id	sid	tid	token	lemma	pos
iajs-966	1	1	ibn	ibn	PROPN
iajs-966	1	2	alhaitham	alhaitham	NOUN
iajs-966	1	3	j.	j.	PROPN
iajs-966	2	1	fo	fo	ADP
iajs-966	2	2	r	r	NOUN
iajs-966	2	3	pure	pure	ADJ
iajs-966	2	4	&	&	CCONJ
iajs-966	2	5	appl	appl	PROPN
iajs-966	2	6	.	.	PUNCT
iajs-966	3	1	sc	sc	PROPN
iajs-966	3	2	i.	i.	PROPN
iajs-966	3	3	vo	vo	PROPN
iajs-966	3	4	l.23	l.23	PROPN
iajs-966	3	5	(	(	PUNCT
iajs-966	3	6	2	2	NUM
iajs-966	3	7	)	)	PUNCT
iajs-966	3	8	2010	2010	NUM
iajs-966	3	9	on	on	ADP
iajs-966	3	10	almost	almost	ADV
iajs-966	3	11	bounded	bound	VERB
iajs-966	3	12	submodules	submodules	PROPN
iajs-966	3	13	b.	b.	PROPN
iajs-966	3	14	n.shihab	n.shihab	PROPN
iajs-966	3	15	department	department	PROPN
iajs-966	3	16	of	of	ADP
iajs-966	3	17	mathematics	mathematics	PROPN
iajs-966	3	18	,	,	PUNCT
iajs-966	3	19	college	college	NOUN
iajs-966	3	20	of	of	ADP
iajs-966	3	21	educationibn	educationibn	PROPN
iajs-966	3	22	-	-	PUNCT
iajs-966	3	23	al	al	PROPN
iajs-966	3	24	-	-	PUNCT
iajs-966	3	25	haitham	haitham	PROPN
iajs-966	3	26	,	,	PUNCT
iajs-966	3	27	univers	univers	PROPN
iajs-966	3	28	ity	ity	PROPN
iajs-966	3	29	of	of	ADP
iajs-966	3	30	baghdad	baghdad	PROPN
iajs-966	3	31	abstract	abstract	ADV
iajs-966	3	32	let	let	VERB
iajs-966	3	33	r	r	PRON
iajs-966	3	34	be	be	AUX
iajs-966	3	35	a	a	DET
iajs-966	3	36	commutative	commutative	ADJ
iajs-966	3	37	ring	ring	NOUN
iajs-966	3	38	with	with	ADP
iajs-966	3	39	identity	identity	NOUN
iajs-966	3	40	,	,	PUNCT
iajs-966	3	41	and	and	CCONJ
iajs-966	3	42	let	let	VERB
iajs-966	3	43	m	m	PRON
iajs-966	3	44	be	be	AUX
iajs-966	3	45	a	a	DET
iajs-966	3	46	unitary	unitary	ADJ
iajs-966	3	47	r	r	NOUN
iajs-966	3	48	-	-	PUNCT
iajs-966	3	49	module	module	NOUN
iajs-966	3	50	.	.	PUNCT
iajs-966	4	1	we	we	PRON
iajs-966	4	2	introduce	introduce	VERB
iajs-966	4	3	a	a	DET
iajs-966	4	4	concept	concept	NOUN
iajs-966	4	5	of	of	ADP
iajs-966	4	6	almost	almost	ADV
iajs-966	4	7	bounded	bound	VERB
iajs-966	4	8	submodules	submodule	NOUN
iajs-966	4	9	as	as	SCONJ
iajs-966	4	10	follows	follow	VERB
iajs-966	4	11	:	:	PUNCT
iajs-966	4	12	a	a	DET
iajs-966	4	13	submodule	submodule	NOUN
iajs-966	4	14	n	n	PROPN
iajs-966	4	15	of	of	ADP
iajs-966	4	16	an	an	DET
iajs-966	4	17	rmodule	rmodule	NOUN
iajs-966	4	18	m	m	VERB
iajs-966	4	19	is	be	AUX
iajs-966	4	20	called	call	VERB
iajs-966	4	21	an	an	DET
iajs-966	4	22	almost	almost	ADV
iajs-966	4	23	bounded	bound	VERB
iajs-966	4	24	submodule	submodule	NOUN
iajs-966	4	25	if	if	SCONJ
iajs-966	4	26	there	there	PRON
iajs-966	4	27	exists	exist	VERB
iajs-966	4	28	xm	xm	NOUN
iajs-966	4	29	,	,	PUNCT
iajs-966	5	1	xn	xn	PROPN
iajs-966	5	2	such	such	ADJ
iajs-966	5	3	that	that	PRON
iajs-966	5	4	annr(n)=annr(x	annr(n)=annr(x	PROPN
iajs-966	5	5	)	)	PUNCT
iajs-966	5	6	.	.	PUNCT
iajs-966	6	1	in	in	ADP
iajs-966	6	2	this	this	DET
iajs-966	6	3	paper	paper	NOUN
iajs-966	6	4	,	,	PUNCT
iajs-966	6	5	some	some	DET
iajs-966	6	6	properties	property	NOUN
iajs-966	6	7	of	of	ADP
iajs-966	6	8	almost	almost	ADV
iajs-966	6	9	bounded	bound	VERB
iajs-966	6	10	submodules	submodule	NOUN
iajs-966	6	11	are	be	AUX
iajs-966	6	12	given	give	VERB
iajs-966	6	13	.	.	PUNCT
iajs-966	7	1	also	also	ADV
iajs-966	7	2	,	,	PUNCT
iajs-966	7	3	various	various	ADJ
iajs-966	7	4	basic	basic	ADJ
iajs-966	7	5	results	result	NOUN
iajs-966	7	6	about	about	ADP
iajs-966	7	7	almost	almost	ADV
iajs-966	7	8	bounded	bound	VERB
iajs-966	7	9	submodules	submodule	NOUN
iajs-966	7	10	are	be	AUX
iajs-966	7	11	considered	consider	VERB
iajs-966	7	12	.	.	PUNCT
iajs-966	8	1	moreover	moreover	ADV
iajs-966	8	2	,	,	PUNCT
iajs-966	8	3	some	some	DET
iajs-966	8	4	relations	relation	NOUN
iajs-966	8	5	between	between	ADP
iajs-966	8	6	almost	almost	ADV
iajs-966	8	7	bounded	bound	VERB
iajs-966	8	8	submodules	submodule	NOUN
iajs-966	8	9	and	and	CCONJ
iajs-966	8	10	other	other	ADJ
iajs-966	8	11	types	type	NOUN
iajs-966	8	12	of	of	ADP
iajs-966	8	13	modules	module	NOUN
iajs-966	8	14	are	be	AUX
iajs-966	8	15	considered	consider	VERB
iajs-966	8	16	.	.	PUNCT
iajs-966	9	1	introduction	introduction	NOUN
iajs-966	9	2	every	every	DET
iajs-966	9	3	ring	ring	NOUN
iajs-966	9	4	considered	consider	VERB
iajs-966	9	5	in	in	ADP
iajs-966	9	6	this	this	DET
iajs-966	9	7	paper	paper	NOUN
iajs-966	9	8	will	will	AUX
iajs-966	9	9	be	be	AUX
iajs-966	9	10	assumed	assume	VERB
iajs-966	9	11	to	to	PART
iajs-966	9	12	be	be	AUX
iajs-966	9	13	commutative	commutative	ADJ
iajs-966	9	14	with	with	ADP
iajs-966	9	15	identity	identity	NOUN
iajs-966	9	16	and	and	CCONJ
iajs-966	9	17	every	every	DET
iajs-966	9	18	module	module	NOUN
iajs-966	9	19	is	be	AUX
iajs-966	9	20	unitary	unitary	ADJ
iajs-966	9	21	.	.	PUNCT
iajs-966	10	1	we	we	PRON
iajs-966	10	2	introduce	introduce	VERB
iajs-966	10	3	the	the	DET
iajs-966	10	4	following	following	NOUN
iajs-966	10	5	:	:	PUNCT
iajs-966	10	6	a	a	DET
iajs-966	10	7	submodule	submodule	NOUN
iajs-966	10	8	n	n	PROPN
iajs-966	10	9	of	of	ADP
iajs-966	10	10	an	an	DET
iajs-966	10	11	r	r	NOUN
iajs-966	10	12	-	-	PUNCT
iajs-966	10	13	module	module	NOUN
iajs-966	10	14	m	m	NOUN
iajs-966	10	15	is	be	AUX
iajs-966	10	16	called	call	VERB
iajs-966	10	17	an	an	DET
iajs-966	10	18	almost	almost	ADV
iajs-966	10	19	bounded	bound	VERB
iajs-966	10	20	submodule	submodule	NOUN
iajs-966	10	21	,	,	PUNCT
iajs-966	10	22	if	if	SCONJ
iajs-966	10	23	there	there	PRON
iajs-966	10	24	exists	exist	VERB
iajs-966	10	25	xm	xm	NOUN
iajs-966	10	26	,	,	PUNCT
iajs-966	11	1	xn	xn	PROPN
iajs-966	11	2	such	such	ADJ
iajs-966	11	3	that	that	PRON
iajs-966	11	4	annr(n)=annr(x	annr(n)=annr(x	PROPN
iajs-966	11	5	)	)	PUNCT
iajs-966	11	6	,	,	PUNCT
iajs-966	11	7	where	where	SCONJ
iajs-966	11	8	annrn={r	annrn={r	VERB
iajs-966	11	9	:	:	PUNCT
iajs-966	11	10	rr	rr	CCONJ
iajs-966	11	11	and	and	CCONJ
iajs-966	11	12	rn=0	rn=0	NUM
iajs-966	11	13	}	}	PUNCT
iajs-966	11	14	.	.	PUNCT
iajs-966	12	1	our	our	PRON
iajs-966	12	2	concern	concern	NOUN
iajs-966	12	3	in	in	ADP
iajs-966	12	4	this	this	DET
iajs-966	12	5	paper	paper	NOUN
iajs-966	12	6	is	be	AUX
iajs-966	12	7	to	to	PART
iajs-966	12	8	study	study	VERB
iajs-966	12	9	almost	almost	ADV
iajs-966	12	10	bounded	bound	VERB
iajs-966	12	11	submodules	submodule	NOUN
iajs-966	12	12	and	and	CCONJ
iajs-966	12	13	to	to	PART
iajs-966	12	14	look	look	VERB
iajs-966	12	15	for	for	ADP
iajs-966	12	16	any	any	DET
iajs-966	12	17	relation	relation	NOUN
iajs-966	12	18	between	between	ADP
iajs-966	12	19	almost	almost	ADV
iajs-966	12	20	bounded	bound	VERB
iajs-966	12	21	submodules	submodule	NOUN
iajs-966	12	22	and	and	CCONJ
iajs-966	12	23	certain	certain	ADJ
iajs-966	12	24	types	type	NOUN
iajs-966	12	25	of	of	ADP
iajs-966	12	26	well	well	ADV
iajs-966	12	27	-	-	PUNCT
iajs-966	12	28	known	know	VERB
iajs-966	12	29	modules	module	NOUN
iajs-966	12	30	especially	especially	ADV
iajs-966	12	31	with	with	ADP
iajs-966	12	32	prime	prime	ADJ
iajs-966	12	33	modules	module	NOUN
iajs-966	12	34	.	.	PUNCT
iajs-966	13	1	this	this	DET
iajs-966	13	2	paper	paper	NOUN
iajs-966	13	3	consists	consist	VERB
iajs-966	13	4	of	of	ADP
iajs-966	13	5	two	two	NUM
iajs-966	13	6	sections	section	NOUN
iajs-966	13	7	.	.	PUNCT
iajs-966	14	1	our	our	PRON
iajs-966	14	2	main	main	ADJ
iajs-966	14	3	concern	concern	NOUN
iajs-966	14	4	in	in	ADP
iajs-966	14	5	section	section	NOUN
iajs-966	14	6	one	one	NUM
iajs-966	14	7	,	,	PUNCT
iajs-966	14	8	is	be	AUX
iajs-966	14	9	to	to	PART
iajs-966	14	10	define	define	VERB
iajs-966	14	11	and	and	CCONJ
iajs-966	14	12	study	study	VERB
iajs-966	14	13	almost	almost	ADV
iajs-966	14	14	bounded	bound	VERB
iajs-966	14	15	submodules	submodule	NOUN
iajs-966	14	16	.	.	PUNCT
iajs-966	15	1	also	also	ADV
iajs-966	15	2	,	,	PUNCT
iajs-966	15	3	we	we	PRON
iajs-966	15	4	give	give	VERB
iajs-966	15	5	some	some	DET
iajs-966	15	6	basic	basic	ADJ
iajs-966	15	7	results	result	NOUN
iajs-966	15	8	for	for	ADP
iajs-966	15	9	this	this	DET
iajs-966	15	10	concept	concept	NOUN
iajs-966	15	11	.	.	PUNCT
iajs-966	16	1	in	in	ADP
iajs-966	16	2	section	section	NOUN
iajs-966	16	3	two	two	NUM
iajs-966	16	4	,	,	PUNCT
iajs-966	16	5	we	we	PRON
iajs-966	16	6	study	study	VERB
iajs-966	16	7	the	the	DET
iajs-966	16	8	relation	relation	NOUN
iajs-966	16	9	between	between	ADP
iajs-966	16	10	almost	almost	ADV
iajs-966	16	11	bounded	bound	VERB
iajs-966	16	12	submodules	submodule	NOUN
iajs-966	16	13	and	and	CCONJ
iajs-966	16	14	bounded	bounded	ADJ
iajs-966	16	15	modules	module	NOUN
iajs-966	16	16	.	.	PUNCT
iajs-966	17	1	we	we	PRON
iajs-966	17	2	show	show	VERB
iajs-966	17	3	that	that	SCONJ
iajs-966	17	4	the	the	DET
iajs-966	17	5	proper	proper	ADJ
iajs-966	17	6	submodule	submodule	NOUN
iajs-966	17	7	of	of	ADP
iajs-966	17	8	bounded	bound	VERB
iajs-966	17	9	module	module	NOUN
iajs-966	17	10	is	be	AUX
iajs-966	17	11	not	not	PART
iajs-966	17	12	necessary	necessary	ADJ
iajs-966	17	13	to	to	PART
iajs-966	17	14	be	be	AUX
iajs-966	17	15	almost	almost	ADV
iajs-966	17	16	bounded	bound	VERB
iajs-966	17	17	submodule	submodule	NOUN
iajs-966	17	18	and	and	CCONJ
iajs-966	17	19	we	we	PRON
iajs-966	17	20	give	give	VERB
iajs-966	17	21	some	some	DET
iajs-966	17	22	conditions	condition	NOUN
iajs-966	17	23	under	under	ADP
iajs-966	17	24	which	which	PRON
iajs-966	17	25	a	a	DET
iajs-966	17	26	proper	proper	ADJ
iajs-966	17	27	submodule	submodule	NOUN
iajs-966	17	28	of	of	ADP
iajs-966	17	29	bounded	bound	VERB
iajs-966	17	30	module	module	NOUN
iajs-966	17	31	is	be	AUX
iajs-966	17	32	an	an	DET
iajs-966	17	33	almost	almost	ADV
iajs-966	17	34	bounded	bound	VERB
iajs-966	17	35	submodule	submodule	NOUN
iajs-966	17	36	.	.	PUNCT
iajs-966	18	1	next	next	ADV
iajs-966	18	2	we	we	PRON
iajs-966	18	3	investigate	investigate	VERB
iajs-966	18	4	the	the	DET
iajs-966	18	5	relationships	relationship	NOUN
iajs-966	18	6	between	between	ADP
iajs-966	18	7	almost	almost	ADV
iajs-966	18	8	bounded	bound	VERB
iajs-966	18	9	submodules	submodule	NOUN
iajs-966	18	10	,	,	PUNCT
iajs-966	18	11	prime	prime	ADJ
iajs-966	18	12	and	and	CCONJ
iajs-966	18	13	fully	fully	ADV
iajs-966	18	14	stable	stable	ADJ
iajs-966	18	15	module	module	NOUN
iajs-966	18	16	.	.	PUNCT
iajs-966	19	1	1basic	1basic	NUM
iajs-966	19	2	properties	property	NOUN
iajs-966	19	3	of	of	ADP
iajs-966	19	4	almost	almost	ADV
iajs-966	19	5	bounded	bound	VERB
iajs-966	19	6	submodules	submodule	NOUN
iajs-966	19	7	in	in	ADP
iajs-966	19	8	this	this	DET
iajs-966	19	9	section	section	NOUN
iajs-966	19	10	,	,	PUNCT
iajs-966	19	11	we	we	PRON
iajs-966	19	12	introduce	introduce	VERB
iajs-966	19	13	the	the	DET
iajs-966	19	14	concept	concept	NOUN
iajs-966	19	15	of	of	ADP
iajs-966	19	16	almost	almost	ADV
iajs-966	19	17	bounded	bound	VERB
iajs-966	19	18	submodule	submodule	NOUN
iajs-966	19	19	.	.	PUNCT
iajs-966	20	1	we	we	PRON
iajs-966	20	2	establishe	establishe	VERB
iajs-966	20	3	some	some	DET
iajs-966	20	4	basic	basic	ADJ
iajs-966	20	5	properties	property	NOUN
iajs-966	20	6	of	of	ADP
iajs-966	20	7	this	this	DET
iajs-966	20	8	concept	concept	NOUN
iajs-966	20	9	.	.	PUNCT
iajs-966	21	1	first	first	ADV
iajs-966	21	2	,	,	PUNCT
iajs-966	21	3	we	we	PRON
iajs-966	21	4	introduce	introduce	VERB
iajs-966	21	5	the	the	DET
iajs-966	21	6	following	following	ADJ
iajs-966	21	7	definition	definition	NOUN
iajs-966	21	8	.	.	PUNCT
iajs-966	22	1	1.1	1.1	NUM
iajs-966	22	2	definition	definition	NOUN
iajs-966	22	3	:	:	PUNCT
iajs-966	22	4	a	a	DET
iajs-966	22	5	proper	proper	ADJ
iajs-966	22	6	submodule	submodule	NOUN
iajs-966	22	7	n	n	PROPN
iajs-966	22	8	of	of	ADP
iajs-966	22	9	an	an	DET
iajs-966	22	10	r	r	NOUN
iajs-966	22	11	-	-	PUNCT
iajs-966	22	12	module	module	NOUN
iajs-966	22	13	m	m	NOUN
iajs-966	22	14	is	be	AUX
iajs-966	22	15	called	call	VERB
iajs-966	22	16	almost	almost	ADV
iajs-966	22	17	bounded	bound	VERB
iajs-966	22	18	submodule	submodule	NOUN
iajs-966	22	19	if	if	SCONJ
iajs-966	22	20	there	there	PRON
iajs-966	22	21	exists	exist	VERB
iajs-966	22	22	xm	xm	NOUN
iajs-966	22	23	,	,	PUNCT
iajs-966	22	24	xn	xn	PROPN
iajs-966	22	25	such	such	ADJ
iajs-966	22	26	that	that	PRON
iajs-966	22	27	annr(n)=annr(x	annr(n)=annr(x	PROPN
iajs-966	22	28	)	)	PUNCT
iajs-966	22	29	.	.	PUNCT
iajs-966	23	1	an	an	DET
iajs-966	23	2	ideal	ideal	ADJ
iajs-966	23	3	i	i	PRON
iajs-966	23	4	of	of	ADP
iajs-966	23	5	a	a	DET
iajs-966	23	6	ring	ring	NOUN
iajs-966	23	7	r	r	NOUN
iajs-966	23	8	is	be	AUX
iajs-966	23	9	an	an	DET
iajs-966	23	10	almost	almost	ADV
iajs-966	23	11	bounded	bound	VERB
iajs-966	23	12	ideal	ideal	NOUN
iajs-966	23	13	if	if	SCONJ
iajs-966	23	14	i	i	PRON
iajs-966	23	15	is	be	AUX
iajs-966	23	16	an	an	DET
iajs-966	23	17	almost	almost	ADV
iajs-966	23	18	bounded	bound	VERB
iajs-966	23	19	rsubmodule	rsubmodule	NOUN
iajs-966	23	20	.	.	PUNCT
iajs-966	24	1	1.2	1.2	NUM
iajs-966	24	2	remarks	remark	NOUN
iajs-966	24	3	and	and	CCONJ
iajs-966	24	4	examples	example	NOUN
iajs-966	24	5	:	:	PUNCT
iajs-966	24	6	1	1	X
iajs-966	24	7	.	.	X
iajs-966	24	8	let	let	VERB
iajs-966	24	9	m	m	NOUN
iajs-966	24	10	=	=	NOUN
iajs-966	24	11	zz	zz	PROPN
iajs-966	24	12	as	as	ADP
iajs-966	24	13	a	a	DET
iajs-966	24	14	z	z	NOUN
iajs-966	24	15	-	-	PUNCT
iajs-966	24	16	module	module	NOUN
iajs-966	24	17	and	and	CCONJ
iajs-966	24	18	n=2z0	n=2z0	NOUN
iajs-966	24	19	be	be	VERB
iajs-966	24	20	a	a	DET
iajs-966	24	21	submodule	submodule	NOUN
iajs-966	24	22	of	of	ADP
iajs-966	24	23	m.	m.	NOUN
iajs-966	24	24	then	then	ADV
iajs-966	24	25	n	n	PRON
iajs-966	24	26	is	be	AUX
iajs-966	24	27	an	an	DET
iajs-966	24	28	almost	almost	ADV
iajs-966	24	29	bounded	bound	VERB
iajs-966	24	30	submodule	submodule	NOUN
iajs-966	24	31	.	.	PUNCT
iajs-966	25	1	2	2	X
iajs-966	25	2	.	.	X
iajs-966	25	3	every	every	DET
iajs-966	25	4	submodule	submodule	NOUN
iajs-966	25	5	of	of	ADP
iajs-966	25	6	the	the	DET
iajs-966	25	7	z	z	NOUN
iajs-966	25	8	-	-	PUNCT
iajs-966	25	9	module	module	NOUN
iajs-966	25	10	z	z	NOUN
iajs-966	25	11	is	be	AUX
iajs-966	25	12	an	an	DET
iajs-966	25	13	almost	almost	ADV
iajs-966	25	14	bounded	bound	VERB
iajs-966	25	15	submodule	submodule	NOUN
iajs-966	25	16	.	.	PUNCT
iajs-966	26	1	key	key	ADJ
iajs-966	26	2	words	word	NOUN
iajs-966	26	3	:	:	PUNCT
iajs-966	26	4	almost	almost	ADV
iajs-966	26	5	bounded	bounded	ADJ
iajs-966	26	6	submodule	submodule	NOUN
iajs-966	26	7	,	,	PUNCT
iajs-966	26	8	bounded	bound	VERB
iajs-966	26	9	module	module	NOUN
iajs-966	26	10	,	,	PUNCT
iajs-966	26	11	prime	prime	ADJ
iajs-966	26	12	module	module	NOUN
iajs-966	26	13	,	,	PUNCT
iajs-966	26	14	quasi	quasi	ADJ
iajs-966	26	15	-	-	ADJ
iajs-966	26	16	prime	prime	ADJ
iajs-966	26	17	module	module	NOUN
iajs-966	26	18	,	,	PUNCT
iajs-966	26	19	fully	fully	ADV
iajs-966	26	20	stable	stable	ADJ
iajs-966	26	21	module	module	NOUN
iajs-966	26	22	.	.	PUNCT
iajs-966	27	1	ihjpas	ihjpa	VERB
iajs-966	27	2	ibn	ibn	PROPN
iajs-966	27	3	alhaitham	alhaitham	PROPN
iajs-966	28	1	j.	j.	PROPN
iajs-966	29	1	fo	fo	ADP
iajs-966	29	2	r	r	NOUN
iajs-966	29	3	pure	pure	ADJ
iajs-966	29	4	&	&	CCONJ
iajs-966	29	5	appl	appl	PROPN
iajs-966	29	6	.	.	PUNCT
iajs-966	30	1	sc	sc	PROPN
iajs-966	30	2	i.	i.	PROPN
iajs-966	30	3	vo	vo	PROPN
iajs-966	30	4	l.23	l.23	PROPN
iajs-966	30	5	(	(	PUNCT
iajs-966	30	6	2	2	NUM
iajs-966	30	7	)	)	PUNCT
iajs-966	30	8	2010	2010	NUM
iajs-966	30	9	3	3	NUM
iajs-966	30	10	.	.	PUNCT
iajs-966	31	1	consider	consider	VERB
iajs-966	31	2	the	the	DET
iajs-966	31	3	z	z	NOUN
iajs-966	31	4	-	-	PUNCT
iajs-966	31	5	module	module	NOUN
iajs-966	31	6	m	m	NOUN
iajs-966	31	7	=	=	NOUN
iajs-966	31	8	zzp	zzp	NOUN
iajs-966	31	9	,	,	PUNCT
iajs-966	31	10	where	where	SCONJ
iajs-966	31	11	p	p	NOUN
iajs-966	31	12	is	be	AUX
iajs-966	31	13	a	a	DET
iajs-966	31	14	prime	prime	ADJ
iajs-966	31	15	number	number	NOUN
iajs-966	31	16	and	and	CCONJ
iajs-966	31	17	the	the	DET
iajs-966	31	18	z	z	NOUN
iajs-966	31	19	-	-	PUNCT
iajs-966	31	20	suubmodule	suubmodule	NOUN
iajs-966	31	21	n	n	CCONJ
iajs-966	31	22	=	=	NOUN
iajs-966	31	23	qzzp	qzzp	PROPN
iajs-966	31	24	,	,	PUNCT
iajs-966	31	25	where	where	SCONJ
iajs-966	31	26	q	q	NOUN
iajs-966	31	27	is	be	AUX
iajs-966	31	28	any	any	DET
iajs-966	31	29	prime	prime	ADJ
iajs-966	31	30	number	number	NOUN
iajs-966	31	31	.	.	PUNCT
iajs-966	32	1	then	then	ADV
iajs-966	32	2	n	n	CCONJ
iajs-966	32	3	an	an	DET
iajs-966	32	4	almost	almost	ADV
iajs-966	32	5	bounded	bound	VERB
iajs-966	32	6	sumodule	sumodule	NOUN
iajs-966	32	7	.	.	PUNCT
iajs-966	33	1	4	4	X
iajs-966	33	2	.	.	X
iajs-966	33	3	for	for	ADP
iajs-966	33	4	each	each	DET
iajs-966	33	5	positive	positive	ADJ
iajs-966	33	6	integer	integer	NOUN
iajs-966	33	7	n	n	NOUN
iajs-966	33	8	and	and	CCONJ
iajs-966	33	9	n	n	PROPN
iajs-966	33	10	is	be	AUX
iajs-966	33	11	not	not	PART
iajs-966	33	12	prime	prime	ADJ
iajs-966	33	13	number	number	NOUN
iajs-966	33	14	,	,	PUNCT
iajs-966	33	15	every	every	DET
iajs-966	33	16	proper	proper	ADJ
iajs-966	33	17	submodule	submodule	NOUN
iajs-966	33	18	of	of	ADP
iajs-966	33	19	a	a	DET
iajs-966	33	20	znmodule	znmodule	NOUN
iajs-966	33	21	zn	zn	PROPN
iajs-966	33	22	is	be	AUX
iajs-966	33	23	not	not	PART
iajs-966	33	24	almost	almost	ADV
iajs-966	33	25	bounded	bound	VERB
iajs-966	33	26	submodule	submodule	NOUN
iajs-966	33	27	.	.	PUNCT
iajs-966	34	1	5	5	X
iajs-966	34	2	.	.	X
iajs-966	34	3	2	2	NUM
iajs-966	34	4			INTJ
iajs-966	34	5	as	as	ADP
iajs-966	34	6	a	a	DET
iajs-966	34	7	z	z	NOUN
iajs-966	34	8	-	-	PUNCT
iajs-966	34	9	submodule	submodule	NOUN
iajs-966	34	10	of	of	ADP
iajs-966	34	11	z12	z12	PROPN
iajs-966	34	12	is	be	AUX
iajs-966	34	13	not	not	PART
iajs-966	34	14	almost	almost	ADV
iajs-966	34	15	bounded	bound	VERB
iajs-966	34	16	.	.	PUNCT
iajs-966	35	1	in	in	ADP
iajs-966	35	2	general	general	ADJ
iajs-966	35	3	,	,	PUNCT
iajs-966	35	4	let	let	VERB
iajs-966	35	5	n	n	PRON
iajs-966	35	6	be	be	AUX
iajs-966	35	7	a	a	DET
iajs-966	35	8	positive	positive	ADJ
iajs-966	35	9	integer	integer	NOUN
iajs-966	35	10	,	,	PUNCT
iajs-966	35	11	then	then	ADV
iajs-966	35	12	the	the	DET
iajs-966	35	13	z	z	NOUN
iajs-966	35	14	-	-	PUNCT
iajs-966	35	15	module	module	NOUN
iajs-966	35	16	zn	zn	NOUN
iajs-966	35	17	has	have	VERB
iajs-966	35	18	no	no	DET
iajs-966	35	19	proper	proper	ADJ
iajs-966	35	20	almost	almost	ADV
iajs-966	35	21	bounded	bound	VERB
iajs-966	35	22	submodule	submodule	NOUN
iajs-966	35	23	.	.	PUNCT
iajs-966	36	1	6	6	X
iajs-966	36	2	.	.	X
iajs-966	36	3	let	let	VERB
iajs-966	36	4	p	p	PRON
iajs-966	36	5	be	be	AUX
iajs-966	36	6	a	a	DET
iajs-966	36	7	prime	prime	ADJ
iajs-966	36	8	number	number	NOUN
iajs-966	36	9	.	.	PUNCT
iajs-966	37	1	the	the	DET
iajs-966	37	2	z	z	NOUN
iajs-966	37	3	-	-	PUNCT
iajs-966	37	4	module	module	NOUN
iajs-966	37	5	zp	zp	PROPN
iajs-966	37	6	dose	dose	AUX
iajs-966	37	7	not	not	PART
iajs-966	37	8	contain	contain	VERB
iajs-966	37	9	any	any	DET
iajs-966	37	10	proper	proper	ADJ
iajs-966	37	11	almost	almost	ADV
iajs-966	37	12	bounded	bound	VERB
iajs-966	37	13	submodule	submodule	NOUN
iajs-966	37	14	.	.	PUNCT
iajs-966	38	1	the	the	DET
iajs-966	38	2	following	follow	VERB
iajs-966	38	3	remark	remark	NOUN
iajs-966	38	4	ensures	ensure	VERB
iajs-966	38	5	that	that	SCONJ
iajs-966	38	6	the	the	DET
iajs-966	38	7	almost	almost	ADV
iajs-966	38	8	boundedness	boundedness	NOUN
iajs-966	38	9	property	property	NOUN
iajs-966	38	10	is	be	AUX
iajs-966	38	11	not	not	PART
iajs-966	38	12	hereditary	hereditary	ADJ
iajs-966	38	13	.	.	PUNCT
iajs-966	39	1	1.3	1.3	NUM
iajs-966	39	2	remark	remark	NOUN
iajs-966	39	3	:	:	PUNCT
iajs-966	39	4	a	a	DET
iajs-966	39	5	submodule	submodule	NOUN
iajs-966	39	6	of	of	ADP
iajs-966	39	7	an	an	DET
iajs-966	39	8	almost	almost	ADV
iajs-966	39	9	bounded	bound	VERB
iajs-966	39	10	submodule	submodule	NOUN
iajs-966	39	11	need	need	AUX
iajs-966	39	12	not	not	PART
iajs-966	39	13	be	be	AUX
iajs-966	39	14	almost	almost	ADV
iajs-966	39	15	bounded	bound	VERB
iajs-966	39	16	in	in	ADP
iajs-966	39	17	general	general	ADJ
iajs-966	39	18	.	.	PUNCT
iajs-966	40	1	for	for	ADP
iajs-966	40	2	example	example	NOUN
iajs-966	40	3	:	:	PUNCT
iajs-966	40	4	m	m	PUNCT
iajs-966	40	5	=	=	NOUN
iajs-966	40	6	zzp	zzp	PROPN
iajs-966	40	7	as	as	ADP
iajs-966	40	8	a	a	DET
iajs-966	40	9	z	z	NOUN
iajs-966	40	10	-	-	PUNCT
iajs-966	40	11	module	module	NOUN
iajs-966	40	12	,	,	PUNCT
iajs-966	40	13	where	where	SCONJ
iajs-966	40	14	p	p	NOUN
iajs-966	40	15	any	any	DET
iajs-966	40	16	prime	prime	ADJ
iajs-966	40	17	number	number	NOUN
iajs-966	40	18	,	,	PUNCT
iajs-966	40	19	n	n	CCONJ
iajs-966	40	20	=	=	NOUN
iajs-966	40	21	qzzp	qzzp	NOUN
iajs-966	40	22	be	be	AUX
iajs-966	40	23	a	a	DET
iajs-966	40	24	submodule	submodule	NOUN
iajs-966	40	25	of	of	ADP
iajs-966	40	26	m	m	PROPN
iajs-966	40	27	,	,	PUNCT
iajs-966	40	28	where	where	SCONJ
iajs-966	40	29	q	q	NOUN
iajs-966	40	30	is	be	AUX
iajs-966	40	31	any	any	DET
iajs-966	40	32	prime	prime	ADJ
iajs-966	40	33	number	number	NOUN
iajs-966	40	34	.	.	PUNCT
iajs-966	41	1	then	then	ADV
iajs-966	41	2	n	n	PRON
iajs-966	41	3	is	be	AUX
iajs-966	41	4	an	an	DET
iajs-966	41	5	almost	almost	ADV
iajs-966	41	6	bounded	bound	VERB
iajs-966	41	7	submodule	submodule	NOUN
iajs-966	41	8	of	of	ADP
iajs-966	41	9	m	m	PROPN
iajs-966	41	10	,	,	PUNCT
iajs-966	41	11	but	but	CCONJ
iajs-966	41	12	k=0zp	k=0zp	NOUN
iajs-966	41	13	as	as	ADP
iajs-966	41	14	a	a	DET
iajs-966	41	15	submodule	submodule	NOUN
iajs-966	41	16	of	of	ADP
iajs-966	41	17	n	n	PRON
iajs-966	41	18	which	which	PRON
iajs-966	41	19	is	be	AUX
iajs-966	41	20	not	not	PART
iajs-966	41	21	almost	almost	ADV
iajs-966	41	22	bounded	bound	VERB
iajs-966	41	23	submodule	submodule	NOUN
iajs-966	41	24	of	of	ADP
iajs-966	41	25	n.	n.	PROPN
iajs-966	41	26	we	we	PRON
iajs-966	41	27	state	state	VERB
iajs-966	41	28	and	and	CCONJ
iajs-966	41	29	prove	prove	VERB
iajs-966	41	30	the	the	DET
iajs-966	41	31	following	follow	VERB
iajs-966	41	32	proposition	proposition	NOUN
iajs-966	41	33	.	.	PUNCT
iajs-966	42	1	1.4	1.4	NUM
iajs-966	42	2	proposition	proposition	NOUN
iajs-966	42	3	:	:	PUNCT
iajs-966	42	4	let	let	VERB
iajs-966	42	5	m	m	PRON
iajs-966	42	6	1	1	NUM
iajs-966	42	7	and	and	CCONJ
iajs-966	42	8	m	m	PROPN
iajs-966	42	9	2	2	NUM
iajs-966	42	10	be	be	VERB
iajs-966	42	11	two	two	NUM
iajs-966	42	12	r	r	NOUN
iajs-966	42	13	-	-	PUNCT
iajs-966	42	14	modules	module	NOUN
iajs-966	42	15	,	,	PUNCT
iajs-966	42	16	m	m	NOUN
iajs-966	42	17	=	=	NOUN
iajs-966	42	18	m	m	PROPN
iajs-966	42	19	1m	1m	NUM
iajs-966	42	20	2	2	NUM
iajs-966	42	21	.	.	PUNCT
iajs-966	43	1	if	if	SCONJ
iajs-966	43	2	n1	n1	PROPN
iajs-966	43	3	and	and	CCONJ
iajs-966	43	4	n2	n2	NOUN
iajs-966	43	5	are	be	AUX
iajs-966	43	6	almost	almost	ADV
iajs-966	43	7	bounded	bound	VERB
iajs-966	43	8	rsubmodules	rsubmodule	NOUN
iajs-966	43	9	of	of	ADP
iajs-966	43	10	m	m	PROPN
iajs-966	43	11	1	1	NUM
iajs-966	43	12	and	and	CCONJ
iajs-966	43	13	m2	m2	PROPN
iajs-966	43	14	respectively	respectively	ADV
iajs-966	43	15	,	,	PUNCT
iajs-966	43	16	then	then	ADV
iajs-966	43	17	n1n2	n1n2	PROPN
iajs-966	43	18	is	be	AUX
iajs-966	43	19	an	an	DET
iajs-966	43	20	almost	almost	ADV
iajs-966	43	21	bounded	bound	VERB
iajs-966	43	22	r	r	NOUN
iajs-966	43	23	-	-	PUNCT
iajs-966	43	24	submodule	submodule	NOUN
iajs-966	43	25	of	of	ADP
iajs-966	43	26	m.	m.	NOUN
iajs-966	43	27	proof	proof	NOUN
iajs-966	43	28	:	:	PUNCT
iajs-966	43	29	we	we	PRON
iajs-966	43	30	have	have	VERB
iajs-966	43	31	n1	n1	NOUN
iajs-966	43	32	and	and	CCONJ
iajs-966	43	33	n2	n2	NOUN
iajs-966	43	34	are	be	AUX
iajs-966	43	35	almost	almost	ADV
iajs-966	43	36	bounded	bound	VERB
iajs-966	43	37	r	r	NOUN
iajs-966	43	38	-	-	PUNCT
iajs-966	43	39	submodules	submodule	NOUN
iajs-966	43	40	of	of	ADP
iajs-966	43	41	m	m	PROPN
iajs-966	43	42	1	1	NUM
iajs-966	43	43	and	and	CCONJ
iajs-966	43	44	m	m	PROPN
iajs-966	43	45	2	2	NUM
iajs-966	43	46	respectively	respectively	ADV
iajs-966	43	47	.	.	PUNCT
iajs-966	44	1	then	then	ADV
iajs-966	44	2	there	there	PRON
iajs-966	44	3	exists	exist	VERB
iajs-966	44	4	xm	xm	X
iajs-966	44	5	1	1	X
iajs-966	44	6	,	,	PUNCT
iajs-966	44	7	xn1	xn1	PUNCT
iajs-966	44	8	such	such	ADJ
iajs-966	44	9	that	that	DET
iajs-966	44	10	annrn1	annrn1	PROPN
iajs-966	44	11	=	=	SYM
iajs-966	44	12	annr(x	annr(x	NOUN
iajs-966	44	13	)	)	PUNCT
iajs-966	44	14	and	and	CCONJ
iajs-966	44	15	also	also	ADV
iajs-966	44	16	there	there	PRON
iajs-966	44	17	exists	exist	VERB
iajs-966	44	18	ym	ym	NOUN
iajs-966	44	19	2	2	NUM
iajs-966	44	20	,	,	PUNCT
iajs-966	44	21	yn2	yn2	NOUN
iajs-966	44	22	such	such	ADJ
iajs-966	44	23	that	that	PRON
iajs-966	44	24	annrn2	annrn2	NOUN
iajs-966	44	25	=	=	SYM
iajs-966	44	26	annr(y	annr(y	VERB
iajs-966	44	27	)	)	PUNCT
iajs-966	44	28	.	.	PUNCT
iajs-966	45	1	therefore	therefore	ADV
iajs-966	45	2	(	(	PUNCT
iajs-966	45	3	x	x	X
iajs-966	45	4	,	,	PUNCT
iajs-966	45	5	y)m	y)m	PROPN
iajs-966	45	6	1m	1m	PROPN
iajs-966	45	7	2	2	NUM
iajs-966	45	8	,	,	PUNCT
iajs-966	45	9	(	(	PUNCT
iajs-966	45	10	x	x	NOUN
iajs-966	45	11	,	,	PUNCT
iajs-966	45	12	y	y	PROPN
iajs-966	45	13	)	)	PUNCT
iajs-966	45	14			NOUN
iajs-966	45	15	n1n2	n1n2	NOUN
iajs-966	45	16	.	.	PUNCT
iajs-966	46	1	now	now	ADV
iajs-966	46	2	,	,	PUNCT
iajs-966	46	3	annr(x	annr(x	PROPN
iajs-966	46	4	,	,	PUNCT
iajs-966	46	5	y	y	NOUN
iajs-966	46	6	)	)	PUNCT
iajs-966	46	7	=	=	SYM
iajs-966	46	8	annr(x	annr(x	NOUN
iajs-966	46	9	)	)	PUNCT
iajs-966	46	10			NOUN
iajs-966	46	11	annr(y	annr(y	VERB
iajs-966	46	12	)	)	PUNCT
iajs-966	46	13	=	=	PUNCT
iajs-966	46	14	annrn1	annrn1	NOUN
iajs-966	46	15			PUNCT
iajs-966	46	16	annrn2	annrn2	NOUN
iajs-966	46	17	=	=	PUNCT
iajs-966	46	18	annr(n1n2	annr(n1n2	NUM
iajs-966	46	19	)	)	PUNCT
iajs-966	46	20	.	.	PUNCT
iajs-966	47	1	hence	hence	ADV
iajs-966	47	2	n1n2	n1n2	PROPN
iajs-966	47	3	is	be	AUX
iajs-966	47	4	an	an	DET
iajs-966	47	5	almost	almost	ADV
iajs-966	47	6	bounded	bound	VERB
iajs-966	47	7	r	r	NOUN
iajs-966	47	8	-	-	PUNCT
iajs-966	47	9	submodule	submodule	NOUN
iajs-966	47	10	of	of	ADP
iajs-966	47	11	m.	m.	NOUN
iajs-966	47	12	the	the	DET
iajs-966	47	13	converse	converse	NOUN
iajs-966	47	14	of	of	ADP
iajs-966	47	15	proposition	proposition	NOUN
iajs-966	47	16	(	(	PUNCT
iajs-966	47	17	1.4	1.4	NUM
iajs-966	47	18	)	)	PUNCT
iajs-966	47	19	is	be	AUX
iajs-966	47	20	not	not	PART
iajs-966	47	21	true	true	ADJ
iajs-966	47	22	in	in	ADP
iajs-966	47	23	general	general	ADJ
iajs-966	47	24	as	as	SCONJ
iajs-966	47	25	the	the	DET
iajs-966	47	26	following	follow	VERB
iajs-966	47	27	example	example	NOUN
iajs-966	47	28	shows	show	NOUN
iajs-966	47	29	.	.	PUNCT
iajs-966	48	1	1.5	1.5	NUM
iajs-966	48	2	example	example	NOUN
iajs-966	48	3	:	:	PUNCT
iajs-966	48	4	consider	consider	VERB
iajs-966	48	5	m	m	NOUN
iajs-966	48	6	=	=	NOUN
iajs-966	48	7	z6z12	z6z12	NOUN
iajs-966	48	8	as	as	ADP
iajs-966	48	9	a	a	DET
iajs-966	48	10	z	z	NOUN
iajs-966	48	11	-	-	PUNCT
iajs-966	48	12	module	module	NOUN
iajs-966	48	13	.	.	PUNCT
iajs-966	49	1	let	let	VERB
iajs-966	49	2	n=	n=	ADJ
iajs-966	49	3	n1n2=	n1n2=	NOUN
iajs-966	49	4	3	3	NUM
iajs-966	50	1	2	2	NUM
iajs-966	50	2			INTJ
iajs-966	50	3			PROPN
iajs-966	50	4			PROPN
iajs-966	51	1			INTJ
iajs-966	51	2	be	be	AUX
iajs-966	51	3	a	a	DET
iajs-966	51	4	zsubmodule	zsubmodule	NOUN
iajs-966	51	5	of	of	ADP
iajs-966	51	6	m.	m.	NOUN
iajs-966	51	7	then	then	ADV
iajs-966	51	8	n	n	PRON
iajs-966	51	9	is	be	AUX
iajs-966	51	10	an	an	DET
iajs-966	51	11	almost	almost	ADV
iajs-966	51	12	bounded	bound	VERB
iajs-966	51	13	submodule	submodule	NOUN
iajs-966	51	14	of	of	ADP
iajs-966	51	15	m.	m.	NOUN
iajs-966	51	16	since	since	SCONJ
iajs-966	51	17	annzn	annzn	NOUN
iajs-966	51	18	=	=	SYM
iajs-966	51	19	annz	annz	NOUN
iajs-966	51	20	(	(	PUNCT
iajs-966	51	21	3	3	NUM
iajs-966	51	22	2	2	NUM
iajs-966	52	1			INTJ
iajs-966	52	2			PROPN
iajs-966	52	3			PROPN
iajs-966	52	4			INTJ
iajs-966	52	5	)	)	PUNCT
iajs-966	53	1	=	=	SYM
iajs-966	53	2	z	z	X
iajs-966	53	3	zann	zann	PROPN
iajs-966	53	4	3	3	NUM
iajs-966	53	5	ann	ann	PROPN
iajs-966	53	6	2	2	NUM
iajs-966	53	7			PROPN
iajs-966	53	8			NOUN
iajs-966	53	9			PROPN
iajs-966	53	10			VERB
iajs-966	54	1	=	=	NOUN
iajs-966	54	2	2z6z=6z	2z6z=6z	NOUN
iajs-966	54	3	and	and	CCONJ
iajs-966	54	4	there	there	PRON
iajs-966	54	5	exists	exist	VERB
iajs-966	54	6	(	(	PUNCT
iajs-966	54	7	2,2)m	2,2)m	NUM
iajs-966	54	8	,	,	PUNCT
iajs-966	54	9	(	(	PUNCT
iajs-966	54	10	2,2)n	2,2)n	NUM
iajs-966	54	11	such	such	ADJ
iajs-966	54	12	that	that	DET
iajs-966	54	13	annzn	annzn	NOUN
iajs-966	54	14	=	=	NOUN
iajs-966	54	15	annz	annz	NOUN
iajs-966	54	16	(	(	PUNCT
iajs-966	54	17	2,2	2,2	NUM
iajs-966	54	18	)	)	PUNCT
iajs-966	54	19	=	=	SYM
iajs-966	54	20	z	z	PROPN
iajs-966	54	21	zann	zann	X
iajs-966	54	22	(	(	PUNCT
iajs-966	54	23	2	2	NUM
iajs-966	54	24	)	)	PUNCT
iajs-966	54	25	ann	ann	NOUN
iajs-966	54	26	(	(	PUNCT
iajs-966	54	27	2)	2)	NUM
iajs-966	54	28	=	=	SYM
iajs-966	54	29	3z6z=6z	3z6z=6z	NUM
iajs-966	54	30	.	.	PUNCT
iajs-966	55	1	but	but	CCONJ
iajs-966	55	2	n1=	n1=	PROPN
iajs-966	55	3	3	3	NUM
iajs-966	55	4			NOUN
iajs-966	55	5	and	and	CCONJ
iajs-966	55	6	n2=	n2=	PROPN
iajs-966	55	7	2	2	NUM
iajs-966	55	8			PROPN
iajs-966	55	9	is	be	AUX
iajs-966	55	10	not	not	PART
iajs-966	55	11	almost	almost	ADV
iajs-966	55	12	bounded	bounded	ADJ
iajs-966	55	13	submodules	submodule	NOUN
iajs-966	55	14	of	of	ADP
iajs-966	55	15	m	m	PROPN
iajs-966	55	16	1	1	NUM
iajs-966	55	17	and	and	CCONJ
iajs-966	55	18	m2	m2	PROPN
iajs-966	55	19	respectively	respectively	ADV
iajs-966	55	20	.	.	PUNCT
iajs-966	56	1	since	since	SCONJ
iajs-966	56	2	for	for	ADP
iajs-966	56	3	each	each	DET
iajs-966	56	4	xz6	xz6	NOUN
iajs-966	56	5	,	,	PUNCT
iajs-966	56	6	x=1,2,4,5n1	x=1,2,4,5n1	NOUN
iajs-966	56	7	,	,	PUNCT
iajs-966	56	8	annz(1)=6z	annz(1)=6z	PROPN
iajs-966	56	9	,	,	PUNCT
iajs-966	56	10	annz	annz	NOUN
iajs-966	56	11	(	(	PUNCT
iajs-966	56	12	2	2	NUM
iajs-966	56	13	)	)	PUNCT
iajs-966	56	14	=	=	SYM
iajs-966	56	15	3z	3z	NUM
iajs-966	56	16	,	,	PUNCT
iajs-966	56	17	annz	annz	NOUN
iajs-966	56	18	(	(	PUNCT
iajs-966	56	19	4	4	NUM
iajs-966	56	20	)	)	PUNCT
iajs-966	56	21	=	=	SYM
iajs-966	56	22	3z	3z	ADJ
iajs-966	56	23	,	,	PUNCT
iajs-966	56	24	annz(5	annz(5	NOUN
iajs-966	56	25	)	)	PUNCT
iajs-966	56	26	=	=	NOUN
iajs-966	56	27	6z	6z	NOUN
iajs-966	56	28	.	.	PUNCT
iajs-966	57	1	therefore	therefore	ADV
iajs-966	57	2	for	for	ADP
iajs-966	57	3	each	each	DET
iajs-966	57	4	xz6	xz6	NOUN
iajs-966	57	5	,	,	PUNCT
iajs-966	57	6	xn1	xn1	ADP
iajs-966	57	7	annz(x	annz(x	ADJ
iajs-966	57	8	)	)	PUNCT
iajs-966	57	9			NOUN
iajs-966	57	10	annzn1	annzn1	ADJ
iajs-966	58	1	=	=	SYM
iajs-966	58	2	annz	annz	NOUN
iajs-966	58	3	3	3	NUM
iajs-966	58	4			PROPN
iajs-966	58	5	=	=	NOUN
iajs-966	58	6	2z	2z	NUM
iajs-966	58	7	.	.	PUNCT
iajs-966	59	1	thus	thus	ADV
iajs-966	59	2	n1	n1	NOUN
iajs-966	59	3	is	be	AUX
iajs-966	59	4	not	not	PART
iajs-966	59	5	almost	almost	ADV
iajs-966	59	6	bounded	bound	VERB
iajs-966	59	7	submodule	submodule	NOUN
iajs-966	59	8	of	of	ADP
iajs-966	59	9	m1	m1	PROPN
iajs-966	59	10	.	.	PUNCT
iajs-966	60	1	in	in	ADP
iajs-966	60	2	the	the	DET
iajs-966	60	3	same	same	ADJ
iajs-966	60	4	way	way	NOUN
iajs-966	60	5	,	,	PUNCT
iajs-966	60	6	n2	n2	ADJ
iajs-966	60	7	is	be	AUX
iajs-966	60	8	not	not	PART
iajs-966	60	9	almost	almost	ADV
iajs-966	60	10	bounded	bound	VERB
iajs-966	60	11	.	.	PUNCT
iajs-966	61	1	using	use	VERB
iajs-966	61	2	the	the	DET
iajs-966	61	3	mathematical	mathematical	ADJ
iajs-966	61	4	induction	induction	NOUN
iajs-966	61	5	,	,	PUNCT
iajs-966	61	6	we	we	PRON
iajs-966	61	7	obtain	obtain	VERB
iajs-966	61	8	the	the	DET
iajs-966	61	9	following	follow	VERB
iajs-966	61	10	corollary	corollary	NOUN
iajs-966	61	11	.	.	PUNCT
iajs-966	62	1	1.6	1.6	NUM
iajs-966	62	2	corollary	corollary	NOUN
iajs-966	62	3	:	:	PUNCT
iajs-966	62	4	let	let	VERB
iajs-966	62	5	m	m	PROPN
iajs-966	62	6	1	1	NUM
iajs-966	62	7	,	,	PUNCT
iajs-966	62	8	m	m	VERB
iajs-966	62	9	2	2	NUM
iajs-966	62	10	,	,	PUNCT
iajs-966	62	11	…	…	PUNCT
iajs-966	62	12	,	,	PUNCT
iajs-966	62	13	m	m	AUX
iajs-966	62	14	n	n	PRON
iajs-966	62	15	be	be	AUX
iajs-966	62	16	a	a	DET
iajs-966	62	17	finite	finite	ADJ
iajs-966	62	18	collection	collection	NOUN
iajs-966	62	19	of	of	ADP
iajs-966	62	20	r	r	NOUN
iajs-966	62	21	-	-	PUNCT
iajs-966	62	22	modules	module	NOUN
iajs-966	62	23	and	and	CCONJ
iajs-966	62	24	m=	m=	ADJ
iajs-966	62	25	m	m	PROPN
iajs-966	62	26	1m	1m	NUM
iajs-966	62	27	2	2	ADJ
iajs-966	62	28	…	…	PUNCT
iajs-966	62	29	m	m	X
iajs-966	62	30	n.	n.	NOUN
iajs-966	62	31	if	if	SCONJ
iajs-966	62	32	n1	n1	NOUN
iajs-966	62	33	,	,	PUNCT
iajs-966	62	34	n2	n2	NOUN
iajs-966	62	35	,	,	PUNCT
iajs-966	62	36	…	…	PUNCT
iajs-966	62	37	and	and	CCONJ
iajs-966	62	38	nn	nn	X
iajs-966	62	39	are	be	AUX
iajs-966	62	40	almost	almost	ADV
iajs-966	62	41	bounded	bound	VERB
iajs-966	62	42	r	r	NOUN
iajs-966	62	43	-	-	PUNCT
iajs-966	62	44	submodules	submodule	NOUN
iajs-966	62	45	of	of	ADP
iajs-966	62	46	m	m	PROPN
iajs-966	62	47	1	1	NUM
iajs-966	62	48	,	,	PUNCT
iajs-966	62	49	m	m	VERB
iajs-966	62	50	2	2	NUM
iajs-966	62	51	,	,	PUNCT
iajs-966	62	52	…	…	PUNCT
iajs-966	62	53	and	and	CCONJ
iajs-966	62	54	mn	mn	PROPN
iajs-966	62	55	respectively	respectively	ADV
iajs-966	62	56	,	,	PUNCT
iajs-966	62	57	then	then	ADV
iajs-966	62	58	n=	n=	ADV
iajs-966	62	59	n1n2	n1n2	X
iajs-966	62	60	…	…	PUNCT
iajs-966	62	61	nn	nn	PROPN
iajs-966	62	62	is	be	AUX
iajs-966	62	63	an	an	DET
iajs-966	62	64	almost	almost	ADV
iajs-966	62	65	bounded	bound	VERB
iajs-966	62	66	submodule	submodule	NOUN
iajs-966	62	67	of	of	ADP
iajs-966	62	68	m.	m.	NOUN
iajs-966	62	69	so	so	ADV
iajs-966	62	70	,	,	PUNCT
iajs-966	62	71	we	we	PRON
iajs-966	62	72	have	have	VERB
iajs-966	62	73	the	the	DET
iajs-966	62	74	following	follow	VERB
iajs-966	62	75	applications	application	NOUN
iajs-966	62	76	of	of	ADP
iajs-966	62	77	(	(	PUNCT
iajs-966	62	78	1.4	1.4	NUM
iajs-966	62	79	)	)	PUNCT
iajs-966	62	80	1.7	1.7	NUM
iajs-966	62	81	corollary	corollary	NOUN
iajs-966	62	82	:	:	PUNCT
iajs-966	62	83	let	let	VERB
iajs-966	62	84	n1	n1	NOUN
iajs-966	62	85	and	and	CCONJ
iajs-966	62	86	n2	n2	ADJ
iajs-966	62	87	be	be	AUX
iajs-966	62	88	two	two	NUM
iajs-966	62	89	almost	almost	ADV
iajs-966	62	90	bounded	bound	VERB
iajs-966	62	91	submodules	submodule	NOUN
iajs-966	62	92	of	of	ADP
iajs-966	62	93	an	an	DET
iajs-966	62	94	r	r	NOUN
iajs-966	62	95	-	-	PUNCT
iajs-966	62	96	module	module	NOUN
iajs-966	62	97	m.	m.	NOUN
iajs-966	62	98	then	then	ADV
iajs-966	62	99	n1n2	n1n2	PROPN
iajs-966	62	100	is	be	AUX
iajs-966	62	101	an	an	DET
iajs-966	62	102	almost	almost	ADV
iajs-966	62	103	bounded	bounded	ADJ
iajs-966	62	104	submodule	submodule	NOUN
iajs-966	62	105	of	of	ADP
iajs-966	62	106	mm	mm	NOUN
iajs-966	62	107	.	.	PUNCT
iajs-966	63	1	proof	proof	NOUN
iajs-966	63	2	:	:	PUNCT
iajs-966	63	3	we	we	PRON
iajs-966	63	4	haven1	haven1	ADJ
iajs-966	63	5	and	and	CCONJ
iajs-966	63	6	n2	n2	NOUN
iajs-966	63	7	are	be	AUX
iajs-966	63	8	almost	almost	ADV
iajs-966	63	9	bounded	bound	VERB
iajs-966	63	10	submodules	submodule	NOUN
iajs-966	63	11	of	of	ADP
iajs-966	63	12	m	m	PROPN
iajs-966	63	13	,	,	PUNCT
iajs-966	63	14	means	mean	VERB
iajs-966	63	15	there	there	PRON
iajs-966	63	16	exists	exist	VERB
iajs-966	63	17	xm	xm	NOUN
iajs-966	63	18	,	,	PUNCT
iajs-966	63	19	xn1	xn1	PUNCT
iajs-966	63	20	such	such	ADJ
iajs-966	63	21	that	that	DET
iajs-966	63	22	annrn1	annrn1	PROPN
iajs-966	63	23	=	=	SYM
iajs-966	63	24	annr(x	annr(x	NOUN
iajs-966	63	25	)	)	PUNCT
iajs-966	63	26	and	and	CCONJ
iajs-966	63	27	there	there	PRON
iajs-966	63	28	exists	exist	VERB
iajs-966	63	29	ym	ym	PROPN
iajs-966	63	30	,	,	PUNCT
iajs-966	63	31	yn2	yn2	NOUN
iajs-966	63	32	such	such	ADJ
iajs-966	63	33	that	that	PRON
iajs-966	63	34	annrn2	annrn2	NOUN
iajs-966	63	35	=	=	SYM
iajs-966	63	36	annr(y	annr(y	VERB
iajs-966	63	37	)	)	PUNCT
iajs-966	63	38	,	,	PUNCT
iajs-966	63	39	implies	imply	VERB
iajs-966	63	40	(	(	PUNCT
iajs-966	63	41	x	x	X
iajs-966	63	42	,	,	PUNCT
iajs-966	63	43	y)n1n2	y)n1n2	NOUN
iajs-966	63	44	.	.	PUNCT
iajs-966	64	1	now	now	ADV
iajs-966	64	2	,	,	PUNCT
iajs-966	64	3	we	we	PRON
iajs-966	64	4	claim	claim	VERB
iajs-966	64	5	that	that	SCONJ
iajs-966	64	6	annr(n1n2)=annr(x	annr(n1n2)=annr(x	PROPN
iajs-966	64	7	,	,	PUNCT
iajs-966	64	8	y	y	PROPN
iajs-966	64	9	)	)	PUNCT
iajs-966	64	10	.	.	PUNCT
iajs-966	65	1	let	let	VERB
iajs-966	65	2	rannr(x	rannr(x	PRON
iajs-966	65	3	,	,	PUNCT
iajs-966	65	4	y	y	PROPN
iajs-966	65	5	)	)	PUNCT
iajs-966	65	6	.	.	PUNCT
iajs-966	66	1	then	then	ADV
iajs-966	66	2	r(x	r(x	PROPN
iajs-966	66	3	,	,	PUNCT
iajs-966	66	4	y)=(0,0	y)=(0,0	NOUN
iajs-966	66	5	)	)	PUNCT
iajs-966	66	6	,	,	PUNCT
iajs-966	66	7	implies	imply	VERB
iajs-966	66	8	rx=0	rx=0	NOUN
iajs-966	66	9	and	and	CCONJ
iajs-966	66	10	ry=0	ry=0	NOUN
iajs-966	66	11	.	.	PUNCT
iajs-966	67	1	therefore	therefore	ADV
iajs-966	67	2	rannr(x)=	rannr(x)=	PROPN
iajs-966	67	3	annrn1	annrn1	NOUN
iajs-966	67	4	and	and	CCONJ
iajs-966	67	5	ihjpas	ihjpa	VERB
iajs-966	67	6	ibn	ibn	PROPN
iajs-966	67	7	alhaitham	alhaitham	PROPN
iajs-966	67	8	j.	j.	PROPN
iajs-966	68	1	fo	fo	ADP
iajs-966	68	2	r	r	NOUN
iajs-966	68	3	pure	pure	ADJ
iajs-966	68	4	&	&	CCONJ
iajs-966	68	5	appl	appl	PROPN
iajs-966	68	6	.	.	PUNCT
iajs-966	69	1	sc	sc	PROPN
iajs-966	69	2	i.	i.	PROPN
iajs-966	69	3	vo	vo	PROPN
iajs-966	69	4	l.23	l.23	PROPN
iajs-966	69	5	(	(	PUNCT
iajs-966	69	6	2	2	NUM
iajs-966	69	7	)	)	PUNCT
iajs-966	69	8	2010	2010	NUM
iajs-966	69	9	rannr(y)=annrn2	rannr(y)=annrn2	NOUN
iajs-966	69	10	.	.	PUNCT
iajs-966	70	1	thus	thus	ADV
iajs-966	70	2	rannrn1	rannrn1	VERB
iajs-966	70	3	annrn2	annrn2	NOUN
iajs-966	70	4	.	.	PUNCT
iajs-966	71	1	but	but	CCONJ
iajs-966	71	2	annr(n1n2)=	annr(n1n2)=	ADP
iajs-966	71	3	annrn1annrn2	annrn1annrn2	NOUN
iajs-966	71	4	,	,	PUNCT
iajs-966	71	5	so	so	SCONJ
iajs-966	71	6	we	we	PRON
iajs-966	71	7	get	get	VERB
iajs-966	71	8	rannr(n1n2	rannr(n1n2	NOUN
iajs-966	71	9	)	)	PUNCT
iajs-966	71	10	.	.	PUNCT
iajs-966	72	1	conversely	conversely	ADV
iajs-966	72	2	,	,	PUNCT
iajs-966	72	3	let	let	VERB
iajs-966	72	4	rannr(n1n2	rannr(n1n2	NOUN
iajs-966	72	5	)	)	PUNCT
iajs-966	72	6	.	.	PUNCT
iajs-966	73	1	then	then	ADV
iajs-966	73	2	r(a	r(a	PROPN
iajs-966	73	3	,	,	PUNCT
iajs-966	73	4	b)=(0,0	b)=(0,0	NOUN
iajs-966	73	5	)	)	PUNCT
iajs-966	73	6	for	for	ADP
iajs-966	73	7	all	all	DET
iajs-966	73	8	(	(	PUNCT
iajs-966	73	9	a	a	PRON
iajs-966	73	10	,	,	PUNCT
iajs-966	73	11	b)n1n2	b)n1n2	NOUN
iajs-966	73	12	which	which	PRON
iajs-966	73	13	implies	imply	VERB
iajs-966	73	14	ra=0	ra=0	VERB
iajs-966	73	15	for	for	ADP
iajs-966	73	16	all	all	DET
iajs-966	73	17	an1	an1	NOUN
iajs-966	73	18	and	and	CCONJ
iajs-966	73	19	rb=0	rb=0	PROPN
iajs-966	73	20	for	for	ADP
iajs-966	73	21	all	all	DET
iajs-966	73	22	bn2	bn2	NOUN
iajs-966	73	23	which	which	PRON
iajs-966	73	24	implies	imply	VERB
iajs-966	73	25	rn1=0	rn1=0	ADJ
iajs-966	73	26	and	and	CCONJ
iajs-966	73	27	rn2=0	rn2=0	PUNCT
iajs-966	73	28	.	.	PUNCT
iajs-966	74	1	thus	thus	ADV
iajs-966	74	2	rannrn1	rannrn1	NUM
iajs-966	74	3	and	and	CCONJ
iajs-966	74	4	rannrn2	rannrn2	PROPN
iajs-966	74	5	.	.	PROPN
iajs-966	74	6	but	but	CCONJ
iajs-966	74	7	annrn1	annrn1	PROPN
iajs-966	74	8	=	=	SYM
iajs-966	74	9	annr(x	annr(x	NOUN
iajs-966	74	10	)	)	PUNCT
iajs-966	74	11	and	and	CCONJ
iajs-966	74	12	annr(y)=annrn2	annr(y)=annrn2	PROPN
iajs-966	74	13	.	.	PUNCT
iajs-966	75	1	this	this	PRON
iajs-966	75	2	implies	imply	VERB
iajs-966	75	3	that	that	SCONJ
iajs-966	75	4	rannr(x	rannr(x	NOUN
iajs-966	75	5	)	)	PUNCT
iajs-966	75	6	and	and	CCONJ
iajs-966	75	7	rannr(y	rannr(y	NUM
iajs-966	75	8	)	)	PUNCT
iajs-966	75	9	,	,	PUNCT
iajs-966	75	10	that	that	ADV
iajs-966	75	11	is	is	ADV
iajs-966	75	12	,	,	PUNCT
iajs-966	75	13	rx=0	rx=0	NOUN
iajs-966	75	14	and	and	CCONJ
iajs-966	75	15	ry=0	ry=0	NOUN
iajs-966	75	16	.	.	PUNCT
iajs-966	76	1	thus	thus	ADV
iajs-966	76	2	(	(	PUNCT
iajs-966	76	3	rx	rx	ADJ
iajs-966	76	4	,	,	PUNCT
iajs-966	76	5	ry)=(0,0	ry)=(0,0	NOUN
iajs-966	76	6	)	)	PUNCT
iajs-966	76	7	,	,	PUNCT
iajs-966	76	8	so	so	SCONJ
iajs-966	76	9	that	that	SCONJ
iajs-966	76	10	r(x	r(x	PROPN
iajs-966	76	11	,	,	PUNCT
iajs-966	76	12	y)=(0,0	y)=(0,0	NOUN
iajs-966	76	13	)	)	PUNCT
iajs-966	76	14	.	.	PUNCT
iajs-966	77	1	hence	hence	ADV
iajs-966	77	2	rannr(x	rannr(x	PROPN
iajs-966	77	3	,	,	PUNCT
iajs-966	77	4	y	y	PROPN
iajs-966	77	5	)	)	PUNCT
iajs-966	77	6	.	.	PUNCT
iajs-966	78	1	this	this	PRON
iajs-966	78	2	completes	complete	VERB
iajs-966	78	3	the	the	DET
iajs-966	78	4	proof	proof	NOUN
iajs-966	78	5	.	.	PUNCT
iajs-966	79	1	the	the	DET
iajs-966	79	2	following	follow	VERB
iajs-966	79	3	corollary	corollary	NOUN
iajs-966	79	4	is	be	AUX
iajs-966	79	5	a	a	DET
iajs-966	79	6	special	special	ADJ
iajs-966	79	7	case	case	NOUN
iajs-966	79	8	of	of	ADP
iajs-966	79	9	proposition	proposition	NOUN
iajs-966	79	10	(	(	PUNCT
iajs-966	79	11	1.4	1.4	NUM
iajs-966	79	12	)	)	PUNCT
iajs-966	79	13	.	.	PUNCT
iajs-966	80	1	1.8	1.8	NUM
iajs-966	80	2	corollary	corollary	NOUN
iajs-966	80	3	:	:	PUNCT
iajs-966	80	4	let	let	VERB
iajs-966	80	5	m	m	PRON
iajs-966	80	6	be	be	AUX
iajs-966	80	7	an	an	DET
iajs-966	80	8	r	r	NOUN
iajs-966	80	9	-	-	PUNCT
iajs-966	80	10	module	module	NOUN
iajs-966	80	11	,	,	PUNCT
iajs-966	80	12	n	n	PRON
iajs-966	80	13	be	be	VERB
iajs-966	80	14	an	an	DET
iajs-966	80	15	almost	almost	ADV
iajs-966	80	16	bounded	bound	VERB
iajs-966	80	17	submodule	submodule	NOUN
iajs-966	80	18	of	of	ADP
iajs-966	80	19	m.	m.	NOUN
iajs-966	80	20	then	then	ADV
iajs-966	80	21	n	n	PROPN
iajs-966	80	22	2	2	NUM
iajs-966	80	23	=	=	NOUN
iajs-966	80	24	nn	nn	NOUN
iajs-966	80	25	is	be	AUX
iajs-966	80	26	an	an	DET
iajs-966	80	27	almost	almost	ADV
iajs-966	80	28	bounded	bounded	ADJ
iajs-966	80	29	submodule	submodule	NOUN
iajs-966	80	30	of	of	ADP
iajs-966	80	31	m2	m2	PROPN
iajs-966	80	32	=	=	NOUN
iajs-966	80	33	mm	mm	NOUN
iajs-966	80	34	.	.	PUNCT
iajs-966	81	1	proof	proof	NOUN
iajs-966	81	2	:	:	PUNCT
iajs-966	81	3	from	from	ADP
iajs-966	81	4	hypothesis	hypothesis	NOUN
iajs-966	81	5	n	n	NOUN
iajs-966	81	6	is	be	AUX
iajs-966	81	7	an	an	DET
iajs-966	81	8	almost	almost	ADV
iajs-966	81	9	bounded	bound	VERB
iajs-966	81	10	submodule	submodule	NOUN
iajs-966	81	11	of	of	ADP
iajs-966	81	12	m.	m.	NOUN
iajs-966	81	13	then	then	ADV
iajs-966	81	14	there	there	PRON
iajs-966	81	15	exists	exist	VERB
iajs-966	81	16	xm	xm	NOUN
iajs-966	81	17	,	,	PUNCT
iajs-966	81	18	xn	xn	PROPN
iajs-966	81	19	such	such	ADJ
iajs-966	81	20	that	that	SCONJ
iajs-966	81	21	annrn	annrn	NOUN
iajs-966	81	22	=	=	SYM
iajs-966	81	23	annr(x	annr(x	NOUN
iajs-966	81	24	)	)	PUNCT
iajs-966	81	25	.	.	PUNCT
iajs-966	82	1	thus	thus	ADV
iajs-966	82	2	(	(	PUNCT
iajs-966	82	3	x	x	X
iajs-966	82	4	,	,	PUNCT
iajs-966	82	5	x)m	x)m	PROPN
iajs-966	82	6	2	2	NUM
iajs-966	82	7	=	=	NOUN
iajs-966	82	8	mm	mm	NOUN
iajs-966	82	9	and	and	CCONJ
iajs-966	82	10	(	(	PUNCT
iajs-966	82	11	x	x	NOUN
iajs-966	82	12	,	,	PUNCT
iajs-966	82	13	x)n2	x)n2	NOUN
iajs-966	82	14	=	=	SYM
iajs-966	82	15	nn	nn	ADJ
iajs-966	82	16	since	since	SCONJ
iajs-966	82	17	annr(x	annr(x	PROPN
iajs-966	82	18	,	,	PUNCT
iajs-966	82	19	x)=annr(x)annr(x)=annrn	x)=annr(x)annr(x)=annrn	PROPN
iajs-966	82	20	=	=	SYM
iajs-966	82	21	annr(nn	annr(nn	PROPN
iajs-966	82	22	)	)	PUNCT
iajs-966	82	23	.	.	PUNCT
iajs-966	83	1	hence	hence	ADV
iajs-966	83	2	annr(x	annr(x	NUM
iajs-966	83	3	,	,	PUNCT
iajs-966	83	4	x)=annr(nn	x)=annr(nn	PROPN
iajs-966	83	5	)	)	PUNCT
iajs-966	83	6	which	which	PRON
iajs-966	83	7	is	be	AUX
iajs-966	83	8	what	what	PRON
iajs-966	83	9	we	we	PRON
iajs-966	83	10	wanted	want	VERB
iajs-966	83	11	.	.	PUNCT
iajs-966	84	1	now	now	ADV
iajs-966	84	2	,	,	PUNCT
iajs-966	84	3	we	we	PRON
iajs-966	84	4	have	have	VERB
iajs-966	84	5	the	the	DET
iajs-966	84	6	following	follow	VERB
iajs-966	84	7	proposition	proposition	NOUN
iajs-966	84	8	:	:	PUNCT
iajs-966	84	9	1.9	1.9	NUM
iajs-966	84	10	proposition	proposition	NOUN
iajs-966	84	11	:	:	PUNCT
iajs-966	84	12	let	let	VERB
iajs-966	84	13	m=	m=	AUX
iajs-966	84	14	m	m	VERB
iajs-966	84	15	1m	1m	NUM
iajs-966	84	16	2	2	NUM
iajs-966	84	17	be	be	AUX
iajs-966	84	18	a	a	DET
iajs-966	84	19	direct	direct	ADJ
iajs-966	84	20	sum	sum	NOUN
iajs-966	84	21	of	of	ADP
iajs-966	84	22	two	two	NUM
iajs-966	84	23	r	r	NOUN
iajs-966	84	24	-	-	PUNCT
iajs-966	84	25	modules	module	NOUN
iajs-966	84	26	m	m	NOUN
iajs-966	84	27	1	1	NUM
iajs-966	84	28	and	and	CCONJ
iajs-966	84	29	m	m	PROPN
iajs-966	84	30	2	2	NUM
iajs-966	84	31	.	.	PUNCT
iajs-966	85	1	if	if	SCONJ
iajs-966	85	2	l1	l1	PROPN
iajs-966	85	3	is	be	AUX
iajs-966	85	4	an	an	DET
iajs-966	85	5	almost	almost	ADV
iajs-966	85	6	bounded	bound	VERB
iajs-966	85	7	submodule	submodule	NOUN
iajs-966	85	8	of	of	ADP
iajs-966	85	9	m	m	PROPN
iajs-966	85	10	1	1	NUM
iajs-966	85	11	and	and	CCONJ
iajs-966	85	12	annr(y)=annrm	annr(y)=annrm	NOUN
iajs-966	85	13	2	2	NUM
iajs-966	85	14	for	for	ADP
iajs-966	85	15	some	some	DET
iajs-966	85	16	ym	ym	NOUN
iajs-966	85	17	2	2	NUM
iajs-966	85	18	,	,	PUNCT
iajs-966	85	19	y≠0	y≠0	NOUN
iajs-966	85	20	,	,	PUNCT
iajs-966	85	21	then	then	ADV
iajs-966	85	22	l1m	l1m	PROPN
iajs-966	85	23	2	2	NUM
iajs-966	85	24	is	be	AUX
iajs-966	85	25	an	an	DET
iajs-966	85	26	almost	almost	ADV
iajs-966	85	27	bounded	bound	VERB
iajs-966	85	28	submodule	submodule	NOUN
iajs-966	85	29	of	of	ADP
iajs-966	85	30	m.	m.	NOUN
iajs-966	85	31	proof	proof	NOUN
iajs-966	85	32	:	:	PUNCT
iajs-966	85	33	we	we	PRON
iajs-966	85	34	have	have	VERB
iajs-966	85	35	l1which	l1which	NOUN
iajs-966	85	36	is	be	AUX
iajs-966	85	37	an	an	DET
iajs-966	85	38	almost	almost	ADV
iajs-966	85	39	bounded	bound	VERB
iajs-966	85	40	submodule	submodule	NOUN
iajs-966	85	41	of	of	ADP
iajs-966	85	42	m	m	PROPN
iajs-966	85	43	1	1	NUM
iajs-966	85	44	,	,	PUNCT
iajs-966	85	45	then	then	ADV
iajs-966	85	46	there	there	PRON
iajs-966	85	47	exists	exist	VERB
iajs-966	85	48	xm	xm	X
iajs-966	86	1	1	1	X
iajs-966	86	2	,	,	PUNCT
iajs-966	86	3	xl1	xl1	PROPN
iajs-966	86	4	such	such	ADJ
iajs-966	86	5	that	that	SCONJ
iajs-966	86	6	annrl1	annrl1	PROPN
iajs-966	86	7	=	=	SYM
iajs-966	86	8	annr(x	annr(x	NOUN
iajs-966	86	9	)	)	PUNCT
iajs-966	86	10	,	,	PUNCT
iajs-966	86	11	ym	ym	PROPN
iajs-966	86	12	2	2	X
iajs-966	86	13	.	.	PUNCT
iajs-966	87	1	then	then	ADV
iajs-966	87	2	(	(	PUNCT
iajs-966	87	3	x	x	X
iajs-966	87	4	,	,	PUNCT
iajs-966	87	5	y)m	y)m	PROPN
iajs-966	87	6	1m	1m	NUM
iajs-966	87	7	2	2	NUM
iajs-966	87	8	and	and	CCONJ
iajs-966	87	9	(	(	PUNCT
iajs-966	87	10	x	x	NOUN
iajs-966	87	11	,	,	PUNCT
iajs-966	87	12	y)l1m	y)l1m	PROPN
iajs-966	87	13	2	2	NUM
iajs-966	87	14	.	.	PUNCT
iajs-966	87	15	we	we	PRON
iajs-966	87	16	claim	claim	VERB
iajs-966	87	17	that	that	SCONJ
iajs-966	87	18	annr(l1m	annr(l1m	PROPN
iajs-966	87	19	2)=annr(x	2)=annr(x	NUM
iajs-966	87	20	,	,	PUNCT
iajs-966	87	21	y	y	NOUN
iajs-966	87	22	)	)	PUNCT
iajs-966	87	23	.	.	PUNCT
iajs-966	88	1	now	now	ADV
iajs-966	88	2	to	to	PART
iajs-966	88	3	prove	prove	VERB
iajs-966	88	4	our	our	PRON
iajs-966	88	5	assumption	assumption	NOUN
iajs-966	88	6	.	.	PUNCT
iajs-966	89	1	let	let	VERB
iajs-966	89	2	rannr(l1m	rannr(l1m	NOUN
iajs-966	89	3	2)=	2)=	NUM
iajs-966	89	4	annrl1annrm	annrl1annrm	NOUN
iajs-966	89	5	2	2	NUM
iajs-966	89	6	.	.	PUNCT
iajs-966	89	7	then	then	ADV
iajs-966	89	8	rannrl1annrm	rannrl1annrm	ADP
iajs-966	89	9	2	2	NUM
iajs-966	89	10	,	,	PUNCT
iajs-966	89	11	so	so	ADV
iajs-966	89	12	rannrl1	rannrl1	PROPN
iajs-966	89	13	and	and	CCONJ
iajs-966	89	14	rannrm	rannrm	PROPN
iajs-966	89	15	2	2	NUM
iajs-966	89	16	=	=	NOUN
iajs-966	89	17	annr(y	annr(y	VERB
iajs-966	89	18	)	)	PUNCT
iajs-966	89	19	.	.	PUNCT
iajs-966	90	1	therefore	therefore	ADV
iajs-966	90	2	rannr(x	rannr(x	NOUN
iajs-966	90	3	)	)	PUNCT
iajs-966	90	4	and	and	CCONJ
iajs-966	90	5	rannr(y	rannr(y	VERB
iajs-966	90	6	)	)	PUNCT
iajs-966	90	7	.	.	PUNCT
iajs-966	91	1	thus	thus	ADV
iajs-966	91	2	rx=0	rx=0	NOUN
iajs-966	91	3	and	and	CCONJ
iajs-966	91	4	ry=0	ry=0	NOUN
iajs-966	91	5	means	mean	NOUN
iajs-966	91	6	(	(	PUNCT
iajs-966	91	7	rx	rx	ADJ
iajs-966	91	8	,	,	PUNCT
iajs-966	91	9	ry)=(0,0	ry)=(0,0	NOUN
iajs-966	91	10	)	)	PUNCT
iajs-966	91	11	,	,	PUNCT
iajs-966	91	12	which	which	PRON
iajs-966	91	13	implies	imply	VERB
iajs-966	91	14	r(x	r(x	PROPN
iajs-966	91	15	,	,	PUNCT
iajs-966	91	16	y)=(0,0	y)=(0,0	NOUN
iajs-966	91	17	)	)	PUNCT
iajs-966	91	18	and	and	CCONJ
iajs-966	91	19	hence	hence	ADV
iajs-966	91	20	rannr(x	rannr(x	PROPN
iajs-966	91	21	,	,	PUNCT
iajs-966	91	22	y	y	NOUN
iajs-966	91	23	)	)	PUNCT
iajs-966	91	24	.	.	PUNCT
iajs-966	92	1	conversely	conversely	ADV
iajs-966	92	2	,	,	PUNCT
iajs-966	92	3	let	let	VERB
iajs-966	92	4	rannr(x	rannr(x	PRON
iajs-966	92	5	,	,	PUNCT
iajs-966	92	6	y	y	PROPN
iajs-966	92	7	)	)	PUNCT
iajs-966	92	8	.	.	PUNCT
iajs-966	93	1	then	then	ADV
iajs-966	93	2	r(x	r(x	PROPN
iajs-966	93	3	,	,	PUNCT
iajs-966	93	4	y)=(0,0	y)=(0,0	NOUN
iajs-966	93	5	)	)	PUNCT
iajs-966	93	6	,	,	PUNCT
iajs-966	93	7	which	which	PRON
iajs-966	93	8	implies	imply	VERB
iajs-966	93	9	(	(	PUNCT
iajs-966	93	10	rx	rx	ADJ
iajs-966	93	11	,	,	PUNCT
iajs-966	93	12	ry)=(0,0	ry)=(0,0	NOUN
iajs-966	93	13	)	)	PUNCT
iajs-966	93	14	.	.	PUNCT
iajs-966	94	1	therefore	therefore	ADV
iajs-966	94	2	rx=0	rx=0	NOUN
iajs-966	94	3	and	and	CCONJ
iajs-966	94	4	ry=0	ry=0	NOUN
iajs-966	94	5	.	.	PUNCT
iajs-966	94	6	thus	thus	ADV
iajs-966	94	7	rannr(x)=annrl1	rannr(x)=annrl1	NOUN
iajs-966	94	8	and	and	CCONJ
iajs-966	94	9	rannr(y)=annrm	rannr(y)=annrm	ADV
iajs-966	94	10	2	2	NUM
iajs-966	94	11	.	.	X
iajs-966	94	12	hence	hence	ADV
iajs-966	94	13	r	r	ADJ
iajs-966	94	14	annrl1annrm	annrl1annrm	PROPN
iajs-966	94	15	2	2	NUM
iajs-966	94	16	,	,	PUNCT
iajs-966	94	17	which	which	PRON
iajs-966	94	18	implies	imply	VERB
iajs-966	94	19	rannr(l1m	rannr(l1m	NOUN
iajs-966	94	20	1	1	NUM
iajs-966	94	21	)	)	PUNCT
iajs-966	94	22	.	.	PUNCT
iajs-966	95	1	therefore	therefore	ADV
iajs-966	95	2	rannr(l1m	rannr(l1m	VERB
iajs-966	95	3	2)=	2)=	NUM
iajs-966	95	4	annr(x	annr(x	NOUN
iajs-966	95	5	,	,	PUNCT
iajs-966	95	6	y	y	PROPN
iajs-966	95	7	)	)	PUNCT
iajs-966	95	8	.	.	PUNCT
iajs-966	96	1	next	next	ADV
iajs-966	96	2	,	,	PUNCT
iajs-966	96	3	we	we	PRON
iajs-966	96	4	have	have	VERB
iajs-966	96	5	the	the	DET
iajs-966	96	6	following	follow	VERB
iajs-966	96	7	remark	remark	NOUN
iajs-966	96	8	.	.	PUNCT
iajs-966	97	1	1.10	1.10	NUM
iajs-966	97	2	remark	remark	NOUN
iajs-966	97	3	:	:	PUNCT
iajs-966	97	4	a	a	DET
iajs-966	97	5	direct	direct	ADJ
iajs-966	97	6	summand	summand	NOUN
iajs-966	97	7	of	of	ADP
iajs-966	97	8	almost	almost	ADV
iajs-966	97	9	bounded	bound	VERB
iajs-966	97	10	need	need	AUX
iajs-966	97	11	not	not	PART
iajs-966	97	12	be	be	AUX
iajs-966	97	13	an	an	DET
iajs-966	97	14	almost	almost	ADV
iajs-966	97	15	bounded	bound	VERB
iajs-966	97	16	.	.	PUNCT
iajs-966	98	1	for	for	ADP
iajs-966	98	2	example	example	NOUN
iajs-966	98	3	:	:	PUNCT
iajs-966	98	4	it	it	PRON
iajs-966	98	5	is	be	AUX
iajs-966	98	6	known	know	VERB
iajs-966	98	7	that	that	SCONJ
iajs-966	98	8	n	n	CCONJ
iajs-966	98	9	=	=	NOUN
iajs-966	98	10	qzzp	qzzp	NOUN
iajs-966	98	11	is	be	AUX
iajs-966	98	12	an	an	DET
iajs-966	98	13	almost	almost	ADV
iajs-966	98	14	bounded	bound	VERB
iajs-966	98	15	submodule	submodule	NOUN
iajs-966	98	16	of	of	ADP
iajs-966	98	17	a	a	DET
iajs-966	98	18	z	z	NOUN
iajs-966	98	19	-	-	PUNCT
iajs-966	98	20	module	module	NOUN
iajs-966	98	21	m	m	NOUN
iajs-966	98	22	,	,	PUNCT
iajs-966	98	23	where	where	SCONJ
iajs-966	98	24	p	p	X
iajs-966	98	25	,	,	PUNCT
iajs-966	98	26	q	q	X
iajs-966	98	27	is	be	AUX
iajs-966	98	28	any	any	DET
iajs-966	98	29	prime	prime	ADJ
iajs-966	98	30	numbers	number	NOUN
iajs-966	98	31	and	and	CCONJ
iajs-966	98	32	m	m	NOUN
iajs-966	98	33	=	=	NOUN
iajs-966	98	34	zzp	zzp	NOUN
iajs-966	98	35	.	.	PUNCT
iajs-966	99	1	but	but	CCONJ
iajs-966	99	2	zp	zp	PROPN
iajs-966	99	3	is	be	AUX
iajs-966	99	4	not	not	PART
iajs-966	99	5	almost	almost	ADV
iajs-966	99	6	bounded	bound	VERB
iajs-966	99	7	because	because	SCONJ
iajs-966	99	8	zp	zp	PROPN
iajs-966	99	9	has	have	VERB
iajs-966	99	10	no	no	DET
iajs-966	99	11	proper	proper	ADJ
iajs-966	99	12	almost	almost	ADV
iajs-966	99	13	bounded	bound	VERB
iajs-966	99	14	submodule	submodule	NOUN
iajs-966	99	15	.	.	PUNCT
iajs-966	100	1	we	we	PRON
iajs-966	100	2	have	have	AUX
iajs-966	100	3	seen	see	VERB
iajs-966	100	4	by	by	ADP
iajs-966	100	5	the	the	DET
iajs-966	100	6	following	follow	VERB
iajs-966	100	7	proposition	proposition	NOUN
iajs-966	100	8	that	that	SCONJ
iajs-966	100	9	the	the	DET
iajs-966	100	10	class	class	NOUN
iajs-966	100	11	of	of	ADP
iajs-966	100	12	almost	almost	ADV
iajs-966	100	13	bounded	bound	VERB
iajs-966	100	14	submodule	submodule	NOUN
iajs-966	100	15	is	be	AUX
iajs-966	100	16	closed	close	VERB
iajs-966	100	17	under	under	ADP
iajs-966	100	18	homomorphic	homomorphic	ADJ
iajs-966	100	19	image	image	NOUN
iajs-966	100	20	and	and	CCONJ
iajs-966	100	21	inverse	inverse	NOUN
iajs-966	100	22	image	image	NOUN
iajs-966	100	23	.	.	PUNCT
iajs-966	101	1	1.11	1.11	NUM
iajs-966	101	2	proposition	proposition	NOUN
iajs-966	101	3	:	:	PUNCT
iajs-966	101	4	let	let	VERB
iajs-966	101	5	m	m	PRON
iajs-966	101	6	and	and	CCONJ
iajs-966	101	7	m	m	VERB
iajs-966	101	8	'	'	PUNCT
iajs-966	101	9	be	be	VERB
iajs-966	101	10	two	two	NUM
iajs-966	101	11	r	r	NOUN
iajs-966	101	12	-	-	PUNCT
iajs-966	101	13	modules	module	NOUN
iajs-966	101	14	and	and	CCONJ
iajs-966	101	15	let	let	VERB
iajs-966	101	16			NOUN
iajs-966	101	17	:	:	PUNCT
iajs-966	101	18	m	m	PROPN
iajs-966	101	19			PROPN
iajs-966	101	20	m	m	VERB
iajs-966	101	21	'	'	PUNCT
iajs-966	101	22	be	be	VERB
iajs-966	101	23	an	an	DET
iajs-966	101	24	isomorphism	isomorphism	NOUN
iajs-966	101	25	.	.	PUNCT
iajs-966	102	1	then	then	ADV
iajs-966	102	2	:	:	PUNCT
iajs-966	102	3	1	1	X
iajs-966	102	4	.	.	X
iajs-966	102	5	if	if	SCONJ
iajs-966	102	6	n	n	CCONJ
iajs-966	102	7	'	'	PUNCT
iajs-966	102	8	is	be	AUX
iajs-966	102	9	an	an	DET
iajs-966	102	10	almost	almost	ADV
iajs-966	102	11	bounded	bound	VERB
iajs-966	102	12	submodule	submodule	NOUN
iajs-966	102	13	of	of	ADP
iajs-966	102	14	m	m	PROPN
iajs-966	102	15	'	'	PUNCT
iajs-966	102	16	,	,	PUNCT
iajs-966	102	17	then	then	ADV
iajs-966	102	18			PROPN
iajs-966	102	19	–	–	PUNCT
iajs-966	102	20	1	1	NUM
iajs-966	102	21	(	(	PUNCT
iajs-966	102	22	n	n	CCONJ
iajs-966	102	23	'	'	CCONJ
iajs-966	102	24	)	)	PUNCT
iajs-966	102	25	is	be	AUX
iajs-966	102	26	also	also	ADV
iajs-966	102	27	almost	almost	ADV
iajs-966	102	28	bounded	bound	VERB
iajs-966	102	29	submodule	submodule	NOUN
iajs-966	102	30	of	of	ADP
iajs-966	102	31	m.	m.	NOUN
iajs-966	102	32	2	2	NUM
iajs-966	102	33	.	.	PUNCT
iajs-966	103	1	if	if	SCONJ
iajs-966	103	2	n	n	PRON
iajs-966	103	3	is	be	AUX
iajs-966	103	4	an	an	DET
iajs-966	103	5	almost	almost	ADV
iajs-966	103	6	bounded	bound	VERB
iajs-966	103	7	submodule	submodule	NOUN
iajs-966	103	8	of	of	ADP
iajs-966	103	9	m	m	PROPN
iajs-966	103	10	,	,	PUNCT
iajs-966	103	11	then	then	ADV
iajs-966	103	12	(n	(n	ADJ
iajs-966	103	13	)	)	PUNCT
iajs-966	103	14	is	be	AUX
iajs-966	103	15	an	an	DET
iajs-966	103	16	almost	almost	ADV
iajs-966	103	17	bounded	bound	VERB
iajs-966	103	18	submodule	submodule	NOUN
iajs-966	103	19	of	of	ADP
iajs-966	103	20	m	m	PROPN
iajs-966	103	21	'	'	PUNCT
iajs-966	103	22	.	.	PUNCT
iajs-966	104	1	proof	proof	NOUN
iajs-966	104	2	:	:	PUNCT
iajs-966	104	3	1	1	X
iajs-966	104	4	.	.	X
iajs-966	104	5	assume	assume	VERB
iajs-966	104	6	that	that	SCONJ
iajs-966	104	7	n	n	X
iajs-966	104	8	'	'	PUNCT
iajs-966	104	9	is	be	AUX
iajs-966	104	10	an	an	DET
iajs-966	104	11	almost	almost	ADV
iajs-966	104	12	bounded	bound	VERB
iajs-966	104	13	submodule	submodule	NOUN
iajs-966	104	14	of	of	ADP
iajs-966	104	15	m	m	PROPN
iajs-966	104	16	'	'	PUNCT
iajs-966	104	17	,	,	PUNCT
iajs-966	104	18	then	then	ADV
iajs-966	104	19	there	there	PRON
iajs-966	104	20	exists	exist	VERB
iajs-966	104	21	yn	yn	PROPN
iajs-966	104	22	'	'	PART
iajs-966	104	23	such	such	ADJ
iajs-966	104	24	that	that	SCONJ
iajs-966	104	25	annr(y)=annrn	annr(y)=annrn	NOUN
iajs-966	104	26	'	'	PART
iajs-966	104	27	.	.	PUNCT
iajs-966	105	1	since	since	SCONJ
iajs-966	105	2			PROPN
iajs-966	105	3	is	be	AUX
iajs-966	105	4	an	an	DET
iajs-966	105	5	epimorphisim	epimorphisim	NOUN
iajs-966	105	6	,	,	PUNCT
iajs-966	105	7	then	then	ADV
iajs-966	105	8	there	there	PRON
iajs-966	105	9	exists	exist	VERB
iajs-966	105	10	xm	xm	PUNCT
iajs-966	105	11	such	such	ADJ
iajs-966	105	12	that	that	DET
iajs-966	105	13	(x)=y	(x)=y	NOUN
iajs-966	105	14	.	.	PUNCT
iajs-966	106	1	it	it	PRON
iajs-966	106	2	is	be	AUX
iajs-966	106	3	clear	clear	ADJ
iajs-966	106	4	that	that	SCONJ
iajs-966	106	5	x	x	PROPN
iajs-966	106	6	–	–	PUNCT
iajs-966	106	7	1	1	NUM
iajs-966	106	8	(	(	PUNCT
iajs-966	106	9	n	n	CCONJ
iajs-966	106	10	'	'	NUM
iajs-966	106	11	)	)	PUNCT
iajs-966	106	12	.	.	PUNCT
iajs-966	107	1	we	we	PRON
iajs-966	107	2	claim	claim	VERB
iajs-966	107	3	that	that	SCONJ
iajs-966	107	4	annr(	annr(	VERB
iajs-966	107	5	–	–	PUNCT
iajs-966	107	6	1	1	NUM
iajs-966	107	7	(	(	PUNCT
iajs-966	107	8	n'))=annr(x	n'))=annr(x	PROPN
iajs-966	107	9	)	)	PUNCT
iajs-966	107	10	,	,	PUNCT
iajs-966	107	11	let	let	VERB
iajs-966	107	12	rannr(x	rannr(x	PRON
iajs-966	107	13	)	)	PUNCT
iajs-966	107	14	.	.	PUNCT
iajs-966	108	1	then	then	ADV
iajs-966	108	2	rx=0	rx=0	PROPN
iajs-966	108	3	,	,	PUNCT
iajs-966	108	4	which	which	PRON
iajs-966	108	5	implies	imply	VERB
iajs-966	108	6	(rx)=0	(rx)=0	PROPN
iajs-966	108	7	.	.	PUNCT
iajs-966	109	1	thus	thus	ADV
iajs-966	109	2	r(x)=0	r(x)=0	VERB
iajs-966	109	3	.	.	PUNCT
iajs-966	110	1	this	this	PRON
iajs-966	110	2	means	mean	VERB
iajs-966	110	3	rannr((x))=annr(y)=annrn	rannr((x))=annr(y)=annrn	NOUN
iajs-966	110	4	'	'	PART
iajs-966	110	5	.	.	PUNCT
iajs-966	111	1	thus	thus	ADV
iajs-966	111	2	rn'=0	rn'=0	VERB
iajs-966	111	3	,	,	PUNCT
iajs-966	111	4	which	which	PRON
iajs-966	111	5	implies	imply	VERB
iajs-966	111	6	–1(rn')=0	–1(rn')=0	PROPN
iajs-966	111	7	.	.	PUNCT
iajs-966	112	1	then	then	ADV
iajs-966	112	2	r–1(n')=0	r–1(n')=0	ADJ
iajs-966	112	3	and	and	CCONJ
iajs-966	112	4	implies	imply	VERB
iajs-966	112	5	rannr(–1(n	rannr(–1(n	NOUN
iajs-966	112	6	'	'	PUNCT
iajs-966	112	7	)	)	PUNCT
iajs-966	112	8	)	)	PUNCT
iajs-966	112	9	.	.	PUNCT
iajs-966	113	1	ihjpas	ihjpa	VERB
iajs-966	113	2	ibn	ibn	PROPN
iajs-966	113	3	alhaitham	alhaitham	PROPN
iajs-966	114	1	j.	j.	PROPN
iajs-966	115	1	fo	fo	ADP
iajs-966	115	2	r	r	NOUN
iajs-966	115	3	pure	pure	ADJ
iajs-966	115	4	&	&	CCONJ
iajs-966	115	5	appl	appl	PROPN
iajs-966	115	6	.	.	PUNCT
iajs-966	116	1	sc	sc	PROPN
iajs-966	116	2	i.	i.	PROPN
iajs-966	116	3	vo	vo	PROPN
iajs-966	116	4	l.23	l.23	PROPN
iajs-966	116	5	(	(	PUNCT
iajs-966	116	6	2	2	NUM
iajs-966	116	7	)	)	PUNCT
iajs-966	116	8	2010	2010	NUM
iajs-966	116	9	on	on	ADP
iajs-966	116	10	the	the	DET
iajs-966	116	11	other	other	ADJ
iajs-966	116	12	hand	hand	NOUN
iajs-966	116	13	,	,	PUNCT
iajs-966	116	14	let	let	VERB
iajs-966	116	15	rannr(	rannr(	PRON
iajs-966	116	16	–	–	PUNCT
iajs-966	116	17	1	1	NUM
iajs-966	116	18	(	(	PUNCT
iajs-966	116	19	n	n	CCONJ
iajs-966	116	20	'	'	NUM
iajs-966	116	21	)	)	PUNCT
iajs-966	116	22	)	)	PUNCT
iajs-966	116	23	.	.	PUNCT
iajs-966	117	1	then	then	ADV
iajs-966	117	2	r	r	PROPN
iajs-966	117	3	–	–	PUNCT
iajs-966	117	4	1	1	NUM
iajs-966	117	5	(	(	PUNCT
iajs-966	117	6	n')=0	n')=0	ADJ
iajs-966	117	7	,	,	PUNCT
iajs-966	117	8	which	which	PRON
iajs-966	117	9	implies	imply	VERB
iajs-966	117	10			X
iajs-966	117	11	–	–	PUNCT
iajs-966	117	12	1	1	NUM
iajs-966	117	13	(	(	PUNCT
iajs-966	117	14	rn')=0	rn')=0	NOUN
iajs-966	117	15	.	.	PUNCT
iajs-966	118	1	this	this	PRON
iajs-966	118	2	means	mean	VERB
iajs-966	118	3	rn'=0	rn'=0	ADJ
iajs-966	118	4	.	.	PUNCT
iajs-966	119	1	therefore	therefore	ADV
iajs-966	119	2	rannrn'=annr(y)=annr((x	rannrn'=annr(y)=annr((x	PROPN
iajs-966	119	3	)	)	PUNCT
iajs-966	119	4	)	)	PUNCT
iajs-966	119	5	.	.	PUNCT
iajs-966	120	1	thus	thus	ADV
iajs-966	120	2	rannr((x	rannr((x	VERB
iajs-966	120	3	)	)	PUNCT
iajs-966	120	4	)	)	PUNCT
iajs-966	120	5	and	and	CCONJ
iajs-966	120	6	from	from	ADP
iajs-966	120	7	this	this	PRON
iajs-966	120	8	,	,	PUNCT
iajs-966	120	9	we	we	PRON
iajs-966	120	10	get	get	VERB
iajs-966	120	11	r(x)=0	r(x)=0	PUNCT
iajs-966	120	12	which	which	PRON
iajs-966	120	13	implies	imply	VERB
iajs-966	120	14	(rx)=0	(rx)=0	PROPN
iajs-966	120	15	.	.	PUNCT
iajs-966	121	1	then	then	ADV
iajs-966	121	2	rx=0	rx=0	PROPN
iajs-966	121	3	and	and	CCONJ
iajs-966	121	4	hence	hence	ADV
iajs-966	121	5	rannr(x	rannr(x	NOUN
iajs-966	121	6	)	)	PUNCT
iajs-966	121	7	.	.	PUNCT
iajs-966	122	1	thus	thus	ADV
iajs-966	122	2	annr(x)=annr(–1(n	annr(x)=annr(–1(n	NOUN
iajs-966	122	3	'	'	PUNCT
iajs-966	122	4	)	)	PUNCT
iajs-966	122	5	)	)	PUNCT
iajs-966	122	6	which	which	PRON
iajs-966	122	7	completes	complete	VERB
iajs-966	122	8	the	the	DET
iajs-966	122	9	proof	proof	NOUN
iajs-966	122	10	.	.	PUNCT
iajs-966	123	1	2	2	X
iajs-966	123	2	.	.	X
iajs-966	123	3	suppose	suppose	VERB
iajs-966	123	4	that	that	SCONJ
iajs-966	123	5	n	n	PRON
iajs-966	123	6	is	be	AUX
iajs-966	123	7	an	an	DET
iajs-966	123	8	almost	almost	ADV
iajs-966	123	9	bounded	bound	VERB
iajs-966	123	10	submodule	submodule	NOUN
iajs-966	123	11	of	of	ADP
iajs-966	123	12	m.	m.	NOUN
iajs-966	123	13	then	then	ADV
iajs-966	123	14	xm	xm	PROPN
iajs-966	123	15	,	,	PUNCT
iajs-966	123	16	xn	xn	PROPN
iajs-966	123	17	such	such	ADJ
iajs-966	123	18	that	that	SCONJ
iajs-966	123	19	annr(x)=annrn	annr(x)=annrn	NOUN
iajs-966	123	20	.	.	PUNCT
iajs-966	124	1	since	since	SCONJ
iajs-966	124	2	xm	xm	NOUN
iajs-966	124	3	,	,	PUNCT
iajs-966	124	4	we	we	PRON
iajs-966	124	5	get	get	VERB
iajs-966	124	6	(x)m	(x)m	PROPN
iajs-966	124	7	'	'	PUNCT
iajs-966	124	8	.	.	PUNCT
iajs-966	125	1	we	we	PRON
iajs-966	125	2	claim	claim	VERB
iajs-966	125	3	that	that	SCONJ
iajs-966	125	4	(x)(n	(x)(n	NOUN
iajs-966	125	5	)	)	PUNCT
iajs-966	125	6	.	.	PUNCT
iajs-966	126	1	suppose	suppose	VERB
iajs-966	126	2	that	that	SCONJ
iajs-966	126	3	(x)(n	(x)(n	PROPN
iajs-966	126	4	)	)	PUNCT
iajs-966	126	5	.	.	PUNCT
iajs-966	127	1	then	then	ADV
iajs-966	127	2	(x)=	(x)=	X
iajs-966	128	1	(n	(n	ADJ
iajs-966	128	2	)	)	PUNCT
iajs-966	128	3	for	for	ADP
iajs-966	128	4	some	some	DET
iajs-966	128	5	nn	nn	NOUN
iajs-966	128	6	,	,	PUNCT
iajs-966	128	7	which	which	PRON
iajs-966	128	8	implies	imply	VERB
iajs-966	128	9	that	that	SCONJ
iajs-966	128	10	(x	(x	NOUN
iajs-966	128	11	)	)	PUNCT
iajs-966	128	12	–	–	PUNCT
iajs-966	128	13	(n)=0	(n)=0	VERB
iajs-966	128	14	,	,	PUNCT
iajs-966	128	15	so	so	SCONJ
iajs-966	128	16	that	that	SCONJ
iajs-966	128	17	(x	(x	NOUN
iajs-966	128	18	–	–	PUNCT
iajs-966	128	19	n)=0	n)=0	NOUN
iajs-966	128	20	.	.	PUNCT
iajs-966	129	1	thus	thus	ADV
iajs-966	129	2	x	x	X
iajs-966	129	3	–	–	PUNCT
iajs-966	129	4	n=–1(0	n=–1(0	NOUN
iajs-966	129	5	)	)	PUNCT
iajs-966	129	6	and	and	CCONJ
iajs-966	129	7	hence	hence	ADV
iajs-966	129	8	x	x	NOUN
iajs-966	129	9	–	–	PUNCT
iajs-966	129	10	n=0	n=0	NUM
iajs-966	129	11	.	.	PUNCT
iajs-966	130	1	then	then	ADV
iajs-966	130	2	x	x	X
iajs-966	130	3	=	=	PUNCT
iajs-966	130	4	nn	nn	X
iajs-966	130	5	.	.	PUNCT
iajs-966	130	6	therefore	therefore	ADV
iajs-966	130	7	xn	xn	X
iajs-966	130	8	which	which	PRON
iajs-966	130	9	is	be	AUX
iajs-966	130	10	a	a	DET
iajs-966	130	11	contradiction	contradiction	NOUN
iajs-966	130	12	.	.	PUNCT
iajs-966	131	1	hence	hence	ADV
iajs-966	131	2	(x)(n	(x)(n	NUM
iajs-966	131	3	)	)	PUNCT
iajs-966	131	4	.	.	PUNCT
iajs-966	132	1	to	to	PART
iajs-966	132	2	show	show	VERB
iajs-966	132	3	that	that	DET
iajs-966	132	4	annr((n))=annr((x	annr((n))=annr((x	NOUN
iajs-966	132	5	)	)	PUNCT
iajs-966	132	6	)	)	PUNCT
iajs-966	132	7	.	.	PUNCT
iajs-966	133	1	let	let	VERB
iajs-966	133	2	rannr((x	rannr((x	NOUN
iajs-966	133	3	)	)	PUNCT
iajs-966	133	4	)	)	PUNCT
iajs-966	133	5	.	.	PUNCT
iajs-966	134	1	then	then	ADV
iajs-966	134	2	r(x)=0	r(x)=0	X
iajs-966	134	3	,	,	PUNCT
iajs-966	134	4	which	which	PRON
iajs-966	134	5	implies	imply	VERB
iajs-966	134	6	(rx)=0	(rx)=0	PROPN
iajs-966	134	7	.	.	PUNCT
iajs-966	135	1	thus	thus	ADV
iajs-966	135	2	rx=0	rx=0	ADP
iajs-966	135	3	,	,	PUNCT
iajs-966	135	4	that	that	PRON
iajs-966	135	5	is	be	AUX
iajs-966	135	6	rannr(x)=annrn	rannr(x)=annrn	PROPN
iajs-966	135	7	.	.	PUNCT
iajs-966	136	1	then	then	ADV
iajs-966	136	2	rannrn	rannrn	PROPN
iajs-966	136	3	,	,	PUNCT
iajs-966	136	4	which	which	PRON
iajs-966	136	5	implies	imply	VERB
iajs-966	136	6	that	that	SCONJ
iajs-966	136	7	rn=0	rn=0	PROPN
iajs-966	136	8	,	,	PUNCT
iajs-966	136	9	so	so	SCONJ
iajs-966	136	10	that	that	SCONJ
iajs-966	136	11	(rn)=0	(rn)=0	PROPN
iajs-966	136	12	.	.	PUNCT
iajs-966	137	1	then	then	ADV
iajs-966	137	2	r(n)=0	r(n)=0	PUNCT
iajs-966	137	3	.	.	PUNCT
iajs-966	138	1	hence	hence	ADV
iajs-966	138	2	rannr(n	rannr(n	NOUN
iajs-966	138	3	)	)	PUNCT
iajs-966	138	4	)	)	PUNCT
iajs-966	138	5	.	.	PUNCT
iajs-966	139	1	therefore	therefore	ADV
iajs-966	139	2	annr((x))annr((n	annr((x))annr((n	PROPN
iajs-966	139	3	)	)	PUNCT
iajs-966	139	4	)	)	PUNCT
iajs-966	139	5	.	.	PUNCT
iajs-966	140	1	by	by	ADP
iajs-966	140	2	using	use	VERB
iajs-966	140	3	the	the	DET
iajs-966	140	4	same	same	ADJ
iajs-966	140	5	way	way	NOUN
iajs-966	140	6	,	,	PUNCT
iajs-966	140	7	we	we	PRON
iajs-966	140	8	can	can	AUX
iajs-966	140	9	prove	prove	VERB
iajs-966	140	10	the	the	DET
iajs-966	140	11	other	other	ADJ
iajs-966	140	12	inclusion	inclusion	NOUN
iajs-966	140	13	.	.	PUNCT
iajs-966	141	1	hence	hence	ADV
iajs-966	141	2	annr((n))=	annr((n))=	PROPN
iajs-966	141	3	annr((x	annr((x	NOUN
iajs-966	141	4	)	)	PUNCT
iajs-966	141	5	)	)	PUNCT
iajs-966	142	1	which	which	PRON
iajs-966	142	2	is	be	AUX
iajs-966	142	3	what	what	PRON
iajs-966	142	4	we	we	PRON
iajs-966	142	5	wanted	want	VERB
iajs-966	142	6	.	.	PUNCT
iajs-966	143	1	the	the	DET
iajs-966	143	2	condition	condition	NOUN
iajs-966	143	3	(	(	PUNCT
iajs-966	143	4			NOUN
iajs-966	143	5	:	:	PUNCT
iajs-966	143	6	m	m	VERB
iajs-966	143	7			PROPN
iajs-966	143	8	m	m	X
iajs-966	143	9	'	'	PUNCT
iajs-966	143	10	is	be	AUX
iajs-966	143	11	an	an	DET
iajs-966	143	12	isomorphism	isomorphism	NOUN
iajs-966	143	13	)	)	PUNCT
iajs-966	143	14	in	in	ADP
iajs-966	143	15	proposition	proposition	NOUN
iajs-966	143	16	(	(	PUNCT
iajs-966	143	17	1.11	1.11	NUM
iajs-966	143	18	)	)	PUNCT
iajs-966	143	19	can	can	AUX
iajs-966	143	20	not	not	PART
iajs-966	143	21	be	be	AUX
iajs-966	143	22	dropped	drop	VERB
iajs-966	143	23	as	as	SCONJ
iajs-966	143	24	the	the	DET
iajs-966	143	25	following	follow	VERB
iajs-966	143	26	example	example	NOUN
iajs-966	143	27	shows	show	VERB
iajs-966	143	28	.	.	PUNCT
iajs-966	144	1	1.12	1.12	NUM
iajs-966	144	2	example	example	NOUN
iajs-966	144	3	:	:	PUNCT
iajs-966	144	4	1	1	X
iajs-966	144	5	.	.	X
iajs-966	144	6	let	let	VERB
iajs-966	144	7	:zz4z4	:zz4z4	PROPN
iajs-966	144	8	be	be	AUX
iajs-966	144	9	a	a	DET
iajs-966	144	10	projection	projection	NOUN
iajs-966	144	11	map	map	NOUN
iajs-966	144	12	such	such	ADJ
iajs-966	144	13	that	that	SCONJ
iajs-966	144	14	(x	(x	NOUN
iajs-966	144	15	,	,	PUNCT
iajs-966	144	16	y)=y	y)=y	NOUN
iajs-966	144	17	for	for	ADP
iajs-966	144	18	all	all	DET
iajs-966	144	19	(	(	PUNCT
iajs-966	144	20	x	x	NOUN
iajs-966	144	21	,	,	PUNCT
iajs-966	144	22	y)zz4	y)zz4	NUM
iajs-966	144	23	.	.	PUNCT
iajs-966	145	1	let	let	VERB
iajs-966	145	2	n=<3>	n=<3>	PRON
iajs-966	145	3	2	2	PRON
iajs-966	145	4			PROPN
iajs-966	145	5	be	be	AUX
iajs-966	145	6	a	a	DET
iajs-966	145	7	submodule	submodule	NOUN
iajs-966	145	8	of	of	ADP
iajs-966	145	9	zz4	zz4	PROPN
iajs-966	145	10	.	.	PUNCT
iajs-966	146	1	it	it	PRON
iajs-966	146	2	is	be	AUX
iajs-966	146	3	easily	easily	ADV
iajs-966	146	4	to	to	PART
iajs-966	146	5	show	show	VERB
iajs-966	146	6	that	that	SCONJ
iajs-966	146	7	n	n	PRON
iajs-966	146	8	is	be	AUX
iajs-966	146	9	an	an	DET
iajs-966	146	10	almost	almost	ADV
iajs-966	146	11	bounded	bounded	ADJ
iajs-966	146	12	submodule	submodule	NOUN
iajs-966	146	13	of	of	ADP
iajs-966	146	14	zz4	zz4	PROPN
iajs-966	146	15	.	.	PUNCT
iajs-966	147	1	but	but	CCONJ
iajs-966	147	2	(n	(n	X
iajs-966	147	3	)	)	PUNCT
iajs-966	147	4	is	be	AUX
iajs-966	147	5	a	a	DET
iajs-966	147	6	submodule	submodule	NOUN
iajs-966	147	7	of	of	ADP
iajs-966	147	8	z4	z4	PROPN
iajs-966	147	9	and	and	CCONJ
iajs-966	147	10	it	it	PRON
iajs-966	147	11	is	be	AUX
iajs-966	147	12	not	not	PART
iajs-966	147	13	almost	almost	ADV
iajs-966	147	14	bounded	bound	VERB
iajs-966	147	15	submodule	submodule	NOUN
iajs-966	147	16	of	of	ADP
iajs-966	147	17	z4	z4	PROPN
iajs-966	147	18	by	by	ADP
iajs-966	147	19	(	(	PUNCT
iajs-966	147	20	remarks	remark	NOUN
iajs-966	147	21	and	and	CCONJ
iajs-966	147	22	examples	example	NOUN
iajs-966	147	23	(	(	PUNCT
iajs-966	147	24	1.2	1.2	NUM
iajs-966	147	25	)	)	PUNCT
iajs-966	147	26	(	(	PUNCT
iajs-966	147	27	5	5	NUM
iajs-966	147	28	)	)	PUNCT
iajs-966	147	29	)	)	PUNCT
iajs-966	147	30	.	.	PUNCT
iajs-966	148	1	2	2	X
iajs-966	148	2	.	.	X
iajs-966	148	3	let	let	VERB
iajs-966	148	4			NOUN
iajs-966	148	5	:	:	PUNCT
iajs-966	148	6	z4	z4	PROPN
iajs-966	148	7	z4z	z4z	PROPN
iajs-966	148	8	be	be	AUX
iajs-966	148	9	an	an	DET
iajs-966	148	10	injection	injection	NOUN
iajs-966	148	11	map	map	NOUN
iajs-966	148	12	such	such	ADJ
iajs-966	148	13	that	that	DET
iajs-966	148	14	(x)=(x,0	(x)=(x,0	NOUN
iajs-966	148	15	)	)	PUNCT
iajs-966	148	16	for	for	ADP
iajs-966	148	17	all	all	DET
iajs-966	148	18	xz4	xz4	NOUN
iajs-966	148	19	,	,	PUNCT
iajs-966	148	20	let	let	VERB
iajs-966	148	21	n'=	n'=	ADV
iajs-966	148	22	2	2	NUM
iajs-966	148	23	<3	<3	NOUN
iajs-966	148	24	>	>	X
iajs-966	148	25	be	be	VERB
iajs-966	148	26	an	an	DET
iajs-966	148	27	almost	almost	ADV
iajs-966	148	28	bounded	bounded	ADJ
iajs-966	148	29	submodule	submodule	NOUN
iajs-966	148	30	of	of	ADP
iajs-966	148	31	z4z	z4z	PROPN
iajs-966	148	32	.	.	PUNCT
iajs-966	149	1	it	it	PRON
iajs-966	149	2	is	be	AUX
iajs-966	149	3	know	know	ADJ
iajs-966	149	4	that	that	SCONJ
iajs-966	149	5	z4	z4	PROPN
iajs-966	149	6	has	have	VERB
iajs-966	149	7	no	no	DET
iajs-966	149	8	proper	proper	ADJ
iajs-966	149	9	almost	almost	ADV
iajs-966	149	10	bounded	bound	VERB
iajs-966	149	11	submodule	submodule	NOUN
iajs-966	149	12	.	.	PUNCT
iajs-966	150	1	since	since	SCONJ
iajs-966	150	2	(	(	PUNCT
iajs-966	150	3			X
iajs-966	150	4	–	–	PUNCT
iajs-966	150	5	1	1	NUM
iajs-966	150	6	(	(	PUNCT
iajs-966	150	7	n	n	CCONJ
iajs-966	150	8	'	'	CCONJ
iajs-966	150	9	)	)	PUNCT
iajs-966	150	10	is	be	AUX
iajs-966	150	11	a	a	DET
iajs-966	150	12	submodule	submodule	NOUN
iajs-966	150	13	of	of	ADP
iajs-966	150	14	z4	z4	PROPN
iajs-966	150	15	,	,	PUNCT
iajs-966	150	16	then	then	ADV
iajs-966	150	17	(	(	PUNCT
iajs-966	150	18			X
iajs-966	150	19	–	–	PUNCT
iajs-966	150	20	1	1	NUM
iajs-966	150	21	(	(	PUNCT
iajs-966	150	22	n	n	CCONJ
iajs-966	150	23	'	'	CCONJ
iajs-966	150	24	)	)	PUNCT
iajs-966	150	25	is	be	AUX
iajs-966	150	26	not	not	PART
iajs-966	150	27	almost	almost	ADV
iajs-966	150	28	bounded	bound	VERB
iajs-966	150	29	submodule	submodule	NOUN
iajs-966	150	30	of	of	ADP
iajs-966	150	31	z4	z4	PROPN
iajs-966	150	32	by	by	ADP
iajs-966	150	33	(	(	PUNCT
iajs-966	150	34	remarks	remark	NOUN
iajs-966	150	35	and	and	CCONJ
iajs-966	150	36	examples	example	NOUN
iajs-966	150	37	(	(	PUNCT
iajs-966	150	38	1.2	1.2	NUM
iajs-966	150	39	)	)	PUNCT
iajs-966	150	40	(	(	PUNCT
iajs-966	150	41	5	5	NUM
iajs-966	150	42	)	)	PUNCT
iajs-966	150	43	)	)	PUNCT
iajs-966	150	44	.	.	PUNCT
iajs-966	151	1	2modules	2modules	NUM
iajs-966	151	2	related	related	ADJ
iajs-966	151	3	to	to	ADP
iajs-966	151	4	almost	almost	ADV
iajs-966	151	5	bounded	bound	VERB
iajs-966	151	6	submodules	submodule	NOUN
iajs-966	151	7	in	in	ADP
iajs-966	151	8	this	this	DET
iajs-966	151	9	section	section	NOUN
iajs-966	151	10	,	,	PUNCT
iajs-966	151	11	we	we	PRON
iajs-966	151	12	study	study	VERB
iajs-966	151	13	the	the	DET
iajs-966	151	14	relationships	relationship	NOUN
iajs-966	151	15	between	between	ADP
iajs-966	151	16	almost	almost	ADV
iajs-966	151	17	bounded	bound	VERB
iajs-966	151	18	submoduls	submoduls	NOUN
iajs-966	151	19	and	and	CCONJ
iajs-966	151	20	bounded	bound	VERB
iajs-966	151	21	modules	module	NOUN
iajs-966	151	22	,	,	PUNCT
iajs-966	151	23	p	p	NOUN
iajs-966	151	24	rime	rime	NOUN
iajs-966	151	25	and	and	CCONJ
iajs-966	151	26	fully	fully	ADV
iajs-966	151	27	stable	stable	ADJ
iajs-966	151	28	modules	module	NOUN
iajs-966	151	29	.	.	PUNCT
iajs-966	152	1	we	we	PRON
iajs-966	152	2	start	start	VERB
iajs-966	152	3	with	with	ADP
iajs-966	152	4	the	the	DET
iajs-966	152	5	following	follow	VERB
iajs-966	152	6	definition	definition	NOUN
iajs-966	152	7	which	which	PRON
iajs-966	152	8	will	will	AUX
iajs-966	152	9	be	be	AUX
iajs-966	152	10	needed	need	VERB
iajs-966	152	11	.	.	PUNCT
iajs-966	153	1	recall	recall	VERB
iajs-966	153	2	that	that	SCONJ
iajs-966	153	3	an	an	DET
iajs-966	153	4	r	r	NOUN
iajs-966	153	5	-	-	PUNCT
iajs-966	153	6	module	module	NOUN
iajs-966	153	7	m	m	NOUN
iajs-966	153	8	is	be	AUX
iajs-966	153	9	said	say	VERB
iajs-966	153	10	to	to	PART
iajs-966	153	11	be	be	AUX
iajs-966	153	12	bounded	bound	VERB
iajs-966	153	13	module	module	NOUN
iajs-966	153	14	,	,	PUNCT
iajs-966	153	15	if	if	SCONJ
iajs-966	153	16	there	there	PRON
iajs-966	153	17	exists	exist	VERB
iajs-966	153	18	an	an	DET
iajs-966	153	19	element	element	NOUN
iajs-966	153	20	xm	xm	PUNCT
iajs-966	153	21	such	such	ADJ
iajs-966	153	22	that	that	DET
iajs-966	153	23	annrm	annrm	NOUN
iajs-966	153	24	=	=	SYM
iajs-966	153	25	annr(x	annr(x	NOUN
iajs-966	153	26	)	)	PUNCT
iajs-966	153	27	,	,	PUNCT
iajs-966	154	1	[	[	X
iajs-966	154	2	1	1	NUM
iajs-966	154	3	]	]	PUNCT
iajs-966	154	4	.	.	PUNCT
iajs-966	155	1	by	by	ADP
iajs-966	155	2	using	use	VERB
iajs-966	155	3	this	this	DET
iajs-966	155	4	concept	concept	NOUN
iajs-966	155	5	,	,	PUNCT
iajs-966	155	6	we	we	PRON
iajs-966	155	7	have	have	VERB
iajs-966	155	8	the	the	DET
iajs-966	155	9	following	following	NOUN
iajs-966	155	10	.	.	PUNCT
iajs-966	156	1	2.1	2.1	NUM
iajs-966	156	2	remark	remark	NOUN
iajs-966	156	3	:	:	PUNCT
iajs-966	156	4	a	a	DET
iajs-966	156	5	submodule	submodule	NOUN
iajs-966	156	6	n	n	PROPN
iajs-966	156	7	of	of	ADP
iajs-966	156	8	a	a	DET
iajs-966	156	9	bounded	bounded	ADJ
iajs-966	156	10	r	r	NOUN
iajs-966	156	11	-	-	PUNCT
iajs-966	156	12	module	module	NOUN
iajs-966	156	13	m	m	NOUN
iajs-966	156	14	is	be	AUX
iajs-966	156	15	not	not	PART
iajs-966	156	16	necessary	necessary	ADJ
iajs-966	156	17	be	be	AUX
iajs-966	156	18	an	an	DET
iajs-966	156	19	almost	almost	ADV
iajs-966	156	20	bounded	bound	VERB
iajs-966	156	21	.	.	PUNCT
iajs-966	157	1	for	for	ADP
iajs-966	157	2	example	example	NOUN
iajs-966	157	3	z4	z4	PROPN
iajs-966	157	4	as	as	ADP
iajs-966	157	5	a	a	DET
iajs-966	157	6	z4	z4	NOUN
iajs-966	157	7	-	-	PUNCT
iajs-966	157	8	module	module	NOUN
iajs-966	157	9	is	be	AUX
iajs-966	157	10	bounded	bound	VERB
iajs-966	157	11	module	module	NOUN
iajs-966	157	12	,	,	PUNCT
iajs-966	157	13	but	but	CCONJ
iajs-966	157	14	2	2	NUM
iajs-966	157	15			INTJ
iajs-966	157	16	is	be	AUX
iajs-966	157	17	not	not	PART
iajs-966	157	18	almost	almost	ADV
iajs-966	157	19	bounded	bounded	ADJ
iajs-966	157	20	submodule	submodule	PROPN
iajs-966	157	21	.	.	PUNCT
iajs-966	158	1	recall	recall	VERB
iajs-966	158	2	that	that	SCONJ
iajs-966	158	3	an	an	DET
iajs-966	158	4	r	r	NOUN
iajs-966	158	5	-	-	PUNCT
iajs-966	158	6	module	module	NOUN
iajs-966	158	7	m	m	NOUN
iajs-966	158	8	is	be	AUX
iajs-966	158	9	called	call	VERB
iajs-966	158	10	a	a	DET
iajs-966	158	11	quasi	quasi	ADJ
iajs-966	158	12	-	-	ADJ
iajs-966	158	13	prime	prime	ADJ
iajs-966	158	14	r	r	NOUN
iajs-966	158	15	-	-	PUNCT
iajs-966	158	16	module	module	NOUN
iajs-966	158	17	if	if	SCONJ
iajs-966	158	18	and	and	CCONJ
iajs-966	158	19	only	only	ADV
iajs-966	158	20	if	if	SCONJ
iajs-966	158	21	annrn	annrn	NOUN
iajs-966	158	22	is	be	AUX
iajs-966	158	23	a	a	DET
iajs-966	158	24	prime	prime	ADJ
iajs-966	158	25	ideal	ideal	NOUN
iajs-966	158	26	for	for	ADP
iajs-966	158	27	each	each	DET
iajs-966	158	28	non	non	ADJ
iajs-966	158	29	-	-	ADJ
iajs-966	158	30	zero	zero	NUM
iajs-966	158	31	submodule	submodule	NOUN
iajs-966	158	32	n	n	PROPN
iajs-966	158	33	of	of	ADP
iajs-966	158	34	m	m	PRON
iajs-966	158	35	,	,	PUNCT
iajs-966	159	1	[	[	X
iajs-966	159	2	2	2	NUM
iajs-966	159	3	]	]	PUNCT
iajs-966	159	4	.	.	PUNCT
iajs-966	160	1	recall	recall	VERB
iajs-966	160	2	that	that	SCONJ
iajs-966	160	3	a	a	DET
iajs-966	160	4	submodule	submodule	NOUN
iajs-966	160	5	n	n	PROPN
iajs-966	160	6	of	of	ADP
iajs-966	160	7	an	an	DET
iajs-966	160	8	r	r	NOUN
iajs-966	160	9	-	-	PUNCT
iajs-966	160	10	module	module	NOUN
iajs-966	160	11	m	m	NOUN
iajs-966	160	12	is	be	AUX
iajs-966	160	13	called	call	VERB
iajs-966	160	14	essential	essential	ADJ
iajs-966	160	15	if	if	SCONJ
iajs-966	160	16	nk≠0	nk≠0	ADJ
iajs-966	160	17	for	for	ADP
iajs-966	160	18	every	every	DET
iajs-966	160	19	non	non	ADJ
iajs-966	160	20	-	-	ADJ
iajs-966	160	21	zero	zero	NUM
iajs-966	160	22	submodule	submodule	NOUN
iajs-966	160	23	k	k	PROPN
iajs-966	160	24	of	of	ADP
iajs-966	160	25	m	m	PRON
iajs-966	160	26	,	,	PUNCT
iajs-966	160	27	[	[	X
iajs-966	160	28	1	1	NUM
iajs-966	160	29	]	]	PUNCT
iajs-966	160	30	.	.	PUNCT
iajs-966	161	1	the	the	DET
iajs-966	161	2	following	follow	VERB
iajs-966	161	3	proposition	proposition	NOUN
iajs-966	161	4	gives	give	VERB
iajs-966	161	5	a	a	DET
iajs-966	161	6	sufficient	sufficient	ADJ
iajs-966	161	7	condition	condition	NOUN
iajs-966	161	8	under	under	ADP
iajs-966	161	9	which	which	PRON
iajs-966	161	10	every	every	DET
iajs-966	161	11	submodule	submodule	NOUN
iajs-966	161	12	of	of	ADP
iajs-966	161	13	a	a	DET
iajs-966	161	14	bounded	bound	VERB
iajs-966	161	15	module	module	NOUN
iajs-966	161	16	is	be	AUX
iajs-966	161	17	an	an	DET
iajs-966	161	18	almost	almost	ADV
iajs-966	161	19	bounded	bound	VERB
iajs-966	161	20	.	.	PUNCT
iajs-966	162	1	2.2	2.2	NUM
iajs-966	162	2	proposition	proposition	NOUN
iajs-966	162	3	:	:	PUNCT
iajs-966	162	4	let	let	VERB
iajs-966	162	5	m	m	PRON
iajs-966	162	6	be	be	AUX
iajs-966	162	7	a	a	DET
iajs-966	162	8	cyclic	cyclic	ADJ
iajs-966	162	9	quasi	quasi	NOUN
iajs-966	162	10	-	-	ADJ
iajs-966	162	11	prime	prime	ADJ
iajs-966	162	12	r	r	NOUN
iajs-966	162	13	-	-	PUNCT
iajs-966	162	14	module	module	NOUN
iajs-966	162	15	and	and	CCONJ
iajs-966	162	16	n	n	CCONJ
iajs-966	162	17	be	be	VERB
iajs-966	162	18	a	a	DET
iajs-966	162	19	proper	proper	ADJ
iajs-966	162	20	essential	essential	ADJ
iajs-966	162	21	submodule	submodule	NOUN
iajs-966	162	22	of	of	ADP
iajs-966	162	23	m.	m.	NOUN
iajs-966	162	24	then	then	ADV
iajs-966	162	25	n	n	PRON
iajs-966	162	26	is	be	AUX
iajs-966	162	27	an	an	DET
iajs-966	162	28	almost	almost	ADV
iajs-966	162	29	bounded	bounded	ADJ
iajs-966	162	30	submodule	submodule	NOUN
iajs-966	162	31	.	.	PUNCT
iajs-966	163	1	proof	proof	NOUN
iajs-966	163	2	:	:	PUNCT
iajs-966	163	3	assume	assume	VERB
iajs-966	163	4	that	that	SCONJ
iajs-966	163	5	n	n	PRON
iajs-966	163	6	is	be	AUX
iajs-966	163	7	p	p	PROPN
iajs-966	163	8	roper	roper	NOUN
iajs-966	163	9	submodule	submodule	NOUN
iajs-966	163	10	of	of	ADP
iajs-966	163	11	an	an	DET
iajs-966	163	12	r	r	NOUN
iajs-966	163	13	-	-	PUNCT
iajs-966	163	14	module	module	NOUN
iajs-966	163	15	m	m	NOUN
iajs-966	163	16	,	,	PUNCT
iajs-966	163	17	then	then	ADV
iajs-966	163	18	there	there	PRON
iajs-966	163	19	exists	exist	VERB
iajs-966	163	20	ym	ym	PROPN
iajs-966	163	21	,	,	PUNCT
iajs-966	163	22	yn	yn	PROPN
iajs-966	163	23	.	.	PROPN
iajs-966	164	1	since	since	SCONJ
iajs-966	164	2	n	n	NUM
iajs-966	164	3	is	be	AUX
iajs-966	164	4	essential	essential	ADJ
iajs-966	164	5	submodule	submodule	NOUN
iajs-966	164	6	of	of	ADP
iajs-966	164	7	m	m	PROPN
iajs-966	164	8	,	,	PUNCT
iajs-966	164	9	thus	thus	ADV
iajs-966	164	10	there	there	PRON
iajs-966	164	11	exists	exist	VERB
iajs-966	164	12	rr	rr	NOUN
iajs-966	164	13	,	,	PUNCT
iajs-966	164	14	r≠0	r≠0	NOUN
iajs-966	164	15	.	.	PUNCT
iajs-966	165	1	thus	thus	ADV
iajs-966	165	2	annrry	annrry	VERB
iajs-966	165	3			PROPN
iajs-966	165	4	annrn	annrn	NOUN
iajs-966	165	5	.	.	PUNCT
iajs-966	166	1	but	but	CCONJ
iajs-966	166	2	m	m	VERB
iajs-966	166	3	quasi	quasi	ADJ
iajs-966	166	4	-	-	ADJ
iajs-966	166	5	prime	prime	ADJ
iajs-966	166	6	,	,	PUNCT
iajs-966	166	7	so	so	ADV
iajs-966	166	8	annrry	annrry	VERB
iajs-966	166	9	=	=	SYM
iajs-966	166	10	annry	annry	NOUN
iajs-966	166	11	.	.	PUNCT
iajs-966	167	1	then	then	ADV
iajs-966	167	2	annry	annry	ADJ
iajs-966	167	3			PROPN
iajs-966	167	4	annrn	annrn	NOUN
iajs-966	167	5			PROPN
iajs-966	167	6	annrm	annrm	NOUN
iajs-966	167	7	.	.	PUNCT
iajs-966	168	1	let	let	VERB
iajs-966	168	2	tannry	tannry	PROPN
iajs-966	168	3	.	.	PUNCT
iajs-966	169	1	then	then	ADV
iajs-966	169	2	ty=0	ty=0	ADV
iajs-966	169	3	,	,	PUNCT
iajs-966	169	4	but	but	CCONJ
iajs-966	169	5	m	m	NOUN
iajs-966	169	6	is	be	AUX
iajs-966	169	7	cyclic	cyclic	ADJ
iajs-966	169	8	.	.	PUNCT
iajs-966	170	1	thus	thus	ADV
iajs-966	170	2	y	y	X
iajs-966	170	3	=	=	PROPN
iajs-966	170	4	cx	cx	PROPN
iajs-966	170	5	for	for	ADP
iajs-966	170	6	some	some	DET
iajs-966	170	7	cr	cr	NOUN
iajs-966	170	8	.	.	PUNCT
iajs-966	171	1	therefore	therefore	ADV
iajs-966	171	2	tcx=0	tcx=0	X
iajs-966	171	3	which	which	PRON
iajs-966	171	4	implies	imply	VERB
iajs-966	171	5	that	that	SCONJ
iajs-966	171	6	tcannr(x	tcannr(x	NOUN
iajs-966	171	7	)	)	PUNCT
iajs-966	171	8	.	.	PUNCT
iajs-966	172	1	thus	thus	ADV
iajs-966	172	2	either	either	ADV
iajs-966	172	3	cannr(x	cannr(x	PUNCT
iajs-966	172	4	)	)	PUNCT
iajs-966	172	5	or	or	CCONJ
iajs-966	172	6	tannr(x	tannr(x	NUM
iajs-966	172	7	)	)	PUNCT
iajs-966	172	8	.	.	PUNCT
iajs-966	173	1	if	if	SCONJ
iajs-966	173	2	cannr(x	cannr(x	PRON
iajs-966	173	3	)	)	PUNCT
iajs-966	173	4	,	,	PUNCT
iajs-966	173	5	then	then	ADV
iajs-966	173	6	cx	cx	X
iajs-966	173	7	=	=	SYM
iajs-966	173	8	y=0	y=0	X
iajs-966	173	9	.	.	PUNCT
iajs-966	174	1	this	this	PRON
iajs-966	174	2	is	be	AUX
iajs-966	174	3	a	a	DET
iajs-966	174	4	contradiction	contradiction	NOUN
iajs-966	174	5	.	.	PUNCT
iajs-966	175	1	thus	thus	ADV
iajs-966	175	2	tannr(x	tannr(x	X
iajs-966	175	3	)	)	PUNCT
iajs-966	175	4	=	=	SYM
iajs-966	175	5	annrm	annrm	NOUN
iajs-966	175	6			PROPN
iajs-966	175	7	annrn	annrn	NOUN
iajs-966	175	8	.	.	PUNCT
iajs-966	176	1	therefore	therefore	ADV
iajs-966	176	2	annr(y	annr(y	VERB
iajs-966	176	3	)	)	PUNCT
iajs-966	176	4	=	=	SYM
iajs-966	176	5	annrn	annrn	NOUN
iajs-966	176	6	and	and	CCONJ
iajs-966	176	7	hence	hence	ADV
iajs-966	176	8	n	n	PRON
iajs-966	176	9	is	be	AUX
iajs-966	176	10	an	an	DET
iajs-966	176	11	almost	almost	ADV
iajs-966	176	12	bounded	bound	VERB
iajs-966	176	13	submodule	submodule	NOUN
iajs-966	176	14	of	of	ADP
iajs-966	176	15	m.	m.	NOUN
iajs-966	176	16	ihjpas	ihjpas	AUX
iajs-966	176	17	ibn	ibn	PROPN
iajs-966	176	18	alhaitham	alhaitham	PROPN
iajs-966	177	1	j.	j.	PROPN
iajs-966	178	1	fo	fo	ADP
iajs-966	178	2	r	r	NOUN
iajs-966	178	3	pure	pure	ADJ
iajs-966	178	4	&	&	CCONJ
iajs-966	178	5	appl	appl	PROPN
iajs-966	178	6	.	.	PUNCT
iajs-966	179	1	sc	sc	PROPN
iajs-966	179	2	i.	i.	PROPN
iajs-966	179	3	vo	vo	PROPN
iajs-966	179	4	l.23	l.23	PROPN
iajs-966	179	5	(	(	PUNCT
iajs-966	179	6	2	2	NUM
iajs-966	179	7	)	)	PUNCT
iajs-966	179	8	2010	2010	NUM
iajs-966	179	9	an	an	DET
iajs-966	179	10	r	r	NOUN
iajs-966	179	11	-	-	PUNCT
iajs-966	179	12	module	module	NOUN
iajs-966	179	13	m	m	NOUN
iajs-966	179	14	is	be	AUX
iajs-966	179	15	said	say	VERB
iajs-966	179	16	to	to	PART
iajs-966	179	17	be	be	AUX
iajs-966	179	18	uniform	uniform	ADJ
iajs-966	179	19	module	module	NOUN
iajs-966	179	20	if	if	SCONJ
iajs-966	179	21	every	every	DET
iajs-966	179	22	nonzero	nonzero	PROPN
iajs-966	179	23	submodule	submodule	NOUN
iajs-966	179	24	of	of	ADP
iajs-966	179	25	m	m	PROPN
iajs-966	179	26	is	be	AUX
iajs-966	179	27	essential	essential	ADJ
iajs-966	179	28	,	,	PUNCT
iajs-966	179	29	[	[	X
iajs-966	179	30	1	1	NUM
iajs-966	179	31	]	]	PUNCT
iajs-966	179	32	.	.	PUNCT
iajs-966	180	1	now	now	ADV
iajs-966	180	2	,	,	PUNCT
iajs-966	180	3	we	we	PRON
iajs-966	180	4	deduce	deduce	VERB
iajs-966	180	5	the	the	DET
iajs-966	180	6	following	follow	VERB
iajs-966	180	7	corollary	corollary	NOUN
iajs-966	180	8	.	.	PUNCT
iajs-966	181	1	2.3	2.3	NUM
iajs-966	181	2	corollary	corollary	NOUN
iajs-966	181	3	:	:	PUNCT
iajs-966	181	4	let	let	VERB
iajs-966	181	5	m	m	PRON
iajs-966	181	6	be	be	AUX
iajs-966	181	7	a	a	DET
iajs-966	181	8	cyclic	cyclic	ADJ
iajs-966	181	9	uniform	uniform	ADJ
iajs-966	181	10	r	r	NOUN
iajs-966	181	11	-	-	PUNCT
iajs-966	181	12	module	module	NOUN
iajs-966	181	13	and	and	CCONJ
iajs-966	181	14	annrm	annrm	NOUN
iajs-966	181	15	is	be	AUX
iajs-966	181	16	prime	prime	ADJ
iajs-966	181	17	ideal	ideal	NOUN
iajs-966	181	18	of	of	ADP
iajs-966	181	19	r.	r.	PROPN
iajs-966	181	20	let	let	VERB
iajs-966	181	21	n	n	PRON
iajs-966	181	22	be	be	AUX
iajs-966	181	23	a	a	DET
iajs-966	181	24	proper	proper	ADJ
iajs-966	181	25	submodule	submodule	NOUN
iajs-966	181	26	of	of	ADP
iajs-966	181	27	m.	m.	NOUN
iajs-966	181	28	then	then	ADV
iajs-966	181	29	n	n	PRON
iajs-966	181	30	is	be	AUX
iajs-966	181	31	an	an	DET
iajs-966	181	32	almost	almost	ADV
iajs-966	181	33	bounded	bounded	ADJ
iajs-966	181	34	submodule	submodule	NOUN
iajs-966	181	35	.	.	PUNCT
iajs-966	182	1	proof	proof	NOUN
iajs-966	182	2	:	:	PUNCT
iajs-966	182	3	the	the	DET
iajs-966	182	4	result	result	NOUN
iajs-966	182	5	follows	follow	VERB
iajs-966	182	6	from	from	ADP
iajs-966	182	7	the	the	DET
iajs-966	182	8	definition	definition	NOUN
iajs-966	182	9	of	of	ADP
iajs-966	182	10	a	a	DET
iajs-966	182	11	uniform	uniform	ADJ
iajs-966	182	12	module	module	NOUN
iajs-966	182	13	,	,	PUNCT
iajs-966	182	14	[	[	X
iajs-966	182	15	2,corollary	2,corollary	NUM
iajs-966	182	16	(	(	PUNCT
iajs-966	182	17	1.2.8	1.2.8	NUM
iajs-966	182	18	)	)	PUNCT
iajs-966	182	19	]	]	PUNCT
iajs-966	182	20	and	and	CCONJ
iajs-966	182	21	proposition	proposition	NOUN
iajs-966	182	22	(	(	PUNCT
iajs-966	182	23	2.2	2.2	NUM
iajs-966	182	24	)	)	PUNCT
iajs-966	182	25	.	.	PUNCT
iajs-966	183	1	recall	recall	VERB
iajs-966	183	2	that	that	SCONJ
iajs-966	183	3	an	an	DET
iajs-966	183	4	r	r	NOUN
iajs-966	183	5	-	-	PUNCT
iajs-966	183	6	module	module	NOUN
iajs-966	183	7	m	m	NOUN
iajs-966	183	8	is	be	AUX
iajs-966	183	9	said	say	VERB
iajs-966	183	10	to	to	PART
iajs-966	183	11	be	be	AUX
iajs-966	183	12	a	a	DET
iajs-966	183	13	multiplication	multiplication	NOUN
iajs-966	183	14	module	module	NOUN
iajs-966	183	15	if	if	SCONJ
iajs-966	183	16	for	for	ADP
iajs-966	183	17	every	every	DET
iajs-966	183	18	submodule	submodule	NOUN
iajs-966	183	19	n	n	PROPN
iajs-966	183	20	of	of	ADP
iajs-966	183	21	m	m	PROPN
iajs-966	183	22	,	,	PUNCT
iajs-966	183	23	there	there	PRON
iajs-966	183	24	exists	exist	VERB
iajs-966	183	25	an	an	DET
iajs-966	183	26	ideal	ideal	NOUN
iajs-966	183	27	i	i	PRON
iajs-966	183	28	of	of	ADP
iajs-966	183	29	r	r	NOUN
iajs-966	183	30	such	such	ADJ
iajs-966	183	31	that	that	PRON
iajs-966	183	32	n	n	NOUN
iajs-966	183	33	=	=	NOUN
iajs-966	183	34	im	im	NOUN
iajs-966	183	35	,	,	PUNCT
iajs-966	184	1	[	[	X
iajs-966	184	2	3	3	NUM
iajs-966	184	3	]	]	PUNCT
iajs-966	184	4	.	.	PUNCT
iajs-966	185	1	an	an	DET
iajs-966	185	2	r	r	NOUN
iajs-966	185	3	-	-	PUNCT
iajs-966	185	4	module	module	NOUN
iajs-966	185	5	m	m	NOUN
iajs-966	185	6	is	be	AUX
iajs-966	185	7	called	call	VERB
iajs-966	185	8	fully	fully	ADV
iajs-966	185	9	stable	stable	ADJ
iajs-966	185	10	in	in	ADP
iajs-966	185	11	case	case	NOUN
iajs-966	185	12	each	each	DET
iajs-966	185	13	submodule	submodule	NOUN
iajs-966	185	14	n	n	PROPN
iajs-966	185	15	of	of	ADP
iajs-966	185	16	m	m	PROPN
iajs-966	185	17	is	be	AUX
iajs-966	185	18	stable	stable	ADJ
iajs-966	185	19	,	,	PUNCT
iajs-966	185	20	where	where	SCONJ
iajs-966	185	21	a	a	DET
iajs-966	185	22	submodule	submodule	NOUN
iajs-966	185	23	n	n	PRON
iajs-966	185	24	is	be	AUX
iajs-966	185	25	said	say	VERB
iajs-966	185	26	to	to	PART
iajs-966	185	27	be	be	AUX
iajs-966	185	28	stable	stable	ADJ
iajs-966	185	29	,	,	PUNCT
iajs-966	185	30	if	if	SCONJ
iajs-966	185	31	f(n)n	f(n)n	VERB
iajs-966	185	32	for	for	ADP
iajs-966	185	33	each	each	DET
iajs-966	185	34	r	r	NOUN
iajs-966	185	35	-	-	PUNCT
iajs-966	185	36	homomorphism	homomorphism	ADJ
iajs-966	185	37	f	f	X
iajs-966	185	38	:	:	PUNCT
iajs-966	185	39	nm	nm	PROPN
iajs-966	185	40	,	,	PUNCT
iajs-966	185	41	[	[	X
iajs-966	185	42	4	4	NUM
iajs-966	185	43	]	]	PUNCT
iajs-966	185	44	.	.	PUNCT
iajs-966	186	1	so	so	ADV
iajs-966	186	2	,	,	PUNCT
iajs-966	186	3	we	we	PRON
iajs-966	186	4	have	have	VERB
iajs-966	186	5	the	the	DET
iajs-966	186	6	following	follow	VERB
iajs-966	186	7	proposition	proposition	NOUN
iajs-966	186	8	.	.	PUNCT
iajs-966	187	1	2.4	2.4	NUM
iajs-966	187	2	proposition	proposition	NOUN
iajs-966	187	3	:	:	PUNCT
iajs-966	187	4	let	let	VERB
iajs-966	187	5	n	n	PRON
iajs-966	187	6	be	be	AUX
iajs-966	187	7	a	a	DET
iajs-966	187	8	proper	proper	ADJ
iajs-966	187	9	submodule	submodule	NOUN
iajs-966	187	10	of	of	ADP
iajs-966	187	11	an	an	DET
iajs-966	187	12	r	r	NOUN
iajs-966	187	13	-	-	PUNCT
iajs-966	187	14	module	module	NOUN
iajs-966	187	15	m	m	NOUN
iajs-966	187	16	such	such	ADJ
iajs-966	187	17	that	that	SCONJ
iajs-966	187	18	,	,	PUNCT
iajs-966	187	19	1	1	X
iajs-966	187	20	.	.	X
iajs-966	188	1	m	m	PROPN
iajs-966	188	2	is	be	AUX
iajs-966	188	3	fully	fully	ADV
iajs-966	188	4	stable	stable	ADJ
iajs-966	188	5	and	and	CCONJ
iajs-966	188	6	bounded	bound	VERB
iajs-966	188	7	r	r	NOUN
iajs-966	188	8	-	-	PUNCT
iajs-966	188	9	module	module	NOUN
iajs-966	188	10	.	.	PUNCT
iajs-966	189	1	2	2	X
iajs-966	189	2	.	.	X
iajs-966	190	1	[	[	X
iajs-966	190	2	n	n	X
iajs-966	190	3	r	r	NOUN
iajs-966	190	4	:	:	PUNCT
iajs-966	190	5	m	m	ADJ
iajs-966	190	6	]	]	X
iajs-966	190	7			PROPN
iajs-966	190	8	annrm	annrm	PROPN
iajs-966	190	9	.	.	PUNCT
iajs-966	191	1	3	3	X
iajs-966	191	2	.	.	X
iajs-966	191	3	annrm	annrm	NOUN
iajs-966	191	4	is	be	AUX
iajs-966	191	5	prime	prime	ADJ
iajs-966	191	6	ideal	ideal	NOUN
iajs-966	191	7	of	of	ADP
iajs-966	191	8	r.	r.	PROPN
iajs-966	191	9	then	then	ADV
iajs-966	191	10	n	n	PRON
iajs-966	191	11	is	be	AUX
iajs-966	191	12	an	an	DET
iajs-966	191	13	almost	almost	ADV
iajs-966	191	14	bounded	bound	VERB
iajs-966	191	15	submodule	submodule	NOUN
iajs-966	191	16	of	of	ADP
iajs-966	191	17	m.	m.	NOUN
iajs-966	191	18	proof	proof	NOUN
iajs-966	191	19	:	:	PUNCT
iajs-966	191	20	from	from	ADP
iajs-966	191	21	[	[	X
iajs-966	191	22	1,corollary	1,corollary	PRON
iajs-966	191	23	(	(	PUNCT
iajs-966	191	24	1.1.9	1.1.9	NUM
iajs-966	191	25	)	)	PUNCT
iajs-966	191	26	]	]	PUNCT
iajs-966	191	27	,	,	PUNCT
iajs-966	191	28	we	we	PRON
iajs-966	191	29	get	get	VERB
iajs-966	191	30	m	m	NOUN
iajs-966	191	31	is	be	AUX
iajs-966	191	32	multiplication	multiplication	NOUN
iajs-966	191	33	r	r	NOUN
iajs-966	191	34	-	-	PUNCT
iajs-966	191	35	module	module	NOUN
iajs-966	191	36	and	and	CCONJ
iajs-966	191	37	by	by	ADP
iajs-966	191	38	[	[	PUNCT
iajs-966	191	39	4,corollary	4,corollary	NUM
iajs-966	191	40	(	(	PUNCT
iajs-966	191	41	2.7	2.7	NUM
iajs-966	191	42	)	)	PUNCT
iajs-966	191	43	]	]	PUNCT
iajs-966	191	44	,	,	PUNCT
iajs-966	191	45	we	we	PRON
iajs-966	191	46	obtain	obtain	VERB
iajs-966	191	47	[	[	X
iajs-966	191	48	annrm	annrm	NOUN
iajs-966	191	49	:	:	PUNCT
iajs-966	191	50	annr(x)][(x	annr(x)][(x	NOUN
iajs-966	191	51	)	)	PUNCT
iajs-966	191	52	r	r	NOUN
iajs-966	191	53	:	:	PUNCT
iajs-966	191	54	m	m	X
iajs-966	191	55	]	]	X
iajs-966	191	56	for	for	ADP
iajs-966	191	57	each	each	DET
iajs-966	191	58	xm	xm	NOUN
iajs-966	191	59	.	.	PUNCT
iajs-966	192	1	now	now	ADV
iajs-966	192	2	,	,	PUNCT
iajs-966	192	3	we	we	PRON
iajs-966	192	4	have	have	AUX
iajs-966	192	5	m	m	PROPN
iajs-966	192	6	is	be	AUX
iajs-966	192	7	bounded	bound	VERB
iajs-966	192	8	.	.	PUNCT
iajs-966	193	1	then	then	ADV
iajs-966	193	2	there	there	PRON
iajs-966	193	3	exists	exist	VERB
iajs-966	193	4	xm	xm	PUNCT
iajs-966	193	5	such	such	ADJ
iajs-966	193	6	that	that	DET
iajs-966	193	7	annrm	annrm	NOUN
iajs-966	193	8	=	=	SYM
iajs-966	193	9	annr(x	annr(x	NOUN
iajs-966	193	10	)	)	PUNCT
iajs-966	193	11	.	.	PUNCT
iajs-966	194	1	therefore	therefore	ADV
iajs-966	194	2	[	[	X
iajs-966	194	3	annr(x	annr(x	NOUN
iajs-966	194	4	)	)	PUNCT
iajs-966	194	5	r	r	NOUN
iajs-966	194	6	:	:	PUNCT
iajs-966	194	7	annr(x)][(x	annr(x)][(x	NOUN
iajs-966	194	8	)	)	PUNCT
iajs-966	194	9	r	r	NOUN
iajs-966	194	10	:	:	PUNCT
iajs-966	194	11	m	m	VERB
iajs-966	194	12	]	]	X
iajs-966	194	13	,	,	PUNCT
iajs-966	194	14	implies	imply	VERB
iajs-966	194	15	r[(x	r[(x	NOUN
iajs-966	194	16	)	)	PUNCT
iajs-966	194	17	r	r	NOUN
iajs-966	194	18	:	:	PUNCT
iajs-966	194	19	m	m	VERB
iajs-966	194	20	]	]	X
iajs-966	194	21	.	.	PUNCT
iajs-966	195	1	thus	thus	ADV
iajs-966	195	2	rm=<x	rm=<x	X
iajs-966	195	3	>	>	X
iajs-966	195	4	is	be	AUX
iajs-966	195	5	cyclic	cyclic	ADJ
iajs-966	195	6	.	.	PUNCT
iajs-966	196	1	to	to	PART
iajs-966	196	2	prove	prove	VERB
iajs-966	196	3	n	n	PRON
iajs-966	196	4	is	be	AUX
iajs-966	196	5	an	an	DET
iajs-966	196	6	almost	almost	ADV
iajs-966	196	7	bounded	bound	VERB
iajs-966	196	8	submodule	submodule	NOUN
iajs-966	196	9	of	of	ADP
iajs-966	196	10	m	m	PROPN
iajs-966	196	11	,	,	PUNCT
iajs-966	196	12	we	we	PRON
iajs-966	196	13	must	must	AUX
iajs-966	196	14	show	show	VERB
iajs-966	196	15	that	that	SCONJ
iajs-966	196	16	annrn	annrn	NOUN
iajs-966	196	17	=	=	SYM
iajs-966	196	18	annr(x	annr(x	NOUN
iajs-966	196	19	)	)	PUNCT
iajs-966	196	20	.	.	PUNCT
iajs-966	197	1	in	in	ADP
iajs-966	197	2	the	the	DET
iajs-966	197	3	first	first	ADJ
iajs-966	197	4	,	,	PUNCT
iajs-966	197	5	we	we	PRON
iajs-966	197	6	claim	claim	VERB
iajs-966	197	7	that	that	SCONJ
iajs-966	197	8	xn	xn	PROPN
iajs-966	197	9	.	.	PUNCT
iajs-966	198	1	if	if	SCONJ
iajs-966	198	2	xn	xn	PROPN
iajs-966	198	3	,	,	PUNCT
iajs-966	198	4	then	then	ADV
iajs-966	198	5	[	[	X
iajs-966	198	6	(	(	PUNCT
iajs-966	198	7	x	x	X
iajs-966	198	8	)	)	PUNCT
iajs-966	198	9	r	r	NOUN
iajs-966	198	10	:	:	PUNCT
iajs-966	198	11	m][n	m][n	ADJ
iajs-966	198	12	r	r	NOUN
iajs-966	198	13	:	:	PUNCT
iajs-966	198	14	m	m	VERB
iajs-966	198	15	]	]	PUNCT
iajs-966	198	16	,	,	PUNCT
iajs-966	198	17	but	but	CCONJ
iajs-966	198	18	[	[	X
iajs-966	198	19	(	(	PUNCT
iajs-966	198	20	x	x	X
iajs-966	198	21	)	)	PUNCT
iajs-966	198	22	r	r	NOUN
iajs-966	198	23	:	:	PUNCT
iajs-966	198	24	m]=r	m]=r	X
iajs-966	198	25	.	.	PUNCT
iajs-966	199	1	therefore	therefore	ADV
iajs-966	199	2	[	[	X
iajs-966	199	3	n	n	X
iajs-966	199	4	r	r	NOUN
iajs-966	199	5	:	:	PUNCT
iajs-966	199	6	m]=r	m]=r	NOUN
iajs-966	199	7	,	,	PUNCT
iajs-966	199	8	implies	imply	VERB
iajs-966	199	9	that	that	SCONJ
iajs-966	199	10	rm=[n	rm=[n	PROPN
iajs-966	199	11	r	r	NOUN
iajs-966	199	12	:	:	PUNCT
iajs-966	199	13	m]m	m]m	X
iajs-966	199	14	=	=	NOUN
iajs-966	199	15	n.	n.	NOUN
iajs-966	199	16	thus	thus	ADV
iajs-966	199	17	n	n	CCONJ
iajs-966	199	18	=	=	NOUN
iajs-966	199	19	m	m	NOUN
iajs-966	199	20	which	which	PRON
iajs-966	199	21	is	be	AUX
iajs-966	199	22	a	a	DET
iajs-966	199	23	contradiction	contradiction	NOUN
iajs-966	199	24	.	.	PUNCT
iajs-966	200	1	hence	hence	ADV
iajs-966	200	2	xn	xn	PROPN
iajs-966	200	3	.	.	PUNCT
iajs-966	201	1	it	it	PRON
iajs-966	201	2	is	be	AUX
iajs-966	201	3	easily	easily	ADV
iajs-966	201	4	to	to	PART
iajs-966	201	5	show	show	VERB
iajs-966	201	6	that	that	DET
iajs-966	201	7	annr(x)annrn	annr(x)annrn	NOUN
iajs-966	201	8	.	.	PUNCT
iajs-966	202	1	on	on	ADP
iajs-966	202	2	the	the	DET
iajs-966	202	3	other	other	ADJ
iajs-966	202	4	hand	hand	NOUN
iajs-966	202	5	,	,	PUNCT
iajs-966	202	6	let	let	VERB
iajs-966	202	7	rannrn	rannrn	NOUN
iajs-966	202	8	.	.	PUNCT
iajs-966	203	1	then	then	ADV
iajs-966	203	2	rn=0	rn=0	PROPN
iajs-966	203	3	but	but	CCONJ
iajs-966	203	4	m	m	NOUN
iajs-966	203	5	is	be	AUX
iajs-966	203	6	multiplication	multiplication	NOUN
iajs-966	204	1	[	[	X
iajs-966	204	2	1,corollary	1,corollary	NUM
iajs-966	204	3	(	(	PUNCT
iajs-966	204	4	1.1.9	1.1.9	NUM
iajs-966	204	5	)	)	PUNCT
iajs-966	204	6	]	]	PUNCT
iajs-966	204	7	,	,	PUNCT
iajs-966	204	8	then	then	ADV
iajs-966	204	9	r[n	r[n	VERB
iajs-966	204	10	r	r	NOUN
iajs-966	204	11	:	:	PUNCT
iajs-966	204	12	m]m=0	m]m=0	NOUN
iajs-966	204	13	implies	imply	VERB
iajs-966	204	14	r[n	r[n	NOUN
iajs-966	204	15	:	:	PUNCT
iajs-966	204	16	m]annrm	m]annrm	ADJ
iajs-966	204	17	.	.	PUNCT
iajs-966	205	1	but	but	CCONJ
iajs-966	205	2	annrm	annrm	NOUN
iajs-966	205	3	is	be	AUX
iajs-966	205	4	prime	prime	ADJ
iajs-966	205	5	ideal	ideal	NOUN
iajs-966	205	6	and	and	CCONJ
iajs-966	205	7	[	[	X
iajs-966	205	8	n	n	X
iajs-966	205	9	r	r	NOUN
iajs-966	205	10	:	:	PUNCT
iajs-966	205	11	m	m	X
iajs-966	205	12	]	]	X
iajs-966	205	13			PROPN
iajs-966	205	14	annrm	annrm	PROPN
iajs-966	205	15	by	by	ADP
iajs-966	205	16	(	(	PUNCT
iajs-966	205	17	2	2	NUM
iajs-966	205	18	)	)	PUNCT
iajs-966	205	19	.	.	PUNCT
iajs-966	206	1	then	then	ADV
iajs-966	206	2	rannrm	rannrm	PROPN
iajs-966	206	3	=	=	SYM
iajs-966	206	4	annr(x	annr(x	NOUN
iajs-966	206	5	)	)	PUNCT
iajs-966	206	6	because	because	SCONJ
iajs-966	206	7	m	m	PROPN
iajs-966	206	8	is	be	AUX
iajs-966	206	9	bounded	bounded	ADJ
iajs-966	206	10	module	module	NOUN
iajs-966	206	11	.	.	PUNCT
iajs-966	207	1	thus	thus	ADV
iajs-966	207	2	annrn	annrn	NOUN
iajs-966	207	3	=	=	SYM
iajs-966	207	4	annr(x	annr(x	NOUN
iajs-966	207	5	)	)	PUNCT
iajs-966	207	6	and	and	CCONJ
iajs-966	207	7	hence	hence	ADV
iajs-966	207	8	n	n	PRON
iajs-966	207	9	is	be	AUX
iajs-966	207	10	an	an	DET
iajs-966	207	11	almost	almost	ADV
iajs-966	207	12	bounded	bound	VERB
iajs-966	207	13	submodule	submodule	NOUN
iajs-966	207	14	of	of	ADP
iajs-966	207	15	m.	m.	NOUN
iajs-966	207	16	the	the	DET
iajs-966	207	17	conditions	condition	NOUN
iajs-966	207	18	[	[	X
iajs-966	207	19	n	n	X
iajs-966	207	20	r	r	NOUN
iajs-966	207	21	:	:	PUNCT
iajs-966	207	22	m	m	X
iajs-966	207	23	]	]	X
iajs-966	207	24			PROPN
iajs-966	207	25	annrm	annrm	PROPN
iajs-966	207	26	and	and	CCONJ
iajs-966	207	27	annrm	annrm	NOUN
iajs-966	207	28	is	be	AUX
iajs-966	207	29	prime	prime	ADJ
iajs-966	207	30	ideal	ideal	NOUN
iajs-966	207	31	can	can	AUX
iajs-966	207	32	not	not	PART
iajs-966	207	33	be	be	AUX
iajs-966	207	34	dropped	drop	VERB
iajs-966	207	35	from	from	ADP
iajs-966	207	36	proposition	proposition	NOUN
iajs-966	207	37	(	(	PUNCT
iajs-966	207	38	2.4	2.4	NUM
iajs-966	207	39	)	)	PUNCT
iajs-966	207	40	as	as	ADP
iajs-966	207	41	in	in	ADP
iajs-966	207	42	the	the	DET
iajs-966	207	43	following	follow	VERB
iajs-966	207	44	example	example	NOUN
iajs-966	207	45	.	.	PUNCT
iajs-966	208	1	2.5	2.5	NUM
iajs-966	208	2	example	example	NOUN
iajs-966	208	3	:	:	PUNCT
iajs-966	208	4	let	let	VERB
iajs-966	208	5	m	m	PRON
iajs-966	208	6	=	=	NOUN
iajs-966	208	7	z6	z6	PROPN
iajs-966	208	8	as	as	ADP
iajs-966	208	9	a	a	DET
iajs-966	208	10	z	z	NOUN
iajs-966	208	11	-	-	PUNCT
iajs-966	208	12	module	module	NOUN
iajs-966	208	13	.	.	PUNCT
iajs-966	209	1	since	since	SCONJ
iajs-966	209	2	m	m	PROPN
iajs-966	209	3	is	be	AUX
iajs-966	209	4	bounded	bound	VERB
iajs-966	209	5	z	z	NOUN
iajs-966	209	6	-	-	PUNCT
iajs-966	209	7	module	module	NOUN
iajs-966	209	8	,	,	PUNCT
iajs-966	209	9	see	see	VERB
iajs-966	209	10	[	[	X
iajs-966	209	11	1	1	X
iajs-966	209	12	]	]	PUNCT
iajs-966	209	13	and	and	CCONJ
iajs-966	209	14	m	m	PROPN
iajs-966	209	15	is	be	AUX
iajs-966	209	16	fully	fully	ADV
iajs-966	209	17	stable	stable	ADJ
iajs-966	209	18	zmodule	zmodule	NOUN
iajs-966	209	19	,	,	PUNCT
iajs-966	209	20	see	see	VERB
iajs-966	210	1	[	[	X
iajs-966	210	2	4,example	4,example	PRON
iajs-966	210	3	and	and	CCONJ
iajs-966	210	4	remarks	remark	VERB
iajs-966	210	5	(	(	PUNCT
iajs-966	210	6	3.7),(c	3.7),(c	NUM
iajs-966	210	7	)	)	PUNCT
iajs-966	210	8	]	]	PUNCT
iajs-966	210	9	,	,	PUNCT
iajs-966	210	10	but	but	CCONJ
iajs-966	210	11	annzm=6z	annzm=6z	NOUN
iajs-966	210	12	is	be	AUX
iajs-966	210	13	not	not	PART
iajs-966	210	14	prime	prime	ADJ
iajs-966	210	15	ideal	ideal	NOUN
iajs-966	210	16	of	of	ADP
iajs-966	210	17	z.	z.	PROPN
iajs-966	210	18	let	let	VERB
iajs-966	211	1	n1=	n1=	PROPN
iajs-966	211	2	2	2	PRON
iajs-966	211	3			NOUN
iajs-966	211	4	and	and	CCONJ
iajs-966	211	5	n2=	n2=	PROPN
iajs-966	211	6	3	3	NUM
iajs-966	211	7			VERB
iajs-966	211	8	.	.	PUNCT
iajs-966	212	1	[	[	X
iajs-966	212	2	n1	n1	NOUN
iajs-966	212	3	z	z	NOUN
iajs-966	212	4	:	:	PUNCT
iajs-966	213	1	m]=	m]=	NOUN
iajs-966	213	2	[	[	PUNCT
iajs-966	213	3	2	2	NUM
iajs-966	213	4			NOUN
iajs-966	213	5	z	z	NOUN
iajs-966	213	6	:	:	PUNCT
iajs-966	213	7	z6]=	z6]=	X
iajs-966	213	8	2z	2z	NUM
iajs-966	213	9			PROPN
iajs-966	213	10	annzm=6z	annzm=6z	NOUN
iajs-966	213	11	and	and	CCONJ
iajs-966	213	12	[	[	X
iajs-966	213	13	n2	n2	ADJ
iajs-966	213	14	z	z	NOUN
iajs-966	213	15	:	:	PUNCT
iajs-966	214	1	m]=	m]=	NOUN
iajs-966	214	2	[	[	PUNCT
iajs-966	214	3	3	3	NUM
iajs-966	214	4			ADJ
iajs-966	214	5	z	z	NOUN
iajs-966	214	6	:	:	PUNCT
iajs-966	214	7	z6]=3z	z6]=3z	PROPN
iajs-966	214	8			PROPN
iajs-966	214	9	annzm	annzm	PROPN
iajs-966	214	10	.	.	PUNCT
iajs-966	215	1	therefore	therefore	ADV
iajs-966	215	2	n1	n1	PROPN
iajs-966	215	3	,	,	PUNCT
iajs-966	215	4	n2	n2	NOUN
iajs-966	215	5	are	be	AUX
iajs-966	215	6	not	not	PART
iajs-966	215	7	almost	almost	ADV
iajs-966	215	8	bounded	bounded	ADJ
iajs-966	215	9	submodules	submodule	NOUN
iajs-966	215	10	of	of	ADP
iajs-966	215	11	m.	m.	NOUN
iajs-966	215	12	an	an	DET
iajs-966	215	13	r	r	NOUN
iajs-966	215	14	-	-	PUNCT
iajs-966	215	15	module	module	NOUN
iajs-966	215	16	m	m	NOUN
iajs-966	215	17	is	be	AUX
iajs-966	215	18	said	say	VERB
iajs-966	215	19	to	to	PART
iajs-966	215	20	be	be	AUX
iajs-966	215	21	i	i	NOUN
iajs-966	215	22	-	-	PUNCT
iajs-966	215	23	multiplication	multiplication	NOUN
iajs-966	215	24	if	if	SCONJ
iajs-966	215	25	each	each	DET
iajs-966	215	26	submodule	submodule	NOUN
iajs-966	215	27	of	of	ADP
iajs-966	215	28	m	m	PROPN
iajs-966	215	29	is	be	AUX
iajs-966	215	30	of	of	ADP
iajs-966	215	31	the	the	DET
iajs-966	215	32	form	form	NOUN
iajs-966	215	33	am	be	AUX
iajs-966	215	34	for	for	ADP
iajs-966	215	35	some	some	DET
iajs-966	215	36	idempotent	idempotent	ADJ
iajs-966	215	37	ideal	ideal	NOUN
iajs-966	215	38	a	a	PRON
iajs-966	215	39	of	of	ADP
iajs-966	215	40	r	r	NOUN
iajs-966	215	41	,	,	PUNCT
iajs-966	215	42	[	[	X
iajs-966	215	43	4	4	NUM
iajs-966	215	44	]	]	PUNCT
iajs-966	215	45	.	.	PUNCT
iajs-966	216	1	as	as	ADP
iajs-966	216	2	an	an	DET
iajs-966	216	3	immediate	immediate	ADJ
iajs-966	216	4	consequence	consequence	NOUN
iajs-966	216	5	of	of	ADP
iajs-966	216	6	proposition	proposition	NOUN
iajs-966	216	7	(	(	PUNCT
iajs-966	216	8	2.4	2.4	NUM
iajs-966	216	9	)	)	PUNCT
iajs-966	216	10	.	.	PUNCT
iajs-966	217	1	2.6	2.6	NUM
iajs-966	217	2	corollary	corollary	NOUN
iajs-966	217	3	:	:	PUNCT
iajs-966	217	4	let	let	VERB
iajs-966	217	5	n	n	PRON
iajs-966	217	6	be	be	AUX
iajs-966	217	7	a	a	DET
iajs-966	217	8	proper	proper	ADJ
iajs-966	217	9	submodule	submodule	NOUN
iajs-966	217	10	n	n	PROPN
iajs-966	217	11	of	of	ADP
iajs-966	217	12	an	an	DET
iajs-966	217	13	r	r	NOUN
iajs-966	217	14	-	-	PUNCT
iajs-966	217	15	module	module	NOUN
iajs-966	217	16	m	m	NOUN
iajs-966	217	17	such	such	ADJ
iajs-966	217	18	that	that	SCONJ
iajs-966	217	19	:	:	PUNCT
iajs-966	217	20	1	1	X
iajs-966	217	21	.	.	X
iajs-966	217	22	m	m	PROPN
iajs-966	217	23	is	be	AUX
iajs-966	217	24	i	i	NOUN
iajs-966	217	25	-	-	PUNCT
iajs-966	217	26	multiplication	multiplication	NOUN
iajs-966	217	27	bounded	bounded	ADJ
iajs-966	217	28	module	module	NOUN
iajs-966	217	29	ihjpas	ihjpa	VERB
iajs-966	217	30	ibn	ibn	PROPN
iajs-966	217	31	alhaitham	alhaitham	NOUN
iajs-966	218	1	j.	j.	PROPN
iajs-966	219	1	fo	fo	ADP
iajs-966	219	2	r	r	NOUN
iajs-966	219	3	pure	pure	ADJ
iajs-966	219	4	&	&	CCONJ
iajs-966	219	5	appl	appl	PROPN
iajs-966	219	6	.	.	PUNCT
iajs-966	220	1	sc	sc	PROPN
iajs-966	220	2	i.	i.	PROPN
iajs-966	220	3	vo	vo	PROPN
iajs-966	220	4	l.23	l.23	PROPN
iajs-966	220	5	(	(	PUNCT
iajs-966	220	6	2	2	NUM
iajs-966	220	7	)	)	PUNCT
iajs-966	220	8	2010	2010	NUM
iajs-966	220	9	2	2	NUM
iajs-966	220	10	.	.	X
iajs-966	220	11	annrm	annrm	NOUN
iajs-966	220	12	is	be	AUX
iajs-966	220	13	prime	prime	ADJ
iajs-966	220	14	ideal	ideal	NOUN
iajs-966	220	15	of	of	ADP
iajs-966	220	16	r.	r.	PROPN
iajs-966	220	17	3	3	NUM
iajs-966	220	18	.	.	PUNCT
iajs-966	221	1	[	[	X
iajs-966	221	2	n	n	X
iajs-966	221	3	r	r	NOUN
iajs-966	221	4	:	:	PUNCT
iajs-966	221	5	m	m	ADJ
iajs-966	221	6	]	]	X
iajs-966	221	7			PROPN
iajs-966	221	8	annrm	annrm	PROPN
iajs-966	221	9	.	.	PUNCT
iajs-966	222	1	then	then	ADV
iajs-966	222	2	n	n	PRON
iajs-966	222	3	is	be	AUX
iajs-966	222	4	an	an	DET
iajs-966	222	5	almost	almost	ADV
iajs-966	222	6	bounded	bound	VERB
iajs-966	222	7	submodule	submodule	NOUN
iajs-966	222	8	of	of	ADP
iajs-966	222	9	m.	m.	NOUN
iajs-966	222	10	proof	proof	NOUN
iajs-966	222	11	:	:	PUNCT
iajs-966	222	12	the	the	DET
iajs-966	222	13	result	result	NOUN
iajs-966	222	14	follows	follow	VERB
iajs-966	222	15	according	accord	VERB
iajs-966	222	16	to	to	ADP
iajs-966	222	17	[	[	X
iajs-966	222	18	4,theorem	4,theorem	NUM
iajs-966	222	19	(	(	PUNCT
iajs-966	222	20	2.9	2.9	NUM
iajs-966	222	21	)	)	PUNCT
iajs-966	222	22	]	]	PUNCT
iajs-966	222	23	and	and	CCONJ
iajs-966	222	24	proposition	proposition	NOUN
iajs-966	222	25	(	(	PUNCT
iajs-966	222	26	2.4	2.4	NUM
iajs-966	222	27	)	)	PUNCT
iajs-966	222	28	.	.	PUNCT
iajs-966	223	1	recall	recall	VERB
iajs-966	223	2	that	that	SCONJ
iajs-966	223	3	an	an	DET
iajs-966	223	4	r	r	NOUN
iajs-966	223	5	-	-	PUNCT
iajs-966	223	6	module	module	NOUN
iajs-966	223	7	m	m	NOUN
iajs-966	223	8	is	be	AUX
iajs-966	223	9	called	call	VERB
iajs-966	223	10	a	a	DET
iajs-966	223	11	prime	prime	ADJ
iajs-966	223	12	module	module	NOUN
iajs-966	223	13	if	if	SCONJ
iajs-966	223	14	annrm=	annrm=	PROPN
iajs-966	223	15	annrn	annrn	NOUN
iajs-966	223	16	for	for	ADP
iajs-966	223	17	every	every	DET
iajs-966	223	18	nonzero	nonzero	PROPN
iajs-966	223	19	submodule	submodule	PROPN
iajs-966	223	20	n	n	PROPN
iajs-966	223	21	of	of	ADP
iajs-966	223	22	m	m	PRON
iajs-966	223	23	,	,	PUNCT
iajs-966	223	24	[	[	X
iajs-966	223	25	5	5	NUM
iajs-966	223	26	]	]	PUNCT
iajs-966	223	27	,	,	PUNCT
iajs-966	223	28	[	[	X
iajs-966	223	29	6	6	NUM
iajs-966	223	30	]	]	PUNCT
iajs-966	223	31	.	.	PUNCT
iajs-966	224	1	2.7	2.7	NUM
iajs-966	224	2	proposition	proposition	NOUN
iajs-966	224	3	:	:	PUNCT
iajs-966	224	4	let	let	VERB
iajs-966	224	5	m	m	PRON
iajs-966	224	6	be	be	AUX
iajs-966	224	7	a	a	DET
iajs-966	224	8	prime	prime	ADJ
iajs-966	224	9	r	r	NOUN
iajs-966	224	10	-	-	PUNCT
iajs-966	224	11	module	module	NOUN
iajs-966	224	12	and	and	CCONJ
iajs-966	224	13	n	n	CCONJ
iajs-966	224	14	,	,	PUNCT
iajs-966	224	15	k	k	X
iajs-966	224	16	be	be	VERB
iajs-966	224	17	two	two	NUM
iajs-966	224	18	submodules	submodule	NOUN
iajs-966	224	19	of	of	ADP
iajs-966	224	20	m	m	NOUN
iajs-966	224	21	such	such	ADJ
iajs-966	224	22	that	that	SCONJ
iajs-966	224	23	nkm	nkm	NOUN
iajs-966	224	24	,	,	PUNCT
iajs-966	224	25	k	k	PROPN
iajs-966	224	26	is	be	AUX
iajs-966	224	27	an	an	DET
iajs-966	224	28	almost	almost	ADV
iajs-966	224	29	bounded	bound	VERB
iajs-966	224	30	submodule	submodule	NOUN
iajs-966	224	31	of	of	ADP
iajs-966	224	32	m.	m.	NOUN
iajs-966	224	33	then	then	ADV
iajs-966	224	34	n	n	PRON
iajs-966	224	35	is	be	AUX
iajs-966	224	36	an	an	DET
iajs-966	224	37	almost	almost	ADV
iajs-966	224	38	bounded	bound	VERB
iajs-966	224	39	submodule	submodule	NOUN
iajs-966	224	40	of	of	ADP
iajs-966	224	41	m.	m.	NOUN
iajs-966	224	42	proof	proof	NOUN
iajs-966	224	43	:	:	PUNCT
iajs-966	224	44	assume	assume	VERB
iajs-966	224	45	that	that	SCONJ
iajs-966	224	46	k	k	PROPN
iajs-966	224	47	is	be	AUX
iajs-966	224	48	almost	almost	ADV
iajs-966	224	49	bounded	bound	VERB
iajs-966	224	50	submodule	submodule	NOUN
iajs-966	224	51	of	of	ADP
iajs-966	224	52	m	m	PROPN
iajs-966	224	53	,	,	PUNCT
iajs-966	224	54	that	that	PRON
iajs-966	224	55	is	be	AUX
iajs-966	224	56	there	there	PRON
iajs-966	224	57	exists	exist	VERB
iajs-966	224	58	xm	xm	NOUN
iajs-966	224	59	,	,	PUNCT
iajs-966	224	60	xk	xk	ADP
iajs-966	224	61	such	such	ADJ
iajs-966	224	62	that	that	DET
iajs-966	224	63	annrk	annrk	NOUN
iajs-966	224	64	=	=	SYM
iajs-966	224	65	annr(x	annr(x	NOUN
iajs-966	224	66	)	)	PUNCT
iajs-966	224	67	.	.	PUNCT
iajs-966	225	1	since	since	SCONJ
iajs-966	225	2	,	,	PUNCT
iajs-966	225	3	xk	xk	ADV
iajs-966	225	4	,	,	PUNCT
iajs-966	225	5	nk	nk	PROPN
iajs-966	225	6	.	.	PUNCT
iajs-966	226	1	then	then	ADV
iajs-966	226	2	we	we	PRON
iajs-966	226	3	obtain	obtain	VERB
iajs-966	226	4	xn	xn	PROPN
iajs-966	226	5	.	.	PUNCT
iajs-966	227	1	to	to	PART
iajs-966	227	2	prove	prove	VERB
iajs-966	227	3	annrn=	annrn=	NUM
iajs-966	227	4	annr(x	annr(x	NOUN
iajs-966	227	5	)	)	PUNCT
iajs-966	227	6	.	.	PUNCT
iajs-966	228	1	annrkannrn	annrkannrn	VERB
iajs-966	228	2	(	(	PUNCT
iajs-966	228	3	since	since	SCONJ
iajs-966	228	4	nkm	nkm	PROPN
iajs-966	228	5	)	)	PUNCT
iajs-966	228	6	,	,	PUNCT
iajs-966	228	7	implies	imply	VERB
iajs-966	228	8	annr(x)	annr(x)	NOUN
iajs-966	228	9	annrn	annrn	NOUN
iajs-966	228	10	.	.	PUNCT
iajs-966	229	1	hence	hence	ADV
iajs-966	229	2	annr(x)annrn	annr(x)annrn	PROPN
iajs-966	229	3	.	.	PUNCT
iajs-966	230	1	now	now	ADV
iajs-966	230	2	,	,	PUNCT
iajs-966	230	3	let	let	VERB
iajs-966	230	4	rr	rr	NUM
iajs-966	230	5	,	,	PUNCT
iajs-966	230	6	rannrn=	rannrn=	NOUN
iajs-966	230	7	annrm	annrm	NOUN
iajs-966	230	8	for	for	ADP
iajs-966	230	9	each	each	DET
iajs-966	230	10	submodule	submodule	NOUN
iajs-966	230	11	n	n	PROPN
iajs-966	230	12	of	of	ADP
iajs-966	230	13	m	m	PROPN
iajs-966	230	14	(	(	PUNCT
iajs-966	230	15	since	since	SCONJ
iajs-966	230	16	m	m	PROPN
iajs-966	230	17	is	be	AUX
iajs-966	230	18	prime	prime	ADJ
iajs-966	230	19	module	module	NOUN
iajs-966	230	20	)	)	PUNCT
iajs-966	230	21	,	,	PUNCT
iajs-966	230	22	but	but	CCONJ
iajs-966	230	23	annrm	annrm	PRON
iajs-966	230	24	annrk=	annrk=	NUM
iajs-966	230	25	annr(x	annr(x	PROPN
iajs-966	230	26	)	)	PUNCT
iajs-966	230	27	.	.	PUNCT
iajs-966	231	1	therefore	therefore	ADV
iajs-966	231	2	r	r	VERB
iajs-966	231	3	annr(x	annr(x	NOUN
iajs-966	231	4	)	)	PUNCT
iajs-966	231	5	.	.	PUNCT
iajs-966	232	1	thus	thus	ADV
iajs-966	232	2	annrn	annrn	ADP
iajs-966	232	3	annr(x	annr(x	NOUN
iajs-966	232	4	)	)	PUNCT
iajs-966	232	5	.	.	PUNCT
iajs-966	233	1	hence	hence	ADV
iajs-966	233	2	n	n	PRON
iajs-966	233	3	is	be	AUX
iajs-966	233	4	an	an	DET
iajs-966	233	5	almost	almost	ADV
iajs-966	233	6	bounded	bound	VERB
iajs-966	233	7	submodule	submodule	NOUN
iajs-966	233	8	of	of	ADP
iajs-966	233	9	m.	m.	NOUN
iajs-966	233	10	so	so	ADV
iajs-966	233	11	,	,	PUNCT
iajs-966	233	12	we	we	PRON
iajs-966	233	13	have	have	VERB
iajs-966	233	14	the	the	DET
iajs-966	233	15	following	follow	VERB
iajs-966	233	16	application	application	NOUN
iajs-966	233	17	of	of	ADP
iajs-966	233	18	(	(	PUNCT
iajs-966	233	19	2.7	2.7	NUM
iajs-966	233	20	)	)	PUNCT
iajs-966	233	21	.	.	PUNCT
iajs-966	234	1	2.8	2.8	NUM
iajs-966	234	2	corollary	corollary	NOUN
iajs-966	234	3	:	:	PUNCT
iajs-966	234	4	let	let	VERB
iajs-966	234	5	m	m	PRON
iajs-966	234	6	be	be	AUX
iajs-966	234	7	a	a	DET
iajs-966	234	8	prime	prime	ADJ
iajs-966	234	9	r	r	NOUN
iajs-966	234	10	-	-	PUNCT
iajs-966	234	11	module	module	NOUN
iajs-966	234	12	and	and	CCONJ
iajs-966	234	13	n	n	CCONJ
iajs-966	234	14	,	,	PUNCT
iajs-966	234	15	k	k	X
iajs-966	234	16	be	be	VERB
iajs-966	234	17	two	two	NUM
iajs-966	234	18	submodules	submodule	NOUN
iajs-966	234	19	of	of	ADP
iajs-966	234	20	m	m	NOUN
iajs-966	234	21	such	such	ADJ
iajs-966	234	22	that	that	SCONJ
iajs-966	234	23	n	n	PRON
iajs-966	234	24	is	be	AUX
iajs-966	234	25	an	an	DET
iajs-966	234	26	almost	almost	ADV
iajs-966	234	27	bounded	bound	VERB
iajs-966	234	28	submodule	submodule	NOUN
iajs-966	234	29	of	of	ADP
iajs-966	234	30	m.	m.	NOUN
iajs-966	234	31	then	then	ADV
iajs-966	234	32	nk	nk	NOUN
iajs-966	234	33	is	be	AUX
iajs-966	234	34	also	also	ADV
iajs-966	234	35	almost	almost	ADV
iajs-966	234	36	bounded	bound	VERB
iajs-966	234	37	submodule	submodule	NOUN
iajs-966	234	38	of	of	ADP
iajs-966	234	39	m.	m.	NOUN
iajs-966	234	40	proof	proof	NOUN
iajs-966	234	41	:	:	PUNCT
iajs-966	234	42	it	it	PRON
iajs-966	234	43	is	be	AUX
iajs-966	234	44	know	know	VERB
iajs-966	234	45	that	that	SCONJ
iajs-966	234	46	nkn	nkn	PROPN
iajs-966	234	47	.	.	PUNCT
iajs-966	235	1	so	so	ADV
iajs-966	235	2	according	accord	VERB
iajs-966	235	3	to	to	ADP
iajs-966	235	4	proposition	proposition	NOUN
iajs-966	235	5	(	(	PUNCT
iajs-966	235	6	2.7	2.7	NUM
iajs-966	235	7	)	)	PUNCT
iajs-966	235	8	,	,	PUNCT
iajs-966	235	9	nk	nk	NOUN
iajs-966	235	10	is	be	AUX
iajs-966	235	11	an	an	DET
iajs-966	235	12	almost	almost	ADV
iajs-966	235	13	bounded	bound	VERB
iajs-966	235	14	submodule	submodule	NOUN
iajs-966	235	15	of	of	ADP
iajs-966	235	16	m.	m.	NOUN
iajs-966	235	17	as	as	ADP
iajs-966	235	18	a	a	DET
iajs-966	235	19	generalization	generalization	NOUN
iajs-966	235	20	of	of	ADP
iajs-966	235	21	corollary	corollary	ADJ
iajs-966	235	22	(	(	PUNCT
iajs-966	235	23	2.8	2.8	NUM
iajs-966	235	24	)	)	PUNCT
iajs-966	235	25	,	,	PUNCT
iajs-966	235	26	we	we	PRON
iajs-966	235	27	give	give	VERB
iajs-966	235	28	the	the	DET
iajs-966	235	29	following	follow	VERB
iajs-966	235	30	corollary	corollary	NOUN
iajs-966	235	31	.	.	PUNCT
iajs-966	236	1	2.9	2.9	NUM
iajs-966	236	2	corollary	corollary	NOUN
iajs-966	236	3	:	:	PUNCT
iajs-966	236	4	let	let	VERB
iajs-966	236	5	m	m	PRON
iajs-966	236	6	be	be	AUX
iajs-966	236	7	a	a	DET
iajs-966	236	8	prime	prime	ADJ
iajs-966	236	9	r	r	NOUN
iajs-966	236	10	-	-	PUNCT
iajs-966	236	11	module	module	NOUN
iajs-966	236	12	and	and	CCONJ
iajs-966	236	13	n	n	NOUN
iajs-966	236	14	i	i	PRON
iajs-966	236	15	i	i	PRON
iajs-966	236	16	1{n	1{n	NUM
iajs-966	236	17	}	}	PUNCT
iajs-966	236	18			PRON
iajs-966	236	19	be	be	AUX
iajs-966	236	20	a	a	DET
iajs-966	236	21	finite	finite	ADJ
iajs-966	236	22	collection	collection	NOUN
iajs-966	236	23	of	of	ADP
iajs-966	236	24	submodules	submodule	NOUN
iajs-966	236	25	of	of	ADP
iajs-966	236	26	m	m	NOUN
iajs-966	236	27	such	such	ADJ
iajs-966	236	28	that	that	SCONJ
iajs-966	236	29	ni	ni	PROPN
iajs-966	236	30	is	be	AUX
iajs-966	236	31	an	an	DET
iajs-966	236	32	almost	almost	ADV
iajs-966	236	33	bounded	bound	VERB
iajs-966	236	34	submodule	submodule	NOUN
iajs-966	236	35	of	of	ADP
iajs-966	236	36	m	m	PROPN
iajs-966	236	37	for	for	ADP
iajs-966	236	38	some	some	DET
iajs-966	236	39	i	i	PROPN
iajs-966	236	40	,	,	PUNCT
iajs-966	236	41	i=1,2,	i=1,2,	NOUN
iajs-966	236	42	…	…	X
iajs-966	236	43	,n	,n	NOUN
iajs-966	236	44	.	.	PUNCT
iajs-966	237	1	then	then	ADV
iajs-966	237	2	n	n	INTJ
iajs-966	237	3	i	i	PRON
iajs-966	237	4	i	i	VERB
iajs-966	238	1	1	1	NUM
iajs-966	238	2	n	n	NOUN
iajs-966	238	3			NOUN
iajs-966	238	4			NOUN
iajs-966	238	5	is	be	AUX
iajs-966	238	6	also	also	ADV
iajs-966	238	7	almost	almost	ADV
iajs-966	238	8	bounded	bound	VERB
iajs-966	238	9	submodule	submodule	PROPN
iajs-966	238	10	f	f	PROPN
iajs-966	238	11	m.	m.	NOUN
iajs-966	238	12	proof	proof	NOUN
iajs-966	238	13	:	:	PUNCT
iajs-966	238	14	the	the	DET
iajs-966	238	15	proof	proof	NOUN
iajs-966	238	16	is	be	AUX
iajs-966	238	17	by	by	ADP
iajs-966	238	18	induction	induction	NOUN
iajs-966	238	19	on	on	ADP
iajs-966	238	20	n	n	PRON
iajs-966	238	21	and	and	CCONJ
iajs-966	238	22	corollary	corollary	ADJ
iajs-966	238	23	(	(	PUNCT
iajs-966	238	24	2.8	2.8	NUM
iajs-966	238	25	)	)	PUNCT
iajs-966	238	26	.	.	PUNCT
iajs-966	239	1	the	the	DET
iajs-966	239	2	following	follow	VERB
iajs-966	239	3	example	example	NOUN
iajs-966	239	4	shows	show	VERB
iajs-966	239	5	that	that	SCONJ
iajs-966	239	6	the	the	DET
iajs-966	239	7	intersection	intersection	NOUN
iajs-966	239	8	of	of	ADP
iajs-966	239	9	an	an	DET
iajs-966	239	10	infinite	infinite	ADJ
iajs-966	239	11	collection	collection	NOUN
iajs-966	239	12	of	of	ADP
iajs-966	239	13	almost	almost	ADV
iajs-966	239	14	bounded	bound	VERB
iajs-966	239	15	submodules	submodule	NOUN
iajs-966	239	16	of	of	ADP
iajs-966	239	17	m	m	PRON
iajs-966	239	18	need	need	AUX
iajs-966	239	19	not	not	PART
iajs-966	239	20	be	be	AUX
iajs-966	239	21	almost	almost	ADV
iajs-966	239	22	bounded	bound	VERB
iajs-966	239	23	submodule	submodule	NOUN
iajs-966	239	24	of	of	ADP
iajs-966	239	25	m.	m.	NOUN
iajs-966	239	26	2.10	2.10	NUM
iajs-966	239	27	example	example	NOUN
iajs-966	239	28	:	:	PUNCT
iajs-966	239	29	consider	consider	VERB
iajs-966	239	30	z	z	NOUN
iajs-966	239	31	as	as	ADP
iajs-966	239	32	a	a	DET
iajs-966	239	33	z	z	NOUN
iajs-966	239	34	-	-	PUNCT
iajs-966	239	35	module	module	NOUN
iajs-966	239	36	,	,	PUNCT
iajs-966	239	37	z	z	PROPN
iajs-966	239	38	is	be	AUX
iajs-966	239	39	prime	prime	ADJ
iajs-966	239	40	z	z	NOUN
iajs-966	239	41	-	-	PUNCT
iajs-966	239	42	module	module	NOUN
iajs-966	239	43	.	.	PUNCT
iajs-966	240	1	since	since	SCONJ
iajs-966	240	2	pz	pz	PROPN
iajs-966	240	3	is	be	AUX
iajs-966	240	4	an	an	DET
iajs-966	240	5	almost	almost	ADV
iajs-966	240	6	bounded	bound	VERB
iajs-966	240	7	of	of	ADP
iajs-966	240	8	z	z	PROPN
iajs-966	240	9	,	,	PUNCT
iajs-966	240	10	for	for	ADP
iajs-966	240	11	each	each	DET
iajs-966	240	12	p	p	NOUN
iajs-966	240	13	where	where	SCONJ
iajs-966	240	14	p	p	NOUN
iajs-966	240	15	is	be	AUX
iajs-966	240	16	a	a	DET
iajs-966	240	17	prime	prime	ADJ
iajs-966	240	18	number	number	NOUN
iajs-966	240	19	.	.	PUNCT
iajs-966	241	1	however	however	ADV
iajs-966	241	2	pisp	pisp	PROPN
iajs-966	241	3	rime	rime	NOUN
iajs-966	241	4	pz	pz	PROPN
iajs-966	241	5	=	=	NOUN
iajs-966	241	6	0	0	NUM
iajs-966	241	7	is	be	AUX
iajs-966	241	8	not	not	PART
iajs-966	241	9	almost	almost	ADV
iajs-966	241	10	bounded	bound	VERB
iajs-966	241	11	submodule	submodule	NOUN
iajs-966	241	12	of	of	ADP
iajs-966	241	13	z.	z.	PROPN
iajs-966	241	14	references	reference	NOUN
iajs-966	241	15	:	:	PUNCT
iajs-966	241	16	1	1	X
iajs-966	241	17	.	.	NUM
iajs-966	241	18	ammen	amman	NOUN
iajs-966	241	19	,	,	PUNCT
iajs-966	241	20	sh.a	sh.a	VERB
iajs-966	241	21	.	.	PROPN
iajs-966	241	22	,	,	PUNCT
iajs-966	241	23	(	(	PUNCT
iajs-966	241	24	2002	2002	NUM
iajs-966	241	25	)	)	PUNCT
iajs-966	241	26	,	,	PUNCT
iajs-966	241	27	bounded	bound	VERB
iajs-966	241	28	modules	module	NOUN
iajs-966	241	29	,	,	PUNCT
iajs-966	241	30	m	m	VERB
iajs-966	241	31	.d.thesis	.d.thesis	ADJ
iajs-966	241	32	,	,	PUNCT
iajs-966	241	33	university	university	NOUN
iajs-966	241	34	of	of	ADP
iajs-966	241	35	baghdad	baghdad	PROPN
iajs-966	241	36	.	.	PUNCT
iajs-966	242	1	2	2	X
iajs-966	242	2	.	.	X
iajs-966	242	3	abdul	abdul	PROPN
iajs-966	242	4	-	-	PUNCT
iajs-966	242	5	razak	razak	PROPN
iajs-966	242	6	,	,	PUNCT
iajs-966	242	7	h.m	h.m	PROPN
iajs-966	242	8	.	.	PROPN
iajs-966	242	9	,	,	PUNCT
iajs-966	242	10	(	(	PUNCT
iajs-966	242	11	1999	1999	NUM
iajs-966	242	12	)	)	PUNCT
iajs-966	242	13	,	,	PUNCT
iajs-966	242	14	quasi	quasi	ADJ
iajs-966	242	15	-	-	ADJ
iajs-966	242	16	prime	prime	ADJ
iajs-966	242	17	modules	module	NOUN
iajs-966	242	18	and	and	CCONJ
iajs-966	242	19	quasi	quasi	ADJ
iajs-966	242	20	-	-	ADJ
iajs-966	242	21	prime	prime	ADJ
iajs-966	242	22	submodules	submodule	NOUN
iajs-966	242	23	,	,	PUNCT
iajs-966	242	24	m.d	m.d	PROPN
iajs-966	242	25	.	.	PROPN
iajs-966	242	26	thesis	thesis	PROPN
iajs-966	242	27	,	,	PUNCT
iajs-966	242	28	university	university	NOUN
iajs-966	242	29	of	of	ADP
iajs-966	242	30	baghdad	baghdad	PROPN
iajs-966	242	31	.	.	PUNCT
iajs-966	243	1	3	3	X
iajs-966	243	2	.	.	X
iajs-966	243	3	ansari	ansari	ADJ
iajs-966	243	4	-	-	PUNCT
iajs-966	243	5	toroghy	toroghy	ADJ
iajs-966	243	6	,	,	PUNCT
iajs-966	243	7	h.	h.	PROPN
iajs-966	243	8	and	and	CCONJ
iajs-966	243	9	farshadifar	farshadifar	PROPN
iajs-966	243	10	,	,	PUNCT
iajs-966	243	11	f.	f.	PROPN
iajs-966	243	12	,	,	PUNCT
iajs-966	243	13	(	(	PUNCT
iajs-966	243	14	2008	2008	NUM
iajs-966	243	15	)	)	PUNCT
iajs-966	243	16	,	,	PUNCT
iajs-966	243	17	on	on	ADP
iajs-966	243	18	endomorphisims	endomorphisim	NOUN
iajs-966	243	19	of	of	ADP
iajs-966	243	20	multiplication	multiplication	NOUN
iajs-966	243	21	and	and	CCONJ
iajs-966	243	22	comultiplication	comultiplication	NOUN
iajs-966	243	23	modules	module	NOUN
iajs-966	243	24	,	,	PUNCT
iajs-966	243	25	archivum	archivum	NOUN
iajs-966	243	26	mathematicum	mathematicum	NOUN
iajs-966	243	27	(	(	PUNCT
iajs-966	243	28	brno	brno	NOUN
iajs-966	243	29	)	)	PUNCT
iajs-966	243	30	,	,	PUNCT
iajs-966	243	31	tomus	tomus	PROPN
iajs-966	243	32	44	44	NUM
iajs-966	243	33	,	,	PUNCT
iajs-966	243	34	9	9	NUM
iajs-966	243	35	-	-	SYM
iajs-966	243	36	15	15	NUM
iajs-966	243	37	.	.	NOUN
iajs-966	244	1	4	4	NUM
iajs-966	244	2	.	.	X
iajs-966	244	3	abass	abass	PROPN
iajs-966	244	4	,	,	PUNCT
iajs-966	244	5	m.s	m.s	PROPN
iajs-966	244	6	.	.	PROPN
iajs-966	244	7	,	,	PUNCT
iajs-966	244	8	(	(	PUNCT
iajs-966	244	9	1990	1990	NUM
iajs-966	244	10	)	)	PUNCT
iajs-966	244	11	,	,	PUNCT
iajs-966	244	12	on	on	ADP
iajs-966	244	13	fully	fully	ADV
iajs-966	244	14	stable	stable	ADJ
iajs-966	244	15	modules	module	NOUN
iajs-966	244	16	,	,	PUNCT
iajs-966	244	17	ph.d	ph.d	PROPN
iajs-966	244	18	.	.	PUNCT
iajs-966	245	1	thesis	thesis	PROPN
iajs-966	245	2	university	university	PROPN
iajs-966	245	3	of	of	ADP
iajs-966	245	4	baghdad	baghdad	PROPN
iajs-966	245	5	.	.	PUNCT
iajs-966	246	1	5	5	X
iajs-966	246	2	.	.	X
iajs-966	246	3	desale	desale	NOUN
iajs-966	246	4	,	,	PUNCT
iajs-966	246	5	g.	g.	PROPN
iajs-966	246	6	,and	,and	PUNCT
iajs-966	246	7	nicholson	nicholson	PROPN
iajs-966	246	8	,	,	PUNCT
iajs-966	246	9	k.w	k.w	PROPN
iajs-966	246	10	.	.	PROPN
iajs-966	246	11	,	,	PUNCT
iajs-966	246	12	(	(	PUNCT
iajs-966	246	13	1981	1981	NUM
iajs-966	246	14	)	)	PUNCT
iajs-966	246	15	,	,	PUNCT
iajs-966	246	16	endomorphisim	endomorphisim	PROPN
iajs-966	246	17	rings	ring	NOUN
iajs-966	246	18	,	,	PUNCT
iajs-966	246	19	j.	j.	PROPN
iajs-966	246	20	algebra,.70	algebra,.70	PROPN
iajs-966	246	21	:	:	PUNCT
iajs-966	246	22	548	548	NUM
iajs-966	246	23	-	-	SYM
iajs-966	246	24	560	560	NUM
iajs-966	246	25	.	.	NOUN
iajs-966	246	26	6	6	NUM
iajs-966	246	27	.	.	X
iajs-966	246	28	ebrahimi	ebrahimi	PROPN
iajs-966	246	29	,	,	PUNCT
iajs-966	246	30	atani	atani	PROPN
iajs-966	246	31	,	,	PUNCT
iajs-966	246	32	s.	s.	PROPN
iajs-966	246	33	,	,	PUNCT
iajs-966	246	34	(	(	PUNCT
iajs-966	246	35	2008	2008	NUM
iajs-966	246	36	)	)	PUNCT
iajs-966	246	37	,	,	PUNCT
iajs-966	246	38	on	on	ADP
iajs-966	246	39	generalized	generalized	ADJ
iajs-966	246	40	distinguished	distinguished	ADJ
iajs-966	246	41	prime	prime	ADJ
iajs-966	246	42	submodules	submodule	NOUN
iajs-966	246	43	,	,	PUNCT
iajs-966	246	44	thai	thai	PROPN
iajs-966	246	45	journal	journal	NOUN
iajs-966	246	46	of	of	ADP
iajs-966	246	47	mathematics,.6(2	mathematics,.6(2	PROPN
iajs-966	246	48	):	):	PUNCT
iajs-966	246	49	369	369	NUM
iajs-966	246	50	-	-	SYM
iajs-966	246	51	376	376	NUM
iajs-966	246	52	.	.	PUNCT
iajs-966	247	1	ihjpas	ihjpas	PROPN
iajs-966	247	2	2010	2010	NUM
iajs-966	247	3	)	)	PUNCT
iajs-966	247	4	2	2	NUM
iajs-966	247	5	(	(	PUNCT
iajs-966	247	6	23مجلة	23مجلة	NUM
iajs-966	247	7	ابن	ابن	PROPN
iajs-966	247	8	الھیثم	الھیثم	PROPN
iajs-966	247	9	للعلوم	للعلوم	PROPN
iajs-966	247	10	الصرفة	الصرفة	PROPN
iajs-966	247	11	والتطبیقیة	والتطبیقیة	PROPN
iajs-966	247	12	المجلد	المجلد	PROPN
iajs-966	247	13	حول	حول	PROPN
iajs-966	247	14	المقاسات	المقاسات	PROPN
iajs-966	247	15	الجزئیة	الجزئیة	PROPN
iajs-966	247	16	المقیدة	المقیدة	PROPN
iajs-966	247	17	تقریبا	تقریبا	VERB
iajs-966	247	18	ً	ً	PROPN
iajs-966	247	19	بثینة	بثینة	PROPN
iajs-966	247	20	نجاد	نجاد	PROPN
iajs-966	247	21	شھاب	شھاب	VERB
iajs-966	247	22	جامعة	جامعة	PROPN
iajs-966	247	23	بغداد	بغداد	PROPN
iajs-966	247	24	،	،	PROPN
iajs-966	247	25	ابن	ابن	AUX
iajs-966	247	26	الھیثم	الھیثم	PROPN
iajs-966	247	27	كلیة	كلیة	PRON
iajs-966	247	28	التربیة،قسم	التربیة،قسم	PROPN
iajs-966	247	29	الریاضیات	الریاضیات	ADJ
iajs-966	247	30	الخالصة	الخالصة	NOUN
iajs-966	248	1	ـاً	ـاً	PUNCT
iajs-966	248	2	أیسـراً	أیسـراً	PROPN
iajs-966	248	3	علــى	علــى	PROPN
iajs-966	248	4	الحلقـة	الحلقـة	PROPN
iajs-966	248	5	mعنصـر	mعنصـر	PROPN
iajs-966	248	6	محایــد	محایــد	PROPN
iajs-966	248	7	،	،	PROPN
iajs-966	248	8	ولـیكن	ولـیكن	PROPN
iajs-966	248	9	يحلقـة	يحلقـة	PROPN
iajs-966	248	10	ابدالیــة	ابدالیــة	ADJ
iajs-966	249	1	ذ	ذ	DET
iajs-966	249	2	rلـتكن	rلـتكن	NOUN
iajs-966	249	3	فـي	فـي	INTJ
iajs-966	249	4	هــذا	هــذا	NOUN
iajs-966	249	5	البحـث	البحـث	NOUN
iajs-966	249	6	قــدمنا	قــدمنا	PROPN
iajs-966	249	7	.	.	PUNCT
iajs-966	250	1	rمقاسـاً	rمقاسـاً	PROPN
iajs-966	250	2	احادیـ	احادیـ	VERB
iajs-966	250	3	ا	ا	ADJ
iajs-966	250	4	یـ	یـ	INTJ
iajs-966	250	5	ـاً	ـاً	INTJ
iajs-966	250	6	كمـ	كمـ	NOUN
iajs-966	250	7	ـاً	ـاً	NOUN
iajs-966	250	8	اذا	اذا	NOUN
iajs-966	250	9	وجـد	وجـد	NOUN
iajs-966	250	10	عنصــر	عنصــر	NOUN
iajs-966	250	11	mمـن	mمـن	NOUN
iajs-966	250	12	المقــاس	المقــاس	VERB
iajs-966	250	13	nیطلــق	nیطلــق	PROPN
iajs-966	250	14	علـى	علـى	PROPN
iajs-966	250	15	المقــاس	المقــاس	PROPN
iajs-966	250	16	الجزئـي	الجزئـي	VERB
iajs-966	250	17	:	:	PUNCT
iajs-966	250	18	اتيمفهـوم	اتيمفهـوم	ADJ
iajs-966	250	19	مقـاس	مقـاس	PROPN
iajs-966	250	20	جزئــي	جزئــي	VERB
iajs-966	250	21	مقیـد	مقیـد	PROPN
iajs-966	250	22	تقریبـ	تقریبـ	PROPN
iajs-966	250	23	مقیـد	مقیـد	PROPN
iajs-966	250	24	تقریبـ	تقریبـ	NOUN
iajs-966	250	25	xm	xm	PUNCT
iajs-966	251	1	وxn	وxn	PROPN
iajs-966	251	2	بحیث	بحیث	X
iajs-966	251	3	انannr(n)=annr(x	انannr(n)=annr(x	PROPN
iajs-966	251	4	)	)	PUNCT
iajs-966	251	5	.ئج	.ئج	PUNCT
iajs-966	252	1	في	في	SCONJ
iajs-966	252	2	هذا	هذا	NOUN
iajs-966	252	3	البحث	البحث	PROPN
iajs-966	252	4	،	،	PROPN
iajs-966	252	5	اعطیت	اعطیت	PROPN
iajs-966	252	6	بعض	بعض	NOUN
iajs-966	252	7	الخـواص	الخـواص	PROPN
iajs-966	252	8	وكـذلك	وكـذلك	PROPN
iajs-966	252	9	ُدرسـت	ُدرسـت	ADV
iajs-966	252	10	العدیـد	العدیـد	PROPN
iajs-966	252	11	مـن	مـن	PROPN
iajs-966	252	12	النتـا	النتـا	NOUN
iajs-966	252	13	ـا	ـا	PROPN
iajs-966	252	14	ً	ً	NOUN
iajs-966	252	15	ـات	ـات	PROPN
iajs-966	252	16	الجزئیـة	الجزئیـة	PROPN
iajs-966	252	17	المقیـدة	المقیـدة	PROPN
iajs-966	252	18	تقریبـ	تقریبـ	NOUN
iajs-966	252	19	الـى	الـى	VERB
iajs-966	252	20	هـذا	هـذا	NOUN
iajs-966	252	21	ُدرســت	ُدرســت	PROPN
iajs-966	252	22	بعـض	بعـض	PROPN
iajs-966	252	23	العالقــات	العالقــات	PROPN
iajs-966	252	24	بینـه	بینـه	PROPN
iajs-966	252	25	وبـین	وبـین	PROPN
iajs-966	252	26	انــواع	انــواع	NOUN
iajs-966	252	27	اخـرى	اخـرى	PROPN
iajs-966	252	28	مــن	مــن	PROPN
iajs-966	252	29	فضـال	فضـال	ADV
iajs-966	252	30	عــن	عــن	NOUN
iajs-966	252	31	.	.	PUNCT
iajs-966	253	1	االساسـیة	االساسـیة	PROPN
iajs-966	253	2	حـول	حـول	ADJ
iajs-966	253	3	المقاسـ	المقاسـ	ADV
iajs-966	253	4	.المقاسات	.المقاسات	PROPN
iajs-966	253	5	ihjpas	ihjpa	VERB