id	sid	tid	token	lemma	pos
iajs-702	1	1	مجلة	مجلة	VERB
iajs-702	1	2	إبن	إبن	VERB
iajs-702	1	3	الھیثم	الھیثم	NOUN
iajs-702	1	4	للعلوم	للعلوم	NOUN
iajs-702	1	5	الصرفة	الصرفة	NOUN
iajs-702	1	6	و	و	PRON
iajs-702	1	7	التطبیقیة	التطبیقیة	PROPN
iajs-702	1	8	2012	2012	NUM
iajs-702	1	9	السنة	السنة	NOUN
iajs-702	1	10	25	25	NUM
iajs-702	1	11	المجلد	المجلد	NOUN
iajs-702	1	12	1	1	NUM
iajs-702	1	13	العدد	العدد	PROPN
iajs-702	1	14	ibn	ibn	PROPN
iajs-702	1	15	al	al	PROPN
iajs-702	1	16	-	-	PUNCT
iajs-702	1	17	haitham	haitham	PROPN
iajs-702	1	18	journal	journal	PROPN
iajs-702	1	19	for	for	ADP
iajs-702	1	20	pure	pure	ADJ
iajs-702	1	21	and	and	CCONJ
iajs-702	1	22	applied	apply	VERB
iajs-702	1	23	science	science	NOUN
iajs-702	1	24	no	no	NOUN
iajs-702	1	25	.	.	NOUN
iajs-702	1	26	1	1	NUM
iajs-702	1	27	vol	vol	NOUN
iajs-702	1	28	.	.	PUNCT
iajs-702	2	1	25	25	NUM
iajs-702	2	2	year	year	NOUN
iajs-702	2	3	2012	2012	NUM
iajs-702	2	4	min	min	NOUN
iajs-702	2	5	(	(	PUNCT
iajs-702	2	6	max)-cs	max)-cs	PROPN
iajs-702	2	7	modules	module	NOUN
iajs-702	2	8	i.	i.	PROPN
iajs-702	2	9	m.	m.	PROPN
iajs-702	2	10	a.	a.	PROPN
iajs-702	2	11	hadi	hadi	PROPN
iajs-702	2	12	,	,	PUNCT
iajs-702	2	13	r	r	NOUN
iajs-702	2	14	.	.	PUNCT
iajs-702	3	1	n.	n.	PROPN
iajs-702	3	2	majeed	majeed	PROPN
iajs-702	3	3	department	department	PROPN
iajs-702	3	4	of	of	ADP
iajs-702	3	5	mathematics	mathematics	PROPN
iajs-702	3	6	,	,	PUNCT
iajs-702	3	7	college	college	NOUN
iajs-702	3	8	of	of	ADP
iajs-702	3	9	ibn	ibn	PROPN
iajs-702	3	10	-	-	PUNCT
iajs-702	3	11	al	al	PROPN
iajs-702	3	12	-	-	PUNCT
iajs-702	3	13	haitham	haitham	PROPN
iajs-702	3	14	,	,	PUNCT
iajs-702	3	15	university	university	PROPN
iajs-702	3	16	of	of	ADP
iajs-702	3	17	baghdad	baghdad	PROPN
iajs-702	3	18	received	receive	VERB
iajs-702	3	19	in:25	in:25	PROPN
iajs-702	3	20	august	august	PROPN
iajs-702	3	21	2011	2011	NUM
iajs-702	3	22	,	,	PUNCT
iajs-702	3	23	accepted	accept	VERB
iajs-702	3	24	in:20	in:20	PROPN
iajs-702	3	25	september	september	PROPN
iajs-702	3	26	2011	2011	NUM
iajs-702	3	27	abstract	abstract	NOUN
iajs-702	3	28	.	.	PUNCT
iajs-702	4	1	in	in	ADP
iajs-702	4	2	this	this	DET
iajs-702	4	3	paper	paper	NOUN
iajs-702	4	4	,	,	PUNCT
iajs-702	4	5	we	we	PRON
iajs-702	4	6	give	give	VERB
iajs-702	4	7	a	a	DET
iajs-702	4	8	comprehensive	comprehensive	ADJ
iajs-702	4	9	study	study	NOUN
iajs-702	4	10	of	of	ADP
iajs-702	4	11	min	min	PROPN
iajs-702	4	12	(	(	PUNCT
iajs-702	4	13	max)-cs	max)-cs	NOUN
iajs-702	4	14	modules	module	NOUN
iajs-702	4	15	such	such	ADJ
iajs-702	4	16	as	as	ADP
iajs-702	4	17	a	a	DET
iajs-702	4	18	closed	closed	ADJ
iajs-702	4	19	submodule	submodule	NOUN
iajs-702	4	20	of	of	ADP
iajs-702	4	21	min	min	PROPN
iajs-702	4	22	-	-	ADJ
iajs-702	4	23	cs	cs	ADJ
iajs-702	4	24	module	module	NOUN
iajs-702	4	25	is	be	AUX
iajs-702	4	26	min	min	NOUN
iajs-702	4	27	-	-	ADJ
iajs-702	4	28	cs	cs	ADJ
iajs-702	4	29	.	.	PROPN
iajs-702	5	1	amongst	amongst	ADP
iajs-702	5	2	other	other	ADJ
iajs-702	5	3	results	result	NOUN
iajs-702	5	4	we	we	PRON
iajs-702	5	5	show	show	VERB
iajs-702	5	6	that	that	SCONJ
iajs-702	5	7	a	a	DET
iajs-702	5	8	direct	direct	ADJ
iajs-702	5	9	summand	summand	NOUN
iajs-702	5	10	of	of	ADP
iajs-702	5	11	min	min	PROPN
iajs-702	5	12	(	(	PUNCT
iajs-702	5	13	max)-cs	max)-cs	NOUN
iajs-702	5	14	module	module	NOUN
iajs-702	5	15	is	be	AUX
iajs-702	5	16	min	min	NOUN
iajs-702	5	17	(	(	PUNCT
iajs-702	5	18	max)-cs	max)-cs	NOUN
iajs-702	5	19	module	module	NOUN
iajs-702	5	20	.	.	PUNCT
iajs-702	6	1	one	one	NUM
iajs-702	6	2	of	of	ADP
iajs-702	6	3	interested	interested	ADJ
iajs-702	6	4	theorems	theorem	NOUN
iajs-702	6	5	in	in	ADP
iajs-702	6	6	this	this	DET
iajs-702	6	7	paper	paper	NOUN
iajs-702	6	8	is	be	AUX
iajs-702	6	9	,	,	PUNCT
iajs-702	6	10	if	if	SCONJ
iajs-702	6	11	r	r	NOUN
iajs-702	6	12	is	be	AUX
iajs-702	6	13	a	a	DET
iajs-702	6	14	nonsingular	nonsingular	ADJ
iajs-702	6	15	ring	ring	NOUN
iajs-702	6	16	then	then	ADV
iajs-702	6	17	r	r	NOUN
iajs-702	6	18	is	be	AUX
iajs-702	6	19	a	a	DET
iajs-702	6	20	max	max	PROPN
iajs-702	6	21	-	-	PUNCT
iajs-702	6	22	cs	cs	ADJ
iajs-702	6	23	ring	ring	NOUN
iajs-702	6	24	if	if	SCONJ
iajs-702	6	25	and	and	CCONJ
iajs-702	6	26	only	only	ADV
iajs-702	6	27	if	if	SCONJ
iajs-702	6	28	r	r	NOUN
iajs-702	6	29	is	be	AUX
iajs-702	6	30	a	a	DET
iajs-702	6	31	min	min	PROPN
iajs-702	6	32	-	-	ADJ
iajs-702	6	33	cs	cs	ADJ
iajs-702	6	34	ring	ring	NOUN
iajs-702	6	35	.	.	PUNCT
iajs-702	7	1	key	key	ADJ
iajs-702	7	2	words	word	NOUN
iajs-702	7	3	:	:	PUNCT
iajs-702	7	4	cs	cs	ADJ
iajs-702	7	5	-	-	NOUN
iajs-702	7	6	module	module	NOUN
iajs-702	7	7	,	,	PUNCT
iajs-702	7	8	min	min	PROPN
iajs-702	7	9	-	-	ADJ
iajs-702	7	10	cs	cs	ADJ
iajs-702	7	11	module	module	NOUN
iajs-702	7	12	,	,	PUNCT
iajs-702	7	13	max	max	PROPN
iajs-702	7	14	-	-	PUNCT
iajs-702	7	15	cs	cs	PROPN
iajs-702	7	16	module	module	NOUN
iajs-702	7	17	,	,	PUNCT
iajs-702	7	18	uniform	uniform	ADJ
iajs-702	7	19	-	-	PUNCT
iajs-702	7	20	cs	cs	NOUN
iajs-702	7	21	module	module	NOUN
iajs-702	7	22	.	.	PUNCT
iajs-702	8	1	1introduction	1introduction	NUM
iajs-702	8	2	throughout	throughout	ADP
iajs-702	8	3	the	the	DET
iajs-702	8	4	paper	paper	NOUN
iajs-702	8	5	all	all	DET
iajs-702	8	6	rings	ring	NOUN
iajs-702	8	7	r	r	NOUN
iajs-702	8	8	are	be	AUX
iajs-702	8	9	commutative	commutative	ADJ
iajs-702	8	10	with	with	ADP
iajs-702	8	11	identity	identity	NOUN
iajs-702	8	12	and	and	CCONJ
iajs-702	8	13	all	all	DET
iajs-702	8	14	r	r	NOUN
iajs-702	8	15	-	-	PUNCT
iajs-702	8	16	modules	module	NOUN
iajs-702	8	17	are	be	AUX
iajs-702	8	18	unitary	unitary	ADJ
iajs-702	8	19	.	.	PUNCT
iajs-702	9	1	we	we	PRON
iajs-702	9	2	write	write	VERB
iajs-702	9	3	a	a	DET
iajs-702	9	4			NUM
iajs-702	9	5	m	m	NOUN
iajs-702	9	6	and	and	CCONJ
iajs-702	9	7	a	a	DET
iajs-702	9	8			NUM
iajs-702	9	9	e	e	NOUN
iajs-702	9	10	m	m	VERB
iajs-702	9	11	to	to	PART
iajs-702	9	12	indicate	indicate	VERB
iajs-702	9	13	that	that	SCONJ
iajs-702	9	14	a	a	PRON
iajs-702	9	15	is	be	AUX
iajs-702	9	16	a	a	DET
iajs-702	9	17	submodule	submodule	NOUN
iajs-702	9	18	of	of	ADP
iajs-702	9	19	m	m	PROPN
iajs-702	9	20	and	and	CCONJ
iajs-702	9	21	a	a	PRON
iajs-702	9	22	is	be	AUX
iajs-702	9	23	an	an	DET
iajs-702	9	24	essential	essential	ADJ
iajs-702	9	25	submodule	submodule	NOUN
iajs-702	9	26	of	of	ADP
iajs-702	9	27	m	m	PROPN
iajs-702	9	28	,	,	PUNCT
iajs-702	9	29	respectively	respectively	ADV
iajs-702	9	30	.	.	PUNCT
iajs-702	10	1	recall	recall	VERB
iajs-702	10	2	that	that	PRON
iajs-702	10	3	anr	anr	NOUN
iajs-702	10	4	-	-	PUNCT
iajs-702	10	5	module	module	NOUN
iajs-702	10	6	m	m	NOUN
iajs-702	10	7	is	be	AUX
iajs-702	10	8	called	call	VERB
iajs-702	10	9	an	an	DET
iajs-702	10	10	extending	extend	VERB
iajs-702	10	11	module	module	NOUN
iajs-702	10	12	(	(	PUNCT
iajs-702	10	13	or	or	CCONJ
iajs-702	10	14	,	,	PUNCT
iajs-702	10	15	cs	cs	ADJ
iajs-702	10	16	-	-	NOUN
iajs-702	10	17	module	module	NOUN
iajs-702	10	18	)	)	PUNCT
iajs-702	10	19	if	if	SCONJ
iajs-702	10	20	every	every	DET
iajs-702	10	21	submodule	submodule	NOUN
iajs-702	10	22	is	be	AUX
iajs-702	10	23	essential	essential	ADJ
iajs-702	10	24	in	in	ADP
iajs-702	10	25	a	a	DET
iajs-702	10	26	direct	direct	ADJ
iajs-702	10	27	summand	summand	NOUN
iajs-702	10	28	of	of	ADP
iajs-702	10	29	m	m	PROPN
iajs-702	10	30	or	or	CCONJ
iajs-702	10	31	m	m	VERB
iajs-702	10	32	is	be	AUX
iajs-702	10	33	extending	extend	VERB
iajs-702	10	34	if	if	SCONJ
iajs-702	10	35	and	and	CCONJ
iajs-702	10	36	only	only	ADV
iajs-702	10	37	if	if	SCONJ
iajs-702	10	38	every	every	DET
iajs-702	10	39	closed	closed	ADJ
iajs-702	10	40	submodule	submodule	NOUN
iajs-702	10	41	is	be	AUX
iajs-702	10	42	a	a	DET
iajs-702	10	43	direct	direct	ADJ
iajs-702	10	44	summand	summand	NOUN
iajs-702	10	45	,	,	PUNCT
iajs-702	10	46	[	[	X
iajs-702	10	47	1	1	NUM
iajs-702	10	48	,	,	PUNCT
iajs-702	10	49	p.55	p.55	NOUN
iajs-702	10	50	]	]	PUNCT
iajs-702	10	51	.	.	PUNCT
iajs-702	11	1	in	in	ADP
iajs-702	11	2	this	this	DET
iajs-702	11	3	paper	paper	NOUN
iajs-702	11	4	definitions	definition	NOUN
iajs-702	11	5	,	,	PUNCT
iajs-702	11	6	notations	notation	NOUN
iajs-702	11	7	,	,	PUNCT
iajs-702	11	8	examples	example	NOUN
iajs-702	11	9	and	and	CCONJ
iajs-702	11	10	fundamental	fundamental	ADJ
iajs-702	11	11	results	result	NOUN
iajs-702	11	12	of	of	ADP
iajs-702	11	13	min	min	NOUN
iajs-702	11	14	(	(	PUNCT
iajs-702	11	15	max)-cs	max)-cs	NOUN
iajs-702	11	16	modules	module	NOUN
iajs-702	11	17	are	be	AUX
iajs-702	11	18	introduced	introduce	VERB
iajs-702	11	19	.	.	PUNCT
iajs-702	12	1	1.1	1.1	NUM
iajs-702	12	2	definition	definition	NOUN
iajs-702	12	3	:	:	PUNCT
iajs-702	12	4	[	[	X
iajs-702	12	5	2	2	X
iajs-702	12	6	]	]	PUNCT
iajs-702	12	7	an	an	DET
iajs-702	12	8	r	r	NOUN
iajs-702	12	9	-	-	PUNCT
iajs-702	12	10	module	module	NOUN
iajs-702	12	11	m	m	NOUN
iajs-702	12	12	is	be	AUX
iajs-702	12	13	called	call	VERB
iajs-702	12	14	min	min	PROPN
iajs-702	12	15	-	-	ADJ
iajs-702	12	16	cs	cs	ADJ
iajs-702	12	17	module	module	NOUN
iajs-702	12	18	if	if	SCONJ
iajs-702	12	19	every	every	DET
iajs-702	12	20	minimal	minimal	ADJ
iajs-702	12	21	closed	closed	ADJ
iajs-702	12	22	submodule	submodule	NOUN
iajs-702	12	23	of	of	ADP
iajs-702	12	24	m	m	PROPN
iajs-702	12	25	is	be	AUX
iajs-702	12	26	a	a	DET
iajs-702	12	27	direct	direct	ADJ
iajs-702	12	28	summand	summand	NOUN
iajs-702	12	29	of	of	ADP
iajs-702	12	30	m.	m.	NOUN
iajs-702	12	31	a	a	DET
iajs-702	12	32	ring	ring	NOUN
iajs-702	12	33	r	r	NOUN
iajs-702	12	34	is	be	AUX
iajs-702	12	35	called	call	VERB
iajs-702	12	36	min	min	NOUN
iajs-702	12	37	-	-	PROPN
iajs-702	12	38	cs	cs	ADJ
iajs-702	12	39	if	if	SCONJ
iajs-702	12	40	it	it	PRON
iajs-702	12	41	is	be	AUX
iajs-702	12	42	min	min	ADJ
iajs-702	12	43	-	-	ADJ
iajs-702	12	44	cs	cs	ADJ
iajs-702	12	45	r	r	NOUN
iajs-702	12	46	-	-	PUNCT
iajs-702	12	47	module	module	NOUN
iajs-702	12	48	.	.	PUNCT
iajs-702	13	1	1.2	1.2	NUM
iajs-702	13	2	definition	definition	NOUN
iajs-702	13	3	:	:	PUNCT
iajs-702	13	4	[	[	X
iajs-702	13	5	2	2	X
iajs-702	13	6	]	]	PUNCT
iajs-702	13	7	an	an	DET
iajs-702	13	8	r	r	NOUN
iajs-702	13	9	-	-	PUNCT
iajs-702	13	10	module	module	NOUN
iajs-702	13	11	m	m	NOUN
iajs-702	13	12	is	be	AUX
iajs-702	13	13	called	call	VERB
iajs-702	13	14	max	max	PROPN
iajs-702	13	15	-	-	PUNCT
iajs-702	13	16	cs	cs	PROPN
iajs-702	13	17	module	module	NOUN
iajs-702	13	18	if	if	SCONJ
iajs-702	13	19	every	every	DET
iajs-702	13	20	maximal	maximal	ADJ
iajs-702	13	21	closed	closed	ADJ
iajs-702	13	22	submodule	submodule	NOUN
iajs-702	13	23	of	of	ADP
iajs-702	13	24	m	m	PROPN
iajs-702	13	25	with	with	ADP
iajs-702	13	26	nonzero	nonzero	PROPN
iajs-702	13	27	annihilator	annihilator	PROPN
iajs-702	13	28	is	be	AUX
iajs-702	13	29	a	a	DET
iajs-702	13	30	direct	direct	ADJ
iajs-702	13	31	summand	summand	NOUN
iajs-702	13	32	of	of	ADP
iajs-702	13	33	m.	m.	NOUN
iajs-702	13	34	a	a	DET
iajs-702	13	35	ring	ring	NOUN
iajs-702	13	36	r	r	NOUN
iajs-702	13	37	is	be	AUX
iajs-702	13	38	max	max	PROPN
iajs-702	13	39	-	-	PUNCT
iajs-702	13	40	cs	cs	PROPN
iajs-702	13	41	if	if	SCONJ
iajs-702	13	42	it	it	PRON
iajs-702	13	43	is	be	AUX
iajs-702	13	44	max	max	PROPN
iajs-702	13	45	-	-	PUNCT
iajs-702	13	46	cs	cs	ADJ
iajs-702	13	47	r	r	NOUN
iajs-702	13	48	-	-	PUNCT
iajs-702	13	49	module	module	NOUN
iajs-702	13	50	.	.	PUNCT
iajs-702	14	1	recall	recall	VERB
iajs-702	14	2	that	that	SCONJ
iajs-702	14	3	an	an	DET
iajs-702	14	4	r	r	NOUN
iajs-702	14	5	-	-	PUNCT
iajs-702	14	6	module	module	NOUN
iajs-702	14	7	m	m	NOUN
iajs-702	14	8	is	be	AUX
iajs-702	14	9	-injective	-injective	ADJ
iajs-702	14	10	(	(	PUNCT
iajs-702	14	11	quasi	quasi	ADJ
iajs-702	14	12	-	-	ADJ
iajs-702	14	13	continuous	continuous	ADJ
iajs-702	14	14	)	)	PUNCT
iajs-702	15	1	if	if	SCONJ
iajs-702	15	2	and	and	CCONJ
iajs-702	15	3	only	only	ADV
iajs-702	15	4	if	if	SCONJ
iajs-702	15	5	m	m	PROPN
iajs-702	15	6	satisfies	satisfie	NOUN
iajs-702	15	7	c1	c1	PROPN
iajs-702	15	8	(	(	PUNCT
iajs-702	15	9	m	m	PROPN
iajs-702	15	10	is	be	AUX
iajs-702	15	11	extending	extend	VERB
iajs-702	15	12	)	)	PUNCT
iajs-702	15	13	and	and	CCONJ
iajs-702	15	14	c3	c3	PROPN
iajs-702	15	15	,	,	PUNCT
iajs-702	15	16	where	where	SCONJ
iajs-702	15	17	m	m	NOUN
iajs-702	15	18	is	be	AUX
iajs-702	15	19	said	say	VERB
iajs-702	15	20	to	to	PART
iajs-702	15	21	satisfy	satisfy	VERB
iajs-702	15	22	the	the	DET
iajs-702	15	23	c3	c3	PROPN
iajs-702	15	24	if	if	SCONJ
iajs-702	15	25	the	the	DET
iajs-702	15	26	sum	sum	NOUN
iajs-702	15	27	of	of	ADP
iajs-702	15	28	any	any	DET
iajs-702	15	29	two	two	NUM
iajs-702	15	30	direct	direct	ADJ
iajs-702	15	31	summands	summand	NOUN
iajs-702	15	32	of	of	ADP
iajs-702	15	33	m	m	PROPN
iajs-702	15	34	with	with	ADP
iajs-702	15	35	zero	zero	NUM
iajs-702	15	36	intersection	intersection	NOUN
iajs-702	15	37	is	be	AUX
iajs-702	15	38	a	a	DET
iajs-702	15	39	direct	direct	ADJ
iajs-702	15	40	summand	summand	NOUN
iajs-702	15	41	of	of	ADP
iajs-702	15	42	m.	m.	NOUN
iajs-702	16	1	[	[	X
iajs-702	16	2	3	3	NUM
iajs-702	16	3	,	,	PUNCT
iajs-702	16	4	p.18	p.18	X
iajs-702	16	5	]	]	X
iajs-702	16	6	1.3	1.3	NUM
iajs-702	16	7	remarks	remark	NOUN
iajs-702	16	8	and	and	CCONJ
iajs-702	16	9	examples	example	NOUN
iajs-702	16	10	1	1	NUM
iajs-702	16	11	.	.	PUNCT
iajs-702	17	1	every	every	DET
iajs-702	17	2	cs	cs	PROPN
iajs-702	17	3	-	-	ADJ
iajs-702	17	4	module	module	NOUN
iajs-702	17	5	is	be	AUX
iajs-702	17	6	min	min	NOUN
iajs-702	17	7	-	-	ADJ
iajs-702	17	8	cs	cs	ADJ
iajs-702	17	9	and	and	CCONJ
iajs-702	17	10	max	max	PROPN
iajs-702	17	11	-	-	PUNCT
iajs-702	17	12	cs	cs	PROPN
iajs-702	17	13	.	.	PROPN
iajs-702	17	14	proof	proof	NOUN
iajs-702	17	15	:	:	PUNCT
iajs-702	17	16	it	it	PRON
iajs-702	17	17	follows	follow	VERB
iajs-702	17	18	directly	directly	ADV
iajs-702	17	19	by	by	ADP
iajs-702	17	20	[	[	X
iajs-702	17	21	1	1	NUM
iajs-702	17	22	,	,	PUNCT
iajs-702	17	23	p.55	p.55	NOUN
iajs-702	17	24	]	]	X
iajs-702	17	25	.	.	PUNCT
iajs-702	18	1	2	2	X
iajs-702	18	2	.	.	X
iajs-702	18	3	every	every	DET
iajs-702	18	4	semisimple	semisimple	NOUN
iajs-702	18	5	module	module	NOUN
iajs-702	18	6	is	be	AUX
iajs-702	18	7	max	max	PROPN
iajs-702	18	8	-	-	PUNCT
iajs-702	18	9	cs	cs	PROPN
iajs-702	18	10	and	and	CCONJ
iajs-702	18	11	min	min	NOUN
iajs-702	18	12	-	-	PROPN
iajs-702	18	13	cs	cs	PROPN
iajs-702	18	14	.	.	PROPN
iajs-702	18	15	in	in	ADP
iajs-702	18	16	particular	particular	ADJ
iajs-702	18	17	ℤ2	ℤ2	PROPN
iajs-702	18	18	,	,	PUNCT
iajs-702	18	19	ℤ3	ℤ3	PROPN
iajs-702	18	20	,	,	PUNCT
iajs-702	18	21	ℤ6	ℤ6	NOUN
iajs-702	18	22	,	,	PUNCT
iajs-702	18	23	ℤ10	ℤ10	PROPN
iajs-702	18	24	,	,	PUNCT
iajs-702	18	25	…	…	PUNCT
iajs-702	18	26	,	,	PUNCT
iajs-702	18	27	ℤ30	ℤ30	PROPN
iajs-702	18	28	as	as	SCONJ
iajs-702	18	29	a	a	DET
iajs-702	18	30	ℤ-module	ℤ-module	PROPN
iajs-702	18	31	is	be	AUX
iajs-702	18	32	max	max	PROPN
iajs-702	18	33	-	-	PUNCT
iajs-702	18	34	cs	cs	PROPN
iajs-702	18	35	and	and	CCONJ
iajs-702	18	36	min	min	NOUN
iajs-702	18	37	-	-	PROPN
iajs-702	18	38	cs	cs	ADJ
iajs-702	18	39	.	.	PROPN
iajs-702	18	40	proof	proof	NOUN
iajs-702	18	41	:	:	PUNCT
iajs-702	18	42	by	by	ADP
iajs-702	18	43	[	[	X
iajs-702	18	44	1	1	NUM
iajs-702	18	45	,	,	PUNCT
iajs-702	18	46	p.55	p.55	PROPN
iajs-702	18	47	]	]	X
iajs-702	18	48	,	,	PUNCT
iajs-702	18	49	every	every	DET
iajs-702	18	50	semisimple	semisimple	NOUN
iajs-702	18	51	module	module	NOUN
iajs-702	18	52	is	be	AUX
iajs-702	18	53	cs	cs	PROPN
iajs-702	18	54	.	.	PROPN
iajs-702	18	55	hence	hence	ADV
iajs-702	18	56	the	the	DET
iajs-702	18	57	result	result	NOUN
iajs-702	18	58	follows	follow	VERB
iajs-702	18	59	by	by	ADP
iajs-702	18	60	remark	remark	NOUN
iajs-702	18	61	1	1	NUM
iajs-702	19	1	.	.	PUNCT
iajs-702	19	2	مجلة	مجلة	VERB
iajs-702	19	3	إبن	إبن	VERB
iajs-702	19	4	الھیثم	الھیثم	NOUN
iajs-702	19	5	للعلوم	للعلوم	NOUN
iajs-702	19	6	الصرفة	الصرفة	NOUN
iajs-702	19	7	و	و	PRON
iajs-702	19	8	التطبیقیة	التطبیقیة	PROPN
iajs-702	19	9	2012	2012	NUM
iajs-702	19	10	السنة	السنة	NOUN
iajs-702	19	11	25	25	NUM
iajs-702	19	12	المجلد	المجلد	NOUN
iajs-702	19	13	1	1	NUM
iajs-702	19	14	العدد	العدد	PROPN
iajs-702	19	15	ibn	ibn	PROPN
iajs-702	19	16	al	al	PROPN
iajs-702	19	17	-	-	PUNCT
iajs-702	19	18	haitham	haitham	PROPN
iajs-702	19	19	journal	journal	PROPN
iajs-702	19	20	for	for	ADP
iajs-702	19	21	pure	pure	ADJ
iajs-702	19	22	and	and	CCONJ
iajs-702	19	23	applied	apply	VERB
iajs-702	19	24	science	science	NOUN
iajs-702	19	25	no	no	NOUN
iajs-702	19	26	.	.	NOUN
iajs-702	19	27	1	1	NUM
iajs-702	19	28	vol	vol	NOUN
iajs-702	19	29	.	.	PUNCT
iajs-702	20	1	25	25	NUM
iajs-702	20	2	year	year	NOUN
iajs-702	20	3	2012	2012	NUM
iajs-702	20	4	3	3	NUM
iajs-702	20	5	.	.	PUNCT
iajs-702	21	1	every	every	DET
iajs-702	21	2	uniform	uniform	ADJ
iajs-702	21	3	r	r	NOUN
iajs-702	21	4	-	-	PUNCT
iajs-702	21	5	module	module	NOUN
iajs-702	21	6	m	m	NOUN
iajs-702	21	7	is	be	AUX
iajs-702	21	8	min	min	NOUN
iajs-702	21	9	-	-	ADJ
iajs-702	21	10	cs	cs	ADJ
iajs-702	21	11	and	and	CCONJ
iajs-702	21	12	max	max	PROPN
iajs-702	21	13	-	-	PUNCT
iajs-702	21	14	cs	cs	PROPN
iajs-702	21	15	.	.	PROPN
iajs-702	22	1	in	in	ADP
iajs-702	22	2	particular	particular	ADJ
iajs-702	22	3	each	each	PRON
iajs-702	22	4	of	of	ADP
iajs-702	22	5	ℤ-module	ℤ-module	PROPN
iajs-702	22	6	ℤ	ℤ	PROPN
iajs-702	22	7	,	,	PUNCT
iajs-702	22	8	ℤ4	ℤ4	PROPN
iajs-702	22	9	,	,	PUNCT
iajs-702	22	10	ℤ8	ℤ8	NOUN
iajs-702	22	11	,	,	PUNCT
iajs-702	22	12	ℤ9	ℤ9	NOUN
iajs-702	22	13	,	,	PUNCT
iajs-702	22	14	ℤ16	ℤ16	PROPN
iajs-702	22	15	is	be	AUX
iajs-702	22	16	min	min	NOUN
iajs-702	22	17	-	-	ADJ
iajs-702	22	18	cs	cs	ADJ
iajs-702	22	19	and	and	CCONJ
iajs-702	22	20	max	max	PROPN
iajs-702	22	21	-	-	PUNCT
iajs-702	22	22	cs	cs	PROPN
iajs-702	22	23	.	.	PROPN
iajs-702	22	24	4	4	NUM
iajs-702	22	25	.	.	PUNCT
iajs-702	23	1	if	if	SCONJ
iajs-702	23	2	r	r	NOUN
iajs-702	23	3	is	be	AUX
iajs-702	23	4	a	a	DET
iajs-702	23	5	semisimple	semisimple	NOUN
iajs-702	23	6	ring	ring	NOUN
iajs-702	23	7	,	,	PUNCT
iajs-702	23	8	then	then	ADV
iajs-702	23	9	every	every	DET
iajs-702	23	10	r	r	NOUN
iajs-702	23	11	-	-	PUNCT
iajs-702	23	12	module	module	NOUN
iajs-702	23	13	m	m	NOUN
iajs-702	23	14	is	be	AUX
iajs-702	23	15	injective	injective	ADJ
iajs-702	23	16	,	,	PUNCT
iajs-702	23	17	by	by	ADP
iajs-702	23	18	[	[	X
iajs-702	23	19	4	4	NUM
iajs-702	23	20	,	,	PUNCT
iajs-702	23	21	theorem	theorem	VERB
iajs-702	23	22	1.18	1.18	NUM
iajs-702	23	23	,	,	PUNCT
iajs-702	23	24	p.29	p.29	NOUN
iajs-702	23	25	]	]	PUNCT
iajs-702	23	26	.	.	PUNCT
iajs-702	24	1	hence	hence	ADV
iajs-702	24	2	m	m	PROPN
iajs-702	24	3	is	be	AUX
iajs-702	24	4	max	max	PROPN
iajs-702	24	5	(	(	PUNCT
iajs-702	24	6	min)-cs	min)-cs	VERB
iajs-702	24	7	module	module	NOUN
iajs-702	24	8	since	since	SCONJ
iajs-702	24	9	every	every	DET
iajs-702	24	10	injective	injective	ADJ
iajs-702	24	11	module	module	NOUN
iajs-702	24	12	is	be	AUX
iajs-702	24	13	cs	cs	PROPN
iajs-702	24	14	.	.	PUNCT
iajs-702	24	15	by	by	ADP
iajs-702	24	16	[	[	X
iajs-702	24	17	1	1	NUM
iajs-702	24	18	,	,	PUNCT
iajs-702	24	19	p.16	p.16	PROPN
iajs-702	24	20	]	]	PUNCT
iajs-702	24	21	.	.	PUNCT
iajs-702	25	1	5	5	X
iajs-702	25	2	.	.	X
iajs-702	25	3	the	the	DET
iajs-702	25	4	ℤ-module	ℤ-module	PROPN
iajs-702	25	5	ℤ12	ℤ12	PROPN
iajs-702	25	6	is	be	AUX
iajs-702	25	7	max	max	PROPN
iajs-702	25	8	-	-	PUNCT
iajs-702	25	9	cs	cs	PROPN
iajs-702	25	10	and	and	CCONJ
iajs-702	25	11	min	min	NOUN
iajs-702	25	12	-	-	PROPN
iajs-702	25	13	cs	cs	PROPN
iajs-702	25	14	.	.	PUNCT
iajs-702	26	1	the	the	DET
iajs-702	26	2	submodules	submodule	NOUN
iajs-702	26	3	of	of	ADP
iajs-702	26	4	ℤ12	ℤ12	PROPN
iajs-702	26	5	are	be	AUX
iajs-702	26	6	2	2	NUM
iajs-702	26	7	,	,	PUNCT
iajs-702	26	8	3	3	NUM
iajs-702	26	9	,	,	PUNCT
iajs-702	26	10	4	4	NUM
iajs-702	26	11	,	,	PUNCT
iajs-702	26	12	6	6	NUM
iajs-702	26	13	,	,	PUNCT
iajs-702	26	14	0	0	PROPN
iajs-702	27	1			INTJ
iajs-702	27	2			PROPN
iajs-702	28	1			INTJ
iajs-702	28	2			PROPN
iajs-702	29	1			INTJ
iajs-702	29	2			PROPN
iajs-702	30	1			INTJ
iajs-702	30	2			PROPN
iajs-702	31	1			PROPN
iajs-702	31	2	and	and	CCONJ
iajs-702	31	3	ℤ12	ℤ12	PROPN
iajs-702	31	4	.	.	PUNCT
iajs-702	32	1	since	since	SCONJ
iajs-702	32	2	123	123	NUM
iajs-702	32	3	4	4	NUM
iajs-702	32	4			INTJ
iajs-702	32	5			PROPN
iajs-702	32	6			PROPN
iajs-702	32	7			PROPN
iajs-702	33	1	¢	¢	NOUN
iajs-702	33	2	.	.	PUNCT
iajs-702	34	1	so	so	ADV
iajs-702	34	2	each	each	PRON
iajs-702	34	3	of	of	ADP
iajs-702	34	4	3	3	NUM
iajs-702	34	5			ADJ
iajs-702	34	6	and	and	CCONJ
iajs-702	34	7	4	4	NUM
iajs-702	34	8			NOUN
iajs-702	34	9	are	be	AUX
iajs-702	34	10	direct	direct	ADJ
iajs-702	34	11	summands	summand	NOUN
iajs-702	34	12	.	.	PUNCT
iajs-702	35	1	hence	hence	ADV
iajs-702	35	2	they	they	PRON
iajs-702	35	3	are	be	AUX
iajs-702	35	4	closed	closed	ADJ
iajs-702	35	5	submodules	submodule	NOUN
iajs-702	35	6	.	.	PUNCT
iajs-702	36	1	2	2	NUM
iajs-702	36	2			PROPN
iajs-702	36	3			NUM
iajs-702	36	4	e	e	NOUN
iajs-702	36	5	ℤ12	ℤ12	PROPN
iajs-702	36	6	and	and	CCONJ
iajs-702	36	7	e6	e6	PROPN
iajs-702	36	8	3	3	PROPN
iajs-702	36	9			PROPN
iajs-702	36	10			NOUN
iajs-702	36	11			PROPN
iajs-702	37	1			PRON
iajs-702	37	2	imply	imply	VERB
iajs-702	37	3	hat	hat	NOUN
iajs-702	37	4	2	2	NUM
iajs-702	37	5			ADJ
iajs-702	37	6	and	and	CCONJ
iajs-702	37	7	6	6	NUM
iajs-702	37	8			NOUN
iajs-702	37	9	are	be	AUX
iajs-702	37	10	not	not	PART
iajs-702	37	11	closed	closed	ADJ
iajs-702	37	12	.	.	PUNCT
iajs-702	38	1	thus	thus	ADV
iajs-702	38	2	m	m	PROPN
iajs-702	38	3	is	be	AUX
iajs-702	38	4	cs	cs	PROPN
iajs-702	38	5	and	and	CCONJ
iajs-702	38	6	so	so	ADV
iajs-702	38	7	max	max	PROPN
iajs-702	38	8	-	-	PUNCT
iajs-702	38	9	cs	cs	PROPN
iajs-702	38	10	and	and	CCONJ
iajs-702	38	11	min	min	NOUN
iajs-702	38	12	-	-	PROPN
iajs-702	38	13	cs	cs	PROPN
iajs-702	38	14	.	.	PROPN
iajs-702	38	15	6	6	NUM
iajs-702	38	16	.	.	PUNCT
iajs-702	39	1	it	it	PRON
iajs-702	39	2	is	be	AUX
iajs-702	39	3	easy	easy	ADJ
iajs-702	39	4	to	to	PART
iajs-702	39	5	check	check	VERB
iajs-702	39	6	that	that	SCONJ
iajs-702	39	7	each	each	PRON
iajs-702	39	8	of	of	ADP
iajs-702	39	9	the	the	DET
iajs-702	39	10	ℤ-modules	ℤ-modules	PROPN
iajs-702	39	11	ℤ18	ℤ18	PROPN
iajs-702	39	12	and	and	CCONJ
iajs-702	39	13	ℤ24	ℤ24	PROPN
iajs-702	39	14	are	be	AUX
iajs-702	39	15	min	min	NOUN
iajs-702	39	16	-	-	ADJ
iajs-702	39	17	cs	cs	ADJ
iajs-702	39	18	and	and	CCONJ
iajs-702	39	19	max	max	PROPN
iajs-702	39	20	-	-	PUNCT
iajs-702	39	21	cs	cs	PROPN
iajs-702	39	22	.	.	PROPN
iajs-702	39	23	7	7	X
iajs-702	39	24	.	.	X
iajs-702	40	1	let	let	VERB
iajs-702	40	2	m	m	PRON
iajs-702	40	3	be	be	AUX
iajs-702	40	4	a	a	DET
iajs-702	40	5	module	module	NOUN
iajs-702	40	6	whose	whose	DET
iajs-702	40	7	lattice	lattice	NOUN
iajs-702	40	8	of	of	ADP
iajs-702	40	9	submodules	submodule	NOUN
iajs-702	40	10	is	be	AUX
iajs-702	40	11	the	the	DET
iajs-702	40	12	following	following	NOUN
iajs-702	40	13	:	:	PUNCT
iajs-702	40	14	2	2	NUM
iajs-702	40	15	1	1	NUM
iajs-702	40	16	2	2	NUM
iajs-702	40	17	m	m	NOUN
iajs-702	40	18	n	n	ADP
iajs-702	40	19	n	n	CCONJ
iajs-702	40	20	n	n	ADV
iajs-702	40	21	0	0	NUM
iajs-702	40	22			PROPN
iajs-702	40	23			PROPN
iajs-702	40	24			PROPN
iajs-702	40	25			PROPN
iajs-702	40	26	/	/	SYM
iajs-702	40	27	\	\	NOUN
iajs-702	40	28	\	\	NOUN
iajs-702	40	29	/	/	PUNCT
iajs-702	40	30	it	it	PRON
iajs-702	40	31	is	be	AUX
iajs-702	40	32	clear	clear	ADJ
iajs-702	40	33	that	that	SCONJ
iajs-702	40	34	n1	n1	PROPN
iajs-702	40	35	is	be	AUX
iajs-702	40	36	closed	close	VERB
iajs-702	40	37	in	in	ADP
iajs-702	40	38	m	m	PROPN
iajs-702	40	39	,	,	PUNCT
iajs-702	40	40	but	but	CCONJ
iajs-702	40	41	it	it	PRON
iajs-702	40	42	is	be	AUX
iajs-702	40	43	not	not	PART
iajs-702	40	44	a	a	DET
iajs-702	40	45	direct	direct	ADJ
iajs-702	40	46	summand	summand	NOUN
iajs-702	40	47	of	of	ADP
iajs-702	40	48	m.	m.	NOUN
iajs-702	41	1	so	so	ADV
iajs-702	41	2	m	m	VERB
iajs-702	41	3	is	be	AUX
iajs-702	41	4	not	not	PART
iajs-702	41	5	cs	cs	PROPN
iajs-702	41	6	.	.	PROPN
iajs-702	41	7	also	also	ADV
iajs-702	41	8	n1	n1	PROPN
iajs-702	41	9	is	be	AUX
iajs-702	41	10	a	a	DET
iajs-702	41	11	minimal	minimal	ADJ
iajs-702	41	12	closed	closed	ADJ
iajs-702	41	13	submodule	submodule	NOUN
iajs-702	41	14	of	of	ADP
iajs-702	41	15	m.	m.	NOUN
iajs-702	41	16	hence	hence	ADV
iajs-702	41	17	m	m	VERB
iajs-702	41	18	is	be	AUX
iajs-702	41	19	not	not	PART
iajs-702	41	20	a	a	DET
iajs-702	41	21	min	min	ADJ
iajs-702	41	22	-	-	ADJ
iajs-702	41	23	cs	cs	ADJ
iajs-702	41	24	module	module	NOUN
iajs-702	41	25	.	.	PUNCT
iajs-702	42	1	notice	notice	VERB
iajs-702	42	2	that	that	SCONJ
iajs-702	42	3	n1	n1	PROPN
iajs-702	42	4			ADJ
iajs-702	42	5	n2	n2	NOUN
iajs-702	42	6			PROPN
iajs-702	42	7	e	e	X
iajs-702	42	8	m	m	PROPN
iajs-702	42	9	,	,	PUNCT
iajs-702	42	10	so	so	SCONJ
iajs-702	42	11	it	it	PRON
iajs-702	42	12	is	be	AUX
iajs-702	42	13	not	not	PART
iajs-702	42	14	closed	close	VERB
iajs-702	42	15	submodule	submodule	NOUN
iajs-702	42	16	of	of	ADP
iajs-702	42	17	m.	m.	NOUN
iajs-702	42	18	it	it	PRON
iajs-702	42	19	follows	follow	VERB
iajs-702	42	20	that	that	SCONJ
iajs-702	42	21	n1	n1	PROPN
iajs-702	42	22	is	be	AUX
iajs-702	42	23	a	a	DET
iajs-702	42	24	max	max	PROPN
iajs-702	42	25	-	-	PUNCT
iajs-702	42	26	closed	close	VERB
iajs-702	42	27	submodule	submodule	NOUN
iajs-702	42	28	of	of	ADP
iajs-702	42	29	m.	m.	NOUN
iajs-702	42	30	hence	hence	ADV
iajs-702	42	31	m	m	VERB
iajs-702	42	32	is	be	AUX
iajs-702	42	33	not	not	PART
iajs-702	42	34	a	a	DET
iajs-702	42	35	max	max	PROPN
iajs-702	42	36	-	-	PUNCT
iajs-702	42	37	cs	cs	PROPN
iajs-702	42	38	module	module	NOUN
iajs-702	42	39	.	.	PUNCT
iajs-702	43	1	8	8	NUM
iajs-702	43	2	.	.	PUNCT
iajs-702	44	1	every	every	DET
iajs-702	44	2	-injective	-injective	NOUN
iajs-702	44	3	is	be	AUX
iajs-702	44	4	min	min	NOUN
iajs-702	44	5	-	-	ADJ
iajs-702	44	6	cs	cs	ADJ
iajs-702	44	7	and	and	CCONJ
iajs-702	44	8	max	max	PROPN
iajs-702	44	9	-	-	PUNCT
iajs-702	44	10	cs	cs	PROPN
iajs-702	44	11	.	.	PROPN
iajs-702	44	12	proof	proof	NOUN
iajs-702	44	13	:	:	PUNCT
iajs-702	44	14	it	it	PRON
iajs-702	44	15	follows	follow	VERB
iajs-702	44	16	by	by	ADP
iajs-702	44	17	the	the	DET
iajs-702	44	18	definition	definition	NOUN
iajs-702	44	19	of	of	ADP
iajs-702	44	20	-injective	-injective	NOUN
iajs-702	44	21	module	module	NOUN
iajs-702	44	22	and	and	CCONJ
iajs-702	44	23	remark	remark	NOUN
iajs-702	44	24	1.3	1.3	NUM
iajs-702	44	25	(	(	PUNCT
iajs-702	44	26	1	1	NUM
iajs-702	44	27	)	)	PUNCT
iajs-702	44	28	.	.	PUNCT
iajs-702	45	1	9	9	X
iajs-702	45	2	.	.	X
iajs-702	45	3	let	let	VERB
iajs-702	45	4	m	m	PRON
iajs-702	45	5	be	be	AUX
iajs-702	45	6	the	the	DET
iajs-702	45	7	ℤ-module	ℤ-module	ADJ
iajs-702	45	8	ℤ8	ℤ8	NOUN
iajs-702	45	9			PROPN
iajs-702	45	10	ℤ2	ℤ2	PROPN
iajs-702	45	11	.	.	PUNCT
iajs-702	46	1	m	m	PROPN
iajs-702	46	2	is	be	AUX
iajs-702	46	3	not	not	PART
iajs-702	46	4	cs	cs	ADJ
iajs-702	46	5	-	-	NOUN
iajs-702	46	6	module	module	NOUN
iajs-702	46	7	,	,	PUNCT
iajs-702	46	8	since	since	SCONJ
iajs-702	46	9	there	there	PRON
iajs-702	46	10	exists	exist	VERB
iajs-702	46	11	a	a	DET
iajs-702	46	12	submodule	submodule	NOUN
iajs-702	46	13	n	n	NOUN
iajs-702	46	14	=	=	PRON
iajs-702	46	15	{	{	PUNCT
iajs-702	46	16	(	(	PUNCT
iajs-702	46	17	2	2	NUM
iajs-702	46	18	,	,	PUNCT
iajs-702	46	19	1	1	NUM
iajs-702	46	20	)	)	PUNCT
iajs-702	46	21	,	,	PUNCT
iajs-702	46	22	(	(	PUNCT
iajs-702	46	23	4	4	NUM
iajs-702	46	24	,	,	PUNCT
iajs-702	46	25	0	0	NUM
iajs-702	46	26	)	)	PUNCT
iajs-702	46	27	,	,	PUNCT
iajs-702	46	28	(	(	PUNCT
iajs-702	46	29	6	6	NUM
iajs-702	46	30	,	,	PUNCT
iajs-702	46	31	1	1	NUM
iajs-702	46	32	)	)	PUNCT
iajs-702	46	33	,	,	PUNCT
iajs-702	46	34	(	(	PUNCT
iajs-702	46	35	0,0	0,0	NOUN
iajs-702	46	36	)	)	PUNCT
iajs-702	46	37	}	}	PUNCT
iajs-702	46	38	which	which	PRON
iajs-702	46	39	is	be	AUX
iajs-702	46	40	closed	close	VERB
iajs-702	46	41	but	but	CCONJ
iajs-702	46	42	not	not	PART
iajs-702	46	43	a	a	DET
iajs-702	46	44	direct	direct	ADJ
iajs-702	46	45	summand	summand	NOUN
iajs-702	46	46	.	.	PUNCT
iajs-702	47	1	moreover	moreover	ADV
iajs-702	47	2	n	n	VERB
iajs-702	47	3	is	be	AUX
iajs-702	47	4	minimal	minimal	ADJ
iajs-702	47	5	closed	closed	ADJ
iajs-702	47	6	,	,	PUNCT
iajs-702	47	7	so	so	ADV
iajs-702	47	8	m	m	NOUN
iajs-702	47	9	is	be	AUX
iajs-702	47	10	not	not	PART
iajs-702	47	11	min	min	ADJ
iajs-702	47	12	-	-	ADJ
iajs-702	47	13	cs	cs	ADJ
iajs-702	47	14	module	module	NOUN
iajs-702	47	15	.	.	PUNCT
iajs-702	48	1	on	on	ADP
iajs-702	48	2	the	the	DET
iajs-702	48	3	other	other	ADJ
iajs-702	48	4	hand	hand	NOUN
iajs-702	48	5	m	m	NOUN
iajs-702	48	6	is	be	AUX
iajs-702	48	7	a	a	DET
iajs-702	48	8	max	max	PROPN
iajs-702	48	9	-	-	PUNCT
iajs-702	48	10	cs	cs	PROPN
iajs-702	48	11	module	module	NOUN
iajs-702	48	12	,	,	PUNCT
iajs-702	48	13	since	since	SCONJ
iajs-702	48	14	the	the	DET
iajs-702	48	15	only	only	ADJ
iajs-702	48	16	maximal	maximal	ADJ
iajs-702	48	17	closed	closed	ADJ
iajs-702	48	18	submodules	submodule	NOUN
iajs-702	48	19	of	of	ADP
iajs-702	48	20	m	m	NOUN
iajs-702	48	21	are	be	AUX
iajs-702	48	22	ℤ8	ℤ8	NOUN
iajs-702	48	23			ADJ
iajs-702	48	24	(	(	PUNCT
iajs-702	48	25	0	0	NUM
iajs-702	48	26	)	)	PUNCT
iajs-702	48	27	and	and	CCONJ
iajs-702	48	28	<	<	X
iajs-702	48	29	(	(	PUNCT
iajs-702	48	30	3,1	3,1	NUM
iajs-702	48	31	)	)	PUNCT
iajs-702	48	32	>	>	PUNCT
iajs-702	48	33	,	,	PUNCT
iajs-702	48	34	and	and	CCONJ
iajs-702	48	35	(	(	PUNCT
iajs-702	48	36	ℤ8	ℤ8	NOUN
iajs-702	48	37			NOUN
iajs-702	48	38	(	(	PUNCT
iajs-702	48	39	0	0	NUM
iajs-702	48	40	)	)	PUNCT
iajs-702	48	41	)	)	PUNCT
iajs-702	49	1			INTJ
iajs-702	49	2	(	(	PUNCT
iajs-702	49	3	(	(	PUNCT
iajs-702	49	4	0	0	NUM
iajs-702	49	5	)	)	PUNCT
iajs-702	49	6	ℤ2	ℤ2	NOUN
iajs-702	49	7	)	)	PUNCT
iajs-702	50	1	=	=	SYM
iajs-702	50	2	m	m	VERB
iajs-702	50	3	and	and	CCONJ
iajs-702	50	4	<	<	X
iajs-702	50	5	(	(	PUNCT
iajs-702	50	6	3,1	3,1	NUM
iajs-702	50	7	)	)	PUNCT
iajs-702	50	8	>	>	X
iajs-702	50	9			PROPN
iajs-702	50	10	(	(	PUNCT
iajs-702	50	11	(	(	PUNCT
iajs-702	50	12	0	0	X
iajs-702	50	13	)	)	PUNCT
iajs-702	50	14			PROPN
iajs-702	50	15	ℤ2	ℤ2	PROPN
iajs-702	50	16	)	)	PUNCT
iajs-702	50	17	=	=	PUNCT
iajs-702	50	18	m	m	VERB
iajs-702	50	19	.	.	PUNCT
iajs-702	51	1	hence	hence	ADV
iajs-702	51	2	we	we	PRON
iajs-702	51	3	deduce	deduce	VERB
iajs-702	51	4	that	that	SCONJ
iajs-702	51	5	m	m	PROPN
iajs-702	51	6	is	be	AUX
iajs-702	51	7	max	max	PROPN
iajs-702	51	8	-	-	PUNCT
iajs-702	51	9	cs	cs	PROPN
iajs-702	51	10	.	.	PROPN
iajs-702	51	11	10	10	NUM
iajs-702	51	12	.	.	PUNCT
iajs-702	52	1	let	let	VERB
iajs-702	52	2	m	m	PRON
iajs-702	52	3	1	1	NUM
iajs-702	52	4	and	and	CCONJ
iajs-702	52	5	m	m	PROPN
iajs-702	52	6	2	2	NUM
iajs-702	52	7	be	be	VERB
iajs-702	52	8	two	two	NUM
iajs-702	52	9	r	r	NOUN
iajs-702	52	10	-	-	PUNCT
iajs-702	52	11	modules	module	NOUN
iajs-702	52	12	such	such	ADJ
iajs-702	52	13	that	that	SCONJ
iajs-702	52	14	m	m	PROPN
iajs-702	52	15	1	1	NUM
iajs-702	52	16	is	be	AUX
iajs-702	52	17	isomorphic	isomorphic	ADJ
iajs-702	52	18	to	to	ADP
iajs-702	52	19	m	m	PROPN
iajs-702	52	20	2	2	NUM
iajs-702	52	21	(	(	PUNCT
iajs-702	52	22	m	m	NOUN
iajs-702	52	23	1≅m	1≅m	ADJ
iajs-702	52	24	2	2	NUM
iajs-702	52	25	)	)	PUNCT
iajs-702	52	26	,	,	PUNCT
iajs-702	53	1	then	then	ADV
iajs-702	53	2	m1	m1	PROPN
iajs-702	53	3	min	min	PROPN
iajs-702	53	4	(	(	PUNCT
iajs-702	53	5	max)-cs	max)-c	VERB
iajs-702	53	6	if	if	SCONJ
iajs-702	53	7	and	and	CCONJ
iajs-702	53	8	only	only	ADV
iajs-702	53	9	if	if	SCONJ
iajs-702	53	10	m2	m2	PROPN
iajs-702	53	11	is	be	AUX
iajs-702	53	12	min	min	PROPN
iajs-702	53	13	(	(	PUNCT
iajs-702	53	14	max)-cs	max)-cs	PROPN
iajs-702	53	15	.	.	PUNCT
iajs-702	54	1	mahmoud	mahmoud	PROPN
iajs-702	54	2	a.kamal	a.kamal	PROPN
iajs-702	54	3	and	and	CCONJ
iajs-702	54	4	amany	amany	VERB
iajs-702	54	5	m.menshawy	m.menshawy	NOUN
iajs-702	54	6	in	in	ADP
iajs-702	54	7	[	[	X
iajs-702	54	8	5	5	NUM
iajs-702	54	9	]	]	PUNCT
iajs-702	54	10	gave	give	VERB
iajs-702	54	11	the	the	DET
iajs-702	54	12	following	following	NOUN
iajs-702	54	13	:	:	PUNCT
iajs-702	54	14	an	an	DET
iajs-702	54	15	r	r	NOUN
iajs-702	54	16	-	-	PUNCT
iajs-702	54	17	module	module	NOUN
iajs-702	54	18	m	m	NOUN
iajs-702	54	19	is	be	AUX
iajs-702	54	20	called	call	VERB
iajs-702	54	21	min	min	NOUN
iajs-702	54	22	-	-	PROPN
iajs-702	54	23	cs	cs	ADJ
iajs-702	54	24	if	if	SCONJ
iajs-702	54	25	every	every	DET
iajs-702	54	26	simple	simple	ADJ
iajs-702	54	27	submodule	submodule	NOUN
iajs-702	54	28	of	of	ADP
iajs-702	54	29	m	m	PROPN
iajs-702	54	30	is	be	AUX
iajs-702	54	31	essential	essential	ADJ
iajs-702	54	32	in	in	ADP
iajs-702	54	33	a	a	DET
iajs-702	54	34	direct	direct	ADJ
iajs-702	54	35	summand	summand	NOUN
iajs-702	54	36	.	.	PUNCT
iajs-702	55	1	however	however	ADV
iajs-702	55	2	this	this	DET
iajs-702	55	3	definition	definition	NOUN
iajs-702	55	4	of	of	ADP
iajs-702	55	5	min	min	PROPN
iajs-702	55	6	-	-	ADJ
iajs-702	55	7	cs	cs	PROPN
iajs-702	55	8	is	be	AUX
iajs-702	55	9	different	different	ADJ
iajs-702	55	10	from	from	ADP
iajs-702	55	11	definition	definition	NOUN
iajs-702	55	12	1.1	1.1	NUM
iajs-702	55	13	,	,	PUNCT
iajs-702	55	14	since	since	SCONJ
iajs-702	55	15	the	the	DET
iajs-702	55	16	ℤ-module	ℤ-module	PROPN
iajs-702	55	17	m	m	NOUN
iajs-702	55	18	=	=	NOUN
iajs-702	55	19	ℤ8	ℤ8	NOUN
iajs-702	55	20			PROPN
iajs-702	55	21	ℤ2	ℤ2	PROPN
iajs-702	55	22	is	be	AUX
iajs-702	55	23	not	not	PART
iajs-702	55	24	min	min	NOUN
iajs-702	55	25	-	-	NOUN
iajs-702	55	26	cs	cs	ADJ
iajs-702	55	27	in	in	ADP
iajs-702	55	28	our	our	PRON
iajs-702	55	29	sense	sense	NOUN
iajs-702	55	30	.	.	PUNCT
iajs-702	56	1	however	however	ADV
iajs-702	56	2	it	it	PRON
iajs-702	56	3	is	be	AUX
iajs-702	56	4	min	min	NOUN
iajs-702	56	5	-	-	ADJ
iajs-702	56	6	cs	cs	ADJ
iajs-702	56	7	(	(	PUNCT
iajs-702	56	8	in	in	ADP
iajs-702	56	9	sense	sense	NOUN
iajs-702	56	10	of	of	ADP
iajs-702	56	11	kamal	kamal	PROPN
iajs-702	56	12	and	and	CCONJ
iajs-702	56	13	menshawy	menshawy	NOUN
iajs-702	56	14	)	)	PUNCT
iajs-702	56	15	since	since	SCONJ
iajs-702	56	16	ℤ8	ℤ8	NOUN
iajs-702	56	17	is	be	AUX
iajs-702	56	18	cs	cs	ADJ
iajs-702	56	19	,	,	PUNCT
iajs-702	56	20	so	so	ADV
iajs-702	56	21	min	min	PROPN
iajs-702	56	22	-	-	ADJ
iajs-702	56	23	cs	cs	X
iajs-702	56	24	(	(	PUNCT
iajs-702	56	25	in	in	ADP
iajs-702	56	26	sense	sense	NOUN
iajs-702	56	27	of	of	ADP
iajs-702	56	28	kamal	kamal	PROPN
iajs-702	56	29	and	and	CCONJ
iajs-702	56	30	menshawy	menshawy	NOUN
iajs-702	56	31	)	)	PUNCT
iajs-702	56	32	and	and	CCONJ
iajs-702	56	33	ℤ2	ℤ2	PROPN
iajs-702	56	34	is	be	AUX
iajs-702	56	35	simple	simple	ADJ
iajs-702	56	36	,	,	PUNCT
iajs-702	56	37	so	so	ADV
iajs-702	56	38	semisimple	semisimple	NOUN
iajs-702	56	39	.	.	PUNCT
iajs-702	57	1	مجلة	مجلة	VERB
iajs-702	57	2	إبن	إبن	VERB
iajs-702	57	3	الھیثم	الھیثم	NOUN
iajs-702	57	4	للعلوم	للعلوم	NOUN
iajs-702	57	5	الصرفة	الصرفة	NOUN
iajs-702	57	6	و	و	PRON
iajs-702	57	7	التطبیقیة	التطبیقیة	PROPN
iajs-702	57	8	2012	2012	NUM
iajs-702	57	9	السنة	السنة	NOUN
iajs-702	58	1	25	25	NUM
iajs-702	58	2	المجلد	المجلد	NOUN
iajs-702	58	3	1	1	NUM
iajs-702	58	4	العدد	العدد	PROPN
iajs-702	58	5	ibn	ibn	PROPN
iajs-702	58	6	al	al	PROPN
iajs-702	58	7	-	-	PUNCT
iajs-702	58	8	haitham	haitham	PROPN
iajs-702	58	9	journal	journal	PROPN
iajs-702	58	10	for	for	ADP
iajs-702	58	11	pure	pure	ADJ
iajs-702	58	12	and	and	CCONJ
iajs-702	58	13	applied	apply	VERB
iajs-702	58	14	science	science	NOUN
iajs-702	58	15	no	no	NOUN
iajs-702	58	16	.	.	NOUN
iajs-702	58	17	1	1	NUM
iajs-702	58	18	vol	vol	NOUN
iajs-702	58	19	.	.	PUNCT
iajs-702	59	1	25	25	NUM
iajs-702	59	2	year	year	NOUN
iajs-702	59	3	2012	2012	NUM
iajs-702	59	4	hence	hence	ADV
iajs-702	59	5	by	by	ADP
iajs-702	59	6	applying	apply	VERB
iajs-702	59	7	[	[	X
iajs-702	59	8	5	5	NUM
iajs-702	59	9	,	,	PUNCT
iajs-702	59	10	lemma	lemma	PROPN
iajs-702	59	11	3	3	NUM
iajs-702	59	12	,	,	PUNCT
iajs-702	59	13	p.166	p.166	ADV
iajs-702	59	14	]	]	PUNCT
iajs-702	59	15	,	,	PUNCT
iajs-702	59	16	m	m	PROPN
iajs-702	59	17	is	be	AUX
iajs-702	59	18	min	min	NOUN
iajs-702	59	19	-	-	ADJ
iajs-702	59	20	cs	cs	ADJ
iajs-702	59	21	(	(	PUNCT
iajs-702	59	22	in	in	ADP
iajs-702	59	23	sense	sense	NOUN
iajs-702	59	24	of	of	ADP
iajs-702	59	25	kamal	kamal	PROPN
iajs-702	59	26	and	and	CCONJ
iajs-702	59	27	menshawy	menshawy	NOUN
iajs-702	59	28	)	)	PUNCT
iajs-702	59	29	.	.	PUNCT
iajs-702	60	1	1.4	1.4	NUM
iajs-702	60	2	proposition	proposition	NOUN
iajs-702	60	3	:	:	PUNCT
iajs-702	60	4	let	let	VERB
iajs-702	60	5	m	m	PRON
iajs-702	60	6	be	be	AUX
iajs-702	60	7	an	an	DET
iajs-702	60	8	r	r	NOUN
iajs-702	60	9	-	-	PUNCT
iajs-702	60	10	module	module	NOUN
iajs-702	60	11	,	,	PUNCT
iajs-702	60	12	and	and	CCONJ
iajs-702	60	13	let	let	VERB
iajs-702	60	14	i	i	PRON
iajs-702	60	15	be	be	AUX
iajs-702	60	16	an	an	DET
iajs-702	60	17	ideal	ideal	NOUN
iajs-702	60	18	of	of	ADP
iajs-702	60	19	r	r	NOUN
iajs-702	60	20	such	such	ADJ
iajs-702	60	21	that	that	SCONJ
iajs-702	60	22	i	i	PRON
iajs-702	60	23	⊆	⊆	NUM
iajs-702	60	24	annm	annm	NOUN
iajs-702	60	25	.	.	PUNCT
iajs-702	61	1	m	m	PROPN
iajs-702	61	2	is	be	AUX
iajs-702	61	3	maxcs	maxcs	ADJ
iajs-702	61	4	r	r	NOUN
iajs-702	61	5	-	-	PUNCT
iajs-702	61	6	module	module	NOUN
iajs-702	61	7	then	then	ADV
iajs-702	61	8	m	m	VERB
iajs-702	61	9	is	be	AUX
iajs-702	61	10	max	max	PROPN
iajs-702	61	11	-	-	PUNCT
iajs-702	61	12	cs	cs	X
iajs-702	61	13	(	(	PUNCT
iajs-702	61	14	r	r	NOUN
iajs-702	61	15	/	/	SYM
iajs-702	61	16	i)-module	i)-module	NOUN
iajs-702	61	17	and	and	CCONJ
iajs-702	61	18	the	the	DET
iajs-702	61	19	converse	converse	NOUN
iajs-702	61	20	is	be	AUX
iajs-702	61	21	true	true	ADJ
iajs-702	61	22	if	if	SCONJ
iajs-702	61	23	annm	annm	ADJ
iajs-702	61	24	≠	≠	PROPN
iajs-702	61	25	annn	annn	NOUN
iajs-702	61	26	,	,	PUNCT
iajs-702	61	27	for	for	ADP
iajs-702	61	28	all	all	DET
iajs-702	61	29	maximal	maximal	ADJ
iajs-702	61	30	closed	closed	ADJ
iajs-702	61	31	submodule	submodule	NOUN
iajs-702	61	32	n	n	CCONJ
iajs-702	61	33	≨	≨	PROPN
iajs-702	61	34	m.	m.	NOUN
iajs-702	61	35	proof	proof	NOUN
iajs-702	61	36	:	:	PUNCT
iajs-702	61	37	let	let	VERB
iajs-702	61	38	n	n	PRON
iajs-702	61	39	be	be	AUX
iajs-702	61	40	a	a	DET
iajs-702	61	41	maximal	maximal	ADV
iajs-702	61	42	-	-	PUNCT
iajs-702	61	43	closed	closed	ADJ
iajs-702	61	44	(	(	PUNCT
iajs-702	61	45	r	r	NOUN
iajs-702	61	46	/	/	SYM
iajs-702	61	47	i)-submodule	i)-submodule	NOUN
iajs-702	61	48	of	of	ADP
iajs-702	61	49	m	m	NOUN
iajs-702	61	50	and	and	CCONJ
iajs-702	61	51	annr	annr	NOUN
iajs-702	61	52	/	/	SYM
iajs-702	61	53	in	in	ADP
iajs-702	61	54	≠	≠	NOUN
iajs-702	62	1	0r	0r	NOUN
iajs-702	62	2	/	/	SYM
iajs-702	63	1	i	i	NOUN
iajs-702	63	2	=	=	PUNCT
iajs-702	63	3	i.	i.	NOUN
iajs-702	64	1	it	it	PRON
iajs-702	64	2	is	be	AUX
iajs-702	64	3	easy	easy	ADJ
iajs-702	64	4	to	to	PART
iajs-702	64	5	see	see	VERB
iajs-702	64	6	that	that	SCONJ
iajs-702	64	7	n	n	NOUN
iajs-702	64	8	is	be	AUX
iajs-702	64	9	a	a	DET
iajs-702	64	10	maximal	maximal	ADJ
iajs-702	64	11	closed	closed	ADJ
iajs-702	64	12	submodule	submodule	NOUN
iajs-702	64	13	of	of	ADP
iajs-702	64	14	m.	m.	NOUN
iajs-702	64	15	since	since	SCONJ
iajs-702	64	16	annr	annr	NOUN
iajs-702	64	17	/	/	SYM
iajs-702	64	18	in	in	ADP
iajs-702	64	19	≠	≠	PROPN
iajs-702	64	20	i	i	NOUN
iajs-702	64	21	=	=	PUNCT
iajs-702	64	22	0r	0r	NUM
iajs-702	64	23	/	/	SYM
iajs-702	65	1	i	i	PRON
iajs-702	65	2	,	,	PUNCT
iajs-702	65	3	so	so	ADV
iajs-702	65	4	there	there	PRON
iajs-702	65	5	exists	exist	VERB
iajs-702	65	6	r	r	NOUN
iajs-702	65	7	+	+	X
iajs-702	65	8	i	i	PRON
iajs-702	65	9			NOUN
iajs-702	65	10	r	r	X
iajs-702	65	11	/	/	SYM
iajs-702	65	12	i	i	PRON
iajs-702	65	13	with	with	ADP
iajs-702	65	14	ri	ri	PROPN
iajs-702	65	15	such	such	ADJ
iajs-702	65	16	that	that	DET
iajs-702	65	17	r	r	NOUN
iajs-702	66	1	+	+	NUM
iajs-702	67	1	i	i	PRON
iajs-702	67	2			NOUN
iajs-702	67	3	annn	annn	NOUN
iajs-702	67	4	,	,	PUNCT
iajs-702	67	5	hence	hence	ADV
iajs-702	67	6	r	r	NOUN
iajs-702	67	7	≠	≠	PROPN
iajs-702	67	8	0	0	NUM
iajs-702	67	9	and	and	CCONJ
iajs-702	67	10	rn	rn	PROPN
iajs-702	67	11	=	=	NOUN
iajs-702	67	12	0	0	PROPN
iajs-702	67	13	.	.	PUNCT
iajs-702	68	1	thus	thus	ADV
iajs-702	68	2	annrn	annrn	NOUN
iajs-702	68	3	≠	≠	PROPN
iajs-702	68	4	0	0	PUNCT
iajs-702	69	1	and	and	CCONJ
iajs-702	69	2	so	so	ADV
iajs-702	69	3	that	that	SCONJ
iajs-702	69	4	n	n	X
iajs-702	69	5	is	be	AUX
iajs-702	69	6	a	a	DET
iajs-702	69	7	direct	direct	ADJ
iajs-702	69	8	summand	summand	NOUN
iajs-702	69	9	of	of	ADP
iajs-702	69	10	m.	m.	NOUN
iajs-702	69	11	conversely	conversely	ADV
iajs-702	69	12	,	,	PUNCT
iajs-702	69	13	let	let	VERB
iajs-702	69	14	n	n	PRON
iajs-702	69	15	be	be	AUX
iajs-702	69	16	a	a	DET
iajs-702	69	17	maximal	maximal	ADJ
iajs-702	69	18	closed	closed	ADJ
iajs-702	69	19	r	r	NOUN
iajs-702	69	20	-	-	PUNCT
iajs-702	69	21	submodule	submodule	NOUN
iajs-702	69	22	of	of	ADP
iajs-702	69	23	m	m	PROPN
iajs-702	69	24	with	with	ADP
iajs-702	69	25	annn	annn	NOUN
iajs-702	69	26	≠	≠	PROPN
iajs-702	69	27	0	0	NUM
iajs-702	69	28	.	.	PUNCT
iajs-702	70	1	hence	hence	ADV
iajs-702	70	2	n	n	ADV
iajs-702	70	3	is	be	AUX
iajs-702	70	4	a	a	DET
iajs-702	70	5	maximal	maximal	ADJ
iajs-702	70	6	closed	close	VERB
iajs-702	70	7	(	(	PUNCT
iajs-702	70	8	r	r	NOUN
iajs-702	70	9	/	/	SYM
iajs-702	70	10	i)-submodule	i)-submodule	NOUN
iajs-702	70	11	.	.	PUNCT
iajs-702	71	1	now	now	ADV
iajs-702	71	2	,	,	PUNCT
iajs-702	71	3	since	since	SCONJ
iajs-702	71	4	ann	ann	PROPN
iajs-702	71	5	m	m	PROPN
iajs-702	71	6			PROPN
iajs-702	71	7	ann	ann	NOUN
iajs-702	71	8	n	n	CCONJ
iajs-702	71	9	,	,	PUNCT
iajs-702	71	10	there	there	PRON
iajs-702	71	11	exists	exist	VERB
iajs-702	71	12	r	r	NOUN
iajs-702	71	13			PROPN
iajs-702	71	14	ann	ann	PROPN
iajs-702	71	15	n	n	PROPN
iajs-702	71	16	and	and	CCONJ
iajs-702	71	17	r	r	PROPN
iajs-702	71	18			PROPN
iajs-702	71	19	ann	ann	PROPN
iajs-702	71	20	m.	m.	NOUN
iajs-702	71	21	thus	thus	ADV
iajs-702	71	22	r	r	NOUN
iajs-702	71	23			PROPN
iajs-702	71	24	i	i	NOUN
iajs-702	71	25	;	;	PUNCT
iajs-702	71	26	that	that	PRON
iajs-702	71	27	is	be	AUX
iajs-702	71	28	0r	0r	X
iajs-702	72	1	/	/	SYM
iajs-702	73	1	i	i	PRON
iajs-702	73	2	=	=	VERB
iajs-702	74	1	i	i	PRON
iajs-702	74	2	≠	≠	VERB
iajs-702	74	3	r	r	NOUN
iajs-702	75	1	+	+	NOUN
iajs-702	75	2	i	i	PRON
iajs-702	75	3	and	and	CCONJ
iajs-702	75	4	(	(	PUNCT
iajs-702	75	5	r	r	NOUN
iajs-702	75	6	+	+	NOUN
iajs-702	75	7	i)n	i)n	ADJ
iajs-702	75	8	=	=	SYM
iajs-702	75	9	rn	rn	NOUN
iajs-702	75	10	=	=	NOUN
iajs-702	75	11	0	0	PROPN
iajs-702	75	12	.	.	PUNCT
iajs-702	76	1	hence	hence	ADV
iajs-702	76	2	ann	ann	PROPN
iajs-702	76	3	r	r	NOUN
iajs-702	76	4	/	/	SYM
iajs-702	76	5	in	in	ADP
iajs-702	76	6	≠	≠	PROPN
iajs-702	76	7	0r	0r	X
iajs-702	76	8	/	/	SYM
iajs-702	76	9	i.	i.	PROPN
iajs-702	76	10	but	but	CCONJ
iajs-702	76	11	m	m	PROPN
iajs-702	76	12	is	be	AUX
iajs-702	76	13	a	a	DET
iajs-702	76	14	max	max	PROPN
iajs-702	76	15	-	-	PUNCT
iajs-702	76	16	cs	cs	PROPN
iajs-702	76	17	(	(	PUNCT
iajs-702	76	18	r/	r/	ADV
iajs-702	76	19	i)-module	i)-module	NOUN
iajs-702	76	20	,	,	PUNCT
iajs-702	76	21	so	so	CCONJ
iajs-702	76	22	n	n	PROPN
iajs-702	76	23	is	be	AUX
iajs-702	76	24	a	a	DET
iajs-702	76	25	direct	direct	ADJ
iajs-702	76	26	summand	summand	NOUN
iajs-702	76	27	.	.	PUNCT
iajs-702	77	1	1.5	1.5	NUM
iajs-702	77	2	proposition	proposition	NOUN
iajs-702	77	3	:	:	PUNCT
iajs-702	77	4	let	let	VERB
iajs-702	77	5	m	m	PRON
iajs-702	77	6	be	be	AUX
iajs-702	77	7	an	an	DET
iajs-702	77	8	r	r	NOUN
iajs-702	77	9	-	-	PUNCT
iajs-702	77	10	module	module	NOUN
iajs-702	77	11	,	,	PUNCT
iajs-702	77	12	let	let	VERB
iajs-702	77	13	i	i	PRON
iajs-702	77	14	be	be	AUX
iajs-702	77	15	an	an	DET
iajs-702	77	16	ideal	ideal	NOUN
iajs-702	77	17	of	of	ADP
iajs-702	77	18	r	r	NOUN
iajs-702	77	19	such	such	ADJ
iajs-702	77	20	that	that	SCONJ
iajs-702	77	21	i	i	PRON
iajs-702	77	22			PROPN
iajs-702	77	23	ann	ann	PROPN
iajs-702	77	24	m.	m.	NOUN
iajs-702	77	25	then	then	ADV
iajs-702	77	26	m	m	VERB
iajs-702	77	27	is	be	AUX
iajs-702	77	28	min	min	ADJ
iajs-702	77	29	-	-	ADJ
iajs-702	77	30	cs	cs	ADJ
iajs-702	77	31	rmodule	rmodule	NOUN
iajs-702	77	32	if	if	SCONJ
iajs-702	77	33	and	and	CCONJ
iajs-702	77	34	only	only	ADV
iajs-702	77	35	if	if	SCONJ
iajs-702	77	36	m	m	NOUN
iajs-702	77	37	is	be	AUX
iajs-702	77	38	min	min	NOUN
iajs-702	77	39	-	-	ADJ
iajs-702	77	40	cs	cs	X
iajs-702	77	41	(	(	PUNCT
iajs-702	77	42	r/	r/	ADV
iajs-702	77	43	i)-module	i)-module	NOUN
iajs-702	77	44	.	.	PUNCT
iajs-702	78	1	proof	proof	NOUN
iajs-702	78	2	:	:	PUNCT
iajs-702	78	3	it	it	PRON
iajs-702	78	4	is	be	AUX
iajs-702	78	5	straight	straight	ADV
iajs-702	78	6	forward	forward	ADV
iajs-702	78	7	,	,	PUNCT
iajs-702	78	8	so	so	ADV
iajs-702	78	9	it	it	PRON
iajs-702	78	10	is	be	AUX
iajs-702	78	11	omitted	omit	VERB
iajs-702	78	12	.	.	PUNCT
iajs-702	79	1	recall	recall	VERB
iajs-702	79	2	that	that	SCONJ
iajs-702	79	3	an	an	DET
iajs-702	79	4	r	r	NOUN
iajs-702	79	5	-	-	PUNCT
iajs-702	79	6	module	module	NOUN
iajs-702	79	7	m	m	NOUN
iajs-702	79	8	is	be	AUX
iajs-702	79	9	called	call	VERB
iajs-702	79	10	a	a	DET
iajs-702	79	11	uniform	uniform	NOUN
iajs-702	79	12	extending	extend	VERB
iajs-702	79	13	(	(	PUNCT
iajs-702	79	14	or	or	CCONJ
iajs-702	79	15	uniform	uniform	NOUN
iajs-702	79	16	-	-	PUNCT
iajs-702	79	17	cs	cs	NOUN
iajs-702	79	18	)	)	PUNCT
iajs-702	79	19	if	if	SCONJ
iajs-702	79	20	every	every	DET
iajs-702	79	21	uniform	uniform	ADJ
iajs-702	79	22	submodule	submodule	PROPN
iajs-702	79	23	is	be	AUX
iajs-702	79	24	essential	essential	ADJ
iajs-702	79	25	in	in	ADP
iajs-702	79	26	a	a	DET
iajs-702	79	27	direct	direct	ADJ
iajs-702	79	28	summand	summand	NOUN
iajs-702	79	29	.	.	PUNCT
iajs-702	80	1	[	[	X
iajs-702	80	2	1	1	NUM
iajs-702	80	3	,	,	PUNCT
iajs-702	80	4	p.55	p.55	NOUN
iajs-702	80	5	]	]	PUNCT
iajs-702	80	6	.	.	PUNCT
iajs-702	81	1	al	al	PROPN
iajs-702	81	2	-	-	PUNCT
iajs-702	81	3	hazmi	hazmi	NOUN
iajs-702	81	4	in	in	ADP
iajs-702	81	5	[	[	X
iajs-702	81	6	2,p.24	2,p.24	NOUN
iajs-702	81	7	]	]	PUNCT
iajs-702	81	8	,	,	PUNCT
iajs-702	81	9	mentioned	mention	VERB
iajs-702	81	10	that	that	SCONJ
iajs-702	81	11	min	min	PROPN
iajs-702	81	12	-	-	ADJ
iajs-702	81	13	cs	cs	ADJ
iajs-702	81	14	and	and	CCONJ
iajs-702	81	15	uniform	uniform	ADJ
iajs-702	81	16	-	-	PUNCT
iajs-702	81	17	cs	c	NOUN
iajs-702	81	18	are	be	AUX
iajs-702	81	19	equivalent	equivalent	ADJ
iajs-702	81	20	concepts	concept	NOUN
iajs-702	81	21	without	without	ADP
iajs-702	81	22	proof	proof	NOUN
iajs-702	81	23	.	.	PUNCT
iajs-702	82	1	we	we	PRON
iajs-702	82	2	shall	shall	AUX
iajs-702	82	3	prove	prove	VERB
iajs-702	82	4	this	this	DET
iajs-702	82	5	equivalence	equivalence	NOUN
iajs-702	82	6	,	,	PUNCT
iajs-702	82	7	but	but	CCONJ
iajs-702	82	8	first	first	ADV
iajs-702	82	9	we	we	PRON
iajs-702	82	10	need	need	VERB
iajs-702	82	11	the	the	DET
iajs-702	82	12	following	follow	VERB
iajs-702	82	13	lemmas	lemmas	NOUN
iajs-702	82	14	.	.	PUNCT
iajs-702	83	1	1.6	1.6	NUM
iajs-702	83	2	lemma	lemma	PROPN
iajs-702	83	3	:	:	PUNCT
iajs-702	83	4	let	let	VERB
iajs-702	83	5	n	n	PRON
iajs-702	83	6	be	be	AUX
iajs-702	83	7	a	a	DET
iajs-702	83	8	submodule	submodule	NOUN
iajs-702	83	9	of	of	ADP
iajs-702	83	10	an	an	DET
iajs-702	83	11	r	r	NOUN
iajs-702	83	12	-	-	PUNCT
iajs-702	83	13	module	module	NOUN
iajs-702	83	14	m.	m.	NOUN
iajs-702	83	15	n	n	PRON
iajs-702	83	16	is	be	AUX
iajs-702	83	17	minimal	minimal	ADJ
iajs-702	83	18	closed	close	VERB
iajs-702	83	19	if	if	SCONJ
iajs-702	83	20	and	and	CCONJ
iajs-702	83	21	only	only	ADV
iajs-702	83	22	if	if	SCONJ
iajs-702	83	23	n	n	NOUN
iajs-702	83	24	is	be	AUX
iajs-702	83	25	uniform	uniform	ADJ
iajs-702	83	26	-	-	PUNCT
iajs-702	83	27	closed	closed	ADJ
iajs-702	83	28	(	(	PUNCT
iajs-702	83	29	that	that	PRON
iajs-702	83	30	is	be	AUX
iajs-702	83	31	every	every	DET
iajs-702	83	32	closed	closed	ADJ
iajs-702	83	33	submodule	submodule	NOUN
iajs-702	83	34	of	of	ADP
iajs-702	83	35	n	n	PROPN
iajs-702	83	36	is	be	AUX
iajs-702	83	37	essential	essential	ADJ
iajs-702	83	38	in	in	ADP
iajs-702	83	39	n	n	CCONJ
iajs-702	83	40	)	)	PUNCT
iajs-702	83	41	.	.	PUNCT
iajs-702	84	1	proof	proof	NOUN
iajs-702	84	2	:	:	PUNCT
iajs-702	84	3	(	(	PUNCT
iajs-702	84	4			NOUN
iajs-702	84	5	)	)	PUNCT
iajs-702	84	6	it	it	PRON
iajs-702	84	7	is	be	AUX
iajs-702	84	8	enough	enough	ADJ
iajs-702	84	9	to	to	PART
iajs-702	84	10	prove	prove	VERB
iajs-702	84	11	that	that	SCONJ
iajs-702	84	12	n	n	PRON
iajs-702	84	13	is	be	AUX
iajs-702	84	14	uniform	uniform	ADJ
iajs-702	84	15	let	let	VERB
iajs-702	84	16	v	v	NOUN
iajs-702	84	17	,	,	PUNCT
iajs-702	84	18	w	w	PROPN
iajs-702	84	19	be	be	AUX
iajs-702	84	20	two	two	NUM
iajs-702	84	21	nonzero	nonzero	ADJ
iajs-702	84	22	submodules	submodule	NOUN
iajs-702	84	23	of	of	ADP
iajs-702	84	24	n.	n.	PROPN
iajs-702	84	25	suppose	suppose	VERB
iajs-702	84	26	v	v	ADP
iajs-702	84	27			NOUN
iajs-702	84	28	w	w	NOUN
iajs-702	84	29	=	=	SYM
iajs-702	84	30	(	(	PUNCT
iajs-702	84	31	0	0	NUM
iajs-702	84	32	)	)	PUNCT
iajs-702	84	33	.	.	PUNCT
iajs-702	85	1	hence	hence	ADV
iajs-702	85	2	,	,	PUNCT
iajs-702	85	3	there	there	PRON
iajs-702	85	4	exists	exist	VERB
iajs-702	85	5	v	v	NOUN
iajs-702	85	6	'	'	PART
iajs-702	85	7			NOUN
iajs-702	85	8	n	n	CCONJ
iajs-702	85	9	such	such	ADJ
iajs-702	85	10	that	that	DET
iajs-702	85	11	v	v	NOUN
iajs-702	85	12	'	'	PUNCT
iajs-702	85	13	is	be	AUX
iajs-702	85	14	a	a	DET
iajs-702	85	15	relative	relative	ADJ
iajs-702	85	16	complement	complement	NOUN
iajs-702	85	17	of	of	ADP
iajs-702	85	18	v	v	NOUN
iajs-702	85	19	and	and	CCONJ
iajs-702	85	20	hence	hence	ADV
iajs-702	85	21	v	v	NOUN
iajs-702	85	22	'	'	PUNCT
iajs-702	85	23	is	be	AUX
iajs-702	85	24	closed	close	VERB
iajs-702	85	25	in	in	ADP
iajs-702	85	26	n.	n.	NOUN
iajs-702	85	27	since	since	SCONJ
iajs-702	85	28	n	n	NUM
iajs-702	85	29	is	be	AUX
iajs-702	85	30	minimal	minimal	ADJ
iajs-702	85	31	closed	close	VERB
iajs-702	85	32	of	of	ADP
iajs-702	85	33	m	m	PRON
iajs-702	85	34	thus	thus	ADV
iajs-702	85	35	n	n	ADV
iajs-702	85	36	is	be	AUX
iajs-702	85	37	closed	close	VERB
iajs-702	85	38	in	in	ADP
iajs-702	85	39	m	m	PROPN
iajs-702	85	40	,	,	PUNCT
iajs-702	85	41	so	so	ADV
iajs-702	85	42	by	by	ADP
iajs-702	85	43	[	[	X
iajs-702	85	44	4	4	NUM
iajs-702	85	45	,	,	PUNCT
iajs-702	85	46	proposition	proposition	NOUN
iajs-702	85	47	1.5	1.5	NUM
iajs-702	85	48	,	,	PUNCT
iajs-702	85	49	p.18	p.18	PROPN
iajs-702	85	50	]	]	X
iajs-702	85	51	,	,	PUNCT
iajs-702	85	52	v	v	NOUN
iajs-702	85	53	'	'	PUNCT
iajs-702	85	54	is	be	AUX
iajs-702	85	55	closed	close	VERB
iajs-702	85	56	in	in	ADP
iajs-702	85	57	m	m	PROPN
iajs-702	85	58	and	and	CCONJ
iajs-702	85	59	0	0	NUM
iajs-702	85	60	≠v	≠v	NOUN
iajs-702	85	61	'	'	PUNCT
iajs-702	85	62			PROPN
iajs-702	85	63	n.	n.	NOUN
iajs-702	85	64	thus	thus	ADV
iajs-702	85	65	n	n	ADV
iajs-702	85	66	is	be	AUX
iajs-702	85	67	not	not	PART
iajs-702	85	68	minimal	minimal	ADJ
iajs-702	85	69	closed	closed	ADJ
iajs-702	85	70	submodule	submodule	NOUN
iajs-702	85	71	of	of	ADP
iajs-702	85	72	m	m	PROPN
iajs-702	85	73	,	,	PUNCT
iajs-702	85	74	which	which	PRON
iajs-702	85	75	is	be	AUX
iajs-702	85	76	a	a	DET
iajs-702	85	77	contradiction	contradiction	NOUN
iajs-702	85	78	.	.	PUNCT
iajs-702	86	1	therefore	therefore	ADV
iajs-702	86	2	,	,	PUNCT
iajs-702	86	3	n	n	X
iajs-702	86	4	is	be	AUX
iajs-702	86	5	uniform	uniform	ADJ
iajs-702	86	6	-	-	PUNCT
iajs-702	86	7	closed	close	VERB
iajs-702	86	8	submodule	submodule	NOUN
iajs-702	86	9	.	.	PUNCT
iajs-702	87	1	(	(	PUNCT
iajs-702	87	2			NOUN
iajs-702	87	3	)	)	PUNCT
iajs-702	87	4	suppose	suppose	VERB
iajs-702	87	5	that	that	SCONJ
iajs-702	87	6	there	there	PRON
iajs-702	87	7	exists	exist	VERB
iajs-702	87	8	a	a	DET
iajs-702	87	9	closed	closed	ADJ
iajs-702	87	10	submodule	submodule	NOUN
iajs-702	87	11	v	v	NOUN
iajs-702	87	12	of	of	ADP
iajs-702	87	13	m	m	PRON
iajs-702	87	14	such	such	ADJ
iajs-702	87	15	that	that	DET
iajs-702	87	16	v	v	NOUN
iajs-702	87	17			PROPN
iajs-702	87	18	n.	n.	NOUN
iajs-702	87	19	but	but	CCONJ
iajs-702	87	20	n	n	PROPN
iajs-702	87	21	is	be	AUX
iajs-702	87	22	uniform	uniform	ADJ
iajs-702	87	23	.	.	PUNCT
iajs-702	88	1	so	so	ADV
iajs-702	88	2	v	v	ADP
iajs-702	88	3			NUM
iajs-702	88	4	e	e	NOUN
iajs-702	88	5	n.	n.	NOUN
iajs-702	88	6	hence	hence	ADV
iajs-702	88	7	v	v	AUX
iajs-702	88	8	=	=	SYM
iajs-702	88	9	n	n	CCONJ
iajs-702	88	10	,	,	PUNCT
iajs-702	88	11	since	since	SCONJ
iajs-702	88	12	v	v	NOUN
iajs-702	88	13	is	be	AUX
iajs-702	88	14	closed	closed	ADJ
iajs-702	88	15	.	.	PUNCT
iajs-702	89	1	thus	thus	ADV
iajs-702	89	2	n	n	ADV
iajs-702	89	3	is	be	AUX
iajs-702	89	4	minimal	minimal	ADJ
iajs-702	89	5	closed	closed	ADJ
iajs-702	89	6	.	.	PUNCT
iajs-702	90	1	1.7	1.7	NUM
iajs-702	90	2	lemma	lemma	PROPN
iajs-702	90	3	:	:	PUNCT
iajs-702	90	4	if	if	SCONJ
iajs-702	90	5	u	u	NOUN
iajs-702	90	6	is	be	AUX
iajs-702	90	7	a	a	DET
iajs-702	90	8	uniform	uniform	ADJ
iajs-702	90	9	submodule	submodule	NOUN
iajs-702	90	10	of	of	ADP
iajs-702	90	11	m	m	PRON
iajs-702	90	12	such	such	ADJ
iajs-702	90	13	that	that	SCONJ
iajs-702	90	14	u	u	PROPN
iajs-702	90	15	e	e	PRON
iajs-702	90	16	k	k	NOUN
iajs-702	90	17			NUM
iajs-702	90	18	m.	m.	NOUN
iajs-702	90	19	then	then	ADV
iajs-702	90	20	k	k	PROPN
iajs-702	90	21	is	be	AUX
iajs-702	90	22	uniform	uniform	ADJ
iajs-702	90	23	.	.	PUNCT
iajs-702	91	1	proof	proof	NOUN
iajs-702	91	2	:	:	PUNCT
iajs-702	91	3	مجلة	مجلة	NOUN
iajs-702	91	4	إبن	إبن	VERB
iajs-702	91	5	الھیثم	الھیثم	NOUN
iajs-702	91	6	للعلوم	للعلوم	NOUN
iajs-702	91	7	الصرفة	الصرفة	NOUN
iajs-702	91	8	و	و	PRON
iajs-702	91	9	التطبیقیة	التطبیقیة	PROPN
iajs-702	91	10	2012	2012	NUM
iajs-702	91	11	السنة	السنة	NOUN
iajs-702	92	1	25	25	NUM
iajs-702	92	2	المجلد	المجلد	NOUN
iajs-702	92	3	1	1	NUM
iajs-702	92	4	العدد	العدد	PROPN
iajs-702	92	5	ibn	ibn	PROPN
iajs-702	92	6	al	al	PROPN
iajs-702	92	7	-	-	PUNCT
iajs-702	92	8	haitham	haitham	PROPN
iajs-702	92	9	journal	journal	PROPN
iajs-702	92	10	for	for	ADP
iajs-702	92	11	pure	pure	ADJ
iajs-702	92	12	and	and	CCONJ
iajs-702	92	13	applied	apply	VERB
iajs-702	92	14	science	science	NOUN
iajs-702	92	15	no	no	NOUN
iajs-702	92	16	.	.	NOUN
iajs-702	92	17	1	1	NUM
iajs-702	92	18	vol	vol	NOUN
iajs-702	92	19	.	.	PUNCT
iajs-702	93	1	25	25	NUM
iajs-702	93	2	year	year	NOUN
iajs-702	93	3	2012	2012	NUM
iajs-702	93	4	let	let	VERB
iajs-702	93	5	v	v	NOUN
iajs-702	93	6	and	and	CCONJ
iajs-702	93	7	w	w	NOUN
iajs-702	93	8	be	be	AUX
iajs-702	93	9	two	two	NUM
iajs-702	93	10	nonzero	nonzero	ADJ
iajs-702	93	11	submodules	submodule	NOUN
iajs-702	93	12	of	of	ADP
iajs-702	93	13	k.	k.	PROPN
iajs-702	93	14	since	since	SCONJ
iajs-702	93	15	u	u	PRON
iajs-702	93	16			PROPN
iajs-702	93	17	e	e	X
iajs-702	93	18	k	k	X
iajs-702	93	19	,	,	PUNCT
iajs-702	93	20	u	u	NOUN
iajs-702	93	21			PUNCT
iajs-702	93	22	w	w	PROPN
iajs-702	93	23	≠	≠	PROPN
iajs-702	93	24	(	(	PUNCT
iajs-702	93	25	0	0	NUM
iajs-702	93	26	)	)	PUNCT
iajs-702	93	27	and	and	CCONJ
iajs-702	93	28	u	u	NOUN
iajs-702	93	29			NOUN
iajs-702	93	30	v	v	ADP
iajs-702	93	31	≠	≠	PROPN
iajs-702	93	32	(	(	PUNCT
iajs-702	93	33	0	0	NUM
iajs-702	93	34	)	)	PUNCT
iajs-702	93	35	.	.	PUNCT
iajs-702	94	1	but	but	CCONJ
iajs-702	94	2	u	u	NOUN
iajs-702	94	3	is	be	AUX
iajs-702	94	4	uniform	uniform	ADJ
iajs-702	94	5	submodule	submodule	NOUN
iajs-702	94	6	of	of	ADP
iajs-702	94	7	m.	m.	NOUN
iajs-702	95	1	so	so	CCONJ
iajs-702	95	2	(	(	PUNCT
iajs-702	95	3	u	u	NOUN
iajs-702	95	4			NOUN
iajs-702	95	5	v	v	NOUN
iajs-702	95	6	)	)	PUNCT
iajs-702	95	7			NOUN
iajs-702	95	8	(	(	PUNCT
iajs-702	95	9	u	u	NOUN
iajs-702	95	10			PROPN
iajs-702	95	11	w	w	NOUN
iajs-702	95	12	)	)	PUNCT
iajs-702	95	13	≠	≠	PROPN
iajs-702	95	14	(	(	PUNCT
iajs-702	95	15	0	0	NUM
iajs-702	95	16	)	)	PUNCT
iajs-702	95	17	.	.	PUNCT
iajs-702	96	1	hence	hence	ADV
iajs-702	96	2	u	u	NOUN
iajs-702	96	3			PUNCT
iajs-702	96	4	(	(	PUNCT
iajs-702	96	5	v	v	X
iajs-702	96	6			PROPN
iajs-702	96	7	w	w	NOUN
iajs-702	96	8	)	)	PUNCT
iajs-702	96	9	≠	≠	PROPN
iajs-702	96	10	(	(	PUNCT
iajs-702	96	11	0	0	NUM
iajs-702	96	12	)	)	PUNCT
iajs-702	96	13	.	.	PUNCT
iajs-702	97	1	thus	thus	ADV
iajs-702	97	2	v	v	ADP
iajs-702	97	3			X
iajs-702	97	4	w	w	NOUN
iajs-702	97	5	≠	≠	PROPN
iajs-702	97	6	(	(	PUNCT
iajs-702	97	7	0	0	NUM
iajs-702	97	8	)	)	PUNCT
iajs-702	97	9	;	;	PUNCT
iajs-702	97	10	that	that	PRON
iajs-702	97	11	is	is	ADV
iajs-702	97	12	k	k	PROPN
iajs-702	97	13	is	be	AUX
iajs-702	97	14	uniform	uniform	ADJ
iajs-702	97	15	.	.	PUNCT
iajs-702	98	1	1.8	1.8	NUM
iajs-702	98	2	lemma	lemma	PROPN
iajs-702	98	3	:	:	PUNCT
iajs-702	98	4	let	let	VERB
iajs-702	98	5	u	u	PRON
iajs-702	98	6	be	be	AUX
iajs-702	98	7	a	a	DET
iajs-702	98	8	submodule	submodule	NOUN
iajs-702	98	9	of	of	ADP
iajs-702	98	10	an	an	DET
iajs-702	98	11	r	r	NOUN
iajs-702	98	12	-	-	PUNCT
iajs-702	98	13	module	module	NOUN
iajs-702	98	14	m.	m.	NOUN
iajs-702	98	15	then	then	ADV
iajs-702	98	16	u	u	NOUN
iajs-702	98	17	is	be	AUX
iajs-702	98	18	uniform	uniform	NOUN
iajs-702	98	19	closed	close	VERB
iajs-702	98	20	if	if	SCONJ
iajs-702	98	21	and	and	CCONJ
iajs-702	98	22	only	only	ADV
iajs-702	98	23	if	if	SCONJ
iajs-702	98	24	u	u	NOUN
iajs-702	98	25	is	be	AUX
iajs-702	98	26	maximal	maximal	ADJ
iajs-702	98	27	uniform	uniform	NOUN
iajs-702	98	28	;	;	PUNCT
iajs-702	98	29	that	that	PRON
iajs-702	98	30	is	be	AUX
iajs-702	98	31	u	u	NOUN
iajs-702	98	32	is	be	AUX
iajs-702	98	33	maximal	maximal	ADJ
iajs-702	98	34	in	in	ADP
iajs-702	98	35	the	the	DET
iajs-702	98	36	collection	collection	NOUN
iajs-702	98	37	of	of	ADP
iajs-702	98	38	uniform	uniform	ADJ
iajs-702	98	39	submodules	submodule	NOUN
iajs-702	98	40	of	of	ADP
iajs-702	98	41	m	m	PROPN
iajs-702	98	42	.	.	PUNCT
iajs-702	99	1	proof	proof	NOUN
iajs-702	99	2	:	:	PUNCT
iajs-702	99	3	(	(	PUNCT
iajs-702	99	4			NOUN
iajs-702	99	5	)	)	PUNCT
iajs-702	99	6	suppose	suppose	VERB
iajs-702	99	7	there	there	PRON
iajs-702	99	8	exists	exist	VERB
iajs-702	99	9	a	a	DET
iajs-702	99	10	uniform	uniform	ADJ
iajs-702	99	11	submodule	submodule	NOUN
iajs-702	99	12	v	v	NOUN
iajs-702	99	13	of	of	ADP
iajs-702	99	14	m	m	PRON
iajs-702	99	15	such	such	ADJ
iajs-702	99	16	that	that	SCONJ
iajs-702	99	17	u	u	PRON
iajs-702	99	18			NOUN
iajs-702	99	19	v.	v.	CCONJ
iajs-702	99	20	since	since	SCONJ
iajs-702	99	21	v	v	NOUN
iajs-702	99	22	is	be	AUX
iajs-702	99	23	uniform	uniform	ADJ
iajs-702	99	24	so	so	SCONJ
iajs-702	99	25	u	u	NOUN
iajs-702	99	26			NUM
iajs-702	99	27	e	e	X
iajs-702	99	28	v.	v.	NOUN
iajs-702	100	1	but	but	CCONJ
iajs-702	100	2	u	u	NOUN
iajs-702	100	3	is	be	AUX
iajs-702	100	4	closed	close	VERB
iajs-702	101	1	so	so	SCONJ
iajs-702	101	2	u	u	X
iajs-702	101	3	=	=	PROPN
iajs-702	101	4	v.	v.	ADP
iajs-702	101	5	thus	thus	ADV
iajs-702	101	6	u	u	NOUN
iajs-702	101	7	is	be	AUX
iajs-702	101	8	a	a	DET
iajs-702	101	9	maximal	maximal	ADJ
iajs-702	101	10	-	-	PUNCT
iajs-702	101	11	uniform	uniform	NOUN
iajs-702	101	12	submodule	submodule	NOUN
iajs-702	101	13	.	.	PUNCT
iajs-702	102	1	(	(	PUNCT
iajs-702	102	2			NOUN
iajs-702	102	3	)	)	PUNCT
iajs-702	102	4	it	it	PRON
iajs-702	102	5	is	be	AUX
iajs-702	102	6	enough	enough	ADJ
iajs-702	102	7	to	to	PART
iajs-702	102	8	show	show	VERB
iajs-702	102	9	that	that	SCONJ
iajs-702	102	10	u	u	PRON
iajs-702	102	11	is	be	AUX
iajs-702	102	12	closed	closed	ADJ
iajs-702	102	13	.	.	PUNCT
iajs-702	103	1	suppose	suppose	VERB
iajs-702	103	2	there	there	PRON
iajs-702	103	3	exists	exist	VERB
iajs-702	103	4	v	v	ADP
iajs-702	103	5			NOUN
iajs-702	103	6	m	m	VERB
iajs-702	103	7	such	such	ADJ
iajs-702	103	8	that	that	SCONJ
iajs-702	103	9	u	u	PRON
iajs-702	103	10			NOUN
iajs-702	103	11	e	e	X
iajs-702	103	12	v.	v.	CCONJ
iajs-702	103	13	hence	hence	ADV
iajs-702	103	14	by	by	ADP
iajs-702	103	15	lemma1.7	lemma1.7	PROPN
iajs-702	103	16	,	,	PUNCT
iajs-702	103	17	v	v	NOUN
iajs-702	103	18	is	be	AUX
iajs-702	103	19	uniform	uniform	ADJ
iajs-702	103	20	.	.	PUNCT
iajs-702	104	1	thus	thus	ADV
iajs-702	104	2	u	u	X
iajs-702	104	3	=	=	PROPN
iajs-702	104	4	v	v	NOUN
iajs-702	104	5	,	,	PUNCT
iajs-702	104	6	since	since	SCONJ
iajs-702	104	7	u	u	NOUN
iajs-702	104	8	is	be	AUX
iajs-702	104	9	maximal	maximal	ADJ
iajs-702	104	10	uniform	uniform	NOUN
iajs-702	104	11	,	,	PUNCT
iajs-702	104	12	so	so	SCONJ
iajs-702	104	13	that	that	SCONJ
iajs-702	104	14	u	u	NOUN
iajs-702	104	15	is	be	AUX
iajs-702	104	16	closed	closed	ADJ
iajs-702	104	17	.	.	PUNCT
iajs-702	105	1	by	by	ADP
iajs-702	105	2	combining	combine	VERB
iajs-702	105	3	lemma	lemma	PROPN
iajs-702	105	4	1.6	1.6	NUM
iajs-702	105	5	and	and	CCONJ
iajs-702	105	6	lemma	lemma	PROPN
iajs-702	105	7	1.8	1.8	NUM
iajs-702	105	8	,	,	PUNCT
iajs-702	105	9	we	we	PRON
iajs-702	105	10	have	have	VERB
iajs-702	105	11	the	the	DET
iajs-702	105	12	following	following	NOUN
iajs-702	105	13	:	:	PUNCT
iajs-702	105	14	1.9	1.9	NUM
iajs-702	105	15	corollary	corollary	NOUN
iajs-702	105	16	:	:	PUNCT
iajs-702	105	17	let	let	VERB
iajs-702	105	18	u	u	PRON
iajs-702	105	19	be	be	AUX
iajs-702	105	20	a	a	DET
iajs-702	105	21	submodule	submodule	NOUN
iajs-702	105	22	of	of	ADP
iajs-702	105	23	an	an	DET
iajs-702	105	24	r	r	NOUN
iajs-702	105	25	-	-	PUNCT
iajs-702	105	26	module	module	NOUN
iajs-702	105	27	.	.	PUNCT
iajs-702	106	1	then	then	ADV
iajs-702	106	2	the	the	DET
iajs-702	106	3	following	follow	VERB
iajs-702	106	4	are	be	AUX
iajs-702	106	5	equivalent	equivalent	ADJ
iajs-702	106	6	:	:	PUNCT
iajs-702	106	7	(	(	PUNCT
iajs-702	106	8	1	1	X
iajs-702	106	9	)	)	PUNCT
iajs-702	106	10	u	u	NOUN
iajs-702	106	11	is	be	AUX
iajs-702	106	12	minimal	minimal	ADJ
iajs-702	106	13	-	-	PUNCT
iajs-702	106	14	closed	closed	ADJ
iajs-702	106	15	.	.	PUNCT
iajs-702	107	1	(	(	PUNCT
iajs-702	107	2	2	2	X
iajs-702	107	3	)	)	PUNCT
iajs-702	107	4	u	u	NOUN
iajs-702	107	5	is	be	AUX
iajs-702	107	6	uniform	uniform	ADJ
iajs-702	107	7	-	-	PUNCT
iajs-702	107	8	closed	closed	ADJ
iajs-702	107	9	.	.	PUNCT
iajs-702	108	1	(	(	PUNCT
iajs-702	108	2	3	3	X
iajs-702	108	3	)	)	PUNCT
iajs-702	108	4	u	u	NOUN
iajs-702	108	5	is	be	AUX
iajs-702	108	6	maximal	maximal	ADJ
iajs-702	108	7	-	-	PUNCT
iajs-702	108	8	uniform	uniform	NOUN
iajs-702	108	9	.	.	PUNCT
iajs-702	109	1	now	now	ADV
iajs-702	109	2	,	,	PUNCT
iajs-702	109	3	we	we	PRON
iajs-702	109	4	can	can	AUX
iajs-702	109	5	give	give	VERB
iajs-702	109	6	the	the	DET
iajs-702	109	7	proof	proof	NOUN
iajs-702	109	8	of	of	ADP
iajs-702	109	9	the	the	DET
iajs-702	109	10	following	follow	VERB
iajs-702	109	11	result	result	NOUN
iajs-702	109	12	.	.	PUNCT
iajs-702	110	1	1.10	1.10	NUM
iajs-702	110	2	proposition	proposition	NOUN
iajs-702	110	3	:	:	PUNCT
iajs-702	111	1	[	[	X
iajs-702	111	2	2	2	NUM
iajs-702	111	3	,	,	PUNCT
iajs-702	111	4	p.24	p.24	PROPN
iajs-702	111	5	]	]	X
iajs-702	111	6	let	let	AUX
iajs-702	111	7	m	m	PRON
iajs-702	111	8	be	be	AUX
iajs-702	111	9	an	an	DET
iajs-702	111	10	r	r	NOUN
iajs-702	111	11	-	-	PUNCT
iajs-702	111	12	module	module	NOUN
iajs-702	111	13	.	.	PUNCT
iajs-702	112	1	m	m	PROPN
iajs-702	112	2	is	be	AUX
iajs-702	112	3	uniform	uniform	ADJ
iajs-702	112	4	-	-	PUNCT
iajs-702	112	5	cs	cs	NOUN
iajs-702	112	6	if	if	SCONJ
iajs-702	113	1	and	and	CCONJ
iajs-702	113	2	only	only	ADV
iajs-702	113	3	if	if	SCONJ
iajs-702	113	4	m	m	NOUN
iajs-702	113	5	is	be	AUX
iajs-702	113	6	min	min	NOUN
iajs-702	113	7	-	-	ADJ
iajs-702	113	8	cs	cs	ADJ
iajs-702	113	9	.	.	PROPN
iajs-702	113	10	proof	proof	NOUN
iajs-702	113	11	:	:	PUNCT
iajs-702	113	12	(	(	PUNCT
iajs-702	113	13			NOUN
iajs-702	113	14	)	)	PUNCT
iajs-702	113	15	let	let	VERB
iajs-702	113	16	u	u	PRON
iajs-702	113	17	be	be	AUX
iajs-702	113	18	a	a	DET
iajs-702	113	19	maximal	maximal	ADJ
iajs-702	113	20	uniform	uniform	ADJ
iajs-702	113	21	submodule	submodule	NOUN
iajs-702	113	22	of	of	ADP
iajs-702	113	23	m.	m.	NOUN
iajs-702	113	24	since	since	SCONJ
iajs-702	113	25	m	m	PROPN
iajs-702	113	26	is	be	AUX
iajs-702	113	27	uniform	uniform	ADJ
iajs-702	113	28	-	-	PUNCT
iajs-702	113	29	cs	cs	ADJ
iajs-702	113	30	,	,	PUNCT
iajs-702	113	31	so	so	CCONJ
iajs-702	113	32	u	u	NOUN
iajs-702	113	33	is	be	AUX
iajs-702	113	34	essential	essential	ADJ
iajs-702	113	35	in	in	ADP
iajs-702	113	36	a	a	DET
iajs-702	113	37	direct	direct	ADJ
iajs-702	113	38	summand	summand	NOUN
iajs-702	114	1	v.	v.	CCONJ
iajs-702	114	2	then	then	ADV
iajs-702	114	3	by	by	ADP
iajs-702	114	4	lemma	lemma	PROPN
iajs-702	114	5	1.7	1.7	NUM
iajs-702	114	6	,	,	PUNCT
iajs-702	114	7	v	v	NOUN
iajs-702	114	8	is	be	AUX
iajs-702	114	9	uniform	uniform	ADJ
iajs-702	114	10	.	.	PUNCT
iajs-702	115	1	hence	hence	ADV
iajs-702	115	2	u	u	NOUN
iajs-702	116	1	=	=	X
iajs-702	116	2	v	v	NOUN
iajs-702	116	3	since	since	SCONJ
iajs-702	116	4	u	u	NOUN
iajs-702	116	5	is	be	AUX
iajs-702	116	6	a	a	DET
iajs-702	116	7	maximal	maximal	ADJ
iajs-702	116	8	uniform	uniform	ADJ
iajs-702	116	9	submodule	submodule	NOUN
iajs-702	116	10	.	.	PUNCT
iajs-702	117	1	thus	thus	ADV
iajs-702	117	2	u	u	PRON
iajs-702	117	3	is	be	AUX
iajs-702	117	4	a	a	DET
iajs-702	117	5	direct	direct	ADJ
iajs-702	117	6	summand	summand	NOUN
iajs-702	117	7	of	of	ADP
iajs-702	117	8	m	m	PROPN
iajs-702	117	9	,	,	PUNCT
iajs-702	117	10	it	it	PRON
iajs-702	117	11	follows	follow	VERB
iajs-702	117	12	that	that	SCONJ
iajs-702	117	13	every	every	DET
iajs-702	117	14	minimal	minimal	ADJ
iajs-702	117	15	-	-	PUNCT
iajs-702	117	16	closed	closed	ADJ
iajs-702	117	17	is	be	AUX
iajs-702	117	18	a	a	DET
iajs-702	117	19	direct	direct	ADJ
iajs-702	117	20	summand	summand	NOUN
iajs-702	117	21	by	by	ADP
iajs-702	117	22	lemma	lemma	PROPN
iajs-702	117	23	1.6	1.6	NUM
iajs-702	117	24	.	.	PUNCT
iajs-702	118	1	so	so	SCONJ
iajs-702	118	2	that	that	SCONJ
iajs-702	118	3	m	m	NOUN
iajs-702	118	4	is	be	AUX
iajs-702	118	5	a	a	DET
iajs-702	118	6	min	min	PROPN
iajs-702	118	7	-	-	ADJ
iajs-702	118	8	cs	cs	ADJ
iajs-702	118	9	module	module	NOUN
iajs-702	118	10	.	.	PUNCT
iajs-702	119	1	(	(	PUNCT
iajs-702	119	2			NOUN
iajs-702	119	3	)	)	PUNCT
iajs-702	119	4	let	let	VERB
iajs-702	119	5	u	u	PRON
iajs-702	119	6	be	be	AUX
iajs-702	119	7	a	a	DET
iajs-702	119	8	uniform	uniform	ADJ
iajs-702	119	9	submodule	submodule	NOUN
iajs-702	119	10	of	of	ADP
iajs-702	119	11	m.	m.	NOUN
iajs-702	119	12	by	by	ADP
iajs-702	119	13	[	[	X
iajs-702	119	14	4	4	NUM
iajs-702	119	15	,	,	PUNCT
iajs-702	119	16	exc.13	exc.13	NOUN
iajs-702	119	17	,	,	PUNCT
iajs-702	119	18	p.20	p.20	PROPN
iajs-702	119	19	]	]	X
iajs-702	119	20	,	,	PUNCT
iajs-702	119	21	there	there	PRON
iajs-702	119	22	exists	exist	VERB
iajs-702	119	23	a	a	DET
iajs-702	119	24	closed	closed	ADJ
iajs-702	119	25	submodule	submodule	NOUN
iajs-702	119	26	v	v	NOUN
iajs-702	119	27	of	of	ADP
iajs-702	119	28	m	m	PRON
iajs-702	119	29	such	such	ADJ
iajs-702	119	30	that	that	SCONJ
iajs-702	119	31	u	u	PRON
iajs-702	119	32			NOUN
iajs-702	119	33	e	e	X
iajs-702	119	34	v.	v.	CCONJ
iajs-702	119	35	hence	hence	ADV
iajs-702	119	36	by	by	ADP
iajs-702	119	37	lemma	lemma	PROPN
iajs-702	119	38	1.7	1.7	NUM
iajs-702	119	39	,	,	PUNCT
iajs-702	119	40	v	v	NOUN
iajs-702	119	41	is	be	AUX
iajs-702	119	42	uniform	uniform	ADJ
iajs-702	119	43	.	.	PUNCT
iajs-702	120	1	thus	thus	ADV
iajs-702	120	2	v	v	NOUN
iajs-702	120	3	is	be	AUX
iajs-702	120	4	a	a	DET
iajs-702	120	5	closed	closed	ADJ
iajs-702	120	6	-	-	PUNCT
iajs-702	120	7	uniform	uniform	NOUN
iajs-702	120	8	submodule	submodule	NOUN
iajs-702	120	9	of	of	ADP
iajs-702	120	10	m	m	PROPN
iajs-702	120	11	,	,	PUNCT
iajs-702	120	12	and	and	CCONJ
iajs-702	120	13	hence	hence	ADV
iajs-702	120	14	by	by	ADP
iajs-702	120	15	lemma	lemma	PROPN
iajs-702	120	16	1.6	1.6	NUM
iajs-702	120	17	,	,	PUNCT
iajs-702	120	18	v	v	NOUN
iajs-702	120	19	is	be	AUX
iajs-702	120	20	minimal	minimal	ADJ
iajs-702	120	21	-	-	PUNCT
iajs-702	120	22	closed	closed	ADJ
iajs-702	120	23	.	.	PUNCT
iajs-702	121	1	so	so	ADV
iajs-702	121	2	that	that	PRON
iajs-702	121	3	v	v	NOUN
iajs-702	121	4	is	be	AUX
iajs-702	121	5	a	a	DET
iajs-702	121	6	direct	direct	ADJ
iajs-702	121	7	summand	summand	NOUN
iajs-702	121	8	of	of	ADP
iajs-702	121	9	m	m	PROPN
iajs-702	121	10	,	,	PUNCT
iajs-702	121	11	since	since	SCONJ
iajs-702	121	12	m	m	PROPN
iajs-702	121	13	is	be	AUX
iajs-702	121	14	min	min	ADJ
iajs-702	121	15	-	-	ADJ
iajs-702	121	16	cs	cs	ADJ
iajs-702	121	17	module	module	NOUN
iajs-702	121	18	.	.	PUNCT
iajs-702	122	1	it	it	PRON
iajs-702	122	2	follows	follow	VERB
iajs-702	122	3	that	that	SCONJ
iajs-702	122	4	u	u	PRON
iajs-702	122	5	is	be	AUX
iajs-702	122	6	essential	essential	ADJ
iajs-702	122	7	in	in	ADP
iajs-702	122	8	a	a	DET
iajs-702	122	9	direct	direct	ADJ
iajs-702	122	10	summand	summand	NOUN
iajs-702	122	11	.	.	PUNCT
iajs-702	123	1	thus	thus	ADV
iajs-702	123	2	m	m	PROPN
iajs-702	123	3	is	be	AUX
iajs-702	123	4	uniform	uniform	ADJ
iajs-702	123	5	-	-	PUNCT
iajs-702	123	6	cs	cs	NOUN
iajs-702	123	7	.	.	PUNCT
iajs-702	124	1	the	the	DET
iajs-702	124	2	following	following	ADJ
iajs-702	124	3	result	result	NOUN
iajs-702	124	4	is	be	AUX
iajs-702	124	5	given	give	VERB
iajs-702	124	6	in	in	ADP
iajs-702	124	7	[	[	PUNCT
iajs-702	124	8	2	2	NUM
iajs-702	124	9	,	,	PUNCT
iajs-702	124	10	lemma	lemma	PROPN
iajs-702	124	11	3.1.1	3.1.1	NUM
iajs-702	124	12	,	,	PUNCT
iajs-702	124	13	p.45	p.45	ADP
iajs-702	124	14	]	]	PUNCT
iajs-702	124	15	,	,	PUNCT
iajs-702	124	16	we	we	PRON
iajs-702	124	17	give	give	VERB
iajs-702	124	18	the	the	DET
iajs-702	124	19	details	detail	NOUN
iajs-702	124	20	of	of	ADP
iajs-702	124	21	proof	proof	NOUN
iajs-702	124	22	.	.	PUNCT
iajs-702	125	1	1.11	1.11	NUM
iajs-702	125	2	proposition	proposition	NOUN
iajs-702	125	3	:	:	PUNCT
iajs-702	125	4	let	let	VERB
iajs-702	125	5	m	m	PRON
iajs-702	125	6	be	be	AUX
iajs-702	125	7	an	an	DET
iajs-702	125	8	indecomposable	indecomposable	ADJ
iajs-702	125	9	r	r	NOUN
iajs-702	125	10	-	-	PUNCT
iajs-702	125	11	module	module	NOUN
iajs-702	125	12	with	with	ADP
iajs-702	125	13	a	a	DET
iajs-702	125	14	uniform	uniform	ADJ
iajs-702	125	15	submodule	submodule	NOUN
iajs-702	125	16	.	.	PUNCT
iajs-702	126	1	if	if	SCONJ
iajs-702	126	2	m	m	NOUN
iajs-702	126	3	is	be	AUX
iajs-702	126	4	a	a	DET
iajs-702	126	5	min	min	PROPN
iajs-702	126	6	-	-	ADJ
iajs-702	126	7	cs	cs	ADJ
iajs-702	126	8	module	module	NOUN
iajs-702	126	9	,	,	PUNCT
iajs-702	126	10	then	then	ADV
iajs-702	126	11	m	m	NOUN
iajs-702	126	12	is	be	AUX
iajs-702	126	13	uniform	uniform	ADJ
iajs-702	126	14	.	.	PUNCT
iajs-702	127	1	proof	proof	NOUN
iajs-702	127	2	:	:	PUNCT
iajs-702	127	3	مجلة	مجلة	NOUN
iajs-702	127	4	إبن	إبن	VERB
iajs-702	127	5	الھیثم	الھیثم	NOUN
iajs-702	127	6	للعلوم	للعلوم	NOUN
iajs-702	127	7	الصرفة	الصرفة	NOUN
iajs-702	127	8	و	و	PRON
iajs-702	127	9	التطبیقیة	التطبیقیة	PROPN
iajs-702	127	10	2012	2012	NUM
iajs-702	127	11	السنة	السنة	NOUN
iajs-702	128	1	25	25	NUM
iajs-702	128	2	المجلد	المجلد	NOUN
iajs-702	128	3	1	1	NUM
iajs-702	128	4	العدد	العدد	PROPN
iajs-702	128	5	ibn	ibn	PROPN
iajs-702	128	6	al	al	PROPN
iajs-702	128	7	-	-	PUNCT
iajs-702	128	8	haitham	haitham	PROPN
iajs-702	128	9	journal	journal	PROPN
iajs-702	128	10	for	for	ADP
iajs-702	128	11	pure	pure	ADJ
iajs-702	128	12	and	and	CCONJ
iajs-702	128	13	applied	apply	VERB
iajs-702	128	14	science	science	NOUN
iajs-702	128	15	no	no	NOUN
iajs-702	128	16	.	.	NOUN
iajs-702	128	17	1	1	NUM
iajs-702	128	18	vol	vol	NOUN
iajs-702	128	19	.	.	PUNCT
iajs-702	129	1	25	25	NUM
iajs-702	129	2	year	year	NOUN
iajs-702	129	3	2012	2012	NUM
iajs-702	129	4	by	by	ADP
iajs-702	129	5	hypothesis	hypothesis	NOUN
iajs-702	129	6	m	m	AUX
iajs-702	129	7	has	have	VERB
iajs-702	129	8	a	a	DET
iajs-702	129	9	uniform	uniform	ADJ
iajs-702	129	10	submodule	submodule	NOUN
iajs-702	129	11	,	,	PUNCT
iajs-702	129	12	say	say	VERB
iajs-702	129	13	u.	u.	NOUN
iajs-702	129	14	by	by	ADP
iajs-702	129	15	[	[	X
iajs-702	129	16	4	4	NUM
iajs-702	129	17	,	,	PUNCT
iajs-702	129	18	exc.13	exc.13	NOUN
iajs-702	129	19	,	,	PUNCT
iajs-702	129	20	p.20	p.20	PROPN
iajs-702	129	21	]	]	X
iajs-702	129	22	,	,	PUNCT
iajs-702	129	23	there	there	PRON
iajs-702	129	24	exists	exist	VERB
iajs-702	129	25	a	a	DET
iajs-702	129	26	closed	closed	ADJ
iajs-702	129	27	submodule	submodule	NOUN
iajs-702	129	28	k	k	PROPN
iajs-702	129	29	of	of	ADP
iajs-702	129	30	m	m	PRON
iajs-702	129	31	such	such	ADJ
iajs-702	129	32	that	that	SCONJ
iajs-702	129	33	u	u	PRON
iajs-702	129	34			NUM
iajs-702	129	35	e	e	X
iajs-702	129	36	k.	k.	NOUN
iajs-702	129	37	hence	hence	ADV
iajs-702	129	38	by	by	ADP
iajs-702	129	39	lemma	lemma	PROPN
iajs-702	129	40	1.7	1.7	NUM
iajs-702	129	41	,	,	PUNCT
iajs-702	129	42	k	k	PROPN
iajs-702	129	43	is	be	AUX
iajs-702	129	44	a	a	DET
iajs-702	129	45	uniform	uniform	ADJ
iajs-702	129	46	submodule	submodule	NOUN
iajs-702	129	47	of	of	ADP
iajs-702	129	48	m.	m.	NOUN
iajs-702	129	49	let	let	VERB
iajs-702	129	50	c	c	NOUN
iajs-702	129	51	=	=	PRON
iajs-702	129	52	{	{	PUNCT
iajs-702	129	53	k	k	NOUN
iajs-702	129	54	:	:	PUNCT
iajs-702	129	55	k	k	X
iajs-702	129	56	is	be	AUX
iajs-702	129	57	a	a	DET
iajs-702	129	58	uniform	uniform	ADJ
iajs-702	129	59	submodule	submodule	NOUN
iajs-702	129	60	of	of	ADP
iajs-702	129	61	m	m	PROPN
iajs-702	129	62	and	and	CCONJ
iajs-702	129	63	u	u	NOUN
iajs-702	129	64			PROPN
iajs-702	129	65	k	k	PROPN
iajs-702	129	66	}	}	PUNCT
iajs-702	129	67	.	.	PUNCT
iajs-702	130	1	hence	hence	ADV
iajs-702	130	2	c	c	NOUN
iajs-702	130	3	≠	≠	PROPN
iajs-702	130	4			NOUN
iajs-702	130	5	,	,	PUNCT
iajs-702	130	6	and	and	CCONJ
iajs-702	130	7	so	so	SCONJ
iajs-702	130	8	that	that	SCONJ
iajs-702	130	9	by	by	ADP
iajs-702	130	10	zorn	zorn	PROPN
iajs-702	130	11	's	's	PART
iajs-702	130	12	lemma	lemma	PROPN
iajs-702	130	13	c	c	PROPN
iajs-702	130	14	has	have	VERB
iajs-702	130	15	a	a	DET
iajs-702	130	16	maximal	maximal	ADJ
iajs-702	130	17	element	element	NOUN
iajs-702	130	18	say	say	VERB
iajs-702	130	19	w.	w.	PROPN
iajs-702	130	20	it	it	PRON
iajs-702	130	21	is	be	AUX
iajs-702	130	22	clear	clear	ADJ
iajs-702	130	23	that	that	SCONJ
iajs-702	130	24	w	w	NOUN
iajs-702	130	25	is	be	AUX
iajs-702	130	26	a	a	DET
iajs-702	130	27	maximal	maximal	ADJ
iajs-702	130	28	-	-	PUNCT
iajs-702	130	29	uniform	uniform	NOUN
iajs-702	130	30	.	.	PUNCT
iajs-702	131	1	so	so	ADV
iajs-702	131	2	it	it	PRON
iajs-702	131	3	is	be	AUX
iajs-702	131	4	a	a	DET
iajs-702	131	5	minimal	minimal	ADJ
iajs-702	131	6	-	-	PUNCT
iajs-702	131	7	closed	closed	ADJ
iajs-702	131	8	submodule	submodule	NOUN
iajs-702	131	9	of	of	ADP
iajs-702	131	10	m.	m.	NOUN
iajs-702	131	11	thus	thus	ADV
iajs-702	131	12	w	w	PROPN
iajs-702	131	13	is	be	AUX
iajs-702	131	14	a	a	DET
iajs-702	131	15	direct	direct	ADJ
iajs-702	131	16	summand	summand	NOUN
iajs-702	131	17	of	of	ADP
iajs-702	131	18	m	m	PROPN
iajs-702	131	19	,	,	PUNCT
iajs-702	131	20	since	since	SCONJ
iajs-702	131	21	m	m	PROPN
iajs-702	131	22	is	be	AUX
iajs-702	131	23	a	a	DET
iajs-702	131	24	min	min	PROPN
iajs-702	131	25	-	-	ADJ
iajs-702	131	26	cs	cs	ADJ
iajs-702	131	27	module	module	NOUN
iajs-702	131	28	.	.	PUNCT
iajs-702	132	1	then	then	ADV
iajs-702	132	2	w	w	PROPN
iajs-702	132	3			PROPN
iajs-702	132	4	v	v	X
iajs-702	132	5	=	=	NOUN
iajs-702	132	6	m	m	VERB
iajs-702	132	7	for	for	ADP
iajs-702	132	8	some	some	DET
iajs-702	132	9	submodule	submodule	NOUN
iajs-702	132	10	v	v	ADP
iajs-702	132	11			NUM
iajs-702	132	12	m.	m.	NOUN
iajs-702	132	13	but	but	CCONJ
iajs-702	132	14	m	m	NOUN
iajs-702	132	15	is	be	AUX
iajs-702	132	16	indecomposable	indecomposable	ADJ
iajs-702	132	17	,	,	PUNCT
iajs-702	132	18	hence	hence	ADV
iajs-702	132	19	v	v	NOUN
iajs-702	132	20	=	=	SYM
iajs-702	132	21	(	(	PUNCT
iajs-702	132	22	0	0	NUM
iajs-702	132	23	)	)	PUNCT
iajs-702	132	24	.	.	PUNCT
iajs-702	133	1	thus	thus	ADV
iajs-702	133	2	w	w	X
iajs-702	133	3	=	=	VERB
iajs-702	133	4	m	m	ADJ
iajs-702	133	5	;	;	PUNCT
iajs-702	133	6	that	that	PRON
iajs-702	133	7	is	is	ADV
iajs-702	133	8	m	m	VERB
iajs-702	133	9	is	be	AUX
iajs-702	133	10	uniform	uniform	ADJ
iajs-702	133	11	.	.	PUNCT
iajs-702	134	1	1.12	1.12	NUM
iajs-702	134	2	corollary	corollary	NOUN
iajs-702	134	3	:	:	PUNCT
iajs-702	134	4	let	let	VERB
iajs-702	134	5	m	m	PRON
iajs-702	134	6	be	be	AUX
iajs-702	134	7	an	an	DET
iajs-702	134	8	indecomposable	indecomposable	ADJ
iajs-702	134	9	r	r	NOUN
iajs-702	134	10	-	-	PUNCT
iajs-702	134	11	module	module	NOUN
iajs-702	134	12	with	with	ADP
iajs-702	134	13	a	a	DET
iajs-702	134	14	uniform	uniform	ADJ
iajs-702	134	15	submodule	submodule	NOUN
iajs-702	134	16	.	.	PUNCT
iajs-702	135	1	then	then	ADV
iajs-702	135	2	m	m	PROPN
iajs-702	135	3	is	be	AUX
iajs-702	135	4	min	min	ADJ
iajs-702	135	5	-	-	ADJ
iajs-702	135	6	cs	cs	ADJ
iajs-702	135	7	module	module	NOUN
iajs-702	135	8	if	if	SCONJ
iajs-702	135	9	and	and	CCONJ
iajs-702	135	10	only	only	ADV
iajs-702	135	11	if	if	SCONJ
iajs-702	135	12	m	m	NOUN
iajs-702	135	13	is	be	AUX
iajs-702	135	14	a	a	DET
iajs-702	135	15	uniform	uniform	ADJ
iajs-702	135	16	module	module	NOUN
iajs-702	135	17	.	.	PUNCT
iajs-702	136	1	1.13	1.13	NUM
iajs-702	136	2	proposition	proposition	NOUN
iajs-702	136	3	:	:	PUNCT
iajs-702	136	4	let	let	VERB
iajs-702	136	5	m	m	PRON
iajs-702	136	6	be	be	AUX
iajs-702	136	7	an	an	DET
iajs-702	136	8	r	r	NOUN
iajs-702	136	9	-	-	PUNCT
iajs-702	136	10	module	module	NOUN
iajs-702	136	11	.	.	PUNCT
iajs-702	137	1	the	the	DET
iajs-702	137	2	following	follow	VERB
iajs-702	137	3	statements	statement	NOUN
iajs-702	137	4	are	be	AUX
iajs-702	137	5	equivalent	equivalent	ADJ
iajs-702	137	6	for	for	ADP
iajs-702	137	7	a	a	DET
iajs-702	137	8	module	module	NOUN
iajs-702	137	9	m	m	NOUN
iajs-702	137	10	:	:	PUNCT
iajs-702	137	11	(	(	PUNCT
iajs-702	137	12	1	1	X
iajs-702	137	13	)	)	PUNCT
iajs-702	138	1	m	m	VERB
iajs-702	138	2	is	be	AUX
iajs-702	138	3	a	a	DET
iajs-702	138	4	min	min	PROPN
iajs-702	138	5	-	-	ADJ
iajs-702	138	6	cs	cs	ADJ
iajs-702	138	7	module	module	NOUN
iajs-702	138	8	.	.	PUNCT
iajs-702	139	1	(	(	PUNCT
iajs-702	139	2	2	2	X
iajs-702	139	3	)	)	PUNCT
iajs-702	139	4	for	for	ADP
iajs-702	139	5	every	every	DET
iajs-702	139	6	minimal	minimal	ADJ
iajs-702	139	7	-	-	PUNCT
iajs-702	139	8	closed	closed	ADJ
iajs-702	139	9	submodule	submodule	NOUN
iajs-702	139	10	a	a	PRON
iajs-702	139	11	of	of	ADP
iajs-702	139	12	m	m	PROPN
iajs-702	139	13	,	,	PUNCT
iajs-702	139	14	there	there	PRON
iajs-702	139	15	is	be	VERB
iajs-702	139	16	a	a	DET
iajs-702	139	17	decomposition	decomposition	NOUN
iajs-702	139	18	m	m	NOUN
iajs-702	139	19	=	=	ADJ
iajs-702	139	20	m	m	VERB
iajs-702	139	21	1	1	NUM
iajs-702	139	22			ADJ
iajs-702	139	23	m	m	VERB
iajs-702	139	24	2	2	NUM
iajs-702	139	25	such	such	ADJ
iajs-702	139	26	that	that	SCONJ
iajs-702	139	27	a	a	PRON
iajs-702	139	28	is	be	AUX
iajs-702	139	29	a	a	DET
iajs-702	139	30	submodule	submodule	NOUN
iajs-702	139	31	of	of	ADP
iajs-702	139	32	m1	m1	PROPN
iajs-702	139	33	and	and	CCONJ
iajs-702	139	34	m2	m2	PROPN
iajs-702	139	35	is	be	AUX
iajs-702	139	36	a	a	DET
iajs-702	139	37	complement	complement	NOUN
iajs-702	139	38	of	of	ADP
iajs-702	139	39	a	a	PRON
iajs-702	139	40	in	in	ADP
iajs-702	139	41	m.	m.	NOUN
iajs-702	139	42	proof	proof	NOUN
iajs-702	139	43	:	:	PUNCT
iajs-702	139	44	(	(	PUNCT
iajs-702	139	45	1	1	X
iajs-702	139	46	)	)	PUNCT
iajs-702	139	47			NOUN
iajs-702	139	48	(	(	PUNCT
iajs-702	139	49	2	2	X
iajs-702	139	50	)	)	PUNCT
iajs-702	139	51	let	let	VERB
iajs-702	139	52	a	a	PRON
iajs-702	139	53	be	be	AUX
iajs-702	139	54	a	a	DET
iajs-702	139	55	minimal	minimal	ADJ
iajs-702	139	56	-	-	PUNCT
iajs-702	139	57	closed	closed	ADJ
iajs-702	139	58	submodule	submodule	NOUN
iajs-702	139	59	in	in	ADP
iajs-702	139	60	m.	m.	NOUN
iajs-702	139	61	therefore	therefore	ADV
iajs-702	139	62	a	a	PRON
iajs-702	139	63	is	be	AUX
iajs-702	139	64	a	a	DET
iajs-702	139	65	direct	direct	ADJ
iajs-702	139	66	summand	summand	NOUN
iajs-702	139	67	of	of	ADP
iajs-702	139	68	m	m	PROPN
iajs-702	139	69	since	since	SCONJ
iajs-702	139	70	m	m	PROPN
iajs-702	139	71	is	be	AUX
iajs-702	139	72	a	a	DET
iajs-702	139	73	min	min	PROPN
iajs-702	139	74	-	-	ADJ
iajs-702	139	75	cs	cs	ADJ
iajs-702	139	76	module	module	NOUN
iajs-702	139	77	.	.	PUNCT
iajs-702	140	1	that	that	PRON
iajs-702	140	2	is	be	AUX
iajs-702	140	3	m	m	NOUN
iajs-702	140	4	=	=	X
iajs-702	140	5	a	a	DET
iajs-702	140	6			PROPN
iajs-702	140	7	m	m	NOUN
iajs-702	140	8	'	'	PUNCT
iajs-702	140	9	for	for	ADP
iajs-702	140	10	some	some	DET
iajs-702	140	11	submodule	submodule	PROPN
iajs-702	140	12	m	m	PROPN
iajs-702	140	13	'	'	PUNCT
iajs-702	140	14	of	of	ADP
iajs-702	140	15	m.	m.	NOUN
iajs-702	140	16	it	it	PRON
iajs-702	140	17	is	be	AUX
iajs-702	140	18	clear	clear	ADJ
iajs-702	140	19	that	that	SCONJ
iajs-702	140	20	a	a	PRON
iajs-702	140	21	is	be	AUX
iajs-702	140	22	a	a	DET
iajs-702	140	23	submodule	submodule	NOUN
iajs-702	140	24	of	of	ADP
iajs-702	140	25	a	a	PRON
iajs-702	140	26	,	,	PUNCT
iajs-702	140	27	and	and	CCONJ
iajs-702	140	28	it	it	PRON
iajs-702	140	29	is	be	AUX
iajs-702	140	30	easy	easy	ADJ
iajs-702	140	31	to	to	PART
iajs-702	140	32	check	check	VERB
iajs-702	140	33	that	that	PRON
iajs-702	140	34	m	m	NOUN
iajs-702	140	35	'	'	PUNCT
iajs-702	140	36	is	be	AUX
iajs-702	140	37	a	a	DET
iajs-702	140	38	complement	complement	NOUN
iajs-702	140	39	of	of	ADP
iajs-702	140	40	a.	a.	NOUN
iajs-702	140	41	(	(	PUNCT
iajs-702	140	42	2	2	NUM
iajs-702	140	43	)	)	PUNCT
iajs-702	140	44			NOUN
iajs-702	140	45	(	(	PUNCT
iajs-702	140	46	1	1	NUM
iajs-702	140	47	)	)	PUNCT
iajs-702	140	48	to	to	PART
iajs-702	140	49	prove	prove	VERB
iajs-702	140	50	m	m	NOUN
iajs-702	140	51	is	be	AUX
iajs-702	140	52	a	a	DET
iajs-702	140	53	min	min	PROPN
iajs-702	140	54	-	-	ADJ
iajs-702	140	55	cs	cs	ADJ
iajs-702	140	56	module	module	NOUN
iajs-702	140	57	.	.	PUNCT
iajs-702	141	1	let	let	VERB
iajs-702	141	2	a	a	PRON
iajs-702	141	3	be	be	AUX
iajs-702	141	4	a	a	DET
iajs-702	141	5	minimal	minimal	ADJ
iajs-702	141	6	-	-	PUNCT
iajs-702	141	7	closed	closed	ADJ
iajs-702	141	8	submodule	submodule	NOUN
iajs-702	141	9	of	of	ADP
iajs-702	141	10	m.	m.	NOUN
iajs-702	141	11	therefore	therefore	ADV
iajs-702	141	12	,	,	PUNCT
iajs-702	141	13	there	there	PRON
iajs-702	141	14	is	be	VERB
iajs-702	141	15	a	a	DET
iajs-702	141	16	decomposition	decomposition	NOUN
iajs-702	141	17	m	m	NOUN
iajs-702	141	18	=	=	ADJ
iajs-702	141	19	m	m	VERB
iajs-702	141	20	1	1	NUM
iajs-702	141	21			ADJ
iajs-702	141	22	m	m	VERB
iajs-702	141	23	2	2	NUM
iajs-702	141	24	,	,	PUNCT
iajs-702	141	25	where	where	SCONJ
iajs-702	141	26	m1	m1	PROPN
iajs-702	141	27	and	and	CCONJ
iajs-702	141	28	m2	m2	PROPN
iajs-702	141	29	are	be	AUX
iajs-702	141	30	two	two	NUM
iajs-702	141	31	submodules	submodule	NOUN
iajs-702	141	32	of	of	ADP
iajs-702	141	33	m	m	PROPN
iajs-702	141	34	and	and	CCONJ
iajs-702	141	35	a	a	PRON
iajs-702	141	36	is	be	AUX
iajs-702	141	37	a	a	DET
iajs-702	141	38	submodule	submodule	NOUN
iajs-702	141	39	of	of	ADP
iajs-702	141	40	m1	m1	PROPN
iajs-702	141	41	and	and	CCONJ
iajs-702	141	42	m2	m2	PROPN
iajs-702	141	43	is	be	AUX
iajs-702	141	44	a	a	DET
iajs-702	141	45	complement	complement	NOUN
iajs-702	141	46	of	of	ADP
iajs-702	141	47	a	a	PRON
iajs-702	141	48	in	in	ADP
iajs-702	141	49	m.	m.	NOUN
iajs-702	141	50	since	since	SCONJ
iajs-702	141	51	m2	m2	PROPN
iajs-702	141	52	is	be	AUX
iajs-702	141	53	a	a	DET
iajs-702	141	54	complement	complement	NOUN
iajs-702	141	55	of	of	ADP
iajs-702	141	56	a	a	PRON
iajs-702	141	57	in	in	ADP
iajs-702	141	58	m	m	PROPN
iajs-702	141	59	,	,	PUNCT
iajs-702	141	60	then	then	ADV
iajs-702	141	61	a	a	DET
iajs-702	141	62			ADJ
iajs-702	141	63	m	m	NOUN
iajs-702	141	64	2	2	NUM
iajs-702	141	65			NUM
iajs-702	141	66	e	e	NOUN
iajs-702	141	67	m.	m.	NOUN
iajs-702	141	68	but	but	CCONJ
iajs-702	141	69	a	a	PRON
iajs-702	141	70	is	be	AUX
iajs-702	141	71	a	a	DET
iajs-702	141	72	closed	closed	ADJ
iajs-702	141	73	submodule	submodule	NOUN
iajs-702	141	74	in	in	ADP
iajs-702	141	75	m	m	PROPN
iajs-702	141	76	and	and	CCONJ
iajs-702	141	77	a	a	PRON
iajs-702	141	78	is	be	AUX
iajs-702	141	79	a	a	DET
iajs-702	141	80	submodule	submodule	NOUN
iajs-702	141	81	in	in	ADP
iajs-702	141	82	m	m	PROPN
iajs-702	141	83	1	1	NUM
iajs-702	141	84	,	,	PUNCT
iajs-702	141	85	therefore	therefore	ADV
iajs-702	141	86	a	a	PRON
iajs-702	141	87	is	be	AUX
iajs-702	141	88	closed	close	VERB
iajs-702	141	89	in	in	ADP
iajs-702	141	90	m1	m1	PROPN
iajs-702	141	91	.	.	PUNCT
iajs-702	142	1	therefore	therefore	ADV
iajs-702	142	2	,	,	PUNCT
iajs-702	142	3	a	a	DET
iajs-702	142	4			PROPN
iajs-702	142	5	m	m	VERB
iajs-702	142	6	2	2	NUM
iajs-702	142	7	is	be	AUX
iajs-702	142	8	closed	close	VERB
iajs-702	142	9	in	in	ADP
iajs-702	142	10	m1	m1	PROPN
iajs-702	142	11			PROPN
iajs-702	142	12	m	m	VERB
iajs-702	142	13	2	2	NUM
iajs-702	142	14	=	=	SYM
iajs-702	142	15	m	m	VERB
iajs-702	142	16	by	by	ADP
iajs-702	142	17	[	[	X
iajs-702	142	18	4	4	NUM
iajs-702	142	19	,	,	PUNCT
iajs-702	142	20	exc.15	exc.15	PRON
iajs-702	142	21	,	,	PUNCT
iajs-702	142	22	p.20	p.20	NOUN
iajs-702	142	23	]	]	X
iajs-702	142	24	.	.	PUNCT
iajs-702	143	1	thus	thus	ADV
iajs-702	143	2	a	a	DET
iajs-702	143	3			ADJ
iajs-702	143	4	m	m	NOUN
iajs-702	143	5	2	2	NUM
iajs-702	143	6	=	=	NOUN
iajs-702	143	7	m	m	NOUN
iajs-702	143	8	.	.	PUNCT
iajs-702	144	1	so	so	ADV
iajs-702	144	2	that	that	SCONJ
iajs-702	144	3	a	a	PRON
iajs-702	144	4	is	be	AUX
iajs-702	144	5	a	a	DET
iajs-702	144	6	direct	direct	ADJ
iajs-702	144	7	summand	summand	NOUN
iajs-702	144	8	in	in	ADP
iajs-702	144	9	m.	m.	NOUN
iajs-702	144	10	hence	hence	ADV
iajs-702	144	11	m	m	VERB
iajs-702	144	12	is	be	AUX
iajs-702	144	13	a	a	DET
iajs-702	144	14	min	min	PROPN
iajs-702	144	15	-	-	ADJ
iajs-702	144	16	cs	cs	ADJ
iajs-702	144	17	module	module	NOUN
iajs-702	144	18	.	.	PUNCT
iajs-702	145	1	1.14	1.14	NUM
iajs-702	145	2	proposition	proposition	NOUN
iajs-702	145	3	:	:	PUNCT
iajs-702	145	4	let	let	VERB
iajs-702	145	5	m	m	PRON
iajs-702	145	6	be	be	AUX
iajs-702	145	7	a	a	DET
iajs-702	145	8	min	min	NOUN
iajs-702	145	9	-	-	ADJ
iajs-702	145	10	cs	cs	ADJ
iajs-702	145	11	r	r	NOUN
iajs-702	145	12	-	-	PUNCT
iajs-702	145	13	module	module	NOUN
iajs-702	145	14	.	.	PUNCT
iajs-702	146	1	if	if	SCONJ
iajs-702	146	2	n	n	PRON
iajs-702	146	3	is	be	AUX
iajs-702	146	4	a	a	DET
iajs-702	146	5	closed	closed	ADJ
iajs-702	146	6	submodule	submodule	NOUN
iajs-702	146	7	,	,	PUNCT
iajs-702	146	8	then	then	ADV
iajs-702	146	9	n	n	PRON
iajs-702	146	10	is	be	AUX
iajs-702	146	11	a	a	DET
iajs-702	146	12	min	min	PROPN
iajs-702	146	13	-	-	ADJ
iajs-702	146	14	cs	cs	ADJ
iajs-702	146	15	module	module	NOUN
iajs-702	146	16	.	.	PUNCT
iajs-702	147	1	proof	proof	NOUN
iajs-702	147	2	:	:	PUNCT
iajs-702	147	3	let	let	VERB
iajs-702	147	4	u	u	PRON
iajs-702	147	5	be	be	AUX
iajs-702	147	6	a	a	DET
iajs-702	147	7	minimal	minimal	ADJ
iajs-702	147	8	closed	closed	ADJ
iajs-702	147	9	submodule	submodule	NOUN
iajs-702	147	10	of	of	ADP
iajs-702	147	11	n.	n.	NOUN
iajs-702	147	12	since	since	SCONJ
iajs-702	147	13	n	n	NUM
iajs-702	147	14	is	be	AUX
iajs-702	147	15	a	a	DET
iajs-702	147	16	closed	closed	ADJ
iajs-702	147	17	submodule	submodule	NOUN
iajs-702	147	18	in	in	ADP
iajs-702	147	19	m	m	PROPN
iajs-702	147	20	,	,	PUNCT
iajs-702	147	21	then	then	ADV
iajs-702	147	22	by	by	ADP
iajs-702	147	23	[	[	X
iajs-702	147	24	4	4	NUM
iajs-702	147	25	,	,	PUNCT
iajs-702	147	26	proposition	proposition	NOUN
iajs-702	147	27	1.5	1.5	NUM
iajs-702	147	28	,	,	PUNCT
iajs-702	147	29	p.18	p.18	PROPN
iajs-702	147	30	]	]	X
iajs-702	147	31	,	,	PUNCT
iajs-702	147	32	u	u	NOUN
iajs-702	147	33	is	be	AUX
iajs-702	147	34	closed	close	VERB
iajs-702	147	35	in	in	ADP
iajs-702	147	36	m.	m.	NOUN
iajs-702	147	37	we	we	PRON
iajs-702	147	38	claim	claim	VERB
iajs-702	147	39	that	that	SCONJ
iajs-702	147	40	u	u	NOUN
iajs-702	147	41	is	be	AUX
iajs-702	147	42	a	a	DET
iajs-702	147	43	minimal	minimal	ADJ
iajs-702	147	44	closed	closed	ADJ
iajs-702	147	45	submodule	submodule	NOUN
iajs-702	147	46	of	of	ADP
iajs-702	147	47	m.	m.	NOUN
iajs-702	147	48	to	to	PART
iajs-702	147	49	prove	prove	VERB
iajs-702	147	50	our	our	PRON
iajs-702	147	51	assertion	assertion	NOUN
iajs-702	147	52	:	:	PUNCT
iajs-702	147	53	suppose	suppose	VERB
iajs-702	147	54	there	there	PRON
iajs-702	147	55	exists	exist	VERB
iajs-702	147	56	a	a	DET
iajs-702	147	57	closed	closed	ADJ
iajs-702	147	58	submodule	submodule	NOUN
iajs-702	147	59	v	v	NOUN
iajs-702	147	60	of	of	ADP
iajs-702	147	61	m	m	PRON
iajs-702	147	62	such	such	ADJ
iajs-702	147	63	that	that	DET
iajs-702	147	64	v	v	ADJ
iajs-702	147	65			PROPN
iajs-702	147	66	u.	u.	PROPN
iajs-702	147	67	but	but	CCONJ
iajs-702	147	68	v	v	ADP
iajs-702	147	69			PROPN
iajs-702	147	70	u	u	PROPN
iajs-702	147	71			PROPN
iajs-702	147	72	n	n	CCONJ
iajs-702	147	73	,	,	PUNCT
iajs-702	147	74	implies	imply	VERB
iajs-702	147	75	v	v	NOUN
iajs-702	147	76	is	be	AUX
iajs-702	147	77	a	a	DET
iajs-702	147	78	closed	closed	ADJ
iajs-702	147	79	submodule	submodule	NOUN
iajs-702	147	80	in	in	ADP
iajs-702	147	81	n	n	NUM
iajs-702	147	82	by	by	ADP
iajs-702	147	83	[	[	X
iajs-702	147	84	4	4	NUM
iajs-702	147	85	,	,	PUNCT
iajs-702	147	86	p.18	p.18	NOUN
iajs-702	147	87	]	]	PUNCT
iajs-702	147	88	.	.	PUNCT
iajs-702	148	1	but	but	CCONJ
iajs-702	148	2	u	u	NOUN
iajs-702	148	3	is	be	AUX
iajs-702	148	4	a	a	DET
iajs-702	148	5	minimal	minimal	ADJ
iajs-702	148	6	closed	closed	ADJ
iajs-702	148	7	submodule	submodule	NOUN
iajs-702	148	8	of	of	ADP
iajs-702	148	9	n	n	PROPN
iajs-702	148	10	so	so	ADV
iajs-702	148	11	u	u	NOUN
iajs-702	148	12	=	=	PROPN
iajs-702	148	13	v.	v.	ADP
iajs-702	148	14	thus	thus	ADV
iajs-702	148	15	u	u	NOUN
iajs-702	148	16	is	be	AUX
iajs-702	148	17	a	a	DET
iajs-702	148	18	minimal	minimal	ADJ
iajs-702	148	19	closed	closed	ADJ
iajs-702	148	20	submodule	submodule	NOUN
iajs-702	148	21	of	of	ADP
iajs-702	148	22	m	m	PROPN
iajs-702	148	23	,	,	PUNCT
iajs-702	148	24	and	and	CCONJ
iajs-702	148	25	hence	hence	ADV
iajs-702	148	26	u	u	NOUN
iajs-702	148	27	is	be	AUX
iajs-702	148	28	a	a	DET
iajs-702	148	29	direct	direct	ADJ
iajs-702	148	30	summand	summand	NOUN
iajs-702	148	31	of	of	ADP
iajs-702	148	32	m	m	PROPN
iajs-702	148	33	,	,	PUNCT
iajs-702	148	34	since	since	SCONJ
iajs-702	148	35	m	m	PROPN
iajs-702	148	36	is	be	AUX
iajs-702	148	37	a	a	DET
iajs-702	148	38	min	min	PROPN
iajs-702	148	39	-	-	ADJ
iajs-702	148	40	cs	cs	ADJ
iajs-702	148	41	module	module	NOUN
iajs-702	148	42	.	.	PUNCT
iajs-702	149	1	hence	hence	ADV
iajs-702	149	2	m	m	VERB
iajs-702	149	3	=	=	SYM
iajs-702	149	4	u	u	NOUN
iajs-702	149	5			PROPN
iajs-702	149	6	w	w	NOUN
iajs-702	149	7	for	for	ADP
iajs-702	149	8	some	some	DET
iajs-702	149	9	w	w	NOUN
iajs-702	149	10			PROPN
iajs-702	149	11	m.	m.	NOUN
iajs-702	149	12	it	it	PRON
iajs-702	149	13	follows	follow	VERB
iajs-702	149	14	that	that	PRON
iajs-702	149	15	n	n	NOUN
iajs-702	149	16	=	=	SYM
iajs-702	149	17	(	(	PUNCT
iajs-702	149	18	u	u	PROPN
iajs-702	149	19			PROPN
iajs-702	149	20	w	w	PROPN
iajs-702	149	21	)	)	PUNCT
iajs-702	149	22			PUNCT
iajs-702	149	23	n	n	CCONJ
iajs-702	149	24	and	and	CCONJ
iajs-702	149	25	by	by	ADP
iajs-702	149	26	modular	modular	ADJ
iajs-702	149	27	law	law	NOUN
iajs-702	149	28	,	,	PUNCT
iajs-702	149	29	n	n	PROPN
iajs-702	149	30	=	=	SYM
iajs-702	149	31	u	u	NOUN
iajs-702	149	32			ADJ
iajs-702	149	33	(	(	PUNCT
iajs-702	149	34	w	w	PROPN
iajs-702	149	35			PUNCT
iajs-702	149	36	n	n	CCONJ
iajs-702	149	37	)	)	PUNCT
iajs-702	149	38	;	;	PUNCT
iajs-702	149	39	that	that	PRON
iajs-702	149	40	is	be	AUX
iajs-702	149	41	u	u	NOUN
iajs-702	149	42	is	be	AUX
iajs-702	149	43	a	a	DET
iajs-702	149	44	direct	direct	ADJ
iajs-702	149	45	summand	summand	NOUN
iajs-702	149	46	of	of	ADP
iajs-702	149	47	n	n	PROPN
iajs-702	149	48	and	and	CCONJ
iajs-702	149	49	so	so	ADV
iajs-702	149	50	n	n	PRON
iajs-702	149	51	is	be	AUX
iajs-702	149	52	a	a	DET
iajs-702	149	53	min	min	PROPN
iajs-702	149	54	-	-	ADJ
iajs-702	149	55	cs	cs	ADJ
iajs-702	149	56	module	module	NOUN
iajs-702	149	57	.	.	PUNCT
iajs-702	150	1	مجلة	مجلة	NOUN
iajs-702	150	2	إبن	إبن	VERB
iajs-702	150	3	الھیثم	الھیثم	NOUN
iajs-702	150	4	للعلوم	للعلوم	NOUN
iajs-702	150	5	الصرفة	الصرفة	NOUN
iajs-702	150	6	و	و	PRON
iajs-702	150	7	التطبیقیة	التطبیقیة	PROPN
iajs-702	150	8	2012	2012	NUM
iajs-702	150	9	السنة	السنة	NOUN
iajs-702	151	1	25	25	NUM
iajs-702	151	2	المجلد	المجلد	NOUN
iajs-702	151	3	1	1	NUM
iajs-702	151	4	العدد	العدد	PROPN
iajs-702	151	5	ibn	ibn	PROPN
iajs-702	151	6	al	al	PROPN
iajs-702	151	7	-	-	PUNCT
iajs-702	151	8	haitham	haitham	PROPN
iajs-702	151	9	journal	journal	PROPN
iajs-702	151	10	for	for	ADP
iajs-702	151	11	pure	pure	ADJ
iajs-702	151	12	and	and	CCONJ
iajs-702	151	13	applied	apply	VERB
iajs-702	151	14	science	science	NOUN
iajs-702	151	15	no	no	NOUN
iajs-702	151	16	.	.	NOUN
iajs-702	151	17	1	1	NUM
iajs-702	151	18	vol	vol	NOUN
iajs-702	151	19	.	.	PUNCT
iajs-702	152	1	25	25	NUM
iajs-702	152	2	year	year	NOUN
iajs-702	152	3	2012	2012	NUM
iajs-702	152	4	1.15	1.15	NUM
iajs-702	152	5	remark	remark	NOUN
iajs-702	152	6	:	:	PUNCT
iajs-702	152	7	the	the	DET
iajs-702	152	8	converse	converse	NOUN
iajs-702	152	9	of	of	ADP
iajs-702	152	10	the	the	DET
iajs-702	152	11	previous	previous	ADJ
iajs-702	152	12	proposition	proposition	NOUN
iajs-702	152	13	is	be	AUX
iajs-702	152	14	not	not	PART
iajs-702	152	15	true	true	ADJ
iajs-702	152	16	in	in	ADP
iajs-702	152	17	general	general	ADJ
iajs-702	152	18	for	for	ADP
iajs-702	152	19	example	example	NOUN
iajs-702	152	20	:	:	PUNCT
iajs-702	152	21	the	the	DET
iajs-702	152	22	ℤ-module	ℤ-module	ADJ
iajs-702	152	23	ℤ8	ℤ8	NOUN
iajs-702	152	24			NOUN
iajs-702	152	25	ℤ2	ℤ2	PROPN
iajs-702	152	26	is	be	AUX
iajs-702	152	27	not	not	PART
iajs-702	152	28	min	min	NOUN
iajs-702	152	29	-	-	NOUN
iajs-702	152	30	cs	cs	ADJ
iajs-702	152	31	,	,	PUNCT
iajs-702	152	32	but	but	CCONJ
iajs-702	152	33	n	n	NOUN
iajs-702	152	34	=	=	NOUN
iajs-702	152	35	ℤ8	ℤ8	NOUN
iajs-702	152	36			NOUN
iajs-702	152	37	(	(	PUNCT
iajs-702	152	38	0	0	NUM
iajs-702	152	39	)	)	PUNCT
iajs-702	152	40	≅	≅	NOUN
iajs-702	152	41	ℤ8	ℤ8	NOUN
iajs-702	152	42	is	be	AUX
iajs-702	152	43	min	min	NOUN
iajs-702	152	44	-	-	ADJ
iajs-702	152	45	cs	cs	ADJ
iajs-702	152	46	.	.	PROPN
iajs-702	152	47	1.16	1.16	NUM
iajs-702	152	48	corollary	corollary	NOUN
iajs-702	152	49	:	:	PUNCT
iajs-702	152	50	a	a	DET
iajs-702	152	51	direct	direct	ADJ
iajs-702	152	52	summand	summand	NOUN
iajs-702	152	53	of	of	ADP
iajs-702	152	54	min	min	PROPN
iajs-702	152	55	-	-	ADJ
iajs-702	152	56	cs	cs	ADJ
iajs-702	152	57	module	module	NOUN
iajs-702	152	58	is	be	AUX
iajs-702	152	59	min	min	NOUN
iajs-702	152	60	-	-	ADJ
iajs-702	152	61	cs	cs	ADJ
iajs-702	152	62	.	.	PROPN
iajs-702	152	63	proof	proof	NOUN
iajs-702	152	64	:	:	PUNCT
iajs-702	152	65	let	let	VERB
iajs-702	152	66	m	m	PRON
iajs-702	152	67	be	be	AUX
iajs-702	152	68	a	a	DET
iajs-702	152	69	min	min	ADJ
iajs-702	152	70	-	-	ADJ
iajs-702	152	71	cs	cs	ADJ
iajs-702	152	72	module	module	NOUN
iajs-702	152	73	and	and	CCONJ
iajs-702	152	74	let	let	VERB
iajs-702	152	75	n	n	PRON
iajs-702	152	76	be	be	AUX
iajs-702	152	77	a	a	DET
iajs-702	152	78	direct	direct	ADJ
iajs-702	152	79	summand	summand	NOUN
iajs-702	152	80	of	of	ADP
iajs-702	152	81	m.	m.	NOUN
iajs-702	152	82	hence	hence	ADV
iajs-702	152	83	by	by	ADP
iajs-702	152	84	[	[	X
iajs-702	152	85	4	4	NUM
iajs-702	152	86	,	,	PUNCT
iajs-702	152	87	exc.3	exc.3	NOUN
iajs-702	152	88	,	,	PUNCT
iajs-702	152	89	p.19	p.19	NOUN
iajs-702	152	90	]	]	PUNCT
iajs-702	152	91	,	,	PUNCT
iajs-702	152	92	n	n	X
iajs-702	152	93	is	be	AUX
iajs-702	152	94	a	a	DET
iajs-702	152	95	closed	closed	ADJ
iajs-702	152	96	submodule	submodule	NOUN
iajs-702	152	97	of	of	ADP
iajs-702	152	98	m.	m.	NOUN
iajs-702	152	99	hence	hence	ADV
iajs-702	152	100	,	,	PUNCT
iajs-702	152	101	n	n	X
iajs-702	152	102	is	be	AUX
iajs-702	152	103	a	a	DET
iajs-702	152	104	min	min	PROPN
iajs-702	152	105	-	-	ADJ
iajs-702	152	106	cs	cs	ADJ
iajs-702	152	107	module	module	NOUN
iajs-702	152	108	by	by	ADP
iajs-702	152	109	proposition	proposition	NOUN
iajs-702	152	110	14	14	NUM
iajs-702	152	111	.	.	PUNCT
iajs-702	153	1	1.17	1.17	NUM
iajs-702	153	2	corollary	corollary	NOUN
iajs-702	153	3	:	:	PUNCT
iajs-702	153	4	let	let	VERB
iajs-702	153	5	m	m	PRON
iajs-702	153	6	be	be	AUX
iajs-702	153	7	an	an	DET
iajs-702	153	8	r	r	NOUN
iajs-702	153	9	-	-	PUNCT
iajs-702	153	10	module	module	NOUN
iajs-702	153	11	.	.	PUNCT
iajs-702	154	1	if	if	SCONJ
iajs-702	154	2	m	m	PROPN
iajs-702	154	3			ADJ
iajs-702	154	4	m	m	VERB
iajs-702	154	5	is	be	AUX
iajs-702	154	6	min	min	NOUN
iajs-702	154	7	-	-	ADJ
iajs-702	154	8	cs	cs	ADJ
iajs-702	154	9	,	,	PUNCT
iajs-702	154	10	then	then	ADV
iajs-702	154	11	m	m	PROPN
iajs-702	154	12	is	be	AUX
iajs-702	154	13	a	a	DET
iajs-702	154	14	min	min	PROPN
iajs-702	154	15	-	-	ADJ
iajs-702	154	16	cs	cs	ADJ
iajs-702	154	17	module	module	NOUN
iajs-702	154	18	.	.	PUNCT
iajs-702	155	1	proof	proof	NOUN
iajs-702	155	2	:	:	PUNCT
iajs-702	155	3	it	it	PRON
iajs-702	155	4	follows	follow	VERB
iajs-702	155	5	by	by	ADP
iajs-702	155	6	corollary	corollary	ADJ
iajs-702	155	7	1.16	1.16	NUM
iajs-702	155	8	.	.	PUNCT
iajs-702	156	1	corollary	corollary	ADJ
iajs-702	156	2	1.16	1.16	NUM
iajs-702	157	1	,	,	PUNCT
iajs-702	157	2	lead	lead	VERB
iajs-702	157	3	us	we	PRON
iajs-702	157	4	to	to	PART
iajs-702	157	5	give	give	VERB
iajs-702	157	6	the	the	DET
iajs-702	157	7	following	follow	VERB
iajs-702	157	8	example	example	NOUN
iajs-702	157	9	:	:	PUNCT
iajs-702	157	10	let	let	VERB
iajs-702	157	11	m	m	PRON
iajs-702	157	12	=	=	VERB
iajs-702	157	13	ℤ8	ℤ8	NOUN
iajs-702	157	14			PROPN
iajs-702	157	15	ℤ2	ℤ2	PROPN
iajs-702	157	16			PROPN
iajs-702	157	17	ℤ3	ℤ3	NOUN
iajs-702	157	18	as	as	ADP
iajs-702	157	19	a	a	DET
iajs-702	157	20	ℤ-module	ℤ-module	PROPN
iajs-702	157	21	.	.	PUNCT
iajs-702	158	1	thus	thus	ADV
iajs-702	158	2	m	m	NOUN
iajs-702	158	3	is	be	AUX
iajs-702	158	4	not	not	PART
iajs-702	158	5	min	min	NOUN
iajs-702	158	6	-	-	NOUN
iajs-702	158	7	cs	cs	ADJ
iajs-702	158	8	because	because	SCONJ
iajs-702	158	9	if	if	SCONJ
iajs-702	158	10	it	it	PRON
iajs-702	158	11	is	be	AUX
iajs-702	158	12	,	,	PUNCT
iajs-702	158	13	then	then	ADV
iajs-702	158	14	n	n	NOUN
iajs-702	158	15	=	=	NOUN
iajs-702	158	16	ℤ8	ℤ8	NOUN
iajs-702	158	17			PROPN
iajs-702	158	18	ℤ2	ℤ2	PROPN
iajs-702	158	19			PROPN
iajs-702	158	20	m	m	VERB
iajs-702	158	21	is	be	AUX
iajs-702	158	22	min	min	NOUN
iajs-702	158	23	-	-	NOUN
iajs-702	158	24	cs	cs	ADJ
iajs-702	158	25	by	by	ADP
iajs-702	158	26	corollary	corollary	ADJ
iajs-702	158	27	1.16	1.16	NUM
iajs-702	158	28	,	,	PUNCT
iajs-702	158	29	which	which	PRON
iajs-702	158	30	is	be	AUX
iajs-702	158	31	a	a	DET
iajs-702	158	32	contradiction	contradiction	NOUN
iajs-702	158	33	.	.	PUNCT
iajs-702	159	1	1.18	1.18	NUM
iajs-702	159	2	remark	remark	NOUN
iajs-702	159	3	:	:	PUNCT
iajs-702	159	4	the	the	DET
iajs-702	159	5	condition	condition	NOUN
iajs-702	159	6	n	n	PRON
iajs-702	159	7	is	be	AUX
iajs-702	159	8	closed	closed	ADJ
iajs-702	159	9	can	can	AUX
iajs-702	159	10	not	not	PART
iajs-702	159	11	be	be	AUX
iajs-702	159	12	dropped	drop	VERB
iajs-702	159	13	from	from	ADP
iajs-702	159	14	proposition	proposition	NOUN
iajs-702	159	15	1.14	1.14	NUM
iajs-702	159	16	as	as	SCONJ
iajs-702	159	17	the	the	DET
iajs-702	159	18	following	follow	VERB
iajs-702	159	19	example	example	NOUN
iajs-702	159	20	shows	show	VERB
iajs-702	159	21	:	:	PUNCT
iajs-702	159	22	let	let	VERB
iajs-702	159	23	m	m	PRON
iajs-702	159	24	be	be	AUX
iajs-702	159	25	the	the	DET
iajs-702	159	26	ℤ-module	ℤ-module	PROPN
iajs-702	159	27	ℤ16	ℤ16	PROPN
iajs-702	159	28			ADJ
iajs-702	159	29	ℤ2	ℤ2	PROPN
iajs-702	159	30	.	.	PUNCT
iajs-702	160	1	m	m	PROPN
iajs-702	160	2	is	be	AUX
iajs-702	160	3	min	min	NOUN
iajs-702	160	4	-	-	PROPN
iajs-702	160	5	cs	cs	ADJ
iajs-702	160	6	.	.	PUNCT
iajs-702	161	1	let	let	VERB
iajs-702	161	2	n	n	NOUN
iajs-702	161	3	=	=	SYM
iajs-702	161	4	(	(	PUNCT
iajs-702	161	5	2	2	X
iajs-702	161	6	)	)	PUNCT
iajs-702	161	7			ADJ
iajs-702	161	8	ℤ2	ℤ2	PROPN
iajs-702	161	9	.	.	PUNCT
iajs-702	162	1	it	it	PRON
iajs-702	162	2	is	be	AUX
iajs-702	162	3	clear	clear	ADJ
iajs-702	162	4	that	that	SCONJ
iajs-702	162	5	n	n	NUM
iajs-702	162	6			NUM
iajs-702	162	7	e	e	NOUN
iajs-702	162	8	m	m	PROPN
iajs-702	162	9	,	,	PUNCT
iajs-702	162	10	so	so	ADV
iajs-702	162	11	n	n	PROPN
iajs-702	162	12	is	be	AUX
iajs-702	162	13	not	not	PART
iajs-702	162	14	closed	close	VERB
iajs-702	162	15	in	in	ADP
iajs-702	162	16	m	m	PROPN
iajs-702	162	17	,	,	PUNCT
iajs-702	162	18	also	also	ADV
iajs-702	162	19	n	n	PROPN
iajs-702	162	20	is	be	AUX
iajs-702	162	21	isomorphic	isomorphic	ADJ
iajs-702	162	22	to	to	AUX
iajs-702	162	23	ℤ8	ℤ8	NOUN
iajs-702	162	24			PROPN
iajs-702	162	25	ℤ2	ℤ2	PROPN
iajs-702	162	26	which	which	PRON
iajs-702	162	27	is	be	AUX
iajs-702	162	28	not	not	PART
iajs-702	162	29	a	a	DET
iajs-702	162	30	min	min	ADJ
iajs-702	162	31	-	-	ADJ
iajs-702	162	32	cs	cs	ADJ
iajs-702	162	33	module	module	NOUN
iajs-702	162	34	.	.	PUNCT
iajs-702	163	1	1.19	1.19	NUM
iajs-702	163	2	proposition	proposition	NOUN
iajs-702	163	3	:	:	PUNCT
iajs-702	163	4	let	let	VERB
iajs-702	163	5	m	m	PRON
iajs-702	163	6	be	be	AUX
iajs-702	163	7	a	a	DET
iajs-702	163	8	finitely	finitely	ADV
iajs-702	163	9	generated	generate	VERB
iajs-702	163	10	or	or	CCONJ
iajs-702	163	11	multiplication	multiplication	NOUN
iajs-702	163	12	r	r	NOUN
iajs-702	163	13	-	-	NOUN
iajs-702	163	14	module	module	NOUN
iajs-702	163	15	.	.	PUNCT
iajs-702	164	1	then	then	ADV
iajs-702	164	2	if	if	SCONJ
iajs-702	164	3	every	every	DET
iajs-702	164	4	maximal	maximal	ADJ
iajs-702	164	5	submodule	submodule	NOUN
iajs-702	164	6	is	be	AUX
iajs-702	164	7	a	a	DET
iajs-702	164	8	direct	direct	ADJ
iajs-702	164	9	summand	summand	NOUN
iajs-702	164	10	,	,	PUNCT
iajs-702	164	11	then	then	ADV
iajs-702	164	12	m	m	VERB
iajs-702	164	13	is	be	AUX
iajs-702	164	14	a	a	DET
iajs-702	164	15	max	max	PROPN
iajs-702	164	16	-	-	PUNCT
iajs-702	164	17	cs	cs	PROPN
iajs-702	164	18	module	module	NOUN
iajs-702	164	19	.	.	PUNCT
iajs-702	165	1	proof	proof	NOUN
iajs-702	165	2	:	:	PUNCT
iajs-702	165	3	let	let	VERB
iajs-702	165	4	a	a	PRON
iajs-702	165	5	be	be	AUX
iajs-702	165	6	a	a	DET
iajs-702	165	7	maximal	maximal	ADJ
iajs-702	165	8	closed	closed	ADJ
iajs-702	165	9	submodule	submodule	NOUN
iajs-702	165	10	of	of	ADP
iajs-702	165	11	m.	m.	NOUN
iajs-702	165	12	since	since	SCONJ
iajs-702	165	13	m	m	PROPN
iajs-702	165	14	is	be	AUX
iajs-702	165	15	a	a	DET
iajs-702	165	16	finitely	finitely	ADV
iajs-702	165	17	generated	generate	VERB
iajs-702	165	18	or	or	CCONJ
iajs-702	165	19	multiplication	multiplication	NOUN
iajs-702	165	20	module	module	NOUN
iajs-702	165	21	.	.	PUNCT
iajs-702	166	1	then	then	ADV
iajs-702	166	2	by	by	ADP
iajs-702	166	3	[	[	X
iajs-702	166	4	6	6	NUM
iajs-702	166	5	,	,	PUNCT
iajs-702	166	6	theorem	theorem	ADJ
iajs-702	166	7	2.3.11	2.3.11	NOUN
iajs-702	166	8	,	,	PUNCT
iajs-702	166	9	p.28	p.28	NOUN
iajs-702	166	10	]	]	PUNCT
iajs-702	166	11	and	and	CCONJ
iajs-702	166	12	by	by	ADP
iajs-702	166	13	[	[	X
iajs-702	166	14	7	7	NUM
iajs-702	166	15	,	,	PUNCT
iajs-702	166	16	theorem	theorem	VERB
iajs-702	166	17	2.5	2.5	NUM
iajs-702	166	18	(	(	PUNCT
iajs-702	166	19	1	1	NUM
iajs-702	166	20	)	)	PUNCT
iajs-702	166	21	]	]	PUNCT
iajs-702	166	22	,	,	PUNCT
iajs-702	166	23	there	there	PRON
iajs-702	166	24	exists	exist	VERB
iajs-702	166	25	a	a	DET
iajs-702	166	26	maximal	maximal	ADJ
iajs-702	166	27	submodule	submodule	NOUN
iajs-702	166	28	b	b	PROPN
iajs-702	166	29	of	of	ADP
iajs-702	166	30	m	m	PRON
iajs-702	166	31	such	such	ADJ
iajs-702	166	32	that	that	SCONJ
iajs-702	166	33	a	a	DET
iajs-702	166	34			NUM
iajs-702	166	35	b.	b.	NOUN
iajs-702	167	1	but	but	CCONJ
iajs-702	167	2	b	b	PROPN
iajs-702	167	3	is	be	AUX
iajs-702	167	4	a	a	DET
iajs-702	167	5	direct	direct	ADJ
iajs-702	167	6	summand	summand	NOUN
iajs-702	167	7	,	,	PUNCT
iajs-702	167	8	by	by	ADP
iajs-702	167	9	hypothesis	hypothesis	NOUN
iajs-702	167	10	;	;	PUNCT
iajs-702	167	11	hence	hence	ADV
iajs-702	167	12	b	b	X
iajs-702	167	13	is	be	AUX
iajs-702	167	14	a	a	DET
iajs-702	167	15	closed	closed	ADJ
iajs-702	167	16	submodule	submodule	NOUN
iajs-702	167	17	by	by	ADP
iajs-702	167	18	[	[	X
iajs-702	167	19	4	4	NUM
iajs-702	167	20	,	,	PUNCT
iajs-702	167	21	exc.3	exc.3	NOUN
iajs-702	167	22	,	,	PUNCT
iajs-702	167	23	p.19	p.19	NOUN
iajs-702	167	24	]	]	PUNCT
iajs-702	167	25	.	.	PUNCT
iajs-702	168	1	it	it	PRON
iajs-702	168	2	follows	follow	VERB
iajs-702	168	3	that	that	SCONJ
iajs-702	168	4	a	a	DET
iajs-702	168	5	=	=	X
iajs-702	168	6	b.	b.	PROPN
iajs-702	168	7	thus	thus	ADV
iajs-702	168	8	a	a	PRON
iajs-702	168	9	is	be	AUX
iajs-702	168	10	a	a	DET
iajs-702	168	11	direct	direct	ADJ
iajs-702	168	12	summand	summand	NOUN
iajs-702	168	13	of	of	ADP
iajs-702	168	14	m.	m.	NOUN
iajs-702	168	15	hence	hence	ADV
iajs-702	168	16	m	m	VERB
iajs-702	168	17	is	be	AUX
iajs-702	168	18	a	a	DET
iajs-702	168	19	max	max	PROPN
iajs-702	168	20	-	-	PUNCT
iajs-702	168	21	cs	cs	PROPN
iajs-702	168	22	module	module	NOUN
iajs-702	168	23	.	.	PUNCT
iajs-702	169	1	the	the	DET
iajs-702	169	2	converse	converse	NOUN
iajs-702	169	3	of	of	ADP
iajs-702	169	4	the	the	DET
iajs-702	169	5	previous	previous	ADJ
iajs-702	169	6	proposition	proposition	NOUN
iajs-702	169	7	is	be	AUX
iajs-702	169	8	not	not	PART
iajs-702	169	9	true	true	ADJ
iajs-702	169	10	in	in	ADP
iajs-702	169	11	general	general	ADJ
iajs-702	169	12	,	,	PUNCT
iajs-702	169	13	as	as	SCONJ
iajs-702	169	14	the	the	DET
iajs-702	169	15	following	follow	VERB
iajs-702	169	16	example	example	NOUN
iajs-702	169	17	shows	show	VERB
iajs-702	169	18	:	:	PUNCT
iajs-702	169	19	1.20	1.20	NUM
iajs-702	169	20	example	example	NOUN
iajs-702	169	21	:	:	PUNCT
iajs-702	169	22	ℤ	ℤ	PROPN
iajs-702	169	23	as	as	ADP
iajs-702	169	24	a	a	DET
iajs-702	169	25	ℤ-module	ℤ-module	PROPN
iajs-702	169	26	is	be	AUX
iajs-702	169	27	a	a	DET
iajs-702	169	28	max	max	PROPN
iajs-702	169	29	-	-	PUNCT
iajs-702	169	30	cs	cs	PROPN
iajs-702	169	31	module	module	NOUN
iajs-702	169	32	;	;	PUNCT
iajs-702	169	33	also	also	ADV
iajs-702	169	34	ℤ	ℤ	PROPN
iajs-702	169	35	is	be	AUX
iajs-702	169	36	a	a	DET
iajs-702	169	37	multiplication	multiplication	NOUN
iajs-702	169	38	ℤ-module	ℤ-module	PROPN
iajs-702	169	39	.	.	PUNCT
iajs-702	169	40	مجلة	مجلة	PROPN
iajs-702	169	41	إبن	إبن	VERB
iajs-702	169	42	الھیثم	الھیثم	NOUN
iajs-702	169	43	للعلوم	للعلوم	NOUN
iajs-702	169	44	الصرفة	الصرفة	NOUN
iajs-702	169	45	و	و	PRON
iajs-702	169	46	التطبیقیة	التطبیقیة	PROPN
iajs-702	169	47	2012	2012	NUM
iajs-702	169	48	السنة	السنة	NOUN
iajs-702	170	1	25	25	NUM
iajs-702	171	1	المجلد	المجلد	NOUN
iajs-702	171	2	1	1	NUM
iajs-702	171	3	العدد	العدد	PROPN
iajs-702	171	4	ibn	ibn	PROPN
iajs-702	171	5	al	al	PROPN
iajs-702	171	6	-	-	PUNCT
iajs-702	171	7	haitham	haitham	PROPN
iajs-702	171	8	journal	journal	PROPN
iajs-702	171	9	for	for	ADP
iajs-702	171	10	pure	pure	ADJ
iajs-702	171	11	and	and	CCONJ
iajs-702	171	12	applied	apply	VERB
iajs-702	171	13	science	science	NOUN
iajs-702	171	14	no	no	NOUN
iajs-702	171	15	.	.	NOUN
iajs-702	171	16	1	1	NUM
iajs-702	171	17	vol	vol	NOUN
iajs-702	171	18	.	.	PUNCT
iajs-702	172	1	25	25	NUM
iajs-702	172	2	year	year	NOUN
iajs-702	172	3	2012	2012	NUM
iajs-702	172	4	but	but	CCONJ
iajs-702	172	5	2ℤ	2ℤ	NOUN
iajs-702	172	6	as	as	ADP
iajs-702	172	7	a	a	DET
iajs-702	172	8	ℤ-module	ℤ-module	PROPN
iajs-702	172	9	is	be	AUX
iajs-702	172	10	a	a	DET
iajs-702	172	11	maximal	maximal	ADJ
iajs-702	172	12	submodule	submodule	NOUN
iajs-702	172	13	of	of	ADP
iajs-702	172	14	ℤ	ℤ	PROPN
iajs-702	172	15	and	and	CCONJ
iajs-702	172	16	it	it	PRON
iajs-702	172	17	is	be	AUX
iajs-702	172	18	not	not	PART
iajs-702	172	19	a	a	DET
iajs-702	172	20	direct	direct	ADJ
iajs-702	172	21	summand	summand	NOUN
iajs-702	172	22	.	.	PUNCT
iajs-702	173	1	note	note	VERB
iajs-702	173	2	that	that	SCONJ
iajs-702	173	3	we	we	PRON
iajs-702	173	4	have	have	VERB
iajs-702	173	5	an	an	DET
iajs-702	173	6	analogous	analogous	ADJ
iajs-702	173	7	result	result	NOUN
iajs-702	173	8	to	to	PART
iajs-702	173	9	corollary	corollary	VERB
iajs-702	173	10	1.16	1.16	NUM
iajs-702	173	11	,	,	PUNCT
iajs-702	173	12	for	for	ADP
iajs-702	173	13	max	max	PROPN
iajs-702	173	14	-	-	PUNCT
iajs-702	173	15	cs	cs	PROPN
iajs-702	173	16	modules	module	NOUN
iajs-702	173	17	,	,	PUNCT
iajs-702	173	18	that	that	PRON
iajs-702	173	19	is	be	AUX
iajs-702	173	20	a	a	DET
iajs-702	173	21	direct	direct	ADJ
iajs-702	173	22	summand	summand	NOUN
iajs-702	173	23	of	of	ADP
iajs-702	173	24	max	max	PROPN
iajs-702	173	25	-	-	PUNCT
iajs-702	173	26	cs	cs	PROPN
iajs-702	173	27	module	module	NOUN
iajs-702	173	28	is	be	AUX
iajs-702	173	29	max	max	PROPN
iajs-702	173	30	-	-	PUNCT
iajs-702	173	31	cs	cs	PROPN
iajs-702	173	32	module	module	NOUN
iajs-702	173	33	,	,	PUNCT
iajs-702	173	34	but	but	CCONJ
iajs-702	173	35	first	first	ADV
iajs-702	173	36	we	we	PRON
iajs-702	173	37	prove	prove	VERB
iajs-702	173	38	the	the	DET
iajs-702	173	39	following	following	NOUN
iajs-702	173	40	:	:	PUNCT
iajs-702	173	41	1.21	1.21	NUM
iajs-702	173	42	proposition	proposition	NOUN
iajs-702	173	43	:	:	PUNCT
iajs-702	173	44	let	let	VERB
iajs-702	173	45	m	m	PRON
iajs-702	173	46	be	be	AUX
iajs-702	173	47	an	an	DET
iajs-702	173	48	r	r	NOUN
iajs-702	173	49	-	-	PUNCT
iajs-702	173	50	module	module	NOUN
iajs-702	173	51	,	,	PUNCT
iajs-702	173	52	and	and	CCONJ
iajs-702	173	53	let	let	VERB
iajs-702	173	54	n	n	PRON
iajs-702	173	55	be	be	AUX
iajs-702	173	56	a	a	DET
iajs-702	173	57	closed	closed	ADJ
iajs-702	173	58	submodule	submodule	NOUN
iajs-702	173	59	of	of	ADP
iajs-702	173	60	m.	m.	NOUN
iajs-702	173	61	if	if	SCONJ
iajs-702	173	62	m	m	NOUN
iajs-702	173	63	is	be	AUX
iajs-702	173	64	a	a	DET
iajs-702	173	65	max	max	PROPN
iajs-702	173	66	-	-	PUNCT
iajs-702	173	67	cs	cs	PROPN
iajs-702	173	68	module	module	NOUN
iajs-702	173	69	,	,	PUNCT
iajs-702	173	70	then	then	ADV
iajs-702	173	71	(	(	PUNCT
iajs-702	173	72	m	m	NOUN
iajs-702	173	73	/	/	SYM
iajs-702	173	74	n	n	CCONJ
iajs-702	173	75	)	)	PUNCT
iajs-702	173	76	is	be	AUX
iajs-702	173	77	a	a	DET
iajs-702	173	78	max	max	PROPN
iajs-702	173	79	-	-	PUNCT
iajs-702	173	80	cs	cs	ADJ
iajs-702	173	81	r	r	NOUN
iajs-702	173	82	-	-	PUNCT
iajs-702	173	83	module	module	NOUN
iajs-702	173	84	,	,	PUNCT
iajs-702	173	85	provided	provide	VERB
iajs-702	173	86	m	m	PROPN
iajs-702	173	87	is	be	AUX
iajs-702	173	88	not	not	PART
iajs-702	173	89	faithful	faithful	ADJ
iajs-702	173	90	.	.	PUNCT
iajs-702	174	1	proof	proof	NOUN
iajs-702	174	2	:	:	PUNCT
iajs-702	174	3	let	let	VERB
iajs-702	174	4	(	(	PUNCT
iajs-702	174	5	b	b	NOUN
iajs-702	174	6	/	/	SYM
iajs-702	174	7	n	n	CCONJ
iajs-702	174	8	)	)	PUNCT
iajs-702	174	9	be	be	AUX
iajs-702	174	10	a	a	DET
iajs-702	174	11	maximal	maximal	ADJ
iajs-702	174	12	closed	closed	ADJ
iajs-702	174	13	submodule	submodule	NOUN
iajs-702	174	14	in	in	ADP
iajs-702	174	15	(	(	PUNCT
iajs-702	174	16	m	m	NOUN
iajs-702	174	17	/	/	SYM
iajs-702	174	18	n	n	CCONJ
iajs-702	174	19	)	)	PUNCT
iajs-702	174	20	,	,	PUNCT
iajs-702	174	21	with	with	ADP
iajs-702	174	22	annr(b	annr(b	PROPN
iajs-702	174	23	/	/	SYM
iajs-702	174	24	n)≠	n)≠	PROPN
iajs-702	174	25	0	0	NUM
iajs-702	174	26	.	.	PUNCT
iajs-702	175	1	we	we	PRON
iajs-702	175	2	claim	claim	VERB
iajs-702	175	3	that	that	SCONJ
iajs-702	175	4	b	b	NOUN
iajs-702	175	5	is	be	AUX
iajs-702	175	6	a	a	DET
iajs-702	175	7	maximal	maximal	ADJ
iajs-702	175	8	closed	closed	ADJ
iajs-702	175	9	submodule	submodule	NOUN
iajs-702	175	10	in	in	ADP
iajs-702	175	11	m.	m.	NOUN
iajs-702	175	12	to	to	PART
iajs-702	175	13	prove	prove	VERB
iajs-702	175	14	our	our	PRON
iajs-702	175	15	assertation	assertation	NOUN
iajs-702	175	16	:	:	PUNCT
iajs-702	175	17	first	first	ADV
iajs-702	175	18	,	,	PUNCT
iajs-702	175	19	assume	assume	VERB
iajs-702	175	20	that	that	SCONJ
iajs-702	175	21	b	b	X
iajs-702	175	22			PROPN
iajs-702	175	23	e	e	NOUN
iajs-702	175	24	l	l	NOUN
iajs-702	175	25			NUM
iajs-702	175	26	m.	m.	NOUN
iajs-702	175	27	but	but	CCONJ
iajs-702	175	28	n	n	PRON
iajs-702	175	29	is	be	AUX
iajs-702	175	30	closed	close	VERB
iajs-702	175	31	in	in	ADP
iajs-702	175	32	m	m	PROPN
iajs-702	175	33	and	and	CCONJ
iajs-702	175	34	n	n	CCONJ
iajs-702	175	35			PROPN
iajs-702	175	36	b	b	PROPN
iajs-702	175	37			PROPN
iajs-702	175	38	l	l	PROPN
iajs-702	175	39	,	,	PUNCT
iajs-702	175	40	so	so	ADV
iajs-702	175	41	n	n	NUM
iajs-702	175	42	is	be	AUX
iajs-702	175	43	closed	close	VERB
iajs-702	175	44	in	in	ADP
iajs-702	175	45	l.	l.	NOUN
iajs-702	175	46	hence	hence	ADV
iajs-702	175	47	by	by	ADP
iajs-702	175	48	[	[	X
iajs-702	175	49	4	4	NUM
iajs-702	175	50	,	,	PUNCT
iajs-702	175	51	proposition	proposition	NOUN
iajs-702	175	52	1.4	1.4	NUM
iajs-702	175	53	,	,	PUNCT
iajs-702	175	54	p.18	p.18	NOUN
iajs-702	175	55	]	]	X
iajs-702	175	56	,	,	PUNCT
iajs-702	175	57	n	n	PRON
iajs-702	175	58			PROPN
iajs-702	175	59	b	b	PROPN
iajs-702	175	60			PROPN
iajs-702	175	61	e	e	NOUN
iajs-702	175	62	l	l	NOUN
iajs-702	175	63	implies	imply	VERB
iajs-702	175	64	b	b	X
iajs-702	175	65	/	/	SYM
iajs-702	175	66	n	n	CCONJ
iajs-702	175	67			NOUN
iajs-702	175	68	e	e	X
iajs-702	175	69	l	l	X
iajs-702	175	70	/	/	SYM
iajs-702	175	71	n	n	CCONJ
iajs-702	175	72			PROPN
iajs-702	175	73	m	m	PROPN
iajs-702	175	74	/	/	SYM
iajs-702	175	75	n.	n.	NOUN
iajs-702	175	76	hence	hence	ADV
iajs-702	175	77	(	(	PUNCT
iajs-702	175	78	b	b	X
iajs-702	175	79	/	/	SYM
iajs-702	175	80	n	n	CCONJ
iajs-702	175	81	)	)	PUNCT
iajs-702	175	82	=	=	SYM
iajs-702	175	83	(	(	PUNCT
iajs-702	175	84	l	l	NOUN
iajs-702	175	85	/	/	SYM
iajs-702	175	86	n	n	CCONJ
iajs-702	175	87	)	)	PUNCT
iajs-702	175	88	,	,	PUNCT
iajs-702	175	89	since	since	SCONJ
iajs-702	175	90	(	(	PUNCT
iajs-702	175	91	b	b	NOUN
iajs-702	175	92	/	/	SYM
iajs-702	175	93	n	n	CCONJ
iajs-702	175	94	)	)	PUNCT
iajs-702	175	95	closed	close	VERB
iajs-702	175	96	in	in	ADP
iajs-702	175	97	(	(	PUNCT
iajs-702	175	98	m	m	NOUN
iajs-702	175	99	/	/	SYM
iajs-702	175	100	n	n	CCONJ
iajs-702	175	101	)	)	PUNCT
iajs-702	175	102	.	.	PUNCT
iajs-702	176	1	thus	thus	ADV
iajs-702	176	2	b	b	X
iajs-702	176	3	=	=	SYM
iajs-702	176	4	l	l	PROPN
iajs-702	176	5	and	and	CCONJ
iajs-702	176	6	b	b	PROPN
iajs-702	176	7	is	be	AUX
iajs-702	176	8	closed	close	VERB
iajs-702	176	9	in	in	ADP
iajs-702	176	10	m.	m.	NOUN
iajs-702	176	11	now	now	ADV
iajs-702	176	12	,	,	PUNCT
iajs-702	176	13	assume	assume	VERB
iajs-702	176	14	there	there	PRON
iajs-702	176	15	exists	exist	VERB
iajs-702	176	16	a	a	DET
iajs-702	176	17	closed	closed	ADJ
iajs-702	176	18	submodule	submodule	NOUN
iajs-702	176	19	b	b	NOUN
iajs-702	176	20	'	'	PUNCT
iajs-702	176	21	of	of	ADP
iajs-702	176	22	m	m	PRON
iajs-702	176	23	such	such	ADJ
iajs-702	177	1	that	that	DET
iajs-702	177	2	b	b	PROPN
iajs-702	177	3			PROPN
iajs-702	177	4	b	b	PROPN
iajs-702	177	5	'	'	NOUN
iajs-702	177	6	,	,	PUNCT
iajs-702	177	7	hence	hence	ADV
iajs-702	177	8	n	n	ADJ
iajs-702	177	9			NUM
iajs-702	177	10	b	b	NOUN
iajs-702	177	11	'	'	PUNCT
iajs-702	177	12			NOUN
iajs-702	177	13	m.	m.	NOUN
iajs-702	177	14	then	then	ADV
iajs-702	177	15	,	,	PUNCT
iajs-702	177	16	by	by	ADP
iajs-702	177	17	[	[	X
iajs-702	177	18	4	4	NUM
iajs-702	177	19	,	,	PUNCT
iajs-702	177	20	exc.16	exc.16	PROPN
iajs-702	177	21	,	,	PUNCT
iajs-702	177	22	p.20	p.20	PROPN
iajs-702	177	23	]	]	X
iajs-702	177	24	,	,	PUNCT
iajs-702	177	25	(	(	PUNCT
iajs-702	177	26	b'/n	b'/n	NOUN
iajs-702	177	27	)	)	PUNCT
iajs-702	177	28	is	be	AUX
iajs-702	177	29	closed	close	VERB
iajs-702	177	30	in	in	ADP
iajs-702	177	31	(	(	PUNCT
iajs-702	177	32	m	m	NOUN
iajs-702	177	33	/	/	SYM
iajs-702	177	34	n	n	CCONJ
iajs-702	177	35	)	)	PUNCT
iajs-702	177	36	and	and	CCONJ
iajs-702	177	37	so	so	ADV
iajs-702	177	38	that	that	SCONJ
iajs-702	177	39	(	(	PUNCT
iajs-702	177	40	b	b	X
iajs-702	177	41	/	/	SYM
iajs-702	177	42	n	n	CCONJ
iajs-702	177	43	)	)	PUNCT
iajs-702	177	44			PROPN
iajs-702	177	45	(	(	PUNCT
iajs-702	177	46	b'/n	b'/n	NOUN
iajs-702	177	47	)	)	PUNCT
iajs-702	177	48	.	.	PUNCT
iajs-702	178	1	this	this	PRON
iajs-702	178	2	implies	imply	VERB
iajs-702	178	3	(	(	PUNCT
iajs-702	178	4	b	b	NOUN
iajs-702	178	5	/	/	SYM
iajs-702	178	6	n	n	CCONJ
iajs-702	178	7	)	)	PUNCT
iajs-702	178	8	=	=	SYM
iajs-702	178	9	(	(	PUNCT
iajs-702	178	10	b'/n	b'/n	NOUN
iajs-702	178	11	)	)	PUNCT
iajs-702	178	12	,	,	PUNCT
iajs-702	178	13	since	since	SCONJ
iajs-702	178	14	(	(	PUNCT
iajs-702	178	15	b	b	NOUN
iajs-702	178	16	/	/	SYM
iajs-702	178	17	n	n	CCONJ
iajs-702	178	18	)	)	PUNCT
iajs-702	178	19	is	be	AUX
iajs-702	178	20	a	a	DET
iajs-702	178	21	maximal	maximal	ADJ
iajs-702	178	22	closed	closed	ADJ
iajs-702	178	23	submodule	submodule	NOUN
iajs-702	178	24	in	in	ADP
iajs-702	178	25	(	(	PUNCT
iajs-702	178	26	m	m	NOUN
iajs-702	178	27	/	/	SYM
iajs-702	178	28	n	n	CCONJ
iajs-702	178	29	)	)	PUNCT
iajs-702	178	30	.	.	PUNCT
iajs-702	179	1	moreover	moreover	ADV
iajs-702	179	2	annrb	annrb	PROPN
iajs-702	179	3			PROPN
iajs-702	179	4	annrm	annrm	NOUN
iajs-702	179	5	≠	≠	PROPN
iajs-702	179	6	0	0	NUM
iajs-702	179	7	,	,	PUNCT
iajs-702	179	8	so	so	ADV
iajs-702	179	9	annb	annb	ADJ
iajs-702	179	10	≠	≠	NOUN
iajs-702	179	11	0	0	X
iajs-702	179	12	.	.	PUNCT
iajs-702	180	1	since	since	SCONJ
iajs-702	180	2	m	m	PROPN
iajs-702	180	3	is	be	AUX
iajs-702	180	4	max	max	PROPN
iajs-702	180	5	-	-	PUNCT
iajs-702	180	6	cs	cs	PROPN
iajs-702	180	7	then	then	ADV
iajs-702	180	8	b	b	PROPN
iajs-702	180	9			PROPN
iajs-702	180	10	k	k	PROPN
iajs-702	181	1	=	=	PUNCT
iajs-702	181	2	m	m	VERB
iajs-702	181	3	for	for	ADP
iajs-702	181	4	some	some	DET
iajs-702	181	5	k	k	PROPN
iajs-702	181	6			NUM
iajs-702	181	7	m	m	NOUN
iajs-702	181	8	,	,	PUNCT
iajs-702	181	9	and	and	CCONJ
iajs-702	181	10	hence	hence	ADV
iajs-702	181	11	(	(	PUNCT
iajs-702	181	12	b	b	X
iajs-702	181	13	/	/	SYM
iajs-702	181	14	n)((k+	n)((k+	ADJ
iajs-702	181	15	n)/n	n)/n	PROPN
iajs-702	181	16	)	)	PUNCT
iajs-702	181	17	=	=	PRON
iajs-702	181	18	(	(	PUNCT
iajs-702	181	19	m	m	NOUN
iajs-702	181	20	/	/	SYM
iajs-702	181	21	n	n	CCONJ
iajs-702	181	22	)	)	PUNCT
iajs-702	181	23	.	.	PUNCT
iajs-702	182	1	it	it	PRON
iajs-702	182	2	follows	follow	VERB
iajs-702	182	3	that	that	SCONJ
iajs-702	182	4	(	(	PUNCT
iajs-702	182	5	m	m	NOUN
iajs-702	182	6	/n	/n	PUNCT
iajs-702	182	7	)	)	PUNCT
iajs-702	182	8	is	be	AUX
iajs-702	182	9	a	a	DET
iajs-702	182	10	max	max	PROPN
iajs-702	182	11	-	-	PUNCT
iajs-702	182	12	cs	cs	PROPN
iajs-702	182	13	module	module	NOUN
iajs-702	182	14	.	.	PUNCT
iajs-702	183	1	1.22	1.22	NUM
iajs-702	183	2	corollary	corollary	NOUN
iajs-702	183	3	:	:	PUNCT
iajs-702	183	4	a	a	DET
iajs-702	183	5	direct	direct	ADJ
iajs-702	183	6	summand	summand	NOUN
iajs-702	183	7	of	of	ADP
iajs-702	183	8	a	a	DET
iajs-702	183	9	max	max	PROPN
iajs-702	183	10	-	-	PROPN
iajs-702	183	11	cs	cs	ADJ
iajs-702	183	12	r	r	NOUN
iajs-702	183	13	-	-	PUNCT
iajs-702	183	14	module	module	NOUN
iajs-702	183	15	m	m	NOUN
iajs-702	183	16	is	be	AUX
iajs-702	183	17	a	a	DET
iajs-702	183	18	max	max	PROPN
iajs-702	183	19	-	-	PUNCT
iajs-702	183	20	cs	cs	PROPN
iajs-702	183	21	module	module	NOUN
iajs-702	183	22	,	,	PUNCT
iajs-702	183	23	provided	provide	VERB
iajs-702	183	24	m	m	PROPN
iajs-702	183	25	is	be	AUX
iajs-702	183	26	not	not	PART
iajs-702	183	27	faithful	faithful	ADJ
iajs-702	183	28	.	.	PUNCT
iajs-702	184	1	proof	proof	NOUN
iajs-702	184	2	:	:	PUNCT
iajs-702	184	3	since	since	SCONJ
iajs-702	184	4	n	n	PRON
iajs-702	184	5	is	be	AUX
iajs-702	184	6	a	a	DET
iajs-702	184	7	direct	direct	ADJ
iajs-702	184	8	summand	summand	NOUN
iajs-702	184	9	of	of	ADP
iajs-702	184	10	m	m	PROPN
iajs-702	184	11	,	,	PUNCT
iajs-702	184	12	so	so	ADV
iajs-702	184	13	m	m	NOUN
iajs-702	184	14	=	=	SYM
iajs-702	184	15	n	n	X
iajs-702	184	16			ADJ
iajs-702	184	17	w	w	NOUN
iajs-702	184	18	for	for	ADP
iajs-702	184	19	some	some	DET
iajs-702	184	20	w	w	NOUN
iajs-702	184	21			PROPN
iajs-702	184	22	m.	m.	NOUN
iajs-702	184	23	hence	hence	ADV
iajs-702	184	24	(	(	PUNCT
iajs-702	184	25	m	m	NOUN
iajs-702	184	26	/	/	SYM
iajs-702	184	27	w	w	NOUN
iajs-702	184	28	)	)	PUNCT
iajs-702	184	29	isomorphic	isomorphic	ADJ
iajs-702	184	30	to	to	ADP
iajs-702	184	31	n	n	NUM
iajs-702	184	32	by	by	ADP
iajs-702	184	33	second	second	ADJ
iajs-702	184	34	isomorphism	isomorphism	NOUN
iajs-702	184	35	theorem	theorem	VERB
iajs-702	184	36	.	.	PUNCT
iajs-702	185	1	but	but	CCONJ
iajs-702	185	2	(	(	PUNCT
iajs-702	185	3	m	m	NOUN
iajs-702	185	4	/	/	SYM
iajs-702	185	5	w	w	NOUN
iajs-702	185	6	)	)	PUNCT
iajs-702	185	7	is	be	AUX
iajs-702	185	8	a	a	DET
iajs-702	185	9	max	max	PROPN
iajs-702	185	10	-	-	PUNCT
iajs-702	185	11	cs	cs	PROPN
iajs-702	185	12	by	by	ADP
iajs-702	185	13	proposition	proposition	NOUN
iajs-702	185	14	1.21	1.21	NUM
iajs-702	185	15	.	.	PUNCT
iajs-702	186	1	so	so	ADV
iajs-702	186	2	n	n	PROPN
iajs-702	186	3	is	be	AUX
iajs-702	186	4	a	a	DET
iajs-702	186	5	max	max	PROPN
iajs-702	186	6	-	-	PUNCT
iajs-702	186	7	cs	cs	PROPN
iajs-702	186	8	module	module	NOUN
iajs-702	186	9	,	,	PUNCT
iajs-702	186	10	by	by	ADP
iajs-702	186	11	remark	remark	NOUN
iajs-702	186	12	1.3	1.3	NUM
iajs-702	186	13	(	(	PUNCT
iajs-702	186	14	10	10	NUM
iajs-702	186	15	)	)	PUNCT
iajs-702	186	16	.	.	PUNCT
iajs-702	187	1	1.23	1.23	NUM
iajs-702	187	2	corollary	corollary	NOUN
iajs-702	187	3	:	:	PUNCT
iajs-702	187	4	let	let	VERB
iajs-702	187	5	m	m	PRON
iajs-702	187	6	be	be	AUX
iajs-702	187	7	an	an	DET
iajs-702	187	8	r	r	NOUN
iajs-702	187	9	-	-	PUNCT
iajs-702	187	10	module	module	NOUN
iajs-702	187	11	.	.	PUNCT
iajs-702	188	1	if	if	SCONJ
iajs-702	188	2	m	m	PROPN
iajs-702	188	3			ADJ
iajs-702	188	4	m	m	VERB
iajs-702	188	5	is	be	AUX
iajs-702	188	6	a	a	DET
iajs-702	188	7	max	max	PROPN
iajs-702	188	8	-	-	PUNCT
iajs-702	188	9	cs	cs	PROPN
iajs-702	188	10	module	module	NOUN
iajs-702	188	11	.	.	PUNCT
iajs-702	189	1	then	then	ADV
iajs-702	189	2	m	m	PROPN
iajs-702	189	3	is	be	AUX
iajs-702	189	4	max	max	PROPN
iajs-702	189	5	-	-	PUNCT
iajs-702	189	6	cs	cs	PROPN
iajs-702	189	7	module	module	NOUN
iajs-702	189	8	provided	provide	VERB
iajs-702	189	9	m	m	PROPN
iajs-702	189	10	is	be	AUX
iajs-702	189	11	not	not	PART
iajs-702	189	12	faithful	faithful	ADJ
iajs-702	189	13	.	.	PUNCT
iajs-702	190	1	proof	proof	NOUN
iajs-702	190	2	:	:	PUNCT
iajs-702	190	3	it	it	PRON
iajs-702	190	4	follows	follow	VERB
iajs-702	190	5	by	by	ADP
iajs-702	190	6	corollary	corollary	ADJ
iajs-702	190	7	1.22	1.22	NUM
iajs-702	190	8	.	.	PUNCT
iajs-702	191	1	recall	recall	VERB
iajs-702	191	2	that	that	SCONJ
iajs-702	191	3	a	a	DET
iajs-702	191	4	proper	proper	ADJ
iajs-702	191	5	submodule	submodule	NOUN
iajs-702	191	6	n	n	PROPN
iajs-702	191	7	of	of	ADP
iajs-702	191	8	an	an	DET
iajs-702	191	9	r	r	NOUN
iajs-702	191	10	-	-	PUNCT
iajs-702	191	11	module	module	NOUN
iajs-702	191	12	m	m	NOUN
iajs-702	191	13	is	be	AUX
iajs-702	191	14	called	call	VERB
iajs-702	191	15	prime	prime	ADJ
iajs-702	191	16	submodule	submodule	NOUN
iajs-702	192	1	if	if	SCONJ
iajs-702	192	2	whenever	whenever	SCONJ
iajs-702	192	3	r	r	NOUN
iajs-702	192	4			NOUN
iajs-702	192	5	r	r	NOUN
iajs-702	192	6	,	,	PUNCT
iajs-702	192	7	x	x	SYM
iajs-702	192	8			NOUN
iajs-702	192	9	m	m	PROPN
iajs-702	192	10	,	,	PUNCT
iajs-702	192	11	r	r	NOUN
iajs-702	192	12	x	x	SYM
iajs-702	192	13			NOUN
iajs-702	192	14	n	n	PRON
iajs-702	192	15	implies	imply	VERB
iajs-702	192	16	x	x	PUNCT
iajs-702	192	17			NOUN
iajs-702	192	18	n	n	CCONJ
iajs-702	192	19	or	or	CCONJ
iajs-702	192	20	r	r	NOUN
iajs-702	192	21			NOUN
iajs-702	192	22	[	[	X
iajs-702	192	23	n	n	CCONJ
iajs-702	192	24	:	:	PUNCT
iajs-702	192	25	m	m	VERB
iajs-702	192	26	]	]	X
iajs-702	192	27	.	.	PUNCT
iajs-702	193	1	an	an	DET
iajs-702	193	2	r	r	NOUN
iajs-702	193	3	-	-	PUNCT
iajs-702	193	4	module	module	NOUN
iajs-702	193	5	m	m	NOUN
iajs-702	193	6	is	be	AUX
iajs-702	193	7	called	call	VERB
iajs-702	193	8	a	a	DET
iajs-702	193	9	prime	prime	ADJ
iajs-702	193	10	module	module	NOUN
iajs-702	193	11	if	if	SCONJ
iajs-702	193	12	annrm	annrm	NOUN
iajs-702	193	13	=	=	NOUN
iajs-702	193	14	annrn	annrn	NOUN
iajs-702	193	15	for	for	ADP
iajs-702	193	16	each	each	DET
iajs-702	193	17	submodule	submodule	NOUN
iajs-702	193	18	n	n	PROPN
iajs-702	193	19	of	of	ADP
iajs-702	193	20	m	m	PROPN
iajs-702	193	21	.	.	PUNCT
iajs-702	194	1	equivalently	equivalently	ADV
iajs-702	194	2	,	,	PUNCT
iajs-702	194	3	m	m	VERB
iajs-702	194	4	is	be	AUX
iajs-702	194	5	called	call	VERB
iajs-702	194	6	a	a	DET
iajs-702	194	7	prime	prime	ADJ
iajs-702	194	8	module	module	NOUN
iajs-702	194	9	if	if	SCONJ
iajs-702	194	10	(	(	PUNCT
iajs-702	194	11	0	0	NUM
iajs-702	194	12	)	)	PUNCT
iajs-702	194	13	is	be	AUX
iajs-702	194	14	a	a	DET
iajs-702	194	15	prime	prime	ADJ
iajs-702	194	16	submodule	submodule	NOUN
iajs-702	194	17	of	of	ADP
iajs-702	194	18	m	m	PRON
iajs-702	194	19	,	,	PUNCT
iajs-702	194	20	[	[	X
iajs-702	194	21	8	8	NUM
iajs-702	194	22	]	]	PUNCT
iajs-702	194	23	.	.	PUNCT
iajs-702	195	1	1.24	1.24	NUM
iajs-702	195	2	corollary	corollary	NOUN
iajs-702	195	3	:	:	PUNCT
iajs-702	195	4	let	let	VERB
iajs-702	195	5	m	m	PRON
iajs-702	195	6	be	be	AUX
iajs-702	195	7	a	a	DET
iajs-702	195	8	not	not	PART
iajs-702	195	9	faithful	faithful	ADJ
iajs-702	195	10	prime	prime	ADJ
iajs-702	195	11	max	max	PROPN
iajs-702	195	12	-	-	PROPN
iajs-702	195	13	cs	cs	ADJ
iajs-702	195	14	r	r	NOUN
iajs-702	195	15	-	-	PUNCT
iajs-702	195	16	module	module	NOUN
iajs-702	195	17	.	.	PUNCT
iajs-702	196	1	if	if	SCONJ
iajs-702	196	2	f	f	PROPN
iajs-702	196	3	:	:	PUNCT
iajs-702	196	4	m	m	VERB
iajs-702	196	5			PROPN
iajs-702	196	6	m	m	X
iajs-702	196	7	'	'	PUNCT
iajs-702	196	8	is	be	AUX
iajs-702	196	9	an	an	DET
iajs-702	196	10	epimorphism	epimorphism	NOUN
iajs-702	196	11	,	,	PUNCT
iajs-702	196	12	then	then	ADV
iajs-702	196	13	m	m	NOUN
iajs-702	196	14	'	'	PUNCT
iajs-702	196	15	is	be	AUX
iajs-702	196	16	a	a	DET
iajs-702	196	17	max	max	PROPN
iajs-702	196	18	-	-	PUNCT
iajs-702	196	19	cs	cs	PROPN
iajs-702	196	20	module	module	NOUN
iajs-702	196	21	.	.	PUNCT
iajs-702	197	1	proof	proof	NOUN
iajs-702	197	2	:	:	PUNCT
iajs-702	197	3	m	m	VERB
iajs-702	197	4	is	be	AUX
iajs-702	197	5	a	a	DET
iajs-702	197	6	max	max	PROPN
iajs-702	197	7	-	-	PUNCT
iajs-702	197	8	cs	cs	PROPN
iajs-702	197	9	module	module	NOUN
iajs-702	197	10	.	.	PUNCT
iajs-702	198	1	then	then	ADV
iajs-702	198	2	(	(	PUNCT
iajs-702	198	3	m	m	NOUN
iajs-702	198	4	/	/	SYM
iajs-702	198	5	ker	ker	PROPN
iajs-702	198	6	f	f	X
iajs-702	198	7	)	)	PUNCT
iajs-702	198	8	≅	≅	PROPN
iajs-702	198	9	m	m	PROPN
iajs-702	198	10	'	'	PUNCT
iajs-702	198	11	,	,	PUNCT
iajs-702	198	12	by	by	ADP
iajs-702	198	13	the	the	DET
iajs-702	198	14	1	1	NUM
iajs-702	198	15	st	st	PROPN
iajs-702	198	16	fundamental	fundamental	ADJ
iajs-702	198	17	theorem	theorem	PROPN
iajs-702	198	18	.	.	PUNCT
iajs-702	199	1	we	we	PRON
iajs-702	199	2	claim	claim	VERB
iajs-702	199	3	that	that	SCONJ
iajs-702	199	4	ker	ker	PROPN
iajs-702	200	1	f	f	PROPN
iajs-702	200	2	is	be	AUX
iajs-702	200	3	clsed	clsed	ADJ
iajs-702	200	4	submodule	submodule	NOUN
iajs-702	200	5	in	in	ADP
iajs-702	200	6	m.	m.	NOUN
iajs-702	200	7	to	to	PART
iajs-702	200	8	show	show	VERB
iajs-702	200	9	this	this	PRON
iajs-702	200	10	,	,	PUNCT
iajs-702	200	11	let	let	VERB
iajs-702	200	12	ker	ker	PROPN
iajs-702	200	13	f	f	PROPN
iajs-702	200	14	≨	≨	PROPN
iajs-702	200	15	e	e	NOUN
iajs-702	200	16	l	l	NOUN
iajs-702	200	17			NUM
iajs-702	200	18	m.	m.	NOUN
iajs-702	200	19	assume	assume	VERB
iajs-702	200	20	there	there	PRON
iajs-702	200	21	exists	exist	VERB
iajs-702	200	22	0	0	NUM
iajs-702	200	23	≠	≠	PROPN
iajs-702	200	24	x	x	X
iajs-702	200	25			NOUN
iajs-702	200	26	l	l	NOUN
iajs-702	200	27	such	such	ADJ
iajs-702	200	28	that	that	SCONJ
iajs-702	200	29	x	x	SYM
iajs-702	200	30			PROPN
iajs-702	200	31	ker	ker	PROPN
iajs-702	200	32	f.	f.	PROPN
iajs-702	200	33	مجلة	مجلة	PROPN
iajs-702	200	34	إبن	إبن	VERB
iajs-702	200	35	الھیثم	الھیثم	NOUN
iajs-702	200	36	للعلوم	للعلوم	NOUN
iajs-702	200	37	الصرفة	الصرفة	NOUN
iajs-702	200	38	و	و	PRON
iajs-702	200	39	التطبیقیة	التطبیقیة	PROPN
iajs-702	200	40	2012	2012	NUM
iajs-702	200	41	السنة	السنة	NOUN
iajs-702	201	1	25	25	NUM
iajs-702	201	2	المجلد	المجلد	NOUN
iajs-702	201	3	1	1	NUM
iajs-702	201	4	العدد	العدد	PROPN
iajs-702	201	5	ibn	ibn	PROPN
iajs-702	201	6	al	al	PROPN
iajs-702	201	7	-	-	PUNCT
iajs-702	201	8	haitham	haitham	PROPN
iajs-702	201	9	journal	journal	PROPN
iajs-702	201	10	for	for	ADP
iajs-702	201	11	pure	pure	ADJ
iajs-702	201	12	and	and	CCONJ
iajs-702	201	13	applied	apply	VERB
iajs-702	201	14	science	science	NOUN
iajs-702	201	15	no	no	NOUN
iajs-702	201	16	.	.	NOUN
iajs-702	201	17	1	1	NUM
iajs-702	201	18	vol	vol	NOUN
iajs-702	201	19	.	.	PUNCT
iajs-702	202	1	25	25	NUM
iajs-702	202	2	year	year	NOUN
iajs-702	202	3	2012	2012	NUM
iajs-702	202	4	then	then	ADV
iajs-702	202	5	there	there	PRON
iajs-702	202	6	exists	exist	VERB
iajs-702	202	7	0	0	NUM
iajs-702	202	8	≠	≠	PROPN
iajs-702	202	9	r	r	NOUN
iajs-702	202	10			NOUN
iajs-702	202	11	r	r	NOUN
iajs-702	203	1	such	such	ADJ
iajs-702	203	2	that	that	DET
iajs-702	203	3	0	0	NUM
iajs-702	203	4	≠	≠	PROPN
iajs-702	203	5	r	r	NOUN
iajs-702	203	6	x	x	X
iajs-702	203	7			NOUN
iajs-702	203	8	ker	ker	PROPN
iajs-702	203	9	f.	f.	PROPN
iajs-702	203	10	hence	hence	ADV
iajs-702	203	11	rf(x	rf(x	PROPN
iajs-702	203	12	)	)	PUNCT
iajs-702	204	1	=	=	SYM
iajs-702	204	2	0	0	X
iajs-702	204	3	.	.	PUNCT
iajs-702	205	1	since	since	SCONJ
iajs-702	205	2	m	m	PROPN
iajs-702	205	3	is	be	AUX
iajs-702	205	4	a	a	DET
iajs-702	205	5	prime	prime	ADJ
iajs-702	205	6	module	module	NOUN
iajs-702	205	7	,	,	PUNCT
iajs-702	205	8	so	so	SCONJ
iajs-702	205	9	either	either	CCONJ
iajs-702	205	10	f(x	f(x	PROPN
iajs-702	205	11	)	)	PUNCT
iajs-702	206	1	=	=	SYM
iajs-702	206	2	0	0	NUM
iajs-702	206	3	or	or	CCONJ
iajs-702	206	4	r	r	NOUN
iajs-702	206	5			NOUN
iajs-702	206	6	annm	annm	NOUN
iajs-702	206	7	.	.	PUNCT
iajs-702	207	1	but	but	CCONJ
iajs-702	207	2	f(x	f(x	PROPN
iajs-702	207	3	)	)	PUNCT
iajs-702	207	4	≠	≠	PROPN
iajs-702	207	5	0	0	NUM
iajs-702	207	6	since	since	SCONJ
iajs-702	207	7	x	x	PROPN
iajs-702	207	8			NOUN
iajs-702	207	9	ker	ker	NOUN
iajs-702	208	1	f	f	X
iajs-702	208	2	,	,	PUNCT
iajs-702	208	3	hence	hence	ADV
iajs-702	208	4	r	r	NOUN
iajs-702	208	5			NOUN
iajs-702	208	6	annm	annm	NOUN
iajs-702	208	7	and	and	CCONJ
iajs-702	208	8	this	this	PRON
iajs-702	208	9	implies	imply	VERB
iajs-702	208	10	r	r	NOUN
iajs-702	208	11	x	x	SYM
iajs-702	208	12	=	=	SYM
iajs-702	208	13	0	0	NUM
iajs-702	208	14	which	which	PRON
iajs-702	208	15	is	be	AUX
iajs-702	208	16	a	a	DET
iajs-702	208	17	contradiction	contradiction	NOUN
iajs-702	208	18	.	.	PUNCT
iajs-702	209	1	thus	thus	ADV
iajs-702	209	2	ker	ker	X
iajs-702	209	3	f	f	PROPN
iajs-702	209	4	=	=	SYM
iajs-702	209	5	l	l	NOUN
iajs-702	210	1	and	and	CCONJ
iajs-702	210	2	so	so	ADV
iajs-702	210	3	that	that	DET
iajs-702	210	4	ker	ker	PROPN
iajs-702	210	5	f	f	PROPN
iajs-702	210	6	is	be	AUX
iajs-702	210	7	a	a	DET
iajs-702	210	8	closed	closed	ADJ
iajs-702	210	9	submodule	submodule	NOUN
iajs-702	210	10	of	of	ADP
iajs-702	210	11	m.	m.	NOUN
iajs-702	210	12	since	since	SCONJ
iajs-702	210	13	annm	annm	NOUN
iajs-702	210	14	≠	≠	PROPN
iajs-702	210	15	0	0	NUM
iajs-702	210	16	by	by	ADP
iajs-702	210	17	hypothesis	hypothesis	NOUN
iajs-702	210	18	.	.	PUNCT
iajs-702	211	1	then	then	ADV
iajs-702	211	2	by	by	ADP
iajs-702	211	3	proposition	proposition	NOUN
iajs-702	211	4	1.21	1.21	NUM
iajs-702	211	5	,	,	PUNCT
iajs-702	211	6	(	(	PUNCT
iajs-702	211	7	m	m	NOUN
iajs-702	211	8	/	/	SYM
iajs-702	211	9	ker	ker	NOUN
iajs-702	211	10	f	f	X
iajs-702	211	11	)	)	PUNCT
iajs-702	211	12	is	be	AUX
iajs-702	211	13	a	a	DET
iajs-702	211	14	max	max	PROPN
iajs-702	211	15	-	-	PUNCT
iajs-702	211	16	cs	cs	PROPN
iajs-702	211	17	module	module	NOUN
iajs-702	211	18	and	and	CCONJ
iajs-702	211	19	hence	hence	ADV
iajs-702	211	20	by	by	ADP
iajs-702	211	21	remark	remark	NOUN
iajs-702	211	22	1.3	1.3	NUM
iajs-702	211	23	(	(	PUNCT
iajs-702	211	24	10	10	NUM
iajs-702	211	25	)	)	PUNCT
iajs-702	211	26	m	m	NOUN
iajs-702	211	27	'	'	PUNCT
iajs-702	211	28	is	be	AUX
iajs-702	211	29	a	a	DET
iajs-702	211	30	max	max	PROPN
iajs-702	211	31	-	-	PUNCT
iajs-702	211	32	cs	cs	PROPN
iajs-702	211	33	module	module	NOUN
iajs-702	211	34	.	.	PUNCT
iajs-702	212	1	1.25	1.25	NUM
iajs-702	212	2	corollary	corollary	NOUN
iajs-702	212	3	:	:	PUNCT
iajs-702	212	4	let	let	VERB
iajs-702	212	5	m	m	PRON
iajs-702	212	6	be	be	AUX
iajs-702	212	7	a	a	DET
iajs-702	212	8	max	max	PROPN
iajs-702	212	9	-	-	PUNCT
iajs-702	212	10	cs	cs	ADJ
iajs-702	212	11	not	not	PART
iajs-702	212	12	faithful	faithful	ADJ
iajs-702	212	13	r	r	NOUN
iajs-702	212	14	-	-	PUNCT
iajs-702	212	15	module	module	NOUN
iajs-702	212	16	.	.	PUNCT
iajs-702	213	1	if	if	SCONJ
iajs-702	213	2	n	n	PRON
iajs-702	213	3	is	be	AUX
iajs-702	213	4	a	a	DET
iajs-702	213	5	prime	prime	ADJ
iajs-702	213	6	submodule	submodule	NOUN
iajs-702	213	7	of	of	ADP
iajs-702	213	8	m	m	PROPN
iajs-702	213	9	and	and	CCONJ
iajs-702	214	1	[	[	X
iajs-702	214	2	n	n	X
iajs-702	214	3	r	r	NOUN
iajs-702	214	4	:	:	PUNCT
iajs-702	214	5	m	m	VERB
iajs-702	214	6	]	]	X
iajs-702	214	7	=	=	PUNCT
iajs-702	214	8	annrm	annrm	NOUN
iajs-702	214	9	.	.	PUNCT
iajs-702	215	1	then	then	ADV
iajs-702	215	2	(	(	PUNCT
iajs-702	215	3	m	m	NOUN
iajs-702	215	4	/	/	SYM
iajs-702	215	5	n	n	CCONJ
iajs-702	215	6	)	)	PUNCT
iajs-702	215	7	is	be	AUX
iajs-702	215	8	a	a	DET
iajs-702	215	9	max	max	PROPN
iajs-702	215	10	-	-	PUNCT
iajs-702	215	11	cs	cs	ADJ
iajs-702	215	12	r	r	NOUN
iajs-702	215	13	-	-	PUNCT
iajs-702	215	14	module	module	NOUN
iajs-702	215	15	.	.	PUNCT
iajs-702	216	1	proof	proof	NOUN
iajs-702	216	2	:	:	PUNCT
iajs-702	216	3	since	since	SCONJ
iajs-702	216	4	n	n	NOUN
iajs-702	216	5	is	be	AUX
iajs-702	216	6	a	a	DET
iajs-702	216	7	prime	prime	ADJ
iajs-702	216	8	r	r	NOUN
iajs-702	216	9	-	-	PUNCT
iajs-702	216	10	submodule	submodule	NOUN
iajs-702	216	11	of	of	ADP
iajs-702	216	12	m	m	PROPN
iajs-702	216	13	,	,	PUNCT
iajs-702	216	14	then	then	ADV
iajs-702	216	15	(	(	PUNCT
iajs-702	216	16	m	m	NOUN
iajs-702	216	17	/	/	SYM
iajs-702	216	18	n	n	CCONJ
iajs-702	216	19	)	)	PUNCT
iajs-702	216	20	is	be	AUX
iajs-702	216	21	a	a	DET
iajs-702	216	22	torsion	torsion	NOUN
iajs-702	216	23	free	free	ADJ
iajs-702	216	24	(	(	PUNCT
iajs-702	216	25	r/[n	r/[n	NOUN
iajs-702	216	26	:	:	PUNCT
iajs-702	216	27	m])-module	m])-module	NOUN
iajs-702	216	28	,	,	PUNCT
iajs-702	216	29	[	[	X
iajs-702	216	30	8	8	NUM
iajs-702	216	31	,	,	PUNCT
iajs-702	216	32	p.61	p.61	NOUN
iajs-702	216	33	-	-	PUNCT
iajs-702	216	34	69	69	NUM
iajs-702	216	35	]	]	PUNCT
iajs-702	216	36	.	.	PUNCT
iajs-702	217	1	hence	hence	ADV
iajs-702	217	2	(	(	PUNCT
iajs-702	217	3	m	m	NOUN
iajs-702	217	4	/	/	SYM
iajs-702	217	5	n	n	CCONJ
iajs-702	217	6	)	)	PUNCT
iajs-702	217	7	is	be	AUX
iajs-702	217	8	a	a	DET
iajs-702	217	9	torsion	torsion	NOUN
iajs-702	217	10	free	free	ADJ
iajs-702	217	11	(	(	PUNCT
iajs-702	217	12	r	r	NOUN
iajs-702	217	13	/	/	SYM
iajs-702	217	14	annm)-module	annm)-module	NOUN
iajs-702	217	15	,	,	PUNCT
iajs-702	217	16	and	and	CCONJ
iajs-702	217	17	so	so	SCONJ
iajs-702	217	18	that	that	SCONJ
iajs-702	217	19	n	n	PRON
iajs-702	217	20	is	be	AUX
iajs-702	217	21	closed	close	VERB
iajs-702	217	22	(	(	PUNCT
iajs-702	217	23	r	r	X
iajs-702	217	24	/	/	SYM
iajs-702	217	25	annm)-module	annm)-module	NOUN
iajs-702	217	26	of	of	ADP
iajs-702	217	27	m	m	VERB
iajs-702	217	28	by	by	ADP
iajs-702	217	29	[	[	X
iajs-702	217	30	9	9	NUM
iajs-702	217	31	,	,	PUNCT
iajs-702	217	32	remark	remark	NOUN
iajs-702	217	33	3.3,p.48	3.3,p.48	NOUN
iajs-702	217	34	]	]	PUNCT
iajs-702	217	35	.	.	PUNCT
iajs-702	218	1	it	it	PRON
iajs-702	218	2	follows	follow	VERB
iajs-702	218	3	that	that	SCONJ
iajs-702	218	4	n	n	PRON
iajs-702	218	5	is	be	AUX
iajs-702	218	6	closed	close	VERB
iajs-702	218	7	r	r	NOUN
iajs-702	218	8	-	-	PUNCT
iajs-702	218	9	submodule	submodule	NOUN
iajs-702	218	10	of	of	ADP
iajs-702	218	11	m.	m.	NOUN
iajs-702	218	12	then	then	ADV
iajs-702	218	13	by	by	ADP
iajs-702	218	14	proposition	proposition	NOUN
iajs-702	218	15	1.15	1.15	NUM
iajs-702	218	16	,	,	PUNCT
iajs-702	218	17	(	(	PUNCT
iajs-702	218	18	m	m	NOUN
iajs-702	218	19	/	/	SYM
iajs-702	218	20	n	n	CCONJ
iajs-702	218	21	)	)	PUNCT
iajs-702	218	22	is	be	AUX
iajs-702	218	23	a	a	DET
iajs-702	218	24	max	max	PROPN
iajs-702	218	25	-	-	PUNCT
iajs-702	218	26	cs	cs	ADJ
iajs-702	218	27	r	r	NOUN
iajs-702	218	28	-	-	PUNCT
iajs-702	218	29	module	module	NOUN
iajs-702	218	30	.	.	PUNCT
iajs-702	219	1	1.26	1.26	NUM
iajs-702	219	2	corollary	corollary	NOUN
iajs-702	219	3	:	:	PUNCT
iajs-702	219	4	if	if	SCONJ
iajs-702	219	5	m	m	NOUN
iajs-702	219	6	is	be	AUX
iajs-702	219	7	a	a	DET
iajs-702	219	8	max	max	PROPN
iajs-702	219	9	-	-	PUNCT
iajs-702	219	10	cs	cs	ADJ
iajs-702	219	11	not	not	PART
iajs-702	219	12	faithful	faithful	ADJ
iajs-702	219	13	r	r	NOUN
iajs-702	219	14	-	-	PUNCT
iajs-702	219	15	module	module	NOUN
iajs-702	219	16	and	and	CCONJ
iajs-702	219	17	n	n	NOUN
iajs-702	219	18	is	be	AUX
iajs-702	219	19	a	a	DET
iajs-702	219	20	submodule	submodule	NOUN
iajs-702	219	21	of	of	ADP
iajs-702	219	22	m	m	PROPN
iajs-702	219	23	,	,	PUNCT
iajs-702	219	24	such	such	ADJ
iajs-702	219	25	that	that	PRON
iajs-702	219	26	(	(	PUNCT
iajs-702	219	27	m	m	NOUN
iajs-702	219	28	/n	/n	PUNCT
iajs-702	219	29	)	)	PUNCT
iajs-702	219	30	is	be	AUX
iajs-702	219	31	torsion	torsion	NOUN
iajs-702	219	32	free	free	ADJ
iajs-702	219	33	,	,	PUNCT
iajs-702	219	34	then	then	ADV
iajs-702	219	35	(	(	PUNCT
iajs-702	219	36	m	m	NOUN
iajs-702	219	37	/	/	SYM
iajs-702	219	38	n	n	CCONJ
iajs-702	219	39	)	)	PUNCT
iajs-702	219	40	is	be	AUX
iajs-702	219	41	max	max	PROPN
iajs-702	219	42	-	-	PUNCT
iajs-702	219	43	cs	cs	PROPN
iajs-702	219	44	.	.	PROPN
iajs-702	219	45	proof	proof	NOUN
iajs-702	219	46	:	:	PUNCT
iajs-702	219	47	since	since	SCONJ
iajs-702	219	48	(	(	PUNCT
iajs-702	219	49	m	m	NOUN
iajs-702	219	50	/	/	SYM
iajs-702	219	51	n	n	CCONJ
iajs-702	219	52	)	)	PUNCT
iajs-702	219	53	is	be	AUX
iajs-702	219	54	torsion	torsion	NOUN
iajs-702	219	55	free	free	ADJ
iajs-702	219	56	r	r	NOUN
iajs-702	219	57	-	-	PUNCT
iajs-702	219	58	module	module	NOUN
iajs-702	219	59	.	.	PUNCT
iajs-702	220	1	then	then	ADV
iajs-702	220	2	n	n	PRON
iajs-702	220	3	is	be	AUX
iajs-702	220	4	closed	close	VERB
iajs-702	220	5	submodule	submodule	NOUN
iajs-702	220	6	in	in	ADP
iajs-702	220	7	m	m	PROPN
iajs-702	220	8	by	by	ADP
iajs-702	220	9	[	[	X
iajs-702	220	10	9	9	NUM
iajs-702	220	11	,	,	PUNCT
iajs-702	220	12	remark	remark	NOUN
iajs-702	220	13	3.3	3.3	NUM
iajs-702	220	14	,	,	PUNCT
iajs-702	220	15	p.48	p.48	PROPN
iajs-702	220	16	]	]	PUNCT
iajs-702	220	17	.	.	PUNCT
iajs-702	221	1	hence	hence	ADV
iajs-702	221	2	the	the	DET
iajs-702	221	3	result	result	NOUN
iajs-702	221	4	follows	follow	VERB
iajs-702	221	5	by	by	ADP
iajs-702	221	6	proposition	proposition	NOUN
iajs-702	221	7	1.21	1.21	NUM
iajs-702	221	8	.	.	PUNCT
iajs-702	222	1	now	now	ADV
iajs-702	222	2	we	we	PRON
iajs-702	222	3	have	have	VERB
iajs-702	222	4	the	the	DET
iajs-702	222	5	following	follow	VERB
iajs-702	222	6	note	note	NOUN
iajs-702	222	7	for	for	ADP
iajs-702	222	8	min	min	PROPN
iajs-702	222	9	-	-	ADJ
iajs-702	222	10	cs	cs	ADJ
iajs-702	222	11	modules	module	NOUN
iajs-702	222	12	.	.	PUNCT
iajs-702	223	1	1.27	1.27	NUM
iajs-702	223	2	remark	remark	NOUN
iajs-702	223	3	:	:	PUNCT
iajs-702	223	4	the	the	DET
iajs-702	223	5	homomorphic	homomorphic	ADJ
iajs-702	223	6	image	image	NOUN
iajs-702	223	7	of	of	ADP
iajs-702	223	8	min	min	NOUN
iajs-702	223	9	-	-	ADJ
iajs-702	223	10	cs	cs	ADJ
iajs-702	223	11	need	need	AUX
iajs-702	223	12	not	not	PART
iajs-702	223	13	be	be	AUX
iajs-702	223	14	a	a	DET
iajs-702	223	15	min	min	NOUN
iajs-702	223	16	-	-	NOUN
iajs-702	223	17	cs	cs	ADJ
iajs-702	223	18	,	,	PUNCT
iajs-702	223	19	as	as	SCONJ
iajs-702	223	20	the	the	DET
iajs-702	223	21	following	follow	VERB
iajs-702	223	22	examples	example	NOUN
iajs-702	223	23	show	show	VERB
iajs-702	223	24	.	.	PUNCT
iajs-702	224	1	1.28	1.28	NUM
iajs-702	224	2	examples	example	NOUN
iajs-702	224	3	:	:	PUNCT
iajs-702	224	4	(	(	PUNCT
iajs-702	224	5	1	1	X
iajs-702	224	6	)	)	PUNCT
iajs-702	224	7	let	let	VERB
iajs-702	224	8	m	m	PRON
iajs-702	224	9	be	be	AUX
iajs-702	224	10	the	the	DET
iajs-702	224	11	ℤ-module	ℤ-module	PROPN
iajs-702	224	12	ℤℤ.	ℤℤ.	PROPN
iajs-702	224	13	it	it	PRON
iajs-702	224	14	is	be	AUX
iajs-702	224	15	clear	clear	ADJ
iajs-702	224	16	that	that	SCONJ
iajs-702	224	17	m	m	PROPN
iajs-702	224	18	is	be	AUX
iajs-702	224	19	a	a	DET
iajs-702	224	20	min	min	PROPN
iajs-702	224	21	-	-	ADJ
iajs-702	224	22	cs	cs	ADJ
iajs-702	224	23	module	module	NOUN
iajs-702	224	24	.	.	PUNCT
iajs-702	225	1	let	let	VERB
iajs-702	225	2	n	n	NOUN
iajs-702	225	3	=	=	SYM
iajs-702	225	4	8ℤ2ℤ	8ℤ2ℤ	NUM
iajs-702	225	5	,	,	PUNCT
iajs-702	225	6	and	and	CCONJ
iajs-702	225	7	let	let	VERB
iajs-702	225	8			ADJ
iajs-702	225	9	:	:	PUNCT
iajs-702	225	10	m	m	PROPN
iajs-702	225	11			PROPN
iajs-702	225	12	(	(	PUNCT
iajs-702	225	13	m	m	PROPN
iajs-702	225	14	/	/	SYM
iajs-702	225	15	n	n	CCONJ
iajs-702	225	16	)	)	PUNCT
iajs-702	225	17	be	be	AUX
iajs-702	225	18	the	the	DET
iajs-702	225	19	natural	natural	ADJ
iajs-702	225	20	projection	projection	NOUN
iajs-702	225	21	.	.	PUNCT
iajs-702	226	1	since	since	SCONJ
iajs-702	226	2	(	(	PUNCT
iajs-702	226	3	m	m	NOUN
iajs-702	226	4	/	/	SYM
iajs-702	226	5	n	n	CCONJ
iajs-702	226	6	)	)	PUNCT
iajs-702	226	7	is	be	AUX
iajs-702	226	8	isomorphic	isomorphic	ADJ
iajs-702	226	9	to	to	ADP
iajs-702	226	10	ℤ8ℤ2	ℤ8ℤ2	PROPN
iajs-702	226	11	,	,	PUNCT
iajs-702	226	12	so	so	CCONJ
iajs-702	226	13	(	(	PUNCT
iajs-702	226	14	m	m	VERB
iajs-702	226	15	/n	/n	PRON
iajs-702	226	16	)	)	PUNCT
iajs-702	226	17	=	=	SYM
iajs-702	226	18	(m	(m	PROPN
iajs-702	226	19	)	)	PUNCT
iajs-702	226	20	is	be	AUX
iajs-702	226	21	not	not	PART
iajs-702	226	22	min	min	ADJ
iajs-702	226	23	-	-	ADJ
iajs-702	226	24	cs	cs	ADJ
iajs-702	226	25	module	module	NOUN
iajs-702	226	26	.	.	PUNCT
iajs-702	227	1	(	(	PUNCT
iajs-702	227	2	2	2	X
iajs-702	227	3	)	)	PUNCT
iajs-702	227	4	let	let	VERB
iajs-702	227	5	m	m	PRON
iajs-702	227	6	be	be	AUX
iajs-702	227	7	the	the	DET
iajs-702	227	8	ℤ-module	ℤ-module	PROPN
iajs-702	227	9	ℤℤ2	ℤℤ2	PROPN
iajs-702	227	10	.	.	PUNCT
iajs-702	228	1	m	m	PROPN
iajs-702	228	2	is	be	AUX
iajs-702	228	3	min	min	NOUN
iajs-702	228	4	-	-	PROPN
iajs-702	228	5	cs	cs	ADJ
iajs-702	228	6	.	.	PUNCT
iajs-702	229	1	let	let	VERB
iajs-702	229	2	n	n	NOUN
iajs-702	229	3	=	=	SYM
iajs-702	229	4	8ℤ	8ℤ	NUM
iajs-702	229	5	(	(	PUNCT
iajs-702	229	6	0	0	NUM
iajs-702	229	7	)	)	PUNCT
iajs-702	229	8	,	,	PUNCT
iajs-702	229	9	n	n	CCONJ
iajs-702	229	10			NUM
iajs-702	229	11	e	e	X
iajs-702	229	12	ℤ	ℤ	PROPN
iajs-702	229	13	(	(	PUNCT
iajs-702	229	14	0	0	NUM
iajs-702	229	15	)	)	PUNCT
iajs-702	229	16	,	,	PUNCT
iajs-702	229	17	but	but	CCONJ
iajs-702	229	18	(	(	PUNCT
iajs-702	229	19	m	m	NOUN
iajs-702	229	20	/	/	SYM
iajs-702	229	21	n	n	CCONJ
iajs-702	229	22	)	)	PUNCT
iajs-702	229	23	=	=	SYM
iajs-702	229	24	ℤ8ℤ2	ℤ8ℤ2	PROPN
iajs-702	229	25	is	be	AUX
iajs-702	229	26	not	not	PART
iajs-702	229	27	min	min	NOUN
iajs-702	229	28	-	-	PROPN
iajs-702	229	29	cs	cs	X
iajs-702	229	30	.	.	PROPN
iajs-702	230	1	(	(	PUNCT
iajs-702	230	2	3	3	X
iajs-702	230	3	)	)	PUNCT
iajs-702	230	4	let	let	VERB
iajs-702	230	5	m	m	AUX
iajs-702	230	6	=	=	VERB
iajs-702	230	7	ℤ8ℤ	ℤ8ℤ	VERB
iajs-702	230	8	be	be	VERB
iajs-702	230	9	a	a	DET
iajs-702	230	10	ℤ-module	ℤ-module	PROPN
iajs-702	230	11	,	,	PUNCT
iajs-702	230	12	m	m	VERB
iajs-702	230	13	is	be	AUX
iajs-702	230	14	a	a	DET
iajs-702	230	15	cs	cs	ADJ
iajs-702	230	16	-	-	NOUN
iajs-702	230	17	module	module	NOUN
iajs-702	230	18	.	.	PUNCT
iajs-702	231	1	so	so	ADV
iajs-702	231	2	m	m	PROPN
iajs-702	231	3	is	be	AUX
iajs-702	231	4	a	a	DET
iajs-702	231	5	min	min	PROPN
iajs-702	231	6	-	-	ADJ
iajs-702	231	7	cs	cs	ADJ
iajs-702	231	8	module	module	NOUN
iajs-702	231	9	.	.	PUNCT
iajs-702	232	1	let	let	VERB
iajs-702	232	2	n	n	NOUN
iajs-702	232	3	=	=	SYM
iajs-702	232	4	(	(	PUNCT
iajs-702	232	5	0	0	NUM
iajs-702	232	6	)	)	PUNCT
iajs-702	232	7	(	(	PUNCT
iajs-702	232	8	2)	2)	NUM
iajs-702	232	9	.	.	PUNCT
iajs-702	233	1	(	(	PUNCT
iajs-702	233	2	m	m	NOUN
iajs-702	233	3	/	/	SYM
iajs-702	233	4	n	n	CCONJ
iajs-702	233	5	)	)	PUNCT
iajs-702	233	6	is	be	AUX
iajs-702	233	7	isomorphic	isomorphic	ADJ
iajs-702	233	8	to	to	ADP
iajs-702	233	9	ℤ8ℤ2	ℤ8ℤ2	PROPN
iajs-702	233	10	which	which	PRON
iajs-702	233	11	is	be	AUX
iajs-702	233	12	not	not	PART
iajs-702	233	13	min	min	PROPN
iajs-702	233	14	-	-	ADJ
iajs-702	233	15	cs	cs	ADJ
iajs-702	233	16	module	module	NOUN
iajs-702	233	17	.	.	PUNCT
iajs-702	234	1	1.29	1.29	NUM
iajs-702	234	2	theorem	theorem	VERB
iajs-702	234	3	:	:	PUNCT
iajs-702	234	4	let	let	VERB
iajs-702	234	5	m	m	PRON
iajs-702	234	6	be	be	AUX
iajs-702	234	7	an	an	DET
iajs-702	234	8	r	r	NOUN
iajs-702	234	9	-	-	PUNCT
iajs-702	234	10	module	module	NOUN
iajs-702	234	11	.	.	PUNCT
iajs-702	235	1	if	if	SCONJ
iajs-702	235	2	m	m	NOUN
iajs-702	235	3	is	be	AUX
iajs-702	235	4	a	a	DET
iajs-702	235	5	faithful	faithful	ADJ
iajs-702	235	6	,	,	PUNCT
iajs-702	235	7	finitely	finitely	ADV
iajs-702	235	8	generating	generating	NOUN
iajs-702	235	9	and	and	CCONJ
iajs-702	235	10	multiplication	multiplication	NOUN
iajs-702	235	11	r	r	NOUN
iajs-702	235	12	-	-	PUNCT
iajs-702	235	13	module	module	NOUN
iajs-702	235	14	,	,	PUNCT
iajs-702	235	15	then	then	ADV
iajs-702	235	16	m	m	NOUN
iajs-702	235	17	is	be	AUX
iajs-702	235	18	a	a	DET
iajs-702	235	19	max	max	PROPN
iajs-702	235	20	-	-	PUNCT
iajs-702	235	21	cs	cs	PROPN
iajs-702	235	22	module	module	NOUN
iajs-702	235	23	if	if	SCONJ
iajs-702	235	24	and	and	CCONJ
iajs-702	235	25	only	only	ADV
iajs-702	235	26	if	if	SCONJ
iajs-702	235	27	r	r	NOUN
iajs-702	235	28	is	be	AUX
iajs-702	235	29	a	a	DET
iajs-702	235	30	max	max	PROPN
iajs-702	235	31	-	-	PUNCT
iajs-702	235	32	cs	cs	PROPN
iajs-702	235	33	ring	ring	NOUN
iajs-702	235	34	.	.	PUNCT
iajs-702	236	1	proof	proof	NOUN
iajs-702	236	2	:	:	PUNCT
iajs-702	236	3	(	(	PUNCT
iajs-702	236	4			NOUN
iajs-702	236	5	)	)	PUNCT
iajs-702	236	6	if	if	SCONJ
iajs-702	236	7	m	m	NOUN
iajs-702	236	8	is	be	AUX
iajs-702	236	9	a	a	DET
iajs-702	236	10	max	max	PROPN
iajs-702	236	11	-	-	PUNCT
iajs-702	236	12	cs	cs	ADJ
iajs-702	236	13	r	r	NOUN
iajs-702	236	14	-	-	PUNCT
iajs-702	236	15	module	module	NOUN
iajs-702	236	16	.	.	PUNCT
iajs-702	237	1	let	let	VERB
iajs-702	237	2	i	i	PRON
iajs-702	237	3	be	be	AUX
iajs-702	237	4	a	a	DET
iajs-702	237	5	maximal	maximal	ADJ
iajs-702	237	6	closed	closed	ADJ
iajs-702	237	7	ideal	ideal	NOUN
iajs-702	237	8	of	of	ADP
iajs-702	237	9	r	r	NOUN
iajs-702	237	10	such	such	ADJ
iajs-702	237	11	that	that	SCONJ
iajs-702	237	12	ann	ann	PROPN
iajs-702	237	13	i≠0	i≠0	NOUN
iajs-702	237	14	.	.	PUNCT
iajs-702	238	1	we	we	PRON
iajs-702	238	2	claim	claim	VERB
iajs-702	238	3	that	that	SCONJ
iajs-702	238	4	n	n	NOUN
iajs-702	238	5	=	=	VERB
iajs-702	239	1	i	i	PRON
iajs-702	239	2	m	m	VERB
iajs-702	239	3	is	be	AUX
iajs-702	239	4	a	a	DET
iajs-702	239	5	maximal	maximal	ADJ
iajs-702	239	6	closed	closed	ADJ
iajs-702	239	7	r	r	NOUN
iajs-702	239	8	-	-	PUNCT
iajs-702	239	9	submodule	submodule	NOUN
iajs-702	239	10	of	of	ADP
iajs-702	239	11	m.	m.	NOUN
iajs-702	239	12	first	first	ADV
iajs-702	239	13	of	of	ADP
iajs-702	239	14	all	all	DET
iajs-702	239	15	n	n	NOUN
iajs-702	239	16	=	=	PUNCT
iajs-702	240	1	[	[	X
iajs-702	240	2	i	i	PRON
iajs-702	240	3	m	m	VERB
iajs-702	240	4	:	:	PUNCT
iajs-702	240	5	m]m	m]m	NOUN
iajs-702	240	6	since	since	SCONJ
iajs-702	240	7	m	m	PROPN
iajs-702	240	8	is	be	AUX
iajs-702	240	9	multiplication	multiplication	NOUN
iajs-702	240	10	and	and	CCONJ
iajs-702	240	11	by	by	ADP
iajs-702	240	12	[	[	X
iajs-702	240	13	10	10	NUM
iajs-702	240	14	,	,	PUNCT
iajs-702	240	15	proposition	proposition	NOUN
iajs-702	240	16	3.31	3.31	NUM
iajs-702	240	17	]	]	PUNCT
iajs-702	241	1	[	[	X
iajs-702	241	2	im	im	X
iajs-702	241	3	:	:	PUNCT
iajs-702	241	4	m	m	VERB
iajs-702	241	5	]	]	X
iajs-702	241	6	is	be	AUX
iajs-702	241	7	a	a	DET
iajs-702	241	8	closed	closed	ADJ
iajs-702	241	9	ideal	ideal	NOUN
iajs-702	241	10	of	of	ADP
iajs-702	241	11	r.	r.	PROPN
iajs-702	241	12	but	but	CCONJ
iajs-702	241	13	m	m	PROPN
iajs-702	241	14	is	be	AUX
iajs-702	241	15	finitely	finitely	ADV
iajs-702	241	16	generating	generate	VERB
iajs-702	241	17	faithful	faithful	ADJ
iajs-702	241	18	multiplication	multiplication	NOUN
iajs-702	241	19	,	,	PUNCT
iajs-702	241	20	then	then	ADV
iajs-702	241	21	by	by	ADP
iajs-702	241	22	[	[	X
iajs-702	241	23	7	7	NUM
iajs-702	241	24	,	,	PUNCT
iajs-702	241	25	theorem	theorem	VERB
iajs-702	241	26	3.1	3.1	NUM
iajs-702	241	27	]	]	PUNCT
iajs-702	242	1	i	i	PRON
iajs-702	242	2	=	=	PUNCT
iajs-702	243	1	[	[	X
iajs-702	243	2	im	im	X
iajs-702	243	3	:	:	PUNCT
iajs-702	243	4	m	m	ADJ
iajs-702	243	5	]	]	X
iajs-702	243	6	.	.	PUNCT
iajs-702	244	1	مجلة	مجلة	PROPN
iajs-702	244	2	إبن	إبن	VERB
iajs-702	244	3	الھیثم	الھیثم	NOUN
iajs-702	244	4	للعلوم	للعلوم	NOUN
iajs-702	244	5	الصرفة	الصرفة	NOUN
iajs-702	244	6	و	و	PRON
iajs-702	244	7	التطبیقیة	التطبیقیة	PROPN
iajs-702	244	8	2012	2012	NUM
iajs-702	244	9	السنة	السنة	NOUN
iajs-702	245	1	25	25	NUM
iajs-702	245	2	المجلد	المجلد	NOUN
iajs-702	245	3	1	1	NUM
iajs-702	245	4	العدد	العدد	PROPN
iajs-702	245	5	ibn	ibn	PROPN
iajs-702	245	6	al	al	PROPN
iajs-702	245	7	-	-	PUNCT
iajs-702	245	8	haitham	haitham	PROPN
iajs-702	245	9	journal	journal	PROPN
iajs-702	245	10	for	for	ADP
iajs-702	245	11	pure	pure	ADJ
iajs-702	245	12	and	and	CCONJ
iajs-702	245	13	applied	apply	VERB
iajs-702	245	14	science	science	NOUN
iajs-702	245	15	no	no	NOUN
iajs-702	245	16	.	.	NOUN
iajs-702	245	17	1	1	NUM
iajs-702	245	18	vol	vol	NOUN
iajs-702	245	19	.	.	PUNCT
iajs-702	246	1	25	25	NUM
iajs-702	246	2	year	year	NOUN
iajs-702	246	3	2012	2012	NUM
iajs-702	246	4	thus	thus	ADV
iajs-702	246	5	by	by	ADP
iajs-702	246	6	[	[	X
iajs-702	246	7	10	10	NUM
iajs-702	246	8	,	,	PUNCT
iajs-702	246	9	proposition	proposition	NOUN
iajs-702	246	10	3.31	3.31	NUM
iajs-702	246	11	,	,	PUNCT
iajs-702	246	12	ch.3	ch.3	PROPN
iajs-702	246	13	]	]	PUNCT
iajs-702	246	14	,	,	PUNCT
iajs-702	246	15	n	n	NOUN
iajs-702	246	16	=	=	PUNCT
iajs-702	247	1	[	[	X
iajs-702	247	2	im	im	X
iajs-702	247	3	:	:	PUNCT
iajs-702	247	4	m]m	m]m	NOUN
iajs-702	247	5	is	be	AUX
iajs-702	247	6	a	a	DET
iajs-702	247	7	closed	closed	ADJ
iajs-702	247	8	submodule	submodule	NOUN
iajs-702	247	9	of	of	ADP
iajs-702	247	10	m.	m.	NOUN
iajs-702	247	11	now	now	ADV
iajs-702	247	12	,	,	PUNCT
iajs-702	247	13	to	to	PART
iajs-702	247	14	prove	prove	VERB
iajs-702	247	15	that	that	SCONJ
iajs-702	247	16	n	n	PRON
iajs-702	247	17	is	be	AUX
iajs-702	247	18	a	a	DET
iajs-702	247	19	maximal	maximal	ADJ
iajs-702	247	20	submodule	submodule	NOUN
iajs-702	247	21	of	of	ADP
iajs-702	247	22	m.	m.	NOUN
iajs-702	247	23	suppose	suppose	VERB
iajs-702	247	24	there	there	PRON
iajs-702	247	25	exists	exist	VERB
iajs-702	247	26	a	a	DET
iajs-702	247	27	closed	closed	ADJ
iajs-702	247	28	submodule	submodule	NOUN
iajs-702	247	29	w	w	PROPN
iajs-702	247	30	of	of	ADP
iajs-702	247	31	m	m	PRON
iajs-702	247	32	such	such	ADJ
iajs-702	247	33	that	that	SCONJ
iajs-702	247	34	n	n	PROPN
iajs-702	247	35			PROPN
iajs-702	247	36	w.	w.	PROPN
iajs-702	247	37	hence	hence	ADV
iajs-702	247	38	n	n	PROPN
iajs-702	247	39	=	=	X
iajs-702	248	1	i	i	NOUN
iajs-702	248	2	m	m	VERB
iajs-702	248	3	=	=	X
iajs-702	249	1	[	[	X
iajs-702	249	2	n	n	CCONJ
iajs-702	249	3	:	:	PUNCT
iajs-702	249	4	m]m	m]m	NOUN
iajs-702	249	5			PROPN
iajs-702	250	1	[	[	X
iajs-702	250	2	w	w	NOUN
iajs-702	250	3	:	:	PUNCT
iajs-702	250	4	m]m	m]m	NOUN
iajs-702	250	5	=	=	SYM
iajs-702	250	6	w	w	NOUN
iajs-702	250	7	and	and	CCONJ
iajs-702	250	8	by	by	ADP
iajs-702	250	9	[	[	X
iajs-702	250	10	7	7	NUM
iajs-702	250	11	,	,	PUNCT
iajs-702	250	12	theorem	theorem	VERB
iajs-702	250	13	3.1	3.1	NUM
iajs-702	250	14	]	]	PUNCT
iajs-702	250	15	,	,	PUNCT
iajs-702	250	16	i	i	PRON
iajs-702	250	17	=	=	PUNCT
iajs-702	251	1	[	[	X
iajs-702	251	2	n	n	NUM
iajs-702	251	3	:	:	PUNCT
iajs-702	251	4	m	m	VERB
iajs-702	251	5	]	]	X
iajs-702	251	6			PROPN
iajs-702	252	1	[	[	X
iajs-702	252	2	w	w	X
iajs-702	252	3	:	:	PUNCT
iajs-702	252	4	m	m	VERB
iajs-702	252	5	]	]	X
iajs-702	252	6	.	.	PUNCT
iajs-702	253	1	on	on	ADP
iajs-702	253	2	the	the	DET
iajs-702	253	3	other	other	ADJ
iajs-702	253	4	hand	hand	NOUN
iajs-702	253	5	by	by	ADP
iajs-702	253	6	[	[	X
iajs-702	253	7	10	10	NUM
iajs-702	253	8	,	,	PUNCT
iajs-702	253	9	proposition	proposition	NOUN
iajs-702	253	10	3.31	3.31	NUM
iajs-702	253	11	,	,	PUNCT
iajs-702	253	12	chapter	chapter	NOUN
iajs-702	253	13	three	three	NUM
iajs-702	253	14	]	]	PUNCT
iajs-702	253	15	,	,	PUNCT
iajs-702	253	16	i	i	PRON
iajs-702	253	17	=	=	PUNCT
iajs-702	254	1	[	[	X
iajs-702	254	2	n	n	NUM
iajs-702	254	3	:	:	PUNCT
iajs-702	254	4	m	m	VERB
iajs-702	254	5	]	]	PUNCT
iajs-702	254	6	and	and	CCONJ
iajs-702	254	7	[	[	X
iajs-702	254	8	w	w	X
iajs-702	254	9	:	:	PUNCT
iajs-702	254	10	m	m	VERB
iajs-702	254	11	]	]	X
iajs-702	254	12	are	be	AUX
iajs-702	254	13	closed	closed	ADJ
iajs-702	254	14	ideals	ideal	NOUN
iajs-702	254	15	of	of	ADP
iajs-702	254	16	r.	r.	PROPN
iajs-702	254	17	hence	hence	ADV
iajs-702	254	18	i	i	PRON
iajs-702	254	19	=	=	PUNCT
iajs-702	255	1	[	[	X
iajs-702	255	2	w	w	X
iajs-702	255	3	:	:	PUNCT
iajs-702	255	4	m	m	VERB
iajs-702	255	5	]	]	X
iajs-702	255	6	,	,	PUNCT
iajs-702	255	7	since	since	SCONJ
iajs-702	255	8	i	i	PRON
iajs-702	255	9	is	be	AUX
iajs-702	255	10	a	a	DET
iajs-702	255	11	maximal	maximal	ADJ
iajs-702	255	12	closed	closed	ADJ
iajs-702	255	13	ideal	ideal	NOUN
iajs-702	255	14	of	of	ADP
iajs-702	255	15	r.	r.	PROPN
iajs-702	255	16	it	it	PRON
iajs-702	255	17	follows	follow	VERB
iajs-702	255	18	n	n	NOUN
iajs-702	255	19	=	=	SYM
iajs-702	255	20	w	w	PROPN
iajs-702	255	21	and	and	CCONJ
iajs-702	255	22	n	n	ADV
iajs-702	255	23	is	be	AUX
iajs-702	255	24	maximal	maximal	ADJ
iajs-702	255	25	closed	closed	ADJ
iajs-702	255	26	submodule	submodule	NOUN
iajs-702	255	27	of	of	ADP
iajs-702	255	28	m.	m.	NOUN
iajs-702	255	29	but	but	CCONJ
iajs-702	255	30	m	m	PROPN
iajs-702	255	31	is	be	AUX
iajs-702	255	32	faithful	faithful	ADJ
iajs-702	255	33	multiplication	multiplication	NOUN
iajs-702	255	34	,	,	PUNCT
iajs-702	255	35	so	so	SCONJ
iajs-702	255	36	annrn	annrn	NOUN
iajs-702	255	37	=	=	SYM
iajs-702	255	38	annri	annri	NOUN
iajs-702	255	39	≠	≠	PROPN
iajs-702	255	40	0	0	NUM
iajs-702	255	41	.	.	PUNCT
iajs-702	256	1	thus	thus	ADV
iajs-702	256	2	n	n	PRON
iajs-702	256	3	is	be	AUX
iajs-702	256	4	a	a	DET
iajs-702	256	5	direct	direct	ADJ
iajs-702	256	6	summand	summand	NOUN
iajs-702	256	7	of	of	ADP
iajs-702	256	8	m.	m.	NOUN
iajs-702	256	9	so	so	SCONJ
iajs-702	256	10	that	that	SCONJ
iajs-702	256	11	n	n	NOUN
iajs-702	256	12			ADJ
iajs-702	256	13	l	l	NOUN
iajs-702	257	1	=	=	PUNCT
iajs-702	257	2	m	m	VERB
iajs-702	257	3	for	for	ADP
iajs-702	257	4	some	some	DET
iajs-702	257	5	l	l	NOUN
iajs-702	257	6			NUM
iajs-702	257	7	m.	m.	NOUN
iajs-702	257	8	but	but	CCONJ
iajs-702	257	9	n	n	NOUN
iajs-702	257	10	=	=	PROPN
iajs-702	257	11	i	i	NOUN
iajs-702	257	12	m	m	PROPN
iajs-702	257	13	,	,	PUNCT
iajs-702	257	14	l	l	X
iajs-702	257	15	=	=	PUNCT
iajs-702	258	1	[	[	X
iajs-702	258	2	l	l	NOUN
iajs-702	258	3	:	:	PUNCT
iajs-702	258	4	m]m	m]m	NOUN
iajs-702	258	5	,	,	PUNCT
iajs-702	258	6	so	so	SCONJ
iajs-702	258	7	that	that	SCONJ
iajs-702	258	8	i	i	PRON
iajs-702	258	9	m	m	VERB
iajs-702	258	10			ADJ
iajs-702	259	1	[	[	X
iajs-702	259	2	l	l	NOUN
iajs-702	259	3	:	:	PUNCT
iajs-702	259	4	m]m	m]m	NOUN
iajs-702	259	5	=	=	NOUN
iajs-702	259	6	m	m	ADJ
iajs-702	259	7	and	and	CCONJ
iajs-702	259	8	hence	hence	ADV
iajs-702	259	9	(	(	PUNCT
iajs-702	259	10	i	i	PRON
iajs-702	259	11			VERB
iajs-702	260	1	[	[	X
iajs-702	260	2	l	l	NOUN
iajs-702	260	3	:	:	PUNCT
iajs-702	260	4	m])m	m])m	PROPN
iajs-702	260	5	=	=	NOUN
iajs-702	260	6	m	m	PROPN
iajs-702	260	7	.	.	PUNCT
iajs-702	261	1	but	but	CCONJ
iajs-702	261	2	by	by	ADP
iajs-702	261	3	[	[	X
iajs-702	261	4	7	7	NUM
iajs-702	261	5	,	,	PUNCT
iajs-702	261	6	theorem	theorem	VERB
iajs-702	261	7	1.6	1.6	NUM
iajs-702	261	8	]	]	PUNCT
iajs-702	261	9	(	(	PUNCT
iajs-702	261	10	i	i	PRON
iajs-702	261	11			PUNCT
iajs-702	262	1	[	[	X
iajs-702	262	2	l	l	NOUN
iajs-702	262	3	:	:	PUNCT
iajs-702	262	4	m])m	m])m	PROPN
iajs-702	262	5	=	=	VERB
iajs-702	263	1	i	i	VERB
iajs-702	263	2	m	m	VERB
iajs-702	263	3			PUNCT
iajs-702	264	1	[	[	X
iajs-702	264	2	l	l	NOUN
iajs-702	264	3	:	:	PUNCT
iajs-702	264	4	m]m	m]m	NOUN
iajs-702	264	5	=	=	SYM
iajs-702	264	6	(	(	PUNCT
iajs-702	264	7	0	0	NUM
iajs-702	264	8	)	)	PUNCT
iajs-702	264	9	,	,	PUNCT
iajs-702	264	10	and	and	CCONJ
iajs-702	264	11	so	so	ADV
iajs-702	264	12	(	(	PUNCT
iajs-702	264	13	i	i	PRON
iajs-702	264	14			PUNCT
iajs-702	265	1	[	[	X
iajs-702	265	2	l	l	NOUN
iajs-702	265	3	:	:	PUNCT
iajs-702	265	4	m])m	m])m	NOUN
iajs-702	265	5	=	=	SYM
iajs-702	265	6	0	0	NUM
iajs-702	265	7	;	;	PUNCT
iajs-702	265	8	that	that	PRON
iajs-702	265	9	is	is	ADV
iajs-702	265	10	(	(	PUNCT
iajs-702	265	11	i	i	PRON
iajs-702	265	12			PUNCT
iajs-702	266	1	[	[	X
iajs-702	266	2	l	l	NOUN
iajs-702	266	3	:	:	PUNCT
iajs-702	266	4	m	m	NOUN
iajs-702	266	5	]	]	X
iajs-702	266	6	)	)	PUNCT
iajs-702	266	7			PROPN
iajs-702	266	8	annm	annm	NOUN
iajs-702	266	9	=	=	SYM
iajs-702	266	10	(	(	PUNCT
iajs-702	266	11	0	0	NUM
iajs-702	266	12	)	)	PUNCT
iajs-702	266	13	.	.	PUNCT
iajs-702	267	1	thus	thus	ADV
iajs-702	267	2	i	i	PRON
iajs-702	267	3			PUNCT
iajs-702	268	1	[	[	X
iajs-702	268	2	l	l	NOUN
iajs-702	268	3	:	:	PUNCT
iajs-702	268	4	m	m	VERB
iajs-702	268	5	]	]	X
iajs-702	268	6	=	=	X
iajs-702	268	7	(	(	PUNCT
iajs-702	268	8	0	0	NUM
iajs-702	268	9	)	)	PUNCT
iajs-702	268	10	,	,	PUNCT
iajs-702	268	11	so	so	SCONJ
iajs-702	268	12	that	that	SCONJ
iajs-702	268	13	m	m	VERB
iajs-702	268	14	=	=	SYM
iajs-702	268	15	rm	rm	NOUN
iajs-702	268	16	=	=	PUNCT
iajs-702	268	17	(	(	PUNCT
iajs-702	268	18	i	i	PRON
iajs-702	268	19			VERB
iajs-702	269	1	[	[	X
iajs-702	269	2	l	l	NOUN
iajs-702	269	3	:	:	PUNCT
iajs-702	269	4	m])m	m])m	NOUN
iajs-702	269	5	and	and	CCONJ
iajs-702	269	6	hence	hence	ADV
iajs-702	269	7	r	r	NOUN
iajs-702	270	1	=	=	NOUN
iajs-702	270	2	i	i	PRON
iajs-702	270	3			VERB
iajs-702	270	4	[	[	X
iajs-702	270	5	w	w	X
iajs-702	270	6	:	:	PUNCT
iajs-702	270	7	m	m	VERB
iajs-702	270	8	]	]	PUNCT
iajs-702	270	9	by	by	ADP
iajs-702	270	10	[	[	X
iajs-702	270	11	7	7	NUM
iajs-702	270	12	,	,	PUNCT
iajs-702	270	13	theorem	theorem	VERB
iajs-702	270	14	6.1	6.1	NUM
iajs-702	270	15	]	]	PUNCT
iajs-702	270	16	.	.	PUNCT
iajs-702	271	1	(	(	PUNCT
iajs-702	271	2			NOUN
iajs-702	271	3	)	)	PUNCT
iajs-702	271	4	let	let	VERB
iajs-702	271	5	n	n	PRON
iajs-702	271	6	be	be	AUX
iajs-702	271	7	a	a	DET
iajs-702	271	8	maximal	maximal	ADJ
iajs-702	271	9	closed	closed	ADJ
iajs-702	271	10	submodule	submodule	NOUN
iajs-702	271	11	of	of	ADP
iajs-702	271	12	m	m	PROPN
iajs-702	271	13	with	with	ADP
iajs-702	271	14	annn	annn	NOUN
iajs-702	271	15	≠	≠	PROPN
iajs-702	271	16	0	0	NUM
iajs-702	271	17	.	.	PUNCT
iajs-702	272	1	since	since	SCONJ
iajs-702	272	2	n	n	PRON
iajs-702	272	3	is	be	AUX
iajs-702	272	4	closed	close	VERB
iajs-702	272	5	,	,	PUNCT
iajs-702	272	6	then	then	ADV
iajs-702	272	7	by	by	ADP
iajs-702	272	8	[	[	X
iajs-702	272	9	10	10	NUM
iajs-702	272	10	,	,	PUNCT
iajs-702	272	11	proposition	proposition	NOUN
iajs-702	272	12	3.1	3.1	NUM
iajs-702	272	13	]	]	PUNCT
iajs-702	273	1	[	[	X
iajs-702	273	2	n	n	X
iajs-702	273	3	:	:	PUNCT
iajs-702	273	4	m	m	VERB
iajs-702	273	5	]	]	X
iajs-702	273	6	is	be	AUX
iajs-702	273	7	closed	close	VERB
iajs-702	273	8	in	in	ADP
iajs-702	273	9	r.	r.	NOUN
iajs-702	273	10	we	we	PRON
iajs-702	273	11	claim	claim	VERB
iajs-702	273	12	that	that	SCONJ
iajs-702	273	13	[	[	X
iajs-702	273	14	n	n	CCONJ
iajs-702	273	15	:	:	PUNCT
iajs-702	273	16	m	m	VERB
iajs-702	273	17	]	]	X
iajs-702	273	18	is	be	AUX
iajs-702	273	19	a	a	DET
iajs-702	273	20	maximal	maximal	ADJ
iajs-702	273	21	closed	closed	ADJ
iajs-702	273	22	ideal	ideal	NOUN
iajs-702	273	23	in	in	ADP
iajs-702	273	24	r.	r.	PROPN
iajs-702	273	25	to	to	PART
iajs-702	273	26	show	show	VERB
iajs-702	273	27	this	this	PRON
iajs-702	273	28	,	,	PUNCT
iajs-702	273	29	assume	assume	VERB
iajs-702	273	30	j	j	PROPN
iajs-702	273	31	is	be	AUX
iajs-702	273	32	a	a	DET
iajs-702	273	33	closed	closed	ADJ
iajs-702	273	34	ideal	ideal	NOUN
iajs-702	273	35	in	in	ADP
iajs-702	273	36	r	r	NOUN
iajs-702	273	37	such	such	ADJ
iajs-702	273	38	that	that	SCONJ
iajs-702	273	39	[	[	X
iajs-702	273	40	n	n	CCONJ
iajs-702	273	41	:	:	PUNCT
iajs-702	273	42	m	m	ADJ
iajs-702	273	43	]	]	X
iajs-702	273	44			PROPN
iajs-702	273	45	j	j	PROPN
iajs-702	273	46	,	,	PUNCT
iajs-702	273	47	hence	hence	ADV
iajs-702	273	48	n	n	NOUN
iajs-702	273	49	=	=	PUNCT
iajs-702	274	1	[	[	X
iajs-702	274	2	n	n	CCONJ
iajs-702	274	3	:	:	PUNCT
iajs-702	274	4	m]m	m]m	NOUN
iajs-702	274	5			PROPN
iajs-702	274	6	jm	jm	PROPN
iajs-702	274	7	.	.	PUNCT
iajs-702	275	1	but	but	CCONJ
iajs-702	275	2	jm	jm	PROPN
iajs-702	275	3	=	=	PUNCT
iajs-702	276	1	[	[	X
iajs-702	276	2	jm	jm	NOUN
iajs-702	276	3	:	:	PUNCT
iajs-702	276	4	m]m	m]m	NOUN
iajs-702	276	5	and	and	CCONJ
iajs-702	276	6	by	by	ADP
iajs-702	276	7	[	[	X
iajs-702	276	8	7	7	NUM
iajs-702	276	9	,	,	PUNCT
iajs-702	276	10	theorem	theorem	VERB
iajs-702	276	11	6.1	6.1	NUM
iajs-702	276	12	]	]	PUNCT
iajs-702	277	1	[	[	X
iajs-702	277	2	jm	jm	NOUN
iajs-702	277	3	:	:	PUNCT
iajs-702	277	4	m	m	VERB
iajs-702	277	5	]	]	X
iajs-702	277	6	=	=	SYM
iajs-702	277	7	j	j	PROPN
iajs-702	277	8	and	and	CCONJ
iajs-702	277	9	by	by	ADP
iajs-702	277	10	[	[	X
iajs-702	277	11	10	10	NUM
iajs-702	277	12	,	,	PUNCT
iajs-702	277	13	proposition	proposition	NOUN
iajs-702	277	14	3.1	3.1	NUM
iajs-702	277	15	]	]	X
iajs-702	277	16	jm	jm	PROPN
iajs-702	277	17	is	be	AUX
iajs-702	277	18	a	a	DET
iajs-702	277	19	closed	closed	ADJ
iajs-702	277	20	submodule	submodule	NOUN
iajs-702	277	21	of	of	ADP
iajs-702	277	22	m.	m.	NOUN
iajs-702	277	23	it	it	PRON
iajs-702	277	24	follows	follow	VERB
iajs-702	277	25	that	that	PRON
iajs-702	277	26	n	n	NOUN
iajs-702	277	27	=	=	PUNCT
iajs-702	278	1	[	[	X
iajs-702	278	2	n	n	CCONJ
iajs-702	278	3	:	:	PUNCT
iajs-702	278	4	m]m	m]m	NOUN
iajs-702	278	5	=	=	SYM
iajs-702	278	6	jm	jm	PROPN
iajs-702	278	7	,	,	PUNCT
iajs-702	278	8	since	since	SCONJ
iajs-702	278	9	n	n	NUM
iajs-702	278	10	is	be	AUX
iajs-702	278	11	a	a	DET
iajs-702	278	12	maximal	maximal	ADJ
iajs-702	278	13	closed	closed	ADJ
iajs-702	278	14	submodule	submodule	NOUN
iajs-702	278	15	of	of	ADP
iajs-702	278	16	m.	m.	NOUN
iajs-702	278	17	then	then	ADV
iajs-702	278	18	by	by	ADP
iajs-702	278	19	[	[	X
iajs-702	278	20	7	7	NUM
iajs-702	278	21	,	,	PUNCT
iajs-702	278	22	theorem	theorem	VERB
iajs-702	278	23	3.1	3.1	NUM
iajs-702	278	24	]	]	PUNCT
iajs-702	278	25	,	,	PUNCT
iajs-702	278	26	[	[	X
iajs-702	278	27	n	n	CCONJ
iajs-702	278	28	:	:	PUNCT
iajs-702	278	29	m	m	VERB
iajs-702	278	30	]	]	X
iajs-702	278	31	=	=	SYM
iajs-702	278	32	j	j	NOUN
iajs-702	278	33	;	;	PUNCT
iajs-702	279	1	that	that	SCONJ
iajs-702	279	2	[	[	X
iajs-702	279	3	n	n	CCONJ
iajs-702	279	4	:	:	PUNCT
iajs-702	279	5	m	m	VERB
iajs-702	279	6	]	]	X
iajs-702	279	7	is	be	AUX
iajs-702	279	8	a	a	DET
iajs-702	279	9	maximal	maximal	ADJ
iajs-702	279	10	closed	closed	ADJ
iajs-702	279	11	.	.	PUNCT
iajs-702	280	1	but	but	CCONJ
iajs-702	280	2	annn	annn	NOUN
iajs-702	280	3	=	=	SYM
iajs-702	280	4	ann[n	ann[n	NOUN
iajs-702	280	5	:	:	PUNCT
iajs-702	280	6	m	m	X
iajs-702	280	7	]	]	X
iajs-702	280	8	since	since	SCONJ
iajs-702	280	9	m	m	PROPN
iajs-702	280	10	is	be	AUX
iajs-702	280	11	faithful	faithful	ADJ
iajs-702	280	12	multiplication	multiplication	NOUN
iajs-702	280	13	.	.	PUNCT
iajs-702	281	1	thus	thus	ADV
iajs-702	281	2	[	[	X
iajs-702	281	3	n	n	X
iajs-702	281	4	:	:	PUNCT
iajs-702	281	5	m	m	VERB
iajs-702	281	6	]	]	X
iajs-702	281	7	is	be	AUX
iajs-702	281	8	a	a	DET
iajs-702	281	9	maximal	maximal	ADJ
iajs-702	281	10	closed	closed	ADJ
iajs-702	281	11	ideal	ideal	NOUN
iajs-702	281	12	of	of	ADP
iajs-702	281	13	r	r	NOUN
iajs-702	281	14	,	,	PUNCT
iajs-702	281	15	with	with	ADP
iajs-702	281	16	annr[n	annr[n	NOUN
iajs-702	281	17	:	:	PUNCT
iajs-702	281	18	m	m	VERB
iajs-702	281	19	]	]	X
iajs-702	281	20	≠	≠	PROPN
iajs-702	281	21	0	0	NUM
iajs-702	281	22	.	.	PUNCT
iajs-702	282	1	on	on	ADP
iajs-702	282	2	the	the	DET
iajs-702	282	3	other	other	ADJ
iajs-702	282	4	hand	hand	NOUN
iajs-702	282	5	r	r	NOUN
iajs-702	282	6	is	be	AUX
iajs-702	282	7	a	a	DET
iajs-702	282	8	max	max	PROPN
iajs-702	282	9	-	-	PUNCT
iajs-702	282	10	cs	cs	PROPN
iajs-702	282	11	ring	ring	NOUN
iajs-702	282	12	,	,	PUNCT
iajs-702	282	13	so	so	CCONJ
iajs-702	282	14	[	[	X
iajs-702	282	15	n	n	CCONJ
iajs-702	282	16	:	:	PUNCT
iajs-702	282	17	m	m	VERB
iajs-702	282	18	]	]	X
iajs-702	282	19	is	be	AUX
iajs-702	282	20	a	a	DET
iajs-702	282	21	direct	direct	ADJ
iajs-702	282	22	summand	summand	NOUN
iajs-702	282	23	of	of	ADP
iajs-702	282	24	r.	r.	PROPN
iajs-702	283	1	thus	thus	ADV
iajs-702	283	2	r	r	NOUN
iajs-702	283	3	=	=	PUNCT
iajs-702	284	1	[	[	X
iajs-702	284	2	n	n	NUM
iajs-702	284	3	:	:	PUNCT
iajs-702	284	4	m	m	VERB
iajs-702	284	5	]	]	X
iajs-702	284	6			PROPN
iajs-702	284	7	t	t	PROPN
iajs-702	284	8	where	where	SCONJ
iajs-702	284	9	t	t	PROPN
iajs-702	284	10	is	be	AUX
iajs-702	284	11	an	an	DET
iajs-702	284	12	ideal	ideal	NOUN
iajs-702	284	13	of	of	ADP
iajs-702	284	14	r	r	NOUN
iajs-702	284	15	,	,	PUNCT
iajs-702	284	16	and	and	CCONJ
iajs-702	284	17	hence	hence	ADV
iajs-702	284	18	m	m	VERB
iajs-702	284	19	=	=	SYM
iajs-702	284	20	rm	rm	NOUN
iajs-702	284	21	=	=	SYM
iajs-702	284	22	(	(	PUNCT
iajs-702	284	23	[	[	X
iajs-702	284	24	n	n	CCONJ
iajs-702	284	25	:	:	PUNCT
iajs-702	284	26	m	m	VERB
iajs-702	284	27	]	]	X
iajs-702	284	28			ADJ
iajs-702	284	29	t)m	t)m	NOUN
iajs-702	284	30	=	=	PUNCT
iajs-702	285	1	[	[	X
iajs-702	285	2	n	n	CCONJ
iajs-702	285	3	:	:	PUNCT
iajs-702	285	4	m]m	m]m	NOUN
iajs-702	285	5			PROPN
iajs-702	285	6	tm	tm	PROPN
iajs-702	285	7	.	.	PUNCT
iajs-702	286	1	but	but	CCONJ
iajs-702	286	2	by	by	ADP
iajs-702	286	3	[	[	X
iajs-702	286	4	7	7	NUM
iajs-702	286	5	,	,	PUNCT
iajs-702	286	6	theorem	theorem	VERB
iajs-702	286	7	1.6	1.6	NUM
iajs-702	286	8	]	]	PUNCT
iajs-702	286	9	,	,	PUNCT
iajs-702	286	10	[	[	X
iajs-702	286	11	n	n	CCONJ
iajs-702	286	12	:	:	PUNCT
iajs-702	286	13	m]m	m]m	NOUN
iajs-702	286	14			PUNCT
iajs-702	286	15	tm	tm	NOUN
iajs-702	286	16	=	=	SYM
iajs-702	286	17	(	(	PUNCT
iajs-702	286	18	[	[	X
iajs-702	286	19	n	n	CCONJ
iajs-702	286	20	:	:	PUNCT
iajs-702	286	21	m	m	PART
iajs-702	286	22	]	]	X
iajs-702	286	23			X
iajs-702	286	24	t)m	t)m	NOUN
iajs-702	286	25	=	=	SYM
iajs-702	286	26	0	0	NUM
iajs-702	286	27	m	m	VERB
iajs-702	286	28	=	=	NOUN
iajs-702	286	29	0	0	NUM
iajs-702	287	1	so	so	ADV
iajs-702	287	2	m	m	VERB
iajs-702	287	3	=	=	PUNCT
iajs-702	288	1	[	[	X
iajs-702	288	2	n	n	CCONJ
iajs-702	288	3	:	:	PUNCT
iajs-702	288	4	m]m	m]m	NOUN
iajs-702	288	5			PROPN
iajs-702	288	6	tm	tm	NOUN
iajs-702	288	7	;	;	PUNCT
iajs-702	288	8	that	that	PRON
iajs-702	288	9	is	be	AUX
iajs-702	288	10	m	m	NOUN
iajs-702	288	11	=	=	SYM
iajs-702	288	12	n	n	PRON
iajs-702	288	13			ADJ
iajs-702	288	14	tm	tm	NOUN
iajs-702	288	15	.	.	PROPN
iajs-702	289	1	by	by	ADP
iajs-702	289	2	the	the	DET
iajs-702	289	3	same	same	ADJ
iajs-702	289	4	argument	argument	NOUN
iajs-702	289	5	of	of	ADP
iajs-702	289	6	proof	proof	NOUN
iajs-702	289	7	of	of	ADP
iajs-702	289	8	theorem	theorem	ADJ
iajs-702	289	9	1.29	1.29	NUM
iajs-702	289	10	,	,	PUNCT
iajs-702	289	11	we	we	PRON
iajs-702	289	12	have	have	VERB
iajs-702	289	13	the	the	DET
iajs-702	289	14	following	following	NOUN
iajs-702	289	15	:	:	PUNCT
iajs-702	289	16	1.30	1.30	NUM
iajs-702	289	17	proposition	proposition	NOUN
iajs-702	289	18	:	:	PUNCT
iajs-702	289	19	let	let	VERB
iajs-702	289	20	m	m	PRON
iajs-702	289	21	be	be	AUX
iajs-702	289	22	a	a	DET
iajs-702	289	23	faithful	faithful	ADJ
iajs-702	289	24	,	,	PUNCT
iajs-702	289	25	multiplication	multiplication	NOUN
iajs-702	289	26	and	and	CCONJ
iajs-702	289	27	finitely	finitely	ADV
iajs-702	289	28	generated	generate	VERB
iajs-702	289	29	r	r	NOUN
iajs-702	289	30	-	-	PUNCT
iajs-702	289	31	module	module	NOUN
iajs-702	289	32	,	,	PUNCT
iajs-702	289	33	then	then	ADV
iajs-702	289	34	m	m	NOUN
iajs-702	289	35	is	be	AUX
iajs-702	289	36	a	a	DET
iajs-702	289	37	min	min	PROPN
iajs-702	289	38	-	-	ADJ
iajs-702	289	39	cs	cs	ADJ
iajs-702	289	40	ring	ring	NOUN
iajs-702	289	41	if	if	SCONJ
iajs-702	289	42	and	and	CCONJ
iajs-702	289	43	only	only	ADV
iajs-702	289	44	if	if	SCONJ
iajs-702	289	45	r	r	NOUN
iajs-702	289	46	is	be	AUX
iajs-702	289	47	min	min	ADJ
iajs-702	289	48	-	-	ADJ
iajs-702	289	49	cs	cs	ADJ
iajs-702	289	50	ring	ring	NOUN
iajs-702	289	51	.	.	PUNCT
iajs-702	290	1	next	next	ADV
iajs-702	290	2	we	we	PRON
iajs-702	290	3	have	have	VERB
iajs-702	290	4	for	for	ADP
iajs-702	290	5	nonsingular	nonsingular	ADJ
iajs-702	290	6	rings	ring	NOUN
iajs-702	290	7	,	,	PUNCT
iajs-702	290	8	the	the	DET
iajs-702	290	9	concepts	concept	NOUN
iajs-702	290	10	min	min	PROPN
iajs-702	290	11	-	-	ADJ
iajs-702	290	12	cs	cs	ADJ
iajs-702	290	13	ring	ring	NOUN
iajs-702	290	14	and	and	CCONJ
iajs-702	290	15	max	max	PROPN
iajs-702	290	16	-	-	PUNCT
iajs-702	290	17	cs	cs	PROPN
iajs-702	290	18	ring	ring	NOUN
iajs-702	290	19	are	be	AUX
iajs-702	290	20	equivalent	equivalent	ADJ
iajs-702	290	21	.	.	PUNCT
iajs-702	291	1	but	but	CCONJ
iajs-702	291	2	first	first	ADV
iajs-702	291	3	we	we	PRON
iajs-702	291	4	need	need	VERB
iajs-702	291	5	the	the	DET
iajs-702	291	6	following	following	NOUN
iajs-702	291	7	which	which	PRON
iajs-702	291	8	is	be	AUX
iajs-702	291	9	given	give	VERB
iajs-702	291	10	in	in	ADP
iajs-702	291	11	[	[	PUNCT
iajs-702	291	12	2	2	NUM
iajs-702	291	13	,	,	PUNCT
iajs-702	291	14	lemma	lemma	PROPN
iajs-702	291	15	2.1.1	2.1.1	NUM
iajs-702	291	16	,	,	PUNCT
iajs-702	291	17	p.31	p.31	X
iajs-702	291	18	]	]	PUNCT
iajs-702	291	19	.	.	PUNCT
iajs-702	292	1	we	we	PRON
iajs-702	292	2	give	give	VERB
iajs-702	292	3	the	the	DET
iajs-702	292	4	details	detail	NOUN
iajs-702	292	5	of	of	ADP
iajs-702	292	6	proof	proof	NOUN
iajs-702	292	7	.	.	PUNCT
iajs-702	293	1	1.31	1.31	NUM
iajs-702	293	2	lemma	lemma	PROPN
iajs-702	293	3	:	:	PUNCT
iajs-702	293	4	for	for	ADP
iajs-702	293	5	a	a	DET
iajs-702	293	6	ring	ring	NOUN
iajs-702	293	7	r	r	NOUN
iajs-702	293	8	,	,	PUNCT
iajs-702	293	9	a	a	DET
iajs-702	293	10	complement	complement	NOUN
iajs-702	293	11	of	of	ADP
iajs-702	293	12	minimal	minimal	ADJ
iajs-702	293	13	(	(	PUNCT
iajs-702	293	14	maximal	maximal	ADJ
iajs-702	293	15	)	)	PUNCT
iajs-702	293	16	closed	closed	ADJ
iajs-702	293	17	ideal	ideal	NOUN
iajs-702	293	18	of	of	ADP
iajs-702	293	19	r	r	NOUN
iajs-702	293	20	is	be	AUX
iajs-702	293	21	a	a	DET
iajs-702	293	22	maximal	maximal	ADJ
iajs-702	293	23	(	(	PUNCT
iajs-702	293	24	minimal	minimal	ADJ
iajs-702	293	25	)	)	PUNCT
iajs-702	293	26	closed	closed	ADJ
iajs-702	293	27	ideal	ideal	NOUN
iajs-702	293	28	of	of	ADP
iajs-702	293	29	r.	r.	PROPN
iajs-702	293	30	proof	proof	NOUN
iajs-702	293	31	:	:	PUNCT
iajs-702	293	32	(	(	PUNCT
iajs-702	293	33			NOUN
iajs-702	293	34	)	)	PUNCT
iajs-702	293	35	let	let	VERB
iajs-702	293	36	i	i	PRON
iajs-702	293	37	be	be	AUX
iajs-702	293	38	a	a	DET
iajs-702	293	39	minimal	minimal	ADJ
iajs-702	293	40	closed	closed	ADJ
iajs-702	293	41	ideal	ideal	NOUN
iajs-702	293	42	of	of	ADP
iajs-702	293	43	r.	r.	PROPN
iajs-702	293	44	let	let	VERB
iajs-702	293	45	j	j	PROPN
iajs-702	293	46	be	be	AUX
iajs-702	293	47	the	the	DET
iajs-702	293	48	relative	relative	ADJ
iajs-702	293	49	complement	complement	NOUN
iajs-702	293	50	of	of	ADP
iajs-702	293	51	i.	i.	PROPN
iajs-702	293	52	then	then	ADV
iajs-702	293	53	by	by	ADP
iajs-702	293	54	[	[	X
iajs-702	293	55	4	4	NUM
iajs-702	293	56	,	,	PUNCT
iajs-702	293	57	proposition	proposition	NOUN
iajs-702	293	58	1.3	1.3	NUM
iajs-702	293	59	,	,	PUNCT
iajs-702	293	60	p.17	p.17	PROPN
iajs-702	293	61	]	]	X
iajs-702	293	62	,	,	PUNCT
iajs-702	293	63	(	(	PUNCT
iajs-702	293	64	i	i	PRON
iajs-702	293	65			VERB
iajs-702	293	66	j	j	PROPN
iajs-702	293	67	)	)	PUNCT
iajs-702	293	68			PROPN
iajs-702	293	69	e	e	NOUN
iajs-702	293	70	r	r	NOUN
iajs-702	293	71	,	,	PUNCT
iajs-702	293	72	and	and	CCONJ
iajs-702	293	73	j	j	PROPN
iajs-702	293	74	is	be	AUX
iajs-702	293	75	closed	close	VERB
iajs-702	293	76	in	in	ADP
iajs-702	293	77	r	r	NOUN
iajs-702	293	78	,	,	PUNCT
iajs-702	293	79	by	by	ADP
iajs-702	293	80	[	[	X
iajs-702	293	81	4	4	NUM
iajs-702	293	82	,	,	PUNCT
iajs-702	293	83	proposition	proposition	NOUN
iajs-702	293	84	1.4	1.4	NUM
iajs-702	293	85	,	,	PUNCT
iajs-702	293	86	p.18	p.18	NOUN
iajs-702	293	87	]	]	PUNCT
iajs-702	293	88	.	.	PUNCT
iajs-702	294	1	now	now	ADV
iajs-702	294	2	,	,	PUNCT
iajs-702	294	3	we	we	PRON
iajs-702	294	4	shall	shall	AUX
iajs-702	294	5	show	show	VERB
iajs-702	294	6	that	that	SCONJ
iajs-702	294	7	j	j	PROPN
iajs-702	294	8	is	be	AUX
iajs-702	294	9	a	a	DET
iajs-702	294	10	maximal	maximal	ADJ
iajs-702	294	11	closed	closed	ADJ
iajs-702	294	12	ideal	ideal	NOUN
iajs-702	294	13	in	in	ADP
iajs-702	294	14	r.	r.	PROPN
iajs-702	294	15	assume	assume	VERB
iajs-702	294	16	that	that	SCONJ
iajs-702	294	17	there	there	PRON
iajs-702	294	18	exists	exist	VERB
iajs-702	294	19	a	a	DET
iajs-702	294	20	closed	closed	ADJ
iajs-702	294	21	ideal	ideal	ADJ
iajs-702	294	22	j	j	NOUN
iajs-702	294	23	*	*	PROPN
iajs-702	294	24	in	in	ADP
iajs-702	294	25	r	r	NOUN
iajs-702	294	26	,	,	PUNCT
iajs-702	294	27	such	such	ADJ
iajs-702	294	28	that	that	SCONJ
iajs-702	294	29	j	j	PROPN
iajs-702	294	30			PROPN
iajs-702	294	31	j	j	AUX
iajs-702	294	32	*	*	PROPN
iajs-702	294	33			NOUN
iajs-702	295	1			PROPN
iajs-702	295	2	r.	r.	PROPN
iajs-702	295	3	it	it	PRON
iajs-702	295	4	follows	follow	VERB
iajs-702	295	5	that	that	SCONJ
iajs-702	295	6	j	j	PROPN
iajs-702	295	7	*	*	X
iajs-702	295	8			PUNCT
iajs-702	295	9	i	i	PRON
iajs-702	295	10	≠	≠	PROPN
iajs-702	295	11	(	(	PUNCT
iajs-702	295	12	0	0	NUM
iajs-702	295	13	)	)	PUNCT
iajs-702	295	14	because	because	SCONJ
iajs-702	295	15	j	j	PROPN
iajs-702	295	16	is	be	AUX
iajs-702	295	17	the	the	DET
iajs-702	295	18	largest	large	ADJ
iajs-702	295	19	ideal	ideal	NOUN
iajs-702	295	20	in	in	ADP
iajs-702	295	21	r	r	NOUN
iajs-702	295	22	such	such	ADJ
iajs-702	295	23	that	that	SCONJ
iajs-702	295	24	i	i	PRON
iajs-702	295	25			PUNCT
iajs-702	295	26	j	j	X
iajs-702	296	1	=	=	PUNCT
iajs-702	296	2	(	(	PUNCT
iajs-702	296	3	0	0	NUM
iajs-702	296	4	)	)	PUNCT
iajs-702	296	5	.	.	PUNCT
iajs-702	297	1	مجلة	مجلة	NOUN
iajs-702	297	2	إبن	إبن	VERB
iajs-702	297	3	الھیثم	الھیثم	NOUN
iajs-702	297	4	للعلوم	للعلوم	NOUN
iajs-702	297	5	الصرفة	الصرفة	NOUN
iajs-702	297	6	و	و	PRON
iajs-702	297	7	التطبیقیة	التطبیقیة	PROPN
iajs-702	297	8	2012	2012	NUM
iajs-702	297	9	السنة	السنة	NOUN
iajs-702	298	1	25	25	NUM
iajs-702	298	2	المجلد	المجلد	NOUN
iajs-702	298	3	1	1	NUM
iajs-702	298	4	العدد	العدد	PROPN
iajs-702	298	5	ibn	ibn	PROPN
iajs-702	298	6	al	al	PROPN
iajs-702	298	7	-	-	PUNCT
iajs-702	298	8	haitham	haitham	PROPN
iajs-702	298	9	journal	journal	PROPN
iajs-702	298	10	for	for	ADP
iajs-702	298	11	pure	pure	ADJ
iajs-702	298	12	and	and	CCONJ
iajs-702	298	13	applied	apply	VERB
iajs-702	298	14	science	science	NOUN
iajs-702	298	15	no	no	NOUN
iajs-702	298	16	.	.	NOUN
iajs-702	298	17	1	1	NUM
iajs-702	298	18	vol	vol	NOUN
iajs-702	298	19	.	.	PUNCT
iajs-702	299	1	25	25	NUM
iajs-702	299	2	year	year	NOUN
iajs-702	299	3	2012	2012	NUM
iajs-702	299	4	on	on	ADP
iajs-702	299	5	the	the	DET
iajs-702	299	6	other	other	ADJ
iajs-702	299	7	hand	hand	NOUN
iajs-702	299	8	i	i	PRON
iajs-702	299	9	is	be	AUX
iajs-702	299	10	a	a	DET
iajs-702	299	11	minimal	minimal	ADJ
iajs-702	299	12	closed	closed	ADJ
iajs-702	299	13	ideal	ideal	NOUN
iajs-702	299	14	in	in	ADP
iajs-702	299	15	r	r	NOUN
iajs-702	299	16	,	,	PUNCT
iajs-702	299	17	so	so	SCONJ
iajs-702	299	18	it	it	PRON
iajs-702	299	19	is	be	AUX
iajs-702	299	20	uniform	uniform	NOUN
iajs-702	299	21	closed	close	VERB
iajs-702	299	22	by	by	ADP
iajs-702	299	23	lemma	lemma	PROPN
iajs-702	299	24	1.6	1.6	NUM
iajs-702	299	25	.	.	PUNCT
iajs-702	300	1	hence	hence	ADV
iajs-702	300	2	j*	j*	PROPN
iajs-702	300	3	i	i	PRON
iajs-702	300	4			NUM
iajs-702	300	5	e	e	NOUN
iajs-702	300	6	i.	i.	NOUN
iajs-702	300	7	but	but	CCONJ
iajs-702	300	8	j	j	PROPN
iajs-702	300	9			PROPN
iajs-702	300	10	e	e	X
iajs-702	300	11	i	i	PRON
iajs-702	300	12	,	,	PUNCT
iajs-702	300	13	hence	hence	ADV
iajs-702	300	14	(	(	PUNCT
iajs-702	300	15	j*	j*	X
iajs-702	300	16	i	i	PRON
iajs-702	300	17	)	)	PUNCT
iajs-702	300	18			PROPN
iajs-702	300	19	j	j	PROPN
iajs-702	300	20			NUM
iajs-702	301	1	e	e	X
iajs-702	301	2	i	i	PRON
iajs-702	301	3			VERB
iajs-702	301	4	j	j	PROPN
iajs-702	301	5			NUM
iajs-702	301	6	e	e	NOUN
iajs-702	301	7	r	r	NOUN
iajs-702	301	8	and	and	CCONJ
iajs-702	301	9	so	so	SCONJ
iajs-702	301	10	that	that	SCONJ
iajs-702	301	11	(	(	PUNCT
iajs-702	301	12	j*	j*	X
iajs-702	301	13	i	i	X
iajs-702	301	14	)	)	PUNCT
iajs-702	301	15			PROPN
iajs-702	301	16	j	j	PROPN
iajs-702	301	17			PROPN
iajs-702	301	18	e	e	NOUN
iajs-702	301	19	r	r	NOUN
iajs-702	301	20	,	,	PUNCT
iajs-702	301	21	by	by	ADP
iajs-702	301	22	[	[	X
iajs-702	301	23	4	4	NUM
iajs-702	301	24	,	,	PUNCT
iajs-702	301	25	proposition	proposition	NOUN
iajs-702	301	26	1.1(a	1.1(a	NUM
iajs-702	301	27	)	)	PUNCT
iajs-702	301	28	,	,	PUNCT
iajs-702	301	29	p.16	p.16	PROPN
iajs-702	301	30	]	]	PUNCT
iajs-702	301	31	.	.	PUNCT
iajs-702	302	1	but	but	CCONJ
iajs-702	302	2	j*	j*	PROPN
iajs-702	302	3	i	i	PROPN
iajs-702	302	4			NUM
iajs-702	302	5	j	j	PROPN
iajs-702	302	6	*	*	PUNCT
iajs-702	302	7	and	and	CCONJ
iajs-702	302	8	j	j	PROPN
iajs-702	302	9	<	<	X
iajs-702	302	10	j	j	PROPN
iajs-702	302	11	*	*	PROPN
iajs-702	302	12	,	,	PUNCT
iajs-702	302	13	hence	hence	ADV
iajs-702	302	14	(	(	PUNCT
iajs-702	302	15	j*	j*	X
iajs-702	302	16	i	i	PRON
iajs-702	302	17	)	)	PUNCT
iajs-702	302	18			PROPN
iajs-702	302	19	j	j	PROPN
iajs-702	302	20			NUM
iajs-702	302	21	j	j	PROPN
iajs-702	302	22	*	*	PUNCT
iajs-702	302	23	.	.	PUNCT
iajs-702	303	1	thus	thus	ADV
iajs-702	303	2	j	j	PROPN
iajs-702	303	3	*	*	NOUN
iajs-702	303	4			PROPN
iajs-702	303	5	e	e	NOUN
iajs-702	303	6	r	r	NOUN
iajs-702	303	7	,	,	PUNCT
iajs-702	303	8	which	which	PRON
iajs-702	303	9	is	be	AUX
iajs-702	303	10	a	a	DET
iajs-702	303	11	contradiction	contradiction	NOUN
iajs-702	303	12	.	.	PUNCT
iajs-702	304	1	(	(	PUNCT
iajs-702	304	2			NOUN
iajs-702	304	3	)	)	PUNCT
iajs-702	304	4	if	if	SCONJ
iajs-702	304	5	t	t	PROPN
iajs-702	304	6	is	be	AUX
iajs-702	304	7	a	a	DET
iajs-702	304	8	maximal	maximal	ADJ
iajs-702	304	9	closed	closed	ADJ
iajs-702	304	10	ideal	ideal	NOUN
iajs-702	304	11	of	of	ADP
iajs-702	304	12	r	r	NOUN
iajs-702	304	13	and	and	CCONJ
iajs-702	304	14	v	v	ADP
iajs-702	304	15	its	its	PRON
iajs-702	304	16	complement	complement	NOUN
iajs-702	304	17	.	.	PUNCT
iajs-702	305	1	to	to	PART
iajs-702	305	2	prove	prove	VERB
iajs-702	305	3	v	v	NOUN
iajs-702	305	4	is	be	AUX
iajs-702	305	5	minimal	minimal	ADJ
iajs-702	305	6	closed	close	VERB
iajs-702	305	7	in	in	ADP
iajs-702	305	8	r	r	NOUN
iajs-702	305	9	,	,	PUNCT
iajs-702	305	10	it	it	PRON
iajs-702	305	11	is	be	AUX
iajs-702	305	12	enough	enough	ADJ
iajs-702	305	13	to	to	PART
iajs-702	305	14	show	show	VERB
iajs-702	305	15	that	that	SCONJ
iajs-702	305	16	v	v	NOUN
iajs-702	305	17	is	be	AUX
iajs-702	305	18	uniform	uniform	ADJ
iajs-702	305	19	by	by	ADP
iajs-702	305	20	lemma	lemma	PROPN
iajs-702	305	21	1.6	1.6	NUM
iajs-702	305	22	.	.	PUNCT
iajs-702	306	1	assume	assume	VERB
iajs-702	306	2	a	a	DET
iajs-702	306	3			NOUN
iajs-702	306	4	b	b	NOUN
iajs-702	306	5	=	=	SYM
iajs-702	306	6	(	(	PUNCT
iajs-702	306	7	0	0	NUM
iajs-702	306	8	)	)	PUNCT
iajs-702	306	9	for	for	ADP
iajs-702	306	10	some	some	DET
iajs-702	306	11	two	two	NUM
iajs-702	306	12	nonzero	nonzero	NOUN
iajs-702	306	13	ideals	ideal	NOUN
iajs-702	306	14	a	a	PRON
iajs-702	306	15	and	and	CCONJ
iajs-702	306	16	b	b	NOUN
iajs-702	306	17	of	of	ADP
iajs-702	306	18	v.	v.	ADV
iajs-702	306	19	now	now	ADV
iajs-702	306	20	,	,	PUNCT
iajs-702	306	21	there	there	PRON
iajs-702	306	22	exists	exist	VERB
iajs-702	306	23	a	a	DET
iajs-702	306	24	closed	closed	ADJ
iajs-702	306	25	ideal	ideal	NOUN
iajs-702	306	26	i	i	PRON
iajs-702	306	27	in	in	ADP
iajs-702	306	28	r	r	NOUN
iajs-702	306	29	such	such	ADJ
iajs-702	306	30	that	that	SCONJ
iajs-702	306	31	a	a	DET
iajs-702	306	32			ADJ
iajs-702	306	33	t	t	NOUN
iajs-702	306	34			NOUN
iajs-702	306	35	e	e	X
iajs-702	306	36	i	i	PRON
iajs-702	306	37	,	,	PUNCT
iajs-702	306	38	by	by	ADP
iajs-702	306	39	[	[	X
iajs-702	306	40	4	4	NUM
iajs-702	306	41	,	,	PUNCT
iajs-702	306	42	exc.13	exc.13	NOUN
iajs-702	306	43	,	,	PUNCT
iajs-702	306	44	p.20	p.20	X
iajs-702	306	45	]	]	X
iajs-702	306	46	.	.	PUNCT
iajs-702	307	1	hence	hence	ADV
iajs-702	307	2	t	t	PROPN
iajs-702	307	3			VERB
iajs-702	307	4			PROPN
iajs-702	307	5	i	i	PRON
iajs-702	307	6	which	which	PRON
iajs-702	307	7	is	be	AUX
iajs-702	307	8	a	a	DET
iajs-702	307	9	contradiction	contradiction	NOUN
iajs-702	307	10	,	,	PUNCT
iajs-702	307	11	since	since	SCONJ
iajs-702	307	12	i	i	PRON
iajs-702	307	13	is	be	AUX
iajs-702	307	14	closed	close	VERB
iajs-702	307	15	in	in	ADP
iajs-702	307	16	r	r	NOUN
iajs-702	307	17	and	and	CCONJ
iajs-702	307	18	t	t	PROPN
iajs-702	307	19	is	be	AUX
iajs-702	307	20	a	a	DET
iajs-702	307	21	maximal	maximal	ADJ
iajs-702	307	22	closed	closed	ADJ
iajs-702	307	23	ideal	ideal	NOUN
iajs-702	307	24	in	in	ADP
iajs-702	307	25	r.	r.	PROPN
iajs-702	307	26	thus	thus	ADV
iajs-702	307	27	v	v	NOUN
iajs-702	307	28	is	be	AUX
iajs-702	307	29	uniform	uniform	NOUN
iajs-702	307	30	closed	close	VERB
iajs-702	307	31	,	,	PUNCT
iajs-702	307	32	that	that	PRON
iajs-702	307	33	is	be	AUX
iajs-702	307	34	v	v	NOUN
iajs-702	307	35	is	be	AUX
iajs-702	307	36	minimal	minimal	ADJ
iajs-702	307	37	closed	closed	ADJ
iajs-702	307	38	.	.	PUNCT
iajs-702	308	1	1.32	1.32	NUM
iajs-702	308	2	note	note	NOUN
iajs-702	308	3	:	:	PUNCT
iajs-702	308	4	by	by	ADP
iajs-702	308	5	a	a	DET
iajs-702	308	6	similar	similar	ADJ
iajs-702	308	7	argument	argument	NOUN
iajs-702	308	8	of	of	ADP
iajs-702	308	9	proof	proof	NOUN
iajs-702	308	10	of	of	ADP
iajs-702	308	11	lemma	lemma	PROPN
iajs-702	308	12	1.31	1.31	NUM
iajs-702	308	13	,	,	PUNCT
iajs-702	308	14	we	we	PRON
iajs-702	308	15	get	get	VERB
iajs-702	308	16	analogous	analogous	ADJ
iajs-702	308	17	results	result	NOUN
iajs-702	308	18	for	for	ADP
iajs-702	308	19	submodules	submodule	NOUN
iajs-702	308	20	.	.	PUNCT
iajs-702	309	1	recall	recall	VERB
iajs-702	309	2	that	that	SCONJ
iajs-702	309	3	a	a	DET
iajs-702	309	4	ring	ring	NOUN
iajs-702	309	5	r	r	NOUN
iajs-702	309	6	is	be	AUX
iajs-702	309	7	semiprime	semiprime	NOUN
iajs-702	309	8	if	if	SCONJ
iajs-702	309	9	for	for	ADP
iajs-702	309	10	each	each	DET
iajs-702	309	11	x	x	ADJ
iajs-702	309	12			NOUN
iajs-702	309	13	r	r	NOUN
iajs-702	309	14	,	,	PUNCT
iajs-702	309	15	x	x	NOUN
iajs-702	309	16	2	2	NUM
iajs-702	309	17	=	=	SYM
iajs-702	309	18	0	0	NUM
iajs-702	309	19	,	,	PUNCT
iajs-702	309	20	then	then	ADV
iajs-702	309	21	x	x	X
iajs-702	309	22	=	=	SYM
iajs-702	309	23	0	0	NUM
iajs-702	309	24	,	,	PUNCT
iajs-702	309	25	[	[	X
iajs-702	309	26	4	4	NUM
iajs-702	309	27	,	,	PUNCT
iajs-702	309	28	p	p	NOUN
iajs-702	309	29	.2	.2	NUM
iajs-702	309	30	]	]	PUNCT
iajs-702	309	31	.	.	PUNCT
iajs-702	310	1	now	now	ADV
iajs-702	310	2	,	,	PUNCT
iajs-702	310	3	we	we	PRON
iajs-702	310	4	can	can	AUX
iajs-702	310	5	give	give	VERB
iajs-702	310	6	the	the	DET
iajs-702	310	7	following	following	NOUN
iajs-702	310	8	theorem	theorem	ADJ
iajs-702	310	9	:	:	PUNCT
iajs-702	310	10	1.33	1.33	NUM
iajs-702	310	11	theorem	theorem	NOUN
iajs-702	310	12	:	:	PUNCT
iajs-702	310	13	let	let	VERB
iajs-702	310	14	r	r	PRON
iajs-702	310	15	be	be	AUX
iajs-702	310	16	a	a	DET
iajs-702	310	17	nonsingular	nonsingular	ADJ
iajs-702	310	18	ring	ring	NOUN
iajs-702	310	19	.	.	PUNCT
iajs-702	311	1	then	then	ADV
iajs-702	311	2	r	r	NOUN
iajs-702	311	3	is	be	AUX
iajs-702	311	4	max	max	PROPN
iajs-702	311	5	-	-	PUNCT
iajs-702	311	6	cs	cs	PROPN
iajs-702	311	7	if	if	SCONJ
iajs-702	312	1	and	and	CCONJ
iajs-702	312	2	only	only	ADV
iajs-702	312	3	if	if	SCONJ
iajs-702	312	4	r	r	NOUN
iajs-702	312	5	is	be	AUX
iajs-702	312	6	min	min	NOUN
iajs-702	312	7	-	-	ADJ
iajs-702	312	8	cs	cs	ADJ
iajs-702	312	9	.	.	PROPN
iajs-702	312	10	proof	proof	NOUN
iajs-702	312	11	:	:	PUNCT
iajs-702	312	12	(	(	PUNCT
iajs-702	312	13			NOUN
iajs-702	312	14	)	)	PUNCT
iajs-702	312	15	if	if	SCONJ
iajs-702	312	16	r	r	NOUN
iajs-702	312	17	is	be	AUX
iajs-702	312	18	a	a	DET
iajs-702	312	19	max	max	PROPN
iajs-702	312	20	-	-	PUNCT
iajs-702	312	21	cs	cs	PROPN
iajs-702	312	22	ring	ring	NOUN
iajs-702	312	23	.	.	PUNCT
iajs-702	313	1	let	let	VERB
iajs-702	313	2	i	i	PRON
iajs-702	313	3	be	be	AUX
iajs-702	313	4	a	a	DET
iajs-702	313	5	minimal	minimal	ADJ
iajs-702	313	6	closed	closed	ADJ
iajs-702	313	7	ideal	ideal	NOUN
iajs-702	313	8	in	in	ADP
iajs-702	313	9	r.	r.	PROPN
iajs-702	313	10	by	by	ADP
iajs-702	313	11	[	[	X
iajs-702	313	12	4	4	NUM
iajs-702	313	13	,	,	PUNCT
iajs-702	313	14	theorem	theorem	VERB
iajs-702	313	15	2.38	2.38	NUM
iajs-702	313	16	,	,	PUNCT
iajs-702	313	17	p.65	p.65	ADP
iajs-702	313	18	]	]	PUNCT
iajs-702	313	19	,	,	PUNCT
iajs-702	313	20	i	i	PRON
iajs-702	313	21	=	=	SYM
iajs-702	313	22	ann	ann	PROPN
iajs-702	313	23	anni	anni	PROPN
iajs-702	313	24	,	,	PUNCT
iajs-702	313	25	hence	hence	ADV
iajs-702	313	26	anni	anni	PROPN
iajs-702	313	27	≠	≠	PROPN
iajs-702	313	28	0	0	NUM
iajs-702	313	29	,	,	PUNCT
iajs-702	313	30	but	but	CCONJ
iajs-702	313	31	r	r	NOUN
iajs-702	313	32	is	be	AUX
iajs-702	313	33	nonsingular	nonsingular	ADJ
iajs-702	313	34	,	,	PUNCT
iajs-702	313	35	so	so	CCONJ
iajs-702	313	36	r	r	NOUN
iajs-702	313	37	is	be	AUX
iajs-702	313	38	semiprime	semiprime	NOUN
iajs-702	313	39	by	by	ADP
iajs-702	313	40	[	[	X
iajs-702	313	41	4	4	NUM
iajs-702	313	42	,	,	PUNCT
iajs-702	313	43	proposition	proposition	NOUN
iajs-702	313	44	1.27(b	1.27(b	NUM
iajs-702	313	45	)	)	PUNCT
iajs-702	313	46	,	,	PUNCT
iajs-702	313	47	p.35	p.35	NOUN
iajs-702	313	48	]	]	X
iajs-702	313	49	,	,	PUNCT
iajs-702	313	50	which	which	PRON
iajs-702	313	51	implies	imply	VERB
iajs-702	313	52	i	i	PRON
iajs-702	313	53			PUNCT
iajs-702	313	54	anni	anni	X
iajs-702	313	55	=	=	PROPN
iajs-702	313	56	0	0	PROPN
iajs-702	313	57	.	.	PUNCT
iajs-702	314	1	let	let	VERB
iajs-702	314	2	j	j	PROPN
iajs-702	314	3	be	be	AUX
iajs-702	314	4	the	the	DET
iajs-702	314	5	relative	relative	ADJ
iajs-702	314	6	complement	complement	NOUN
iajs-702	314	7	of	of	ADP
iajs-702	314	8	i	i	PRON
iajs-702	314	9	,	,	PUNCT
iajs-702	314	10	so	so	ADV
iajs-702	314	11	by	by	ADP
iajs-702	314	12	lemma	lemma	PROPN
iajs-702	314	13	1.31	1.31	NUM
iajs-702	314	14	,	,	PUNCT
iajs-702	314	15	j	j	PROPN
iajs-702	314	16	is	be	AUX
iajs-702	314	17	a	a	DET
iajs-702	314	18	maximal	maximal	ADJ
iajs-702	314	19	closed	closed	ADJ
iajs-702	314	20	ideal	ideal	NOUN
iajs-702	314	21	in	in	ADP
iajs-702	314	22	r.	r.	PROPN
iajs-702	314	23	also	also	ADV
iajs-702	314	24	by	by	ADP
iajs-702	314	25	[	[	X
iajs-702	314	26	4	4	NUM
iajs-702	314	27	,	,	PUNCT
iajs-702	314	28	theorem	theorem	VERB
iajs-702	314	29	2.38	2.38	NUM
iajs-702	314	30	,	,	PUNCT
iajs-702	314	31	p.65	p.65	ADP
iajs-702	314	32	]	]	PUNCT
iajs-702	314	33	,	,	PUNCT
iajs-702	314	34	j	j	PROPN
iajs-702	314	35	=	=	SYM
iajs-702	314	36	ann	ann	PROPN
iajs-702	314	37	annj	annj	PROPN
iajs-702	314	38	,	,	PUNCT
iajs-702	314	39	so	so	SCONJ
iajs-702	314	40	annj	annj	VERB
iajs-702	314	41	≠	≠	PROPN
iajs-702	314	42	0	0	X
iajs-702	314	43	.	.	PUNCT
iajs-702	315	1	therefore	therefore	ADV
iajs-702	315	2	j	j	PROPN
iajs-702	315	3	is	be	AUX
iajs-702	315	4	a	a	DET
iajs-702	315	5	direct	direct	ADJ
iajs-702	315	6	summand	summand	NOUN
iajs-702	315	7	of	of	ADP
iajs-702	315	8	r	r	NOUN
iajs-702	315	9	since	since	SCONJ
iajs-702	315	10	r	r	NOUN
iajs-702	315	11	is	be	AUX
iajs-702	315	12	max	max	PROPN
iajs-702	315	13	-	-	PUNCT
iajs-702	315	14	cs	cs	PROPN
iajs-702	315	15	.	.	PUNCT
iajs-702	316	1	it	it	PRON
iajs-702	316	2	follows	follow	VERB
iajs-702	316	3	that	that	PRON
iajs-702	316	4	j	j	PROPN
iajs-702	316	5	=	=	PUNCT
iajs-702	317	1	<	<	X
iajs-702	317	2	e	e	X
iajs-702	317	3	>	>	X
iajs-702	317	4	for	for	ADP
iajs-702	317	5	some	some	DET
iajs-702	317	6	idempotent	idempotent	ADJ
iajs-702	317	7	e	e	NOUN
iajs-702	317	8	in	in	ADP
iajs-702	317	9	r.	r.	PROPN
iajs-702	317	10	on	on	ADP
iajs-702	317	11	the	the	DET
iajs-702	317	12	other	other	ADJ
iajs-702	317	13	hand	hand	NOUN
iajs-702	317	14	,	,	PUNCT
iajs-702	317	15	ij	ij	INTJ
iajs-702	317	16			PROPN
iajs-702	317	17	i	i	PROPN
iajs-702	317	18			PUNCT
iajs-702	317	19	j	j	X
iajs-702	317	20	=	=	PUNCT
iajs-702	317	21	(	(	PUNCT
iajs-702	317	22	0	0	NUM
iajs-702	317	23	)	)	PUNCT
iajs-702	317	24	,	,	PUNCT
iajs-702	317	25	implies	imply	VERB
iajs-702	317	26	that	that	SCONJ
iajs-702	317	27	j	j	PROPN
iajs-702	317	28			PROPN
iajs-702	317	29	anni	anni	PROPN
iajs-702	317	30	.	.	PUNCT
iajs-702	318	1	but	but	CCONJ
iajs-702	318	2	i	i	PRON
iajs-702	318	3			PUNCT
iajs-702	318	4	anni	anni	X
iajs-702	318	5	=	=	SYM
iajs-702	318	6	(	(	PUNCT
iajs-702	318	7	0	0	NUM
iajs-702	318	8	)	)	PUNCT
iajs-702	318	9	,	,	PUNCT
iajs-702	318	10	so	so	ADV
iajs-702	318	11	j	j	PROPN
iajs-702	318	12	=	=	PROPN
iajs-702	318	13	anni	anni	PROPN
iajs-702	318	14	since	since	SCONJ
iajs-702	318	15	j	j	PROPN
iajs-702	318	16	is	be	AUX
iajs-702	318	17	the	the	DET
iajs-702	318	18	largest	large	ADJ
iajs-702	318	19	ideal	ideal	NOUN
iajs-702	318	20	in	in	ADP
iajs-702	318	21	r	r	NOUN
iajs-702	318	22	such	such	ADJ
iajs-702	318	23	that	that	SCONJ
iajs-702	318	24	i	i	PRON
iajs-702	318	25			PUNCT
iajs-702	318	26	j	j	X
iajs-702	319	1	=	=	PUNCT
iajs-702	320	1	(	(	PUNCT
iajs-702	320	2	0	0	NUM
iajs-702	320	3	)	)	PUNCT
iajs-702	320	4	.	.	PUNCT
iajs-702	321	1	thus	thus	ADV
iajs-702	321	2	annj	annj	VERB
iajs-702	321	3	=	=	SYM
iajs-702	321	4	ann	ann	PROPN
iajs-702	321	5	anni	anni	PROPN
iajs-702	321	6	=	=	PROPN
iajs-702	321	7	i	i	PROPN
iajs-702	321	8	and	and	CCONJ
iajs-702	321	9	so	so	ADV
iajs-702	321	10	<	<	X
iajs-702	321	11	1	1	NUM
iajs-702	321	12	–	–	PUNCT
iajs-702	321	13	e	e	X
iajs-702	321	14	>	>	PUNCT
iajs-702	321	15	=	=	PUNCT
iajs-702	321	16	i.	i.	NOUN
iajs-702	321	17	it	it	PRON
iajs-702	321	18	follows	follow	VERB
iajs-702	321	19	that	that	SCONJ
iajs-702	321	20	i	i	PRON
iajs-702	321	21	is	be	AUX
iajs-702	321	22	a	a	DET
iajs-702	321	23	direct	direct	ADJ
iajs-702	321	24	summand	summand	NOUN
iajs-702	321	25	of	of	ADP
iajs-702	321	26	r.	r.	PROPN
iajs-702	321	27	(	(	PUNCT
iajs-702	321	28			NOUN
iajs-702	321	29	)	)	PUNCT
iajs-702	321	30	if	if	SCONJ
iajs-702	321	31	r	r	NOUN
iajs-702	321	32	is	be	AUX
iajs-702	321	33	a	a	DET
iajs-702	321	34	min	min	PROPN
iajs-702	321	35	-	-	ADJ
iajs-702	321	36	cs	cs	ADJ
iajs-702	321	37	ring	ring	NOUN
iajs-702	321	38	.	.	PUNCT
iajs-702	322	1	let	let	VERB
iajs-702	322	2	i	i	PRON
iajs-702	322	3	be	be	AUX
iajs-702	322	4	a	a	DET
iajs-702	322	5	maximal	maximal	ADJ
iajs-702	322	6	closed	closed	ADJ
iajs-702	322	7	ideal	ideal	NOUN
iajs-702	322	8	in	in	ADP
iajs-702	322	9	r	r	NOUN
iajs-702	322	10	,	,	PUNCT
iajs-702	322	11	with	with	ADP
iajs-702	322	12	anni	anni	PROPN
iajs-702	322	13	≠	≠	PROPN
iajs-702	322	14	0	0	NUM
iajs-702	322	15	.	.	PUNCT
iajs-702	323	1	by	by	ADP
iajs-702	323	2	[	[	X
iajs-702	323	3	4	4	NUM
iajs-702	323	4	,	,	PUNCT
iajs-702	323	5	theorem	theorem	VERB
iajs-702	323	6	2.38	2.38	NUM
iajs-702	323	7	,	,	PUNCT
iajs-702	323	8	p.65	p.65	ADP
iajs-702	323	9	]	]	PUNCT
iajs-702	323	10	,	,	PUNCT
iajs-702	323	11	i	i	PRON
iajs-702	323	12	=	=	SYM
iajs-702	323	13	ann	ann	PROPN
iajs-702	323	14	anni	anni	PROPN
iajs-702	323	15	.	.	PUNCT
iajs-702	324	1	let	let	VERB
iajs-702	324	2	j	j	PROPN
iajs-702	324	3	be	be	AUX
iajs-702	324	4	a	a	DET
iajs-702	324	5	relative	relative	ADJ
iajs-702	324	6	complement	complement	NOUN
iajs-702	324	7	of	of	ADP
iajs-702	324	8	i.	i.	PROPN
iajs-702	324	9	so	so	ADV
iajs-702	324	10	j	j	PROPN
iajs-702	324	11	is	be	AUX
iajs-702	324	12	a	a	DET
iajs-702	324	13	minimal	minimal	ADJ
iajs-702	324	14	closed	closed	ADJ
iajs-702	324	15	ideal	ideal	NOUN
iajs-702	324	16	of	of	ADP
iajs-702	324	17	r	r	NOUN
iajs-702	324	18	,	,	PUNCT
iajs-702	324	19	by	by	ADP
iajs-702	324	20	lemma	lemma	PROPN
iajs-702	324	21	1.31	1.31	NUM
iajs-702	324	22	.	.	PUNCT
iajs-702	325	1	hence	hence	ADV
iajs-702	325	2	j	j	PROPN
iajs-702	325	3	is	be	AUX
iajs-702	325	4	a	a	DET
iajs-702	325	5	direct	direct	ADJ
iajs-702	325	6	summand	summand	NOUN
iajs-702	325	7	of	of	ADP
iajs-702	325	8	r	r	NOUN
iajs-702	325	9	,	,	PUNCT
iajs-702	325	10	since	since	SCONJ
iajs-702	325	11	r	r	NOUN
iajs-702	325	12	is	be	AUX
iajs-702	325	13	min	min	NOUN
iajs-702	325	14	-	-	PROPN
iajs-702	325	15	cs	cs	ADJ
iajs-702	325	16	.	.	PUNCT
iajs-702	326	1	thus	thus	ADV
iajs-702	326	2	j	j	X
iajs-702	327	1	=	=	PUNCT
iajs-702	327	2	<	<	X
iajs-702	327	3	e	e	X
iajs-702	327	4	>	>	X
iajs-702	327	5	for	for	ADP
iajs-702	327	6	some	some	DET
iajs-702	327	7	idempotent	idempotent	ADJ
iajs-702	327	8	e	e	NOUN
iajs-702	327	9			PROPN
iajs-702	327	10	r.	r.	PROPN
iajs-702	327	11	but	but	CCONJ
iajs-702	327	12	ij	ij	INTJ
iajs-702	327	13			PROPN
iajs-702	327	14	i	i	PRON
iajs-702	327	15			PUNCT
iajs-702	327	16	j	j	X
iajs-702	327	17	=	=	PUNCT
iajs-702	327	18	(	(	PUNCT
iajs-702	327	19	0	0	NUM
iajs-702	327	20	)	)	PUNCT
iajs-702	327	21	,	,	PUNCT
iajs-702	327	22	so	so	ADV
iajs-702	327	23	i	i	PRON
iajs-702	327	24			PROPN
iajs-702	327	25	ann	ann	PROPN
iajs-702	327	26	j	j	PROPN
iajs-702	328	1	=	=	PUNCT
iajs-702	328	2	ann	ann	PROPN
iajs-702	329	1	<	<	X
iajs-702	329	2	e	e	X
iajs-702	329	3	>	>	X
iajs-702	329	4	=	=	PUNCT
iajs-702	330	1	<	<	X
iajs-702	330	2	1	1	NUM
iajs-702	330	3	–	–	PUNCT
iajs-702	330	4	e	e	NOUN
iajs-702	330	5	>	>	PUNCT
iajs-702	330	6	.	.	PUNCT
iajs-702	331	1	but	but	CCONJ
iajs-702	331	2	<	<	X
iajs-702	331	3	1	1	NUM
iajs-702	331	4	–	–	PUNCT
iajs-702	331	5	e	e	X
iajs-702	331	6	>	>	X
iajs-702	331	7	is	be	AUX
iajs-702	331	8	closed	close	VERB
iajs-702	331	9	ideal	ideal	NOUN
iajs-702	331	10	of	of	ADP
iajs-702	331	11	r	r	NOUN
iajs-702	331	12	,	,	PUNCT
iajs-702	331	13	since	since	SCONJ
iajs-702	331	14	it	it	PRON
iajs-702	331	15	is	be	AUX
iajs-702	331	16	a	a	DET
iajs-702	331	17	direct	direct	ADJ
iajs-702	331	18	summand	summand	NOUN
iajs-702	331	19	of	of	ADP
iajs-702	331	20	r.	r.	PROPN
iajs-702	331	21	it	it	PRON
iajs-702	331	22	follows	follow	VERB
iajs-702	331	23	that	that	SCONJ
iajs-702	331	24	i	i	PRON
iajs-702	331	25	=	=	PUNCT
iajs-702	332	1	<	<	X
iajs-702	332	2	1	1	NUM
iajs-702	332	3	–	–	PUNCT
iajs-702	332	4	e	e	X
iajs-702	332	5	>	>	PUNCT
iajs-702	332	6	because	because	SCONJ
iajs-702	332	7	i	i	PRON
iajs-702	332	8	is	be	AUX
iajs-702	332	9	a	a	DET
iajs-702	332	10	maximal	maximal	ADJ
iajs-702	332	11	closed	closed	ADJ
iajs-702	332	12	ideal	ideal	NOUN
iajs-702	332	13	of	of	ADP
iajs-702	332	14	r.	r.	PROPN
iajs-702	332	15	thus	thus	ADV
iajs-702	332	16	i	i	PRON
iajs-702	332	17	is	be	AUX
iajs-702	332	18	a	a	DET
iajs-702	332	19	direct	direct	ADJ
iajs-702	332	20	summand	summand	NOUN
iajs-702	332	21	of	of	ADP
iajs-702	332	22	r.	r.	PROPN
iajs-702	332	23	recall	recall	PROPN
iajs-702	332	24	that	that	PRON
iajs-702	332	25	,	,	PUNCT
iajs-702	332	26	a	a	DET
iajs-702	332	27	ring	ring	NOUN
iajs-702	332	28	r	r	NOUN
iajs-702	332	29	is	be	AUX
iajs-702	332	30	called	call	VERB
iajs-702	332	31	semihereditary	semihereditary	ADJ
iajs-702	332	32	if	if	SCONJ
iajs-702	332	33	every	every	DET
iajs-702	332	34	finitely	finitely	ADV
iajs-702	332	35	generating	generate	VERB
iajs-702	332	36	ideal	ideal	NOUN
iajs-702	332	37	of	of	ADP
iajs-702	332	38	r	r	NOUN
iajs-702	332	39	is	be	AUX
iajs-702	332	40	projective	projective	ADJ
iajs-702	332	41	,	,	PUNCT
iajs-702	332	42	[	[	X
iajs-702	332	43	4	4	NUM
iajs-702	332	44	,	,	PUNCT
iajs-702	332	45	p.10	p.10	ADP
iajs-702	332	46	]	]	X
iajs-702	332	47	.	.	PUNCT
iajs-702	333	1	1.34	1.34	NUM
iajs-702	333	2	corollary	corollary	NOUN
iajs-702	333	3	:	:	PUNCT
iajs-702	333	4	let	let	VERB
iajs-702	333	5	r	r	PRON
iajs-702	333	6	be	be	AUX
iajs-702	333	7	a	a	DET
iajs-702	333	8	semiheriditary	semiheriditary	ADJ
iajs-702	333	9	ring	ring	NOUN
iajs-702	333	10	.	.	PUNCT
iajs-702	334	1	then	then	ADV
iajs-702	334	2	r	r	NOUN
iajs-702	334	3	is	be	AUX
iajs-702	334	4	a	a	DET
iajs-702	334	5	min	min	PROPN
iajs-702	334	6	-	-	ADJ
iajs-702	334	7	cs	cs	ADJ
iajs-702	334	8	ring	ring	NOUN
iajs-702	334	9	if	if	SCONJ
iajs-702	334	10	and	and	CCONJ
iajs-702	334	11	only	only	ADV
iajs-702	334	12	if	if	SCONJ
iajs-702	334	13	r	r	NOUN
iajs-702	334	14	is	be	AUX
iajs-702	334	15	a	a	DET
iajs-702	334	16	max	max	PROPN
iajs-702	334	17	-	-	PUNCT
iajs-702	334	18	cs	cs	PROPN
iajs-702	334	19	ring	ring	NOUN
iajs-702	334	20	.	.	PUNCT
iajs-702	335	1	proof	proof	NOUN
iajs-702	335	2	:	:	PUNCT
iajs-702	335	3	since	since	SCONJ
iajs-702	335	4	r	r	NOUN
iajs-702	335	5	is	be	AUX
iajs-702	335	6	a	a	DET
iajs-702	335	7	semiheriditary	semiheriditary	ADJ
iajs-702	335	8	ring	ring	NOUN
iajs-702	335	9	.	.	PUNCT
iajs-702	336	1	then	then	ADV
iajs-702	336	2	r	r	NOUN
iajs-702	336	3	is	be	AUX
iajs-702	336	4	a	a	DET
iajs-702	336	5	nonsingular	nonsingular	ADJ
iajs-702	336	6	ring	ring	NOUN
iajs-702	336	7	,	,	PUNCT
iajs-702	336	8	[	[	X
iajs-702	336	9	4	4	NUM
iajs-702	336	10	,	,	PUNCT
iajs-702	336	11	p.36	p.36	NOUN
iajs-702	336	12	]	]	X
iajs-702	336	13	.	.	PUNCT
iajs-702	337	1	so	so	ADV
iajs-702	337	2	we	we	PRON
iajs-702	337	3	get	get	VERB
iajs-702	337	4	the	the	DET
iajs-702	337	5	result	result	NOUN
iajs-702	337	6	by	by	ADP
iajs-702	337	7	theorem	theorem	NOUN
iajs-702	337	8	1.33	1.33	NUM
iajs-702	337	9	.	.	PUNCT
iajs-702	338	1	مجلة	مجلة	NOUN
iajs-702	338	2	إبن	إبن	VERB
iajs-702	338	3	الھیثم	الھیثم	NOUN
iajs-702	338	4	للعلوم	للعلوم	NOUN
iajs-702	338	5	الصرفة	الصرفة	NOUN
iajs-702	338	6	و	و	PRON
iajs-702	338	7	التطبیقیة	التطبیقیة	PROPN
iajs-702	338	8	2012	2012	NUM
iajs-702	338	9	السنة	السنة	NOUN
iajs-702	339	1	25	25	NUM
iajs-702	339	2	المجلد	المجلد	NOUN
iajs-702	339	3	1	1	NUM
iajs-702	339	4	العدد	العدد	PROPN
iajs-702	339	5	ibn	ibn	PROPN
iajs-702	339	6	al	al	PROPN
iajs-702	339	7	-	-	PUNCT
iajs-702	339	8	haitham	haitham	PROPN
iajs-702	339	9	journal	journal	PROPN
iajs-702	339	10	for	for	ADP
iajs-702	339	11	pure	pure	ADJ
iajs-702	339	12	and	and	CCONJ
iajs-702	339	13	applied	apply	VERB
iajs-702	339	14	science	science	NOUN
iajs-702	339	15	no	no	NOUN
iajs-702	339	16	.	.	NOUN
iajs-702	339	17	1	1	NUM
iajs-702	339	18	vol	vol	NOUN
iajs-702	339	19	.	.	PUNCT
iajs-702	340	1	25	25	NUM
iajs-702	340	2	year	year	NOUN
iajs-702	340	3	2012	2012	NUM
iajs-702	340	4	recall	recall	VERB
iajs-702	340	5	that	that	SCONJ
iajs-702	340	6	a	a	DET
iajs-702	340	7	ring	ring	NOUN
iajs-702	340	8	r	r	NOUN
iajs-702	340	9	is	be	AUX
iajs-702	340	10	called	call	VERB
iajs-702	340	11	regular	regular	ADJ
iajs-702	340	12	(	(	PUNCT
iajs-702	340	13	von	von	PROPN
iajs-702	340	14	neumann	neumann	PROPN
iajs-702	340	15	)	)	PUNCT
iajs-702	340	16	if	if	SCONJ
iajs-702	340	17	for	for	ADP
iajs-702	340	18	every	every	PRON
iajs-702	340	19	ar	ar	ADP
iajs-702	340	20	there	there	PRON
iajs-702	340	21	is	be	VERB
iajs-702	340	22	an	an	DET
iajs-702	340	23	x	x	ADJ
iajs-702	340	24			NOUN
iajs-702	340	25	r	r	NOUN
iajs-702	340	26	such	such	DET
iajs-702	340	27	that	that	SCONJ
iajs-702	340	28	a	a	DET
iajs-702	340	29	x	x	X
iajs-702	340	30	a	a	DET
iajs-702	340	31	=	=	X
iajs-702	340	32	a	a	NOUN
iajs-702	340	33	,	,	PUNCT
iajs-702	340	34	that	that	PRON
iajs-702	340	35	is	be	AUX
iajs-702	340	36	a	a	DET
iajs-702	340	37	=	=	NOUN
iajs-702	340	38	a2	a2	PROPN
iajs-702	340	39	x	x	X
iajs-702	340	40	,	,	PUNCT
iajs-702	340	41	[	[	X
iajs-702	340	42	4	4	NUM
iajs-702	340	43	,	,	PUNCT
iajs-702	340	44	p.10	p.10	ADP
iajs-702	340	45	]	]	X
iajs-702	340	46	.	.	PUNCT
iajs-702	341	1	1.35	1.35	NUM
iajs-702	341	2	corollary	corollary	NOUN
iajs-702	341	3	:	:	PUNCT
iajs-702	341	4	if	if	SCONJ
iajs-702	341	5	r	r	NOUN
iajs-702	341	6	is	be	AUX
iajs-702	341	7	a	a	DET
iajs-702	341	8	regular	regular	ADJ
iajs-702	341	9	ring	ring	NOUN
iajs-702	341	10	.	.	PUNCT
iajs-702	342	1	then	then	ADV
iajs-702	342	2	r	r	NOUN
iajs-702	342	3	is	be	AUX
iajs-702	342	4	min	min	NOUN
iajs-702	342	5	-	-	ADJ
iajs-702	342	6	cs	cs	ADJ
iajs-702	342	7	if	if	SCONJ
iajs-702	343	1	and	and	CCONJ
iajs-702	343	2	only	only	ADV
iajs-702	343	3	if	if	SCONJ
iajs-702	343	4	r	r	NOUN
iajs-702	343	5	is	be	AUX
iajs-702	343	6	a	a	DET
iajs-702	343	7	max	max	PROPN
iajs-702	343	8	-	-	PUNCT
iajs-702	343	9	cs	cs	PROPN
iajs-702	343	10	.	.	PUNCT
iajs-702	344	1	the	the	DET
iajs-702	344	2	following	follow	VERB
iajs-702	344	3	lemma	lemma	PROPN
iajs-702	344	4	is	be	AUX
iajs-702	344	5	needed	need	VERB
iajs-702	344	6	for	for	ADP
iajs-702	344	7	the	the	DET
iajs-702	344	8	following	follow	VERB
iajs-702	344	9	corollary	corollary	NOUN
iajs-702	344	10	.	.	PUNCT
iajs-702	345	1	1.36	1.36	NUM
iajs-702	345	2	lemma	lemma	NOUN
iajs-702	345	3	:	:	PUNCT
iajs-702	346	1	[	[	X
iajs-702	346	2	4	4	NUM
iajs-702	346	3	,	,	PUNCT
iajs-702	346	4	exc.5	exc.5	PROPN
iajs-702	346	5	,	,	PUNCT
iajs-702	346	6	p.36	p.36	X
iajs-702	346	7	]	]	X
iajs-702	346	8	for	for	ADP
iajs-702	346	9	a	a	DET
iajs-702	346	10	commutative	commutative	ADJ
iajs-702	346	11	ring	ring	NOUN
iajs-702	346	12	,	,	PUNCT
iajs-702	346	13	r	r	NOUN
iajs-702	346	14	/	/	SYM
iajs-702	346	15	z(r	z(r	NOUN
iajs-702	346	16	)	)	PUNCT
iajs-702	346	17	is	be	AUX
iajs-702	346	18	a	a	DET
iajs-702	346	19	nonsingular	nonsingular	ADJ
iajs-702	346	20	ring	ring	NOUN
iajs-702	346	21	.	.	PUNCT
iajs-702	347	1	proof	proof	NOUN
iajs-702	347	2	:	:	PUNCT
iajs-702	347	3	suppose	suppose	VERB
iajs-702	347	4	there	there	PRON
iajs-702	347	5	exists	exist	VERB
iajs-702	347	6	x	x	X
iajs-702	347	7	=	=	PUNCT
iajs-702	347	8	x	x	SYM
iajs-702	347	9	+	+	NUM
iajs-702	347	10	z(r	z(r	NOUN
iajs-702	347	11	)	)	PUNCT
iajs-702	347	12			NOUN
iajs-702	347	13	z(r	z(r	NOUN
iajs-702	347	14	/	/	SYM
iajs-702	347	15	z(r	z(r	NOUN
iajs-702	347	16	)	)	PUNCT
iajs-702	347	17	)	)	PUNCT
iajs-702	348	1	and	and	CCONJ
iajs-702	348	2	x	x	X
iajs-702	348	3			PROPN
iajs-702	348	4	z(r	z(r	NOUN
iajs-702	348	5	)	)	PUNCT
iajs-702	348	6	,	,	PUNCT
iajs-702	348	7	so	so	ADV
iajs-702	348	8	x	x	SYM
iajs-702	348	9	≠	≠	PROPN
iajs-702	348	10	0	0	NUM
iajs-702	348	11	and	and	CCONJ
iajs-702	348	12	ann(x	ann(x	PROPN
iajs-702	348	13	)	)	PUNCT
iajs-702	348	14	is	be	AUX
iajs-702	348	15	not	not	PART
iajs-702	348	16	essential	essential	ADJ
iajs-702	348	17	in	in	ADP
iajs-702	348	18	r	r	NOUN
iajs-702	348	19	,	,	PUNCT
iajs-702	348	20	also	also	ADV
iajs-702	348	21	annr	annr	NOUN
iajs-702	348	22	/	/	SYM
iajs-702	348	23	z(r	z(r	NOUN
iajs-702	348	24	)	)	PUNCT
iajs-702	348	25	(	(	PUNCT
iajs-702	348	26	x	x	X
iajs-702	348	27	+	+	NUM
iajs-702	348	28	z(r	z(r	NOUN
iajs-702	348	29	)	)	PUNCT
iajs-702	348	30	)	)	PUNCT
iajs-702	348	31			NOUN
iajs-702	348	32	e	e	X
iajs-702	348	33	r	r	NOUN
iajs-702	348	34	/	/	SYM
iajs-702	348	35	z(r	z(r	NOUN
iajs-702	348	36	)	)	PUNCT
iajs-702	348	37	.	.	PUNCT
iajs-702	349	1	then	then	ADV
iajs-702	349	2	for	for	ADP
iajs-702	349	3	each	each	DET
iajs-702	349	4	i	i	NOUN
iajs-702	349	5	/	/	SYM
iajs-702	349	6	z(r	z(r	PROPN
iajs-702	349	7	)	)	PUNCT
iajs-702	349	8	with	with	ADP
iajs-702	349	9	i	i	PRON
iajs-702	349	10			VERB
iajs-702	349	11	z(r	z(r	PROPN
iajs-702	349	12	)	)	PUNCT
iajs-702	349	13	,	,	PUNCT
iajs-702	349	14	(	(	PUNCT
iajs-702	349	15	i	i	NOUN
iajs-702	349	16	/	/	SYM
iajs-702	349	17	z(r	z(r	NOUN
iajs-702	349	18	)	)	PUNCT
iajs-702	349	19	)	)	PUNCT
iajs-702	349	20	annr	annr	ADP
iajs-702	349	21	/	/	SYM
iajs-702	349	22	z(r	z(r	NOUN
iajs-702	349	23	)	)	PUNCT
iajs-702	349	24	(	(	PUNCT
iajs-702	349	25	x+z(r))≠or	x+z(r))≠or	PROPN
iajs-702	349	26	/	/	SYM
iajs-702	349	27	z(r)=z(r	z(r)=z(r	NUM
iajs-702	349	28	)	)	PUNCT
iajs-702	349	29	.	.	PUNCT
iajs-702	350	1	so	so	ADV
iajs-702	350	2	there	there	PRON
iajs-702	350	3	exists	exist	VERB
iajs-702	350	4	a	a	DET
iajs-702	350	5	+	+	NUM
iajs-702	350	6	z(r	z(r	NOUN
iajs-702	350	7	)	)	PUNCT
iajs-702	350	8			NOUN
iajs-702	350	9	ann	ann	PROPN
iajs-702	350	10	r	r	X
iajs-702	350	11	/	/	SYM
iajs-702	350	12	z(r	z(r	NOUN
iajs-702	350	13	)	)	PUNCT
iajs-702	350	14	(	(	PUNCT
iajs-702	350	15	x	x	X
iajs-702	350	16	+	+	NUM
iajs-702	350	17	z(r	z(r	NOUN
iajs-702	350	18	)	)	PUNCT
iajs-702	350	19	)	)	PUNCT
iajs-702	350	20	,	,	PUNCT
iajs-702	350	21	a	a	DET
iajs-702	350	22			NOUN
iajs-702	350	23	i	i	PRON
iajs-702	350	24	such	such	ADJ
iajs-702	350	25	that	that	SCONJ
iajs-702	350	26	a	a	DET
iajs-702	350	27			NOUN
iajs-702	350	28	z(r	z(r	NOUN
iajs-702	350	29	)	)	PUNCT
iajs-702	350	30	.	.	PUNCT
iajs-702	351	1	then	then	ADV
iajs-702	351	2	a	a	DET
iajs-702	351	3	≠	≠	PROPN
iajs-702	351	4	0	0	NUM
iajs-702	351	5	.	.	PUNCT
iajs-702	352	1	thus	thus	ADV
iajs-702	352	2	ax	ax	X
iajs-702	352	3	+	+	CCONJ
iajs-702	352	4	z(r	z(r	NOUN
iajs-702	352	5	)	)	PUNCT
iajs-702	352	6	=	=	SYM
iajs-702	352	7	z(r	z(r	NOUN
iajs-702	352	8	)	)	PUNCT
iajs-702	352	9	,	,	PUNCT
iajs-702	352	10	annr(a	annr(a	ADJ
iajs-702	352	11	)	)	PUNCT
iajs-702	352	12	is	be	AUX
iajs-702	352	13	not	not	PART
iajs-702	352	14	essential	essential	ADJ
iajs-702	352	15	in	in	ADP
iajs-702	352	16	r	r	NOUN
iajs-702	352	17	and	and	CCONJ
iajs-702	352	18	a	a	DET
iajs-702	352	19			NOUN
iajs-702	352	20	i.	i.	NOUN
iajs-702	352	21	hence	hence	ADV
iajs-702	352	22	ax	ax	VERB
iajs-702	352	23			PROPN
iajs-702	352	24	z(r	z(r	NUM
iajs-702	352	25	)	)	PUNCT
iajs-702	352	26	and	and	CCONJ
iajs-702	352	27	annr(a	annr(a	ADJ
iajs-702	352	28	)	)	PUNCT
iajs-702	352	29	is	be	AUX
iajs-702	352	30	not	not	PART
iajs-702	352	31	essential	essential	ADJ
iajs-702	352	32	in	in	ADP
iajs-702	352	33	r.	r.	PROPN
iajs-702	353	1	so	so	SCONJ
iajs-702	353	2	that	that	PRON
iajs-702	353	3	ann(ax	ann(ax	VERB
iajs-702	353	4	)	)	PUNCT
iajs-702	353	5			NOUN
iajs-702	353	6	e	e	NOUN
iajs-702	353	7	r	r	NOUN
iajs-702	353	8	and	and	CCONJ
iajs-702	353	9	ann(a	ann(a	PROPN
iajs-702	353	10	)	)	PUNCT
iajs-702	353	11	is	be	AUX
iajs-702	353	12	not	not	PART
iajs-702	353	13	essential	essential	ADJ
iajs-702	353	14	in	in	ADP
iajs-702	353	15	r.	r.	PROPN
iajs-702	353	16	now	now	ADV
iajs-702	353	17	,	,	PUNCT
iajs-702	353	18	since	since	SCONJ
iajs-702	353	19	ann(ax	ann(ax	NOUN
iajs-702	353	20	)	)	PUNCT
iajs-702	353	21			NOUN
iajs-702	353	22	e	e	NOUN
iajs-702	353	23	r	r	NOUN
iajs-702	353	24	,	,	PUNCT
iajs-702	353	25	then	then	ADV
iajs-702	353	26	for	for	ADP
iajs-702	353	27	each	each	DET
iajs-702	353	28	nonzero	nonzero	NOUN
iajs-702	353	29	ideal	ideal	PROPN
iajs-702	353	30	j	j	PROPN
iajs-702	353	31	of	of	ADP
iajs-702	353	32	r	r	PROPN
iajs-702	353	33	,	,	PUNCT
iajs-702	353	34	j	j	PROPN
iajs-702	353	35			X
iajs-702	353	36	ann(ax	ann(ax	NOUN
iajs-702	353	37	)	)	PUNCT
iajs-702	353	38	≠	≠	PROPN
iajs-702	353	39	0	0	NUM
iajs-702	353	40	.	.	PUNCT
iajs-702	354	1	hence	hence	ADV
iajs-702	354	2	there	there	PRON
iajs-702	354	3	exists	exist	VERB
iajs-702	354	4	y	y	PROPN
iajs-702	354	5			PROPN
iajs-702	354	6	ann(ax	ann(ax	PROPN
iajs-702	354	7	)	)	PUNCT
iajs-702	354	8	and	and	CCONJ
iajs-702	354	9	y	y	PROPN
iajs-702	354	10	≠	≠	PROPN
iajs-702	354	11	0	0	NUM
iajs-702	354	12	.	.	PUNCT
iajs-702	355	1	thus	thus	ADV
iajs-702	355	2	yax	yax	X
iajs-702	356	1	=	=	SYM
iajs-702	356	2	0	0	X
iajs-702	356	3	.	.	PUNCT
iajs-702	357	1	(	(	PUNCT
iajs-702	357	2	1	1	X
iajs-702	357	3	)	)	PUNCT
iajs-702	357	4	if	if	SCONJ
iajs-702	357	5	ya	ya	PRON
iajs-702	357	6	=	=	NOUN
iajs-702	357	7	0	0	PUNCT
iajs-702	357	8	then	then	ADV
iajs-702	357	9	0	0	NUM
iajs-702	357	10	≠	≠	PROPN
iajs-702	357	11	y	y	PROPN
iajs-702	357	12			NOUN
iajs-702	357	13	ann(a	ann(a	ADP
iajs-702	357	14	)	)	PUNCT
iajs-702	357	15	thus	thus	ADV
iajs-702	357	16	0	0	NUM
iajs-702	357	17	≠	≠	PROPN
iajs-702	357	18	y	y	PROPN
iajs-702	357	19			PROPN
iajs-702	357	20	jann(a	jann(a	PROPN
iajs-702	357	21	)	)	PUNCT
iajs-702	357	22	so	so	SCONJ
iajs-702	357	23	that	that	SCONJ
iajs-702	357	24	ann(a	ann(a	ADP
iajs-702	357	25	)	)	PUNCT
iajs-702	357	26			NOUN
iajs-702	357	27	e	e	NOUN
iajs-702	357	28	r	r	NOUN
iajs-702	357	29	which	which	PRON
iajs-702	357	30	is	be	AUX
iajs-702	357	31	a	a	DET
iajs-702	357	32	contradiction	contradiction	NOUN
iajs-702	357	33	.	.	PUNCT
iajs-702	358	1	(	(	PUNCT
iajs-702	358	2	2	2	X
iajs-702	358	3	)	)	PUNCT
iajs-702	358	4	if	if	SCONJ
iajs-702	358	5	ya	ya	PRON
iajs-702	358	6	≠	≠	PROPN
iajs-702	358	7	0	0	NUM
iajs-702	358	8	,	,	PUNCT
iajs-702	358	9	then	then	ADV
iajs-702	358	10	0≠yaann(x)j	0≠yaann(x)j	NOUN
iajs-702	358	11	.	.	PUNCT
iajs-702	359	1	hence	hence	ADV
iajs-702	359	2	ann(x	ann(x	ADJ
iajs-702	359	3	)	)	PUNCT
iajs-702	359	4			NOUN
iajs-702	359	5	e	e	NOUN
iajs-702	359	6	r	r	NOUN
iajs-702	359	7	which	which	PRON
iajs-702	359	8	is	be	AUX
iajs-702	359	9	a	a	DET
iajs-702	359	10	contradiction	contradiction	NOUN
iajs-702	359	11	.	.	PUNCT
iajs-702	360	1	(	(	PUNCT
iajs-702	360	2	3	3	X
iajs-702	360	3	)	)	PUNCT
iajs-702	360	4	if	if	SCONJ
iajs-702	360	5	yx	yx	NOUN
iajs-702	360	6	=	=	NOUN
iajs-702	360	7	0	0	PUNCT
iajs-702	361	1	then	then	ADV
iajs-702	361	2	0	0	NUM
iajs-702	361	3	≠	≠	PROPN
iajs-702	361	4	y	y	PROPN
iajs-702	361	5			NOUN
iajs-702	361	6	ann(x	ann(x	PROPN
iajs-702	361	7	)	)	PUNCT
iajs-702	361	8	,	,	PUNCT
iajs-702	361	9	so	so	SCONJ
iajs-702	361	10	that	that	SCONJ
iajs-702	361	11	0	0	NUM
iajs-702	362	1	≠	≠	PROPN
iajs-702	362	2	y	y	PROPN
iajs-702	362	3			NOUN
iajs-702	362	4	ann(x	ann(x	PROPN
iajs-702	362	5	)	)	PUNCT
iajs-702	362	6			PUNCT
iajs-702	362	7	j.	j.	PROPN
iajs-702	362	8	thus	thus	ADV
iajs-702	362	9	ann(x	ann(x	PROPN
iajs-702	362	10	)	)	PUNCT
iajs-702	362	11			NOUN
iajs-702	362	12	e	e	NOUN
iajs-702	362	13	r	r	NOUN
iajs-702	362	14	which	which	PRON
iajs-702	362	15	is	be	AUX
iajs-702	362	16	a	a	DET
iajs-702	362	17	contradiction	contradiction	NOUN
iajs-702	362	18	.	.	PUNCT
iajs-702	363	1	(	(	PUNCT
iajs-702	363	2	4	4	X
iajs-702	363	3	)	)	PUNCT
iajs-702	363	4	if	if	SCONJ
iajs-702	363	5	yx	yx	ADP
iajs-702	363	6	≠	≠	PROPN
iajs-702	363	7	0	0	NUM
iajs-702	363	8	,	,	PUNCT
iajs-702	363	9	then	then	ADV
iajs-702	363	10	0≠	0≠	NUM
iajs-702	363	11	yxann(a)j	yxann(a)j	NOUN
iajs-702	363	12	,	,	PUNCT
iajs-702	363	13	so	so	SCONJ
iajs-702	363	14	that	that	SCONJ
iajs-702	363	15	ann(a)	ann(a)	PROPN
iajs-702	363	16	e	e	NOUN
iajs-702	363	17	r	r	NOUN
iajs-702	363	18	which	which	PRON
iajs-702	363	19	is	be	AUX
iajs-702	363	20	a	a	DET
iajs-702	363	21	contradiction	contradiction	NOUN
iajs-702	363	22	.	.	PUNCT
iajs-702	364	1	(	(	PUNCT
iajs-702	364	2	5	5	X
iajs-702	364	3	)	)	PUNCT
iajs-702	364	4	if	if	SCONJ
iajs-702	364	5	ax	ax	NOUN
iajs-702	364	6	=	=	NOUN
iajs-702	364	7	0	0	NUM
iajs-702	364	8	then	then	ADV
iajs-702	364	9	0	0	NUM
iajs-702	364	10	≠	≠	PROPN
iajs-702	364	11	a	a	VERB
iajs-702	364	12	ann(x	ann(x	NOUN
iajs-702	364	13	)	)	PUNCT
iajs-702	364	14			PUNCT
iajs-702	365	1	i	i	PRON
iajs-702	365	2	,	,	PUNCT
iajs-702	365	3	that	that	PRON
iajs-702	365	4	is	be	AUX
iajs-702	365	5	ann(x	ann(x	PROPN
iajs-702	365	6	)	)	PUNCT
iajs-702	365	7			NOUN
iajs-702	365	8	e	e	NOUN
iajs-702	365	9	r	r	NOUN
iajs-702	365	10	which	which	PRON
iajs-702	365	11	is	be	AUX
iajs-702	365	12	a	a	DET
iajs-702	365	13	contradiction	contradiction	NOUN
iajs-702	365	14	.	.	PUNCT
iajs-702	366	1	thus	thus	ADV
iajs-702	366	2	our	our	PRON
iajs-702	366	3	assumption	assumption	NOUN
iajs-702	366	4	is	be	AUX
iajs-702	366	5	false	false	ADJ
iajs-702	366	6	and	and	CCONJ
iajs-702	366	7	so	so	ADV
iajs-702	366	8	z(r	z(r	PROPN
iajs-702	366	9	/	/	SYM
iajs-702	366	10	z(r	z(r	NOUN
iajs-702	366	11	)	)	PUNCT
iajs-702	366	12	)	)	PUNCT
iajs-702	367	1	=	=	PUNCT
iajs-702	367	2	z(r	z(r	NOUN
iajs-702	367	3	)	)	PUNCT
iajs-702	367	4	=	=	PUNCT
iajs-702	367	5	or	or	CCONJ
iajs-702	367	6	/	/	SYM
iajs-702	367	7	z(r	z(r	NOUN
iajs-702	367	8	)	)	PUNCT
iajs-702	367	9	.	.	PUNCT
iajs-702	368	1	1.37	1.37	NUM
iajs-702	368	2	corollary	corollary	NOUN
iajs-702	368	3	let	let	VERB
iajs-702	368	4	r	r	PRON
iajs-702	368	5	be	be	AUX
iajs-702	368	6	a	a	DET
iajs-702	368	7	ring	ring	NOUN
iajs-702	368	8	.	.	PUNCT
iajs-702	369	1	then	then	ADV
iajs-702	369	2	r	r	X
iajs-702	369	3	/	/	SYM
iajs-702	369	4	z(r	z(r	NOUN
iajs-702	369	5	)	)	PUNCT
iajs-702	369	6	is	be	AUX
iajs-702	369	7	min	min	NOUN
iajs-702	369	8	-	-	ADJ
iajs-702	369	9	cs	cs	ADJ
iajs-702	369	10	if	if	SCONJ
iajs-702	370	1	and	and	CCONJ
iajs-702	370	2	only	only	ADV
iajs-702	370	3	if	if	SCONJ
iajs-702	370	4	r	r	NOUN
iajs-702	370	5	/	/	SYM
iajs-702	370	6	z(r	z(r	NOUN
iajs-702	370	7	)	)	PUNCT
iajs-702	370	8	is	be	AUX
iajs-702	370	9	max	max	PROPN
iajs-702	370	10	-	-	PUNCT
iajs-702	370	11	cs	cs	PROPN
iajs-702	370	12	.	.	PROPN
iajs-702	370	13	proof	proof	NOUN
iajs-702	370	14	:	:	PUNCT
iajs-702	370	15	by	by	ADP
iajs-702	370	16	lemma	lemma	PROPN
iajs-702	370	17	1.36	1.36	NUM
iajs-702	370	18	r	r	NOUN
iajs-702	370	19	/	/	SYM
iajs-702	370	20	z(r	z(r	NOUN
iajs-702	370	21	)	)	PUNCT
iajs-702	370	22	is	be	AUX
iajs-702	370	23	nonsingular	nonsingular	ADJ
iajs-702	370	24	,	,	PUNCT
iajs-702	370	25	so	so	ADV
iajs-702	370	26	the	the	DET
iajs-702	370	27	result	result	NOUN
iajs-702	370	28	follows	follow	VERB
iajs-702	370	29	by	by	ADP
iajs-702	370	30	theorem	theorem	NOUN
iajs-702	370	31	1.33	1.33	NUM
iajs-702	370	32	.	.	PUNCT
iajs-702	371	1	before	before	SCONJ
iajs-702	371	2	we	we	PRON
iajs-702	371	3	give	give	VERB
iajs-702	371	4	the	the	DET
iajs-702	371	5	following	follow	VERB
iajs-702	371	6	corollary	corollary	NOUN
iajs-702	371	7	,	,	PUNCT
iajs-702	371	8	we	we	PRON
iajs-702	371	9	need	need	VERB
iajs-702	371	10	the	the	DET
iajs-702	371	11	following	follow	VERB
iajs-702	371	12	lemma	lemma	PROPN
iajs-702	371	13	.	.	PROPN
iajs-702	372	1	1.38	1.38	NUM
iajs-702	372	2	lemma	lemma	PROPN
iajs-702	372	3	:	:	PUNCT
iajs-702	373	1	[	[	X
iajs-702	373	2	4	4	NUM
iajs-702	373	3	,	,	PUNCT
iajs-702	373	4	exc.13	exc.13	NOUN
iajs-702	373	5	,	,	PUNCT
iajs-702	373	6	p.37	p.37	X
iajs-702	373	7	]	]	X
iajs-702	373	8	if	if	SCONJ
iajs-702	373	9	r	r	NOUN
iajs-702	373	10	is	be	AUX
iajs-702	373	11	a	a	DET
iajs-702	373	12	nonsingular	nonsingular	ADJ
iajs-702	373	13	ring	ring	NOUN
iajs-702	373	14	,	,	PUNCT
iajs-702	373	15	then	then	ADV
iajs-702	373	16	r[x1	r[x1	NOUN
iajs-702	373	17	,	,	PUNCT
iajs-702	373	18	x2	x2	PROPN
iajs-702	373	19	,	,	PUNCT
iajs-702	373	20	…	…	PUNCT
iajs-702	373	21	,	,	PUNCT
iajs-702	373	22	xn	xn	PROPN
iajs-702	373	23	]	]	X
iajs-702	373	24	is	be	AUX
iajs-702	373	25	a	a	DET
iajs-702	373	26	nonsingular	nonsingular	ADJ
iajs-702	373	27	ring	ring	NOUN
iajs-702	373	28	.	.	PUNCT
iajs-702	374	1	proof	proof	NOUN
iajs-702	374	2	:	:	PUNCT
iajs-702	374	3	first	first	ADV
iajs-702	374	4	we	we	PRON
iajs-702	374	5	shall	shall	AUX
iajs-702	374	6	prove	prove	VERB
iajs-702	374	7	that	that	PRON
iajs-702	374	8	r[x	r[x	NOUN
iajs-702	374	9	]	]	PUNCT
iajs-702	374	10	is	be	AUX
iajs-702	374	11	a	a	DET
iajs-702	374	12	nonsingular	nonsingular	ADJ
iajs-702	374	13	.	.	PUNCT
iajs-702	375	1	let	let	VERB
iajs-702	375	2	f(x	f(x	PROPN
iajs-702	375	3	)	)	PUNCT
iajs-702	375	4			NOUN
iajs-702	375	5	z(r[x	z(r[x	PROPN
iajs-702	375	6	]	]	PUNCT
iajs-702	375	7	)	)	PUNCT
iajs-702	375	8	,	,	PUNCT
iajs-702	375	9	then	then	ADV
iajs-702	375	10	annr[x]f(x	annr[x]f(x	ADP
iajs-702	375	11	)	)	PUNCT
iajs-702	375	12			PROPN
iajs-702	375	13	e	e	ADP
iajs-702	375	14	r[x	r[x	NOUN
iajs-702	375	15	]	]	PUNCT
iajs-702	375	16	.	.	PUNCT
iajs-702	376	1	assume	assume	VERB
iajs-702	376	2	f(x	f(x	PROPN
iajs-702	376	3	)	)	PUNCT
iajs-702	377	1	=	=	SYM
iajs-702	377	2	a0	a0	PROPN
iajs-702	377	3	+	+	CCONJ
iajs-702	377	4	a1x	a1x	NOUN
iajs-702	377	5	+	+	PUNCT
iajs-702	377	6	…	…	PUNCT
iajs-702	377	7	+	+	CCONJ
iajs-702	377	8	anx	anx	ADJ
iajs-702	377	9	n	n	CCONJ
iajs-702	377	10	,	,	PUNCT
iajs-702	377	11	where	where	SCONJ
iajs-702	377	12	ai	ai	VERB
iajs-702	377	13			PROPN
iajs-702	377	14	r	r	NOUN
iajs-702	377	15	,	,	PUNCT
iajs-702	377	16	for	for	ADP
iajs-702	377	17	each	each	DET
iajs-702	377	18	i	i	NOUN
iajs-702	377	19	=	=	NOUN
iajs-702	377	20	0	0	NUM
iajs-702	377	21	,	,	PUNCT
iajs-702	377	22	1	1	NUM
iajs-702	377	23	,	,	PUNCT
iajs-702	377	24	…	…	PUNCT
iajs-702	377	25	,	,	PUNCT
iajs-702	377	26	n.	n.	NOUN
iajs-702	377	27	annr[x]f(x	annr[x]f(x	NOUN
iajs-702	377	28	)	)	PUNCT
iajs-702	377	29	=	=	PRON
iajs-702	377	30	annr[x]a0	annr[x]a0	VERB
iajs-702	377	31			PUNCT
iajs-702	377	32	annr[x]a1x	annr[x]a1x	NOUN
iajs-702	377	33			NOUN
iajs-702	377	34	…	…	PUNCT
iajs-702	377	35			PUNCT
iajs-702	377	36	annr[x]anx	annr[x]anx	PROPN
iajs-702	377	37	n	n	NUM
iajs-702	377	38			NUM
iajs-702	377	39	e	e	NOUN
iajs-702	377	40	r[x	r[x	NOUN
iajs-702	377	41	]	]	PUNCT
iajs-702	377	42	.	.	PUNCT
iajs-702	378	1	so	so	ADV
iajs-702	378	2	,	,	PUNCT
iajs-702	378	3	annr[x]a0	annr[x]a0	NOUN
iajs-702	378	4	e	e	PROPN
iajs-702	378	5	r[x	r[x	PROPN
iajs-702	378	6	]	]	X
iajs-702	378	7	,	,	PUNCT
iajs-702	378	8	annr[x]a1x	annr[x]a1x	ADP
iajs-702	378	9			NOUN
iajs-702	378	10	e	e	ADP
iajs-702	378	11	r[x	r[x	NOUN
iajs-702	378	12	]	]	X
iajs-702	378	13	,	,	PUNCT
iajs-702	378	14	…	…	PUNCT
iajs-702	378	15	,	,	PUNCT
iajs-702	378	16	annr[x]anx	annr[x]anx	PROPN
iajs-702	378	17	n	n	NUM
iajs-702	378	18			NUM
iajs-702	378	19	e	e	NOUN
iajs-702	378	20	r[x	r[x	NOUN
iajs-702	378	21	]	]	PUNCT
iajs-702	378	22	.	.	PUNCT
iajs-702	379	1	we	we	PRON
iajs-702	379	2	claim	claim	VERB
iajs-702	379	3	that	that	SCONJ
iajs-702	379	4	annr(ai	annr(ai	ADJ
iajs-702	379	5	)	)	PUNCT
iajs-702	379	6			PROPN
iajs-702	379	7	e	e	NOUN
iajs-702	379	8	r	r	NOUN
iajs-702	379	9	,	,	PUNCT
iajs-702	379	10	for	for	ADP
iajs-702	379	11	all	all	DET
iajs-702	379	12	i	i	PRON
iajs-702	379	13	=	=	NOUN
iajs-702	379	14	0	0	NUM
iajs-702	379	15	,	,	PUNCT
iajs-702	379	16	1	1	NUM
iajs-702	379	17	,	,	PUNCT
iajs-702	379	18	…	…	PUNCT
iajs-702	379	19	,	,	PUNCT
iajs-702	379	20	n.	n.	NOUN
iajs-702	379	21	to	to	PART
iajs-702	379	22	prove	prove	VERB
iajs-702	379	23	this	this	PRON
iajs-702	379	24	.	.	PUNCT
iajs-702	380	1	suppose	suppose	VERB
iajs-702	380	2	there	there	PRON
iajs-702	380	3	exists	exist	VERB
iajs-702	380	4	j	j	PROPN
iajs-702	380	5			NUM
iajs-702	380	6	r	r	NOUN
iajs-702	380	7	and	and	CCONJ
iajs-702	380	8	j	j	PROPN
iajs-702	380	9	≠	≠	PROPN
iajs-702	380	10	0	0	NUM
iajs-702	380	11	such	such	ADJ
iajs-702	380	12	that	that	SCONJ
iajs-702	380	13	annra0	annra0	NOUN
iajs-702	380	14			PUNCT
iajs-702	380	15	j	j	PROPN
iajs-702	381	1	=	=	SYM
iajs-702	381	2	0	0	PROPN
iajs-702	381	3	.	.	PUNCT
iajs-702	382	1	j	j	PROPN
iajs-702	382	2	≠	≠	PROPN
iajs-702	382	3	0	0	NUM
iajs-702	382	4	so	so	CCONJ
iajs-702	382	5	j[x	j[x	PROPN
iajs-702	382	6	]	]	PUNCT
iajs-702	382	7	is	be	AUX
iajs-702	382	8	an	an	DET
iajs-702	382	9	ideal	ideal	NOUN
iajs-702	382	10	in	in	ADP
iajs-702	382	11	r[x	r[x	NOUN
iajs-702	382	12	]	]	PUNCT
iajs-702	382	13	and	and	CCONJ
iajs-702	382	14	j[x	j[x	NOUN
iajs-702	382	15	]	]	PUNCT
iajs-702	382	16	≠	≠	PROPN
iajs-702	382	17	0	0	X
iajs-702	382	18	.	.	PUNCT
iajs-702	383	1	hence	hence	ADV
iajs-702	383	2	j[x	j[x	X
iajs-702	383	3	]	]	PUNCT
iajs-702	383	4			PUNCT
iajs-702	383	5	annr[x]a0	annr[x]a0	VERB
iajs-702	383	6	≠	≠	PROPN
iajs-702	383	7	0	0	X
iajs-702	383	8	.	.	PUNCT
iajs-702	384	1	let	let	VERB
iajs-702	384	2	0	0	NUM
iajs-702	384	3	≠	≠	PROPN
iajs-702	384	4	g(x	g(x	PROPN
iajs-702	384	5	)	)	PUNCT
iajs-702	384	6			NOUN
iajs-702	384	7	annr[x]a0	annr[x]a0	NOUN
iajs-702	384	8	and	and	CCONJ
iajs-702	384	9	g(x	g(x	NOUN
iajs-702	384	10	)	)	PUNCT
iajs-702	384	11			NOUN
iajs-702	384	12	j[x	j[x	PROPN
iajs-702	384	13	]	]	PUNCT
iajs-702	384	14	,	,	PUNCT
iajs-702	384	15	such	such	ADJ
iajs-702	384	16	that	that	SCONJ
iajs-702	384	17	g(x	g(x	NOUN
iajs-702	384	18	)	)	PUNCT
iajs-702	385	1	=	=	SYM
iajs-702	385	2	b0	b0	NOUN
iajs-702	385	3	+	+	NOUN
iajs-702	385	4	b1x	b1x	PROPN
iajs-702	385	5	+	+	CCONJ
iajs-702	385	6	…	…	PUNCT
iajs-702	385	7	+	+	CCONJ
iajs-702	385	8	bmxm	bmxm	ADJ
iajs-702	385	9	,	,	PUNCT
iajs-702	385	10	where	where	SCONJ
iajs-702	385	11	bi	bi	ADJ
iajs-702	385	12			PROPN
iajs-702	385	13	j	j	PROPN
iajs-702	385	14	,	,	PUNCT
iajs-702	385	15	for	for	ADP
iajs-702	385	16	all	all	DET
iajs-702	385	17	i	i	PRON
iajs-702	385	18	=	=	NOUN
iajs-702	385	19	0	0	NUM
iajs-702	385	20	,	,	PUNCT
iajs-702	385	21	1	1	NUM
iajs-702	385	22	,	,	PUNCT
iajs-702	385	23	…	…	PUNCT
iajs-702	385	24	,	,	PUNCT
iajs-702	385	25	m	m	VERB
iajs-702	385	26	and	and	CCONJ
iajs-702	385	27	there	there	PRON
iajs-702	385	28	exists	exist	VERB
iajs-702	385	29	k	k	PROPN
iajs-702	385	30			PROPN
iajs-702	385	31	{	{	PUNCT
iajs-702	385	32	0,1,	0,1,	NOUN
iajs-702	385	33	…	…	SYM
iajs-702	385	34	,m	,m	PUNCT
iajs-702	385	35	}	}	PUNCT
iajs-702	385	36	with	with	ADP
iajs-702	385	37	bk	bk	ADP
iajs-702	385	38	≠	≠	PROPN
iajs-702	385	39	0	0	NUM
iajs-702	385	40	.	.	PUNCT
iajs-702	386	1	since	since	SCONJ
iajs-702	386	2	g(x	g(x	NOUN
iajs-702	386	3	)	)	PUNCT
iajs-702	386	4			NOUN
iajs-702	386	5	annr[x](a	annr[x](a	PROPN
iajs-702	386	6	)	)	PUNCT
iajs-702	386	7	,	,	PUNCT
iajs-702	386	8	then	then	ADV
iajs-702	386	9	(	(	PUNCT
iajs-702	386	10	b0	b0	NOUN
iajs-702	386	11	+	+	NOUN
iajs-702	386	12	b1x	b1x	PROPN
iajs-702	386	13	+	+	CCONJ
iajs-702	386	14	…	…	PUNCT
iajs-702	386	15	+	+	CCONJ
iajs-702	386	16	bk	bk	NOUN
iajs-702	386	17	x	x	PUNCT
iajs-702	386	18	k	k	PROPN
iajs-702	386	19	+	+	PROPN
iajs-702	386	20	…	…	PUNCT
iajs-702	386	21	+	+	CCONJ
iajs-702	386	22	bmxm)a0	bmxm)a0	NOUN
iajs-702	386	23	=	=	SYM
iajs-702	386	24	0	0	X
iajs-702	386	25	.	.	PUNCT
iajs-702	386	26	مجلة	مجلة	VERB
iajs-702	387	1	إبن	إبن	VERB
iajs-702	387	2	الھیثم	الھیثم	NOUN
iajs-702	387	3	للعلوم	للعلوم	NOUN
iajs-702	387	4	الصرفة	الصرفة	NOUN
iajs-702	387	5	و	و	PRON
iajs-702	387	6	التطبیقیة	التطبیقیة	PROPN
iajs-702	387	7	2012	2012	NUM
iajs-702	387	8	السنة	السنة	NOUN
iajs-702	388	1	25	25	NUM
iajs-702	388	2	المجلد	المجلد	NOUN
iajs-702	388	3	1	1	NUM
iajs-702	388	4	العدد	العدد	PROPN
iajs-702	388	5	ibn	ibn	PROPN
iajs-702	388	6	al	al	PROPN
iajs-702	388	7	-	-	PUNCT
iajs-702	388	8	haitham	haitham	PROPN
iajs-702	388	9	journal	journal	PROPN
iajs-702	388	10	for	for	ADP
iajs-702	388	11	pure	pure	ADJ
iajs-702	388	12	and	and	CCONJ
iajs-702	388	13	applied	apply	VERB
iajs-702	388	14	science	science	NOUN
iajs-702	388	15	no	no	NOUN
iajs-702	388	16	.	.	NOUN
iajs-702	388	17	1	1	NUM
iajs-702	388	18	vol	vol	NOUN
iajs-702	388	19	.	.	PUNCT
iajs-702	389	1	25	25	NUM
iajs-702	389	2	year	year	NOUN
iajs-702	389	3	2012	2012	NUM
iajs-702	389	4	thus	thus	ADV
iajs-702	389	5	b0	b0	VERB
iajs-702	389	6	a0	a0	PROPN
iajs-702	389	7	+	+	CCONJ
iajs-702	389	8	b1	b1	PROPN
iajs-702	389	9	a0	a0	NOUN
iajs-702	389	10	x	x	PUNCT
iajs-702	390	1	+	+	PUNCT
iajs-702	390	2	…	…	PUNCT
iajs-702	390	3	+	+	NUM
iajs-702	390	4	bk	bk	VERB
iajs-702	390	5	a0	a0	NOUN
iajs-702	390	6	x	x	PUNCT
iajs-702	391	1	k	k	PROPN
iajs-702	391	2	+	+	PUNCT
iajs-702	391	3	…	…	PUNCT
iajs-702	391	4	+	+	CCONJ
iajs-702	391	5	bm	bm	PROPN
iajs-702	391	6	a0	a0	PROPN
iajs-702	391	7	x	x	PROPN
iajs-702	391	8	m	m	VERB
iajs-702	391	9	=	=	SYM
iajs-702	391	10	0	0	NUM
iajs-702	391	11	which	which	PRON
iajs-702	391	12	implies	imply	VERB
iajs-702	391	13	b0	b0	VERB
iajs-702	391	14	a0	a0	NOUN
iajs-702	391	15	=	=	SYM
iajs-702	391	16	0	0	PROPN
iajs-702	391	17	,	,	PUNCT
iajs-702	391	18	b1	b1	NOUN
iajs-702	391	19	a0	a0	NOUN
iajs-702	391	20	=	=	SYM
iajs-702	391	21	0	0	NUM
iajs-702	391	22	,	,	PUNCT
iajs-702	391	23	…	…	PUNCT
iajs-702	391	24	,	,	PUNCT
iajs-702	391	25	bk	bk	VERB
iajs-702	391	26	a0	a0	NOUN
iajs-702	391	27	=	=	SYM
iajs-702	391	28	0	0	NUM
iajs-702	391	29	,	,	PUNCT
iajs-702	391	30	…	…	PUNCT
iajs-702	391	31	,	,	PUNCT
iajs-702	391	32	bm	bm	PROPN
iajs-702	391	33	a0	a0	PROPN
iajs-702	391	34	=	=	PUNCT
iajs-702	391	35	0	0	PROPN
iajs-702	391	36	.	.	PUNCT
iajs-702	392	1	but	but	CCONJ
iajs-702	392	2	bk	bk	VERB
iajs-702	392	3	a0	a0	NOUN
iajs-702	392	4	=	=	PUNCT
iajs-702	392	5	0	0	NUM
iajs-702	392	6	and	and	CCONJ
iajs-702	392	7	0	0	NUM
iajs-702	392	8	≠	≠	PROPN
iajs-702	392	9	bk	bk	VERB
iajs-702	392	10			NOUN
iajs-702	392	11	j	j	PROPN
iajs-702	392	12	implies	imply	VERB
iajs-702	392	13	that	that	SCONJ
iajs-702	392	14	j	j	PROPN
iajs-702	392	15			PUNCT
iajs-702	392	16	annra0	annra0	NOUN
iajs-702	392	17	≠	≠	PROPN
iajs-702	392	18	0	0	NUM
iajs-702	392	19	,	,	PUNCT
iajs-702	392	20	which	which	PRON
iajs-702	392	21	is	be	AUX
iajs-702	392	22	a	a	DET
iajs-702	392	23	contradiction	contradiction	NOUN
iajs-702	392	24	with	with	ADP
iajs-702	392	25	our	our	PRON
iajs-702	392	26	assumption	assumption	NOUN
iajs-702	392	27	.	.	PUNCT
iajs-702	393	1	hence	hence	ADV
iajs-702	393	2	annra0	annra0	VERB
iajs-702	393	3			NOUN
iajs-702	393	4	er	er	INTJ
iajs-702	393	5	,	,	PUNCT
iajs-702	393	6	so	so	SCONJ
iajs-702	393	7	that	that	SCONJ
iajs-702	393	8	a0	a0	PROPN
iajs-702	393	9			PROPN
iajs-702	393	10	z(r	z(r	PROPN
iajs-702	393	11	)	)	PUNCT
iajs-702	393	12	.	.	PUNCT
iajs-702	394	1	but	but	CCONJ
iajs-702	394	2	r	r	NOUN
iajs-702	394	3	is	be	AUX
iajs-702	394	4	nonsingular	nonsingular	ADJ
iajs-702	394	5	,	,	PUNCT
iajs-702	394	6	hence	hence	ADV
iajs-702	394	7	a0	a0	NOUN
iajs-702	394	8	=	=	SYM
iajs-702	394	9	0	0	PROPN
iajs-702	394	10	.	.	PUNCT
iajs-702	395	1	by	by	ADP
iajs-702	395	2	the	the	DET
iajs-702	395	3	same	same	ADJ
iajs-702	395	4	way	way	NOUN
iajs-702	395	5	we	we	PRON
iajs-702	395	6	can	can	AUX
iajs-702	395	7	prove	prove	VERB
iajs-702	395	8	that	that	SCONJ
iajs-702	395	9	each	each	PRON
iajs-702	395	10	of	of	ADP
iajs-702	395	11	a1	a1	NOUN
iajs-702	395	12	,	,	PUNCT
iajs-702	395	13	…	…	PUNCT
iajs-702	395	14	,	,	PUNCT
iajs-702	395	15	am	be	AUX
iajs-702	395	16	belongs	belong	VERB
iajs-702	395	17	to	to	ADP
iajs-702	395	18	z(r)=(0	z(r)=(0	NUM
iajs-702	395	19	)	)	PUNCT
iajs-702	395	20	.	.	PUNCT
iajs-702	396	1	thus	thus	ADV
iajs-702	396	2	f(x	f(x	PROPN
iajs-702	396	3	)	)	PUNCT
iajs-702	396	4	=	=	SYM
iajs-702	396	5	0	0	NUM
iajs-702	396	6	and	and	CCONJ
iajs-702	396	7	z(r[x	z(r[x	NOUN
iajs-702	396	8	]	]	PUNCT
iajs-702	396	9	)	)	PUNCT
iajs-702	396	10	=	=	SYM
iajs-702	396	11	(	(	PUNCT
iajs-702	396	12	0	0	NUM
iajs-702	396	13	)	)	PUNCT
iajs-702	396	14	;	;	PUNCT
iajs-702	396	15	that	that	DET
iajs-702	396	16	r[x	r[x	NOUN
iajs-702	396	17	]	]	PUNCT
iajs-702	396	18	is	be	AUX
iajs-702	396	19	nonsingular	nonsingular	ADJ
iajs-702	396	20	.	.	PUNCT
iajs-702	397	1	then	then	ADV
iajs-702	397	2	by	by	ADP
iajs-702	397	3	induction	induction	NOUN
iajs-702	397	4	r[x1	r[x1	NOUN
iajs-702	397	5	,	,	PUNCT
iajs-702	397	6	x2	x2	PROPN
iajs-702	397	7	,	,	PUNCT
iajs-702	397	8	…	…	PUNCT
iajs-702	397	9	,	,	PUNCT
iajs-702	397	10	xn	xn	PROPN
iajs-702	397	11	]	]	X
iajs-702	397	12	is	be	AUX
iajs-702	397	13	a	a	DET
iajs-702	397	14	nonsingular	nonsingular	ADJ
iajs-702	397	15	ring	ring	NOUN
iajs-702	397	16	.	.	PUNCT
iajs-702	398	1	1.39	1.39	NUM
iajs-702	398	2	corollary	corollary	NOUN
iajs-702	398	3	:	:	PUNCT
iajs-702	398	4	let	let	VERB
iajs-702	398	5	r	r	PRON
iajs-702	398	6	be	be	AUX
iajs-702	398	7	a	a	DET
iajs-702	398	8	nonsingular	nonsingular	ADJ
iajs-702	398	9	ring	ring	NOUN
iajs-702	398	10	.	.	PUNCT
iajs-702	399	1	then	then	ADV
iajs-702	399	2	r[x1	r[x1	NOUN
iajs-702	399	3	,	,	PUNCT
iajs-702	399	4	x2	x2	PROPN
iajs-702	399	5	,	,	PUNCT
iajs-702	399	6	…	…	PUNCT
iajs-702	399	7	,	,	PUNCT
iajs-702	399	8	xn	xn	PROPN
iajs-702	399	9	]	]	X
iajs-702	399	10	is	be	AUX
iajs-702	399	11	min	min	NOUN
iajs-702	399	12	-	-	ADJ
iajs-702	399	13	cs	cs	ADJ
iajs-702	399	14	if	if	SCONJ
iajs-702	400	1	and	and	CCONJ
iajs-702	400	2	only	only	ADV
iajs-702	400	3	if	if	SCONJ
iajs-702	400	4	r[x1	r[x1	X
iajs-702	400	5	,	,	PUNCT
iajs-702	400	6	x2	x2	PROPN
iajs-702	400	7	,	,	PUNCT
iajs-702	400	8	…	…	PUNCT
iajs-702	400	9	,	,	PUNCT
iajs-702	400	10	xn	xn	PROPN
iajs-702	400	11	]	]	X
iajs-702	400	12	is	be	AUX
iajs-702	400	13	a	a	DET
iajs-702	400	14	max	max	PROPN
iajs-702	400	15	-	-	PUNCT
iajs-702	400	16	cs	cs	PROPN
iajs-702	400	17	ring	ring	NOUN
iajs-702	400	18	.	.	PUNCT
iajs-702	401	1	proof	proof	NOUN
iajs-702	401	2	:	:	PUNCT
iajs-702	401	3	by	by	ADP
iajs-702	401	4	lemma	lemma	PROPN
iajs-702	401	5	38	38	NUM
iajs-702	401	6	if	if	SCONJ
iajs-702	401	7	r	r	NOUN
iajs-702	401	8	is	be	AUX
iajs-702	401	9	a	a	DET
iajs-702	401	10	nonsingular	nonsingular	ADJ
iajs-702	401	11	ring	ring	NOUN
iajs-702	401	12	,	,	PUNCT
iajs-702	401	13	then	then	ADV
iajs-702	401	14	r[x1	r[x1	NOUN
iajs-702	401	15	,	,	PUNCT
iajs-702	401	16	x2	x2	PROPN
iajs-702	401	17	,	,	PUNCT
iajs-702	401	18	…	…	PUNCT
iajs-702	401	19	,	,	PUNCT
iajs-702	401	20	xn	xn	PROPN
iajs-702	401	21	]	]	X
iajs-702	401	22	is	be	AUX
iajs-702	401	23	a	a	DET
iajs-702	401	24	nonsingular	nonsingular	ADJ
iajs-702	401	25	ring	ring	NOUN
iajs-702	401	26	.	.	PUNCT
iajs-702	402	1	hence	hence	ADV
iajs-702	402	2	the	the	DET
iajs-702	402	3	result	result	NOUN
iajs-702	402	4	follows	follow	VERB
iajs-702	402	5	by	by	ADP
iajs-702	402	6	theorem	theorem	ADJ
iajs-702	402	7	33	33	NUM
iajs-702	402	8	.	.	NOUN
iajs-702	402	9	1.40	1.40	NUM
iajs-702	402	10	note	note	NOUN
iajs-702	402	11	:	:	PUNCT
iajs-702	402	12	a	a	DET
iajs-702	402	13	direct	direct	ADJ
iajs-702	402	14	sum	sum	NOUN
iajs-702	402	15	of	of	ADP
iajs-702	402	16	min	min	NOUN
iajs-702	402	17	(	(	PUNCT
iajs-702	402	18	max)-cs	max)-cs	NOUN
iajs-702	402	19	modules	module	NOUN
iajs-702	402	20	will	will	AUX
iajs-702	402	21	be	be	AUX
iajs-702	402	22	discussed	discuss	VERB
iajs-702	402	23	in	in	ADP
iajs-702	402	24	another	another	DET
iajs-702	402	25	paper	paper	NOUN
iajs-702	402	26	.	.	PUNCT
iajs-702	403	1	references	reference	NOUN
iajs-702	403	2	1	1	NUM
iajs-702	403	3	.	.	PUNCT
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iajs-702	403	5	,	,	PUNCT
iajs-702	403	6	n.v	n.v	PROPN
iajs-702	403	7	.	.	PROPN
iajs-702	403	8	;	;	PUNCT
iajs-702	404	1	huynh	huynh	PROPN
iajs-702	404	2	,	,	PUNCT
iajs-702	404	3	d.v	d.v	PROPN
iajs-702	404	4	.	.	PROPN
iajs-702	404	5	;	;	PUNCT
iajs-702	404	6	smith	smith	PROPN
iajs-702	404	7	,	,	PUNCT
iajs-702	404	8	p.f	p.f	PROPN
iajs-702	404	9	.	.	PROPN
iajs-702	404	10	and	and	CCONJ
iajs-702	404	11	wisbauer	wisbauer	NOUN
iajs-702	404	12	,	,	PUNCT
iajs-702	404	13	r.	r.	PROPN
iajs-702	404	14	,	,	PUNCT
iajs-702	404	15	(	(	PUNCT
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iajs-702	404	17	)	)	PUNCT
iajs-702	404	18	,	,	PUNCT
iajs-702	404	19	extending	extend	VERB
iajs-702	404	20	modules	module	NOUN
iajs-702	404	21	,	,	PUNCT
iajs-702	404	22	john	john	PROPN
iajs-702	404	23	wiley	wiley	PROPN
iajs-702	404	24	and	and	CCONJ
iajs-702	404	25	sons	son	NOUN
iajs-702	404	26	,	,	PUNCT
iajs-702	404	27	inc	inc	PROPN
iajs-702	404	28	.	.	PROPN
iajs-702	404	29	new	new	PROPN
iajs-702	404	30	york	york	PROPN
iajs-702	404	31	.	.	PUNCT
iajs-702	405	1	2	2	X
iajs-702	405	2	.	.	X
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iajs-702	405	5	-	-	PUNCT
iajs-702	405	6	hazmi	hazmi	NOUN
iajs-702	405	7	,	,	PUNCT
iajs-702	405	8	(	(	PUNCT
iajs-702	405	9	2005	2005	NUM
iajs-702	405	10	)	)	PUNCT
iajs-702	405	11	,	,	PUNCT
iajs-702	405	12	a	a	DET
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iajs-702	405	14	of	of	ADP
iajs-702	405	15	cs	cs	PROPN
iajs-702	405	16	and	and	CCONJ
iajs-702	405	17	-cs	-cs	NUM
iajs-702	405	18	rings	ring	NOUN
iajs-702	405	19	and	and	CCONJ
iajs-702	405	20	modules	module	NOUN
iajs-702	405	21	,	,	PUNCT
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iajs-702	405	23	,	,	PUNCT
iajs-702	405	24	college	college	NOUN
iajs-702	405	25	of	of	ADP
iajs-702	405	26	arts	art	NOUN
iajs-702	405	27	and	and	CCONJ
iajs-702	405	28	sciences	science	NOUN
iajs-702	405	29	of	of	ADP
iajs-702	405	30	ohio	ohio	PROPN
iajs-702	405	31	university	university	PROPN
iajs-702	405	32	.	.	PUNCT
iajs-702	406	1	3	3	X
iajs-702	406	2	.	.	X
iajs-702	406	3	saad.h.m	saad.h.m	PROPN
iajs-702	406	4	ohamed	ohame	VERB
iajs-702	406	5	,	,	PUNCT
iajs-702	406	6	bruno	bruno	PROPN
iajs-702	406	7	j.	j.	PROPN
iajs-702	406	8	muller	muller	PROPN
iajs-702	406	9	,	,	PUNCT
iajs-702	406	10	(	(	PUNCT
iajs-702	406	11	1990	1990	NUM
iajs-702	406	12	)	)	PUNCT
iajs-702	406	13	,	,	PUNCT
iajs-702	406	14	continuous	continuous	ADJ
iajs-702	406	15	and	and	CCONJ
iajs-702	406	16	discrete	discrete	ADJ
iajs-702	406	17	modules	module	NOUN
iajs-702	406	18	,	,	PUNCT
iajs-702	406	19	combridge	combridge	PROPN
iajs-702	406	20	univ	univ	PROPN
iajs-702	406	21	.	.	PUNCT
iajs-702	407	1	press	press	PROPN
iajs-702	407	2	,	,	PUNCT
iajs-702	407	3	new	new	PROPN
iajs-702	407	4	york	york	PROPN
iajs-702	407	5	,	,	PUNCT
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iajs-702	407	7	chester	chester	PROPN
iajs-702	407	8	melbourne	melbourne	PROPN
iajs-702	407	9	sydney	sydney	PROPN
iajs-702	407	10	.	.	PUNCT
iajs-702	408	1	4	4	X
iajs-702	408	2	.	.	X
iajs-702	408	3	goodearl	goodearl	PROPN
iajs-702	408	4	,	,	PUNCT
iajs-702	408	5	k.r	k.r	PROPN
iajs-702	408	6	.	.	PROPN
iajs-702	408	7	(	(	PUNCT
iajs-702	408	8	1976	1976	NUM
iajs-702	408	9	)	)	PUNCT
iajs-702	408	10	,	,	PUNCT
iajs-702	408	11	ring	ring	NOUN
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iajs-702	408	13	,	,	PUNCT
iajs-702	408	14	nonsingular	nonsingular	ADJ
iajs-702	408	15	rings	ring	NOUN
iajs-702	408	16	and	and	CCONJ
iajs-702	408	17	modules	module	NOUN
iajs-702	408	18	,	,	PUNCT
iajs-702	408	19	marcel	marcel	PROPN
iajs-702	408	20	dekker	dekker	PROPN
iajs-702	408	21	,	,	PUNCT
iajs-702	408	22	inc	inc	PROPN
iajs-702	408	23	.	.	PROPN
iajs-702	408	24	newyork	newyork	PROPN
iajs-702	408	25	and	and	CCONJ
iajs-702	408	26	basel	basel	PROPN
iajs-702	408	27	.	.	PUNCT
iajs-702	409	1	5	5	X
iajs-702	409	2	.	.	X
iajs-702	409	3	mahmoud	mahmoud	PROPN
iajs-702	409	4	a.	a.	PROPN
iajs-702	409	5	kamal	kamal	PROPN
iajs-702	409	6	and	and	CCONJ
iajs-702	409	7	amany	amany	PROPN
iajs-702	409	8	m.	m.	NOUN
iajs-702	409	9	menshawy	menshawy	PROPN
iajs-702	409	10	,	,	PUNCT
iajs-702	409	11	(	(	PUNCT
iajs-702	409	12	2007	2007	NUM
iajs-702	409	13	)	)	PUNCT
iajs-702	409	14	,	,	PUNCT
iajs-702	409	15	j.	j.	PROPN
iajs-702	409	16	egypt	egypt	PROPN
iajs-702	409	17	.	.	PUNCT
iajs-702	410	1	math.soc	math.soc	X
iajs-702	410	2	.	.	NOUN
iajs-702	410	3	,	,	PUNCT
iajs-702	410	4	15(2	15(2	NUM
iajs-702	410	5	)	)	PUNCT
iajs-702	410	6	,	,	PUNCT
iajs-702	410	7	.	.	PUNCT
iajs-702	411	1	157	157	NUM
iajs-702	411	2	-	-	SYM
iajs-702	411	3	168	168	NUM
iajs-702	411	4	.	.	PUNCT
iajs-702	412	1	6	6	NUM
iajs-702	412	2	.	.	X
iajs-702	412	3	kasch	kasch	PROPN
iajs-702	412	4	,	,	PUNCT
iajs-702	412	5	f.	f.	PROPN
iajs-702	412	6	,	,	PUNCT
iajs-702	412	7	(	(	PUNCT
iajs-702	412	8	1982	1982	NUM
iajs-702	412	9	)	)	PUNCT
iajs-702	412	10	,	,	PUNCT
iajs-702	412	11	m	m	VERB
iajs-702	412	12	odules	odule	NOUN
iajs-702	412	13	and	and	CCONJ
iajs-702	412	14	rings	ring	NOUN
iajs-702	412	15	,	,	PUNCT
iajs-702	412	16	academic	academic	ADJ
iajs-702	412	17	press	press	PROPN
iajs-702	412	18	,	,	PUNCT
iajs-702	412	19	inc	inc	PROPN
iajs-702	412	20	.	.	PROPN
iajs-702	412	21	london	london	PROPN
iajs-702	412	22	.	.	PUNCT
iajs-702	413	1	7	7	NUM
iajs-702	413	2	.	.	X
iajs-702	413	3	z.a.el	z.a.el	NOUN
iajs-702	413	4	-	-	PUNCT
iajs-702	413	5	bast	bast	NOUN
iajs-702	413	6	and	and	CCONJ
iajs-702	413	7	p.f.smith	p.f.smith	NOUN
iajs-702	413	8	,	,	PUNCT
iajs-702	413	9	(	(	PUNCT
iajs-702	413	10	1988	1988	NUM
iajs-702	413	11	)	)	PUNCT
iajs-702	413	12	,	,	PUNCT
iajs-702	413	13	multiplication	multiplication	NOUN
iajs-702	413	14	modules	module	NOUN
iajs-702	413	15	,	,	PUNCT
iajs-702	413	16	communication	communication	NOUN
iajs-702	413	17	in	in	ADP
iajs-702	413	18	algebra	algebra	NOUN
iajs-702	413	19	,	,	PUNCT
iajs-702	413	20	10(4),.755	10(4),.755	PROPN
iajs-702	413	21	-	-	SYM
iajs-702	413	22	779	779	NUM
iajs-702	413	23	.	.	NOUN
iajs-702	413	24	8	8	NUM
iajs-702	413	25	.	.	PUNCT
iajs-702	414	1	c.p.lu	c.p.lu	PROPN
iajs-702	414	2	,	,	PUNCT
iajs-702	414	3	(	(	PUNCT
iajs-702	414	4	1984	1984	NUM
iajs-702	414	5	)	)	PUNCT
iajs-702	414	6	,	,	PUNCT
iajs-702	414	7	prime	prime	ADJ
iajs-702	414	8	submodules	submodule	NOUN
iajs-702	414	9	of	of	ADP
iajs-702	414	10	modules	module	NOUN
iajs-702	414	11	,	,	PUNCT
iajs-702	414	12	comment.m	comment.m	PROPN
iajs-702	414	13	ath	ath	NOUN
iajs-702	414	14	.	.	PUNCT
iajs-702	415	1	univ.st.paul	univ.st.paul	PROPN
iajs-702	415	2	,	,	PUNCT
iajs-702	415	3	33	33	NUM
iajs-702	415	4	,	,	PUNCT
iajs-702	415	5	.	.	PUNCT
iajs-702	416	1	61	61	NUM
iajs-702	416	2	-	-	SYM
iajs-702	416	3	69	69	NUM
iajs-702	416	4	.	.	PUNCT
iajs-702	417	1	9	9	NUM
iajs-702	417	2	.	.	X
iajs-702	417	3	zeinab	zeinab	PROPN
iajs-702	417	4	talib	talib	PROPN
iajs-702	417	5	salman	salman	PROPN
iajs-702	417	6	al	al	PROPN
iajs-702	417	7	-	-	PUNCT
iajs-702	417	8	zubaidey	zubaidey	PROPN
iajs-702	417	9	,	,	PUNCT
iajs-702	417	10	(	(	PUNCT
iajs-702	417	11	2005	2005	NUM
iajs-702	417	12	)	)	PUNCT
iajs-702	417	13	,	,	PUNCT
iajs-702	417	14	on	on	ADP
iajs-702	417	15	purely	purely	ADV
iajs-702	417	16	extending	extend	VERB
iajs-702	417	17	modules	module	NOUN
iajs-702	417	18	,	,	PUNCT
iajs-702	417	19	m.sc	m.sc	PROPN
iajs-702	417	20	.	.	PUNCT
iajs-702	418	1	thesis	thesis	NOUN
iajs-702	418	2	,	,	PUNCT
iajs-702	418	3	college	college	NOUN
iajs-702	418	4	of	of	ADP
iajs-702	418	5	science	science	NOUN
iajs-702	418	6	,	,	PUNCT
iajs-702	418	7	university	university	NOUN
iajs-702	418	8	of	of	ADP
iajs-702	418	9	baghdad	baghdad	PROPN
iajs-702	418	10	.	.	PUNCT
iajs-702	419	1	10	10	NUM
iajs-702	419	2	.	.	PUNCT
iajs-702	420	1	ahmed	ahmed	PROPN
iajs-702	420	2	,	,	PUNCT
iajs-702	420	3	a.a	a.a	PROPN
iajs-702	420	4	.	.	PROPN
iajs-702	420	5	(	(	PUNCT
iajs-702	420	6	1992	1992	NUM
iajs-702	420	7	)	)	PUNCT
iajs-702	420	8	,	,	PUNCT
iajs-702	420	9	on	on	ADP
iajs-702	420	10	submodules	submodule	NOUN
iajs-702	420	11	of	of	ADP
iajs-702	420	12	multiplication	multiplication	NOUN
iajs-702	420	13	modules	module	NOUN
iajs-702	420	14	,	,	PUNCT
iajs-702	420	15	m.sc	m.sc	PROPN
iajs-702	420	16	.	.	PUNCT
iajs-702	421	1	thesis	thesis	NOUN
iajs-702	421	2	,	,	PUNCT
iajs-702	421	3	university	university	NOUN
iajs-702	421	4	of	of	ADP
iajs-702	421	5	baghdad	baghdad	PROPN
iajs-702	421	6	.	.	PUNCT
iajs-702	422	1	مقاسات	مقاسات	PROPN
iajs-702	422	2	التوسع	التوسع	PROPN
iajs-702	422	3	)	)	PUNCT
iajs-702	422	4	أعظم(أصغر	أعظم(أصغر	NOUN
iajs-702	422	5	مجلة	مجلة	VERB
iajs-702	422	6	إبن	إبن	NOUN
iajs-702	422	7	الھیثم	الھیثم	NOUN
iajs-702	422	8	للعلوم	للعلوم	NOUN
iajs-702	422	9	الصرفة	الصرفة	NOUN
iajs-702	422	10	و	و	PRON
iajs-702	422	11	التطبیقیة	التطبیقیة	PROPN
iajs-702	422	12	2012	2012	NUM
iajs-702	422	13	السنة	السنة	NOUN
iajs-702	422	14	25	25	NUM
iajs-702	422	15	المجلد	المجلد	NOUN
iajs-702	422	16	1	1	NUM
iajs-702	422	17	العدد	العدد	PROPN
iajs-702	422	18	ibn	ibn	PROPN
iajs-702	422	19	al	al	PROPN
iajs-702	422	20	-	-	PUNCT
iajs-702	422	21	haitham	haitham	PROPN
iajs-702	422	22	journal	journal	PROPN
iajs-702	422	23	for	for	ADP
iajs-702	422	24	pure	pure	ADJ
iajs-702	422	25	and	and	CCONJ
iajs-702	422	26	applied	apply	VERB
iajs-702	422	27	science	science	NOUN
iajs-702	422	28	no	no	NOUN
iajs-702	422	29	.	.	NOUN
iajs-702	422	30	1	1	NUM
iajs-702	422	31	vol	vol	NOUN
iajs-702	422	32	.	.	PUNCT
iajs-702	423	1	25	25	NUM
iajs-702	423	2	year	year	NOUN
iajs-702	423	3	2012	2012	NUM
iajs-702	423	4	،	،	NOUN
iajs-702	423	5	رنا	رنا	PROPN
iajs-702	423	6	نوري	نوري	VERB
iajs-702	423	7	مجید	مجید	INTJ
iajs-702	423	8	ھاديانعام	ھاديانعام	PROPN
iajs-702	423	9	محمد	محمد	PROPN
iajs-702	423	10	علي	علي	NOUN
iajs-702	423	11	جامعة	جامعة	NOUN
iajs-702	423	12	بغدادابن	بغدادابن	VERB
iajs-702	423	13	الھیثم	الھیثم	VERB
iajs-702	423	14	-كلیة	-كلیة	PROPN
iajs-702	423	15	التربیة	التربیة	NOUN
iajs-702	423	16	-قسم	-قسم	PUNCT
iajs-702	423	17	الریاضیات	الریاضیات	ADJ
iajs-702	423	18	2011	2011	NUM
iajs-702	423	19	ایلول20	ایلول20	NOUN
iajs-702	423	20	:	:	PUNCT
iajs-702	423	21	قبل	قبل	PROPN
iajs-702	423	22	البحث	البحث	VERB
iajs-702	423	23	في	في	PRON
iajs-702	423	24	،	،	PROPN
iajs-702	423	25	2011	2011	NUM
iajs-702	423	26	أب	أب	ADP
iajs-702	423	27	25	25	NUM
iajs-702	423	28	:	:	PUNCT
iajs-702	423	29	استلم	استلم	PROPN
iajs-702	423	30	البحث	البحث	VERB
iajs-702	423	31	في	في	ADP
iajs-702	423	32	الخالصة	الخالصة	PROPN
iajs-702	423	33	ن	ن	PROPN
iajs-702	423	34	أصـغر	أصـغر	VERB
iajs-702	423	35	مقـاس	مقـاس	VERB
iajs-702	423	36	)	)	PUNCT
iajs-702	423	37	أعظـم	أعظـم	PROPN
iajs-702	423	38	(	(	PUNCT
iajs-702	423	39	في	في	ADP
iajs-702	423	40	ھذا	ھذا	NOUN
iajs-702	423	41	البحث	البحث	VERB
iajs-702	423	42	نعطي	نعطي	PROPN
iajs-702	423	43	دراسة	دراسة	PROPN
iajs-702	423	44	واسعة	واسعة	PROPN
iajs-702	423	45	ألصغر	ألصغر	PROPN
iajs-702	423	46	ي	ي	X
iajs-702	423	47	المغلـق	المغلـق	NOUN
iajs-702	423	48	ـم	ـم	NOUN
iajs-702	423	49	اس	اس	ADP
iajs-702	423	50	الجزـئ	الجزـئ	PROPN
iajs-702	423	51	ل	ل	PROPN
iajs-702	423	52	المـق	المـق	ADV
iajs-702	423	53	مقاسـات	مقاسـات	PROPN
iajs-702	423	54	التوسـع	التوسـع	PROPN
iajs-702	423	55	مـث	مـث	PROPN
iajs-702	423	56	مقـاس	مقـاس	PROPN
iajs-702	423	57	توسـع	توسـع	NOUN
iajs-702	423	58	)	)	PUNCT
iajs-702	423	59	أعظم(ومن	أعظم(ومن	PROPN
iajs-702	423	60	بین	بین	PROPN
iajs-702	423	61	النتائج	النتائج	PROPN
iajs-702	423	62	االخرى	االخرى	PROPN
iajs-702	423	63	نستعرض	نستعرض	PROPN
iajs-702	423	64	أنھ	أنھ	PROPN
iajs-702	423	65	مركبة	مركبة	PROPN
iajs-702	423	66	المجموع	المجموع	PROPN
iajs-702	423	67	المباشر	المباشر	PROPN
iajs-702	423	68	ألصغر	ألصغر	PROPN
iajs-702	423	69	.	.	PUNCT
iajs-702	424	1	توسع	توسع	PROPN
iajs-702	424	2	ھو	ھو	PROPN
iajs-702	424	3	أصغر	أصغر	PROPN
iajs-702	424	4	مقاس	مقاس	PROPN
iajs-702	424	5	توسع	توسع	PROPN
iajs-702	424	6	ت	ت	DET
iajs-702	424	7	الحلقـة	الحلقـة	NOUN
iajs-702	424	8	)	)	PUNCT
iajs-702	425	1	أعظم(ھو	أعظم(ھو	PROPN
iajs-702	425	2	أصغر	أصغر	NOUN
iajs-702	425	3	ط	ط	PROPN
iajs-702	425	4	إذا	إذا	NOUN
iajs-702	425	5	كانـت	كانـت	NOUN
iajs-702	425	6	rیـر	rیـر	PROPN
iajs-702	425	7	مفـرده	مفـرده	PROPN
iajs-702	425	8	فـان	فـان	PROPN
iajs-702	425	9	الحلقـة	الحلقـة	PROPN
iajs-702	425	10	غrمقاس	غrمقاس	PROPN
iajs-702	425	11	توسـع	توسـع	PROPN
iajs-702	425	12	وكـذلك	وكـذلك	PROPN
iajs-702	425	13	اذا	اذا	PROPN
iajs-702	425	14	كاـن	كاـن	VERB
iajs-702	425	15	ـع	ـع	PART
iajs-702	425	16	إذا	إذا	VERB
iajs-702	425	17	وفـق	وفـق	NOUN
iajs-702	425	18	أعظـم	أعظـم	NOUN
iajs-702	425	19	حلقـة	حلقـة	PROPN
iajs-702	425	20	توس	توس	PROPN
iajs-702	425	21	.	.	PUNCT
iajs-702	426	1	أصغر	أصغر	PROPN
iajs-702	426	2	حلقة	حلقة	PROPN
iajs-702	426	3	توسع	توسع	PROPN
iajs-702	426	4	والتي	والتي	PROPN
iajs-702	426	5	ھي	ھي	PROPN
iajs-702	426	6	واحدة	واحدة	PROPN
iajs-702	427	1	من	من	PROPN
iajs-702	427	2	المبرھنات	المبرھنات	ADV
iajs-702	427	3	المھمة	المھمة	PROPN
iajs-702	427	4	في	في	ADP
iajs-702	427	5	ھذا	ھذا	PROPN
iajs-702	427	6	البحثrالحلقة	البحثrالحلقة	PROPN
iajs-702	427	7	.	.	PUNCT
iajs-702	428	1	مقاس	مقاس	VERB
iajs-702	428	2	توسع	توسع	PROPN
iajs-702	428	3	،	،	PROPN
iajs-702	428	4	أصغر	أصغر	PROPN
iajs-702	428	5	مقاس	مقاس	PROPN
iajs-702	428	6	توسع	توسع	PROPN
iajs-702	428	7	،	،	PROPN
iajs-702	428	8	أعظم	أعظم	PROPN
iajs-702	428	9	مقاس	مقاس	PROPN
iajs-702	428	10	توسع	توسع	PROPN
iajs-702	428	11	،	،	PROPN
iajs-702	428	12	مقاس	مقاس	PROPN
iajs-702	428	13	توسع	توسع	NOUN
iajs-702	428	14	منتظم	منتظم	PROPN
iajs-702	428	15	:	:	PUNCT
iajs-702	428	16	الكلمات	الكلمات	VERB
iajs-702	428	17	المفتاحیة	المفتاحیة	ADV