id	sid	tid	token	lemma	pos
ejpam-3275	1	1	european	european	PROPN
ejpam-3275	1	2	journal	journal	PROPN
ejpam-3275	1	3	of	of	ADP
ejpam-3275	1	4	pure	pure	ADJ
ejpam-3275	1	5	and	and	CCONJ
ejpam-3275	1	6	applied	apply	VERB
ejpam-3275	1	7	mathematics	mathematic	NOUN
ejpam-3275	1	8	vol	vol	NOUN
ejpam-3275	1	9	.	.	PUNCT
ejpam-3275	2	1	11	11	NUM
ejpam-3275	2	2	,	,	PUNCT
ejpam-3275	2	3	no	no	INTJ
ejpam-3275	2	4	.	.	NOUN
ejpam-3275	2	5	3	3	NUM
ejpam-3275	2	6	,	,	PUNCT
ejpam-3275	2	7	2018	2018	NUM
ejpam-3275	2	8	,	,	PUNCT
ejpam-3275	2	9	730	730	NUM
ejpam-3275	2	10	-	-	SYM
ejpam-3275	2	11	739	739	NUM
ejpam-3275	2	12	issn	issn	PROPN
ejpam-3275	2	13	1307	1307	NUM
ejpam-3275	2	14	-	-	SYM
ejpam-3275	2	15	5543	5543	NUM
ejpam-3275	2	16	–	–	PUNCT
ejpam-3275	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3275	2	18	published	publish	VERB
ejpam-3275	2	19	by	by	ADP
ejpam-3275	2	20	new	new	PROPN
ejpam-3275	2	21	york	york	PROPN
ejpam-3275	2	22	business	business	PROPN
ejpam-3275	2	23	global	global	PROPN
ejpam-3275	2	24	on	on	ADP
ejpam-3275	2	25	α	α	NOUN
ejpam-3275	2	26	-	-	ADJ
ejpam-3275	2	27	prime	prime	ADJ
ejpam-3275	2	28	and	and	CCONJ
ejpam-3275	2	29	weakly	weakly	ADJ
ejpam-3275	2	30	α	α	NOUN
ejpam-3275	2	31	-	-	ADJ
ejpam-3275	2	32	prime	prime	ADJ
ejpam-3275	2	33	submodules	submodule	NOUN
ejpam-3275	2	34	thawatchai	thawatchai	PROPN
ejpam-3275	2	35	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	2	36	department	department	PROPN
ejpam-3275	2	37	of	of	ADP
ejpam-3275	2	38	mathematics	mathematic	NOUN
ejpam-3275	2	39	,	,	PUNCT
ejpam-3275	2	40	faculty	faculty	NOUN
ejpam-3275	2	41	of	of	ADP
ejpam-3275	2	42	science	science	PROPN
ejpam-3275	2	43	king	king	PROPN
ejpam-3275	2	44	mongkut	mongkut	PROPN
ejpam-3275	2	45	’s	’s	PROPN
ejpam-3275	2	46	institute	institute	PROPN
ejpam-3275	2	47	of	of	ADP
ejpam-3275	2	48	technology	technology	PROPN
ejpam-3275	2	49	ladkrabang	ladkrabang	PROPN
ejpam-3275	2	50	,	,	PUNCT
ejpam-3275	2	51	bangkok	bangkok	PROPN
ejpam-3275	2	52	10520	10520	NUM
ejpam-3275	2	53	,	,	PUNCT
ejpam-3275	2	54	thailand	thailand	PROPN
ejpam-3275	2	55	abstract	abstract	NOUN
ejpam-3275	2	56	.	.	PUNCT
ejpam-3275	3	1	we	we	PRON
ejpam-3275	3	2	have	have	AUX
ejpam-3275	3	3	introduced	introduce	VERB
ejpam-3275	3	4	the	the	DET
ejpam-3275	3	5	notion	notion	NOUN
ejpam-3275	3	6	of	of	ADP
ejpam-3275	3	7	α	α	NOUN
ejpam-3275	3	8	-	-	ADJ
ejpam-3275	3	9	prime	prime	ADJ
ejpam-3275	3	10	and	and	CCONJ
ejpam-3275	3	11	weakly	weakly	ADJ
ejpam-3275	3	12	α	α	NOUN
ejpam-3275	3	13	-	-	ADJ
ejpam-3275	3	14	prime	prime	ADJ
ejpam-3275	3	15	submodules	submodule	NOUN
ejpam-3275	3	16	as	as	ADP
ejpam-3275	3	17	a	a	DET
ejpam-3275	3	18	generalization	generalization	NOUN
ejpam-3275	3	19	of	of	ADP
ejpam-3275	3	20	prime	prime	ADJ
ejpam-3275	3	21	submodules	submodule	NOUN
ejpam-3275	3	22	.	.	PUNCT
ejpam-3275	4	1	some	some	DET
ejpam-3275	4	2	basic	basic	ADJ
ejpam-3275	4	3	properties	property	NOUN
ejpam-3275	4	4	of	of	ADP
ejpam-3275	4	5	α	α	NOUN
ejpam-3275	4	6	-	-	ADJ
ejpam-3275	4	7	prime	prime	ADJ
ejpam-3275	4	8	and	and	CCONJ
ejpam-3275	4	9	weakly	weakly	ADJ
ejpam-3275	4	10	α	α	NOUN
ejpam-3275	4	11	-	-	ADJ
ejpam-3275	4	12	prime	prime	ADJ
ejpam-3275	4	13	submodules	submodule	NOUN
ejpam-3275	4	14	are	be	AUX
ejpam-3275	4	15	the	the	DET
ejpam-3275	4	16	extension	extension	NOUN
ejpam-3275	4	17	of	of	ADP
ejpam-3275	4	18	prime	prime	ADJ
ejpam-3275	4	19	submodules	submodule	NOUN
ejpam-3275	4	20	.	.	PUNCT
ejpam-3275	5	1	finally	finally	ADV
ejpam-3275	5	2	,	,	PUNCT
ejpam-3275	5	3	after	after	ADP
ejpam-3275	5	4	introducing	introduce	VERB
ejpam-3275	5	5	the	the	DET
ejpam-3275	5	6	notion	notion	NOUN
ejpam-3275	5	7	of	of	ADP
ejpam-3275	5	8	α	α	NOUN
ejpam-3275	5	9	-	-	ADJ
ejpam-3275	5	10	prime	prime	ADJ
ejpam-3275	5	11	submodules	submodule	NOUN
ejpam-3275	5	12	,	,	PUNCT
ejpam-3275	5	13	we	we	PRON
ejpam-3275	5	14	also	also	ADV
ejpam-3275	5	15	define	define	VERB
ejpam-3275	5	16	and	and	CCONJ
ejpam-3275	5	17	study	study	VERB
ejpam-3275	5	18	the	the	DET
ejpam-3275	5	19	concept	concept	NOUN
ejpam-3275	5	20	of	of	ADP
ejpam-3275	5	21	α	α	NOUN
ejpam-3275	5	22	-	-	ADJ
ejpam-3275	5	23	prime	prime	ADJ
ejpam-3275	5	24	ideals	ideal	NOUN
ejpam-3275	5	25	in	in	ADP
ejpam-3275	5	26	a	a	DET
ejpam-3275	5	27	ring	ring	NOUN
ejpam-3275	5	28	.	.	PUNCT
ejpam-3275	6	1	2010	2010	NUM
ejpam-3275	6	2	mathematics	mathematic	NOUN
ejpam-3275	6	3	subject	subject	NOUN
ejpam-3275	6	4	classifications	classification	NOUN
ejpam-3275	6	5	:	:	PUNCT
ejpam-3275	6	6	13c99	13c99	NUM
ejpam-3275	6	7	key	key	ADJ
ejpam-3275	6	8	words	word	NOUN
ejpam-3275	6	9	and	and	CCONJ
ejpam-3275	6	10	phrases	phrase	NOUN
ejpam-3275	6	11	:	:	PUNCT
ejpam-3275	6	12	α	α	NUM
ejpam-3275	6	13	-	-	ADJ
ejpam-3275	6	14	prime	prime	ADJ
ejpam-3275	6	15	submodules	submodule	NOUN
ejpam-3275	6	16	,	,	PUNCT
ejpam-3275	6	17	weakly	weakly	ADJ
ejpam-3275	6	18	α	α	NOUN
ejpam-3275	6	19	-	-	ADJ
ejpam-3275	6	20	prime	prime	ADJ
ejpam-3275	6	21	submodules	submodule	NOUN
ejpam-3275	6	22	,	,	PUNCT
ejpam-3275	6	23	α	α	NOUN
ejpam-3275	6	24	-	-	ADJ
ejpam-3275	6	25	prime	prime	ADJ
ejpam-3275	6	26	ideals	ideal	NOUN
ejpam-3275	6	27	,	,	PUNCT
ejpam-3275	6	28	weakly	weakly	ADJ
ejpam-3275	6	29	α	α	NOUN
ejpam-3275	6	30	-	-	ADJ
ejpam-3275	6	31	prime	prime	ADJ
ejpam-3275	6	32	ideals	ideal	NOUN
ejpam-3275	6	33	1	1	NUM
ejpam-3275	6	34	.	.	PUNCT
ejpam-3275	7	1	introduction	introduction	NOUN
ejpam-3275	7	2	all	all	DET
ejpam-3275	7	3	rings	ring	NOUN
ejpam-3275	7	4	are	be	AUX
ejpam-3275	7	5	assumed	assume	VERB
ejpam-3275	7	6	to	to	PART
ejpam-3275	7	7	be	be	AUX
ejpam-3275	7	8	commutative	commutative	ADJ
ejpam-3275	7	9	with	with	ADP
ejpam-3275	7	10	nonzero	nonzero	PROPN
ejpam-3275	7	11	identity	identity	NOUN
ejpam-3275	7	12	and	and	CCONJ
ejpam-3275	7	13	all	all	DET
ejpam-3275	7	14	modules	module	NOUN
ejpam-3275	7	15	are	be	AUX
ejpam-3275	7	16	left	leave	VERB
ejpam-3275	7	17	unital	unital	ADJ
ejpam-3275	7	18	.	.	PUNCT
ejpam-3275	8	1	let	let	VERB
ejpam-3275	8	2	(	(	PUNCT
ejpam-3275	8	3	g,+	g,+	PROPN
ejpam-3275	8	4	)	)	PUNCT
ejpam-3275	8	5	be	be	AUX
ejpam-3275	8	6	a	a	DET
ejpam-3275	8	7	group	group	NOUN
ejpam-3275	8	8	.	.	PUNCT
ejpam-3275	9	1	for	for	ADP
ejpam-3275	9	2	a	a	DET
ejpam-3275	9	3	subset	subset	ADJ
ejpam-3275	9	4	h	h	NOUN
ejpam-3275	9	5	of	of	ADP
ejpam-3275	9	6	g	g	NOUN
ejpam-3275	9	7	,	,	PUNCT
ejpam-3275	9	8	denote	denote	NOUN
ejpam-3275	9	9	α(h	α(h	NOUN
ejpam-3275	9	10	)	)	PUNCT
ejpam-3275	9	11	=	=	PRON
ejpam-3275	9	12	{	{	PUNCT
ejpam-3275	9	13	h	h	NOUN
ejpam-3275	9	14	∈	∈	PROPN
ejpam-3275	9	15	g	g	PROPN
ejpam-3275	10	1	|	|	ADV
ejpam-3275	10	2	h+	h+	PUNCT
ejpam-3275	11	1	h	h	PROPN
ejpam-3275	11	2	∈	∈	PROPN
ejpam-3275	11	3	h	h	NOUN
ejpam-3275	11	4	}	}	PUNCT
ejpam-3275	11	5	and	and	CCONJ
ejpam-3275	11	6	β(h	β(h	ADJ
ejpam-3275	11	7	)	)	PUNCT
ejpam-3275	11	8	=	=	PRON
ejpam-3275	11	9	{	{	PUNCT
ejpam-3275	11	10	h+	h+	PROPN
ejpam-3275	11	11	h	h	NOUN
ejpam-3275	11	12	|	|	ADV
ejpam-3275	11	13	h	h	NOUN
ejpam-3275	11	14	∈	∈	PROPN
ejpam-3275	11	15	h	h	NOUN
ejpam-3275	11	16	}	}	PUNCT
ejpam-3275	11	17	.	.	PUNCT
ejpam-3275	12	1	it	it	PRON
ejpam-3275	12	2	is	be	AUX
ejpam-3275	12	3	clear	clear	ADJ
ejpam-3275	12	4	that	that	SCONJ
ejpam-3275	12	5	β(h	β(h	X
ejpam-3275	12	6	)	)	PUNCT
ejpam-3275	12	7	⊆	⊆	NUM
ejpam-3275	12	8	h	h	NOUN
ejpam-3275	12	9	⊆	⊆	NUM
ejpam-3275	12	10	α(h	α(h	NOUN
ejpam-3275	12	11	)	)	PUNCT
ejpam-3275	12	12	.	.	PUNCT
ejpam-3275	13	1	if	if	SCONJ
ejpam-3275	13	2	i	i	PRON
ejpam-3275	13	3	is	be	AUX
ejpam-3275	13	4	an	an	DET
ejpam-3275	13	5	ideal	ideal	NOUN
ejpam-3275	13	6	of	of	ADP
ejpam-3275	13	7	a	a	DET
ejpam-3275	13	8	ring	ring	NOUN
ejpam-3275	13	9	r	r	NOUN
ejpam-3275	13	10	,	,	PUNCT
ejpam-3275	13	11	then	then	ADV
ejpam-3275	13	12	α(i	α(i	PROPN
ejpam-3275	13	13	)	)	PUNCT
ejpam-3275	13	14	and	and	CCONJ
ejpam-3275	13	15	β(i	β(i	NOUN
ejpam-3275	13	16	)	)	PUNCT
ejpam-3275	13	17	are	be	AUX
ejpam-3275	13	18	ideals	ideal	NOUN
ejpam-3275	13	19	of	of	ADP
ejpam-3275	13	20	r.	r.	PROPN
ejpam-3275	13	21	if	if	SCONJ
ejpam-3275	13	22	n	n	PROPN
ejpam-3275	13	23	is	be	AUX
ejpam-3275	13	24	a	a	DET
ejpam-3275	13	25	submodule	submodule	NOUN
ejpam-3275	13	26	of	of	ADP
ejpam-3275	13	27	a	a	DET
ejpam-3275	13	28	module	module	NOUN
ejpam-3275	13	29	m	m	NOUN
ejpam-3275	13	30	,	,	PUNCT
ejpam-3275	13	31	then	then	ADV
ejpam-3275	13	32	α(n	α(n	NOUN
ejpam-3275	13	33	)	)	PUNCT
ejpam-3275	13	34	and	and	CCONJ
ejpam-3275	13	35	β(n	β(n	NUM
ejpam-3275	13	36	)	)	PUNCT
ejpam-3275	13	37	are	be	AUX
ejpam-3275	13	38	submodules	submodule	NOUN
ejpam-3275	13	39	of	of	ADP
ejpam-3275	13	40	m	m	PROPN
ejpam-3275	13	41	.	.	PUNCT
ejpam-3275	14	1	we	we	PRON
ejpam-3275	14	2	recall	recall	VERB
ejpam-3275	14	3	the	the	DET
ejpam-3275	14	4	definition	definition	NOUN
ejpam-3275	14	5	of	of	ADP
ejpam-3275	14	6	prime	prime	ADJ
ejpam-3275	14	7	submodules	submodule	NOUN
ejpam-3275	14	8	from	from	ADP
ejpam-3275	14	9	[	[	X
ejpam-3275	14	10	1	1	NUM
ejpam-3275	14	11	]	]	PUNCT
ejpam-3275	14	12	.	.	PUNCT
ejpam-3275	15	1	a	a	DET
ejpam-3275	15	2	proper	proper	ADJ
ejpam-3275	15	3	submodule	submodule	NOUN
ejpam-3275	15	4	p	p	NOUN
ejpam-3275	15	5	of	of	ADP
ejpam-3275	15	6	a	a	DET
ejpam-3275	15	7	left	left	ADJ
ejpam-3275	15	8	r	r	NOUN
ejpam-3275	15	9	-	-	PUNCT
ejpam-3275	15	10	module	module	NOUN
ejpam-3275	15	11	m	m	NOUN
ejpam-3275	15	12	is	be	AUX
ejpam-3275	15	13	called	call	VERB
ejpam-3275	15	14	prime	prime	ADJ
ejpam-3275	15	15	if	if	SCONJ
ejpam-3275	15	16	rm	rm	PROPN
ejpam-3275	15	17	∈	∈	PROPN
ejpam-3275	15	18	p	p	PROPN
ejpam-3275	15	19	for	for	ADP
ejpam-3275	15	20	some	some	DET
ejpam-3275	15	21	r	r	NOUN
ejpam-3275	15	22	∈	∈	NOUN
ejpam-3275	15	23	r	r	NOUN
ejpam-3275	15	24	and	and	CCONJ
ejpam-3275	15	25	m	m	NOUN
ejpam-3275	15	26	∈m	∈m	NOUN
ejpam-3275	15	27	,	,	PUNCT
ejpam-3275	15	28	then	then	ADV
ejpam-3275	15	29	r	r	NOUN
ejpam-3275	15	30	∈	∈	PROPN
ejpam-3275	15	31	(	(	PUNCT
ejpam-3275	15	32	p	p	X
ejpam-3275	15	33	:	:	PUNCT
ejpam-3275	15	34	m	m	NUM
ejpam-3275	15	35	)	)	PUNCT
ejpam-3275	15	36	or	or	CCONJ
ejpam-3275	15	37	m	m	PROPN
ejpam-3275	15	38	∈	∈	NOUN
ejpam-3275	15	39	p	p	NOUN
ejpam-3275	15	40	where	where	SCONJ
ejpam-3275	15	41	(	(	PUNCT
ejpam-3275	15	42	n	n	NUM
ejpam-3275	15	43	:	:	PUNCT
ejpam-3275	15	44	m	m	X
ejpam-3275	15	45	)	)	PUNCT
ejpam-3275	15	46	=	=	PRON
ejpam-3275	15	47	{	{	PUNCT
ejpam-3275	15	48	r	r	NOUN
ejpam-3275	15	49	∈	∈	NOUN
ejpam-3275	15	50	r	r	NOUN
ejpam-3275	15	51	|	|	NOUN
ejpam-3275	15	52	rm	rm	NOUN
ejpam-3275	15	53	⊆	⊆	NUM
ejpam-3275	15	54	n	n	CCONJ
ejpam-3275	15	55	}	}	PUNCT
ejpam-3275	15	56	.	.	PUNCT
ejpam-3275	16	1	let	let	VERB
ejpam-3275	16	2	m	m	PRON
ejpam-3275	16	3	be	be	AUX
ejpam-3275	16	4	a	a	DET
ejpam-3275	16	5	left	left	ADJ
ejpam-3275	16	6	r	r	NOUN
ejpam-3275	16	7	-	-	PUNCT
ejpam-3275	16	8	module	module	NOUN
ejpam-3275	16	9	,	,	PUNCT
ejpam-3275	16	10	m	m	VERB
ejpam-3275	16	11	∈	∈	PROPN
ejpam-3275	16	12	m	m	NOUN
ejpam-3275	16	13	and	and	CCONJ
ejpam-3275	16	14	n	n	ADV
ejpam-3275	16	15	be	be	VERB
ejpam-3275	16	16	a	a	DET
ejpam-3275	16	17	submodule	submodule	NOUN
ejpam-3275	16	18	of	of	ADP
ejpam-3275	16	19	m	m	PROPN
ejpam-3275	16	20	.	.	PUNCT
ejpam-3275	17	1	for	for	ADP
ejpam-3275	17	2	convenience	convenience	NOUN
ejpam-3275	17	3	,	,	PUNCT
ejpam-3275	17	4	we	we	PRON
ejpam-3275	17	5	denote	denote	VERB
ejpam-3275	17	6	(	(	PUNCT
ejpam-3275	17	7	0	0	NUM
ejpam-3275	17	8	:	:	PUNCT
ejpam-3275	17	9	m	m	X
ejpam-3275	17	10	)	)	PUNCT
ejpam-3275	18	1	=	=	PRON
ejpam-3275	18	2	{	{	PUNCT
ejpam-3275	18	3	r	r	NOUN
ejpam-3275	18	4	∈	∈	PROPN
ejpam-3275	18	5	r	r	NOUN
ejpam-3275	18	6	|	|	NOUN
ejpam-3275	18	7	rm	rm	NOUN
ejpam-3275	18	8	=	=	PUNCT
ejpam-3275	18	9	0	0	NUM
ejpam-3275	18	10	}	}	PUNCT
ejpam-3275	18	11	and	and	CCONJ
ejpam-3275	18	12	(	(	PUNCT
ejpam-3275	18	13	n	n	X
ejpam-3275	18	14	:	:	PUNCT
ejpam-3275	18	15	m	m	X
ejpam-3275	18	16	)	)	PUNCT
ejpam-3275	18	17	=	=	PRON
ejpam-3275	19	1	{	{	PUNCT
ejpam-3275	19	2	r	r	NOUN
ejpam-3275	19	3	∈	∈	NOUN
ejpam-3275	19	4	r	r	NOUN
ejpam-3275	19	5	|	|	NOUN
ejpam-3275	19	6	rm	rm	PROPN
ejpam-3275	19	7	∈	∈	PROPN
ejpam-3275	19	8	n	n	CCONJ
ejpam-3275	19	9	}	}	PUNCT
ejpam-3275	19	10	.	.	PUNCT
ejpam-3275	20	1	with	with	ADP
ejpam-3275	20	2	these	these	DET
ejpam-3275	20	3	notations	notation	NOUN
ejpam-3275	20	4	,	,	PUNCT
ejpam-3275	20	5	we	we	PRON
ejpam-3275	20	6	have	have	VERB
ejpam-3275	20	7	both	both	PRON
ejpam-3275	20	8	of	of	ADP
ejpam-3275	20	9	(	(	PUNCT
ejpam-3275	20	10	n	n	NUM
ejpam-3275	20	11	:	:	PUNCT
ejpam-3275	20	12	m	m	X
ejpam-3275	20	13	)	)	PUNCT
ejpam-3275	20	14	and	and	CCONJ
ejpam-3275	20	15	(	(	PUNCT
ejpam-3275	20	16	0	0	NUM
ejpam-3275	20	17	:	:	PUNCT
ejpam-3275	20	18	m	m	X
ejpam-3275	20	19	)	)	PUNCT
ejpam-3275	20	20	are	be	AUX
ejpam-3275	20	21	ideals	ideal	NOUN
ejpam-3275	20	22	of	of	ADP
ejpam-3275	20	23	r.	r.	PROPN
ejpam-3275	20	24	it	it	PRON
ejpam-3275	20	25	is	be	AUX
ejpam-3275	20	26	well	well	ADV
ejpam-3275	20	27	known	know	VERB
ejpam-3275	20	28	that	that	SCONJ
ejpam-3275	20	29	there	there	PRON
ejpam-3275	20	30	are	be	VERB
ejpam-3275	20	31	several	several	ADJ
ejpam-3275	20	32	authors	author	NOUN
ejpam-3275	20	33	have	have	AUX
ejpam-3275	20	34	extended	extend	VERB
ejpam-3275	20	35	the	the	DET
ejpam-3275	20	36	notion	notion	NOUN
ejpam-3275	20	37	of	of	ADP
ejpam-3275	20	38	prime	prime	ADJ
ejpam-3275	20	39	submodules	submodule	NOUN
ejpam-3275	20	40	.	.	PUNCT
ejpam-3275	21	1	all	all	PRON
ejpam-3275	21	2	of	of	ADP
ejpam-3275	21	3	those	those	DET
ejpam-3275	21	4	definitions	definition	NOUN
ejpam-3275	21	5	focus	focus	VERB
ejpam-3275	21	6	on	on	ADP
ejpam-3275	21	7	multiplication	multiplication	NOUN
ejpam-3275	21	8	between	between	ADP
ejpam-3275	21	9	element	element	NOUN
ejpam-3275	21	10	of	of	ADP
ejpam-3275	21	11	rings	ring	NOUN
ejpam-3275	21	12	and	and	CCONJ
ejpam-3275	21	13	of	of	ADP
ejpam-3275	21	14	modules	module	NOUN
ejpam-3275	21	15	.	.	PUNCT
ejpam-3275	22	1	this	this	PRON
ejpam-3275	22	2	motivates	motivate	VERB
ejpam-3275	22	3	us	we	PRON
ejpam-3275	22	4	to	to	PART
ejpam-3275	22	5	study	study	VERB
ejpam-3275	22	6	α	α	NUM
ejpam-3275	22	7	-	-	ADJ
ejpam-3275	22	8	prime	prime	ADJ
ejpam-3275	22	9	submodules	submodule	NOUN
ejpam-3275	22	10	by	by	ADP
ejpam-3275	22	11	taking	take	VERB
ejpam-3275	22	12	care	care	NOUN
ejpam-3275	22	13	on	on	ADP
ejpam-3275	22	14	all	all	DET
ejpam-3275	22	15	operations	operation	NOUN
ejpam-3275	22	16	of	of	ADP
ejpam-3275	22	17	a	a	DET
ejpam-3275	22	18	left	left	ADJ
ejpam-3275	22	19	module	module	NOUN
ejpam-3275	22	20	structure	structure	NOUN
ejpam-3275	22	21	.	.	PUNCT
ejpam-3275	23	1	our	our	PRON
ejpam-3275	23	2	extension	extension	NOUN
ejpam-3275	23	3	obtains	obtain	VERB
ejpam-3275	23	4	a	a	DET
ejpam-3275	23	5	generalization	generalization	NOUN
ejpam-3275	23	6	of	of	ADP
ejpam-3275	23	7	prime	prime	ADJ
ejpam-3275	23	8	submodules	submodule	NOUN
ejpam-3275	23	9	which	which	PRON
ejpam-3275	23	10	call	call	VERB
ejpam-3275	23	11	α	α	NUM
ejpam-3275	23	12	-	-	PUNCT
ejpam-3275	23	13	prime	prime	ADJ
ejpam-3275	23	14	submodules	submodule	NOUN
ejpam-3275	23	15	.	.	PUNCT
ejpam-3275	24	1	its	its	PRON
ejpam-3275	24	2	definition	definition	NOUN
ejpam-3275	24	3	and	and	CCONJ
ejpam-3275	24	4	results	result	NOUN
ejpam-3275	24	5	appear	appear	VERB
ejpam-3275	24	6	in	in	ADP
ejpam-3275	24	7	section	section	NOUN
ejpam-3275	24	8	1	1	NUM
ejpam-3275	24	9	.	.	PUNCT
ejpam-3275	25	1	in	in	ADP
ejpam-3275	25	2	section	section	NOUN
ejpam-3275	25	3	2	2	NUM
ejpam-3275	25	4	,	,	PUNCT
ejpam-3275	25	5	we	we	PRON
ejpam-3275	25	6	introduce	introduce	VERB
ejpam-3275	25	7	α	α	PRON
ejpam-3275	25	8	-	-	ADJ
ejpam-3275	25	9	prime	prime	ADJ
ejpam-3275	25	10	submodules	submodule	NOUN
ejpam-3275	25	11	and	and	CCONJ
ejpam-3275	25	12	also	also	ADV
ejpam-3275	25	13	give	give	VERB
ejpam-3275	25	14	some	some	DET
ejpam-3275	25	15	examples	example	NOUN
ejpam-3275	25	16	of	of	ADP
ejpam-3275	25	17	an	an	DET
ejpam-3275	25	18	α	α	PRON
ejpam-3275	25	19	-	-	ADJ
ejpam-3275	25	20	prime	prime	ADJ
ejpam-3275	25	21	submodule	submodule	NOUN
ejpam-3275	25	22	which	which	PRON
ejpam-3275	25	23	is	be	AUX
ejpam-3275	25	24	not	not	PART
ejpam-3275	25	25	a	a	DET
ejpam-3275	25	26	prime	prime	ADJ
ejpam-3275	25	27	submodule	submodule	NOUN
ejpam-3275	25	28	.	.	PUNCT
ejpam-3275	26	1	characterization	characterization	NOUN
ejpam-3275	26	2	of	of	ADP
ejpam-3275	26	3	α	α	NOUN
ejpam-3275	26	4	-	-	ADJ
ejpam-3275	26	5	prime	prime	ADJ
ejpam-3275	26	6	submodules	submodule	NOUN
ejpam-3275	26	7	of	of	ADP
ejpam-3275	26	8	z	z	NOUN
ejpam-3275	26	9	-	-	PUNCT
ejpam-3275	26	10	module	module	NOUN
ejpam-3275	26	11	z	z	NOUN
ejpam-3275	26	12	is	be	AUX
ejpam-3275	26	13	completely	completely	ADV
ejpam-3275	26	14	given	give	VERB
ejpam-3275	26	15	.	.	PUNCT
ejpam-3275	27	1	doi	doi	NOUN
ejpam-3275	27	2	:	:	PUNCT
ejpam-3275	27	3	https://doi.org/10.29020/nybg.ejpam.v11i3.3275	https://doi.org/10.29020/nybg.ejpam.v11i3.3275	PROPN
ejpam-3275	27	4	email	email	NOUN
ejpam-3275	27	5	address	address	NOUN
ejpam-3275	27	6	:	:	PUNCT
ejpam-3275	27	7	thawatchai.kh@kmitl.ac.th	thawatchai.kh@kmitl.ac.th	PROPN
ejpam-3275	27	8	(	(	PUNCT
ejpam-3275	27	9	thawatchai	thawatchai	PROPN
ejpam-3275	27	10	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	27	11	)	)	PUNCT
ejpam-3275	27	12	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3275	28	1	730	730	NUM
ejpam-3275	28	2	c	c	NOUN
ejpam-3275	28	3	©	©	PROPN
ejpam-3275	28	4	2018	2018	NUM
ejpam-3275	28	5	ejpam	ejpam	VERB
ejpam-3275	28	6	all	all	DET
ejpam-3275	28	7	rights	right	NOUN
ejpam-3275	28	8	reserved	reserve	VERB
ejpam-3275	28	9	.	.	PUNCT
ejpam-3275	29	1	t.	t.	PROPN
ejpam-3275	29	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	29	3	/	/	SYM
ejpam-3275	29	4	eur	eur	PROPN
ejpam-3275	29	5	.	.	PUNCT
ejpam-3275	30	1	j.	j.	PROPN
ejpam-3275	30	2	pure	pure	PROPN
ejpam-3275	30	3	appl	appl	PROPN
ejpam-3275	30	4	.	.	PROPN
ejpam-3275	30	5	math	math	PROPN
ejpam-3275	30	6	,	,	PUNCT
ejpam-3275	30	7	11	11	NUM
ejpam-3275	30	8	(	(	PUNCT
ejpam-3275	30	9	3	3	NUM
ejpam-3275	30	10	)	)	PUNCT
ejpam-3275	30	11	(	(	PUNCT
ejpam-3275	30	12	2018	2018	NUM
ejpam-3275	30	13	)	)	PUNCT
ejpam-3275	30	14	,	,	PUNCT
ejpam-3275	30	15	730	730	NUM
ejpam-3275	30	16	-	-	SYM
ejpam-3275	30	17	739	739	NUM
ejpam-3275	30	18	731	731	NUM
ejpam-3275	30	19	in	in	ADP
ejpam-3275	30	20	section	section	NOUN
ejpam-3275	30	21	3	3	NUM
ejpam-3275	31	1	,	,	PUNCT
ejpam-3275	31	2	we	we	PRON
ejpam-3275	31	3	extend	extend	VERB
ejpam-3275	31	4	the	the	DET
ejpam-3275	31	5	notion	notion	NOUN
ejpam-3275	31	6	of	of	ADP
ejpam-3275	31	7	α	α	NOUN
ejpam-3275	31	8	-	-	ADJ
ejpam-3275	31	9	prime	prime	ADJ
ejpam-3275	31	10	submodules	submodule	NOUN
ejpam-3275	31	11	to	to	ADP
ejpam-3275	31	12	weakly	weakly	ADJ
ejpam-3275	31	13	α	α	PRON
ejpam-3275	31	14	-	-	ADJ
ejpam-3275	31	15	prime	prime	ADJ
ejpam-3275	31	16	submodules	submodule	NOUN
ejpam-3275	31	17	.	.	PUNCT
ejpam-3275	32	1	we	we	PRON
ejpam-3275	32	2	study	study	VERB
ejpam-3275	32	3	properties	property	NOUN
ejpam-3275	32	4	the	the	DET
ejpam-3275	32	5	product	product	NOUN
ejpam-3275	32	6	of	of	ADP
ejpam-3275	32	7	submodules	submodule	NOUN
ejpam-3275	32	8	in	in	ADP
ejpam-3275	32	9	the	the	DET
ejpam-3275	32	10	cartesian	cartesian	ADJ
ejpam-3275	32	11	product	product	NOUN
ejpam-3275	32	12	of	of	ADP
ejpam-3275	32	13	modules	module	NOUN
ejpam-3275	32	14	.	.	PUNCT
ejpam-3275	33	1	in	in	ADP
ejpam-3275	33	2	section	section	NOUN
ejpam-3275	33	3	4	4	NUM
ejpam-3275	33	4	,	,	PUNCT
ejpam-3275	33	5	we	we	PRON
ejpam-3275	33	6	move	move	VERB
ejpam-3275	33	7	the	the	DET
ejpam-3275	33	8	investigation	investigation	NOUN
ejpam-3275	33	9	of	of	ADP
ejpam-3275	33	10	α	α	NOUN
ejpam-3275	33	11	-	-	ADJ
ejpam-3275	33	12	prime	prime	ADJ
ejpam-3275	33	13	submodules	submodule	NOUN
ejpam-3275	33	14	to	to	ADP
ejpam-3275	33	15	α	α	NOUN
ejpam-3275	33	16	-	-	PUNCT
ejpam-3275	33	17	prime	prime	ADJ
ejpam-3275	33	18	ideals	ideal	NOUN
ejpam-3275	33	19	.	.	PUNCT
ejpam-3275	34	1	2	2	X
ejpam-3275	34	2	.	.	X
ejpam-3275	34	3	α	α	X
ejpam-3275	34	4	-	-	ADJ
ejpam-3275	34	5	prime	prime	ADJ
ejpam-3275	34	6	submodules	submodule	NOUN
ejpam-3275	34	7	first	first	ADV
ejpam-3275	34	8	,	,	PUNCT
ejpam-3275	34	9	we	we	PRON
ejpam-3275	34	10	present	present	VERB
ejpam-3275	34	11	fundamental	fundamental	ADJ
ejpam-3275	34	12	definitions	definition	NOUN
ejpam-3275	34	13	of	of	ADP
ejpam-3275	34	14	α	α	NOUN
ejpam-3275	34	15	-	-	ADJ
ejpam-3275	34	16	prime	prime	ADJ
ejpam-3275	34	17	submodules	submodule	NOUN
ejpam-3275	34	18	which	which	PRON
ejpam-3275	34	19	will	will	AUX
ejpam-3275	34	20	be	be	AUX
ejpam-3275	34	21	studied	study	VERB
ejpam-3275	34	22	in	in	ADP
ejpam-3275	34	23	this	this	DET
ejpam-3275	34	24	paper	paper	NOUN
ejpam-3275	34	25	.	.	PUNCT
ejpam-3275	35	1	definition	definition	NOUN
ejpam-3275	35	2	1	1	NUM
ejpam-3275	35	3	.	.	PUNCT
ejpam-3275	36	1	let	let	VERB
ejpam-3275	36	2	p	p	PRON
ejpam-3275	36	3	be	be	AUX
ejpam-3275	36	4	a	a	DET
ejpam-3275	36	5	proper	proper	ADJ
ejpam-3275	36	6	submodule	submodule	NOUN
ejpam-3275	36	7	of	of	ADP
ejpam-3275	36	8	m	m	PROPN
ejpam-3275	36	9	.	.	PUNCT
ejpam-3275	37	1	we	we	PRON
ejpam-3275	37	2	call	call	VERB
ejpam-3275	37	3	p	p	NOUN
ejpam-3275	37	4	is	be	AUX
ejpam-3275	37	5	α	α	NOUN
ejpam-3275	37	6	-	-	ADJ
ejpam-3275	37	7	prime	prime	NOUN
ejpam-3275	37	8	if	if	SCONJ
ejpam-3275	37	9	for	for	ADP
ejpam-3275	37	10	any	any	DET
ejpam-3275	37	11	element	element	NOUN
ejpam-3275	37	12	r	r	NOUN
ejpam-3275	37	13	∈	∈	NOUN
ejpam-3275	37	14	r	r	NOUN
ejpam-3275	37	15	and	and	CCONJ
ejpam-3275	37	16	m	m	PROPN
ejpam-3275	37	17	∈	∈	NOUN
ejpam-3275	37	18	m	m	VERB
ejpam-3275	37	19	such	such	ADJ
ejpam-3275	37	20	that	that	SCONJ
ejpam-3275	37	21	r(m	r(m	PROPN
ejpam-3275	37	22	+	+	NOUN
ejpam-3275	37	23	m	m	NOUN
ejpam-3275	37	24	)	)	PUNCT
ejpam-3275	37	25	∈	∈	PROPN
ejpam-3275	37	26	p	p	NOUN
ejpam-3275	37	27	,	,	PUNCT
ejpam-3275	37	28	we	we	PRON
ejpam-3275	37	29	have	have	VERB
ejpam-3275	37	30	r	r	NOUN
ejpam-3275	37	31	+	+	NOUN
ejpam-3275	37	32	r	r	NOUN
ejpam-3275	37	33	∈	∈	NOUN
ejpam-3275	37	34	(	(	PUNCT
ejpam-3275	37	35	p	p	X
ejpam-3275	37	36	:	:	PUNCT
ejpam-3275	37	37	m	m	NUM
ejpam-3275	37	38	)	)	PUNCT
ejpam-3275	37	39	or	or	CCONJ
ejpam-3275	37	40	m+m	m+m	PROPN
ejpam-3275	37	41	∈	∈	PROPN
ejpam-3275	37	42	p	p	NOUN
ejpam-3275	37	43	.	.	PUNCT
ejpam-3275	38	1	by	by	ADP
ejpam-3275	38	2	this	this	DET
ejpam-3275	38	3	definition	definition	NOUN
ejpam-3275	38	4	,	,	PUNCT
ejpam-3275	38	5	every	every	DET
ejpam-3275	38	6	prime	prime	ADJ
ejpam-3275	38	7	submodule	submodule	NOUN
ejpam-3275	38	8	is	be	AUX
ejpam-3275	38	9	an	an	DET
ejpam-3275	38	10	α	α	NOUN
ejpam-3275	38	11	-	-	PUNCT
ejpam-3275	38	12	prime	prime	ADJ
ejpam-3275	38	13	submodule	submodule	NOUN
ejpam-3275	38	14	,	,	PUNCT
ejpam-3275	38	15	but	but	CCONJ
ejpam-3275	38	16	the	the	DET
ejpam-3275	38	17	converse	converse	NOUN
ejpam-3275	38	18	is	be	AUX
ejpam-3275	38	19	not	not	PART
ejpam-3275	38	20	true	true	ADJ
ejpam-3275	38	21	in	in	ADP
ejpam-3275	38	22	general	general	ADJ
ejpam-3275	38	23	.	.	PUNCT
ejpam-3275	39	1	example	example	NOUN
ejpam-3275	40	1	1	1	NUM
ejpam-3275	40	2	.	.	PUNCT
ejpam-3275	40	3	let	let	VERB
ejpam-3275	40	4	z	z	PRON
ejpam-3275	40	5	be	be	AUX
ejpam-3275	40	6	an	an	DET
ejpam-3275	40	7	z	z	NOUN
ejpam-3275	40	8	-	-	PUNCT
ejpam-3275	40	9	module	module	NOUN
ejpam-3275	40	10	and	and	CCONJ
ejpam-3275	40	11	p	p	NOUN
ejpam-3275	40	12	∈	∈	PROPN
ejpam-3275	40	13	z.	z.	PROPN
ejpam-3275	40	14	then	then	ADV
ejpam-3275	40	15	pz	pz	PROPN
ejpam-3275	40	16	is	be	AUX
ejpam-3275	40	17	an	an	DET
ejpam-3275	40	18	α	α	NOUN
ejpam-3275	40	19	-	-	ADJ
ejpam-3275	40	20	prime	prime	ADJ
ejpam-3275	40	21	submodule	submodule	NOUN
ejpam-3275	40	22	of	of	ADP
ejpam-3275	40	23	z	z	NOUN
ejpam-3275	40	24	if	if	SCONJ
ejpam-3275	41	1	and	and	CCONJ
ejpam-3275	41	2	only	only	ADV
ejpam-3275	41	3	if	if	SCONJ
ejpam-3275	41	4	p	p	X
ejpam-3275	41	5	=	=	NOUN
ejpam-3275	41	6	0	0	NUM
ejpam-3275	41	7	or	or	CCONJ
ejpam-3275	41	8	p	p	NOUN
ejpam-3275	41	9	is	be	AUX
ejpam-3275	41	10	a	a	DET
ejpam-3275	41	11	prime	prime	ADJ
ejpam-3275	41	12	number	number	NOUN
ejpam-3275	41	13	or	or	CCONJ
ejpam-3275	41	14	p	p	NOUN
ejpam-3275	41	15	=	=	NOUN
ejpam-3275	41	16	2q	2q	NUM
ejpam-3275	41	17	where	where	SCONJ
ejpam-3275	41	18	q	q	NOUN
ejpam-3275	41	19	is	be	AUX
ejpam-3275	41	20	a	a	DET
ejpam-3275	41	21	prime	prime	ADJ
ejpam-3275	41	22	number	number	NOUN
ejpam-3275	41	23	.	.	PUNCT
ejpam-3275	42	1	proof	proof	NOUN
ejpam-3275	42	2	.	.	PUNCT
ejpam-3275	43	1	(	(	PUNCT
ejpam-3275	43	2	→	→	NOUN
ejpam-3275	43	3	)	)	PUNCT
ejpam-3275	43	4	assume	assume	VERB
ejpam-3275	43	5	that	that	SCONJ
ejpam-3275	43	6	pz	pz	PROPN
ejpam-3275	43	7	is	be	AUX
ejpam-3275	43	8	an	an	DET
ejpam-3275	43	9	α	α	NOUN
ejpam-3275	43	10	-	-	ADJ
ejpam-3275	43	11	prime	prime	ADJ
ejpam-3275	43	12	submodule	submodule	NOUN
ejpam-3275	43	13	of	of	ADP
ejpam-3275	43	14	z.	z.	PROPN
ejpam-3275	43	15	suppose	suppose	VERB
ejpam-3275	43	16	that	that	SCONJ
ejpam-3275	43	17	p	p	PROPN
ejpam-3275	43	18	6=	6=	ADP
ejpam-3275	43	19	0	0	NUM
ejpam-3275	43	20	and	and	CCONJ
ejpam-3275	43	21	p	p	NOUN
ejpam-3275	43	22	is	be	AUX
ejpam-3275	43	23	not	not	PART
ejpam-3275	43	24	prime	prime	ADJ
ejpam-3275	43	25	number	number	NOUN
ejpam-3275	43	26	.	.	PUNCT
ejpam-3275	44	1	then	then	ADV
ejpam-3275	44	2	p	p	PROPN
ejpam-3275	44	3	=	=	PUNCT
ejpam-3275	44	4	ab	ab	PROPN
ejpam-3275	44	5	for	for	ADP
ejpam-3275	44	6	some	some	DET
ejpam-3275	44	7	integers	integer	NOUN
ejpam-3275	44	8	a	a	PRON
ejpam-3275	44	9	and	and	CCONJ
ejpam-3275	44	10	b	b	NOUN
ejpam-3275	44	11	with	with	ADP
ejpam-3275	44	12	1	1	NUM
ejpam-3275	44	13	<	<	X
ejpam-3275	44	14	a	a	PROPN
ejpam-3275	44	15	,	,	PUNCT
ejpam-3275	45	1	b	b	X
ejpam-3275	45	2	<	<	X
ejpam-3275	45	3	p.	p.	NOUN
ejpam-3275	45	4	we	we	PRON
ejpam-3275	45	5	see	see	VERB
ejpam-3275	45	6	that	that	SCONJ
ejpam-3275	45	7	p	p	NOUN
ejpam-3275	45	8	|	|	ADV
ejpam-3275	45	9	a(b	a(b	PROPN
ejpam-3275	45	10	+	+	CCONJ
ejpam-3275	45	11	b	b	X
ejpam-3275	45	12	)	)	PUNCT
ejpam-3275	45	13	.	.	PUNCT
ejpam-3275	46	1	this	this	PRON
ejpam-3275	46	2	implies	imply	VERB
ejpam-3275	46	3	that	that	SCONJ
ejpam-3275	46	4	p	p	PROPN
ejpam-3275	47	1	|	|	ADV
ejpam-3275	47	2	a	a	PRON
ejpam-3275	47	3	+	+	NOUN
ejpam-3275	47	4	a	a	PRON
ejpam-3275	47	5	or	or	CCONJ
ejpam-3275	47	6	p	p	ADJ
ejpam-3275	47	7	|	|	NOUN
ejpam-3275	47	8	b	b	PROPN
ejpam-3275	47	9	+	+	CCONJ
ejpam-3275	47	10	b.	b.	NOUN
ejpam-3275	47	11	now	now	ADV
ejpam-3275	47	12	,	,	PUNCT
ejpam-3275	47	13	we	we	PRON
ejpam-3275	47	14	assume	assume	VERB
ejpam-3275	47	15	that	that	SCONJ
ejpam-3275	47	16	p	p	PROPN
ejpam-3275	48	1	|	|	ADV
ejpam-3275	48	2	a	a	DET
ejpam-3275	48	3	+	+	NOUN
ejpam-3275	48	4	a.	a.	NOUN
ejpam-3275	48	5	this	this	PRON
ejpam-3275	48	6	means	mean	VERB
ejpam-3275	48	7	p	p	NOUN
ejpam-3275	48	8	≤	≤	ADJ
ejpam-3275	48	9	2a	2a	NUM
ejpam-3275	48	10	.	.	PUNCT
ejpam-3275	49	1	hence	hence	ADV
ejpam-3275	49	2	ab	ab	PROPN
ejpam-3275	49	3	≤	≤	NUM
ejpam-3275	49	4	2a	2a	NUM
ejpam-3275	49	5	.	.	PUNCT
ejpam-3275	50	1	therefore	therefore	ADV
ejpam-3275	50	2	b	b	X
ejpam-3275	50	3	≤	≤	ADV
ejpam-3275	50	4	2	2	NUM
ejpam-3275	50	5	.	.	PUNCT
ejpam-3275	51	1	that	that	PRON
ejpam-3275	51	2	is	be	AUX
ejpam-3275	51	3	b	b	NOUN
ejpam-3275	51	4	=	=	SYM
ejpam-3275	51	5	2	2	NUM
ejpam-3275	51	6	.	.	PUNCT
ejpam-3275	52	1	next	next	ADV
ejpam-3275	52	2	,	,	PUNCT
ejpam-3275	52	3	suppose	suppose	VERB
ejpam-3275	52	4	that	that	SCONJ
ejpam-3275	52	5	a	a	PRON
ejpam-3275	52	6	is	be	AUX
ejpam-3275	52	7	not	not	PART
ejpam-3275	52	8	a	a	DET
ejpam-3275	52	9	prime	prime	ADJ
ejpam-3275	52	10	number	number	NOUN
ejpam-3275	52	11	.	.	PUNCT
ejpam-3275	53	1	then	then	ADV
ejpam-3275	53	2	a	a	DET
ejpam-3275	53	3	=	=	X
ejpam-3275	53	4	cd	cd	NOUN
ejpam-3275	53	5	for	for	ADP
ejpam-3275	53	6	some	some	DET
ejpam-3275	53	7	integers	integer	NOUN
ejpam-3275	53	8	c	c	NOUN
ejpam-3275	53	9	and	and	CCONJ
ejpam-3275	53	10	d	d	X
ejpam-3275	53	11	with	with	ADP
ejpam-3275	53	12	1	1	NUM
ejpam-3275	53	13	<	<	X
ejpam-3275	53	14	c	c	NOUN
ejpam-3275	53	15	,	,	PUNCT
ejpam-3275	53	16	d	d	X
ejpam-3275	53	17	<	<	X
ejpam-3275	53	18	a.	a.	NOUN
ejpam-3275	53	19	we	we	PRON
ejpam-3275	53	20	have	have	VERB
ejpam-3275	53	21	p	p	NOUN
ejpam-3275	53	22	=	=	SYM
ejpam-3275	53	23	2a	2a	NUM
ejpam-3275	53	24	=	=	SYM
ejpam-3275	53	25	2cd	2cd	X
ejpam-3275	53	26	=	=	PUNCT
ejpam-3275	53	27	c(d	c(d	PROPN
ejpam-3275	53	28	+	+	CCONJ
ejpam-3275	53	29	d	d	NOUN
ejpam-3275	53	30	)	)	PUNCT
ejpam-3275	53	31	.	.	PUNCT
ejpam-3275	54	1	since	since	SCONJ
ejpam-3275	54	2	pz	pz	PROPN
ejpam-3275	54	3	is	be	AUX
ejpam-3275	54	4	an	an	DET
ejpam-3275	54	5	α	α	NOUN
ejpam-3275	54	6	-	-	ADJ
ejpam-3275	54	7	prime	prime	ADJ
ejpam-3275	54	8	submodule	submodule	NOUN
ejpam-3275	54	9	of	of	ADP
ejpam-3275	54	10	z	z	PROPN
ejpam-3275	54	11	,	,	PUNCT
ejpam-3275	54	12	p	p	NOUN
ejpam-3275	55	1	|	|	NOUN
ejpam-3275	55	2	c	c	NOUN
ejpam-3275	56	1	+	+	NOUN
ejpam-3275	56	2	c	c	PROPN
ejpam-3275	56	3	or	or	CCONJ
ejpam-3275	56	4	p	p	NOUN
ejpam-3275	57	1	|	|	NOUN
ejpam-3275	57	2	d	d	PROPN
ejpam-3275	57	3	+	+	CCONJ
ejpam-3275	57	4	d.	d.	NOUN
ejpam-3275	57	5	hence	hence	ADV
ejpam-3275	57	6	a	a	DET
ejpam-3275	57	7	|	|	NOUN
ejpam-3275	57	8	c	c	NOUN
ejpam-3275	57	9	or	or	CCONJ
ejpam-3275	57	10	a	a	DET
ejpam-3275	57	11	|	|	NOUN
ejpam-3275	57	12	d.	d.	NOUN
ejpam-3275	57	13	this	this	PRON
ejpam-3275	57	14	implies	imply	VERB
ejpam-3275	57	15	that	that	SCONJ
ejpam-3275	57	16	a	a	DET
ejpam-3275	57	17	≤	≤	ADJ
ejpam-3275	57	18	c	c	NOUN
ejpam-3275	57	19	or	or	CCONJ
ejpam-3275	57	20	a	a	DET
ejpam-3275	57	21	≤	≤	NUM
ejpam-3275	57	22	d	d	NOUN
ejpam-3275	57	23	which	which	PRON
ejpam-3275	57	24	is	be	AUX
ejpam-3275	57	25	a	a	DET
ejpam-3275	57	26	contradiction	contradiction	NOUN
ejpam-3275	57	27	.	.	PUNCT
ejpam-3275	58	1	this	this	PRON
ejpam-3275	58	2	prove	prove	VERB
ejpam-3275	58	3	that	that	SCONJ
ejpam-3275	58	4	p	p	NOUN
ejpam-3275	58	5	=	=	X
ejpam-3275	58	6	2q	2q	NUM
ejpam-3275	58	7	for	for	ADP
ejpam-3275	58	8	some	some	DET
ejpam-3275	58	9	prime	prime	ADJ
ejpam-3275	58	10	numbers	number	NOUN
ejpam-3275	58	11	q.	q.	PROPN
ejpam-3275	58	12	(	(	PUNCT
ejpam-3275	58	13	←	←	PROPN
ejpam-3275	58	14	)	)	PUNCT
ejpam-3275	58	15	it	it	PRON
ejpam-3275	58	16	is	be	AUX
ejpam-3275	58	17	clear	clear	ADJ
ejpam-3275	58	18	that	that	SCONJ
ejpam-3275	58	19	pz	pz	PROPN
ejpam-3275	58	20	is	be	AUX
ejpam-3275	58	21	an	an	DET
ejpam-3275	58	22	α	α	NOUN
ejpam-3275	58	23	-	-	ADJ
ejpam-3275	58	24	prime	prime	ADJ
ejpam-3275	58	25	submodule	submodule	NOUN
ejpam-3275	58	26	of	of	ADP
ejpam-3275	58	27	z	z	PROPN
ejpam-3275	58	28	where	where	SCONJ
ejpam-3275	58	29	p	p	NOUN
ejpam-3275	58	30	=	=	NOUN
ejpam-3275	58	31	0	0	NUM
ejpam-3275	58	32	or	or	CCONJ
ejpam-3275	58	33	p	p	NOUN
ejpam-3275	58	34	is	be	AUX
ejpam-3275	58	35	a	a	DET
ejpam-3275	58	36	prime	prime	ADJ
ejpam-3275	58	37	number	number	NOUN
ejpam-3275	58	38	or	or	CCONJ
ejpam-3275	58	39	p	p	NOUN
ejpam-3275	58	40	=	=	NOUN
ejpam-3275	58	41	2q	2q	NUM
ejpam-3275	58	42	for	for	ADP
ejpam-3275	58	43	some	some	DET
ejpam-3275	58	44	prime	prime	ADJ
ejpam-3275	58	45	numbers	number	NOUN
ejpam-3275	58	46	q.	q.	PROPN
ejpam-3275	58	47	example	example	NOUN
ejpam-3275	58	48	1	1	NUM
ejpam-3275	58	49	obtains	obtain	VERB
ejpam-3275	58	50	that	that	SCONJ
ejpam-3275	58	51	4z	4z	NOUN
ejpam-3275	58	52	is	be	AUX
ejpam-3275	58	53	α	α	NOUN
ejpam-3275	58	54	-	-	ADJ
ejpam-3275	58	55	prime	prime	NOUN
ejpam-3275	58	56	but	but	CCONJ
ejpam-3275	58	57	is	be	AUX
ejpam-3275	58	58	not	not	PART
ejpam-3275	58	59	prime	prime	ADJ
ejpam-3275	58	60	submodule	submodule	NOUN
ejpam-3275	58	61	of	of	ADP
ejpam-3275	58	62	z.	z.	PROPN
ejpam-3275	59	1	the	the	DET
ejpam-3275	59	2	following	follow	VERB
ejpam-3275	59	3	first	first	ADJ
ejpam-3275	59	4	result	result	NOUN
ejpam-3275	59	5	gives	give	VERB
ejpam-3275	59	6	the	the	DET
ejpam-3275	59	7	characterization	characterization	NOUN
ejpam-3275	59	8	of	of	ADP
ejpam-3275	59	9	α	α	NOUN
ejpam-3275	59	10	-	-	ADJ
ejpam-3275	59	11	prime	prime	ADJ
ejpam-3275	59	12	submodules	submodule	NOUN
ejpam-3275	59	13	.	.	PUNCT
ejpam-3275	60	1	theorem	theorem	NOUN
ejpam-3275	60	2	1	1	NUM
ejpam-3275	60	3	.	.	PUNCT
ejpam-3275	61	1	let	let	VERB
ejpam-3275	61	2	p	p	PRON
ejpam-3275	61	3	be	be	AUX
ejpam-3275	61	4	a	a	DET
ejpam-3275	61	5	proper	proper	ADJ
ejpam-3275	61	6	submodule	submodule	NOUN
ejpam-3275	61	7	of	of	ADP
ejpam-3275	61	8	an	an	DET
ejpam-3275	61	9	r	r	NOUN
ejpam-3275	61	10	-	-	PUNCT
ejpam-3275	61	11	module	module	NOUN
ejpam-3275	61	12	m	m	NOUN
ejpam-3275	61	13	.	.	PUNCT
ejpam-3275	62	1	the	the	DET
ejpam-3275	62	2	following	follow	VERB
ejpam-3275	62	3	statements	statement	NOUN
ejpam-3275	62	4	are	be	AUX
ejpam-3275	62	5	equivalent	equivalent	ADJ
ejpam-3275	62	6	.	.	PUNCT
ejpam-3275	63	1	(	(	PUNCT
ejpam-3275	63	2	i	i	NOUN
ejpam-3275	63	3	)	)	PUNCT
ejpam-3275	63	4	p	p	NOUN
ejpam-3275	63	5	is	be	AUX
ejpam-3275	63	6	an	an	DET
ejpam-3275	63	7	α	α	NOUN
ejpam-3275	63	8	-	-	ADJ
ejpam-3275	63	9	prime	prime	ADJ
ejpam-3275	63	10	submodule	submodule	NOUN
ejpam-3275	63	11	of	of	ADP
ejpam-3275	63	12	m	m	PROPN
ejpam-3275	63	13	.	.	PUNCT
ejpam-3275	64	1	(	(	PUNCT
ejpam-3275	64	2	ii	ii	NOUN
ejpam-3275	64	3	)	)	PUNCT
ejpam-3275	64	4	for	for	ADP
ejpam-3275	64	5	all	all	DET
ejpam-3275	64	6	ideals	ideal	NOUN
ejpam-3275	64	7	i	i	PRON
ejpam-3275	64	8	of	of	ADP
ejpam-3275	64	9	r	r	NOUN
ejpam-3275	64	10	and	and	CCONJ
ejpam-3275	64	11	for	for	ADP
ejpam-3275	64	12	all	all	DET
ejpam-3275	64	13	submodules	submodule	NOUN
ejpam-3275	64	14	n	n	PROPN
ejpam-3275	64	15	of	of	ADP
ejpam-3275	64	16	m	m	PRON
ejpam-3275	64	17	,	,	PUNCT
ejpam-3275	64	18	if	if	SCONJ
ejpam-3275	64	19	iβ(n	iβ(n	VERB
ejpam-3275	64	20	)	)	PUNCT
ejpam-3275	64	21	⊆	⊆	NUM
ejpam-3275	64	22	p	p	NOUN
ejpam-3275	64	23	,	,	PUNCT
ejpam-3275	64	24	then	then	ADV
ejpam-3275	64	25	i	i	PRON
ejpam-3275	64	26	⊆	⊆	NUM
ejpam-3275	64	27	α((p	α((p	NOUN
ejpam-3275	64	28	:	:	PUNCT
ejpam-3275	64	29	m	m	X
ejpam-3275	64	30	)	)	PUNCT
ejpam-3275	64	31	)	)	PUNCT
ejpam-3275	64	32	or	or	CCONJ
ejpam-3275	64	33	n	n	PRON
ejpam-3275	64	34	⊆	⊆	NUM
ejpam-3275	64	35	α(p	α(p	NUM
ejpam-3275	64	36	)	)	PUNCT
ejpam-3275	64	37	.	.	PUNCT
ejpam-3275	65	1	(	(	PUNCT
ejpam-3275	65	2	iii	iii	X
ejpam-3275	65	3	)	)	PUNCT
ejpam-3275	65	4	for	for	ADP
ejpam-3275	65	5	all	all	DET
ejpam-3275	65	6	a	a	DET
ejpam-3275	65	7	∈	∈	NOUN
ejpam-3275	65	8	r	r	NOUN
ejpam-3275	65	9	and	and	CCONJ
ejpam-3275	65	10	for	for	ADP
ejpam-3275	65	11	all	all	DET
ejpam-3275	65	12	submodules	submodule	NOUN
ejpam-3275	65	13	n	n	PROPN
ejpam-3275	65	14	of	of	ADP
ejpam-3275	65	15	m	m	PRON
ejpam-3275	65	16	,	,	PUNCT
ejpam-3275	65	17	if	if	SCONJ
ejpam-3275	65	18	aβ(n	aβ(n	VERB
ejpam-3275	65	19	)	)	PUNCT
ejpam-3275	65	20	⊆	⊆	NUM
ejpam-3275	65	21	p	p	NOUN
ejpam-3275	65	22	,	,	PUNCT
ejpam-3275	65	23	then	then	ADV
ejpam-3275	65	24	a	a	DET
ejpam-3275	65	25	∈	∈	PROPN
ejpam-3275	65	26	α((p	α((p	NOUN
ejpam-3275	65	27	:	:	PUNCT
ejpam-3275	66	1	m	m	X
ejpam-3275	66	2	)	)	PUNCT
ejpam-3275	66	3	)	)	PUNCT
ejpam-3275	66	4	or	or	CCONJ
ejpam-3275	66	5	n	n	PRON
ejpam-3275	66	6	⊆	⊆	NUM
ejpam-3275	66	7	α(p	α(p	NUM
ejpam-3275	66	8	)	)	PUNCT
ejpam-3275	66	9	.	.	PUNCT
ejpam-3275	67	1	t.	t.	PROPN
ejpam-3275	67	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	67	3	/	/	SYM
ejpam-3275	67	4	eur	eur	PROPN
ejpam-3275	67	5	.	.	PUNCT
ejpam-3275	68	1	j.	j.	PROPN
ejpam-3275	68	2	pure	pure	PROPN
ejpam-3275	68	3	appl	appl	PROPN
ejpam-3275	68	4	.	.	PROPN
ejpam-3275	68	5	math	math	PROPN
ejpam-3275	68	6	,	,	PUNCT
ejpam-3275	68	7	11	11	NUM
ejpam-3275	68	8	(	(	PUNCT
ejpam-3275	68	9	3	3	NUM
ejpam-3275	68	10	)	)	PUNCT
ejpam-3275	68	11	(	(	PUNCT
ejpam-3275	68	12	2018	2018	NUM
ejpam-3275	68	13	)	)	PUNCT
ejpam-3275	68	14	,	,	PUNCT
ejpam-3275	68	15	730	730	NUM
ejpam-3275	68	16	-	-	SYM
ejpam-3275	68	17	739	739	NUM
ejpam-3275	68	18	732	732	NUM
ejpam-3275	68	19	(	(	PUNCT
ejpam-3275	68	20	iv	iv	NOUN
ejpam-3275	68	21	)	)	PUNCT
ejpam-3275	68	22	for	for	ADP
ejpam-3275	68	23	all	all	DET
ejpam-3275	68	24	ideals	ideal	NOUN
ejpam-3275	68	25	i	i	PRON
ejpam-3275	68	26	of	of	ADP
ejpam-3275	68	27	r	r	NOUN
ejpam-3275	68	28	and	and	CCONJ
ejpam-3275	68	29	for	for	ADP
ejpam-3275	68	30	all	all	DET
ejpam-3275	68	31	m	m	NOUN
ejpam-3275	68	32	∈m	∈m	NOUN
ejpam-3275	68	33	,	,	PUNCT
ejpam-3275	68	34	if	if	SCONJ
ejpam-3275	68	35	i(m+m	i(m+m	NOUN
ejpam-3275	68	36	)	)	PUNCT
ejpam-3275	68	37	⊆	⊆	NUM
ejpam-3275	68	38	p	p	NOUN
ejpam-3275	68	39	,	,	PUNCT
ejpam-3275	68	40	then	then	ADV
ejpam-3275	68	41	i	i	PRON
ejpam-3275	68	42	⊆	⊆	NUM
ejpam-3275	68	43	α((p	α((p	NOUN
ejpam-3275	68	44	:	:	PUNCT
ejpam-3275	68	45	m	m	X
ejpam-3275	68	46	)	)	PUNCT
ejpam-3275	68	47	)	)	PUNCT
ejpam-3275	68	48	or	or	CCONJ
ejpam-3275	68	49	m	m	PROPN
ejpam-3275	68	50	∈	∈	PROPN
ejpam-3275	68	51	α(p	α(p	PROPN
ejpam-3275	68	52	)	)	PUNCT
ejpam-3275	68	53	.	.	PUNCT
ejpam-3275	69	1	(	(	PUNCT
ejpam-3275	69	2	v	v	NOUN
ejpam-3275	69	3	)	)	PUNCT
ejpam-3275	69	4	for	for	ADP
ejpam-3275	69	5	all	all	DET
ejpam-3275	69	6	a	a	DET
ejpam-3275	69	7	∈	∈	NOUN
ejpam-3275	69	8	r	r	NOUN
ejpam-3275	69	9	and	and	CCONJ
ejpam-3275	69	10	for	for	ADP
ejpam-3275	69	11	all	all	DET
ejpam-3275	69	12	m	m	NOUN
ejpam-3275	69	13	∈m	∈m	NOUN
ejpam-3275	69	14	,	,	PUNCT
ejpam-3275	69	15	if	if	SCONJ
ejpam-3275	69	16	ar(m+m	ar(m+m	PROPN
ejpam-3275	69	17	)	)	PUNCT
ejpam-3275	69	18	⊆	⊆	NUM
ejpam-3275	69	19	p	p	NOUN
ejpam-3275	69	20	,	,	PUNCT
ejpam-3275	69	21	then	then	ADV
ejpam-3275	69	22	a	a	DET
ejpam-3275	69	23	∈	∈	PROPN
ejpam-3275	69	24	α((p	α((p	NOUN
ejpam-3275	69	25	:	:	PUNCT
ejpam-3275	69	26	m	m	X
ejpam-3275	69	27	)	)	PUNCT
ejpam-3275	69	28	)	)	PUNCT
ejpam-3275	69	29	or	or	CCONJ
ejpam-3275	69	30	m	m	PROPN
ejpam-3275	69	31	∈	∈	PROPN
ejpam-3275	69	32	α(p	α(p	PROPN
ejpam-3275	69	33	)	)	PUNCT
ejpam-3275	69	34	.	.	PUNCT
ejpam-3275	70	1	(	(	PUNCT
ejpam-3275	70	2	vi	vi	X
ejpam-3275	70	3	)	)	PUNCT
ejpam-3275	70	4	for	for	ADP
ejpam-3275	70	5	all	all	DET
ejpam-3275	70	6	m	m	NOUN
ejpam-3275	70	7	∈m	∈m	NOUN
ejpam-3275	70	8	,	,	PUNCT
ejpam-3275	70	9	if	if	SCONJ
ejpam-3275	70	10	m+m	m+m	PROPN
ejpam-3275	70	11	/∈	/∈	PUNCT
ejpam-3275	71	1	p	p	NOUN
ejpam-3275	71	2	,	,	PUNCT
ejpam-3275	71	3	then	then	ADV
ejpam-3275	71	4	α((p	α((p	PUNCT
ejpam-3275	71	5	:	:	PUNCT
ejpam-3275	71	6	m	m	X
ejpam-3275	71	7	)	)	PUNCT
ejpam-3275	71	8	)	)	PUNCT
ejpam-3275	72	1	=	=	PUNCT
ejpam-3275	72	2	α((p	α((p	NOUN
ejpam-3275	72	3	:	:	PUNCT
ejpam-3275	72	4	m	m	X
ejpam-3275	72	5	)	)	PUNCT
ejpam-3275	72	6	)	)	PUNCT
ejpam-3275	72	7	.	.	PUNCT
ejpam-3275	73	1	proof	proof	NOUN
ejpam-3275	73	2	.	.	PUNCT
ejpam-3275	74	1	(	(	PUNCT
ejpam-3275	74	2	i	i	NOUN
ejpam-3275	74	3	)	)	PUNCT
ejpam-3275	74	4	→	→	SYM
ejpam-3275	74	5	(	(	PUNCT
ejpam-3275	74	6	ii	ii	NOUN
ejpam-3275	74	7	)	)	PUNCT
ejpam-3275	74	8	assume	assume	VERB
ejpam-3275	74	9	that	that	SCONJ
ejpam-3275	74	10	p	p	NOUN
ejpam-3275	74	11	is	be	AUX
ejpam-3275	74	12	an	an	DET
ejpam-3275	74	13	α	α	NOUN
ejpam-3275	74	14	-	-	ADJ
ejpam-3275	74	15	prime	prime	ADJ
ejpam-3275	74	16	submodule	submodule	NOUN
ejpam-3275	74	17	of	of	ADP
ejpam-3275	74	18	m	m	PROPN
ejpam-3275	74	19	.	.	PUNCT
ejpam-3275	75	1	let	let	VERB
ejpam-3275	75	2	i	i	PRON
ejpam-3275	75	3	be	be	AUX
ejpam-3275	75	4	an	an	DET
ejpam-3275	75	5	ideal	ideal	NOUN
ejpam-3275	75	6	of	of	ADP
ejpam-3275	75	7	r	r	NOUN
ejpam-3275	75	8	and	and	CCONJ
ejpam-3275	75	9	n	n	CCONJ
ejpam-3275	75	10	be	be	VERB
ejpam-3275	75	11	a	a	DET
ejpam-3275	75	12	submodule	submodule	NOUN
ejpam-3275	75	13	of	of	ADP
ejpam-3275	75	14	m	m	PRON
ejpam-3275	75	15	such	such	ADJ
ejpam-3275	75	16	that	that	PRON
ejpam-3275	75	17	iβ(n	iβ(n	NOUN
ejpam-3275	75	18	)	)	PUNCT
ejpam-3275	75	19	⊆	⊆	NUM
ejpam-3275	75	20	p	p	NOUN
ejpam-3275	75	21	and	and	CCONJ
ejpam-3275	75	22	n	n	PROPN
ejpam-3275	75	23	*	*	PROPN
ejpam-3275	75	24	α(p	α(p	PROPN
ejpam-3275	75	25	)	)	PUNCT
ejpam-3275	75	26	.	.	PUNCT
ejpam-3275	76	1	to	to	PART
ejpam-3275	76	2	show	show	VERB
ejpam-3275	76	3	that	that	SCONJ
ejpam-3275	76	4	i	i	PRON
ejpam-3275	76	5	⊆	⊆	NUM
ejpam-3275	76	6	α((p	α((p	NOUN
ejpam-3275	76	7	:	:	PUNCT
ejpam-3275	76	8	m	m	X
ejpam-3275	76	9	)	)	PUNCT
ejpam-3275	76	10	)	)	PUNCT
ejpam-3275	76	11	,	,	PUNCT
ejpam-3275	76	12	let	let	VERB
ejpam-3275	76	13	r	r	PRON
ejpam-3275	76	14	∈	∈	PROPN
ejpam-3275	76	15	i	i	PRON
ejpam-3275	76	16	and	and	CCONJ
ejpam-3275	76	17	n	n	PRON
ejpam-3275	76	18	∈	∈	PROPN
ejpam-3275	76	19	n	n	AUX
ejpam-3275	76	20	be	be	AUX
ejpam-3275	76	21	such	such	ADJ
ejpam-3275	76	22	that	that	SCONJ
ejpam-3275	76	23	n	n	NUM
ejpam-3275	76	24	/∈	/∈	PUNCT
ejpam-3275	76	25	α(p	α(p	PROPN
ejpam-3275	76	26	)	)	PUNCT
ejpam-3275	76	27	.	.	PUNCT
ejpam-3275	77	1	then	then	ADV
ejpam-3275	77	2	n	n	PROPN
ejpam-3275	77	3	+	+	CCONJ
ejpam-3275	77	4	n	n	NUM
ejpam-3275	77	5	/∈	/∈	PUNCT
ejpam-3275	77	6	p	p	NOUN
ejpam-3275	77	7	and	and	CCONJ
ejpam-3275	77	8	n	n	PROPN
ejpam-3275	77	9	+	+	CCONJ
ejpam-3275	77	10	n	n	CCONJ
ejpam-3275	77	11	∈	∈	PROPN
ejpam-3275	77	12	β(n	β(n	NUM
ejpam-3275	77	13	)	)	PUNCT
ejpam-3275	77	14	.	.	PUNCT
ejpam-3275	78	1	this	this	PRON
ejpam-3275	78	2	implies	imply	VERB
ejpam-3275	78	3	that	that	SCONJ
ejpam-3275	78	4	r(n	r(n	PROPN
ejpam-3275	78	5	+	+	CCONJ
ejpam-3275	78	6	n	n	CCONJ
ejpam-3275	78	7	)	)	PUNCT
ejpam-3275	78	8	∈	∈	PROPN
ejpam-3275	78	9	p	p	NOUN
ejpam-3275	78	10	.	.	PUNCT
ejpam-3275	79	1	since	since	SCONJ
ejpam-3275	79	2	p	p	NOUN
ejpam-3275	79	3	is	be	AUX
ejpam-3275	79	4	an	an	DET
ejpam-3275	79	5	α	α	NOUN
ejpam-3275	79	6	-	-	ADJ
ejpam-3275	79	7	prime	prime	ADJ
ejpam-3275	79	8	submodule	submodule	NOUN
ejpam-3275	79	9	of	of	ADP
ejpam-3275	79	10	m	m	PROPN
ejpam-3275	79	11	and	and	CCONJ
ejpam-3275	79	12	n+	n+	PUNCT
ejpam-3275	79	13	n	n	ADV
ejpam-3275	79	14	/∈	/∈	PUNCT
ejpam-3275	80	1	p	p	NOUN
ejpam-3275	80	2	,	,	PUNCT
ejpam-3275	80	3	r	r	NOUN
ejpam-3275	80	4	+	+	CCONJ
ejpam-3275	80	5	r	r	NOUN
ejpam-3275	80	6	∈	∈	NOUN
ejpam-3275	80	7	(	(	PUNCT
ejpam-3275	80	8	p	p	X
ejpam-3275	80	9	:	:	PUNCT
ejpam-3275	80	10	m	m	PROPN
ejpam-3275	80	11	)	)	PUNCT
ejpam-3275	80	12	.	.	PUNCT
ejpam-3275	81	1	hence	hence	ADV
ejpam-3275	81	2	i	i	PRON
ejpam-3275	81	3	⊆	⊆	NUM
ejpam-3275	81	4	α((p	α((p	NOUN
ejpam-3275	81	5	:	:	PUNCT
ejpam-3275	81	6	m	m	X
ejpam-3275	81	7	)	)	PUNCT
ejpam-3275	81	8	)	)	PUNCT
ejpam-3275	81	9	.	.	PUNCT
ejpam-3275	82	1	(	(	PUNCT
ejpam-3275	82	2	ii)→	ii)→	NOUN
ejpam-3275	82	3	(	(	PUNCT
ejpam-3275	82	4	iii	iii	NOUN
ejpam-3275	82	5	)	)	PUNCT
ejpam-3275	82	6	assume	assume	VERB
ejpam-3275	82	7	that	that	SCONJ
ejpam-3275	82	8	(	(	PUNCT
ejpam-3275	82	9	ii	ii	NOUN
ejpam-3275	82	10	)	)	PUNCT
ejpam-3275	82	11	holds	hold	VERB
ejpam-3275	82	12	.	.	PUNCT
ejpam-3275	83	1	let	let	VERB
ejpam-3275	83	2	a	a	DET
ejpam-3275	83	3	∈	∈	NOUN
ejpam-3275	83	4	r	r	NOUN
ejpam-3275	83	5	and	and	CCONJ
ejpam-3275	83	6	n	n	CCONJ
ejpam-3275	83	7	be	be	VERB
ejpam-3275	83	8	a	a	DET
ejpam-3275	83	9	submodule	submodule	NOUN
ejpam-3275	83	10	of	of	ADP
ejpam-3275	83	11	m	m	PRON
ejpam-3275	83	12	such	such	ADJ
ejpam-3275	83	13	that	that	PRON
ejpam-3275	83	14	aβ(n	aβ(n	NUM
ejpam-3275	83	15	)	)	PUNCT
ejpam-3275	83	16	⊆	⊆	NUM
ejpam-3275	83	17	p	p	NOUN
ejpam-3275	83	18	.	.	PUNCT
ejpam-3275	84	1	then	then	ADV
ejpam-3275	84	2	(	(	PUNCT
ejpam-3275	84	3	ra)β(n	ra)β(n	NOUN
ejpam-3275	84	4	)	)	PUNCT
ejpam-3275	84	5	=	=	SYM
ejpam-3275	84	6	r(aβ(n	r(aβ(n	NOUN
ejpam-3275	84	7	)	)	PUNCT
ejpam-3275	84	8	)	)	PUNCT
ejpam-3275	85	1	⊆	⊆	NUM
ejpam-3275	85	2	rp	rp	NOUN
ejpam-3275	85	3	⊆	⊆	NUM
ejpam-3275	85	4	p	p	NOUN
ejpam-3275	85	5	.	.	PUNCT
ejpam-3275	86	1	by	by	ADP
ejpam-3275	86	2	(	(	PUNCT
ejpam-3275	86	3	ii	ii	NOUN
ejpam-3275	86	4	)	)	PUNCT
ejpam-3275	86	5	,	,	PUNCT
ejpam-3275	86	6	we	we	PRON
ejpam-3275	86	7	have	have	VERB
ejpam-3275	86	8	ra	ra	PROPN
ejpam-3275	86	9	⊆	⊆	NUM
ejpam-3275	86	10	α((p	α((p	NOUN
ejpam-3275	86	11	:	:	PUNCT
ejpam-3275	86	12	m	m	X
ejpam-3275	86	13	)	)	PUNCT
ejpam-3275	86	14	)	)	PUNCT
ejpam-3275	86	15	or	or	CCONJ
ejpam-3275	86	16	n	n	PRON
ejpam-3275	86	17	⊆	⊆	NUM
ejpam-3275	86	18	α(p	α(p	NUM
ejpam-3275	86	19	)	)	PUNCT
ejpam-3275	86	20	.	.	PUNCT
ejpam-3275	87	1	therefore	therefore	ADV
ejpam-3275	87	2	a	a	DET
ejpam-3275	87	3	∈	∈	PROPN
ejpam-3275	87	4	α((p	α((p	NOUN
ejpam-3275	87	5	:	:	PUNCT
ejpam-3275	87	6	m	m	X
ejpam-3275	87	7	)	)	PUNCT
ejpam-3275	87	8	)	)	PUNCT
ejpam-3275	87	9	or	or	CCONJ
ejpam-3275	87	10	n	n	PRON
ejpam-3275	87	11	⊆	⊆	NUM
ejpam-3275	87	12	α(p	α(p	NUM
ejpam-3275	87	13	)	)	PUNCT
ejpam-3275	87	14	.	.	PUNCT
ejpam-3275	88	1	(	(	PUNCT
ejpam-3275	88	2	iii	iii	NOUN
ejpam-3275	88	3	)	)	PUNCT
ejpam-3275	88	4	→	→	SYM
ejpam-3275	88	5	(	(	PUNCT
ejpam-3275	88	6	iv	iv	X
ejpam-3275	88	7	)	)	PUNCT
ejpam-3275	88	8	assume	assume	VERB
ejpam-3275	88	9	that	that	SCONJ
ejpam-3275	88	10	(	(	PUNCT
ejpam-3275	88	11	iii	iii	NOUN
ejpam-3275	88	12	)	)	PUNCT
ejpam-3275	88	13	holds	hold	VERB
ejpam-3275	88	14	.	.	PUNCT
ejpam-3275	89	1	to	to	PART
ejpam-3275	89	2	prove	prove	VERB
ejpam-3275	89	3	that	that	SCONJ
ejpam-3275	89	4	(	(	PUNCT
ejpam-3275	89	5	iv	iv	X
ejpam-3275	89	6	)	)	PUNCT
ejpam-3275	89	7	holds	hold	NOUN
ejpam-3275	89	8	,	,	PUNCT
ejpam-3275	89	9	let	let	VERB
ejpam-3275	89	10	i	i	PRON
ejpam-3275	89	11	be	be	AUX
ejpam-3275	89	12	an	an	DET
ejpam-3275	89	13	ideal	ideal	NOUN
ejpam-3275	89	14	of	of	ADP
ejpam-3275	89	15	r	r	NOUN
ejpam-3275	89	16	and	and	CCONJ
ejpam-3275	89	17	m	m	PROPN
ejpam-3275	89	18	∈	∈	NOUN
ejpam-3275	89	19	m	m	VERB
ejpam-3275	89	20	such	such	ADJ
ejpam-3275	89	21	that	that	DET
ejpam-3275	89	22	i(m+m	i(m+m	NOUN
ejpam-3275	89	23	)	)	PUNCT
ejpam-3275	89	24	⊆	⊆	NUM
ejpam-3275	89	25	p	p	NOUN
ejpam-3275	89	26	and	and	CCONJ
ejpam-3275	89	27	m	m	PROPN
ejpam-3275	89	28	/∈	/∈	PUNCT
ejpam-3275	90	1	α(p	α(p	PROPN
ejpam-3275	90	2	)	)	PUNCT
ejpam-3275	90	3	.	.	PUNCT
ejpam-3275	91	1	let	let	VERB
ejpam-3275	91	2	a	a	DET
ejpam-3275	91	3	∈	∈	PROPN
ejpam-3275	91	4	i.	i.	NOUN
ejpam-3275	91	5	then	then	ADV
ejpam-3275	91	6	aβ(rm	aβ(rm	NOUN
ejpam-3275	91	7	)	)	PUNCT
ejpam-3275	92	1	⊆	⊆	NUM
ejpam-3275	92	2	p	p	NOUN
ejpam-3275	92	3	.	.	PUNCT
ejpam-3275	93	1	by	by	ADP
ejpam-3275	93	2	(	(	PUNCT
ejpam-3275	93	3	iii	iii	NOUN
ejpam-3275	93	4	)	)	PUNCT
ejpam-3275	93	5	and	and	CCONJ
ejpam-3275	93	6	m	m	PROPN
ejpam-3275	93	7	/∈	/∈	PUNCT
ejpam-3275	93	8	α(p	α(p	PROPN
ejpam-3275	93	9	)	)	PUNCT
ejpam-3275	93	10	,	,	PUNCT
ejpam-3275	93	11	a	a	DET
ejpam-3275	93	12	∈	∈	PROPN
ejpam-3275	93	13	α((p	α((p	NOUN
ejpam-3275	93	14	:	:	PUNCT
ejpam-3275	93	15	m	m	X
ejpam-3275	93	16	)	)	PUNCT
ejpam-3275	93	17	)	)	PUNCT
ejpam-3275	93	18	.	.	PUNCT
ejpam-3275	94	1	hence	hence	ADV
ejpam-3275	94	2	i	i	PRON
ejpam-3275	94	3	⊆	⊆	NUM
ejpam-3275	94	4	α((p	α((p	NOUN
ejpam-3275	94	5	:	:	PUNCT
ejpam-3275	94	6	m	m	X
ejpam-3275	94	7	)	)	PUNCT
ejpam-3275	94	8	)	)	PUNCT
ejpam-3275	94	9	.	.	PUNCT
ejpam-3275	95	1	(	(	PUNCT
ejpam-3275	95	2	iv)→	iv)→	ADP
ejpam-3275	95	3	(	(	PUNCT
ejpam-3275	95	4	v	v	NOUN
ejpam-3275	95	5	)	)	PUNCT
ejpam-3275	95	6	,	,	PUNCT
ejpam-3275	95	7	(	(	PUNCT
ejpam-3275	95	8	v)→	v)→	X
ejpam-3275	95	9	(	(	PUNCT
ejpam-3275	95	10	i	i	NOUN
ejpam-3275	95	11	)	)	PUNCT
ejpam-3275	95	12	and	and	CCONJ
ejpam-3275	95	13	(	(	PUNCT
ejpam-3275	95	14	vi)→	vi)→	X
ejpam-3275	95	15	(	(	PUNCT
ejpam-3275	95	16	i	i	NOUN
ejpam-3275	95	17	)	)	PUNCT
ejpam-3275	95	18	are	be	AUX
ejpam-3275	95	19	obvious	obvious	ADJ
ejpam-3275	95	20	.	.	PUNCT
ejpam-3275	96	1	(	(	PUNCT
ejpam-3275	96	2	i	i	NOUN
ejpam-3275	96	3	)	)	PUNCT
ejpam-3275	96	4	→	→	SYM
ejpam-3275	96	5	(	(	PUNCT
ejpam-3275	96	6	vi	vi	NOUN
ejpam-3275	96	7	)	)	PUNCT
ejpam-3275	96	8	assume	assume	VERB
ejpam-3275	96	9	that	that	SCONJ
ejpam-3275	96	10	p	p	NOUN
ejpam-3275	96	11	is	be	AUX
ejpam-3275	96	12	an	an	DET
ejpam-3275	96	13	α	α	NOUN
ejpam-3275	96	14	-	-	ADJ
ejpam-3275	96	15	prime	prime	ADJ
ejpam-3275	96	16	submodule	submodule	NOUN
ejpam-3275	96	17	of	of	ADP
ejpam-3275	96	18	m	m	PROPN
ejpam-3275	96	19	.	.	PUNCT
ejpam-3275	97	1	let	let	VERB
ejpam-3275	97	2	m	m	PRON
ejpam-3275	97	3	∈	∈	VERB
ejpam-3275	97	4	m	m	AUX
ejpam-3275	97	5	be	be	VERB
ejpam-3275	97	6	such	such	ADJ
ejpam-3275	97	7	that	that	SCONJ
ejpam-3275	97	8	m	m	VERB
ejpam-3275	97	9	+	+	NOUN
ejpam-3275	97	10	m	m	VERB
ejpam-3275	97	11	/∈	/∈	PUNCT
ejpam-3275	98	1	p	p	X
ejpam-3275	98	2	.	.	PUNCT
ejpam-3275	99	1	it	it	PRON
ejpam-3275	99	2	is	be	AUX
ejpam-3275	99	3	clear	clear	ADJ
ejpam-3275	99	4	that	that	SCONJ
ejpam-3275	99	5	α((p	α((p	NOUN
ejpam-3275	99	6	:	:	PUNCT
ejpam-3275	99	7	m	m	X
ejpam-3275	99	8	)	)	PUNCT
ejpam-3275	99	9	)	)	PUNCT
ejpam-3275	100	1	⊆	⊆	NUM
ejpam-3275	100	2	α((p	α((p	NOUN
ejpam-3275	100	3	:	:	PUNCT
ejpam-3275	100	4	m	m	X
ejpam-3275	100	5	)	)	PUNCT
ejpam-3275	100	6	)	)	PUNCT
ejpam-3275	100	7	.	.	PUNCT
ejpam-3275	101	1	let	let	VERB
ejpam-3275	101	2	r	r	NOUN
ejpam-3275	101	3	∈	∈	PROPN
ejpam-3275	101	4	α((p	α((p	NOUN
ejpam-3275	101	5	:	:	PUNCT
ejpam-3275	101	6	m	m	X
ejpam-3275	101	7	)	)	PUNCT
ejpam-3275	101	8	)	)	PUNCT
ejpam-3275	101	9	.	.	PUNCT
ejpam-3275	102	1	then	then	ADV
ejpam-3275	102	2	r	r	NOUN
ejpam-3275	102	3	+	+	CCONJ
ejpam-3275	102	4	r	r	NOUN
ejpam-3275	102	5	∈	∈	NOUN
ejpam-3275	102	6	(	(	PUNCT
ejpam-3275	102	7	p	p	X
ejpam-3275	102	8	:	:	PUNCT
ejpam-3275	102	9	m	m	PROPN
ejpam-3275	102	10	)	)	PUNCT
ejpam-3275	102	11	.	.	PUNCT
ejpam-3275	103	1	hence	hence	ADV
ejpam-3275	103	2	r(m	r(m	PROPN
ejpam-3275	103	3	+	+	NUM
ejpam-3275	103	4	m	m	VERB
ejpam-3275	103	5	)	)	PUNCT
ejpam-3275	104	1	=	=	SYM
ejpam-3275	104	2	(	(	PUNCT
ejpam-3275	104	3	r	r	NOUN
ejpam-3275	104	4	+	+	NUM
ejpam-3275	104	5	r)m	r)m	NOUN
ejpam-3275	104	6	∈	∈	PROPN
ejpam-3275	104	7	p	p	NOUN
ejpam-3275	104	8	.	.	PUNCT
ejpam-3275	105	1	since	since	SCONJ
ejpam-3275	105	2	p	p	PROPN
ejpam-3275	105	3	is	be	AUX
ejpam-3275	105	4	α	α	NOUN
ejpam-3275	105	5	-	-	ADJ
ejpam-3275	105	6	prime	prime	NOUN
ejpam-3275	105	7	and	and	CCONJ
ejpam-3275	105	8	m	m	PROPN
ejpam-3275	105	9	+	+	NOUN
ejpam-3275	105	10	m	m	VERB
ejpam-3275	105	11	/∈	/∈	ADJ
ejpam-3275	106	1	p	p	NOUN
ejpam-3275	106	2	,	,	PUNCT
ejpam-3275	106	3	r	r	NOUN
ejpam-3275	106	4	+	+	CCONJ
ejpam-3275	106	5	r	r	NOUN
ejpam-3275	106	6	∈	∈	NOUN
ejpam-3275	106	7	(	(	PUNCT
ejpam-3275	106	8	p	p	X
ejpam-3275	106	9	:	:	PUNCT
ejpam-3275	106	10	m	m	PROPN
ejpam-3275	106	11	)	)	PUNCT
ejpam-3275	106	12	.	.	PUNCT
ejpam-3275	107	1	that	that	PRON
ejpam-3275	107	2	is	be	AUX
ejpam-3275	107	3	r	r	PROPN
ejpam-3275	107	4	∈	∈	PROPN
ejpam-3275	107	5	α((p	α((p	NOUN
ejpam-3275	107	6	:	:	PUNCT
ejpam-3275	107	7	m	m	X
ejpam-3275	107	8	)	)	PUNCT
ejpam-3275	107	9	)	)	PUNCT
ejpam-3275	107	10	.	.	PUNCT
ejpam-3275	108	1	therefore	therefore	ADV
ejpam-3275	108	2	α((p	α((p	PUNCT
ejpam-3275	108	3	:	:	PUNCT
ejpam-3275	108	4	m	m	X
ejpam-3275	108	5	)	)	PUNCT
ejpam-3275	108	6	)	)	PUNCT
ejpam-3275	109	1	=	=	PUNCT
ejpam-3275	109	2	α((p	α((p	NOUN
ejpam-3275	109	3	:	:	PUNCT
ejpam-3275	109	4	m	m	X
ejpam-3275	109	5	)	)	PUNCT
ejpam-3275	109	6	)	)	PUNCT
ejpam-3275	109	7	.	.	PUNCT
ejpam-3275	110	1	lemma	lemma	PROPN
ejpam-3275	110	2	1	1	X
ejpam-3275	110	3	.	.	PUNCT
ejpam-3275	111	1	let	let	VERB
ejpam-3275	111	2	φ	φ	NOUN
ejpam-3275	111	3	:	:	PUNCT
ejpam-3275	111	4	m1	m1	PROPN
ejpam-3275	111	5	→	→	SYM
ejpam-3275	111	6	m2	m2	PROPN
ejpam-3275	111	7	be	be	AUX
ejpam-3275	111	8	an	an	DET
ejpam-3275	111	9	r	r	NOUN
ejpam-3275	111	10	-	-	PUNCT
ejpam-3275	111	11	module	module	NOUN
ejpam-3275	111	12	homomorphism	homomorphism	NOUN
ejpam-3275	111	13	,	,	PUNCT
ejpam-3275	111	14	p	p	NOUN
ejpam-3275	111	15	be	be	VERB
ejpam-3275	111	16	a	a	DET
ejpam-3275	111	17	submodule	submodule	NOUN
ejpam-3275	111	18	of	of	ADP
ejpam-3275	111	19	m1	m1	PROPN
ejpam-3275	111	20	and	and	CCONJ
ejpam-3275	111	21	k	k	PROPN
ejpam-3275	111	22	be	be	AUX
ejpam-3275	111	23	a	a	DET
ejpam-3275	111	24	submodule	submodule	NOUN
ejpam-3275	111	25	of	of	ADP
ejpam-3275	111	26	m2	m2	PROPN
ejpam-3275	111	27	.	.	PUNCT
ejpam-3275	112	1	then	then	ADV
ejpam-3275	112	2	(	(	PUNCT
ejpam-3275	112	3	i	i	NOUN
ejpam-3275	112	4	)	)	PUNCT
ejpam-3275	112	5	if	if	SCONJ
ejpam-3275	112	6	φ	φ	PROPN
ejpam-3275	112	7	is	be	AUX
ejpam-3275	112	8	an	an	DET
ejpam-3275	112	9	epimorphism	epimorphism	NOUN
ejpam-3275	112	10	and	and	CCONJ
ejpam-3275	112	11	r	r	NOUN
ejpam-3275	112	12	+	+	CCONJ
ejpam-3275	112	13	r	r	NOUN
ejpam-3275	112	14	∈	∈	NOUN
ejpam-3275	112	15	(	(	PUNCT
ejpam-3275	112	16	p	p	NOUN
ejpam-3275	112	17	:	:	PUNCT
ejpam-3275	112	18	m1	m1	PROPN
ejpam-3275	112	19	)	)	PUNCT
ejpam-3275	112	20	,	,	PUNCT
ejpam-3275	112	21	then	then	ADV
ejpam-3275	112	22	r	r	NOUN
ejpam-3275	112	23	+	+	CCONJ
ejpam-3275	112	24	r	r	NOUN
ejpam-3275	112	25	∈	∈	NOUN
ejpam-3275	112	26	(	(	PUNCT
ejpam-3275	112	27	φ(p	φ(p	PROPN
ejpam-3275	112	28	)	)	PUNCT
ejpam-3275	112	29	:	:	PUNCT
ejpam-3275	112	30	m2	m2	PROPN
ejpam-3275	112	31	)	)	PUNCT
ejpam-3275	112	32	.	.	PUNCT
ejpam-3275	113	1	(	(	PUNCT
ejpam-3275	113	2	ii	ii	NOUN
ejpam-3275	113	3	)	)	PUNCT
ejpam-3275	113	4	if	if	SCONJ
ejpam-3275	113	5	r	r	NOUN
ejpam-3275	113	6	+	+	NOUN
ejpam-3275	113	7	r	r	NOUN
ejpam-3275	113	8	∈	∈	NOUN
ejpam-3275	113	9	(	(	PUNCT
ejpam-3275	113	10	k	k	NOUN
ejpam-3275	113	11	:	:	PUNCT
ejpam-3275	113	12	m2	m2	PROPN
ejpam-3275	113	13	)	)	PUNCT
ejpam-3275	113	14	,	,	PUNCT
ejpam-3275	113	15	then	then	ADV
ejpam-3275	113	16	r	r	NOUN
ejpam-3275	113	17	+	+	CCONJ
ejpam-3275	113	18	r	r	NOUN
ejpam-3275	113	19	∈	∈	PROPN
ejpam-3275	113	20	(	(	PUNCT
ejpam-3275	113	21	φ−1(k	φ−1(k	PROPN
ejpam-3275	113	22	)	)	PUNCT
ejpam-3275	113	23	:	:	PUNCT
ejpam-3275	113	24	m1	m1	NOUN
ejpam-3275	113	25	)	)	PUNCT
ejpam-3275	113	26	.	.	PUNCT
ejpam-3275	114	1	proof	proof	NOUN
ejpam-3275	114	2	.	.	PUNCT
ejpam-3275	115	1	(	(	PUNCT
ejpam-3275	115	2	i	i	NOUN
ejpam-3275	115	3	)	)	PUNCT
ejpam-3275	115	4	assume	assume	VERB
ejpam-3275	115	5	that	that	SCONJ
ejpam-3275	115	6	φ	φ	PROPN
ejpam-3275	115	7	is	be	AUX
ejpam-3275	115	8	an	an	DET
ejpam-3275	115	9	epimorphism	epimorphism	NOUN
ejpam-3275	115	10	and	and	CCONJ
ejpam-3275	115	11	(	(	PUNCT
ejpam-3275	115	12	r	r	NOUN
ejpam-3275	115	13	+	+	CCONJ
ejpam-3275	115	14	r)m1	r)m1	PROPN
ejpam-3275	115	15	⊆	⊆	NUM
ejpam-3275	115	16	p	p	NOUN
ejpam-3275	115	17	.	.	PUNCT
ejpam-3275	116	1	let	let	VERB
ejpam-3275	116	2	m2	m2	PROPN
ejpam-3275	116	3	∈m2	∈m2	PROPN
ejpam-3275	116	4	.	.	PUNCT
ejpam-3275	117	1	then	then	ADV
ejpam-3275	117	2	φ(m1	φ(m1	VERB
ejpam-3275	117	3	)	)	PUNCT
ejpam-3275	118	1	=	=	SYM
ejpam-3275	118	2	m2	m2	PROPN
ejpam-3275	118	3	for	for	ADP
ejpam-3275	118	4	some	some	DET
ejpam-3275	118	5	m1	m1	PROPN
ejpam-3275	118	6	∈	∈	PROPN
ejpam-3275	118	7	m1	m1	NOUN
ejpam-3275	118	8	.	.	PUNCT
ejpam-3275	119	1	thus	thus	ADV
ejpam-3275	119	2	(	(	PUNCT
ejpam-3275	119	3	r	r	NOUN
ejpam-3275	119	4	+	+	CCONJ
ejpam-3275	119	5	r)m1	r)m1	PROPN
ejpam-3275	119	6	∈	∈	PROPN
ejpam-3275	119	7	p	p	NOUN
ejpam-3275	119	8	.	.	PUNCT
ejpam-3275	120	1	this	this	PRON
ejpam-3275	120	2	implies	imply	VERB
ejpam-3275	120	3	that	that	SCONJ
ejpam-3275	120	4	(	(	PUNCT
ejpam-3275	120	5	r	r	NOUN
ejpam-3275	120	6	+	+	NUM
ejpam-3275	120	7	r)m2	r)m2	PROPN
ejpam-3275	120	8	=	=	PUNCT
ejpam-3275	120	9	(	(	PUNCT
ejpam-3275	120	10	r	r	NOUN
ejpam-3275	120	11	+	+	NOUN
ejpam-3275	120	12	r)φ(m1	r)φ(m1	NOUN
ejpam-3275	120	13	)	)	PUNCT
ejpam-3275	120	14	∈	∈	PROPN
ejpam-3275	120	15	φ(p	φ(p	PROPN
ejpam-3275	120	16	)	)	PUNCT
ejpam-3275	120	17	.	.	PUNCT
ejpam-3275	121	1	that	that	PRON
ejpam-3275	121	2	is	be	AUX
ejpam-3275	121	3	r	r	NOUN
ejpam-3275	121	4	+	+	CCONJ
ejpam-3275	121	5	r	r	NOUN
ejpam-3275	121	6	∈	∈	NOUN
ejpam-3275	121	7	(	(	PUNCT
ejpam-3275	121	8	φ(p	φ(p	PROPN
ejpam-3275	121	9	)	)	PUNCT
ejpam-3275	121	10	:	:	PUNCT
ejpam-3275	121	11	m2	m2	PROPN
ejpam-3275	121	12	)	)	PUNCT
ejpam-3275	121	13	.	.	PUNCT
ejpam-3275	122	1	(	(	PUNCT
ejpam-3275	122	2	ii	ii	NOUN
ejpam-3275	122	3	)	)	PUNCT
ejpam-3275	122	4	assume	assume	VERB
ejpam-3275	122	5	that	that	SCONJ
ejpam-3275	122	6	(	(	PUNCT
ejpam-3275	122	7	r+r)m2	r+r)m2	PROPN
ejpam-3275	122	8	⊆	⊆	NUM
ejpam-3275	122	9	k.	k.	NOUN
ejpam-3275	122	10	let	let	VERB
ejpam-3275	122	11	m1	m1	PROPN
ejpam-3275	122	12	∈m1	∈m1	PROPN
ejpam-3275	122	13	.	.	PUNCT
ejpam-3275	123	1	then	then	ADV
ejpam-3275	123	2	φ((r+r)m1	φ((r+r)m1	NUM
ejpam-3275	123	3	)	)	PUNCT
ejpam-3275	123	4	=	=	NOUN
ejpam-3275	124	1	(	(	PUNCT
ejpam-3275	124	2	r+r)φ(m1	r+r)φ(m1	NOUN
ejpam-3275	124	3	)	)	PUNCT
ejpam-3275	124	4	∈	∈	PROPN
ejpam-3275	124	5	k.	k.	NOUN
ejpam-3275	125	1	hence	hence	ADV
ejpam-3275	125	2	(	(	PUNCT
ejpam-3275	125	3	r	r	NOUN
ejpam-3275	125	4	+	+	CCONJ
ejpam-3275	125	5	r)m1	r)m1	PROPN
ejpam-3275	125	6	∈	∈	PROPN
ejpam-3275	125	7	φ−1(k	φ−1(k	NOUN
ejpam-3275	125	8	)	)	PUNCT
ejpam-3275	125	9	.	.	PUNCT
ejpam-3275	126	1	therefore	therefore	ADV
ejpam-3275	126	2	r	r	NOUN
ejpam-3275	126	3	+	+	CCONJ
ejpam-3275	126	4	r	r	NOUN
ejpam-3275	126	5	∈	∈	PROPN
ejpam-3275	126	6	(	(	PUNCT
ejpam-3275	126	7	φ−1(k	φ−1(k	PROPN
ejpam-3275	126	8	)	)	PUNCT
ejpam-3275	126	9	:	:	PUNCT
ejpam-3275	126	10	m1	m1	NOUN
ejpam-3275	126	11	)	)	PUNCT
ejpam-3275	126	12	.	.	PUNCT
ejpam-3275	127	1	proposition	proposition	NOUN
ejpam-3275	127	2	1	1	NUM
ejpam-3275	127	3	.	.	PUNCT
ejpam-3275	128	1	let	let	VERB
ejpam-3275	128	2	φ	φ	NOUN
ejpam-3275	128	3	:	:	PUNCT
ejpam-3275	128	4	m1	m1	PROPN
ejpam-3275	128	5	→m2	→m2	PART
ejpam-3275	128	6	be	be	AUX
ejpam-3275	128	7	an	an	DET
ejpam-3275	128	8	r	r	NOUN
ejpam-3275	128	9	-	-	PUNCT
ejpam-3275	128	10	module	module	NOUN
ejpam-3275	128	11	homomorphism	homomorphism	NOUN
ejpam-3275	128	12	.	.	PUNCT
ejpam-3275	129	1	then	then	ADV
ejpam-3275	129	2	(	(	PUNCT
ejpam-3275	129	3	i	i	NOUN
ejpam-3275	129	4	)	)	PUNCT
ejpam-3275	129	5	if	if	SCONJ
ejpam-3275	129	6	φ	φ	PROPN
ejpam-3275	129	7	is	be	AUX
ejpam-3275	129	8	an	an	DET
ejpam-3275	129	9	epimorphism	epimorphism	NOUN
ejpam-3275	129	10	and	and	CCONJ
ejpam-3275	129	11	p	p	NOUN
ejpam-3275	129	12	is	be	AUX
ejpam-3275	129	13	an	an	DET
ejpam-3275	129	14	α	α	NOUN
ejpam-3275	129	15	-	-	ADJ
ejpam-3275	129	16	prime	prime	ADJ
ejpam-3275	129	17	submodule	submodule	NOUN
ejpam-3275	129	18	of	of	ADP
ejpam-3275	129	19	m1	m1	PROPN
ejpam-3275	129	20	containing	contain	VERB
ejpam-3275	129	21	kerφ	kerφ	PROPN
ejpam-3275	129	22	,	,	PUNCT
ejpam-3275	129	23	then	then	ADV
ejpam-3275	129	24	φ(p	φ(p	PROPN
ejpam-3275	129	25	)	)	PUNCT
ejpam-3275	129	26	is	be	AUX
ejpam-3275	129	27	an	an	DET
ejpam-3275	129	28	α	α	NOUN
ejpam-3275	129	29	-	-	ADJ
ejpam-3275	129	30	prime	prime	ADJ
ejpam-3275	129	31	submodule	submodule	NOUN
ejpam-3275	129	32	of	of	ADP
ejpam-3275	129	33	m2	m2	PROPN
ejpam-3275	129	34	.	.	PUNCT
ejpam-3275	130	1	(	(	PUNCT
ejpam-3275	130	2	ii	ii	NOUN
ejpam-3275	130	3	)	)	PUNCT
ejpam-3275	130	4	if	if	SCONJ
ejpam-3275	130	5	k	k	PROPN
ejpam-3275	130	6	is	be	AUX
ejpam-3275	130	7	an	an	DET
ejpam-3275	130	8	α	α	NOUN
ejpam-3275	130	9	-	-	ADJ
ejpam-3275	130	10	prime	prime	ADJ
ejpam-3275	130	11	submodule	submodule	NOUN
ejpam-3275	130	12	of	of	ADP
ejpam-3275	130	13	m2	m2	PROPN
ejpam-3275	130	14	,	,	PUNCT
ejpam-3275	130	15	then	then	ADV
ejpam-3275	130	16	φ−1(k	φ−1(k	PROPN
ejpam-3275	130	17	)	)	PUNCT
ejpam-3275	130	18	is	be	AUX
ejpam-3275	130	19	an	an	DET
ejpam-3275	130	20	α	α	NOUN
ejpam-3275	130	21	-	-	ADJ
ejpam-3275	130	22	prime	prime	ADJ
ejpam-3275	130	23	submodule	submodule	NOUN
ejpam-3275	130	24	of	of	ADP
ejpam-3275	130	25	m1	m1	PROPN
ejpam-3275	130	26	.	.	PUNCT
ejpam-3275	131	1	t.	t.	PROPN
ejpam-3275	131	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	131	3	/	/	SYM
ejpam-3275	131	4	eur	eur	PROPN
ejpam-3275	131	5	.	.	PUNCT
ejpam-3275	132	1	j.	j.	PROPN
ejpam-3275	132	2	pure	pure	PROPN
ejpam-3275	132	3	appl	appl	PROPN
ejpam-3275	132	4	.	.	PROPN
ejpam-3275	132	5	math	math	PROPN
ejpam-3275	132	6	,	,	PUNCT
ejpam-3275	132	7	11	11	NUM
ejpam-3275	132	8	(	(	PUNCT
ejpam-3275	132	9	3	3	NUM
ejpam-3275	132	10	)	)	PUNCT
ejpam-3275	132	11	(	(	PUNCT
ejpam-3275	132	12	2018	2018	NUM
ejpam-3275	132	13	)	)	PUNCT
ejpam-3275	132	14	,	,	PUNCT
ejpam-3275	132	15	730	730	NUM
ejpam-3275	132	16	-	-	SYM
ejpam-3275	132	17	739	739	NUM
ejpam-3275	132	18	733	733	NUM
ejpam-3275	132	19	proof	proof	NOUN
ejpam-3275	132	20	.	.	PUNCT
ejpam-3275	133	1	(	(	PUNCT
ejpam-3275	133	2	i	i	NOUN
ejpam-3275	133	3	)	)	PUNCT
ejpam-3275	133	4	assume	assume	VERB
ejpam-3275	133	5	that	that	SCONJ
ejpam-3275	133	6	φ	φ	PROPN
ejpam-3275	133	7	is	be	AUX
ejpam-3275	133	8	an	an	DET
ejpam-3275	133	9	epimorphism	epimorphism	NOUN
ejpam-3275	133	10	and	and	CCONJ
ejpam-3275	133	11	p	p	NOUN
ejpam-3275	133	12	is	be	AUX
ejpam-3275	133	13	an	an	DET
ejpam-3275	133	14	α	α	NOUN
ejpam-3275	133	15	-	-	ADJ
ejpam-3275	133	16	prime	prime	ADJ
ejpam-3275	133	17	submodule	submodule	NOUN
ejpam-3275	133	18	of	of	ADP
ejpam-3275	133	19	m1	m1	PROPN
ejpam-3275	133	20	containing	contain	VERB
ejpam-3275	133	21	kerφ	kerφ	PROPN
ejpam-3275	133	22	.	.	PUNCT
ejpam-3275	134	1	let	let	VERB
ejpam-3275	134	2	r	r	NOUN
ejpam-3275	134	3	∈	∈	NOUN
ejpam-3275	134	4	r	r	NOUN
ejpam-3275	134	5	and	and	CCONJ
ejpam-3275	134	6	m	m	PROPN
ejpam-3275	134	7	∈	∈	PROPN
ejpam-3275	134	8	m2	m2	PROPN
ejpam-3275	134	9	be	be	VERB
ejpam-3275	134	10	such	such	ADJ
ejpam-3275	134	11	that	that	SCONJ
ejpam-3275	134	12	r(m	r(m	PROPN
ejpam-3275	134	13	+	+	NOUN
ejpam-3275	134	14	m	m	NOUN
ejpam-3275	134	15	)	)	PUNCT
ejpam-3275	134	16	∈	∈	PROPN
ejpam-3275	134	17	φ(p	φ(p	PROPN
ejpam-3275	134	18	)	)	PUNCT
ejpam-3275	134	19	.	.	PUNCT
ejpam-3275	135	1	there	there	PRON
ejpam-3275	135	2	exist	exist	VERB
ejpam-3275	135	3	elements	element	NOUN
ejpam-3275	135	4	n	n	PRON
ejpam-3275	135	5	∈	∈	PROPN
ejpam-3275	135	6	m1	m1	NOUN
ejpam-3275	135	7	and	and	CCONJ
ejpam-3275	135	8	p	p	NOUN
ejpam-3275	135	9	∈	∈	PROPN
ejpam-3275	135	10	p	p	NOUN
ejpam-3275	135	11	such	such	ADJ
ejpam-3275	135	12	that	that	DET
ejpam-3275	135	13	r(m	r(m	PROPN
ejpam-3275	135	14	+	+	NOUN
ejpam-3275	135	15	m	m	VERB
ejpam-3275	135	16	)	)	PUNCT
ejpam-3275	135	17	=	=	SYM
ejpam-3275	135	18	φ(p	φ(p	PROPN
ejpam-3275	135	19	)	)	PUNCT
ejpam-3275	135	20	and	and	CCONJ
ejpam-3275	135	21	φ(n	φ(n	ADJ
ejpam-3275	135	22	)	)	PUNCT
ejpam-3275	135	23	=	=	VERB
ejpam-3275	135	24	m.	m.	NOUN
ejpam-3275	135	25	then	then	ADV
ejpam-3275	135	26	φ(p	φ(p	PROPN
ejpam-3275	135	27	)	)	PUNCT
ejpam-3275	135	28	=	=	PUNCT
ejpam-3275	136	1	r(m	r(m	PROPN
ejpam-3275	136	2	+	+	NUM
ejpam-3275	136	3	m	m	VERB
ejpam-3275	136	4	)	)	PUNCT
ejpam-3275	136	5	=	=	SYM
ejpam-3275	136	6	r(φ(n	r(φ(n	NOUN
ejpam-3275	136	7	)	)	PUNCT
ejpam-3275	137	1	+	+	CCONJ
ejpam-3275	137	2	φ(n	φ(n	NOUN
ejpam-3275	137	3	)	)	PUNCT
ejpam-3275	137	4	)	)	PUNCT
ejpam-3275	138	1	=	=	PUNCT
ejpam-3275	138	2	r(φ(n	r(φ(n	NOUN
ejpam-3275	138	3	+	+	CCONJ
ejpam-3275	138	4	n	n	CCONJ
ejpam-3275	138	5	)	)	PUNCT
ejpam-3275	138	6	)	)	PUNCT
ejpam-3275	139	1	=	=	PUNCT
ejpam-3275	139	2	φ(r(n	φ(r(n	NOUN
ejpam-3275	139	3	+	+	CCONJ
ejpam-3275	139	4	n	n	CCONJ
ejpam-3275	139	5	)	)	PUNCT
ejpam-3275	139	6	)	)	PUNCT
ejpam-3275	139	7	.	.	PUNCT
ejpam-3275	140	1	this	this	PRON
ejpam-3275	140	2	implies	imply	VERB
ejpam-3275	140	3	that	that	SCONJ
ejpam-3275	140	4	r(n+	r(n+	VERB
ejpam-3275	141	1	n)−	n)−	PROPN
ejpam-3275	141	2	p	p	PROPN
ejpam-3275	141	3	∈	∈	PROPN
ejpam-3275	141	4	kerφ	kerφ	PROPN
ejpam-3275	141	5	.	.	PUNCT
ejpam-3275	142	1	since	since	SCONJ
ejpam-3275	142	2	kerφ	kerφ	PROPN
ejpam-3275	142	3	⊆	⊆	NUM
ejpam-3275	142	4	p	p	NOUN
ejpam-3275	142	5	,	,	PUNCT
ejpam-3275	142	6	r(n+	r(n+	NOUN
ejpam-3275	142	7	n	n	CCONJ
ejpam-3275	142	8	)	)	PUNCT
ejpam-3275	142	9	∈	∈	PROPN
ejpam-3275	142	10	p	p	NOUN
ejpam-3275	142	11	.	.	PUNCT
ejpam-3275	143	1	since	since	SCONJ
ejpam-3275	143	2	p	p	NOUN
ejpam-3275	143	3	is	be	AUX
ejpam-3275	143	4	an	an	DET
ejpam-3275	143	5	α	α	NOUN
ejpam-3275	143	6	-	-	ADJ
ejpam-3275	143	7	prime	prime	ADJ
ejpam-3275	143	8	submodule	submodule	NOUN
ejpam-3275	143	9	of	of	ADP
ejpam-3275	143	10	m1	m1	PROPN
ejpam-3275	143	11	,	,	PUNCT
ejpam-3275	143	12	r+	r+	PUNCT
ejpam-3275	143	13	r	r	NOUN
ejpam-3275	143	14	∈	∈	PROPN
ejpam-3275	143	15	(	(	PUNCT
ejpam-3275	143	16	p	p	NOUN
ejpam-3275	143	17	:	:	PUNCT
ejpam-3275	143	18	m1	m1	NOUN
ejpam-3275	143	19	)	)	PUNCT
ejpam-3275	143	20	or	or	CCONJ
ejpam-3275	143	21	n+n	n+n	PUNCT
ejpam-3275	143	22	∈	∈	PROPN
ejpam-3275	143	23	p	p	NOUN
ejpam-3275	143	24	.	.	PUNCT
ejpam-3275	144	1	since	since	SCONJ
ejpam-3275	144	2	φ	φ	PROPN
ejpam-3275	144	3	is	be	AUX
ejpam-3275	144	4	onto	onto	ADP
ejpam-3275	144	5	,	,	PUNCT
ejpam-3275	144	6	r+	r+	PUNCT
ejpam-3275	144	7	r	r	NOUN
ejpam-3275	144	8	∈	∈	PROPN
ejpam-3275	144	9	(	(	PUNCT
ejpam-3275	144	10	φ(p	φ(p	PROPN
ejpam-3275	144	11	)	)	PUNCT
ejpam-3275	144	12	:	:	PUNCT
ejpam-3275	144	13	m2	m2	PROPN
ejpam-3275	144	14	)	)	PUNCT
ejpam-3275	144	15	or	or	CCONJ
ejpam-3275	144	16	m+m	m+m	PROPN
ejpam-3275	144	17	∈	∈	PROPN
ejpam-3275	144	18	φ(p	φ(p	PROPN
ejpam-3275	144	19	)	)	PUNCT
ejpam-3275	144	20	.	.	PUNCT
ejpam-3275	145	1	hence	hence	ADV
ejpam-3275	145	2	φ(p	φ(p	PROPN
ejpam-3275	145	3	)	)	PUNCT
ejpam-3275	145	4	is	be	AUX
ejpam-3275	145	5	an	an	DET
ejpam-3275	145	6	α	α	NOUN
ejpam-3275	145	7	-	-	ADJ
ejpam-3275	145	8	prime	prime	ADJ
ejpam-3275	145	9	submodule	submodule	NOUN
ejpam-3275	145	10	of	of	ADP
ejpam-3275	145	11	m2	m2	PROPN
ejpam-3275	145	12	.	.	PUNCT
ejpam-3275	146	1	(	(	PUNCT
ejpam-3275	146	2	ii	ii	NOUN
ejpam-3275	146	3	)	)	PUNCT
ejpam-3275	146	4	assume	assume	VERB
ejpam-3275	146	5	that	that	SCONJ
ejpam-3275	146	6	k	k	PROPN
ejpam-3275	146	7	is	be	AUX
ejpam-3275	146	8	an	an	DET
ejpam-3275	146	9	α	α	NOUN
ejpam-3275	146	10	-	-	ADJ
ejpam-3275	146	11	prime	prime	ADJ
ejpam-3275	146	12	submodule	submodule	NOUN
ejpam-3275	146	13	of	of	ADP
ejpam-3275	146	14	m2	m2	PROPN
ejpam-3275	146	15	.	.	PUNCT
ejpam-3275	147	1	let	let	VERB
ejpam-3275	147	2	r	r	NOUN
ejpam-3275	147	3	∈	∈	NOUN
ejpam-3275	147	4	r	r	NOUN
ejpam-3275	147	5	and	and	CCONJ
ejpam-3275	147	6	m	m	PROPN
ejpam-3275	147	7	∈	∈	NOUN
ejpam-3275	147	8	m	m	AUX
ejpam-3275	147	9	be	be	VERB
ejpam-3275	147	10	such	such	ADJ
ejpam-3275	147	11	that	that	DET
ejpam-3275	147	12	r(m+m	r(m+m	NOUN
ejpam-3275	147	13	)	)	PUNCT
ejpam-3275	147	14	∈	∈	PROPN
ejpam-3275	147	15	φ−1(k	φ−1(k	NOUN
ejpam-3275	147	16	)	)	PUNCT
ejpam-3275	147	17	.	.	PUNCT
ejpam-3275	148	1	then	then	ADV
ejpam-3275	148	2	r(φ(m	r(φ(m	PROPN
ejpam-3275	148	3	)	)	PUNCT
ejpam-3275	149	1	+	+	CCONJ
ejpam-3275	149	2	φ(m	φ(m	NOUN
ejpam-3275	149	3	)	)	PUNCT
ejpam-3275	149	4	)	)	PUNCT
ejpam-3275	150	1	∈	∈	PROPN
ejpam-3275	150	2	k.	k.	PROPN
ejpam-3275	150	3	since	since	SCONJ
ejpam-3275	150	4	k	k	PROPN
ejpam-3275	150	5	is	be	AUX
ejpam-3275	150	6	an	an	DET
ejpam-3275	150	7	α	α	NOUN
ejpam-3275	150	8	-	-	ADJ
ejpam-3275	150	9	prime	prime	ADJ
ejpam-3275	150	10	submodule	submodule	NOUN
ejpam-3275	150	11	of	of	ADP
ejpam-3275	150	12	m2	m2	PROPN
ejpam-3275	150	13	,	,	PUNCT
ejpam-3275	150	14	r	r	NOUN
ejpam-3275	150	15	+	+	CCONJ
ejpam-3275	150	16	r	r	NOUN
ejpam-3275	150	17	∈	∈	NOUN
ejpam-3275	150	18	(	(	PUNCT
ejpam-3275	150	19	k	k	NOUN
ejpam-3275	150	20	:	:	PUNCT
ejpam-3275	150	21	m2	m2	PROPN
ejpam-3275	150	22	)	)	PUNCT
ejpam-3275	150	23	or	or	CCONJ
ejpam-3275	150	24	φ(m	φ(m	NOUN
ejpam-3275	150	25	)	)	PUNCT
ejpam-3275	150	26	+	+	CCONJ
ejpam-3275	150	27	φ(m	φ(m	NOUN
ejpam-3275	150	28	)	)	PUNCT
ejpam-3275	150	29	∈	∈	PROPN
ejpam-3275	150	30	k.	k.	NOUN
ejpam-3275	151	1	this	this	PRON
ejpam-3275	151	2	implies	imply	VERB
ejpam-3275	151	3	that	that	SCONJ
ejpam-3275	151	4	r	r	NOUN
ejpam-3275	151	5	+	+	CCONJ
ejpam-3275	151	6	r	r	NOUN
ejpam-3275	151	7	∈	∈	PROPN
ejpam-3275	151	8	(	(	PUNCT
ejpam-3275	151	9	φ−1(k	φ−1(k	PROPN
ejpam-3275	151	10	)	)	PUNCT
ejpam-3275	151	11	:	:	PUNCT
ejpam-3275	151	12	m1	m1	NOUN
ejpam-3275	151	13	)	)	PUNCT
ejpam-3275	151	14	or	or	CCONJ
ejpam-3275	151	15	m+m	m+m	PROPN
ejpam-3275	151	16	∈	∈	PROPN
ejpam-3275	151	17	φ−1(k	φ−1(k	NOUN
ejpam-3275	151	18	)	)	PUNCT
ejpam-3275	151	19	.	.	PUNCT
ejpam-3275	152	1	hence	hence	ADV
ejpam-3275	152	2	φ−1(k	φ−1(k	PROPN
ejpam-3275	152	3	)	)	PUNCT
ejpam-3275	152	4	is	be	AUX
ejpam-3275	152	5	an	an	DET
ejpam-3275	152	6	α	α	NOUN
ejpam-3275	152	7	-	-	ADJ
ejpam-3275	152	8	prime	prime	ADJ
ejpam-3275	152	9	submodule	submodule	NOUN
ejpam-3275	152	10	of	of	ADP
ejpam-3275	152	11	m1	m1	PROPN
ejpam-3275	152	12	.	.	PUNCT
ejpam-3275	153	1	corollary	corollary	ADJ
ejpam-3275	153	2	1	1	NUM
ejpam-3275	153	3	.	.	PUNCT
ejpam-3275	154	1	let	let	VERB
ejpam-3275	154	2	n	n	PRON
ejpam-3275	154	3	be	be	AUX
ejpam-3275	154	4	a	a	DET
ejpam-3275	154	5	submodule	submodule	NOUN
ejpam-3275	154	6	of	of	ADP
ejpam-3275	154	7	m	m	PROPN
ejpam-3275	154	8	.	.	PUNCT
ejpam-3275	155	1	then	then	ADV
ejpam-3275	155	2	(	(	PUNCT
ejpam-3275	155	3	i	i	NOUN
ejpam-3275	155	4	)	)	PUNCT
ejpam-3275	155	5	if	if	SCONJ
ejpam-3275	155	6	p	p	NOUN
ejpam-3275	155	7	is	be	AUX
ejpam-3275	155	8	an	an	DET
ejpam-3275	155	9	α	α	NOUN
ejpam-3275	155	10	-	-	ADJ
ejpam-3275	155	11	prime	prime	ADJ
ejpam-3275	155	12	submodule	submodule	NOUN
ejpam-3275	155	13	of	of	ADP
ejpam-3275	155	14	m	m	PROPN
ejpam-3275	155	15	and	and	CCONJ
ejpam-3275	155	16	k	k	PROPN
ejpam-3275	155	17	is	be	AUX
ejpam-3275	155	18	a	a	DET
ejpam-3275	155	19	submodule	submodule	NOUN
ejpam-3275	155	20	of	of	ADP
ejpam-3275	155	21	m	m	PROPN
ejpam-3275	155	22	contained	contain	VERB
ejpam-3275	155	23	in	in	ADP
ejpam-3275	155	24	p	p	NOUN
ejpam-3275	155	25	,	,	PUNCT
ejpam-3275	155	26	then	then	ADV
ejpam-3275	155	27	p	p	X
ejpam-3275	155	28	/	/	SYM
ejpam-3275	155	29	k	k	PROPN
ejpam-3275	155	30	is	be	AUX
ejpam-3275	155	31	an	an	DET
ejpam-3275	155	32	α	α	NOUN
ejpam-3275	155	33	-	-	ADJ
ejpam-3275	155	34	prime	prime	ADJ
ejpam-3275	155	35	submodule	submodule	NOUN
ejpam-3275	155	36	of	of	ADP
ejpam-3275	155	37	m	m	PROPN
ejpam-3275	155	38	/	/	SYM
ejpam-3275	155	39	k	k	PROPN
ejpam-3275	155	40	.	.	PUNCT
ejpam-3275	156	1	(	(	PUNCT
ejpam-3275	156	2	ii	ii	NOUN
ejpam-3275	156	3	)	)	PUNCT
ejpam-3275	156	4	if	if	SCONJ
ejpam-3275	156	5	k	k	PROPN
ejpam-3275	156	6	′	′	NOUN
ejpam-3275	156	7	is	be	AUX
ejpam-3275	156	8	an	an	DET
ejpam-3275	156	9	α	α	NOUN
ejpam-3275	156	10	-	-	ADJ
ejpam-3275	156	11	prime	prime	ADJ
ejpam-3275	156	12	submodule	submodule	NOUN
ejpam-3275	156	13	of	of	ADP
ejpam-3275	156	14	m	m	PROPN
ejpam-3275	156	15	/	/	SYM
ejpam-3275	156	16	n	n	PROPN
ejpam-3275	156	17	,	,	PUNCT
ejpam-3275	156	18	then	then	ADV
ejpam-3275	156	19	k	k	PROPN
ejpam-3275	157	1	′	′	NOUN
ejpam-3275	158	1	=	=	PUNCT
ejpam-3275	159	1	k	k	NOUN
ejpam-3275	159	2	/	/	SYM
ejpam-3275	159	3	n	n	PROPN
ejpam-3275	159	4	.	.	PUNCT
ejpam-3275	160	1	for	for	ADP
ejpam-3275	160	2	some	some	DET
ejpam-3275	160	3	α	α	ADJ
ejpam-3275	160	4	-	-	ADJ
ejpam-3275	160	5	prime	prime	ADJ
ejpam-3275	160	6	submodule	submodule	PROPN
ejpam-3275	160	7	k	k	PROPN
ejpam-3275	160	8	of	of	ADP
ejpam-3275	160	9	m	m	PROPN
ejpam-3275	160	10	.	.	PUNCT
ejpam-3275	161	1	proof	proof	NOUN
ejpam-3275	161	2	.	.	PUNCT
ejpam-3275	162	1	(	(	PUNCT
ejpam-3275	162	2	i	i	NOUN
ejpam-3275	162	3	)	)	PUNCT
ejpam-3275	162	4	assume	assume	VERB
ejpam-3275	162	5	that	that	SCONJ
ejpam-3275	162	6	p	p	NOUN
ejpam-3275	162	7	is	be	AUX
ejpam-3275	162	8	an	an	DET
ejpam-3275	162	9	α	α	NOUN
ejpam-3275	162	10	-	-	ADJ
ejpam-3275	162	11	prime	prime	ADJ
ejpam-3275	162	12	submodule	submodule	NOUN
ejpam-3275	162	13	of	of	ADP
ejpam-3275	162	14	m	m	PROPN
ejpam-3275	162	15	and	and	CCONJ
ejpam-3275	162	16	k	k	PROPN
ejpam-3275	162	17	is	be	AUX
ejpam-3275	162	18	a	a	DET
ejpam-3275	162	19	submodule	submodule	NOUN
ejpam-3275	162	20	of	of	ADP
ejpam-3275	162	21	m	m	PROPN
ejpam-3275	162	22	contained	contain	VERB
ejpam-3275	162	23	in	in	ADP
ejpam-3275	162	24	p	p	PROPN
ejpam-3275	162	25	.	.	PUNCT
ejpam-3275	163	1	define	define	VERB
ejpam-3275	163	2	a	a	DET
ejpam-3275	163	3	homomorphism	homomorphism	NOUN
ejpam-3275	163	4	ϕ	ϕ	X
ejpam-3275	163	5	:	:	PUNCT
ejpam-3275	163	6	m	m	PROPN
ejpam-3275	163	7	→	→	SYM
ejpam-3275	163	8	m	m	PROPN
ejpam-3275	163	9	/	/	SYM
ejpam-3275	163	10	k	k	X
ejpam-3275	163	11	by	by	ADP
ejpam-3275	163	12	ϕ(m	ϕ(m	NOUN
ejpam-3275	163	13	)	)	PUNCT
ejpam-3275	164	1	=	=	SYM
ejpam-3275	164	2	m+k	m+k	PROPN
ejpam-3275	164	3	for	for	ADP
ejpam-3275	164	4	all	all	DET
ejpam-3275	164	5	m	m	NOUN
ejpam-3275	164	6	∈m	∈m	NOUN
ejpam-3275	164	7	.	.	PUNCT
ejpam-3275	165	1	then	then	ADV
ejpam-3275	165	2	ϕ	ϕ	PROPN
ejpam-3275	165	3	is	be	AUX
ejpam-3275	165	4	an	an	DET
ejpam-3275	165	5	epimorphism	epimorphism	NOUN
ejpam-3275	165	6	and	and	CCONJ
ejpam-3275	165	7	kerϕ	kerϕ	NOUN
ejpam-3275	165	8	=	=	PROPN
ejpam-3275	165	9	k.	k.	PROPN
ejpam-3275	165	10	by	by	ADP
ejpam-3275	165	11	proposition	proposition	NOUN
ejpam-3275	165	12	1	1	NUM
ejpam-3275	165	13	(	(	PUNCT
ejpam-3275	165	14	i	i	NOUN
ejpam-3275	165	15	)	)	PUNCT
ejpam-3275	165	16	,	,	PUNCT
ejpam-3275	165	17	ϕ(p	ϕ(p	PROPN
ejpam-3275	165	18	)	)	PUNCT
ejpam-3275	166	1	=	=	PUNCT
ejpam-3275	167	1	p	p	X
ejpam-3275	167	2	/	/	SYM
ejpam-3275	167	3	k	k	PROPN
ejpam-3275	167	4	is	be	AUX
ejpam-3275	167	5	an	an	DET
ejpam-3275	167	6	α	α	NOUN
ejpam-3275	167	7	-	-	ADJ
ejpam-3275	167	8	prime	prime	ADJ
ejpam-3275	167	9	submodule	submodule	NOUN
ejpam-3275	167	10	of	of	ADP
ejpam-3275	167	11	m	m	PROPN
ejpam-3275	167	12	/	/	SYM
ejpam-3275	167	13	k	k	PROPN
ejpam-3275	167	14	.	.	PUNCT
ejpam-3275	168	1	(	(	PUNCT
ejpam-3275	168	2	ii	ii	NOUN
ejpam-3275	168	3	)	)	PUNCT
ejpam-3275	168	4	assume	assume	VERB
ejpam-3275	168	5	that	that	SCONJ
ejpam-3275	168	6	k	k	PROPN
ejpam-3275	168	7	′	′	NOUN
ejpam-3275	168	8	is	be	AUX
ejpam-3275	168	9	an	an	DET
ejpam-3275	168	10	α	α	NOUN
ejpam-3275	168	11	-	-	ADJ
ejpam-3275	168	12	prime	prime	ADJ
ejpam-3275	168	13	submodule	submodule	NOUN
ejpam-3275	168	14	of	of	ADP
ejpam-3275	168	15	m	m	PROPN
ejpam-3275	168	16	/	/	SYM
ejpam-3275	168	17	n	n	PROPN
ejpam-3275	168	18	.	.	PUNCT
ejpam-3275	169	1	then	then	ADV
ejpam-3275	169	2	the	the	DET
ejpam-3275	169	3	set	set	NOUN
ejpam-3275	169	4	k	k	PROPN
ejpam-3275	169	5	=	=	PRON
ejpam-3275	169	6	{	{	PUNCT
ejpam-3275	169	7	x	x	SYM
ejpam-3275	169	8	∈	∈	NOUN
ejpam-3275	169	9	m	m	VERB
ejpam-3275	169	10	|	|	ADV
ejpam-3275	169	11	x+n	x+n	PUNCT
ejpam-3275	169	12	∈	∈	PROPN
ejpam-3275	169	13	k	k	NOUN
ejpam-3275	169	14	′	′	NOUN
ejpam-3275	169	15	}	}	PUNCT
ejpam-3275	169	16	is	be	AUX
ejpam-3275	169	17	an	an	DET
ejpam-3275	169	18	α	α	NOUN
ejpam-3275	169	19	-	-	ADJ
ejpam-3275	169	20	prime	prime	ADJ
ejpam-3275	169	21	submodule	submodule	NOUN
ejpam-3275	169	22	of	of	ADP
ejpam-3275	169	23	m	m	PROPN
ejpam-3275	169	24	.	.	PUNCT
ejpam-3275	170	1	clearly	clearly	ADV
ejpam-3275	170	2	,	,	PUNCT
ejpam-3275	170	3	k	k	PROPN
ejpam-3275	170	4	′	′	NOUN
ejpam-3275	171	1	=	=	PUNCT
ejpam-3275	171	2	k	k	NOUN
ejpam-3275	171	3	/	/	SYM
ejpam-3275	171	4	n	n	PROPN
ejpam-3275	171	5	.	.	PUNCT
ejpam-3275	172	1	for	for	ADP
ejpam-3275	172	2	subgroups	subgroup	NOUN
ejpam-3275	172	3	a	a	PRON
ejpam-3275	172	4	and	and	CCONJ
ejpam-3275	172	5	b	b	NOUN
ejpam-3275	172	6	of	of	ADP
ejpam-3275	172	7	a	a	DET
ejpam-3275	172	8	group	group	NOUN
ejpam-3275	172	9	(	(	PUNCT
ejpam-3275	172	10	g,+	g,+	PROPN
ejpam-3275	172	11	)	)	PUNCT
ejpam-3275	172	12	,	,	PUNCT
ejpam-3275	172	13	we	we	PRON
ejpam-3275	172	14	have	have	VERB
ejpam-3275	172	15	a	a	DET
ejpam-3275	172	16	⊆	⊆	NUM
ejpam-3275	172	17	α(b	α(b	NOUN
ejpam-3275	172	18	)	)	PUNCT
ejpam-3275	173	1	if	if	SCONJ
ejpam-3275	173	2	and	and	CCONJ
ejpam-3275	173	3	only	only	ADV
ejpam-3275	173	4	if	if	SCONJ
ejpam-3275	173	5	β(a	β(a	NOUN
ejpam-3275	173	6	)	)	PUNCT
ejpam-3275	173	7	⊆	⊆	NUM
ejpam-3275	173	8	b.	b.	NOUN
ejpam-3275	173	9	definition	definition	NOUN
ejpam-3275	173	10	2	2	NUM
ejpam-3275	173	11	.	.	PUNCT
ejpam-3275	174	1	let	let	VERB
ejpam-3275	174	2	r	r	PRON
ejpam-3275	174	3	be	be	AUX
ejpam-3275	174	4	a	a	DET
ejpam-3275	174	5	ring	ring	NOUN
ejpam-3275	174	6	and	and	CCONJ
ejpam-3275	174	7	m	m	AUX
ejpam-3275	174	8	be	be	AUX
ejpam-3275	174	9	an	an	DET
ejpam-3275	174	10	r	r	NOUN
ejpam-3275	174	11	-	-	PUNCT
ejpam-3275	174	12	module	module	NOUN
ejpam-3275	174	13	.	.	PUNCT
ejpam-3275	175	1	a	a	DET
ejpam-3275	175	2	nonempty	nonempty	ADV
ejpam-3275	175	3	set	set	VERB
ejpam-3275	175	4	s	s	NOUN
ejpam-3275	175	5	⊆	⊆	NUM
ejpam-3275	175	6	m\{0	m\{0	NOUN
ejpam-3275	175	7	}	}	PUNCT
ejpam-3275	175	8	is	be	AUX
ejpam-3275	175	9	called	call	VERB
ejpam-3275	175	10	an	an	DET
ejpam-3275	175	11	α	α	NOUN
ejpam-3275	175	12	-	-	PUNCT
ejpam-3275	175	13	multiplicative	multiplicative	ADJ
ejpam-3275	175	14	system	system	NOUN
ejpam-3275	175	15	if	if	SCONJ
ejpam-3275	175	16	for	for	ADP
ejpam-3275	175	17	all	all	DET
ejpam-3275	175	18	ideal	ideal	ADJ
ejpam-3275	175	19	i	i	PRON
ejpam-3275	175	20	of	of	ADP
ejpam-3275	175	21	r	r	NOUN
ejpam-3275	175	22	and	and	CCONJ
ejpam-3275	175	23	for	for	ADP
ejpam-3275	175	24	all	all	DET
ejpam-3275	175	25	submodules	submodule	NOUN
ejpam-3275	175	26	k	k	PROPN
ejpam-3275	175	27	and	and	CCONJ
ejpam-3275	175	28	n	n	PROPN
ejpam-3275	175	29	of	of	ADP
ejpam-3275	175	30	m	m	PRON
ejpam-3275	175	31	,	,	PUNCT
ejpam-3275	175	32	if	if	SCONJ
ejpam-3275	175	33	(	(	PUNCT
ejpam-3275	175	34	k+β(i)m	k+β(i)m	NOUN
ejpam-3275	175	35	)	)	PUNCT
ejpam-3275	175	36	∩s	∩s	PROPN
ejpam-3275	175	37	6=	6=	PROPN
ejpam-3275	175	38	∅	∅	NOUN
ejpam-3275	175	39	and	and	CCONJ
ejpam-3275	175	40	(	(	PUNCT
ejpam-3275	175	41	k+β(n	k+β(n	NOUN
ejpam-3275	175	42	)	)	PUNCT
ejpam-3275	175	43	)	)	PUNCT
ejpam-3275	176	1	∩s	∩s	PROPN
ejpam-3275	176	2	6=	6=	NUM
ejpam-3275	176	3	∅	∅	NOUN
ejpam-3275	176	4	,	,	PUNCT
ejpam-3275	176	5	then	then	ADV
ejpam-3275	176	6	(	(	PUNCT
ejpam-3275	176	7	k+iβ(n	k+iβ(n	NOUN
ejpam-3275	176	8	)	)	PUNCT
ejpam-3275	176	9	)	)	PUNCT
ejpam-3275	177	1	∩s	∩s	PROPN
ejpam-3275	177	2	6=	6=	ADP
ejpam-3275	177	3	∅.	∅.	NOUN
ejpam-3275	177	4	proposition	proposition	NOUN
ejpam-3275	177	5	2	2	NUM
ejpam-3275	177	6	.	.	PUNCT
ejpam-3275	177	7	let	let	VERB
ejpam-3275	177	8	p	p	PRON
ejpam-3275	177	9	be	be	AUX
ejpam-3275	177	10	a	a	DET
ejpam-3275	177	11	submodule	submodule	NOUN
ejpam-3275	177	12	of	of	ADP
ejpam-3275	177	13	an	an	DET
ejpam-3275	177	14	r	r	NOUN
ejpam-3275	177	15	-	-	PUNCT
ejpam-3275	177	16	module	module	NOUN
ejpam-3275	177	17	m	m	NOUN
ejpam-3275	177	18	.	.	PUNCT
ejpam-3275	178	1	then	then	ADV
ejpam-3275	178	2	p	p	NOUN
ejpam-3275	178	3	is	be	AUX
ejpam-3275	178	4	an	an	DET
ejpam-3275	178	5	α	α	NOUN
ejpam-3275	178	6	-	-	ADJ
ejpam-3275	178	7	prime	prime	ADJ
ejpam-3275	178	8	submodule	submodule	NOUN
ejpam-3275	178	9	of	of	ADP
ejpam-3275	178	10	m	m	PROPN
ejpam-3275	178	11	if	if	SCONJ
ejpam-3275	179	1	and	and	CCONJ
ejpam-3275	179	2	only	only	ADV
ejpam-3275	179	3	if	if	SCONJ
ejpam-3275	179	4	m\p	m\p	PROPN
ejpam-3275	179	5	is	be	AUX
ejpam-3275	179	6	an	an	DET
ejpam-3275	179	7	α	α	NOUN
ejpam-3275	179	8	-	-	PUNCT
ejpam-3275	179	9	multiplicative	multiplicative	ADJ
ejpam-3275	179	10	system	system	NOUN
ejpam-3275	179	11	.	.	PUNCT
ejpam-3275	180	1	proof	proof	NOUN
ejpam-3275	180	2	.	.	PUNCT
ejpam-3275	181	1	(	(	PUNCT
ejpam-3275	181	2	→	→	NOUN
ejpam-3275	181	3	)	)	PUNCT
ejpam-3275	181	4	assume	assume	VERB
ejpam-3275	181	5	that	that	SCONJ
ejpam-3275	181	6	p	p	NOUN
ejpam-3275	181	7	is	be	AUX
ejpam-3275	181	8	an	an	DET
ejpam-3275	181	9	α	α	NOUN
ejpam-3275	181	10	-	-	ADJ
ejpam-3275	181	11	prime	prime	ADJ
ejpam-3275	181	12	submodule	submodule	NOUN
ejpam-3275	181	13	of	of	ADP
ejpam-3275	181	14	m	m	PROPN
ejpam-3275	181	15	.	.	PUNCT
ejpam-3275	182	1	let	let	VERB
ejpam-3275	182	2	i	i	PRON
ejpam-3275	182	3	be	be	AUX
ejpam-3275	182	4	an	an	DET
ejpam-3275	182	5	ideal	ideal	NOUN
ejpam-3275	182	6	of	of	ADP
ejpam-3275	182	7	r	r	NOUN
ejpam-3275	182	8	and	and	CCONJ
ejpam-3275	182	9	let	let	VERB
ejpam-3275	182	10	k	k	NOUN
ejpam-3275	182	11	and	and	CCONJ
ejpam-3275	182	12	n	n	CCONJ
ejpam-3275	182	13	be	be	VERB
ejpam-3275	182	14	submodules	submodule	NOUN
ejpam-3275	182	15	of	of	ADP
ejpam-3275	182	16	m	m	NOUN
ejpam-3275	182	17	such	such	ADJ
ejpam-3275	182	18	that	that	SCONJ
ejpam-3275	182	19	(	(	PUNCT
ejpam-3275	182	20	k	k	X
ejpam-3275	182	21	+	+	CCONJ
ejpam-3275	182	22	iβ(n	iβ(n	NUM
ejpam-3275	182	23	)	)	PUNCT
ejpam-3275	182	24	)	)	PUNCT
ejpam-3275	182	25	∩m\p	∩m\p	NOUN
ejpam-3275	182	26	=	=	PUNCT
ejpam-3275	182	27	∅.	∅.	NOUN
ejpam-3275	182	28	then	then	ADV
ejpam-3275	182	29	k	k	PROPN
ejpam-3275	182	30	+	+	PUNCT
ejpam-3275	182	31	iβ(n	iβ(n	NOUN
ejpam-3275	182	32	)	)	PUNCT
ejpam-3275	182	33	⊆	⊆	NUM
ejpam-3275	182	34	p	p	NOUN
ejpam-3275	182	35	.	.	PUNCT
ejpam-3275	183	1	it	it	PRON
ejpam-3275	183	2	follows	follow	VERB
ejpam-3275	183	3	that	that	SCONJ
ejpam-3275	183	4	k	k	PROPN
ejpam-3275	183	5	⊆	⊆	NUM
ejpam-3275	183	6	p	p	NOUN
ejpam-3275	183	7	and	and	CCONJ
ejpam-3275	183	8	iβ(n	iβ(n	NOUN
ejpam-3275	183	9	)	)	PUNCT
ejpam-3275	183	10	⊆	⊆	NUM
ejpam-3275	183	11	p	p	NOUN
ejpam-3275	183	12	.	.	PUNCT
ejpam-3275	184	1	since	since	SCONJ
ejpam-3275	184	2	p	p	NOUN
ejpam-3275	184	3	is	be	AUX
ejpam-3275	184	4	an	an	DET
ejpam-3275	184	5	α	α	NOUN
ejpam-3275	184	6	-	-	ADJ
ejpam-3275	184	7	prime	prime	ADJ
ejpam-3275	184	8	submodule	submodule	NOUN
ejpam-3275	184	9	of	of	ADP
ejpam-3275	184	10	m	m	PROPN
ejpam-3275	184	11	,	,	PUNCT
ejpam-3275	184	12	i	i	PRON
ejpam-3275	184	13	⊆	⊆	NUM
ejpam-3275	184	14	α((p	α((p	NOUN
ejpam-3275	184	15	:	:	PUNCT
ejpam-3275	184	16	m	m	X
ejpam-3275	184	17	)	)	PUNCT
ejpam-3275	184	18	)	)	PUNCT
ejpam-3275	184	19	or	or	CCONJ
ejpam-3275	184	20	n	n	PRON
ejpam-3275	184	21	⊆	⊆	NUM
ejpam-3275	184	22	α(p	α(p	NUM
ejpam-3275	184	23	)	)	PUNCT
ejpam-3275	184	24	.	.	PUNCT
ejpam-3275	185	1	this	this	PRON
ejpam-3275	185	2	implies	imply	VERB
ejpam-3275	185	3	that	that	SCONJ
ejpam-3275	185	4	β(i	β(i	PRON
ejpam-3275	185	5	)	)	PUNCT
ejpam-3275	185	6	⊆	⊆	NUM
ejpam-3275	185	7	(	(	PUNCT
ejpam-3275	185	8	p	p	X
ejpam-3275	185	9	:	:	PUNCT
ejpam-3275	185	10	m	m	NOUN
ejpam-3275	185	11	)	)	PUNCT
ejpam-3275	185	12	or	or	CCONJ
ejpam-3275	185	13	β(n	β(n	NUM
ejpam-3275	185	14	)	)	PUNCT
ejpam-3275	185	15	⊆	⊆	NUM
ejpam-3275	185	16	p	p	NOUN
ejpam-3275	185	17	.	.	PUNCT
ejpam-3275	186	1	hence	hence	ADV
ejpam-3275	186	2	k	k	PROPN
ejpam-3275	187	1	+	+	CCONJ
ejpam-3275	187	2	β(i)m	β(i)m	PROPN
ejpam-3275	187	3	⊆	⊆	NUM
ejpam-3275	187	4	p	p	NOUN
ejpam-3275	187	5	or	or	CCONJ
ejpam-3275	187	6	k	k	NOUN
ejpam-3275	187	7	+	+	CCONJ
ejpam-3275	187	8	β(n	β(n	NUM
ejpam-3275	187	9	)	)	PUNCT
ejpam-3275	187	10	⊆	⊆	NUM
ejpam-3275	187	11	p	p	NOUN
ejpam-3275	187	12	.	.	PUNCT
ejpam-3275	188	1	hence	hence	ADV
ejpam-3275	188	2	(	(	PUNCT
ejpam-3275	188	3	k	k	PROPN
ejpam-3275	188	4	+	+	CCONJ
ejpam-3275	188	5	β(i)m	β(i)m	ADJ
ejpam-3275	188	6	)	)	PUNCT
ejpam-3275	188	7	∩m\p	∩m\p	NOUN
ejpam-3275	188	8	=	=	NOUN
ejpam-3275	188	9	∅	∅	NOUN
ejpam-3275	188	10	or	or	CCONJ
ejpam-3275	188	11	(	(	PUNCT
ejpam-3275	188	12	k	k	PROPN
ejpam-3275	188	13	+	+	PROPN
ejpam-3275	188	14	β(n	β(n	NUM
ejpam-3275	188	15	)	)	PUNCT
ejpam-3275	188	16	)	)	PUNCT
ejpam-3275	189	1	∩m\p	∩m\p	NOUN
ejpam-3275	189	2	=	=	PUNCT
ejpam-3275	189	3	∅.	∅.	ADP
ejpam-3275	189	4	this	this	PRON
ejpam-3275	189	5	shows	show	VERB
ejpam-3275	189	6	that	that	SCONJ
ejpam-3275	189	7	m\p	m\p	PROPN
ejpam-3275	189	8	is	be	AUX
ejpam-3275	189	9	an	an	DET
ejpam-3275	189	10	α	α	NOUN
ejpam-3275	189	11	-	-	PUNCT
ejpam-3275	189	12	multiplicative	multiplicative	ADJ
ejpam-3275	189	13	system	system	NOUN
ejpam-3275	189	14	.	.	PUNCT
ejpam-3275	190	1	t.	t.	PROPN
ejpam-3275	190	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	190	3	/	/	SYM
ejpam-3275	190	4	eur	eur	PROPN
ejpam-3275	190	5	.	.	PUNCT
ejpam-3275	191	1	j.	j.	PROPN
ejpam-3275	191	2	pure	pure	PROPN
ejpam-3275	191	3	appl	appl	PROPN
ejpam-3275	191	4	.	.	PROPN
ejpam-3275	191	5	math	math	PROPN
ejpam-3275	191	6	,	,	PUNCT
ejpam-3275	191	7	11	11	NUM
ejpam-3275	191	8	(	(	PUNCT
ejpam-3275	191	9	3	3	NUM
ejpam-3275	191	10	)	)	PUNCT
ejpam-3275	191	11	(	(	PUNCT
ejpam-3275	191	12	2018	2018	NUM
ejpam-3275	191	13	)	)	PUNCT
ejpam-3275	191	14	,	,	PUNCT
ejpam-3275	191	15	730	730	NUM
ejpam-3275	191	16	-	-	SYM
ejpam-3275	191	17	739	739	NUM
ejpam-3275	191	18	734	734	NUM
ejpam-3275	191	19	(	(	PUNCT
ejpam-3275	191	20	←	←	PROPN
ejpam-3275	191	21	)	)	PUNCT
ejpam-3275	191	22	assume	assume	VERB
ejpam-3275	191	23	that	that	SCONJ
ejpam-3275	191	24	m\p	m\p	PROPN
ejpam-3275	191	25	is	be	AUX
ejpam-3275	191	26	an	an	DET
ejpam-3275	191	27	α	α	NOUN
ejpam-3275	191	28	-	-	PUNCT
ejpam-3275	191	29	multiplicative	multiplicative	ADJ
ejpam-3275	191	30	system	system	NOUN
ejpam-3275	191	31	.	.	PUNCT
ejpam-3275	192	1	let	let	VERB
ejpam-3275	192	2	i	i	PRON
ejpam-3275	192	3	be	be	AUX
ejpam-3275	192	4	an	an	DET
ejpam-3275	192	5	ideals	ideal	NOUN
ejpam-3275	192	6	of	of	ADP
ejpam-3275	192	7	r	r	NOUN
ejpam-3275	192	8	and	and	CCONJ
ejpam-3275	192	9	n	n	CCONJ
ejpam-3275	192	10	be	be	VERB
ejpam-3275	192	11	a	a	DET
ejpam-3275	192	12	submodule	submodule	NOUN
ejpam-3275	192	13	of	of	ADP
ejpam-3275	192	14	m	m	PRON
ejpam-3275	192	15	such	such	ADJ
ejpam-3275	192	16	that	that	PRON
ejpam-3275	192	17	iβ(n	iβ(n	NOUN
ejpam-3275	192	18	)	)	PUNCT
ejpam-3275	192	19	⊆	⊆	NUM
ejpam-3275	192	20	p	p	NOUN
ejpam-3275	192	21	.	.	PUNCT
ejpam-3275	193	1	hence	hence	ADV
ejpam-3275	193	2	(	(	PUNCT
ejpam-3275	193	3	iβ(n	iβ(n	NOUN
ejpam-3275	193	4	)	)	PUNCT
ejpam-3275	193	5	)	)	PUNCT
ejpam-3275	194	1	∩m\p	∩m\p	NOUN
ejpam-3275	194	2	=	=	PUNCT
ejpam-3275	194	3	∅.	∅.	NOUN
ejpam-3275	194	4	since	since	SCONJ
ejpam-3275	194	5	m\p	m\p	PROPN
ejpam-3275	194	6	is	be	AUX
ejpam-3275	194	7	an	an	DET
ejpam-3275	194	8	α	α	NOUN
ejpam-3275	194	9	-	-	PUNCT
ejpam-3275	194	10	multiplicative	multiplicative	ADJ
ejpam-3275	194	11	system	system	NOUN
ejpam-3275	194	12	,	,	PUNCT
ejpam-3275	194	13	(	(	PUNCT
ejpam-3275	194	14	β(i)m	β(i)m	NOUN
ejpam-3275	194	15	)	)	PUNCT
ejpam-3275	194	16	∩m\p	∩m\p	NOUN
ejpam-3275	194	17	=	=	NOUN
ejpam-3275	194	18	∅	∅	NOUN
ejpam-3275	194	19	or	or	CCONJ
ejpam-3275	194	20	(	(	PUNCT
ejpam-3275	194	21	β(n	β(n	NUM
ejpam-3275	194	22	)	)	PUNCT
ejpam-3275	194	23	)	)	PUNCT
ejpam-3275	195	1	∩m\p	∩m\p	NOUN
ejpam-3275	195	2	=	=	PUNCT
ejpam-3275	195	3	∅.	∅.	NOUN
ejpam-3275	195	4	that	that	PRON
ejpam-3275	195	5	is	be	AUX
ejpam-3275	195	6	,	,	PUNCT
ejpam-3275	195	7	β(i)m	β(i)m	PROPN
ejpam-3275	195	8	⊆	⊆	NUM
ejpam-3275	195	9	p	p	NOUN
ejpam-3275	195	10	or	or	CCONJ
ejpam-3275	195	11	β(n	β(n	NUM
ejpam-3275	195	12	)	)	PUNCT
ejpam-3275	195	13	⊆	⊆	NUM
ejpam-3275	195	14	p	p	NOUN
ejpam-3275	195	15	.	.	PUNCT
ejpam-3275	196	1	we	we	PRON
ejpam-3275	196	2	already	already	ADV
ejpam-3275	196	3	show	show	VERB
ejpam-3275	196	4	that	that	SCONJ
ejpam-3275	196	5	β(i	β(i	PRON
ejpam-3275	196	6	)	)	PUNCT
ejpam-3275	196	7	⊆	⊆	NUM
ejpam-3275	196	8	(	(	PUNCT
ejpam-3275	196	9	p	p	X
ejpam-3275	196	10	:	:	PUNCT
ejpam-3275	196	11	m	m	NOUN
ejpam-3275	196	12	)	)	PUNCT
ejpam-3275	196	13	or	or	CCONJ
ejpam-3275	196	14	β(n	β(n	NUM
ejpam-3275	196	15	)	)	PUNCT
ejpam-3275	196	16	⊆	⊆	NUM
ejpam-3275	196	17	p	p	NOUN
ejpam-3275	196	18	.	.	PUNCT
ejpam-3275	197	1	this	this	PRON
ejpam-3275	197	2	means	mean	VERB
ejpam-3275	197	3	i	i	PROPN
ejpam-3275	197	4	⊆	⊆	NUM
ejpam-3275	197	5	α((p	α((p	NOUN
ejpam-3275	197	6	:	:	PUNCT
ejpam-3275	197	7	m	m	X
ejpam-3275	197	8	)	)	PUNCT
ejpam-3275	197	9	)	)	PUNCT
ejpam-3275	197	10	or	or	CCONJ
ejpam-3275	197	11	n	n	PRON
ejpam-3275	197	12	⊆	⊆	NUM
ejpam-3275	197	13	α(p	α(p	NUM
ejpam-3275	197	14	)	)	PUNCT
ejpam-3275	197	15	.	.	PUNCT
ejpam-3275	198	1	therefore	therefore	ADV
ejpam-3275	198	2	p	p	PROPN
ejpam-3275	198	3	is	be	AUX
ejpam-3275	198	4	an	an	DET
ejpam-3275	198	5	α	α	NOUN
ejpam-3275	198	6	-	-	ADJ
ejpam-3275	198	7	prime	prime	ADJ
ejpam-3275	198	8	submodule	submodule	NOUN
ejpam-3275	198	9	of	of	ADP
ejpam-3275	198	10	m	m	PROPN
ejpam-3275	198	11	.	.	PUNCT
ejpam-3275	199	1	proposition	proposition	NOUN
ejpam-3275	199	2	3	3	X
ejpam-3275	199	3	.	.	PUNCT
ejpam-3275	200	1	let	let	VERB
ejpam-3275	200	2	m	m	PRON
ejpam-3275	200	3	be	be	AUX
ejpam-3275	200	4	an	an	DET
ejpam-3275	200	5	r	r	NOUN
ejpam-3275	200	6	-	-	PUNCT
ejpam-3275	200	7	module	module	NOUN
ejpam-3275	200	8	and	and	CCONJ
ejpam-3275	200	9	x	x	AUX
ejpam-3275	200	10	be	be	AUX
ejpam-3275	200	11	an	an	DET
ejpam-3275	200	12	α	α	NOUN
ejpam-3275	200	13	-	-	PUNCT
ejpam-3275	200	14	multiplicative	multiplicative	ADJ
ejpam-3275	200	15	system	system	NOUN
ejpam-3275	200	16	.	.	PUNCT
ejpam-3275	201	1	if	if	SCONJ
ejpam-3275	201	2	p	p	NOUN
ejpam-3275	201	3	is	be	AUX
ejpam-3275	201	4	a	a	DET
ejpam-3275	201	5	submodule	submodule	NOUN
ejpam-3275	201	6	of	of	ADP
ejpam-3275	201	7	m	m	PRON
ejpam-3275	201	8	maximal	maximal	ADJ
ejpam-3275	201	9	with	with	ADP
ejpam-3275	201	10	respect	respect	NOUN
ejpam-3275	201	11	to	to	ADP
ejpam-3275	201	12	the	the	DET
ejpam-3275	201	13	property	property	NOUN
ejpam-3275	201	14	that	that	PRON
ejpam-3275	201	15	p	p	ADJ
ejpam-3275	201	16	∩	∩	NOUN
ejpam-3275	201	17	x	x	NOUN
ejpam-3275	201	18	=	=	SYM
ejpam-3275	201	19	∅	∅	NOUN
ejpam-3275	201	20	,	,	PUNCT
ejpam-3275	201	21	then	then	ADV
ejpam-3275	201	22	p	p	PROPN
ejpam-3275	201	23	is	be	AUX
ejpam-3275	201	24	an	an	DET
ejpam-3275	201	25	α	α	NOUN
ejpam-3275	201	26	-	-	ADJ
ejpam-3275	201	27	prime	prime	ADJ
ejpam-3275	201	28	submodule	submodule	NOUN
ejpam-3275	201	29	of	of	ADP
ejpam-3275	201	30	m	m	PROPN
ejpam-3275	201	31	.	.	PUNCT
ejpam-3275	202	1	proof	proof	NOUN
ejpam-3275	202	2	.	.	PUNCT
ejpam-3275	203	1	assume	assume	VERB
ejpam-3275	203	2	that	that	SCONJ
ejpam-3275	203	3	p	p	NOUN
ejpam-3275	203	4	is	be	AUX
ejpam-3275	203	5	a	a	DET
ejpam-3275	203	6	submodule	submodule	NOUN
ejpam-3275	203	7	of	of	ADP
ejpam-3275	203	8	m	m	PRON
ejpam-3275	203	9	maximal	maximal	ADJ
ejpam-3275	203	10	with	with	ADP
ejpam-3275	203	11	respect	respect	NOUN
ejpam-3275	203	12	to	to	ADP
ejpam-3275	203	13	the	the	DET
ejpam-3275	203	14	property	property	NOUN
ejpam-3275	203	15	that	that	PRON
ejpam-3275	203	16	p	p	X
ejpam-3275	203	17	∩	∩	NOUN
ejpam-3275	203	18	x	x	SYM
ejpam-3275	203	19	=	=	PUNCT
ejpam-3275	203	20	∅.	∅.	NOUN
ejpam-3275	203	21	let	let	VERB
ejpam-3275	203	22	i	i	PRON
ejpam-3275	203	23	be	be	AUX
ejpam-3275	203	24	an	an	DET
ejpam-3275	203	25	ideal	ideal	NOUN
ejpam-3275	203	26	of	of	ADP
ejpam-3275	203	27	r	r	NOUN
ejpam-3275	203	28	and	and	CCONJ
ejpam-3275	203	29	n	n	CCONJ
ejpam-3275	203	30	be	be	VERB
ejpam-3275	203	31	a	a	DET
ejpam-3275	203	32	submodule	submodule	NOUN
ejpam-3275	203	33	of	of	ADP
ejpam-3275	203	34	m	m	PROPN
ejpam-3275	203	35	.	.	PUNCT
ejpam-3275	204	1	now	now	ADV
ejpam-3275	204	2	,	,	PUNCT
ejpam-3275	204	3	assume	assume	VERB
ejpam-3275	204	4	that	that	SCONJ
ejpam-3275	205	1	i	i	PRON
ejpam-3275	205	2	*	*	PUNCT
ejpam-3275	205	3	α((p	α((p	NOUN
ejpam-3275	205	4	:	:	PUNCT
ejpam-3275	205	5	m	m	X
ejpam-3275	205	6	)	)	PUNCT
ejpam-3275	205	7	)	)	PUNCT
ejpam-3275	205	8	and	and	CCONJ
ejpam-3275	205	9	n	n	PROPN
ejpam-3275	205	10	*	*	PROPN
ejpam-3275	205	11	α(p	α(p	PROPN
ejpam-3275	205	12	)	)	PUNCT
ejpam-3275	205	13	.	.	PUNCT
ejpam-3275	206	1	hence	hence	ADV
ejpam-3275	206	2	β(i)m	β(i)m	PROPN
ejpam-3275	206	3	*	*	PUNCT
ejpam-3275	206	4	p	p	NOUN
ejpam-3275	206	5	and	and	CCONJ
ejpam-3275	206	6	β(n	β(n	NUM
ejpam-3275	206	7	)	)	PUNCT
ejpam-3275	207	1	*	*	PUNCT
ejpam-3275	208	1	p	p	NOUN
ejpam-3275	208	2	.	.	PUNCT
ejpam-3275	209	1	then	then	ADV
ejpam-3275	209	2	(	(	PUNCT
ejpam-3275	209	3	p	p	X
ejpam-3275	209	4	+	+	NOUN
ejpam-3275	209	5	β(i)m	β(i)m	ADJ
ejpam-3275	209	6	)	)	PUNCT
ejpam-3275	209	7	∩	∩	NOUN
ejpam-3275	209	8	x	x	SYM
ejpam-3275	209	9	6=	6=	ADP
ejpam-3275	209	10	∅	∅	NOUN
ejpam-3275	209	11	and	and	CCONJ
ejpam-3275	209	12	(	(	PUNCT
ejpam-3275	209	13	p	p	X
ejpam-3275	209	14	+	+	NOUN
ejpam-3275	209	15	β(n	β(n	NUM
ejpam-3275	209	16	)	)	PUNCT
ejpam-3275	209	17	)	)	PUNCT
ejpam-3275	210	1	∩	∩	NOUN
ejpam-3275	210	2	x	x	SYM
ejpam-3275	210	3	6=	6=	ADP
ejpam-3275	210	4	∅.	∅.	NOUN
ejpam-3275	210	5	since	since	SCONJ
ejpam-3275	210	6	x	x	PRON
ejpam-3275	210	7	is	be	AUX
ejpam-3275	210	8	an	an	DET
ejpam-3275	210	9	α	α	NOUN
ejpam-3275	210	10	-	-	PUNCT
ejpam-3275	210	11	multiplicative	multiplicative	ADJ
ejpam-3275	210	12	system	system	NOUN
ejpam-3275	210	13	,	,	PUNCT
ejpam-3275	210	14	(	(	PUNCT
ejpam-3275	210	15	p	p	X
ejpam-3275	210	16	+	+	NOUN
ejpam-3275	210	17	iβ(n	iβ(n	NOUN
ejpam-3275	210	18	)	)	PUNCT
ejpam-3275	210	19	)	)	PUNCT
ejpam-3275	211	1	∩x	∩x	X
ejpam-3275	211	2	6=	6=	PUNCT
ejpam-3275	211	3	∅.	∅.	VERB
ejpam-3275	211	4	since	since	SCONJ
ejpam-3275	211	5	p	p	NOUN
ejpam-3275	211	6	∩x	∩x	NOUN
ejpam-3275	211	7	=	=	SYM
ejpam-3275	211	8	∅	∅	NOUN
ejpam-3275	211	9	,	,	PUNCT
ejpam-3275	211	10	iβ(n	iβ(n	NOUN
ejpam-3275	211	11	)	)	PUNCT
ejpam-3275	211	12	*	*	PUNCT
ejpam-3275	212	1	p	p	NOUN
ejpam-3275	212	2	.	.	PUNCT
ejpam-3275	213	1	this	this	PRON
ejpam-3275	213	2	implies	imply	VERB
ejpam-3275	213	3	that	that	SCONJ
ejpam-3275	213	4	p	p	PROPN
ejpam-3275	213	5	is	be	AUX
ejpam-3275	213	6	an	an	DET
ejpam-3275	213	7	α	α	NOUN
ejpam-3275	213	8	-	-	ADJ
ejpam-3275	213	9	prime	prime	ADJ
ejpam-3275	213	10	submodule	submodule	NOUN
ejpam-3275	213	11	of	of	ADP
ejpam-3275	213	12	m	m	PROPN
ejpam-3275	213	13	.	.	PUNCT
ejpam-3275	214	1	definition	definition	NOUN
ejpam-3275	214	2	3	3	NUM
ejpam-3275	214	3	.	.	PUNCT
ejpam-3275	215	1	let	let	VERB
ejpam-3275	215	2	m	m	PRON
ejpam-3275	215	3	be	be	AUX
ejpam-3275	215	4	an	an	DET
ejpam-3275	215	5	r	r	NOUN
ejpam-3275	215	6	-	-	PUNCT
ejpam-3275	215	7	module	module	NOUN
ejpam-3275	215	8	and	and	CCONJ
ejpam-3275	215	9	n	n	CCONJ
ejpam-3275	215	10	be	be	VERB
ejpam-3275	215	11	a	a	DET
ejpam-3275	215	12	submodule	submodule	NOUN
ejpam-3275	215	13	of	of	ADP
ejpam-3275	215	14	m	m	PROPN
ejpam-3275	215	15	.	.	PUNCT
ejpam-3275	216	1	if	if	SCONJ
ejpam-3275	216	2	there	there	PRON
ejpam-3275	216	3	is	be	VERB
ejpam-3275	216	4	an	an	DET
ejpam-3275	216	5	α	α	NOUN
ejpam-3275	216	6	-	-	ADJ
ejpam-3275	216	7	prime	prime	ADJ
ejpam-3275	216	8	submodule	submodule	NOUN
ejpam-3275	216	9	of	of	ADP
ejpam-3275	216	10	m	m	PROPN
ejpam-3275	216	11	containing	contain	VERB
ejpam-3275	216	12	n	n	NOUN
ejpam-3275	216	13	,	,	PUNCT
ejpam-3275	216	14	then	then	ADV
ejpam-3275	216	15	we	we	PRON
ejpam-3275	216	16	define	define	VERB
ejpam-3275	216	17	α	α	PRON
ejpam-3275	216	18	√	√	NOUN
ejpam-3275	216	19	n	n	NOUN
ejpam-3275	216	20	=	=	PRON
ejpam-3275	216	21	{	{	PUNCT
ejpam-3275	216	22	x	x	NOUN
ejpam-3275	216	23	∈m	∈m	NOUN
ejpam-3275	216	24	|	|	ADV
ejpam-3275	216	25	every	every	DET
ejpam-3275	216	26	α	α	NOUN
ejpam-3275	216	27	-	-	ADJ
ejpam-3275	216	28	multiplicative	multiplicative	ADJ
ejpam-3275	216	29	system	system	NOUN
ejpam-3275	216	30	containing	contain	VERB
ejpam-3275	216	31	x	x	X
ejpam-3275	216	32	meets	meet	VERB
ejpam-3275	216	33	n	n	X
ejpam-3275	216	34	}	}	PUNCT
ejpam-3275	216	35	.	.	PUNCT
ejpam-3275	217	1	if	if	SCONJ
ejpam-3275	217	2	there	there	PRON
ejpam-3275	217	3	is	be	VERB
ejpam-3275	217	4	no	no	DET
ejpam-3275	217	5	a	a	DET
ejpam-3275	217	6	α	α	NUM
ejpam-3275	217	7	-	-	ADJ
ejpam-3275	217	8	prime	prime	ADJ
ejpam-3275	217	9	submodule	submodule	NOUN
ejpam-3275	217	10	of	of	ADP
ejpam-3275	217	11	m	m	PROPN
ejpam-3275	217	12	containing	contain	VERB
ejpam-3275	217	13	n	n	NOUN
ejpam-3275	217	14	,	,	PUNCT
ejpam-3275	217	15	then	then	ADV
ejpam-3275	217	16	we	we	PRON
ejpam-3275	217	17	define	define	VERB
ejpam-3275	217	18	α	α	PRON
ejpam-3275	217	19	√	√	NOUN
ejpam-3275	217	20	n	n	NOUN
ejpam-3275	217	21	=	=	NOUN
ejpam-3275	217	22	m	m	PROPN
ejpam-3275	217	23	.	.	PUNCT
ejpam-3275	218	1	theorem	theorem	NOUN
ejpam-3275	218	2	2	2	X
ejpam-3275	218	3	.	.	PUNCT
ejpam-3275	219	1	let	let	VERB
ejpam-3275	219	2	m	m	PRON
ejpam-3275	219	3	be	be	AUX
ejpam-3275	219	4	an	an	DET
ejpam-3275	219	5	r	r	NOUN
ejpam-3275	219	6	-	-	PUNCT
ejpam-3275	219	7	module	module	NOUN
ejpam-3275	219	8	and	and	CCONJ
ejpam-3275	219	9	n	n	CCONJ
ejpam-3275	219	10	be	be	VERB
ejpam-3275	219	11	a	a	DET
ejpam-3275	219	12	submodule	submodule	NOUN
ejpam-3275	219	13	of	of	ADP
ejpam-3275	219	14	m	m	PROPN
ejpam-3275	219	15	.	.	PUNCT
ejpam-3275	220	1	then	then	ADV
ejpam-3275	220	2	either	either	CCONJ
ejpam-3275	220	3	α	α	NOUN
ejpam-3275	220	4	√	√	VERB
ejpam-3275	220	5	n	n	NOUN
ejpam-3275	220	6	=	=	SYM
ejpam-3275	220	7	m	m	PROPN
ejpam-3275	220	8	or	or	CCONJ
ejpam-3275	220	9	α	α	PRON
ejpam-3275	220	10	√	√	NOUN
ejpam-3275	220	11	n	n	NUM
ejpam-3275	220	12	is	be	AUX
ejpam-3275	220	13	the	the	DET
ejpam-3275	220	14	intersection	intersection	NOUN
ejpam-3275	220	15	of	of	ADP
ejpam-3275	220	16	all	all	DET
ejpam-3275	220	17	α	α	PRON
ejpam-3275	220	18	-	-	ADJ
ejpam-3275	220	19	prime	prime	ADJ
ejpam-3275	220	20	submodule	submodule	NOUN
ejpam-3275	220	21	of	of	ADP
ejpam-3275	220	22	m	m	PROPN
ejpam-3275	220	23	containing	contain	VERB
ejpam-3275	220	24	n	n	NOUN
ejpam-3275	220	25	.	.	PUNCT
ejpam-3275	221	1	proof	proof	NOUN
ejpam-3275	221	2	.	.	PUNCT
ejpam-3275	222	1	assume	assume	VERB
ejpam-3275	222	2	that	that	SCONJ
ejpam-3275	222	3	α	α	NOUN
ejpam-3275	222	4	√	√	VERB
ejpam-3275	222	5	n	n	PROPN
ejpam-3275	222	6	6=	6=	NOUN
ejpam-3275	222	7	m	m	VERB
ejpam-3275	222	8	.	.	PUNCT
ejpam-3275	223	1	let	let	VERB
ejpam-3275	223	2	x	x	PUNCT
ejpam-3275	223	3	∈	∈	VERB
ejpam-3275	223	4	α	α	NOUN
ejpam-3275	223	5	√	√	NOUN
ejpam-3275	223	6	n	n	PROPN
ejpam-3275	223	7	and	and	CCONJ
ejpam-3275	223	8	p	p	NOUN
ejpam-3275	223	9	be	be	AUX
ejpam-3275	223	10	an	an	DET
ejpam-3275	223	11	α	α	NOUN
ejpam-3275	223	12	-	-	ADJ
ejpam-3275	223	13	prime	prime	ADJ
ejpam-3275	223	14	submodule	submodule	NOUN
ejpam-3275	223	15	of	of	ADP
ejpam-3275	223	16	m	m	PROPN
ejpam-3275	223	17	containing	contain	VERB
ejpam-3275	223	18	n	n	PRON
ejpam-3275	223	19	.	.	PUNCT
ejpam-3275	224	1	by	by	ADP
ejpam-3275	224	2	proposition	proposition	NOUN
ejpam-3275	224	3	2	2	NUM
ejpam-3275	224	4	,	,	PUNCT
ejpam-3275	224	5	m\p	m\p	PROPN
ejpam-3275	224	6	is	be	AUX
ejpam-3275	224	7	an	an	DET
ejpam-3275	224	8	α	α	NOUN
ejpam-3275	224	9	-	-	PUNCT
ejpam-3275	224	10	multiplicative	multiplicative	ADJ
ejpam-3275	224	11	system	system	NOUN
ejpam-3275	224	12	and	and	CCONJ
ejpam-3275	224	13	n	n	NOUN
ejpam-3275	224	14	∩	∩	NOUN
ejpam-3275	224	15	(	(	PUNCT
ejpam-3275	224	16	m\p	m\p	NOUN
ejpam-3275	224	17	)	)	PUNCT
ejpam-3275	225	1	=	=	PUNCT
ejpam-3275	225	2	∅.	∅.	VERB
ejpam-3275	225	3	hence	hence	ADV
ejpam-3275	225	4	x	x	PUNCT
ejpam-3275	225	5	∈	∈	PROPN
ejpam-3275	225	6	p	p	NOUN
ejpam-3275	225	7	.	.	PUNCT
ejpam-3275	226	1	conversely	conversely	ADV
ejpam-3275	226	2	,	,	PUNCT
ejpam-3275	226	3	let	let	VERB
ejpam-3275	226	4	x	x	PART
ejpam-3275	226	5	∈m	∈m	NOUN
ejpam-3275	226	6	be	be	AUX
ejpam-3275	226	7	such	such	ADJ
ejpam-3275	226	8	that	that	PRON
ejpam-3275	226	9	x	x	PUNCT
ejpam-3275	226	10	/∈	/∈	PUNCT
ejpam-3275	227	1	β	β	NOUN
ejpam-3275	227	2	√	√	NOUN
ejpam-3275	227	3	n	n	ADV
ejpam-3275	227	4	.	.	PUNCT
ejpam-3275	228	1	let	let	VERB
ejpam-3275	228	2	s	s	PRON
ejpam-3275	228	3	be	be	AUX
ejpam-3275	228	4	an	an	DET
ejpam-3275	228	5	α	α	PRON
ejpam-3275	228	6	-	-	PUNCT
ejpam-3275	228	7	multiplicative	multiplicative	ADJ
ejpam-3275	228	8	system	system	NOUN
ejpam-3275	228	9	such	such	ADJ
ejpam-3275	228	10	that	that	SCONJ
ejpam-3275	228	11	x	x	SYM
ejpam-3275	228	12	∈	∈	NOUN
ejpam-3275	228	13	s	s	X
ejpam-3275	228	14	and	and	CCONJ
ejpam-3275	228	15	s	s	X
ejpam-3275	228	16	∩	∩	ADJ
ejpam-3275	228	17	n	n	NOUN
ejpam-3275	228	18	=	=	PUNCT
ejpam-3275	228	19	∅.	∅.	NOUN
ejpam-3275	228	20	by	by	ADP
ejpam-3275	228	21	zorn	zorn	PROPN
ejpam-3275	228	22	’s	’s	PART
ejpam-3275	228	23	lemma	lemma	PROPN
ejpam-3275	228	24	on	on	ADP
ejpam-3275	228	25	the	the	DET
ejpam-3275	228	26	set	set	NOUN
ejpam-3275	228	27	of	of	ADP
ejpam-3275	228	28	submodule	submodule	PROPN
ejpam-3275	228	29	j	j	PROPN
ejpam-3275	228	30	of	of	AUX
ejpam-3275	228	31	m	m	PROPN
ejpam-3275	228	32	containing	contain	VERB
ejpam-3275	228	33	n	n	ADV
ejpam-3275	228	34	and	and	CCONJ
ejpam-3275	228	35	s	s	PROPN
ejpam-3275	228	36	∩	∩	ADJ
ejpam-3275	228	37	j	j	NOUN
ejpam-3275	228	38	=	=	SYM
ejpam-3275	228	39	∅	∅	NOUN
ejpam-3275	228	40	,	,	PUNCT
ejpam-3275	228	41	there	there	PRON
ejpam-3275	228	42	exists	exist	VERB
ejpam-3275	228	43	a	a	DET
ejpam-3275	228	44	maximal	maximal	ADJ
ejpam-3275	228	45	submodule	submodule	NOUN
ejpam-3275	228	46	k	k	PROPN
ejpam-3275	228	47	of	of	ADP
ejpam-3275	228	48	m	m	PRON
ejpam-3275	228	49	such	such	ADJ
ejpam-3275	228	50	that	that	PRON
ejpam-3275	228	51	s	s	VERB
ejpam-3275	228	52	∩k	∩k	NOUN
ejpam-3275	228	53	=	=	PUNCT
ejpam-3275	228	54	∅.	∅.	NOUN
ejpam-3275	228	55	by	by	ADP
ejpam-3275	228	56	proposition	proposition	NOUN
ejpam-3275	228	57	3	3	NUM
ejpam-3275	228	58	,	,	PUNCT
ejpam-3275	228	59	k	k	PROPN
ejpam-3275	228	60	is	be	AUX
ejpam-3275	228	61	a	a	DET
ejpam-3275	228	62	α	α	NOUN
ejpam-3275	228	63	-	-	ADJ
ejpam-3275	228	64	prime	prime	ADJ
ejpam-3275	228	65	submodule	submodule	NOUN
ejpam-3275	228	66	of	of	ADP
ejpam-3275	228	67	m	m	PROPN
ejpam-3275	228	68	.	.	PUNCT
ejpam-3275	229	1	hence	hence	ADV
ejpam-3275	229	2	x	x	PROPN
ejpam-3275	229	3	/∈	/∈	PUNCT
ejpam-3275	229	4	k.	k.	NOUN
ejpam-3275	230	1	3	3	X
ejpam-3275	230	2	.	.	X
ejpam-3275	230	3	weakly	weakly	ADJ
ejpam-3275	230	4	α	α	NOUN
ejpam-3275	230	5	-	-	ADJ
ejpam-3275	230	6	prime	prime	ADJ
ejpam-3275	230	7	submodules	submodule	NOUN
ejpam-3275	230	8	in	in	ADP
ejpam-3275	230	9	this	this	DET
ejpam-3275	230	10	section	section	NOUN
ejpam-3275	230	11	we	we	PRON
ejpam-3275	230	12	begin	begin	VERB
ejpam-3275	230	13	with	with	ADP
ejpam-3275	230	14	the	the	DET
ejpam-3275	230	15	definition	definition	NOUN
ejpam-3275	230	16	of	of	ADP
ejpam-3275	230	17	weakly	weakly	ADJ
ejpam-3275	230	18	α	α	NOUN
ejpam-3275	230	19	-	-	ADJ
ejpam-3275	230	20	prime	prime	ADJ
ejpam-3275	230	21	submodules	submodule	NOUN
ejpam-3275	230	22	which	which	PRON
ejpam-3275	230	23	is	be	AUX
ejpam-3275	230	24	a	a	DET
ejpam-3275	230	25	generalization	generalization	NOUN
ejpam-3275	230	26	of	of	ADP
ejpam-3275	230	27	α	α	NOUN
ejpam-3275	230	28	-	-	ADJ
ejpam-3275	230	29	prime	prime	ADJ
ejpam-3275	230	30	submodules	submodule	NOUN
ejpam-3275	230	31	.	.	PUNCT
ejpam-3275	231	1	in	in	ADP
ejpam-3275	231	2	[	[	X
ejpam-3275	231	3	2	2	NUM
ejpam-3275	231	4	]	]	PUNCT
ejpam-3275	231	5	,	,	PUNCT
ejpam-3275	231	6	s.e	s.e	PROPN
ejpam-3275	231	7	.	.	PROPN
ejpam-3275	231	8	atani	atani	PROPN
ejpam-3275	231	9	and	and	CCONJ
ejpam-3275	231	10	f.	f.	PROPN
ejpam-3275	231	11	farzalipour	farzalipour	PROPN
ejpam-3275	231	12	gave	give	VERB
ejpam-3275	231	13	the	the	DET
ejpam-3275	231	14	notion	notion	NOUN
ejpam-3275	231	15	of	of	ADP
ejpam-3275	231	16	weakly	weakly	ADJ
ejpam-3275	231	17	prime	prime	ADJ
ejpam-3275	231	18	submodules	submodule	NOUN
ejpam-3275	231	19	stated	state	VERB
ejpam-3275	231	20	that	that	SCONJ
ejpam-3275	231	21	a	a	DET
ejpam-3275	231	22	proper	proper	ADJ
ejpam-3275	231	23	submodule	submodule	NOUN
ejpam-3275	231	24	p	p	NOUN
ejpam-3275	231	25	of	of	ADP
ejpam-3275	231	26	a	a	DET
ejpam-3275	231	27	left	left	ADJ
ejpam-3275	231	28	r	r	NOUN
ejpam-3275	231	29	-	-	PUNCT
ejpam-3275	231	30	module	module	NOUN
ejpam-3275	231	31	m	m	NOUN
ejpam-3275	231	32	is	be	AUX
ejpam-3275	231	33	called	call	VERB
ejpam-3275	231	34	weakly	weakly	ADJ
ejpam-3275	231	35	prime	prime	ADJ
ejpam-3275	231	36	if	if	SCONJ
ejpam-3275	231	37	0	0	NUM
ejpam-3275	231	38	6=	6=	NUM
ejpam-3275	231	39	rm	rm	PROPN
ejpam-3275	231	40	∈	∈	PROPN
ejpam-3275	231	41	p	p	PROPN
ejpam-3275	231	42	for	for	ADP
ejpam-3275	231	43	some	some	DET
ejpam-3275	231	44	r	r	NOUN
ejpam-3275	231	45	∈	∈	NOUN
ejpam-3275	231	46	r	r	NOUN
ejpam-3275	231	47	and	and	CCONJ
ejpam-3275	231	48	m	m	NOUN
ejpam-3275	231	49	∈m	∈m	NOUN
ejpam-3275	231	50	,	,	PUNCT
ejpam-3275	231	51	then	then	ADV
ejpam-3275	231	52	r	r	NOUN
ejpam-3275	231	53	∈	∈	PROPN
ejpam-3275	231	54	(	(	PUNCT
ejpam-3275	231	55	p	p	X
ejpam-3275	231	56	:	:	PUNCT
ejpam-3275	231	57	m	m	NUM
ejpam-3275	231	58	)	)	PUNCT
ejpam-3275	231	59	or	or	CCONJ
ejpam-3275	231	60	m	m	PROPN
ejpam-3275	231	61	∈	∈	NOUN
ejpam-3275	231	62	p	p	NOUN
ejpam-3275	231	63	where	where	SCONJ
ejpam-3275	231	64	(	(	PUNCT
ejpam-3275	231	65	n	n	NUM
ejpam-3275	231	66	:	:	PUNCT
ejpam-3275	231	67	m	m	X
ejpam-3275	231	68	)	)	PUNCT
ejpam-3275	231	69	=	=	PRON
ejpam-3275	231	70	{	{	PUNCT
ejpam-3275	231	71	r	r	NOUN
ejpam-3275	231	72	∈	∈	NOUN
ejpam-3275	231	73	r	r	NOUN
ejpam-3275	231	74	|	|	NOUN
ejpam-3275	231	75	rm	rm	NOUN
ejpam-3275	231	76	⊆	⊆	NUM
ejpam-3275	231	77	n	n	CCONJ
ejpam-3275	231	78	}	}	PUNCT
ejpam-3275	231	79	.	.	PUNCT
ejpam-3275	232	1	t.	t.	PROPN
ejpam-3275	232	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	232	3	/	/	SYM
ejpam-3275	232	4	eur	eur	PROPN
ejpam-3275	232	5	.	.	PUNCT
ejpam-3275	233	1	j.	j.	PROPN
ejpam-3275	233	2	pure	pure	PROPN
ejpam-3275	233	3	appl	appl	PROPN
ejpam-3275	233	4	.	.	PROPN
ejpam-3275	233	5	math	math	PROPN
ejpam-3275	233	6	,	,	PUNCT
ejpam-3275	233	7	11	11	NUM
ejpam-3275	233	8	(	(	PUNCT
ejpam-3275	233	9	3	3	NUM
ejpam-3275	233	10	)	)	PUNCT
ejpam-3275	233	11	(	(	PUNCT
ejpam-3275	233	12	2018	2018	NUM
ejpam-3275	233	13	)	)	PUNCT
ejpam-3275	233	14	,	,	PUNCT
ejpam-3275	233	15	730	730	NUM
ejpam-3275	233	16	-	-	SYM
ejpam-3275	233	17	739	739	NUM
ejpam-3275	233	18	735	735	NUM
ejpam-3275	233	19	definition	definition	NOUN
ejpam-3275	233	20	4	4	NUM
ejpam-3275	233	21	.	.	PUNCT
ejpam-3275	234	1	let	let	VERB
ejpam-3275	234	2	p	p	PRON
ejpam-3275	234	3	be	be	AUX
ejpam-3275	234	4	a	a	DET
ejpam-3275	234	5	proper	proper	ADJ
ejpam-3275	234	6	submodule	submodule	NOUN
ejpam-3275	234	7	of	of	ADP
ejpam-3275	234	8	m	m	PROPN
ejpam-3275	234	9	.	.	PUNCT
ejpam-3275	235	1	we	we	PRON
ejpam-3275	235	2	call	call	VERB
ejpam-3275	235	3	p	p	NOUN
ejpam-3275	235	4	is	be	AUX
ejpam-3275	235	5	weakly	weakly	ADJ
ejpam-3275	235	6	α	α	NOUN
ejpam-3275	235	7	-	-	NOUN
ejpam-3275	235	8	prime	prime	NOUN
ejpam-3275	235	9	if	if	SCONJ
ejpam-3275	235	10	for	for	ADP
ejpam-3275	235	11	any	any	DET
ejpam-3275	235	12	elements	element	NOUN
ejpam-3275	235	13	r	r	NOUN
ejpam-3275	235	14	∈	∈	NOUN
ejpam-3275	235	15	r	r	NOUN
ejpam-3275	235	16	and	and	CCONJ
ejpam-3275	235	17	m	m	PROPN
ejpam-3275	235	18	∈	∈	NOUN
ejpam-3275	235	19	m	m	VERB
ejpam-3275	235	20	such	such	ADJ
ejpam-3275	235	21	that	that	SCONJ
ejpam-3275	235	22	r(m	r(m	PROPN
ejpam-3275	235	23	+	+	NOUN
ejpam-3275	235	24	m	m	NOUN
ejpam-3275	235	25	)	)	PUNCT
ejpam-3275	235	26	∈	∈	PROPN
ejpam-3275	235	27	p\{0	p\{0	NOUN
ejpam-3275	235	28	}	}	PUNCT
ejpam-3275	235	29	,	,	PUNCT
ejpam-3275	235	30	we	we	PRON
ejpam-3275	235	31	have	have	VERB
ejpam-3275	235	32	r	r	NOUN
ejpam-3275	235	33	+	+	NOUN
ejpam-3275	235	34	r	r	NOUN
ejpam-3275	235	35	∈	∈	NOUN
ejpam-3275	235	36	(	(	PUNCT
ejpam-3275	235	37	p	p	X
ejpam-3275	235	38	:	:	PUNCT
ejpam-3275	235	39	m	m	NUM
ejpam-3275	235	40	)	)	PUNCT
ejpam-3275	235	41	or	or	CCONJ
ejpam-3275	235	42	m+m	m+m	PROPN
ejpam-3275	235	43	∈	∈	PROPN
ejpam-3275	235	44	p	p	NOUN
ejpam-3275	235	45	.	.	PUNCT
ejpam-3275	236	1	every	every	DET
ejpam-3275	236	2	α	α	X
ejpam-3275	236	3	-	-	ADJ
ejpam-3275	236	4	prime	prime	ADJ
ejpam-3275	236	5	submodule	submodule	NOUN
ejpam-3275	236	6	is	be	AUX
ejpam-3275	236	7	weakly	weakly	ADJ
ejpam-3275	236	8	α	α	PRON
ejpam-3275	236	9	-	-	ADJ
ejpam-3275	236	10	prime	prime	ADJ
ejpam-3275	236	11	submodule	submodule	NOUN
ejpam-3275	236	12	.	.	PUNCT
ejpam-3275	237	1	but	but	CCONJ
ejpam-3275	237	2	the	the	DET
ejpam-3275	237	3	converse	converse	NOUN
ejpam-3275	237	4	need	need	AUX
ejpam-3275	237	5	not	not	PART
ejpam-3275	237	6	be	be	AUX
ejpam-3275	237	7	true	true	ADJ
ejpam-3275	237	8	.	.	PUNCT
ejpam-3275	238	1	for	for	ADP
ejpam-3275	238	2	example	example	NOUN
ejpam-3275	238	3	,	,	PUNCT
ejpam-3275	238	4	{	{	PUNCT
ejpam-3275	238	5	0̄	0̄	NOUN
ejpam-3275	238	6	}	}	PUNCT
ejpam-3275	238	7	is	be	AUX
ejpam-3275	238	8	weakly	weakly	ADJ
ejpam-3275	238	9	α	α	NOUN
ejpam-3275	238	10	-	-	ADJ
ejpam-3275	238	11	prime	prime	NOUN
ejpam-3275	238	12	but	but	CCONJ
ejpam-3275	238	13	is	be	AUX
ejpam-3275	238	14	not	not	PART
ejpam-3275	238	15	α	α	NOUN
ejpam-3275	238	16	-	-	ADJ
ejpam-3275	238	17	prime	prime	ADJ
ejpam-3275	238	18	submodule	submodule	NOUN
ejpam-3275	238	19	of	of	ADP
ejpam-3275	238	20	z	z	NOUN
ejpam-3275	238	21	-	-	PUNCT
ejpam-3275	238	22	module	module	NOUN
ejpam-3275	238	23	z8	z8	NOUN
ejpam-3275	238	24	because	because	SCONJ
ejpam-3275	238	25	2	2	NUM
ejpam-3275	238	26	·	·	PUNCT
ejpam-3275	238	27	(	(	PUNCT
ejpam-3275	238	28	2̄	2̄	NOUN
ejpam-3275	238	29	+	+	NOUN
ejpam-3275	238	30	2̄	2̄	NUM
ejpam-3275	238	31	)	)	PUNCT
ejpam-3275	238	32	=	=	SYM
ejpam-3275	238	33	2	2	X
ejpam-3275	238	34	·	·	PUNCT
ejpam-3275	238	35	4̄	4̄	NOUN
ejpam-3275	239	1	=	=	PUNCT
ejpam-3275	239	2	8̄	8̄	NOUN
ejpam-3275	239	3	=	=	SYM
ejpam-3275	240	1	0̄	0̄	NUM
ejpam-3275	240	2	and	and	CCONJ
ejpam-3275	240	3	(	(	PUNCT
ejpam-3275	240	4	2	2	NUM
ejpam-3275	240	5	+	+	NUM
ejpam-3275	240	6	2)z8	2)z8	NOUN
ejpam-3275	240	7	*	*	PUNCT
ejpam-3275	240	8	{	{	PUNCT
ejpam-3275	240	9	0̄	0̄	NUM
ejpam-3275	240	10	}	}	PUNCT
ejpam-3275	240	11	and	and	CCONJ
ejpam-3275	240	12	2̄	2̄	PROPN
ejpam-3275	240	13	+	+	CCONJ
ejpam-3275	240	14	2̄	2̄	PROPN
ejpam-3275	240	15	6=	6=	NUM
ejpam-3275	241	1	0̄.	0̄.	ADP
ejpam-3275	241	2	next	next	ADV
ejpam-3275	241	3	we	we	PRON
ejpam-3275	241	4	give	give	VERB
ejpam-3275	241	5	several	several	ADJ
ejpam-3275	241	6	characterizations	characterization	NOUN
ejpam-3275	241	7	of	of	ADP
ejpam-3275	241	8	weakly	weakly	ADJ
ejpam-3275	241	9	α	α	NOUN
ejpam-3275	241	10	-	-	ADJ
ejpam-3275	241	11	prime	prime	ADJ
ejpam-3275	241	12	submodules	submodule	NOUN
ejpam-3275	241	13	.	.	PUNCT
ejpam-3275	242	1	theorem	theorem	NOUN
ejpam-3275	242	2	3	3	X
ejpam-3275	242	3	.	.	PUNCT
ejpam-3275	243	1	let	let	VERB
ejpam-3275	243	2	m	m	PRON
ejpam-3275	243	3	be	be	AUX
ejpam-3275	243	4	an	an	DET
ejpam-3275	243	5	r	r	NOUN
ejpam-3275	243	6	-	-	PUNCT
ejpam-3275	243	7	module	module	NOUN
ejpam-3275	243	8	and	and	CCONJ
ejpam-3275	243	9	p	p	NOUN
ejpam-3275	243	10	be	be	AUX
ejpam-3275	243	11	a	a	DET
ejpam-3275	243	12	submodule	submodule	NOUN
ejpam-3275	243	13	of	of	ADP
ejpam-3275	243	14	m	m	PROPN
ejpam-3275	243	15	.	.	PUNCT
ejpam-3275	244	1	the	the	DET
ejpam-3275	244	2	following	follow	VERB
ejpam-3275	244	3	statements	statement	NOUN
ejpam-3275	244	4	are	be	AUX
ejpam-3275	244	5	equivalent	equivalent	ADJ
ejpam-3275	244	6	.	.	PUNCT
ejpam-3275	245	1	(	(	PUNCT
ejpam-3275	245	2	i	i	NOUN
ejpam-3275	245	3	)	)	PUNCT
ejpam-3275	245	4	p	p	NOUN
ejpam-3275	245	5	is	be	AUX
ejpam-3275	245	6	a	a	DET
ejpam-3275	245	7	weakly	weakly	ADJ
ejpam-3275	245	8	α	α	NOUN
ejpam-3275	245	9	-	-	ADJ
ejpam-3275	245	10	prime	prime	ADJ
ejpam-3275	245	11	submodule	submodule	NOUN
ejpam-3275	245	12	of	of	ADP
ejpam-3275	245	13	m	m	PROPN
ejpam-3275	245	14	.	.	PUNCT
ejpam-3275	246	1	(	(	PUNCT
ejpam-3275	246	2	ii	ii	NOUN
ejpam-3275	246	3	)	)	PUNCT
ejpam-3275	246	4	for	for	ADP
ejpam-3275	246	5	any	any	DET
ejpam-3275	246	6	m	m	NOUN
ejpam-3275	246	7	∈m	∈m	NOUN
ejpam-3275	246	8	,	,	PUNCT
ejpam-3275	246	9	if	if	SCONJ
ejpam-3275	246	10	m+m	m+m	PROPN
ejpam-3275	246	11	/∈	/∈	PUNCT
ejpam-3275	247	1	p	p	NOUN
ejpam-3275	247	2	,	,	PUNCT
ejpam-3275	247	3	then	then	ADV
ejpam-3275	247	4	(	(	PUNCT
ejpam-3275	247	5	p	p	X
ejpam-3275	247	6	:	:	PUNCT
ejpam-3275	247	7	m+m	m+m	NUM
ejpam-3275	247	8	)	)	PUNCT
ejpam-3275	248	1	=	=	SYM
ejpam-3275	248	2	α((p	α((p	NOUN
ejpam-3275	248	3	:	:	PUNCT
ejpam-3275	248	4	m	m	X
ejpam-3275	248	5	)	)	PUNCT
ejpam-3275	248	6	)	)	PUNCT
ejpam-3275	248	7	∪	∪	ADP
ejpam-3275	248	8	α((0	α((0	PROPN
ejpam-3275	248	9	:	:	PUNCT
ejpam-3275	248	10	m	m	NOUN
ejpam-3275	248	11	)	)	PUNCT
ejpam-3275	248	12	)	)	PUNCT
ejpam-3275	248	13	.	.	PUNCT
ejpam-3275	249	1	(	(	PUNCT
ejpam-3275	249	2	iii	iii	X
ejpam-3275	249	3	)	)	PUNCT
ejpam-3275	249	4	for	for	ADP
ejpam-3275	249	5	any	any	DET
ejpam-3275	249	6	m	m	NOUN
ejpam-3275	249	7	∈	∈	NOUN
ejpam-3275	249	8	m	m	NOUN
ejpam-3275	249	9	,	,	PUNCT
ejpam-3275	249	10	if	if	SCONJ
ejpam-3275	249	11	m+m	m+m	PROPN
ejpam-3275	249	12	/∈	/∈	PUNCT
ejpam-3275	250	1	p	p	NOUN
ejpam-3275	250	2	,	,	PUNCT
ejpam-3275	250	3	then	then	ADV
ejpam-3275	250	4	(	(	PUNCT
ejpam-3275	250	5	p	p	X
ejpam-3275	250	6	:	:	PUNCT
ejpam-3275	250	7	m+m	m+m	NUM
ejpam-3275	250	8	)	)	PUNCT
ejpam-3275	251	1	=	=	SYM
ejpam-3275	251	2	α((p	α((p	NOUN
ejpam-3275	251	3	:	:	PUNCT
ejpam-3275	251	4	m	m	X
ejpam-3275	251	5	)	)	PUNCT
ejpam-3275	251	6	)	)	PUNCT
ejpam-3275	251	7	or	or	CCONJ
ejpam-3275	251	8	(	(	PUNCT
ejpam-3275	251	9	p	p	X
ejpam-3275	251	10	:	:	PUNCT
ejpam-3275	251	11	m+m	m+m	NUM
ejpam-3275	251	12	)	)	PUNCT
ejpam-3275	251	13	=	=	PUNCT
ejpam-3275	252	1	α((0	α((0	PROPN
ejpam-3275	252	2	:	:	PUNCT
ejpam-3275	252	3	m	m	X
ejpam-3275	252	4	)	)	PUNCT
ejpam-3275	252	5	)	)	PUNCT
ejpam-3275	252	6	.	.	PUNCT
ejpam-3275	253	1	proof	proof	NOUN
ejpam-3275	253	2	.	.	PUNCT
ejpam-3275	254	1	(	(	PUNCT
ejpam-3275	254	2	i)→	i)→	PROPN
ejpam-3275	254	3	(	(	PUNCT
ejpam-3275	254	4	ii	ii	NOUN
ejpam-3275	254	5	)	)	PUNCT
ejpam-3275	254	6	assume	assume	VERB
ejpam-3275	254	7	that	that	SCONJ
ejpam-3275	254	8	p	p	NOUN
ejpam-3275	254	9	is	be	AUX
ejpam-3275	254	10	a	a	DET
ejpam-3275	254	11	weakly	weakly	ADJ
ejpam-3275	254	12	α	α	NOUN
ejpam-3275	254	13	-	-	ADJ
ejpam-3275	254	14	prime	prime	ADJ
ejpam-3275	254	15	submodule	submodule	NOUN
ejpam-3275	254	16	of	of	ADP
ejpam-3275	254	17	m	m	PROPN
ejpam-3275	254	18	.	.	PUNCT
ejpam-3275	255	1	let	let	VERB
ejpam-3275	255	2	m	m	PRON
ejpam-3275	255	3	∈m	∈m	ADJ
ejpam-3275	255	4	be	be	AUX
ejpam-3275	255	5	such	such	ADJ
ejpam-3275	255	6	that	that	DET
ejpam-3275	255	7	m+m	m+m	PROPN
ejpam-3275	255	8	/∈	/∈	PUNCT
ejpam-3275	256	1	p	p	X
ejpam-3275	256	2	.	.	PUNCT
ejpam-3275	257	1	let	let	VERB
ejpam-3275	257	2	r	r	NOUN
ejpam-3275	257	3	∈	∈	PROPN
ejpam-3275	257	4	(	(	PUNCT
ejpam-3275	257	5	p	p	NOUN
ejpam-3275	257	6	:	:	PUNCT
ejpam-3275	257	7	m+m	m+m	NUM
ejpam-3275	257	8	)	)	PUNCT
ejpam-3275	257	9	.	.	PUNCT
ejpam-3275	258	1	then	then	ADV
ejpam-3275	258	2	r(m+m	r(m+m	NOUN
ejpam-3275	258	3	)	)	PUNCT
ejpam-3275	258	4	∈	∈	PROPN
ejpam-3275	258	5	p	p	NOUN
ejpam-3275	258	6	.	.	PUNCT
ejpam-3275	259	1	if	if	SCONJ
ejpam-3275	259	2	r(m+m	r(m+m	NOUN
ejpam-3275	259	3	)	)	PUNCT
ejpam-3275	260	1	=	=	SYM
ejpam-3275	260	2	0	0	NUM
ejpam-3275	260	3	,	,	PUNCT
ejpam-3275	260	4	then	then	ADV
ejpam-3275	260	5	r	r	NOUN
ejpam-3275	260	6	∈	∈	PROPN
ejpam-3275	260	7	α((0	α((0	PROPN
ejpam-3275	260	8	:	:	PUNCT
ejpam-3275	260	9	m	m	X
ejpam-3275	260	10	)	)	PUNCT
ejpam-3275	260	11	)	)	PUNCT
ejpam-3275	260	12	.	.	PUNCT
ejpam-3275	261	1	suppose	suppose	VERB
ejpam-3275	261	2	that	that	SCONJ
ejpam-3275	261	3	r(m+m	r(m+m	NOUN
ejpam-3275	261	4	)	)	PUNCT
ejpam-3275	261	5	6=	6=	ADP
ejpam-3275	261	6	0	0	X
ejpam-3275	261	7	.	.	PUNCT
ejpam-3275	262	1	since	since	SCONJ
ejpam-3275	262	2	p	p	NOUN
ejpam-3275	262	3	is	be	AUX
ejpam-3275	262	4	weakly	weakly	ADJ
ejpam-3275	262	5	α	α	NOUN
ejpam-3275	262	6	-	-	ADJ
ejpam-3275	262	7	prime	prime	NOUN
ejpam-3275	262	8	and	and	CCONJ
ejpam-3275	262	9	m+m	m+m	PROPN
ejpam-3275	262	10	/∈	/∈	PUNCT
ejpam-3275	263	1	p	p	NOUN
ejpam-3275	263	2	,	,	PUNCT
ejpam-3275	263	3	r	r	NOUN
ejpam-3275	263	4	+	+	CCONJ
ejpam-3275	263	5	r	r	NOUN
ejpam-3275	263	6	∈	∈	NOUN
ejpam-3275	263	7	(	(	PUNCT
ejpam-3275	263	8	p	p	X
ejpam-3275	263	9	:	:	PUNCT
ejpam-3275	263	10	m	m	PROPN
ejpam-3275	263	11	)	)	PUNCT
ejpam-3275	263	12	.	.	PUNCT
ejpam-3275	264	1	that	that	PRON
ejpam-3275	264	2	is	be	AUX
ejpam-3275	264	3	r	r	PROPN
ejpam-3275	264	4	∈	∈	PROPN
ejpam-3275	264	5	α((p	α((p	NOUN
ejpam-3275	264	6	:	:	PUNCT
ejpam-3275	264	7	m	m	X
ejpam-3275	264	8	)	)	PUNCT
ejpam-3275	264	9	)	)	PUNCT
ejpam-3275	264	10	.	.	PUNCT
ejpam-3275	265	1	conversely	conversely	ADV
ejpam-3275	265	2	,	,	PUNCT
ejpam-3275	265	3	let	let	VERB
ejpam-3275	265	4	r	r	NOUN
ejpam-3275	265	5	∈	∈	PROPN
ejpam-3275	265	6	α((p	α((p	NOUN
ejpam-3275	265	7	:	:	PUNCT
ejpam-3275	265	8	m	m	X
ejpam-3275	265	9	)	)	PUNCT
ejpam-3275	265	10	)	)	PUNCT
ejpam-3275	265	11	∪	∪	ADP
ejpam-3275	265	12	α((0	α((0	PROPN
ejpam-3275	265	13	:	:	PUNCT
ejpam-3275	265	14	m	m	NOUN
ejpam-3275	265	15	)	)	PUNCT
ejpam-3275	265	16	)	)	PUNCT
ejpam-3275	265	17	.	.	PUNCT
ejpam-3275	266	1	then	then	ADV
ejpam-3275	266	2	r	r	NOUN
ejpam-3275	266	3	+	+	CCONJ
ejpam-3275	266	4	r	r	NOUN
ejpam-3275	266	5	∈	∈	NOUN
ejpam-3275	266	6	(	(	PUNCT
ejpam-3275	266	7	p	p	X
ejpam-3275	266	8	:	:	PUNCT
ejpam-3275	266	9	m	m	NUM
ejpam-3275	266	10	)	)	PUNCT
ejpam-3275	266	11	or	or	CCONJ
ejpam-3275	266	12	rm+	rm+	NOUN
ejpam-3275	266	13	rm	rm	PROPN
ejpam-3275	266	14	=	=	NOUN
ejpam-3275	266	15	0	0	PROPN
ejpam-3275	266	16	.	.	PUNCT
ejpam-3275	267	1	these	these	PRON
ejpam-3275	267	2	implie	implie	VERB
ejpam-3275	267	3	that	that	SCONJ
ejpam-3275	267	4	r	r	NOUN
ejpam-3275	267	5	∈	∈	PROPN
ejpam-3275	267	6	(	(	PUNCT
ejpam-3275	267	7	p	p	NOUN
ejpam-3275	267	8	:	:	PUNCT
ejpam-3275	267	9	m+m	m+m	NUM
ejpam-3275	267	10	)	)	PUNCT
ejpam-3275	267	11	.	.	PUNCT
ejpam-3275	268	1	(	(	PUNCT
ejpam-3275	268	2	ii)→	ii)→	NOUN
ejpam-3275	268	3	(	(	PUNCT
ejpam-3275	268	4	iii	iii	NOUN
ejpam-3275	268	5	)	)	PUNCT
ejpam-3275	268	6	obvious	obvious	ADJ
ejpam-3275	268	7	.	.	PUNCT
ejpam-3275	269	1	(	(	PUNCT
ejpam-3275	269	2	iii	iii	NOUN
ejpam-3275	269	3	)	)	PUNCT
ejpam-3275	269	4	→	→	SYM
ejpam-3275	269	5	(	(	PUNCT
ejpam-3275	269	6	i	i	NOUN
ejpam-3275	269	7	)	)	PUNCT
ejpam-3275	269	8	assume	assume	VERB
ejpam-3275	269	9	that	that	SCONJ
ejpam-3275	269	10	(	(	PUNCT
ejpam-3275	269	11	iii	iii	NOUN
ejpam-3275	269	12	)	)	PUNCT
ejpam-3275	269	13	holds	hold	VERB
ejpam-3275	269	14	.	.	PUNCT
ejpam-3275	270	1	let	let	VERB
ejpam-3275	270	2	r	r	NOUN
ejpam-3275	270	3	∈	∈	NOUN
ejpam-3275	270	4	r	r	NOUN
ejpam-3275	270	5	and	and	CCONJ
ejpam-3275	270	6	m	m	PROPN
ejpam-3275	270	7	∈	∈	NOUN
ejpam-3275	270	8	m	m	AUX
ejpam-3275	270	9	be	be	VERB
ejpam-3275	270	10	such	such	ADJ
ejpam-3275	270	11	that	that	SCONJ
ejpam-3275	270	12	r(m	r(m	PROPN
ejpam-3275	270	13	+	+	NOUN
ejpam-3275	270	14	m	m	NOUN
ejpam-3275	270	15	)	)	PUNCT
ejpam-3275	270	16	∈	∈	PROPN
ejpam-3275	270	17	p\{0	p\{0	NOUN
ejpam-3275	270	18	}	}	PUNCT
ejpam-3275	270	19	and	and	CCONJ
ejpam-3275	270	20	m+m	m+m	PROPN
ejpam-3275	270	21	/∈	/∈	PUNCT
ejpam-3275	271	1	p	p	X
ejpam-3275	271	2	.	.	PUNCT
ejpam-3275	272	1	then	then	ADV
ejpam-3275	272	2	r	r	NOUN
ejpam-3275	272	3	∈	∈	PROPN
ejpam-3275	272	4	(	(	PUNCT
ejpam-3275	272	5	p	p	NOUN
ejpam-3275	272	6	:	:	PUNCT
ejpam-3275	272	7	m+m	m+m	NUM
ejpam-3275	272	8	)	)	PUNCT
ejpam-3275	272	9	.	.	PUNCT
ejpam-3275	273	1	since	since	SCONJ
ejpam-3275	273	2	r(m+m	r(m+m	NOUN
ejpam-3275	273	3	)	)	PUNCT
ejpam-3275	273	4	6=	6=	ADP
ejpam-3275	273	5	0	0	NUM
ejpam-3275	273	6	,	,	PUNCT
ejpam-3275	273	7	r	r	NOUN
ejpam-3275	273	8	/∈	/∈	PUNCT
ejpam-3275	273	9	α((0	α((0	PROPN
ejpam-3275	273	10	:	:	PUNCT
ejpam-3275	273	11	m	m	X
ejpam-3275	273	12	)	)	PUNCT
ejpam-3275	273	13	)	)	PUNCT
ejpam-3275	273	14	.	.	PUNCT
ejpam-3275	274	1	by	by	ADP
ejpam-3275	274	2	(	(	PUNCT
ejpam-3275	274	3	iii	iii	NOUN
ejpam-3275	274	4	)	)	PUNCT
ejpam-3275	274	5	,	,	PUNCT
ejpam-3275	274	6	(	(	PUNCT
ejpam-3275	274	7	p	p	X
ejpam-3275	274	8	:	:	PUNCT
ejpam-3275	274	9	m+m	m+m	NUM
ejpam-3275	274	10	)	)	PUNCT
ejpam-3275	274	11	=	=	SYM
ejpam-3275	274	12	α((p	α((p	NOUN
ejpam-3275	274	13	:	:	PUNCT
ejpam-3275	274	14	m	m	X
ejpam-3275	274	15	)	)	PUNCT
ejpam-3275	274	16	)	)	PUNCT
ejpam-3275	274	17	.	.	PUNCT
ejpam-3275	275	1	hence	hence	ADV
ejpam-3275	275	2	r	r	NOUN
ejpam-3275	275	3	∈	∈	PROPN
ejpam-3275	275	4	α((p	α((p	NOUN
ejpam-3275	275	5	:	:	PUNCT
ejpam-3275	275	6	m	m	X
ejpam-3275	275	7	)	)	PUNCT
ejpam-3275	275	8	)	)	PUNCT
ejpam-3275	275	9	.	.	PUNCT
ejpam-3275	276	1	therefore	therefore	ADV
ejpam-3275	276	2	r+	r+	VERB
ejpam-3275	276	3	r	r	NOUN
ejpam-3275	276	4	∈	∈	PROPN
ejpam-3275	276	5	(	(	PUNCT
ejpam-3275	276	6	p	p	X
ejpam-3275	276	7	:	:	PUNCT
ejpam-3275	276	8	m	m	PROPN
ejpam-3275	276	9	)	)	PUNCT
ejpam-3275	276	10	.	.	PUNCT
ejpam-3275	277	1	this	this	PRON
ejpam-3275	277	2	proves	prove	VERB
ejpam-3275	277	3	that	that	SCONJ
ejpam-3275	277	4	p	p	NOUN
ejpam-3275	277	5	is	be	AUX
ejpam-3275	277	6	a	a	DET
ejpam-3275	277	7	weakly	weakly	ADJ
ejpam-3275	277	8	α	α	NOUN
ejpam-3275	277	9	-	-	ADJ
ejpam-3275	277	10	prime	prime	ADJ
ejpam-3275	277	11	submodule	submodule	NOUN
ejpam-3275	277	12	of	of	ADP
ejpam-3275	277	13	m	m	PROPN
ejpam-3275	277	14	.	.	PUNCT
ejpam-3275	278	1	let	let	VERB
ejpam-3275	278	2	m1	m1	PROPN
ejpam-3275	278	3	and	and	CCONJ
ejpam-3275	278	4	m2	m2	PROPN
ejpam-3275	278	5	be	be	VERB
ejpam-3275	278	6	r	r	NOUN
ejpam-3275	278	7	-	-	PUNCT
ejpam-3275	278	8	modules	module	NOUN
ejpam-3275	278	9	.	.	PUNCT
ejpam-3275	279	1	then	then	ADV
ejpam-3275	279	2	m1	m1	PROPN
ejpam-3275	279	3	×m2	×m2	PROPN
ejpam-3275	279	4	is	be	AUX
ejpam-3275	279	5	an	an	DET
ejpam-3275	279	6	r	r	NOUN
ejpam-3275	279	7	-	-	PUNCT
ejpam-3275	279	8	module	module	NOUN
ejpam-3275	279	9	under	under	ADP
ejpam-3275	279	10	the	the	DET
ejpam-3275	279	11	operation	operation	NOUN
ejpam-3275	279	12	(	(	PUNCT
ejpam-3275	279	13	a	a	DET
ejpam-3275	279	14	,	,	PUNCT
ejpam-3275	279	15	b	b	NOUN
ejpam-3275	279	16	)	)	PUNCT
ejpam-3275	280	1	+	+	CCONJ
ejpam-3275	280	2	(	(	PUNCT
ejpam-3275	280	3	c	c	X
ejpam-3275	280	4	,	,	PUNCT
ejpam-3275	280	5	d	d	NOUN
ejpam-3275	280	6	)	)	PUNCT
ejpam-3275	280	7	=	=	SYM
ejpam-3275	280	8	(	(	PUNCT
ejpam-3275	280	9	a+	a+	PUNCT
ejpam-3275	280	10	c	c	NOUN
ejpam-3275	280	11	,	,	PUNCT
ejpam-3275	280	12	b+	b+	X
ejpam-3275	280	13	d	d	X
ejpam-3275	280	14	)	)	PUNCT
ejpam-3275	280	15	and	and	CCONJ
ejpam-3275	280	16	r(a	r(a	PROPN
ejpam-3275	280	17	,	,	PUNCT
ejpam-3275	280	18	b	b	NOUN
ejpam-3275	280	19	)	)	PUNCT
ejpam-3275	280	20	=	=	SYM
ejpam-3275	280	21	(	(	PUNCT
ejpam-3275	280	22	ra	ra	PROPN
ejpam-3275	280	23	,	,	PUNCT
ejpam-3275	280	24	rb	rb	PROPN
ejpam-3275	280	25	)	)	PUNCT
ejpam-3275	280	26	for	for	ADP
ejpam-3275	280	27	all	all	DET
ejpam-3275	280	28	a	a	DET
ejpam-3275	280	29	,	,	PUNCT
ejpam-3275	280	30	c	c	PROPN
ejpam-3275	280	31	∈m1	∈m1	PROPN
ejpam-3275	280	32	,	,	PUNCT
ejpam-3275	280	33	b	b	NOUN
ejpam-3275	280	34	,	,	PUNCT
ejpam-3275	280	35	d	d	NOUN
ejpam-3275	280	36	∈m2	∈m2	NOUN
ejpam-3275	280	37	and	and	CCONJ
ejpam-3275	280	38	r	r	NOUN
ejpam-3275	280	39	∈	∈	PROPN
ejpam-3275	280	40	r.	r.	NOUN
ejpam-3275	280	41	we	we	PRON
ejpam-3275	280	42	denote	denote	VERB
ejpam-3275	280	43	this	this	DET
ejpam-3275	280	44	module	module	NOUN
ejpam-3275	280	45	by	by	ADP
ejpam-3275	280	46	m1	m1	PROPN
ejpam-3275	280	47	⊕m2	⊕m2	PROPN
ejpam-3275	280	48	.	.	PUNCT
ejpam-3275	281	1	proposition	proposition	NOUN
ejpam-3275	281	2	4	4	NUM
ejpam-3275	281	3	.	.	PUNCT
ejpam-3275	282	1	let	let	VERB
ejpam-3275	282	2	n1	n1	PROPN
ejpam-3275	282	3	be	be	AUX
ejpam-3275	282	4	a	a	DET
ejpam-3275	282	5	submodule	submodule	NOUN
ejpam-3275	282	6	of	of	ADP
ejpam-3275	282	7	m1	m1	PROPN
ejpam-3275	282	8	and	and	CCONJ
ejpam-3275	282	9	n2	n2	PROPN
ejpam-3275	282	10	be	be	AUX
ejpam-3275	282	11	a	a	DET
ejpam-3275	282	12	submodule	submodule	NOUN
ejpam-3275	282	13	of	of	ADP
ejpam-3275	282	14	m2	m2	PROPN
ejpam-3275	282	15	.	.	PUNCT
ejpam-3275	283	1	if	if	SCONJ
ejpam-3275	283	2	n1	n1	PROPN
ejpam-3275	283	3	×n2	×n2	PROPN
ejpam-3275	283	4	is	be	AUX
ejpam-3275	283	5	a	a	DET
ejpam-3275	283	6	weakly	weakly	ADJ
ejpam-3275	283	7	α	α	NOUN
ejpam-3275	283	8	-	-	ADJ
ejpam-3275	283	9	prime	prime	ADJ
ejpam-3275	283	10	submodule	submodule	NOUN
ejpam-3275	283	11	of	of	ADP
ejpam-3275	283	12	m1	m1	PROPN
ejpam-3275	283	13	⊕m2	⊕m2	PROPN
ejpam-3275	283	14	,	,	PUNCT
ejpam-3275	283	15	then	then	ADV
ejpam-3275	283	16	n1	n1	PROPN
ejpam-3275	283	17	is	be	AUX
ejpam-3275	283	18	a	a	DET
ejpam-3275	283	19	weakly	weakly	ADJ
ejpam-3275	283	20	α	α	NOUN
ejpam-3275	283	21	-	-	ADJ
ejpam-3275	283	22	prime	prime	ADJ
ejpam-3275	283	23	submodule	submodule	NOUN
ejpam-3275	283	24	of	of	ADP
ejpam-3275	283	25	m1	m1	PROPN
ejpam-3275	283	26	and	and	CCONJ
ejpam-3275	283	27	n2	n2	NOUN
ejpam-3275	283	28	is	be	AUX
ejpam-3275	283	29	a	a	DET
ejpam-3275	283	30	weakly	weakly	ADJ
ejpam-3275	283	31	α	α	NOUN
ejpam-3275	283	32	-	-	ADJ
ejpam-3275	283	33	prime	prime	ADJ
ejpam-3275	283	34	submodule	submodule	NOUN
ejpam-3275	283	35	of	of	ADP
ejpam-3275	283	36	m2	m2	PROPN
ejpam-3275	283	37	.	.	PUNCT
ejpam-3275	284	1	proof	proof	NOUN
ejpam-3275	284	2	.	.	PUNCT
ejpam-3275	285	1	it	it	PRON
ejpam-3275	285	2	is	be	AUX
ejpam-3275	285	3	straightforward	straightforward	ADJ
ejpam-3275	285	4	.	.	PUNCT
ejpam-3275	286	1	let	let	VERB
ejpam-3275	286	2	r1	r1	PROPN
ejpam-3275	286	3	and	and	CCONJ
ejpam-3275	286	4	r2	r2	PROPN
ejpam-3275	286	5	be	be	AUX
ejpam-3275	286	6	commutative	commutative	ADJ
ejpam-3275	286	7	rings	ring	NOUN
ejpam-3275	286	8	with	with	ADP
ejpam-3275	286	9	identity	identity	NOUN
ejpam-3275	286	10	,	,	PUNCT
ejpam-3275	286	11	mi	mi	PROPN
ejpam-3275	286	12	be	be	AUX
ejpam-3275	286	13	a	a	DET
ejpam-3275	286	14	unital	unital	ADJ
ejpam-3275	286	15	ri	ri	NOUN
ejpam-3275	286	16	-	-	PUNCT
ejpam-3275	286	17	module	module	NOUN
ejpam-3275	286	18	where	where	SCONJ
ejpam-3275	286	19	i	i	PRON
ejpam-3275	286	20	=	=	NOUN
ejpam-3275	286	21	1	1	NUM
ejpam-3275	286	22	,	,	PUNCT
ejpam-3275	286	23	2	2	NUM
ejpam-3275	286	24	.	.	PUNCT
ejpam-3275	286	25	then	then	ADV
ejpam-3275	286	26	m1	m1	PROPN
ejpam-3275	286	27	×m2	×m2	PROPN
ejpam-3275	286	28	is	be	AUX
ejpam-3275	286	29	an	an	DET
ejpam-3275	286	30	(	(	PUNCT
ejpam-3275	286	31	r1	r1	NOUN
ejpam-3275	286	32	×	×	PROPN
ejpam-3275	286	33	r2)-module	r2)-module	NOUN
ejpam-3275	286	34	under	under	ADP
ejpam-3275	286	35	the	the	DET
ejpam-3275	286	36	operation	operation	NOUN
ejpam-3275	286	37	(	(	PUNCT
ejpam-3275	286	38	r1	r1	PROPN
ejpam-3275	286	39	,	,	PUNCT
ejpam-3275	286	40	r2)(m1,m1	r2)(m1,m1	NOUN
ejpam-3275	286	41	)	)	PUNCT
ejpam-3275	287	1	=	=	SYM
ejpam-3275	287	2	(	(	PUNCT
ejpam-3275	287	3	r1m1	r1m1	NOUN
ejpam-3275	287	4	,	,	PUNCT
ejpam-3275	287	5	r2m2	r2m2	NOUN
ejpam-3275	287	6	)	)	PUNCT
ejpam-3275	287	7	for	for	ADP
ejpam-3275	287	8	all	all	PRON
ejpam-3275	287	9	(	(	PUNCT
ejpam-3275	287	10	r1	r1	PROPN
ejpam-3275	287	11	,	,	PUNCT
ejpam-3275	287	12	r2	r2	PROPN
ejpam-3275	287	13	)	)	PUNCT
ejpam-3275	287	14	∈	∈	PROPN
ejpam-3275	287	15	r1×r2	r1×r2	PROPN
ejpam-3275	287	16	and	and	CCONJ
ejpam-3275	287	17	(	(	PUNCT
ejpam-3275	287	18	m1,m2	m1,m2	PROPN
ejpam-3275	287	19	)	)	PUNCT
ejpam-3275	287	20	∈m1×m2	∈m1×m2	PROPN
ejpam-3275	287	21	.	.	PUNCT
ejpam-3275	288	1	we	we	PRON
ejpam-3275	288	2	set	set	VERB
ejpam-3275	288	3	up	up	ADP
ejpam-3275	288	4	these	these	DET
ejpam-3275	288	5	notation	notation	NOUN
ejpam-3275	288	6	for	for	ADP
ejpam-3275	288	7	the	the	DET
ejpam-3275	288	8	next	next	ADJ
ejpam-3275	288	9	two	two	NUM
ejpam-3275	288	10	results	result	NOUN
ejpam-3275	288	11	.	.	PUNCT
ejpam-3275	289	1	proposition	proposition	NOUN
ejpam-3275	289	2	5	5	NUM
ejpam-3275	289	3	.	.	PUNCT
ejpam-3275	290	1	let	let	VERB
ejpam-3275	290	2	r	r	NOUN
ejpam-3275	290	3	=	=	SYM
ejpam-3275	290	4	r1	r1	PROPN
ejpam-3275	290	5	×r2	×r2	PROPN
ejpam-3275	290	6	and	and	CCONJ
ejpam-3275	290	7	m	m	NOUN
ejpam-3275	290	8	=	=	NOUN
ejpam-3275	290	9	m1	m1	NOUN
ejpam-3275	290	10	×m2	×m2	NOUN
ejpam-3275	290	11	and	and	CCONJ
ejpam-3275	290	12	let	let	VERB
ejpam-3275	290	13	n1	n1	PROPN
ejpam-3275	290	14	be	be	AUX
ejpam-3275	290	15	an	an	DET
ejpam-3275	290	16	r1	r1	NOUN
ejpam-3275	290	17	-	-	PUNCT
ejpam-3275	290	18	submodule	submodule	NOUN
ejpam-3275	290	19	of	of	ADP
ejpam-3275	290	20	m1	m1	PROPN
ejpam-3275	290	21	.	.	PUNCT
ejpam-3275	291	1	consider	consider	VERB
ejpam-3275	291	2	the	the	DET
ejpam-3275	291	3	following	follow	VERB
ejpam-3275	291	4	statements	statement	NOUN
ejpam-3275	291	5	.	.	PUNCT
ejpam-3275	292	1	t.	t.	PROPN
ejpam-3275	292	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	292	3	/	/	SYM
ejpam-3275	292	4	eur	eur	PROPN
ejpam-3275	292	5	.	.	PUNCT
ejpam-3275	293	1	j.	j.	PROPN
ejpam-3275	293	2	pure	pure	PROPN
ejpam-3275	293	3	appl	appl	PROPN
ejpam-3275	293	4	.	.	PROPN
ejpam-3275	293	5	math	math	PROPN
ejpam-3275	293	6	,	,	PUNCT
ejpam-3275	293	7	11	11	NUM
ejpam-3275	293	8	(	(	PUNCT
ejpam-3275	293	9	3	3	NUM
ejpam-3275	293	10	)	)	PUNCT
ejpam-3275	293	11	(	(	PUNCT
ejpam-3275	293	12	2018	2018	NUM
ejpam-3275	293	13	)	)	PUNCT
ejpam-3275	293	14	,	,	PUNCT
ejpam-3275	293	15	730	730	NUM
ejpam-3275	293	16	-	-	SYM
ejpam-3275	293	17	739	739	NUM
ejpam-3275	293	18	736	736	NUM
ejpam-3275	293	19	(	(	PUNCT
ejpam-3275	293	20	i	i	NOUN
ejpam-3275	293	21	)	)	PUNCT
ejpam-3275	293	22	n1	n1	PROPN
ejpam-3275	293	23	is	be	AUX
ejpam-3275	293	24	an	an	DET
ejpam-3275	293	25	α	α	NOUN
ejpam-3275	293	26	-	-	ADJ
ejpam-3275	293	27	prime	prime	ADJ
ejpam-3275	293	28	submodule	submodule	NOUN
ejpam-3275	293	29	of	of	ADP
ejpam-3275	293	30	m1	m1	PROPN
ejpam-3275	293	31	.	.	PUNCT
ejpam-3275	294	1	(	(	PUNCT
ejpam-3275	294	2	ii	ii	NOUN
ejpam-3275	294	3	)	)	PUNCT
ejpam-3275	294	4	n1	n1	NOUN
ejpam-3275	294	5	×m2	×m2	NOUN
ejpam-3275	294	6	is	be	AUX
ejpam-3275	294	7	an	an	DET
ejpam-3275	294	8	α	α	NOUN
ejpam-3275	294	9	-	-	ADJ
ejpam-3275	294	10	prime	prime	ADJ
ejpam-3275	294	11	submodule	submodule	NOUN
ejpam-3275	294	12	of	of	ADP
ejpam-3275	294	13	m1	m1	PROPN
ejpam-3275	294	14	×m2	×m2	PROPN
ejpam-3275	294	15	.	.	PUNCT
ejpam-3275	295	1	(	(	PUNCT
ejpam-3275	295	2	iii	iii	X
ejpam-3275	295	3	)	)	PUNCT
ejpam-3275	295	4	n1	n1	NOUN
ejpam-3275	295	5	×m2	×m2	NOUN
ejpam-3275	295	6	is	be	AUX
ejpam-3275	295	7	a	a	DET
ejpam-3275	295	8	weakly	weakly	ADJ
ejpam-3275	295	9	α	α	NOUN
ejpam-3275	295	10	-	-	ADJ
ejpam-3275	295	11	prime	prime	ADJ
ejpam-3275	295	12	submodule	submodule	NOUN
ejpam-3275	295	13	of	of	ADP
ejpam-3275	295	14	m1	m1	PROPN
ejpam-3275	295	15	×m2	×m2	PROPN
ejpam-3275	295	16	.	.	PUNCT
ejpam-3275	296	1	then	then	ADV
ejpam-3275	296	2	(	(	PUNCT
ejpam-3275	296	3	i)→	i)→	ADJ
ejpam-3275	296	4	(	(	PUNCT
ejpam-3275	296	5	ii)→	ii)→	NOUN
ejpam-3275	296	6	(	(	PUNCT
ejpam-3275	296	7	iii	iii	NOUN
ejpam-3275	296	8	)	)	PUNCT
ejpam-3275	296	9	.	.	PUNCT
ejpam-3275	297	1	moreover	moreover	ADV
ejpam-3275	297	2	,	,	PUNCT
ejpam-3275	297	3	if	if	SCONJ
ejpam-3275	297	4	β(m2	β(m2	X
ejpam-3275	297	5	)	)	PUNCT
ejpam-3275	297	6	6=	6=	PUNCT
ejpam-3275	297	7	{	{	PUNCT
ejpam-3275	297	8	0	0	NUM
ejpam-3275	297	9	}	}	PUNCT
ejpam-3275	297	10	,	,	PUNCT
ejpam-3275	297	11	then	then	ADV
ejpam-3275	297	12	(	(	PUNCT
ejpam-3275	297	13	i	i	NOUN
ejpam-3275	297	14	)	)	PUNCT
ejpam-3275	297	15	,	,	PUNCT
ejpam-3275	297	16	(	(	PUNCT
ejpam-3275	297	17	ii	ii	NOUN
ejpam-3275	297	18	)	)	PUNCT
ejpam-3275	297	19	and	and	CCONJ
ejpam-3275	297	20	(	(	PUNCT
ejpam-3275	297	21	iii	iii	X
ejpam-3275	297	22	)	)	PUNCT
ejpam-3275	297	23	are	be	AUX
ejpam-3275	297	24	equivalent	equivalent	ADJ
ejpam-3275	297	25	.	.	PUNCT
ejpam-3275	298	1	proof	proof	NOUN
ejpam-3275	298	2	.	.	PUNCT
ejpam-3275	299	1	(	(	PUNCT
ejpam-3275	299	2	i	i	NOUN
ejpam-3275	299	3	)	)	PUNCT
ejpam-3275	299	4	→	→	SYM
ejpam-3275	299	5	(	(	PUNCT
ejpam-3275	299	6	ii	ii	NOUN
ejpam-3275	299	7	)	)	PUNCT
ejpam-3275	299	8	assume	assume	VERB
ejpam-3275	299	9	that	that	SCONJ
ejpam-3275	299	10	n1	n1	PROPN
ejpam-3275	299	11	is	be	AUX
ejpam-3275	299	12	an	an	DET
ejpam-3275	299	13	α	α	NOUN
ejpam-3275	299	14	-	-	ADJ
ejpam-3275	299	15	prime	prime	ADJ
ejpam-3275	299	16	submodule	submodule	NOUN
ejpam-3275	299	17	of	of	ADP
ejpam-3275	299	18	m1	m1	PROPN
ejpam-3275	299	19	.	.	PUNCT
ejpam-3275	300	1	let	let	VERB
ejpam-3275	300	2	(	(	PUNCT
ejpam-3275	300	3	a	a	PRON
ejpam-3275	300	4	,	,	PUNCT
ejpam-3275	300	5	b	b	NOUN
ejpam-3275	300	6	)	)	PUNCT
ejpam-3275	300	7	∈	∈	PROPN
ejpam-3275	300	8	r1	r1	NOUN
ejpam-3275	300	9	×	×	NOUN
ejpam-3275	300	10	r2	r2	PROPN
ejpam-3275	300	11	and	and	CCONJ
ejpam-3275	300	12	(	(	PUNCT
ejpam-3275	300	13	x	x	NOUN
ejpam-3275	300	14	,	,	PUNCT
ejpam-3275	300	15	y	y	NOUN
ejpam-3275	300	16	)	)	PUNCT
ejpam-3275	300	17	∈	∈	PROPN
ejpam-3275	300	18	m1	m1	PROPN
ejpam-3275	300	19	×	×	PROPN
ejpam-3275	300	20	m2	m2	PROPN
ejpam-3275	300	21	be	be	VERB
ejpam-3275	300	22	such	such	ADJ
ejpam-3275	300	23	that	that	SCONJ
ejpam-3275	300	24	(	(	PUNCT
ejpam-3275	300	25	a	a	PRON
ejpam-3275	300	26	,	,	PUNCT
ejpam-3275	300	27	b)[(x	b)[(x	NOUN
ejpam-3275	300	28	,	,	PUNCT
ejpam-3275	300	29	y	y	PROPN
ejpam-3275	300	30	)	)	PUNCT
ejpam-3275	301	1	+	+	CCONJ
ejpam-3275	301	2	(	(	PUNCT
ejpam-3275	301	3	x	x	X
ejpam-3275	301	4	,	,	PUNCT
ejpam-3275	301	5	y	y	PROPN
ejpam-3275	301	6	)	)	PUNCT
ejpam-3275	301	7	]	]	PUNCT
ejpam-3275	302	1	∈	∈	PROPN
ejpam-3275	302	2	n1	n1	PROPN
ejpam-3275	302	3	×	×	PROPN
ejpam-3275	302	4	m2	m2	PROPN
ejpam-3275	302	5	.	.	PUNCT
ejpam-3275	303	1	then	then	ADV
ejpam-3275	303	2	[	[	X
ejpam-3275	303	3	a(x+	a(x+	ADP
ejpam-3275	303	4	x	x	NOUN
ejpam-3275	303	5	)	)	PUNCT
ejpam-3275	303	6	,	,	PUNCT
ejpam-3275	303	7	b(y+	b(y+	PROPN
ejpam-3275	303	8	y	y	NOUN
ejpam-3275	303	9	)	)	PUNCT
ejpam-3275	303	10	]	]	PUNCT
ejpam-3275	304	1	∈	∈	PROPN
ejpam-3275	304	2	n1×m2	n1×m2	PROPN
ejpam-3275	304	3	.	.	PUNCT
ejpam-3275	305	1	thus	thus	ADV
ejpam-3275	305	2	a(x+	a(x+	ADP
ejpam-3275	305	3	x	x	X
ejpam-3275	305	4	)	)	PUNCT
ejpam-3275	305	5	∈	∈	PROPN
ejpam-3275	305	6	n1	n1	NOUN
ejpam-3275	305	7	.	.	PUNCT
ejpam-3275	306	1	since	since	SCONJ
ejpam-3275	306	2	n1	n1	PROPN
ejpam-3275	306	3	is	be	AUX
ejpam-3275	306	4	an	an	DET
ejpam-3275	306	5	α	α	NOUN
ejpam-3275	306	6	-	-	ADJ
ejpam-3275	306	7	prime	prime	ADJ
ejpam-3275	306	8	submodule	submodule	NOUN
ejpam-3275	306	9	of	of	ADP
ejpam-3275	306	10	m1	m1	PROPN
ejpam-3275	306	11	,	,	PUNCT
ejpam-3275	306	12	a+a	a+a	X
ejpam-3275	306	13	∈	∈	PROPN
ejpam-3275	306	14	(	(	PUNCT
ejpam-3275	306	15	n1	n1	NOUN
ejpam-3275	306	16	:	:	PUNCT
ejpam-3275	306	17	m1	m1	NOUN
ejpam-3275	306	18	)	)	PUNCT
ejpam-3275	306	19	or	or	CCONJ
ejpam-3275	306	20	x+x	x+x	PROPN
ejpam-3275	306	21	∈	∈	PROPN
ejpam-3275	306	22	n1	n1	PROPN
ejpam-3275	306	23	.	.	PUNCT
ejpam-3275	307	1	this	this	PRON
ejpam-3275	307	2	leads	lead	VERB
ejpam-3275	307	3	to	to	ADP
ejpam-3275	307	4	(	(	PUNCT
ejpam-3275	307	5	a+a	a+a	NUM
ejpam-3275	307	6	,	,	PUNCT
ejpam-3275	307	7	b+	b+	X
ejpam-3275	307	8	b	b	X
ejpam-3275	307	9	)	)	PUNCT
ejpam-3275	307	10	∈	∈	PROPN
ejpam-3275	307	11	(	(	PUNCT
ejpam-3275	307	12	n1×m2	n1×m2	NOUN
ejpam-3275	307	13	:	:	PUNCT
ejpam-3275	307	14	m1×m2	m1×m2	PROPN
ejpam-3275	307	15	)	)	PUNCT
ejpam-3275	307	16	or	or	CCONJ
ejpam-3275	307	17	(	(	PUNCT
ejpam-3275	307	18	x	x	NOUN
ejpam-3275	307	19	,	,	PUNCT
ejpam-3275	307	20	y	y	PROPN
ejpam-3275	307	21	)	)	PUNCT
ejpam-3275	308	1	+	+	CCONJ
ejpam-3275	308	2	(	(	PUNCT
ejpam-3275	308	3	x	x	NOUN
ejpam-3275	308	4	,	,	PUNCT
ejpam-3275	308	5	y	y	NOUN
ejpam-3275	308	6	)	)	PUNCT
ejpam-3275	308	7	∈	∈	PROPN
ejpam-3275	308	8	n1×m2	n1×m2	PROPN
ejpam-3275	308	9	.	.	PUNCT
ejpam-3275	309	1	therefore	therefore	ADV
ejpam-3275	309	2	n1×m2	n1×m2	PROPN
ejpam-3275	309	3	is	be	AUX
ejpam-3275	309	4	an	an	DET
ejpam-3275	309	5	α	α	NOUN
ejpam-3275	309	6	-	-	ADJ
ejpam-3275	309	7	prime	prime	ADJ
ejpam-3275	309	8	prime	prime	ADJ
ejpam-3275	309	9	submodule	submodule	NOUN
ejpam-3275	309	10	of	of	ADP
ejpam-3275	309	11	m1×m2	m1×m2	PROPN
ejpam-3275	309	12	.	.	PUNCT
ejpam-3275	310	1	(	(	PUNCT
ejpam-3275	310	2	ii)→	ii)→	NOUN
ejpam-3275	310	3	(	(	PUNCT
ejpam-3275	310	4	iii	iii	X
ejpam-3275	310	5	)	)	PUNCT
ejpam-3275	310	6	it	it	PRON
ejpam-3275	310	7	is	be	AUX
ejpam-3275	310	8	obvious	obvious	ADJ
ejpam-3275	310	9	.	.	PUNCT
ejpam-3275	311	1	next	next	ADV
ejpam-3275	311	2	,	,	PUNCT
ejpam-3275	311	3	let	let	VERB
ejpam-3275	311	4	w	w	PROPN
ejpam-3275	311	5	∈	∈	PROPN
ejpam-3275	311	6	m2	m2	PROPN
ejpam-3275	311	7	be	be	VERB
ejpam-3275	311	8	such	such	ADJ
ejpam-3275	311	9	that	that	SCONJ
ejpam-3275	311	10	w	w	PROPN
ejpam-3275	311	11	+	+	NUM
ejpam-3275	311	12	w	w	PROPN
ejpam-3275	311	13	6=	6=	ADP
ejpam-3275	311	14	0	0	NUM
ejpam-3275	311	15	and	and	CCONJ
ejpam-3275	311	16	assume	assume	VERB
ejpam-3275	311	17	that	that	SCONJ
ejpam-3275	311	18	n1	n1	NOUN
ejpam-3275	311	19	×m2	×m2	NOUN
ejpam-3275	311	20	is	be	AUX
ejpam-3275	311	21	a	a	DET
ejpam-3275	311	22	weakly	weakly	ADJ
ejpam-3275	311	23	α	α	NOUN
ejpam-3275	311	24	-	-	ADJ
ejpam-3275	311	25	prime	prime	ADJ
ejpam-3275	311	26	submodule	submodule	NOUN
ejpam-3275	311	27	of	of	ADP
ejpam-3275	311	28	m1	m1	PROPN
ejpam-3275	311	29	×m2	×m2	PROPN
ejpam-3275	311	30	.	.	PUNCT
ejpam-3275	312	1	let	let	VERB
ejpam-3275	312	2	r	r	NOUN
ejpam-3275	312	3	∈	∈	PROPN
ejpam-3275	312	4	r1	r1	NOUN
ejpam-3275	312	5	and	and	CCONJ
ejpam-3275	312	6	m	m	PROPN
ejpam-3275	312	7	∈	∈	PROPN
ejpam-3275	312	8	m1	m1	NOUN
ejpam-3275	312	9	such	such	ADJ
ejpam-3275	312	10	that	that	SCONJ
ejpam-3275	312	11	r(m	r(m	PROPN
ejpam-3275	312	12	+	+	NOUN
ejpam-3275	312	13	m	m	NOUN
ejpam-3275	312	14	)	)	PUNCT
ejpam-3275	312	15	∈	∈	PROPN
ejpam-3275	312	16	n1	n1	NOUN
ejpam-3275	312	17	.	.	PUNCT
ejpam-3275	313	1	then	then	ADV
ejpam-3275	313	2	(	(	PUNCT
ejpam-3275	313	3	r	r	NOUN
ejpam-3275	313	4	,	,	PUNCT
ejpam-3275	313	5	1)[(m	1)[(m	NUM
ejpam-3275	313	6	,	,	PUNCT
ejpam-3275	313	7	w	w	NOUN
ejpam-3275	313	8	)	)	PUNCT
ejpam-3275	313	9	+	+	CCONJ
ejpam-3275	313	10	(	(	PUNCT
ejpam-3275	313	11	m	m	PROPN
ejpam-3275	313	12	,	,	PUNCT
ejpam-3275	313	13	w	w	NOUN
ejpam-3275	313	14	)	)	PUNCT
ejpam-3275	313	15	]	]	PUNCT
ejpam-3275	314	1	=	=	SYM
ejpam-3275	314	2	(	(	PUNCT
ejpam-3275	314	3	r(m+m	r(m+m	NOUN
ejpam-3275	314	4	)	)	PUNCT
ejpam-3275	314	5	,	,	PUNCT
ejpam-3275	314	6	w+w	w+w	NUM
ejpam-3275	314	7	)	)	PUNCT
ejpam-3275	314	8	∈	∈	PROPN
ejpam-3275	314	9	n1×m2\{(0	n1×m2\{(0	NOUN
ejpam-3275	314	10	,	,	PUNCT
ejpam-3275	314	11	0	0	NUM
ejpam-3275	314	12	)	)	PUNCT
ejpam-3275	314	13	}	}	PUNCT
ejpam-3275	314	14	.	.	PUNCT
ejpam-3275	315	1	since	since	SCONJ
ejpam-3275	315	2	n1×m2	n1×m2	NOUN
ejpam-3275	315	3	is	be	AUX
ejpam-3275	315	4	a	a	DET
ejpam-3275	315	5	weakly	weakly	ADJ
ejpam-3275	315	6	α	α	NOUN
ejpam-3275	315	7	-	-	ADJ
ejpam-3275	315	8	prime	prime	ADJ
ejpam-3275	315	9	submodule	submodule	NOUN
ejpam-3275	315	10	of	of	ADP
ejpam-3275	315	11	m1	m1	PROPN
ejpam-3275	315	12	×m2	×m2	PROPN
ejpam-3275	315	13	,	,	PUNCT
ejpam-3275	315	14	we	we	PRON
ejpam-3275	315	15	have	have	AUX
ejpam-3275	315	16	(	(	PUNCT
ejpam-3275	315	17	r+	r+	VERB
ejpam-3275	315	18	r	r	NOUN
ejpam-3275	315	19	,	,	PUNCT
ejpam-3275	315	20	1	1	NUM
ejpam-3275	315	21	+	+	NUM
ejpam-3275	315	22	1	1	X
ejpam-3275	315	23	)	)	PUNCT
ejpam-3275	315	24	∈	∈	NOUN
ejpam-3275	315	25	(	(	PUNCT
ejpam-3275	315	26	n1	n1	PROPN
ejpam-3275	315	27	×m1	×m1	NOUN
ejpam-3275	315	28	:	:	PUNCT
ejpam-3275	315	29	m1	m1	PROPN
ejpam-3275	315	30	×m2	×m2	PROPN
ejpam-3275	315	31	)	)	PUNCT
ejpam-3275	315	32	or	or	CCONJ
ejpam-3275	315	33	(	(	PUNCT
ejpam-3275	315	34	m	m	PROPN
ejpam-3275	315	35	,	,	PUNCT
ejpam-3275	315	36	w	w	NOUN
ejpam-3275	315	37	)	)	PUNCT
ejpam-3275	315	38	+	+	CCONJ
ejpam-3275	315	39	(	(	PUNCT
ejpam-3275	315	40	m	m	PROPN
ejpam-3275	315	41	,	,	PUNCT
ejpam-3275	315	42	w	w	NOUN
ejpam-3275	315	43	)	)	PUNCT
ejpam-3275	315	44	∈	∈	NOUN
ejpam-3275	315	45	n1	n1	NOUN
ejpam-3275	315	46	×m2	×m2	PROPN
ejpam-3275	315	47	.	.	PUNCT
ejpam-3275	316	1	this	this	PRON
ejpam-3275	316	2	implies	imply	VERB
ejpam-3275	316	3	that	that	SCONJ
ejpam-3275	316	4	r	r	NOUN
ejpam-3275	316	5	+	+	CCONJ
ejpam-3275	316	6	r	r	NOUN
ejpam-3275	316	7	∈	∈	NOUN
ejpam-3275	316	8	(	(	PUNCT
ejpam-3275	316	9	n1	n1	NOUN
ejpam-3275	316	10	:	:	PUNCT
ejpam-3275	316	11	m1	m1	NOUN
ejpam-3275	316	12	)	)	PUNCT
ejpam-3275	316	13	or	or	CCONJ
ejpam-3275	316	14	m+m	m+m	PROPN
ejpam-3275	316	15	∈	∈	PROPN
ejpam-3275	316	16	n1	n1	NOUN
ejpam-3275	316	17	.	.	PUNCT
ejpam-3275	317	1	hence	hence	ADV
ejpam-3275	317	2	n1	n1	PROPN
ejpam-3275	317	3	is	be	AUX
ejpam-3275	317	4	an	an	DET
ejpam-3275	317	5	α	α	NOUN
ejpam-3275	317	6	-	-	ADJ
ejpam-3275	317	7	prime	prime	ADJ
ejpam-3275	317	8	submodule	submodule	NOUN
ejpam-3275	317	9	of	of	ADP
ejpam-3275	317	10	m1	m1	PROPN
ejpam-3275	317	11	.	.	PUNCT
ejpam-3275	318	1	the	the	DET
ejpam-3275	318	2	following	follow	VERB
ejpam-3275	318	3	example	example	NOUN
ejpam-3275	318	4	shows	show	VERB
ejpam-3275	318	5	that	that	SCONJ
ejpam-3275	318	6	,	,	PUNCT
ejpam-3275	318	7	in	in	ADP
ejpam-3275	318	8	general	general	ADJ
ejpam-3275	318	9	,	,	PUNCT
ejpam-3275	318	10	the	the	DET
ejpam-3275	318	11	condition	condition	NOUN
ejpam-3275	318	12	β(m2	β(m2	NOUN
ejpam-3275	318	13	)	)	PUNCT
ejpam-3275	318	14	6=	6=	ADP
ejpam-3275	318	15	{	{	PUNCT
ejpam-3275	318	16	0	0	NUM
ejpam-3275	318	17	}	}	PUNCT
ejpam-3275	318	18	in	in	ADP
ejpam-3275	318	19	proposition	proposition	NOUN
ejpam-3275	318	20	5	5	NUM
ejpam-3275	318	21	can	can	AUX
ejpam-3275	318	22	not	not	PART
ejpam-3275	318	23	be	be	AUX
ejpam-3275	318	24	omitted	omit	VERB
ejpam-3275	318	25	.	.	PUNCT
ejpam-3275	318	26	example	example	NOUN
ejpam-3275	319	1	2	2	NUM
ejpam-3275	319	2	.	.	PUNCT
ejpam-3275	319	3	let	let	VERB
ejpam-3275	319	4	m1	m1	PROPN
ejpam-3275	319	5	=	=	SYM
ejpam-3275	319	6	z8	z8	PROPN
ejpam-3275	319	7	,	,	PUNCT
ejpam-3275	319	8	m2	m2	PROPN
ejpam-3275	319	9	=	=	PROPN
ejpam-3275	319	10	{	{	PUNCT
ejpam-3275	319	11	0	0	NUM
ejpam-3275	319	12	}	}	PUNCT
ejpam-3275	319	13	,	,	PUNCT
ejpam-3275	319	14	r1	r1	NOUN
ejpam-3275	319	15	=	=	PUNCT
ejpam-3275	319	16	r2	r2	PROPN
ejpam-3275	319	17	=	=	PUNCT
ejpam-3275	320	1	z.	z.	PROPN
ejpam-3275	321	1	it	it	PRON
ejpam-3275	321	2	is	be	AUX
ejpam-3275	321	3	clear	clear	ADJ
ejpam-3275	321	4	that	that	SCONJ
ejpam-3275	321	5	{	{	PUNCT
ejpam-3275	321	6	0̄}×{0	0̄}×{0	ADJ
ejpam-3275	321	7	}	}	PUNCT
ejpam-3275	321	8	is	be	AUX
ejpam-3275	321	9	a	a	DET
ejpam-3275	321	10	weakly	weakly	ADJ
ejpam-3275	321	11	α	α	NOUN
ejpam-3275	321	12	-	-	ADJ
ejpam-3275	321	13	prime	prime	ADJ
ejpam-3275	321	14	submodule	submodule	NOUN
ejpam-3275	321	15	of	of	ADP
ejpam-3275	321	16	m1	m1	PROPN
ejpam-3275	321	17	×m2	×m2	PROPN
ejpam-3275	321	18	.	.	PUNCT
ejpam-3275	322	1	however	however	ADV
ejpam-3275	322	2	,	,	PUNCT
ejpam-3275	322	3	{	{	PUNCT
ejpam-3275	322	4	0̄	0̄	NOUN
ejpam-3275	322	5	}	}	PUNCT
ejpam-3275	322	6	is	be	AUX
ejpam-3275	322	7	not	not	PART
ejpam-3275	322	8	an	an	DET
ejpam-3275	322	9	α	α	NOUN
ejpam-3275	322	10	-	-	ADJ
ejpam-3275	322	11	prime	prime	ADJ
ejpam-3275	322	12	submodule	submodule	NOUN
ejpam-3275	322	13	of	of	ADP
ejpam-3275	322	14	z	z	NOUN
ejpam-3275	322	15	-	-	PUNCT
ejpam-3275	322	16	module	module	NOUN
ejpam-3275	322	17	z8	z8	NOUN
ejpam-3275	322	18	.	.	PUNCT
ejpam-3275	323	1	proposition	proposition	NOUN
ejpam-3275	323	2	6	6	NUM
ejpam-3275	323	3	.	.	PUNCT
ejpam-3275	324	1	let	let	VERB
ejpam-3275	324	2	m1,m2	m1,m2	PROPN
ejpam-3275	324	3	be	be	AUX
ejpam-3275	324	4	r1	r1	VERB
ejpam-3275	324	5	,	,	PUNCT
ejpam-3275	324	6	r2	r2	NOUN
ejpam-3275	324	7	-	-	PUNCT
ejpam-3275	324	8	modules	module	NOUN
ejpam-3275	324	9	respectively	respectively	ADV
ejpam-3275	324	10	and	and	CCONJ
ejpam-3275	324	11	n1	n1	PROPN
ejpam-3275	324	12	×n2	×n2	PROPN
ejpam-3275	324	13	be	be	VERB
ejpam-3275	324	14	a	a	DET
ejpam-3275	324	15	submodule	submodule	NOUN
ejpam-3275	324	16	of	of	ADP
ejpam-3275	324	17	m1	m1	PROPN
ejpam-3275	324	18	×m2	×m2	PROPN
ejpam-3275	324	19	.	.	PUNCT
ejpam-3275	325	1	then	then	ADV
ejpam-3275	325	2	β(n1	β(n1	VERB
ejpam-3275	325	3	×n2	×n2	NOUN
ejpam-3275	325	4	)	)	PUNCT
ejpam-3275	325	5	=	=	PRON
ejpam-3275	325	6	{	{	PUNCT
ejpam-3275	325	7	(	(	PUNCT
ejpam-3275	325	8	0	0	NUM
ejpam-3275	325	9	,	,	PUNCT
ejpam-3275	325	10	0	0	NUM
ejpam-3275	325	11	)	)	PUNCT
ejpam-3275	325	12	}	}	PUNCT
ejpam-3275	325	13	if	if	SCONJ
ejpam-3275	325	14	and	and	CCONJ
ejpam-3275	325	15	only	only	ADV
ejpam-3275	325	16	if	if	SCONJ
ejpam-3275	325	17	β(n1	β(n1	NOUN
ejpam-3275	325	18	)	)	PUNCT
ejpam-3275	325	19	=	=	PRON
ejpam-3275	325	20	{	{	PUNCT
ejpam-3275	325	21	0	0	NUM
ejpam-3275	325	22	}	}	PUNCT
ejpam-3275	325	23	and	and	CCONJ
ejpam-3275	325	24	β(n2	β(n2	NUM
ejpam-3275	325	25	)	)	PUNCT
ejpam-3275	325	26	=	=	PRON
ejpam-3275	325	27	{	{	PUNCT
ejpam-3275	325	28	0	0	NUM
ejpam-3275	325	29	}	}	PUNCT
ejpam-3275	325	30	.	.	PUNCT
ejpam-3275	326	1	proof	proof	NOUN
ejpam-3275	326	2	.	.	PUNCT
ejpam-3275	327	1	it	it	PRON
ejpam-3275	327	2	is	be	AUX
ejpam-3275	327	3	evident	evident	ADJ
ejpam-3275	327	4	.	.	PUNCT
ejpam-3275	328	1	proposition	proposition	NOUN
ejpam-3275	328	2	7	7	NUM
ejpam-3275	328	3	.	.	PUNCT
ejpam-3275	329	1	let	let	VERB
ejpam-3275	329	2	m1,m2	m1,m2	PROPN
ejpam-3275	329	3	be	be	AUX
ejpam-3275	329	4	r1	r1	VERB
ejpam-3275	329	5	,	,	PUNCT
ejpam-3275	329	6	r2	r2	NOUN
ejpam-3275	329	7	-	-	PUNCT
ejpam-3275	329	8	modules	module	NOUN
ejpam-3275	329	9	respectively	respectively	ADV
ejpam-3275	329	10	.	.	PUNCT
ejpam-3275	330	1	then	then	ADV
ejpam-3275	330	2	(	(	PUNCT
ejpam-3275	330	3	i	i	NOUN
ejpam-3275	330	4	)	)	PUNCT
ejpam-3275	330	5	if	if	SCONJ
ejpam-3275	330	6	n1	n1	PROPN
ejpam-3275	330	7	×n2	×n2	PROPN
ejpam-3275	330	8	is	be	AUX
ejpam-3275	330	9	a	a	DET
ejpam-3275	330	10	weakly	weakly	ADJ
ejpam-3275	330	11	α	α	NOUN
ejpam-3275	330	12	-	-	ADJ
ejpam-3275	330	13	prime	prime	ADJ
ejpam-3275	330	14	submodule	submodule	NOUN
ejpam-3275	330	15	of	of	ADP
ejpam-3275	330	16	m1	m1	PROPN
ejpam-3275	330	17	×m2	×m2	PROPN
ejpam-3275	330	18	,	,	PUNCT
ejpam-3275	330	19	then	then	ADV
ejpam-3275	330	20	either	either	CCONJ
ejpam-3275	330	21	β(n1	β(n1	NOUN
ejpam-3275	330	22	)	)	PUNCT
ejpam-3275	330	23	=	=	PRON
ejpam-3275	330	24	{	{	PUNCT
ejpam-3275	330	25	0	0	NUM
ejpam-3275	330	26	}	}	PUNCT
ejpam-3275	330	27	or	or	CCONJ
ejpam-3275	330	28	α(n1	α(n1	NOUN
ejpam-3275	330	29	)	)	PUNCT
ejpam-3275	331	1	=	=	SYM
ejpam-3275	331	2	m1	m1	PROPN
ejpam-3275	331	3	or	or	CCONJ
ejpam-3275	331	4	α(n2	α(n2	NUM
ejpam-3275	331	5	)	)	PUNCT
ejpam-3275	331	6	=	=	SYM
ejpam-3275	331	7	m2	m2	PROPN
ejpam-3275	331	8	.	.	PUNCT
ejpam-3275	331	9	(	(	PUNCT
ejpam-3275	331	10	ii	ii	NOUN
ejpam-3275	331	11	)	)	PUNCT
ejpam-3275	331	12	if	if	SCONJ
ejpam-3275	331	13	n1	n1	PROPN
ejpam-3275	331	14	×n2	×n2	PROPN
ejpam-3275	331	15	is	be	AUX
ejpam-3275	331	16	a	a	DET
ejpam-3275	331	17	weakly	weakly	ADJ
ejpam-3275	331	18	α	α	NOUN
ejpam-3275	331	19	-	-	ADJ
ejpam-3275	331	20	prime	prime	ADJ
ejpam-3275	331	21	submodule	submodule	NOUN
ejpam-3275	331	22	of	of	ADP
ejpam-3275	331	23	m1	m1	PROPN
ejpam-3275	331	24	×m2	×m2	PROPN
ejpam-3275	331	25	,	,	PUNCT
ejpam-3275	331	26	then	then	ADV
ejpam-3275	331	27	either	either	CCONJ
ejpam-3275	331	28	β(n2	β(n2	NOUN
ejpam-3275	331	29	)	)	PUNCT
ejpam-3275	331	30	=	=	PRON
ejpam-3275	331	31	{	{	PUNCT
ejpam-3275	331	32	0	0	NUM
ejpam-3275	331	33	}	}	PUNCT
ejpam-3275	331	34	or	or	CCONJ
ejpam-3275	331	35	α(n1	α(n1	NOUN
ejpam-3275	331	36	)	)	PUNCT
ejpam-3275	332	1	=	=	SYM
ejpam-3275	332	2	m1	m1	PROPN
ejpam-3275	332	3	or	or	CCONJ
ejpam-3275	332	4	α(n2	α(n2	NUM
ejpam-3275	332	5	)	)	PUNCT
ejpam-3275	332	6	=	=	SYM
ejpam-3275	332	7	m2	m2	PROPN
ejpam-3275	332	8	.	.	PUNCT
ejpam-3275	333	1	(	(	PUNCT
ejpam-3275	333	2	iii	iii	X
ejpam-3275	333	3	)	)	PUNCT
ejpam-3275	333	4	if	if	SCONJ
ejpam-3275	333	5	n1×n2	n1×n2	NOUN
ejpam-3275	333	6	is	be	AUX
ejpam-3275	333	7	a	a	DET
ejpam-3275	333	8	weakly	weakly	ADJ
ejpam-3275	333	9	α	α	NOUN
ejpam-3275	333	10	-	-	ADJ
ejpam-3275	333	11	prime	prime	ADJ
ejpam-3275	333	12	submodule	submodule	NOUN
ejpam-3275	333	13	of	of	ADP
ejpam-3275	333	14	m1×m2	m1×m2	PROPN
ejpam-3275	333	15	,	,	PUNCT
ejpam-3275	333	16	then	then	ADV
ejpam-3275	333	17	β(n1	β(n1	NOUN
ejpam-3275	333	18	)	)	PUNCT
ejpam-3275	333	19	=	=	PRON
ejpam-3275	333	20	{	{	PUNCT
ejpam-3275	333	21	0	0	NUM
ejpam-3275	333	22	}	}	PUNCT
ejpam-3275	333	23	or	or	CCONJ
ejpam-3275	333	24	α(n2	α(n2	NUM
ejpam-3275	333	25	)	)	PUNCT
ejpam-3275	334	1	=	=	SYM
ejpam-3275	334	2	m2	m2	PROPN
ejpam-3275	334	3	or	or	CCONJ
ejpam-3275	334	4	n1	n1	PROPN
ejpam-3275	334	5	×n2	×n2	PROPN
ejpam-3275	334	6	is	be	AUX
ejpam-3275	334	7	an	an	DET
ejpam-3275	334	8	α	α	NOUN
ejpam-3275	334	9	-	-	ADJ
ejpam-3275	334	10	prime	prime	ADJ
ejpam-3275	334	11	submodule	submodule	NOUN
ejpam-3275	334	12	of	of	ADP
ejpam-3275	334	13	m1	m1	PROPN
ejpam-3275	334	14	×m2	×m2	PROPN
ejpam-3275	334	15	.	.	PUNCT
ejpam-3275	335	1	(	(	PUNCT
ejpam-3275	335	2	iv	iv	X
ejpam-3275	335	3	)	)	PUNCT
ejpam-3275	335	4	if	if	SCONJ
ejpam-3275	335	5	n1×n2	n1×n2	NOUN
ejpam-3275	335	6	is	be	AUX
ejpam-3275	335	7	a	a	DET
ejpam-3275	335	8	weakly	weakly	ADJ
ejpam-3275	335	9	α	α	NOUN
ejpam-3275	335	10	-	-	ADJ
ejpam-3275	335	11	prime	prime	ADJ
ejpam-3275	335	12	submodule	submodule	NOUN
ejpam-3275	335	13	of	of	ADP
ejpam-3275	335	14	m1×m2	m1×m2	PROPN
ejpam-3275	335	15	,	,	PUNCT
ejpam-3275	335	16	then	then	ADV
ejpam-3275	335	17	β(n2	β(n2	PUNCT
ejpam-3275	335	18	)	)	PUNCT
ejpam-3275	336	1	=	=	PRON
ejpam-3275	336	2	{	{	PUNCT
ejpam-3275	336	3	0	0	NUM
ejpam-3275	336	4	}	}	PUNCT
ejpam-3275	336	5	or	or	CCONJ
ejpam-3275	336	6	α(n1	α(n1	NOUN
ejpam-3275	336	7	)	)	PUNCT
ejpam-3275	337	1	=	=	SYM
ejpam-3275	337	2	m1	m1	PROPN
ejpam-3275	337	3	or	or	CCONJ
ejpam-3275	337	4	n1	n1	PROPN
ejpam-3275	337	5	×n2	×n2	PROPN
ejpam-3275	337	6	is	be	AUX
ejpam-3275	337	7	an	an	DET
ejpam-3275	337	8	α	α	NOUN
ejpam-3275	337	9	-	-	ADJ
ejpam-3275	337	10	prime	prime	ADJ
ejpam-3275	337	11	submodule	submodule	NOUN
ejpam-3275	337	12	of	of	ADP
ejpam-3275	337	13	m1	m1	PROPN
ejpam-3275	337	14	×m2	×m2	PROPN
ejpam-3275	337	15	.	.	PUNCT
ejpam-3275	338	1	t.	t.	PROPN
ejpam-3275	338	2	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	338	3	/	/	SYM
ejpam-3275	338	4	eur	eur	PROPN
ejpam-3275	338	5	.	.	PUNCT
ejpam-3275	339	1	j.	j.	PROPN
ejpam-3275	339	2	pure	pure	PROPN
ejpam-3275	339	3	appl	appl	PROPN
ejpam-3275	339	4	.	.	PROPN
ejpam-3275	339	5	math	math	PROPN
ejpam-3275	339	6	,	,	PUNCT
ejpam-3275	339	7	11	11	NUM
ejpam-3275	339	8	(	(	PUNCT
ejpam-3275	339	9	3	3	NUM
ejpam-3275	339	10	)	)	PUNCT
ejpam-3275	339	11	(	(	PUNCT
ejpam-3275	339	12	2018	2018	NUM
ejpam-3275	339	13	)	)	PUNCT
ejpam-3275	339	14	,	,	PUNCT
ejpam-3275	339	15	730	730	NUM
ejpam-3275	339	16	-	-	SYM
ejpam-3275	339	17	739	739	NUM
ejpam-3275	339	18	737	737	NUM
ejpam-3275	339	19	proof	proof	NOUN
ejpam-3275	339	20	.	.	PUNCT
ejpam-3275	340	1	(	(	PUNCT
ejpam-3275	340	2	i	i	NOUN
ejpam-3275	340	3	)	)	PUNCT
ejpam-3275	340	4	assume	assume	VERB
ejpam-3275	340	5	that	that	SCONJ
ejpam-3275	340	6	n1	n1	ADJ
ejpam-3275	340	7	×	×	NOUN
ejpam-3275	340	8	n2	n2	NOUN
ejpam-3275	340	9	is	be	AUX
ejpam-3275	340	10	a	a	DET
ejpam-3275	340	11	weakly	weakly	ADJ
ejpam-3275	340	12	α	α	NOUN
ejpam-3275	340	13	-	-	ADJ
ejpam-3275	340	14	prime	prime	ADJ
ejpam-3275	340	15	submodule	submodule	NOUN
ejpam-3275	340	16	of	of	ADP
ejpam-3275	340	17	m1	m1	PROPN
ejpam-3275	340	18	×	×	PROPN
ejpam-3275	340	19	m2	m2	PROPN
ejpam-3275	340	20	and	and	CCONJ
ejpam-3275	340	21	β(n1	β(n1	NOUN
ejpam-3275	340	22	)	)	PUNCT
ejpam-3275	340	23	6=	6=	ADP
ejpam-3275	340	24	{	{	PUNCT
ejpam-3275	340	25	0	0	NUM
ejpam-3275	340	26	}	}	PUNCT
ejpam-3275	340	27	and	and	CCONJ
ejpam-3275	340	28	α(n1	α(n1	NOUN
ejpam-3275	340	29	)	)	PUNCT
ejpam-3275	340	30	6=	6=	ADP
ejpam-3275	341	1	m1	m1	PROPN
ejpam-3275	341	2	.	.	PUNCT
ejpam-3275	342	1	let	let	VERB
ejpam-3275	342	2	a	a	DET
ejpam-3275	342	3	∈	∈	PROPN
ejpam-3275	342	4	n1	n1	NOUN
ejpam-3275	342	5	be	be	AUX
ejpam-3275	342	6	such	such	ADJ
ejpam-3275	342	7	that	that	SCONJ
ejpam-3275	342	8	a+	a+	PUNCT
ejpam-3275	342	9	a	a	PRON
ejpam-3275	342	10	6=	6=	NUM
ejpam-3275	342	11	0	0	NUM
ejpam-3275	342	12	.	.	PUNCT
ejpam-3275	343	1	let	let	VERB
ejpam-3275	343	2	r	r	NOUN
ejpam-3275	343	3	∈	∈	PROPN
ejpam-3275	343	4	(	(	PUNCT
ejpam-3275	343	5	n2	n2	NOUN
ejpam-3275	343	6	:	:	PUNCT
ejpam-3275	343	7	m2	m2	PROPN
ejpam-3275	343	8	)	)	PUNCT
ejpam-3275	343	9	and	and	CCONJ
ejpam-3275	343	10	y	y	PROPN
ejpam-3275	343	11	∈m2	∈m2	NOUN
ejpam-3275	343	12	.	.	PUNCT
ejpam-3275	344	1	then	then	ADV
ejpam-3275	344	2	(	(	PUNCT
ejpam-3275	344	3	0	0	NUM
ejpam-3275	344	4	,	,	PUNCT
ejpam-3275	344	5	0	0	NUM
ejpam-3275	344	6	)	)	PUNCT
ejpam-3275	344	7	6=	6=	NUM
ejpam-3275	344	8	(	(	PUNCT
ejpam-3275	344	9	a+	a+	X
ejpam-3275	344	10	a	a	PRON
ejpam-3275	344	11	,	,	PUNCT
ejpam-3275	344	12	r(y	r(y	ADJ
ejpam-3275	344	13	+	+	PROPN
ejpam-3275	344	14	y	y	NOUN
ejpam-3275	344	15	)	)	PUNCT
ejpam-3275	344	16	)	)	PUNCT
ejpam-3275	345	1	=	=	PUNCT
ejpam-3275	345	2	(	(	PUNCT
ejpam-3275	345	3	1	1	NUM
ejpam-3275	345	4	,	,	PUNCT
ejpam-3275	345	5	r)[(a	r)[(a	NOUN
ejpam-3275	345	6	,	,	PUNCT
ejpam-3275	345	7	y	y	PROPN
ejpam-3275	345	8	)	)	PUNCT
ejpam-3275	346	1	+	+	CCONJ
ejpam-3275	346	2	(	(	PUNCT
ejpam-3275	346	3	a	a	DET
ejpam-3275	346	4	,	,	PUNCT
ejpam-3275	346	5	y	y	NOUN
ejpam-3275	346	6	)	)	PUNCT
ejpam-3275	346	7	]	]	PUNCT
ejpam-3275	347	1	∈	∈	PROPN
ejpam-3275	347	2	n1	n1	PROPN
ejpam-3275	347	3	×n2	×n2	PROPN
ejpam-3275	347	4	.	.	PUNCT
ejpam-3275	348	1	since	since	SCONJ
ejpam-3275	348	2	n1	n1	PROPN
ejpam-3275	348	3	×n2	×n2	PROPN
ejpam-3275	348	4	is	be	AUX
ejpam-3275	348	5	a	a	DET
ejpam-3275	348	6	weakly	weakly	ADJ
ejpam-3275	348	7	α	α	NOUN
ejpam-3275	348	8	-	-	ADJ
ejpam-3275	348	9	prime	prime	ADJ
ejpam-3275	348	10	submodule	submodule	NOUN
ejpam-3275	348	11	of	of	ADP
ejpam-3275	348	12	m1	m1	PROPN
ejpam-3275	348	13	×m2	×m2	PROPN
ejpam-3275	348	14	,	,	PUNCT
ejpam-3275	348	15	we	we	PRON
ejpam-3275	348	16	have	have	VERB
ejpam-3275	348	17	(	(	PUNCT
ejpam-3275	348	18	1	1	NUM
ejpam-3275	348	19	+	+	NUM
ejpam-3275	348	20	1	1	NUM
ejpam-3275	348	21	,	,	PUNCT
ejpam-3275	348	22	r+	r+	PUNCT
ejpam-3275	348	23	r	r	NOUN
ejpam-3275	348	24	)	)	PUNCT
ejpam-3275	348	25	(	(	PUNCT
ejpam-3275	348	26	m1	m1	NOUN
ejpam-3275	348	27	×m2	×m2	PROPN
ejpam-3275	348	28	)	)	PUNCT
ejpam-3275	348	29	⊆	⊆	NUM
ejpam-3275	348	30	n1	n1	PROPN
ejpam-3275	348	31	×n2	×n2	PROPN
ejpam-3275	348	32	or	or	CCONJ
ejpam-3275	348	33	(	(	PUNCT
ejpam-3275	348	34	a	a	PRON
ejpam-3275	348	35	,	,	PUNCT
ejpam-3275	348	36	y	y	PROPN
ejpam-3275	348	37	)	)	PUNCT
ejpam-3275	349	1	+	+	CCONJ
ejpam-3275	349	2	(	(	PUNCT
ejpam-3275	349	3	a	a	PRON
ejpam-3275	349	4	,	,	PUNCT
ejpam-3275	349	5	y	y	NOUN
ejpam-3275	349	6	)	)	PUNCT
ejpam-3275	349	7	∈	∈	PROPN
ejpam-3275	349	8	n1	n1	PROPN
ejpam-3275	349	9	×	×	PROPN
ejpam-3275	349	10	n2	n2	NOUN
ejpam-3275	349	11	.	.	PUNCT
ejpam-3275	350	1	this	this	PRON
ejpam-3275	350	2	implies	imply	VERB
ejpam-3275	350	3	that	that	SCONJ
ejpam-3275	350	4	(	(	PUNCT
ejpam-3275	350	5	1	1	NUM
ejpam-3275	350	6	+	+	NUM
ejpam-3275	350	7	1)m1	1)m1	NUM
ejpam-3275	350	8	⊆	⊆	NUM
ejpam-3275	350	9	n1	n1	NOUN
ejpam-3275	350	10	or	or	CCONJ
ejpam-3275	350	11	y	y	PROPN
ejpam-3275	350	12	+	+	CCONJ
ejpam-3275	350	13	y	y	PROPN
ejpam-3275	350	14	∈	∈	PROPN
ejpam-3275	350	15	n2	n2	NOUN
ejpam-3275	350	16	.	.	PUNCT
ejpam-3275	351	1	since	since	SCONJ
ejpam-3275	351	2	α(n1	α(n1	NOUN
ejpam-3275	351	3	)	)	PUNCT
ejpam-3275	351	4	6=	6=	ADP
ejpam-3275	351	5	m1	m1	PROPN
ejpam-3275	351	6	,	,	PUNCT
ejpam-3275	351	7	there	there	PRON
ejpam-3275	351	8	is	be	VERB
ejpam-3275	351	9	m	m	PROPN
ejpam-3275	351	10	∈	∈	PROPN
ejpam-3275	351	11	m1	m1	NOUN
ejpam-3275	351	12	such	such	ADJ
ejpam-3275	351	13	that	that	SCONJ
ejpam-3275	351	14	m	m	VERB
ejpam-3275	351	15	+	+	ADJ
ejpam-3275	351	16	m	m	VERB
ejpam-3275	351	17	/∈	/∈	ADJ
ejpam-3275	352	1	n1	n1	PROPN
ejpam-3275	352	2	.	.	PUNCT
ejpam-3275	353	1	this	this	PRON
ejpam-3275	353	2	means	mean	VERB
ejpam-3275	353	3	(	(	PUNCT
ejpam-3275	353	4	1	1	NUM
ejpam-3275	353	5	+	+	NUM
ejpam-3275	353	6	1)m1	1)m1	NUM
ejpam-3275	353	7	*	*	SYM
ejpam-3275	353	8	n1	n1	PROPN
ejpam-3275	353	9	.	.	PUNCT
ejpam-3275	354	1	therefore	therefore	ADV
ejpam-3275	354	2	y	y	PROPN
ejpam-3275	354	3	∈	∈	PROPN
ejpam-3275	354	4	α(n2	α(n2	PRON
ejpam-3275	354	5	)	)	PUNCT
ejpam-3275	354	6	.	.	PUNCT
ejpam-3275	355	1	(	(	PUNCT
ejpam-3275	355	2	ii	ii	X
ejpam-3275	355	3	)	)	PUNCT
ejpam-3275	355	4	the	the	DET
ejpam-3275	355	5	proof	proof	NOUN
ejpam-3275	355	6	is	be	AUX
ejpam-3275	355	7	similar	similar	ADJ
ejpam-3275	355	8	to	to	ADP
ejpam-3275	355	9	(	(	PUNCT
ejpam-3275	355	10	i	i	NOUN
ejpam-3275	355	11	)	)	PUNCT
ejpam-3275	355	12	.	.	PUNCT
ejpam-3275	356	1	(	(	PUNCT
ejpam-3275	356	2	iii	iii	X
ejpam-3275	356	3	)	)	PUNCT
ejpam-3275	356	4	assume	assume	VERB
ejpam-3275	356	5	that	that	SCONJ
ejpam-3275	356	6	n1×n2	n1×n2	NOUN
ejpam-3275	356	7	is	be	AUX
ejpam-3275	356	8	a	a	DET
ejpam-3275	356	9	weakly	weakly	ADJ
ejpam-3275	356	10	α	α	NOUN
ejpam-3275	356	11	-	-	ADJ
ejpam-3275	356	12	prime	prime	ADJ
ejpam-3275	356	13	submodule	submodule	NOUN
ejpam-3275	356	14	of	of	ADP
ejpam-3275	356	15	m1×m2	m1×m2	PROPN
ejpam-3275	356	16	and	and	CCONJ
ejpam-3275	356	17	β(n1	β(n1	NOUN
ejpam-3275	356	18	)	)	PUNCT
ejpam-3275	356	19	6=	6=	ADP
ejpam-3275	356	20	{	{	PUNCT
ejpam-3275	356	21	0	0	NUM
ejpam-3275	356	22	}	}	PUNCT
ejpam-3275	356	23	and	and	CCONJ
ejpam-3275	356	24	α(n2	α(n2	NUM
ejpam-3275	356	25	)	)	PUNCT
ejpam-3275	356	26	6=	6=	NUM
ejpam-3275	357	1	m2	m2	PROPN
ejpam-3275	357	2	.	.	PUNCT
ejpam-3275	358	1	by	by	ADP
ejpam-3275	358	2	(	(	PUNCT
ejpam-3275	358	3	i	i	NOUN
ejpam-3275	358	4	)	)	PUNCT
ejpam-3275	358	5	,	,	PUNCT
ejpam-3275	358	6	α(n1	α(n1	INTJ
ejpam-3275	358	7	)	)	PUNCT
ejpam-3275	358	8	=	=	SYM
ejpam-3275	358	9	m1	m1	NOUN
ejpam-3275	358	10	.	.	PUNCT
ejpam-3275	359	1	let	let	VERB
ejpam-3275	359	2	(	(	PUNCT
ejpam-3275	359	3	r1	r1	NOUN
ejpam-3275	359	4	,	,	PUNCT
ejpam-3275	359	5	r2	r2	PROPN
ejpam-3275	359	6	)	)	PUNCT
ejpam-3275	359	7	∈	∈	PROPN
ejpam-3275	359	8	r1	r1	NOUN
ejpam-3275	359	9	×	×	NOUN
ejpam-3275	359	10	r2	r2	PROPN
ejpam-3275	359	11	and	and	CCONJ
ejpam-3275	359	12	(	(	PUNCT
ejpam-3275	359	13	m1,m2	m1,m2	PROPN
ejpam-3275	359	14	)	)	PUNCT
ejpam-3275	359	15	∈	∈	PROPN
ejpam-3275	359	16	m1	m1	NOUN
ejpam-3275	359	17	×m2	×m2	NOUN
ejpam-3275	359	18	be	be	VERB
ejpam-3275	359	19	such	such	ADJ
ejpam-3275	359	20	that	that	SCONJ
ejpam-3275	359	21	(	(	PUNCT
ejpam-3275	359	22	r1	r1	PROPN
ejpam-3275	359	23	,	,	PUNCT
ejpam-3275	359	24	r2)[(m1,m2	r2)[(m1,m2	PROPN
ejpam-3275	359	25	)	)	PUNCT
ejpam-3275	360	1	+	+	CCONJ
ejpam-3275	360	2	(	(	PUNCT
ejpam-3275	360	3	m1,m2	m1,m2	PROPN
ejpam-3275	360	4	)	)	PUNCT
ejpam-3275	360	5	]	]	PUNCT
ejpam-3275	361	1	∈	∈	PROPN
ejpam-3275	361	2	n1	n1	PROPN
ejpam-3275	361	3	×	×	PROPN
ejpam-3275	361	4	n2	n2	NOUN
ejpam-3275	361	5	.	.	PUNCT
ejpam-3275	362	1	then	then	ADV
ejpam-3275	362	2	r1(m1	r1(m1	NOUN
ejpam-3275	362	3	+	+	CCONJ
ejpam-3275	362	4	m1	m1	NOUN
ejpam-3275	362	5	)	)	PUNCT
ejpam-3275	362	6	∈	∈	PROPN
ejpam-3275	362	7	n1	n1	NOUN
ejpam-3275	362	8	and	and	CCONJ
ejpam-3275	362	9	r2(m2+m2	r2(m2+m2	NOUN
ejpam-3275	362	10	)	)	PUNCT
ejpam-3275	362	11	∈	∈	PROPN
ejpam-3275	362	12	n2	n2	NOUN
ejpam-3275	362	13	.	.	PUNCT
ejpam-3275	363	1	let	let	VERB
ejpam-3275	363	2	a	a	DET
ejpam-3275	363	3	∈	∈	PROPN
ejpam-3275	363	4	n1	n1	NOUN
ejpam-3275	363	5	be	be	AUX
ejpam-3275	363	6	such	such	ADJ
ejpam-3275	363	7	that	that	DET
ejpam-3275	363	8	a+a	a+a	NUM
ejpam-3275	363	9	6=	6=	ADP
ejpam-3275	363	10	0	0	NUM
ejpam-3275	363	11	.	.	PUNCT
ejpam-3275	364	1	then	then	ADV
ejpam-3275	364	2	(	(	PUNCT
ejpam-3275	364	3	0	0	NUM
ejpam-3275	364	4	,	,	PUNCT
ejpam-3275	364	5	0	0	NUM
ejpam-3275	364	6	)	)	PUNCT
ejpam-3275	364	7	6=	6=	ADP
ejpam-3275	364	8	(	(	PUNCT
ejpam-3275	364	9	a+a	a+a	NOUN
ejpam-3275	364	10	,	,	PUNCT
ejpam-3275	364	11	r2(m2+m2	r2(m2+m2	NOUN
ejpam-3275	364	12	)	)	PUNCT
ejpam-3275	364	13	)	)	PUNCT
ejpam-3275	365	1	=	=	PUNCT
ejpam-3275	365	2	(	(	PUNCT
ejpam-3275	365	3	1	1	NUM
ejpam-3275	365	4	,	,	PUNCT
ejpam-3275	365	5	r2)[(a	r2)[(a	PROPN
ejpam-3275	365	6	,	,	PUNCT
ejpam-3275	365	7	m2	m2	PROPN
ejpam-3275	365	8	)	)	PUNCT
ejpam-3275	365	9	+	+	CCONJ
ejpam-3275	365	10	(	(	PUNCT
ejpam-3275	365	11	a	a	PRON
ejpam-3275	365	12	,	,	PUNCT
ejpam-3275	365	13	m2	m2	PROPN
ejpam-3275	365	14	)	)	PUNCT
ejpam-3275	365	15	]	]	PUNCT
ejpam-3275	366	1	∈	∈	PROPN
ejpam-3275	366	2	n1	n1	PROPN
ejpam-3275	366	3	×	×	PROPN
ejpam-3275	366	4	n2	n2	NOUN
ejpam-3275	366	5	.	.	PUNCT
ejpam-3275	367	1	since	since	SCONJ
ejpam-3275	367	2	n1	n1	PROPN
ejpam-3275	367	3	×	×	PROPN
ejpam-3275	367	4	n2	n2	NOUN
ejpam-3275	367	5	is	be	AUX
ejpam-3275	367	6	a	a	DET
ejpam-3275	367	7	weakly	weakly	ADJ
ejpam-3275	367	8	α	α	NOUN
ejpam-3275	367	9	-	-	ADJ
ejpam-3275	367	10	prime	prime	ADJ
ejpam-3275	367	11	submodule	submodule	NOUN
ejpam-3275	367	12	of	of	ADP
ejpam-3275	367	13	m1×m2	m1×m2	PROPN
ejpam-3275	367	14	,	,	PUNCT
ejpam-3275	367	15	we	we	PRON
ejpam-3275	367	16	have	have	VERB
ejpam-3275	367	17	(	(	PUNCT
ejpam-3275	367	18	1	1	NUM
ejpam-3275	367	19	+	+	NOUN
ejpam-3275	367	20	1	1	NUM
ejpam-3275	367	21	,	,	PUNCT
ejpam-3275	367	22	r2+r2	r2+r2	NOUN
ejpam-3275	367	23	)	)	PUNCT
ejpam-3275	367	24	(	(	PUNCT
ejpam-3275	367	25	m1×m2	m1×m2	PROPN
ejpam-3275	367	26	)	)	PUNCT
ejpam-3275	367	27	⊆	⊆	NUM
ejpam-3275	367	28	n1×n2	n1×n2	NOUN
ejpam-3275	367	29	or	or	CCONJ
ejpam-3275	367	30	(	(	PUNCT
ejpam-3275	367	31	a	a	PRON
ejpam-3275	367	32	,	,	PUNCT
ejpam-3275	367	33	m2)+(a	m2)+(a	PROPN
ejpam-3275	367	34	,	,	PUNCT
ejpam-3275	367	35	m2	m2	PROPN
ejpam-3275	367	36	)	)	PUNCT
ejpam-3275	367	37	∈	∈	PROPN
ejpam-3275	367	38	n1×n2	n1×n2	NOUN
ejpam-3275	367	39	.	.	PUNCT
ejpam-3275	368	1	since	since	SCONJ
ejpam-3275	368	2	n1×n2	n1×n2	NOUN
ejpam-3275	368	3	is	be	AUX
ejpam-3275	368	4	a	a	DET
ejpam-3275	368	5	submodule	submodule	NOUN
ejpam-3275	368	6	of	of	ADP
ejpam-3275	368	7	m1×m2	m1×m2	PROPN
ejpam-3275	368	8	and	and	CCONJ
ejpam-3275	368	9	α(n1	α(n1	NOUN
ejpam-3275	368	10	)	)	PUNCT
ejpam-3275	368	11	=	=	SYM
ejpam-3275	368	12	m1	m1	NOUN
ejpam-3275	368	13	,	,	PUNCT
ejpam-3275	368	14	(	(	PUNCT
ejpam-3275	368	15	r1+r1	r1+r1	NOUN
ejpam-3275	368	16	,	,	PUNCT
ejpam-3275	368	17	r2+r2	r2+r2	NOUN
ejpam-3275	368	18	)	)	PUNCT
ejpam-3275	368	19	(	(	PUNCT
ejpam-3275	368	20	m1×m2	m1×m2	PROPN
ejpam-3275	368	21	)	)	PUNCT
ejpam-3275	368	22	⊆	⊆	NUM
ejpam-3275	368	23	n1×n2	n1×n2	NOUN
ejpam-3275	368	24	or	or	CCONJ
ejpam-3275	368	25	(	(	PUNCT
ejpam-3275	368	26	m1,m2	m1,m2	PROPN
ejpam-3275	368	27	)	)	PUNCT
ejpam-3275	369	1	+	+	CCONJ
ejpam-3275	369	2	(	(	PUNCT
ejpam-3275	369	3	m1,m2	m1,m2	PROPN
ejpam-3275	369	4	)	)	PUNCT
ejpam-3275	369	5	∈	∈	PROPN
ejpam-3275	369	6	n1	n1	PROPN
ejpam-3275	369	7	×n2	×n2	PROPN
ejpam-3275	369	8	.	.	PUNCT
ejpam-3275	370	1	this	this	PRON
ejpam-3275	370	2	implies	imply	VERB
ejpam-3275	370	3	that	that	SCONJ
ejpam-3275	370	4	n1	n1	PROPN
ejpam-3275	370	5	×n2	×n2	PROPN
ejpam-3275	370	6	is	be	AUX
ejpam-3275	370	7	an	an	DET
ejpam-3275	370	8	α	α	NOUN
ejpam-3275	370	9	-	-	ADJ
ejpam-3275	370	10	prime	prime	ADJ
ejpam-3275	370	11	submodule	submodule	NOUN
ejpam-3275	370	12	of	of	ADP
ejpam-3275	370	13	m1	m1	PROPN
ejpam-3275	370	14	×m2	×m2	PROPN
ejpam-3275	370	15	.	.	PUNCT
ejpam-3275	371	1	(	(	PUNCT
ejpam-3275	371	2	iv	iv	X
ejpam-3275	371	3	)	)	PUNCT
ejpam-3275	371	4	the	the	DET
ejpam-3275	371	5	proof	proof	NOUN
ejpam-3275	371	6	is	be	AUX
ejpam-3275	371	7	similar	similar	ADJ
ejpam-3275	371	8	to	to	ADP
ejpam-3275	371	9	(	(	PUNCT
ejpam-3275	371	10	iii	iii	NOUN
ejpam-3275	371	11	)	)	PUNCT
ejpam-3275	371	12	.	.	PUNCT
ejpam-3275	372	1	the	the	DET
ejpam-3275	372	2	following	follow	VERB
ejpam-3275	372	3	example	example	NOUN
ejpam-3275	372	4	obtains	obtain	VERB
ejpam-3275	372	5	that	that	SCONJ
ejpam-3275	372	6	the	the	DET
ejpam-3275	372	7	assumption	assumption	NOUN
ejpam-3275	372	8	α(n2	α(n2	ADV
ejpam-3275	372	9	)	)	PUNCT
ejpam-3275	372	10	6=	6=	PUNCT
ejpam-3275	373	1	m2	m2	PROPN
ejpam-3275	373	2	in	in	ADP
ejpam-3275	373	3	the	the	DET
ejpam-3275	373	4	proof	proof	NOUN
ejpam-3275	373	5	of	of	ADP
ejpam-3275	373	6	proposition	proposition	NOUN
ejpam-3275	373	7	7	7	NUM
ejpam-3275	373	8	(	(	PUNCT
ejpam-3275	373	9	iii	iii	NOUN
ejpam-3275	373	10	)	)	PUNCT
ejpam-3275	373	11	is	be	AUX
ejpam-3275	373	12	necessary	necessary	ADJ
ejpam-3275	373	13	.	.	PUNCT
ejpam-3275	373	14	example	example	NOUN
ejpam-3275	373	15	3	3	X
ejpam-3275	373	16	.	.	X
ejpam-3275	374	1	consider	consider	VERB
ejpam-3275	374	2	a	a	DET
ejpam-3275	374	3	submodule	submodule	NOUN
ejpam-3275	374	4	4z×	4z×	DET
ejpam-3275	374	5	3z	3z	NUM
ejpam-3275	374	6	of	of	ADP
ejpam-3275	374	7	a	a	DET
ejpam-3275	374	8	z×z	z×z	NUM
ejpam-3275	374	9	-	-	PUNCT
ejpam-3275	374	10	module	module	NOUN
ejpam-3275	374	11	z×	z×	NOUN
ejpam-3275	374	12	3z	3z	NUM
ejpam-3275	374	13	,	,	PUNCT
ejpam-3275	374	14	by	by	ADP
ejpam-3275	374	15	proposition	proposition	NOUN
ejpam-3275	374	16	5	5	NUM
ejpam-3275	374	17	,	,	PUNCT
ejpam-3275	374	18	4z×	4z×	DET
ejpam-3275	374	19	3z	3z	NUM
ejpam-3275	374	20	is	be	AUX
ejpam-3275	374	21	a	a	DET
ejpam-3275	374	22	weakly	weakly	ADJ
ejpam-3275	374	23	α	α	NOUN
ejpam-3275	374	24	-	-	ADJ
ejpam-3275	374	25	prime	prime	ADJ
ejpam-3275	374	26	submodule	submodule	NOUN
ejpam-3275	374	27	of	of	ADP
ejpam-3275	374	28	z×	z×	NOUN
ejpam-3275	374	29	3z	3z	NUM
ejpam-3275	374	30	.	.	PUNCT
ejpam-3275	375	1	however	however	ADV
ejpam-3275	375	2	,	,	PUNCT
ejpam-3275	375	3	4z×	4z×	DET
ejpam-3275	375	4	3z	3z	NUM
ejpam-3275	375	5	is	be	AUX
ejpam-3275	375	6	not	not	PART
ejpam-3275	375	7	an	an	DET
ejpam-3275	375	8	α	α	NOUN
ejpam-3275	375	9	-	-	ADJ
ejpam-3275	375	10	prime	prime	ADJ
ejpam-3275	375	11	submodule	submodule	NOUN
ejpam-3275	375	12	of	of	ADP
ejpam-3275	375	13	z×	z×	NOUN
ejpam-3275	375	14	3z	3z	NUM
ejpam-3275	375	15	because	because	SCONJ
ejpam-3275	375	16	(	(	PUNCT
ejpam-3275	375	17	1	1	NUM
ejpam-3275	375	18	,	,	PUNCT
ejpam-3275	375	19	3)[(2	3)[(2	NOUN
ejpam-3275	375	20	,	,	PUNCT
ejpam-3275	375	21	1	1	NUM
ejpam-3275	375	22	)	)	PUNCT
ejpam-3275	375	23	+	+	CCONJ
ejpam-3275	375	24	(	(	PUNCT
ejpam-3275	375	25	2	2	NUM
ejpam-3275	375	26	,	,	PUNCT
ejpam-3275	375	27	1	1	NUM
ejpam-3275	375	28	)	)	PUNCT
ejpam-3275	375	29	]	]	PUNCT
ejpam-3275	376	1	=	=	PUNCT
ejpam-3275	376	2	(	(	PUNCT
ejpam-3275	376	3	4	4	NUM
ejpam-3275	376	4	,	,	PUNCT
ejpam-3275	376	5	6	6	NUM
ejpam-3275	376	6	)	)	PUNCT
ejpam-3275	376	7	∈	∈	NOUN
ejpam-3275	376	8	4z×	4z×	PRON
ejpam-3275	376	9	3z	3z	NUM
ejpam-3275	376	10	and	and	CCONJ
ejpam-3275	376	11	(	(	PUNCT
ejpam-3275	376	12	2	2	NUM
ejpam-3275	376	13	,	,	PUNCT
ejpam-3275	376	14	6	6	NUM
ejpam-3275	376	15	)	)	PUNCT
ejpam-3275	376	16	(	(	PUNCT
ejpam-3275	376	17	z×	z×	NUM
ejpam-3275	376	18	3z	3z	NUM
ejpam-3275	376	19	)	)	PUNCT
ejpam-3275	376	20	*	*	PUNCT
ejpam-3275	377	1	4z×	4z×	PRON
ejpam-3275	377	2	3z	3z	NUM
ejpam-3275	377	3	and	and	CCONJ
ejpam-3275	377	4	(	(	PUNCT
ejpam-3275	377	5	4	4	NUM
ejpam-3275	377	6	,	,	PUNCT
ejpam-3275	377	7	2	2	NUM
ejpam-3275	377	8	)	)	PUNCT
ejpam-3275	377	9	/∈	/∈	PUNCT
ejpam-3275	378	1	4z×	4z×	PRON
ejpam-3275	378	2	3z	3z	NUM
ejpam-3275	378	3	.	.	PUNCT
ejpam-3275	379	1	in	in	ADP
ejpam-3275	379	2	particular	particular	ADJ
ejpam-3275	379	3	,	,	PUNCT
ejpam-3275	379	4	α(3z	α(3z	NUM
ejpam-3275	379	5	)	)	PUNCT
ejpam-3275	379	6	=	=	SYM
ejpam-3275	379	7	3z	3z	NUM
ejpam-3275	379	8	.	.	PUNCT
ejpam-3275	380	1	4	4	X
ejpam-3275	380	2	.	.	X
ejpam-3275	380	3	the	the	DET
ejpam-3275	380	4	traveling	traveling	NOUN
ejpam-3275	380	5	of	of	ADP
ejpam-3275	380	6	α	α	NOUN
ejpam-3275	380	7	-	-	NOUN
ejpam-3275	380	8	prime	prime	NOUN
ejpam-3275	380	9	from	from	ADP
ejpam-3275	380	10	modules	module	NOUN
ejpam-3275	380	11	to	to	ADP
ejpam-3275	380	12	rings	ring	NOUN
ejpam-3275	380	13	in	in	ADP
ejpam-3275	380	14	this	this	DET
ejpam-3275	380	15	section	section	NOUN
ejpam-3275	380	16	we	we	PRON
ejpam-3275	380	17	apply	apply	VERB
ejpam-3275	380	18	the	the	DET
ejpam-3275	380	19	notion	notion	NOUN
ejpam-3275	380	20	of	of	ADP
ejpam-3275	380	21	(	(	PUNCT
ejpam-3275	380	22	weakly	weakly	ADJ
ejpam-3275	380	23	)	)	PUNCT
ejpam-3275	380	24	α	α	NOUN
ejpam-3275	380	25	-	-	ADJ
ejpam-3275	380	26	prime	prime	ADJ
ejpam-3275	380	27	submodules	submodule	NOUN
ejpam-3275	380	28	to	to	ADP
ejpam-3275	380	29	(	(	PUNCT
ejpam-3275	380	30	weakly	weakly	ADJ
ejpam-3275	380	31	)	)	PUNCT
ejpam-3275	380	32	αprime	αprime	ADJ
ejpam-3275	380	33	ideals	ideal	NOUN
ejpam-3275	380	34	.	.	PUNCT
ejpam-3275	381	1	definition	definition	NOUN
ejpam-3275	381	2	5	5	NUM
ejpam-3275	381	3	.	.	PUNCT
ejpam-3275	382	1	a	a	DET
ejpam-3275	382	2	proper	proper	ADJ
ejpam-3275	382	3	ideal	ideal	NOUN
ejpam-3275	382	4	p	p	NOUN
ejpam-3275	382	5	of	of	ADP
ejpam-3275	382	6	a	a	DET
ejpam-3275	382	7	ring	ring	NOUN
ejpam-3275	382	8	r	r	NOUN
ejpam-3275	382	9	is	be	AUX
ejpam-3275	382	10	called	call	VERB
ejpam-3275	382	11	an	an	DET
ejpam-3275	382	12	α	α	NUM
ejpam-3275	382	13	-	-	ADJ
ejpam-3275	382	14	prime	prime	ADJ
ejpam-3275	382	15	ideal	ideal	NOUN
ejpam-3275	382	16	of	of	ADP
ejpam-3275	382	17	r	r	NOUN
ejpam-3275	382	18	if	if	SCONJ
ejpam-3275	382	19	p	p	NOUN
ejpam-3275	382	20	is	be	AUX
ejpam-3275	382	21	an	an	DET
ejpam-3275	382	22	α	α	NOUN
ejpam-3275	382	23	-	-	ADJ
ejpam-3275	382	24	prime	prime	ADJ
ejpam-3275	382	25	submodule	submodule	NOUN
ejpam-3275	382	26	of	of	ADP
ejpam-3275	382	27	an	an	DET
ejpam-3275	382	28	r	r	NOUN
ejpam-3275	382	29	-	-	PUNCT
ejpam-3275	382	30	modules	module	NOUN
ejpam-3275	382	31	r.	r.	NOUN
ejpam-3275	382	32	similarly	similarly	ADV
ejpam-3275	382	33	,	,	PUNCT
ejpam-3275	382	34	a	a	DET
ejpam-3275	382	35	proper	proper	ADJ
ejpam-3275	382	36	ideal	ideal	NOUN
ejpam-3275	382	37	p	p	NOUN
ejpam-3275	382	38	of	of	ADP
ejpam-3275	382	39	a	a	DET
ejpam-3275	382	40	ring	ring	NOUN
ejpam-3275	382	41	r	r	NOUN
ejpam-3275	382	42	is	be	AUX
ejpam-3275	382	43	called	call	VERB
ejpam-3275	382	44	a	a	DET
ejpam-3275	382	45	weakly	weakly	ADJ
ejpam-3275	382	46	α	α	NOUN
ejpam-3275	382	47	-	-	ADJ
ejpam-3275	382	48	prime	prime	ADJ
ejpam-3275	382	49	ideal	ideal	NOUN
ejpam-3275	382	50	of	of	ADP
ejpam-3275	382	51	r	r	NOUN
ejpam-3275	382	52	if	if	SCONJ
ejpam-3275	382	53	p	p	NOUN
ejpam-3275	382	54	is	be	AUX
ejpam-3275	382	55	an	an	DET
ejpam-3275	382	56	weakly	weakly	ADJ
ejpam-3275	382	57	α	α	NOUN
ejpam-3275	382	58	-	-	ADJ
ejpam-3275	382	59	prime	prime	ADJ
ejpam-3275	382	60	submodule	submodule	NOUN
ejpam-3275	382	61	of	of	ADP
ejpam-3275	382	62	an	an	DET
ejpam-3275	382	63	r	r	NOUN
ejpam-3275	382	64	-	-	PUNCT
ejpam-3275	382	65	modules	module	NOUN
ejpam-3275	382	66	r.	r.	NOUN
ejpam-3275	382	67	it	it	PRON
ejpam-3275	382	68	is	be	AUX
ejpam-3275	382	69	easy	easy	ADJ
ejpam-3275	382	70	to	to	PART
ejpam-3275	382	71	show	show	VERB
ejpam-3275	382	72	that	that	SCONJ
ejpam-3275	382	73	for	for	ADP
ejpam-3275	382	74	an	an	DET
ejpam-3275	382	75	ideal	ideal	ADJ
ejpam-3275	382	76	p	p	NOUN
ejpam-3275	382	77	of	of	ADP
ejpam-3275	382	78	r	r	NOUN
ejpam-3275	382	79	,	,	PUNCT
ejpam-3275	382	80	p	p	PRON
ejpam-3275	382	81	is	be	AUX
ejpam-3275	382	82	an	an	DET
ejpam-3275	382	83	α	α	NOUN
ejpam-3275	382	84	-	-	ADJ
ejpam-3275	382	85	prime	prime	ADJ
ejpam-3275	382	86	ideal	ideal	NOUN
ejpam-3275	382	87	of	of	ADP
ejpam-3275	382	88	r	r	NOUN
ejpam-3275	382	89	if	if	SCONJ
ejpam-3275	382	90	and	and	CCONJ
ejpam-3275	382	91	only	only	ADV
ejpam-3275	382	92	if	if	SCONJ
ejpam-3275	382	93	for	for	ADP
ejpam-3275	382	94	all	all	DET
ejpam-3275	382	95	a	a	PRON
ejpam-3275	382	96	,	,	PUNCT
ejpam-3275	382	97	b	b	X
ejpam-3275	382	98	∈	∈	PROPN
ejpam-3275	382	99	r	r	NOUN
ejpam-3275	382	100	,	,	PUNCT
ejpam-3275	382	101	if	if	SCONJ
ejpam-3275	382	102	a(b+	a(b+	ADP
ejpam-3275	382	103	b	b	X
ejpam-3275	382	104	)	)	PUNCT
ejpam-3275	382	105	∈	∈	PROPN
ejpam-3275	382	106	p	p	NOUN
ejpam-3275	382	107	,	,	PUNCT
ejpam-3275	382	108	then	then	ADV
ejpam-3275	382	109	a+	a+	PUNCT
ejpam-3275	382	110	a	a	DET
ejpam-3275	382	111	∈	∈	PROPN
ejpam-3275	382	112	p	p	NOUN
ejpam-3275	382	113	or	or	CCONJ
ejpam-3275	382	114	b+	b+	X
ejpam-3275	382	115	b	b	X
ejpam-3275	382	116	∈	∈	PROPN
ejpam-3275	382	117	p	p	NOUN
ejpam-3275	382	118	.	.	PUNCT
ejpam-3275	383	1	similarly	similarly	ADV
ejpam-3275	383	2	,	,	PUNCT
ejpam-3275	383	3	p	p	PRON
ejpam-3275	383	4	is	be	AUX
ejpam-3275	383	5	a	a	DET
ejpam-3275	383	6	weakly	weakly	ADJ
ejpam-3275	383	7	α	α	NOUN
ejpam-3275	383	8	-	-	ADJ
ejpam-3275	383	9	prime	prime	ADJ
ejpam-3275	383	10	ideal	ideal	NOUN
ejpam-3275	383	11	of	of	ADP
ejpam-3275	383	12	r	r	NOUN
ejpam-3275	383	13	if	if	SCONJ
ejpam-3275	384	1	and	and	CCONJ
ejpam-3275	384	2	only	only	ADV
ejpam-3275	384	3	if	if	SCONJ
ejpam-3275	384	4	for	for	ADP
ejpam-3275	384	5	all	all	DET
ejpam-3275	384	6	a	a	PRON
ejpam-3275	384	7	,	,	PUNCT
ejpam-3275	384	8	b	b	X
ejpam-3275	384	9	∈	∈	PROPN
ejpam-3275	384	10	r	r	NOUN
ejpam-3275	384	11	,	,	PUNCT
ejpam-3275	384	12	if	if	SCONJ
ejpam-3275	384	13	a(b+	a(b+	ADP
ejpam-3275	384	14	b	b	X
ejpam-3275	384	15	)	)	PUNCT
ejpam-3275	384	16	∈	∈	PROPN
ejpam-3275	384	17	p\{0	p\{0	NOUN
ejpam-3275	384	18	}	}	PUNCT
ejpam-3275	384	19	,	,	PUNCT
ejpam-3275	384	20	then	then	ADV
ejpam-3275	384	21	a+	a+	PUNCT
ejpam-3275	384	22	a	a	DET
ejpam-3275	384	23	∈	∈	PROPN
ejpam-3275	384	24	p	p	NOUN
ejpam-3275	384	25	or	or	CCONJ
ejpam-3275	384	26	b+	b+	X
ejpam-3275	384	27	b	b	PROPN
ejpam-3275	384	28	∈	∈	PROPN
ejpam-3275	384	29	p	p	NOUN
ejpam-3275	384	30	.	.	PUNCT
ejpam-3275	385	1	proposition	proposition	NOUN
ejpam-3275	385	2	8	8	NUM
ejpam-3275	385	3	.	.	PUNCT
ejpam-3275	386	1	if	if	SCONJ
ejpam-3275	386	2	p	p	NOUN
ejpam-3275	386	3	is	be	AUX
ejpam-3275	386	4	an	an	DET
ejpam-3275	386	5	α	α	NOUN
ejpam-3275	386	6	-	-	ADJ
ejpam-3275	386	7	prime	prime	ADJ
ejpam-3275	386	8	submodule	submodule	NOUN
ejpam-3275	386	9	of	of	ADP
ejpam-3275	386	10	an	an	DET
ejpam-3275	386	11	r	r	NOUN
ejpam-3275	386	12	-	-	PUNCT
ejpam-3275	386	13	module	module	NOUN
ejpam-3275	386	14	m	m	NOUN
ejpam-3275	386	15	,	,	PUNCT
ejpam-3275	386	16	then	then	ADV
ejpam-3275	386	17	(	(	PUNCT
ejpam-3275	386	18	p	p	X
ejpam-3275	386	19	:	:	PUNCT
ejpam-3275	386	20	m	m	X
ejpam-3275	386	21	)	)	PUNCT
ejpam-3275	386	22	is	be	AUX
ejpam-3275	386	23	an	an	DET
ejpam-3275	386	24	α	α	NOUN
ejpam-3275	386	25	-	-	PUNCT
ejpam-3275	386	26	prime	prime	ADJ
ejpam-3275	386	27	ideal	ideal	NOUN
ejpam-3275	386	28	of	of	ADP
ejpam-3275	386	29	r.	r.	PROPN
ejpam-3275	386	30	t.	t.	PROPN
ejpam-3275	386	31	khumprapussorn	khumprapussorn	PROPN
ejpam-3275	386	32	/	/	SYM
ejpam-3275	386	33	eur	eur	PROPN
ejpam-3275	386	34	.	.	PUNCT
ejpam-3275	387	1	j.	j.	PROPN
ejpam-3275	387	2	pure	pure	PROPN
ejpam-3275	387	3	appl	appl	PROPN
ejpam-3275	387	4	.	.	PROPN
ejpam-3275	387	5	math	math	PROPN
ejpam-3275	387	6	,	,	PUNCT
ejpam-3275	387	7	11	11	NUM
ejpam-3275	387	8	(	(	PUNCT
ejpam-3275	387	9	3	3	NUM
ejpam-3275	387	10	)	)	PUNCT
ejpam-3275	387	11	(	(	PUNCT
ejpam-3275	387	12	2018	2018	NUM
ejpam-3275	387	13	)	)	PUNCT
ejpam-3275	387	14	,	,	PUNCT
ejpam-3275	387	15	730	730	NUM
ejpam-3275	387	16	-	-	SYM
ejpam-3275	387	17	739	739	NUM
ejpam-3275	387	18	738	738	NUM
ejpam-3275	387	19	proof	proof	NOUN
ejpam-3275	387	20	.	.	PUNCT
ejpam-3275	388	1	assume	assume	VERB
ejpam-3275	388	2	that	that	SCONJ
ejpam-3275	388	3	p	p	NOUN
ejpam-3275	388	4	is	be	AUX
ejpam-3275	388	5	an	an	DET
ejpam-3275	388	6	α	α	NOUN
ejpam-3275	388	7	-	-	ADJ
ejpam-3275	388	8	prime	prime	ADJ
ejpam-3275	388	9	submodule	submodule	NOUN
ejpam-3275	388	10	of	of	ADP
ejpam-3275	388	11	an	an	DET
ejpam-3275	388	12	r	r	NOUN
ejpam-3275	388	13	-	-	PUNCT
ejpam-3275	388	14	module	module	NOUN
ejpam-3275	388	15	m	m	NOUN
ejpam-3275	388	16	.	.	PUNCT
ejpam-3275	389	1	let	let	VERB
ejpam-3275	389	2	a	a	PRON
ejpam-3275	389	3	,	,	PUNCT
ejpam-3275	389	4	b	b	X
ejpam-3275	389	5	∈	∈	NOUN
ejpam-3275	389	6	r	r	NOUN
ejpam-3275	389	7	be	be	VERB
ejpam-3275	389	8	such	such	ADJ
ejpam-3275	389	9	that	that	SCONJ
ejpam-3275	389	10	a(b	a(b	PROPN
ejpam-3275	389	11	+	+	CCONJ
ejpam-3275	389	12	b	b	X
ejpam-3275	389	13	)	)	PUNCT
ejpam-3275	389	14	∈	∈	PROPN
ejpam-3275	389	15	(	(	PUNCT
ejpam-3275	389	16	p	p	X
ejpam-3275	389	17	:	:	PUNCT
ejpam-3275	389	18	m	m	NUM
ejpam-3275	389	19	)	)	PUNCT
ejpam-3275	389	20	and	and	CCONJ
ejpam-3275	389	21	b	b	X
ejpam-3275	390	1	+	+	NUM
ejpam-3275	390	2	b	b	NOUN
ejpam-3275	390	3	/∈	/∈	PUNCT
ejpam-3275	390	4	(	(	PUNCT
ejpam-3275	390	5	p	p	X
ejpam-3275	390	6	:	:	PUNCT
ejpam-3275	390	7	m	m	PROPN
ejpam-3275	390	8	)	)	PUNCT
ejpam-3275	390	9	.	.	PUNCT
ejpam-3275	391	1	then	then	ADV
ejpam-3275	391	2	there	there	PRON
ejpam-3275	391	3	exists	exist	VERB
ejpam-3275	391	4	an	an	DET
ejpam-3275	391	5	element	element	NOUN
ejpam-3275	391	6	m	m	NOUN
ejpam-3275	391	7	∈m	∈m	NOUN
ejpam-3275	391	8	such	such	ADJ
ejpam-3275	391	9	that	that	SCONJ
ejpam-3275	391	10	(	(	PUNCT
ejpam-3275	391	11	b+	b+	X
ejpam-3275	391	12	b)m	b)m	X
ejpam-3275	391	13	/∈	/∈	PUNCT
ejpam-3275	392	1	p	p	NOUN
ejpam-3275	393	1	and	and	CCONJ
ejpam-3275	393	2	a(b+	a(b+	ADV
ejpam-3275	393	3	b)m	b)m	X
ejpam-3275	393	4	∈	∈	PROPN
ejpam-3275	393	5	p	p	X
ejpam-3275	393	6	.	.	PUNCT
ejpam-3275	394	1	since	since	SCONJ
ejpam-3275	394	2	p	p	PROPN
ejpam-3275	394	3	is	be	AUX
ejpam-3275	394	4	α	α	NOUN
ejpam-3275	394	5	-	-	ADJ
ejpam-3275	394	6	prime	prime	NOUN
ejpam-3275	394	7	and	and	CCONJ
ejpam-3275	394	8	(	(	PUNCT
ejpam-3275	394	9	b+	b+	X
ejpam-3275	394	10	b)m	b)m	X
ejpam-3275	394	11	/∈	/∈	PUNCT
ejpam-3275	395	1	p	p	NOUN
ejpam-3275	395	2	,	,	PUNCT
ejpam-3275	395	3	a+	a+	PUNCT
ejpam-3275	395	4	a	a	DET
ejpam-3275	395	5	∈	∈	NOUN
ejpam-3275	395	6	(	(	PUNCT
ejpam-3275	395	7	p	p	X
ejpam-3275	395	8	:	:	PUNCT
ejpam-3275	395	9	m	m	PROPN
ejpam-3275	395	10	)	)	PUNCT
ejpam-3275	395	11	.	.	PUNCT
ejpam-3275	396	1	therefore	therefore	ADV
ejpam-3275	396	2	(	(	PUNCT
ejpam-3275	396	3	p	p	X
ejpam-3275	396	4	:	:	PUNCT
ejpam-3275	396	5	m	m	X
ejpam-3275	396	6	)	)	PUNCT
ejpam-3275	396	7	is	be	AUX
ejpam-3275	396	8	an	an	DET
ejpam-3275	396	9	α	α	NOUN
ejpam-3275	396	10	-	-	PUNCT
ejpam-3275	396	11	prime	prime	ADJ
ejpam-3275	396	12	ideal	ideal	NOUN
ejpam-3275	396	13	of	of	ADP
ejpam-3275	396	14	r.	r.	PROPN
ejpam-3275	396	15	let	let	VERB
ejpam-3275	396	16	r	r	PRON
ejpam-3275	396	17	be	be	AUX
ejpam-3275	396	18	a	a	DET
ejpam-3275	396	19	ring	ring	NOUN
ejpam-3275	396	20	.	.	PUNCT
ejpam-3275	397	1	the	the	DET
ejpam-3275	397	2	cartesian	cartesian	ADJ
ejpam-3275	397	3	product	product	NOUN
ejpam-3275	397	4	r×r	r×r	PROPN
ejpam-3275	397	5	is	be	AUX
ejpam-3275	397	6	a	a	DET
ejpam-3275	397	7	ring	ring	NOUN
ejpam-3275	397	8	under	under	ADP
ejpam-3275	397	9	componentwise	componentwise	NOUN
ejpam-3275	397	10	addition	addition	NOUN
ejpam-3275	397	11	and	and	CCONJ
ejpam-3275	397	12	the	the	DET
ejpam-3275	397	13	multiplication	multiplication	NOUN
ejpam-3275	397	14	(	(	PUNCT
ejpam-3275	397	15	a	a	DET
ejpam-3275	397	16	,	,	PUNCT
ejpam-3275	397	17	b	b	NOUN
ejpam-3275	397	18	)	)	PUNCT
ejpam-3275	397	19	∗	∗	NOUN
ejpam-3275	397	20	(	(	PUNCT
ejpam-3275	397	21	c	c	X
ejpam-3275	397	22	,	,	PUNCT
ejpam-3275	397	23	d	d	NOUN
ejpam-3275	397	24	)	)	PUNCT
ejpam-3275	397	25	=	=	SYM
ejpam-3275	397	26	(	(	PUNCT
ejpam-3275	397	27	ac	ac	PROPN
ejpam-3275	397	28	,	,	PUNCT
ejpam-3275	397	29	ad	ad	NOUN
ejpam-3275	397	30	+	+	CCONJ
ejpam-3275	397	31	bc	bc	PROPN
ejpam-3275	397	32	)	)	PUNCT
ejpam-3275	397	33	.	.	PUNCT
ejpam-3275	398	1	we	we	PRON
ejpam-3275	398	2	use	use	VERB
ejpam-3275	398	3	the	the	DET
ejpam-3275	398	4	notation	notation	NOUN
ejpam-3275	398	5	r(+)r	r(+)r	PROPN
ejpam-3275	398	6	for	for	ADP
ejpam-3275	398	7	this	this	DET
ejpam-3275	398	8	ring	ring	NOUN
ejpam-3275	398	9	.	.	PUNCT
ejpam-3275	399	1	proposition	proposition	NOUN
ejpam-3275	399	2	9	9	NUM
ejpam-3275	399	3	.	.	PUNCT
ejpam-3275	400	1	if	if	SCONJ
ejpam-3275	400	2	i	i	PRON
ejpam-3275	400	3	is	be	AUX
ejpam-3275	400	4	an	an	DET
ejpam-3275	400	5	α	α	NOUN
ejpam-3275	400	6	-	-	ADJ
ejpam-3275	400	7	prime	prime	ADJ
ejpam-3275	400	8	ideal	ideal	NOUN
ejpam-3275	400	9	of	of	ADP
ejpam-3275	400	10	a	a	DET
ejpam-3275	400	11	ring	ring	NOUN
ejpam-3275	400	12	r	r	NOUN
ejpam-3275	400	13	,	,	PUNCT
ejpam-3275	400	14	then	then	ADV
ejpam-3275	400	15	i	i	PRON
ejpam-3275	400	16	×	×	VERB
ejpam-3275	400	17	r	r	NOUN
ejpam-3275	400	18	is	be	AUX
ejpam-3275	400	19	an	an	DET
ejpam-3275	400	20	α	α	NOUN
ejpam-3275	400	21	-	-	ADJ
ejpam-3275	400	22	prime	prime	ADJ
ejpam-3275	400	23	ideal	ideal	NOUN
ejpam-3275	400	24	of	of	ADP
ejpam-3275	400	25	r(+)r	r(+)r	PROPN
ejpam-3275	400	26	.	.	PUNCT
ejpam-3275	401	1	proof	proof	NOUN
ejpam-3275	401	2	.	.	PUNCT
ejpam-3275	402	1	it	it	PRON
ejpam-3275	402	2	is	be	AUX
ejpam-3275	402	3	straightforward	straightforward	ADJ
ejpam-3275	402	4	.	.	PUNCT
ejpam-3275	403	1	example	example	NOUN
ejpam-3275	404	1	4	4	X
ejpam-3275	404	2	.	.	PUNCT
ejpam-3275	405	1	we	we	PRON
ejpam-3275	405	2	know	know	VERB
ejpam-3275	405	3	that	that	SCONJ
ejpam-3275	405	4	4z	4z	NOUN
ejpam-3275	405	5	and	and	CCONJ
ejpam-3275	405	6	6z	6z	NOUN
ejpam-3275	405	7	are	be	AUX
ejpam-3275	405	8	α	α	DET
ejpam-3275	405	9	-	-	ADJ
ejpam-3275	405	10	prime	prime	ADJ
ejpam-3275	405	11	ideal	ideal	NOUN
ejpam-3275	405	12	of	of	ADP
ejpam-3275	405	13	z.	z.	PROPN
ejpam-3275	405	14	in	in	ADP
ejpam-3275	405	15	z(+)z	z(+)z	PROPN
ejpam-3275	405	16	,	,	PUNCT
ejpam-3275	405	17	we	we	PRON
ejpam-3275	405	18	have	have	VERB
ejpam-3275	405	19	(	(	PUNCT
ejpam-3275	405	20	2	2	NUM
ejpam-3275	405	21	,	,	PUNCT
ejpam-3275	405	22	1)[(1	1)[(1	NOUN
ejpam-3275	405	23	,	,	PUNCT
ejpam-3275	405	24	1	1	NUM
ejpam-3275	405	25	)	)	PUNCT
ejpam-3275	405	26	+	+	CCONJ
ejpam-3275	405	27	(	(	PUNCT
ejpam-3275	405	28	1	1	NUM
ejpam-3275	405	29	,	,	PUNCT
ejpam-3275	405	30	1	1	NUM
ejpam-3275	405	31	)	)	PUNCT
ejpam-3275	405	32	]	]	PUNCT
ejpam-3275	406	1	=	=	PUNCT
ejpam-3275	406	2	(	(	PUNCT
ejpam-3275	406	3	2	2	NUM
ejpam-3275	406	4	,	,	PUNCT
ejpam-3275	406	5	1)(2	1)(2	NUM
ejpam-3275	406	6	,	,	PUNCT
ejpam-3275	406	7	2	2	NUM
ejpam-3275	406	8	)	)	PUNCT
ejpam-3275	406	9	=	=	NOUN
ejpam-3275	406	10	(	(	PUNCT
ejpam-3275	406	11	4	4	NUM
ejpam-3275	406	12	,	,	PUNCT
ejpam-3275	406	13	6	6	NUM
ejpam-3275	406	14	)	)	PUNCT
ejpam-3275	406	15	∈	∈	PROPN
ejpam-3275	406	16	4z	4z	NOUN
ejpam-3275	406	17	×	×	NOUN
ejpam-3275	406	18	6z	6z	NOUN
ejpam-3275	406	19	.	.	PUNCT
ejpam-3275	407	1	however	however	ADV
ejpam-3275	407	2	,	,	PUNCT
ejpam-3275	407	3	(	(	PUNCT
ejpam-3275	407	4	4	4	NUM
ejpam-3275	407	5	,	,	PUNCT
ejpam-3275	407	6	2	2	NUM
ejpam-3275	407	7	)	)	PUNCT
ejpam-3275	407	8	/∈	/∈	PUNCT
ejpam-3275	408	1	4z	4z	NOUN
ejpam-3275	408	2	×	×	NOUN
ejpam-3275	408	3	6z	6z	NOUN
ejpam-3275	408	4	and	and	CCONJ
ejpam-3275	408	5	(	(	PUNCT
ejpam-3275	408	6	2	2	NUM
ejpam-3275	408	7	,	,	PUNCT
ejpam-3275	408	8	2	2	NUM
ejpam-3275	408	9	)	)	PUNCT
ejpam-3275	408	10	/∈	/∈	PUNCT
ejpam-3275	409	1	4z	4z	NOUN
ejpam-3275	409	2	×	×	NOUN
ejpam-3275	409	3	6z	6z	NOUN
ejpam-3275	409	4	.	.	PUNCT
ejpam-3275	410	1	this	this	PRON
ejpam-3275	410	2	is	be	AUX
ejpam-3275	410	3	an	an	DET
ejpam-3275	410	4	example	example	NOUN
ejpam-3275	410	5	shows	show	VERB
ejpam-3275	410	6	that	that	SCONJ
ejpam-3275	410	7	i	i	PRON
ejpam-3275	410	8	×	×	VERB
ejpam-3275	410	9	j	j	PROPN
ejpam-3275	410	10	may	may	AUX
ejpam-3275	410	11	be	be	AUX
ejpam-3275	410	12	not	not	PART
ejpam-3275	410	13	an	an	DET
ejpam-3275	410	14	α	α	NUM
ejpam-3275	410	15	-	-	ADJ
ejpam-3275	410	16	prime	prime	ADJ
ejpam-3275	410	17	ideal	ideal	NOUN
ejpam-3275	410	18	of	of	ADP
ejpam-3275	410	19	r(+)r	r(+)r	PROPN
ejpam-3275	410	20	even	even	ADV
ejpam-3275	410	21	if	if	SCONJ
ejpam-3275	410	22	i	i	PRON
ejpam-3275	410	23	and	and	CCONJ
ejpam-3275	410	24	j	j	PROPN
ejpam-3275	410	25	are	be	AUX
ejpam-3275	410	26	α	α	DET
ejpam-3275	410	27	-	-	ADJ
ejpam-3275	410	28	prime	prime	ADJ
ejpam-3275	410	29	ideals	ideal	NOUN
ejpam-3275	410	30	of	of	ADP
ejpam-3275	410	31	r.	r.	PROPN
ejpam-3275	410	32	proposition	proposition	PROPN
ejpam-3275	410	33	10	10	NUM
ejpam-3275	410	34	.	.	PUNCT
ejpam-3275	411	1	if	if	SCONJ
ejpam-3275	411	2	p	p	NOUN
ejpam-3275	411	3	is	be	AUX
ejpam-3275	411	4	a	a	DET
ejpam-3275	411	5	weakly	weakly	ADJ
ejpam-3275	411	6	α	α	NOUN
ejpam-3275	411	7	-	-	ADJ
ejpam-3275	411	8	prime	prime	ADJ
ejpam-3275	411	9	submodule	submodule	NOUN
ejpam-3275	411	10	of	of	ADP
ejpam-3275	411	11	m	m	PROPN
ejpam-3275	411	12	and	and	CCONJ
ejpam-3275	411	13	(	(	PUNCT
ejpam-3275	411	14	p	p	X
ejpam-3275	411	15	:	:	PUNCT
ejpam-3275	411	16	m)β(p	m)β(p	ADJ
ejpam-3275	411	17	)	)	PUNCT
ejpam-3275	411	18	6=	6=	ADP
ejpam-3275	411	19	0	0	NUM
ejpam-3275	411	20	,	,	PUNCT
ejpam-3275	411	21	then	then	ADV
ejpam-3275	411	22	p	p	NOUN
ejpam-3275	411	23	is	be	AUX
ejpam-3275	411	24	an	an	DET
ejpam-3275	411	25	α	α	NOUN
ejpam-3275	411	26	-	-	ADJ
ejpam-3275	411	27	prime	prime	ADJ
ejpam-3275	411	28	submodule	submodule	NOUN
ejpam-3275	411	29	of	of	ADP
ejpam-3275	411	30	m	m	PROPN
ejpam-3275	411	31	proof	proof	NOUN
ejpam-3275	411	32	.	.	PUNCT
ejpam-3275	412	1	assume	assume	VERB
ejpam-3275	412	2	that	that	SCONJ
ejpam-3275	412	3	p	p	NOUN
ejpam-3275	412	4	is	be	AUX
ejpam-3275	412	5	a	a	DET
ejpam-3275	412	6	weakly	weakly	ADJ
ejpam-3275	412	7	α	α	NOUN
ejpam-3275	412	8	-	-	ADJ
ejpam-3275	412	9	prime	prime	ADJ
ejpam-3275	412	10	submodule	submodule	NOUN
ejpam-3275	412	11	of	of	ADP
ejpam-3275	412	12	m	m	PROPN
ejpam-3275	412	13	and	and	CCONJ
ejpam-3275	412	14	(	(	PUNCT
ejpam-3275	412	15	p	p	X
ejpam-3275	412	16	:	:	PUNCT
ejpam-3275	412	17	m)β(p	m)β(p	ADJ
ejpam-3275	412	18	)	)	PUNCT
ejpam-3275	413	1	6=	6=	ADP
ejpam-3275	413	2	0	0	X
ejpam-3275	413	3	.	.	PUNCT
ejpam-3275	414	1	let	let	VERB
ejpam-3275	414	2	r	r	NOUN
ejpam-3275	414	3	∈	∈	NOUN
ejpam-3275	414	4	r	r	NOUN
ejpam-3275	414	5	and	and	CCONJ
ejpam-3275	414	6	m	m	PROPN
ejpam-3275	414	7	∈	∈	NOUN
ejpam-3275	414	8	m	m	AUX
ejpam-3275	414	9	be	be	VERB
ejpam-3275	414	10	such	such	ADJ
ejpam-3275	414	11	that	that	PRON
ejpam-3275	414	12	r(m+m	r(m+m	NOUN
ejpam-3275	414	13	)	)	PUNCT
ejpam-3275	414	14	∈	∈	PROPN
ejpam-3275	414	15	p	p	NOUN
ejpam-3275	414	16	.	.	PUNCT
ejpam-3275	415	1	if	if	SCONJ
ejpam-3275	415	2	r(m+m	r(m+m	NOUN
ejpam-3275	415	3	)	)	PUNCT
ejpam-3275	416	1	6=	6=	ADP
ejpam-3275	416	2	0	0	NUM
ejpam-3275	416	3	,	,	PUNCT
ejpam-3275	416	4	r	r	NOUN
ejpam-3275	416	5	+	+	NOUN
ejpam-3275	416	6	r	r	NOUN
ejpam-3275	416	7	∈	∈	NOUN
ejpam-3275	416	8	(	(	PUNCT
ejpam-3275	416	9	p	p	X
ejpam-3275	416	10	:	:	PUNCT
ejpam-3275	416	11	m	m	NUM
ejpam-3275	416	12	)	)	PUNCT
ejpam-3275	416	13	or	or	CCONJ
ejpam-3275	416	14	m+m	m+m	PROPN
ejpam-3275	416	15	∈	∈	PROPN
ejpam-3275	416	16	p	p	PROPN
ejpam-3275	416	17	.	.	PUNCT
ejpam-3275	417	1	assume	assume	VERB
ejpam-3275	417	2	that	that	SCONJ
ejpam-3275	417	3	r(m+m	r(m+m	NOUN
ejpam-3275	417	4	)	)	PUNCT
ejpam-3275	417	5	=	=	SYM
ejpam-3275	418	1	0	0	X
ejpam-3275	418	2	.	.	PUNCT
ejpam-3275	419	1	we	we	PRON
ejpam-3275	419	2	consider	consider	VERB
ejpam-3275	419	3	the	the	DET
ejpam-3275	419	4	following	follow	VERB
ejpam-3275	419	5	two	two	NUM
ejpam-3275	419	6	cases	case	NOUN
ejpam-3275	419	7	.	.	PUNCT
ejpam-3275	420	1	case	case	NOUN
ejpam-3275	420	2	1	1	NUM
ejpam-3275	420	3	.	.	NUM
ejpam-3275	420	4	rβ(p	rβ(p	NUM
ejpam-3275	420	5	)	)	PUNCT
ejpam-3275	421	1	6=	6=	ADP
ejpam-3275	421	2	0	0	X
ejpam-3275	421	3	.	.	PUNCT
ejpam-3275	421	4	then	then	ADV
ejpam-3275	421	5	r(n0	r(n0	VERB
ejpam-3275	421	6	+	+	PROPN
ejpam-3275	421	7	n0	n0	ADJ
ejpam-3275	421	8	)	)	PUNCT
ejpam-3275	421	9	6=	6=	ADP
ejpam-3275	421	10	0	0	NUM
ejpam-3275	421	11	for	for	ADP
ejpam-3275	421	12	some	some	DET
ejpam-3275	421	13	n0	n0	NOUN
ejpam-3275	421	14	∈	∈	PROPN
ejpam-3275	421	15	p	p	NOUN
ejpam-3275	421	16	.	.	PUNCT
ejpam-3275	422	1	hence	hence	ADV
ejpam-3275	422	2	r(m+m+n0	r(m+m+n0	VERB
ejpam-3275	422	3	+	+	PROPN
ejpam-3275	422	4	n0	n0	ADJ
ejpam-3275	422	5	)	)	PUNCT
ejpam-3275	422	6	=	=	NOUN
ejpam-3275	422	7	r(n0	r(n0	VERB
ejpam-3275	422	8	+	+	ADJ
ejpam-3275	422	9	n0	n0	ADJ
ejpam-3275	422	10	)	)	PUNCT
ejpam-3275	422	11	∈	∈	PROPN
ejpam-3275	422	12	p	p	NOUN
ejpam-3275	422	13	.	.	PUNCT
ejpam-3275	423	1	since	since	SCONJ
ejpam-3275	423	2	p	p	NOUN
ejpam-3275	423	3	is	be	AUX
ejpam-3275	423	4	a	a	DET
ejpam-3275	423	5	weakly	weakly	ADJ
ejpam-3275	423	6	α	α	NOUN
ejpam-3275	423	7	-	-	ADJ
ejpam-3275	423	8	prime	prime	ADJ
ejpam-3275	423	9	submodule	submodule	NOUN
ejpam-3275	423	10	of	of	ADP
ejpam-3275	423	11	m	m	PROPN
ejpam-3275	423	12	,	,	PUNCT
ejpam-3275	423	13	r	r	NOUN
ejpam-3275	423	14	+	+	CCONJ
ejpam-3275	423	15	r	r	NOUN
ejpam-3275	423	16	∈	∈	NOUN
ejpam-3275	423	17	(	(	PUNCT
ejpam-3275	423	18	p	p	X
ejpam-3275	423	19	:	:	PUNCT
ejpam-3275	423	20	m	m	NUM
ejpam-3275	423	21	)	)	PUNCT
ejpam-3275	423	22	or	or	CCONJ
ejpam-3275	423	23	m+m+	m+m+	PROPN
ejpam-3275	423	24	n0	n0	X
ejpam-3275	423	25	+	+	CCONJ
ejpam-3275	423	26	n0	n0	PROPN
ejpam-3275	423	27	∈	∈	PROPN
ejpam-3275	423	28	p	p	X
ejpam-3275	423	29	.	.	PUNCT
ejpam-3275	424	1	since	since	SCONJ
ejpam-3275	424	2	no	no	DET
ejpam-3275	424	3	∈	∈	PROPN
ejpam-3275	424	4	p	p	NOUN
ejpam-3275	424	5	,	,	PUNCT
ejpam-3275	424	6	r	r	NOUN
ejpam-3275	424	7	+	+	CCONJ
ejpam-3275	424	8	r	r	NOUN
ejpam-3275	424	9	∈	∈	NOUN
ejpam-3275	424	10	(	(	PUNCT
ejpam-3275	424	11	p	p	X
ejpam-3275	424	12	:	:	PUNCT
ejpam-3275	424	13	m	m	NUM
ejpam-3275	424	14	)	)	PUNCT
ejpam-3275	424	15	or	or	CCONJ
ejpam-3275	424	16	m+m	m+m	PROPN
ejpam-3275	424	17	∈	∈	PROPN
ejpam-3275	424	18	p	p	NOUN
ejpam-3275	424	19	.	.	PUNCT
ejpam-3275	425	1	hence	hence	ADV
ejpam-3275	425	2	p	p	NOUN
ejpam-3275	425	3	is	be	AUX
ejpam-3275	425	4	an	an	DET
ejpam-3275	425	5	α	α	NOUN
ejpam-3275	425	6	-	-	ADJ
ejpam-3275	425	7	prime	prime	ADJ
ejpam-3275	425	8	submodule	submodule	NOUN
ejpam-3275	425	9	of	of	ADP
ejpam-3275	425	10	m	m	PROPN
ejpam-3275	425	11	.	.	PUNCT
ejpam-3275	426	1	case	case	NOUN
ejpam-3275	426	2	2	2	NUM
ejpam-3275	426	3	.	.	NUM
ejpam-3275	426	4	rβ(p	rβ(p	NUM
ejpam-3275	426	5	)	)	PUNCT
ejpam-3275	427	1	=	=	SYM
ejpam-3275	427	2	0	0	X
ejpam-3275	427	3	.	.	PUNCT
ejpam-3275	427	4	subcase	subcase	PROPN
ejpam-3275	427	5	2.1	2.1	NUM
ejpam-3275	427	6	.	.	PUNCT
ejpam-3275	428	1	(	(	PUNCT
ejpam-3275	428	2	p	p	X
ejpam-3275	428	3	:	:	PUNCT
ejpam-3275	428	4	m)(m+m	m)(m+m	PROPN
ejpam-3275	428	5	)	)	PUNCT
ejpam-3275	428	6	6=	6=	ADP
ejpam-3275	429	1	0	0	X
ejpam-3275	429	2	.	.	PUNCT
ejpam-3275	430	1	let	let	VERB
ejpam-3275	430	2	k	k	PROPN
ejpam-3275	430	3	∈	∈	PROPN
ejpam-3275	430	4	(	(	PUNCT
ejpam-3275	430	5	p	p	X
ejpam-3275	430	6	:	:	PUNCT
ejpam-3275	430	7	m	m	VERB
ejpam-3275	430	8	)	)	PUNCT
ejpam-3275	430	9	be	be	AUX
ejpam-3275	430	10	such	such	ADJ
ejpam-3275	430	11	that	that	SCONJ
ejpam-3275	430	12	k(m+m	k(m+m	NOUN
ejpam-3275	430	13	)	)	PUNCT
ejpam-3275	431	1	6=	6=	ADP
ejpam-3275	431	2	0	0	X
ejpam-3275	431	3	.	.	PUNCT
ejpam-3275	432	1	then	then	ADV
ejpam-3275	432	2	(	(	PUNCT
ejpam-3275	432	3	r	r	NOUN
ejpam-3275	432	4	+	+	NUM
ejpam-3275	432	5	k)(m+m	k)(m+m	NOUN
ejpam-3275	432	6	)	)	PUNCT
ejpam-3275	432	7	=	=	PUNCT
ejpam-3275	432	8	k(m+m	k(m+m	NOUN
ejpam-3275	432	9	)	)	PUNCT
ejpam-3275	432	10	∈	∈	PROPN
ejpam-3275	432	11	p	p	NOUN
ejpam-3275	432	12	.	.	PUNCT
ejpam-3275	433	1	since	since	SCONJ
ejpam-3275	433	2	p	p	NOUN
ejpam-3275	433	3	is	be	AUX
ejpam-3275	433	4	a	a	DET
ejpam-3275	433	5	weakly	weakly	ADJ
ejpam-3275	433	6	α	α	NOUN
ejpam-3275	433	7	-	-	ADJ
ejpam-3275	433	8	prime	prime	ADJ
ejpam-3275	433	9	submodule	submodule	NOUN
ejpam-3275	433	10	of	of	ADP
ejpam-3275	433	11	m	m	PROPN
ejpam-3275	433	12	,	,	PUNCT
ejpam-3275	434	1	r	r	NOUN
ejpam-3275	434	2	+	+	PROPN
ejpam-3275	435	1	k	k	NOUN
ejpam-3275	436	1	+	+	CCONJ
ejpam-3275	436	2	r	r	NOUN
ejpam-3275	436	3	+	+	CCONJ
ejpam-3275	436	4	k	k	PROPN
ejpam-3275	436	5	∈	∈	PROPN
ejpam-3275	436	6	(	(	PUNCT
ejpam-3275	436	7	p	p	X
ejpam-3275	436	8	:	:	PUNCT
ejpam-3275	436	9	m	m	X
ejpam-3275	436	10	)	)	PUNCT
ejpam-3275	436	11	or	or	CCONJ
ejpam-3275	436	12	m	m	PROPN
ejpam-3275	436	13	+	+	NUM
ejpam-3275	436	14	m	m	VERB
ejpam-3275	436	15	∈	∈	NOUN
ejpam-3275	436	16	p	p	NOUN
ejpam-3275	436	17	.	.	PUNCT
ejpam-3275	437	1	since	since	SCONJ
ejpam-3275	437	2	k	k	PROPN
ejpam-3275	437	3	∈	∈	PROPN
ejpam-3275	437	4	(	(	PUNCT
ejpam-3275	437	5	p	p	X
ejpam-3275	437	6	:	:	PUNCT
ejpam-3275	437	7	m	m	NUM
ejpam-3275	437	8	)	)	PUNCT
ejpam-3275	437	9	,	,	PUNCT
ejpam-3275	437	10	r	r	NOUN
ejpam-3275	437	11	+	+	NOUN
ejpam-3275	437	12	r	r	NOUN
ejpam-3275	437	13	∈	∈	NOUN
ejpam-3275	437	14	(	(	PUNCT
ejpam-3275	437	15	p	p	X
ejpam-3275	437	16	:	:	PUNCT
ejpam-3275	437	17	m	m	NUM
ejpam-3275	437	18	)	)	PUNCT
ejpam-3275	437	19	or	or	CCONJ
ejpam-3275	437	20	m+m	m+m	PROPN
ejpam-3275	437	21	∈	∈	PROPN
ejpam-3275	437	22	p	p	NOUN
ejpam-3275	437	23	.	.	PUNCT
ejpam-3275	438	1	hence	hence	ADV
ejpam-3275	438	2	p	p	NOUN
ejpam-3275	438	3	is	be	AUX
ejpam-3275	438	4	an	an	DET
ejpam-3275	438	5	α	α	NOUN
ejpam-3275	438	6	-	-	ADJ
ejpam-3275	438	7	prime	prime	ADJ
ejpam-3275	438	8	submodule	submodule	NOUN
ejpam-3275	438	9	of	of	ADP
ejpam-3275	438	10	m	m	PROPN
ejpam-3275	438	11	.	.	PUNCT
ejpam-3275	439	1	subcase	subcase	PROPN
ejpam-3275	439	2	2.2	2.2	NUM
ejpam-3275	439	3	.	.	PUNCT
ejpam-3275	440	1	(	(	PUNCT
ejpam-3275	440	2	p	p	X
ejpam-3275	440	3	:	:	PUNCT
ejpam-3275	440	4	m)(m+m	m)(m+m	NOUN
ejpam-3275	440	5	)	)	PUNCT
ejpam-3275	440	6	=	=	PUNCT
ejpam-3275	441	1	0	0	X
ejpam-3275	441	2	.	.	PUNCT
ejpam-3275	442	1	since	since	SCONJ
ejpam-3275	442	2	(	(	PUNCT
ejpam-3275	442	3	p	p	X
ejpam-3275	442	4	:	:	PUNCT
ejpam-3275	442	5	m)β(p	m)β(p	ADJ
ejpam-3275	442	6	)	)	PUNCT
ejpam-3275	442	7	6=	6=	ADP
ejpam-3275	442	8	0	0	NUM
ejpam-3275	442	9	,	,	PUNCT
ejpam-3275	442	10	we	we	PRON
ejpam-3275	442	11	have	have	VERB
ejpam-3275	442	12	k(n	k(n	PROPN
ejpam-3275	442	13	+	+	CCONJ
ejpam-3275	442	14	n	n	CCONJ
ejpam-3275	442	15	)	)	PUNCT
ejpam-3275	442	16	6=	6=	ADP
ejpam-3275	442	17	0	0	NUM
ejpam-3275	442	18	for	for	ADP
ejpam-3275	442	19	some	some	DET
ejpam-3275	442	20	k	k	PROPN
ejpam-3275	442	21	∈	∈	PROPN
ejpam-3275	442	22	(	(	PUNCT
ejpam-3275	442	23	p	p	X
ejpam-3275	442	24	:	:	PUNCT
ejpam-3275	442	25	m	m	NUM
ejpam-3275	442	26	)	)	PUNCT
ejpam-3275	442	27	and	and	CCONJ
ejpam-3275	442	28	n	n	PRON
ejpam-3275	442	29	∈	∈	PROPN
ejpam-3275	442	30	p	p	NOUN
ejpam-3275	442	31	.	.	PUNCT
ejpam-3275	443	1	then	then	ADV
ejpam-3275	443	2	(	(	PUNCT
ejpam-3275	443	3	r+k)(m+m+n+n	r+k)(m+m+n+n	PROPN
ejpam-3275	443	4	)	)	PUNCT
ejpam-3275	443	5	=	=	SYM
ejpam-3275	443	6	r(m+m)+r(n+n)+k(m+m)+k(n+n	r(m+m)+r(n+n)+k(m+m)+k(n+n	NOUN
ejpam-3275	443	7	)	)	PUNCT
ejpam-3275	443	8	=	=	SYM
ejpam-3275	443	9	k(n+n	k(n+n	X
ejpam-3275	443	10	)	)	PUNCT
ejpam-3275	443	11	∈	∈	PROPN
ejpam-3275	443	12	p	p	NOUN
ejpam-3275	443	13	.	.	PUNCT
ejpam-3275	444	1	since	since	SCONJ
ejpam-3275	444	2	p	p	NOUN
ejpam-3275	444	3	is	be	AUX
ejpam-3275	444	4	a	a	DET
ejpam-3275	444	5	weakly	weakly	ADJ
ejpam-3275	444	6	α	α	NOUN
ejpam-3275	444	7	-	-	ADJ
ejpam-3275	444	8	prime	prime	ADJ
ejpam-3275	444	9	submodule	submodule	NOUN
ejpam-3275	444	10	of	of	ADP
ejpam-3275	444	11	m	m	PROPN
ejpam-3275	444	12	,	,	PUNCT
ejpam-3275	445	1	r	r	NOUN
ejpam-3275	445	2	+	+	PROPN
ejpam-3275	446	1	k	k	NOUN
ejpam-3275	447	1	+	+	CCONJ
ejpam-3275	447	2	r	r	NOUN
ejpam-3275	447	3	+	+	CCONJ
ejpam-3275	447	4	k	k	PROPN
ejpam-3275	447	5	∈	∈	PROPN
ejpam-3275	447	6	(	(	PUNCT
ejpam-3275	447	7	p	p	X
ejpam-3275	447	8	:	:	PUNCT
ejpam-3275	447	9	m	m	NUM
ejpam-3275	447	10	)	)	PUNCT
ejpam-3275	447	11	or	or	CCONJ
ejpam-3275	447	12	m+m+	m+m+	NOUN
ejpam-3275	447	13	n+	n+	NUM
ejpam-3275	447	14	n	n	PRON
ejpam-3275	447	15	∈	∈	PROPN
ejpam-3275	447	16	p	p	NOUN
ejpam-3275	447	17	.	.	PUNCT
ejpam-3275	448	1	since	since	SCONJ
ejpam-3275	448	2	k	k	PROPN
ejpam-3275	448	3	∈	∈	PROPN
ejpam-3275	448	4	(	(	PUNCT
ejpam-3275	448	5	p	p	X
ejpam-3275	448	6	:	:	PUNCT
ejpam-3275	448	7	m	m	NUM
ejpam-3275	448	8	)	)	PUNCT
ejpam-3275	448	9	and	and	CCONJ
ejpam-3275	448	10	n	n	PRON
ejpam-3275	448	11	∈	∈	PROPN
ejpam-3275	448	12	p	p	NOUN
ejpam-3275	448	13	,	,	PUNCT
ejpam-3275	448	14	r	r	NOUN
ejpam-3275	448	15	+	+	CCONJ
ejpam-3275	448	16	r	r	NOUN
ejpam-3275	448	17	∈	∈	NOUN
ejpam-3275	448	18	(	(	PUNCT
ejpam-3275	448	19	p	p	X
ejpam-3275	448	20	:	:	PUNCT
ejpam-3275	448	21	m	m	X
ejpam-3275	448	22	)	)	PUNCT
ejpam-3275	448	23	or	or	CCONJ
ejpam-3275	448	24	m	m	PROPN
ejpam-3275	448	25	+	+	NUM
ejpam-3275	448	26	m	m	VERB
ejpam-3275	448	27	∈	∈	NOUN
ejpam-3275	448	28	p	p	NOUN
ejpam-3275	448	29	.	.	PUNCT
ejpam-3275	449	1	hence	hence	ADV
ejpam-3275	449	2	p	p	NOUN
ejpam-3275	449	3	is	be	AUX
ejpam-3275	449	4	an	an	DET
ejpam-3275	449	5	α	α	NOUN
ejpam-3275	449	6	-	-	ADJ
ejpam-3275	449	7	prime	prime	ADJ
ejpam-3275	449	8	submodule	submodule	NOUN
ejpam-3275	449	9	of	of	ADP
ejpam-3275	449	10	m	m	PROPN
ejpam-3275	449	11	.	.	PUNCT
ejpam-3275	450	1	the	the	DET
ejpam-3275	450	2	following	follow	VERB
ejpam-3275	450	3	result	result	NOUN
ejpam-3275	450	4	directly	directly	ADV
ejpam-3275	450	5	implies	imply	VERB
ejpam-3275	450	6	from	from	ADP
ejpam-3275	450	7	proposition	proposition	NOUN
ejpam-3275	450	8	8	8	NUM
ejpam-3275	450	9	and	and	CCONJ
ejpam-3275	450	10	10	10	NUM
ejpam-3275	450	11	.	.	PUNCT
ejpam-3275	451	1	corollary	corollary	ADJ
ejpam-3275	451	2	2	2	NUM
ejpam-3275	451	3	.	.	PUNCT
ejpam-3275	452	1	if	if	SCONJ
ejpam-3275	452	2	p	p	NOUN
ejpam-3275	452	3	is	be	AUX
ejpam-3275	452	4	a	a	DET
ejpam-3275	452	5	weakly	weakly	ADJ
ejpam-3275	452	6	α	α	NOUN
ejpam-3275	452	7	-	-	ADJ
ejpam-3275	452	8	prime	prime	ADJ
ejpam-3275	452	9	submodule	submodule	NOUN
ejpam-3275	452	10	of	of	ADP
ejpam-3275	452	11	m	m	PROPN
ejpam-3275	452	12	and	and	CCONJ
ejpam-3275	452	13	(	(	PUNCT
ejpam-3275	452	14	p	p	X
ejpam-3275	452	15	:	:	PUNCT
ejpam-3275	452	16	m)β(p	m)β(p	ADJ
ejpam-3275	452	17	)	)	PUNCT
ejpam-3275	452	18	6=	6=	ADP
ejpam-3275	452	19	0	0	NUM
ejpam-3275	452	20	,	,	PUNCT
ejpam-3275	452	21	then	then	ADV
ejpam-3275	452	22	(	(	PUNCT
ejpam-3275	452	23	p	p	X
ejpam-3275	452	24	:	:	PUNCT
ejpam-3275	452	25	m	m	X
ejpam-3275	452	26	)	)	PUNCT
ejpam-3275	452	27	is	be	AUX
ejpam-3275	452	28	a	a	DET
ejpam-3275	452	29	weakly	weakly	ADJ
ejpam-3275	452	30	α	α	NOUN
ejpam-3275	452	31	-	-	ADJ
ejpam-3275	452	32	prime	prime	ADJ
ejpam-3275	452	33	ideal	ideal	NOUN
ejpam-3275	452	34	of	of	ADP
ejpam-3275	452	35	r.	r.	PROPN
ejpam-3275	452	36	references	reference	NOUN
ejpam-3275	452	37	739	739	NUM
ejpam-3275	452	38	we	we	PRON
ejpam-3275	452	39	prove	prove	VERB
ejpam-3275	452	40	in	in	ADP
ejpam-3275	452	41	proposition	proposition	NOUN
ejpam-3275	452	42	8	8	NUM
ejpam-3275	452	43	that	that	SCONJ
ejpam-3275	452	44	if	if	SCONJ
ejpam-3275	452	45	p	p	NOUN
ejpam-3275	452	46	is	be	AUX
ejpam-3275	452	47	an	an	DET
ejpam-3275	452	48	α	α	NOUN
ejpam-3275	452	49	-	-	ADJ
ejpam-3275	452	50	prime	prime	ADJ
ejpam-3275	452	51	submodule	submodule	NOUN
ejpam-3275	452	52	of	of	ADP
ejpam-3275	452	53	an	an	DET
ejpam-3275	452	54	r	r	NOUN
ejpam-3275	452	55	-	-	PUNCT
ejpam-3275	452	56	module	module	NOUN
ejpam-3275	452	57	m	m	NOUN
ejpam-3275	452	58	,	,	PUNCT
ejpam-3275	452	59	then	then	ADV
ejpam-3275	452	60	(	(	PUNCT
ejpam-3275	452	61	p	p	X
ejpam-3275	452	62	:	:	PUNCT
ejpam-3275	452	63	m	m	X
ejpam-3275	452	64	)	)	PUNCT
ejpam-3275	452	65	is	be	AUX
ejpam-3275	452	66	an	an	DET
ejpam-3275	452	67	α	α	NOUN
ejpam-3275	452	68	-	-	PUNCT
ejpam-3275	452	69	prime	prime	ADJ
ejpam-3275	452	70	ideal	ideal	NOUN
ejpam-3275	452	71	of	of	ADP
ejpam-3275	452	72	r.	r.	PROPN
ejpam-3275	452	73	however	however	ADV
ejpam-3275	452	74	,	,	PUNCT
ejpam-3275	452	75	this	this	DET
ejpam-3275	452	76	situation	situation	NOUN
ejpam-3275	452	77	is	be	AUX
ejpam-3275	452	78	false	false	ADJ
ejpam-3275	452	79	for	for	ADP
ejpam-3275	452	80	weakly	weakly	ADJ
ejpam-3275	452	81	α	α	NOUN
ejpam-3275	452	82	-	-	ADJ
ejpam-3275	452	83	prime	prime	ADJ
ejpam-3275	452	84	submodules	submodule	NOUN
ejpam-3275	452	85	.	.	PUNCT
ejpam-3275	453	1	example	example	NOUN
ejpam-3275	453	2	5	5	NUM
ejpam-3275	453	3	.	.	PUNCT
ejpam-3275	454	1	in	in	ADP
ejpam-3275	454	2	z/8z	z/8z	NUM
ejpam-3275	454	3	as	as	ADP
ejpam-3275	454	4	a	a	DET
ejpam-3275	454	5	z	z	NOUN
ejpam-3275	454	6	-	-	PUNCT
ejpam-3275	454	7	module	module	NOUN
ejpam-3275	454	8	,	,	PUNCT
ejpam-3275	454	9	we	we	PRON
ejpam-3275	454	10	have	have	AUX
ejpam-3275	454	11	{	{	PUNCT
ejpam-3275	454	12	0̄	0̄	NOUN
ejpam-3275	454	13	}	}	PUNCT
ejpam-3275	454	14	is	be	AUX
ejpam-3275	454	15	a	a	DET
ejpam-3275	454	16	weakly	weakly	ADJ
ejpam-3275	454	17	α	α	NOUN
ejpam-3275	454	18	-	-	ADJ
ejpam-3275	454	19	prime	prime	ADJ
ejpam-3275	454	20	submodule	submodule	NOUN
ejpam-3275	454	21	of	of	ADP
ejpam-3275	454	22	z/8z	z/8z	NUM
ejpam-3275	454	23	.	.	PUNCT
ejpam-3275	455	1	however	however	ADV
ejpam-3275	455	2	,	,	PUNCT
ejpam-3275	455	3	(	(	PUNCT
ejpam-3275	455	4	{	{	PUNCT
ejpam-3275	455	5	0̄	0̄	NOUN
ejpam-3275	455	6	}	}	PUNCT
ejpam-3275	455	7	:	:	PUNCT
ejpam-3275	455	8	z/8z	z/8z	NUM
ejpam-3275	455	9	)	)	PUNCT
ejpam-3275	455	10	=	=	NOUN
ejpam-3275	456	1	8z	8z	NOUN
ejpam-3275	456	2	is	be	AUX
ejpam-3275	456	3	not	not	PART
ejpam-3275	456	4	a	a	DET
ejpam-3275	456	5	α	α	NUM
ejpam-3275	456	6	-	-	ADJ
ejpam-3275	456	7	prime	prime	ADJ
ejpam-3275	456	8	ideal	ideal	NOUN
ejpam-3275	456	9	of	of	ADP
ejpam-3275	456	10	z.	z.	PROPN
ejpam-3275	456	11	references	reference	NOUN
ejpam-3275	456	12	[	[	X
ejpam-3275	456	13	1	1	NUM
ejpam-3275	456	14	]	]	X
ejpam-3275	456	15	r.	r.	PROPN
ejpam-3275	456	16	ameri	ameri	PROPN
ejpam-3275	456	17	,	,	PUNCT
ejpam-3275	456	18	on	on	ADP
ejpam-3275	456	19	the	the	DET
ejpam-3275	456	20	prime	prime	ADJ
ejpam-3275	456	21	submodules	submodule	NOUN
ejpam-3275	456	22	of	of	ADP
ejpam-3275	456	23	multiplication	multiplication	NOUN
ejpam-3275	456	24	modules	module	NOUN
ejpam-3275	456	25	,	,	PUNCT
ejpam-3275	456	26	international	international	ADJ
ejpam-3275	456	27	journal	journal	NOUN
ejpam-3275	456	28	of	of	ADP
ejpam-3275	456	29	mathematics	mathematics	PROPN
ejpam-3275	456	30	and	and	CCONJ
ejpam-3275	456	31	mathematical	mathematical	ADJ
ejpam-3275	456	32	sciences	science	NOUN
ejpam-3275	456	33	,	,	PUNCT
ejpam-3275	456	34	27	27	NUM
ejpam-3275	456	35	:	:	SYM
ejpam-3275	456	36	1715	1715	NUM
ejpam-3275	456	37	-	-	SYM
ejpam-3275	456	38	1724	1724	NUM
ejpam-3275	456	39	,	,	PUNCT
ejpam-3275	456	40	(	(	PUNCT
ejpam-3275	456	41	2003	2003	NUM
ejpam-3275	456	42	)	)	PUNCT
ejpam-3275	456	43	.	.	PUNCT
ejpam-3275	457	1	[	[	X
ejpam-3275	457	2	2	2	NUM
ejpam-3275	457	3	]	]	X
ejpam-3275	457	4	s.e	s.e	PROPN
ejpam-3275	457	5	.	.	PROPN
ejpam-3275	457	6	atani	atani	PROPN
ejpam-3275	457	7	and	and	CCONJ
ejpam-3275	457	8	f.	f.	PROPN
ejpam-3275	457	9	farzalipour	farzalipour	PROPN
ejpam-3275	457	10	,	,	PUNCT
ejpam-3275	457	11	on	on	ADP
ejpam-3275	457	12	weakly	weakly	ADJ
ejpam-3275	457	13	prime	prime	ADJ
ejpam-3275	457	14	submodules	submodule	NOUN
ejpam-3275	457	15	,	,	PUNCT
ejpam-3275	457	16	tamkang	tamkang	PROPN
ejpam-3275	457	17	journal	journal	PROPN
ejpam-3275	457	18	of	of	ADP
ejpam-3275	457	19	mathematics	mathematic	NOUN
ejpam-3275	457	20	,	,	PUNCT
ejpam-3275	457	21	38(3	38(3	NUM
ejpam-3275	457	22	):	):	PUNCT
ejpam-3275	457	23	247	247	NUM
ejpam-3275	457	24	-	-	SYM
ejpam-3275	457	25	252	252	NUM
ejpam-3275	457	26	,	,	PUNCT
ejpam-3275	457	27	(	(	PUNCT
ejpam-3275	457	28	2007	2007	NUM
ejpam-3275	457	29	)	)	PUNCT
ejpam-3275	457	30	.	.	PUNCT