id	sid	tid	token	lemma	pos
iajs-901	1	1	ibn	ibn	PROPN
iajs-901	1	2	alhaitham	alhaitham	NOUN
iajs-901	1	3	j.	j.	PROPN
iajs-901	2	1	fo	fo	ADP
iajs-901	2	2	r	r	NOUN
iajs-901	2	3	pure	pure	ADJ
iajs-901	2	4	&	&	CCONJ
iajs-901	2	5	appl	appl	PROPN
iajs-901	2	6	.	.	PUNCT
iajs-901	3	1	sc	sc	PROPN
iajs-901	3	2	i.	i.	PROPN
iajs-901	3	3	vo	vo	PROPN
iajs-901	3	4	l.23	l.23	PROPN
iajs-901	3	5	(	(	PUNCT
iajs-901	3	6	3	3	NUM
iajs-901	3	7	)	)	PUNCT
iajs-901	3	8	2010	2010	NUM
iajs-901	3	9	dual	dual	ADJ
iajs-901	3	10	notions	notion	NOUN
iajs-901	3	11	of	of	ADP
iajs-901	3	12	prime	prime	ADJ
iajs-901	3	13	modules	module	NOUN
iajs-901	3	14	i.	i.	PROPN
iajs-901	3	15	m.	m.	PROPN
iajs-901	3	16	ali	ali	PROPN
iajs-901	3	17	,	,	PUNCT
iajs-901	3	18	r.	r.	PROPN
iajs-901	3	19	i.	i.	PROPN
iajs-901	3	20	khalaf	khalaf	PROPN
iajs-901	3	21	department	department	PROPN
iajs-901	3	22	of	of	ADP
iajs-901	3	23	mathematics	mathematics	PROPN
iajs-901	3	24	,	,	PUNCT
iajs-901	3	25	college	college	NOUN
iajs-901	3	26	of	of	ADP
iajs-901	3	27	education	education	PROPN
iajs-901	3	28	ibn	ibn	PROPN
iajs-901	3	29	-	-	PUNCT
iajs-901	3	30	al	al	PROPN
iajs-901	3	31	-	-	PUNCT
iajs-901	3	32	haitham	haitham	PROPN
iajs-901	3	33	,	,	PUNCT
iajs-901	3	34	university	university	PROPN
iajs-901	3	35	of	of	ADP
iajs-901	3	36	baghdad	baghdad	PROPN
iajs-901	3	37	abstract	abstract	ADV
iajs-901	3	38	let	let	VERB
iajs-901	3	39	r	r	PRON
iajs-901	3	40	be	be	AUX
iajs-901	3	41	a	a	DET
iajs-901	3	42	commutative	commutative	ADJ
iajs-901	3	43	ring	ring	NOUN
iajs-901	3	44	with	with	ADP
iajs-901	3	45	unity	unity	NOUN
iajs-901	3	46	.m	.m	NOUN
iajs-901	4	1	an	an	DET
iajs-901	4	2	r	r	NOUN
iajs-901	4	3	-	-	PUNCT
iajs-901	4	4	module	module	NOUN
iajs-901	4	5	.	.	PUNCT
iajs-901	5	1	m	m	PROPN
iajs-901	5	2	is	be	AUX
iajs-901	5	3	called	call	VERB
iajs-901	5	4	coprime	coprime	NOUN
iajs-901	5	5	module	module	NOUN
iajs-901	5	6	(	(	PUNCT
iajs-901	5	7	dual	dual	ADJ
iajs-901	5	8	notion	notion	NOUN
iajs-901	5	9	of	of	ADP
iajs-901	5	10	prime	prime	ADJ
iajs-901	5	11	module	module	NOUN
iajs-901	5	12	)	)	PUNCT
iajs-901	5	13	if	if	SCONJ
iajs-901	5	14	ann	ann	PROPN
iajs-901	5	15	m	m	PROPN
iajs-901	5	16	=	=	PROPN
iajs-901	5	17	ann	ann	PROPN
iajs-901	5	18	m	m	PROPN
iajs-901	5	19	/	/	SYM
iajs-901	5	20	n	n	PROPN
iajs-901	5	21	for	for	ADP
iajs-901	5	22	every	every	DET
iajs-901	5	23	proper	proper	ADJ
iajs-901	5	24	submodule	submodule	NOUN
iajs-901	5	25	n	n	PROPN
iajs-901	5	26	of	of	ADP
iajs-901	5	27	m	m	PRON
iajs-901	5	28	in	in	ADP
iajs-901	5	29	this	this	DET
iajs-901	5	30	paper	paper	NOUN
iajs-901	5	31	we	we	PRON
iajs-901	5	32	study	study	VERB
iajs-901	5	33	coprime	coprime	NOUN
iajs-901	5	34	modules	module	NOUN
iajs-901	5	35	we	we	PRON
iajs-901	5	36	give	give	VERB
iajs-901	5	37	many	many	ADJ
iajs-901	5	38	basic	basic	ADJ
iajs-901	5	39	properties	property	NOUN
iajs-901	5	40	of	of	ADP
iajs-901	5	41	this	this	DET
iajs-901	5	42	concept	concept	NOUN
iajs-901	5	43	.	.	PUNCT
iajs-901	6	1	also	also	ADV
iajs-901	6	2	we	we	PRON
iajs-901	6	3	give	give	VERB
iajs-901	6	4	many	many	ADJ
iajs-901	6	5	characterization	characterization	NOUN
iajs-901	6	6	of	of	ADP
iajs-901	6	7	it	it	PRON
iajs-901	6	8	under	under	ADP
iajs-901	6	9	certain	certain	ADJ
iajs-901	6	10	of	of	ADP
iajs-901	6	11	module	module	NOUN
iajs-901	6	12	.	.	PUNCT
iajs-901	7	1	introduction	introduction	NOUN
iajs-901	7	2	let	let	VERB
iajs-901	7	3	r	r	PRON
iajs-901	7	4	be	be	AUX
iajs-901	7	5	a	a	DET
iajs-901	7	6	commutative	commutative	ADJ
iajs-901	7	7	ring	ring	NOUN
iajs-901	7	8	with	with	ADP
iajs-901	7	9	unity	unity	NOUN
iajs-901	7	10	.	.	PUNCT
iajs-901	8	1	m	m	VERB
iajs-901	8	2	an	an	DET
iajs-901	8	3	r	r	NOUN
iajs-901	8	4	-	-	PUNCT
iajs-901	8	5	module	module	NOUN
iajs-901	8	6	.	.	PUNCT
iajs-901	9	1	n	n	PRON
iajs-901	9	2	is	be	AUX
iajs-901	9	3	called	call	VERB
iajs-901	9	4	prime	prime	ADJ
iajs-901	9	5	submodule	submodule	NOUN
iajs-901	9	6	of	of	ADP
iajs-901	9	7	an	an	DET
iajs-901	9	8	r	r	NOUN
iajs-901	9	9	-	-	PUNCT
iajs-901	9	10	modules	module	NOUN
iajs-901	9	11	m	m	NOUN
iajs-901	9	12	if	if	SCONJ
iajs-901	9	13	n	n	PRON
iajs-901	9	14	≠	≠	PROPN
iajs-901	9	15	m	m	VERB
iajs-901	9	16	and	and	CCONJ
iajs-901	9	17	whenever	whenever	SCONJ
iajs-901	9	18	rx	rx	VERB
iajs-901	9	19			NOUN
iajs-901	9	20	n	n	CCONJ
iajs-901	9	21	such	such	ADJ
iajs-901	9	22	that	that	DET
iajs-901	9	23	r	r	NOUN
iajs-901	9	24			NOUN
iajs-901	9	25	r	r	NOUN
iajs-901	9	26	,	,	PUNCT
iajs-901	9	27	x	x	SYM
iajs-901	9	28			NOUN
iajs-901	9	29	m	m	PROPN
iajs-901	9	30	,	,	PUNCT
iajs-901	9	31	then	then	ADV
iajs-901	9	32	either	either	CCONJ
iajs-901	9	33	x	x	SYM
iajs-901	9	34	n	n	NOUN
iajs-901	9	35	or	or	CCONJ
iajs-901	9	36	r	r	NOUN
iajs-901	9	37			NOUN
iajs-901	9	38	[	[	X
iajs-901	9	39	n	n	CCONJ
iajs-901	9	40	:	:	PUNCT
iajs-901	9	41	m	m	VERB
iajs-901	9	42	]	]	PUNCT
iajs-901	9	43	,	,	PUNCT
iajs-901	9	44	see	see	VERB
iajs-901	9	45	[	[	X
iajs-901	9	46	1	1	NUM
iajs-901	9	47	]	]	PUNCT
iajs-901	9	48	,	,	PUNCT
iajs-901	9	49	[	[	X
iajs-901	9	50	2	2	NUM
iajs-901	9	51	]	]	PUNCT
iajs-901	9	52	.m	.m	NOUN
iajs-901	9	53	is	be	AUX
iajs-901	9	54	called	call	VERB
iajs-901	9	55	a	a	DET
iajs-901	9	56	prime	prime	ADJ
iajs-901	9	57	module	module	NOUN
iajs-901	9	58	if	if	SCONJ
iajs-901	9	59	ann	ann	PROPN
iajs-901	9	60	m	m	PROPN
iajs-901	9	61	=	=	PUNCT
iajs-901	9	62	ann	ann	PROPN
iajs-901	9	63	n	n	PROPN
iajs-901	9	64	for	for	ADP
iajs-901	9	65	every	every	DET
iajs-901	9	66	non	non	ADJ
iajs-901	9	67	-	-	ADJ
iajs-901	9	68	zero	zero	NUM
iajs-901	9	69	r	r	NOUN
iajs-901	9	70	r	r	NOUN
iajs-901	9	71	submodule	submodule	NOUN
iajs-901	9	72	n	n	PROPN
iajs-901	9	73	of	of	ADP
iajs-901	9	74	m	m	PRON
iajs-901	9	75	,	,	PUNCT
iajs-901	9	76	see	see	VERB
iajs-901	9	77	[	[	X
iajs-901	9	78	3],[4	3],[4	NUM
iajs-901	9	79	]	]	PUNCT
iajs-901	9	80	.it	.it	PUNCT
iajs-901	9	81	is	be	AUX
iajs-901	9	82	clear	clear	ADJ
iajs-901	9	83	that	that	SCONJ
iajs-901	9	84	m	m	NOUN
iajs-901	9	85	is	be	AUX
iajs-901	9	86	prime	prime	ADJ
iajs-901	9	87	iff	iff	PROPN
iajs-901	9	88	<	<	X
iajs-901	9	89	0	0	NUM
iajs-901	9	90	>	>	X
iajs-901	9	91	is	be	AUX
iajs-901	9	92	prime	prime	ADJ
iajs-901	9	93	submodule	submodule	NOUN
iajs-901	9	94	of	of	ADP
iajs-901	9	95	m.	m.	NOUN
iajs-901	10	1	yassemi.s	yassemi.s	PROPN
iajs-901	10	2	.	.	PUNCT
iajs-901	11	1	in	in	ADP
iajs-901	11	2	[	[	X
iajs-901	11	3	5	5	NUM
iajs-901	11	4	]	]	PUNCT
iajs-901	11	5	,	,	PUNCT
iajs-901	11	6	introduced	introduce	VERB
iajs-901	11	7	the	the	DET
iajs-901	11	8	notion	notion	NOUN
iajs-901	11	9	of	of	ADP
iajs-901	11	10	second	second	ADJ
iajs-901	11	11	submodule	submodule	NOUN
iajs-901	11	12	(	(	PUNCT
iajs-901	11	13	as	as	ADP
iajs-901	11	14	dual	dual	ADJ
iajs-901	11	15	notion	notion	NOUN
iajs-901	11	16	of	of	ADP
iajs-901	11	17	prime	prime	ADJ
iajs-901	11	18	submodule	submodule	NOUN
iajs-901	11	19	)	)	PUNCT
iajs-901	11	20	as	as	SCONJ
iajs-901	11	21	follows	follow	VERB
iajs-901	11	22	:	:	PUNCT
iajs-901	11	23	n	n	PRON
iajs-901	11	24	is	be	AUX
iajs-901	11	25	second	second	ADJ
iajs-901	11	26	submodule	submodule	NOUN
iajs-901	11	27	of	of	ADP
iajs-901	11	28	an	an	DET
iajs-901	11	29	r	r	NOUN
iajs-901	11	30	-	-	PUNCT
iajs-901	11	31	module	module	NOUN
iajs-901	11	32	m	m	NOUN
iajs-901	11	33	if	if	SCONJ
iajs-901	11	34	for	for	ADP
iajs-901	11	35	every	every	DET
iajs-901	11	36	r	r	NOUN
iajs-901	11	37			NOUN
iajs-901	11	38	r	r	NOUN
iajs-901	11	39	,	,	PUNCT
iajs-901	11	40	the	the	DET
iajs-901	11	41	homothety	homothety	NOUN
iajs-901	12	1	r	r	PROPN
iajs-901	12	2	*	*	NOUN
iajs-901	12	3	on	on	ADP
iajs-901	12	4	n	n	X
iajs-901	12	5	is	be	AUX
iajs-901	12	6	either	either	DET
iajs-901	12	7	zero	zero	NUM
iajs-901	12	8	or	or	CCONJ
iajs-901	12	9	surjective	surjective	ADJ
iajs-901	12	10	,	,	PUNCT
iajs-901	12	11	where	where	SCONJ
iajs-901	12	12	if	if	SCONJ
iajs-901	12	13	m	m	NOUN
iajs-901	12	14	is	be	AUX
iajs-901	12	15	an	an	DET
iajs-901	12	16	r	r	NOUN
iajs-901	12	17	-	-	PUNCT
iajs-901	12	18	module	module	NOUN
iajs-901	12	19	and	and	CCONJ
iajs-901	12	20	r	r	NOUN
iajs-901	12	21	r	r	NOUN
iajs-901	12	22	,	,	PUNCT
iajs-901	12	23	then	then	ADV
iajs-901	12	24	an	an	DET
iajs-901	12	25	r	r	NOUN
iajs-901	12	26	-	-	PUNCT
iajs-901	12	27	endomorphism	endomorphism	NOUN
iajs-901	12	28	r	r	PROPN
iajs-901	12	29	*	*	PROPN
iajs-901	12	30	is	be	AUX
iajs-901	12	31	called	call	VERB
iajs-901	12	32	homothety	homothety	NOUN
iajs-901	12	33	if	if	SCONJ
iajs-901	12	34	r*(x	r*(x	NOUN
iajs-901	12	35	)	)	PUNCT
iajs-901	12	36	=	=	PUNCT
iajs-901	12	37	rx	rx	VERB
iajs-901	12	38	for	for	ADP
iajs-901	12	39	all	all	DET
iajs-901	12	40	x	x	ADJ
iajs-901	12	41			NOUN
iajs-901	12	42	m.	m.	NOUN
iajs-901	12	43	m	m	VERB
iajs-901	12	44	is	be	AUX
iajs-901	12	45	second	second	ADJ
iajs-901	12	46	module	module	NOUN
iajs-901	12	47	if	if	SCONJ
iajs-901	12	48	it	it	PRON
iajs-901	12	49	is	be	AUX
iajs-901	12	50	second	second	ADJ
iajs-901	12	51	submodule	submodule	NOUN
iajs-901	12	52	of	of	ADP
iajs-901	12	53	m.	m.	NOUN
iajs-901	12	54	annin	annin	PROPN
iajs-901	12	55	.	.	PUNCT
iajs-901	13	1	s.	s.	PROPN
iajs-901	13	2	in	in	ADP
iajs-901	13	3	[	[	X
iajs-901	13	4	6	6	NUM
iajs-901	13	5	]	]	PUNCT
iajs-901	13	6	introduced	introduce	VERB
iajs-901	13	7	the	the	DET
iajs-901	13	8	notion	notion	NOUN
iajs-901	13	9	of	of	ADP
iajs-901	13	10	coprime	coprime	NOUN
iajs-901	13	11	modules	module	NOUN
iajs-901	13	12	(	(	PUNCT
iajs-901	13	13	as	as	ADP
iajs-901	13	14	dual	dual	ADJ
iajs-901	13	15	notion	notion	NOUN
iajs-901	13	16	prime	prime	ADJ
iajs-901	13	17	modules	module	NOUN
iajs-901	13	18	)	)	PUNCT
iajs-901	13	19	as	as	SCONJ
iajs-901	13	20	follows	follow	VERB
iajs-901	13	21	:	:	PUNCT
iajs-901	13	22	an	an	DET
iajs-901	13	23	r	r	NOUN
iajs-901	13	24	-	-	PUNCT
iajs-901	13	25	modules	module	NOUN
iajs-901	13	26	m	m	VERB
iajs-901	13	27	is	be	AUX
iajs-901	13	28	called	call	VERB
iajs-901	13	29	coprime	coprime	ADV
iajs-901	13	30	if	if	SCONJ
iajs-901	13	31	ann	ann	PROPN
iajs-901	13	32	m	m	PROPN
iajs-901	13	33	=	=	PROPN
iajs-901	13	34	ann	ann	PROPN
iajs-901	13	35	m	m	PROPN
iajs-901	13	36	/	/	SYM
iajs-901	13	37	n	n	PROPN
iajs-901	13	38	for	for	ADP
iajs-901	13	39	every	every	DET
iajs-901	13	40	r	r	NOUN
iajs-901	13	41	r	r	NOUN
iajs-901	13	42	proper	proper	ADJ
iajs-901	13	43	submodule	submodule	NOUN
iajs-901	13	44	n	n	PROPN
iajs-901	13	45	of	of	ADP
iajs-901	13	46	m	m	PROPN
iajs-901	13	47	.not	.not	NOUN
iajs-901	14	1	that	that	PRON
iajs-901	15	1	[	[	X
iajs-901	15	2	n	n	X
iajs-901	15	3	:	:	PUNCT
iajs-901	15	4	m	m	VERB
iajs-901	15	5	]	]	X
iajs-901	15	6	=	=	PUNCT
iajs-901	15	7	ann	ann	PROPN
iajs-901	15	8	m	m	PROPN
iajs-901	15	9	/	/	SYM
iajs-901	15	10	n	n	PROPN
iajs-901	15	11	.specially	.specially	ADV
iajs-901	15	12	a	a	DET
iajs-901	15	13	ring	ring	NOUN
iajs-901	15	14	r	r	NOUN
iajs-901	15	15	is	be	AUX
iajs-901	15	16	coprime	coprime	ADJ
iajs-901	15	17	iff	iff	PROPN
iajs-901	15	18	r	r	NOUN
iajs-901	15	19	is	be	AUX
iajs-901	15	20	r	r	NOUN
iajs-901	15	21	r	r	NOUN
iajs-901	15	22	coprime	coprime	NOUN
iajs-901	15	23	r	r	NOUN
iajs-901	15	24	-	-	PUNCT
iajs-901	15	25	module	module	NOUN
iajs-901	15	26	.	.	PUNCT
iajs-901	16	1	abuhilail	abuhilail	VERB
iajs-901	17	1	j.in	j.in	PROPN
iajs-901	18	1	[	[	X
iajs-901	18	2	7	7	X
iajs-901	18	3	]	]	PUNCT
iajs-901	18	4	in	in	SCONJ
iajs-901	18	5	introduced	introduce	VERB
iajs-901	18	6	a	a	DET
iajs-901	18	7	notion	notion	NOUN
iajs-901	18	8	coprime	coprime	NOUN
iajs-901	18	9	submodules	submodule	NOUN
iajs-901	18	10	(	(	PUNCT
iajs-901	18	11	as	as	ADP
iajs-901	18	12	dual	dual	ADJ
iajs-901	18	13	notion	notion	NOUN
iajs-901	18	14	prime	prime	ADJ
iajs-901	18	15	submodules	submodule	NOUN
iajs-901	18	16	)	)	PUNCT
iajs-901	18	17	as	as	SCONJ
iajs-901	18	18	follows	follow	VERB
iajs-901	18	19	:	:	PUNCT
iajs-901	18	20	n	n	PRON
iajs-901	18	21	is	be	AUX
iajs-901	18	22	called	call	VERB
iajs-901	18	23	coprime	coprime	ADJ
iajs-901	18	24	submodule	submodule	NOUN
iajs-901	18	25	of	of	ADP
iajs-901	18	26	an	an	DET
iajs-901	18	27	r	r	NOUN
iajs-901	18	28	-	-	PUNCT
iajs-901	18	29	module	module	NOUN
iajs-901	18	30	m	m	NOUN
iajs-901	18	31	if	if	SCONJ
iajs-901	18	32	ann	ann	PROPN
iajs-901	18	33	m	m	PROPN
iajs-901	18	34	=	=	PROPN
iajs-901	18	35	w	w	NOUN
iajs-901	18	36	r	r	NOUN
iajs-901	18	37	(	(	PUNCT
iajs-901	18	38	m	m	NOUN
iajs-901	18	39	/	/	SYM
iajs-901	18	40	n	n	CCONJ
iajs-901	18	41	)	)	PUNCT
iajs-901	18	42	=	=	PRON
iajs-901	18	43	{	{	PUNCT
iajs-901	18	44	a	a	DET
iajs-901	18	45			NOUN
iajs-901	18	46	r	r	NOUN
iajs-901	18	47	;	;	PUNCT
iajs-901	18	48	the	the	DET
iajs-901	18	49	homothety	homothety	NOUN
iajs-901	18	50	r	r	PROPN
iajs-901	18	51	*	*	X
iajs-901	18	52	on	on	ADP
iajs-901	18	53	m	m	NOUN
iajs-901	18	54	/	/	SYM
iajs-901	18	55	n	n	PROPN
iajs-901	18	56	is	be	AUX
iajs-901	18	57	not	not	PART
iajs-901	18	58	surjective	surjective	ADJ
iajs-901	18	59	}	}	PUNCT
iajs-901	18	60	,	,	PUNCT
iajs-901	18	61	see	see	VERB
iajs-901	18	62	[	[	X
iajs-901	18	63	5].we	5].we	NUM
iajs-901	18	64	notice	notice	VERB
iajs-901	18	65	that	that	SCONJ
iajs-901	18	66	m	m	NOUN
iajs-901	18	67	is	be	AUX
iajs-901	18	68	coprime	coprime	NOUN
iajs-901	18	69	module	module	NOUN
iajs-901	18	70	iff	iff	PROPN
iajs-901	18	71	m	m	VERB
iajs-901	18	72	is	be	AUX
iajs-901	18	73	second	second	ADJ
iajs-901	18	74	module	module	NOUN
iajs-901	18	75	iff	iff	PROPN
iajs-901	18	76	(	(	PUNCT
iajs-901	18	77	0	0	NUM
iajs-901	18	78	)	)	PUNCT
iajs-901	18	79	is	be	AUX
iajs-901	18	80	coprime	coprime	ADJ
iajs-901	18	81	submodule	submodule	NOUN
iajs-901	18	82	.	.	PUNCT
iajs-901	19	1	the	the	DET
iajs-901	19	2	main	main	ADJ
iajs-901	19	3	purpose	purpose	NOUN
iajs-901	19	4	of	of	ADP
iajs-901	19	5	this	this	DET
iajs-901	19	6	paper	paper	NOUN
iajs-901	19	7	is	be	AUX
iajs-901	19	8	to	to	PART
iajs-901	19	9	give	give	VERB
iajs-901	19	10	basic	basic	ADJ
iajs-901	19	11	properties	property	NOUN
iajs-901	19	12	of	of	ADP
iajs-901	19	13	coprime	coprime	NOUN
iajs-901	19	14	(	(	PUNCT
iajs-901	19	15	second	second	ADJ
iajs-901	19	16	)	)	PUNCT
iajs-901	19	17	modules	module	NOUN
iajs-901	19	18	and	and	CCONJ
iajs-901	19	19	study	study	VERB
iajs-901	19	20	the	the	DET
iajs-901	19	21	relationships	relationship	NOUN
iajs-901	19	22	between	between	ADP
iajs-901	19	23	coprime	coprime	NOUN
iajs-901	19	24	modules	module	NOUN
iajs-901	19	25	and	and	CCONJ
iajs-901	19	26	other	other	ADJ
iajs-901	19	27	modules	module	NOUN
iajs-901	19	28	.	.	PUNCT
iajs-901	20	1	we	we	PRON
iajs-901	20	2	show	show	VERB
iajs-901	20	3	that	that	SCONJ
iajs-901	20	4	a	a	DET
iajs-901	20	5	submodule	submodule	NOUN
iajs-901	20	6	of	of	ADP
iajs-901	20	7	coprime	coprime	NOUN
iajs-901	20	8	module	module	NOUN
iajs-901	20	9	need	need	AUX
iajs-901	20	10	not	not	PART
iajs-901	20	11	be	be	AUX
iajs-901	20	12	coprime	coprime	NOUN
iajs-901	20	13	module	module	NOUN
iajs-901	20	14	and	and	CCONJ
iajs-901	20	15	we	we	PRON
iajs-901	20	16	give	give	VERB
iajs-901	20	17	certain	certain	ADJ
iajs-901	20	18	conditions	condition	NOUN
iajs-901	20	19	to	to	PART
iajs-901	20	20	make	make	VERB
iajs-901	20	21	a	a	DET
iajs-901	20	22	submodule	submodule	NOUN
iajs-901	20	23	of	of	ADP
iajs-901	20	24	coprime	coprime	NOUN
iajs-901	20	25	module	module	NOUN
iajs-901	20	26	is	be	AUX
iajs-901	20	27	coprime	coprime	ADJ
iajs-901	20	28	.	.	PUNCT
iajs-901	21	1	moreover	moreover	ADV
iajs-901	21	2	we	we	PRON
iajs-901	21	3	obtained	obtain	VERB
iajs-901	21	4	the	the	DET
iajs-901	21	5	relationships	relationship	NOUN
iajs-901	21	6	between	between	ADP
iajs-901	21	7	coprime	coprime	NOUN
iajs-901	21	8	modules	module	NOUN
iajs-901	21	9	and	and	CCONJ
iajs-901	21	10	divisible	divisible	ADJ
iajs-901	21	11	modules	module	NOUN
iajs-901	21	12	,	,	PUNCT
iajs-901	21	13	principally	principally	ADV
iajs-901	21	14	injective	injective	ADJ
iajs-901	21	15	and	and	CCONJ
iajs-901	21	16	injective	injective	ADJ
iajs-901	21	17	modules	module	NOUN
iajs-901	21	18	.	.	PUNCT
iajs-901	22	1	finally	finally	ADV
iajs-901	22	2	we	we	PRON
iajs-901	22	3	investigate	investigate	VERB
iajs-901	22	4	the	the	DET
iajs-901	22	5	behavior	behavior	NOUN
iajs-901	22	6	of	of	ADP
iajs-901	22	7	coprime	coprime	NOUN
iajs-901	22	8	modules	module	NOUN
iajs-901	22	9	under	under	ADP
iajs-901	22	10	localization	localization	NOUN
iajs-901	22	11	.	.	PUNCT
iajs-901	23	1	definition	definition	NOUN
iajs-901	23	2	1:[6	1:[6	NUM
iajs-901	23	3	]	]	X
iajs-901	23	4	an	an	DET
iajs-901	23	5	r	r	NOUN
iajs-901	23	6	-	-	PUNCT
iajs-901	23	7	module	module	NOUN
iajs-901	23	8	m	m	NOUN
iajs-901	23	9	is	be	AUX
iajs-901	23	10	called	call	VERB
iajs-901	23	11	coprime	coprime	ADV
iajs-901	23	12	if	if	SCONJ
iajs-901	23	13	for	for	ADP
iajs-901	23	14	every	every	DET
iajs-901	23	15	proper	proper	ADJ
iajs-901	23	16	submodule	submodule	NOUN
iajs-901	23	17	n	n	PROPN
iajs-901	23	18	of	of	ADP
iajs-901	23	19	m	m	PRON
iajs-901	23	20	,	,	PUNCT
iajs-901	23	21	r	r	NOUN
iajs-901	23	22	ann	ann	PROPN
iajs-901	23	23	n	n	PROPN
iajs-901	23	24	=	=	SYM
iajs-901	23	25	r	r	NOUN
iajs-901	23	26	ann	ann	PROPN
iajs-901	23	27	(	(	PUNCT
iajs-901	23	28	m	m	PROPN
iajs-901	23	29	/	/	SYM
iajs-901	23	30	n	n	CCONJ
iajs-901	23	31	)	)	PUNCT
iajs-901	23	32	.	.	PUNCT
iajs-901	24	1	ihjpas	ihjpa	VERB
iajs-901	24	2	ibn	ibn	PROPN
iajs-901	24	3	alhaitham	alhaitham	PROPN
iajs-901	25	1	j.	j.	PROPN
iajs-901	26	1	fo	fo	ADP
iajs-901	26	2	r	r	NOUN
iajs-901	26	3	pure	pure	ADJ
iajs-901	26	4	&	&	CCONJ
iajs-901	26	5	appl	appl	PROPN
iajs-901	26	6	.	.	PUNCT
iajs-901	27	1	sc	sc	PROPN
iajs-901	27	2	i.	i.	PROPN
iajs-901	27	3	vo	vo	PROPN
iajs-901	27	4	l.23	l.23	PROPN
iajs-901	27	5	(	(	PUNCT
iajs-901	27	6	3	3	NUM
iajs-901	27	7	)	)	PUNCT
iajs-901	27	8	2010	2010	NUM
iajs-901	27	9	specially	specially	ADV
iajs-901	27	10	,	,	PUNCT
iajs-901	27	11	a	a	DET
iajs-901	27	12	ring	ring	NOUN
iajs-901	27	13	r	r	NOUN
iajs-901	27	14	is	be	AUX
iajs-901	27	15	coprime	coprime	ADJ
iajs-901	27	16	iff	iff	PROPN
iajs-901	27	17	r	r	NOUN
iajs-901	27	18	is	be	AUX
iajs-901	27	19	a	a	DET
iajs-901	27	20	coprime	coprime	ADJ
iajs-901	27	21	r	r	NOUN
iajs-901	27	22	-	-	PUNCT
iajs-901	27	23	module	module	NOUN
iajs-901	27	24	.	.	PUNCT
iajs-901	28	1	recall	recall	VERB
iajs-901	28	2	that	that	SCONJ
iajs-901	28	3	a	a	DET
iajs-901	28	4	proper	proper	ADJ
iajs-901	28	5	submodule	submodule	NOUN
iajs-901	28	6	n	n	PROPN
iajs-901	28	7	of	of	ADP
iajs-901	28	8	an	an	DET
iajs-901	28	9	r	r	NOUN
iajs-901	28	10	-	-	PUNCT
iajs-901	28	11	module	module	NOUN
iajs-901	28	12	m	m	NOUN
iajs-901	28	13	is	be	AUX
iajs-901	28	14	called	call	VERB
iajs-901	28	15	invariant	invariant	ADJ
iajs-901	28	16	if	if	SCONJ
iajs-901	28	17	for	for	ADP
iajs-901	28	18	each	each	DET
iajs-901	28	19	f	f	PROPN
iajs-901	28	20			PROPN
iajs-901	28	21	r	r	NOUN
iajs-901	28	22	nd	nd	NOUN
iajs-901	28	23	(	(	PUNCT
iajs-901	28	24	m	m	NOUN
iajs-901	28	25	)	)	PUNCT
iajs-901	28	26	,	,	PUNCT
iajs-901	28	27	f	f	PROPN
iajs-901	28	28	(	(	PUNCT
iajs-901	28	29	n	n	CCONJ
iajs-901	28	30	)	)	PUNCT
iajs-901	28	31			PROPN
iajs-901	29	1	n.	n.	PROPN
iajs-901	29	2	m	m	PROPN
iajs-901	29	3	is	be	AUX
iajs-901	29	4	called	call	VERB
iajs-901	29	5	fully	fully	ADV
iajs-901	29	6	invariant	invariant	ADJ
iajs-901	29	7	if	if	SCONJ
iajs-901	29	8	every	every	DET
iajs-901	29	9	submodule	submodule	NOUN
iajs-901	29	10	of	of	ADP
iajs-901	29	11	m	m	PROPN
iajs-901	29	12	is	be	AUX
iajs-901	29	13	invariant	invariant	ADJ
iajs-901	29	14	,	,	PUNCT
iajs-901	29	15	see	see	VERB
iajs-901	29	16	[	[	X
iajs-901	29	17	8	8	NUM
iajs-901	29	18	]	]	PUNCT
iajs-901	29	19	.	.	PUNCT
iajs-901	30	1	.wijayanti	.wijayanti	INTJ
iajs-901	31	1	i.e	i.e	X
iajs-901	31	2	in	in	ADP
iajs-901	31	3	[	[	PUNCT
iajs-901	31	4	9	9	NUM
iajs-901	31	5	]	]	PUNCT
iajs-901	31	6	,	,	PUNCT
iajs-901	31	7	gave	give	VERB
iajs-901	31	8	the	the	DET
iajs-901	31	9	following	follow	VERB
iajs-901	31	10	characterization	characterization	NOUN
iajs-901	31	11	for	for	ADP
iajs-901	31	12	coprime	coprime	NOUN
iajs-901	31	13	modules	module	NOUN
iajs-901	31	14	.	.	PUNCT
iajs-901	32	1	theorem	theorem	VERB
iajs-901	32	2	2	2	NUM
iajs-901	32	3	:	:	PUNCT
iajs-901	33	1	[	[	X
iajs-901	33	2	9	9	NUM
iajs-901	33	3	]	]	PUNCT
iajs-901	33	4	let	let	VERB
iajs-901	33	5	m	m	PRON
iajs-901	33	6	be	be	AUX
iajs-901	33	7	an	an	DET
iajs-901	33	8	r	r	NOUN
iajs-901	33	9	-	-	PUNCT
iajs-901	33	10	module	module	NOUN
iajs-901	33	11	,	,	PUNCT
iajs-901	33	12	then	then	ADV
iajs-901	33	13	the	the	DET
iajs-901	33	14	following	following	ADJ
iajs-901	33	15	statements	statement	NOUN
iajs-901	33	16	are	be	AUX
iajs-901	33	17	equivalent	equivalent	ADJ
iajs-901	33	18	:	:	PUNCT
iajs-901	33	19	1	1	X
iajs-901	33	20	.	.	X
iajs-901	33	21	m	m	PROPN
iajs-901	33	22	is	be	AUX
iajs-901	33	23	a	a	DET
iajs-901	33	24	coprime	coprime	ADJ
iajs-901	33	25	r	r	NOUN
iajs-901	33	26	-	-	NOUN
iajs-901	33	27	module	module	NOUN
iajs-901	33	28	.	.	PUNCT
iajs-901	34	1	2	2	X
iajs-901	34	2	.	.	X
iajs-901	34	3	r	r	NOUN
iajs-901	34	4	ann	ann	PROPN
iajs-901	34	5	m	m	NOUN
iajs-901	34	6	=	=	SYM
iajs-901	34	7	r	r	NOUN
iajs-901	34	8	ann	ann	PROPN
iajs-901	34	9	(	(	PUNCT
iajs-901	34	10	m	m	PROPN
iajs-901	34	11	/	/	SYM
iajs-901	34	12	n	n	CCONJ
iajs-901	34	13	)	)	PUNCT
iajs-901	34	14	for	for	ADP
iajs-901	34	15	every	every	DET
iajs-901	34	16	proper	proper	ADJ
iajs-901	34	17	invariant	invariant	ADJ
iajs-901	34	18	submodule	submodule	NOUN
iajs-901	34	19	n	n	PROPN
iajs-901	34	20	of	of	ADP
iajs-901	34	21	m.	m.	NOUN
iajs-901	34	22	proof	proof	NOUN
iajs-901	34	23	:	:	PUNCT
iajs-901	34	24	(	(	PUNCT
iajs-901	34	25	1	1	X
iajs-901	34	26	)	)	PUNCT
iajs-901	34	27			NOUN
iajs-901	34	28	(	(	PUNCT
iajs-901	34	29	2	2	NUM
iajs-901	34	30	)	)	PUNCT
iajs-901	34	31	is	be	AUX
iajs-901	34	32	obvious	obvious	ADJ
iajs-901	34	33	.	.	PUNCT
iajs-901	35	1	(	(	PUNCT
iajs-901	35	2	2	2	X
iajs-901	35	3	)	)	PUNCT
iajs-901	35	4			NOUN
iajs-901	35	5	(	(	PUNCT
iajs-901	35	6	1	1	NUM
iajs-901	35	7	)	)	PUNCT
iajs-901	35	8	.	.	PUNCT
iajs-901	36	1	to	to	PART
iajs-901	36	2	prove	prove	VERB
iajs-901	36	3	m	m	NOUN
iajs-901	36	4	is	be	AUX
iajs-901	36	5	a	a	DET
iajs-901	36	6	coprime	coprime	ADJ
iajs-901	36	7	r	r	NOUN
iajs-901	36	8	-	-	PUNCT
iajs-901	36	9	module	module	NOUN
iajs-901	36	10	.	.	PUNCT
iajs-901	37	1	assume	assume	VERB
iajs-901	37	2	that	that	SCONJ
iajs-901	37	3	r	r	NOUN
iajs-901	37	4	ann	ann	PROPN
iajs-901	37	5	(	(	PUNCT
iajs-901	37	6	m	m	PROPN
iajs-901	37	7	/	/	SYM
iajs-901	37	8	n	n	CCONJ
iajs-901	37	9	)	)	PUNCT
iajs-901	38	1			PROPN
iajs-901	38	2	r	r	PROPN
iajs-901	38	3	ann	ann	PROPN
iajs-901	38	4	m	m	PROPN
iajs-901	38	5	for	for	ADP
iajs-901	38	6	some	some	DET
iajs-901	38	7	proper	proper	ADJ
iajs-901	38	8	submodule	submodule	NOUN
iajs-901	38	9	n	n	PROPN
iajs-901	38	10	of	of	ADP
iajs-901	38	11	m	m	PROPN
iajs-901	38	12	.	.	PUNCT
iajs-901	39	1	let	let	VERB
iajs-901	40	1	i	i	PRON
iajs-901	40	2	=	=	SYM
iajs-901	40	3	r	r	NOUN
iajs-901	40	4	ann	ann	PROPN
iajs-901	40	5	(	(	PUNCT
iajs-901	40	6	m	m	PROPN
iajs-901	40	7	/	/	SYM
iajs-901	40	8	n	n	CCONJ
iajs-901	40	9	)	)	PUNCT
iajs-901	40	10	.	.	PUNCT
iajs-901	41	1	then	then	ADV
iajs-901	41	2	i	i	PRON
iajs-901	41	3			PROPN
iajs-901	41	4	r	r	PROPN
iajs-901	41	5	ann	ann	PROPN
iajs-901	41	6	m	m	PROPN
iajs-901	42	1	and	and	CCONJ
iajs-901	42	2	i	i	PRON
iajs-901	42	3	m	m	VERB
iajs-901	43	1			PROPN
iajs-901	43	2	n.	n.	NOUN
iajs-901	44	1	but	but	CCONJ
iajs-901	44	2	i	i	PRON
iajs-901	44	3	m	m	VERB
iajs-901	44	4	is	be	AUX
iajs-901	44	5	invariant	invariant	ADJ
iajs-901	44	6	,	,	PUNCT
iajs-901	44	7	since	since	SCONJ
iajs-901	44	8	for	for	ADP
iajs-901	44	9	each	each	DET
iajs-901	44	10	f	f	PROPN
iajs-901	44	11			PROPN
iajs-901	44	12	r	r	NOUN
iajs-901	44	13	nd	nd	NOUN
iajs-901	44	14	(	(	PUNCT
iajs-901	44	15	m	m	NOUN
iajs-901	44	16	)	)	PUNCT
iajs-901	44	17	,	,	PUNCT
iajs-901	44	18	f	f	PROPN
iajs-901	44	19	(	(	PUNCT
iajs-901	44	20	i	i	NOUN
iajs-901	44	21	m	m	VERB
iajs-901	44	22	)	)	PUNCT
iajs-901	45	1	=	=	PUNCT
iajs-901	46	1	i	i	PRON
iajs-901	46	2	f	f	X
iajs-901	46	3	(	(	PUNCT
iajs-901	46	4	m	m	PROPN
iajs-901	46	5	)	)	PUNCT
iajs-901	46	6			PROPN
iajs-901	47	1	i	i	PRON
iajs-901	47	2	m.	m.	VERB
iajs-901	47	3	hence	hence	ADV
iajs-901	48	1	i	i	PRON
iajs-901	48	2			PROPN
iajs-901	48	3	r	r	PROPN
iajs-901	48	4	ann	ann	PROPN
iajs-901	48	5	(	(	PUNCT
iajs-901	48	6	m	m	PROPN
iajs-901	48	7	/	/	SYM
iajs-901	48	8	i	i	NOUN
iajs-901	48	9	m	m	VERB
iajs-901	48	10	)	)	PUNCT
iajs-901	49	1	=	=	SYM
iajs-901	49	2	r	r	NOUN
iajs-901	49	3	ann	ann	PROPN
iajs-901	49	4	m	m	PROPN
iajs-901	49	5	,	,	PUNCT
iajs-901	49	6	which	which	PRON
iajs-901	49	7	is	be	AUX
iajs-901	49	8	a	a	DET
iajs-901	49	9	contradiction	contradiction	NOUN
iajs-901	49	10	.	.	PUNCT
iajs-901	50	1	■	■	PUNCT
iajs-901	50	2	note	note	VERB
iajs-901	50	3	that	that	SCONJ
iajs-901	50	4	,	,	PUNCT
iajs-901	50	5	statement	statement	NOUN
iajs-901	50	6	(	(	PUNCT
iajs-901	50	7	2	2	NUM
iajs-901	50	8	)	)	PUNCT
iajs-901	50	9	in	in	ADP
iajs-901	50	10	theorem	theorem	NOUN
iajs-901	50	11	(	(	PUNCT
iajs-901	50	12	2	2	NUM
iajs-901	50	13	)	)	PUNCT
iajs-901	50	14	is	be	AUX
iajs-901	50	15	used	use	VERB
iajs-901	50	16	to	to	PART
iajs-901	50	17	define	define	VERB
iajs-901	50	18	coprime	coprime	NOUN
iajs-901	50	19	modules	module	NOUN
iajs-901	50	20	in	in	ADP
iajs-901	50	21	[	[	X
iajs-901	50	22	27	27	NUM
iajs-901	50	23	,	,	PUNCT
iajs-901	50	24	definition	definition	NOUN
iajs-901	50	25	1.3.1	1.3.1	NUM
iajs-901	50	26	]	]	PUNCT
iajs-901	50	27	.	.	PUNCT
iajs-901	51	1	first	first	ADV
iajs-901	51	2	,	,	PUNCT
iajs-901	51	3	we	we	PRON
iajs-901	51	4	give	give	VERB
iajs-901	51	5	some	some	DET
iajs-901	51	6	remarks	remark	NOUN
iajs-901	51	7	and	and	CCONJ
iajs-901	51	8	examples	example	NOUN
iajs-901	51	9	of	of	ADP
iajs-901	51	10	coprime	coprime	NOUN
iajs-901	51	11	modules	module	NOUN
iajs-901	51	12	.	.	PUNCT
iajs-901	52	1	remarks	remark	NOUN
iajs-901	52	2	and	and	CCONJ
iajs-901	52	3	examples	example	NOUN
iajs-901	52	4	3	3	NUM
iajs-901	52	5	:	:	SYM
iajs-901	52	6	1	1	NUM
iajs-901	52	7	.	.	X
iajs-901	52	8	z	z	NOUN
iajs-901	52	9	as	as	SCONJ
iajs-901	52	10	z	z	NOUN
iajs-901	52	11	-	-	PUNCT
iajs-901	52	12	module	module	NOUN
iajs-901	52	13	is	be	AUX
iajs-901	52	14	not	not	PART
iajs-901	52	15	coprime	coprime	ADJ
iajs-901	52	16	.	.	PUNCT
iajs-901	53	1	2	2	X
iajs-901	53	2	.	.	X
iajs-901	53	3	q	q	PUNCT
iajs-901	53	4	as	as	SCONJ
iajs-901	53	5	z	z	NOUN
iajs-901	53	6	-	-	PUNCT
iajs-901	53	7	module	module	NOUN
iajs-901	53	8	is	be	AUX
iajs-901	53	9	coprime	coprime	NOUN
iajs-901	53	10	module	module	NOUN
iajs-901	53	11	.	.	PUNCT
iajs-901	54	1	3	3	X
iajs-901	54	2	.	.	X
iajs-901	54	3	p	p	X
iajs-901	54	4	z	z	NOUN
iajs-901	54	5			NOUN
iajs-901	54	6	as	as	ADP
iajs-901	54	7	a	a	DET
iajs-901	54	8	z	z	NOUN
iajs-901	54	9	-	-	PUNCT
iajs-901	54	10	module	module	NOUN
iajs-901	54	11	is	be	AUX
iajs-901	54	12	a	a	DET
iajs-901	54	13	coprime	coprime	NOUN
iajs-901	54	14	module	module	NOUN
iajs-901	54	15	[	[	X
iajs-901	54	16	9	9	NUM
iajs-901	54	17	]	]	SYM
iajs-901	54	18	.	.	PUNCT
iajs-901	55	1	4	4	X
iajs-901	55	2	.	.	X
iajs-901	55	3	every	every	DET
iajs-901	55	4	simple	simple	ADJ
iajs-901	55	5	r	r	NOUN
iajs-901	55	6	-	-	PUNCT
iajs-901	55	7	module	module	NOUN
iajs-901	55	8	is	be	AUX
iajs-901	55	9	a	a	DET
iajs-901	55	10	coprime	coprime	NOUN
iajs-901	55	11	module	module	NOUN
iajs-901	55	12	,	,	PUNCT
iajs-901	55	13	however	however	ADV
iajs-901	55	14	the	the	DET
iajs-901	55	15	converse	converse	NOUN
iajs-901	55	16	is	be	AUX
iajs-901	55	17	not	not	PART
iajs-901	55	18	true	true	ADJ
iajs-901	55	19	for	for	ADP
iajs-901	55	20	example	example	NOUN
iajs-901	55	21	:	:	PUNCT
iajs-901	55	22	q	q	X
iajs-901	55	23	as	as	SCONJ
iajs-901	55	24	z	z	NOUN
iajs-901	55	25	-	-	PUNCT
iajs-901	55	26	module	module	NOUN
iajs-901	55	27	is	be	AUX
iajs-901	55	28	coprime	coprime	ADJ
iajs-901	55	29	and	and	CCONJ
iajs-901	55	30	not	not	PART
iajs-901	55	31	simple	simple	ADJ
iajs-901	55	32	.	.	PUNCT
iajs-901	56	1	recall	recall	VERB
iajs-901	56	2	that	that	SCONJ
iajs-901	56	3	an	an	DET
iajs-901	56	4	r	r	NOUN
iajs-901	56	5	-	-	PUNCT
iajs-901	56	6	module	module	NOUN
iajs-901	56	7	m	m	NOUN
iajs-901	56	8	is	be	AUX
iajs-901	56	9	called	call	VERB
iajs-901	56	10	multiplication	multiplication	NOUN
iajs-901	56	11	if	if	SCONJ
iajs-901	56	12	for	for	ADP
iajs-901	56	13	any	any	DET
iajs-901	56	14	submodule	submodule	NOUN
iajs-901	56	15	n	n	PROPN
iajs-901	56	16	of	of	ADP
iajs-901	56	17	m	m	PRON
iajs-901	56	18	,	,	PUNCT
iajs-901	56	19	there	there	PRON
iajs-901	56	20	exist	exist	VERB
iajs-901	56	21	an	an	DET
iajs-901	56	22	ideal	ideal	NOUN
iajs-901	56	23	i	i	PRON
iajs-901	56	24	of	of	ADP
iajs-901	56	25	r	r	NOUN
iajs-901	57	1	such	such	ADJ
iajs-901	57	2	that	that	SCONJ
iajs-901	57	3	im	im	PROPN
iajs-901	57	4	=	=	NOUN
iajs-901	57	5	n	n	CCONJ
iajs-901	57	6	,	,	PUNCT
iajs-901	57	7	equivalently	equivalently	ADV
iajs-901	57	8	for	for	ADP
iajs-901	57	9	every	every	DET
iajs-901	57	10	submodule	submodule	NOUN
iajs-901	57	11	n	n	PROPN
iajs-901	57	12	of	of	ADP
iajs-901	57	13	m	m	PROPN
iajs-901	57	14	,	,	PUNCT
iajs-901	57	15	n=	n=	X
iajs-901	57	16	[	[	X
iajs-901	57	17	n	n	X
iajs-901	57	18	:	:	PUNCT
iajs-901	57	19	m	m	VERB
iajs-901	57	20	]	]	X
iajs-901	57	21	m	m	VERB
iajs-901	57	22	,	,	PUNCT
iajs-901	57	23	see	see	VERB
iajs-901	57	24	[	[	X
iajs-901	57	25	10	10	NUM
iajs-901	57	26	]	]	PUNCT
iajs-901	57	27	.	.	PUNCT
iajs-901	58	1	5	5	X
iajs-901	58	2	.	.	X
iajs-901	59	1	if	if	SCONJ
iajs-901	59	2	m	m	NOUN
iajs-901	59	3	is	be	AUX
iajs-901	59	4	a	a	DET
iajs-901	59	5	multiplication	multiplication	NOUN
iajs-901	59	6	coprime	coprime	NOUN
iajs-901	59	7	module	module	NOUN
iajs-901	59	8	,	,	PUNCT
iajs-901	59	9	then	then	ADV
iajs-901	59	10	m	m	NOUN
iajs-901	59	11	is	be	AUX
iajs-901	59	12	simple	simple	ADJ
iajs-901	59	13	,	,	PUNCT
iajs-901	59	14	and	and	CCONJ
iajs-901	59	15	hence	hence	ADV
iajs-901	59	16	m	m	VERB
iajs-901	59	17	is	be	AUX
iajs-901	59	18	prime	prime	ADJ
iajs-901	59	19	.	.	PUNCT
iajs-901	60	1	proof	proof	NOUN
iajs-901	60	2	:	:	PUNCT
iajs-901	60	3	since	since	SCONJ
iajs-901	60	4	m	m	PROPN
iajs-901	60	5	is	be	AUX
iajs-901	60	6	a	a	DET
iajs-901	60	7	coprime	coprime	ADJ
iajs-901	60	8	r	r	NOUN
iajs-901	60	9	-	-	NOUN
iajs-901	60	10	module	module	NOUN
iajs-901	60	11	,	,	PUNCT
iajs-901	60	12	then	then	ADV
iajs-901	60	13	r	r	NOUN
iajs-901	60	14	ann	ann	PROPN
iajs-901	60	15	m	m	NOUN
iajs-901	60	16	=	=	PUNCT
iajs-901	61	1	[	[	X
iajs-901	61	2	n	n	X
iajs-901	61	3	r	r	NOUN
iajs-901	61	4	:	:	PUNCT
iajs-901	61	5	m	m	X
iajs-901	61	6	]	]	X
iajs-901	61	7	for	for	ADP
iajs-901	61	8	all	all	DET
iajs-901	61	9	proper	proper	ADJ
iajs-901	61	10	submodule	submodule	NOUN
iajs-901	61	11	n	n	PROPN
iajs-901	61	12	of	of	ADP
iajs-901	61	13	m	m	PROPN
iajs-901	61	14	.	.	PUNCT
iajs-901	62	1	hence	hence	ADV
iajs-901	62	2	(	(	PUNCT
iajs-901	62	3	r	r	NOUN
iajs-901	62	4	ann	ann	PROPN
iajs-901	62	5	m)m	m)m	PROPN
iajs-901	62	6	=	=	PUNCT
iajs-901	63	1	[	[	X
iajs-901	63	2	n	n	X
iajs-901	63	3	r	r	NOUN
iajs-901	63	4	:	:	PUNCT
iajs-901	63	5	m]m	m]m	NOUN
iajs-901	63	6	.	.	PUNCT
iajs-901	64	1	thus	thus	ADV
iajs-901	64	2	(	(	PUNCT
iajs-901	64	3	0	0	NUM
iajs-901	64	4	)	)	PUNCT
iajs-901	64	5	=	=	NOUN
iajs-901	65	1	[	[	X
iajs-901	65	2	n	n	X
iajs-901	65	3	r	r	NOUN
iajs-901	65	4	:	:	PUNCT
iajs-901	65	5	m]m	m]m	NOUN
iajs-901	65	6	.	.	PUNCT
iajs-901	66	1	but	but	CCONJ
iajs-901	66	2	m	m	PROPN
iajs-901	66	3	is	be	AUX
iajs-901	66	4	a	a	DET
iajs-901	66	5	multiplication	multiplication	NOUN
iajs-901	66	6	r	r	NOUN
iajs-901	66	7	-	-	PUNCT
iajs-901	66	8	module	module	NOUN
iajs-901	66	9	,	,	PUNCT
iajs-901	66	10	so	so	SCONJ
iajs-901	67	1	[	[	X
iajs-901	67	2	n	n	X
iajs-901	67	3	r	r	NOUN
iajs-901	67	4	:	:	PUNCT
iajs-901	67	5	m]m	m]m	PROPN
iajs-901	67	6	=	=	SYM
iajs-901	67	7	n	n	CCONJ
iajs-901	67	8	,	,	PUNCT
iajs-901	67	9	and	and	CCONJ
iajs-901	67	10	hence	hence	ADV
iajs-901	67	11	n	n	NOUN
iajs-901	67	12	=	=	SYM
iajs-901	67	13	(	(	PUNCT
iajs-901	67	14	0	0	NUM
iajs-901	67	15	)	)	PUNCT
iajs-901	67	16	.	.	PUNCT
iajs-901	68	1	thus	thus	ADV
iajs-901	68	2	m	m	NOUN
iajs-901	68	3	is	be	AUX
iajs-901	68	4	a	a	DET
iajs-901	68	5	simple	simple	ADJ
iajs-901	68	6	r	r	NOUN
iajs-901	68	7	-	-	PUNCT
iajs-901	68	8	module	module	NOUN
iajs-901	68	9	,	,	PUNCT
iajs-901	68	10	and	and	CCONJ
iajs-901	68	11	m	m	PROPN
iajs-901	68	12	is	be	AUX
iajs-901	68	13	a	a	DET
iajs-901	68	14	prime	prime	ADJ
iajs-901	68	15	r	r	NOUN
iajs-901	68	16	-	-	PUNCT
iajs-901	68	17	module	module	NOUN
iajs-901	68	18	.	.	PUNCT
iajs-901	69	1	■	■	PUNCT
iajs-901	69	2	the	the	DET
iajs-901	69	3	condition	condition	NOUN
iajs-901	69	4	"	"	PUNCT
iajs-901	69	5	m	m	NOUN
iajs-901	69	6	is	be	AUX
iajs-901	69	7	multiplication	multiplication	NOUN
iajs-901	69	8	"	"	PUNCT
iajs-901	69	9	can	can	AUX
iajs-901	69	10	not	not	PART
iajs-901	69	11	be	be	AUX
iajs-901	69	12	dropped	drop	VERB
iajs-901	69	13	from	from	ADP
iajs-901	69	14	previous	previous	ADJ
iajs-901	69	15	remark	remark	NOUN
iajs-901	69	16	for	for	ADP
iajs-901	69	17	example	example	NOUN
iajs-901	69	18	:	:	PUNCT
iajs-901	69	19	p	p	X
iajs-901	69	20	z	z	NOUN
iajs-901	69	21			NOUN
iajs-901	69	22	is	be	AUX
iajs-901	69	23	a	a	DET
iajs-901	69	24	coprime	coprime	ADJ
iajs-901	69	25	z	z	NOUN
iajs-901	69	26	-	-	PUNCT
iajs-901	69	27	module	module	NOUN
iajs-901	69	28	,	,	PUNCT
iajs-901	69	29	and	and	CCONJ
iajs-901	69	30	by	by	ADP
iajs-901	69	31	[	[	X
iajs-901	69	32	2	2	NUM
iajs-901	69	33	,	,	PUNCT
iajs-901	69	34	remark	remark	NOUN
iajs-901	69	35	1.1.3(11	1.1.3(11	NUM
iajs-901	69	36	)	)	PUNCT
iajs-901	69	37	]	]	PUNCT
iajs-901	70	1	p	p	X
iajs-901	70	2	z	z	NOUN
iajs-901	70	3			NOUN
iajs-901	70	4	is	be	AUX
iajs-901	70	5	not	not	PART
iajs-901	70	6	prime	prime	ADJ
iajs-901	70	7	and	and	CCONJ
iajs-901	70	8	not	not	PART
iajs-901	70	9	multiplication	multiplication	NOUN
iajs-901	70	10	.	.	PUNCT
iajs-901	71	1	6	6	X
iajs-901	71	2	.	.	X
iajs-901	72	1	if	if	SCONJ
iajs-901	72	2	m	m	NOUN
iajs-901	72	3	is	be	AUX
iajs-901	72	4	a	a	DET
iajs-901	72	5	cyclic	cyclic	ADJ
iajs-901	72	6	coprime	coprime	NOUN
iajs-901	72	7	r	r	NOUN
iajs-901	72	8	-	-	PUNCT
iajs-901	72	9	module	module	NOUN
iajs-901	72	10	,	,	PUNCT
iajs-901	72	11	then	then	ADV
iajs-901	72	12	m	m	NOUN
iajs-901	72	13	is	be	AUX
iajs-901	72	14	simple	simple	ADJ
iajs-901	72	15	and	and	CCONJ
iajs-901	72	16	so	so	ADV
iajs-901	72	17	prime	prime	ADJ
iajs-901	72	18	.	.	PUNCT
iajs-901	73	1	7	7	X
iajs-901	73	2	.	.	X
iajs-901	73	3	r	r	NOUN
iajs-901	73	4	is	be	AUX
iajs-901	73	5	a	a	DET
iajs-901	73	6	coprime	coprime	NOUN
iajs-901	73	7	ring	ring	NOUN
iajs-901	73	8	if	if	SCONJ
iajs-901	73	9	and	and	CCONJ
iajs-901	73	10	only	only	ADV
iajs-901	73	11	if	if	SCONJ
iajs-901	73	12	r	r	NOUN
iajs-901	73	13	is	be	AUX
iajs-901	73	14	a	a	DET
iajs-901	73	15	field	field	NOUN
iajs-901	73	16	.	.	PUNCT
iajs-901	74	1	proof	proof	NOUN
iajs-901	74	2	:	:	PUNCT
iajs-901	74	3	the	the	DET
iajs-901	74	4	proof	proof	NOUN
iajs-901	74	5	follows	follow	VERB
iajs-901	74	6	by	by	ADP
iajs-901	74	7	(	(	PUNCT
iajs-901	74	8	rem	rem	PROPN
iajs-901	74	9	.	.	PROPN
iajs-901	74	10	and	and	CCONJ
iajs-901	74	11	ex	ex	NOUN
iajs-901	74	12	.	.	NOUN
iajs-901	74	13	2	2	NUM
iajs-901	74	14	.	.	PUNCT
iajs-901	75	1	(	(	PUNCT
iajs-901	75	2	5	5	NUM
iajs-901	75	3	)	)	PUNCT
iajs-901	75	4	,	,	PUNCT
iajs-901	75	5	(	(	PUNCT
iajs-901	75	6	4	4	NUM
iajs-901	75	7	)	)	PUNCT
iajs-901	75	8	)	)	PUNCT
iajs-901	75	9	.	.	PUNCT
iajs-901	76	1	■	■	PUNCT
iajs-901	76	2	8	8	X
iajs-901	76	3	.	.	X
iajs-901	76	4	zn	zn	PROPN
iajs-901	76	5	is	be	AUX
iajs-901	76	6	a	a	DET
iajs-901	76	7	coprime	coprime	ADJ
iajs-901	76	8	z	z	NOUN
iajs-901	76	9	-	-	PUNCT
iajs-901	76	10	module	module	NOUN
iajs-901	76	11	iff	iff	PROPN
iajs-901	76	12	n	n	PART
iajs-901	76	13	is	be	AUX
iajs-901	76	14	a	a	DET
iajs-901	76	15	prime	prime	ADJ
iajs-901	76	16	number	number	NOUN
iajs-901	76	17	.	.	PUNCT
iajs-901	77	1	proof	proof	NOUN
iajs-901	77	2	:	:	PUNCT
iajs-901	77	3	the	the	DET
iajs-901	77	4	proof	proof	NOUN
iajs-901	77	5	follows	follow	VERB
iajs-901	77	6	directly	directly	ADV
iajs-901	77	7	by	by	ADP
iajs-901	77	8	(	(	PUNCT
iajs-901	77	9	rem	rem	PROPN
iajs-901	77	10	.	.	PROPN
iajs-901	77	11	and	and	CCONJ
iajs-901	77	12	ex	ex	PROPN
iajs-901	77	13	.	.	NOUN
iajs-901	77	14	3	3	NUM
iajs-901	77	15	(	(	PUNCT
iajs-901	77	16	4	4	NUM
iajs-901	77	17	)	)	PUNCT
iajs-901	77	18	,	,	PUNCT
iajs-901	77	19	(	(	PUNCT
iajs-901	77	20	5	5	NUM
iajs-901	77	21	)	)	PUNCT
iajs-901	77	22	)	)	PUNCT
iajs-901	77	23	.	.	PUNCT
iajs-901	78	1	■	■	PUNCT
iajs-901	78	2	9	9	X
iajs-901	78	3	.	.	X
iajs-901	78	4	for	for	ADP
iajs-901	78	5	any	any	DET
iajs-901	78	6	n	n	CCONJ
iajs-901	78	7	,	,	PUNCT
iajs-901	78	8	m	m	VERB
iajs-901	78	9			NOUN
iajs-901	78	10	z	z	NOUN
iajs-901	78	11	;	;	PUNCT
iajs-901	78	12	n	n	NUM
iajs-901	78	13			NOUN
iajs-901	78	14	m	m	PROPN
iajs-901	78	15	,	,	PUNCT
iajs-901	78	16	the	the	DET
iajs-901	78	17	z	z	NOUN
iajs-901	78	18	-	-	PUNCT
iajs-901	78	19	module	module	NOUN
iajs-901	78	20	m	m	NOUN
iajs-901	78	21	=	=	PUNCT
iajs-901	78	22	zn	zn	AUX
iajs-901	78	23	zm	zm	PROPN
iajs-901	78	24	is	be	AUX
iajs-901	78	25	not	not	PART
iajs-901	78	26	coprime	coprime	ADJ
iajs-901	78	27	.	.	PUNCT
iajs-901	79	1	10	10	NUM
iajs-901	79	2	.	.	PUNCT
iajs-901	80	1	every	every	DET
iajs-901	80	2	vector	vector	NOUN
iajs-901	80	3	space	space	NOUN
iajs-901	80	4	m	m	VERB
iajs-901	80	5	over	over	ADP
iajs-901	80	6	a	a	DET
iajs-901	80	7	field	field	NOUN
iajs-901	80	8	r	r	NOUN
iajs-901	80	9	is	be	AUX
iajs-901	80	10	a	a	DET
iajs-901	80	11	coprime	coprime	ADJ
iajs-901	80	12	r	r	NOUN
iajs-901	80	13	-	-	NOUN
iajs-901	80	14	module	module	NOUN
iajs-901	80	15	.	.	PUNCT
iajs-901	81	1	ihjpas	ihjpa	VERB
iajs-901	81	2	ibn	ibn	PROPN
iajs-901	81	3	alhaitham	alhaitham	PROPN
iajs-901	82	1	j.	j.	PROPN
iajs-901	83	1	fo	fo	ADP
iajs-901	83	2	r	r	NOUN
iajs-901	83	3	pure	pure	ADJ
iajs-901	83	4	&	&	CCONJ
iajs-901	83	5	appl	appl	PROPN
iajs-901	83	6	.	.	PUNCT
iajs-901	84	1	sc	sc	PROPN
iajs-901	84	2	i.	i.	PROPN
iajs-901	84	3	vo	vo	PROPN
iajs-901	84	4	l.23	l.23	PROPN
iajs-901	84	5	(	(	PUNCT
iajs-901	84	6	3	3	NUM
iajs-901	84	7	)	)	PUNCT
iajs-901	84	8	2010	2010	NUM
iajs-901	84	9	we	we	PRON
iajs-901	84	10	have	have	VERB
iajs-901	84	11	the	the	DET
iajs-901	84	12	following	follow	VERB
iajs-901	84	13	proposition	proposition	NOUN
iajs-901	84	14	:	:	PUNCT
iajs-901	84	15	proposition	proposition	NOUN
iajs-901	84	16	(	(	PUNCT
iajs-901	84	17	4	4	NUM
iajs-901	84	18	):	):	PUNCT
iajs-901	84	19	let	let	VERB
iajs-901	84	20	m	m	PRON
iajs-901	84	21	be	be	AUX
iajs-901	84	22	an	an	DET
iajs-901	84	23	r	r	NOUN
iajs-901	84	24	-	-	PUNCT
iajs-901	84	25	module	module	NOUN
iajs-901	84	26	.	.	PUNCT
iajs-901	85	1	consider	consider	VERB
iajs-901	85	2	the	the	DET
iajs-901	85	3	following	follow	VERB
iajs-901	85	4	statements	statement	NOUN
iajs-901	85	5	:	:	PUNCT
iajs-901	85	6	1	1	X
iajs-901	85	7	.	.	X
iajs-901	85	8	m	m	PROPN
iajs-901	85	9	is	be	AUX
iajs-901	85	10	a	a	DET
iajs-901	85	11	coprime	coprime	ADJ
iajs-901	85	12	r	r	NOUN
iajs-901	85	13	-	-	NOUN
iajs-901	85	14	module	module	NOUN
iajs-901	85	15	.	.	PUNCT
iajs-901	86	1	2	2	X
iajs-901	86	2	.	.	X
iajs-901	86	3	r	r	NOUN
iajs-901	86	4	ann	ann	PROPN
iajs-901	86	5	m	m	NOUN
iajs-901	86	6	=	=	PUNCT
iajs-901	87	1	[	[	X
iajs-901	87	2	x	x	X
iajs-901	87	3	r	r	NOUN
iajs-901	87	4	:	:	PUNCT
iajs-901	87	5	m	m	X
iajs-901	87	6	]	]	X
iajs-901	87	7	for	for	ADP
iajs-901	87	8	every	every	DET
iajs-901	87	9	x	x	PRON
iajs-901	87	10			NOUN
iajs-901	87	11	m	m	VERB
iajs-901	87	12	such	such	ADJ
iajs-901	87	13	that	that	SCONJ
iajs-901	87	14	(	(	PUNCT
iajs-901	87	15	x	x	X
iajs-901	87	16	)	)	PUNCT
iajs-901	87	17	is	be	AUX
iajs-901	87	18	a	a	DET
iajs-901	87	19	proper	proper	ADJ
iajs-901	87	20	submodule	submodule	NOUN
iajs-901	87	21	of	of	ADP
iajs-901	87	22	m.	m.	NOUN
iajs-901	87	23	3	3	NUM
iajs-901	87	24	.	.	PUNCT
iajs-901	88	1	for	for	ADP
iajs-901	88	2	every	every	DET
iajs-901	88	3	ideal	ideal	NOUN
iajs-901	88	4	i	i	PRON
iajs-901	88	5	of	of	ADP
iajs-901	88	6	r	r	NOUN
iajs-901	88	7	and	and	CCONJ
iajs-901	88	8	for	for	ADP
iajs-901	88	9	every	every	DET
iajs-901	88	10	x	x	PROPN
iajs-901	88	11			NOUN
iajs-901	88	12	m	m	PROPN
iajs-901	88	13	,	,	PUNCT
iajs-901	88	14	(	(	PUNCT
iajs-901	88	15	x	x	X
iajs-901	88	16	)	)	PUNCT
iajs-901	88	17	is	be	AUX
iajs-901	88	18	a	a	DET
iajs-901	88	19	proper	proper	ADJ
iajs-901	88	20	submodule	submodule	NOUN
iajs-901	88	21	of	of	ADP
iajs-901	88	22	m	m	PRON
iajs-901	88	23	such	such	ADJ
iajs-901	88	24	that	that	SCONJ
iajs-901	88	25	i	i	PRON
iajs-901	88	26			PROPN
iajs-901	88	27	(	(	PUNCT
iajs-901	88	28	)	)	PUNCT
iajs-901	88	29	x	x	SYM
iajs-901	88	30			PROPN
iajs-901	88	31	=	=	SYM
iajs-901	88	32	(	(	PUNCT
iajs-901	88	33	x	x	NOUN
iajs-901	88	34	)	)	PUNCT
iajs-901	88	35	,	,	PUNCT
iajs-901	88	36	implies	imply	VERB
iajs-901	88	37	i	i	PRON
iajs-901	88	38	=	=	SYM
iajs-901	88	39	0	0	PUNCT
iajs-901	88	40	or	or	CCONJ
iajs-901	88	41	i	i	PRON
iajs-901	88	42	m	m	VERB
iajs-901	88	43	=	=	ADJ
iajs-901	88	44	0	0	X
iajs-901	88	45	.	.	PUNCT
iajs-901	89	1	then	then	ADV
iajs-901	89	2	(	(	PUNCT
iajs-901	89	3	1	1	X
iajs-901	89	4	)	)	PUNCT
iajs-901	89	5			NOUN
iajs-901	89	6	(	(	PUNCT
iajs-901	89	7	2	2	NUM
iajs-901	89	8	)	)	PUNCT
iajs-901	89	9			NOUN
iajs-901	89	10	(	(	PUNCT
iajs-901	89	11	3	3	NUM
iajs-901	89	12	)	)	PUNCT
iajs-901	89	13	,	,	PUNCT
iajs-901	89	14	and	and	CCONJ
iajs-901	89	15	(	(	PUNCT
iajs-901	89	16	2	2	X
iajs-901	89	17	)	)	PUNCT
iajs-901	89	18			NOUN
iajs-901	89	19	(	(	PUNCT
iajs-901	89	20	1	1	X
iajs-901	89	21	)	)	PUNCT
iajs-901	89	22	if	if	SCONJ
iajs-901	89	23	i	i	PRON
iajs-901	89	24	r	r	VERB
iajs-901	89	25	r	r	NOUN
iajs-901	89	26	i	i	PRON
iajs-901	89	27	[	[	PUNCT
iajs-901	89	28	:	:	PUNCT
iajs-901	89	29	]	]	PUNCT
iajs-901	89	30	[	[	PUNCT
iajs-901	89	31	r	r	NOUN
iajs-901	89	32	:	:	PUNCT
iajs-901	89	33	]	]	X
iajs-901	89	34	i	i	PRON
iajs-901	89	35	ix	ix	ADP
iajs-901	89	36	x	x	PUNCT
iajs-901	89	37			NOUN
iajs-901	89	38			NOUN
iajs-901	89	39			PUNCT
iajs-901	89	40			VERB
iajs-901	89	41			NUM
iajs-901	89	42			PROPN
iajs-901	89	43	,	,	PUNCT
iajs-901	89	44	where	where	SCONJ
iajs-901	89	45	xi	xi	X
iajs-901	89	46	m	m	X
iajs-901	89	47	and	and	CCONJ
iajs-901	89	48			PROPN
iajs-901	89	49	is	be	AUX
iajs-901	89	50	any	any	DET
iajs-901	89	51	index	index	NOUN
iajs-901	89	52	set	set	NOUN
iajs-901	89	53	.	.	PUNCT
iajs-901	90	1	proof	proof	NOUN
iajs-901	90	2	:	:	PUNCT
iajs-901	90	3	(	(	PUNCT
iajs-901	90	4	1	1	X
iajs-901	90	5	)	)	PUNCT
iajs-901	90	6			NOUN
iajs-901	90	7	(	(	PUNCT
iajs-901	90	8	2	2	NUM
iajs-901	90	9	)	)	PUNCT
iajs-901	90	10	.	.	PUNCT
iajs-901	91	1	it	it	PRON
iajs-901	91	2	is	be	AUX
iajs-901	91	3	clear	clear	ADJ
iajs-901	91	4	.	.	PUNCT
iajs-901	92	1	(	(	PUNCT
iajs-901	92	2	2	2	X
iajs-901	92	3	)	)	PUNCT
iajs-901	92	4			NOUN
iajs-901	92	5	(	(	PUNCT
iajs-901	92	6	3	3	NUM
iajs-901	92	7	)	)	PUNCT
iajs-901	92	8	.	.	PUNCT
iajs-901	93	1	let	let	VERB
iajs-901	93	2	i	i	PRON
iajs-901	93	3			PROPN
iajs-901	93	4	(	(	PUNCT
iajs-901	93	5	)	)	PUNCT
iajs-901	93	6	x	x	SYM
iajs-901	93	7			PROPN
iajs-901	93	8	=	=	SYM
iajs-901	93	9	(	(	PUNCT
iajs-901	93	10	x	x	NOUN
iajs-901	93	11	)	)	PUNCT
iajs-901	93	12	and	and	CCONJ
iajs-901	93	13	assume	assume	VERB
iajs-901	93	14	i	i	PRON
iajs-901	93	15			NOUN
iajs-901	93	16	0	0	NUM
iajs-901	93	17	,	,	PUNCT
iajs-901	93	18	then	then	ADV
iajs-901	93	19	i	i	PRON
iajs-901	93	20	m	m	VERB
iajs-901	93	21			ADJ
iajs-901	93	22	(	(	PUNCT
iajs-901	93	23	x	x	NOUN
iajs-901	93	24	)	)	PUNCT
iajs-901	93	25	,	,	PUNCT
iajs-901	93	26	that	that	PRON
iajs-901	93	27	is	is	ADV
iajs-901	93	28	i	i	PRON
iajs-901	93	29			PROPN
iajs-901	94	1	[	[	X
iajs-901	94	2	x	x	X
iajs-901	94	3	r	r	NOUN
iajs-901	94	4	:	:	PUNCT
iajs-901	94	5	m	m	X
iajs-901	94	6	]	]	PUNCT
iajs-901	94	7	and	and	CCONJ
iajs-901	94	8	by	by	ADP
iajs-901	94	9	(	(	PUNCT
iajs-901	94	10	2	2	X
iajs-901	94	11	)	)	PUNCT
iajs-901	94	12	i	i	PRON
iajs-901	94	13			VERB
iajs-901	94	14	r	r	PROPN
iajs-901	94	15	ann	ann	PROPN
iajs-901	94	16	m.	m.	NOUN
iajs-901	95	1	thus	thus	ADV
iajs-901	95	2	i	i	PRON
iajs-901	95	3	m	m	VERB
iajs-901	95	4	=	=	PUNCT
iajs-901	95	5	(	(	PUNCT
iajs-901	95	6	0	0	NUM
iajs-901	95	7	)	)	PUNCT
iajs-901	95	8	.	.	PUNCT
iajs-901	96	1	(	(	PUNCT
iajs-901	96	2	3	3	X
iajs-901	96	3	)	)	PUNCT
iajs-901	96	4			NOUN
iajs-901	96	5	(	(	PUNCT
iajs-901	96	6	2	2	NUM
iajs-901	96	7	)	)	PUNCT
iajs-901	96	8	.	.	PUNCT
iajs-901	97	1	let	let	VERB
iajs-901	98	1	r	r	NOUN
iajs-901	98	2			NOUN
iajs-901	98	3	[	[	X
iajs-901	98	4	x	x	X
iajs-901	98	5	r	r	NOUN
iajs-901	98	6	:	:	PUNCT
iajs-901	98	7	m	m	VERB
iajs-901	98	8	]	]	X
iajs-901	98	9	;	;	PUNCT
iajs-901	98	10	x	x	SYM
iajs-901	98	11			NOUN
iajs-901	98	12	m	m	VERB
iajs-901	98	13	and	and	CCONJ
iajs-901	98	14	(	(	PUNCT
iajs-901	98	15	x	x	X
iajs-901	98	16	)	)	PUNCT
iajs-901	98	17	proper	proper	ADJ
iajs-901	98	18	submodule	submodule	NOUN
iajs-901	98	19	of	of	ADP
iajs-901	98	20	m.	m.	NOUN
iajs-901	98	21	then	then	ADV
iajs-901	98	22	r	r	VERB
iajs-901	98	23			PROPN
iajs-901	98	24	r	r	PROPN
iajs-901	98	25	ann	ann	PROPN
iajs-901	98	26	(	(	PUNCT
iajs-901	98	27	)	)	PUNCT
iajs-901	98	28	x	x	PUNCT
iajs-901	98	29			VERB
iajs-901	98	30	,	,	PUNCT
iajs-901	98	31	that	that	ADV
iajs-901	98	32	is	is	ADV
iajs-901	98	33	(	(	PUNCT
iajs-901	98	34	r	r	NOUN
iajs-901	98	35	)	)	PUNCT
iajs-901	98	36	(	(	PUNCT
iajs-901	98	37	)	)	PUNCT
iajs-901	98	38	x	x	SYM
iajs-901	98	39			PROPN
iajs-901	98	40	=	=	SYM
iajs-901	98	41	0	0	NUM
iajs-901	98	42	,	,	PUNCT
iajs-901	98	43	hence	hence	ADV
iajs-901	98	44	by	by	ADP
iajs-901	98	45	(	(	PUNCT
iajs-901	98	46	3	3	NUM
iajs-901	98	47	)	)	PUNCT
iajs-901	98	48	,	,	PUNCT
iajs-901	98	49	either	either	CCONJ
iajs-901	98	50	r	r	NOUN
iajs-901	98	51	=	=	SYM
iajs-901	98	52	0	0	NUM
iajs-901	98	53	or	or	CCONJ
iajs-901	98	54	(	(	PUNCT
iajs-901	98	55	r	r	NOUN
iajs-901	98	56	)	)	PUNCT
iajs-901	98	57	m	m	PROPN
iajs-901	98	58	=	=	NOUN
iajs-901	98	59	0	0	NUM
iajs-901	98	60	.	.	PUNCT
iajs-901	99	1	thus	thus	ADV
iajs-901	99	2	r	r	VERB
iajs-901	99	3			PROPN
iajs-901	99	4	r	r	PROPN
iajs-901	99	5	ann	ann	PROPN
iajs-901	99	6	m.	m.	NOUN
iajs-901	99	7	(	(	PUNCT
iajs-901	99	8	2	2	NUM
iajs-901	99	9	)	)	PUNCT
iajs-901	99	10			NOUN
iajs-901	99	11	(	(	PUNCT
iajs-901	99	12	1	1	NUM
iajs-901	99	13	)	)	PUNCT
iajs-901	99	14	.	.	PUNCT
iajs-901	100	1	let	let	VERB
iajs-901	100	2	n	n	PRON
iajs-901	100	3	be	be	AUX
iajs-901	100	4	a	a	DET
iajs-901	100	5	proper	proper	ADJ
iajs-901	100	6	submodule	submodule	NOUN
iajs-901	100	7	of	of	ADP
iajs-901	100	8	m.	m.	NOUN
iajs-901	100	9	then	then	ADV
iajs-901	100	10	n	n	PROPN
iajs-901	100	11	=	=	PROPN
iajs-901	100	12	r	r	PROPN
iajs-901	100	13	xi	xi	PROPN
iajs-901	100	14	,	,	PUNCT
iajs-901	100	15	xi	xi	PROPN
iajs-901	100	16			PROPN
iajs-901	100	17	n.	n.	PROPN
iajs-901	101	1	so	so	SCONJ
iajs-901	101	2	that	that	SCONJ
iajs-901	101	3	[	[	X
iajs-901	101	4	n	n	X
iajs-901	101	5	r	r	NOUN
iajs-901	101	6	:	:	PUNCT
iajs-901	101	7	m	m	VERB
iajs-901	101	8	]	]	X
iajs-901	101	9	=	=	PUNCT
iajs-901	102	1	[	[	X
iajs-901	102	2	r	r	NOUN
iajs-901	102	3	xi	xi	ADP
iajs-901	102	4	r	r	NOUN
iajs-901	102	5	:	:	PUNCT
iajs-901	102	6	m	m	VERB
iajs-901	102	7	]	]	X
iajs-901	103	1	=	=	PUNCT
iajs-901	103	2	i	i	PRON
iajs-901	103	3	r	r	NOUN
iajs-901	103	4	[	[	PUNCT
iajs-901	103	5	:	:	PUNCT
iajs-901	103	6	]	]	X
iajs-901	103	7	ix	ix	ADP
iajs-901	103	8	n	n	PROPN
iajs-901	103	9	x	x	PROPN
iajs-901	103	10			NOUN
iajs-901	103	11			PUNCT
iajs-901	103	12			NUM
iajs-901	103	13	=	=	SYM
iajs-901	103	14	r	r	NOUN
iajs-901	103	15	ann	ann	PROPN
iajs-901	103	16	m.	m.	NOUN
iajs-901	104	1	■	■	PUNCT
iajs-901	104	2	the	the	DET
iajs-901	104	3	following	follow	VERB
iajs-901	104	4	proposition	proposition	NOUN
iajs-901	104	5	is	be	AUX
iajs-901	104	6	a	a	DET
iajs-901	104	7	characterization	characterization	NOUN
iajs-901	104	8	of	of	ADP
iajs-901	104	9	coprime	coprime	NOUN
iajs-901	104	10	module	module	NOUN
iajs-901	104	11	under	under	ADP
iajs-901	104	12	the	the	DET
iajs-901	104	13	class	class	NOUN
iajs-901	104	14	of	of	ADP
iajs-901	104	15	finitely	finitely	ADV
iajs-901	104	16	generated	generate	VERB
iajs-901	104	17	(	(	PUNCT
iajs-901	104	18	multiplication	multiplication	NOUN
iajs-901	104	19	)	)	PUNCT
iajs-901	104	20	modules	module	NOUN
iajs-901	104	21	.	.	PUNCT
iajs-901	105	1	proposition	proposition	NOUN
iajs-901	105	2	(	(	PUNCT
iajs-901	105	3	5	5	NUM
iajs-901	105	4	):	):	PUNCT
iajs-901	105	5	let	let	VERB
iajs-901	105	6	m	m	PRON
iajs-901	105	7	be	be	AUX
iajs-901	105	8	a	a	DET
iajs-901	105	9	finitely	finitely	ADV
iajs-901	105	10	generated	generate	VERB
iajs-901	105	11	(	(	PUNCT
iajs-901	105	12	or	or	CCONJ
iajs-901	105	13	multiplication	multiplication	NOUN
iajs-901	105	14	)	)	PUNCT
iajs-901	105	15	r	r	NOUN
iajs-901	105	16	-	-	PUNCT
iajs-901	105	17	module	module	NOUN
iajs-901	105	18	,	,	PUNCT
iajs-901	105	19	then	then	ADV
iajs-901	105	20	m	m	NOUN
iajs-901	105	21	is	be	AUX
iajs-901	105	22	a	a	DET
iajs-901	105	23	coprime	coprime	ADJ
iajs-901	105	24	rmodule	rmodule	NOUN
iajs-901	105	25	if	if	SCONJ
iajs-901	105	26	and	and	CCONJ
iajs-901	105	27	only	only	ADV
iajs-901	105	28	if	if	SCONJ
iajs-901	105	29	r	r	NOUN
iajs-901	105	30	ann	ann	NOUN
iajs-901	105	31	m	m	NOUN
iajs-901	105	32	=	=	PUNCT
iajs-901	106	1	[	[	X
iajs-901	106	2	n	n	X
iajs-901	106	3	r	r	NOUN
iajs-901	106	4	:	:	PUNCT
iajs-901	106	5	m	m	VERB
iajs-901	106	6	]	]	X
iajs-901	106	7	,	,	PUNCT
iajs-901	106	8	for	for	ADP
iajs-901	106	9	every	every	DET
iajs-901	106	10	prime	prime	ADJ
iajs-901	106	11	submodule	submodule	PROPN
iajs-901	106	12	n	n	PROPN
iajs-901	106	13	of	of	ADP
iajs-901	106	14	m.	m.	NOUN
iajs-901	106	15	proof	proof	NOUN
iajs-901	106	16	:	:	PUNCT
iajs-901	106	17	(	(	PUNCT
iajs-901	106	18			NOUN
iajs-901	106	19	)	)	PUNCT
iajs-901	106	20	it	it	PRON
iajs-901	106	21	is	be	AUX
iajs-901	106	22	clear	clear	ADJ
iajs-901	106	23	.	.	PUNCT
iajs-901	107	1	to	to	PART
iajs-901	107	2	prove	prove	VERB
iajs-901	107	3	the	the	DET
iajs-901	107	4	converse	converse	NOUN
iajs-901	107	5	,	,	PUNCT
iajs-901	107	6	let	let	VERB
iajs-901	107	7	w	w	NOUN
iajs-901	107	8	be	be	AUX
iajs-901	107	9	a	a	DET
iajs-901	107	10	proper	proper	ADJ
iajs-901	107	11	submodule	submodule	NOUN
iajs-901	107	12	of	of	ADP
iajs-901	107	13	m.	m.	NOUN
iajs-901	107	14	since	since	SCONJ
iajs-901	107	15	m	m	PROPN
iajs-901	107	16	is	be	AUX
iajs-901	107	17	finitely	finitely	ADV
iajs-901	107	18	generated	generate	VERB
iajs-901	107	19	(	(	PUNCT
iajs-901	107	20	or	or	CCONJ
iajs-901	107	21	multiplication	multiplication	NOUN
iajs-901	107	22	)	)	PUNCT
iajs-901	108	1	r	r	NOUN
iajs-901	108	2	-	-	PUNCT
iajs-901	108	3	module	module	NOUN
iajs-901	108	4	,	,	PUNCT
iajs-901	108	5	then	then	ADV
iajs-901	108	6	by	by	ADP
iajs-901	108	7	[	[	X
iajs-901	108	8	15	15	NUM
iajs-901	108	9	]	]	PUNCT
iajs-901	108	10	,	,	PUNCT
iajs-901	108	11	[	[	X
iajs-901	108	12	1	1	X
iajs-901	108	13	]	]	PUNCT
iajs-901	108	14	there	there	PRON
iajs-901	108	15	exists	exist	VERB
iajs-901	108	16	a	a	DET
iajs-901	108	17	maximal	maximal	ADJ
iajs-901	108	18	submodule	submodule	NOUN
iajs-901	108	19	n	n	PROPN
iajs-901	108	20	of	of	ADP
iajs-901	108	21	m(which	m(which	NOUN
iajs-901	108	22	is	be	AUX
iajs-901	108	23	prime	prime	ADJ
iajs-901	108	24	by	by	ADP
iajs-901	108	25	[	[	X
iajs-901	108	26	30	30	NUM
iajs-901	108	27	,	,	PUNCT
iajs-901	108	28	corollary	corollary	ADJ
iajs-901	108	29	2.5	2.5	NUM
iajs-901	108	30	,	,	PUNCT
iajs-901	108	31	ch.1	ch.1	PROPN
iajs-901	108	32	]	]	X
iajs-901	108	33	)	)	PUNCT
iajs-901	108	34	such	such	ADJ
iajs-901	108	35	that	that	DET
iajs-901	108	36	w	w	PROPN
iajs-901	108	37			PROPN
iajs-901	108	38	n	n	PROPN
iajs-901	108	39	m.	m.	NOUN
iajs-901	109	1	hence	hence	ADV
iajs-901	109	2	[	[	X
iajs-901	109	3	w	w	NOUN
iajs-901	109	4	r	r	NOUN
iajs-901	109	5	:	:	PUNCT
iajs-901	109	6	m	m	VERB
iajs-901	109	7	]	]	X
iajs-901	109	8			PROPN
iajs-901	110	1	[	[	X
iajs-901	110	2	n	n	X
iajs-901	110	3	r	r	NOUN
iajs-901	110	4	:	:	PUNCT
iajs-901	110	5	m	m	VERB
iajs-901	110	6	]	]	X
iajs-901	110	7	.	.	PUNCT
iajs-901	111	1	but	but	CCONJ
iajs-901	111	2	[	[	X
iajs-901	111	3	n	n	X
iajs-901	111	4	r	r	NOUN
iajs-901	111	5	:	:	PUNCT
iajs-901	111	6	m	m	VERB
iajs-901	111	7	]	]	X
iajs-901	111	8	=	=	PUNCT
iajs-901	111	9	r	r	NOUN
iajs-901	111	10	ann	ann	PROPN
iajs-901	111	11	m	m	VERB
iajs-901	111	12	by	by	ADP
iajs-901	111	13	assumption	assumption	NOUN
iajs-901	111	14	.	.	PUNCT
iajs-901	112	1	thus	thus	ADV
iajs-901	112	2	[	[	X
iajs-901	112	3	w	w	NOUN
iajs-901	112	4	r	r	NOUN
iajs-901	112	5	:	:	PUNCT
iajs-901	112	6	m	m	X
iajs-901	112	7	]	]	X
iajs-901	112	8			PROPN
iajs-901	112	9	r	r	PROPN
iajs-901	112	10	ann	ann	PROPN
iajs-901	112	11	m	m	PROPN
iajs-901	112	12	,	,	PUNCT
iajs-901	112	13	and	and	CCONJ
iajs-901	112	14	so	so	ADV
iajs-901	112	15	r	r	X
iajs-901	112	16	ann	ann	PROPN
iajs-901	112	17	m	m	NOUN
iajs-901	112	18	=	=	PUNCT
iajs-901	113	1	[	[	X
iajs-901	113	2	w	w	NOUN
iajs-901	113	3	r	r	NOUN
iajs-901	113	4	:	:	PUNCT
iajs-901	113	5	m	m	VERB
iajs-901	113	6	]	]	X
iajs-901	113	7	.	.	PUNCT
iajs-901	114	1	recall	recall	VERB
iajs-901	114	2	that	that	SCONJ
iajs-901	114	3	a	a	DET
iajs-901	114	4	submodule	submodule	NOUN
iajs-901	114	5	n	n	PROPN
iajs-901	114	6	of	of	ADP
iajs-901	114	7	an	an	DET
iajs-901	114	8	r	r	NOUN
iajs-901	114	9	-	-	PUNCT
iajs-901	114	10	module	module	NOUN
iajs-901	114	11	m	m	NOUN
iajs-901	114	12	is	be	AUX
iajs-901	114	13	called	call	VERB
iajs-901	114	14	second	second	ADJ
iajs-901	114	15	submodule	submodule	NOUN
iajs-901	114	16	if	if	SCONJ
iajs-901	114	17	for	for	ADP
iajs-901	114	18	every	every	DET
iajs-901	114	19	r	r	NOUN
iajs-901	114	20			NOUN
iajs-901	114	21	r	r	NOUN
iajs-901	114	22	,	,	PUNCT
iajs-901	114	23	the	the	DET
iajs-901	114	24	homothety	homothety	NOUN
iajs-901	114	25	r	r	PROPN
iajs-901	114	26	*	*	NOUN
iajs-901	114	27	on	on	ADP
iajs-901	114	28	n	n	X
iajs-901	114	29	is	be	AUX
iajs-901	114	30	either	either	DET
iajs-901	114	31	zero	zero	NUM
iajs-901	114	32	or	or	CCONJ
iajs-901	114	33	surjective	surjective	ADJ
iajs-901	114	34	,	,	PUNCT
iajs-901	114	35	where	where	SCONJ
iajs-901	114	36	if	if	SCONJ
iajs-901	114	37	m	m	NOUN
iajs-901	114	38	is	be	AUX
iajs-901	114	39	an	an	DET
iajs-901	114	40	r	r	NOUN
iajs-901	114	41	-	-	PUNCT
iajs-901	114	42	module	module	NOUN
iajs-901	114	43	and	and	CCONJ
iajs-901	114	44	r	r	NOUN
iajs-901	114	45	r	r	NOUN
iajs-901	114	46	,	,	PUNCT
iajs-901	114	47	then	then	ADV
iajs-901	114	48	an	an	DET
iajs-901	114	49	r	r	NOUN
iajs-901	114	50	-	-	PUNCT
iajs-901	114	51	endomorphism	endomorphism	NOUN
iajs-901	114	52	r	r	PROPN
iajs-901	114	53	*	*	PROPN
iajs-901	114	54	is	be	AUX
iajs-901	114	55	called	call	VERB
iajs-901	114	56	homothety	homothety	NOUN
iajs-901	114	57	if	if	SCONJ
iajs-901	114	58	r*(x	r*(x	NOUN
iajs-901	114	59	)	)	PUNCT
iajs-901	114	60	=	=	PUNCT
iajs-901	114	61	rx	rx	VERB
iajs-901	114	62	for	for	ADP
iajs-901	114	63	all	all	DET
iajs-901	114	64	x	x	PRON
iajs-901	114	65			NOUN
iajs-901	114	66	m	m	ADP
iajs-901	114	67	;	;	PUNCT
iajs-901	114	68	see	see	VERB
iajs-901	114	69	[	[	X
iajs-901	114	70	29	29	NUM
iajs-901	114	71	,	,	PUNCT
iajs-901	114	72	definition	definition	NOUN
iajs-901	114	73	2.1	2.1	NUM
iajs-901	114	74	.	.	PUNCT
iajs-901	115	1	(	(	PUNCT
iajs-901	115	2	b	b	NOUN
iajs-901	115	3	)	)	PUNCT
iajs-901	115	4	]	]	PUNCT
iajs-901	115	5	.	.	PUNCT
iajs-901	116	1	m	m	PROPN
iajs-901	116	2	is	be	AUX
iajs-901	116	3	second	second	ADJ
iajs-901	116	4	module	module	NOUN
iajs-901	116	5	if	if	SCONJ
iajs-901	116	6	it	it	PRON
iajs-901	116	7	is	be	AUX
iajs-901	116	8	second	second	ADJ
iajs-901	116	9	submodule	submodule	NOUN
iajs-901	116	10	of	of	ADP
iajs-901	116	11	m	m	PROPN
iajs-901	116	12	remark	remark	NOUN
iajs-901	116	13	(	(	PUNCT
iajs-901	116	14	6	6	NUM
iajs-901	116	15	):	):	PUNCT
iajs-901	116	16	it	it	PRON
iajs-901	116	17	is	be	AUX
iajs-901	116	18	clear	clear	ADJ
iajs-901	116	19	that	that	SCONJ
iajs-901	116	20	n	n	X
iajs-901	116	21	is	be	AUX
iajs-901	116	22	a	a	DET
iajs-901	116	23	second	second	ADJ
iajs-901	116	24	submodule	submodule	NOUN
iajs-901	116	25	of	of	ADP
iajs-901	116	26	an	an	DET
iajs-901	116	27	r	r	NOUN
iajs-901	116	28	-	-	PUNCT
iajs-901	116	29	module	module	NOUN
iajs-901	116	30	iff	iff	NOUN
iajs-901	116	31	for	for	ADP
iajs-901	116	32	every	every	DET
iajs-901	116	33	rr	rr	NUM
iajs-901	116	34	,	,	PUNCT
iajs-901	116	35	r≠0	r≠0	NOUN
iajs-901	116	36	,	,	PUNCT
iajs-901	116	37	either	either	CCONJ
iajs-901	116	38	rnn	rnn	PROPN
iajs-901	116	39	or	or	CCONJ
iajs-901	116	40	rn0	rn0	PROPN
iajs-901	116	41	.	.	PUNCT
iajs-901	117	1	the	the	DET
iajs-901	117	2	following	following	ADJ
iajs-901	117	3	result	result	NOUN
iajs-901	117	4	is	be	AUX
iajs-901	117	5	an	an	DET
iajs-901	117	6	interesting	interesting	ADJ
iajs-901	117	7	characterization	characterization	NOUN
iajs-901	117	8	of	of	ADP
iajs-901	117	9	coprime	coprime	NOUN
iajs-901	117	10	modules	module	NOUN
iajs-901	117	11	.	.	PUNCT
iajs-901	118	1	theorem	theorem	NOUN
iajs-901	118	2	(	(	PUNCT
iajs-901	118	3	7	7	NUM
iajs-901	118	4	):	):	PUNCT
iajs-901	118	5	ihjpas	ihjpas	PROPN
iajs-901	118	6	ibn	ibn	PROPN
iajs-901	118	7	alhaitham	alhaitham	PROPN
iajs-901	119	1	j.	j.	PROPN
iajs-901	120	1	fo	fo	ADP
iajs-901	120	2	r	r	NOUN
iajs-901	120	3	pure	pure	ADJ
iajs-901	120	4	&	&	CCONJ
iajs-901	120	5	appl	appl	PROPN
iajs-901	120	6	.	.	PUNCT
iajs-901	121	1	sc	sc	PROPN
iajs-901	121	2	i.	i.	PROPN
iajs-901	121	3	vo	vo	PROPN
iajs-901	121	4	l.23	l.23	PROPN
iajs-901	121	5	(	(	PUNCT
iajs-901	121	6	3	3	NUM
iajs-901	121	7	)	)	PUNCT
iajs-901	121	8	2010	2010	NUM
iajs-901	121	9	let	let	VERB
iajs-901	121	10	m	m	PRON
iajs-901	121	11	be	be	AUX
iajs-901	121	12	an	an	DET
iajs-901	121	13	r	r	NOUN
iajs-901	121	14	-	-	PUNCT
iajs-901	121	15	module	module	NOUN
iajs-901	121	16	,	,	PUNCT
iajs-901	121	17	then	then	ADV
iajs-901	121	18	m	m	NOUN
iajs-901	121	19	is	be	AUX
iajs-901	121	20	coprime	coprime	ADJ
iajs-901	121	21	r	r	NOUN
iajs-901	121	22	-	-	PUNCT
iajs-901	121	23	module	module	NOUN
iajs-901	121	24	iff	iff	PROPN
iajs-901	121	25	m	m	VERB
iajs-901	121	26	is	be	AUX
iajs-901	121	27	a	a	DET
iajs-901	121	28	second	second	ADJ
iajs-901	121	29	r	r	NOUN
iajs-901	121	30	-	-	PUNCT
iajs-901	121	31	module	module	NOUN
iajs-901	121	32	.	.	PUNCT
iajs-901	122	1	proof	proof	NOUN
iajs-901	122	2	:	:	PUNCT
iajs-901	122	3	(	(	PUNCT
iajs-901	122	4			NOUN
iajs-901	122	5	)	)	PUNCT
iajs-901	122	6	let	let	VERB
iajs-901	122	7	m	m	PRON
iajs-901	122	8	be	be	AUX
iajs-901	122	9	a	a	DET
iajs-901	122	10	coprime	coprime	ADJ
iajs-901	122	11	r	r	NOUN
iajs-901	122	12	-	-	NOUN
iajs-901	122	13	module	module	NOUN
iajs-901	122	14	.	.	PUNCT
iajs-901	123	1	let	let	VERB
iajs-901	123	2	r	r	NOUN
iajs-901	123	3			NOUN
iajs-901	123	4	r	r	NOUN
iajs-901	123	5	such	such	ADJ
iajs-901	123	6	that	that	SCONJ
iajs-901	123	7	r	r	NOUN
iajs-901	123	8			NOUN
iajs-901	123	9	0	0	NUM
iajs-901	123	10	.	.	PUNCT
iajs-901	123	11	suppose	suppose	VERB
iajs-901	123	12	the	the	DET
iajs-901	123	13	homothety	homothety	NOUN
iajs-901	123	14	r	r	PROPN
iajs-901	123	15	*	*	NOUN
iajs-901	123	16	on	on	ADP
iajs-901	123	17	m	m	PROPN
iajs-901	123	18	is	be	AUX
iajs-901	123	19	not	not	PART
iajs-901	123	20	surjective	surjective	ADJ
iajs-901	123	21	,	,	PUNCT
iajs-901	123	22	so	so	SCONJ
iajs-901	123	23	r	r	NOUN
iajs-901	123	24	m	m	NOUN
iajs-901	123	25			NOUN
iajs-901	123	26	m.	m.	NOUN
iajs-901	123	27	let	let	VERB
iajs-901	123	28	r	r	NOUN
iajs-901	123	29	m	m	VERB
iajs-901	123	30	=	=	SYM
iajs-901	123	31	n	n	CCONJ
iajs-901	123	32	,	,	PUNCT
iajs-901	123	33	then	then	ADV
iajs-901	123	34	it	it	PRON
iajs-901	123	35	is	be	AUX
iajs-901	123	36	clear	clear	ADJ
iajs-901	123	37	that	that	SCONJ
iajs-901	123	38	r	r	NOUN
iajs-901	123	39			NOUN
iajs-901	123	40	[	[	X
iajs-901	123	41	n	n	X
iajs-901	123	42	r	r	NOUN
iajs-901	123	43	:	:	PUNCT
iajs-901	123	44	m	m	VERB
iajs-901	123	45	]	]	X
iajs-901	123	46	.	.	PUNCT
iajs-901	124	1	since	since	SCONJ
iajs-901	124	2	m	m	PROPN
iajs-901	124	3	is	be	AUX
iajs-901	124	4	a	a	DET
iajs-901	124	5	coprime	coprime	NOUN
iajs-901	124	6	,	,	PUNCT
iajs-901	124	7	then	then	ADV
iajs-901	124	8	[	[	X
iajs-901	124	9	n	n	X
iajs-901	124	10	r	r	NOUN
iajs-901	124	11	:	:	PUNCT
iajs-901	124	12	m	m	VERB
iajs-901	124	13	]	]	X
iajs-901	124	14	=	=	SYM
iajs-901	124	15	r	r	NOUN
iajs-901	124	16	ann	ann	PROPN
iajs-901	124	17	m.	m.	NOUN
iajs-901	124	18	hence	hence	ADV
iajs-901	124	19	r	r	VERB
iajs-901	124	20			PROPN
iajs-901	124	21	r	r	NOUN
iajs-901	124	22	ann	ann	PROPN
iajs-901	124	23	m	m	PROPN
iajs-901	124	24	,	,	PUNCT
iajs-901	124	25	that	that	ADV
iajs-901	124	26	is	be	AUX
iajs-901	124	27	r	r	NOUN
iajs-901	124	28	m	m	NOUN
iajs-901	124	29	=	=	NOUN
iajs-901	124	30	0	0	NUM
iajs-901	124	31	.	.	PUNCT
iajs-901	125	1	thus	thus	ADV
iajs-901	125	2	r	r	NOUN
iajs-901	125	3	*	*	PUNCT
iajs-901	125	4	=	=	NOUN
iajs-901	125	5	0	0	X
iajs-901	125	6	.	.	PUNCT
iajs-901	125	7	(	(	PUNCT
iajs-901	125	8			NOUN
iajs-901	125	9	)	)	PUNCT
iajs-901	125	10	to	to	PART
iajs-901	125	11	prove	prove	VERB
iajs-901	125	12	r	r	PRON
iajs-901	125	13	ann	ann	PROPN
iajs-901	125	14	(	(	PUNCT
iajs-901	125	15	m	m	PROPN
iajs-901	125	16	/	/	SYM
iajs-901	125	17	n	n	CCONJ
iajs-901	125	18	)	)	PUNCT
iajs-901	125	19	=	=	SYM
iajs-901	126	1	r	r	NOUN
iajs-901	126	2	ann	ann	PROPN
iajs-901	126	3	m	m	PROPN
iajs-901	126	4	for	for	ADP
iajs-901	126	5	every	every	DET
iajs-901	126	6	proper	proper	ADJ
iajs-901	126	7	submodule	submodule	NOUN
iajs-901	126	8	n	n	PROPN
iajs-901	126	9	of	of	ADP
iajs-901	126	10	m	m	PROPN
iajs-901	126	11	.	.	PUNCT
iajs-901	127	1	let	let	VERB
iajs-901	127	2	r	r	NOUN
iajs-901	127	3			NOUN
iajs-901	127	4	[	[	X
iajs-901	127	5	n	n	X
iajs-901	127	6	r	r	NOUN
iajs-901	127	7	:	:	PUNCT
iajs-901	127	8	m	m	VERB
iajs-901	127	9	]	]	X
iajs-901	127	10	,	,	PUNCT
iajs-901	127	11	then	then	ADV
iajs-901	127	12	r	r	NOUN
iajs-901	127	13	m	m	NOUN
iajs-901	127	14			ADJ
iajs-901	127	15	n	n	PRON
iajs-901	127	16			ADJ
iajs-901	127	17	m.	m.	NOUN
iajs-901	127	18	since	since	SCONJ
iajs-901	127	19	m	m	PROPN
iajs-901	127	20	is	be	AUX
iajs-901	127	21	a	a	DET
iajs-901	127	22	second	second	ADJ
iajs-901	127	23	submodule	submodule	NOUN
iajs-901	127	24	,	,	PUNCT
iajs-901	127	25	then	then	ADV
iajs-901	127	26	the	the	DET
iajs-901	127	27	homothety	homothety	NOUN
iajs-901	127	28	r	r	NOUN
iajs-901	127	29	*	*	PUNCT
iajs-901	127	30	on	on	ADP
iajs-901	127	31	m	m	PROPN
iajs-901	127	32	is	be	AUX
iajs-901	127	33	either	either	DET
iajs-901	127	34	zero	zero	NUM
iajs-901	127	35	or	or	CCONJ
iajs-901	127	36	surjective	surjective	ADJ
iajs-901	127	37	.	.	PUNCT
iajs-901	128	1	if	if	SCONJ
iajs-901	128	2	r	r	NOUN
iajs-901	128	3	*	*	VERB
iajs-901	128	4	is	be	AUX
iajs-901	128	5	surjective	surjective	ADJ
iajs-901	128	6	,	,	PUNCT
iajs-901	128	7	then	then	ADV
iajs-901	128	8	r	r	NOUN
iajs-901	128	9	*	*	PUNCT
iajs-901	128	10	(	(	PUNCT
iajs-901	128	11	m	m	NOUN
iajs-901	128	12	)	)	PUNCT
iajs-901	129	1	=	=	SYM
iajs-901	129	2	r	r	NOUN
iajs-901	129	3	m	m	NOUN
iajs-901	129	4	=	=	SYM
iajs-901	129	5	m	m	PROPN
iajs-901	129	6	,	,	PUNCT
iajs-901	129	7	implies	imply	VERB
iajs-901	129	8	m	m	PROPN
iajs-901	129	9			PROPN
iajs-901	129	10	n	n	PRON
iajs-901	129	11	which	which	PRON
iajs-901	129	12	is	be	AUX
iajs-901	129	13	a	a	DET
iajs-901	129	14	contradiction	contradiction	NOUN
iajs-901	129	15	.	.	PUNCT
iajs-901	130	1	thus	thus	ADV
iajs-901	130	2	r	r	NOUN
iajs-901	130	3	m	m	NOUN
iajs-901	130	4	=	=	SYM
iajs-901	130	5	0	0	NUM
iajs-901	131	1	and	and	CCONJ
iajs-901	131	2	so	so	ADV
iajs-901	131	3	r	r	NOUN
iajs-901	131	4			PROPN
iajs-901	131	5	r	r	PROPN
iajs-901	131	6	ann	ann	PROPN
iajs-901	131	7	m.	m.	NOUN
iajs-901	131	8	therefore	therefore	ADV
iajs-901	131	9	r	r	PROPN
iajs-901	131	10	ann	ann	PROPN
iajs-901	131	11			PROPN
iajs-901	131	12			NOUN
iajs-901	131	13	=	=	PROPN
iajs-901	131	14	r	r	NOUN
iajs-901	131	15	ann	ann	PROPN
iajs-901	131	16	m.	m.	NOUN
iajs-901	132	1	■	■	PUNCT
iajs-901	132	2	the	the	DET
iajs-901	132	3	following	following	ADJ
iajs-901	132	4	result	result	NOUN
iajs-901	132	5	is	be	AUX
iajs-901	132	6	an	an	DET
iajs-901	132	7	immediate	immediate	ADJ
iajs-901	132	8	consequence	consequence	NOUN
iajs-901	132	9	of	of	ADP
iajs-901	132	10	theorem	theorem	NOUN
iajs-901	132	11	(	(	PUNCT
iajs-901	132	12	7	7	NUM
iajs-901	132	13	)	)	PUNCT
iajs-901	132	14	and	and	CCONJ
iajs-901	132	15	[	[	X
iajs-901	132	16	16	16	NUM
iajs-901	132	17	,	,	PUNCT
iajs-901	132	18	rem	rem	X
iajs-901	132	19	.	.	PROPN
iajs-901	132	20	and	and	CCONJ
iajs-901	132	21	ex	ex	ADJ
iajs-901	132	22	.	.	NOUN
iajs-901	133	1	1.1.4	1.1.4	NUM
iajs-901	133	2	(	(	PUNCT
iajs-901	133	3	3	3	NUM
iajs-901	133	4	)	)	PUNCT
iajs-901	133	5	)	)	PUNCT
iajs-901	133	6	.	.	PUNCT
iajs-901	134	1	note	note	NOUN
iajs-901	134	2	(	(	PUNCT
iajs-901	134	3	8)	8)	NUM
iajs-901	134	4	:	:	PUNCT
iajs-901	134	5	if	if	SCONJ
iajs-901	134	6	m	m	NOUN
iajs-901	134	7	is	be	AUX
iajs-901	134	8	a	a	DET
iajs-901	134	9	coprime	coprime	ADJ
iajs-901	134	10	r	r	NOUN
iajs-901	134	11	-	-	NOUN
iajs-901	134	12	module	module	NOUN
iajs-901	134	13	,	,	PUNCT
iajs-901	134	14	then	then	ADV
iajs-901	134	15	r	r	PROPN
iajs-901	134	16	ann	ann	PROPN
iajs-901	134	17	m	m	NOUN
iajs-901	134	18	is	be	AUX
iajs-901	134	19	a	a	DET
iajs-901	134	20	prime	prime	ADJ
iajs-901	134	21	ideal	ideal	NOUN
iajs-901	134	22	,	,	PUNCT
iajs-901	134	23	and	and	CCONJ
iajs-901	134	24	r	r	NOUN
iajs-901	134	25	/	/	SYM
iajs-901	134	26	r	r	NOUN
iajs-901	134	27	ann	ann	PROPN
iajs-901	134	28	m	m	NOUN
iajs-901	134	29	is	be	AUX
iajs-901	134	30	an	an	DET
iajs-901	134	31	integral	integral	ADJ
iajs-901	134	32	domain	domain	NOUN
iajs-901	134	33	.	.	PUNCT
iajs-901	135	1	so	so	SCONJ
iajs-901	135	2	that	that	SCONJ
iajs-901	135	3	we	we	PRON
iajs-901	135	4	shall	shall	AUX
iajs-901	135	5	say	say	VERB
iajs-901	135	6	that	that	SCONJ
iajs-901	135	7	m	m	PROPN
iajs-901	135	8	is	be	AUX
iajs-901	135	9	p	p	NOUN
iajs-901	135	10	-	-	PUNCT
iajs-901	135	11	coprime	coprime	NOUN
iajs-901	135	12	if	if	SCONJ
iajs-901	135	13	m	m	NOUN
iajs-901	135	14	is	be	AUX
iajs-901	135	15	coprime	coprime	ADJ
iajs-901	135	16	with	with	ADP
iajs-901	135	17	r	r	NOUN
iajs-901	135	18	ann	ann	NOUN
iajs-901	135	19	m	m	PROPN
iajs-901	135	20	=	=	PROPN
iajs-901	136	1	p.	p.	NOUN
iajs-901	136	2	a	a	DET
iajs-901	136	3	series	series	NOUN
iajs-901	136	4	of	of	ADP
iajs-901	136	5	results	result	NOUN
iajs-901	136	6	follows	follow	VERB
iajs-901	136	7	by	by	ADP
iajs-901	136	8	using	use	VERB
iajs-901	136	9	theorem	theorem	NOUN
iajs-901	136	10	(	(	PUNCT
iajs-901	136	11	7	7	NUM
iajs-901	136	12	)	)	PUNCT
iajs-901	136	13	.	.	PUNCT
iajs-901	137	1	corollary	corollary	ADJ
iajs-901	137	2	(	(	PUNCT
iajs-901	137	3	9	9	NUM
iajs-901	137	4	):	):	PUNCT
iajs-901	137	5	[	[	X
iajs-901	137	6	28	28	NUM
iajs-901	137	7	]	]	X
iajs-901	137	8	let	let	VERB
iajs-901	137	9	m	m	PRON
iajs-901	137	10	be	be	AUX
iajs-901	137	11	an	an	DET
iajs-901	137	12	r	r	NOUN
iajs-901	137	13	-	-	PUNCT
iajs-901	137	14	module	module	NOUN
iajs-901	137	15	,	,	PUNCT
iajs-901	137	16	then	then	ADV
iajs-901	137	17	m	m	NOUN
iajs-901	137	18	is	be	AUX
iajs-901	137	19	coprime	coprime	ADJ
iajs-901	137	20	iff	iff	NOUN
iajs-901	137	21	for	for	ADP
iajs-901	137	22	every	every	DET
iajs-901	137	23	r	r	NOUN
iajs-901	137	24			NOUN
iajs-901	137	25	r	r	NOUN
iajs-901	137	26	,	,	PUNCT
iajs-901	137	27	r	r	NOUN
iajs-901	137	28			NOUN
iajs-901	137	29	0	0	NUM
iajs-901	137	30	,	,	PUNCT
iajs-901	137	31	either	either	CCONJ
iajs-901	138	1	r	r	NOUN
iajs-901	138	2	m	m	NOUN
iajs-901	138	3	=	=	NOUN
iajs-901	138	4	0	0	NUM
iajs-901	138	5	or	or	CCONJ
iajs-901	138	6	r	r	NOUN
iajs-901	138	7	m	m	NOUN
iajs-901	138	8	=	=	VERB
iajs-901	138	9	m	m	ADJ
iajs-901	138	10	(	(	PUNCT
iajs-901	138	11	i.e.	i.e.	X
iajs-901	138	12	m	m	VERB
iajs-901	138	13	is	be	AUX
iajs-901	138	14	a	a	DET
iajs-901	138	15	second	second	ADJ
iajs-901	138	16	module	module	NOUN
iajs-901	138	17	)	)	PUNCT
iajs-901	138	18	.	.	PUNCT
iajs-901	139	1	proof	proof	NOUN
iajs-901	139	2	:	:	PUNCT
iajs-901	139	3	it	it	PRON
iajs-901	139	4	follows	follow	VERB
iajs-901	139	5	directly	directly	ADV
iajs-901	139	6	by	by	ADP
iajs-901	139	7	theorem	theorem	NOUN
iajs-901	139	8	(	(	PUNCT
iajs-901	139	9	7	7	NUM
iajs-901	139	10	)	)	PUNCT
iajs-901	139	11	and	and	CCONJ
iajs-901	139	12	remark	remark	NOUN
iajs-901	139	13	(	(	PUNCT
iajs-901	139	14	6	6	NUM
iajs-901	139	15	)	)	PUNCT
iajs-901	139	16	.	.	PUNCT
iajs-901	140	1	■	■	PUNCT
iajs-901	140	2	corollary	corollary	ADJ
iajs-901	140	3	(	(	PUNCT
iajs-901	140	4	10	10	NUM
iajs-901	140	5	):	):	PUNCT
iajs-901	140	6	let	let	VERB
iajs-901	140	7	m	m	PRON
iajs-901	140	8	be	be	AUX
iajs-901	140	9	an	an	DET
iajs-901	140	10	r	r	NOUN
iajs-901	140	11	-	-	PUNCT
iajs-901	140	12	module	module	NOUN
iajs-901	140	13	,	,	PUNCT
iajs-901	140	14	let	let	VERB
iajs-901	140	15	i	i	PRON
iajs-901	140	16	be	be	AUX
iajs-901	140	17	an	an	DET
iajs-901	140	18	ideal	ideal	NOUN
iajs-901	140	19	of	of	ADP
iajs-901	140	20	r	r	NOUN
iajs-901	140	21	such	such	ADJ
iajs-901	140	22	that	that	SCONJ
iajs-901	140	23	i	i	PRON
iajs-901	140	24			VERB
iajs-901	140	25	r	r	PROPN
iajs-901	140	26	ann	ann	PROPN
iajs-901	140	27	m.	m.	NOUN
iajs-901	140	28	thus	thus	ADV
iajs-901	140	29	m	m	VERB
iajs-901	140	30	is	be	AUX
iajs-901	140	31	a	a	DET
iajs-901	140	32	coprime	coprime	ADJ
iajs-901	140	33	r	r	NOUN
iajs-901	140	34	-	-	PUNCT
iajs-901	140	35	module	module	NOUN
iajs-901	140	36	if	if	SCONJ
iajs-901	140	37	and	and	CCONJ
iajs-901	140	38	only	only	ADV
iajs-901	140	39	if	if	SCONJ
iajs-901	140	40	m	m	NOUN
iajs-901	140	41	is	be	AUX
iajs-901	140	42	a	a	DET
iajs-901	140	43	coprime	coprime	NOUN
iajs-901	140	44	r	r	NOUN
iajs-901	140	45	/	/	SYM
iajs-901	140	46	i	i	NOUN
iajs-901	140	47	-	-	PUNCT
iajs-901	140	48	module	module	NOUN
iajs-901	140	49	.	.	PUNCT
iajs-901	141	1	proof	proof	NOUN
iajs-901	141	2	:	:	PUNCT
iajs-901	141	3	it	it	PRON
iajs-901	141	4	follows	follow	VERB
iajs-901	141	5	by	by	ADP
iajs-901	141	6	theorem	theorem	NOUN
iajs-901	141	7	(	(	PUNCT
iajs-901	141	8	7	7	NUM
iajs-901	141	9	)	)	PUNCT
iajs-901	141	10	and	and	CCONJ
iajs-901	141	11	[	[	X
iajs-901	141	12	16	16	NUM
iajs-901	141	13	,	,	PUNCT
iajs-901	141	14	rem	rem	X
iajs-901	141	15	.	.	PROPN
iajs-901	141	16	and	and	CCONJ
iajs-901	141	17	ex	ex	ADJ
iajs-901	141	18	.	.	PUNCT
iajs-901	142	1	(	(	PUNCT
iajs-901	142	2	1.1.4(8	1.1.4(8	NUM
iajs-901	142	3	)	)	PUNCT
iajs-901	142	4	)	)	PUNCT
iajs-901	142	5	.	.	PUNCT
iajs-901	143	1	■	■	PUNCT
iajs-901	143	2	corollary	corollary	ADJ
iajs-901	143	3	(	(	PUNCT
iajs-901	143	4	11	11	NUM
iajs-901	143	5	):	):	PUNCT
iajs-901	143	6	let	let	VERB
iajs-901	143	7	m	m	PRON
iajs-901	143	8	be	be	AUX
iajs-901	143	9	an	an	DET
iajs-901	143	10	r	r	NOUN
iajs-901	143	11	-	-	PUNCT
iajs-901	143	12	module	module	NOUN
iajs-901	143	13	.	.	PUNCT
iajs-901	144	1	then	then	ADV
iajs-901	144	2	m	m	PROPN
iajs-901	144	3	is	be	AUX
iajs-901	144	4	a	a	DET
iajs-901	144	5	coprime	coprime	ADJ
iajs-901	144	6	r	r	NOUN
iajs-901	144	7	-	-	PUNCT
iajs-901	144	8	module	module	NOUN
iajs-901	144	9	iff	iff	PROPN
iajs-901	144	10	m	m	VERB
iajs-901	144	11	is	be	AUX
iajs-901	144	12	a	a	DET
iajs-901	144	13	coprime	coprime	NOUN
iajs-901	144	14	r	r	NOUN
iajs-901	144	15	=	=	NOUN
iajs-901	144	16	r	r	NOUN
iajs-901	144	17	/	/	SYM
iajs-901	144	18	r	r	NOUN
iajs-901	144	19	ann	ann	PROPN
iajs-901	144	20	m	m	NOUN
iajs-901	144	21	-	-	NOUN
iajs-901	144	22	module	module	NOUN
iajs-901	144	23	.	.	PUNCT
iajs-901	145	1	proof	proof	NOUN
iajs-901	145	2	:	:	PUNCT
iajs-901	145	3	it	it	PRON
iajs-901	145	4	follows	follow	VERB
iajs-901	145	5	directly	directly	ADV
iajs-901	145	6	by	by	ADP
iajs-901	145	7	previous	previous	ADJ
iajs-901	145	8	corollary	corollary	NOUN
iajs-901	145	9	.	.	PUNCT
iajs-901	146	1	■	■	PUNCT
iajs-901	146	2	corollary	corollary	ADJ
iajs-901	146	3	(	(	PUNCT
iajs-901	146	4	12	12	NUM
iajs-901	146	5	):	):	PUNCT
iajs-901	146	6	if	if	SCONJ
iajs-901	146	7	r	r	NOUN
iajs-901	146	8	is	be	AUX
iajs-901	146	9	an	an	DET
iajs-901	146	10	integral	integral	ADJ
iajs-901	146	11	domain	domain	NOUN
iajs-901	146	12	,	,	PUNCT
iajs-901	146	13	then	then	ADV
iajs-901	146	14	q(r	q(r	PROPN
iajs-901	146	15	)	)	PUNCT
iajs-901	147	1	[	[	X
iajs-901	147	2	the	the	DET
iajs-901	147	3	total	total	ADJ
iajs-901	147	4	quotient	quotient	NOUN
iajs-901	147	5	field	field	NOUN
iajs-901	147	6	of	of	ADP
iajs-901	147	7	r	r	NOUN
iajs-901	147	8	]	]	X
iajs-901	147	9	is	be	AUX
iajs-901	147	10	a	a	DET
iajs-901	147	11	coprime	coprime	ADJ
iajs-901	147	12	rmodule	rmodule	NOUN
iajs-901	147	13	.	.	PUNCT
iajs-901	148	1	proof	proof	NOUN
iajs-901	148	2	:	:	PUNCT
iajs-901	148	3	is	be	AUX
iajs-901	148	4	obvious	obvious	ADJ
iajs-901	148	5	.	.	PUNCT
iajs-901	149	1	corollary	corollary	ADJ
iajs-901	149	2	(	(	PUNCT
iajs-901	149	3	13	13	NUM
iajs-901	149	4	):	):	PUNCT
iajs-901	149	5	the	the	DET
iajs-901	149	6	homomorphic	homomorphic	ADJ
iajs-901	149	7	image	image	NOUN
iajs-901	149	8	of	of	ADP
iajs-901	149	9	coprime	coprime	ADJ
iajs-901	149	10	r	r	NOUN
iajs-901	149	11	-	-	PUNCT
iajs-901	149	12	module	module	NOUN
iajs-901	149	13	is	be	AUX
iajs-901	149	14	coprime	coprime	ADJ
iajs-901	149	15	.	.	PUNCT
iajs-901	150	1	proof	proof	NOUN
iajs-901	150	2	:	:	PUNCT
iajs-901	150	3	it	it	PRON
iajs-901	150	4	follows	follow	VERB
iajs-901	150	5	by	by	ADP
iajs-901	150	6	theorem	theorem	NOUN
iajs-901	150	7	(	(	PUNCT
iajs-901	150	8	7	7	NUM
iajs-901	150	9	)	)	PUNCT
iajs-901	150	10	and	and	CCONJ
iajs-901	151	1	[	[	X
iajs-901	151	2	16.rem	16.rem	PROPN
iajs-901	151	3	.	.	PUNCT
iajs-901	151	4	and	and	CCONJ
iajs-901	151	5	ex	ex	ADJ
iajs-901	151	6	.	.	PROPN
iajs-901	151	7	(	(	PUNCT
iajs-901	151	8	1.1.4(5	1.1.4(5	NOUN
iajs-901	151	9	)	)	PUNCT
iajs-901	151	10	]	]	PUNCT
iajs-901	151	11	.	.	PUNCT
iajs-901	152	1	■	■	PUNCT
iajs-901	152	2	note	note	VERB
iajs-901	152	3	that	that	SCONJ
iajs-901	152	4	,	,	PUNCT
iajs-901	152	5	wijayanti	wijayanti	X
iajs-901	152	6	i.e.	i.e.	X
iajs-901	152	7	proved	prove	VERB
iajs-901	152	8	that	that	SCONJ
iajs-901	152	9	if	if	SCONJ
iajs-901	152	10	m	m	NOUN
iajs-901	152	11	is	be	AUX
iajs-901	152	12	a	a	DET
iajs-901	152	13	coprime	coprime	ADJ
iajs-901	152	14	r	r	NOUN
iajs-901	152	15	-	-	PUNCT
iajs-901	152	16	module	module	NOUN
iajs-901	152	17	and	and	CCONJ
iajs-901	152	18	n	n	NOUN
iajs-901	152	19	is	be	AUX
iajs-901	152	20	an	an	DET
iajs-901	152	21	invariant	invariant	ADJ
iajs-901	152	22	submodule	submodule	NOUN
iajs-901	152	23	of	of	ADP
iajs-901	152	24	m	m	PROPN
iajs-901	152	25	,	,	PUNCT
iajs-901	152	26	then	then	ADV
iajs-901	152	27	m	m	VERB
iajs-901	152	28	/	/	SYM
iajs-901	152	29	n	n	PROPN
iajs-901	152	30	is	be	AUX
iajs-901	152	31	a	a	DET
iajs-901	152	32	coprime	coprime	ADJ
iajs-901	152	33	r	r	NOUN
iajs-901	152	34	-	-	NOUN
iajs-901	152	35	module	module	NOUN
iajs-901	152	36	,	,	PUNCT
iajs-901	152	37	see	see	VERB
iajs-901	152	38	[	[	X
iajs-901	152	39	9	9	NUM
iajs-901	152	40	,	,	PUNCT
iajs-901	152	41	prop	prop	NOUN
iajs-901	152	42	.	.	PUNCT
iajs-901	153	1	1.3.8	1.3.8	NUM
iajs-901	153	2	]	]	PUNCT
iajs-901	153	3	.	.	PUNCT
iajs-901	154	1	hence	hence	ADV
iajs-901	154	2	the	the	DET
iajs-901	154	3	following	follow	VERB
iajs-901	154	4	corollary	corollary	NOUN
iajs-901	154	5	is	be	AUX
iajs-901	154	6	a	a	DET
iajs-901	154	7	stronger	strong	ADJ
iajs-901	154	8	result	result	NOUN
iajs-901	154	9	.	.	PUNCT
iajs-901	155	1	corollary	corollary	ADJ
iajs-901	155	2	(	(	PUNCT
iajs-901	155	3	14	14	NUM
iajs-901	155	4	):	):	PUNCT
iajs-901	155	5	if	if	SCONJ
iajs-901	155	6	m	m	NOUN
iajs-901	155	7	is	be	AUX
iajs-901	155	8	a	a	DET
iajs-901	155	9	coprime	coprime	ADJ
iajs-901	155	10	r	r	NOUN
iajs-901	155	11	-	-	NOUN
iajs-901	155	12	module	module	NOUN
iajs-901	155	13	,	,	PUNCT
iajs-901	155	14	then	then	ADV
iajs-901	155	15	m	m	VERB
iajs-901	155	16	/	/	SYM
iajs-901	155	17	w	w	PROPN
iajs-901	155	18	is	be	AUX
iajs-901	155	19	a	a	DET
iajs-901	155	20	coprime	coprime	ADJ
iajs-901	155	21	r	r	NOUN
iajs-901	155	22	-	-	PUNCT
iajs-901	155	23	module	module	NOUN
iajs-901	155	24	.	.	PUNCT
iajs-901	156	1	proof	proof	NOUN
iajs-901	156	2	:	:	PUNCT
iajs-901	156	3	let	let	VERB
iajs-901	156	4			ADJ
iajs-901	156	5	:	:	PUNCT
iajs-901	156	6	m	m	PROPN
iajs-901	156	7			PROPN
iajs-901	156	8	m	m	VERB
iajs-901	156	9	/	/	SYM
iajs-901	156	10	w	w	PROPN
iajs-901	156	11	be	be	VERB
iajs-901	156	12	the	the	DET
iajs-901	156	13	natural	natural	ADJ
iajs-901	156	14	projection	projection	NOUN
iajs-901	156	15	.	.	PUNCT
iajs-901	157	1	hence	hence	ADV
iajs-901	157	2	the	the	DET
iajs-901	157	3	result	result	NOUN
iajs-901	157	4	follows	follow	VERB
iajs-901	157	5	by	by	ADP
iajs-901	157	6	previous	previous	ADJ
iajs-901	157	7	corollary	corollary	NOUN
iajs-901	157	8	.	.	PUNCT
iajs-901	158	1	■	■	PUNCT
iajs-901	158	2	corollary	corollary	ADJ
iajs-901	158	3	(	(	PUNCT
iajs-901	158	4	15	15	NUM
iajs-901	158	5	):	):	PUNCT
iajs-901	158	6	ihjpas	ihjpa	VERB
iajs-901	158	7	ibn	ibn	PROPN
iajs-901	158	8	alhaitham	alhaitham	PROPN
iajs-901	159	1	j.	j.	PROPN
iajs-901	160	1	fo	fo	ADP
iajs-901	160	2	r	r	NOUN
iajs-901	160	3	pure	pure	ADJ
iajs-901	160	4	&	&	CCONJ
iajs-901	160	5	appl	appl	PROPN
iajs-901	160	6	.	.	PUNCT
iajs-901	161	1	sc	sc	PROPN
iajs-901	161	2	i.	i.	PROPN
iajs-901	161	3	vo	vo	PROPN
iajs-901	161	4	l.23	l.23	PROPN
iajs-901	161	5	(	(	PUNCT
iajs-901	161	6	3	3	NUM
iajs-901	161	7	)	)	PUNCT
iajs-901	161	8	2010	2010	NUM
iajs-901	161	9	let	let	VERB
iajs-901	161	10	m	m	PRON
iajs-901	161	11	,	,	PUNCT
iajs-901	161	12	w	w	PROPN
iajs-901	161	13	be	be	AUX
iajs-901	161	14	two	two	NUM
iajs-901	161	15	r	r	NOUN
iajs-901	161	16	-	-	PUNCT
iajs-901	161	17	modules	module	NOUN
iajs-901	161	18	such	such	ADJ
iajs-901	161	19	that	that	SCONJ
iajs-901	161	20	m	m	VERB
iajs-901	161	21			PROPN
iajs-901	161	22	w	w	PROPN
iajs-901	161	23	,	,	PUNCT
iajs-901	161	24	then	then	ADV
iajs-901	161	25	m	m	VERB
iajs-901	161	26	is	be	AUX
iajs-901	161	27	coprime	coprime	ADJ
iajs-901	161	28	iff	iff	PROPN
iajs-901	161	29	w	w	PROPN
iajs-901	161	30	is	be	AUX
iajs-901	161	31	coprime	coprime	ADJ
iajs-901	161	32	.	.	PUNCT
iajs-901	162	1	proof	proof	NOUN
iajs-901	162	2	:	:	PUNCT
iajs-901	162	3	it	it	PRON
iajs-901	162	4	is	be	AUX
iajs-901	162	5	immediate	immediate	ADJ
iajs-901	162	6	by	by	ADP
iajs-901	162	7	corollary	corollary	ADJ
iajs-901	162	8	(	(	PUNCT
iajs-901	162	9	14	14	NUM
iajs-901	162	10	)	)	PUNCT
iajs-901	162	11	.	.	PUNCT
iajs-901	163	1	■	■	PUNCT
iajs-901	163	2	by	by	ADP
iajs-901	163	3	considering	consider	VERB
iajs-901	163	4	(	(	PUNCT
iajs-901	163	5	rem	rem	X
iajs-901	163	6	.	.	NOUN
iajs-901	163	7	and	and	CCONJ
iajs-901	163	8	ex	ex	PROPN
iajs-901	163	9	.	.	NOUN
iajs-901	163	10	3	3	NUM
iajs-901	163	11	(	(	PUNCT
iajs-901	163	12	1	1	NUM
iajs-901	163	13	)	)	PUNCT
iajs-901	163	14	,	,	PUNCT
iajs-901	163	15	(	(	PUNCT
iajs-901	163	16	2	2	NUM
iajs-901	163	17	)	)	PUNCT
iajs-901	163	18	)	)	PUNCT
iajs-901	163	19	a	a	DET
iajs-901	163	20	submodule	submodule	NOUN
iajs-901	163	21	of	of	ADP
iajs-901	163	22	coprime	coprime	ADJ
iajs-901	163	23	r	r	NOUN
iajs-901	163	24	-	-	PUNCT
iajs-901	163	25	module	module	NOUN
iajs-901	163	26	(	(	PUNCT
iajs-901	163	27	second	second	ADJ
iajs-901	163	28	module	module	NOUN
iajs-901	163	29	)	)	PUNCT
iajs-901	163	30	need	need	AUX
iajs-901	163	31	not	not	PART
iajs-901	163	32	be	be	AUX
iajs-901	163	33	coprime	coprime	ADJ
iajs-901	163	34	r	r	NOUN
iajs-901	163	35	-	-	NOUN
iajs-901	163	36	module	module	NOUN
iajs-901	163	37	.	.	PUNCT
iajs-901	164	1	however	however	ADV
iajs-901	164	2	,	,	PUNCT
iajs-901	164	3	in	in	ADP
iajs-901	164	4	the	the	DET
iajs-901	164	5	following	follow	VERB
iajs-901	164	6	proposition	proposition	NOUN
iajs-901	164	7	,	,	PUNCT
iajs-901	164	8	this	this	PRON
iajs-901	164	9	is	be	AUX
iajs-901	164	10	true	true	ADJ
iajs-901	164	11	under	under	ADP
iajs-901	164	12	certain	certain	ADJ
iajs-901	164	13	condition	condition	NOUN
iajs-901	164	14	.	.	PUNCT
iajs-901	165	1	proposition	proposition	NOUN
iajs-901	165	2	(	(	PUNCT
iajs-901	165	3	16	16	NUM
iajs-901	165	4	):	):	PUNCT
iajs-901	165	5	let	let	VERB
iajs-901	165	6	n	n	PRON
iajs-901	165	7	be	be	AUX
iajs-901	165	8	a	a	DET
iajs-901	165	9	non	non	ADJ
iajs-901	165	10	-	-	ADJ
iajs-901	165	11	zero	zero	ADJ
iajs-901	165	12	proper	proper	ADJ
iajs-901	165	13	submodule	submodule	NOUN
iajs-901	165	14	of	of	ADP
iajs-901	165	15	an	an	DET
iajs-901	165	16	r	r	NOUN
iajs-901	165	17	-	-	PUNCT
iajs-901	165	18	module	module	NOUN
iajs-901	165	19	m	m	NOUN
iajs-901	165	20	such	such	ADJ
iajs-901	165	21	that	that	SCONJ
iajs-901	165	22	r	r	NOUN
iajs-901	165	23	m	m	VERB
iajs-901	165	24			PUNCT
iajs-901	165	25	n	n	NOUN
iajs-901	165	26	=	=	SYM
iajs-901	165	27	r	r	NOUN
iajs-901	165	28	n	n	CCONJ
iajs-901	165	29	,	,	PUNCT
iajs-901	165	30	for	for	ADP
iajs-901	165	31	every	every	DET
iajs-901	165	32	r	r	NOUN
iajs-901	165	33			NOUN
iajs-901	165	34	r	r	NOUN
iajs-901	165	35	,	,	PUNCT
iajs-901	165	36	then	then	ADV
iajs-901	165	37	m	m	PROPN
iajs-901	165	38	is	be	AUX
iajs-901	165	39	p	p	ADJ
iajs-901	165	40	-	-	PUNCT
iajs-901	165	41	coprime	coprime	NOUN
iajs-901	165	42	iff	iff	PROPN
iajs-901	165	43	n	n	PROPN
iajs-901	165	44	and	and	CCONJ
iajs-901	165	45	m	m	PROPN
iajs-901	165	46	/	/	SYM
iajs-901	165	47	n	n	PROPN
iajs-901	165	48	are	be	AUX
iajs-901	165	49	p	p	NOUN
iajs-901	165	50	-	-	PUNCT
iajs-901	165	51	coprime	coprime	NOUN
iajs-901	165	52	r	r	NOUN
iajs-901	165	53	-	-	PUNCT
iajs-901	165	54	modules	module	NOUN
iajs-901	165	55	.	.	PUNCT
iajs-901	166	1	proof	proof	NOUN
iajs-901	166	2	:	:	PUNCT
iajs-901	166	3	if	if	SCONJ
iajs-901	166	4	m	m	NOUN
iajs-901	166	5	is	be	AUX
iajs-901	166	6	a	a	DET
iajs-901	166	7	p	p	NOUN
iajs-901	166	8	-	-	PUNCT
iajs-901	166	9	coprime	coprime	NOUN
iajs-901	166	10	r	r	NOUN
iajs-901	166	11	-	-	PUNCT
iajs-901	166	12	module	module	NOUN
iajs-901	166	13	,	,	PUNCT
iajs-901	166	14	then	then	ADV
iajs-901	166	15	m	m	NOUN
iajs-901	166	16	is	be	AUX
iajs-901	166	17	coprime	coprime	ADJ
iajs-901	166	18	with	with	ADP
iajs-901	166	19	r	r	NOUN
iajs-901	166	20	ann	ann	PROPN
iajs-901	166	21	m	m	PROPN
iajs-901	166	22	=	=	PROPN
iajs-901	167	1	p.	p.	NOUN
iajs-901	167	2	thus	thus	ADV
iajs-901	167	3	by	by	ADP
iajs-901	167	4	corollary	corollary	ADJ
iajs-901	167	5	(	(	PUNCT
iajs-901	167	6	14	14	NUM
iajs-901	167	7	)	)	PUNCT
iajs-901	167	8	m	m	VERB
iajs-901	167	9	/	/	SYM
iajs-901	167	10	n	n	PROPN
iajs-901	167	11	is	be	AUX
iajs-901	167	12	coprime	coprime	ADJ
iajs-901	167	13	.	.	PUNCT
iajs-901	168	1	since	since	SCONJ
iajs-901	168	2	p	p	NOUN
iajs-901	168	3	=	=	PROPN
iajs-901	168	4	r	r	NOUN
iajs-901	168	5	ann	ann	PROPN
iajs-901	168	6	m	m	NOUN
iajs-901	168	7	=	=	SYM
iajs-901	168	8	r	r	NOUN
iajs-901	168	9	ann	ann	PROPN
iajs-901	168	10	(	(	PUNCT
iajs-901	168	11	m	m	PROPN
iajs-901	168	12	/	/	SYM
iajs-901	168	13	n	n	CCONJ
iajs-901	168	14	)	)	PUNCT
iajs-901	168	15	,	,	PUNCT
iajs-901	168	16	then	then	ADV
iajs-901	168	17	m	m	VERB
iajs-901	168	18	/n	/n	PUNCT
iajs-901	168	19	is	be	AUX
iajs-901	168	20	p	p	NOUN
iajs-901	168	21	-	-	PUNCT
iajs-901	168	22	coprime	coprime	NOUN
iajs-901	168	23	.	.	PUNCT
iajs-901	169	1	now	now	ADV
iajs-901	169	2	,	,	PUNCT
iajs-901	169	3	to	to	PART
iajs-901	169	4	prove	prove	VERB
iajs-901	169	5	n	n	PRON
iajs-901	169	6	is	be	AUX
iajs-901	169	7	a	a	DET
iajs-901	169	8	p	p	ADJ
iajs-901	169	9	-	-	PUNCT
iajs-901	169	10	coprime	coprime	NOUN
iajs-901	169	11	module	module	NOUN
iajs-901	169	12	.	.	PUNCT
iajs-901	170	1	since	since	SCONJ
iajs-901	170	2	m	m	PROPN
iajs-901	170	3	is	be	AUX
iajs-901	170	4	coprime	coprime	ADJ
iajs-901	170	5	,	,	PUNCT
iajs-901	170	6	then	then	ADV
iajs-901	170	7	for	for	ADP
iajs-901	170	8	any	any	DET
iajs-901	170	9	r	r	NOUN
iajs-901	170	10			NOUN
iajs-901	170	11	r	r	NOUN
iajs-901	170	12	,	,	PUNCT
iajs-901	170	13	r	r	NOUN
iajs-901	170	14			NOUN
iajs-901	170	15	0	0	NUM
iajs-901	170	16	,	,	PUNCT
iajs-901	171	1	either	either	CCONJ
iajs-901	171	2	r	r	NOUN
iajs-901	171	3	m	m	NOUN
iajs-901	171	4	=	=	NOUN
iajs-901	171	5	0	0	NUM
iajs-901	171	6	or	or	CCONJ
iajs-901	171	7	r	r	NOUN
iajs-901	171	8	m	m	NOUN
iajs-901	171	9	=	=	NOUN
iajs-901	171	10	m	m	VERB
iajs-901	171	11	.	.	PUNCT
iajs-901	172	1	if	if	SCONJ
iajs-901	172	2	r	r	NOUN
iajs-901	172	3	m	m	VERB
iajs-901	172	4	=	=	SYM
iajs-901	172	5	0	0	NUM
iajs-901	172	6	,	,	PUNCT
iajs-901	172	7	then	then	ADV
iajs-901	172	8	r	r	NOUN
iajs-901	172	9	m	m	VERB
iajs-901	172	10			PUNCT
iajs-901	172	11	n	n	NOUN
iajs-901	172	12	=	=	SYM
iajs-901	172	13	0	0	PROPN
iajs-901	172	14	.	.	PUNCT
iajs-901	173	1	but	but	CCONJ
iajs-901	173	2	r	r	NOUN
iajs-901	173	3	m	m	VERB
iajs-901	173	4			PUNCT
iajs-901	173	5	n	n	NOUN
iajs-901	173	6	=	=	SYM
iajs-901	173	7	r	r	NOUN
iajs-901	173	8	n	n	CCONJ
iajs-901	173	9	,	,	PUNCT
iajs-901	173	10	then	then	ADV
iajs-901	173	11	r	r	NOUN
iajs-901	173	12	n	n	NOUN
iajs-901	173	13	=	=	SYM
iajs-901	173	14	0	0	PROPN
iajs-901	173	15	.	.	PUNCT
iajs-901	174	1	if	if	SCONJ
iajs-901	174	2	r	r	NOUN
iajs-901	174	3	m	m	VERB
iajs-901	174	4	=	=	VERB
iajs-901	174	5	m	m	ADJ
iajs-901	174	6	,	,	PUNCT
iajs-901	174	7	so	so	ADV
iajs-901	175	1	r	r	NOUN
iajs-901	175	2	m	m	VERB
iajs-901	175	3			PUNCT
iajs-901	175	4	n	n	PROPN
iajs-901	175	5	=	=	SYM
iajs-901	175	6	n.	n.	NOUN
iajs-901	176	1	but	but	CCONJ
iajs-901	176	2	r	r	NOUN
iajs-901	176	3	m	m	VERB
iajs-901	176	4			PUNCT
iajs-901	176	5	n	n	NOUN
iajs-901	176	6	=	=	SYM
iajs-901	176	7	r	r	NOUN
iajs-901	176	8	n	n	CCONJ
iajs-901	176	9	,	,	PUNCT
iajs-901	176	10	hence	hence	ADV
iajs-901	176	11	r	r	NOUN
iajs-901	176	12	n	n	NOUN
iajs-901	176	13	=	=	SYM
iajs-901	176	14	n.	n.	NOUN
iajs-901	176	15	thus	thus	ADV
iajs-901	176	16	n	n	ADV
iajs-901	176	17	is	be	AUX
iajs-901	176	18	second	second	ADJ
iajs-901	176	19	.	.	PUNCT
iajs-901	177	1	to	to	PART
iajs-901	177	2	prove	prove	VERB
iajs-901	177	3	n	n	PRON
iajs-901	177	4	=	=	SYM
iajs-901	177	5	p	p	NOUN
iajs-901	177	6	=	=	PUNCT
iajs-901	177	7	r	r	NOUN
iajs-901	177	8	ann	ann	PROPN
iajs-901	177	9	m.	m.	NOUN
iajs-901	177	10	it	it	PRON
iajs-901	177	11	is	be	AUX
iajs-901	177	12	clear	clear	ADJ
iajs-901	177	13	that	that	SCONJ
iajs-901	177	14	r	r	NOUN
iajs-901	177	15	ann	ann	PROPN
iajs-901	177	16	m	m	PROPN
iajs-901	177	17			PROPN
iajs-901	177	18	r	r	PROPN
iajs-901	177	19	ann	ann	PROPN
iajs-901	177	20	n.	n.	PROPN
iajs-901	177	21	let	let	VERB
iajs-901	177	22	r	r	NOUN
iajs-901	177	23			PROPN
iajs-901	177	24	r	r	PROPN
iajs-901	177	25	ann	ann	PROPN
iajs-901	177	26	n	n	CCONJ
iajs-901	177	27	,	,	PUNCT
iajs-901	177	28	thus	thus	ADV
iajs-901	177	29	r	r	NOUN
iajs-901	177	30	n	n	NOUN
iajs-901	177	31	=	=	SYM
iajs-901	177	32	0	0	PROPN
iajs-901	177	33	.	.	PUNCT
iajs-901	178	1	if	if	SCONJ
iajs-901	178	2	r	r	NOUN
iajs-901	178	3	m	m	VERB
iajs-901	178	4	=	=	SYM
iajs-901	178	5	0	0	NUM
iajs-901	178	6	,	,	PUNCT
iajs-901	178	7	there	there	PRON
iajs-901	178	8	is	be	VERB
iajs-901	178	9	nothing	nothing	PRON
iajs-901	178	10	to	to	PART
iajs-901	178	11	prove	prove	VERB
iajs-901	178	12	.	.	PUNCT
iajs-901	179	1	if	if	SCONJ
iajs-901	179	2	r	r	NOUN
iajs-901	179	3	m	m	VERB
iajs-901	179	4	=	=	VERB
iajs-901	179	5	m	m	ADJ
iajs-901	179	6	,	,	PUNCT
iajs-901	179	7	then	then	ADV
iajs-901	179	8	r	r	NOUN
iajs-901	179	9	m	m	VERB
iajs-901	179	10			PUNCT
iajs-901	179	11	n	n	PROPN
iajs-901	179	12	=	=	SYM
iajs-901	179	13	n.	n.	NOUN
iajs-901	180	1	but	but	CCONJ
iajs-901	180	2	r	r	NOUN
iajs-901	180	3	m	m	VERB
iajs-901	180	4			PUNCT
iajs-901	180	5	n	n	NOUN
iajs-901	180	6	=	=	SYM
iajs-901	180	7	r	r	NOUN
iajs-901	180	8	n.	n.	NOUN
iajs-901	180	9	thus	thus	ADV
iajs-901	180	10	r	r	NOUN
iajs-901	180	11	n	n	NOUN
iajs-901	180	12	=	=	SYM
iajs-901	180	13	n	n	NOUN
iajs-901	180	14	and	and	CCONJ
iajs-901	180	15	so	so	ADV
iajs-901	180	16	n	n	ADV
iajs-901	180	17	=	=	SYM
iajs-901	180	18	(	(	PUNCT
iajs-901	180	19	0	0	NUM
iajs-901	180	20	)	)	PUNCT
iajs-901	180	21	which	which	PRON
iajs-901	180	22	is	be	AUX
iajs-901	180	23	a	a	DET
iajs-901	180	24	contradiction	contradiction	NOUN
iajs-901	180	25	.	.	PUNCT
iajs-901	181	1	conversely	conversely	ADV
iajs-901	181	2	,	,	PUNCT
iajs-901	181	3	if	if	SCONJ
iajs-901	181	4	n	n	PROPN
iajs-901	181	5	and	and	CCONJ
iajs-901	181	6	m	m	PROPN
iajs-901	181	7	/	/	SYM
iajs-901	181	8	n	n	PROPN
iajs-901	181	9	are	be	AUX
iajs-901	181	10	p	p	NOUN
iajs-901	181	11	-	-	PUNCT
iajs-901	181	12	coprime	coprime	NOUN
iajs-901	181	13	.	.	PUNCT
iajs-901	182	1	then	then	ADV
iajs-901	182	2	p	p	X
iajs-901	182	3	=	=	PUNCT
iajs-901	182	4	r	r	NOUN
iajs-901	182	5	ann	ann	PROPN
iajs-901	182	6	n	n	PROPN
iajs-901	182	7	=	=	SYM
iajs-901	182	8	r	r	PROPN
iajs-901	182	9	ann	ann	PROPN
iajs-901	182	10			PROPN
iajs-901	182	11			PROPN
iajs-901	182	12	and	and	CCONJ
iajs-901	182	13	r	r	NOUN
iajs-901	182	14	n	n	NOUN
iajs-901	182	15	=	=	SYM
iajs-901	182	16	n	n	CCONJ
iajs-901	182	17	,	,	PUNCT
iajs-901	182	18	r	r	PROPN
iajs-901	182	19			PROPN
iajs-901	182	20			NOUN
iajs-901	182	21	=	=	SYM
iajs-901	182	22			PROPN
iajs-901	182	23			NOUN
iajs-901	182	24	for	for	ADP
iajs-901	182	25	every	every	DET
iajs-901	182	26	r	r	NOUN
iajs-901	182	27			NOUN
iajs-901	182	28	p.	p.	NOUN
iajs-901	182	29	to	to	PART
iajs-901	182	30	prove	prove	VERB
iajs-901	182	31	m	m	NOUN
iajs-901	182	32	is	be	AUX
iajs-901	182	33	p	p	NOUN
iajs-901	182	34	-	-	PUNCT
iajs-901	182	35	coprime	coprime	NOUN
iajs-901	182	36	.	.	PUNCT
iajs-901	183	1	it	it	PRON
iajs-901	183	2	is	be	AUX
iajs-901	183	3	clear	clear	ADJ
iajs-901	183	4	r	r	NOUN
iajs-901	183	5	ann	ann	PROPN
iajs-901	183	6	m	m	PROPN
iajs-901	183	7			PROPN
iajs-901	183	8	r	r	PROPN
iajs-901	183	9	ann	ann	PROPN
iajs-901	183	10	n	n	PROPN
iajs-901	183	11	=	=	SYM
iajs-901	183	12	r	r	NOUN
iajs-901	183	13	ann	ann	PROPN
iajs-901	183	14			PROPN
iajs-901	183	15			PROPN
iajs-901	184	1	=	=	PROPN
iajs-901	185	1	p.	p.	NOUN
iajs-901	185	2	let	let	VERB
iajs-901	185	3	r	r	NOUN
iajs-901	185	4			PROPN
iajs-901	185	5	p	p	NOUN
iajs-901	185	6	,	,	PUNCT
iajs-901	186	1	so	so	SCONJ
iajs-901	186	2	r	r	NOUN
iajs-901	187	1	n	n	NOUN
iajs-901	187	2	=	=	SYM
iajs-901	187	3	0	0	NUM
iajs-901	187	4	and	and	CCONJ
iajs-901	187	5	r	r	NOUN
iajs-901	187	6	m	m	NOUN
iajs-901	188	1			PROPN
iajs-901	188	2	n.	n.	PROPN
iajs-901	189	1	hence	hence	ADV
iajs-901	190	1	r	r	NOUN
iajs-901	190	2	m	m	NOUN
iajs-901	190	3			PUNCT
iajs-901	190	4	n	n	NOUN
iajs-901	190	5	=	=	SYM
iajs-901	190	6	r	r	NOUN
iajs-901	190	7	m	m	PROPN
iajs-901	190	8	,	,	PUNCT
iajs-901	190	9	but	but	CCONJ
iajs-901	190	10	r	r	NOUN
iajs-901	190	11	m	m	VERB
iajs-901	190	12			PUNCT
iajs-901	190	13	n	n	NOUN
iajs-901	190	14	=	=	SYM
iajs-901	190	15	r	r	NOUN
iajs-901	190	16	n	n	NOUN
iajs-901	190	17	,	,	PUNCT
iajs-901	190	18	so	so	SCONJ
iajs-901	191	1	r	r	NOUN
iajs-901	191	2	m	m	NOUN
iajs-901	191	3	=	=	NOUN
iajs-901	191	4	r	r	NOUN
iajs-901	191	5	n	n	NOUN
iajs-901	191	6	=	=	SYM
iajs-901	191	7	0	0	NUM
iajs-901	191	8	.	.	PUNCT
iajs-901	192	1	thus	thus	ADV
iajs-901	192	2	r	r	NOUN
iajs-901	192	3			PROPN
iajs-901	192	4	r	r	PROPN
iajs-901	192	5	ann	ann	PROPN
iajs-901	192	6	m	m	PROPN
iajs-901	192	7	,	,	PUNCT
iajs-901	192	8	hence	hence	ADV
iajs-901	192	9	p	p	NOUN
iajs-901	192	10	=	=	PUNCT
iajs-901	192	11	r	r	NOUN
iajs-901	192	12	ann	ann	PROPN
iajs-901	192	13	m.	m.	NOUN
iajs-901	192	14	let	let	VERB
iajs-901	192	15	r	r	NOUN
iajs-901	192	16			NOUN
iajs-901	192	17	r	r	NOUN
iajs-901	192	18	ann	ann	PROPN
iajs-901	192	19	m	m	NOUN
iajs-901	192	20	=	=	SYM
iajs-901	192	21	p	p	NOUN
iajs-901	192	22	and	and	CCONJ
iajs-901	192	23	let	let	VERB
iajs-901	192	24	m	m	PROPN
iajs-901	192	25			PROPN
iajs-901	192	26	m	m	PROPN
iajs-901	192	27	,	,	PUNCT
iajs-901	192	28	then	then	ADV
iajs-901	192	29	m	m	VERB
iajs-901	192	30	+	+	ADJ
iajs-901	192	31	n	n	CCONJ
iajs-901	192	32			NOUN
iajs-901	192	33			NUM
iajs-901	192	34			NOUN
iajs-901	192	35	=	=	PUNCT
iajs-901	192	36	r	r	NOUN
iajs-901	192	37			PROPN
iajs-901	192	38			NOUN
iajs-901	192	39	,	,	PUNCT
iajs-901	192	40	hence	hence	ADV
iajs-901	192	41	m	m	VERB
iajs-901	192	42	+	+	SYM
iajs-901	192	43	n	n	CCONJ
iajs-901	192	44	=	=	SYM
iajs-901	192	45	r	r	NOUN
iajs-901	192	46	(	(	PUNCT
iajs-901	192	47	m+n	m+n	NOUN
iajs-901	192	48	)	)	PUNCT
iajs-901	192	49	for	for	ADP
iajs-901	192	50	some	some	DET
iajs-901	192	51	m	m	NOUN
iajs-901	192	52			PROPN
iajs-901	192	53	m.	m.	NOUN
iajs-901	193	1	thus	thus	ADV
iajs-901	193	2	m	m	VERB
iajs-901	193	3	–	–	PUNCT
iajs-901	193	4	r	r	NOUN
iajs-901	193	5	m	m	NOUN
iajs-901	193	6			NOUN
iajs-901	193	7	n	n	PROPN
iajs-901	193	8	=	=	SYM
iajs-901	193	9	r	r	NOUN
iajs-901	193	10	n	n	CCONJ
iajs-901	193	11	,	,	PUNCT
iajs-901	193	12	so	so	ADV
iajs-901	193	13	m	m	VERB
iajs-901	193	14	–	–	PUNCT
iajs-901	193	15	r	r	NOUN
iajs-901	193	16	m	m	NOUN
iajs-901	193	17	=	=	NOUN
iajs-901	193	18	r	r	NOUN
iajs-901	193	19	n	n	NOUN
iajs-901	193	20	for	for	ADP
iajs-901	193	21	some	some	DET
iajs-901	193	22	n	n	ADJ
iajs-901	193	23			NOUN
iajs-901	193	24	n	n	CCONJ
iajs-901	193	25	and	and	CCONJ
iajs-901	193	26	hence	hence	ADV
iajs-901	193	27	m	m	VERB
iajs-901	193	28	=	=	SYM
iajs-901	193	29	r	r	NOUN
iajs-901	193	30	(	(	PUNCT
iajs-901	193	31	m	m	NOUN
iajs-901	193	32	+	+	NOUN
iajs-901	193	33	n	n	CCONJ
iajs-901	193	34	)	)	PUNCT
iajs-901	193	35			NOUN
iajs-901	193	36	r	r	NOUN
iajs-901	193	37	m.	m.	NOUN
iajs-901	193	38	thus	thus	ADV
iajs-901	193	39	m	m	VERB
iajs-901	193	40	=	=	NOUN
iajs-901	194	1	r	r	NOUN
iajs-901	194	2	m	m	VERB
iajs-901	194	3	for	for	ADP
iajs-901	194	4	every	every	DET
iajs-901	194	5	r	r	NOUN
iajs-901	194	6			NOUN
iajs-901	194	7	p	p	NOUN
iajs-901	194	8	,	,	PUNCT
iajs-901	194	9	and	and	CCONJ
iajs-901	194	10	so	so	ADV
iajs-901	194	11	m	m	VERB
iajs-901	194	12	is	be	AUX
iajs-901	194	13	p	p	NOUN
iajs-901	194	14	-	-	PUNCT
iajs-901	194	15	coprime	coprime	NOUN
iajs-901	194	16	.	.	PUNCT
iajs-901	195	1	■	■	PUNCT
iajs-901	195	2	recall	recall	VERB
iajs-901	195	3	that	that	SCONJ
iajs-901	195	4	a	a	DET
iajs-901	195	5	ring	ring	NOUN
iajs-901	195	6	r	r	NOUN
iajs-901	195	7	is	be	AUX
iajs-901	195	8	said	say	VERB
iajs-901	195	9	to	to	PART
iajs-901	195	10	be	be	AUX
iajs-901	195	11	regular	regular	ADJ
iajs-901	195	12	(	(	PUNCT
iajs-901	195	13	in	in	ADP
iajs-901	195	14	sense	sense	NOUN
iajs-901	195	15	of	of	ADP
iajs-901	195	16	von	von	PROPN
iajs-901	195	17	neumann	neumann	PROPN
iajs-901	195	18	)	)	PUNCT
iajs-901	195	19	if	if	SCONJ
iajs-901	195	20	for	for	ADP
iajs-901	195	21	each	each	DET
iajs-901	195	22	x	x	ADJ
iajs-901	195	23			NOUN
iajs-901	195	24	r	r	NOUN
iajs-901	195	25	,	,	PUNCT
iajs-901	195	26	there	there	PRON
iajs-901	195	27	exists	exist	VERB
iajs-901	195	28	a	a	DET
iajs-901	195	29			NOUN
iajs-901	195	30	r	r	NOUN
iajs-901	195	31	such	such	ADJ
iajs-901	195	32	that	that	PRON
iajs-901	195	33	x	x	X
iajs-901	196	1	=	=	SYM
iajs-901	196	2	x2	x2	PROPN
iajs-901	196	3			PROPN
iajs-901	196	4	a	a	PRON
iajs-901	196	5	,	,	PUNCT
iajs-901	196	6	see	see	VERB
iajs-901	196	7	[	[	X
iajs-901	196	8	6	6	NUM
iajs-901	196	9	]	]	PUNCT
iajs-901	196	10	.	.	PUNCT
iajs-901	197	1	corollary	corollary	ADJ
iajs-901	197	2	(	(	PUNCT
iajs-901	197	3	17	17	NUM
iajs-901	197	4	):	):	PUNCT
iajs-901	197	5	let	let	VERB
iajs-901	197	6	m	m	PRON
iajs-901	197	7	be	be	AUX
iajs-901	197	8	a	a	DET
iajs-901	197	9	module	module	NOUN
iajs-901	197	10	over	over	ADP
iajs-901	197	11	a	a	DET
iajs-901	197	12	regular	regular	ADJ
iajs-901	197	13	ring	ring	NOUN
iajs-901	197	14	(	(	PUNCT
iajs-901	197	15	in	in	ADP
iajs-901	197	16	sense	sense	NOUN
iajs-901	197	17	of	of	ADP
iajs-901	197	18	von	von	PROPN
iajs-901	197	19	neumann	neumann	PROPN
iajs-901	197	20	)	)	PUNCT
iajs-901	197	21	and	and	CCONJ
iajs-901	197	22	let	let	VERB
iajs-901	197	23	n	n	PRON
iajs-901	197	24	be	be	AUX
iajs-901	197	25	a	a	DET
iajs-901	197	26	submodule	submodule	NOUN
iajs-901	197	27	of	of	ADP
iajs-901	197	28	m.	m.	NOUN
iajs-901	197	29	then	then	ADV
iajs-901	197	30	m	m	VERB
iajs-901	197	31	is	be	AUX
iajs-901	197	32	a	a	DET
iajs-901	197	33	p	p	ADJ
iajs-901	197	34	-	-	PUNCT
iajs-901	197	35	coprime	coprime	NOUN
iajs-901	197	36	module	module	NOUN
iajs-901	197	37	iff	iff	PROPN
iajs-901	197	38	n	n	PROPN
iajs-901	197	39	and	and	CCONJ
iajs-901	197	40	m	m	VERB
iajs-901	197	41	/n	/n	PUNCT
iajs-901	197	42	are	be	AUX
iajs-901	197	43	p	p	ADJ
iajs-901	197	44	-	-	PUNCT
iajs-901	197	45	coprime	coprime	NOUN
iajs-901	197	46	rmodules	rmodule	NOUN
iajs-901	197	47	.	.	PUNCT
iajs-901	198	1	proof	proof	NOUN
iajs-901	198	2	:	:	PUNCT
iajs-901	198	3	since	since	SCONJ
iajs-901	198	4	r	r	NOUN
iajs-901	198	5	is	be	AUX
iajs-901	198	6	a	a	DET
iajs-901	198	7	regular	regular	ADJ
iajs-901	198	8	ring	ring	NOUN
iajs-901	198	9	,	,	PUNCT
iajs-901	198	10	then	then	ADV
iajs-901	198	11	for	for	ADP
iajs-901	198	12	every	every	DET
iajs-901	198	13	r	r	NOUN
iajs-901	198	14			NOUN
iajs-901	198	15	r	r	NOUN
iajs-901	198	16	,	,	PUNCT
iajs-901	198	17	r	r	NOUN
iajs-901	198	18	m	m	NOUN
iajs-901	198	19			PUNCT
iajs-901	198	20	n	n	NOUN
iajs-901	198	21	=	=	SYM
iajs-901	198	22	r	r	NOUN
iajs-901	198	23	n.	n.	NOUN
iajs-901	198	24	hence	hence	ADV
iajs-901	198	25	the	the	DET
iajs-901	198	26	result	result	NOUN
iajs-901	198	27	follows	follow	VERB
iajs-901	198	28	by	by	ADP
iajs-901	198	29	proposition	proposition	NOUN
iajs-901	198	30	(	(	PUNCT
iajs-901	198	31	16	16	NUM
iajs-901	198	32	)	)	PUNCT
iajs-901	198	33	.	.	PUNCT
iajs-901	199	1	■	■	PUNCT
iajs-901	199	2	now	now	ADV
iajs-901	199	3	,	,	PUNCT
iajs-901	199	4	we	we	PRON
iajs-901	199	5	have	have	VERB
iajs-901	199	6	the	the	DET
iajs-901	199	7	following	follow	VERB
iajs-901	199	8	result	result	NOUN
iajs-901	199	9	.	.	PUNCT
iajs-901	200	1	proposition	proposition	NOUN
iajs-901	200	2	(	(	PUNCT
iajs-901	200	3	18	18	NUM
iajs-901	200	4	):	):	PUNCT
iajs-901	200	5	let	let	VERB
iajs-901	200	6	n	n	PRON
iajs-901	200	7	be	be	AUX
iajs-901	200	8	a	a	DET
iajs-901	200	9	finitely	finitely	ADV
iajs-901	200	10	generated	generate	VERB
iajs-901	200	11	submodule	submodule	NOUN
iajs-901	200	12	of	of	ADP
iajs-901	200	13	an	an	DET
iajs-901	200	14	r	r	NOUN
iajs-901	200	15	-	-	PUNCT
iajs-901	200	16	module	module	NOUN
iajs-901	200	17	m	m	NOUN
iajs-901	200	18	such	such	ADJ
iajs-901	200	19	that	that	SCONJ
iajs-901	200	20	r	r	NOUN
iajs-901	200	21	ann	ann	PROPN
iajs-901	200	22	n	n	PROPN
iajs-901	200	23	=	=	SYM
iajs-901	200	24	r	r	NOUN
iajs-901	200	25	ann	ann	PROPN
iajs-901	200	26	m.	m.	NOUN
iajs-901	200	27	if	if	SCONJ
iajs-901	200	28	n	n	PRON
iajs-901	200	29	is	be	AUX
iajs-901	200	30	a	a	DET
iajs-901	200	31	coprime	coprime	ADJ
iajs-901	200	32	r	r	NOUN
iajs-901	200	33	-	-	NOUN
iajs-901	200	34	module	module	NOUN
iajs-901	200	35	,	,	PUNCT
iajs-901	200	36	then	then	ADV
iajs-901	200	37	m	m	NOUN
iajs-901	200	38	is	be	AUX
iajs-901	200	39	a	a	DET
iajs-901	200	40	coprime	coprime	ADJ
iajs-901	200	41	r	r	NOUN
iajs-901	200	42	-	-	PUNCT
iajs-901	200	43	module	module	NOUN
iajs-901	200	44	.	.	PUNCT
iajs-901	201	1	proof	proof	NOUN
iajs-901	201	2	:	:	PUNCT
iajs-901	201	3	let	let	VERB
iajs-901	201	4	r	r	NOUN
iajs-901	201	5			NOUN
iajs-901	201	6	r	r	NOUN
iajs-901	201	7	and	and	CCONJ
iajs-901	201	8	r	r	NOUN
iajs-901	201	9			NOUN
iajs-901	201	10	r	r	NOUN
iajs-901	201	11	ann	ann	PROPN
iajs-901	201	12	m.	m.	NOUN
iajs-901	201	13	since	since	SCONJ
iajs-901	201	14	n	n	NUM
iajs-901	201	15	is	be	AUX
iajs-901	201	16	a	a	DET
iajs-901	201	17	coprime	coprime	NOUN
iajs-901	201	18	module	module	NOUN
iajs-901	201	19	and	and	CCONJ
iajs-901	201	20	r	r	NOUN
iajs-901	201	21	ann	ann	PROPN
iajs-901	201	22	n	n	PROPN
iajs-901	201	23	=	=	SYM
iajs-901	201	24	r	r	NOUN
iajs-901	201	25	ann	ann	PROPN
iajs-901	201	26	m	m	PROPN
iajs-901	201	27	,	,	PUNCT
iajs-901	201	28	r	r	NOUN
iajs-901	201	29	n	n	NOUN
iajs-901	201	30	=	=	SYM
iajs-901	201	31	n.	n.	NOUN
iajs-901	201	32	but	but	CCONJ
iajs-901	201	33	n	n	PRON
iajs-901	201	34	is	be	AUX
iajs-901	201	35	a	a	DET
iajs-901	201	36	finitely	finitely	ADV
iajs-901	201	37	generated	generate	VERB
iajs-901	201	38	submodule	submodule	NOUN
iajs-901	201	39	,	,	PUNCT
iajs-901	201	40	so	so	ADV
iajs-901	201	41	by	by	ADP
iajs-901	201	42	[	[	X
iajs-901	201	43	25	25	NUM
iajs-901	201	44	,	,	PUNCT
iajs-901	201	45	p.50	p.50	X
iajs-901	201	46	]	]	X
iajs-901	201	47	,	,	PUNCT
iajs-901	201	48	there	there	PRON
iajs-901	201	49	exists	exist	VERB
iajs-901	201	50	r	r	NOUN
iajs-901	201	51			ADJ
iajs-901	201	52			NOUN
iajs-901	201	53	r	r	NOUN
iajs-901	202	1	such	such	ADJ
iajs-901	202	2	that	that	SCONJ
iajs-901	202	3	(	(	PUNCT
iajs-901	202	4	1	1	NUM
iajs-901	202	5	–	–	PUNCT
iajs-901	202	6	r	r	NOUN
iajs-901	202	7	r	r	NOUN
iajs-901	202	8			NOUN
iajs-901	202	9	)	)	PUNCT
iajs-901	202	10	n	n	NOUN
iajs-901	202	11	=	=	SYM
iajs-901	202	12	0	0	X
iajs-901	202	13	.	.	PUNCT
iajs-901	203	1	hence	hence	ADV
iajs-901	203	2	(	(	PUNCT
iajs-901	203	3	1	1	X
iajs-901	203	4	–	–	PUNCT
iajs-901	203	5	r	r	NOUN
iajs-901	203	6	r	r	NOUN
iajs-901	203	7			NOUN
iajs-901	203	8	)	)	PUNCT
iajs-901	203	9	m	m	VERB
iajs-901	203	10	=	=	SYM
iajs-901	203	11	0	0	NUM
iajs-901	203	12	,	,	PUNCT
iajs-901	203	13	and	and	CCONJ
iajs-901	203	14	so	so	ADV
iajs-901	203	15	m	m	VERB
iajs-901	203	16	=	=	SYM
iajs-901	203	17	r	r	NOUN
iajs-901	203	18	m.	m.	NOUN
iajs-901	203	19	therefore	therefore	ADV
iajs-901	203	20	m	m	VERB
iajs-901	203	21	is	be	AUX
iajs-901	203	22	coprime	coprime	ADJ
iajs-901	203	23	.	.	PUNCT
iajs-901	204	1	■	■	PUNCT
iajs-901	204	2	note	note	VERB
iajs-901	204	3	that	that	SCONJ
iajs-901	204	4	,	,	PUNCT
iajs-901	204	5	the	the	DET
iajs-901	204	6	condition	condition	NOUN
iajs-901	204	7	r	r	NOUN
iajs-901	204	8	ann	ann	PROPN
iajs-901	204	9	n	n	PROPN
iajs-901	204	10	=	=	SYM
iajs-901	204	11	r	r	NOUN
iajs-901	204	12	ann	ann	PROPN
iajs-901	204	13	m	m	PROPN
iajs-901	204	14	can	can	AUX
iajs-901	204	15	not	not	PART
iajs-901	204	16	dropped	drop	VERB
iajs-901	204	17	from	from	ADP
iajs-901	204	18	the	the	DET
iajs-901	204	19	previous	previous	ADJ
iajs-901	204	20	proposition	proposition	NOUN
iajs-901	204	21	as	as	ADP
iajs-901	204	22	the	the	DET
iajs-901	204	23	following	follow	VERB
iajs-901	204	24	example	example	NOUN
iajs-901	204	25	shows	show	VERB
iajs-901	204	26	:	:	PUNCT
iajs-901	204	27	ihjpas	ihjpa	VERB
iajs-901	204	28	ibn	ibn	PROPN
iajs-901	204	29	alhaitham	alhaitham	PROPN
iajs-901	205	1	j.	j.	PROPN
iajs-901	206	1	fo	fo	ADP
iajs-901	206	2	r	r	NOUN
iajs-901	206	3	pure	pure	ADJ
iajs-901	206	4	&	&	CCONJ
iajs-901	206	5	appl	appl	PROPN
iajs-901	206	6	.	.	PUNCT
iajs-901	207	1	sc	sc	PROPN
iajs-901	207	2	i.	i.	PROPN
iajs-901	207	3	vo	vo	PROPN
iajs-901	207	4	l.23	l.23	PROPN
iajs-901	207	5	(	(	PUNCT
iajs-901	207	6	3	3	NUM
iajs-901	207	7	)	)	PUNCT
iajs-901	207	8	2010	2010	NUM
iajs-901	207	9	(	(	PUNCT
iajs-901	207	10	2	2	NUM
iajs-901	207	11	)	)	PUNCT
iajs-901	207	12	in	in	ADP
iajs-901	207	13	z6	z6	PROPN
iajs-901	207	14	as	as	SCONJ
iajs-901	207	15	z	z	NOUN
iajs-901	207	16	-	-	PUNCT
iajs-901	207	17	module	module	NOUN
iajs-901	207	18	is	be	AUX
iajs-901	207	19	coprime	coprime	NOUN
iajs-901	207	20	module	module	NOUN
iajs-901	207	21	,	,	PUNCT
iajs-901	207	22	and	and	CCONJ
iajs-901	207	23	ann	ann	PROPN
iajs-901	207	24			PROPN
iajs-901	207	25	(	(	PUNCT
iajs-901	207	26	2	2	X
iajs-901	207	27	)	)	PUNCT
iajs-901	207	28	=	=	SYM
iajs-901	207	29	3z	3z	NUM
iajs-901	207	30			PROPN
iajs-901	207	31	ann	ann	PROPN
iajs-901	207	32			PROPN
iajs-901	207	33	(	(	PUNCT
iajs-901	207	34	z6	z6	PROPN
iajs-901	207	35	)	)	PUNCT
iajs-901	208	1	=	=	SYM
iajs-901	208	2	6z	6z	NOUN
iajs-901	208	3	.	.	PUNCT
iajs-901	209	1	however	however	ADV
iajs-901	209	2	z6	z6	PROPN
iajs-901	209	3	is	be	AUX
iajs-901	209	4	not	not	PART
iajs-901	209	5	coprime	coprime	ADJ
iajs-901	209	6	z	z	NOUN
iajs-901	209	7	-	-	PUNCT
iajs-901	209	8	module	module	NOUN
iajs-901	209	9	,	,	PUNCT
iajs-901	209	10	see	see	VERB
iajs-901	209	11	(	(	PUNCT
iajs-901	209	12	rem	rem	X
iajs-901	209	13	.	.	NOUN
iajs-901	209	14	and	and	CCONJ
iajs-901	209	15	ex	ex	PROPN
iajs-901	209	16	.	.	NOUN
iajs-901	209	17	3	3	NUM
iajs-901	209	18	(	(	PUNCT
iajs-901	209	19	8)	8)	NUM
iajs-901	209	20	)	)	PUNCT
iajs-901	209	21	.	.	PUNCT
iajs-901	210	1	next	next	ADV
iajs-901	210	2	,	,	PUNCT
iajs-901	210	3	we	we	PRON
iajs-901	210	4	can	can	AUX
iajs-901	210	5	give	give	VERB
iajs-901	210	6	the	the	DET
iajs-901	210	7	following	follow	VERB
iajs-901	210	8	proposition	proposition	NOUN
iajs-901	210	9	.	.	PUNCT
iajs-901	211	1	proposition	proposition	NOUN
iajs-901	211	2	(	(	PUNCT
iajs-901	211	3	19	19	NUM
iajs-901	211	4	):	):	PUNCT
iajs-901	211	5	let	let	VERB
iajs-901	211	6	n	n	PRON
iajs-901	211	7	be	be	AUX
iajs-901	211	8	a	a	DET
iajs-901	211	9	finitely	finitely	ADV
iajs-901	211	10	generated	generate	VERB
iajs-901	211	11	submodule	submodule	NOUN
iajs-901	211	12	of	of	ADP
iajs-901	211	13	m	m	PROPN
iajs-901	211	14	and	and	CCONJ
iajs-901	211	15	r	r	PROPN
iajs-901	211	16	ann	ann	PROPN
iajs-901	211	17	m	m	NOUN
iajs-901	211	18	is	be	AUX
iajs-901	211	19	a	a	DET
iajs-901	211	20	prime	prime	ADJ
iajs-901	211	21	ideal	ideal	NOUN
iajs-901	211	22	.	.	PUNCT
iajs-901	212	1	if	if	SCONJ
iajs-901	212	2	n	n	PRON
iajs-901	212	3	is	be	AUX
iajs-901	212	4	a	a	DET
iajs-901	212	5	coprime	coprime	NOUN
iajs-901	212	6	module	module	NOUN
iajs-901	212	7	,	,	PUNCT
iajs-901	212	8	then	then	ADV
iajs-901	212	9	r	r	NOUN
iajs-901	212	10	ann	ann	PROPN
iajs-901	212	11	m	m	NOUN
iajs-901	212	12	=	=	PUNCT
iajs-901	213	1	[	[	X
iajs-901	213	2	n	n	X
iajs-901	213	3	r	r	NOUN
iajs-901	213	4	:	:	PUNCT
iajs-901	213	5	m	m	VERB
iajs-901	213	6	]	]	X
iajs-901	213	7	.	.	PUNCT
iajs-901	214	1	proof	proof	NOUN
iajs-901	214	2	:	:	PUNCT
iajs-901	214	3	let	let	VERB
iajs-901	214	4	r	r	NOUN
iajs-901	214	5			NOUN
iajs-901	214	6	[	[	X
iajs-901	214	7	n	n	X
iajs-901	214	8	r	r	NOUN
iajs-901	214	9	:	:	PUNCT
iajs-901	214	10	m	m	VERB
iajs-901	214	11	]	]	X
iajs-901	214	12	,	,	PUNCT
iajs-901	214	13	then	then	ADV
iajs-901	214	14	r	r	NOUN
iajs-901	214	15	m	m	PROPN
iajs-901	214	16			PROPN
iajs-901	214	17	n.	n.	PROPN
iajs-901	214	18	it	it	PRON
iajs-901	214	19	is	be	AUX
iajs-901	214	20	clear	clear	ADJ
iajs-901	214	21	that	that	SCONJ
iajs-901	214	22	r	r	NOUN
iajs-901	214	23			PROPN
iajs-901	214	24	r	r	PROPN
iajs-901	214	25	ann	ann	PROPN
iajs-901	214	26	n	n	PROPN
iajs-901	214	27	or	or	CCONJ
iajs-901	214	28	r	r	NOUN
iajs-901	214	29			NOUN
iajs-901	214	30	r	r	NOUN
iajs-901	214	31	ann	ann	PROPN
iajs-901	214	32	n.	n.	NOUN
iajs-901	214	33	if	if	SCONJ
iajs-901	214	34	r	r	NOUN
iajs-901	214	35			PROPN
iajs-901	214	36	r	r	PROPN
iajs-901	214	37	ann	ann	PROPN
iajs-901	214	38	n	n	CCONJ
iajs-901	214	39	,	,	PUNCT
iajs-901	214	40	then	then	ADV
iajs-901	214	41	r	r	NOUN
iajs-901	214	42	n	n	NOUN
iajs-901	214	43	=	=	SYM
iajs-901	214	44	0	0	PUNCT
iajs-901	214	45	and	and	CCONJ
iajs-901	214	46	hence	hence	ADV
iajs-901	214	47	r	r	NOUN
iajs-901	214	48	2	2	NUM
iajs-901	214	49	m	m	NOUN
iajs-901	214	50			PROPN
iajs-901	214	51	r	r	NOUN
iajs-901	214	52	n	n	NOUN
iajs-901	214	53	=	=	SYM
iajs-901	214	54	0	0	NUM
iajs-901	214	55	;	;	PUNCT
iajs-901	214	56	that	that	PRON
iajs-901	214	57	is	is	ADV
iajs-901	214	58	r	r	NOUN
iajs-901	214	59	2	2	NUM
iajs-901	214	60			NOUN
iajs-901	214	61	r	r	NOUN
iajs-901	214	62	ann	ann	PROPN
iajs-901	214	63	m.	m.	NOUN
iajs-901	214	64	since	since	SCONJ
iajs-901	214	65	r	r	PROPN
iajs-901	214	66	ann	ann	PROPN
iajs-901	214	67	m	m	NOUN
iajs-901	214	68	is	be	AUX
iajs-901	214	69	prime	prime	ADJ
iajs-901	214	70	,	,	PUNCT
iajs-901	214	71	then	then	ADV
iajs-901	214	72	r	r	NOUN
iajs-901	214	73			PROPN
iajs-901	214	74	r	r	PROPN
iajs-901	214	75	ann	ann	PROPN
iajs-901	214	76	m.	m.	NOUN
iajs-901	214	77	if	if	SCONJ
iajs-901	214	78	r	r	NOUN
iajs-901	214	79			NOUN
iajs-901	214	80	r	r	NOUN
iajs-901	214	81	ann	ann	PROPN
iajs-901	214	82	n	n	CCONJ
iajs-901	214	83	,	,	PUNCT
iajs-901	214	84	then	then	ADV
iajs-901	214	85	r	r	NOUN
iajs-901	214	86	n	n	NOUN
iajs-901	214	87	=	=	SYM
iajs-901	214	88	n	n	PROPN
iajs-901	214	89	because	because	SCONJ
iajs-901	214	90	n	n	X
iajs-901	214	91	is	be	AUX
iajs-901	214	92	a	a	DET
iajs-901	214	93	coprime	coprime	ADJ
iajs-901	214	94	r	r	NOUN
iajs-901	214	95	-	-	NOUN
iajs-901	214	96	module	module	NOUN
iajs-901	214	97	.	.	PUNCT
iajs-901	215	1	on	on	ADP
iajs-901	215	2	the	the	DET
iajs-901	215	3	other	other	ADJ
iajs-901	215	4	hand	hand	NOUN
iajs-901	215	5	,	,	PUNCT
iajs-901	215	6	n	n	PRON
iajs-901	215	7	is	be	AUX
iajs-901	215	8	finitely	finitely	ADV
iajs-901	215	9	generated	generate	VERB
iajs-901	215	10	,	,	PUNCT
iajs-901	215	11	there	there	PRON
iajs-901	215	12	exists	exist	VERB
iajs-901	215	13	r	r	NOUN
iajs-901	215	14			ADJ
iajs-901	215	15			NOUN
iajs-901	215	16	r	r	NOUN
iajs-901	215	17	such	such	ADJ
iajs-901	215	18	that	that	SCONJ
iajs-901	215	19	(	(	PUNCT
iajs-901	215	20	1	1	NUM
iajs-901	215	21	–	–	PUNCT
iajs-901	215	22	r	r	NOUN
iajs-901	215	23	r	r	NOUN
iajs-901	215	24	)	)	PUNCT
iajs-901	215	25	n	n	NOUN
iajs-901	215	26	=	=	SYM
iajs-901	215	27	0	0	NUM
iajs-901	215	28	,	,	PUNCT
iajs-901	215	29	see	see	VERB
iajs-901	215	30	[	[	X
iajs-901	215	31	25	25	NUM
iajs-901	215	32	,	,	PUNCT
iajs-901	215	33	p.50	p.50	X
iajs-901	215	34	]	]	X
iajs-901	215	35	.	.	PUNCT
iajs-901	216	1	it	it	PRON
iajs-901	216	2	follows	follow	VERB
iajs-901	216	3	that	that	SCONJ
iajs-901	216	4	r	r	NOUN
iajs-901	216	5	(	(	PUNCT
iajs-901	216	6	1	1	NUM
iajs-901	216	7	–	–	PUNCT
iajs-901	216	8	r	r	NOUN
iajs-901	216	9	r	r	NOUN
iajs-901	216	10			PROPN
iajs-901	216	11	)	)	PUNCT
iajs-901	216	12	m	m	VERB
iajs-901	216	13			ADJ
iajs-901	216	14	(	(	PUNCT
iajs-901	216	15	1	1	NUM
iajs-901	216	16	–	–	PUNCT
iajs-901	216	17	r	r	NOUN
iajs-901	216	18	r	r	NOUN
iajs-901	216	19			NOUN
iajs-901	216	20	)	)	PUNCT
iajs-901	216	21	n	n	PROPN
iajs-901	216	22	=	=	SYM
iajs-901	216	23	0	0	NUM
iajs-901	216	24	.	.	PUNCT
iajs-901	217	1	thus	thus	ADV
iajs-901	217	2	r	r	NOUN
iajs-901	217	3	(	(	PUNCT
iajs-901	217	4	1	1	NUM
iajs-901	217	5	–	–	PUNCT
iajs-901	217	6	r	r	NOUN
iajs-901	217	7	r	r	NOUN
iajs-901	217	8			PROPN
iajs-901	217	9	)	)	PUNCT
iajs-901	217	10	m	m	VERB
iajs-901	217	11	=	=	NOUN
iajs-901	217	12	0	0	NUM
iajs-901	218	1	and	and	CCONJ
iajs-901	218	2	so	so	ADV
iajs-901	218	3	r	r	NOUN
iajs-901	218	4	(	(	PUNCT
iajs-901	218	5	1	1	NUM
iajs-901	218	6	–	–	PUNCT
iajs-901	218	7	r	r	NOUN
iajs-901	218	8	r	r	NOUN
iajs-901	218	9			ADJ
iajs-901	218	10	)	)	PUNCT
iajs-901	218	11			PROPN
iajs-901	218	12	r	r	PROPN
iajs-901	218	13	ann	ann	PROPN
iajs-901	218	14	m	m	PROPN
iajs-901	218	15	,	,	PUNCT
iajs-901	218	16	which	which	PRON
iajs-901	218	17	implies	imply	VERB
iajs-901	218	18	either	either	CCONJ
iajs-901	218	19	r	r	NOUN
iajs-901	218	20			PROPN
iajs-901	218	21	r	r	NOUN
iajs-901	218	22	ann	ann	PROPN
iajs-901	218	23	m	m	PROPN
iajs-901	218	24	or	or	CCONJ
iajs-901	218	25	(	(	PUNCT
iajs-901	218	26	1	1	NUM
iajs-901	218	27	–	–	PUNCT
iajs-901	218	28	r	r	NOUN
iajs-901	218	29	r	r	NOUN
iajs-901	218	30			ADJ
iajs-901	218	31	)	)	PUNCT
iajs-901	218	32			NOUN
iajs-901	218	33	r	r	PROPN
iajs-901	218	34	ann	ann	PROPN
iajs-901	218	35	m.	m.	NOUN
iajs-901	218	36	if	if	SCONJ
iajs-901	218	37	(	(	PUNCT
iajs-901	218	38	1	1	NUM
iajs-901	218	39	–	–	PUNCT
iajs-901	218	40	r	r	NOUN
iajs-901	218	41	r	r	NOUN
iajs-901	218	42			ADJ
iajs-901	218	43	)	)	PUNCT
iajs-901	218	44			NOUN
iajs-901	218	45	r	r	PROPN
iajs-901	218	46	ann	ann	PROPN
iajs-901	218	47	m	m	PROPN
iajs-901	218	48	,	,	PUNCT
iajs-901	218	49	then	then	ADV
iajs-901	218	50	m	m	VERB
iajs-901	218	51	=	=	SYM
iajs-901	218	52	r	r	NOUN
iajs-901	218	53	m	m	NOUN
iajs-901	218	54			PROPN
iajs-901	218	55	n	n	CCONJ
iajs-901	218	56	which	which	PRON
iajs-901	218	57	is	be	AUX
iajs-901	218	58	a	a	DET
iajs-901	218	59	contradiction	contradiction	NOUN
iajs-901	218	60	.	.	PUNCT
iajs-901	219	1	thus	thus	ADV
iajs-901	219	2	r	r	NOUN
iajs-901	219	3			PROPN
iajs-901	219	4	r	r	NOUN
iajs-901	219	5	ann	ann	PROPN
iajs-901	219	6	m	m	PROPN
iajs-901	219	7	and	and	CCONJ
iajs-901	219	8	so	so	ADV
iajs-901	220	1	[	[	X
iajs-901	220	2	n	n	X
iajs-901	220	3	r	r	NOUN
iajs-901	220	4	:	:	PUNCT
iajs-901	220	5	m	m	VERB
iajs-901	220	6	]	]	X
iajs-901	220	7	=	=	SYM
iajs-901	220	8	r	r	NOUN
iajs-901	220	9	ann	ann	PROPN
iajs-901	220	10	m.	m.	NOUN
iajs-901	220	11	recall	recall	VERB
iajs-901	220	12	that	that	SCONJ
iajs-901	220	13	an	an	DET
iajs-901	220	14	r	r	NOUN
iajs-901	220	15	-	-	PUNCT
iajs-901	220	16	module	module	NOUN
iajs-901	220	17	m	m	NOUN
iajs-901	220	18	is	be	AUX
iajs-901	220	19	said	say	VERB
iajs-901	220	20	to	to	PART
iajs-901	220	21	be	be	AUX
iajs-901	220	22	noetherian	noetherian	ADJ
iajs-901	220	23	if	if	SCONJ
iajs-901	220	24	every	every	DET
iajs-901	220	25	submodule	submodule	NOUN
iajs-901	220	26	of	of	ADP
iajs-901	220	27	m	m	PROPN
iajs-901	220	28	is	be	AUX
iajs-901	220	29	finitely	finitely	ADV
iajs-901	220	30	generated	generate	VERB
iajs-901	220	31	,	,	PUNCT
iajs-901	220	32	see	see	VERB
iajs-901	220	33	[	[	X
iajs-901	220	34	19	19	NUM
iajs-901	220	35	,	,	PUNCT
iajs-901	220	36	prop	prop	NOUN
iajs-901	220	37	.	.	PUNCT
iajs-901	220	38	6.2	6.2	NUM
iajs-901	220	39	,	,	PUNCT
iajs-901	220	40	p.75	p.75	PROPN
iajs-901	220	41	]	]	X
iajs-901	220	42	.	.	PUNCT
iajs-901	221	1	the	the	DET
iajs-901	221	2	following	following	ADJ
iajs-901	221	3	result	result	NOUN
iajs-901	221	4	is	be	AUX
iajs-901	221	5	an	an	DET
iajs-901	221	6	immediate	immediate	ADJ
iajs-901	221	7	consequence	consequence	NOUN
iajs-901	221	8	of	of	ADP
iajs-901	221	9	proposition	proposition	NOUN
iajs-901	221	10	(	(	PUNCT
iajs-901	221	11	19	19	NUM
iajs-901	221	12	)	)	PUNCT
iajs-901	221	13	.	.	PUNCT
iajs-901	222	1	corollary	corollary	ADJ
iajs-901	222	2	(	(	PUNCT
iajs-901	222	3	20	20	NUM
iajs-901	222	4	):	):	PUNCT
iajs-901	222	5	let	let	VERB
iajs-901	222	6	m	m	PRON
iajs-901	222	7	be	be	AUX
iajs-901	222	8	a	a	DET
iajs-901	222	9	noetherian	noetherian	ADJ
iajs-901	222	10	r	r	NOUN
iajs-901	222	11	-	-	PUNCT
iajs-901	222	12	module	module	NOUN
iajs-901	222	13	such	such	ADJ
iajs-901	222	14	that	that	SCONJ
iajs-901	222	15	r	r	NOUN
iajs-901	222	16	ann	ann	PROPN
iajs-901	222	17	m	m	NOUN
iajs-901	222	18	is	be	AUX
iajs-901	222	19	a	a	DET
iajs-901	222	20	prime	prime	ADJ
iajs-901	222	21	ideal	ideal	NOUN
iajs-901	222	22	of	of	ADP
iajs-901	222	23	r	r	NOUN
iajs-901	222	24	,	,	PUNCT
iajs-901	222	25	if	if	SCONJ
iajs-901	222	26	every	every	DET
iajs-901	222	27	submodule	submodule	NOUN
iajs-901	222	28	of	of	ADP
iajs-901	222	29	m	m	PROPN
iajs-901	222	30	is	be	AUX
iajs-901	222	31	a	a	DET
iajs-901	222	32	coprime	coprime	ADJ
iajs-901	222	33	r	r	NOUN
iajs-901	222	34	-	-	NOUN
iajs-901	222	35	module	module	NOUN
iajs-901	222	36	,	,	PUNCT
iajs-901	222	37	then	then	ADV
iajs-901	222	38	m	m	NOUN
iajs-901	222	39	is	be	AUX
iajs-901	222	40	coprime	coprime	ADJ
iajs-901	222	41	.	.	PUNCT
iajs-901	223	1	recall	recall	VERB
iajs-901	223	2	that	that	SCONJ
iajs-901	223	3	a	a	DET
iajs-901	223	4	submodule	submodule	NOUN
iajs-901	223	5	n	n	PROPN
iajs-901	223	6	of	of	ADP
iajs-901	223	7	an	an	DET
iajs-901	223	8	rmodule	rmodule	NOUN
iajs-901	223	9	m	m	VERB
iajs-901	223	10	is	be	AUX
iajs-901	223	11	called	call	VERB
iajs-901	223	12	primary	primary	ADJ
iajs-901	223	13	if	if	SCONJ
iajs-901	223	14	for	for	ADP
iajs-901	223	15	every	every	DET
iajs-901	223	16	rr	rr	NOUN
iajs-901	223	17	,	,	PUNCT
iajs-901	223	18	xm	xm	NOUN
iajs-901	223	19	such	such	ADJ
iajs-901	223	20	that	that	SCONJ
iajs-901	223	21	rxn	rxn	NOUN
iajs-901	223	22	,	,	PUNCT
iajs-901	223	23	then	then	ADV
iajs-901	223	24	either	either	CCONJ
iajs-901	223	25	xn	xn	PROPN
iajs-901	223	26	or	or	CCONJ
iajs-901	223	27	r	r	NOUN
iajs-901	223	28			NOUN
iajs-901	223	29	r	r	NOUN
iajs-901	223	30	[	[	PUNCT
iajs-901	223	31	:	:	PUNCT
iajs-901	223	32	]	]	PUNCT
iajs-901	223	33			NOUN
iajs-901	223	34			PROPN
iajs-901	223	35	=	=	PUNCT
iajs-901	223	36	{	{	PUNCT
iajs-901	223	37	s	s	PROPN
iajs-901	223	38			NOUN
iajs-901	223	39	r	r	NOUN
iajs-901	223	40	;	;	PUNCT
iajs-901	223	41	s	s	PART
iajs-901	223	42	n	n	PRON
iajs-901	223	43			NOUN
iajs-901	223	44	r	r	NOUN
iajs-901	223	45	[	[	PUNCT
iajs-901	223	46	:	:	PUNCT
iajs-901	223	47	]	]	PUNCT
iajs-901	223	48			NOUN
iajs-901	223	49			VERB
iajs-901	223	50	for	for	ADP
iajs-901	223	51	some	some	DET
iajs-901	223	52	n	n	DET
iajs-901	223	53			NOUN
iajs-901	223	54	z+	z+	NUM
iajs-901	223	55	}	}	PUNCT
iajs-901	223	56	,	,	PUNCT
iajs-901	223	57	see	see	VERB
iajs-901	223	58	[	[	X
iajs-901	223	59	20	20	NUM
iajs-901	223	60	]	]	PUNCT
iajs-901	223	61	.	.	PUNCT
iajs-901	224	1	we	we	PRON
iajs-901	224	2	know	know	VERB
iajs-901	224	3	that	that	SCONJ
iajs-901	224	4	every	every	DET
iajs-901	224	5	prime	prime	ADJ
iajs-901	224	6	submodule	submodule	NOUN
iajs-901	224	7	is	be	AUX
iajs-901	224	8	primary	primary	ADJ
iajs-901	224	9	;	;	PUNCT
iajs-901	224	10	however	however	ADV
iajs-901	224	11	the	the	DET
iajs-901	224	12	converse	converse	NOUN
iajs-901	224	13	is	be	AUX
iajs-901	224	14	not	not	PART
iajs-901	224	15	true	true	ADJ
iajs-901	224	16	in	in	ADP
iajs-901	224	17	general	general	ADJ
iajs-901	224	18	.	.	PUNCT
iajs-901	225	1	the	the	DET
iajs-901	225	2	following	follow	VERB
iajs-901	225	3	result	result	NOUN
iajs-901	225	4	shows	show	VERB
iajs-901	225	5	that	that	SCONJ
iajs-901	225	6	the	the	DET
iajs-901	225	7	two	two	NUM
iajs-901	225	8	concepts	concept	NOUN
iajs-901	225	9	are	be	AUX
iajs-901	225	10	equivalent	equivalent	ADJ
iajs-901	225	11	in	in	ADP
iajs-901	225	12	the	the	DET
iajs-901	225	13	class	class	NOUN
iajs-901	225	14	of	of	ADP
iajs-901	225	15	coprime	coprime	NOUN
iajs-901	225	16	modules	module	NOUN
iajs-901	225	17	.	.	PUNCT
iajs-901	226	1	proposition	proposition	NOUN
iajs-901	226	2	(	(	PUNCT
iajs-901	226	3	21	21	NUM
iajs-901	226	4	):	):	PUNCT
iajs-901	226	5	let	let	VERB
iajs-901	226	6	m	m	PRON
iajs-901	226	7	be	be	AUX
iajs-901	226	8	a	a	DET
iajs-901	226	9	coprime	coprime	ADJ
iajs-901	226	10	r	r	NOUN
iajs-901	226	11	-	-	PUNCT
iajs-901	226	12	module	module	NOUN
iajs-901	226	13	,	,	PUNCT
iajs-901	226	14	let	let	VERB
iajs-901	226	15	n	n	PRON
iajs-901	226	16	be	be	AUX
iajs-901	226	17	a	a	DET
iajs-901	226	18	proper	proper	ADJ
iajs-901	226	19	submodule	submodule	NOUN
iajs-901	226	20	of	of	ADP
iajs-901	226	21	m	m	PROPN
iajs-901	226	22	,	,	PUNCT
iajs-901	226	23	then	then	ADV
iajs-901	226	24	n	n	PROPN
iajs-901	226	25	is	be	AUX
iajs-901	226	26	primary	primary	ADJ
iajs-901	226	27	iff	iff	PROPN
iajs-901	226	28	n	n	PART
iajs-901	226	29	is	be	AUX
iajs-901	226	30	prime	prime	ADJ
iajs-901	226	31	.	.	PUNCT
iajs-901	227	1	proof	proof	NOUN
iajs-901	227	2	:	:	PUNCT
iajs-901	227	3	let	let	VERB
iajs-901	227	4	n	n	PRON
iajs-901	227	5	be	be	AUX
iajs-901	227	6	a	a	DET
iajs-901	227	7	primary	primary	ADJ
iajs-901	227	8	submodule	submodule	NOUN
iajs-901	227	9	of	of	ADP
iajs-901	227	10	m	m	PROPN
iajs-901	227	11	,	,	PUNCT
iajs-901	227	12	since	since	SCONJ
iajs-901	227	13	m	m	PROPN
iajs-901	227	14	is	be	AUX
iajs-901	227	15	coprime	coprime	ADJ
iajs-901	227	16	,	,	PUNCT
iajs-901	227	17	then	then	ADV
iajs-901	227	18	r	r	NOUN
iajs-901	227	19	ann	ann	PROPN
iajs-901	227	20	m	m	NOUN
iajs-901	227	21	=	=	PUNCT
iajs-901	228	1	[	[	X
iajs-901	228	2	n	n	X
iajs-901	228	3	r	r	NOUN
iajs-901	228	4	:	:	PUNCT
iajs-901	228	5	m	m	VERB
iajs-901	228	6	]	]	X
iajs-901	228	7	,	,	PUNCT
iajs-901	228	8	so	so	CCONJ
iajs-901	229	1	[	[	X
iajs-901	229	2	n	n	X
iajs-901	229	3	r	r	NOUN
iajs-901	229	4	:	:	PUNCT
iajs-901	229	5	m	m	VERB
iajs-901	229	6	]	]	X
iajs-901	229	7	is	be	AUX
iajs-901	229	8	a	a	DET
iajs-901	229	9	prime	prime	ADJ
iajs-901	229	10	ideal	ideal	NOUN
iajs-901	229	11	,	,	PUNCT
iajs-901	229	12	hence	hence	ADV
iajs-901	229	13	by	by	ADP
iajs-901	229	14	[	[	X
iajs-901	229	15	14	14	NUM
iajs-901	229	16	,	,	PUNCT
iajs-901	229	17	prop	prop	NOUN
iajs-901	229	18	.	.	PUNCT
iajs-901	230	1	2.10	2.10	NUM
iajs-901	230	2	,	,	PUNCT
iajs-901	230	3	ch.1	ch.1	PROPN
iajs-901	230	4	]	]	PUNCT
iajs-901	230	5	,	,	PUNCT
iajs-901	230	6	we	we	PRON
iajs-901	230	7	have	have	VERB
iajs-901	230	8	n	n	ADV
iajs-901	230	9	is	be	AUX
iajs-901	230	10	prime	prime	ADJ
iajs-901	230	11	.	.	PUNCT
iajs-901	231	1	■	■	PUNCT
iajs-901	231	2	s.yassem	s.yassem	NOUN
iajs-901	231	3	in	in	ADP
iajs-901	231	4	[	[	X
iajs-901	231	5	5	5	NUM
iajs-901	231	6	]	]	PUNCT
iajs-901	231	7	gave	give	VERB
iajs-901	231	8	the	the	DET
iajs-901	231	9	following	following	ADJ
iajs-901	231	10	result	result	NOUN
iajs-901	231	11	.	.	PUNCT
iajs-901	232	1	we	we	PRON
iajs-901	232	2	give	give	VERB
iajs-901	232	3	its	its	PRON
iajs-901	232	4	proof	proof	NOUN
iajs-901	232	5	for	for	ADP
iajs-901	232	6	the	the	DET
iajs-901	232	7	sake	sake	NOUN
iajs-901	232	8	of	of	ADP
iajs-901	232	9	completeness	completeness	NOUN
iajs-901	232	10	.	.	PUNCT
iajs-901	233	1	recall	recall	VERB
iajs-901	233	2	that	that	SCONJ
iajs-901	233	3	a	a	DET
iajs-901	233	4	submodule	submodule	NOUN
iajs-901	233	5	n	n	PROPN
iajs-901	233	6	of	of	ADP
iajs-901	233	7	an	an	DET
iajs-901	233	8	r	r	NOUN
iajs-901	233	9	-	-	PUNCT
iajs-901	233	10	module	module	NOUN
iajs-901	233	11	m	m	NOUN
iajs-901	233	12	is	be	AUX
iajs-901	233	13	called	call	VERB
iajs-901	233	14	secondary	secondary	ADJ
iajs-901	233	15	submodule	submodule	NOUN
iajs-901	233	16	if	if	SCONJ
iajs-901	233	17	for	for	ADP
iajs-901	233	18	each	each	DET
iajs-901	233	19	rr	rr	NUM
iajs-901	233	20	,	,	PUNCT
iajs-901	233	21	the	the	DET
iajs-901	233	22	homothety	homothety	NOUN
iajs-901	233	23	r	r	PROPN
iajs-901	233	24	*	*	NOUN
iajs-901	233	25	on	on	ADP
iajs-901	233	26	n	n	X
iajs-901	233	27	is	be	AUX
iajs-901	233	28	either	either	CCONJ
iajs-901	233	29	surjective	surjective	ADJ
iajs-901	233	30	or	or	CCONJ
iajs-901	233	31	nilpotent	nilpotent	ADJ
iajs-901	233	32	,	,	PUNCT
iajs-901	233	33	where	where	SCONJ
iajs-901	233	34	r	r	NOUN
iajs-901	233	35	*	*	NOUN
iajs-901	233	36	is	be	AUX
iajs-901	233	37	nilpotent	nilpotent	ADJ
iajs-901	233	38	if	if	SCONJ
iajs-901	233	39	there	there	PRON
iajs-901	233	40	exist	exist	VERB
iajs-901	233	41	kz+	kz+	NOUN
iajs-901	233	42	such	such	ADJ
iajs-901	233	43	that	that	SCONJ
iajs-901	233	44	(	(	PUNCT
iajs-901	233	45	r	r	NOUN
iajs-901	233	46	*	*	NOUN
iajs-901	233	47	)	)	PUNCT
iajs-901	234	1	=	=	SYM
iajs-901	234	2	0,see[21	0,see[21	NOUN
iajs-901	234	3	]	]	PUNCT
iajs-901	234	4	.	.	PUNCT
iajs-901	235	1	proposition	proposition	NOUN
iajs-901	235	2	(	(	PUNCT
iajs-901	235	3	22	22	NUM
iajs-901	235	4	):	):	PUNCT
iajs-901	235	5	let	let	VERB
iajs-901	235	6	n	n	PRON
iajs-901	235	7	be	be	AUX
iajs-901	235	8	a	a	DET
iajs-901	235	9	submodule	submodule	NOUN
iajs-901	235	10	of	of	ADP
iajs-901	235	11	an	an	DET
iajs-901	235	12	r	r	NOUN
iajs-901	235	13	-	-	PUNCT
iajs-901	235	14	module	module	NOUN
iajs-901	235	15	m	m	NOUN
iajs-901	235	16	such	such	ADJ
iajs-901	235	17	that	that	SCONJ
iajs-901	235	18	m	m	PROPN
iajs-901	235	19	is	be	AUX
iajs-901	235	20	p	p	ADJ
iajs-901	235	21	-	-	PUNCT
iajs-901	235	22	second	second	ADJ
iajs-901	235	23	(	(	PUNCT
iajs-901	235	24	pcoprime	pcoprime	NOUN
iajs-901	235	25	)	)	PUNCT
iajs-901	235	26	,	,	PUNCT
iajs-901	235	27	then	then	ADV
iajs-901	235	28	n	n	X
iajs-901	235	29	is	be	AUX
iajs-901	235	30	p	p	ADJ
iajs-901	235	31	-	-	PUNCT
iajs-901	235	32	secondary	secondary	ADJ
iajs-901	235	33	iff	iff	PROPN
iajs-901	235	34	n	n	PART
iajs-901	235	35	is	be	AUX
iajs-901	235	36	p	p	NOUN
iajs-901	235	37	-	-	PUNCT
iajs-901	235	38	second	second	NOUN
iajs-901	235	39	(	(	PUNCT
iajs-901	235	40	p	p	NOUN
iajs-901	235	41	-	-	PUNCT
iajs-901	235	42	coprime	coprime	NOUN
iajs-901	235	43	)	)	PUNCT
iajs-901	235	44	.	.	PUNCT
iajs-901	236	1	proof	proof	NOUN
iajs-901	236	2	:	:	PUNCT
iajs-901	236	3	(	(	PUNCT
iajs-901	236	4			NOUN
iajs-901	236	5	)	)	PUNCT
iajs-901	236	6	since	since	SCONJ
iajs-901	236	7	m	m	PROPN
iajs-901	236	8	is	be	AUX
iajs-901	236	9	p	p	NOUN
iajs-901	236	10	-	-	PUNCT
iajs-901	236	11	coprime	coprime	NOUN
iajs-901	236	12	,	,	PUNCT
iajs-901	236	13	r	r	NOUN
iajs-901	236	14	ann	ann	PROPN
iajs-901	236	15	m	m	PROPN
iajs-901	236	16	=	=	SYM
iajs-901	237	1	p.	p.	NOUN
iajs-901	237	2	suppose	suppose	VERB
iajs-901	237	3	that	that	SCONJ
iajs-901	237	4	n	n	PRON
iajs-901	237	5	is	be	AUX
iajs-901	237	6	p	p	NOUN
iajs-901	237	7	-	-	PUNCT
iajs-901	237	8	secondary	secondary	ADJ
iajs-901	237	9	,	,	PUNCT
iajs-901	237	10	then	then	ADV
iajs-901	237	11	p	p	NOUN
iajs-901	237	12	=	=	PUNCT
iajs-901	237	13	r	r	PROPN
iajs-901	237	14	ann	ann	PROPN
iajs-901	237	15	.	.	PUNCT
iajs-901	238	1	thus	thus	ADV
iajs-901	238	2	p	p	X
iajs-901	238	3	=	=	PUNCT
iajs-901	238	4	r	r	NOUN
iajs-901	238	5	ann	ann	PUNCT
iajs-901	238	6	=	=	SYM
iajs-901	238	7	r	r	NOUN
iajs-901	238	8	ann	ann	PROPN
iajs-901	238	9			PROPN
iajs-901	238	10	.	.	PUNCT
iajs-901	239	1	to	to	PART
iajs-901	239	2	prove	prove	VERB
iajs-901	239	3	n	n	PRON
iajs-901	239	4	is	be	AUX
iajs-901	239	5	p	p	NOUN
iajs-901	239	6	-	-	PUNCT
iajs-901	239	7	second	second	NOUN
iajs-901	239	8	.	.	PUNCT
iajs-901	240	1	since	since	SCONJ
iajs-901	240	2	n	n	NUM
iajs-901	240	3	is	be	AUX
iajs-901	240	4	a	a	DET
iajs-901	240	5	submodule	submodule	NOUN
iajs-901	240	6	of	of	ADP
iajs-901	240	7	ihjpas	ihjpas	PROPN
iajs-901	240	8	ibn	ibn	PROPN
iajs-901	240	9	alhaitham	alhaitham	PROPN
iajs-901	241	1	j.	j.	PROPN
iajs-901	242	1	fo	fo	ADP
iajs-901	242	2	r	r	NOUN
iajs-901	242	3	pure	pure	ADJ
iajs-901	242	4	&	&	CCONJ
iajs-901	242	5	appl	appl	PROPN
iajs-901	242	6	.	.	PUNCT
iajs-901	243	1	sc	sc	PROPN
iajs-901	243	2	i.	i.	PROPN
iajs-901	243	3	vo	vo	PROPN
iajs-901	244	1	l.23	l.23	PROPN
iajs-901	244	2	(	(	PUNCT
iajs-901	244	3	3	3	NUM
iajs-901	244	4	)	)	PUNCT
iajs-901	244	5	2010	2010	NUM
iajs-901	244	6	m	m	NOUN
iajs-901	244	7	,	,	PUNCT
iajs-901	244	8	then	then	ADV
iajs-901	244	9	p	p	NOUN
iajs-901	244	10	=	=	SYM
iajs-901	244	11	r	r	NOUN
iajs-901	244	12	ann	ann	PROPN
iajs-901	244	13			PROPN
iajs-901	244	14	r	r	PROPN
iajs-901	244	15	ann	ann	PROPN
iajs-901	244	16	n	n	CCONJ
iajs-901	244	17	,	,	PUNCT
iajs-901	244	18	but	but	CCONJ
iajs-901	244	19	r	r	NOUN
iajs-901	244	20	ann	ann	PROPN
iajs-901	245	1	n	n	CCONJ
iajs-901	245	2			PROPN
iajs-901	245	3	r	r	X
iajs-901	245	4	ann	ann	PUNCT
iajs-901	245	5	=	=	PUNCT
iajs-901	245	6	p	p	NOUN
iajs-901	245	7	which	which	PRON
iajs-901	245	8	is	be	AUX
iajs-901	245	9	a	a	DET
iajs-901	245	10	prime	prime	ADJ
iajs-901	245	11	ideal	ideal	NOUN
iajs-901	245	12	.	.	PUNCT
iajs-901	246	1	thus	thus	ADV
iajs-901	246	2	by	by	ADP
iajs-901	246	3	proposition	proposition	NOUN
iajs-901	246	4	(	(	PUNCT
iajs-901	246	5	7	7	NUM
iajs-901	246	6	)	)	PUNCT
iajs-901	246	7	n	n	PRON
iajs-901	246	8	is	be	AUX
iajs-901	246	9	a	a	DET
iajs-901	246	10	second	second	ADJ
iajs-901	246	11	submodule	submodule	NOUN
iajs-901	246	12	with	with	ADP
iajs-901	246	13	p	p	NOUN
iajs-901	246	14	=	=	PUNCT
iajs-901	246	15	r	r	NOUN
iajs-901	246	16	ann	ann	PROPN
iajs-901	246	17	n	n	CCONJ
iajs-901	246	18	;	;	PUNCT
iajs-901	246	19	that	that	PRON
iajs-901	246	20	is	be	AUX
iajs-901	246	21	n	n	PRON
iajs-901	246	22	is	be	AUX
iajs-901	246	23	p	p	ADJ
iajs-901	246	24	-	-	PUNCT
iajs-901	246	25	second	second	NOUN
iajs-901	246	26	submodule	submodule	NOUN
iajs-901	246	27	.	.	PUNCT
iajs-901	247	1	the	the	DET
iajs-901	247	2	converse	converse	NOUN
iajs-901	247	3	is	be	AUX
iajs-901	247	4	obvious	obvious	ADJ
iajs-901	247	5	by	by	ADP
iajs-901	247	6	[	[	X
iajs-901	247	7	15	15	NUM
iajs-901	247	8	,	,	PUNCT
iajs-901	247	9	rem	rem	X
iajs-901	247	10	.	.	PROPN
iajs-901	247	11	and	and	CCONJ
iajs-901	247	12	ex	ex	X
iajs-901	247	13	.	.	NOUN
iajs-901	248	1	1.2.2	1.2.2	NUM
iajs-901	248	2	(	(	PUNCT
iajs-901	248	3	1	1	NUM
iajs-901	248	4	)	)	PUNCT
iajs-901	248	5	]	]	PUNCT
iajs-901	248	6	.	.	PUNCT
iajs-901	249	1	■	■	PUNCT
iajs-901	249	2	recall	recall	VERB
iajs-901	249	3	that	that	SCONJ
iajs-901	249	4	if	if	SCONJ
iajs-901	249	5	m	m	NOUN
iajs-901	249	6	is	be	AUX
iajs-901	249	7	an	an	DET
iajs-901	249	8	r	r	NOUN
iajs-901	249	9	-	-	PUNCT
iajs-901	249	10	module	module	NOUN
iajs-901	249	11	,	,	PUNCT
iajs-901	249	12	then	then	ADV
iajs-901	249	13	j(m	j(m	PROPN
iajs-901	249	14	)	)	PUNCT
iajs-901	249	15	is	be	AUX
iajs-901	249	16	the	the	DET
iajs-901	249	17	intersection	intersection	NOUN
iajs-901	249	18	of	of	ADP
iajs-901	249	19	all	all	DET
iajs-901	249	20	maximal	maximal	ADJ
iajs-901	249	21	submodules	submodule	NOUN
iajs-901	249	22	of	of	ADP
iajs-901	249	23	m	m	PROPN
iajs-901	249	24	(	(	PUNCT
iajs-901	249	25	if	if	SCONJ
iajs-901	249	26	exist	exist	VERB
iajs-901	249	27	)	)	PUNCT
iajs-901	249	28	or	or	CCONJ
iajs-901	249	29	j(m	j(m	PROPN
iajs-901	249	30	)	)	PUNCT
iajs-901	250	1	=	=	PUNCT
iajs-901	250	2	m	m	VERB
iajs-901	250	3	if	if	SCONJ
iajs-901	250	4	m	m	NOUN
iajs-901	250	5	has	have	VERB
iajs-901	250	6	no	no	DET
iajs-901	250	7	maximal	maximal	ADJ
iajs-901	250	8	submodule	submodule	NOUN
iajs-901	250	9	,	,	PUNCT
iajs-901	250	10	see	see	VERB
iajs-901	250	11	[	[	X
iajs-901	250	12	12	12	NUM
iajs-901	250	13	,	,	PUNCT
iajs-901	250	14	definition	definition	NOUN
iajs-901	250	15	9.1.2,p.214	9.1.2,p.214	NOUN
iajs-901	250	16	]	]	PUNCT
iajs-901	250	17	.	.	PUNCT
iajs-901	251	1	the	the	DET
iajs-901	251	2	following	following	ADJ
iajs-901	251	3	result	result	NOUN
iajs-901	251	4	appeared	appear	VERB
iajs-901	251	5	in	in	ADP
iajs-901	251	6	[	[	X
iajs-901	251	7	27	27	NUM
iajs-901	251	8	]	]	PUNCT
iajs-901	251	9	.	.	PUNCT
iajs-901	252	1	however	however	ADV
iajs-901	252	2	we	we	PRON
iajs-901	252	3	give	give	VERB
iajs-901	252	4	another	another	DET
iajs-901	252	5	proof	proof	NOUN
iajs-901	252	6	.	.	PUNCT
iajs-901	253	1	proposition	proposition	NOUN
iajs-901	253	2	(	(	PUNCT
iajs-901	253	3	23	23	NUM
iajs-901	253	4	):	):	PUNCT
iajs-901	253	5	if	if	SCONJ
iajs-901	253	6	m	m	NOUN
iajs-901	253	7	is	be	AUX
iajs-901	253	8	a	a	DET
iajs-901	253	9	coprime	coprime	ADJ
iajs-901	253	10	r	r	NOUN
iajs-901	253	11	-	-	PUNCT
iajs-901	253	12	module	module	NOUN
iajs-901	253	13	and	and	CCONJ
iajs-901	253	14	j(m	j(m	PROPN
iajs-901	253	15	)	)	PUNCT
iajs-901	254	1			PROPN
iajs-901	254	2	m	m	PROPN
iajs-901	254	3	,	,	PUNCT
iajs-901	254	4	then	then	ADV
iajs-901	254	5	r	r	NOUN
iajs-901	254	6	is	be	AUX
iajs-901	254	7	a	a	DET
iajs-901	254	8	coprime	coprime	NOUN
iajs-901	254	9	ring	ring	NOUN
iajs-901	254	10	,	,	PUNCT
iajs-901	254	11	where	where	SCONJ
iajs-901	254	12	r	r	NOUN
iajs-901	254	13	=	=	NOUN
iajs-901	254	14	r	r	NOUN
iajs-901	254	15	/	/	SYM
iajs-901	254	16	r	r	NOUN
iajs-901	254	17	ann	ann	PROPN
iajs-901	254	18	m.	m.	NOUN
iajs-901	254	19	proof	proof	NOUN
iajs-901	254	20	:	:	PUNCT
iajs-901	254	21	since	since	SCONJ
iajs-901	254	22	j(m	j(m	PROPN
iajs-901	254	23	)	)	PUNCT
iajs-901	255	1			PROPN
iajs-901	255	2	m	m	PROPN
iajs-901	255	3	,	,	PUNCT
iajs-901	255	4	then	then	ADV
iajs-901	255	5	there	there	PRON
iajs-901	255	6	exists	exist	VERB
iajs-901	255	7	a	a	DET
iajs-901	255	8	maximal	maximal	ADJ
iajs-901	255	9	submodule	submodule	NOUN
iajs-901	255	10	n	n	PROPN
iajs-901	255	11	of	of	ADP
iajs-901	255	12	m.	m.	NOUN
iajs-901	255	13	by	by	ADP
iajs-901	255	14	[	[	X
iajs-901	255	15	15	15	NUM
iajs-901	255	16	,	,	PUNCT
iajs-901	255	17	propo	propo	NOUN
iajs-901	255	18	.	.	PUNCT
iajs-901	256	1	2.2	2.2	NUM
iajs-901	256	2	,	,	PUNCT
iajs-901	256	3	ch	ch	NOUN
iajs-901	256	4	.	.	PROPN
iajs-901	256	5	1	1	NUM
iajs-901	256	6	]	]	PUNCT
iajs-901	256	7	the	the	DET
iajs-901	256	8	ideal	ideal	NOUN
iajs-901	257	1	[	[	X
iajs-901	257	2	n	n	X
iajs-901	257	3	r	r	NOUN
iajs-901	257	4	:	:	PUNCT
iajs-901	257	5	m	m	VERB
iajs-901	257	6	]	]	X
iajs-901	257	7	is	be	AUX
iajs-901	257	8	a	a	DET
iajs-901	257	9	maximal	maximal	ADJ
iajs-901	257	10	ideal	ideal	NOUN
iajs-901	257	11	of	of	ADP
iajs-901	257	12	r.	r.	PROPN
iajs-901	257	13	since	since	SCONJ
iajs-901	257	14	m	m	PROPN
iajs-901	257	15	is	be	AUX
iajs-901	257	16	coprime	coprime	ADJ
iajs-901	257	17	,	,	PUNCT
iajs-901	257	18	then	then	ADV
iajs-901	257	19	[	[	X
iajs-901	257	20	n	n	X
iajs-901	257	21	r	r	NOUN
iajs-901	257	22	:	:	PUNCT
iajs-901	257	23	m	m	VERB
iajs-901	257	24	]	]	X
iajs-901	257	25	=	=	PUNCT
iajs-901	257	26	r	r	NOUN
iajs-901	257	27	ann	ann	PROPN
iajs-901	257	28	m	m	PROPN
iajs-901	257	29	and	and	CCONJ
iajs-901	257	30	so	so	ADV
iajs-901	257	31	r	r	VERB
iajs-901	257	32	ann	ann	PROPN
iajs-901	257	33	m	m	NOUN
iajs-901	257	34	is	be	AUX
iajs-901	257	35	a	a	DET
iajs-901	257	36	maximal	maximal	ADJ
iajs-901	257	37	ideal	ideal	NOUN
iajs-901	257	38	of	of	ADP
iajs-901	257	39	r	r	NOUN
iajs-901	257	40	,	,	PUNCT
iajs-901	257	41	so	so	ADV
iajs-901	257	42	r	r	NOUN
iajs-901	257	43	=	=	SYM
iajs-901	257	44	r	r	NOUN
iajs-901	257	45	/	/	SYM
iajs-901	257	46	r	r	NOUN
iajs-901	257	47	ann	ann	PROPN
iajs-901	257	48	m	m	NOUN
iajs-901	257	49	is	be	AUX
iajs-901	257	50	a	a	DET
iajs-901	257	51	field	field	NOUN
iajs-901	257	52	.	.	PUNCT
iajs-901	258	1	thus	thus	ADV
iajs-901	258	2	r	r	NOUN
iajs-901	258	3	is	be	AUX
iajs-901	258	4	a	a	DET
iajs-901	258	5	coprime	coprime	ADJ
iajs-901	258	6	ring	ring	NOUN
iajs-901	258	7	.	.	PUNCT
iajs-901	259	1	■	■	PUNCT
iajs-901	259	2	we	we	PRON
iajs-901	259	3	note	note	VERB
iajs-901	259	4	that	that	SCONJ
iajs-901	259	5	the	the	DET
iajs-901	259	6	condition	condition	NOUN
iajs-901	259	7	j(m	j(m	PROPN
iajs-901	259	8	)	)	PUNCT
iajs-901	259	9			PROPN
iajs-901	259	10	m	m	VERB
iajs-901	259	11	in	in	ADP
iajs-901	259	12	proposition	proposition	NOUN
iajs-901	259	13	(	(	PUNCT
iajs-901	259	14	23	23	NUM
iajs-901	259	15	)	)	PUNCT
iajs-901	259	16	is	be	AUX
iajs-901	259	17	necessary	necessary	ADJ
iajs-901	259	18	for	for	ADP
iajs-901	259	19	example	example	NOUN
iajs-901	259	20	:	:	PUNCT
iajs-901	259	21	q	q	X
iajs-901	259	22	as	as	SCONJ
iajs-901	259	23	z	z	NOUN
iajs-901	259	24	-	-	PUNCT
iajs-901	259	25	module	module	NOUN
iajs-901	259	26	is	be	AUX
iajs-901	259	27	coprime	coprime	NOUN
iajs-901	259	28	module	module	NOUN
iajs-901	259	29	,	,	PUNCT
iajs-901	259	30	which	which	PRON
iajs-901	259	31	has	have	VERB
iajs-901	259	32	no	no	DET
iajs-901	259	33	maximal	maximal	ADJ
iajs-901	259	34	submodule	submodule	NOUN
iajs-901	259	35	;	;	PUNCT
iajs-901	259	36	that	that	PRON
iajs-901	259	37	is	is	ADV
iajs-901	259	38	j(q	j(q	NOUN
iajs-901	259	39	)	)	PUNCT
iajs-901	259	40	=	=	SYM
iajs-901	260	1	q	q	NOUN
iajs-901	261	1	and	and	CCONJ
iajs-901	261	2	z	z	NOUN
iajs-901	261	3	is	be	AUX
iajs-901	261	4	not	not	PART
iajs-901	261	5	a	a	DET
iajs-901	261	6	field	field	NOUN
iajs-901	261	7	.	.	PUNCT
iajs-901	262	1	recall	recall	VERB
iajs-901	262	2	that	that	SCONJ
iajs-901	262	3	an	an	DET
iajs-901	262	4	r	r	NOUN
iajs-901	262	5	-	-	PUNCT
iajs-901	262	6	module	module	NOUN
iajs-901	262	7	m	m	NOUN
iajs-901	262	8	is	be	AUX
iajs-901	262	9	called	call	VERB
iajs-901	262	10	divisible	divisible	ADJ
iajs-901	262	11	if	if	SCONJ
iajs-901	262	12	for	for	ADP
iajs-901	262	13	each	each	DET
iajs-901	262	14	non	non	ADJ
iajs-901	262	15	zero	zero	NUM
iajs-901	262	16	divisor	divisor	NOUN
iajs-901	262	17	r	r	NOUN
iajs-901	262	18	of	of	ADP
iajs-901	262	19	r	r	NOUN
iajs-901	262	20	,	,	PUNCT
iajs-901	262	21	rm	rm	PROPN
iajs-901	262	22	=	=	NOUN
iajs-901	262	23	m	m	PROPN
iajs-901	262	24	,	,	PUNCT
iajs-901	262	25	see[22	see[22	ADJ
iajs-901	262	26	]	]	PUNCT
iajs-901	262	27	.	.	PUNCT
iajs-901	263	1	the	the	DET
iajs-901	263	2	following	follow	VERB
iajs-901	263	3	proposition	proposition	NOUN
iajs-901	263	4	is	be	AUX
iajs-901	263	5	an	an	DET
iajs-901	263	6	immediate	immediate	ADJ
iajs-901	263	7	result	result	NOUN
iajs-901	263	8	by	by	ADP
iajs-901	263	9	theorem	theorem	NOUN
iajs-901	263	10	(	(	PUNCT
iajs-901	263	11	7	7	NUM
iajs-901	263	12	)	)	PUNCT
iajs-901	263	13	and	and	CCONJ
iajs-901	263	14	[	[	X
iajs-901	263	15	16	16	NUM
iajs-901	263	16	,	,	PUNCT
iajs-901	263	17	prop	prop	VERB
iajs-901	263	18	1.1.7	1.1.7	NUM
iajs-901	263	19	]	]	PUNCT
iajs-901	263	20	.	.	PUNCT
iajs-901	264	1	proposition	proposition	NOUN
iajs-901	264	2	(	(	PUNCT
iajs-901	264	3	24	24	NUM
iajs-901	264	4	):	):	PUNCT
iajs-901	264	5	let	let	VERB
iajs-901	264	6	m	m	PRON
iajs-901	264	7	be	be	AUX
iajs-901	264	8	an	an	DET
iajs-901	264	9	r	r	NOUN
iajs-901	264	10	-	-	PUNCT
iajs-901	264	11	module	module	NOUN
iajs-901	264	12	,	,	PUNCT
iajs-901	264	13	then	then	ADV
iajs-901	264	14	the	the	DET
iajs-901	264	15	following	following	ADJ
iajs-901	264	16	statements	statement	NOUN
iajs-901	264	17	are	be	AUX
iajs-901	264	18	equivalent	equivalent	ADJ
iajs-901	264	19	:	:	PUNCT
iajs-901	264	20	1	1	X
iajs-901	264	21	.	.	X
iajs-901	264	22	m	m	PROPN
iajs-901	264	23	is	be	AUX
iajs-901	264	24	a	a	DET
iajs-901	264	25	coprime	coprime	ADJ
iajs-901	264	26	r	r	NOUN
iajs-901	264	27	-	-	NOUN
iajs-901	264	28	module	module	NOUN
iajs-901	264	29	.	.	PUNCT
iajs-901	265	1	2	2	X
iajs-901	265	2	.	.	X
iajs-901	265	3	m	m	PROPN
iajs-901	265	4	is	be	AUX
iajs-901	265	5	a	a	DET
iajs-901	265	6	divisible	divisible	ADJ
iajs-901	265	7	r	r	NOUN
iajs-901	265	8	/	/	SYM
iajs-901	265	9	r	r	NOUN
iajs-901	265	10	ann	ann	PROPN
iajs-901	265	11	m	m	NOUN
iajs-901	265	12	-	-	NOUN
iajs-901	265	13	module	module	NOUN
iajs-901	265	14	.	.	PUNCT
iajs-901	266	1	3	3	X
iajs-901	266	2	.	.	X
iajs-901	266	3	r	r	NOUN
iajs-901	266	4	m	m	NOUN
iajs-901	266	5	=	=	VERB
iajs-901	266	6	m	m	VERB
iajs-901	266	7	for	for	ADP
iajs-901	266	8	every	every	DET
iajs-901	266	9	r	r	NOUN
iajs-901	266	10			NOUN
iajs-901	266	11	r	r	NOUN
iajs-901	266	12	\	\	NOUN
iajs-901	266	13	r	r	NOUN
iajs-901	266	14	ann	ann	PROPN
iajs-901	266	15	m.	m.	NOUN
iajs-901	266	16	4	4	NUM
iajs-901	266	17	.	.	PUNCT
iajs-901	267	1	i	i	PRON
iajs-901	267	2	m	m	VERB
iajs-901	267	3	=	=	VERB
iajs-901	267	4	m	m	VERB
iajs-901	267	5	for	for	ADP
iajs-901	267	6	every	every	DET
iajs-901	267	7	ideal	ideal	NOUN
iajs-901	268	1	i	i	PRON
iajs-901	268	2			PROPN
iajs-901	268	3	r	r	PROPN
iajs-901	268	4	ann	ann	PROPN
iajs-901	268	5	m.	m.	NOUN
iajs-901	268	6	5	5	NUM
iajs-901	268	7	.	.	PUNCT
iajs-901	268	8	w(m	w(m	PROPN
iajs-901	268	9	)	)	PUNCT
iajs-901	269	1	=	=	SYM
iajs-901	269	2	r	r	NOUN
iajs-901	269	3	ann	ann	PROPN
iajs-901	269	4	m.	m.	NOUN
iajs-901	269	5	note	note	VERB
iajs-901	269	6	that	that	DET
iajs-901	269	7	statement	statement	NOUN
iajs-901	269	8	(	(	PUNCT
iajs-901	269	9	5	5	NUM
iajs-901	269	10	)	)	PUNCT
iajs-901	269	11	in	in	ADP
iajs-901	269	12	proposition	proposition	NOUN
iajs-901	269	13	(	(	PUNCT
iajs-901	269	14	24	24	NUM
iajs-901	269	15	)	)	PUNCT
iajs-901	269	16	was	be	AUX
iajs-901	269	17	given	give	VERB
iajs-901	269	18	as	as	ADP
iajs-901	269	19	definition	definition	NOUN
iajs-901	269	20	of	of	ADP
iajs-901	269	21	coprime	coprime	NOUN
iajs-901	269	22	module	module	NOUN
iajs-901	269	23	by	by	ADP
iajs-901	269	24	abuihlial	abuihlial	ADJ
iajs-901	269	25	j.	j.	PROPN
iajs-901	269	26	in	in	ADP
iajs-901	269	27	[	[	X
iajs-901	269	28	7].he	7].he	NUM
iajs-901	269	29	defined	define	VERB
iajs-901	269	30	that	that	SCONJ
iajs-901	269	31	an	an	DET
iajs-901	269	32	r	r	NOUN
iajs-901	269	33	-	-	PUNCT
iajs-901	269	34	submodule	submodule	NOUN
iajs-901	269	35	n	n	PROPN
iajs-901	269	36	of	of	ADP
iajs-901	269	37	m	m	PROPN
iajs-901	269	38	is	be	AUX
iajs-901	269	39	a	a	DET
iajs-901	269	40	coprime	coprime	ADJ
iajs-901	269	41	submodule	submodule	NOUN
iajs-901	269	42	if	if	SCONJ
iajs-901	269	43	for	for	ADP
iajs-901	269	44	each	each	DET
iajs-901	269	45	homothety	homothety	NOUN
iajs-901	269	46	r	r	NOUN
iajs-901	269	47	*	*	PUNCT
iajs-901	269	48	on	on	ADP
iajs-901	269	49	m	m	PROPN
iajs-901	269	50	/	/	SYM
iajs-901	269	51	n	n	PROPN
iajs-901	269	52	is	be	AUX
iajs-901	269	53	either	either	DET
iajs-901	269	54	zero	zero	NUM
iajs-901	269	55	or	or	CCONJ
iajs-901	269	56	surjective	surjective	ADJ
iajs-901	269	57	.	.	PUNCT
iajs-901	270	1	then	then	ADV
iajs-901	270	2	it	it	PRON
iajs-901	270	3	is	be	AUX
iajs-901	270	4	clear	clear	ADJ
iajs-901	270	5	that	that	SCONJ
iajs-901	270	6	m	m	PROPN
iajs-901	270	7	is	be	AUX
iajs-901	270	8	a	a	DET
iajs-901	270	9	coprime	coprime	ADJ
iajs-901	270	10	rmodule	rmodule	NOUN
iajs-901	270	11	if	if	SCONJ
iajs-901	270	12	and	and	CCONJ
iajs-901	270	13	only	only	ADV
iajs-901	270	14	if	if	SCONJ
iajs-901	270	15	(	(	PUNCT
iajs-901	270	16	0	0	NUM
iajs-901	270	17	)	)	PUNCT
iajs-901	270	18	is	be	AUX
iajs-901	270	19	a	a	DET
iajs-901	270	20	coprime	coprime	NOUN
iajs-901	270	21	submodule	submodule	NOUN
iajs-901	270	22	.	.	PUNCT
iajs-901	271	1	by	by	ADP
iajs-901	271	2	combining	combine	VERB
iajs-901	271	3	this	this	DET
iajs-901	271	4	result	result	NOUN
iajs-901	271	5	with	with	ADP
iajs-901	271	6	theorem	theorem	NOUN
iajs-901	271	7	(	(	PUNCT
iajs-901	271	8	7	7	X
iajs-901	271	9	)	)	PUNCT
iajs-901	271	10	we	we	PRON
iajs-901	271	11	have	have	VERB
iajs-901	271	12	the	the	DET
iajs-901	271	13	following	follow	VERB
iajs-901	271	14	corollary	corollary	NOUN
iajs-901	271	15	.	.	PUNCT
iajs-901	272	1	corollary	corollary	NOUN
iajs-901	272	2	(	(	PUNCT
iajs-901	272	3	25	25	NUM
iajs-901	272	4	):	):	PUNCT
iajs-901	272	5	let	let	VERB
iajs-901	272	6	m	m	PRON
iajs-901	272	7	be	be	AUX
iajs-901	272	8	an	an	DET
iajs-901	272	9	r	r	NOUN
iajs-901	272	10	-	-	PUNCT
iajs-901	272	11	module	module	NOUN
iajs-901	272	12	,	,	PUNCT
iajs-901	272	13	then	then	ADV
iajs-901	272	14	the	the	DET
iajs-901	272	15	following	following	ADJ
iajs-901	272	16	statements	statement	NOUN
iajs-901	272	17	are	be	AUX
iajs-901	272	18	equivalent	equivalent	ADJ
iajs-901	272	19	:	:	PUNCT
iajs-901	272	20	1	1	X
iajs-901	272	21	.	.	X
iajs-901	272	22	m	m	PROPN
iajs-901	272	23	is	be	AUX
iajs-901	272	24	a	a	DET
iajs-901	272	25	coprime	coprime	ADJ
iajs-901	272	26	r	r	NOUN
iajs-901	272	27	-	-	NOUN
iajs-901	272	28	module	module	NOUN
iajs-901	272	29	.	.	PUNCT
iajs-901	273	1	2	2	X
iajs-901	273	2	.	.	X
iajs-901	273	3	m	m	PROPN
iajs-901	273	4	is	be	AUX
iajs-901	273	5	a	a	DET
iajs-901	273	6	second	second	ADJ
iajs-901	273	7	submodule	submodule	NOUN
iajs-901	273	8	of	of	ADP
iajs-901	273	9	itself	itself	PRON
iajs-901	273	10	.	.	PUNCT
iajs-901	274	1	3	3	X
iajs-901	274	2	.	.	X
iajs-901	274	3	(	(	PUNCT
iajs-901	274	4	0	0	NUM
iajs-901	274	5	)	)	PUNCT
iajs-901	274	6	is	be	AUX
iajs-901	274	7	a	a	DET
iajs-901	274	8	coprime	coprime	ADJ
iajs-901	274	9	submodule	submodule	NOUN
iajs-901	274	10	of	of	ADP
iajs-901	274	11	m.	m.	NOUN
iajs-901	274	12	the	the	DET
iajs-901	274	13	following	follow	VERB
iajs-901	274	14	remark	remark	NOUN
iajs-901	274	15	is	be	AUX
iajs-901	274	16	clear	clear	ADJ
iajs-901	274	17	.	.	PUNCT
iajs-901	275	1	remark	remark	NOUN
iajs-901	275	2	(	(	PUNCT
iajs-901	275	3	26	26	NUM
iajs-901	275	4	):	):	PUNCT
iajs-901	275	5	let	let	VERB
iajs-901	275	6	n	n	PRON
iajs-901	275	7	be	be	AUX
iajs-901	275	8	a	a	DET
iajs-901	275	9	proper	proper	ADJ
iajs-901	275	10	submodule	submodule	NOUN
iajs-901	275	11	of	of	ADP
iajs-901	275	12	an	an	DET
iajs-901	275	13	r	r	NOUN
iajs-901	275	14	-	-	PUNCT
iajs-901	275	15	module	module	NOUN
iajs-901	275	16	m.	m.	NOUN
iajs-901	275	17	then	then	ADV
iajs-901	275	18	n	n	PRON
iajs-901	275	19	is	be	AUX
iajs-901	275	20	a	a	DET
iajs-901	275	21	coprime	coprime	ADJ
iajs-901	275	22	submodule	submodule	NOUN
iajs-901	275	23	of	of	ADP
iajs-901	275	24	m	m	PROPN
iajs-901	275	25	iff	iff	PROPN
iajs-901	275	26	m	m	PROPN
iajs-901	275	27	/	/	SYM
iajs-901	275	28	n	n	PROPN
iajs-901	275	29	is	be	AUX
iajs-901	275	30	a	a	DET
iajs-901	275	31	coprime	coprime	ADJ
iajs-901	275	32	r	r	NOUN
iajs-901	275	33	-	-	NOUN
iajs-901	275	34	module	module	NOUN
iajs-901	275	35	.	.	PUNCT
iajs-901	276	1	the	the	DET
iajs-901	276	2	following	following	ADJ
iajs-901	276	3	result	result	NOUN
iajs-901	276	4	follows	follow	VERB
iajs-901	276	5	by	by	ADP
iajs-901	276	6	remark	remark	NOUN
iajs-901	276	7	(	(	PUNCT
iajs-901	276	8	26	26	NUM
iajs-901	276	9	)	)	PUNCT
iajs-901	276	10	and	and	CCONJ
iajs-901	276	11	note	note	VERB
iajs-901	276	12	(	(	PUNCT
iajs-901	276	13	8)	8)	NUM
iajs-901	276	14	.	.	PUNCT
iajs-901	276	15	corollary	corollary	NOUN
iajs-901	276	16	(	(	PUNCT
iajs-901	276	17	27	27	NUM
iajs-901	276	18	):	):	PUNCT
iajs-901	276	19	let	let	VERB
iajs-901	276	20	n	n	PRON
iajs-901	276	21	be	be	AUX
iajs-901	276	22	a	a	DET
iajs-901	276	23	proper	proper	ADJ
iajs-901	276	24	submodule	submodule	NOUN
iajs-901	276	25	of	of	ADP
iajs-901	276	26	an	an	DET
iajs-901	276	27	r	r	NOUN
iajs-901	276	28	-	-	PUNCT
iajs-901	276	29	module	module	NOUN
iajs-901	276	30	m	m	NOUN
iajs-901	276	31	,	,	PUNCT
iajs-901	276	32	if	if	SCONJ
iajs-901	276	33	n	n	PRON
iajs-901	276	34	is	be	AUX
iajs-901	276	35	a	a	DET
iajs-901	276	36	coprime	coprime	ADJ
iajs-901	276	37	submodule	submodule	NOUN
iajs-901	276	38	then	then	ADV
iajs-901	276	39	r	r	PROPN
iajs-901	276	40	ann	ann	PROPN
iajs-901	276	41	(	(	PUNCT
iajs-901	276	42	m	m	PROPN
iajs-901	276	43	/	/	SYM
iajs-901	276	44	n	n	CCONJ
iajs-901	276	45	)	)	PUNCT
iajs-901	276	46	is	be	AUX
iajs-901	276	47	a	a	DET
iajs-901	276	48	prime	prime	ADJ
iajs-901	276	49	ideal	ideal	NOUN
iajs-901	276	50	.	.	PUNCT
iajs-901	277	1	ihjpas	ihjpa	VERB
iajs-901	277	2	ibn	ibn	PROPN
iajs-901	277	3	alhaitham	alhaitham	PROPN
iajs-901	278	1	j.	j.	PROPN
iajs-901	279	1	fo	fo	ADP
iajs-901	279	2	r	r	NOUN
iajs-901	279	3	pure	pure	ADJ
iajs-901	279	4	&	&	CCONJ
iajs-901	279	5	appl	appl	PROPN
iajs-901	279	6	.	.	PUNCT
iajs-901	280	1	sc	sc	PROPN
iajs-901	280	2	i.	i.	PROPN
iajs-901	280	3	vo	vo	PROPN
iajs-901	280	4	l.23	l.23	PROPN
iajs-901	280	5	(	(	PUNCT
iajs-901	280	6	3	3	NUM
iajs-901	280	7	)	)	PUNCT
iajs-901	280	8	2010	2010	NUM
iajs-901	280	9	recall	recall	VERB
iajs-901	280	10	that	that	SCONJ
iajs-901	280	11	an	an	DET
iajs-901	280	12	r	r	NOUN
iajs-901	280	13	-	-	PUNCT
iajs-901	280	14	module	module	NOUN
iajs-901	280	15	m	m	NOUN
iajs-901	280	16	is	be	AUX
iajs-901	280	17	called	call	VERB
iajs-901	280	18	divisible	divisible	ADJ
iajs-901	280	19	if	if	SCONJ
iajs-901	280	20	for	for	ADP
iajs-901	280	21	each	each	DET
iajs-901	280	22	non	non	ADJ
iajs-901	280	23	zero	zero	NUM
iajs-901	280	24	divisor	divisor	NOUN
iajs-901	280	25	r	r	NOUN
iajs-901	280	26	of	of	ADP
iajs-901	280	27	r	r	NOUN
iajs-901	280	28	,	,	PUNCT
iajs-901	280	29	rm	rm	PROPN
iajs-901	280	30	=	=	NOUN
iajs-901	280	31	m	m	PROPN
iajs-901	280	32	,	,	PUNCT
iajs-901	280	33	see	see	VERB
iajs-901	280	34	[	[	X
iajs-901	280	35	22	22	NUM
iajs-901	280	36	]	]	PUNCT
iajs-901	280	37	.	.	PUNCT
iajs-901	281	1	now	now	ADV
iajs-901	281	2	,	,	PUNCT
iajs-901	281	3	we	we	PRON
iajs-901	281	4	list	list	VERB
iajs-901	281	5	some	some	DET
iajs-901	281	6	consequences	consequence	NOUN
iajs-901	281	7	of	of	ADP
iajs-901	281	8	proposition	proposition	NOUN
iajs-901	281	9	(	(	PUNCT
iajs-901	281	10	24	24	NUM
iajs-901	281	11	)	)	PUNCT
iajs-901	281	12	.	.	PUNCT
iajs-901	282	1	corollary	corollary	ADJ
iajs-901	282	2	(	(	PUNCT
iajs-901	282	3	28	28	NUM
iajs-901	282	4	):	):	PUNCT
iajs-901	282	5	every	every	DET
iajs-901	282	6	faithful	faithful	ADJ
iajs-901	282	7	coprime	coprime	NOUN
iajs-901	282	8	r	r	NOUN
iajs-901	282	9	-	-	PUNCT
iajs-901	282	10	module	module	NOUN
iajs-901	282	11	is	be	AUX
iajs-901	282	12	divisible	divisible	ADJ
iajs-901	282	13	.	.	PUNCT
iajs-901	283	1	proof	proof	NOUN
iajs-901	283	2	:	:	PUNCT
iajs-901	283	3	is	be	AUX
iajs-901	283	4	clear	clear	ADJ
iajs-901	283	5	.	.	PUNCT
iajs-901	284	1	■	■	PUNCT
iajs-901	284	2	remark	remark	NOUN
iajs-901	284	3	(	(	PUNCT
iajs-901	284	4	29	29	NUM
iajs-901	284	5	):	):	PUNCT
iajs-901	284	6	every	every	DET
iajs-901	284	7	divisible	divisible	ADJ
iajs-901	284	8	module	module	NOUN
iajs-901	284	9	over	over	ADP
iajs-901	284	10	an	an	DET
iajs-901	284	11	integral	integral	ADJ
iajs-901	284	12	domain	domain	NOUN
iajs-901	284	13	is	be	AUX
iajs-901	284	14	faithful	faithful	ADJ
iajs-901	284	15	coprime	coprime	NOUN
iajs-901	284	16	.	.	PUNCT
iajs-901	285	1	by	by	ADP
iajs-901	285	2	combining	combine	VERB
iajs-901	285	3	corollary	corollary	ADJ
iajs-901	285	4	(	(	PUNCT
iajs-901	285	5	28	28	NUM
iajs-901	285	6	)	)	PUNCT
iajs-901	285	7	and	and	CCONJ
iajs-901	285	8	remark	remark	NOUN
iajs-901	285	9	(	(	PUNCT
iajs-901	285	10	29	29	NUM
iajs-901	285	11	)	)	PUNCT
iajs-901	285	12	we	we	PRON
iajs-901	285	13	get	get	VERB
iajs-901	285	14	:	:	PUNCT
iajs-901	285	15	corollary	corollary	ADJ
iajs-901	285	16	(	(	PUNCT
iajs-901	285	17	30	30	NUM
iajs-901	285	18	):	):	PUNCT
iajs-901	285	19	let	let	VERB
iajs-901	285	20	m	m	PRON
iajs-901	285	21	be	be	AUX
iajs-901	285	22	a	a	DET
iajs-901	285	23	module	module	NOUN
iajs-901	285	24	over	over	ADP
iajs-901	285	25	an	an	DET
iajs-901	285	26	integral	integral	ADJ
iajs-901	285	27	domain	domain	NOUN
iajs-901	285	28	r.	r.	NOUN
iajs-901	285	29	then	then	ADV
iajs-901	285	30	m	m	VERB
iajs-901	285	31	is	be	AUX
iajs-901	285	32	a	a	DET
iajs-901	285	33	faithful	faithful	ADJ
iajs-901	285	34	coprime	coprime	NOUN
iajs-901	285	35	r	r	NOUN
iajs-901	285	36	-	-	PUNCT
iajs-901	285	37	module	module	NOUN
iajs-901	285	38	iff	iff	PROPN
iajs-901	285	39	m	m	VERB
iajs-901	285	40	is	be	AUX
iajs-901	285	41	a	a	DET
iajs-901	285	42	divisible	divisible	ADJ
iajs-901	285	43	r	r	NOUN
iajs-901	285	44	-	-	PUNCT
iajs-901	285	45	module	module	NOUN
iajs-901	285	46	.	.	PUNCT
iajs-901	286	1	recall	recall	VERB
iajs-901	286	2	that	that	SCONJ
iajs-901	286	3	an	an	DET
iajs-901	286	4	r	r	NOUN
iajs-901	286	5	-	-	PUNCT
iajs-901	286	6	module	module	NOUN
iajs-901	286	7	m	m	NOUN
iajs-901	286	8	is	be	AUX
iajs-901	286	9	called	call	VERB
iajs-901	286	10	principally	principally	ADV
iajs-901	286	11	injective	injective	ADJ
iajs-901	286	12	if	if	SCONJ
iajs-901	286	13	for	for	ADP
iajs-901	286	14	every	every	DET
iajs-901	286	15	principal	principal	ADJ
iajs-901	286	16	ideal	ideal	NOUN
iajs-901	286	17	i	i	PRON
iajs-901	286	18	of	of	ADP
iajs-901	286	19	r	r	NOUN
iajs-901	286	20	and	and	CCONJ
iajs-901	286	21	every	every	DET
iajs-901	286	22	monomorphism	monomorphism	NOUN
iajs-901	287	1	f	f	X
iajs-901	287	2	:	:	PUNCT
iajs-901	287	3	i	i	PRON
iajs-901	287	4			PROPN
iajs-901	287	5	m	m	VERB
iajs-901	287	6	,	,	PUNCT
iajs-901	287	7	there	there	PRON
iajs-901	287	8	is	be	VERB
iajs-901	287	9	a	a	DET
iajs-901	287	10	homomorphism	homomorphism	NOUN
iajs-901	287	11	g	g	NOUN
iajs-901	287	12	:	:	PUNCT
iajs-901	287	13	r	r	NOUN
iajs-901	287	14			PROPN
iajs-901	287	15	m	m	VERB
iajs-901	287	16	such	such	ADJ
iajs-901	287	17	that	that	SCONJ
iajs-901	287	18	g	g	PROPN
iajs-901	287	19	/	/	SYM
iajs-901	287	20	i	i	NOUN
iajs-901	287	21	=	=	SYM
iajs-901	287	22	f	f	X
iajs-901	287	23	,	,	PUNCT
iajs-901	287	24	see	see	VERB
iajs-901	287	25	[	[	X
iajs-901	287	26	23	23	NUM
iajs-901	287	27	]	]	PUNCT
iajs-901	287	28	.	.	PUNCT
iajs-901	288	1	proposition	proposition	NOUN
iajs-901	288	2	(	(	PUNCT
iajs-901	288	3	31	31	NUM
iajs-901	288	4	):	):	PUNCT
iajs-901	288	5	let	let	VERB
iajs-901	288	6	m	m	PRON
iajs-901	288	7	be	be	AUX
iajs-901	288	8	a	a	DET
iajs-901	288	9	module	module	NOUN
iajs-901	288	10	over	over	ADP
iajs-901	288	11	an	an	DET
iajs-901	288	12	integral	integral	ADJ
iajs-901	288	13	domain	domain	NOUN
iajs-901	288	14	r.	r.	NOUN
iajs-901	288	15	then	then	ADV
iajs-901	288	16	the	the	DET
iajs-901	288	17	following	follow	VERB
iajs-901	288	18	statements	statement	NOUN
iajs-901	288	19	are	be	AUX
iajs-901	288	20	equivalent	equivalent	ADJ
iajs-901	288	21	:	:	PUNCT
iajs-901	288	22	1	1	X
iajs-901	288	23	.	.	X
iajs-901	288	24	m	m	PROPN
iajs-901	288	25	is	be	AUX
iajs-901	288	26	a	a	DET
iajs-901	288	27	faithful	faithful	ADJ
iajs-901	288	28	coprime	coprime	NOUN
iajs-901	288	29	r	r	NOUN
iajs-901	288	30	-	-	NOUN
iajs-901	288	31	module	module	NOUN
iajs-901	288	32	.	.	PUNCT
iajs-901	289	1	2	2	X
iajs-901	289	2	.	.	X
iajs-901	289	3	m	m	PROPN
iajs-901	289	4	is	be	AUX
iajs-901	289	5	a	a	DET
iajs-901	289	6	divisible	divisible	ADJ
iajs-901	289	7	r	r	NOUN
iajs-901	289	8	-	-	PUNCT
iajs-901	289	9	module	module	NOUN
iajs-901	289	10	.	.	PUNCT
iajs-901	290	1	3	3	X
iajs-901	290	2	.	.	X
iajs-901	290	3	m	m	PROPN
iajs-901	290	4	is	be	AUX
iajs-901	290	5	a	a	DET
iajs-901	290	6	principally	principally	ADV
iajs-901	290	7	injective	injective	ADJ
iajs-901	290	8	r	r	NOUN
iajs-901	290	9	-	-	PUNCT
iajs-901	290	10	module	module	NOUN
iajs-901	290	11	.	.	PUNCT
iajs-901	291	1	proof	proof	NOUN
iajs-901	291	2	:	:	PUNCT
iajs-901	291	3	(	(	PUNCT
iajs-901	291	4	1	1	X
iajs-901	291	5	)	)	PUNCT
iajs-901	291	6			NOUN
iajs-901	291	7	(	(	PUNCT
iajs-901	291	8	2	2	NUM
iajs-901	291	9	)	)	PUNCT
iajs-901	291	10	follows	follow	VERB
iajs-901	291	11	by	by	ADP
iajs-901	291	12	corollary	corollary	ADJ
iajs-901	291	13	(	(	PUNCT
iajs-901	291	14	30	30	NUM
iajs-901	291	15	)	)	PUNCT
iajs-901	291	16	.	.	PUNCT
iajs-901	292	1	(	(	PUNCT
iajs-901	292	2	2	2	X
iajs-901	292	3	)	)	PUNCT
iajs-901	292	4			ADJ
iajs-901	292	5	(	(	PUNCT
iajs-901	292	6	3	3	NUM
iajs-901	292	7	)	)	PUNCT
iajs-901	292	8	follows	follow	VERB
iajs-901	292	9	by	by	ADP
iajs-901	292	10	[	[	X
iajs-901	292	11	12	12	NUM
iajs-901	292	12	,	,	PUNCT
iajs-901	292	13	exc	exc	NOUN
iajs-901	292	14	.	.	PUNCT
iajs-901	293	1	9(a	9(a	NUM
iajs-901	293	2	)	)	PUNCT
iajs-901	294	1	,	,	PUNCT
iajs-901	294	2	p.104	p.104	NOUN
iajs-901	294	3	]	]	PUNCT
iajs-901	294	4	.	.	PUNCT
iajs-901	295	1	recall	recall	VERB
iajs-901	295	2	that	that	SCONJ
iajs-901	295	3	an	an	DET
iajs-901	295	4	r	r	NOUN
iajs-901	295	5	-	-	PUNCT
iajs-901	295	6	module	module	NOUN
iajs-901	295	7	m	m	NOUN
iajs-901	295	8	is	be	AUX
iajs-901	295	9	said	say	VERB
iajs-901	295	10	to	to	PART
iajs-901	295	11	be	be	AUX
iajs-901	295	12	injective	injective	ADJ
iajs-901	295	13	if	if	SCONJ
iajs-901	295	14	and	and	CCONJ
iajs-901	295	15	only	only	ADV
iajs-901	295	16	if	if	SCONJ
iajs-901	295	17	for	for	ADP
iajs-901	295	18	any	any	DET
iajs-901	295	19	monomorphism	monomorphism	NOUN
iajs-901	295	20	f	f	NOUN
iajs-901	295	21	:	:	PUNCT
iajs-901	295	22	a	a	DET
iajs-901	295	23			PROPN
iajs-901	295	24	b	b	NOUN
iajs-901	295	25	where	where	SCONJ
iajs-901	295	26	a	a	PRON
iajs-901	295	27	and	and	CCONJ
iajs-901	295	28	b	b	NOUN
iajs-901	295	29	are	be	AUX
iajs-901	295	30	any	any	DET
iajs-901	295	31	two	two	NUM
iajs-901	295	32	r	r	NOUN
iajs-901	295	33	-	-	PUNCT
iajs-901	295	34	modules	module	NOUN
iajs-901	295	35	and	and	CCONJ
iajs-901	295	36	for	for	ADP
iajs-901	295	37	any	any	DET
iajs-901	295	38	homomorphism	homomorphism	NOUN
iajs-901	295	39	g	g	NOUN
iajs-901	295	40	:	:	PUNCT
iajs-901	295	41	a	a	DET
iajs-901	295	42			NOUN
iajs-901	295	43	m	m	VERB
iajs-901	295	44	there	there	ADV
iajs-901	295	45	exists	exist	VERB
iajs-901	295	46	a	a	DET
iajs-901	295	47	homomorphism	homomorphism	NOUN
iajs-901	295	48	h	h	NOUN
iajs-901	295	49	:	:	PUNCT
iajs-901	295	50	b	b	X
iajs-901	295	51			X
iajs-901	295	52	m	m	VERB
iajs-901	295	53	such	such	ADJ
iajs-901	295	54	that	that	SCONJ
iajs-901	295	55	h	h	PROPN
iajs-901	295	56	○	○	PROPN
iajs-901	295	57	f	f	PROPN
iajs-901	295	58	=	=	SYM
iajs-901	295	59	g	g	PROPN
iajs-901	295	60	,	,	PUNCT
iajs-901	295	61	see	see	VERB
iajs-901	295	62	[	[	X
iajs-901	295	63	24	24	NUM
iajs-901	295	64	,	,	PUNCT
iajs-901	295	65	p.28	p.28	NOUN
iajs-901	295	66	]	]	PUNCT
iajs-901	295	67	.	.	PUNCT
iajs-901	296	1	corollary	corollary	ADJ
iajs-901	296	2	(	(	PUNCT
iajs-901	296	3	32	32	NUM
iajs-901	296	4	):	):	PUNCT
iajs-901	296	5	if	if	SCONJ
iajs-901	296	6	r	r	NOUN
iajs-901	296	7	is	be	AUX
iajs-901	296	8	pid	pid	NOUN
iajs-901	296	9	and	and	CCONJ
iajs-901	296	10	m	m	PROPN
iajs-901	296	11	is	be	AUX
iajs-901	296	12	an	an	DET
iajs-901	296	13	r	r	NOUN
iajs-901	296	14	-	-	PUNCT
iajs-901	296	15	module	module	NOUN
iajs-901	296	16	,	,	PUNCT
iajs-901	296	17	then	then	ADV
iajs-901	296	18	the	the	DET
iajs-901	296	19	following	following	ADJ
iajs-901	296	20	statements	statement	NOUN
iajs-901	296	21	are	be	AUX
iajs-901	296	22	equivalent	equivalent	ADJ
iajs-901	296	23	:	:	PUNCT
iajs-901	296	24	1	1	X
iajs-901	296	25	.	.	X
iajs-901	296	26	m	m	PROPN
iajs-901	296	27	is	be	AUX
iajs-901	296	28	a	a	DET
iajs-901	296	29	faithful	faithful	ADJ
iajs-901	296	30	coprime	coprime	NOUN
iajs-901	296	31	r	r	NOUN
iajs-901	296	32	-	-	NOUN
iajs-901	296	33	module	module	NOUN
iajs-901	296	34	.	.	PUNCT
iajs-901	297	1	2	2	X
iajs-901	297	2	.	.	X
iajs-901	297	3	m	m	PROPN
iajs-901	297	4	is	be	AUX
iajs-901	297	5	a	a	DET
iajs-901	297	6	divisible	divisible	ADJ
iajs-901	297	7	r	r	NOUN
iajs-901	297	8	-	-	PUNCT
iajs-901	297	9	module	module	NOUN
iajs-901	297	10	.	.	PUNCT
iajs-901	298	1	3	3	X
iajs-901	298	2	.	.	X
iajs-901	298	3	m	m	PROPN
iajs-901	298	4	is	be	AUX
iajs-901	298	5	an	an	DET
iajs-901	298	6	injective	injective	ADJ
iajs-901	298	7	r	r	NOUN
iajs-901	298	8	-	-	PUNCT
iajs-901	298	9	module	module	NOUN
iajs-901	298	10	.	.	PUNCT
iajs-901	299	1	proof	proof	NOUN
iajs-901	299	2	:	:	PUNCT
iajs-901	299	3	(	(	PUNCT
iajs-901	299	4	1	1	X
iajs-901	299	5	)	)	PUNCT
iajs-901	299	6			NOUN
iajs-901	299	7	(	(	PUNCT
iajs-901	299	8	2	2	X
iajs-901	299	9	)	)	PUNCT
iajs-901	299	10	follows	follow	VERB
iajs-901	299	11	by	by	ADP
iajs-901	299	12	corollary	corollary	ADJ
iajs-901	299	13	(	(	PUNCT
iajs-901	299	14	.30	.30	NUM
iajs-901	299	15	)	)	PUNCT
iajs-901	299	16	.	.	PUNCT
iajs-901	300	1	(	(	PUNCT
iajs-901	300	2	2	2	X
iajs-901	300	3	)	)	PUNCT
iajs-901	300	4			NOUN
iajs-901	300	5	(	(	PUNCT
iajs-901	300	6	3	3	NUM
iajs-901	300	7	)	)	PUNCT
iajs-901	300	8	.	.	PUNCT
iajs-901	301	1	by	by	ADP
iajs-901	301	2	[	[	X
iajs-901	301	3	26	26	NUM
iajs-901	301	4	,	,	PUNCT
iajs-901	301	5	th	th	X
iajs-901	301	6	2.8	2.8	NUM
iajs-901	301	7	,	,	PUNCT
iajs-901	301	8	p.35	p.35	NOUN
iajs-901	301	9	]	]	PUNCT
iajs-901	301	10	.	.	PUNCT
iajs-901	302	1	■	■	PUNCT
iajs-901	302	2	corollary	corollary	ADJ
iajs-901	302	3	(	(	PUNCT
iajs-901	302	4	33	33	NUM
iajs-901	302	5	):	):	PUNCT
iajs-901	302	6	let	let	VERB
iajs-901	302	7	m	m	PRON
iajs-901	302	8	be	be	AUX
iajs-901	302	9	an	an	DET
iajs-901	302	10	injective	injective	ADJ
iajs-901	302	11	module	module	NOUN
iajs-901	302	12	over	over	ADP
iajs-901	302	13	an	an	DET
iajs-901	302	14	integral	integral	ADJ
iajs-901	302	15	domain	domain	NOUN
iajs-901	302	16	,	,	PUNCT
iajs-901	302	17	then	then	ADV
iajs-901	302	18	m	m	VERB
iajs-901	302	19	is	be	AUX
iajs-901	302	20	coprime	coprime	ADJ
iajs-901	302	21	.	.	PUNCT
iajs-901	303	1	proof	proof	NOUN
iajs-901	303	2	:	:	PUNCT
iajs-901	303	3	since	since	SCONJ
iajs-901	303	4	every	every	DET
iajs-901	303	5	injective	injective	ADJ
iajs-901	303	6	r	r	NOUN
iajs-901	303	7	-	-	PUNCT
iajs-901	303	8	module	module	NOUN
iajs-901	303	9	is	be	AUX
iajs-901	303	10	divisible	divisible	ADJ
iajs-901	303	11	by	by	ADP
iajs-901	303	12	[	[	X
iajs-901	303	13	26	26	NUM
iajs-901	303	14	,	,	PUNCT
iajs-901	303	15	theorem	theorem	VERB
iajs-901	303	16	2.6	2.6	NUM
iajs-901	303	17	,	,	PUNCT
iajs-901	303	18	p.	p.	NOUN
iajs-901	303	19	33	33	NUM
iajs-901	303	20	]	]	PUNCT
iajs-901	303	21	.	.	PUNCT
iajs-901	304	1	hence	hence	ADV
iajs-901	304	2	the	the	DET
iajs-901	304	3	result	result	NOUN
iajs-901	304	4	obtained	obtain	VERB
iajs-901	304	5	by	by	ADP
iajs-901	304	6	remark	remark	NOUN
iajs-901	304	7	(	(	PUNCT
iajs-901	304	8	29	29	NUM
iajs-901	304	9	)	)	PUNCT
iajs-901	304	10	.	.	PUNCT
iajs-901	305	1	■	■	PUNCT
iajs-901	305	2	recall	recall	VERB
iajs-901	305	3	that	that	SCONJ
iajs-901	305	4	a	a	DET
iajs-901	305	5	module	module	NOUN
iajs-901	305	6	m	m	VERB
iajs-901	305	7	over	over	ADP
iajs-901	305	8	an	an	DET
iajs-901	305	9	integral	integral	ADJ
iajs-901	305	10	domain	domain	NOUN
iajs-901	305	11	is	be	AUX
iajs-901	305	12	called	call	VERB
iajs-901	305	13	torsion	torsion	NOUN
iajs-901	305	14	free	free	ADJ
iajs-901	305	15	if	if	SCONJ
iajs-901	305	16	(m	(m	NOUN
iajs-901	305	17	)	)	PUNCT
iajs-901	305	18	=	=	SYM
iajs-901	305	19	0	0	NUM
iajs-901	305	20	,	,	PUNCT
iajs-901	305	21	where	where	SCONJ
iajs-901	305	22	(m	(m	NOUN
iajs-901	305	23	)	)	PUNCT
iajs-901	305	24	=	=	PRON
iajs-901	305	25	{	{	PUNCT
iajs-901	305	26	m	m	PROPN
iajs-901	305	27			NOUN
iajs-901	305	28	m	m	NOUN
iajs-901	305	29	;	;	PUNCT
iajs-901	305	30			NUM
iajs-901	305	31	r	r	NOUN
iajs-901	305	32			NOUN
iajs-901	305	33	r	r	NOUN
iajs-901	305	34	,	,	PUNCT
iajs-901	305	35	r	r	NOUN
iajs-901	305	36			NOUN
iajs-901	305	37	0	0	NUM
iajs-901	305	38	;	;	PUNCT
iajs-901	305	39	r	r	NOUN
iajs-901	305	40	m	m	NOUN
iajs-901	305	41	=	=	NOUN
iajs-901	305	42	0	0	NUM
iajs-901	305	43	}	}	PUNCT
iajs-901	305	44	,	,	PUNCT
iajs-901	305	45	see	see	VERB
iajs-901	305	46	[	[	X
iajs-901	305	47	19	19	NUM
iajs-901	305	48	,	,	PUNCT
iajs-901	305	49	p.45	p.45	X
iajs-901	305	50	]	]	PUNCT
iajs-901	305	51	.	.	PUNCT
iajs-901	306	1	under	under	ADP
iajs-901	306	2	the	the	DET
iajs-901	306	3	class	class	NOUN
iajs-901	306	4	of	of	ADP
iajs-901	306	5	torsion	torsion	NOUN
iajs-901	306	6	free	free	ADJ
iajs-901	306	7	modules	module	NOUN
iajs-901	306	8	over	over	ADP
iajs-901	306	9	an	an	DET
iajs-901	306	10	integral	integral	ADJ
iajs-901	306	11	domain	domain	NOUN
iajs-901	306	12	,	,	PUNCT
iajs-901	306	13	we	we	PRON
iajs-901	306	14	have	have	VERB
iajs-901	306	15	the	the	DET
iajs-901	306	16	following	follow	VERB
iajs-901	306	17	result	result	NOUN
iajs-901	306	18	.	.	PUNCT
iajs-901	307	1	proposition	proposition	NOUN
iajs-901	307	2	(	(	PUNCT
iajs-901	307	3	34	34	NUM
iajs-901	307	4	):	):	PUNCT
iajs-901	307	5	let	let	VERB
iajs-901	307	6	m	m	PRON
iajs-901	307	7	be	be	AUX
iajs-901	307	8	torsion	torsion	NOUN
iajs-901	307	9	free	free	ADJ
iajs-901	307	10	over	over	ADP
iajs-901	307	11	an	an	DET
iajs-901	307	12	integral	integral	ADJ
iajs-901	307	13	domain	domain	NOUN
iajs-901	307	14	r.	r.	NOUN
iajs-901	307	15	then	then	ADV
iajs-901	307	16	the	the	DET
iajs-901	307	17	following	follow	VERB
iajs-901	307	18	statements	statement	NOUN
iajs-901	307	19	are	be	AUX
iajs-901	307	20	equivalent	equivalent	ADJ
iajs-901	307	21	:	:	PUNCT
iajs-901	307	22	1	1	X
iajs-901	307	23	.	.	X
iajs-901	307	24	m	m	PROPN
iajs-901	307	25	is	be	AUX
iajs-901	307	26	a	a	DET
iajs-901	307	27	coprime	coprime	ADJ
iajs-901	307	28	r	r	NOUN
iajs-901	307	29	-	-	NOUN
iajs-901	307	30	module	module	NOUN
iajs-901	307	31	.	.	PUNCT
iajs-901	308	1	2	2	X
iajs-901	308	2	.	.	X
iajs-901	308	3	m	m	PROPN
iajs-901	308	4	is	be	AUX
iajs-901	308	5	a	a	DET
iajs-901	308	6	divisible	divisible	ADJ
iajs-901	308	7	r	r	NOUN
iajs-901	308	8	-	-	PUNCT
iajs-901	308	9	module	module	NOUN
iajs-901	308	10	.	.	PUNCT
iajs-901	309	1	3	3	X
iajs-901	309	2	.	.	X
iajs-901	309	3	m	m	PROPN
iajs-901	309	4	is	be	AUX
iajs-901	309	5	an	an	DET
iajs-901	309	6	injective	injective	ADJ
iajs-901	309	7	r	r	NOUN
iajs-901	309	8	-	-	PUNCT
iajs-901	309	9	module	module	NOUN
iajs-901	309	10	.	.	PUNCT
iajs-901	310	1	proof	proof	NOUN
iajs-901	310	2	:	:	PUNCT
iajs-901	310	3	(	(	PUNCT
iajs-901	310	4	1	1	X
iajs-901	310	5	)	)	PUNCT
iajs-901	310	6			NOUN
iajs-901	310	7	(	(	PUNCT
iajs-901	310	8	2	2	NUM
iajs-901	310	9	)	)	PUNCT
iajs-901	310	10	.	.	PUNCT
iajs-901	311	1	since	since	SCONJ
iajs-901	311	2	m	m	PROPN
iajs-901	311	3	is	be	AUX
iajs-901	311	4	torsion	torsion	NOUN
iajs-901	311	5	free	free	ADJ
iajs-901	311	6	,	,	PUNCT
iajs-901	311	7	then	then	ADV
iajs-901	311	8	m	m	VERB
iajs-901	311	9	is	be	AUX
iajs-901	311	10	faithful	faithful	ADJ
iajs-901	311	11	.	.	PUNCT
iajs-901	312	1	thus	thus	ADV
iajs-901	312	2	the	the	DET
iajs-901	312	3	result	result	NOUN
iajs-901	312	4	follows	follow	VERB
iajs-901	312	5	by	by	ADP
iajs-901	312	6	corollary	corollary	ADJ
iajs-901	312	7	(	(	PUNCT
iajs-901	312	8	30	30	NUM
iajs-901	312	9	)	)	PUNCT
iajs-901	312	10	.	.	PUNCT
iajs-901	313	1	(	(	PUNCT
iajs-901	313	2	2	2	X
iajs-901	313	3	)	)	PUNCT
iajs-901	313	4			NOUN
iajs-901	313	5	(	(	PUNCT
iajs-901	313	6	3	3	X
iajs-901	313	7	)	)	PUNCT
iajs-901	313	8	follows	follow	VERB
iajs-901	313	9	by	by	ADP
iajs-901	313	10	[	[	X
iajs-901	313	11	26	26	NUM
iajs-901	313	12	,	,	PUNCT
iajs-901	313	13	th	th	X
iajs-901	313	14	2.7	2.7	NUM
iajs-901	313	15	,	,	PUNCT
iajs-901	313	16	p.34	p.34	NOUN
iajs-901	313	17	]	]	X
iajs-901	313	18	.	.	PUNCT
iajs-901	314	1	■	■	PUNCT
iajs-901	314	2	ihjpas	ihjpa	VERB
iajs-901	314	3	ibn	ibn	PROPN
iajs-901	314	4	alhaitham	alhaitham	NOUN
iajs-901	314	5	j.	j.	PROPN
iajs-901	315	1	fo	fo	ADP
iajs-901	315	2	r	r	NOUN
iajs-901	315	3	pure	pure	ADJ
iajs-901	315	4	&	&	CCONJ
iajs-901	315	5	appl	appl	PROPN
iajs-901	315	6	.	.	PUNCT
iajs-901	316	1	sc	sc	PROPN
iajs-901	316	2	i.	i.	PROPN
iajs-901	316	3	vo	vo	PROPN
iajs-901	316	4	l.23	l.23	PROPN
iajs-901	316	5	(	(	PUNCT
iajs-901	316	6	3	3	NUM
iajs-901	316	7	)	)	PUNCT
iajs-901	316	8	2010	2010	NUM
iajs-901	316	9	recall	recall	VERB
iajs-901	316	10	that	that	SCONJ
iajs-901	316	11	an	an	DET
iajs-901	316	12	integral	integral	ADJ
iajs-901	316	13	domain	domain	NOUN
iajs-901	316	14	r	r	NOUN
iajs-901	316	15	is	be	AUX
iajs-901	316	16	called	call	VERB
iajs-901	316	17	a	a	DET
iajs-901	316	18	dedekind	dedekind	ADJ
iajs-901	316	19	domain	domain	NOUN
iajs-901	316	20	if	if	SCONJ
iajs-901	316	21	every	every	DET
iajs-901	316	22	non	non	ADJ
iajs-901	316	23	-	-	ADJ
iajs-901	316	24	zero	zero	NUM
iajs-901	316	25	ideal	ideal	NOUN
iajs-901	316	26	of	of	ADP
iajs-901	316	27	r	r	NOUN
iajs-901	316	28	is	be	AUX
iajs-901	316	29	invertible	invertible	ADJ
iajs-901	316	30	and	and	CCONJ
iajs-901	316	31	an	an	DET
iajs-901	316	32	ideal	ideal	NOUN
iajs-901	316	33	i	i	PRON
iajs-901	316	34	is	be	AUX
iajs-901	316	35	called	call	VERB
iajs-901	316	36	invertible	invertible	ADJ
iajs-901	316	37	,	,	PUNCT
iajs-901	316	38	when	when	SCONJ
iajs-901	316	39	i	i	PRON
iajs-901	316	40	–	–	PUNCT
iajs-901	316	41	1	1	NUM
iajs-901	316	42	=	=	SYM
iajs-901	316	43	{	{	PUNCT
iajs-901	316	44	x	x	PROPN
iajs-901	316	45			NOUN
iajs-901	316	46	rs	rs	NOUN
iajs-901	316	47	:	:	PUNCT
iajs-901	316	48	x	x	X
iajs-901	316	49	i	i	PRON
iajs-901	316	50			VERB
iajs-901	316	51	r	r	X
iajs-901	316	52	}	}	PUNCT
iajs-901	316	53	and	and	CCONJ
iajs-901	316	54	s	s	VERB
iajs-901	316	55	is	be	AUX
iajs-901	316	56	the	the	DET
iajs-901	316	57	set	set	NOUN
iajs-901	316	58	of	of	ADP
iajs-901	316	59	non	non	ADJ
iajs-901	316	60	-	-	ADJ
iajs-901	316	61	zero	zero	ADJ
iajs-901	316	62	divisors	divisor	NOUN
iajs-901	316	63	of	of	ADP
iajs-901	316	64	r	r	NOUN
iajs-901	316	65	,	,	PUNCT
iajs-901	316	66	then	then	ADV
iajs-901	316	67	i	i	PRON
iajs-901	316	68	–	–	PUNCT
iajs-901	316	69	1	1	INTJ
iajs-901	317	1	i	i	X
iajs-901	317	2	=	=	SYM
iajs-901	317	3	r	r	NOUN
iajs-901	317	4	,	,	PUNCT
iajs-901	317	5	see	see	VERB
iajs-901	317	6	[	[	X
iajs-901	317	7	25	25	NUM
iajs-901	317	8	]	]	PUNCT
iajs-901	317	9	.	.	PUNCT
iajs-901	318	1	next	next	ADV
iajs-901	318	2	,	,	PUNCT
iajs-901	318	3	we	we	PRON
iajs-901	318	4	prove	prove	VERB
iajs-901	318	5	the	the	DET
iajs-901	318	6	following	following	NOUN
iajs-901	318	7	.	.	PUNCT
iajs-901	319	1	proposition	proposition	NOUN
iajs-901	319	2	(	(	PUNCT
iajs-901	319	3	35	35	NUM
iajs-901	319	4	):	):	PUNCT
iajs-901	319	5	if	if	SCONJ
iajs-901	319	6	r	r	NOUN
iajs-901	319	7	is	be	AUX
iajs-901	319	8	a	a	DET
iajs-901	319	9	dedekind	dedekind	ADJ
iajs-901	319	10	domain	domain	NOUN
iajs-901	319	11	and	and	CCONJ
iajs-901	319	12	m	m	NOUN
iajs-901	319	13	is	be	AUX
iajs-901	319	14	an	an	DET
iajs-901	319	15	r	r	NOUN
iajs-901	319	16	-	-	PUNCT
iajs-901	319	17	module	module	NOUN
iajs-901	319	18	.	.	PUNCT
iajs-901	320	1	then	then	ADV
iajs-901	320	2	the	the	DET
iajs-901	320	3	following	follow	VERB
iajs-901	320	4	statements	statement	NOUN
iajs-901	320	5	are	be	AUX
iajs-901	320	6	equivalent	equivalent	ADJ
iajs-901	320	7	:	:	PUNCT
iajs-901	320	8	1	1	X
iajs-901	320	9	.	.	X
iajs-901	320	10	m	m	PROPN
iajs-901	320	11	is	be	AUX
iajs-901	320	12	a	a	DET
iajs-901	320	13	faithful	faithful	ADJ
iajs-901	320	14	coprime	coprime	NOUN
iajs-901	320	15	r	r	NOUN
iajs-901	320	16	-	-	NOUN
iajs-901	320	17	module	module	NOUN
iajs-901	320	18	.	.	PUNCT
iajs-901	321	1	2	2	X
iajs-901	321	2	.	.	X
iajs-901	321	3	m	m	PROPN
iajs-901	321	4	is	be	AUX
iajs-901	321	5	a	a	DET
iajs-901	321	6	divisible	divisible	ADJ
iajs-901	321	7	r	r	NOUN
iajs-901	321	8	-	-	PUNCT
iajs-901	321	9	module	module	NOUN
iajs-901	321	10	.	.	PUNCT
iajs-901	322	1	3	3	X
iajs-901	322	2	.	.	X
iajs-901	322	3	m	m	PROPN
iajs-901	322	4	is	be	AUX
iajs-901	322	5	an	an	DET
iajs-901	322	6	injective	injective	ADJ
iajs-901	322	7	r	r	NOUN
iajs-901	322	8	-	-	PUNCT
iajs-901	322	9	module	module	NOUN
iajs-901	322	10	.	.	PUNCT
iajs-901	323	1	proof	proof	NOUN
iajs-901	323	2	:	:	PUNCT
iajs-901	323	3	(	(	PUNCT
iajs-901	323	4	1	1	X
iajs-901	323	5	)	)	PUNCT
iajs-901	323	6			NOUN
iajs-901	323	7	(	(	PUNCT
iajs-901	323	8	2	2	NUM
iajs-901	323	9	)	)	PUNCT
iajs-901	323	10	follows	follow	VERB
iajs-901	323	11	by	by	ADP
iajs-901	323	12	corollary	corollary	ADJ
iajs-901	323	13	(	(	PUNCT
iajs-901	323	14	30	30	NUM
iajs-901	323	15	)	)	PUNCT
iajs-901	323	16	.	.	PUNCT
iajs-901	324	1	(	(	PUNCT
iajs-901	324	2	2	2	X
iajs-901	324	3	)	)	PUNCT
iajs-901	324	4			NOUN
iajs-901	324	5	(	(	PUNCT
iajs-901	324	6	3	3	X
iajs-901	324	7	)	)	PUNCT
iajs-901	324	8	follows	follow	VERB
iajs-901	324	9	by	by	ADP
iajs-901	324	10	[	[	X
iajs-901	324	11	24	24	NUM
iajs-901	324	12	,	,	PUNCT
iajs-901	324	13	prop	prop	NOUN
iajs-901	324	14	.	.	PUNCT
iajs-901	325	1	2.10	2.10	NUM
iajs-901	325	2	,	,	PUNCT
iajs-901	325	3	p.36	p.36	NOUN
iajs-901	325	4	]	]	X
iajs-901	325	5	.	.	PUNCT
iajs-901	326	1	■	■	PUNCT
iajs-901	326	2	yassemi	yassemi	NOUN
iajs-901	326	3	s.	s.	PROPN
iajs-901	326	4	in	in	ADP
iajs-901	326	5	[	[	X
iajs-901	326	6	5	5	NUM
iajs-901	326	7	]	]	PUNCT
iajs-901	326	8	introduced	introduce	VERB
iajs-901	326	9	the	the	DET
iajs-901	326	10	following	following	ADJ
iajs-901	326	11	result	result	NOUN
iajs-901	326	12	without	without	ADP
iajs-901	326	13	proof	proof	NOUN
iajs-901	326	14	.	.	PUNCT
iajs-901	327	1	we	we	PRON
iajs-901	327	2	give	give	VERB
iajs-901	327	3	its	its	PRON
iajs-901	327	4	proof	proof	NOUN
iajs-901	327	5	for	for	ADP
iajs-901	327	6	sake	sake	NOUN
iajs-901	327	7	of	of	ADP
iajs-901	327	8	completeness	completeness	NOUN
iajs-901	327	9	.	.	PUNCT
iajs-901	328	1	theorem	theorem	NOUN
iajs-901	328	2	(	(	PUNCT
iajs-901	328	3	36	36	NUM
iajs-901	328	4	):	):	PUNCT
iajs-901	328	5	let	let	VERB
iajs-901	328	6	m	m	PRON
iajs-901	328	7	be	be	AUX
iajs-901	328	8	a	a	DET
iajs-901	328	9	prime	prime	ADJ
iajs-901	328	10	r	r	NOUN
iajs-901	328	11	-	-	PUNCT
iajs-901	328	12	module	module	NOUN
iajs-901	328	13	.	.	PUNCT
iajs-901	329	1	the	the	DET
iajs-901	329	2	following	follow	VERB
iajs-901	329	3	statements	statement	NOUN
iajs-901	329	4	are	be	AUX
iajs-901	329	5	equivalent	equivalent	ADJ
iajs-901	329	6	:	:	PUNCT
iajs-901	329	7	1	1	X
iajs-901	329	8	.	.	X
iajs-901	329	9	m	m	PROPN
iajs-901	329	10	is	be	AUX
iajs-901	329	11	a	a	DET
iajs-901	329	12	coprime	coprime	ADJ
iajs-901	329	13	r	r	NOUN
iajs-901	329	14	-	-	NOUN
iajs-901	329	15	module	module	NOUN
iajs-901	329	16	.	.	PUNCT
iajs-901	330	1	2	2	X
iajs-901	330	2	.	.	X
iajs-901	330	3	m	m	PROPN
iajs-901	330	4	is	be	AUX
iajs-901	330	5	an	an	DET
iajs-901	330	6	injective	injective	ADJ
iajs-901	330	7	r/	r/	NOUN
iajs-901	330	8	r	r	PROPN
iajs-901	330	9	ann	ann	PROPN
iajs-901	331	1	m	m	NOUN
iajs-901	331	2	-	-	NOUN
iajs-901	331	3	module	module	NOUN
iajs-901	331	4	.	.	PUNCT
iajs-901	332	1	proof	proof	NOUN
iajs-901	332	2	:	:	PUNCT
iajs-901	332	3	since	since	SCONJ
iajs-901	332	4	m	m	PROPN
iajs-901	332	5	is	be	AUX
iajs-901	332	6	a	a	DET
iajs-901	332	7	prime	prime	ADJ
iajs-901	332	8	r	r	NOUN
iajs-901	332	9	-	-	PUNCT
iajs-901	332	10	module	module	NOUN
iajs-901	332	11	,	,	PUNCT
iajs-901	332	12	then	then	ADV
iajs-901	332	13	by	by	ADP
iajs-901	332	14	[	[	X
iajs-901	332	15	2	2	NUM
iajs-901	332	16	,	,	PUNCT
iajs-901	332	17	rem	rem	X
iajs-901	332	18	.	.	PROPN
iajs-901	332	19	and	and	CCONJ
iajs-901	332	20	ex	ex	ADJ
iajs-901	332	21	.	.	PUNCT
iajs-901	333	1	(	(	PUNCT
iajs-901	333	2	1.1.3	1.1.3	NUM
iajs-901	333	3	(	(	PUNCT
iajs-901	333	4	3	3	NUM
iajs-901	333	5	)	)	PUNCT
iajs-901	333	6	]	]	PUNCT
iajs-901	333	7	,	,	PUNCT
iajs-901	333	8	r	r	PROPN
iajs-901	333	9	ann	ann	PROPN
iajs-901	333	10	m	m	NOUN
iajs-901	333	11	is	be	AUX
iajs-901	333	12	a	a	DET
iajs-901	333	13	prime	prime	ADJ
iajs-901	333	14	ideal	ideal	NOUN
iajs-901	333	15	,	,	PUNCT
iajs-901	333	16	so	so	CCONJ
iajs-901	333	17	r	r	NOUN
iajs-901	333	18	is	be	AUX
iajs-901	333	19	an	an	DET
iajs-901	333	20	integral	integral	ADJ
iajs-901	333	21	domain	domain	NOUN
iajs-901	333	22	and	and	CCONJ
iajs-901	333	23	m	m	NOUN
iajs-901	333	24	is	be	AUX
iajs-901	333	25	a	a	DET
iajs-901	333	26	torsion	torsion	NOUN
iajs-901	333	27	free	free	ADJ
iajs-901	333	28	r	r	NOUN
iajs-901	333	29	=	=	SYM
iajs-901	333	30	r	r	NOUN
iajs-901	333	31	/	/	SYM
iajs-901	333	32	r	r	NOUN
iajs-901	333	33	ann	ann	PROPN
iajs-901	333	34	m	m	NOUN
iajs-901	333	35	-	-	NOUN
iajs-901	333	36	module	module	NOUN
iajs-901	333	37	,	,	PUNCT
iajs-901	333	38	see	see	VERB
iajs-901	333	39	[	[	X
iajs-901	333	40	20	20	NUM
iajs-901	333	41	]	]	PUNCT
iajs-901	333	42	,	,	PUNCT
iajs-901	333	43	[	[	X
iajs-901	333	44	11	11	NUM
iajs-901	333	45	]	]	PUNCT
iajs-901	333	46	.	.	PUNCT
iajs-901	334	1	(	(	PUNCT
iajs-901	334	2	1	1	X
iajs-901	334	3	)	)	PUNCT
iajs-901	334	4			NOUN
iajs-901	334	5	(	(	PUNCT
iajs-901	334	6	2	2	NUM
iajs-901	334	7	)	)	PUNCT
iajs-901	334	8	.	.	PUNCT
iajs-901	335	1	since	since	SCONJ
iajs-901	335	2	m	m	PROPN
iajs-901	335	3	is	be	AUX
iajs-901	335	4	a	a	DET
iajs-901	335	5	coprime	coprime	ADJ
iajs-901	335	6	r	r	NOUN
iajs-901	335	7	-	-	NOUN
iajs-901	335	8	module	module	NOUN
iajs-901	335	9	,	,	PUNCT
iajs-901	335	10	then	then	ADV
iajs-901	335	11	by	by	ADP
iajs-901	335	12	corollary	corollary	ADJ
iajs-901	335	13	(	(	PUNCT
iajs-901	335	14	10	10	NUM
iajs-901	335	15	)	)	PUNCT
iajs-901	335	16	m	m	VERB
iajs-901	335	17	is	be	AUX
iajs-901	335	18	a	a	DET
iajs-901	335	19	coprime	coprime	NOUN
iajs-901	335	20	r	r	NOUN
iajs-901	335	21	module	module	NOUN
iajs-901	335	22	.	.	PUNCT
iajs-901	336	1	hence	hence	ADV
iajs-901	336	2	by	by	ADP
iajs-901	336	3	proposition	proposition	NOUN
iajs-901	336	4	(	(	PUNCT
iajs-901	336	5	34	34	NUM
iajs-901	336	6	)	)	PUNCT
iajs-901	336	7	m	m	VERB
iajs-901	336	8	is	be	AUX
iajs-901	336	9	an	an	DET
iajs-901	336	10	injective	injective	ADJ
iajs-901	336	11	r	r	NOUN
iajs-901	336	12	-module	-module	NOUN
iajs-901	336	13	.	.	PUNCT
iajs-901	337	1	(	(	PUNCT
iajs-901	337	2	2	2	X
iajs-901	337	3	)	)	PUNCT
iajs-901	337	4			NOUN
iajs-901	337	5	(	(	PUNCT
iajs-901	337	6	1	1	NUM
iajs-901	337	7	)	)	PUNCT
iajs-901	337	8	.	.	PUNCT
iajs-901	338	1	if	if	SCONJ
iajs-901	338	2	m	m	NOUN
iajs-901	338	3	is	be	AUX
iajs-901	338	4	an	an	DET
iajs-901	338	5	injective	injective	ADJ
iajs-901	338	6	r	r	NOUN
iajs-901	338	7	-module	-module	NOUN
iajs-901	338	8	,	,	PUNCT
iajs-901	338	9	then	then	ADV
iajs-901	338	10	by	by	ADP
iajs-901	338	11	corollary	corollary	ADJ
iajs-901	338	12	(	(	PUNCT
iajs-901	338	13	33	33	NUM
iajs-901	338	14	)	)	PUNCT
iajs-901	338	15	m	m	VERB
iajs-901	338	16	is	be	AUX
iajs-901	338	17	a	a	DET
iajs-901	338	18	coprime	coprime	NOUN
iajs-901	338	19	r	r	NOUN
iajs-901	338	20	module	module	NOUN
iajs-901	338	21	.	.	PUNCT
iajs-901	339	1	hence	hence	ADV
iajs-901	339	2	by	by	ADP
iajs-901	339	3	corollary	corollary	ADJ
iajs-901	339	4	(	(	PUNCT
iajs-901	339	5	10	10	NUM
iajs-901	339	6	)	)	PUNCT
iajs-901	339	7	m	m	VERB
iajs-901	339	8	is	be	AUX
iajs-901	339	9	a	a	DET
iajs-901	339	10	coprime	coprime	ADJ
iajs-901	339	11	r	r	NOUN
iajs-901	339	12	-	-	NOUN
iajs-901	339	13	module	module	NOUN
iajs-901	339	14	.	.	PUNCT
iajs-901	340	1	■	■	PUNCT
iajs-901	340	2	recall	recall	VERB
iajs-901	340	3	that	that	SCONJ
iajs-901	340	4	an	an	DET
iajs-901	340	5	r	r	NOUN
iajs-901	340	6	-	-	PUNCT
iajs-901	340	7	module	module	NOUN
iajs-901	340	8	m	m	NOUN
iajs-901	340	9	is	be	AUX
iajs-901	340	10	called	call	VERB
iajs-901	340	11	flat	flat	ADJ
iajs-901	340	12	if	if	SCONJ
iajs-901	340	13	n	n	ADV
iajs-901	340	14	k	k	NOUN
iajs-901	340	15	k=1	k=1	PUNCT
iajs-901	341	1	ka	ka	NOUN
iajs-901	342	1	=	=	SYM
iajs-901	342	2	0	0	NUM
iajs-901	342	3	,	,	PUNCT
iajs-901	342	4	where	where	SCONJ
iajs-901	342	5	k	k	PROPN
iajs-901	342	6			PROPN
iajs-901	342	7	r	r	PROPN
iajs-901	342	8	,	,	PUNCT
iajs-901	342	9	ka	ka	PROPN
iajs-901	342	10			PROPN
iajs-901	342	11	m	m	PROPN
iajs-901	342	12	,	,	PUNCT
iajs-901	342	13	then	then	ADV
iajs-901	342	14	there	there	PRON
iajs-901	342	15	exist	exist	VERB
iajs-901	342	16	b1	b1	NOUN
iajs-901	342	17	,	,	PUNCT
iajs-901	342	18	…	…	PUNCT
iajs-901	342	19	,	,	PUNCT
iajs-901	342	20	bn	bn	ADP
iajs-901	342	21			PROPN
iajs-901	342	22	m	m	VERB
iajs-901	342	23	and	and	CCONJ
iajs-901	342	24	{	{	PUNCT
iajs-901	342	25	u	u	NOUN
iajs-901	342	26	i	i	PROPN
iajs-901	342	27	k	k	PROPN
iajs-901	342	28	}	}	PUNCT
iajs-901	342	29			PROPN
iajs-901	342	30	r	r	PROPN
iajs-901	342	31	,	,	PUNCT
iajs-901	342	32	i	i	NOUN
iajs-901	342	33	=	=	NOUN
iajs-901	342	34	1	1	NUM
iajs-901	342	35	,	,	PUNCT
iajs-901	342	36	…	…	PUNCT
iajs-901	342	37	,	,	PUNCT
iajs-901	342	38	n	n	CCONJ
iajs-901	342	39	,	,	PUNCT
iajs-901	342	40	k	k	PROPN
iajs-901	342	41	=	=	SYM
iajs-901	342	42	1,2,	1,2,	NUM
iajs-901	342	43	…	…	PUNCT
iajs-901	342	44	,r	,r	PUNCT
iajs-901	342	45	such	such	ADJ
iajs-901	343	1	that	that	SCONJ
iajs-901	343	2	n	n	PROPN
iajs-901	343	3	ik	ik	PROPN
iajs-901	343	4	i=1	i=1	PROPN
iajs-901	343	5	iu	iu	ADP
iajs-901	343	6	b	b	PROPN
iajs-901	343	7	=	=	SYM
iajs-901	343	8	ka	ka	PROPN
iajs-901	343	9	and	and	CCONJ
iajs-901	343	10	r	r	PROPN
iajs-901	343	11	ik	ik	PROPN
iajs-901	343	12	k=1	k=1	PROPN
iajs-901	343	13	ku	ku	PROPN
iajs-901	343	14			PROPN
iajs-901	343	15	=	=	SYM
iajs-901	343	16	0	0	NUM
iajs-901	343	17	,	,	PUNCT
iajs-901	343	18	see	see	VERB
iajs-901	343	19	[	[	X
iajs-901	343	20	2	2	NUM
iajs-901	343	21	]	]	PUNCT
iajs-901	343	22	.	.	PUNCT
iajs-901	344	1	the	the	DET
iajs-901	344	2	following	follow	VERB
iajs-901	344	3	theorem	theorem	NOUN
iajs-901	344	4	appeared	appear	VERB
iajs-901	344	5	in	in	ADP
iajs-901	344	6	[	[	X
iajs-901	344	7	18	18	NUM
iajs-901	344	8	]	]	PUNCT
iajs-901	344	9	,	,	PUNCT
iajs-901	344	10	we	we	PRON
iajs-901	344	11	give	give	VERB
iajs-901	344	12	the	the	DET
iajs-901	344	13	details	detail	NOUN
iajs-901	344	14	of	of	ADP
iajs-901	344	15	the	the	DET
iajs-901	344	16	proof	proof	NOUN
iajs-901	344	17	for	for	ADP
iajs-901	344	18	completeness	completeness	NOUN
iajs-901	344	19	.	.	PUNCT
iajs-901	345	1	recall	recall	VERB
iajs-901	345	2	that	that	SCONJ
iajs-901	345	3	a	a	DET
iajs-901	345	4	subset	subset	NOUN
iajs-901	345	5	a	a	PRON
iajs-901	345	6	of	of	ADP
iajs-901	345	7	an	an	DET
iajs-901	345	8	r	r	NOUN
iajs-901	345	9	-	-	PUNCT
iajs-901	345	10	module	module	NOUN
iajs-901	345	11	m	m	NOUN
iajs-901	345	12	is	be	AUX
iajs-901	345	13	called	call	VERB
iajs-901	345	14	a	a	DET
iajs-901	345	15	basis	basis	NOUN
iajs-901	345	16	of	of	ADP
iajs-901	345	17	m	m	PROPN
iajs-901	345	18	,	,	PUNCT
iajs-901	345	19	if	if	SCONJ
iajs-901	345	20	a	a	DET
iajs-901	345	21	generates	generate	VERB
iajs-901	345	22	m	m	PRON
iajs-901	345	23	and	and	CCONJ
iajs-901	345	24	a	a	PRON
iajs-901	345	25	is	be	AUX
iajs-901	345	26	r	r	NOUN
iajs-901	345	27	-	-	PUNCT
iajs-901	345	28	linearly	linearly	ADV
iajs-901	345	29	independent	independent	ADJ
iajs-901	345	30	.	.	PUNCT
iajs-901	346	1	m	m	PROPN
iajs-901	346	2	is	be	AUX
iajs-901	346	3	said	say	VERB
iajs-901	346	4	to	to	PART
iajs-901	346	5	be	be	AUX
iajs-901	346	6	a	a	DET
iajs-901	346	7	free	free	ADJ
iajs-901	346	8	r	r	NOUN
iajs-901	346	9	-	-	PUNCT
iajs-901	346	10	module	module	NOUN
iajs-901	346	11	,	,	PUNCT
iajs-901	346	12	if	if	SCONJ
iajs-901	346	13	m	m	PROPN
iajs-901	346	14	has	have	VERB
iajs-901	346	15	a	a	DET
iajs-901	346	16	basis	basis	NOUN
iajs-901	346	17	[	[	X
iajs-901	346	18	25	25	NUM
iajs-901	346	19	,	,	PUNCT
iajs-901	346	20	p	p	NOUN
iajs-901	346	21	.190	.190	NUM
iajs-901	346	22	]	]	PUNCT
iajs-901	346	23	.	.	PUNCT
iajs-901	347	1	theorem	theorem	NOUN
iajs-901	347	2	(	(	PUNCT
iajs-901	347	3	37	37	NUM
iajs-901	347	4	):	):	PUNCT
iajs-901	347	5	let	let	VERB
iajs-901	347	6	m	m	PRON
iajs-901	347	7	be	be	AUX
iajs-901	347	8	a	a	DET
iajs-901	347	9	coprime	coprime	ADJ
iajs-901	347	10	r	r	NOUN
iajs-901	347	11	-	-	NOUN
iajs-901	347	12	module	module	NOUN
iajs-901	347	13	.	.	PUNCT
iajs-901	348	1	then	then	ADV
iajs-901	348	2	the	the	DET
iajs-901	348	3	following	follow	VERB
iajs-901	348	4	statements	statement	NOUN
iajs-901	348	5	are	be	AUX
iajs-901	348	6	equivalent	equivalent	ADJ
iajs-901	348	7	:	:	PUNCT
iajs-901	348	8	1	1	X
iajs-901	348	9	.	.	X
iajs-901	348	10	m	m	PROPN
iajs-901	348	11	is	be	AUX
iajs-901	348	12	a	a	DET
iajs-901	348	13	prime	prime	ADJ
iajs-901	348	14	r	r	NOUN
iajs-901	348	15	-	-	PUNCT
iajs-901	348	16	module	module	NOUN
iajs-901	348	17	.	.	PUNCT
iajs-901	349	1	2	2	X
iajs-901	349	2	.	.	X
iajs-901	349	3	-m	-m	PUNCT
iajs-901	349	4	is	be	AUX
iajs-901	349	5	a	a	DET
iajs-901	349	6	flat	flat	ADJ
iajs-901	349	7	r	r	NOUN
iajs-901	349	8	-module	-module	NOUN
iajs-901	349	9	,	,	PUNCT
iajs-901	349	10	where	where	SCONJ
iajs-901	349	11	r	r	NOUN
iajs-901	349	12	=	=	NOUN
iajs-901	349	13	r	r	NOUN
iajs-901	349	14	/	/	SYM
iajs-901	349	15	r	r	NOUN
iajs-901	349	16	ann	ann	PROPN
iajs-901	349	17	m.	m.	NOUN
iajs-901	349	18	proof	proof	NOUN
iajs-901	349	19	:	:	PUNCT
iajs-901	349	20	since	since	SCONJ
iajs-901	349	21	m	m	PROPN
iajs-901	349	22	is	be	AUX
iajs-901	349	23	a	a	DET
iajs-901	349	24	coprime	coprime	ADJ
iajs-901	349	25	r	r	NOUN
iajs-901	349	26	-	-	NOUN
iajs-901	349	27	module	module	NOUN
iajs-901	349	28	,	,	PUNCT
iajs-901	349	29	then	then	ADV
iajs-901	349	30	r	r	NOUN
iajs-901	349	31	=	=	PUNCT
iajs-901	349	32	r/	r/	ADV
iajs-901	349	33	r	r	NOUN
iajs-901	349	34	ann	ann	PROPN
iajs-901	349	35	m	m	NOUN
iajs-901	349	36	is	be	AUX
iajs-901	349	37	an	an	DET
iajs-901	349	38	integral	integral	ADJ
iajs-901	349	39	domain	domain	NOUN
iajs-901	349	40	.	.	PUNCT
iajs-901	350	1	also	also	ADV
iajs-901	350	2	by	by	ADP
iajs-901	350	3	proposition	proposition	NOUN
iajs-901	350	4	(	(	PUNCT
iajs-901	350	5	24	24	NUM
iajs-901	350	6	)	)	PUNCT
iajs-901	350	7	,	,	PUNCT
iajs-901	350	8	m	m	VERB
iajs-901	350	9	is	be	AUX
iajs-901	350	10	a	a	DET
iajs-901	350	11	divisible	divisible	ADJ
iajs-901	350	12	r	r	NOUN
iajs-901	350	13	-module	-module	NOUN
iajs-901	350	14	.	.	PUNCT
iajs-901	351	1	ihjpas	ihjpa	VERB
iajs-901	351	2	ibn	ibn	PROPN
iajs-901	351	3	alhaitham	alhaitham	PROPN
iajs-901	352	1	j.	j.	PROPN
iajs-901	353	1	fo	fo	ADP
iajs-901	353	2	r	r	NOUN
iajs-901	353	3	pure	pure	ADJ
iajs-901	353	4	&	&	CCONJ
iajs-901	353	5	appl	appl	PROPN
iajs-901	353	6	.	.	PUNCT
iajs-901	354	1	sc	sc	PROPN
iajs-901	354	2	i.	i.	PROPN
iajs-901	354	3	vo	vo	PROPN
iajs-901	354	4	l.23	l.23	PROPN
iajs-901	354	5	(	(	PUNCT
iajs-901	354	6	3	3	NUM
iajs-901	354	7	)	)	PUNCT
iajs-901	354	8	2010	2010	NUM
iajs-901	354	9	(	(	PUNCT
iajs-901	354	10	1	1	NUM
iajs-901	354	11	)	)	PUNCT
iajs-901	354	12			NOUN
iajs-901	354	13	(	(	PUNCT
iajs-901	354	14	2	2	NUM
iajs-901	354	15	)	)	PUNCT
iajs-901	354	16	.	.	PUNCT
iajs-901	355	1	since	since	SCONJ
iajs-901	355	2	m	m	PROPN
iajs-901	355	3	is	be	AUX
iajs-901	355	4	a	a	DET
iajs-901	355	5	prime	prime	ADJ
iajs-901	355	6	r	r	NOUN
iajs-901	355	7	-	-	PUNCT
iajs-901	355	8	module	module	NOUN
iajs-901	355	9	,	,	PUNCT
iajs-901	355	10	then	then	ADV
iajs-901	355	11	m	m	NOUN
iajs-901	355	12	is	be	AUX
iajs-901	355	13	a	a	DET
iajs-901	355	14	torsion	torsion	NOUN
iajs-901	355	15	free	free	ADJ
iajs-901	355	16	r	r	NOUN
iajs-901	355	17	-module	-module	NOUN
iajs-901	355	18	,	,	PUNCT
iajs-901	355	19	where	where	SCONJ
iajs-901	355	20	r	r	NOUN
iajs-901	355	21	=	=	NOUN
iajs-901	355	22	r	r	NOUN
iajs-901	355	23	/	/	SYM
iajs-901	355	24	r	r	NOUN
iajs-901	355	25	ann	ann	PROPN
iajs-901	355	26	m.	m.	NOUN
iajs-901	355	27	now	now	ADV
iajs-901	355	28	,	,	PUNCT
iajs-901	355	29	we	we	PRON
iajs-901	355	30	can	can	AUX
iajs-901	355	31	show	show	VERB
iajs-901	355	32	that	that	SCONJ
iajs-901	355	33	m	m	PROPN
iajs-901	355	34	is	be	AUX
iajs-901	355	35	a	a	DET
iajs-901	355	36	vector	vector	NOUN
iajs-901	355	37	space	space	NOUN
iajs-901	355	38	over	over	ADP
iajs-901	355	39	k	k	PROPN
iajs-901	356	1	=	=	PUNCT
iajs-901	357	1	the	the	DET
iajs-901	357	2	total	total	ADJ
iajs-901	357	3	quotient	quotient	NOUN
iajs-901	357	4	field	field	NOUN
iajs-901	357	5	of	of	ADP
iajs-901	357	6	r	r	NOUN
iajs-901	357	7	as	as	SCONJ
iajs-901	357	8	follows	follow	VERB
iajs-901	357	9	:	:	PUNCT
iajs-901	357	10	let	let	VERB
iajs-901	357	11	+	+	CCONJ
iajs-901	357	12	ann	ann	PROPN
iajs-901	357	13	m	m	PROPN
iajs-901	357	14	s	s	PART
iajs-901	357	15	+	+	CCONJ
iajs-901	357	16	ann	ann	PROPN
iajs-901	357	17	m	m	PROPN
iajs-901	357	18	r	r	NOUN
iajs-901	357	19			PROPN
iajs-901	357	20	k	k	PROPN
iajs-901	357	21	,	,	PUNCT
iajs-901	357	22	where	where	SCONJ
iajs-901	357	23	r	r	NOUN
iajs-901	357	24	,	,	PUNCT
iajs-901	357	25	s	s	PART
iajs-901	357	26			NOUN
iajs-901	357	27	r	r	NOUN
iajs-901	357	28	,	,	PUNCT
iajs-901	357	29	s	s	PART
iajs-901	357	30			CCONJ
iajs-901	357	31	r	r	NOUN
iajs-901	357	32	ann	ann	PROPN
iajs-901	357	33	m.	m.	NOUN
iajs-901	357	34	let	let	VERB
iajs-901	357	35	m	m	PRON
iajs-901	357	36			PROPN
iajs-901	357	37	m	m	NOUN
iajs-901	357	38	,	,	PUNCT
iajs-901	357	39	m	m	VERB
iajs-901	357	40	=	=	X
iajs-901	358	1	(	(	PUNCT
iajs-901	358	2	s	s	X
iajs-901	358	3	+	+	X
iajs-901	358	4	r	r	X
iajs-901	358	5	ann	ann	PROPN
iajs-901	358	6	m	m	PROPN
iajs-901	358	7	)	)	PUNCT
iajs-901	358	8	m	m	NOUN
iajs-901	358	9	,	,	PUNCT
iajs-901	358	10	for	for	ADP
iajs-901	358	11	some	some	DET
iajs-901	358	12	m	m	NOUN
iajs-901	358	13	m	m	VERB
iajs-901	358	14	since	since	SCONJ
iajs-901	358	15	m	m	PROPN
iajs-901	358	16	is	be	AUX
iajs-901	358	17	a	a	DET
iajs-901	358	18	divisible	divisible	ADJ
iajs-901	358	19	r	r	NOUN
iajs-901	358	20	-module	-module	NOUN
iajs-901	358	21	.	.	PUNCT
iajs-901	359	1	it	it	PRON
iajs-901	359	2	follows	follow	VERB
iajs-901	359	3	that	that	PRON
iajs-901	359	4	+	+	CCONJ
iajs-901	360	1	ann	ann	PROPN
iajs-901	360	2	m	m	PROPN
iajs-901	360	3	s	s	PART
iajs-901	360	4	+	+	CCONJ
iajs-901	360	5	ann	ann	PROPN
iajs-901	360	6	m	m	VERB
iajs-901	360	7	r	r	NOUN
iajs-901	360	8			PROPN
iajs-901	360	9	m	m	NOUN
iajs-901	360	10	=	=	PUNCT
iajs-901	360	11	(	(	PUNCT
iajs-901	360	12	r	r	NOUN
iajs-901	360	13	+	+	NOUN
iajs-901	360	14	r	r	NOUN
iajs-901	360	15	ann	ann	PROPN
iajs-901	360	16	m	m	PROPN
iajs-901	360	17	)	)	PUNCT
iajs-901	360	18	m	m	NOUN
iajs-901	360	19	=	=	NOUN
iajs-901	360	20	r	r	NOUN
iajs-901	360	21	m	m	NOUN
iajs-901	360	22			PROPN
iajs-901	360	23	m.	m.	NOUN
iajs-901	360	24	thus	thus	ADV
iajs-901	360	25	m	m	VERB
iajs-901	360	26	is	be	AUX
iajs-901	360	27	a	a	DET
iajs-901	360	28	vector	vector	NOUN
iajs-901	360	29	space	space	NOUN
iajs-901	360	30	over	over	ADP
iajs-901	360	31	k	k	PROPN
iajs-901	360	32	,	,	PUNCT
iajs-901	360	33	so	so	SCONJ
iajs-901	360	34	it	it	PRON
iajs-901	360	35	has	have	VERB
iajs-901	360	36	a	a	DET
iajs-901	360	37	basis	basis	NOUN
iajs-901	360	38	.	.	PUNCT
iajs-901	361	1	it	it	PRON
iajs-901	361	2	follows	follow	VERB
iajs-901	361	3	that	that	SCONJ
iajs-901	361	4	m	m	PROPN
iajs-901	361	5	is	be	AUX
iajs-901	361	6	a	a	DET
iajs-901	361	7	free	free	ADJ
iajs-901	361	8	k	k	NOUN
iajs-901	361	9	-	-	NOUN
iajs-901	361	10	module	module	NOUN
iajs-901	361	11	,	,	PUNCT
iajs-901	361	12	hence	hence	ADV
iajs-901	361	13	m	m	VERB
iajs-901	361	14	is	be	AUX
iajs-901	361	15	a	a	DET
iajs-901	361	16	flat	flat	ADJ
iajs-901	361	17	k	k	NOUN
iajs-901	361	18	-	-	NOUN
iajs-901	361	19	module	module	NOUN
iajs-901	361	20	[	[	X
iajs-901	361	21	25	25	NUM
iajs-901	361	22	,	,	PUNCT
iajs-901	361	23	prop1.26	prop1.26	ADJ
iajs-901	361	24	,	,	PUNCT
iajs-901	361	25	p.22	p.22	NOUN
iajs-901	361	26	]	]	X
iajs-901	361	27	.	.	PUNCT
iajs-901	362	1	then	then	ADV
iajs-901	362	2	k	k	PROPN
iajs-901	362	3	is	be	AUX
iajs-901	362	4	a	a	DET
iajs-901	362	5	flat	flat	ADJ
iajs-901	362	6	r	r	NOUN
iajs-901	362	7	-module	-module	NOUN
iajs-901	362	8	by	by	ADP
iajs-901	362	9	[	[	X
iajs-901	362	10	27	27	NUM
iajs-901	362	11	,	,	PUNCT
iajs-901	362	12	ex	ex	NOUN
iajs-901	362	13	.	.	NOUN
iajs-901	362	14	20	20	NUM
iajs-901	362	15	,	,	PUNCT
iajs-901	362	16	p.319	p.319	NOUN
iajs-901	362	17	]	]	PUNCT
iajs-901	362	18	.	.	PUNCT
iajs-901	363	1	thus	thus	ADV
iajs-901	363	2	by	by	ADP
iajs-901	363	3	[	[	X
iajs-901	363	4	19	19	NUM
iajs-901	363	5	,	,	PUNCT
iajs-901	363	6	exc	exc	NOUN
iajs-901	363	7	.	.	PUNCT
iajs-901	363	8	9(a	9(a	NUM
iajs-901	363	9	)	)	PUNCT
iajs-901	363	10	,	,	PUNCT
iajs-901	363	11	p.32	p.32	AUX
iajs-901	363	12	]	]	X
iajs-901	363	13	m	m	VERB
iajs-901	363	14	is	be	AUX
iajs-901	363	15	flat	flat	ADJ
iajs-901	363	16	r	r	NOUN
iajs-901	363	17	-module	-module	NOUN
iajs-901	363	18	.	.	PUNCT
iajs-901	364	1	(	(	PUNCT
iajs-901	364	2	2	2	X
iajs-901	364	3	)	)	PUNCT
iajs-901	364	4			NOUN
iajs-901	364	5	(	(	PUNCT
iajs-901	364	6	1	1	NUM
iajs-901	364	7	)	)	PUNCT
iajs-901	364	8	.	.	PUNCT
iajs-901	365	1	[	[	X
iajs-901	365	2	the	the	DET
iajs-901	365	3	proof	proof	NOUN
iajs-901	365	4	is	be	AUX
iajs-901	365	5	different	different	ADJ
iajs-901	365	6	from	from	ADP
iajs-901	365	7	the	the	DET
iajs-901	365	8	proof	proof	NOUN
iajs-901	365	9	given	give	VERB
iajs-901	365	10	in	in	ADP
iajs-901	365	11	[	[	NOUN
iajs-901	365	12	29	29	NUM
iajs-901	365	13	]	]	PUNCT
iajs-901	365	14	.	.	PUNCT
iajs-901	366	1	since	since	SCONJ
iajs-901	366	2	r	r	NOUN
iajs-901	366	3	is	be	AUX
iajs-901	366	4	an	an	DET
iajs-901	366	5	integral	integral	ADJ
iajs-901	366	6	domain	domain	NOUN
iajs-901	366	7	,	,	PUNCT
iajs-901	366	8	(	(	PUNCT
iajs-901	366	9	0	0	X
iajs-901	366	10	)	)	PUNCT
iajs-901	366	11	is	be	AUX
iajs-901	366	12	a	a	DET
iajs-901	366	13	prime	prime	ADJ
iajs-901	366	14	ideal	ideal	NOUN
iajs-901	366	15	of	of	ADP
iajs-901	366	16	r	r	NOUN
iajs-901	366	17	.	.	PUNCT
iajs-901	367	1	hence	hence	ADV
iajs-901	367	2	by	by	ADP
iajs-901	367	3	[	[	X
iajs-901	367	4	30	30	NUM
iajs-901	367	5	,	,	PUNCT
iajs-901	367	6	corollary	corollary	ADJ
iajs-901	367	7	4.9	4.9	NUM
iajs-901	367	8	,	,	PUNCT
iajs-901	367	9	ch.1	ch.1	PROPN
iajs-901	367	10	]	]	X
iajs-901	367	11	(	(	PUNCT
iajs-901	367	12	0	0	NUM
iajs-901	367	13	)	)	PUNCT
iajs-901	367	14	m	m	VERB
iajs-901	367	15	=	=	SYM
iajs-901	367	16	(	(	PUNCT
iajs-901	367	17	0	0	NUM
iajs-901	367	18	)	)	PUNCT
iajs-901	367	19	is	be	AUX
iajs-901	367	20	a	a	DET
iajs-901	367	21	prime	prime	ADJ
iajs-901	367	22	r	r	NOUN
iajs-901	367	23	-submodule	-submodule	NOUN
iajs-901	367	24	of	of	ADP
iajs-901	367	25	m.	m.	NOUN
iajs-901	367	26	thus	thus	ADV
iajs-901	367	27	(	(	PUNCT
iajs-901	367	28	0	0	X
iajs-901	367	29	)	)	PUNCT
iajs-901	368	1	is	be	AUX
iajs-901	368	2	a	a	DET
iajs-901	368	3	prime	prime	ADJ
iajs-901	368	4	r	r	NOUN
iajs-901	368	5	-submodule	-submodule	NOUN
iajs-901	368	6	.	.	PUNCT
iajs-901	369	1	then	then	ADV
iajs-901	369	2	it	it	PRON
iajs-901	369	3	is	be	AUX
iajs-901	369	4	easy	easy	ADJ
iajs-901	369	5	to	to	PART
iajs-901	369	6	check	check	VERB
iajs-901	369	7	that	that	PRON
iajs-901	369	8	(	(	PUNCT
iajs-901	369	9	0	0	X
iajs-901	369	10	)	)	PUNCT
iajs-901	369	11	is	be	AUX
iajs-901	369	12	a	a	DET
iajs-901	369	13	prime	prime	ADJ
iajs-901	369	14	r	r	NOUN
iajs-901	369	15	-	-	PUNCT
iajs-901	369	16	submodule	submodule	NOUN
iajs-901	369	17	of	of	ADP
iajs-901	369	18	m	m	PROPN
iajs-901	369	19	and	and	CCONJ
iajs-901	369	20	hence	hence	ADV
iajs-901	369	21	a	a	DET
iajs-901	369	22	prime	prime	ADJ
iajs-901	369	23	r	r	NOUN
iajs-901	369	24	-	-	PUNCT
iajs-901	369	25	module	module	NOUN
iajs-901	369	26	.	.	PUNCT
iajs-901	370	1	finally	finally	ADV
iajs-901	370	2	,	,	PUNCT
iajs-901	370	3	we	we	PRON
iajs-901	370	4	turn	turn	VERB
iajs-901	370	5	our	our	PRON
iajs-901	370	6	attention	attention	NOUN
iajs-901	370	7	to	to	ADP
iajs-901	370	8	the	the	DET
iajs-901	370	9	localization	localization	NOUN
iajs-901	370	10	of	of	ADP
iajs-901	370	11	coprime	coprime	NOUN
iajs-901	370	12	modules	module	NOUN
iajs-901	370	13	.	.	PUNCT
iajs-901	371	1	first	first	ADV
iajs-901	371	2	we	we	PRON
iajs-901	371	3	have	have	VERB
iajs-901	371	4	.	.	PUNCT
iajs-901	372	1	proposition	proposition	NOUN
iajs-901	372	2	(	(	PUNCT
iajs-901	372	3	38	38	NUM
iajs-901	372	4	):	):	PUNCT
iajs-901	372	5	let	let	VERB
iajs-901	372	6	m	m	PRON
iajs-901	372	7	be	be	AUX
iajs-901	372	8	a	a	DET
iajs-901	372	9	coprime	coprime	ADJ
iajs-901	372	10	r	r	NOUN
iajs-901	372	11	-	-	NOUN
iajs-901	372	12	module	module	NOUN
iajs-901	372	13	,	,	PUNCT
iajs-901	372	14	then	then	ADV
iajs-901	372	15	ms	ms	PROPN
iajs-901	372	16	is	be	AUX
iajs-901	372	17	a	a	DET
iajs-901	372	18	coprime	coprime	NOUN
iajs-901	372	19	r	r	NOUN
iajs-901	372	20	s	s	NOUN
iajs-901	372	21	-module	-module	NOUN
iajs-901	372	22	,	,	PUNCT
iajs-901	372	23	where	where	SCONJ
iajs-901	372	24	s	s	NOUN
iajs-901	372	25	is	be	AUX
iajs-901	372	26	a	a	DET
iajs-901	372	27	multiplicatively	multiplicatively	ADV
iajs-901	372	28	closed	close	VERB
iajs-901	372	29	subset	subset	NOUN
iajs-901	372	30	of	of	ADP
iajs-901	372	31	r.	r.	PROPN
iajs-901	372	32	proof	proof	NOUN
iajs-901	372	33	:	:	PUNCT
iajs-901	372	34	it	it	PRON
iajs-901	372	35	follows	follow	VERB
iajs-901	372	36	by	by	ADP
iajs-901	372	37	theorem	theorem	NOUN
iajs-901	372	38	(	(	PUNCT
iajs-901	372	39	)	)	PUNCT
iajs-901	372	40	and	and	CCONJ
iajs-901	372	41	[	[	X
iajs-901	372	42	16	16	NUM
iajs-901	372	43	,	,	PUNCT
iajs-901	372	44	prop1.1.20	prop1.1.20	X
iajs-901	372	45	]	]	X
iajs-901	372	46	.	.	PUNCT
iajs-901	373	1	we	we	PRON
iajs-901	373	2	notice	notice	VERB
iajs-901	373	3	that	that	SCONJ
iajs-901	373	4	the	the	DET
iajs-901	373	5	converse	converse	NOUN
iajs-901	373	6	of	of	ADP
iajs-901	373	7	proposition	proposition	NOUN
iajs-901	373	8	(	(	PUNCT
iajs-901	373	9	38	38	NUM
iajs-901	373	10	)	)	PUNCT
iajs-901	373	11	is	be	AUX
iajs-901	373	12	not	not	PART
iajs-901	373	13	true	true	ADJ
iajs-901	373	14	as	as	SCONJ
iajs-901	373	15	the	the	DET
iajs-901	373	16	following	follow	VERB
iajs-901	373	17	example	example	NOUN
iajs-901	373	18	shows	show	VERB
iajs-901	373	19	:	:	PUNCT
iajs-901	373	20	let	let	VERB
iajs-901	373	21	m	m	PRON
iajs-901	373	22	be	be	AUX
iajs-901	373	23	z	z	NOUN
iajs-901	373	24	-	-	PUNCT
iajs-901	373	25	module	module	NOUN
iajs-901	373	26	z	z	NOUN
iajs-901	373	27	and	and	CCONJ
iajs-901	373	28	s	s	NOUN
iajs-901	373	29	=	=	SYM
iajs-901	373	30	z	z	NOUN
iajs-901	373	31	–	–	PUNCT
iajs-901	373	32	{	{	PUNCT
iajs-901	373	33	0	0	NUM
iajs-901	373	34	}	}	PUNCT
iajs-901	373	35	,	,	PUNCT
iajs-901	373	36	it	it	PRON
iajs-901	373	37	is	be	AUX
iajs-901	373	38	clear	clear	ADJ
iajs-901	373	39	that	that	SCONJ
iajs-901	373	40	s	s	VERB
iajs-901	373	41	is	be	AUX
iajs-901	373	42	a	a	DET
iajs-901	373	43	multiplicatively	multiplicatively	ADV
iajs-901	373	44	closed	close	VERB
iajs-901	373	45	subset	subset	NOUN
iajs-901	373	46	of	of	ADP
iajs-901	373	47	z.	z.	PROPN
iajs-901	373	48	z	z	PROPN
iajs-901	373	49	s	s	PART
iajs-901	374	1	=	=	X
iajs-901	374	2	q	q	X
iajs-901	374	3	and	and	CCONJ
iajs-901	374	4	q	q	NOUN
iajs-901	374	5	as	as	SCONJ
iajs-901	374	6	q	q	NOUN
iajs-901	374	7	-	-	PUNCT
iajs-901	374	8	module	module	NOUN
iajs-901	374	9	is	be	AUX
iajs-901	374	10	coprime	coprime	ADJ
iajs-901	374	11	,	,	PUNCT
iajs-901	374	12	but	but	CCONJ
iajs-901	374	13	z	z	NOUN
iajs-901	374	14	is	be	AUX
iajs-901	374	15	not	not	PART
iajs-901	374	16	coprime	coprime	ADJ
iajs-901	374	17	z	z	NOUN
iajs-901	374	18	-	-	PUNCT
iajs-901	374	19	module	module	NOUN
iajs-901	374	20	.	.	PUNCT
iajs-901	375	1	the	the	DET
iajs-901	375	2	following	follow	VERB
iajs-901	375	3	corollary	corollary	NOUN
iajs-901	375	4	follows	follow	VERB
iajs-901	375	5	immediately	immediately	ADV
iajs-901	375	6	from	from	ADP
iajs-901	375	7	proposition	proposition	NOUN
iajs-901	375	8	(	(	PUNCT
iajs-901	375	9	38	38	NUM
iajs-901	375	10	)	)	PUNCT
iajs-901	375	11	.	.	PUNCT
iajs-901	376	1	corollary	corollary	ADJ
iajs-901	376	2	(	(	PUNCT
iajs-901	376	3	39	39	NUM
iajs-901	376	4	):	):	PUNCT
iajs-901	376	5	let	let	VERB
iajs-901	376	6	m	m	PRON
iajs-901	376	7	be	be	AUX
iajs-901	376	8	a	a	DET
iajs-901	376	9	coprime	coprime	ADJ
iajs-901	376	10	r	r	NOUN
iajs-901	376	11	-	-	NOUN
iajs-901	376	12	module	module	NOUN
iajs-901	376	13	,	,	PUNCT
iajs-901	376	14	then	then	ADV
iajs-901	376	15	mp	mp	PROPN
iajs-901	376	16	is	be	AUX
iajs-901	376	17	a	a	DET
iajs-901	376	18	coprime	coprime	ADJ
iajs-901	376	19	rp	rp	NOUN
iajs-901	376	20	-	-	PUNCT
iajs-901	376	21	module	module	NOUN
iajs-901	376	22	for	for	ADP
iajs-901	376	23	any	any	DET
iajs-901	376	24	prime	prime	ADJ
iajs-901	376	25	ideal	ideal	NOUN
iajs-901	376	26	p	p	PROPN
iajs-901	376	27	of	of	ADP
iajs-901	376	28	r.	r.	PROPN
iajs-901	376	29	next	next	ADV
iajs-901	376	30	,	,	PUNCT
iajs-901	376	31	we	we	PRON
iajs-901	376	32	have	have	VERB
iajs-901	376	33	the	the	DET
iajs-901	376	34	following	follow	VERB
iajs-901	376	35	result	result	NOUN
iajs-901	376	36	.	.	PUNCT
iajs-901	377	1	corollary	corollary	ADJ
iajs-901	377	2	(	(	PUNCT
iajs-901	377	3	40	40	NUM
iajs-901	377	4	):	):	PUNCT
iajs-901	377	5	let	let	VERB
iajs-901	377	6	m	m	PRON
iajs-901	377	7	be	be	AUX
iajs-901	377	8	a	a	DET
iajs-901	377	9	finitely	finitely	ADV
iajs-901	377	10	generated	generate	VERB
iajs-901	377	11	r	r	NOUN
iajs-901	377	12	-	-	PUNCT
iajs-901	377	13	module	module	NOUN
iajs-901	377	14	.	.	PUNCT
iajs-901	378	1	let	let	VERB
iajs-901	378	2	s	s	PRON
iajs-901	378	3	be	be	AUX
iajs-901	378	4	a	a	DET
iajs-901	378	5	multiplicatively	multiplicatively	ADV
iajs-901	378	6	closed	close	VERB
iajs-901	378	7	subset	subset	NOUN
iajs-901	378	8	of	of	ADP
iajs-901	378	9	r.	r.	PROPN
iajs-901	378	10	if	if	SCONJ
iajs-901	378	11	m	m	PROPN
iajs-901	378	12	is	be	AUX
iajs-901	378	13	p	p	ADJ
iajs-901	378	14	-	-	PUNCT
iajs-901	378	15	coprime	coprime	NOUN
iajs-901	378	16	r	r	NOUN
iajs-901	378	17	-	-	PUNCT
iajs-901	378	18	module	module	NOUN
iajs-901	378	19	,	,	PUNCT
iajs-901	378	20	then	then	ADV
iajs-901	378	21	ms	ms	PROPN
iajs-901	378	22	is	be	AUX
iajs-901	378	23	ps	ps	PROPN
iajs-901	378	24	coprime	coprime	ADJ
iajs-901	378	25	rs	rs	NOUN
iajs-901	378	26	–	–	PUNCT
iajs-901	378	27	module	module	NOUN
iajs-901	378	28	.	.	PUNCT
iajs-901	379	1	proof	proof	NOUN
iajs-901	379	2	:	:	PUNCT
iajs-901	379	3	since	since	SCONJ
iajs-901	379	4	m	m	PROPN
iajs-901	379	5	is	be	AUX
iajs-901	379	6	p	p	NOUN
iajs-901	379	7	-	-	PUNCT
iajs-901	379	8	coprime	coprime	NOUN
iajs-901	379	9	,	,	PUNCT
iajs-901	379	10	then	then	ADV
iajs-901	379	11	m	m	VERB
iajs-901	379	12	is	be	AUX
iajs-901	379	13	coprime	coprime	ADJ
iajs-901	379	14	with	with	ADP
iajs-901	379	15	r	r	NOUN
iajs-901	379	16	ann	ann	NOUN
iajs-901	379	17	m	m	NOUN
iajs-901	379	18	=	=	PUNCT
iajs-901	379	19	p.therefore	p.therefore	VERB
iajs-901	379	20	by	by	ADP
iajs-901	379	21	proposition	proposition	NOUN
iajs-901	379	22	(	(	PUNCT
iajs-901	379	23	2.1.38	2.1.38	NUM
iajs-901	379	24	)	)	PUNCT
iajs-901	379	25	,	,	PUNCT
iajs-901	379	26	m	m	PROPN
iajs-901	379	27	s	s	NOUN
iajs-901	379	28	is	be	AUX
iajs-901	379	29	coprime	coprime	ADJ
iajs-901	379	30	rs	rs	NOUN
iajs-901	379	31	–	–	PUNCT
iajs-901	379	32	module	module	NOUN
iajs-901	379	33	.	.	PUNCT
iajs-901	380	1	now	now	ADV
iajs-901	380	2	,	,	PUNCT
iajs-901	380	3	r	r	NOUN
iajs-901	380	4	ann	ann	PROPN
iajs-901	380	5	m	m	NOUN
iajs-901	380	6	=	=	PUNCT
iajs-901	381	1	[	[	X
iajs-901	381	2	0	0	X
iajs-901	381	3	r	r	NOUN
iajs-901	381	4	:	:	PUNCT
iajs-901	381	5	m	m	VERB
iajs-901	381	6	]	]	X
iajs-901	381	7	=	=	SYM
iajs-901	381	8	p	p	X
iajs-901	381	9	,	,	PUNCT
iajs-901	381	10	that	that	PRON
iajs-901	381	11	is	be	AUX
iajs-901	381	12	[	[	X
iajs-901	381	13	0	0	NUM
iajs-901	381	14	r	r	NOUN
iajs-901	381	15	:	:	PUNCT
iajs-901	381	16	m	m	X
iajs-901	381	17	]	]	X
iajs-901	381	18	s	s	PART
iajs-901	381	19	=	=	X
iajs-901	381	20	ps	ps	NOUN
iajs-901	381	21	and	and	CCONJ
iajs-901	381	22	by	by	ADP
iajs-901	381	23	[	[	X
iajs-901	381	24	28	28	NUM
iajs-901	381	25	,	,	PUNCT
iajs-901	381	26	p.152	p.152	NOUN
iajs-901	381	27	]	]	PUNCT
iajs-901	381	28	,	,	PUNCT
iajs-901	381	29	[	[	X
iajs-901	381	30	0s	0s	NOUN
iajs-901	381	31	:	:	PUNCT
iajs-901	381	32	ms	ms	NOUN
iajs-901	381	33	]	]	X
iajs-901	381	34	=	=	PUNCT
iajs-901	381	35	ps	ps	PROPN
iajs-901	381	36	.	.	PUNCT
iajs-901	381	37	thus	thus	ADV
iajs-901	381	38	sr	sr	PROPN
iajs-901	381	39	ann	ann	PROPN
iajs-901	381	40	m	m	PROPN
iajs-901	381	41	s	s	PART
iajs-901	381	42	=	=	X
iajs-901	381	43	ps	ps	PROPN
iajs-901	381	44	.	.	PROPN
iajs-901	382	1	therefore	therefore	ADV
iajs-901	382	2	ms	ms	PROPN
iajs-901	382	3	is	be	AUX
iajs-901	382	4	ps	ps	NOUN
iajs-901	382	5	-	-	PUNCT
iajs-901	382	6	coprime	coprime	ADJ
iajs-901	382	7	rs	rs	NOUN
iajs-901	382	8	-	-	PUNCT
iajs-901	382	9	module	module	NOUN
iajs-901	382	10	.	.	PUNCT
iajs-901	383	1	■	■	PUNCT
iajs-901	383	2	refrences	refrence	VERB
iajs-901	383	3	1c.p.lu	1c.p.lu	PRON
iajs-901	383	4	,	,	PUNCT
iajs-901	383	5	(	(	PUNCT
iajs-901	383	6	1984),"prime	1984),"prime	NUM
iajs-901	383	7	submodule	submodule	NOUN
iajs-901	383	8	of	of	ADP
iajs-901	383	9	modules	module	NOUN
iajs-901	383	10	"	"	PUNCT
iajs-901	383	11	comment	comment	NOUN
iajs-901	383	12	,	,	PUNCT
iajs-901	383	13	math.univ.st.paul	math.univ.st.paul	PROPN
iajs-901	383	14	,	,	PUNCT
iajs-901	383	15	vol.23	vol.23	NOUN
iajs-901	383	16	,	,	PUNCT
iajs-901	383	17	1	1	NUM
iajs-901	383	18	,	,	PUNCT
iajs-901	383	19	(	(	PUNCT
iajs-901	383	20	61	61	NUM
iajs-901	383	21	-	-	SYM
iajs-901	383	22	69	69	NUM
iajs-901	383	23	)	)	PUNCT
iajs-901	383	24	.	.	PUNCT
iajs-901	384	1	2	2	NUM
iajs-901	384	2	-	-	PUNCT
iajs-901	384	3	bhraany	bhraany	NOUN
iajs-901	384	4	,	,	PUNCT
iajs-901	384	5	b.al	b.al	PROPN
iajs-901	384	6	(	(	PUNCT
iajs-901	384	7	1969)"a	1969)"a	PROPN
iajs-901	384	8	note	note	NOUN
iajs-901	384	9	on	on	ADP
iajs-901	384	10	prime	prime	ADJ
iajs-901	384	11	modules	module	NOUN
iajs-901	384	12	and	and	CCONJ
iajs-901	384	13	pure	pure	ADJ
iajs-901	384	14	submodules	submodule	NOUN
iajs-901	384	15	"	"	PUNCT
iajs-901	384	16	,	,	PUNCT
iajs-901	384	17	j.science	j.science	NOUN
iajs-901	384	18	,	,	PUNCT
iajs-901	384	19	vol.37	vol.37	VERB
iajs-901	384	20	,	,	PUNCT
iajs-901	384	21	(	(	PUNCT
iajs-901	384	22	1431	1431	NUM
iajs-901	384	23	-	-	SYM
iajs-901	384	24	1441	1441	NUM
iajs-901	384	25	)	)	PUNCT
iajs-901	384	26	.	.	PUNCT
iajs-901	385	1	3desale	3desale	NUM
iajs-901	385	2	g.	g.	PROPN
iajs-901	385	3	,	,	PUNCT
iajs-901	385	4	nicholson	nicholson	PROPN
iajs-901	385	5	w.k	w.k	PROPN
iajs-901	385	6	.	.	PROPN
iajs-901	385	7	,(1981),"endoprimitive	,(1981),"endoprimitive	PUNCT
iajs-901	385	8	rings",j.algebra,.70	rings",j.algebra,.70	X
iajs-901	385	9	,	,	PUNCT
iajs-901	385	10	(	(	PUNCT
iajs-901	385	11	548	548	NUM
iajs-901	385	12	-	-	SYM
iajs-901	385	13	560	560	NUM
iajs-901	385	14	)	)	PUNCT
iajs-901	385	15	.	.	PUNCT
iajs-901	386	1	4beachy	4beachy	NUM
iajs-901	386	2	j.a	j.a	PROPN
iajs-901	386	3	.	.	PROPN
iajs-901	386	4	,	,	PUNCT
iajs-901	386	5	(	(	PUNCT
iajs-901	386	6	1976	1976	NUM
iajs-901	386	7	)	)	PUNCT
iajs-901	386	8	"	"	PUNCT
iajs-901	386	9	some	some	DET
iajs-901	386	10	aspects	aspect	NOUN
iajs-901	386	11	of	of	ADP
iajs-901	386	12	non	non	ADJ
iajs-901	386	13	-	-	ADJ
iajs-901	386	14	commutative	commutative	ADJ
iajs-901	386	15	localization	localization	NOUN
iajs-901	386	16	"	"	PUNCT
iajs-901	386	17	lecture	lecture	NOUN
iajs-901	386	18	notes	note	NOUN
iajs-901	386	19	in	in	ADP
iajs-901	386	20	mathematics,.545	mathematics,.545	ADJ
iajs-901	386	21	,	,	PUNCT
iajs-901	386	22	springer	springer	NOUN
iajs-901	386	23	verlage	verlage	NOUN
iajs-901	386	24	,	,	PUNCT
iajs-901	386	25	heildelberg	heildelberg	PROPN
iajs-901	386	26	,	,	PUNCT
iajs-901	386	27	new	new	PROPN
iajs-901	386	28	york	york	PROPN
iajs-901	386	29	.	.	PUNCT
iajs-901	387	1	ihjpas	ihjpa	VERB
iajs-901	387	2	ibn	ibn	PROPN
iajs-901	387	3	alhaitham	alhaitham	PROPN
iajs-901	388	1	j.	j.	PROPN
iajs-901	389	1	fo	fo	ADP
iajs-901	389	2	r	r	NOUN
iajs-901	389	3	pure	pure	ADJ
iajs-901	389	4	&	&	CCONJ
iajs-901	389	5	appl	appl	PROPN
iajs-901	389	6	.	.	PUNCT
iajs-901	390	1	sc	sc	PROPN
iajs-901	390	2	i.	i.	PROPN
iajs-901	390	3	vo	vo	PROPN
iajs-901	390	4	l.23	l.23	PROPN
iajs-901	390	5	(	(	PUNCT
iajs-901	390	6	3	3	NUM
iajs-901	390	7	)	)	PUNCT
iajs-901	390	8	2010	2010	NUM
iajs-901	390	9	5yassemi	5yassemi	NUM
iajs-901	390	10	s.	s.	PROPN
iajs-901	390	11	,	,	PUNCT
iajs-901	390	12	(	(	PUNCT
iajs-901	390	13	2001),"the	2001),"the	DET
iajs-901	390	14	dual	dual	ADJ
iajs-901	390	15	notion	notion	NOUN
iajs-901	390	16	of	of	ADP
iajs-901	390	17	prime	prime	ADJ
iajs-901	390	18	submodule	submodule	NOUN
iajs-901	390	19	"	"	PUNCT
iajs-901	390	20	,	,	PUNCT
iajs-901	390	21	arch.math	arch.math	PROPN
iajs-901	390	22	.	.	PUNCT
iajs-901	390	23	(	(	PUNCT
iajs-901	390	24	born	bear	VERB
iajs-901	390	25	)	)	PUNCT
iajs-901	390	26	,	,	PUNCT
iajs-901	390	27	37	37	NUM
iajs-901	390	28	(	(	PUNCT
iajs-901	390	29	273278	273278	NUM
iajs-901	390	30	)	)	PUNCT
iajs-901	390	31	.	.	PUNCT
iajs-901	391	1	6annin	6annin	NUM
iajs-901	391	2	s.	s.	PROPN
iajs-901	391	3	,	,	PUNCT
iajs-901	391	4	(	(	PUNCT
iajs-901	391	5	2002),"associated	2002),"associate	VERB
iajs-901	391	6	and	and	CCONJ
iajs-901	391	7	attached	attach	VERB
iajs-901	391	8	primes	prime	NOUN
iajs-901	391	9	over	over	ADP
iajs-901	391	10	non	non	ADJ
iajs-901	391	11	commutative	commutative	ADJ
iajs-901	391	12	rings	ring	NOUN
iajs-901	391	13	"	"	PUNCT
iajs-901	391	14	,	,	PUNCT
iajs-901	391	15	ph.d.thesis	ph.d.thesis	PROPN
iajs-901	391	16	,	,	PUNCT
iajs-901	391	17	univ	univ	PROPN
iajs-901	391	18	.	.	PROPN
iajs-901	391	19	of	of	ADP
iajs-901	391	20	baghdad	baghdad	PROPN
iajs-901	391	21	7abuihlail	7abuihlail	NUM
iajs-901	391	22	j.	j.	PROPN
iajs-901	391	23	,"zariski	,"zariski	PROPN
iajs-901	391	24	-	-	PUNCT
iajs-901	391	25	like	like	ADJ
iajs-901	391	26	topologies	topology	NOUN
iajs-901	391	27	for	for	ADP
iajs-901	391	28	m	m	NOUN
iajs-901	391	29	odules	odule	NOUN
iajs-901	391	30	over	over	ADP
iajs-901	391	31	commutative	commutative	ADJ
iajs-901	391	32	rings	ring	NOUN
iajs-901	391	33	"	"	PUNCT
iajs-901	391	34	,	,	PUNCT
iajs-901	391	35	internet	internet	NOUN
iajs-901	391	36	.	.	PUNCT
iajs-901	392	1	8faith	8faith	NUM
iajs-901	392	2	c.	c.	NOUN
iajs-901	392	3	,	,	PUNCT
iajs-901	392	4	(	(	PUNCT
iajs-901	392	5	1973),"algebra	1973),"algebra	NOUN
iajs-901	392	6	:	:	PUNCT
iajs-901	392	7	rings	ring	NOUN
iajs-901	392	8	:	:	PUNCT
iajs-901	392	9	modules	module	NOUN
iajs-901	392	10	and	and	CCONJ
iajs-901	392	11	catgories	catgorie	NOUN
iajs-901	392	12	1	1	NUM
iajs-901	392	13	"	"	PUNCT
iajs-901	392	14	,	,	PUNCT
iajs-901	392	15	sp	sp	ADP
iajs-901	392	16	ringer	ringer	NOUN
iajs-901	392	17	-	-	PUNCT
iajs-901	392	18	verlag	verlag	NOUN
iajs-901	392	19	,	,	PUNCT
iajs-901	392	20	berline	berline	PROPN
iajs-901	392	21	,	,	PUNCT
iajs-901	392	22	heidelberg	heidelberg	PROPN
iajs-901	392	23	,	,	PUNCT
iajs-901	392	24	new	new	PROPN
iajs-901	392	25	york	york	PROPN
iajs-901	392	26	.	.	PUNCT
iajs-901	393	1	9wijayanti	9wijayanti	NUM
iajs-901	393	2	i.e.	i.e.	X
iajs-901	393	3	,	,	PUNCT
iajs-901	393	4	(	(	PUNCT
iajs-901	393	5	2006),"coprime	2006),"coprime	NUM
iajs-901	393	6	modules	module	NOUN
iajs-901	393	7	and	and	CCONJ
iajs-901	393	8	comodules",ph.d.thesis	comodules",ph.d.thesis	NOUN
iajs-901	393	9	,	,	PUNCT
iajs-901	393	10	heinrich	heinrich	NOUN
iajs-901	393	11	-	-	PUNCT
iajs-901	393	12	heine	heine	PROPN
iajs-901	393	13	univ.dusseldorf	univ.dusseldorf	PROPN
iajs-901	393	14	.	.	PUNCT
iajs-901	394	1	10barand	10barand	NUM
iajs-901	394	2	a.	a.	NOUN
iajs-901	394	3	,	,	PUNCT
iajs-901	394	4	(	(	PUNCT
iajs-901	394	5	1981),"m	1981),"m	NUM
iajs-901	394	6	ultiplication	ultiplication	NOUN
iajs-901	394	7	modules	module	NOUN
iajs-901	394	8	"	"	PUNCT
iajs-901	394	9	,	,	PUNCT
iajs-901	394	10	j.algebra	j.algebra	PROPN
iajs-901	394	11	,	,	PUNCT
iajs-901	394	12	71	71	NUM
iajs-901	394	13	,	,	PUNCT
iajs-901	394	14	(	(	PUNCT
iajs-901	394	15	174	174	NUM
iajs-901	394	16	-	-	SYM
iajs-901	394	17	178	178	NUM
iajs-901	394	18	)	)	PUNCT
iajs-901	394	19	.	.	PUNCT
iajs-901	395	1	11h.m.abbdule	11h.m.abbdule	NUM
iajs-901	395	2	-	-	SYM
iajs-901	395	3	razak	razak	NOUN
iajs-901	395	4	,	,	PUNCT
iajs-901	395	5	(	(	PUNCT
iajs-901	395	6	1999),"quasi	1999),"quasi	NUM
iajs-901	395	7	-	-	PUNCT
iajs-901	395	8	prime	prime	ADJ
iajs-901	395	9	modules	module	NOUN
iajs-901	395	10	and	and	CCONJ
iajs-901	395	11	quasi	quasi	ADJ
iajs-901	395	12	prime	prime	PROPN
iajs-901	395	13	submodules"m	submodules"m	PROPN
iajs-901	395	14	.d.thesis	.d.thesis	PROPN
iajs-901	395	15	,	,	PUNCT
iajs-901	395	16	univ	univ	PROPN
iajs-901	395	17	.	.	PROPN
iajs-901	395	18	of	of	ADP
iajs-901	395	19	baghdad	baghdad	PROPN
iajs-901	395	20	12kasch	12kasch	PROPN
iajs-901	395	21	f.	f.	PROPN
iajs-901	395	22	,(1982),"modules	,(1982),"modules	PUNCT
iajs-901	395	23	and	and	CCONJ
iajs-901	395	24	rings",acadimic	rings",acadimic	ADJ
iajs-901	395	25	press	press	NOUN
iajs-901	395	26	,	,	PUNCT
iajs-901	395	27	london	london	PROPN
iajs-901	395	28	.	.	PUNCT
iajs-901	396	1	13abdul	13abdul	NUM
iajs-901	396	2	-	-	PUNCT
iajs-901	396	3	baste	baste	NOUN
iajs-901	396	4	z.	z.	PROPN
iajs-901	396	5	and	and	CCONJ
iajs-901	396	6	smith	smith	PROPN
iajs-901	396	7	p.f	p.f	PROPN
iajs-901	396	8	.	.	PROPN
iajs-901	396	9	,	,	PUNCT
iajs-901	396	10	(	(	PUNCT
iajs-901	396	11	1988),"multiplication	1988),"multiplication	NOUN
iajs-901	396	12	modules",comm	modules",comm	NOUN
iajs-901	396	13	.	.	PUNCT
iajs-901	397	1	in	in	ADP
iajs-901	397	2	algebra	algebra	NOUN
iajs-901	397	3	,	,	PUNCT
iajs-901	397	4	16	16	NUM
iajs-901	397	5	,	,	PUNCT
iajs-901	397	6	(	(	PUNCT
iajs-901	397	7	755	755	NUM
iajs-901	397	8	-	-	SYM
iajs-901	397	9	779	779	NUM
iajs-901	397	10	)	)	PUNCT
iajs-901	397	11	،	،	NOUN
iajs-901	397	12	رسالة	رسالة	NOUN
iajs-901	397	13	الماجستیر،جامعة	الماجستیر،جامعة	PROPN
iajs-901	397	14	"	"	PUNCT
iajs-901	397	15	المودیوالت	المودیوالت	PROPN
iajs-901	397	16	الجزئیة	الجزئیة	PROPN
iajs-901	397	17	األولیة	األولیة	PROPN
iajs-901	397	18	والمودیوالت	والمودیوالت	PROPN
iajs-901	397	19	الجزئیة	الجزئیة	PROPN
iajs-901	397	20	الشبھ	الشبھ	PROPN
iajs-901	397	21	أولیة)"1969(،إیمان	أولیة)"1969(،إیمان	PROPN
iajs-901	397	22	علي	علي	NOUN
iajs-901	397	23	؛	؛	PROPN
iajs-901	397	24	عذاب-14	عذاب-14	PROPN
iajs-901	397	25	.	.	PUNCT
iajs-901	397	26	بغداد	بغداد	PROPN
iajs-901	397	27	15khakaf	15khakaf	PROPN
iajs-901	397	28	r.i	r.i	PROPN
iajs-901	397	29	.	.	PROPN
iajs-901	397	30	,"dual	,"dual	PUNCT
iajs-901	397	31	notions	notion	NOUN
iajs-901	397	32	of	of	ADP
iajs-901	397	33	prime	prime	ADJ
iajs-901	397	34	submodules	submodule	NOUN
iajs-901	397	35	and	and	CCONJ
iajs-901	397	36	prime	prime	ADJ
iajs-901	397	37	modules	module	NOUN
iajs-901	397	38	"	"	PUNCT
iajs-901	397	39	,	,	PUNCT
iajs-901	397	40	m.d.thesis	m.d.thesis	ADJ
iajs-901	397	41	,	,	PUNCT
iajs-901	397	42	univ.of	univ.of	PROPN
iajs-901	397	43	baghdad	baghdad	PROPN
iajs-901	397	44	.	.	PUNCT
iajs-901	398	1	16yaseen	16yaseen	NUM
iajs-901	399	1	s.m	s.m	PROPN
iajs-901	399	2	.	.	PROPN
iajs-901	399	3	,	,	PUNCT
iajs-901	399	4	(	(	PUNCT
iajs-901	399	5	2003),coquasi	2003),coquasi	NUM
iajs-901	399	6	-	-	PUNCT
iajs-901	399	7	dedekind	dedekind	NOUN
iajs-901	399	8	modules",ph.d.thesis	modules",ph.d.thesis	NOUN
iajs-901	399	9	,	,	PUNCT
iajs-901	399	10	univ.of	univ.of	PROPN
iajs-901	399	11	baghdad	baghdad	PROPN
iajs-901	399	12	17anderson	17anderson	PROPN
iajs-901	400	1	f.w	f.w	PROPN
iajs-901	400	2	and	and	CCONJ
iajs-901	400	3	fuller	full	ADJ
iajs-901	400	4	k.r	k.r	PROPN
iajs-901	400	5	.	.	PROPN
iajs-901	400	6	,	,	PUNCT
iajs-901	400	7	(	(	PUNCT
iajs-901	400	8	1973),"ring	1973),"ring	NOUN
iajs-901	400	9	and	and	CCONJ
iajs-901	400	10	categories	category	NOUN
iajs-901	400	11	of	of	ADP
iajs-901	400	12	modules	module	NOUN
iajs-901	400	13	"	"	PUNCT
iajs-901	400	14	,	,	PUNCT
iajs-901	400	15	ph.d.thesis	ph.d.thesis	PROPN
iajs-901	400	16	,	,	PUNCT
iajs-901	400	17	univ.of	univ.of	PROPN
iajs-901	400	18	berkeley	berkeley	PROPN
iajs-901	400	19	.	.	PUNCT
iajs-901	401	1	18sahai	18sahai	NUM
iajs-901	402	1	v.	v.	CCONJ
iajs-901	402	2	and	and	CCONJ
iajs-901	402	3	bist	bist	NOUN
iajs-901	402	4	,	,	PUNCT
iajs-901	402	5	v.	v.	CCONJ
iajs-901	402	6	"	"	PUNCT
iajs-901	402	7	algebra",narosa	algebra",narosa	PROPN
iajs-901	402	8	publishing	publishing	PROPN
iajs-901	402	9	gouse	gouse	PROPN
iajs-901	402	10	,	,	PUNCT
iajs-901	402	11	new	new	PROPN
iajs-901	402	12	delhi	delhi	PROPN
iajs-901	402	13	madras	madras	PROPN
iajs-901	402	14	bombay	bombay	PROPN
iajs-901	402	15	calculate	calculate	NOUN
iajs-901	402	16	..	..	PUNCT
iajs-901	403	1	19atiyah	19atiyah	NUM
iajs-901	404	1	m.f	m.f	PROPN
iajs-901	404	2	.	.	PROPN
iajs-901	405	1	and	and	CCONJ
iajs-901	405	2	macdonald	macdonald	PROPN
iajs-901	405	3	i.g	i.g	PROPN
iajs-901	405	4	.	.	PROPN
iajs-901	405	5	,	,	PUNCT
iajs-901	405	6	(	(	PUNCT
iajs-901	405	7	1969),"introduction	1969),"introduction	NOUN
iajs-901	405	8	to	to	PART
iajs-901	405	9	commutative	commutative	VERB
iajs-901	405	10	algebra",univ.of	algebra",univ.of	PROPN
iajs-901	405	11	oxford	oxford	NOUN
iajs-901	405	12	20c.p.lu	20c.p.lu	ADP
iajs-901	405	13	,	,	PUNCT
iajs-901	405	14	(	(	PUNCT
iajs-901	405	15	1989),"m	1989),"m	NUM
iajs-901	405	16	-	-	PUNCT
iajs-901	405	17	radicals	radical	NOUN
iajs-901	405	18	of	of	ADP
iajs-901	405	19	submodules	submodule	NOUN
iajs-901	405	20	in	in	ADP
iajs-901	405	21	modules	module	NOUN
iajs-901	405	22	"	"	PUNCT
iajs-901	405	23	,	,	PUNCT
iajs-901	405	24	math.japonica	math.japonica	PROPN
iajs-901	405	25	,	,	PUNCT
iajs-901	405	26	34	34	NUM
iajs-901	405	27	,	,	PUNCT
iajs-901	405	28	(	(	PUNCT
iajs-901	405	29	211	211	NUM
iajs-901	405	30	-	-	SYM
iajs-901	405	31	219	219	NUM
iajs-901	405	32	)	)	PUNCT
iajs-901	405	33	21macdonald	21macdonald	NUM
iajs-901	405	34	,	,	PUNCT
iajs-901	405	35	l.g	l.g	PROPN
iajs-901	405	36	.	.	PUNCT
iajs-901	406	1	(	(	PUNCT
iajs-901	406	2	1973),"secondary	1973),"secondary	NUM
iajs-901	406	3	representation	representation	NOUN
iajs-901	406	4	of	of	ADP
iajs-901	406	5	modules	module	NOUN
iajs-901	406	6	over	over	ADP
iajs-901	406	7	a	a	DET
iajs-901	406	8	commutative	commutative	ADJ
iajs-901	406	9	ring"sympos	ring"sympos	NOUN
iajs-901	406	10	.	.	PUNCT
iajs-901	407	1	math.xi	math.xi	NUM
iajs-901	407	2	,	,	PUNCT
iajs-901	407	3	pp.(23	pp.(23	NOUN
iajs-901	407	4	-	-	PUNCT
iajs-901	407	5	43	43	NUM
iajs-901	407	6	)	)	PUNCT
iajs-901	407	7	.	.	PUNCT
iajs-901	408	1	22c.p.lu	22c.p.lu	X
iajs-901	408	2	,	,	PUNCT
iajs-901	408	3	(	(	PUNCT
iajs-901	408	4	1995),"comm	1995),"comm	NOUN
iajs-901	408	5	in	in	ADP
iajs-901	408	6	alg	alg	PROPN
iajs-901	408	7	.	.	PUNCT
iajs-901	409	1	"	"	PUNCT
iajs-901	409	2	,	,	PUNCT
iajs-901	409	3	23	23	NUM
iajs-901	409	4	,	,	PUNCT
iajs-901	409	5	no.10	no.10	ADJ
iajs-901	409	6	,	,	PUNCT
iajs-901	409	7	(	(	PUNCT
iajs-901	409	8	3741	3741	NUM
iajs-901	409	9	-	-	SYM
iajs-901	409	10	3752	3752	NUM
iajs-901	409	11	)	)	PUNCT
iajs-901	409	12	.	.	PUNCT
iajs-901	410	1	23nicholson	23nicholson	NUM
iajs-901	411	1	w.n	w.n	PROPN
iajs-901	411	2	.	.	PROPN
iajs-901	412	1	and	and	CCONJ
iajs-901	412	2	yousif	yousif	PROPN
iajs-901	412	3	m.f	m.f	PROPN
iajs-901	412	4	.	.	PROPN
iajs-901	412	5	,	,	PUNCT
iajs-901	412	6	(	(	PUNCT
iajs-901	412	7	1995	1995	NUM
iajs-901	412	8	)	)	PUNCT
iajs-901	412	9	,	,	PUNCT
iajs-901	412	10	"	"	PUNCT
iajs-901	412	11	principally	principally	ADV
iajs-901	412	12	injective	injective	ADJ
iajs-901	412	13	rings	ring	NOUN
iajs-901	412	14	"	"	PUNCT
iajs-901	412	15	,	,	PUNCT
iajs-901	412	16	j.algebra	j.algebra	PROPN
iajs-901	412	17	,	,	PUNCT
iajs-901	412	18	174	174	NUM
iajs-901	412	19	,	,	PUNCT
iajs-901	412	20	(	(	PUNCT
iajs-901	412	21	77	77	NUM
iajs-901	412	22	-	-	SYM
iajs-901	412	23	93	93	NUM
iajs-901	412	24	)	)	PUNCT
iajs-901	412	25	.	.	PUNCT
iajs-901	413	1	24sharp	24sharp	NUM
iajs-901	413	2	d.w	d.w	NOUN
iajs-901	413	3	.	.	PROPN
iajs-901	413	4	and	and	CCONJ
iajs-901	413	5	vamous	vamous	ADJ
iajs-901	413	6	p.	p.	NOUN
iajs-901	413	7	,	,	PUNCT
iajs-901	413	8	(	(	PUNCT
iajs-901	413	9	1972)"injective	1972)"injective	NUM
iajs-901	413	10	modules",cambridge	modules",cambridge	NOUN
iajs-901	413	11	univ.press	univ.press	PRON
iajs-901	413	12	,	,	PUNCT
iajs-901	413	13	london	london	PROPN
iajs-901	413	14	25larsen	25larsen	NUM
iajs-901	414	1	m.d	m.d	PROPN
iajs-901	414	2	.	.	PROPN
iajs-901	414	3	,	,	PUNCT
iajs-901	414	4	mc	mc	PROPN
iajs-901	414	5	carthy	carthy	PROPN
iajs-901	414	6	pj	pj	PROPN
iajs-901	414	7	.	.	PROPN
iajs-901	414	8	,(1971),"multiplication	,(1971),"multiplication	PUNCT
iajs-901	414	9	theory	theory	NOUN
iajs-901	414	10	of	of	ADP
iajs-901	414	11	ideals	ideal	NOUN
iajs-901	414	12	,	,	PUNCT
iajs-901	414	13	acadimec	acadimec	PROPN
iajs-901	414	14	pres	pre	NOUN
iajs-901	414	15	,	,	PUNCT
iajs-901	414	16	new	new	PROPN
iajs-901	414	17	york	york	PROPN
iajs-901	414	18	and	and	CCONJ
iajs-901	414	19	london	london	PROPN
iajs-901	414	20	26dauns	26dauns	PROPN
iajs-901	414	21	j.	j.	PROPN
iajs-901	414	22	,	,	PUNCT
iajs-901	414	23	(	(	PUNCT
iajs-901	414	24	1980)"prime	1980)"prime	NOUN
iajs-901	414	25	modules	module	NOUN
iajs-901	414	26	and	and	CCONJ
iajs-901	414	27	one	one	NUM
iajs-901	414	28	-	-	PUNCT
iajs-901	414	29	sided	sided	ADJ
iajs-901	414	30	ideals	ideal	NOUN
iajs-901	414	31	in	in	ADP
iajs-901	414	32	ring	ring	NOUN
iajs-901	414	33	theory	theory	NOUN
iajs-901	414	34	and	and	CCONJ
iajs-901	414	35	agebra	agebra	NOUN
iajs-901	414	36	iii""(proccedings	iii""(procceding	NOUN
iajs-901	414	37	of	of	ADP
iajs-901	414	38	the	the	DET
iajs-901	414	39	third	third	PROPN
iajs-901	414	40	oklahoma	oklahoma	PROPN
iajs-901	414	41	conference),b.r.mcdonald	conference),b.r.mcdonald	PROPN
iajs-901	414	42	(	(	PUNCT
iajs-901	414	43	editor	editor	NOUN
iajs-901	414	44	)	)	PUNCT
iajs-901	414	45	(	(	PUNCT
iajs-901	414	46	dekker	dekker	PROPN
iajs-901	414	47	,	,	PUNCT
iajs-901	414	48	new	new	PROPN
iajs-901	414	49	york	york	PROPN
iajs-901	414	50	)	)	PUNCT
iajs-901	414	51	,	,	PUNCT
iajs-901	414	52	(	(	PUNCT
iajs-901	414	53	301	301	NUM
iajs-901	414	54	-	-	NUM
iajs-901	414	55	344	344	NUM
iajs-901	414	56	)	)	PUNCT
iajs-901	414	57	.	.	PUNCT
iajs-901	415	1	14kaplanskj	14kaplanskj	INTJ
iajs-901	416	1	i	i	PRON
iajs-901	416	2	,	,	PUNCT
iajs-901	416	3	(	(	PUNCT
iajs-901	416	4	1974),"commutative	1974),"commutative	NUM
iajs-901	416	5	rings",the	rings",the	PROPN
iajs-901	416	6	univ	univ	PROPN
iajs-901	416	7	.	.	PUNCT
iajs-901	416	8	of	of	ADP
iajs-901	416	9	chicago	chicago	PROPN
iajs-901	416	10	press	press	PROPN
iajs-901	416	11	,	,	PUNCT
iajs-901	416	12	chicago	chicago	PROPN
iajs-901	416	13	&	&	CCONJ
iajs-901	416	14	london	london	PROPN
iajs-901	416	15	.	.	PUNCT
iajs-901	417	1	27.lam	27.lam	PROPN
iajs-901	417	2	t.y	t.y	PROPN
iajs-901	417	3	,	,	PUNCT
iajs-901	417	4	(	(	PUNCT
iajs-901	417	5	1999)"lecture	1999)"lecture	NOUN
iajs-901	417	6	on	on	ADP
iajs-901	417	7	modules	module	NOUN
iajs-901	417	8	and	and	CCONJ
iajs-901	417	9	rings"gtm	rings"gtm	PROPN
iajs-901	417	10	189,sipringer	189,sipringer	NUM
iajs-901	417	11	verlage	verlage	NOUN
iajs-901	417	12	,	,	PUNCT
iajs-901	417	13	new	new	PROPN
iajs-901	417	14	york	york	PROPN
iajs-901	417	15	.	.	PUNCT
iajs-901	418	1	28northeoit	28northeoit	PROPN
iajs-901	419	1	d.g	d.g	PROPN
iajs-901	419	2	.	.	PROPN
iajs-901	419	3	,	,	PUNCT
iajs-901	419	4	(	(	PUNCT
iajs-901	419	5	1968),"lessons	1968),"lesson	NOUN
iajs-901	419	6	on	on	ADP
iajs-901	419	7	ring	ring	NOUN
iajs-901	419	8	,	,	PUNCT
iajs-901	419	9	modules	module	NOUN
iajs-901	419	10	and	and	CCONJ
iajs-901	419	11	multiplication",cambridge	multiplication",cambridge	VERB
iajs-901	419	12	univ.press	univ.press	PROPN
iajs-901	419	13	,	,	PUNCT
iajs-901	419	14	london	london	PROPN
iajs-901	419	15	.	.	PUNCT
iajs-901	420	1	ihjpas	ihjpas	PROPN
iajs-901	420	2	2010	2010	NUM
iajs-901	420	3	)	)	PUNCT
iajs-901	421	1	3	3	NUM
iajs-901	421	2	(	(	PUNCT
iajs-901	421	3	23مجلة	23مجلة	NUM
iajs-901	421	4	ابن	ابن	PROPN
iajs-901	421	5	الھیثم	الھیثم	PROPN
iajs-901	421	6	للعلوم	للعلوم	PROPN
iajs-901	421	7	الصرفة	الصرفة	PROPN
iajs-901	421	8	والتطبیقیة	والتطبیقیة	PROPN
iajs-901	421	9	المجلد	المجلد	PROPN
iajs-901	421	10	المفھوم	المفھوم	PROPN
iajs-901	421	11	الردیف	الردیف	VERB
iajs-901	422	1	للمقاسات	للمقاسات	NOUN
iajs-901	422	2	االولیة	االولیة	PROPN
iajs-901	422	3	رشا	رشا	PROPN
iajs-901	422	4	ابراھیم	ابراھیم	PROPN
iajs-901	422	5	خلف	خلف	PROPN
iajs-901	422	6	،	،	PROPN
iajs-901	422	7	انعام	انعام	PROPN
iajs-901	422	8	محمد	محمد	PROPN
iajs-901	422	9	علي	علي	NOUN
iajs-901	422	10	جامعة	جامعة	PROPN
iajs-901	422	11	بغداد	بغداد	PROPN
iajs-901	422	12	،	،	PROPN
iajs-901	422	13	ابن	ابن	PROPN
iajs-901	422	14	الھیثم	الھیثم	PROPN
iajs-901	422	15	–	–	PUNCT
iajs-901	422	16	كلیة	كلیة	PROPN
iajs-901	422	17	التربیة	التربیة	NOUN
iajs-901	422	18	،	،	PROPN
iajs-901	422	19	قسم	قسم	PROPN
iajs-901	422	20	الریاضیات	الریاضیات	PROPN
iajs-901	422	21	خالصةال	خالصةال	VERB
iajs-901	422	22	؛	؛	PROPN
iajs-901	422	23	r	r	NOUN
iajs-901	422	24	لتكن	لتكن	NOUN
iajs-901	422	25	دالیة	دالیة	NOUN
iajs-901	422	26	ذات	ذات	VERB
iajs-901	422	27	محاید	محاید	PROPN
iajs-901	422	28	إذا	إذا	NOUN
iajs-901	422	29	كان	كان	NOUN
iajs-901	422	30	)	)	PUNCT
iajs-901	423	1	د	د	DET
iajs-901	423	2	للمقاس	للمقاس	ADJ
iajs-901	423	3	األولي	األولي	NOUN
iajs-901	423	4	مضا	مضا	VERB
iajs-901	423	5	كمفھوم(أولیا	كمفھوم(أولیا	PROPN
iajs-901	423	6	مضادا	مضادا	NOUN
iajs-901	423	7	مقاسا	مقاسا	NOUN
iajs-901	423	8	mسمى	mسمى	NOUN
iajs-901	424	1	یr.على	یr.على	PROPN
iajs-901	424	2	مقاسا	مقاسا	VERB
iajs-901	424	3	mحلقة	mحلقة	INTJ
iajs-901	425	1	إب	إب	PROPN
iajs-901	425	2	m	m	PROPN
iajs-901	425	3	تآلف	تآلف	PROPN
iajs-901	425	4	r=	r=	PROPN
iajs-901	425	5	تآلف	تآلف	PROPN
iajs-901	426	1	r	r	PROPN
iajs-901	426	2			PROPN
iajs-901	426	3			PROPN
iajs-901	426	4	.	.	PUNCT
iajs-901	427	1	m	m	VERB
iajs-901	427	2	في	في	ADP
iajs-901	427	3	n	n	ADV
iajs-901	427	4	لكل	لكل	NOUN
iajs-901	427	5	مقاس	مقاس	PROPN
iajs-901	427	6	جزئي	جزئي	NOUN
iajs-901	427	7	فعلي	فعلي	PROPN
iajs-901	427	8	لھذا	لھذا	VERB
iajs-901	427	9	المفھوم	المفھوم	PROPN
iajs-901	427	10	وكذلك	وكذلك	PROPN
iajs-901	427	11	أعطینا	أعطینا	NOUN
iajs-901	427	12	عددا	عددا	VERB
iajs-901	427	13	ساسیة	ساسیة	SCONJ
iajs-901	427	14	في	في	INTJ
iajs-901	427	15	ھذا	ھذا	NOUN
iajs-901	427	16	البحث	البحث	VERB
iajs-901	427	17	درسنا	درسنا	PROPN
iajs-901	427	18	المقاسات	المقاسات	PROPN
iajs-901	427	19	األولیة	األولیة	PROPN
iajs-901	427	20	المضادة	المضادة	PROPN
iajs-901	427	21	ةأعطینا	ةأعطینا	NOUN
iajs-901	427	22	عددا	عددا	VERB
iajs-901	427	23	من	من	DET
iajs-901	427	24	الخواص	الخواص	NOUN
iajs-901	427	25	األ	األ	PROPN
iajs-901	428	1	ن	ن	PROPN
iajs-901	428	2	الممیزات	الممیزات	PROPN
iajs-901	428	3	لھ	لھ	ADP
iajs-901	428	4	تحت	تحت	PROPN
iajs-901	428	5	أصناف	أصناف	PROPN
iajs-901	428	6	معینة	معینة	PROPN
iajs-901	428	7	من	من	PROPN
iajs-901	428	8	المقاسات	المقاسات	PROPN
iajs-901	428	9	م	م	PROPN
iajs-901	428	10	ihjpas	ihjpa	NOUN