id	sid	tid	token	lemma	pos
ejpam-492	1	1	3_xxx_jayaram.dvi	3_xxx_jayaram.dvi	NUM
ejpam-492	1	2	european	european	ADJ
ejpam-492	1	3	journal	journal	NOUN
ejpam-492	1	4	of	of	ADP
ejpam-492	1	5	pure	pure	ADJ
ejpam-492	1	6	and	and	CCONJ
ejpam-492	1	7	applied	apply	VERB
ejpam-492	1	8	mathematics	mathematic	NOUN
ejpam-492	1	9	vol	vol	NOUN
ejpam-492	1	10	.	.	PROPN
ejpam-492	2	1	2	2	NUM
ejpam-492	2	2	,	,	PUNCT
ejpam-492	2	3	no	no	INTJ
ejpam-492	2	4	.	.	NOUN
ejpam-492	2	5	4	4	NUM
ejpam-492	2	6	,	,	PUNCT
ejpam-492	2	7	2009	2009	NUM
ejpam-492	2	8	,	,	PUNCT
ejpam-492	2	9	(	(	PUNCT
ejpam-492	2	10	508	508	NUM
ejpam-492	2	11	-	-	NUM
ejpam-492	2	12	519	519	NUM
ejpam-492	2	13	)	)	PUNCT
ejpam-492	2	14	issn	issn	PROPN
ejpam-492	2	15	1307	1307	NUM
ejpam-492	2	16	-	-	SYM
ejpam-492	2	17	5543	5543	NUM
ejpam-492	2	18	–	–	PUNCT
ejpam-492	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-492	2	20	π	π	NOUN
ejpam-492	2	21	-	-	PUNCT
ejpam-492	2	22	modules	module	NOUN
ejpam-492	2	23	c.	c.	PROPN
ejpam-492	2	24	jayaram	jayaram	PROPN
ejpam-492	2	25	the	the	DET
ejpam-492	2	26	university	university	PROPN
ejpam-492	2	27	of	of	ADP
ejpam-492	2	28	the	the	DET
ejpam-492	2	29	west	west	PROPN
ejpam-492	2	30	indies	indies	PROPN
ejpam-492	2	31	,	,	PUNCT
ejpam-492	2	32	department	department	NOUN
ejpam-492	2	33	of	of	ADP
ejpam-492	2	34	mathematics	mathematics	PROPN
ejpam-492	2	35	,	,	PUNCT
ejpam-492	2	36	p.o	p.o	PROPN
ejpam-492	2	37	.	.	PROPN
ejpam-492	2	38	box	box	PROPN
ejpam-492	2	39	64	64	NUM
ejpam-492	2	40	,	,	PUNCT
ejpam-492	2	41	bridgetown	bridgetown	PROPN
ejpam-492	2	42	,	,	PUNCT
ejpam-492	2	43	barbados	barbado	NOUN
ejpam-492	2	44	.	.	PUNCT
ejpam-492	3	1	abstract	abstract	ADJ
ejpam-492	3	2	.	.	PUNCT
ejpam-492	4	1	in	in	ADP
ejpam-492	4	2	this	this	DET
ejpam-492	4	3	paper	paper	NOUN
ejpam-492	4	4	we	we	PRON
ejpam-492	4	5	characterize	characterize	VERB
ejpam-492	4	6	π	π	NOUN
ejpam-492	4	7	-	-	NOUN
ejpam-492	4	8	modules	module	NOUN
ejpam-492	4	9	.	.	PUNCT
ejpam-492	5	1	next	next	ADV
ejpam-492	5	2	we	we	PRON
ejpam-492	5	3	establish	establish	VERB
ejpam-492	5	4	some	some	DET
ejpam-492	5	5	equivalent	equivalent	ADJ
ejpam-492	5	6	conditions	condition	NOUN
ejpam-492	5	7	for	for	SCONJ
ejpam-492	5	8	an	an	DET
ejpam-492	5	9	almost	almost	ADV
ejpam-492	5	10	π	π	NOUN
ejpam-492	5	11	-	-	NOUN
ejpam-492	5	12	module	module	NOUN
ejpam-492	5	13	to	to	PART
ejpam-492	5	14	be	be	AUX
ejpam-492	5	15	a	a	DET
ejpam-492	5	16	π	π	NOUN
ejpam-492	5	17	-	-	NOUN
ejpam-492	5	18	module	module	NOUN
ejpam-492	5	19	.	.	PUNCT
ejpam-492	6	1	2000	2000	NUM
ejpam-492	6	2	mathematics	mathematic	NOUN
ejpam-492	6	3	subject	subject	NOUN
ejpam-492	6	4	classifications	classification	NOUN
ejpam-492	6	5	:	:	PUNCT
ejpam-492	6	6	primary	primary	ADJ
ejpam-492	6	7	05c38	05c38	NOUN
ejpam-492	6	8	,	,	PUNCT
ejpam-492	6	9	15a15	15a15	NUM
ejpam-492	6	10	;	;	PUNCT
ejpam-492	6	11	secondary	secondary	ADJ
ejpam-492	6	12	05a15	05a15	NUM
ejpam-492	6	13	,	,	PUNCT
ejpam-492	6	14	15a18	15a18	NUM
ejpam-492	6	15	key	key	ADJ
ejpam-492	6	16	words	word	NOUN
ejpam-492	6	17	and	and	CCONJ
ejpam-492	6	18	phrases	phrase	NOUN
ejpam-492	6	19	:	:	PUNCT
ejpam-492	6	20	π	π	NOUN
ejpam-492	6	21	-	-	NOUN
ejpam-492	6	22	module	module	NOUN
ejpam-492	6	23	,	,	PUNCT
ejpam-492	6	24	almost	almost	ADV
ejpam-492	6	25	π	π	NOUN
ejpam-492	6	26	-	-	NOUN
ejpam-492	6	27	module	module	NOUN
ejpam-492	6	28	,	,	PUNCT
ejpam-492	6	29	multiplication	multiplication	NOUN
ejpam-492	6	30	module	module	NOUN
ejpam-492	6	31	,	,	PUNCT
ejpam-492	6	32	π	π	PROPN
ejpam-492	6	33	-	-	NOUN
ejpam-492	6	34	ring	ring	NOUN
ejpam-492	6	35	,	,	PUNCT
ejpam-492	6	36	almost	almost	ADV
ejpam-492	6	37	π	π	NOUN
ejpam-492	6	38	-	-	NOUN
ejpam-492	6	39	ring	ring	ADJ
ejpam-492	6	40	,	,	PUNCT
ejpam-492	6	41	multiplication	multiplication	NOUN
ejpam-492	6	42	ideal	ideal	NOUN
ejpam-492	6	43	and	and	CCONJ
ejpam-492	6	44	quasi	quasi	ADJ
ejpam-492	6	45	-	-	ADJ
ejpam-492	6	46	principal	principal	ADJ
ejpam-492	6	47	ideal	ideal	NOUN
ejpam-492	6	48	.	.	PUNCT
ejpam-492	7	1	1	1	X
ejpam-492	7	2	.	.	X
ejpam-492	7	3	introduction	introduction	NOUN
ejpam-492	7	4	throughout	throughout	ADP
ejpam-492	7	5	this	this	DET
ejpam-492	7	6	paper	paper	NOUN
ejpam-492	7	7	r	r	NOUN
ejpam-492	7	8	denotes	denote	VERB
ejpam-492	7	9	a	a	DET
ejpam-492	7	10	commutative	commutative	ADJ
ejpam-492	7	11	ring	ring	NOUN
ejpam-492	7	12	with	with	ADP
ejpam-492	7	13	identity	identity	NOUN
ejpam-492	7	14	and	and	CCONJ
ejpam-492	7	15	all	all	DET
ejpam-492	7	16	modules	module	NOUN
ejpam-492	7	17	are	be	AUX
ejpam-492	7	18	unital	unital	ADJ
ejpam-492	7	19	r	r	NOUN
ejpam-492	7	20	-	-	PUNCT
ejpam-492	7	21	modules	module	NOUN
ejpam-492	7	22	.	.	PUNCT
ejpam-492	8	1	l(r	l(r	PROPN
ejpam-492	8	2	)	)	PUNCT
ejpam-492	8	3	denotes	denote	VERB
ejpam-492	8	4	the	the	DET
ejpam-492	8	5	lattice	lattice	NOUN
ejpam-492	8	6	of	of	ADP
ejpam-492	8	7	all	all	DET
ejpam-492	8	8	ideals	ideal	NOUN
ejpam-492	8	9	of	of	ADP
ejpam-492	8	10	r.	r.	PROPN
ejpam-492	8	11	throughout	throughout	ADP
ejpam-492	8	12	this	this	DET
ejpam-492	8	13	paper	paper	NOUN
ejpam-492	9	1	m	m	VERB
ejpam-492	9	2	denotes	denote	VERB
ejpam-492	9	3	a	a	DET
ejpam-492	9	4	unital	unital	ADJ
ejpam-492	9	5	r	r	NOUN
ejpam-492	9	6	-	-	PUNCT
ejpam-492	9	7	module	module	NOUN
ejpam-492	9	8	.	.	PUNCT
ejpam-492	10	1	in	in	ADP
ejpam-492	10	2	this	this	DET
ejpam-492	10	3	paper	paper	NOUN
ejpam-492	10	4	we	we	PRON
ejpam-492	10	5	introduce	introduce	VERB
ejpam-492	10	6	and	and	CCONJ
ejpam-492	10	7	study	study	VERB
ejpam-492	10	8	the	the	DET
ejpam-492	10	9	concepts	concept	NOUN
ejpam-492	10	10	of	of	ADP
ejpam-492	10	11	π	π	NOUN
ejpam-492	10	12	-	-	NOUN
ejpam-492	10	13	module	module	NOUN
ejpam-492	10	14	and	and	CCONJ
ejpam-492	10	15	almost	almost	ADV
ejpam-492	10	16	π	π	NOUN
ejpam-492	10	17	-	-	NOUN
ejpam-492	10	18	module	module	NOUN
ejpam-492	10	19	.	.	PUNCT
ejpam-492	11	1	in	in	ADP
ejpam-492	11	2	section	section	NOUN
ejpam-492	11	3	3	3	NUM
ejpam-492	11	4	,	,	PUNCT
ejpam-492	11	5	we	we	PRON
ejpam-492	11	6	prove	prove	VERB
ejpam-492	11	7	that	that	SCONJ
ejpam-492	11	8	a	a	DET
ejpam-492	11	9	faithful	faithful	ADJ
ejpam-492	11	10	r	r	NOUN
ejpam-492	11	11	-	-	PUNCT
ejpam-492	11	12	module	module	NOUN
ejpam-492	11	13	m	m	NOUN
ejpam-492	11	14	is	be	AUX
ejpam-492	11	15	a	a	DET
ejpam-492	11	16	π	π	NOUN
ejpam-492	11	17	-	-	NOUN
ejpam-492	11	18	module	module	NOUN
ejpam-492	11	19	if	if	SCONJ
ejpam-492	11	20	and	and	CCONJ
ejpam-492	11	21	only	only	ADV
ejpam-492	11	22	if	if	SCONJ
ejpam-492	11	23	r	r	NOUN
ejpam-492	11	24	is	be	AUX
ejpam-492	11	25	a	a	DET
ejpam-492	11	26	π	π	NOUN
ejpam-492	11	27	-	-	NOUN
ejpam-492	11	28	ring	ring	NOUN
ejpam-492	11	29	and	and	CCONJ
ejpam-492	11	30	m	m	NOUN
ejpam-492	11	31	is	be	AUX
ejpam-492	11	32	a	a	DET
ejpam-492	11	33	multiplication	multiplication	NOUN
ejpam-492	11	34	module	module	NOUN
ejpam-492	11	35	if	if	SCONJ
ejpam-492	11	36	and	and	CCONJ
ejpam-492	11	37	only	only	ADV
ejpam-492	11	38	if	if	SCONJ
ejpam-492	11	39	every	every	DET
ejpam-492	11	40	cyclic	cyclic	ADJ
ejpam-492	11	41	submodule	submodule	NOUN
ejpam-492	11	42	of	of	ADP
ejpam-492	11	43	m	m	PROPN
ejpam-492	11	44	is	be	AUX
ejpam-492	11	45	of	of	ADP
ejpam-492	11	46	the	the	DET
ejpam-492	11	47	form	form	NOUN
ejpam-492	11	48	i	i	PRON
ejpam-492	11	49	m	m	VERB
ejpam-492	11	50	,	,	PUNCT
ejpam-492	11	51	where	where	SCONJ
ejpam-492	11	52	i	i	PRON
ejpam-492	11	53	is	be	AUX
ejpam-492	11	54	a	a	DET
ejpam-492	11	55	finite	finite	ADJ
ejpam-492	11	56	product	product	NOUN
ejpam-492	11	57	of	of	ADP
ejpam-492	11	58	email	email	NOUN
ejpam-492	11	59	address	address	NOUN
ejpam-492	11	60	:	:	PUNCT
ejpam-492	12	1	jayaram	jayaram	PROPN
ejpam-492	12	2	.	.	PROPN
ejpam-492	12	3	hillumu	hillumu	PROPN
ejpam-492	12	4	�	�	PROPN
ejpam-492	12	5	avehill.uwi.edu	avehill.uwi.edu	PROPN
ejpam-492	12	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-492	12	7	508	508	NUM
ejpam-492	12	8	c	c	NOUN
ejpam-492	12	9	©	©	PROPN
ejpam-492	12	10	2009	2009	NUM
ejpam-492	12	11	ejpam	ejpam	NOUN
ejpam-492	12	12	all	all	DET
ejpam-492	12	13	rights	right	NOUN
ejpam-492	12	14	reserved	reserve	VERB
ejpam-492	12	15	.	.	PUNCT
ejpam-492	13	1	c.	c.	PROPN
ejpam-492	13	2	jayaram	jayaram	PROPN
ejpam-492	13	3	/	/	SYM
ejpam-492	13	4	eur	eur	PROPN
ejpam-492	13	5	.	.	PUNCT
ejpam-492	14	1	j.	j.	PROPN
ejpam-492	14	2	pure	pure	PROPN
ejpam-492	14	3	appl	appl	PROPN
ejpam-492	14	4	.	.	PROPN
ejpam-492	14	5	math	math	PROPN
ejpam-492	14	6	,	,	PUNCT
ejpam-492	14	7	2	2	NUM
ejpam-492	14	8	(	(	PUNCT
ejpam-492	14	9	2009	2009	NUM
ejpam-492	14	10	)	)	PUNCT
ejpam-492	14	11	,	,	PUNCT
ejpam-492	14	12	(	(	PUNCT
ejpam-492	14	13	508	508	NUM
ejpam-492	14	14	-	-	NUM
ejpam-492	14	15	519	519	NUM
ejpam-492	14	16	)	)	PUNCT
ejpam-492	14	17	509	509	NUM
ejpam-492	14	18	quasi	quasi	ADJ
ejpam-492	14	19	-	-	ADJ
ejpam-492	14	20	principal	principal	ADJ
ejpam-492	14	21	prime	prime	ADJ
ejpam-492	14	22	ideals	ideal	NOUN
ejpam-492	14	23	of	of	ADP
ejpam-492	14	24	rank	rank	NOUN
ejpam-492	14	25	less	less	ADJ
ejpam-492	14	26	than	than	ADP
ejpam-492	14	27	or	or	CCONJ
ejpam-492	14	28	equal	equal	ADJ
ejpam-492	14	29	to	to	ADP
ejpam-492	14	30	one	one	NUM
ejpam-492	14	31	(	(	PUNCT
ejpam-492	14	32	see	see	VERB
ejpam-492	14	33	theorem	theorem	NOUN
ejpam-492	14	34	1	1	NUM
ejpam-492	14	35	)	)	PUNCT
ejpam-492	14	36	.	.	PUNCT
ejpam-492	15	1	using	use	VERB
ejpam-492	15	2	these	these	DET
ejpam-492	15	3	results	result	NOUN
ejpam-492	15	4	,	,	PUNCT
ejpam-492	15	5	we	we	PRON
ejpam-492	15	6	establish	establish	VERB
ejpam-492	15	7	some	some	DET
ejpam-492	15	8	equivalent	equivalent	ADJ
ejpam-492	15	9	conditions	condition	NOUN
ejpam-492	15	10	for	for	SCONJ
ejpam-492	15	11	an	an	DET
ejpam-492	15	12	almost	almost	ADV
ejpam-492	15	13	π	π	NOUN
ejpam-492	15	14	-	-	NOUN
ejpam-492	15	15	module	module	NOUN
ejpam-492	15	16	to	to	PART
ejpam-492	15	17	be	be	AUX
ejpam-492	15	18	a	a	DET
ejpam-492	15	19	π	π	NOUN
ejpam-492	15	20	-	-	NOUN
ejpam-492	15	21	module	module	NOUN
ejpam-492	15	22	(	(	PUNCT
ejpam-492	15	23	see	see	VERB
ejpam-492	15	24	theorem	theorem	NOUN
ejpam-492	15	25	2	2	NUM
ejpam-492	15	26	)	)	PUNCT
ejpam-492	15	27	.	.	PUNCT
ejpam-492	16	1	2	2	X
ejpam-492	16	2	.	.	X
ejpam-492	16	3	basic	basic	ADJ
ejpam-492	16	4	notions	notion	NOUN
ejpam-492	16	5	for	for	ADP
ejpam-492	16	6	any	any	DET
ejpam-492	16	7	x	x	SYM
ejpam-492	16	8	∈	∈	PROPN
ejpam-492	16	9	r	r	NOUN
ejpam-492	16	10	,	,	PUNCT
ejpam-492	16	11	the	the	DET
ejpam-492	16	12	principal	principal	ADJ
ejpam-492	16	13	ideal	ideal	NOUN
ejpam-492	16	14	generated	generate	VERB
ejpam-492	16	15	by	by	ADP
ejpam-492	16	16	x	x	SYM
ejpam-492	16	17	is	be	AUX
ejpam-492	16	18	denoted	denote	VERB
ejpam-492	16	19	by	by	ADP
ejpam-492	16	20	(	(	PUNCT
ejpam-492	16	21	x	x	NOUN
ejpam-492	16	22	)	)	PUNCT
ejpam-492	16	23	.	.	PUNCT
ejpam-492	17	1	recall	recall	VERB
ejpam-492	17	2	that	that	SCONJ
ejpam-492	17	3	an	an	DET
ejpam-492	17	4	ideal	ideal	NOUN
ejpam-492	17	5	i	i	PRON
ejpam-492	17	6	of	of	ADP
ejpam-492	17	7	r	r	NOUN
ejpam-492	17	8	is	be	AUX
ejpam-492	17	9	called	call	VERB
ejpam-492	17	10	a	a	DET
ejpam-492	17	11	multiplication	multiplication	NOUN
ejpam-492	17	12	ideal	ideal	ADJ
ejpam-492	17	13	,	,	PUNCT
ejpam-492	17	14	if	if	SCONJ
ejpam-492	17	15	for	for	ADP
ejpam-492	17	16	every	every	DET
ejpam-492	17	17	ideal	ideal	NOUN
ejpam-492	17	18	j	j	PROPN
ejpam-492	17	19	⊆	⊆	NUM
ejpam-492	17	20	i	i	PRON
ejpam-492	17	21	,	,	PUNCT
ejpam-492	17	22	there	there	PRON
ejpam-492	17	23	exists	exist	VERB
ejpam-492	17	24	an	an	DET
ejpam-492	17	25	ideal	ideal	NOUN
ejpam-492	17	26	k	k	PROPN
ejpam-492	17	27	with	with	ADP
ejpam-492	17	28	j	j	PROPN
ejpam-492	17	29	=	=	SYM
ejpam-492	17	30	ki	ki	PROPN
ejpam-492	17	31	.	.	PUNCT
ejpam-492	18	1	multiplication	multiplication	NOUN
ejpam-492	18	2	ideals	ideal	NOUN
ejpam-492	18	3	have	have	AUX
ejpam-492	18	4	been	be	AUX
ejpam-492	18	5	extensively	extensively	ADV
ejpam-492	18	6	studied	study	VERB
ejpam-492	18	7	for	for	ADP
ejpam-492	18	8	example	example	NOUN
ejpam-492	18	9	,	,	PUNCT
ejpam-492	18	10	see	see	VERB
ejpam-492	18	11	[	[	X
ejpam-492	18	12	2	2	NUM
ejpam-492	18	13	]	]	PUNCT
ejpam-492	18	14	,	,	PUNCT
ejpam-492	18	15	[	[	X
ejpam-492	18	16	5	5	NUM
ejpam-492	18	17	]	]	PUNCT
ejpam-492	18	18	and	and	CCONJ
ejpam-492	18	19	[	[	X
ejpam-492	18	20	6	6	NUM
ejpam-492	18	21	]	]	PUNCT
ejpam-492	18	22	.	.	PUNCT
ejpam-492	19	1	an	an	DET
ejpam-492	19	2	ideal	ideal	ADJ
ejpam-492	19	3	i	i	PRON
ejpam-492	19	4	of	of	ADP
ejpam-492	19	5	r	r	NOUN
ejpam-492	19	6	is	be	AUX
ejpam-492	19	7	called	call	VERB
ejpam-492	19	8	a	a	DET
ejpam-492	19	9	quasi	quasi	ADJ
ejpam-492	19	10	-	-	ADJ
ejpam-492	19	11	principal	principal	ADJ
ejpam-492	19	12	ideal	ideal	NOUN
ejpam-492	19	13	[	[	X
ejpam-492	19	14	16	16	NUM
ejpam-492	19	15	,	,	PUNCT
ejpam-492	19	16	exercise	exercise	VERB
ejpam-492	19	17	10	10	NUM
ejpam-492	19	18	,	,	PUNCT
ejpam-492	19	19	page	page	NOUN
ejpam-492	19	20	147	147	NUM
ejpam-492	19	21	]	]	PUNCT
ejpam-492	19	22	(	(	PUNCT
ejpam-492	19	23	or	or	CCONJ
ejpam-492	19	24	a	a	DET
ejpam-492	19	25	principal	principal	ADJ
ejpam-492	19	26	element	element	NOUN
ejpam-492	19	27	of	of	ADP
ejpam-492	19	28	l(r	l(r	PROPN
ejpam-492	19	29	)	)	PUNCT
ejpam-492	20	1	[	[	X
ejpam-492	20	2	19	19	NUM
ejpam-492	20	3	]	]	PUNCT
ejpam-492	20	4	)	)	PUNCT
ejpam-492	20	5	if	if	SCONJ
ejpam-492	20	6	it	it	PRON
ejpam-492	20	7	satisfies	satisfy	VERB
ejpam-492	20	8	the	the	DET
ejpam-492	20	9	following	follow	VERB
ejpam-492	20	10	identities	identity	NOUN
ejpam-492	20	11	(	(	PUNCT
ejpam-492	20	12	i	i	NOUN
ejpam-492	20	13	)	)	PUNCT
ejpam-492	20	14	(	(	PUNCT
ejpam-492	20	15	a∩	a∩	PROPN
ejpam-492	20	16	(	(	PUNCT
ejpam-492	20	17	b	b	X
ejpam-492	20	18	:	:	PUNCT
ejpam-492	20	19	i))i	i))i	NOUN
ejpam-492	20	20	=	=	SYM
ejpam-492	20	21	ai∩b	ai∩b	PROPN
ejpam-492	20	22	and	and	CCONJ
ejpam-492	20	23	(	(	PUNCT
ejpam-492	20	24	ii	ii	NOUN
ejpam-492	20	25	)	)	PUNCT
ejpam-492	20	26	(	(	PUNCT
ejpam-492	20	27	a+bi	a+bi	PROPN
ejpam-492	20	28	)	)	PUNCT
ejpam-492	20	29	:	:	PUNCT
ejpam-492	20	30	i	i	PRON
ejpam-492	20	31	=	=	PUNCT
ejpam-492	20	32	(	(	PUNCT
ejpam-492	20	33	a	a	PRON
ejpam-492	20	34	:	:	PUNCT
ejpam-492	20	35	i)+b	i)+b	PROPN
ejpam-492	20	36	,	,	PUNCT
ejpam-492	20	37	for	for	ADP
ejpam-492	20	38	all	all	DET
ejpam-492	20	39	a	a	PRON
ejpam-492	20	40	,	,	PUNCT
ejpam-492	20	41	b	b	NOUN
ejpam-492	20	42	∈	∈	PROPN
ejpam-492	20	43	l(r	l(r	PROPN
ejpam-492	20	44	)	)	PUNCT
ejpam-492	20	45	.	.	PUNCT
ejpam-492	21	1	it	it	PRON
ejpam-492	21	2	is	be	AUX
ejpam-492	21	3	well	well	ADV
ejpam-492	21	4	known	know	VERB
ejpam-492	21	5	that	that	SCONJ
ejpam-492	21	6	an	an	DET
ejpam-492	21	7	ideal	ideal	NOUN
ejpam-492	21	8	i	i	PRON
ejpam-492	21	9	of	of	ADP
ejpam-492	21	10	r	r	NOUN
ejpam-492	21	11	is	be	AUX
ejpam-492	21	12	quasi	quasi	ADJ
ejpam-492	21	13	-	-	NOUN
ejpam-492	21	14	principal	principal	ADJ
ejpam-492	21	15	if	if	SCONJ
ejpam-492	22	1	and	and	CCONJ
ejpam-492	22	2	only	only	ADV
ejpam-492	22	3	if	if	SCONJ
ejpam-492	22	4	it	it	PRON
ejpam-492	22	5	is	be	AUX
ejpam-492	22	6	finitely	finitely	ADV
ejpam-492	22	7	generated	generate	VERB
ejpam-492	22	8	and	and	CCONJ
ejpam-492	22	9	locally	locally	ADV
ejpam-492	22	10	principal	principal	ADJ
ejpam-492	22	11	if	if	SCONJ
ejpam-492	22	12	and	and	CCONJ
ejpam-492	22	13	only	only	ADV
ejpam-492	22	14	if	if	SCONJ
ejpam-492	22	15	it	it	PRON
ejpam-492	22	16	is	be	AUX
ejpam-492	22	17	a	a	DET
ejpam-492	22	18	finitely	finitely	ADV
ejpam-492	22	19	generated	generate	VERB
ejpam-492	22	20	multiplication	multiplication	NOUN
ejpam-492	22	21	ideal	ideal	NOUN
ejpam-492	23	1	[	[	X
ejpam-492	23	2	8	8	NUM
ejpam-492	23	3	,	,	PUNCT
ejpam-492	23	4	theorem	theorem	VERB
ejpam-492	23	5	3	3	NUM
ejpam-492	23	6	]	]	PUNCT
ejpam-492	23	7	.	.	PUNCT
ejpam-492	24	1	r	r	NOUN
ejpam-492	24	2	is	be	AUX
ejpam-492	24	3	a	a	DET
ejpam-492	24	4	π	π	NOUN
ejpam-492	24	5	-	-	NOUN
ejpam-492	24	6	ring	ring	NOUN
ejpam-492	24	7	if	if	SCONJ
ejpam-492	24	8	every	every	DET
ejpam-492	24	9	principal	principal	ADJ
ejpam-492	24	10	ideal	ideal	NOUN
ejpam-492	24	11	is	be	AUX
ejpam-492	24	12	a	a	DET
ejpam-492	24	13	finite	finite	ADJ
ejpam-492	24	14	product	product	NOUN
ejpam-492	24	15	of	of	ADP
ejpam-492	24	16	prime	prime	ADJ
ejpam-492	24	17	ideals	ideal	NOUN
ejpam-492	24	18	of	of	ADP
ejpam-492	24	19	r.	r.	PROPN
ejpam-492	24	20	r	r	PROPN
ejpam-492	24	21	is	be	AUX
ejpam-492	24	22	an	an	DET
ejpam-492	24	23	almost	almost	ADV
ejpam-492	24	24	π	π	NOUN
ejpam-492	24	25	-	-	NOUN
ejpam-492	24	26	ring	ring	NOUN
ejpam-492	24	27	if	if	SCONJ
ejpam-492	24	28	rp	rp	NOUN
ejpam-492	24	29	is	be	AUX
ejpam-492	24	30	a	a	DET
ejpam-492	24	31	π	π	NOUN
ejpam-492	24	32	-	-	NOUN
ejpam-492	24	33	ring	ring	NOUN
ejpam-492	24	34	,	,	PUNCT
ejpam-492	24	35	for	for	ADP
ejpam-492	24	36	every	every	DET
ejpam-492	24	37	maximal	maximal	ADJ
ejpam-492	24	38	ideal	ideal	NOUN
ejpam-492	24	39	p	p	PROPN
ejpam-492	24	40	of	of	ADP
ejpam-492	24	41	r.	r.	PROPN
ejpam-492	24	42	π	π	PROPN
ejpam-492	24	43	-	-	PUNCT
ejpam-492	24	44	rings	ring	NOUN
ejpam-492	24	45	have	have	AUX
ejpam-492	24	46	been	be	AUX
ejpam-492	24	47	extensively	extensively	ADV
ejpam-492	24	48	studied	study	VERB
ejpam-492	24	49	for	for	ADP
ejpam-492	24	50	example	example	NOUN
ejpam-492	24	51	,	,	PUNCT
ejpam-492	24	52	see	see	VERB
ejpam-492	24	53	[	[	X
ejpam-492	24	54	13	13	NUM
ejpam-492	24	55	]	]	PUNCT
ejpam-492	24	56	,	,	PUNCT
ejpam-492	24	57	[	[	X
ejpam-492	24	58	15	15	NUM
ejpam-492	24	59	]	]	PUNCT
ejpam-492	24	60	and	and	CCONJ
ejpam-492	24	61	[	[	X
ejpam-492	24	62	17	17	NUM
ejpam-492	24	63	]	]	PUNCT
ejpam-492	24	64	.	.	PUNCT
ejpam-492	25	1	by	by	ADP
ejpam-492	25	2	a	a	DET
ejpam-492	25	3	special	special	ADJ
ejpam-492	25	4	principal	principal	ADJ
ejpam-492	25	5	ideal	ideal	NOUN
ejpam-492	25	6	ring	ring	NOUN
ejpam-492	25	7	,	,	PUNCT
ejpam-492	25	8	we	we	PRON
ejpam-492	25	9	mean	mean	VERB
ejpam-492	25	10	a	a	DET
ejpam-492	25	11	principal	principal	ADJ
ejpam-492	25	12	ideal	ideal	NOUN
ejpam-492	25	13	ring	ring	NOUN
ejpam-492	25	14	r	r	NOUN
ejpam-492	25	15	with	with	ADP
ejpam-492	25	16	exactly	exactly	ADV
ejpam-492	25	17	one	one	NUM
ejpam-492	25	18	prime	prime	ADJ
ejpam-492	25	19	ideal	ideal	NOUN
ejpam-492	25	20	p	p	PROPN
ejpam-492	25	21	6=	6=	PROPN
ejpam-492	25	22	r	r	NOUN
ejpam-492	25	23	,	,	PUNCT
ejpam-492	25	24	pn	pn	NOUN
ejpam-492	25	25	=	=	SYM
ejpam-492	25	26	(	(	PUNCT
ejpam-492	25	27	0	0	NUM
ejpam-492	25	28	)	)	PUNCT
ejpam-492	25	29	for	for	ADP
ejpam-492	25	30	some	some	DET
ejpam-492	25	31	positive	positive	ADJ
ejpam-492	25	32	integer	integer	NOUN
ejpam-492	25	33	n	n	CCONJ
ejpam-492	25	34	,	,	PUNCT
ejpam-492	25	35	so	so	ADV
ejpam-492	25	36	the	the	DET
ejpam-492	25	37	only	only	ADJ
ejpam-492	25	38	ideals	ideal	NOUN
ejpam-492	25	39	of	of	ADP
ejpam-492	25	40	r	r	NOUN
ejpam-492	25	41	are	be	AUX
ejpam-492	25	42	r	r	NOUN
ejpam-492	25	43	,	,	PUNCT
ejpam-492	25	44	p	p	NOUN
ejpam-492	25	45	,	,	PUNCT
ejpam-492	25	46	p2	p2	NOUN
ejpam-492	25	47	,	,	PUNCT
ejpam-492	25	48	...	...	PUNCT
ejpam-492	25	49	,	,	PUNCT
ejpam-492	25	50	pn	pn	PROPN
ejpam-492	25	51	=	=	SYM
ejpam-492	25	52	(	(	PUNCT
ejpam-492	25	53	0	0	NUM
ejpam-492	25	54	)	)	PUNCT
ejpam-492	25	55	.	.	PUNCT
ejpam-492	26	1	a	a	DET
ejpam-492	26	2	submodule	submodule	PROPN
ejpam-492	26	3	n	n	PROPN
ejpam-492	26	4	of	of	ADP
ejpam-492	26	5	m	m	PROPN
ejpam-492	26	6	is	be	AUX
ejpam-492	26	7	proper	proper	ADJ
ejpam-492	26	8	if	if	SCONJ
ejpam-492	26	9	n	n	PROPN
ejpam-492	26	10	6=	6=	ADP
ejpam-492	26	11	m	m	VERB
ejpam-492	26	12	.	.	PUNCT
ejpam-492	27	1	for	for	ADP
ejpam-492	27	2	any	any	DET
ejpam-492	27	3	two	two	NUM
ejpam-492	27	4	submodules	submodule	NOUN
ejpam-492	27	5	n	n	PRON
ejpam-492	27	6	and	and	CCONJ
ejpam-492	27	7	k	k	PROPN
ejpam-492	27	8	of	of	ADP
ejpam-492	27	9	m	m	PROPN
ejpam-492	27	10	,	,	PUNCT
ejpam-492	27	11	the	the	DET
ejpam-492	27	12	ideal	ideal	NOUN
ejpam-492	27	13	{	{	PUNCT
ejpam-492	27	14	a	a	DET
ejpam-492	27	15	∈	∈	PROPN
ejpam-492	27	16	r	r	NOUN
ejpam-492	27	17	|	|	NOUN
ejpam-492	27	18	ak	ak	PROPN
ejpam-492	27	19	⊆	⊆	NUM
ejpam-492	27	20	n	n	CCONJ
ejpam-492	27	21	}	}	PUNCT
ejpam-492	27	22	will	will	AUX
ejpam-492	27	23	be	be	AUX
ejpam-492	27	24	denoted	denote	VERB
ejpam-492	27	25	by	by	ADP
ejpam-492	27	26	(	(	PUNCT
ejpam-492	27	27	n	n	NUM
ejpam-492	27	28	:	:	PUNCT
ejpam-492	27	29	k	k	X
ejpam-492	27	30	)	)	PUNCT
ejpam-492	27	31	.	.	PUNCT
ejpam-492	28	1	thus	thus	ADV
ejpam-492	28	2	(	(	PUNCT
ejpam-492	28	3	o	o	NOUN
ejpam-492	28	4	:	:	PUNCT
ejpam-492	28	5	m	m	VERB
ejpam-492	28	6	)	)	PUNCT
ejpam-492	28	7	is	be	AUX
ejpam-492	28	8	the	the	DET
ejpam-492	28	9	annihilator	annihilator	NOUN
ejpam-492	28	10	of	of	ADP
ejpam-492	28	11	m	m	PROPN
ejpam-492	28	12	.	.	PUNCT
ejpam-492	29	1	m	m	PROPN
ejpam-492	29	2	is	be	AUX
ejpam-492	29	3	said	say	VERB
ejpam-492	29	4	to	to	PART
ejpam-492	29	5	be	be	AUX
ejpam-492	29	6	a	a	DET
ejpam-492	29	7	faithful	faithful	ADJ
ejpam-492	29	8	module	module	NOUN
ejpam-492	29	9	if	if	SCONJ
ejpam-492	29	10	(	(	PUNCT
ejpam-492	29	11	o	o	NOUN
ejpam-492	29	12	:	:	PUNCT
ejpam-492	29	13	m	m	VERB
ejpam-492	29	14	)	)	PUNCT
ejpam-492	29	15	is	be	AUX
ejpam-492	29	16	the	the	DET
ejpam-492	29	17	zero	zero	NUM
ejpam-492	29	18	ideal	ideal	NOUN
ejpam-492	29	19	of	of	ADP
ejpam-492	29	20	r.	r.	PROPN
ejpam-492	29	21	m	m	PROPN
ejpam-492	29	22	is	be	AUX
ejpam-492	29	23	said	say	VERB
ejpam-492	29	24	to	to	PART
ejpam-492	29	25	be	be	AUX
ejpam-492	29	26	a	a	DET
ejpam-492	29	27	multiplication	multiplication	NOUN
ejpam-492	29	28	module	module	NOUN
ejpam-492	30	1	[	[	X
ejpam-492	30	2	9	9	NUM
ejpam-492	30	3	]	]	PUNCT
ejpam-492	30	4	if	if	SCONJ
ejpam-492	30	5	every	every	DET
ejpam-492	30	6	submodule	submodule	NOUN
ejpam-492	30	7	of	of	ADP
ejpam-492	30	8	m	m	PROPN
ejpam-492	30	9	is	be	AUX
ejpam-492	30	10	of	of	ADP
ejpam-492	30	11	the	the	DET
ejpam-492	30	12	form	form	NOUN
ejpam-492	31	1	i	i	PRON
ejpam-492	31	2	m	m	VERB
ejpam-492	31	3	,	,	PUNCT
ejpam-492	31	4	for	for	ADP
ejpam-492	31	5	some	some	DET
ejpam-492	31	6	ideal	ideal	ADJ
ejpam-492	31	7	i	i	PRON
ejpam-492	31	8	of	of	ADP
ejpam-492	31	9	r.	r.	PROPN
ejpam-492	31	10	a	a	DET
ejpam-492	31	11	submodule	submodule	PROPN
ejpam-492	31	12	n	n	PROPN
ejpam-492	31	13	of	of	ADP
ejpam-492	31	14	m	m	PROPN
ejpam-492	31	15	is	be	AUX
ejpam-492	31	16	said	say	VERB
ejpam-492	31	17	to	to	PART
ejpam-492	31	18	be	be	AUX
ejpam-492	31	19	a	a	DET
ejpam-492	31	20	multiplication	multiplication	NOUN
ejpam-492	31	21	submodule	submodule	NOUN
ejpam-492	31	22	if	if	SCONJ
ejpam-492	31	23	for	for	ADP
ejpam-492	31	24	every	every	DET
ejpam-492	31	25	submodule	submodule	NOUN
ejpam-492	31	26	n1	n1	NOUN
ejpam-492	31	27	⊆	⊆	NUM
ejpam-492	31	28	n	n	NOUN
ejpam-492	31	29	,	,	PUNCT
ejpam-492	31	30	there	there	PRON
ejpam-492	31	31	exists	exist	VERB
ejpam-492	31	32	an	an	DET
ejpam-492	31	33	ideal	ideal	ADJ
ejpam-492	31	34	j	j	NOUN
ejpam-492	31	35	of	of	ADP
ejpam-492	31	36	r	r	NOUN
ejpam-492	31	37	such	such	ADJ
ejpam-492	31	38	that	that	DET
ejpam-492	31	39	n1	n1	NOUN
ejpam-492	31	40	=	=	SYM
ejpam-492	31	41	jn	jn	PROPN
ejpam-492	31	42	.	.	PUNCT
ejpam-492	32	1	an	an	DET
ejpam-492	32	2	r	r	NOUN
ejpam-492	32	3	-	-	PUNCT
ejpam-492	32	4	module	module	NOUN
ejpam-492	32	5	m	m	NOUN
ejpam-492	32	6	is	be	AUX
ejpam-492	32	7	said	say	VERB
ejpam-492	32	8	to	to	PART
ejpam-492	32	9	be	be	AUX
ejpam-492	32	10	locally	locally	ADV
ejpam-492	32	11	cyclic	cyclic	ADJ
ejpam-492	32	12	if	if	SCONJ
ejpam-492	32	13	mp	mp	PROPN
ejpam-492	32	14	is	be	AUX
ejpam-492	32	15	a	a	DET
ejpam-492	32	16	cyclic	cyclic	ADJ
ejpam-492	32	17	rp	rp	NOUN
ejpam-492	32	18	-	-	PUNCT
ejpam-492	32	19	module	module	NOUN
ejpam-492	32	20	for	for	ADP
ejpam-492	32	21	all	all	DET
ejpam-492	32	22	maximal	maximal	ADJ
ejpam-492	32	23	ideals	ideal	NOUN
ejpam-492	32	24	p	p	PROPN
ejpam-492	32	25	of	of	ADP
ejpam-492	32	26	r.	r.	PROPN
ejpam-492	32	27	a	a	DET
ejpam-492	32	28	proper	proper	ADJ
ejpam-492	32	29	submodule	submodule	NOUN
ejpam-492	32	30	n	n	PROPN
ejpam-492	32	31	of	of	ADP
ejpam-492	32	32	m	m	PROPN
ejpam-492	32	33	is	be	AUX
ejpam-492	32	34	said	say	VERB
ejpam-492	32	35	to	to	PART
ejpam-492	32	36	be	be	AUX
ejpam-492	32	37	a	a	DET
ejpam-492	32	38	maximal	maximal	ADJ
ejpam-492	32	39	submodule	submodule	NOUN
ejpam-492	32	40	,	,	PUNCT
ejpam-492	32	41	if	if	SCONJ
ejpam-492	32	42	it	it	PRON
ejpam-492	32	43	is	be	AUX
ejpam-492	32	44	not	not	PART
ejpam-492	32	45	properly	properly	ADV
ejpam-492	32	46	c.	c.	PROPN
ejpam-492	32	47	jayaram	jayaram	PROPN
ejpam-492	32	48	/	/	SYM
ejpam-492	32	49	eur	eur	PROPN
ejpam-492	32	50	.	.	PUNCT
ejpam-492	33	1	j.	j.	PROPN
ejpam-492	33	2	pure	pure	PROPN
ejpam-492	33	3	appl	appl	PROPN
ejpam-492	33	4	.	.	PROPN
ejpam-492	33	5	math	math	PROPN
ejpam-492	33	6	,	,	PUNCT
ejpam-492	33	7	2	2	NUM
ejpam-492	33	8	(	(	PUNCT
ejpam-492	33	9	2009	2009	NUM
ejpam-492	33	10	)	)	PUNCT
ejpam-492	33	11	,	,	PUNCT
ejpam-492	33	12	(	(	PUNCT
ejpam-492	33	13	508	508	NUM
ejpam-492	33	14	-	-	NUM
ejpam-492	33	15	519	519	NUM
ejpam-492	33	16	)	)	PUNCT
ejpam-492	33	17	510	510	NUM
ejpam-492	33	18	contained	contain	VERB
ejpam-492	33	19	in	in	ADP
ejpam-492	33	20	any	any	DET
ejpam-492	33	21	other	other	ADJ
ejpam-492	33	22	proper	proper	ADJ
ejpam-492	33	23	submodule	submodule	NOUN
ejpam-492	33	24	of	of	ADP
ejpam-492	33	25	m	m	PROPN
ejpam-492	33	26	.	.	PUNCT
ejpam-492	34	1	a	a	DET
ejpam-492	34	2	proper	proper	ADJ
ejpam-492	34	3	submodule	submodule	NOUN
ejpam-492	34	4	n	n	PROPN
ejpam-492	34	5	of	of	ADP
ejpam-492	34	6	m	m	PROPN
ejpam-492	34	7	is	be	AUX
ejpam-492	34	8	a	a	DET
ejpam-492	34	9	prime	prime	ADJ
ejpam-492	34	10	submodule	submodule	NOUN
ejpam-492	34	11	,	,	PUNCT
ejpam-492	34	12	if	if	SCONJ
ejpam-492	34	13	for	for	ADP
ejpam-492	34	14	any	any	DET
ejpam-492	34	15	r	r	NOUN
ejpam-492	34	16	∈	∈	NOUN
ejpam-492	34	17	r	r	NOUN
ejpam-492	34	18	and	and	CCONJ
ejpam-492	34	19	m	m	PROPN
ejpam-492	34	20	∈	∈	PROPN
ejpam-492	34	21	m	m	PROPN
ejpam-492	34	22	,	,	PUNCT
ejpam-492	34	23	rm	rm	PROPN
ejpam-492	34	24	∈	∈	PROPN
ejpam-492	34	25	n	n	PRON
ejpam-492	34	26	implies	imply	VERB
ejpam-492	34	27	either	either	CCONJ
ejpam-492	34	28	m	m	PROPN
ejpam-492	34	29	∈	∈	PROPN
ejpam-492	34	30	n	n	NOUN
ejpam-492	34	31	or	or	CCONJ
ejpam-492	34	32	r	r	NOUN
ejpam-492	34	33	∈	∈	PROPN
ejpam-492	34	34	(	(	PUNCT
ejpam-492	34	35	n	n	NOUN
ejpam-492	34	36	:	:	PUNCT
ejpam-492	34	37	m	m	X
ejpam-492	34	38	)	)	PUNCT
ejpam-492	34	39	.	.	PUNCT
ejpam-492	35	1	a	a	DET
ejpam-492	35	2	proper	proper	ADJ
ejpam-492	35	3	submodule	submodule	NOUN
ejpam-492	35	4	n	n	PROPN
ejpam-492	35	5	of	of	ADP
ejpam-492	35	6	m	m	PROPN
ejpam-492	35	7	is	be	AUX
ejpam-492	35	8	a	a	DET
ejpam-492	35	9	primary	primary	ADJ
ejpam-492	35	10	submodule	submodule	NOUN
ejpam-492	35	11	,	,	PUNCT
ejpam-492	35	12	if	if	SCONJ
ejpam-492	35	13	for	for	ADP
ejpam-492	35	14	any	any	DET
ejpam-492	35	15	r	r	NOUN
ejpam-492	35	16	∈	∈	NOUN
ejpam-492	35	17	r	r	NOUN
ejpam-492	35	18	and	and	CCONJ
ejpam-492	35	19	m	m	PROPN
ejpam-492	35	20	∈	∈	PROPN
ejpam-492	35	21	m	m	PROPN
ejpam-492	35	22	,	,	PUNCT
ejpam-492	35	23	rm	rm	PROPN
ejpam-492	35	24	∈	∈	PROPN
ejpam-492	35	25	n	n	PRON
ejpam-492	35	26	implies	imply	VERB
ejpam-492	35	27	either	either	CCONJ
ejpam-492	35	28	m	m	PROPN
ejpam-492	35	29	∈	∈	PROPN
ejpam-492	35	30	n	n	NOUN
ejpam-492	35	31	or	or	CCONJ
ejpam-492	35	32	rn	rn	PROPN
ejpam-492	35	33	∈	∈	PROPN
ejpam-492	35	34	(	(	PUNCT
ejpam-492	35	35	n	n	NOUN
ejpam-492	35	36	:	:	PUNCT
ejpam-492	35	37	m	m	X
ejpam-492	35	38	)	)	PUNCT
ejpam-492	35	39	for	for	ADP
ejpam-492	35	40	some	some	DET
ejpam-492	35	41	positive	positive	ADJ
ejpam-492	35	42	integer	integer	NOUN
ejpam-492	35	43	n.	n.	NOUN
ejpam-492	35	44	by	by	ADP
ejpam-492	35	45	a	a	DET
ejpam-492	35	46	minimal	minimal	ADJ
ejpam-492	35	47	prime	prime	ADJ
ejpam-492	35	48	submodule	submodule	NOUN
ejpam-492	35	49	over	over	ADP
ejpam-492	35	50	a	a	DET
ejpam-492	35	51	submodule	submodule	NOUN
ejpam-492	35	52	n	n	PROPN
ejpam-492	35	53	of	of	ADP
ejpam-492	35	54	m	m	PROPN
ejpam-492	35	55	(	(	PUNCT
ejpam-492	35	56	or	or	CCONJ
ejpam-492	35	57	a	a	DET
ejpam-492	35	58	prime	prime	ADJ
ejpam-492	35	59	submodule	submodule	NOUN
ejpam-492	35	60	minimal	minimal	ADJ
ejpam-492	35	61	over	over	ADP
ejpam-492	35	62	n	n	CCONJ
ejpam-492	35	63	)	)	PUNCT
ejpam-492	35	64	,	,	PUNCT
ejpam-492	35	65	we	we	PRON
ejpam-492	35	66	mean	mean	VERB
ejpam-492	35	67	a	a	DET
ejpam-492	35	68	prime	prime	ADJ
ejpam-492	35	69	submodule	submodule	NOUN
ejpam-492	35	70	which	which	PRON
ejpam-492	35	71	is	be	AUX
ejpam-492	35	72	minimal	minimal	ADJ
ejpam-492	35	73	in	in	ADP
ejpam-492	35	74	the	the	DET
ejpam-492	35	75	collection	collection	NOUN
ejpam-492	35	76	of	of	ADP
ejpam-492	35	77	all	all	DET
ejpam-492	35	78	prime	prime	ADJ
ejpam-492	35	79	submodules	submodule	NOUN
ejpam-492	35	80	containing	contain	VERB
ejpam-492	35	81	n	n	NOUN
ejpam-492	35	82	.	.	PUNCT
ejpam-492	36	1	minimal	minimal	ADJ
ejpam-492	36	2	prime	prime	ADJ
ejpam-492	36	3	submodules	submodule	NOUN
ejpam-492	36	4	over	over	ADP
ejpam-492	36	5	the	the	DET
ejpam-492	36	6	zero	zero	NUM
ejpam-492	36	7	submodule	submodule	NOUN
ejpam-492	36	8	are	be	AUX
ejpam-492	36	9	simply	simply	ADV
ejpam-492	36	10	called	call	VERB
ejpam-492	36	11	the	the	DET
ejpam-492	36	12	minimal	minimal	ADJ
ejpam-492	36	13	prime	prime	ADJ
ejpam-492	36	14	submodules	submodule	NOUN
ejpam-492	36	15	.	.	PUNCT
ejpam-492	37	1	it	it	PRON
ejpam-492	37	2	is	be	AUX
ejpam-492	37	3	well	well	ADV
ejpam-492	37	4	known	know	VERB
ejpam-492	37	5	that	that	SCONJ
ejpam-492	37	6	maximal	maximal	ADJ
ejpam-492	37	7	submodules	submodule	NOUN
ejpam-492	37	8	and	and	CCONJ
ejpam-492	37	9	prime	prime	ADJ
ejpam-492	37	10	submodules	submodule	NOUN
ejpam-492	37	11	exist	exist	VERB
ejpam-492	37	12	in	in	ADP
ejpam-492	37	13	multiplication	multiplication	NOUN
ejpam-492	37	14	modules	module	NOUN
ejpam-492	37	15	(	(	PUNCT
ejpam-492	37	16	for	for	ADP
ejpam-492	37	17	details	detail	NOUN
ejpam-492	37	18	,	,	PUNCT
ejpam-492	37	19	see	see	VERB
ejpam-492	37	20	[	[	X
ejpam-492	37	21	11	11	NUM
ejpam-492	37	22	]	]	NUM
ejpam-492	37	23	)	)	PUNCT
ejpam-492	37	24	.	.	PUNCT
ejpam-492	38	1	it	it	PRON
ejpam-492	38	2	is	be	AUX
ejpam-492	38	3	well	well	ADV
ejpam-492	38	4	known	know	VERB
ejpam-492	38	5	that	that	SCONJ
ejpam-492	38	6	if	if	SCONJ
ejpam-492	38	7	m	m	NOUN
ejpam-492	38	8	is	be	AUX
ejpam-492	38	9	a	a	DET
ejpam-492	38	10	faithful	faithful	ADJ
ejpam-492	38	11	multiplication	multiplication	NOUN
ejpam-492	38	12	r	r	NOUN
ejpam-492	38	13	-	-	PUNCT
ejpam-492	38	14	module	module	NOUN
ejpam-492	38	15	and	and	CCONJ
ejpam-492	38	16	p	p	NOUN
ejpam-492	38	17	is	be	AUX
ejpam-492	38	18	a	a	DET
ejpam-492	38	19	prime	prime	ADJ
ejpam-492	38	20	ideal	ideal	NOUN
ejpam-492	38	21	of	of	ADP
ejpam-492	38	22	r	r	NOUN
ejpam-492	38	23	such	such	ADJ
ejpam-492	38	24	that	that	SCONJ
ejpam-492	38	25	m	m	PROPN
ejpam-492	38	26	6=	6=	ADP
ejpam-492	38	27	pm	pm	NOUN
ejpam-492	38	28	,	,	PUNCT
ejpam-492	38	29	then	then	ADV
ejpam-492	38	30	pm	pm	NOUN
ejpam-492	38	31	is	be	AUX
ejpam-492	38	32	a	a	DET
ejpam-492	38	33	prime	prime	ADJ
ejpam-492	38	34	submodule	submodule	NOUN
ejpam-492	38	35	of	of	ADP
ejpam-492	38	36	m	m	PROPN
ejpam-492	38	37	and	and	CCONJ
ejpam-492	38	38	every	every	DET
ejpam-492	38	39	prime	prime	ADJ
ejpam-492	38	40	submodule	submodule	NOUN
ejpam-492	38	41	of	of	ADP
ejpam-492	38	42	m	m	PROPN
ejpam-492	38	43	is	be	AUX
ejpam-492	38	44	of	of	ADP
ejpam-492	38	45	the	the	DET
ejpam-492	38	46	form	form	NOUN
ejpam-492	38	47	pm	pm	NOUN
ejpam-492	38	48	for	for	ADP
ejpam-492	38	49	some	some	DET
ejpam-492	38	50	prime	prime	ADJ
ejpam-492	38	51	ideal	ideal	NOUN
ejpam-492	38	52	p	p	NOUN
ejpam-492	38	53	of	of	ADP
ejpam-492	38	54	r	r	NOUN
ejpam-492	38	55	(	(	PUNCT
ejpam-492	38	56	see	see	VERB
ejpam-492	38	57	[	[	X
ejpam-492	38	58	11	11	NUM
ejpam-492	38	59	,	,	PUNCT
ejpam-492	38	60	corollary	corollary	NOUN
ejpam-492	38	61	2.11	2.11	NUM
ejpam-492	38	62	]	]	PUNCT
ejpam-492	38	63	)	)	PUNCT
ejpam-492	38	64	.	.	PUNCT
ejpam-492	39	1	also	also	ADV
ejpam-492	39	2	if	if	SCONJ
ejpam-492	39	3	m	m	NOUN
ejpam-492	39	4	is	be	AUX
ejpam-492	39	5	a	a	DET
ejpam-492	39	6	faithful	faithful	ADJ
ejpam-492	39	7	and	and	CCONJ
ejpam-492	39	8	finitely	finitely	ADV
ejpam-492	39	9	generated	generate	VERB
ejpam-492	39	10	multiplication	multiplication	NOUN
ejpam-492	39	11	r	r	NOUN
ejpam-492	39	12	-	-	PUNCT
ejpam-492	39	13	module	module	NOUN
ejpam-492	39	14	and	and	CCONJ
ejpam-492	39	15	p	p	NOUN
ejpam-492	39	16	is	be	AUX
ejpam-492	39	17	a	a	DET
ejpam-492	39	18	prime	prime	ADJ
ejpam-492	39	19	ideal	ideal	NOUN
ejpam-492	39	20	of	of	ADP
ejpam-492	39	21	r	r	NOUN
ejpam-492	39	22	,	,	PUNCT
ejpam-492	39	23	then	then	ADV
ejpam-492	39	24	by	by	ADP
ejpam-492	39	25	[	[	PUNCT
ejpam-492	39	26	11	11	NUM
ejpam-492	39	27	,	,	PUNCT
ejpam-492	39	28	theorem	theorem	VERB
ejpam-492	39	29	3.l	3.l	NUM
ejpam-492	39	30	]	]	PUNCT
ejpam-492	39	31	,	,	PUNCT
ejpam-492	39	32	pm	pm	NOUN
ejpam-492	39	33	is	be	AUX
ejpam-492	39	34	a	a	DET
ejpam-492	39	35	proper	proper	ADJ
ejpam-492	39	36	prime	prime	ADJ
ejpam-492	39	37	submodule	submodule	NOUN
ejpam-492	39	38	of	of	ADP
ejpam-492	39	39	m	m	PROPN
ejpam-492	39	40	.	.	PUNCT
ejpam-492	40	1	further	further	ADJ
ejpam-492	40	2	pm	pm	NOUN
ejpam-492	40	3	is	be	AUX
ejpam-492	40	4	minimal	minimal	ADJ
ejpam-492	40	5	over	over	ADP
ejpam-492	40	6	a	a	DET
ejpam-492	40	7	submodule	submodule	NOUN
ejpam-492	40	8	n	n	PROPN
ejpam-492	40	9	of	of	ADP
ejpam-492	40	10	m	m	PROPN
ejpam-492	40	11	if	if	SCONJ
ejpam-492	41	1	and	and	CCONJ
ejpam-492	41	2	only	only	ADV
ejpam-492	41	3	if	if	SCONJ
ejpam-492	41	4	p	p	NOUN
ejpam-492	41	5	is	be	AUX
ejpam-492	41	6	minimal	minimal	ADJ
ejpam-492	41	7	over	over	ADP
ejpam-492	41	8	the	the	DET
ejpam-492	41	9	ideal	ideal	NOUN
ejpam-492	41	10	(	(	PUNCT
ejpam-492	41	11	n	n	NUM
ejpam-492	41	12	:	:	PUNCT
ejpam-492	41	13	m	m	X
ejpam-492	41	14	)	)	PUNCT
ejpam-492	41	15	of	of	ADP
ejpam-492	41	16	r.	r.	PROPN
ejpam-492	41	17	by	by	ADP
ejpam-492	41	18	a	a	DET
ejpam-492	41	19	multiplicative	multiplicative	ADJ
ejpam-492	41	20	lattice	lattice	NOUN
ejpam-492	41	21	we	we	PRON
ejpam-492	41	22	mean	mean	VERB
ejpam-492	41	23	a	a	DET
ejpam-492	41	24	complete	complete	ADJ
ejpam-492	41	25	lattice	lattice	NOUN
ejpam-492	41	26	l	l	NOUN
ejpam-492	41	27	on	on	ADP
ejpam-492	41	28	which	which	PRON
ejpam-492	41	29	there	there	PRON
ejpam-492	41	30	is	be	VERB
ejpam-492	41	31	defined	define	VERB
ejpam-492	41	32	a	a	DET
ejpam-492	41	33	commutative	commutative	ADJ
ejpam-492	41	34	,	,	PUNCT
ejpam-492	41	35	associative	associative	ADJ
ejpam-492	41	36	multiplication	multiplication	NOUN
ejpam-492	41	37	which	which	PRON
ejpam-492	41	38	distributes	distribute	VERB
ejpam-492	41	39	over	over	ADP
ejpam-492	41	40	arbitrary	arbitrary	ADJ
ejpam-492	41	41	joins	join	NOUN
ejpam-492	41	42	(	(	PUNCT
ejpam-492	41	43	i.e.	i.e.	X
ejpam-492	41	44	,	,	PUNCT
ejpam-492	41	45	a(∨b	a(∨b	VERB
ejpam-492	41	46	α	α	NOUN
ejpam-492	41	47	)	)	PUNCT
ejpam-492	42	1	=	=	PUNCT
ejpam-492	42	2	∨	∨	NUM
ejpam-492	42	3	α	α	PROPN
ejpam-492	42	4	ab	ab	PROPN
ejpam-492	42	5	α	α	PROPN
ejpam-492	42	6	)	)	PUNCT
ejpam-492	43	1	and	and	CCONJ
ejpam-492	43	2	has	have	VERB
ejpam-492	43	3	compact	compact	ADJ
ejpam-492	43	4	greatest	great	ADJ
ejpam-492	43	5	element	element	NOUN
ejpam-492	43	6	1	1	NUM
ejpam-492	43	7	as	as	ADP
ejpam-492	43	8	a	a	DET
ejpam-492	43	9	multiplicative	multiplicative	ADJ
ejpam-492	43	10	identity	identity	NOUN
ejpam-492	43	11	[	[	X
ejpam-492	43	12	1	1	NUM
ejpam-492	43	13	]	]	PUNCT
ejpam-492	43	14	.	.	PUNCT
ejpam-492	44	1	an	an	DET
ejpam-492	44	2	element	element	NOUN
ejpam-492	44	3	e	e	NOUN
ejpam-492	44	4	of	of	ADP
ejpam-492	44	5	a	a	DET
ejpam-492	44	6	multiplicative	multiplicative	ADJ
ejpam-492	44	7	lattice	lattice	NOUN
ejpam-492	44	8	l	l	NOUN
ejpam-492	44	9	is	be	AUX
ejpam-492	44	10	said	say	VERB
ejpam-492	44	11	to	to	PART
ejpam-492	44	12	be	be	AUX
ejpam-492	44	13	principal	principal	ADJ
ejpam-492	44	14	if	if	SCONJ
ejpam-492	44	15	it	it	PRON
ejpam-492	44	16	satisfies	satisfy	VERB
ejpam-492	44	17	the	the	DET
ejpam-492	44	18	dual	dual	ADJ
ejpam-492	44	19	identities	identity	NOUN
ejpam-492	44	20	(	(	PUNCT
ejpam-492	44	21	i	i	NOUN
ejpam-492	44	22	)	)	PUNCT
ejpam-492	44	23	a	a	DET
ejpam-492	44	24	∧	∧	NOUN
ejpam-492	44	25	be	be	AUX
ejpam-492	44	26	=	=	PUNCT
ejpam-492	44	27	(	(	PUNCT
ejpam-492	44	28	(	(	PUNCT
ejpam-492	44	29	a	a	DET
ejpam-492	44	30	:	:	PUNCT
ejpam-492	44	31	e	e	X
ejpam-492	44	32	)	)	PUNCT
ejpam-492	44	33	∧	∧	PROPN
ejpam-492	44	34	b)e	b)e	ADJ
ejpam-492	44	35	and	and	CCONJ
ejpam-492	44	36	(	(	PUNCT
ejpam-492	44	37	ii	ii	NOUN
ejpam-492	44	38	)	)	PUNCT
ejpam-492	44	39	(	(	PUNCT
ejpam-492	44	40	a	a	DET
ejpam-492	44	41	∨	∨	NOUN
ejpam-492	44	42	be	be	NOUN
ejpam-492	44	43	)	)	PUNCT
ejpam-492	44	44	:	:	PUNCT
ejpam-492	45	1	e	e	X
ejpam-492	45	2	=	=	SYM
ejpam-492	45	3	(	(	PUNCT
ejpam-492	45	4	a	a	DET
ejpam-492	45	5	:	:	PUNCT
ejpam-492	45	6	e	e	X
ejpam-492	45	7	)	)	PUNCT
ejpam-492	45	8	∨	∨	PROPN
ejpam-492	45	9	b.	b.	PROPN
ejpam-492	45	10	a	a	DET
ejpam-492	45	11	principally	principally	ADV
ejpam-492	45	12	generated	generate	VERB
ejpam-492	45	13	,	,	PUNCT
ejpam-492	45	14	compactly	compactly	ADV
ejpam-492	45	15	generated	generate	VERB
ejpam-492	45	16	modular	modular	ADJ
ejpam-492	45	17	multiplicative	multiplicative	ADJ
ejpam-492	45	18	lattice	lattice	NOUN
ejpam-492	45	19	is	be	AUX
ejpam-492	45	20	called	call	VERB
ejpam-492	45	21	an	an	DET
ejpam-492	45	22	r	r	NOUN
ejpam-492	45	23	-	-	PUNCT
ejpam-492	45	24	lattice	lattice	NOUN
ejpam-492	45	25	.	.	PUNCT
ejpam-492	46	1	an	an	DET
ejpam-492	46	2	r	r	NOUN
ejpam-492	46	3	-	-	PUNCT
ejpam-492	46	4	lattice	lattice	NOUN
ejpam-492	46	5	l	l	NOUN
ejpam-492	46	6	is	be	AUX
ejpam-492	46	7	said	say	VERB
ejpam-492	46	8	to	to	PART
ejpam-492	46	9	be	be	AUX
ejpam-492	46	10	a	a	DET
ejpam-492	46	11	π	π	NOUN
ejpam-492	46	12	-	-	NOUN
ejpam-492	46	13	lattice	lattice	NOUN
ejpam-492	46	14	[	[	X
ejpam-492	46	15	1	1	NUM
ejpam-492	46	16	]	]	X
ejpam-492	46	17	if	if	SCONJ
ejpam-492	46	18	l	l	NOUN
ejpam-492	46	19	is	be	AUX
ejpam-492	46	20	generated	generate	VERB
ejpam-492	46	21	by	by	ADP
ejpam-492	46	22	a	a	DET
ejpam-492	46	23	set	set	NOUN
ejpam-492	46	24	s	s	NOUN
ejpam-492	46	25	of	of	ADP
ejpam-492	46	26	elements	element	NOUN
ejpam-492	46	27	(	(	PUNCT
ejpam-492	46	28	not	not	PART
ejpam-492	46	29	necessarily	necessarily	ADV
ejpam-492	46	30	principal	principal	ADJ
ejpam-492	46	31	)	)	PUNCT
ejpam-492	46	32	each	each	PRON
ejpam-492	46	33	of	of	ADP
ejpam-492	46	34	which	which	PRON
ejpam-492	46	35	is	be	AUX
ejpam-492	46	36	a	a	DET
ejpam-492	46	37	finite	finite	ADJ
ejpam-492	46	38	product	product	NOUN
ejpam-492	46	39	of	of	ADP
ejpam-492	46	40	prime	prime	ADJ
ejpam-492	46	41	elements	element	NOUN
ejpam-492	46	42	.	.	PUNCT
ejpam-492	47	1	it	it	PRON
ejpam-492	47	2	should	should	AUX
ejpam-492	47	3	be	be	AUX
ejpam-492	47	4	mentioned	mention	VERB
ejpam-492	47	5	that	that	SCONJ
ejpam-492	47	6	every	every	DET
ejpam-492	47	7	principal	principal	ADJ
ejpam-492	47	8	ideal	ideal	NOUN
ejpam-492	47	9	of	of	ADP
ejpam-492	47	10	r	r	NOUN
ejpam-492	47	11	is	be	AUX
ejpam-492	47	12	quasi	quasi	ADJ
ejpam-492	47	13	-	-	NOUN
ejpam-492	47	14	principal	principal	ADJ
ejpam-492	47	15	and	and	CCONJ
ejpam-492	47	16	hence	hence	ADV
ejpam-492	47	17	l(r	l(r	PROPN
ejpam-492	47	18	)	)	PUNCT
ejpam-492	47	19	,	,	PUNCT
ejpam-492	47	20	the	the	DET
ejpam-492	47	21	lattice	lattice	NOUN
ejpam-492	47	22	of	of	ADP
ejpam-492	47	23	all	all	DET
ejpam-492	47	24	ideals	ideal	NOUN
ejpam-492	47	25	of	of	ADP
ejpam-492	47	26	r	r	NOUN
ejpam-492	47	27	,	,	PUNCT
ejpam-492	47	28	is	be	AUX
ejpam-492	47	29	an	an	DET
ejpam-492	47	30	r	r	NOUN
ejpam-492	47	31	-	-	PUNCT
ejpam-492	47	32	lattice	lattice	NOUN
ejpam-492	47	33	.	.	PUNCT
ejpam-492	48	1	note	note	VERB
ejpam-492	48	2	that	that	SCONJ
ejpam-492	48	3	if	if	SCONJ
ejpam-492	48	4	r	r	NOUN
ejpam-492	48	5	is	be	AUX
ejpam-492	48	6	a	a	DET
ejpam-492	48	7	π	π	NOUN
ejpam-492	48	8	-	-	NOUN
ejpam-492	48	9	ring	ring	NOUN
ejpam-492	48	10	,	,	PUNCT
ejpam-492	48	11	then	then	ADV
ejpam-492	48	12	l(r	l(r	PROPN
ejpam-492	48	13	)	)	PUNCT
ejpam-492	48	14	is	be	AUX
ejpam-492	48	15	a	a	DET
ejpam-492	48	16	π	π	PROPN
ejpam-492	48	17	-	-	NOUN
ejpam-492	48	18	lattice	lattice	NOUN
ejpam-492	48	19	.	.	PUNCT
ejpam-492	49	1	for	for	ADP
ejpam-492	49	2	general	general	ADJ
ejpam-492	49	3	background	background	NOUN
ejpam-492	49	4	and	and	CCONJ
ejpam-492	49	5	terminology	terminology	NOUN
ejpam-492	49	6	,	,	PUNCT
ejpam-492	49	7	the	the	DET
ejpam-492	49	8	reader	reader	NOUN
ejpam-492	49	9	is	be	AUX
ejpam-492	49	10	referred	refer	VERB
ejpam-492	49	11	to	to	ADP
ejpam-492	49	12	[	[	X
ejpam-492	49	13	16	16	NUM
ejpam-492	49	14	]	]	PUNCT
ejpam-492	49	15	and	and	CCONJ
ejpam-492	49	16	[	[	X
ejpam-492	49	17	20	20	NUM
ejpam-492	49	18	]	]	PUNCT
ejpam-492	49	19	.	.	PUNCT
ejpam-492	50	1	c.	c.	PROPN
ejpam-492	50	2	jayaram	jayaram	PROPN
ejpam-492	50	3	/	/	SYM
ejpam-492	50	4	eur	eur	PROPN
ejpam-492	50	5	.	.	PUNCT
ejpam-492	51	1	j.	j.	PROPN
ejpam-492	51	2	pure	pure	PROPN
ejpam-492	51	3	appl	appl	PROPN
ejpam-492	51	4	.	.	PROPN
ejpam-492	51	5	math	math	PROPN
ejpam-492	51	6	,	,	PUNCT
ejpam-492	51	7	2	2	NUM
ejpam-492	51	8	(	(	PUNCT
ejpam-492	51	9	2009	2009	NUM
ejpam-492	51	10	)	)	PUNCT
ejpam-492	51	11	,	,	PUNCT
ejpam-492	51	12	(	(	PUNCT
ejpam-492	51	13	508	508	NUM
ejpam-492	51	14	-	-	NUM
ejpam-492	51	15	519	519	NUM
ejpam-492	51	16	)	)	PUNCT
ejpam-492	51	17	511	511	NUM
ejpam-492	51	18	3	3	NUM
ejpam-492	51	19	.	.	PUNCT
ejpam-492	52	1	π	π	NOUN
ejpam-492	52	2	-	-	NOUN
ejpam-492	52	3	modules	module	NOUN
ejpam-492	52	4	in	in	ADP
ejpam-492	52	5	this	this	DET
ejpam-492	52	6	section	section	NOUN
ejpam-492	52	7	,	,	PUNCT
ejpam-492	52	8	we	we	PRON
ejpam-492	52	9	characterize	characterize	VERB
ejpam-492	52	10	π	π	NOUN
ejpam-492	52	11	-	-	NOUN
ejpam-492	52	12	modules	module	NOUN
ejpam-492	52	13	.	.	PUNCT
ejpam-492	53	1	next	next	ADV
ejpam-492	53	2	we	we	PRON
ejpam-492	53	3	establish	establish	VERB
ejpam-492	53	4	several	several	ADJ
ejpam-492	53	5	equivalent	equivalent	ADJ
ejpam-492	53	6	conditions	condition	NOUN
ejpam-492	53	7	for	for	SCONJ
ejpam-492	53	8	an	an	DET
ejpam-492	53	9	almost	almost	ADV
ejpam-492	53	10	π	π	NOUN
ejpam-492	53	11	-	-	NOUN
ejpam-492	53	12	module	module	NOUN
ejpam-492	53	13	to	to	PART
ejpam-492	53	14	be	be	AUX
ejpam-492	53	15	a	a	DET
ejpam-492	53	16	π	π	NOUN
ejpam-492	53	17	-	-	NOUN
ejpam-492	53	18	module	module	NOUN
ejpam-492	53	19	.	.	PUNCT
ejpam-492	54	1	we	we	PRON
ejpam-492	54	2	shall	shall	AUX
ejpam-492	54	3	begin	begin	VERB
ejpam-492	54	4	with	with	ADP
ejpam-492	54	5	the	the	DET
ejpam-492	54	6	following	follow	VERB
ejpam-492	54	7	definition	definition	NOUN
ejpam-492	54	8	.	.	PUNCT
ejpam-492	55	1	definition	definition	NOUN
ejpam-492	55	2	1	1	NUM
ejpam-492	55	3	.	.	PUNCT
ejpam-492	56	1	an	an	DET
ejpam-492	56	2	r	r	NOUN
ejpam-492	56	3	-	-	PUNCT
ejpam-492	56	4	module	module	NOUN
ejpam-492	56	5	m	m	NOUN
ejpam-492	56	6	is	be	AUX
ejpam-492	56	7	said	say	VERB
ejpam-492	56	8	to	to	PART
ejpam-492	56	9	be	be	AUX
ejpam-492	56	10	a	a	DET
ejpam-492	56	11	π	π	NOUN
ejpam-492	56	12	-	-	NOUN
ejpam-492	56	13	module	module	NOUN
ejpam-492	56	14	if	if	SCONJ
ejpam-492	56	15	every	every	DET
ejpam-492	56	16	proper	proper	ADJ
ejpam-492	56	17	cyclic	cyclic	ADJ
ejpam-492	56	18	submodule	submodule	NOUN
ejpam-492	56	19	n	n	PROPN
ejpam-492	56	20	of	of	ADP
ejpam-492	56	21	m	m	PROPN
ejpam-492	56	22	is	be	AUX
ejpam-492	56	23	of	of	ADP
ejpam-492	56	24	the	the	DET
ejpam-492	56	25	form	form	NOUN
ejpam-492	56	26	i	i	PRON
ejpam-492	56	27	m	m	VERB
ejpam-492	56	28	,	,	PUNCT
ejpam-492	56	29	where	where	SCONJ
ejpam-492	56	30	i	i	PRON
ejpam-492	56	31	is	be	AUX
ejpam-492	56	32	a	a	DET
ejpam-492	56	33	finite	finite	ADJ
ejpam-492	56	34	product	product	NOUN
ejpam-492	56	35	of	of	ADP
ejpam-492	56	36	prime	prime	ADJ
ejpam-492	56	37	ideals	ideal	NOUN
ejpam-492	56	38	of	of	ADP
ejpam-492	56	39	r.	r.	PROPN
ejpam-492	56	40	definition	definition	NOUN
ejpam-492	56	41	2	2	NUM
ejpam-492	56	42	.	.	PUNCT
ejpam-492	56	43	an	an	DET
ejpam-492	56	44	r	r	NOUN
ejpam-492	56	45	-	-	PUNCT
ejpam-492	56	46	module	module	NOUN
ejpam-492	56	47	m	m	NOUN
ejpam-492	56	48	is	be	AUX
ejpam-492	56	49	said	say	VERB
ejpam-492	56	50	to	to	PART
ejpam-492	56	51	be	be	AUX
ejpam-492	56	52	an	an	DET
ejpam-492	56	53	almost	almost	ADV
ejpam-492	56	54	π	π	NOUN
ejpam-492	56	55	-	-	NOUN
ejpam-492	56	56	module	module	NOUN
ejpam-492	56	57	if	if	SCONJ
ejpam-492	56	58	for	for	ADP
ejpam-492	56	59	any	any	DET
ejpam-492	56	60	maximal	maximal	ADJ
ejpam-492	56	61	ideal	ideal	NOUN
ejpam-492	56	62	p	p	NOUN
ejpam-492	56	63	of	of	ADP
ejpam-492	56	64	r	r	NOUN
ejpam-492	56	65	,	,	PUNCT
ejpam-492	56	66	the	the	DET
ejpam-492	56	67	rp	rp	PROPN
ejpam-492	56	68	-module	-module	PROPN
ejpam-492	56	69	mp	mp	NOUN
ejpam-492	56	70	is	be	AUX
ejpam-492	56	71	a	a	DET
ejpam-492	56	72	π	π	NOUN
ejpam-492	56	73	-	-	NOUN
ejpam-492	56	74	module	module	NOUN
ejpam-492	56	75	.	.	PUNCT
ejpam-492	57	1	observe	observe	VERB
ejpam-492	57	2	that	that	SCONJ
ejpam-492	57	3	π	π	PROPN
ejpam-492	57	4	-	-	PUNCT
ejpam-492	57	5	rings	ring	NOUN
ejpam-492	57	6	and	and	CCONJ
ejpam-492	57	7	cyclic	cyclic	ADJ
ejpam-492	57	8	modules	module	NOUN
ejpam-492	57	9	over	over	ADP
ejpam-492	57	10	π	π	NOUN
ejpam-492	57	11	-	-	PUNCT
ejpam-492	57	12	rings	ring	NOUN
ejpam-492	57	13	are	be	AUX
ejpam-492	57	14	examples	example	NOUN
ejpam-492	57	15	of	of	ADP
ejpam-492	57	16	π	π	NOUN
ejpam-492	57	17	-	-	NOUN
ejpam-492	57	18	modules	module	NOUN
ejpam-492	57	19	.	.	PUNCT
ejpam-492	58	1	almost	almost	ADV
ejpam-492	58	2	π	π	NOUN
ejpam-492	58	3	-	-	PUNCT
ejpam-492	58	4	rings	ring	NOUN
ejpam-492	58	5	are	be	AUX
ejpam-492	58	6	almost	almost	ADV
ejpam-492	58	7	π	π	NOUN
ejpam-492	58	8	-	-	NOUN
ejpam-492	58	9	modules	module	NOUN
ejpam-492	58	10	.	.	PUNCT
ejpam-492	59	1	again	again	ADV
ejpam-492	59	2	note	note	VERB
ejpam-492	59	3	that	that	SCONJ
ejpam-492	59	4	π	π	NOUN
ejpam-492	59	5	-	-	PUNCT
ejpam-492	59	6	modules	module	NOUN
ejpam-492	59	7	are	be	AUX
ejpam-492	59	8	almost	almost	ADV
ejpam-492	59	9	πmodules	πmodule	NOUN
ejpam-492	59	10	,	,	PUNCT
ejpam-492	59	11	but	but	CCONJ
ejpam-492	59	12	the	the	DET
ejpam-492	59	13	converse	converse	NOUN
ejpam-492	59	14	need	need	AUX
ejpam-492	59	15	not	not	PART
ejpam-492	59	16	be	be	AUX
ejpam-492	59	17	true	true	ADJ
ejpam-492	59	18	.	.	PUNCT
ejpam-492	60	1	lemma	lemma	PROPN
ejpam-492	60	2	1	1	X
ejpam-492	60	3	.	.	PUNCT
ejpam-492	60	4	suppose	suppose	VERB
ejpam-492	60	5	m	m	PRON
ejpam-492	60	6	is	be	AUX
ejpam-492	60	7	a	a	DET
ejpam-492	60	8	π	π	NOUN
ejpam-492	60	9	-	-	NOUN
ejpam-492	60	10	module	module	NOUN
ejpam-492	60	11	.	.	PUNCT
ejpam-492	61	1	then	then	ADV
ejpam-492	61	2	m	m	PROPN
ejpam-492	61	3	is	be	AUX
ejpam-492	61	4	a	a	DET
ejpam-492	61	5	multiplication	multiplication	NOUN
ejpam-492	61	6	module	module	NOUN
ejpam-492	61	7	.	.	PUNCT
ejpam-492	62	1	proof	proof	NOUN
ejpam-492	62	2	.	.	PUNCT
ejpam-492	63	1	the	the	DET
ejpam-492	63	2	proof	proof	NOUN
ejpam-492	63	3	of	of	ADP
ejpam-492	63	4	the	the	DET
ejpam-492	63	5	lemma	lemma	PROPN
ejpam-492	63	6	follows	follow	VERB
ejpam-492	63	7	from	from	ADP
ejpam-492	63	8	[	[	X
ejpam-492	63	9	11	11	NUM
ejpam-492	63	10	,	,	PUNCT
ejpam-492	63	11	proposition	proposition	NOUN
ejpam-492	63	12	1.1	1.1	NUM
ejpam-492	63	13	]	]	PUNCT
ejpam-492	63	14	.	.	PUNCT
ejpam-492	64	1	lemma	lemma	PROPN
ejpam-492	64	2	2	2	X
ejpam-492	64	3	.	.	PUNCT
ejpam-492	64	4	suppose	suppose	VERB
ejpam-492	64	5	m	m	PRON
ejpam-492	64	6	is	be	AUX
ejpam-492	64	7	a	a	DET
ejpam-492	64	8	faithful	faithful	ADJ
ejpam-492	64	9	π	π	NOUN
ejpam-492	64	10	-	-	NOUN
ejpam-492	64	11	module	module	NOUN
ejpam-492	64	12	.	.	PUNCT
ejpam-492	65	1	then	then	ADV
ejpam-492	65	2	(	(	PUNCT
ejpam-492	65	3	i	i	NOUN
ejpam-492	65	4	)	)	PUNCT
ejpam-492	65	5	r	r	NOUN
ejpam-492	65	6	contains	contain	VERB
ejpam-492	65	7	only	only	ADV
ejpam-492	65	8	finitely	finitely	ADV
ejpam-492	65	9	many	many	ADJ
ejpam-492	65	10	minimal	minimal	ADJ
ejpam-492	65	11	prime	prime	ADJ
ejpam-492	65	12	ideals	ideal	NOUN
ejpam-492	65	13	of	of	ADP
ejpam-492	65	14	r.	r.	PROPN
ejpam-492	65	15	(	(	PUNCT
ejpam-492	65	16	ii	ii	PROPN
ejpam-492	65	17	)	)	PUNCT
ejpam-492	65	18	m	m	VERB
ejpam-492	65	19	contains	contain	VERB
ejpam-492	65	20	only	only	ADV
ejpam-492	65	21	finitely	finitely	ADV
ejpam-492	65	22	many	many	ADJ
ejpam-492	65	23	minimal	minimal	ADJ
ejpam-492	65	24	prime	prime	ADJ
ejpam-492	65	25	submodules	submodule	NOUN
ejpam-492	65	26	.	.	PUNCT
ejpam-492	66	1	(	(	PUNCT
ejpam-492	66	2	iii	iii	X
ejpam-492	66	3	)	)	PUNCT
ejpam-492	66	4	m	m	VERB
ejpam-492	66	5	is	be	AUX
ejpam-492	66	6	finitely	finitely	ADV
ejpam-492	66	7	generated	generate	VERB
ejpam-492	66	8	.	.	PUNCT
ejpam-492	67	1	proof	proof	NOUN
ejpam-492	67	2	.	.	PUNCT
ejpam-492	68	1	(	(	PUNCT
ejpam-492	68	2	i	i	NOUN
ejpam-492	68	3	)	)	PUNCT
ejpam-492	68	4	.	.	PUNCT
ejpam-492	69	1	as	as	SCONJ
ejpam-492	69	2	m	m	PROPN
ejpam-492	69	3	is	be	AUX
ejpam-492	69	4	a	a	DET
ejpam-492	69	5	faithful	faithful	ADJ
ejpam-492	69	6	π	π	NOUN
ejpam-492	69	7	-	-	NOUN
ejpam-492	69	8	module	module	NOUN
ejpam-492	69	9	,	,	PUNCT
ejpam-492	69	10	the	the	DET
ejpam-492	69	11	zero	zero	NUM
ejpam-492	69	12	ideal	ideal	NOUN
ejpam-492	69	13	is	be	AUX
ejpam-492	69	14	a	a	DET
ejpam-492	69	15	finite	finite	ADJ
ejpam-492	69	16	product	product	NOUN
ejpam-492	69	17	of	of	ADP
ejpam-492	69	18	prime	prime	ADJ
ejpam-492	69	19	ideals	ideal	NOUN
ejpam-492	69	20	and	and	CCONJ
ejpam-492	69	21	hence	hence	ADV
ejpam-492	69	22	r	r	NOUN
ejpam-492	69	23	contains	contain	VERB
ejpam-492	69	24	only	only	ADV
ejpam-492	69	25	finitely	finitely	ADV
ejpam-492	69	26	many	many	ADJ
ejpam-492	69	27	minimal	minimal	ADJ
ejpam-492	69	28	prime	prime	ADJ
ejpam-492	69	29	ideals	ideal	NOUN
ejpam-492	69	30	.	.	PUNCT
ejpam-492	70	1	(	(	PUNCT
ejpam-492	70	2	ii	ii	NOUN
ejpam-492	70	3	)	)	PUNCT
ejpam-492	70	4	.	.	PUNCT
ejpam-492	71	1	by	by	ADP
ejpam-492	71	2	lemma	lemma	PROPN
ejpam-492	71	3	1	1	NUM
ejpam-492	71	4	,	,	PUNCT
ejpam-492	71	5	m	m	VERB
ejpam-492	71	6	is	be	AUX
ejpam-492	71	7	a	a	DET
ejpam-492	71	8	multiplication	multiplication	NOUN
ejpam-492	71	9	module	module	NOUN
ejpam-492	71	10	.	.	PUNCT
ejpam-492	72	1	as	as	SCONJ
ejpam-492	72	2	m	m	PROPN
ejpam-492	72	3	is	be	AUX
ejpam-492	72	4	faithful	faithful	ADJ
ejpam-492	72	5	,	,	PUNCT
ejpam-492	72	6	it	it	PRON
ejpam-492	72	7	follows	follow	VERB
ejpam-492	72	8	that	that	SCONJ
ejpam-492	72	9	every	every	DET
ejpam-492	72	10	minimal	minimal	ADJ
ejpam-492	72	11	prime	prime	ADJ
ejpam-492	72	12	submodule	submodule	NOUN
ejpam-492	72	13	is	be	AUX
ejpam-492	72	14	of	of	ADP
ejpam-492	72	15	the	the	DET
ejpam-492	72	16	form	form	NOUN
ejpam-492	72	17	pm	pm	NOUN
ejpam-492	72	18	for	for	ADP
ejpam-492	72	19	some	some	DET
ejpam-492	72	20	minimal	minimal	ADJ
ejpam-492	72	21	prime	prime	ADJ
ejpam-492	72	22	ideal	ideal	NOUN
ejpam-492	72	23	p	p	PROPN
ejpam-492	72	24	of	of	ADP
ejpam-492	72	25	r.	r.	PROPN
ejpam-492	72	26	so	so	ADV
ejpam-492	72	27	by	by	ADP
ejpam-492	72	28	(	(	PUNCT
ejpam-492	72	29	i	i	NOUN
ejpam-492	72	30	)	)	PUNCT
ejpam-492	72	31	,	,	PUNCT
ejpam-492	72	32	m	m	PROPN
ejpam-492	72	33	contains	contain	VERB
ejpam-492	72	34	only	only	ADV
ejpam-492	72	35	finitely	finitely	ADV
ejpam-492	72	36	many	many	ADJ
ejpam-492	72	37	minimal	minimal	ADJ
ejpam-492	72	38	prime	prime	ADJ
ejpam-492	72	39	submodules	submodule	NOUN
ejpam-492	72	40	.	.	PUNCT
ejpam-492	73	1	(	(	PUNCT
ejpam-492	73	2	iii	iii	NOUN
ejpam-492	73	3	)	)	PUNCT
ejpam-492	73	4	.	.	PUNCT
ejpam-492	74	1	the	the	DET
ejpam-492	74	2	result	result	NOUN
ejpam-492	74	3	follows	follow	VERB
ejpam-492	74	4	from	from	ADP
ejpam-492	74	5	(	(	PUNCT
ejpam-492	74	6	ii	ii	NOUN
ejpam-492	74	7	)	)	PUNCT
ejpam-492	74	8	and	and	CCONJ
ejpam-492	74	9	[	[	X
ejpam-492	74	10	11	11	NUM
ejpam-492	74	11	,	,	PUNCT
ejpam-492	74	12	theorem	theorem	VERB
ejpam-492	74	13	3.7	3.7	NUM
ejpam-492	74	14	]	]	PUNCT
ejpam-492	74	15	.	.	PUNCT
ejpam-492	75	1	c.	c.	PROPN
ejpam-492	75	2	jayaram	jayaram	PROPN
ejpam-492	75	3	/	/	SYM
ejpam-492	75	4	eur	eur	PROPN
ejpam-492	75	5	.	.	PUNCT
ejpam-492	76	1	j.	j.	PROPN
ejpam-492	76	2	pure	pure	PROPN
ejpam-492	76	3	appl	appl	PROPN
ejpam-492	76	4	.	.	PROPN
ejpam-492	76	5	math	math	PROPN
ejpam-492	76	6	,	,	PUNCT
ejpam-492	76	7	2	2	NUM
ejpam-492	76	8	(	(	PUNCT
ejpam-492	76	9	2009	2009	NUM
ejpam-492	76	10	)	)	PUNCT
ejpam-492	76	11	,	,	PUNCT
ejpam-492	76	12	(	(	PUNCT
ejpam-492	76	13	508	508	NUM
ejpam-492	76	14	-	-	NUM
ejpam-492	76	15	519	519	NUM
ejpam-492	76	16	)	)	PUNCT
ejpam-492	76	17	512	512	NUM
ejpam-492	76	18	lemma	lemma	PROPN
ejpam-492	76	19	3	3	X
ejpam-492	76	20	.	.	PUNCT
ejpam-492	76	21	suppose	suppose	VERB
ejpam-492	76	22	m	m	PRON
ejpam-492	76	23	is	be	AUX
ejpam-492	76	24	a	a	DET
ejpam-492	76	25	faithful	faithful	ADJ
ejpam-492	76	26	π	π	NOUN
ejpam-492	76	27	-	-	NOUN
ejpam-492	76	28	module	module	NOUN
ejpam-492	76	29	.	.	PUNCT
ejpam-492	77	1	then	then	ADV
ejpam-492	77	2	every	every	DET
ejpam-492	77	3	proper	proper	ADJ
ejpam-492	77	4	cyclic	cyclic	ADJ
ejpam-492	77	5	submodule	submodule	NOUN
ejpam-492	77	6	of	of	ADP
ejpam-492	77	7	m	m	PROPN
ejpam-492	77	8	has	have	VERB
ejpam-492	77	9	only	only	ADV
ejpam-492	77	10	finitely	finitely	ADV
ejpam-492	77	11	many	many	ADJ
ejpam-492	77	12	minimal	minimal	ADJ
ejpam-492	77	13	primes	prime	NOUN
ejpam-492	77	14	.	.	PUNCT
ejpam-492	78	1	proof	proof	NOUN
ejpam-492	78	2	.	.	PUNCT
ejpam-492	79	1	let	let	VERB
ejpam-492	79	2	x	x	PUNCT
ejpam-492	79	3	∈	∈	PROPN
ejpam-492	79	4	m	m	VERB
ejpam-492	79	5	.	.	PUNCT
ejpam-492	80	1	as	as	SCONJ
ejpam-492	80	2	m	m	PROPN
ejpam-492	80	3	is	be	AUX
ejpam-492	80	4	a	a	DET
ejpam-492	80	5	π	π	NOUN
ejpam-492	80	6	-	-	NOUN
ejpam-492	80	7	module	module	NOUN
ejpam-492	80	8	,	,	PUNCT
ejpam-492	80	9	by	by	ADP
ejpam-492	80	10	definition	definition	NOUN
ejpam-492	80	11	,	,	PUNCT
ejpam-492	80	12	rx	rx	VERB
ejpam-492	80	13	=	=	PUNCT
ejpam-492	80	14	p1p2	p1p2	X
ejpam-492	80	15	·	·	PUNCT
ejpam-492	80	16	·	·	PUNCT
ejpam-492	80	17	·	·	PUNCT
ejpam-492	80	18	pnm	pnm	NOUN
ejpam-492	80	19	for	for	ADP
ejpam-492	80	20	some	some	DET
ejpam-492	80	21	prime	prime	ADJ
ejpam-492	80	22	ideals	ideal	NOUN
ejpam-492	80	23	p1	p1	NOUN
ejpam-492	80	24	,	,	PUNCT
ejpam-492	80	25	p2	p2	NOUN
ejpam-492	80	26	,	,	PUNCT
ejpam-492	80	27	·	·	PUNCT
ejpam-492	80	28	·	·	PUNCT
ejpam-492	80	29	·	·	PUNCT
ejpam-492	80	30	,	,	PUNCT
ejpam-492	80	31	pn	pn	PROPN
ejpam-492	80	32	of	of	ADP
ejpam-492	80	33	r.	r.	PROPN
ejpam-492	80	34	let	let	VERB
ejpam-492	80	35	n	n	PRON
ejpam-492	80	36	be	be	AUX
ejpam-492	80	37	a	a	DET
ejpam-492	80	38	prime	prime	ADJ
ejpam-492	80	39	submodule	submodule	NOUN
ejpam-492	80	40	minimal	minimal	ADJ
ejpam-492	80	41	over	over	ADP
ejpam-492	80	42	rx	rx	ADJ
ejpam-492	80	43	.	.	PUNCT
ejpam-492	81	1	note	note	VERB
ejpam-492	81	2	that	that	SCONJ
ejpam-492	81	3	by	by	ADP
ejpam-492	81	4	lemma	lemma	PROPN
ejpam-492	81	5	1	1	NUM
ejpam-492	81	6	and	and	CCONJ
ejpam-492	81	7	lemma	lemma	PROPN
ejpam-492	81	8	2	2	NUM
ejpam-492	81	9	,	,	PUNCT
ejpam-492	81	10	m	m	VERB
ejpam-492	81	11	is	be	AUX
ejpam-492	81	12	a	a	DET
ejpam-492	81	13	faithful	faithful	ADJ
ejpam-492	81	14	and	and	CCONJ
ejpam-492	81	15	finitely	finitely	ADV
ejpam-492	81	16	generated	generate	VERB
ejpam-492	81	17	multiplication	multiplication	NOUN
ejpam-492	81	18	r	r	NOUN
ejpam-492	81	19	-	-	NOUN
ejpam-492	81	20	module	module	NOUN
ejpam-492	81	21	.	.	PUNCT
ejpam-492	82	1	so	so	ADV
ejpam-492	82	2	n	n	NOUN
ejpam-492	82	3	=	=	PRON
ejpam-492	82	4	pm	pm	NOUN
ejpam-492	82	5	for	for	ADP
ejpam-492	82	6	some	some	DET
ejpam-492	82	7	prime	prime	ADJ
ejpam-492	82	8	ideal	ideal	NOUN
ejpam-492	82	9	p	p	PROPN
ejpam-492	82	10	of	of	ADP
ejpam-492	82	11	r.	r.	PROPN
ejpam-492	82	12	as	as	SCONJ
ejpam-492	82	13	rx	rx	VERB
ejpam-492	82	14	⊆	⊆	NUM
ejpam-492	82	15	n	n	NOUN
ejpam-492	82	16	,	,	PUNCT
ejpam-492	82	17	by	by	ADP
ejpam-492	82	18	[	[	PUNCT
ejpam-492	82	19	11	11	NUM
ejpam-492	82	20	,	,	PUNCT
ejpam-492	82	21	theorem	theorem	VERB
ejpam-492	82	22	3.1	3.1	NUM
ejpam-492	82	23	]	]	PUNCT
ejpam-492	82	24	,	,	PUNCT
ejpam-492	82	25	it	it	PRON
ejpam-492	82	26	follows	follow	VERB
ejpam-492	82	27	that	that	PRON
ejpam-492	82	28	pi	pi	NOUN
ejpam-492	82	29	⊆	⊆	NUM
ejpam-492	82	30	p	p	NOUN
ejpam-492	82	31	for	for	ADP
ejpam-492	82	32	some	some	DET
ejpam-492	82	33	i	i	PRON
ejpam-492	82	34	,	,	PUNCT
ejpam-492	82	35	so	so	ADV
ejpam-492	82	36	rx	rx	VERB
ejpam-492	82	37	⊆	⊆	NUM
ejpam-492	82	38	pi	pi	NOUN
ejpam-492	82	39	m	m	NOUN
ejpam-492	82	40	⊆	⊆	NUM
ejpam-492	82	41	pm	pm	NOUN
ejpam-492	82	42	=	=	SYM
ejpam-492	82	43	n	n	NOUN
ejpam-492	82	44	.	.	PUNCT
ejpam-492	83	1	as	as	SCONJ
ejpam-492	83	2	m	m	PROPN
ejpam-492	83	3	is	be	AUX
ejpam-492	83	4	a	a	DET
ejpam-492	83	5	faithful	faithful	ADJ
ejpam-492	83	6	and	and	CCONJ
ejpam-492	83	7	finitely	finitely	ADV
ejpam-492	83	8	generated	generate	VERB
ejpam-492	83	9	multiplication	multiplication	NOUN
ejpam-492	83	10	r	r	NOUN
ejpam-492	83	11	-	-	PUNCT
ejpam-492	83	12	module	module	NOUN
ejpam-492	83	13	,	,	PUNCT
ejpam-492	83	14	it	it	PRON
ejpam-492	83	15	follows	follow	VERB
ejpam-492	83	16	that	that	SCONJ
ejpam-492	83	17	pi	pi	PROPN
ejpam-492	83	18	m	m	VERB
ejpam-492	83	19	is	be	AUX
ejpam-492	83	20	a	a	DET
ejpam-492	83	21	prime	prime	ADJ
ejpam-492	83	22	submodule	submodule	NOUN
ejpam-492	83	23	and	and	CCONJ
ejpam-492	83	24	hence	hence	ADV
ejpam-492	83	25	pi	pi	NOUN
ejpam-492	83	26	m	m	NOUN
ejpam-492	83	27	=	=	SYM
ejpam-492	83	28	n	n	PROPN
ejpam-492	83	29	.	.	PUNCT
ejpam-492	84	1	therefore	therefore	ADV
ejpam-492	84	2	rx	rx	X
ejpam-492	84	3	has	have	VERB
ejpam-492	84	4	only	only	ADV
ejpam-492	84	5	finitely	finitely	ADV
ejpam-492	84	6	many	many	ADJ
ejpam-492	84	7	minimal	minimal	ADJ
ejpam-492	84	8	primes	prime	NOUN
ejpam-492	84	9	.	.	PUNCT
ejpam-492	85	1	this	this	PRON
ejpam-492	85	2	completes	complete	VERB
ejpam-492	85	3	the	the	DET
ejpam-492	85	4	proof	proof	NOUN
ejpam-492	85	5	of	of	ADP
ejpam-492	85	6	the	the	DET
ejpam-492	85	7	lemma	lemma	PROPN
ejpam-492	85	8	.	.	PUNCT
ejpam-492	86	1	lemma	lemma	PROPN
ejpam-492	86	2	4	4	X
ejpam-492	86	3	.	.	PUNCT
ejpam-492	86	4	suppose	suppose	VERB
ejpam-492	86	5	m	m	PRON
ejpam-492	86	6	is	be	AUX
ejpam-492	86	7	a	a	DET
ejpam-492	86	8	faithful	faithful	ADJ
ejpam-492	86	9	cyclic	cyclic	ADJ
ejpam-492	86	10	r	r	NOUN
ejpam-492	86	11	-	-	PUNCT
ejpam-492	86	12	module	module	NOUN
ejpam-492	86	13	.	.	PUNCT
ejpam-492	87	1	then	then	ADV
ejpam-492	87	2	r	r	NOUN
ejpam-492	87	3	is	be	AUX
ejpam-492	87	4	a	a	DET
ejpam-492	87	5	π	π	NOUN
ejpam-492	87	6	-	-	NOUN
ejpam-492	87	7	ring	ring	NOUN
ejpam-492	87	8	if	if	SCONJ
ejpam-492	88	1	and	and	CCONJ
ejpam-492	88	2	only	only	ADV
ejpam-492	88	3	if	if	SCONJ
ejpam-492	88	4	m	m	NOUN
ejpam-492	88	5	is	be	AUX
ejpam-492	88	6	a	a	DET
ejpam-492	88	7	π	π	NOUN
ejpam-492	88	8	-	-	NOUN
ejpam-492	88	9	module	module	NOUN
ejpam-492	88	10	.	.	PUNCT
ejpam-492	89	1	proof	proof	NOUN
ejpam-492	89	2	.	.	PUNCT
ejpam-492	90	1	the	the	DET
ejpam-492	90	2	proof	proof	NOUN
ejpam-492	90	3	of	of	ADP
ejpam-492	90	4	the	the	DET
ejpam-492	90	5	lemma	lemma	PROPN
ejpam-492	90	6	follows	follow	VERB
ejpam-492	90	7	from	from	ADP
ejpam-492	90	8	[	[	X
ejpam-492	90	9	14	14	NUM
ejpam-492	90	10	,	,	PUNCT
ejpam-492	90	11	lemma	lemma	PROPN
ejpam-492	90	12	6	6	NUM
ejpam-492	90	13	]	]	PUNCT
ejpam-492	90	14	and	and	CCONJ
ejpam-492	90	15	[	[	X
ejpam-492	90	16	11	11	NUM
ejpam-492	90	17	,	,	PUNCT
ejpam-492	90	18	theorem	theorem	VERB
ejpam-492	90	19	3.1	3.1	NUM
ejpam-492	90	20	]	]	PUNCT
ejpam-492	90	21	.	.	PUNCT
ejpam-492	91	1	lemma	lemma	PROPN
ejpam-492	91	2	5	5	X
ejpam-492	91	3	.	.	PUNCT
ejpam-492	91	4	suppose	suppose	VERB
ejpam-492	91	5	m	m	PRON
ejpam-492	91	6	is	be	AUX
ejpam-492	91	7	a	a	DET
ejpam-492	91	8	faithful	faithful	ADJ
ejpam-492	91	9	and	and	CCONJ
ejpam-492	91	10	finitely	finitely	ADV
ejpam-492	91	11	generated	generate	VERB
ejpam-492	91	12	multiplication	multiplication	NOUN
ejpam-492	91	13	r	r	NOUN
ejpam-492	91	14	-	-	PUNCT
ejpam-492	91	15	module	module	NOUN
ejpam-492	91	16	.	.	PUNCT
ejpam-492	92	1	then	then	ADV
ejpam-492	92	2	m	m	PROPN
ejpam-492	92	3	is	be	AUX
ejpam-492	92	4	an	an	DET
ejpam-492	92	5	almost	almost	ADV
ejpam-492	92	6	π	π	NOUN
ejpam-492	92	7	-	-	NOUN
ejpam-492	92	8	module	module	NOUN
ejpam-492	92	9	if	if	SCONJ
ejpam-492	92	10	and	and	CCONJ
ejpam-492	92	11	only	only	ADV
ejpam-492	92	12	if	if	SCONJ
ejpam-492	92	13	r	r	NOUN
ejpam-492	92	14	is	be	AUX
ejpam-492	92	15	an	an	DET
ejpam-492	92	16	almost	almost	ADV
ejpam-492	92	17	π	π	NOUN
ejpam-492	92	18	-	-	NOUN
ejpam-492	92	19	ring	ring	NOUN
ejpam-492	92	20	.	.	PUNCT
ejpam-492	93	1	proof	proof	NOUN
ejpam-492	93	2	.	.	PUNCT
ejpam-492	94	1	let	let	VERB
ejpam-492	94	2	p	p	PRON
ejpam-492	94	3	be	be	AUX
ejpam-492	94	4	a	a	DET
ejpam-492	94	5	maximal	maximal	ADJ
ejpam-492	94	6	ideal	ideal	NOUN
ejpam-492	94	7	of	of	ADP
ejpam-492	94	8	r.	r.	PROPN
ejpam-492	94	9	consider	consider	VERB
ejpam-492	94	10	the	the	DET
ejpam-492	94	11	rp	rp	NOUN
ejpam-492	94	12	-module	-module	PROPN
ejpam-492	94	13	mp	mp	PROPN
ejpam-492	94	14	.	.	PUNCT
ejpam-492	95	1	as	as	SCONJ
ejpam-492	95	2	m	m	PROPN
ejpam-492	95	3	is	be	AUX
ejpam-492	95	4	a	a	DET
ejpam-492	95	5	finitely	finitely	ADV
ejpam-492	95	6	generated	generate	VERB
ejpam-492	95	7	faithful	faithful	ADJ
ejpam-492	95	8	multiplication	multiplication	NOUN
ejpam-492	95	9	r	r	NOUN
ejpam-492	95	10	-	-	PUNCT
ejpam-492	95	11	module	module	NOUN
ejpam-492	95	12	,	,	PUNCT
ejpam-492	95	13	it	it	PRON
ejpam-492	95	14	follows	follow	VERB
ejpam-492	95	15	that	that	SCONJ
ejpam-492	95	16	mp	mp	PROPN
ejpam-492	95	17	is	be	AUX
ejpam-492	95	18	a	a	DET
ejpam-492	95	19	faithful	faithful	ADJ
ejpam-492	95	20	cyclic	cyclic	NOUN
ejpam-492	95	21	rp	rp	NOUN
ejpam-492	95	22	module	module	NOUN
ejpam-492	95	23	.	.	PUNCT
ejpam-492	96	1	so	so	ADV
ejpam-492	96	2	by	by	ADP
ejpam-492	96	3	lemma	lemma	PROPN
ejpam-492	96	4	4	4	NUM
ejpam-492	96	5	,	,	PUNCT
ejpam-492	96	6	mp	mp	PROPN
ejpam-492	96	7	is	be	AUX
ejpam-492	96	8	a	a	DET
ejpam-492	96	9	π	π	NOUN
ejpam-492	96	10	-	-	NOUN
ejpam-492	96	11	module	module	NOUN
ejpam-492	96	12	if	if	SCONJ
ejpam-492	96	13	and	and	CCONJ
ejpam-492	96	14	only	only	ADV
ejpam-492	96	15	if	if	SCONJ
ejpam-492	96	16	rp	rp	NOUN
ejpam-492	96	17	is	be	AUX
ejpam-492	96	18	a	a	DET
ejpam-492	96	19	π	π	NOUN
ejpam-492	96	20	-	-	NOUN
ejpam-492	96	21	ring	ring	NOUN
ejpam-492	96	22	.	.	PUNCT
ejpam-492	97	1	therefore	therefore	ADV
ejpam-492	97	2	r	r	NOUN
ejpam-492	97	3	is	be	AUX
ejpam-492	97	4	an	an	DET
ejpam-492	97	5	almost	almost	ADV
ejpam-492	97	6	π	π	NOUN
ejpam-492	97	7	-	-	NOUN
ejpam-492	97	8	ring	ring	NOUN
ejpam-492	97	9	if	if	SCONJ
ejpam-492	97	10	and	and	CCONJ
ejpam-492	97	11	only	only	ADV
ejpam-492	97	12	if	if	SCONJ
ejpam-492	97	13	m	m	NOUN
ejpam-492	97	14	is	be	AUX
ejpam-492	97	15	an	an	DET
ejpam-492	97	16	almost	almost	ADV
ejpam-492	97	17	π	π	NOUN
ejpam-492	97	18	-	-	NOUN
ejpam-492	97	19	module	module	NOUN
ejpam-492	97	20	.	.	PUNCT
ejpam-492	98	1	this	this	PRON
ejpam-492	98	2	completes	complete	VERB
ejpam-492	98	3	the	the	DET
ejpam-492	98	4	proof	proof	NOUN
ejpam-492	98	5	of	of	ADP
ejpam-492	98	6	the	the	DET
ejpam-492	98	7	theorem	theorem	PROPN
ejpam-492	98	8	.	.	PUNCT
ejpam-492	99	1	lemma	lemma	PROPN
ejpam-492	99	2	6	6	NUM
ejpam-492	99	3	.	.	PUNCT
ejpam-492	100	1	let	let	VERB
ejpam-492	100	2	m	m	PRON
ejpam-492	100	3	be	be	AUX
ejpam-492	100	4	a	a	DET
ejpam-492	100	5	faithful	faithful	ADJ
ejpam-492	100	6	π	π	NOUN
ejpam-492	100	7	-	-	NOUN
ejpam-492	100	8	module	module	NOUN
ejpam-492	100	9	.	.	PUNCT
ejpam-492	101	1	if	if	SCONJ
ejpam-492	101	2	n	n	PRON
ejpam-492	101	3	is	be	AUX
ejpam-492	101	4	a	a	DET
ejpam-492	101	5	minimal	minimal	ADJ
ejpam-492	101	6	prime	prime	ADJ
ejpam-492	101	7	submodule	submodule	NOUN
ejpam-492	101	8	,	,	PUNCT
ejpam-492	101	9	then	then	ADV
ejpam-492	101	10	n	n	PRON
ejpam-492	101	11	is	be	AUX
ejpam-492	101	12	a	a	DET
ejpam-492	101	13	multiplication	multiplication	NOUN
ejpam-492	101	14	submodule	submodule	NOUN
ejpam-492	101	15	.	.	PUNCT
ejpam-492	102	1	c.	c.	PROPN
ejpam-492	102	2	jayaram	jayaram	PROPN
ejpam-492	102	3	/	/	SYM
ejpam-492	102	4	eur	eur	PROPN
ejpam-492	102	5	.	.	PUNCT
ejpam-492	103	1	j.	j.	PROPN
ejpam-492	103	2	pure	pure	PROPN
ejpam-492	103	3	appl	appl	PROPN
ejpam-492	103	4	.	.	PROPN
ejpam-492	103	5	math	math	PROPN
ejpam-492	103	6	,	,	PUNCT
ejpam-492	103	7	2	2	NUM
ejpam-492	103	8	(	(	PUNCT
ejpam-492	103	9	2009	2009	NUM
ejpam-492	103	10	)	)	PUNCT
ejpam-492	103	11	,	,	PUNCT
ejpam-492	103	12	(	(	PUNCT
ejpam-492	103	13	508	508	NUM
ejpam-492	103	14	-	-	NUM
ejpam-492	103	15	519	519	NUM
ejpam-492	103	16	)	)	PUNCT
ejpam-492	103	17	513	513	NUM
ejpam-492	103	18	proof	proof	NOUN
ejpam-492	103	19	.	.	PUNCT
ejpam-492	104	1	note	note	VERB
ejpam-492	104	2	that	that	SCONJ
ejpam-492	104	3	by	by	ADP
ejpam-492	104	4	lemma	lemma	PROPN
ejpam-492	104	5	1	1	NUM
ejpam-492	104	6	and	and	CCONJ
ejpam-492	104	7	lemma	lemma	PROPN
ejpam-492	104	8	2	2	NUM
ejpam-492	104	9	,	,	PUNCT
ejpam-492	104	10	m	m	VERB
ejpam-492	104	11	is	be	AUX
ejpam-492	104	12	a	a	DET
ejpam-492	104	13	faithful	faithful	ADJ
ejpam-492	104	14	and	and	CCONJ
ejpam-492	104	15	finitely	finitely	ADV
ejpam-492	104	16	generated	generate	VERB
ejpam-492	104	17	multiplication	multiplication	NOUN
ejpam-492	104	18	module	module	NOUN
ejpam-492	104	19	.	.	PUNCT
ejpam-492	105	1	suppose	suppose	VERB
ejpam-492	105	2	n	n	PRON
ejpam-492	105	3	is	be	AUX
ejpam-492	105	4	a	a	DET
ejpam-492	105	5	minimal	minimal	ADJ
ejpam-492	105	6	prime	prime	ADJ
ejpam-492	105	7	submodule	submodule	NOUN
ejpam-492	105	8	.	.	PUNCT
ejpam-492	106	1	then	then	ADV
ejpam-492	106	2	n	n	NOUN
ejpam-492	106	3	=	=	PUNCT
ejpam-492	106	4	pm	pm	NOUN
ejpam-492	106	5	for	for	ADP
ejpam-492	106	6	some	some	DET
ejpam-492	106	7	minimal	minimal	ADJ
ejpam-492	106	8	prime	prime	ADJ
ejpam-492	106	9	ideal	ideal	NOUN
ejpam-492	106	10	p	p	PROPN
ejpam-492	106	11	of	of	ADP
ejpam-492	106	12	r.	r.	PROPN
ejpam-492	106	13	suppose	suppose	VERB
ejpam-492	106	14	rx	rx	VERB
ejpam-492	106	15	⊆	⊆	NUM
ejpam-492	106	16	pm	pm	NOUN
ejpam-492	106	17	for	for	ADP
ejpam-492	106	18	some	some	DET
ejpam-492	106	19	x	x	SYM
ejpam-492	106	20	∈	∈	PROPN
ejpam-492	106	21	m	m	NOUN
ejpam-492	106	22	.	.	PUNCT
ejpam-492	107	1	as	as	SCONJ
ejpam-492	107	2	m	m	PROPN
ejpam-492	107	3	is	be	AUX
ejpam-492	107	4	a	a	DET
ejpam-492	107	5	π	π	NOUN
ejpam-492	107	6	-	-	NOUN
ejpam-492	107	7	module	module	NOUN
ejpam-492	107	8	,	,	PUNCT
ejpam-492	107	9	it	it	PRON
ejpam-492	107	10	follows	follow	VERB
ejpam-492	107	11	that	that	SCONJ
ejpam-492	107	12	rx	rx	NOUN
ejpam-492	107	13	=	=	NOUN
ejpam-492	107	14	i	i	PRON
ejpam-492	107	15	m	m	VERB
ejpam-492	107	16	,	,	PUNCT
ejpam-492	107	17	where	where	SCONJ
ejpam-492	107	18	i	i	PRON
ejpam-492	107	19	=	=	SYM
ejpam-492	107	20	p1p2	p1p2	NOUN
ejpam-492	107	21	...	...	PUNCT
ejpam-492	107	22	pn	pn	NOUN
ejpam-492	107	23	and	and	CCONJ
ejpam-492	107	24	pi	pi	PROPN
ejpam-492	107	25	′s	′s	PROPN
ejpam-492	107	26	are	be	AUX
ejpam-492	107	27	prime	prime	ADJ
ejpam-492	107	28	ideals	ideal	NOUN
ejpam-492	107	29	of	of	ADP
ejpam-492	107	30	r.	r.	PROPN
ejpam-492	107	31	since	since	SCONJ
ejpam-492	107	32	rx	rx	VERB
ejpam-492	107	33	⊆	⊆	NUM
ejpam-492	107	34	pm	pm	NOUN
ejpam-492	107	35	,	,	PUNCT
ejpam-492	107	36	by	by	ADP
ejpam-492	107	37	[	[	PUNCT
ejpam-492	107	38	11	11	NUM
ejpam-492	107	39	,	,	PUNCT
ejpam-492	107	40	theorem	theorem	VERB
ejpam-492	107	41	3.1	3.1	NUM
ejpam-492	107	42	]	]	PUNCT
ejpam-492	107	43	,	,	PUNCT
ejpam-492	107	44	it	it	PRON
ejpam-492	107	45	follows	follow	VERB
ejpam-492	107	46	that	that	PRON
ejpam-492	107	47	pi	pi	NOUN
ejpam-492	107	48	⊆	⊆	NUM
ejpam-492	107	49	p	p	NOUN
ejpam-492	107	50	for	for	ADP
ejpam-492	107	51	some	some	DET
ejpam-492	107	52	i.	i.	NOUN
ejpam-492	107	53	as	as	SCONJ
ejpam-492	107	54	p	p	PROPN
ejpam-492	107	55	is	be	AUX
ejpam-492	107	56	a	a	DET
ejpam-492	107	57	minimal	minimal	ADJ
ejpam-492	107	58	prime	prime	ADJ
ejpam-492	107	59	ideal	ideal	NOUN
ejpam-492	107	60	,	,	PUNCT
ejpam-492	107	61	it	it	PRON
ejpam-492	107	62	follows	follow	VERB
ejpam-492	107	63	that	that	SCONJ
ejpam-492	107	64	p	p	PROPN
ejpam-492	107	65	=	=	ADJ
ejpam-492	107	66	pi	pi	NOUN
ejpam-492	107	67	.	.	PUNCT
ejpam-492	108	1	therefore	therefore	ADV
ejpam-492	108	2	rx	rx	VERB
ejpam-492	108	3	=	=	SYM
ejpam-492	108	4	j(pm	j(pm	PROPN
ejpam-492	108	5	)	)	PUNCT
ejpam-492	108	6	=	=	SYM
ejpam-492	108	7	jn	jn	PROPN
ejpam-492	108	8	for	for	ADP
ejpam-492	108	9	some	some	DET
ejpam-492	108	10	j	j	PROPN
ejpam-492	108	11	∈	∈	PROPN
ejpam-492	108	12	l(r	l(r	PROPN
ejpam-492	108	13	)	)	PUNCT
ejpam-492	108	14	.	.	PUNCT
ejpam-492	109	1	consequently	consequently	ADV
ejpam-492	109	2	,	,	PUNCT
ejpam-492	109	3	n	n	PRON
ejpam-492	109	4	is	be	AUX
ejpam-492	109	5	a	a	DET
ejpam-492	109	6	multiplication	multiplication	NOUN
ejpam-492	109	7	submodule	submodule	NOUN
ejpam-492	109	8	.	.	PUNCT
ejpam-492	110	1	lemma	lemma	PROPN
ejpam-492	110	2	7	7	X
ejpam-492	110	3	.	.	PUNCT
ejpam-492	111	1	let	let	VERB
ejpam-492	111	2	m	m	PRON
ejpam-492	111	3	be	be	AUX
ejpam-492	111	4	a	a	DET
ejpam-492	111	5	faithful	faithful	ADJ
ejpam-492	111	6	π	π	NOUN
ejpam-492	111	7	-	-	NOUN
ejpam-492	111	8	module	module	NOUN
ejpam-492	111	9	.	.	PUNCT
ejpam-492	112	1	if	if	SCONJ
ejpam-492	112	2	n	n	PRON
ejpam-492	112	3	is	be	AUX
ejpam-492	112	4	a	a	DET
ejpam-492	112	5	prime	prime	ADJ
ejpam-492	112	6	submodule	submodule	NOUN
ejpam-492	112	7	minimal	minimal	ADJ
ejpam-492	112	8	over	over	ADP
ejpam-492	112	9	a	a	DET
ejpam-492	112	10	cyclic	cyclic	ADJ
ejpam-492	112	11	submodule	submodule	NOUN
ejpam-492	112	12	of	of	ADP
ejpam-492	112	13	m	m	PROPN
ejpam-492	112	14	,	,	PUNCT
ejpam-492	112	15	then	then	ADV
ejpam-492	112	16	n	n	PRON
ejpam-492	112	17	is	be	AUX
ejpam-492	112	18	either	either	CCONJ
ejpam-492	112	19	minimal	minimal	ADJ
ejpam-492	112	20	or	or	CCONJ
ejpam-492	112	21	a	a	DET
ejpam-492	112	22	multiplication	multiplication	NOUN
ejpam-492	112	23	submodule	submodule	NOUN
ejpam-492	112	24	with	with	ADP
ejpam-492	112	25	rankn	rankn	NOUN
ejpam-492	112	26	=	=	SYM
ejpam-492	112	27	1	1	X
ejpam-492	112	28	.	.	PUNCT
ejpam-492	113	1	proof	proof	NOUN
ejpam-492	113	2	.	.	PUNCT
ejpam-492	114	1	observe	observe	VERB
ejpam-492	114	2	that	that	SCONJ
ejpam-492	114	3	m	m	PROPN
ejpam-492	114	4	is	be	AUX
ejpam-492	114	5	a	a	DET
ejpam-492	114	6	faithful	faithful	ADJ
ejpam-492	114	7	and	and	CCONJ
ejpam-492	114	8	finitely	finitely	ADV
ejpam-492	114	9	generated	generate	VERB
ejpam-492	114	10	multiplication	multiplication	NOUN
ejpam-492	114	11	module	module	NOUN
ejpam-492	114	12	.	.	PUNCT
ejpam-492	115	1	suppose	suppose	VERB
ejpam-492	115	2	n	n	PRON
ejpam-492	115	3	is	be	AUX
ejpam-492	115	4	a	a	DET
ejpam-492	115	5	prime	prime	ADJ
ejpam-492	115	6	submodule	submodule	NOUN
ejpam-492	115	7	minimal	minimal	ADJ
ejpam-492	115	8	over	over	ADP
ejpam-492	115	9	a	a	DET
ejpam-492	115	10	cyclic	cyclic	ADJ
ejpam-492	115	11	submodule	submodule	NOUN
ejpam-492	115	12	of	of	ADP
ejpam-492	115	13	m	m	PROPN
ejpam-492	115	14	.	.	PUNCT
ejpam-492	116	1	then	then	ADV
ejpam-492	116	2	n	n	NOUN
ejpam-492	116	3	=	=	PUNCT
ejpam-492	116	4	pm	pm	NOUN
ejpam-492	116	5	for	for	ADP
ejpam-492	116	6	some	some	DET
ejpam-492	116	7	prime	prime	ADJ
ejpam-492	116	8	ideal	ideal	NOUN
ejpam-492	116	9	p	p	PROPN
ejpam-492	116	10	of	of	ADP
ejpam-492	116	11	r.	r.	PROPN
ejpam-492	116	12	suppose	suppose	VERB
ejpam-492	116	13	pm	pm	NOUN
ejpam-492	116	14	is	be	AUX
ejpam-492	116	15	non	non	ADJ
ejpam-492	116	16	-	-	ADJ
ejpam-492	116	17	minimal	minimal	ADJ
ejpam-492	116	18	.	.	PUNCT
ejpam-492	117	1	then	then	ADV
ejpam-492	117	2	p	p	PROPN
ejpam-492	117	3	is	be	AUX
ejpam-492	117	4	non	non	ADJ
ejpam-492	117	5	-	-	ADJ
ejpam-492	117	6	minimal	minimal	ADJ
ejpam-492	117	7	.	.	PUNCT
ejpam-492	118	1	let	let	VERB
ejpam-492	118	2	p0	p0	PROPN
ejpam-492	118	3	⊇	⊇	PROPN
ejpam-492	118	4	p	p	PROPN
ejpam-492	118	5	be	be	AUX
ejpam-492	118	6	a	a	DET
ejpam-492	118	7	maximal	maximal	ADJ
ejpam-492	118	8	ideal	ideal	NOUN
ejpam-492	118	9	of	of	ADP
ejpam-492	118	10	r.	r.	PROPN
ejpam-492	118	11	suppose	suppose	VERB
ejpam-492	118	12	pm	pm	NOUN
ejpam-492	118	13	is	be	AUX
ejpam-492	118	14	minimal	minimal	ADJ
ejpam-492	118	15	over	over	ADP
ejpam-492	118	16	a	a	DET
ejpam-492	118	17	cyclic	cyclic	ADJ
ejpam-492	118	18	submodule	submodule	NOUN
ejpam-492	118	19	ry	ry	PROPN
ejpam-492	118	20	of	of	ADP
ejpam-492	118	21	m	m	PROPN
ejpam-492	118	22	.	.	PUNCT
ejpam-492	119	1	then	then	ADV
ejpam-492	119	2	by	by	ADP
ejpam-492	119	3	[	[	X
ejpam-492	119	4	18	18	NUM
ejpam-492	119	5	,	,	PUNCT
ejpam-492	119	6	lemma	lemma	PROPN
ejpam-492	119	7	1.4	1.4	NUM
ejpam-492	119	8	]	]	PUNCT
ejpam-492	119	9	,	,	PUNCT
ejpam-492	119	10	p	p	NOUN
ejpam-492	119	11	is	be	AUX
ejpam-492	119	12	minimal	minimal	ADJ
ejpam-492	119	13	over	over	ADP
ejpam-492	119	14	a	a	DET
ejpam-492	119	15	quasi	quasi	ADJ
ejpam-492	119	16	-	-	ADJ
ejpam-492	119	17	principal	principal	ADJ
ejpam-492	119	18	ideal	ideal	NOUN
ejpam-492	119	19	(	(	PUNCT
ejpam-492	119	20	ry	ry	NOUN
ejpam-492	119	21	:	:	PUNCT
ejpam-492	119	22	m	m	X
ejpam-492	119	23	)	)	PUNCT
ejpam-492	119	24	of	of	ADP
ejpam-492	119	25	r.	r.	PROPN
ejpam-492	119	26	note	note	PROPN
ejpam-492	119	27	that	that	SCONJ
ejpam-492	119	28	by	by	ADP
ejpam-492	119	29	lemma	lemma	PROPN
ejpam-492	119	30	5	5	NUM
ejpam-492	119	31	,	,	PUNCT
ejpam-492	119	32	r	r	NOUN
ejpam-492	119	33	is	be	AUX
ejpam-492	119	34	an	an	DET
ejpam-492	119	35	almost	almost	ADV
ejpam-492	119	36	π	π	NOUN
ejpam-492	119	37	-	-	NOUN
ejpam-492	119	38	ring	ring	NOUN
ejpam-492	119	39	.	.	PUNCT
ejpam-492	120	1	therefore	therefore	ADV
ejpam-492	120	2	by	by	ADP
ejpam-492	120	3	[	[	X
ejpam-492	120	4	12	12	NUM
ejpam-492	120	5	,	,	PUNCT
ejpam-492	120	6	theorem	theorem	VERB
ejpam-492	120	7	46.8	46.8	NUM
ejpam-492	120	8	and	and	CCONJ
ejpam-492	120	9	corollary	corollary	ADJ
ejpam-492	120	10	46.10	46.10	NUM
ejpam-492	120	11	,	,	PUNCT
ejpam-492	120	12	page	page	NOUN
ejpam-492	120	13	576	576	NUM
ejpam-492	120	14	-	-	SYM
ejpam-492	120	15	577	577	NUM
ejpam-492	120	16	]	]	PUNCT
ejpam-492	120	17	,	,	PUNCT
ejpam-492	120	18	for	for	ADP
ejpam-492	120	19	every	every	DET
ejpam-492	120	20	maximal	maximal	ADJ
ejpam-492	120	21	ideal	ideal	NOUN
ejpam-492	120	22	q	q	NOUN
ejpam-492	120	23	of	of	ADP
ejpam-492	120	24	r	r	NOUN
ejpam-492	120	25	,	,	PUNCT
ejpam-492	120	26	rq	rq	X
ejpam-492	120	27	is	be	AUX
ejpam-492	120	28	either	either	CCONJ
ejpam-492	120	29	a	a	DET
ejpam-492	120	30	π	π	NOUN
ejpam-492	120	31	-	-	NOUN
ejpam-492	120	32	domain	domain	NOUN
ejpam-492	120	33	or	or	CCONJ
ejpam-492	120	34	a	a	DET
ejpam-492	120	35	special	special	ADJ
ejpam-492	120	36	principal	principal	ADJ
ejpam-492	120	37	ideal	ideal	ADJ
ejpam-492	120	38	ring	ring	NOUN
ejpam-492	120	39	.	.	PUNCT
ejpam-492	121	1	as	as	SCONJ
ejpam-492	121	2	rp0	rp0	NOUN
ejpam-492	121	3	is	be	AUX
ejpam-492	121	4	a	a	DET
ejpam-492	121	5	π	π	NOUN
ejpam-492	121	6	-	-	NOUN
ejpam-492	121	7	domain	domain	NOUN
ejpam-492	121	8	and	and	CCONJ
ejpam-492	121	9	pp0	pp0	NOUN
ejpam-492	121	10	is	be	AUX
ejpam-492	121	11	a	a	DET
ejpam-492	121	12	prime	prime	NOUN
ejpam-492	121	13	minimal	minimal	ADJ
ejpam-492	121	14	over	over	ADP
ejpam-492	121	15	a	a	DET
ejpam-492	121	16	non	non	ADJ
ejpam-492	121	17	-	-	ADJ
ejpam-492	121	18	zero	zero	ADJ
ejpam-492	121	19	principal	principal	ADJ
ejpam-492	121	20	element	element	NOUN
ejpam-492	121	21	of	of	ADP
ejpam-492	121	22	rp0	rp0	NOUN
ejpam-492	121	23	,	,	PUNCT
ejpam-492	121	24	by	by	ADP
ejpam-492	121	25	[	[	X
ejpam-492	121	26	15	15	NUM
ejpam-492	121	27	,	,	PUNCT
ejpam-492	121	28	theorem	theorem	VERB
ejpam-492	121	29	4.2	4.2	NUM
ejpam-492	121	30	and	and	CCONJ
ejpam-492	121	31	corollary	corollary	ADJ
ejpam-492	121	32	4.3	4.3	NUM
ejpam-492	121	33	]	]	PUNCT
ejpam-492	121	34	,	,	PUNCT
ejpam-492	121	35	pp0	pp0	NOUN
ejpam-492	121	36	is	be	AUX
ejpam-492	121	37	principal	principal	ADJ
ejpam-492	121	38	and	and	CCONJ
ejpam-492	121	39	rank	rank	NOUN
ejpam-492	121	40	p	p	NOUN
ejpam-492	121	41	=	=	NOUN
ejpam-492	122	1	1	1	X
ejpam-492	122	2	.	.	PUNCT
ejpam-492	122	3	again	again	ADV
ejpam-492	122	4	note	note	VERB
ejpam-492	122	5	that	that	SCONJ
ejpam-492	122	6	p	p	NOUN
ejpam-492	122	7	is	be	AUX
ejpam-492	122	8	locally	locally	ADV
ejpam-492	122	9	principal	principal	ADJ
ejpam-492	122	10	.	.	PUNCT
ejpam-492	123	1	observe	observe	VERB
ejpam-492	123	2	that	that	SCONJ
ejpam-492	123	3	by	by	ADP
ejpam-492	123	4	[	[	X
ejpam-492	123	5	12	12	NUM
ejpam-492	123	6	,	,	PUNCT
ejpam-492	123	7	corollary	corollary	NOUN
ejpam-492	123	8	46.9	46.9	NUM
ejpam-492	123	9	,	,	PUNCT
ejpam-492	123	10	page	page	NOUN
ejpam-492	123	11	577	577	NUM
ejpam-492	123	12	]	]	PUNCT
ejpam-492	123	13	,	,	PUNCT
ejpam-492	123	14	every	every	DET
ejpam-492	123	15	prime	prime	ADJ
ejpam-492	123	16	ideal	ideal	NOUN
ejpam-492	123	17	contains	contain	VERB
ejpam-492	123	18	a	a	DET
ejpam-492	123	19	unique	unique	ADJ
ejpam-492	123	20	minimal	minimal	ADJ
ejpam-492	123	21	prime	prime	ADJ
ejpam-492	123	22	ideal	ideal	NOUN
ejpam-492	123	23	.	.	PUNCT
ejpam-492	124	1	let	let	VERB
ejpam-492	124	2	p	p	PRON
ejpam-492	124	3	′	′	NOUN
ejpam-492	124	4	be	be	AUX
ejpam-492	124	5	a	a	DET
ejpam-492	124	6	minimal	minimal	ADJ
ejpam-492	124	7	prime	prime	ADJ
ejpam-492	124	8	ideal	ideal	NOUN
ejpam-492	124	9	contained	contain	VERB
ejpam-492	124	10	in	in	ADP
ejpam-492	124	11	p.	p.	NOUN
ejpam-492	124	12	then	then	ADV
ejpam-492	125	1	p	p	X
ejpam-492	125	2	′p	′p	ADJ
ejpam-492	125	3	=	=	PUNCT
ejpam-492	126	1	p	p	NOUN
ejpam-492	126	2	′	′	INTJ
ejpam-492	127	1	locally	locally	ADV
ejpam-492	127	2	and	and	CCONJ
ejpam-492	127	3	hence	hence	ADV
ejpam-492	127	4	globally	globally	ADV
ejpam-492	127	5	.	.	PUNCT
ejpam-492	128	1	now	now	ADV
ejpam-492	128	2	we	we	PRON
ejpam-492	128	3	show	show	VERB
ejpam-492	128	4	that	that	SCONJ
ejpam-492	128	5	pm	pm	NOUN
ejpam-492	128	6	is	be	AUX
ejpam-492	128	7	a	a	DET
ejpam-492	128	8	multiplication	multiplication	NOUN
ejpam-492	128	9	submodule	submodule	NOUN
ejpam-492	128	10	.	.	PUNCT
ejpam-492	129	1	suppose	suppose	VERB
ejpam-492	129	2	rx	rx	VERB
ejpam-492	129	3	⊆	⊆	NUM
ejpam-492	129	4	pm	pm	NOUN
ejpam-492	129	5	for	for	ADP
ejpam-492	129	6	some	some	DET
ejpam-492	129	7	x	x	SYM
ejpam-492	129	8	∈	∈	PROPN
ejpam-492	129	9	m	m	NOUN
ejpam-492	129	10	.	.	PUNCT
ejpam-492	130	1	as	as	SCONJ
ejpam-492	130	2	m	m	PROPN
ejpam-492	130	3	is	be	AUX
ejpam-492	130	4	a	a	DET
ejpam-492	130	5	π	π	NOUN
ejpam-492	130	6	-	-	NOUN
ejpam-492	130	7	module	module	NOUN
ejpam-492	130	8	,	,	PUNCT
ejpam-492	130	9	it	it	PRON
ejpam-492	130	10	follows	follow	VERB
ejpam-492	130	11	that	that	SCONJ
ejpam-492	130	12	rx	rx	NOUN
ejpam-492	130	13	=	=	NOUN
ejpam-492	130	14	i	i	PRON
ejpam-492	130	15	m	m	VERB
ejpam-492	130	16	,	,	PUNCT
ejpam-492	130	17	where	where	SCONJ
ejpam-492	130	18	i	i	PRON
ejpam-492	130	19	=	=	SYM
ejpam-492	130	20	p1p2	p1p2	NOUN
ejpam-492	130	21	...	...	PUNCT
ejpam-492	130	22	pn	pn	NOUN
ejpam-492	130	23	and	and	CCONJ
ejpam-492	130	24	pi	pi	PROPN
ejpam-492	130	25	′s	′s	PROPN
ejpam-492	130	26	are	be	AUX
ejpam-492	130	27	prime	prime	ADJ
ejpam-492	130	28	ideals	ideal	NOUN
ejpam-492	130	29	of	of	ADP
ejpam-492	130	30	r.	r.	PROPN
ejpam-492	130	31	since	since	SCONJ
ejpam-492	130	32	rx	rx	VERB
ejpam-492	130	33	⊆	⊆	NUM
ejpam-492	130	34	pm	pm	NOUN
ejpam-492	130	35	,	,	PUNCT
ejpam-492	130	36	it	it	PRON
ejpam-492	130	37	follows	follow	VERB
ejpam-492	130	38	that	that	PRON
ejpam-492	130	39	pi	pi	NOUN
ejpam-492	130	40	⊆	⊆	NUM
ejpam-492	130	41	p	p	NOUN
ejpam-492	130	42	,	,	PUNCT
ejpam-492	130	43	for	for	ADP
ejpam-492	130	44	some	some	DET
ejpam-492	130	45	i.	i.	NOUN
ejpam-492	131	1	so	so	ADV
ejpam-492	131	2	either	either	CCONJ
ejpam-492	131	3	p	p	X
ejpam-492	131	4	′	′	NOUN
ejpam-492	132	1	=	=	PUNCT
ejpam-492	132	2	pi	pi	NOUN
ejpam-492	132	3	or	or	CCONJ
ejpam-492	132	4	p	p	NOUN
ejpam-492	132	5	=	=	ADJ
ejpam-492	132	6	pi	pi	NOUN
ejpam-492	132	7	.	.	PUNCT
ejpam-492	133	1	therefore	therefore	ADV
ejpam-492	133	2	rx	rx	VERB
ejpam-492	133	3	=	=	SYM
ejpam-492	133	4	j(pm	j(pm	PROPN
ejpam-492	133	5	)	)	PUNCT
ejpam-492	133	6	=	=	SYM
ejpam-492	133	7	jn	jn	PROPN
ejpam-492	133	8	for	for	ADP
ejpam-492	133	9	some	some	DET
ejpam-492	133	10	j	j	PROPN
ejpam-492	133	11	∈	∈	PROPN
ejpam-492	133	12	l(r	l(r	PROPN
ejpam-492	133	13	)	)	PUNCT
ejpam-492	133	14	.	.	PUNCT
ejpam-492	134	1	c.	c.	PROPN
ejpam-492	134	2	jayaram	jayaram	PROPN
ejpam-492	134	3	/	/	SYM
ejpam-492	134	4	eur	eur	PROPN
ejpam-492	134	5	.	.	PUNCT
ejpam-492	135	1	j.	j.	PROPN
ejpam-492	135	2	pure	pure	PROPN
ejpam-492	135	3	appl	appl	PROPN
ejpam-492	135	4	.	.	PROPN
ejpam-492	135	5	math	math	PROPN
ejpam-492	135	6	,	,	PUNCT
ejpam-492	135	7	2	2	NUM
ejpam-492	135	8	(	(	PUNCT
ejpam-492	135	9	2009	2009	NUM
ejpam-492	135	10	)	)	PUNCT
ejpam-492	135	11	,	,	PUNCT
ejpam-492	135	12	(	(	PUNCT
ejpam-492	135	13	508	508	NUM
ejpam-492	135	14	-	-	NUM
ejpam-492	135	15	519	519	NUM
ejpam-492	135	16	)	)	PUNCT
ejpam-492	135	17	514	514	NUM
ejpam-492	135	18	as	as	SCONJ
ejpam-492	135	19	m	m	PROPN
ejpam-492	135	20	is	be	AUX
ejpam-492	135	21	a	a	DET
ejpam-492	135	22	π	π	NOUN
ejpam-492	135	23	-	-	NOUN
ejpam-492	135	24	module	module	NOUN
ejpam-492	136	1	,	,	PUNCT
ejpam-492	136	2	it	it	PRON
ejpam-492	136	3	follows	follow	VERB
ejpam-492	136	4	that	that	SCONJ
ejpam-492	136	5	n	n	NOUN
ejpam-492	136	6	=	=	PRON
ejpam-492	136	7	pm	pm	NOUN
ejpam-492	136	8	is	be	AUX
ejpam-492	136	9	a	a	DET
ejpam-492	136	10	multiplication	multiplication	NOUN
ejpam-492	136	11	submodule	submodule	NOUN
ejpam-492	136	12	.	.	PUNCT
ejpam-492	137	1	since	since	SCONJ
ejpam-492	137	2	rank	rank	NOUN
ejpam-492	137	3	p	p	NOUN
ejpam-492	137	4	=	=	NOUN
ejpam-492	137	5	1	1	NUM
ejpam-492	137	6	,	,	PUNCT
ejpam-492	137	7	by	by	ADP
ejpam-492	137	8	[	[	PUNCT
ejpam-492	137	9	11	11	NUM
ejpam-492	137	10	,	,	PUNCT
ejpam-492	137	11	theorem	theorem	VERB
ejpam-492	137	12	3.1	3.1	NUM
ejpam-492	137	13	]	]	PUNCT
ejpam-492	137	14	,	,	PUNCT
ejpam-492	137	15	rank	rank	NOUN
ejpam-492	137	16	n	n	NOUN
ejpam-492	137	17	=	=	SYM
ejpam-492	137	18	1	1	X
ejpam-492	137	19	.	.	PUNCT
ejpam-492	138	1	this	this	PRON
ejpam-492	138	2	completes	complete	VERB
ejpam-492	138	3	the	the	DET
ejpam-492	138	4	proof	proof	NOUN
ejpam-492	138	5	of	of	ADP
ejpam-492	138	6	the	the	DET
ejpam-492	138	7	lemma	lemma	PROPN
ejpam-492	138	8	.	.	PUNCT
ejpam-492	139	1	lemma	lemma	PROPN
ejpam-492	139	2	8	8	NUM
ejpam-492	139	3	.	.	PUNCT
ejpam-492	140	1	let	let	VERB
ejpam-492	140	2	m	m	PRON
ejpam-492	140	3	be	be	AUX
ejpam-492	140	4	a	a	DET
ejpam-492	140	5	faithful	faithful	ADJ
ejpam-492	140	6	π	π	NOUN
ejpam-492	140	7	-	-	NOUN
ejpam-492	140	8	module	module	NOUN
ejpam-492	140	9	.	.	PUNCT
ejpam-492	141	1	then	then	ADV
ejpam-492	141	2	every	every	DET
ejpam-492	141	3	cyclic	cyclic	ADJ
ejpam-492	141	4	submodule	submodule	NOUN
ejpam-492	141	5	is	be	AUX
ejpam-492	141	6	a	a	DET
ejpam-492	141	7	finite	finite	ADJ
ejpam-492	141	8	intersection	intersection	NOUN
ejpam-492	141	9	of	of	ADP
ejpam-492	141	10	primary	primary	ADJ
ejpam-492	141	11	submodules	submodule	NOUN
ejpam-492	141	12	.	.	PUNCT
ejpam-492	142	1	proof	proof	NOUN
ejpam-492	142	2	.	.	PUNCT
ejpam-492	143	1	let	let	VERB
ejpam-492	143	2	rx	rx	AUX
ejpam-492	143	3	be	be	AUX
ejpam-492	143	4	a	a	DET
ejpam-492	143	5	cyclic	cyclic	ADJ
ejpam-492	143	6	submodule	submodule	NOUN
ejpam-492	143	7	of	of	ADP
ejpam-492	143	8	m	m	PRON
ejpam-492	143	9	and	and	CCONJ
ejpam-492	143	10	let	let	VERB
ejpam-492	143	11	i	i	PRON
ejpam-492	143	12	=	=	PUNCT
ejpam-492	143	13	(	(	PUNCT
ejpam-492	143	14	rx	rx	VERB
ejpam-492	143	15	:	:	PUNCT
ejpam-492	143	16	m	m	NOUN
ejpam-492	143	17	)	)	PUNCT
ejpam-492	143	18	.	.	PUNCT
ejpam-492	144	1	as	as	SCONJ
ejpam-492	144	2	m	m	PROPN
ejpam-492	144	3	is	be	AUX
ejpam-492	144	4	a	a	DET
ejpam-492	144	5	faithful	faithful	ADJ
ejpam-492	144	6	π	π	NOUN
ejpam-492	144	7	-	-	NOUN
ejpam-492	144	8	module	module	NOUN
ejpam-492	144	9	,	,	PUNCT
ejpam-492	144	10	by	by	ADP
ejpam-492	144	11	lemma	lemma	PROPN
ejpam-492	144	12	3	3	NUM
ejpam-492	144	13	,	,	PUNCT
ejpam-492	144	14	it	it	PRON
ejpam-492	144	15	follows	follow	VERB
ejpam-492	144	16	that	that	SCONJ
ejpam-492	144	17	rx	rx	NOUN
ejpam-492	144	18	has	have	VERB
ejpam-492	144	19	only	only	ADV
ejpam-492	144	20	finitely	finitely	ADV
ejpam-492	144	21	many	many	ADJ
ejpam-492	144	22	minimal	minimal	ADJ
ejpam-492	144	23	primes	prime	NOUN
ejpam-492	144	24	.	.	PUNCT
ejpam-492	145	1	as	as	SCONJ
ejpam-492	145	2	m	m	PROPN
ejpam-492	145	3	is	be	AUX
ejpam-492	145	4	a	a	DET
ejpam-492	145	5	faithful	faithful	ADJ
ejpam-492	145	6	and	and	CCONJ
ejpam-492	145	7	finitely	finitely	ADV
ejpam-492	145	8	generated	generate	VERB
ejpam-492	145	9	multiplication	multiplication	NOUN
ejpam-492	145	10	module	module	NOUN
ejpam-492	145	11	,	,	PUNCT
ejpam-492	145	12	it	it	PRON
ejpam-492	145	13	follows	follow	VERB
ejpam-492	145	14	that	that	SCONJ
ejpam-492	145	15	every	every	DET
ejpam-492	145	16	prime	prime	ADJ
ejpam-492	145	17	submodule	submodule	NOUN
ejpam-492	145	18	is	be	AUX
ejpam-492	145	19	of	of	ADP
ejpam-492	145	20	the	the	DET
ejpam-492	145	21	form	form	NOUN
ejpam-492	145	22	pm	pm	NOUN
ejpam-492	145	23	for	for	ADP
ejpam-492	145	24	some	some	DET
ejpam-492	145	25	prime	prime	ADJ
ejpam-492	145	26	ideal	ideal	NOUN
ejpam-492	145	27	p	p	PROPN
ejpam-492	145	28	of	of	ADP
ejpam-492	145	29	r.	r.	PROPN
ejpam-492	145	30	let	let	VERB
ejpam-492	145	31	pi	pi	NOUN
ejpam-492	145	32	′s	′s	PROPN
ejpam-492	145	33	for	for	ADP
ejpam-492	145	34	i	i	PROPN
ejpam-492	145	35	=	=	NOUN
ejpam-492	145	36	1	1	NUM
ejpam-492	145	37	,	,	PUNCT
ejpam-492	145	38	2	2	NUM
ejpam-492	145	39	,	,	PUNCT
ejpam-492	145	40	...	...	PUNCT
ejpam-492	145	41	,	,	PUNCT
ejpam-492	145	42	m	m	VERB
ejpam-492	145	43	be	be	VERB
ejpam-492	145	44	the	the	DET
ejpam-492	145	45	distinct	distinct	ADJ
ejpam-492	145	46	prime	prime	ADJ
ejpam-492	145	47	ideals	ideal	NOUN
ejpam-492	145	48	of	of	ADP
ejpam-492	145	49	r	r	NOUN
ejpam-492	145	50	such	such	ADJ
ejpam-492	145	51	that	that	DET
ejpam-492	145	52	p1	p1	PROPN
ejpam-492	145	53	m	m	PROPN
ejpam-492	145	54	,	,	PUNCT
ejpam-492	145	55	p2	p2	PROPN
ejpam-492	145	56	m	m	PROPN
ejpam-492	145	57	,	,	PUNCT
ejpam-492	145	58	...	...	PUNCT
ejpam-492	145	59	,	,	PUNCT
ejpam-492	145	60	pmm	pmm	PROPN
ejpam-492	145	61	are	be	AUX
ejpam-492	145	62	the	the	DET
ejpam-492	145	63	distinct	distinct	ADJ
ejpam-492	145	64	prime	prime	ADJ
ejpam-492	145	65	submodules	submodule	NOUN
ejpam-492	145	66	which	which	PRON
ejpam-492	145	67	are	be	AUX
ejpam-492	145	68	minimal	minimal	ADJ
ejpam-492	145	69	over	over	ADP
ejpam-492	145	70	rx	rx	ADJ
ejpam-492	145	71	.	.	PUNCT
ejpam-492	146	1	it	it	PRON
ejpam-492	146	2	can	can	AUX
ejpam-492	146	3	be	be	AUX
ejpam-492	146	4	easily	easily	ADV
ejpam-492	146	5	seen	see	VERB
ejpam-492	146	6	that	that	SCONJ
ejpam-492	146	7	a	a	DET
ejpam-492	146	8	prime	prime	ADJ
ejpam-492	146	9	ideal	ideal	NOUN
ejpam-492	146	10	p	p	NOUN
ejpam-492	146	11	of	of	ADP
ejpam-492	146	12	r	r	NOUN
ejpam-492	146	13	is	be	AUX
ejpam-492	146	14	minimal	minimal	ADJ
ejpam-492	146	15	over	over	ADP
ejpam-492	146	16	i	i	PRON
ejpam-492	146	17	if	if	SCONJ
ejpam-492	147	1	and	and	CCONJ
ejpam-492	147	2	only	only	ADV
ejpam-492	147	3	if	if	SCONJ
ejpam-492	147	4	p	p	NOUN
ejpam-492	147	5	=	=	NOUN
ejpam-492	147	6	pi	pi	NOUN
ejpam-492	147	7	for	for	ADP
ejpam-492	147	8	some	some	DET
ejpam-492	147	9	i.	i.	NOUN
ejpam-492	147	10	as	as	SCONJ
ejpam-492	147	11	r	r	NOUN
ejpam-492	147	12	is	be	AUX
ejpam-492	147	13	an	an	DET
ejpam-492	147	14	almost	almost	ADV
ejpam-492	147	15	π	π	NOUN
ejpam-492	147	16	-	-	NOUN
ejpam-492	147	17	ring	ring	NOUN
ejpam-492	147	18	,	,	PUNCT
ejpam-492	147	19	by	by	ADP
ejpam-492	147	20	[	[	X
ejpam-492	147	21	12	12	NUM
ejpam-492	147	22	,	,	PUNCT
ejpam-492	147	23	corollary	corollary	ADJ
ejpam-492	147	24	46.10	46.10	NUM
ejpam-492	147	25	,	,	PUNCT
ejpam-492	147	26	page	page	NOUN
ejpam-492	147	27	577	577	NUM
ejpam-492	147	28	]	]	PUNCT
ejpam-492	147	29	,	,	PUNCT
ejpam-492	147	30	it	it	PRON
ejpam-492	147	31	follows	follow	VERB
ejpam-492	147	32	that	that	SCONJ
ejpam-492	147	33	the	the	DET
ejpam-492	147	34	non	non	ADJ
ejpam-492	147	35	-	-	ADJ
ejpam-492	147	36	maximal	maximal	ADJ
ejpam-492	147	37	minimal	minimal	ADJ
ejpam-492	147	38	primes	prime	NOUN
ejpam-492	147	39	are	be	AUX
ejpam-492	147	40	unbranched	unbranche	VERB
ejpam-492	147	41	and	and	CCONJ
ejpam-492	147	42	idempotent	idempotent	NOUN
ejpam-492	147	43	.	.	PUNCT
ejpam-492	148	1	by	by	ADP
ejpam-492	148	2	lemma	lemma	PROPN
ejpam-492	148	3	6	6	NUM
ejpam-492	148	4	and	and	CCONJ
ejpam-492	148	5	lemma	lemma	PROPN
ejpam-492	148	6	7	7	NUM
ejpam-492	148	7	,	,	PUNCT
ejpam-492	148	8	each	each	DET
ejpam-492	148	9	pi	pi	NOUN
ejpam-492	148	10	m	m	VERB
ejpam-492	148	11	is	be	AUX
ejpam-492	148	12	a	a	DET
ejpam-492	148	13	multiplication	multiplication	NOUN
ejpam-492	148	14	submodule	submodule	NOUN
ejpam-492	148	15	and	and	CCONJ
ejpam-492	148	16	rank	rank	NOUN
ejpam-492	148	17	pi	pi	NOUN
ejpam-492	148	18	≤	≤	NUM
ejpam-492	148	19	1	1	NUM
ejpam-492	148	20	for	for	ADP
ejpam-492	148	21	i	i	PRON
ejpam-492	148	22	=	=	NOUN
ejpam-492	148	23	1	1	NUM
ejpam-492	148	24	,	,	PUNCT
ejpam-492	148	25	2	2	NUM
ejpam-492	148	26	,	,	PUNCT
ejpam-492	148	27	...	...	PUNCT
ejpam-492	148	28	,	,	PUNCT
ejpam-492	148	29	m.	m.	NOUN
ejpam-492	148	30	again	again	ADV
ejpam-492	148	31	by	by	ADP
ejpam-492	148	32	[	[	PUNCT
ejpam-492	148	33	11	11	NUM
ejpam-492	148	34	,	,	PUNCT
ejpam-492	148	35	theorem	theorem	VERB
ejpam-492	148	36	3.1	3.1	NUM
ejpam-492	148	37	]	]	PUNCT
ejpam-492	148	38	and	and	CCONJ
ejpam-492	148	39	[	[	X
ejpam-492	148	40	18	18	NUM
ejpam-492	148	41	,	,	PUNCT
ejpam-492	148	42	lemma	lemma	PROPN
ejpam-492	148	43	1.4	1.4	NUM
ejpam-492	148	44	]	]	PUNCT
ejpam-492	148	45	,	,	PUNCT
ejpam-492	148	46	each	each	DET
ejpam-492	148	47	pi	pi	NOUN
ejpam-492	148	48	is	be	AUX
ejpam-492	148	49	a	a	DET
ejpam-492	148	50	multiplication	multiplication	NOUN
ejpam-492	148	51	ideal	ideal	ADJ
ejpam-492	148	52	.	.	PUNCT
ejpam-492	149	1	without	without	ADP
ejpam-492	149	2	loss	loss	NOUN
ejpam-492	149	3	of	of	ADP
ejpam-492	149	4	generality	generality	NOUN
ejpam-492	149	5	,	,	PUNCT
ejpam-492	149	6	assume	assume	VERB
ejpam-492	149	7	that	that	SCONJ
ejpam-492	149	8	p1	p1	NOUN
ejpam-492	149	9	,	,	PUNCT
ejpam-492	149	10	p2	p2	NOUN
ejpam-492	149	11	,	,	PUNCT
ejpam-492	149	12	...	...	PUNCT
ejpam-492	149	13	,	,	PUNCT
ejpam-492	149	14	ps	ps	PROPN
ejpam-492	149	15	are	be	AUX
ejpam-492	149	16	the	the	DET
ejpam-492	149	17	rank	rank	NOUN
ejpam-492	149	18	one	one	NUM
ejpam-492	149	19	multiplication	multiplication	NOUN
ejpam-492	149	20	prime	prime	ADJ
ejpam-492	149	21	ideals	ideal	NOUN
ejpam-492	149	22	,	,	PUNCT
ejpam-492	149	23	ps+1	ps+1	NOUN
ejpam-492	149	24	,	,	PUNCT
ejpam-492	149	25	ps+2	ps+2	NOUN
ejpam-492	149	26	,	,	PUNCT
ejpam-492	149	27	...	...	PUNCT
ejpam-492	149	28	,	,	PUNCT
ejpam-492	149	29	ps+t	ps+t	PROPN
ejpam-492	149	30	are	be	AUX
ejpam-492	149	31	the	the	DET
ejpam-492	149	32	non	non	ADJ
ejpam-492	149	33	maximal	maximal	ADJ
ejpam-492	149	34	minimal	minimal	ADJ
ejpam-492	149	35	primes	prime	NOUN
ejpam-492	149	36	and	and	CCONJ
ejpam-492	149	37	ps+t+1	ps+t+1	NOUN
ejpam-492	149	38	,	,	PUNCT
ejpam-492	149	39	ps+t+2	ps+t+2	NOUN
ejpam-492	149	40	,	,	PUNCT
ejpam-492	149	41	...	...	PUNCT
ejpam-492	149	42	,	,	PUNCT
ejpam-492	149	43	pm	pm	NOUN
ejpam-492	149	44	are	be	AUX
ejpam-492	149	45	the	the	DET
ejpam-492	149	46	minimal	minimal	ADJ
ejpam-492	149	47	primes	prime	NOUN
ejpam-492	149	48	which	which	PRON
ejpam-492	149	49	are	be	AUX
ejpam-492	149	50	also	also	ADV
ejpam-492	149	51	maximal	maximal	ADJ
ejpam-492	149	52	.	.	PUNCT
ejpam-492	150	1	since	since	SCONJ
ejpam-492	150	2	p1	p1	NOUN
ejpam-492	150	3	,	,	PUNCT
ejpam-492	150	4	p2	p2	NOUN
ejpam-492	150	5	,	,	PUNCT
ejpam-492	150	6	...	...	PUNCT
ejpam-492	150	7	,	,	PUNCT
ejpam-492	150	8	ps	ps	PROPN
ejpam-492	150	9	are	be	AUX
ejpam-492	150	10	the	the	DET
ejpam-492	150	11	rank	rank	NOUN
ejpam-492	150	12	one	one	NUM
ejpam-492	150	13	multiplication	multiplication	NOUN
ejpam-492	150	14	prime	prime	ADJ
ejpam-492	150	15	ideals	ideal	NOUN
ejpam-492	150	16	,	,	PUNCT
ejpam-492	150	17	by	by	ADP
ejpam-492	150	18	[	[	X
ejpam-492	150	19	5	5	NUM
ejpam-492	150	20	,	,	PUNCT
ejpam-492	150	21	theorem	theorem	VERB
ejpam-492	150	22	3	3	NUM
ejpam-492	150	23	]	]	PUNCT
ejpam-492	150	24	,	,	PUNCT
ejpam-492	150	25	these	these	PRON
ejpam-492	150	26	are	be	AUX
ejpam-492	150	27	quasi	quasi	ADJ
ejpam-492	150	28	-	-	ADJ
ejpam-492	150	29	principal	principal	ADJ
ejpam-492	150	30	ideals	ideal	NOUN
ejpam-492	150	31	.	.	PUNCT
ejpam-492	151	1	therefore	therefore	ADV
ejpam-492	151	2	by	by	ADP
ejpam-492	151	3	[	[	X
ejpam-492	151	4	3	3	NUM
ejpam-492	151	5	,	,	PUNCT
ejpam-492	151	6	theorem	theorem	VERB
ejpam-492	151	7	2.2	2.2	NUM
ejpam-492	151	8	]	]	PUNCT
ejpam-492	151	9	,	,	PUNCT
ejpam-492	151	10	there	there	PRON
ejpam-492	151	11	exist	exist	VERB
ejpam-492	151	12	positive	positive	ADJ
ejpam-492	151	13	integers	integer	NOUN
ejpam-492	151	14	ni	ni	PROPN
ejpam-492	151	15	′s	′s	PROPN
ejpam-492	151	16	for	for	ADP
ejpam-492	151	17	i	i	PROPN
ejpam-492	151	18	=	=	NOUN
ejpam-492	151	19	1	1	NUM
ejpam-492	151	20	,	,	PUNCT
ejpam-492	151	21	2	2	NUM
ejpam-492	151	22	,	,	PUNCT
ejpam-492	151	23	...	...	PUNCT
ejpam-492	151	24	,	,	PUNCT
ejpam-492	151	25	s	s	AUX
ejpam-492	151	26	,	,	PUNCT
ejpam-492	151	27	such	such	ADJ
ejpam-492	151	28	that	that	SCONJ
ejpam-492	151	29	i	i	PRON
ejpam-492	151	30	⊆	⊆	NUM
ejpam-492	151	31	p	p	X
ejpam-492	151	32	ni	ni	PROPN
ejpam-492	152	1	i	i	PROPN
ejpam-492	153	1	and	and	CCONJ
ejpam-492	153	2	i	i	PRON
ejpam-492	153	3	6⊆	6⊆	VERB
ejpam-492	154	1	p	p	NOUN
ejpam-492	155	1	ni+1	ni+1	PRON
ejpam-492	156	1	i	i	PRON
ejpam-492	156	2	.	.	PUNCT
ejpam-492	157	1	since	since	SCONJ
ejpam-492	157	2	each	each	DET
ejpam-492	157	3	rpi	rpi	NOUN
ejpam-492	157	4	(	(	PUNCT
ejpam-492	157	5	s+	s+	NUM
ejpam-492	157	6	t	t	X
ejpam-492	157	7	+	+	CCONJ
ejpam-492	157	8	1	1	NUM
ejpam-492	157	9	≤	≤	NUM
ejpam-492	157	10	i	i	PRON
ejpam-492	157	11	≤	≤	NUM
ejpam-492	157	12	m	m	VERB
ejpam-492	157	13	)	)	PUNCT
ejpam-492	157	14	is	be	AUX
ejpam-492	157	15	a	a	DET
ejpam-492	157	16	special	special	ADJ
ejpam-492	157	17	principal	principal	ADJ
ejpam-492	157	18	ideal	ideal	NOUN
ejpam-492	157	19	ring	ring	NOUN
ejpam-492	157	20	,	,	PUNCT
ejpam-492	157	21	there	there	PRON
ejpam-492	157	22	exist	exist	VERB
ejpam-492	157	23	positive	positive	ADJ
ejpam-492	157	24	integers	integer	NOUN
ejpam-492	157	25	n	n	ADP
ejpam-492	157	26	j	j	PROPN
ejpam-492	157	27	′s	′s	PROPN
ejpam-492	157	28	for	for	ADP
ejpam-492	157	29	(	(	PUNCT
ejpam-492	157	30	s	s	PART
ejpam-492	157	31	+	+	X
ejpam-492	157	32	t	t	NOUN
ejpam-492	158	1	+	+	CCONJ
ejpam-492	158	2	1	1	NUM
ejpam-492	158	3	≤	≤	NUM
ejpam-492	158	4	j	j	PROPN
ejpam-492	158	5	≤	≤	PROPN
ejpam-492	158	6	m	m	PROPN
ejpam-492	158	7	)	)	PUNCT
ejpam-492	158	8	such	such	ADJ
ejpam-492	158	9	that	that	DET
ejpam-492	158	10	ipj	ipj	NOUN
ejpam-492	158	11	=	=	SYM
ejpam-492	158	12	(	(	PUNCT
ejpam-492	158	13	p	p	NOUN
ejpam-492	158	14	n	n	PROPN
ejpam-492	158	15	j	j	PROPN
ejpam-492	158	16	j	j	PROPN
ejpam-492	158	17	)	)	PUNCT
ejpam-492	158	18	pj	pj	PROPN
ejpam-492	158	19	.	.	PUNCT
ejpam-492	159	1	observe	observe	VERB
ejpam-492	159	2	that	that	SCONJ
ejpam-492	159	3	by	by	ADP
ejpam-492	159	4	[	[	X
ejpam-492	159	5	5	5	NUM
ejpam-492	159	6	,	,	PUNCT
ejpam-492	159	7	corollary	corollary	NOUN
ejpam-492	159	8	]	]	PUNCT
ejpam-492	159	9	and	and	CCONJ
ejpam-492	159	10	[	[	X
ejpam-492	159	11	6	6	NUM
ejpam-492	159	12	,	,	PUNCT
ejpam-492	159	13	lemma	lemma	PROPN
ejpam-492	159	14	1	1	NUM
ejpam-492	159	15	]	]	PUNCT
ejpam-492	159	16	,	,	PUNCT
ejpam-492	159	17	the	the	DET
ejpam-492	159	18	powers	power	NOUN
ejpam-492	159	19	of	of	ADP
ejpam-492	159	20	pi	pi	NOUN
ejpam-492	159	21	(	(	PUNCT
ejpam-492	159	22	1	1	NUM
ejpam-492	159	23	≤	≤	NUM
ejpam-492	159	24	i	i	PRON
ejpam-492	159	25	≤	≤	PROPN
ejpam-492	159	26	m	m	VERB
ejpam-492	159	27	)	)	PUNCT
ejpam-492	159	28	are	be	AUX
ejpam-492	159	29	multiplication	multiplication	NOUN
ejpam-492	159	30	piprimary	piprimary	ADJ
ejpam-492	159	31	ideals	ideal	NOUN
ejpam-492	159	32	.	.	PUNCT
ejpam-492	160	1	let	let	VERB
ejpam-492	160	2	j	j	NOUN
ejpam-492	160	3	=	=	PUNCT
ejpam-492	160	4	p	p	PROPN
ejpam-492	160	5	n1	n1	PROPN
ejpam-492	160	6	1	1	NUM
ejpam-492	160	7	∩	∩	NOUN
ejpam-492	160	8	p	p	NOUN
ejpam-492	160	9	n2	n2	ADJ
ejpam-492	160	10	2	2	NUM
ejpam-492	160	11	∩	∩	NOUN
ejpam-492	160	12	...	...	PUNCT
ejpam-492	160	13	∩	∩	PROPN
ejpam-492	160	14	pns	pns	PROPN
ejpam-492	160	15	s	s	PART
ejpam-492	160	16	∩	∩	PROPN
ejpam-492	160	17	ps+1	ps+1	NOUN
ejpam-492	160	18	∩	∩	NOUN
ejpam-492	160	19	...	...	PUNCT
ejpam-492	160	20	∩	∩	ADJ
ejpam-492	160	21	ps+t	ps+t	PROPN
ejpam-492	160	22	∩	∩	NOUN
ejpam-492	160	23	p	p	PROPN
ejpam-492	160	24	ns+t+1	ns+t+1	PROPN
ejpam-492	160	25	s+t+1	s+t+1	PROPN
ejpam-492	160	26	∩	∩	NOUN
ejpam-492	160	27	...	...	PUNCT
ejpam-492	160	28	∩	∩	PROPN
ejpam-492	160	29	pnm	pnm	PROPN
ejpam-492	160	30	m	m	PROPN
ejpam-492	160	31	.	.	PUNCT
ejpam-492	161	1	now	now	ADV
ejpam-492	161	2	we	we	PRON
ejpam-492	161	3	claim	claim	VERB
ejpam-492	161	4	that	that	SCONJ
ejpam-492	161	5	i	i	PRON
ejpam-492	162	1	=	=	SYM
ejpam-492	162	2	j	j	PROPN
ejpam-492	162	3	.	.	PUNCT
ejpam-492	163	1	let	let	VERB
ejpam-492	163	2	q	q	PRON
ejpam-492	163	3	be	be	AUX
ejpam-492	163	4	a	a	DET
ejpam-492	163	5	maximal	maximal	ADJ
ejpam-492	163	6	prime	prime	ADJ
ejpam-492	163	7	ideal	ideal	NOUN
ejpam-492	163	8	of	of	ADP
ejpam-492	163	9	r.	r.	PROPN
ejpam-492	163	10	if	if	SCONJ
ejpam-492	163	11	pj	pj	PROPN
ejpam-492	163	12	⊆	⊆	NUM
ejpam-492	163	13	q	q	NOUN
ejpam-492	163	14	for	for	ADP
ejpam-492	163	15	some	some	DET
ejpam-492	163	16	j	j	PROPN
ejpam-492	163	17	∈	∈	PROPN
ejpam-492	163	18	{	{	PUNCT
ejpam-492	163	19	s	s	NOUN
ejpam-492	163	20	+	+	NOUN
ejpam-492	163	21	1	1	NUM
ejpam-492	163	22	,	,	PUNCT
ejpam-492	163	23	s	s	PART
ejpam-492	163	24	+	+	ADJ
ejpam-492	163	25	2	2	NUM
ejpam-492	163	26	,	,	PUNCT
ejpam-492	163	27	....	....	PUNCT
ejpam-492	163	28	,	,	PUNCT
ejpam-492	163	29	s+	s+	X
ejpam-492	163	30	t	t	PROPN
ejpam-492	163	31	}	}	PUNCT
ejpam-492	163	32	,	,	PUNCT
ejpam-492	163	33	then	then	ADV
ejpam-492	163	34	iq	iq	VERB
ejpam-492	163	35	=	=	PUNCT
ejpam-492	163	36	jq	jq	PROPN
ejpam-492	164	1	=	=	PRON
ejpam-492	164	2	0q	0q	PROPN
ejpam-492	164	3	as	as	SCONJ
ejpam-492	164	4	rq	rq	X
ejpam-492	164	5	is	be	AUX
ejpam-492	164	6	a	a	DET
ejpam-492	164	7	π	π	NOUN
ejpam-492	164	8	-	-	NOUN
ejpam-492	164	9	domain	domain	NOUN
ejpam-492	164	10	.	.	PUNCT
ejpam-492	165	1	without	without	ADP
ejpam-492	165	2	loss	loss	NOUN
ejpam-492	165	3	of	of	ADP
ejpam-492	165	4	c.	c.	PROPN
ejpam-492	165	5	jayaram	jayaram	PROPN
ejpam-492	165	6	/	/	SYM
ejpam-492	165	7	eur	eur	PROPN
ejpam-492	165	8	.	.	PUNCT
ejpam-492	166	1	j.	j.	PROPN
ejpam-492	166	2	pure	pure	PROPN
ejpam-492	166	3	appl	appl	PROPN
ejpam-492	166	4	.	.	PROPN
ejpam-492	166	5	math	math	PROPN
ejpam-492	166	6	,	,	PUNCT
ejpam-492	166	7	2	2	NUM
ejpam-492	166	8	(	(	PUNCT
ejpam-492	166	9	2009	2009	NUM
ejpam-492	166	10	)	)	PUNCT
ejpam-492	166	11	,	,	PUNCT
ejpam-492	166	12	(	(	PUNCT
ejpam-492	166	13	508	508	NUM
ejpam-492	166	14	-	-	NUM
ejpam-492	166	15	519	519	NUM
ejpam-492	166	16	)	)	PUNCT
ejpam-492	166	17	515	515	NUM
ejpam-492	166	18	generality	generality	NOUN
ejpam-492	166	19	,	,	PUNCT
ejpam-492	166	20	assume	assume	VERB
ejpam-492	166	21	that	that	SCONJ
ejpam-492	166	22	p1	p1	NOUN
ejpam-492	166	23	,	,	PUNCT
ejpam-492	166	24	p2	p2	NOUN
ejpam-492	166	25	,	,	PUNCT
ejpam-492	166	26	...	...	PUNCT
ejpam-492	166	27	,	,	PUNCT
ejpam-492	166	28	pt	pt	X
ejpam-492	166	29	⊆q	⊆q	NOUN
ejpam-492	166	30	for	for	ADP
ejpam-492	166	31	(	(	PUNCT
ejpam-492	166	32	1	1	NUM
ejpam-492	166	33	≤	≤	NOUN
ejpam-492	166	34	t	t	PROPN
ejpam-492	166	35	<	<	X
ejpam-492	166	36	s	s	X
ejpam-492	166	37	)	)	PUNCT
ejpam-492	166	38	and	and	CCONJ
ejpam-492	166	39	pj	pj	PROPN
ejpam-492	166	40	6⊆q	6⊆q	PROPN
ejpam-492	166	41	for	for	ADP
ejpam-492	166	42	(	(	PUNCT
ejpam-492	166	43	t+1	t+1	PROPN
ejpam-492	166	44	≤	≤	NUM
ejpam-492	166	45	j	j	PROPN
ejpam-492	166	46	≤	≤	PROPN
ejpam-492	166	47	s	s	PART
ejpam-492	166	48	)	)	PUNCT
ejpam-492	166	49	.	.	PUNCT
ejpam-492	167	1	note	note	VERB
ejpam-492	167	2	that	that	SCONJ
ejpam-492	167	3	rq	rq	NOUN
ejpam-492	167	4	is	be	AUX
ejpam-492	167	5	a	a	DET
ejpam-492	167	6	π	π	NOUN
ejpam-492	167	7	-	-	NOUN
ejpam-492	167	8	domain	domain	NOUN
ejpam-492	167	9	and	and	CCONJ
ejpam-492	167	10	iq	iq	PROPN
ejpam-492	167	11	is	be	AUX
ejpam-492	167	12	a	a	DET
ejpam-492	167	13	non	non	ADJ
ejpam-492	167	14	zero	zero	NUM
ejpam-492	167	15	principal	principal	ADJ
ejpam-492	167	16	ideal	ideal	NOUN
ejpam-492	167	17	of	of	ADP
ejpam-492	167	18	rq	rq	PROPN
ejpam-492	167	19	.	.	PUNCT
ejpam-492	168	1	therefore	therefore	ADV
ejpam-492	168	2	by	by	ADP
ejpam-492	168	3	[	[	X
ejpam-492	168	4	15	15	NUM
ejpam-492	168	5	,	,	PUNCT
ejpam-492	168	6	theorem	theorem	VERB
ejpam-492	168	7	4.2	4.2	NUM
ejpam-492	168	8	and	and	CCONJ
ejpam-492	168	9	corollary	corollary	ADJ
ejpam-492	168	10	4.3	4.3	NUM
ejpam-492	168	11	]	]	PUNCT
ejpam-492	168	12	,	,	PUNCT
ejpam-492	168	13	iq	iq	PROPN
ejpam-492	168	14	is	be	AUX
ejpam-492	168	15	a	a	DET
ejpam-492	168	16	finite	finite	ADJ
ejpam-492	168	17	product	product	NOUN
ejpam-492	168	18	of	of	ADP
ejpam-492	168	19	the	the	DET
ejpam-492	168	20	rank	rank	NOUN
ejpam-492	168	21	one	one	NUM
ejpam-492	168	22	principal	principal	ADJ
ejpam-492	168	23	prime	prime	ADJ
ejpam-492	168	24	ideals	ideal	NOUN
ejpam-492	168	25	minimal	minimal	ADJ
ejpam-492	168	26	over	over	ADP
ejpam-492	168	27	it	it	PRON
ejpam-492	168	28	.	.	PUNCT
ejpam-492	169	1	again	again	ADV
ejpam-492	169	2	using	use	VERB
ejpam-492	169	3	theorem	theorem	NOUN
ejpam-492	169	4	3	3	NUM
ejpam-492	169	5	of	of	ADP
ejpam-492	169	6	[	[	X
ejpam-492	169	7	7	7	NUM
ejpam-492	169	8	]	]	PUNCT
ejpam-492	169	9	,	,	PUNCT
ejpam-492	169	10	it	it	PRON
ejpam-492	169	11	can	can	AUX
ejpam-492	169	12	be	be	AUX
ejpam-492	169	13	easily	easily	ADV
ejpam-492	169	14	shown	show	VERB
ejpam-492	169	15	that	that	SCONJ
ejpam-492	169	16	iq	iq	NOUN
ejpam-492	169	17	=	=	PUNCT
ejpam-492	169	18	(	(	PUNCT
ejpam-492	169	19	p1	p1	PROPN
ejpam-492	169	20	n1)q	n1)q	NOUN
ejpam-492	169	21	∩	∩	NOUN
ejpam-492	169	22	(	(	PUNCT
ejpam-492	169	23	p2	p2	PROPN
ejpam-492	169	24	n2)q	n2)q	ADJ
ejpam-492	169	25	∩	∩	NOUN
ejpam-492	169	26	...	...	PUNCT
ejpam-492	169	27	∩	∩	NOUN
ejpam-492	169	28	(	(	PUNCT
ejpam-492	169	29	pt	pt	INTJ
ejpam-492	169	30	nt	not	PART
ejpam-492	169	31	)	)	PUNCT
ejpam-492	169	32	q.	q.	PROPN
ejpam-492	169	33	therefore	therefore	ADV
ejpam-492	169	34	iq	iq	VERB
ejpam-492	169	35	=	=	PROPN
ejpam-492	169	36	jq	jq	PROPN
ejpam-492	169	37	since	since	SCONJ
ejpam-492	169	38	(	(	PUNCT
ejpam-492	169	39	pj	pj	PROPN
ejpam-492	169	40	n	n	PROPN
ejpam-492	169	41	j	j	PROPN
ejpam-492	169	42	)	)	PUNCT
ejpam-492	169	43	q	q	PUNCT
ejpam-492	170	1	=	=	PRON
ejpam-492	170	2	rq	rq	NOUN
ejpam-492	170	3	for	for	ADP
ejpam-492	170	4	(	(	PUNCT
ejpam-492	170	5	t+1≤	t+1≤	SYM
ejpam-492	170	6	j	j	X
ejpam-492	170	7	≤	≤	NUM
ejpam-492	170	8	s	s	PART
ejpam-492	170	9	)	)	PUNCT
ejpam-492	170	10	and	and	CCONJ
ejpam-492	170	11	(	(	PUNCT
ejpam-492	170	12	pk)q	pk)q	PROPN
ejpam-492	170	13	=	=	PUNCT
ejpam-492	170	14	rq	rq	VERB
ejpam-492	170	15	for	for	ADP
ejpam-492	170	16	(	(	PUNCT
ejpam-492	170	17	s+1	s+1	PROPN
ejpam-492	170	18	≤	≤	PROPN
ejpam-492	170	19	k	k	X
ejpam-492	170	20	≤	≤	NUM
ejpam-492	170	21	m	m	PROPN
ejpam-492	170	22	)	)	PUNCT
ejpam-492	170	23	.	.	PUNCT
ejpam-492	171	1	if	if	SCONJ
ejpam-492	171	2	pj	pj	PROPN
ejpam-492	171	3	⊆q	⊆q	NOUN
ejpam-492	171	4	for	for	ADP
ejpam-492	171	5	(	(	PUNCT
ejpam-492	171	6	s+t+1	s+t+1	PROPN
ejpam-492	171	7	≤	≤	PROPN
ejpam-492	171	8	j	j	PROPN
ejpam-492	171	9	≤	≤	PROPN
ejpam-492	171	10	m	m	PROPN
ejpam-492	171	11	)	)	PUNCT
ejpam-492	171	12	,	,	PUNCT
ejpam-492	171	13	then	then	ADV
ejpam-492	171	14	iq	iq	VERB
ejpam-492	171	15	=	=	PROPN
ejpam-492	171	16	jq	jq	PROPN
ejpam-492	171	17	.	.	PUNCT
ejpam-492	172	1	this	this	PRON
ejpam-492	172	2	shows	show	VERB
ejpam-492	172	3	that	that	SCONJ
ejpam-492	172	4	iq	iq	NOUN
ejpam-492	172	5	=	=	PROPN
ejpam-492	172	6	jq	jq	PROPN
ejpam-492	172	7	for	for	ADP
ejpam-492	172	8	all	all	DET
ejpam-492	172	9	maximal	maximal	ADJ
ejpam-492	172	10	prime	prime	ADJ
ejpam-492	172	11	ideals	ideal	NOUN
ejpam-492	172	12	q	q	NOUN
ejpam-492	172	13	containing	contain	VERB
ejpam-492	172	14	i	i	PRON
ejpam-492	172	15	.	.	PUNCT
ejpam-492	173	1	further	far	ADV
ejpam-492	173	2	,	,	PUNCT
ejpam-492	173	3	if	if	SCONJ
ejpam-492	173	4	i	i	PRON
ejpam-492	173	5	6⊆	6⊆	VERB
ejpam-492	173	6	q	q	PROPN
ejpam-492	173	7	,	,	PUNCT
ejpam-492	173	8	then	then	ADV
ejpam-492	173	9	iq	iq	VERB
ejpam-492	173	10	=	=	PUNCT
ejpam-492	173	11	jq	jq	PROPN
ejpam-492	173	12	=	=	PUNCT
ejpam-492	173	13	rq	rq	PROPN
ejpam-492	173	14	.	.	PUNCT
ejpam-492	174	1	consequently	consequently	ADV
ejpam-492	174	2	,	,	PUNCT
ejpam-492	174	3	i	i	PROPN
ejpam-492	174	4	=	=	SYM
ejpam-492	174	5	j	j	PROPN
ejpam-492	174	6	and	and	CCONJ
ejpam-492	174	7	hence	hence	ADV
ejpam-492	174	8	rx	rx	VERB
ejpam-492	174	9	=	=	NOUN
ejpam-492	175	1	i	i	NOUN
ejpam-492	175	2	m	m	VERB
ejpam-492	175	3	=	=	VERB
ejpam-492	175	4	j	j	PROPN
ejpam-492	175	5	m	m	NOUN
ejpam-492	175	6	.	.	PUNCT
ejpam-492	176	1	since	since	SCONJ
ejpam-492	176	2	j	j	PROPN
ejpam-492	176	3	is	be	AUX
ejpam-492	176	4	a	a	DET
ejpam-492	176	5	finite	finite	ADJ
ejpam-492	176	6	intersection	intersection	NOUN
ejpam-492	176	7	of	of	ADP
ejpam-492	176	8	primary	primary	ADJ
ejpam-492	176	9	ideals	ideal	NOUN
ejpam-492	176	10	,	,	PUNCT
ejpam-492	176	11	by	by	ADP
ejpam-492	176	12	[	[	PUNCT
ejpam-492	176	13	11	11	NUM
ejpam-492	176	14	,	,	PUNCT
ejpam-492	176	15	theorem	theorem	VERB
ejpam-492	176	16	1.6	1.6	NUM
ejpam-492	176	17	]	]	PUNCT
ejpam-492	176	18	and	and	CCONJ
ejpam-492	176	19	[	[	X
ejpam-492	176	20	21	21	NUM
ejpam-492	176	21	,	,	PUNCT
ejpam-492	176	22	corollary	corollary	ADJ
ejpam-492	176	23	1	1	NUM
ejpam-492	176	24	]	]	PUNCT
ejpam-492	176	25	,	,	PUNCT
ejpam-492	176	26	rx	rx	VERB
ejpam-492	176	27	is	be	AUX
ejpam-492	176	28	a	a	DET
ejpam-492	176	29	finite	finite	ADJ
ejpam-492	176	30	intersection	intersection	NOUN
ejpam-492	176	31	of	of	ADP
ejpam-492	176	32	primary	primary	ADJ
ejpam-492	176	33	submodules	submodule	NOUN
ejpam-492	176	34	.	.	PUNCT
ejpam-492	177	1	this	this	PRON
ejpam-492	177	2	completes	complete	VERB
ejpam-492	177	3	the	the	DET
ejpam-492	177	4	proof	proof	NOUN
ejpam-492	177	5	of	of	ADP
ejpam-492	177	6	the	the	DET
ejpam-492	177	7	lemma	lemma	PROPN
ejpam-492	177	8	.	.	PUNCT
ejpam-492	178	1	theorem	theorem	PROPN
ejpam-492	178	2	1	1	NUM
ejpam-492	178	3	.	.	PUNCT
ejpam-492	179	1	suppose	suppose	VERB
ejpam-492	179	2	m	m	PRON
ejpam-492	179	3	is	be	AUX
ejpam-492	179	4	a	a	DET
ejpam-492	179	5	faithful	faithful	ADJ
ejpam-492	179	6	r	r	NOUN
ejpam-492	179	7	-	-	PUNCT
ejpam-492	179	8	module	module	NOUN
ejpam-492	179	9	.	.	PUNCT
ejpam-492	180	1	then	then	ADV
ejpam-492	180	2	the	the	DET
ejpam-492	180	3	following	following	ADJ
ejpam-492	180	4	statements	statement	NOUN
ejpam-492	180	5	on	on	ADP
ejpam-492	180	6	m	m	NOUN
ejpam-492	180	7	are	be	AUX
ejpam-492	180	8	equivalent	equivalent	ADJ
ejpam-492	180	9	:	:	PUNCT
ejpam-492	180	10	(	(	PUNCT
ejpam-492	180	11	i	i	NOUN
ejpam-492	180	12	)	)	PUNCT
ejpam-492	180	13	m	m	VERB
ejpam-492	180	14	is	be	AUX
ejpam-492	180	15	a	a	DET
ejpam-492	180	16	π	π	NOUN
ejpam-492	180	17	-	-	NOUN
ejpam-492	180	18	module	module	NOUN
ejpam-492	180	19	.	.	PUNCT
ejpam-492	181	1	(	(	PUNCT
ejpam-492	181	2	ii	ii	NOUN
ejpam-492	181	3	)	)	PUNCT
ejpam-492	181	4	r	r	NOUN
ejpam-492	181	5	is	be	AUX
ejpam-492	181	6	a	a	DET
ejpam-492	181	7	π	π	NOUN
ejpam-492	181	8	-	-	NOUN
ejpam-492	181	9	ring	ring	NOUN
ejpam-492	181	10	and	and	CCONJ
ejpam-492	181	11	m	m	NOUN
ejpam-492	181	12	is	be	AUX
ejpam-492	181	13	a	a	DET
ejpam-492	181	14	multiplication	multiplication	NOUN
ejpam-492	181	15	module	module	NOUN
ejpam-492	181	16	.	.	PUNCT
ejpam-492	182	1	(	(	PUNCT
ejpam-492	182	2	iii	iii	NOUN
ejpam-492	182	3	)	)	PUNCT
ejpam-492	182	4	every	every	DET
ejpam-492	182	5	cyclic	cyclic	ADJ
ejpam-492	182	6	submodule	submodule	NOUN
ejpam-492	182	7	of	of	ADP
ejpam-492	182	8	m	m	PROPN
ejpam-492	182	9	is	be	AUX
ejpam-492	182	10	of	of	ADP
ejpam-492	182	11	the	the	DET
ejpam-492	182	12	form	form	NOUN
ejpam-492	182	13	i	i	PRON
ejpam-492	182	14	m	m	VERB
ejpam-492	182	15	,	,	PUNCT
ejpam-492	182	16	where	where	SCONJ
ejpam-492	182	17	i	i	PRON
ejpam-492	182	18	is	be	AUX
ejpam-492	182	19	a	a	DET
ejpam-492	182	20	finite	finite	ADJ
ejpam-492	182	21	product	product	NOUN
ejpam-492	182	22	of	of	ADP
ejpam-492	182	23	quasi	quasi	ADJ
ejpam-492	182	24	-	-	ADJ
ejpam-492	182	25	principal	principal	ADJ
ejpam-492	182	26	prime	prime	ADJ
ejpam-492	182	27	ideals	ideal	NOUN
ejpam-492	182	28	of	of	ADP
ejpam-492	182	29	rank	rank	NOUN
ejpam-492	182	30	less	less	ADJ
ejpam-492	182	31	than	than	ADP
ejpam-492	182	32	or	or	CCONJ
ejpam-492	182	33	equal	equal	ADJ
ejpam-492	182	34	to	to	ADP
ejpam-492	182	35	one	one	NUM
ejpam-492	182	36	.	.	PUNCT
ejpam-492	183	1	proof	proof	NOUN
ejpam-492	183	2	.	.	PUNCT
ejpam-492	184	1	(	(	PUNCT
ejpam-492	184	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-492	184	3	)	)	PUNCT
ejpam-492	184	4	.	.	PUNCT
ejpam-492	185	1	suppose	suppose	VERB
ejpam-492	185	2	(	(	PUNCT
ejpam-492	185	3	i	i	NOUN
ejpam-492	185	4	)	)	PUNCT
ejpam-492	185	5	holds	hold	VERB
ejpam-492	185	6	.	.	PUNCT
ejpam-492	186	1	then	then	ADV
ejpam-492	186	2	m	m	VERB
ejpam-492	186	3	is	be	AUX
ejpam-492	186	4	a	a	DET
ejpam-492	186	5	faithful	faithful	ADJ
ejpam-492	186	6	and	and	CCONJ
ejpam-492	186	7	finitely	finitely	ADV
ejpam-492	186	8	generated	generate	VERB
ejpam-492	186	9	multiplication	multiplication	NOUN
ejpam-492	186	10	module	module	NOUN
ejpam-492	186	11	.	.	PUNCT
ejpam-492	187	1	by	by	ADP
ejpam-492	187	2	lemma	lemma	PROPN
ejpam-492	187	3	5	5	NUM
ejpam-492	187	4	,	,	PUNCT
ejpam-492	187	5	r	r	NOUN
ejpam-492	187	6	is	be	AUX
ejpam-492	187	7	an	an	DET
ejpam-492	187	8	almost	almost	ADV
ejpam-492	187	9	π	π	NOUN
ejpam-492	187	10	-	-	NOUN
ejpam-492	187	11	ring	ring	NOUN
ejpam-492	187	12	.	.	PUNCT
ejpam-492	188	1	by	by	ADP
ejpam-492	188	2	[	[	X
ejpam-492	188	3	13	13	NUM
ejpam-492	188	4	,	,	PUNCT
ejpam-492	188	5	theorem	theorem	VERB
ejpam-492	188	6	6	6	NUM
ejpam-492	188	7	]	]	PUNCT
ejpam-492	188	8	,	,	PUNCT
ejpam-492	188	9	it	it	PRON
ejpam-492	188	10	is	be	AUX
ejpam-492	188	11	enough	enough	ADJ
ejpam-492	188	12	if	if	SCONJ
ejpam-492	188	13	we	we	PRON
ejpam-492	188	14	show	show	VERB
ejpam-492	188	15	that	that	SCONJ
ejpam-492	188	16	every	every	DET
ejpam-492	188	17	prime	prime	ADJ
ejpam-492	188	18	ideal	ideal	NOUN
ejpam-492	188	19	of	of	ADP
ejpam-492	188	20	r	r	NOUN
ejpam-492	188	21	of	of	ADP
ejpam-492	188	22	rank	rank	NOUN
ejpam-492	188	23	less	less	ADV
ejpam-492	188	24	than	than	ADP
ejpam-492	188	25	or	or	CCONJ
ejpam-492	188	26	equal	equal	ADJ
ejpam-492	188	27	to	to	ADP
ejpam-492	188	28	one	one	NUM
ejpam-492	188	29	is	be	AUX
ejpam-492	188	30	finitely	finitely	ADV
ejpam-492	188	31	generated	generate	VERB
ejpam-492	188	32	.	.	PUNCT
ejpam-492	189	1	note	note	VERB
ejpam-492	189	2	that	that	SCONJ
ejpam-492	189	3	if	if	SCONJ
ejpam-492	189	4	p	p	NOUN
ejpam-492	189	5	is	be	AUX
ejpam-492	189	6	a	a	DET
ejpam-492	189	7	minimal	minimal	ADJ
ejpam-492	189	8	prime	prime	ADJ
ejpam-492	189	9	ideal	ideal	NOUN
ejpam-492	189	10	of	of	ADP
ejpam-492	189	11	r	r	NOUN
ejpam-492	189	12	,	,	PUNCT
ejpam-492	189	13	then	then	ADV
ejpam-492	189	14	pm	pm	NOUN
ejpam-492	189	15	is	be	AUX
ejpam-492	189	16	a	a	DET
ejpam-492	189	17	minimal	minimal	ADJ
ejpam-492	189	18	prime	prime	ADJ
ejpam-492	189	19	submodule	submodule	NOUN
ejpam-492	189	20	,	,	PUNCT
ejpam-492	189	21	so	so	ADV
ejpam-492	189	22	by	by	ADP
ejpam-492	189	23	[	[	X
ejpam-492	189	24	14	14	NUM
ejpam-492	189	25	,	,	PUNCT
ejpam-492	189	26	lemma	lemma	PROPN
ejpam-492	189	27	7	7	NUM
ejpam-492	189	28	]	]	PUNCT
ejpam-492	189	29	,	,	PUNCT
ejpam-492	189	30	lemma	lemma	PROPN
ejpam-492	189	31	6	6	NUM
ejpam-492	189	32	and	and	CCONJ
ejpam-492	189	33	lemma	lemma	PROPN
ejpam-492	189	34	8	8	NUM
ejpam-492	189	35	,	,	PUNCT
ejpam-492	189	36	pm	pm	NOUN
ejpam-492	189	37	is	be	AUX
ejpam-492	189	38	a	a	DET
ejpam-492	189	39	finitely	finitely	ADV
ejpam-492	189	40	generated	generate	VERB
ejpam-492	189	41	multiplication	multiplication	NOUN
ejpam-492	189	42	submodule	submodule	NOUN
ejpam-492	189	43	.	.	PUNCT
ejpam-492	190	1	therefore	therefore	ADV
ejpam-492	190	2	by	by	ADP
ejpam-492	190	3	[	[	X
ejpam-492	190	4	18	18	NUM
ejpam-492	190	5	,	,	PUNCT
ejpam-492	190	6	lemma	lemma	PROPN
ejpam-492	190	7	1.4	1.4	NUM
ejpam-492	190	8	]	]	PUNCT
ejpam-492	190	9	,	,	PUNCT
ejpam-492	190	10	p	p	NOUN
ejpam-492	190	11	=	=	X
ejpam-492	190	12	(	(	PUNCT
ejpam-492	190	13	pm	pm	NOUN
ejpam-492	190	14	:	:	PUNCT
ejpam-492	190	15	m	m	X
ejpam-492	190	16	)	)	PUNCT
ejpam-492	190	17	is	be	AUX
ejpam-492	190	18	a	a	DET
ejpam-492	190	19	finitely	finitely	ADV
ejpam-492	190	20	generated	generate	VERB
ejpam-492	190	21	multiplication	multiplication	NOUN
ejpam-492	190	22	ideal	ideal	ADJ
ejpam-492	190	23	.	.	PUNCT
ejpam-492	191	1	let	let	VERB
ejpam-492	191	2	p	p	PRON
ejpam-492	191	3	be	be	AUX
ejpam-492	191	4	a	a	DET
ejpam-492	191	5	rank	rank	NOUN
ejpam-492	191	6	one	one	NUM
ejpam-492	191	7	prime	prime	ADJ
ejpam-492	191	8	ideal	ideal	NOUN
ejpam-492	191	9	.	.	PUNCT
ejpam-492	192	1	as	as	SCONJ
ejpam-492	192	2	r	r	NOUN
ejpam-492	192	3	is	be	AUX
ejpam-492	192	4	an	an	DET
ejpam-492	192	5	almost	almost	ADV
ejpam-492	192	6	π	π	NOUN
ejpam-492	192	7	-	-	NOUN
ejpam-492	192	8	ring	ring	NOUN
ejpam-492	192	9	,	,	PUNCT
ejpam-492	192	10	it	it	PRON
ejpam-492	192	11	follows	follow	VERB
ejpam-492	192	12	that	that	SCONJ
ejpam-492	192	13	p	p	NOUN
ejpam-492	192	14	is	be	AUX
ejpam-492	192	15	locally	locally	ADV
ejpam-492	192	16	principal	principal	ADJ
ejpam-492	192	17	.	.	PUNCT
ejpam-492	193	1	so	so	ADV
ejpam-492	193	2	by	by	ADP
ejpam-492	193	3	[	[	X
ejpam-492	193	4	14	14	NUM
ejpam-492	193	5	,	,	PUNCT
ejpam-492	193	6	lemma	lemma	PROPN
ejpam-492	193	7	6	6	NUM
ejpam-492	193	8	]	]	PUNCT
ejpam-492	193	9	,	,	PUNCT
ejpam-492	193	10	pm	pm	PROPN
ejpam-492	193	11	is	be	AUX
ejpam-492	193	12	locally	locally	ADV
ejpam-492	193	13	cyclic	cyclic	ADJ
ejpam-492	193	14	and	and	CCONJ
ejpam-492	193	15	hence	hence	ADV
ejpam-492	193	16	by	by	ADP
ejpam-492	193	17	[	[	X
ejpam-492	193	18	14	14	NUM
ejpam-492	193	19	,	,	PUNCT
ejpam-492	193	20	lemma	lemma	PROPN
ejpam-492	193	21	7	7	NUM
ejpam-492	193	22	]	]	PUNCT
ejpam-492	193	23	and	and	CCONJ
ejpam-492	193	24	lemma	lemma	PROPN
ejpam-492	193	25	8	8	NUM
ejpam-492	193	26	,	,	PUNCT
ejpam-492	193	27	pm	pm	NOUN
ejpam-492	193	28	is	be	AUX
ejpam-492	193	29	a	a	DET
ejpam-492	193	30	finitely	finitely	ADV
ejpam-492	193	31	generated	generate	VERB
ejpam-492	193	32	c.	c.	PROPN
ejpam-492	193	33	jayaram	jayaram	PROPN
ejpam-492	193	34	/	/	SYM
ejpam-492	193	35	eur	eur	PROPN
ejpam-492	193	36	.	.	PUNCT
ejpam-492	194	1	j.	j.	PROPN
ejpam-492	194	2	pure	pure	PROPN
ejpam-492	194	3	appl	appl	PROPN
ejpam-492	194	4	.	.	PROPN
ejpam-492	194	5	math	math	PROPN
ejpam-492	194	6	,	,	PUNCT
ejpam-492	194	7	2	2	NUM
ejpam-492	194	8	(	(	PUNCT
ejpam-492	194	9	2009	2009	NUM
ejpam-492	194	10	)	)	PUNCT
ejpam-492	194	11	,	,	PUNCT
ejpam-492	194	12	(	(	PUNCT
ejpam-492	194	13	508	508	NUM
ejpam-492	194	14	-	-	NUM
ejpam-492	194	15	519	519	NUM
ejpam-492	194	16	)	)	PUNCT
ejpam-492	194	17	516	516	NUM
ejpam-492	194	18	multiplication	multiplication	NOUN
ejpam-492	194	19	submodule	submodule	NOUN
ejpam-492	194	20	.	.	PUNCT
ejpam-492	195	1	consequently	consequently	ADV
ejpam-492	195	2	,	,	PUNCT
ejpam-492	195	3	p	p	PROPN
ejpam-492	195	4	is	be	AUX
ejpam-492	195	5	a	a	DET
ejpam-492	195	6	finitely	finitely	ADV
ejpam-492	195	7	generated	generate	VERB
ejpam-492	195	8	multiplication	multiplication	NOUN
ejpam-492	195	9	ideal	ideal	ADJ
ejpam-492	195	10	.	.	PUNCT
ejpam-492	196	1	therefore	therefore	ADV
ejpam-492	196	2	r	r	NOUN
ejpam-492	196	3	is	be	AUX
ejpam-492	196	4	a	a	DET
ejpam-492	196	5	π	π	NOUN
ejpam-492	196	6	-	-	NOUN
ejpam-492	196	7	ring	ring	NOUN
ejpam-492	196	8	.	.	PUNCT
ejpam-492	197	1	(	(	PUNCT
ejpam-492	197	2	ii)⇒(iii	ii)⇒(iii	X
ejpam-492	197	3	)	)	PUNCT
ejpam-492	197	4	.	.	PUNCT
ejpam-492	198	1	suppose	suppose	VERB
ejpam-492	198	2	(	(	PUNCT
ejpam-492	198	3	ii	ii	NOUN
ejpam-492	198	4	)	)	PUNCT
ejpam-492	198	5	holds	hold	VERB
ejpam-492	198	6	.	.	PUNCT
ejpam-492	199	1	let	let	VERB
ejpam-492	199	2	x	x	SYM
ejpam-492	199	3	∈	∈	PROPN
ejpam-492	199	4	m	m	VERB
ejpam-492	199	5	.	.	PUNCT
ejpam-492	200	1	as	as	SCONJ
ejpam-492	200	2	m	m	PROPN
ejpam-492	200	3	is	be	AUX
ejpam-492	200	4	a	a	DET
ejpam-492	200	5	multiplication	multiplication	NOUN
ejpam-492	200	6	module	module	NOUN
ejpam-492	200	7	,	,	PUNCT
ejpam-492	200	8	it	it	PRON
ejpam-492	200	9	follows	follow	VERB
ejpam-492	200	10	that	that	SCONJ
ejpam-492	200	11	rx	rx	NOUN
ejpam-492	200	12	=	=	VERB
ejpam-492	200	13	(	(	PUNCT
ejpam-492	200	14	rx	rx	ADJ
ejpam-492	200	15	:	:	PUNCT
ejpam-492	200	16	m)m	m)m	X
ejpam-492	200	17	.	.	PUNCT
ejpam-492	201	1	since	since	SCONJ
ejpam-492	201	2	r	r	NOUN
ejpam-492	201	3	is	be	AUX
ejpam-492	201	4	a	a	DET
ejpam-492	201	5	π	π	NOUN
ejpam-492	201	6	-	-	NOUN
ejpam-492	201	7	ring	ring	NOUN
ejpam-492	201	8	,	,	PUNCT
ejpam-492	201	9	it	it	PRON
ejpam-492	201	10	follows	follow	VERB
ejpam-492	201	11	that	that	SCONJ
ejpam-492	201	12	r	r	NOUN
ejpam-492	201	13	contains	contain	VERB
ejpam-492	201	14	only	only	ADV
ejpam-492	201	15	finitely	finitely	ADV
ejpam-492	201	16	many	many	ADJ
ejpam-492	201	17	minimal	minimal	ADJ
ejpam-492	201	18	prime	prime	ADJ
ejpam-492	201	19	ideals	ideal	NOUN
ejpam-492	201	20	and	and	CCONJ
ejpam-492	201	21	so	so	ADV
ejpam-492	201	22	m	m	VERB
ejpam-492	201	23	contains	contain	VERB
ejpam-492	201	24	only	only	ADV
ejpam-492	201	25	finitely	finitely	ADV
ejpam-492	201	26	many	many	ADJ
ejpam-492	201	27	minimal	minimal	ADJ
ejpam-492	201	28	prime	prime	ADJ
ejpam-492	201	29	submodules	submodule	NOUN
ejpam-492	201	30	.	.	PUNCT
ejpam-492	202	1	so	so	ADV
ejpam-492	202	2	by	by	ADP
ejpam-492	202	3	[	[	PUNCT
ejpam-492	202	4	11	11	NUM
ejpam-492	202	5	,	,	PUNCT
ejpam-492	202	6	theorem	theorem	VERB
ejpam-492	202	7	3.7	3.7	NUM
ejpam-492	202	8	]	]	PUNCT
ejpam-492	202	9	,	,	PUNCT
ejpam-492	202	10	m	m	VERB
ejpam-492	202	11	is	be	AUX
ejpam-492	202	12	finitely	finitely	ADV
ejpam-492	202	13	generated	generate	VERB
ejpam-492	202	14	.	.	PUNCT
ejpam-492	203	1	again	again	ADV
ejpam-492	203	2	by	by	ADP
ejpam-492	203	3	[	[	X
ejpam-492	203	4	18	18	NUM
ejpam-492	203	5	,	,	PUNCT
ejpam-492	203	6	lemma	lemma	PROPN
ejpam-492	203	7	1.4	1.4	NUM
ejpam-492	203	8	]	]	PUNCT
ejpam-492	203	9	,	,	PUNCT
ejpam-492	203	10	(	(	PUNCT
ejpam-492	203	11	rx	rx	VERB
ejpam-492	203	12	:	:	PUNCT
ejpam-492	203	13	m	m	X
ejpam-492	203	14	)	)	PUNCT
ejpam-492	203	15	is	be	AUX
ejpam-492	203	16	a	a	DET
ejpam-492	203	17	quasi	quasi	ADJ
ejpam-492	203	18	-	-	ADJ
ejpam-492	203	19	principal	principal	ADJ
ejpam-492	203	20	ideal	ideal	NOUN
ejpam-492	203	21	of	of	ADP
ejpam-492	203	22	r	r	NOUN
ejpam-492	203	23	(	(	PUNCT
ejpam-492	203	24	i.e.	i.e.	X
ejpam-492	203	25	,	,	PUNCT
ejpam-492	203	26	a	a	DET
ejpam-492	203	27	principal	principal	ADJ
ejpam-492	203	28	element	element	NOUN
ejpam-492	203	29	of	of	ADP
ejpam-492	203	30	r	r	NOUN
ejpam-492	203	31	)	)	PUNCT
ejpam-492	203	32	.	.	PUNCT
ejpam-492	204	1	as	as	SCONJ
ejpam-492	204	2	r	r	NOUN
ejpam-492	204	3	is	be	AUX
ejpam-492	204	4	a	a	DET
ejpam-492	204	5	π	π	NOUN
ejpam-492	204	6	-	-	NOUN
ejpam-492	204	7	ring	ring	NOUN
ejpam-492	204	8	,	,	PUNCT
ejpam-492	204	9	it	it	PRON
ejpam-492	204	10	follows	follow	VERB
ejpam-492	204	11	that	that	SCONJ
ejpam-492	204	12	l(r	l(r	PROPN
ejpam-492	204	13	)	)	PUNCT
ejpam-492	204	14	is	be	AUX
ejpam-492	204	15	a	a	DET
ejpam-492	204	16	π	π	PROPN
ejpam-492	204	17	-	-	NOUN
ejpam-492	204	18	lattice	lattice	ADJ
ejpam-492	204	19	,	,	PUNCT
ejpam-492	204	20	so	so	ADV
ejpam-492	204	21	by	by	ADP
ejpam-492	204	22	[	[	X
ejpam-492	204	23	4	4	NUM
ejpam-492	204	24	,	,	PUNCT
ejpam-492	204	25	theorem	theorem	VERB
ejpam-492	204	26	2	2	NUM
ejpam-492	204	27	]	]	PUNCT
ejpam-492	204	28	,	,	PUNCT
ejpam-492	204	29	(	(	PUNCT
ejpam-492	204	30	rx	rx	VERB
ejpam-492	204	31	:	:	PUNCT
ejpam-492	204	32	m	m	X
ejpam-492	204	33	)	)	PUNCT
ejpam-492	204	34	is	be	AUX
ejpam-492	204	35	a	a	DET
ejpam-492	204	36	finite	finite	ADJ
ejpam-492	204	37	product	product	NOUN
ejpam-492	204	38	of	of	ADP
ejpam-492	204	39	quasi	quasi	ADJ
ejpam-492	204	40	-	-	ADJ
ejpam-492	204	41	principal	principal	ADJ
ejpam-492	204	42	prime	prime	ADJ
ejpam-492	204	43	ideals	ideal	NOUN
ejpam-492	204	44	of	of	ADP
ejpam-492	204	45	rank	rank	NOUN
ejpam-492	204	46	less	less	ADJ
ejpam-492	204	47	than	than	ADP
ejpam-492	204	48	or	or	CCONJ
ejpam-492	204	49	equal	equal	ADJ
ejpam-492	204	50	to	to	ADP
ejpam-492	204	51	one	one	NUM
ejpam-492	204	52	.	.	PUNCT
ejpam-492	205	1	therefore	therefore	ADV
ejpam-492	205	2	(	(	PUNCT
ejpam-492	205	3	iii	iii	NOUN
ejpam-492	205	4	)	)	PUNCT
ejpam-492	205	5	holds	hold	VERB
ejpam-492	205	6	and	and	CCONJ
ejpam-492	205	7	(	(	PUNCT
ejpam-492	205	8	iii)⇒(i	iii)⇒(i	NOUN
ejpam-492	205	9	)	)	PUNCT
ejpam-492	205	10	follows	follow	VERB
ejpam-492	205	11	from	from	ADP
ejpam-492	205	12	the	the	DET
ejpam-492	205	13	definition	definition	NOUN
ejpam-492	205	14	.	.	PUNCT
ejpam-492	206	1	this	this	PRON
ejpam-492	206	2	completes	complete	VERB
ejpam-492	206	3	the	the	DET
ejpam-492	206	4	proof	proof	NOUN
ejpam-492	206	5	of	of	ADP
ejpam-492	206	6	the	the	DET
ejpam-492	206	7	theorem	theorem	PROPN
ejpam-492	206	8	.	.	PUNCT
ejpam-492	206	9	theorem	theorem	NOUN
ejpam-492	206	10	2	2	NUM
ejpam-492	206	11	.	.	PUNCT
ejpam-492	206	12	suppose	suppose	VERB
ejpam-492	206	13	m	m	PRON
ejpam-492	206	14	is	be	AUX
ejpam-492	206	15	a	a	DET
ejpam-492	206	16	faithful	faithful	ADJ
ejpam-492	206	17	r	r	NOUN
ejpam-492	206	18	-	-	PUNCT
ejpam-492	206	19	module	module	NOUN
ejpam-492	206	20	.	.	PUNCT
ejpam-492	207	1	then	then	ADV
ejpam-492	207	2	the	the	DET
ejpam-492	207	3	following	following	ADJ
ejpam-492	207	4	statements	statement	NOUN
ejpam-492	207	5	on	on	ADP
ejpam-492	207	6	m	m	NOUN
ejpam-492	207	7	are	be	AUX
ejpam-492	207	8	equivalent	equivalent	ADJ
ejpam-492	207	9	:	:	PUNCT
ejpam-492	207	10	(	(	PUNCT
ejpam-492	207	11	i	i	NOUN
ejpam-492	207	12	)	)	PUNCT
ejpam-492	207	13	m	m	VERB
ejpam-492	207	14	is	be	AUX
ejpam-492	207	15	a	a	DET
ejpam-492	207	16	π	π	NOUN
ejpam-492	207	17	-	-	NOUN
ejpam-492	207	18	module	module	NOUN
ejpam-492	207	19	.	.	PUNCT
ejpam-492	208	1	(	(	PUNCT
ejpam-492	208	2	ii	ii	NOUN
ejpam-492	208	3	)	)	PUNCT
ejpam-492	208	4	m	m	VERB
ejpam-492	208	5	is	be	AUX
ejpam-492	208	6	an	an	DET
ejpam-492	208	7	almost	almost	ADV
ejpam-492	208	8	π	π	NOUN
ejpam-492	208	9	-	-	NOUN
ejpam-492	208	10	module	module	NOUN
ejpam-492	208	11	in	in	ADP
ejpam-492	208	12	which	which	PRON
ejpam-492	208	13	every	every	DET
ejpam-492	208	14	finitely	finitely	ADV
ejpam-492	208	15	generated	generate	VERB
ejpam-492	208	16	multiplication	multiplication	NOUN
ejpam-492	208	17	submodule	submodule	NOUN
ejpam-492	208	18	is	be	AUX
ejpam-492	208	19	a	a	DET
ejpam-492	208	20	finite	finite	ADJ
ejpam-492	208	21	intersection	intersection	NOUN
ejpam-492	208	22	of	of	ADP
ejpam-492	208	23	primary	primary	ADJ
ejpam-492	208	24	submodules	submodule	NOUN
ejpam-492	208	25	.	.	PUNCT
ejpam-492	209	1	(	(	PUNCT
ejpam-492	209	2	iii	iii	X
ejpam-492	209	3	)	)	PUNCT
ejpam-492	209	4	m	m	VERB
ejpam-492	209	5	is	be	AUX
ejpam-492	209	6	an	an	DET
ejpam-492	209	7	almost	almost	ADV
ejpam-492	209	8	π	π	NOUN
ejpam-492	209	9	-	-	NOUN
ejpam-492	209	10	module	module	NOUN
ejpam-492	209	11	in	in	ADP
ejpam-492	209	12	which	which	PRON
ejpam-492	209	13	every	every	DET
ejpam-492	209	14	cyclic	cyclic	ADJ
ejpam-492	209	15	submodule	submodule	NOUN
ejpam-492	209	16	is	be	AUX
ejpam-492	209	17	a	a	DET
ejpam-492	209	18	finite	finite	ADJ
ejpam-492	209	19	intersection	intersection	NOUN
ejpam-492	209	20	of	of	ADP
ejpam-492	209	21	primary	primary	ADJ
ejpam-492	209	22	submodules	submodule	NOUN
ejpam-492	209	23	.	.	PUNCT
ejpam-492	210	1	(	(	PUNCT
ejpam-492	210	2	iv	iv	X
ejpam-492	210	3	)	)	PUNCT
ejpam-492	210	4	m	m	VERB
ejpam-492	210	5	is	be	AUX
ejpam-492	210	6	finitely	finitely	ADV
ejpam-492	210	7	generated	generate	VERB
ejpam-492	210	8	and	and	CCONJ
ejpam-492	210	9	an	an	DET
ejpam-492	210	10	almost	almost	ADV
ejpam-492	210	11	π	π	NOUN
ejpam-492	210	12	-	-	NOUN
ejpam-492	210	13	module	module	NOUN
ejpam-492	210	14	in	in	ADP
ejpam-492	210	15	which	which	PRON
ejpam-492	210	16	every	every	DET
ejpam-492	210	17	prime	prime	ADJ
ejpam-492	210	18	submodule	submodule	NOUN
ejpam-492	210	19	of	of	ADP
ejpam-492	210	20	rank	rank	NOUN
ejpam-492	210	21	less	less	ADJ
ejpam-492	210	22	than	than	ADP
ejpam-492	210	23	or	or	CCONJ
ejpam-492	210	24	equal	equal	ADJ
ejpam-492	210	25	to	to	ADP
ejpam-492	210	26	one	one	NUM
ejpam-492	210	27	is	be	AUX
ejpam-492	210	28	finitely	finitely	ADV
ejpam-492	210	29	generated	generate	VERB
ejpam-492	210	30	.	.	PUNCT
ejpam-492	211	1	(	(	PUNCT
ejpam-492	211	2	v	v	NOUN
ejpam-492	211	3	)	)	PUNCT
ejpam-492	211	4	m	m	VERB
ejpam-492	211	5	is	be	AUX
ejpam-492	211	6	a	a	DET
ejpam-492	211	7	multiplication	multiplication	NOUN
ejpam-492	211	8	module	module	NOUN
ejpam-492	211	9	in	in	ADP
ejpam-492	211	10	which	which	PRON
ejpam-492	211	11	every	every	DET
ejpam-492	211	12	minimal	minimal	ADJ
ejpam-492	211	13	prime	prime	ADJ
ejpam-492	211	14	submodule	submodule	NOUN
ejpam-492	211	15	is	be	AUX
ejpam-492	211	16	a	a	DET
ejpam-492	211	17	finitely	finitely	ADV
ejpam-492	211	18	generated	generate	VERB
ejpam-492	211	19	multiplication	multiplication	NOUN
ejpam-492	211	20	submodule	submodule	NOUN
ejpam-492	211	21	and	and	CCONJ
ejpam-492	211	22	every	every	DET
ejpam-492	211	23	non	non	NOUN
ejpam-492	211	24	minimal	minimal	ADJ
ejpam-492	211	25	prime	prime	ADJ
ejpam-492	211	26	submodule	submodule	PROPN
ejpam-492	211	27	contains	contain	VERB
ejpam-492	211	28	a	a	DET
ejpam-492	211	29	non	non	ADJ
ejpam-492	211	30	minimal	minimal	ADJ
ejpam-492	211	31	finitely	finitely	ADV
ejpam-492	211	32	generated	generate	VERB
ejpam-492	211	33	multiplication	multiplication	NOUN
ejpam-492	211	34	prime	prime	NOUN
ejpam-492	211	35	submodule	submodule	NOUN
ejpam-492	211	36	.	.	PUNCT
ejpam-492	212	1	proof	proof	NOUN
ejpam-492	212	2	.	.	PUNCT
ejpam-492	213	1	(	(	PUNCT
ejpam-492	213	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-492	213	3	)	)	PUNCT
ejpam-492	213	4	.	.	PUNCT
ejpam-492	214	1	suppose	suppose	VERB
ejpam-492	214	2	(	(	PUNCT
ejpam-492	214	3	i	i	NOUN
ejpam-492	214	4	)	)	PUNCT
ejpam-492	214	5	holds	hold	VERB
ejpam-492	214	6	.	.	PUNCT
ejpam-492	215	1	clearly	clearly	ADV
ejpam-492	215	2	,	,	PUNCT
ejpam-492	215	3	m	m	VERB
ejpam-492	215	4	is	be	AUX
ejpam-492	215	5	an	an	DET
ejpam-492	215	6	almost	almost	ADV
ejpam-492	215	7	π	π	NOUN
ejpam-492	215	8	-	-	NOUN
ejpam-492	215	9	module	module	NOUN
ejpam-492	215	10	.	.	PUNCT
ejpam-492	216	1	let	let	VERB
ejpam-492	216	2	n	n	PRON
ejpam-492	216	3	be	be	AUX
ejpam-492	216	4	a	a	DET
ejpam-492	216	5	finitely	finitely	ADV
ejpam-492	216	6	generated	generate	VERB
ejpam-492	216	7	multiplication	multiplication	NOUN
ejpam-492	216	8	submodule	submodule	NOUN
ejpam-492	216	9	.	.	PUNCT
ejpam-492	217	1	as	as	SCONJ
ejpam-492	217	2	m	m	PROPN
ejpam-492	217	3	is	be	AUX
ejpam-492	217	4	a	a	DET
ejpam-492	217	5	faithful	faithful	ADJ
ejpam-492	217	6	and	and	CCONJ
ejpam-492	217	7	finitely	finitely	ADV
ejpam-492	217	8	generated	generate	VERB
ejpam-492	217	9	multiplication	multiplication	NOUN
ejpam-492	217	10	module	module	NOUN
ejpam-492	217	11	,	,	PUNCT
ejpam-492	217	12	by	by	ADP
ejpam-492	217	13	[	[	X
ejpam-492	217	14	18	18	NUM
ejpam-492	217	15	,	,	PUNCT
ejpam-492	217	16	lemma	lemma	PROPN
ejpam-492	217	17	1.4	1.4	NUM
ejpam-492	217	18	]	]	PUNCT
ejpam-492	217	19	,	,	PUNCT
ejpam-492	217	20	(	(	PUNCT
ejpam-492	217	21	n	n	X
ejpam-492	217	22	:	:	PUNCT
ejpam-492	217	23	m	m	X
ejpam-492	217	24	)	)	PUNCT
ejpam-492	217	25	is	be	AUX
ejpam-492	217	26	a	a	DET
ejpam-492	217	27	principal	principal	ADJ
ejpam-492	217	28	element	element	NOUN
ejpam-492	217	29	of	of	ADP
ejpam-492	217	30	l(r	l(r	PROPN
ejpam-492	217	31	)	)	PUNCT
ejpam-492	217	32	.	.	PUNCT
ejpam-492	218	1	c.	c.	PROPN
ejpam-492	218	2	jayaram	jayaram	PROPN
ejpam-492	218	3	/	/	SYM
ejpam-492	218	4	eur	eur	PROPN
ejpam-492	218	5	.	.	PUNCT
ejpam-492	219	1	j.	j.	PROPN
ejpam-492	219	2	pure	pure	PROPN
ejpam-492	219	3	appl	appl	PROPN
ejpam-492	219	4	.	.	PROPN
ejpam-492	219	5	math	math	PROPN
ejpam-492	219	6	,	,	PUNCT
ejpam-492	219	7	2	2	NUM
ejpam-492	219	8	(	(	PUNCT
ejpam-492	219	9	2009	2009	NUM
ejpam-492	219	10	)	)	PUNCT
ejpam-492	219	11	,	,	PUNCT
ejpam-492	219	12	(	(	PUNCT
ejpam-492	219	13	508	508	NUM
ejpam-492	219	14	-	-	NUM
ejpam-492	219	15	519	519	NUM
ejpam-492	219	16	)	)	PUNCT
ejpam-492	219	17	517	517	NUM
ejpam-492	219	18	so	so	ADV
ejpam-492	219	19	by	by	ADP
ejpam-492	219	20	theorem	theorem	NOUN
ejpam-492	219	21	1	1	NUM
ejpam-492	219	22	and	and	CCONJ
ejpam-492	219	23	[	[	X
ejpam-492	219	24	13	13	NUM
ejpam-492	219	25	,	,	PUNCT
ejpam-492	219	26	theorem	theorem	VERB
ejpam-492	219	27	6	6	NUM
ejpam-492	219	28	]	]	PUNCT
ejpam-492	219	29	,	,	PUNCT
ejpam-492	219	30	(	(	PUNCT
ejpam-492	219	31	n	n	X
ejpam-492	219	32	:	:	PUNCT
ejpam-492	219	33	m	m	X
ejpam-492	219	34	)	)	PUNCT
ejpam-492	219	35	is	be	AUX
ejpam-492	219	36	a	a	DET
ejpam-492	219	37	finite	finite	ADJ
ejpam-492	219	38	intersection	intersection	NOUN
ejpam-492	219	39	of	of	ADP
ejpam-492	219	40	primary	primary	ADJ
ejpam-492	219	41	ideals	ideal	NOUN
ejpam-492	219	42	and	and	CCONJ
ejpam-492	219	43	hence	hence	ADV
ejpam-492	219	44	by	by	ADP
ejpam-492	219	45	[	[	X
ejpam-492	219	46	11	11	NUM
ejpam-492	219	47	,	,	PUNCT
ejpam-492	219	48	theorem	theorem	VERB
ejpam-492	219	49	1.6	1.6	NUM
ejpam-492	219	50	]	]	PUNCT
ejpam-492	219	51	and	and	CCONJ
ejpam-492	219	52	[	[	X
ejpam-492	219	53	21	21	NUM
ejpam-492	219	54	,	,	PUNCT
ejpam-492	219	55	corollary	corollary	ADJ
ejpam-492	219	56	1	1	NUM
ejpam-492	219	57	]	]	PUNCT
ejpam-492	219	58	,	,	PUNCT
ejpam-492	219	59	n	n	NOUN
ejpam-492	219	60	=	=	SYM
ejpam-492	219	61	(	(	PUNCT
ejpam-492	219	62	n	n	CCONJ
ejpam-492	219	63	:	:	PUNCT
ejpam-492	219	64	m)m	m)m	X
ejpam-492	219	65	is	be	AUX
ejpam-492	219	66	a	a	DET
ejpam-492	219	67	finite	finite	ADJ
ejpam-492	219	68	intersection	intersection	NOUN
ejpam-492	219	69	of	of	ADP
ejpam-492	219	70	primary	primary	ADJ
ejpam-492	219	71	submodules	submodule	NOUN
ejpam-492	219	72	.	.	PUNCT
ejpam-492	220	1	(	(	PUNCT
ejpam-492	220	2	ii)⇒(iii	ii)⇒(iii	X
ejpam-492	220	3	)	)	PUNCT
ejpam-492	220	4	is	be	AUX
ejpam-492	220	5	obvious	obvious	ADJ
ejpam-492	220	6	.	.	PUNCT
ejpam-492	221	1	(	(	PUNCT
ejpam-492	221	2	iii)⇒(iv	iii)⇒(iv	NOUN
ejpam-492	221	3	)	)	PUNCT
ejpam-492	221	4	.	.	PUNCT
ejpam-492	222	1	suppose	suppose	VERB
ejpam-492	222	2	(	(	PUNCT
ejpam-492	222	3	iii	iii	NOUN
ejpam-492	222	4	)	)	PUNCT
ejpam-492	222	5	holds	hold	VERB
ejpam-492	222	6	.	.	PUNCT
ejpam-492	223	1	by	by	ADP
ejpam-492	223	2	(	(	PUNCT
ejpam-492	223	3	iii	iii	NOUN
ejpam-492	223	4	)	)	PUNCT
ejpam-492	223	5	and	and	CCONJ
ejpam-492	223	6	lemma	lemma	PROPN
ejpam-492	223	7	1	1	NUM
ejpam-492	223	8	,	,	PUNCT
ejpam-492	223	9	m	m	VERB
ejpam-492	223	10	is	be	AUX
ejpam-492	223	11	locally	locally	ADV
ejpam-492	223	12	cyclic	cyclic	ADJ
ejpam-492	223	13	.	.	PUNCT
ejpam-492	224	1	so	so	ADV
ejpam-492	224	2	by	by	ADP
ejpam-492	224	3	[	[	X
ejpam-492	224	4	14	14	NUM
ejpam-492	224	5	,	,	PUNCT
ejpam-492	224	6	lemma	lemma	PROPN
ejpam-492	224	7	7	7	NUM
ejpam-492	224	8	]	]	PUNCT
ejpam-492	224	9	,	,	PUNCT
ejpam-492	224	10	m	m	VERB
ejpam-492	224	11	is	be	AUX
ejpam-492	224	12	a	a	DET
ejpam-492	224	13	finitely	finitely	ADV
ejpam-492	224	14	generated	generate	VERB
ejpam-492	224	15	multiplication	multiplication	NOUN
ejpam-492	224	16	module	module	NOUN
ejpam-492	224	17	.	.	PUNCT
ejpam-492	225	1	let	let	VERB
ejpam-492	225	2	n	n	PRON
ejpam-492	225	3	be	be	AUX
ejpam-492	225	4	a	a	DET
ejpam-492	225	5	prime	prime	ADJ
ejpam-492	225	6	submodule	submodule	NOUN
ejpam-492	225	7	of	of	ADP
ejpam-492	225	8	rank	rank	NOUN
ejpam-492	225	9	less	less	ADJ
ejpam-492	225	10	than	than	ADP
ejpam-492	225	11	or	or	CCONJ
ejpam-492	225	12	equal	equal	ADJ
ejpam-492	225	13	to	to	ADP
ejpam-492	225	14	one	one	NUM
ejpam-492	225	15	.	.	PUNCT
ejpam-492	226	1	as	as	SCONJ
ejpam-492	226	2	m	m	PROPN
ejpam-492	226	3	is	be	AUX
ejpam-492	226	4	a	a	DET
ejpam-492	226	5	faithful	faithful	ADJ
ejpam-492	226	6	and	and	CCONJ
ejpam-492	226	7	finitely	finitely	ADV
ejpam-492	226	8	generated	generate	VERB
ejpam-492	226	9	multiplication	multiplication	NOUN
ejpam-492	226	10	module	module	NOUN
ejpam-492	226	11	,	,	PUNCT
ejpam-492	226	12	n	n	NOUN
ejpam-492	226	13	=	=	NOUN
ejpam-492	226	14	pm	pm	NOUN
ejpam-492	226	15	for	for	ADP
ejpam-492	226	16	some	some	DET
ejpam-492	226	17	prime	prime	ADJ
ejpam-492	226	18	ideal	ideal	NOUN
ejpam-492	226	19	p	p	PROPN
ejpam-492	226	20	of	of	ADP
ejpam-492	226	21	rank	rank	NOUN
ejpam-492	226	22	less	less	ADJ
ejpam-492	226	23	than	than	ADP
ejpam-492	226	24	or	or	CCONJ
ejpam-492	226	25	equal	equal	ADJ
ejpam-492	226	26	to	to	ADP
ejpam-492	226	27	one	one	NUM
ejpam-492	226	28	.	.	PUNCT
ejpam-492	227	1	by	by	ADP
ejpam-492	227	2	the	the	DET
ejpam-492	227	3	proof	proof	NOUN
ejpam-492	227	4	of	of	ADP
ejpam-492	227	5	(	(	PUNCT
ejpam-492	227	6	i)⇒(ii	i)⇒(ii	ADV
ejpam-492	227	7	)	)	PUNCT
ejpam-492	227	8	of	of	ADP
ejpam-492	227	9	theorem	theorem	NOUN
ejpam-492	227	10	1	1	NUM
ejpam-492	227	11	,	,	PUNCT
ejpam-492	227	12	p	p	NOUN
ejpam-492	227	13	is	be	AUX
ejpam-492	227	14	finitely	finitely	ADV
ejpam-492	227	15	generated	generate	VERB
ejpam-492	227	16	and	and	CCONJ
ejpam-492	227	17	hence	hence	ADV
ejpam-492	227	18	n	n	ADV
ejpam-492	227	19	is	be	AUX
ejpam-492	227	20	finitely	finitely	ADV
ejpam-492	227	21	generated	generate	VERB
ejpam-492	227	22	.	.	PUNCT
ejpam-492	228	1	therefore	therefore	ADV
ejpam-492	228	2	(	(	PUNCT
ejpam-492	228	3	iv	iv	X
ejpam-492	228	4	)	)	PUNCT
ejpam-492	228	5	holds	hold	NOUN
ejpam-492	228	6	.	.	PUNCT
ejpam-492	229	1	(	(	PUNCT
ejpam-492	229	2	iv)⇒(v	iv)⇒(v	NUM
ejpam-492	229	3	)	)	PUNCT
ejpam-492	229	4	.	.	PUNCT
ejpam-492	230	1	suppose	suppose	VERB
ejpam-492	230	2	(	(	PUNCT
ejpam-492	230	3	iv	iv	X
ejpam-492	230	4	)	)	PUNCT
ejpam-492	230	5	holds	hold	NOUN
ejpam-492	230	6	.	.	PUNCT
ejpam-492	231	1	by	by	ADP
ejpam-492	231	2	(	(	PUNCT
ejpam-492	231	3	iv	iv	X
ejpam-492	231	4	)	)	PUNCT
ejpam-492	231	5	,	,	PUNCT
ejpam-492	231	6	m	m	VERB
ejpam-492	231	7	is	be	AUX
ejpam-492	231	8	finitely	finitely	ADV
ejpam-492	231	9	generated	generate	VERB
ejpam-492	231	10	and	and	CCONJ
ejpam-492	231	11	locally	locally	ADV
ejpam-492	231	12	cyclic	cyclic	ADJ
ejpam-492	231	13	.	.	PUNCT
ejpam-492	232	1	so	so	ADV
ejpam-492	232	2	by	by	ADP
ejpam-492	232	3	[	[	PUNCT
ejpam-492	232	4	9	9	NUM
ejpam-492	232	5	,	,	PUNCT
ejpam-492	232	6	proposition	proposition	NOUN
ejpam-492	232	7	5	5	NUM
ejpam-492	232	8	]	]	PUNCT
ejpam-492	232	9	,	,	PUNCT
ejpam-492	232	10	m	m	VERB
ejpam-492	232	11	is	be	AUX
ejpam-492	232	12	a	a	DET
ejpam-492	232	13	finitely	finitely	ADV
ejpam-492	232	14	generated	generate	VERB
ejpam-492	232	15	multiplication	multiplication	NOUN
ejpam-492	232	16	module	module	NOUN
ejpam-492	232	17	.	.	PUNCT
ejpam-492	233	1	so	so	ADV
ejpam-492	233	2	by	by	ADP
ejpam-492	233	3	lemma	lemma	PROPN
ejpam-492	233	4	5	5	NUM
ejpam-492	233	5	,	,	PUNCT
ejpam-492	233	6	r	r	NOUN
ejpam-492	233	7	is	be	AUX
ejpam-492	233	8	an	an	DET
ejpam-492	233	9	almost	almost	ADV
ejpam-492	233	10	π	π	NOUN
ejpam-492	233	11	-	-	NOUN
ejpam-492	233	12	ring	ring	NOUN
ejpam-492	233	13	.	.	PUNCT
ejpam-492	234	1	as	as	SCONJ
ejpam-492	234	2	m	m	PROPN
ejpam-492	234	3	is	be	AUX
ejpam-492	234	4	a	a	DET
ejpam-492	234	5	faithful	faithful	ADJ
ejpam-492	234	6	and	and	CCONJ
ejpam-492	234	7	finitely	finitely	ADV
ejpam-492	234	8	generated	generate	VERB
ejpam-492	234	9	multiplication	multiplication	NOUN
ejpam-492	234	10	module	module	NOUN
ejpam-492	234	11	,	,	PUNCT
ejpam-492	234	12	by	by	ADP
ejpam-492	234	13	hypothesis	hypothesis	NOUN
ejpam-492	234	14	and	and	CCONJ
ejpam-492	234	15	[	[	X
ejpam-492	234	16	11	11	NUM
ejpam-492	234	17	,	,	PUNCT
ejpam-492	234	18	theorem	theorem	VERB
ejpam-492	234	19	3.1	3.1	NUM
ejpam-492	234	20	]	]	PUNCT
ejpam-492	234	21	,	,	PUNCT
ejpam-492	234	22	every	every	DET
ejpam-492	234	23	prime	prime	ADJ
ejpam-492	234	24	ideal	ideal	NOUN
ejpam-492	234	25	of	of	ADP
ejpam-492	234	26	rank	rank	NOUN
ejpam-492	234	27	less	less	ADJ
ejpam-492	234	28	than	than	ADP
ejpam-492	234	29	or	or	CCONJ
ejpam-492	234	30	equal	equal	ADJ
ejpam-492	234	31	to	to	ADP
ejpam-492	234	32	one	one	NUM
ejpam-492	234	33	is	be	AUX
ejpam-492	234	34	finitely	finitely	ADV
ejpam-492	234	35	generated	generate	VERB
ejpam-492	234	36	.	.	PUNCT
ejpam-492	235	1	so	so	ADV
ejpam-492	235	2	by	by	ADP
ejpam-492	235	3	[	[	PUNCT
ejpam-492	235	4	13	13	NUM
ejpam-492	235	5	,	,	PUNCT
ejpam-492	235	6	theorem	theorem	VERB
ejpam-492	235	7	6	6	NUM
ejpam-492	235	8	]	]	PUNCT
ejpam-492	235	9	,	,	PUNCT
ejpam-492	235	10	the	the	DET
ejpam-492	235	11	minimal	minimal	ADJ
ejpam-492	235	12	prime	prime	ADJ
ejpam-492	235	13	ideals	ideal	NOUN
ejpam-492	235	14	of	of	ADP
ejpam-492	235	15	r	r	NOUN
ejpam-492	235	16	are	be	AUX
ejpam-492	235	17	quasi	quasi	ADJ
ejpam-492	235	18	-	-	ADJ
ejpam-492	235	19	principal	principal	ADJ
ejpam-492	235	20	ideals	ideal	NOUN
ejpam-492	235	21	and	and	CCONJ
ejpam-492	235	22	every	every	DET
ejpam-492	235	23	non	non	NOUN
ejpam-492	235	24	minimal	minimal	ADJ
ejpam-492	235	25	prime	prime	ADJ
ejpam-492	235	26	ideal	ideal	NOUN
ejpam-492	235	27	contains	contain	VERB
ejpam-492	235	28	a	a	DET
ejpam-492	235	29	non	non	ADJ
ejpam-492	235	30	minimal	minimal	ADJ
ejpam-492	235	31	quasi	quasi	ADJ
ejpam-492	235	32	-	-	ADJ
ejpam-492	235	33	principal	principal	ADJ
ejpam-492	235	34	prime	prime	ADJ
ejpam-492	235	35	ideal	ideal	NOUN
ejpam-492	235	36	.	.	PUNCT
ejpam-492	236	1	now	now	ADV
ejpam-492	236	2	the	the	DET
ejpam-492	236	3	result	result	NOUN
ejpam-492	236	4	follows	follow	VERB
ejpam-492	236	5	from	from	ADP
ejpam-492	236	6	[	[	X
ejpam-492	236	7	18	18	NUM
ejpam-492	236	8	,	,	PUNCT
ejpam-492	236	9	lemma	lemma	PROPN
ejpam-492	236	10	1.4	1.4	NUM
ejpam-492	236	11	]	]	PUNCT
ejpam-492	236	12	and	and	CCONJ
ejpam-492	236	13	[	[	X
ejpam-492	236	14	11	11	NUM
ejpam-492	236	15	,	,	PUNCT
ejpam-492	236	16	theorem	theorem	VERB
ejpam-492	236	17	3.1	3.1	NUM
ejpam-492	236	18	]	]	PUNCT
ejpam-492	236	19	.	.	PUNCT
ejpam-492	237	1	therefore	therefore	ADV
ejpam-492	237	2	(	(	PUNCT
ejpam-492	237	3	v	v	NOUN
ejpam-492	237	4	)	)	PUNCT
ejpam-492	237	5	holds	hold	VERB
ejpam-492	237	6	.	.	PUNCT
ejpam-492	238	1	(	(	PUNCT
ejpam-492	238	2	v)⇒(i	v)⇒(i	NOUN
ejpam-492	238	3	)	)	PUNCT
ejpam-492	238	4	.	.	PUNCT
ejpam-492	239	1	suppose	suppose	VERB
ejpam-492	239	2	(	(	PUNCT
ejpam-492	239	3	v	v	NOUN
ejpam-492	239	4	)	)	PUNCT
ejpam-492	239	5	holds	hold	NOUN
ejpam-492	239	6	.	.	PUNCT
ejpam-492	240	1	by	by	ADP
ejpam-492	240	2	hypothesis	hypothesis	NOUN
ejpam-492	240	3	,	,	PUNCT
ejpam-492	240	4	the	the	DET
ejpam-492	240	5	minimal	minimal	ADJ
ejpam-492	240	6	prime	prime	ADJ
ejpam-492	240	7	submodules	submodule	NOUN
ejpam-492	240	8	are	be	AUX
ejpam-492	240	9	finitely	finitely	ADV
ejpam-492	240	10	generated	generate	VERB
ejpam-492	240	11	,	,	PUNCT
ejpam-492	240	12	so	so	SCONJ
ejpam-492	240	13	m	m	VERB
ejpam-492	240	14	contains	contain	VERB
ejpam-492	240	15	only	only	ADV
ejpam-492	240	16	finitely	finitely	ADV
ejpam-492	240	17	many	many	ADJ
ejpam-492	240	18	minimal	minimal	ADJ
ejpam-492	240	19	prime	prime	ADJ
ejpam-492	240	20	submodules	submodule	NOUN
ejpam-492	240	21	and	and	CCONJ
ejpam-492	240	22	hence	hence	ADV
ejpam-492	240	23	m	m	VERB
ejpam-492	240	24	is	be	AUX
ejpam-492	240	25	finitely	finitely	ADV
ejpam-492	240	26	generated	generate	VERB
ejpam-492	240	27	(	(	PUNCT
ejpam-492	240	28	for	for	ADP
ejpam-492	240	29	details	detail	NOUN
ejpam-492	240	30	,	,	PUNCT
ejpam-492	240	31	see	see	VERB
ejpam-492	240	32	[	[	X
ejpam-492	240	33	10	10	NUM
ejpam-492	240	34	,	,	PUNCT
ejpam-492	240	35	theorem	theorem	VERB
ejpam-492	240	36	2	2	NUM
ejpam-492	240	37	]	]	PUNCT
ejpam-492	240	38	)	)	PUNCT
ejpam-492	240	39	.	.	PUNCT
ejpam-492	241	1	as	as	SCONJ
ejpam-492	241	2	m	m	PROPN
ejpam-492	241	3	is	be	AUX
ejpam-492	241	4	a	a	DET
ejpam-492	241	5	faithful	faithful	ADJ
ejpam-492	241	6	and	and	CCONJ
ejpam-492	241	7	finitely	finitely	ADV
ejpam-492	241	8	generated	generate	VERB
ejpam-492	241	9	multiplication	multiplication	NOUN
ejpam-492	241	10	module	module	NOUN
ejpam-492	241	11	,	,	PUNCT
ejpam-492	241	12	by	by	ADP
ejpam-492	241	13	(	(	PUNCT
ejpam-492	241	14	v	v	NOUN
ejpam-492	241	15	)	)	PUNCT
ejpam-492	241	16	,	,	PUNCT
ejpam-492	241	17	[	[	X
ejpam-492	241	18	11	11	NUM
ejpam-492	241	19	,	,	PUNCT
ejpam-492	241	20	theorem	theorem	VERB
ejpam-492	241	21	3.1	3.1	NUM
ejpam-492	241	22	]	]	PUNCT
ejpam-492	241	23	and	and	CCONJ
ejpam-492	241	24	[	[	X
ejpam-492	241	25	18	18	NUM
ejpam-492	241	26	,	,	PUNCT
ejpam-492	241	27	lemma	lemma	PROPN
ejpam-492	241	28	1.4	1.4	NUM
ejpam-492	241	29	]	]	PUNCT
ejpam-492	241	30	,	,	PUNCT
ejpam-492	241	31	the	the	DET
ejpam-492	241	32	minimal	minimal	ADJ
ejpam-492	241	33	prime	prime	ADJ
ejpam-492	241	34	ideals	ideal	NOUN
ejpam-492	241	35	of	of	ADP
ejpam-492	241	36	r	r	NOUN
ejpam-492	241	37	are	be	AUX
ejpam-492	241	38	quasi	quasi	ADJ
ejpam-492	241	39	-	-	NOUN
ejpam-492	241	40	principal	principal	ADJ
ejpam-492	241	41	and	and	CCONJ
ejpam-492	241	42	every	every	DET
ejpam-492	241	43	non	non	NOUN
ejpam-492	241	44	minimal	minimal	ADJ
ejpam-492	241	45	prime	prime	ADJ
ejpam-492	241	46	ideal	ideal	NOUN
ejpam-492	241	47	contains	contain	VERB
ejpam-492	241	48	a	a	DET
ejpam-492	241	49	non	non	ADJ
ejpam-492	241	50	minimal	minimal	ADJ
ejpam-492	241	51	quasi	quasi	ADJ
ejpam-492	241	52	-	-	ADJ
ejpam-492	241	53	principal	principal	ADJ
ejpam-492	241	54	prime	prime	ADJ
ejpam-492	241	55	ideal	ideal	NOUN
ejpam-492	241	56	.	.	PUNCT
ejpam-492	242	1	now	now	ADV
ejpam-492	242	2	the	the	DET
ejpam-492	242	3	result	result	NOUN
ejpam-492	242	4	follows	follow	VERB
ejpam-492	242	5	from	from	ADP
ejpam-492	242	6	theorem	theorem	ADJ
ejpam-492	242	7	1	1	NUM
ejpam-492	242	8	and	and	CCONJ
ejpam-492	242	9	[	[	X
ejpam-492	242	10	13	13	NUM
ejpam-492	242	11	,	,	PUNCT
ejpam-492	242	12	theorem	theorem	VERB
ejpam-492	242	13	6	6	NUM
ejpam-492	242	14	]	]	PUNCT
ejpam-492	242	15	.	.	PUNCT
ejpam-492	243	1	this	this	PRON
ejpam-492	243	2	completes	complete	VERB
ejpam-492	243	3	the	the	DET
ejpam-492	243	4	proof	proof	NOUN
ejpam-492	243	5	of	of	ADP
ejpam-492	243	6	the	the	DET
ejpam-492	243	7	theorem	theorem	PROPN
ejpam-492	243	8	.	.	PROPN
ejpam-492	243	9	references	reference	NOUN
ejpam-492	243	10	518	518	NUM
ejpam-492	243	11	acknowledgements	acknowledgement	NOUN
ejpam-492	243	12	the	the	DET
ejpam-492	243	13	author	author	NOUN
ejpam-492	243	14	wishes	wish	VERB
ejpam-492	243	15	to	to	PART
ejpam-492	243	16	thank	thank	VERB
ejpam-492	243	17	the	the	DET
ejpam-492	243	18	referee	referee	NOUN
ejpam-492	243	19	for	for	ADP
ejpam-492	243	20	his	his	PRON
ejpam-492	243	21	helpful	helpful	ADJ
ejpam-492	243	22	comments	comment	NOUN
ejpam-492	243	23	and	and	CCONJ
ejpam-492	243	24	suggestions	suggestion	NOUN
ejpam-492	243	25	.	.	PUNCT
ejpam-492	244	1	references	reference	NOUN
ejpam-492	244	2	[	[	X
ejpam-492	244	3	1	1	NUM
ejpam-492	244	4	]	]	X
ejpam-492	244	5	d.d	d.d	PROPN
ejpam-492	244	6	.	.	PROPN
ejpam-492	244	7	anderson	anderson	PROPN
ejpam-492	244	8	,	,	PUNCT
ejpam-492	244	9	abstract	abstract	PROPN
ejpam-492	244	10	commutative	commutative	ADJ
ejpam-492	244	11	ideal	ideal	ADJ
ejpam-492	244	12	theory	theory	NOUN
ejpam-492	244	13	without	without	ADP
ejpam-492	244	14	chain	chain	NOUN
ejpam-492	244	15	condition	condition	NOUN
ejpam-492	244	16	,	,	PUNCT
ejpam-492	244	17	algebra	algebra	NOUN
ejpam-492	244	18	universalis	universali	VERB
ejpam-492	244	19	,	,	PUNCT
ejpam-492	244	20	6	6	NUM
ejpam-492	244	21	:	:	SYM
ejpam-492	244	22	131	131	NUM
ejpam-492	244	23	-	-	SYM
ejpam-492	244	24	145	145	NUM
ejpam-492	244	25	(	(	PUNCT
ejpam-492	244	26	1976	1976	NUM
ejpam-492	244	27	)	)	PUNCT
ejpam-492	244	28	.	.	PUNCT
ejpam-492	245	1	[	[	X
ejpam-492	245	2	2	2	NUM
ejpam-492	245	3	]	]	X
ejpam-492	245	4	d.d	d.d	PROPN
ejpam-492	245	5	.	.	PROPN
ejpam-492	245	6	anderson	anderson	PROPN
ejpam-492	245	7	,	,	PUNCT
ejpam-492	245	8	multiplication	multiplication	NOUN
ejpam-492	245	9	ideals	ideal	NOUN
ejpam-492	245	10	,	,	PUNCT
ejpam-492	245	11	multiplication	multiplication	NOUN
ejpam-492	245	12	rings	ring	NOUN
ejpam-492	245	13	,	,	PUNCT
ejpam-492	245	14	and	and	CCONJ
ejpam-492	245	15	the	the	DET
ejpam-492	245	16	ring	ring	NOUN
ejpam-492	245	17	r(x	r(x	PROPN
ejpam-492	245	18	)	)	PUNCT
ejpam-492	245	19	,	,	PUNCT
ejpam-492	245	20	canad	canad	PROPN
ejpam-492	245	21	.	.	PUNCT
ejpam-492	246	1	jour	jour	PROPN
ejpam-492	246	2	.	.	PROPN
ejpam-492	246	3	of	of	ADP
ejpam-492	246	4	mathematics	mathematic	NOUN
ejpam-492	246	5	.	.	PUNCT
ejpam-492	247	1	28	28	NUM
ejpam-492	247	2	:	:	PUNCT
ejpam-492	247	3	760	760	NUM
ejpam-492	247	4	-	-	SYM
ejpam-492	247	5	768	768	NUM
ejpam-492	247	6	(	(	PUNCT
ejpam-492	247	7	1976	1976	NUM
ejpam-492	247	8	)	)	PUNCT
ejpam-492	247	9	.	.	PUNCT
ejpam-492	248	1	[	[	X
ejpam-492	248	2	3	3	X
ejpam-492	248	3	]	]	X
ejpam-492	248	4	d.d	d.d	PROPN
ejpam-492	248	5	.	.	PROPN
ejpam-492	248	6	anderson	anderson	PROPN
ejpam-492	248	7	,	,	PUNCT
ejpam-492	248	8	j.	j.	PROPN
ejpam-492	248	9	matijevic	matijevic	PROPN
ejpam-492	248	10	and	and	CCONJ
ejpam-492	248	11	w.	w.	PROPN
ejpam-492	248	12	nichols	nichols	PROPN
ejpam-492	248	13	,	,	PUNCT
ejpam-492	248	14	the	the	DET
ejpam-492	248	15	krull	krull	PROPN
ejpam-492	248	16	intersection	intersection	PROPN
ejpam-492	248	17	theorem	theorem	PROPN
ejpam-492	248	18	ii	ii	PROPN
ejpam-492	248	19	,	,	PUNCT
ejpam-492	248	20	pacific	pacific	PROPN
ejpam-492	248	21	journal	journal	PROPN
ejpam-492	248	22	of	of	ADP
ejpam-492	248	23	mathematics	mathematic	NOUN
ejpam-492	248	24	,	,	PUNCT
ejpam-492	248	25	66	66	NUM
ejpam-492	248	26	:	:	SYM
ejpam-492	248	27	15	15	NUM
ejpam-492	248	28	-	-	SYM
ejpam-492	248	29	22	22	NUM
ejpam-492	248	30	(	(	PUNCT
ejpam-492	248	31	1976	1976	NUM
ejpam-492	248	32	)	)	PUNCT
ejpam-492	248	33	.	.	PUNCT
ejpam-492	249	1	[	[	X
ejpam-492	249	2	4	4	NUM
ejpam-492	249	3	]	]	X
ejpam-492	249	4	d.d	d.d	PROPN
ejpam-492	249	5	.	.	PROPN
ejpam-492	249	6	anderson	anderson	PROPN
ejpam-492	249	7	,	,	PUNCT
ejpam-492	249	8	multiplicative	multiplicative	ADJ
ejpam-492	249	9	lattices	lattice	NOUN
ejpam-492	249	10	in	in	ADP
ejpam-492	249	11	which	which	PRON
ejpam-492	249	12	every	every	DET
ejpam-492	249	13	principal	principal	ADJ
ejpam-492	249	14	element	element	NOUN
ejpam-492	249	15	is	be	AUX
ejpam-492	249	16	a	a	DET
ejpam-492	249	17	product	product	NOUN
ejpam-492	249	18	of	of	ADP
ejpam-492	249	19	prime	prime	ADJ
ejpam-492	249	20	elements	element	NOUN
ejpam-492	249	21	,	,	PUNCT
ejpam-492	249	22	algebra	algebra	NOUN
ejpam-492	249	23	universalis	universali	VERB
ejpam-492	249	24	,	,	PUNCT
ejpam-492	249	25	8	8	NUM
ejpam-492	249	26	:	:	SYM
ejpam-492	249	27	330	330	NUM
ejpam-492	249	28	-	-	SYM
ejpam-492	249	29	335	335	NUM
ejpam-492	249	30	(	(	PUNCT
ejpam-492	249	31	1978	1978	NUM
ejpam-492	249	32	)	)	PUNCT
ejpam-492	249	33	.	.	PUNCT
ejpam-492	250	1	[	[	X
ejpam-492	250	2	5	5	NUM
ejpam-492	250	3	]	]	X
ejpam-492	250	4	d.d	d.d	PROPN
ejpam-492	250	5	.	.	PROPN
ejpam-492	250	6	anderson	anderson	PROPN
ejpam-492	250	7	,	,	PUNCT
ejpam-492	250	8	some	some	DET
ejpam-492	250	9	remarks	remark	NOUN
ejpam-492	250	10	on	on	ADP
ejpam-492	250	11	multiplication	multiplication	NOUN
ejpam-492	250	12	ideals	ideal	NOUN
ejpam-492	250	13	,	,	PUNCT
ejpam-492	250	14	math	math	NOUN
ejpam-492	250	15	.	.	PUNCT
ejpam-492	251	1	japon	japon	PROPN
ejpam-492	251	2	.	.	PUNCT
ejpam-492	252	1	25	25	NUM
ejpam-492	252	2	:	:	PUNCT
ejpam-492	252	3	463	463	NUM
ejpam-492	252	4	-	-	SYM
ejpam-492	252	5	469	469	NUM
ejpam-492	252	6	(	(	PUNCT
ejpam-492	252	7	1980	1980	NUM
ejpam-492	252	8	)	)	PUNCT
ejpam-492	252	9	.	.	PUNCT
ejpam-492	253	1	[	[	X
ejpam-492	253	2	6	6	NUM
ejpam-492	253	3	]	]	X
ejpam-492	253	4	d.d	d.d	PROPN
ejpam-492	253	5	.	.	PROPN
ejpam-492	253	6	anderson	anderson	PROPN
ejpam-492	253	7	,	,	PUNCT
ejpam-492	253	8	noetherian	noetherian	ADJ
ejpam-492	253	9	rings	ring	NOUN
ejpam-492	253	10	in	in	ADP
ejpam-492	253	11	which	which	PRON
ejpam-492	253	12	every	every	DET
ejpam-492	253	13	ideal	ideal	NOUN
ejpam-492	253	14	is	be	AUX
ejpam-492	253	15	a	a	DET
ejpam-492	253	16	product	product	NOUN
ejpam-492	253	17	of	of	ADP
ejpam-492	253	18	primary	primary	ADJ
ejpam-492	253	19	ideals	ideal	NOUN
ejpam-492	253	20	,	,	PUNCT
ejpam-492	253	21	canad	canad	PROPN
ejpam-492	253	22	.	.	PUNCT
ejpam-492	254	1	math	math	NOUN
ejpam-492	254	2	.	.	PUNCT
ejpam-492	255	1	bull	bull	NOUN
ejpam-492	255	2	.	.	PUNCT
ejpam-492	256	1	23(4	23(4	NUM
ejpam-492	256	2	):	):	PUNCT
ejpam-492	256	3	457	457	NUM
ejpam-492	256	4	-	-	SYM
ejpam-492	256	5	459	459	NUM
ejpam-492	256	6	(	(	PUNCT
ejpam-492	256	7	1980	1980	NUM
ejpam-492	256	8	)	)	PUNCT
ejpam-492	256	9	.	.	PUNCT
ejpam-492	257	1	[	[	X
ejpam-492	257	2	7	7	X
ejpam-492	257	3	]	]	X
ejpam-492	257	4	d.d	d.d	PROPN
ejpam-492	257	5	.	.	PROPN
ejpam-492	257	6	anderson	anderson	PROPN
ejpam-492	257	7	and	and	CCONJ
ejpam-492	257	8	l.a	l.a	PROPN
ejpam-492	257	9	.	.	PROPN
ejpam-492	257	10	mahaney	mahaney	PROPN
ejpam-492	257	11	,	,	PUNCT
ejpam-492	257	12	on	on	ADP
ejpam-492	257	13	primary	primary	ADJ
ejpam-492	257	14	factorizations	factorization	NOUN
ejpam-492	257	15	,	,	PUNCT
ejpam-492	257	16	journal	journal	NOUN
ejpam-492	257	17	of	of	ADP
ejpam-492	257	18	pure	pure	ADJ
ejpam-492	257	19	and	and	CCONJ
ejpam-492	257	20	applied	applied	ADJ
ejpam-492	257	21	algebra	algebra	NOUN
ejpam-492	257	22	,	,	PUNCT
ejpam-492	257	23	54	54	NUM
ejpam-492	257	24	:	:	SYM
ejpam-492	257	25	141	141	NUM
ejpam-492	257	26	-	-	SYM
ejpam-492	257	27	154	154	NUM
ejpam-492	257	28	(	(	PUNCT
ejpam-492	257	29	1988	1988	NUM
ejpam-492	257	30	)	)	PUNCT
ejpam-492	257	31	.	.	PUNCT
ejpam-492	258	1	[	[	X
ejpam-492	258	2	8	8	NUM
ejpam-492	258	3	]	]	X
ejpam-492	258	4	d.d	d.d	PROPN
ejpam-492	258	5	.	.	PROPN
ejpam-492	258	6	anderson	anderson	PROPN
ejpam-492	258	7	and	and	CCONJ
ejpam-492	258	8	e.w	e.w	PROPN
ejpam-492	258	9	.	.	PROPN
ejpam-492	258	10	johnson	johnson	PROPN
ejpam-492	258	11	,	,	PUNCT
ejpam-492	258	12	dilworth	dilworth	PROPN
ejpam-492	258	13	’s	’s	PART
ejpam-492	258	14	principal	principal	ADJ
ejpam-492	258	15	elements	element	NOUN
ejpam-492	258	16	,	,	PUNCT
ejpam-492	258	17	algebra	algebra	NOUN
ejpam-492	258	18	universalis	universali	VERB
ejpam-492	258	19	,	,	PUNCT
ejpam-492	258	20	36	36	NUM
ejpam-492	258	21	:	:	SYM
ejpam-492	258	22	99	99	NUM
ejpam-492	258	23	-	-	SYM
ejpam-492	258	24	109	109	NUM
ejpam-492	258	25	(	(	PUNCT
ejpam-492	258	26	1996	1996	NUM
ejpam-492	258	27	)	)	PUNCT
ejpam-492	258	28	.	.	PUNCT
ejpam-492	259	1	[	[	X
ejpam-492	259	2	9	9	NUM
ejpam-492	259	3	]	]	PUNCT
ejpam-492	259	4	a.	a.	NOUN
ejpam-492	259	5	barnard	barnard	PROPN
ejpam-492	259	6	,	,	PUNCT
ejpam-492	259	7	multiplication	multiplication	NOUN
ejpam-492	259	8	modules	module	NOUN
ejpam-492	259	9	,	,	PUNCT
ejpam-492	259	10	journal	journal	NOUN
ejpam-492	259	11	of	of	ADP
ejpam-492	259	12	algebra	algebra	PROPN
ejpam-492	259	13	,	,	PUNCT
ejpam-492	259	14	71	71	NUM
ejpam-492	259	15	:	:	SYM
ejpam-492	259	16	174	174	NUM
ejpam-492	259	17	-	-	SYM
ejpam-492	259	18	178	178	NUM
ejpam-492	259	19	(	(	PUNCT
ejpam-492	259	20	1981	1981	NUM
ejpam-492	259	21	)	)	PUNCT
ejpam-492	259	22	.	.	PUNCT
ejpam-492	260	1	[	[	X
ejpam-492	260	2	10	10	NUM
ejpam-492	260	3	]	]	PUNCT
ejpam-492	260	4	m.	m.	NOUN
ejpam-492	260	5	behboodi	behboodi	NOUN
ejpam-492	260	6	and	and	CCONJ
ejpam-492	260	7	h.	h.	PROPN
ejpam-492	260	8	koohy	koohy	PROPN
ejpam-492	260	9	,	,	PUNCT
ejpam-492	260	10	on	on	ADP
ejpam-492	260	11	minimal	minimal	ADJ
ejpam-492	260	12	prime	prime	ADJ
ejpam-492	260	13	submodules	submodule	NOUN
ejpam-492	260	14	,	,	PUNCT
ejpam-492	260	15	far	far	PROPN
ejpam-492	260	16	east	east	PROPN
ejpam-492	260	17	j.	j.	PROPN
ejpam-492	260	18	math	math	PROPN
ejpam-492	260	19	.	.	PUNCT
ejpam-492	261	1	sci	sci	PROPN
ejpam-492	261	2	.	.	PUNCT
ejpam-492	261	3	(	(	PUNCT
ejpam-492	261	4	fjms	fjms	NOUN
ejpam-492	261	5	)	)	PUNCT
ejpam-492	261	6	.	.	PUNCT
ejpam-492	262	1	6	6	NUM
ejpam-492	262	2	:	:	SYM
ejpam-492	262	3	83	83	NUM
ejpam-492	262	4	-	-	SYM
ejpam-492	262	5	88	88	NUM
ejpam-492	262	6	(	(	PUNCT
ejpam-492	262	7	2002	2002	NUM
ejpam-492	262	8	)	)	PUNCT
ejpam-492	262	9	.	.	PUNCT
ejpam-492	263	1	[	[	X
ejpam-492	263	2	11	11	NUM
ejpam-492	263	3	]	]	PUNCT
ejpam-492	263	4	z.	z.	PROPN
ejpam-492	263	5	el	el	PROPN
ejpam-492	263	6	-	-	PUNCT
ejpam-492	263	7	bast	bast	NOUN
ejpam-492	263	8	and	and	CCONJ
ejpam-492	263	9	p.f	p.f	PROPN
ejpam-492	263	10	.	.	PROPN
ejpam-492	263	11	smith	smith	PROPN
ejpam-492	263	12	,	,	PUNCT
ejpam-492	263	13	multiplication	multiplication	NOUN
ejpam-492	263	14	modules	module	NOUN
ejpam-492	263	15	,	,	PUNCT
ejpam-492	263	16	communications	communication	NOUN
ejpam-492	263	17	in	in	ADP
ejpam-492	263	18	algebra	algebra	NOUN
ejpam-492	263	19	,	,	PUNCT
ejpam-492	263	20	16(4	16(4	NUM
ejpam-492	263	21	):	):	PUNCT
ejpam-492	263	22	755	755	NUM
ejpam-492	263	23	-	-	SYM
ejpam-492	263	24	779	779	NUM
ejpam-492	263	25	(	(	PUNCT
ejpam-492	263	26	1988	1988	NUM
ejpam-492	263	27	)	)	PUNCT
ejpam-492	263	28	.	.	PUNCT
ejpam-492	264	1	[	[	X
ejpam-492	264	2	12	12	NUM
ejpam-492	264	3	]	]	X
ejpam-492	264	4	r.w	r.w	PROPN
ejpam-492	264	5	.	.	PROPN
ejpam-492	264	6	gilmer	gilmer	PROPN
ejpam-492	264	7	,	,	PUNCT
ejpam-492	264	8	multiplicative	multiplicative	PROPN
ejpam-492	264	9	ideal	ideal	PROPN
ejpam-492	264	10	theory	theory	NOUN
ejpam-492	264	11	,	,	PUNCT
ejpam-492	264	12	marcel	marcel	PROPN
ejpam-492	264	13	decker	decker	PROPN
ejpam-492	264	14	,	,	PUNCT
ejpam-492	264	15	1972	1972	NUM
ejpam-492	264	16	.	.	PUNCT
ejpam-492	265	1	[	[	X
ejpam-492	265	2	13	13	NUM
ejpam-492	265	3	]	]	X
ejpam-492	265	4	c.	c.	PROPN
ejpam-492	265	5	jayaram	jayaram	PROPN
ejpam-492	265	6	,	,	PUNCT
ejpam-492	265	7	almost	almost	ADV
ejpam-492	265	8	π	π	NOUN
ejpam-492	265	9	-	-	NOUN
ejpam-492	265	10	lattices	lattice	NOUN
ejpam-492	265	11	,	,	PUNCT
ejpam-492	265	12	czechoslovak	czechoslovak	ADJ
ejpam-492	265	13	mathematical	mathematical	ADJ
ejpam-492	265	14	journal	journal	PROPN
ejpam-492	265	15	,	,	PUNCT
ejpam-492	265	16	54(129	54(129	NUM
ejpam-492	265	17	):	):	PUNCT
ejpam-492	265	18	119	119	NUM
ejpam-492	265	19	-	-	SYM
ejpam-492	265	20	130	130	NUM
ejpam-492	265	21	(	(	PUNCT
ejpam-492	265	22	2004	2004	NUM
ejpam-492	265	23	)	)	PUNCT
ejpam-492	265	24	.	.	PUNCT
ejpam-492	266	1	references	reference	NOUN
ejpam-492	266	2	519	519	NUM
ejpam-492	266	3	[	[	X
ejpam-492	266	4	14	14	NUM
ejpam-492	266	5	]	]	X
ejpam-492	266	6	c.	c.	PROPN
ejpam-492	266	7	jayaram	jayaram	PROPN
ejpam-492	266	8	and	and	CCONJ
ejpam-492	266	9	ünsal	ünsal	PROPN
ejpam-492	266	10	tekir	tekir	NOUN
ejpam-492	266	11	,	,	PUNCT
ejpam-492	266	12	q	q	NOUN
ejpam-492	266	13	-	-	PUNCT
ejpam-492	266	14	modules	module	NOUN
ejpam-492	266	15	,	,	PUNCT
ejpam-492	266	16	turkish	turkish	ADJ
ejpam-492	266	17	journal	journal	NOUN
ejpam-492	266	18	of	of	ADP
ejpam-492	266	19	mathematics	mathematic	NOUN
ejpam-492	266	20	,	,	PUNCT
ejpam-492	266	21	33	33	NUM
ejpam-492	266	22	:	:	SYM
ejpam-492	266	23	215	215	NUM
ejpam-492	266	24	-	-	SYM
ejpam-492	266	25	225	225	NUM
ejpam-492	266	26	(	(	PUNCT
ejpam-492	266	27	2009	2009	NUM
ejpam-492	266	28	)	)	PUNCT
ejpam-492	266	29	.	.	PUNCT
ejpam-492	267	1	[	[	X
ejpam-492	267	2	15	15	NUM
ejpam-492	267	3	]	]	X
ejpam-492	267	4	b.g	b.g	PROPN
ejpam-492	267	5	.	.	PROPN
ejpam-492	267	6	kang	kang	PROPN
ejpam-492	267	7	,	,	PUNCT
ejpam-492	267	8	on	on	ADP
ejpam-492	267	9	the	the	DET
ejpam-492	267	10	converse	converse	NOUN
ejpam-492	267	11	of	of	ADP
ejpam-492	267	12	a	a	DET
ejpam-492	267	13	well	well	ADV
ejpam-492	267	14	-	-	PUNCT
ejpam-492	267	15	known	know	VERB
ejpam-492	267	16	fact	fact	NOUN
ejpam-492	267	17	about	about	ADP
ejpam-492	267	18	krull	krull	PROPN
ejpam-492	267	19	domains	domain	NOUN
ejpam-492	267	20	,	,	PUNCT
ejpam-492	267	21	journal	journal	NOUN
ejpam-492	267	22	of	of	ADP
ejpam-492	267	23	algebra	algebra	PROPN
ejpam-492	267	24	,	,	PUNCT
ejpam-492	267	25	124	124	NUM
ejpam-492	267	26	:	:	SYM
ejpam-492	267	27	284	284	NUM
ejpam-492	267	28	-	-	SYM
ejpam-492	267	29	299	299	NUM
ejpam-492	267	30	(	(	PUNCT
ejpam-492	267	31	1989	1989	NUM
ejpam-492	267	32	)	)	PUNCT
ejpam-492	267	33	.	.	PUNCT
ejpam-492	268	1	[	[	X
ejpam-492	268	2	16	16	NUM
ejpam-492	268	3	]	]	X
ejpam-492	268	4	m.d	m.d	PROPN
ejpam-492	268	5	.	.	PROPN
ejpam-492	268	6	larsen	larsen	PROPN
ejpam-492	268	7	and	and	CCONJ
ejpam-492	268	8	p.j	p.j	PROPN
ejpam-492	268	9	.	.	PROPN
ejpam-492	268	10	mccarthy	mccarthy	PROPN
ejpam-492	268	11	,	,	PUNCT
ejpam-492	268	12	multiplicative	multiplicative	ADJ
ejpam-492	268	13	theory	theory	NOUN
ejpam-492	268	14	of	of	ADP
ejpam-492	268	15	ideals	ideal	NOUN
ejpam-492	268	16	,	,	PUNCT
ejpam-492	268	17	academic	academic	ADJ
ejpam-492	268	18	press	press	NOUN
ejpam-492	268	19	,	,	PUNCT
ejpam-492	268	20	new	new	PROPN
ejpam-492	268	21	york	york	PROPN
ejpam-492	268	22	,	,	PUNCT
ejpam-492	268	23	1971	1971	NUM
ejpam-492	268	24	.	.	PUNCT
ejpam-492	269	1	[	[	X
ejpam-492	269	2	17	17	NUM
ejpam-492	269	3	]	]	X
ejpam-492	269	4	k.b	k.b	PROPN
ejpam-492	269	5	.	.	PROPN
ejpam-492	269	6	levitz	levitz	PROPN
ejpam-492	269	7	,	,	PUNCT
ejpam-492	269	8	a	a	DET
ejpam-492	269	9	characterization	characterization	NOUN
ejpam-492	269	10	of	of	ADP
ejpam-492	269	11	general	general	ADJ
ejpam-492	269	12	zpi	zpi	NOUN
ejpam-492	269	13	-	-	PUNCT
ejpam-492	269	14	rings	ring	NOUN
ejpam-492	269	15	,	,	PUNCT
ejpam-492	269	16	proc	proc	NOUN
ejpam-492	269	17	.	.	PUNCT
ejpam-492	270	1	amer	amer	PROPN
ejpam-492	270	2	.	.	PUNCT
ejpam-492	270	3	math	math	PROPN
ejpam-492	270	4	.	.	PUNCT
ejpam-492	271	1	soc	soc	PROPN
ejpam-492	271	2	.	.	PUNCT
ejpam-492	272	1	32	32	NUM
ejpam-492	272	2	:	:	PUNCT
ejpam-492	272	3	376	376	NUM
ejpam-492	272	4	-	-	SYM
ejpam-492	272	5	380	380	NUM
ejpam-492	272	6	(	(	PUNCT
ejpam-492	272	7	1972	1972	NUM
ejpam-492	272	8	)	)	PUNCT
ejpam-492	272	9	.	.	PUNCT
ejpam-492	273	1	[	[	X
ejpam-492	273	2	18	18	NUM
ejpam-492	273	3	]	]	X
ejpam-492	273	4	g.m	g.m	PROPN
ejpam-492	273	5	.	.	PROPN
ejpam-492	273	6	low	low	PROPN
ejpam-492	273	7	and	and	CCONJ
ejpam-492	273	8	p.f	p.f	PROPN
ejpam-492	273	9	.	.	PROPN
ejpam-492	273	10	smith	smith	PROPN
ejpam-492	273	11	,	,	PUNCT
ejpam-492	273	12	multiplication	multiplication	NOUN
ejpam-492	273	13	modules	module	NOUN
ejpam-492	273	14	and	and	CCONJ
ejpam-492	273	15	ideals	ideal	NOUN
ejpam-492	273	16	,	,	PUNCT
ejpam-492	273	17	communications	communication	NOUN
ejpam-492	273	18	in	in	ADP
ejpam-492	273	19	algebra	algebra	NOUN
ejpam-492	273	20	,	,	PUNCT
ejpam-492	273	21	18(12	18(12	NUM
ejpam-492	273	22	):	):	PUNCT
ejpam-492	273	23	4353	4353	NUM
ejpam-492	273	24	-	-	SYM
ejpam-492	273	25	4375	4375	NUM
ejpam-492	273	26	(	(	PUNCT
ejpam-492	273	27	1990	1990	NUM
ejpam-492	273	28	)	)	PUNCT
ejpam-492	273	29	.	.	PUNCT
ejpam-492	274	1	[	[	X
ejpam-492	274	2	19	19	NUM
ejpam-492	274	3	]	]	X
ejpam-492	274	4	p.j	p.j	PROPN
ejpam-492	274	5	.	.	PROPN
ejpam-492	274	6	mccarthy	mccarthy	PROPN
ejpam-492	274	7	,	,	PUNCT
ejpam-492	274	8	principal	principal	ADJ
ejpam-492	274	9	elements	element	NOUN
ejpam-492	274	10	of	of	ADP
ejpam-492	274	11	lattices	lattice	NOUN
ejpam-492	274	12	of	of	ADP
ejpam-492	274	13	ideals	ideal	NOUN
ejpam-492	274	14	,	,	PUNCT
ejpam-492	274	15	proc	proc	NOUN
ejpam-492	274	16	.	.	PUNCT
ejpam-492	275	1	amer	amer	PROPN
ejpam-492	275	2	.	.	PUNCT
ejpam-492	275	3	math	math	PROPN
ejpam-492	275	4	.	.	PUNCT
ejpam-492	276	1	soc	soc	PROPN
ejpam-492	276	2	.	.	PUNCT
ejpam-492	277	1	30	30	NUM
ejpam-492	277	2	:	:	PUNCT
ejpam-492	277	3	43	43	NUM
ejpam-492	277	4	-	-	SYM
ejpam-492	277	5	45	45	NUM
ejpam-492	277	6	(	(	PUNCT
ejpam-492	277	7	1971	1971	NUM
ejpam-492	277	8	)	)	PUNCT
ejpam-492	277	9	.	.	PUNCT
ejpam-492	278	1	[	[	X
ejpam-492	278	2	20	20	NUM
ejpam-492	278	3	]	]	X
ejpam-492	278	4	d.g	d.g	PROPN
ejpam-492	278	5	.	.	PROPN
ejpam-492	278	6	northcott	northcott	PROPN
ejpam-492	278	7	,	,	PUNCT
ejpam-492	278	8	lessons	lesson	NOUN
ejpam-492	278	9	on	on	ADP
ejpam-492	278	10	rings	ring	NOUN
ejpam-492	278	11	,	,	PUNCT
ejpam-492	278	12	modules	module	NOUN
ejpam-492	278	13	and	and	CCONJ
ejpam-492	278	14	multiplicities	multiplicity	NOUN
ejpam-492	278	15	,	,	PUNCT
ejpam-492	278	16	cambridge	cambridge	PROPN
ejpam-492	278	17	university	university	PROPN
ejpam-492	278	18	press	press	PROPN
ejpam-492	278	19	,	,	PUNCT
ejpam-492	278	20	london	london	PROPN
ejpam-492	278	21	,	,	PUNCT
ejpam-492	278	22	1968	1968	NUM
ejpam-492	278	23	.	.	PUNCT
ejpam-492	279	1	[	[	X
ejpam-492	279	2	21	21	NUM
ejpam-492	279	3	]	]	X
ejpam-492	279	4	shahabaddin	shahabaddin	VERB
ejpam-492	279	5	ebrahimi	ebrahimi	PROPN
ejpam-492	279	6	atani	atani	PROPN
ejpam-492	279	7	,	,	PUNCT
ejpam-492	279	8	fethi	fethi	ADJ
ejpam-492	279	9	çallıalp	çallıalp	NOUN
ejpam-492	279	10	and	and	CCONJ
ejpam-492	279	11	ünsal	ünsal	PROPN
ejpam-492	279	12	tekir	tekir	NOUN
ejpam-492	279	13	,	,	PUNCT
ejpam-492	279	14	a	a	DET
ejpam-492	279	15	short	short	ADJ
ejpam-492	279	16	note	note	NOUN
ejpam-492	279	17	on	on	ADP
ejpam-492	279	18	primary	primary	ADJ
ejpam-492	279	19	submodules	submodule	NOUN
ejpam-492	279	20	of	of	ADP
ejpam-492	279	21	multiplication	multiplication	NOUN
ejpam-492	279	22	modules	module	NOUN
ejpam-492	279	23	,	,	PUNCT
ejpam-492	279	24	international	international	ADJ
ejpam-492	279	25	journal	journal	NOUN
ejpam-492	279	26	of	of	ADP
ejpam-492	279	27	algebra	algebra	PROPN
ejpam-492	279	28	,	,	PUNCT
ejpam-492	279	29	1	1	NUM
ejpam-492	279	30	:	:	SYM
ejpam-492	279	31	381	381	NUM
ejpam-492	279	32	-	-	SYM
ejpam-492	279	33	384	384	NUM
ejpam-492	279	34	(	(	PUNCT
ejpam-492	279	35	2007	2007	NUM
ejpam-492	279	36	)	)	PUNCT
ejpam-492	279	37	.	.	PUNCT