id	sid	tid	token	lemma	pos
ejpam-2534	1	1	compile	compile	NOUN
ejpam-2534	1	2	/	/	SYM
ejpam-2534	1	3	output.dvi	output.dvi	NOUN
ejpam-2534	1	4	european	european	ADJ
ejpam-2534	1	5	journal	journal	NOUN
ejpam-2534	1	6	of	of	ADP
ejpam-2534	1	7	pure	pure	ADJ
ejpam-2534	1	8	and	and	CCONJ
ejpam-2534	1	9	applied	apply	VERB
ejpam-2534	1	10	mathematics	mathematic	NOUN
ejpam-2534	1	11	vol	vol	NOUN
ejpam-2534	1	12	.	.	PROPN
ejpam-2534	2	1	9	9	NUM
ejpam-2534	2	2	,	,	PUNCT
ejpam-2534	2	3	no	no	INTJ
ejpam-2534	2	4	.	.	NOUN
ejpam-2534	2	5	1	1	NUM
ejpam-2534	2	6	,	,	PUNCT
ejpam-2534	2	7	2016	2016	NUM
ejpam-2534	2	8	,	,	PUNCT
ejpam-2534	2	9	48	48	NUM
ejpam-2534	2	10	-	-	SYM
ejpam-2534	2	11	56	56	NUM
ejpam-2534	2	12	issn	issn	PROPN
ejpam-2534	2	13	1307	1307	NUM
ejpam-2534	2	14	-	-	SYM
ejpam-2534	2	15	5543	5543	NUM
ejpam-2534	2	16	–	–	PUNCT
ejpam-2534	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2534	2	18	a	a	DET
ejpam-2534	2	19	note	note	NOUN
ejpam-2534	2	20	on	on	ADP
ejpam-2534	2	21	primary	primary	ADJ
ejpam-2534	2	22	and	and	CCONJ
ejpam-2534	2	23	weakly	weakly	ADJ
ejpam-2534	2	24	primary	primary	ADJ
ejpam-2534	2	25	submodules	submodule	NOUN
ejpam-2534	2	26	gulsen	gulsen	VERB
ejpam-2534	2	27	ulucak1,∗and	ulucak1,∗and	PRON
ejpam-2534	2	28	rabia	rabia	PROPN
ejpam-2534	2	29	nagehan	nagehan	ADV
ejpam-2534	2	30	uregen2	uregen2	PROPN
ejpam-2534	3	1	1	1	NUM
ejpam-2534	3	2	department	department	NOUN
ejpam-2534	3	3	of	of	ADP
ejpam-2534	3	4	mathematics	mathematic	NOUN
ejpam-2534	3	5	,	,	PUNCT
ejpam-2534	3	6	faculty	faculty	NOUN
ejpam-2534	3	7	of	of	ADP
ejpam-2534	3	8	science	science	NOUN
ejpam-2534	3	9	,	,	PUNCT
ejpam-2534	3	10	gebze	gebze	NOUN
ejpam-2534	3	11	technical	technical	PROPN
ejpam-2534	3	12	university	university	PROPN
ejpam-2534	3	13	,	,	PUNCT
ejpam-2534	3	14	141	141	NUM
ejpam-2534	3	15	41400	41400	NUM
ejpam-2534	3	16	,	,	PUNCT
ejpam-2534	3	17	kocaeli	kocaeli	VERB
ejpam-2534	3	18	,	,	PUNCT
ejpam-2534	3	19	turkey	turkey	PROPN
ejpam-2534	3	20	2	2	NUM
ejpam-2534	3	21	yildiz	yildiz	PROPN
ejpam-2534	3	22	technical	technical	PROPN
ejpam-2534	3	23	university	university	PROPN
ejpam-2534	3	24	,	,	PUNCT
ejpam-2534	3	25	graduate	graduate	NOUN
ejpam-2534	3	26	school	school	NOUN
ejpam-2534	3	27	of	of	ADP
ejpam-2534	3	28	natural	natural	ADJ
ejpam-2534	3	29	and	and	CCONJ
ejpam-2534	3	30	applied	applied	ADJ
ejpam-2534	3	31	sciences	science	NOUN
ejpam-2534	3	32	,	,	PUNCT
ejpam-2534	3	33	34349	34349	NUM
ejpam-2534	3	34	,	,	PUNCT
ejpam-2534	3	35	istanbul	istanbul	PROPN
ejpam-2534	3	36	,	,	PUNCT
ejpam-2534	3	37	turkey	turkey	PROPN
ejpam-2534	3	38	abstract	abstract	NOUN
ejpam-2534	3	39	.	.	PUNCT
ejpam-2534	4	1	in	in	ADP
ejpam-2534	4	2	this	this	DET
ejpam-2534	4	3	paper	paper	NOUN
ejpam-2534	4	4	,	,	PUNCT
ejpam-2534	4	5	the	the	DET
ejpam-2534	4	6	generalizations	generalization	NOUN
ejpam-2534	4	7	of	of	ADP
ejpam-2534	4	8	primary	primary	ADJ
ejpam-2534	4	9	submodules	submodule	NOUN
ejpam-2534	4	10	and	and	CCONJ
ejpam-2534	4	11	weakly	weakly	ADJ
ejpam-2534	4	12	primary	primary	ADJ
ejpam-2534	4	13	submodules	submodule	NOUN
ejpam-2534	4	14	are	be	AUX
ejpam-2534	4	15	proposed	propose	VERB
ejpam-2534	4	16	as	as	ADP
ejpam-2534	4	17	p(n)-locally	p(n)-locally	ADV
ejpam-2534	4	18	primary	primary	ADJ
ejpam-2534	4	19	submodules	submodule	NOUN
ejpam-2534	4	20	and	and	CCONJ
ejpam-2534	4	21	p(n)-locally	p(n)-locally	ADV
ejpam-2534	4	22	weakly	weakly	ADJ
ejpam-2534	4	23	primary	primary	ADJ
ejpam-2534	4	24	submodules	submodule	NOUN
ejpam-2534	4	25	,	,	PUNCT
ejpam-2534	4	26	respectively	respectively	ADV
ejpam-2534	4	27	.	.	PUNCT
ejpam-2534	5	1	the	the	DET
ejpam-2534	5	2	relationships	relationship	NOUN
ejpam-2534	5	3	of	of	ADP
ejpam-2534	5	4	these	these	DET
ejpam-2534	5	5	submodules	submodule	NOUN
ejpam-2534	5	6	are	be	AUX
ejpam-2534	5	7	investigated	investigate	VERB
ejpam-2534	5	8	extensively	extensively	ADV
ejpam-2534	5	9	.	.	PUNCT
ejpam-2534	6	1	2010	2010	NUM
ejpam-2534	6	2	mathematics	mathematic	NOUN
ejpam-2534	6	3	subject	subject	NOUN
ejpam-2534	6	4	classifications	classification	NOUN
ejpam-2534	6	5	:	:	PUNCT
ejpam-2534	6	6	13a15	13a15	NUM
ejpam-2534	6	7	,	,	PUNCT
ejpam-2534	6	8	13c99	13c99	NUM
ejpam-2534	6	9	,	,	PUNCT
ejpam-2534	6	10	13f05	13f05	NUM
ejpam-2534	6	11	key	key	ADJ
ejpam-2534	6	12	words	word	NOUN
ejpam-2534	6	13	and	and	CCONJ
ejpam-2534	6	14	phrases	phrase	NOUN
ejpam-2534	6	15	:	:	PUNCT
ejpam-2534	6	16	primary	primary	ADJ
ejpam-2534	6	17	,	,	PUNCT
ejpam-2534	6	18	weakly	weakly	ADJ
ejpam-2534	6	19	primary	primary	ADJ
ejpam-2534	6	20	,	,	PUNCT
ejpam-2534	6	21	p(n)-locally	p(n)-locally	ADV
ejpam-2534	6	22	primary	primary	ADJ
ejpam-2534	6	23	,	,	PUNCT
ejpam-2534	6	24	p(n)-locally	p(n)-locally	ADV
ejpam-2534	6	25	weakly	weakly	ADJ
ejpam-2534	6	26	primary	primary	ADJ
ejpam-2534	6	27	.	.	PUNCT
ejpam-2534	7	1	1	1	X
ejpam-2534	7	2	.	.	X
ejpam-2534	7	3	introduction	introduction	NOUN
ejpam-2534	7	4	throughout	throughout	ADP
ejpam-2534	7	5	this	this	DET
ejpam-2534	7	6	paper	paper	NOUN
ejpam-2534	7	7	,	,	PUNCT
ejpam-2534	7	8	we	we	PRON
ejpam-2534	7	9	assume	assume	VERB
ejpam-2534	7	10	that	that	SCONJ
ejpam-2534	7	11	all	all	DET
ejpam-2534	7	12	rings	ring	NOUN
ejpam-2534	7	13	are	be	AUX
ejpam-2534	7	14	commutative	commutative	ADJ
ejpam-2534	7	15	with	with	ADP
ejpam-2534	7	16	identity	identity	NOUN
ejpam-2534	7	17	1	1	NUM
ejpam-2534	7	18	6=	6=	ADP
ejpam-2534	7	19	0	0	NUM
ejpam-2534	7	20	.	.	PUNCT
ejpam-2534	8	1	an	an	DET
ejpam-2534	8	2	ideal	ideal	ADJ
ejpam-2534	8	3	i	i	PRON
ejpam-2534	8	4	of	of	ADP
ejpam-2534	8	5	r	r	NOUN
ejpam-2534	8	6	is	be	AUX
ejpam-2534	8	7	called	call	VERB
ejpam-2534	8	8	a	a	DET
ejpam-2534	8	9	proper	proper	ADJ
ejpam-2534	8	10	ideal	ideal	NOUN
ejpam-2534	8	11	if	if	SCONJ
ejpam-2534	8	12	i	i	PRON
ejpam-2534	8	13	6=	6=	PROPN
ejpam-2534	8	14	r.	r.	PROPN
ejpam-2534	8	15	then	then	ADV
ejpam-2534	8	16	the	the	DET
ejpam-2534	8	17	radical	radical	NOUN
ejpam-2534	8	18	of	of	ADP
ejpam-2534	8	19	a	a	DET
ejpam-2534	8	20	proper	proper	ADJ
ejpam-2534	8	21	ideal	ideal	NOUN
ejpam-2534	8	22	i	i	PRON
ejpam-2534	8	23	of	of	ADP
ejpam-2534	8	24	r	r	NOUN
ejpam-2534	8	25	is	be	AUX
ejpam-2534	8	26	denoted	denote	VERB
ejpam-2534	8	27	by	by	ADP
ejpam-2534	8	28	rad(i	rad(i	PROPN
ejpam-2534	8	29	)	)	PUNCT
ejpam-2534	8	30	and	and	CCONJ
ejpam-2534	8	31	rad(i	rad(i	NOUN
ejpam-2534	8	32	)	)	PUNCT
ejpam-2534	9	1	=	=	PRON
ejpam-2534	9	2	{	{	PUNCT
ejpam-2534	9	3	x	x	PUNCT
ejpam-2534	9	4	∈	∈	NOUN
ejpam-2534	9	5	r	r	NOUN
ejpam-2534	9	6	|	|	NOUN
ejpam-2534	9	7	xn	xn	PROPN
ejpam-2534	9	8	∈	∈	PROPN
ejpam-2534	9	9	i	i	PRON
ejpam-2534	9	10	for	for	ADP
ejpam-2534	9	11	some	some	DET
ejpam-2534	9	12	positive	positive	ADJ
ejpam-2534	9	13	integer	integer	NOUN
ejpam-2534	9	14	n	n	CCONJ
ejpam-2534	9	15	}	}	PUNCT
ejpam-2534	9	16	.	.	PUNCT
ejpam-2534	10	1	a	a	DET
ejpam-2534	10	2	proper	proper	ADJ
ejpam-2534	10	3	ideal	ideal	NOUN
ejpam-2534	10	4	p	p	NOUN
ejpam-2534	10	5	of	of	ADP
ejpam-2534	10	6	r	r	NOUN
ejpam-2534	10	7	is	be	AUX
ejpam-2534	10	8	called	call	VERB
ejpam-2534	10	9	prime	prime	ADJ
ejpam-2534	10	10	(	(	PUNCT
ejpam-2534	10	11	primary	primary	NOUN
ejpam-2534	10	12	)	)	PUNCT
ejpam-2534	10	13	if	if	SCONJ
ejpam-2534	10	14	ab	ab	PROPN
ejpam-2534	10	15	∈	∈	PROPN
ejpam-2534	10	16	p	p	PROPN
ejpam-2534	10	17	for	for	ADP
ejpam-2534	10	18	some	some	PRON
ejpam-2534	10	19	a	a	PRON
ejpam-2534	10	20	,	,	PUNCT
ejpam-2534	10	21	b	b	X
ejpam-2534	10	22	∈	∈	NOUN
ejpam-2534	10	23	r	r	NOUN
ejpam-2534	10	24	implies	imply	VERB
ejpam-2534	10	25	that	that	SCONJ
ejpam-2534	10	26	either	either	CCONJ
ejpam-2534	10	27	a	a	DET
ejpam-2534	10	28	∈	∈	PROPN
ejpam-2534	10	29	p	p	NOUN
ejpam-2534	10	30	or	or	CCONJ
ejpam-2534	10	31	b	b	NOUN
ejpam-2534	10	32	∈	∈	PROPN
ejpam-2534	10	33	p	p	X
ejpam-2534	10	34	(	(	PUNCT
ejpam-2534	10	35	either	either	CCONJ
ejpam-2534	10	36	a	a	DET
ejpam-2534	10	37	∈	∈	PROPN
ejpam-2534	10	38	p	p	NOUN
ejpam-2534	10	39	or	or	CCONJ
ejpam-2534	10	40	bn	bn	ADP
ejpam-2534	10	41	∈	∈	PROPN
ejpam-2534	10	42	p	p	NOUN
ejpam-2534	10	43	for	for	ADP
ejpam-2534	10	44	some	some	DET
ejpam-2534	10	45	positive	positive	ADJ
ejpam-2534	10	46	integer	integer	NOUN
ejpam-2534	10	47	n	n	CCONJ
ejpam-2534	10	48	)	)	PUNCT
ejpam-2534	10	49	.	.	PUNCT
ejpam-2534	11	1	a	a	DET
ejpam-2534	11	2	proper	proper	ADJ
ejpam-2534	11	3	ideal	ideal	NOUN
ejpam-2534	11	4	p	p	NOUN
ejpam-2534	11	5	of	of	ADP
ejpam-2534	11	6	r	r	NOUN
ejpam-2534	11	7	is	be	AUX
ejpam-2534	11	8	said	say	VERB
ejpam-2534	11	9	to	to	PART
ejpam-2534	11	10	be	be	AUX
ejpam-2534	11	11	a	a	DET
ejpam-2534	11	12	weakly	weakly	ADJ
ejpam-2534	11	13	prime	prime	ADJ
ejpam-2534	11	14	ideal	ideal	NOUN
ejpam-2534	11	15	if	if	SCONJ
ejpam-2534	11	16	0	0	NUM
ejpam-2534	11	17	6=	6=	NUM
ejpam-2534	11	18	ab	ab	PROPN
ejpam-2534	11	19	∈	∈	PROPN
ejpam-2534	11	20	p	p	PROPN
ejpam-2534	11	21	for	for	ADP
ejpam-2534	11	22	some	some	PRON
ejpam-2534	11	23	a	a	PRON
ejpam-2534	11	24	,	,	PUNCT
ejpam-2534	11	25	b	b	X
ejpam-2534	11	26	∈	∈	NOUN
ejpam-2534	11	27	r	r	NOUN
ejpam-2534	11	28	implies	imply	VERB
ejpam-2534	11	29	that	that	SCONJ
ejpam-2534	11	30	either	either	CCONJ
ejpam-2534	11	31	a	a	DET
ejpam-2534	11	32	∈	∈	PROPN
ejpam-2534	11	33	p	p	NOUN
ejpam-2534	11	34	or	or	CCONJ
ejpam-2534	11	35	b	b	NOUN
ejpam-2534	11	36	∈	∈	PROPN
ejpam-2534	11	37	p	p	NOUN
ejpam-2534	11	38	,	,	PUNCT
ejpam-2534	11	39	and	and	CCONJ
ejpam-2534	11	40	it	it	PRON
ejpam-2534	11	41	is	be	AUX
ejpam-2534	11	42	called	call	VERB
ejpam-2534	11	43	a	a	DET
ejpam-2534	11	44	weakly	weakly	ADJ
ejpam-2534	11	45	primary	primary	ADJ
ejpam-2534	11	46	ideal	ideal	NOUN
ejpam-2534	11	47	if	if	SCONJ
ejpam-2534	11	48	0	0	NUM
ejpam-2534	11	49	6=	6=	NUM
ejpam-2534	11	50	ab	ab	PROPN
ejpam-2534	11	51	∈	∈	PROPN
ejpam-2534	11	52	p	p	PROPN
ejpam-2534	11	53	for	for	ADP
ejpam-2534	11	54	some	some	PRON
ejpam-2534	11	55	a	a	PRON
ejpam-2534	11	56	,	,	PUNCT
ejpam-2534	11	57	b	b	X
ejpam-2534	11	58	∈	∈	NOUN
ejpam-2534	11	59	r	r	NOUN
ejpam-2534	11	60	implies	imply	VERB
ejpam-2534	11	61	that	that	SCONJ
ejpam-2534	11	62	either	either	CCONJ
ejpam-2534	11	63	a	a	DET
ejpam-2534	11	64	∈	∈	PROPN
ejpam-2534	11	65	p	p	NOUN
ejpam-2534	11	66	or	or	CCONJ
ejpam-2534	11	67	bn	bn	ADP
ejpam-2534	11	68	∈	∈	PROPN
ejpam-2534	11	69	p	p	NOUN
ejpam-2534	11	70	for	for	ADP
ejpam-2534	11	71	some	some	DET
ejpam-2534	11	72	positive	positive	ADJ
ejpam-2534	11	73	integer	integer	NOUN
ejpam-2534	11	74	n	n	CCONJ
ejpam-2534	11	75	(	(	PUNCT
ejpam-2534	11	76	see	see	VERB
ejpam-2534	11	77	[	[	X
ejpam-2534	11	78	2	2	NUM
ejpam-2534	11	79	,	,	PUNCT
ejpam-2534	11	80	3	3	NUM
ejpam-2534	11	81	]	]	NUM
ejpam-2534	11	82	)	)	PUNCT
ejpam-2534	11	83	.	.	PUNCT
ejpam-2534	12	1	let	let	VERB
ejpam-2534	12	2	m	m	PRON
ejpam-2534	12	3	be	be	AUX
ejpam-2534	12	4	an	an	DET
ejpam-2534	12	5	r	r	NOUN
ejpam-2534	12	6	-	-	PUNCT
ejpam-2534	12	7	module	module	NOUN
ejpam-2534	12	8	.	.	PUNCT
ejpam-2534	13	1	a	a	DET
ejpam-2534	13	2	submodule	submodule	NOUN
ejpam-2534	13	3	n	n	PROPN
ejpam-2534	13	4	of	of	ADP
ejpam-2534	13	5	m	m	PROPN
ejpam-2534	13	6	is	be	AUX
ejpam-2534	13	7	called	call	VERB
ejpam-2534	13	8	a	a	DET
ejpam-2534	13	9	proper	proper	ADJ
ejpam-2534	13	10	submodule	submodule	NOUN
ejpam-2534	13	11	if	if	SCONJ
ejpam-2534	13	12	n	n	PROPN
ejpam-2534	13	13	6=	6=	ADP
ejpam-2534	13	14	m	m	VERB
ejpam-2534	13	15	.	.	PUNCT
ejpam-2534	14	1	a	a	DET
ejpam-2534	14	2	proper	proper	ADJ
ejpam-2534	14	3	submodule	submodule	NOUN
ejpam-2534	14	4	n	n	PROPN
ejpam-2534	14	5	of	of	ADP
ejpam-2534	14	6	m	m	PROPN
ejpam-2534	14	7	is	be	AUX
ejpam-2534	14	8	called	call	VERB
ejpam-2534	14	9	a	a	DET
ejpam-2534	14	10	prime	prime	ADJ
ejpam-2534	14	11	submodule	submodule	NOUN
ejpam-2534	14	12	if	if	SCONJ
ejpam-2534	14	13	rm	rm	PROPN
ejpam-2534	14	14	∈	∈	PROPN
ejpam-2534	14	15	n	n	PROPN
ejpam-2534	14	16	for	for	ADP
ejpam-2534	14	17	some	some	DET
ejpam-2534	14	18	r	r	NOUN
ejpam-2534	14	19	∈	∈	NOUN
ejpam-2534	14	20	r	r	NOUN
ejpam-2534	14	21	and	and	CCONJ
ejpam-2534	14	22	m	m	PROPN
ejpam-2534	14	23	∈	∈	NOUN
ejpam-2534	14	24	m	m	VERB
ejpam-2534	14	25	implies	imply	VERB
ejpam-2534	14	26	that	that	SCONJ
ejpam-2534	14	27	either	either	CCONJ
ejpam-2534	14	28	m	m	PROPN
ejpam-2534	14	29	∈	∈	PROPN
ejpam-2534	14	30	n	n	NOUN
ejpam-2534	14	31	or	or	CCONJ
ejpam-2534	14	32	rm	rm	PROPN
ejpam-2534	14	33	⊆	⊆	NUM
ejpam-2534	14	34	n	n	PROPN
ejpam-2534	14	35	and	and	CCONJ
ejpam-2534	14	36	it	it	PRON
ejpam-2534	14	37	is	be	AUX
ejpam-2534	14	38	said	say	VERB
ejpam-2534	14	39	to	to	PART
ejpam-2534	14	40	be	be	AUX
ejpam-2534	14	41	a	a	DET
ejpam-2534	14	42	weakly	weakly	ADJ
ejpam-2534	14	43	prime	prime	ADJ
ejpam-2534	14	44	submodule	submodule	NOUN
ejpam-2534	14	45	if	if	SCONJ
ejpam-2534	14	46	0	0	NUM
ejpam-2534	14	47	6=	6=	NUM
ejpam-2534	14	48	rm	rm	PROPN
ejpam-2534	14	49	∈	∈	PROPN
ejpam-2534	15	1	n	n	PROPN
ejpam-2534	15	2	for	for	ADP
ejpam-2534	15	3	some	some	DET
ejpam-2534	15	4	r	r	NOUN
ejpam-2534	15	5	∈	∈	NOUN
ejpam-2534	15	6	r	r	NOUN
ejpam-2534	15	7	and	and	CCONJ
ejpam-2534	15	8	m	m	PROPN
ejpam-2534	15	9	∈	∈	NOUN
ejpam-2534	15	10	m	m	VERB
ejpam-2534	15	11	implies	imply	VERB
ejpam-2534	15	12	that	that	SCONJ
ejpam-2534	15	13	either	either	CCONJ
ejpam-2534	15	14	m	m	PROPN
ejpam-2534	15	15	∈	∈	PROPN
ejpam-2534	15	16	n	n	NOUN
ejpam-2534	15	17	or	or	CCONJ
ejpam-2534	15	18	rm	rm	PROPN
ejpam-2534	15	19	⊆	⊆	NUM
ejpam-2534	15	20	n	n	NOUN
ejpam-2534	15	21	.	.	PUNCT
ejpam-2534	16	1	a	a	DET
ejpam-2534	16	2	non	non	X
ejpam-2534	16	3	empty	empty	ADJ
ejpam-2534	16	4	subset	subset	NOUN
ejpam-2534	16	5	s	s	NOUN
ejpam-2534	16	6	of	of	ADP
ejpam-2534	16	7	r	r	NOUN
ejpam-2534	16	8	is	be	AUX
ejpam-2534	16	9	said	say	VERB
ejpam-2534	16	10	to	to	PART
ejpam-2534	16	11	be	be	AUX
ejpam-2534	16	12	multiplicative	multiplicative	ADJ
ejpam-2534	16	13	closed	close	VERB
ejpam-2534	16	14	set	set	VERB
ejpam-2534	16	15	if	if	SCONJ
ejpam-2534	16	16	0	0	NUM
ejpam-2534	16	17	/∈	/∈	SYM
ejpam-2534	16	18	s	s	PART
ejpam-2534	16	19	and	and	CCONJ
ejpam-2534	16	20	whenever	whenever	SCONJ
ejpam-2534	16	21	a	a	DET
ejpam-2534	16	22	,	,	PUNCT
ejpam-2534	16	23	b	b	PROPN
ejpam-2534	16	24	∈	∈	PROPN
ejpam-2534	16	25	s	s	PROPN
ejpam-2534	16	26	,	,	PUNCT
ejpam-2534	16	27	then	then	ADV
ejpam-2534	16	28	ab	ab	PROPN
ejpam-2534	16	29	∈	∈	PROPN
ejpam-2534	16	30	s.	s.	PROPN
ejpam-2534	16	31	let	let	VERB
ejpam-2534	16	32	s	s	PRON
ejpam-2534	16	33	be	be	AUX
ejpam-2534	16	34	a	a	DET
ejpam-2534	16	35	multiplicative	multiplicative	ADJ
ejpam-2534	16	36	closed	close	VERB
ejpam-2534	16	37	set	set	VERB
ejpam-2534	16	38	in	in	ADP
ejpam-2534	16	39	r.	r.	PROPN
ejpam-2534	16	40	it	it	PRON
ejpam-2534	16	41	can	can	AUX
ejpam-2534	16	42	be	be	AUX
ejpam-2534	16	43	easily	easily	ADV
ejpam-2534	16	44	seen	see	VERB
ejpam-2534	16	45	that	that	SCONJ
ejpam-2534	16	46	ms	ms	PROPN
ejpam-2534	16	47	is	be	AUX
ejpam-2534	16	48	an	an	DET
ejpam-2534	16	49	rs	rs	NOUN
ejpam-2534	16	50	-	-	PUNCT
ejpam-2534	16	51	module	module	NOUN
ejpam-2534	16	52	under	under	ADP
ejpam-2534	16	53	the	the	DET
ejpam-2534	16	54	operations	operation	NOUN
ejpam-2534	16	55	a	a	DET
ejpam-2534	16	56	s	s	NOUN
ejpam-2534	16	57	+	+	NOUN
ejpam-2534	16	58	b	b	NOUN
ejpam-2534	16	59	u	u	NOUN
ejpam-2534	16	60	=	=	PROPN
ejpam-2534	16	61	ua+sb	ua+sb	PROPN
ejpam-2534	16	62	su	su	NOUN
ejpam-2534	16	63	and	and	CCONJ
ejpam-2534	16	64	r	r	NOUN
ejpam-2534	16	65	v	v	ADP
ejpam-2534	16	66	a	a	DET
ejpam-2534	16	67	s	s	NOUN
ejpam-2534	16	68	=	=	X
ejpam-2534	16	69	ra	ra	PROPN
ejpam-2534	16	70	vs	vs	ADP
ejpam-2534	16	71	for	for	ADP
ejpam-2534	16	72	any	any	DET
ejpam-2534	16	73	r	r	NOUN
ejpam-2534	16	74	v	v	NOUN
ejpam-2534	16	75	∈	∈	NOUN
ejpam-2534	16	76	rs	rs	NOUN
ejpam-2534	16	77	and	and	CCONJ
ejpam-2534	16	78	a	a	DET
ejpam-2534	16	79	s	s	X
ejpam-2534	16	80	,	,	PUNCT
ejpam-2534	16	81	b	b	PROPN
ejpam-2534	16	82	u	u	X
ejpam-2534	16	83	∈	∈	PROPN
ejpam-2534	16	84	ms	ms	NOUN
ejpam-2534	17	1	[	[	X
ejpam-2534	17	2	5	5	NUM
ejpam-2534	17	3	]	]	PUNCT
ejpam-2534	17	4	.	.	PUNCT
ejpam-2534	18	1	a	a	DET
ejpam-2534	18	2	proper	proper	ADJ
ejpam-2534	18	3	submodule	submodule	NOUN
ejpam-2534	18	4	n	n	PROPN
ejpam-2534	18	5	of	of	ADP
ejpam-2534	18	6	m	m	PROPN
ejpam-2534	18	7	is	be	AUX
ejpam-2534	18	8	said	say	VERB
ejpam-2534	18	9	to	to	PART
ejpam-2534	18	10	be	be	AUX
ejpam-2534	18	11	s(n)-locally	s(n)-locally	ADV
ejpam-2534	18	12	prime	prime	ADJ
ejpam-2534	18	13	(	(	PUNCT
ejpam-2534	18	14	s(n)-weakly	s(n)-weakly	ADJ
ejpam-2534	18	15	prime	prime	ADJ
ejpam-2534	18	16	)	)	PUNCT
ejpam-2534	18	17	submodule	submodule	NOUN
ejpam-2534	18	18	if	if	SCONJ
ejpam-2534	18	19	nm	nm	PRON
ejpam-2534	18	20	is	be	AUX
ejpam-2534	18	21	a	a	DET
ejpam-2534	18	22	prime	prime	NOUN
ejpam-2534	18	23	(	(	PUNCT
ejpam-2534	18	24	a	a	DET
ejpam-2534	18	25	weakly	weakly	ADJ
ejpam-2534	18	26	prime	prime	ADJ
ejpam-2534	18	27	)	)	PUNCT
ejpam-2534	18	28	submodule	submodule	NOUN
ejpam-2534	18	29	of	of	ADP
ejpam-2534	18	30	mm	mm	PROPN
ejpam-2534	18	31	for	for	ADP
ejpam-2534	18	32	each	each	DET
ejpam-2534	18	33	maximal	maximal	ADJ
ejpam-2534	18	34	ideal	ideal	NOUN
ejpam-2534	18	35	m	m	VERB
ejpam-2534	18	36	with	with	ADP
ejpam-2534	18	37	s(n	s(n	NOUN
ejpam-2534	18	38	)	)	PUNCT
ejpam-2534	18	39	⊆m	⊆m	NOUN
ejpam-2534	19	1	[	[	X
ejpam-2534	19	2	4	4	NUM
ejpam-2534	19	3	]	]	PUNCT
ejpam-2534	19	4	.	.	PUNCT
ejpam-2534	20	1	∗corresponding	∗corresponde	VERB
ejpam-2534	20	2	author	author	NOUN
ejpam-2534	20	3	.	.	PUNCT
ejpam-2534	21	1	email	email	NOUN
ejpam-2534	21	2	addresses	address	NOUN
ejpam-2534	21	3	:	:	PUNCT
ejpam-2534	21	4	gulsenulucak@gtu.edu.tr	gulsenulucak@gtu.edu.tr	PROPN
ejpam-2534	21	5	(	(	PUNCT
ejpam-2534	21	6	g.	g.	PROPN
ejpam-2534	21	7	ulucak	ulucak	PROPN
ejpam-2534	21	8	)	)	PUNCT
ejpam-2534	21	9	,	,	PUNCT
ejpam-2534	21	10	rnuregen@yildiz.edu.tr	rnuregen@yildiz.edu.tr	X
ejpam-2534	21	11	(	(	PUNCT
ejpam-2534	21	12	r.	r.	PROPN
ejpam-2534	21	13	uregen	uregen	PROPN
ejpam-2534	21	14	)	)	PUNCT
ejpam-2534	21	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2534	22	1	48	48	NUM
ejpam-2534	22	2	c	c	NOUN
ejpam-2534	22	3	©	©	PROPN
ejpam-2534	22	4	2016	2016	NUM
ejpam-2534	22	5	ejpam	ejpam	VERB
ejpam-2534	22	6	all	all	DET
ejpam-2534	22	7	rights	right	NOUN
ejpam-2534	22	8	reserved	reserve	VERB
ejpam-2534	22	9	.	.	PUNCT
ejpam-2534	23	1	g.	g.	PROPN
ejpam-2534	23	2	ulucak	ulucak	PROPN
ejpam-2534	23	3	and	and	CCONJ
ejpam-2534	23	4	r.	r.	PROPN
ejpam-2534	23	5	uregen	uregen	PROPN
ejpam-2534	23	6	/	/	SYM
ejpam-2534	23	7	eur	eur	PROPN
ejpam-2534	23	8	.	.	PUNCT
ejpam-2534	24	1	j.	j.	PROPN
ejpam-2534	24	2	pure	pure	PROPN
ejpam-2534	24	3	appl	appl	PROPN
ejpam-2534	24	4	.	.	PROPN
ejpam-2534	24	5	math	math	PROPN
ejpam-2534	24	6	,	,	PUNCT
ejpam-2534	24	7	9	9	NUM
ejpam-2534	24	8	(	(	PUNCT
ejpam-2534	24	9	2016	2016	NUM
ejpam-2534	24	10	)	)	PUNCT
ejpam-2534	24	11	,	,	PUNCT
ejpam-2534	24	12	48	48	NUM
ejpam-2534	24	13	-	-	SYM
ejpam-2534	24	14	56	56	NUM
ejpam-2534	24	15	49	49	NUM
ejpam-2534	24	16	a	a	DET
ejpam-2534	24	17	proper	proper	ADJ
ejpam-2534	24	18	submodule	submodule	NOUN
ejpam-2534	24	19	n	n	PROPN
ejpam-2534	24	20	of	of	ADP
ejpam-2534	24	21	m	m	PROPN
ejpam-2534	24	22	is	be	AUX
ejpam-2534	24	23	said	say	VERB
ejpam-2534	24	24	to	to	PART
ejpam-2534	24	25	be	be	AUX
ejpam-2534	24	26	a	a	DET
ejpam-2534	24	27	primary	primary	ADJ
ejpam-2534	24	28	submodule	submodule	NOUN
ejpam-2534	24	29	if	if	SCONJ
ejpam-2534	24	30	rm	rm	PROPN
ejpam-2534	24	31	∈	∈	PROPN
ejpam-2534	24	32	n	n	PROPN
ejpam-2534	24	33	for	for	ADP
ejpam-2534	24	34	some	some	DET
ejpam-2534	24	35	r	r	NOUN
ejpam-2534	24	36	∈	∈	NOUN
ejpam-2534	24	37	r	r	NOUN
ejpam-2534	24	38	,	,	PUNCT
ejpam-2534	24	39	m	m	VERB
ejpam-2534	24	40	∈	∈	NOUN
ejpam-2534	24	41	m	m	NOUN
ejpam-2534	24	42	implies	imply	VERB
ejpam-2534	24	43	that	that	SCONJ
ejpam-2534	24	44	either	either	CCONJ
ejpam-2534	24	45	m	m	PROPN
ejpam-2534	24	46	∈	∈	PROPN
ejpam-2534	24	47	n	n	NOUN
ejpam-2534	24	48	or	or	CCONJ
ejpam-2534	24	49	rnm	rnm	VERB
ejpam-2534	24	50	⊆	⊆	NUM
ejpam-2534	24	51	n	n	NOUN
ejpam-2534	24	52	for	for	ADP
ejpam-2534	24	53	some	some	DET
ejpam-2534	24	54	positive	positive	ADJ
ejpam-2534	24	55	integer	integer	NOUN
ejpam-2534	24	56	n	n	NOUN
ejpam-2534	24	57	and	and	CCONJ
ejpam-2534	24	58	it	it	PRON
ejpam-2534	24	59	is	be	AUX
ejpam-2534	24	60	said	say	VERB
ejpam-2534	24	61	to	to	PART
ejpam-2534	24	62	be	be	AUX
ejpam-2534	24	63	a	a	DET
ejpam-2534	24	64	weakly	weakly	ADJ
ejpam-2534	24	65	primary	primary	ADJ
ejpam-2534	24	66	submodule	submodule	NOUN
ejpam-2534	24	67	if	if	SCONJ
ejpam-2534	24	68	0	0	NUM
ejpam-2534	24	69	6=	6=	NUM
ejpam-2534	24	70	rm	rm	PROPN
ejpam-2534	24	71	∈	∈	PROPN
ejpam-2534	24	72	n	n	PROPN
ejpam-2534	24	73	for	for	ADP
ejpam-2534	24	74	some	some	DET
ejpam-2534	24	75	r	r	NOUN
ejpam-2534	24	76	∈	∈	NOUN
ejpam-2534	24	77	r	r	NOUN
ejpam-2534	24	78	,	,	PUNCT
ejpam-2534	24	79	m	m	VERB
ejpam-2534	24	80	∈	∈	NOUN
ejpam-2534	24	81	m	m	NOUN
ejpam-2534	24	82	implies	imply	VERB
ejpam-2534	24	83	that	that	SCONJ
ejpam-2534	24	84	either	either	CCONJ
ejpam-2534	24	85	m	m	PROPN
ejpam-2534	24	86	∈	∈	PROPN
ejpam-2534	24	87	n	n	NOUN
ejpam-2534	24	88	or	or	CCONJ
ejpam-2534	24	89	rnm	rnm	VERB
ejpam-2534	24	90	⊆	⊆	NUM
ejpam-2534	24	91	n	n	NOUN
ejpam-2534	24	92	for	for	ADP
ejpam-2534	24	93	some	some	DET
ejpam-2534	24	94	positive	positive	ADJ
ejpam-2534	24	95	integer	integer	NOUN
ejpam-2534	24	96	n.	n.	NOUN
ejpam-2534	24	97	the	the	DET
ejpam-2534	24	98	ideal	ideal	NOUN
ejpam-2534	24	99	{	{	PUNCT
ejpam-2534	24	100	r	r	NOUN
ejpam-2534	24	101	∈	∈	NOUN
ejpam-2534	24	102	r	r	NOUN
ejpam-2534	24	103	|	|	NOUN
ejpam-2534	24	104	rm	rm	NOUN
ejpam-2534	24	105	⊆	⊆	NUM
ejpam-2534	24	106	n	n	CCONJ
ejpam-2534	24	107	}	}	PUNCT
ejpam-2534	24	108	will	will	AUX
ejpam-2534	24	109	be	be	AUX
ejpam-2534	24	110	denoted	denote	VERB
ejpam-2534	24	111	by	by	ADP
ejpam-2534	24	112	(	(	PUNCT
ejpam-2534	24	113	n	n	NUM
ejpam-2534	24	114	:	:	PUNCT
ejpam-2534	24	115	m	m	X
ejpam-2534	24	116	)	)	PUNCT
ejpam-2534	24	117	and	and	CCONJ
ejpam-2534	24	118	(	(	PUNCT
ejpam-2534	24	119	0	0	NUM
ejpam-2534	24	120	:	:	PUNCT
ejpam-2534	24	121	n	n	X
ejpam-2534	24	122	)	)	PUNCT
ejpam-2534	24	123	=	=	PRON
ejpam-2534	24	124	{	{	PUNCT
ejpam-2534	24	125	r	r	NOUN
ejpam-2534	24	126	∈	∈	PROPN
ejpam-2534	24	127	r	r	NOUN
ejpam-2534	24	128	|	|	NOUN
ejpam-2534	24	129	rn	rn	PROPN
ejpam-2534	24	130	=	=	NOUN
ejpam-2534	24	131	0	0	NUM
ejpam-2534	24	132	}	}	PUNCT
ejpam-2534	24	133	where	where	SCONJ
ejpam-2534	24	134	n	n	PRON
ejpam-2534	24	135	is	be	AUX
ejpam-2534	24	136	a	a	DET
ejpam-2534	24	137	submodule	submodule	NOUN
ejpam-2534	24	138	of	of	ADP
ejpam-2534	24	139	m	m	PROPN
ejpam-2534	24	140	.	.	PUNCT
ejpam-2534	25	1	then	then	ADV
ejpam-2534	25	2	the	the	DET
ejpam-2534	25	3	annihilator	annihilator	NOUN
ejpam-2534	25	4	of	of	ADP
ejpam-2534	25	5	m	m	PROPN
ejpam-2534	25	6	is	be	AUX
ejpam-2534	25	7	(	(	PUNCT
ejpam-2534	25	8	0	0	NUM
ejpam-2534	25	9	:	:	PUNCT
ejpam-2534	25	10	m	m	X
ejpam-2534	25	11	)	)	PUNCT
ejpam-2534	25	12	where	where	SCONJ
ejpam-2534	25	13	(	(	PUNCT
ejpam-2534	25	14	0	0	NUM
ejpam-2534	25	15	:	:	PUNCT
ejpam-2534	25	16	m	m	X
ejpam-2534	25	17	)	)	PUNCT
ejpam-2534	25	18	=	=	PRON
ejpam-2534	26	1	{	{	PUNCT
ejpam-2534	26	2	r	r	NOUN
ejpam-2534	26	3	∈	∈	PROPN
ejpam-2534	26	4	r	r	NOUN
ejpam-2534	26	5	|	|	NOUN
ejpam-2534	26	6	rm	rm	NOUN
ejpam-2534	26	7	=	=	PUNCT
ejpam-2534	26	8	0	0	NUM
ejpam-2534	26	9	}	}	PUNCT
ejpam-2534	26	10	.	.	PUNCT
ejpam-2534	27	1	an	an	DET
ejpam-2534	27	2	r	r	NOUN
ejpam-2534	27	3	-	-	PUNCT
ejpam-2534	27	4	module	module	NOUN
ejpam-2534	27	5	m	m	NOUN
ejpam-2534	27	6	is	be	AUX
ejpam-2534	27	7	called	call	VERB
ejpam-2534	27	8	a	a	DET
ejpam-2534	27	9	faithful	faithful	ADJ
ejpam-2534	27	10	module	module	NOUN
ejpam-2534	27	11	if	if	SCONJ
ejpam-2534	27	12	(	(	PUNCT
ejpam-2534	27	13	0	0	NUM
ejpam-2534	27	14	:	:	PUNCT
ejpam-2534	27	15	m	m	X
ejpam-2534	27	16	)	)	PUNCT
ejpam-2534	27	17	=	=	SYM
ejpam-2534	28	1	(	(	PUNCT
ejpam-2534	28	2	0	0	NUM
ejpam-2534	28	3	)	)	PUNCT
ejpam-2534	28	4	.	.	PUNCT
ejpam-2534	29	1	note	note	VERB
ejpam-2534	29	2	that	that	SCONJ
ejpam-2534	29	3	if	if	SCONJ
ejpam-2534	29	4	n	n	PRON
ejpam-2534	29	5	is	be	AUX
ejpam-2534	29	6	a	a	DET
ejpam-2534	29	7	primary	primary	ADJ
ejpam-2534	29	8	submodule	submodule	NOUN
ejpam-2534	29	9	of	of	ADP
ejpam-2534	29	10	m	m	PROPN
ejpam-2534	29	11	,	,	PUNCT
ejpam-2534	29	12	then	then	ADV
ejpam-2534	29	13	(	(	PUNCT
ejpam-2534	29	14	n	n	X
ejpam-2534	29	15	:	:	PUNCT
ejpam-2534	29	16	m	m	X
ejpam-2534	29	17	)	)	PUNCT
ejpam-2534	29	18	is	be	AUX
ejpam-2534	29	19	a	a	DET
ejpam-2534	29	20	primary	primary	ADJ
ejpam-2534	29	21	ideal	ideal	NOUN
ejpam-2534	29	22	of	of	ADP
ejpam-2534	29	23	r	r	NOUN
ejpam-2534	29	24	and	and	CCONJ
ejpam-2534	29	25	rad(n	rad(n	NOUN
ejpam-2534	29	26	:	:	PUNCT
ejpam-2534	29	27	m	m	X
ejpam-2534	29	28	)	)	PUNCT
ejpam-2534	30	1	=	=	PRON
ejpam-2534	30	2	{	{	PUNCT
ejpam-2534	30	3	r	r	NOUN
ejpam-2534	30	4	∈	∈	NOUN
ejpam-2534	30	5	r	r	NOUN
ejpam-2534	30	6	|	|	ADV
ejpam-2534	30	7	rnm	rnm	VERB
ejpam-2534	30	8	⊆	⊆	NUM
ejpam-2534	30	9	n	n	NOUN
ejpam-2534	30	10	for	for	ADP
ejpam-2534	30	11	some	some	DET
ejpam-2534	30	12	positive	positive	ADJ
ejpam-2534	30	13	integer	integer	NOUN
ejpam-2534	30	14	n	n	CCONJ
ejpam-2534	30	15	}	}	PUNCT
ejpam-2534	30	16	is	be	AUX
ejpam-2534	30	17	a	a	DET
ejpam-2534	30	18	prime	prime	ADJ
ejpam-2534	30	19	ideal	ideal	NOUN
ejpam-2534	30	20	of	of	ADP
ejpam-2534	30	21	r	r	NOUN
ejpam-2534	30	22	(	(	PUNCT
ejpam-2534	30	23	[	[	X
ejpam-2534	30	24	1	1	NUM
ejpam-2534	30	25	,	,	PUNCT
ejpam-2534	30	26	6	6	NUM
ejpam-2534	30	27	,	,	PUNCT
ejpam-2534	30	28	7	7	NUM
ejpam-2534	30	29	]	]	NUM
ejpam-2534	30	30	)	)	PUNCT
ejpam-2534	30	31	.	.	PUNCT
ejpam-2534	31	1	main	main	ADJ
ejpam-2534	31	2	aim	aim	NOUN
ejpam-2534	31	3	is	be	AUX
ejpam-2534	31	4	to	to	PART
ejpam-2534	31	5	obtain	obtain	VERB
ejpam-2534	31	6	the	the	DET
ejpam-2534	31	7	two	two	NUM
ejpam-2534	31	8	generalization	generalization	NOUN
ejpam-2534	31	9	on	on	ADP
ejpam-2534	31	10	primary	primary	ADJ
ejpam-2534	31	11	submodules	submodule	NOUN
ejpam-2534	31	12	and	and	CCONJ
ejpam-2534	31	13	weakly	weakly	ADJ
ejpam-2534	31	14	primary	primary	ADJ
ejpam-2534	31	15	submodules	submodule	NOUN
ejpam-2534	31	16	of	of	ADP
ejpam-2534	31	17	an	an	DET
ejpam-2534	31	18	r	r	NOUN
ejpam-2534	31	19	-	-	PUNCT
ejpam-2534	31	20	module	module	NOUN
ejpam-2534	31	21	m.let	m.let	NOUN
ejpam-2534	31	22	n	n	CCONJ
ejpam-2534	31	23	be	be	AUX
ejpam-2534	31	24	a	a	DET
ejpam-2534	31	25	proper	proper	ADJ
ejpam-2534	31	26	submodule	submodule	NOUN
ejpam-2534	31	27	of	of	ADP
ejpam-2534	31	28	m	m	PROPN
ejpam-2534	31	29	.	.	PUNCT
ejpam-2534	32	1	an	an	DET
ejpam-2534	32	2	element	element	NOUN
ejpam-2534	32	3	r	r	NOUN
ejpam-2534	32	4	∈	∈	NOUN
ejpam-2534	32	5	r	r	NOUN
ejpam-2534	32	6	is	be	AUX
ejpam-2534	32	7	said	say	VERB
ejpam-2534	32	8	to	to	PART
ejpam-2534	32	9	be	be	AUX
ejpam-2534	32	10	primary	primary	ADJ
ejpam-2534	32	11	to	to	ADP
ejpam-2534	32	12	n	n	PRON
ejpam-2534	32	13	if	if	SCONJ
ejpam-2534	32	14	rnm	rnm	NOUN
ejpam-2534	32	15	∈	∈	PROPN
ejpam-2534	32	16	n	n	NOUN
ejpam-2534	32	17	,	,	PUNCT
ejpam-2534	32	18	where	where	SCONJ
ejpam-2534	32	19	m	m	VERB
ejpam-2534	32	20	∈	∈	PROPN
ejpam-2534	32	21	m	m	NOUN
ejpam-2534	32	22	and	and	CCONJ
ejpam-2534	32	23	n	n	PROPN
ejpam-2534	32	24	is	be	AUX
ejpam-2534	32	25	a	a	DET
ejpam-2534	32	26	positive	positive	ADJ
ejpam-2534	32	27	integer	integer	NOUN
ejpam-2534	32	28	,	,	PUNCT
ejpam-2534	32	29	then	then	ADV
ejpam-2534	32	30	m	m	VERB
ejpam-2534	32	31	∈	∈	PROPN
ejpam-2534	32	32	n	n	NOUN
ejpam-2534	32	33	.	.	PUNCT
ejpam-2534	33	1	then	then	ADV
ejpam-2534	33	2	r	r	NOUN
ejpam-2534	33	3	∈	∈	PROPN
ejpam-2534	33	4	r	r	NOUN
ejpam-2534	33	5	is	be	AUX
ejpam-2534	33	6	said	say	VERB
ejpam-2534	33	7	to	to	PART
ejpam-2534	33	8	be	be	AUX
ejpam-2534	33	9	not	not	PART
ejpam-2534	33	10	primary	primary	ADJ
ejpam-2534	33	11	to	to	ADP
ejpam-2534	33	12	n	n	PRON
ejpam-2534	33	13	if	if	SCONJ
ejpam-2534	33	14	rnm	rnm	NOUN
ejpam-2534	33	15	∈	∈	PROPN
ejpam-2534	33	16	n	n	NOUN
ejpam-2534	33	17	for	for	ADP
ejpam-2534	33	18	some	some	DET
ejpam-2534	33	19	positive	positive	ADJ
ejpam-2534	33	20	integer	integer	NOUN
ejpam-2534	33	21	n	n	NOUN
ejpam-2534	33	22	and	and	CCONJ
ejpam-2534	33	23	for	for	ADP
ejpam-2534	33	24	some	some	DET
ejpam-2534	33	25	m	m	NOUN
ejpam-2534	33	26	∈	∈	NOUN
ejpam-2534	33	27	m	m	VERB
ejpam-2534	33	28	\	\	NOUN
ejpam-2534	33	29	n	n	NOUN
ejpam-2534	33	30	.	.	PUNCT
ejpam-2534	34	1	the	the	DET
ejpam-2534	34	2	set	set	NOUN
ejpam-2534	34	3	of	of	ADP
ejpam-2534	34	4	all	all	DET
ejpam-2534	34	5	elements	element	NOUN
ejpam-2534	34	6	of	of	ADP
ejpam-2534	34	7	r	r	NOUN
ejpam-2534	34	8	that	that	PRON
ejpam-2534	34	9	are	be	AUX
ejpam-2534	34	10	not	not	PART
ejpam-2534	34	11	primary	primary	ADJ
ejpam-2534	34	12	to	to	ADP
ejpam-2534	34	13	n	n	PROPN
ejpam-2534	34	14	is	be	AUX
ejpam-2534	34	15	denoted	denote	VERB
ejpam-2534	34	16	by	by	ADP
ejpam-2534	34	17	p(n	p(n	PROPN
ejpam-2534	34	18	)	)	PUNCT
ejpam-2534	34	19	.	.	PUNCT
ejpam-2534	35	1	then	then	ADV
ejpam-2534	35	2	we	we	PRON
ejpam-2534	35	3	get	get	VERB
ejpam-2534	35	4	p(n	p(n	NOUN
ejpam-2534	35	5	)	)	PUNCT
ejpam-2534	35	6	=	=	PRON
ejpam-2534	35	7	{	{	PUNCT
ejpam-2534	35	8	r	r	NOUN
ejpam-2534	35	9	∈	∈	NOUN
ejpam-2534	35	10	r	r	NOUN
ejpam-2534	35	11	|	|	NOUN
ejpam-2534	35	12	rnm	rnm	NOUN
ejpam-2534	35	13	∈	∈	PROPN
ejpam-2534	35	14	n	n	NOUN
ejpam-2534	35	15	for	for	ADP
ejpam-2534	35	16	some	some	DET
ejpam-2534	35	17	positive	positive	ADJ
ejpam-2534	35	18	integer	integer	NOUN
ejpam-2534	35	19	n	n	CCONJ
ejpam-2534	35	20	,	,	PUNCT
ejpam-2534	35	21	for	for	ADP
ejpam-2534	35	22	some	some	DET
ejpam-2534	35	23	element	element	NOUN
ejpam-2534	35	24	m	m	NOUN
ejpam-2534	35	25	∈	∈	NOUN
ejpam-2534	35	26	m	m	NOUN
ejpam-2534	35	27	\n	\n	NUM
ejpam-2534	35	28	}	}	PUNCT
ejpam-2534	35	29	.	.	PUNCT
ejpam-2534	36	1	if	if	SCONJ
ejpam-2534	36	2	n	n	PRON
ejpam-2534	36	3	=	=	SYM
ejpam-2534	36	4	(	(	PUNCT
ejpam-2534	36	5	0	0	NUM
ejpam-2534	36	6	)	)	PUNCT
ejpam-2534	36	7	,	,	PUNCT
ejpam-2534	36	8	then	then	ADV
ejpam-2534	36	9	p((0	p((0	PROPN
ejpam-2534	36	10	)	)	PUNCT
ejpam-2534	36	11	)	)	PUNCT
ejpam-2534	37	1	=	=	PRON
ejpam-2534	37	2	{	{	PUNCT
ejpam-2534	37	3	r	r	NOUN
ejpam-2534	37	4	∈	∈	NOUN
ejpam-2534	37	5	r	r	NOUN
ejpam-2534	37	6	|	|	ADV
ejpam-2534	37	7	rnm	rnm	NOUN
ejpam-2534	37	8	=	=	NOUN
ejpam-2534	37	9	0	0	NUM
ejpam-2534	37	10	for	for	ADP
ejpam-2534	37	11	some	some	DET
ejpam-2534	37	12	positive	positive	ADJ
ejpam-2534	37	13	integer	integer	NOUN
ejpam-2534	37	14	n	n	CCONJ
ejpam-2534	37	15	,	,	PUNCT
ejpam-2534	37	16	for	for	ADP
ejpam-2534	37	17	some	some	DET
ejpam-2534	37	18	0	0	NUM
ejpam-2534	37	19	6=	6=	ADP
ejpam-2534	37	20	m	m	PROPN
ejpam-2534	37	21	∈	∈	PROPN
ejpam-2534	37	22	m	m	PRON
ejpam-2534	37	23	}	}	PUNCT
ejpam-2534	37	24	.	.	PUNCT
ejpam-2534	38	1	a	a	DET
ejpam-2534	38	2	proper	proper	ADJ
ejpam-2534	38	3	submodule	submodule	NOUN
ejpam-2534	38	4	n	n	PROPN
ejpam-2534	38	5	of	of	ADP
ejpam-2534	38	6	m	m	PROPN
ejpam-2534	38	7	is	be	AUX
ejpam-2534	38	8	said	say	VERB
ejpam-2534	38	9	to	to	PART
ejpam-2534	38	10	be	be	AUX
ejpam-2534	38	11	an	an	DET
ejpam-2534	38	12	m	m	NOUN
ejpam-2534	38	13	-	-	NOUN
ejpam-2534	38	14	primal	primal	ADJ
ejpam-2534	38	15	if	if	SCONJ
ejpam-2534	38	16	p(n	p(n	NOUN
ejpam-2534	38	17	)	)	PUNCT
ejpam-2534	38	18	forms	form	VERB
ejpam-2534	38	19	an	an	DET
ejpam-2534	38	20	ideal	ideal	NOUN
ejpam-2534	38	21	of	of	ADP
ejpam-2534	38	22	r.	r.	PROPN
ejpam-2534	38	23	2	2	NUM
ejpam-2534	38	24	.	.	PUNCT
ejpam-2534	39	1	p(n)-locally	p(n)-locally	ADV
ejpam-2534	39	2	primary	primary	ADJ
ejpam-2534	39	3	and	and	CCONJ
ejpam-2534	39	4	p(n)-locally	p(n)-locally	ADV
ejpam-2534	39	5	weakly	weakly	ADJ
ejpam-2534	39	6	primary	primary	ADJ
ejpam-2534	39	7	submodules	submodules	NOUN
ejpam-2534	39	8	definition	definition	NOUN
ejpam-2534	39	9	1	1	X
ejpam-2534	39	10	.	.	PUNCT
ejpam-2534	40	1	let	let	VERB
ejpam-2534	40	2	n	n	PRON
ejpam-2534	40	3	be	be	AUX
ejpam-2534	40	4	a	a	DET
ejpam-2534	40	5	proper	proper	ADJ
ejpam-2534	40	6	submodule	submodule	NOUN
ejpam-2534	40	7	of	of	ADP
ejpam-2534	40	8	an	an	DET
ejpam-2534	40	9	r	r	NOUN
ejpam-2534	40	10	-	-	PUNCT
ejpam-2534	40	11	module	module	NOUN
ejpam-2534	40	12	m.	m.	NOUN
ejpam-2534	40	13	then	then	ADV
ejpam-2534	40	14	n	n	PRON
ejpam-2534	40	15	is	be	AUX
ejpam-2534	40	16	called	call	VERB
ejpam-2534	40	17	a	a	DET
ejpam-2534	40	18	p(n)-locally	p(n)-locally	ADV
ejpam-2534	40	19	primary	primary	ADJ
ejpam-2534	40	20	submodule	submodule	NOUN
ejpam-2534	40	21	of	of	ADP
ejpam-2534	40	22	m	m	PROPN
ejpam-2534	40	23	if	if	SCONJ
ejpam-2534	40	24	nm	nm	ADJ
ejpam-2534	40	25	is	be	AUX
ejpam-2534	40	26	a	a	DET
ejpam-2534	40	27	primary	primary	ADJ
ejpam-2534	40	28	submodule	submodule	NOUN
ejpam-2534	40	29	of	of	ADP
ejpam-2534	40	30	mm	mm	PROPN
ejpam-2534	40	31	for	for	ADP
ejpam-2534	40	32	all	all	DET
ejpam-2534	40	33	maximal	maximal	ADJ
ejpam-2534	40	34	ideal	ideal	NOUN
ejpam-2534	40	35	m	m	VERB
ejpam-2534	40	36	where	where	SCONJ
ejpam-2534	40	37	p(n	p(n	NOUN
ejpam-2534	40	38	)	)	PUNCT
ejpam-2534	40	39	⊆m	⊆m	NOUN
ejpam-2534	40	40	.	.	PUNCT
ejpam-2534	41	1	definition	definition	NOUN
ejpam-2534	41	2	2	2	NUM
ejpam-2534	41	3	.	.	PUNCT
ejpam-2534	42	1	a	a	DET
ejpam-2534	42	2	proper	proper	ADJ
ejpam-2534	42	3	submodule	submodule	NOUN
ejpam-2534	42	4	n	n	PROPN
ejpam-2534	42	5	of	of	ADP
ejpam-2534	42	6	an	an	DET
ejpam-2534	42	7	r	r	NOUN
ejpam-2534	42	8	-	-	PUNCT
ejpam-2534	42	9	module	module	NOUN
ejpam-2534	42	10	m	m	NOUN
ejpam-2534	42	11	is	be	AUX
ejpam-2534	42	12	called	call	VERB
ejpam-2534	42	13	a	a	DET
ejpam-2534	42	14	p(n)-locally	p(n)-locally	ADV
ejpam-2534	42	15	weakly	weakly	ADJ
ejpam-2534	42	16	primary	primary	ADJ
ejpam-2534	42	17	submodule	submodule	NOUN
ejpam-2534	42	18	of	of	ADP
ejpam-2534	42	19	m	m	PROPN
ejpam-2534	42	20	if	if	SCONJ
ejpam-2534	42	21	nm	nm	ADJ
ejpam-2534	42	22	is	be	AUX
ejpam-2534	42	23	a	a	DET
ejpam-2534	42	24	weakly	weakly	ADJ
ejpam-2534	42	25	primary	primary	ADJ
ejpam-2534	42	26	submodule	submodule	NOUN
ejpam-2534	42	27	of	of	ADP
ejpam-2534	42	28	mm	mm	PROPN
ejpam-2534	42	29	for	for	ADP
ejpam-2534	42	30	every	every	DET
ejpam-2534	42	31	maximal	maximal	ADJ
ejpam-2534	42	32	ideal	ideal	NOUN
ejpam-2534	42	33	m	m	VERB
ejpam-2534	42	34	where	where	SCONJ
ejpam-2534	42	35	p(n	p(n	NOUN
ejpam-2534	42	36	)	)	PUNCT
ejpam-2534	42	37	⊆m	⊆m	NOUN
ejpam-2534	42	38	.	.	PUNCT
ejpam-2534	43	1	lemma	lemma	PROPN
ejpam-2534	43	2	1	1	X
ejpam-2534	43	3	.	.	PUNCT
ejpam-2534	44	1	let	let	VERB
ejpam-2534	44	2	n	n	PRON
ejpam-2534	44	3	be	be	AUX
ejpam-2534	44	4	a	a	DET
ejpam-2534	44	5	proper	proper	ADJ
ejpam-2534	44	6	submodule	submodule	NOUN
ejpam-2534	44	7	of	of	ADP
ejpam-2534	44	8	an	an	DET
ejpam-2534	44	9	r	r	NOUN
ejpam-2534	44	10	-	-	PUNCT
ejpam-2534	44	11	module	module	NOUN
ejpam-2534	44	12	m.	m.	NOUN
ejpam-2534	44	13	then	then	ADV
ejpam-2534	44	14	rad(n	rad(n	VERB
ejpam-2534	44	15	:	:	PUNCT
ejpam-2534	44	16	m	m	X
ejpam-2534	44	17	)	)	PUNCT
ejpam-2534	45	1	⊆	⊆	NUM
ejpam-2534	45	2	p(n	p(n	PROPN
ejpam-2534	45	3	)	)	PUNCT
ejpam-2534	45	4	.	.	PUNCT
ejpam-2534	46	1	proof	proof	NOUN
ejpam-2534	46	2	.	.	PUNCT
ejpam-2534	47	1	let	let	VERB
ejpam-2534	47	2	r	r	NOUN
ejpam-2534	47	3	∈	∈	PROPN
ejpam-2534	47	4	rad(n	rad(n	NOUN
ejpam-2534	47	5	:	:	PUNCT
ejpam-2534	47	6	m	m	NUM
ejpam-2534	47	7	)	)	PUNCT
ejpam-2534	47	8	.	.	PUNCT
ejpam-2534	48	1	then	then	ADV
ejpam-2534	48	2	rnm	rnm	VERB
ejpam-2534	48	3	⊆	⊆	NUM
ejpam-2534	48	4	n	n	NOUN
ejpam-2534	48	5	for	for	ADP
ejpam-2534	48	6	some	some	DET
ejpam-2534	48	7	positive	positive	ADJ
ejpam-2534	48	8	integer	integer	NOUN
ejpam-2534	48	9	n.	n.	NOUN
ejpam-2534	48	10	there	there	PRON
ejpam-2534	48	11	exists	exist	VERB
ejpam-2534	48	12	m	m	VERB
ejpam-2534	48	13	∈	∈	PROPN
ejpam-2534	48	14	m	m	VERB
ejpam-2534	48	15	\	\	NOUN
ejpam-2534	49	1	n	n	CCONJ
ejpam-2534	49	2	such	such	ADJ
ejpam-2534	49	3	that	that	DET
ejpam-2534	49	4	rnm	rnm	NOUN
ejpam-2534	49	5	∈	∈	PROPN
ejpam-2534	49	6	n	n	NOUN
ejpam-2534	49	7	.	.	PUNCT
ejpam-2534	50	1	then	then	ADV
ejpam-2534	50	2	r	r	PROPN
ejpam-2534	50	3	∈	∈	PROPN
ejpam-2534	50	4	p(n	p(n	PROPN
ejpam-2534	50	5	)	)	PUNCT
ejpam-2534	50	6	.	.	PUNCT
ejpam-2534	51	1	thus	thus	ADV
ejpam-2534	51	2	rad(n	rad(n	NOUN
ejpam-2534	51	3	:	:	PUNCT
ejpam-2534	51	4	m	m	X
ejpam-2534	51	5	)	)	PUNCT
ejpam-2534	51	6	⊆	⊆	NUM
ejpam-2534	51	7	p(n	p(n	PROPN
ejpam-2534	51	8	)	)	PUNCT
ejpam-2534	51	9	.	.	PUNCT
ejpam-2534	52	1	every	every	DET
ejpam-2534	52	2	primary	primary	ADJ
ejpam-2534	52	3	submodule	submodule	NOUN
ejpam-2534	52	4	n	n	NUM
ejpam-2534	52	5	is	be	AUX
ejpam-2534	52	6	proposed	propose	VERB
ejpam-2534	52	7	as	as	ADP
ejpam-2534	52	8	p(n)-locally	p(n)-locally	ADV
ejpam-2534	52	9	primary	primary	ADJ
ejpam-2534	52	10	submodule	submodule	NOUN
ejpam-2534	52	11	and	and	CCONJ
ejpam-2534	52	12	every	every	DET
ejpam-2534	52	13	weakly	weakly	ADJ
ejpam-2534	52	14	primary	primary	ADJ
ejpam-2534	52	15	submodule	submodule	NOUN
ejpam-2534	52	16	n	n	NUM
ejpam-2534	52	17	is	be	AUX
ejpam-2534	52	18	proposed	propose	VERB
ejpam-2534	52	19	as	as	ADP
ejpam-2534	52	20	p(n)-locally	p(n)-locally	ADV
ejpam-2534	52	21	weakly	weakly	ADJ
ejpam-2534	52	22	primary	primary	ADJ
ejpam-2534	52	23	submodule	submodule	NOUN
ejpam-2534	52	24	in	in	ADP
ejpam-2534	52	25	the	the	DET
ejpam-2534	52	26	following	follow	VERB
ejpam-2534	52	27	propositions	proposition	NOUN
ejpam-2534	52	28	,	,	PUNCT
ejpam-2534	52	29	respectively	respectively	ADV
ejpam-2534	52	30	.	.	PUNCT
ejpam-2534	53	1	proposition	proposition	NOUN
ejpam-2534	53	2	1	1	NUM
ejpam-2534	53	3	.	.	PUNCT
ejpam-2534	54	1	a	a	DET
ejpam-2534	54	2	primary	primary	ADJ
ejpam-2534	54	3	submodule	submodule	NOUN
ejpam-2534	54	4	n	n	PROPN
ejpam-2534	54	5	of	of	ADP
ejpam-2534	54	6	an	an	DET
ejpam-2534	54	7	r	r	NOUN
ejpam-2534	54	8	-	-	PUNCT
ejpam-2534	54	9	module	module	NOUN
ejpam-2534	54	10	m	m	NOUN
ejpam-2534	54	11	is	be	AUX
ejpam-2534	54	12	a	a	DET
ejpam-2534	54	13	p(n)-locally	p(n)-locally	ADV
ejpam-2534	54	14	primary	primary	ADJ
ejpam-2534	54	15	submodule	submodule	NOUN
ejpam-2534	54	16	.	.	PUNCT
ejpam-2534	55	1	proof	proof	NOUN
ejpam-2534	55	2	.	.	PUNCT
ejpam-2534	56	1	let	let	VERB
ejpam-2534	56	2	m	m	PRON
ejpam-2534	56	3	be	be	AUX
ejpam-2534	56	4	a	a	DET
ejpam-2534	56	5	maximal	maximal	ADJ
ejpam-2534	56	6	ideal	ideal	NOUN
ejpam-2534	56	7	of	of	ADP
ejpam-2534	56	8	r	r	NOUN
ejpam-2534	56	9	where	where	SCONJ
ejpam-2534	56	10	p(n	p(n	NOUN
ejpam-2534	56	11	)	)	PUNCT
ejpam-2534	56	12	⊆m	⊆m	NOUN
ejpam-2534	56	13	.	.	PUNCT
ejpam-2534	57	1	by	by	ADP
ejpam-2534	57	2	the	the	DET
ejpam-2534	57	3	previous	previous	ADJ
ejpam-2534	57	4	lemma	lemma	PROPN
ejpam-2534	57	5	,	,	PUNCT
ejpam-2534	57	6	we	we	PRON
ejpam-2534	57	7	say	say	VERB
ejpam-2534	57	8	that	that	SCONJ
ejpam-2534	57	9	rad(n	rad(n	VERB
ejpam-2534	57	10	:	:	PUNCT
ejpam-2534	57	11	m	m	X
ejpam-2534	57	12	)	)	PUNCT
ejpam-2534	58	1	⊆	⊆	X
ejpam-2534	58	2	p(n	p(n	PROPN
ejpam-2534	58	3	)	)	PUNCT
ejpam-2534	58	4	⊆	⊆	NUM
ejpam-2534	58	5	m.	m.	NOUN
ejpam-2534	58	6	nm	nm	NOUN
ejpam-2534	58	7	is	be	AUX
ejpam-2534	58	8	a	a	DET
ejpam-2534	58	9	proper	proper	ADJ
ejpam-2534	58	10	submodule	submodule	NOUN
ejpam-2534	58	11	of	of	ADP
ejpam-2534	58	12	mm	mm	PROPN
ejpam-2534	58	13	.	.	PUNCT
ejpam-2534	59	1	indeed	indeed	ADV
ejpam-2534	59	2	,	,	PUNCT
ejpam-2534	59	3	if	if	SCONJ
ejpam-2534	59	4	nm	nm	ADV
ejpam-2534	59	5	=	=	SYM
ejpam-2534	59	6	mm	mm	PROPN
ejpam-2534	59	7	,	,	PUNCT
ejpam-2534	59	8	then	then	ADV
ejpam-2534	59	9	m	m	VERB
ejpam-2534	59	10	1	1	NUM
ejpam-2534	59	11	∈	∈	NOUN
ejpam-2534	59	12	mm	mm	NOUN
ejpam-2534	59	13	for	for	ADP
ejpam-2534	59	14	any	any	DET
ejpam-2534	59	15	m	m	NOUN
ejpam-2534	59	16	∈	∈	NOUN
ejpam-2534	59	17	m	m	NOUN
ejpam-2534	59	18	.	.	PUNCT
ejpam-2534	60	1	then	then	ADV
ejpam-2534	60	2	rm	rm	PROPN
ejpam-2534	60	3	∈	∈	PROPN
ejpam-2534	60	4	n	n	PROPN
ejpam-2534	60	5	for	for	ADP
ejpam-2534	60	6	some	some	DET
ejpam-2534	60	7	r	r	NOUN
ejpam-2534	60	8	/∈	/∈	PUNCT
ejpam-2534	60	9	m.	m.	NOUN
ejpam-2534	60	10	we	we	PRON
ejpam-2534	60	11	get	get	VERB
ejpam-2534	60	12	rn	rn	PROPN
ejpam-2534	60	13	/∈	/∈	PUNCT
ejpam-2534	60	14	rad(n	rad(n	NOUN
ejpam-2534	60	15	:	:	PUNCT
ejpam-2534	60	16	m	m	NUM
ejpam-2534	60	17	)	)	PUNCT
ejpam-2534	60	18	.	.	PUNCT
ejpam-2534	61	1	since	since	SCONJ
ejpam-2534	61	2	n	n	NUM
ejpam-2534	61	3	is	be	AUX
ejpam-2534	61	4	a	a	DET
ejpam-2534	61	5	primary	primary	ADJ
ejpam-2534	61	6	submodule	submodule	NOUN
ejpam-2534	61	7	of	of	ADP
ejpam-2534	61	8	m	m	PROPN
ejpam-2534	61	9	,	,	PUNCT
ejpam-2534	61	10	then	then	ADV
ejpam-2534	61	11	m	m	VERB
ejpam-2534	61	12	∈	∈	PROPN
ejpam-2534	61	13	n	n	NOUN
ejpam-2534	61	14	.	.	PUNCT
ejpam-2534	62	1	thus	thus	ADV
ejpam-2534	62	2	n	n	PROPN
ejpam-2534	62	3	=	=	SYM
ejpam-2534	62	4	m	m	PROPN
ejpam-2534	62	5	,	,	PUNCT
ejpam-2534	62	6	a	a	DET
ejpam-2534	62	7	contradiction	contradiction	NOUN
ejpam-2534	62	8	.	.	PUNCT
ejpam-2534	63	1	since	since	SCONJ
ejpam-2534	63	2	rad(n	rad(n	NOUN
ejpam-2534	63	3	:	:	PUNCT
ejpam-2534	63	4	m	m	X
ejpam-2534	63	5	)	)	PUNCT
ejpam-2534	63	6	∩	∩	NOUN
ejpam-2534	63	7	(	(	PUNCT
ejpam-2534	63	8	r\m	r\m	X
ejpam-2534	63	9	)	)	PUNCT
ejpam-2534	63	10	=	=	PUNCT
ejpam-2534	63	11	;	;	PUNCT
ejpam-2534	63	12	,	,	PUNCT
ejpam-2534	63	13	then	then	ADV
ejpam-2534	63	14	nm	nm	PRON
ejpam-2534	63	15	is	be	AUX
ejpam-2534	63	16	a	a	DET
ejpam-2534	63	17	primary	primary	ADJ
ejpam-2534	63	18	submodule	submodule	NOUN
ejpam-2534	63	19	of	of	ADP
ejpam-2534	63	20	mm	mm	PROPN
ejpam-2534	63	21	.	.	PUNCT
ejpam-2534	64	1	consequently	consequently	ADV
ejpam-2534	64	2	,	,	PUNCT
ejpam-2534	64	3	n	n	PRON
ejpam-2534	64	4	is	be	AUX
ejpam-2534	64	5	a	a	DET
ejpam-2534	64	6	p(n)-locally	p(n)-locally	ADV
ejpam-2534	64	7	primary	primary	ADJ
ejpam-2534	64	8	submodule	submodule	NOUN
ejpam-2534	64	9	.	.	PUNCT
ejpam-2534	65	1	g.	g.	PROPN
ejpam-2534	65	2	ulucak	ulucak	PROPN
ejpam-2534	65	3	and	and	CCONJ
ejpam-2534	65	4	r.	r.	PROPN
ejpam-2534	65	5	uregen	uregen	PROPN
ejpam-2534	65	6	/	/	SYM
ejpam-2534	65	7	eur	eur	PROPN
ejpam-2534	65	8	.	.	PUNCT
ejpam-2534	66	1	j.	j.	PROPN
ejpam-2534	66	2	pure	pure	PROPN
ejpam-2534	66	3	appl	appl	PROPN
ejpam-2534	66	4	.	.	PROPN
ejpam-2534	66	5	math	math	PROPN
ejpam-2534	66	6	,	,	PUNCT
ejpam-2534	66	7	9	9	NUM
ejpam-2534	66	8	(	(	PUNCT
ejpam-2534	66	9	2016	2016	NUM
ejpam-2534	66	10	)	)	PUNCT
ejpam-2534	66	11	,	,	PUNCT
ejpam-2534	66	12	48	48	NUM
ejpam-2534	66	13	-	-	SYM
ejpam-2534	66	14	56	56	NUM
ejpam-2534	66	15	50	50	NUM
ejpam-2534	66	16	proposition	proposition	NOUN
ejpam-2534	66	17	2	2	NUM
ejpam-2534	66	18	.	.	PUNCT
ejpam-2534	66	19	a	a	DET
ejpam-2534	66	20	weakly	weakly	ADJ
ejpam-2534	66	21	primary	primary	ADJ
ejpam-2534	66	22	submodule	submodule	NOUN
ejpam-2534	66	23	n	n	PROPN
ejpam-2534	66	24	of	of	ADP
ejpam-2534	66	25	an	an	DET
ejpam-2534	66	26	r	r	NOUN
ejpam-2534	66	27	-	-	PUNCT
ejpam-2534	66	28	module	module	NOUN
ejpam-2534	66	29	m	m	NOUN
ejpam-2534	66	30	is	be	AUX
ejpam-2534	66	31	a	a	DET
ejpam-2534	66	32	p(n)-locally	p(n)-locally	ADV
ejpam-2534	66	33	weakly	weakly	ADJ
ejpam-2534	66	34	primary	primary	ADJ
ejpam-2534	66	35	submodule	submodule	NOUN
ejpam-2534	66	36	.	.	PUNCT
ejpam-2534	67	1	proof	proof	NOUN
ejpam-2534	67	2	.	.	PUNCT
ejpam-2534	68	1	suppose	suppose	VERB
ejpam-2534	68	2	that	that	SCONJ
ejpam-2534	68	3	m	m	PROPN
ejpam-2534	68	4	is	be	AUX
ejpam-2534	68	5	a	a	DET
ejpam-2534	68	6	maximal	maximal	ADJ
ejpam-2534	68	7	ideal	ideal	NOUN
ejpam-2534	68	8	of	of	ADP
ejpam-2534	68	9	r	r	NOUN
ejpam-2534	68	10	where	where	SCONJ
ejpam-2534	68	11	p(n	p(n	NOUN
ejpam-2534	68	12	)	)	PUNCT
ejpam-2534	68	13	⊆m	⊆m	NOUN
ejpam-2534	68	14	.	.	PUNCT
ejpam-2534	69	1	in	in	ADP
ejpam-2534	69	2	the	the	DET
ejpam-2534	69	3	same	same	ADJ
ejpam-2534	69	4	manner	manner	NOUN
ejpam-2534	69	5	as	as	ADP
ejpam-2534	69	6	in	in	ADP
ejpam-2534	69	7	the	the	DET
ejpam-2534	69	8	proof	proof	NOUN
ejpam-2534	69	9	of	of	ADP
ejpam-2534	69	10	the	the	DET
ejpam-2534	69	11	previous	previous	ADJ
ejpam-2534	69	12	proposition	proposition	NOUN
ejpam-2534	69	13	,	,	PUNCT
ejpam-2534	69	14	we	we	PRON
ejpam-2534	69	15	have	have	VERB
ejpam-2534	69	16	that	that	SCONJ
ejpam-2534	69	17	nm	nm	NOUN
ejpam-2534	69	18	is	be	AUX
ejpam-2534	69	19	a	a	DET
ejpam-2534	69	20	proper	proper	ADJ
ejpam-2534	69	21	submodule	submodule	NOUN
ejpam-2534	69	22	of	of	ADP
ejpam-2534	69	23	mm	mm	PROPN
ejpam-2534	69	24	.	.	PUNCT
ejpam-2534	70	1	let	let	VERB
ejpam-2534	70	2	0	0	NUM
ejpam-2534	70	3	m	m	NOUN
ejpam-2534	70	4	6=	6=	ADP
ejpam-2534	70	5	r	r	NOUN
ejpam-2534	70	6	s	s	NOUN
ejpam-2534	70	7	m	m	NOUN
ejpam-2534	70	8	p	p	NOUN
ejpam-2534	70	9	∈	∈	ADJ
ejpam-2534	70	10	nm	nm	NOUN
ejpam-2534	70	11	for	for	ADP
ejpam-2534	70	12	some	some	DET
ejpam-2534	70	13	r	r	NOUN
ejpam-2534	70	14	s	s	NOUN
ejpam-2534	70	15	∈	∈	PROPN
ejpam-2534	70	16	rm	rm	NOUN
ejpam-2534	70	17	and	and	CCONJ
ejpam-2534	70	18	m	m	PROPN
ejpam-2534	70	19	p	p	NOUN
ejpam-2534	70	20	∈	∈	ADJ
ejpam-2534	70	21	mm	mm	NOUN
ejpam-2534	70	22	(	(	PUNCT
ejpam-2534	70	23	for	for	ADP
ejpam-2534	70	24	some	some	DET
ejpam-2534	70	25	r	r	NOUN
ejpam-2534	70	26	∈	∈	NOUN
ejpam-2534	70	27	r	r	NOUN
ejpam-2534	70	28	,	,	PUNCT
ejpam-2534	70	29	m	m	VERB
ejpam-2534	70	30	∈	∈	NOUN
ejpam-2534	70	31	m	m	NOUN
ejpam-2534	70	32	and	and	CCONJ
ejpam-2534	70	33	s	s	PROPN
ejpam-2534	70	34	,	,	PUNCT
ejpam-2534	70	35	p	p	PROPN
ejpam-2534	70	36	∈	∈	PROPN
ejpam-2534	70	37	r\m	r\m	NOUN
ejpam-2534	70	38	)	)	PUNCT
ejpam-2534	70	39	.	.	PUNCT
ejpam-2534	71	1	then	then	ADV
ejpam-2534	71	2	there	there	PRON
ejpam-2534	71	3	is	be	VERB
ejpam-2534	71	4	a	a	DET
ejpam-2534	71	5	q	q	PUNCT
ejpam-2534	71	6	∈	∈	PROPN
ejpam-2534	71	7	r	r	NOUN
ejpam-2534	71	8	\m	\m	NOUN
ejpam-2534	71	9	such	such	ADJ
ejpam-2534	71	10	that	that	SCONJ
ejpam-2534	71	11	qrm	qrm	PROPN
ejpam-2534	71	12	∈	∈	PROPN
ejpam-2534	71	13	n	n	X
ejpam-2534	71	14	.	.	PUNCT
ejpam-2534	72	1	assume	assume	VERB
ejpam-2534	72	2	that	that	SCONJ
ejpam-2534	72	3	qrm=	qrm=	PROPN
ejpam-2534	72	4	0	0	X
ejpam-2534	72	5	.	.	PUNCT
ejpam-2534	73	1	then	then	ADV
ejpam-2534	73	2	r	r	NOUN
ejpam-2534	73	3	s	s	VERB
ejpam-2534	73	4	m	m	NOUN
ejpam-2534	73	5	p	p	NOUN
ejpam-2534	73	6	=	=	X
ejpam-2534	73	7	q	q	NOUN
ejpam-2534	74	1	q	q	X
ejpam-2534	74	2	r	r	NOUN
ejpam-2534	74	3	s	s	NOUN
ejpam-2534	74	4	m	m	NOUN
ejpam-2534	74	5	p	p	NOUN
ejpam-2534	74	6	=	=	X
ejpam-2534	74	7	qrm	qrm	PROPN
ejpam-2534	74	8	qsp	qsp	NOUN
ejpam-2534	74	9	=	=	SYM
ejpam-2534	74	10	0	0	NUM
ejpam-2534	74	11	m	m	NOUN
ejpam-2534	74	12	,	,	PUNCT
ejpam-2534	74	13	this	this	PRON
ejpam-2534	74	14	is	be	AUX
ejpam-2534	74	15	a	a	DET
ejpam-2534	74	16	contradiction	contradiction	NOUN
ejpam-2534	74	17	.	.	PUNCT
ejpam-2534	75	1	so	so	ADV
ejpam-2534	75	2	0	0	NUM
ejpam-2534	75	3	6=	6=	NUM
ejpam-2534	75	4	qrm	qrm	PROPN
ejpam-2534	75	5	∈	∈	PROPN
ejpam-2534	75	6	n	n	ADV
ejpam-2534	75	7	.	.	PUNCT
ejpam-2534	76	1	as	as	ADP
ejpam-2534	76	2	rad(n	rad(n	NOUN
ejpam-2534	76	3	:	:	PUNCT
ejpam-2534	76	4	m	m	X
ejpam-2534	76	5	)	)	PUNCT
ejpam-2534	76	6	⊆	⊆	NUM
ejpam-2534	76	7	p(n	p(n	NOUN
ejpam-2534	76	8	)	)	PUNCT
ejpam-2534	76	9	⊆m	⊆m	NOUN
ejpam-2534	76	10	,	,	PUNCT
ejpam-2534	76	11	then	then	ADV
ejpam-2534	76	12	q	q	X
ejpam-2534	76	13	/∈	/∈	PUNCT
ejpam-2534	76	14	rad(n	rad(n	NOUN
ejpam-2534	76	15	:	:	PUNCT
ejpam-2534	76	16	m	m	NUM
ejpam-2534	76	17	)	)	PUNCT
ejpam-2534	76	18	.	.	PUNCT
ejpam-2534	77	1	thus	thus	ADV
ejpam-2534	77	2	rm	rm	PROPN
ejpam-2534	77	3	∈	∈	PROPN
ejpam-2534	77	4	n	n	CCONJ
ejpam-2534	77	5	since	since	SCONJ
ejpam-2534	77	6	n	n	NUM
ejpam-2534	77	7	is	be	AUX
ejpam-2534	77	8	a	a	DET
ejpam-2534	77	9	weakly	weakly	ADJ
ejpam-2534	77	10	primary	primary	ADJ
ejpam-2534	77	11	submodule	submodule	NOUN
ejpam-2534	77	12	.	.	PUNCT
ejpam-2534	78	1	it	it	PRON
ejpam-2534	78	2	is	be	AUX
ejpam-2534	78	3	clear	clear	ADJ
ejpam-2534	78	4	that	that	SCONJ
ejpam-2534	78	5	rm	rm	PROPN
ejpam-2534	78	6	6=	6=	PROPN
ejpam-2534	78	7	0	0	NUM
ejpam-2534	78	8	.	.	PUNCT
ejpam-2534	79	1	hence	hence	ADV
ejpam-2534	79	2	0	0	NUM
ejpam-2534	79	3	6=	6=	NUM
ejpam-2534	79	4	rm	rm	PROPN
ejpam-2534	79	5	∈	∈	PROPN
ejpam-2534	79	6	n	n	PRON
ejpam-2534	79	7	implies	imply	VERB
ejpam-2534	79	8	that	that	SCONJ
ejpam-2534	79	9	m	m	VERB
ejpam-2534	79	10	∈	∈	ADJ
ejpam-2534	79	11	n	n	NOUN
ejpam-2534	79	12	or	or	CCONJ
ejpam-2534	79	13	rnm	rnm	VERB
ejpam-2534	79	14	⊆	⊆	NUM
ejpam-2534	79	15	n	n	NOUN
ejpam-2534	79	16	for	for	ADP
ejpam-2534	79	17	some	some	DET
ejpam-2534	79	18	positive	positive	ADJ
ejpam-2534	79	19	integer	integer	NOUN
ejpam-2534	79	20	n.	n.	NOUN
ejpam-2534	79	21	thus	thus	ADV
ejpam-2534	79	22	we	we	PRON
ejpam-2534	79	23	get	get	VERB
ejpam-2534	79	24	m	m	PRON
ejpam-2534	79	25	p	p	NOUN
ejpam-2534	79	26	∈	∈	PROPN
ejpam-2534	79	27	nm	nm	NOUN
ejpam-2534	79	28	or	or	CCONJ
ejpam-2534	79	29	rn	rn	PROPN
ejpam-2534	79	30	sn	sn	PROPN
ejpam-2534	79	31	mm	mm	PROPN
ejpam-2534	79	32	⊆	⊆	NUM
ejpam-2534	79	33	nm	nm	NOUN
ejpam-2534	79	34	for	for	ADP
ejpam-2534	79	35	some	some	DET
ejpam-2534	79	36	positive	positive	ADJ
ejpam-2534	79	37	integer	integer	NOUN
ejpam-2534	79	38	n	n	X
ejpam-2534	79	39	by	by	ADP
ejpam-2534	79	40	[	[	X
ejpam-2534	79	41	4	4	NUM
ejpam-2534	79	42	,	,	PUNCT
ejpam-2534	79	43	corollary	corollary	ADJ
ejpam-2534	79	44	2.9	2.9	NUM
ejpam-2534	79	45	]	]	PUNCT
ejpam-2534	79	46	.	.	PUNCT
ejpam-2534	80	1	then	then	ADV
ejpam-2534	80	2	we	we	PRON
ejpam-2534	80	3	get	get	VERB
ejpam-2534	80	4	that	that	SCONJ
ejpam-2534	80	5	nm	nm	NOUN
ejpam-2534	80	6	is	be	AUX
ejpam-2534	80	7	a	a	DET
ejpam-2534	80	8	weakly	weakly	ADJ
ejpam-2534	80	9	primary	primary	ADJ
ejpam-2534	80	10	submodule	submodule	NOUN
ejpam-2534	80	11	of	of	ADP
ejpam-2534	80	12	mm	mm	PROPN
ejpam-2534	80	13	.	.	PUNCT
ejpam-2534	81	1	consequently	consequently	ADV
ejpam-2534	81	2	,	,	PUNCT
ejpam-2534	81	3	n	n	PRON
ejpam-2534	81	4	is	be	AUX
ejpam-2534	81	5	a	a	DET
ejpam-2534	81	6	p(n)-locally	p(n)-locally	ADV
ejpam-2534	81	7	weakly	weakly	ADJ
ejpam-2534	81	8	primary	primary	ADJ
ejpam-2534	81	9	submodule	submodule	NOUN
ejpam-2534	81	10	.	.	PUNCT
ejpam-2534	82	1	corollary	corollary	ADJ
ejpam-2534	82	2	1	1	NUM
ejpam-2534	82	3	.	.	PUNCT
ejpam-2534	83	1	let	let	VERB
ejpam-2534	83	2	n	n	PRON
ejpam-2534	83	3	be	be	AUX
ejpam-2534	83	4	a	a	DET
ejpam-2534	83	5	proper	proper	ADJ
ejpam-2534	83	6	submodule	submodule	NOUN
ejpam-2534	83	7	of	of	ADP
ejpam-2534	83	8	an	an	DET
ejpam-2534	83	9	r	r	NOUN
ejpam-2534	83	10	-	-	PUNCT
ejpam-2534	83	11	module	module	NOUN
ejpam-2534	83	12	m.	m.	NOUN
ejpam-2534	83	13	if	if	SCONJ
ejpam-2534	83	14	n	n	PRON
ejpam-2534	83	15	is	be	AUX
ejpam-2534	83	16	primary	primary	ADJ
ejpam-2534	83	17	,	,	PUNCT
ejpam-2534	83	18	then	then	ADV
ejpam-2534	83	19	n	n	PRON
ejpam-2534	83	20	is	be	AUX
ejpam-2534	83	21	p(n)locally	p(n)locally	ADV
ejpam-2534	83	22	weakly	weakly	ADJ
ejpam-2534	83	23	primary	primary	ADJ
ejpam-2534	83	24	.	.	PUNCT
ejpam-2534	84	1	proof	proof	NOUN
ejpam-2534	84	2	.	.	PUNCT
ejpam-2534	85	1	assume	assume	VERB
ejpam-2534	85	2	that	that	SCONJ
ejpam-2534	85	3	n	n	PRON
ejpam-2534	85	4	is	be	AUX
ejpam-2534	85	5	a	a	DET
ejpam-2534	85	6	primary	primary	ADJ
ejpam-2534	85	7	submodule	submodule	NOUN
ejpam-2534	85	8	.	.	PUNCT
ejpam-2534	86	1	then	then	ADV
ejpam-2534	86	2	n	n	PRON
ejpam-2534	86	3	is	be	AUX
ejpam-2534	86	4	a	a	DET
ejpam-2534	86	5	weakly	weakly	ADJ
ejpam-2534	86	6	primary	primary	ADJ
ejpam-2534	86	7	submodule	submodule	NOUN
ejpam-2534	86	8	.	.	PUNCT
ejpam-2534	87	1	thus	thus	ADV
ejpam-2534	87	2	,	,	PUNCT
ejpam-2534	87	3	n	n	PRON
ejpam-2534	87	4	is	be	AUX
ejpam-2534	87	5	a	a	DET
ejpam-2534	87	6	p(n)-locally	p(n)-locally	ADV
ejpam-2534	87	7	weakly	weakly	ADJ
ejpam-2534	87	8	primary	primary	ADJ
ejpam-2534	87	9	submodule	submodule	NOUN
ejpam-2534	87	10	by	by	ADP
ejpam-2534	87	11	proposition	proposition	NOUN
ejpam-2534	87	12	2	2	NUM
ejpam-2534	87	13	.	.	X
ejpam-2534	87	14	note	note	VERB
ejpam-2534	87	15	that	that	SCONJ
ejpam-2534	87	16	if	if	SCONJ
ejpam-2534	87	17	n	n	PRON
ejpam-2534	87	18	is	be	AUX
ejpam-2534	87	19	a	a	DET
ejpam-2534	87	20	p(n)-locally	p(n)-locally	ADV
ejpam-2534	87	21	primary	primary	ADJ
ejpam-2534	87	22	submodule	submodule	NOUN
ejpam-2534	87	23	of	of	ADP
ejpam-2534	87	24	m	m	PROPN
ejpam-2534	87	25	,	,	PUNCT
ejpam-2534	87	26	then	then	ADV
ejpam-2534	87	27	n	n	PRON
ejpam-2534	87	28	is	be	AUX
ejpam-2534	87	29	a	a	DET
ejpam-2534	87	30	p(n)-locally	p(n)-locally	ADV
ejpam-2534	87	31	weakly	weakly	ADJ
ejpam-2534	87	32	primary	primary	ADJ
ejpam-2534	87	33	submodule	submodule	NOUN
ejpam-2534	87	34	of	of	ADP
ejpam-2534	87	35	m	m	PROPN
ejpam-2534	87	36	.	.	PUNCT
ejpam-2534	88	1	we	we	PRON
ejpam-2534	88	2	give	give	VERB
ejpam-2534	88	3	an	an	DET
ejpam-2534	88	4	example	example	NOUN
ejpam-2534	88	5	to	to	PART
ejpam-2534	88	6	show	show	VERB
ejpam-2534	88	7	the	the	DET
ejpam-2534	88	8	converse	converse	NOUN
ejpam-2534	88	9	is	be	AUX
ejpam-2534	88	10	not	not	PART
ejpam-2534	88	11	true	true	ADJ
ejpam-2534	88	12	.	.	PUNCT
ejpam-2534	89	1	example	example	NOUN
ejpam-2534	89	2	1	1	NUM
ejpam-2534	89	3	.	.	X
ejpam-2534	89	4	consider	consider	VERB
ejpam-2534	89	5	r=	r=	PROPN
ejpam-2534	89	6	f[x	f[x	PROPN
ejpam-2534	89	7	,	,	PUNCT
ejpam-2534	89	8	y	y	PROPN
ejpam-2534	89	9	,	,	PUNCT
ejpam-2534	89	10	z]−module	z]−module	NOUN
ejpam-2534	89	11	m	m	PROPN
ejpam-2534	89	12	=	=	PROPN
ejpam-2534	89	13	f[x	f[x	PROPN
ejpam-2534	89	14	,	,	PUNCT
ejpam-2534	89	15	y	y	PROPN
ejpam-2534	89	16	,	,	PUNCT
ejpam-2534	89	17	z]	z]	PROPN
ejpam-2534	89	18	�	�	PROPN
ejpam-2534	89	19	(x	(x	PROPN
ejpam-2534	89	20	2	2	NUM
ejpam-2534	89	21	,	,	PUNCT
ejpam-2534	89	22	y	y	PROPN
ejpam-2534	89	23	z	z	PROPN
ejpam-2534	89	24	)	)	PUNCT
ejpam-2534	89	25	and	and	CCONJ
ejpam-2534	89	26	the	the	DET
ejpam-2534	89	27	zero	zero	NUM
ejpam-2534	89	28	submodule	submodule	NOUN
ejpam-2534	89	29	n	n	NOUN
ejpam-2534	89	30	=	=	SYM
ejpam-2534	89	31	(	(	PUNCT
ejpam-2534	89	32	0	0	NUM
ejpam-2534	89	33	)	)	PUNCT
ejpam-2534	89	34	of	of	ADP
ejpam-2534	89	35	m.	m.	NOUN
ejpam-2534	89	36	one	one	PRON
ejpam-2534	89	37	can	can	AUX
ejpam-2534	89	38	easily	easily	ADV
ejpam-2534	89	39	see	see	VERB
ejpam-2534	89	40	that	that	SCONJ
ejpam-2534	89	41	p(n	p(n	NOUN
ejpam-2534	89	42	)	)	PUNCT
ejpam-2534	89	43	=	=	SYM
ejpam-2534	89	44	{	{	PUNCT
ejpam-2534	89	45	0	0	NUM
ejpam-2534	89	46	,	,	PUNCT
ejpam-2534	89	47	x	x	SYM
ejpam-2534	89	48	,	,	PUNCT
ejpam-2534	89	49	y	y	PROPN
ejpam-2534	89	50	,	,	PUNCT
ejpam-2534	89	51	z	z	NOUN
ejpam-2534	89	52	,	,	PUNCT
ejpam-2534	89	53	.	.	PUNCT
ejpam-2534	89	54	.	.	PUNCT
ejpam-2534	90	1	.	.	PUNCT
ejpam-2534	90	2	}	}	PUNCT
ejpam-2534	90	3	.	.	PUNCT
ejpam-2534	91	1	note	note	VERB
ejpam-2534	91	2	that	that	SCONJ
ejpam-2534	91	3	p(n	p(n	NOUN
ejpam-2534	91	4	)	)	PUNCT
ejpam-2534	91	5	⊆m=	⊆m=	NOUN
ejpam-2534	91	6	(	(	PUNCT
ejpam-2534	91	7	x	x	PROPN
ejpam-2534	91	8	,	,	PUNCT
ejpam-2534	91	9	y	y	PROPN
ejpam-2534	91	10	,	,	PUNCT
ejpam-2534	91	11	z	z	NOUN
ejpam-2534	91	12	)	)	PUNCT
ejpam-2534	91	13	which	which	PRON
ejpam-2534	91	14	is	be	AUX
ejpam-2534	91	15	the	the	DET
ejpam-2534	91	16	unique	unique	ADJ
ejpam-2534	91	17	maximal	maximal	ADJ
ejpam-2534	91	18	ideal	ideal	NOUN
ejpam-2534	91	19	of	of	ADP
ejpam-2534	91	20	r.	r.	PROPN
ejpam-2534	91	21	then	then	ADV
ejpam-2534	91	22	nm	nm	VERB
ejpam-2534	91	23	=	=	SYM
ejpam-2534	91	24	(	(	PUNCT
ejpam-2534	91	25	0	0	NUM
ejpam-2534	91	26	)	)	PUNCT
ejpam-2534	91	27	is	be	AUX
ejpam-2534	91	28	weakly	weakly	ADJ
ejpam-2534	91	29	primary	primary	ADJ
ejpam-2534	91	30	submodule	submodule	NOUN
ejpam-2534	91	31	of	of	ADP
ejpam-2534	91	32	rm−	rm−	PROPN
ejpam-2534	91	33	module	module	NOUN
ejpam-2534	91	34	mm	mm	PROPN
ejpam-2534	91	35	.	.	PUNCT
ejpam-2534	92	1	thus	thus	ADV
ejpam-2534	92	2	n	n	ADV
ejpam-2534	92	3	is	be	AUX
ejpam-2534	92	4	p(n)−weakly	p(n)−weakly	ADV
ejpam-2534	92	5	primary	primary	ADJ
ejpam-2534	92	6	submodule	submodule	NOUN
ejpam-2534	92	7	.	.	PUNCT
ejpam-2534	93	1	but	but	CCONJ
ejpam-2534	93	2	nm	nm	ADV
ejpam-2534	93	3	is	be	AUX
ejpam-2534	93	4	not	not	PART
ejpam-2534	93	5	primary	primary	ADJ
ejpam-2534	93	6	submodule	submodule	NOUN
ejpam-2534	93	7	since	since	SCONJ
ejpam-2534	93	8	,	,	PUNCT
ejpam-2534	93	9	y	y	PROPN
ejpam-2534	93	10	1	1	NUM
ejpam-2534	93	11	·	·	PUNCT
ejpam-2534	93	12	z̄	z̄	NOUN
ejpam-2534	93	13	1	1	NUM
ejpam-2534	93	14	=	=	SYM
ejpam-2534	93	15	0̄	0̄	NUM
ejpam-2534	93	16	1	1	NUM
ejpam-2534	93	17	∈	∈	NOUN
ejpam-2534	93	18	nm	nm	NOUN
ejpam-2534	93	19	but	but	CCONJ
ejpam-2534	93	20	�	�	PROPN
ejpam-2534	93	21	y	y	PROPN
ejpam-2534	93	22	1	1	NUM
ejpam-2534	93	23	�	�	PROPN
ejpam-2534	93	24	n	n	NUM
ejpam-2534	93	25	/∈	/∈	PUNCT
ejpam-2534	94	1	(	(	PUNCT
ejpam-2534	94	2	nm	nm	INTJ
ejpam-2534	94	3	:	:	PUNCT
ejpam-2534	94	4	mm	mm	X
ejpam-2534	94	5	)	)	PUNCT
ejpam-2534	94	6	,	,	PUNCT
ejpam-2534	94	7	�	�	PROPN
ejpam-2534	94	8	z̄	z̄	PROPN
ejpam-2534	94	9	1	1	NUM
ejpam-2534	94	10	�	�	PROPN
ejpam-2534	94	11	/∈	/∈	PUNCT
ejpam-2534	94	12	nm	nm	NOUN
ejpam-2534	94	13	for	for	ADP
ejpam-2534	94	14	all	all	DET
ejpam-2534	94	15	positive	positive	ADJ
ejpam-2534	94	16	integer	integer	NOUN
ejpam-2534	94	17	n.	n.	NOUN
ejpam-2534	94	18	thus	thus	ADV
ejpam-2534	94	19	n	n	ADV
ejpam-2534	94	20	is	be	AUX
ejpam-2534	94	21	not	not	PART
ejpam-2534	94	22	p(n)−primary	p(n)−primary	ADJ
ejpam-2534	94	23	.	.	PUNCT
ejpam-2534	95	1	in	in	ADP
ejpam-2534	95	2	the	the	DET
ejpam-2534	95	3	following	follow	VERB
ejpam-2534	95	4	example	example	NOUN
ejpam-2534	95	5	,	,	PUNCT
ejpam-2534	95	6	it	it	PRON
ejpam-2534	95	7	is	be	AUX
ejpam-2534	95	8	illustrated	illustrate	VERB
ejpam-2534	95	9	that	that	SCONJ
ejpam-2534	95	10	a	a	DET
ejpam-2534	95	11	submodule	submodule	NOUN
ejpam-2534	95	12	n	n	PRON
ejpam-2534	95	13	can	can	AUX
ejpam-2534	95	14	be	be	AUX
ejpam-2534	95	15	both	both	PRON
ejpam-2534	95	16	p(n)-locally	p(n)-locally	ADV
ejpam-2534	95	17	primary	primary	ADJ
ejpam-2534	95	18	submodule	submodule	NOUN
ejpam-2534	95	19	of	of	ADP
ejpam-2534	95	20	m	m	PROPN
ejpam-2534	95	21	and	and	CCONJ
ejpam-2534	95	22	p(n)-locally	p(n)-locally	ADV
ejpam-2534	95	23	weakly	weakly	ADJ
ejpam-2534	95	24	primary	primary	ADJ
ejpam-2534	95	25	submodule	submodule	NOUN
ejpam-2534	95	26	of	of	ADP
ejpam-2534	95	27	m	m	PROPN
ejpam-2534	96	1	but	but	CCONJ
ejpam-2534	96	2	it	it	PRON
ejpam-2534	96	3	is	be	AUX
ejpam-2534	96	4	neither	neither	CCONJ
ejpam-2534	96	5	primary	primary	ADJ
ejpam-2534	96	6	submodule	submodule	NOUN
ejpam-2534	96	7	of	of	ADP
ejpam-2534	96	8	m	m	NOUN
ejpam-2534	96	9	nor	nor	CCONJ
ejpam-2534	96	10	weakly	weakly	ADJ
ejpam-2534	96	11	primary	primary	ADJ
ejpam-2534	96	12	submodule	submodule	NOUN
ejpam-2534	96	13	of	of	ADP
ejpam-2534	96	14	m	m	PROPN
ejpam-2534	96	15	.	.	PUNCT
ejpam-2534	97	1	example	example	NOUN
ejpam-2534	98	1	2	2	NUM
ejpam-2534	98	2	.	.	PUNCT
ejpam-2534	98	3	let	let	VERB
ejpam-2534	98	4	r	r	NOUN
ejpam-2534	98	5	=	=	SYM
ejpam-2534	98	6	z	z	NOUN
ejpam-2534	98	7	and	and	CCONJ
ejpam-2534	98	8	consider	consider	VERB
ejpam-2534	98	9	the	the	DET
ejpam-2534	98	10	r	r	NOUN
ejpam-2534	98	11	−	−	NOUN
ejpam-2534	98	12	module	module	NOUN
ejpam-2534	98	13	m	m	NOUN
ejpam-2534	98	14	=	=	NOUN
ejpam-2534	98	15	z12	z12	NUM
ejpam-2534	98	16	.	.	PUNCT
ejpam-2534	99	1	let	let	VERB
ejpam-2534	99	2	n	n	PRON
ejpam-2534	99	3	be	be	AUX
ejpam-2534	99	4	the	the	DET
ejpam-2534	99	5	submodule	submodule	NOUN
ejpam-2534	99	6	of	of	ADP
ejpam-2534	99	7	z12	z12	PROPN
ejpam-2534	99	8	generated	generate	VERB
ejpam-2534	99	9	by	by	ADP
ejpam-2534	99	10	6	6	NUM
ejpam-2534	99	11	.	.	PUNCT
ejpam-2534	100	1	it	it	PRON
ejpam-2534	100	2	is	be	AUX
ejpam-2534	100	3	easily	easily	ADV
ejpam-2534	100	4	seen	see	VERB
ejpam-2534	100	5	that	that	SCONJ
ejpam-2534	100	6	0	0	NUM
ejpam-2534	101	1	6=	6=	NUM
ejpam-2534	101	2	23(=	23(=	PROPN
ejpam-2534	101	3	32	32	NUM
ejpam-2534	101	4	)	)	PUNCT
ejpam-2534	101	5	∈	∈	PROPN
ejpam-2534	101	6	n	n	NOUN
ejpam-2534	101	7	but	but	CCONJ
ejpam-2534	101	8	2	2	NUM
ejpam-2534	101	9	/∈	/∈	INTJ
ejpam-2534	101	10	(	(	PUNCT
ejpam-2534	101	11	n	n	NOUN
ejpam-2534	101	12	:	:	PUNCT
ejpam-2534	101	13	m	m	X
ejpam-2534	101	14	)	)	PUNCT
ejpam-2534	101	15	and	and	CCONJ
ejpam-2534	101	16	3	3	NUM
ejpam-2534	101	17	/∈	/∈	SYM
ejpam-2534	102	1	n	n	CCONJ
ejpam-2534	102	2	(	(	PUNCT
ejpam-2534	102	3	3	3	NUM
ejpam-2534	102	4	/∈	/∈	PUNCT
ejpam-2534	102	5	(	(	PUNCT
ejpam-2534	102	6	n	n	NOUN
ejpam-2534	102	7	:	:	PUNCT
ejpam-2534	102	8	m	m	X
ejpam-2534	102	9	)	)	PUNCT
ejpam-2534	102	10	and	and	CCONJ
ejpam-2534	102	11	2	2	NUM
ejpam-2534	102	12	/∈	/∈	NUM
ejpam-2534	102	13	n	n	CCONJ
ejpam-2534	102	14	)	)	PUNCT
ejpam-2534	102	15	,	,	PUNCT
ejpam-2534	102	16	that	that	ADV
ejpam-2534	102	17	is	is	ADV
ejpam-2534	102	18	,	,	PUNCT
ejpam-2534	102	19	n	n	PRON
ejpam-2534	102	20	is	be	AUX
ejpam-2534	102	21	not	not	PART
ejpam-2534	102	22	a	a	DET
ejpam-2534	102	23	weakly	weakly	ADJ
ejpam-2534	102	24	primary	primary	ADJ
ejpam-2534	102	25	submodule	submodule	NOUN
ejpam-2534	102	26	of	of	ADP
ejpam-2534	102	27	m	m	PROPN
ejpam-2534	102	28	,	,	PUNCT
ejpam-2534	102	29	hence	hence	ADV
ejpam-2534	102	30	n	n	ADV
ejpam-2534	102	31	is	be	AUX
ejpam-2534	102	32	not	not	PART
ejpam-2534	102	33	a	a	DET
ejpam-2534	102	34	primary	primary	ADJ
ejpam-2534	102	35	submodule	submodule	NOUN
ejpam-2534	102	36	of	of	ADP
ejpam-2534	102	37	m.	m.	NOUN
ejpam-2534	102	38	assume	assume	VERB
ejpam-2534	102	39	that	that	SCONJ
ejpam-2534	102	40	n	n	PRON
ejpam-2534	102	41	is	be	AUX
ejpam-2534	102	42	not	not	PART
ejpam-2534	102	43	a	a	DET
ejpam-2534	102	44	p(n)-locally	p(n)-locally	ADV
ejpam-2534	102	45	primary	primary	ADJ
ejpam-2534	102	46	submodule	submodule	NOUN
ejpam-2534	102	47	of	of	ADP
ejpam-2534	102	48	m.	m.	NOUN
ejpam-2534	102	49	then	then	ADV
ejpam-2534	102	50	there	there	PRON
ejpam-2534	102	51	exists	exist	VERB
ejpam-2534	102	52	a	a	DET
ejpam-2534	102	53	maximal	maximal	ADJ
ejpam-2534	102	54	ideal	ideal	NOUN
ejpam-2534	102	55	m	m	NOUN
ejpam-2534	102	56	of	of	ADP
ejpam-2534	102	57	r	r	NOUN
ejpam-2534	102	58	with	with	ADP
ejpam-2534	102	59	p(n	p(n	PROPN
ejpam-2534	102	60	)	)	PUNCT
ejpam-2534	102	61	⊆	⊆	NUM
ejpam-2534	102	62	m	m	NOUN
ejpam-2534	102	63	where	where	SCONJ
ejpam-2534	102	64	nm	nm	ADV
ejpam-2534	102	65	is	be	AUX
ejpam-2534	102	66	not	not	PART
ejpam-2534	102	67	a	a	DET
ejpam-2534	102	68	primary	primary	ADJ
ejpam-2534	102	69	submodule	submodule	NOUN
ejpam-2534	102	70	of	of	ADP
ejpam-2534	102	71	mm	mm	PROPN
ejpam-2534	102	72	.	.	PUNCT
ejpam-2534	102	73	note	note	VERB
ejpam-2534	102	74	that	that	SCONJ
ejpam-2534	102	75	2,3	2,3	NUM
ejpam-2534	102	76	∈	∈	NOUN
ejpam-2534	102	77	p(n	p(n	NOUN
ejpam-2534	102	78	)	)	PUNCT
ejpam-2534	102	79	.	.	PUNCT
ejpam-2534	103	1	thus	thus	ADV
ejpam-2534	103	2	1	1	NUM
ejpam-2534	103	3	∈	∈	PROPN
ejpam-2534	103	4	m	m	NOUN
ejpam-2534	103	5	,	,	PUNCT
ejpam-2534	103	6	a	a	DET
ejpam-2534	103	7	contradiction	contradiction	NOUN
ejpam-2534	103	8	.	.	PUNCT
ejpam-2534	104	1	therefore	therefore	ADV
ejpam-2534	104	2	,	,	PUNCT
ejpam-2534	104	3	n	n	PRON
ejpam-2534	104	4	is	be	AUX
ejpam-2534	104	5	a	a	DET
ejpam-2534	104	6	p(n)-locally	p(n)-locally	ADV
ejpam-2534	104	7	primary	primary	ADJ
ejpam-2534	104	8	submodule	submodule	NOUN
ejpam-2534	104	9	of	of	ADP
ejpam-2534	104	10	m.	m.	NOUN
ejpam-2534	104	11	hence	hence	ADV
ejpam-2534	104	12	n	n	ADV
ejpam-2534	104	13	is	be	AUX
ejpam-2534	104	14	a	a	DET
ejpam-2534	104	15	p(n)-locally	p(n)-locally	ADV
ejpam-2534	104	16	weakly	weakly	ADJ
ejpam-2534	104	17	primary	primary	ADJ
ejpam-2534	104	18	submodule	submodule	NOUN
ejpam-2534	104	19	of	of	ADP
ejpam-2534	104	20	m.	m.	NOUN
ejpam-2534	104	21	theorem	theorem	NOUN
ejpam-2534	104	22	1	1	X
ejpam-2534	104	23	.	.	PUNCT
ejpam-2534	105	1	let	let	VERB
ejpam-2534	105	2	n	n	PRON
ejpam-2534	105	3	be	be	AUX
ejpam-2534	105	4	a	a	DET
ejpam-2534	105	5	proper	proper	ADJ
ejpam-2534	105	6	submodule	submodule	NOUN
ejpam-2534	105	7	of	of	ADP
ejpam-2534	105	8	an	an	DET
ejpam-2534	105	9	r	r	NOUN
ejpam-2534	105	10	-	-	PUNCT
ejpam-2534	105	11	module	module	NOUN
ejpam-2534	105	12	m.	m.	NOUN
ejpam-2534	105	13	then	then	ADV
ejpam-2534	105	14	the	the	DET
ejpam-2534	105	15	following	follow	VERB
ejpam-2534	105	16	statements	statement	NOUN
ejpam-2534	105	17	hold	hold	VERB
ejpam-2534	105	18	:	:	PUNCT
ejpam-2534	106	1	g.	g.	PROPN
ejpam-2534	106	2	ulucak	ulucak	PROPN
ejpam-2534	106	3	and	and	CCONJ
ejpam-2534	106	4	r.	r.	PROPN
ejpam-2534	106	5	uregen	uregen	PROPN
ejpam-2534	106	6	/	/	SYM
ejpam-2534	106	7	eur	eur	PROPN
ejpam-2534	106	8	.	.	PUNCT
ejpam-2534	107	1	j.	j.	PROPN
ejpam-2534	107	2	pure	pure	PROPN
ejpam-2534	107	3	appl	appl	PROPN
ejpam-2534	107	4	.	.	PROPN
ejpam-2534	107	5	math	math	PROPN
ejpam-2534	107	6	,	,	PUNCT
ejpam-2534	107	7	9	9	NUM
ejpam-2534	107	8	(	(	PUNCT
ejpam-2534	107	9	2016	2016	NUM
ejpam-2534	107	10	)	)	PUNCT
ejpam-2534	107	11	,	,	PUNCT
ejpam-2534	107	12	48	48	NUM
ejpam-2534	107	13	-	-	SYM
ejpam-2534	107	14	56	56	NUM
ejpam-2534	107	15	51	51	NUM
ejpam-2534	107	16	(	(	PUNCT
ejpam-2534	107	17	i	i	NOUN
ejpam-2534	107	18	)	)	PUNCT
ejpam-2534	107	19	n	n	PRON
ejpam-2534	107	20	is	be	AUX
ejpam-2534	107	21	a	a	DET
ejpam-2534	107	22	primary	primary	ADJ
ejpam-2534	107	23	submodule	submodule	NOUN
ejpam-2534	107	24	if	if	SCONJ
ejpam-2534	107	25	and	and	CCONJ
ejpam-2534	107	26	only	only	ADV
ejpam-2534	107	27	if	if	SCONJ
ejpam-2534	107	28	p(n	p(n	NUM
ejpam-2534	107	29	)	)	PUNCT
ejpam-2534	107	30	=	=	PUNCT
ejpam-2534	107	31	rad(n	rad(n	NOUN
ejpam-2534	107	32	:	:	PUNCT
ejpam-2534	107	33	m	m	NOUN
ejpam-2534	107	34	)	)	PUNCT
ejpam-2534	107	35	.	.	PUNCT
ejpam-2534	108	1	(	(	PUNCT
ejpam-2534	108	2	ii	ii	NOUN
ejpam-2534	108	3	)	)	PUNCT
ejpam-2534	108	4	let	let	VERB
ejpam-2534	108	5	p(0	p(0	NOUN
ejpam-2534	108	6	)	)	PUNCT
ejpam-2534	108	7	⊆	⊆	NUM
ejpam-2534	108	8	rad(n	rad(n	NOUN
ejpam-2534	108	9	:	:	PUNCT
ejpam-2534	108	10	m	m	NUM
ejpam-2534	108	11	)	)	PUNCT
ejpam-2534	108	12	.	.	PUNCT
ejpam-2534	109	1	then	then	ADV
ejpam-2534	109	2	n	n	PRON
ejpam-2534	109	3	is	be	AUX
ejpam-2534	109	4	a	a	DET
ejpam-2534	109	5	primary	primary	ADJ
ejpam-2534	109	6	submodule	submodule	NOUN
ejpam-2534	109	7	if	if	SCONJ
ejpam-2534	109	8	and	and	CCONJ
ejpam-2534	109	9	only	only	ADV
ejpam-2534	109	10	if	if	SCONJ
ejpam-2534	109	11	n	n	PRON
ejpam-2534	109	12	is	be	AUX
ejpam-2534	109	13	a	a	DET
ejpam-2534	109	14	weakly	weakly	ADJ
ejpam-2534	109	15	primary	primary	ADJ
ejpam-2534	109	16	submodule	submodule	NOUN
ejpam-2534	109	17	.	.	PUNCT
ejpam-2534	110	1	proof	proof	NOUN
ejpam-2534	110	2	.	.	PUNCT
ejpam-2534	111	1	(	(	PUNCT
ejpam-2534	111	2	i	i	NOUN
ejpam-2534	111	3	)	)	PUNCT
ejpam-2534	111	4	assume	assume	VERB
ejpam-2534	111	5	that	that	SCONJ
ejpam-2534	111	6	n	n	PRON
ejpam-2534	111	7	is	be	AUX
ejpam-2534	111	8	a	a	DET
ejpam-2534	111	9	primary	primary	ADJ
ejpam-2534	111	10	submodule	submodule	NOUN
ejpam-2534	111	11	.	.	PUNCT
ejpam-2534	112	1	let	let	VERB
ejpam-2534	112	2	r	r	NOUN
ejpam-2534	112	3	∈	∈	NOUN
ejpam-2534	112	4	p(n	p(n	PROPN
ejpam-2534	112	5	)	)	PUNCT
ejpam-2534	112	6	.	.	PUNCT
ejpam-2534	113	1	then	then	ADV
ejpam-2534	113	2	rnm	rnm	VERB
ejpam-2534	113	3	∈	∈	PROPN
ejpam-2534	113	4	n	n	NOUN
ejpam-2534	113	5	for	for	ADP
ejpam-2534	113	6	some	some	DET
ejpam-2534	113	7	positive	positive	ADJ
ejpam-2534	113	8	integer	integer	NOUN
ejpam-2534	113	9	n	n	NOUN
ejpam-2534	113	10	and	and	CCONJ
ejpam-2534	113	11	for	for	ADP
ejpam-2534	113	12	some	some	DET
ejpam-2534	113	13	m	m	NOUN
ejpam-2534	113	14	∈	∈	NOUN
ejpam-2534	113	15	m	m	VERB
ejpam-2534	113	16	\	\	PROPN
ejpam-2534	113	17	n	n	X
ejpam-2534	113	18	.	.	PUNCT
ejpam-2534	114	1	since	since	SCONJ
ejpam-2534	114	2	n	n	NUM
ejpam-2534	114	3	is	be	AUX
ejpam-2534	114	4	a	a	DET
ejpam-2534	114	5	primary	primary	ADJ
ejpam-2534	114	6	submodule	submodule	NOUN
ejpam-2534	114	7	,	,	PUNCT
ejpam-2534	114	8	then	then	ADV
ejpam-2534	114	9	(	(	PUNCT
ejpam-2534	114	10	rn)km	rn)km	X
ejpam-2534	114	11	=	=	NOUN
ejpam-2534	114	12	rnkm	rnkm	NOUN
ejpam-2534	114	13	⊆	⊆	NUM
ejpam-2534	114	14	n	n	NOUN
ejpam-2534	114	15	for	for	ADP
ejpam-2534	114	16	some	some	DET
ejpam-2534	114	17	positive	positive	ADJ
ejpam-2534	114	18	integer	integer	NOUN
ejpam-2534	114	19	k	k	NOUN
ejpam-2534	114	20	,	,	PUNCT
ejpam-2534	114	21	that	that	ADV
ejpam-2534	114	22	is	is	ADV
ejpam-2534	114	23	,	,	PUNCT
ejpam-2534	114	24	r	r	NOUN
ejpam-2534	114	25	∈	∈	PROPN
ejpam-2534	114	26	rad(n	rad(n	NOUN
ejpam-2534	114	27	:	:	PUNCT
ejpam-2534	114	28	m	m	NUM
ejpam-2534	114	29	)	)	PUNCT
ejpam-2534	114	30	.	.	PUNCT
ejpam-2534	115	1	hence	hence	ADV
ejpam-2534	115	2	p(n	p(n	PROPN
ejpam-2534	115	3	)	)	PUNCT
ejpam-2534	116	1	⊆	⊆	NUM
ejpam-2534	116	2	rad(n	rad(n	NOUN
ejpam-2534	116	3	:	:	PUNCT
ejpam-2534	116	4	m	m	NUM
ejpam-2534	116	5	)	)	PUNCT
ejpam-2534	116	6	.	.	PUNCT
ejpam-2534	117	1	by	by	ADP
ejpam-2534	117	2	lemma	lemma	PROPN
ejpam-2534	117	3	1	1	NUM
ejpam-2534	117	4	,	,	PUNCT
ejpam-2534	117	5	we	we	PRON
ejpam-2534	117	6	get	get	VERB
ejpam-2534	117	7	p(n	p(n	NOUN
ejpam-2534	117	8	)	)	PUNCT
ejpam-2534	117	9	=	=	PUNCT
ejpam-2534	117	10	rad(n	rad(n	NOUN
ejpam-2534	117	11	:	:	PUNCT
ejpam-2534	117	12	m	m	NUM
ejpam-2534	117	13	)	)	PUNCT
ejpam-2534	117	14	.	.	PUNCT
ejpam-2534	118	1	suppose	suppose	VERB
ejpam-2534	118	2	that	that	SCONJ
ejpam-2534	118	3	p(n	p(n	NOUN
ejpam-2534	118	4	)	)	PUNCT
ejpam-2534	118	5	=	=	PUNCT
ejpam-2534	118	6	rad(n	rad(n	NOUN
ejpam-2534	118	7	:	:	PUNCT
ejpam-2534	118	8	m	m	NUM
ejpam-2534	118	9	)	)	PUNCT
ejpam-2534	118	10	.	.	PUNCT
ejpam-2534	119	1	let	let	VERB
ejpam-2534	119	2	rm	rm	PROPN
ejpam-2534	119	3	∈	∈	PROPN
ejpam-2534	119	4	n	n	PROPN
ejpam-2534	119	5	and	and	CCONJ
ejpam-2534	119	6	m	m	PROPN
ejpam-2534	119	7	∈	∈	NOUN
ejpam-2534	119	8	m	m	VERB
ejpam-2534	119	9	\	\	NOUN
ejpam-2534	119	10	n	n	CCONJ
ejpam-2534	119	11	where	where	SCONJ
ejpam-2534	119	12	r	r	NOUN
ejpam-2534	119	13	∈	∈	PROPN
ejpam-2534	119	14	r	r	NOUN
ejpam-2534	119	15	,	,	PUNCT
ejpam-2534	119	16	m	m	VERB
ejpam-2534	119	17	∈	∈	NOUN
ejpam-2534	119	18	m	m	NOUN
ejpam-2534	119	19	.	.	PUNCT
ejpam-2534	120	1	then	then	ADV
ejpam-2534	120	2	r	r	PROPN
ejpam-2534	120	3	∈	∈	PROPN
ejpam-2534	120	4	p(n	p(n	PROPN
ejpam-2534	120	5	)	)	PUNCT
ejpam-2534	120	6	.	.	PUNCT
ejpam-2534	121	1	thus	thus	ADV
ejpam-2534	121	2	r	r	NOUN
ejpam-2534	121	3	∈	∈	PROPN
ejpam-2534	121	4	rad(n	rad(n	NOUN
ejpam-2534	121	5	:	:	PUNCT
ejpam-2534	121	6	m	m	X
ejpam-2534	121	7	)	)	PUNCT
ejpam-2534	121	8	,	,	PUNCT
ejpam-2534	121	9	that	that	ADV
ejpam-2534	121	10	is	is	ADV
ejpam-2534	121	11	,	,	PUNCT
ejpam-2534	121	12	rkm	rkm	PROPN
ejpam-2534	121	13	⊆	⊆	NUM
ejpam-2534	121	14	n	n	PROPN
ejpam-2534	121	15	for	for	ADP
ejpam-2534	121	16	some	some	DET
ejpam-2534	121	17	positive	positive	ADJ
ejpam-2534	121	18	integer	integer	NOUN
ejpam-2534	121	19	k.	k.	PROPN
ejpam-2534	122	1	consequently	consequently	ADV
ejpam-2534	122	2	,	,	PUNCT
ejpam-2534	122	3	n	n	PRON
ejpam-2534	122	4	is	be	AUX
ejpam-2534	122	5	a	a	DET
ejpam-2534	122	6	primary	primary	ADJ
ejpam-2534	122	7	submodule	submodule	NOUN
ejpam-2534	122	8	.	.	PUNCT
ejpam-2534	123	1	(	(	PUNCT
ejpam-2534	123	2	ii	ii	X
ejpam-2534	123	3	)	)	PUNCT
ejpam-2534	123	4	it	it	PRON
ejpam-2534	123	5	is	be	AUX
ejpam-2534	123	6	clear	clear	ADJ
ejpam-2534	123	7	that	that	SCONJ
ejpam-2534	123	8	every	every	DET
ejpam-2534	123	9	primary	primary	ADJ
ejpam-2534	123	10	submodule	submodule	NOUN
ejpam-2534	123	11	is	be	AUX
ejpam-2534	123	12	a	a	DET
ejpam-2534	123	13	weakly	weakly	ADJ
ejpam-2534	123	14	primary	primary	ADJ
ejpam-2534	123	15	submodule	submodule	NOUN
ejpam-2534	123	16	.	.	PUNCT
ejpam-2534	124	1	now	now	ADV
ejpam-2534	124	2	,	,	PUNCT
ejpam-2534	124	3	assume	assume	VERB
ejpam-2534	124	4	that	that	SCONJ
ejpam-2534	124	5	n	n	PRON
ejpam-2534	124	6	is	be	AUX
ejpam-2534	124	7	a	a	DET
ejpam-2534	124	8	weakly	weakly	ADJ
ejpam-2534	124	9	primary	primary	ADJ
ejpam-2534	124	10	submodule	submodule	NOUN
ejpam-2534	124	11	.	.	PUNCT
ejpam-2534	125	1	let	let	VERB
ejpam-2534	125	2	r	r	NOUN
ejpam-2534	125	3	∈	∈	NOUN
ejpam-2534	125	4	p(n	p(n	PROPN
ejpam-2534	125	5	)	)	PUNCT
ejpam-2534	125	6	.	.	PUNCT
ejpam-2534	126	1	then	then	ADV
ejpam-2534	126	2	rnm	rnm	VERB
ejpam-2534	126	3	∈	∈	PROPN
ejpam-2534	126	4	n	n	NOUN
ejpam-2534	126	5	for	for	ADP
ejpam-2534	126	6	some	some	DET
ejpam-2534	126	7	positive	positive	ADJ
ejpam-2534	126	8	integer	integer	NOUN
ejpam-2534	126	9	n	n	NOUN
ejpam-2534	126	10	and	and	CCONJ
ejpam-2534	126	11	for	for	ADP
ejpam-2534	126	12	some	some	DET
ejpam-2534	126	13	m	m	NOUN
ejpam-2534	126	14	∈	∈	NOUN
ejpam-2534	126	15	m	m	NOUN
ejpam-2534	126	16	\n	\n	PUNCT
ejpam-2534	126	17	.	.	PUNCT
ejpam-2534	127	1	suppose	suppose	VERB
ejpam-2534	127	2	that	that	SCONJ
ejpam-2534	127	3	rnm=	rnm=	NOUN
ejpam-2534	127	4	0	0	NUM
ejpam-2534	127	5	.	.	PUNCT
ejpam-2534	128	1	since	since	SCONJ
ejpam-2534	128	2	m	m	PROPN
ejpam-2534	128	3	∈	∈	PROPN
ejpam-2534	128	4	m	m	NOUN
ejpam-2534	128	5	\n	\n	NOUN
ejpam-2534	128	6	,	,	PUNCT
ejpam-2534	128	7	then	then	ADV
ejpam-2534	128	8	we	we	PRON
ejpam-2534	128	9	get	get	VERB
ejpam-2534	128	10	m	m	PRON
ejpam-2534	128	11	6=	6=	NUM
ejpam-2534	128	12	0	0	NUM
ejpam-2534	128	13	.	.	PUNCT
ejpam-2534	129	1	so	so	ADV
ejpam-2534	129	2	r	r	NOUN
ejpam-2534	129	3	∈	∈	PROPN
ejpam-2534	129	4	p(0	p(0	PROPN
ejpam-2534	129	5	)	)	PUNCT
ejpam-2534	129	6	.	.	PUNCT
ejpam-2534	130	1	thus	thus	ADV
ejpam-2534	130	2	r	r	NOUN
ejpam-2534	130	3	∈	∈	PROPN
ejpam-2534	130	4	rad(n	rad(n	NOUN
ejpam-2534	130	5	:	:	PUNCT
ejpam-2534	130	6	m	m	X
ejpam-2534	130	7	)	)	PUNCT
ejpam-2534	130	8	,	,	PUNCT
ejpam-2534	130	9	by	by	ADP
ejpam-2534	130	10	assumption	assumption	NOUN
ejpam-2534	130	11	.	.	PUNCT
ejpam-2534	131	1	hence	hence	ADV
ejpam-2534	131	2	p(n	p(n	PROPN
ejpam-2534	131	3	)	)	PUNCT
ejpam-2534	131	4	=	=	PUNCT
ejpam-2534	131	5	rad(n	rad(n	NOUN
ejpam-2534	131	6	:	:	PUNCT
ejpam-2534	131	7	m	m	VERB
ejpam-2534	131	8	)	)	PUNCT
ejpam-2534	131	9	by	by	ADP
ejpam-2534	131	10	lemma	lemma	PROPN
ejpam-2534	131	11	1	1	NUM
ejpam-2534	131	12	.	.	PUNCT
ejpam-2534	131	13	suppose	suppose	VERB
ejpam-2534	131	14	that	that	SCONJ
ejpam-2534	131	15	0	0	NUM
ejpam-2534	131	16	6=	6=	NUM
ejpam-2534	131	17	rnm	rnm	NOUN
ejpam-2534	131	18	∈	∈	PROPN
ejpam-2534	131	19	n	n	NOUN
ejpam-2534	131	20	.	.	PUNCT
ejpam-2534	132	1	since	since	SCONJ
ejpam-2534	132	2	m	m	PROPN
ejpam-2534	132	3	∈	∈	PROPN
ejpam-2534	132	4	m	m	VERB
ejpam-2534	132	5	\	\	NOUN
ejpam-2534	132	6	n	n	PROPN
ejpam-2534	132	7	and	and	CCONJ
ejpam-2534	132	8	n	n	PROPN
ejpam-2534	132	9	is	be	AUX
ejpam-2534	132	10	a	a	DET
ejpam-2534	132	11	weakly	weakly	ADJ
ejpam-2534	132	12	primary	primary	ADJ
ejpam-2534	132	13	submodule	submodule	NOUN
ejpam-2534	132	14	,	,	PUNCT
ejpam-2534	132	15	then	then	ADV
ejpam-2534	132	16	(	(	PUNCT
ejpam-2534	132	17	rn)km	rn)km	VERB
ejpam-2534	132	18	⊆	⊆	NUM
ejpam-2534	132	19	n	n	NOUN
ejpam-2534	132	20	for	for	ADP
ejpam-2534	132	21	some	some	DET
ejpam-2534	132	22	positive	positive	ADJ
ejpam-2534	132	23	integer	integer	NOUN
ejpam-2534	132	24	k	k	NOUN
ejpam-2534	132	25	,	,	PUNCT
ejpam-2534	132	26	that	that	ADV
ejpam-2534	132	27	is	is	ADV
ejpam-2534	132	28	,	,	PUNCT
ejpam-2534	132	29	r	r	NOUN
ejpam-2534	132	30	∈	∈	PROPN
ejpam-2534	132	31	rad(n	rad(n	NOUN
ejpam-2534	132	32	:	:	PUNCT
ejpam-2534	132	33	m	m	X
ejpam-2534	132	34	)	)	PUNCT
ejpam-2534	132	35	and	and	CCONJ
ejpam-2534	132	36	so	so	ADV
ejpam-2534	132	37	p(n	p(n	PROPN
ejpam-2534	132	38	)	)	PUNCT
ejpam-2534	132	39	=	=	PUNCT
ejpam-2534	132	40	rad(n	rad(n	NOUN
ejpam-2534	132	41	:	:	PUNCT
ejpam-2534	132	42	m	m	NOUN
ejpam-2534	132	43	)	)	PUNCT
ejpam-2534	132	44	.	.	PUNCT
ejpam-2534	133	1	by	by	ADP
ejpam-2534	133	2	(	(	PUNCT
ejpam-2534	133	3	i	i	NOUN
ejpam-2534	133	4	)	)	PUNCT
ejpam-2534	133	5	,	,	PUNCT
ejpam-2534	133	6	n	n	PRON
ejpam-2534	133	7	is	be	AUX
ejpam-2534	133	8	a	a	DET
ejpam-2534	133	9	primary	primary	ADJ
ejpam-2534	133	10	submodule	submodule	NOUN
ejpam-2534	133	11	.	.	PUNCT
ejpam-2534	134	1	corollary	corollary	ADJ
ejpam-2534	134	2	2	2	NUM
ejpam-2534	134	3	.	.	PUNCT
ejpam-2534	135	1	let	let	VERB
ejpam-2534	135	2	n	n	PRON
ejpam-2534	135	3	be	be	AUX
ejpam-2534	135	4	a	a	DET
ejpam-2534	135	5	proper	proper	ADJ
ejpam-2534	135	6	submodule	submodule	NOUN
ejpam-2534	135	7	of	of	ADP
ejpam-2534	135	8	an	an	DET
ejpam-2534	135	9	r	r	NOUN
ejpam-2534	135	10	-	-	PUNCT
ejpam-2534	135	11	module	module	NOUN
ejpam-2534	135	12	m	m	NOUN
ejpam-2534	135	13	with	with	ADP
ejpam-2534	135	14	p(n	p(n	NOUN
ejpam-2534	135	15	)	)	PUNCT
ejpam-2534	135	16	=	=	PUNCT
ejpam-2534	135	17	rad(n	rad(n	NOUN
ejpam-2534	135	18	:	:	PUNCT
ejpam-2534	135	19	m	m	NUM
ejpam-2534	135	20	)	)	PUNCT
ejpam-2534	135	21	.	.	PUNCT
ejpam-2534	136	1	then	then	ADV
ejpam-2534	136	2	n	n	PRON
ejpam-2534	136	3	is	be	AUX
ejpam-2534	136	4	a	a	DET
ejpam-2534	136	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	136	6	primary	primary	ADJ
ejpam-2534	136	7	submodule	submodule	NOUN
ejpam-2534	136	8	and	and	CCONJ
ejpam-2534	136	9	p(n)-locally	p(n)-locally	ADV
ejpam-2534	136	10	weakly	weakly	ADJ
ejpam-2534	136	11	primary	primary	ADJ
ejpam-2534	136	12	submodule	submodule	NOUN
ejpam-2534	136	13	.	.	PUNCT
ejpam-2534	137	1	proof	proof	NOUN
ejpam-2534	137	2	.	.	PUNCT
ejpam-2534	138	1	we	we	PRON
ejpam-2534	138	2	get	get	VERB
ejpam-2534	138	3	that	that	PRON
ejpam-2534	138	4	n	n	NOUN
ejpam-2534	138	5	is	be	AUX
ejpam-2534	138	6	a	a	DET
ejpam-2534	138	7	primary	primary	ADJ
ejpam-2534	138	8	submodule	submodule	NOUN
ejpam-2534	138	9	by	by	ADP
ejpam-2534	138	10	theorem	theorem	NOUN
ejpam-2534	138	11	1(i	1(i	NUM
ejpam-2534	138	12	)	)	PUNCT
ejpam-2534	138	13	.	.	PUNCT
ejpam-2534	139	1	then	then	ADV
ejpam-2534	139	2	n	n	PRON
ejpam-2534	139	3	is	be	AUX
ejpam-2534	139	4	a	a	DET
ejpam-2534	139	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	139	6	primary	primary	ADJ
ejpam-2534	139	7	submodule	submodule	NOUN
ejpam-2534	139	8	by	by	ADP
ejpam-2534	139	9	proposition	proposition	NOUN
ejpam-2534	139	10	1	1	NUM
ejpam-2534	139	11	.	.	PUNCT
ejpam-2534	140	1	since	since	SCONJ
ejpam-2534	140	2	n	n	NUM
ejpam-2534	140	3	is	be	AUX
ejpam-2534	140	4	primary	primary	ADJ
ejpam-2534	140	5	submodule	submodule	NOUN
ejpam-2534	140	6	,	,	PUNCT
ejpam-2534	140	7	then	then	ADV
ejpam-2534	140	8	n	n	PRON
ejpam-2534	140	9	is	be	AUX
ejpam-2534	140	10	weakly	weakly	ADJ
ejpam-2534	140	11	primary	primary	ADJ
ejpam-2534	140	12	submodule	submodule	NOUN
ejpam-2534	140	13	.	.	PUNCT
ejpam-2534	141	1	therefore	therefore	ADV
ejpam-2534	141	2	,	,	PUNCT
ejpam-2534	141	3	n	n	PRON
ejpam-2534	141	4	is	be	AUX
ejpam-2534	141	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	141	6	weakly	weakly	ADJ
ejpam-2534	141	7	primary	primary	ADJ
ejpam-2534	141	8	submodule	submodule	NOUN
ejpam-2534	141	9	by	by	ADP
ejpam-2534	141	10	proposition	proposition	NOUN
ejpam-2534	141	11	2	2	NUM
ejpam-2534	141	12	.	.	X
ejpam-2534	141	13	note	note	VERB
ejpam-2534	141	14	that	that	SCONJ
ejpam-2534	141	15	,	,	PUNCT
ejpam-2534	141	16	by	by	ADP
ejpam-2534	141	17	[	[	X
ejpam-2534	141	18	4	4	NUM
ejpam-2534	141	19	,	,	PUNCT
ejpam-2534	141	20	lemma	lemma	PROPN
ejpam-2534	141	21	2.19	2.19	NUM
ejpam-2534	141	22	]	]	PUNCT
ejpam-2534	141	23	,	,	PUNCT
ejpam-2534	141	24	if	if	SCONJ
ejpam-2534	141	25	m	m	NOUN
ejpam-2534	141	26	is	be	AUX
ejpam-2534	141	27	a	a	DET
ejpam-2534	141	28	maximal	maximal	ADJ
ejpam-2534	141	29	ideal	ideal	NOUN
ejpam-2534	141	30	of	of	ADP
ejpam-2534	141	31	r	r	NOUN
ejpam-2534	141	32	,	,	PUNCT
ejpam-2534	141	33	then	then	ADV
ejpam-2534	141	34	(	(	PUNCT
ejpam-2534	141	35	n	n	X
ejpam-2534	141	36	:	:	PUNCT
ejpam-2534	141	37	m)m	m)m	X
ejpam-2534	141	38	⊆	⊆	NUM
ejpam-2534	141	39	(	(	PUNCT
ejpam-2534	141	40	nm	nm	INTJ
ejpam-2534	141	41	:	:	PUNCT
ejpam-2534	141	42	mm	mm	PROPN
ejpam-2534	141	43	)	)	PUNCT
ejpam-2534	141	44	.	.	PUNCT
ejpam-2534	142	1	now	now	ADV
ejpam-2534	142	2	,	,	PUNCT
ejpam-2534	142	3	we	we	PRON
ejpam-2534	142	4	explain	explain	VERB
ejpam-2534	142	5	that	that	DET
ejpam-2534	142	6	rad((n	rad((n	NOUN
ejpam-2534	142	7	:	:	PUNCT
ejpam-2534	142	8	m)m	m)m	X
ejpam-2534	142	9	)	)	PUNCT
ejpam-2534	142	10	=	=	NOUN
ejpam-2534	142	11	rad(nm	rad(nm	NOUN
ejpam-2534	142	12	:	:	PUNCT
ejpam-2534	142	13	mm	mm	X
ejpam-2534	142	14	)	)	PUNCT
ejpam-2534	142	15	when	when	SCONJ
ejpam-2534	142	16	m	m	PROPN
ejpam-2534	142	17	is	be	AUX
ejpam-2534	142	18	a	a	DET
ejpam-2534	142	19	maximal	maximal	ADJ
ejpam-2534	142	20	ideal	ideal	NOUN
ejpam-2534	142	21	of	of	ADP
ejpam-2534	142	22	r	r	NOUN
ejpam-2534	142	23	with	with	ADP
ejpam-2534	142	24	p(n	p(n	NOUN
ejpam-2534	142	25	)	)	PUNCT
ejpam-2534	142	26	⊆m	⊆m	NOUN
ejpam-2534	142	27	.	.	PUNCT
ejpam-2534	143	1	proposition	proposition	NOUN
ejpam-2534	143	2	3	3	X
ejpam-2534	143	3	.	.	PUNCT
ejpam-2534	144	1	let	let	VERB
ejpam-2534	144	2	n	n	PRON
ejpam-2534	144	3	be	be	AUX
ejpam-2534	144	4	a	a	DET
ejpam-2534	144	5	proper	proper	ADJ
ejpam-2534	144	6	submodule	submodule	NOUN
ejpam-2534	144	7	of	of	ADP
ejpam-2534	144	8	an	an	DET
ejpam-2534	144	9	r	r	NOUN
ejpam-2534	144	10	-	-	PUNCT
ejpam-2534	144	11	module	module	NOUN
ejpam-2534	144	12	m.	m.	NOUN
ejpam-2534	144	13	then	then	ADV
ejpam-2534	144	14	rad((n	rad((n	NOUN
ejpam-2534	144	15	:	:	PUNCT
ejpam-2534	144	16	m)m	m)m	X
ejpam-2534	144	17	)	)	PUNCT
ejpam-2534	144	18	=	=	NOUN
ejpam-2534	144	19	rad(nm	rad(nm	NOUN
ejpam-2534	144	20	:	:	PUNCT
ejpam-2534	144	21	mm	mm	X
ejpam-2534	144	22	)	)	PUNCT
ejpam-2534	144	23	for	for	ADP
ejpam-2534	144	24	any	any	DET
ejpam-2534	144	25	maximal	maximal	ADJ
ejpam-2534	144	26	ideal	ideal	NOUN
ejpam-2534	144	27	m	m	NOUN
ejpam-2534	144	28	of	of	ADP
ejpam-2534	144	29	r	r	NOUN
ejpam-2534	144	30	with	with	ADP
ejpam-2534	144	31	p(n	p(n	NOUN
ejpam-2534	144	32	)	)	PUNCT
ejpam-2534	144	33	⊆m	⊆m	NOUN
ejpam-2534	144	34	.	.	PUNCT
ejpam-2534	145	1	proof	proof	NOUN
ejpam-2534	145	2	.	.	PUNCT
ejpam-2534	146	1	since	since	SCONJ
ejpam-2534	146	2	s(n	s(n	PROPN
ejpam-2534	146	3	)	)	PUNCT
ejpam-2534	146	4	⊆	⊆	NUM
ejpam-2534	146	5	p(n	p(n	PROPN
ejpam-2534	146	6	)	)	PUNCT
ejpam-2534	146	7	for	for	ADP
ejpam-2534	146	8	any	any	DET
ejpam-2534	146	9	proper	proper	ADJ
ejpam-2534	146	10	submodule	submodule	NOUN
ejpam-2534	146	11	n	n	PROPN
ejpam-2534	146	12	of	of	ADP
ejpam-2534	146	13	m	m	PRON
ejpam-2534	146	14	,	,	PUNCT
ejpam-2534	146	15	it	it	PRON
ejpam-2534	146	16	is	be	AUX
ejpam-2534	146	17	clear	clear	ADJ
ejpam-2534	146	18	from	from	ADP
ejpam-2534	146	19	[	[	X
ejpam-2534	146	20	4	4	NUM
ejpam-2534	146	21	,	,	PUNCT
ejpam-2534	146	22	lemma	lemma	PROPN
ejpam-2534	146	23	2.19	2.19	NUM
ejpam-2534	146	24	and	and	CCONJ
ejpam-2534	146	25	lemma	lemma	PROPN
ejpam-2534	146	26	2.20	2.20	NUM
ejpam-2534	146	27	]	]	PUNCT
ejpam-2534	146	28	.	.	PUNCT
ejpam-2534	147	1	lemma	lemma	PROPN
ejpam-2534	147	2	2	2	X
ejpam-2534	147	3	.	.	PUNCT
ejpam-2534	148	1	let	let	VERB
ejpam-2534	148	2	n	n	PRON
ejpam-2534	148	3	be	be	AUX
ejpam-2534	148	4	a	a	DET
ejpam-2534	148	5	proper	proper	ADJ
ejpam-2534	148	6	submodule	submodule	NOUN
ejpam-2534	148	7	of	of	ADP
ejpam-2534	148	8	an	an	DET
ejpam-2534	148	9	r	r	NOUN
ejpam-2534	148	10	-	-	PUNCT
ejpam-2534	148	11	module	module	NOUN
ejpam-2534	148	12	m.	m.	NOUN
ejpam-2534	148	13	then	then	ADV
ejpam-2534	148	14	rad((n	rad((n	NOUN
ejpam-2534	148	15	:	:	PUNCT
ejpam-2534	148	16	m)m	m)m	X
ejpam-2534	148	17	)	)	PUNCT
ejpam-2534	148	18	=	=	SYM
ejpam-2534	148	19	(	(	PUNCT
ejpam-2534	148	20	rad(n	rad(n	NOUN
ejpam-2534	148	21	:	:	PUNCT
ejpam-2534	148	22	m))m	m))m	NOUN
ejpam-2534	148	23	for	for	ADP
ejpam-2534	148	24	any	any	DET
ejpam-2534	148	25	maximal	maximal	ADJ
ejpam-2534	148	26	ideal	ideal	NOUN
ejpam-2534	148	27	m	m	NOUN
ejpam-2534	148	28	of	of	ADP
ejpam-2534	148	29	r	r	NOUN
ejpam-2534	148	30	with	with	ADP
ejpam-2534	148	31	p(n	p(n	NOUN
ejpam-2534	148	32	)	)	PUNCT
ejpam-2534	148	33	⊆m	⊆m	NOUN
ejpam-2534	148	34	.	.	PUNCT
ejpam-2534	149	1	proof	proof	NOUN
ejpam-2534	149	2	.	.	PUNCT
ejpam-2534	150	1	let	let	VERB
ejpam-2534	150	2	r	r	NOUN
ejpam-2534	150	3	p	p	X
ejpam-2534	150	4	∈	∈	PROPN
ejpam-2534	150	5	rad((n	rad((n	NOUN
ejpam-2534	150	6	:	:	PUNCT
ejpam-2534	150	7	m)m	m)m	X
ejpam-2534	150	8	)	)	PUNCT
ejpam-2534	150	9	for	for	ADP
ejpam-2534	150	10	some	some	DET
ejpam-2534	150	11	r	r	NOUN
ejpam-2534	150	12	∈	∈	NOUN
ejpam-2534	150	13	r	r	NOUN
ejpam-2534	150	14	and	and	CCONJ
ejpam-2534	150	15	p	p	NOUN
ejpam-2534	150	16	∈	∈	PROPN
ejpam-2534	150	17	r	r	NOUN
ejpam-2534	150	18	\m	\m	NOUN
ejpam-2534	150	19	.	.	PUNCT
ejpam-2534	151	1	then	then	ADV
ejpam-2534	151	2	(	(	PUNCT
ejpam-2534	151	3	r	r	NOUN
ejpam-2534	151	4	p	p	NOUN
ejpam-2534	151	5	)	)	PUNCT
ejpam-2534	151	6	n	n	PROPN
ejpam-2534	151	7	=	=	SYM
ejpam-2534	151	8	rn	rn	PROPN
ejpam-2534	151	9	pn	pn	PROPN
ejpam-2534	151	10	∈	∈	PROPN
ejpam-2534	151	11	(	(	PUNCT
ejpam-2534	151	12	n	n	NUM
ejpam-2534	151	13	:	:	PUNCT
ejpam-2534	151	14	m)m	m)m	NOUN
ejpam-2534	151	15	for	for	ADP
ejpam-2534	151	16	some	some	DET
ejpam-2534	151	17	positive	positive	ADJ
ejpam-2534	151	18	integer	integer	NOUN
ejpam-2534	151	19	n.	n.	NOUN
ejpam-2534	151	20	there	there	PRON
ejpam-2534	151	21	exists	exist	VERB
ejpam-2534	151	22	an	an	DET
ejpam-2534	151	23	element	element	NOUN
ejpam-2534	151	24	q	q	PROPN
ejpam-2534	151	25	∈	∈	PROPN
ejpam-2534	151	26	r	r	NOUN
ejpam-2534	151	27	\m	\m	NOUN
ejpam-2534	151	28	such	such	ADJ
ejpam-2534	151	29	that	that	SCONJ
ejpam-2534	151	30	qrn	qrn	PROPN
ejpam-2534	151	31	∈	∈	PROPN
ejpam-2534	151	32	(	(	PUNCT
ejpam-2534	151	33	n	n	NOUN
ejpam-2534	151	34	:	:	PUNCT
ejpam-2534	151	35	m	m	X
ejpam-2534	151	36	)	)	PUNCT
ejpam-2534	151	37	,	,	PUNCT
ejpam-2534	151	38	that	that	ADV
ejpam-2534	151	39	is	is	ADV
ejpam-2534	151	40	,	,	PUNCT
ejpam-2534	151	41	qrnm	qrnm	NOUN
ejpam-2534	151	42	∈	∈	PROPN
ejpam-2534	151	43	n	n	CCONJ
ejpam-2534	151	44	for	for	ADP
ejpam-2534	151	45	every	every	DET
ejpam-2534	151	46	m	m	NOUN
ejpam-2534	151	47	∈	∈	NOUN
ejpam-2534	151	48	m	m	NOUN
ejpam-2534	151	49	.	.	PUNCT
ejpam-2534	152	1	then	then	ADV
ejpam-2534	152	2	rnm	rnm	VERB
ejpam-2534	152	3	∈	∈	PROPN
ejpam-2534	152	4	n	n	NOUN
ejpam-2534	152	5	for	for	ADP
ejpam-2534	152	6	every	every	DET
ejpam-2534	152	7	m	m	NOUN
ejpam-2534	152	8	∈	∈	NOUN
ejpam-2534	152	9	m	m	NOUN
ejpam-2534	152	10	since	since	SCONJ
ejpam-2534	152	11	q	q	PROPN
ejpam-2534	152	12	/∈	/∈	PUNCT
ejpam-2534	152	13	p(n	p(n	NOUN
ejpam-2534	152	14	)	)	PUNCT
ejpam-2534	152	15	.	.	PUNCT
ejpam-2534	153	1	thus	thus	ADV
ejpam-2534	153	2	r	r	NOUN
ejpam-2534	153	3	∈	∈	PROPN
ejpam-2534	153	4	rad(n	rad(n	NOUN
ejpam-2534	153	5	:	:	PUNCT
ejpam-2534	153	6	m	m	NUM
ejpam-2534	153	7	)	)	PUNCT
ejpam-2534	153	8	.	.	PUNCT
ejpam-2534	154	1	then	then	ADV
ejpam-2534	154	2	r	r	NOUN
ejpam-2534	154	3	p	p	X
ejpam-2534	154	4	∈	∈	PROPN
ejpam-2534	154	5	rad((n	rad((n	NOUN
ejpam-2534	154	6	:	:	PUNCT
ejpam-2534	154	7	m)m	m)m	X
ejpam-2534	154	8	)	)	PUNCT
ejpam-2534	154	9	.	.	PUNCT
ejpam-2534	155	1	conversely	conversely	ADV
ejpam-2534	155	2	,	,	PUNCT
ejpam-2534	155	3	assume	assume	VERB
ejpam-2534	155	4	that	that	SCONJ
ejpam-2534	155	5	r	r	NOUN
ejpam-2534	155	6	p	p	X
ejpam-2534	155	7	∈	∈	PROPN
ejpam-2534	155	8	rad((n	rad((n	NOUN
ejpam-2534	155	9	:	:	PUNCT
ejpam-2534	155	10	m)m	m)m	X
ejpam-2534	155	11	)	)	PUNCT
ejpam-2534	155	12	.	.	PUNCT
ejpam-2534	156	1	there	there	PRON
ejpam-2534	156	2	is	be	VERB
ejpam-2534	156	3	an	an	DET
ejpam-2534	156	4	u	u	NOUN
ejpam-2534	156	5	∈	∈	PROPN
ejpam-2534	156	6	r	r	NOUN
ejpam-2534	156	7	\m	\m	NOUN
ejpam-2534	156	8	such	such	ADJ
ejpam-2534	156	9	that	that	SCONJ
ejpam-2534	156	10	ur	ur	PROPN
ejpam-2534	156	11	∈	∈	PROPN
ejpam-2534	156	12	rad(n	rad(n	PROPN
ejpam-2534	156	13	:	:	PUNCT
ejpam-2534	156	14	m	m	NUM
ejpam-2534	156	15	)	)	PUNCT
ejpam-2534	156	16	.	.	PUNCT
ejpam-2534	157	1	then	then	ADV
ejpam-2534	157	2	(	(	PUNCT
ejpam-2534	157	3	ur)n	ur)n	NOUN
ejpam-2534	157	4	=	=	SYM
ejpam-2534	157	5	unrn	unrn	NOUN
ejpam-2534	157	6	∈	∈	NOUN
ejpam-2534	157	7	(	(	PUNCT
ejpam-2534	157	8	n	n	NOUN
ejpam-2534	157	9	:	:	PUNCT
ejpam-2534	157	10	m	m	X
ejpam-2534	157	11	)	)	PUNCT
ejpam-2534	157	12	.	.	PUNCT
ejpam-2534	158	1	hence	hence	ADV
ejpam-2534	158	2	un	un	PROPN
ejpam-2534	158	3	un	un	PROPN
ejpam-2534	158	4	rn	rn	PROPN
ejpam-2534	158	5	pn	pn	PROPN
ejpam-2534	158	6	∈	∈	PROPN
ejpam-2534	158	7	(	(	PUNCT
ejpam-2534	158	8	n	n	NOUN
ejpam-2534	158	9	:	:	PUNCT
ejpam-2534	158	10	m)m	m)m	X
ejpam-2534	158	11	.	.	PUNCT
ejpam-2534	159	1	consequently	consequently	ADV
ejpam-2534	159	2	,	,	PUNCT
ejpam-2534	159	3	rn	rn	PROPN
ejpam-2534	159	4	pn	pn	PROPN
ejpam-2534	159	5	=	=	PUNCT
ejpam-2534	159	6	(	(	PUNCT
ejpam-2534	159	7	r	r	NOUN
ejpam-2534	159	8	p	p	NOUN
ejpam-2534	159	9	)	)	PUNCT
ejpam-2534	159	10	n	n	CCONJ
ejpam-2534	159	11	∈	∈	PROPN
ejpam-2534	159	12	(	(	PUNCT
ejpam-2534	159	13	n	n	NOUN
ejpam-2534	159	14	:	:	PUNCT
ejpam-2534	159	15	m)m	m)m	NOUN
ejpam-2534	159	16	and	and	CCONJ
ejpam-2534	159	17	so	so	ADV
ejpam-2534	159	18	r	r	NOUN
ejpam-2534	159	19	p	p	X
ejpam-2534	159	20	∈	∈	PROPN
ejpam-2534	159	21	rad((n	rad((n	NOUN
ejpam-2534	159	22	:	:	PUNCT
ejpam-2534	159	23	m)m	m)m	X
ejpam-2534	159	24	)	)	PUNCT
ejpam-2534	159	25	.	.	PUNCT
ejpam-2534	159	26	g.	g.	PROPN
ejpam-2534	159	27	ulucak	ulucak	PROPN
ejpam-2534	159	28	and	and	CCONJ
ejpam-2534	159	29	r.	r.	PROPN
ejpam-2534	159	30	uregen	uregen	PROPN
ejpam-2534	159	31	/	/	SYM
ejpam-2534	159	32	eur	eur	PROPN
ejpam-2534	159	33	.	.	PUNCT
ejpam-2534	160	1	j.	j.	PROPN
ejpam-2534	160	2	pure	pure	PROPN
ejpam-2534	160	3	appl	appl	PROPN
ejpam-2534	160	4	.	.	PROPN
ejpam-2534	160	5	math	math	PROPN
ejpam-2534	160	6	,	,	PUNCT
ejpam-2534	160	7	9	9	NUM
ejpam-2534	160	8	(	(	PUNCT
ejpam-2534	160	9	2016	2016	NUM
ejpam-2534	160	10	)	)	PUNCT
ejpam-2534	160	11	,	,	PUNCT
ejpam-2534	160	12	48	48	NUM
ejpam-2534	160	13	-	-	SYM
ejpam-2534	160	14	56	56	NUM
ejpam-2534	160	15	52	52	NUM
ejpam-2534	160	16	corollary	corollary	ADJ
ejpam-2534	160	17	3	3	NUM
ejpam-2534	160	18	.	.	PUNCT
ejpam-2534	161	1	let	let	VERB
ejpam-2534	161	2	n	n	PRON
ejpam-2534	161	3	be	be	AUX
ejpam-2534	161	4	a	a	DET
ejpam-2534	161	5	proper	proper	ADJ
ejpam-2534	161	6	submodule	submodule	NOUN
ejpam-2534	161	7	of	of	ADP
ejpam-2534	161	8	an	an	DET
ejpam-2534	161	9	r	r	NOUN
ejpam-2534	161	10	-	-	PUNCT
ejpam-2534	161	11	module	module	NOUN
ejpam-2534	161	12	.	.	PUNCT
ejpam-2534	162	1	if	if	SCONJ
ejpam-2534	162	2	m	m	NOUN
ejpam-2534	162	3	is	be	AUX
ejpam-2534	162	4	any	any	DET
ejpam-2534	162	5	maximal	maximal	ADJ
ejpam-2534	162	6	ideal	ideal	NOUN
ejpam-2534	162	7	of	of	ADP
ejpam-2534	162	8	r	r	NOUN
ejpam-2534	162	9	with	with	ADP
ejpam-2534	162	10	p(n	p(n	NOUN
ejpam-2534	162	11	)	)	PUNCT
ejpam-2534	162	12	⊆m	⊆m	NOUN
ejpam-2534	162	13	,	,	PUNCT
ejpam-2534	162	14	then	then	ADV
ejpam-2534	162	15	rad((n	rad((n	NOUN
ejpam-2534	162	16	:	:	PUNCT
ejpam-2534	162	17	m)m	m)m	X
ejpam-2534	162	18	)	)	PUNCT
ejpam-2534	162	19	=	=	NOUN
ejpam-2534	162	20	rad(nm	rad(nm	NOUN
ejpam-2534	162	21	:	:	PUNCT
ejpam-2534	162	22	mm	mm	X
ejpam-2534	162	23	)	)	PUNCT
ejpam-2534	162	24	.	.	PUNCT
ejpam-2534	163	1	proof	proof	NOUN
ejpam-2534	163	2	.	.	PUNCT
ejpam-2534	164	1	it	it	PRON
ejpam-2534	164	2	is	be	AUX
ejpam-2534	164	3	clear	clear	ADJ
ejpam-2534	164	4	from	from	ADP
ejpam-2534	164	5	proposition	proposition	NOUN
ejpam-2534	164	6	3	3	NUM
ejpam-2534	164	7	and	and	CCONJ
ejpam-2534	164	8	lemma	lemma	PROPN
ejpam-2534	164	9	2	2	NUM
ejpam-2534	164	10	.	.	X
ejpam-2534	164	11	proposition	proposition	NOUN
ejpam-2534	164	12	4	4	NUM
ejpam-2534	164	13	.	.	PUNCT
ejpam-2534	165	1	let	let	VERB
ejpam-2534	165	2	n	n	PRON
ejpam-2534	165	3	be	be	AUX
ejpam-2534	165	4	a	a	DET
ejpam-2534	165	5	proper	proper	ADJ
ejpam-2534	165	6	submodule	submodule	NOUN
ejpam-2534	165	7	of	of	ADP
ejpam-2534	165	8	an	an	DET
ejpam-2534	165	9	r	r	NOUN
ejpam-2534	165	10	-	-	PUNCT
ejpam-2534	165	11	module	module	NOUN
ejpam-2534	165	12	m	m	NOUN
ejpam-2534	165	13	and	and	CCONJ
ejpam-2534	165	14	m	m	PROPN
ejpam-2534	165	15	∈	∈	PROPN
ejpam-2534	165	16	m.	m.	NOUN
ejpam-2534	165	17	then	then	ADV
ejpam-2534	165	18	rad((n	rad((n	NOUN
ejpam-2534	165	19	:	:	PUNCT
ejpam-2534	165	20	rm)m	rm)m	PROPN
ejpam-2534	165	21	)	)	PUNCT
ejpam-2534	165	22	=	=	NOUN
ejpam-2534	165	23	rad(nm	rad(nm	NOUN
ejpam-2534	165	24	:	:	PUNCT
ejpam-2534	165	25	(	(	PUNCT
ejpam-2534	165	26	rm)m	rm)m	PROPN
ejpam-2534	165	27	)	)	PUNCT
ejpam-2534	165	28	for	for	ADP
ejpam-2534	165	29	any	any	DET
ejpam-2534	165	30	maximal	maximal	ADJ
ejpam-2534	165	31	ideal	ideal	NOUN
ejpam-2534	165	32	m	m	NOUN
ejpam-2534	165	33	of	of	ADP
ejpam-2534	165	34	r	r	NOUN
ejpam-2534	165	35	with	with	ADP
ejpam-2534	165	36	p(n	p(n	NOUN
ejpam-2534	165	37	)	)	PUNCT
ejpam-2534	165	38	⊆m	⊆m	NOUN
ejpam-2534	165	39	.	.	PUNCT
ejpam-2534	166	1	proof	proof	NOUN
ejpam-2534	166	2	.	.	PUNCT
ejpam-2534	167	1	it	it	PRON
ejpam-2534	167	2	is	be	AUX
ejpam-2534	167	3	clear	clear	ADJ
ejpam-2534	167	4	.	.	PUNCT
ejpam-2534	168	1	if	if	SCONJ
ejpam-2534	168	2	we	we	PRON
ejpam-2534	168	3	put	put	VERB
ejpam-2534	168	4	n	n	NOUN
ejpam-2534	168	5	=	=	SYM
ejpam-2534	168	6	0	0	NUM
ejpam-2534	168	7	in	in	ADP
ejpam-2534	168	8	proposition	proposition	NOUN
ejpam-2534	168	9	4	4	NUM
ejpam-2534	168	10	,	,	PUNCT
ejpam-2534	168	11	we	we	PRON
ejpam-2534	168	12	have	have	VERB
ejpam-2534	168	13	the	the	DET
ejpam-2534	168	14	following	follow	VERB
ejpam-2534	168	15	corollary	corollary	NOUN
ejpam-2534	168	16	.	.	PUNCT
ejpam-2534	169	1	corollary	corollary	ADJ
ejpam-2534	169	2	4	4	NUM
ejpam-2534	169	3	.	.	PUNCT
ejpam-2534	170	1	let	let	VERB
ejpam-2534	170	2	m	m	PRON
ejpam-2534	170	3	be	be	AUX
ejpam-2534	170	4	an	an	DET
ejpam-2534	170	5	r	r	NOUN
ejpam-2534	170	6	-	-	PUNCT
ejpam-2534	170	7	module	module	NOUN
ejpam-2534	170	8	and	and	CCONJ
ejpam-2534	170	9	m	m	NOUN
ejpam-2534	170	10	∈	∈	NOUN
ejpam-2534	170	11	m.	m.	NOUN
ejpam-2534	170	12	then	then	ADV
ejpam-2534	170	13	rad((0	rad((0	ADJ
ejpam-2534	170	14	:	:	PUNCT
ejpam-2534	170	15	rm)m	rm)m	PROPN
ejpam-2534	170	16	)	)	PUNCT
ejpam-2534	170	17	=	=	PUNCT
ejpam-2534	170	18	rad(0	rad(0	NOUN
ejpam-2534	170	19	m	m	VERB
ejpam-2534	170	20	:	:	PUNCT
ejpam-2534	170	21	(	(	PUNCT
ejpam-2534	170	22	rm)m	rm)m	PROPN
ejpam-2534	170	23	)	)	PUNCT
ejpam-2534	170	24	for	for	ADP
ejpam-2534	170	25	any	any	DET
ejpam-2534	170	26	maximal	maximal	ADJ
ejpam-2534	170	27	ideal	ideal	NOUN
ejpam-2534	170	28	m	m	NOUN
ejpam-2534	170	29	of	of	ADP
ejpam-2534	170	30	r	r	NOUN
ejpam-2534	170	31	with	with	ADP
ejpam-2534	170	32	p(0	p(0	NOUN
ejpam-2534	170	33	)	)	PUNCT
ejpam-2534	170	34	⊆m	⊆m	NOUN
ejpam-2534	170	35	.	.	PUNCT
ejpam-2534	171	1	proposition	proposition	NOUN
ejpam-2534	171	2	5	5	NUM
ejpam-2534	171	3	.	.	PUNCT
ejpam-2534	172	1	let	let	VERB
ejpam-2534	172	2	n	n	PRON
ejpam-2534	172	3	be	be	AUX
ejpam-2534	172	4	a	a	DET
ejpam-2534	172	5	proper	proper	ADJ
ejpam-2534	172	6	submodule	submodule	NOUN
ejpam-2534	172	7	of	of	ADP
ejpam-2534	172	8	an	an	DET
ejpam-2534	172	9	r	r	NOUN
ejpam-2534	172	10	-	-	PUNCT
ejpam-2534	172	11	module	module	NOUN
ejpam-2534	172	12	m	m	NOUN
ejpam-2534	172	13	and	and	CCONJ
ejpam-2534	172	14	m	m	AUX
ejpam-2534	172	15	be	be	AUX
ejpam-2534	172	16	a	a	DET
ejpam-2534	172	17	maximal	maximal	ADJ
ejpam-2534	172	18	ideal	ideal	NOUN
ejpam-2534	172	19	of	of	ADP
ejpam-2534	172	20	r	r	NOUN
ejpam-2534	172	21	with	with	ADP
ejpam-2534	172	22	p(n	p(n	NOUN
ejpam-2534	172	23	)	)	PUNCT
ejpam-2534	172	24	⊆m	⊆m	NOUN
ejpam-2534	172	25	.	.	PUNCT
ejpam-2534	173	1	then	then	ADV
ejpam-2534	173	2	the	the	DET
ejpam-2534	173	3	following	following	ADJ
ejpam-2534	173	4	statements	statement	NOUN
ejpam-2534	173	5	hold	hold	VERB
ejpam-2534	173	6	:	:	PUNCT
ejpam-2534	173	7	(	(	PUNCT
ejpam-2534	173	8	i	i	NOUN
ejpam-2534	173	9	)	)	PUNCT
ejpam-2534	173	10	let	let	VERB
ejpam-2534	173	11	p(0	p(0	NOUN
ejpam-2534	173	12	)	)	PUNCT
ejpam-2534	173	13	⊆	⊆	NUM
ejpam-2534	173	14	p(n	p(n	PROPN
ejpam-2534	173	15	)	)	PUNCT
ejpam-2534	173	16	.	.	PUNCT
ejpam-2534	174	1	then	then	ADV
ejpam-2534	174	2	rad(n	rad(n	VERB
ejpam-2534	174	3	:	:	PUNCT
ejpam-2534	174	4	m	m	X
ejpam-2534	174	5	)	)	PUNCT
ejpam-2534	174	6	is	be	AUX
ejpam-2534	174	7	a	a	DET
ejpam-2534	174	8	weakly	weakly	ADJ
ejpam-2534	174	9	prime	prime	ADJ
ejpam-2534	174	10	ideal	ideal	NOUN
ejpam-2534	174	11	of	of	ADP
ejpam-2534	174	12	r	r	NOUN
ejpam-2534	174	13	if	if	SCONJ
ejpam-2534	175	1	and	and	CCONJ
ejpam-2534	175	2	only	only	ADV
ejpam-2534	175	3	if	if	SCONJ
ejpam-2534	175	4	rad((n	rad((n	NOUN
ejpam-2534	175	5	:	:	PUNCT
ejpam-2534	175	6	m)m	m)m	X
ejpam-2534	175	7	)	)	PUNCT
ejpam-2534	175	8	is	be	AUX
ejpam-2534	175	9	a	a	DET
ejpam-2534	175	10	weakly	weakly	ADJ
ejpam-2534	175	11	prime	prime	ADJ
ejpam-2534	175	12	ideal	ideal	NOUN
ejpam-2534	175	13	of	of	ADP
ejpam-2534	175	14	rm	rm	PROPN
ejpam-2534	175	15	.	.	PUNCT
ejpam-2534	176	1	(	(	PUNCT
ejpam-2534	176	2	ii	ii	X
ejpam-2534	176	3	)	)	PUNCT
ejpam-2534	176	4	rad(n	rad(n	PROPN
ejpam-2534	176	5	:	:	PUNCT
ejpam-2534	176	6	m	m	X
ejpam-2534	176	7	)	)	PUNCT
ejpam-2534	176	8	is	be	AUX
ejpam-2534	176	9	a	a	DET
ejpam-2534	176	10	prime	prime	ADJ
ejpam-2534	176	11	ideal	ideal	NOUN
ejpam-2534	176	12	of	of	ADP
ejpam-2534	176	13	r	r	NOUN
ejpam-2534	176	14	if	if	SCONJ
ejpam-2534	177	1	and	and	CCONJ
ejpam-2534	177	2	only	only	ADV
ejpam-2534	177	3	if	if	SCONJ
ejpam-2534	177	4	rad((n	rad((n	NOUN
ejpam-2534	177	5	:	:	PUNCT
ejpam-2534	177	6	m)m	m)m	X
ejpam-2534	177	7	)	)	PUNCT
ejpam-2534	177	8	is	be	AUX
ejpam-2534	177	9	a	a	DET
ejpam-2534	177	10	prime	prime	ADJ
ejpam-2534	177	11	ideal	ideal	NOUN
ejpam-2534	177	12	of	of	ADP
ejpam-2534	177	13	rm	rm	PROPN
ejpam-2534	177	14	.	.	PUNCT
ejpam-2534	178	1	proof	proof	NOUN
ejpam-2534	178	2	.	.	PUNCT
ejpam-2534	179	1	(	(	PUNCT
ejpam-2534	179	2	i	i	NOUN
ejpam-2534	179	3	)	)	PUNCT
ejpam-2534	179	4	suppose	suppose	VERB
ejpam-2534	179	5	that	that	SCONJ
ejpam-2534	179	6	rad(n	rad(n	NOUN
ejpam-2534	179	7	:	:	PUNCT
ejpam-2534	179	8	m	m	X
ejpam-2534	179	9	)	)	PUNCT
ejpam-2534	179	10	is	be	AUX
ejpam-2534	179	11	a	a	DET
ejpam-2534	179	12	weakly	weakly	ADJ
ejpam-2534	179	13	prime	prime	ADJ
ejpam-2534	179	14	ideal	ideal	NOUN
ejpam-2534	179	15	of	of	ADP
ejpam-2534	179	16	r.	r.	PROPN
ejpam-2534	179	17	if	if	SCONJ
ejpam-2534	179	18	rad(n	rad(n	NOUN
ejpam-2534	179	19	:	:	PUNCT
ejpam-2534	179	20	m)m	m)m	X
ejpam-2534	179	21	=	=	SYM
ejpam-2534	179	22	rm	rm	PROPN
ejpam-2534	179	23	,	,	PUNCT
ejpam-2534	179	24	then	then	ADV
ejpam-2534	179	25	1	1	NUM
ejpam-2534	179	26	1	1	NUM
ejpam-2534	179	27	∈	∈	NOUN
ejpam-2534	179	28	rad((n	rad((n	NOUN
ejpam-2534	179	29	:	:	PUNCT
ejpam-2534	179	30	m)m	m)m	X
ejpam-2534	179	31	)	)	PUNCT
ejpam-2534	179	32	=	=	SYM
ejpam-2534	179	33	(	(	PUNCT
ejpam-2534	179	34	rad(n	rad(n	NOUN
ejpam-2534	179	35	:	:	PUNCT
ejpam-2534	179	36	m))m	m))m	ADJ
ejpam-2534	179	37	and	and	CCONJ
ejpam-2534	179	38	so	so	ADV
ejpam-2534	179	39	q1	q1	PROPN
ejpam-2534	179	40	=	=	SYM
ejpam-2534	179	41	q	q	PROPN
ejpam-2534	179	42	∈	∈	PROPN
ejpam-2534	179	43	rad(n	rad(n	NOUN
ejpam-2534	179	44	:	:	PUNCT
ejpam-2534	179	45	m	m	X
ejpam-2534	179	46	)	)	PUNCT
ejpam-2534	179	47	for	for	ADP
ejpam-2534	179	48	some	some	DET
ejpam-2534	179	49	q	q	NOUN
ejpam-2534	179	50	∈	∈	PROPN
ejpam-2534	179	51	r	r	NOUN
ejpam-2534	179	52	\m	\m	NOUN
ejpam-2534	179	53	.	.	PUNCT
ejpam-2534	180	1	but	but	CCONJ
ejpam-2534	180	2	by	by	ADP
ejpam-2534	180	3	lemma	lemma	PROPN
ejpam-2534	180	4	1	1	NUM
ejpam-2534	180	5	,	,	PUNCT
ejpam-2534	180	6	rad(n	rad(n	NOUN
ejpam-2534	180	7	:	:	PUNCT
ejpam-2534	180	8	m	m	X
ejpam-2534	180	9	)	)	PUNCT
ejpam-2534	180	10	⊆	⊆	NUM
ejpam-2534	180	11	p(n	p(n	NOUN
ejpam-2534	180	12	)	)	PUNCT
ejpam-2534	180	13	⊆m	⊆m	NOUN
ejpam-2534	180	14	,	,	PUNCT
ejpam-2534	180	15	which	which	PRON
ejpam-2534	180	16	is	be	AUX
ejpam-2534	180	17	a	a	DET
ejpam-2534	180	18	contradiction	contradiction	NOUN
ejpam-2534	180	19	.	.	PUNCT
ejpam-2534	181	1	so	so	ADV
ejpam-2534	181	2	rad((n	rad((n	NOUN
ejpam-2534	181	3	:	:	PUNCT
ejpam-2534	181	4	m)m	m)m	X
ejpam-2534	181	5	)	)	PUNCT
ejpam-2534	181	6	6=	6=	X
ejpam-2534	182	1	rm	rm	PROPN
ejpam-2534	182	2	,	,	PUNCT
ejpam-2534	182	3	that	that	ADV
ejpam-2534	182	4	is	is	ADV
ejpam-2534	182	5	,	,	PUNCT
ejpam-2534	182	6	rad((n	rad((n	NOUN
ejpam-2534	182	7	:	:	PUNCT
ejpam-2534	182	8	m)m	m)m	X
ejpam-2534	182	9	)	)	PUNCT
ejpam-2534	182	10	is	be	AUX
ejpam-2534	182	11	a	a	DET
ejpam-2534	182	12	proper	proper	ADJ
ejpam-2534	182	13	ideal	ideal	NOUN
ejpam-2534	182	14	of	of	ADP
ejpam-2534	182	15	rm	rm	PROPN
ejpam-2534	182	16	.	.	PUNCT
ejpam-2534	183	1	let	let	VERB
ejpam-2534	183	2	0	0	NUM
ejpam-2534	184	1	6=	6=	ADP
ejpam-2534	184	2	r	r	NOUN
ejpam-2534	184	3	p	p	X
ejpam-2534	184	4	s	s	X
ejpam-2534	184	5	q	q	NOUN
ejpam-2534	184	6	∈	∈	PROPN
ejpam-2534	184	7	rad((n	rad((n	NOUN
ejpam-2534	184	8	:	:	PUNCT
ejpam-2534	184	9	m)m	m)m	X
ejpam-2534	184	10	)	)	PUNCT
ejpam-2534	184	11	,	,	PUNCT
ejpam-2534	184	12	where	where	SCONJ
ejpam-2534	184	13	r	r	NOUN
ejpam-2534	184	14	,	,	PUNCT
ejpam-2534	184	15	s	s	NOUN
ejpam-2534	184	16	∈	∈	PROPN
ejpam-2534	184	17	r	r	NOUN
ejpam-2534	184	18	and	and	CCONJ
ejpam-2534	184	19	p	p	NOUN
ejpam-2534	184	20	,	,	PUNCT
ejpam-2534	184	21	q	q	PROPN
ejpam-2534	184	22	∈	∈	PROPN
ejpam-2534	184	23	r	r	NOUN
ejpam-2534	184	24	\m	\m	NOUN
ejpam-2534	184	25	.	.	PUNCT
ejpam-2534	185	1	then	then	ADV
ejpam-2534	185	2	we	we	PRON
ejpam-2534	185	3	have	have	VERB
ejpam-2534	185	4	r	r	NOUN
ejpam-2534	185	5	p	p	NOUN
ejpam-2534	185	6	s	s	PART
ejpam-2534	185	7	q	q	NOUN
ejpam-2534	185	8	=	=	PUNCT
ejpam-2534	185	9	rs	rs	PROPN
ejpam-2534	185	10	pq	pq	INTJ
ejpam-2534	185	11	∈	∈	PROPN
ejpam-2534	185	12	(	(	PUNCT
ejpam-2534	185	13	rad(n	rad(n	NOUN
ejpam-2534	185	14	:	:	PUNCT
ejpam-2534	185	15	m))m	m))m	PROPN
ejpam-2534	185	16	,	,	PUNCT
ejpam-2534	185	17	then	then	ADV
ejpam-2534	185	18	there	there	PRON
ejpam-2534	185	19	exists	exist	VERB
ejpam-2534	185	20	an	an	DET
ejpam-2534	185	21	u	u	NOUN
ejpam-2534	185	22	∈	∈	PROPN
ejpam-2534	185	23	r	r	NOUN
ejpam-2534	185	24	\m	\m	NOUN
ejpam-2534	185	25	such	such	ADJ
ejpam-2534	185	26	that	that	SCONJ
ejpam-2534	185	27	urs	urs	PROPN
ejpam-2534	185	28	∈	∈	PROPN
ejpam-2534	185	29	rad(n	rad(n	PROPN
ejpam-2534	185	30	:	:	PUNCT
ejpam-2534	185	31	m	m	NUM
ejpam-2534	185	32	)	)	PUNCT
ejpam-2534	185	33	.	.	PUNCT
ejpam-2534	186	1	if	if	SCONJ
ejpam-2534	186	2	urs	urs	PRON
ejpam-2534	186	3	=	=	SYM
ejpam-2534	186	4	0	0	NUM
ejpam-2534	186	5	,	,	PUNCT
ejpam-2534	186	6	then	then	ADV
ejpam-2534	186	7	r	r	NOUN
ejpam-2534	186	8	p	p	X
ejpam-2534	186	9	s	s	PART
ejpam-2534	186	10	q	q	X
ejpam-2534	186	11	=	=	PUNCT
ejpam-2534	186	12	u	u	NOUN
ejpam-2534	186	13	u	u	NOUN
ejpam-2534	186	14	r	r	NOUN
ejpam-2534	186	15	p	p	NOUN
ejpam-2534	186	16	s	s	NOUN
ejpam-2534	186	17	q	q	NOUN
ejpam-2534	186	18	=	=	X
ejpam-2534	186	19	urs	urs	X
ejpam-2534	186	20	upq	upq	NOUN
ejpam-2534	186	21	=	=	NOUN
ejpam-2534	186	22	0	0	NUM
ejpam-2534	186	23	,	,	PUNCT
ejpam-2534	186	24	this	this	PRON
ejpam-2534	186	25	is	be	AUX
ejpam-2534	186	26	a	a	DET
ejpam-2534	186	27	contradiction	contradiction	NOUN
ejpam-2534	186	28	.	.	PUNCT
ejpam-2534	187	1	so	so	ADV
ejpam-2534	187	2	urs	urs	PROPN
ejpam-2534	187	3	6=	6=	ADP
ejpam-2534	187	4	0	0	NUM
ejpam-2534	187	5	.	.	PUNCT
ejpam-2534	188	1	since	since	SCONJ
ejpam-2534	188	2	0	0	NUM
ejpam-2534	188	3	6=	6=	NUM
ejpam-2534	188	4	urs	urs	PROPN
ejpam-2534	188	5	∈	∈	PROPN
ejpam-2534	188	6	rad(n	rad(n	PROPN
ejpam-2534	188	7	:	:	PUNCT
ejpam-2534	188	8	m	m	X
ejpam-2534	188	9	)	)	PUNCT
ejpam-2534	188	10	and	and	CCONJ
ejpam-2534	188	11	rad(n	rad(n	NOUN
ejpam-2534	188	12	:	:	PUNCT
ejpam-2534	188	13	m	m	X
ejpam-2534	188	14	)	)	PUNCT
ejpam-2534	188	15	is	be	AUX
ejpam-2534	188	16	a	a	DET
ejpam-2534	188	17	weakly	weakly	ADJ
ejpam-2534	188	18	prime	prime	ADJ
ejpam-2534	188	19	ideal	ideal	NOUN
ejpam-2534	188	20	of	of	ADP
ejpam-2534	188	21	r	r	NOUN
ejpam-2534	188	22	,	,	PUNCT
ejpam-2534	188	23	then	then	ADV
ejpam-2534	188	24	ur	ur	PROPN
ejpam-2534	188	25	∈	∈	PROPN
ejpam-2534	188	26	rad(n	rad(n	PROPN
ejpam-2534	188	27	:	:	PUNCT
ejpam-2534	188	28	m	m	X
ejpam-2534	188	29	)	)	PUNCT
ejpam-2534	188	30	or	or	CCONJ
ejpam-2534	188	31	s	s	PROPN
ejpam-2534	188	32	∈	∈	PROPN
ejpam-2534	188	33	rad(n	rad(n	PROPN
ejpam-2534	188	34	:	:	PUNCT
ejpam-2534	188	35	m	m	NUM
ejpam-2534	188	36	)	)	PUNCT
ejpam-2534	188	37	.	.	PUNCT
ejpam-2534	189	1	hence	hence	ADV
ejpam-2534	189	2	r	r	NOUN
ejpam-2534	189	3	p	p	X
ejpam-2534	189	4	=	=	PUNCT
ejpam-2534	189	5	u	u	NOUN
ejpam-2534	189	6	u	u	NOUN
ejpam-2534	189	7	r	r	NOUN
ejpam-2534	189	8	p	p	X
ejpam-2534	189	9	∈	∈	PROPN
ejpam-2534	189	10	(	(	PUNCT
ejpam-2534	189	11	rad(n	rad(n	NOUN
ejpam-2534	189	12	:	:	PUNCT
ejpam-2534	189	13	m))m	m))m	ADJ
ejpam-2534	189	14	or	or	CCONJ
ejpam-2534	189	15	s	s	PRON
ejpam-2534	189	16	q	q	X
ejpam-2534	189	17	∈	∈	PROPN
ejpam-2534	189	18	(	(	PUNCT
ejpam-2534	189	19	rad(n	rad(n	NOUN
ejpam-2534	189	20	:	:	PUNCT
ejpam-2534	189	21	m))m	m))m	ADJ
ejpam-2534	189	22	,	,	PUNCT
ejpam-2534	189	23	that	that	ADV
ejpam-2534	189	24	is	is	ADV
ejpam-2534	189	25	,	,	PUNCT
ejpam-2534	189	26	r	r	PROPN
ejpam-2534	189	27	p	p	X
ejpam-2534	189	28	∈	∈	PROPN
ejpam-2534	189	29	rad((n	rad((n	NOUN
ejpam-2534	189	30	:	:	PUNCT
ejpam-2534	189	31	m)m	m)m	X
ejpam-2534	189	32	)	)	PUNCT
ejpam-2534	189	33	or	or	CCONJ
ejpam-2534	189	34	s	s	PRON
ejpam-2534	189	35	q	q	PROPN
ejpam-2534	189	36	∈	∈	PROPN
ejpam-2534	189	37	rad((n	rad((n	NOUN
ejpam-2534	189	38	:	:	PUNCT
ejpam-2534	189	39	m)m	m)m	X
ejpam-2534	189	40	)	)	PUNCT
ejpam-2534	189	41	.	.	PUNCT
ejpam-2534	189	42	assume	assume	VERB
ejpam-2534	189	43	that	that	SCONJ
ejpam-2534	189	44	rad((n	rad((n	NOUN
ejpam-2534	189	45	:	:	PUNCT
ejpam-2534	189	46	m)m	m)m	X
ejpam-2534	189	47	)	)	PUNCT
ejpam-2534	189	48	is	be	AUX
ejpam-2534	189	49	a	a	DET
ejpam-2534	189	50	weakly	weakly	ADJ
ejpam-2534	189	51	prime	prime	ADJ
ejpam-2534	189	52	ideal	ideal	NOUN
ejpam-2534	189	53	of	of	ADP
ejpam-2534	189	54	rm	rm	PROPN
ejpam-2534	189	55	.	.	PUNCT
ejpam-2534	190	1	if	if	SCONJ
ejpam-2534	190	2	rad(n	rad(n	NOUN
ejpam-2534	190	3	:	:	PUNCT
ejpam-2534	190	4	m	m	X
ejpam-2534	190	5	)	)	PUNCT
ejpam-2534	191	1	=	=	SYM
ejpam-2534	191	2	r	r	NOUN
ejpam-2534	191	3	,	,	PUNCT
ejpam-2534	191	4	then	then	ADV
ejpam-2534	191	5	rad((n	rad((n	NOUN
ejpam-2534	191	6	:	:	PUNCT
ejpam-2534	191	7	m)m	m)m	X
ejpam-2534	191	8	)	)	PUNCT
ejpam-2534	191	9	=	=	SYM
ejpam-2534	191	10	rm	rm	PROPN
ejpam-2534	191	11	,	,	PUNCT
ejpam-2534	191	12	a	a	DET
ejpam-2534	191	13	contradiction	contradiction	NOUN
ejpam-2534	191	14	.	.	PUNCT
ejpam-2534	192	1	so	so	ADV
ejpam-2534	192	2	rad(n	rad(n	NOUN
ejpam-2534	192	3	:	:	PUNCT
ejpam-2534	192	4	m	m	X
ejpam-2534	192	5	)	)	PUNCT
ejpam-2534	192	6	is	be	AUX
ejpam-2534	192	7	a	a	DET
ejpam-2534	192	8	proper	proper	ADJ
ejpam-2534	192	9	ideal	ideal	NOUN
ejpam-2534	192	10	of	of	ADP
ejpam-2534	192	11	r.	r.	PROPN
ejpam-2534	192	12	let	let	VERB
ejpam-2534	192	13	0	0	NUM
ejpam-2534	192	14	6=	6=	NUM
ejpam-2534	192	15	ab	ab	PROPN
ejpam-2534	192	16	∈	∈	PROPN
ejpam-2534	192	17	rad(n	rad(n	PROPN
ejpam-2534	192	18	:	:	PUNCT
ejpam-2534	192	19	m	m	X
ejpam-2534	192	20	)	)	PUNCT
ejpam-2534	192	21	for	for	ADP
ejpam-2534	192	22	some	some	DET
ejpam-2534	192	23	a	a	PRON
ejpam-2534	192	24	,	,	PUNCT
ejpam-2534	192	25	b	b	PROPN
ejpam-2534	192	26	∈	∈	PROPN
ejpam-2534	192	27	r.	r.	PROPN
ejpam-2534	192	28	then	then	ADV
ejpam-2534	192	29	ab	ab	PROPN
ejpam-2534	192	30	1	1	NUM
ejpam-2534	192	31	=	=	PUNCT
ejpam-2534	192	32	a	a	DET
ejpam-2534	192	33	1	1	NUM
ejpam-2534	192	34	b	b	SYM
ejpam-2534	192	35	1	1	NUM
ejpam-2534	192	36	∈	∈	PROPN
ejpam-2534	192	37	rad((n	rad((n	NOUN
ejpam-2534	192	38	:	:	PUNCT
ejpam-2534	192	39	m)m	m)m	X
ejpam-2534	192	40	)	)	PUNCT
ejpam-2534	192	41	.	.	PUNCT
ejpam-2534	193	1	if	if	SCONJ
ejpam-2534	193	2	a	a	DET
ejpam-2534	193	3	1	1	NUM
ejpam-2534	193	4	b	b	SYM
ejpam-2534	193	5	1	1	NUM
ejpam-2534	193	6	=	=	SYM
ejpam-2534	193	7	0	0	NUM
ejpam-2534	193	8	,	,	PUNCT
ejpam-2534	193	9	then	then	ADV
ejpam-2534	193	10	qab	qab	NOUN
ejpam-2534	193	11	=	=	NOUN
ejpam-2534	193	12	0	0	NUM
ejpam-2534	193	13	for	for	ADP
ejpam-2534	193	14	some	some	DET
ejpam-2534	193	15	q	q	NOUN
ejpam-2534	193	16	∈	∈	PROPN
ejpam-2534	193	17	r	r	NOUN
ejpam-2534	193	18	\m	\m	NOUN
ejpam-2534	193	19	.	.	PUNCT
ejpam-2534	194	1	as	as	ADP
ejpam-2534	194	2	0	0	NUM
ejpam-2534	194	3	6=	6=	NUM
ejpam-2534	194	4	ab	ab	PROPN
ejpam-2534	194	5	,	,	PUNCT
ejpam-2534	194	6	then	then	ADV
ejpam-2534	194	7	q	q	PROPN
ejpam-2534	194	8	∈	∈	PROPN
ejpam-2534	194	9	p(0	p(0	PROPN
ejpam-2534	194	10	)	)	PUNCT
ejpam-2534	194	11	.	.	PUNCT
ejpam-2534	195	1	thus	thus	ADV
ejpam-2534	195	2	q	q	DET
ejpam-2534	195	3	∈m	∈m	NOUN
ejpam-2534	195	4	,	,	PUNCT
ejpam-2534	195	5	which	which	PRON
ejpam-2534	195	6	is	be	AUX
ejpam-2534	195	7	a	a	DET
ejpam-2534	195	8	contradiction	contradiction	NOUN
ejpam-2534	195	9	.	.	PUNCT
ejpam-2534	196	1	so	so	ADV
ejpam-2534	196	2	0	0	NUM
ejpam-2534	196	3	6=	6=	ADP
ejpam-2534	196	4	a	a	DET
ejpam-2534	196	5	1	1	NUM
ejpam-2534	196	6	b	b	SYM
ejpam-2534	196	7	1	1	NUM
ejpam-2534	196	8	∈	∈	PROPN
ejpam-2534	196	9	rad((n	rad((n	NOUN
ejpam-2534	196	10	:	:	PUNCT
ejpam-2534	196	11	m)m	m)m	X
ejpam-2534	196	12	)	)	PUNCT
ejpam-2534	196	13	.	.	PUNCT
ejpam-2534	197	1	since	since	SCONJ
ejpam-2534	197	2	rad((n	rad((n	NOUN
ejpam-2534	197	3	:	:	PUNCT
ejpam-2534	197	4	m)m	m)m	X
ejpam-2534	197	5	)	)	PUNCT
ejpam-2534	197	6	is	be	AUX
ejpam-2534	197	7	a	a	DET
ejpam-2534	197	8	weakly	weakly	ADJ
ejpam-2534	197	9	prime	prime	ADJ
ejpam-2534	197	10	ideal	ideal	NOUN
ejpam-2534	197	11	of	of	ADP
ejpam-2534	197	12	rm	rm	PROPN
ejpam-2534	197	13	,	,	PUNCT
ejpam-2534	197	14	then	then	ADV
ejpam-2534	197	15	a	a	DET
ejpam-2534	197	16	1	1	NUM
ejpam-2534	197	17	∈	∈	PROPN
ejpam-2534	197	18	rad((n	rad((n	NOUN
ejpam-2534	197	19	:	:	PUNCT
ejpam-2534	197	20	m)m	m)m	X
ejpam-2534	197	21	)	)	PUNCT
ejpam-2534	197	22	or	or	CCONJ
ejpam-2534	197	23	b	b	X
ejpam-2534	197	24	1	1	NUM
ejpam-2534	197	25	∈	∈	PROPN
ejpam-2534	197	26	rad((n	rad((n	NOUN
ejpam-2534	197	27	:	:	PUNCT
ejpam-2534	197	28	m)m	m)m	X
ejpam-2534	197	29	)	)	PUNCT
ejpam-2534	197	30	.	.	PUNCT
ejpam-2534	198	1	therefore	therefore	ADV
ejpam-2534	198	2	pa	pa	PROPN
ejpam-2534	198	3	∈	∈	PROPN
ejpam-2534	198	4	rad(n	rad(n	PROPN
ejpam-2534	198	5	:	:	PUNCT
ejpam-2534	198	6	m	m	X
ejpam-2534	198	7	)	)	PUNCT
ejpam-2534	198	8	for	for	ADP
ejpam-2534	198	9	some	some	DET
ejpam-2534	198	10	p	p	NOUN
ejpam-2534	198	11	∈	∈	PROPN
ejpam-2534	198	12	r\m	r\m	PRON
ejpam-2534	198	13	or	or	CCONJ
ejpam-2534	198	14	sb	sb	PROPN
ejpam-2534	198	15	∈	∈	PROPN
ejpam-2534	198	16	rad(n	rad(n	PROPN
ejpam-2534	198	17	:	:	PUNCT
ejpam-2534	198	18	m	m	X
ejpam-2534	198	19	)	)	PUNCT
ejpam-2534	198	20	for	for	ADP
ejpam-2534	198	21	some	some	PRON
ejpam-2534	198	22	s	s	VERB
ejpam-2534	198	23	/∈m	/∈m	PROPN
ejpam-2534	198	24	.	.	PUNCT
ejpam-2534	199	1	as	as	ADP
ejpam-2534	199	2	p	p	X
ejpam-2534	199	3	∈	∈	PROPN
ejpam-2534	199	4	r\m	r\m	NOUN
ejpam-2534	199	5	and	and	CCONJ
ejpam-2534	199	6	s	s	PROPN
ejpam-2534	199	7	∈	∈	PROPN
ejpam-2534	199	8	r\m	r\m	NOUN
ejpam-2534	199	9	,	,	PUNCT
ejpam-2534	199	10	then	then	ADV
ejpam-2534	199	11	p	p	X
ejpam-2534	199	12	,	,	PUNCT
ejpam-2534	199	13	s	s	PART
ejpam-2534	199	14	/∈	/∈	PUNCT
ejpam-2534	199	15	p(n	p(n	NOUN
ejpam-2534	199	16	)	)	PUNCT
ejpam-2534	199	17	.	.	PUNCT
ejpam-2534	200	1	consequently	consequently	ADV
ejpam-2534	200	2	,	,	PUNCT
ejpam-2534	200	3	a	a	DET
ejpam-2534	200	4	∈	∈	PROPN
ejpam-2534	200	5	rad(n	rad(n	NOUN
ejpam-2534	200	6	:	:	PUNCT
ejpam-2534	200	7	m	m	X
ejpam-2534	200	8	)	)	PUNCT
ejpam-2534	200	9	or	or	CCONJ
ejpam-2534	200	10	b	b	X
ejpam-2534	200	11	∈	∈	PROPN
ejpam-2534	200	12	rad(n	rad(n	PROPN
ejpam-2534	200	13	:	:	PUNCT
ejpam-2534	200	14	m	m	NUM
ejpam-2534	200	15	)	)	PUNCT
ejpam-2534	200	16	.	.	PUNCT
ejpam-2534	201	1	(	(	PUNCT
ejpam-2534	201	2	ii	ii	NOUN
ejpam-2534	201	3	)	)	PUNCT
ejpam-2534	201	4	assume	assume	VERB
ejpam-2534	201	5	that	that	SCONJ
ejpam-2534	201	6	rad(n	rad(n	NOUN
ejpam-2534	201	7	:	:	PUNCT
ejpam-2534	201	8	m	m	X
ejpam-2534	201	9	)	)	PUNCT
ejpam-2534	201	10	is	be	AUX
ejpam-2534	201	11	a	a	DET
ejpam-2534	201	12	prime	prime	ADJ
ejpam-2534	201	13	ideal	ideal	NOUN
ejpam-2534	201	14	of	of	ADP
ejpam-2534	201	15	r.	r.	PROPN
ejpam-2534	201	16	in	in	ADP
ejpam-2534	201	17	a	a	DET
ejpam-2534	201	18	similar	similar	ADJ
ejpam-2534	201	19	way	way	NOUN
ejpam-2534	201	20	,	,	PUNCT
ejpam-2534	201	21	we	we	PRON
ejpam-2534	201	22	get	get	VERB
ejpam-2534	201	23	rad((n	rad((n	NOUN
ejpam-2534	201	24	:	:	PUNCT
ejpam-2534	201	25	m)m	m)m	X
ejpam-2534	201	26	)	)	PUNCT
ejpam-2534	201	27	is	be	AUX
ejpam-2534	201	28	a	a	DET
ejpam-2534	201	29	proper	proper	ADJ
ejpam-2534	201	30	ideal	ideal	NOUN
ejpam-2534	201	31	of	of	ADP
ejpam-2534	201	32	rm	rm	PROPN
ejpam-2534	201	33	.	.	PUNCT
ejpam-2534	202	1	now	now	ADV
ejpam-2534	202	2	,	,	PUNCT
ejpam-2534	202	3	let	let	VERB
ejpam-2534	202	4	r	r	PRON
ejpam-2534	202	5	p	p	X
ejpam-2534	202	6	s	s	PART
ejpam-2534	202	7	q	q	NOUN
ejpam-2534	202	8	∈	∈	PROPN
ejpam-2534	202	9	rad((n	rad((n	NOUN
ejpam-2534	202	10	:	:	PUNCT
ejpam-2534	202	11	m)m	m)m	X
ejpam-2534	202	12	)	)	PUNCT
ejpam-2534	202	13	,	,	PUNCT
ejpam-2534	202	14	where	where	SCONJ
ejpam-2534	202	15	r	r	NOUN
ejpam-2534	202	16	,	,	PUNCT
ejpam-2534	202	17	s	s	NOUN
ejpam-2534	202	18	∈	∈	PROPN
ejpam-2534	202	19	r	r	NOUN
ejpam-2534	202	20	and	and	CCONJ
ejpam-2534	202	21	p	p	NOUN
ejpam-2534	202	22	,	,	PUNCT
ejpam-2534	202	23	q	q	PROPN
ejpam-2534	202	24	∈	∈	PROPN
ejpam-2534	202	25	r	r	NOUN
ejpam-2534	202	26	\m	\m	NOUN
ejpam-2534	202	27	.	.	PUNCT
ejpam-2534	203	1	then	then	ADV
ejpam-2534	203	2	we	we	PRON
ejpam-2534	203	3	have	have	VERB
ejpam-2534	203	4	rs	rs	PROPN
ejpam-2534	203	5	pq	pq	PROPN
ejpam-2534	203	6	∈	∈	PROPN
ejpam-2534	203	7	(	(	PUNCT
ejpam-2534	203	8	rad(n	rad(n	NOUN
ejpam-2534	203	9	:	:	PUNCT
ejpam-2534	203	10	m))m	m))m	ADJ
ejpam-2534	204	1	and	and	CCONJ
ejpam-2534	204	2	so	so	ADV
ejpam-2534	204	3	we	we	PRON
ejpam-2534	204	4	have	have	VERB
ejpam-2534	204	5	urs	urs	PROPN
ejpam-2534	204	6	∈	∈	PROPN
ejpam-2534	204	7	rad(n	rad(n	NOUN
ejpam-2534	204	8	:	:	PUNCT
ejpam-2534	204	9	m	m	X
ejpam-2534	204	10	)	)	PUNCT
ejpam-2534	204	11	for	for	ADP
ejpam-2534	204	12	some	some	DET
ejpam-2534	204	13	u	u	NOUN
ejpam-2534	204	14	∈	∈	PROPN
ejpam-2534	204	15	r	r	NOUN
ejpam-2534	204	16	\m	\m	NOUN
ejpam-2534	204	17	.	.	PUNCT
ejpam-2534	205	1	since	since	SCONJ
ejpam-2534	205	2	rad(n	rad(n	NOUN
ejpam-2534	205	3	:	:	PUNCT
ejpam-2534	205	4	m	m	X
ejpam-2534	205	5	)	)	PUNCT
ejpam-2534	205	6	is	be	AUX
ejpam-2534	205	7	a	a	DET
ejpam-2534	205	8	prime	prime	ADJ
ejpam-2534	205	9	ideal	ideal	NOUN
ejpam-2534	205	10	of	of	ADP
ejpam-2534	205	11	r	r	NOUN
ejpam-2534	205	12	,	,	PUNCT
ejpam-2534	205	13	then	then	ADV
ejpam-2534	205	14	ur	ur	PROPN
ejpam-2534	205	15	∈	∈	PROPN
ejpam-2534	205	16	rad(n	rad(n	PROPN
ejpam-2534	205	17	:	:	PUNCT
ejpam-2534	205	18	m	m	X
ejpam-2534	205	19	)	)	PUNCT
ejpam-2534	205	20	or	or	CCONJ
ejpam-2534	205	21	s	s	PROPN
ejpam-2534	205	22	∈	∈	PROPN
ejpam-2534	205	23	rad(n	rad(n	PROPN
ejpam-2534	205	24	:	:	PUNCT
ejpam-2534	205	25	m	m	NUM
ejpam-2534	205	26	)	)	PUNCT
ejpam-2534	205	27	.	.	PUNCT
ejpam-2534	206	1	consequently	consequently	ADV
ejpam-2534	206	2	,	,	PUNCT
ejpam-2534	206	3	r	r	NOUN
ejpam-2534	206	4	p	p	X
ejpam-2534	206	5	=	=	PUNCT
ejpam-2534	206	6	u	u	NOUN
ejpam-2534	206	7	u	u	NOUN
ejpam-2534	206	8	r	r	NOUN
ejpam-2534	206	9	p	p	X
ejpam-2534	206	10	∈	∈	PROPN
ejpam-2534	206	11	(	(	PUNCT
ejpam-2534	206	12	rad(n	rad(n	NOUN
ejpam-2534	206	13	:	:	PUNCT
ejpam-2534	206	14	m))m	m))m	ADJ
ejpam-2534	206	15	or	or	CCONJ
ejpam-2534	206	16	s	s	PRON
ejpam-2534	206	17	q	q	X
ejpam-2534	206	18	∈	∈	PROPN
ejpam-2534	206	19	(	(	PUNCT
ejpam-2534	206	20	rad(n	rad(n	NOUN
ejpam-2534	206	21	:	:	PUNCT
ejpam-2534	206	22	m))m	m))m	ADJ
ejpam-2534	206	23	,	,	PUNCT
ejpam-2534	206	24	that	that	ADV
ejpam-2534	206	25	is	is	ADV
ejpam-2534	206	26	,	,	PUNCT
ejpam-2534	206	27	r	r	PROPN
ejpam-2534	206	28	p	p	X
ejpam-2534	206	29	∈	∈	PROPN
ejpam-2534	206	30	rad((n	rad((n	NOUN
ejpam-2534	206	31	:	:	PUNCT
ejpam-2534	206	32	m)m	m)m	X
ejpam-2534	206	33	)	)	PUNCT
ejpam-2534	206	34	or	or	CCONJ
ejpam-2534	206	35	s	s	PRON
ejpam-2534	206	36	q	q	PROPN
ejpam-2534	206	37	∈	∈	PROPN
ejpam-2534	206	38	rad((n	rad((n	NOUN
ejpam-2534	206	39	:	:	PUNCT
ejpam-2534	206	40	m)m	m)m	X
ejpam-2534	206	41	)	)	PUNCT
ejpam-2534	206	42	.	.	PUNCT
ejpam-2534	206	43	g.	g.	PROPN
ejpam-2534	206	44	ulucak	ulucak	PROPN
ejpam-2534	206	45	and	and	CCONJ
ejpam-2534	206	46	r.	r.	PROPN
ejpam-2534	206	47	uregen	uregen	PROPN
ejpam-2534	206	48	/	/	SYM
ejpam-2534	206	49	eur	eur	PROPN
ejpam-2534	206	50	.	.	PUNCT
ejpam-2534	207	1	j.	j.	PROPN
ejpam-2534	207	2	pure	pure	PROPN
ejpam-2534	207	3	appl	appl	PROPN
ejpam-2534	207	4	.	.	PROPN
ejpam-2534	207	5	math	math	PROPN
ejpam-2534	207	6	,	,	PUNCT
ejpam-2534	207	7	9	9	NUM
ejpam-2534	207	8	(	(	PUNCT
ejpam-2534	207	9	2016	2016	NUM
ejpam-2534	207	10	)	)	PUNCT
ejpam-2534	207	11	,	,	PUNCT
ejpam-2534	207	12	48	48	NUM
ejpam-2534	207	13	-	-	SYM
ejpam-2534	207	14	56	56	NUM
ejpam-2534	207	15	53	53	NUM
ejpam-2534	207	16	suppose	suppose	VERB
ejpam-2534	207	17	that	that	SCONJ
ejpam-2534	207	18	rad((n	rad((n	NOUN
ejpam-2534	207	19	:	:	PUNCT
ejpam-2534	207	20	m)m	m)m	X
ejpam-2534	207	21	)	)	PUNCT
ejpam-2534	207	22	is	be	AUX
ejpam-2534	207	23	a	a	DET
ejpam-2534	207	24	prime	prime	ADJ
ejpam-2534	207	25	ideal	ideal	NOUN
ejpam-2534	207	26	of	of	ADP
ejpam-2534	207	27	rm	rm	PROPN
ejpam-2534	207	28	.	.	PUNCT
ejpam-2534	208	1	from	from	ADP
ejpam-2534	208	2	(	(	PUNCT
ejpam-2534	208	3	i	i	NOUN
ejpam-2534	208	4	)	)	PUNCT
ejpam-2534	208	5	,	,	PUNCT
ejpam-2534	208	6	it	it	PRON
ejpam-2534	208	7	is	be	AUX
ejpam-2534	208	8	clear	clear	ADJ
ejpam-2534	208	9	that	that	SCONJ
ejpam-2534	208	10	rad(n	rad(n	NOUN
ejpam-2534	208	11	:	:	PUNCT
ejpam-2534	208	12	m	m	X
ejpam-2534	208	13	)	)	PUNCT
ejpam-2534	208	14	is	be	AUX
ejpam-2534	208	15	a	a	DET
ejpam-2534	208	16	proper	proper	ADJ
ejpam-2534	208	17	ideal	ideal	NOUN
ejpam-2534	208	18	of	of	ADP
ejpam-2534	208	19	r.	r.	PROPN
ejpam-2534	208	20	then	then	ADV
ejpam-2534	208	21	ab	ab	PROPN
ejpam-2534	208	22	1	1	NUM
ejpam-2534	208	23	=	=	PUNCT
ejpam-2534	208	24	a	a	DET
ejpam-2534	208	25	1	1	NUM
ejpam-2534	208	26	b	b	SYM
ejpam-2534	208	27	1	1	NUM
ejpam-2534	208	28	∈	∈	PROPN
ejpam-2534	208	29	rad((n	rad((n	NOUN
ejpam-2534	208	30	:	:	PUNCT
ejpam-2534	208	31	m)m	m)m	X
ejpam-2534	208	32	)	)	PUNCT
ejpam-2534	208	33	for	for	ADP
ejpam-2534	208	34	some	some	DET
ejpam-2534	208	35	a	a	PRON
ejpam-2534	208	36	,	,	PUNCT
ejpam-2534	208	37	b	b	X
ejpam-2534	208	38	∈	∈	NOUN
ejpam-2534	208	39	r	r	NOUN
ejpam-2534	208	40	and	and	CCONJ
ejpam-2534	208	41	since	since	SCONJ
ejpam-2534	208	42	rad(n	rad(n	NOUN
ejpam-2534	208	43	:	:	PUNCT
ejpam-2534	208	44	m)m	m)m	X
ejpam-2534	208	45	is	be	AUX
ejpam-2534	208	46	a	a	DET
ejpam-2534	208	47	prime	prime	ADJ
ejpam-2534	208	48	ideal	ideal	NOUN
ejpam-2534	208	49	of	of	ADP
ejpam-2534	208	50	rm	rm	PROPN
ejpam-2534	208	51	,	,	PUNCT
ejpam-2534	208	52	then	then	ADV
ejpam-2534	208	53	a	a	DET
ejpam-2534	208	54	1	1	NUM
ejpam-2534	208	55	∈	∈	PROPN
ejpam-2534	208	56	rad((n	rad((n	NOUN
ejpam-2534	208	57	:	:	PUNCT
ejpam-2534	208	58	m)m	m)m	X
ejpam-2534	208	59	)	)	PUNCT
ejpam-2534	208	60	or	or	CCONJ
ejpam-2534	208	61	b	b	X
ejpam-2534	208	62	1	1	NUM
ejpam-2534	208	63	∈	∈	PROPN
ejpam-2534	208	64	rad((n	rad((n	NOUN
ejpam-2534	208	65	:	:	PUNCT
ejpam-2534	208	66	m)m	m)m	X
ejpam-2534	208	67	)	)	PUNCT
ejpam-2534	208	68	.	.	PUNCT
ejpam-2534	209	1	thus	thus	ADV
ejpam-2534	209	2	pa	pa	PROPN
ejpam-2534	209	3	∈	∈	PROPN
ejpam-2534	209	4	rad(n	rad(n	PROPN
ejpam-2534	209	5	:	:	PUNCT
ejpam-2534	209	6	m	m	X
ejpam-2534	209	7	)	)	PUNCT
ejpam-2534	209	8	for	for	ADP
ejpam-2534	209	9	some	some	DET
ejpam-2534	209	10	p	p	NOUN
ejpam-2534	209	11	∈	∈	PROPN
ejpam-2534	209	12	r	r	NOUN
ejpam-2534	209	13	\m	\m	NOUN
ejpam-2534	209	14	or	or	CCONJ
ejpam-2534	209	15	sb	sb	PROPN
ejpam-2534	209	16	∈	∈	PROPN
ejpam-2534	209	17	rad(n	rad(n	PROPN
ejpam-2534	209	18	:	:	PUNCT
ejpam-2534	209	19	m	m	X
ejpam-2534	209	20	)	)	PUNCT
ejpam-2534	209	21	for	for	ADP
ejpam-2534	209	22	some	some	DET
ejpam-2534	209	23	s	s	PART
ejpam-2534	209	24	∈	∈	PROPN
ejpam-2534	209	25	r	r	NOUN
ejpam-2534	209	26	\m	\m	NOUN
ejpam-2534	209	27	.	.	PUNCT
ejpam-2534	210	1	as	as	ADP
ejpam-2534	210	2	p	p	PROPN
ejpam-2534	210	3	∈	∈	PROPN
ejpam-2534	210	4	r	r	NOUN
ejpam-2534	210	5	\m	\m	NOUN
ejpam-2534	210	6	and	and	CCONJ
ejpam-2534	210	7	s	s	PROPN
ejpam-2534	210	8	∈	∈	PROPN
ejpam-2534	210	9	r	r	NOUN
ejpam-2534	210	10	\m	\m	NOUN
ejpam-2534	210	11	,	,	PUNCT
ejpam-2534	210	12	then	then	ADV
ejpam-2534	210	13	p	p	X
ejpam-2534	210	14	,	,	PUNCT
ejpam-2534	210	15	s	s	PART
ejpam-2534	210	16	/∈	/∈	PUNCT
ejpam-2534	210	17	p(n	p(n	NOUN
ejpam-2534	210	18	)	)	PUNCT
ejpam-2534	210	19	.	.	PUNCT
ejpam-2534	211	1	therefore	therefore	ADV
ejpam-2534	211	2	,	,	PUNCT
ejpam-2534	211	3	a	a	DET
ejpam-2534	211	4	∈	∈	PROPN
ejpam-2534	211	5	rad(n	rad(n	NOUN
ejpam-2534	211	6	:	:	PUNCT
ejpam-2534	211	7	m	m	X
ejpam-2534	211	8	)	)	PUNCT
ejpam-2534	211	9	or	or	CCONJ
ejpam-2534	211	10	b	b	X
ejpam-2534	211	11	∈	∈	PROPN
ejpam-2534	211	12	rad(n	rad(n	PROPN
ejpam-2534	211	13	:	:	PUNCT
ejpam-2534	211	14	m	m	NUM
ejpam-2534	211	15	)	)	PUNCT
ejpam-2534	211	16	.	.	PUNCT
ejpam-2534	212	1	proposition	proposition	NOUN
ejpam-2534	212	2	6	6	NUM
ejpam-2534	212	3	.	.	PUNCT
ejpam-2534	213	1	let	let	VERB
ejpam-2534	213	2	m	m	PRON
ejpam-2534	213	3	be	be	AUX
ejpam-2534	213	4	a	a	DET
ejpam-2534	213	5	faithful	faithful	ADJ
ejpam-2534	213	6	cyclic	cyclic	ADJ
ejpam-2534	213	7	r	r	NOUN
ejpam-2534	213	8	-	-	PUNCT
ejpam-2534	213	9	module	module	NOUN
ejpam-2534	213	10	and	and	CCONJ
ejpam-2534	213	11	n	n	CCONJ
ejpam-2534	213	12	be	be	VERB
ejpam-2534	213	13	a	a	DET
ejpam-2534	213	14	proper	proper	ADJ
ejpam-2534	213	15	submodule	submodule	NOUN
ejpam-2534	213	16	of	of	ADP
ejpam-2534	213	17	m	m	PROPN
ejpam-2534	213	18	with	with	ADP
ejpam-2534	213	19	p(0	p(0	PROPN
ejpam-2534	213	20	)	)	PUNCT
ejpam-2534	213	21	⊆	⊆	NUM
ejpam-2534	213	22	p(n	p(n	PROPN
ejpam-2534	213	23	)	)	PUNCT
ejpam-2534	213	24	.	.	PUNCT
ejpam-2534	214	1	if	if	SCONJ
ejpam-2534	214	2	n	n	PRON
ejpam-2534	214	3	is	be	AUX
ejpam-2534	214	4	a	a	DET
ejpam-2534	214	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	214	6	weakly	weakly	ADJ
ejpam-2534	214	7	primary	primary	ADJ
ejpam-2534	214	8	submodule	submodule	NOUN
ejpam-2534	214	9	of	of	ADP
ejpam-2534	214	10	m	m	PROPN
ejpam-2534	214	11	,	,	PUNCT
ejpam-2534	214	12	then	then	ADV
ejpam-2534	214	13	rad(n	rad(n	PROPN
ejpam-2534	214	14	:	:	PUNCT
ejpam-2534	214	15	m	m	X
ejpam-2534	214	16	)	)	PUNCT
ejpam-2534	214	17	is	be	AUX
ejpam-2534	214	18	a	a	DET
ejpam-2534	214	19	weakly	weakly	ADJ
ejpam-2534	214	20	prime	prime	ADJ
ejpam-2534	214	21	ideal	ideal	NOUN
ejpam-2534	214	22	of	of	ADP
ejpam-2534	214	23	r.	r.	PROPN
ejpam-2534	214	24	proof	proof	NOUN
ejpam-2534	214	25	.	.	PUNCT
ejpam-2534	215	1	let	let	VERB
ejpam-2534	215	2	m	m	PRON
ejpam-2534	215	3	be	be	AUX
ejpam-2534	215	4	a	a	DET
ejpam-2534	215	5	maximal	maximal	ADJ
ejpam-2534	215	6	ideal	ideal	NOUN
ejpam-2534	215	7	of	of	ADP
ejpam-2534	215	8	r	r	NOUN
ejpam-2534	215	9	with	with	ADP
ejpam-2534	215	10	p(n	p(n	PROPN
ejpam-2534	215	11	)	)	PUNCT
ejpam-2534	215	12	⊆	⊆	NUM
ejpam-2534	215	13	m.	m.	NOUN
ejpam-2534	215	14	by	by	ADP
ejpam-2534	215	15	[	[	X
ejpam-2534	215	16	4	4	NUM
ejpam-2534	215	17	,	,	PUNCT
ejpam-2534	215	18	proposition	proposition	NOUN
ejpam-2534	215	19	2.18	2.18	NUM
ejpam-2534	215	20	]	]	PUNCT
ejpam-2534	215	21	,	,	PUNCT
ejpam-2534	215	22	mm	mm	PROPN
ejpam-2534	215	23	is	be	AUX
ejpam-2534	215	24	a	a	DET
ejpam-2534	215	25	faithful	faithful	ADJ
ejpam-2534	215	26	cyclic	cyclic	ADJ
ejpam-2534	215	27	rm	rm	NOUN
ejpam-2534	215	28	-	-	PUNCT
ejpam-2534	215	29	module	module	NOUN
ejpam-2534	215	30	.	.	PUNCT
ejpam-2534	216	1	then	then	ADV
ejpam-2534	216	2	nm	nm	PRON
ejpam-2534	216	3	is	be	AUX
ejpam-2534	216	4	a	a	DET
ejpam-2534	216	5	weakly	weakly	ADJ
ejpam-2534	216	6	primary	primary	ADJ
ejpam-2534	216	7	submodule	submodule	NOUN
ejpam-2534	216	8	of	of	ADP
ejpam-2534	216	9	mm	mm	PROPN
ejpam-2534	216	10	.	.	PUNCT
ejpam-2534	217	1	thus	thus	ADV
ejpam-2534	217	2	by	by	ADP
ejpam-2534	217	3	[	[	X
ejpam-2534	217	4	1	1	NUM
ejpam-2534	217	5	,	,	PUNCT
ejpam-2534	217	6	proposition	proposition	NOUN
ejpam-2534	217	7	2.3	2.3	NUM
ejpam-2534	217	8	]	]	PUNCT
ejpam-2534	217	9	,	,	PUNCT
ejpam-2534	217	10	rad(nm	rad(nm	VERB
ejpam-2534	217	11	:	:	PUNCT
ejpam-2534	217	12	mm	mm	X
ejpam-2534	217	13	)	)	PUNCT
ejpam-2534	217	14	is	be	AUX
ejpam-2534	217	15	a	a	DET
ejpam-2534	217	16	weakly	weakly	ADJ
ejpam-2534	217	17	prime	prime	ADJ
ejpam-2534	217	18	submodule	submodule	NOUN
ejpam-2534	217	19	of	of	ADP
ejpam-2534	217	20	mm	mm	PROPN
ejpam-2534	217	21	.	.	PUNCT
ejpam-2534	218	1	by	by	ADP
ejpam-2534	218	2	proposition	proposition	NOUN
ejpam-2534	218	3	3	3	NUM
ejpam-2534	218	4	,	,	PUNCT
ejpam-2534	218	5	rad((n	rad((n	NOUN
ejpam-2534	218	6	:	:	PUNCT
ejpam-2534	218	7	m)m	m)m	X
ejpam-2534	218	8	)	)	PUNCT
ejpam-2534	218	9	is	be	AUX
ejpam-2534	218	10	a	a	DET
ejpam-2534	218	11	weakly	weakly	ADJ
ejpam-2534	218	12	prime	prime	ADJ
ejpam-2534	218	13	submodule	submodule	NOUN
ejpam-2534	218	14	of	of	ADP
ejpam-2534	218	15	mm	mm	PROPN
ejpam-2534	218	16	.	.	PUNCT
ejpam-2534	219	1	by	by	ADP
ejpam-2534	219	2	proposition	proposition	NOUN
ejpam-2534	219	3	5	5	NUM
ejpam-2534	219	4	i	i	NOUN
ejpam-2534	219	5	)	)	PUNCT
ejpam-2534	219	6	,	,	PUNCT
ejpam-2534	219	7	rad(n	rad(n	PROPN
ejpam-2534	219	8	:	:	PUNCT
ejpam-2534	219	9	m	m	X
ejpam-2534	219	10	)	)	PUNCT
ejpam-2534	219	11	is	be	AUX
ejpam-2534	219	12	a	a	DET
ejpam-2534	219	13	weakly	weakly	ADJ
ejpam-2534	219	14	prime	prime	ADJ
ejpam-2534	219	15	ideal	ideal	NOUN
ejpam-2534	219	16	of	of	ADP
ejpam-2534	219	17	r.	r.	PROPN
ejpam-2534	219	18	proposition	proposition	PROPN
ejpam-2534	219	19	7	7	NUM
ejpam-2534	219	20	.	.	PUNCT
ejpam-2534	220	1	let	let	VERB
ejpam-2534	220	2	m	m	PRON
ejpam-2534	220	3	be	be	AUX
ejpam-2534	220	4	an	an	DET
ejpam-2534	220	5	r	r	NOUN
ejpam-2534	220	6	-	-	PUNCT
ejpam-2534	220	7	module	module	NOUN
ejpam-2534	220	8	.	.	PUNCT
ejpam-2534	221	1	suppose	suppose	VERB
ejpam-2534	221	2	that	that	SCONJ
ejpam-2534	221	3	n	n	PRON
ejpam-2534	221	4	is	be	AUX
ejpam-2534	221	5	an	an	DET
ejpam-2534	221	6	m	m	NOUN
ejpam-2534	221	7	-	-	NOUN
ejpam-2534	221	8	primal	primal	ADJ
ejpam-2534	221	9	and	and	CCONJ
ejpam-2534	221	10	a	a	DET
ejpam-2534	221	11	p(n)-locally	p(n)-locally	ADV
ejpam-2534	221	12	weakly	weakly	ADJ
ejpam-2534	221	13	primary	primary	ADJ
ejpam-2534	221	14	submodule	submodule	NOUN
ejpam-2534	221	15	of	of	ADP
ejpam-2534	221	16	m	m	PRON
ejpam-2534	221	17	not	not	PART
ejpam-2534	221	18	primary	primary	ADJ
ejpam-2534	221	19	submodule	submodule	NOUN
ejpam-2534	221	20	of	of	ADP
ejpam-2534	221	21	m.	m.	NOUN
ejpam-2534	221	22	if	if	SCONJ
ejpam-2534	221	23	p(0	p(0	PROPN
ejpam-2534	221	24	)	)	PUNCT
ejpam-2534	221	25	⊆	⊆	NUM
ejpam-2534	221	26	p(n	p(n	PROPN
ejpam-2534	221	27	)	)	PUNCT
ejpam-2534	221	28	and	and	CCONJ
ejpam-2534	221	29	i	i	PRON
ejpam-2534	221	30	is	be	AUX
ejpam-2534	221	31	an	an	DET
ejpam-2534	221	32	ideal	ideal	NOUN
ejpam-2534	221	33	of	of	ADP
ejpam-2534	221	34	r	r	NOUN
ejpam-2534	221	35	such	such	ADJ
ejpam-2534	221	36	that	that	SCONJ
ejpam-2534	221	37	i	i	PRON
ejpam-2534	221	38	⊆	⊆	NUM
ejpam-2534	221	39	rad(n	rad(n	NOUN
ejpam-2534	221	40	:	:	PUNCT
ejpam-2534	221	41	m	m	NUM
ejpam-2534	221	42	)	)	PUNCT
ejpam-2534	221	43	,	,	PUNCT
ejpam-2534	221	44	then	then	ADV
ejpam-2534	221	45	in	in	ADP
ejpam-2534	221	46	=	=	PROPN
ejpam-2534	221	47	0	0	X
ejpam-2534	221	48	.	.	PUNCT
ejpam-2534	222	1	particularly	particularly	ADV
ejpam-2534	222	2	,	,	PUNCT
ejpam-2534	222	3	rad(n	rad(n	NOUN
ejpam-2534	222	4	:	:	PUNCT
ejpam-2534	222	5	m)n	m)n	X
ejpam-2534	222	6	=	=	SYM
ejpam-2534	222	7	0	0	X
ejpam-2534	222	8	.	.	PUNCT
ejpam-2534	223	1	proof	proof	NOUN
ejpam-2534	223	2	.	.	PUNCT
ejpam-2534	224	1	suppose	suppose	VERB
ejpam-2534	224	2	that	that	SCONJ
ejpam-2534	224	3	p(0	p(0	NOUN
ejpam-2534	224	4	)	)	PUNCT
ejpam-2534	224	5	⊆	⊆	NUM
ejpam-2534	224	6	p(n	p(n	PROPN
ejpam-2534	224	7	)	)	PUNCT
ejpam-2534	224	8	and	and	CCONJ
ejpam-2534	224	9	i	i	PRON
ejpam-2534	224	10	is	be	AUX
ejpam-2534	224	11	an	an	DET
ejpam-2534	224	12	ideal	ideal	NOUN
ejpam-2534	224	13	of	of	ADP
ejpam-2534	224	14	r	r	NOUN
ejpam-2534	224	15	such	such	ADJ
ejpam-2534	224	16	that	that	SCONJ
ejpam-2534	224	17	i	i	PRON
ejpam-2534	224	18	⊆	⊆	NUM
ejpam-2534	224	19	rad(n	rad(n	NOUN
ejpam-2534	224	20	:	:	PUNCT
ejpam-2534	224	21	m	m	NUM
ejpam-2534	224	22	)	)	PUNCT
ejpam-2534	224	23	.	.	PUNCT
ejpam-2534	225	1	since	since	SCONJ
ejpam-2534	225	2	n	n	NUM
ejpam-2534	225	3	is	be	AUX
ejpam-2534	225	4	m	m	NOUN
ejpam-2534	225	5	-	-	ADJ
ejpam-2534	225	6	primal	primal	ADJ
ejpam-2534	225	7	,	,	PUNCT
ejpam-2534	225	8	then	then	ADV
ejpam-2534	225	9	p(n	p(n	PROPN
ejpam-2534	225	10	)	)	PUNCT
ejpam-2534	225	11	is	be	AUX
ejpam-2534	225	12	an	an	DET
ejpam-2534	225	13	ideal	ideal	NOUN
ejpam-2534	225	14	of	of	ADP
ejpam-2534	225	15	r.	r.	PROPN
ejpam-2534	225	16	as	as	ADP
ejpam-2534	225	17	1	1	NUM
ejpam-2534	225	18	/∈	/∈	PUNCT
ejpam-2534	225	19	p(n	p(n	PROPN
ejpam-2534	225	20	)	)	PUNCT
ejpam-2534	225	21	,	,	PUNCT
ejpam-2534	225	22	then	then	ADV
ejpam-2534	225	23	p(n	p(n	PROPN
ejpam-2534	225	24	)	)	PUNCT
ejpam-2534	225	25	is	be	AUX
ejpam-2534	225	26	a	a	DET
ejpam-2534	225	27	proper	proper	ADJ
ejpam-2534	225	28	ideal	ideal	NOUN
ejpam-2534	225	29	.	.	PUNCT
ejpam-2534	226	1	hence	hence	ADV
ejpam-2534	226	2	there	there	PRON
ejpam-2534	226	3	is	be	VERB
ejpam-2534	226	4	a	a	DET
ejpam-2534	226	5	maximal	maximal	ADJ
ejpam-2534	226	6	ideal	ideal	NOUN
ejpam-2534	226	7	m	m	NOUN
ejpam-2534	226	8	of	of	ADP
ejpam-2534	226	9	r	r	NOUN
ejpam-2534	226	10	such	such	ADJ
ejpam-2534	226	11	that	that	DET
ejpam-2534	226	12	p(n	p(n	NOUN
ejpam-2534	226	13	)	)	PUNCT
ejpam-2534	226	14	⊆m	⊆m	NOUN
ejpam-2534	226	15	.	.	PUNCT
ejpam-2534	227	1	then	then	ADV
ejpam-2534	227	2	,	,	PUNCT
ejpam-2534	227	3	nm	nm	ADV
ejpam-2534	227	4	is	be	AUX
ejpam-2534	227	5	a	a	DET
ejpam-2534	227	6	weakly	weakly	ADJ
ejpam-2534	227	7	primary	primary	ADJ
ejpam-2534	227	8	submodule	submodule	NOUN
ejpam-2534	227	9	of	of	ADP
ejpam-2534	227	10	mm	mm	PROPN
ejpam-2534	227	11	because	because	SCONJ
ejpam-2534	227	12	n	n	PROPN
ejpam-2534	227	13	is	be	AUX
ejpam-2534	227	14	a	a	DET
ejpam-2534	227	15	p(n)-locally	p(n)-locally	ADV
ejpam-2534	227	16	weakly	weakly	ADJ
ejpam-2534	227	17	primary	primary	ADJ
ejpam-2534	227	18	submodule	submodule	NOUN
ejpam-2534	227	19	of	of	ADP
ejpam-2534	227	20	m	m	PROPN
ejpam-2534	227	21	.	.	PUNCT
ejpam-2534	228	1	our	our	PRON
ejpam-2534	228	2	aim	aim	NOUN
ejpam-2534	228	3	is	be	AUX
ejpam-2534	228	4	to	to	PART
ejpam-2534	228	5	show	show	VERB
ejpam-2534	228	6	that	that	SCONJ
ejpam-2534	228	7	nm	nm	NOUN
ejpam-2534	228	8	is	be	AUX
ejpam-2534	228	9	not	not	PART
ejpam-2534	228	10	a	a	DET
ejpam-2534	228	11	primary	primary	ADJ
ejpam-2534	228	12	submodule	submodule	NOUN
ejpam-2534	228	13	of	of	ADP
ejpam-2534	228	14	mm	mm	PROPN
ejpam-2534	228	15	.	.	PUNCT
ejpam-2534	229	1	assume	assume	VERB
ejpam-2534	229	2	that	that	SCONJ
ejpam-2534	229	3	nm	nm	NOUN
ejpam-2534	229	4	is	be	AUX
ejpam-2534	229	5	a	a	DET
ejpam-2534	229	6	primary	primary	ADJ
ejpam-2534	229	7	submodule	submodule	NOUN
ejpam-2534	229	8	of	of	ADP
ejpam-2534	229	9	mm	mm	PROPN
ejpam-2534	229	10	.	.	PUNCT
ejpam-2534	230	1	let	let	VERB
ejpam-2534	230	2	rm	rm	PROPN
ejpam-2534	230	3	∈	∈	PROPN
ejpam-2534	230	4	n	n	PROPN
ejpam-2534	230	5	for	for	ADP
ejpam-2534	230	6	some	some	DET
ejpam-2534	230	7	r	r	NOUN
ejpam-2534	230	8	∈	∈	NOUN
ejpam-2534	230	9	r	r	NOUN
ejpam-2534	230	10	,	,	PUNCT
ejpam-2534	230	11	m	m	VERB
ejpam-2534	230	12	∈	∈	NOUN
ejpam-2534	230	13	m	m	NOUN
ejpam-2534	230	14	.	.	PUNCT
ejpam-2534	231	1	then	then	ADV
ejpam-2534	231	2	rm	rm	PROPN
ejpam-2534	231	3	1	1	NUM
ejpam-2534	231	4	=	=	SYM
ejpam-2534	231	5	r	r	NOUN
ejpam-2534	231	6	1	1	NUM
ejpam-2534	231	7	m	m	NUM
ejpam-2534	231	8	1	1	NUM
ejpam-2534	231	9	∈	∈	PROPN
ejpam-2534	231	10	nm	nm	NOUN
ejpam-2534	231	11	.	.	PUNCT
ejpam-2534	232	1	by	by	ADP
ejpam-2534	232	2	assumption	assumption	NOUN
ejpam-2534	232	3	,	,	PUNCT
ejpam-2534	232	4	m	m	PROPN
ejpam-2534	232	5	1	1	NUM
ejpam-2534	232	6	∈	∈	NOUN
ejpam-2534	232	7	nm	nm	NOUN
ejpam-2534	232	8	or	or	CCONJ
ejpam-2534	232	9	(	(	PUNCT
ejpam-2534	232	10	r	r	NOUN
ejpam-2534	232	11	1	1	NUM
ejpam-2534	232	12	)	)	PUNCT
ejpam-2534	232	13	nmm	nmm	PROPN
ejpam-2534	232	14	⊆	⊆	NUM
ejpam-2534	232	15	nm	nm	NOUN
ejpam-2534	232	16	for	for	ADP
ejpam-2534	232	17	some	some	DET
ejpam-2534	232	18	positive	positive	ADJ
ejpam-2534	232	19	integer	integer	NOUN
ejpam-2534	232	20	n.	n.	NOUN
ejpam-2534	232	21	by	by	ADP
ejpam-2534	232	22	using	use	VERB
ejpam-2534	232	23	a	a	DET
ejpam-2534	232	24	similar	similar	ADJ
ejpam-2534	232	25	technique	technique	NOUN
ejpam-2534	232	26	in	in	ADP
ejpam-2534	232	27	the	the	DET
ejpam-2534	232	28	previous	previous	ADJ
ejpam-2534	232	29	proofs	proof	NOUN
ejpam-2534	232	30	,	,	PUNCT
ejpam-2534	232	31	m	m	VERB
ejpam-2534	232	32	∈	∈	PROPN
ejpam-2534	232	33	n	n	NOUN
ejpam-2534	232	34	or	or	CCONJ
ejpam-2534	232	35	rnm	rnm	VERB
ejpam-2534	232	36	⊆	⊆	NUM
ejpam-2534	232	37	n	n	NOUN
ejpam-2534	232	38	for	for	ADP
ejpam-2534	232	39	some	some	DET
ejpam-2534	232	40	positive	positive	ADJ
ejpam-2534	232	41	integer	integer	NOUN
ejpam-2534	232	42	n	n	CCONJ
ejpam-2534	232	43	since	since	SCONJ
ejpam-2534	232	44	p(n	p(n	PROPN
ejpam-2534	232	45	)	)	PUNCT
ejpam-2534	232	46	⊆	⊆	NUM
ejpam-2534	232	47	m	m	NOUN
ejpam-2534	232	48	,	,	PUNCT
ejpam-2534	232	49	but	but	CCONJ
ejpam-2534	232	50	this	this	PRON
ejpam-2534	232	51	contradicts	contradict	VERB
ejpam-2534	232	52	with	with	ADP
ejpam-2534	232	53	n	n	PRON
ejpam-2534	232	54	which	which	PRON
ejpam-2534	232	55	is	be	AUX
ejpam-2534	232	56	not	not	PART
ejpam-2534	232	57	a	a	DET
ejpam-2534	232	58	primary	primary	ADJ
ejpam-2534	232	59	submodule	submodule	NOUN
ejpam-2534	232	60	of	of	ADP
ejpam-2534	232	61	m	m	PROPN
ejpam-2534	232	62	.	.	PUNCT
ejpam-2534	233	1	by	by	ADP
ejpam-2534	233	2	[	[	X
ejpam-2534	233	3	4	4	NUM
ejpam-2534	233	4	,	,	PUNCT
ejpam-2534	233	5	lemma	lemma	PROPN
ejpam-2534	233	6	2.19	2.19	NUM
ejpam-2534	233	7	]	]	PUNCT
ejpam-2534	233	8	,	,	PUNCT
ejpam-2534	233	9	i	i	PRON
ejpam-2534	233	10	m	m	VERB
ejpam-2534	233	11	⊆	⊆	NUM
ejpam-2534	233	12	rad((n	rad((n	NOUN
ejpam-2534	233	13	:	:	PUNCT
ejpam-2534	233	14	m)m	m)m	X
ejpam-2534	233	15	)	)	PUNCT
ejpam-2534	233	16	⊆	⊆	NUM
ejpam-2534	233	17	rad(nm	rad(nm	NOUN
ejpam-2534	233	18	:	:	PUNCT
ejpam-2534	233	19	mm	mm	X
ejpam-2534	233	20	)	)	PUNCT
ejpam-2534	233	21	.	.	PUNCT
ejpam-2534	234	1	by	by	ADP
ejpam-2534	234	2	[	[	X
ejpam-2534	234	3	1	1	NUM
ejpam-2534	234	4	,	,	PUNCT
ejpam-2534	234	5	corollary	corollary	ADJ
ejpam-2534	234	6	3.4	3.4	NUM
ejpam-2534	234	7	]	]	PUNCT
ejpam-2534	234	8	,	,	PUNCT
ejpam-2534	234	9	imnm	imnm	PROPN
ejpam-2534	234	10	=	=	PUNCT
ejpam-2534	234	11	0	0	X
ejpam-2534	234	12	.	.	PUNCT
ejpam-2534	235	1	we	we	PRON
ejpam-2534	235	2	get	get	VERB
ejpam-2534	235	3	r	r	NOUN
ejpam-2534	235	4	1	1	NUM
ejpam-2534	235	5	m	m	NOUN
ejpam-2534	235	6	1	1	NUM
ejpam-2534	235	7	=	=	SYM
ejpam-2534	235	8	rm	rm	NOUN
ejpam-2534	235	9	1	1	NUM
ejpam-2534	235	10	=	=	SYM
ejpam-2534	235	11	0	0	NUM
ejpam-2534	235	12	for	for	ADP
ejpam-2534	235	13	every	every	DET
ejpam-2534	235	14	r	r	NOUN
ejpam-2534	235	15	∈	∈	NOUN
ejpam-2534	236	1	i	i	PRON
ejpam-2534	236	2	and	and	CCONJ
ejpam-2534	236	3	every	every	DET
ejpam-2534	236	4	m	m	PROPN
ejpam-2534	236	5	∈	∈	PROPN
ejpam-2534	236	6	n	n	NOUN
ejpam-2534	236	7	.	.	PUNCT
ejpam-2534	237	1	therefore	therefore	ADV
ejpam-2534	237	2	qrm	qrm	PROPN
ejpam-2534	237	3	=	=	PUNCT
ejpam-2534	237	4	0	0	NUM
ejpam-2534	237	5	for	for	ADP
ejpam-2534	237	6	some	some	DET
ejpam-2534	237	7	q	q	NOUN
ejpam-2534	237	8	∈	∈	PROPN
ejpam-2534	237	9	r	r	NOUN
ejpam-2534	237	10	\m	\m	NOUN
ejpam-2534	237	11	.	.	PUNCT
ejpam-2534	238	1	if	if	SCONJ
ejpam-2534	238	2	rm	rm	PROPN
ejpam-2534	238	3	6=	6=	PROPN
ejpam-2534	238	4	0	0	NUM
ejpam-2534	238	5	,	,	PUNCT
ejpam-2534	238	6	then	then	ADV
ejpam-2534	238	7	q	q	PROPN
ejpam-2534	238	8	∈	∈	PROPN
ejpam-2534	238	9	p(0	p(0	PROPN
ejpam-2534	238	10	)	)	PUNCT
ejpam-2534	238	11	and	and	CCONJ
ejpam-2534	238	12	so	so	ADV
ejpam-2534	238	13	q	q	PUNCT
ejpam-2534	238	14	∈	∈	PROPN
ejpam-2534	238	15	m	m	PROPN
ejpam-2534	238	16	,	,	PUNCT
ejpam-2534	238	17	which	which	PRON
ejpam-2534	238	18	is	be	AUX
ejpam-2534	238	19	a	a	DET
ejpam-2534	238	20	contradiction	contradiction	NOUN
ejpam-2534	238	21	.	.	PUNCT
ejpam-2534	239	1	hence	hence	ADV
ejpam-2534	239	2	rm	rm	PROPN
ejpam-2534	239	3	=	=	SYM
ejpam-2534	239	4	0	0	PROPN
ejpam-2534	239	5	,	,	PUNCT
ejpam-2534	239	6	that	that	ADV
ejpam-2534	239	7	is	is	ADV
ejpam-2534	239	8	,	,	PUNCT
ejpam-2534	239	9	in	in	ADP
ejpam-2534	239	10	=	=	NOUN
ejpam-2534	239	11	0	0	X
ejpam-2534	239	12	.	.	PUNCT
ejpam-2534	240	1	particularly	particularly	ADV
ejpam-2534	240	2	,	,	PUNCT
ejpam-2534	240	3	by	by	ADP
ejpam-2534	240	4	putting	put	VERB
ejpam-2534	240	5	i	i	PRON
ejpam-2534	240	6	=	=	SYM
ejpam-2534	240	7	rad(n	rad(n	PROPN
ejpam-2534	240	8	:	:	PUNCT
ejpam-2534	240	9	m	m	X
ejpam-2534	240	10	)	)	PUNCT
ejpam-2534	240	11	,	,	PUNCT
ejpam-2534	240	12	we	we	PRON
ejpam-2534	240	13	have	have	VERB
ejpam-2534	240	14	rad(n	rad(n	NOUN
ejpam-2534	240	15	:	:	PUNCT
ejpam-2534	240	16	m)n	m)n	X
ejpam-2534	240	17	=	=	SYM
ejpam-2534	240	18	0	0	X
ejpam-2534	240	19	.	.	PUNCT
ejpam-2534	241	1	proposition	proposition	NOUN
ejpam-2534	241	2	8	8	NUM
ejpam-2534	241	3	(	(	PUNCT
ejpam-2534	241	4	[	[	X
ejpam-2534	241	5	4	4	NUM
ejpam-2534	241	6	,	,	PUNCT
ejpam-2534	241	7	proposition	proposition	NOUN
ejpam-2534	241	8	2.16	2.16	NUM
ejpam-2534	241	9	]	]	PUNCT
ejpam-2534	241	10	)	)	PUNCT
ejpam-2534	241	11	.	.	PUNCT
ejpam-2534	242	1	let	let	VERB
ejpam-2534	242	2	m	m	PRON
ejpam-2534	242	3	be	be	AUX
ejpam-2534	242	4	an	an	DET
ejpam-2534	242	5	r	r	NOUN
ejpam-2534	242	6	-	-	PUNCT
ejpam-2534	242	7	module	module	NOUN
ejpam-2534	242	8	and	and	CCONJ
ejpam-2534	242	9	m	m	AUX
ejpam-2534	242	10	be	be	AUX
ejpam-2534	242	11	a	a	DET
ejpam-2534	242	12	maximal	maximal	ADJ
ejpam-2534	242	13	ideal	ideal	NOUN
ejpam-2534	242	14	of	of	ADP
ejpam-2534	242	15	r.	r.	PROPN
ejpam-2534	242	16	if	if	SCONJ
ejpam-2534	242	17	i	i	PRON
ejpam-2534	242	18	is	be	AUX
ejpam-2534	242	19	an	an	DET
ejpam-2534	242	20	ideal	ideal	NOUN
ejpam-2534	242	21	of	of	ADP
ejpam-2534	242	22	rm	rm	PROPN
ejpam-2534	242	23	and	and	CCONJ
ejpam-2534	242	24	n	n	PROPN
ejpam-2534	242	25	is	be	AUX
ejpam-2534	242	26	a	a	DET
ejpam-2534	242	27	submodule	submodule	NOUN
ejpam-2534	242	28	of	of	ADP
ejpam-2534	242	29	mm	mm	PROPN
ejpam-2534	242	30	,	,	PUNCT
ejpam-2534	242	31	then	then	ADV
ejpam-2534	242	32	(	(	PUNCT
ejpam-2534	242	33	i	i	NOUN
ejpam-2534	242	34	)	)	PUNCT
ejpam-2534	243	1	i	i	PRON
ejpam-2534	243	2	=	=	PRON
ejpam-2534	243	3	{	{	PUNCT
ejpam-2534	243	4	a	a	PRON
ejpam-2534	243	5	∈	∈	NOUN
ejpam-2534	243	6	r	r	NOUN
ejpam-2534	243	7	|	|	ADV
ejpam-2534	243	8	a	a	DET
ejpam-2534	243	9	1	1	NUM
ejpam-2534	243	10	∈	∈	NOUN
ejpam-2534	243	11	i	i	PRON
ejpam-2534	243	12	}	}	PUNCT
ejpam-2534	243	13	is	be	AUX
ejpam-2534	243	14	an	an	DET
ejpam-2534	243	15	ideal	ideal	NOUN
ejpam-2534	243	16	of	of	ADP
ejpam-2534	243	17	r	r	NOUN
ejpam-2534	243	18	and	and	CCONJ
ejpam-2534	243	19	i	i	PRON
ejpam-2534	243	20	=	=	PROPN
ejpam-2534	244	1	i	i	PRON
ejpam-2534	244	2	m.	m.	NOUN
ejpam-2534	244	3	(	(	PUNCT
ejpam-2534	244	4	ii	ii	NOUN
ejpam-2534	244	5	)	)	PUNCT
ejpam-2534	244	6	n	n	NOUN
ejpam-2534	244	7	=	=	PRON
ejpam-2534	244	8	{	{	PUNCT
ejpam-2534	244	9	m	m	VERB
ejpam-2534	244	10	∈	∈	NOUN
ejpam-2534	244	11	m	m	VERB
ejpam-2534	244	12	|	|	ADV
ejpam-2534	244	13	m	m	VERB
ejpam-2534	244	14	1	1	NUM
ejpam-2534	244	15	∈	∈	PROPN
ejpam-2534	244	16	n	n	CCONJ
ejpam-2534	244	17	}	}	PUNCT
ejpam-2534	244	18	is	be	AUX
ejpam-2534	244	19	a	a	DET
ejpam-2534	244	20	submodule	submodule	NOUN
ejpam-2534	244	21	of	of	ADP
ejpam-2534	244	22	m	m	PROPN
ejpam-2534	244	23	and	and	CCONJ
ejpam-2534	244	24	n	n	PROPN
ejpam-2534	244	25	=	=	SYM
ejpam-2534	244	26	nm	nm	PROPN
ejpam-2534	244	27	.	.	PUNCT
ejpam-2534	244	28	theorem	theorem	NOUN
ejpam-2534	244	29	2	2	NUM
ejpam-2534	244	30	.	.	PUNCT
ejpam-2534	245	1	let	let	VERB
ejpam-2534	245	2	n	n	PRON
ejpam-2534	245	3	be	be	AUX
ejpam-2534	245	4	an	an	DET
ejpam-2534	245	5	m	m	ADJ
ejpam-2534	245	6	-	-	ADJ
ejpam-2534	245	7	primal	primal	ADJ
ejpam-2534	245	8	submodule	submodule	NOUN
ejpam-2534	245	9	of	of	ADP
ejpam-2534	245	10	an	an	DET
ejpam-2534	245	11	r	r	NOUN
ejpam-2534	245	12	-	-	PUNCT
ejpam-2534	245	13	module	module	NOUN
ejpam-2534	245	14	m	m	NOUN
ejpam-2534	245	15	with	with	ADP
ejpam-2534	245	16	p(0	p(0	PROPN
ejpam-2534	245	17	)	)	PUNCT
ejpam-2534	246	1	⊆	⊆	NUM
ejpam-2534	246	2	p(n	p(n	PROPN
ejpam-2534	246	3	)	)	PUNCT
ejpam-2534	246	4	.	.	PUNCT
ejpam-2534	247	1	then	then	ADV
ejpam-2534	247	2	n	n	PRON
ejpam-2534	247	3	is	be	AUX
ejpam-2534	247	4	a	a	DET
ejpam-2534	247	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	247	6	weakly	weakly	ADJ
ejpam-2534	247	7	primary	primary	ADJ
ejpam-2534	247	8	submodule	submodule	NOUN
ejpam-2534	247	9	of	of	ADP
ejpam-2534	247	10	m	m	PROPN
ejpam-2534	247	11	if	if	SCONJ
ejpam-2534	248	1	and	and	CCONJ
ejpam-2534	248	2	only	only	ADV
ejpam-2534	248	3	if	if	SCONJ
ejpam-2534	248	4	0	0	NUM
ejpam-2534	248	5	6=	6=	NUM
ejpam-2534	248	6	i	i	PROPN
ejpam-2534	248	7	d	d	PROPN
ejpam-2534	248	8	⊆	⊆	NUM
ejpam-2534	248	9	n	n	PRON
ejpam-2534	248	10	for	for	ADP
ejpam-2534	248	11	some	some	DET
ejpam-2534	248	12	ideal	ideal	ADJ
ejpam-2534	248	13	i	i	PRON
ejpam-2534	248	14	of	of	ADP
ejpam-2534	248	15	r	r	NOUN
ejpam-2534	248	16	and	and	CCONJ
ejpam-2534	248	17	some	some	DET
ejpam-2534	248	18	submodule	submodule	NOUN
ejpam-2534	248	19	d	d	PROPN
ejpam-2534	248	20	of	of	ADP
ejpam-2534	248	21	m	m	PROPN
ejpam-2534	248	22	implies	imply	VERB
ejpam-2534	248	23	i	i	PRON
ejpam-2534	248	24	⊆	⊆	NUM
ejpam-2534	248	25	rad(n	rad(n	NOUN
ejpam-2534	248	26	:	:	PUNCT
ejpam-2534	248	27	m	m	X
ejpam-2534	248	28	)	)	PUNCT
ejpam-2534	248	29	or	or	CCONJ
ejpam-2534	248	30	d	d	PROPN
ejpam-2534	248	31	⊆	⊆	NUM
ejpam-2534	248	32	n.	n.	PROPN
ejpam-2534	248	33	g.	g.	PROPN
ejpam-2534	248	34	ulucak	ulucak	PROPN
ejpam-2534	248	35	and	and	CCONJ
ejpam-2534	248	36	r.	r.	PROPN
ejpam-2534	248	37	uregen	uregen	PROPN
ejpam-2534	248	38	/	/	SYM
ejpam-2534	248	39	eur	eur	PROPN
ejpam-2534	248	40	.	.	PUNCT
ejpam-2534	249	1	j.	j.	PROPN
ejpam-2534	249	2	pure	pure	PROPN
ejpam-2534	249	3	appl	appl	PROPN
ejpam-2534	249	4	.	.	PROPN
ejpam-2534	249	5	math	math	PROPN
ejpam-2534	249	6	,	,	PUNCT
ejpam-2534	249	7	9	9	NUM
ejpam-2534	249	8	(	(	PUNCT
ejpam-2534	249	9	2016	2016	NUM
ejpam-2534	249	10	)	)	PUNCT
ejpam-2534	249	11	,	,	PUNCT
ejpam-2534	249	12	48	48	NUM
ejpam-2534	249	13	-	-	SYM
ejpam-2534	249	14	56	56	NUM
ejpam-2534	249	15	54	54	NUM
ejpam-2534	249	16	proof	proof	NOUN
ejpam-2534	249	17	.	.	PUNCT
ejpam-2534	250	1	assume	assume	VERB
ejpam-2534	250	2	that	that	SCONJ
ejpam-2534	250	3	n	n	PRON
ejpam-2534	250	4	is	be	AUX
ejpam-2534	250	5	a	a	DET
ejpam-2534	250	6	p(n)-locally	p(n)-locally	ADV
ejpam-2534	250	7	weakly	weakly	ADJ
ejpam-2534	250	8	primary	primary	ADJ
ejpam-2534	250	9	submodule	submodule	NOUN
ejpam-2534	250	10	of	of	ADP
ejpam-2534	250	11	m	m	PROPN
ejpam-2534	250	12	.	.	PUNCT
ejpam-2534	251	1	let	let	VERB
ejpam-2534	251	2	0	0	NUM
ejpam-2534	252	1	6=	6=	NUM
ejpam-2534	252	2	i	i	PROPN
ejpam-2534	253	1	d	d	PROPN
ejpam-2534	253	2	⊆	⊆	NUM
ejpam-2534	253	3	n	n	PRON
ejpam-2534	253	4	for	for	ADP
ejpam-2534	253	5	some	some	DET
ejpam-2534	253	6	ideal	ideal	ADJ
ejpam-2534	253	7	i	i	PRON
ejpam-2534	253	8	of	of	ADP
ejpam-2534	253	9	r	r	NOUN
ejpam-2534	253	10	and	and	CCONJ
ejpam-2534	253	11	some	some	DET
ejpam-2534	253	12	submodule	submodule	NOUN
ejpam-2534	253	13	d	d	PROPN
ejpam-2534	253	14	of	of	ADP
ejpam-2534	253	15	m	m	PROPN
ejpam-2534	253	16	.	.	PUNCT
ejpam-2534	254	1	since	since	SCONJ
ejpam-2534	254	2	n	n	NUM
ejpam-2534	254	3	is	be	AUX
ejpam-2534	254	4	m	m	NOUN
ejpam-2534	254	5	-	-	ADJ
ejpam-2534	254	6	primal	primal	ADJ
ejpam-2534	254	7	,	,	PUNCT
ejpam-2534	254	8	then	then	ADV
ejpam-2534	254	9	p(n	p(n	PROPN
ejpam-2534	254	10	)	)	PUNCT
ejpam-2534	254	11	is	be	AUX
ejpam-2534	254	12	an	an	DET
ejpam-2534	254	13	ideal	ideal	NOUN
ejpam-2534	254	14	of	of	ADP
ejpam-2534	254	15	r.	r.	PROPN
ejpam-2534	254	16	as	as	ADP
ejpam-2534	254	17	1	1	NUM
ejpam-2534	254	18	/∈	/∈	PUNCT
ejpam-2534	254	19	p(n	p(n	PROPN
ejpam-2534	254	20	)	)	PUNCT
ejpam-2534	254	21	,	,	PUNCT
ejpam-2534	254	22	then	then	ADV
ejpam-2534	254	23	p(n	p(n	PROPN
ejpam-2534	254	24	)	)	PUNCT
ejpam-2534	254	25	is	be	AUX
ejpam-2534	254	26	a	a	DET
ejpam-2534	254	27	proper	proper	ADJ
ejpam-2534	254	28	ideal	ideal	NOUN
ejpam-2534	254	29	.	.	PUNCT
ejpam-2534	255	1	so	so	ADV
ejpam-2534	255	2	we	we	PRON
ejpam-2534	255	3	have	have	AUX
ejpam-2534	255	4	p(n	p(n	NOUN
ejpam-2534	255	5	)	)	PUNCT
ejpam-2534	255	6	⊆	⊆	NUM
ejpam-2534	255	7	m	m	NOUN
ejpam-2534	255	8	for	for	ADP
ejpam-2534	255	9	some	some	DET
ejpam-2534	255	10	maximal	maximal	ADJ
ejpam-2534	255	11	ideal	ideal	NOUN
ejpam-2534	255	12	m	m	PROPN
ejpam-2534	255	13	of	of	ADP
ejpam-2534	255	14	r.	r.	PROPN
ejpam-2534	255	15	thus	thus	ADV
ejpam-2534	255	16	nm	nm	ADV
ejpam-2534	255	17	is	be	AUX
ejpam-2534	255	18	a	a	DET
ejpam-2534	255	19	weakly	weakly	ADJ
ejpam-2534	255	20	primary	primary	ADJ
ejpam-2534	255	21	submodule	submodule	NOUN
ejpam-2534	255	22	of	of	ADP
ejpam-2534	255	23	mm	mm	PROPN
ejpam-2534	255	24	.	.	PUNCT
ejpam-2534	256	1	now	now	ADV
ejpam-2534	256	2	,	,	PUNCT
ejpam-2534	256	3	i	i	PRON
ejpam-2534	256	4	m	m	VERB
ejpam-2534	256	5	is	be	AUX
ejpam-2534	256	6	an	an	DET
ejpam-2534	256	7	ideal	ideal	NOUN
ejpam-2534	256	8	of	of	ADP
ejpam-2534	256	9	rm	rm	NOUN
ejpam-2534	256	10	and	and	CCONJ
ejpam-2534	256	11	dm	dm	PROPN
ejpam-2534	256	12	is	be	AUX
ejpam-2534	256	13	a	a	DET
ejpam-2534	256	14	submodule	submodule	NOUN
ejpam-2534	256	15	of	of	ADP
ejpam-2534	256	16	mm	mm	PROPN
ejpam-2534	256	17	with	with	ADP
ejpam-2534	256	18	(	(	PUNCT
ejpam-2534	256	19	i	i	PRON
ejpam-2534	256	20	d)m	d)m	NOUN
ejpam-2534	256	21	=	=	PUNCT
ejpam-2534	256	22	imdm	imdm	PROPN
ejpam-2534	256	23	⊆	⊆	NUM
ejpam-2534	256	24	nm	nm	NOUN
ejpam-2534	256	25	.	.	PUNCT
ejpam-2534	257	1	suppose	suppose	VERB
ejpam-2534	257	2	that	that	SCONJ
ejpam-2534	257	3	imdm	imdm	NOUN
ejpam-2534	257	4	=	=	SYM
ejpam-2534	257	5	0	0	NUM
ejpam-2534	257	6	m.	m.	NOUN
ejpam-2534	257	7	then	then	ADV
ejpam-2534	257	8	r	r	NOUN
ejpam-2534	257	9	1	1	NUM
ejpam-2534	257	10	m	m	NOUN
ejpam-2534	257	11	1	1	NUM
ejpam-2534	257	12	=	=	SYM
ejpam-2534	257	13	rm	rm	NOUN
ejpam-2534	257	14	1	1	NUM
ejpam-2534	257	15	=	=	SYM
ejpam-2534	257	16	0	0	NUM
ejpam-2534	257	17	for	for	ADP
ejpam-2534	257	18	every	every	DET
ejpam-2534	257	19	r	r	NOUN
ejpam-2534	257	20	∈	∈	NOUN
ejpam-2534	258	1	i	i	PRON
ejpam-2534	258	2	and	and	CCONJ
ejpam-2534	258	3	every	every	DET
ejpam-2534	258	4	m	m	PROPN
ejpam-2534	258	5	∈	∈	PROPN
ejpam-2534	258	6	d.	d.	NOUN
ejpam-2534	259	1	so	so	ADV
ejpam-2534	259	2	there	there	PRON
ejpam-2534	259	3	exists	exist	VERB
ejpam-2534	259	4	a	a	DET
ejpam-2534	259	5	q	q	NOUN
ejpam-2534	259	6	∈	∈	PROPN
ejpam-2534	259	7	r	r	NOUN
ejpam-2534	259	8	\m	\m	NOUN
ejpam-2534	259	9	such	such	ADJ
ejpam-2534	259	10	that	that	SCONJ
ejpam-2534	259	11	qrm=	qrm=	PROPN
ejpam-2534	259	12	0	0	X
ejpam-2534	259	13	.	.	PUNCT
ejpam-2534	260	1	if	if	SCONJ
ejpam-2534	260	2	rm	rm	PROPN
ejpam-2534	260	3	6=	6=	PROPN
ejpam-2534	260	4	0	0	NUM
ejpam-2534	260	5	,	,	PUNCT
ejpam-2534	260	6	then	then	ADV
ejpam-2534	260	7	q	q	PROPN
ejpam-2534	260	8	∈	∈	PROPN
ejpam-2534	260	9	p(0	p(0	PROPN
ejpam-2534	260	10	)	)	PUNCT
ejpam-2534	260	11	.	.	PUNCT
ejpam-2534	261	1	thus	thus	ADV
ejpam-2534	261	2	q	q	DET
ejpam-2534	261	3	∈m	∈m	NOUN
ejpam-2534	261	4	,	,	PUNCT
ejpam-2534	261	5	which	which	PRON
ejpam-2534	261	6	is	be	AUX
ejpam-2534	261	7	a	a	DET
ejpam-2534	261	8	contradiction	contradiction	NOUN
ejpam-2534	261	9	.	.	PUNCT
ejpam-2534	262	1	so	so	ADV
ejpam-2534	262	2	rm	rm	PROPN
ejpam-2534	262	3	=	=	SYM
ejpam-2534	262	4	0	0	PROPN
ejpam-2534	262	5	,	,	PUNCT
ejpam-2534	262	6	hence	hence	ADV
ejpam-2534	262	7	i	i	NOUN
ejpam-2534	262	8	d	d	NOUN
ejpam-2534	262	9	=	=	SYM
ejpam-2534	262	10	0	0	NUM
ejpam-2534	262	11	,	,	PUNCT
ejpam-2534	262	12	that	that	PRON
ejpam-2534	262	13	is	be	AUX
ejpam-2534	262	14	a	a	DET
ejpam-2534	262	15	contradiction	contradiction	NOUN
ejpam-2534	262	16	.	.	PUNCT
ejpam-2534	263	1	then	then	ADV
ejpam-2534	263	2	0	0	NUM
ejpam-2534	263	3	m	m	PROPN
ejpam-2534	263	4	6=	6=	NUM
ejpam-2534	263	5	imdm	imdm	PROPN
ejpam-2534	263	6	⊆	⊆	NUM
ejpam-2534	263	7	nm	nm	NOUN
ejpam-2534	263	8	.	.	PUNCT
ejpam-2534	264	1	since	since	SCONJ
ejpam-2534	264	2	n	n	NUM
ejpam-2534	264	3	is	be	AUX
ejpam-2534	264	4	a	a	DET
ejpam-2534	264	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	264	6	weakly	weakly	ADJ
ejpam-2534	264	7	primary	primary	ADJ
ejpam-2534	264	8	submodule	submodule	NOUN
ejpam-2534	264	9	of	of	ADP
ejpam-2534	264	10	m	m	PROPN
ejpam-2534	264	11	,	,	PUNCT
ejpam-2534	264	12	then	then	ADV
ejpam-2534	264	13	nm	nm	PRON
ejpam-2534	264	14	is	be	AUX
ejpam-2534	264	15	a	a	DET
ejpam-2534	264	16	weakly	weakly	ADJ
ejpam-2534	264	17	primary	primary	ADJ
ejpam-2534	264	18	submodule	submodule	NOUN
ejpam-2534	264	19	of	of	ADP
ejpam-2534	264	20	mm	mm	PROPN
ejpam-2534	264	21	.	.	PUNCT
ejpam-2534	265	1	by	by	ADP
ejpam-2534	265	2	[	[	X
ejpam-2534	265	3	1	1	NUM
ejpam-2534	265	4	,	,	PUNCT
ejpam-2534	265	5	theorem	theorem	VERB
ejpam-2534	265	6	3.6	3.6	NUM
ejpam-2534	265	7	]	]	PUNCT
ejpam-2534	265	8	,	,	PUNCT
ejpam-2534	265	9	either	either	CCONJ
ejpam-2534	266	1	i	i	PRON
ejpam-2534	266	2	m	m	VERB
ejpam-2534	266	3	⊆	⊆	NUM
ejpam-2534	266	4	rad(nm	rad(nm	NOUN
ejpam-2534	266	5	:	:	PUNCT
ejpam-2534	266	6	mm	mm	X
ejpam-2534	266	7	)	)	PUNCT
ejpam-2534	266	8	or	or	CCONJ
ejpam-2534	266	9	dm	dm	VERB
ejpam-2534	266	10	⊆	⊆	NUM
ejpam-2534	266	11	nm	nm	NOUN
ejpam-2534	266	12	.	.	PUNCT
ejpam-2534	267	1	since	since	SCONJ
ejpam-2534	267	2	p(n	p(n	NOUN
ejpam-2534	267	3	)	)	PUNCT
ejpam-2534	267	4	⊆m	⊆m	NOUN
ejpam-2534	267	5	,	,	PUNCT
ejpam-2534	267	6	then	then	ADV
ejpam-2534	267	7	i	i	PROPN
ejpam-2534	267	8	⊆	⊆	NUM
ejpam-2534	267	9	rad(n	rad(n	NOUN
ejpam-2534	267	10	:	:	PUNCT
ejpam-2534	267	11	m	m	X
ejpam-2534	267	12	)	)	PUNCT
ejpam-2534	267	13	or	or	CCONJ
ejpam-2534	267	14	d	d	PROPN
ejpam-2534	267	15	⊆	⊆	NUM
ejpam-2534	267	16	n	n	NOUN
ejpam-2534	267	17	.	.	PUNCT
ejpam-2534	268	1	let	let	VERB
ejpam-2534	268	2	m	m	PRON
ejpam-2534	268	3	be	be	AUX
ejpam-2534	268	4	a	a	DET
ejpam-2534	268	5	maximal	maximal	ADJ
ejpam-2534	268	6	ideal	ideal	NOUN
ejpam-2534	268	7	of	of	ADP
ejpam-2534	268	8	r	r	NOUN
ejpam-2534	268	9	with	with	ADP
ejpam-2534	268	10	p(n	p(n	NOUN
ejpam-2534	268	11	)	)	PUNCT
ejpam-2534	268	12	⊆m	⊆m	NOUN
ejpam-2534	268	13	.	.	PUNCT
ejpam-2534	269	1	since	since	SCONJ
ejpam-2534	269	2	n	n	NUM
ejpam-2534	269	3	is	be	AUX
ejpam-2534	269	4	a	a	DET
ejpam-2534	269	5	proper	proper	ADJ
ejpam-2534	269	6	ideal	ideal	NOUN
ejpam-2534	269	7	of	of	ADP
ejpam-2534	269	8	r	r	NOUN
ejpam-2534	269	9	,	,	PUNCT
ejpam-2534	269	10	then	then	ADV
ejpam-2534	269	11	there	there	PRON
ejpam-2534	269	12	is	be	VERB
ejpam-2534	269	13	an	an	DET
ejpam-2534	269	14	a	a	DET
ejpam-2534	269	15	∈	∈	NOUN
ejpam-2534	269	16	m	m	NOUN
ejpam-2534	269	17	\n	\n	NOUN
ejpam-2534	269	18	,	,	PUNCT
ejpam-2534	269	19	but	but	CCONJ
ejpam-2534	269	20	a	a	DET
ejpam-2534	269	21	1	1	NUM
ejpam-2534	269	22	∈	∈	NOUN
ejpam-2534	269	23	mm	mm	INTJ
ejpam-2534	269	24	.	.	PUNCT
ejpam-2534	270	1	if	if	SCONJ
ejpam-2534	270	2	a	a	DET
ejpam-2534	270	3	1	1	NUM
ejpam-2534	270	4	∈	∈	NOUN
ejpam-2534	270	5	nm	nm	NOUN
ejpam-2534	270	6	,	,	PUNCT
ejpam-2534	270	7	then	then	ADV
ejpam-2534	270	8	qa	qa	PROPN
ejpam-2534	270	9	∈	∈	PROPN
ejpam-2534	270	10	n	n	PRON
ejpam-2534	270	11	such	such	ADJ
ejpam-2534	270	12	that	that	DET
ejpam-2534	270	13	q	q	PROPN
ejpam-2534	270	14	∈	∈	PROPN
ejpam-2534	270	15	r	r	NOUN
ejpam-2534	270	16	\m	\m	NOUN
ejpam-2534	270	17	.	.	PUNCT
ejpam-2534	271	1	as	as	ADP
ejpam-2534	271	2	a	a	DET
ejpam-2534	271	3	∈	∈	NOUN
ejpam-2534	271	4	m	m	NOUN
ejpam-2534	271	5	\n	\n	NOUN
ejpam-2534	271	6	,	,	PUNCT
ejpam-2534	271	7	then	then	ADV
ejpam-2534	271	8	q	q	X
ejpam-2534	271	9	∈	∈	PROPN
ejpam-2534	271	10	p(n	p(n	PROPN
ejpam-2534	271	11	)	)	PUNCT
ejpam-2534	271	12	,	,	PUNCT
ejpam-2534	271	13	that	that	ADV
ejpam-2534	271	14	is	is	ADV
ejpam-2534	271	15	,	,	PUNCT
ejpam-2534	271	16	q	q	NOUN
ejpam-2534	271	17	∈m	∈m	NOUN
ejpam-2534	271	18	,	,	PUNCT
ejpam-2534	271	19	which	which	PRON
ejpam-2534	271	20	is	be	AUX
ejpam-2534	271	21	a	a	DET
ejpam-2534	271	22	contradiction	contradiction	NOUN
ejpam-2534	271	23	.	.	PUNCT
ejpam-2534	272	1	so	so	ADV
ejpam-2534	272	2	a	a	DET
ejpam-2534	272	3	1	1	NUM
ejpam-2534	272	4	∈	∈	NOUN
ejpam-2534	272	5	mm	mm	PROPN
ejpam-2534	272	6	\	\	PROPN
ejpam-2534	272	7	nm	nm	PROPN
ejpam-2534	272	8	.	.	PUNCT
ejpam-2534	273	1	hence	hence	ADV
ejpam-2534	273	2	nm	nm	PRON
ejpam-2534	273	3	is	be	AUX
ejpam-2534	273	4	a	a	DET
ejpam-2534	273	5	proper	proper	ADJ
ejpam-2534	273	6	ideal	ideal	NOUN
ejpam-2534	273	7	of	of	ADP
ejpam-2534	273	8	rm	rm	PROPN
ejpam-2534	273	9	.	.	PUNCT
ejpam-2534	274	1	let	let	VERB
ejpam-2534	274	2	i	i	PRON
ejpam-2534	274	3	be	be	AUX
ejpam-2534	274	4	an	an	DET
ejpam-2534	274	5	ideal	ideal	NOUN
ejpam-2534	274	6	of	of	ADP
ejpam-2534	274	7	rm	rm	PROPN
ejpam-2534	274	8	and	and	CCONJ
ejpam-2534	274	9	d	d	NOUN
ejpam-2534	274	10	be	be	VERB
ejpam-2534	274	11	a	a	DET
ejpam-2534	274	12	submodule	submodule	NOUN
ejpam-2534	274	13	of	of	ADP
ejpam-2534	274	14	mm	mm	PROPN
ejpam-2534	274	15	with	with	ADP
ejpam-2534	274	16	0	0	NUM
ejpam-2534	274	17	m	m	NOUN
ejpam-2534	274	18	6=	6=	NUM
ejpam-2534	275	1	i	i	PROPN
ejpam-2534	275	2	d	d	PROPN
ejpam-2534	275	3	⊆	⊆	NUM
ejpam-2534	275	4	nm	nm	NOUN
ejpam-2534	275	5	.	.	PUNCT
ejpam-2534	276	1	by	by	ADP
ejpam-2534	276	2	[	[	X
ejpam-2534	276	3	4	4	NUM
ejpam-2534	276	4	,	,	PUNCT
ejpam-2534	276	5	proposition	proposition	NOUN
ejpam-2534	276	6	2.16	2.16	NUM
ejpam-2534	276	7	]	]	PUNCT
ejpam-2534	276	8	,	,	PUNCT
ejpam-2534	276	9	i	i	PRON
ejpam-2534	276	10	=	=	VERB
ejpam-2534	276	11	i	i	PRON
ejpam-2534	276	12	m	m	PROPN
ejpam-2534	276	13	,	,	PUNCT
ejpam-2534	276	14	for	for	ADP
ejpam-2534	276	15	some	some	DET
ejpam-2534	276	16	ideal	ideal	ADJ
ejpam-2534	276	17	i	i	PRON
ejpam-2534	276	18	of	of	ADP
ejpam-2534	276	19	r	r	NOUN
ejpam-2534	276	20	and	and	CCONJ
ejpam-2534	276	21	d	d	NOUN
ejpam-2534	276	22	=	=	SYM
ejpam-2534	276	23	dm	dm	PROPN
ejpam-2534	276	24	,	,	PUNCT
ejpam-2534	276	25	for	for	ADP
ejpam-2534	276	26	some	some	DET
ejpam-2534	276	27	submodule	submodule	NOUN
ejpam-2534	276	28	d	d	PROPN
ejpam-2534	276	29	of	of	ADP
ejpam-2534	276	30	m	m	PROPN
ejpam-2534	276	31	.	.	PUNCT
ejpam-2534	277	1	so	so	ADV
ejpam-2534	277	2	0	0	NUM
ejpam-2534	277	3	m	m	PROPN
ejpam-2534	277	4	6=	6=	NUM
ejpam-2534	277	5	imdm	imdm	PROPN
ejpam-2534	277	6	⊆	⊆	NUM
ejpam-2534	277	7	nm	nm	NOUN
ejpam-2534	277	8	,	,	PUNCT
ejpam-2534	277	9	that	that	ADV
ejpam-2534	277	10	is	is	ADV
ejpam-2534	277	11	,	,	PUNCT
ejpam-2534	277	12	0	0	NUM
ejpam-2534	277	13	m	m	NOUN
ejpam-2534	277	14	6=	6=	NUM
ejpam-2534	277	15	(	(	PUNCT
ejpam-2534	277	16	i	i	PRON
ejpam-2534	277	17	d)m	d)m	NOUN
ejpam-2534	277	18	⊆	⊆	NUM
ejpam-2534	277	19	nm	nm	NOUN
ejpam-2534	277	20	.	.	PUNCT
ejpam-2534	278	1	since	since	SCONJ
ejpam-2534	278	2	p(n	p(n	NOUN
ejpam-2534	278	3	)	)	PUNCT
ejpam-2534	278	4	⊆m	⊆m	NOUN
ejpam-2534	278	5	,	,	PUNCT
ejpam-2534	278	6	then	then	ADV
ejpam-2534	278	7	i	i	X
ejpam-2534	278	8	d	d	PROPN
ejpam-2534	278	9	⊆	⊆	NUM
ejpam-2534	278	10	n	n	NOUN
ejpam-2534	278	11	.	.	PUNCT
ejpam-2534	279	1	also	also	ADV
ejpam-2534	279	2	0	0	NUM
ejpam-2534	280	1	6=	6=	NUM
ejpam-2534	281	1	i	i	PRON
ejpam-2534	281	2	d.	d.	PROPN
ejpam-2534	281	3	on	on	ADP
ejpam-2534	281	4	the	the	DET
ejpam-2534	281	5	contrary	contrary	NOUN
ejpam-2534	281	6	,	,	PUNCT
ejpam-2534	281	7	(	(	PUNCT
ejpam-2534	281	8	i	i	PRON
ejpam-2534	281	9	d)m	d)m	NOUN
ejpam-2534	281	10	=	=	SYM
ejpam-2534	281	11	0	0	NUM
ejpam-2534	281	12	m.	m.	NOUN
ejpam-2534	281	13	by	by	ADP
ejpam-2534	281	14	the	the	DET
ejpam-2534	281	15	hypothesis	hypothesis	NOUN
ejpam-2534	281	16	,	,	PUNCT
ejpam-2534	281	17	we	we	PRON
ejpam-2534	281	18	have	have	VERB
ejpam-2534	281	19	either	either	CCONJ
ejpam-2534	281	20	i	i	PRON
ejpam-2534	281	21	⊆	⊆	NUM
ejpam-2534	281	22	rad(n	rad(n	NOUN
ejpam-2534	281	23	:	:	PUNCT
ejpam-2534	281	24	m	m	X
ejpam-2534	281	25	)	)	PUNCT
ejpam-2534	281	26	or	or	CCONJ
ejpam-2534	281	27	d	d	PROPN
ejpam-2534	281	28	⊆	⊆	NUM
ejpam-2534	281	29	n	n	NOUN
ejpam-2534	281	30	.	.	PUNCT
ejpam-2534	282	1	if	if	SCONJ
ejpam-2534	282	2	i	i	PRON
ejpam-2534	282	3	⊆	⊆	NUM
ejpam-2534	282	4	rad(n	rad(n	NOUN
ejpam-2534	282	5	:	:	PUNCT
ejpam-2534	282	6	m	m	NUM
ejpam-2534	282	7	)	)	PUNCT
ejpam-2534	282	8	,	,	PUNCT
ejpam-2534	282	9	then	then	ADV
ejpam-2534	282	10	i	i	PRON
ejpam-2534	282	11	=	=	VERB
ejpam-2534	283	1	i	i	PRON
ejpam-2534	283	2	m	m	VERB
ejpam-2534	283	3	⊆	⊆	NUM
ejpam-2534	283	4	rad((n	rad((n	NOUN
ejpam-2534	283	5	:	:	PUNCT
ejpam-2534	283	6	m)m	m)m	X
ejpam-2534	283	7	)	)	PUNCT
ejpam-2534	283	8	.	.	PUNCT
ejpam-2534	284	1	if	if	SCONJ
ejpam-2534	284	2	d	d	PROPN
ejpam-2534	284	3	⊆	⊆	NUM
ejpam-2534	284	4	n	n	NOUN
ejpam-2534	284	5	,	,	PUNCT
ejpam-2534	284	6	then	then	ADV
ejpam-2534	284	7	d	d	X
ejpam-2534	284	8	=	=	SYM
ejpam-2534	284	9	dm	dm	PROPN
ejpam-2534	284	10	⊆	⊆	NUM
ejpam-2534	284	11	nm	nm	NOUN
ejpam-2534	284	12	.	.	PUNCT
ejpam-2534	285	1	from	from	ADP
ejpam-2534	285	2	[	[	X
ejpam-2534	285	3	1	1	NUM
ejpam-2534	285	4	,	,	PUNCT
ejpam-2534	285	5	theorem	theorem	VERB
ejpam-2534	285	6	3.6	3.6	NUM
ejpam-2534	285	7	]	]	PUNCT
ejpam-2534	285	8	,	,	PUNCT
ejpam-2534	285	9	nm	nm	PRON
ejpam-2534	285	10	is	be	AUX
ejpam-2534	285	11	a	a	DET
ejpam-2534	285	12	weakly	weakly	ADJ
ejpam-2534	285	13	primary	primary	ADJ
ejpam-2534	285	14	submodule	submodule	NOUN
ejpam-2534	285	15	of	of	ADP
ejpam-2534	285	16	mm	mm	PROPN
ejpam-2534	285	17	.	.	PUNCT
ejpam-2534	286	1	therefore	therefore	ADV
ejpam-2534	286	2	,	,	PUNCT
ejpam-2534	286	3	n	n	PRON
ejpam-2534	286	4	is	be	AUX
ejpam-2534	286	5	a	a	DET
ejpam-2534	286	6	p(n)-locally	p(n)-locally	ADV
ejpam-2534	286	7	weakly	weakly	ADJ
ejpam-2534	286	8	primary	primary	ADJ
ejpam-2534	286	9	submodule	submodule	NOUN
ejpam-2534	286	10	of	of	ADP
ejpam-2534	286	11	m	m	PROPN
ejpam-2534	286	12	.	.	PUNCT
ejpam-2534	287	1	corollary	corollary	ADJ
ejpam-2534	287	2	5	5	NUM
ejpam-2534	287	3	.	.	PUNCT
ejpam-2534	288	1	let	let	VERB
ejpam-2534	288	2	n	n	PRON
ejpam-2534	288	3	be	be	AUX
ejpam-2534	288	4	an	an	DET
ejpam-2534	288	5	m	m	ADJ
ejpam-2534	288	6	-	-	ADJ
ejpam-2534	288	7	primal	primal	ADJ
ejpam-2534	288	8	submodule	submodule	NOUN
ejpam-2534	288	9	of	of	ADP
ejpam-2534	288	10	an	an	DET
ejpam-2534	288	11	r	r	NOUN
ejpam-2534	288	12	-	-	PUNCT
ejpam-2534	288	13	module	module	NOUN
ejpam-2534	288	14	m	m	NOUN
ejpam-2534	288	15	with	with	ADP
ejpam-2534	288	16	p(0	p(0	PROPN
ejpam-2534	288	17	)	)	PUNCT
ejpam-2534	289	1	⊆	⊆	NUM
ejpam-2534	289	2	p(n	p(n	PROPN
ejpam-2534	289	3	)	)	PUNCT
ejpam-2534	289	4	.	.	PUNCT
ejpam-2534	290	1	then	then	ADV
ejpam-2534	290	2	n	n	PRON
ejpam-2534	290	3	is	be	AUX
ejpam-2534	290	4	a	a	DET
ejpam-2534	290	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	290	6	weakly	weakly	ADJ
ejpam-2534	290	7	primary	primary	ADJ
ejpam-2534	290	8	submodule	submodule	NOUN
ejpam-2534	290	9	of	of	ADP
ejpam-2534	290	10	m	m	PROPN
ejpam-2534	290	11	if	if	SCONJ
ejpam-2534	291	1	and	and	CCONJ
ejpam-2534	291	2	only	only	ADV
ejpam-2534	291	3	if	if	SCONJ
ejpam-2534	291	4	n	n	PRON
ejpam-2534	291	5	is	be	AUX
ejpam-2534	291	6	a	a	DET
ejpam-2534	291	7	weakly	weakly	ADJ
ejpam-2534	291	8	primary	primary	ADJ
ejpam-2534	291	9	submodule	submodule	NOUN
ejpam-2534	291	10	of	of	ADP
ejpam-2534	291	11	m.	m.	NOUN
ejpam-2534	291	12	proof	proof	NOUN
ejpam-2534	291	13	.	.	PUNCT
ejpam-2534	292	1	it	it	PRON
ejpam-2534	292	2	is	be	AUX
ejpam-2534	292	3	clear	clear	ADJ
ejpam-2534	292	4	from	from	ADP
ejpam-2534	292	5	theorem	theorem	ADJ
ejpam-2534	292	6	2	2	NUM
ejpam-2534	292	7	and	and	CCONJ
ejpam-2534	292	8	[	[	X
ejpam-2534	292	9	1	1	NUM
ejpam-2534	292	10	,	,	PUNCT
ejpam-2534	292	11	theorem	theorem	VERB
ejpam-2534	292	12	3.6	3.6	NUM
ejpam-2534	292	13	]	]	PUNCT
ejpam-2534	292	14	.	.	PUNCT
ejpam-2534	293	1	theorem	theorem	NOUN
ejpam-2534	293	2	3	3	X
ejpam-2534	293	3	.	.	PUNCT
ejpam-2534	294	1	let	let	VERB
ejpam-2534	294	2	m	m	PRON
ejpam-2534	294	3	be	be	AUX
ejpam-2534	294	4	an	an	DET
ejpam-2534	294	5	r	r	NOUN
ejpam-2534	294	6	-	-	PUNCT
ejpam-2534	294	7	module	module	NOUN
ejpam-2534	294	8	and	and	CCONJ
ejpam-2534	294	9	n	n	CCONJ
ejpam-2534	294	10	be	be	AUX
ejpam-2534	294	11	an	an	DET
ejpam-2534	294	12	m	m	ADJ
ejpam-2534	294	13	-	-	ADJ
ejpam-2534	294	14	primal	primal	ADJ
ejpam-2534	294	15	submodule	submodule	NOUN
ejpam-2534	294	16	of	of	ADP
ejpam-2534	294	17	m	m	PROPN
ejpam-2534	294	18	with	with	ADP
ejpam-2534	294	19	p(0	p(0	PROPN
ejpam-2534	294	20	)	)	PUNCT
ejpam-2534	294	21	⊆	⊆	NUM
ejpam-2534	294	22	p(n	p(n	PROPN
ejpam-2534	294	23	)	)	PUNCT
ejpam-2534	294	24	.	.	PUNCT
ejpam-2534	295	1	then	then	ADV
ejpam-2534	295	2	the	the	DET
ejpam-2534	295	3	following	follow	VERB
ejpam-2534	295	4	statements	statement	NOUN
ejpam-2534	295	5	are	be	AUX
ejpam-2534	295	6	equivalent	equivalent	ADJ
ejpam-2534	295	7	:	:	PUNCT
ejpam-2534	295	8	(	(	PUNCT
ejpam-2534	295	9	i	i	NOUN
ejpam-2534	295	10	)	)	PUNCT
ejpam-2534	295	11	n	n	PRON
ejpam-2534	295	12	is	be	AUX
ejpam-2534	295	13	a	a	DET
ejpam-2534	295	14	p(n)-locally	p(n)-locally	ADV
ejpam-2534	295	15	weakly	weakly	ADJ
ejpam-2534	295	16	primary	primary	ADJ
ejpam-2534	295	17	submodule	submodule	NOUN
ejpam-2534	295	18	of	of	ADP
ejpam-2534	295	19	m.	m.	NOUN
ejpam-2534	295	20	(	(	PUNCT
ejpam-2534	295	21	ii	ii	NOUN
ejpam-2534	295	22	)	)	PUNCT
ejpam-2534	295	23	for	for	ADP
ejpam-2534	295	24	any	any	DET
ejpam-2534	295	25	m	m	NOUN
ejpam-2534	295	26	∈	∈	PROPN
ejpam-2534	295	27	m	m	VERB
ejpam-2534	295	28	\	\	PROPN
ejpam-2534	295	29	n	n	CCONJ
ejpam-2534	295	30	,	,	PUNCT
ejpam-2534	295	31	rad(n	rad(n	PROPN
ejpam-2534	295	32	:	:	PUNCT
ejpam-2534	295	33	rm	rm	NOUN
ejpam-2534	295	34	)	)	PUNCT
ejpam-2534	295	35	=	=	PUNCT
ejpam-2534	295	36	rad(n	rad(n	NOUN
ejpam-2534	295	37	:	:	PUNCT
ejpam-2534	295	38	m)∪	m)∪	NOUN
ejpam-2534	295	39	(	(	PUNCT
ejpam-2534	295	40	0	0	NUM
ejpam-2534	295	41	:	:	PUNCT
ejpam-2534	295	42	rm	rm	PROPN
ejpam-2534	295	43	)	)	PUNCT
ejpam-2534	295	44	.	.	PUNCT
ejpam-2534	296	1	(	(	PUNCT
ejpam-2534	296	2	iii	iii	X
ejpam-2534	296	3	)	)	PUNCT
ejpam-2534	296	4	for	for	ADP
ejpam-2534	296	5	any	any	DET
ejpam-2534	296	6	m	m	NOUN
ejpam-2534	296	7	∈	∈	PROPN
ejpam-2534	296	8	m	m	VERB
ejpam-2534	296	9	\	\	PROPN
ejpam-2534	296	10	n	n	CCONJ
ejpam-2534	296	11	,	,	PUNCT
ejpam-2534	296	12	rad(n	rad(n	PROPN
ejpam-2534	296	13	:	:	PUNCT
ejpam-2534	296	14	rm	rm	NOUN
ejpam-2534	296	15	)	)	PUNCT
ejpam-2534	296	16	=	=	PUNCT
ejpam-2534	296	17	rad(n	rad(n	NOUN
ejpam-2534	296	18	:	:	PUNCT
ejpam-2534	296	19	m	m	X
ejpam-2534	296	20	)	)	PUNCT
ejpam-2534	296	21	or	or	CCONJ
ejpam-2534	296	22	rad(n	rad(n	PROPN
ejpam-2534	296	23	:	:	PUNCT
ejpam-2534	296	24	rm	rm	NOUN
ejpam-2534	296	25	)	)	PUNCT
ejpam-2534	296	26	=	=	PUNCT
ejpam-2534	297	1	(	(	PUNCT
ejpam-2534	297	2	0	0	NUM
ejpam-2534	297	3	:	:	PUNCT
ejpam-2534	297	4	rm	rm	PROPN
ejpam-2534	297	5	)	)	PUNCT
ejpam-2534	297	6	.	.	PUNCT
ejpam-2534	298	1	proof	proof	NOUN
ejpam-2534	298	2	.	.	PUNCT
ejpam-2534	299	1	(	(	PUNCT
ejpam-2534	299	2	i	i	NOUN
ejpam-2534	299	3	)	)	PUNCT
ejpam-2534	300	1	=	=	NOUN
ejpam-2534	300	2	⇒	⇒	NOUN
ejpam-2534	300	3	(	(	PUNCT
ejpam-2534	300	4	ii	ii	NOUN
ejpam-2534	300	5	):	):	PUNCT
ejpam-2534	300	6	let	let	VERB
ejpam-2534	300	7	n	n	PRON
ejpam-2534	300	8	be	be	AUX
ejpam-2534	300	9	a	a	DET
ejpam-2534	300	10	p(n)-locally	p(n)-locally	ADV
ejpam-2534	300	11	weakly	weakly	ADJ
ejpam-2534	300	12	primary	primary	ADJ
ejpam-2534	300	13	submodule	submodule	NOUN
ejpam-2534	300	14	of	of	ADP
ejpam-2534	300	15	m	m	PRON
ejpam-2534	300	16	and	and	CCONJ
ejpam-2534	300	17	let	let	VERB
ejpam-2534	300	18	m	m	PRON
ejpam-2534	300	19	∈	∈	PROPN
ejpam-2534	300	20	m	m	VERB
ejpam-2534	300	21	\	\	PROPN
ejpam-2534	300	22	n	n	X
ejpam-2534	300	23	.	.	PUNCT
ejpam-2534	301	1	since	since	SCONJ
ejpam-2534	301	2	n	n	NUM
ejpam-2534	301	3	is	be	AUX
ejpam-2534	301	4	m	m	NOUN
ejpam-2534	301	5	-	-	ADJ
ejpam-2534	301	6	primal	primal	ADJ
ejpam-2534	301	7	,	,	PUNCT
ejpam-2534	301	8	then	then	ADV
ejpam-2534	301	9	p(n	p(n	PROPN
ejpam-2534	301	10	)	)	PUNCT
ejpam-2534	301	11	is	be	AUX
ejpam-2534	301	12	an	an	DET
ejpam-2534	301	13	ideal	ideal	NOUN
ejpam-2534	301	14	of	of	ADP
ejpam-2534	301	15	r.	r.	PROPN
ejpam-2534	301	16	as	as	ADP
ejpam-2534	301	17	i	i	PRON
ejpam-2534	301	18	/∈	/∈	PUNCT
ejpam-2534	302	1	p(n	p(n	PROPN
ejpam-2534	302	2	)	)	PUNCT
ejpam-2534	302	3	,	,	PUNCT
ejpam-2534	302	4	then	then	ADV
ejpam-2534	302	5	p(n	p(n	PROPN
ejpam-2534	302	6	)	)	PUNCT
ejpam-2534	302	7	is	be	AUX
ejpam-2534	302	8	a	a	DET
ejpam-2534	302	9	proper	proper	ADJ
ejpam-2534	302	10	ideal	ideal	NOUN
ejpam-2534	302	11	.	.	PUNCT
ejpam-2534	303	1	so	so	ADV
ejpam-2534	303	2	we	we	PRON
ejpam-2534	303	3	have	have	VERB
ejpam-2534	303	4	p(n	p(n	NOUN
ejpam-2534	303	5	)	)	PUNCT
ejpam-2534	303	6	⊆m	⊆m	NOUN
ejpam-2534	303	7	for	for	ADP
ejpam-2534	303	8	some	some	DET
ejpam-2534	303	9	maximal	maximal	ADJ
ejpam-2534	303	10	ideal	ideal	NOUN
ejpam-2534	303	11	m	m	PROPN
ejpam-2534	303	12	of	of	ADP
ejpam-2534	303	13	r.	r.	PROPN
ejpam-2534	303	14	hence	hence	ADV
ejpam-2534	303	15	nm	nm	PROPN
ejpam-2534	303	16	is	be	AUX
ejpam-2534	303	17	a	a	DET
ejpam-2534	303	18	weakly	weakly	ADJ
ejpam-2534	303	19	primary	primary	ADJ
ejpam-2534	303	20	submodule	submodule	NOUN
ejpam-2534	303	21	of	of	ADP
ejpam-2534	303	22	mm	mm	PROPN
ejpam-2534	303	23	.	.	PUNCT
ejpam-2534	304	1	as	as	ADP
ejpam-2534	304	2	m	m	PROPN
ejpam-2534	304	3	∈	∈	PROPN
ejpam-2534	304	4	m	m	NOUN
ejpam-2534	304	5	,	,	PUNCT
ejpam-2534	304	6	then	then	ADV
ejpam-2534	304	7	m	m	VERB
ejpam-2534	304	8	1	1	NUM
ejpam-2534	304	9	∈	∈	NOUN
ejpam-2534	304	10	mm	mm	NOUN
ejpam-2534	304	11	,	,	PUNCT
ejpam-2534	304	12	but	but	CCONJ
ejpam-2534	304	13	m	m	VERB
ejpam-2534	304	14	1	1	NUM
ejpam-2534	304	15	∈	∈	PROPN
ejpam-2534	304	16	mm	mm	PROPN
ejpam-2534	304	17	\	\	PROPN
ejpam-2534	304	18	nm	nm	PROPN
ejpam-2534	304	19	.	.	PUNCT
ejpam-2534	305	1	if	if	SCONJ
ejpam-2534	305	2	m	m	PROPN
ejpam-2534	305	3	1	1	NUM
ejpam-2534	305	4	∈	∈	PROPN
ejpam-2534	305	5	nm	nm	NOUN
ejpam-2534	305	6	,	,	PUNCT
ejpam-2534	305	7	then	then	ADV
ejpam-2534	305	8	pm	pm	VERB
ejpam-2534	305	9	∈	∈	PROPN
ejpam-2534	305	10	n	n	CCONJ
ejpam-2534	305	11	for	for	ADP
ejpam-2534	305	12	some	some	DET
ejpam-2534	305	13	p	p	NOUN
ejpam-2534	305	14	∈	∈	PROPN
ejpam-2534	305	15	r	r	NOUN
ejpam-2534	305	16	\m	\m	NOUN
ejpam-2534	305	17	.	.	PUNCT
ejpam-2534	306	1	since	since	SCONJ
ejpam-2534	306	2	p	p	NOUN
ejpam-2534	306	3	/∈	/∈	PUNCT
ejpam-2534	306	4	p(n	p(n	NOUN
ejpam-2534	306	5	)	)	PUNCT
ejpam-2534	306	6	,	,	PUNCT
ejpam-2534	306	7	then	then	ADV
ejpam-2534	306	8	m	m	VERB
ejpam-2534	306	9	∈	∈	PROPN
ejpam-2534	306	10	n	n	NOUN
ejpam-2534	306	11	,	,	PUNCT
ejpam-2534	306	12	this	this	PRON
ejpam-2534	306	13	is	be	AUX
ejpam-2534	306	14	a	a	DET
ejpam-2534	306	15	contradiction	contradiction	NOUN
ejpam-2534	306	16	.	.	PUNCT
ejpam-2534	307	1	by	by	ADP
ejpam-2534	307	2	[	[	X
ejpam-2534	307	3	2	2	NUM
ejpam-2534	307	4	,	,	PUNCT
ejpam-2534	307	5	theorem	theorem	VERB
ejpam-2534	307	6	2.15	2.15	NUM
ejpam-2534	307	7	]	]	PUNCT
ejpam-2534	307	8	,	,	PUNCT
ejpam-2534	307	9	rad(nm	rad(nm	VERB
ejpam-2534	307	10	:	:	PUNCT
ejpam-2534	308	1	rm	rm	PROPN
ejpam-2534	308	2	m	m	PROPN
ejpam-2534	308	3	1	1	NUM
ejpam-2534	308	4	)	)	PUNCT
ejpam-2534	308	5	=	=	NOUN
ejpam-2534	308	6	rad(nm	rad(nm	NOUN
ejpam-2534	308	7	:	:	PUNCT
ejpam-2534	308	8	mm)∪	mm)∪	NOUN
ejpam-2534	308	9	(	(	PUNCT
ejpam-2534	308	10	0	0	NUM
ejpam-2534	308	11	m	m	VERB
ejpam-2534	308	12	:	:	PUNCT
ejpam-2534	308	13	rm	rm	PROPN
ejpam-2534	308	14	m	m	PROPN
ejpam-2534	308	15	1	1	NUM
ejpam-2534	308	16	)	)	PUNCT
ejpam-2534	308	17	and	and	CCONJ
ejpam-2534	308	18	from	from	ADP
ejpam-2534	308	19	[	[	X
ejpam-2534	308	20	4	4	NUM
ejpam-2534	308	21	,	,	PUNCT
ejpam-2534	308	22	corollary	corollary	ADJ
ejpam-2534	308	23	2.9	2.9	NUM
ejpam-2534	308	24	]	]	PUNCT
ejpam-2534	308	25	,	,	PUNCT
ejpam-2534	308	26	rad(nm	rad(nm	VERB
ejpam-2534	308	27	:	:	PUNCT
ejpam-2534	308	28	(	(	PUNCT
ejpam-2534	308	29	rm)m	rm)m	SYM
ejpam-2534	308	30	)	)	PUNCT
ejpam-2534	308	31	=	=	NOUN
ejpam-2534	308	32	rad(nm	rad(nm	NOUN
ejpam-2534	308	33	:	:	PUNCT
ejpam-2534	308	34	mm	mm	X
ejpam-2534	308	35	)	)	PUNCT
ejpam-2534	308	36	∪	∪	NOUN
ejpam-2534	308	37	(	(	PUNCT
ejpam-2534	308	38	0	0	NUM
ejpam-2534	308	39	m	m	VERB
ejpam-2534	308	40	:	:	PUNCT
ejpam-2534	308	41	(	(	PUNCT
ejpam-2534	308	42	rm)m	rm)m	PROPN
ejpam-2534	308	43	)	)	PUNCT
ejpam-2534	308	44	.	.	PUNCT
ejpam-2534	309	1	then	then	ADV
ejpam-2534	309	2	by	by	ADP
ejpam-2534	309	3	proposition	proposition	NOUN
ejpam-2534	309	4	3	3	NUM
ejpam-2534	309	5	,	,	PUNCT
ejpam-2534	309	6	proposition	proposition	NOUN
ejpam-2534	309	7	4	4	NUM
ejpam-2534	309	8	and	and	CCONJ
ejpam-2534	309	9	corollary	corollary	ADJ
ejpam-2534	309	10	4	4	NUM
ejpam-2534	309	11	,	,	PUNCT
ejpam-2534	309	12	rad((n	rad((n	NOUN
ejpam-2534	309	13	:	:	PUNCT
ejpam-2534	309	14	rm)m	rm)m	PROPN
ejpam-2534	309	15	)	)	PUNCT
ejpam-2534	310	1	=	=	SYM
ejpam-2534	310	2	rad((n	rad((n	NOUN
ejpam-2534	310	3	:	:	PUNCT
ejpam-2534	310	4	m)m)∪	m)m)∪	X
ejpam-2534	310	5	(	(	PUNCT
ejpam-2534	310	6	0	0	NUM
ejpam-2534	310	7	:	:	PUNCT
ejpam-2534	310	8	rm)m	rm)m	PROPN
ejpam-2534	310	9	.	.	PUNCT
ejpam-2534	311	1	let	let	VERB
ejpam-2534	311	2	r	r	NOUN
ejpam-2534	311	3	∈	∈	PROPN
ejpam-2534	311	4	rad(n	rad(n	PROPN
ejpam-2534	311	5	:	:	PUNCT
ejpam-2534	311	6	rm	rm	PROPN
ejpam-2534	311	7	)	)	PUNCT
ejpam-2534	311	8	.	.	PUNCT
ejpam-2534	312	1	then	then	ADV
ejpam-2534	312	2	r	r	NOUN
ejpam-2534	312	3	1	1	NUM
ejpam-2534	312	4	∈	∈	NOUN
ejpam-2534	312	5	rad((n	rad((n	NOUN
ejpam-2534	312	6	:	:	PUNCT
ejpam-2534	312	7	rm)m	rm)m	PROPN
ejpam-2534	312	8	)	)	PUNCT
ejpam-2534	312	9	and	and	CCONJ
ejpam-2534	312	10	so	so	ADV
ejpam-2534	312	11	r	r	NOUN
ejpam-2534	312	12	1	1	NUM
ejpam-2534	312	13	∈	∈	PROPN
ejpam-2534	312	14	rad((n	rad((n	NOUN
ejpam-2534	312	15	:	:	PUNCT
ejpam-2534	312	16	m)m	m)m	X
ejpam-2534	312	17	)	)	PUNCT
ejpam-2534	312	18	or	or	CCONJ
ejpam-2534	312	19	r	r	NOUN
ejpam-2534	312	20	1	1	NUM
ejpam-2534	312	21	∈	∈	NOUN
ejpam-2534	312	22	(	(	PUNCT
ejpam-2534	312	23	0	0	NUM
ejpam-2534	312	24	:	:	PUNCT
ejpam-2534	312	25	rm)m	rm)m	PROPN
ejpam-2534	312	26	.	.	PUNCT
ejpam-2534	313	1	if	if	SCONJ
ejpam-2534	313	2	r	r	NOUN
ejpam-2534	313	3	1	1	NUM
ejpam-2534	313	4	∈	∈	PROPN
ejpam-2534	313	5	rad((n	rad((n	NOUN
ejpam-2534	313	6	:	:	PUNCT
ejpam-2534	313	7	m)m	m)m	X
ejpam-2534	313	8	)	)	PUNCT
ejpam-2534	313	9	,	,	PUNCT
ejpam-2534	313	10	then	then	ADV
ejpam-2534	313	11	references	reference	VERB
ejpam-2534	313	12	55	55	NUM
ejpam-2534	313	13	rn	rn	PROPN
ejpam-2534	313	14	1	1	NUM
ejpam-2534	313	15	∈	∈	PROPN
ejpam-2534	313	16	(	(	PUNCT
ejpam-2534	313	17	n	n	NUM
ejpam-2534	313	18	:	:	PUNCT
ejpam-2534	313	19	m)m	m)m	NOUN
ejpam-2534	313	20	for	for	ADP
ejpam-2534	313	21	some	some	DET
ejpam-2534	313	22	positive	positive	ADJ
ejpam-2534	313	23	integer	integer	NOUN
ejpam-2534	313	24	n	n	NOUN
ejpam-2534	313	25	and	and	CCONJ
ejpam-2534	313	26	thus	thus	ADV
ejpam-2534	313	27	qrn	qrn	VERB
ejpam-2534	313	28	∈	∈	PROPN
ejpam-2534	313	29	(	(	PUNCT
ejpam-2534	313	30	n	n	NOUN
ejpam-2534	313	31	:	:	PUNCT
ejpam-2534	313	32	m	m	X
ejpam-2534	313	33	)	)	PUNCT
ejpam-2534	313	34	for	for	ADP
ejpam-2534	313	35	some	some	DET
ejpam-2534	313	36	q	q	NOUN
ejpam-2534	313	37	∈	∈	PROPN
ejpam-2534	313	38	r	r	NOUN
ejpam-2534	313	39	\m	\m	NOUN
ejpam-2534	313	40	,	,	PUNCT
ejpam-2534	313	41	that	that	ADV
ejpam-2534	313	42	is	be	AUX
ejpam-2534	313	43	,	,	PUNCT
ejpam-2534	313	44	qrnm	qrnm	NOUN
ejpam-2534	313	45	⊆	⊆	NUM
ejpam-2534	313	46	n	n	NOUN
ejpam-2534	313	47	.	.	PUNCT
ejpam-2534	314	1	assume	assume	VERB
ejpam-2534	314	2	that	that	SCONJ
ejpam-2534	314	3	rnm	rnm	NOUN
ejpam-2534	314	4	6⊆	6⊆	PROPN
ejpam-2534	314	5	n	n	PROPN
ejpam-2534	314	6	.	.	PUNCT
ejpam-2534	315	1	then	then	ADV
ejpam-2534	315	2	rnm	rnm	NOUN
ejpam-2534	315	3	/∈	/∈	PUNCT
ejpam-2534	316	1	n	n	PROPN
ejpam-2534	316	2	for	for	ADP
ejpam-2534	316	3	some	some	DET
ejpam-2534	316	4	m	m	NOUN
ejpam-2534	316	5	∈	∈	ADJ
ejpam-2534	316	6	m	m	NOUN
ejpam-2534	316	7	,	,	PUNCT
ejpam-2534	316	8	however	however	ADV
ejpam-2534	316	9	qrnm	qrnm	NOUN
ejpam-2534	316	10	∈	∈	PROPN
ejpam-2534	316	11	n	n	ADV
ejpam-2534	316	12	.	.	PUNCT
ejpam-2534	317	1	hence	hence	ADV
ejpam-2534	317	2	q	q	X
ejpam-2534	317	3	∈	∈	PROPN
ejpam-2534	317	4	p(n	p(n	PROPN
ejpam-2534	317	5	)	)	PUNCT
ejpam-2534	317	6	.	.	PUNCT
ejpam-2534	318	1	then	then	ADV
ejpam-2534	318	2	q	q	PROPN
ejpam-2534	318	3	∈	∈	PROPN
ejpam-2534	318	4	m	m	PROPN
ejpam-2534	318	5	,	,	PUNCT
ejpam-2534	318	6	which	which	PRON
ejpam-2534	318	7	is	be	AUX
ejpam-2534	318	8	a	a	DET
ejpam-2534	318	9	contradiction	contradiction	NOUN
ejpam-2534	318	10	.	.	PUNCT
ejpam-2534	319	1	so	so	ADV
ejpam-2534	319	2	rnm	rnm	VERB
ejpam-2534	319	3	⊆	⊆	NUM
ejpam-2534	319	4	n	n	NOUN
ejpam-2534	319	5	for	for	ADP
ejpam-2534	319	6	some	some	DET
ejpam-2534	319	7	positive	positive	ADJ
ejpam-2534	319	8	integer	integer	NOUN
ejpam-2534	319	9	n	n	CCONJ
ejpam-2534	319	10	,	,	PUNCT
ejpam-2534	319	11	that	that	ADV
ejpam-2534	319	12	is	is	ADV
ejpam-2534	319	13	,	,	PUNCT
ejpam-2534	319	14	r	r	NOUN
ejpam-2534	319	15	∈	∈	PROPN
ejpam-2534	319	16	rad(n	rad(n	NOUN
ejpam-2534	319	17	:	:	PUNCT
ejpam-2534	319	18	m	m	NUM
ejpam-2534	319	19	)	)	PUNCT
ejpam-2534	319	20	.	.	PUNCT
ejpam-2534	320	1	if	if	SCONJ
ejpam-2534	320	2	r	r	NOUN
ejpam-2534	320	3	1	1	NUM
ejpam-2534	320	4	∈	∈	NOUN
ejpam-2534	320	5	(	(	PUNCT
ejpam-2534	320	6	0	0	NUM
ejpam-2534	320	7	:	:	PUNCT
ejpam-2534	320	8	rm)m	rm)m	PROPN
ejpam-2534	320	9	,	,	PUNCT
ejpam-2534	320	10	then	then	ADV
ejpam-2534	320	11	pr	pr	NOUN
ejpam-2534	320	12	∈	∈	PROPN
ejpam-2534	320	13	(	(	PUNCT
ejpam-2534	320	14	0	0	NUM
ejpam-2534	320	15	:	:	PUNCT
ejpam-2534	320	16	rm	rm	NOUN
ejpam-2534	320	17	)	)	PUNCT
ejpam-2534	320	18	for	for	ADP
ejpam-2534	320	19	some	some	DET
ejpam-2534	320	20	p	p	NOUN
ejpam-2534	320	21	∈	∈	PROPN
ejpam-2534	320	22	r	r	NOUN
ejpam-2534	320	23	\m	\m	NOUN
ejpam-2534	320	24	.	.	PUNCT
ejpam-2534	321	1	thus	thus	ADV
ejpam-2534	321	2	prrm	prrm	ADJ
ejpam-2534	321	3	=	=	NOUN
ejpam-2534	321	4	0	0	X
ejpam-2534	321	5	.	.	PUNCT
ejpam-2534	321	6	assume	assume	VERB
ejpam-2534	321	7	that	that	SCONJ
ejpam-2534	321	8	rrm	rrm	PROPN
ejpam-2534	321	9	6=	6=	PROPN
ejpam-2534	321	10	0	0	NUM
ejpam-2534	321	11	.	.	PUNCT
ejpam-2534	322	1	then	then	ADV
ejpam-2534	322	2	rsm	rsm	PROPN
ejpam-2534	322	3	6=	6=	PRON
ejpam-2534	322	4	0	0	NUM
ejpam-2534	322	5	for	for	ADP
ejpam-2534	322	6	some	some	DET
ejpam-2534	322	7	s	s	PART
ejpam-2534	322	8	∈	∈	PROPN
ejpam-2534	322	9	r	r	NOUN
ejpam-2534	322	10	,	,	PUNCT
ejpam-2534	322	11	but	but	CCONJ
ejpam-2534	322	12	prsm	prsm	PROPN
ejpam-2534	322	13	=	=	NOUN
ejpam-2534	323	1	0	0	X
ejpam-2534	323	2	.	.	PUNCT
ejpam-2534	324	1	therefore	therefore	ADV
ejpam-2534	324	2	p	p	PROPN
ejpam-2534	324	3	∈	∈	PROPN
ejpam-2534	324	4	p(0	p(0	PROPN
ejpam-2534	324	5	)	)	PUNCT
ejpam-2534	324	6	.	.	PUNCT
ejpam-2534	325	1	as	as	ADP
ejpam-2534	325	2	p(0	p(0	NOUN
ejpam-2534	325	3	)	)	PUNCT
ejpam-2534	325	4	⊆m	⊆m	NOUN
ejpam-2534	325	5	,	,	PUNCT
ejpam-2534	325	6	then	then	ADV
ejpam-2534	325	7	p	p	ADJ
ejpam-2534	325	8	∈m	∈m	NOUN
ejpam-2534	325	9	,	,	PUNCT
ejpam-2534	325	10	which	which	PRON
ejpam-2534	325	11	is	be	AUX
ejpam-2534	325	12	a	a	DET
ejpam-2534	325	13	contradiction	contradiction	NOUN
ejpam-2534	325	14	.	.	PUNCT
ejpam-2534	326	1	so	so	ADV
ejpam-2534	326	2	rrm	rrm	PROPN
ejpam-2534	326	3	=	=	PROPN
ejpam-2534	326	4	0	0	PROPN
ejpam-2534	326	5	.	.	PUNCT
ejpam-2534	327	1	then	then	ADV
ejpam-2534	327	2	r	r	NOUN
ejpam-2534	327	3	∈	∈	PROPN
ejpam-2534	327	4	(	(	PUNCT
ejpam-2534	327	5	0	0	NUM
ejpam-2534	327	6	:	:	PUNCT
ejpam-2534	327	7	rm	rm	PROPN
ejpam-2534	327	8	)	)	PUNCT
ejpam-2534	327	9	.	.	PUNCT
ejpam-2534	328	1	hence	hence	ADV
ejpam-2534	328	2	r	r	NOUN
ejpam-2534	328	3	∈	∈	PROPN
ejpam-2534	328	4	rad(n	rad(n	NOUN
ejpam-2534	328	5	:	:	PUNCT
ejpam-2534	328	6	m)∪	m)∪	NOUN
ejpam-2534	328	7	(	(	PUNCT
ejpam-2534	328	8	0	0	NUM
ejpam-2534	328	9	:	:	PUNCT
ejpam-2534	328	10	rm	rm	PROPN
ejpam-2534	328	11	)	)	PUNCT
ejpam-2534	328	12	.	.	PUNCT
ejpam-2534	329	1	conversely	conversely	ADV
ejpam-2534	329	2	,	,	PUNCT
ejpam-2534	329	3	let	let	VERB
ejpam-2534	329	4	r	r	NOUN
ejpam-2534	329	5	∈	∈	PROPN
ejpam-2534	329	6	rad(n	rad(n	NOUN
ejpam-2534	329	7	:	:	PUNCT
ejpam-2534	329	8	m)∪	m)∪	NOUN
ejpam-2534	329	9	(	(	PUNCT
ejpam-2534	329	10	0	0	NUM
ejpam-2534	329	11	:	:	PUNCT
ejpam-2534	329	12	rm	rm	NOUN
ejpam-2534	329	13	)	)	PUNCT
ejpam-2534	329	14	.	.	PUNCT
ejpam-2534	330	1	if	if	SCONJ
ejpam-2534	330	2	r	r	NOUN
ejpam-2534	330	3	∈	∈	PROPN
ejpam-2534	330	4	rad(n	rad(n	NOUN
ejpam-2534	330	5	:	:	PUNCT
ejpam-2534	330	6	m	m	X
ejpam-2534	330	7	)	)	PUNCT
ejpam-2534	330	8	,	,	PUNCT
ejpam-2534	330	9	then	then	ADV
ejpam-2534	330	10	rnm	rnm	VERB
ejpam-2534	330	11	⊆	⊆	X
ejpam-2534	330	12	n	n	NOUN
ejpam-2534	330	13	for	for	ADP
ejpam-2534	330	14	some	some	DET
ejpam-2534	330	15	positive	positive	ADJ
ejpam-2534	330	16	integer	integer	NOUN
ejpam-2534	330	17	n	n	NOUN
ejpam-2534	331	1	and	and	CCONJ
ejpam-2534	331	2	so	so	ADV
ejpam-2534	331	3	we	we	PRON
ejpam-2534	331	4	get	get	VERB
ejpam-2534	331	5	rnrm	rnrm	ADJ
ejpam-2534	331	6	⊆	⊆	NUM
ejpam-2534	331	7	rnm	rnm	NOUN
ejpam-2534	331	8	⊆	⊆	NUM
ejpam-2534	331	9	n	n	NOUN
ejpam-2534	331	10	.	.	PUNCT
ejpam-2534	332	1	thus	thus	ADV
ejpam-2534	332	2	r	r	NOUN
ejpam-2534	332	3	∈	∈	PROPN
ejpam-2534	332	4	rad(n	rad(n	PROPN
ejpam-2534	332	5	:	:	PUNCT
ejpam-2534	332	6	rm	rm	NOUN
ejpam-2534	332	7	)	)	PUNCT
ejpam-2534	332	8	.	.	PUNCT
ejpam-2534	333	1	if	if	SCONJ
ejpam-2534	333	2	r	r	NOUN
ejpam-2534	333	3	∈	∈	PROPN
ejpam-2534	333	4	(	(	PUNCT
ejpam-2534	333	5	0	0	NUM
ejpam-2534	333	6	:	:	PUNCT
ejpam-2534	333	7	rm	rm	PROPN
ejpam-2534	333	8	)	)	PUNCT
ejpam-2534	333	9	,	,	PUNCT
ejpam-2534	333	10	then	then	ADV
ejpam-2534	333	11	rrm=	rrm=	PROPN
ejpam-2534	333	12	0	0	NUM
ejpam-2534	333	13	⊆	⊆	NUM
ejpam-2534	333	14	n	n	NOUN
ejpam-2534	333	15	.	.	PUNCT
ejpam-2534	334	1	thus	thus	ADV
ejpam-2534	334	2	r	r	X
ejpam-2534	334	3	∈	∈	PROPN
ejpam-2534	334	4	(	(	PUNCT
ejpam-2534	334	5	n	n	NOUN
ejpam-2534	334	6	:	:	PUNCT
ejpam-2534	334	7	rm	rm	PROPN
ejpam-2534	334	8	)	)	PUNCT
ejpam-2534	334	9	⊆	⊆	NUM
ejpam-2534	334	10	rad(n	rad(n	PROPN
ejpam-2534	334	11	:	:	PUNCT
ejpam-2534	334	12	rm	rm	PROPN
ejpam-2534	334	13	)	)	PUNCT
ejpam-2534	334	14	.	.	PUNCT
ejpam-2534	335	1	(	(	PUNCT
ejpam-2534	335	2	ii)⇒	ii)⇒	PROPN
ejpam-2534	335	3	(	(	PUNCT
ejpam-2534	335	4	iii	iii	NOUN
ejpam-2534	335	5	):	):	PUNCT
ejpam-2534	335	6	clear	clear	ADJ
ejpam-2534	335	7	.	.	PUNCT
ejpam-2534	336	1	(	(	PUNCT
ejpam-2534	336	2	iii)⇒	iii)⇒	PROPN
ejpam-2534	336	3	(	(	PUNCT
ejpam-2534	336	4	i	i	NOUN
ejpam-2534	336	5	):	):	PUNCT
ejpam-2534	336	6	let	let	VERB
ejpam-2534	336	7	m	m	PRON
ejpam-2534	336	8	be	be	AUX
ejpam-2534	336	9	a	a	DET
ejpam-2534	336	10	maximal	maximal	ADJ
ejpam-2534	336	11	ideal	ideal	NOUN
ejpam-2534	336	12	of	of	ADP
ejpam-2534	336	13	r	r	NOUN
ejpam-2534	336	14	with	with	ADP
ejpam-2534	336	15	p(n	p(n	NOUN
ejpam-2534	336	16	)	)	PUNCT
ejpam-2534	336	17	⊆m	⊆m	NOUN
ejpam-2534	336	18	.	.	PUNCT
ejpam-2534	337	1	let	let	VERB
ejpam-2534	337	2	m	m	PRON
ejpam-2534	337	3	p	p	NOUN
ejpam-2534	337	4	∈	∈	PROPN
ejpam-2534	337	5	mm\nm	mm\nm	PROPN
ejpam-2534	337	6	where	where	SCONJ
ejpam-2534	337	7	m	m	VERB
ejpam-2534	337	8	∈	∈	PROPN
ejpam-2534	337	9	m	m	NOUN
ejpam-2534	337	10	,	,	PUNCT
ejpam-2534	337	11	p	p	PROPN
ejpam-2534	337	12	∈	∈	PROPN
ejpam-2534	337	13	r	r	NOUN
ejpam-2534	337	14	\m	\m	NOUN
ejpam-2534	337	15	.	.	PUNCT
ejpam-2534	338	1	then	then	ADV
ejpam-2534	338	2	m	m	VERB
ejpam-2534	338	3	∈	∈	PROPN
ejpam-2534	338	4	m	m	VERB
ejpam-2534	338	5	\	\	PROPN
ejpam-2534	338	6	n	n	X
ejpam-2534	338	7	.	.	PUNCT
ejpam-2534	339	1	by	by	ADP
ejpam-2534	339	2	the	the	DET
ejpam-2534	339	3	condition	condition	NOUN
ejpam-2534	339	4	of	of	ADP
ejpam-2534	339	5	the	the	DET
ejpam-2534	339	6	theorem	theorem	NOUN
ejpam-2534	339	7	,	,	PUNCT
ejpam-2534	339	8	rad(n	rad(n	PROPN
ejpam-2534	339	9	:	:	PUNCT
ejpam-2534	339	10	rm	rm	NOUN
ejpam-2534	339	11	)	)	PUNCT
ejpam-2534	339	12	=	=	PUNCT
ejpam-2534	339	13	rad(n	rad(n	NOUN
ejpam-2534	339	14	:	:	PUNCT
ejpam-2534	339	15	m	m	X
ejpam-2534	339	16	)	)	PUNCT
ejpam-2534	339	17	or	or	CCONJ
ejpam-2534	339	18	rad(n	rad(n	PROPN
ejpam-2534	339	19	:	:	PUNCT
ejpam-2534	339	20	rm	rm	NOUN
ejpam-2534	339	21	)	)	PUNCT
ejpam-2534	339	22	=	=	PUNCT
ejpam-2534	340	1	(	(	PUNCT
ejpam-2534	340	2	0	0	NUM
ejpam-2534	340	3	:	:	PUNCT
ejpam-2534	340	4	rm	rm	NOUN
ejpam-2534	340	5	)	)	PUNCT
ejpam-2534	340	6	for	for	ADP
ejpam-2534	340	7	some	some	DET
ejpam-2534	340	8	m	m	NOUN
ejpam-2534	340	9	∈	∈	NOUN
ejpam-2534	340	10	m	m	VERB
ejpam-2534	340	11	\	\	NOUN
ejpam-2534	341	1	n	n	NOUN
ejpam-2534	341	2	.	.	PUNCT
ejpam-2534	342	1	if	if	SCONJ
ejpam-2534	342	2	rad(n	rad(n	NOUN
ejpam-2534	342	3	:	:	PUNCT
ejpam-2534	342	4	rm	rm	NOUN
ejpam-2534	342	5	)	)	PUNCT
ejpam-2534	342	6	=	=	PUNCT
ejpam-2534	342	7	rad(n	rad(n	NOUN
ejpam-2534	342	8	:	:	PUNCT
ejpam-2534	342	9	m	m	NUM
ejpam-2534	342	10	)	)	PUNCT
ejpam-2534	342	11	,	,	PUNCT
ejpam-2534	342	12	then	then	ADV
ejpam-2534	342	13	rad((n	rad((n	NOUN
ejpam-2534	342	14	:	:	PUNCT
ejpam-2534	342	15	rm)m	rm)m	PROPN
ejpam-2534	342	16	)	)	PUNCT
ejpam-2534	342	17	=	=	SYM
ejpam-2534	342	18	rad((n	rad((n	NOUN
ejpam-2534	342	19	:	:	PUNCT
ejpam-2534	342	20	m)m	m)m	X
ejpam-2534	342	21	)	)	PUNCT
ejpam-2534	342	22	and	and	CCONJ
ejpam-2534	342	23	from	from	ADP
ejpam-2534	342	24	proposition	proposition	NOUN
ejpam-2534	342	25	3	3	NUM
ejpam-2534	342	26	and	and	CCONJ
ejpam-2534	342	27	proposition	proposition	NOUN
ejpam-2534	342	28	4	4	NUM
ejpam-2534	342	29	rad(nm	rad(nm	NOUN
ejpam-2534	342	30	:	:	PUNCT
ejpam-2534	342	31	(	(	PUNCT
ejpam-2534	342	32	rm)m	rm)m	SYM
ejpam-2534	342	33	)	)	PUNCT
ejpam-2534	342	34	=	=	NOUN
ejpam-2534	342	35	rad(nm	rad(nm	NOUN
ejpam-2534	342	36	:	:	PUNCT
ejpam-2534	342	37	mm	mm	X
ejpam-2534	342	38	)	)	PUNCT
ejpam-2534	342	39	.	.	PUNCT
ejpam-2534	343	1	by	by	ADP
ejpam-2534	343	2	[	[	X
ejpam-2534	343	3	4	4	NUM
ejpam-2534	343	4	,	,	PUNCT
ejpam-2534	343	5	proposition	proposition	NOUN
ejpam-2534	343	6	2.8	2.8	NUM
ejpam-2534	343	7	]	]	PUNCT
ejpam-2534	343	8	,	,	PUNCT
ejpam-2534	343	9	rad(nm	rad(nm	VERB
ejpam-2534	343	10	:	:	PUNCT
ejpam-2534	343	11	rm	rm	PROPN
ejpam-2534	343	12	m	m	PROPN
ejpam-2534	343	13	p	p	NOUN
ejpam-2534	343	14	)	)	PUNCT
ejpam-2534	344	1	=	=	NOUN
ejpam-2534	344	2	rad(nm	rad(nm	NOUN
ejpam-2534	344	3	:	:	PUNCT
ejpam-2534	344	4	mm	mm	X
ejpam-2534	344	5	)	)	PUNCT
ejpam-2534	344	6	.	.	PUNCT
ejpam-2534	345	1	if	if	SCONJ
ejpam-2534	345	2	rad(n	rad(n	NOUN
ejpam-2534	345	3	:	:	PUNCT
ejpam-2534	345	4	rm	rm	NOUN
ejpam-2534	345	5	)	)	PUNCT
ejpam-2534	345	6	=	=	PUNCT
ejpam-2534	346	1	(	(	PUNCT
ejpam-2534	346	2	0	0	NUM
ejpam-2534	346	3	:	:	PUNCT
ejpam-2534	346	4	rm	rm	PROPN
ejpam-2534	346	5	)	)	PUNCT
ejpam-2534	346	6	,	,	PUNCT
ejpam-2534	346	7	then	then	ADV
ejpam-2534	346	8	rad((n	rad((n	NOUN
ejpam-2534	346	9	:	:	PUNCT
ejpam-2534	346	10	rm)m	rm)m	PROPN
ejpam-2534	346	11	)	)	PUNCT
ejpam-2534	346	12	=	=	PUNCT
ejpam-2534	346	13	(	(	PUNCT
ejpam-2534	346	14	0	0	NUM
ejpam-2534	346	15	:	:	PUNCT
ejpam-2534	346	16	rm)m	rm)m	PROPN
ejpam-2534	346	17	and	and	CCONJ
ejpam-2534	346	18	by	by	ADP
ejpam-2534	346	19	proposition	proposition	NOUN
ejpam-2534	346	20	4	4	NUM
ejpam-2534	346	21	and	and	CCONJ
ejpam-2534	346	22	corollary	corollary	ADJ
ejpam-2534	346	23	4	4	NUM
ejpam-2534	346	24	,	,	PUNCT
ejpam-2534	346	25	rad(nm	rad(nm	VERB
ejpam-2534	346	26	:	:	PUNCT
ejpam-2534	346	27	(	(	PUNCT
ejpam-2534	346	28	rm)m	rm)m	PROPN
ejpam-2534	346	29	)	)	PUNCT
ejpam-2534	346	30	=	=	SYM
ejpam-2534	346	31	(	(	PUNCT
ejpam-2534	346	32	0	0	NUM
ejpam-2534	346	33	m	m	VERB
ejpam-2534	346	34	:	:	PUNCT
ejpam-2534	346	35	(	(	PUNCT
ejpam-2534	346	36	rm)m	rm)m	PROPN
ejpam-2534	346	37	)	)	PUNCT
ejpam-2534	346	38	.	.	PUNCT
ejpam-2534	347	1	by	by	ADP
ejpam-2534	347	2	[	[	X
ejpam-2534	347	3	4	4	NUM
ejpam-2534	347	4	,	,	PUNCT
ejpam-2534	347	5	proposition	proposition	NOUN
ejpam-2534	347	6	2.8	2.8	NUM
ejpam-2534	347	7	]	]	PUNCT
ejpam-2534	347	8	,	,	PUNCT
ejpam-2534	347	9	rad(nm	rad(nm	VERB
ejpam-2534	347	10	:	:	PUNCT
ejpam-2534	348	1	rm	rm	PROPN
ejpam-2534	348	2	m	m	PROPN
ejpam-2534	348	3	p	p	NOUN
ejpam-2534	348	4	)	)	PUNCT
ejpam-2534	348	5	=	=	SYM
ejpam-2534	349	1	(	(	PUNCT
ejpam-2534	349	2	0	0	NUM
ejpam-2534	349	3	m	m	VERB
ejpam-2534	349	4	:	:	PUNCT
ejpam-2534	350	1	rm	rm	PROPN
ejpam-2534	350	2	m	m	PROPN
ejpam-2534	350	3	p	p	NOUN
ejpam-2534	350	4	)	)	PUNCT
ejpam-2534	350	5	.	.	PUNCT
ejpam-2534	351	1	by	by	ADP
ejpam-2534	351	2	[	[	X
ejpam-2534	351	3	2	2	NUM
ejpam-2534	351	4	,	,	PUNCT
ejpam-2534	351	5	theorem	theorem	VERB
ejpam-2534	351	6	2.15	2.15	NUM
ejpam-2534	351	7	]	]	PUNCT
ejpam-2534	351	8	,	,	PUNCT
ejpam-2534	351	9	nm	nm	PRON
ejpam-2534	351	10	is	be	AUX
ejpam-2534	351	11	a	a	DET
ejpam-2534	351	12	weakly	weakly	ADJ
ejpam-2534	351	13	primary	primary	ADJ
ejpam-2534	351	14	submodule	submodule	NOUN
ejpam-2534	351	15	of	of	ADP
ejpam-2534	351	16	mm	mm	PROPN
ejpam-2534	351	17	.	.	PUNCT
ejpam-2534	352	1	thus	thus	ADV
ejpam-2534	352	2	n	n	PRON
ejpam-2534	352	3	is	be	AUX
ejpam-2534	352	4	a	a	DET
ejpam-2534	352	5	p(n)-locally	p(n)-locally	ADV
ejpam-2534	352	6	weakly	weakly	ADJ
ejpam-2534	352	7	primary	primary	ADJ
ejpam-2534	352	8	submodule	submodule	NOUN
ejpam-2534	352	9	of	of	ADP
ejpam-2534	352	10	m	m	PROPN
ejpam-2534	352	11	.	.	PUNCT
ejpam-2534	353	1	theorem	theorem	ADJ
ejpam-2534	353	2	4	4	NUM
ejpam-2534	353	3	.	.	PUNCT
ejpam-2534	354	1	let	let	VERB
ejpam-2534	354	2	m	m	PRON
ejpam-2534	354	3	be	be	AUX
ejpam-2534	354	4	an	an	DET
ejpam-2534	354	5	r	r	NOUN
ejpam-2534	354	6	-	-	PUNCT
ejpam-2534	354	7	module	module	NOUN
ejpam-2534	354	8	and	and	CCONJ
ejpam-2534	354	9	n	n	CCONJ
ejpam-2534	354	10	be	be	AUX
ejpam-2534	354	11	an	an	DET
ejpam-2534	354	12	m	m	ADJ
ejpam-2534	354	13	-	-	ADJ
ejpam-2534	354	14	primal	primal	ADJ
ejpam-2534	354	15	submodule	submodule	NOUN
ejpam-2534	354	16	of	of	ADP
ejpam-2534	354	17	m	m	PROPN
ejpam-2534	354	18	with	with	ADP
ejpam-2534	354	19	p(0	p(0	PROPN
ejpam-2534	354	20	)	)	PUNCT
ejpam-2534	354	21	⊆	⊆	NUM
ejpam-2534	354	22	p(n	p(n	PROPN
ejpam-2534	354	23	)	)	PUNCT
ejpam-2534	354	24	.	.	PUNCT
ejpam-2534	355	1	then	then	ADV
ejpam-2534	355	2	the	the	DET
ejpam-2534	355	3	following	follow	VERB
ejpam-2534	355	4	statements	statement	NOUN
ejpam-2534	355	5	are	be	AUX
ejpam-2534	355	6	equivalent	equivalent	ADJ
ejpam-2534	355	7	:	:	PUNCT
ejpam-2534	355	8	(	(	PUNCT
ejpam-2534	355	9	i	i	NOUN
ejpam-2534	355	10	)	)	PUNCT
ejpam-2534	355	11	n	n	PRON
ejpam-2534	355	12	is	be	AUX
ejpam-2534	355	13	a	a	DET
ejpam-2534	355	14	p(n)-locally	p(n)-locally	ADV
ejpam-2534	355	15	weakly	weakly	ADJ
ejpam-2534	355	16	primary	primary	ADJ
ejpam-2534	355	17	submodule	submodule	NOUN
ejpam-2534	355	18	of	of	ADP
ejpam-2534	355	19	m.	m.	NOUN
ejpam-2534	355	20	(	(	PUNCT
ejpam-2534	355	21	ii	ii	PROPN
ejpam-2534	355	22	)	)	PUNCT
ejpam-2534	355	23	0	0	NUM
ejpam-2534	356	1	6=	6=	NUM
ejpam-2534	357	1	i	i	PROPN
ejpam-2534	357	2	d	d	PROPN
ejpam-2534	357	3	⊆	⊆	NUM
ejpam-2534	357	4	n	n	PROPN
ejpam-2534	357	5	for	for	ADP
ejpam-2534	357	6	any	any	DET
ejpam-2534	357	7	ideal	ideal	NOUN
ejpam-2534	357	8	i	i	PRON
ejpam-2534	357	9	of	of	ADP
ejpam-2534	357	10	r	r	NOUN
ejpam-2534	357	11	and	and	CCONJ
ejpam-2534	357	12	any	any	DET
ejpam-2534	357	13	submodule	submodule	NOUN
ejpam-2534	357	14	d	d	PROPN
ejpam-2534	357	15	of	of	ADP
ejpam-2534	357	16	m	m	PROPN
ejpam-2534	357	17	implies	imply	VERB
ejpam-2534	357	18	either	either	CCONJ
ejpam-2534	357	19	i	i	PROPN
ejpam-2534	357	20	⊆	⊆	NUM
ejpam-2534	357	21	rad(n	rad(n	NOUN
ejpam-2534	357	22	:	:	PUNCT
ejpam-2534	357	23	m	m	X
ejpam-2534	357	24	)	)	PUNCT
ejpam-2534	357	25	or	or	CCONJ
ejpam-2534	357	26	d	d	PROPN
ejpam-2534	357	27	⊆	⊆	NUM
ejpam-2534	357	28	n.	n.	NOUN
ejpam-2534	357	29	(	(	PUNCT
ejpam-2534	357	30	iii	iii	NOUN
ejpam-2534	357	31	)	)	PUNCT
ejpam-2534	357	32	rad(n	rad(n	PROPN
ejpam-2534	357	33	:	:	PUNCT
ejpam-2534	357	34	rm	rm	NOUN
ejpam-2534	357	35	)	)	PUNCT
ejpam-2534	357	36	=	=	PUNCT
ejpam-2534	357	37	rad(n	rad(n	NOUN
ejpam-2534	357	38	:	:	PUNCT
ejpam-2534	357	39	m)∪	m)∪	NOUN
ejpam-2534	357	40	(	(	PUNCT
ejpam-2534	357	41	0	0	NUM
ejpam-2534	357	42	:	:	PUNCT
ejpam-2534	357	43	rm	rm	NOUN
ejpam-2534	357	44	)	)	PUNCT
ejpam-2534	357	45	for	for	ADP
ejpam-2534	357	46	any	any	DET
ejpam-2534	357	47	m	m	NOUN
ejpam-2534	357	48	∈	∈	PROPN
ejpam-2534	357	49	m	m	NOUN
ejpam-2534	357	50	\	\	PROPN
ejpam-2534	357	51	n.	n.	NOUN
ejpam-2534	357	52	(	(	PUNCT
ejpam-2534	357	53	iv	iv	X
ejpam-2534	357	54	)	)	PUNCT
ejpam-2534	357	55	rad(n	rad(n	PROPN
ejpam-2534	357	56	:	:	PUNCT
ejpam-2534	357	57	rm	rm	NOUN
ejpam-2534	357	58	)	)	PUNCT
ejpam-2534	358	1	=	=	PUNCT
ejpam-2534	358	2	rad(n	rad(n	NOUN
ejpam-2534	358	3	:	:	PUNCT
ejpam-2534	358	4	m	m	X
ejpam-2534	358	5	)	)	PUNCT
ejpam-2534	358	6	or	or	CCONJ
ejpam-2534	358	7	rad(n	rad(n	PROPN
ejpam-2534	358	8	:	:	PUNCT
ejpam-2534	358	9	rm	rm	NOUN
ejpam-2534	358	10	)	)	PUNCT
ejpam-2534	358	11	=	=	PUNCT
ejpam-2534	359	1	(	(	PUNCT
ejpam-2534	359	2	0	0	NUM
ejpam-2534	359	3	:	:	PUNCT
ejpam-2534	359	4	rm	rm	NOUN
ejpam-2534	359	5	)	)	PUNCT
ejpam-2534	359	6	for	for	ADP
ejpam-2534	359	7	any	any	DET
ejpam-2534	359	8	m	m	NOUN
ejpam-2534	359	9	∈	∈	PROPN
ejpam-2534	359	10	m	m	VERB
ejpam-2534	359	11	\	\	NOUN
ejpam-2534	359	12	n	n	CCONJ
ejpam-2534	359	13	,	,	PUNCT
ejpam-2534	359	14	proof	proof	NOUN
ejpam-2534	359	15	.	.	PUNCT
ejpam-2534	360	1	it	it	PRON
ejpam-2534	360	2	is	be	AUX
ejpam-2534	360	3	clear	clear	ADJ
ejpam-2534	360	4	from	from	ADP
ejpam-2534	360	5	theorem	theorem	ADJ
ejpam-2534	360	6	2	2	NUM
ejpam-2534	360	7	and	and	CCONJ
ejpam-2534	360	8	theorem	theorem	VERB
ejpam-2534	360	9	3	3	NUM
ejpam-2534	360	10	.	.	PUNCT
ejpam-2534	360	11	references	reference	NOUN
ejpam-2534	360	12	[	[	X
ejpam-2534	360	13	1	1	NUM
ejpam-2534	360	14	]	]	X
ejpam-2534	360	15	a.e	a.e	PROPN
ejpam-2534	360	16	.	.	PROPN
ejpam-2534	360	17	ashour	ashour	NOUN
ejpam-2534	360	18	.	.	PUNCT
ejpam-2534	361	1	on	on	ADP
ejpam-2534	361	2	weakly	weakly	ADJ
ejpam-2534	361	3	primary	primary	ADJ
ejpam-2534	361	4	submodules	submodule	NOUN
ejpam-2534	361	5	,	,	PUNCT
ejpam-2534	361	6	journal	journal	NOUN
ejpam-2534	361	7	of	of	ADP
ejpam-2534	361	8	al	al	PROPN
ejpam-2534	361	9	azhar	azhar	PROPN
ejpam-2534	361	10	universitygaza(natural	universitygaza(natural	PROPN
ejpam-2534	361	11	sciences	science	NOUN
ejpam-2534	361	12	)	)	PUNCT
ejpam-2534	361	13	,	,	PUNCT
ejpam-2534	361	14	13	13	NUM
ejpam-2534	361	15	,	,	PUNCT
ejpam-2534	361	16	31	31	NUM
ejpam-2534	361	17	-	-	SYM
ejpam-2534	361	18	40	40	NUM
ejpam-2534	361	19	.	.	PUNCT
ejpam-2534	361	20	2011	2011	NUM
ejpam-2534	361	21	.	.	PUNCT
ejpam-2534	362	1	[	[	X
ejpam-2534	362	2	2	2	NUM
ejpam-2534	362	3	]	]	X
ejpam-2534	362	4	s.e	s.e	PROPN
ejpam-2534	362	5	.	.	PROPN
ejpam-2534	362	6	atani	atani	PROPN
ejpam-2534	362	7	and	and	CCONJ
ejpam-2534	362	8	f.	f.	PROPN
ejpam-2534	362	9	farzalipour	farzalipour	PROPN
ejpam-2534	362	10	.	.	PUNCT
ejpam-2534	363	1	on	on	ADP
ejpam-2534	363	2	weakly	weakly	ADJ
ejpam-2534	363	3	primary	primary	ADJ
ejpam-2534	363	4	ideals	ideal	NOUN
ejpam-2534	363	5	,	,	PUNCT
ejpam-2534	363	6	georgian	georgian	PROPN
ejpam-2534	363	7	mathematical	mathematical	ADJ
ejpam-2534	363	8	journal	journal	NOUN
ejpam-2534	363	9	,	,	PUNCT
ejpam-2534	363	10	12	12	NUM
ejpam-2534	363	11	,	,	PUNCT
ejpam-2534	363	12	423	423	NUM
ejpam-2534	363	13	-	-	SYM
ejpam-2534	363	14	429	429	NUM
ejpam-2534	363	15	.	.	PUNCT
ejpam-2534	363	16	2005	2005	NUM
ejpam-2534	363	17	.	.	PUNCT
ejpam-2534	364	1	[	[	X
ejpam-2534	364	2	3	3	NUM
ejpam-2534	364	3	]	]	X
ejpam-2534	364	4	s.e	s.e	PROPN
ejpam-2534	364	5	.	.	PROPN
ejpam-2534	364	6	atani	atani	PROPN
ejpam-2534	364	7	and	and	CCONJ
ejpam-2534	364	8	f.	f.	PROPN
ejpam-2534	364	9	farzalipour	farzalipour	PROPN
ejpam-2534	364	10	.	.	PUNCT
ejpam-2534	365	1	on	on	ADP
ejpam-2534	365	2	weakly	weakly	ADJ
ejpam-2534	365	3	prime	prime	ADJ
ejpam-2534	365	4	submodules	submodule	NOUN
ejpam-2534	365	5	,	,	PUNCT
ejpam-2534	365	6	tamkang	tamkang	PROPN
ejpam-2534	365	7	journal	journal	PROPN
ejpam-2534	365	8	of	of	ADP
ejpam-2534	365	9	mathematics	mathematic	NOUN
ejpam-2534	365	10	,	,	PUNCT
ejpam-2534	365	11	38	38	NUM
ejpam-2534	365	12	,	,	PUNCT
ejpam-2534	365	13	247	247	NUM
ejpam-2534	365	14	-	-	SYM
ejpam-2534	365	15	252	252	NUM
ejpam-2534	365	16	.	.	PUNCT
ejpam-2534	365	17	2007	2007	NUM
ejpam-2534	365	18	.	.	PUNCT
ejpam-2534	366	1	references	reference	NOUN
ejpam-2534	366	2	56	56	NUM
ejpam-2534	366	3	[	[	X
ejpam-2534	366	4	4	4	NUM
ejpam-2534	366	5	]	]	X
ejpam-2534	366	6	a.k	a.k	PROPN
ejpam-2534	366	7	.	.	PROPN
ejpam-2534	366	8	jabbar	jabbar	PROPN
ejpam-2534	366	9	.	.	PUNCT
ejpam-2534	367	1	a	a	DET
ejpam-2534	367	2	generalization	generalization	NOUN
ejpam-2534	367	3	of	of	ADP
ejpam-2534	367	4	prime	prime	ADJ
ejpam-2534	367	5	and	and	CCONJ
ejpam-2534	367	6	weakly	weakly	ADJ
ejpam-2534	367	7	prime	prime	ADJ
ejpam-2534	367	8	submodules	submodule	NOUN
ejpam-2534	367	9	,	,	PUNCT
ejpam-2534	367	10	pure	pure	ADJ
ejpam-2534	367	11	mathematical	mathematical	ADJ
ejpam-2534	367	12	sciences	science	NOUN
ejpam-2534	367	13	,	,	PUNCT
ejpam-2534	367	14	2	2	NUM
ejpam-2534	367	15	,	,	PUNCT
ejpam-2534	367	16	1	1	NUM
ejpam-2534	367	17	-	-	SYM
ejpam-2534	367	18	11	11	NUM
ejpam-2534	367	19	.	.	PUNCT
ejpam-2534	367	20	2013	2013	NUM
ejpam-2534	367	21	.	.	PUNCT
ejpam-2534	368	1	[	[	X
ejpam-2534	368	2	5	5	NUM
ejpam-2534	368	3	]	]	X
ejpam-2534	368	4	r.y	r.y	PROPN
ejpam-2534	368	5	.	.	PROPN
ejpam-2534	368	6	sharp	sharp	PROPN
ejpam-2534	368	7	.	.	PUNCT
ejpam-2534	369	1	steps	step	NOUN
ejpam-2534	369	2	in	in	ADP
ejpam-2534	369	3	commutative	commutative	ADJ
ejpam-2534	369	4	algebra	algebra	NOUN
ejpam-2534	369	5	,	,	PUNCT
ejpam-2534	369	6	cambridge	cambridge	PROPN
ejpam-2534	369	7	university	university	PROPN
ejpam-2534	369	8	press	press	PROPN
ejpam-2534	369	9	,	,	PUNCT
ejpam-2534	369	10	cambridge	cambridge	PROPN
ejpam-2534	369	11	,	,	PUNCT
ejpam-2534	369	12	1990	1990	NUM
ejpam-2534	369	13	.	.	PUNCT
ejpam-2534	370	1	[	[	X
ejpam-2534	370	2	6	6	NUM
ejpam-2534	370	3	]	]	X
ejpam-2534	370	4	u.	u.	PROPN
ejpam-2534	370	5	tekir	tekir	PROPN
ejpam-2534	370	6	.	.	PUNCT
ejpam-2534	371	1	a	a	DET
ejpam-2534	371	2	note	note	NOUN
ejpam-2534	371	3	on	on	ADP
ejpam-2534	371	4	multiplication	multiplication	NOUN
ejpam-2534	371	5	modules	module	NOUN
ejpam-2534	371	6	,	,	PUNCT
ejpam-2534	371	7	international	international	ADJ
ejpam-2534	371	8	journal	journal	NOUN
ejpam-2534	371	9	of	of	ADP
ejpam-2534	371	10	pure	pure	ADJ
ejpam-2534	371	11	and	and	CCONJ
ejpam-2534	371	12	applied	applied	ADJ
ejpam-2534	371	13	mathematics	mathematic	NOUN
ejpam-2534	371	14	,	,	PUNCT
ejpam-2534	371	15	27	27	NUM
ejpam-2534	371	16	,	,	PUNCT
ejpam-2534	371	17	103	103	NUM
ejpam-2534	371	18	-	-	SYM
ejpam-2534	371	19	107	107	NUM
ejpam-2534	371	20	.	.	PUNCT
ejpam-2534	372	1	2006	2006	NUM
ejpam-2534	372	2	.	.	PUNCT
ejpam-2534	373	1	[	[	X
ejpam-2534	373	2	7	7	X
ejpam-2534	373	3	]	]	X
ejpam-2534	373	4	u.	u.	PROPN
ejpam-2534	373	5	tekir	tekir	PROPN
ejpam-2534	373	6	.	.	PUNCT
ejpam-2534	374	1	on	on	ADP
ejpam-2534	374	2	primary	primary	ADJ
ejpam-2534	374	3	submodules	submodule	NOUN
ejpam-2534	374	4	,	,	PUNCT
ejpam-2534	374	5	international	international	ADJ
ejpam-2534	374	6	journal	journal	NOUN
ejpam-2534	374	7	of	of	ADP
ejpam-2534	374	8	pure	pure	ADJ
ejpam-2534	374	9	and	and	CCONJ
ejpam-2534	374	10	applied	applied	ADJ
ejpam-2534	374	11	mathematics	mathematic	NOUN
ejpam-2534	374	12	,	,	PUNCT
ejpam-2534	374	13	27	27	NUM
ejpam-2534	374	14	,	,	PUNCT
ejpam-2534	374	15	283	283	NUM
ejpam-2534	374	16	-	-	SYM
ejpam-2534	374	17	289	289	NUM
ejpam-2534	374	18	.	.	PUNCT
ejpam-2534	374	19	2006	2006	NUM
ejpam-2534	374	20	.	.	PUNCT