id	sid	tid	token	lemma	pos
ejpam-1242	1	1	6_xxx_tekir.dvi	6_xxx_tekir.dvi	NUM
ejpam-1242	1	2	european	european	ADJ
ejpam-1242	1	3	journal	journal	NOUN
ejpam-1242	1	4	of	of	ADP
ejpam-1242	1	5	pure	pure	ADJ
ejpam-1242	1	6	and	and	CCONJ
ejpam-1242	1	7	applied	apply	VERB
ejpam-1242	1	8	mathematics	mathematic	NOUN
ejpam-1242	1	9	vol	vol	NOUN
ejpam-1242	1	10	.	.	PROPN
ejpam-1242	2	1	4	4	NUM
ejpam-1242	2	2	,	,	PUNCT
ejpam-1242	2	3	no	no	INTJ
ejpam-1242	2	4	.	.	NOUN
ejpam-1242	2	5	3	3	NUM
ejpam-1242	2	6	,	,	PUNCT
ejpam-1242	2	7	2011	2011	NUM
ejpam-1242	2	8	,	,	PUNCT
ejpam-1242	2	9	251	251	NUM
ejpam-1242	2	10	-	-	SYM
ejpam-1242	2	11	265	265	NUM
ejpam-1242	2	12	issn	issn	PROPN
ejpam-1242	2	13	1307	1307	NUM
ejpam-1242	2	14	-	-	SYM
ejpam-1242	2	15	5543	5543	NUM
ejpam-1242	2	16	–	–	PUNCT
ejpam-1242	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1242	2	18	a	a	DET
ejpam-1242	2	19	zariski	zariski	ADJ
ejpam-1242	2	20	topology	topology	NOUN
ejpam-1242	2	21	for	for	ADP
ejpam-1242	2	22	semimodules	semimodule	NOUN
ejpam-1242	2	23	shahabaddin	shahabaddin	VERB
ejpam-1242	2	24	ebrahimi	ebrahimi	PROPN
ejpam-1242	2	25	atani1	atani1	PROPN
ejpam-1242	2	26	,	,	PUNCT
ejpam-1242	2	27	reza	reza	PROPN
ejpam-1242	2	28	ebrahimi	ebrahimi	PROPN
ejpam-1242	2	29	atani2	atani2	PROPN
ejpam-1242	2	30	,	,	PUNCT
ejpam-1242	2	31	ünsal	ünsal	PROPN
ejpam-1242	2	32	tekir3,∗	tekir3,∗	VERB
ejpam-1242	2	33	1	1	NUM
ejpam-1242	2	34	faculty	faculty	NOUN
ejpam-1242	2	35	of	of	ADP
ejpam-1242	2	36	mathematical	mathematical	ADJ
ejpam-1242	2	37	sciences	sciences	PROPN
ejpam-1242	2	38	,	,	PUNCT
ejpam-1242	2	39	university	university	NOUN
ejpam-1242	2	40	of	of	ADP
ejpam-1242	2	41	guilan	guilan	PROPN
ejpam-1242	2	42	,	,	PUNCT
ejpam-1242	2	43	p.	p.	PROPN
ejpam-1242	2	44	o.	o.	PROPN
ejpam-1242	2	45	box	box	PROPN
ejpam-1242	2	46	1914	1914	NUM
ejpam-1242	2	47	,	,	PUNCT
ejpam-1242	2	48	rasht	rasht	NOUN
ejpam-1242	2	49	,	,	PUNCT
ejpam-1242	2	50	iran	iran	PROPN
ejpam-1242	2	51	2	2	NUM
ejpam-1242	2	52	department	department	NOUN
ejpam-1242	2	53	of	of	ADP
ejpam-1242	2	54	computer	computer	NOUN
ejpam-1242	2	55	engineering	engineering	NOUN
ejpam-1242	2	56	,	,	PUNCT
ejpam-1242	2	57	university	university	NOUN
ejpam-1242	2	58	of	of	ADP
ejpam-1242	2	59	guilan	guilan	PROPN
ejpam-1242	2	60	,	,	PUNCT
ejpam-1242	2	61	p.o	p.o	PROPN
ejpam-1242	2	62	.	.	PROPN
ejpam-1242	2	63	box	box	PROPN
ejpam-1242	2	64	3756	3756	NUM
ejpam-1242	2	65	,	,	PUNCT
ejpam-1242	2	66	rasht	rasht	NOUN
ejpam-1242	2	67	,	,	PUNCT
ejpam-1242	2	68	iran	iran	PROPN
ejpam-1242	2	69	3	3	NUM
ejpam-1242	2	70	department	department	NOUN
ejpam-1242	2	71	of	of	ADP
ejpam-1242	2	72	mathematics	mathematic	NOUN
ejpam-1242	2	73	,	,	PUNCT
ejpam-1242	2	74	the	the	DET
ejpam-1242	2	75	university	university	NOUN
ejpam-1242	2	76	of	of	ADP
ejpam-1242	2	77	marmara	marmara	PROPN
ejpam-1242	2	78	,	,	PUNCT
ejpam-1242	2	79	goztepe	goztepe	NOUN
ejpam-1242	2	80	,	,	PUNCT
ejpam-1242	2	81	istanbul	istanbul	PROPN
ejpam-1242	2	82	,	,	PUNCT
ejpam-1242	2	83	turkey	turkey	PROPN
ejpam-1242	2	84	abstract	abstract	NOUN
ejpam-1242	2	85	.	.	PUNCT
ejpam-1242	3	1	given	give	VERB
ejpam-1242	3	2	a	a	DET
ejpam-1242	3	3	very	very	ADV
ejpam-1242	3	4	strong	strong	ADJ
ejpam-1242	3	5	multiplication	multiplication	NOUN
ejpam-1242	3	6	semimidule	semimidule	NOUN
ejpam-1242	3	7	m	m	VERB
ejpam-1242	3	8	over	over	ADP
ejpam-1242	3	9	a	a	DET
ejpam-1242	3	10	commutative	commutative	ADJ
ejpam-1242	3	11	semiring	semiring	NOUN
ejpam-1242	3	12	r	r	NOUN
ejpam-1242	3	13	,	,	PUNCT
ejpam-1242	3	14	a	a	DET
ejpam-1242	3	15	zariski	zariski	NOUN
ejpam-1242	3	16	topology	topology	NOUN
ejpam-1242	3	17	is	be	AUX
ejpam-1242	3	18	defined	define	VERB
ejpam-1242	3	19	on	on	ADP
ejpam-1242	3	20	the	the	DET
ejpam-1242	3	21	spectrum	spectrum	NOUN
ejpam-1242	3	22	speck(m	speck(m	NOUN
ejpam-1242	3	23	)	)	PUNCT
ejpam-1242	3	24	of	of	ADP
ejpam-1242	3	25	prime	prime	ADJ
ejpam-1242	3	26	k	k	NOUN
ejpam-1242	3	27	-	-	NOUN
ejpam-1242	3	28	subsemimodules	subsemimodules	NOUN
ejpam-1242	3	29	of	of	ADP
ejpam-1242	3	30	m	m	PROPN
ejpam-1242	3	31	.	.	PUNCT
ejpam-1242	4	1	the	the	DET
ejpam-1242	4	2	properties	property	NOUN
ejpam-1242	4	3	and	and	CCONJ
ejpam-1242	4	4	possible	possible	ADJ
ejpam-1242	4	5	structures	structure	NOUN
ejpam-1242	4	6	of	of	ADP
ejpam-1242	4	7	this	this	DET
ejpam-1242	4	8	topology	topology	NOUN
ejpam-1242	4	9	are	be	AUX
ejpam-1242	4	10	studied	study	VERB
ejpam-1242	4	11	.	.	PUNCT
ejpam-1242	5	1	2000	2000	NUM
ejpam-1242	5	2	mathematics	mathematic	NOUN
ejpam-1242	5	3	subject	subject	NOUN
ejpam-1242	5	4	classifications	classification	NOUN
ejpam-1242	5	5	:	:	PUNCT
ejpam-1242	5	6	16y60	16y60	NUM
ejpam-1242	5	7	key	key	ADJ
ejpam-1242	5	8	words	word	NOUN
ejpam-1242	5	9	and	and	CCONJ
ejpam-1242	5	10	phrases	phrase	NOUN
ejpam-1242	5	11	:	:	PUNCT
ejpam-1242	5	12	prime	prime	ADJ
ejpam-1242	5	13	subsemimodule	subsemimodule	NOUN
ejpam-1242	5	14	,	,	PUNCT
ejpam-1242	5	15	strong	strong	ADJ
ejpam-1242	5	16	multiplication	multiplication	NOUN
ejpam-1242	5	17	semimodule	semimodule	NOUN
ejpam-1242	5	18	,	,	PUNCT
ejpam-1242	5	19	very	very	ADV
ejpam-1242	5	20	strong	strong	ADJ
ejpam-1242	5	21	multiplication	multiplication	NOUN
ejpam-1242	5	22	semimodule	semimodule	NOUN
ejpam-1242	5	23	,	,	PUNCT
ejpam-1242	5	24	prime	prime	ADJ
ejpam-1242	5	25	spectrum	spectrum	NOUN
ejpam-1242	5	26	1	1	NUM
ejpam-1242	5	27	.	.	PUNCT
ejpam-1242	6	1	introduction	introduction	NOUN
ejpam-1242	6	2	as	as	ADP
ejpam-1242	6	3	a	a	DET
ejpam-1242	6	4	generalization	generalization	NOUN
ejpam-1242	6	5	of	of	ADP
ejpam-1242	6	6	rings	ring	NOUN
ejpam-1242	6	7	,	,	PUNCT
ejpam-1242	6	8	semirings	semiring	NOUN
ejpam-1242	6	9	have	have	AUX
ejpam-1242	6	10	been	be	AUX
ejpam-1242	6	11	found	find	VERB
ejpam-1242	6	12	useful	useful	ADJ
ejpam-1242	6	13	for	for	ADP
ejpam-1242	6	14	solving	solve	VERB
ejpam-1242	6	15	problems	problem	NOUN
ejpam-1242	6	16	in	in	ADP
ejpam-1242	6	17	different	different	ADJ
ejpam-1242	6	18	areas	area	NOUN
ejpam-1242	6	19	of	of	ADP
ejpam-1242	6	20	applied	applied	ADJ
ejpam-1242	6	21	mathematics	mathematic	NOUN
ejpam-1242	6	22	and	and	CCONJ
ejpam-1242	6	23	information	information	NOUN
ejpam-1242	6	24	sciences	science	NOUN
ejpam-1242	6	25	,	,	PUNCT
ejpam-1242	6	26	since	since	SCONJ
ejpam-1242	6	27	the	the	DET
ejpam-1242	6	28	structure	structure	NOUN
ejpam-1242	6	29	of	of	ADP
ejpam-1242	6	30	a	a	DET
ejpam-1242	6	31	semiring	semiring	NOUN
ejpam-1242	6	32	provides	provide	VERB
ejpam-1242	6	33	an	an	DET
ejpam-1242	6	34	algebraic	algebraic	ADJ
ejpam-1242	6	35	framework	framework	NOUN
ejpam-1242	6	36	for	for	ADP
ejpam-1242	6	37	modelling	modelling	NOUN
ejpam-1242	6	38	and	and	CCONJ
ejpam-1242	6	39	studying	study	VERB
ejpam-1242	6	40	the	the	DET
ejpam-1242	6	41	key	key	ADJ
ejpam-1242	6	42	factors	factor	NOUN
ejpam-1242	6	43	in	in	ADP
ejpam-1242	6	44	these	these	DET
ejpam-1242	6	45	applied	apply	VERB
ejpam-1242	6	46	areas	area	NOUN
ejpam-1242	6	47	.	.	PUNCT
ejpam-1242	7	1	they	they	PRON
ejpam-1242	7	2	play	play	VERB
ejpam-1242	7	3	an	an	DET
ejpam-1242	7	4	important	important	ADJ
ejpam-1242	7	5	role	role	NOUN
ejpam-1242	7	6	in	in	ADP
ejpam-1242	7	7	studying	study	VERB
ejpam-1242	7	8	optimization	optimization	NOUN
ejpam-1242	7	9	theory	theory	NOUN
ejpam-1242	7	10	,	,	PUNCT
ejpam-1242	7	11	graph	graph	NOUN
ejpam-1242	7	12	theory	theory	NOUN
ejpam-1242	7	13	,	,	PUNCT
ejpam-1242	7	14	theory	theory	NOUN
ejpam-1242	7	15	of	of	ADP
ejpam-1242	7	16	discrete	discrete	ADJ
ejpam-1242	7	17	event	event	NOUN
ejpam-1242	7	18	dynamical	dynamical	ADJ
ejpam-1242	7	19	systems	system	NOUN
ejpam-1242	7	20	,	,	PUNCT
ejpam-1242	7	21	generalized	generalize	VERB
ejpam-1242	7	22	fuzzy	fuzzy	ADJ
ejpam-1242	7	23	computation	computation	NOUN
ejpam-1242	7	24	,	,	PUNCT
ejpam-1242	7	25	automata	automata	NOUN
ejpam-1242	7	26	theory	theory	NOUN
ejpam-1242	7	27	,	,	PUNCT
ejpam-1242	7	28	coding	code	VERB
ejpam-1242	7	29	theory	theory	NOUN
ejpam-1242	7	30	,	,	PUNCT
ejpam-1242	7	31	cryptography	cryptography	NOUN
ejpam-1242	7	32	theory	theory	NOUN
ejpam-1242	7	33	,	,	PUNCT
ejpam-1242	7	34	and	and	CCONJ
ejpam-1242	7	35	so	so	ADV
ejpam-1242	7	36	on	on	ADV
ejpam-1242	7	37	(	(	PUNCT
ejpam-1242	7	38	see	see	VERB
ejpam-1242	7	39	golan	golan	PROPN
ejpam-1242	7	40	[	[	X
ejpam-1242	7	41	16	16	NUM
ejpam-1242	7	42	]	]	PUNCT
ejpam-1242	7	43	,	,	PUNCT
ejpam-1242	7	44	glazek	glazek	PROPN
ejpam-1242	8	1	[	[	X
ejpam-1242	8	2	17	17	NUM
ejpam-1242	8	3	]	]	PUNCT
ejpam-1242	8	4	,	,	PUNCT
ejpam-1242	8	5	hebisch	hebisch	NOUN
ejpam-1242	8	6	and	and	CCONJ
ejpam-1242	8	7	weinert	weinert	ADJ
ejpam-1242	9	1	[	[	X
ejpam-1242	9	2	18	18	NUM
ejpam-1242	9	3	]	]	PUNCT
ejpam-1242	9	4	,	,	PUNCT
ejpam-1242	9	5	simon	simon	PROPN
ejpam-1242	9	6	[	[	X
ejpam-1242	9	7	23	23	NUM
ejpam-1242	9	8	]	]	PUNCT
ejpam-1242	9	9	)	)	PUNCT
ejpam-1242	9	10	.	.	PUNCT
ejpam-1242	10	1	ideals	ideal	NOUN
ejpam-1242	10	2	of	of	ADP
ejpam-1242	10	3	semirings	semiring	NOUN
ejpam-1242	10	4	play	play	VERB
ejpam-1242	10	5	a	a	DET
ejpam-1242	10	6	central	central	ADJ
ejpam-1242	10	7	role	role	NOUN
ejpam-1242	10	8	in	in	ADP
ejpam-1242	10	9	the	the	DET
ejpam-1242	10	10	structure	structure	NOUN
ejpam-1242	10	11	theory	theory	NOUN
ejpam-1242	10	12	and	and	CCONJ
ejpam-1242	10	13	are	be	AUX
ejpam-1242	10	14	useful	useful	ADJ
ejpam-1242	10	15	for	for	ADP
ejpam-1242	10	16	many	many	ADJ
ejpam-1242	10	17	purposes	purpose	NOUN
ejpam-1242	10	18	.	.	PUNCT
ejpam-1242	11	1	however	however	ADV
ejpam-1242	11	2	,	,	PUNCT
ejpam-1242	11	3	they	they	PRON
ejpam-1242	11	4	do	do	AUX
ejpam-1242	11	5	not	not	PART
ejpam-1242	11	6	in	in	ADP
ejpam-1242	11	7	general	general	ADJ
ejpam-1242	11	8	coincide	coincide	NOUN
ejpam-1242	11	9	with	with	ADP
ejpam-1242	11	10	the	the	DET
ejpam-1242	11	11	usual	usual	ADJ
ejpam-1242	11	12	ring	ring	NOUN
ejpam-1242	11	13	ideals	ideal	NOUN
ejpam-1242	11	14	and	and	CCONJ
ejpam-1242	11	15	,	,	PUNCT
ejpam-1242	11	16	for	for	ADP
ejpam-1242	11	17	this	this	DET
ejpam-1242	11	18	reason	reason	NOUN
ejpam-1242	11	19	,	,	PUNCT
ejpam-1242	11	20	their	their	PRON
ejpam-1242	11	21	use	use	NOUN
ejpam-1242	11	22	is	be	AUX
ejpam-1242	11	23	somewhat	somewhat	ADV
ejpam-1242	11	24	limited	limited	ADJ
ejpam-1242	11	25	in	in	ADP
ejpam-1242	11	26	trying	try	VERB
ejpam-1242	11	27	to	to	PART
ejpam-1242	11	28	obtain	obtain	VERB
ejpam-1242	11	29	analogues	analogue	NOUN
ejpam-1242	11	30	of	of	ADP
ejpam-1242	11	31	ring	ring	NOUN
ejpam-1242	11	32	theorems	theorem	NOUN
ejpam-1242	11	33	for	for	ADP
ejpam-1242	11	34	semirings	semiring	NOUN
ejpam-1242	11	35	.	.	PUNCT
ejpam-1242	12	1	indeed	indeed	ADV
ejpam-1242	12	2	,	,	PUNCT
ejpam-1242	12	3	many	many	ADJ
ejpam-1242	12	4	results	result	NOUN
ejpam-1242	12	5	in	in	ADP
ejpam-1242	12	6	rings	ring	NOUN
ejpam-1242	12	7	apparently	apparently	ADV
ejpam-1242	12	8	have	have	VERB
ejpam-1242	12	9	no	no	DET
ejpam-1242	12	10	analogues	analogue	NOUN
ejpam-1242	12	11	in	in	ADP
ejpam-1242	12	12	semirings	semiring	NOUN
ejpam-1242	12	13	using	use	VERB
ejpam-1242	12	14	only	only	ADJ
ejpam-1242	12	15	ideals	ideal	NOUN
ejpam-1242	12	16	.	.	PUNCT
ejpam-1242	13	1	in	in	ADP
ejpam-1242	13	2	order	order	NOUN
ejpam-1242	13	3	to	to	PART
ejpam-1242	13	4	overcome	overcome	VERB
ejpam-1242	13	5	this	this	DET
ejpam-1242	13	6	deficiency	deficiency	NOUN
ejpam-1242	13	7	,	,	PUNCT
ejpam-1242	13	8	the	the	DET
ejpam-1242	13	9	present	present	ADJ
ejpam-1242	13	10	authors	author	NOUN
ejpam-1242	13	11	[	[	X
ejpam-1242	13	12	14	14	NUM
ejpam-1242	13	13	]	]	PUNCT
ejpam-1242	13	14	defined	define	VERB
ejpam-1242	13	15	a	a	DET
ejpam-1242	13	16	more	more	ADV
ejpam-1242	13	17	resttricted	resttricted	ADJ
ejpam-1242	13	18	class	class	NOUN
ejpam-1242	13	19	of	of	ADP
ejpam-1242	13	20	ideals	ideal	NOUN
ejpam-1242	13	21	in	in	ADP
ejpam-1242	13	22	semirings	semiring	NOUN
ejpam-1242	13	23	,	,	PUNCT
ejpam-1242	13	24	which	which	PRON
ejpam-1242	13	25	is	be	AUX
ejpam-1242	13	26	called	call	VERB
ejpam-1242	13	27	the	the	DET
ejpam-1242	13	28	class	class	NOUN
ejpam-1242	13	29	of	of	ADP
ejpam-1242	13	30	“	"	PUNCT
ejpam-1242	13	31	strong	strong	ADJ
ejpam-1242	13	32	ideals	ideal	NOUN
ejpam-1242	13	33	”	"	PUNCT
ejpam-1242	13	34	,	,	PUNCT
ejpam-1242	13	35	with	with	ADP
ejpam-1242	13	36	the	the	DET
ejpam-1242	13	37	property	property	NOUN
ejpam-1242	13	38	that	that	PRON
ejpam-1242	13	39	they	they	PRON
ejpam-1242	13	40	are	be	AUX
ejpam-1242	13	41	k	k	NOUN
ejpam-1242	13	42	-	-	NOUN
ejpam-1242	13	43	ideals	ideal	NOUN
ejpam-1242	13	44	.	.	PUNCT
ejpam-1242	14	1	let	let	VERB
ejpam-1242	14	2	m	m	PRON
ejpam-1242	14	3	be	be	AUX
ejpam-1242	14	4	a	a	DET
ejpam-1242	14	5	module	module	NOUN
ejpam-1242	14	6	over	over	ADP
ejpam-1242	14	7	a	a	DET
ejpam-1242	14	8	commutative	commutative	ADJ
ejpam-1242	14	9	ring	ring	NOUN
ejpam-1242	14	10	r	r	NOUN
ejpam-1242	14	11	with	with	ADP
ejpam-1242	14	12	identity	identity	NOUN
ejpam-1242	14	13	.	.	PUNCT
ejpam-1242	15	1	the	the	DET
ejpam-1242	15	2	prime	prime	PROPN
ejpam-1242	15	3	spectrum	spectrum	NOUN
ejpam-1242	15	4	spec(r	spec(r	PROPN
ejpam-1242	15	5	)	)	PUNCT
ejpam-1242	15	6	and	and	CCONJ
ejpam-1242	15	7	the	the	DET
ejpam-1242	15	8	topological	topological	ADJ
ejpam-1242	15	9	space	space	NOUN
ejpam-1242	15	10	obtained	obtain	VERB
ejpam-1242	15	11	by	by	ADP
ejpam-1242	15	12	introducing	introduce	VERB
ejpam-1242	15	13	zariski	zariski	NOUN
ejpam-1242	15	14	topology	topology	NOUN
ejpam-1242	15	15	on	on	ADP
ejpam-1242	15	16	the	the	DET
ejpam-1242	15	17	set	set	NOUN
ejpam-1242	15	18	of	of	ADP
ejpam-1242	15	19	prime	prime	ADJ
ejpam-1242	15	20	ideals	ideal	NOUN
ejpam-1242	15	21	of	of	ADP
ejpam-1242	15	22	r	r	NOUN
ejpam-1242	15	23	play	play	VERB
ejpam-1242	15	24	an	an	DET
ejpam-1242	15	25	important	important	ADJ
ejpam-1242	15	26	role	role	NOUN
ejpam-1242	15	27	in	in	ADP
ejpam-1242	15	28	the	the	DET
ejpam-1242	15	29	fields	field	NOUN
ejpam-1242	15	30	of	of	ADP
ejpam-1242	15	31	commutative	commutative	ADJ
ejpam-1242	15	32	algebra	algebra	NOUN
ejpam-1242	15	33	,	,	PUNCT
ejpam-1242	15	34	algebraic	algebraic	ADJ
ejpam-1242	15	35	geometry	geometry	NOUN
ejpam-1242	15	36	and	and	CCONJ
ejpam-1242	15	37	lattice	lattice	NOUN
ejpam-1242	15	38	∗corresponding	∗corresponde	VERB
ejpam-1242	15	39	author	author	NOUN
ejpam-1242	15	40	.	.	PUNCT
ejpam-1242	16	1	email	email	NOUN
ejpam-1242	16	2	address	address	NOUN
ejpam-1242	16	3	:	:	PUNCT
ejpam-1242	16	4	utekir�marmara.edu.tr	utekir�marmara.edu.tr	PROPN
ejpam-1242	16	5	(	(	PUNCT
ejpam-1242	16	6	ü.	ü.	NOUN
ejpam-1242	16	7	tekir	tekir	NOUN
ejpam-1242	16	8	)	)	PUNCT
ejpam-1242	16	9	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1242	17	1	251	251	NUM
ejpam-1242	17	2	c	c	X
ejpam-1242	17	3	©	©	PROPN
ejpam-1242	17	4	2011	2011	NUM
ejpam-1242	17	5	ejpam	ejpam	VERB
ejpam-1242	17	6	all	all	DET
ejpam-1242	17	7	rights	right	NOUN
ejpam-1242	17	8	reserved	reserve	VERB
ejpam-1242	17	9	.	.	PUNCT
ejpam-1242	18	1	s.	s.	PROPN
ejpam-1242	18	2	atani	atani	PROPN
ejpam-1242	18	3	,	,	PUNCT
ejpam-1242	18	4	r.	r.	PROPN
ejpam-1242	18	5	atrani	atrani	PROPN
ejpam-1242	18	6	,	,	PUNCT
ejpam-1242	18	7	ü.	ü.	NOUN
ejpam-1242	18	8	tekir	tekir	PROPN
ejpam-1242	18	9	/	/	SYM
ejpam-1242	18	10	eur	eur	PROPN
ejpam-1242	18	11	.	.	PUNCT
ejpam-1242	19	1	j.	j.	PROPN
ejpam-1242	19	2	pure	pure	PROPN
ejpam-1242	19	3	appl	appl	PROPN
ejpam-1242	19	4	.	.	PROPN
ejpam-1242	19	5	math	math	PROPN
ejpam-1242	19	6	,	,	PUNCT
ejpam-1242	19	7	4	4	NUM
ejpam-1242	19	8	(	(	PUNCT
ejpam-1242	19	9	2011	2011	NUM
ejpam-1242	19	10	)	)	PUNCT
ejpam-1242	19	11	,	,	PUNCT
ejpam-1242	19	12	251	251	NUM
ejpam-1242	19	13	-	-	SYM
ejpam-1242	19	14	265	265	NUM
ejpam-1242	19	15	252	252	NUM
ejpam-1242	19	16	theory	theory	NOUN
ejpam-1242	19	17	.	.	PUNCT
ejpam-1242	20	1	also	also	ADV
ejpam-1242	20	2	,	,	PUNCT
ejpam-1242	20	3	recently	recently	ADV
ejpam-1242	20	4	the	the	DET
ejpam-1242	20	5	notion	notion	NOUN
ejpam-1242	20	6	of	of	ADP
ejpam-1242	20	7	prime	prime	ADJ
ejpam-1242	20	8	submodules	submodule	NOUN
ejpam-1242	20	9	and	and	CCONJ
ejpam-1242	20	10	zariski	zariski	NOUN
ejpam-1242	20	11	topology	topology	NOUN
ejpam-1242	20	12	on	on	ADP
ejpam-1242	20	13	spec(m	spec(m	PROPN
ejpam-1242	20	14	)	)	PUNCT
ejpam-1242	20	15	,	,	PUNCT
ejpam-1242	20	16	the	the	DET
ejpam-1242	20	17	set	set	NOUN
ejpam-1242	20	18	of	of	ADP
ejpam-1242	20	19	all	all	DET
ejpam-1242	20	20	prime	prime	ADJ
ejpam-1242	20	21	r	r	NOUN
ejpam-1242	20	22	-	-	PUNCT
ejpam-1242	20	23	submodules	submodule	NOUN
ejpam-1242	20	24	of	of	ADP
ejpam-1242	20	25	m	m	PRON
ejpam-1242	20	26	,	,	PUNCT
ejpam-1242	20	27	are	be	AUX
ejpam-1242	20	28	studied	study	VERB
ejpam-1242	20	29	by	by	ADP
ejpam-1242	20	30	many	many	ADJ
ejpam-1242	20	31	authors	author	NOUN
ejpam-1242	20	32	(	(	PUNCT
ejpam-1242	20	33	for	for	ADP
ejpam-1242	20	34	example	example	NOUN
ejpam-1242	20	35	see	see	VERB
ejpam-1242	20	36	lu[19	lu[19	X
ejpam-1242	20	37	]	]	X
ejpam-1242	20	38	,	,	PUNCT
ejpam-1242	20	39	lu	lu	PROPN
ejpam-1242	21	1	[	[	X
ejpam-1242	21	2	20	20	NUM
ejpam-1242	21	3	]	]	PUNCT
ejpam-1242	21	4	,	,	PUNCT
ejpam-1242	21	5	lu	lu	PROPN
ejpam-1242	22	1	[	[	X
ejpam-1242	22	2	21	21	NUM
ejpam-1242	22	3	]	]	X
ejpam-1242	22	4	,	,	PUNCT
ejpam-1242	22	5	maccasland	maccasland	PROPN
ejpam-1242	22	6	,	,	PUNCT
ejpam-1242	22	7	moore	moore	PROPN
ejpam-1242	22	8	and	and	CCONJ
ejpam-1242	22	9	smith	smith	PROPN
ejpam-1242	23	1	[	[	X
ejpam-1242	23	2	22	22	NUM
ejpam-1242	23	3	]	]	PUNCT
ejpam-1242	23	4	)	)	PUNCT
ejpam-1242	23	5	.	.	PUNCT
ejpam-1242	24	1	the	the	DET
ejpam-1242	24	2	main	main	ADJ
ejpam-1242	24	3	dificulty	dificulty	NOUN
ejpam-1242	24	4	is	be	AUX
ejpam-1242	24	5	figuring	figure	VERB
ejpam-1242	24	6	out	out	ADP
ejpam-1242	24	7	what	what	PRON
ejpam-1242	24	8	additional	additional	ADJ
ejpam-1242	24	9	hypotheses	hypothesis	NOUN
ejpam-1242	24	10	the	the	DET
ejpam-1242	24	11	semimodule	semimodule	NOUN
ejpam-1242	24	12	must	must	AUX
ejpam-1242	24	13	satisfy	satisfy	VERB
ejpam-1242	24	14	to	to	PART
ejpam-1242	24	15	get	get	VERB
ejpam-1242	24	16	similar	similar	ADJ
ejpam-1242	24	17	results	result	NOUN
ejpam-1242	24	18	of	of	ADP
ejpam-1242	24	19	modules	module	NOUN
ejpam-1242	24	20	.	.	PUNCT
ejpam-1242	25	1	the	the	DET
ejpam-1242	25	2	two	two	NUM
ejpam-1242	25	3	new	new	ADJ
ejpam-1242	25	4	key	key	ADJ
ejpam-1242	25	5	notions	notion	NOUN
ejpam-1242	25	6	are	be	AUX
ejpam-1242	25	7	that	that	SCONJ
ejpam-1242	25	8	a	a	DET
ejpam-1242	25	9	“	"	PUNCT
ejpam-1242	25	10	strong	strong	ADJ
ejpam-1242	25	11	subsemimodule	subsemimodule	NOUN
ejpam-1242	25	12	”	"	PUNCT
ejpam-1242	25	13	and	and	CCONJ
ejpam-1242	25	14	a	a	DET
ejpam-1242	25	15	“	"	PUNCT
ejpam-1242	25	16	very	very	ADV
ejpam-1242	25	17	strong	strong	ADJ
ejpam-1242	25	18	subsemimodule	subsemimodule	NOUN
ejpam-1242	25	19	”	"	PUNCT
ejpam-1242	25	20	.	.	PUNCT
ejpam-1242	26	1	in	in	ADP
ejpam-1242	26	2	this	this	DET
ejpam-1242	26	3	paper	paper	NOUN
ejpam-1242	26	4	,	,	PUNCT
ejpam-1242	26	5	section	section	NOUN
ejpam-1242	26	6	3	3	NUM
ejpam-1242	26	7	,	,	PUNCT
ejpam-1242	26	8	we	we	PRON
ejpam-1242	26	9	list	list	VERB
ejpam-1242	26	10	some	some	DET
ejpam-1242	26	11	basic	basic	ADJ
ejpam-1242	26	12	properties	property	NOUN
ejpam-1242	26	13	concerning	concern	VERB
ejpam-1242	26	14	very	very	ADV
ejpam-1242	26	15	strong	strong	ADJ
ejpam-1242	26	16	multiplication	multiplication	NOUN
ejpam-1242	26	17	semimodules	semimodule	NOUN
ejpam-1242	26	18	.	.	PUNCT
ejpam-1242	27	1	for	for	ADP
ejpam-1242	27	2	example	example	NOUN
ejpam-1242	27	3	,	,	PUNCT
ejpam-1242	27	4	we	we	PRON
ejpam-1242	27	5	give	give	VERB
ejpam-1242	27	6	some	some	DET
ejpam-1242	27	7	results	result	NOUN
ejpam-1242	27	8	to	to	PART
ejpam-1242	27	9	characterize	characterize	VERB
ejpam-1242	27	10	the	the	DET
ejpam-1242	27	11	prime	prime	ADJ
ejpam-1242	27	12	k	k	NOUN
ejpam-1242	27	13	-	-	NOUN
ejpam-1242	27	14	subsemimodules	subsemimodule	NOUN
ejpam-1242	27	15	of	of	ADP
ejpam-1242	27	16	very	very	ADV
ejpam-1242	27	17	strong	strong	ADJ
ejpam-1242	27	18	multiplication	multiplication	NOUN
ejpam-1242	27	19	semimodules	semimodule	NOUN
ejpam-1242	27	20	.	.	PUNCT
ejpam-1242	28	1	in	in	ADP
ejpam-1242	28	2	section	section	NOUN
ejpam-1242	28	3	4	4	NUM
ejpam-1242	28	4	,	,	PUNCT
ejpam-1242	28	5	we	we	PRON
ejpam-1242	28	6	concentrate	concentrate	VERB
ejpam-1242	28	7	on	on	ADP
ejpam-1242	28	8	zariski	zariski	ADJ
ejpam-1242	28	9	topology	topology	NOUN
ejpam-1242	28	10	of	of	ADP
ejpam-1242	28	11	very	very	ADV
ejpam-1242	28	12	strong	strong	ADJ
ejpam-1242	28	13	multiplication	multiplication	NOUN
ejpam-1242	28	14	semimodules	semimodule	NOUN
ejpam-1242	28	15	m	m	VERB
ejpam-1242	28	16	over	over	ADP
ejpam-1242	28	17	commutative	commutative	ADJ
ejpam-1242	28	18	semirings	semiring	NOUN
ejpam-1242	28	19	r	r	NOUN
ejpam-1242	28	20	with	with	ADP
ejpam-1242	28	21	identity	identity	NOUN
ejpam-1242	28	22	and	and	CCONJ
ejpam-1242	28	23	generalize	generalize	VERB
ejpam-1242	28	24	the	the	DET
ejpam-1242	28	25	some	some	DET
ejpam-1242	28	26	well	well	ADV
ejpam-1242	28	27	known	know	VERB
ejpam-1242	28	28	results	result	NOUN
ejpam-1242	28	29	of	of	ADP
ejpam-1242	28	30	zariski	zariski	ADJ
ejpam-1242	28	31	topoloy	topoloy	NOUN
ejpam-1242	28	32	on	on	ADP
ejpam-1242	28	33	the	the	DET
ejpam-1242	28	34	sets	set	NOUN
ejpam-1242	28	35	of	of	ADP
ejpam-1242	28	36	prime	prime	ADJ
ejpam-1242	28	37	ideals	ideal	NOUN
ejpam-1242	28	38	of	of	ADP
ejpam-1242	28	39	a	a	DET
ejpam-1242	28	40	commutative	commutative	ADJ
ejpam-1242	28	41	ring	ring	NOUN
ejpam-1242	28	42	to	to	ADP
ejpam-1242	28	43	spec(m	spec(m	PROPN
ejpam-1242	28	44	)	)	PUNCT
ejpam-1242	28	45	,	,	PUNCT
ejpam-1242	28	46	the	the	DET
ejpam-1242	28	47	sets	set	NOUN
ejpam-1242	28	48	of	of	ADP
ejpam-1242	28	49	prime	prime	ADJ
ejpam-1242	28	50	k	k	NOUN
ejpam-1242	28	51	-	-	NOUN
ejpam-1242	28	52	subsemimodules	subsemimodules	NOUN
ejpam-1242	28	53	of	of	ADP
ejpam-1242	28	54	m	m	PRON
ejpam-1242	28	55	and	and	CCONJ
ejpam-1242	28	56	investigate	investigate	VERB
ejpam-1242	28	57	the	the	DET
ejpam-1242	28	58	basic	basic	ADJ
ejpam-1242	28	59	properties	property	NOUN
ejpam-1242	28	60	of	of	ADP
ejpam-1242	28	61	this	this	DET
ejpam-1242	28	62	topology	topology	NOUN
ejpam-1242	28	63	.	.	PUNCT
ejpam-1242	29	1	in	in	ADP
ejpam-1242	29	2	this	this	DET
ejpam-1242	29	3	regard	regard	NOUN
ejpam-1242	29	4	we	we	PRON
ejpam-1242	29	5	strongly	strongly	ADV
ejpam-1242	29	6	use	use	VERB
ejpam-1242	29	7	the	the	DET
ejpam-1242	29	8	notion	notion	NOUN
ejpam-1242	29	9	of	of	ADP
ejpam-1242	29	10	product	product	NOUN
ejpam-1242	29	11	of	of	ADP
ejpam-1242	29	12	subsemimodules	subsemimodule	NOUN
ejpam-1242	29	13	of	of	ADP
ejpam-1242	29	14	very	very	ADV
ejpam-1242	29	15	strong	strong	ADJ
ejpam-1242	29	16	multiplication	multiplication	NOUN
ejpam-1242	29	17	semimodules	semimodule	NOUN
ejpam-1242	29	18	.	.	PUNCT
ejpam-1242	30	1	for	for	ADP
ejpam-1242	30	2	example	example	NOUN
ejpam-1242	30	3	,	,	PUNCT
ejpam-1242	30	4	we	we	PRON
ejpam-1242	30	5	prove	prove	VERB
ejpam-1242	30	6	that	that	SCONJ
ejpam-1242	30	7	an	an	DET
ejpam-1242	30	8	open	open	ADJ
ejpam-1242	30	9	set	set	NOUN
ejpam-1242	30	10	x	x	NOUN
ejpam-1242	30	11	=	=	SYM
ejpam-1242	30	12	speck(m	speck(m	NOUN
ejpam-1242	30	13	)	)	PUNCT
ejpam-1242	30	14	is	be	AUX
ejpam-1242	30	15	compact	compact	ADJ
ejpam-1242	30	16	if	if	SCONJ
ejpam-1242	30	17	and	and	CCONJ
ejpam-1242	30	18	only	only	ADV
ejpam-1242	30	19	if	if	SCONJ
ejpam-1242	30	20	it	it	PRON
ejpam-1242	30	21	is	be	AUX
ejpam-1242	30	22	a	a	DET
ejpam-1242	30	23	finite	finite	ADJ
ejpam-1242	30	24	union	union	NOUN
ejpam-1242	30	25	of	of	ADP
ejpam-1242	30	26	basic	basic	ADJ
ejpam-1242	30	27	open	open	ADJ
ejpam-1242	30	28	sets	set	NOUN
ejpam-1242	30	29	..	..	PUNCT
ejpam-1242	31	1	2	2	X
ejpam-1242	31	2	.	.	X
ejpam-1242	31	3	preliminaries	preliminary	NOUN
ejpam-1242	31	4	throughout	throughout	ADP
ejpam-1242	31	5	this	this	DET
ejpam-1242	31	6	paper	paper	NOUN
ejpam-1242	31	7	r	r	NOUN
ejpam-1242	31	8	is	be	AUX
ejpam-1242	31	9	a	a	DET
ejpam-1242	31	10	commutative	commutative	ADJ
ejpam-1242	31	11	semiring	semiring	NOUN
ejpam-1242	31	12	with	with	ADP
ejpam-1242	31	13	identity	identity	NOUN
ejpam-1242	31	14	.	.	PUNCT
ejpam-1242	32	1	in	in	ADP
ejpam-1242	32	2	order	order	NOUN
ejpam-1242	32	3	to	to	PART
ejpam-1242	32	4	make	make	VERB
ejpam-1242	32	5	this	this	DET
ejpam-1242	32	6	paper	paper	NOUN
ejpam-1242	32	7	easier	easy	ADJ
ejpam-1242	32	8	to	to	PART
ejpam-1242	32	9	follow	follow	VERB
ejpam-1242	32	10	,	,	PUNCT
ejpam-1242	32	11	we	we	PRON
ejpam-1242	32	12	recall	recall	VERB
ejpam-1242	32	13	in	in	ADP
ejpam-1242	32	14	this	this	DET
ejpam-1242	32	15	section	section	NOUN
ejpam-1242	32	16	various	various	ADJ
ejpam-1242	32	17	notions	notion	NOUN
ejpam-1242	32	18	from	from	ADP
ejpam-1242	32	19	semimodule	semimodule	NOUN
ejpam-1242	32	20	theory	theory	NOUN
ejpam-1242	32	21	which	which	PRON
ejpam-1242	32	22	will	will	AUX
ejpam-1242	32	23	be	be	AUX
ejpam-1242	32	24	used	use	VERB
ejpam-1242	32	25	in	in	ADP
ejpam-1242	32	26	the	the	DET
ejpam-1242	32	27	sequel	sequel	NOUN
ejpam-1242	32	28	.	.	PUNCT
ejpam-1242	33	1	for	for	ADP
ejpam-1242	33	2	the	the	DET
ejpam-1242	33	3	definitions	definition	NOUN
ejpam-1242	33	4	of	of	ADP
ejpam-1242	33	5	monoid	monoid	PROPN
ejpam-1242	33	6	,	,	PUNCT
ejpam-1242	33	7	semirings	semiring	NOUN
ejpam-1242	33	8	,	,	PUNCT
ejpam-1242	33	9	semimodules	semimodule	NOUN
ejpam-1242	33	10	and	and	CCONJ
ejpam-1242	33	11	subsemimodules	subsemimodule	NOUN
ejpam-1242	33	12	we	we	PRON
ejpam-1242	33	13	refer	refer	VERB
ejpam-1242	33	14	[	[	X
ejpam-1242	33	15	16	16	NUM
ejpam-1242	33	16	,	,	PUNCT
ejpam-1242	33	17	18	18	NUM
ejpam-1242	33	18	,	,	PUNCT
ejpam-1242	33	19	13	13	NUM
ejpam-1242	33	20	,	,	PUNCT
ejpam-1242	33	21	7	7	NUM
ejpam-1242	33	22	]	]	PUNCT
ejpam-1242	33	23	.	.	PUNCT
ejpam-1242	34	1	all	all	PRON
ejpam-1242	34	2	semiring	semire	VERB
ejpam-1242	34	3	in	in	ADP
ejpam-1242	34	4	this	this	DET
ejpam-1242	34	5	paper	paper	NOUN
ejpam-1242	34	6	are	be	AUX
ejpam-1242	34	7	commutative	commutative	ADJ
ejpam-1242	34	8	with	with	ADP
ejpam-1242	34	9	non	non	ADJ
ejpam-1242	34	10	-	-	ADJ
ejpam-1242	34	11	zero	zero	NUM
ejpam-1242	34	12	identity	identity	NOUN
ejpam-1242	34	13	.	.	PUNCT
ejpam-1242	35	1	definition	definition	NOUN
ejpam-1242	35	2	1	1	NUM
ejpam-1242	35	3	.	.	PUNCT
ejpam-1242	36	1	(	(	PUNCT
ejpam-1242	36	2	1	1	X
ejpam-1242	36	3	)	)	PUNCT
ejpam-1242	36	4	let	let	VERB
ejpam-1242	36	5	m	m	PRON
ejpam-1242	36	6	be	be	AUX
ejpam-1242	36	7	a	a	DET
ejpam-1242	36	8	semimodule	semimodule	NOUN
ejpam-1242	36	9	over	over	ADP
ejpam-1242	36	10	a	a	DET
ejpam-1242	36	11	semiring	semire	VERB
ejpam-1242	36	12	r.	r.	NOUN
ejpam-1242	36	13	a	a	DET
ejpam-1242	36	14	subtractive	subtractive	NOUN
ejpam-1242	36	15	subsemimodule	subsemimodule	NOUN
ejpam-1242	36	16	(=	(=	ADP
ejpam-1242	36	17	k	k	ADJ
ejpam-1242	36	18	-	-	NOUN
ejpam-1242	36	19	subsemimodule	subsemimodule	NOUN
ejpam-1242	36	20	)	)	PUNCT
ejpam-1242	36	21	n	n	CCONJ
ejpam-1242	36	22	is	be	AUX
ejpam-1242	36	23	a	a	DET
ejpam-1242	36	24	subsemimodule	subsemimodule	NOUN
ejpam-1242	36	25	of	of	ADP
ejpam-1242	36	26	m	m	NOUN
ejpam-1242	36	27	such	such	ADJ
ejpam-1242	36	28	that	that	SCONJ
ejpam-1242	36	29	if	if	SCONJ
ejpam-1242	36	30	x	x	X
ejpam-1242	36	31	,	,	PUNCT
ejpam-1242	36	32	x	x	PROPN
ejpam-1242	37	1	+	+	CCONJ
ejpam-1242	37	2	y	y	PROPN
ejpam-1242	37	3	∈	∈	PROPN
ejpam-1242	37	4	n	n	CCONJ
ejpam-1242	37	5	,	,	PUNCT
ejpam-1242	37	6	then	then	ADV
ejpam-1242	37	7	y	y	PROPN
ejpam-1242	37	8	∈	∈	PROPN
ejpam-1242	37	9	n	n	CCONJ
ejpam-1242	37	10	(	(	PUNCT
ejpam-1242	37	11	so	so	CCONJ
ejpam-1242	37	12	{	{	PUNCT
ejpam-1242	37	13	0	0	NUM
ejpam-1242	37	14	m	m	VERB
ejpam-1242	37	15	}	}	PUNCT
ejpam-1242	37	16	is	be	AUX
ejpam-1242	37	17	a	a	DET
ejpam-1242	37	18	k	k	NOUN
ejpam-1242	37	19	-	-	NOUN
ejpam-1242	37	20	subsemimodule	subsemimodule	NOUN
ejpam-1242	37	21	of	of	ADP
ejpam-1242	37	22	m	m	PROPN
ejpam-1242	37	23	)	)	PUNCT
ejpam-1242	37	24	.	.	PUNCT
ejpam-1242	38	1	(	(	PUNCT
ejpam-1242	38	2	2	2	X
ejpam-1242	38	3	)	)	PUNCT
ejpam-1242	38	4	a	a	DET
ejpam-1242	38	5	prime	prime	ADJ
ejpam-1242	38	6	subsemimodule	subsemimodule	NOUN
ejpam-1242	38	7	of	of	ADP
ejpam-1242	38	8	m	m	PROPN
ejpam-1242	38	9	is	be	AUX
ejpam-1242	38	10	a	a	DET
ejpam-1242	38	11	proper	proper	ADJ
ejpam-1242	38	12	subsemimodule	subsemimodule	NOUN
ejpam-1242	38	13	n	n	PROPN
ejpam-1242	38	14	of	of	ADP
ejpam-1242	38	15	m	m	PRON
ejpam-1242	38	16	in	in	ADP
ejpam-1242	38	17	which	which	PRON
ejpam-1242	38	18	x	x	SYM
ejpam-1242	38	19	∈	∈	PROPN
ejpam-1242	38	20	n	n	NOUN
ejpam-1242	38	21	or	or	CCONJ
ejpam-1242	38	22	rm	rm	PROPN
ejpam-1242	38	23	⊆	⊆	NUM
ejpam-1242	38	24	n	n	NOUN
ejpam-1242	38	25	whenever	whenever	SCONJ
ejpam-1242	38	26	r	r	NOUN
ejpam-1242	38	27	x	x	SYM
ejpam-1242	38	28	∈	∈	PROPN
ejpam-1242	38	29	n.	n.	NOUN
ejpam-1242	38	30	the	the	DET
ejpam-1242	38	31	collection	collection	NOUN
ejpam-1242	38	32	of	of	ADP
ejpam-1242	38	33	all	all	DET
ejpam-1242	38	34	prime	prime	ADJ
ejpam-1242	38	35	k	k	NOUN
ejpam-1242	38	36	-	-	NOUN
ejpam-1242	38	37	subsemimodules	subsemimodule	NOUN
ejpam-1242	38	38	of	of	ADP
ejpam-1242	38	39	m	m	PROPN
ejpam-1242	38	40	is	be	AUX
ejpam-1242	38	41	called	call	VERB
ejpam-1242	38	42	the	the	DET
ejpam-1242	38	43	k	k	NOUN
ejpam-1242	38	44	-	-	NOUN
ejpam-1242	38	45	spectrum	spectrum	NOUN
ejpam-1242	38	46	of	of	ADP
ejpam-1242	38	47	m	m	PRON
ejpam-1242	38	48	and	and	CCONJ
ejpam-1242	38	49	denoted	denote	VERB
ejpam-1242	38	50	by	by	ADP
ejpam-1242	38	51	speck(m	speck(m	NOUN
ejpam-1242	38	52	)	)	PUNCT
ejpam-1242	38	53	.	.	PUNCT
ejpam-1242	39	1	we	we	PRON
ejpam-1242	39	2	define	define	VERB
ejpam-1242	39	3	k	k	NOUN
ejpam-1242	39	4	-	-	NOUN
ejpam-1242	39	5	ideals	ideal	NOUN
ejpam-1242	39	6	and	and	CCONJ
ejpam-1242	39	7	prime	prime	ADJ
ejpam-1242	39	8	k	k	NOUN
ejpam-1242	39	9	-	-	NOUN
ejpam-1242	39	10	ideals	ideal	NOUN
ejpam-1242	39	11	of	of	ADP
ejpam-1242	39	12	a	a	DET
ejpam-1242	39	13	semiring	semire	VERB
ejpam-1242	39	14	r	r	NOUN
ejpam-1242	39	15	in	in	ADP
ejpam-1242	39	16	a	a	DET
ejpam-1242	39	17	similar	similar	ADJ
ejpam-1242	39	18	fashion	fashion	NOUN
ejpam-1242	39	19	.	.	PUNCT
ejpam-1242	40	1	(	(	PUNCT
ejpam-1242	40	2	3	3	X
ejpam-1242	40	3	)	)	PUNCT
ejpam-1242	40	4	a	a	DET
ejpam-1242	40	5	subsemimodule	subsemimodule	NOUN
ejpam-1242	40	6	l	l	NOUN
ejpam-1242	40	7	of	of	ADP
ejpam-1242	40	8	m	m	PROPN
ejpam-1242	40	9	is	be	AUX
ejpam-1242	40	10	said	say	VERB
ejpam-1242	40	11	to	to	PART
ejpam-1242	40	12	be	be	AUX
ejpam-1242	40	13	semiprime	semiprime	NOUN
ejpam-1242	40	14	if	if	SCONJ
ejpam-1242	40	15	l	l	NOUN
ejpam-1242	40	16	is	be	AUX
ejpam-1242	40	17	an	an	DET
ejpam-1242	40	18	intersection	intersection	NOUN
ejpam-1242	40	19	of	of	ADP
ejpam-1242	40	20	prime	prime	ADJ
ejpam-1242	40	21	ksubsemimodules	ksubsemimodule	NOUN
ejpam-1242	40	22	of	of	ADP
ejpam-1242	40	23	m.	m.	NOUN
ejpam-1242	40	24	(	(	PUNCT
ejpam-1242	40	25	4	4	X
ejpam-1242	40	26	)	)	PUNCT
ejpam-1242	40	27	let	let	VERB
ejpam-1242	40	28	t	t	NOUN
ejpam-1242	40	29	be	be	AUX
ejpam-1242	40	30	a	a	DET
ejpam-1242	40	31	proper	proper	ADJ
ejpam-1242	40	32	subsemimodule	subsemimodule	NOUN
ejpam-1242	40	33	of	of	ADP
ejpam-1242	40	34	an	an	DET
ejpam-1242	40	35	r	r	NOUN
ejpam-1242	40	36	-	-	PUNCT
ejpam-1242	40	37	semimodule	semimodule	NOUN
ejpam-1242	40	38	m.	m.	NOUN
ejpam-1242	40	39	then	then	ADV
ejpam-1242	40	40	the	the	DET
ejpam-1242	40	41	prime	prime	ADJ
ejpam-1242	40	42	radical	radical	ADJ
ejpam-1242	40	43	rad(t	rad(t	PROPN
ejpam-1242	40	44	)	)	PUNCT
ejpam-1242	40	45	of	of	ADP
ejpam-1242	40	46	t	t	PROPN
ejpam-1242	40	47	(	(	PUNCT
ejpam-1242	40	48	in	in	ADP
ejpam-1242	40	49	m	m	NOUN
ejpam-1242	40	50	)	)	PUNCT
ejpam-1242	40	51	is	be	AUX
ejpam-1242	40	52	the	the	DET
ejpam-1242	40	53	intersection	intersection	NOUN
ejpam-1242	40	54	of	of	ADP
ejpam-1242	40	55	all	all	DET
ejpam-1242	40	56	prime	prime	ADJ
ejpam-1242	40	57	k	k	NOUN
ejpam-1242	40	58	-	-	NOUN
ejpam-1242	40	59	subsemimodules	subsemimodules	NOUN
ejpam-1242	40	60	of	of	ADP
ejpam-1242	40	61	m	m	AUX
ejpam-1242	40	62	containing	contain	VERB
ejpam-1242	40	63	t	t	PROPN
ejpam-1242	40	64	or	or	CCONJ
ejpam-1242	40	65	,	,	PUNCT
ejpam-1242	40	66	in	in	ADP
ejpam-1242	40	67	case	case	NOUN
ejpam-1242	40	68	there	there	PRON
ejpam-1242	40	69	are	be	VERB
ejpam-1242	40	70	no	no	DET
ejpam-1242	40	71	such	such	ADJ
ejpam-1242	40	72	prime	prime	ADJ
ejpam-1242	40	73	k	k	NOUN
ejpam-1242	40	74	-	-	NOUN
ejpam-1242	40	75	subsemimodules	subsemimodule	NOUN
ejpam-1242	40	76	,	,	PUNCT
ejpam-1242	40	77	rad(t	rad(t	PROPN
ejpam-1242	40	78	)	)	PUNCT
ejpam-1242	40	79	is	be	AUX
ejpam-1242	40	80	m.	m.	NOUN
ejpam-1242	40	81	note	note	NOUN
ejpam-1242	40	82	that	that	SCONJ
ejpam-1242	40	83	t	t	PROPN
ejpam-1242	40	84	⊆	⊆	NUM
ejpam-1242	40	85	rad(t	rad(t	PROPN
ejpam-1242	40	86	)	)	PUNCT
ejpam-1242	40	87	and	and	CCONJ
ejpam-1242	40	88	that	that	SCONJ
ejpam-1242	40	89	rad(t	rad(t	ADJ
ejpam-1242	40	90	)	)	PUNCT
ejpam-1242	41	1	=	=	SYM
ejpam-1242	41	2	m	m	NOUN
ejpam-1242	41	3	or	or	CCONJ
ejpam-1242	41	4	rad(t	rad(t	ADJ
ejpam-1242	41	5	)	)	PUNCT
ejpam-1242	41	6	is	be	AUX
ejpam-1242	41	7	semiprime	semiprime	NOUN
ejpam-1242	41	8	k	k	X
ejpam-1242	41	9	-	-	PUNCT
ejpam-1242	41	10	subsemimodule	subsemimodule	NOUN
ejpam-1242	41	11	of	of	ADP
ejpam-1242	41	12	m.	m.	NOUN
ejpam-1242	41	13	(	(	PUNCT
ejpam-1242	41	14	5	5	NUM
ejpam-1242	41	15	)	)	PUNCT
ejpam-1242	41	16	a	a	DET
ejpam-1242	41	17	prime	prime	ADJ
ejpam-1242	41	18	subsemimodule	subsemimodule	NOUN
ejpam-1242	41	19	n	n	PROPN
ejpam-1242	41	20	of	of	ADP
ejpam-1242	41	21	m	m	PROPN
ejpam-1242	41	22	is	be	AUX
ejpam-1242	41	23	called	call	VERB
ejpam-1242	41	24	extraordinary	extraordinary	ADJ
ejpam-1242	41	25	if	if	SCONJ
ejpam-1242	41	26	whenever	whenever	SCONJ
ejpam-1242	41	27	a	a	PRON
ejpam-1242	41	28	and	and	CCONJ
ejpam-1242	41	29	b	b	NOUN
ejpam-1242	41	30	are	be	AUX
ejpam-1242	41	31	semiprime	semiprime	NOUN
ejpam-1242	41	32	k	k	X
ejpam-1242	41	33	-	-	PUNCT
ejpam-1242	41	34	subsemimodules	subsemimodule	NOUN
ejpam-1242	41	35	of	of	ADP
ejpam-1242	41	36	m	m	PROPN
ejpam-1242	41	37	with	with	ADP
ejpam-1242	41	38	a∩	a∩	PROPN
ejpam-1242	41	39	b	b	PROPN
ejpam-1242	41	40	⊆	⊆	NUM
ejpam-1242	41	41	n	n	CCONJ
ejpam-1242	41	42	,	,	PUNCT
ejpam-1242	41	43	then	then	ADV
ejpam-1242	41	44	a⊆	a⊆	VERB
ejpam-1242	41	45	n	n	PRON
ejpam-1242	41	46	or	or	CCONJ
ejpam-1242	41	47	b	b	NOUN
ejpam-1242	41	48	⊆	⊆	NUM
ejpam-1242	41	49	n.	n.	NOUN
ejpam-1242	41	50	(	(	PUNCT
ejpam-1242	41	51	6	6	NUM
ejpam-1242	41	52	)	)	PUNCT
ejpam-1242	41	53	an	an	DET
ejpam-1242	41	54	r	r	NOUN
ejpam-1242	41	55	-	-	PUNCT
ejpam-1242	41	56	semimodule	semimodule	NOUN
ejpam-1242	41	57	m	m	NOUN
ejpam-1242	41	58	is	be	AUX
ejpam-1242	41	59	called	call	VERB
ejpam-1242	41	60	multiplication	multiplication	NOUN
ejpam-1242	41	61	semimodule	semimodule	NOUN
ejpam-1242	41	62	provided	provide	VERB
ejpam-1242	41	63	that	that	SCONJ
ejpam-1242	41	64	for	for	ADP
ejpam-1242	41	65	every	every	DET
ejpam-1242	41	66	subsemimodule	subsemimodule	NOUN
ejpam-1242	41	67	n	n	PROPN
ejpam-1242	41	68	of	of	ADP
ejpam-1242	41	69	m	m	PRON
ejpam-1242	41	70	there	there	PRON
ejpam-1242	41	71	exists	exist	VERB
ejpam-1242	41	72	an	an	DET
ejpam-1242	41	73	ideal	ideal	NOUN
ejpam-1242	41	74	i	i	PRON
ejpam-1242	41	75	of	of	ADP
ejpam-1242	41	76	r	r	NOUN
ejpam-1242	42	1	such	such	ADJ
ejpam-1242	42	2	that	that	SCONJ
ejpam-1242	42	3	n	n	NOUN
ejpam-1242	42	4	=	=	SYM
ejpam-1242	42	5	i	i	PROPN
ejpam-1242	42	6	m.	m.	PROPN
ejpam-1242	42	7	s.	s.	PROPN
ejpam-1242	42	8	atani	atani	PROPN
ejpam-1242	42	9	,	,	PUNCT
ejpam-1242	42	10	r.	r.	PROPN
ejpam-1242	42	11	atrani	atrani	PROPN
ejpam-1242	42	12	,	,	PUNCT
ejpam-1242	42	13	ü.	ü.	NOUN
ejpam-1242	42	14	tekir	tekir	PROPN
ejpam-1242	42	15	/	/	SYM
ejpam-1242	42	16	eur	eur	PROPN
ejpam-1242	42	17	.	.	PUNCT
ejpam-1242	43	1	j.	j.	PROPN
ejpam-1242	43	2	pure	pure	PROPN
ejpam-1242	43	3	appl	appl	PROPN
ejpam-1242	43	4	.	.	PROPN
ejpam-1242	43	5	math	math	PROPN
ejpam-1242	43	6	,	,	PUNCT
ejpam-1242	43	7	4	4	NUM
ejpam-1242	43	8	(	(	PUNCT
ejpam-1242	43	9	2011	2011	NUM
ejpam-1242	43	10	)	)	PUNCT
ejpam-1242	43	11	,	,	PUNCT
ejpam-1242	43	12	251	251	NUM
ejpam-1242	43	13	-	-	SYM
ejpam-1242	43	14	265	265	NUM
ejpam-1242	43	15	253	253	NUM
ejpam-1242	43	16	(	(	PUNCT
ejpam-1242	43	17	7	7	NUM
ejpam-1242	43	18	)	)	PUNCT
ejpam-1242	43	19	a	a	DET
ejpam-1242	43	20	proper	proper	ADJ
ejpam-1242	43	21	ideal	ideal	NOUN
ejpam-1242	43	22	i	i	PRON
ejpam-1242	43	23	of	of	ADP
ejpam-1242	43	24	a	a	DET
ejpam-1242	43	25	semiring	semiring	NOUN
ejpam-1242	43	26	r	r	NOUN
ejpam-1242	43	27	is	be	AUX
ejpam-1242	43	28	said	say	VERB
ejpam-1242	43	29	to	to	PART
ejpam-1242	43	30	be	be	AUX
ejpam-1242	43	31	strong	strong	ADJ
ejpam-1242	43	32	ideal	ideal	NOUN
ejpam-1242	43	33	(	(	PUNCT
ejpam-1242	43	34	or	or	CCONJ
ejpam-1242	43	35	strongly	strongly	ADV
ejpam-1242	43	36	zero	zero	NUM
ejpam-1242	43	37	-	-	PUNCT
ejpam-1242	43	38	sum	sum	NOUN
ejpam-1242	43	39	ideal	ideal	NOUN
ejpam-1242	43	40	)	)	PUNCT
ejpam-1242	43	41	,	,	PUNCT
ejpam-1242	43	42	if	if	SCONJ
ejpam-1242	43	43	for	for	ADP
ejpam-1242	43	44	each	each	DET
ejpam-1242	43	45	a	a	DET
ejpam-1242	43	46	∈	∈	NOUN
ejpam-1242	43	47	i	i	PRON
ejpam-1242	43	48	there	there	PRON
ejpam-1242	43	49	exists	exist	VERB
ejpam-1242	43	50	b	b	X
ejpam-1242	43	51	∈	∈	PROPN
ejpam-1242	43	52	i	i	PRON
ejpam-1242	43	53	such	such	ADJ
ejpam-1242	43	54	that	that	SCONJ
ejpam-1242	43	55	a+	a+	PRON
ejpam-1242	43	56	b	b	X
ejpam-1242	43	57	=	=	SYM
ejpam-1242	43	58	0	0	PROPN
ejpam-1242	43	59	(	(	PUNCT
ejpam-1242	43	60	see	see	VERB
ejpam-1242	43	61	[	[	X
ejpam-1242	43	62	14	14	NUM
ejpam-1242	43	63	,	,	PUNCT
ejpam-1242	43	64	example	example	NOUN
ejpam-1242	43	65	2.3	2.3	NUM
ejpam-1242	43	66	]	]	PUNCT
ejpam-1242	43	67	and	and	CCONJ
ejpam-1242	44	1	[	[	X
ejpam-1242	44	2	11	11	NUM
ejpam-1242	44	3	]	]	NUM
ejpam-1242	44	4	)	)	PUNCT
ejpam-1242	44	5	.	.	PUNCT
ejpam-1242	45	1	quotient	quotient	PROPN
ejpam-1242	45	2	semirings	semiring	NOUN
ejpam-1242	45	3	are	be	AUX
ejpam-1242	45	4	determined	determine	VERB
ejpam-1242	45	5	by	by	ADP
ejpam-1242	45	6	equivalence	equivalence	NOUN
ejpam-1242	45	7	relations	relation	NOUN
ejpam-1242	45	8	rather	rather	ADV
ejpam-1242	45	9	than	than	ADP
ejpam-1242	45	10	by	by	ADP
ejpam-1242	45	11	ideals	ideal	NOUN
ejpam-1242	45	12	as	as	ADP
ejpam-1242	45	13	in	in	ADP
ejpam-1242	45	14	the	the	DET
ejpam-1242	45	15	ring	ring	NOUN
ejpam-1242	45	16	case	case	NOUN
ejpam-1242	45	17	.	.	PUNCT
ejpam-1242	46	1	allen	allen	PROPN
ejpam-1242	47	1	[	[	X
ejpam-1242	47	2	1	1	NUM
ejpam-1242	47	3	]	]	PUNCT
ejpam-1242	47	4	has	have	AUX
ejpam-1242	47	5	presented	present	VERB
ejpam-1242	47	6	the	the	DET
ejpam-1242	47	7	notion	notion	NOUN
ejpam-1242	47	8	of	of	ADP
ejpam-1242	47	9	a	a	DET
ejpam-1242	47	10	partitioning	partition	VERB
ejpam-1242	47	11	ideal	ideal	NOUN
ejpam-1242	47	12	(=	(=	ADP
ejpam-1242	47	13	q	q	ADJ
ejpam-1242	47	14	-	-	PUNCT
ejpam-1242	47	15	ideal	ideal	ADJ
ejpam-1242	47	16	)	)	PUNCT
ejpam-1242	48	1	i	i	PRON
ejpam-1242	48	2	in	in	ADP
ejpam-1242	48	3	the	the	DET
ejpam-1242	48	4	semiring	semire	VERB
ejpam-1242	48	5	r	r	NOUN
ejpam-1242	48	6	and	and	CCONJ
ejpam-1242	48	7	constructed	construct	VERB
ejpam-1242	48	8	the	the	DET
ejpam-1242	48	9	quotient	quotient	NOUN
ejpam-1242	48	10	semiring	semire	VERB
ejpam-1242	48	11	r	r	NOUN
ejpam-1242	48	12	/	/	SYM
ejpam-1242	48	13	i	i	NOUN
ejpam-1242	48	14	.	.	PUNCT
ejpam-1242	49	1	if	if	SCONJ
ejpam-1242	49	2	i	i	PRON
ejpam-1242	49	3	is	be	AUX
ejpam-1242	49	4	an	an	DET
ejpam-1242	49	5	ideal	ideal	NOUN
ejpam-1242	49	6	of	of	ADP
ejpam-1242	49	7	a	a	DET
ejpam-1242	49	8	semiring	semiring	NOUN
ejpam-1242	49	9	r	r	NOUN
ejpam-1242	49	10	,	,	PUNCT
ejpam-1242	49	11	we	we	PRON
ejpam-1242	49	12	define	define	VERB
ejpam-1242	49	13	a	a	DET
ejpam-1242	49	14	relation	relation	NOUN
ejpam-1242	49	15	∼	∼	NOUN
ejpam-1242	49	16	on	on	ADP
ejpam-1242	49	17	r	r	NOUN
ejpam-1242	49	18	,	,	PUNCT
ejpam-1242	49	19	given	give	VERB
ejpam-1242	49	20	by	by	ADP
ejpam-1242	49	21	r1	r1	NOUN
ejpam-1242	49	22	∼	∼	NOUN
ejpam-1242	49	23	r2	r2	NOUN
ejpam-1242	49	24	if	if	SCONJ
ejpam-1242	49	25	and	and	CCONJ
ejpam-1242	49	26	only	only	ADV
ejpam-1242	49	27	if	if	SCONJ
ejpam-1242	49	28	there	there	PRON
ejpam-1242	49	29	exist	exist	VERB
ejpam-1242	49	30	a1	a1	NOUN
ejpam-1242	49	31	,	,	PUNCT
ejpam-1242	49	32	a2	a2	PROPN
ejpam-1242	49	33	∈	∈	PROPN
ejpam-1242	49	34	i	i	PRON
ejpam-1242	49	35	satisfying	satisfy	VERB
ejpam-1242	49	36	r1+a1	r1+a1	PROPN
ejpam-1242	49	37	=	=	SYM
ejpam-1242	49	38	r2+a2	r2+a2	PROPN
ejpam-1242	49	39	.	.	PUNCT
ejpam-1242	50	1	then	then	ADV
ejpam-1242	50	2	∼	∼	NOUN
ejpam-1242	50	3	is	be	AUX
ejpam-1242	50	4	an	an	DET
ejpam-1242	50	5	equivalence	equivalence	NOUN
ejpam-1242	50	6	relation	relation	NOUN
ejpam-1242	50	7	on	on	ADP
ejpam-1242	50	8	r	r	NOUN
ejpam-1242	50	9	,	,	PUNCT
ejpam-1242	50	10	and	and	CCONJ
ejpam-1242	50	11	we	we	PRON
ejpam-1242	50	12	denote	denote	VERB
ejpam-1242	50	13	the	the	DET
ejpam-1242	50	14	equivalence	equivalence	NOUN
ejpam-1242	50	15	class	class	NOUN
ejpam-1242	50	16	of	of	ADP
ejpam-1242	50	17	r	r	NOUN
ejpam-1242	50	18	by	by	ADP
ejpam-1242	50	19	r+	r+	PUNCT
ejpam-1242	50	20	i	i	PRON
ejpam-1242	50	21	and	and	CCONJ
ejpam-1242	50	22	these	these	DET
ejpam-1242	50	23	collection	collection	NOUN
ejpam-1242	50	24	of	of	ADP
ejpam-1242	50	25	all	all	DET
ejpam-1242	50	26	equivalence	equivalence	NOUN
ejpam-1242	50	27	classes	class	NOUN
ejpam-1242	50	28	by	by	ADP
ejpam-1242	50	29	r	r	PROPN
ejpam-1242	50	30	/	/	SYM
ejpam-1242	50	31	i	i	PROPN
ejpam-1242	50	32	.	.	PUNCT
ejpam-1242	51	1	golan	golan	PROPN
ejpam-1242	51	2	shows	show	VERB
ejpam-1242	51	3	that	that	SCONJ
ejpam-1242	51	4	r	r	NOUN
ejpam-1242	51	5	/	/	SYM
ejpam-1242	51	6	i	i	PRON
ejpam-1242	51	7	is	be	AUX
ejpam-1242	51	8	a	a	DET
ejpam-1242	51	9	semiring	semiring	NOUN
ejpam-1242	51	10	with	with	ADP
ejpam-1242	51	11	(	(	PUNCT
ejpam-1242	51	12	r	r	NOUN
ejpam-1242	51	13	+	+	NOUN
ejpam-1242	51	14	i)+	i)+	X
ejpam-1242	51	15	(	(	PUNCT
ejpam-1242	51	16	s+	s+	X
ejpam-1242	51	17	i	i	X
ejpam-1242	51	18	)	)	PUNCT
ejpam-1242	52	1	=	=	PUNCT
ejpam-1242	52	2	r	r	NOUN
ejpam-1242	53	1	+	+	PUNCT
ejpam-1242	53	2	s+	s+	PUNCT
ejpam-1242	53	3	i	i	PRON
ejpam-1242	53	4	and	and	CCONJ
ejpam-1242	53	5	(	(	PUNCT
ejpam-1242	53	6	r	r	NOUN
ejpam-1242	53	7	+	+	CCONJ
ejpam-1242	53	8	i)(s+	i)(s+	PROPN
ejpam-1242	53	9	i	i	NOUN
ejpam-1242	53	10	)	)	PUNCT
ejpam-1242	54	1	=	=	VERB
ejpam-1242	54	2	rs+	rs+	VERB
ejpam-1242	54	3	i	i	PRON
ejpam-1242	54	4	.	.	PUNCT
ejpam-1242	55	1	the	the	DET
ejpam-1242	55	2	semiring	semire	VERB
ejpam-1242	55	3	r	r	NOUN
ejpam-1242	55	4	/	/	SYM
ejpam-1242	55	5	i	i	PRON
ejpam-1242	55	6	has	have	VERB
ejpam-1242	55	7	additive	additive	ADJ
ejpam-1242	55	8	identity	identity	NOUN
ejpam-1242	55	9	0	0	NUM
ejpam-1242	56	1	+	+	NUM
ejpam-1242	56	2	i	i	PRON
ejpam-1242	56	3	and	and	CCONJ
ejpam-1242	56	4	multiplicative	multiplicative	ADJ
ejpam-1242	56	5	identity	identity	NOUN
ejpam-1242	56	6	1	1	NUM
ejpam-1242	56	7	+	+	NUM
ejpam-1242	57	1	i	i	PRON
ejpam-1242	57	2	(	(	PUNCT
ejpam-1242	57	3	see	see	VERB
ejpam-1242	57	4	golan	golan	PROPN
ejpam-1242	57	5	[	[	X
ejpam-1242	57	6	16	16	NUM
ejpam-1242	57	7	]	]	PUNCT
ejpam-1242	57	8	)	)	PUNCT
ejpam-1242	57	9	.	.	PUNCT
ejpam-1242	58	1	in	in	ADP
ejpam-1242	58	2	this	this	DET
ejpam-1242	58	3	paper	paper	NOUN
ejpam-1242	58	4	,	,	PUNCT
ejpam-1242	58	5	we	we	PRON
ejpam-1242	58	6	will	will	AUX
ejpam-1242	58	7	follow	follow	VERB
ejpam-1242	58	8	golan	golan	PROPN
ejpam-1242	58	9	’s	’s	PART
ejpam-1242	58	10	terminology	terminology	NOUN
ejpam-1242	58	11	for	for	ADP
ejpam-1242	58	12	quotient	quotient	NOUN
ejpam-1242	58	13	semirings	semiring	NOUN
ejpam-1242	58	14	.	.	PUNCT
ejpam-1242	59	1	quotient	quotient	PROPN
ejpam-1242	59	2	semimodules	semimodule	NOUN
ejpam-1242	59	3	over	over	ADP
ejpam-1242	59	4	a	a	DET
ejpam-1242	59	5	semiring	semiring	NOUN
ejpam-1242	59	6	r	r	NOUN
ejpam-1242	59	7	have	have	AUX
ejpam-1242	59	8	already	already	ADV
ejpam-1242	59	9	been	be	AUX
ejpam-1242	59	10	introduced	introduce	VERB
ejpam-1242	59	11	and	and	CCONJ
ejpam-1242	59	12	studied	study	VERB
ejpam-1242	59	13	by	by	ADP
ejpam-1242	59	14	present	present	ADJ
ejpam-1242	59	15	authors	author	NOUN
ejpam-1242	59	16	in	in	ADP
ejpam-1242	59	17	[	[	X
ejpam-1242	59	18	13	13	NUM
ejpam-1242	59	19	]	]	PUNCT
ejpam-1242	59	20	.	.	PUNCT
ejpam-1242	60	1	chaudhari	chaudhari	PROPN
ejpam-1242	60	2	and	and	CCONJ
ejpam-1242	60	3	bonde	bonde	VERB
ejpam-1242	60	4	[	[	X
ejpam-1242	60	5	7	7	NUM
ejpam-1242	60	6	]	]	PUNCT
ejpam-1242	60	7	extended	extend	VERB
ejpam-1242	60	8	the	the	DET
ejpam-1242	60	9	definition	definition	NOUN
ejpam-1242	60	10	of	of	ADP
ejpam-1242	60	11	qm	qm	PROPN
ejpam-1242	60	12	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	60	13	of	of	ADP
ejpam-1242	60	14	a	a	DET
ejpam-1242	60	15	semimodule	semimodule	NOUN
ejpam-1242	60	16	to	to	ADP
ejpam-1242	60	17	a	a	DET
ejpam-1242	60	18	more	more	ADV
ejpam-1242	60	19	general	general	ADJ
ejpam-1242	60	20	case	case	NOUN
ejpam-1242	60	21	:	:	PUNCT
ejpam-1242	60	22	a	a	DET
ejpam-1242	60	23	subsemimodule	subsemimodule	NOUN
ejpam-1242	60	24	n	n	NOUN
ejpam-1242	60	25	of	of	ADP
ejpam-1242	60	26	a	a	DET
ejpam-1242	60	27	semimodule	semimodule	NOUN
ejpam-1242	60	28	m	m	VERB
ejpam-1242	60	29	over	over	ADP
ejpam-1242	60	30	a	a	DET
ejpam-1242	60	31	semiring	semire	VERB
ejpam-1242	60	32	r	r	NOUN
ejpam-1242	60	33	is	be	AUX
ejpam-1242	60	34	called	call	VERB
ejpam-1242	60	35	a	a	DET
ejpam-1242	60	36	partitioning	partition	VERB
ejpam-1242	60	37	subsemimodule	subsemimodule	NOUN
ejpam-1242	60	38	(=	(=	NOUN
ejpam-1242	60	39	qm	qm	PROPN
ejpam-1242	60	40	-subsemimodule	-subsemimodule	PROPN
ejpam-1242	60	41	)	)	PUNCT
ejpam-1242	60	42	if	if	SCONJ
ejpam-1242	60	43	there	there	PRON
ejpam-1242	60	44	exists	exist	VERB
ejpam-1242	60	45	a	a	DET
ejpam-1242	60	46	subset	subset	ADJ
ejpam-1242	60	47	qm	qm	NOUN
ejpam-1242	60	48	of	of	ADP
ejpam-1242	60	49	m	m	PROPN
ejpam-1242	60	50	such	such	ADJ
ejpam-1242	60	51	that	that	SCONJ
ejpam-1242	60	52	m	m	VERB
ejpam-1242	60	53	=	=	SYM
ejpam-1242	60	54	∪{q+	∪{q+	PROPN
ejpam-1242	60	55	n	n	X
ejpam-1242	60	56	:	:	PUNCT
ejpam-1242	60	57	q	q	PROPN
ejpam-1242	60	58	∈	∈	PROPN
ejpam-1242	60	59	qm	qm	PROPN
ejpam-1242	60	60	}	}	PUNCT
ejpam-1242	60	61	and	and	CCONJ
ejpam-1242	60	62	if	if	SCONJ
ejpam-1242	60	63	q1,q2	q1,q2	PROPN
ejpam-1242	60	64	∈	∈	PROPN
ejpam-1242	60	65	qm	qm	NOUN
ejpam-1242	60	66	then	then	ADV
ejpam-1242	60	67	(	(	PUNCT
ejpam-1242	60	68	q1	q1	NOUN
ejpam-1242	60	69	+	+	CCONJ
ejpam-1242	60	70	n)∩	n)∩	NOUN
ejpam-1242	60	71	(	(	PUNCT
ejpam-1242	60	72	q2	q2	NOUN
ejpam-1242	60	73	+	+	NOUN
ejpam-1242	60	74	n	n	CCONJ
ejpam-1242	60	75	)	)	PUNCT
ejpam-1242	60	76	6=	6=	NUM
ejpam-1242	60	77	;	;	PUNCT
ejpam-1242	60	78	if	if	SCONJ
ejpam-1242	60	79	and	and	CCONJ
ejpam-1242	60	80	only	only	ADV
ejpam-1242	60	81	if	if	SCONJ
ejpam-1242	60	82	q1	q1	PROPN
ejpam-1242	60	83	=	=	SYM
ejpam-1242	60	84	q2	q2	PROPN
ejpam-1242	60	85	.	.	PUNCT
ejpam-1242	61	1	let	let	VERB
ejpam-1242	61	2	n	n	PRON
ejpam-1242	61	3	be	be	AUX
ejpam-1242	61	4	a	a	DET
ejpam-1242	61	5	qm	qm	PROPN
ejpam-1242	61	6	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	61	7	of	of	ADP
ejpam-1242	61	8	m	m	PRON
ejpam-1242	61	9	and	and	CCONJ
ejpam-1242	61	10	let	let	VERB
ejpam-1242	61	11	m	m	NOUN
ejpam-1242	61	12	/	/	SYM
ejpam-1242	61	13	n	n	NOUN
ejpam-1242	61	14	=	=	PRON
ejpam-1242	61	15	{	{	PUNCT
ejpam-1242	61	16	q	q	PROPN
ejpam-1242	61	17	+	+	CCONJ
ejpam-1242	61	18	n	n	X
ejpam-1242	61	19	:	:	PUNCT
ejpam-1242	61	20	q	q	PROPN
ejpam-1242	61	21	∈	∈	PROPN
ejpam-1242	61	22	qm	qm	PROPN
ejpam-1242	61	23	}	}	PUNCT
ejpam-1242	61	24	.	.	PUNCT
ejpam-1242	62	1	then	then	ADV
ejpam-1242	62	2	m	m	PROPN
ejpam-1242	62	3	/	/	SYM
ejpam-1242	62	4	n	n	PRON
ejpam-1242	62	5	forms	form	VERB
ejpam-1242	62	6	an	an	DET
ejpam-1242	62	7	r	r	NOUN
ejpam-1242	62	8	-	-	PUNCT
ejpam-1242	62	9	semimodule	semimodule	NOUN
ejpam-1242	62	10	under	under	ADP
ejpam-1242	62	11	the	the	DET
ejpam-1242	62	12	operations	operation	NOUN
ejpam-1242	62	13	⊕	⊕	PROPN
ejpam-1242	62	14	and	and	CCONJ
ejpam-1242	62	15	⊙	⊙	NOUN
ejpam-1242	62	16	defined	define	VERB
ejpam-1242	62	17	as	as	SCONJ
ejpam-1242	62	18	follows	follow	VERB
ejpam-1242	62	19	:	:	PUNCT
ejpam-1242	62	20	(	(	PUNCT
ejpam-1242	62	21	q1	q1	NOUN
ejpam-1242	62	22	+	+	CCONJ
ejpam-1242	62	23	n)⊕	n)⊕	NOUN
ejpam-1242	62	24	(	(	PUNCT
ejpam-1242	62	25	q2	q2	NOUN
ejpam-1242	62	26	+	+	CCONJ
ejpam-1242	62	27	n	n	CCONJ
ejpam-1242	62	28	)	)	PUNCT
ejpam-1242	62	29	=	=	SYM
ejpam-1242	62	30	q3	q3	NOUN
ejpam-1242	62	31	+	+	CCONJ
ejpam-1242	62	32	n	n	CCONJ
ejpam-1242	62	33	,	,	PUNCT
ejpam-1242	62	34	where	where	SCONJ
ejpam-1242	62	35	q3	q3	PROPN
ejpam-1242	62	36	∈	∈	PROPN
ejpam-1242	62	37	qm	qm	PROPN
ejpam-1242	62	38	is	be	AUX
ejpam-1242	62	39	the	the	DET
ejpam-1242	62	40	unique	unique	ADJ
ejpam-1242	62	41	element	element	NOUN
ejpam-1242	62	42	such	such	ADJ
ejpam-1242	62	43	that	that	DET
ejpam-1242	62	44	q1	q1	PROPN
ejpam-1242	62	45	+	+	CCONJ
ejpam-1242	62	46	q2	q2	NOUN
ejpam-1242	62	47	+	+	CCONJ
ejpam-1242	62	48	n	n	CCONJ
ejpam-1242	62	49	⊆	⊆	NUM
ejpam-1242	62	50	q3	q3	NOUN
ejpam-1242	62	51	+	+	CCONJ
ejpam-1242	62	52	n	n	PROPN
ejpam-1242	62	53	and	and	CCONJ
ejpam-1242	62	54	r	r	NOUN
ejpam-1242	62	55	⊙	⊙	X
ejpam-1242	62	56	(	(	PUNCT
ejpam-1242	62	57	q1	q1	PROPN
ejpam-1242	62	58	+	+	CCONJ
ejpam-1242	62	59	n	n	CCONJ
ejpam-1242	62	60	)	)	PUNCT
ejpam-1242	62	61	=	=	SYM
ejpam-1242	62	62	q4	q4	PROPN
ejpam-1242	62	63	+	+	CCONJ
ejpam-1242	62	64	i	i	PRON
ejpam-1242	62	65	,	,	PUNCT
ejpam-1242	62	66	where	where	SCONJ
ejpam-1242	62	67	r	r	NOUN
ejpam-1242	62	68	∈	∈	PROPN
ejpam-1242	62	69	r	r	NOUN
ejpam-1242	62	70	and	and	CCONJ
ejpam-1242	62	71	q4	q4	PROPN
ejpam-1242	62	72	∈	∈	PROPN
ejpam-1242	62	73	qm	qm	PROPN
ejpam-1242	62	74	is	be	AUX
ejpam-1242	62	75	the	the	DET
ejpam-1242	62	76	unique	unique	ADJ
ejpam-1242	62	77	element	element	NOUN
ejpam-1242	62	78	such	such	DET
ejpam-1242	62	79	that	that	DET
ejpam-1242	62	80	rq1	rq1	NOUN
ejpam-1242	62	81	+	+	CCONJ
ejpam-1242	62	82	n	n	CCONJ
ejpam-1242	62	83	⊆	⊆	NUM
ejpam-1242	62	84	q4	q4	PROPN
ejpam-1242	62	85	+	+	CCONJ
ejpam-1242	62	86	n	n	CCONJ
ejpam-1242	62	87	.	.	PUNCT
ejpam-1242	63	1	this	this	DET
ejpam-1242	63	2	r	r	NOUN
ejpam-1242	63	3	-	-	PUNCT
ejpam-1242	63	4	semimodule	semimodule	NOUN
ejpam-1242	63	5	m	m	PROPN
ejpam-1242	63	6	/	/	SYM
ejpam-1242	63	7	n	n	PROPN
ejpam-1242	63	8	is	be	AUX
ejpam-1242	63	9	called	call	VERB
ejpam-1242	63	10	the	the	DET
ejpam-1242	63	11	quotient	quotient	NOUN
ejpam-1242	63	12	semimodule	semimodule	NOUN
ejpam-1242	63	13	of	of	ADP
ejpam-1242	63	14	m	m	PROPN
ejpam-1242	63	15	by	by	ADP
ejpam-1242	63	16	n	n	PRON
ejpam-1242	63	17	[	[	X
ejpam-1242	63	18	7	7	NUM
ejpam-1242	63	19	]	]	PUNCT
ejpam-1242	63	20	.	.	PUNCT
ejpam-1242	64	1	by	by	ADP
ejpam-1242	64	2	[	[	X
ejpam-1242	64	3	7	7	NUM
ejpam-1242	64	4	,	,	PUNCT
ejpam-1242	64	5	lemma	lemma	PROPN
ejpam-1242	64	6	2.3	2.3	NUM
ejpam-1242	64	7	]	]	PUNCT
ejpam-1242	64	8	,	,	PUNCT
ejpam-1242	64	9	there	there	PRON
ejpam-1242	64	10	exists	exist	VERB
ejpam-1242	64	11	a	a	DET
ejpam-1242	64	12	unique	unique	ADJ
ejpam-1242	64	13	element	element	NOUN
ejpam-1242	64	14	q0	q0	NOUN
ejpam-1242	64	15	∈	∈	PROPN
ejpam-1242	64	16	qm	qm	PROPN
ejpam-1242	64	17	such	such	ADJ
ejpam-1242	64	18	that	that	SCONJ
ejpam-1242	64	19	q0	q0	PROPN
ejpam-1242	64	20	+	+	CCONJ
ejpam-1242	64	21	n	n	NOUN
ejpam-1242	64	22	=	=	SYM
ejpam-1242	64	23	n	n	NOUN
ejpam-1242	64	24	.	.	PUNCT
ejpam-1242	65	1	thus	thus	ADV
ejpam-1242	65	2	q0	q0	VERB
ejpam-1242	65	3	+	+	CCONJ
ejpam-1242	65	4	n	n	NUM
ejpam-1242	65	5	is	be	AUX
ejpam-1242	65	6	the	the	DET
ejpam-1242	65	7	zero	zero	NUM
ejpam-1242	65	8	element	element	NOUN
ejpam-1242	65	9	of	of	ADP
ejpam-1242	65	10	m	m	PROPN
ejpam-1242	65	11	/	/	SYM
ejpam-1242	65	12	n	n	PROPN
ejpam-1242	65	13	.	.	PUNCT
ejpam-1242	66	1	also	also	ADV
ejpam-1242	66	2	,	,	PUNCT
ejpam-1242	66	3	[	[	X
ejpam-1242	66	4	7	7	NUM
ejpam-1242	66	5	,	,	PUNCT
ejpam-1242	66	6	theorem	theorem	VERB
ejpam-1242	66	7	2.4	2.4	NUM
ejpam-1242	66	8	]	]	PUNCT
ejpam-1242	66	9	show	show	VERB
ejpam-1242	66	10	that	that	SCONJ
ejpam-1242	66	11	the	the	DET
ejpam-1242	66	12	structure	structure	NOUN
ejpam-1242	66	13	(	(	PUNCT
ejpam-1242	66	14	m	m	NOUN
ejpam-1242	66	15	/	/	SYM
ejpam-1242	66	16	n	n	CCONJ
ejpam-1242	66	17	,	,	PUNCT
ejpam-1242	66	18	⊕,⊙	⊕,⊙	NUM
ejpam-1242	66	19	)	)	PUNCT
ejpam-1242	66	20	is	be	AUX
ejpam-1242	66	21	essentially	essentially	ADV
ejpam-1242	66	22	independent	independent	ADJ
ejpam-1242	66	23	of	of	ADP
ejpam-1242	66	24	qm	qm	PROPN
ejpam-1242	66	25	(	(	PUNCT
ejpam-1242	66	26	see	see	VERB
ejpam-1242	66	27	[	[	X
ejpam-1242	66	28	7	7	NUM
ejpam-1242	66	29	,	,	PUNCT
ejpam-1242	66	30	example	example	NOUN
ejpam-1242	66	31	2.6	2.6	NUM
ejpam-1242	66	32	]	]	PUNCT
ejpam-1242	66	33	)	)	PUNCT
ejpam-1242	66	34	.	.	PUNCT
ejpam-1242	67	1	lemma	lemma	PROPN
ejpam-1242	67	2	1	1	X
ejpam-1242	67	3	.	.	PUNCT
ejpam-1242	68	1	let	let	VERB
ejpam-1242	68	2	m	m	PRON
ejpam-1242	68	3	be	be	AUX
ejpam-1242	68	4	a	a	DET
ejpam-1242	68	5	semimodule	semimodule	NOUN
ejpam-1242	68	6	over	over	ADP
ejpam-1242	68	7	a	a	DET
ejpam-1242	68	8	semiring	semiring	NOUN
ejpam-1242	68	9	r.	r.	NOUN
ejpam-1242	68	10	if	if	SCONJ
ejpam-1242	68	11	{	{	PUNCT
ejpam-1242	68	12	mi}i∈λ	mi}i∈λ	PROPN
ejpam-1242	68	13	is	be	AUX
ejpam-1242	68	14	a	a	DET
ejpam-1242	68	15	collection	collection	NOUN
ejpam-1242	68	16	of	of	ADP
ejpam-1242	68	17	subsemimodules	subsemimodule	NOUN
ejpam-1242	68	18	of	of	ADP
ejpam-1242	68	19	m	m	PROPN
ejpam-1242	68	20	,	,	PUNCT
ejpam-1242	68	21	then	then	ADV
ejpam-1242	68	22	∑	∑	ADP
ejpam-1242	68	23	i∈λmi	i∈λmi	PROPN
ejpam-1242	68	24	and	and	CCONJ
ejpam-1242	68	25	⋂	⋂	PROPN
ejpam-1242	68	26	i∈λmi	i∈λmi	PROPN
ejpam-1242	68	27	are	be	AUX
ejpam-1242	68	28	subsemimodules	subsemimodule	NOUN
ejpam-1242	68	29	of	of	ADP
ejpam-1242	68	30	m.	m.	NOUN
ejpam-1242	68	31	3	3	NUM
ejpam-1242	68	32	.	.	PUNCT
ejpam-1242	69	1	properties	property	NOUN
ejpam-1242	69	2	of	of	ADP
ejpam-1242	69	3	strong	strong	ADJ
ejpam-1242	69	4	multiplication	multiplication	NOUN
ejpam-1242	69	5	semimodules	semimodule	NOUN
ejpam-1242	69	6	in	in	ADP
ejpam-1242	69	7	this	this	DET
ejpam-1242	69	8	section	section	NOUN
ejpam-1242	69	9	,	,	PUNCT
ejpam-1242	69	10	we	we	PRON
ejpam-1242	69	11	list	list	VERB
ejpam-1242	69	12	some	some	DET
ejpam-1242	69	13	basic	basic	ADJ
ejpam-1242	69	14	properties	property	NOUN
ejpam-1242	69	15	concerning	concern	VERB
ejpam-1242	69	16	very	very	ADV
ejpam-1242	69	17	strong	strong	ADJ
ejpam-1242	69	18	multiplication	multiplication	NOUN
ejpam-1242	69	19	semimodules	semimodule	NOUN
ejpam-1242	69	20	.	.	PUNCT
ejpam-1242	70	1	our	our	PRON
ejpam-1242	70	2	starting	starting	NOUN
ejpam-1242	70	3	point	point	NOUN
ejpam-1242	70	4	is	be	AUX
ejpam-1242	70	5	the	the	DET
ejpam-1242	70	6	following	follow	VERB
ejpam-1242	70	7	lemma	lemma	PROPN
ejpam-1242	70	8	.	.	PUNCT
ejpam-1242	71	1	lemma	lemma	PROPN
ejpam-1242	71	2	2	2	X
ejpam-1242	71	3	.	.	PUNCT
ejpam-1242	72	1	let	let	VERB
ejpam-1242	72	2	n	n	PRON
ejpam-1242	72	3	be	be	AUX
ejpam-1242	72	4	a	a	DET
ejpam-1242	72	5	qm	qm	PROPN
ejpam-1242	72	6	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	72	7	of	of	ADP
ejpam-1242	72	8	a	a	DET
ejpam-1242	72	9	semimodule	semimodule	NOUN
ejpam-1242	72	10	m	m	VERB
ejpam-1242	72	11	over	over	ADP
ejpam-1242	72	12	a	a	DET
ejpam-1242	72	13	semiring	semiring	NOUN
ejpam-1242	72	14	r.	r.	NOUN
ejpam-1242	72	15	if	if	SCONJ
ejpam-1242	72	16	t	t	PROPN
ejpam-1242	72	17	is	be	AUX
ejpam-1242	72	18	a	a	DET
ejpam-1242	72	19	k	k	NOUN
ejpam-1242	72	20	-	-	NOUN
ejpam-1242	72	21	subsemimodule	subsemimodule	NOUN
ejpam-1242	72	22	of	of	ADP
ejpam-1242	72	23	m	m	AUX
ejpam-1242	72	24	containing	contain	VERB
ejpam-1242	72	25	n	n	CCONJ
ejpam-1242	72	26	,	,	PUNCT
ejpam-1242	72	27	then	then	ADV
ejpam-1242	72	28	(	(	PUNCT
ejpam-1242	72	29	t	t	NOUN
ejpam-1242	72	30	:	:	PUNCT
ejpam-1242	72	31	r	r	NOUN
ejpam-1242	72	32	m	m	NOUN
ejpam-1242	72	33	)	)	PUNCT
ejpam-1242	73	1	=	=	SYM
ejpam-1242	73	2	(	(	PUNCT
ejpam-1242	73	3	t	t	PROPN
ejpam-1242	73	4	/	/	SYM
ejpam-1242	73	5	n	n	PROPN
ejpam-1242	73	6	:	:	PUNCT
ejpam-1242	73	7	r	r	NOUN
ejpam-1242	73	8	m	m	PROPN
ejpam-1242	73	9	/	/	SYM
ejpam-1242	73	10	n	n	CCONJ
ejpam-1242	73	11	)	)	PUNCT
ejpam-1242	73	12	.	.	PUNCT
ejpam-1242	74	1	proof	proof	NOUN
ejpam-1242	74	2	.	.	PUNCT
ejpam-1242	75	1	let	let	VERB
ejpam-1242	75	2	r	r	NOUN
ejpam-1242	75	3	∈	∈	PROPN
ejpam-1242	75	4	(	(	PUNCT
ejpam-1242	75	5	t	t	NOUN
ejpam-1242	75	6	:	:	PUNCT
ejpam-1242	75	7	m	m	PROPN
ejpam-1242	75	8	)	)	PUNCT
ejpam-1242	75	9	.	.	PUNCT
ejpam-1242	76	1	if	if	SCONJ
ejpam-1242	76	2	q	q	PROPN
ejpam-1242	76	3	+	+	CCONJ
ejpam-1242	76	4	n	n	CCONJ
ejpam-1242	76	5	∈	∈	PROPN
ejpam-1242	76	6	m	m	PROPN
ejpam-1242	76	7	/	/	SYM
ejpam-1242	76	8	n	n	PROPN
ejpam-1242	76	9	,	,	PUNCT
ejpam-1242	76	10	then	then	ADV
ejpam-1242	76	11	there	there	PRON
ejpam-1242	76	12	exists	exist	VERB
ejpam-1242	76	13	a	a	DET
ejpam-1242	76	14	unique	unique	ADJ
ejpam-1242	76	15	element	element	NOUN
ejpam-1242	76	16	q′	q′	NOUN
ejpam-1242	76	17	of	of	ADP
ejpam-1242	76	18	qm	qm	PROPN
ejpam-1242	76	19	such	such	ADJ
ejpam-1242	76	20	that	that	SCONJ
ejpam-1242	76	21	r(q	r(q	PROPN
ejpam-1242	76	22	+	+	NUM
ejpam-1242	76	23	n	n	CCONJ
ejpam-1242	76	24	)	)	PUNCT
ejpam-1242	77	1	=	=	SYM
ejpam-1242	77	2	q′	q′	NOUN
ejpam-1242	78	1	+	+	NUM
ejpam-1242	78	2	n	n	CCONJ
ejpam-1242	78	3	,	,	PUNCT
ejpam-1242	78	4	where	where	SCONJ
ejpam-1242	78	5	rq	rq	VERB
ejpam-1242	78	6	+	+	CCONJ
ejpam-1242	78	7	n	n	NUM
ejpam-1242	78	8	⊆	⊆	NUM
ejpam-1242	78	9	q′	q′	NOUN
ejpam-1242	78	10	+	+	CCONJ
ejpam-1242	78	11	n	n	CCONJ
ejpam-1242	78	12	;	;	PUNCT
ejpam-1242	78	13	so	so	CCONJ
ejpam-1242	78	14	q′	q′	NOUN
ejpam-1242	78	15	∈	∈	PROPN
ejpam-1242	78	16	t	t	PROPN
ejpam-1242	78	17	∩qm	∩qm	NOUN
ejpam-1242	78	18	since	since	SCONJ
ejpam-1242	78	19	rq	rq	INTJ
ejpam-1242	78	20	∈	∈	PROPN
ejpam-1242	78	21	t	t	PROPN
ejpam-1242	78	22	and	and	CCONJ
ejpam-1242	78	23	t	t	PROPN
ejpam-1242	78	24	is	be	AUX
ejpam-1242	78	25	a	a	DET
ejpam-1242	78	26	k	k	NOUN
ejpam-1242	78	27	-	-	NOUN
ejpam-1242	78	28	subsemimodule	subsemimodule	NOUN
ejpam-1242	78	29	.	.	PUNCT
ejpam-1242	79	1	thus	thus	ADV
ejpam-1242	79	2	(	(	PUNCT
ejpam-1242	79	3	t	t	NOUN
ejpam-1242	79	4	:	:	PUNCT
ejpam-1242	79	5	m	m	X
ejpam-1242	79	6	)	)	PUNCT
ejpam-1242	80	1	⊆	⊆	NUM
ejpam-1242	80	2	(	(	PUNCT
ejpam-1242	80	3	t	t	PROPN
ejpam-1242	80	4	/	/	SYM
ejpam-1242	80	5	n	n	PROPN
ejpam-1242	80	6	:	:	PUNCT
ejpam-1242	80	7	m	m	X
ejpam-1242	80	8	/	/	SYM
ejpam-1242	80	9	n	n	CCONJ
ejpam-1242	80	10	)	)	PUNCT
ejpam-1242	80	11	.	.	PUNCT
ejpam-1242	81	1	for	for	ADP
ejpam-1242	81	2	the	the	DET
ejpam-1242	81	3	other	other	ADJ
ejpam-1242	81	4	inclusion	inclusion	NOUN
ejpam-1242	81	5	,	,	PUNCT
ejpam-1242	81	6	assume	assume	VERB
ejpam-1242	81	7	that	that	SCONJ
ejpam-1242	81	8	a	a	DET
ejpam-1242	81	9	∈	∈	PROPN
ejpam-1242	81	10	(	(	PUNCT
ejpam-1242	81	11	t	t	PROPN
ejpam-1242	81	12	/	/	SYM
ejpam-1242	81	13	n	n	PROPN
ejpam-1242	81	14	:	:	PUNCT
ejpam-1242	81	15	m	m	X
ejpam-1242	81	16	/	/	SYM
ejpam-1242	81	17	n	n	CCONJ
ejpam-1242	81	18	)	)	PUNCT
ejpam-1242	81	19	and	and	CCONJ
ejpam-1242	81	20	m	m	PROPN
ejpam-1242	81	21	∈	∈	ADJ
ejpam-1242	81	22	m	m	NOUN
ejpam-1242	81	23	.	.	PUNCT
ejpam-1242	82	1	then	then	ADV
ejpam-1242	82	2	m	m	VERB
ejpam-1242	82	3	=	=	PROPN
ejpam-1242	82	4	q1	q1	PROPN
ejpam-1242	82	5	+	+	CCONJ
ejpam-1242	82	6	n	n	NOUN
ejpam-1242	82	7	for	for	ADP
ejpam-1242	82	8	some	some	DET
ejpam-1242	82	9	q1	q1	PROPN
ejpam-1242	82	10	∈	∈	PROPN
ejpam-1242	82	11	qm	qm	PROPN
ejpam-1242	82	12	and	and	CCONJ
ejpam-1242	82	13	n	n	CCONJ
ejpam-1242	82	14	∈	∈	PROPN
ejpam-1242	82	15	n	n	NOUN
ejpam-1242	82	16	;	;	PUNCT
ejpam-1242	82	17	so	so	CCONJ
ejpam-1242	82	18	there	there	PRON
ejpam-1242	82	19	is	be	VERB
ejpam-1242	82	20	a	a	DET
ejpam-1242	82	21	unique	unique	ADJ
ejpam-1242	82	22	element	element	NOUN
ejpam-1242	82	23	q2	q2	NOUN
ejpam-1242	82	24	of	of	ADP
ejpam-1242	82	25	qm	qm	PROPN
ejpam-1242	82	26	with	with	ADP
ejpam-1242	82	27	a(q1	a(q1	NOUN
ejpam-1242	82	28	+	+	CCONJ
ejpam-1242	82	29	n	n	CCONJ
ejpam-1242	82	30	)	)	PUNCT
ejpam-1242	82	31	=	=	SYM
ejpam-1242	82	32	q2	q2	NOUN
ejpam-1242	82	33	+	+	CCONJ
ejpam-1242	82	34	n	n	CCONJ
ejpam-1242	82	35	∈	∈	PROPN
ejpam-1242	82	36	t	t	PROPN
ejpam-1242	82	37	/	/	SYM
ejpam-1242	82	38	n	n	PROPN
ejpam-1242	82	39	,	,	PUNCT
ejpam-1242	82	40	where	where	SCONJ
ejpam-1242	82	41	aq1	aq1	NOUN
ejpam-1242	82	42	+	+	CCONJ
ejpam-1242	82	43	n	n	CCONJ
ejpam-1242	82	44	⊆	⊆	NUM
ejpam-1242	82	45	q2	q2	NOUN
ejpam-1242	82	46	+	+	CCONJ
ejpam-1242	82	47	n	n	CCONJ
ejpam-1242	82	48	.	.	PUNCT
ejpam-1242	83	1	thus	thus	ADV
ejpam-1242	83	2	t	t	AUX
ejpam-1242	83	3	being	be	AUX
ejpam-1242	83	4	a	a	DET
ejpam-1242	83	5	k	k	NOUN
ejpam-1242	83	6	-	-	NOUN
ejpam-1242	83	7	subsemimodule	subsemimodule	NOUN
ejpam-1242	83	8	gives	give	VERB
ejpam-1242	83	9	aq1	aq1	VERB
ejpam-1242	83	10	∈	∈	PROPN
ejpam-1242	83	11	t	t	PROPN
ejpam-1242	83	12	.	.	PUNCT
ejpam-1242	84	1	as	as	SCONJ
ejpam-1242	84	2	am	be	AUX
ejpam-1242	84	3	=	=	PUNCT
ejpam-1242	84	4	aq1	aq1	VERB
ejpam-1242	84	5	+	+	CCONJ
ejpam-1242	84	6	an	an	DET
ejpam-1242	84	7	∈	∈	PROPN
ejpam-1242	84	8	t	t	NOUN
ejpam-1242	84	9	,	,	PUNCT
ejpam-1242	84	10	we	we	PRON
ejpam-1242	84	11	have	have	VERB
ejpam-1242	84	12	a	a	DET
ejpam-1242	84	13	∈	∈	NOUN
ejpam-1242	84	14	(	(	PUNCT
ejpam-1242	84	15	t	t	NOUN
ejpam-1242	84	16	:	:	PUNCT
ejpam-1242	84	17	m	m	PROPN
ejpam-1242	84	18	)	)	PUNCT
ejpam-1242	84	19	.	.	PUNCT
ejpam-1242	85	1	s.	s.	PROPN
ejpam-1242	85	2	atani	atani	PROPN
ejpam-1242	85	3	,	,	PUNCT
ejpam-1242	85	4	r.	r.	PROPN
ejpam-1242	85	5	atrani	atrani	PROPN
ejpam-1242	85	6	,	,	PUNCT
ejpam-1242	85	7	ü.	ü.	NOUN
ejpam-1242	85	8	tekir	tekir	PROPN
ejpam-1242	85	9	/	/	SYM
ejpam-1242	85	10	eur	eur	PROPN
ejpam-1242	85	11	.	.	PUNCT
ejpam-1242	86	1	j.	j.	PROPN
ejpam-1242	86	2	pure	pure	PROPN
ejpam-1242	86	3	appl	appl	PROPN
ejpam-1242	86	4	.	.	PROPN
ejpam-1242	86	5	math	math	PROPN
ejpam-1242	86	6	,	,	PUNCT
ejpam-1242	86	7	4	4	NUM
ejpam-1242	86	8	(	(	PUNCT
ejpam-1242	86	9	2011	2011	NUM
ejpam-1242	86	10	)	)	PUNCT
ejpam-1242	86	11	,	,	PUNCT
ejpam-1242	86	12	251	251	NUM
ejpam-1242	86	13	-	-	SYM
ejpam-1242	86	14	265	265	NUM
ejpam-1242	86	15	254	254	NUM
ejpam-1242	86	16	theorem	theorem	NOUN
ejpam-1242	86	17	1	1	NUM
ejpam-1242	86	18	.	.	PUNCT
ejpam-1242	87	1	let	let	VERB
ejpam-1242	87	2	r	r	PRON
ejpam-1242	87	3	be	be	AUX
ejpam-1242	87	4	a	a	DET
ejpam-1242	87	5	semiring	semiring	NOUN
ejpam-1242	87	6	with	with	ADP
ejpam-1242	87	7	identity	identity	NOUN
ejpam-1242	87	8	,	,	PUNCT
ejpam-1242	87	9	m	m	VERB
ejpam-1242	87	10	an	an	DET
ejpam-1242	87	11	r	r	NOUN
ejpam-1242	87	12	-	-	PUNCT
ejpam-1242	87	13	semimodule	semimodule	NOUN
ejpam-1242	87	14	and	and	CCONJ
ejpam-1242	87	15	n	n	DET
ejpam-1242	87	16	an	an	DET
ejpam-1242	87	17	qm	qm	PROPN
ejpam-1242	87	18	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	87	19	of	of	ADP
ejpam-1242	87	20	m.	m.	NOUN
ejpam-1242	87	21	then	then	ADV
ejpam-1242	87	22	there	there	PRON
ejpam-1242	87	23	is	be	VERB
ejpam-1242	87	24	a	a	DET
ejpam-1242	87	25	one	one	NUM
ejpam-1242	87	26	-	-	PUNCT
ejpam-1242	87	27	to	to	ADP
ejpam-1242	87	28	-	-	PUNCT
ejpam-1242	87	29	one	one	NUM
ejpam-1242	87	30	correspondence	correspondence	NOUN
ejpam-1242	87	31	between	between	ADP
ejpam-1242	87	32	prime	prime	ADJ
ejpam-1242	87	33	k	k	NOUN
ejpam-1242	87	34	-	-	NOUN
ejpam-1242	87	35	subsemimodules	subsemimodules	NOUN
ejpam-1242	87	36	of	of	ADP
ejpam-1242	87	37	r	r	NOUN
ejpam-1242	87	38	-	-	PUNCT
ejpam-1242	87	39	semimodule	semimodule	NOUN
ejpam-1242	87	40	m	m	PROPN
ejpam-1242	87	41	/	/	SYM
ejpam-1242	87	42	n	n	PROPN
ejpam-1242	87	43	and	and	CCONJ
ejpam-1242	87	44	prime	prime	ADJ
ejpam-1242	87	45	k	k	NOUN
ejpam-1242	87	46	-	-	NOUN
ejpam-1242	87	47	subsemimodules	subsemimodules	NOUN
ejpam-1242	87	48	of	of	ADP
ejpam-1242	87	49	m	m	AUX
ejpam-1242	87	50	containing	contain	VERB
ejpam-1242	87	51	n.	n.	NOUN
ejpam-1242	87	52	proof	proof	NOUN
ejpam-1242	87	53	.	.	PUNCT
ejpam-1242	88	1	let	let	VERB
ejpam-1242	88	2	t	t	NOUN
ejpam-1242	88	3	be	be	AUX
ejpam-1242	88	4	a	a	DET
ejpam-1242	88	5	prime	prime	ADJ
ejpam-1242	88	6	k	k	NOUN
ejpam-1242	88	7	-	-	NOUN
ejpam-1242	88	8	subsemimodule	subsemimodule	NOUN
ejpam-1242	88	9	of	of	ADP
ejpam-1242	88	10	m	m	AUX
ejpam-1242	88	11	containing	contain	VERB
ejpam-1242	88	12	n	n	NOUN
ejpam-1242	88	13	.	.	PUNCT
ejpam-1242	89	1	it	it	PRON
ejpam-1242	89	2	then	then	ADV
ejpam-1242	89	3	follows	follow	VERB
ejpam-1242	89	4	from	from	ADP
ejpam-1242	89	5	[	[	X
ejpam-1242	89	6	7	7	NUM
ejpam-1242	89	7	,	,	PUNCT
ejpam-1242	89	8	theorem	theorem	VERB
ejpam-1242	89	9	3.6	3.6	NUM
ejpam-1242	89	10	]	]	PUNCT
ejpam-1242	89	11	that	that	SCONJ
ejpam-1242	89	12	t	t	PROPN
ejpam-1242	89	13	/	/	SYM
ejpam-1242	89	14	n	n	PROPN
ejpam-1242	89	15	is	be	AUX
ejpam-1242	89	16	a	a	DET
ejpam-1242	89	17	proper	proper	ADJ
ejpam-1242	89	18	k	k	NOUN
ejpam-1242	89	19	-	-	NOUN
ejpam-1242	89	20	susemimodule	susemimodule	NOUN
ejpam-1242	89	21	of	of	ADP
ejpam-1242	89	22	m	m	PROPN
ejpam-1242	89	23	/	/	SYM
ejpam-1242	89	24	n	n	PROPN
ejpam-1242	89	25	.	.	PUNCT
ejpam-1242	90	1	let	let	VERB
ejpam-1242	90	2	a(q1+n	a(q1+n	PRON
ejpam-1242	90	3	)	)	PUNCT
ejpam-1242	91	1	=	=	PUNCT
ejpam-1242	92	1	q2+n	q2+n	ADP
ejpam-1242	92	2	∈	∈	PROPN
ejpam-1242	92	3	t	t	PROPN
ejpam-1242	92	4	/	/	SYM
ejpam-1242	92	5	n	n	PROPN
ejpam-1242	92	6	,	,	PUNCT
ejpam-1242	92	7	where	where	SCONJ
ejpam-1242	92	8	q2	q2	PROPN
ejpam-1242	92	9	∈	∈	PROPN
ejpam-1242	92	10	qm	qm	PROPN
ejpam-1242	92	11	∩	∩	PROPN
ejpam-1242	92	12	t	t	PROPN
ejpam-1242	92	13	and	and	CCONJ
ejpam-1242	92	14	aq1	aq1	VERB
ejpam-1242	92	15	+	+	CCONJ
ejpam-1242	92	16	n	n	CCONJ
ejpam-1242	92	17	⊆	⊆	NUM
ejpam-1242	92	18	q2	q2	NOUN
ejpam-1242	92	19	+	+	CCONJ
ejpam-1242	92	20	n	n	CCONJ
ejpam-1242	92	21	,	,	PUNCT
ejpam-1242	92	22	so	so	ADV
ejpam-1242	92	23	aq1	aq1	PROPN
ejpam-1242	92	24	∈	∈	PROPN
ejpam-1242	92	25	t	t	PROPN
ejpam-1242	92	26	since	since	SCONJ
ejpam-1242	92	27	t	t	PROPN
ejpam-1242	92	28	is	be	AUX
ejpam-1242	92	29	a	a	DET
ejpam-1242	92	30	k	k	NOUN
ejpam-1242	92	31	-	-	NOUN
ejpam-1242	92	32	subsemimodule	subsemimodule	NOUN
ejpam-1242	92	33	.	.	PUNCT
ejpam-1242	93	1	then	then	ADV
ejpam-1242	93	2	t	t	PROPN
ejpam-1242	93	3	prime	prime	NOUN
ejpam-1242	93	4	gives	give	VERB
ejpam-1242	93	5	either	either	CCONJ
ejpam-1242	93	6	q1	q1	PROPN
ejpam-1242	93	7	∈	∈	PROPN
ejpam-1242	93	8	t	t	PROPN
ejpam-1242	93	9	(	(	PUNCT
ejpam-1242	93	10	so	so	ADV
ejpam-1242	93	11	q1	q1	PROPN
ejpam-1242	93	12	+	+	CCONJ
ejpam-1242	93	13	n	n	CCONJ
ejpam-1242	93	14	∈	∈	PROPN
ejpam-1242	93	15	t	t	PROPN
ejpam-1242	93	16	/	/	SYM
ejpam-1242	93	17	n	n	CCONJ
ejpam-1242	93	18	)	)	PUNCT
ejpam-1242	93	19	or	or	CCONJ
ejpam-1242	93	20	a	a	DET
ejpam-1242	93	21	∈	∈	NOUN
ejpam-1242	93	22	(	(	PUNCT
ejpam-1242	93	23	t	t	NOUN
ejpam-1242	93	24	:	:	PUNCT
ejpam-1242	93	25	m	m	X
ejpam-1242	93	26	)	)	PUNCT
ejpam-1242	93	27	=	=	SYM
ejpam-1242	93	28	(	(	PUNCT
ejpam-1242	93	29	t	t	PROPN
ejpam-1242	93	30	/	/	SYM
ejpam-1242	93	31	n	n	PROPN
ejpam-1242	93	32	:	:	PUNCT
ejpam-1242	93	33	m	m	X
ejpam-1242	93	34	/	/	SYM
ejpam-1242	93	35	n	n	CCONJ
ejpam-1242	93	36	)	)	PUNCT
ejpam-1242	93	37	by	by	ADP
ejpam-1242	93	38	lemma	lemma	PROPN
ejpam-1242	93	39	2	2	NUM
ejpam-1242	93	40	.	.	PUNCT
ejpam-1242	94	1	thus	thus	ADV
ejpam-1242	94	2	,	,	PUNCT
ejpam-1242	94	3	t	t	PROPN
ejpam-1242	94	4	/	/	SYM
ejpam-1242	94	5	n	n	PROPN
ejpam-1242	94	6	is	be	AUX
ejpam-1242	94	7	a	a	DET
ejpam-1242	94	8	prime	prime	ADJ
ejpam-1242	94	9	k	k	NOUN
ejpam-1242	94	10	-	-	NOUN
ejpam-1242	94	11	subsemimodule	subsemimodule	NOUN
ejpam-1242	94	12	of	of	ADP
ejpam-1242	94	13	m	m	PROPN
ejpam-1242	94	14	/	/	SYM
ejpam-1242	94	15	n	n	PROPN
ejpam-1242	94	16	.	.	PUNCT
ejpam-1242	95	1	conversely	conversely	ADV
ejpam-1242	95	2	,	,	PUNCT
ejpam-1242	95	3	assume	assume	VERB
ejpam-1242	95	4	that	that	SCONJ
ejpam-1242	95	5	t	t	PROPN
ejpam-1242	95	6	/	/	SYM
ejpam-1242	95	7	n	n	PROPN
ejpam-1242	95	8	is	be	AUX
ejpam-1242	95	9	a	a	DET
ejpam-1242	95	10	prime	prime	ADJ
ejpam-1242	95	11	k	k	NOUN
ejpam-1242	95	12	-	-	NOUN
ejpam-1242	95	13	subsemimodule	subsemimodule	NOUN
ejpam-1242	95	14	of	of	ADP
ejpam-1242	95	15	m	m	PROPN
ejpam-1242	95	16	/	/	SYM
ejpam-1242	95	17	n	n	PROPN
ejpam-1242	95	18	.	.	PUNCT
ejpam-1242	96	1	to	to	PART
ejpam-1242	96	2	show	show	VERB
ejpam-1242	96	3	that	that	SCONJ
ejpam-1242	96	4	t	t	PROPN
ejpam-1242	96	5	is	be	AUX
ejpam-1242	96	6	a	a	DET
ejpam-1242	96	7	prime	prime	ADJ
ejpam-1242	96	8	k	k	NOUN
ejpam-1242	96	9	-	-	NOUN
ejpam-1242	96	10	subsemimodule	subsemimodule	NOUN
ejpam-1242	96	11	of	of	ADP
ejpam-1242	96	12	m	m	PRON
ejpam-1242	96	13	,	,	PUNCT
ejpam-1242	96	14	suppose	suppose	VERB
ejpam-1242	96	15	that	that	SCONJ
ejpam-1242	96	16	rm	rm	PROPN
ejpam-1242	96	17	∈	∈	PROPN
ejpam-1242	96	18	t	t	PROPN
ejpam-1242	96	19	,	,	PUNCT
ejpam-1242	96	20	where	where	SCONJ
ejpam-1242	96	21	r	r	NOUN
ejpam-1242	96	22	∈	∈	PROPN
ejpam-1242	96	23	r	r	NOUN
ejpam-1242	96	24	and	and	CCONJ
ejpam-1242	96	25	m	m	PROPN
ejpam-1242	96	26	∈	∈	NOUN
ejpam-1242	96	27	m	m	NOUN
ejpam-1242	96	28	.	.	PUNCT
ejpam-1242	97	1	we	we	PRON
ejpam-1242	97	2	may	may	AUX
ejpam-1242	97	3	assume	assume	VERB
ejpam-1242	97	4	that	that	SCONJ
ejpam-1242	97	5	r	r	NOUN
ejpam-1242	97	6	6=	6=	ADP
ejpam-1242	97	7	0	0	NUM
ejpam-1242	97	8	.	.	PUNCT
ejpam-1242	98	1	there	there	PRON
ejpam-1242	98	2	are	be	VERB
ejpam-1242	98	3	elements	element	NOUN
ejpam-1242	98	4	q	q	PROPN
ejpam-1242	98	5	∈	∈	PROPN
ejpam-1242	98	6	qm	qm	PROPN
ejpam-1242	98	7	and	and	CCONJ
ejpam-1242	98	8	n	n	PRON
ejpam-1242	98	9	∈	∈	PROPN
ejpam-1242	98	10	n	n	PRON
ejpam-1242	98	11	such	such	ADJ
ejpam-1242	98	12	that	that	SCONJ
ejpam-1242	98	13	m	m	VERB
ejpam-1242	98	14	=	=	SYM
ejpam-1242	98	15	q+	q+	NUM
ejpam-1242	98	16	n	n	CCONJ
ejpam-1242	98	17	,	,	PUNCT
ejpam-1242	98	18	so	so	ADV
ejpam-1242	98	19	rm	rm	PROPN
ejpam-1242	98	20	=	=	PROPN
ejpam-1242	98	21	rq+	rq+	PROPN
ejpam-1242	98	22	rn	rn	PROPN
ejpam-1242	98	23	∈	∈	PROPN
ejpam-1242	98	24	t	t	PROPN
ejpam-1242	98	25	;	;	PUNCT
ejpam-1242	98	26	hence	hence	ADV
ejpam-1242	98	27	rq	rq	VERB
ejpam-1242	98	28	∈	∈	PROPN
ejpam-1242	98	29	t	t	PROPN
ejpam-1242	98	30	since	since	SCONJ
ejpam-1242	98	31	t	t	PROPN
ejpam-1242	98	32	is	be	AUX
ejpam-1242	98	33	a	a	DET
ejpam-1242	98	34	ksubsemimodule	ksubsemimodule	NOUN
ejpam-1242	98	35	.	.	PUNCT
ejpam-1242	99	1	therefore	therefore	ADV
ejpam-1242	99	2	,	,	PUNCT
ejpam-1242	99	3	there	there	PRON
ejpam-1242	99	4	exists	exist	VERB
ejpam-1242	99	5	a	a	DET
ejpam-1242	99	6	unique	unique	ADJ
ejpam-1242	99	7	element	element	NOUN
ejpam-1242	99	8	q′	q′	NOUN
ejpam-1242	99	9	∈qm	∈qm	NOUN
ejpam-1242	99	10	such	such	ADJ
ejpam-1242	99	11	that	that	SCONJ
ejpam-1242	99	12	r(q+n	r(q+n	NOUN
ejpam-1242	99	13	)	)	PUNCT
ejpam-1242	99	14	=	=	SYM
ejpam-1242	99	15	q′+n	q′+n	PROPN
ejpam-1242	99	16	,	,	PUNCT
ejpam-1242	99	17	where	where	SCONJ
ejpam-1242	99	18	rq+	rq+	ADJ
ejpam-1242	99	19	n	n	PROPN
ejpam-1242	99	20	⊆	⊆	NUM
ejpam-1242	99	21	q′	q′	NOUN
ejpam-1242	99	22	+	+	CCONJ
ejpam-1242	99	23	n	n	CCONJ
ejpam-1242	99	24	;	;	PUNCT
ejpam-1242	99	25	hence	hence	ADV
ejpam-1242	99	26	q′	q′	NOUN
ejpam-1242	99	27	∈	∈	PROPN
ejpam-1242	99	28	t	t	PROPN
ejpam-1242	99	29	.	.	PUNCT
ejpam-1242	100	1	thus	thus	ADV
ejpam-1242	100	2	r(q+	r(q+	X
ejpam-1242	100	3	n	n	CCONJ
ejpam-1242	100	4	)	)	PUNCT
ejpam-1242	100	5	∈	∈	PROPN
ejpam-1242	100	6	t	t	PROPN
ejpam-1242	100	7	/	/	SYM
ejpam-1242	100	8	n	n	PROPN
ejpam-1242	100	9	.	.	PUNCT
ejpam-1242	101	1	then	then	ADV
ejpam-1242	101	2	t	t	PROPN
ejpam-1242	101	3	/	/	SYM
ejpam-1242	101	4	n	n	NOUN
ejpam-1242	101	5	prime	prime	NOUN
ejpam-1242	101	6	gives	give	VERB
ejpam-1242	101	7	either	either	DET
ejpam-1242	101	8	q+	q+	ADV
ejpam-1242	101	9	n	n	PROPN
ejpam-1242	101	10	∈	∈	PROPN
ejpam-1242	101	11	t	t	PROPN
ejpam-1242	101	12	/	/	SYM
ejpam-1242	101	13	n	n	PROPN
ejpam-1242	101	14	(	(	PUNCT
ejpam-1242	101	15	so	so	ADV
ejpam-1242	101	16	m	m	PROPN
ejpam-1242	101	17	∈	∈	PROPN
ejpam-1242	101	18	t	t	PROPN
ejpam-1242	101	19	)	)	PUNCT
ejpam-1242	101	20	or	or	CCONJ
ejpam-1242	101	21	r	r	NOUN
ejpam-1242	101	22	∈	∈	PROPN
ejpam-1242	101	23	(	(	PUNCT
ejpam-1242	101	24	t	t	PROPN
ejpam-1242	101	25	/	/	SYM
ejpam-1242	101	26	n	n	PROPN
ejpam-1242	101	27	:	:	PUNCT
ejpam-1242	101	28	m	m	X
ejpam-1242	101	29	/	/	SYM
ejpam-1242	101	30	n	n	CCONJ
ejpam-1242	101	31	)	)	PUNCT
ejpam-1242	101	32	=	=	PUNCT
ejpam-1242	102	1	(	(	PUNCT
ejpam-1242	102	2	t	t	X
ejpam-1242	102	3	:	:	PUNCT
ejpam-1242	102	4	m	m	X
ejpam-1242	102	5	)	)	PUNCT
ejpam-1242	102	6	,	,	PUNCT
ejpam-1242	102	7	and	and	CCONJ
ejpam-1242	102	8	the	the	DET
ejpam-1242	102	9	proof	proof	NOUN
ejpam-1242	102	10	is	be	AUX
ejpam-1242	102	11	complete	complete	ADJ
ejpam-1242	102	12	.	.	PUNCT
ejpam-1242	103	1	corollary	corollary	ADJ
ejpam-1242	103	2	1	1	NUM
ejpam-1242	103	3	.	.	PUNCT
ejpam-1242	104	1	let	let	VERB
ejpam-1242	104	2	r	r	PRON
ejpam-1242	104	3	be	be	AUX
ejpam-1242	104	4	a	a	DET
ejpam-1242	104	5	semiring	semiring	NOUN
ejpam-1242	104	6	with	with	ADP
ejpam-1242	104	7	identity	identity	NOUN
ejpam-1242	104	8	,	,	PUNCT
ejpam-1242	104	9	m	m	VERB
ejpam-1242	104	10	an	an	DET
ejpam-1242	104	11	r	r	NOUN
ejpam-1242	104	12	-	-	PUNCT
ejpam-1242	104	13	semimodule	semimodule	NOUN
ejpam-1242	104	14	and	and	CCONJ
ejpam-1242	104	15	n	n	DET
ejpam-1242	104	16	an	an	DET
ejpam-1242	104	17	qm	qm	PROPN
ejpam-1242	104	18	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	104	19	of	of	ADP
ejpam-1242	104	20	m.	m.	NOUN
ejpam-1242	104	21	then	then	ADV
ejpam-1242	104	22	there	there	PRON
ejpam-1242	104	23	is	be	VERB
ejpam-1242	104	24	a	a	DET
ejpam-1242	104	25	one	one	NUM
ejpam-1242	104	26	-	-	PUNCT
ejpam-1242	104	27	to	to	ADP
ejpam-1242	104	28	-	-	PUNCT
ejpam-1242	104	29	one	one	NUM
ejpam-1242	104	30	correspondence	correspondence	NOUN
ejpam-1242	104	31	between	between	ADP
ejpam-1242	104	32	semiprime	semiprime	NOUN
ejpam-1242	104	33	k	k	X
ejpam-1242	104	34	-	-	NOUN
ejpam-1242	104	35	subsemimodules	subsemimodule	NOUN
ejpam-1242	104	36	of	of	ADP
ejpam-1242	104	37	rsemimodule	rsemimodule	NOUN
ejpam-1242	104	38	m	m	PROPN
ejpam-1242	104	39	/	/	SYM
ejpam-1242	104	40	n	n	PROPN
ejpam-1242	104	41	and	and	CCONJ
ejpam-1242	104	42	semiprime	semiprime	NOUN
ejpam-1242	105	1	k	k	X
ejpam-1242	105	2	-	-	NOUN
ejpam-1242	105	3	subsemimodules	subsemimodule	NOUN
ejpam-1242	105	4	of	of	ADP
ejpam-1242	105	5	m	m	AUX
ejpam-1242	105	6	containing	contain	VERB
ejpam-1242	105	7	n.	n.	NOUN
ejpam-1242	105	8	proof	proof	NOUN
ejpam-1242	105	9	.	.	PUNCT
ejpam-1242	106	1	apply	apply	VERB
ejpam-1242	106	2	theorem	theorem	NOUN
ejpam-1242	106	3	1	1	NUM
ejpam-1242	106	4	.	.	PUNCT
ejpam-1242	106	5	definition	definition	NOUN
ejpam-1242	106	6	2	2	NUM
ejpam-1242	106	7	.	.	PUNCT
ejpam-1242	107	1	let	let	VERB
ejpam-1242	107	2	m	m	PRON
ejpam-1242	107	3	be	be	AUX
ejpam-1242	107	4	a	a	DET
ejpam-1242	107	5	semimodule	semimodule	NOUN
ejpam-1242	107	6	over	over	ADP
ejpam-1242	107	7	a	a	DET
ejpam-1242	107	8	semiring	semire	VERB
ejpam-1242	107	9	r.	r.	NOUN
ejpam-1242	107	10	a	a	DET
ejpam-1242	107	11	subsemimodule	subsemimodule	PROPN
ejpam-1242	107	12	n	n	PROPN
ejpam-1242	107	13	of	of	ADP
ejpam-1242	107	14	m	m	PROPN
ejpam-1242	107	15	is	be	AUX
ejpam-1242	107	16	said	say	VERB
ejpam-1242	107	17	to	to	PART
ejpam-1242	107	18	be	be	AUX
ejpam-1242	107	19	a	a	DET
ejpam-1242	107	20	strong	strong	ADJ
ejpam-1242	107	21	subsemimodule	subsemimodule	NOUN
ejpam-1242	107	22	if	if	SCONJ
ejpam-1242	107	23	for	for	ADP
ejpam-1242	107	24	each	each	DET
ejpam-1242	107	25	x	x	SYM
ejpam-1242	107	26	∈	∈	PROPN
ejpam-1242	107	27	n	n	CCONJ
ejpam-1242	107	28	there	there	ADV
ejpam-1242	107	29	exists	exist	VERB
ejpam-1242	107	30	y	y	PROPN
ejpam-1242	107	31	∈	∈	PROPN
ejpam-1242	107	32	n	n	PRON
ejpam-1242	108	1	such	such	ADJ
ejpam-1242	108	2	that	that	SCONJ
ejpam-1242	108	3	x	x	X
ejpam-1242	108	4	+	+	NUM
ejpam-1242	108	5	y	y	PROPN
ejpam-1242	108	6	=	=	SYM
ejpam-1242	108	7	0	0	PROPN
ejpam-1242	108	8	.	.	PUNCT
ejpam-1242	108	9	example	example	NOUN
ejpam-1242	109	1	1	1	NUM
ejpam-1242	109	2	.	.	PUNCT
ejpam-1242	109	3	(	(	PUNCT
ejpam-1242	109	4	1	1	X
ejpam-1242	109	5	)	)	PUNCT
ejpam-1242	109	6	clearly	clearly	ADV
ejpam-1242	109	7	,	,	PUNCT
ejpam-1242	109	8	every	every	DET
ejpam-1242	109	9	submodule	submodule	NOUN
ejpam-1242	109	10	of	of	ADP
ejpam-1242	109	11	a	a	DET
ejpam-1242	109	12	module	module	NOUN
ejpam-1242	109	13	over	over	ADP
ejpam-1242	109	14	a	a	DET
ejpam-1242	109	15	ring	ring	NOUN
ejpam-1242	109	16	r	r	NOUN
ejpam-1242	109	17	is	be	AUX
ejpam-1242	109	18	a	a	DET
ejpam-1242	109	19	strong	strong	ADJ
ejpam-1242	109	20	subsemimodule	subsemimodule	NOUN
ejpam-1242	109	21	.	.	PUNCT
ejpam-1242	110	1	(	(	PUNCT
ejpam-1242	110	2	2	2	X
ejpam-1242	110	3	)	)	PUNCT
ejpam-1242	110	4	let	let	VERB
ejpam-1242	110	5	r	r	NOUN
ejpam-1242	110	6	denote	denote	VERB
ejpam-1242	110	7	the	the	DET
ejpam-1242	110	8	semiring	semiring	NOUN
ejpam-1242	110	9	of	of	ADP
ejpam-1242	110	10	non	non	ADJ
ejpam-1242	110	11	-	-	ADJ
ejpam-1242	110	12	negative	negative	ADJ
ejpam-1242	110	13	integers	integer	NOUN
ejpam-1242	110	14	with	with	ADP
ejpam-1242	110	15	the	the	DET
ejpam-1242	110	16	usual	usual	ADJ
ejpam-1242	110	17	operations	operation	NOUN
ejpam-1242	110	18	of	of	ADP
ejpam-1242	110	19	addition	addition	NOUN
ejpam-1242	110	20	and	and	CCONJ
ejpam-1242	110	21	multiplication	multiplication	NOUN
ejpam-1242	110	22	,	,	PUNCT
ejpam-1242	110	23	and	and	CCONJ
ejpam-1242	110	24	let	let	VERB
ejpam-1242	110	25	m	m	PROPN
ejpam-1242	110	26	=	=	PROPN
ejpam-1242	110	27	z6	z6	PROPN
ejpam-1242	110	28	denote	denote	VERB
ejpam-1242	110	29	the	the	DET
ejpam-1242	110	30	monoid	monoid	NOUN
ejpam-1242	110	31	of	of	ADP
ejpam-1242	110	32	integers	integer	NOUN
ejpam-1242	110	33	modulo	modulo	VERB
ejpam-1242	110	34	6	6	NUM
ejpam-1242	110	35	.	.	PUNCT
ejpam-1242	111	1	then	then	ADV
ejpam-1242	111	2	m	m	PROPN
ejpam-1242	111	3	is	be	AUX
ejpam-1242	111	4	a	a	DET
ejpam-1242	111	5	semimodule	semimodule	NOUN
ejpam-1242	111	6	over	over	ADP
ejpam-1242	111	7	r	r	NOUN
ejpam-1242	111	8	by	by	ADP
ejpam-1242	111	9	[	[	X
ejpam-1242	111	10	16	16	NUM
ejpam-1242	111	11	,	,	PUNCT
ejpam-1242	111	12	p.	p.	NOUN
ejpam-1242	111	13	151	151	NUM
ejpam-1242	111	14	]	]	PUNCT
ejpam-1242	111	15	,	,	PUNCT
ejpam-1242	111	16	and	and	CCONJ
ejpam-1242	111	17	an	an	DET
ejpam-1242	111	18	inspection	inspection	NOUN
ejpam-1242	111	19	will	will	AUX
ejpam-1242	111	20	show	show	VERB
ejpam-1242	111	21	that	that	SCONJ
ejpam-1242	111	22	n	n	NOUN
ejpam-1242	111	23	=	=	PRON
ejpam-1242	111	24	{	{	PUNCT
ejpam-1242	111	25	0̄	0̄	PROPN
ejpam-1242	111	26	,	,	PUNCT
ejpam-1242	111	27	2̄	2̄	NUM
ejpam-1242	111	28	,	,	PUNCT
ejpam-1242	111	29	4̄	4̄	PROPN
ejpam-1242	111	30	}	}	PUNCT
ejpam-1242	111	31	and	and	CCONJ
ejpam-1242	111	32	m	m	PROPN
ejpam-1242	111	33	are	be	AUX
ejpam-1242	111	34	strong	strong	ADJ
ejpam-1242	111	35	subsemimodules	subsemimodule	NOUN
ejpam-1242	111	36	of	of	ADP
ejpam-1242	111	37	m.	m.	NOUN
ejpam-1242	111	38	definition	definition	NOUN
ejpam-1242	111	39	3	3	NUM
ejpam-1242	111	40	.	.	PUNCT
ejpam-1242	112	1	a	a	DET
ejpam-1242	112	2	semimodule	semimodule	NOUN
ejpam-1242	112	3	m	m	VERB
ejpam-1242	112	4	over	over	ADP
ejpam-1242	112	5	a	a	DET
ejpam-1242	112	6	semiring	semire	VERB
ejpam-1242	112	7	r	r	NOUN
ejpam-1242	112	8	is	be	AUX
ejpam-1242	112	9	called	call	VERB
ejpam-1242	112	10	a	a	DET
ejpam-1242	112	11	strong	strong	ADJ
ejpam-1242	112	12	multiplication	multiplication	NOUN
ejpam-1242	112	13	semimodule	semimodule	NOUN
ejpam-1242	112	14	whenever	whenever	SCONJ
ejpam-1242	112	15	n	n	PRON
ejpam-1242	112	16	is	be	AUX
ejpam-1242	112	17	a	a	DET
ejpam-1242	112	18	k	k	NOUN
ejpam-1242	112	19	-	-	NOUN
ejpam-1242	112	20	subsemimodule	subsemimodule	NOUN
ejpam-1242	112	21	of	of	ADP
ejpam-1242	112	22	m	m	PROPN
ejpam-1242	112	23	,	,	PUNCT
ejpam-1242	112	24	then	then	ADV
ejpam-1242	112	25	there	there	PRON
ejpam-1242	112	26	exists	exist	VERB
ejpam-1242	112	27	a	a	DET
ejpam-1242	112	28	strong	strong	ADJ
ejpam-1242	112	29	ideal	ideal	NOUN
ejpam-1242	112	30	i	i	PRON
ejpam-1242	112	31	of	of	ADP
ejpam-1242	112	32	r	r	NOUN
ejpam-1242	112	33	such	such	ADJ
ejpam-1242	112	34	that	that	SCONJ
ejpam-1242	112	35	n	n	NOUN
ejpam-1242	112	36	=	=	SYM
ejpam-1242	112	37	i	i	PRON
ejpam-1242	112	38	m.	m.	NOUN
ejpam-1242	112	39	definition	definition	NOUN
ejpam-1242	112	40	4	4	NUM
ejpam-1242	112	41	.	.	PUNCT
ejpam-1242	113	1	a	a	DET
ejpam-1242	113	2	semimodule	semimodule	NOUN
ejpam-1242	113	3	m	m	VERB
ejpam-1242	113	4	over	over	ADP
ejpam-1242	113	5	a	a	DET
ejpam-1242	113	6	semiring	semire	VERB
ejpam-1242	113	7	r	r	NOUN
ejpam-1242	113	8	is	be	AUX
ejpam-1242	113	9	called	call	VERB
ejpam-1242	113	10	a	a	DET
ejpam-1242	113	11	very	very	ADV
ejpam-1242	113	12	strong	strong	ADJ
ejpam-1242	113	13	semimodule	semimodule	NOUN
ejpam-1242	113	14	if	if	SCONJ
ejpam-1242	113	15	i	i	PRON
ejpam-1242	113	16	is	be	AUX
ejpam-1242	113	17	an	an	DET
ejpam-1242	113	18	ideal	ideal	NOUN
ejpam-1242	113	19	of	of	ADP
ejpam-1242	113	20	r	r	NOUN
ejpam-1242	113	21	and	and	CCONJ
ejpam-1242	113	22	m	m	PROPN
ejpam-1242	113	23	∈	∈	PROPN
ejpam-1242	113	24	m	m	NOUN
ejpam-1242	113	25	,	,	PUNCT
ejpam-1242	113	26	then	then	ADV
ejpam-1242	113	27	the	the	DET
ejpam-1242	113	28	ideal	ideal	NOUN
ejpam-1242	113	29	{	{	PUNCT
ejpam-1242	113	30	r	r	NOUN
ejpam-1242	113	31	∈	∈	PROPN
ejpam-1242	113	32	r	r	NOUN
ejpam-1242	113	33	:	:	PUNCT
ejpam-1242	113	34	rm	rm	PROPN
ejpam-1242	113	35	∈	∈	PROPN
ejpam-1242	114	1	i	i	PRON
ejpam-1242	114	2	m	m	VERB
ejpam-1242	114	3	}	}	PUNCT
ejpam-1242	114	4	is	be	AUX
ejpam-1242	114	5	a	a	DET
ejpam-1242	114	6	strong	strong	ADJ
ejpam-1242	114	7	k	k	NOUN
ejpam-1242	114	8	-	-	NOUN
ejpam-1242	114	9	ideal	ideal	NOUN
ejpam-1242	114	10	of	of	ADP
ejpam-1242	114	11	r.	r.	PROPN
ejpam-1242	114	12	definition	definition	NOUN
ejpam-1242	114	13	5	5	NUM
ejpam-1242	114	14	.	.	PUNCT
ejpam-1242	114	15	a	a	DET
ejpam-1242	114	16	very	very	ADV
ejpam-1242	114	17	strong	strong	ADJ
ejpam-1242	114	18	semimodule	semimodule	NOUN
ejpam-1242	114	19	m	m	VERB
ejpam-1242	114	20	over	over	ADP
ejpam-1242	114	21	a	a	DET
ejpam-1242	114	22	semiring	semire	VERB
ejpam-1242	114	23	r	r	NOUN
ejpam-1242	114	24	is	be	AUX
ejpam-1242	114	25	called	call	VERB
ejpam-1242	114	26	a	a	DET
ejpam-1242	114	27	very	very	ADV
ejpam-1242	114	28	strong	strong	ADJ
ejpam-1242	114	29	multiplication	multiplication	NOUN
ejpam-1242	114	30	semimodule	semimodule	NOUN
ejpam-1242	114	31	whenever	whenever	SCONJ
ejpam-1242	114	32	n	n	PRON
ejpam-1242	114	33	is	be	AUX
ejpam-1242	114	34	a	a	DET
ejpam-1242	114	35	k	k	NOUN
ejpam-1242	114	36	-	-	NOUN
ejpam-1242	114	37	subsemimodule	subsemimodule	NOUN
ejpam-1242	114	38	of	of	ADP
ejpam-1242	114	39	m	m	PROPN
ejpam-1242	114	40	,	,	PUNCT
ejpam-1242	114	41	then	then	ADV
ejpam-1242	114	42	there	there	PRON
ejpam-1242	114	43	exists	exist	VERB
ejpam-1242	114	44	a	a	DET
ejpam-1242	114	45	strong	strong	ADJ
ejpam-1242	114	46	ideal	ideal	NOUN
ejpam-1242	114	47	i	i	PRON
ejpam-1242	114	48	of	of	ADP
ejpam-1242	114	49	r	r	NOUN
ejpam-1242	114	50	such	such	ADJ
ejpam-1242	114	51	that	that	SCONJ
ejpam-1242	114	52	n	n	NOUN
ejpam-1242	114	53	=	=	SYM
ejpam-1242	114	54	i	i	PRON
ejpam-1242	114	55	m.	m.	NOUN
ejpam-1242	114	56	proposition	proposition	NOUN
ejpam-1242	114	57	1	1	NUM
ejpam-1242	114	58	.	.	PUNCT
ejpam-1242	115	1	let	let	VERB
ejpam-1242	115	2	m	m	PRON
ejpam-1242	115	3	be	be	AUX
ejpam-1242	115	4	a	a	DET
ejpam-1242	115	5	semimodule	semimodule	NOUN
ejpam-1242	115	6	over	over	ADP
ejpam-1242	115	7	a	a	DET
ejpam-1242	115	8	semirig	semirig	NOUN
ejpam-1242	115	9	r.	r.	PROPN
ejpam-1242	115	10	then	then	ADV
ejpam-1242	115	11	the	the	DET
ejpam-1242	115	12	following	follow	VERB
ejpam-1242	115	13	statements	statement	NOUN
ejpam-1242	115	14	hold	hold	VERB
ejpam-1242	115	15	:	:	PUNCT
ejpam-1242	115	16	(	(	PUNCT
ejpam-1242	115	17	i	i	NOUN
ejpam-1242	115	18	)	)	PUNCT
ejpam-1242	115	19	if	if	SCONJ
ejpam-1242	115	20	n	n	PRON
ejpam-1242	115	21	is	be	AUX
ejpam-1242	115	22	a	a	DET
ejpam-1242	115	23	strong	strong	ADJ
ejpam-1242	115	24	subsemimodule	subsemimodule	NOUN
ejpam-1242	115	25	of	of	ADP
ejpam-1242	115	26	m	m	PROPN
ejpam-1242	115	27	,	,	PUNCT
ejpam-1242	115	28	then	then	ADV
ejpam-1242	115	29	n	n	PRON
ejpam-1242	115	30	is	be	AUX
ejpam-1242	115	31	a	a	DET
ejpam-1242	115	32	k	k	NOUN
ejpam-1242	115	33	-	-	NOUN
ejpam-1242	115	34	subsemimodule	subsemimodule	NOUN
ejpam-1242	115	35	.	.	PUNCT
ejpam-1242	116	1	(	(	PUNCT
ejpam-1242	116	2	ii	ii	NOUN
ejpam-1242	116	3	)	)	PUNCT
ejpam-1242	116	4	if	if	SCONJ
ejpam-1242	116	5	i	i	PRON
ejpam-1242	116	6	is	be	AUX
ejpam-1242	116	7	a	a	DET
ejpam-1242	116	8	strong	strong	ADJ
ejpam-1242	116	9	ideal	ideal	NOUN
ejpam-1242	116	10	of	of	ADP
ejpam-1242	116	11	r	r	NOUN
ejpam-1242	116	12	,	,	PUNCT
ejpam-1242	116	13	then	then	ADV
ejpam-1242	116	14	i	i	PRON
ejpam-1242	116	15	m	m	VERB
ejpam-1242	116	16	is	be	AUX
ejpam-1242	116	17	a	a	DET
ejpam-1242	116	18	strong	strong	ADJ
ejpam-1242	116	19	k	k	NOUN
ejpam-1242	116	20	-	-	NOUN
ejpam-1242	116	21	subsemimodule	subsemimodule	NOUN
ejpam-1242	116	22	.	.	PUNCT
ejpam-1242	117	1	s.	s.	PROPN
ejpam-1242	117	2	atani	atani	PROPN
ejpam-1242	117	3	,	,	PUNCT
ejpam-1242	117	4	r.	r.	PROPN
ejpam-1242	117	5	atrani	atrani	PROPN
ejpam-1242	117	6	,	,	PUNCT
ejpam-1242	117	7	ü.	ü.	NOUN
ejpam-1242	117	8	tekir	tekir	PROPN
ejpam-1242	117	9	/	/	SYM
ejpam-1242	117	10	eur	eur	PROPN
ejpam-1242	117	11	.	.	PUNCT
ejpam-1242	118	1	j.	j.	PROPN
ejpam-1242	118	2	pure	pure	PROPN
ejpam-1242	118	3	appl	appl	PROPN
ejpam-1242	118	4	.	.	PROPN
ejpam-1242	118	5	math	math	PROPN
ejpam-1242	118	6	,	,	PUNCT
ejpam-1242	118	7	4	4	NUM
ejpam-1242	118	8	(	(	PUNCT
ejpam-1242	118	9	2011	2011	NUM
ejpam-1242	118	10	)	)	PUNCT
ejpam-1242	118	11	,	,	PUNCT
ejpam-1242	118	12	251	251	NUM
ejpam-1242	118	13	-	-	SYM
ejpam-1242	118	14	265	265	NUM
ejpam-1242	118	15	255	255	NUM
ejpam-1242	118	16	(	(	PUNCT
ejpam-1242	118	17	iii	iii	NOUN
ejpam-1242	118	18	)	)	PUNCT
ejpam-1242	118	19	if	if	SCONJ
ejpam-1242	118	20	n	n	PRON
ejpam-1242	118	21	is	be	AUX
ejpam-1242	118	22	a	a	DET
ejpam-1242	118	23	strong	strong	ADJ
ejpam-1242	118	24	subsemimodule	subsemimodule	NOUN
ejpam-1242	118	25	of	of	ADP
ejpam-1242	118	26	m	m	PROPN
ejpam-1242	118	27	,	,	PUNCT
ejpam-1242	118	28	then	then	ADV
ejpam-1242	118	29	n	n	PROPN
ejpam-1242	118	30	+	+	NUM
ejpam-1242	118	31	l	l	NOUN
ejpam-1242	118	32	is	be	AUX
ejpam-1242	118	33	a	a	DET
ejpam-1242	118	34	strong	strong	ADJ
ejpam-1242	118	35	k	k	NOUN
ejpam-1242	118	36	-	-	NOUN
ejpam-1242	118	37	subsemimodule	subsemimodule	NOUN
ejpam-1242	118	38	of	of	ADP
ejpam-1242	118	39	m	m	PRON
ejpam-1242	118	40	for	for	ADP
ejpam-1242	118	41	every	every	DET
ejpam-1242	118	42	strong	strong	ADJ
ejpam-1242	118	43	subsemimodule	subsemimodule	NOUN
ejpam-1242	118	44	l	l	NOUN
ejpam-1242	118	45	of	of	ADP
ejpam-1242	118	46	m.	m.	NOUN
ejpam-1242	118	47	(	(	PUNCT
ejpam-1242	118	48	iv	iv	X
ejpam-1242	118	49	)	)	PUNCT
ejpam-1242	118	50	if	if	SCONJ
ejpam-1242	118	51	i	i	PRON
ejpam-1242	118	52	is	be	AUX
ejpam-1242	118	53	a	a	DET
ejpam-1242	118	54	strong	strong	ADJ
ejpam-1242	118	55	ideal	ideal	NOUN
ejpam-1242	118	56	of	of	ADP
ejpam-1242	118	57	r	r	NOUN
ejpam-1242	118	58	and	and	CCONJ
ejpam-1242	118	59	n	n	NOUN
ejpam-1242	118	60	is	be	AUX
ejpam-1242	118	61	a	a	DET
ejpam-1242	118	62	strong	strong	ADJ
ejpam-1242	118	63	qm	qm	NOUN
ejpam-1242	118	64	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	118	65	of	of	ADP
ejpam-1242	118	66	m	m	PRON
ejpam-1242	118	67	,	,	PUNCT
ejpam-1242	118	68	then	then	ADV
ejpam-1242	118	69	i(m	i(m	NOUN
ejpam-1242	118	70	/	/	SYM
ejpam-1242	118	71	n	n	CCONJ
ejpam-1242	118	72	)	)	PUNCT
ejpam-1242	118	73	=	=	PUNCT
ejpam-1242	118	74	(	(	PUNCT
ejpam-1242	118	75	i	i	PRON
ejpam-1242	118	76	m	m	VERB
ejpam-1242	118	77	+	+	X
ejpam-1242	118	78	n)/n	n)/n	PROPN
ejpam-1242	118	79	.	.	PUNCT
ejpam-1242	119	1	proof	proof	NOUN
ejpam-1242	119	2	.	.	PUNCT
ejpam-1242	120	1	(	(	PUNCT
ejpam-1242	120	2	i	i	NOUN
ejpam-1242	120	3	)	)	PUNCT
ejpam-1242	120	4	let	let	VERB
ejpam-1242	120	5	a	a	PRON
ejpam-1242	120	6	,	,	PUNCT
ejpam-1242	120	7	a+	a+	PRON
ejpam-1242	120	8	b	b	X
ejpam-1242	120	9	∈	∈	PROPN
ejpam-1242	120	10	n	n	X
ejpam-1242	120	11	for	for	ADP
ejpam-1242	120	12	some	some	DET
ejpam-1242	120	13	a	a	PRON
ejpam-1242	120	14	,	,	PUNCT
ejpam-1242	120	15	b	b	X
ejpam-1242	120	16	∈	∈	ADV
ejpam-1242	120	17	m	m	VERB
ejpam-1242	120	18	.	.	PUNCT
ejpam-1242	121	1	then	then	ADV
ejpam-1242	121	2	a+	a+	PUNCT
ejpam-1242	121	3	a′	a′	PROPN
ejpam-1242	121	4	=	=	SYM
ejpam-1242	121	5	0	0	NUM
ejpam-1242	122	1	for	for	ADP
ejpam-1242	122	2	some	some	DET
ejpam-1242	122	3	a′	a′	NOUN
ejpam-1242	122	4	∈	∈	PROPN
ejpam-1242	122	5	n	n	NOUN
ejpam-1242	122	6	;	;	PUNCT
ejpam-1242	122	7	hence	hence	ADV
ejpam-1242	122	8	b	b	X
ejpam-1242	122	9	=	=	PRON
ejpam-1242	122	10	a+	a+	PUNCT
ejpam-1242	122	11	a′	a′	PROPN
ejpam-1242	122	12	+	+	SYM
ejpam-1242	122	13	b	b	X
ejpam-1242	122	14	∈	∈	PROPN
ejpam-1242	122	15	n	n	X
ejpam-1242	122	16	.	.	PUNCT
ejpam-1242	123	1	(	(	PUNCT
ejpam-1242	123	2	ii	ii	NOUN
ejpam-1242	123	3	)	)	PUNCT
ejpam-1242	123	4	let	let	VERB
ejpam-1242	123	5	z	z	NOUN
ejpam-1242	123	6	=	=	SYM
ejpam-1242	124	1	∑n	∑n	PROPN
ejpam-1242	124	2	i=1	i=1	PROPN
ejpam-1242	124	3	rimi	rimi	PROPN
ejpam-1242	124	4	∈	∈	PROPN
ejpam-1242	125	1	i	i	PRON
ejpam-1242	125	2	m	m	VERB
ejpam-1242	125	3	.	.	PUNCT
ejpam-1242	126	1	then	then	ADV
ejpam-1242	126	2	there	there	PRON
ejpam-1242	126	3	exists	exist	VERB
ejpam-1242	126	4	si	si	PROPN
ejpam-1242	126	5	∈	∈	PROPN
ejpam-1242	126	6	i	i	PRON
ejpam-1242	126	7	such	such	VERB
ejpam-1242	126	8	that	that	DET
ejpam-1242	126	9	ri	ri	PROPN
ejpam-1242	127	1	+	+	CCONJ
ejpam-1242	127	2	si	si	X
ejpam-1242	127	3	=	=	SYM
ejpam-1242	127	4	0	0	NUM
ejpam-1242	128	1	for	for	ADP
ejpam-1242	128	2	every	every	DET
ejpam-1242	128	3	i	i	NOUN
ejpam-1242	128	4	=	=	NOUN
ejpam-1242	128	5	1	1	NUM
ejpam-1242	128	6	,	,	PUNCT
ejpam-1242	128	7	.	.	PUNCT
ejpam-1242	128	8	.	.	PUNCT
ejpam-1242	128	9	.	.	PUNCT
ejpam-1242	129	1	,	,	PUNCT
ejpam-1242	129	2	n	n	CCONJ
ejpam-1242	129	3	;	;	PUNCT
ejpam-1242	130	1	so	so	SCONJ
ejpam-1242	130	2	z	z	X
ejpam-1242	130	3	+	+	NUM
ejpam-1242	131	1	∑n	∑n	PROPN
ejpam-1242	131	2	i=1	i=1	PROPN
ejpam-1242	131	3	simi	simi	PROPN
ejpam-1242	131	4	=	=	SYM
ejpam-1242	131	5	0	0	X
ejpam-1242	131	6	.	.	PUNCT
ejpam-1242	132	1	now	now	ADV
ejpam-1242	132	2	the	the	DET
ejpam-1242	132	3	assertion	assertion	NOUN
ejpam-1242	132	4	follows	follow	VERB
ejpam-1242	132	5	from	from	ADP
ejpam-1242	132	6	(	(	PUNCT
ejpam-1242	132	7	i	i	NOUN
ejpam-1242	132	8	)	)	PUNCT
ejpam-1242	132	9	.	.	PUNCT
ejpam-1242	133	1	(	(	PUNCT
ejpam-1242	133	2	iii	iii	X
ejpam-1242	133	3	)	)	PUNCT
ejpam-1242	133	4	let	let	VERB
ejpam-1242	133	5	a	a	DET
ejpam-1242	133	6	+	+	NOUN
ejpam-1242	133	7	b	b	NOUN
ejpam-1242	133	8	∈	∈	ADJ
ejpam-1242	133	9	n	n	NOUN
ejpam-1242	133	10	+	+	CCONJ
ejpam-1242	133	11	l	l	NOUN
ejpam-1242	133	12	,	,	PUNCT
ejpam-1242	133	13	where	where	SCONJ
ejpam-1242	133	14	a	a	DET
ejpam-1242	133	15	∈	∈	PROPN
ejpam-1242	133	16	n	n	NOUN
ejpam-1242	133	17	and	and	CCONJ
ejpam-1242	133	18	b	b	PROPN
ejpam-1242	133	19	∈	∈	PROPN
ejpam-1242	133	20	l.	l.	NOUN
ejpam-1242	133	21	then	then	ADV
ejpam-1242	133	22	a	a	DET
ejpam-1242	133	23	+	+	PUNCT
ejpam-1242	133	24	a′	a′	NOUN
ejpam-1242	133	25	=	=	SYM
ejpam-1242	133	26	0	0	NUM
ejpam-1242	133	27	for	for	ADP
ejpam-1242	133	28	some	some	DET
ejpam-1242	133	29	a′	a′	NOUN
ejpam-1242	133	30	∈	∈	PROPN
ejpam-1242	133	31	n	n	PROPN
ejpam-1242	133	32	and	and	CCONJ
ejpam-1242	133	33	b+	b+	VERB
ejpam-1242	133	34	b′	b′	X
ejpam-1242	133	35	=	=	NOUN
ejpam-1242	133	36	0	0	NUM
ejpam-1242	133	37	for	for	ADP
ejpam-1242	133	38	some	some	DET
ejpam-1242	133	39	b′	b′	NUM
ejpam-1242	133	40	∈	∈	PROPN
ejpam-1242	133	41	l	l	NOUN
ejpam-1242	133	42	;	;	PUNCT
ejpam-1242	134	1	hence	hence	ADV
ejpam-1242	134	2	(	(	PUNCT
ejpam-1242	134	3	a+	a+	PUNCT
ejpam-1242	134	4	b	b	X
ejpam-1242	134	5	)	)	PUNCT
ejpam-1242	134	6	+	+	CCONJ
ejpam-1242	134	7	(	(	PUNCT
ejpam-1242	134	8	a′	a′	ADJ
ejpam-1242	134	9	+	+	CCONJ
ejpam-1242	134	10	b′	b′	NUM
ejpam-1242	134	11	)	)	PUNCT
ejpam-1242	134	12	=	=	SYM
ejpam-1242	135	1	0	0	X
ejpam-1242	135	2	.	.	PUNCT
ejpam-1242	136	1	thus	thus	ADV
ejpam-1242	136	2	n	n	PROPN
ejpam-1242	136	3	+	+	NUM
ejpam-1242	136	4	l	l	NOUN
ejpam-1242	136	5	is	be	AUX
ejpam-1242	136	6	a	a	DET
ejpam-1242	136	7	strong	strong	ADJ
ejpam-1242	136	8	ideal	ideal	NOUN
ejpam-1242	136	9	.	.	PUNCT
ejpam-1242	137	1	now	now	ADV
ejpam-1242	137	2	the	the	DET
ejpam-1242	137	3	assertion	assertion	NOUN
ejpam-1242	137	4	follows	follow	VERB
ejpam-1242	137	5	from	from	ADP
ejpam-1242	137	6	(	(	PUNCT
ejpam-1242	137	7	i	i	NOUN
ejpam-1242	137	8	)	)	PUNCT
ejpam-1242	137	9	.	.	PUNCT
ejpam-1242	138	1	(	(	PUNCT
ejpam-1242	138	2	iv	iv	X
ejpam-1242	138	3	)	)	PUNCT
ejpam-1242	138	4	first	first	ADV
ejpam-1242	138	5	we	we	PRON
ejpam-1242	138	6	show	show	VERB
ejpam-1242	138	7	that	that	SCONJ
ejpam-1242	138	8	i(m	i(m	NOUN
ejpam-1242	138	9	/	/	SYM
ejpam-1242	138	10	n	n	CCONJ
ejpam-1242	138	11	)	)	PUNCT
ejpam-1242	138	12	⊆	⊆	NUM
ejpam-1242	138	13	(	(	PUNCT
ejpam-1242	138	14	i	i	PRON
ejpam-1242	138	15	m	m	VERB
ejpam-1242	138	16	+	+	X
ejpam-1242	138	17	n)/n	n)/n	PROPN
ejpam-1242	138	18	.	.	PUNCT
ejpam-1242	139	1	it	it	PRON
ejpam-1242	139	2	is	be	AUX
ejpam-1242	139	3	enough	enough	ADJ
ejpam-1242	139	4	to	to	PART
ejpam-1242	139	5	show	show	VERB
ejpam-1242	139	6	that	that	SCONJ
ejpam-1242	139	7	for	for	ADP
ejpam-1242	139	8	each	each	DET
ejpam-1242	139	9	r	r	NOUN
ejpam-1242	139	10	∈	∈	NOUN
ejpam-1242	139	11	i	i	PRON
ejpam-1242	139	12	and	and	CCONJ
ejpam-1242	139	13	for	for	ADP
ejpam-1242	139	14	each	each	DET
ejpam-1242	139	15	q+	q+	NOUN
ejpam-1242	139	16	n	n	PROPN
ejpam-1242	139	17	∈	∈	PROPN
ejpam-1242	139	18	m	m	PROPN
ejpam-1242	139	19	/	/	SYM
ejpam-1242	139	20	n	n	PRON
ejpam-1242	139	21	we	we	PRON
ejpam-1242	139	22	have	have	VERB
ejpam-1242	139	23	r.(q	r.(q	NOUN
ejpam-1242	139	24	+	+	CCONJ
ejpam-1242	139	25	n	n	CCONJ
ejpam-1242	139	26	)	)	PUNCT
ejpam-1242	139	27	∈	∈	PROPN
ejpam-1242	139	28	(	(	PUNCT
ejpam-1242	139	29	i	i	NOUN
ejpam-1242	139	30	m	m	VERB
ejpam-1242	139	31	+	+	X
ejpam-1242	139	32	n)/n	n)/n	PROPN
ejpam-1242	139	33	.	.	PUNCT
ejpam-1242	140	1	let	let	VERB
ejpam-1242	140	2	r.(q+	r.(q+	PRON
ejpam-1242	140	3	n	n	CCONJ
ejpam-1242	140	4	)	)	PUNCT
ejpam-1242	140	5	=	=	SYM
ejpam-1242	141	1	q′	q′	NOUN
ejpam-1242	141	2	+	+	NUM
ejpam-1242	141	3	n	n	CCONJ
ejpam-1242	141	4	,	,	PUNCT
ejpam-1242	141	5	where	where	SCONJ
ejpam-1242	141	6	q′	q′	NOUN
ejpam-1242	141	7	∈qm	∈qm	PROPN
ejpam-1242	141	8	is	be	AUX
ejpam-1242	141	9	the	the	DET
ejpam-1242	141	10	unique	unique	ADJ
ejpam-1242	141	11	element	element	NOUN
ejpam-1242	142	1	such	such	ADJ
ejpam-1242	142	2	that	that	SCONJ
ejpam-1242	142	3	rq+	rq+	ADJ
ejpam-1242	142	4	n	n	PROPN
ejpam-1242	142	5	⊆	⊆	NUM
ejpam-1242	142	6	q′+	q′+	NOUN
ejpam-1242	142	7	n	n	NOUN
ejpam-1242	142	8	,	,	PUNCT
ejpam-1242	142	9	so	so	ADV
ejpam-1242	142	10	rq+	rq+	ADJ
ejpam-1242	142	11	n=	n=	ADJ
ejpam-1242	142	12	q′	q′	NOUN
ejpam-1242	142	13	+	+	CCONJ
ejpam-1242	142	14	n′	n′	PROPN
ejpam-1242	142	15	for	for	ADP
ejpam-1242	142	16	some	some	DET
ejpam-1242	142	17	n	n	NOUN
ejpam-1242	142	18	,	,	PUNCT
ejpam-1242	142	19	n′	n′	PROPN
ejpam-1242	142	20	∈	∈	PROPN
ejpam-1242	142	21	n	n	ADV
ejpam-1242	142	22	.	.	PUNCT
ejpam-1242	143	1	it	it	PRON
ejpam-1242	143	2	follows	follow	VERB
ejpam-1242	143	3	that	that	SCONJ
ejpam-1242	143	4	q′	q′	NOUN
ejpam-1242	143	5	∈qm	∈qm	NOUN
ejpam-1242	143	6	∩	∩	NOUN
ejpam-1242	143	7	(	(	PUNCT
ejpam-1242	143	8	i	i	NOUN
ejpam-1242	143	9	m	m	VERB
ejpam-1242	143	10	+	+	NOUN
ejpam-1242	143	11	n	n	CCONJ
ejpam-1242	143	12	)	)	PUNCT
ejpam-1242	143	13	since	since	SCONJ
ejpam-1242	143	14	i	i	PRON
ejpam-1242	143	15	m	m	VERB
ejpam-1242	143	16	+	+	NOUN
ejpam-1242	143	17	n	n	VERB
ejpam-1242	143	18	is	be	AUX
ejpam-1242	143	19	a	a	DET
ejpam-1242	143	20	k	k	NOUN
ejpam-1242	143	21	-	-	NOUN
ejpam-1242	143	22	subsemimodule	subsemimodule	NOUN
ejpam-1242	143	23	by	by	ADP
ejpam-1242	143	24	(	(	PUNCT
ejpam-1242	143	25	ii	ii	NOUN
ejpam-1242	143	26	)	)	PUNCT
ejpam-1242	143	27	and	and	CCONJ
ejpam-1242	143	28	(	(	PUNCT
ejpam-1242	143	29	iii	iii	NOUN
ejpam-1242	143	30	)	)	PUNCT
ejpam-1242	143	31	.	.	PUNCT
ejpam-1242	144	1	thus	thus	ADV
ejpam-1242	144	2	r.(q+	r.(q+	PRON
ejpam-1242	144	3	n	n	CCONJ
ejpam-1242	144	4	)	)	PUNCT
ejpam-1242	144	5	∈	∈	PROPN
ejpam-1242	144	6	(	(	PUNCT
ejpam-1242	144	7	i	i	NOUN
ejpam-1242	144	8	m	m	VERB
ejpam-1242	144	9	+	+	X
ejpam-1242	144	10	n)/n	n)/n	PROPN
ejpam-1242	144	11	.	.	PUNCT
ejpam-1242	145	1	for	for	ADP
ejpam-1242	145	2	the	the	DET
ejpam-1242	145	3	reverse	reverse	ADJ
ejpam-1242	145	4	inclusion	inclusion	NOUN
ejpam-1242	145	5	,	,	PUNCT
ejpam-1242	145	6	assume	assume	VERB
ejpam-1242	145	7	that	that	SCONJ
ejpam-1242	145	8	q1	q1	PROPN
ejpam-1242	145	9	+	+	CCONJ
ejpam-1242	145	10	n	n	CCONJ
ejpam-1242	145	11	∈	∈	NOUN
ejpam-1242	145	12	(	(	PUNCT
ejpam-1242	145	13	i	i	NOUN
ejpam-1242	145	14	m	m	VERB
ejpam-1242	145	15	+	+	CCONJ
ejpam-1242	145	16	n)/n	n)/n	PROPN
ejpam-1242	145	17	,	,	PUNCT
ejpam-1242	145	18	where	where	SCONJ
ejpam-1242	145	19	q1	q1	PROPN
ejpam-1242	145	20	∈	∈	PROPN
ejpam-1242	145	21	qm	qm	PROPN
ejpam-1242	145	22	∩	∩	PROPN
ejpam-1242	145	23	(	(	PUNCT
ejpam-1242	145	24	i	i	NOUN
ejpam-1242	145	25	m	m	VERB
ejpam-1242	145	26	+	+	NOUN
ejpam-1242	145	27	n	n	CCONJ
ejpam-1242	145	28	)	)	PUNCT
ejpam-1242	145	29	.	.	PUNCT
ejpam-1242	146	1	then	then	ADV
ejpam-1242	146	2	there	there	PRON
ejpam-1242	146	3	are	be	VERB
ejpam-1242	146	4	elements	element	NOUN
ejpam-1242	146	5	q′	q′	NOUN
ejpam-1242	147	1	i	i	PRON
ejpam-1242	147	2	∈	∈	PROPN
ejpam-1242	147	3	qm	qm	PROPN
ejpam-1242	147	4	,	,	PUNCT
ejpam-1242	147	5	t	t	PROPN
ejpam-1242	148	1	i	i	PRON
ejpam-1242	148	2	∈	∈	PROPN
ejpam-1242	148	3	n	n	X
ejpam-1242	148	4	,	,	PUNCT
ejpam-1242	148	5	ri	ri	PROPN
ejpam-1242	148	6	∈	∈	PROPN
ejpam-1242	148	7	r	r	NOUN
ejpam-1242	148	8	and	and	CCONJ
ejpam-1242	148	9	n	n	CCONJ
ejpam-1242	148	10	∈	∈	PROPN
ejpam-1242	149	1	n	n	PRON
ejpam-1242	149	2	such	such	ADJ
ejpam-1242	149	3	that	that	DET
ejpam-1242	149	4	q1	q1	PROPN
ejpam-1242	149	5	=	=	PUNCT
ejpam-1242	149	6	∑s	∑s	PROPN
ejpam-1242	149	7	i=1	i=1	PROPN
ejpam-1242	149	8	ri(q	ri(q	VERB
ejpam-1242	150	1	′	′	NUM
ejpam-1242	150	2	i	i	PRON
ejpam-1242	150	3	+	+	CCONJ
ejpam-1242	150	4	ni	ni	PROPN
ejpam-1242	150	5	)	)	PUNCT
ejpam-1242	150	6	+	+	NUM
ejpam-1242	150	7	n	n	CCONJ
ejpam-1242	150	8	;	;	PUNCT
ejpam-1242	150	9	hence	hence	ADV
ejpam-1242	150	10	q1	q1	VERB
ejpam-1242	151	1	=	=	PUNCT
ejpam-1242	151	2	∑s	∑s	PROPN
ejpam-1242	151	3	i=1	i=1	PROPN
ejpam-1242	152	1	riq	riq	NOUN
ejpam-1242	153	1	′	′	NUM
ejpam-1242	154	1	i	i	PRON
ejpam-1242	155	1	+	+	CCONJ
ejpam-1242	155	2	x	x	X
ejpam-1242	155	3	,	,	PUNCT
ejpam-1242	155	4	where	where	SCONJ
ejpam-1242	156	1	x	x	SYM
ejpam-1242	156	2	∈	∈	PROPN
ejpam-1242	156	3	n	n	ADV
ejpam-1242	156	4	.	.	PUNCT
ejpam-1242	157	1	let	let	VERB
ejpam-1242	157	2	q0	q0	PROPN
ejpam-1242	157	3	+	+	CCONJ
ejpam-1242	157	4	n	n	NUM
ejpam-1242	157	5	is	be	AUX
ejpam-1242	157	6	the	the	DET
ejpam-1242	157	7	zero	zero	NUM
ejpam-1242	157	8	in	in	ADP
ejpam-1242	157	9	m	m	PROPN
ejpam-1242	157	10	/	/	SYM
ejpam-1242	157	11	n	n	PROPN
ejpam-1242	157	12	.	.	PUNCT
ejpam-1242	158	1	clearly	clearly	ADV
ejpam-1242	158	2	x	x	X
ejpam-1242	158	3	+	+	NUM
ejpam-1242	158	4	n	n	CCONJ
ejpam-1242	158	5	⊆	⊆	NUM
ejpam-1242	158	6	n	n	NOUN
ejpam-1242	158	7	.	.	PUNCT
ejpam-1242	159	1	assume	assume	VERB
ejpam-1242	159	2	that	that	SCONJ
ejpam-1242	159	3	y	y	PROPN
ejpam-1242	159	4	∈	∈	PROPN
ejpam-1242	159	5	n	n	ADV
ejpam-1242	159	6	.	.	PUNCT
ejpam-1242	160	1	since	since	SCONJ
ejpam-1242	160	2	n	n	NOUN
ejpam-1242	160	3	=	=	SYM
ejpam-1242	160	4	q0+n	q0+n	PROPN
ejpam-1242	160	5	by	by	ADP
ejpam-1242	160	6	[	[	PUNCT
ejpam-1242	160	7	7	7	NUM
ejpam-1242	160	8	,	,	PUNCT
ejpam-1242	160	9	lemma	lemma	PROPN
ejpam-1242	160	10	2.3	2.3	NUM
ejpam-1242	160	11	]	]	PUNCT
ejpam-1242	160	12	,	,	PUNCT
ejpam-1242	160	13	there	there	PRON
ejpam-1242	160	14	exist	exist	VERB
ejpam-1242	160	15	a	a	DET
ejpam-1242	160	16	,	,	PUNCT
ejpam-1242	160	17	b	b	NOUN
ejpam-1242	160	18	,	,	PUNCT
ejpam-1242	160	19	c	c	PROPN
ejpam-1242	160	20	∈	∈	PROPN
ejpam-1242	160	21	n	n	INTJ
ejpam-1242	160	22	with	with	ADP
ejpam-1242	161	1	y	y	PROPN
ejpam-1242	161	2	=	=	PUNCT
ejpam-1242	161	3	q0	q0	PROPN
ejpam-1242	161	4	+	+	CCONJ
ejpam-1242	161	5	a	a	PRON
ejpam-1242	161	6	,	,	PUNCT
ejpam-1242	161	7	x	x	X
ejpam-1242	162	1	=	=	PRON
ejpam-1242	162	2	q0	q0	PROPN
ejpam-1242	162	3	+	+	CCONJ
ejpam-1242	162	4	b	b	NOUN
ejpam-1242	162	5	and	and	CCONJ
ejpam-1242	162	6	b+	b+	NOUN
ejpam-1242	162	7	c	c	X
ejpam-1242	162	8	=	=	SYM
ejpam-1242	162	9	0	0	NUM
ejpam-1242	162	10	;	;	PUNCT
ejpam-1242	162	11	so	so	ADV
ejpam-1242	163	1	y	y	PROPN
ejpam-1242	163	2	=	=	NOUN
ejpam-1242	164	1	q0	q0	PROPN
ejpam-1242	165	1	+	+	X
ejpam-1242	165	2	b+a+	b+a+	NOUN
ejpam-1242	165	3	c	c	NOUN
ejpam-1242	166	1	=	=	PUNCT
ejpam-1242	166	2	x+a+	x+a+	PUNCT
ejpam-1242	166	3	c	c	X
ejpam-1242	166	4	∈	∈	PROPN
ejpam-1242	166	5	x+n	x+n	X
ejpam-1242	166	6	;	;	PUNCT
ejpam-1242	166	7	hence	hence	ADV
ejpam-1242	166	8	x+n	x+n	PUNCT
ejpam-1242	166	9	=	=	PUNCT
ejpam-1242	166	10	n	n	PROPN
ejpam-1242	166	11	.	.	PUNCT
ejpam-1242	167	1	an	an	DET
ejpam-1242	167	2	inspection	inspection	NOUN
ejpam-1242	167	3	will	will	AUX
ejpam-1242	167	4	show	show	VERB
ejpam-1242	167	5	that	that	DET
ejpam-1242	167	6	q1	q1	PROPN
ejpam-1242	167	7	+	+	CCONJ
ejpam-1242	167	8	n	n	PROPN
ejpam-1242	167	9	=	=	NOUN
ejpam-1242	167	10	∑s	∑s	PROPN
ejpam-1242	167	11	i=1	i=1	PROPN
ejpam-1242	167	12	ri.(q	ri.(q	NOUN
ejpam-1242	168	1	′	′	NUM
ejpam-1242	168	2	i	i	PRON
ejpam-1242	169	1	+	+	NOUN
ejpam-1242	169	2	n	n	CCONJ
ejpam-1242	169	3	)	)	PUNCT
ejpam-1242	169	4	∈	∈	NOUN
ejpam-1242	169	5	i(m	i(m	NOUN
ejpam-1242	169	6	/	/	SYM
ejpam-1242	169	7	n	n	CCONJ
ejpam-1242	169	8	)	)	PUNCT
ejpam-1242	169	9	.	.	PUNCT
ejpam-1242	170	1	thus	thus	ADV
ejpam-1242	170	2	(	(	PUNCT
ejpam-1242	170	3	i	i	PRON
ejpam-1242	170	4	m	m	VERB
ejpam-1242	170	5	+	+	VERB
ejpam-1242	170	6	n)/n	n)/n	PROPN
ejpam-1242	170	7	⊆	⊆	NUM
ejpam-1242	170	8	i(m	i(m	NOUN
ejpam-1242	170	9	/	/	SYM
ejpam-1242	170	10	n	n	CCONJ
ejpam-1242	170	11	)	)	PUNCT
ejpam-1242	170	12	,	,	PUNCT
ejpam-1242	170	13	and	and	CCONJ
ejpam-1242	170	14	so	so	ADV
ejpam-1242	170	15	we	we	PRON
ejpam-1242	170	16	have	have	VERB
ejpam-1242	170	17	equality	equality	NOUN
ejpam-1242	170	18	.	.	PUNCT
ejpam-1242	171	1	theorem	theorem	NOUN
ejpam-1242	171	2	2	2	NUM
ejpam-1242	171	3	.	.	PUNCT
ejpam-1242	172	1	let	let	VERB
ejpam-1242	172	2	n	n	PRON
ejpam-1242	172	3	be	be	AUX
ejpam-1242	172	4	a	a	DET
ejpam-1242	172	5	strong	strong	ADJ
ejpam-1242	172	6	qm	qm	NOUN
ejpam-1242	172	7	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	172	8	of	of	ADP
ejpam-1242	172	9	a	a	DET
ejpam-1242	172	10	strong	strong	ADJ
ejpam-1242	172	11	multiplication	multiplication	NOUN
ejpam-1242	172	12	semimodule	semimodule	NOUN
ejpam-1242	172	13	m	m	VERB
ejpam-1242	172	14	over	over	ADP
ejpam-1242	172	15	a	a	DET
ejpam-1242	172	16	semiring	semire	VERB
ejpam-1242	172	17	r.	r.	PROPN
ejpam-1242	172	18	then	then	ADV
ejpam-1242	172	19	m	m	PROPN
ejpam-1242	172	20	/	/	SYM
ejpam-1242	172	21	n	n	PROPN
ejpam-1242	172	22	is	be	AUX
ejpam-1242	172	23	a	a	DET
ejpam-1242	172	24	strong	strong	ADJ
ejpam-1242	172	25	multiplication	multiplication	NOUN
ejpam-1242	172	26	r	r	NOUN
ejpam-1242	172	27	-	-	PUNCT
ejpam-1242	172	28	semimodule	semimodule	NOUN
ejpam-1242	172	29	.	.	PUNCT
ejpam-1242	173	1	proof	proof	NOUN
ejpam-1242	173	2	.	.	PUNCT
ejpam-1242	174	1	let	let	VERB
ejpam-1242	174	2	l	l	NOUN
ejpam-1242	174	3	be	be	AUX
ejpam-1242	174	4	a	a	DET
ejpam-1242	174	5	k	k	NOUN
ejpam-1242	174	6	-	-	NOUN
ejpam-1242	174	7	subsemimodule	subsemimodule	NOUN
ejpam-1242	174	8	of	of	ADP
ejpam-1242	174	9	m	m	PROPN
ejpam-1242	174	10	/	/	SYM
ejpam-1242	174	11	n	n	PROPN
ejpam-1242	174	12	.	.	PUNCT
ejpam-1242	175	1	then	then	ADV
ejpam-1242	175	2	by	by	ADP
ejpam-1242	175	3	[	[	X
ejpam-1242	175	4	7	7	NUM
ejpam-1242	175	5	,	,	PUNCT
ejpam-1242	175	6	theorem	theorem	VERB
ejpam-1242	175	7	3.6	3.6	NUM
ejpam-1242	175	8	]	]	PUNCT
ejpam-1242	175	9	,	,	PUNCT
ejpam-1242	175	10	l	l	PROPN
ejpam-1242	175	11	=	=	SYM
ejpam-1242	175	12	t	t	PROPN
ejpam-1242	175	13	/	/	SYM
ejpam-1242	175	14	n	n	PROPN
ejpam-1242	175	15	for	for	ADP
ejpam-1242	175	16	some	some	DET
ejpam-1242	175	17	k	k	ADJ
ejpam-1242	175	18	-	-	PUNCT
ejpam-1242	175	19	subsemimodule	subsemimodule	ADJ
ejpam-1242	175	20	t	t	PROPN
ejpam-1242	175	21	of	of	ADP
ejpam-1242	175	22	m	m	PROPN
ejpam-1242	175	23	with	with	ADP
ejpam-1242	175	24	n	n	PRON
ejpam-1242	175	25	⊆	⊆	NUM
ejpam-1242	175	26	t	t	NOUN
ejpam-1242	175	27	,	,	PUNCT
ejpam-1242	175	28	so	so	CCONJ
ejpam-1242	175	29	there	there	PRON
ejpam-1242	175	30	exists	exist	VERB
ejpam-1242	175	31	a	a	DET
ejpam-1242	175	32	strong	strong	ADJ
ejpam-1242	175	33	ideal	ideal	NOUN
ejpam-1242	175	34	i	i	PRON
ejpam-1242	175	35	of	of	ADP
ejpam-1242	175	36	r	r	NOUN
ejpam-1242	175	37	such	such	ADJ
ejpam-1242	175	38	that	that	DET
ejpam-1242	175	39	t	t	NOUN
ejpam-1242	175	40	=	=	PUNCT
ejpam-1242	176	1	i	i	PRON
ejpam-1242	176	2	m	m	VERB
ejpam-1242	176	3	.	.	PUNCT
ejpam-1242	177	1	therefore	therefore	ADV
ejpam-1242	177	2	i(m	i(m	NOUN
ejpam-1242	177	3	/	/	SYM
ejpam-1242	177	4	n	n	CCONJ
ejpam-1242	177	5	)	)	PUNCT
ejpam-1242	177	6	=	=	PUNCT
ejpam-1242	177	7	(	(	PUNCT
ejpam-1242	177	8	i	i	PRON
ejpam-1242	177	9	m	m	VERB
ejpam-1242	177	10	+	+	CCONJ
ejpam-1242	177	11	n)/n	n)/n	PROPN
ejpam-1242	177	12	=	=	SYM
ejpam-1242	177	13	t	t	PROPN
ejpam-1242	177	14	/	/	SYM
ejpam-1242	177	15	n	n	NOUN
ejpam-1242	177	16	=	=	SYM
ejpam-1242	177	17	l	l	NOUN
ejpam-1242	177	18	by	by	ADP
ejpam-1242	177	19	proposition	proposition	NOUN
ejpam-1242	177	20	1	1	NUM
ejpam-1242	177	21	(	(	PUNCT
ejpam-1242	177	22	iv	iv	NUM
ejpam-1242	177	23	)	)	PUNCT
ejpam-1242	177	24	,	,	PUNCT
ejpam-1242	177	25	as	as	SCONJ
ejpam-1242	177	26	needed	need	VERB
ejpam-1242	177	27	.	.	PUNCT
ejpam-1242	178	1	theorem	theorem	NOUN
ejpam-1242	178	2	3	3	X
ejpam-1242	178	3	.	.	PUNCT
ejpam-1242	179	1	let	let	VERB
ejpam-1242	179	2	n	n	PRON
ejpam-1242	179	3	be	be	AUX
ejpam-1242	179	4	a	a	DET
ejpam-1242	179	5	strong	strong	ADJ
ejpam-1242	179	6	qm	qm	NOUN
ejpam-1242	179	7	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	179	8	of	of	ADP
ejpam-1242	179	9	a	a	DET
ejpam-1242	179	10	very	very	ADV
ejpam-1242	179	11	strong	strong	ADJ
ejpam-1242	179	12	multiplication	multiplication	NOUN
ejpam-1242	179	13	semimodule	semimodule	NOUN
ejpam-1242	179	14	m	m	VERB
ejpam-1242	179	15	over	over	ADP
ejpam-1242	179	16	a	a	DET
ejpam-1242	179	17	semiring	semire	VERB
ejpam-1242	179	18	r.	r.	PROPN
ejpam-1242	179	19	then	then	ADV
ejpam-1242	179	20	m	m	PROPN
ejpam-1242	179	21	/	/	SYM
ejpam-1242	179	22	n	n	PROPN
ejpam-1242	179	23	is	be	AUX
ejpam-1242	179	24	a	a	DET
ejpam-1242	179	25	very	very	ADV
ejpam-1242	179	26	strong	strong	ADJ
ejpam-1242	179	27	multiplication	multiplication	NOUN
ejpam-1242	179	28	r	r	NOUN
ejpam-1242	179	29	-	-	PUNCT
ejpam-1242	179	30	semimodule	semimodule	NOUN
ejpam-1242	179	31	.	.	PUNCT
ejpam-1242	180	1	proof	proof	NOUN
ejpam-1242	180	2	.	.	PUNCT
ejpam-1242	181	1	by	by	ADP
ejpam-1242	181	2	theorem	theorem	NOUN
ejpam-1242	181	3	2	2	NUM
ejpam-1242	181	4	and	and	CCONJ
ejpam-1242	181	5	definition	definition	NOUN
ejpam-1242	181	6	5	5	NUM
ejpam-1242	181	7	,	,	PUNCT
ejpam-1242	181	8	it	it	PRON
ejpam-1242	181	9	suffices	suffice	VERB
ejpam-1242	181	10	to	to	PART
ejpam-1242	181	11	show	show	VERB
ejpam-1242	181	12	that	that	SCONJ
ejpam-1242	181	13	m	m	NOUN
ejpam-1242	181	14	/	/	SYM
ejpam-1242	181	15	n	n	PROPN
ejpam-1242	181	16	is	be	AUX
ejpam-1242	181	17	a	a	DET
ejpam-1242	181	18	very	very	ADV
ejpam-1242	181	19	strong	strong	ADJ
ejpam-1242	181	20	semimodule	semimodule	NOUN
ejpam-1242	181	21	.	.	PUNCT
ejpam-1242	182	1	let	let	VERB
ejpam-1242	182	2	i	i	PRON
ejpam-1242	182	3	be	be	AUX
ejpam-1242	182	4	an	an	DET
ejpam-1242	182	5	ideal	ideal	NOUN
ejpam-1242	182	6	of	of	ADP
ejpam-1242	182	7	r	r	NOUN
ejpam-1242	182	8	and	and	CCONJ
ejpam-1242	182	9	q+	q+	NUM
ejpam-1242	182	10	n	n	CCONJ
ejpam-1242	182	11	∈	∈	PROPN
ejpam-1242	182	12	m	m	PROPN
ejpam-1242	182	13	/	/	SYM
ejpam-1242	182	14	n	n	PROPN
ejpam-1242	182	15	,	,	PUNCT
ejpam-1242	182	16	where	where	SCONJ
ejpam-1242	182	17	q	q	PROPN
ejpam-1242	182	18	∈	∈	PROPN
ejpam-1242	182	19	qm	qm	PROPN
ejpam-1242	182	20	and	and	CCONJ
ejpam-1242	182	21	set	set	VERB
ejpam-1242	182	22	j	j	PROPN
ejpam-1242	182	23	=	=	PUNCT
ejpam-1242	182	24	{	{	PUNCT
ejpam-1242	182	25	r	r	NOUN
ejpam-1242	182	26	∈	∈	PROPN
ejpam-1242	182	27	r	r	NOUN
ejpam-1242	182	28	:	:	PUNCT
ejpam-1242	182	29	r.(q+	r.(q+	PROPN
ejpam-1242	182	30	n	n	CCONJ
ejpam-1242	182	31	)	)	PUNCT
ejpam-1242	182	32	∈	∈	PROPN
ejpam-1242	182	33	i(m	i(m	NOUN
ejpam-1242	182	34	/	/	SYM
ejpam-1242	182	35	n	n	CCONJ
ejpam-1242	182	36	)	)	PUNCT
ejpam-1242	182	37	}	}	PUNCT
ejpam-1242	182	38	;	;	PUNCT
ejpam-1242	182	39	we	we	PRON
ejpam-1242	182	40	show	show	VERB
ejpam-1242	182	41	that	that	SCONJ
ejpam-1242	182	42	j	j	PROPN
ejpam-1242	182	43	is	be	AUX
ejpam-1242	182	44	a	a	DET
ejpam-1242	182	45	strong	strong	ADJ
ejpam-1242	182	46	k	k	NOUN
ejpam-1242	182	47	-	-	NOUN
ejpam-1242	182	48	ideal	ideal	NOUN
ejpam-1242	182	49	of	of	ADP
ejpam-1242	182	50	r.	r.	PROPN
ejpam-1242	182	51	let	let	VERB
ejpam-1242	182	52	r	r	NOUN
ejpam-1242	182	53	,	,	PUNCT
ejpam-1242	182	54	r	r	NOUN
ejpam-1242	182	55	′	′	NUM
ejpam-1242	182	56	∈	∈	PROPN
ejpam-1242	182	57	j	j	NOUN
ejpam-1242	182	58	.	.	PUNCT
ejpam-1242	183	1	there	there	PRON
ejpam-1242	183	2	are	be	VERB
ejpam-1242	183	3	unique	unique	ADJ
ejpam-1242	183	4	elements	element	NOUN
ejpam-1242	183	5	q1,q2	q1,q2	PROPN
ejpam-1242	183	6	∈qm	∈qm	NOUN
ejpam-1242	183	7	such	such	ADJ
ejpam-1242	183	8	that	that	SCONJ
ejpam-1242	183	9	r.(q+	r.(q+	PROPN
ejpam-1242	183	10	n	n	CCONJ
ejpam-1242	183	11	)	)	PUNCT
ejpam-1242	183	12	=	=	SYM
ejpam-1242	183	13	q1	q1	PROPN
ejpam-1242	183	14	+	+	CCONJ
ejpam-1242	183	15	n	n	CCONJ
ejpam-1242	183	16	∈	∈	NOUN
ejpam-1242	183	17	i(m	i(m	NOUN
ejpam-1242	183	18	/	/	SYM
ejpam-1242	183	19	n	n	CCONJ
ejpam-1242	183	20	)	)	PUNCT
ejpam-1242	183	21	and	and	CCONJ
ejpam-1242	183	22	s.	s.	PROPN
ejpam-1242	183	23	atani	atani	PROPN
ejpam-1242	183	24	,	,	PUNCT
ejpam-1242	183	25	r.	r.	PROPN
ejpam-1242	183	26	atrani	atrani	PROPN
ejpam-1242	183	27	,	,	PUNCT
ejpam-1242	183	28	ü.	ü.	NOUN
ejpam-1242	183	29	tekir	tekir	PROPN
ejpam-1242	183	30	/	/	SYM
ejpam-1242	183	31	eur	eur	PROPN
ejpam-1242	183	32	.	.	PUNCT
ejpam-1242	184	1	j.	j.	PROPN
ejpam-1242	184	2	pure	pure	PROPN
ejpam-1242	184	3	appl	appl	PROPN
ejpam-1242	184	4	.	.	PROPN
ejpam-1242	184	5	math	math	PROPN
ejpam-1242	184	6	,	,	PUNCT
ejpam-1242	184	7	4	4	NUM
ejpam-1242	184	8	(	(	PUNCT
ejpam-1242	184	9	2011	2011	NUM
ejpam-1242	184	10	)	)	PUNCT
ejpam-1242	184	11	,	,	PUNCT
ejpam-1242	184	12	251	251	NUM
ejpam-1242	184	13	-	-	SYM
ejpam-1242	184	14	265	265	NUM
ejpam-1242	184	15	256	256	NUM
ejpam-1242	184	16	r	r	NOUN
ejpam-1242	184	17	′.(q+n	′.(q+n	PROPN
ejpam-1242	184	18	)	)	PUNCT
ejpam-1242	184	19	=	=	SYM
ejpam-1242	184	20	q2+n	q2+n	ADP
ejpam-1242	184	21	∈	∈	NOUN
ejpam-1242	184	22	i(m	i(m	NOUN
ejpam-1242	184	23	/	/	SYM
ejpam-1242	184	24	n	n	CCONJ
ejpam-1242	184	25	)	)	PUNCT
ejpam-1242	184	26	,	,	PUNCT
ejpam-1242	184	27	where	where	SCONJ
ejpam-1242	184	28	rq+n	rq+n	NOUN
ejpam-1242	184	29	⊆	⊆	NUM
ejpam-1242	184	30	q1+n	q1+n	INTJ
ejpam-1242	184	31	,	,	PUNCT
ejpam-1242	184	32	r	r	NOUN
ejpam-1242	184	33	′q+n	′q+n	NUM
ejpam-1242	184	34	⊆	⊆	NUM
ejpam-1242	184	35	q2+n	q2+n	NOUN
ejpam-1242	184	36	.	.	PUNCT
ejpam-1242	185	1	then	then	ADV
ejpam-1242	185	2	there	there	PRON
ejpam-1242	185	3	exists	exist	VERB
ejpam-1242	185	4	a	a	DET
ejpam-1242	185	5	unique	unique	ADJ
ejpam-1242	185	6	element	element	NOUN
ejpam-1242	185	7	q3	q3	NOUN
ejpam-1242	185	8	∈qm	∈qm	NOUN
ejpam-1242	185	9	such	such	ADJ
ejpam-1242	185	10	that	that	SCONJ
ejpam-1242	185	11	(	(	PUNCT
ejpam-1242	185	12	q1+n)⊕	q1+n)⊕	PROPN
ejpam-1242	185	13	(	(	PUNCT
ejpam-1242	185	14	q2+n	q2+n	NOUN
ejpam-1242	185	15	)	)	PUNCT
ejpam-1242	185	16	=	=	PUNCT
ejpam-1242	186	1	q3+n	q3+n	PROPN
ejpam-1242	186	2	,	,	PUNCT
ejpam-1242	186	3	where	where	SCONJ
ejpam-1242	186	4	q1	q1	VERB
ejpam-1242	186	5	+	+	CCONJ
ejpam-1242	186	6	q2+n	q2+n	PROPN
ejpam-1242	186	7	⊆	⊆	NUM
ejpam-1242	186	8	q3+n	q3+n	ADV
ejpam-1242	186	9	,	,	PUNCT
ejpam-1242	186	10	so	so	CCONJ
ejpam-1242	186	11	(	(	PUNCT
ejpam-1242	186	12	r	r	NOUN
ejpam-1242	186	13	+	+	NUM
ejpam-1242	186	14	r	r	NOUN
ejpam-1242	186	15	′)q	′)q	PROPN
ejpam-1242	186	16	+	+	CCONJ
ejpam-1242	186	17	n	n	NUM
ejpam-1242	186	18	⊆	⊆	NUM
ejpam-1242	186	19	q1	q1	NOUN
ejpam-1242	186	20	+	+	CCONJ
ejpam-1242	186	21	q2	q2	NOUN
ejpam-1242	186	22	+	+	CCONJ
ejpam-1242	186	23	n	n	CCONJ
ejpam-1242	186	24	⊆	⊆	NUM
ejpam-1242	186	25	q3	q3	NOUN
ejpam-1242	186	26	+	+	CCONJ
ejpam-1242	186	27	n	n	CCONJ
ejpam-1242	186	28	∈	∈	NOUN
ejpam-1242	186	29	i(m	i(m	NOUN
ejpam-1242	186	30	/	/	SYM
ejpam-1242	186	31	n	n	CCONJ
ejpam-1242	186	32	)	)	PUNCT
ejpam-1242	186	33	;	;	PUNCT
ejpam-1242	186	34	hence	hence	ADV
ejpam-1242	186	35	r	r	NOUN
ejpam-1242	186	36	+	+	NUM
ejpam-1242	186	37	r	r	NOUN
ejpam-1242	186	38	′	′	NUM
ejpam-1242	187	1	∈	∈	PROPN
ejpam-1242	187	2	j	j	PROPN
ejpam-1242	187	3	.	.	PUNCT
ejpam-1242	188	1	similarly	similarly	ADV
ejpam-1242	188	2	,	,	PUNCT
ejpam-1242	188	3	if	if	SCONJ
ejpam-1242	188	4	s	s	X
ejpam-1242	188	5	∈	∈	PROPN
ejpam-1242	188	6	r	r	NOUN
ejpam-1242	188	7	,	,	PUNCT
ejpam-1242	188	8	then	then	ADV
ejpam-1242	188	9	sr	sr	PROPN
ejpam-1242	188	10	∈	∈	PROPN
ejpam-1242	188	11	j	j	PROPN
ejpam-1242	188	12	.	.	PUNCT
ejpam-1242	189	1	thus	thus	ADV
ejpam-1242	189	2	j	j	PROPN
ejpam-1242	189	3	is	be	AUX
ejpam-1242	189	4	an	an	DET
ejpam-1242	189	5	ideal	ideal	NOUN
ejpam-1242	189	6	of	of	ADP
ejpam-1242	189	7	r.	r.	PROPN
ejpam-1242	189	8	by	by	ADP
ejpam-1242	189	9	assumption	assumption	NOUN
ejpam-1242	189	10	,	,	PUNCT
ejpam-1242	189	11	there	there	PRON
ejpam-1242	189	12	must	must	AUX
ejpam-1242	189	13	exists	exist	VERB
ejpam-1242	189	14	a	a	DET
ejpam-1242	189	15	strong	strong	ADJ
ejpam-1242	189	16	ideal	ideal	NOUN
ejpam-1242	189	17	i	i	PRON
ejpam-1242	189	18	′	′	VERB
ejpam-1242	189	19	of	of	ADP
ejpam-1242	189	20	r	r	NOUN
ejpam-1242	189	21	such	such	ADJ
ejpam-1242	189	22	that	that	SCONJ
ejpam-1242	189	23	n	n	NOUN
ejpam-1242	189	24	=	=	SYM
ejpam-1242	189	25	i	i	PRON
ejpam-1242	189	26	′m	′m	PROPN
ejpam-1242	189	27	,	,	PUNCT
ejpam-1242	189	28	and	and	CCONJ
ejpam-1242	189	29	the	the	DET
ejpam-1242	189	30	set	set	VERB
ejpam-1242	189	31	j1	j1	NOUN
ejpam-1242	189	32	=	=	PUNCT
ejpam-1242	189	33	{	{	PUNCT
ejpam-1242	189	34	r	r	NOUN
ejpam-1242	189	35	∈	∈	PROPN
ejpam-1242	189	36	r	r	NOUN
ejpam-1242	189	37	:	:	PUNCT
ejpam-1242	190	1	rq	rq	VERB
ejpam-1242	190	2	∈	∈	PROPN
ejpam-1242	190	3	(	(	PUNCT
ejpam-1242	190	4	i	i	PRON
ejpam-1242	191	1	+	+	NUM
ejpam-1242	191	2	i	i	PRON
ejpam-1242	191	3	′)m	′)m	AUX
ejpam-1242	191	4	}	}	PUNCT
ejpam-1242	191	5	is	be	AUX
ejpam-1242	191	6	a	a	DET
ejpam-1242	191	7	strong	strong	ADJ
ejpam-1242	191	8	k	k	NOUN
ejpam-1242	191	9	-	-	NOUN
ejpam-1242	191	10	ideal	ideal	NOUN
ejpam-1242	191	11	of	of	ADP
ejpam-1242	191	12	r	r	NOUN
ejpam-1242	191	13	since	since	SCONJ
ejpam-1242	191	14	m	m	PROPN
ejpam-1242	191	15	is	be	AUX
ejpam-1242	191	16	a	a	DET
ejpam-1242	191	17	very	very	ADV
ejpam-1242	191	18	strong	strong	ADJ
ejpam-1242	191	19	semimodule	semimodule	NOUN
ejpam-1242	191	20	.	.	PUNCT
ejpam-1242	192	1	let	let	VERB
ejpam-1242	192	2	t	t	PROPN
ejpam-1242	192	3	∈	∈	PROPN
ejpam-1242	192	4	j	j	PROPN
ejpam-1242	192	5	.	.	PUNCT
ejpam-1242	193	1	then	then	ADV
ejpam-1242	193	2	tq	tq	INTJ
ejpam-1242	193	3	+	+	NUM
ejpam-1242	193	4	n	n	NUM
ejpam-1242	193	5	⊆	⊆	NUM
ejpam-1242	193	6	q′	q′	NOUN
ejpam-1242	193	7	+	+	CCONJ
ejpam-1242	193	8	n	n	CCONJ
ejpam-1242	193	9	∈	∈	NOUN
ejpam-1242	193	10	i(m	i(m	NOUN
ejpam-1242	193	11	/	/	SYM
ejpam-1242	193	12	n	n	CCONJ
ejpam-1242	193	13	)	)	PUNCT
ejpam-1242	193	14	for	for	ADP
ejpam-1242	193	15	some	some	DET
ejpam-1242	193	16	q′	q′	NOUN
ejpam-1242	193	17	∈	∈	PROPN
ejpam-1242	193	18	(	(	PUNCT
ejpam-1242	193	19	i	i	NOUN
ejpam-1242	193	20	m	m	VERB
ejpam-1242	193	21	+	+	VERB
ejpam-1242	193	22	i	i	PRON
ejpam-1242	193	23	′m)∩qm	′m)∩qm	VERB
ejpam-1242	193	24	,	,	PUNCT
ejpam-1242	193	25	so	so	SCONJ
ejpam-1242	193	26	tq	tq	ADV
ejpam-1242	193	27	∈	∈	PROPN
ejpam-1242	193	28	(	(	PUNCT
ejpam-1242	193	29	i	i	PRON
ejpam-1242	193	30	+	+	NUM
ejpam-1242	193	31	i	i	PRON
ejpam-1242	193	32	′)m	′)m	VERB
ejpam-1242	193	33	since	since	SCONJ
ejpam-1242	193	34	it	it	PRON
ejpam-1242	193	35	is	be	AUX
ejpam-1242	193	36	a	a	DET
ejpam-1242	193	37	ksubsemimodule	ksubsemimodule	NOUN
ejpam-1242	193	38	by	by	ADP
ejpam-1242	193	39	proposition	proposition	NOUN
ejpam-1242	193	40	1	1	NUM
ejpam-1242	193	41	(	(	PUNCT
ejpam-1242	193	42	iii	iii	NOUN
ejpam-1242	193	43	)	)	PUNCT
ejpam-1242	193	44	;	;	PUNCT
ejpam-1242	193	45	hence	hence	ADV
ejpam-1242	193	46	t	t	PROPN
ejpam-1242	193	47	∈	∈	PROPN
ejpam-1242	193	48	j1	j1	PROPN
ejpam-1242	193	49	.	.	PUNCT
ejpam-1242	194	1	thus	thus	ADV
ejpam-1242	194	2	j	j	PROPN
ejpam-1242	194	3	⊆	⊆	NUM
ejpam-1242	194	4	j1	j1	PROPN
ejpam-1242	194	5	.	.	PUNCT
ejpam-1242	195	1	it	it	PRON
ejpam-1242	195	2	follows	follow	VERB
ejpam-1242	195	3	that	that	SCONJ
ejpam-1242	195	4	j	j	PROPN
ejpam-1242	195	5	is	be	AUX
ejpam-1242	195	6	a	a	DET
ejpam-1242	195	7	strong	strong	ADJ
ejpam-1242	195	8	k	k	NOUN
ejpam-1242	195	9	-	-	NOUN
ejpam-1242	195	10	ideal	ideal	NOUN
ejpam-1242	195	11	of	of	ADP
ejpam-1242	195	12	r	r	NOUN
ejpam-1242	195	13	,	,	PUNCT
ejpam-1242	195	14	and	and	CCONJ
ejpam-1242	195	15	this	this	PRON
ejpam-1242	195	16	completes	complete	VERB
ejpam-1242	195	17	the	the	DET
ejpam-1242	195	18	proof	proof	NOUN
ejpam-1242	195	19	.	.	PUNCT
ejpam-1242	196	1	proposition	proposition	NOUN
ejpam-1242	196	2	2	2	NUM
ejpam-1242	196	3	.	.	PUNCT
ejpam-1242	197	1	let	let	VERB
ejpam-1242	197	2	m	m	PRON
ejpam-1242	197	3	be	be	AUX
ejpam-1242	197	4	a	a	DET
ejpam-1242	197	5	non	non	ADJ
ejpam-1242	197	6	-	-	ADJ
ejpam-1242	197	7	strong	strong	ADJ
ejpam-1242	197	8	semimodule	semimodule	NOUN
ejpam-1242	197	9	over	over	ADP
ejpam-1242	197	10	a	a	DET
ejpam-1242	197	11	semiring	semire	VERB
ejpam-1242	197	12	r	r	NOUN
ejpam-1242	197	13	with	with	ADP
ejpam-1242	197	14	m	m	PROPN
ejpam-1242	197	15	6=	6=	NUM
ejpam-1242	197	16	0	0	NUM
ejpam-1242	197	17	.	.	PUNCT
ejpam-1242	198	1	then	then	ADV
ejpam-1242	198	2	m	m	PROPN
ejpam-1242	198	3	has	have	VERB
ejpam-1242	198	4	at	at	ADV
ejpam-1242	198	5	least	least	ADJ
ejpam-1242	198	6	one	one	NUM
ejpam-1242	198	7	strong	strong	ADJ
ejpam-1242	198	8	maximal	maximal	ADJ
ejpam-1242	198	9	qm	qm	NOUN
ejpam-1242	198	10	-subsemimodule	-subsemimodule	PROPN
ejpam-1242	198	11	.	.	PUNCT
ejpam-1242	199	1	proof	proof	NOUN
ejpam-1242	199	2	.	.	PUNCT
ejpam-1242	200	1	since	since	SCONJ
ejpam-1242	200	2	{	{	PUNCT
ejpam-1242	200	3	0	0	NUM
ejpam-1242	200	4	m	m	VERB
ejpam-1242	200	5	}	}	PUNCT
ejpam-1242	200	6	is	be	AUX
ejpam-1242	200	7	a	a	DET
ejpam-1242	200	8	proper	proper	ADJ
ejpam-1242	200	9	strong	strong	ADJ
ejpam-1242	200	10	qm	qm	PROPN
ejpam-1242	200	11	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	200	12	of	of	ADP
ejpam-1242	200	13	m	m	NOUN
ejpam-1242	200	14	with	with	ADP
ejpam-1242	200	15	respect	respect	NOUN
ejpam-1242	200	16	to	to	ADP
ejpam-1242	200	17	the	the	DET
ejpam-1242	200	18	set	set	NOUN
ejpam-1242	200	19	qm	qm	PROPN
ejpam-1242	200	20	=	=	PROPN
ejpam-1242	200	21	m	m	VERB
ejpam-1242	200	22	−	−	NOUN
ejpam-1242	200	23	{	{	PUNCT
ejpam-1242	200	24	0	0	NUM
ejpam-1242	200	25	m	m	NOUN
ejpam-1242	200	26	}	}	PUNCT
ejpam-1242	200	27	,	,	PUNCT
ejpam-1242	200	28	the	the	DET
ejpam-1242	200	29	set	set	ADJ
ejpam-1242	200	30	∆	∆	PROPN
ejpam-1242	200	31	of	of	ADP
ejpam-1242	200	32	all	all	DET
ejpam-1242	200	33	proper	proper	ADJ
ejpam-1242	200	34	strong	strong	ADJ
ejpam-1242	200	35	qm	qm	PROPN
ejpam-1242	200	36	-subsemimodules	-subsemimodule	NOUN
ejpam-1242	200	37	of	of	ADP
ejpam-1242	200	38	m	m	PROPN
ejpam-1242	200	39	is	be	AUX
ejpam-1242	200	40	not	not	PART
ejpam-1242	200	41	empty	empty	ADJ
ejpam-1242	200	42	.	.	PUNCT
ejpam-1242	201	1	of	of	ADP
ejpam-1242	201	2	course	course	NOUN
ejpam-1242	201	3	,	,	PUNCT
ejpam-1242	201	4	the	the	DET
ejpam-1242	201	5	relation	relation	NOUN
ejpam-1242	201	6	of	of	ADP
ejpam-1242	201	7	inclusion	inclusion	NOUN
ejpam-1242	201	8	,	,	PUNCT
ejpam-1242	201	9	⊆	⊆	NUM
ejpam-1242	201	10	,	,	PUNCT
ejpam-1242	201	11	is	be	AUX
ejpam-1242	201	12	a	a	DET
ejpam-1242	201	13	partial	partial	ADJ
ejpam-1242	201	14	order	order	NOUN
ejpam-1242	201	15	on	on	ADP
ejpam-1242	201	16	∆.	∆.	NOUN
ejpam-1242	201	17	if	if	SCONJ
ejpam-1242	201	18	{	{	PUNCT
ejpam-1242	201	19	mi}i∈i	mi}i∈i	X
ejpam-1242	201	20	is	be	AUX
ejpam-1242	201	21	a	a	DET
ejpam-1242	201	22	chain	chain	NOUN
ejpam-1242	201	23	of	of	ADP
ejpam-1242	201	24	strong	strong	ADJ
ejpam-1242	201	25	qm	qm	PROPN
ejpam-1242	201	26	subsemimidules	subsemimidule	NOUN
ejpam-1242	201	27	of	of	ADP
ejpam-1242	201	28	m	m	PRON
ejpam-1242	201	29	,	,	PUNCT
ejpam-1242	201	30	then	then	ADV
ejpam-1242	201	31	n	n	NOUN
ejpam-1242	201	32	=	=	SYM
ejpam-1242	201	33	⋃	⋃	PROPN
ejpam-1242	201	34	i∈i	i∈i	ADJ
ejpam-1242	201	35	mi	mi	PROPN
ejpam-1242	201	36	is	be	AUX
ejpam-1242	201	37	a	a	DET
ejpam-1242	201	38	strong	strong	ADJ
ejpam-1242	201	39	qm	qm	NOUN
ejpam-1242	201	40	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	201	41	of	of	ADP
ejpam-1242	201	42	m	m	PRON
ejpam-1242	201	43	.	.	PUNCT
ejpam-1242	202	1	furthermore	furthermore	ADV
ejpam-1242	202	2	,	,	PUNCT
ejpam-1242	202	3	n	n	PRON
ejpam-1242	202	4	is	be	AUX
ejpam-1242	202	5	proper	proper	ADJ
ejpam-1242	202	6	since	since	SCONJ
ejpam-1242	202	7	m	m	PROPN
ejpam-1242	202	8	is	be	AUX
ejpam-1242	202	9	not	not	PART
ejpam-1242	202	10	strong	strong	ADJ
ejpam-1242	202	11	.	.	PUNCT
ejpam-1242	203	1	so	so	ADV
ejpam-1242	203	2	by	by	ADP
ejpam-1242	203	3	zorn	zorn	PROPN
ejpam-1242	203	4	’s	’s	PART
ejpam-1242	203	5	lemma	lemma	PROPN
ejpam-1242	203	6	∆	∆	PROPN
ejpam-1242	203	7	has	have	VERB
ejpam-1242	203	8	a	a	DET
ejpam-1242	203	9	maximal	maximal	ADJ
ejpam-1242	203	10	element	element	NOUN
ejpam-1242	203	11	,	,	PUNCT
ejpam-1242	203	12	i.e.	i.e.	X
ejpam-1242	203	13	,	,	PUNCT
ejpam-1242	203	14	m	m	VERB
ejpam-1242	203	15	has	have	VERB
ejpam-1242	203	16	a	a	DET
ejpam-1242	203	17	strong	strong	ADJ
ejpam-1242	203	18	maximal	maximal	ADJ
ejpam-1242	203	19	qm	qm	PROPN
ejpam-1242	203	20	-subsemimodule	-subsemimodule	PROPN
ejpam-1242	203	21	.	.	PUNCT
ejpam-1242	204	1	theorem	theorem	ADJ
ejpam-1242	204	2	4	4	NUM
ejpam-1242	204	3	.	.	PUNCT
ejpam-1242	205	1	let	let	VERB
ejpam-1242	205	2	m	m	PRON
ejpam-1242	205	3	be	be	AUX
ejpam-1242	205	4	a	a	DET
ejpam-1242	205	5	non	non	ADJ
ejpam-1242	205	6	-	-	ADJ
ejpam-1242	205	7	strong	strong	ADJ
ejpam-1242	205	8	semimodule	semimodule	NOUN
ejpam-1242	205	9	over	over	ADP
ejpam-1242	205	10	a	a	DET
ejpam-1242	205	11	semiring	semire	VERB
ejpam-1242	205	12	r	r	NOUN
ejpam-1242	205	13	with	with	ADP
ejpam-1242	205	14	m	m	PROPN
ejpam-1242	205	15	6=	6=	NUM
ejpam-1242	205	16	0	0	NUM
ejpam-1242	205	17	.	.	PUNCT
ejpam-1242	206	1	then	then	ADV
ejpam-1242	206	2	speck(m	speck(m	NOUN
ejpam-1242	206	3	)	)	PUNCT
ejpam-1242	206	4	6=	6=	NUM
ejpam-1242	206	5	;	;	PUNCT
ejpam-1242	206	6	.	.	PUNCT
ejpam-1242	207	1	proof	proof	NOUN
ejpam-1242	207	2	.	.	PUNCT
ejpam-1242	208	1	by	by	ADP
ejpam-1242	208	2	proposition	proposition	NOUN
ejpam-1242	208	3	2	2	NUM
ejpam-1242	208	4	,	,	PUNCT
ejpam-1242	208	5	there	there	PRON
ejpam-1242	208	6	exists	exist	VERB
ejpam-1242	208	7	a	a	DET
ejpam-1242	208	8	strong	strong	ADJ
ejpam-1242	208	9	maximal	maximal	ADJ
ejpam-1242	208	10	qm	qm	NOUN
ejpam-1242	208	11	-subsemimodule	-subsemimodule	PROPN
ejpam-1242	208	12	n	n	PROPN
ejpam-1242	208	13	of	of	ADP
ejpam-1242	208	14	m	m	PRON
ejpam-1242	208	15	;	;	PUNCT
ejpam-1242	208	16	so	so	CCONJ
ejpam-1242	208	17	it	it	PRON
ejpam-1242	208	18	is	be	AUX
ejpam-1242	208	19	a	a	DET
ejpam-1242	208	20	k	k	ADJ
ejpam-1242	208	21	-	-	ADJ
ejpam-1242	208	22	prime	prime	ADJ
ejpam-1242	208	23	subsemimodule	subsemimodule	NOUN
ejpam-1242	208	24	of	of	ADP
ejpam-1242	208	25	m	m	PRON
ejpam-1242	208	26	by	by	ADP
ejpam-1242	208	27	[	[	X
ejpam-1242	208	28	13	13	NUM
ejpam-1242	208	29	,	,	PUNCT
ejpam-1242	208	30	theorem	theorem	VERB
ejpam-1242	208	31	14	14	NUM
ejpam-1242	208	32	]	]	PUNCT
ejpam-1242	208	33	(	(	PUNCT
ejpam-1242	208	34	since	since	SCONJ
ejpam-1242	208	35	every	every	DET
ejpam-1242	208	36	qm	qm	NOUN
ejpam-1242	208	37	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	208	38	is	be	AUX
ejpam-1242	208	39	a	a	DET
ejpam-1242	208	40	k	k	NOUN
ejpam-1242	208	41	-	-	NOUN
ejpam-1242	208	42	subsemimodule	subsemimodule	NOUN
ejpam-1242	208	43	by	by	ADP
ejpam-1242	208	44	[	[	X
ejpam-1242	208	45	7	7	NUM
ejpam-1242	208	46	,	,	PUNCT
ejpam-1242	208	47	theorem	theorem	VERB
ejpam-1242	208	48	3.2	3.2	NUM
ejpam-1242	208	49	]	]	PUNCT
ejpam-1242	208	50	)	)	PUNCT
ejpam-1242	208	51	,	,	PUNCT
ejpam-1242	208	52	as	as	SCONJ
ejpam-1242	208	53	required	require	VERB
ejpam-1242	208	54	.	.	PUNCT
ejpam-1242	209	1	recall	recall	NOUN
ejpam-1242	209	2	that	that	SCONJ
ejpam-1242	209	3	we	we	PRON
ejpam-1242	209	4	follows	follow	VERB
ejpam-1242	209	5	golan	golan	PROPN
ejpam-1242	209	6	’s	’s	PART
ejpam-1242	209	7	terminology	terminology	NOUN
ejpam-1242	209	8	for	for	ADP
ejpam-1242	209	9	quotient	quotient	NOUN
ejpam-1242	209	10	semirings	semiring	NOUN
ejpam-1242	209	11	in	in	ADP
ejpam-1242	209	12	the	the	DET
ejpam-1242	209	13	following	follow	VERB
ejpam-1242	209	14	lemma	lemma	PROPN
ejpam-1242	209	15	.	.	PUNCT
ejpam-1242	210	1	lemma	lemma	PROPN
ejpam-1242	210	2	3	3	X
ejpam-1242	210	3	.	.	PUNCT
ejpam-1242	211	1	let	let	VERB
ejpam-1242	211	2	i	i	PRON
ejpam-1242	211	3	be	be	AUX
ejpam-1242	211	4	an	an	DET
ejpam-1242	211	5	ideal	ideal	NOUN
ejpam-1242	211	6	of	of	ADP
ejpam-1242	211	7	a	a	DET
ejpam-1242	211	8	semiring	semire	VERB
ejpam-1242	211	9	r	r	NOUN
ejpam-1242	211	10	with	with	ADP
ejpam-1242	211	11	1	1	NUM
ejpam-1242	211	12	6=	6=	SYM
ejpam-1242	211	13	0	0	NUM
ejpam-1242	211	14	.	.	PUNCT
ejpam-1242	212	1	then	then	ADV
ejpam-1242	212	2	the	the	DET
ejpam-1242	212	3	following	follow	VERB
ejpam-1242	212	4	hold	hold	NOUN
ejpam-1242	212	5	:	:	PUNCT
ejpam-1242	212	6	(	(	PUNCT
ejpam-1242	212	7	i	i	NOUN
ejpam-1242	212	8	)	)	PUNCT
ejpam-1242	212	9	if	if	SCONJ
ejpam-1242	212	10	l	l	NOUN
ejpam-1242	212	11	is	be	AUX
ejpam-1242	212	12	a	a	DET
ejpam-1242	212	13	k	k	NOUN
ejpam-1242	212	14	-	-	NOUN
ejpam-1242	212	15	ideal	ideal	NOUN
ejpam-1242	212	16	of	of	ADP
ejpam-1242	212	17	r	r	NOUN
ejpam-1242	212	18	/	/	SYM
ejpam-1242	212	19	i	i	NOUN
ejpam-1242	212	20	,	,	PUNCT
ejpam-1242	212	21	then	then	ADV
ejpam-1242	212	22	l	l	PROPN
ejpam-1242	212	23	=	=	PUNCT
ejpam-1242	212	24	j	j	PROPN
ejpam-1242	212	25	/	/	SYM
ejpam-1242	212	26	i	i	PRON
ejpam-1242	212	27	for	for	ADP
ejpam-1242	212	28	some	some	DET
ejpam-1242	212	29	k	k	ADJ
ejpam-1242	212	30	-	-	PUNCT
ejpam-1242	212	31	ideal	ideal	ADJ
ejpam-1242	212	32	j	j	PROPN
ejpam-1242	212	33	of	of	ADP
ejpam-1242	212	34	r.	r.	PROPN
ejpam-1242	212	35	(	(	PUNCT
ejpam-1242	212	36	ii	ii	PROPN
ejpam-1242	212	37	)	)	PUNCT
ejpam-1242	212	38	if	if	SCONJ
ejpam-1242	212	39	i	i	PRON
ejpam-1242	212	40	⊆	⊆	NUM
ejpam-1242	212	41	p	p	NOUN
ejpam-1242	212	42	,	,	PUNCT
ejpam-1242	212	43	then	then	ADV
ejpam-1242	212	44	p	p	NOUN
ejpam-1242	212	45	is	be	AUX
ejpam-1242	212	46	a	a	DET
ejpam-1242	212	47	maximal	maximal	ADJ
ejpam-1242	212	48	k	k	NOUN
ejpam-1242	212	49	-	-	NOUN
ejpam-1242	212	50	ideal	ideal	NOUN
ejpam-1242	212	51	of	of	ADP
ejpam-1242	212	52	r	r	NOUN
ejpam-1242	212	53	if	if	SCONJ
ejpam-1242	212	54	and	and	CCONJ
ejpam-1242	212	55	only	only	ADV
ejpam-1242	212	56	p	p	X
ejpam-1242	212	57	/	/	SYM
ejpam-1242	212	58	i	i	PRON
ejpam-1242	212	59	is	be	AUX
ejpam-1242	212	60	a	a	DET
ejpam-1242	212	61	maximal	maximal	ADJ
ejpam-1242	212	62	k	k	NOUN
ejpam-1242	212	63	-	-	NOUN
ejpam-1242	212	64	ideal	ideal	NOUN
ejpam-1242	212	65	of	of	ADP
ejpam-1242	212	66	r	r	NOUN
ejpam-1242	212	67	/	/	SYM
ejpam-1242	212	68	i	i	PRON
ejpam-1242	212	69	.	.	PUNCT
ejpam-1242	213	1	(	(	PUNCT
ejpam-1242	213	2	iii	iii	X
ejpam-1242	213	3	)	)	PUNCT
ejpam-1242	213	4	r	r	NOUN
ejpam-1242	213	5	has	have	VERB
ejpam-1242	213	6	at	at	ADV
ejpam-1242	213	7	least	least	ADJ
ejpam-1242	213	8	one	one	NUM
ejpam-1242	213	9	strong	strong	ADJ
ejpam-1242	213	10	maximal	maximal	ADJ
ejpam-1242	213	11	k	k	NOUN
ejpam-1242	213	12	-	-	NOUN
ejpam-1242	213	13	ideal	ideal	ADJ
ejpam-1242	213	14	.	.	PUNCT
ejpam-1242	214	1	(	(	PUNCT
ejpam-1242	214	2	iv	iv	X
ejpam-1242	214	3	)	)	PUNCT
ejpam-1242	214	4	if	if	SCONJ
ejpam-1242	214	5	j	j	PROPN
ejpam-1242	214	6	is	be	AUX
ejpam-1242	214	7	a	a	DET
ejpam-1242	214	8	proper	proper	ADJ
ejpam-1242	214	9	strong	strong	ADJ
ejpam-1242	214	10	k	k	NOUN
ejpam-1242	214	11	-	-	NOUN
ejpam-1242	214	12	ideal	ideal	NOUN
ejpam-1242	214	13	of	of	ADP
ejpam-1242	214	14	r	r	NOUN
ejpam-1242	214	15	,	,	PUNCT
ejpam-1242	214	16	then	then	ADV
ejpam-1242	214	17	j	j	PROPN
ejpam-1242	214	18	⊆	⊆	NUM
ejpam-1242	214	19	p	p	NOUN
ejpam-1242	214	20	for	for	ADP
ejpam-1242	214	21	some	some	DET
ejpam-1242	214	22	strong	strong	ADJ
ejpam-1242	214	23	maximal	maximal	ADJ
ejpam-1242	214	24	k	k	ADJ
ejpam-1242	214	25	-	-	NOUN
ejpam-1242	214	26	ideal	ideal	ADJ
ejpam-1242	214	27	p	p	NOUN
ejpam-1242	214	28	of	of	ADP
ejpam-1242	214	29	r.	r.	PROPN
ejpam-1242	214	30	proof	proof	NOUN
ejpam-1242	214	31	.	.	PUNCT
ejpam-1242	215	1	(	(	PUNCT
ejpam-1242	215	2	i	i	NOUN
ejpam-1242	215	3	)	)	PUNCT
ejpam-1242	215	4	assume	assume	VERB
ejpam-1242	215	5	that	that	SCONJ
ejpam-1242	215	6	j	j	PROPN
ejpam-1242	215	7	=	=	PRON
ejpam-1242	215	8	{	{	PUNCT
ejpam-1242	215	9	r	r	NOUN
ejpam-1242	215	10	∈	∈	PROPN
ejpam-1242	215	11	r	r	NOUN
ejpam-1242	215	12	:	:	PUNCT
ejpam-1242	215	13	r+	r+	NOUN
ejpam-1242	215	14	i	i	PRON
ejpam-1242	215	15	∈	∈	PROPN
ejpam-1242	215	16	l	l	NOUN
ejpam-1242	215	17	}	}	PUNCT
ejpam-1242	215	18	and	and	CCONJ
ejpam-1242	215	19	let	let	VERB
ejpam-1242	215	20	a	a	DET
ejpam-1242	215	21	∈	∈	NOUN
ejpam-1242	216	1	i	i	PRON
ejpam-1242	216	2	.	.	PUNCT
ejpam-1242	217	1	since	since	SCONJ
ejpam-1242	217	2	a+	a+	PRON
ejpam-1242	217	3	i	i	PRON
ejpam-1242	217	4	=	=	NOUN
ejpam-1242	217	5	0	0	PUNCT
ejpam-1242	217	6	+	+	NUM
ejpam-1242	217	7	i	i	PROPN
ejpam-1242	217	8	∈	∈	PROPN
ejpam-1242	217	9	l	l	NOUN
ejpam-1242	217	10	,	,	PUNCT
ejpam-1242	217	11	we	we	PRON
ejpam-1242	217	12	have	have	VERB
ejpam-1242	217	13	i	i	PROPN
ejpam-1242	217	14	⊆	⊆	NUM
ejpam-1242	217	15	j	j	PROPN
ejpam-1242	217	16	.	.	PUNCT
ejpam-1242	218	1	let	let	VERB
ejpam-1242	218	2	a	a	DET
ejpam-1242	218	3	,	,	PUNCT
ejpam-1242	218	4	b	b	PROPN
ejpam-1242	218	5	∈	∈	PROPN
ejpam-1242	218	6	j	j	PROPN
ejpam-1242	218	7	and	and	CCONJ
ejpam-1242	218	8	r	r	PROPN
ejpam-1242	218	9	∈	∈	PROPN
ejpam-1242	218	10	r.	r.	NOUN
ejpam-1242	218	11	then	then	ADV
ejpam-1242	218	12	(	(	PUNCT
ejpam-1242	218	13	a+	a+	X
ejpam-1242	218	14	i	i	NOUN
ejpam-1242	218	15	)	)	PUNCT
ejpam-1242	219	1	+	+	CCONJ
ejpam-1242	219	2	(	(	PUNCT
ejpam-1242	219	3	b+	b+	X
ejpam-1242	219	4	i	i	X
ejpam-1242	219	5	)	)	PUNCT
ejpam-1242	219	6	=	=	SYM
ejpam-1242	219	7	a+	a+	PUNCT
ejpam-1242	219	8	b+	b+	X
ejpam-1242	219	9	i	i	NOUN
ejpam-1242	219	10	∈	∈	PROPN
ejpam-1242	219	11	l	l	NOUN
ejpam-1242	219	12	;	;	PUNCT
ejpam-1242	219	13	so	so	SCONJ
ejpam-1242	219	14	a+	a+	PUNCT
ejpam-1242	219	15	b	b	X
ejpam-1242	219	16	∈	∈	PROPN
ejpam-1242	219	17	j	j	PROPN
ejpam-1242	219	18	.	.	PUNCT
ejpam-1242	220	1	similarly	similarly	ADV
ejpam-1242	220	2	,	,	PUNCT
ejpam-1242	220	3	ra	ra	PROPN
ejpam-1242	220	4	∈	∈	PROPN
ejpam-1242	220	5	j	j	PROPN
ejpam-1242	220	6	.	.	PUNCT
ejpam-1242	221	1	thus	thus	ADV
ejpam-1242	221	2	j	j	PROPN
ejpam-1242	221	3	is	be	AUX
ejpam-1242	221	4	a	a	DET
ejpam-1242	221	5	k	k	NOUN
ejpam-1242	221	6	-	-	NOUN
ejpam-1242	221	7	ideal	ideal	NOUN
ejpam-1242	221	8	of	of	ADP
ejpam-1242	221	9	r.	r.	PROPN
ejpam-1242	221	10	finally	finally	ADV
ejpam-1242	221	11	,	,	PUNCT
ejpam-1242	221	12	it	it	PRON
ejpam-1242	221	13	is	be	AUX
ejpam-1242	221	14	easy	easy	ADJ
ejpam-1242	221	15	to	to	PART
ejpam-1242	221	16	see	see	VERB
ejpam-1242	221	17	that	that	DET
ejpam-1242	221	18	l	l	NOUN
ejpam-1242	222	1	=	=	PUNCT
ejpam-1242	222	2	j	j	PROPN
ejpam-1242	222	3	/	/	SYM
ejpam-1242	222	4	i	i	PROPN
ejpam-1242	222	5	.	.	PUNCT
ejpam-1242	223	1	(	(	PUNCT
ejpam-1242	223	2	ii	ii	NOUN
ejpam-1242	223	3	)	)	PUNCT
ejpam-1242	223	4	suppose	suppose	VERB
ejpam-1242	223	5	that	that	SCONJ
ejpam-1242	223	6	p	p	PROPN
ejpam-1242	223	7	is	be	AUX
ejpam-1242	223	8	a	a	DET
ejpam-1242	223	9	maximal	maximal	ADJ
ejpam-1242	223	10	k	k	NOUN
ejpam-1242	223	11	-	-	NOUN
ejpam-1242	223	12	ideal	ideal	NOUN
ejpam-1242	223	13	of	of	ADP
ejpam-1242	223	14	r	r	NOUN
ejpam-1242	223	15	and	and	CCONJ
ejpam-1242	223	16	let	let	VERB
ejpam-1242	223	17	l	l	NOUN
ejpam-1242	223	18	be	be	AUX
ejpam-1242	223	19	a	a	DET
ejpam-1242	223	20	k	k	NOUN
ejpam-1242	223	21	-	-	NOUN
ejpam-1242	223	22	ideal	ideal	NOUN
ejpam-1242	223	23	of	of	ADP
ejpam-1242	223	24	r	r	NOUN
ejpam-1242	223	25	/	/	SYM
ejpam-1242	223	26	i	i	PRON
ejpam-1242	223	27	such	such	ADJ
ejpam-1242	223	28	that	that	SCONJ
ejpam-1242	223	29	p	p	X
ejpam-1242	223	30	/	/	SYM
ejpam-1242	223	31	i	i	PRON
ejpam-1242	223	32	$	$	PROPN
ejpam-1242	223	33	l.	l.	NOUN
ejpam-1242	223	34	there	there	PRON
ejpam-1242	223	35	exists	exist	VERB
ejpam-1242	223	36	a	a	DET
ejpam-1242	223	37	k	k	ADJ
ejpam-1242	223	38	-	-	PUNCT
ejpam-1242	223	39	ideal	ideal	ADJ
ejpam-1242	223	40	j	j	PROPN
ejpam-1242	223	41	of	of	ADP
ejpam-1242	223	42	r	r	NOUN
ejpam-1242	223	43	such	such	ADJ
ejpam-1242	223	44	that	that	SCONJ
ejpam-1242	223	45	p	p	NOUN
ejpam-1242	223	46	/	/	SYM
ejpam-1242	223	47	i	i	PROPN
ejpam-1242	223	48	$	$	SYM
ejpam-1242	223	49	l	l	NOUN
ejpam-1242	223	50	=	=	PUNCT
ejpam-1242	223	51	j	j	PROPN
ejpam-1242	223	52	/	/	SYM
ejpam-1242	223	53	i	i	PRON
ejpam-1242	223	54	by	by	ADP
ejpam-1242	223	55	(	(	PUNCT
ejpam-1242	223	56	i	i	NOUN
ejpam-1242	223	57	)	)	PUNCT
ejpam-1242	223	58	,	,	PUNCT
ejpam-1242	223	59	so	so	ADV
ejpam-1242	223	60	p	p	X
ejpam-1242	223	61	$	$	PROPN
ejpam-1242	223	62	j	j	NOUN
ejpam-1242	223	63	;	;	PUNCT
ejpam-1242	223	64	hence	hence	ADV
ejpam-1242	223	65	j	j	PROPN
ejpam-1242	223	66	=	=	SYM
ejpam-1242	223	67	r.	r.	PROPN
ejpam-1242	223	68	thus	thus	ADV
ejpam-1242	223	69	l	l	NOUN
ejpam-1242	224	1	=	=	PUNCT
ejpam-1242	224	2	r	r	X
ejpam-1242	224	3	/	/	SYM
ejpam-1242	224	4	i	i	NOUN
ejpam-1242	224	5	.	.	PUNCT
ejpam-1242	225	1	the	the	DET
ejpam-1242	225	2	other	other	ADJ
ejpam-1242	225	3	implication	implication	NOUN
ejpam-1242	225	4	is	be	AUX
ejpam-1242	225	5	similar	similar	ADJ
ejpam-1242	225	6	.	.	PUNCT
ejpam-1242	226	1	s.	s.	PROPN
ejpam-1242	226	2	atani	atani	PROPN
ejpam-1242	226	3	,	,	PUNCT
ejpam-1242	226	4	r.	r.	PROPN
ejpam-1242	226	5	atrani	atrani	PROPN
ejpam-1242	226	6	,	,	PUNCT
ejpam-1242	226	7	ü.	ü.	NOUN
ejpam-1242	226	8	tekir	tekir	PROPN
ejpam-1242	226	9	/	/	SYM
ejpam-1242	226	10	eur	eur	PROPN
ejpam-1242	226	11	.	.	PUNCT
ejpam-1242	227	1	j.	j.	PROPN
ejpam-1242	227	2	pure	pure	PROPN
ejpam-1242	227	3	appl	appl	PROPN
ejpam-1242	227	4	.	.	PROPN
ejpam-1242	227	5	math	math	PROPN
ejpam-1242	227	6	,	,	PUNCT
ejpam-1242	227	7	4	4	NUM
ejpam-1242	227	8	(	(	PUNCT
ejpam-1242	227	9	2011	2011	NUM
ejpam-1242	227	10	)	)	PUNCT
ejpam-1242	227	11	,	,	PUNCT
ejpam-1242	227	12	251	251	NUM
ejpam-1242	227	13	-	-	SYM
ejpam-1242	227	14	265	265	NUM
ejpam-1242	227	15	257	257	NUM
ejpam-1242	227	16	(	(	PUNCT
ejpam-1242	227	17	iii	iii	NOUN
ejpam-1242	227	18	)	)	PUNCT
ejpam-1242	227	19	since	since	SCONJ
ejpam-1242	227	20	{	{	PUNCT
ejpam-1242	227	21	0	0	NUM
ejpam-1242	227	22	}	}	PUNCT
ejpam-1242	227	23	is	be	AUX
ejpam-1242	227	24	a	a	DET
ejpam-1242	227	25	proper	proper	ADJ
ejpam-1242	227	26	strong	strong	ADJ
ejpam-1242	227	27	k	k	NOUN
ejpam-1242	227	28	-	-	NOUN
ejpam-1242	227	29	ideal	ideal	NOUN
ejpam-1242	227	30	of	of	ADP
ejpam-1242	227	31	r	r	NOUN
ejpam-1242	227	32	,	,	PUNCT
ejpam-1242	227	33	the	the	DET
ejpam-1242	227	34	set	set	ADJ
ejpam-1242	227	35	∆	∆	PROPN
ejpam-1242	227	36	of	of	ADP
ejpam-1242	227	37	all	all	DET
ejpam-1242	227	38	proper	proper	ADJ
ejpam-1242	227	39	strong	strong	ADJ
ejpam-1242	227	40	k	k	NOUN
ejpam-1242	227	41	-	-	NOUN
ejpam-1242	227	42	ideals	ideal	NOUN
ejpam-1242	227	43	of	of	ADP
ejpam-1242	227	44	r	r	NOUN
ejpam-1242	227	45	is	be	AUX
ejpam-1242	227	46	not	not	PART
ejpam-1242	227	47	empty	empty	ADJ
ejpam-1242	227	48	.	.	PUNCT
ejpam-1242	228	1	so	so	ADV
ejpam-1242	228	2	by	by	ADP
ejpam-1242	228	3	zorn	zorn	PROPN
ejpam-1242	228	4	’s	’s	PART
ejpam-1242	228	5	lemma	lemma	PROPN
ejpam-1242	228	6	∆	∆	PROPN
ejpam-1242	228	7	has	have	VERB
ejpam-1242	228	8	a	a	DET
ejpam-1242	228	9	maximal	maximal	ADJ
ejpam-1242	228	10	element	element	NOUN
ejpam-1242	228	11	(	(	PUNCT
ejpam-1242	228	12	with	with	ADP
ejpam-1242	228	13	respect	respect	NOUN
ejpam-1242	228	14	to	to	ADP
ejpam-1242	228	15	⊆	⊆	NOUN
ejpam-1242	228	16	)	)	PUNCT
ejpam-1242	228	17	,	,	PUNCT
ejpam-1242	228	18	i.e.	i.e.	X
ejpam-1242	228	19	,	,	PUNCT
ejpam-1242	228	20	r	r	NOUN
ejpam-1242	228	21	has	have	VERB
ejpam-1242	228	22	a	a	DET
ejpam-1242	228	23	proper	proper	ADJ
ejpam-1242	228	24	strong	strong	ADJ
ejpam-1242	228	25	maximal	maximal	ADJ
ejpam-1242	228	26	k	k	NOUN
ejpam-1242	228	27	-	-	NOUN
ejpam-1242	228	28	ideal	ideal	ADJ
ejpam-1242	228	29	.	.	PUNCT
ejpam-1242	229	1	(	(	PUNCT
ejpam-1242	229	2	iv	iv	X
ejpam-1242	229	3	)	)	PUNCT
ejpam-1242	229	4	since	since	SCONJ
ejpam-1242	229	5	r	r	NOUN
ejpam-1242	229	6	/	/	SYM
ejpam-1242	229	7	i	i	PRON
ejpam-1242	229	8	is	be	AUX
ejpam-1242	229	9	non	non	ADJ
ejpam-1242	229	10	-	-	ADJ
ejpam-1242	229	11	trivial	trivial	ADJ
ejpam-1242	229	12	,	,	PUNCT
ejpam-1242	229	13	and	and	CCONJ
ejpam-1242	229	14	so	so	ADV
ejpam-1242	229	15	,	,	PUNCT
ejpam-1242	229	16	by	by	ADP
ejpam-1242	229	17	(	(	PUNCT
ejpam-1242	229	18	iii	iii	NOUN
ejpam-1242	229	19	)	)	PUNCT
ejpam-1242	229	20	,	,	PUNCT
ejpam-1242	229	21	has	have	VERB
ejpam-1242	229	22	a	a	DET
ejpam-1242	229	23	strong	strong	ADJ
ejpam-1242	229	24	maximal	maximal	ADJ
ejpam-1242	229	25	k	k	ADJ
ejpam-1242	229	26	-	-	PUNCT
ejpam-1242	229	27	ideal	ideal	ADJ
ejpam-1242	229	28	l	l	NOUN
ejpam-1242	229	29	,	,	PUNCT
ejpam-1242	229	30	which	which	PRON
ejpam-1242	229	31	,	,	PUNCT
ejpam-1242	229	32	by	by	ADP
ejpam-1242	229	33	(	(	PUNCT
ejpam-1242	229	34	i	i	NOUN
ejpam-1242	229	35	)	)	PUNCT
ejpam-1242	229	36	,	,	PUNCT
ejpam-1242	229	37	will	will	AUX
ejpam-1242	229	38	have	have	VERB
ejpam-1242	229	39	to	to	PART
ejpam-1242	229	40	have	have	VERB
ejpam-1242	229	41	the	the	DET
ejpam-1242	229	42	form	form	NOUN
ejpam-1242	229	43	p	p	X
ejpam-1242	229	44	/	/	SYM
ejpam-1242	229	45	i	i	PRON
ejpam-1242	229	46	for	for	ADP
ejpam-1242	229	47	some	some	DET
ejpam-1242	229	48	k	k	ADJ
ejpam-1242	229	49	-	-	NOUN
ejpam-1242	229	50	ideal	ideal	ADJ
ejpam-1242	229	51	p	p	NOUN
ejpam-1242	229	52	of	of	ADP
ejpam-1242	229	53	r	r	NOUN
ejpam-1242	229	54	with	with	ADP
ejpam-1242	229	55	i	i	PRON
ejpam-1242	229	56	⊆	⊆	NUM
ejpam-1242	229	57	p.	p.	NOUN
ejpam-1242	229	58	it	it	PRON
ejpam-1242	229	59	now	now	ADV
ejpam-1242	229	60	follows	follow	VERB
ejpam-1242	229	61	from	from	ADP
ejpam-1242	229	62	(	(	PUNCT
ejpam-1242	229	63	ii	ii	NOUN
ejpam-1242	229	64	)	)	PUNCT
ejpam-1242	229	65	that	that	SCONJ
ejpam-1242	229	66	p	p	NOUN
ejpam-1242	229	67	is	be	AUX
ejpam-1242	229	68	a	a	DET
ejpam-1242	229	69	maximal	maximal	ADJ
ejpam-1242	229	70	k	k	NOUN
ejpam-1242	229	71	-	-	NOUN
ejpam-1242	229	72	ideal	ideal	NOUN
ejpam-1242	229	73	of	of	ADP
ejpam-1242	229	74	r.	r.	PROPN
ejpam-1242	229	75	it	it	PRON
ejpam-1242	229	76	remains	remain	VERB
ejpam-1242	229	77	to	to	PART
ejpam-1242	229	78	show	show	VERB
ejpam-1242	229	79	that	that	SCONJ
ejpam-1242	229	80	p	p	NOUN
ejpam-1242	229	81	is	be	AUX
ejpam-1242	229	82	a	a	DET
ejpam-1242	229	83	strong	strong	ADJ
ejpam-1242	229	84	ideal	ideal	NOUN
ejpam-1242	229	85	of	of	ADP
ejpam-1242	229	86	r.	r.	PROPN
ejpam-1242	229	87	let	let	VERB
ejpam-1242	229	88	a	a	DET
ejpam-1242	229	89	∈	∈	PROPN
ejpam-1242	229	90	p.	p.	NOUN
ejpam-1242	230	1	then	then	ADV
ejpam-1242	230	2	a+	a+	PUNCT
ejpam-1242	230	3	i	i	PRON
ejpam-1242	230	4	∈	∈	PROPN
ejpam-1242	230	5	p	p	X
ejpam-1242	230	6	/	/	SYM
ejpam-1242	230	7	i	i	PRON
ejpam-1242	230	8	.	.	PUNCT
ejpam-1242	231	1	by	by	ADP
ejpam-1242	231	2	assumption	assumption	NOUN
ejpam-1242	231	3	,	,	PUNCT
ejpam-1242	231	4	(	(	PUNCT
ejpam-1242	231	5	a+	a+	X
ejpam-1242	231	6	i	i	NOUN
ejpam-1242	231	7	)	)	PUNCT
ejpam-1242	232	1	+	+	CCONJ
ejpam-1242	232	2	(	(	PUNCT
ejpam-1242	232	3	b+	b+	X
ejpam-1242	232	4	i	i	X
ejpam-1242	232	5	)	)	PUNCT
ejpam-1242	232	6	=	=	SYM
ejpam-1242	232	7	a+	a+	PUNCT
ejpam-1242	232	8	b+	b+	X
ejpam-1242	232	9	i	i	NOUN
ejpam-1242	232	10	=	=	NOUN
ejpam-1242	232	11	0	0	PUNCT
ejpam-1242	233	1	+	+	NUM
ejpam-1242	233	2	i	i	PRON
ejpam-1242	233	3	=	=	VERB
ejpam-1242	234	1	i	i	PRON
ejpam-1242	234	2	for	for	ADP
ejpam-1242	234	3	some	some	PRON
ejpam-1242	234	4	b+	b+	NOUN
ejpam-1242	234	5	i	i	NOUN
ejpam-1242	234	6	∈	∈	VERB
ejpam-1242	234	7	p	p	X
ejpam-1242	234	8	/	/	SYM
ejpam-1242	234	9	i	i	NOUN
ejpam-1242	234	10	,	,	PUNCT
ejpam-1242	234	11	so	so	CCONJ
ejpam-1242	234	12	a+	a+	PUNCT
ejpam-1242	234	13	b	b	X
ejpam-1242	234	14	∈	∈	PROPN
ejpam-1242	235	1	i	i	PRON
ejpam-1242	235	2	.	.	PUNCT
ejpam-1242	236	1	then	then	ADV
ejpam-1242	236	2	there	there	PRON
ejpam-1242	236	3	is	be	VERB
ejpam-1242	236	4	an	an	DET
ejpam-1242	236	5	element	element	NOUN
ejpam-1242	236	6	c	c	NOUN
ejpam-1242	236	7	∈	∈	PROPN
ejpam-1242	236	8	i	i	PRON
ejpam-1242	236	9	such	such	ADJ
ejpam-1242	236	10	that	that	SCONJ
ejpam-1242	236	11	a+	a+	PUNCT
ejpam-1242	236	12	b+	b+	X
ejpam-1242	236	13	c	c	X
ejpam-1242	236	14	=	=	SYM
ejpam-1242	236	15	0	0	NUM
ejpam-1242	236	16	,	,	PUNCT
ejpam-1242	236	17	as	as	SCONJ
ejpam-1242	236	18	required	require	VERB
ejpam-1242	236	19	.	.	PUNCT
ejpam-1242	237	1	theorem	theorem	NOUN
ejpam-1242	237	2	5	5	NUM
ejpam-1242	237	3	.	.	PUNCT
ejpam-1242	238	1	let	let	VERB
ejpam-1242	238	2	m	m	PRON
ejpam-1242	238	3	be	be	AUX
ejpam-1242	238	4	a	a	DET
ejpam-1242	238	5	non	non	ADJ
ejpam-1242	238	6	-	-	ADJ
ejpam-1242	238	7	zero	zero	ADJ
ejpam-1242	238	8	very	very	ADV
ejpam-1242	238	9	strong	strong	ADJ
ejpam-1242	238	10	multiplication	multiplication	NOUN
ejpam-1242	238	11	semimodule	semimodule	NOUN
ejpam-1242	238	12	over	over	ADP
ejpam-1242	238	13	a	a	DET
ejpam-1242	238	14	semiring	semire	VERB
ejpam-1242	238	15	r.	r.	PROPN
ejpam-1242	238	16	then	then	ADV
ejpam-1242	238	17	every	every	DET
ejpam-1242	238	18	proper	proper	ADJ
ejpam-1242	238	19	strong	strong	ADJ
ejpam-1242	238	20	qm	qm	PROPN
ejpam-1242	238	21	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	238	22	of	of	ADP
ejpam-1242	238	23	m	m	VERB
ejpam-1242	238	24	is	be	AUX
ejpam-1242	238	25	contained	contain	VERB
ejpam-1242	238	26	in	in	ADP
ejpam-1242	238	27	a	a	DET
ejpam-1242	238	28	strong	strong	ADJ
ejpam-1242	238	29	maximal	maximal	ADJ
ejpam-1242	238	30	k	k	NOUN
ejpam-1242	238	31	-	-	NOUN
ejpam-1242	238	32	subsemimodule	subsemimodule	NOUN
ejpam-1242	238	33	of	of	ADP
ejpam-1242	238	34	m.	m.	NOUN
ejpam-1242	238	35	proof	proof	NOUN
ejpam-1242	238	36	.	.	PUNCT
ejpam-1242	239	1	assume	assume	VERB
ejpam-1242	239	2	that	that	SCONJ
ejpam-1242	239	3	n	n	PRON
ejpam-1242	239	4	is	be	AUX
ejpam-1242	239	5	a	a	DET
ejpam-1242	239	6	proper	proper	ADJ
ejpam-1242	239	7	strong	strong	ADJ
ejpam-1242	239	8	qm	qm	PROPN
ejpam-1242	239	9	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	239	10	of	of	ADP
ejpam-1242	239	11	m	m	PRON
ejpam-1242	239	12	and	and	CCONJ
ejpam-1242	239	13	let	let	VERB
ejpam-1242	239	14	q0	q0	PROPN
ejpam-1242	239	15	+	+	CCONJ
ejpam-1242	239	16	n	n	NUM
ejpam-1242	239	17	is	be	AUX
ejpam-1242	239	18	the	the	DET
ejpam-1242	239	19	zero	zero	NUM
ejpam-1242	239	20	in	in	ADP
ejpam-1242	239	21	m	m	PROPN
ejpam-1242	239	22	/	/	SYM
ejpam-1242	239	23	n	n	PROPN
ejpam-1242	239	24	.	.	PUNCT
ejpam-1242	240	1	then	then	ADV
ejpam-1242	240	2	there	there	PRON
ejpam-1242	240	3	exists	exist	VERB
ejpam-1242	240	4	x	x	X
ejpam-1242	240	5	∈	∈	PROPN
ejpam-1242	240	6	m	m	VERB
ejpam-1242	240	7	−	−	NOUN
ejpam-1242	240	8	n	n	PRON
ejpam-1242	240	9	such	such	ADJ
ejpam-1242	240	10	that	that	SCONJ
ejpam-1242	240	11	x	x	X
ejpam-1242	240	12	=	=	PUNCT
ejpam-1242	240	13	q+	q+	PUNCT
ejpam-1242	240	14	n	n	X
ejpam-1242	240	15	for	for	ADP
ejpam-1242	240	16	some	some	DET
ejpam-1242	240	17	q	q	PROPN
ejpam-1242	240	18	∈	∈	PROPN
ejpam-1242	240	19	qm	qm	PROPN
ejpam-1242	240	20	with	with	ADP
ejpam-1242	240	21	q	q	PROPN
ejpam-1242	240	22	/∈	/∈	PUNCT
ejpam-1242	241	1	n	n	CCONJ
ejpam-1242	241	2	(	(	PUNCT
ejpam-1242	241	3	since	since	SCONJ
ejpam-1242	241	4	every	every	DET
ejpam-1242	241	5	qm	qm	NOUN
ejpam-1242	241	6	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	241	7	is	be	AUX
ejpam-1242	241	8	a	a	DET
ejpam-1242	241	9	k	k	NOUN
ejpam-1242	241	10	-	-	NOUN
ejpam-1242	241	11	subsemimodule	subsemimodule	NOUN
ejpam-1242	241	12	by	by	ADP
ejpam-1242	241	13	[	[	X
ejpam-1242	241	14	7	7	NUM
ejpam-1242	241	15	,	,	PUNCT
ejpam-1242	241	16	theorem	theorem	VERB
ejpam-1242	241	17	3.2	3.2	NUM
ejpam-1242	241	18	]	]	PUNCT
ejpam-1242	241	19	)	)	PUNCT
ejpam-1242	241	20	and	and	CCONJ
ejpam-1242	241	21	n	n	PRON
ejpam-1242	241	22	∈	∈	PROPN
ejpam-1242	241	23	n	n	NOUN
ejpam-1242	241	24	;	;	PUNCT
ejpam-1242	241	25	hence	hence	ADV
ejpam-1242	241	26	q0+n	q0+n	PROPN
ejpam-1242	241	27	6=	6=	PROPN
ejpam-1242	241	28	q+n	q+n	ADJ
ejpam-1242	241	29	∈	∈	PROPN
ejpam-1242	241	30	m	m	PROPN
ejpam-1242	241	31	/	/	SYM
ejpam-1242	241	32	n	n	PROPN
ejpam-1242	241	33	.	.	PUNCT
ejpam-1242	242	1	then	then	ADV
ejpam-1242	242	2	m	m	PROPN
ejpam-1242	242	3	/	/	SYM
ejpam-1242	242	4	n	n	PROPN
ejpam-1242	242	5	is	be	AUX
ejpam-1242	242	6	a	a	DET
ejpam-1242	242	7	non	non	ADJ
ejpam-1242	242	8	-	-	ADJ
ejpam-1242	242	9	zero	zero	ADJ
ejpam-1242	242	10	very	very	ADV
ejpam-1242	242	11	strong	strong	ADJ
ejpam-1242	242	12	multiplication	multiplication	NOUN
ejpam-1242	242	13	semimodule	semimodule	NOUN
ejpam-1242	242	14	.	.	PUNCT
ejpam-1242	243	1	thus	thus	ADV
ejpam-1242	243	2	it	it	PRON
ejpam-1242	243	3	is	be	AUX
ejpam-1242	243	4	sufficient	sufficient	ADJ
ejpam-1242	243	5	to	to	PART
ejpam-1242	243	6	prove	prove	VERB
ejpam-1242	243	7	that	that	SCONJ
ejpam-1242	243	8	any	any	DET
ejpam-1242	243	9	non	non	ADJ
ejpam-1242	243	10	-	-	ADJ
ejpam-1242	243	11	zero	zero	ADJ
ejpam-1242	243	12	very	very	ADV
ejpam-1242	243	13	strong	strong	ADJ
ejpam-1242	243	14	multiplication	multiplication	NOUN
ejpam-1242	243	15	semimodule	semimodule	NOUN
ejpam-1242	243	16	contains	contain	VERB
ejpam-1242	243	17	a	a	DET
ejpam-1242	243	18	maximal	maximal	ADJ
ejpam-1242	243	19	k	k	NOUN
ejpam-1242	243	20	-	-	NOUN
ejpam-1242	243	21	subsemimodule	subsemimodule	NOUN
ejpam-1242	243	22	.	.	PUNCT
ejpam-1242	244	1	let	let	VERB
ejpam-1242	244	2	0	0	NUM
ejpam-1242	245	1	6=	6=	ADP
ejpam-1242	245	2	m	m	PROPN
ejpam-1242	245	3	∈	∈	NOUN
ejpam-1242	245	4	m	m	NOUN
ejpam-1242	245	5	.	.	PUNCT
ejpam-1242	246	1	if	if	SCONJ
ejpam-1242	246	2	i	i	PRON
ejpam-1242	246	3	=	=	PUNCT
ejpam-1242	246	4	{	{	PUNCT
ejpam-1242	246	5	0	0	NUM
ejpam-1242	246	6	}	}	PUNCT
ejpam-1242	246	7	,	,	PUNCT
ejpam-1242	246	8	then	then	ADV
ejpam-1242	246	9	the	the	DET
ejpam-1242	246	10	ideal	ideal	NOUN
ejpam-1242	246	11	j	j	PROPN
ejpam-1242	246	12	=	=	PUNCT
ejpam-1242	246	13	{	{	PUNCT
ejpam-1242	246	14	r	r	NOUN
ejpam-1242	246	15	∈	∈	PROPN
ejpam-1242	246	16	r	r	NOUN
ejpam-1242	246	17	:	:	PUNCT
ejpam-1242	246	18	rm	rm	NOUN
ejpam-1242	246	19	=	=	SYM
ejpam-1242	246	20	0	0	NUM
ejpam-1242	246	21	}	}	PUNCT
ejpam-1242	246	22	is	be	AUX
ejpam-1242	246	23	a	a	DET
ejpam-1242	246	24	proper	proper	ADJ
ejpam-1242	246	25	strong	strong	ADJ
ejpam-1242	246	26	k	k	NOUN
ejpam-1242	246	27	-	-	NOUN
ejpam-1242	246	28	ideal	ideal	NOUN
ejpam-1242	246	29	of	of	ADP
ejpam-1242	246	30	r	r	NOUN
ejpam-1242	246	31	since	since	SCONJ
ejpam-1242	246	32	m	m	PROPN
ejpam-1242	246	33	is	be	AUX
ejpam-1242	246	34	a	a	DET
ejpam-1242	246	35	very	very	ADV
ejpam-1242	246	36	strong	strong	ADJ
ejpam-1242	246	37	semimodule	semimodule	NOUN
ejpam-1242	246	38	and	and	CCONJ
ejpam-1242	246	39	hence	hence	ADV
ejpam-1242	246	40	j	j	PROPN
ejpam-1242	246	41	⊆	⊆	NUM
ejpam-1242	246	42	p	p	NOUN
ejpam-1242	246	43	for	for	ADP
ejpam-1242	246	44	some	some	DET
ejpam-1242	246	45	strong	strong	ADJ
ejpam-1242	246	46	maximal	maximal	ADJ
ejpam-1242	246	47	k	k	ADJ
ejpam-1242	246	48	-	-	NOUN
ejpam-1242	246	49	ideal	ideal	ADJ
ejpam-1242	246	50	p	p	NOUN
ejpam-1242	246	51	of	of	ADP
ejpam-1242	246	52	r	r	NOUN
ejpam-1242	246	53	by	by	ADP
ejpam-1242	246	54	lemma	lemma	PROPN
ejpam-1242	246	55	3	3	NUM
ejpam-1242	246	56	(	(	PUNCT
ejpam-1242	246	57	iv	iv	NUM
ejpam-1242	246	58	)	)	PUNCT
ejpam-1242	246	59	.	.	PUNCT
ejpam-1242	247	1	if	if	SCONJ
ejpam-1242	247	2	m	m	VERB
ejpam-1242	247	3	=	=	VERB
ejpam-1242	247	4	pm	pm	NOUN
ejpam-1242	247	5	,	,	PUNCT
ejpam-1242	247	6	then	then	ADV
ejpam-1242	247	7	rm	rm	PROPN
ejpam-1242	247	8	=	=	PROPN
ejpam-1242	247	9	t	t	PROPN
ejpam-1242	247	10	m	m	VERB
ejpam-1242	247	11	for	for	ADP
ejpam-1242	247	12	some	some	DET
ejpam-1242	247	13	ideal	ideal	ADJ
ejpam-1242	247	14	t	t	PROPN
ejpam-1242	247	15	of	of	ADP
ejpam-1242	247	16	r	r	NOUN
ejpam-1242	247	17	,	,	PUNCT
ejpam-1242	247	18	so	so	ADV
ejpam-1242	247	19	rm	rm	PROPN
ejpam-1242	247	20	=	=	PROPN
ejpam-1242	247	21	t	t	PROPN
ejpam-1242	247	22	pm	pm	NOUN
ejpam-1242	247	23	=	=	SYM
ejpam-1242	247	24	prm	prm	NOUN
ejpam-1242	247	25	=	=	NOUN
ejpam-1242	247	26	pm	pm	NOUN
ejpam-1242	247	27	and	and	CCONJ
ejpam-1242	247	28	hence	hence	ADV
ejpam-1242	247	29	m	m	VERB
ejpam-1242	247	30	=	=	VERB
ejpam-1242	247	31	pm	pm	NOUN
ejpam-1242	247	32	for	for	ADP
ejpam-1242	247	33	some	some	DET
ejpam-1242	247	34	p	p	NOUN
ejpam-1242	247	35	∈	∈	PROPN
ejpam-1242	247	36	p.	p.	NOUN
ejpam-1242	247	37	there	there	PRON
ejpam-1242	247	38	exists	exist	VERB
ejpam-1242	247	39	p′	p′	NOUN
ejpam-1242	247	40	∈	∈	PROPN
ejpam-1242	247	41	p	p	NOUN
ejpam-1242	247	42	such	such	ADJ
ejpam-1242	247	43	that	that	PRON
ejpam-1242	247	44	p+	p+	VERB
ejpam-1242	247	45	p′	p′	NOUN
ejpam-1242	247	46	=	=	SYM
ejpam-1242	247	47	0	0	PROPN
ejpam-1242	247	48	.	.	PUNCT
ejpam-1242	248	1	but	but	CCONJ
ejpam-1242	248	2	this	this	PRON
ejpam-1242	248	3	implies	imply	VERB
ejpam-1242	248	4	(	(	PUNCT
ejpam-1242	248	5	1	1	NUM
ejpam-1242	248	6	+	+	NUM
ejpam-1242	248	7	p′)m	p′)m	NOUN
ejpam-1242	248	8	=	=	SYM
ejpam-1242	248	9	0	0	NUM
ejpam-1242	248	10	;	;	PUNCT
ejpam-1242	248	11	so	so	ADV
ejpam-1242	248	12	1	1	NUM
ejpam-1242	248	13	+	+	NUM
ejpam-1242	248	14	p′	p′	NOUN
ejpam-1242	248	15	∈	∈	NOUN
ejpam-1242	248	16	i	i	NOUN
ejpam-1242	248	17	⊆	⊆	NUM
ejpam-1242	248	18	p	p	NOUN
ejpam-1242	248	19	,	,	PUNCT
ejpam-1242	248	20	a	a	DET
ejpam-1242	248	21	contradiction	contradiction	NOUN
ejpam-1242	248	22	.	.	PUNCT
ejpam-1242	249	1	thus	thus	ADV
ejpam-1242	249	2	m	m	VERB
ejpam-1242	249	3	6=	6=	NOUN
ejpam-1242	249	4	pm	pm	NOUN
ejpam-1242	249	5	.	.	PUNCT
ejpam-1242	250	1	since	since	SCONJ
ejpam-1242	250	2	p	p	NOUN
ejpam-1242	250	3	is	be	AUX
ejpam-1242	250	4	a	a	DET
ejpam-1242	250	5	strong	strong	ADJ
ejpam-1242	250	6	ideal	ideal	NOUN
ejpam-1242	250	7	of	of	ADP
ejpam-1242	250	8	r	r	NOUN
ejpam-1242	250	9	,	,	PUNCT
ejpam-1242	250	10	we	we	PRON
ejpam-1242	250	11	have	have	AUX
ejpam-1242	250	12	pm	pm	NOUN
ejpam-1242	250	13	is	be	AUX
ejpam-1242	250	14	a	a	DET
ejpam-1242	250	15	strong	strong	ADJ
ejpam-1242	250	16	subsemimodule	subsemimodule	NOUN
ejpam-1242	250	17	of	of	ADP
ejpam-1242	250	18	m	m	PRON
ejpam-1242	250	19	by	by	ADP
ejpam-1242	250	20	proposition	proposition	NOUN
ejpam-1242	250	21	1	1	NUM
ejpam-1242	250	22	(	(	PUNCT
ejpam-1242	250	23	i	i	NOUN
ejpam-1242	250	24	)	)	PUNCT
ejpam-1242	250	25	.	.	PUNCT
ejpam-1242	251	1	let	let	VERB
ejpam-1242	251	2	pm	pm	VERB
ejpam-1242	251	3	$	$	SYM
ejpam-1242	251	4	n	n	NOUN
ejpam-1242	251	5	=	=	SYM
ejpam-1242	251	6	lm	lm	PROPN
ejpam-1242	251	7	⊆	⊆	NUM
ejpam-1242	251	8	m	m	NOUN
ejpam-1242	251	9	for	for	ADP
ejpam-1242	251	10	some	some	DET
ejpam-1242	251	11	ideal	ideal	ADJ
ejpam-1242	251	12	l	l	NOUN
ejpam-1242	251	13	of	of	ADP
ejpam-1242	251	14	r.	r.	PROPN
ejpam-1242	251	15	it	it	PRON
ejpam-1242	251	16	follws	follws	ADJ
ejpam-1242	251	17	that	that	SCONJ
ejpam-1242	251	18	there	there	PRON
ejpam-1242	251	19	is	be	VERB
ejpam-1242	251	20	an	an	DET
ejpam-1242	251	21	element	element	NOUN
ejpam-1242	251	22	a	a	DET
ejpam-1242	251	23	∈	∈	NOUN
ejpam-1242	251	24	l	l	NOUN
ejpam-1242	251	25	with	with	ADP
ejpam-1242	251	26	a	a	DET
ejpam-1242	251	27	/∈	/∈	NOUN
ejpam-1242	251	28	p	p	X
ejpam-1242	251	29	;	;	PUNCT
ejpam-1242	251	30	so	so	SCONJ
ejpam-1242	251	31	p	p	X
ejpam-1242	252	1	+	+	PROPN
ejpam-1242	252	2	ra	ra	PROPN
ejpam-1242	252	3	=	=	SYM
ejpam-1242	252	4	r.	r.	PROPN
ejpam-1242	252	5	therefore	therefore	ADV
ejpam-1242	252	6	there	there	PRON
ejpam-1242	252	7	exist	exist	VERB
ejpam-1242	252	8	t	t	PROPN
ejpam-1242	252	9	∈	∈	PROPN
ejpam-1242	252	10	p	p	PROPN
ejpam-1242	252	11	and	and	CCONJ
ejpam-1242	252	12	r	r	NOUN
ejpam-1242	252	13	∈	∈	NOUN
ejpam-1242	252	14	r	r	NOUN
ejpam-1242	252	15	such	such	ADJ
ejpam-1242	252	16	that	that	DET
ejpam-1242	252	17	t	t	PROPN
ejpam-1242	252	18	+	+	CCONJ
ejpam-1242	252	19	ra	ra	PROPN
ejpam-1242	252	20	=	=	SYM
ejpam-1242	252	21	1	1	NUM
ejpam-1242	252	22	;	;	PUNCT
ejpam-1242	252	23	whence	whence	ADP
ejpam-1242	252	24	m	m	PROPN
ejpam-1242	252	25	=	=	SYM
ejpam-1242	252	26	tm+	tm+	PROPN
ejpam-1242	252	27	ram	ram	NOUN
ejpam-1242	252	28	∈	∈	PROPN
ejpam-1242	252	29	pm	pm	NOUN
ejpam-1242	252	30	+	+	CCONJ
ejpam-1242	252	31	lm	lm	X
ejpam-1242	252	32	=	=	SYM
ejpam-1242	252	33	n	n	NOUN
ejpam-1242	252	34	.	.	PUNCT
ejpam-1242	253	1	thus	thus	ADV
ejpam-1242	253	2	m	m	PROPN
ejpam-1242	253	3	=	=	SYM
ejpam-1242	253	4	n	n	CCONJ
ejpam-1242	253	5	and	and	CCONJ
ejpam-1242	253	6	hence	hence	ADV
ejpam-1242	253	7	pm	pm	NOUN
ejpam-1242	253	8	is	be	AUX
ejpam-1242	253	9	a	a	DET
ejpam-1242	253	10	strong	strong	ADJ
ejpam-1242	253	11	maximal	maximal	ADJ
ejpam-1242	253	12	k	k	NOUN
ejpam-1242	253	13	-	-	NOUN
ejpam-1242	253	14	subsemimodule	subsemimodule	NOUN
ejpam-1242	253	15	of	of	ADP
ejpam-1242	253	16	m	m	PROPN
ejpam-1242	253	17	.	.	PUNCT
ejpam-1242	254	1	definition	definition	NOUN
ejpam-1242	254	2	6	6	NUM
ejpam-1242	254	3	.	.	PUNCT
ejpam-1242	255	1	let	let	VERB
ejpam-1242	255	2	m	m	PRON
ejpam-1242	255	3	be	be	AUX
ejpam-1242	255	4	a	a	DET
ejpam-1242	255	5	non	non	ADJ
ejpam-1242	255	6	-	-	ADJ
ejpam-1242	255	7	zero	zero	NUM
ejpam-1242	255	8	semimodule	semimodule	NOUN
ejpam-1242	255	9	over	over	ADP
ejpam-1242	255	10	a	a	DET
ejpam-1242	255	11	semiring	semire	VERB
ejpam-1242	255	12	r.	r.	NOUN
ejpam-1242	255	13	an	an	DET
ejpam-1242	255	14	element	element	NOUN
ejpam-1242	255	15	u	u	NOUN
ejpam-1242	255	16	of	of	ADP
ejpam-1242	255	17	m	m	PROPN
ejpam-1242	255	18	is	be	AUX
ejpam-1242	255	19	said	say	VERB
ejpam-1242	255	20	to	to	PART
ejpam-1242	255	21	be	be	AUX
ejpam-1242	255	22	unit	unit	NOUN
ejpam-1242	255	23	provided	provide	VERB
ejpam-1242	255	24	that	that	SCONJ
ejpam-1242	255	25	u	u	NOUN
ejpam-1242	255	26	is	be	AUX
ejpam-1242	255	27	not	not	PART
ejpam-1242	255	28	contained	contain	VERB
ejpam-1242	255	29	in	in	ADP
ejpam-1242	255	30	any	any	DET
ejpam-1242	255	31	strong	strong	ADJ
ejpam-1242	255	32	maximal	maximal	ADJ
ejpam-1242	255	33	k	k	NOUN
ejpam-1242	255	34	-	-	NOUN
ejpam-1242	255	35	subsemimodule	subsemimodule	NOUN
ejpam-1242	255	36	of	of	ADP
ejpam-1242	255	37	m.	m.	NOUN
ejpam-1242	255	38	theorem	theorem	VERB
ejpam-1242	255	39	6	6	NUM
ejpam-1242	255	40	.	.	PUNCT
ejpam-1242	256	1	let	let	VERB
ejpam-1242	256	2	m	m	PRON
ejpam-1242	256	3	be	be	AUX
ejpam-1242	256	4	a	a	DET
ejpam-1242	256	5	non	non	ADJ
ejpam-1242	256	6	-	-	ADJ
ejpam-1242	256	7	zero	zero	ADJ
ejpam-1242	256	8	very	very	ADV
ejpam-1242	256	9	strong	strong	ADJ
ejpam-1242	256	10	multiplication	multiplication	NOUN
ejpam-1242	256	11	semimodule	semimodule	NOUN
ejpam-1242	256	12	over	over	ADP
ejpam-1242	256	13	a	a	DET
ejpam-1242	256	14	semiring	semire	VERB
ejpam-1242	256	15	r.	r.	PROPN
ejpam-1242	256	16	then	then	ADV
ejpam-1242	256	17	u	u	PROPN
ejpam-1242	256	18	∈	∈	PROPN
ejpam-1242	256	19	m	m	VERB
ejpam-1242	256	20	is	be	AUX
ejpam-1242	256	21	unit	unit	NOUN
ejpam-1242	256	22	if	if	SCONJ
ejpam-1242	256	23	and	and	CCONJ
ejpam-1242	256	24	only	only	ADV
ejpam-1242	256	25	if	if	SCONJ
ejpam-1242	256	26	m	m	PROPN
ejpam-1242	256	27	=	=	SYM
ejpam-1242	256	28	ru	ru	PROPN
ejpam-1242	256	29	.	.	PROPN
ejpam-1242	256	30	proof	proof	NOUN
ejpam-1242	256	31	.	.	PUNCT
ejpam-1242	257	1	the	the	DET
ejpam-1242	257	2	sufficiency	sufficiency	NOUN
ejpam-1242	257	3	is	be	AUX
ejpam-1242	257	4	clear	clear	ADJ
ejpam-1242	257	5	.	.	PUNCT
ejpam-1242	258	1	conversely	conversely	ADV
ejpam-1242	258	2	,	,	PUNCT
ejpam-1242	258	3	suppose	suppose	VERB
ejpam-1242	258	4	that	that	SCONJ
ejpam-1242	258	5	u	u	PROPN
ejpam-1242	258	6	is	be	AUX
ejpam-1242	258	7	an	an	DET
ejpam-1242	258	8	unit	unit	NOUN
ejpam-1242	258	9	element	element	NOUN
ejpam-1242	258	10	of	of	ADP
ejpam-1242	258	11	m	m	PROPN
ejpam-1242	258	12	.	.	PUNCT
ejpam-1242	259	1	then	then	ADV
ejpam-1242	259	2	ru	ru	PROPN
ejpam-1242	259	3	is	be	AUX
ejpam-1242	259	4	not	not	PART
ejpam-1242	259	5	contained	contain	VERB
ejpam-1242	259	6	in	in	ADP
ejpam-1242	259	7	any	any	DET
ejpam-1242	259	8	strong	strong	ADJ
ejpam-1242	259	9	maximal	maximal	ADJ
ejpam-1242	259	10	k	k	NOUN
ejpam-1242	259	11	-	-	NOUN
ejpam-1242	259	12	subsemimodule	subsemimodule	NOUN
ejpam-1242	259	13	of	of	ADP
ejpam-1242	259	14	m	m	PROPN
ejpam-1242	259	15	;	;	PUNCT
ejpam-1242	259	16	hence	hence	ADV
ejpam-1242	259	17	m	m	VERB
ejpam-1242	259	18	=	=	VERB
ejpam-1242	259	19	ru	ru	NOUN
ejpam-1242	259	20	by	by	ADP
ejpam-1242	259	21	theorem	theorem	NOUN
ejpam-1242	259	22	5	5	NUM
ejpam-1242	259	23	.	.	PUNCT
ejpam-1242	259	24	assume	assume	VERB
ejpam-1242	259	25	that	that	SCONJ
ejpam-1242	259	26	p	p	NOUN
ejpam-1242	259	27	is	be	AUX
ejpam-1242	259	28	a	a	DET
ejpam-1242	259	29	strong	strong	ADJ
ejpam-1242	259	30	maximal	maximal	ADJ
ejpam-1242	259	31	k	k	NOUN
ejpam-1242	259	32	-	-	NOUN
ejpam-1242	259	33	ideal	ideal	NOUN
ejpam-1242	259	34	of	of	ADP
ejpam-1242	259	35	a	a	DET
ejpam-1242	259	36	semiring	semiring	NOUN
ejpam-1242	259	37	r	r	NOUN
ejpam-1242	259	38	and	and	CCONJ
ejpam-1242	259	39	let	let	VERB
ejpam-1242	259	40	m	m	PRON
ejpam-1242	259	41	be	be	AUX
ejpam-1242	259	42	a	a	DET
ejpam-1242	259	43	semimodule	semimodule	NOUN
ejpam-1242	259	44	over	over	ADP
ejpam-1242	259	45	r.	r.	PROPN
ejpam-1242	259	46	we	we	PRON
ejpam-1242	259	47	say	say	VERB
ejpam-1242	259	48	that	that	SCONJ
ejpam-1242	259	49	m	m	PROPN
ejpam-1242	259	50	is	be	AUX
ejpam-1242	259	51	a	a	DET
ejpam-1242	259	52	p	p	NOUN
ejpam-1242	259	53	-	-	PUNCT
ejpam-1242	259	54	cyclic	cyclic	NOUN
ejpam-1242	259	55	provided	provide	VERB
ejpam-1242	259	56	there	there	PRON
ejpam-1242	259	57	exist	exist	VERB
ejpam-1242	259	58	p	p	PROPN
ejpam-1242	259	59	∈	∈	PROPN
ejpam-1242	259	60	p	p	NOUN
ejpam-1242	259	61	and	and	CCONJ
ejpam-1242	259	62	m	m	PROPN
ejpam-1242	259	63	∈	∈	NOUN
ejpam-1242	259	64	m	m	VERB
ejpam-1242	259	65	such	such	ADJ
ejpam-1242	259	66	that	that	SCONJ
ejpam-1242	259	67	(	(	PUNCT
ejpam-1242	259	68	1+p)m	1+p)m	NUM
ejpam-1242	259	69	⊆	⊆	NUM
ejpam-1242	259	70	rm	rm	NOUN
ejpam-1242	259	71	.	.	PUNCT
ejpam-1242	260	1	we	we	PRON
ejpam-1242	260	2	say	say	VERB
ejpam-1242	260	3	that	that	SCONJ
ejpam-1242	260	4	a	a	DET
ejpam-1242	260	5	subset	subset	NOUN
ejpam-1242	260	6	tp(m	tp(m	NOUN
ejpam-1242	260	7	)	)	PUNCT
ejpam-1242	260	8	of	of	ADP
ejpam-1242	260	9	m	m	PROPN
ejpam-1242	260	10	is	be	AUX
ejpam-1242	260	11	p	p	ADJ
ejpam-1242	260	12	-	-	PUNCT
ejpam-1242	260	13	torsion	torsion	NOUN
ejpam-1242	260	14	precisely	precisely	ADV
ejpam-1242	260	15	when	when	SCONJ
ejpam-1242	260	16	tp(m	tp(m	NOUN
ejpam-1242	260	17	)	)	PUNCT
ejpam-1242	260	18	=	=	PRON
ejpam-1242	260	19	{	{	PUNCT
ejpam-1242	260	20	m	m	VERB
ejpam-1242	260	21	∈	∈	ADJ
ejpam-1242	260	22	m	m	VERB
ejpam-1242	260	23	:	:	PUNCT
ejpam-1242	260	24	(	(	PUNCT
ejpam-1242	260	25	1	1	NUM
ejpam-1242	260	26	+	+	CCONJ
ejpam-1242	260	27	p)m	p)m	X
ejpam-1242	260	28	=	=	SYM
ejpam-1242	260	29	0	0	NUM
ejpam-1242	260	30	for	for	ADP
ejpam-1242	260	31	some	some	DET
ejpam-1242	260	32	p	p	NOUN
ejpam-1242	260	33	∈	∈	PROPN
ejpam-1242	260	34	p	p	X
ejpam-1242	260	35	}	}	PUNCT
ejpam-1242	260	36	.	.	PUNCT
ejpam-1242	261	1	the	the	DET
ejpam-1242	261	2	definition	definition	NOUN
ejpam-1242	261	3	is	be	AUX
ejpam-1242	261	4	the	the	DET
ejpam-1242	261	5	same	same	ADJ
ejpam-1242	261	6	as	as	ADP
ejpam-1242	261	7	that	that	PRON
ejpam-1242	261	8	introduced	introduce	VERB
ejpam-1242	261	9	by	by	ADP
ejpam-1242	261	10	z.	z.	PROPN
ejpam-1242	261	11	el	el	PROPN
ejpam-1242	261	12	-	-	PUNCT
ejpam-1242	261	13	bast	bast	NOUN
ejpam-1242	261	14	and	and	CCONJ
ejpam-1242	261	15	p.	p.	NOUN
ejpam-1242	261	16	f.	f.	PROPN
ejpam-1242	261	17	smith	smith	PROPN
ejpam-1242	261	18	in	in	ADP
ejpam-1242	261	19	[	[	X
ejpam-1242	261	20	6	6	NUM
ejpam-1242	261	21	]	]	PUNCT
ejpam-1242	261	22	.	.	PUNCT
ejpam-1242	262	1	it	it	PRON
ejpam-1242	262	2	is	be	AUX
ejpam-1242	262	3	easy	easy	ADJ
ejpam-1242	262	4	to	to	PART
ejpam-1242	262	5	see	see	VERB
ejpam-1242	262	6	that	that	PRON
ejpam-1242	262	7	tp(m	tp(m	PUNCT
ejpam-1242	262	8	)	)	PUNCT
ejpam-1242	262	9	is	be	AUX
ejpam-1242	262	10	a	a	DET
ejpam-1242	262	11	subsemimodule	subsemimodule	NOUN
ejpam-1242	262	12	of	of	ADP
ejpam-1242	262	13	m	m	PROPN
ejpam-1242	262	14	.	.	PUNCT
ejpam-1242	263	1	s.	s.	PROPN
ejpam-1242	263	2	atani	atani	PROPN
ejpam-1242	263	3	,	,	PUNCT
ejpam-1242	263	4	r.	r.	PROPN
ejpam-1242	263	5	atrani	atrani	PROPN
ejpam-1242	263	6	,	,	PUNCT
ejpam-1242	263	7	ü.	ü.	NOUN
ejpam-1242	263	8	tekir	tekir	PROPN
ejpam-1242	263	9	/	/	SYM
ejpam-1242	263	10	eur	eur	PROPN
ejpam-1242	263	11	.	.	PUNCT
ejpam-1242	264	1	j.	j.	PROPN
ejpam-1242	264	2	pure	pure	PROPN
ejpam-1242	264	3	appl	appl	PROPN
ejpam-1242	264	4	.	.	PROPN
ejpam-1242	264	5	math	math	PROPN
ejpam-1242	264	6	,	,	PUNCT
ejpam-1242	264	7	4	4	NUM
ejpam-1242	264	8	(	(	PUNCT
ejpam-1242	264	9	2011	2011	NUM
ejpam-1242	264	10	)	)	PUNCT
ejpam-1242	264	11	,	,	PUNCT
ejpam-1242	264	12	251	251	NUM
ejpam-1242	264	13	-	-	SYM
ejpam-1242	264	14	265	265	NUM
ejpam-1242	264	15	258	258	NUM
ejpam-1242	264	16	proposition	proposition	NOUN
ejpam-1242	264	17	3	3	NUM
ejpam-1242	264	18	.	.	PUNCT
ejpam-1242	265	1	(	(	PUNCT
ejpam-1242	265	2	i	i	NOUN
ejpam-1242	265	3	)	)	PUNCT
ejpam-1242	265	4	if	if	SCONJ
ejpam-1242	265	5	m	m	NOUN
ejpam-1242	265	6	is	be	AUX
ejpam-1242	265	7	a	a	DET
ejpam-1242	265	8	strong	strong	ADJ
ejpam-1242	265	9	multiplication	multiplication	NOUN
ejpam-1242	265	10	semimodule	semimodule	NOUN
ejpam-1242	265	11	over	over	ADP
ejpam-1242	265	12	a	a	DET
ejpam-1242	265	13	semiring	semiring	NOUN
ejpam-1242	265	14	r	r	NOUN
ejpam-1242	265	15	,	,	PUNCT
ejpam-1242	265	16	then	then	ADV
ejpam-1242	265	17	for	for	ADP
ejpam-1242	265	18	every	every	DET
ejpam-1242	265	19	strong	strong	ADJ
ejpam-1242	265	20	maximal	maximal	ADJ
ejpam-1242	265	21	k	k	ADJ
ejpam-1242	265	22	-	-	NOUN
ejpam-1242	265	23	ideal	ideal	ADJ
ejpam-1242	265	24	p	p	NOUN
ejpam-1242	265	25	of	of	ADP
ejpam-1242	265	26	r	r	NOUN
ejpam-1242	265	27	either	either	CCONJ
ejpam-1242	265	28	m	m	VERB
ejpam-1242	265	29	=	=	ADJ
ejpam-1242	265	30	tp	tp	X
ejpam-1242	265	31	(	(	PUNCT
ejpam-1242	265	32	m	m	NOUN
ejpam-1242	265	33	)	)	PUNCT
ejpam-1242	265	34	or	or	CCONJ
ejpam-1242	265	35	m	m	PROPN
ejpam-1242	265	36	is	be	AUX
ejpam-1242	265	37	p	p	ADJ
ejpam-1242	265	38	-	-	PUNCT
ejpam-1242	265	39	cyclic	cyclic	NOUN
ejpam-1242	265	40	.	.	PUNCT
ejpam-1242	266	1	(	(	PUNCT
ejpam-1242	266	2	ii	ii	NOUN
ejpam-1242	266	3	)	)	PUNCT
ejpam-1242	266	4	if	if	SCONJ
ejpam-1242	266	5	m	m	NOUN
ejpam-1242	266	6	is	be	AUX
ejpam-1242	266	7	a	a	DET
ejpam-1242	266	8	faithful	faithful	ADJ
ejpam-1242	266	9	very	very	ADV
ejpam-1242	266	10	strong	strong	ADJ
ejpam-1242	266	11	multiplication	multiplication	NOUN
ejpam-1242	266	12	semimodule	semimodule	NOUN
ejpam-1242	266	13	over	over	ADP
ejpam-1242	266	14	a	a	DET
ejpam-1242	266	15	semiring	semire	VERB
ejpam-1242	266	16	r	r	NOUN
ejpam-1242	266	17	,	,	PUNCT
ejpam-1242	266	18	then⋂	then⋂	PROPN
ejpam-1242	266	19	i∈λ(ii	i∈λ(ii	PROPN
ejpam-1242	266	20	m	m	PROPN
ejpam-1242	266	21	)	)	PUNCT
ejpam-1242	267	1	=	=	PRON
ejpam-1242	267	2	(	(	PUNCT
ejpam-1242	267	3	⋂	⋂	PROPN
ejpam-1242	267	4	i∈λ	i∈λ	PROPN
ejpam-1242	267	5	ii)m	ii)m	PROPN
ejpam-1242	267	6	for	for	ADP
ejpam-1242	267	7	any	any	DET
ejpam-1242	267	8	non	non	ADJ
ejpam-1242	267	9	-	-	ADJ
ejpam-1242	267	10	empty	empty	ADJ
ejpam-1242	267	11	collection	collection	NOUN
ejpam-1242	267	12	of	of	ADP
ejpam-1242	267	13	strong	strong	ADJ
ejpam-1242	267	14	ideals	ideal	NOUN
ejpam-1242	267	15	ii	ii	NOUN
ejpam-1242	267	16	(	(	PUNCT
ejpam-1242	267	17	i	i	NOUN
ejpam-1242	267	18	∈	∈	PROPN
ejpam-1242	267	19	λ	λ	PROPN
ejpam-1242	267	20	)	)	PUNCT
ejpam-1242	267	21	of	of	ADP
ejpam-1242	267	22	r.	r.	PROPN
ejpam-1242	267	23	(	(	PUNCT
ejpam-1242	267	24	iii	iii	NOUN
ejpam-1242	267	25	)	)	PUNCT
ejpam-1242	267	26	let	let	VERB
ejpam-1242	267	27	p	p	PRON
ejpam-1242	267	28	be	be	AUX
ejpam-1242	267	29	a	a	DET
ejpam-1242	267	30	strong	strong	ADJ
ejpam-1242	267	31	prime	prime	ADJ
ejpam-1242	267	32	k	k	NOUN
ejpam-1242	267	33	-	-	NOUN
ejpam-1242	267	34	ideal	ideal	NOUN
ejpam-1242	267	35	of	of	ADP
ejpam-1242	267	36	a	a	DET
ejpam-1242	267	37	semiring	semire	VERB
ejpam-1242	267	38	r	r	NOUN
ejpam-1242	267	39	and	and	CCONJ
ejpam-1242	267	40	m	m	VERB
ejpam-1242	267	41	a	a	DET
ejpam-1242	267	42	faithful	faithful	ADJ
ejpam-1242	267	43	very	very	ADV
ejpam-1242	267	44	strong	strong	ADJ
ejpam-1242	267	45	multiplication	multiplication	NOUN
ejpam-1242	267	46	semimodule	semimodule	NOUN
ejpam-1242	267	47	over	over	ADP
ejpam-1242	267	48	r.	r.	PROPN
ejpam-1242	267	49	let	let	VERB
ejpam-1242	267	50	a	a	DET
ejpam-1242	267	51	∈	∈	ADJ
ejpam-1242	267	52	r	r	NOUN
ejpam-1242	267	53	,	,	PUNCT
ejpam-1242	267	54	x	x	SYM
ejpam-1242	267	55	∈	∈	PROPN
ejpam-1242	267	56	m	m	VERB
ejpam-1242	267	57	satisfy	satisfy	NOUN
ejpam-1242	267	58	ax	ax	X
ejpam-1242	267	59	∈	∈	PROPN
ejpam-1242	267	60	pm	pm	NOUN
ejpam-1242	267	61	.	.	PUNCT
ejpam-1242	268	1	then	then	ADV
ejpam-1242	268	2	a	a	DET
ejpam-1242	268	3	∈	∈	PROPN
ejpam-1242	268	4	p	p	NOUN
ejpam-1242	268	5	or	or	CCONJ
ejpam-1242	268	6	x	x	PROPN
ejpam-1242	268	7	∈	∈	PROPN
ejpam-1242	268	8	pm	pm	NOUN
ejpam-1242	268	9	.	.	PUNCT
ejpam-1242	269	1	in	in	ADP
ejpam-1242	269	2	particular	particular	ADJ
ejpam-1242	269	3	,	,	PUNCT
ejpam-1242	269	4	if	if	SCONJ
ejpam-1242	269	5	m	m	PROPN
ejpam-1242	269	6	6=	6=	ADP
ejpam-1242	269	7	pm	pm	NOUN
ejpam-1242	269	8	,	,	PUNCT
ejpam-1242	269	9	then	then	ADV
ejpam-1242	269	10	pm	pm	NOUN
ejpam-1242	269	11	is	be	AUX
ejpam-1242	269	12	a	a	DET
ejpam-1242	269	13	strong	strong	ADJ
ejpam-1242	269	14	prime	prime	ADJ
ejpam-1242	269	15	subsemimodule	subsemimodule	NOUN
ejpam-1242	269	16	of	of	ADP
ejpam-1242	269	17	m.	m.	NOUN
ejpam-1242	269	18	proof	proof	NOUN
ejpam-1242	269	19	.	.	PUNCT
ejpam-1242	270	1	(	(	PUNCT
ejpam-1242	270	2	i	i	NOUN
ejpam-1242	270	3	)	)	PUNCT
ejpam-1242	270	4	let	let	VERB
ejpam-1242	270	5	p	p	PRON
ejpam-1242	270	6	be	be	AUX
ejpam-1242	270	7	a	a	DET
ejpam-1242	270	8	strong	strong	ADJ
ejpam-1242	270	9	maximal	maximal	ADJ
ejpam-1242	270	10	k	k	NOUN
ejpam-1242	270	11	-	-	NOUN
ejpam-1242	270	12	ideal	ideal	NOUN
ejpam-1242	270	13	of	of	ADP
ejpam-1242	270	14	r.	r.	PROPN
ejpam-1242	270	15	suppose	suppose	VERB
ejpam-1242	270	16	m	m	VERB
ejpam-1242	270	17	=	=	VERB
ejpam-1242	270	18	pm	pm	NOUN
ejpam-1242	270	19	.	.	PUNCT
ejpam-1242	271	1	let	let	VERB
ejpam-1242	271	2	m	m	PRON
ejpam-1242	271	3	∈	∈	VERB
ejpam-1242	271	4	m	m	NOUN
ejpam-1242	271	5	.	.	PUNCT
ejpam-1242	272	1	then	then	ADV
ejpam-1242	272	2	rm	rm	NOUN
ejpam-1242	272	3	=	=	PUNCT
ejpam-1242	273	1	i	i	PRON
ejpam-1242	273	2	m	m	VERB
ejpam-1242	273	3	for	for	ADP
ejpam-1242	273	4	some	some	DET
ejpam-1242	273	5	strong	strong	ADJ
ejpam-1242	273	6	ideal	ideal	NOUN
ejpam-1242	273	7	i	i	PRON
ejpam-1242	273	8	of	of	ADP
ejpam-1242	273	9	r	r	NOUN
ejpam-1242	273	10	by	by	ADP
ejpam-1242	273	11	[	[	X
ejpam-1242	273	12	15	15	NUM
ejpam-1242	273	13	,	,	PUNCT
ejpam-1242	273	14	proposition	proposition	NOUN
ejpam-1242	273	15	2.4	2.4	NUM
ejpam-1242	273	16	]	]	PUNCT
ejpam-1242	273	17	.	.	PUNCT
ejpam-1242	274	1	hence	hence	ADV
ejpam-1242	274	2	rm	rm	NOUN
ejpam-1242	275	1	=	=	PUNCT
ejpam-1242	276	1	i	i	PRON
ejpam-1242	276	2	m	m	VERB
ejpam-1242	276	3	=	=	VERB
ejpam-1242	277	1	i	i	PRON
ejpam-1242	277	2	pm	pm	VERB
ejpam-1242	277	3	=	=	VERB
ejpam-1242	278	1	pm	pm	NOUN
ejpam-1242	278	2	and	and	CCONJ
ejpam-1242	278	3	m=	m=	ADJ
ejpam-1242	278	4	pm	pm	VERB
ejpam-1242	278	5	for	for	ADP
ejpam-1242	278	6	some	some	DET
ejpam-1242	278	7	p	p	NOUN
ejpam-1242	278	8	∈	∈	PROPN
ejpam-1242	278	9	p.	p.	NOUN
ejpam-1242	278	10	by	by	ADP
ejpam-1242	278	11	assumption	assumption	NOUN
ejpam-1242	278	12	,	,	PUNCT
ejpam-1242	278	13	there	there	PRON
ejpam-1242	278	14	exists	exist	VERB
ejpam-1242	278	15	p′	p′	NOUN
ejpam-1242	278	16	∈	∈	PROPN
ejpam-1242	278	17	p	p	NOUN
ejpam-1242	278	18	such	such	ADJ
ejpam-1242	278	19	that	that	DET
ejpam-1242	278	20	pm+	pm+	NOUN
ejpam-1242	278	21	p′m	p′m	NOUN
ejpam-1242	278	22	=	=	PUNCT
ejpam-1242	278	23	(	(	PUNCT
ejpam-1242	278	24	1	1	NUM
ejpam-1242	278	25	+	+	NUM
ejpam-1242	278	26	p′)m	p′)m	NOUN
ejpam-1242	278	27	=	=	SYM
ejpam-1242	278	28	0	0	NUM
ejpam-1242	278	29	and	and	CCONJ
ejpam-1242	278	30	m	m	PROPN
ejpam-1242	278	31	∈	∈	NOUN
ejpam-1242	278	32	tp(m	tp(m	NOUN
ejpam-1242	278	33	)	)	PUNCT
ejpam-1242	278	34	.	.	PUNCT
ejpam-1242	279	1	it	it	PRON
ejpam-1242	279	2	follows	follow	VERB
ejpam-1242	279	3	that	that	SCONJ
ejpam-1242	279	4	tp	tp	ADP
ejpam-1242	279	5	(	(	PUNCT
ejpam-1242	279	6	m	m	NOUN
ejpam-1242	279	7	)	)	PUNCT
ejpam-1242	279	8	=	=	SYM
ejpam-1242	280	1	m	m	VERB
ejpam-1242	280	2	.	.	PUNCT
ejpam-1242	281	1	now	now	ADV
ejpam-1242	281	2	suppose	suppose	VERB
ejpam-1242	281	3	that	that	SCONJ
ejpam-1242	281	4	pm	pm	NOUN
ejpam-1242	281	5	6=	6=	ADP
ejpam-1242	281	6	m	m	VERB
ejpam-1242	281	7	.	.	PUNCT
ejpam-1242	282	1	there	there	PRON
ejpam-1242	282	2	exists	exist	VERB
ejpam-1242	282	3	y	y	PROPN
ejpam-1242	282	4	∈	∈	PROPN
ejpam-1242	282	5	m	m	VERB
ejpam-1242	283	1	such	such	ADJ
ejpam-1242	283	2	that	that	SCONJ
ejpam-1242	283	3	y	y	PROPN
ejpam-1242	283	4	/∈	/∈	PUNCT
ejpam-1242	283	5	pm	pm	NOUN
ejpam-1242	283	6	.	.	PUNCT
ejpam-1242	284	1	there	there	PRON
ejpam-1242	284	2	is	be	VERB
ejpam-1242	284	3	a	a	DET
ejpam-1242	284	4	strong	strong	ADJ
ejpam-1242	284	5	ideal	ideal	ADJ
ejpam-1242	284	6	j	j	PROPN
ejpam-1242	284	7	of	of	ADP
ejpam-1242	284	8	r	r	NOUN
ejpam-1242	284	9	such	such	ADJ
ejpam-1242	284	10	that	that	DET
ejpam-1242	284	11	ry	ry	PROPN
ejpam-1242	284	12	=	=	PROPN
ejpam-1242	284	13	j	j	PROPN
ejpam-1242	284	14	m	m	PROPN
ejpam-1242	284	15	.	.	PUNCT
ejpam-1242	285	1	clearly	clearly	ADV
ejpam-1242	285	2	,	,	PUNCT
ejpam-1242	285	3	j	j	PROPN
ejpam-1242	285	4	*	*	PROPN
ejpam-1242	285	5	p.	p.	NOUN
ejpam-1242	285	6	since	since	SCONJ
ejpam-1242	285	7	,	,	PUNCT
ejpam-1242	285	8	j	j	PROPN
ejpam-1242	286	1	+	+	CCONJ
ejpam-1242	286	2	p	p	NOUN
ejpam-1242	286	3	is	be	AUX
ejpam-1242	286	4	a	a	DET
ejpam-1242	286	5	strong	strong	ADJ
ejpam-1242	286	6	ideal	ideal	NOUN
ejpam-1242	286	7	of	of	ADP
ejpam-1242	286	8	r	r	NOUN
ejpam-1242	286	9	,	,	PUNCT
ejpam-1242	286	10	we	we	PRON
ejpam-1242	286	11	must	must	AUX
ejpam-1242	286	12	have	have	VERB
ejpam-1242	286	13	j+p	j+p	NOUN
ejpam-1242	286	14	=	=	SYM
ejpam-1242	286	15	r	r	NOUN
ejpam-1242	286	16	,	,	PUNCT
ejpam-1242	286	17	so	so	SCONJ
ejpam-1242	286	18	1=	1=	X
ejpam-1242	286	19	e+q	e+q	PROPN
ejpam-1242	286	20	for	for	ADP
ejpam-1242	286	21	some	some	DET
ejpam-1242	286	22	e	e	PROPN
ejpam-1242	286	23	∈	∈	PROPN
ejpam-1242	286	24	j	j	PROPN
ejpam-1242	286	25	and	and	CCONJ
ejpam-1242	287	1	q	q	PROPN
ejpam-1242	287	2	∈	∈	PROPN
ejpam-1242	288	1	p.	p.	NOUN
ejpam-1242	288	2	there	there	PRON
ejpam-1242	288	3	exists	exist	VERB
ejpam-1242	288	4	q′	q′	NOUN
ejpam-1242	288	5	∈	∈	PROPN
ejpam-1242	288	6	p	p	NOUN
ejpam-1242	288	7	such	such	ADJ
ejpam-1242	288	8	that	that	DET
ejpam-1242	288	9	q+q′	q+q′	PROPN
ejpam-1242	288	10	=	=	NOUN
ejpam-1242	288	11	0	0	NUM
ejpam-1242	288	12	;	;	PUNCT
ejpam-1242	288	13	hence	hence	ADV
ejpam-1242	288	14	1	1	NUM
ejpam-1242	288	15	+	+	NUM
ejpam-1242	288	16	q′	q′	NOUN
ejpam-1242	288	17	∈	∈	PROPN
ejpam-1242	288	18	j	j	PROPN
ejpam-1242	288	19	.	.	PUNCT
ejpam-1242	289	1	it	it	PRON
ejpam-1242	289	2	follows	follow	VERB
ejpam-1242	289	3	that	that	SCONJ
ejpam-1242	289	4	(	(	PUNCT
ejpam-1242	289	5	1	1	NUM
ejpam-1242	289	6	+	+	NUM
ejpam-1242	289	7	q′)m	q′)m	PROPN
ejpam-1242	289	8	⊆	⊆	NUM
ejpam-1242	289	9	ry	ry	NOUN
ejpam-1242	289	10	and	and	CCONJ
ejpam-1242	289	11	m	m	PROPN
ejpam-1242	289	12	is	be	AUX
ejpam-1242	289	13	p	p	ADJ
ejpam-1242	289	14	-	-	PUNCT
ejpam-1242	289	15	cyclic	cyclic	NOUN
ejpam-1242	289	16	.	.	PUNCT
ejpam-1242	290	1	(	(	PUNCT
ejpam-1242	290	2	ii	ii	NOUN
ejpam-1242	290	3	)	)	PUNCT
ejpam-1242	290	4	let	let	VERB
ejpam-1242	290	5	ii	ii	NOUN
ejpam-1242	290	6	(	(	PUNCT
ejpam-1242	290	7	i	i	NOUN
ejpam-1242	290	8	∈	∈	PROPN
ejpam-1242	290	9	λ	λ	NOUN
ejpam-1242	290	10	)	)	PUNCT
ejpam-1242	290	11	be	be	VERB
ejpam-1242	290	12	any	any	DET
ejpam-1242	290	13	non	non	ADJ
ejpam-1242	290	14	-	-	ADJ
ejpam-1242	290	15	empty	empty	ADJ
ejpam-1242	290	16	collection	collection	NOUN
ejpam-1242	290	17	of	of	ADP
ejpam-1242	290	18	strong	strong	ADJ
ejpam-1242	290	19	ideals	ideal	NOUN
ejpam-1242	290	20	of	of	ADP
ejpam-1242	290	21	r.	r.	PROPN
ejpam-1242	290	22	set	set	PROPN
ejpam-1242	291	1	i	i	PROPN
ejpam-1242	291	2	=	=	SYM
ejpam-1242	291	3	⋂	⋂	PROPN
ejpam-1242	291	4	i∈λ	i∈λ	PROPN
ejpam-1242	291	5	ii	ii	PROPN
ejpam-1242	291	6	.	.	PUNCT
ejpam-1242	292	1	clearly	clearly	ADV
ejpam-1242	292	2	,	,	PUNCT
ejpam-1242	292	3	i	i	PRON
ejpam-1242	292	4	m	m	VERB
ejpam-1242	292	5	⊆	⊆	NUM
ejpam-1242	292	6	⋂	⋂	PROPN
ejpam-1242	292	7	i∈λ(ii	i∈λ(ii	PROPN
ejpam-1242	292	8	m	m	PROPN
ejpam-1242	292	9	)	)	PUNCT
ejpam-1242	292	10	.	.	PUNCT
ejpam-1242	293	1	for	for	ADP
ejpam-1242	293	2	the	the	DET
ejpam-1242	293	3	reverse	reverse	ADJ
ejpam-1242	293	4	inclusion	inclusion	NOUN
ejpam-1242	293	5	,	,	PUNCT
ejpam-1242	293	6	assume	assume	VERB
ejpam-1242	293	7	that	that	SCONJ
ejpam-1242	293	8	x	x	SYM
ejpam-1242	293	9	∈	∈	PROPN
ejpam-1242	293	10	⋂	⋂	PROPN
ejpam-1242	293	11	i∈λ(ii	i∈λ(ii	PROPN
ejpam-1242	293	12	m	m	PROPN
ejpam-1242	293	13	)	)	PUNCT
ejpam-1242	293	14	.	.	PUNCT
ejpam-1242	294	1	then	then	ADV
ejpam-1242	294	2	k	k	PROPN
ejpam-1242	294	3	=	=	PRON
ejpam-1242	294	4	{	{	PUNCT
ejpam-1242	294	5	r	r	NOUN
ejpam-1242	294	6	∈	∈	NOUN
ejpam-1242	294	7	r	r	NOUN
ejpam-1242	294	8	:	:	PUNCT
ejpam-1242	294	9	r	r	NOUN
ejpam-1242	294	10	x	x	SYM
ejpam-1242	294	11	∈	∈	PROPN
ejpam-1242	294	12	i	i	PRON
ejpam-1242	294	13	m	m	VERB
ejpam-1242	294	14	}	}	PUNCT
ejpam-1242	294	15	is	be	AUX
ejpam-1242	294	16	a	a	DET
ejpam-1242	294	17	strong	strong	ADJ
ejpam-1242	294	18	k	k	NOUN
ejpam-1242	294	19	-	-	NOUN
ejpam-1242	294	20	ideal	ideal	NOUN
ejpam-1242	294	21	of	of	ADP
ejpam-1242	294	22	r.	r.	PROPN
ejpam-1242	294	23	suppose	suppose	VERB
ejpam-1242	294	24	k	k	PROPN
ejpam-1242	294	25	6=	6=	PROPN
ejpam-1242	294	26	r.	r.	PROPN
ejpam-1242	294	27	then	then	ADV
ejpam-1242	294	28	by	by	ADP
ejpam-1242	294	29	lemma	lemma	PROPN
ejpam-1242	294	30	3	3	NUM
ejpam-1242	294	31	(	(	PUNCT
ejpam-1242	294	32	iv	iv	NUM
ejpam-1242	294	33	)	)	PUNCT
ejpam-1242	294	34	,	,	PUNCT
ejpam-1242	294	35	there	there	PRON
ejpam-1242	294	36	exists	exist	VERB
ejpam-1242	294	37	a	a	DET
ejpam-1242	294	38	strong	strong	ADJ
ejpam-1242	294	39	maximal	maximal	ADJ
ejpam-1242	294	40	k	k	ADJ
ejpam-1242	294	41	-	-	NOUN
ejpam-1242	294	42	ideal	ideal	ADJ
ejpam-1242	294	43	p	p	NOUN
ejpam-1242	294	44	of	of	ADP
ejpam-1242	294	45	r	r	NOUN
ejpam-1242	294	46	such	such	ADJ
ejpam-1242	294	47	that	that	SCONJ
ejpam-1242	294	48	k	k	PROPN
ejpam-1242	294	49	⊆	⊆	NUM
ejpam-1242	294	50	p.	p.	NOUN
ejpam-1242	294	51	clearly	clearly	ADV
ejpam-1242	294	52	,	,	PUNCT
ejpam-1242	294	53	x	x	X
ejpam-1242	294	54	/∈	/∈	INTJ
ejpam-1242	294	55	tp(m	tp(m	NOUN
ejpam-1242	294	56	)	)	PUNCT
ejpam-1242	294	57	.	.	PUNCT
ejpam-1242	295	1	for	for	ADP
ejpam-1242	295	2	if	if	SCONJ
ejpam-1242	295	3	x	x	SYM
ejpam-1242	295	4	∈	∈	NOUN
ejpam-1242	295	5	tp(m	tp(m	NOUN
ejpam-1242	295	6	)	)	PUNCT
ejpam-1242	295	7	,	,	PUNCT
ejpam-1242	295	8	then	then	ADV
ejpam-1242	295	9	(	(	PUNCT
ejpam-1242	295	10	1	1	NUM
ejpam-1242	295	11	+	+	X
ejpam-1242	295	12	p)x	p)x	ADJ
ejpam-1242	295	13	=	=	SYM
ejpam-1242	295	14	0	0	SYM
ejpam-1242	295	15	∈	∈	PROPN
ejpam-1242	295	16	i	i	PRON
ejpam-1242	295	17	m	m	VERB
ejpam-1242	295	18	for	for	ADP
ejpam-1242	295	19	some	some	DET
ejpam-1242	295	20	p	p	NOUN
ejpam-1242	295	21	∈	∈	PROPN
ejpam-1242	295	22	p	p	X
ejpam-1242	295	23	;	;	PUNCT
ejpam-1242	295	24	hence	hence	ADV
ejpam-1242	295	25	(	(	PUNCT
ejpam-1242	295	26	1	1	NUM
ejpam-1242	295	27	+	+	NUM
ejpam-1242	295	28	p	p	X
ejpam-1242	295	29	)	)	PUNCT
ejpam-1242	295	30	∈	∈	PROPN
ejpam-1242	295	31	k	k	NOUN
ejpam-1242	296	1	⊆	⊆	NUM
ejpam-1242	296	2	p	p	NOUN
ejpam-1242	296	3	,	,	PUNCT
ejpam-1242	296	4	a	a	DET
ejpam-1242	296	5	contradiction	contradiction	NOUN
ejpam-1242	296	6	.	.	PUNCT
ejpam-1242	297	1	therefore	therefore	ADV
ejpam-1242	297	2	,	,	PUNCT
ejpam-1242	297	3	m	m	VERB
ejpam-1242	297	4	is	be	AUX
ejpam-1242	297	5	p	p	NOUN
ejpam-1242	297	6	-	-	PUNCT
ejpam-1242	297	7	cyclic	cyclic	NOUN
ejpam-1242	297	8	by	by	ADP
ejpam-1242	297	9	(	(	PUNCT
ejpam-1242	297	10	i	i	NOUN
ejpam-1242	297	11	)	)	PUNCT
ejpam-1242	297	12	.	.	PUNCT
ejpam-1242	298	1	there	there	PRON
ejpam-1242	298	2	exist	exist	VERB
ejpam-1242	298	3	p	p	PROPN
ejpam-1242	298	4	∈	∈	PROPN
ejpam-1242	298	5	p	p	NOUN
ejpam-1242	298	6	and	and	CCONJ
ejpam-1242	298	7	m	m	PROPN
ejpam-1242	298	8	∈	∈	NOUN
ejpam-1242	298	9	m	m	VERB
ejpam-1242	298	10	such	such	ADJ
ejpam-1242	298	11	that	that	SCONJ
ejpam-1242	298	12	(	(	PUNCT
ejpam-1242	298	13	1	1	NUM
ejpam-1242	298	14	+	+	CCONJ
ejpam-1242	298	15	p)m	p)m	X
ejpam-1242	298	16	⊆	⊆	NUM
ejpam-1242	298	17	rm	rm	NOUN
ejpam-1242	298	18	.	.	PUNCT
ejpam-1242	299	1	then	then	ADV
ejpam-1242	299	2	(	(	PUNCT
ejpam-1242	299	3	1	1	NUM
ejpam-1242	299	4	+	+	NUM
ejpam-1242	299	5	p)x	p)x	ADJ
ejpam-1242	299	6	∈	∈	PROPN
ejpam-1242	299	7	⋂	⋂	PROPN
ejpam-1242	299	8	i∈λ(iim	i∈λ(iim	NOUN
ejpam-1242	299	9	)	)	PUNCT
ejpam-1242	299	10	.	.	PUNCT
ejpam-1242	300	1	for	for	ADP
ejpam-1242	300	2	each	each	DET
ejpam-1242	300	3	i	i	PROPN
ejpam-1242	300	4	∈	∈	PROPN
ejpam-1242	300	5	λ	λ	NOUN
ejpam-1242	300	6	,	,	PUNCT
ejpam-1242	300	7	there	there	PRON
ejpam-1242	300	8	is	be	VERB
ejpam-1242	300	9	an	an	DET
ejpam-1242	300	10	element	element	NOUN
ejpam-1242	300	11	ai	ai	PROPN
ejpam-1242	300	12	∈	∈	PROPN
ejpam-1242	300	13	ii	ii	NOUN
ejpam-1242	300	14	such	such	ADJ
ejpam-1242	300	15	that	that	SCONJ
ejpam-1242	300	16	(	(	PUNCT
ejpam-1242	300	17	1	1	NUM
ejpam-1242	300	18	+	+	X
ejpam-1242	300	19	p)x	p)x	ADJ
ejpam-1242	300	20	=	=	PUNCT
ejpam-1242	300	21	aim	aim	NOUN
ejpam-1242	300	22	.	.	PUNCT
ejpam-1242	301	1	choose	choose	VERB
ejpam-1242	301	2	j	j	PROPN
ejpam-1242	301	3	∈	∈	PROPN
ejpam-1242	301	4	λ	λ	PROPN
ejpam-1242	301	5	.	.	PUNCT
ejpam-1242	302	1	then	then	ADV
ejpam-1242	302	2	for	for	ADP
ejpam-1242	302	3	each	each	DET
ejpam-1242	302	4	i	i	PROPN
ejpam-1242	302	5	∈	∈	PROPN
ejpam-1242	302	6	λ	λ	PROPN
ejpam-1242	302	7	,	,	PUNCT
ejpam-1242	302	8	a	a	DET
ejpam-1242	302	9	jm	jm	NOUN
ejpam-1242	302	10	=	=	PUNCT
ejpam-1242	302	11	aim	aim	NOUN
ejpam-1242	302	12	.	.	PUNCT
ejpam-1242	303	1	by	by	ADP
ejpam-1242	303	2	assumption	assumption	NOUN
ejpam-1242	303	3	,	,	PUNCT
ejpam-1242	303	4	ai	ai	VERB
ejpam-1242	303	5	+	+	SYM
ejpam-1242	303	6	a′i	a′i	X
ejpam-1242	304	1	=	=	PUNCT
ejpam-1242	304	2	0	0	NUM
ejpam-1242	304	3	for	for	ADP
ejpam-1242	304	4	some	some	DET
ejpam-1242	304	5	a′i	a′i	PROPN
ejpam-1242	304	6	∈	∈	PROPN
ejpam-1242	304	7	ii	ii	PROPN
ejpam-1242	304	8	;	;	PUNCT
ejpam-1242	304	9	hence	hence	ADV
ejpam-1242	304	10	a	a	DET
ejpam-1242	304	11	jm+	jm+	NOUN
ejpam-1242	304	12	a′im	a′im	ADV
ejpam-1242	304	13	=	=	NOUN
ejpam-1242	304	14	0	0	X
ejpam-1242	304	15	.	.	PUNCT
ejpam-1242	305	1	now	now	ADV
ejpam-1242	305	2	(	(	PUNCT
ejpam-1242	305	3	1	1	NUM
ejpam-1242	305	4	+	+	NUM
ejpam-1242	305	5	p)(a	p)(a	NUM
ejpam-1242	305	6	j	j	PROPN
ejpam-1242	305	7	+	+	CCONJ
ejpam-1242	305	8	a′i)m	a′i)m	X
ejpam-1242	305	9	⊆	⊆	NUM
ejpam-1242	305	10	(	(	PUNCT
ejpam-1242	305	11	a	a	DET
ejpam-1242	305	12	j	j	PROPN
ejpam-1242	306	1	+	+	CCONJ
ejpam-1242	306	2	a′i)rm	a′i)rm	NOUN
ejpam-1242	306	3	=	=	SYM
ejpam-1242	306	4	0	0	NUM
ejpam-1242	306	5	implies	imply	VERB
ejpam-1242	306	6	(	(	PUNCT
ejpam-1242	306	7	1	1	NUM
ejpam-1242	306	8	+	+	NUM
ejpam-1242	306	9	p)(a	p)(a	NUM
ejpam-1242	306	10	j	j	PROPN
ejpam-1242	306	11	+	+	NUM
ejpam-1242	306	12	a′i	a′i	PROPN
ejpam-1242	306	13	)	)	PUNCT
ejpam-1242	307	1	=	=	SYM
ejpam-1242	307	2	0	0	PUNCT
ejpam-1242	308	1	since	since	SCONJ
ejpam-1242	308	2	m	m	PROPN
ejpam-1242	308	3	is	be	AUX
ejpam-1242	308	4	faithful	faithful	ADJ
ejpam-1242	308	5	.	.	PUNCT
ejpam-1242	309	1	therefore	therefore	ADV
ejpam-1242	309	2	,	,	PUNCT
ejpam-1242	309	3	a	a	DET
ejpam-1242	309	4	j	j	PROPN
ejpam-1242	309	5	+	+	CCONJ
ejpam-1242	309	6	pa	pa	PROPN
ejpam-1242	309	7	j	j	PROPN
ejpam-1242	309	8	+	+	CCONJ
ejpam-1242	309	9	a′i	a′i	PROPN
ejpam-1242	310	1	+	+	CCONJ
ejpam-1242	310	2	pa′i	pa′i	ADJ
ejpam-1242	310	3	=	=	SYM
ejpam-1242	310	4	0	0	NUM
ejpam-1242	310	5	,	,	PUNCT
ejpam-1242	310	6	so	so	CCONJ
ejpam-1242	310	7	(	(	PUNCT
ejpam-1242	310	8	1	1	NUM
ejpam-1242	310	9	+	+	NUM
ejpam-1242	310	10	p)a	p)a	X
ejpam-1242	310	11	j	j	NOUN
ejpam-1242	310	12	=	=	PUNCT
ejpam-1242	310	13	(	(	PUNCT
ejpam-1242	310	14	1	1	NUM
ejpam-1242	310	15	+	+	NUM
ejpam-1242	310	16	p)ai	p)ai	PROPN
ejpam-1242	310	17	∈	∈	PROPN
ejpam-1242	310	18	ii	ii	NOUN
ejpam-1242	310	19	;	;	PUNCT
ejpam-1242	310	20	hence	hence	ADV
ejpam-1242	310	21	(	(	PUNCT
ejpam-1242	310	22	1	1	NUM
ejpam-1242	310	23	+	+	NUM
ejpam-1242	310	24	p)a	p)a	X
ejpam-1242	310	25	j	j	PROPN
ejpam-1242	310	26	∈	∈	PROPN
ejpam-1242	311	1	i	i	PRON
ejpam-1242	311	2	.	.	PUNCT
ejpam-1242	312	1	thus	thus	ADV
ejpam-1242	312	2	(	(	PUNCT
ejpam-1242	312	3	1	1	NUM
ejpam-1242	312	4	+	+	NUM
ejpam-1242	312	5	p)2	p)2	NOUN
ejpam-1242	312	6	x	x	X
ejpam-1242	312	7	=	=	SYM
ejpam-1242	312	8	(	(	PUNCT
ejpam-1242	312	9	1	1	NUM
ejpam-1242	312	10	+	+	CCONJ
ejpam-1242	312	11	p)(aim	p)(aim	NOUN
ejpam-1242	312	12	)	)	PUNCT
ejpam-1242	312	13	∈	∈	PROPN
ejpam-1242	313	1	i	i	PRON
ejpam-1242	313	2	m	m	VERB
ejpam-1242	313	3	.	.	PUNCT
ejpam-1242	314	1	it	it	PRON
ejpam-1242	314	2	follows	follow	VERB
ejpam-1242	314	3	that	that	SCONJ
ejpam-1242	314	4	(	(	PUNCT
ejpam-1242	314	5	1	1	NUM
ejpam-1242	314	6	+	+	NUM
ejpam-1242	314	7	p)2	p)2	NOUN
ejpam-1242	314	8	∈	∈	NOUN
ejpam-1242	314	9	k	k	PROPN
ejpam-1242	314	10	⊆	⊆	NUM
ejpam-1242	314	11	p	p	NOUN
ejpam-1242	314	12	,	,	PUNCT
ejpam-1242	314	13	a	a	DET
ejpam-1242	314	14	contradiction	contradiction	NOUN
ejpam-1242	314	15	.	.	PUNCT
ejpam-1242	315	1	so	so	ADV
ejpam-1242	315	2	k	k	NOUN
ejpam-1242	315	3	=	=	SYM
ejpam-1242	315	4	r	r	NOUN
ejpam-1242	315	5	;	;	PUNCT
ejpam-1242	315	6	hence	hence	ADV
ejpam-1242	315	7	x	x	SYM
ejpam-1242	315	8	∈	∈	PROPN
ejpam-1242	316	1	i	i	PRON
ejpam-1242	316	2	m	m	VERB
ejpam-1242	316	3	,	,	PUNCT
ejpam-1242	316	4	and	and	CCONJ
ejpam-1242	316	5	(	(	PUNCT
ejpam-1242	316	6	ii	ii	NOUN
ejpam-1242	316	7	)	)	PUNCT
ejpam-1242	316	8	is	be	AUX
ejpam-1242	316	9	proved	prove	VERB
ejpam-1242	316	10	.	.	PUNCT
ejpam-1242	317	1	(	(	PUNCT
ejpam-1242	317	2	iii	iii	X
ejpam-1242	317	3	)	)	PUNCT
ejpam-1242	317	4	let	let	VERB
ejpam-1242	317	5	a	a	PRON
ejpam-1242	317	6	/∈	/∈	PUNCT
ejpam-1242	318	1	p.	p.	NOUN
ejpam-1242	318	2	then	then	ADV
ejpam-1242	318	3	the	the	DET
ejpam-1242	318	4	ideal	ideal	NOUN
ejpam-1242	318	5	k	k	PROPN
ejpam-1242	319	1	=	=	PRON
ejpam-1242	319	2	{	{	PUNCT
ejpam-1242	319	3	r	r	NOUN
ejpam-1242	319	4	∈	∈	NOUN
ejpam-1242	319	5	r	r	NOUN
ejpam-1242	319	6	:	:	PUNCT
ejpam-1242	319	7	r	r	NOUN
ejpam-1242	319	8	x	x	SYM
ejpam-1242	319	9	∈	∈	NOUN
ejpam-1242	319	10	pm	pm	NOUN
ejpam-1242	319	11	}	}	PUNCT
ejpam-1242	319	12	is	be	AUX
ejpam-1242	319	13	a	a	DET
ejpam-1242	319	14	strong	strong	ADJ
ejpam-1242	319	15	k	k	NOUN
ejpam-1242	319	16	-	-	NOUN
ejpam-1242	319	17	ideal	ideal	NOUN
ejpam-1242	319	18	of	of	ADP
ejpam-1242	319	19	r.	r.	PROPN
ejpam-1242	319	20	suppose	suppose	VERB
ejpam-1242	319	21	k	k	PROPN
ejpam-1242	319	22	6=	6=	PROPN
ejpam-1242	319	23	r.	r.	PROPN
ejpam-1242	319	24	then	then	ADV
ejpam-1242	319	25	there	there	PRON
ejpam-1242	319	26	exists	exist	VERB
ejpam-1242	319	27	a	a	DET
ejpam-1242	319	28	strong	strong	ADJ
ejpam-1242	319	29	maximal	maximal	ADJ
ejpam-1242	319	30	k	k	ADJ
ejpam-1242	319	31	-	-	NOUN
ejpam-1242	319	32	ideal	ideal	NOUN
ejpam-1242	319	33	p	p	NOUN
ejpam-1242	319	34	′	′	NOUN
ejpam-1242	319	35	of	of	ADP
ejpam-1242	319	36	r	r	NOUN
ejpam-1242	319	37	such	such	ADJ
ejpam-1242	319	38	that	that	SCONJ
ejpam-1242	319	39	k	k	PROPN
ejpam-1242	319	40	⊆	⊆	NUM
ejpam-1242	319	41	p	p	PRON
ejpam-1242	319	42	′.	′.	NOUN
ejpam-1242	319	43	clearly	clearly	ADV
ejpam-1242	319	44	,	,	PUNCT
ejpam-1242	319	45	x	x	PROPN
ejpam-1242	319	46	/∈	/∈	PUNCT
ejpam-1242	319	47	tp	tp	ADP
ejpam-1242	319	48	′(m	′(m	NOUN
ejpam-1242	319	49	)	)	PUNCT
ejpam-1242	319	50	.	.	PUNCT
ejpam-1242	320	1	by	by	ADP
ejpam-1242	320	2	(	(	PUNCT
ejpam-1242	320	3	i	i	NOUN
ejpam-1242	320	4	)	)	PUNCT
ejpam-1242	320	5	,	,	PUNCT
ejpam-1242	320	6	m	m	PROPN
ejpam-1242	320	7	is	be	AUX
ejpam-1242	320	8	p	p	PROPN
ejpam-1242	320	9	′-cyclic	′-cyclic	NOUN
ejpam-1242	320	10	,	,	PUNCT
ejpam-1242	320	11	that	that	ADV
ejpam-1242	320	12	is	is	ADV
ejpam-1242	320	13	,	,	PUNCT
ejpam-1242	320	14	there	there	PRON
ejpam-1242	320	15	exist	exist	VERB
ejpam-1242	320	16	m	m	PROPN
ejpam-1242	320	17	∈	∈	NOUN
ejpam-1242	320	18	m	m	NOUN
ejpam-1242	320	19	and	and	CCONJ
ejpam-1242	320	20	q	q	PROPN
ejpam-1242	320	21	∈	∈	PROPN
ejpam-1242	321	1	p	p	NOUN
ejpam-1242	321	2	′	′	NUM
ejpam-1242	322	1	such	such	ADJ
ejpam-1242	322	2	that	that	SCONJ
ejpam-1242	322	3	(	(	PUNCT
ejpam-1242	322	4	1	1	NUM
ejpam-1242	322	5	+	+	NUM
ejpam-1242	322	6	q)m	q)m	NUM
ejpam-1242	322	7	⊆	⊆	NUM
ejpam-1242	322	8	rm	rm	NOUN
ejpam-1242	322	9	.	.	PUNCT
ejpam-1242	323	1	in	in	ADP
ejpam-1242	323	2	particular	particular	ADJ
ejpam-1242	323	3	,	,	PUNCT
ejpam-1242	323	4	(	(	PUNCT
ejpam-1242	323	5	1	1	NUM
ejpam-1242	323	6	+	+	CCONJ
ejpam-1242	323	7	q)x	q)x	ADJ
ejpam-1242	323	8	=	=	PUNCT
ejpam-1242	323	9	sm	sm	VERB
ejpam-1242	323	10	for	for	ADP
ejpam-1242	323	11	some	some	DET
ejpam-1242	323	12	s	s	PROPN
ejpam-1242	323	13	∈	∈	PROPN
ejpam-1242	323	14	r.	r.	NOUN
ejpam-1242	323	15	therefore	therefore	ADV
ejpam-1242	323	16	we	we	PRON
ejpam-1242	323	17	have	have	VERB
ejpam-1242	323	18	(	(	PUNCT
ejpam-1242	323	19	1	1	NUM
ejpam-1242	323	20	+	+	NUM
ejpam-1242	323	21	q)ax	q)ax	PROPN
ejpam-1242	323	22	∈	∈	PROPN
ejpam-1242	323	23	(	(	PUNCT
ejpam-1242	323	24	1	1	NUM
ejpam-1242	323	25	+	+	CCONJ
ejpam-1242	323	26	q)pm	q)pm	PROPN
ejpam-1242	323	27	⊆	⊆	NUM
ejpam-1242	323	28	prm	prm	X
ejpam-1242	323	29	=	=	PUNCT
ejpam-1242	323	30	pm	pm	NOUN
ejpam-1242	323	31	;	;	PUNCT
ejpam-1242	323	32	hence	hence	ADV
ejpam-1242	323	33	asm	asm	NOUN
ejpam-1242	323	34	=	=	PUNCT
ejpam-1242	323	35	pm	pm	NOUN
ejpam-1242	323	36	for	for	ADP
ejpam-1242	323	37	some	some	DET
ejpam-1242	323	38	p	p	NOUN
ejpam-1242	323	39	∈	∈	PROPN
ejpam-1242	323	40	p.	p.	NOUN
ejpam-1242	323	41	by	by	ADP
ejpam-1242	323	42	assumption	assumption	NOUN
ejpam-1242	323	43	,	,	PUNCT
ejpam-1242	323	44	p+	p+	NOUN
ejpam-1242	323	45	p′	p′	NOUN
ejpam-1242	323	46	=	=	SYM
ejpam-1242	323	47	0	0	NUM
ejpam-1242	323	48	for	for	ADP
ejpam-1242	323	49	some	some	DET
ejpam-1242	323	50	p	p	NOUN
ejpam-1242	323	51	∈	∈	PROPN
ejpam-1242	323	52	p	p	PROPN
ejpam-1242	323	53	′	′	NOUN
ejpam-1242	323	54	;	;	PUNCT
ejpam-1242	323	55	hence	hence	ADV
ejpam-1242	323	56	(	(	PUNCT
ejpam-1242	323	57	as+	as+	PROPN
ejpam-1242	323	58	p′)m=	p′)m=	PROPN
ejpam-1242	323	59	0	0	PROPN
ejpam-1242	323	60	.	.	PUNCT
ejpam-1242	324	1	since	since	SCONJ
ejpam-1242	324	2	(	(	PUNCT
ejpam-1242	324	3	1	1	NUM
ejpam-1242	324	4	+	+	NUM
ejpam-1242	324	5	q)ann(m)m	q)ann(m)m	NOUN
ejpam-1242	324	6	⊆	⊆	NUM
ejpam-1242	324	7	rann(m)m	rann(m)m	X
ejpam-1242	324	8	=	=	SYM
ejpam-1242	324	9	0	0	NUM
ejpam-1242	324	10	,	,	PUNCT
ejpam-1242	324	11	we	we	PRON
ejpam-1242	324	12	must	must	AUX
ejpam-1242	324	13	have	have	VERB
ejpam-1242	324	14	(	(	PUNCT
ejpam-1242	324	15	1	1	NUM
ejpam-1242	324	16	+	+	CCONJ
ejpam-1242	324	17	q)(as	q)(as	ADJ
ejpam-1242	324	18	+	+	NUM
ejpam-1242	324	19	p′	p′	NOUN
ejpam-1242	324	20	)	)	PUNCT
ejpam-1242	324	21	=	=	SYM
ejpam-1242	325	1	0	0	NUM
ejpam-1242	325	2	,	,	PUNCT
ejpam-1242	325	3	(	(	PUNCT
ejpam-1242	325	4	1	1	NUM
ejpam-1242	325	5	+	+	CCONJ
ejpam-1242	326	1	q)as	q)as	PROPN
ejpam-1242	326	2	=	=	SYM
ejpam-1242	326	3	(	(	PUNCT
ejpam-1242	326	4	1	1	NUM
ejpam-1242	326	5	+	+	NUM
ejpam-1242	326	6	q)p	q)p	NOUN
ejpam-1242	326	7	∈	∈	PROPN
ejpam-1242	326	8	p.	p.	NOUN
ejpam-1242	326	9	but	but	CCONJ
ejpam-1242	326	10	p	p	NOUN
ejpam-1242	327	1	⊆	⊆	NUM
ejpam-1242	327	2	k	k	PROPN
ejpam-1242	327	3	⊆	⊆	NUM
ejpam-1242	327	4	p	p	NOUN
ejpam-1242	327	5	′	′	NOUN
ejpam-1242	328	1	so	so	SCONJ
ejpam-1242	328	2	that	that	PRON
ejpam-1242	328	3	s	s	VERB
ejpam-1242	328	4	∈	∈	PROPN
ejpam-1242	328	5	p	p	NOUN
ejpam-1242	328	6	and	and	CCONJ
ejpam-1242	328	7	(	(	PUNCT
ejpam-1242	328	8	1	1	NUM
ejpam-1242	328	9	+	+	NUM
ejpam-1242	328	10	q)x	q)x	ADJ
ejpam-1242	328	11	=	=	PUNCT
ejpam-1242	328	12	sm	sm	PROPN
ejpam-1242	328	13	∈	∈	PROPN
ejpam-1242	328	14	pm	pm	NOUN
ejpam-1242	328	15	.	.	PUNCT
ejpam-1242	329	1	thus	thus	ADV
ejpam-1242	329	2	(	(	PUNCT
ejpam-1242	329	3	1	1	NUM
ejpam-1242	329	4	+	+	NUM
ejpam-1242	329	5	q	q	NOUN
ejpam-1242	329	6	)	)	PUNCT
ejpam-1242	329	7	∈	∈	PROPN
ejpam-1242	329	8	k	k	NOUN
ejpam-1242	330	1	⊆	⊆	NUM
ejpam-1242	330	2	p	p	PROPN
ejpam-1242	330	3	′	′	NOUN
ejpam-1242	330	4	,	,	PUNCT
ejpam-1242	330	5	which	which	PRON
ejpam-1242	330	6	is	be	AUX
ejpam-1242	330	7	a	a	DET
ejpam-1242	330	8	contradiction	contradiction	NOUN
ejpam-1242	330	9	.	.	PUNCT
ejpam-1242	331	1	it	it	PRON
ejpam-1242	331	2	follows	follow	VERB
ejpam-1242	331	3	that	that	SCONJ
ejpam-1242	331	4	k	k	PROPN
ejpam-1242	331	5	=	=	PUNCT
ejpam-1242	331	6	r	r	NOUN
ejpam-1242	331	7	and	and	CCONJ
ejpam-1242	331	8	x	x	PROPN
ejpam-1242	331	9	∈	∈	PROPN
ejpam-1242	331	10	pm	pm	NOUN
ejpam-1242	331	11	,	,	PUNCT
ejpam-1242	331	12	as	as	SCONJ
ejpam-1242	331	13	required	require	VERB
ejpam-1242	331	14	.	.	PUNCT
ejpam-1242	332	1	s.	s.	PROPN
ejpam-1242	332	2	atani	atani	PROPN
ejpam-1242	332	3	,	,	PUNCT
ejpam-1242	332	4	r.	r.	PROPN
ejpam-1242	332	5	atrani	atrani	PROPN
ejpam-1242	332	6	,	,	PUNCT
ejpam-1242	332	7	ü.	ü.	NOUN
ejpam-1242	332	8	tekir	tekir	PROPN
ejpam-1242	332	9	/	/	SYM
ejpam-1242	332	10	eur	eur	PROPN
ejpam-1242	332	11	.	.	PUNCT
ejpam-1242	333	1	j.	j.	PROPN
ejpam-1242	333	2	pure	pure	PROPN
ejpam-1242	333	3	appl	appl	PROPN
ejpam-1242	333	4	.	.	PROPN
ejpam-1242	333	5	math	math	PROPN
ejpam-1242	333	6	,	,	PUNCT
ejpam-1242	333	7	4	4	NUM
ejpam-1242	333	8	(	(	PUNCT
ejpam-1242	333	9	2011	2011	NUM
ejpam-1242	333	10	)	)	PUNCT
ejpam-1242	333	11	,	,	PUNCT
ejpam-1242	333	12	251	251	NUM
ejpam-1242	333	13	-	-	SYM
ejpam-1242	333	14	265	265	NUM
ejpam-1242	333	15	259	259	NUM
ejpam-1242	333	16	remark	remark	NOUN
ejpam-1242	333	17	1	1	NUM
ejpam-1242	333	18	.	.	PUNCT
ejpam-1242	334	1	(	(	PUNCT
ejpam-1242	334	2	i	i	NOUN
ejpam-1242	334	3	)	)	PUNCT
ejpam-1242	334	4	(	(	PUNCT
ejpam-1242	334	5	change	change	NOUN
ejpam-1242	334	6	of	of	ADP
ejpam-1242	334	7	semirings	semiring	NOUN
ejpam-1242	334	8	)	)	PUNCT
ejpam-1242	334	9	assume	assume	VERB
ejpam-1242	334	10	that	that	SCONJ
ejpam-1242	334	11	i	i	PRON
ejpam-1242	334	12	is	be	AUX
ejpam-1242	334	13	an	an	DET
ejpam-1242	334	14	ideal	ideal	NOUN
ejpam-1242	334	15	of	of	ADP
ejpam-1242	334	16	a	a	DET
ejpam-1242	334	17	semiring	semire	VERB
ejpam-1242	334	18	r	r	NOUN
ejpam-1242	334	19	with	with	ADP
ejpam-1242	334	20	i	i	PRON
ejpam-1242	334	21	⊆	⊆	NUM
ejpam-1242	334	22	(	(	PUNCT
ejpam-1242	334	23	0	0	NUM
ejpam-1242	334	24	:	:	PUNCT
ejpam-1242	334	25	m	m	X
ejpam-1242	334	26	)	)	PUNCT
ejpam-1242	334	27	and	and	CCONJ
ejpam-1242	334	28	let	let	VERB
ejpam-1242	334	29	m	m	PRON
ejpam-1242	334	30	be	be	AUX
ejpam-1242	334	31	an	an	DET
ejpam-1242	334	32	r	r	NOUN
ejpam-1242	334	33	-	-	PUNCT
ejpam-1242	334	34	semimodule	semimodule	NOUN
ejpam-1242	334	35	.	.	PUNCT
ejpam-1242	335	1	we	we	PRON
ejpam-1242	335	2	show	show	VERB
ejpam-1242	335	3	now	now	ADV
ejpam-1242	335	4	how	how	SCONJ
ejpam-1242	335	5	m	m	NOUN
ejpam-1242	335	6	can	can	AUX
ejpam-1242	335	7	be	be	AUX
ejpam-1242	335	8	given	give	VERB
ejpam-1242	335	9	a	a	DET
ejpam-1242	335	10	natural	natural	ADJ
ejpam-1242	335	11	structure	structure	NOUN
ejpam-1242	335	12	as	as	ADP
ejpam-1242	335	13	a	a	DET
ejpam-1242	335	14	semimodule	semimodule	NOUN
ejpam-1242	335	15	over	over	ADP
ejpam-1242	335	16	r	r	PROPN
ejpam-1242	335	17	/	/	SYM
ejpam-1242	335	18	i	i	PRON
ejpam-1242	335	19	.	.	PUNCT
ejpam-1242	336	1	let	let	VERB
ejpam-1242	336	2	r	r	NOUN
ejpam-1242	336	3	,	,	PUNCT
ejpam-1242	336	4	s	s	PART
ejpam-1242	336	5	∈	∈	NOUN
ejpam-1242	336	6	r	r	NOUN
ejpam-1242	336	7	such	such	ADJ
ejpam-1242	336	8	that	that	DET
ejpam-1242	336	9	r	r	NOUN
ejpam-1242	337	1	+	+	NUM
ejpam-1242	337	2	i	i	NOUN
ejpam-1242	337	3	=	=	SYM
ejpam-1242	337	4	s	s	PART
ejpam-1242	338	1	+	+	NOUN
ejpam-1242	338	2	i	i	PRON
ejpam-1242	338	3	,	,	PUNCT
ejpam-1242	338	4	and	and	CCONJ
ejpam-1242	338	5	let	let	VERB
ejpam-1242	338	6	m	m	PRON
ejpam-1242	338	7	∈	∈	PROPN
ejpam-1242	338	8	m.	m.	NOUN
ejpam-1242	338	9	then	then	ADV
ejpam-1242	338	10	r	r	NOUN
ejpam-1242	338	11	+	+	CCONJ
ejpam-1242	338	12	a	a	DET
ejpam-1242	338	13	=	=	PUNCT
ejpam-1242	338	14	s+	s+	NOUN
ejpam-1242	338	15	b	b	X
ejpam-1242	338	16	for	for	ADP
ejpam-1242	338	17	some	some	DET
ejpam-1242	338	18	a	a	PRON
ejpam-1242	338	19	,	,	PUNCT
ejpam-1242	338	20	b	b	X
ejpam-1242	338	21	∈	∈	NOUN
ejpam-1242	338	22	i	i	PRON
ejpam-1242	338	23	,	,	PUNCT
ejpam-1242	338	24	and	and	CCONJ
ejpam-1242	338	25	rm	rm	NOUN
ejpam-1242	338	26	=	=	SYM
ejpam-1242	339	1	sm	sm	PROPN
ejpam-1242	339	2	.	.	PUNCT
ejpam-1242	340	1	hence	hence	ADV
ejpam-1242	340	2	we	we	PRON
ejpam-1242	340	3	can	can	AUX
ejpam-1242	340	4	unambiguously	unambiguously	ADV
ejpam-1242	340	5	define	define	VERB
ejpam-1242	340	6	a	a	DET
ejpam-1242	340	7	mapping	mapping	NOUN
ejpam-1242	340	8	r	r	NOUN
ejpam-1242	340	9	/	/	SYM
ejpam-1242	340	10	i	i	PROPN
ejpam-1242	340	11	×m	×m	NOUN
ejpam-1242	340	12	into	into	ADP
ejpam-1242	340	13	m	m	PROPN
ejpam-1242	340	14	(	(	PUNCT
ejpam-1242	340	15	sending	send	VERB
ejpam-1242	340	16	(	(	PUNCT
ejpam-1242	340	17	r	r	NOUN
ejpam-1242	340	18	+	+	CCONJ
ejpam-1242	340	19	i	i	PROPN
ejpam-1242	340	20	,	,	PUNCT
ejpam-1242	340	21	m	m	VERB
ejpam-1242	340	22	)	)	PUNCT
ejpam-1242	340	23	to	to	PART
ejpam-1242	340	24	rm	rm	VERB
ejpam-1242	340	25	)	)	PUNCT
ejpam-1242	340	26	and	and	CCONJ
ejpam-1242	340	27	it	it	PRON
ejpam-1242	340	28	is	be	AUX
ejpam-1242	340	29	routine	routine	ADJ
ejpam-1242	340	30	to	to	PART
ejpam-1242	340	31	check	check	VERB
ejpam-1242	340	32	that	that	SCONJ
ejpam-1242	340	33	this	this	PRON
ejpam-1242	340	34	turns	turn	VERB
ejpam-1242	340	35	the	the	DET
ejpam-1242	340	36	commutative	commutative	ADJ
ejpam-1242	340	37	additive	additive	NOUN
ejpam-1242	340	38	semigroup	semigroup	NOUN
ejpam-1242	340	39	with	with	ADP
ejpam-1242	340	40	a	a	DET
ejpam-1242	340	41	zero	zero	NUM
ejpam-1242	340	42	element	element	NOUN
ejpam-1242	340	43	m	m	VERB
ejpam-1242	340	44	into	into	ADP
ejpam-1242	340	45	an	an	DET
ejpam-1242	340	46	r	r	NOUN
ejpam-1242	340	47	/	/	SYM
ejpam-1242	340	48	i	i	NOUN
ejpam-1242	340	49	-	-	PUNCT
ejpam-1242	340	50	semimodule	semimodule	NOUN
ejpam-1242	340	51	.	.	PUNCT
ejpam-1242	341	1	it	it	PRON
ejpam-1242	341	2	should	should	AUX
ejpam-1242	341	3	be	be	AUX
ejpam-1242	341	4	noted	note	VERB
ejpam-1242	341	5	that	that	SCONJ
ejpam-1242	341	6	a	a	DET
ejpam-1242	341	7	subset	subset	NOUN
ejpam-1242	341	8	of	of	ADP
ejpam-1242	341	9	m	m	PROPN
ejpam-1242	341	10	is	be	AUX
ejpam-1242	341	11	an	an	DET
ejpam-1242	341	12	r	r	NOUN
ejpam-1242	341	13	-	-	PUNCT
ejpam-1242	341	14	subsemimodule	subsemimodule	NOUN
ejpam-1242	341	15	if	if	SCONJ
ejpam-1242	341	16	and	and	CCONJ
ejpam-1242	341	17	only	only	ADV
ejpam-1242	341	18	if	if	SCONJ
ejpam-1242	341	19	it	it	PRON
ejpam-1242	341	20	is	be	AUX
ejpam-1242	341	21	an	an	DET
ejpam-1242	341	22	r	r	NOUN
ejpam-1242	341	23	/	/	SYM
ejpam-1242	341	24	isubsemimodule	isubsemimodule	NOUN
ejpam-1242	341	25	.	.	PUNCT
ejpam-1242	342	1	(	(	PUNCT
ejpam-1242	342	2	ii	ii	NOUN
ejpam-1242	342	3	)	)	PUNCT
ejpam-1242	342	4	assume	assume	VERB
ejpam-1242	342	5	that	that	SCONJ
ejpam-1242	342	6	n	n	PRON
ejpam-1242	342	7	is	be	AUX
ejpam-1242	342	8	a	a	DET
ejpam-1242	342	9	proper	proper	ADJ
ejpam-1242	342	10	k	k	NOUN
ejpam-1242	342	11	-	-	NOUN
ejpam-1242	342	12	subsemimodule	subsemimodule	NOUN
ejpam-1242	342	13	of	of	ADP
ejpam-1242	342	14	a	a	DET
ejpam-1242	342	15	semimodule	semimodule	NOUN
ejpam-1242	342	16	m	m	VERB
ejpam-1242	342	17	over	over	ADP
ejpam-1242	342	18	a	a	DET
ejpam-1242	342	19	semiring	semire	VERB
ejpam-1242	342	20	r	r	NOUN
ejpam-1242	342	21	and	and	CCONJ
ejpam-1242	342	22	let	let	VERB
ejpam-1242	342	23	i	i	PRON
ejpam-1242	342	24	be	be	AUX
ejpam-1242	342	25	an	an	DET
ejpam-1242	342	26	ideal	ideal	NOUN
ejpam-1242	342	27	of	of	ADP
ejpam-1242	342	28	r	r	NOUN
ejpam-1242	342	29	with	with	ADP
ejpam-1242	342	30	i	i	PRON
ejpam-1242	342	31	⊆	⊆	NUM
ejpam-1242	342	32	(	(	PUNCT
ejpam-1242	342	33	0	0	NUM
ejpam-1242	342	34	:	:	PUNCT
ejpam-1242	342	35	m	m	NUM
ejpam-1242	342	36	)	)	PUNCT
ejpam-1242	342	37	.	.	PUNCT
ejpam-1242	343	1	then	then	ADV
ejpam-1242	343	2	n	n	PRON
ejpam-1242	343	3	is	be	AUX
ejpam-1242	343	4	a	a	DET
ejpam-1242	343	5	prime	prime	ADJ
ejpam-1242	343	6	r	r	NOUN
ejpam-1242	343	7	-	-	PUNCT
ejpam-1242	343	8	subsemimodule	subsemimodule	NOUN
ejpam-1242	343	9	of	of	ADP
ejpam-1242	343	10	m	m	PRON
ejpam-1242	343	11	if	if	SCONJ
ejpam-1242	344	1	and	and	CCONJ
ejpam-1242	344	2	only	only	ADV
ejpam-1242	344	3	if	if	SCONJ
ejpam-1242	344	4	n	n	PRON
ejpam-1242	344	5	is	be	AUX
ejpam-1242	344	6	a	a	DET
ejpam-1242	344	7	prime	prime	ADJ
ejpam-1242	344	8	subsemimodule	subsemimodule	NOUN
ejpam-1242	344	9	of	of	ADP
ejpam-1242	344	10	m	m	PRON
ejpam-1242	344	11	as	as	ADP
ejpam-1242	344	12	an	an	DET
ejpam-1242	344	13	r	r	NOUN
ejpam-1242	344	14	/	/	SYM
ejpam-1242	344	15	i	i	NOUN
ejpam-1242	344	16	-	-	PUNCT
ejpam-1242	344	17	semimodule	semimodule	NOUN
ejpam-1242	344	18	.	.	PUNCT
ejpam-1242	345	1	theorem	theorem	VERB
ejpam-1242	345	2	7	7	NUM
ejpam-1242	345	3	.	.	PUNCT
ejpam-1242	346	1	the	the	DET
ejpam-1242	346	2	following	follow	VERB
ejpam-1242	346	3	statements	statement	NOUN
ejpam-1242	346	4	are	be	AUX
ejpam-1242	346	5	equivalent	equivalent	ADJ
ejpam-1242	346	6	for	for	ADP
ejpam-1242	346	7	a	a	DET
ejpam-1242	346	8	proper	proper	ADJ
ejpam-1242	346	9	k	k	ADJ
ejpam-1242	346	10	-	-	ADJ
ejpam-1242	346	11	subsemimodule	subsemimodule	NOUN
ejpam-1242	346	12	n	n	NOUN
ejpam-1242	346	13	of	of	ADP
ejpam-1242	346	14	a	a	DET
ejpam-1242	346	15	very	very	ADV
ejpam-1242	346	16	strong	strong	ADJ
ejpam-1242	346	17	multiplication	multiplication	NOUN
ejpam-1242	346	18	semimodule	semimodule	NOUN
ejpam-1242	346	19	m	m	VERB
ejpam-1242	346	20	over	over	ADP
ejpam-1242	346	21	a	a	DET
ejpam-1242	346	22	semiring	semire	VERB
ejpam-1242	346	23	r.	r.	PROPN
ejpam-1242	346	24	(	(	PUNCT
ejpam-1242	346	25	i	i	NOUN
ejpam-1242	346	26	)	)	PUNCT
ejpam-1242	346	27	n	n	PRON
ejpam-1242	346	28	is	be	AUX
ejpam-1242	346	29	a	a	DET
ejpam-1242	346	30	strong	strong	ADJ
ejpam-1242	346	31	prime	prime	ADJ
ejpam-1242	346	32	subsemimodule	subsemimodule	NOUN
ejpam-1242	346	33	of	of	ADP
ejpam-1242	346	34	m.	m.	NOUN
ejpam-1242	346	35	(	(	PUNCT
ejpam-1242	346	36	ii	ii	NOUN
ejpam-1242	346	37	)	)	PUNCT
ejpam-1242	346	38	(	(	PUNCT
ejpam-1242	346	39	n	n	X
ejpam-1242	346	40	:	:	PUNCT
ejpam-1242	346	41	m	m	X
ejpam-1242	346	42	)	)	PUNCT
ejpam-1242	346	43	is	be	AUX
ejpam-1242	346	44	a	a	DET
ejpam-1242	346	45	strong	strong	ADJ
ejpam-1242	346	46	prime	prime	ADJ
ejpam-1242	346	47	k	k	NOUN
ejpam-1242	346	48	-	-	NOUN
ejpam-1242	346	49	ideal	ideal	NOUN
ejpam-1242	346	50	of	of	ADP
ejpam-1242	346	51	r.	r.	PROPN
ejpam-1242	346	52	item	item	PROPN
ejpam-1242	346	53	n	n	PROPN
ejpam-1242	346	54	=	=	PUNCT
ejpam-1242	346	55	pm	pm	NOUN
ejpam-1242	346	56	for	for	ADP
ejpam-1242	346	57	some	some	DET
ejpam-1242	346	58	strong	strong	ADJ
ejpam-1242	346	59	prime	prime	ADJ
ejpam-1242	346	60	ideal	ideal	NOUN
ejpam-1242	346	61	p	p	NOUN
ejpam-1242	346	62	of	of	ADP
ejpam-1242	346	63	r	r	NOUN
ejpam-1242	346	64	with	with	ADP
ejpam-1242	346	65	(	(	PUNCT
ejpam-1242	346	66	0	0	NUM
ejpam-1242	346	67	:	:	PUNCT
ejpam-1242	346	68	m	m	X
ejpam-1242	346	69	)	)	PUNCT
ejpam-1242	347	1	⊆	⊆	NUM
ejpam-1242	347	2	p.	p.	NOUN
ejpam-1242	347	3	proof	proof	NOUN
ejpam-1242	347	4	.	.	PUNCT
ejpam-1242	348	1	(	(	PUNCT
ejpam-1242	348	2	i)⇒	i)⇒	PROPN
ejpam-1242	348	3	(	(	PUNCT
ejpam-1242	348	4	ii	ii	NOUN
ejpam-1242	348	5	)	)	PUNCT
ejpam-1242	348	6	.	.	PUNCT
ejpam-1242	349	1	by	by	ADP
ejpam-1242	349	2	[	[	X
ejpam-1242	349	3	13	13	NUM
ejpam-1242	349	4	,	,	PUNCT
ejpam-1242	349	5	lemma	lemma	PROPN
ejpam-1242	349	6	4	4	NUM
ejpam-1242	349	7	]	]	PUNCT
ejpam-1242	349	8	,	,	PUNCT
ejpam-1242	349	9	(	(	PUNCT
ejpam-1242	349	10	n	n	X
ejpam-1242	349	11	:	:	PUNCT
ejpam-1242	349	12	m	m	X
ejpam-1242	349	13	)	)	PUNCT
ejpam-1242	349	14	is	be	AUX
ejpam-1242	349	15	a	a	DET
ejpam-1242	349	16	prime	prime	ADJ
ejpam-1242	349	17	ideal	ideal	NOUN
ejpam-1242	349	18	of	of	ADP
ejpam-1242	349	19	r.	r.	PROPN
ejpam-1242	349	20	if	if	SCONJ
ejpam-1242	349	21	0	0	NUM
ejpam-1242	349	22	6=	6=	NUM
ejpam-1242	349	23	m	m	PROPN
ejpam-1242	349	24	∈	∈	NOUN
ejpam-1242	349	25	m	m	NOUN
ejpam-1242	349	26	,	,	PUNCT
ejpam-1242	349	27	then	then	ADV
ejpam-1242	349	28	j	j	PROPN
ejpam-1242	349	29	=	=	PRON
ejpam-1242	349	30	{	{	PUNCT
ejpam-1242	349	31	r	r	NOUN
ejpam-1242	349	32	∈	∈	PROPN
ejpam-1242	349	33	r	r	NOUN
ejpam-1242	349	34	:	:	PUNCT
ejpam-1242	349	35	rm	rm	PROPN
ejpam-1242	349	36	∈	∈	PROPN
ejpam-1242	349	37	(	(	PUNCT
ejpam-1242	349	38	n	n	NOUN
ejpam-1242	349	39	:	:	PUNCT
ejpam-1242	349	40	m)m	m)m	X
ejpam-1242	349	41	}	}	PUNCT
ejpam-1242	349	42	is	be	AUX
ejpam-1242	349	43	a	a	DET
ejpam-1242	349	44	proper	proper	ADJ
ejpam-1242	349	45	strong	strong	ADJ
ejpam-1242	349	46	k	k	NOUN
ejpam-1242	349	47	-	-	NOUN
ejpam-1242	349	48	ideal	ideal	NOUN
ejpam-1242	349	49	of	of	ADP
ejpam-1242	349	50	r	r	NOUN
ejpam-1242	349	51	(	(	PUNCT
ejpam-1242	349	52	since	since	SCONJ
ejpam-1242	349	53	m	m	PROPN
ejpam-1242	349	54	is	be	AUX
ejpam-1242	349	55	very	very	ADV
ejpam-1242	349	56	strong	strong	ADJ
ejpam-1242	349	57	multiplication	multiplication	NOUN
ejpam-1242	349	58	)	)	PUNCT
ejpam-1242	349	59	with	with	ADP
ejpam-1242	349	60	(	(	PUNCT
ejpam-1242	349	61	n	n	NUM
ejpam-1242	349	62	:	:	PUNCT
ejpam-1242	349	63	m	m	X
ejpam-1242	349	64	)	)	PUNCT
ejpam-1242	349	65	⊆	⊆	NUM
ejpam-1242	349	66	j	j	NOUN
ejpam-1242	349	67	;	;	PUNCT
ejpam-1242	349	68	hence	hence	ADV
ejpam-1242	349	69	(	(	PUNCT
ejpam-1242	349	70	n	n	X
ejpam-1242	349	71	:	:	PUNCT
ejpam-1242	349	72	m	m	X
ejpam-1242	349	73	)	)	PUNCT
ejpam-1242	349	74	is	be	AUX
ejpam-1242	349	75	a	a	DET
ejpam-1242	349	76	strong	strong	ADJ
ejpam-1242	349	77	prime	prime	ADJ
ejpam-1242	349	78	k	k	NOUN
ejpam-1242	349	79	-	-	NOUN
ejpam-1242	349	80	ideal	ideal	NOUN
ejpam-1242	349	81	of	of	ADP
ejpam-1242	349	82	r.	r.	PROPN
ejpam-1242	349	83	(	(	PUNCT
ejpam-1242	349	84	ii)⇒	ii)⇒	PROPN
ejpam-1242	349	85	(	(	PUNCT
ejpam-1242	349	86	iii	iii	NOUN
ejpam-1242	349	87	)	)	PUNCT
ejpam-1242	349	88	is	be	AUX
ejpam-1242	349	89	clear	clear	ADJ
ejpam-1242	349	90	.	.	PUNCT
ejpam-1242	350	1	(	(	PUNCT
ejpam-1242	350	2	iii	iii	X
ejpam-1242	350	3	)	)	PUNCT
ejpam-1242	350	4	⇒	⇒	NOUN
ejpam-1242	350	5	(	(	PUNCT
ejpam-1242	350	6	i	i	NOUN
ejpam-1242	350	7	)	)	PUNCT
ejpam-1242	350	8	.	.	PUNCT
ejpam-1242	351	1	since	since	SCONJ
ejpam-1242	351	2	n	n	NOUN
ejpam-1242	351	3	=	=	VERB
ejpam-1242	351	4	pm	pm	NOUN
ejpam-1242	351	5	6=	6=	ADP
ejpam-1242	351	6	m	m	ADJ
ejpam-1242	351	7	and	and	CCONJ
ejpam-1242	351	8	as	as	ADP
ejpam-1242	351	9	an	an	DET
ejpam-1242	351	10	r/(0	r/(0	NOUN
ejpam-1242	351	11	:	:	PUNCT
ejpam-1242	351	12	m)-semimodule	m)-semimodule	NOUN
ejpam-1242	351	13	,	,	PUNCT
ejpam-1242	351	14	n	n	PRON
ejpam-1242	351	15	is	be	AUX
ejpam-1242	351	16	a	a	DET
ejpam-1242	351	17	strong	strong	ADJ
ejpam-1242	351	18	prime	prime	ADJ
ejpam-1242	351	19	subsemimodule	subsemimodule	NOUN
ejpam-1242	351	20	by	by	ADP
ejpam-1242	351	21	proposition	proposition	NOUN
ejpam-1242	351	22	3	3	NUM
ejpam-1242	351	23	(	(	PUNCT
ejpam-1242	351	24	iii	iii	NOUN
ejpam-1242	351	25	)	)	PUNCT
ejpam-1242	351	26	,	,	PUNCT
ejpam-1242	351	27	so	so	ADV
ejpam-1242	351	28	is	be	AUX
ejpam-1242	351	29	a	a	DET
ejpam-1242	351	30	strong	strong	ADJ
ejpam-1242	351	31	prime	prime	NOUN
ejpam-1242	351	32	as	as	ADP
ejpam-1242	351	33	an	an	DET
ejpam-1242	351	34	r	r	NOUN
ejpam-1242	351	35	-	-	PUNCT
ejpam-1242	351	36	subsemimodule	subsemimodule	NOUN
ejpam-1242	351	37	of	of	ADP
ejpam-1242	351	38	m	m	PRON
ejpam-1242	351	39	by	by	ADP
ejpam-1242	351	40	remark	remark	NOUN
ejpam-1242	351	41	1	1	NUM
ejpam-1242	351	42	.	.	PUNCT
ejpam-1242	351	43	theorem	theorem	NOUN
ejpam-1242	351	44	8	8	NUM
ejpam-1242	351	45	.	.	PUNCT
ejpam-1242	352	1	let	let	VERB
ejpam-1242	352	2	r	r	PRON
ejpam-1242	352	3	be	be	AUX
ejpam-1242	352	4	a	a	DET
ejpam-1242	352	5	semiring	semiring	NOUN
ejpam-1242	352	6	,	,	PUNCT
ejpam-1242	352	7	n	n	CCONJ
ejpam-1242	352	8	a	a	DET
ejpam-1242	352	9	proper	proper	ADJ
ejpam-1242	352	10	subsemimodule	subsemimodule	NOUN
ejpam-1242	352	11	of	of	ADP
ejpam-1242	352	12	a	a	DET
ejpam-1242	352	13	very	very	ADV
ejpam-1242	352	14	strong	strong	ADJ
ejpam-1242	352	15	multiplication	multiplication	NOUN
ejpam-1242	352	16	r	r	NOUN
ejpam-1242	352	17	-	-	PUNCT
ejpam-1242	352	18	semimodule	semimodule	NOUN
ejpam-1242	352	19	m	m	NOUN
ejpam-1242	352	20	and	and	CCONJ
ejpam-1242	352	21	a=	a=	ADJ
ejpam-1242	352	22	(	(	PUNCT
ejpam-1242	352	23	n	n	NUM
ejpam-1242	352	24	:	:	PUNCT
ejpam-1242	352	25	m	m	X
ejpam-1242	352	26	)	)	PUNCT
ejpam-1242	352	27	.	.	PUNCT
ejpam-1242	353	1	then	then	ADV
ejpam-1242	353	2	rad(n	rad(n	NOUN
ejpam-1242	353	3	)	)	PUNCT
ejpam-1242	353	4	=	=	SYM
ejpam-1242	353	5	rad(a)m	rad(a)m	NOUN
ejpam-1242	353	6	.	.	PUNCT
ejpam-1242	354	1	proof	proof	NOUN
ejpam-1242	354	2	.	.	PUNCT
ejpam-1242	355	1	without	without	ADP
ejpam-1242	355	2	loss	loss	NOUN
ejpam-1242	355	3	of	of	ADP
ejpam-1242	355	4	generality	generality	NOUN
ejpam-1242	355	5	m	m	AUX
ejpam-1242	355	6	is	be	AUX
ejpam-1242	355	7	a	a	DET
ejpam-1242	355	8	faithful	faithful	ADJ
ejpam-1242	355	9	r	r	NOUN
ejpam-1242	355	10	-	-	PUNCT
ejpam-1242	355	11	semimodule	semimodule	NOUN
ejpam-1242	355	12	.	.	PUNCT
ejpam-1242	356	1	letb	letb	PROPN
ejpam-1242	356	2	denote	denote	VERB
ejpam-1242	356	3	the	the	DET
ejpam-1242	356	4	collection	collection	NOUN
ejpam-1242	356	5	of	of	ADP
ejpam-1242	356	6	all	all	DET
ejpam-1242	356	7	strong	strong	ADJ
ejpam-1242	356	8	prime	prime	ADJ
ejpam-1242	356	9	ideals	ideal	NOUN
ejpam-1242	356	10	p	p	NOUN
ejpam-1242	356	11	of	of	ADP
ejpam-1242	356	12	r	r	NOUN
ejpam-1242	356	13	such	such	ADJ
ejpam-1242	356	14	that	that	SCONJ
ejpam-1242	356	15	a	a	DET
ejpam-1242	356	16	⊆	⊆	NUM
ejpam-1242	356	17	p	p	NOUN
ejpam-1242	356	18	and	and	CCONJ
ejpam-1242	356	19	c	c	PROPN
ejpam-1242	356	20	denote	denote	VERB
ejpam-1242	356	21	the	the	DET
ejpam-1242	356	22	collection	collection	NOUN
ejpam-1242	356	23	of	of	ADP
ejpam-1242	356	24	all	all	DET
ejpam-1242	356	25	prime	prime	ADJ
ejpam-1242	356	26	ideals	ideal	NOUN
ejpam-1242	356	27	p	p	NOUN
ejpam-1242	356	28	of	of	ADP
ejpam-1242	356	29	r	r	NOUN
ejpam-1242	356	30	such	such	ADJ
ejpam-1242	356	31	that	that	SCONJ
ejpam-1242	356	32	a	a	DET
ejpam-1242	356	33	⊆	⊆	NUM
ejpam-1242	356	34	p.	p.	NOUN
ejpam-1242	356	35	clearly	clearly	ADV
ejpam-1242	356	36	,	,	PUNCT
ejpam-1242	356	37	c	c	PROPN
ejpam-1242	356	38	⊆	⊆	NUM
ejpam-1242	356	39	b	b	NOUN
ejpam-1242	356	40	.	.	PUNCT
ejpam-1242	357	1	if	if	SCONJ
ejpam-1242	357	2	b	b	PROPN
ejpam-1242	357	3	=	=	SYM
ejpam-1242	357	4	rad(a	rad(a	PROPN
ejpam-1242	357	5	)	)	PUNCT
ejpam-1242	357	6	,	,	PUNCT
ejpam-1242	357	7	then	then	ADV
ejpam-1242	357	8	b	b	X
ejpam-1242	357	9	=	=	SYM
ejpam-1242	357	10	⋂	⋂	PROPN
ejpam-1242	358	1	p∈c	p∈c	NOUN
ejpam-1242	358	2	p	p	NOUN
ejpam-1242	359	1	[	[	X
ejpam-1242	359	2	see	see	INTJ
ejpam-1242	359	3	,	,	PUNCT
ejpam-1242	359	4	1	1	NUM
ejpam-1242	359	5	]	]	PUNCT
ejpam-1242	359	6	,	,	PUNCT
ejpam-1242	359	7	and	and	CCONJ
ejpam-1242	359	8	hence	hence	ADV
ejpam-1242	359	9	by	by	ADP
ejpam-1242	359	10	proposition	proposition	NOUN
ejpam-1242	359	11	3	3	NUM
ejpam-1242	359	12	(	(	PUNCT
ejpam-1242	359	13	ii	ii	NOUN
ejpam-1242	359	14	)	)	PUNCT
ejpam-1242	359	15	,	,	PUNCT
ejpam-1242	359	16	bm	bm	PROPN
ejpam-1242	359	17	=	=	SYM
ejpam-1242	359	18	⋂	⋂	PROPN
ejpam-1242	359	19	p∈c	p∈c	NOUN
ejpam-1242	359	20	(	(	PUNCT
ejpam-1242	359	21	pm	pm	NOUN
ejpam-1242	359	22	)	)	PUNCT
ejpam-1242	359	23	⊆	⊆	NUM
ejpam-1242	359	24	⋂	⋂	PROPN
ejpam-1242	359	25	p∈b	p∈b	NOUN
ejpam-1242	359	26	pm	pm	NOUN
ejpam-1242	359	27	.	.	PUNCT
ejpam-1242	360	1	let	let	VERB
ejpam-1242	360	2	p	p	PROPN
ejpam-1242	360	3	∈	∈	PROPN
ejpam-1242	360	4	b	b	PROPN
ejpam-1242	360	5	.	.	PUNCT
ejpam-1242	361	1	if	if	SCONJ
ejpam-1242	361	2	m	m	VERB
ejpam-1242	361	3	=	=	VERB
ejpam-1242	361	4	pm	pm	NOUN
ejpam-1242	361	5	,	,	PUNCT
ejpam-1242	361	6	then	then	ADV
ejpam-1242	361	7	rad(n	rad(n	NOUN
ejpam-1242	361	8	)	)	PUNCT
ejpam-1242	361	9	⊆	⊆	NUM
ejpam-1242	361	10	pm	pm	NOUN
ejpam-1242	361	11	.	.	PUNCT
ejpam-1242	362	1	if	if	SCONJ
ejpam-1242	362	2	m	m	PROPN
ejpam-1242	362	3	6=	6=	ADP
ejpam-1242	362	4	pm	pm	NOUN
ejpam-1242	362	5	,	,	PUNCT
ejpam-1242	362	6	then	then	ADV
ejpam-1242	362	7	n	n	CCONJ
ejpam-1242	362	8	=	=	PRON
ejpam-1242	362	9	am	be	AUX
ejpam-1242	362	10	⊆	⊆	NUM
ejpam-1242	362	11	pm	pm	NOUN
ejpam-1242	362	12	implies	imply	VERB
ejpam-1242	362	13	rad(n	rad(n	NOUN
ejpam-1242	362	14	)	)	PUNCT
ejpam-1242	362	15	⊆	⊆	NUM
ejpam-1242	362	16	pm	pm	NOUN
ejpam-1242	362	17	by	by	ADP
ejpam-1242	362	18	theorem	theorem	NOUN
ejpam-1242	362	19	7	7	NUM
ejpam-1242	362	20	.	.	PUNCT
ejpam-1242	363	1	it	it	PRON
ejpam-1242	363	2	follows	follow	VERB
ejpam-1242	363	3	that	that	SCONJ
ejpam-1242	363	4	rad(n)⊆	rad(n)⊆	PROPN
ejpam-1242	363	5	bm	bm	X
ejpam-1242	363	6	.	.	PUNCT
ejpam-1242	364	1	conversely	conversely	ADV
ejpam-1242	364	2	,	,	PUNCT
ejpam-1242	364	3	suppose	suppose	VERB
ejpam-1242	364	4	that	that	SCONJ
ejpam-1242	364	5	l	l	NOUN
ejpam-1242	364	6	is	be	AUX
ejpam-1242	364	7	a	a	DET
ejpam-1242	364	8	strong	strong	ADJ
ejpam-1242	364	9	prime	prime	ADJ
ejpam-1242	364	10	subsemimodule	subsemimodule	NOUN
ejpam-1242	364	11	of	of	ADP
ejpam-1242	364	12	m	m	AUX
ejpam-1242	364	13	containing	contain	VERB
ejpam-1242	364	14	n	n	PRON
ejpam-1242	364	15	.	.	PUNCT
ejpam-1242	365	1	by	by	ADP
ejpam-1242	365	2	theorem	theorem	NOUN
ejpam-1242	365	3	7	7	NUM
ejpam-1242	365	4	,	,	PUNCT
ejpam-1242	365	5	there	there	PRON
ejpam-1242	365	6	exists	exist	VERB
ejpam-1242	365	7	a	a	DET
ejpam-1242	365	8	strong	strong	ADJ
ejpam-1242	365	9	prime	prime	ADJ
ejpam-1242	365	10	ideal	ideal	NOUN
ejpam-1242	365	11	p	p	NOUN
ejpam-1242	365	12	′	′	NOUN
ejpam-1242	365	13	of	of	ADP
ejpam-1242	365	14	r	r	NOUN
ejpam-1242	365	15	such	such	ADJ
ejpam-1242	365	16	that	that	DET
ejpam-1242	365	17	l	l	NOUN
ejpam-1242	365	18	=	=	PUNCT
ejpam-1242	365	19	p	p	X
ejpam-1242	365	20	′m	′m	PROPN
ejpam-1242	365	21	.	.	PUNCT
ejpam-1242	366	1	since	since	SCONJ
ejpam-1242	366	2	am	be	AUX
ejpam-1242	366	3	=	=	NUM
ejpam-1242	366	4	n	n	CCONJ
ejpam-1242	366	5	⊆	⊆	NUM
ejpam-1242	366	6	l	l	NOUN
ejpam-1242	366	7	=	=	PUNCT
ejpam-1242	366	8	p	p	X
ejpam-1242	366	9	′m	′m	PROPN
ejpam-1242	366	10	6=	6=	ADP
ejpam-1242	366	11	m	m	VERB
ejpam-1242	366	12	it	it	PRON
ejpam-1242	366	13	follows	follow	VERB
ejpam-1242	366	14	that	that	SCONJ
ejpam-1242	366	15	a⊆	a⊆	VERB
ejpam-1242	366	16	p	p	NOUN
ejpam-1242	366	17	′	′	NOUN
ejpam-1242	366	18	by	by	ADP
ejpam-1242	366	19	[	[	X
ejpam-1242	366	20	13	13	NUM
ejpam-1242	366	21	,	,	PUNCT
ejpam-1242	366	22	theorem	theorem	VERB
ejpam-1242	366	23	7	7	NUM
ejpam-1242	366	24	]	]	PUNCT
ejpam-1242	366	25	,	,	PUNCT
ejpam-1242	366	26	and	and	CCONJ
ejpam-1242	366	27	hence	hence	ADV
ejpam-1242	366	28	b	b	NOUN
ejpam-1242	366	29	⊆	⊆	NUM
ejpam-1242	366	30	p	p	PRON
ejpam-1242	366	31	′.	′.	NOUN
ejpam-1242	366	32	thus	thus	ADV
ejpam-1242	366	33	bm	bm	PROPN
ejpam-1242	366	34	⊆	⊆	NUM
ejpam-1242	366	35	l.	l.	NOUN
ejpam-1242	366	36	it	it	PRON
ejpam-1242	366	37	follows	follow	VERB
ejpam-1242	366	38	that	that	SCONJ
ejpam-1242	366	39	bm	bm	PROPN
ejpam-1242	366	40	⊆	⊆	NUM
ejpam-1242	366	41	rad(n	rad(n	NOUN
ejpam-1242	366	42	)	)	PUNCT
ejpam-1242	366	43	,	,	PUNCT
ejpam-1242	366	44	and	and	CCONJ
ejpam-1242	366	45	so	so	ADV
ejpam-1242	366	46	we	we	PRON
ejpam-1242	366	47	have	have	VERB
ejpam-1242	366	48	equality	equality	NOUN
ejpam-1242	366	49	.	.	PUNCT
ejpam-1242	367	1	s.	s.	PROPN
ejpam-1242	367	2	atani	atani	PROPN
ejpam-1242	367	3	,	,	PUNCT
ejpam-1242	367	4	r.	r.	PROPN
ejpam-1242	367	5	atrani	atrani	PROPN
ejpam-1242	367	6	,	,	PUNCT
ejpam-1242	367	7	ü.	ü.	NOUN
ejpam-1242	367	8	tekir	tekir	PROPN
ejpam-1242	367	9	/	/	SYM
ejpam-1242	367	10	eur	eur	PROPN
ejpam-1242	367	11	.	.	PUNCT
ejpam-1242	368	1	j.	j.	PROPN
ejpam-1242	368	2	pure	pure	PROPN
ejpam-1242	368	3	appl	appl	PROPN
ejpam-1242	368	4	.	.	PROPN
ejpam-1242	368	5	math	math	PROPN
ejpam-1242	368	6	,	,	PUNCT
ejpam-1242	368	7	4	4	NUM
ejpam-1242	368	8	(	(	PUNCT
ejpam-1242	368	9	2011	2011	NUM
ejpam-1242	368	10	)	)	PUNCT
ejpam-1242	368	11	,	,	PUNCT
ejpam-1242	368	12	251	251	NUM
ejpam-1242	368	13	-	-	SYM
ejpam-1242	368	14	265	265	NUM
ejpam-1242	368	15	260	260	NUM
ejpam-1242	368	16	4	4	NUM
ejpam-1242	368	17	.	.	PUNCT
ejpam-1242	368	18	prime	prime	PROPN
ejpam-1242	368	19	spectrum	spectrum	PROPN
ejpam-1242	368	20	assume	assume	VERB
ejpam-1242	368	21	that	that	SCONJ
ejpam-1242	368	22	r	r	NOUN
ejpam-1242	368	23	is	be	AUX
ejpam-1242	368	24	a	a	DET
ejpam-1242	368	25	semiring	semiring	NOUN
ejpam-1242	368	26	and	and	CCONJ
ejpam-1242	368	27	let	let	VERB
ejpam-1242	368	28	m	m	PRON
ejpam-1242	368	29	be	be	AUX
ejpam-1242	368	30	an	an	DET
ejpam-1242	368	31	r	r	NOUN
ejpam-1242	368	32	-	-	PUNCT
ejpam-1242	368	33	semimodule	semimodule	NOUN
ejpam-1242	368	34	and	and	CCONJ
ejpam-1242	368	35	n	n	PRON
ejpam-1242	368	36	be	be	VERB
ejpam-1242	368	37	a	a	DET
ejpam-1242	368	38	subsemimodule	subsemimodule	NOUN
ejpam-1242	368	39	of	of	ADP
ejpam-1242	368	40	m	m	NOUN
ejpam-1242	368	41	such	such	ADJ
ejpam-1242	368	42	that	that	SCONJ
ejpam-1242	368	43	n	n	NOUN
ejpam-1242	368	44	=	=	VERB
ejpam-1242	369	1	i	i	PRON
ejpam-1242	369	2	m	m	VERB
ejpam-1242	369	3	for	for	ADP
ejpam-1242	369	4	some	some	DET
ejpam-1242	369	5	ideal	ideal	NOUN
ejpam-1242	370	1	i	i	PRON
ejpam-1242	370	2	of	of	ADP
ejpam-1242	370	3	r.	r.	PROPN
ejpam-1242	370	4	then	then	ADV
ejpam-1242	370	5	we	we	PRON
ejpam-1242	370	6	say	say	VERB
ejpam-1242	370	7	that	that	SCONJ
ejpam-1242	370	8	i	i	PRON
ejpam-1242	370	9	is	be	AUX
ejpam-1242	370	10	a	a	DET
ejpam-1242	370	11	presentation	presentation	NOUN
ejpam-1242	370	12	ideal	ideal	NOUN
ejpam-1242	370	13	of	of	ADP
ejpam-1242	370	14	n	n	PROPN
ejpam-1242	370	15	.	.	PUNCT
ejpam-1242	371	1	clearly	clearly	ADV
ejpam-1242	371	2	,	,	PUNCT
ejpam-1242	371	3	every	every	DET
ejpam-1242	371	4	subsemimodule	subsemimodule	NOUN
ejpam-1242	371	5	of	of	ADP
ejpam-1242	371	6	m	m	PROPN
ejpam-1242	371	7	has	have	VERB
ejpam-1242	371	8	a	a	DET
ejpam-1242	371	9	presentation	presentation	NOUN
ejpam-1242	371	10	ideal	ideal	NOUN
ejpam-1242	371	11	if	if	SCONJ
ejpam-1242	371	12	and	and	CCONJ
ejpam-1242	371	13	only	only	ADV
ejpam-1242	371	14	if	if	SCONJ
ejpam-1242	371	15	m	m	NOUN
ejpam-1242	371	16	is	be	AUX
ejpam-1242	371	17	a	a	DET
ejpam-1242	371	18	multiplication	multiplication	NOUN
ejpam-1242	371	19	semimodule	semimodule	NOUN
ejpam-1242	371	20	.	.	PUNCT
ejpam-1242	372	1	let	let	VERB
ejpam-1242	372	2	n	n	NOUN
ejpam-1242	372	3	and	and	CCONJ
ejpam-1242	372	4	k	k	PROPN
ejpam-1242	372	5	be	be	AUX
ejpam-1242	372	6	subsemimodules	subsemimodule	NOUN
ejpam-1242	372	7	of	of	ADP
ejpam-1242	372	8	a	a	DET
ejpam-1242	372	9	multiplication	multiplication	NOUN
ejpam-1242	372	10	r	r	NOUN
ejpam-1242	372	11	-	-	PUNCT
ejpam-1242	372	12	semimodule	semimodule	NOUN
ejpam-1242	372	13	m	m	VERB
ejpam-1242	372	14	with	with	ADP
ejpam-1242	372	15	n	n	PROPN
ejpam-1242	372	16	=	=	PROPN
ejpam-1242	372	17	i1	i1	PROPN
ejpam-1242	372	18	m	m	PROPN
ejpam-1242	372	19	and	and	CCONJ
ejpam-1242	372	20	k	k	PROPN
ejpam-1242	372	21	=	=	PROPN
ejpam-1242	372	22	i2	i2	PROPN
ejpam-1242	372	23	m	m	PROPN
ejpam-1242	372	24	for	for	ADP
ejpam-1242	372	25	some	some	DET
ejpam-1242	372	26	ideals	ideal	NOUN
ejpam-1242	372	27	i1	i1	PROPN
ejpam-1242	372	28	and	and	CCONJ
ejpam-1242	372	29	i2	i2	PROPN
ejpam-1242	372	30	of	of	ADP
ejpam-1242	372	31	r.	r.	PROPN
ejpam-1242	372	32	the	the	DET
ejpam-1242	372	33	product	product	NOUN
ejpam-1242	372	34	n	n	PROPN
ejpam-1242	372	35	and	and	CCONJ
ejpam-1242	372	36	k	k	PROPN
ejpam-1242	372	37	denoted	denote	VERB
ejpam-1242	372	38	by	by	ADP
ejpam-1242	372	39	n	n	CCONJ
ejpam-1242	372	40	k	k	PROPN
ejpam-1242	372	41	is	be	AUX
ejpam-1242	372	42	defined	define	VERB
ejpam-1242	372	43	by	by	ADP
ejpam-1242	372	44	n	n	PRON
ejpam-1242	372	45	k	k	PROPN
ejpam-1242	372	46	=	=	PROPN
ejpam-1242	372	47	i1	i1	PROPN
ejpam-1242	372	48	i2	i2	PROPN
ejpam-1242	372	49	m	m	PROPN
ejpam-1242	372	50	.	.	PUNCT
ejpam-1242	373	1	let	let	VERB
ejpam-1242	373	2	n	n	PROPN
ejpam-1242	373	3	=	=	PROPN
ejpam-1242	373	4	i1	i1	PROPN
ejpam-1242	373	5	m	m	PROPN
ejpam-1242	373	6	=	=	PROPN
ejpam-1242	373	7	i2	i2	PROPN
ejpam-1242	373	8	m	m	NOUN
ejpam-1242	373	9	=	=	SYM
ejpam-1242	373	10	n	n	NUM
ejpam-1242	373	11	′	′	NUM
ejpam-1242	373	12	and	and	CCONJ
ejpam-1242	373	13	k	k	PROPN
ejpam-1242	373	14	=	=	SYM
ejpam-1242	373	15	j1	j1	PROPN
ejpam-1242	373	16	m	m	NOUN
ejpam-1242	373	17	=	=	SYM
ejpam-1242	373	18	j2	j2	PROPN
ejpam-1242	373	19	m	m	PROPN
ejpam-1242	373	20	=	=	SYM
ejpam-1242	373	21	k	k	NOUN
ejpam-1242	373	22	′	′	NOUN
ejpam-1242	373	23	for	for	ADP
ejpam-1242	373	24	some	some	DET
ejpam-1242	373	25	ideals	ideal	NOUN
ejpam-1242	373	26	i1	i1	PROPN
ejpam-1242	373	27	,	,	PUNCT
ejpam-1242	373	28	i2	i2	PROPN
ejpam-1242	373	29	,	,	PUNCT
ejpam-1242	373	30	j1	j1	PROPN
ejpam-1242	373	31	and	and	CCONJ
ejpam-1242	373	32	j2	j2	PROPN
ejpam-1242	373	33	of	of	ADP
ejpam-1242	373	34	r.	r.	PROPN
ejpam-1242	373	35	it	it	PRON
ejpam-1242	373	36	is	be	AUX
ejpam-1242	373	37	easy	easy	ADJ
ejpam-1242	373	38	to	to	PART
ejpam-1242	373	39	show	show	VERB
ejpam-1242	373	40	that	that	SCONJ
ejpam-1242	373	41	n	n	NOUN
ejpam-1242	373	42	k	k	NOUN
ejpam-1242	373	43	=	=	PUNCT
ejpam-1242	373	44	n	n	NUM
ejpam-1242	373	45	′k	′k	NOUN
ejpam-1242	373	46	′	′	NUM
ejpam-1242	373	47	,	,	PUNCT
ejpam-1242	373	48	that	that	ADV
ejpam-1242	373	49	is	is	ADV
ejpam-1242	373	50	,	,	PUNCT
ejpam-1242	373	51	n	n	CCONJ
ejpam-1242	373	52	k	k	PROPN
ejpam-1242	373	53	is	be	AUX
ejpam-1242	373	54	independent	independent	ADJ
ejpam-1242	373	55	of	of	ADP
ejpam-1242	373	56	presentation	presentation	NOUN
ejpam-1242	373	57	ideals	ideal	NOUN
ejpam-1242	373	58	of	of	ADP
ejpam-1242	373	59	n	n	PROPN
ejpam-1242	373	60	and	and	CCONJ
ejpam-1242	373	61	k	k	PROPN
ejpam-1242	373	62	(	(	PUNCT
ejpam-1242	373	63	the	the	DET
ejpam-1242	373	64	proof	proof	NOUN
ejpam-1242	373	65	is	be	AUX
ejpam-1242	373	66	similar	similar	ADJ
ejpam-1242	373	67	to	to	ADP
ejpam-1242	373	68	ameri	ameri	PROPN
ejpam-1242	373	69	[	[	X
ejpam-1242	373	70	4	4	NUM
ejpam-1242	373	71	,	,	PUNCT
ejpam-1242	373	72	theorem	theorem	VERB
ejpam-1242	373	73	3.4	3.4	NUM
ejpam-1242	373	74	]	]	PUNCT
ejpam-1242	373	75	)	)	PUNCT
ejpam-1242	373	76	.	.	PUNCT
ejpam-1242	374	1	it	it	PRON
ejpam-1242	374	2	is	be	AUX
ejpam-1242	374	3	easy	easy	ADJ
ejpam-1242	374	4	to	to	PART
ejpam-1242	374	5	see	see	VERB
ejpam-1242	374	6	that	that	SCONJ
ejpam-1242	374	7	n	n	PROPN
ejpam-1242	374	8	k	k	PROPN
ejpam-1242	374	9	is	be	AUX
ejpam-1242	374	10	a	a	DET
ejpam-1242	374	11	subsemimodule	subsemimodule	NOUN
ejpam-1242	374	12	of	of	ADP
ejpam-1242	374	13	m	m	PROPN
ejpam-1242	374	14	and	and	CCONJ
ejpam-1242	374	15	n	n	CCONJ
ejpam-1242	374	16	k	k	PROPN
ejpam-1242	374	17	⊆	⊆	NUM
ejpam-1242	374	18	n	n	PRON
ejpam-1242	374	19	∩	∩	X
ejpam-1242	374	20	k	k	PROPN
ejpam-1242	374	21	.	.	PUNCT
ejpam-1242	375	1	for	for	ADP
ejpam-1242	375	2	m	m	PROPN
ejpam-1242	375	3	,	,	PUNCT
ejpam-1242	375	4	m′	m′	NOUN
ejpam-1242	375	5	∈	∈	NOUN
ejpam-1242	375	6	m	m	VERB
ejpam-1242	375	7	by	by	ADP
ejpam-1242	375	8	mm′	mm′	NOUN
ejpam-1242	375	9	,	,	PUNCT
ejpam-1242	375	10	we	we	PRON
ejpam-1242	375	11	mean	mean	VERB
ejpam-1242	375	12	the	the	DET
ejpam-1242	375	13	product	product	NOUN
ejpam-1242	375	14	of	of	ADP
ejpam-1242	375	15	rm	rm	PROPN
ejpam-1242	375	16	and	and	CCONJ
ejpam-1242	375	17	rm′	rm′	PROPN
ejpam-1242	375	18	,	,	PUNCT
ejpam-1242	375	19	that	that	ADV
ejpam-1242	375	20	is	is	ADV
ejpam-1242	375	21	,	,	PUNCT
ejpam-1242	375	22	mm′	mm′	ADJ
ejpam-1242	375	23	=	=	SYM
ejpam-1242	375	24	ij	ij	NOUN
ejpam-1242	375	25	m	m	PROPN
ejpam-1242	375	26	,	,	PUNCT
ejpam-1242	375	27	where	where	SCONJ
ejpam-1242	375	28	i	i	PRON
ejpam-1242	375	29	and	and	CCONJ
ejpam-1242	375	30	j	j	PROPN
ejpam-1242	375	31	are	be	AUX
ejpam-1242	375	32	presentations	presentation	NOUN
ejpam-1242	375	33	for	for	ADP
ejpam-1242	375	34	m	m	PROPN
ejpam-1242	375	35	and	and	CCONJ
ejpam-1242	375	36	m′	m′	NUM
ejpam-1242	375	37	,	,	PUNCT
ejpam-1242	375	38	respectively	respectively	ADV
ejpam-1242	375	39	.	.	PUNCT
ejpam-1242	376	1	lemma	lemma	PROPN
ejpam-1242	376	2	4	4	X
ejpam-1242	376	3	.	.	PUNCT
ejpam-1242	377	1	let	let	VERB
ejpam-1242	377	2	n	n	PRON
ejpam-1242	377	3	be	be	AUX
ejpam-1242	377	4	a	a	DET
ejpam-1242	377	5	proper	proper	ADJ
ejpam-1242	377	6	subsemimodule	subsemimodule	NOUN
ejpam-1242	377	7	of	of	ADP
ejpam-1242	377	8	a	a	DET
ejpam-1242	377	9	multiplication	multiplication	NOUN
ejpam-1242	377	10	semimodule	semimodule	NOUN
ejpam-1242	377	11	m	m	VERB
ejpam-1242	377	12	over	over	ADP
ejpam-1242	377	13	a	a	DET
ejpam-1242	377	14	semiring	semire	VERB
ejpam-1242	377	15	r.	r.	PROPN
ejpam-1242	377	16	then	then	ADV
ejpam-1242	377	17	the	the	DET
ejpam-1242	377	18	follwing	follwe	VERB
ejpam-1242	377	19	statements	statement	NOUN
ejpam-1242	377	20	hold	hold	VERB
ejpam-1242	377	21	:	:	PUNCT
ejpam-1242	377	22	(	(	PUNCT
ejpam-1242	377	23	i	i	NOUN
ejpam-1242	377	24	)	)	PUNCT
ejpam-1242	377	25	n	n	PRON
ejpam-1242	377	26	is	be	AUX
ejpam-1242	377	27	prime	prime	ADJ
ejpam-1242	377	28	if	if	SCONJ
ejpam-1242	378	1	and	and	CCONJ
ejpam-1242	378	2	only	only	ADV
ejpam-1242	378	3	if	if	SCONJ
ejpam-1242	378	4	whenever	whenever	SCONJ
ejpam-1242	378	5	uv	uv	NOUN
ejpam-1242	378	6	⊆	⊆	NUM
ejpam-1242	378	7	n	n	NOUN
ejpam-1242	378	8	for	for	ADP
ejpam-1242	378	9	some	some	DET
ejpam-1242	378	10	subsemimodules	subsemimodule	NOUN
ejpam-1242	378	11	u	u	NOUN
ejpam-1242	378	12	and	and	CCONJ
ejpam-1242	378	13	v	v	NOUN
ejpam-1242	378	14	of	of	ADP
ejpam-1242	378	15	m	m	PROPN
ejpam-1242	378	16	,	,	PUNCT
ejpam-1242	378	17	then	then	ADV
ejpam-1242	378	18	u	u	NOUN
ejpam-1242	378	19	⊆	⊆	NUM
ejpam-1242	378	20	n	n	NOUN
ejpam-1242	378	21	or	or	CCONJ
ejpam-1242	378	22	v	v	ADP
ejpam-1242	378	23	⊆	⊆	NUM
ejpam-1242	378	24	n.	n.	NOUN
ejpam-1242	378	25	(	(	PUNCT
ejpam-1242	378	26	ii	ii	NOUN
ejpam-1242	378	27	)	)	PUNCT
ejpam-1242	378	28	n	n	PROPN
ejpam-1242	378	29	is	be	AUX
ejpam-1242	378	30	prime	prime	ADJ
ejpam-1242	378	31	if	if	SCONJ
ejpam-1242	379	1	and	and	CCONJ
ejpam-1242	379	2	only	only	ADV
ejpam-1242	379	3	if	if	SCONJ
ejpam-1242	379	4	whenever	whenever	SCONJ
ejpam-1242	379	5	m.m′	m.m′	VERB
ejpam-1242	379	6	⊆	⊆	NUM
ejpam-1242	379	7	n	n	NOUN
ejpam-1242	379	8	for	for	ADP
ejpam-1242	379	9	some	some	DET
ejpam-1242	379	10	m	m	NOUN
ejpam-1242	379	11	,	,	PUNCT
ejpam-1242	379	12	m′	m′	NOUN
ejpam-1242	379	13	∈	∈	PROPN
ejpam-1242	379	14	m	m	PROPN
ejpam-1242	379	15	,	,	PUNCT
ejpam-1242	379	16	then	then	ADV
ejpam-1242	379	17	rm	rm	PROPN
ejpam-1242	379	18	⊆	⊆	NUM
ejpam-1242	379	19	n	n	ADV
ejpam-1242	379	20	or	or	CCONJ
ejpam-1242	379	21	rm′	rm′	VERB
ejpam-1242	379	22	⊆	⊆	NUM
ejpam-1242	379	23	n.	n.	NOUN
ejpam-1242	379	24	proof	proof	NOUN
ejpam-1242	379	25	.	.	PUNCT
ejpam-1242	380	1	the	the	DET
ejpam-1242	380	2	proofs	proof	NOUN
ejpam-1242	380	3	are	be	AUX
ejpam-1242	380	4	straightforward	straightforward	ADJ
ejpam-1242	380	5	(	(	PUNCT
ejpam-1242	380	6	the	the	DET
ejpam-1242	380	7	proofs	proof	NOUN
ejpam-1242	380	8	are	be	AUX
ejpam-1242	380	9	similar	similar	ADJ
ejpam-1242	380	10	to	to	ADP
ejpam-1242	380	11	ameri	ameri	PROPN
ejpam-1242	380	12	[	[	X
ejpam-1242	380	13	4	4	NUM
ejpam-1242	380	14	,	,	PUNCT
ejpam-1242	380	15	theorem	theorem	VERB
ejpam-1242	380	16	3.16	3.16	NUM
ejpam-1242	380	17	and	and	CCONJ
ejpam-1242	380	18	corollary	corollary	ADJ
ejpam-1242	380	19	3.17	3.17	NUM
ejpam-1242	380	20	]	]	PUNCT
ejpam-1242	380	21	)	)	PUNCT
ejpam-1242	380	22	.	.	PUNCT
ejpam-1242	381	1	let	let	VERB
ejpam-1242	381	2	m	m	PRON
ejpam-1242	381	3	be	be	AUX
ejpam-1242	381	4	a	a	DET
ejpam-1242	381	5	non	non	ADJ
ejpam-1242	381	6	-	-	ADJ
ejpam-1242	381	7	strong	strong	ADJ
ejpam-1242	381	8	semimodule	semimodule	NOUN
ejpam-1242	381	9	over	over	ADP
ejpam-1242	381	10	a	a	DET
ejpam-1242	381	11	semiring	semire	VERB
ejpam-1242	381	12	r	r	NOUN
ejpam-1242	381	13	with	with	ADP
ejpam-1242	381	14	m	m	PROPN
ejpam-1242	381	15	6=	6=	NUM
ejpam-1242	381	16	0	0	NUM
ejpam-1242	381	17	.	.	PUNCT
ejpam-1242	382	1	then	then	ADV
ejpam-1242	382	2	by	by	ADP
ejpam-1242	382	3	theorem	theorem	NOUN
ejpam-1242	382	4	4	4	NUM
ejpam-1242	382	5	,	,	PUNCT
ejpam-1242	382	6	the	the	DET
ejpam-1242	382	7	k	k	NOUN
ejpam-1242	382	8	-	-	NOUN
ejpam-1242	382	9	spectrum	spectrum	NOUN
ejpam-1242	382	10	x	x	X
ejpam-1242	382	11	=	=	SYM
ejpam-1242	382	12	speck(m	speck(m	NOUN
ejpam-1242	382	13	)	)	PUNCT
ejpam-1242	382	14	is	be	AUX
ejpam-1242	382	15	non	non	ADJ
ejpam-1242	382	16	-	-	ADJ
ejpam-1242	382	17	empty	empty	ADJ
ejpam-1242	382	18	.	.	PUNCT
ejpam-1242	383	1	for	for	ADP
ejpam-1242	383	2	any	any	DET
ejpam-1242	383	3	subsemimodule	subsemimodule	NOUN
ejpam-1242	383	4	n	n	PROPN
ejpam-1242	383	5	of	of	ADP
ejpam-1242	383	6	a	a	DET
ejpam-1242	383	7	semimodule	semimodule	NOUN
ejpam-1242	383	8	m	m	VERB
ejpam-1242	383	9	by	by	ADP
ejpam-1242	383	10	v	v	NOUN
ejpam-1242	383	11	(	(	PUNCT
ejpam-1242	383	12	n	n	CCONJ
ejpam-1242	383	13	)	)	PUNCT
ejpam-1242	383	14	we	we	PRON
ejpam-1242	383	15	mean	mean	VERB
ejpam-1242	383	16	the	the	DET
ejpam-1242	383	17	set	set	NOUN
ejpam-1242	383	18	of	of	ADP
ejpam-1242	383	19	all	all	DET
ejpam-1242	383	20	prime	prime	ADJ
ejpam-1242	383	21	k	k	NOUN
ejpam-1242	383	22	-	-	NOUN
ejpam-1242	383	23	subsemimodules	subsemimodules	NOUN
ejpam-1242	383	24	of	of	ADP
ejpam-1242	383	25	m	m	AUX
ejpam-1242	383	26	containing	contain	VERB
ejpam-1242	383	27	n	n	NOUN
ejpam-1242	383	28	.	.	PUNCT
ejpam-1242	384	1	clearly	clearly	ADV
ejpam-1242	384	2	,	,	PUNCT
ejpam-1242	384	3	v	v	X
ejpam-1242	384	4	(	(	PUNCT
ejpam-1242	384	5	m	m	NOUN
ejpam-1242	384	6	)	)	PUNCT
ejpam-1242	384	7	=	=	SYM
ejpam-1242	384	8	;	;	PUNCT
ejpam-1242	384	9	and	and	CCONJ
ejpam-1242	384	10	v	v	X
ejpam-1242	384	11	(	(	PUNCT
ejpam-1242	384	12	{	{	PUNCT
ejpam-1242	384	13	0	0	NUM
ejpam-1242	384	14	}	}	PUNCT
ejpam-1242	384	15	)	)	PUNCT
ejpam-1242	384	16	=	=	SYM
ejpam-1242	384	17	speck(m	speck(m	NOUN
ejpam-1242	384	18	)	)	PUNCT
ejpam-1242	384	19	=	=	SYM
ejpam-1242	384	20	x	x	X
ejpam-1242	384	21	.	.	PUNCT
ejpam-1242	385	1	throughout	throughout	ADP
ejpam-1242	385	2	this	this	DET
ejpam-1242	385	3	section	section	NOUN
ejpam-1242	385	4	we	we	PRON
ejpam-1242	385	5	may	may	AUX
ejpam-1242	385	6	assume	assume	VERB
ejpam-1242	385	7	that	that	SCONJ
ejpam-1242	385	8	speck(m	speck(m	NOUN
ejpam-1242	385	9	)	)	PUNCT
ejpam-1242	385	10	is	be	AUX
ejpam-1242	385	11	non	non	ADJ
ejpam-1242	385	12	-	-	ADJ
ejpam-1242	385	13	empty	empty	ADJ
ejpam-1242	385	14	.	.	PUNCT
ejpam-1242	386	1	lemma	lemma	PROPN
ejpam-1242	386	2	5	5	X
ejpam-1242	386	3	.	.	PUNCT
ejpam-1242	387	1	let	let	VERB
ejpam-1242	387	2	m	m	PRON
ejpam-1242	387	3	be	be	AUX
ejpam-1242	387	4	a	a	DET
ejpam-1242	387	5	semimodule	semimodule	NOUN
ejpam-1242	387	6	over	over	ADP
ejpam-1242	387	7	a	a	DET
ejpam-1242	387	8	semiring	semire	VERB
ejpam-1242	387	9	r.	r.	PROPN
ejpam-1242	387	10	then	then	ADV
ejpam-1242	387	11	the	the	DET
ejpam-1242	387	12	following	following	ADJ
ejpam-1242	387	13	statements	statement	NOUN
ejpam-1242	387	14	hold	hold	VERB
ejpam-1242	387	15	:	:	PUNCT
ejpam-1242	387	16	(	(	PUNCT
ejpam-1242	387	17	i	i	NOUN
ejpam-1242	387	18	)	)	PUNCT
ejpam-1242	387	19	if	if	SCONJ
ejpam-1242	387	20	n	n	PRON
ejpam-1242	387	21	is	be	AUX
ejpam-1242	387	22	a	a	DET
ejpam-1242	387	23	subsemimodule	subsemimodule	NOUN
ejpam-1242	387	24	of	of	ADP
ejpam-1242	387	25	m	m	PROPN
ejpam-1242	387	26	,	,	PUNCT
ejpam-1242	387	27	then	then	ADV
ejpam-1242	387	28	v	v	NOUN
ejpam-1242	387	29	(	(	PUNCT
ejpam-1242	387	30	n	n	CCONJ
ejpam-1242	387	31	)	)	PUNCT
ejpam-1242	387	32	=	=	SYM
ejpam-1242	387	33	v	v	X
ejpam-1242	387	34	(	(	PUNCT
ejpam-1242	387	35	rad(n	rad(n	NOUN
ejpam-1242	387	36	)	)	PUNCT
ejpam-1242	387	37	)	)	PUNCT
ejpam-1242	387	38	.	.	PUNCT
ejpam-1242	388	1	(	(	PUNCT
ejpam-1242	388	2	ii	ii	NOUN
ejpam-1242	388	3	)	)	PUNCT
ejpam-1242	388	4	if	if	SCONJ
ejpam-1242	388	5	{	{	PUNCT
ejpam-1242	388	6	ni}i∈i	ni}i∈i	INTJ
ejpam-1242	388	7	is	be	AUX
ejpam-1242	388	8	a	a	DET
ejpam-1242	388	9	family	family	NOUN
ejpam-1242	388	10	of	of	ADP
ejpam-1242	388	11	subsemimodules	subsemimodule	NOUN
ejpam-1242	388	12	of	of	ADP
ejpam-1242	388	13	m	m	PROPN
ejpam-1242	388	14	,	,	PUNCT
ejpam-1242	388	15	then	then	ADV
ejpam-1242	388	16	v	v	X
ejpam-1242	388	17	(	(	PUNCT
ejpam-1242	388	18	∑	∑	PROPN
ejpam-1242	388	19	i∈i	i∈i	ADJ
ejpam-1242	388	20	ni	ni	PROPN
ejpam-1242	388	21	)	)	PUNCT
ejpam-1242	388	22	=	=	PUNCT
ejpam-1242	389	1	⋂	⋂	PROPN
ejpam-1242	389	2	i∈i	i∈i	ADJ
ejpam-1242	389	3	v	v	PROPN
ejpam-1242	389	4	(	(	PUNCT
ejpam-1242	389	5	ni	ni	NOUN
ejpam-1242	389	6	)	)	PUNCT
ejpam-1242	389	7	.	.	PUNCT
ejpam-1242	390	1	proof	proof	NOUN
ejpam-1242	390	2	.	.	PUNCT
ejpam-1242	391	1	(	(	PUNCT
ejpam-1242	391	2	i	i	NOUN
ejpam-1242	391	3	)	)	PUNCT
ejpam-1242	391	4	since	since	SCONJ
ejpam-1242	391	5	n	n	ADV
ejpam-1242	391	6	⊆	⊆	NUM
ejpam-1242	391	7	rad(n	rad(n	NOUN
ejpam-1242	391	8	)	)	PUNCT
ejpam-1242	391	9	,	,	PUNCT
ejpam-1242	391	10	we	we	PRON
ejpam-1242	391	11	have	have	VERB
ejpam-1242	391	12	v	v	NUM
ejpam-1242	391	13	(	(	PUNCT
ejpam-1242	391	14	rad(n	rad(n	NOUN
ejpam-1242	391	15	)	)	PUNCT
ejpam-1242	391	16	)	)	PUNCT
ejpam-1242	392	1	⊆	⊆	NUM
ejpam-1242	392	2	v	v	NOUN
ejpam-1242	392	3	(	(	PUNCT
ejpam-1242	392	4	n	n	CCONJ
ejpam-1242	392	5	)	)	PUNCT
ejpam-1242	392	6	.	.	PUNCT
ejpam-1242	393	1	for	for	ADP
ejpam-1242	393	2	the	the	DET
ejpam-1242	393	3	reverse	reverse	ADJ
ejpam-1242	393	4	inclusion	inclusion	NOUN
ejpam-1242	393	5	,	,	PUNCT
ejpam-1242	393	6	assume	assume	VERB
ejpam-1242	393	7	that	that	SCONJ
ejpam-1242	393	8	p	p	PROPN
ejpam-1242	393	9	∈	∈	PROPN
ejpam-1242	393	10	v	v	ADP
ejpam-1242	393	11	(	(	PUNCT
ejpam-1242	393	12	n	n	CCONJ
ejpam-1242	393	13	)	)	PUNCT
ejpam-1242	393	14	.	.	PUNCT
ejpam-1242	394	1	then	then	ADV
ejpam-1242	394	2	n	n	PROPN
ejpam-1242	394	3	⊆	⊆	NUM
ejpam-1242	394	4	p	p	NOUN
ejpam-1242	394	5	;	;	PUNCT
ejpam-1242	394	6	hence	hence	ADV
ejpam-1242	394	7	rad(n)⊆	rad(n)⊆	NOUN
ejpam-1242	394	8	p	p	X
ejpam-1242	394	9	,	,	PUNCT
ejpam-1242	394	10	and	and	CCONJ
ejpam-1242	394	11	so	so	ADV
ejpam-1242	394	12	we	we	PRON
ejpam-1242	394	13	have	have	VERB
ejpam-1242	394	14	equality	equality	NOUN
ejpam-1242	394	15	.	.	PUNCT
ejpam-1242	395	1	(	(	PUNCT
ejpam-1242	395	2	ii	ii	NOUN
ejpam-1242	395	3	)	)	PUNCT
ejpam-1242	395	4	let	let	VERB
ejpam-1242	395	5	p	p	PROPN
ejpam-1242	395	6	∈	∈	PROPN
ejpam-1242	395	7	⋂	⋂	PROPN
ejpam-1242	395	8	i∈i	i∈i	ADJ
ejpam-1242	395	9	v	v	PROPN
ejpam-1242	395	10	(	(	PUNCT
ejpam-1242	395	11	ni	ni	PROPN
ejpam-1242	395	12	)	)	PUNCT
ejpam-1242	395	13	.	.	PUNCT
ejpam-1242	396	1	then	then	ADV
ejpam-1242	396	2	ni	ni	PROPN
ejpam-1242	396	3	⊆	⊆	NUM
ejpam-1242	396	4	p	p	NOUN
ejpam-1242	396	5	for	for	ADP
ejpam-1242	396	6	every	every	DET
ejpam-1242	396	7	i	i	NOUN
ejpam-1242	396	8	∈	∈	PROPN
ejpam-1242	397	1	i	i	PRON
ejpam-1242	397	2	,	,	PUNCT
ejpam-1242	397	3	so	so	ADV
ejpam-1242	397	4	∑	∑	ADP
ejpam-1242	397	5	i∈i	i∈i	ADJ
ejpam-1242	397	6	ni	ni	PROPN
ejpam-1242	397	7	⊆	⊆	NUM
ejpam-1242	397	8	p	p	PROPN
ejpam-1242	397	9	,	,	PUNCT
ejpam-1242	397	10	which	which	PRON
ejpam-1242	397	11	implies	imply	VERB
ejpam-1242	397	12	that⋂	that⋂	PRON
ejpam-1242	397	13	i∈i	i∈i	ADJ
ejpam-1242	397	14	v	v	NOUN
ejpam-1242	397	15	(	(	PUNCT
ejpam-1242	397	16	ni)⊆	ni)⊆	PROPN
ejpam-1242	397	17	v	v	NOUN
ejpam-1242	397	18	(	(	PUNCT
ejpam-1242	397	19	∑	∑	PROPN
ejpam-1242	397	20	i∈i	i∈i	ADJ
ejpam-1242	397	21	ni	ni	PROPN
ejpam-1242	397	22	)	)	PUNCT
ejpam-1242	397	23	.	.	PUNCT
ejpam-1242	398	1	the	the	DET
ejpam-1242	398	2	reverse	reverse	ADJ
ejpam-1242	398	3	inclusion	inclusion	NOUN
ejpam-1242	398	4	is	be	AUX
ejpam-1242	398	5	similar	similar	ADJ
ejpam-1242	398	6	.	.	PUNCT
ejpam-1242	399	1	s.	s.	PROPN
ejpam-1242	399	2	atani	atani	PROPN
ejpam-1242	399	3	,	,	PUNCT
ejpam-1242	399	4	r.	r.	PROPN
ejpam-1242	399	5	atrani	atrani	PROPN
ejpam-1242	399	6	,	,	PUNCT
ejpam-1242	399	7	ü.	ü.	NOUN
ejpam-1242	399	8	tekir	tekir	PROPN
ejpam-1242	399	9	/	/	SYM
ejpam-1242	399	10	eur	eur	PROPN
ejpam-1242	399	11	.	.	PUNCT
ejpam-1242	400	1	j.	j.	PROPN
ejpam-1242	400	2	pure	pure	PROPN
ejpam-1242	400	3	appl	appl	PROPN
ejpam-1242	400	4	.	.	PROPN
ejpam-1242	400	5	math	math	PROPN
ejpam-1242	400	6	,	,	PUNCT
ejpam-1242	400	7	4	4	NUM
ejpam-1242	400	8	(	(	PUNCT
ejpam-1242	400	9	2011	2011	NUM
ejpam-1242	400	10	)	)	PUNCT
ejpam-1242	400	11	,	,	PUNCT
ejpam-1242	400	12	251	251	NUM
ejpam-1242	400	13	-	-	SYM
ejpam-1242	400	14	265	265	NUM
ejpam-1242	400	15	261	261	NUM
ejpam-1242	400	16	if	if	SCONJ
ejpam-1242	400	17	ζ(m	ζ(m	ADJ
ejpam-1242	400	18	)	)	PUNCT
ejpam-1242	400	19	denotes	denote	VERB
ejpam-1242	400	20	the	the	DET
ejpam-1242	400	21	collection	collection	NOUN
ejpam-1242	400	22	of	of	ADP
ejpam-1242	400	23	all	all	DET
ejpam-1242	400	24	subsets	subset	NOUN
ejpam-1242	400	25	v	v	ADP
ejpam-1242	400	26	(	(	PUNCT
ejpam-1242	400	27	n	n	CCONJ
ejpam-1242	400	28	)	)	PUNCT
ejpam-1242	400	29	of	of	ADP
ejpam-1242	400	30	speck(m	speck(m	NOUN
ejpam-1242	400	31	)	)	PUNCT
ejpam-1242	400	32	,	,	PUNCT
ejpam-1242	400	33	then	then	ADV
ejpam-1242	400	34	ζ(m	ζ(m	ADJ
ejpam-1242	400	35	)	)	PUNCT
ejpam-1242	400	36	contains	contain	VERB
ejpam-1242	400	37	the	the	DET
ejpam-1242	400	38	empty	empty	ADJ
ejpam-1242	400	39	set	set	NOUN
ejpam-1242	400	40	and	and	CCONJ
ejpam-1242	400	41	spec(m	spec(m	PROPN
ejpam-1242	400	42	)	)	PUNCT
ejpam-1242	400	43	and	and	CCONJ
ejpam-1242	400	44	is	be	AUX
ejpam-1242	400	45	closed	close	VERB
ejpam-1242	400	46	under	under	ADP
ejpam-1242	400	47	arbitrary	arbitrary	ADJ
ejpam-1242	400	48	intersection	intersection	NOUN
ejpam-1242	400	49	by	by	ADP
ejpam-1242	400	50	lemma	lemma	PROPN
ejpam-1242	400	51	5	5	NUM
ejpam-1242	400	52	(	(	PUNCT
ejpam-1242	400	53	ii	ii	NOUN
ejpam-1242	400	54	)	)	PUNCT
ejpam-1242	400	55	.	.	PUNCT
ejpam-1242	401	1	if	if	SCONJ
ejpam-1242	401	2	also	also	ADV
ejpam-1242	401	3	ζ(m	ζ(m	ADJ
ejpam-1242	401	4	)	)	PUNCT
ejpam-1242	401	5	is	be	AUX
ejpam-1242	401	6	closed	close	VERB
ejpam-1242	401	7	under	under	ADP
ejpam-1242	401	8	finite	finite	PROPN
ejpam-1242	401	9	union	union	NOUN
ejpam-1242	401	10	,	,	PUNCT
ejpam-1242	401	11	that	that	ADV
ejpam-1242	401	12	is	is	ADV
ejpam-1242	401	13	,	,	PUNCT
ejpam-1242	401	14	for	for	ADP
ejpam-1242	401	15	every	every	DET
ejpam-1242	401	16	subsemimodules	subsemimodule	NOUN
ejpam-1242	401	17	n	n	NOUN
ejpam-1242	401	18	and	and	CCONJ
ejpam-1242	401	19	l	l	NOUN
ejpam-1242	401	20	of	of	ADP
ejpam-1242	401	21	m	m	PRON
ejpam-1242	401	22	such	such	ADJ
ejpam-1242	401	23	that	that	PRON
ejpam-1242	401	24	v	v	NOUN
ejpam-1242	401	25	(	(	PUNCT
ejpam-1242	401	26	n	n	CCONJ
ejpam-1242	401	27	)	)	PUNCT
ejpam-1242	401	28	∪	∪	NOUN
ejpam-1242	401	29	v	v	NOUN
ejpam-1242	401	30	(	(	PUNCT
ejpam-1242	401	31	l	l	NOUN
ejpam-1242	401	32	)	)	PUNCT
ejpam-1242	401	33	=	=	SYM
ejpam-1242	401	34	v	v	X
ejpam-1242	401	35	(	(	PUNCT
ejpam-1242	401	36	t	t	PROPN
ejpam-1242	401	37	)	)	PUNCT
ejpam-1242	401	38	for	for	ADP
ejpam-1242	401	39	some	some	DET
ejpam-1242	401	40	subsemimodule	subsemimodule	NOUN
ejpam-1242	401	41	t	t	PROPN
ejpam-1242	401	42	of	of	ADP
ejpam-1242	401	43	m	m	PRON
ejpam-1242	401	44	,	,	PUNCT
ejpam-1242	401	45	for	for	ADP
ejpam-1242	401	46	in	in	ADP
ejpam-1242	401	47	this	this	DET
ejpam-1242	401	48	case	case	NOUN
ejpam-1242	401	49	ζ(m	ζ(m	ADJ
ejpam-1242	401	50	)	)	PUNCT
ejpam-1242	401	51	satisfies	satisfy	VERB
ejpam-1242	401	52	the	the	DET
ejpam-1242	401	53	axioms	axiom	NOUN
ejpam-1242	401	54	of	of	ADP
ejpam-1242	401	55	closed	closed	ADJ
ejpam-1242	401	56	subsetes	subsete	NOUN
ejpam-1242	401	57	of	of	ADP
ejpam-1242	401	58	a	a	DET
ejpam-1242	401	59	topological	topological	ADJ
ejpam-1242	401	60	spaces	space	NOUN
ejpam-1242	401	61	,	,	PUNCT
ejpam-1242	401	62	which	which	PRON
ejpam-1242	401	63	is	be	AUX
ejpam-1242	401	64	called	call	VERB
ejpam-1242	401	65	zariski	zariski	ADJ
ejpam-1242	401	66	topology	topology	NOUN
ejpam-1242	401	67	.	.	PUNCT
ejpam-1242	402	1	in	in	ADP
ejpam-1242	402	2	maccasland	maccasland	PROPN
ejpam-1242	402	3	,	,	PUNCT
ejpam-1242	402	4	moore	moore	PROPN
ejpam-1242	402	5	and	and	CCONJ
ejpam-1242	402	6	smith	smith	PROPN
ejpam-1242	402	7	[	[	X
ejpam-1242	402	8	22	22	NUM
ejpam-1242	402	9	]	]	PUNCT
ejpam-1242	402	10	a	a	DET
ejpam-1242	402	11	module	module	NOUN
ejpam-1242	402	12	with	with	ADP
ejpam-1242	402	13	zariski	zariski	NOUN
ejpam-1242	402	14	topology	topology	NOUN
ejpam-1242	402	15	is	be	AUX
ejpam-1242	402	16	called	call	VERB
ejpam-1242	402	17	a	a	DET
ejpam-1242	402	18	top	top	ADJ
ejpam-1242	402	19	module	module	NOUN
ejpam-1242	402	20	.	.	PUNCT
ejpam-1242	403	1	lemma	lemma	PROPN
ejpam-1242	403	2	6	6	NUM
ejpam-1242	403	3	.	.	PUNCT
ejpam-1242	404	1	the	the	DET
ejpam-1242	404	2	following	follow	VERB
ejpam-1242	404	3	statements	statement	NOUN
ejpam-1242	404	4	are	be	AUX
ejpam-1242	404	5	equivalent	equivalent	ADJ
ejpam-1242	404	6	for	for	ADP
ejpam-1242	404	7	a	a	DET
ejpam-1242	404	8	semimodule	semimodule	NOUN
ejpam-1242	404	9	m	m	VERB
ejpam-1242	404	10	over	over	ADP
ejpam-1242	404	11	a	a	DET
ejpam-1242	404	12	semiring	semire	VERB
ejpam-1242	404	13	r.	r.	PROPN
ejpam-1242	404	14	(	(	PUNCT
ejpam-1242	404	15	i	i	NOUN
ejpam-1242	404	16	)	)	PUNCT
ejpam-1242	404	17	m	m	VERB
ejpam-1242	404	18	is	be	AUX
ejpam-1242	404	19	a	a	DET
ejpam-1242	404	20	top	top	ADJ
ejpam-1242	404	21	semimodule	semimodule	NOUN
ejpam-1242	404	22	.	.	PUNCT
ejpam-1242	405	1	(	(	PUNCT
ejpam-1242	405	2	ii	ii	NOUN
ejpam-1242	405	3	)	)	PUNCT
ejpam-1242	405	4	every	every	DET
ejpam-1242	405	5	prime	prime	ADJ
ejpam-1242	405	6	k	k	NOUN
ejpam-1242	405	7	-	-	NOUN
ejpam-1242	405	8	subsemimodule	subsemimodule	NOUN
ejpam-1242	405	9	of	of	ADP
ejpam-1242	405	10	m	m	PROPN
ejpam-1242	405	11	is	be	AUX
ejpam-1242	405	12	extraordinary	extraordinary	ADJ
ejpam-1242	405	13	.	.	PUNCT
ejpam-1242	406	1	(	(	PUNCT
ejpam-1242	406	2	iii	iii	X
ejpam-1242	406	3	)	)	PUNCT
ejpam-1242	406	4	v	v	NOUN
ejpam-1242	406	5	(	(	PUNCT
ejpam-1242	406	6	t	t	NOUN
ejpam-1242	406	7	)	)	PUNCT
ejpam-1242	406	8	∪	∪	ADP
ejpam-1242	406	9	v	v	NOUN
ejpam-1242	406	10	(	(	PUNCT
ejpam-1242	406	11	l	l	NOUN
ejpam-1242	406	12	)	)	PUNCT
ejpam-1242	406	13	=	=	SYM
ejpam-1242	406	14	v	v	X
ejpam-1242	406	15	(	(	PUNCT
ejpam-1242	406	16	t	t	PROPN
ejpam-1242	406	17	∩	∩	ADJ
ejpam-1242	406	18	l	l	NOUN
ejpam-1242	406	19	)	)	PUNCT
ejpam-1242	406	20	for	for	ADP
ejpam-1242	406	21	any	any	DET
ejpam-1242	406	22	semiprime	semiprime	NOUN
ejpam-1242	406	23	subsemimodules	subsemimodules	PROPN
ejpam-1242	406	24	t	t	PROPN
ejpam-1242	406	25	and	and	CCONJ
ejpam-1242	406	26	l	l	PROPN
ejpam-1242	406	27	of	of	ADP
ejpam-1242	406	28	m.	m.	NOUN
ejpam-1242	406	29	proof	proof	NOUN
ejpam-1242	406	30	.	.	PUNCT
ejpam-1242	407	1	(	(	PUNCT
ejpam-1242	407	2	i)⇒	i)⇒	PROPN
ejpam-1242	407	3	(	(	PUNCT
ejpam-1242	407	4	ii	ii	NOUN
ejpam-1242	407	5	)	)	PUNCT
ejpam-1242	407	6	.	.	PUNCT
ejpam-1242	408	1	let	let	VERB
ejpam-1242	408	2	n	n	PRON
ejpam-1242	408	3	be	be	AUX
ejpam-1242	408	4	any	any	DET
ejpam-1242	408	5	prime	prime	ADJ
ejpam-1242	408	6	k	k	NOUN
ejpam-1242	408	7	-	-	NOUN
ejpam-1242	408	8	subsemimodule	subsemimodule	NOUN
ejpam-1242	408	9	of	of	ADP
ejpam-1242	408	10	m	m	PRON
ejpam-1242	408	11	and	and	CCONJ
ejpam-1242	408	12	let	let	VERB
ejpam-1242	408	13	t	t	PROPN
ejpam-1242	408	14	and	and	CCONJ
ejpam-1242	408	15	l	l	PROPN
ejpam-1242	408	16	be	be	AUX
ejpam-1242	408	17	semiprime	semiprime	NOUN
ejpam-1242	408	18	subsemimodules	subsemimodule	NOUN
ejpam-1242	408	19	of	of	ADP
ejpam-1242	408	20	m	m	PRON
ejpam-1242	408	21	such	such	ADJ
ejpam-1242	408	22	that	that	SCONJ
ejpam-1242	408	23	t	t	PROPN
ejpam-1242	408	24	∩	∩	PROPN
ejpam-1242	408	25	l	l	PROPN
ejpam-1242	408	26	⊆	⊆	NUM
ejpam-1242	408	27	n	n	NOUN
ejpam-1242	408	28	.	.	PUNCT
ejpam-1242	409	1	by	by	ADP
ejpam-1242	409	2	(	(	PUNCT
ejpam-1242	409	3	i	i	NOUN
ejpam-1242	409	4	)	)	PUNCT
ejpam-1242	409	5	,	,	PUNCT
ejpam-1242	409	6	there	there	PRON
ejpam-1242	409	7	exists	exist	VERB
ejpam-1242	409	8	a	a	DET
ejpam-1242	409	9	subsemimodule	subsemimodule	NOUN
ejpam-1242	409	10	u	u	NOUN
ejpam-1242	409	11	of	of	ADP
ejpam-1242	409	12	m	m	PRON
ejpam-1242	409	13	such	such	ADJ
ejpam-1242	409	14	that	that	PRON
ejpam-1242	409	15	v	v	NOUN
ejpam-1242	409	16	(	(	PUNCT
ejpam-1242	409	17	t	t	NOUN
ejpam-1242	409	18	)	)	PUNCT
ejpam-1242	409	19	∪	∪	ADP
ejpam-1242	409	20	v	v	NOUN
ejpam-1242	409	21	(	(	PUNCT
ejpam-1242	409	22	l	l	NOUN
ejpam-1242	409	23	)	)	PUNCT
ejpam-1242	409	24	=	=	SYM
ejpam-1242	409	25	v	v	X
ejpam-1242	409	26	(	(	PUNCT
ejpam-1242	409	27	u	u	NOUN
ejpam-1242	409	28	)	)	PUNCT
ejpam-1242	409	29	.	.	PUNCT
ejpam-1242	410	1	now	now	ADV
ejpam-1242	410	2	set	set	VERB
ejpam-1242	410	3	t	t	PROPN
ejpam-1242	410	4	=	=	SYM
ejpam-1242	410	5	⋂	⋂	PROPN
ejpam-1242	410	6	i∈i	i∈i	PROPN
ejpam-1242	410	7	ni	ni	PROPN
ejpam-1242	410	8	,	,	PUNCT
ejpam-1242	410	9	where	where	SCONJ
ejpam-1242	410	10	ni	ni	PROPN
ejpam-1242	410	11	is	be	AUX
ejpam-1242	410	12	a	a	DET
ejpam-1242	410	13	prime	prime	ADJ
ejpam-1242	410	14	k	k	NOUN
ejpam-1242	410	15	-	-	NOUN
ejpam-1242	410	16	subsemimodule	subsemimodule	NOUN
ejpam-1242	410	17	of	of	ADP
ejpam-1242	410	18	m	m	PROPN
ejpam-1242	410	19	(	(	PUNCT
ejpam-1242	410	20	i	i	PRON
ejpam-1242	410	21	∈	∈	PROPN
ejpam-1242	410	22	i	i	PROPN
ejpam-1242	410	23	)	)	PUNCT
ejpam-1242	410	24	.	.	PUNCT
ejpam-1242	411	1	for	for	ADP
ejpam-1242	411	2	each	each	DET
ejpam-1242	411	3	i	i	PRON
ejpam-1242	411	4	∈	∈	PROPN
ejpam-1242	411	5	i	i	PRON
ejpam-1242	411	6	,	,	PUNCT
ejpam-1242	411	7	ni	ni	PROPN
ejpam-1242	411	8	∈	∈	PROPN
ejpam-1242	411	9	v	v	PROPN
ejpam-1242	411	10	(	(	PUNCT
ejpam-1242	411	11	t	t	PROPN
ejpam-1242	411	12	)	)	PUNCT
ejpam-1242	411	13	⊆	⊆	NUM
ejpam-1242	411	14	v	v	X
ejpam-1242	411	15	(	(	PUNCT
ejpam-1242	411	16	u	u	NOUN
ejpam-1242	411	17	)	)	PUNCT
ejpam-1242	411	18	,	,	PUNCT
ejpam-1242	411	19	so	so	SCONJ
ejpam-1242	411	20	that	that	SCONJ
ejpam-1242	411	21	u	u	PROPN
ejpam-1242	411	22	⊆	⊆	NUM
ejpam-1242	411	23	ni	ni	PROPN
ejpam-1242	411	24	.	.	PROPN
ejpam-1242	412	1	thus	thus	ADV
ejpam-1242	412	2	u	u	PROPN
ejpam-1242	412	3	⊆	⊆	NUM
ejpam-1242	412	4	t	t	NOUN
ejpam-1242	412	5	.	.	PUNCT
ejpam-1242	413	1	similarly	similarly	ADV
ejpam-1242	413	2	,	,	PUNCT
ejpam-1242	413	3	u	u	PROPN
ejpam-1242	413	4	⊆	⊆	NUM
ejpam-1242	413	5	l.	l.	NOUN
ejpam-1242	413	6	thus	thus	ADV
ejpam-1242	413	7	u	u	PROPN
ejpam-1242	413	8	⊆	⊆	PROPN
ejpam-1242	413	9	t	t	NOUN
ejpam-1242	413	10	∩	∩	NOUN
ejpam-1242	413	11	l.	l.	PROPN
ejpam-1242	413	12	now	now	ADV
ejpam-1242	413	13	we	we	PRON
ejpam-1242	413	14	have	have	VERB
ejpam-1242	413	15	v	v	NUM
ejpam-1242	413	16	(	(	PUNCT
ejpam-1242	413	17	t	t	NOUN
ejpam-1242	413	18	)	)	PUNCT
ejpam-1242	413	19	∪	∪	ADP
ejpam-1242	413	20	v	v	NOUN
ejpam-1242	413	21	(	(	PUNCT
ejpam-1242	413	22	l	l	NOUN
ejpam-1242	413	23	)	)	PUNCT
ejpam-1242	413	24	⊆	⊆	NUM
ejpam-1242	413	25	v	v	NOUN
ejpam-1242	413	26	(	(	PUNCT
ejpam-1242	413	27	t	t	NOUN
ejpam-1242	413	28	∩	∩	ADJ
ejpam-1242	413	29	l	l	NOUN
ejpam-1242	413	30	)	)	PUNCT
ejpam-1242	413	31	⊆	⊆	NUM
ejpam-1242	413	32	v	v	X
ejpam-1242	413	33	(	(	PUNCT
ejpam-1242	413	34	u	u	NOUN
ejpam-1242	413	35	)	)	PUNCT
ejpam-1242	413	36	=	=	SYM
ejpam-1242	413	37	v	v	X
ejpam-1242	413	38	(	(	PUNCT
ejpam-1242	413	39	t	t	PROPN
ejpam-1242	413	40	)	)	PUNCT
ejpam-1242	413	41	∪	∪	ADP
ejpam-1242	413	42	v	v	NOUN
ejpam-1242	413	43	(	(	PUNCT
ejpam-1242	413	44	l	l	NOUN
ejpam-1242	413	45	)	)	PUNCT
ejpam-1242	413	46	,	,	PUNCT
ejpam-1242	413	47	that	that	ADV
ejpam-1242	413	48	is	is	ADV
ejpam-1242	413	49	,	,	PUNCT
ejpam-1242	413	50	v	v	X
ejpam-1242	413	51	(	(	PUNCT
ejpam-1242	413	52	t	t	NOUN
ejpam-1242	413	53	)	)	PUNCT
ejpam-1242	413	54	∪	∪	ADP
ejpam-1242	413	55	v	v	NOUN
ejpam-1242	413	56	(	(	PUNCT
ejpam-1242	413	57	l	l	NOUN
ejpam-1242	413	58	)	)	PUNCT
ejpam-1242	413	59	=	=	SYM
ejpam-1242	413	60	v	v	X
ejpam-1242	413	61	(	(	PUNCT
ejpam-1242	413	62	t	t	PROPN
ejpam-1242	413	63	∩	∩	PROPN
ejpam-1242	413	64	l	l	NOUN
ejpam-1242	413	65	)	)	PUNCT
ejpam-1242	413	66	.	.	PUNCT
ejpam-1242	414	1	now	now	ADV
ejpam-1242	414	2	n	n	CCONJ
ejpam-1242	414	3	∈	∈	NOUN
ejpam-1242	414	4	v	v	NOUN
ejpam-1242	414	5	(	(	PUNCT
ejpam-1242	414	6	t	t	PROPN
ejpam-1242	414	7	∩	∩	ADJ
ejpam-1242	414	8	l	l	NOUN
ejpam-1242	414	9	)	)	PUNCT
ejpam-1242	414	10	gives	give	VERB
ejpam-1242	414	11	t	t	PROPN
ejpam-1242	414	12	⊆	⊆	NUM
ejpam-1242	414	13	n	n	ADP
ejpam-1242	414	14	or	or	CCONJ
ejpam-1242	414	15	l	l	NOUN
ejpam-1242	414	16	⊆	⊆	NUM
ejpam-1242	414	17	n	n	X
ejpam-1242	414	18	by	by	ADP
ejpam-1242	414	19	lemma	lemma	PROPN
ejpam-1242	414	20	5	5	NUM
ejpam-1242	414	21	.	.	PUNCT
ejpam-1242	414	22	(	(	PUNCT
ejpam-1242	414	23	ii)⇒	ii)⇒	X
ejpam-1242	414	24	(	(	PUNCT
ejpam-1242	414	25	iii	iii	NOUN
ejpam-1242	414	26	)	)	PUNCT
ejpam-1242	414	27	is	be	AUX
ejpam-1242	414	28	clear	clear	ADJ
ejpam-1242	414	29	.	.	PUNCT
ejpam-1242	415	1	(	(	PUNCT
ejpam-1242	415	2	iii)⇒	iii)⇒	PROPN
ejpam-1242	415	3	(	(	PUNCT
ejpam-1242	415	4	i	i	NOUN
ejpam-1242	415	5	)	)	PUNCT
ejpam-1242	415	6	.	.	PUNCT
ejpam-1242	415	7	let	let	VERB
ejpam-1242	415	8	a	a	PRON
ejpam-1242	415	9	and	and	CCONJ
ejpam-1242	415	10	b	b	NOUN
ejpam-1242	415	11	be	be	AUX
ejpam-1242	415	12	any	any	DET
ejpam-1242	415	13	k	k	NOUN
ejpam-1242	415	14	-	-	NOUN
ejpam-1242	415	15	subsemimodules	subsemimodules	NOUN
ejpam-1242	415	16	of	of	ADP
ejpam-1242	415	17	m	m	PROPN
ejpam-1242	415	18	.	.	PUNCT
ejpam-1242	416	1	if	if	SCONJ
ejpam-1242	416	2	v	v	INTJ
ejpam-1242	416	3	(	(	PUNCT
ejpam-1242	416	4	a	a	NOUN
ejpam-1242	416	5	)	)	PUNCT
ejpam-1242	416	6	=	=	SYM
ejpam-1242	416	7	;	;	PUNCT
ejpam-1242	416	8	,	,	PUNCT
ejpam-1242	416	9	we	we	PRON
ejpam-1242	416	10	are	be	AUX
ejpam-1242	416	11	done	do	VERB
ejpam-1242	416	12	.	.	PUNCT
ejpam-1242	417	1	so	so	ADV
ejpam-1242	417	2	we	we	PRON
ejpam-1242	417	3	may	may	AUX
ejpam-1242	417	4	assume	assume	VERB
ejpam-1242	417	5	that	that	SCONJ
ejpam-1242	417	6	v	v	X
ejpam-1242	417	7	(	(	PUNCT
ejpam-1242	417	8	a	a	NOUN
ejpam-1242	417	9	)	)	PUNCT
ejpam-1242	417	10	and	and	CCONJ
ejpam-1242	417	11	v	v	NOUN
ejpam-1242	417	12	(	(	PUNCT
ejpam-1242	417	13	b	b	NOUN
ejpam-1242	417	14	)	)	PUNCT
ejpam-1242	417	15	are	be	AUX
ejpam-1242	417	16	both	both	PRON
ejpam-1242	417	17	non	non	ADJ
ejpam-1242	417	18	-	-	ADJ
ejpam-1242	417	19	empty	empty	ADJ
ejpam-1242	417	20	.	.	PUNCT
ejpam-1242	418	1	then	then	ADV
ejpam-1242	418	2	v	v	X
ejpam-1242	418	3	(	(	PUNCT
ejpam-1242	418	4	a)∪v	a)∪v	X
ejpam-1242	418	5	(	(	PUNCT
ejpam-1242	418	6	b	b	NOUN
ejpam-1242	418	7	)	)	PUNCT
ejpam-1242	418	8	=	=	NOUN
ejpam-1242	418	9	v	v	NOUN
ejpam-1242	418	10	(	(	PUNCT
ejpam-1242	418	11	rad(a))∪v	rad(a))∪v	PROPN
ejpam-1242	418	12	(	(	PUNCT
ejpam-1242	418	13	rad(b	rad(b	PROPN
ejpam-1242	418	14	)	)	PUNCT
ejpam-1242	418	15	)	)	PUNCT
ejpam-1242	419	1	=	=	SYM
ejpam-1242	419	2	v	v	X
ejpam-1242	419	3	(	(	PUNCT
ejpam-1242	419	4	rad(a)∩	rad(a)∩	X
ejpam-1242	419	5	rad(b	rad(b	PROPN
ejpam-1242	419	6	)	)	PUNCT
ejpam-1242	419	7	)	)	PUNCT
ejpam-1242	419	8	by	by	ADP
ejpam-1242	419	9	lemma	lemma	PROPN
ejpam-1242	419	10	5	5	NUM
ejpam-1242	419	11	and	and	CCONJ
ejpam-1242	419	12	(	(	PUNCT
ejpam-1242	419	13	iii	iii	NOUN
ejpam-1242	419	14	)	)	PUNCT
ejpam-1242	419	15	,	,	PUNCT
ejpam-1242	419	16	as	as	SCONJ
ejpam-1242	419	17	required	require	VERB
ejpam-1242	419	18	.	.	PUNCT
ejpam-1242	420	1	theorem	theorem	NOUN
ejpam-1242	420	2	9	9	NUM
ejpam-1242	420	3	.	.	PUNCT
ejpam-1242	421	1	if	if	SCONJ
ejpam-1242	421	2	n	n	PRON
ejpam-1242	421	3	is	be	AUX
ejpam-1242	421	4	a	a	DET
ejpam-1242	421	5	qm	qm	PROPN
ejpam-1242	421	6	-subsemimodule	-subsemimodule	NOUN
ejpam-1242	421	7	of	of	ADP
ejpam-1242	421	8	a	a	DET
ejpam-1242	421	9	top	top	ADJ
ejpam-1242	421	10	semimodule	semimodule	NOUN
ejpam-1242	421	11	m	m	VERB
ejpam-1242	421	12	over	over	ADP
ejpam-1242	421	13	a	a	DET
ejpam-1242	421	14	semiring	semire	VERB
ejpam-1242	421	15	r	r	NOUN
ejpam-1242	421	16	,	,	PUNCT
ejpam-1242	421	17	then	then	ADV
ejpam-1242	421	18	m	m	PROPN
ejpam-1242	421	19	/	/	SYM
ejpam-1242	421	20	n	n	PROPN
ejpam-1242	421	21	is	be	AUX
ejpam-1242	421	22	a	a	DET
ejpam-1242	421	23	top	top	ADJ
ejpam-1242	421	24	semimodule	semimodule	NOUN
ejpam-1242	421	25	.	.	PUNCT
ejpam-1242	422	1	proof	proof	NOUN
ejpam-1242	422	2	.	.	PUNCT
ejpam-1242	423	1	note	note	VERB
ejpam-1242	423	2	that	that	SCONJ
ejpam-1242	423	3	any	any	DET
ejpam-1242	423	4	semiprime	semiprime	NOUN
ejpam-1242	423	5	k	k	NOUN
ejpam-1242	423	6	-	-	NOUN
ejpam-1242	423	7	subsemimodule	subsemimodule	NOUN
ejpam-1242	423	8	of	of	ADP
ejpam-1242	423	9	m	m	PROPN
ejpam-1242	423	10	/	/	SYM
ejpam-1242	423	11	n	n	PROPN
ejpam-1242	423	12	has	have	VERB
ejpam-1242	423	13	the	the	DET
ejpam-1242	423	14	form	form	NOUN
ejpam-1242	423	15	u	u	NOUN
ejpam-1242	423	16	/	/	SYM
ejpam-1242	423	17	n	n	PROPN
ejpam-1242	423	18	where	where	SCONJ
ejpam-1242	423	19	u	u	NOUN
ejpam-1242	423	20	is	be	AUX
ejpam-1242	423	21	a	a	DET
ejpam-1242	423	22	semiprime	semiprime	NOUN
ejpam-1242	423	23	k	k	NOUN
ejpam-1242	423	24	-	-	NOUN
ejpam-1242	423	25	subsemimodule	subsemimodule	NOUN
ejpam-1242	423	26	of	of	ADP
ejpam-1242	423	27	m	m	AUX
ejpam-1242	423	28	containing	contain	VERB
ejpam-1242	423	29	n	n	INTJ
ejpam-1242	423	30	by	by	ADP
ejpam-1242	423	31	corollary	corollary	ADJ
ejpam-1242	423	32	1	1	NUM
ejpam-1242	423	33	.	.	PUNCT
ejpam-1242	424	1	let	let	VERB
ejpam-1242	424	2	t	t	PROPN
ejpam-1242	424	3	/	/	SYM
ejpam-1242	424	4	n	n	CCONJ
ejpam-1242	424	5	be	be	AUX
ejpam-1242	424	6	any	any	DET
ejpam-1242	424	7	prime	prime	ADJ
ejpam-1242	424	8	k	k	NOUN
ejpam-1242	424	9	-	-	NOUN
ejpam-1242	424	10	subsemimodule	subsemimodule	NOUN
ejpam-1242	424	11	of	of	ADP
ejpam-1242	424	12	m	m	PROPN
ejpam-1242	424	13	/	/	SYM
ejpam-1242	424	14	n	n	PROPN
ejpam-1242	424	15	and	and	CCONJ
ejpam-1242	424	16	let	let	VERB
ejpam-1242	424	17	u	u	NOUN
ejpam-1242	424	18	/	/	SYM
ejpam-1242	424	19	n	n	PROPN
ejpam-1242	424	20	and	and	CCONJ
ejpam-1242	424	21	l	l	NOUN
ejpam-1242	424	22	/	/	SYM
ejpam-1242	424	23	n	n	CCONJ
ejpam-1242	424	24	be	be	VERB
ejpam-1242	424	25	semiprime	semiprime	NOUN
ejpam-1242	424	26	k	k	X
ejpam-1242	424	27	-	-	PUNCT
ejpam-1242	424	28	subsemimodules	subsemimodule	NOUN
ejpam-1242	424	29	of	of	ADP
ejpam-1242	424	30	m	m	NOUN
ejpam-1242	424	31	/	/	SYM
ejpam-1242	424	32	n	n	PRON
ejpam-1242	424	33	such	such	ADJ
ejpam-1242	424	34	that	that	SCONJ
ejpam-1242	424	35	(	(	PUNCT
ejpam-1242	424	36	l	l	NOUN
ejpam-1242	424	37	/	/	SYM
ejpam-1242	424	38	n)∩	n)∩	NOUN
ejpam-1242	424	39	(	(	PUNCT
ejpam-1242	424	40	u	u	NOUN
ejpam-1242	424	41	/	/	SYM
ejpam-1242	424	42	n)⊆	n)⊆	PROPN
ejpam-1242	424	43	t	t	PROPN
ejpam-1242	424	44	/	/	SYM
ejpam-1242	424	45	n	n	PROPN
ejpam-1242	424	46	.	.	PUNCT
ejpam-1242	425	1	then	then	ADV
ejpam-1242	425	2	(	(	PUNCT
ejpam-1242	425	3	l	l	NOUN
ejpam-1242	425	4	∩	∩	X
ejpam-1242	425	5	u)/n	u)/n	X
ejpam-1242	425	6	⊆	⊆	NUM
ejpam-1242	425	7	(	(	PUNCT
ejpam-1242	425	8	l	l	NOUN
ejpam-1242	425	9	/	/	SYM
ejpam-1242	425	10	n)∩	n)∩	NOUN
ejpam-1242	425	11	(	(	PUNCT
ejpam-1242	425	12	u	u	NOUN
ejpam-1242	425	13	/	/	SYM
ejpam-1242	425	14	n)⊆	n)⊆	PROPN
ejpam-1242	425	15	t	t	PROPN
ejpam-1242	425	16	/	/	SYM
ejpam-1242	425	17	n	n	PROPN
ejpam-1242	425	18	,	,	PUNCT
ejpam-1242	425	19	so	so	CCONJ
ejpam-1242	425	20	u	u	NOUN
ejpam-1242	425	21	∩	∩	NOUN
ejpam-1242	425	22	l	l	PROPN
ejpam-1242	425	23	⊆	⊆	NUM
ejpam-1242	425	24	t	t	NOUN
ejpam-1242	425	25	;	;	PUNCT
ejpam-1242	425	26	hence	hence	ADV
ejpam-1242	425	27	either	either	CCONJ
ejpam-1242	425	28	u	u	PROPN
ejpam-1242	425	29	⊆	⊆	NUM
ejpam-1242	425	30	t	t	NOUN
ejpam-1242	425	31	or	or	CCONJ
ejpam-1242	425	32	l	l	NOUN
ejpam-1242	425	33	⊆	⊆	NUM
ejpam-1242	425	34	t	t	NOUN
ejpam-1242	425	35	since	since	SCONJ
ejpam-1242	425	36	t	t	PROPN
ejpam-1242	425	37	is	be	AUX
ejpam-1242	425	38	extraordinary	extraordinary	ADJ
ejpam-1242	425	39	by	by	ADP
ejpam-1242	425	40	lemma	lemma	PROPN
ejpam-1242	425	41	6	6	NUM
ejpam-1242	425	42	.	.	PUNCT
ejpam-1242	425	43	thus	thus	ADV
ejpam-1242	425	44	either	either	CCONJ
ejpam-1242	425	45	u	u	NOUN
ejpam-1242	425	46	/	/	SYM
ejpam-1242	425	47	n	n	CCONJ
ejpam-1242	425	48	⊆	⊆	NUM
ejpam-1242	425	49	t	t	PROPN
ejpam-1242	425	50	/	/	SYM
ejpam-1242	425	51	n	n	PROPN
ejpam-1242	425	52	or	or	CCONJ
ejpam-1242	425	53	l	l	NOUN
ejpam-1242	425	54	/	/	SYM
ejpam-1242	425	55	n	n	CCONJ
ejpam-1242	425	56	⊆	⊆	NUM
ejpam-1242	425	57	t	t	PROPN
ejpam-1242	425	58	/	/	SYM
ejpam-1242	425	59	n	n	PROPN
ejpam-1242	425	60	.	.	PUNCT
ejpam-1242	426	1	now	now	ADV
ejpam-1242	426	2	the	the	DET
ejpam-1242	426	3	assertion	assertion	NOUN
ejpam-1242	426	4	follows	follow	VERB
ejpam-1242	426	5	from	from	ADP
ejpam-1242	426	6	lemma	lemma	PROPN
ejpam-1242	426	7	6	6	NUM
ejpam-1242	426	8	(	(	PUNCT
ejpam-1242	426	9	ii	ii	NOUN
ejpam-1242	426	10	)	)	PUNCT
ejpam-1242	426	11	.	.	PUNCT
ejpam-1242	427	1	theorem	theorem	ADJ
ejpam-1242	427	2	10	10	NUM
ejpam-1242	427	3	.	.	PUNCT
ejpam-1242	428	1	let	let	VERB
ejpam-1242	428	2	n	n	PRON
ejpam-1242	428	3	,	,	PUNCT
ejpam-1242	428	4	l	l	X
ejpam-1242	428	5	be	be	AUX
ejpam-1242	428	6	k	k	NOUN
ejpam-1242	428	7	-	-	NOUN
ejpam-1242	428	8	subsemimodules	subsemimodule	NOUN
ejpam-1242	428	9	of	of	ADP
ejpam-1242	428	10	a	a	DET
ejpam-1242	428	11	semimodule	semimodule	NOUN
ejpam-1242	428	12	m	m	VERB
ejpam-1242	428	13	over	over	ADP
ejpam-1242	428	14	a	a	DET
ejpam-1242	428	15	semiring	semire	VERB
ejpam-1242	428	16	r.	r.	PROPN
ejpam-1242	428	17	then	then	ADV
ejpam-1242	428	18	the	the	DET
ejpam-1242	428	19	following	following	ADJ
ejpam-1242	428	20	statements	statement	NOUN
ejpam-1242	428	21	hold	hold	VERB
ejpam-1242	428	22	:	:	PUNCT
ejpam-1242	428	23	(	(	PUNCT
ejpam-1242	428	24	i	i	NOUN
ejpam-1242	428	25	)	)	PUNCT
ejpam-1242	428	26	if	if	SCONJ
ejpam-1242	428	27	s	s	VERB
ejpam-1242	428	28	is	be	AUX
ejpam-1242	428	29	a	a	DET
ejpam-1242	428	30	subset	subset	NOUN
ejpam-1242	428	31	of	of	ADP
ejpam-1242	428	32	m	m	PROPN
ejpam-1242	428	33	,	,	PUNCT
ejpam-1242	428	34	then	then	ADV
ejpam-1242	428	35	v	v	X
ejpam-1242	428	36	(	(	PUNCT
ejpam-1242	428	37	s	s	NOUN
ejpam-1242	428	38	)	)	PUNCT
ejpam-1242	428	39	=	=	SYM
ejpam-1242	428	40	v	v	X
ejpam-1242	428	41	(	(	PUNCT
ejpam-1242	428	42	<	<	X
ejpam-1242	428	43	s	s	X
ejpam-1242	428	44	>	>	X
ejpam-1242	428	45	)	)	PUNCT
ejpam-1242	428	46	.	.	PUNCT
ejpam-1242	429	1	(	(	PUNCT
ejpam-1242	429	2	ii	ii	NOUN
ejpam-1242	429	3	)	)	PUNCT
ejpam-1242	429	4	v	v	NOUN
ejpam-1242	429	5	(	(	PUNCT
ejpam-1242	429	6	n)∪	n)∪	NOUN
ejpam-1242	429	7	v	v	X
ejpam-1242	429	8	(	(	PUNCT
ejpam-1242	429	9	i	i	NOUN
ejpam-1242	429	10	m	m	VERB
ejpam-1242	429	11	)	)	PUNCT
ejpam-1242	430	1	=	=	SYM
ejpam-1242	430	2	v	v	NOUN
ejpam-1242	430	3	(	(	PUNCT
ejpam-1242	430	4	in	in	ADP
ejpam-1242	430	5	)	)	PUNCT
ejpam-1242	430	6	=	=	SYM
ejpam-1242	430	7	v	v	NOUN
ejpam-1242	430	8	(	(	PUNCT
ejpam-1242	430	9	n	n	CCONJ
ejpam-1242	430	10	∩	∩	X
ejpam-1242	430	11	i	i	X
ejpam-1242	430	12	m	m	PROPN
ejpam-1242	430	13	)	)	PUNCT
ejpam-1242	430	14	for	for	ADP
ejpam-1242	430	15	every	every	DET
ejpam-1242	430	16	strong	strong	ADJ
ejpam-1242	430	17	ideal	ideal	NOUN
ejpam-1242	430	18	i	i	PRON
ejpam-1242	430	19	of	of	ADP
ejpam-1242	430	20	r	r	NOUN
ejpam-1242	430	21	.	.	PUNCT
ejpam-1242	431	1	(	(	PUNCT
ejpam-1242	431	2	iii	iii	X
ejpam-1242	431	3	)	)	PUNCT
ejpam-1242	431	4	v	v	NOUN
ejpam-1242	431	5	(	(	PUNCT
ejpam-1242	431	6	im)∪	im)∪	NOUN
ejpam-1242	431	7	v	v	X
ejpam-1242	431	8	(	(	PUNCT
ejpam-1242	431	9	j	j	PROPN
ejpam-1242	431	10	m	m	PROPN
ejpam-1242	431	11	)	)	PUNCT
ejpam-1242	432	1	=	=	SYM
ejpam-1242	432	2	v	v	X
ejpam-1242	432	3	(	(	PUNCT
ejpam-1242	432	4	ij	ij	NOUN
ejpam-1242	432	5	m	m	NOUN
ejpam-1242	432	6	)	)	PUNCT
ejpam-1242	433	1	=	=	SYM
ejpam-1242	433	2	v	v	X
ejpam-1242	433	3	(	(	PUNCT
ejpam-1242	433	4	i	i	NOUN
ejpam-1242	433	5	m	m	PROPN
ejpam-1242	433	6	∩	∩	ADJ
ejpam-1242	433	7	j	j	PROPN
ejpam-1242	433	8	m	m	PROPN
ejpam-1242	433	9	)	)	PUNCT
ejpam-1242	433	10	for	for	ADP
ejpam-1242	433	11	every	every	DET
ejpam-1242	433	12	strong	strong	ADJ
ejpam-1242	433	13	ideals	ideal	NOUN
ejpam-1242	433	14	i	i	PRON
ejpam-1242	433	15	and	and	CCONJ
ejpam-1242	433	16	j	j	PROPN
ejpam-1242	433	17	of	of	ADP
ejpam-1242	433	18	r.	r.	PROPN
ejpam-1242	433	19	(	(	PUNCT
ejpam-1242	433	20	iv	iv	X
ejpam-1242	433	21	)	)	PUNCT
ejpam-1242	433	22	if	if	SCONJ
ejpam-1242	433	23	v	v	X
ejpam-1242	433	24	(	(	PUNCT
ejpam-1242	433	25	n)⊆	n)⊆	NOUN
ejpam-1242	433	26	v	v	NOUN
ejpam-1242	433	27	(	(	PUNCT
ejpam-1242	433	28	l	l	NOUN
ejpam-1242	433	29	)	)	PUNCT
ejpam-1242	433	30	,	,	PUNCT
ejpam-1242	433	31	then	then	ADV
ejpam-1242	433	32	l	l	PROPN
ejpam-1242	433	33	⊆	⊆	NUM
ejpam-1242	433	34	rad(n	rad(n	NOUN
ejpam-1242	433	35	)	)	PUNCT
ejpam-1242	433	36	.	.	PUNCT
ejpam-1242	434	1	s.	s.	PROPN
ejpam-1242	434	2	atani	atani	PROPN
ejpam-1242	434	3	,	,	PUNCT
ejpam-1242	434	4	r.	r.	PROPN
ejpam-1242	434	5	atrani	atrani	PROPN
ejpam-1242	434	6	,	,	PUNCT
ejpam-1242	434	7	ü.	ü.	NOUN
ejpam-1242	434	8	tekir	tekir	PROPN
ejpam-1242	434	9	/	/	SYM
ejpam-1242	434	10	eur	eur	PROPN
ejpam-1242	434	11	.	.	PUNCT
ejpam-1242	435	1	j.	j.	PROPN
ejpam-1242	435	2	pure	pure	PROPN
ejpam-1242	435	3	appl	appl	PROPN
ejpam-1242	435	4	.	.	PROPN
ejpam-1242	435	5	math	math	PROPN
ejpam-1242	435	6	,	,	PUNCT
ejpam-1242	435	7	4	4	NUM
ejpam-1242	435	8	(	(	PUNCT
ejpam-1242	435	9	2011	2011	NUM
ejpam-1242	435	10	)	)	PUNCT
ejpam-1242	435	11	,	,	PUNCT
ejpam-1242	435	12	251	251	NUM
ejpam-1242	435	13	-	-	SYM
ejpam-1242	435	14	265	265	NUM
ejpam-1242	435	15	262	262	NUM
ejpam-1242	435	16	(	(	PUNCT
ejpam-1242	435	17	v	v	NOUN
ejpam-1242	435	18	)	)	PUNCT
ejpam-1242	435	19	v	v	NOUN
ejpam-1242	435	20	(	(	PUNCT
ejpam-1242	435	21	n	n	CCONJ
ejpam-1242	435	22	)	)	PUNCT
ejpam-1242	435	23	=	=	NOUN
ejpam-1242	435	24	v	v	X
ejpam-1242	435	25	(	(	PUNCT
ejpam-1242	435	26	l	l	NOUN
ejpam-1242	435	27	)	)	PUNCT
ejpam-1242	435	28	if	if	SCONJ
ejpam-1242	435	29	and	and	CCONJ
ejpam-1242	435	30	only	only	ADV
ejpam-1242	435	31	if	if	SCONJ
ejpam-1242	435	32	rad(n	rad(n	NOUN
ejpam-1242	435	33	)	)	PUNCT
ejpam-1242	435	34	=	=	PUNCT
ejpam-1242	435	35	rad(l	rad(l	PROPN
ejpam-1242	435	36	)	)	PUNCT
ejpam-1242	435	37	(	(	PUNCT
ejpam-1242	435	38	vi	vi	NOUN
ejpam-1242	435	39	)	)	PUNCT
ejpam-1242	435	40	if	if	SCONJ
ejpam-1242	435	41	m	m	NOUN
ejpam-1242	435	42	is	be	AUX
ejpam-1242	435	43	a	a	DET
ejpam-1242	435	44	strong	strong	ADJ
ejpam-1242	435	45	multiplication	multiplication	NOUN
ejpam-1242	435	46	semimodule	semimodule	NOUN
ejpam-1242	435	47	,	,	PUNCT
ejpam-1242	435	48	then	then	ADV
ejpam-1242	435	49	v	v	X
ejpam-1242	435	50	(	(	PUNCT
ejpam-1242	435	51	n)∪	n)∪	NOUN
ejpam-1242	435	52	v	v	NOUN
ejpam-1242	435	53	(	(	PUNCT
ejpam-1242	435	54	l	l	NOUN
ejpam-1242	435	55	)	)	PUNCT
ejpam-1242	435	56	=	=	SYM
ejpam-1242	435	57	v	v	NOUN
ejpam-1242	435	58	(	(	PUNCT
ejpam-1242	435	59	n	n	X
ejpam-1242	435	60	l	l	NOUN
ejpam-1242	435	61	)	)	PUNCT
ejpam-1242	436	1	=	=	SYM
ejpam-1242	436	2	v	v	NOUN
ejpam-1242	436	3	(	(	PUNCT
ejpam-1242	436	4	n	n	CCONJ
ejpam-1242	436	5	∩	∩	X
ejpam-1242	436	6	l	l	NOUN
ejpam-1242	436	7	)	)	PUNCT
ejpam-1242	436	8	.	.	PUNCT
ejpam-1242	437	1	(	(	PUNCT
ejpam-1242	437	2	vii	vii	PROPN
ejpam-1242	437	3	)	)	PUNCT
ejpam-1242	437	4	every	every	DET
ejpam-1242	437	5	strong	strong	ADJ
ejpam-1242	437	6	multiplication	multiplication	NOUN
ejpam-1242	437	7	semimodule	semimodule	NOUN
ejpam-1242	437	8	is	be	AUX
ejpam-1242	437	9	a	a	DET
ejpam-1242	437	10	top	top	ADJ
ejpam-1242	437	11	module	module	NOUN
ejpam-1242	437	12	.	.	PUNCT
ejpam-1242	438	1	(	(	PUNCT
ejpam-1242	438	2	viii	viii	NOUN
ejpam-1242	438	3	)	)	PUNCT
ejpam-1242	438	4	if	if	SCONJ
ejpam-1242	438	5	m	m	NOUN
ejpam-1242	438	6	is	be	AUX
ejpam-1242	438	7	a	a	DET
ejpam-1242	438	8	strong	strong	ADJ
ejpam-1242	438	9	multiplication	multiplication	NOUN
ejpam-1242	438	10	semimodule	semimodule	NOUN
ejpam-1242	438	11	,	,	PUNCT
ejpam-1242	438	12	then	then	ADV
ejpam-1242	438	13	every	every	DET
ejpam-1242	438	14	prime	prime	ADJ
ejpam-1242	438	15	k	k	NOUN
ejpam-1242	438	16	-	-	NOUN
ejpam-1242	438	17	subsemimodule	subsemimodule	NOUN
ejpam-1242	438	18	of	of	ADP
ejpam-1242	438	19	m	m	PROPN
ejpam-1242	438	20	is	be	AUX
ejpam-1242	438	21	extraordinary	extraordinary	ADJ
ejpam-1242	438	22	.	.	PUNCT
ejpam-1242	439	1	proof	proof	NOUN
ejpam-1242	439	2	.	.	PUNCT
ejpam-1242	440	1	(	(	PUNCT
ejpam-1242	440	2	i	i	NOUN
ejpam-1242	440	3	)	)	PUNCT
ejpam-1242	440	4	obvious	obvious	ADJ
ejpam-1242	440	5	.	.	PUNCT
ejpam-1242	441	1	(	(	PUNCT
ejpam-1242	441	2	ii	ii	X
ejpam-1242	441	3	)	)	PUNCT
ejpam-1242	441	4	it	it	PRON
ejpam-1242	441	5	is	be	AUX
ejpam-1242	441	6	clear	clear	ADJ
ejpam-1242	441	7	that	that	SCONJ
ejpam-1242	441	8	v	v	X
ejpam-1242	441	9	(	(	PUNCT
ejpam-1242	441	10	n	n	CCONJ
ejpam-1242	441	11	)	)	PUNCT
ejpam-1242	441	12	∪	∪	NOUN
ejpam-1242	441	13	v	v	NOUN
ejpam-1242	441	14	(	(	PUNCT
ejpam-1242	441	15	i	i	NOUN
ejpam-1242	441	16	m	m	VERB
ejpam-1242	441	17	)	)	PUNCT
ejpam-1242	442	1	⊆	⊆	NUM
ejpam-1242	442	2	v	v	NOUN
ejpam-1242	442	3	(	(	PUNCT
ejpam-1242	442	4	n	n	CCONJ
ejpam-1242	442	5	∩	∩	X
ejpam-1242	442	6	i	i	PRON
ejpam-1242	442	7	m	m	PROPN
ejpam-1242	442	8	)	)	PUNCT
ejpam-1242	442	9	⊆	⊆	NUM
ejpam-1242	442	10	v	v	NOUN
ejpam-1242	442	11	(	(	PUNCT
ejpam-1242	442	12	in	in	ADP
ejpam-1242	442	13	)	)	PUNCT
ejpam-1242	442	14	.	.	PUNCT
ejpam-1242	443	1	let	let	VERB
ejpam-1242	443	2	p	p	PRON
ejpam-1242	443	3	∈	∈	PROPN
ejpam-1242	443	4	v	v	NOUN
ejpam-1242	443	5	(	(	PUNCT
ejpam-1242	443	6	in	in	ADP
ejpam-1242	443	7	)	)	PUNCT
ejpam-1242	443	8	.	.	PUNCT
ejpam-1242	444	1	then	then	ADV
ejpam-1242	444	2	in	in	ADP
ejpam-1242	444	3	⊆	⊆	NUM
ejpam-1242	444	4	p	p	NOUN
ejpam-1242	444	5	and	and	CCONJ
ejpam-1242	444	6	hence	hence	ADV
ejpam-1242	444	7	n	n	CCONJ
ejpam-1242	444	8	⊆	⊆	NUM
ejpam-1242	444	9	p	p	NOUN
ejpam-1242	444	10	or	or	CCONJ
ejpam-1242	444	11	i	i	PRON
ejpam-1242	444	12	m	m	PROPN
ejpam-1242	444	13	⊆	⊆	NUM
ejpam-1242	444	14	p	p	NOUN
ejpam-1242	444	15	by	by	ADP
ejpam-1242	444	16	[	[	PUNCT
ejpam-1242	444	17	13	13	NUM
ejpam-1242	444	18	,	,	PUNCT
ejpam-1242	444	19	theorem	theorem	VERB
ejpam-1242	444	20	7	7	NUM
ejpam-1242	444	21	]	]	PUNCT
ejpam-1242	444	22	.	.	PUNCT
ejpam-1242	445	1	thus	thus	ADV
ejpam-1242	445	2	p	p	X
ejpam-1242	445	3	∈	∈	PROPN
ejpam-1242	445	4	v	v	ADP
ejpam-1242	445	5	(	(	PUNCT
ejpam-1242	445	6	n	n	CCONJ
ejpam-1242	445	7	)	)	PUNCT
ejpam-1242	445	8	or	or	CCONJ
ejpam-1242	445	9	p	p	NOUN
ejpam-1242	445	10	∈	∈	PROPN
ejpam-1242	445	11	v	v	ADP
ejpam-1242	445	12	(	(	PUNCT
ejpam-1242	445	13	i	i	NOUN
ejpam-1242	445	14	m	m	PROPN
ejpam-1242	445	15	)	)	PUNCT
ejpam-1242	445	16	,	,	PUNCT
ejpam-1242	445	17	i.e.	i.e.	X
ejpam-1242	445	18	p	p	X
ejpam-1242	445	19	∈	∈	PROPN
ejpam-1242	445	20	v	v	NOUN
ejpam-1242	445	21	(	(	PUNCT
ejpam-1242	445	22	n)∪	n)∪	NOUN
ejpam-1242	445	23	v	v	X
ejpam-1242	445	24	(	(	PUNCT
ejpam-1242	445	25	i	i	NOUN
ejpam-1242	445	26	m	m	PROPN
ejpam-1242	445	27	)	)	PUNCT
ejpam-1242	445	28	.	.	PUNCT
ejpam-1242	446	1	hence	hence	ADV
ejpam-1242	446	2	v	v	NOUN
ejpam-1242	446	3	(	(	PUNCT
ejpam-1242	446	4	in)⊆	in)⊆	PROPN
ejpam-1242	446	5	v	v	NOUN
ejpam-1242	446	6	(	(	PUNCT
ejpam-1242	446	7	n)∪	n)∪	NOUN
ejpam-1242	446	8	v	v	X
ejpam-1242	446	9	(	(	PUNCT
ejpam-1242	446	10	i	i	NOUN
ejpam-1242	446	11	m	m	PROPN
ejpam-1242	446	12	)	)	PUNCT
ejpam-1242	446	13	.	.	PUNCT
ejpam-1242	447	1	(	(	PUNCT
ejpam-1242	447	2	iii	iii	NOUN
ejpam-1242	447	3	)	)	PUNCT
ejpam-1242	447	4	follows	follow	VERB
ejpam-1242	447	5	from	from	ADP
ejpam-1242	447	6	(	(	PUNCT
ejpam-1242	447	7	ii	ii	NOUN
ejpam-1242	447	8	)	)	PUNCT
ejpam-1242	447	9	.	.	PUNCT
ejpam-1242	448	1	(	(	PUNCT
ejpam-1242	448	2	iv	iv	X
ejpam-1242	448	3	)	)	PUNCT
ejpam-1242	448	4	obvious	obvious	ADJ
ejpam-1242	448	5	.	.	PUNCT
ejpam-1242	449	1	(	(	PUNCT
ejpam-1242	449	2	v	v	NOUN
ejpam-1242	449	3	)	)	PUNCT
ejpam-1242	449	4	let	let	VERB
ejpam-1242	449	5	v	v	NOUN
ejpam-1242	449	6	(	(	PUNCT
ejpam-1242	449	7	n	n	CCONJ
ejpam-1242	449	8	)	)	PUNCT
ejpam-1242	449	9	=	=	NOUN
ejpam-1242	449	10	v	v	X
ejpam-1242	449	11	(	(	PUNCT
ejpam-1242	449	12	l	l	NOUN
ejpam-1242	449	13	)	)	PUNCT
ejpam-1242	449	14	.	.	PUNCT
ejpam-1242	450	1	by	by	ADP
ejpam-1242	450	2	lemma	lemma	PROPN
ejpam-1242	450	3	5	5	NUM
ejpam-1242	450	4	,	,	PUNCT
ejpam-1242	450	5	we	we	PRON
ejpam-1242	450	6	have	have	VERB
ejpam-1242	450	7	v	v	NUM
ejpam-1242	450	8	(	(	PUNCT
ejpam-1242	450	9	n	n	CCONJ
ejpam-1242	450	10	)	)	PUNCT
ejpam-1242	450	11	⊆	⊆	NUM
ejpam-1242	450	12	v	v	NOUN
ejpam-1242	450	13	(	(	PUNCT
ejpam-1242	450	14	rad(l	rad(l	PROPN
ejpam-1242	450	15	)	)	PUNCT
ejpam-1242	450	16	)	)	PUNCT
ejpam-1242	450	17	;	;	PUNCT
ejpam-1242	450	18	hence	hence	ADV
ejpam-1242	450	19	rad(l	rad(l	PROPN
ejpam-1242	450	20	)	)	PUNCT
ejpam-1242	450	21	⊆	⊆	NUM
ejpam-1242	450	22	rad(n	rad(n	NOUN
ejpam-1242	450	23	)	)	PUNCT
ejpam-1242	450	24	by	by	ADP
ejpam-1242	450	25	(	(	PUNCT
ejpam-1242	450	26	iv	iv	X
ejpam-1242	450	27	)	)	PUNCT
ejpam-1242	450	28	.	.	PUNCT
ejpam-1242	451	1	similarly	similarly	ADV
ejpam-1242	451	2	,	,	PUNCT
ejpam-1242	451	3	rad(n	rad(n	NOUN
ejpam-1242	451	4	)	)	PUNCT
ejpam-1242	451	5	⊆	⊆	NUM
ejpam-1242	451	6	rad(l	rad(l	NOUN
ejpam-1242	451	7	)	)	PUNCT
ejpam-1242	451	8	,	,	PUNCT
ejpam-1242	451	9	and	and	CCONJ
ejpam-1242	451	10	so	so	ADV
ejpam-1242	451	11	we	we	PRON
ejpam-1242	451	12	have	have	VERB
ejpam-1242	451	13	equality	equality	NOUN
ejpam-1242	451	14	.	.	PUNCT
ejpam-1242	452	1	the	the	DET
ejpam-1242	452	2	other	other	ADJ
ejpam-1242	452	3	implication	implication	NOUN
ejpam-1242	452	4	is	be	AUX
ejpam-1242	452	5	similar	similar	ADJ
ejpam-1242	452	6	.	.	PUNCT
ejpam-1242	453	1	(	(	PUNCT
ejpam-1242	453	2	vi	vi	NOUN
ejpam-1242	453	3	)	)	PUNCT
ejpam-1242	453	4	apply	apply	NOUN
ejpam-1242	453	5	(	(	PUNCT
ejpam-1242	453	6	iii	iii	NOUN
ejpam-1242	453	7	)	)	PUNCT
ejpam-1242	453	8	.	.	PUNCT
ejpam-1242	454	1	(	(	PUNCT
ejpam-1242	454	2	vii	vii	PROPN
ejpam-1242	454	3	)	)	PUNCT
ejpam-1242	454	4	follows	follow	VERB
ejpam-1242	454	5	from	from	ADP
ejpam-1242	454	6	(	(	PUNCT
ejpam-1242	454	7	vi	vi	NOUN
ejpam-1242	454	8	)	)	PUNCT
ejpam-1242	454	9	.	.	PUNCT
ejpam-1242	455	1	(	(	PUNCT
ejpam-1242	455	2	viii	viii	NOUN
ejpam-1242	455	3	)	)	PUNCT
ejpam-1242	455	4	follows	follow	VERB
ejpam-1242	455	5	from	from	ADP
ejpam-1242	455	6	(	(	PUNCT
ejpam-1242	455	7	vii	vii	PROPN
ejpam-1242	455	8	)	)	PUNCT
ejpam-1242	455	9	and	and	CCONJ
ejpam-1242	455	10	lemma	lemma	PROPN
ejpam-1242	455	11	6	6	NUM
ejpam-1242	455	12	.	.	PUNCT
ejpam-1242	455	13	remark	remark	PROPN
ejpam-1242	455	14	2	2	NUM
ejpam-1242	455	15	.	.	PUNCT
ejpam-1242	455	16	assume	assume	VERB
ejpam-1242	455	17	that	that	SCONJ
ejpam-1242	455	18	m	m	PROPN
ejpam-1242	455	19	is	be	AUX
ejpam-1242	455	20	a	a	DET
ejpam-1242	455	21	semimodule	semimodule	NOUN
ejpam-1242	455	22	over	over	ADP
ejpam-1242	455	23	a	a	DET
ejpam-1242	455	24	semiring	semire	VERB
ejpam-1242	455	25	r	r	NOUN
ejpam-1242	455	26	and	and	CCONJ
ejpam-1242	455	27	let	let	VERB
ejpam-1242	455	28	x	x	PUNCT
ejpam-1242	455	29	=	=	SYM
ejpam-1242	455	30	speck(m	speck(m	NOUN
ejpam-1242	455	31	)	)	PUNCT
ejpam-1242	455	32	.	.	PUNCT
ejpam-1242	456	1	for	for	ADP
ejpam-1242	456	2	each	each	DET
ejpam-1242	456	3	subset	subset	NOUN
ejpam-1242	456	4	s	s	PROPN
ejpam-1242	456	5	of	of	ADP
ejpam-1242	456	6	m	m	PRON
ejpam-1242	456	7	,	,	PUNCT
ejpam-1242	456	8	by	by	ADP
ejpam-1242	456	9	xs	xs	PROPN
ejpam-1242	456	10	we	we	PRON
ejpam-1242	456	11	mean	mean	VERB
ejpam-1242	456	12	x	x	PUNCT
ejpam-1242	456	13	−	−	NOUN
ejpam-1242	456	14	v	v	X
ejpam-1242	456	15	(	(	PUNCT
ejpam-1242	456	16	s	s	NOUN
ejpam-1242	456	17	)	)	PUNCT
ejpam-1242	456	18	=	=	SYM
ejpam-1242	456	19	{	{	PUNCT
ejpam-1242	456	20	p	p	X
ejpam-1242	456	21	∈	∈	PROPN
ejpam-1242	456	22	x	x	X
ejpam-1242	456	23	:	:	PUNCT
ejpam-1242	456	24	s	s	X
ejpam-1242	456	25	*	*	PUNCT
ejpam-1242	456	26	p	p	X
ejpam-1242	456	27	}	}	PUNCT
ejpam-1242	456	28	.	.	PUNCT
ejpam-1242	457	1	if	if	SCONJ
ejpam-1242	457	2	s	s	VERB
ejpam-1242	457	3	=	=	X
ejpam-1242	457	4	{	{	PUNCT
ejpam-1242	457	5	m	m	NOUN
ejpam-1242	457	6	}	}	PUNCT
ejpam-1242	457	7	,	,	PUNCT
ejpam-1242	457	8	we	we	PRON
ejpam-1242	457	9	denote	denote	VERB
ejpam-1242	457	10	by	by	ADP
ejpam-1242	457	11	xm	xm	PROPN
ejpam-1242	457	12	=	=	PUNCT
ejpam-1242	457	13	{	{	PUNCT
ejpam-1242	457	14	p	p	X
ejpam-1242	457	15	∈	∈	PROPN
ejpam-1242	457	16	x	x	X
ejpam-1242	457	17	:	:	PUNCT
ejpam-1242	458	1	rm	rm	X
ejpam-1242	458	2	*	*	PUNCT
ejpam-1242	459	1	p	p	X
ejpam-1242	459	2	}	}	PUNCT
ejpam-1242	459	3	=	=	PUNCT
ejpam-1242	459	4	{	{	PUNCT
ejpam-1242	459	5	p	p	X
ejpam-1242	459	6	∈	∈	PROPN
ejpam-1242	459	7	x	x	X
ejpam-1242	459	8	:	:	PUNCT
ejpam-1242	460	1	m	m	VERB
ejpam-1242	460	2	*	*	PUNCT
ejpam-1242	460	3	p	p	X
ejpam-1242	460	4	}	}	PUNCT
ejpam-1242	460	5	.	.	PUNCT
ejpam-1242	461	1	clearly	clearly	ADV
ejpam-1242	461	2	,	,	PUNCT
ejpam-1242	461	3	the	the	DET
ejpam-1242	461	4	sets	set	NOUN
ejpam-1242	461	5	xm	xm	PROPN
ejpam-1242	461	6	are	be	AUX
ejpam-1242	461	7	open	open	ADJ
ejpam-1242	461	8	,	,	PUNCT
ejpam-1242	461	9	and	and	CCONJ
ejpam-1242	461	10	they	they	PRON
ejpam-1242	461	11	are	be	AUX
ejpam-1242	461	12	called	call	VERB
ejpam-1242	461	13	basic	basic	ADJ
ejpam-1242	461	14	open	open	ADJ
ejpam-1242	461	15	sets	set	NOUN
ejpam-1242	461	16	.	.	PUNCT
ejpam-1242	462	1	lemma	lemma	PROPN
ejpam-1242	462	2	7	7	X
ejpam-1242	462	3	.	.	PUNCT
ejpam-1242	463	1	let	let	VERB
ejpam-1242	463	2	m	m	PRON
ejpam-1242	463	3	be	be	AUX
ejpam-1242	463	4	a	a	DET
ejpam-1242	463	5	strong	strong	ADJ
ejpam-1242	463	6	multiplication	multiplication	NOUN
ejpam-1242	463	7	semimodule	semimodule	NOUN
ejpam-1242	463	8	over	over	ADP
ejpam-1242	463	9	a	a	DET
ejpam-1242	463	10	semiring	semire	VERB
ejpam-1242	463	11	r.	r.	PROPN
ejpam-1242	463	12	then	then	ADV
ejpam-1242	463	13	the	the	DET
ejpam-1242	463	14	following	following	ADJ
ejpam-1242	463	15	statements	statement	NOUN
ejpam-1242	463	16	hold	hold	VERB
ejpam-1242	463	17	:	:	PUNCT
ejpam-1242	463	18	(	(	PUNCT
ejpam-1242	463	19	i	i	NOUN
ejpam-1242	463	20	)	)	PUNCT
ejpam-1242	463	21	x	x	VERB
ejpam-1242	464	1	i	i	PRON
ejpam-1242	464	2	m	m	VERB
ejpam-1242	464	3	∩	∩	NOUN
ejpam-1242	464	4	x	x	X
ejpam-1242	464	5	j	j	NOUN
ejpam-1242	464	6	m	m	VERB
ejpam-1242	464	7	=	=	NOUN
ejpam-1242	464	8	x	x	PUNCT
ejpam-1242	464	9	i	i	PRON
ejpam-1242	464	10	j	j	PROPN
ejpam-1242	464	11	m	m	VERB
ejpam-1242	464	12	for	for	ADP
ejpam-1242	464	13	every	every	DET
ejpam-1242	464	14	strong	strong	ADJ
ejpam-1242	464	15	ideals	ideal	NOUN
ejpam-1242	465	1	i	i	PRON
ejpam-1242	465	2	and	and	CCONJ
ejpam-1242	465	3	j	j	PROPN
ejpam-1242	465	4	of	of	ADP
ejpam-1242	465	5	r.	r.	PROPN
ejpam-1242	465	6	(	(	PUNCT
ejpam-1242	465	7	ii	ii	PROPN
ejpam-1242	465	8	)	)	PUNCT
ejpam-1242	465	9	the	the	DET
ejpam-1242	465	10	seta	seta	PROPN
ejpam-1242	465	11	=	=	PUNCT
ejpam-1242	465	12	{	{	PUNCT
ejpam-1242	465	13	xm	xm	NOUN
ejpam-1242	465	14	:	:	PUNCT
ejpam-1242	465	15	m	m	VERB
ejpam-1242	465	16	∈	∈	PROPN
ejpam-1242	465	17	m	m	PRON
ejpam-1242	465	18	}	}	PUNCT
ejpam-1242	465	19	forms	form	VERB
ejpam-1242	465	20	a	a	DET
ejpam-1242	465	21	base	base	NOUN
ejpam-1242	465	22	for	for	ADP
ejpam-1242	465	23	the	the	DET
ejpam-1242	465	24	zariski	zariski	ADJ
ejpam-1242	465	25	topology	topology	NOUN
ejpam-1242	465	26	on	on	ADP
ejpam-1242	465	27	x	x	X
ejpam-1242	465	28	.	.	PUNCT
ejpam-1242	466	1	proof	proof	NOUN
ejpam-1242	466	2	.	.	PUNCT
ejpam-1242	467	1	(	(	PUNCT
ejpam-1242	467	2	i	i	NOUN
ejpam-1242	467	3	)	)	PUNCT
ejpam-1242	467	4	immediately	immediately	ADV
ejpam-1242	467	5	follows	follow	VERB
ejpam-1242	467	6	from	from	ADP
ejpam-1242	467	7	theorem	theorem	ADJ
ejpam-1242	467	8	10	10	NUM
ejpam-1242	467	9	(	(	PUNCT
ejpam-1242	467	10	iii	iii	NOUN
ejpam-1242	467	11	)	)	PUNCT
ejpam-1242	467	12	(	(	PUNCT
ejpam-1242	467	13	taking	take	VERB
ejpam-1242	467	14	complements	complement	NOUN
ejpam-1242	467	15	)	)	PUNCT
ejpam-1242	467	16	.	.	PUNCT
ejpam-1242	468	1	references	reference	NOUN
ejpam-1242	468	2	263	263	NUM
ejpam-1242	468	3	(	(	PUNCT
ejpam-1242	468	4	ii	ii	NOUN
ejpam-1242	468	5	)	)	PUNCT
ejpam-1242	468	6	suppose	suppose	VERB
ejpam-1242	468	7	that	that	SCONJ
ejpam-1242	468	8	u	u	PROPN
ejpam-1242	468	9	is	be	AUX
ejpam-1242	468	10	an	an	DET
ejpam-1242	468	11	open	open	ADJ
ejpam-1242	468	12	set	set	NOUN
ejpam-1242	468	13	in	in	ADP
ejpam-1242	468	14	x	x	X
ejpam-1242	468	15	.	.	PUNCT
ejpam-1242	469	1	then	then	ADV
ejpam-1242	469	2	u	u	X
ejpam-1242	469	3	=	=	NOUN
ejpam-1242	469	4	x	x	PROPN
ejpam-1242	469	5	−	−	PROPN
ejpam-1242	469	6	v	v	NOUN
ejpam-1242	469	7	(	(	PUNCT
ejpam-1242	469	8	n	n	CCONJ
ejpam-1242	469	9	)	)	PUNCT
ejpam-1242	469	10	for	for	ADP
ejpam-1242	469	11	some	some	DET
ejpam-1242	469	12	k	k	ADJ
ejpam-1242	469	13	-	-	ADJ
ejpam-1242	469	14	subsemimodule	subsemimodule	ADJ
ejpam-1242	469	15	n	n	PROPN
ejpam-1242	469	16	of	of	ADP
ejpam-1242	469	17	m	m	PROPN
ejpam-1242	469	18	.	.	PUNCT
ejpam-1242	470	1	let	let	VERB
ejpam-1242	470	2	n	n	PRON
ejpam-1242	470	3	=	=	NOUN
ejpam-1242	470	4	<	<	X
ejpam-1242	470	5	{	{	PUNCT
ejpam-1242	470	6	mi	mi	NOUN
ejpam-1242	470	7	:	:	PUNCT
ejpam-1242	470	8	i	i	PROPN
ejpam-1242	470	9	∈	∈	PROPN
ejpam-1242	470	10	i	i	X
ejpam-1242	470	11	}	}	PUNCT
ejpam-1242	470	12	>	>	X
ejpam-1242	470	13	,	,	PUNCT
ejpam-1242	470	14	where	where	SCONJ
ejpam-1242	470	15	{	{	PUNCT
ejpam-1242	470	16	mi	mi	NOUN
ejpam-1242	470	17	:	:	PUNCT
ejpam-1242	470	18	i	i	PRON
ejpam-1242	470	19	∈	∈	PROPN
ejpam-1242	470	20	i	i	PRON
ejpam-1242	470	21	}	}	PUNCT
ejpam-1242	470	22	is	be	AUX
ejpam-1242	470	23	a	a	DET
ejpam-1242	470	24	generator	generator	NOUN
ejpam-1242	470	25	set	set	NOUN
ejpam-1242	470	26	of	of	ADP
ejpam-1242	470	27	n	n	PROPN
ejpam-1242	470	28	.	.	PUNCT
ejpam-1242	471	1	then	then	ADV
ejpam-1242	471	2	v	v	X
ejpam-1242	471	3	(	(	PUNCT
ejpam-1242	471	4	n	n	CCONJ
ejpam-1242	471	5	)	)	PUNCT
ejpam-1242	471	6	=	=	SYM
ejpam-1242	471	7	v	v	NOUN
ejpam-1242	471	8	(	(	PUNCT
ejpam-1242	471	9	∑	∑	PROPN
ejpam-1242	471	10	i∈i	i∈i	ADJ
ejpam-1242	471	11	rmi	rmi	NOUN
ejpam-1242	471	12	)	)	PUNCT
ejpam-1242	471	13	=	=	PUNCT
ejpam-1242	472	1	⋂	⋂	PROPN
ejpam-1242	472	2	i∈i	i∈i	ADJ
ejpam-1242	472	3	v	v	PROPN
ejpam-1242	472	4	(	(	PUNCT
ejpam-1242	472	5	rmi	rmi	NOUN
ejpam-1242	472	6	)	)	PUNCT
ejpam-1242	472	7	by	by	ADP
ejpam-1242	472	8	lemma	lemma	PROPN
ejpam-1242	472	9	5	5	NUM
ejpam-1242	472	10	(	(	PUNCT
ejpam-1242	472	11	ii	ii	NOUN
ejpam-1242	472	12	)	)	PUNCT
ejpam-1242	472	13	.	.	PUNCT
ejpam-1242	473	1	it	it	PRON
ejpam-1242	473	2	follows	follow	VERB
ejpam-1242	473	3	that	that	SCONJ
ejpam-1242	473	4	u	u	PROPN
ejpam-1242	473	5	=	=	SYM
ejpam-1242	473	6	x−v	x−v	PROPN
ejpam-1242	473	7	(	(	PUNCT
ejpam-1242	473	8	n	n	CCONJ
ejpam-1242	473	9	)	)	PUNCT
ejpam-1242	474	1	=	=	SYM
ejpam-1242	474	2	x−	x−	PROPN
ejpam-1242	474	3	⋂	⋂	PROPN
ejpam-1242	474	4	i∈i	i∈i	ADJ
ejpam-1242	474	5	v	v	PROPN
ejpam-1242	474	6	(	(	PUNCT
ejpam-1242	474	7	rmi	rmi	NOUN
ejpam-1242	474	8	)	)	PUNCT
ejpam-1242	474	9	=	=	PUNCT
ejpam-1242	475	1	⋃	⋃	ADP
ejpam-1242	475	2	i∈i	i∈i	ADJ
ejpam-1242	475	3	xmi	xmi	PROPN
ejpam-1242	475	4	.	.	PUNCT
ejpam-1242	476	1	thusa	thusa	PROPN
ejpam-1242	476	2	is	be	AUX
ejpam-1242	476	3	a	a	DET
ejpam-1242	476	4	base	base	NOUN
ejpam-1242	476	5	for	for	ADP
ejpam-1242	476	6	the	the	DET
ejpam-1242	476	7	zariski	zariski	ADJ
ejpam-1242	476	8	topology	topology	NOUN
ejpam-1242	476	9	on	on	ADP
ejpam-1242	476	10	x	x	X
ejpam-1242	476	11	.	.	PUNCT
ejpam-1242	477	1	proposition	proposition	NOUN
ejpam-1242	477	2	4	4	NUM
ejpam-1242	477	3	.	.	PUNCT
ejpam-1242	478	1	let	let	VERB
ejpam-1242	478	2	m	m	PRON
ejpam-1242	478	3	be	be	AUX
ejpam-1242	478	4	a	a	DET
ejpam-1242	478	5	very	very	ADV
ejpam-1242	478	6	strong	strong	ADJ
ejpam-1242	478	7	multiplication	multiplication	NOUN
ejpam-1242	478	8	semimodule	semimodule	NOUN
ejpam-1242	478	9	over	over	ADP
ejpam-1242	478	10	a	a	DET
ejpam-1242	478	11	semiring	semire	VERB
ejpam-1242	478	12	r.	r.	PROPN
ejpam-1242	478	13	then	then	ADV
ejpam-1242	478	14	every	every	DET
ejpam-1242	478	15	basic	basic	ADJ
ejpam-1242	478	16	open	open	ADJ
ejpam-1242	478	17	set	set	NOUN
ejpam-1242	478	18	of	of	ADP
ejpam-1242	478	19	x	x	PUNCT
ejpam-1242	478	20	is	be	AUX
ejpam-1242	478	21	compact	compact	ADJ
ejpam-1242	478	22	.	.	PUNCT
ejpam-1242	479	1	proof	proof	NOUN
ejpam-1242	479	2	.	.	PUNCT
ejpam-1242	480	1	by	by	ADP
ejpam-1242	480	2	lemma	lemma	PROPN
ejpam-1242	480	3	7	7	NUM
ejpam-1242	480	4	(	(	PUNCT
ejpam-1242	480	5	ii	ii	NOUN
ejpam-1242	480	6	)	)	PUNCT
ejpam-1242	480	7	,	,	PUNCT
ejpam-1242	480	8	it	it	PRON
ejpam-1242	480	9	suffices	suffice	VERB
ejpam-1242	480	10	to	to	PART
ejpam-1242	480	11	show	show	VERB
ejpam-1242	480	12	that	that	SCONJ
ejpam-1242	480	13	every	every	DET
ejpam-1242	480	14	cover	cover	NOUN
ejpam-1242	480	15	of	of	ADP
ejpam-1242	480	16	basic	basic	ADJ
ejpam-1242	480	17	open	open	ADJ
ejpam-1242	480	18	sets	set	NOUN
ejpam-1242	480	19	has	have	VERB
ejpam-1242	480	20	a	a	DET
ejpam-1242	480	21	finite	finite	ADJ
ejpam-1242	480	22	subcover	subcover	PROPN
ejpam-1242	480	23	.	.	PUNCT
ejpam-1242	481	1	suppose	suppose	VERB
ejpam-1242	481	2	that	that	SCONJ
ejpam-1242	481	3	xm	xm	PROPN
ejpam-1242	481	4	⊆	⊆	NUM
ejpam-1242	481	5	⋃	⋃	NOUN
ejpam-1242	481	6	t∈i	t∈i	NOUN
ejpam-1242	481	7	xmt	xmt	NOUN
ejpam-1242	481	8	,	,	PUNCT
ejpam-1242	481	9	and	and	CCONJ
ejpam-1242	481	10	let	let	VERB
ejpam-1242	481	11	n	n	PRON
ejpam-1242	481	12	be	be	AUX
ejpam-1242	481	13	the	the	DET
ejpam-1242	481	14	subsemimodule	subsemimodule	NOUN
ejpam-1242	481	15	of	of	ADP
ejpam-1242	481	16	m	m	AUX
ejpam-1242	481	17	generated	generate	VERB
ejpam-1242	481	18	by	by	ADP
ejpam-1242	481	19	{	{	PUNCT
ejpam-1242	481	20	mt	mt	PROPN
ejpam-1242	481	21	:	:	PUNCT
ejpam-1242	481	22	t	t	PROPN
ejpam-1242	481	23	∈	∈	PROPN
ejpam-1242	482	1	i	i	PRON
ejpam-1242	482	2	}	}	PUNCT
ejpam-1242	482	3	.	.	PUNCT
ejpam-1242	483	1	it	it	PRON
ejpam-1242	483	2	follows	follow	VERB
ejpam-1242	483	3	that	that	SCONJ
ejpam-1242	483	4	⋂	⋂	PROPN
ejpam-1242	483	5	t∈i	t∈i	NOUN
ejpam-1242	483	6	v	v	NOUN
ejpam-1242	483	7	(	(	PUNCT
ejpam-1242	483	8	rmt	rmt	NOUN
ejpam-1242	483	9	)	)	PUNCT
ejpam-1242	483	10	=	=	SYM
ejpam-1242	483	11	v	v	X
ejpam-1242	483	12	(	(	PUNCT
ejpam-1242	483	13	n)⊆	n)⊆	NOUN
ejpam-1242	483	14	v	v	NOUN
ejpam-1242	483	15	(	(	PUNCT
ejpam-1242	483	16	rm	rm	PROPN
ejpam-1242	483	17	)	)	PUNCT
ejpam-1242	483	18	,	,	PUNCT
ejpam-1242	483	19	so	so	ADV
ejpam-1242	483	20	v	v	PROPN
ejpam-1242	483	21	(	(	PUNCT
ejpam-1242	483	22	rad(n	rad(n	NOUN
ejpam-1242	483	23	)	)	PUNCT
ejpam-1242	483	24	)	)	PUNCT
ejpam-1242	484	1	⊆	⊆	NUM
ejpam-1242	484	2	v	v	X
ejpam-1242	484	3	(	(	PUNCT
ejpam-1242	484	4	rad	rad	PROPN
ejpam-1242	484	5	(	(	PUNCT
ejpam-1242	484	6	<	<	X
ejpam-1242	484	7	m	m	X
ejpam-1242	484	8	>	>	PUNCT
ejpam-1242	484	9	)	)	PUNCT
ejpam-1242	484	10	)	)	PUNCT
ejpam-1242	484	11	by	by	ADP
ejpam-1242	484	12	lemma	lemma	PROPN
ejpam-1242	484	13	5	5	NUM
ejpam-1242	484	14	(	(	PUNCT
ejpam-1242	484	15	i	i	NOUN
ejpam-1242	484	16	)	)	PUNCT
ejpam-1242	484	17	;	;	PUNCT
ejpam-1242	484	18	hence	hence	ADV
ejpam-1242	484	19	rad	rad	PROPN
ejpam-1242	484	20	(	(	PUNCT
ejpam-1242	484	21	<	<	X
ejpam-1242	484	22	m	m	PROPN
ejpam-1242	484	23	>	>	X
ejpam-1242	484	24	)	)	PUNCT
ejpam-1242	484	25	⊆	⊆	NUM
ejpam-1242	484	26	rad(n	rad(n	NOUN
ejpam-1242	484	27	)	)	PUNCT
ejpam-1242	484	28	by	by	ADP
ejpam-1242	484	29	theorem	theorem	NOUN
ejpam-1242	484	30	10	10	NUM
ejpam-1242	484	31	(	(	PUNCT
ejpam-1242	484	32	iv	iv	NUM
ejpam-1242	484	33	)	)	PUNCT
ejpam-1242	484	34	.	.	PUNCT
ejpam-1242	485	1	moreover	moreover	ADV
ejpam-1242	485	2	,	,	PUNCT
ejpam-1242	485	3	by	by	ADP
ejpam-1242	485	4	theorem	theorem	ADJ
ejpam-1242	485	5	8	8	NUM
ejpam-1242	485	6	,	,	PUNCT
ejpam-1242	485	7	rad(n	rad(n	NOUN
ejpam-1242	485	8	)	)	PUNCT
ejpam-1242	485	9	=	=	SYM
ejpam-1242	485	10	rad(a)m	rad(a)m	NOUN
ejpam-1242	485	11	,	,	PUNCT
ejpam-1242	485	12	where	where	SCONJ
ejpam-1242	485	13	a	a	DET
ejpam-1242	485	14	=	=	X
ejpam-1242	485	15	(	(	PUNCT
ejpam-1242	485	16	n	n	NOUN
ejpam-1242	485	17	:	:	PUNCT
ejpam-1242	485	18	m	m	X
ejpam-1242	485	19	)	)	PUNCT
ejpam-1242	485	20	.	.	PUNCT
ejpam-1242	486	1	by	by	ADP
ejpam-1242	486	2	assumption	assumption	NOUN
ejpam-1242	486	3	,	,	PUNCT
ejpam-1242	486	4	there	there	PRON
ejpam-1242	486	5	exists	exist	VERB
ejpam-1242	486	6	a	a	DET
ejpam-1242	486	7	finite	finite	NOUN
ejpam-1242	486	8	subset	subset	VERB
ejpam-1242	486	9	j	j	PROPN
ejpam-1242	486	10	of	of	ADP
ejpam-1242	486	11	i	i	PROPN
ejpam-1242	486	12	and	and	CCONJ
ejpam-1242	486	13	ri	ri	PROPN
ejpam-1242	486	14	∈	∈	PROPN
ejpam-1242	486	15	rad(a	rad(a	PROPN
ejpam-1242	486	16	)	)	PUNCT
ejpam-1242	486	17	(	(	PUNCT
ejpam-1242	486	18	i	i	PROPN
ejpam-1242	486	19	∈	∈	PROPN
ejpam-1242	486	20	j	j	PROPN
ejpam-1242	486	21	)	)	PUNCT
ejpam-1242	486	22	such	such	ADJ
ejpam-1242	486	23	that	that	SCONJ
ejpam-1242	486	24	m	m	VERB
ejpam-1242	486	25	=	=	PUNCT
ejpam-1242	486	26	∑	∑	PUNCT
ejpam-1242	486	27	t∈j	t∈j	PROPN
ejpam-1242	486	28	rt	rt	PROPN
ejpam-1242	486	29	mt	mt	PROPN
ejpam-1242	486	30	.	.	PUNCT
ejpam-1242	487	1	for	for	ADP
ejpam-1242	487	2	ri	ri	PROPN
ejpam-1242	487	3	∈	∈	PROPN
ejpam-1242	487	4	rad(a	rad(a	PROPN
ejpam-1242	487	5	)	)	PUNCT
ejpam-1242	487	6	,	,	PUNCT
ejpam-1242	487	7	there	there	PRON
ejpam-1242	487	8	is	be	VERB
ejpam-1242	487	9	a	a	DET
ejpam-1242	487	10	positive	positive	ADJ
ejpam-1242	487	11	integer	integer	NOUN
ejpam-1242	487	12	si	si	INTJ
ejpam-1242	488	1	such	such	ADJ
ejpam-1242	488	2	that	that	SCONJ
ejpam-1242	488	3	r	r	NOUN
ejpam-1242	489	1	si	si	NOUN
ejpam-1242	490	1	i	i	NOUN
ejpam-1242	490	2	∈	∈	PROPN
ejpam-1242	490	3	a.	a.	NOUN
ejpam-1242	490	4	if	if	SCONJ
ejpam-1242	490	5	s	s	VERB
ejpam-1242	490	6	=	=	PUNCT
ejpam-1242	490	7	∑	∑	PUNCT
ejpam-1242	490	8	i∈j	i∈j	NOUN
ejpam-1242	490	9	si	si	NOUN
ejpam-1242	490	10	,	,	PUNCT
ejpam-1242	490	11	then	then	ADV
ejpam-1242	490	12	r	r	NOUN
ejpam-1242	490	13	s	s	VERB
ejpam-1242	490	14	i	i	PRON
ejpam-1242	490	15	∈	∈	PROPN
ejpam-1242	490	16	a	a	PRON
ejpam-1242	490	17	for	for	ADP
ejpam-1242	490	18	every	every	DET
ejpam-1242	490	19	i	i	PROPN
ejpam-1242	490	20	∈	∈	PROPN
ejpam-1242	490	21	j	j	PROPN
ejpam-1242	490	22	.	.	PUNCT
ejpam-1242	491	1	for	for	ADP
ejpam-1242	491	2	each	each	DET
ejpam-1242	491	3	i	i	PRON
ejpam-1242	491	4	∈	∈	PROPN
ejpam-1242	491	5	j	j	NOUN
ejpam-1242	491	6	,	,	PUNCT
ejpam-1242	491	7	there	there	PRON
ejpam-1242	491	8	exists	exist	VERB
ejpam-1242	491	9	a	a	DET
ejpam-1242	491	10	strong	strong	ADJ
ejpam-1242	491	11	ideal	ideal	ADJ
ejpam-1242	491	12	ii	ii	NOUN
ejpam-1242	491	13	of	of	ADP
ejpam-1242	491	14	r	r	NOUN
ejpam-1242	492	1	such	such	ADJ
ejpam-1242	492	2	that	that	DET
ejpam-1242	492	3	rmi	rmi	NOUN
ejpam-1242	492	4	=	=	SYM
ejpam-1242	492	5	ii	ii	NOUN
ejpam-1242	492	6	m	m	NOUN
ejpam-1242	492	7	;	;	PUNCT
ejpam-1242	492	8	so	so	ADV
ejpam-1242	492	9	by	by	ADP
ejpam-1242	492	10	proposition	proposition	NOUN
ejpam-1242	492	11	3	3	NUM
ejpam-1242	492	12	(	(	PUNCT
ejpam-1242	492	13	ii	ii	NOUN
ejpam-1242	492	14	)	)	PUNCT
ejpam-1242	492	15	,	,	PUNCT
ejpam-1242	492	16	m	m	VERB
ejpam-1242	492	17	∈	∈	PRON
ejpam-1242	492	18	∑	∑	PUNCT
ejpam-1242	492	19	i∈j	i∈j	NOUN
ejpam-1242	492	20	ri	ri	PROPN
ejpam-1242	492	21	ii	ii	PROPN
ejpam-1242	492	22	m	m	VERB
ejpam-1242	492	23	=	=	PUNCT
ejpam-1242	493	1	(	(	PUNCT
ejpam-1242	493	2	∑	∑	INTJ
ejpam-1242	493	3	i∈j	i∈j	NOUN
ejpam-1242	493	4	(	(	PUNCT
ejpam-1242	493	5	ri	ri	PROPN
ejpam-1242	493	6	ii))m	ii))m	PROPN
ejpam-1242	493	7	.	.	PUNCT
ejpam-1242	494	1	thus	thus	ADV
ejpam-1242	494	2	ms	ms	PROPN
ejpam-1242	494	3	⊆	⊆	NUM
ejpam-1242	494	4	(	(	PUNCT
ejpam-1242	494	5	∑	∑	PUNCT
ejpam-1242	494	6	i∈j	i∈j	NOUN
ejpam-1242	494	7	(	(	PUNCT
ejpam-1242	494	8	ri	ri	PROPN
ejpam-1242	494	9	ii	ii	PROPN
ejpam-1242	494	10	)	)	PUNCT
ejpam-1242	494	11	s)m	s)m	ADJ
ejpam-1242	494	12	⊆	⊆	NUM
ejpam-1242	494	13	am	am	NOUN
ejpam-1242	494	14	.	.	PUNCT
ejpam-1242	495	1	therefore	therefore	ADV
ejpam-1242	495	2	theorem	theorem	ADJ
ejpam-1242	495	3	10	10	NUM
ejpam-1242	495	4	gives	give	VERB
ejpam-1242	495	5	v	v	NOUN
ejpam-1242	495	6	(	(	PUNCT
ejpam-1242	495	7	n	n	CCONJ
ejpam-1242	495	8	)	)	PUNCT
ejpam-1242	496	1	=	=	SYM
ejpam-1242	496	2	⋂	⋂	PROPN
ejpam-1242	496	3	i∈i	i∈i	ADJ
ejpam-1242	496	4	v	v	NOUN
ejpam-1242	496	5	(	(	PUNCT
ejpam-1242	496	6	rmi)⊆	rmi)⊆	NUM
ejpam-1242	496	7	⋂	⋂	PROPN
ejpam-1242	496	8	i∈j	i∈j	NOUN
ejpam-1242	496	9	v	v	NOUN
ejpam-1242	496	10	(	(	PUNCT
ejpam-1242	496	11	rmi)⊆	rmi)⊆	NUM
ejpam-1242	496	12	v	v	NOUN
ejpam-1242	496	13	(	(	PUNCT
ejpam-1242	496	14	m	m	NOUN
ejpam-1242	496	15	)	)	PUNCT
ejpam-1242	497	1	=	=	SYM
ejpam-1242	497	2	v	v	X
ejpam-1242	497	3	(	(	PUNCT
ejpam-1242	497	4	rm	rm	NOUN
ejpam-1242	497	5	)	)	PUNCT
ejpam-1242	497	6	=	=	NOUN
ejpam-1242	497	7	v	v	X
ejpam-1242	497	8	(	(	PUNCT
ejpam-1242	497	9	ms	ms	NOUN
ejpam-1242	497	10	)	)	PUNCT
ejpam-1242	497	11	.	.	PUNCT
ejpam-1242	498	1	taking	take	VERB
ejpam-1242	498	2	complements	complement	NOUN
ejpam-1242	498	3	,	,	PUNCT
ejpam-1242	498	4	we	we	PRON
ejpam-1242	498	5	have	have	VERB
ejpam-1242	498	6	xm	xm	PROPN
ejpam-1242	498	7	⊆	⊆	NUM
ejpam-1242	498	8	⋃	⋃	NOUN
ejpam-1242	498	9	i∈j	i∈j	NOUN
ejpam-1242	498	10	xmi	xmi	NOUN
ejpam-1242	498	11	,	,	PUNCT
ejpam-1242	498	12	and	and	CCONJ
ejpam-1242	498	13	so	so	ADV
ejpam-1242	498	14	the	the	DET
ejpam-1242	498	15	proof	proof	NOUN
ejpam-1242	498	16	is	be	AUX
ejpam-1242	498	17	complete	complete	ADJ
ejpam-1242	498	18	.	.	PUNCT
ejpam-1242	499	1	theorem	theorem	VERB
ejpam-1242	499	2	11	11	NUM
ejpam-1242	499	3	.	.	PUNCT
ejpam-1242	500	1	let	let	VERB
ejpam-1242	500	2	m	m	PRON
ejpam-1242	500	3	be	be	AUX
ejpam-1242	500	4	a	a	DET
ejpam-1242	500	5	very	very	ADV
ejpam-1242	500	6	strong	strong	ADJ
ejpam-1242	500	7	multiplication	multiplication	NOUN
ejpam-1242	500	8	semimodule	semimodule	NOUN
ejpam-1242	500	9	over	over	ADP
ejpam-1242	500	10	a	a	DET
ejpam-1242	500	11	semiring	semire	VERB
ejpam-1242	500	12	r.	r.	PROPN
ejpam-1242	500	13	then	then	ADV
ejpam-1242	500	14	an	an	DET
ejpam-1242	500	15	open	open	ADJ
ejpam-1242	500	16	set	set	NOUN
ejpam-1242	500	17	of	of	ADP
ejpam-1242	500	18	x	x	PUNCT
ejpam-1242	500	19	is	be	AUX
ejpam-1242	500	20	compact	compact	ADJ
ejpam-1242	500	21	if	if	SCONJ
ejpam-1242	501	1	and	and	CCONJ
ejpam-1242	501	2	only	only	ADV
ejpam-1242	501	3	if	if	SCONJ
ejpam-1242	501	4	it	it	PRON
ejpam-1242	501	5	is	be	AUX
ejpam-1242	501	6	a	a	DET
ejpam-1242	501	7	finite	finite	ADJ
ejpam-1242	501	8	union	union	NOUN
ejpam-1242	501	9	of	of	ADP
ejpam-1242	501	10	basic	basic	ADJ
ejpam-1242	501	11	open	open	ADJ
ejpam-1242	501	12	sets	set	NOUN
ejpam-1242	501	13	.	.	PUNCT
ejpam-1242	502	1	proof	proof	NOUN
ejpam-1242	502	2	.	.	PUNCT
ejpam-1242	503	1	apply	apply	VERB
ejpam-1242	503	2	lemma	lemma	PROPN
ejpam-1242	503	3	7	7	NUM
ejpam-1242	503	4	and	and	CCONJ
ejpam-1242	503	5	proposition	proposition	NOUN
ejpam-1242	503	6	4	4	NUM
ejpam-1242	503	7	.	.	PUNCT
ejpam-1242	503	8	corollary	corollary	ADJ
ejpam-1242	503	9	2	2	NUM
ejpam-1242	503	10	.	.	PUNCT
ejpam-1242	504	1	let	let	VERB
ejpam-1242	504	2	m	m	PRON
ejpam-1242	504	3	be	be	AUX
ejpam-1242	504	4	a	a	DET
ejpam-1242	504	5	finitely	finitely	ADV
ejpam-1242	504	6	generated	generate	VERB
ejpam-1242	504	7	very	very	ADV
ejpam-1242	504	8	strong	strong	ADJ
ejpam-1242	504	9	multiplication	multiplication	NOUN
ejpam-1242	504	10	semimodule	semimodule	NOUN
ejpam-1242	504	11	over	over	ADP
ejpam-1242	504	12	a	a	DET
ejpam-1242	504	13	semiring	semire	VERB
ejpam-1242	504	14	r.	r.	PROPN
ejpam-1242	504	15	then	then	ADV
ejpam-1242	504	16	x	x	PRON
ejpam-1242	504	17	is	be	AUX
ejpam-1242	504	18	compact	compact	ADJ
ejpam-1242	504	19	.	.	PUNCT
ejpam-1242	505	1	proof	proof	NOUN
ejpam-1242	505	2	.	.	PUNCT
ejpam-1242	506	1	let	let	VERB
ejpam-1242	506	2	m	m	PRON
ejpam-1242	506	3	=	=	ADJ
ejpam-1242	506	4	∑n	∑n	PROPN
ejpam-1242	506	5	i=1	i=1	PROPN
ejpam-1242	506	6	rmi	rmi	PROPN
ejpam-1242	506	7	.	.	PUNCT
ejpam-1242	507	1	then	then	ADV
ejpam-1242	507	2	v	v	X
ejpam-1242	507	3	(	(	PUNCT
ejpam-1242	507	4	m	m	NOUN
ejpam-1242	507	5	)	)	PUNCT
ejpam-1242	507	6	=	=	SYM
ejpam-1242	507	7	;	;	PUNCT
ejpam-1242	507	8	;	;	PUNCT
ejpam-1242	507	9	hence	hence	ADV
ejpam-1242	507	10	xm	xm	PUNCT
ejpam-1242	508	1	=	=	PUNCT
ejpam-1242	508	2	x	x	X
ejpam-1242	508	3	,	,	PUNCT
ejpam-1242	508	4	that	that	ADV
ejpam-1242	508	5	is	is	ADV
ejpam-1242	508	6	,	,	PUNCT
ejpam-1242	508	7	x	x	SYM
ejpam-1242	508	8	=	=	SYM
ejpam-1242	508	9	⋃n	⋃n	PROPN
ejpam-1242	508	10	i=1	i=1	PROPN
ejpam-1242	508	11	xmi	xmi	PROPN
ejpam-1242	508	12	.	.	PUNCT
ejpam-1242	509	1	thus	thus	ADV
ejpam-1242	509	2	x	x	X
ejpam-1242	509	3	is	be	AUX
ejpam-1242	509	4	compact	compact	ADJ
ejpam-1242	509	5	.	.	PUNCT
ejpam-1242	510	1	question	question	NOUN
ejpam-1242	510	2	:	:	PUNCT
ejpam-1242	510	3	assume	assume	VERB
ejpam-1242	510	4	that	that	SCONJ
ejpam-1242	510	5	m	m	PROPN
ejpam-1242	510	6	is	be	AUX
ejpam-1242	510	7	a	a	DET
ejpam-1242	510	8	very	very	ADV
ejpam-1242	510	9	strong	strong	ADJ
ejpam-1242	510	10	multiplication	multiplication	NOUN
ejpam-1242	510	11	semimodule	semimodule	NOUN
ejpam-1242	510	12	over	over	ADP
ejpam-1242	510	13	a	a	DET
ejpam-1242	510	14	semiring	semire	VERB
ejpam-1242	510	15	r	r	NOUN
ejpam-1242	510	16	and	and	CCONJ
ejpam-1242	510	17	let	let	VERB
ejpam-1242	510	18	x	x	PRON
ejpam-1242	510	19	be	be	AUX
ejpam-1242	510	20	compact	compact	ADJ
ejpam-1242	510	21	.	.	PUNCT
ejpam-1242	511	1	is	be	AUX
ejpam-1242	511	2	m	m	AUX
ejpam-1242	511	3	finitely	finitely	ADV
ejpam-1242	511	4	generated	generate	VERB
ejpam-1242	511	5	?	?	PUNCT
ejpam-1242	512	1	references	reference	NOUN
ejpam-1242	512	2	[	[	X
ejpam-1242	512	3	1	1	X
ejpam-1242	512	4	]	]	PUNCT
ejpam-1242	512	5	p.	p.	NOUN
ejpam-1242	512	6	a.	a.	PROPN
ejpam-1242	512	7	allen	allen	PROPN
ejpam-1242	512	8	,	,	PUNCT
ejpam-1242	512	9	ideal	ideal	ADJ
ejpam-1242	512	10	theory	theory	NOUN
ejpam-1242	512	11	in	in	ADP
ejpam-1242	512	12	semirings	semiring	NOUN
ejpam-1242	512	13	,	,	PUNCT
ejpam-1242	512	14	dissertation	dissertation	NOUN
ejpam-1242	512	15	,	,	PUNCT
ejpam-1242	512	16	texas	texas	PROPN
ejpam-1242	512	17	christian	christian	PROPN
ejpam-1242	512	18	university	university	PROPN
ejpam-1242	512	19	,	,	PUNCT
ejpam-1242	512	20	1967	1967	NUM
ejpam-1242	512	21	.	.	PUNCT
ejpam-1242	513	1	[	[	X
ejpam-1242	513	2	2	2	X
ejpam-1242	513	3	]	]	PUNCT
ejpam-1242	513	4	p.	p.	NOUN
ejpam-1242	513	5	a.	a.	NOUN
ejpam-1242	513	6	allen	allen	PROPN
ejpam-1242	513	7	,	,	PUNCT
ejpam-1242	513	8	a	a	DET
ejpam-1242	513	9	fundamental	fundamental	ADJ
ejpam-1242	513	10	theorem	theorem	NOUN
ejpam-1242	513	11	of	of	ADP
ejpam-1242	513	12	homomorphisms	homomorphism	NOUN
ejpam-1242	513	13	for	for	ADP
ejpam-1242	513	14	semirings	semiring	NOUN
ejpam-1242	513	15	,	,	PUNCT
ejpam-1242	513	16	proc	proc	PROPN
ejpam-1242	513	17	.	.	PUNCT
ejpam-1242	514	1	amer	amer	PROPN
ejpam-1242	514	2	.	.	PUNCT
ejpam-1242	514	3	math	math	PROPN
ejpam-1242	514	4	.	.	PUNCT
ejpam-1242	515	1	soc	soc	PROPN
ejpam-1242	515	2	.	.	PUNCT
ejpam-1242	516	1	21	21	NUM
ejpam-1242	516	2	,	,	PUNCT
ejpam-1242	516	3	412	412	NUM
ejpam-1242	516	4	-	-	SYM
ejpam-1242	516	5	416	416	NUM
ejpam-1242	516	6	.	.	PUNCT
ejpam-1242	517	1	1969	1969	NUM
ejpam-1242	517	2	.	.	PUNCT
ejpam-1242	518	1	[	[	X
ejpam-1242	518	2	3	3	NUM
ejpam-1242	518	3	]	]	X
ejpam-1242	518	4	r.	r.	PROPN
ejpam-1242	518	5	ameri	ameri	PROPN
ejpam-1242	518	6	,	,	PUNCT
ejpam-1242	518	7	some	some	DET
ejpam-1242	518	8	properties	property	NOUN
ejpam-1242	518	9	of	of	ADP
ejpam-1242	518	10	zariski	zariski	ADJ
ejpam-1242	518	11	topology	topology	NOUN
ejpam-1242	518	12	of	of	ADP
ejpam-1242	518	13	multiplication	multiplication	NOUN
ejpam-1242	518	14	modules	module	NOUN
ejpam-1242	518	15	,	,	PUNCT
ejpam-1242	518	16	houston	houston	PROPN
ejpam-1242	518	17	j.	j.	PROPN
ejpam-1242	518	18	of	of	ADP
ejpam-1242	518	19	math	math	NOUN
ejpam-1242	518	20	.	.	PUNCT
ejpam-1242	519	1	38(2	38(2	NUM
ejpam-1242	519	2	)	)	PUNCT
ejpam-1242	519	3	,	,	PUNCT
ejpam-1242	519	4	337	337	NUM
ejpam-1242	519	5	-	-	SYM
ejpam-1242	519	6	344	344	NUM
ejpam-1242	519	7	.	.	PUNCT
ejpam-1242	519	8	2010	2010	NUM
ejpam-1242	519	9	.	.	PUNCT
ejpam-1242	520	1	references	reference	NOUN
ejpam-1242	520	2	264	264	NUM
ejpam-1242	520	3	[	[	X
ejpam-1242	520	4	4	4	NUM
ejpam-1242	520	5	]	]	X
ejpam-1242	520	6	r.	r.	PROPN
ejpam-1242	520	7	ameri	ameri	PROPN
ejpam-1242	520	8	,	,	PUNCT
ejpam-1242	520	9	on	on	ADP
ejpam-1242	520	10	the	the	DET
ejpam-1242	520	11	prime	prime	ADJ
ejpam-1242	520	12	submodules	submodule	NOUN
ejpam-1242	520	13	of	of	ADP
ejpam-1242	520	14	multiplication	multiplication	NOUN
ejpam-1242	520	15	modules	module	NOUN
ejpam-1242	520	16	,	,	PUNCT
ejpam-1242	520	17	inter	inter	PROPN
ejpam-1242	520	18	.	.	PUNCT
ejpam-1242	521	1	j.	j.	PROPN
ejpam-1242	521	2	of	of	ADP
ejpam-1242	521	3	mathematics	mathematics	PROPN
ejpam-1242	521	4	and	and	CCONJ
ejpam-1242	521	5	mathematical	mathematical	ADJ
ejpam-1242	521	6	sciences	science	NOUN
ejpam-1242	521	7	27	27	NUM
ejpam-1242	521	8	,	,	PUNCT
ejpam-1242	521	9	1715	1715	NUM
ejpam-1242	521	10	-	-	SYM
ejpam-1242	521	11	1724	1724	NUM
ejpam-1242	521	12	.	.	PUNCT
ejpam-1242	522	1	2003	2003	NUM
ejpam-1242	522	2	.	.	PUNCT
ejpam-1242	523	1	[	[	X
ejpam-1242	523	2	5	5	NUM
ejpam-1242	523	3	]	]	PUNCT
ejpam-1242	523	4	m.	m.	PROPN
ejpam-1242	523	5	f.	f.	PROPN
ejpam-1242	523	6	atiyah	atiyah	PROPN
ejpam-1242	523	7	and	and	CCONJ
ejpam-1242	523	8	i.	i.	PROPN
ejpam-1242	523	9	g.	g.	PROPN
ejpam-1242	523	10	macdonald	macdonald	PROPN
ejpam-1242	523	11	,	,	PUNCT
ejpam-1242	523	12	introduction	introduction	NOUN
ejpam-1242	523	13	to	to	ADP
ejpam-1242	523	14	commutative	commutative	ADJ
ejpam-1242	523	15	algebra	algebra	PROPN
ejpam-1242	523	16	,	,	PUNCT
ejpam-1242	523	17	addison	addison	PROPN
ejpam-1242	523	18	wesley	wesley	PROPN
ejpam-1242	523	19	publishing	publishing	PROPN
ejpam-1242	523	20	company	company	NOUN
ejpam-1242	523	21	,	,	PUNCT
ejpam-1242	523	22	1969	1969	NUM
ejpam-1242	523	23	.	.	PUNCT
ejpam-1242	524	1	[	[	X
ejpam-1242	524	2	6	6	NUM
ejpam-1242	524	3	]	]	PUNCT
ejpam-1242	524	4	z.	z.	PROPN
ejpam-1242	524	5	el	el	PROPN
ejpam-1242	524	6	-	-	PUNCT
ejpam-1242	524	7	bast	bast	NOUN
ejpam-1242	524	8	and	and	CCONJ
ejpam-1242	524	9	p.	p.	PROPN
ejpam-1242	524	10	f.	f.	PROPN
ejpam-1242	524	11	smith	smith	PROPN
ejpam-1242	524	12	,	,	PUNCT
ejpam-1242	524	13	multiplication	multiplication	NOUN
ejpam-1242	524	14	modules	module	NOUN
ejpam-1242	524	15	,	,	PUNCT
ejpam-1242	524	16	comm	comm	NOUN
ejpam-1242	524	17	.	.	PUNCT
ejpam-1242	525	1	algebra	algebra	NOUN
ejpam-1242	525	2	16	16	NUM
ejpam-1242	525	3	,	,	PUNCT
ejpam-1242	525	4	755	755	NUM
ejpam-1242	525	5	-	-	SYM
ejpam-1242	525	6	779	779	NUM
ejpam-1242	525	7	.	.	NUM
ejpam-1242	525	8	1988	1988	NUM
ejpam-1242	525	9	.	.	PUNCT
ejpam-1242	526	1	[	[	X
ejpam-1242	526	2	7	7	X
ejpam-1242	526	3	]	]	PUNCT
ejpam-1242	526	4	j.	j.	PROPN
ejpam-1242	526	5	n.	n.	PROPN
ejpam-1242	526	6	chaudhari	chaudhari	PROPN
ejpam-1242	526	7	and	and	CCONJ
ejpam-1242	526	8	d.	d.	PROPN
ejpam-1242	526	9	bonde	bonde	PROPN
ejpam-1242	526	10	,	,	PUNCT
ejpam-1242	526	11	on	on	ADP
ejpam-1242	526	12	partitioning	partition	VERB
ejpam-1242	526	13	and	and	CCONJ
ejpam-1242	526	14	subtractive	subtractive	ADJ
ejpam-1242	526	15	subsemimodules	subsemimodule	NOUN
ejpam-1242	526	16	of	of	ADP
ejpam-1242	526	17	semimodules	semimodule	NOUN
ejpam-1242	526	18	over	over	ADP
ejpam-1242	526	19	semirings	semiring	NOUN
ejpam-1242	526	20	,	,	PUNCT
ejpam-1242	526	21	kyungpook	kyungpook	PROPN
ejpam-1242	526	22	math	math	NOUN
ejpam-1242	526	23	.	.	PUNCT
ejpam-1242	527	1	j.	j.	PROPN
ejpam-1242	527	2	50	50	NUM
ejpam-1242	527	3	,	,	PUNCT
ejpam-1242	527	4	329	329	NUM
ejpam-1242	527	5	-	-	SYM
ejpam-1242	527	6	336	336	NUM
ejpam-1242	527	7	.	.	PUNCT
ejpam-1242	527	8	2010	2010	NUM
ejpam-1242	527	9	.	.	PUNCT
ejpam-1242	528	1	[	[	X
ejpam-1242	528	2	8	8	X
ejpam-1242	528	3	]	]	PUNCT
ejpam-1242	528	4	s.	s.	PROPN
ejpam-1242	528	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	528	6	atani	atani	PROPN
ejpam-1242	528	7	,	,	PUNCT
ejpam-1242	528	8	multiplication	multiplication	NOUN
ejpam-1242	528	9	modules	module	NOUN
ejpam-1242	528	10	and	and	CCONJ
ejpam-1242	528	11	related	related	ADJ
ejpam-1242	528	12	results	result	NOUN
ejpam-1242	528	13	,	,	PUNCT
ejpam-1242	528	14	archivium	archivium	NOUN
ejpam-1242	528	15	mathematicum	mathematicum	NOUN
ejpam-1242	528	16	40	40	NUM
ejpam-1242	528	17	,	,	PUNCT
ejpam-1242	528	18	407	407	NUM
ejpam-1242	528	19	-	-	SYM
ejpam-1242	528	20	414	414	NUM
ejpam-1242	528	21	.	.	PUNCT
ejpam-1242	528	22	2004	2004	NUM
ejpam-1242	528	23	.	.	PUNCT
ejpam-1242	529	1	[	[	X
ejpam-1242	529	2	9	9	NUM
ejpam-1242	529	3	]	]	PUNCT
ejpam-1242	529	4	s.	s.	PROPN
ejpam-1242	529	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	529	6	atani	atani	PROPN
ejpam-1242	529	7	,	,	PUNCT
ejpam-1242	529	8	submodules	submodule	NOUN
ejpam-1242	529	9	of	of	ADP
ejpam-1242	529	10	multiplication	multiplication	NOUN
ejpam-1242	529	11	modules	module	NOUN
ejpam-1242	529	12	,	,	PUNCT
ejpam-1242	529	13	taiwanese	taiwanese	ADJ
ejpam-1242	529	14	j.	j.	PROPN
ejpam-1242	529	15	of	of	ADP
ejpam-1242	529	16	math	math	PROPN
ejpam-1242	529	17	.	.	PUNCT
ejpam-1242	530	1	9(3	9(3	NUM
ejpam-1242	530	2	)	)	PUNCT
ejpam-1242	530	3	,	,	PUNCT
ejpam-1242	530	4	385	385	NUM
ejpam-1242	530	5	-	-	SYM
ejpam-1242	530	6	396	396	NUM
ejpam-1242	530	7	.	.	PUNCT
ejpam-1242	531	1	2005	2005	NUM
ejpam-1242	531	2	.	.	PUNCT
ejpam-1242	532	1	[	[	X
ejpam-1242	532	2	10	10	NUM
ejpam-1242	532	3	]	]	X
ejpam-1242	532	4	s.	s.	PROPN
ejpam-1242	532	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	532	6	atani	atani	PROPN
ejpam-1242	532	7	,	,	PUNCT
ejpam-1242	532	8	the	the	DET
ejpam-1242	532	9	ideal	ideal	ADJ
ejpam-1242	532	10	theory	theory	NOUN
ejpam-1242	532	11	in	in	ADP
ejpam-1242	532	12	quotients	quotient	NOUN
ejpam-1242	532	13	of	of	ADP
ejpam-1242	532	14	commutative	commutative	ADJ
ejpam-1242	532	15	semirings	semiring	NOUN
ejpam-1242	532	16	.	.	PUNCT
ejpam-1242	533	1	glas	glas	PROPN
ejpam-1242	533	2	.	.	PUNCT
ejpam-1242	534	1	math	math	NOUN
ejpam-1242	534	2	.	.	PUNCT
ejpam-1242	535	1	42	42	NUM
ejpam-1242	535	2	,	,	PUNCT
ejpam-1242	535	3	301	301	NUM
ejpam-1242	535	4	-	-	SYM
ejpam-1242	535	5	308	308	NUM
ejpam-1242	535	6	.	.	PUNCT
ejpam-1242	536	1	2007	2007	NUM
ejpam-1242	536	2	.	.	PUNCT
ejpam-1242	537	1	[	[	X
ejpam-1242	537	2	11	11	NUM
ejpam-1242	537	3	]	]	PUNCT
ejpam-1242	537	4	s.	s.	PROPN
ejpam-1242	537	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	537	6	atani	atani	PROPN
ejpam-1242	537	7	and	and	CCONJ
ejpam-1242	537	8	r.	r.	PROPN
ejpam-1242	537	9	ebrahimi	ebrahimi	PROPN
ejpam-1242	537	10	atani	atani	PROPN
ejpam-1242	537	11	,	,	PUNCT
ejpam-1242	537	12	very	very	ADV
ejpam-1242	537	13	strong	strong	ADJ
ejpam-1242	537	14	multiplication	multiplication	NOUN
ejpam-1242	537	15	ideals	ideal	NOUN
ejpam-1242	537	16	and	and	CCONJ
ejpam-1242	537	17	the	the	DET
ejpam-1242	537	18	ideal	ideal	NOUN
ejpam-1242	537	19	θ(i	θ(i	PROPN
ejpam-1242	537	20	)	)	PUNCT
ejpam-1242	537	21	.	.	PUNCT
ejpam-1242	538	1	glas	glas	PROPN
ejpam-1242	538	2	.	.	PUNCT
ejpam-1242	539	1	math	math	NOUN
ejpam-1242	539	2	.	.	PUNCT
ejpam-1242	540	1	45	45	NUM
ejpam-1242	540	2	,	,	PUNCT
ejpam-1242	540	3	395	395	NUM
ejpam-1242	540	4	-	-	SYM
ejpam-1242	540	5	406	406	NUM
ejpam-1242	540	6	.	.	PUNCT
ejpam-1242	541	1	2010	2010	NUM
ejpam-1242	541	2	.	.	PUNCT
ejpam-1242	542	1	[	[	X
ejpam-1242	542	2	12	12	NUM
ejpam-1242	542	3	]	]	PUNCT
ejpam-1242	542	4	r.	r.	PROPN
ejpam-1242	542	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	542	6	atani	atani	PROPN
ejpam-1242	542	7	and	and	CCONJ
ejpam-1242	542	8	s.	s.	PROPN
ejpam-1242	542	9	ebrahimi	ebrahimi	PROPN
ejpam-1242	542	10	atani	atani	PROPN
ejpam-1242	542	11	,	,	PUNCT
ejpam-1242	542	12	ideal	ideal	ADJ
ejpam-1242	542	13	theory	theory	NOUN
ejpam-1242	542	14	in	in	ADP
ejpam-1242	542	15	commutative	commutative	ADJ
ejpam-1242	542	16	semirings	semiring	NOUN
ejpam-1242	542	17	.	.	PUNCT
ejpam-1242	543	1	bul	bul	PROPN
ejpam-1242	543	2	.	.	PUNCT
ejpam-1242	544	1	acad	acad	PROPN
ejpam-1242	544	2	.	.	PUNCT
ejpam-1242	545	1	stiinte	stiinte	PROPN
ejpam-1242	545	2	repub	repub	PROPN
ejpam-1242	545	3	.	.	PUNCT
ejpam-1242	546	1	mold	mold	NOUN
ejpam-1242	546	2	.	.	PUNCT
ejpam-1242	547	1	mat	mat	NOUN
ejpam-1242	547	2	.	.	NOUN
ejpam-1242	547	3	2	2	NUM
ejpam-1242	547	4	,	,	PUNCT
ejpam-1242	547	5	14	14	NUM
ejpam-1242	547	6	-	-	SYM
ejpam-1242	547	7	23	23	NUM
ejpam-1242	547	8	.	.	PUNCT
ejpam-1242	548	1	2008	2008	NUM
ejpam-1242	548	2	.	.	PUNCT
ejpam-1242	549	1	[	[	X
ejpam-1242	549	2	13	13	NUM
ejpam-1242	549	3	]	]	PUNCT
ejpam-1242	549	4	r.	r.	PROPN
ejpam-1242	549	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	549	6	atani	atani	PROPN
ejpam-1242	549	7	and	and	CCONJ
ejpam-1242	549	8	s.	s.	PROPN
ejpam-1242	549	9	ebrahimi	ebrahimi	PROPN
ejpam-1242	549	10	atani	atani	PROPN
ejpam-1242	549	11	,	,	PUNCT
ejpam-1242	549	12	subsemimodules	subsemimodule	NOUN
ejpam-1242	549	13	of	of	ADP
ejpam-1242	549	14	semimodules	semimodule	NOUN
ejpam-1242	549	15	.	.	PUNCT
ejpam-1242	550	1	bul	bul	PROPN
ejpam-1242	550	2	.	.	PUNCT
ejpam-1242	551	1	acad	acad	PROPN
ejpam-1242	551	2	.	.	PUNCT
ejpam-1242	552	1	stiinte	stiinte	PROPN
ejpam-1242	552	2	repub	repub	PROPN
ejpam-1242	552	3	.	.	PUNCT
ejpam-1242	553	1	mold	mold	NOUN
ejpam-1242	553	2	.	.	PUNCT
ejpam-1242	554	1	mat	mat	NOUN
ejpam-1242	554	2	.	.	NOUN
ejpam-1242	554	3	63	63	NUM
ejpam-1242	554	4	,	,	PUNCT
ejpam-1242	554	5	20	20	NUM
ejpam-1242	554	6	-	-	SYM
ejpam-1242	554	7	30	30	NUM
ejpam-1242	554	8	.	.	PUNCT
ejpam-1242	555	1	2010	2010	NUM
ejpam-1242	555	2	.	.	PUNCT
ejpam-1242	556	1	[	[	X
ejpam-1242	556	2	14	14	NUM
ejpam-1242	556	3	]	]	X
ejpam-1242	556	4	s.	s.	PROPN
ejpam-1242	556	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	556	6	atani	atani	PROPN
ejpam-1242	556	7	and	and	CCONJ
ejpam-1242	556	8	r.	r.	PROPN
ejpam-1242	556	9	ebrahimi	ebrahimi	PROPN
ejpam-1242	556	10	atani	atani	PROPN
ejpam-1242	556	11	,	,	PUNCT
ejpam-1242	556	12	some	some	DET
ejpam-1242	556	13	remarks	remark	NOUN
ejpam-1242	556	14	on	on	ADP
ejpam-1242	556	15	partitioning	partitioning	PROPN
ejpam-1242	556	16	semirings	semiring	NOUN
ejpam-1242	556	17	.	.	PUNCT
ejpam-1242	557	1	an	an	DET
ejpam-1242	557	2	.	.	PUNCT
ejpam-1242	557	3	st	st	PROPN
ejpam-1242	557	4	.	.	PROPN
ejpam-1242	557	5	univ	univ	PROPN
ejpam-1242	557	6	.	.	PUNCT
ejpam-1242	558	1	ovidius	ovidius	PROPN
ejpam-1242	558	2	constanta	constanta	PROPN
ejpam-1242	558	3	18(1	18(1	NOUN
ejpam-1242	558	4	)	)	PUNCT
ejpam-1242	558	5	,	,	PUNCT
ejpam-1242	558	6	49	49	NUM
ejpam-1242	558	7	-	-	SYM
ejpam-1242	558	8	62	62	NUM
ejpam-1242	558	9	.	.	PUNCT
ejpam-1242	559	1	2010	2010	NUM
ejpam-1242	559	2	.	.	PUNCT
ejpam-1242	560	1	[	[	X
ejpam-1242	560	2	15	15	NUM
ejpam-1242	560	3	]	]	X
ejpam-1242	560	4	s.	s.	PROPN
ejpam-1242	560	5	ebrahimi	ebrahimi	PROPN
ejpam-1242	560	6	atani	atani	PROPN
ejpam-1242	560	7	and	and	CCONJ
ejpam-1242	560	8	m.	m.	PROPN
ejpam-1242	560	9	shajari	shajari	PROPN
ejpam-1242	560	10	kohan	kohan	PROPN
ejpam-1242	560	11	,	,	PUNCT
ejpam-1242	560	12	a	a	DET
ejpam-1242	560	13	note	note	NOUN
ejpam-1242	560	14	on	on	ADP
ejpam-1242	560	15	finitely	finitely	ADV
ejpam-1242	560	16	generated	generate	VERB
ejpam-1242	560	17	multiplication	multiplication	NOUN
ejpam-1242	560	18	semimodules	semimodule	NOUN
ejpam-1242	560	19	over	over	ADP
ejpam-1242	560	20	commutative	commutative	ADJ
ejpam-1242	560	21	semirings	semiring	NOUN
ejpam-1242	560	22	.	.	PUNCT
ejpam-1242	561	1	inter	inter	PROPN
ejpam-1242	561	2	.	.	PUNCT
ejpam-1242	562	1	j.	j.	PROPN
ejpam-1242	562	2	of	of	ADP
ejpam-1242	562	3	algebra	algebra	PROPN
ejpam-1242	562	4	4(8	4(8	NUM
ejpam-1242	562	5	)	)	PUNCT
ejpam-1242	562	6	,	,	PUNCT
ejpam-1242	562	7	389	389	NUM
ejpam-1242	562	8	-	-	SYM
ejpam-1242	562	9	396	396	NUM
ejpam-1242	562	10	.	.	PUNCT
ejpam-1242	563	1	2010	2010	NUM
ejpam-1242	563	2	.	.	PUNCT
ejpam-1242	564	1	[	[	X
ejpam-1242	564	2	16	16	NUM
ejpam-1242	564	3	]	]	PUNCT
ejpam-1242	564	4	j.	j.	PROPN
ejpam-1242	564	5	s.	s.	PROPN
ejpam-1242	564	6	golan	golan	PROPN
ejpam-1242	564	7	,	,	PUNCT
ejpam-1242	564	8	semirings	semiring	NOUN
ejpam-1242	564	9	and	and	CCONJ
ejpam-1242	564	10	their	their	PRON
ejpam-1242	564	11	applications	application	NOUN
ejpam-1242	564	12	,	,	PUNCT
ejpam-1242	564	13	kluwer	kluwer	PROPN
ejpam-1242	564	14	academic	academic	PROPN
ejpam-1242	564	15	publisher	publisher	NOUN
ejpam-1242	564	16	dordrecht	dordrecht	PROPN
ejpam-1242	564	17	,	,	PUNCT
ejpam-1242	564	18	1999	1999	NUM
ejpam-1242	564	19	.	.	PUNCT
ejpam-1242	565	1	[	[	X
ejpam-1242	565	2	17	17	NUM
ejpam-1242	565	3	]	]	PUNCT
ejpam-1242	565	4	k.	k.	PROPN
ejpam-1242	565	5	glazek	glazek	PROPN
ejpam-1242	565	6	,	,	PUNCT
ejpam-1242	565	7	a	a	DET
ejpam-1242	565	8	guide	guide	NOUN
ejpam-1242	565	9	to	to	ADP
ejpam-1242	565	10	the	the	DET
ejpam-1242	565	11	literature	literature	NOUN
ejpam-1242	565	12	on	on	ADP
ejpam-1242	565	13	semirings	semiring	NOUN
ejpam-1242	565	14	and	and	CCONJ
ejpam-1242	565	15	their	their	PRON
ejpam-1242	565	16	applications	application	NOUN
ejpam-1242	565	17	in	in	ADP
ejpam-1242	565	18	mathematics	mathematic	NOUN
ejpam-1242	565	19	and	and	CCONJ
ejpam-1242	565	20	information	information	NOUN
ejpam-1242	565	21	sciences	science	NOUN
ejpam-1242	565	22	:	:	PUNCT
ejpam-1242	565	23	with	with	ADP
ejpam-1242	565	24	computer	computer	NOUN
ejpam-1242	565	25	bibliography	bibliography	NOUN
ejpam-1242	565	26	,	,	PUNCT
ejpam-1242	565	27	kluwer	kluwer	NOUN
ejpam-1242	565	28	acad	acad	PROPN
ejpam-1242	565	29	.	.	PUNCT
ejpam-1242	566	1	publ	publ	PROPN
ejpam-1242	566	2	.	.	PUNCT
ejpam-1242	566	3	,	,	PUNCT
ejpam-1242	566	4	dodrecht	dodrecht	PROPN
ejpam-1242	566	5	,	,	PUNCT
ejpam-1242	566	6	2002	2002	NUM
ejpam-1242	566	7	.	.	PUNCT
ejpam-1242	567	1	[	[	X
ejpam-1242	567	2	18	18	NUM
ejpam-1242	567	3	]	]	X
ejpam-1242	567	4	u.	u.	PROPN
ejpam-1242	567	5	hebisch	hebisch	PROPN
ejpam-1242	567	6	and	and	CCONJ
ejpam-1242	567	7	h.	h.	PROPN
ejpam-1242	567	8	j.	j.	PROPN
ejpam-1242	567	9	weinert	weinert	PROPN
ejpam-1242	567	10	,	,	PUNCT
ejpam-1242	567	11	semirings	semiring	NOUN
ejpam-1242	567	12	:	:	PUNCT
ejpam-1242	567	13	algebraic	algebraic	ADJ
ejpam-1242	567	14	theory	theory	NOUN
ejpam-1242	567	15	and	and	CCONJ
ejpam-1242	567	16	applications	application	NOUN
ejpam-1242	567	17	in	in	ADP
ejpam-1242	567	18	the	the	DET
ejpam-1242	567	19	computer	computer	NOUN
ejpam-1242	567	20	science	science	NOUN
ejpam-1242	567	21	,	,	PUNCT
ejpam-1242	567	22	world	world	NOUN
ejpam-1242	567	23	scientific	scientific	NOUN
ejpam-1242	567	24	,	,	PUNCT
ejpam-1242	567	25	1998	1998	NUM
ejpam-1242	567	26	.	.	PUNCT
ejpam-1242	568	1	[	[	X
ejpam-1242	568	2	19	19	NUM
ejpam-1242	568	3	]	]	X
ejpam-1242	568	4	c.	c.	PROPN
ejpam-1242	568	5	p.	p.	PROPN
ejpam-1242	568	6	lu	lu	PROPN
ejpam-1242	568	7	,	,	PUNCT
ejpam-1242	568	8	prime	prime	ADJ
ejpam-1242	568	9	submodules	submodule	NOUN
ejpam-1242	568	10	of	of	ADP
ejpam-1242	568	11	modules	module	NOUN
ejpam-1242	568	12	,	,	PUNCT
ejpam-1242	568	13	comment	comment	NOUN
ejpam-1242	568	14	.	.	PUNCT
ejpam-1242	569	1	math	math	NOUN
ejpam-1242	569	2	.	.	PUNCT
ejpam-1242	570	1	univ	univ	PROPN
ejpam-1242	570	2	.	.	PUNCT
ejpam-1242	571	1	st	st	PROPN
ejpam-1242	571	2	.	.	PROPN
ejpam-1242	571	3	paul	paul	PROPN
ejpam-1242	571	4	33	33	NUM
ejpam-1242	571	5	,	,	PUNCT
ejpam-1242	571	6	61	61	NUM
ejpam-1242	571	7	-	-	SYM
ejpam-1242	571	8	69	69	NUM
ejpam-1242	571	9	.	.	PUNCT
ejpam-1242	571	10	19840	19840	NUM
ejpam-1242	572	1	[	[	X
ejpam-1242	572	2	20	20	NUM
ejpam-1242	572	3	]	]	PUNCT
ejpam-1242	572	4	c.	c.	PROPN
ejpam-1242	572	5	p.	p.	PROPN
ejpam-1242	572	6	lu	lu	PROPN
ejpam-1242	572	7	,	,	PUNCT
ejpam-1242	572	8	spectra	spectra	NOUN
ejpam-1242	572	9	of	of	ADP
ejpam-1242	572	10	modules	module	NOUN
ejpam-1242	572	11	,	,	PUNCT
ejpam-1242	572	12	comm	comm	NOUN
ejpam-1242	572	13	.	.	PUNCT
ejpam-1242	573	1	algebra	algebra	PROPN
ejpam-1242	573	2	23(10	23(10	NUM
ejpam-1242	573	3	)	)	PUNCT
ejpam-1242	573	4	,	,	PUNCT
ejpam-1242	573	5	3741	3741	NUM
ejpam-1242	573	6	-	-	SYM
ejpam-1242	573	7	3752	3752	NUM
ejpam-1242	573	8	.	.	PUNCT
ejpam-1242	574	1	1995	1995	NUM
ejpam-1242	574	2	references	reference	NOUN
ejpam-1242	574	3	265	265	NUM
ejpam-1242	574	4	[	[	X
ejpam-1242	574	5	21	21	NUM
ejpam-1242	574	6	]	]	X
ejpam-1242	574	7	c.	c.	PROPN
ejpam-1242	574	8	p.	p.	PROPN
ejpam-1242	574	9	lu	lu	PROPN
ejpam-1242	574	10	,	,	PUNCT
ejpam-1242	574	11	a	a	DET
ejpam-1242	574	12	module	module	NOUN
ejpam-1242	574	13	whose	whose	DET
ejpam-1242	574	14	prime	prime	ADJ
ejpam-1242	574	15	spectrum	spectrum	NOUN
ejpam-1242	574	16	has	have	VERB
ejpam-1242	574	17	the	the	DET
ejpam-1242	574	18	surjective	surjective	ADJ
ejpam-1242	574	19	natural	natural	ADJ
ejpam-1242	574	20	map	map	NOUN
ejpam-1242	574	21	,	,	PUNCT
ejpam-1242	574	22	houston	houston	PROPN
ejpam-1242	574	23	j.	j.	PROPN
ejpam-1242	574	24	of	of	ADP
ejpam-1242	574	25	math	math	NOUN
ejpam-1242	574	26	.	.	PUNCT
ejpam-1242	575	1	33(1	33(1	NUM
ejpam-1242	575	2	)	)	PUNCT
ejpam-1242	575	3	,	,	PUNCT
ejpam-1242	575	4	125	125	NUM
ejpam-1242	575	5	-	-	SYM
ejpam-1242	575	6	143	143	NUM
ejpam-1242	575	7	.	.	PUNCT
ejpam-1242	576	1	2007	2007	NUM
ejpam-1242	576	2	.	.	PUNCT
ejpam-1242	577	1	[	[	X
ejpam-1242	577	2	22	22	NUM
ejpam-1242	577	3	]	]	PUNCT
ejpam-1242	577	4	r.	r.	PROPN
ejpam-1242	577	5	y.	y.	PROPN
ejpam-1242	577	6	mccasland	mccasland	PROPN
ejpam-1242	577	7	,	,	PUNCT
ejpam-1242	577	8	m.	m.	PROPN
ejpam-1242	577	9	e.	e.	PROPN
ejpam-1242	577	10	moore	moore	PROPN
ejpam-1242	577	11	and	and	CCONJ
ejpam-1242	577	12	p.	p.	PROPN
ejpam-1242	577	13	f.	f.	PROPN
ejpam-1242	577	14	smith	smith	PROPN
ejpam-1242	577	15	,	,	PUNCT
ejpam-1242	577	16	on	on	ADP
ejpam-1242	577	17	the	the	DET
ejpam-1242	577	18	spectrum	spectrum	NOUN
ejpam-1242	577	19	of	of	ADP
ejpam-1242	577	20	a	a	DET
ejpam-1242	577	21	module	module	NOUN
ejpam-1242	577	22	over	over	ADP
ejpam-1242	577	23	a	a	DET
ejpam-1242	577	24	commutative	commutative	ADJ
ejpam-1242	577	25	ring	ring	NOUN
ejpam-1242	577	26	,	,	PUNCT
ejpam-1242	577	27	comm	comm	NOUN
ejpam-1242	577	28	.	.	PUNCT
ejpam-1242	578	1	algebra	algebra	PROPN
ejpam-1242	578	2	25(1	25(1	NUM
ejpam-1242	578	3	)	)	PUNCT
ejpam-1242	578	4	,	,	PUNCT
ejpam-1242	578	5	79	79	NUM
ejpam-1242	578	6	-	-	SYM
ejpam-1242	578	7	103	103	NUM
ejpam-1242	578	8	.	.	PUNCT
ejpam-1242	578	9	1997	1997	NUM
ejpam-1242	578	10	.	.	PUNCT
ejpam-1242	579	1	[	[	X
ejpam-1242	579	2	23	23	NUM
ejpam-1242	579	3	]	]	X
ejpam-1242	579	4	i.	i.	PROPN
ejpam-1242	579	5	simon	simon	PROPN
ejpam-1242	579	6	,	,	PUNCT
ejpam-1242	579	7	the	the	DET
ejpam-1242	579	8	nondeterministic	nondeterministic	ADJ
ejpam-1242	579	9	complexity	complexity	NOUN
ejpam-1242	579	10	of	of	ADP
ejpam-1242	579	11	finite	finite	PROPN
ejpam-1242	579	12	automaton	automaton	PROPN
ejpam-1242	579	13	,	,	PUNCT
ejpam-1242	579	14	in	in	ADP
ejpam-1242	579	15	:	:	PUNCT
ejpam-1242	579	16	notes	note	NOUN
ejpam-1242	579	17	,	,	PUNCT
ejpam-1242	579	18	hermes	herme	NOUN
ejpam-1242	579	19	,	,	PUNCT
ejpam-1242	579	20	paris	paris	PROPN
ejpam-1242	579	21	,	,	PUNCT
ejpam-1242	579	22	384	384	NUM
ejpam-1242	579	23	-	-	SYM
ejpam-1242	579	24	400	400	NUM
ejpam-1242	579	25	.	.	PUNCT
ejpam-1242	579	26	1990	1990	NUM
ejpam-1242	579	27	.	.	PUNCT