs-778	157	15			PROPN
iajs-778	157	16			PROPN
iajs-778	157	17			NUM
iajs-778	157	18			NOUN
iajs-778	157	19			NOUN
iajs-778	157	20			NOUN
iajs-778	157	21			PROPN
iajs-778	157	22			PROPN
iajs-778	157	23			ADJ
iajs-778	157	24			NOUN
iajs-778	157	25	,	,	PUNCT
iajs-778	157	26	so	so	ADV
iajs-778	157	27	1	1	NUM
iajs-778	157	28	1	1	NUM
iajs-778	157	29	s	s	NOUN
iajs-778	157	30	s	s	PART
iajs-778	157	31			PROPN
iajs-778	157	32			PROPN
iajs-778	157	33			NUM
iajs-778	157	34			PROPN
iajs-778	157	35	is	be	AUX
iajs-778	157	36	a	a	DET
iajs-778	157	37	coprime	coprime	NOUN
iajs-778	157	38	s	s	NOUN
iajs-778	157	39	–	–	PUNCT
iajs-778	157	40	1r	1r	NUM
iajs-778	157	41	-	-	PUNCT
iajs-778	157	42	module	module	NOUN
iajs-778	157	43	.	.	PUNCT
iajs-778	158	1	hence	hence	ADV
iajs-778	158	2	s	s	PROPN
iajs-778	158	3	–	–	PUNCT
iajs-778	158	4	1n	1n	NUM
iajs-778	158	5	is	be	AUX
iajs-778	158	6	a	a	DET
iajs-778	158	7	coprime	coprime	ADJ
iajs-778	158	8	submodule	submodule	NOUN
iajs-778	158	9	of	of	ADP
iajs-778	158	10	s	s	PROPN
iajs-778	158	11	–	–	PUNCT
iajs-778	158	12	1	1	NUM
iajs-778	158	13	m.	m.	NOUN
iajs-778	158	14	recall	recall	NOUN
iajs-778	158	15	that	that	SCONJ
iajs-778	158	16	an	an	DET
iajs-778	158	17	r	r	NOUN
iajs-778	158	18	-	-	PUNCT
iajs-778	158	19	module	module	NOUN
iajs-778	158	20	m	m	NOUN
iajs-778	158	21	is	be	AUX
iajs-778	158	22	antihopfian	antihopfian	ADJ
iajs-778	158	23	if	if	SCONJ
iajs-778	158	24	m	m	VERB
iajs-778	158	25	=	=	VERB
iajs-778	158	26	m	m	VERB
iajs-778	158	27	/n	/n	PUNCT
iajs-778	158	28	for	for	ADP
iajs-778	158	29	all	all	DET
iajs-778	158	30	n	n	NOUN
iajs-778	158	31	<	<	X
iajs-778	158	32	m	m	PROPN
iajs-778	158	33	(	(	PUNCT
iajs-778	158	34	4	4	NUM
iajs-778	158	35	)	)	PUNCT
iajs-778	158	36	.	.	PUNCT
iajs-778	159	1	hence	hence	ADV
iajs-778	159	2	we	we	PRON
iajs-778	159	3	get	get	VERB
iajs-778	159	4	the	the	DET
iajs-778	159	5	following	following	NOUN
iajs-778	159	6	result	result	NOUN
iajs-778	159	7	directly	directly	ADV
iajs-778	159	8	.	.	PUNCT
iajs-778	160	1	1.19	1.19	NUM
iajs-778	160	2	remark	remark	NOUN
iajs-778	160	3	:	:	PUNCT
iajs-778	160	4	let	let	VERB
iajs-778	160	5	m	m	PRON
iajs-778	160	6	be	be	AUX
iajs-778	160	7	an	an	DET
iajs-778	160	8	antihopfian	antihopfian	ADJ
iajs-778	160	9	r	r	NOUN
iajs-778	160	10	-	-	PUNCT
iajs-778	160	11	module	module	NOUN
iajs-778	160	12	.	.	PUNCT
iajs-778	161	1	then	then	ADV
iajs-778	161	2	every	every	DET
iajs-778	161	3	submodule	submodule	NOUN
iajs-778	161	4	of	of	ADP
iajs-778	161	5	m	m	PROPN
iajs-778	161	6	is	be	AUX
iajs-778	161	7	coprime	coprime	ADJ
iajs-778	161	8	submodule	submodule	NOUN
iajs-778	161	9	.	.	PUNCT
iajs-778	162	1	proof	proof	NOUN
iajs-778	162	2	:	:	PUNCT
iajs-778	162	3	since	since	SCONJ
iajs-778	162	4	m	m	PROPN
iajs-778	162	5			VERB
iajs-778	162	6			PROPN
iajs-778	162	7	�	�	PROPN
iajs-778	162	8	,	,	PUNCT
iajs-778	162	9	ann	ann	PROPN
iajs-778	162	10	m	m	PROPN
iajs-778	162	11	=	=	PROPN
iajs-778	162	12	ann	ann	PROPN
iajs-778	162	13			PROPN
iajs-778	162	14			PROPN
iajs-778	162	15	,	,	PUNCT
iajs-778	162	16	that	that	PRON
iajs-778	162	17	is	is	ADV
iajs-778	162	18	m	m	VERB
iajs-778	162	19	is	be	AUX
iajs-778	162	20	coprime	coprime	ADJ
iajs-778	162	21	r	r	NOUN
iajs-778	162	22	-	-	PUNCT
iajs-778	162	23	module	module	NOUN
iajs-778	162	24	.	.	PUNCT
iajs-778	163	1	then	then	ADV
iajs-778	163	2	by	by	ADP
iajs-778	163	3	(	(	PUNCT
iajs-778	163	4	rem	rem	PROPN
iajs-778	163	5	.	.	NOUN
iajs-778	163	6	and	and	CCONJ
iajs-778	163	7	ex	ex	ADJ
iajs-778	163	8	.	.	NOUN
iajs-778	163	9	1.1(4	1.1(4	NUM
iajs-778	163	10	)	)	PUNCT
iajs-778	163	11	)	)	PUNCT
iajs-778	164	1	every	every	DET
iajs-778	164	2	proper	proper	ADJ
iajs-778	164	3	submodule	submodule	NOUN
iajs-778	164	4	is	be	AUX
iajs-778	164	5	coprime	coprime	ADJ
iajs-778	164	6	submodule	submodule	NOUN
iajs-778	164	7	.	.	PUNCT
iajs-778	165	1	1.20	1.20	NUM
iajs-778	165	2	proposition	proposition	NOUN
iajs-778	165	3	:	:	PUNCT
iajs-778	165	4	let	let	VERB
iajs-778	165	5	m	m	PRON
iajs-778	165	6	be	be	AUX
iajs-778	165	7	a	a	DET
iajs-778	165	8	finitely	finitely	ADV
iajs-778	165	9	generated	generate	VERB
iajs-778	165	10	r	r	NOUN
iajs-778	165	11	-	-	PUNCT
iajs-778	165	12	module	module	NOUN
iajs-778	165	13	,	,	PUNCT
iajs-778	165	14	let	let	VERB
iajs-778	165	15	n	n	CCONJ
iajs-778	165	16	<	<	X
iajs-778	165	17	m.	m.	NOUN
iajs-778	165	18	if	if	SCONJ
iajs-778	165	19	n	n	PRON
iajs-778	165	20	is	be	AUX
iajs-778	165	21	a	a	DET
iajs-778	165	22	coprime	coprime	NOUN
iajs-778	165	23	submodule	submodule	NOUN
iajs-778	165	24	,	,	PUNCT
iajs-778	165	25	then	then	ADV
iajs-778	165	26	n	n	PROPN
iajs-778	165	27	is	be	AUX
iajs-778	165	28	prime	prime	ADJ
iajs-778	165	29	.	.	PUNCT
iajs-778	166	1	proof	proof	NOUN
iajs-778	166	2	:	:	PUNCT
iajs-778	166	3	since	since	SCONJ
iajs-778	166	4	n	n	PRON
iajs-778	166	5	is	be	AUX
iajs-778	166	6	a	a	DET
iajs-778	166	7	coprime	coprime	NOUN
iajs-778	166	8	submodule	submodule	NOUN
iajs-778	166	9	,	,	PUNCT
iajs-778	166	10	m	m	PROPN
iajs-778	166	11	/	/	SYM
iajs-778	166	12	n	n	PROPN
iajs-778	166	13	is	be	AUX
iajs-778	166	14	a	a	DET
iajs-778	166	15	coprime	coprime	ADJ
iajs-778	166	16	r	r	NOUN
iajs-778	166	17	-	-	NOUN
iajs-778	166	18	module	module	NOUN
iajs-778	166	19	.	.	PUNCT
iajs-778	167	1	but	but	CCONJ
iajs-778	167	2	m	m	PROPN
iajs-778	167	3	is	be	AUX
iajs-778	167	4	a	a	DET
iajs-778	167	5	finitely	finitely	ADV
iajs-778	167	6	generated	generate	VERB
iajs-778	167	7	r	r	NOUN
iajs-778	167	8	-	-	PUNCT
iajs-778	167	9	module	module	NOUN
iajs-778	167	10	,	,	PUNCT
iajs-778	167	11	so	so	ADV
iajs-778	167	12	m	m	VERB
iajs-778	167	13	/	/	SYM
iajs-778	167	14	n	n	PROPN
iajs-778	167	15	is	be	AUX
iajs-778	167	16	finitely	finitely	ADV
iajs-778	167	17	generated	generate	VERB
iajs-778	167	18	.	.	PUNCT
iajs-778	168	1	hence	hence	ADV
iajs-778	168	2	by	by	ADP
iajs-778	168	3	[	[	X
iajs-778	168	4	3,th	3,th	NUM
iajs-778	168	5	.	.	PUNCT
iajs-778	169	1	2.4.8	2.4.8	NUM
iajs-778	169	2	]	]	PUNCT
iajs-778	169	3	,	,	PUNCT
iajs-778	169	4	m	m	PROPN
iajs-778	169	5	/	/	SYM
iajs-778	169	6	n	n	PROPN
iajs-778	169	7	is	be	AUX
iajs-778	169	8	a	a	DET
iajs-778	169	9	prime	prime	ADJ
iajs-778	169	10	rmodule	rmodule	NOUN
iajs-778	169	11	and	and	CCONJ
iajs-778	169	12	hence	hence	ADV
iajs-778	169	13	om	om	PROPN
iajs-778	169	14	/	/	SYM
iajs-778	169	15	n	n	NOUN
iajs-778	169	16	=	=	PRON
iajs-778	169	17	n	n	X
iajs-778	169	18	is	be	AUX
iajs-778	169	19	a	a	DET
iajs-778	169	20	prime	prime	ADJ
iajs-778	169	21	submodule	submodule	NOUN
iajs-778	169	22	of	of	ADP
iajs-778	169	23			PROPN
iajs-778	169	24			PROPN
iajs-778	169	25	.	.	PUNCT
iajs-778	170	1	it	it	PRON
iajs-778	170	2	follows	follow	VERB
iajs-778	170	3	that	that	SCONJ
iajs-778	170	4	n	n	PRON
iajs-778	170	5	is	be	AUX
iajs-778	170	6	a	a	DET
iajs-778	170	7	prime	prime	ADJ
iajs-778	170	8	submodule	submodule	NOUN
iajs-778	170	9	of	of	ADP
iajs-778	170	10	m.	m.	NOUN
iajs-778	170	11	1.21	1.21	NUM
iajs-778	170	12	remark	remark	NOUN
iajs-778	170	13	:	:	PUNCT
iajs-778	170	14	the	the	DET
iajs-778	170	15	condition	condition	NOUN
iajs-778	170	16	m	m	VERB
iajs-778	170	17	is	be	AUX
iajs-778	170	18	finitely	finitely	ADV
iajs-778	170	19	generated	generate	VERB
iajs-778	170	20	in	in	ADP
iajs-778	170	21	prop	prop	NOUN
iajs-778	170	22	.	.	PUNCT
iajs-778	171	1	2.1	2.1	NUM
iajs-778	171	2	is	be	AUX
iajs-778	171	3	necessary	necessary	ADJ
iajs-778	171	4	condition	condition	NOUN
iajs-778	171	5	,	,	PUNCT
iajs-778	171	6	as	as	SCONJ
iajs-778	171	7	the	the	DET
iajs-778	171	8	following	follow	VERB
iajs-778	171	9	example	example	NOUN
iajs-778	171	10	shows	show	VERB
iajs-778	171	11	.	.	PUNCT
iajs-778	172	1	z	z	NOUN
iajs-778	172	2	is	be	AUX
iajs-778	172	3	a	a	DET
iajs-778	172	4	coprime	coprime	ADJ
iajs-778	172	5	submodule	submodule	NOUN
iajs-778	172	6	of	of	ADP
iajs-778	172	7	the	the	DET
iajs-778	172	8	z	z	NOUN
iajs-778	172	9	-	-	PUNCT
iajs-778	172	10	module	module	NOUN
iajs-778	172	11	q	q	NOUN
iajs-778	172	12	and	and	CCONJ
iajs-778	172	13	q	q	NOUN
iajs-778	172	14	is	be	AUX
iajs-778	172	15	not	not	PART
iajs-778	172	16	finitely	finitely	ADV
iajs-778	172	17	generated	generate	VERB
iajs-778	172	18	.	.	PUNCT
iajs-778	173	1	also	also	ADV
iajs-778	173	2	z	z	PROPN
iajs-778	173	3	is	be	AUX
iajs-778	173	4	not	not	PART
iajs-778	173	5	a	a	DET
iajs-778	173	6	prime	prime	ADJ
iajs-778	173	7	submodule	submodule	NOUN
iajs-778	173	8	of	of	ADP
iajs-778	173	9	q.	q.	PROPN
iajs-778	173	10	1.22	1.22	NUM
iajs-778	173	11	corollary	corollary	NOUN
iajs-778	173	12	:	:	PUNCT
iajs-778	173	13	let	let	VERB
iajs-778	173	14	m	m	PRON
iajs-778	173	15	be	be	AUX
iajs-778	173	16	a	a	DET
iajs-778	173	17	noetherian	noetherian	ADJ
iajs-778	173	18	coprime	coprime	NOUN
iajs-778	173	19	r	r	NOUN
iajs-778	173	20	-	-	PUNCT
iajs-778	173	21	module	module	NOUN
iajs-778	173	22	,	,	PUNCT
iajs-778	173	23	then	then	ADV
iajs-778	173	24	every	every	DET
iajs-778	173	25	proper	proper	ADJ
iajs-778	173	26	submodule	submodule	NOUN
iajs-778	173	27	of	of	ADP
iajs-778	173	28	m	m	PROPN
iajs-778	173	29	is	be	AUX
iajs-778	173	30	prime	prime	ADJ
iajs-778	173	31	.	.	PUNCT
iajs-778	174	1	proof	proof	NOUN
iajs-778	174	2	:	:	PUNCT
iajs-778	174	3	it	it	PRON
iajs-778	174	4	follows	follow	VERB
iajs-778	174	5	directly	directly	ADV
iajs-778	174	6	by	by	ADP
iajs-778	174	7	prop	prop	NOUN
iajs-778	174	8	.	.	PUNCT
iajs-778	175	1	1.20	1.20	NUM
iajs-778	175	2	.	.	PUNCT
iajs-778	176	1	1.23	1.23	NUM
iajs-778	176	2	proposition	proposition	NOUN
iajs-778	176	3	:	:	PUNCT
iajs-778	176	4	let	let	VERB
iajs-778	176	5	m	m	PRON
iajs-778	176	6	be	be	AUX
iajs-778	176	7	an	an	DET
iajs-778	176	8	r	r	NOUN
iajs-778	176	9	-	-	PUNCT
iajs-778	176	10	module	module	NOUN
iajs-778	176	11	such	such	ADJ
iajs-778	176	12	that	that	DET
iajs-778	176	13	rm	rm	PROPN
iajs-778	176	14			PUNCT
iajs-778	176	15	n	n	PROPN
iajs-778	176	16	=	=	SYM
iajs-778	176	17	rn	rn	PROPN
iajs-778	176	18	for	for	ADP
iajs-778	176	19	all	all	DET
iajs-778	176	20	r	r	NOUN
iajs-778	176	21			NOUN
iajs-778	176	22	r	r	NOUN
iajs-778	176	23	and	and	CCONJ
iajs-778	176	24	for	for	ADP
iajs-778	176	25	all	all	DET
iajs-778	176	26	n	n	NOUN
iajs-778	176	27	<	<	X
iajs-778	176	28	m.	m.	NOUN
iajs-778	176	29	then	then	ADV
iajs-778	176	30	every	every	DET
iajs-778	176	31	prime	prime	ADJ
iajs-778	176	32	submodule	submodule	NOUN
iajs-778	176	33	is	be	AUX
iajs-778	176	34	a	a	DET
iajs-778	176	35	coprime	coprime	ADJ
iajs-778	176	36	submodule	submodule	NOUN
iajs-778	176	37	.	.	PUNCT
iajs-778	177	1	proof	proof	NOUN
iajs-778	177	2	:	:	PUNCT
iajs-778	177	3	let	let	VERB
iajs-778	177	4	n	n	PRON
iajs-778	177	5	be	be	AUX
iajs-778	177	6	a	a	DET
iajs-778	177	7	prime	prime	ADJ
iajs-778	177	8	submodule	submodule	NOUN
iajs-778	177	9	of	of	ADP
iajs-778	177	10	m.	m.	NOUN
iajs-778	177	11	let	let	VERB
iajs-778	177	12	w	w	PROPN
iajs-778	177	13			PROPN
iajs-778	177	14	n.	n.	PROPN
iajs-778	177	15	we	we	PRON
iajs-778	177	16	shall	shall	AUX
iajs-778	177	17	prove	prove	VERB
iajs-778	177	18	that	that	SCONJ
iajs-778	177	19	:	:	PUNCT
iajs-778	177	20	w	w	X
iajs-778	177	21	w	w	PROPN
iajs-778	177	22			PUNCT
iajs-778	177	23			NUM
iajs-778	177	24			NUM
iajs-778	177	25			PROPN
iajs-778	177	26			NOUN
iajs-778	177	27			NOUN
iajs-778	177	28	r	r	NOUN
iajs-778	177	29	r	r	NOUN
iajs-778	177	30	as	as	SCONJ
iajs-778	177	31	follows	follow	VERB
iajs-778	177	32	:	:	PUNCT
iajs-778	177	33	let	let	VERB
iajs-778	177	34	x	x	PROPN
iajs-778	177	35	w	w	PROPN
iajs-778	177	36			PUNCT
iajs-778	177	37			NUM
iajs-778	177	38			NOUN
iajs-778	177	39			NOUN
iajs-778	177	40	r	r	NOUN
iajs-778	177	41	,	,	PUNCT
iajs-778	177	42	so	so	CCONJ
iajs-778	177	43	x	x	SYM
iajs-778	177	44	=	=	SYM
iajs-778	177	45	w	w	PROPN
iajs-778	177	46	+	+	NUM
iajs-778	177	47	n	n	CCONJ
iajs-778	177	48	=	=	SYM
iajs-778	177	49	r	r	NOUN
iajs-778	177	50	(	(	PUNCT
iajs-778	177	51	m	m	PROPN
iajs-778	177	52	+	+	NOUN
iajs-778	177	53	n	n	CCONJ
iajs-778	177	54	)	)	PUNCT
iajs-778	177	55	for	for	ADP
iajs-778	177	56	some	some	DET
iajs-778	177	57	ww	ww	PROPN
iajs-778	177	58	,	,	PUNCT
iajs-778	177	59	m	m	VERB
iajs-778	177	60			NOUN
iajs-778	177	61	m.	m.	NOUN
iajs-778	177	62	hence	hence	ADV
iajs-778	177	63	r	r	NOUN
iajs-778	177	64	m	m	NOUN
iajs-778	177	65	–	–	PUNCT
iajs-778	177	66	wnw	wnw	NOUN
iajs-778	177	67	.	.	PUNCT
iajs-778	178	1	thus	thus	ADV
iajs-778	178	2	r	r	NOUN
iajs-778	178	3	m	m	PROPN
iajs-778	178	4			PROPN
iajs-778	178	5	w	w	PROPN
iajs-778	178	6	,	,	PUNCT
iajs-778	178	7	which	which	PRON
iajs-778	178	8	implies	imply	VERB
iajs-778	178	9	that	that	SCONJ
iajs-778	178	10	rmrmw	rmrmw	PROPN
iajs-778	178	11	=	=	NOUN
iajs-778	178	12	rw	rw	NOUN
iajs-778	178	13	and	and	CCONJ
iajs-778	178	14	hence	hence	ADV
iajs-778	178	15	r	r	NOUN
iajs-778	178	16	m	m	NOUN
iajs-778	178	17	=	=	NOUN
iajs-778	178	18	r	r	NOUN
iajs-778	178	19	y	y	NOUN
iajs-778	178	20	for	for	ADP
iajs-778	178	21	some	some	DET
iajs-778	178	22	yw	yw	NOUN
iajs-778	178	23	.	.	PUNCT
iajs-778	179	1	then	then	ADV
iajs-778	179	2	rm	rm	PROPN
iajs-778	179	3	+	+	CCONJ
iajs-778	179	4	n	n	CCONJ
iajs-778	179	5	=	=	PROPN
iajs-778	179	6	r	r	NOUN
iajs-778	179	7	y	y	NOUN
iajs-778	179	8	+	+	CCONJ
iajs-778	179	9	n	n	CCONJ
iajs-778	179	10	,	,	PUNCT
iajs-778	179	11	that	that	ADV
iajs-778	179	12	is	is	ADV
iajs-778	179	13	r	r	NOUN
iajs-778	179	14	(	(	PUNCT
iajs-778	179	15	m	m	PROPN
iajs-778	179	16	+	+	NOUN
iajs-778	179	17	n	n	CCONJ
iajs-778	179	18	)	)	PUNCT
iajs-778	180	1	=	=	SYM
iajs-778	180	2	r	r	NOUN
iajs-778	180	3	(	(	PUNCT
iajs-778	180	4	y	y	PROPN
iajs-778	180	5	+	+	NOUN
iajs-778	180	6	n)	n)	PROPN
iajs-778	180	7	w	w	PROPN
iajs-778	180	8			NOUN
iajs-778	180	9	r	r	NOUN
iajs-778	180	10	.	.	PUNCT
iajs-778	181	1	thus	thus	ADV
iajs-778	181	2	w	w	PROPN
iajs-778	181	3	w	w	PROPN
iajs-778	181	4			PUNCT
iajs-778	181	5			NUM
iajs-778	181	6			NUM
iajs-778	181	7			PROPN
iajs-778	181	8			NOUN
iajs-778	181	9			NOUN
iajs-778	181	10	r	r	NOUN
iajs-778	181	11	r	r	NOUN
iajs-778	181	12	.	.	PUNCT
iajs-778	182	1	on	on	ADP
iajs-778	182	2	the	the	DET
iajs-778	182	3	other	other	ADJ
iajs-778	182	4	hand	hand	NOUN
iajs-778	182	5	,	,	PUNCT
iajs-778	182	6	n	n	X
iajs-778	182	7	is	be	AUX
iajs-778	182	8	a	a	DET
iajs-778	182	9	prime	prime	ADJ
iajs-778	182	10	submodule	submodule	NOUN
iajs-778	182	11	of	of	ADP
iajs-778	182	12	m	m	PROPN
iajs-778	182	13	implies	imply	VERB
iajs-778	182	14			PROPN
iajs-778	182	15			PROPN
iajs-778	182	16	is	be	AUX
iajs-778	182	17	a	a	DET
iajs-778	182	18	prime	prime	ADJ
iajs-778	182	19	rmodule	rmodule	NOUN
iajs-778	182	20	.	.	PUNCT
iajs-778	183	1	then	then	ADV
iajs-778	183	2	by	by	ADP
iajs-778	183	3	[	[	X
iajs-778	183	4	3,prop	3,prop	NUM
iajs-778	183	5	.	.	PUNCT
iajs-778	183	6	2.4.1,p.54	2.4.1,p.54	NUM
iajs-778	183	7	]	]	X
iajs-778	183	8			PROPN
iajs-778	183	9			PROPN
iajs-778	183	10	is	be	AUX
iajs-778	183	11	a	a	DET
iajs-778	183	12	coprime	coprime	ADJ
iajs-778	183	13	r	r	NOUN
iajs-778	183	14	-	-	PUNCT
iajs-778	183	15	module	module	NOUN
iajs-778	183	16	and	and	CCONJ
iajs-778	183	17	hence	hence	ADV
iajs-778	183	18	n	n	PRON
iajs-778	183	19	is	be	AUX
iajs-778	183	20	a	a	DET
iajs-778	183	21	coprime	coprime	NOUN
iajs-778	183	22	submodule	submodule	NOUN
iajs-778	183	23	.	.	PUNCT
iajs-778	184	1	1.24	1.24	NUM
iajs-778	184	2	corollary	corollary	NOUN
iajs-778	184	3	:	:	PUNCT
iajs-778	184	4	let	let	VERB
iajs-778	184	5	r	r	PRON
iajs-778	184	6	be	be	AUX
iajs-778	184	7	a	a	DET
iajs-778	184	8	regular	regular	ADJ
iajs-778	184	9	ring	ring	NOUN
iajs-778	184	10	(	(	PUNCT
iajs-778	184	11	in	in	ADP
iajs-778	184	12	sence	sence	NOUN
iajs-778	184	13	of	of	ADP
iajs-778	184	14	von	von	PROPN
iajs-778	184	15	neumann	neumann	PROPN
iajs-778	184	16	)	)	PUNCT
iajs-778	184	17	,	,	PUNCT
iajs-778	184	18	let	let	VERB
iajs-778	184	19	m	m	PRON
iajs-778	184	20	be	be	AUX
iajs-778	184	21	an	an	DET
iajs-778	184	22	r	r	NOUN
iajs-778	184	23	-	-	PUNCT
iajs-778	184	24	module	module	NOUN
iajs-778	184	25	.	.	PUNCT
iajs-778	185	1	then	then	ADV
iajs-778	185	2	every	every	DET
iajs-778	185	3	prime	prime	ADJ
iajs-778	185	4	submodule	submodule	NOUN
iajs-778	185	5	of	of	ADP
iajs-778	185	6	m	m	PROPN
iajs-778	185	7	is	be	AUX
iajs-778	185	8	a	a	DET
iajs-778	185	9	coprime	coprime	ADJ
iajs-778	185	10	submodule	submodule	NOUN
iajs-778	185	11	of	of	ADP
iajs-778	185	12	m.	m.	NOUN
iajs-778	185	13			PROPN
iajs-778	185	14			PROPN
iajs-778	185	15			NOUN
iajs-778	185	16	ibn	ibn	PROPN
iajs-778	185	17	alhaitham	alhaitham	NOUN
iajs-778	185	18	j.	j.	PROPN
iajs-778	185	19	for	for	ADP
iajs-778	185	20	pure	pure	ADJ
iajs-778	185	21	&	&	CCONJ
iajs-778	185	22	appl	appl	PROPN
iajs-778	185	23	.	.	PUNCT
iajs-778	186	1	sci	sci	PROPN
iajs-778	186	2	.	.	PUNCT
iajs-778	186	3	vol.24	vol.24	NOUN
iajs-778	186	4	(	(	PUNCT
iajs-778	186	5	2	2	NUM
iajs-778	186	6	)	)	PUNCT
iajs-778	186	7	2011	2011	NUM
iajs-778	186	8	proof	proof	NOUN
iajs-778	186	9	:	:	PUNCT
iajs-778	186	10	since	since	SCONJ
iajs-778	186	11	r	r	NOUN
iajs-778	186	12	is	be	AUX
iajs-778	186	13	a	a	DET
iajs-778	186	14	regular	regular	ADJ
iajs-778	186	15	ring	ring	NOUN
iajs-778	186	16	,	,	PUNCT
iajs-778	186	17	implies	imply	VERB
iajs-778	186	18	rmn	rmn	PROPN
iajs-778	186	19	=	=	PROPN
iajs-778	186	20	rn	rn	NOUN
iajs-778	186	21	for	for	ADP
iajs-778	186	22	all	all	DET
iajs-778	186	23	r	r	NOUN
iajs-778	186	24	r	r	NOUN
iajs-778	186	25	and	and	CCONJ
iajs-778	186	26	for	for	ADP
iajs-778	186	27	all	all	DET
iajs-778	186	28	n	n	CCONJ
iajs-778	186	29	<	<	X
iajs-778	186	30	m	m	PROPN
iajs-778	186	31	,	,	PUNCT
iajs-778	186	32	then	then	ADV
iajs-778	186	33	the	the	DET
iajs-778	186	34	result	result	NOUN
iajs-778	186	35	is	be	AUX
iajs-778	186	36	obtained	obtain	VERB
iajs-778	186	37	by	by	ADP
iajs-778	186	38	prop.1.23	prop.1.23	PROPN
iajs-778	186	39	.	.	PROPN
iajs-778	186	40	references	reference	NOUN
iajs-778	186	41	1	1	NUM
iajs-778	186	42	.	.	PUNCT
iajs-778	186	43	abuihlail	abuihlail	PROPN
iajs-778	186	44	,	,	PUNCT
iajs-778	186	45	j.	j.	PROPN
iajs-778	186	46	(	(	PUNCT
iajs-778	186	47	2007	2007	NUM
iajs-778	186	48	)	)	PUNCT
iajs-778	186	49	,	,	PUNCT
iajs-778	186	50	zariski	zariski	NOUN
iajs-778	186	51	-	-	PUNCT
iajs-778	186	52	like	like	ADJ
iajs-778	186	53	topologies	topology	NOUN
iajs-778	186	54	for	for	ADP
iajs-778	186	55	modules	module	NOUN
iajs-778	186	56	over	over	ADP
iajs-778	186	57	commutative	commutative	ADJ
iajs-778	186	58	ring	ring	NOUN
iajs-778	186	59	,	,	PUNCT
iajs-778	186	60	accepted	accept	VERB
iajs-778	186	61	(	(	PUNCT
iajs-778	186	62	ar	ar	NOUN
iajs-778	186	63	x	x	PROPN
iajs-778	186	64	iv	iv	PROPN
iajs-778	186	65	,	,	PUNCT
iajs-778	186	66	math.ra	math.ra	NUM
iajs-778	186	67	)	)	PUNCT
iajs-778	186	68	,	,	PUNCT
iajs-778	186	69	pp.1	pp.1	NOUN
iajs-778	186	70	-	-	PUNCT
iajs-778	186	71	19	19	NUM
iajs-778	186	72	.	.	NOUN
iajs-778	186	73	2	2	NUM
iajs-778	186	74	.	.	PUNCT
iajs-778	186	75	annin	annin	PROPN
iajs-778	186	76	,	,	PUNCT
iajs-778	186	77	s.	s.	PROPN
iajs-778	186	78	(	(	PUNCT
iajs-778	186	79	2002	2002	NUM
iajs-778	186	80	)	)	PUNCT
iajs-778	186	81	associated	associate	VERB
iajs-778	186	82	and	and	CCONJ
iajs-778	186	83	attached	attach	VERB
iajs-778	186	84	primes	prime	NOUN
iajs-778	186	85	over	over	ADP
iajs-778	186	86	non	non	ADJ
iajs-778	186	87	commutative	commutative	ADJ
iajs-778	186	88	rings	ring	NOUN
iajs-778	186	89	,	,	PUNCT
iajs-778	186	90	ph.d	ph.d	PROPN
iajs-778	186	91	.	.	PUNCT
iajs-778	187	1	thesis	thesis	PROPN
iajs-778	187	2	university	university	PROPN
iajs-778	187	3	of	of	ADP
iajs-778	187	4	berkeley	berkeley	PROPN
iajs-778	187	5	.	.	PUNCT
iajs-778	188	1	3	3	X
iajs-778	188	2	.	.	X
iajs-778	188	3	khalaf	khalaf	PROPN
iajs-778	188	4	,	,	PUNCT
iajs-778	188	5	r.	r.	PROPN
iajs-778	188	6	i.	i.	PROPN
iajs-778	188	7	(	(	PUNCT
iajs-778	188	8	2009	2009	NUM
iajs-778	188	9	)	)	PUNCT
iajs-778	188	10	,	,	PUNCT
iajs-778	188	11	dual	dual	ADJ
iajs-778	188	12	notions	notion	NOUN
iajs-778	188	13	of	of	ADP
iajs-778	188	14	prime	prime	ADJ
iajs-778	188	15	submodules	submodule	NOUN
iajs-778	188	16	and	and	CCONJ
iajs-778	188	17	prime	prime	ADJ
iajs-778	188	18	modules	module	NOUN
iajs-778	188	19	,	,	PUNCT
iajs-778	188	20	m	m	PROPN
iajs-778	188	21	.sc	.sc	NOUN
iajs-778	188	22	.	.	PUNCT
iajs-778	189	1	thesis	thesis	NOUN
iajs-778	189	2	,	,	PUNCT
iajs-778	189	3	university	university	NOUN
iajs-778	189	4	of	of	ADP
iajs-778	189	5	baghdad	baghdad	PROPN
iajs-778	189	6	.	.	PUNCT
iajs-778	190	1	4	4	X
iajs-778	190	2	.	.	X
iajs-778	191	1	hirano	hirano	PROPN
iajs-778	191	2	,	,	PUNCT
iajs-778	191	3	y.and	y.and	PROPN
iajs-778	191	4	mogami	mogami	PROPN
iajs-778	191	5	,	,	PUNCT
iajs-778	191	6	i.	i.	PROPN
iajs-778	191	7	(	(	PUNCT
iajs-778	191	8	1987	1987	NUM
iajs-778	191	9	)	)	PUNCT
iajs-778	191	10	,	,	PUNCT
iajs-778	191	11	modules	module	NOUN
iajs-778	191	12	whose	whose	DET
iajs-778	191	13	proper	proper	ADJ
iajs-778	191	14	submodules	submodule	NOUN
iajs-778	191	15	are	be	AUX
iajs-778	191	16	non	non	ADJ
iajs-778	191	17	-	-	ADJ
iajs-778	191	18	hopf	hopf	ADJ
iajs-778	191	19	kernels	kernel	NOUN
iajs-778	191	20	"	"	PUNCT
iajs-778	191	21	,	,	PUNCT
iajs-778	191	22	communications	communication	NOUN
iajs-778	191	23	in	in	ADP
iajs-778	191	24	algebra	algebra	NOUN
iajs-778	191	25	,	,	PUNCT
iajs-778	191	26	15:1549	15:1549	NUM
iajs-778	191	27	-	-	SYM
iajs-778	191	28	1567	1567	NUM
iajs-778	191	29	.	.	PUNCT
iajs-778	192	1	5	5	X
iajs-778	192	2	.	.	X
iajs-778	192	3	sharp	sharp	ADJ
iajs-778	192	4	,	,	PUNCT
iajs-778	192	5	r.y	r.y	PROPN
iajs-778	192	6	.	.	PROPN
iajs-778	193	1	(	(	PUNCT
iajs-778	193	2	1990	1990	NUM
iajs-778	193	3	)	)	PUNCT
iajs-778	194	1	,	,	PUNCT
iajs-778	194	2	steps	step	NOUN
iajs-778	194	3	in	in	ADP
iajs-778	194	4	commutative	commutative	ADJ
iajs-778	194	5	algebra	algebra	NOUN
iajs-778	194	6	,	,	PUNCT
iajs-778	194	7	cambridge	cambridge	PROPN
iajs-778	194	8	university	university	PROPN
iajs-778	194	9	press	press	NOUN
iajs-778	194	10	.	.	PUNCT
iajs-778	195	1	6	6	X
iajs-778	195	2	.	.	X
iajs-778	195	3	wijayanti	wijayanti	PROPN
iajs-778	195	4	,	,	PUNCT
iajs-778	195	5	i.	i.	PROPN
iajs-778	195	6	e.	e.	PROPN
iajs-778	195	7	(	(	PUNCT
iajs-778	195	8	2006	2006	NUM
iajs-778	195	9	)	)	PUNCT
iajs-778	195	10	,	,	PUNCT
iajs-778	195	11	coprime	coprime	NOUN
iajs-778	195	12	modules	module	NOUN
iajs-778	195	13	and	and	CCONJ
iajs-778	195	14	comodules	comodule	NOUN
iajs-778	195	15	,	,	PUNCT
iajs-778	195	16	ph.d	ph.d	PROPN
iajs-778	195	17	.	.	PUNCT
iajs-778	196	1	thesis	thesis	NOUN
iajs-778	196	2	,	,	PUNCT
iajs-778	196	3	heinrich	heinrich	PROPN
iajs-778	196	4	-	-	PUNCT
iajs-778	196	5	heine	heine	PROPN
iajs-778	196	6	universität	universität	PROPN
iajs-778	196	7	,	,	PUNCT
iajs-778	196	8	düsseldorf	düsseldorf	PROPN
iajs-778	196	9	.	.	PUNCT
iajs-778	197	1	7	7	NUM
iajs-778	197	2	.	.	NUM
iajs-778	197	3	yassemi	yassemi	NOUN
iajs-778	197	4	,	,	PUNCT
iajs-778	197	5	s.m	s.m	PROPN
iajs-778	197	6	.	.	PROPN
iajs-778	197	7	(	(	PUNCT
iajs-778	197	8	2001	2001	NUM
iajs-778	197	9	)	)	PUNCT
iajs-778	197	10	,	,	PUNCT
iajs-778	197	11	the	the	DET
iajs-778	197	12	dual	dual	ADJ
iajs-778	197	13	notion	notion	NOUN
iajs-778	197	14	of	of	ADP
iajs-778	197	15	prime	prime	ADJ
iajs-778	197	16	submodules	submodule	NOUN
iajs-778	197	17	,	,	PUNCT
iajs-778	197	18	arch	arch	NOUN
iajs-778	197	19	.	.	PUNCT
iajs-778	198	1	math	math	NOUN
iajs-778	198	2	.	.	PUNCT
iajs-778	199	1	(	(	PUNCT
iajs-778	199	2	bron	bron	PROPN
iajs-778	199	3	)	)	PUNCT
iajs-778	199	4	,	,	PUNCT
iajs-778	199	5	37	37	NUM
iajs-778	199	6	:	:	PUNCT
iajs-778	199	7	273	273	NUM
iajs-778	199	8	-	-	SYM
iajs-778	199	9	278	278	NUM
iajs-778	199	10	.	.	PUNCT