id	sid	tid	token	lemma	pos
ejpam-2272	1	1	compile	compile	NOUN
ejpam-2272	1	2	/	/	SYM
ejpam-2272	1	3	output.dvi	output.dvi	NOUN
ejpam-2272	1	4	on	on	ADP
ejpam-2272	1	5	2	2	NUM
ejpam-2272	1	6	-	-	PUNCT
ejpam-2272	1	7	absorbing	absorbing	ADJ
ejpam-2272	1	8	primary	primary	ADJ
ejpam-2272	1	9	ideals	ideal	NOUN
ejpam-2272	1	10	in	in	ADP
ejpam-2272	1	11	commutative	commutative	ADJ
ejpam-2272	1	12	semirings	semiring	NOUN
ejpam-2272	1	13	pratibha	pratibha	PROPN
ejpam-2272	1	14	kumar1	kumar1	PROPN
ejpam-2272	1	15	,	,	PUNCT
ejpam-2272	1	16	manish	manish	PROPN
ejpam-2272	1	17	kant	kant	PROPN
ejpam-2272	1	18	dubey2	dubey2	PROPN
ejpam-2272	1	19	,	,	PUNCT
ejpam-2272	1	20	poonam	poonam	PROPN
ejpam-2272	1	21	sarohe	sarohe	PROPN
ejpam-2272	1	22	3,∗	3,∗	PROPN
ejpam-2272	1	23	1	1	NUM
ejpam-2272	1	24	department	department	NOUN
ejpam-2272	1	25	of	of	ADP
ejpam-2272	1	26	mathematics	mathematic	NOUN
ejpam-2272	1	27	,	,	PUNCT
ejpam-2272	1	28	kirori	kirori	PROPN
ejpam-2272	1	29	mal	mal	PROPN
ejpam-2272	1	30	college	college	PROPN
ejpam-2272	1	31	,	,	PUNCT
ejpam-2272	1	32	university	university	NOUN
ejpam-2272	1	33	of	of	ADP
ejpam-2272	1	34	delhi	delhi	PROPN
ejpam-2272	1	35	,	,	PUNCT
ejpam-2272	1	36	delhi	delhi	PROPN
ejpam-2272	1	37	110007	110007	NUM
ejpam-2272	1	38	,	,	PUNCT
ejpam-2272	1	39	india	india	PROPN
ejpam-2272	1	40	.	.	PROPN
ejpam-2272	1	41	2	2	NUM
ejpam-2272	1	42	sag	sag	NOUN
ejpam-2272	1	43	,	,	PUNCT
ejpam-2272	1	44	drdo	drdo	PROPN
ejpam-2272	1	45	,	,	PUNCT
ejpam-2272	1	46	metcalf	metcalf	PROPN
ejpam-2272	1	47	house	house	PROPN
ejpam-2272	1	48	,	,	PUNCT
ejpam-2272	1	49	delhi	delhi	PROPN
ejpam-2272	1	50	110054	110054	NUM
ejpam-2272	1	51	,	,	PUNCT
ejpam-2272	1	52	india	india	PROPN
ejpam-2272	1	53	.	.	PROPN
ejpam-2272	1	54	3	3	NUM
ejpam-2272	1	55	department	department	NOUN
ejpam-2272	1	56	of	of	ADP
ejpam-2272	1	57	mathematics	mathematics	PROPN
ejpam-2272	1	58	,	,	PUNCT
ejpam-2272	1	59	lakshmibai	lakshmibai	PROPN
ejpam-2272	1	60	college	college	NOUN
ejpam-2272	1	61	,	,	PUNCT
ejpam-2272	1	62	university	university	NOUN
ejpam-2272	1	63	of	of	ADP
ejpam-2272	1	64	delhi	delhi	PROPN
ejpam-2272	1	65	,	,	PUNCT
ejpam-2272	1	66	delhi	delhi	PROPN
ejpam-2272	1	67	110052	110052	NUM
ejpam-2272	1	68	,	,	PUNCT
ejpam-2272	1	69	india	india	PROPN
ejpam-2272	1	70	.	.	PUNCT
ejpam-2272	2	1	abstract	abstract	PROPN
ejpam-2272	2	2	.	.	PUNCT
ejpam-2272	3	1	in	in	ADP
ejpam-2272	3	2	this	this	DET
ejpam-2272	3	3	paper	paper	NOUN
ejpam-2272	3	4	,	,	PUNCT
ejpam-2272	3	5	we	we	PRON
ejpam-2272	3	6	define	define	VERB
ejpam-2272	3	7	2	2	NUM
ejpam-2272	3	8	-	-	PUNCT
ejpam-2272	3	9	absorbing	absorbing	ADJ
ejpam-2272	3	10	and	and	CCONJ
ejpam-2272	3	11	weakly	weakly	ADJ
ejpam-2272	3	12	2	2	NUM
ejpam-2272	3	13	-	-	PUNCT
ejpam-2272	3	14	absorbing	absorbing	ADJ
ejpam-2272	3	15	primary	primary	ADJ
ejpam-2272	3	16	ideals	ideal	NOUN
ejpam-2272	3	17	in	in	ADP
ejpam-2272	3	18	a	a	DET
ejpam-2272	3	19	commutative	commutative	ADJ
ejpam-2272	3	20	semiring	semiring	NOUN
ejpam-2272	3	21	s	s	NOUN
ejpam-2272	3	22	with	with	ADP
ejpam-2272	3	23	1	1	NUM
ejpam-2272	3	24	≠	≠	NOUN
ejpam-2272	3	25	0	0	NUM
ejpam-2272	3	26	which	which	PRON
ejpam-2272	3	27	are	be	AUX
ejpam-2272	3	28	generalization	generalization	NOUN
ejpam-2272	3	29	of	of	ADP
ejpam-2272	3	30	primary	primary	ADJ
ejpam-2272	3	31	ideals	ideal	NOUN
ejpam-2272	3	32	of	of	ADP
ejpam-2272	3	33	commutative	commutative	ADJ
ejpam-2272	3	34	ring	ring	NOUN
ejpam-2272	3	35	.	.	PUNCT
ejpam-2272	4	1	a	a	DET
ejpam-2272	4	2	proper	proper	ADJ
ejpam-2272	4	3	ideal	ideal	NOUN
ejpam-2272	4	4	i	i	PRON
ejpam-2272	4	5	of	of	ADP
ejpam-2272	4	6	a	a	DET
ejpam-2272	4	7	commutative	commutative	ADJ
ejpam-2272	4	8	semiring	semiring	NOUN
ejpam-2272	4	9	s	s	NOUN
ejpam-2272	4	10	is	be	AUX
ejpam-2272	4	11	said	say	VERB
ejpam-2272	4	12	to	to	PART
ejpam-2272	4	13	be	be	AUX
ejpam-2272	4	14	a	a	DET
ejpam-2272	4	15	2	2	NUM
ejpam-2272	4	16	-	-	PUNCT
ejpam-2272	4	17	absorbing	absorbing	ADJ
ejpam-2272	4	18	primary	primary	NOUN
ejpam-2272	4	19	(	(	PUNCT
ejpam-2272	4	20	weakly	weakly	ADJ
ejpam-2272	4	21	2	2	NUM
ejpam-2272	4	22	-	-	PUNCT
ejpam-2272	4	23	absorbing	absorbing	ADJ
ejpam-2272	4	24	primary	primary	ADJ
ejpam-2272	4	25	)	)	PUNCT
ejpam-2272	4	26	ideal	ideal	NOUN
ejpam-2272	4	27	of	of	ADP
ejpam-2272	4	28	s	s	PRON
ejpam-2272	4	29	if	if	SCONJ
ejpam-2272	4	30	abc	abc	PROPN
ejpam-2272	4	31	∈	∈	PROPN
ejpam-2272	4	32	i	i	PRON
ejpam-2272	4	33	(	(	PUNCT
ejpam-2272	4	34	0	0	NUM
ejpam-2272	4	35	≠	≠	PROPN
ejpam-2272	4	36	abc	abc	PROPN
ejpam-2272	4	37	∈	∈	PROPN
ejpam-2272	4	38	i	i	PROPN
ejpam-2272	4	39	)	)	PUNCT
ejpam-2272	4	40	implies	imply	VERB
ejpam-2272	4	41	ab	ab	PROPN
ejpam-2272	4	42	∈	∈	PROPN
ejpam-2272	5	1	i	i	PRON
ejpam-2272	5	2	or	or	CCONJ
ejpam-2272	5	3	bc	bc	PROPN
ejpam-2272	5	4	∈	∈	PROPN
ejpam-2272	6	1	√	√	VERB
ejpam-2272	6	2	i	i	PRON
ejpam-2272	6	3	or	or	CCONJ
ejpam-2272	6	4	ac	ac	PROPN
ejpam-2272	6	5	∈	∈	PROPN
ejpam-2272	7	1	√	√	VERB
ejpam-2272	7	2	i	i	PRON
ejpam-2272	7	3	.	.	PUNCT
ejpam-2272	8	1	some	some	DET
ejpam-2272	8	2	results	result	NOUN
ejpam-2272	8	3	concerning	concern	VERB
ejpam-2272	8	4	2absorbing	2absorbing	NUM
ejpam-2272	8	5	primary	primary	ADJ
ejpam-2272	8	6	and	and	CCONJ
ejpam-2272	8	7	weakly	weakly	ADJ
ejpam-2272	8	8	2	2	NUM
ejpam-2272	8	9	-	-	PUNCT
ejpam-2272	8	10	absorbing	absorbing	ADJ
ejpam-2272	8	11	primary	primary	ADJ
ejpam-2272	8	12	ideals	ideal	NOUN
ejpam-2272	8	13	are	be	AUX
ejpam-2272	8	14	given	give	VERB
ejpam-2272	8	15	.	.	PUNCT
ejpam-2272	9	1	it	it	PRON
ejpam-2272	9	2	is	be	AUX
ejpam-2272	9	3	proved	prove	VERB
ejpam-2272	9	4	that	that	SCONJ
ejpam-2272	9	5	a	a	DET
ejpam-2272	9	6	subtractive	subtractive	NOUN
ejpam-2272	9	7	weakly	weakly	ADJ
ejpam-2272	9	8	2	2	NUM
ejpam-2272	9	9	-	-	PUNCT
ejpam-2272	9	10	absorbing	absorbing	ADJ
ejpam-2272	9	11	primary	primary	ADJ
ejpam-2272	9	12	ideal	ideal	NOUN
ejpam-2272	9	13	i	i	PRON
ejpam-2272	9	14	that	that	PRON
ejpam-2272	9	15	is	be	AUX
ejpam-2272	9	16	not	not	PART
ejpam-2272	9	17	a	a	DET
ejpam-2272	9	18	2	2	NUM
ejpam-2272	9	19	-	-	PUNCT
ejpam-2272	9	20	absorbing	absorbing	ADJ
ejpam-2272	9	21	primary	primary	ADJ
ejpam-2272	9	22	ideal	ideal	ADJ
ejpam-2272	9	23	satisfies	satisfie	NOUN
ejpam-2272	9	24	√	√	VERB
ejpam-2272	9	25	i	i	NOUN
ejpam-2272	9	26	=	=	PUNCT
ejpam-2272	10	1	√	√	PROPN
ejpam-2272	10	2	0	0	NUM
ejpam-2272	10	3	.	.	X
ejpam-2272	10	4	2010	2010	NUM
ejpam-2272	10	5	mathematics	mathematic	NOUN
ejpam-2272	10	6	subject	subject	NOUN
ejpam-2272	10	7	classifications	classification	NOUN
ejpam-2272	10	8	:	:	PUNCT
ejpam-2272	10	9	16y30	16y30	NUM
ejpam-2272	10	10	,	,	PUNCT
ejpam-2272	10	11	16y60	16y60	NUM
ejpam-2272	10	12	key	key	ADJ
ejpam-2272	10	13	words	word	NOUN
ejpam-2272	10	14	and	and	CCONJ
ejpam-2272	10	15	phrases	phrase	NOUN
ejpam-2272	10	16	:	:	PUNCT
ejpam-2272	10	17	semiring	semiring	NOUN
ejpam-2272	10	18	,	,	PUNCT
ejpam-2272	10	19	subtractive	subtractive	NOUN
ejpam-2272	10	20	ideal	ideal	NOUN
ejpam-2272	10	21	,	,	PUNCT
ejpam-2272	10	22	2	2	NUM
ejpam-2272	10	23	-	-	PUNCT
ejpam-2272	10	24	absorbing	absorbing	ADJ
ejpam-2272	10	25	primary	primary	ADJ
ejpam-2272	10	26	ideal	ideal	NOUN
ejpam-2272	10	27	,	,	PUNCT
ejpam-2272	10	28	weakly	weakly	ADJ
ejpam-2272	10	29	2	2	NUM
ejpam-2272	10	30	-	-	PUNCT
ejpam-2272	10	31	absorbing	absorbing	ADJ
ejpam-2272	10	32	primary	primary	ADJ
ejpam-2272	10	33	ideal	ideal	NOUN
ejpam-2272	10	34	,	,	PUNCT
ejpam-2272	10	35	q	q	NOUN
ejpam-2272	10	36	-	-	PUNCT
ejpam-2272	10	37	ideal	ideal	ADJ
ejpam-2272	10	38	1	1	NUM
ejpam-2272	10	39	.	.	PUNCT
ejpam-2272	11	1	introduction	introduction	NOUN
ejpam-2272	11	2	the	the	DET
ejpam-2272	11	3	algebraic	algebraic	ADJ
ejpam-2272	11	4	structure	structure	NOUN
ejpam-2272	11	5	of	of	ADP
ejpam-2272	11	6	semiring	semiring	NOUN
ejpam-2272	11	7	plays	play	VERB
ejpam-2272	11	8	a	a	DET
ejpam-2272	11	9	prominent	prominent	ADJ
ejpam-2272	11	10	role	role	NOUN
ejpam-2272	11	11	in	in	ADP
ejpam-2272	11	12	various	various	ADJ
ejpam-2272	11	13	branches	branch	NOUN
ejpam-2272	11	14	of	of	ADP
ejpam-2272	11	15	mathematics	mathematic	NOUN
ejpam-2272	11	16	as	as	ADV
ejpam-2272	11	17	well	well	ADV
ejpam-2272	11	18	as	as	ADP
ejpam-2272	11	19	some	some	DET
ejpam-2272	11	20	other	other	ADJ
ejpam-2272	11	21	branches	branch	NOUN
ejpam-2272	11	22	of	of	ADP
ejpam-2272	11	23	applied	apply	VERB
ejpam-2272	11	24	science	science	NOUN
ejpam-2272	11	25	.	.	PUNCT
ejpam-2272	12	1	the	the	DET
ejpam-2272	12	2	concept	concept	NOUN
ejpam-2272	12	3	of	of	ADP
ejpam-2272	12	4	semiring	semiring	NOUN
ejpam-2272	12	5	was	be	AUX
ejpam-2272	12	6	first	first	ADV
ejpam-2272	12	7	introduced	introduce	VERB
ejpam-2272	12	8	by	by	ADP
ejpam-2272	12	9	h.	h.	PROPN
ejpam-2272	12	10	s.	s.	PROPN
ejpam-2272	12	11	vandiver	vandiver	VERB
ejpam-2272	13	1	[	[	X
ejpam-2272	13	2	13	13	NUM
ejpam-2272	13	3	]	]	PUNCT
ejpam-2272	13	4	in	in	ADP
ejpam-2272	13	5	1934	1934	NUM
ejpam-2272	13	6	and	and	CCONJ
ejpam-2272	13	7	has	have	AUX
ejpam-2272	13	8	since	since	SCONJ
ejpam-2272	13	9	then	then	ADV
ejpam-2272	13	10	been	be	AUX
ejpam-2272	13	11	studied	study	VERB
ejpam-2272	13	12	by	by	ADP
ejpam-2272	13	13	many	many	ADJ
ejpam-2272	13	14	authors	author	NOUN
ejpam-2272	13	15	.	.	PUNCT
ejpam-2272	14	1	the	the	DET
ejpam-2272	14	2	structure	structure	NOUN
ejpam-2272	14	3	of	of	ADP
ejpam-2272	14	4	prime	prime	ADJ
ejpam-2272	14	5	ideals	ideal	NOUN
ejpam-2272	14	6	in	in	ADP
ejpam-2272	14	7	semiring	semiring	NOUN
ejpam-2272	14	8	theory	theory	NOUN
ejpam-2272	14	9	have	have	AUX
ejpam-2272	14	10	gained	gain	VERB
ejpam-2272	14	11	importance	importance	NOUN
ejpam-2272	14	12	and	and	CCONJ
ejpam-2272	14	13	many	many	ADJ
ejpam-2272	14	14	mathematicians	mathematician	NOUN
ejpam-2272	14	15	have	have	AUX
ejpam-2272	14	16	exploited	exploit	VERB
ejpam-2272	14	17	its	its	PRON
ejpam-2272	14	18	usefulness	usefulness	NOUN
ejpam-2272	14	19	in	in	ADP
ejpam-2272	14	20	algebraic	algebraic	ADJ
ejpam-2272	14	21	systems	system	NOUN
ejpam-2272	14	22	over	over	ADP
ejpam-2272	14	23	the	the	DET
ejpam-2272	14	24	decades	decade	NOUN
ejpam-2272	14	25	.	.	PUNCT
ejpam-2272	15	1	anderson	anderson	PROPN
ejpam-2272	15	2	and	and	CCONJ
ejpam-2272	15	3	smith	smith	PROPN
ejpam-2272	16	1	[	[	X
ejpam-2272	16	2	2	2	NUM
ejpam-2272	16	3	]	]	PUNCT
ejpam-2272	16	4	introduced	introduce	VERB
ejpam-2272	16	5	the	the	DET
ejpam-2272	16	6	notion	notion	NOUN
ejpam-2272	16	7	of	of	ADP
ejpam-2272	16	8	weakly	weakly	ADJ
ejpam-2272	16	9	prime	prime	ADJ
ejpam-2272	16	10	ideals	ideal	NOUN
ejpam-2272	16	11	in	in	ADP
ejpam-2272	16	12	commutative	commutative	ADJ
ejpam-2272	16	13	ring	ring	NOUN
ejpam-2272	16	14	for	for	ADP
ejpam-2272	16	15	the	the	DET
ejpam-2272	16	16	study	study	NOUN
ejpam-2272	16	17	of	of	ADP
ejpam-2272	16	18	factorization	factorization	NOUN
ejpam-2272	16	19	in	in	ADP
ejpam-2272	16	20	commutative	commutative	ADJ
ejpam-2272	16	21	rings	ring	NOUN
ejpam-2272	16	22	with	with	ADP
ejpam-2272	16	23	zero	zero	NUM
ejpam-2272	16	24	divisors	divisor	NOUN
ejpam-2272	16	25	.	.	PUNCT
ejpam-2272	17	1	the	the	DET
ejpam-2272	17	2	concepts	concept	NOUN
ejpam-2272	17	3	of	of	ADP
ejpam-2272	17	4	2	2	NUM
ejpam-2272	17	5	-	-	PUNCT
ejpam-2272	17	6	absorbing	absorbing	ADJ
ejpam-2272	17	7	and	and	CCONJ
ejpam-2272	17	8	weakly	weakly	ADJ
ejpam-2272	17	9	2	2	NUM
ejpam-2272	17	10	-	-	PUNCT
ejpam-2272	17	11	absorbing	absorbing	ADJ
ejpam-2272	17	12	ideals	ideal	NOUN
ejpam-2272	17	13	of	of	ADP
ejpam-2272	17	14	commutative	commutative	ADJ
ejpam-2272	17	15	ring	ring	NOUN
ejpam-2272	17	16	with	with	ADP
ejpam-2272	17	17	nonzero	nonzero	ADJ
ejpam-2272	17	18	unity	unity	NOUN
ejpam-2272	17	19	have	have	AUX
ejpam-2272	17	20	been	be	AUX
ejpam-2272	17	21	introduced	introduce	VERB
ejpam-2272	17	22	by	by	ADP
ejpam-2272	17	23	badawi	badawi	PROPN
ejpam-2272	17	24	[	[	X
ejpam-2272	17	25	7	7	NUM
ejpam-2272	17	26	]	]	PUNCT
ejpam-2272	17	27	and	and	CCONJ
ejpam-2272	17	28	badawi	badawi	ADJ
ejpam-2272	17	29	and	and	CCONJ
ejpam-2272	17	30	darani	darani	PROPN
ejpam-2272	18	1	[	[	X
ejpam-2272	18	2	8	8	NUM
ejpam-2272	18	3	]	]	PUNCT
ejpam-2272	18	4	respectively	respectively	ADV
ejpam-2272	18	5	which	which	PRON
ejpam-2272	18	6	are	be	AUX
ejpam-2272	18	7	generalizations	generalization	NOUN
ejpam-2272	18	8	of	of	ADP
ejpam-2272	18	9	prime	prime	ADJ
ejpam-2272	18	10	and	and	CCONJ
ejpam-2272	18	11	weakly	weakly	ADJ
ejpam-2272	18	12	prime	prime	ADJ
ejpam-2272	18	13	ideals	ideal	NOUN
ejpam-2272	18	14	in	in	ADP
ejpam-2272	18	15	commutative	commutative	ADJ
ejpam-2272	18	16	rings	ring	NOUN
ejpam-2272	18	17	.	.	PUNCT
ejpam-2272	19	1	recently	recently	ADV
ejpam-2272	19	2	,	,	PUNCT
ejpam-2272	19	3	badawi	badawi	PROPN
ejpam-2272	19	4	et	et	PROPN
ejpam-2272	19	5	al	al	PROPN
ejpam-2272	19	6	.	.	PUNCT
ejpam-2272	20	1	[	[	X
ejpam-2272	20	2	9	9	NUM
ejpam-2272	20	3	]	]	PUNCT
ejpam-2272	20	4	introduced	introduce	VERB
ejpam-2272	20	5	the	the	DET
ejpam-2272	20	6	concept	concept	NOUN
ejpam-2272	20	7	of	of	ADP
ejpam-2272	20	8	2absorbing	2absorbing	NUM
ejpam-2272	20	9	primary	primary	ADJ
ejpam-2272	20	10	ideals	ideal	NOUN
ejpam-2272	20	11	in	in	ADP
ejpam-2272	20	12	commutative	commutative	ADJ
ejpam-2272	20	13	rings	ring	NOUN
ejpam-2272	20	14	with	with	ADP
ejpam-2272	20	15	1	1	NUM
ejpam-2272	20	16	≠	≠	NOUN
ejpam-2272	20	17	0	0	NUM
ejpam-2272	20	18	and	and	CCONJ
ejpam-2272	20	19	gave	give	VERB
ejpam-2272	20	20	some	some	DET
ejpam-2272	20	21	characterizations	characterization	NOUN
ejpam-2272	20	22	related	relate	VERB
ejpam-2272	20	23	to	to	ADP
ejpam-2272	20	24	it	it	PRON
ejpam-2272	20	25	.	.	PUNCT
ejpam-2272	21	1	∗corresponding	∗corresponde	VERB
ejpam-2272	21	2	author	author	NOUN
ejpam-2272	21	3	.	.	PUNCT
ejpam-2272	22	1	email	email	NOUN
ejpam-2272	22	2	addresses	address	NOUN
ejpam-2272	22	3	:	:	PUNCT
ejpam-2272	22	4	pratibhakumar313@gmail.com	pratibhakumar313@gmail.com	X
ejpam-2272	22	5	(	(	PUNCT
ejpam-2272	22	6	p.	p.	PROPN
ejpam-2272	22	7	kumar	kumar	PROPN
ejpam-2272	22	8	)	)	PUNCT
ejpam-2272	22	9	,	,	PUNCT
ejpam-2272	22	10	kantmanish@yahoo.com	kantmanish@yahoo.com	PROPN
ejpam-2272	22	11	(	(	PUNCT
ejpam-2272	22	12	m.	m.	PROPN
ejpam-2272	22	13	k.	k.	PROPN
ejpam-2272	22	14	dubey	dubey	PROPN
ejpam-2272	22	15	)	)	PUNCT
ejpam-2272	22	16	,	,	PUNCT
ejpam-2272	22	17	poonamsarohe@gmail.com(poonam	poonamsarohe@gmail.com(poonam	NOUN
ejpam-2272	22	18	sarohe	sarohe	NOUN
ejpam-2272	22	19	)	)	PUNCT
ejpam-2272	22	20	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2272	23	1	186	186	NUM
ejpam-2272	23	2	©	©	PROPN
ejpam-2272	23	3	2016	2016	NUM
ejpam-2272	23	4	ejpam	ejpam	VERB
ejpam-2272	23	5	all	all	DET
ejpam-2272	23	6	rights	right	NOUN
ejpam-2272	23	7	reserved	reserve	VERB
ejpam-2272	23	8	.	.	PUNCT
ejpam-2272	24	1	european	european	ADJ
ejpam-2272	24	2	journal	journal	PROPN
ejpam-2272	24	3	of	of	ADP
ejpam-2272	24	4	pure	pure	ADJ
ejpam-2272	24	5	and	and	CCONJ
ejpam-2272	24	6	applied	apply	VERB
ejpam-2272	24	7	mathematics	mathematic	NOUN
ejpam-2272	24	8	vol	vol	NOUN
ejpam-2272	24	9	.	.	PROPN
ejpam-2272	25	1	9	9	NUM
ejpam-2272	25	2	,	,	PUNCT
ejpam-2272	25	3	no	no	INTJ
ejpam-2272	25	4	.	.	NOUN
ejpam-2272	25	5	2	2	NUM
ejpam-2272	25	6	,	,	PUNCT
ejpam-2272	25	7	2016	2016	NUM
ejpam-2272	25	8	,	,	PUNCT
ejpam-2272	25	9	186	186	NUM
ejpam-2272	25	10	-	-	SYM
ejpam-2272	25	11	195	195	NUM
ejpam-2272	25	12	issn	issn	PROPN
ejpam-2272	25	13	1307	1307	NUM
ejpam-2272	25	14	-	-	SYM
ejpam-2272	25	15	5543	5543	NUM
ejpam-2272	25	16	–	–	PUNCT
ejpam-2272	25	17	www.ejpam.com	www.ejpam.com	X
ejpam-2272	25	18	p.	p.	PROPN
ejpam-2272	25	19	kumar	kumar	PROPN
ejpam-2272	25	20	,	,	PUNCT
ejpam-2272	25	21	m.	m.	NOUN
ejpam-2272	25	22	dubey	dubey	PROPN
ejpam-2272	25	23	,	,	PUNCT
ejpam-2272	25	24	p.	p.	NOUN
ejpam-2272	25	25	sarohe	sarohe	PROPN
ejpam-2272	25	26	/	/	SYM
ejpam-2272	25	27	eur	eur	PROPN
ejpam-2272	25	28	.	.	PUNCT
ejpam-2272	26	1	j.	j.	PROPN
ejpam-2272	26	2	pure	pure	PROPN
ejpam-2272	26	3	appl	appl	PROPN
ejpam-2272	26	4	.	.	PROPN
ejpam-2272	26	5	math	math	PROPN
ejpam-2272	26	6	,	,	PUNCT
ejpam-2272	26	7	9	9	NUM
ejpam-2272	26	8	(	(	PUNCT
ejpam-2272	26	9	2016	2016	NUM
ejpam-2272	26	10	)	)	PUNCT
ejpam-2272	26	11	,	,	PUNCT
ejpam-2272	26	12	186	186	NUM
ejpam-2272	26	13	-	-	SYM
ejpam-2272	26	14	195	195	NUM
ejpam-2272	26	15	187	187	NUM
ejpam-2272	26	16	a	a	DET
ejpam-2272	26	17	commutative	commutative	ADJ
ejpam-2272	26	18	semiring	semiring	NOUN
ejpam-2272	26	19	is	be	AUX
ejpam-2272	26	20	a	a	DET
ejpam-2272	26	21	commutative	commutative	ADJ
ejpam-2272	26	22	semigroup	semigroup	NOUN
ejpam-2272	26	23	(	(	PUNCT
ejpam-2272	26	24	s	s	PROPN
ejpam-2272	26	25	,	,	PUNCT
ejpam-2272	26	26	⋅	⋅	PROPN
ejpam-2272	26	27	)	)	PUNCT
ejpam-2272	26	28	and	and	CCONJ
ejpam-2272	26	29	a	a	DET
ejpam-2272	26	30	commutative	commutative	ADJ
ejpam-2272	26	31	monoid	monoid	NOUN
ejpam-2272	26	32	(	(	PUNCT
ejpam-2272	26	33	s,+	s,+	PROPN
ejpam-2272	26	34	,	,	PUNCT
ejpam-2272	26	35	0s	0s	NUM
ejpam-2272	26	36	)	)	PUNCT
ejpam-2272	26	37	in	in	ADP
ejpam-2272	26	38	which	which	PRON
ejpam-2272	26	39	0s	0s	NOUN
ejpam-2272	26	40	is	be	AUX
ejpam-2272	26	41	the	the	DET
ejpam-2272	26	42	additive	additive	ADJ
ejpam-2272	26	43	identity	identity	NOUN
ejpam-2272	26	44	and	and	CCONJ
ejpam-2272	26	45	0s	0s	NOUN
ejpam-2272	26	46	⋅	⋅	PROPN
ejpam-2272	26	47	x	x	PUNCT
ejpam-2272	27	1	=	=	PUNCT
ejpam-2272	27	2	x	x	SYM
ejpam-2272	27	3	⋅	⋅	PROPN
ejpam-2272	27	4	0s	0s	NOUN
ejpam-2272	27	5	=	=	PUNCT
ejpam-2272	27	6	0s	0s	NOUN
ejpam-2272	27	7	for	for	ADP
ejpam-2272	27	8	all	all	DET
ejpam-2272	27	9	x	x	SYM
ejpam-2272	27	10	∈	∈	PROPN
ejpam-2272	27	11	s	s	NOUN
ejpam-2272	27	12	,	,	PUNCT
ejpam-2272	27	13	both	both	PRON
ejpam-2272	27	14	are	be	AUX
ejpam-2272	27	15	connected	connect	VERB
ejpam-2272	27	16	by	by	ADP
ejpam-2272	27	17	ring	ring	NOUN
ejpam-2272	27	18	like	like	ADP
ejpam-2272	27	19	distributivity	distributivity	NOUN
ejpam-2272	27	20	.	.	PUNCT
ejpam-2272	28	1	a	a	DET
ejpam-2272	28	2	nonempty	nonempty	NOUN
ejpam-2272	28	3	subset	subset	VERB
ejpam-2272	28	4	i	i	PRON
ejpam-2272	28	5	of	of	ADP
ejpam-2272	28	6	a	a	DET
ejpam-2272	28	7	semiring	semiring	NOUN
ejpam-2272	28	8	s	s	X
ejpam-2272	28	9	is	be	AUX
ejpam-2272	28	10	called	call	VERB
ejpam-2272	28	11	an	an	DET
ejpam-2272	28	12	ideal	ideal	NOUN
ejpam-2272	28	13	of	of	ADP
ejpam-2272	28	14	s	s	PRON
ejpam-2272	28	15	if	if	SCONJ
ejpam-2272	28	16	a	a	PRON
ejpam-2272	28	17	,	,	PUNCT
ejpam-2272	28	18	b	b	X
ejpam-2272	28	19	∈	∈	NOUN
ejpam-2272	29	1	i	i	PRON
ejpam-2272	29	2	and	and	CCONJ
ejpam-2272	29	3	r	r	NOUN
ejpam-2272	29	4	∈	∈	PROPN
ejpam-2272	29	5	s	s	PROPN
ejpam-2272	29	6	,	,	PUNCT
ejpam-2272	29	7	a	a	DET
ejpam-2272	29	8	+	+	X
ejpam-2272	29	9	b	b	NOUN
ejpam-2272	29	10	∈	∈	NOUN
ejpam-2272	29	11	i	i	PRON
ejpam-2272	29	12	and	and	CCONJ
ejpam-2272	29	13	ra	ra	PROPN
ejpam-2272	29	14	,	,	PUNCT
ejpam-2272	29	15	ar	ar	PROPN
ejpam-2272	29	16	∈	∈	PROPN
ejpam-2272	29	17	i	i	PRON
ejpam-2272	29	18	.	.	PUNCT
ejpam-2272	30	1	an	an	DET
ejpam-2272	30	2	ideal	ideal	ADJ
ejpam-2272	30	3	i	i	PRON
ejpam-2272	30	4	of	of	ADP
ejpam-2272	30	5	a	a	DET
ejpam-2272	30	6	semiring	semiring	NOUN
ejpam-2272	30	7	s	s	X
ejpam-2272	30	8	is	be	AUX
ejpam-2272	30	9	called	call	VERB
ejpam-2272	30	10	subtractive	subtractive	NOUN
ejpam-2272	30	11	if	if	SCONJ
ejpam-2272	30	12	a	a	PRON
ejpam-2272	30	13	,	,	PUNCT
ejpam-2272	30	14	a	a	DET
ejpam-2272	30	15	+	+	X
ejpam-2272	30	16	b	b	NOUN
ejpam-2272	30	17	∈	∈	NOUN
ejpam-2272	31	1	i	i	PRON
ejpam-2272	31	2	,	,	PUNCT
ejpam-2272	31	3	b	b	PROPN
ejpam-2272	31	4	∈	∈	PROPN
ejpam-2272	31	5	s	s	NOUN
ejpam-2272	31	6	,	,	PUNCT
ejpam-2272	31	7	then	then	ADV
ejpam-2272	31	8	b	b	X
ejpam-2272	31	9	∈	∈	PROPN
ejpam-2272	32	1	i	i	PRON
ejpam-2272	32	2	.	.	PUNCT
ejpam-2272	33	1	let	let	VERB
ejpam-2272	33	2	i	i	PRON
ejpam-2272	33	3	be	be	AUX
ejpam-2272	33	4	an	an	DET
ejpam-2272	33	5	ideal	ideal	NOUN
ejpam-2272	33	6	of	of	ADP
ejpam-2272	33	7	s.	s.	PROPN
ejpam-2272	33	8	then	then	ADV
ejpam-2272	33	9	,	,	PUNCT
ejpam-2272	33	10	the	the	DET
ejpam-2272	33	11	radical	radical	NOUN
ejpam-2272	33	12	of	of	ADP
ejpam-2272	33	13	i	i	PRON
ejpam-2272	33	14	is	be	AUX
ejpam-2272	33	15	defined	define	VERB
ejpam-2272	33	16	as	as	ADP
ejpam-2272	33	17	rad(i	rad(i	NOUN
ejpam-2272	33	18	)	)	PUNCT
ejpam-2272	33	19	=	=	PUNCT
ejpam-2272	34	1	√	√	PUNCT
ejpam-2272	35	1	i	i	PRON
ejpam-2272	35	2	=	=	PUNCT
ejpam-2272	35	3	{	{	PUNCT
ejpam-2272	35	4	a	a	DET
ejpam-2272	35	5	∈	∈	NOUN
ejpam-2272	35	6	s	s	PART
ejpam-2272	35	7	∶	∶	NOUN
ejpam-2272	35	8	an	an	DET
ejpam-2272	35	9	∈	∈	NOUN
ejpam-2272	35	10	i	i	PRON
ejpam-2272	35	11	for	for	ADP
ejpam-2272	35	12	some	some	DET
ejpam-2272	35	13	positive	positive	ADJ
ejpam-2272	35	14	integer.}n	integer.}n	PROPN
ejpam-2272	35	15	annihilator	annihilator	NOUN
ejpam-2272	35	16	of	of	ADP
ejpam-2272	35	17	a	a	DET
ejpam-2272	35	18	semiring	semiring	NOUN
ejpam-2272	35	19	s	s	X
ejpam-2272	35	20	is	be	AUX
ejpam-2272	35	21	defined	define	VERB
ejpam-2272	35	22	as	as	ADP
ejpam-2272	35	23	ann(a	ann(a	PROPN
ejpam-2272	35	24	)	)	PUNCT
ejpam-2272	35	25	=	=	PRON
ejpam-2272	36	1	{	{	PUNCT
ejpam-2272	36	2	x	x	PUNCT
ejpam-2272	36	3	∈	∈	PROPN
ejpam-2272	36	4	s	s	PART
ejpam-2272	36	5	∶	∶	NOUN
ejpam-2272	36	6	ax	ax	NOUN
ejpam-2272	36	7	=	=	NOUN
ejpam-2272	36	8	0	0	NUM
ejpam-2272	36	9	}	}	PUNCT
ejpam-2272	36	10	.	.	PUNCT
ejpam-2272	37	1	recall	recall	VERB
ejpam-2272	37	2	from	from	ADP
ejpam-2272	37	3	[	[	X
ejpam-2272	37	4	10	10	NUM
ejpam-2272	37	5	]	]	PUNCT
ejpam-2272	37	6	,	,	PUNCT
ejpam-2272	37	7	that	that	SCONJ
ejpam-2272	37	8	a	a	DET
ejpam-2272	37	9	proper	proper	ADJ
ejpam-2272	37	10	ideal	ideal	NOUN
ejpam-2272	37	11	i	i	PRON
ejpam-2272	37	12	of	of	ADP
ejpam-2272	37	13	a	a	DET
ejpam-2272	37	14	commutative	commutative	ADJ
ejpam-2272	37	15	semiring	semiring	NOUN
ejpam-2272	37	16	s	s	NOUN
ejpam-2272	37	17	is	be	AUX
ejpam-2272	37	18	said	say	VERB
ejpam-2272	37	19	to	to	PART
ejpam-2272	37	20	be	be	AUX
ejpam-2272	37	21	a	a	DET
ejpam-2272	37	22	2	2	NUM
ejpam-2272	37	23	-	-	PUNCT
ejpam-2272	37	24	absorbing	absorbing	ADJ
ejpam-2272	37	25	(	(	PUNCT
ejpam-2272	37	26	weakly	weakly	ADJ
ejpam-2272	37	27	2absorbing)ideal	2absorbing)ideal	NUM
ejpam-2272	37	28	of	of	ADP
ejpam-2272	37	29	s	s	PRON
ejpam-2272	37	30	if	if	SCONJ
ejpam-2272	37	31	whenever	whenever	SCONJ
ejpam-2272	37	32	a	a	DET
ejpam-2272	37	33	,	,	PUNCT
ejpam-2272	37	34	b	b	NOUN
ejpam-2272	37	35	,	,	PUNCT
ejpam-2272	37	36	c	c	PROPN
ejpam-2272	37	37	∈	∈	PROPN
ejpam-2272	37	38	s	s	PART
ejpam-2272	37	39	and	and	CCONJ
ejpam-2272	38	1	abc	abc	PROPN
ejpam-2272	38	2	∈	∈	PROPN
ejpam-2272	38	3	i	i	PRON
ejpam-2272	38	4	(	(	PUNCT
ejpam-2272	38	5	0	0	NUM
ejpam-2272	38	6	≠	≠	PROPN
ejpam-2272	38	7	abc	abc	PROPN
ejpam-2272	38	8	∈	∈	PROPN
ejpam-2272	38	9	i	i	PROPN
ejpam-2272	38	10	)	)	PUNCT
ejpam-2272	38	11	,	,	PUNCT
ejpam-2272	38	12	then	then	ADV
ejpam-2272	38	13	ab	ab	PROPN
ejpam-2272	38	14	∈	∈	PROPN
ejpam-2272	39	1	i	i	PRON
ejpam-2272	39	2	or	or	CCONJ
ejpam-2272	39	3	ac	ac	PROPN
ejpam-2272	39	4	∈	∈	PROPN
ejpam-2272	40	1	i	i	PRON
ejpam-2272	40	2	or	or	CCONJ
ejpam-2272	40	3	bc	bc	PROPN
ejpam-2272	40	4	∈	∈	PROPN
ejpam-2272	41	1	i	i	PRON
ejpam-2272	41	2	.	.	PUNCT
ejpam-2272	42	1	it	it	PRON
ejpam-2272	42	2	is	be	AUX
ejpam-2272	42	3	easy	easy	ADJ
ejpam-2272	42	4	to	to	PART
ejpam-2272	42	5	see	see	VERB
ejpam-2272	42	6	that	that	SCONJ
ejpam-2272	42	7	every	every	DET
ejpam-2272	42	8	2	2	NUM
ejpam-2272	42	9	-	-	PUNCT
ejpam-2272	42	10	absorbing	absorbing	ADJ
ejpam-2272	42	11	ideal	ideal	NOUN
ejpam-2272	42	12	of	of	ADP
ejpam-2272	42	13	a	a	DET
ejpam-2272	42	14	semiring	semiring	NOUN
ejpam-2272	42	15	s	s	X
ejpam-2272	42	16	is	be	AUX
ejpam-2272	42	17	a	a	DET
ejpam-2272	42	18	weakly	weakly	ADJ
ejpam-2272	42	19	2absorbing	2absorbing	NUM
ejpam-2272	42	20	ideal	ideal	NOUN
ejpam-2272	42	21	of	of	ADP
ejpam-2272	42	22	s	s	NOUN
ejpam-2272	42	23	but	but	CCONJ
ejpam-2272	42	24	converse	converse	NOUN
ejpam-2272	42	25	need	need	AUX
ejpam-2272	42	26	not	not	PART
ejpam-2272	42	27	be	be	AUX
ejpam-2272	42	28	true	true	ADJ
ejpam-2272	42	29	.	.	PUNCT
ejpam-2272	43	1	for	for	ADP
ejpam-2272	43	2	further	far	ADV
ejpam-2272	43	3	understanding	understand	VERB
ejpam-2272	43	4	the	the	DET
ejpam-2272	43	5	concept	concept	NOUN
ejpam-2272	43	6	of	of	ADP
ejpam-2272	43	7	semiring	semiring	NOUN
ejpam-2272	43	8	,	,	PUNCT
ejpam-2272	43	9	refer	refer	VERB
ejpam-2272	43	10	[	[	X
ejpam-2272	43	11	11	11	NUM
ejpam-2272	43	12	]	]	PUNCT
ejpam-2272	43	13	and	and	CCONJ
ejpam-2272	43	14	the	the	DET
ejpam-2272	43	15	properties	property	NOUN
ejpam-2272	43	16	of	of	ADP
ejpam-2272	43	17	a	a	DET
ejpam-2272	43	18	2	2	NUM
ejpam-2272	43	19	-	-	PUNCT
ejpam-2272	43	20	absorbing	absorbing	ADJ
ejpam-2272	43	21	and	and	CCONJ
ejpam-2272	43	22	weakly	weakly	ADJ
ejpam-2272	43	23	2	2	NUM
ejpam-2272	43	24	-	-	PUNCT
ejpam-2272	43	25	absorbing	absorbing	ADJ
ejpam-2272	43	26	ideals	ideal	NOUN
ejpam-2272	43	27	in	in	ADP
ejpam-2272	43	28	commutative	commutative	ADJ
ejpam-2272	43	29	semirings	semiring	NOUN
ejpam-2272	43	30	,	,	PUNCT
ejpam-2272	43	31	we	we	PRON
ejpam-2272	43	32	refer[10	refer[10	VERB
ejpam-2272	43	33	]	]	PUNCT
ejpam-2272	43	34	.	.	PUNCT
ejpam-2272	44	1	the	the	DET
ejpam-2272	44	2	paper	paper	NOUN
ejpam-2272	44	3	is	be	AUX
ejpam-2272	44	4	organized	organize	VERB
ejpam-2272	44	5	as	as	SCONJ
ejpam-2272	44	6	follows	follow	VERB
ejpam-2272	44	7	:	:	PUNCT
ejpam-2272	44	8	in	in	ADP
ejpam-2272	44	9	section	section	NOUN
ejpam-2272	44	10	2	2	NUM
ejpam-2272	44	11	,	,	PUNCT
ejpam-2272	44	12	we	we	PRON
ejpam-2272	44	13	introduce	introduce	VERB
ejpam-2272	44	14	the	the	DET
ejpam-2272	44	15	concepts	concept	NOUN
ejpam-2272	44	16	of	of	ADP
ejpam-2272	44	17	2	2	NUM
ejpam-2272	44	18	-	-	PUNCT
ejpam-2272	44	19	absorbing	absorbing	ADJ
ejpam-2272	44	20	primary	primary	ADJ
ejpam-2272	44	21	ideal	ideal	NOUN
ejpam-2272	44	22	of	of	ADP
ejpam-2272	44	23	a	a	DET
ejpam-2272	44	24	commutative	commutative	ADJ
ejpam-2272	44	25	semiring	semiring	NOUN
ejpam-2272	44	26	and	and	CCONJ
ejpam-2272	44	27	prove	prove	VERB
ejpam-2272	44	28	some	some	DET
ejpam-2272	44	29	results	result	NOUN
ejpam-2272	44	30	corresponding	correspond	VERB
ejpam-2272	44	31	to	to	ADP
ejpam-2272	44	32	ring	ring	NOUN
ejpam-2272	44	33	theory	theory	NOUN
ejpam-2272	44	34	.	.	PUNCT
ejpam-2272	45	1	in	in	ADP
ejpam-2272	45	2	section	section	NOUN
ejpam-2272	45	3	3	3	NUM
ejpam-2272	45	4	,	,	PUNCT
ejpam-2272	45	5	we	we	PRON
ejpam-2272	45	6	introduce	introduce	VERB
ejpam-2272	45	7	the	the	DET
ejpam-2272	45	8	concept	concept	NOUN
ejpam-2272	45	9	of	of	ADP
ejpam-2272	45	10	weakly	weakly	ADJ
ejpam-2272	45	11	2absorbing	2absorbing	NUM
ejpam-2272	45	12	primary	primary	ADJ
ejpam-2272	45	13	ideal	ideal	NOUN
ejpam-2272	45	14	of	of	ADP
ejpam-2272	45	15	a	a	DET
ejpam-2272	45	16	commutative	commutative	ADJ
ejpam-2272	45	17	semiring	semiring	NOUN
ejpam-2272	45	18	and	and	CCONJ
ejpam-2272	45	19	give	give	VERB
ejpam-2272	45	20	some	some	DET
ejpam-2272	45	21	generalizations	generalization	NOUN
ejpam-2272	45	22	of	of	ADP
ejpam-2272	45	23	[	[	X
ejpam-2272	45	24	5	5	NUM
ejpam-2272	45	25	,	,	PUNCT
ejpam-2272	45	26	6	6	NUM
ejpam-2272	45	27	,	,	PUNCT
ejpam-2272	45	28	10	10	NUM
ejpam-2272	45	29	]	]	PUNCT
ejpam-2272	45	30	and	and	CCONJ
ejpam-2272	45	31	[	[	X
ejpam-2272	45	32	12	12	NUM
ejpam-2272	45	33	]	]	PUNCT
ejpam-2272	45	34	which	which	PRON
ejpam-2272	45	35	are	be	AUX
ejpam-2272	45	36	analogous	analogous	ADJ
ejpam-2272	45	37	to	to	ADP
ejpam-2272	45	38	commutative	commutative	ADJ
ejpam-2272	45	39	ring	ring	NOUN
ejpam-2272	45	40	theory	theory	NOUN
ejpam-2272	45	41	.	.	PUNCT
ejpam-2272	46	1	throughout	throughout	ADP
ejpam-2272	46	2	this	this	DET
ejpam-2272	46	3	paper	paper	NOUN
ejpam-2272	46	4	,	,	PUNCT
ejpam-2272	46	5	semiring	semire	VERB
ejpam-2272	46	6	s	s	X
ejpam-2272	46	7	is	be	AUX
ejpam-2272	46	8	considered	consider	VERB
ejpam-2272	46	9	as	as	ADP
ejpam-2272	46	10	commutative	commutative	ADJ
ejpam-2272	46	11	with	with	ADP
ejpam-2272	46	12	identity	identity	NOUN
ejpam-2272	46	13	1	1	NUM
ejpam-2272	46	14	≠	≠	PROPN
ejpam-2272	46	15	0	0	NUM
ejpam-2272	46	16	.	.	NOUN
ejpam-2272	46	17	2	2	NUM
ejpam-2272	46	18	.	.	NOUN
ejpam-2272	46	19	2	2	NUM
ejpam-2272	46	20	-	-	PUNCT
ejpam-2272	46	21	absorbing	absorbing	ADJ
ejpam-2272	46	22	primary	primary	ADJ
ejpam-2272	46	23	ideals	ideal	NOUN
ejpam-2272	46	24	in	in	ADP
ejpam-2272	46	25	this	this	DET
ejpam-2272	46	26	section	section	NOUN
ejpam-2272	46	27	,	,	PUNCT
ejpam-2272	46	28	we	we	PRON
ejpam-2272	46	29	introduce	introduce	VERB
ejpam-2272	46	30	the	the	DET
ejpam-2272	46	31	concept	concept	NOUN
ejpam-2272	46	32	of	of	ADP
ejpam-2272	46	33	2	2	NUM
ejpam-2272	46	34	-	-	PUNCT
ejpam-2272	46	35	absorbing	absorbing	ADJ
ejpam-2272	46	36	primary	primary	ADJ
ejpam-2272	46	37	ideal	ideal	NOUN
ejpam-2272	46	38	of	of	ADP
ejpam-2272	46	39	a	a	DET
ejpam-2272	46	40	commutative	commutative	ADJ
ejpam-2272	46	41	semiring	semiring	NOUN
ejpam-2272	46	42	and	and	CCONJ
ejpam-2272	46	43	prove	prove	VERB
ejpam-2272	46	44	some	some	DET
ejpam-2272	46	45	results	result	NOUN
ejpam-2272	46	46	related	relate	VERB
ejpam-2272	46	47	to	to	ADP
ejpam-2272	46	48	it	it	PRON
ejpam-2272	46	49	.	.	PUNCT
ejpam-2272	47	1	definition	definition	NOUN
ejpam-2272	47	2	1	1	NUM
ejpam-2272	47	3	.	.	PUNCT
ejpam-2272	48	1	let	let	VERB
ejpam-2272	48	2	s	s	PRON
ejpam-2272	48	3	be	be	AUX
ejpam-2272	48	4	a	a	DET
ejpam-2272	48	5	commutative	commutative	ADJ
ejpam-2272	48	6	semiring	semiring	NOUN
ejpam-2272	48	7	and	and	CCONJ
ejpam-2272	48	8	i	i	PRON
ejpam-2272	48	9	be	be	VERB
ejpam-2272	48	10	a	a	DET
ejpam-2272	48	11	proper	proper	ADJ
ejpam-2272	48	12	ideal	ideal	NOUN
ejpam-2272	48	13	of	of	ADP
ejpam-2272	48	14	s.	s.	PROPN
ejpam-2272	49	1	then	then	ADV
ejpam-2272	49	2	i	i	PRON
ejpam-2272	49	3	is	be	AUX
ejpam-2272	49	4	said	say	VERB
ejpam-2272	49	5	to	to	PART
ejpam-2272	49	6	be	be	AUX
ejpam-2272	49	7	a	a	DET
ejpam-2272	49	8	2	2	NUM
ejpam-2272	49	9	-	-	PUNCT
ejpam-2272	49	10	absorbing	absorbing	ADJ
ejpam-2272	49	11	primary	primary	ADJ
ejpam-2272	49	12	ideal	ideal	NOUN
ejpam-2272	49	13	of	of	ADP
ejpam-2272	49	14	s	s	PRON
ejpam-2272	49	15	if	if	SCONJ
ejpam-2272	49	16	whenever	whenever	SCONJ
ejpam-2272	49	17	a	a	DET
ejpam-2272	49	18	,	,	PUNCT
ejpam-2272	49	19	b	b	NOUN
ejpam-2272	49	20	,	,	PUNCT
ejpam-2272	49	21	c	c	PROPN
ejpam-2272	49	22	∈	∈	PROPN
ejpam-2272	49	23	s	s	PART
ejpam-2272	49	24	and	and	CCONJ
ejpam-2272	49	25	abc	abc	PROPN
ejpam-2272	49	26	∈	∈	PROPN
ejpam-2272	50	1	i	i	PRON
ejpam-2272	50	2	,	,	PUNCT
ejpam-2272	50	3	then	then	ADV
ejpam-2272	50	4	ab	ab	PROPN
ejpam-2272	50	5	∈	∈	PROPN
ejpam-2272	51	1	i	i	PRON
ejpam-2272	51	2	or	or	CCONJ
ejpam-2272	51	3	ac	ac	PROPN
ejpam-2272	51	4	∈	∈	PROPN
ejpam-2272	52	1	√	√	VERB
ejpam-2272	52	2	i	i	PRON
ejpam-2272	52	3	or	or	CCONJ
ejpam-2272	52	4	bc	bc	PROPN
ejpam-2272	52	5	∈	∈	PROPN
ejpam-2272	53	1	√	√	VERB
ejpam-2272	53	2	i	i	PRON
ejpam-2272	53	3	.	.	PUNCT
ejpam-2272	54	1	it	it	PRON
ejpam-2272	54	2	is	be	AUX
ejpam-2272	54	3	easy	easy	ADJ
ejpam-2272	54	4	to	to	PART
ejpam-2272	54	5	see	see	VERB
ejpam-2272	54	6	that	that	SCONJ
ejpam-2272	54	7	every	every	DET
ejpam-2272	54	8	2	2	NUM
ejpam-2272	54	9	-	-	PUNCT
ejpam-2272	54	10	absorbing	absorbing	ADJ
ejpam-2272	54	11	ideal	ideal	NOUN
ejpam-2272	54	12	of	of	ADP
ejpam-2272	54	13	a	a	DET
ejpam-2272	54	14	commutative	commutative	ADJ
ejpam-2272	54	15	semiring	semiring	NOUN
ejpam-2272	54	16	s	s	VERB
ejpam-2272	54	17	is	be	AUX
ejpam-2272	54	18	a	a	DET
ejpam-2272	54	19	2	2	NUM
ejpam-2272	54	20	-	-	PUNCT
ejpam-2272	54	21	absorbing	absorbing	ADJ
ejpam-2272	54	22	primary	primary	ADJ
ejpam-2272	54	23	ideal	ideal	NOUN
ejpam-2272	54	24	of	of	ADP
ejpam-2272	54	25	s	s	NOUN
ejpam-2272	54	26	but	but	CCONJ
ejpam-2272	54	27	converse	converse	NOUN
ejpam-2272	54	28	need	need	AUX
ejpam-2272	54	29	not	not	PART
ejpam-2272	54	30	be	be	AUX
ejpam-2272	54	31	true	true	ADJ
ejpam-2272	54	32	.	.	PUNCT
ejpam-2272	55	1	for	for	ADP
ejpam-2272	55	2	instance	instance	NOUN
ejpam-2272	55	3	,	,	PUNCT
ejpam-2272	55	4	consider	consider	VERB
ejpam-2272	55	5	a	a	DET
ejpam-2272	55	6	semiring	semiring	NOUN
ejpam-2272	55	7	s	s	PART
ejpam-2272	55	8	=	=	NOUN
ejpam-2272	55	9	z+∪{0	z+∪{0	NOUN
ejpam-2272	55	10	}	}	PUNCT
ejpam-2272	55	11	and	and	CCONJ
ejpam-2272	55	12	an	an	DET
ejpam-2272	55	13	ideal	ideal	NOUN
ejpam-2272	56	1	i	i	X
ejpam-2272	56	2	=	=	SYM
ejpam-2272	56	3	⟨8⟩	⟨8⟩	PROPN
ejpam-2272	56	4	of	of	ADP
ejpam-2272	56	5	s.	s.	PROPN
ejpam-2272	57	1	then	then	ADV
ejpam-2272	57	2	i	i	PRON
ejpam-2272	57	3	is	be	AUX
ejpam-2272	57	4	a	a	DET
ejpam-2272	57	5	2	2	NUM
ejpam-2272	57	6	-	-	PUNCT
ejpam-2272	57	7	absorbing	absorbing	ADJ
ejpam-2272	57	8	primary	primary	ADJ
ejpam-2272	57	9	ideal	ideal	NOUN
ejpam-2272	57	10	of	of	ADP
ejpam-2272	57	11	s	s	PRON
ejpam-2272	58	1	but	but	CCONJ
ejpam-2272	58	2	it	it	PRON
ejpam-2272	58	3	is	be	AUX
ejpam-2272	58	4	not	not	PART
ejpam-2272	58	5	a	a	DET
ejpam-2272	58	6	2	2	NUM
ejpam-2272	58	7	-	-	PUNCT
ejpam-2272	58	8	absorbing	absorbing	ADJ
ejpam-2272	58	9	ideal	ideal	NOUN
ejpam-2272	58	10	of	of	ADP
ejpam-2272	58	11	s	s	PROPN
ejpam-2272	58	12	,	,	PUNCT
ejpam-2272	58	13	as	as	SCONJ
ejpam-2272	58	14	2.2.2	2.2.2	NUM
ejpam-2272	58	15	∈	∈	NOUN
ejpam-2272	58	16	⟨8⟩	⟨8⟩	NUM
ejpam-2272	58	17	but	but	CCONJ
ejpam-2272	58	18	2.2	2.2	NUM
ejpam-2272	58	19	∉	∉	PROPN
ejpam-2272	58	20	⟨8⟩.	⟨8⟩.	NOUN
ejpam-2272	58	21	also	also	ADV
ejpam-2272	58	22	,	,	PUNCT
ejpam-2272	58	23	every	every	DET
ejpam-2272	58	24	primary	primary	ADJ
ejpam-2272	58	25	ideal	ideal	NOUN
ejpam-2272	58	26	of	of	ADP
ejpam-2272	58	27	s	s	PROPN
ejpam-2272	58	28	is	be	AUX
ejpam-2272	58	29	a	a	DET
ejpam-2272	58	30	2	2	NUM
ejpam-2272	58	31	-	-	PUNCT
ejpam-2272	58	32	absorbing	absorbing	ADJ
ejpam-2272	58	33	primary	primary	ADJ
ejpam-2272	58	34	ideal	ideal	NOUN
ejpam-2272	58	35	of	of	ADP
ejpam-2272	58	36	s	s	NOUN
ejpam-2272	58	37	but	but	CCONJ
ejpam-2272	58	38	converse	converse	NOUN
ejpam-2272	58	39	is	be	AUX
ejpam-2272	58	40	not	not	PART
ejpam-2272	58	41	true	true	ADJ
ejpam-2272	58	42	,	,	PUNCT
ejpam-2272	58	43	as	as	SCONJ
ejpam-2272	58	44	⟨10⟩	⟨10⟩	NOUN
ejpam-2272	58	45	is	be	AUX
ejpam-2272	58	46	a	a	DET
ejpam-2272	58	47	2	2	NUM
ejpam-2272	58	48	-	-	PUNCT
ejpam-2272	58	49	absorbing	absorbing	ADJ
ejpam-2272	58	50	primary	primary	ADJ
ejpam-2272	58	51	ideal	ideal	NOUN
ejpam-2272	58	52	of	of	ADP
ejpam-2272	58	53	s	s	PRON
ejpam-2272	59	1	but	but	CCONJ
ejpam-2272	59	2	it	it	PRON
ejpam-2272	59	3	is	be	AUX
ejpam-2272	59	4	not	not	PART
ejpam-2272	59	5	a	a	DET
ejpam-2272	59	6	primary	primary	ADJ
ejpam-2272	59	7	ideal	ideal	NOUN
ejpam-2272	59	8	of	of	ADP
ejpam-2272	59	9	s.	s.	PROPN
ejpam-2272	59	10	theorem	theorem	VERB
ejpam-2272	59	11	1	1	X
ejpam-2272	59	12	.	.	PUNCT
ejpam-2272	60	1	let	let	VERB
ejpam-2272	60	2	f	f	PROPN
ejpam-2272	60	3	∶	∶	NOUN
ejpam-2272	60	4	s	s	PART
ejpam-2272	60	5	↦	↦	PROPN
ejpam-2272	60	6	s′	s′	NUM
ejpam-2272	60	7	be	be	VERB
ejpam-2272	60	8	a	a	DET
ejpam-2272	60	9	homomorphism	homomorphism	NOUN
ejpam-2272	60	10	of	of	ADP
ejpam-2272	60	11	commutative	commutative	ADJ
ejpam-2272	60	12	semirings	semiring	NOUN
ejpam-2272	60	13	.	.	PUNCT
ejpam-2272	61	1	then	then	ADV
ejpam-2272	61	2	,	,	PUNCT
ejpam-2272	61	3	if	if	SCONJ
ejpam-2272	61	4	i	i	PRON
ejpam-2272	61	5	′	′	VERB
ejpam-2272	61	6	is	be	AUX
ejpam-2272	61	7	a	a	DET
ejpam-2272	61	8	2	2	NUM
ejpam-2272	61	9	-	-	PUNCT
ejpam-2272	61	10	absorbing	absorbing	ADJ
ejpam-2272	61	11	primary	primary	ADJ
ejpam-2272	61	12	ideal	ideal	NOUN
ejpam-2272	61	13	of	of	ADP
ejpam-2272	61	14	s′	s′	PROPN
ejpam-2272	61	15	,	,	PUNCT
ejpam-2272	61	16	then	then	ADV
ejpam-2272	61	17	f	f	PROPN
ejpam-2272	61	18	−1(i	−1(i	PROPN
ejpam-2272	61	19	′	′	NOUN
ejpam-2272	61	20	)	)	PUNCT
ejpam-2272	61	21	is	be	AUX
ejpam-2272	61	22	a	a	DET
ejpam-2272	61	23	2	2	NUM
ejpam-2272	61	24	-	-	PUNCT
ejpam-2272	61	25	absorbing	absorbing	ADJ
ejpam-2272	61	26	primary	primary	ADJ
ejpam-2272	61	27	ideal	ideal	NOUN
ejpam-2272	61	28	of	of	ADP
ejpam-2272	61	29	s.	s.	PROPN
ejpam-2272	61	30	proof	proof	PROPN
ejpam-2272	61	31	.	.	PUNCT
ejpam-2272	62	1	let	let	VERB
ejpam-2272	62	2	abc	abc	PROPN
ejpam-2272	62	3	∈	∈	PROPN
ejpam-2272	62	4	f	f	PROPN
ejpam-2272	62	5	−1(i	−1(i	PROPN
ejpam-2272	62	6	′	′	NOUN
ejpam-2272	62	7	)	)	PUNCT
ejpam-2272	62	8	for	for	ADP
ejpam-2272	62	9	some	some	DET
ejpam-2272	62	10	a	a	DET
ejpam-2272	62	11	,	,	PUNCT
ejpam-2272	62	12	b	b	NOUN
ejpam-2272	62	13	,	,	PUNCT
ejpam-2272	62	14	c	c	PROPN
ejpam-2272	62	15	∈	∈	PROPN
ejpam-2272	62	16	s.	s.	PROPN
ejpam-2272	63	1	then	then	ADV
ejpam-2272	63	2	f	f	PROPN
ejpam-2272	63	3	(	(	PUNCT
ejpam-2272	63	4	abc	abc	PROPN
ejpam-2272	63	5	)	)	PUNCT
ejpam-2272	63	6	∈	∈	PROPN
ejpam-2272	64	1	i	i	PRON
ejpam-2272	64	2	′	′	VERB
ejpam-2272	64	3	,	,	PUNCT
ejpam-2272	64	4	that	that	ADV
ejpam-2272	64	5	is	is	ADV
ejpam-2272	64	6	,	,	PUNCT
ejpam-2272	64	7	f	f	PROPN
ejpam-2272	64	8	(	(	PUNCT
ejpam-2272	64	9	a	a	NOUN
ejpam-2272	64	10	)	)	PUNCT
ejpam-2272	64	11	f	f	NOUN
ejpam-2272	64	12	(	(	PUNCT
ejpam-2272	64	13	b	b	NOUN
ejpam-2272	64	14	)	)	PUNCT
ejpam-2272	64	15	f	f	NOUN
ejpam-2272	64	16	(	(	PUNCT
ejpam-2272	64	17	c	c	X
ejpam-2272	64	18	)	)	PUNCT
ejpam-2272	64	19	∈	∈	PROPN
ejpam-2272	64	20	i	i	PRON
ejpam-2272	64	21	′.	′.	NOUN
ejpam-2272	64	22	since	since	SCONJ
ejpam-2272	64	23	i	i	PRON
ejpam-2272	64	24	′	′	VERB
ejpam-2272	64	25	is	be	AUX
ejpam-2272	64	26	a	a	DET
ejpam-2272	64	27	2	2	NUM
ejpam-2272	64	28	-	-	PUNCT
ejpam-2272	64	29	absorbing	absorbing	ADJ
ejpam-2272	64	30	primary	primary	ADJ
ejpam-2272	64	31	ideal	ideal	NOUN
ejpam-2272	64	32	of	of	ADP
ejpam-2272	64	33	s′	s′	PROPN
ejpam-2272	64	34	,	,	PUNCT
ejpam-2272	64	35	therefore	therefore	ADV
ejpam-2272	64	36	f	f	X
ejpam-2272	64	37	(	(	PUNCT
ejpam-2272	64	38	a	a	PROPN
ejpam-2272	64	39	)	)	PUNCT
ejpam-2272	64	40	f	f	NOUN
ejpam-2272	64	41	(	(	PUNCT
ejpam-2272	64	42	b	b	X
ejpam-2272	64	43	)	)	PUNCT
ejpam-2272	64	44	∈	∈	NOUN
ejpam-2272	65	1	i	i	PRON
ejpam-2272	65	2	′	′	VERB
ejpam-2272	65	3	or	or	CCONJ
ejpam-2272	65	4	f	f	X
ejpam-2272	65	5	(	(	PUNCT
ejpam-2272	65	6	b	b	NOUN
ejpam-2272	65	7	)	)	PUNCT
ejpam-2272	65	8	f	f	NOUN
ejpam-2272	65	9	(	(	PUNCT
ejpam-2272	65	10	c	c	X
ejpam-2272	65	11	)	)	PUNCT
ejpam-2272	65	12	∈	∈	NOUN
ejpam-2272	66	1	√	√	VERB
ejpam-2272	67	1	i	i	PRON
ejpam-2272	67	2	′	′	VERB
ejpam-2272	67	3	or	or	CCONJ
ejpam-2272	67	4	f	f	X
ejpam-2272	67	5	(	(	PUNCT
ejpam-2272	67	6	c	c	NOUN
ejpam-2272	67	7	)	)	PUNCT
ejpam-2272	67	8	f	f	NOUN
ejpam-2272	67	9	(	(	PUNCT
ejpam-2272	67	10	a	a	X
ejpam-2272	67	11	)	)	PUNCT
ejpam-2272	67	12	∈	∈	NOUN
ejpam-2272	68	1	√	√	VERB
ejpam-2272	68	2	i	i	PRON
ejpam-2272	68	3	′.	′.	NOUN
ejpam-2272	68	4	hence	hence	ADV
ejpam-2272	68	5	,	,	PUNCT
ejpam-2272	68	6	ab	ab	PROPN
ejpam-2272	68	7	∈	∈	PROPN
ejpam-2272	68	8	f	f	PROPN
ejpam-2272	68	9	−1(i	−1(i	PROPN
ejpam-2272	68	10	′	′	NOUN
ejpam-2272	68	11	)	)	PUNCT
ejpam-2272	68	12	or	or	CCONJ
ejpam-2272	68	13	bc	bc	PROPN
ejpam-2272	68	14	∈	∈	PROPN
ejpam-2272	68	15	f	f	PROPN
ejpam-2272	68	16	−1	−1	NOUN
ejpam-2272	68	17	(	(	PUNCT
ejpam-2272	68	18	√	√	PROPN
ejpam-2272	68	19	i	i	PRON
ejpam-2272	68	20	′	′	NOUN
ejpam-2272	68	21	)	)	PUNCT
ejpam-2272	68	22	or	or	CCONJ
ejpam-2272	68	23	ca	ca	NOUN
ejpam-2272	68	24	∈	∈	PROPN
ejpam-2272	68	25	f	f	PROPN
ejpam-2272	68	26	−1	−1	NOUN
ejpam-2272	68	27	(	(	PUNCT
ejpam-2272	68	28	√	√	PROPN
ejpam-2272	68	29	i	i	PRON
ejpam-2272	68	30	′	′	NOUN
ejpam-2272	68	31	)	)	PUNCT
ejpam-2272	68	32	.	.	PUNCT
ejpam-2272	69	1	since	since	SCONJ
ejpam-2272	69	2	f	f	PROPN
ejpam-2272	69	3	−1	−1	VERB
ejpam-2272	69	4	(	(	PUNCT
ejpam-2272	69	5	√	√	PROPN
ejpam-2272	69	6	i	i	PRON
ejpam-2272	69	7	′	′	NUM
ejpam-2272	69	8	)	)	PUNCT
ejpam-2272	69	9	⊆	⊆	NUM
ejpam-2272	69	10	√	√	PROPN
ejpam-2272	69	11	f	f	PROPN
ejpam-2272	69	12	−1(i	−1(i	NUM
ejpam-2272	69	13	′	′	NUM
ejpam-2272	69	14	)	)	PUNCT
ejpam-2272	69	15	,	,	PUNCT
ejpam-2272	69	16	we	we	PRON
ejpam-2272	69	17	have	have	AUX
ejpam-2272	69	18	f	f	PROPN
ejpam-2272	69	19	−1(i	−1(i	NOUN
ejpam-2272	69	20	′	′	NOUN
ejpam-2272	69	21	)	)	PUNCT
ejpam-2272	69	22	is	be	AUX
ejpam-2272	69	23	a	a	DET
ejpam-2272	69	24	2	2	NUM
ejpam-2272	69	25	-	-	PUNCT
ejpam-2272	69	26	absorbing	absorbing	ADJ
ejpam-2272	69	27	primary	primary	ADJ
ejpam-2272	69	28	ideal	ideal	NOUN
ejpam-2272	69	29	of	of	ADP
ejpam-2272	69	30	s.	s.	PROPN
ejpam-2272	69	31	p.	p.	PROPN
ejpam-2272	69	32	kumar	kumar	PROPN
ejpam-2272	69	33	,	,	PUNCT
ejpam-2272	69	34	m.	m.	NOUN
ejpam-2272	69	35	dubey	dubey	PROPN
ejpam-2272	69	36	,	,	PUNCT
ejpam-2272	69	37	p.	p.	NOUN
ejpam-2272	69	38	sarohe	sarohe	PROPN
ejpam-2272	69	39	/	/	SYM
ejpam-2272	69	40	eur	eur	PROPN
ejpam-2272	69	41	.	.	PUNCT
ejpam-2272	70	1	j.	j.	PROPN
ejpam-2272	70	2	pure	pure	PROPN
ejpam-2272	70	3	appl	appl	PROPN
ejpam-2272	70	4	.	.	PROPN
ejpam-2272	70	5	math	math	PROPN
ejpam-2272	70	6	,	,	PUNCT
ejpam-2272	70	7	9	9	NUM
ejpam-2272	70	8	(	(	PUNCT
ejpam-2272	70	9	2016	2016	NUM
ejpam-2272	70	10	)	)	PUNCT
ejpam-2272	70	11	,	,	PUNCT
ejpam-2272	70	12	186	186	NUM
ejpam-2272	70	13	-	-	SYM
ejpam-2272	70	14	195	195	NUM
ejpam-2272	70	15	188	188	NUM
ejpam-2272	70	16	theorem	theorem	NOUN
ejpam-2272	70	17	2	2	NUM
ejpam-2272	70	18	.	.	PUNCT
ejpam-2272	71	1	if	if	SCONJ
ejpam-2272	71	2	i	i	PRON
ejpam-2272	71	3	is	be	AUX
ejpam-2272	71	4	a	a	DET
ejpam-2272	71	5	2	2	NUM
ejpam-2272	71	6	-	-	PUNCT
ejpam-2272	71	7	absorbing	absorbing	ADJ
ejpam-2272	71	8	primary	primary	ADJ
ejpam-2272	71	9	ideal	ideal	NOUN
ejpam-2272	71	10	of	of	ADP
ejpam-2272	71	11	a	a	DET
ejpam-2272	71	12	semiring	semiring	NOUN
ejpam-2272	71	13	s	s	PART
ejpam-2272	71	14	,	,	PUNCT
ejpam-2272	71	15	then	then	ADV
ejpam-2272	71	16	√	√	VERB
ejpam-2272	71	17	i	i	PRON
ejpam-2272	71	18	is	be	AUX
ejpam-2272	71	19	a	a	DET
ejpam-2272	71	20	2	2	NUM
ejpam-2272	71	21	-	-	PUNCT
ejpam-2272	71	22	absorbing	absorbing	ADJ
ejpam-2272	71	23	ideal	ideal	NOUN
ejpam-2272	71	24	of	of	ADP
ejpam-2272	71	25	s.	s.	PROPN
ejpam-2272	71	26	proof	proof	PROPN
ejpam-2272	71	27	.	.	PUNCT
ejpam-2272	72	1	let	let	VERB
ejpam-2272	72	2	abc	abc	PROPN
ejpam-2272	72	3	∈	∈	PROPN
ejpam-2272	72	4	√	√	VERB
ejpam-2272	72	5	i	i	PRON
ejpam-2272	72	6	for	for	ADP
ejpam-2272	72	7	some	some	DET
ejpam-2272	72	8	a	a	DET
ejpam-2272	72	9	,	,	PUNCT
ejpam-2272	72	10	b	b	NOUN
ejpam-2272	72	11	,	,	PUNCT
ejpam-2272	72	12	c	c	PROPN
ejpam-2272	72	13	∈	∈	PROPN
ejpam-2272	72	14	s.	s.	PROPN
ejpam-2272	72	15	suppose	suppose	VERB
ejpam-2272	72	16	that	that	SCONJ
ejpam-2272	72	17	ac	ac	PROPN
ejpam-2272	72	18	∉	∉	PROPN
ejpam-2272	72	19	√	√	PROPN
ejpam-2272	72	20	i	i	PRON
ejpam-2272	72	21	and	and	CCONJ
ejpam-2272	72	22	bc	bc	PROPN
ejpam-2272	72	23	∉	∉	PROPN
ejpam-2272	72	24	√	√	PROPN
ejpam-2272	72	25	i	i	PRON
ejpam-2272	72	26	.	.	PUNCT
ejpam-2272	73	1	since	since	SCONJ
ejpam-2272	73	2	abc	abc	PROPN
ejpam-2272	73	3	∈	∈	PROPN
ejpam-2272	73	4	√	√	VERB
ejpam-2272	73	5	i	i	PRON
ejpam-2272	73	6	,	,	PUNCT
ejpam-2272	73	7	then	then	ADV
ejpam-2272	73	8	there	there	PRON
ejpam-2272	73	9	exists	exist	VERB
ejpam-2272	73	10	a	a	DET
ejpam-2272	73	11	positive	positive	ADJ
ejpam-2272	73	12	integer	integer	NOUN
ejpam-2272	73	13	n	n	CCONJ
ejpam-2272	73	14	such	such	ADJ
ejpam-2272	73	15	that	that	PRON
ejpam-2272	73	16	(	(	PUNCT
ejpam-2272	74	1	abc)n	abc)n	NOUN
ejpam-2272	74	2	=	=	PUNCT
ejpam-2272	74	3	an	an	DET
ejpam-2272	74	4	bncn	bncn	NOUN
ejpam-2272	74	5	∈	∈	NOUN
ejpam-2272	74	6	i	i	PRON
ejpam-2272	74	7	.	.	PUNCT
ejpam-2272	75	1	this	this	PRON
ejpam-2272	75	2	gives	give	VERB
ejpam-2272	75	3	an	an	DET
ejpam-2272	75	4	bn	bn	NOUN
ejpam-2272	75	5	∈	∈	NOUN
ejpam-2272	76	1	i	i	PRON
ejpam-2272	76	2	,	,	PUNCT
ejpam-2272	76	3	since	since	SCONJ
ejpam-2272	76	4	i	i	PRON
ejpam-2272	76	5	is	be	AUX
ejpam-2272	76	6	a	a	DET
ejpam-2272	76	7	2	2	NUM
ejpam-2272	76	8	-	-	PUNCT
ejpam-2272	76	9	absorbing	absorbing	ADJ
ejpam-2272	76	10	primary	primary	ADJ
ejpam-2272	76	11	ideal	ideal	NOUN
ejpam-2272	76	12	of	of	ADP
ejpam-2272	76	13	s	s	PROPN
ejpam-2272	76	14	and	and	CCONJ
ejpam-2272	76	15	ac	ac	PROPN
ejpam-2272	76	16	∉	∉	PROPN
ejpam-2272	76	17	√	√	PROPN
ejpam-2272	76	18	i	i	PRON
ejpam-2272	76	19	and	and	CCONJ
ejpam-2272	76	20	bc	bc	PROPN
ejpam-2272	76	21	∉	∉	PROPN
ejpam-2272	76	22	√	√	PROPN
ejpam-2272	76	23	i	i	PRON
ejpam-2272	76	24	.	.	PUNCT
ejpam-2272	77	1	hence	hence	ADV
ejpam-2272	77	2	,	,	PUNCT
ejpam-2272	77	3	ab	ab	PROPN
ejpam-2272	77	4	∈	∈	PROPN
ejpam-2272	77	5	√	√	VERB
ejpam-2272	77	6	i	i	PRON
ejpam-2272	77	7	.	.	PUNCT
ejpam-2272	78	1	thus	thus	ADV
ejpam-2272	78	2	,	,	PUNCT
ejpam-2272	78	3	√	√	VERB
ejpam-2272	78	4	i	i	PRON
ejpam-2272	78	5	is	be	AUX
ejpam-2272	78	6	a	a	DET
ejpam-2272	78	7	2	2	NUM
ejpam-2272	78	8	-	-	PUNCT
ejpam-2272	78	9	absorbing	absorbing	ADJ
ejpam-2272	78	10	ideal	ideal	NOUN
ejpam-2272	78	11	of	of	ADP
ejpam-2272	78	12	s.	s.	PROPN
ejpam-2272	78	13	corollary	corollary	PROPN
ejpam-2272	78	14	1	1	NUM
ejpam-2272	78	15	.	.	PUNCT
ejpam-2272	79	1	let	let	VERB
ejpam-2272	79	2	i	i	PRON
ejpam-2272	79	3	be	be	AUX
ejpam-2272	79	4	an	an	DET
ejpam-2272	79	5	ideal	ideal	NOUN
ejpam-2272	79	6	of	of	ADP
ejpam-2272	79	7	a	a	DET
ejpam-2272	79	8	semiring	semire	VERB
ejpam-2272	79	9	s.	s.	PROPN
ejpam-2272	79	10	then	then	ADV
ejpam-2272	79	11	the	the	DET
ejpam-2272	79	12	following	follow	VERB
ejpam-2272	79	13	statements	statement	NOUN
ejpam-2272	79	14	are	be	AUX
ejpam-2272	79	15	equivalent	equivalent	ADJ
ejpam-2272	79	16	:	:	PUNCT
ejpam-2272	79	17	(	(	PUNCT
ejpam-2272	79	18	1	1	X
ejpam-2272	79	19	)	)	PUNCT
ejpam-2272	79	20	i	i	PRON
ejpam-2272	79	21	is	be	AUX
ejpam-2272	79	22	a	a	DET
ejpam-2272	79	23	2	2	NUM
ejpam-2272	79	24	-	-	PUNCT
ejpam-2272	79	25	absorbing	absorbing	ADJ
ejpam-2272	79	26	primary	primary	ADJ
ejpam-2272	79	27	ideal	ideal	NOUN
ejpam-2272	79	28	of	of	ADP
ejpam-2272	79	29	s.	s.	PROPN
ejpam-2272	79	30	(	(	PUNCT
ejpam-2272	79	31	2	2	NUM
ejpam-2272	79	32	)	)	PUNCT
ejpam-2272	79	33	√	√	NOUN
ejpam-2272	80	1	i	i	PRON
ejpam-2272	80	2	is	be	AUX
ejpam-2272	80	3	a	a	DET
ejpam-2272	80	4	2	2	NUM
ejpam-2272	80	5	-	-	PUNCT
ejpam-2272	80	6	absorbing	absorbing	ADJ
ejpam-2272	80	7	ideal	ideal	NOUN
ejpam-2272	80	8	of	of	ADP
ejpam-2272	80	9	s	s	PRON
ejpam-2272	80	10	and	and	CCONJ
ejpam-2272	80	11	if	if	SCONJ
ejpam-2272	80	12	abc	abc	PROPN
ejpam-2272	80	13	∈	∈	PROPN
ejpam-2272	80	14	i	i	PRON
ejpam-2272	80	15	with	with	ADP
ejpam-2272	80	16	bc	bc	PROPN
ejpam-2272	80	17	∉	∉	PROPN
ejpam-2272	80	18	√	√	PROPN
ejpam-2272	81	1	i	i	PRON
ejpam-2272	81	2	and	and	CCONJ
ejpam-2272	81	3	ca	can	AUX
ejpam-2272	81	4	∉	∉	PROPN
ejpam-2272	81	5	√	√	PROPN
ejpam-2272	82	1	i	i	PRON
ejpam-2272	82	2	then	then	ADV
ejpam-2272	82	3	ab	ab	PROPN
ejpam-2272	82	4	∈	∈	PROPN
ejpam-2272	83	1	i	i	PRON
ejpam-2272	83	2	.	.	PUNCT
ejpam-2272	84	1	definition	definition	NOUN
ejpam-2272	84	2	2	2	NUM
ejpam-2272	84	3	.	.	PUNCT
ejpam-2272	85	1	let	let	VERB
ejpam-2272	85	2	i	i	PRON
ejpam-2272	85	3	be	be	AUX
ejpam-2272	85	4	a	a	DET
ejpam-2272	85	5	2	2	NUM
ejpam-2272	85	6	-	-	PUNCT
ejpam-2272	85	7	absorbing	absorbing	ADJ
ejpam-2272	85	8	primary	primary	ADJ
ejpam-2272	85	9	ideal	ideal	NOUN
ejpam-2272	85	10	of	of	ADP
ejpam-2272	85	11	a	a	DET
ejpam-2272	85	12	semiring	semire	VERB
ejpam-2272	85	13	s.	s.	PROPN
ejpam-2272	85	14	then	then	ADV
ejpam-2272	85	15	by	by	ADP
ejpam-2272	85	16	above	above	ADP
ejpam-2272	85	17	theorem	theorem	ADJ
ejpam-2272	85	18	p	p	PROPN
ejpam-2272	85	19	=	=	PUNCT
ejpam-2272	85	20	√	√	PROPN
ejpam-2272	86	1	i	i	PRON
ejpam-2272	86	2	is	be	AUX
ejpam-2272	86	3	a	a	DET
ejpam-2272	86	4	2	2	NUM
ejpam-2272	86	5	-	-	PUNCT
ejpam-2272	86	6	absorbing	absorbing	ADJ
ejpam-2272	86	7	ideal	ideal	NOUN
ejpam-2272	86	8	of	of	ADP
ejpam-2272	86	9	s.	s.	PROPN
ejpam-2272	86	10	in	in	ADP
ejpam-2272	86	11	this	this	DET
ejpam-2272	86	12	case	case	NOUN
ejpam-2272	86	13	,	,	PUNCT
ejpam-2272	86	14	i	i	PRON
ejpam-2272	86	15	is	be	AUX
ejpam-2272	86	16	said	say	VERB
ejpam-2272	86	17	to	to	PART
ejpam-2272	86	18	be	be	AUX
ejpam-2272	86	19	a	a	DET
ejpam-2272	86	20	p	p	NOUN
ejpam-2272	86	21	−	−	PROPN
ejpam-2272	86	22	2	2	NUM
ejpam-2272	86	23	-	-	PUNCT
ejpam-2272	86	24	absorbing	absorbing	ADJ
ejpam-2272	86	25	primary	primary	ADJ
ejpam-2272	86	26	ideal	ideal	NOUN
ejpam-2272	86	27	of	of	ADP
ejpam-2272	86	28	s.	s.	PROPN
ejpam-2272	86	29	theorem	theorem	VERB
ejpam-2272	86	30	3	3	X
ejpam-2272	86	31	.	.	PUNCT
ejpam-2272	87	1	let	let	VERB
ejpam-2272	87	2	i1	i1	PROPN
ejpam-2272	87	3	,	,	PUNCT
ejpam-2272	87	4	i2	i2	PROPN
ejpam-2272	87	5	,	,	PUNCT
ejpam-2272	87	6	.	.	PUNCT
ejpam-2272	87	7	.	.	PUNCT
ejpam-2272	88	1	.	.	PUNCT
ejpam-2272	89	1	,	,	PUNCT
ejpam-2272	89	2	in	in	ADP
ejpam-2272	89	3	be	be	AUX
ejpam-2272	89	4	p	p	ADJ
ejpam-2272	89	5	−	−	PROPN
ejpam-2272	89	6	2	2	NUM
ejpam-2272	89	7	-	-	PUNCT
ejpam-2272	89	8	absorbing	absorbing	ADJ
ejpam-2272	89	9	primary	primary	ADJ
ejpam-2272	89	10	ideals	ideal	NOUN
ejpam-2272	89	11	of	of	ADP
ejpam-2272	89	12	s	s	NOUN
ejpam-2272	89	13	,	,	PUNCT
ejpam-2272	89	14	where	where	SCONJ
ejpam-2272	89	15	p	p	NOUN
ejpam-2272	89	16	is	be	AUX
ejpam-2272	89	17	a	a	DET
ejpam-2272	89	18	2	2	NUM
ejpam-2272	89	19	-	-	PUNCT
ejpam-2272	89	20	absorbing	absorbing	ADJ
ejpam-2272	89	21	ideal	ideal	NOUN
ejpam-2272	89	22	of	of	ADP
ejpam-2272	89	23	s.	s.	PROPN
ejpam-2272	90	1	then	then	ADV
ejpam-2272	90	2	i	i	PRON
ejpam-2272	90	3	=	=	SYM
ejpam-2272	90	4	n	n	PROPN
ejpam-2272	90	5	⋂	⋂	PROPN
ejpam-2272	90	6	i	i	PRON
ejpam-2272	90	7	=	=	NOUN
ejpam-2272	90	8	i	i	NOUN
ejpam-2272	90	9	ii	ii	NOUN
ejpam-2272	90	10	is	be	AUX
ejpam-2272	90	11	a	a	DET
ejpam-2272	90	12	p	p	NOUN
ejpam-2272	90	13	−	−	PROPN
ejpam-2272	90	14	2−absorbing	2−absorbing	NUM
ejpam-2272	90	15	primary	primary	ADJ
ejpam-2272	90	16	ideal	ideal	NOUN
ejpam-2272	90	17	of	of	ADP
ejpam-2272	90	18	s.	s.	PROPN
ejpam-2272	90	19	proof	proof	PROPN
ejpam-2272	90	20	.	.	PUNCT
ejpam-2272	91	1	proof	proof	NOUN
ejpam-2272	91	2	is	be	AUX
ejpam-2272	91	3	similar	similar	ADJ
ejpam-2272	91	4	to	to	ADP
ejpam-2272	91	5	[	[	X
ejpam-2272	91	6	9	9	NUM
ejpam-2272	91	7	,	,	PUNCT
ejpam-2272	91	8	theorem	theorem	VERB
ejpam-2272	91	9	2.16	2.16	NUM
ejpam-2272	91	10	]	]	PUNCT
ejpam-2272	91	11	.	.	PUNCT
ejpam-2272	92	1	theorem	theorem	ADJ
ejpam-2272	92	2	4	4	NUM
ejpam-2272	92	3	.	.	PUNCT
ejpam-2272	93	1	let	let	VERB
ejpam-2272	93	2	s	s	PRON
ejpam-2272	93	3	be	be	AUX
ejpam-2272	93	4	a	a	DET
ejpam-2272	93	5	semiring	semiring	NOUN
ejpam-2272	93	6	.	.	PUNCT
ejpam-2272	94	1	suppose	suppose	VERB
ejpam-2272	94	2	that	that	SCONJ
ejpam-2272	94	3	i1	i1	PROPN
ejpam-2272	94	4	is	be	AUX
ejpam-2272	94	5	a	a	DET
ejpam-2272	94	6	p1−primary	p1−primary	ADJ
ejpam-2272	94	7	ideal	ideal	NOUN
ejpam-2272	94	8	of	of	ADP
ejpam-2272	94	9	s	s	PRON
ejpam-2272	94	10	for	for	ADP
ejpam-2272	94	11	some	some	DET
ejpam-2272	94	12	prime	prime	ADJ
ejpam-2272	94	13	ideal	ideal	NOUN
ejpam-2272	94	14	p1	p1	NOUN
ejpam-2272	94	15	of	of	ADP
ejpam-2272	94	16	s	s	PROPN
ejpam-2272	94	17	,	,	PUNCT
ejpam-2272	94	18	and	and	CCONJ
ejpam-2272	94	19	i2	i2	PROPN
ejpam-2272	94	20	is	be	AUX
ejpam-2272	94	21	a	a	DET
ejpam-2272	94	22	p2−primary	p2−primary	ADJ
ejpam-2272	94	23	ideal	ideal	NOUN
ejpam-2272	94	24	of	of	ADP
ejpam-2272	94	25	s	s	PRON
ejpam-2272	94	26	for	for	ADP
ejpam-2272	94	27	some	some	DET
ejpam-2272	94	28	prime	prime	ADJ
ejpam-2272	94	29	ideal	ideal	NOUN
ejpam-2272	94	30	p2	p2	PROPN
ejpam-2272	94	31	of	of	ADP
ejpam-2272	94	32	s.	s.	PROPN
ejpam-2272	94	33	then	then	ADV
ejpam-2272	94	34	the	the	DET
ejpam-2272	94	35	following	follow	VERB
ejpam-2272	94	36	statements	statement	NOUN
ejpam-2272	94	37	hold	hold	VERB
ejpam-2272	94	38	:	:	PUNCT
ejpam-2272	94	39	(	(	PUNCT
ejpam-2272	94	40	1	1	X
ejpam-2272	94	41	)	)	PUNCT
ejpam-2272	94	42	i1	i1	PROPN
ejpam-2272	94	43	i2	i2	PROPN
ejpam-2272	94	44	is	be	AUX
ejpam-2272	94	45	a	a	DET
ejpam-2272	94	46	2	2	NUM
ejpam-2272	94	47	-	-	PUNCT
ejpam-2272	94	48	absorbing	absorbing	ADJ
ejpam-2272	94	49	primary	primary	ADJ
ejpam-2272	94	50	ideal	ideal	NOUN
ejpam-2272	94	51	of	of	ADP
ejpam-2272	94	52	s.	s.	PROPN
ejpam-2272	94	53	(	(	PUNCT
ejpam-2272	94	54	2	2	X
ejpam-2272	94	55	)	)	PUNCT
ejpam-2272	94	56	i1	i1	PROPN
ejpam-2272	94	57	∩	∩	PROPN
ejpam-2272	94	58	i2	i2	PROPN
ejpam-2272	94	59	is	be	AUX
ejpam-2272	94	60	a	a	DET
ejpam-2272	94	61	2	2	NUM
ejpam-2272	94	62	-	-	PUNCT
ejpam-2272	94	63	absorbing	absorbing	ADJ
ejpam-2272	94	64	primary	primary	ADJ
ejpam-2272	94	65	ideal	ideal	NOUN
ejpam-2272	94	66	of	of	ADP
ejpam-2272	94	67	s.	s.	PROPN
ejpam-2272	94	68	proof	proof	PROPN
ejpam-2272	94	69	.	.	PUNCT
ejpam-2272	95	1	proof	proof	NOUN
ejpam-2272	95	2	is	be	AUX
ejpam-2272	95	3	similar	similar	ADJ
ejpam-2272	95	4	to	to	ADP
ejpam-2272	95	5	[	[	X
ejpam-2272	95	6	9	9	NUM
ejpam-2272	95	7	,	,	PUNCT
ejpam-2272	95	8	theorem	theorem	VERB
ejpam-2272	95	9	2.4	2.4	NUM
ejpam-2272	95	10	]	]	PUNCT
ejpam-2272	95	11	.	.	PUNCT
ejpam-2272	96	1	theorem	theorem	NOUN
ejpam-2272	96	2	5	5	NUM
ejpam-2272	96	3	.	.	PUNCT
ejpam-2272	97	1	let	let	VERB
ejpam-2272	97	2	i	i	PRON
ejpam-2272	97	3	be	be	AUX
ejpam-2272	97	4	a	a	DET
ejpam-2272	97	5	2	2	NUM
ejpam-2272	97	6	-	-	PUNCT
ejpam-2272	97	7	absorbing	absorbing	ADJ
ejpam-2272	97	8	primary	primary	ADJ
ejpam-2272	97	9	ideal	ideal	NOUN
ejpam-2272	97	10	of	of	ADP
ejpam-2272	97	11	s	s	PRON
ejpam-2272	97	12	such	such	ADJ
ejpam-2272	97	13	that	that	PRON
ejpam-2272	97	14	√	√	PROPN
ejpam-2272	98	1	i	i	PRON
ejpam-2272	98	2	=	=	PRON
ejpam-2272	99	1	p	p	NOUN
ejpam-2272	99	2	is	be	AUX
ejpam-2272	99	3	a	a	DET
ejpam-2272	99	4	prime	prime	ADJ
ejpam-2272	99	5	ideal	ideal	NOUN
ejpam-2272	99	6	of	of	ADP
ejpam-2272	99	7	s.	s.	PROPN
ejpam-2272	99	8	then	then	ADV
ejpam-2272	99	9	(	(	PUNCT
ejpam-2272	99	10	i	i	PRON
ejpam-2272	99	11	∶	∶	VERB
ejpam-2272	99	12	x	x	VERB
ejpam-2272	99	13	)	)	PUNCT
ejpam-2272	99	14	is	be	AUX
ejpam-2272	99	15	a	a	DET
ejpam-2272	99	16	2	2	NUM
ejpam-2272	99	17	-	-	PUNCT
ejpam-2272	99	18	absorbing	absorbing	ADJ
ejpam-2272	99	19	primary	primary	ADJ
ejpam-2272	99	20	ideal	ideal	NOUN
ejpam-2272	99	21	of	of	ADP
ejpam-2272	99	22	s	s	PRON
ejpam-2272	99	23	with	with	ADP
ejpam-2272	99	24	√	√	PROPN
ejpam-2272	99	25	(	(	PUNCT
ejpam-2272	99	26	i	i	PRON
ejpam-2272	99	27	∶	∶	VERB
ejpam-2272	99	28	x	x	NOUN
ejpam-2272	99	29	)	)	PUNCT
ejpam-2272	100	1	=	=	SYM
ejpam-2272	100	2	p	p	NOUN
ejpam-2272	100	3	for	for	ADP
ejpam-2272	100	4	all	all	DET
ejpam-2272	100	5	x	x	PART
ejpam-2272	100	6	∈	∈	PROPN
ejpam-2272	100	7	s	s	PART
ejpam-2272	100	8	∖	∖	NOUN
ejpam-2272	100	9	√	√	VERB
ejpam-2272	100	10	i	i	PRON
ejpam-2272	100	11	,	,	PUNCT
ejpam-2272	100	12	where	where	SCONJ
ejpam-2272	100	13	(	(	PUNCT
ejpam-2272	100	14	i	i	PRON
ejpam-2272	100	15	∶	∶	VERB
ejpam-2272	100	16	x	x	NOUN
ejpam-2272	100	17	)	)	PUNCT
ejpam-2272	100	18	=	=	PRON
ejpam-2272	101	1	{	{	PUNCT
ejpam-2272	101	2	r	r	NOUN
ejpam-2272	101	3	∈	∈	PROPN
ejpam-2272	101	4	s	s	PART
ejpam-2272	101	5	∶	∶	NOUN
ejpam-2272	101	6	x	x	PUNCT
ejpam-2272	101	7	r	r	NOUN
ejpam-2272	101	8	∈	∈	PROPN
ejpam-2272	101	9	i	i	NOUN
ejpam-2272	101	10	}	}	PUNCT
ejpam-2272	101	11	.	.	PUNCT
ejpam-2272	102	1	proof	proof	NOUN
ejpam-2272	102	2	.	.	PUNCT
ejpam-2272	103	1	let	let	VERB
ejpam-2272	103	2	x	x	PUNCT
ejpam-2272	103	3	∈	∈	PROPN
ejpam-2272	103	4	s	s	PART
ejpam-2272	103	5	∖	∖	NOUN
ejpam-2272	103	6	√	√	VERB
ejpam-2272	103	7	i	i	PRON
ejpam-2272	103	8	and	and	CCONJ
ejpam-2272	103	9	a	a	DET
ejpam-2272	103	10	∈	∈	NOUN
ejpam-2272	103	11	(	(	PUNCT
ejpam-2272	103	12	i	i	PRON
ejpam-2272	103	13	∶	∶	VERB
ejpam-2272	103	14	x	x	NOUN
ejpam-2272	103	15	)	)	PUNCT
ejpam-2272	103	16	.	.	PUNCT
ejpam-2272	104	1	then	then	ADV
ejpam-2272	104	2	ax	ax	NOUN
ejpam-2272	104	3	∈	∈	PROPN
ejpam-2272	104	4	i	i	PRON
ejpam-2272	104	5	⊆	⊆	NUM
ejpam-2272	104	6	√	√	VERB
ejpam-2272	104	7	i	i	PRON
ejpam-2272	104	8	,	,	PUNCT
ejpam-2272	104	9	gives	give	VERB
ejpam-2272	104	10	a	a	DET
ejpam-2272	104	11	∈	∈	NOUN
ejpam-2272	104	12	√	√	NOUN
ejpam-2272	105	1	i	i	PRON
ejpam-2272	105	2	,	,	PUNCT
ejpam-2272	105	3	since	since	SCONJ
ejpam-2272	105	4	x	x	PROPN
ejpam-2272	105	5	∉	∉	PROPN
ejpam-2272	105	6	√	√	VERB
ejpam-2272	105	7	i	i	PRON
ejpam-2272	105	8	and	and	CCONJ
ejpam-2272	105	9	√	√	INTJ
ejpam-2272	105	10	i	i	PRON
ejpam-2272	105	11	is	be	AUX
ejpam-2272	105	12	prime	prime	ADJ
ejpam-2272	105	13	.	.	PUNCT
ejpam-2272	106	1	hence	hence	ADV
ejpam-2272	106	2	,	,	PUNCT
ejpam-2272	106	3	a	a	DET
ejpam-2272	106	4	∈	∈	NOUN
ejpam-2272	106	5	√	√	VERB
ejpam-2272	106	6	i	i	PRON
ejpam-2272	106	7	,	,	PUNCT
ejpam-2272	106	8	gives	give	VERB
ejpam-2272	106	9	i	i	PRON
ejpam-2272	106	10	⊆	⊆	NUM
ejpam-2272	106	11	(	(	PUNCT
ejpam-2272	106	12	i	i	PRON
ejpam-2272	106	13	∶	∶	VERB
ejpam-2272	106	14	x	x	SYM
ejpam-2272	106	15	)	)	PUNCT
ejpam-2272	106	16	⊆	⊆	NUM
ejpam-2272	106	17	√	√	NUM
ejpam-2272	107	1	i	i	NOUN
ejpam-2272	107	2	=	=	SYM
ejpam-2272	107	3	p	p	NOUN
ejpam-2272	107	4	,	,	PUNCT
ejpam-2272	107	5	which	which	PRON
ejpam-2272	107	6	implies	imply	VERB
ejpam-2272	107	7	that	that	SCONJ
ejpam-2272	107	8	p	p	PROPN
ejpam-2272	107	9	=	=	PUNCT
ejpam-2272	108	1	√	√	PROPN
ejpam-2272	108	2	i	i	PRON
ejpam-2272	108	3	⊆	⊆	NUM
ejpam-2272	108	4	√	√	NUM
ejpam-2272	108	5	(	(	PUNCT
ejpam-2272	108	6	i	i	PRON
ejpam-2272	108	7	∶	∶	VERB
ejpam-2272	108	8	x	x	SYM
ejpam-2272	108	9	)	)	PUNCT
ejpam-2272	108	10	⊆	⊆	NUM
ejpam-2272	109	1	√	√	NUM
ejpam-2272	109	2	i	i	NOUN
ejpam-2272	109	3	=	=	PUNCT
ejpam-2272	110	1	p.	p.	NOUN
ejpam-2272	110	2	thus	thus	ADV
ejpam-2272	110	3	,	,	PUNCT
ejpam-2272	110	4	we	we	PRON
ejpam-2272	110	5	have	have	VERB
ejpam-2272	110	6	√	√	NUM
ejpam-2272	110	7	(	(	PUNCT
ejpam-2272	110	8	i	i	PRON
ejpam-2272	110	9	∶	∶	VERB
ejpam-2272	110	10	x	x	NOUN
ejpam-2272	110	11	)	)	PUNCT
ejpam-2272	111	1	=	=	VERB
ejpam-2272	112	1	p.	p.	NOUN
ejpam-2272	112	2	now	now	ADV
ejpam-2272	112	3	,	,	PUNCT
ejpam-2272	112	4	let	let	VERB
ejpam-2272	112	5	a	a	DET
ejpam-2272	112	6	,	,	PUNCT
ejpam-2272	112	7	b	b	NOUN
ejpam-2272	112	8	,	,	PUNCT
ejpam-2272	112	9	c	c	PROPN
ejpam-2272	112	10	∈	∈	PROPN
ejpam-2272	112	11	s	s	AUX
ejpam-2272	112	12	be	be	AUX
ejpam-2272	112	13	such	such	ADJ
ejpam-2272	112	14	that	that	SCONJ
ejpam-2272	112	15	abc	abc	PROPN
ejpam-2272	112	16	∈	∈	PROPN
ejpam-2272	112	17	(	(	PUNCT
ejpam-2272	112	18	i	i	PRON
ejpam-2272	112	19	∶	∶	VERB
ejpam-2272	112	20	x	x	NOUN
ejpam-2272	112	21	)	)	PUNCT
ejpam-2272	112	22	.	.	PUNCT
ejpam-2272	113	1	then	then	ADV
ejpam-2272	113	2	abcx	abcx	PROPN
ejpam-2272	113	3	∈	∈	PROPN
ejpam-2272	114	1	i	i	PRON
ejpam-2272	114	2	,	,	PUNCT
ejpam-2272	114	3	implies	imply	VERB
ejpam-2272	114	4	that	that	SCONJ
ejpam-2272	114	5	either	either	CCONJ
ejpam-2272	114	6	abc	abc	PROPN
ejpam-2272	114	7	∈	∈	PROPN
ejpam-2272	114	8	i	i	PRON
ejpam-2272	114	9	or	or	CCONJ
ejpam-2272	114	10	ax	ax	NOUN
ejpam-2272	114	11	∈	∈	PROPN
ejpam-2272	114	12	√	√	VERB
ejpam-2272	115	1	i	i	PRON
ejpam-2272	115	2	or	or	CCONJ
ejpam-2272	115	3	bcx	bcx	PROPN
ejpam-2272	115	4	∈	∈	PROPN
ejpam-2272	116	1	√	√	VERB
ejpam-2272	117	1	i	i	PRON
ejpam-2272	117	2	.	.	PUNCT
ejpam-2272	118	1	if	if	SCONJ
ejpam-2272	118	2	ax	ax	NOUN
ejpam-2272	118	3	∈	∈	PROPN
ejpam-2272	118	4	√	√	VERB
ejpam-2272	118	5	i	i	PRON
ejpam-2272	118	6	or	or	CCONJ
ejpam-2272	118	7	bcx	bcx	PROPN
ejpam-2272	118	8	∈	∈	PROPN
ejpam-2272	118	9	√	√	VERB
ejpam-2272	119	1	i	i	PRON
ejpam-2272	119	2	,	,	PUNCT
ejpam-2272	119	3	we	we	PRON
ejpam-2272	119	4	get	get	VERB
ejpam-2272	119	5	ac	ac	PROPN
ejpam-2272	119	6	∈	∈	PROPN
ejpam-2272	119	7	√	√	NUM
ejpam-2272	120	1	(	(	PUNCT
ejpam-2272	120	2	i	i	PRON
ejpam-2272	120	3	∶	∶	VERB
ejpam-2272	120	4	x	x	SYM
ejpam-2272	120	5	)	)	PUNCT
ejpam-2272	120	6	or	or	CCONJ
ejpam-2272	120	7	bc	bc	PROPN
ejpam-2272	120	8	∈	∈	PROPN
ejpam-2272	120	9	√	√	NUM
ejpam-2272	121	1	(	(	PUNCT
ejpam-2272	121	2	i	i	PRON
ejpam-2272	121	3	∶	∶	VERB
ejpam-2272	121	4	x	x	X
ejpam-2272	121	5	)	)	PUNCT
ejpam-2272	121	6	,	,	PUNCT
ejpam-2272	121	7	since	since	SCONJ
ejpam-2272	121	8	√	√	PROPN
ejpam-2272	121	9	(	(	PUNCT
ejpam-2272	121	10	i	i	PRON
ejpam-2272	121	11	∶	∶	VERB
ejpam-2272	121	12	x	x	NOUN
ejpam-2272	121	13	)	)	PUNCT
ejpam-2272	121	14	=	=	SYM
ejpam-2272	122	1	√	√	VERB
ejpam-2272	122	2	i	i	PRON
ejpam-2272	122	3	and	and	CCONJ
ejpam-2272	122	4	x	x	PROPN
ejpam-2272	122	5	∉	∉	PROPN
ejpam-2272	122	6	√	√	INTJ
ejpam-2272	122	7	i	i	PRON
ejpam-2272	122	8	.	.	PUNCT
ejpam-2272	123	1	next	next	ADV
ejpam-2272	123	2	,	,	PUNCT
ejpam-2272	123	3	if	if	SCONJ
ejpam-2272	123	4	abc	abc	PROPN
ejpam-2272	123	5	∈	∈	PROPN
ejpam-2272	123	6	i	i	PRON
ejpam-2272	123	7	,	,	PUNCT
ejpam-2272	123	8	we	we	PRON
ejpam-2272	123	9	have	have	VERB
ejpam-2272	123	10	either	either	CCONJ
ejpam-2272	123	11	ab	ab	PROPN
ejpam-2272	123	12	∈	∈	PROPN
ejpam-2272	124	1	i	i	PRON
ejpam-2272	124	2	or	or	CCONJ
ejpam-2272	124	3	bc	bc	PROPN
ejpam-2272	124	4	∈	∈	PROPN
ejpam-2272	125	1	√	√	VERB
ejpam-2272	125	2	i	i	PRON
ejpam-2272	125	3	or	or	CCONJ
ejpam-2272	125	4	ca	ca	NOUN
ejpam-2272	125	5	∈	∈	PROPN
ejpam-2272	126	1	√	√	PROPN
ejpam-2272	126	2	i	i	PRON
ejpam-2272	126	3	,	,	PUNCT
ejpam-2272	126	4	since	since	SCONJ
ejpam-2272	126	5	i	i	PRON
ejpam-2272	126	6	is	be	AUX
ejpam-2272	126	7	a	a	DET
ejpam-2272	126	8	2	2	NUM
ejpam-2272	126	9	-	-	PUNCT
ejpam-2272	126	10	absorbing	absorbing	ADJ
ejpam-2272	126	11	primary	primary	ADJ
ejpam-2272	126	12	ideal	ideal	NOUN
ejpam-2272	126	13	of	of	ADP
ejpam-2272	126	14	s.	s.	PROPN
ejpam-2272	126	15	thus	thus	ADV
ejpam-2272	126	16	,	,	PUNCT
ejpam-2272	126	17	ab	ab	PROPN
ejpam-2272	126	18	∈	∈	PROPN
ejpam-2272	126	19	(	(	PUNCT
ejpam-2272	126	20	i	i	PRON
ejpam-2272	126	21	∶	∶	VERB
ejpam-2272	126	22	x	x	SYM
ejpam-2272	126	23	)	)	PUNCT
ejpam-2272	126	24	or	or	CCONJ
ejpam-2272	126	25	bc	bc	PROPN
ejpam-2272	126	26	∈	∈	PROPN
ejpam-2272	126	27	√	√	NUM
ejpam-2272	126	28	(	(	PUNCT
ejpam-2272	126	29	i	i	PRON
ejpam-2272	126	30	∶	∶	VERB
ejpam-2272	126	31	x	x	SYM
ejpam-2272	126	32	)	)	PUNCT
ejpam-2272	126	33	or	or	CCONJ
ejpam-2272	126	34	ca	ca	NOUN
ejpam-2272	126	35	∈	∈	PROPN
ejpam-2272	126	36	√	√	PROPN
ejpam-2272	126	37	(	(	PUNCT
ejpam-2272	126	38	i	i	PRON
ejpam-2272	126	39	∶	∶	VERB
ejpam-2272	126	40	x	x	NOUN
ejpam-2272	126	41	)	)	PUNCT
ejpam-2272	126	42	.	.	PUNCT
ejpam-2272	127	1	therefore	therefore	ADV
ejpam-2272	127	2	(	(	PUNCT
ejpam-2272	127	3	i	i	PRON
ejpam-2272	127	4	∶	∶	VERB
ejpam-2272	127	5	x	x	VERB
ejpam-2272	127	6	)	)	PUNCT
ejpam-2272	127	7	is	be	AUX
ejpam-2272	127	8	a	a	DET
ejpam-2272	127	9	2	2	NUM
ejpam-2272	127	10	-	-	PUNCT
ejpam-2272	127	11	absorbing	absorbing	ADJ
ejpam-2272	127	12	primary	primary	ADJ
ejpam-2272	127	13	ideal	ideal	NOUN
ejpam-2272	127	14	of	of	ADP
ejpam-2272	127	15	s.	s.	PROPN
ejpam-2272	127	16	p.	p.	PROPN
ejpam-2272	127	17	kumar	kumar	PROPN
ejpam-2272	127	18	,	,	PUNCT
ejpam-2272	127	19	m.	m.	NOUN
ejpam-2272	127	20	dubey	dubey	PROPN
ejpam-2272	127	21	,	,	PUNCT
ejpam-2272	127	22	p.	p.	NOUN
ejpam-2272	127	23	sarohe	sarohe	PROPN
ejpam-2272	127	24	/	/	SYM
ejpam-2272	127	25	eur	eur	PROPN
ejpam-2272	127	26	.	.	PUNCT
ejpam-2272	128	1	j.	j.	PROPN
ejpam-2272	128	2	pure	pure	PROPN
ejpam-2272	128	3	appl	appl	PROPN
ejpam-2272	128	4	.	.	PROPN
ejpam-2272	128	5	math	math	PROPN
ejpam-2272	128	6	,	,	PUNCT
ejpam-2272	128	7	9	9	NUM
ejpam-2272	128	8	(	(	PUNCT
ejpam-2272	128	9	2016	2016	NUM
ejpam-2272	128	10	)	)	PUNCT
ejpam-2272	128	11	,	,	PUNCT
ejpam-2272	128	12	186	186	NUM
ejpam-2272	128	13	-	-	SYM
ejpam-2272	128	14	195	195	NUM
ejpam-2272	128	15	189	189	NUM
ejpam-2272	128	16	theorem	theorem	NOUN
ejpam-2272	128	17	6	6	NUM
ejpam-2272	128	18	.	.	PUNCT
ejpam-2272	129	1	if	if	SCONJ
ejpam-2272	129	2	i	i	PRON
ejpam-2272	129	3	is	be	AUX
ejpam-2272	129	4	a	a	DET
ejpam-2272	129	5	2	2	NUM
ejpam-2272	129	6	-	-	PUNCT
ejpam-2272	129	7	absorbing	absorbing	ADJ
ejpam-2272	129	8	primary	primary	ADJ
ejpam-2272	129	9	ideal	ideal	NOUN
ejpam-2272	129	10	of	of	ADP
ejpam-2272	129	11	a	a	DET
ejpam-2272	129	12	semiring	semiring	NOUN
ejpam-2272	129	13	s	s	NOUN
ejpam-2272	129	14	,	,	PUNCT
ejpam-2272	129	15	then	then	ADV
ejpam-2272	129	16	the	the	DET
ejpam-2272	129	17	following	follow	VERB
ejpam-2272	129	18	holds	hold	VERB
ejpam-2272	129	19	:	:	PUNCT
ejpam-2272	129	20	(	(	PUNCT
ejpam-2272	129	21	1	1	X
ejpam-2272	129	22	)	)	PUNCT
ejpam-2272	129	23	(	(	PUNCT
ejpam-2272	129	24	√	√	PUNCT
ejpam-2272	129	25	i	i	PRON
ejpam-2272	129	26	∶	∶	PROPN
ejpam-2272	129	27	x	x	VERB
ejpam-2272	129	28	)	)	PUNCT
ejpam-2272	129	29	is	be	AUX
ejpam-2272	129	30	a	a	DET
ejpam-2272	129	31	2	2	NUM
ejpam-2272	129	32	-	-	PUNCT
ejpam-2272	129	33	absorbing	absorbing	ADJ
ejpam-2272	129	34	ideal	ideal	NOUN
ejpam-2272	129	35	of	of	ADP
ejpam-2272	129	36	s	s	PRON
ejpam-2272	129	37	for	for	ADP
ejpam-2272	129	38	all	all	DET
ejpam-2272	129	39	x	x	SYM
ejpam-2272	129	40	∈	∈	PROPN
ejpam-2272	129	41	s	s	PART
ejpam-2272	129	42	∖	∖	NOUN
ejpam-2272	129	43	√	√	VERB
ejpam-2272	129	44	i	i	PRON
ejpam-2272	129	45	.	.	PUNCT
ejpam-2272	130	1	(	(	PUNCT
ejpam-2272	130	2	2	2	X
ejpam-2272	130	3	)	)	PUNCT
ejpam-2272	130	4	(	(	PUNCT
ejpam-2272	130	5	√	√	PUNCT
ejpam-2272	130	6	i	i	PRON
ejpam-2272	130	7	∶	∶	NOUN
ejpam-2272	130	8	x	x	NOUN
ejpam-2272	130	9	)	)	PUNCT
ejpam-2272	130	10	=	=	SYM
ejpam-2272	131	1	(	(	PUNCT
ejpam-2272	131	2	√	√	INTJ
ejpam-2272	131	3	i	i	PRON
ejpam-2272	131	4	∶	∶	VERB
ejpam-2272	131	5	x2	x2	NUM
ejpam-2272	131	6	)	)	PUNCT
ejpam-2272	131	7	for	for	ADP
ejpam-2272	131	8	all	all	DET
ejpam-2272	131	9	x	x	SYM
ejpam-2272	131	10	∈	∈	PROPN
ejpam-2272	131	11	s	s	PART
ejpam-2272	131	12	∖	∖	NOUN
ejpam-2272	131	13	√	√	VERB
ejpam-2272	131	14	i	i	PRON
ejpam-2272	131	15	.	.	PUNCT
ejpam-2272	132	1	proof	proof	NOUN
ejpam-2272	132	2	.	.	PUNCT
ejpam-2272	133	1	(	(	PUNCT
ejpam-2272	133	2	1	1	X
ejpam-2272	133	3	)	)	PUNCT
ejpam-2272	133	4	let	let	VERB
ejpam-2272	133	5	a	a	DET
ejpam-2272	133	6	,	,	PUNCT
ejpam-2272	133	7	b	b	NOUN
ejpam-2272	133	8	,	,	PUNCT
ejpam-2272	133	9	c	c	PROPN
ejpam-2272	133	10	∈	∈	PROPN
ejpam-2272	133	11	s	s	AUX
ejpam-2272	133	12	be	be	AUX
ejpam-2272	133	13	such	such	ADJ
ejpam-2272	133	14	that	that	SCONJ
ejpam-2272	133	15	abc	abc	PROPN
ejpam-2272	133	16	∈	∈	PROPN
ejpam-2272	133	17	(	(	PUNCT
ejpam-2272	133	18	√	√	PROPN
ejpam-2272	133	19	i	i	PRON
ejpam-2272	133	20	∶	∶	PROPN
ejpam-2272	133	21	x	x	NOUN
ejpam-2272	133	22	)	)	PUNCT
ejpam-2272	133	23	.	.	PUNCT
ejpam-2272	134	1	then	then	ADV
ejpam-2272	134	2	abcx	abcx	PROPN
ejpam-2272	134	3	∈	∈	PROPN
ejpam-2272	135	1	√	√	VERB
ejpam-2272	135	2	i	i	PRON
ejpam-2272	135	3	.	.	PUNCT
ejpam-2272	136	1	since	since	SCONJ
ejpam-2272	136	2	√	√	PROPN
ejpam-2272	136	3	i	i	PRON
ejpam-2272	136	4	is	be	AUX
ejpam-2272	136	5	a	a	DET
ejpam-2272	136	6	2	2	NUM
ejpam-2272	136	7	-	-	PUNCT
ejpam-2272	136	8	absorbing	absorbing	ADJ
ejpam-2272	136	9	ideal	ideal	NOUN
ejpam-2272	136	10	of	of	ADP
ejpam-2272	136	11	s	s	PRON
ejpam-2272	136	12	therefore	therefore	ADV
ejpam-2272	136	13	ab	ab	PROPN
ejpam-2272	136	14	∈	∈	PROPN
ejpam-2272	137	1	√	√	VERB
ejpam-2272	138	1	i	i	PRON
ejpam-2272	138	2	or	or	CCONJ
ejpam-2272	138	3	bcx	bcx	PROPN
ejpam-2272	138	4	∈	∈	PROPN
ejpam-2272	139	1	√	√	VERB
ejpam-2272	139	2	i	i	PRON
ejpam-2272	139	3	or	or	CCONJ
ejpam-2272	139	4	cax	cax	PROPN
ejpam-2272	139	5	∈	∈	PROPN
ejpam-2272	140	1	√	√	VERB
ejpam-2272	140	2	i	i	PRON
ejpam-2272	140	3	,	,	PUNCT
ejpam-2272	140	4	that	that	ADV
ejpam-2272	140	5	is	is	ADV
ejpam-2272	140	6	,	,	PUNCT
ejpam-2272	140	7	ab	ab	PROPN
ejpam-2272	140	8	∈	∈	PROPN
ejpam-2272	140	9	(	(	PUNCT
ejpam-2272	140	10	√	√	PROPN
ejpam-2272	140	11	i	i	PRON
ejpam-2272	140	12	∶	∶	PROPN
ejpam-2272	140	13	x	x	NOUN
ejpam-2272	140	14	)	)	PUNCT
ejpam-2272	140	15	or	or	CCONJ
ejpam-2272	140	16	bc	bc	PROPN
ejpam-2272	140	17	∈	∈	PROPN
ejpam-2272	140	18	(	(	PUNCT
ejpam-2272	140	19	√	√	PROPN
ejpam-2272	140	20	i	i	PRON
ejpam-2272	140	21	∶	∶	PROPN
ejpam-2272	140	22	x	x	SYM
ejpam-2272	140	23	)	)	PUNCT
ejpam-2272	140	24	or	or	CCONJ
ejpam-2272	140	25	ca	ca	NOUN
ejpam-2272	140	26	∈	∈	PROPN
ejpam-2272	140	27	(	(	PUNCT
ejpam-2272	140	28	√	√	PROPN
ejpam-2272	140	29	i	i	PRON
ejpam-2272	140	30	∶	∶	PROPN
ejpam-2272	140	31	x	x	NOUN
ejpam-2272	140	32	)	)	PUNCT
ejpam-2272	140	33	.	.	PUNCT
ejpam-2272	141	1	hence	hence	ADV
ejpam-2272	141	2	(	(	PUNCT
ejpam-2272	141	3	√	√	INTJ
ejpam-2272	141	4	i	i	PRON
ejpam-2272	141	5	∶	∶	PROPN
ejpam-2272	141	6	x	x	VERB
ejpam-2272	141	7	)	)	PUNCT
ejpam-2272	141	8	is	be	AUX
ejpam-2272	141	9	a	a	DET
ejpam-2272	141	10	2	2	NUM
ejpam-2272	141	11	-	-	PUNCT
ejpam-2272	141	12	absorbing	absorbing	ADJ
ejpam-2272	141	13	ideal	ideal	NOUN
ejpam-2272	141	14	of	of	ADP
ejpam-2272	141	15	s.	s.	PROPN
ejpam-2272	141	16	(	(	PUNCT
ejpam-2272	141	17	2	2	X
ejpam-2272	141	18	)	)	PUNCT
ejpam-2272	141	19	it	it	PRON
ejpam-2272	141	20	is	be	AUX
ejpam-2272	141	21	clear	clear	ADJ
ejpam-2272	141	22	that	that	SCONJ
ejpam-2272	141	23	(	(	PUNCT
ejpam-2272	141	24	√	√	INTJ
ejpam-2272	141	25	i	i	PRON
ejpam-2272	141	26	∶	∶	PROPN
ejpam-2272	141	27	x	x	NOUN
ejpam-2272	141	28	)	)	PUNCT
ejpam-2272	141	29	⊆	⊆	NUM
ejpam-2272	141	30	(	(	PUNCT
ejpam-2272	141	31	√	√	VERB
ejpam-2272	141	32	i	i	PRON
ejpam-2272	141	33	∶	∶	VERB
ejpam-2272	141	34	x2	x2	NUM
ejpam-2272	141	35	)	)	PUNCT
ejpam-2272	141	36	.	.	PUNCT
ejpam-2272	142	1	let	let	VERB
ejpam-2272	142	2	y	y	PROPN
ejpam-2272	142	3	∈	∈	PROPN
ejpam-2272	142	4	(	(	PUNCT
ejpam-2272	142	5	√	√	PROPN
ejpam-2272	142	6	i	i	PRON
ejpam-2272	142	7	∶	∶	VERB
ejpam-2272	142	8	x2	x2	NUM
ejpam-2272	142	9	)	)	PUNCT
ejpam-2272	142	10	.	.	PUNCT
ejpam-2272	143	1	then	then	ADV
ejpam-2272	143	2	x2	x2	PROPN
ejpam-2272	143	3	y	y	PROPN
ejpam-2272	143	4	∈	∈	PROPN
ejpam-2272	143	5	√	√	VERB
ejpam-2272	144	1	i	i	PRON
ejpam-2272	144	2	.	.	PUNCT
ejpam-2272	145	1	since√	since√	NOUN
ejpam-2272	145	2	i	i	PRON
ejpam-2272	145	3	is	be	AUX
ejpam-2272	145	4	2	2	NUM
ejpam-2272	145	5	-	-	PUNCT
ejpam-2272	145	6	absorbing	absorbing	ADJ
ejpam-2272	145	7	ideal	ideal	NOUN
ejpam-2272	145	8	of	of	ADP
ejpam-2272	145	9	s	s	PROPN
ejpam-2272	145	10	,	,	PUNCT
ejpam-2272	145	11	therefore	therefore	ADV
ejpam-2272	145	12	we	we	PRON
ejpam-2272	145	13	have	have	VERB
ejpam-2272	145	14	either	either	CCONJ
ejpam-2272	145	15	x2	x2	PROPN
ejpam-2272	145	16	∈	∈	PROPN
ejpam-2272	145	17	√	√	VERB
ejpam-2272	145	18	i	i	PRON
ejpam-2272	145	19	or	or	CCONJ
ejpam-2272	145	20	x	x	SYM
ejpam-2272	145	21	y	y	PROPN
ejpam-2272	145	22	∈	∈	PROPN
ejpam-2272	146	1	√	√	VERB
ejpam-2272	146	2	i	i	PRON
ejpam-2272	146	3	.	.	PUNCT
ejpam-2272	147	1	if	if	SCONJ
ejpam-2272	147	2	x	x	X
ejpam-2272	147	3	y	y	PROPN
ejpam-2272	147	4	∈	∈	PROPN
ejpam-2272	147	5	√	√	VERB
ejpam-2272	147	6	i	i	PRON
ejpam-2272	147	7	,	,	PUNCT
ejpam-2272	147	8	then	then	ADV
ejpam-2272	147	9	y	y	PROPN
ejpam-2272	147	10	∈	∈	PROPN
ejpam-2272	147	11	(	(	PUNCT
ejpam-2272	147	12	√	√	PROPN
ejpam-2272	147	13	i	i	PRON
ejpam-2272	147	14	∶	∶	PROPN
ejpam-2272	147	15	x	x	PUNCT
ejpam-2272	147	16	)	)	PUNCT
ejpam-2272	147	17	and	and	CCONJ
ejpam-2272	147	18	we	we	PRON
ejpam-2272	147	19	are	be	AUX
ejpam-2272	147	20	done	do	VERB
ejpam-2272	147	21	.	.	PUNCT
ejpam-2272	148	1	if	if	SCONJ
ejpam-2272	148	2	x2	x2	PROPN
ejpam-2272	148	3	∈	∈	PROPN
ejpam-2272	148	4	√	√	VERB
ejpam-2272	148	5	i	i	PRON
ejpam-2272	148	6	,	,	PUNCT
ejpam-2272	148	7	then	then	ADV
ejpam-2272	148	8	x	x	SYM
ejpam-2272	148	9	∈	∈	NOUN
ejpam-2272	148	10	√	√	VERB
ejpam-2272	148	11	i	i	PRON
ejpam-2272	148	12	,	,	PUNCT
ejpam-2272	148	13	a	a	DET
ejpam-2272	148	14	contradiction	contradiction	NOUN
ejpam-2272	148	15	.	.	PUNCT
ejpam-2272	149	1	hence	hence	ADV
ejpam-2272	149	2	,	,	PUNCT
ejpam-2272	149	3	(	(	PUNCT
ejpam-2272	149	4	√	√	VERB
ejpam-2272	149	5	i	i	PRON
ejpam-2272	149	6	∶	∶	NOUN
ejpam-2272	149	7	x	x	NOUN
ejpam-2272	149	8	)	)	PUNCT
ejpam-2272	149	9	=	=	SYM
ejpam-2272	150	1	(	(	PUNCT
ejpam-2272	150	2	√	√	INTJ
ejpam-2272	150	3	i	i	PRON
ejpam-2272	150	4	∶	∶	VERB
ejpam-2272	150	5	x2	x2	NUM
ejpam-2272	150	6	)	)	PUNCT
ejpam-2272	150	7	.	.	PUNCT
ejpam-2272	151	1	let	let	VERB
ejpam-2272	151	2	s	s	PRON
ejpam-2272	151	3	be	be	AUX
ejpam-2272	151	4	a	a	DET
ejpam-2272	151	5	semiring	semiring	NOUN
ejpam-2272	151	6	and	and	CCONJ
ejpam-2272	151	7	a	a	DET
ejpam-2272	151	8	be	be	AUX
ejpam-2272	151	9	the	the	DET
ejpam-2272	151	10	set	set	NOUN
ejpam-2272	151	11	of	of	ADP
ejpam-2272	151	12	all	all	DET
ejpam-2272	151	13	multiplicatively	multiplicatively	ADV
ejpam-2272	151	14	cancellable	cancellable	ADJ
ejpam-2272	151	15	elements	element	NOUN
ejpam-2272	151	16	of	of	ADP
ejpam-2272	151	17	s	s	NOUN
ejpam-2272	151	18	(	(	PUNCT
ejpam-2272	151	19	so	so	ADV
ejpam-2272	151	20	1	1	NUM
ejpam-2272	151	21	∈	∈	NOUN
ejpam-2272	151	22	s	s	NOUN
ejpam-2272	151	23	)	)	PUNCT
ejpam-2272	151	24	.	.	PUNCT
ejpam-2272	152	1	for	for	ADP
ejpam-2272	152	2	further	further	ADJ
ejpam-2272	152	3	understanding	understanding	NOUN
ejpam-2272	152	4	of	of	ADP
ejpam-2272	152	5	the	the	DET
ejpam-2272	152	6	structure	structure	NOUN
ejpam-2272	152	7	of	of	ADP
ejpam-2272	152	8	the	the	DET
ejpam-2272	152	9	semiring	semiring	NOUN
ejpam-2272	152	10	of	of	ADP
ejpam-2272	152	11	fractions	fraction	NOUN
ejpam-2272	152	12	sa	sa	NOUN
ejpam-2272	152	13	of	of	ADP
ejpam-2272	152	14	s	s	PRON
ejpam-2272	152	15	with	with	ADP
ejpam-2272	152	16	respect	respect	NOUN
ejpam-2272	152	17	to	to	ADP
ejpam-2272	152	18	a	a	PRON
ejpam-2272	152	19	,	,	PUNCT
ejpam-2272	152	20	refer	refer	VERB
ejpam-2272	152	21	[	[	X
ejpam-2272	152	22	3	3	NUM
ejpam-2272	152	23	]	]	PUNCT
ejpam-2272	152	24	.	.	PUNCT
ejpam-2272	153	1	theorem	theorem	ADJ
ejpam-2272	153	2	7	7	NUM
ejpam-2272	153	3	.	.	PUNCT
ejpam-2272	154	1	let	let	VERB
ejpam-2272	154	2	i	i	PRON
ejpam-2272	154	3	be	be	AUX
ejpam-2272	154	4	a	a	DET
ejpam-2272	154	5	2	2	NUM
ejpam-2272	154	6	-	-	PUNCT
ejpam-2272	154	7	absorbing	absorbing	ADJ
ejpam-2272	154	8	primary	primary	ADJ
ejpam-2272	154	9	ideal	ideal	NOUN
ejpam-2272	154	10	of	of	ADP
ejpam-2272	154	11	a	a	DET
ejpam-2272	154	12	semiring	semiring	NOUN
ejpam-2272	154	13	s	s	X
ejpam-2272	154	14	and	and	CCONJ
ejpam-2272	154	15	a	a	PRON
ejpam-2272	154	16	be	be	AUX
ejpam-2272	154	17	the	the	DET
ejpam-2272	154	18	multiplicatively	multiplicatively	ADV
ejpam-2272	154	19	cancellable	cancellable	ADJ
ejpam-2272	154	20	subset	subset	NOUN
ejpam-2272	154	21	of	of	ADP
ejpam-2272	154	22	s.	s.	PROPN
ejpam-2272	154	23	then	then	ADV
ejpam-2272	154	24	isa	isa	PROPN
ejpam-2272	154	25	is	be	AUX
ejpam-2272	154	26	a	a	DET
ejpam-2272	154	27	2	2	NUM
ejpam-2272	154	28	-	-	PUNCT
ejpam-2272	154	29	absorbing	absorbing	ADJ
ejpam-2272	154	30	primary	primary	ADJ
ejpam-2272	154	31	ideal	ideal	NOUN
ejpam-2272	154	32	of	of	ADP
ejpam-2272	154	33	sa	sa	PROPN
ejpam-2272	154	34	.	.	PUNCT
ejpam-2272	155	1	proof	proof	NOUN
ejpam-2272	155	2	.	.	PUNCT
ejpam-2272	156	1	let	let	VERB
ejpam-2272	156	2	a	a	PRON
ejpam-2272	156	3	/	/	SYM
ejpam-2272	156	4	s	s	PROPN
ejpam-2272	156	5	,	,	PUNCT
ejpam-2272	156	6	b	b	X
ejpam-2272	156	7	/	/	SYM
ejpam-2272	156	8	t	t	PROPN
ejpam-2272	156	9	,	,	PUNCT
ejpam-2272	156	10	c	c	NOUN
ejpam-2272	156	11	/	/	SYM
ejpam-2272	156	12	r	r	NOUN
ejpam-2272	156	13	∈	∈	PROPN
ejpam-2272	156	14	sa	sa	NOUN
ejpam-2272	156	15	,	,	PUNCT
ejpam-2272	156	16	where	where	SCONJ
ejpam-2272	156	17	a	a	DET
ejpam-2272	156	18	,	,	PUNCT
ejpam-2272	156	19	b	b	NOUN
ejpam-2272	156	20	,	,	PUNCT
ejpam-2272	156	21	c	c	PROPN
ejpam-2272	156	22	∈	∈	PROPN
ejpam-2272	156	23	s	s	X
ejpam-2272	156	24	and	and	CCONJ
ejpam-2272	156	25	s	s	PROPN
ejpam-2272	156	26	,	,	PUNCT
ejpam-2272	156	27	t	t	PROPN
ejpam-2272	156	28	,	,	PUNCT
ejpam-2272	156	29	r	r	NOUN
ejpam-2272	156	30	∈	∈	PROPN
ejpam-2272	156	31	a	a	DET
ejpam-2272	156	32	be	be	AUX
ejpam-2272	156	33	such	such	ADJ
ejpam-2272	156	34	that	that	SCONJ
ejpam-2272	156	35	abc	abc	PROPN
ejpam-2272	156	36	/	/	SYM
ejpam-2272	156	37	st	st	PROPN
ejpam-2272	156	38	r	r	PROPN
ejpam-2272	156	39	∈	∈	PROPN
ejpam-2272	156	40	isa	isa	NOUN
ejpam-2272	156	41	but	but	CCONJ
ejpam-2272	156	42	bc	bc	PROPN
ejpam-2272	156	43	/	/	SYM
ejpam-2272	156	44	t	t	PROPN
ejpam-2272	156	45	r	r	NOUN
ejpam-2272	156	46	∉	∉	PROPN
ejpam-2272	156	47	√	√	PROPN
ejpam-2272	156	48	isa	isa	NOUN
ejpam-2272	156	49	and	and	CCONJ
ejpam-2272	156	50	ca	ca	NOUN
ejpam-2272	156	51	/	/	SYM
ejpam-2272	156	52	rs	rs	NOUN
ejpam-2272	156	53	∉	∉	PROPN
ejpam-2272	157	1	√	√	PROPN
ejpam-2272	157	2	isa	isa	PROPN
ejpam-2272	157	3	.	.	PUNCT
ejpam-2272	158	1	then	then	ADV
ejpam-2272	158	2	there	there	PRON
ejpam-2272	158	3	exist	exist	VERB
ejpam-2272	158	4	p	p	PRON
ejpam-2272	158	5	∈	∈	PROPN
ejpam-2272	158	6	i	i	PRON
ejpam-2272	158	7	and	and	CCONJ
ejpam-2272	158	8	z	z	NOUN
ejpam-2272	158	9	∈	∈	PROPN
ejpam-2272	158	10	a	a	DET
ejpam-2272	158	11	such	such	ADJ
ejpam-2272	158	12	that	that	DET
ejpam-2272	158	13	abcz	abcz	NOUN
ejpam-2272	159	1	=	=	SYM
ejpam-2272	159	2	st	st	PROPN
ejpam-2272	159	3	rp	rp	PROPN
ejpam-2272	159	4	∈	∈	PROPN
ejpam-2272	160	1	i	i	PRON
ejpam-2272	160	2	but	but	CCONJ
ejpam-2272	160	3	bcz	bcz	VERB
ejpam-2272	160	4	∉	∉	PROPN
ejpam-2272	160	5	i	i	PROPN
ejpam-2272	160	6	and	and	CCONJ
ejpam-2272	160	7	caz	caz	VERB
ejpam-2272	160	8	∉	∉	PROPN
ejpam-2272	161	1	i	i	PRON
ejpam-2272	161	2	since	since	SCONJ
ejpam-2272	161	3	if	if	SCONJ
ejpam-2272	161	4	bcz	bcz	VERB
ejpam-2272	161	5	∈	∈	PRON
ejpam-2272	161	6	i	i	PRON
ejpam-2272	161	7	and	and	CCONJ
ejpam-2272	161	8	caz	caz	X
ejpam-2272	161	9	∈	∈	PROPN
ejpam-2272	162	1	i	i	PRON
ejpam-2272	162	2	,	,	PUNCT
ejpam-2272	162	3	we	we	PRON
ejpam-2272	162	4	get	get	VERB
ejpam-2272	162	5	bc	bc	PROPN
ejpam-2272	162	6	/	/	SYM
ejpam-2272	162	7	t	t	PROPN
ejpam-2272	162	8	r	r	NOUN
ejpam-2272	162	9	∈	∈	PROPN
ejpam-2272	162	10	√	√	NOUN
ejpam-2272	162	11	isa	isa	NOUN
ejpam-2272	162	12	and	and	CCONJ
ejpam-2272	162	13	ca	ca	NOUN
ejpam-2272	162	14	/	/	SYM
ejpam-2272	162	15	rs	rs	NOUN
ejpam-2272	162	16	∈	∈	PROPN
ejpam-2272	163	1	√	√	NUM
ejpam-2272	163	2	isa	isa	PROPN
ejpam-2272	163	3	,	,	PUNCT
ejpam-2272	163	4	which	which	PRON
ejpam-2272	163	5	leads	lead	VERB
ejpam-2272	163	6	to	to	ADP
ejpam-2272	163	7	a	a	DET
ejpam-2272	163	8	contradiction	contradiction	NOUN
ejpam-2272	163	9	.	.	PUNCT
ejpam-2272	164	1	since	since	SCONJ
ejpam-2272	164	2	abcz	abcz	NOUN
ejpam-2272	164	3	∈	∈	PROPN
ejpam-2272	164	4	i	i	PRON
ejpam-2272	164	5	and	and	CCONJ
ejpam-2272	164	6	i	i	PRON
ejpam-2272	164	7	is	be	AUX
ejpam-2272	164	8	a	a	DET
ejpam-2272	164	9	2	2	NUM
ejpam-2272	164	10	-	-	PUNCT
ejpam-2272	164	11	absorbing	absorbing	ADJ
ejpam-2272	164	12	primary	primary	ADJ
ejpam-2272	164	13	ideal	ideal	NOUN
ejpam-2272	164	14	of	of	ADP
ejpam-2272	164	15	s	s	PROPN
ejpam-2272	164	16	,	,	PUNCT
ejpam-2272	164	17	we	we	PRON
ejpam-2272	164	18	have	have	VERB
ejpam-2272	164	19	ab	ab	PROPN
ejpam-2272	164	20	∈	∈	PROPN
ejpam-2272	164	21	i	i	PRON
ejpam-2272	164	22	,	,	PUNCT
ejpam-2272	164	23	implies	imply	VERB
ejpam-2272	164	24	ab	ab	PROPN
ejpam-2272	164	25	/	/	SYM
ejpam-2272	164	26	st	st	PROPN
ejpam-2272	164	27	∈	∈	PROPN
ejpam-2272	164	28	isa	isa	PROPN
ejpam-2272	164	29	.	.	PUNCT
ejpam-2272	165	1	hence	hence	ADV
ejpam-2272	165	2	,	,	PUNCT
ejpam-2272	165	3	isa	isa	NOUN
ejpam-2272	165	4	is	be	AUX
ejpam-2272	165	5	a	a	DET
ejpam-2272	165	6	2	2	NUM
ejpam-2272	165	7	-	-	PUNCT
ejpam-2272	165	8	absorbing	absorbing	ADJ
ejpam-2272	165	9	primary	primary	ADJ
ejpam-2272	165	10	ideal	ideal	NOUN
ejpam-2272	165	11	of	of	ADP
ejpam-2272	165	12	sa	sa	PROPN
ejpam-2272	165	13	.	.	PUNCT
ejpam-2272	166	1	lemma	lemma	PROPN
ejpam-2272	166	2	1	1	X
ejpam-2272	166	3	.	.	PUNCT
ejpam-2272	167	1	let	let	AUX
ejpam-2272	167	2	i	i	PRON
ejpam-2272	167	3	be	be	AUX
ejpam-2272	167	4	a	a	DET
ejpam-2272	167	5	2	2	NUM
ejpam-2272	167	6	-	-	PUNCT
ejpam-2272	167	7	absorbing	absorbing	ADJ
ejpam-2272	167	8	primary	primary	ADJ
ejpam-2272	167	9	ideal	ideal	NOUN
ejpam-2272	167	10	of	of	ADP
ejpam-2272	167	11	s.	s.	PROPN
ejpam-2272	167	12	suppose	suppose	VERB
ejpam-2272	167	13	that	that	SCONJ
ejpam-2272	167	14	i	i	PRON
ejpam-2272	167	15	and	and	CCONJ
ejpam-2272	167	16	√	√	INTJ
ejpam-2272	167	17	i	i	PRON
ejpam-2272	167	18	be	be	VERB
ejpam-2272	167	19	subtractive	subtractive	ADJ
ejpam-2272	167	20	ideals	ideal	NOUN
ejpam-2272	167	21	of	of	ADP
ejpam-2272	167	22	s	s	PRON
ejpam-2272	167	23	and	and	CCONJ
ejpam-2272	167	24	abj	abj	PROPN
ejpam-2272	167	25	⊆	⊆	NUM
ejpam-2272	167	26	i	i	PRON
ejpam-2272	167	27	for	for	ADP
ejpam-2272	167	28	some	some	PRON
ejpam-2272	167	29	a	a	PRON
ejpam-2272	167	30	,	,	PUNCT
ejpam-2272	167	31	b	b	PROPN
ejpam-2272	167	32	∈	∈	PROPN
ejpam-2272	167	33	s	s	X
ejpam-2272	167	34	and	and	CCONJ
ejpam-2272	167	35	an	an	DET
ejpam-2272	167	36	ideal	ideal	ADJ
ejpam-2272	167	37	j	j	PROPN
ejpam-2272	167	38	of	of	ADP
ejpam-2272	167	39	s.	s.	PROPN
ejpam-2272	167	40	if	if	SCONJ
ejpam-2272	167	41	ab	ab	PROPN
ejpam-2272	167	42	∉	∉	PROPN
ejpam-2272	167	43	i	i	PRON
ejpam-2272	167	44	,	,	PUNCT
ejpam-2272	167	45	then	then	ADV
ejpam-2272	167	46	either	either	CCONJ
ejpam-2272	167	47	aj	aj	PROPN
ejpam-2272	167	48	⊆	⊆	NUM
ejpam-2272	167	49	√	√	PROPN
ejpam-2272	167	50	i	i	PRON
ejpam-2272	167	51	or	or	CCONJ
ejpam-2272	167	52	bj	bj	VERB
ejpam-2272	167	53	⊆	⊆	NUM
ejpam-2272	167	54	√	√	PROPN
ejpam-2272	167	55	i	i	PRON
ejpam-2272	167	56	.	.	PUNCT
ejpam-2272	168	1	proof	proof	NOUN
ejpam-2272	168	2	.	.	PUNCT
ejpam-2272	169	1	suppose	suppose	VERB
ejpam-2272	169	2	that	that	SCONJ
ejpam-2272	169	3	aj	aj	PROPN
ejpam-2272	169	4	/⊆	/⊆	PUNCT
ejpam-2272	169	5	√	√	INTJ
ejpam-2272	169	6	i	i	PRON
ejpam-2272	169	7	and	and	CCONJ
ejpam-2272	169	8	bj	bj	VERB
ejpam-2272	169	9	/⊆	/⊆	PUNCT
ejpam-2272	170	1	√	√	INTJ
ejpam-2272	171	1	i	i	PRON
ejpam-2272	171	2	.	.	PUNCT
ejpam-2272	172	1	therefore	therefore	ADV
ejpam-2272	172	2	,	,	PUNCT
ejpam-2272	172	3	there	there	PRON
ejpam-2272	172	4	are	be	VERB
ejpam-2272	172	5	some	some	PRON
ejpam-2272	172	6	x	x	NOUN
ejpam-2272	172	7	,	,	PUNCT
ejpam-2272	172	8	y	y	PROPN
ejpam-2272	172	9	∈	∈	PROPN
ejpam-2272	172	10	j	j	PROPN
ejpam-2272	172	11	such	such	ADJ
ejpam-2272	172	12	that	that	DET
ejpam-2272	172	13	ax	ax	NOUN
ejpam-2272	172	14	∉	∉	PROPN
ejpam-2272	172	15	√	√	PROPN
ejpam-2272	173	1	i	i	PRON
ejpam-2272	173	2	and	and	CCONJ
ejpam-2272	173	3	b	b	PROPN
ejpam-2272	173	4	y	y	PROPN
ejpam-2272	173	5	∉	∉	PROPN
ejpam-2272	173	6	√	√	PROPN
ejpam-2272	173	7	i	i	PRON
ejpam-2272	173	8	.	.	PUNCT
ejpam-2272	174	1	since	since	SCONJ
ejpam-2272	174	2	abx	abx	NOUN
ejpam-2272	174	3	∈	∈	PROPN
ejpam-2272	174	4	i	i	PRON
ejpam-2272	174	5	and	and	CCONJ
ejpam-2272	174	6	ab	ab	PROPN
ejpam-2272	174	7	∉	∉	PROPN
ejpam-2272	174	8	i	i	PRON
ejpam-2272	174	9	and	and	CCONJ
ejpam-2272	174	10	ax	ax	NOUN
ejpam-2272	174	11	∉	∉	PROPN
ejpam-2272	174	12	√	√	PROPN
ejpam-2272	174	13	i	i	PRON
ejpam-2272	174	14	,	,	PUNCT
ejpam-2272	174	15	we	we	PRON
ejpam-2272	174	16	have	have	VERB
ejpam-2272	174	17	bx	bx	NOUN
ejpam-2272	174	18	∈	∈	PROPN
ejpam-2272	175	1	√	√	PROPN
ejpam-2272	176	1	i	i	PRON
ejpam-2272	176	2	.	.	PUNCT
ejpam-2272	177	1	since	since	SCONJ
ejpam-2272	177	2	ab	ab	PROPN
ejpam-2272	177	3	y	y	PROPN
ejpam-2272	177	4	∈	∈	PROPN
ejpam-2272	178	1	i	i	PRON
ejpam-2272	178	2	and	and	CCONJ
ejpam-2272	178	3	ab	ab	PROPN
ejpam-2272	178	4	∉	∉	PROPN
ejpam-2272	178	5	i	i	PROPN
ejpam-2272	178	6	and	and	CCONJ
ejpam-2272	178	7	b	b	PROPN
ejpam-2272	178	8	y	y	PROPN
ejpam-2272	178	9	∉	∉	PROPN
ejpam-2272	179	1	√	√	PROPN
ejpam-2272	180	1	i	i	PRON
ejpam-2272	180	2	,	,	PUNCT
ejpam-2272	180	3	we	we	PRON
ejpam-2272	180	4	have	have	VERB
ejpam-2272	180	5	a	a	DET
ejpam-2272	180	6	y	y	PROPN
ejpam-2272	180	7	∈	∈	PROPN
ejpam-2272	180	8	√	√	VERB
ejpam-2272	181	1	i	i	PRON
ejpam-2272	181	2	.	.	PUNCT
ejpam-2272	182	1	now	now	ADV
ejpam-2272	182	2	,	,	PUNCT
ejpam-2272	182	3	since	since	SCONJ
ejpam-2272	182	4	ab(x	ab(x	CCONJ
ejpam-2272	182	5	+	+	CCONJ
ejpam-2272	182	6	y	y	X
ejpam-2272	182	7	)	)	PUNCT
ejpam-2272	182	8	∈	∈	PROPN
ejpam-2272	183	1	i	i	PRON
ejpam-2272	183	2	and	and	CCONJ
ejpam-2272	183	3	ab	ab	PROPN
ejpam-2272	183	4	∉	∉	PROPN
ejpam-2272	184	1	i	i	PRON
ejpam-2272	184	2	,	,	PUNCT
ejpam-2272	184	3	we	we	PRON
ejpam-2272	184	4	have	have	VERB
ejpam-2272	184	5	a(x	a(x	NOUN
ejpam-2272	184	6	+	+	CCONJ
ejpam-2272	184	7	y	y	NOUN
ejpam-2272	184	8	)	)	PUNCT
ejpam-2272	184	9	∈	∈	NOUN
ejpam-2272	184	10	√	√	VERB
ejpam-2272	185	1	i	i	PRON
ejpam-2272	185	2	or	or	CCONJ
ejpam-2272	185	3	b(x	b(x	PROPN
ejpam-2272	185	4	+	+	CCONJ
ejpam-2272	185	5	y	y	NOUN
ejpam-2272	185	6	)	)	PUNCT
ejpam-2272	185	7	∈	∈	NOUN
ejpam-2272	185	8	√	√	VERB
ejpam-2272	186	1	i	i	PRON
ejpam-2272	186	2	,	,	PUNCT
ejpam-2272	186	3	since	since	SCONJ
ejpam-2272	186	4	i	i	PRON
ejpam-2272	186	5	is	be	AUX
ejpam-2272	186	6	a	a	DET
ejpam-2272	186	7	2	2	NUM
ejpam-2272	186	8	-	-	PUNCT
ejpam-2272	186	9	absorbing	absorbing	ADJ
ejpam-2272	186	10	primary	primary	ADJ
ejpam-2272	186	11	ideal	ideal	NOUN
ejpam-2272	186	12	of	of	ADP
ejpam-2272	186	13	s.	s.	PROPN
ejpam-2272	186	14	if	if	SCONJ
ejpam-2272	186	15	a(x	a(x	PROPN
ejpam-2272	186	16	+	+	NUM
ejpam-2272	186	17	y	y	NOUN
ejpam-2272	186	18	)	)	PUNCT
ejpam-2272	186	19	∈	∈	NOUN
ejpam-2272	187	1	√	√	VERB
ejpam-2272	187	2	i	i	PRON
ejpam-2272	187	3	and	and	CCONJ
ejpam-2272	187	4	a	a	DET
ejpam-2272	187	5	y	y	PROPN
ejpam-2272	187	6	∈	∈	PROPN
ejpam-2272	188	1	√	√	VERB
ejpam-2272	188	2	i	i	PRON
ejpam-2272	188	3	,	,	PUNCT
ejpam-2272	188	4	then	then	ADV
ejpam-2272	188	5	ax	ax	NOUN
ejpam-2272	188	6	∈	∈	PROPN
ejpam-2272	189	1	√	√	VERB
ejpam-2272	189	2	i	i	PRON
ejpam-2272	189	3	,	,	PUNCT
ejpam-2272	189	4	since	since	SCONJ
ejpam-2272	189	5	√	√	INTJ
ejpam-2272	189	6	i	i	PRON
ejpam-2272	189	7	is	be	AUX
ejpam-2272	189	8	subtractive	subtractive	NOUN
ejpam-2272	189	9	,	,	PUNCT
ejpam-2272	189	10	which	which	PRON
ejpam-2272	189	11	is	be	AUX
ejpam-2272	189	12	a	a	DET
ejpam-2272	189	13	contradiction	contradiction	NOUN
ejpam-2272	189	14	.	.	PUNCT
ejpam-2272	190	1	similarly	similarly	ADV
ejpam-2272	190	2	,	,	PUNCT
ejpam-2272	190	3	if	if	SCONJ
ejpam-2272	190	4	b(x	b(x	VERB
ejpam-2272	190	5	+	+	CCONJ
ejpam-2272	190	6	y	y	NOUN
ejpam-2272	190	7	)	)	PUNCT
ejpam-2272	190	8	∈	∈	NOUN
ejpam-2272	190	9	√	√	VERB
ejpam-2272	191	1	i	i	PRON
ejpam-2272	191	2	and	and	CCONJ
ejpam-2272	191	3	bx	bx	NOUN
ejpam-2272	191	4	∈	∈	PROPN
ejpam-2272	192	1	√	√	VERB
ejpam-2272	192	2	i	i	PRON
ejpam-2272	192	3	,	,	PUNCT
ejpam-2272	192	4	we	we	PRON
ejpam-2272	192	5	get	get	VERB
ejpam-2272	192	6	b	b	NUM
ejpam-2272	192	7	y	y	PROPN
ejpam-2272	192	8	∈	∈	PROPN
ejpam-2272	192	9	√	√	VERB
ejpam-2272	192	10	i	i	PRON
ejpam-2272	192	11	,	,	PUNCT
ejpam-2272	192	12	a	a	DET
ejpam-2272	192	13	contradiction	contradiction	NOUN
ejpam-2272	192	14	.	.	PUNCT
ejpam-2272	193	1	hence	hence	ADV
ejpam-2272	193	2	,	,	PUNCT
ejpam-2272	193	3	either	either	CCONJ
ejpam-2272	193	4	aj	aj	PROPN
ejpam-2272	193	5	⊆	⊆	NUM
ejpam-2272	193	6	√	√	PROPN
ejpam-2272	193	7	i	i	PRON
ejpam-2272	193	8	or	or	CCONJ
ejpam-2272	193	9	bj	bj	VERB
ejpam-2272	193	10	⊆	⊆	NUM
ejpam-2272	193	11	√	√	PROPN
ejpam-2272	193	12	i	i	PRON
ejpam-2272	193	13	.	.	PUNCT
ejpam-2272	194	1	theorem	theorem	ADJ
ejpam-2272	194	2	8	8	NUM
ejpam-2272	194	3	.	.	PUNCT
ejpam-2272	195	1	let	let	VERB
ejpam-2272	195	2	i	i	PRON
ejpam-2272	195	3	be	be	AUX
ejpam-2272	195	4	a	a	DET
ejpam-2272	195	5	proper	proper	ADJ
ejpam-2272	195	6	subtractive	subtractive	NOUN
ejpam-2272	195	7	ideal	ideal	NOUN
ejpam-2272	195	8	of	of	ADP
ejpam-2272	195	9	s	s	PRON
ejpam-2272	195	10	and	and	CCONJ
ejpam-2272	195	11	suppose	suppose	VERB
ejpam-2272	195	12	that	that	SCONJ
ejpam-2272	195	13	√	√	VERB
ejpam-2272	195	14	i	i	PRON
ejpam-2272	195	15	is	be	AUX
ejpam-2272	195	16	a	a	DET
ejpam-2272	195	17	subtractive	subtractive	NOUN
ejpam-2272	195	18	ideal	ideal	NOUN
ejpam-2272	195	19	of	of	ADP
ejpam-2272	195	20	s.	s.	PROPN
ejpam-2272	196	1	then	then	ADV
ejpam-2272	196	2	i	i	PRON
ejpam-2272	196	3	is	be	AUX
ejpam-2272	196	4	a	a	DET
ejpam-2272	196	5	2	2	NUM
ejpam-2272	196	6	-	-	PUNCT
ejpam-2272	196	7	absorbing	absorbing	ADJ
ejpam-2272	196	8	primary	primary	ADJ
ejpam-2272	196	9	ideal	ideal	NOUN
ejpam-2272	196	10	of	of	ADP
ejpam-2272	196	11	s	s	PRON
ejpam-2272	196	12	if	if	SCONJ
ejpam-2272	197	1	and	and	CCONJ
ejpam-2272	197	2	only	only	ADV
ejpam-2272	197	3	if	if	SCONJ
ejpam-2272	197	4	whenever	whenever	SCONJ
ejpam-2272	197	5	i1	i1	PROPN
ejpam-2272	197	6	i2	i2	PROPN
ejpam-2272	197	7	i3	i3	PROPN
ejpam-2272	197	8	⊆	⊆	PROPN
ejpam-2272	197	9	i	i	PRON
ejpam-2272	197	10	for	for	ADP
ejpam-2272	197	11	some	some	DET
ejpam-2272	197	12	ideals	ideal	NOUN
ejpam-2272	197	13	i1	i1	PROPN
ejpam-2272	197	14	,	,	PUNCT
ejpam-2272	197	15	i2	i2	PROPN
ejpam-2272	197	16	,	,	PUNCT
ejpam-2272	197	17	i3	i3	NOUN
ejpam-2272	197	18	of	of	ADP
ejpam-2272	197	19	s	s	PROPN
ejpam-2272	197	20	,	,	PUNCT
ejpam-2272	197	21	then	then	ADV
ejpam-2272	197	22	either	either	CCONJ
ejpam-2272	197	23	i1	i1	PROPN
ejpam-2272	197	24	i2	i2	PROPN
ejpam-2272	197	25	⊆	⊆	NUM
ejpam-2272	197	26	i	i	PROPN
ejpam-2272	197	27	or	or	CCONJ
ejpam-2272	197	28	i2	i2	PROPN
ejpam-2272	197	29	i3	i3	NOUN
ejpam-2272	197	30	⊆	⊆	PROPN
ejpam-2272	197	31	√	√	NUM
ejpam-2272	197	32	i	i	PRON
ejpam-2272	197	33	or	or	CCONJ
ejpam-2272	197	34	i3	i3	PROPN
ejpam-2272	197	35	i1	i1	PROPN
ejpam-2272	198	1	⊆	⊆	NUM
ejpam-2272	198	2	√	√	PROPN
ejpam-2272	199	1	i	i	PRON
ejpam-2272	199	2	.	.	PUNCT
ejpam-2272	200	1	proof	proof	NOUN
ejpam-2272	200	2	.	.	PUNCT
ejpam-2272	201	1	proof	proof	NOUN
ejpam-2272	201	2	is	be	AUX
ejpam-2272	201	3	similar	similar	ADJ
ejpam-2272	201	4	to	to	ADP
ejpam-2272	201	5	the	the	DET
ejpam-2272	201	6	proof	proof	NOUN
ejpam-2272	201	7	of	of	ADP
ejpam-2272	201	8	[	[	X
ejpam-2272	201	9	9	9	NUM
ejpam-2272	201	10	,	,	PUNCT
ejpam-2272	201	11	theorem	theorem	VERB
ejpam-2272	201	12	2.19	2.19	NUM
ejpam-2272	201	13	]	]	PUNCT
ejpam-2272	201	14	definition	definition	NOUN
ejpam-2272	201	15	3	3	NUM
ejpam-2272	201	16	(	(	PUNCT
ejpam-2272	201	17	[	[	X
ejpam-2272	201	18	1	1	NUM
ejpam-2272	201	19	,	,	PUNCT
ejpam-2272	201	20	definition	definition	NOUN
ejpam-2272	201	21	(	(	PUNCT
ejpam-2272	201	22	4	4	NUM
ejpam-2272	201	23	)	)	PUNCT
ejpam-2272	201	24	]	]	PUNCT
ejpam-2272	201	25	)	)	PUNCT
ejpam-2272	201	26	.	.	PUNCT
ejpam-2272	202	1	an	an	DET
ejpam-2272	202	2	ideal	ideal	ADJ
ejpam-2272	202	3	i	i	PRON
ejpam-2272	202	4	of	of	ADP
ejpam-2272	202	5	a	a	DET
ejpam-2272	202	6	semiring	semiring	NOUN
ejpam-2272	202	7	s	s	X
ejpam-2272	202	8	is	be	AUX
ejpam-2272	202	9	called	call	VERB
ejpam-2272	202	10	a	a	DET
ejpam-2272	202	11	q	q	NOUN
ejpam-2272	202	12	-	-	PUNCT
ejpam-2272	202	13	ideal	ideal	ADJ
ejpam-2272	202	14	(	(	PUNCT
ejpam-2272	202	15	partitioning	partition	VERB
ejpam-2272	202	16	ideal	ideal	NOUN
ejpam-2272	202	17	)	)	PUNCT
ejpam-2272	202	18	if	if	SCONJ
ejpam-2272	202	19	there	there	PRON
ejpam-2272	202	20	exists	exist	VERB
ejpam-2272	202	21	a	a	DET
ejpam-2272	202	22	subset	subset	NOUN
ejpam-2272	202	23	q	q	NOUN
ejpam-2272	202	24	of	of	ADP
ejpam-2272	202	25	s	s	PRON
ejpam-2272	203	1	such	such	ADJ
ejpam-2272	203	2	that	that	SCONJ
ejpam-2272	203	3	p.	p.	PROPN
ejpam-2272	203	4	kumar	kumar	PROPN
ejpam-2272	203	5	,	,	PUNCT
ejpam-2272	203	6	m.	m.	NOUN
ejpam-2272	203	7	dubey	dubey	PROPN
ejpam-2272	203	8	,	,	PUNCT
ejpam-2272	203	9	p.	p.	NOUN
ejpam-2272	203	10	sarohe	sarohe	PROPN
ejpam-2272	203	11	/	/	SYM
ejpam-2272	203	12	eur	eur	PROPN
ejpam-2272	203	13	.	.	PUNCT
ejpam-2272	204	1	j.	j.	PROPN
ejpam-2272	204	2	pure	pure	PROPN
ejpam-2272	204	3	appl	appl	PROPN
ejpam-2272	204	4	.	.	PROPN
ejpam-2272	204	5	math	math	PROPN
ejpam-2272	204	6	,	,	PUNCT
ejpam-2272	204	7	9	9	NUM
ejpam-2272	204	8	(	(	PUNCT
ejpam-2272	204	9	2016	2016	NUM
ejpam-2272	204	10	)	)	PUNCT
ejpam-2272	204	11	,	,	PUNCT
ejpam-2272	204	12	186	186	NUM
ejpam-2272	204	13	-	-	SYM
ejpam-2272	204	14	195	195	NUM
ejpam-2272	204	15	190	190	NUM
ejpam-2272	204	16	(	(	PUNCT
ejpam-2272	204	17	1	1	NUM
ejpam-2272	204	18	)	)	PUNCT
ejpam-2272	204	19	s	s	PART
ejpam-2272	205	1	=	=	SYM
ejpam-2272	205	2	∪{q	∪{q	NOUN
ejpam-2272	205	3	+	+	NOUN
ejpam-2272	205	4	i	i	PRON
ejpam-2272	205	5	∶	∶	VERB
ejpam-2272	205	6	q	q	X
ejpam-2272	205	7	∈	∈	ADJ
ejpam-2272	205	8	q	q	X
ejpam-2272	205	9	}	}	PUNCT
ejpam-2272	205	10	(	(	PUNCT
ejpam-2272	205	11	2	2	X
ejpam-2272	205	12	)	)	PUNCT
ejpam-2272	205	13	if	if	SCONJ
ejpam-2272	205	14	q1	q1	PROPN
ejpam-2272	205	15	,	,	PUNCT
ejpam-2272	205	16	q2	q2	PROPN
ejpam-2272	205	17	∈	∈	PROPN
ejpam-2272	206	1	q	q	NOUN
ejpam-2272	206	2	,	,	PUNCT
ejpam-2272	206	3	then	then	ADV
ejpam-2272	206	4	(	(	PUNCT
ejpam-2272	206	5	q1	q1	PROPN
ejpam-2272	206	6	+	+	CCONJ
ejpam-2272	206	7	i	i	NOUN
ejpam-2272	206	8	)	)	PUNCT
ejpam-2272	206	9	∩	∩	NOUN
ejpam-2272	206	10	(	(	PUNCT
ejpam-2272	206	11	q2	q2	NOUN
ejpam-2272	206	12	+	+	CCONJ
ejpam-2272	206	13	i	i	NOUN
ejpam-2272	206	14	)	)	PUNCT
ejpam-2272	206	15	≠	≠	PROPN
ejpam-2272	206	16	∅	∅	NOUN
ejpam-2272	206	17	⇔	⇔	PROPN
ejpam-2272	206	18	q1	q1	PROPN
ejpam-2272	206	19	=	=	PROPN
ejpam-2272	206	20	q2	q2	PROPN
ejpam-2272	206	21	.	.	PUNCT
ejpam-2272	207	1	let	let	VERB
ejpam-2272	207	2	i	i	PRON
ejpam-2272	207	3	be	be	AUX
ejpam-2272	207	4	a	a	DET
ejpam-2272	207	5	q	q	NOUN
ejpam-2272	207	6	-	-	PUNCT
ejpam-2272	207	7	ideal	ideal	NOUN
ejpam-2272	207	8	of	of	ADP
ejpam-2272	207	9	a	a	DET
ejpam-2272	207	10	semiring	semire	VERB
ejpam-2272	207	11	s.	s.	PROPN
ejpam-2272	207	12	then	then	ADV
ejpam-2272	207	13	s	s	PROPN
ejpam-2272	207	14	/	/	SYM
ejpam-2272	207	15	iq	iq	NOUN
ejpam-2272	207	16	=	=	PUNCT
ejpam-2272	207	17	{	{	PUNCT
ejpam-2272	207	18	q	q	PROPN
ejpam-2272	208	1	+	+	CCONJ
ejpam-2272	208	2	i	i	PRON
ejpam-2272	208	3	∶	∶	VERB
ejpam-2272	208	4	q	q	X
ejpam-2272	208	5	∈	∈	ADJ
ejpam-2272	208	6	q	q	NOUN
ejpam-2272	208	7	}	}	PUNCT
ejpam-2272	208	8	forms	form	VERB
ejpam-2272	208	9	a	a	DET
ejpam-2272	208	10	semiring	semiring	NOUN
ejpam-2272	208	11	under	under	ADP
ejpam-2272	208	12	the	the	DET
ejpam-2272	208	13	following	follow	VERB
ejpam-2272	208	14	addition	addition	NOUN
ejpam-2272	208	15	‘	'	PUNCT
ejpam-2272	208	16	⊕	⊕	NOUN
ejpam-2272	208	17	’	'	PUNCT
ejpam-2272	208	18	and	and	CCONJ
ejpam-2272	208	19	multiplication	multiplication	NOUN
ejpam-2272	208	20	’	'	PUNCT
ejpam-2272	208	21	⊙	⊙	PROPN
ejpam-2272	208	22	’	'	PUNCT
ejpam-2272	208	23	,	,	PUNCT
ejpam-2272	208	24	(	(	PUNCT
ejpam-2272	208	25	q1	q1	VERB
ejpam-2272	208	26	+	+	CCONJ
ejpam-2272	209	1	i	i	PROPN
ejpam-2272	209	2	)	)	PUNCT
ejpam-2272	209	3	⊕	⊕	PROPN
ejpam-2272	209	4	(	(	PUNCT
ejpam-2272	209	5	q2	q2	PROPN
ejpam-2272	209	6	+	+	CCONJ
ejpam-2272	209	7	i	i	NOUN
ejpam-2272	209	8	)	)	PUNCT
ejpam-2272	210	1	=	=	SYM
ejpam-2272	210	2	q3	q3	PROPN
ejpam-2272	210	3	+	+	CCONJ
ejpam-2272	210	4	i	i	PRON
ejpam-2272	210	5	,	,	PUNCT
ejpam-2272	210	6	where	where	SCONJ
ejpam-2272	210	7	q3	q3	PROPN
ejpam-2272	210	8	∈	∈	PROPN
ejpam-2272	211	1	q	q	NOUN
ejpam-2272	211	2	is	be	AUX
ejpam-2272	211	3	unique	unique	ADJ
ejpam-2272	211	4	such	such	ADJ
ejpam-2272	211	5	that	that	DET
ejpam-2272	211	6	q1	q1	PROPN
ejpam-2272	211	7	+	+	CCONJ
ejpam-2272	211	8	q2	q2	NOUN
ejpam-2272	212	1	+	+	CCONJ
ejpam-2272	212	2	i	i	PROPN
ejpam-2272	212	3	⊆	⊆	NUM
ejpam-2272	212	4	q3	q3	NOUN
ejpam-2272	213	1	+	+	CCONJ
ejpam-2272	213	2	i	i	PRON
ejpam-2272	213	3	and	and	CCONJ
ejpam-2272	213	4	(	(	PUNCT
ejpam-2272	213	5	q1	q1	PROPN
ejpam-2272	213	6	+	+	CCONJ
ejpam-2272	213	7	i	i	PROPN
ejpam-2272	213	8	)	)	PUNCT
ejpam-2272	213	9	⊙	⊙	PROPN
ejpam-2272	213	10	(	(	PUNCT
ejpam-2272	213	11	q2	q2	PROPN
ejpam-2272	213	12	+	+	CCONJ
ejpam-2272	213	13	i	i	NOUN
ejpam-2272	213	14	)	)	PUNCT
ejpam-2272	214	1	=	=	SYM
ejpam-2272	214	2	q4	q4	PROPN
ejpam-2272	214	3	+	+	CCONJ
ejpam-2272	214	4	i	i	PRON
ejpam-2272	214	5	,	,	PUNCT
ejpam-2272	214	6	where	where	SCONJ
ejpam-2272	214	7	q4	q4	PROPN
ejpam-2272	214	8	∈	∈	PROPN
ejpam-2272	214	9	q	q	NOUN
ejpam-2272	214	10	is	be	AUX
ejpam-2272	214	11	unique	unique	ADJ
ejpam-2272	215	1	such	such	ADJ
ejpam-2272	215	2	that	that	SCONJ
ejpam-2272	215	3	q1q2	q1q2	NOUN
ejpam-2272	216	1	+	+	NUM
ejpam-2272	216	2	i	i	PROPN
ejpam-2272	216	3	⊆	⊆	NUM
ejpam-2272	216	4	q4	q4	PROPN
ejpam-2272	216	5	+	+	CCONJ
ejpam-2272	216	6	i	i	NOUN
ejpam-2272	216	7	.	.	PUNCT
ejpam-2272	217	1	this	this	DET
ejpam-2272	217	2	semiring	semire	VERB
ejpam-2272	217	3	s	s	AUX
ejpam-2272	217	4	/	/	SYM
ejpam-2272	217	5	iq	iq	PROPN
ejpam-2272	217	6	is	be	AUX
ejpam-2272	217	7	called	call	VERB
ejpam-2272	217	8	the	the	DET
ejpam-2272	217	9	quotient	quotient	NOUN
ejpam-2272	217	10	semiring	semire	VERB
ejpam-2272	217	11	of	of	ADP
ejpam-2272	217	12	s	s	PRON
ejpam-2272	217	13	and	and	CCONJ
ejpam-2272	217	14	denoted	denote	VERB
ejpam-2272	217	15	by	by	ADP
ejpam-2272	217	16	(	(	PUNCT
ejpam-2272	217	17	s	s	X
ejpam-2272	217	18	/	/	SYM
ejpam-2272	217	19	iq,⊕,⊙	iq,⊕,⊙	VERB
ejpam-2272	217	20	)	)	PUNCT
ejpam-2272	217	21	or	or	CCONJ
ejpam-2272	217	22	s	s	PROPN
ejpam-2272	217	23	/	/	SYM
ejpam-2272	217	24	iq	iq	NOUN
ejpam-2272	217	25	.	.	PUNCT
ejpam-2272	218	1	by	by	ADP
ejpam-2272	218	2	definition	definition	NOUN
ejpam-2272	218	3	of	of	ADP
ejpam-2272	218	4	q	q	NOUN
ejpam-2272	218	5	-	-	PUNCT
ejpam-2272	218	6	ideal	ideal	ADJ
ejpam-2272	218	7	,	,	PUNCT
ejpam-2272	218	8	there	there	PRON
ejpam-2272	218	9	exists	exist	VERB
ejpam-2272	218	10	a	a	DET
ejpam-2272	218	11	unique	unique	ADJ
ejpam-2272	218	12	q0	q0	NOUN
ejpam-2272	218	13	∈	∈	NOUN
ejpam-2272	218	14	q	q	NOUN
ejpam-2272	218	15	such	such	ADJ
ejpam-2272	218	16	that	that	DET
ejpam-2272	218	17	0	0	NUM
ejpam-2272	219	1	+	+	NUM
ejpam-2272	219	2	i	i	PRON
ejpam-2272	219	3	⊆	⊆	NUM
ejpam-2272	219	4	q0	q0	NOUN
ejpam-2272	220	1	+	+	X
ejpam-2272	220	2	i	i	PRON
ejpam-2272	220	3	.	.	PUNCT
ejpam-2272	221	1	then	then	ADV
ejpam-2272	221	2	q0	q0	PROPN
ejpam-2272	222	1	+	+	CCONJ
ejpam-2272	222	2	i	i	PRON
ejpam-2272	222	3	is	be	AUX
ejpam-2272	222	4	a	a	DET
ejpam-2272	222	5	zero	zero	NUM
ejpam-2272	222	6	element	element	NOUN
ejpam-2272	222	7	of	of	ADP
ejpam-2272	222	8	s	s	PROPN
ejpam-2272	222	9	/	/	SYM
ejpam-2272	222	10	iq	iq	NOUN
ejpam-2272	222	11	.	.	PUNCT
ejpam-2272	223	1	clearly	clearly	ADV
ejpam-2272	223	2	,	,	PUNCT
ejpam-2272	223	3	if	if	SCONJ
ejpam-2272	223	4	s	s	VERB
ejpam-2272	223	5	is	be	AUX
ejpam-2272	223	6	commutative	commutative	ADJ
ejpam-2272	223	7	then	then	ADV
ejpam-2272	223	8	s	s	PART
ejpam-2272	223	9	/	/	SYM
ejpam-2272	223	10	iq	iq	PROPN
ejpam-2272	223	11	is	be	AUX
ejpam-2272	223	12	commutative	commutative	ADJ
ejpam-2272	223	13	.	.	PUNCT
ejpam-2272	224	1	theorem	theorem	NOUN
ejpam-2272	224	2	9	9	NUM
ejpam-2272	224	3	.	.	PUNCT
ejpam-2272	225	1	let	let	VERB
ejpam-2272	225	2	s	s	PRON
ejpam-2272	225	3	be	be	AUX
ejpam-2272	225	4	a	a	DET
ejpam-2272	225	5	semiring	semiring	NOUN
ejpam-2272	225	6	,	,	PUNCT
ejpam-2272	225	7	i	i	PRON
ejpam-2272	225	8	a	a	DET
ejpam-2272	225	9	q	q	NOUN
ejpam-2272	225	10	-	-	PUNCT
ejpam-2272	225	11	ideal	ideal	NOUN
ejpam-2272	225	12	of	of	ADP
ejpam-2272	225	13	s	s	PRON
ejpam-2272	225	14	and	and	CCONJ
ejpam-2272	225	15	p	p	X
ejpam-2272	225	16	a	a	DET
ejpam-2272	225	17	subtractive	subtractive	NOUN
ejpam-2272	225	18	ideal	ideal	NOUN
ejpam-2272	225	19	of	of	ADP
ejpam-2272	225	20	s	s	PRON
ejpam-2272	225	21	such	such	ADJ
ejpam-2272	225	22	that	that	SCONJ
ejpam-2272	225	23	i	i	PRON
ejpam-2272	225	24	⊆	⊆	NUM
ejpam-2272	226	1	p.	p.	NOUN
ejpam-2272	226	2	then	then	ADV
ejpam-2272	226	3	p	p	NOUN
ejpam-2272	226	4	is	be	AUX
ejpam-2272	226	5	a	a	DET
ejpam-2272	226	6	2	2	NUM
ejpam-2272	226	7	-	-	PUNCT
ejpam-2272	226	8	absorbing	absorbing	ADJ
ejpam-2272	226	9	primary	primary	ADJ
ejpam-2272	226	10	ideal	ideal	NOUN
ejpam-2272	226	11	of	of	ADP
ejpam-2272	226	12	s	s	PRON
ejpam-2272	226	13	if	if	SCONJ
ejpam-2272	227	1	and	and	CCONJ
ejpam-2272	227	2	only	only	ADV
ejpam-2272	227	3	if	if	SCONJ
ejpam-2272	227	4	p	p	X
ejpam-2272	227	5	/	/	SYM
ejpam-2272	227	6	iq∩p	iq∩p	PROPN
ejpam-2272	227	7	is	be	AUX
ejpam-2272	227	8	a	a	DET
ejpam-2272	227	9	2	2	NUM
ejpam-2272	227	10	-	-	PUNCT
ejpam-2272	227	11	absorbing	absorbing	ADJ
ejpam-2272	227	12	primary	primary	ADJ
ejpam-2272	227	13	ideal	ideal	NOUN
ejpam-2272	227	14	of	of	ADP
ejpam-2272	227	15	s	s	PROPN
ejpam-2272	227	16	/	/	SYM
ejpam-2272	227	17	iq	iq	NOUN
ejpam-2272	227	18	.	.	PUNCT
ejpam-2272	228	1	proof	proof	NOUN
ejpam-2272	228	2	.	.	PUNCT
ejpam-2272	229	1	let	let	VERB
ejpam-2272	229	2	p	p	PRON
ejpam-2272	229	3	be	be	AUX
ejpam-2272	229	4	a	a	DET
ejpam-2272	229	5	2	2	NUM
ejpam-2272	229	6	-	-	PUNCT
ejpam-2272	229	7	absorbing	absorbing	ADJ
ejpam-2272	229	8	primary	primary	ADJ
ejpam-2272	229	9	ideal	ideal	NOUN
ejpam-2272	229	10	of	of	ADP
ejpam-2272	229	11	s.	s.	PROPN
ejpam-2272	229	12	suppose	suppose	VERB
ejpam-2272	229	13	that	that	SCONJ
ejpam-2272	229	14	q1	q1	PROPN
ejpam-2272	229	15	+	+	CCONJ
ejpam-2272	229	16	i	i	PROPN
ejpam-2272	229	17	,	,	PUNCT
ejpam-2272	229	18	q2	q2	PROPN
ejpam-2272	229	19	+	+	CCONJ
ejpam-2272	229	20	i	i	PROPN
ejpam-2272	229	21	,	,	PUNCT
ejpam-2272	229	22	q3	q3	PROPN
ejpam-2272	230	1	+	+	CCONJ
ejpam-2272	230	2	i	i	PRON
ejpam-2272	230	3	∈	∈	PROPN
ejpam-2272	230	4	s	s	X
ejpam-2272	230	5	/	/	SYM
ejpam-2272	230	6	iq	iq	NOUN
ejpam-2272	230	7	are	be	AUX
ejpam-2272	230	8	such	such	ADJ
ejpam-2272	230	9	that	that	SCONJ
ejpam-2272	230	10	(	(	PUNCT
ejpam-2272	230	11	q1	q1	PROPN
ejpam-2272	230	12	+	+	CCONJ
ejpam-2272	230	13	i	i	PROPN
ejpam-2272	230	14	)	)	PUNCT
ejpam-2272	230	15	⊙	⊙	PROPN
ejpam-2272	230	16	(	(	PUNCT
ejpam-2272	230	17	q2	q2	PROPN
ejpam-2272	230	18	+	+	CCONJ
ejpam-2272	230	19	i	i	PROPN
ejpam-2272	230	20	)	)	PUNCT
ejpam-2272	230	21	⊙	⊙	PROPN
ejpam-2272	230	22	(	(	PUNCT
ejpam-2272	230	23	q3	q3	PROPN
ejpam-2272	230	24	+	+	PROPN
ejpam-2272	230	25	i	i	NOUN
ejpam-2272	230	26	)	)	PUNCT
ejpam-2272	231	1	=	=	SYM
ejpam-2272	231	2	q4	q4	PROPN
ejpam-2272	231	3	+	+	CCONJ
ejpam-2272	232	1	i	i	PRON
ejpam-2272	232	2	∈	∈	PROPN
ejpam-2272	232	3	p	p	X
ejpam-2272	232	4	/	/	SYM
ejpam-2272	232	5	iq∩p	iq∩p	PROPN
ejpam-2272	232	6	where	where	SCONJ
ejpam-2272	232	7	q4	q4	PROPN
ejpam-2272	232	8	∈	∈	PROPN
ejpam-2272	232	9	q	q	PROPN
ejpam-2272	232	10	∩	∩	PROPN
ejpam-2272	232	11	p	p	NOUN
ejpam-2272	232	12	is	be	AUX
ejpam-2272	232	13	a	a	DET
ejpam-2272	232	14	unique	unique	ADJ
ejpam-2272	232	15	element	element	NOUN
ejpam-2272	233	1	such	such	ADJ
ejpam-2272	233	2	that	that	SCONJ
ejpam-2272	233	3	q1q2q3	q1q2q3	NOUN
ejpam-2272	233	4	+	+	PROPN
ejpam-2272	233	5	i	i	PROPN
ejpam-2272	233	6	⊆	⊆	NUM
ejpam-2272	233	7	q4	q4	PROPN
ejpam-2272	234	1	+	+	CCONJ
ejpam-2272	234	2	i	i	PRON
ejpam-2272	234	3	∈	∈	PROPN
ejpam-2272	234	4	p	p	X
ejpam-2272	234	5	/	/	SYM
ejpam-2272	234	6	iq∩p	iq∩p	PROPN
ejpam-2272	234	7	.	.	PUNCT
ejpam-2272	235	1	so	so	ADV
ejpam-2272	235	2	q1q2q3	q1q2q3	PROPN
ejpam-2272	235	3	=	=	SYM
ejpam-2272	235	4	q4	q4	PROPN
ejpam-2272	235	5	+	+	CCONJ
ejpam-2272	235	6	i	i	PRON
ejpam-2272	235	7	for	for	ADP
ejpam-2272	235	8	some	some	PRON
ejpam-2272	235	9	i	i	PRON
ejpam-2272	235	10	∈	∈	PROPN
ejpam-2272	236	1	i	i	PRON
ejpam-2272	236	2	.	.	PUNCT
ejpam-2272	237	1	since	since	SCONJ
ejpam-2272	237	2	p	p	NOUN
ejpam-2272	237	3	is	be	AUX
ejpam-2272	237	4	a	a	DET
ejpam-2272	237	5	2absorbing	2absorbing	NUM
ejpam-2272	237	6	primary	primary	ADJ
ejpam-2272	237	7	ideal	ideal	NOUN
ejpam-2272	237	8	of	of	ADP
ejpam-2272	237	9	s	s	PRON
ejpam-2272	237	10	and	and	CCONJ
ejpam-2272	237	11	q1q2q3	q1q2q3	PROPN
ejpam-2272	237	12	∈	∈	PROPN
ejpam-2272	237	13	p	p	NOUN
ejpam-2272	237	14	,	,	PUNCT
ejpam-2272	237	15	therefore	therefore	ADV
ejpam-2272	237	16	q1q2	q1q2	X
ejpam-2272	237	17	∈	∈	PROPN
ejpam-2272	237	18	p	p	NOUN
ejpam-2272	237	19	or	or	CCONJ
ejpam-2272	237	20	(	(	PUNCT
ejpam-2272	237	21	q2q3)m	q2q3)m	NOUN
ejpam-2272	237	22	∈	∈	PROPN
ejpam-2272	237	23	p	p	NOUN
ejpam-2272	237	24	or	or	CCONJ
ejpam-2272	237	25	(	(	PUNCT
ejpam-2272	237	26	q3q1)n	q3q1)n	PROPN
ejpam-2272	237	27	∈	∈	PROPN
ejpam-2272	237	28	p	p	NOUN
ejpam-2272	237	29	for	for	ADP
ejpam-2272	237	30	some	some	DET
ejpam-2272	237	31	positive	positive	ADJ
ejpam-2272	237	32	integers	integer	NOUN
ejpam-2272	237	33	m	m	VERB
ejpam-2272	237	34	,	,	PUNCT
ejpam-2272	237	35	n.	n.	PROPN
ejpam-2272	237	36	consider	consider	VERB
ejpam-2272	237	37	the	the	DET
ejpam-2272	237	38	case	case	NOUN
ejpam-2272	238	1	q1q2	q1q2	X
ejpam-2272	238	2	∈	∈	PROPN
ejpam-2272	239	1	p.	p.	NOUN
ejpam-2272	239	2	if	if	SCONJ
ejpam-2272	239	3	(	(	PUNCT
ejpam-2272	239	4	q1	q1	NOUN
ejpam-2272	239	5	+	+	CCONJ
ejpam-2272	239	6	i	i	PROPN
ejpam-2272	239	7	)	)	PUNCT
ejpam-2272	239	8	⊙	⊙	PROPN
ejpam-2272	239	9	(	(	PUNCT
ejpam-2272	239	10	q2	q2	PROPN
ejpam-2272	239	11	+	+	CCONJ
ejpam-2272	239	12	i	i	NOUN
ejpam-2272	239	13	)	)	PUNCT
ejpam-2272	240	1	=	=	SYM
ejpam-2272	240	2	i1	i1	PROPN
ejpam-2272	240	3	+	+	CCONJ
ejpam-2272	240	4	i	i	PRON
ejpam-2272	240	5	where	where	SCONJ
ejpam-2272	240	6	i1	i1	PROPN
ejpam-2272	240	7	∈	∈	PROPN
ejpam-2272	241	1	q	q	NOUN
ejpam-2272	241	2	is	be	AUX
ejpam-2272	241	3	a	a	DET
ejpam-2272	241	4	unique	unique	ADJ
ejpam-2272	241	5	element	element	NOUN
ejpam-2272	241	6	such	such	ADJ
ejpam-2272	241	7	that	that	SCONJ
ejpam-2272	241	8	q1q2	q1q2	NOUN
ejpam-2272	242	1	+	+	NUM
ejpam-2272	242	2	i	i	PROPN
ejpam-2272	242	3	⊆	⊆	NUM
ejpam-2272	242	4	i1	i1	PROPN
ejpam-2272	242	5	+	+	CCONJ
ejpam-2272	242	6	i	i	PRON
ejpam-2272	242	7	.	.	PUNCT
ejpam-2272	243	1	so	so	ADV
ejpam-2272	243	2	i1	i1	PROPN
ejpam-2272	243	3	+	+	CCONJ
ejpam-2272	244	1	f	f	X
ejpam-2272	244	2	=	=	SYM
ejpam-2272	244	3	q1q2	q1q2	PROPN
ejpam-2272	244	4	+	+	NUM
ejpam-2272	244	5	e	e	NOUN
ejpam-2272	244	6	for	for	ADP
ejpam-2272	244	7	some	some	DET
ejpam-2272	244	8	e	e	NOUN
ejpam-2272	244	9	,	,	PUNCT
ejpam-2272	244	10	f	f	PROPN
ejpam-2272	244	11	∈	∈	PROPN
ejpam-2272	244	12	i	i	PRON
ejpam-2272	244	13	.	.	PUNCT
ejpam-2272	245	1	since	since	SCONJ
ejpam-2272	245	2	p	p	NOUN
ejpam-2272	245	3	is	be	AUX
ejpam-2272	245	4	subtractive	subtractive	NOUN
ejpam-2272	245	5	and	and	CCONJ
ejpam-2272	245	6	i	i	PRON
ejpam-2272	245	7	⊆	⊆	NUM
ejpam-2272	245	8	p	p	NOUN
ejpam-2272	245	9	,	,	PUNCT
ejpam-2272	245	10	we	we	PRON
ejpam-2272	245	11	have	have	VERB
ejpam-2272	245	12	i1	i1	PROPN
ejpam-2272	245	13	∈	∈	PROPN
ejpam-2272	245	14	p	p	X
ejpam-2272	245	15	,	,	PUNCT
ejpam-2272	245	16	therefore	therefore	ADV
ejpam-2272	245	17	i1	i1	PROPN
ejpam-2272	245	18	∈	∈	PROPN
ejpam-2272	245	19	q	q	PROPN
ejpam-2272	245	20	∩	∩	PROPN
ejpam-2272	245	21	p.	p.	NOUN
ejpam-2272	245	22	thus	thus	ADV
ejpam-2272	245	23	,	,	PUNCT
ejpam-2272	245	24	p	p	X
ejpam-2272	245	25	/	/	SYM
ejpam-2272	245	26	iq∩p	iq∩p	PROPN
ejpam-2272	245	27	is	be	AUX
ejpam-2272	245	28	a	a	DET
ejpam-2272	245	29	2	2	NUM
ejpam-2272	245	30	-	-	PUNCT
ejpam-2272	245	31	absorbing	absorbing	ADJ
ejpam-2272	245	32	primary	primary	ADJ
ejpam-2272	245	33	ideal	ideal	NOUN
ejpam-2272	245	34	of	of	ADP
ejpam-2272	245	35	s	s	PROPN
ejpam-2272	245	36	/	/	SYM
ejpam-2272	245	37	iq	iq	NOUN
ejpam-2272	245	38	.	.	PUNCT
ejpam-2272	246	1	next	next	ADV
ejpam-2272	246	2	,	,	PUNCT
ejpam-2272	246	3	if	if	SCONJ
ejpam-2272	246	4	qm	qm	PROPN
ejpam-2272	246	5	2	2	NUM
ejpam-2272	246	6	qm	qm	PROPN
ejpam-2272	246	7	3	3	NUM
ejpam-2272	246	8	∈	∈	PROPN
ejpam-2272	246	9	p	p	NOUN
ejpam-2272	246	10	for	for	ADP
ejpam-2272	246	11	some	some	DET
ejpam-2272	246	12	positive	positive	ADJ
ejpam-2272	246	13	integer	integer	NOUN
ejpam-2272	246	14	m.	m.	NOUN
ejpam-2272	246	15	let	let	VERB
ejpam-2272	246	16	(	(	PUNCT
ejpam-2272	246	17	qm	qm	PROPN
ejpam-2272	246	18	2	2	NUM
ejpam-2272	246	19	+	+	NUM
ejpam-2272	246	20	i	i	PROPN
ejpam-2272	246	21	)	)	PUNCT
ejpam-2272	246	22	⊙	⊙	PROPN
ejpam-2272	246	23	(	(	PUNCT
ejpam-2272	246	24	qm	qm	PROPN
ejpam-2272	246	25	3	3	NUM
ejpam-2272	246	26	+	+	CCONJ
ejpam-2272	246	27	i	i	NOUN
ejpam-2272	246	28	)	)	PUNCT
ejpam-2272	247	1	=	=	SYM
ejpam-2272	247	2	i2	i2	PROPN
ejpam-2272	247	3	+	+	CCONJ
ejpam-2272	247	4	i	i	PRON
ejpam-2272	247	5	where	where	SCONJ
ejpam-2272	247	6	i2	i2	PROPN
ejpam-2272	247	7	∈	∈	PROPN
ejpam-2272	247	8	q	q	NOUN
ejpam-2272	247	9	is	be	AUX
ejpam-2272	247	10	a	a	DET
ejpam-2272	247	11	unique	unique	ADJ
ejpam-2272	247	12	element	element	NOUN
ejpam-2272	247	13	such	such	ADJ
ejpam-2272	247	14	that	that	SCONJ
ejpam-2272	247	15	qm	qm	PROPN
ejpam-2272	247	16	2	2	NUM
ejpam-2272	247	17	qm	qm	PROPN
ejpam-2272	247	18	3	3	NUM
ejpam-2272	247	19	+	+	CCONJ
ejpam-2272	247	20	i	i	PROPN
ejpam-2272	247	21	⊆	⊆	NUM
ejpam-2272	247	22	i2	i2	PROPN
ejpam-2272	247	23	+	+	X
ejpam-2272	247	24	i	i	PRON
ejpam-2272	247	25	.	.	PUNCT
ejpam-2272	248	1	so	so	ADV
ejpam-2272	248	2	,	,	PUNCT
ejpam-2272	248	3	i2	i2	PROPN
ejpam-2272	248	4	+	+	CCONJ
ejpam-2272	248	5	f1	f1	NOUN
ejpam-2272	248	6	=	=	SYM
ejpam-2272	248	7	qm	qm	PROPN
ejpam-2272	248	8	2	2	NUM
ejpam-2272	248	9	qm	qm	PROPN
ejpam-2272	248	10	3	3	NUM
ejpam-2272	248	11	+	+	NOUN
ejpam-2272	248	12	e1	e1	VERB
ejpam-2272	248	13	for	for	ADP
ejpam-2272	248	14	some	some	DET
ejpam-2272	248	15	f1	f1	NOUN
ejpam-2272	248	16	,	,	PUNCT
ejpam-2272	248	17	e1	e1	NOUN
ejpam-2272	248	18	∈	∈	NOUN
ejpam-2272	249	1	i	i	PRON
ejpam-2272	249	2	.	.	PUNCT
ejpam-2272	250	1	since	since	SCONJ
ejpam-2272	250	2	p	p	NOUN
ejpam-2272	250	3	is	be	AUX
ejpam-2272	250	4	subtractive	subtractive	NOUN
ejpam-2272	250	5	and	and	CCONJ
ejpam-2272	250	6	i	i	PRON
ejpam-2272	250	7	⊆	⊆	NUM
ejpam-2272	250	8	p	p	NOUN
ejpam-2272	250	9	,	,	PUNCT
ejpam-2272	250	10	we	we	PRON
ejpam-2272	250	11	have	have	VERB
ejpam-2272	250	12	i2	i2	PROPN
ejpam-2272	250	13	∈	∈	PROPN
ejpam-2272	250	14	p	p	X
ejpam-2272	250	15	,	,	PUNCT
ejpam-2272	250	16	therefore	therefore	ADV
ejpam-2272	250	17	i2	i2	PROPN
ejpam-2272	250	18	∈	∈	PROPN
ejpam-2272	251	1	q	q	PROPN
ejpam-2272	251	2	∩	∩	ADJ
ejpam-2272	251	3	p.	p.	NOUN
ejpam-2272	251	4	this	this	PRON
ejpam-2272	251	5	gives	give	VERB
ejpam-2272	251	6	,	,	PUNCT
ejpam-2272	251	7	(	(	PUNCT
ejpam-2272	251	8	q2	q2	NOUN
ejpam-2272	251	9	+	+	CCONJ
ejpam-2272	251	10	i)m	i)m	X
ejpam-2272	251	11	⊙	⊙	NOUN
ejpam-2272	251	12	(	(	PUNCT
ejpam-2272	251	13	q3	q3	NOUN
ejpam-2272	251	14	+	+	CCONJ
ejpam-2272	251	15	i)m	i)m	NOUN
ejpam-2272	252	1	=	=	SYM
ejpam-2272	252	2	(	(	PUNCT
ejpam-2272	252	3	qm	qm	PROPN
ejpam-2272	252	4	2	2	NUM
ejpam-2272	252	5	+	+	CCONJ
ejpam-2272	252	6	i)⊙	i)⊙	PROPN
ejpam-2272	252	7	(	(	PUNCT
ejpam-2272	252	8	qm	qm	PROPN
ejpam-2272	252	9	3	3	NUM
ejpam-2272	252	10	+	+	CCONJ
ejpam-2272	252	11	i	i	NOUN
ejpam-2272	252	12	)	)	PUNCT
ejpam-2272	253	1	=	=	SYM
ejpam-2272	253	2	qm	qm	PROPN
ejpam-2272	253	3	2	2	NUM
ejpam-2272	253	4	qm	qm	PROPN
ejpam-2272	253	5	3	3	NUM
ejpam-2272	253	6	+	+	CCONJ
ejpam-2272	253	7	i	i	PROPN
ejpam-2272	253	8	⊆	⊆	NUM
ejpam-2272	253	9	i2	i2	NOUN
ejpam-2272	254	1	+	+	X
ejpam-2272	254	2	i	i	PRON
ejpam-2272	254	3	where	where	SCONJ
ejpam-2272	254	4	i2	i2	PROPN
ejpam-2272	254	5	∈	∈	PROPN
ejpam-2272	255	1	q	q	PROPN
ejpam-2272	255	2	∩	∩	ADJ
ejpam-2272	255	3	p.	p.	NOUN
ejpam-2272	255	4	hence	hence	ADV
ejpam-2272	255	5	,	,	PUNCT
ejpam-2272	255	6	p	p	PROPN
ejpam-2272	255	7	/	/	SYM
ejpam-2272	255	8	iq∩p	iq∩p	PROPN
ejpam-2272	255	9	is	be	AUX
ejpam-2272	255	10	a	a	DET
ejpam-2272	255	11	2	2	NUM
ejpam-2272	255	12	-	-	PUNCT
ejpam-2272	255	13	absorbing	absorbing	ADJ
ejpam-2272	255	14	primary	primary	ADJ
ejpam-2272	255	15	ideal	ideal	NOUN
ejpam-2272	255	16	of	of	ADP
ejpam-2272	255	17	s	s	PROPN
ejpam-2272	255	18	/	/	SYM
ejpam-2272	255	19	iq	iq	NOUN
ejpam-2272	255	20	.	.	PUNCT
ejpam-2272	256	1	similarly	similarly	ADV
ejpam-2272	256	2	,	,	PUNCT
ejpam-2272	256	3	if	if	SCONJ
ejpam-2272	256	4	(	(	PUNCT
ejpam-2272	256	5	q3q1)n	q3q1)n	PROPN
ejpam-2272	256	6	∈	∈	PROPN
ejpam-2272	256	7	p	p	NOUN
ejpam-2272	256	8	for	for	ADP
ejpam-2272	256	9	some	some	DET
ejpam-2272	256	10	positive	positive	ADJ
ejpam-2272	256	11	integer	integer	NOUN
ejpam-2272	256	12	n	n	CCONJ
ejpam-2272	256	13	,	,	PUNCT
ejpam-2272	256	14	we	we	PRON
ejpam-2272	256	15	get	get	VERB
ejpam-2272	256	16	p	p	PRON
ejpam-2272	256	17	/	/	SYM
ejpam-2272	256	18	iq∩p	iq∩p	PROPN
ejpam-2272	256	19	is	be	AUX
ejpam-2272	256	20	a	a	DET
ejpam-2272	256	21	2	2	NUM
ejpam-2272	256	22	-	-	PUNCT
ejpam-2272	256	23	absorbing	absorbing	ADJ
ejpam-2272	256	24	primary	primary	ADJ
ejpam-2272	256	25	ideal	ideal	NOUN
ejpam-2272	256	26	of	of	ADP
ejpam-2272	256	27	s	s	PROPN
ejpam-2272	256	28	/	/	SYM
ejpam-2272	256	29	iq	iq	NOUN
ejpam-2272	256	30	.	.	PUNCT
ejpam-2272	257	1	conversely	conversely	ADV
ejpam-2272	257	2	,	,	PUNCT
ejpam-2272	257	3	if	if	SCONJ
ejpam-2272	257	4	p	p	X
ejpam-2272	257	5	/	/	SYM
ejpam-2272	257	6	iq∩p	iq∩p	PROPN
ejpam-2272	257	7	is	be	AUX
ejpam-2272	257	8	a	a	DET
ejpam-2272	257	9	2	2	NUM
ejpam-2272	257	10	-	-	PUNCT
ejpam-2272	257	11	absorbing	absorbing	ADJ
ejpam-2272	257	12	primary	primary	ADJ
ejpam-2272	257	13	ideal	ideal	NOUN
ejpam-2272	257	14	of	of	ADP
ejpam-2272	257	15	s	s	PROPN
ejpam-2272	257	16	/	/	SYM
ejpam-2272	257	17	iq	iq	NOUN
ejpam-2272	257	18	.	.	PUNCT
ejpam-2272	258	1	let	let	VERB
ejpam-2272	258	2	abc	abc	PROPN
ejpam-2272	258	3	∈	∈	PROPN
ejpam-2272	258	4	p	p	PROPN
ejpam-2272	258	5	for	for	ADP
ejpam-2272	258	6	some	some	DET
ejpam-2272	258	7	a	a	DET
ejpam-2272	258	8	,	,	PUNCT
ejpam-2272	258	9	b	b	NOUN
ejpam-2272	258	10	,	,	PUNCT
ejpam-2272	258	11	c	c	PROPN
ejpam-2272	258	12	∈	∈	PROPN
ejpam-2272	258	13	s.	s.	PROPN
ejpam-2272	258	14	since	since	SCONJ
ejpam-2272	258	15	i	i	PRON
ejpam-2272	258	16	is	be	AUX
ejpam-2272	258	17	a	a	DET
ejpam-2272	258	18	q	q	NOUN
ejpam-2272	258	19	-	-	PUNCT
ejpam-2272	258	20	ideal	ideal	NOUN
ejpam-2272	258	21	of	of	ADP
ejpam-2272	258	22	s	s	PRON
ejpam-2272	258	23	therefore	therefore	ADV
ejpam-2272	258	24	there	there	PRON
ejpam-2272	258	25	exist	exist	VERB
ejpam-2272	258	26	q1	q1	PROPN
ejpam-2272	258	27	,	,	PUNCT
ejpam-2272	258	28	q2	q2	NOUN
ejpam-2272	258	29	,	,	PUNCT
ejpam-2272	258	30	q3	q3	PROPN
ejpam-2272	258	31	,	,	PUNCT
ejpam-2272	258	32	q4	q4	PROPN
ejpam-2272	258	33	∈	∈	PROPN
ejpam-2272	258	34	q	q	NOUN
ejpam-2272	258	35	such	such	ADJ
ejpam-2272	258	36	that	that	SCONJ
ejpam-2272	258	37	a	a	DET
ejpam-2272	258	38	∈	∈	PROPN
ejpam-2272	258	39	q1	q1	NOUN
ejpam-2272	258	40	+	+	CCONJ
ejpam-2272	258	41	i	i	PROPN
ejpam-2272	258	42	,	,	PUNCT
ejpam-2272	258	43	b	b	PROPN
ejpam-2272	258	44	∈	∈	PROPN
ejpam-2272	258	45	q2	q2	NOUN
ejpam-2272	258	46	+	+	CCONJ
ejpam-2272	258	47	i	i	PRON
ejpam-2272	258	48	,	,	PUNCT
ejpam-2272	258	49	c	c	PROPN
ejpam-2272	258	50	∈	∈	PROPN
ejpam-2272	258	51	q3	q3	NOUN
ejpam-2272	259	1	+	+	CCONJ
ejpam-2272	260	1	i	i	PRON
ejpam-2272	260	2	.	.	PUNCT
ejpam-2272	261	1	now	now	ADV
ejpam-2272	261	2	,	,	PUNCT
ejpam-2272	261	3	abc	abc	PROPN
ejpam-2272	261	4	∈	∈	PROPN
ejpam-2272	261	5	(	(	PUNCT
ejpam-2272	261	6	q1	q1	PROPN
ejpam-2272	261	7	+	+	CCONJ
ejpam-2272	261	8	i	i	PROPN
ejpam-2272	261	9	)	)	PUNCT
ejpam-2272	261	10	⊙	⊙	PROPN
ejpam-2272	261	11	(	(	PUNCT
ejpam-2272	261	12	q2	q2	PROPN
ejpam-2272	261	13	+	+	CCONJ
ejpam-2272	261	14	i	i	PROPN
ejpam-2272	261	15	)	)	PUNCT
ejpam-2272	261	16	⊙	⊙	PROPN
ejpam-2272	261	17	(	(	PUNCT
ejpam-2272	261	18	q3	q3	PROPN
ejpam-2272	261	19	+	+	PROPN
ejpam-2272	261	20	i	i	NOUN
ejpam-2272	261	21	)	)	PUNCT
ejpam-2272	262	1	=	=	SYM
ejpam-2272	262	2	q4	q4	PROPN
ejpam-2272	262	3	+	+	CCONJ
ejpam-2272	262	4	i	i	PRON
ejpam-2272	262	5	.	.	PUNCT
ejpam-2272	263	1	so	so	ADV
ejpam-2272	263	2	,	,	PUNCT
ejpam-2272	263	3	abc	abc	PROPN
ejpam-2272	263	4	=	=	SYM
ejpam-2272	263	5	q4	q4	PROPN
ejpam-2272	263	6	+	+	CCONJ
ejpam-2272	263	7	i3	i3	PROPN
ejpam-2272	263	8	∈	∈	PROPN
ejpam-2272	263	9	p	p	NOUN
ejpam-2272	263	10	for	for	ADP
ejpam-2272	263	11	some	some	DET
ejpam-2272	263	12	i3	i3	NOUN
ejpam-2272	263	13	∈	∈	NOUN
ejpam-2272	264	1	i	i	PRON
ejpam-2272	264	2	.	.	PUNCT
ejpam-2272	265	1	since	since	SCONJ
ejpam-2272	265	2	p	p	NOUN
ejpam-2272	265	3	is	be	AUX
ejpam-2272	265	4	a	a	DET
ejpam-2272	265	5	subtractive	subtractive	NOUN
ejpam-2272	265	6	ideal	ideal	NOUN
ejpam-2272	265	7	of	of	ADP
ejpam-2272	265	8	s	s	PRON
ejpam-2272	265	9	and	and	CCONJ
ejpam-2272	265	10	i	i	PRON
ejpam-2272	265	11	⊆	⊆	NUM
ejpam-2272	265	12	p	p	NOUN
ejpam-2272	265	13	,	,	PUNCT
ejpam-2272	265	14	we	we	PRON
ejpam-2272	265	15	have	have	VERB
ejpam-2272	265	16	q4	q4	PROPN
ejpam-2272	265	17	∈	∈	PROPN
ejpam-2272	265	18	p.	p.	NOUN
ejpam-2272	266	1	so	so	ADV
ejpam-2272	266	2	,	,	PUNCT
ejpam-2272	266	3	(	(	PUNCT
ejpam-2272	266	4	q1	q1	PROPN
ejpam-2272	266	5	+	+	CCONJ
ejpam-2272	266	6	i)⊙	i)⊙	PROPN
ejpam-2272	266	7	(	(	PUNCT
ejpam-2272	266	8	q2	q2	NOUN
ejpam-2272	266	9	+	+	CCONJ
ejpam-2272	266	10	i)⊙	i)⊙	PROPN
ejpam-2272	266	11	(	(	PUNCT
ejpam-2272	266	12	q3	q3	PROPN
ejpam-2272	266	13	+	+	PROPN
ejpam-2272	266	14	i	i	NOUN
ejpam-2272	266	15	)	)	PUNCT
ejpam-2272	267	1	=	=	SYM
ejpam-2272	267	2	q4	q4	PROPN
ejpam-2272	267	3	+	+	CCONJ
ejpam-2272	267	4	i	i	PRON
ejpam-2272	267	5	∈	∈	PROPN
ejpam-2272	267	6	p	p	X
ejpam-2272	267	7	/	/	SYM
ejpam-2272	267	8	iq∩p	iq∩p	PROPN
ejpam-2272	267	9	,	,	PUNCT
ejpam-2272	267	10	which	which	PRON
ejpam-2272	267	11	gives	give	VERB
ejpam-2272	267	12	(	(	PUNCT
ejpam-2272	267	13	q1+i)⊙(q2+i	q1+i)⊙(q2+i	INTJ
ejpam-2272	267	14	)	)	PUNCT
ejpam-2272	267	15	∈	∈	PROPN
ejpam-2272	267	16	p	p	PROPN
ejpam-2272	267	17	/	/	SYM
ejpam-2272	267	18	iq∩p	iq∩p	PROPN
ejpam-2272	267	19	or	or	CCONJ
ejpam-2272	267	20	(	(	PUNCT
ejpam-2272	267	21	qr	qr	PROPN
ejpam-2272	267	22	2+i)⊙(qr	2+i)⊙(qr	NUM
ejpam-2272	267	23	3+i	3+i	NUM
ejpam-2272	267	24	)	)	PUNCT
ejpam-2272	267	25	∈	∈	PROPN
ejpam-2272	267	26	p	p	PROPN
ejpam-2272	267	27	/	/	SYM
ejpam-2272	267	28	iq∩p	iq∩p	PROPN
ejpam-2272	267	29	or	or	CCONJ
ejpam-2272	267	30	(	(	PUNCT
ejpam-2272	267	31	qt	qt	PROPN
ejpam-2272	267	32	3+i)⊙(qt	3+i)⊙(qt	NUM
ejpam-2272	267	33	1+i	1+i	NUM
ejpam-2272	267	34	)	)	PUNCT
ejpam-2272	267	35	∈	∈	PROPN
ejpam-2272	267	36	p	p	X
ejpam-2272	267	37	/	/	SYM
ejpam-2272	267	38	iq∩p	iq∩p	PROPN
ejpam-2272	267	39	for	for	ADP
ejpam-2272	267	40	some	some	DET
ejpam-2272	267	41	positive	positive	ADJ
ejpam-2272	267	42	integers	integer	NOUN
ejpam-2272	267	43	r	r	NOUN
ejpam-2272	267	44	,	,	PUNCT
ejpam-2272	267	45	t	t	PROPN
ejpam-2272	267	46	,	,	PUNCT
ejpam-2272	267	47	since	since	SCONJ
ejpam-2272	267	48	p	p	PROPN
ejpam-2272	267	49	/	/	SYM
ejpam-2272	267	50	iq∩p	iq∩p	PROPN
ejpam-2272	267	51	is	be	AUX
ejpam-2272	267	52	a	a	DET
ejpam-2272	267	53	2	2	NUM
ejpam-2272	267	54	-	-	PUNCT
ejpam-2272	267	55	absorbing	absorbing	ADJ
ejpam-2272	267	56	primary	primary	ADJ
ejpam-2272	267	57	ideal	ideal	NOUN
ejpam-2272	267	58	of	of	ADP
ejpam-2272	267	59	s	s	NOUN
ejpam-2272	267	60	/	/	SYM
ejpam-2272	267	61	iq	iq	NOUN
ejpam-2272	267	62	.	.	PUNCT
ejpam-2272	268	1	if	if	SCONJ
ejpam-2272	268	2	(	(	PUNCT
ejpam-2272	268	3	q1	q1	PROPN
ejpam-2272	268	4	+	+	X
ejpam-2272	268	5	i)⊙	i)⊙	PROPN
ejpam-2272	268	6	(	(	PUNCT
ejpam-2272	268	7	q2+i	q2+i	PROPN
ejpam-2272	268	8	)	)	PUNCT
ejpam-2272	268	9	∈	∈	NOUN
ejpam-2272	268	10	p	p	PROPN
ejpam-2272	268	11	/	/	SYM
ejpam-2272	268	12	iq∩p	iq∩p	PROPN
ejpam-2272	268	13	,	,	PUNCT
ejpam-2272	268	14	then	then	ADV
ejpam-2272	268	15	there	there	PRON
ejpam-2272	268	16	exists	exist	VERB
ejpam-2272	268	17	q5	q5	PROPN
ejpam-2272	268	18	∈	∈	PROPN
ejpam-2272	268	19	q	q	PROPN
ejpam-2272	268	20	∩	∩	PROPN
ejpam-2272	268	21	p	p	X
ejpam-2272	268	22	such	such	ADJ
ejpam-2272	268	23	that	that	SCONJ
ejpam-2272	268	24	ab	ab	PROPN
ejpam-2272	268	25	∈	∈	PROPN
ejpam-2272	268	26	(	(	PUNCT
ejpam-2272	268	27	q1+i)⊙(q2+i	q1+i)⊙(q2+i	PROPN
ejpam-2272	268	28	)	)	PUNCT
ejpam-2272	268	29	=	=	SYM
ejpam-2272	268	30	q5+i	q5+i	NOUN
ejpam-2272	268	31	.	.	PUNCT
ejpam-2272	269	1	this	this	PRON
ejpam-2272	269	2	gives	give	VERB
ejpam-2272	269	3	,	,	PUNCT
ejpam-2272	269	4	ab	ab	PROPN
ejpam-2272	269	5	=	=	PROPN
ejpam-2272	269	6	q5	q5	PROPN
ejpam-2272	269	7	+	+	CCONJ
ejpam-2272	269	8	i4	i4	PROPN
ejpam-2272	269	9	for	for	ADP
ejpam-2272	269	10	some	some	DET
ejpam-2272	269	11	i4	i4	PROPN
ejpam-2272	269	12	∈	∈	PROPN
ejpam-2272	270	1	i	i	PRON
ejpam-2272	270	2	.	.	PUNCT
ejpam-2272	271	1	this	this	PRON
ejpam-2272	271	2	implies	imply	VERB
ejpam-2272	271	3	ab	ab	PROPN
ejpam-2272	271	4	∈	∈	PROPN
ejpam-2272	271	5	p.	p.	NOUN
ejpam-2272	272	1	thus	thus	ADV
ejpam-2272	272	2	p	p	NOUN
ejpam-2272	272	3	is	be	AUX
ejpam-2272	272	4	a	a	DET
ejpam-2272	272	5	2	2	NUM
ejpam-2272	272	6	-	-	PUNCT
ejpam-2272	272	7	absorbing	absorbing	ADJ
ejpam-2272	272	8	primary	primary	ADJ
ejpam-2272	272	9	ideal	ideal	NOUN
ejpam-2272	272	10	of	of	ADP
ejpam-2272	272	11	s.	s.	PROPN
ejpam-2272	272	12	if	if	SCONJ
ejpam-2272	272	13	(	(	PUNCT
ejpam-2272	272	14	qr	qr	PROPN
ejpam-2272	272	15	2+i)⊙(qr	2+i)⊙(qr	NUM
ejpam-2272	272	16	3+i	3+i	NUM
ejpam-2272	272	17	)	)	PUNCT
ejpam-2272	272	18	∈	∈	PROPN
ejpam-2272	272	19	p	p	PROPN
ejpam-2272	272	20	/	/	SYM
ejpam-2272	272	21	iq∩p	iq∩p	PROPN
ejpam-2272	272	22	,	,	PUNCT
ejpam-2272	272	23	then	then	ADV
ejpam-2272	272	24	there	there	PRON
ejpam-2272	272	25	exists	exist	VERB
ejpam-2272	272	26	q6	q6	PROPN
ejpam-2272	272	27	∈	∈	PROPN
ejpam-2272	272	28	q	q	PROPN
ejpam-2272	272	29	∩	∩	PROPN
ejpam-2272	272	30	p	p	VERB
ejpam-2272	272	31	such	such	ADJ
ejpam-2272	272	32	that	that	SCONJ
ejpam-2272	272	33	br	br	PROPN
ejpam-2272	272	34	cr	cr	X
ejpam-2272	272	35	∈	∈	PROPN
ejpam-2272	272	36	(	(	PUNCT
ejpam-2272	272	37	qr	qr	NOUN
ejpam-2272	272	38	2+i)⊙(qr	2+i)⊙(qr	NUM
ejpam-2272	272	39	3+i	3+i	NUM
ejpam-2272	272	40	)	)	PUNCT
ejpam-2272	273	1	=	=	SYM
ejpam-2272	273	2	q6+i	q6+i	INTJ
ejpam-2272	273	3	.	.	PUNCT
ejpam-2272	274	1	this	this	PRON
ejpam-2272	274	2	gives	give	VERB
ejpam-2272	274	3	,	,	PUNCT
ejpam-2272	274	4	br	br	NOUN
ejpam-2272	274	5	cr	cr	NOUN
ejpam-2272	275	1	=	=	SYM
ejpam-2272	276	1	q6+i5	q6+i5	PROPN
ejpam-2272	276	2	for	for	ADP
ejpam-2272	276	3	some	some	DET
ejpam-2272	276	4	i5	i5	NOUN
ejpam-2272	276	5	∈	∈	PROPN
ejpam-2272	277	1	i	i	PRON
ejpam-2272	277	2	.	.	PUNCT
ejpam-2272	278	1	this	this	PRON
ejpam-2272	278	2	implies	imply	VERB
ejpam-2272	278	3	,	,	PUNCT
ejpam-2272	278	4	(	(	PUNCT
ejpam-2272	278	5	bc)r	bc)r	PROPN
ejpam-2272	278	6	∈	∈	PROPN
ejpam-2272	278	7	p.	p.	NOUN
ejpam-2272	278	8	therefore	therefore	ADV
ejpam-2272	278	9	,	,	PUNCT
ejpam-2272	278	10	bc	bc	PROPN
ejpam-2272	278	11	∈	∈	PROPN
ejpam-2272	278	12	√	√	PROPN
ejpam-2272	279	1	p.	p.	NOUN
ejpam-2272	280	1	similarly	similarly	ADV
ejpam-2272	280	2	,	,	PUNCT
ejpam-2272	280	3	we	we	PRON
ejpam-2272	280	4	can	can	AUX
ejpam-2272	280	5	prove	prove	VERB
ejpam-2272	280	6	that	that	SCONJ
ejpam-2272	280	7	ca	can	AUX
ejpam-2272	280	8	∈	∈	VERB
ejpam-2272	280	9	√	√	PROPN
ejpam-2272	280	10	p.	p.	NOUN
ejpam-2272	280	11	hence	hence	ADV
ejpam-2272	280	12	,	,	PUNCT
ejpam-2272	280	13	p	p	PROPN
ejpam-2272	280	14	is	be	AUX
ejpam-2272	280	15	a	a	DET
ejpam-2272	280	16	2	2	NUM
ejpam-2272	280	17	-	-	PUNCT
ejpam-2272	280	18	absorbing	absorbing	ADJ
ejpam-2272	280	19	primary	primary	ADJ
ejpam-2272	280	20	ideal	ideal	NOUN
ejpam-2272	280	21	of	of	ADP
ejpam-2272	280	22	s.	s.	PROPN
ejpam-2272	280	23	p.	p.	PROPN
ejpam-2272	280	24	kumar	kumar	PROPN
ejpam-2272	280	25	,	,	PUNCT
ejpam-2272	280	26	m.	m.	NOUN
ejpam-2272	280	27	dubey	dubey	PROPN
ejpam-2272	280	28	,	,	PUNCT
ejpam-2272	280	29	p.	p.	NOUN
ejpam-2272	280	30	sarohe	sarohe	PROPN
ejpam-2272	280	31	/	/	SYM
ejpam-2272	280	32	eur	eur	PROPN
ejpam-2272	280	33	.	.	PUNCT
ejpam-2272	281	1	j.	j.	PROPN
ejpam-2272	281	2	pure	pure	PROPN
ejpam-2272	281	3	appl	appl	PROPN
ejpam-2272	281	4	.	.	PROPN
ejpam-2272	281	5	math	math	PROPN
ejpam-2272	281	6	,	,	PUNCT
ejpam-2272	281	7	9	9	NUM
ejpam-2272	281	8	(	(	PUNCT
ejpam-2272	281	9	2016	2016	NUM
ejpam-2272	281	10	)	)	PUNCT
ejpam-2272	281	11	,	,	PUNCT
ejpam-2272	281	12	186	186	NUM
ejpam-2272	281	13	-	-	SYM
ejpam-2272	281	14	195	195	NUM
ejpam-2272	281	15	191	191	NUM
ejpam-2272	281	16	3	3	NUM
ejpam-2272	281	17	.	.	PUNCT
ejpam-2272	281	18	weakly	weakly	ADJ
ejpam-2272	281	19	2	2	NUM
ejpam-2272	281	20	-	-	PUNCT
ejpam-2272	281	21	absorbing	absorbing	ADJ
ejpam-2272	281	22	primary	primary	ADJ
ejpam-2272	281	23	ideals	ideal	NOUN
ejpam-2272	281	24	in	in	ADP
ejpam-2272	281	25	this	this	DET
ejpam-2272	281	26	section	section	NOUN
ejpam-2272	281	27	,	,	PUNCT
ejpam-2272	281	28	we	we	PRON
ejpam-2272	281	29	introduce	introduce	VERB
ejpam-2272	281	30	the	the	DET
ejpam-2272	281	31	notion	notion	NOUN
ejpam-2272	281	32	of	of	ADP
ejpam-2272	281	33	weakly	weakly	ADJ
ejpam-2272	281	34	2	2	NUM
ejpam-2272	281	35	-	-	PUNCT
ejpam-2272	281	36	absorbing	absorbing	ADJ
ejpam-2272	281	37	primary	primary	ADJ
ejpam-2272	281	38	ideal	ideal	NOUN
ejpam-2272	281	39	of	of	ADP
ejpam-2272	281	40	a	a	DET
ejpam-2272	281	41	commutative	commutative	ADJ
ejpam-2272	281	42	semiring	semiring	NOUN
ejpam-2272	281	43	and	and	CCONJ
ejpam-2272	281	44	prove	prove	VERB
ejpam-2272	281	45	some	some	DET
ejpam-2272	281	46	results	result	NOUN
ejpam-2272	281	47	related	relate	VERB
ejpam-2272	281	48	to	to	ADP
ejpam-2272	281	49	it	it	PRON
ejpam-2272	281	50	.	.	PUNCT
ejpam-2272	282	1	definition	definition	NOUN
ejpam-2272	282	2	4	4	NUM
ejpam-2272	282	3	.	.	PUNCT
ejpam-2272	283	1	let	let	VERB
ejpam-2272	283	2	s	s	PRON
ejpam-2272	283	3	be	be	AUX
ejpam-2272	283	4	a	a	DET
ejpam-2272	283	5	commutative	commutative	ADJ
ejpam-2272	283	6	semiring	semiring	NOUN
ejpam-2272	283	7	and	and	CCONJ
ejpam-2272	283	8	i	i	PRON
ejpam-2272	283	9	be	be	VERB
ejpam-2272	283	10	a	a	DET
ejpam-2272	283	11	proper	proper	ADJ
ejpam-2272	283	12	ideal	ideal	NOUN
ejpam-2272	283	13	of	of	ADP
ejpam-2272	283	14	s.	s.	PROPN
ejpam-2272	284	1	then	then	ADV
ejpam-2272	284	2	i	i	PRON
ejpam-2272	284	3	is	be	AUX
ejpam-2272	284	4	said	say	VERB
ejpam-2272	284	5	to	to	PART
ejpam-2272	284	6	be	be	AUX
ejpam-2272	284	7	a	a	DET
ejpam-2272	284	8	weakly	weakly	ADJ
ejpam-2272	284	9	2	2	NUM
ejpam-2272	284	10	-	-	PUNCT
ejpam-2272	284	11	absorbing	absorbing	ADJ
ejpam-2272	284	12	primary	primary	ADJ
ejpam-2272	284	13	ideal	ideal	NOUN
ejpam-2272	284	14	of	of	ADP
ejpam-2272	284	15	s	s	PRON
ejpam-2272	284	16	if	if	SCONJ
ejpam-2272	284	17	whenever	whenever	SCONJ
ejpam-2272	284	18	a	a	DET
ejpam-2272	284	19	,	,	PUNCT
ejpam-2272	284	20	b	b	NOUN
ejpam-2272	284	21	,	,	PUNCT
ejpam-2272	284	22	c	c	PROPN
ejpam-2272	284	23	∈	∈	PROPN
ejpam-2272	284	24	s	s	X
ejpam-2272	284	25	and	and	CCONJ
ejpam-2272	284	26	0	0	NUM
ejpam-2272	285	1	≠	≠	PROPN
ejpam-2272	285	2	abc	abc	PROPN
ejpam-2272	285	3	∈	∈	PROPN
ejpam-2272	285	4	i	i	PRON
ejpam-2272	285	5	,	,	PUNCT
ejpam-2272	285	6	then	then	ADV
ejpam-2272	285	7	ab	ab	PROPN
ejpam-2272	285	8	∈	∈	PROPN
ejpam-2272	286	1	i	i	PRON
ejpam-2272	286	2	or	or	CCONJ
ejpam-2272	286	3	ac	ac	PROPN
ejpam-2272	286	4	∈	∈	PROPN
ejpam-2272	287	1	√	√	VERB
ejpam-2272	287	2	i	i	PRON
ejpam-2272	287	3	or	or	CCONJ
ejpam-2272	287	4	bc	bc	PROPN
ejpam-2272	287	5	∈	∈	PROPN
ejpam-2272	288	1	√	√	VERB
ejpam-2272	288	2	i	i	PRON
ejpam-2272	288	3	.	.	PUNCT
ejpam-2272	289	1	it	it	PRON
ejpam-2272	289	2	is	be	AUX
ejpam-2272	289	3	clear	clear	ADJ
ejpam-2272	289	4	that	that	SCONJ
ejpam-2272	289	5	every	every	DET
ejpam-2272	289	6	2	2	NUM
ejpam-2272	289	7	-	-	PUNCT
ejpam-2272	289	8	absorbing	absorbing	ADJ
ejpam-2272	289	9	primary	primary	ADJ
ejpam-2272	289	10	ideal	ideal	NOUN
ejpam-2272	289	11	of	of	ADP
ejpam-2272	289	12	s	s	PROPN
ejpam-2272	289	13	is	be	AUX
ejpam-2272	289	14	a	a	DET
ejpam-2272	289	15	weakly	weakly	ADJ
ejpam-2272	289	16	2	2	NUM
ejpam-2272	289	17	-	-	PUNCT
ejpam-2272	289	18	absorbing	absorbing	ADJ
ejpam-2272	289	19	primary	primary	ADJ
ejpam-2272	289	20	ideal	ideal	NOUN
ejpam-2272	289	21	of	of	ADP
ejpam-2272	289	22	s	s	NOUN
ejpam-2272	289	23	but	but	CCONJ
ejpam-2272	289	24	converse	converse	NOUN
ejpam-2272	289	25	is	be	AUX
ejpam-2272	289	26	not	not	PART
ejpam-2272	289	27	true	true	ADJ
ejpam-2272	289	28	,	,	PUNCT
ejpam-2272	289	29	as	as	SCONJ
ejpam-2272	289	30	⟨0⟩	⟨0⟩	PROPN
ejpam-2272	289	31	is	be	AUX
ejpam-2272	289	32	a	a	DET
ejpam-2272	289	33	weakly	weakly	ADJ
ejpam-2272	289	34	2	2	NUM
ejpam-2272	289	35	-	-	PUNCT
ejpam-2272	289	36	absorbing	absorbing	ADJ
ejpam-2272	289	37	primary	primary	ADJ
ejpam-2272	289	38	ideal	ideal	NOUN
ejpam-2272	289	39	of	of	ADP
ejpam-2272	289	40	s	s	PROPN
ejpam-2272	289	41	but	but	CCONJ
ejpam-2272	289	42	not	not	PART
ejpam-2272	289	43	a	a	DET
ejpam-2272	289	44	2absorbing	2absorbing	NUM
ejpam-2272	289	45	primary	primary	ADJ
ejpam-2272	289	46	ideal	ideal	NOUN
ejpam-2272	289	47	of	of	ADP
ejpam-2272	289	48	s.	s.	PROPN
ejpam-2272	289	49	consider	consider	VERB
ejpam-2272	289	50	the	the	DET
ejpam-2272	289	51	set	set	NOUN
ejpam-2272	289	52	s	s	PART
ejpam-2272	289	53	=	=	NOUN
ejpam-2272	289	54	z16	z16	NOUN
ejpam-2272	289	55	=	=	SYM
ejpam-2272	289	56	{	{	PUNCT
ejpam-2272	289	57	0	0	NUM
ejpam-2272	289	58	,	,	PUNCT
ejpam-2272	289	59	1	1	NUM
ejpam-2272	289	60	,	,	PUNCT
ejpam-2272	289	61	2	2	NUM
ejpam-2272	289	62	,	,	PUNCT
ejpam-2272	289	63	.	.	PUNCT
ejpam-2272	289	64	.	.	PUNCT
ejpam-2272	289	65	.	.	PUNCT
ejpam-2272	290	1	,	,	PUNCT
ejpam-2272	290	2	15	15	NUM
ejpam-2272	290	3	}	}	PUNCT
ejpam-2272	290	4	.	.	PUNCT
ejpam-2272	291	1	then	then	ADV
ejpam-2272	291	2	s	s	VERB
ejpam-2272	291	3	forms	form	NOUN
ejpam-2272	291	4	a	a	DET
ejpam-2272	291	5	semiring	semiring	NOUN
ejpam-2272	291	6	under	under	ADP
ejpam-2272	291	7	addition	addition	NOUN
ejpam-2272	291	8	and	and	CCONJ
ejpam-2272	291	9	multiplication	multiplication	NOUN
ejpam-2272	291	10	modulo	modulo	VERB
ejpam-2272	291	11	16	16	NUM
ejpam-2272	291	12	.	.	PUNCT
ejpam-2272	292	1	if	if	SCONJ
ejpam-2272	292	2	we	we	PRON
ejpam-2272	292	3	take	take	VERB
ejpam-2272	292	4	the	the	DET
ejpam-2272	292	5	set	set	NOUN
ejpam-2272	292	6	i	i	PRON
ejpam-2272	292	7	=	=	PUNCT
ejpam-2272	292	8	{	{	PUNCT
ejpam-2272	292	9	0	0	NUM
ejpam-2272	292	10	,	,	PUNCT
ejpam-2272	292	11	8	8	NUM
ejpam-2272	292	12	}	}	PUNCT
ejpam-2272	292	13	.	.	PUNCT
ejpam-2272	293	1	then	then	ADV
ejpam-2272	293	2	it	it	PRON
ejpam-2272	293	3	is	be	AUX
ejpam-2272	293	4	easy	easy	ADJ
ejpam-2272	293	5	to	to	PART
ejpam-2272	293	6	check	check	VERB
ejpam-2272	293	7	that	that	SCONJ
ejpam-2272	293	8	i	i	PRON
ejpam-2272	293	9	is	be	AUX
ejpam-2272	293	10	a	a	DET
ejpam-2272	293	11	weakly	weakly	ADJ
ejpam-2272	293	12	2	2	NUM
ejpam-2272	293	13	-	-	PUNCT
ejpam-2272	293	14	absorbing	absorbing	ADJ
ejpam-2272	293	15	primary	primary	ADJ
ejpam-2272	293	16	ideal	ideal	NOUN
ejpam-2272	293	17	of	of	ADP
ejpam-2272	293	18	s	s	PRON
ejpam-2272	294	1	but	but	CCONJ
ejpam-2272	294	2	it	it	PRON
ejpam-2272	294	3	not	not	PART
ejpam-2272	294	4	a	a	DET
ejpam-2272	294	5	weakly	weakly	ADJ
ejpam-2272	294	6	2	2	NUM
ejpam-2272	294	7	-	-	PUNCT
ejpam-2272	294	8	absorbing	absorbing	ADJ
ejpam-2272	294	9	ideal	ideal	NOUN
ejpam-2272	294	10	of	of	ADP
ejpam-2272	294	11	s	s	PRON
ejpam-2272	294	12	because	because	SCONJ
ejpam-2272	294	13	0	0	NUM
ejpam-2272	294	14	≠	≠	PROPN
ejpam-2272	294	15	2.2.2	2.2.2	NUM
ejpam-2272	294	16	∈	∈	NOUN
ejpam-2272	295	1	i	i	PRON
ejpam-2272	295	2	but	but	CCONJ
ejpam-2272	295	3	2.2	2.2	NUM
ejpam-2272	295	4	∉	∉	X
ejpam-2272	295	5	i	i	PRON
ejpam-2272	295	6	.	.	PUNCT
ejpam-2272	296	1	for	for	ADP
ejpam-2272	296	2	any	any	DET
ejpam-2272	296	3	ideal	ideal	NOUN
ejpam-2272	296	4	,	,	PUNCT
ejpam-2272	296	5	the	the	DET
ejpam-2272	296	6	following	follow	VERB
ejpam-2272	296	7	implications	implication	NOUN
ejpam-2272	296	8	hold	hold	VERB
ejpam-2272	296	9	:	:	PUNCT
ejpam-2272	296	10	prime	prime	ADJ
ejpam-2272	296	11	⇒	⇒	NOUN
ejpam-2272	296	12	2	2	NUM
ejpam-2272	296	13	-	-	PUNCT
ejpam-2272	296	14	absorbing	absorbing	ADJ
ejpam-2272	296	15	⇒	⇒	NOUN
ejpam-2272	296	16	weakly	weakly	ADJ
ejpam-2272	296	17	2	2	NUM
ejpam-2272	296	18	-	-	PUNCT
ejpam-2272	296	19	absorbing	absorbing	ADJ
ejpam-2272	296	20	ideal	ideal	NOUN
ejpam-2272	296	21	/	/	SYM
ejpam-2272	296	22	⇐	⇐	NOUN
ejpam-2272	296	23	ideal	ideal	ADJ
ejpam-2272	296	24	/	/	SYM
ejpam-2272	296	25	⇐	⇐	ADJ
ejpam-2272	296	26	ideal	ideal	NOUN
ejpam-2272	296	27	⇓	⇓	PROPN
ejpam-2272	296	28	⇓	⇓	PROPN
ejpam-2272	296	29	⇓	⇓	PROPN
ejpam-2272	296	30	primary	primary	ADJ
ejpam-2272	296	31	⇒	⇒	NOUN
ejpam-2272	296	32	2	2	NUM
ejpam-2272	296	33	-	-	PUNCT
ejpam-2272	296	34	absorbing	absorbing	ADJ
ejpam-2272	296	35	primary	primary	ADJ
ejpam-2272	296	36	⇒	⇒	NOUN
ejpam-2272	296	37	weakly	weakly	ADJ
ejpam-2272	296	38	2	2	NUM
ejpam-2272	296	39	-	-	PUNCT
ejpam-2272	296	40	absorbing	absorbing	ADJ
ejpam-2272	296	41	primary	primary	ADJ
ejpam-2272	296	42	ideal	ideal	NOUN
ejpam-2272	296	43	/	/	SYM
ejpam-2272	296	44	⇐	⇐	NOUN
ejpam-2272	296	45	ideal	ideal	ADJ
ejpam-2272	296	46	/	/	SYM
ejpam-2272	296	47	⇐	⇐	PROPN
ejpam-2272	296	48	ideal	ideal	ADJ
ejpam-2272	296	49	lemma	lemma	PROPN
ejpam-2272	296	50	2	2	NUM
ejpam-2272	296	51	(	(	PUNCT
ejpam-2272	296	52	[	[	X
ejpam-2272	296	53	6	6	NUM
ejpam-2272	296	54	,	,	PUNCT
ejpam-2272	296	55	lemma	lemma	PROPN
ejpam-2272	296	56	2.5	2.5	NUM
ejpam-2272	296	57	]	]	PUNCT
ejpam-2272	296	58	)	)	PUNCT
ejpam-2272	296	59	.	.	PUNCT
ejpam-2272	297	1	let	let	VERB
ejpam-2272	297	2	i	i	PRON
ejpam-2272	297	3	be	be	AUX
ejpam-2272	297	4	a	a	DET
ejpam-2272	297	5	subtractive	subtractive	NOUN
ejpam-2272	297	6	ideal	ideal	NOUN
ejpam-2272	297	7	of	of	ADP
ejpam-2272	297	8	a	a	DET
ejpam-2272	297	9	semiring	semiring	NOUN
ejpam-2272	297	10	s	s	X
ejpam-2272	297	11	and	and	CCONJ
ejpam-2272	297	12	let	let	VERB
ejpam-2272	297	13	a	a	DET
ejpam-2272	297	14	∈	∈	ADJ
ejpam-2272	298	1	i	i	PRON
ejpam-2272	298	2	and	and	CCONJ
ejpam-2272	298	3	a	a	DET
ejpam-2272	298	4	+	+	NOUN
ejpam-2272	298	5	b	b	NOUN
ejpam-2272	298	6	∈	∈	NOUN
ejpam-2272	298	7	√	√	VERB
ejpam-2272	298	8	i	i	PRON
ejpam-2272	298	9	.	.	PUNCT
ejpam-2272	299	1	then	then	ADV
ejpam-2272	299	2	b	b	X
ejpam-2272	299	3	∈	∈	PROPN
ejpam-2272	299	4	√	√	VERB
ejpam-2272	299	5	i	i	PRON
ejpam-2272	299	6	.	.	PUNCT
ejpam-2272	300	1	proof	proof	NOUN
ejpam-2272	300	2	.	.	PUNCT
ejpam-2272	301	1	let	let	VERB
ejpam-2272	301	2	a	a	DET
ejpam-2272	301	3	∈	∈	ADJ
ejpam-2272	302	1	i	i	PRON
ejpam-2272	302	2	and	and	CCONJ
ejpam-2272	302	3	a	a	DET
ejpam-2272	302	4	+	+	NOUN
ejpam-2272	302	5	b	b	NOUN
ejpam-2272	302	6	∈	∈	NOUN
ejpam-2272	302	7	√	√	VERB
ejpam-2272	302	8	i	i	PRON
ejpam-2272	302	9	.	.	PUNCT
ejpam-2272	303	1	then	then	ADV
ejpam-2272	303	2	,	,	PUNCT
ejpam-2272	303	3	we	we	PRON
ejpam-2272	303	4	can	can	AUX
ejpam-2272	303	5	assume	assume	VERB
ejpam-2272	303	6	that	that	SCONJ
ejpam-2272	303	7	there	there	PRON
ejpam-2272	303	8	exists	exist	VERB
ejpam-2272	303	9	a	a	DET
ejpam-2272	303	10	positive	positive	ADJ
ejpam-2272	303	11	integer	integer	NOUN
ejpam-2272	303	12	m	m	VERB
ejpam-2272	304	1	such	such	ADJ
ejpam-2272	304	2	that	that	SCONJ
ejpam-2272	304	3	(	(	PUNCT
ejpam-2272	304	4	a	a	DET
ejpam-2272	304	5	+	+	NOUN
ejpam-2272	304	6	b)m	b)m	X
ejpam-2272	304	7	=	=	SYM
ejpam-2272	304	8	c	c	X
ejpam-2272	304	9	+	+	CCONJ
ejpam-2272	304	10	bm	bm	PROPN
ejpam-2272	304	11	∈	∈	PROPN
ejpam-2272	304	12	i	i	PRON
ejpam-2272	304	13	,	,	PUNCT
ejpam-2272	304	14	where	where	SCONJ
ejpam-2272	304	15	c	c	PROPN
ejpam-2272	304	16	∈	∈	PROPN
ejpam-2272	304	17	i	i	PRON
ejpam-2272	304	18	(	(	PUNCT
ejpam-2272	304	19	as	as	ADP
ejpam-2272	304	20	a	a	DET
ejpam-2272	304	21	∈	∈	NOUN
ejpam-2272	304	22	i	i	NOUN
ejpam-2272	304	23	)	)	PUNCT
ejpam-2272	304	24	.	.	PUNCT
ejpam-2272	305	1	this	this	PRON
ejpam-2272	305	2	gives	give	VERB
ejpam-2272	305	3	bm	bm	PROPN
ejpam-2272	305	4	∈	∈	PROPN
ejpam-2272	305	5	i	i	PRON
ejpam-2272	305	6	since	since	SCONJ
ejpam-2272	305	7	i	i	PRON
ejpam-2272	305	8	is	be	AUX
ejpam-2272	305	9	subtractive	subtractive	NOUN
ejpam-2272	305	10	.	.	PUNCT
ejpam-2272	306	1	hence	hence	ADV
ejpam-2272	306	2	b	b	X
ejpam-2272	306	3	∈	∈	NOUN
ejpam-2272	306	4	√	√	VERB
ejpam-2272	306	5	i	i	PRON
ejpam-2272	306	6	.	.	PUNCT
ejpam-2272	307	1	theorem	theorem	ADJ
ejpam-2272	307	2	10	10	NUM
ejpam-2272	307	3	.	.	PUNCT
ejpam-2272	308	1	let	let	VERB
ejpam-2272	308	2	s	s	PRON
ejpam-2272	308	3	be	be	AUX
ejpam-2272	308	4	a	a	DET
ejpam-2272	308	5	semiring	semiring	NOUN
ejpam-2272	309	1	and	and	CCONJ
ejpam-2272	309	2	i	i	PRON
ejpam-2272	309	3	be	be	VERB
ejpam-2272	309	4	a	a	DET
ejpam-2272	309	5	subtractive	subtractive	NOUN
ejpam-2272	309	6	weakly	weakly	ADJ
ejpam-2272	309	7	2	2	NUM
ejpam-2272	309	8	-	-	PUNCT
ejpam-2272	309	9	absorbing	absorbing	ADJ
ejpam-2272	309	10	primary	primary	ADJ
ejpam-2272	309	11	ideal	ideal	NOUN
ejpam-2272	309	12	that	that	PRON
ejpam-2272	309	13	is	be	AUX
ejpam-2272	309	14	not	not	PART
ejpam-2272	309	15	a	a	DET
ejpam-2272	309	16	2	2	NUM
ejpam-2272	309	17	-	-	PUNCT
ejpam-2272	309	18	absorbing	absorbing	ADJ
ejpam-2272	309	19	primary	primary	ADJ
ejpam-2272	309	20	ideal	ideal	NOUN
ejpam-2272	309	21	of	of	ADP
ejpam-2272	309	22	s.	s.	PROPN
ejpam-2272	309	23	then	then	ADV
ejpam-2272	309	24	√	√	VERB
ejpam-2272	310	1	i	i	NOUN
ejpam-2272	311	1	=	=	PUNCT
ejpam-2272	312	1	√	√	PROPN
ejpam-2272	312	2	0	0	NUM
ejpam-2272	312	3	.	.	PUNCT
ejpam-2272	313	1	proof	proof	NOUN
ejpam-2272	313	2	.	.	PUNCT
ejpam-2272	314	1	we	we	PRON
ejpam-2272	314	2	first	first	ADV
ejpam-2272	314	3	prove	prove	VERB
ejpam-2272	314	4	that	that	SCONJ
ejpam-2272	314	5	i3	i3	NOUN
ejpam-2272	314	6	=	=	NOUN
ejpam-2272	314	7	0	0	X
ejpam-2272	314	8	.	.	PUNCT
ejpam-2272	314	9	suppose	suppose	VERB
ejpam-2272	314	10	that	that	SCONJ
ejpam-2272	314	11	i3	i3	NOUN
ejpam-2272	314	12	≠	≠	PROPN
ejpam-2272	314	13	0	0	NUM
ejpam-2272	314	14	.	.	PUNCT
ejpam-2272	315	1	then	then	ADV
ejpam-2272	315	2	,	,	PUNCT
ejpam-2272	315	3	we	we	PRON
ejpam-2272	315	4	prove	prove	VERB
ejpam-2272	315	5	that	that	SCONJ
ejpam-2272	315	6	i	i	PRON
ejpam-2272	315	7	is	be	AUX
ejpam-2272	315	8	a	a	DET
ejpam-2272	315	9	2absorbing	2absorbing	NUM
ejpam-2272	315	10	primary	primary	ADJ
ejpam-2272	315	11	ideal	ideal	NOUN
ejpam-2272	315	12	of	of	ADP
ejpam-2272	315	13	s.	s.	PROPN
ejpam-2272	315	14	let	let	VERB
ejpam-2272	315	15	abc	abc	PROPN
ejpam-2272	315	16	∈	∈	PROPN
ejpam-2272	315	17	i	i	PRON
ejpam-2272	315	18	for	for	ADP
ejpam-2272	315	19	some	some	DET
ejpam-2272	315	20	a	a	DET
ejpam-2272	315	21	,	,	PUNCT
ejpam-2272	315	22	b	b	NOUN
ejpam-2272	315	23	,	,	PUNCT
ejpam-2272	315	24	c	c	PROPN
ejpam-2272	315	25	∈	∈	PROPN
ejpam-2272	315	26	s.	s.	PROPN
ejpam-2272	315	27	suppose	suppose	VERB
ejpam-2272	316	1	that	that	SCONJ
ejpam-2272	316	2	abc	abc	PROPN
ejpam-2272	316	3	≠	≠	PROPN
ejpam-2272	316	4	0	0	NUM
ejpam-2272	316	5	,	,	PUNCT
ejpam-2272	316	6	then	then	ADV
ejpam-2272	316	7	ab	ab	PROPN
ejpam-2272	316	8	∈	∈	PROPN
ejpam-2272	316	9	i	i	PRON
ejpam-2272	316	10	or	or	CCONJ
ejpam-2272	316	11	bc	bc	PROPN
ejpam-2272	316	12	∈	∈	PROPN
ejpam-2272	317	1	√	√	VERB
ejpam-2272	317	2	i	i	PRON
ejpam-2272	317	3	or	or	CCONJ
ejpam-2272	317	4	ac	ac	PROPN
ejpam-2272	317	5	∈	∈	PROPN
ejpam-2272	318	1	√	√	VERB
ejpam-2272	318	2	i	i	PRON
ejpam-2272	318	3	since	since	SCONJ
ejpam-2272	318	4	i	i	PRON
ejpam-2272	318	5	is	be	AUX
ejpam-2272	318	6	a	a	DET
ejpam-2272	318	7	weakly	weakly	ADJ
ejpam-2272	318	8	2	2	NUM
ejpam-2272	318	9	-	-	PUNCT
ejpam-2272	318	10	absorbing	absorbing	ADJ
ejpam-2272	318	11	primary	primary	ADJ
ejpam-2272	318	12	ideal	ideal	NOUN
ejpam-2272	318	13	of	of	ADP
ejpam-2272	318	14	s.	s.	PROPN
ejpam-2272	318	15	so	so	ADV
ejpam-2272	319	1	,	,	PUNCT
ejpam-2272	319	2	assume	assume	VERB
ejpam-2272	319	3	that	that	SCONJ
ejpam-2272	319	4	abc	abc	PROPN
ejpam-2272	319	5	=	=	SYM
ejpam-2272	319	6	0	0	PROPN
ejpam-2272	319	7	.	.	PUNCT
ejpam-2272	320	1	if	if	SCONJ
ejpam-2272	320	2	abi	abi	PROPN
ejpam-2272	320	3	≠	≠	PROPN
ejpam-2272	320	4	0	0	NUM
ejpam-2272	320	5	,	,	PUNCT
ejpam-2272	320	6	then	then	ADV
ejpam-2272	320	7	there	there	PRON
ejpam-2272	320	8	exists	exist	VERB
ejpam-2272	320	9	an	an	DET
ejpam-2272	320	10	element	element	NOUN
ejpam-2272	320	11	a′	a′	NOUN
ejpam-2272	320	12	in	in	ADP
ejpam-2272	320	13	i	i	PRON
ejpam-2272	320	14	such	such	ADJ
ejpam-2272	320	15	that	that	SCONJ
ejpam-2272	320	16	aba′	aba′	PROPN
ejpam-2272	320	17	≠	≠	PROPN
ejpam-2272	320	18	0	0	NUM
ejpam-2272	320	19	,	,	PUNCT
ejpam-2272	320	20	which	which	PRON
ejpam-2272	320	21	implies	imply	VERB
ejpam-2272	320	22	0	0	NUM
ejpam-2272	320	23	≠	≠	PROPN
ejpam-2272	321	1	aba′	aba′	ADJ
ejpam-2272	321	2	=	=	PUNCT
ejpam-2272	321	3	ab(c	ab(c	X
ejpam-2272	321	4	+	+	CCONJ
ejpam-2272	321	5	a′	a′	PROPN
ejpam-2272	321	6	)	)	PUNCT
ejpam-2272	321	7	∈	∈	PROPN
ejpam-2272	321	8	i	i	PRON
ejpam-2272	321	9	.	.	PUNCT
ejpam-2272	322	1	since	since	SCONJ
ejpam-2272	322	2	i	i	PRON
ejpam-2272	322	3	is	be	AUX
ejpam-2272	322	4	a	a	DET
ejpam-2272	322	5	weakly	weakly	ADJ
ejpam-2272	322	6	2	2	NUM
ejpam-2272	322	7	-	-	PUNCT
ejpam-2272	322	8	absorbing	absorbing	ADJ
ejpam-2272	322	9	primary	primary	ADJ
ejpam-2272	322	10	ideal	ideal	NOUN
ejpam-2272	322	11	of	of	ADP
ejpam-2272	322	12	s	s	PROPN
ejpam-2272	322	13	,	,	PUNCT
ejpam-2272	322	14	therefore	therefore	ADV
ejpam-2272	322	15	either	either	CCONJ
ejpam-2272	322	16	ab	ab	PROPN
ejpam-2272	322	17	∈	∈	PROPN
ejpam-2272	322	18	i	i	PROPN
ejpam-2272	322	19	or	or	CCONJ
ejpam-2272	322	20	b(c	b(c	NOUN
ejpam-2272	322	21	+	+	CCONJ
ejpam-2272	322	22	a′	a′	PROPN
ejpam-2272	322	23	)	)	PUNCT
ejpam-2272	322	24	∈	∈	NOUN
ejpam-2272	322	25	√	√	VERB
ejpam-2272	323	1	i	i	PRON
ejpam-2272	323	2	or	or	CCONJ
ejpam-2272	323	3	a(c	a(c	PROPN
ejpam-2272	323	4	+	+	CCONJ
ejpam-2272	323	5	a′	a′	PROPN
ejpam-2272	323	6	)	)	PUNCT
ejpam-2272	323	7	∈	∈	NOUN
ejpam-2272	324	1	√	√	VERB
ejpam-2272	324	2	i	i	PRON
ejpam-2272	324	3	.	.	PUNCT
ejpam-2272	325	1	by	by	ADP
ejpam-2272	325	2	lemma	lemma	PROPN
ejpam-2272	325	3	2	2	NUM
ejpam-2272	325	4	,	,	PUNCT
ejpam-2272	325	5	we	we	PRON
ejpam-2272	325	6	have	have	VERB
ejpam-2272	325	7	ab	ab	PROPN
ejpam-2272	325	8	∈	∈	PROPN
ejpam-2272	326	1	i	i	PRON
ejpam-2272	326	2	or	or	CCONJ
ejpam-2272	326	3	bc	bc	PROPN
ejpam-2272	326	4	∈	∈	PROPN
ejpam-2272	327	1	√	√	VERB
ejpam-2272	327	2	i	i	PRON
ejpam-2272	327	3	or	or	CCONJ
ejpam-2272	327	4	ac	ac	PROPN
ejpam-2272	327	5	∈	∈	PROPN
ejpam-2272	328	1	√	√	VERB
ejpam-2272	329	1	i	i	PRON
ejpam-2272	329	2	.	.	PUNCT
ejpam-2272	330	1	so	so	ADV
ejpam-2272	330	2	,	,	PUNCT
ejpam-2272	330	3	we	we	PRON
ejpam-2272	330	4	assume	assume	VERB
ejpam-2272	330	5	that	that	SCONJ
ejpam-2272	330	6	abi	abi	PROPN
ejpam-2272	330	7	=	=	PUNCT
ejpam-2272	330	8	0	0	X
ejpam-2272	330	9	.	.	PUNCT
ejpam-2272	331	1	similarly	similarly	ADV
ejpam-2272	331	2	,	,	PUNCT
ejpam-2272	331	3	we	we	PRON
ejpam-2272	331	4	can	can	AUX
ejpam-2272	331	5	assume	assume	VERB
ejpam-2272	331	6	that	that	SCONJ
ejpam-2272	331	7	aic	aic	PROPN
ejpam-2272	331	8	=	=	PROPN
ejpam-2272	331	9	0	0	PROPN
ejpam-2272	331	10	and	and	CCONJ
ejpam-2272	331	11	i	i	PRON
ejpam-2272	331	12	bc	bc	VERB
ejpam-2272	331	13	=	=	SYM
ejpam-2272	332	1	0	0	PROPN
ejpam-2272	332	2	.	.	PUNCT
ejpam-2272	333	1	now	now	ADV
ejpam-2272	333	2	,	,	PUNCT
ejpam-2272	333	3	let	let	VERB
ejpam-2272	333	4	ai2	ai2	NOUN
ejpam-2272	333	5	≠	≠	PROPN
ejpam-2272	333	6	0	0	NUM
ejpam-2272	333	7	.	.	PUNCT
ejpam-2272	334	1	then	then	ADV
ejpam-2272	334	2	there	there	PRON
ejpam-2272	334	3	exist	exist	VERB
ejpam-2272	334	4	i1	i1	PROPN
ejpam-2272	334	5	,	,	PUNCT
ejpam-2272	334	6	i2	i2	PROPN
ejpam-2272	334	7	∈	∈	PROPN
ejpam-2272	335	1	i	i	PRON
ejpam-2272	335	2	such	such	ADJ
ejpam-2272	335	3	that	that	SCONJ
ejpam-2272	335	4	ai1i2	ai1i2	PROPN
ejpam-2272	335	5	≠	≠	PROPN
ejpam-2272	335	6	0	0	X
ejpam-2272	335	7	.	.	PUNCT
ejpam-2272	336	1	since	since	SCONJ
ejpam-2272	336	2	abi	abi	PROPN
ejpam-2272	336	3	=	=	PUNCT
ejpam-2272	336	4	aic	aic	PROPN
ejpam-2272	336	5	=	=	PROPN
ejpam-2272	336	6	i	i	PRON
ejpam-2272	336	7	bc	bc	VERB
ejpam-2272	336	8	=	=	SYM
ejpam-2272	336	9	0	0	PROPN
ejpam-2272	336	10	,	,	PUNCT
ejpam-2272	336	11	we	we	PRON
ejpam-2272	336	12	have	have	VERB
ejpam-2272	336	13	0	0	NUM
ejpam-2272	336	14	≠	≠	PROPN
ejpam-2272	336	15	a(b	a(b	PROPN
ejpam-2272	336	16	+	+	CCONJ
ejpam-2272	336	17	i1)(c	i1)(c	PROPN
ejpam-2272	336	18	+	+	CCONJ
ejpam-2272	336	19	i2	i2	NOUN
ejpam-2272	336	20	)	)	PUNCT
ejpam-2272	336	21	=	=	PUNCT
ejpam-2272	337	1	ai1i2	ai1i2	PROPN
ejpam-2272	337	2	∈	∈	PROPN
ejpam-2272	338	1	i	i	PRON
ejpam-2272	338	2	.	.	PUNCT
ejpam-2272	339	1	therefore	therefore	ADV
ejpam-2272	339	2	,	,	PUNCT
ejpam-2272	339	3	either	either	CCONJ
ejpam-2272	339	4	a(b	a(b	PROPN
ejpam-2272	339	5	+	+	CCONJ
ejpam-2272	339	6	i1	i1	PROPN
ejpam-2272	339	7	)	)	PUNCT
ejpam-2272	339	8	∈	∈	PROPN
ejpam-2272	339	9	i	i	PRON
ejpam-2272	339	10	or	or	CCONJ
ejpam-2272	339	11	a(c	a(c	PROPN
ejpam-2272	339	12	+	+	CCONJ
ejpam-2272	339	13	i2	i2	PROPN
ejpam-2272	339	14	)	)	PUNCT
ejpam-2272	339	15	∈	∈	NOUN
ejpam-2272	340	1	√	√	VERB
ejpam-2272	340	2	i	i	PRON
ejpam-2272	340	3	or	or	CCONJ
ejpam-2272	340	4	(	(	PUNCT
ejpam-2272	340	5	b	b	PROPN
ejpam-2272	340	6	+	+	CCONJ
ejpam-2272	340	7	i1)(c	i1)(c	ADJ
ejpam-2272	340	8	+	+	CCONJ
ejpam-2272	340	9	i2	i2	PROPN
ejpam-2272	340	10	)	)	PUNCT
ejpam-2272	340	11	∈	∈	NOUN
ejpam-2272	341	1	√	√	VERB
ejpam-2272	342	1	i	i	PRON
ejpam-2272	342	2	.	.	PUNCT
ejpam-2272	343	1	hence	hence	ADV
ejpam-2272	343	2	,	,	PUNCT
ejpam-2272	343	3	we	we	PRON
ejpam-2272	343	4	have	have	VERB
ejpam-2272	343	5	either	either	CCONJ
ejpam-2272	343	6	ab	ab	PROPN
ejpam-2272	343	7	∈	∈	PROPN
ejpam-2272	344	1	i	i	PRON
ejpam-2272	344	2	or	or	CCONJ
ejpam-2272	344	3	ac	ac	PROPN
ejpam-2272	344	4	∈	∈	PROPN
ejpam-2272	345	1	√	√	VERB
ejpam-2272	345	2	i	i	PRON
ejpam-2272	345	3	or	or	CCONJ
ejpam-2272	345	4	bc	bc	PROPN
ejpam-2272	345	5	∈	∈	PROPN
ejpam-2272	346	1	√	√	VERB
ejpam-2272	346	2	i	i	PRON
ejpam-2272	346	3	.	.	PUNCT
ejpam-2272	347	1	so	so	ADV
ejpam-2272	347	2	,	,	PUNCT
ejpam-2272	347	3	we	we	PRON
ejpam-2272	347	4	can	can	AUX
ejpam-2272	347	5	assume	assume	VERB
ejpam-2272	347	6	that	that	SCONJ
ejpam-2272	347	7	ai2	ai2	NOUN
ejpam-2272	347	8	=	=	NOUN
ejpam-2272	347	9	0	0	X
ejpam-2272	347	10	.	.	PUNCT
ejpam-2272	348	1	likewise	likewise	ADV
ejpam-2272	348	2	,	,	PUNCT
ejpam-2272	348	3	we	we	PRON
ejpam-2272	348	4	can	can	AUX
ejpam-2272	348	5	assume	assume	VERB
ejpam-2272	348	6	that	that	SCONJ
ejpam-2272	348	7	bi2	bi2	PROPN
ejpam-2272	349	1	=	=	SYM
ejpam-2272	349	2	0	0	NUM
ejpam-2272	349	3	and	and	CCONJ
ejpam-2272	349	4	ci2	ci2	PROPN
ejpam-2272	349	5	=	=	SYM
ejpam-2272	349	6	0	0	NUM
ejpam-2272	349	7	.	.	PUNCT
ejpam-2272	350	1	since	since	SCONJ
ejpam-2272	350	2	i3	i3	NOUN
ejpam-2272	350	3	≠	≠	PROPN
ejpam-2272	350	4	0	0	NUM
ejpam-2272	350	5	,	,	PUNCT
ejpam-2272	350	6	there	there	PRON
ejpam-2272	350	7	exist	exist	VERB
ejpam-2272	350	8	p	p	PRON
ejpam-2272	350	9	,	,	PUNCT
ejpam-2272	350	10	q	q	ADJ
ejpam-2272	350	11	,	,	PUNCT
ejpam-2272	350	12	r	r	NOUN
ejpam-2272	350	13	∈	∈	PROPN
ejpam-2272	350	14	i	i	PRON
ejpam-2272	350	15	such	such	ADJ
ejpam-2272	350	16	that	that	SCONJ
ejpam-2272	350	17	pqr	pqr	PROPN
ejpam-2272	350	18	≠	≠	PROPN
ejpam-2272	350	19	0	0	X
ejpam-2272	350	20	.	.	PUNCT
ejpam-2272	351	1	again	again	ADV
ejpam-2272	351	2	,	,	PUNCT
ejpam-2272	351	3	(	(	PUNCT
ejpam-2272	351	4	a	a	DET
ejpam-2272	351	5	+	+	NOUN
ejpam-2272	351	6	p)(b	p)(b	ADJ
ejpam-2272	351	7	+	+	CCONJ
ejpam-2272	351	8	q)(c	q)(c	X
ejpam-2272	352	1	+	+	CCONJ
ejpam-2272	352	2	r	r	X
ejpam-2272	352	3	)	)	PUNCT
ejpam-2272	352	4	=	=	PUNCT
ejpam-2272	352	5	pqr	pqr	PROPN
ejpam-2272	352	6	∈	∈	PROPN
ejpam-2272	353	1	i	i	PRON
ejpam-2272	353	2	,	,	PUNCT
ejpam-2272	353	3	so	so	ADV
ejpam-2272	353	4	either	either	CCONJ
ejpam-2272	353	5	(	(	PUNCT
ejpam-2272	353	6	a	a	DET
ejpam-2272	353	7	+	+	NOUN
ejpam-2272	353	8	p)(b	p)(b	ADJ
ejpam-2272	353	9	+	+	CCONJ
ejpam-2272	353	10	q	q	X
ejpam-2272	353	11	)	)	PUNCT
ejpam-2272	353	12	∈	∈	PROPN
ejpam-2272	354	1	i	i	PRON
ejpam-2272	354	2	or	or	CCONJ
ejpam-2272	354	3	(	(	PUNCT
ejpam-2272	354	4	b+q)(c+	b+q)(c+	NOUN
ejpam-2272	354	5	r	r	NOUN
ejpam-2272	354	6	)	)	PUNCT
ejpam-2272	354	7	∈	∈	NOUN
ejpam-2272	355	1	√	√	VERB
ejpam-2272	355	2	i	i	PRON
ejpam-2272	355	3	or	or	CCONJ
ejpam-2272	355	4	(	(	PUNCT
ejpam-2272	355	5	a+p)(c+	a+p)(c+	ADJ
ejpam-2272	355	6	r	r	NOUN
ejpam-2272	355	7	)	)	PUNCT
ejpam-2272	355	8	∈	∈	NOUN
ejpam-2272	356	1	√	√	VERB
ejpam-2272	357	1	i	i	PRON
ejpam-2272	357	2	,	,	PUNCT
ejpam-2272	357	3	that	that	ADV
ejpam-2272	357	4	is	is	ADV
ejpam-2272	357	5	,	,	PUNCT
ejpam-2272	357	6	ab+aq+pb+pq	ab+aq+pb+pq	PROPN
ejpam-2272	357	7	∈	∈	PROPN
ejpam-2272	358	1	i	i	PRON
ejpam-2272	358	2	or	or	CCONJ
ejpam-2272	358	3	bc+br+qc+qr	bc+br+qc+qr	PROPN
ejpam-2272	358	4	∈	∈	NOUN
ejpam-2272	358	5	√	√	VERB
ejpam-2272	359	1	i	i	PROPN
ejpam-2272	359	2	p.	p.	PROPN
ejpam-2272	359	3	kumar	kumar	PROPN
ejpam-2272	359	4	,	,	PUNCT
ejpam-2272	359	5	m.	m.	NOUN
ejpam-2272	359	6	dubey	dubey	PROPN
ejpam-2272	359	7	,	,	PUNCT
ejpam-2272	359	8	p.	p.	NOUN
ejpam-2272	359	9	sarohe	sarohe	PROPN
ejpam-2272	359	10	/	/	SYM
ejpam-2272	359	11	eur	eur	PROPN
ejpam-2272	359	12	.	.	PUNCT
ejpam-2272	360	1	j.	j.	PROPN
ejpam-2272	360	2	pure	pure	PROPN
ejpam-2272	360	3	appl	appl	PROPN
ejpam-2272	360	4	.	.	PROPN
ejpam-2272	360	5	math	math	PROPN
ejpam-2272	360	6	,	,	PUNCT
ejpam-2272	360	7	9	9	NUM
ejpam-2272	360	8	(	(	PUNCT
ejpam-2272	360	9	2016	2016	NUM
ejpam-2272	360	10	)	)	PUNCT
ejpam-2272	360	11	,	,	PUNCT
ejpam-2272	360	12	186	186	NUM
ejpam-2272	360	13	-	-	SYM
ejpam-2272	360	14	195	195	NUM
ejpam-2272	360	15	192	192	NUM
ejpam-2272	360	16	or	or	CCONJ
ejpam-2272	360	17	ac	ac	PROPN
ejpam-2272	360	18	+	+	CCONJ
ejpam-2272	360	19	ar	ar	PROPN
ejpam-2272	360	20	+	+	NOUN
ejpam-2272	360	21	pc	pc	NOUN
ejpam-2272	360	22	+	+	CCONJ
ejpam-2272	360	23	pr	pr	NOUN
ejpam-2272	360	24	∈	∈	NOUN
ejpam-2272	360	25	√	√	VERB
ejpam-2272	361	1	i	i	PRON
ejpam-2272	361	2	.	.	PUNCT
ejpam-2272	362	1	hence	hence	ADV
ejpam-2272	362	2	,	,	PUNCT
ejpam-2272	362	3	either	either	CCONJ
ejpam-2272	362	4	ab	ab	PROPN
ejpam-2272	362	5	∈	∈	PROPN
ejpam-2272	362	6	i	i	PRON
ejpam-2272	362	7	or	or	CCONJ
ejpam-2272	362	8	bc	bc	PROPN
ejpam-2272	362	9	∈	∈	PROPN
ejpam-2272	362	10	√	√	VERB
ejpam-2272	362	11	i	i	PRON
ejpam-2272	362	12	or	or	CCONJ
ejpam-2272	362	13	ac	ac	PROPN
ejpam-2272	362	14	∈	∈	PROPN
ejpam-2272	363	1	√	√	VERB
ejpam-2272	363	2	i	i	PRON
ejpam-2272	363	3	.	.	PUNCT
ejpam-2272	364	1	this	this	PRON
ejpam-2272	364	2	implies	imply	VERB
ejpam-2272	364	3	that	that	SCONJ
ejpam-2272	364	4	i	i	PRON
ejpam-2272	364	5	is	be	AUX
ejpam-2272	364	6	a	a	DET
ejpam-2272	364	7	2	2	NUM
ejpam-2272	364	8	-	-	PUNCT
ejpam-2272	364	9	absorbing	absorbing	ADJ
ejpam-2272	364	10	primary	primary	ADJ
ejpam-2272	364	11	ideal	ideal	NOUN
ejpam-2272	364	12	of	of	ADP
ejpam-2272	364	13	s	s	PROPN
ejpam-2272	364	14	,	,	PUNCT
ejpam-2272	364	15	which	which	PRON
ejpam-2272	364	16	is	be	AUX
ejpam-2272	364	17	a	a	DET
ejpam-2272	364	18	contradiction	contradiction	NOUN
ejpam-2272	364	19	.	.	PUNCT
ejpam-2272	365	1	therefore	therefore	ADV
ejpam-2272	365	2	,	,	PUNCT
ejpam-2272	365	3	i3	i3	NOUN
ejpam-2272	365	4	=	=	SYM
ejpam-2272	365	5	0	0	X
ejpam-2272	365	6	.	.	PUNCT
ejpam-2272	366	1	clearly	clearly	ADV
ejpam-2272	366	2	,	,	PUNCT
ejpam-2272	366	3	√	√	ADV
ejpam-2272	366	4	0	0	NUM
ejpam-2272	366	5	⊆	⊆	NUM
ejpam-2272	366	6	√	√	NUM
ejpam-2272	366	7	i	i	PRON
ejpam-2272	366	8	.	.	PUNCT
ejpam-2272	367	1	as	as	ADP
ejpam-2272	367	2	i3	i3	NOUN
ejpam-2272	367	3	=	=	SYM
ejpam-2272	367	4	0	0	NUM
ejpam-2272	367	5	,	,	PUNCT
ejpam-2272	367	6	we	we	PRON
ejpam-2272	367	7	get	get	VERB
ejpam-2272	367	8	i	i	PRON
ejpam-2272	367	9	⊆	⊆	NUM
ejpam-2272	367	10	√	√	NUM
ejpam-2272	367	11	0	0	NUM
ejpam-2272	367	12	.	.	PUNCT
ejpam-2272	368	1	this	this	PRON
ejpam-2272	368	2	concludes	conclude	VERB
ejpam-2272	368	3	that	that	SCONJ
ejpam-2272	368	4	√	√	VERB
ejpam-2272	368	5	i	i	PRON
ejpam-2272	368	6	⊆	⊆	NUM
ejpam-2272	368	7	√	√	PROPN
ejpam-2272	368	8	0	0	NUM
ejpam-2272	368	9	.	.	PUNCT
ejpam-2272	369	1	thus	thus	ADV
ejpam-2272	369	2	,	,	PUNCT
ejpam-2272	369	3	√	√	PROPN
ejpam-2272	369	4	i	i	NOUN
ejpam-2272	369	5	=	=	PUNCT
ejpam-2272	370	1	√	√	PROPN
ejpam-2272	370	2	0	0	NUM
ejpam-2272	370	3	.	.	PUNCT
ejpam-2272	371	1	theorem	theorem	NOUN
ejpam-2272	371	2	11	11	NUM
ejpam-2272	371	3	.	.	PUNCT
ejpam-2272	372	1	let	let	VERB
ejpam-2272	372	2	s	s	PRON
ejpam-2272	372	3	be	be	AUX
ejpam-2272	372	4	a	a	DET
ejpam-2272	372	5	semiring	semiring	NOUN
ejpam-2272	372	6	and	and	CCONJ
ejpam-2272	372	7	{	{	PUNCT
ejpam-2272	372	8	ii}i∈∆	ii}i∈∆	NOUN
ejpam-2272	372	9	be	be	AUX
ejpam-2272	372	10	a	a	DET
ejpam-2272	372	11	family	family	NOUN
ejpam-2272	372	12	of	of	ADP
ejpam-2272	372	13	subtractive	subtractive	NOUN
ejpam-2272	372	14	weakly	weakly	ADJ
ejpam-2272	372	15	2	2	NUM
ejpam-2272	372	16	-	-	PUNCT
ejpam-2272	372	17	absorbing	absorbing	ADJ
ejpam-2272	372	18	primary	primary	ADJ
ejpam-2272	372	19	ideals	ideal	NOUN
ejpam-2272	372	20	of	of	ADP
ejpam-2272	372	21	s	s	PRON
ejpam-2272	372	22	that	that	PRON
ejpam-2272	372	23	are	be	AUX
ejpam-2272	372	24	not	not	PART
ejpam-2272	372	25	2	2	NUM
ejpam-2272	372	26	-	-	PUNCT
ejpam-2272	372	27	absorbing	absorbing	ADJ
ejpam-2272	372	28	primary	primary	ADJ
ejpam-2272	372	29	ideals	ideal	NOUN
ejpam-2272	372	30	of	of	ADP
ejpam-2272	372	31	s.	s.	PROPN
ejpam-2272	373	1	then	then	ADV
ejpam-2272	373	2	i	i	PRON
ejpam-2272	373	3	=	=	SYM
ejpam-2272	373	4	⋂	⋂	PROPN
ejpam-2272	373	5	i∈∆	i∈∆	PROPN
ejpam-2272	373	6	ii	ii	PROPN
ejpam-2272	373	7	is	be	AUX
ejpam-2272	373	8	a	a	DET
ejpam-2272	373	9	weakly	weakly	ADJ
ejpam-2272	373	10	2	2	NUM
ejpam-2272	373	11	-	-	PUNCT
ejpam-2272	373	12	absorbing	absorbing	ADJ
ejpam-2272	373	13	primary	primary	ADJ
ejpam-2272	373	14	ideal	ideal	NOUN
ejpam-2272	373	15	of	of	ADP
ejpam-2272	373	16	s.	s.	PROPN
ejpam-2272	373	17	proof	proof	PROPN
ejpam-2272	373	18	.	.	PUNCT
ejpam-2272	374	1	let	let	VERB
ejpam-2272	374	2	{	{	PUNCT
ejpam-2272	374	3	ii}i∈∆	ii}i∈∆	NOUN
ejpam-2272	374	4	be	be	AUX
ejpam-2272	374	5	a	a	DET
ejpam-2272	374	6	family	family	NOUN
ejpam-2272	374	7	of	of	ADP
ejpam-2272	374	8	weakly	weakly	ADJ
ejpam-2272	374	9	2	2	NUM
ejpam-2272	374	10	-	-	PUNCT
ejpam-2272	374	11	absorbing	absorbing	ADJ
ejpam-2272	374	12	primary	primary	ADJ
ejpam-2272	374	13	ideals	ideal	NOUN
ejpam-2272	374	14	of	of	ADP
ejpam-2272	374	15	s	s	PRON
ejpam-2272	374	16	that	that	PRON
ejpam-2272	374	17	are	be	AUX
ejpam-2272	374	18	not	not	PART
ejpam-2272	374	19	2absorbing	2absorbing	NUM
ejpam-2272	374	20	primary	primary	ADJ
ejpam-2272	374	21	ideals	ideal	NOUN
ejpam-2272	374	22	of	of	ADP
ejpam-2272	374	23	s.	s.	PROPN
ejpam-2272	374	24	therefore	therefore	ADV
ejpam-2272	374	25	,	,	PUNCT
ejpam-2272	374	26	by	by	ADP
ejpam-2272	374	27	theorem	theorem	NOUN
ejpam-2272	374	28	10	10	NUM
ejpam-2272	374	29	,	,	PUNCT
ejpam-2272	374	30	we	we	PRON
ejpam-2272	374	31	have	have	VERB
ejpam-2272	374	32	√	√	NUM
ejpam-2272	374	33	ii	ii	NOUN
ejpam-2272	374	34	=	=	PUNCT
ejpam-2272	374	35	√	√	ADP
ejpam-2272	374	36	0	0	NUM
ejpam-2272	374	37	for	for	ADP
ejpam-2272	374	38	all	all	PRON
ejpam-2272	374	39	i	i	PRON
ejpam-2272	374	40	∈	∈	VERB
ejpam-2272	375	1	∆.	∆.	NOUN
ejpam-2272	375	2	this	this	PRON
ejpam-2272	375	3	gives	give	VERB
ejpam-2272	375	4	⋂	⋂	PROPN
ejpam-2272	375	5	i∈∆	i∈∆	PROPN
ejpam-2272	375	6	√	√	PROPN
ejpam-2272	375	7	ii	ii	NOUN
ejpam-2272	375	8	=	=	PUNCT
ejpam-2272	375	9	√	√	ADP
ejpam-2272	375	10	0	0	NUM
ejpam-2272	375	11	.	.	PUNCT
ejpam-2272	376	1	thus	thus	ADV
ejpam-2272	376	2	we	we	PRON
ejpam-2272	376	3	have	have	VERB
ejpam-2272	376	4	√	√	VERB
ejpam-2272	377	1	i	i	PRON
ejpam-2272	377	2	=	=	PUNCT
ejpam-2272	377	3	√	√	NUM
ejpam-2272	377	4	0	0	NUM
ejpam-2272	377	5	,	,	PUNCT
ejpam-2272	377	6	since	since	SCONJ
ejpam-2272	377	7	⋂	⋂	PROPN
ejpam-2272	377	8	i∈∆	i∈∆	PROPN
ejpam-2272	377	9	√	√	ADV
ejpam-2272	377	10	ii	ii	NOUN
ejpam-2272	377	11	=	=	PUNCT
ejpam-2272	378	1	√	√	PROPN
ejpam-2272	378	2	i	i	PRON
ejpam-2272	378	3	.	.	PUNCT
ejpam-2272	379	1	next	next	ADV
ejpam-2272	379	2	,	,	PUNCT
ejpam-2272	379	3	let	let	VERB
ejpam-2272	379	4	a	a	DET
ejpam-2272	379	5	,	,	PUNCT
ejpam-2272	379	6	b	b	NOUN
ejpam-2272	379	7	,	,	PUNCT
ejpam-2272	379	8	c	c	PROPN
ejpam-2272	379	9	∈	∈	PROPN
ejpam-2272	379	10	s	s	AUX
ejpam-2272	379	11	be	be	AUX
ejpam-2272	379	12	such	such	ADJ
ejpam-2272	379	13	that	that	SCONJ
ejpam-2272	379	14	0	0	NUM
ejpam-2272	379	15	≠	≠	PROPN
ejpam-2272	379	16	abc	abc	PROPN
ejpam-2272	379	17	∈	∈	PROPN
ejpam-2272	380	1	i	i	PRON
ejpam-2272	380	2	but	but	CCONJ
ejpam-2272	380	3	ab	ab	PROPN
ejpam-2272	380	4	∉	∉	PROPN
ejpam-2272	380	5	i	i	PRON
ejpam-2272	380	6	.	.	PUNCT
ejpam-2272	381	1	then	then	ADV
ejpam-2272	381	2	there	there	PRON
ejpam-2272	381	3	exists	exist	VERB
ejpam-2272	381	4	i	i	PRON
ejpam-2272	381	5	∈	∈	PROPN
ejpam-2272	381	6	∆	∆	PROPN
ejpam-2272	381	7	such	such	ADJ
ejpam-2272	381	8	that	that	SCONJ
ejpam-2272	381	9	ab	ab	PROPN
ejpam-2272	381	10	∉	∉	PROPN
ejpam-2272	381	11	ii	ii	PROPN
ejpam-2272	381	12	and	and	CCONJ
ejpam-2272	381	13	0	0	NUM
ejpam-2272	381	14	≠	≠	PROPN
ejpam-2272	381	15	abc	abc	PROPN
ejpam-2272	381	16	∈	∈	PROPN
ejpam-2272	381	17	ii	ii	PROPN
ejpam-2272	381	18	.	.	PUNCT
ejpam-2272	382	1	this	this	PRON
ejpam-2272	382	2	gives	give	VERB
ejpam-2272	382	3	bc	bc	PROPN
ejpam-2272	382	4	∈	∈	PROPN
ejpam-2272	382	5	√ii	√ii	PROPN
ejpam-2272	382	6	or	or	CCONJ
ejpam-2272	382	7	ac	ac	PROPN
ejpam-2272	382	8	∈	∈	PROPN
ejpam-2272	382	9	√ii	√ii	PROPN
ejpam-2272	382	10	since	since	SCONJ
ejpam-2272	382	11	ii	ii	PROPN
ejpam-2272	382	12	is	be	AUX
ejpam-2272	382	13	a	a	DET
ejpam-2272	382	14	weakly	weakly	ADJ
ejpam-2272	382	15	2	2	NUM
ejpam-2272	382	16	-	-	PUNCT
ejpam-2272	382	17	absorbing	absorbing	ADJ
ejpam-2272	382	18	primary	primary	ADJ
ejpam-2272	382	19	ideal	ideal	NOUN
ejpam-2272	382	20	of	of	ADP
ejpam-2272	382	21	s	s	PRON
ejpam-2272	382	22	and	and	CCONJ
ejpam-2272	382	23	ab	ab	PROPN
ejpam-2272	382	24	∉	∉	PROPN
ejpam-2272	382	25	ii	ii	PROPN
ejpam-2272	382	26	.	.	PUNCT
ejpam-2272	383	1	thus	thus	ADV
ejpam-2272	383	2	,	,	PUNCT
ejpam-2272	383	3	either	either	CCONJ
ejpam-2272	383	4	bc	bc	PROPN
ejpam-2272	383	5	∈	∈	PROPN
ejpam-2272	383	6	√ii	√ii	PROPN
ejpam-2272	383	7	=	=	PUNCT
ejpam-2272	383	8	√	√	NUM
ejpam-2272	383	9	0	0	NUM
ejpam-2272	384	1	=	=	PUNCT
ejpam-2272	384	2	√	√	VERB
ejpam-2272	384	3	i	i	PRON
ejpam-2272	384	4	or	or	CCONJ
ejpam-2272	384	5	ca	ca	NOUN
ejpam-2272	384	6	∈	∈	PROPN
ejpam-2272	385	1	√ii	√ii	PROPN
ejpam-2272	385	2	=	=	PUNCT
ejpam-2272	385	3	√	√	NUM
ejpam-2272	385	4	0	0	NUM
ejpam-2272	386	1	=	=	PUNCT
ejpam-2272	386	2	√	√	INTJ
ejpam-2272	387	1	i	i	PRON
ejpam-2272	387	2	.	.	PUNCT
ejpam-2272	388	1	hence	hence	ADV
ejpam-2272	388	2	i	i	PRON
ejpam-2272	388	3	is	be	AUX
ejpam-2272	388	4	a	a	DET
ejpam-2272	388	5	weakly	weakly	ADJ
ejpam-2272	388	6	2	2	NUM
ejpam-2272	388	7	-	-	PUNCT
ejpam-2272	388	8	absorbing	absorbing	ADJ
ejpam-2272	388	9	primary	primary	ADJ
ejpam-2272	388	10	ideal	ideal	NOUN
ejpam-2272	388	11	of	of	ADP
ejpam-2272	388	12	s.	s.	PROPN
ejpam-2272	388	13	definition	definition	NOUN
ejpam-2272	388	14	5	5	NUM
ejpam-2272	388	15	(	(	PUNCT
ejpam-2272	388	16	[	[	X
ejpam-2272	388	17	4	4	NUM
ejpam-2272	388	18	,	,	PUNCT
ejpam-2272	388	19	definition	definition	NOUN
ejpam-2272	388	20	1(i	1(i	NUM
ejpam-2272	388	21	)	)	PUNCT
ejpam-2272	388	22	]	]	PUNCT
ejpam-2272	388	23	)	)	PUNCT
ejpam-2272	388	24	.	.	PUNCT
ejpam-2272	389	1	a	a	DET
ejpam-2272	389	2	proper	proper	ADJ
ejpam-2272	389	3	ideal	ideal	NOUN
ejpam-2272	389	4	i	i	PRON
ejpam-2272	389	5	of	of	ADP
ejpam-2272	389	6	a	a	DET
ejpam-2272	389	7	semiring	semiring	NOUN
ejpam-2272	389	8	s	s	NOUN
ejpam-2272	389	9	is	be	AUX
ejpam-2272	389	10	said	say	VERB
ejpam-2272	389	11	to	to	PART
ejpam-2272	389	12	be	be	AUX
ejpam-2272	389	13	a	a	DET
ejpam-2272	389	14	strong	strong	ADJ
ejpam-2272	389	15	ideal	ideal	NOUN
ejpam-2272	389	16	,	,	PUNCT
ejpam-2272	389	17	if	if	SCONJ
ejpam-2272	389	18	for	for	ADP
ejpam-2272	389	19	each	each	DET
ejpam-2272	389	20	a	a	DET
ejpam-2272	389	21	∈	∈	NOUN
ejpam-2272	389	22	i	i	PRON
ejpam-2272	389	23	there	there	PRON
ejpam-2272	389	24	exists	exist	VERB
ejpam-2272	389	25	b	b	X
ejpam-2272	389	26	∈	∈	PROPN
ejpam-2272	389	27	i	i	PRON
ejpam-2272	389	28	such	such	ADJ
ejpam-2272	390	1	that	that	SCONJ
ejpam-2272	390	2	a	a	DET
ejpam-2272	390	3	+	+	NOUN
ejpam-2272	390	4	b	b	NOUN
ejpam-2272	390	5	=	=	SYM
ejpam-2272	390	6	0	0	PROPN
ejpam-2272	390	7	.	.	PUNCT
ejpam-2272	391	1	proposition	proposition	NOUN
ejpam-2272	391	2	1	1	NUM
ejpam-2272	391	3	.	.	PUNCT
ejpam-2272	392	1	let	let	VERB
ejpam-2272	392	2	s	s	PRON
ejpam-2272	392	3	and	and	CCONJ
ejpam-2272	392	4	s′	s′	ADJ
ejpam-2272	392	5	be	be	VERB
ejpam-2272	392	6	semirings	semiring	NOUN
ejpam-2272	392	7	,	,	PUNCT
ejpam-2272	392	8	f	f	PROPN
ejpam-2272	392	9	∶	∶	PROPN
ejpam-2272	392	10	s	s	PART
ejpam-2272	392	11	↦	↦	PROPN
ejpam-2272	392	12	s′	s′	NOUN
ejpam-2272	392	13	be	be	VERB
ejpam-2272	392	14	an	an	DET
ejpam-2272	392	15	epimorphism	epimorphism	NOUN
ejpam-2272	392	16	such	such	ADJ
ejpam-2272	392	17	that	that	SCONJ
ejpam-2272	392	18	f	f	PROPN
ejpam-2272	392	19	(	(	PUNCT
ejpam-2272	392	20	0	0	NUM
ejpam-2272	392	21	)	)	PUNCT
ejpam-2272	392	22	=	=	SYM
ejpam-2272	392	23	0	0	PUNCT
ejpam-2272	393	1	and	and	CCONJ
ejpam-2272	393	2	i	i	PRON
ejpam-2272	393	3	be	be	VERB
ejpam-2272	393	4	a	a	DET
ejpam-2272	393	5	subtractive	subtractive	NOUN
ejpam-2272	393	6	strong	strong	ADJ
ejpam-2272	393	7	ideal	ideal	NOUN
ejpam-2272	393	8	of	of	ADP
ejpam-2272	393	9	s.	s.	PROPN
ejpam-2272	393	10	then	then	ADV
ejpam-2272	393	11	the	the	DET
ejpam-2272	393	12	following	follow	VERB
ejpam-2272	393	13	holds	hold	VERB
ejpam-2272	393	14	:	:	PUNCT
ejpam-2272	393	15	(	(	PUNCT
ejpam-2272	393	16	1	1	X
ejpam-2272	393	17	)	)	PUNCT
ejpam-2272	393	18	if	if	SCONJ
ejpam-2272	393	19	i	i	PRON
ejpam-2272	393	20	is	be	AUX
ejpam-2272	393	21	a	a	DET
ejpam-2272	393	22	weakly	weakly	ADJ
ejpam-2272	393	23	2	2	NUM
ejpam-2272	393	24	-	-	PUNCT
ejpam-2272	393	25	absorbing	absorbing	ADJ
ejpam-2272	393	26	primary	primary	ADJ
ejpam-2272	393	27	ideal	ideal	NOUN
ejpam-2272	393	28	of	of	ADP
ejpam-2272	393	29	s	s	PRON
ejpam-2272	393	30	such	such	ADJ
ejpam-2272	393	31	that	that	DET
ejpam-2272	393	32	ker	ker	NOUN
ejpam-2272	394	1	f	f	PROPN
ejpam-2272	395	1	⊆	⊆	NUM
ejpam-2272	396	1	i	i	PRON
ejpam-2272	396	2	,	,	PUNCT
ejpam-2272	396	3	then	then	ADV
ejpam-2272	396	4	f	f	X
ejpam-2272	396	5	(	(	PUNCT
ejpam-2272	396	6	i	i	NOUN
ejpam-2272	396	7	)	)	PUNCT
ejpam-2272	396	8	is	be	AUX
ejpam-2272	396	9	a	a	DET
ejpam-2272	396	10	weakly	weakly	ADJ
ejpam-2272	396	11	2	2	NUM
ejpam-2272	396	12	-	-	PUNCT
ejpam-2272	396	13	absorbing	absorbing	ADJ
ejpam-2272	396	14	primary	primary	ADJ
ejpam-2272	396	15	ideal	ideal	NOUN
ejpam-2272	396	16	of	of	ADP
ejpam-2272	396	17	s′.	s′.	PROPN
ejpam-2272	396	18	(	(	PUNCT
ejpam-2272	396	19	2	2	NUM
ejpam-2272	396	20	)	)	PUNCT
ejpam-2272	396	21	if	if	SCONJ
ejpam-2272	396	22	i	i	PRON
ejpam-2272	396	23	is	be	AUX
ejpam-2272	396	24	a	a	DET
ejpam-2272	396	25	2	2	NUM
ejpam-2272	396	26	-	-	PUNCT
ejpam-2272	396	27	absorbing	absorbing	ADJ
ejpam-2272	396	28	primary	primary	ADJ
ejpam-2272	396	29	ideal	ideal	NOUN
ejpam-2272	396	30	of	of	ADP
ejpam-2272	396	31	s	s	PRON
ejpam-2272	396	32	such	such	ADJ
ejpam-2272	396	33	that	that	DET
ejpam-2272	396	34	ker	ker	NOUN
ejpam-2272	397	1	f	f	PROPN
ejpam-2272	397	2	⊆	⊆	NUM
ejpam-2272	397	3	i	i	PRON
ejpam-2272	397	4	,	,	PUNCT
ejpam-2272	397	5	then	then	ADV
ejpam-2272	397	6	f	f	X
ejpam-2272	397	7	(	(	PUNCT
ejpam-2272	397	8	i	i	NOUN
ejpam-2272	397	9	)	)	PUNCT
ejpam-2272	397	10	is	be	AUX
ejpam-2272	397	11	a	a	DET
ejpam-2272	397	12	2	2	NUM
ejpam-2272	397	13	-	-	PUNCT
ejpam-2272	397	14	absorbing	absorbing	ADJ
ejpam-2272	397	15	primary	primary	ADJ
ejpam-2272	397	16	ideal	ideal	NOUN
ejpam-2272	397	17	of	of	ADP
ejpam-2272	397	18	s′.	s′.	PROPN
ejpam-2272	397	19	proof	proof	NOUN
ejpam-2272	397	20	.	.	PUNCT
ejpam-2272	398	1	(	(	PUNCT
ejpam-2272	398	2	1	1	X
ejpam-2272	398	3	)	)	PUNCT
ejpam-2272	398	4	let	let	VERB
ejpam-2272	398	5	a	a	DET
ejpam-2272	398	6	,	,	PUNCT
ejpam-2272	398	7	b	b	NOUN
ejpam-2272	398	8	,	,	PUNCT
ejpam-2272	398	9	c	c	PROPN
ejpam-2272	398	10	∈	∈	PROPN
ejpam-2272	398	11	s′	s′	VERB
ejpam-2272	398	12	be	be	VERB
ejpam-2272	398	13	such	such	ADJ
ejpam-2272	398	14	that	that	SCONJ
ejpam-2272	398	15	0	0	NUM
ejpam-2272	398	16	≠	≠	PROPN
ejpam-2272	398	17	abc	abc	PROPN
ejpam-2272	398	18	∈	∈	PROPN
ejpam-2272	398	19	f	f	X
ejpam-2272	398	20	(	(	PUNCT
ejpam-2272	398	21	i	i	NOUN
ejpam-2272	398	22	)	)	PUNCT
ejpam-2272	398	23	.	.	PUNCT
ejpam-2272	399	1	then	then	ADV
ejpam-2272	399	2	there	there	PRON
ejpam-2272	399	3	exists	exist	VERB
ejpam-2272	399	4	an	an	DET
ejpam-2272	399	5	element	element	NOUN
ejpam-2272	399	6	m	m	NOUN
ejpam-2272	399	7	∈	∈	NOUN
ejpam-2272	399	8	i	i	PRON
ejpam-2272	399	9	such	such	ADJ
ejpam-2272	399	10	that	that	SCONJ
ejpam-2272	399	11	0	0	NUM
ejpam-2272	399	12	≠	≠	PROPN
ejpam-2272	399	13	abc	abc	NOUN
ejpam-2272	399	14	=	=	SYM
ejpam-2272	399	15	f	f	PROPN
ejpam-2272	399	16	(	(	PUNCT
ejpam-2272	399	17	m	m	PROPN
ejpam-2272	399	18	)	)	PUNCT
ejpam-2272	399	19	.	.	PUNCT
ejpam-2272	400	1	since	since	SCONJ
ejpam-2272	400	2	f	f	PROPN
ejpam-2272	400	3	is	be	AUX
ejpam-2272	400	4	an	an	DET
ejpam-2272	400	5	epimorphism	epimorphism	NOUN
ejpam-2272	400	6	,	,	PUNCT
ejpam-2272	400	7	therefore	therefore	ADV
ejpam-2272	400	8	there	there	PRON
ejpam-2272	400	9	exist	exist	VERB
ejpam-2272	400	10	p	p	PRON
ejpam-2272	400	11	,	,	PUNCT
ejpam-2272	400	12	q	q	INTJ
ejpam-2272	400	13	,	,	PUNCT
ejpam-2272	400	14	r	r	NOUN
ejpam-2272	400	15	∈	∈	PROPN
ejpam-2272	400	16	s	s	VERB
ejpam-2272	400	17	such	such	ADJ
ejpam-2272	400	18	that	that	SCONJ
ejpam-2272	400	19	f	f	PROPN
ejpam-2272	400	20	(	(	PUNCT
ejpam-2272	400	21	p	p	X
ejpam-2272	400	22	)	)	PUNCT
ejpam-2272	400	23	=	=	SYM
ejpam-2272	400	24	a	a	PROPN
ejpam-2272	400	25	,	,	PUNCT
ejpam-2272	400	26	f	f	PROPN
ejpam-2272	400	27	(	(	PUNCT
ejpam-2272	400	28	q	q	X
ejpam-2272	400	29	)	)	PUNCT
ejpam-2272	400	30	=	=	SYM
ejpam-2272	400	31	b	b	PROPN
ejpam-2272	400	32	,	,	PUNCT
ejpam-2272	400	33	f	f	PROPN
ejpam-2272	400	34	(	(	PUNCT
ejpam-2272	400	35	r	r	NOUN
ejpam-2272	400	36	)	)	PUNCT
ejpam-2272	400	37	=	=	SYM
ejpam-2272	400	38	c.	c.	NOUN
ejpam-2272	400	39	also	also	ADV
ejpam-2272	400	40	,	,	PUNCT
ejpam-2272	400	41	since	since	SCONJ
ejpam-2272	400	42	i	i	PRON
ejpam-2272	400	43	is	be	AUX
ejpam-2272	400	44	a	a	DET
ejpam-2272	400	45	strong	strong	ADJ
ejpam-2272	400	46	ideal	ideal	NOUN
ejpam-2272	400	47	of	of	ADP
ejpam-2272	400	48	s	s	PRON
ejpam-2272	400	49	and	and	CCONJ
ejpam-2272	400	50	m	m	PROPN
ejpam-2272	400	51	∈	∈	PROPN
ejpam-2272	401	1	i	i	PRON
ejpam-2272	401	2	,	,	PUNCT
ejpam-2272	401	3	therefore	therefore	ADV
ejpam-2272	401	4	there	there	PRON
ejpam-2272	401	5	exists	exist	VERB
ejpam-2272	401	6	n	n	PRON
ejpam-2272	401	7	∈	∈	NOUN
ejpam-2272	401	8	i	i	PRON
ejpam-2272	401	9	such	such	ADJ
ejpam-2272	401	10	that	that	SCONJ
ejpam-2272	401	11	m	m	VERB
ejpam-2272	401	12	+	+	CCONJ
ejpam-2272	401	13	n	n	CCONJ
ejpam-2272	401	14	=	=	SYM
ejpam-2272	401	15	0	0	PROPN
ejpam-2272	401	16	.	.	PUNCT
ejpam-2272	402	1	this	this	PRON
ejpam-2272	402	2	implies	imply	VERB
ejpam-2272	402	3	f	f	PROPN
ejpam-2272	402	4	(	(	PUNCT
ejpam-2272	402	5	n	n	PROPN
ejpam-2272	402	6	+	+	NUM
ejpam-2272	402	7	m	m	NOUN
ejpam-2272	402	8	)	)	PUNCT
ejpam-2272	402	9	=	=	SYM
ejpam-2272	402	10	0	0	NUM
ejpam-2272	402	11	,	,	PUNCT
ejpam-2272	402	12	that	that	ADV
ejpam-2272	402	13	is	is	ADV
ejpam-2272	402	14	,	,	PUNCT
ejpam-2272	402	15	f	f	PROPN
ejpam-2272	402	16	(	(	PUNCT
ejpam-2272	402	17	pqr	pqr	PROPN
ejpam-2272	402	18	+	+	CCONJ
ejpam-2272	402	19	n	n	CCONJ
ejpam-2272	402	20	)	)	PUNCT
ejpam-2272	402	21	=	=	SYM
ejpam-2272	402	22	0	0	NUM
ejpam-2272	402	23	,	,	PUNCT
ejpam-2272	402	24	implies	imply	VERB
ejpam-2272	402	25	pqr	pqr	PROPN
ejpam-2272	402	26	+	+	CCONJ
ejpam-2272	402	27	n	n	CCONJ
ejpam-2272	402	28	∈	∈	NOUN
ejpam-2272	402	29	ker	ker	NOUN
ejpam-2272	403	1	f	f	PROPN
ejpam-2272	403	2	⊆	⊆	NUM
ejpam-2272	403	3	i	i	PRON
ejpam-2272	403	4	.	.	PUNCT
ejpam-2272	404	1	so	so	ADV
ejpam-2272	404	2	,	,	PUNCT
ejpam-2272	404	3	0	0	NUM
ejpam-2272	404	4	≠	≠	PROPN
ejpam-2272	404	5	pqr	pqr	PROPN
ejpam-2272	404	6	∈	∈	PROPN
ejpam-2272	405	1	i	i	PRON
ejpam-2272	405	2	(	(	PUNCT
ejpam-2272	405	3	as	as	SCONJ
ejpam-2272	405	4	i	i	PRON
ejpam-2272	405	5	is	be	AUX
ejpam-2272	405	6	a	a	DET
ejpam-2272	405	7	subtractive	subtractive	NOUN
ejpam-2272	405	8	ideal	ideal	NOUN
ejpam-2272	405	9	of	of	ADP
ejpam-2272	405	10	s	s	NOUN
ejpam-2272	405	11	)	)	PUNCT
ejpam-2272	405	12	because	because	SCONJ
ejpam-2272	405	13	if	if	SCONJ
ejpam-2272	405	14	pqr	pqr	PROPN
ejpam-2272	405	15	=	=	NOUN
ejpam-2272	405	16	0	0	PROPN
ejpam-2272	405	17	,	,	PUNCT
ejpam-2272	405	18	then	then	ADV
ejpam-2272	405	19	f	f	PROPN
ejpam-2272	405	20	(	(	PUNCT
ejpam-2272	405	21	m	m	PROPN
ejpam-2272	405	22	)	)	PUNCT
ejpam-2272	405	23	=	=	SYM
ejpam-2272	405	24	0	0	NUM
ejpam-2272	405	25	,	,	PUNCT
ejpam-2272	405	26	a	a	DET
ejpam-2272	405	27	contradiction	contradiction	NOUN
ejpam-2272	405	28	.	.	PUNCT
ejpam-2272	406	1	since	since	SCONJ
ejpam-2272	406	2	i	i	PRON
ejpam-2272	406	3	is	be	AUX
ejpam-2272	406	4	a	a	DET
ejpam-2272	406	5	weakly	weakly	ADJ
ejpam-2272	406	6	2	2	NUM
ejpam-2272	406	7	-	-	PUNCT
ejpam-2272	406	8	absorbing	absorbing	ADJ
ejpam-2272	406	9	primary	primary	ADJ
ejpam-2272	406	10	ideal	ideal	NOUN
ejpam-2272	406	11	of	of	ADP
ejpam-2272	406	12	s	s	PROPN
ejpam-2272	406	13	,	,	PUNCT
ejpam-2272	406	14	therefore	therefore	ADV
ejpam-2272	406	15	either	either	CCONJ
ejpam-2272	406	16	pq	pq	INTJ
ejpam-2272	406	17	∈	∈	PROPN
ejpam-2272	407	1	i	i	PRON
ejpam-2272	407	2	or	or	CCONJ
ejpam-2272	407	3	qr	qr	ADP
ejpam-2272	407	4	∈	∈	PROPN
ejpam-2272	407	5	√	√	VERB
ejpam-2272	407	6	i	i	PRON
ejpam-2272	407	7	or	or	CCONJ
ejpam-2272	407	8	rp	rp	NOUN
ejpam-2272	407	9	∈	∈	NOUN
ejpam-2272	407	10	√	√	VERB
ejpam-2272	407	11	i	i	PRON
ejpam-2272	407	12	.	.	PUNCT
ejpam-2272	408	1	thus	thus	ADV
ejpam-2272	408	2	ab	ab	PROPN
ejpam-2272	408	3	∈	∈	PROPN
ejpam-2272	408	4	f	f	X
ejpam-2272	408	5	(	(	PUNCT
ejpam-2272	408	6	i	i	NOUN
ejpam-2272	408	7	)	)	PUNCT
ejpam-2272	408	8	or	or	CCONJ
ejpam-2272	408	9	bc	bc	PROPN
ejpam-2272	408	10	∈	∈	PROPN
ejpam-2272	408	11	f	f	PROPN
ejpam-2272	408	12	(	(	PUNCT
ejpam-2272	408	13	√	√	PROPN
ejpam-2272	408	14	i	i	PRON
ejpam-2272	408	15	)	)	PUNCT
ejpam-2272	409	1	⊆	⊆	NUM
ejpam-2272	409	2	√	√	NUM
ejpam-2272	409	3	f	f	NOUN
ejpam-2272	409	4	(	(	PUNCT
ejpam-2272	409	5	i	i	NOUN
ejpam-2272	409	6	)	)	PUNCT
ejpam-2272	409	7	or	or	CCONJ
ejpam-2272	409	8	ac	ac	PROPN
ejpam-2272	409	9	∈	∈	PROPN
ejpam-2272	409	10	f	f	PROPN
ejpam-2272	409	11	(	(	PUNCT
ejpam-2272	409	12	√	√	PROPN
ejpam-2272	409	13	i	i	PRON
ejpam-2272	409	14	)	)	PUNCT
ejpam-2272	410	1	⊆	⊆	NUM
ejpam-2272	410	2	√	√	NUM
ejpam-2272	410	3	f	f	NOUN
ejpam-2272	410	4	(	(	PUNCT
ejpam-2272	410	5	i	i	NOUN
ejpam-2272	410	6	)	)	PUNCT
ejpam-2272	410	7	.	.	PUNCT
ejpam-2272	411	1	hence	hence	ADV
ejpam-2272	411	2	,	,	PUNCT
ejpam-2272	411	3	f	f	PROPN
ejpam-2272	411	4	(	(	PUNCT
ejpam-2272	411	5	i	i	NOUN
ejpam-2272	411	6	)	)	PUNCT
ejpam-2272	411	7	is	be	AUX
ejpam-2272	411	8	a	a	DET
ejpam-2272	411	9	weakly	weakly	ADJ
ejpam-2272	411	10	2	2	NUM
ejpam-2272	411	11	-	-	PUNCT
ejpam-2272	411	12	absorbing	absorbing	ADJ
ejpam-2272	411	13	primary	primary	ADJ
ejpam-2272	411	14	ideal	ideal	NOUN
ejpam-2272	411	15	of	of	ADP
ejpam-2272	411	16	s′.	s′.	PROPN
ejpam-2272	411	17	(	(	PUNCT
ejpam-2272	411	18	2	2	X
ejpam-2272	411	19	)	)	PUNCT
ejpam-2272	411	20	it	it	PRON
ejpam-2272	411	21	follows	follow	VERB
ejpam-2272	411	22	from	from	ADP
ejpam-2272	411	23	(	(	PUNCT
ejpam-2272	411	24	1	1	NUM
ejpam-2272	411	25	)	)	PUNCT
ejpam-2272	411	26	.	.	PUNCT
ejpam-2272	412	1	proposition	proposition	NOUN
ejpam-2272	412	2	2	2	NUM
ejpam-2272	412	3	.	.	PUNCT
ejpam-2272	413	1	let	let	VERB
ejpam-2272	413	2	a	a	DET
ejpam-2272	413	3	,	,	PUNCT
ejpam-2272	413	4	x	x	SYM
ejpam-2272	413	5	∈	∈	PROPN
ejpam-2272	413	6	s.	s.	PROPN
ejpam-2272	413	7	then	then	ADV
ejpam-2272	413	8	the	the	DET
ejpam-2272	413	9	following	follow	VERB
ejpam-2272	413	10	holds	hold	VERB
ejpam-2272	413	11	:	:	PUNCT
ejpam-2272	413	12	(	(	PUNCT
ejpam-2272	413	13	1	1	X
ejpam-2272	413	14	)	)	PUNCT
ejpam-2272	413	15	suppose	suppose	VERB
ejpam-2272	413	16	sx	sx	PROPN
ejpam-2272	413	17	be	be	AUX
ejpam-2272	413	18	a	a	DET
ejpam-2272	413	19	subtractive	subtractive	NOUN
ejpam-2272	413	20	ideal	ideal	NOUN
ejpam-2272	413	21	s	s	PROPN
ejpam-2272	413	22	and	and	CCONJ
ejpam-2272	413	23	if	if	SCONJ
ejpam-2272	413	24	ann(x	ann(x	PROPN
ejpam-2272	413	25	)	)	PUNCT
ejpam-2272	413	26	⊆	⊆	NUM
ejpam-2272	413	27	sx	sx	PROPN
ejpam-2272	413	28	.	.	PUNCT
ejpam-2272	414	1	then	then	ADV
ejpam-2272	414	2	sx	sx	PROPN
ejpam-2272	414	3	is	be	AUX
ejpam-2272	414	4	a	a	DET
ejpam-2272	414	5	2	2	NUM
ejpam-2272	414	6	-	-	PUNCT
ejpam-2272	414	7	absorbing	absorbing	ADJ
ejpam-2272	414	8	primary	primary	ADJ
ejpam-2272	414	9	ideal	ideal	NOUN
ejpam-2272	414	10	of	of	ADP
ejpam-2272	414	11	s	s	PRON
ejpam-2272	414	12	if	if	SCONJ
ejpam-2272	415	1	and	and	CCONJ
ejpam-2272	415	2	only	only	ADV
ejpam-2272	415	3	if	if	SCONJ
ejpam-2272	415	4	sx	sx	PROPN
ejpam-2272	415	5	is	be	AUX
ejpam-2272	415	6	a	a	DET
ejpam-2272	415	7	weakly	weakly	ADJ
ejpam-2272	415	8	2	2	NUM
ejpam-2272	415	9	-	-	PUNCT
ejpam-2272	415	10	absorbing	absorbing	ADJ
ejpam-2272	415	11	primary	primary	ADJ
ejpam-2272	415	12	ideal	ideal	NOUN
ejpam-2272	415	13	of	of	ADP
ejpam-2272	415	14	s.	s.	PROPN
ejpam-2272	415	15	(	(	PUNCT
ejpam-2272	415	16	2	2	X
ejpam-2272	415	17	)	)	PUNCT
ejpam-2272	415	18	suppose	suppose	VERB
ejpam-2272	415	19	ai	ai	AUX
ejpam-2272	415	20	be	be	AUX
ejpam-2272	415	21	a	a	DET
ejpam-2272	415	22	subtractive	subtractive	NOUN
ejpam-2272	415	23	ideal	ideal	NOUN
ejpam-2272	415	24	s	s	PROPN
ejpam-2272	415	25	and	and	CCONJ
ejpam-2272	415	26	if	if	SCONJ
ejpam-2272	415	27	ann(a	ann(a	PROPN
ejpam-2272	415	28	)	)	PUNCT
ejpam-2272	415	29	⊆	⊆	NUM
ejpam-2272	415	30	ai	ai	NOUN
ejpam-2272	415	31	.	.	PUNCT
ejpam-2272	416	1	then	then	ADV
ejpam-2272	416	2	ai	ai	VERB
ejpam-2272	416	3	is	be	AUX
ejpam-2272	416	4	a	a	DET
ejpam-2272	416	5	2	2	NUM
ejpam-2272	416	6	-	-	PUNCT
ejpam-2272	416	7	absorbing	absorbing	ADJ
ejpam-2272	416	8	primary	primary	ADJ
ejpam-2272	416	9	ideal	ideal	NOUN
ejpam-2272	416	10	of	of	ADP
ejpam-2272	416	11	s	s	PRON
ejpam-2272	416	12	if	if	SCONJ
ejpam-2272	417	1	and	and	CCONJ
ejpam-2272	417	2	only	only	ADV
ejpam-2272	417	3	if	if	SCONJ
ejpam-2272	417	4	it	it	PRON
ejpam-2272	417	5	is	be	AUX
ejpam-2272	417	6	a	a	DET
ejpam-2272	417	7	weakly	weakly	ADJ
ejpam-2272	417	8	2	2	NUM
ejpam-2272	417	9	-	-	PUNCT
ejpam-2272	417	10	absorbing	absorbing	ADJ
ejpam-2272	417	11	primary	primary	ADJ
ejpam-2272	417	12	ideal	ideal	NOUN
ejpam-2272	417	13	of	of	ADP
ejpam-2272	417	14	s.	s.	PROPN
ejpam-2272	417	15	p.	p.	PROPN
ejpam-2272	417	16	kumar	kumar	PROPN
ejpam-2272	417	17	,	,	PUNCT
ejpam-2272	417	18	m.	m.	NOUN
ejpam-2272	417	19	dubey	dubey	PROPN
ejpam-2272	417	20	,	,	PUNCT
ejpam-2272	417	21	p.	p.	NOUN
ejpam-2272	417	22	sarohe	sarohe	PROPN
ejpam-2272	417	23	/	/	SYM
ejpam-2272	417	24	eur	eur	PROPN
ejpam-2272	417	25	.	.	PUNCT
ejpam-2272	418	1	j.	j.	PROPN
ejpam-2272	418	2	pure	pure	PROPN
ejpam-2272	418	3	appl	appl	PROPN
ejpam-2272	418	4	.	.	PROPN
ejpam-2272	418	5	math	math	PROPN
ejpam-2272	418	6	,	,	PUNCT
ejpam-2272	418	7	9	9	NUM
ejpam-2272	418	8	(	(	PUNCT
ejpam-2272	418	9	2016	2016	NUM
ejpam-2272	418	10	)	)	PUNCT
ejpam-2272	418	11	,	,	PUNCT
ejpam-2272	418	12	186	186	NUM
ejpam-2272	418	13	-	-	SYM
ejpam-2272	418	14	195	195	NUM
ejpam-2272	418	15	193	193	NUM
ejpam-2272	418	16	proof	proof	NOUN
ejpam-2272	418	17	.	.	PUNCT
ejpam-2272	419	1	(	(	PUNCT
ejpam-2272	419	2	1	1	X
ejpam-2272	419	3	)	)	PUNCT
ejpam-2272	419	4	let	let	VERB
ejpam-2272	419	5	sx	sx	PROPN
ejpam-2272	419	6	be	be	AUX
ejpam-2272	419	7	a	a	DET
ejpam-2272	419	8	weakly	weakly	ADJ
ejpam-2272	419	9	2	2	NUM
ejpam-2272	419	10	-	-	PUNCT
ejpam-2272	419	11	absorbing	absorbing	ADJ
ejpam-2272	419	12	primary	primary	ADJ
ejpam-2272	419	13	ideal	ideal	NOUN
ejpam-2272	419	14	of	of	ADP
ejpam-2272	419	15	s	s	PRON
ejpam-2272	419	16	and	and	CCONJ
ejpam-2272	419	17	r	r	NOUN
ejpam-2272	419	18	,	,	PUNCT
ejpam-2272	419	19	s	s	PROPN
ejpam-2272	419	20	,	,	PUNCT
ejpam-2272	419	21	t	t	PROPN
ejpam-2272	419	22	∈	∈	PROPN
ejpam-2272	419	23	s	s	PART
ejpam-2272	419	24	with	with	ADP
ejpam-2272	419	25	rst	rst	PROPN
ejpam-2272	419	26	∈	∈	PROPN
ejpam-2272	419	27	sx	sx	PROPN
ejpam-2272	419	28	.	.	PUNCT
ejpam-2272	420	1	if	if	SCONJ
ejpam-2272	420	2	rst	rst	PROPN
ejpam-2272	420	3	≠	≠	PROPN
ejpam-2272	420	4	0	0	NUM
ejpam-2272	420	5	,	,	PUNCT
ejpam-2272	420	6	then	then	ADV
ejpam-2272	420	7	rs	rs	PROPN
ejpam-2272	420	8	∈	∈	PROPN
ejpam-2272	420	9	sx	sx	PROPN
ejpam-2272	420	10	or	or	CCONJ
ejpam-2272	420	11	r	r	NOUN
ejpam-2272	420	12	t	t	NOUN
ejpam-2272	420	13	∈	∈	NOUN
ejpam-2272	420	14	√	√	PROPN
ejpam-2272	420	15	sx	sx	PROPN
ejpam-2272	420	16	or	or	CCONJ
ejpam-2272	420	17	st	st	PROPN
ejpam-2272	420	18	∈	∈	PROPN
ejpam-2272	420	19	√	√	PROPN
ejpam-2272	420	20	sx	sx	PROPN
ejpam-2272	420	21	,	,	PUNCT
ejpam-2272	420	22	which	which	PRON
ejpam-2272	420	23	implies	imply	VERB
ejpam-2272	420	24	sx	sx	PROPN
ejpam-2272	420	25	is	be	AUX
ejpam-2272	420	26	a	a	DET
ejpam-2272	420	27	2	2	NUM
ejpam-2272	420	28	-	-	PUNCT
ejpam-2272	420	29	absorbing	absorbing	ADJ
ejpam-2272	420	30	primary	primary	ADJ
ejpam-2272	420	31	ideal	ideal	NOUN
ejpam-2272	420	32	of	of	ADP
ejpam-2272	420	33	s.	s.	PROPN
ejpam-2272	421	1	so	so	SCONJ
ejpam-2272	421	2	we	we	PRON
ejpam-2272	421	3	assume	assume	VERB
ejpam-2272	421	4	that	that	SCONJ
ejpam-2272	421	5	rst	rst	PRON
ejpam-2272	421	6	=	=	SYM
ejpam-2272	421	7	0	0	X
ejpam-2272	421	8	.	.	PUNCT
ejpam-2272	422	1	evidently	evidently	ADV
ejpam-2272	422	2	,	,	PUNCT
ejpam-2272	422	3	rs(x	rs(x	PUNCT
ejpam-2272	422	4	+	+	CCONJ
ejpam-2272	422	5	t	t	X
ejpam-2272	422	6	)	)	PUNCT
ejpam-2272	422	7	∈	∈	PROPN
ejpam-2272	422	8	sx	sx	PROPN
ejpam-2272	422	9	.	.	PUNCT
ejpam-2272	423	1	if	if	SCONJ
ejpam-2272	423	2	rs(x	rs(x	PRON
ejpam-2272	423	3	+	+	SYM
ejpam-2272	423	4	t	t	X
ejpam-2272	423	5	)	)	PUNCT
ejpam-2272	423	6	≠	≠	PROPN
ejpam-2272	423	7	0	0	NUM
ejpam-2272	423	8	,	,	PUNCT
ejpam-2272	423	9	we	we	PRON
ejpam-2272	423	10	have	have	VERB
ejpam-2272	423	11	rs	rs	PROPN
ejpam-2272	423	12	∈	∈	PROPN
ejpam-2272	423	13	sx	sx	NOUN
ejpam-2272	423	14	or	or	CCONJ
ejpam-2272	423	15	r(x	r(x	PROPN
ejpam-2272	423	16	+	+	CCONJ
ejpam-2272	423	17	t	t	PROPN
ejpam-2272	423	18	)	)	PUNCT
ejpam-2272	423	19	∈	∈	PROPN
ejpam-2272	423	20	√	√	NUM
ejpam-2272	423	21	sx	sx	PROPN
ejpam-2272	423	22	or	or	CCONJ
ejpam-2272	423	23	s(x	s(x	PROPN
ejpam-2272	423	24	+	+	CCONJ
ejpam-2272	423	25	t	t	PROPN
ejpam-2272	423	26	)	)	PUNCT
ejpam-2272	423	27	∈	∈	PROPN
ejpam-2272	423	28	√	√	NUM
ejpam-2272	423	29	sx	sx	PROPN
ejpam-2272	423	30	,	,	PUNCT
ejpam-2272	423	31	as	as	SCONJ
ejpam-2272	423	32	sx	sx	PROPN
ejpam-2272	423	33	is	be	AUX
ejpam-2272	423	34	a	a	DET
ejpam-2272	423	35	weakly	weakly	ADJ
ejpam-2272	423	36	2	2	NUM
ejpam-2272	423	37	-	-	PUNCT
ejpam-2272	423	38	absorbing	absorbing	ADJ
ejpam-2272	423	39	primary	primary	ADJ
ejpam-2272	423	40	ideal	ideal	NOUN
ejpam-2272	423	41	of	of	ADP
ejpam-2272	423	42	s.	s.	PROPN
ejpam-2272	423	43	by	by	ADP
ejpam-2272	423	44	lemma	lemma	PROPN
ejpam-2272	423	45	2	2	NUM
ejpam-2272	423	46	,	,	PUNCT
ejpam-2272	423	47	we	we	PRON
ejpam-2272	423	48	have	have	VERB
ejpam-2272	423	49	either	either	CCONJ
ejpam-2272	423	50	rs	rs	PROPN
ejpam-2272	423	51	∈	∈	PROPN
ejpam-2272	423	52	sx	sx	NOUN
ejpam-2272	423	53	or	or	CCONJ
ejpam-2272	423	54	r	r	NOUN
ejpam-2272	423	55	t	t	NOUN
ejpam-2272	423	56	∈	∈	NOUN
ejpam-2272	423	57	√	√	PROPN
ejpam-2272	423	58	sx	sx	PROPN
ejpam-2272	423	59	or	or	CCONJ
ejpam-2272	423	60	st	st	PROPN
ejpam-2272	423	61	∈	∈	PROPN
ejpam-2272	423	62	√	√	PROPN
ejpam-2272	423	63	sx	sx	PROPN
ejpam-2272	423	64	.	.	PUNCT
ejpam-2272	424	1	therefore	therefore	ADV
ejpam-2272	424	2	,	,	PUNCT
ejpam-2272	424	3	we	we	PRON
ejpam-2272	424	4	have	have	AUX
ejpam-2272	424	5	rs(x	rs(x	NUM
ejpam-2272	424	6	+	+	CCONJ
ejpam-2272	424	7	t	t	X
ejpam-2272	424	8	)	)	PUNCT
ejpam-2272	424	9	=	=	SYM
ejpam-2272	424	10	0	0	NUM
ejpam-2272	424	11	implies	imply	VERB
ejpam-2272	424	12	rsx	rsx	PROPN
ejpam-2272	424	13	=	=	SYM
ejpam-2272	424	14	0	0	NUM
ejpam-2272	424	15	and	and	CCONJ
ejpam-2272	424	16	so	so	ADV
ejpam-2272	424	17	rs	rs	PROPN
ejpam-2272	424	18	∈	∈	PROPN
ejpam-2272	424	19	ann(x	ann(x	PROPN
ejpam-2272	424	20	)	)	PUNCT
ejpam-2272	424	21	⊆	⊆	NUM
ejpam-2272	424	22	sx	sx	NOUN
ejpam-2272	424	23	and	and	CCONJ
ejpam-2272	424	24	thus	thus	ADV
ejpam-2272	424	25	rs	rs	ADP
ejpam-2272	424	26	∈	∈	PROPN
ejpam-2272	424	27	sx	sx	PROPN
ejpam-2272	424	28	.	.	PUNCT
ejpam-2272	425	1	hence	hence	ADV
ejpam-2272	425	2	sx	sx	PROPN
ejpam-2272	425	3	is	be	AUX
ejpam-2272	425	4	a	a	DET
ejpam-2272	425	5	2	2	NUM
ejpam-2272	425	6	-	-	PUNCT
ejpam-2272	425	7	absorbing	absorbing	ADJ
ejpam-2272	425	8	primary	primary	ADJ
ejpam-2272	425	9	ideal	ideal	NOUN
ejpam-2272	425	10	of	of	ADP
ejpam-2272	425	11	s.	s.	PROPN
ejpam-2272	425	12	(	(	PUNCT
ejpam-2272	425	13	2	2	X
ejpam-2272	425	14	)	)	PUNCT
ejpam-2272	425	15	let	let	VERB
ejpam-2272	425	16	ai	ai	AUX
ejpam-2272	425	17	be	be	AUX
ejpam-2272	425	18	a	a	DET
ejpam-2272	425	19	weakly	weakly	ADJ
ejpam-2272	425	20	2	2	NUM
ejpam-2272	425	21	-	-	PUNCT
ejpam-2272	425	22	absorbing	absorbing	ADJ
ejpam-2272	425	23	primary	primary	ADJ
ejpam-2272	425	24	ideal	ideal	NOUN
ejpam-2272	425	25	and	and	CCONJ
ejpam-2272	425	26	r	r	NOUN
ejpam-2272	425	27	,	,	PUNCT
ejpam-2272	425	28	s	s	PROPN
ejpam-2272	425	29	,	,	PUNCT
ejpam-2272	425	30	t	t	PROPN
ejpam-2272	425	31	∈	∈	PROPN
ejpam-2272	425	32	s	s	VERB
ejpam-2272	425	33	such	such	ADJ
ejpam-2272	425	34	that	that	SCONJ
ejpam-2272	425	35	rst	rst	PROPN
ejpam-2272	425	36	∈	∈	PROPN
ejpam-2272	425	37	ai	ai	VERB
ejpam-2272	425	38	.	.	PUNCT
ejpam-2272	426	1	if	if	SCONJ
ejpam-2272	426	2	rst	rst	PROPN
ejpam-2272	426	3	≠	≠	PROPN
ejpam-2272	426	4	0	0	NUM
ejpam-2272	427	1	then	then	ADV
ejpam-2272	427	2	rs	rs	PROPN
ejpam-2272	427	3	∈	∈	PROPN
ejpam-2272	427	4	ai	ai	VERB
ejpam-2272	427	5	or	or	CCONJ
ejpam-2272	427	6	r	r	NOUN
ejpam-2272	427	7	t	t	NOUN
ejpam-2272	427	8	∈	∈	NOUN
ejpam-2272	427	9	√	√	VERB
ejpam-2272	427	10	ai	ai	VERB
ejpam-2272	427	11	or	or	CCONJ
ejpam-2272	427	12	st	st	PROPN
ejpam-2272	427	13	∈	∈	PROPN
ejpam-2272	427	14	√	√	NUM
ejpam-2272	427	15	ai	ai	VERB
ejpam-2272	427	16	,	,	PUNCT
ejpam-2272	427	17	which	which	PRON
ejpam-2272	427	18	implies	imply	VERB
ejpam-2272	427	19	ai	ai	VERB
ejpam-2272	427	20	is	be	AUX
ejpam-2272	427	21	a	a	DET
ejpam-2272	427	22	2	2	NUM
ejpam-2272	427	23	-	-	PUNCT
ejpam-2272	427	24	absorbing	absorbing	ADJ
ejpam-2272	427	25	primary	primary	ADJ
ejpam-2272	427	26	ideal	ideal	NOUN
ejpam-2272	427	27	of	of	ADP
ejpam-2272	427	28	s.	s.	PROPN
ejpam-2272	427	29	so	so	ADV
ejpam-2272	427	30	,	,	PUNCT
ejpam-2272	427	31	we	we	PRON
ejpam-2272	427	32	assume	assume	VERB
ejpam-2272	427	33	rst	rst	PROPN
ejpam-2272	427	34	=	=	SYM
ejpam-2272	427	35	0	0	X
ejpam-2272	427	36	.	.	PUNCT
ejpam-2272	428	1	clearly	clearly	ADV
ejpam-2272	428	2	,	,	PUNCT
ejpam-2272	428	3	r(s	r(s	PROPN
ejpam-2272	428	4	+	+	NOUN
ejpam-2272	428	5	a)t	a)t	X
ejpam-2272	428	6	=	=	SYM
ejpam-2272	428	7	rst	rst	X
ejpam-2272	428	8	+	+	CCONJ
ejpam-2272	428	9	rat	rat	NOUN
ejpam-2272	428	10	∈	∈	NOUN
ejpam-2272	428	11	ai	ai	VERB
ejpam-2272	428	12	.	.	PUNCT
ejpam-2272	429	1	if	if	SCONJ
ejpam-2272	429	2	r(s	r(s	PROPN
ejpam-2272	429	3	+	+	CCONJ
ejpam-2272	429	4	a)t	a)t	ADJ
ejpam-2272	429	5	≠	≠	PROPN
ejpam-2272	429	6	0	0	NUM
ejpam-2272	429	7	,	,	PUNCT
ejpam-2272	429	8	then	then	ADV
ejpam-2272	429	9	r(s	r(s	PROPN
ejpam-2272	429	10	+	+	CCONJ
ejpam-2272	429	11	a	a	X
ejpam-2272	429	12	)	)	PUNCT
ejpam-2272	429	13	∈	∈	NOUN
ejpam-2272	429	14	ai	ai	VERB
ejpam-2272	429	15	or	or	CCONJ
ejpam-2272	429	16	r	r	NOUN
ejpam-2272	429	17	t	t	NOUN
ejpam-2272	429	18	∈	∈	NOUN
ejpam-2272	429	19	√	√	AUX
ejpam-2272	429	20	ai	ai	VERB
ejpam-2272	429	21	or	or	CCONJ
ejpam-2272	429	22	(	(	PUNCT
ejpam-2272	429	23	s	s	PART
ejpam-2272	429	24	+	+	NOUN
ejpam-2272	429	25	a)t	a)t	ADJ
ejpam-2272	429	26	∈	∈	NOUN
ejpam-2272	429	27	√	√	NUM
ejpam-2272	429	28	ai	ai	VERB
ejpam-2272	429	29	.	.	PUNCT
ejpam-2272	430	1	by	by	ADP
ejpam-2272	430	2	lemma	lemma	PROPN
ejpam-2272	430	3	2	2	NUM
ejpam-2272	430	4	,	,	PUNCT
ejpam-2272	430	5	we	we	PRON
ejpam-2272	430	6	get	get	VERB
ejpam-2272	430	7	either	either	CCONJ
ejpam-2272	430	8	rs	rs	PROPN
ejpam-2272	430	9	∈	∈	PROPN
ejpam-2272	430	10	ai	ai	VERB
ejpam-2272	430	11	or	or	CCONJ
ejpam-2272	430	12	r	r	NOUN
ejpam-2272	430	13	t	t	NOUN
ejpam-2272	430	14	∈	∈	NOUN
ejpam-2272	431	1	√	√	VERB
ejpam-2272	431	2	ai	ai	VERB
ejpam-2272	431	3	or	or	CCONJ
ejpam-2272	431	4	st	st	PROPN
ejpam-2272	431	5	∈	∈	PROPN
ejpam-2272	431	6	√	√	AUX
ejpam-2272	431	7	ai	ai	VERB
ejpam-2272	431	8	.	.	PUNCT
ejpam-2272	432	1	so	so	ADV
ejpam-2272	432	2	,	,	PUNCT
ejpam-2272	432	3	we	we	PRON
ejpam-2272	432	4	assume	assume	VERB
ejpam-2272	432	5	that	that	SCONJ
ejpam-2272	432	6	r(s	r(s	PROPN
ejpam-2272	432	7	+	+	NOUN
ejpam-2272	432	8	a)t	a)t	ADJ
ejpam-2272	432	9	=	=	SYM
ejpam-2272	432	10	0	0	NUM
ejpam-2272	432	11	implies	imply	VERB
ejpam-2272	432	12	rat	rat	NOUN
ejpam-2272	432	13	=	=	SYM
ejpam-2272	432	14	0	0	NUM
ejpam-2272	432	15	,	,	PUNCT
ejpam-2272	432	16	as	as	ADP
ejpam-2272	432	17	rst	rst	PROPN
ejpam-2272	432	18	=	=	SYM
ejpam-2272	432	19	0	0	X
ejpam-2272	432	20	.	.	PUNCT
ejpam-2272	433	1	hence	hence	ADV
ejpam-2272	433	2	r	r	NOUN
ejpam-2272	433	3	t	t	X
ejpam-2272	433	4	∈	∈	PROPN
ejpam-2272	433	5	ann(a	ann(a	PROPN
ejpam-2272	433	6	)	)	PUNCT
ejpam-2272	433	7	⊆	⊆	NUM
ejpam-2272	433	8	ai	ai	NOUN
ejpam-2272	433	9	.	.	PUNCT
ejpam-2272	434	1	thus	thus	ADV
ejpam-2272	434	2	r	r	NOUN
ejpam-2272	434	3	t	t	NOUN
ejpam-2272	434	4	∈	∈	NOUN
ejpam-2272	434	5	ai	ai	VERB
ejpam-2272	434	6	and	and	CCONJ
ejpam-2272	434	7	hence	hence	ADV
ejpam-2272	434	8	ai	ai	VERB
ejpam-2272	434	9	is	be	AUX
ejpam-2272	434	10	a	a	DET
ejpam-2272	434	11	2	2	NUM
ejpam-2272	434	12	-	-	PUNCT
ejpam-2272	434	13	absorbing	absorbing	ADJ
ejpam-2272	434	14	primary	primary	ADJ
ejpam-2272	434	15	ideal	ideal	NOUN
ejpam-2272	434	16	of	of	ADP
ejpam-2272	434	17	s.	s.	PROPN
ejpam-2272	434	18	consider	consider	VERB
ejpam-2272	434	19	s	s	PART
ejpam-2272	434	20	=	=	PUNCT
ejpam-2272	434	21	s1	s1	PROPN
ejpam-2272	434	22	×	×	NOUN
ejpam-2272	434	23	s2	s2	NOUN
ejpam-2272	434	24	where	where	SCONJ
ejpam-2272	434	25	each	each	DET
ejpam-2272	434	26	si	si	X
ejpam-2272	434	27	,	,	PUNCT
ejpam-2272	434	28	i	i	PRON
ejpam-2272	434	29	=	=	NOUN
ejpam-2272	434	30	1	1	NUM
ejpam-2272	434	31	,	,	PUNCT
ejpam-2272	434	32	2	2	NUM
ejpam-2272	434	33	is	be	AUX
ejpam-2272	434	34	a	a	DET
ejpam-2272	434	35	commutative	commutative	ADJ
ejpam-2272	434	36	semiring	semiring	NOUN
ejpam-2272	434	37	with	with	ADP
ejpam-2272	434	38	unity	unity	NOUN
ejpam-2272	434	39	and	and	CCONJ
ejpam-2272	434	40	(	(	PUNCT
ejpam-2272	434	41	a1	a1	NOUN
ejpam-2272	434	42	,	,	PUNCT
ejpam-2272	434	43	a2)(b1	a2)(b1	NOUN
ejpam-2272	434	44	,	,	PUNCT
ejpam-2272	434	45	b2	b2	NOUN
ejpam-2272	434	46	)	)	PUNCT
ejpam-2272	434	47	=	=	PUNCT
ejpam-2272	435	1	(	(	PUNCT
ejpam-2272	435	2	a1	a1	NOUN
ejpam-2272	435	3	b1	b1	PROPN
ejpam-2272	435	4	,	,	PUNCT
ejpam-2272	435	5	a2	a2	PROPN
ejpam-2272	435	6	b2	b2	NOUN
ejpam-2272	435	7	)	)	PUNCT
ejpam-2272	435	8	for	for	ADP
ejpam-2272	435	9	all	all	DET
ejpam-2272	435	10	a1	a1	NOUN
ejpam-2272	435	11	,	,	PUNCT
ejpam-2272	435	12	b1	b1	NOUN
ejpam-2272	435	13	∈	∈	PROPN
ejpam-2272	435	14	s1	s1	NOUN
ejpam-2272	435	15	and	and	CCONJ
ejpam-2272	435	16	a2	a2	PROPN
ejpam-2272	435	17	,	,	PUNCT
ejpam-2272	435	18	b2	b2	NOUN
ejpam-2272	435	19	∈	∈	PROPN
ejpam-2272	435	20	s2	s2	PROPN
ejpam-2272	435	21	.	.	PUNCT
ejpam-2272	436	1	proposition	proposition	NOUN
ejpam-2272	436	2	3	3	NUM
ejpam-2272	436	3	.	.	PUNCT
ejpam-2272	437	1	let	let	VERB
ejpam-2272	437	2	i	i	PRON
ejpam-2272	437	3	be	be	AUX
ejpam-2272	437	4	a	a	DET
ejpam-2272	437	5	proper	proper	ADJ
ejpam-2272	437	6	ideal	ideal	NOUN
ejpam-2272	437	7	of	of	ADP
ejpam-2272	437	8	a	a	DET
ejpam-2272	437	9	semiring	semiring	NOUN
ejpam-2272	437	10	s1	s1	NOUN
ejpam-2272	437	11	.	.	PUNCT
ejpam-2272	438	1	then	then	ADV
ejpam-2272	438	2	the	the	DET
ejpam-2272	438	3	following	follow	VERB
ejpam-2272	438	4	statements	statement	NOUN
ejpam-2272	438	5	are	be	AUX
ejpam-2272	438	6	equivalent	equivalent	ADJ
ejpam-2272	438	7	:	:	PUNCT
ejpam-2272	438	8	(	(	PUNCT
ejpam-2272	438	9	1	1	X
ejpam-2272	438	10	)	)	PUNCT
ejpam-2272	438	11	i	i	PRON
ejpam-2272	438	12	is	be	AUX
ejpam-2272	438	13	a	a	DET
ejpam-2272	438	14	2	2	NUM
ejpam-2272	438	15	-	-	PUNCT
ejpam-2272	438	16	absorbing	absorbing	ADJ
ejpam-2272	438	17	primary	primary	ADJ
ejpam-2272	438	18	ideal	ideal	NOUN
ejpam-2272	438	19	of	of	ADP
ejpam-2272	438	20	s1	s1	PROPN
ejpam-2272	438	21	.	.	PUNCT
ejpam-2272	439	1	(	(	PUNCT
ejpam-2272	439	2	2	2	X
ejpam-2272	439	3	)	)	PUNCT
ejpam-2272	439	4	i	i	PRON
ejpam-2272	439	5	×	×	PROPN
ejpam-2272	439	6	s2	s2	NOUN
ejpam-2272	439	7	is	be	AUX
ejpam-2272	439	8	a	a	DET
ejpam-2272	439	9	2	2	NUM
ejpam-2272	439	10	-	-	PUNCT
ejpam-2272	439	11	absorbing	absorbing	ADJ
ejpam-2272	439	12	primary	primary	ADJ
ejpam-2272	439	13	ideal	ideal	NOUN
ejpam-2272	439	14	of	of	ADP
ejpam-2272	439	15	s	s	NOUN
ejpam-2272	439	16	=	=	NOUN
ejpam-2272	439	17	s1	s1	PROPN
ejpam-2272	439	18	×	×	PROPN
ejpam-2272	439	19	s2	s2	PROPN
ejpam-2272	439	20	.	.	PUNCT
ejpam-2272	440	1	(	(	PUNCT
ejpam-2272	440	2	3	3	X
ejpam-2272	440	3	)	)	PUNCT
ejpam-2272	440	4	i	i	PRON
ejpam-2272	440	5	×	×	PROPN
ejpam-2272	440	6	s2	s2	NOUN
ejpam-2272	440	7	is	be	AUX
ejpam-2272	440	8	a	a	DET
ejpam-2272	440	9	weakly	weakly	ADJ
ejpam-2272	440	10	2	2	NUM
ejpam-2272	440	11	-	-	PUNCT
ejpam-2272	440	12	absorbing	absorbing	ADJ
ejpam-2272	440	13	primary	primary	ADJ
ejpam-2272	440	14	ideal	ideal	NOUN
ejpam-2272	440	15	of	of	ADP
ejpam-2272	440	16	s	s	NOUN
ejpam-2272	440	17	=	=	NOUN
ejpam-2272	440	18	s1	s1	PROPN
ejpam-2272	440	19	×	×	PROPN
ejpam-2272	440	20	s2	s2	PROPN
ejpam-2272	440	21	.	.	PUNCT
ejpam-2272	441	1	proof	proof	NOUN
ejpam-2272	441	2	.	.	PUNCT
ejpam-2272	442	1	(	(	PUNCT
ejpam-2272	442	2	1	1	X
ejpam-2272	442	3	)	)	PUNCT
ejpam-2272	442	4	⇒	⇒	NOUN
ejpam-2272	442	5	(	(	PUNCT
ejpam-2272	442	6	2	2	X
ejpam-2272	442	7	)	)	PUNCT
ejpam-2272	442	8	let	let	NOUN
ejpam-2272	442	9	(	(	PUNCT
ejpam-2272	442	10	a1	a1	NOUN
ejpam-2272	442	11	,	,	PUNCT
ejpam-2272	442	12	a2	a2	PROPN
ejpam-2272	442	13	)	)	PUNCT
ejpam-2272	442	14	,	,	PUNCT
ejpam-2272	442	15	(	(	PUNCT
ejpam-2272	442	16	b1	b1	NOUN
ejpam-2272	442	17	,	,	PUNCT
ejpam-2272	442	18	b2	b2	NOUN
ejpam-2272	442	19	)	)	PUNCT
ejpam-2272	442	20	,	,	PUNCT
ejpam-2272	442	21	(	(	PUNCT
ejpam-2272	442	22	c1	c1	PROPN
ejpam-2272	442	23	,	,	PUNCT
ejpam-2272	442	24	c2	c2	PROPN
ejpam-2272	442	25	)	)	PUNCT
ejpam-2272	442	26	∈	∈	PROPN
ejpam-2272	442	27	s	s	AUX
ejpam-2272	442	28	be	be	AUX
ejpam-2272	442	29	such	such	ADJ
ejpam-2272	442	30	that	that	SCONJ
ejpam-2272	442	31	(	(	PUNCT
ejpam-2272	442	32	a1	a1	NOUN
ejpam-2272	442	33	,	,	PUNCT
ejpam-2272	442	34	a2)(b1	a2)(b1	NOUN
ejpam-2272	442	35	,	,	PUNCT
ejpam-2272	442	36	b2)(c1	b2)(c1	PROPN
ejpam-2272	442	37	,	,	PUNCT
ejpam-2272	442	38	c2	c2	PROPN
ejpam-2272	442	39	)	)	PUNCT
ejpam-2272	442	40	∈	∈	PROPN
ejpam-2272	443	1	i	i	PRON
ejpam-2272	443	2	×	×	PROPN
ejpam-2272	443	3	s2	s2	PROPN
ejpam-2272	443	4	.	.	PUNCT
ejpam-2272	444	1	then	then	ADV
ejpam-2272	444	2	(	(	PUNCT
ejpam-2272	444	3	a1	a1	NOUN
ejpam-2272	444	4	b1c1	b1c1	X
ejpam-2272	444	5	,	,	PUNCT
ejpam-2272	444	6	a2	a2	PROPN
ejpam-2272	444	7	b2c2	b2c2	NOUN
ejpam-2272	444	8	)	)	PUNCT
ejpam-2272	444	9	∈	∈	PROPN
ejpam-2272	444	10	i	i	PRON
ejpam-2272	444	11	×	×	PROPN
ejpam-2272	444	12	s2	s2	PROPN
ejpam-2272	444	13	implies	imply	VERB
ejpam-2272	444	14	a1	a1	NOUN
ejpam-2272	444	15	b1c1	b1c1	X
ejpam-2272	444	16	∈	∈	PROPN
ejpam-2272	444	17	i	i	PRON
ejpam-2272	444	18	.	.	PUNCT
ejpam-2272	445	1	this	this	PRON
ejpam-2272	445	2	gives	give	VERB
ejpam-2272	445	3	either	either	CCONJ
ejpam-2272	445	4	a1	a1	NOUN
ejpam-2272	445	5	b1	b1	NOUN
ejpam-2272	445	6	∈	∈	PROPN
ejpam-2272	446	1	i	i	PRON
ejpam-2272	446	2	or	or	CCONJ
ejpam-2272	446	3	(	(	PUNCT
ejpam-2272	446	4	b1c1)m	b1c1)m	PROPN
ejpam-2272	446	5	∈	∈	PROPN
ejpam-2272	446	6	i	i	PRON
ejpam-2272	446	7	or	or	CCONJ
ejpam-2272	446	8	(	(	PUNCT
ejpam-2272	446	9	a1c1)n	a1c1)n	PROPN
ejpam-2272	446	10	∈	∈	PROPN
ejpam-2272	447	1	i	i	PRON
ejpam-2272	447	2	for	for	ADP
ejpam-2272	447	3	some	some	DET
ejpam-2272	447	4	positive	positive	ADJ
ejpam-2272	447	5	integers	integer	NOUN
ejpam-2272	447	6	m	m	PRON
ejpam-2272	447	7	,	,	PUNCT
ejpam-2272	447	8	n	n	CCONJ
ejpam-2272	447	9	,	,	PUNCT
ejpam-2272	447	10	since	since	SCONJ
ejpam-2272	447	11	i	i	PRON
ejpam-2272	447	12	is	be	AUX
ejpam-2272	447	13	a	a	DET
ejpam-2272	447	14	2absorbing	2absorbing	NUM
ejpam-2272	447	15	primary	primary	ADJ
ejpam-2272	447	16	ideal	ideal	NOUN
ejpam-2272	447	17	of	of	ADP
ejpam-2272	447	18	s1	s1	NOUN
ejpam-2272	447	19	.	.	PUNCT
ejpam-2272	448	1	if	if	SCONJ
ejpam-2272	448	2	a1	a1	NOUN
ejpam-2272	448	3	b1	b1	NOUN
ejpam-2272	448	4	∈	∈	PROPN
ejpam-2272	448	5	i	i	PRON
ejpam-2272	448	6	,	,	PUNCT
ejpam-2272	448	7	then	then	ADV
ejpam-2272	448	8	(	(	PUNCT
ejpam-2272	448	9	a1	a1	NOUN
ejpam-2272	448	10	,	,	PUNCT
ejpam-2272	448	11	a2)(b1	a2)(b1	NOUN
ejpam-2272	448	12	,	,	PUNCT
ejpam-2272	448	13	b2	b2	NOUN
ejpam-2272	448	14	)	)	PUNCT
ejpam-2272	448	15	∈	∈	PROPN
ejpam-2272	449	1	i	i	PRON
ejpam-2272	449	2	×	×	PROPN
ejpam-2272	449	3	s2	s2	PROPN
ejpam-2272	449	4	.	.	PUNCT
ejpam-2272	450	1	if	if	SCONJ
ejpam-2272	450	2	bm	bm	PROPN
ejpam-2272	450	3	1	1	NUM
ejpam-2272	450	4	cm	cm	NOUN
ejpam-2272	450	5	1	1	NUM
ejpam-2272	450	6	∈	∈	NOUN
ejpam-2272	450	7	i	i	PRON
ejpam-2272	450	8	for	for	ADP
ejpam-2272	450	9	some	some	DET
ejpam-2272	450	10	positive	positive	ADJ
ejpam-2272	450	11	integer	integer	NOUN
ejpam-2272	450	12	m	m	NOUN
ejpam-2272	450	13	,	,	PUNCT
ejpam-2272	450	14	then	then	ADV
ejpam-2272	450	15	(	(	PUNCT
ejpam-2272	450	16	bm	bm	PROPN
ejpam-2272	450	17	1	1	NUM
ejpam-2272	450	18	,	,	PUNCT
ejpam-2272	450	19	bm	bm	PROPN
ejpam-2272	450	20	2	2	NUM
ejpam-2272	450	21	)	)	PUNCT
ejpam-2272	450	22	(	(	PUNCT
ejpam-2272	450	23	cm	cm	NOUN
ejpam-2272	450	24	1	1	NUM
ejpam-2272	450	25	,	,	PUNCT
ejpam-2272	450	26	cm	cm	NOUN
ejpam-2272	450	27	2	2	NUM
ejpam-2272	450	28	)	)	PUNCT
ejpam-2272	450	29	∈	∈	PROPN
ejpam-2272	450	30	i	i	PRON
ejpam-2272	450	31	×s2	×s2	PROPN
ejpam-2272	450	32	,	,	PUNCT
ejpam-2272	450	33	that	that	ADV
ejpam-2272	450	34	is	is	ADV
ejpam-2272	450	35	,	,	PUNCT
ejpam-2272	450	36	(	(	PUNCT
ejpam-2272	450	37	bm	bm	PROPN
ejpam-2272	450	38	1	1	NUM
ejpam-2272	450	39	cm	cm	NOUN
ejpam-2272	450	40	1	1	NUM
ejpam-2272	450	41	,	,	PUNCT
ejpam-2272	450	42	bm	bm	PROPN
ejpam-2272	450	43	2	2	NUM
ejpam-2272	450	44	cm	cm	NOUN
ejpam-2272	450	45	2	2	NUM
ejpam-2272	450	46	)	)	PUNCT
ejpam-2272	450	47	∈	∈	PROPN
ejpam-2272	450	48	i	i	PRON
ejpam-2272	450	49	×s2	×s2	PROPN
ejpam-2272	450	50	.	.	PUNCT
ejpam-2272	451	1	similarly	similarly	ADV
ejpam-2272	451	2	,	,	PUNCT
ejpam-2272	451	3	we	we	PRON
ejpam-2272	451	4	can	can	AUX
ejpam-2272	451	5	prove	prove	VERB
ejpam-2272	451	6	the	the	DET
ejpam-2272	451	7	case	case	NOUN
ejpam-2272	451	8	when	when	SCONJ
ejpam-2272	451	9	(	(	PUNCT
ejpam-2272	451	10	a1c1)n	a1c1)n	PROPN
ejpam-2272	451	11	∈	∈	NOUN
ejpam-2272	451	12	i	i	PRON
ejpam-2272	451	13	for	for	ADP
ejpam-2272	451	14	some	some	DET
ejpam-2272	451	15	positive	positive	ADJ
ejpam-2272	451	16	integer	integer	NOUN
ejpam-2272	451	17	n.	n.	NOUN
ejpam-2272	451	18	hence	hence	ADV
ejpam-2272	451	19	,	,	PUNCT
ejpam-2272	451	20	i	i	PRON
ejpam-2272	451	21	×s2	×s2	PROPN
ejpam-2272	451	22	is	be	AUX
ejpam-2272	451	23	a	a	DET
ejpam-2272	451	24	2	2	NUM
ejpam-2272	451	25	-	-	PUNCT
ejpam-2272	451	26	absorbing	absorbing	ADJ
ejpam-2272	451	27	primary	primary	ADJ
ejpam-2272	451	28	ideal	ideal	NOUN
ejpam-2272	451	29	of	of	ADP
ejpam-2272	451	30	s.	s.	PROPN
ejpam-2272	451	31	(	(	PUNCT
ejpam-2272	451	32	2	2	NUM
ejpam-2272	451	33	)	)	PUNCT
ejpam-2272	451	34	⇒	⇒	NOUN
ejpam-2272	451	35	(	(	PUNCT
ejpam-2272	451	36	3	3	X
ejpam-2272	451	37	)	)	PUNCT
ejpam-2272	451	38	it	it	PRON
ejpam-2272	451	39	is	be	AUX
ejpam-2272	451	40	obvious	obvious	ADJ
ejpam-2272	451	41	.	.	PUNCT
ejpam-2272	452	1	(	(	PUNCT
ejpam-2272	452	2	3	3	X
ejpam-2272	452	3	)	)	PUNCT
ejpam-2272	452	4	⇒	⇒	NOUN
ejpam-2272	452	5	(	(	PUNCT
ejpam-2272	452	6	1	1	X
ejpam-2272	452	7	)	)	PUNCT
ejpam-2272	452	8	let	let	VERB
ejpam-2272	452	9	abc	abc	PROPN
ejpam-2272	452	10	∈	∈	PROPN
ejpam-2272	452	11	i	i	PRON
ejpam-2272	452	12	for	for	ADP
ejpam-2272	452	13	some	some	DET
ejpam-2272	452	14	a	a	DET
ejpam-2272	452	15	,	,	PUNCT
ejpam-2272	452	16	b	b	NOUN
ejpam-2272	452	17	,	,	PUNCT
ejpam-2272	452	18	c	c	PROPN
ejpam-2272	452	19	∈	∈	PROPN
ejpam-2272	452	20	s1	s1	PROPN
ejpam-2272	452	21	.	.	PUNCT
ejpam-2272	453	1	then	then	ADV
ejpam-2272	453	2	for	for	ADP
ejpam-2272	453	3	each	each	DET
ejpam-2272	453	4	0	0	NUM
ejpam-2272	453	5	≠	≠	PROPN
ejpam-2272	453	6	r	r	NOUN
ejpam-2272	453	7	∈	∈	PROPN
ejpam-2272	453	8	s2	s2	NOUN
ejpam-2272	453	9	,	,	PUNCT
ejpam-2272	453	10	we	we	PRON
ejpam-2272	453	11	have	have	VERB
ejpam-2272	453	12	(	(	PUNCT
ejpam-2272	453	13	0	0	NUM
ejpam-2272	453	14	,	,	PUNCT
ejpam-2272	453	15	0	0	NUM
ejpam-2272	453	16	)	)	PUNCT
ejpam-2272	453	17	≠	≠	PROPN
ejpam-2272	453	18	(	(	PUNCT
ejpam-2272	453	19	a	a	PRON
ejpam-2272	453	20	,	,	PUNCT
ejpam-2272	453	21	1)(b	1)(b	NUM
ejpam-2272	453	22	,	,	PUNCT
ejpam-2272	453	23	1)(c	1)(c	NUM
ejpam-2272	453	24	,	,	PUNCT
ejpam-2272	453	25	r	r	NOUN
ejpam-2272	453	26	)	)	PUNCT
ejpam-2272	453	27	∈	∈	NOUN
ejpam-2272	454	1	i	i	PRON
ejpam-2272	454	2	×	×	PROPN
ejpam-2272	454	3	s2	s2	PROPN
ejpam-2272	454	4	.	.	PUNCT
ejpam-2272	455	1	this	this	PRON
ejpam-2272	455	2	gives	give	VERB
ejpam-2272	455	3	(	(	PUNCT
ejpam-2272	455	4	a	a	DET
ejpam-2272	455	5	,	,	PUNCT
ejpam-2272	455	6	1)(b	1)(b	NUM
ejpam-2272	455	7	,	,	PUNCT
ejpam-2272	455	8	1	1	NUM
ejpam-2272	455	9	)	)	PUNCT
ejpam-2272	455	10	∈	∈	NOUN
ejpam-2272	455	11	i	i	PRON
ejpam-2272	455	12	×	×	PROPN
ejpam-2272	455	13	s2	s2	NOUN
ejpam-2272	455	14	or	or	CCONJ
ejpam-2272	455	15	(	(	PUNCT
ejpam-2272	455	16	bm	bm	PROPN
ejpam-2272	455	17	,	,	PUNCT
ejpam-2272	455	18	1)(cm	1)(cm	NUM
ejpam-2272	455	19	,	,	PUNCT
ejpam-2272	455	20	rm	rm	PROPN
ejpam-2272	455	21	)	)	PUNCT
ejpam-2272	455	22	∈	∈	PROPN
ejpam-2272	456	1	i	i	PRON
ejpam-2272	456	2	×	×	PROPN
ejpam-2272	456	3	s2	s2	NOUN
ejpam-2272	456	4	or	or	CCONJ
ejpam-2272	456	5	(	(	PUNCT
ejpam-2272	456	6	cn	cn	PROPN
ejpam-2272	456	7	,	,	PUNCT
ejpam-2272	456	8	rn)(an	rn)(an	PROPN
ejpam-2272	456	9	,	,	PUNCT
ejpam-2272	456	10	1	1	NUM
ejpam-2272	456	11	)	)	PUNCT
ejpam-2272	456	12	∈	∈	PROPN
ejpam-2272	457	1	i	i	PRON
ejpam-2272	457	2	×	×	PROPN
ejpam-2272	457	3	s2	s2	PROPN
ejpam-2272	457	4	,	,	PUNCT
ejpam-2272	457	5	since	since	SCONJ
ejpam-2272	457	6	i	i	PRON
ejpam-2272	457	7	×	×	PROPN
ejpam-2272	457	8	s2	s2	PROPN
ejpam-2272	457	9	is	be	AUX
ejpam-2272	457	10	a	a	DET
ejpam-2272	457	11	weakly	weakly	ADJ
ejpam-2272	457	12	2	2	NUM
ejpam-2272	457	13	-	-	PUNCT
ejpam-2272	457	14	absorbing	absorbing	ADJ
ejpam-2272	457	15	primary	primary	ADJ
ejpam-2272	457	16	ideal	ideal	NOUN
ejpam-2272	457	17	of	of	ADP
ejpam-2272	457	18	s.	s.	PROPN
ejpam-2272	457	19	that	that	PRON
ejpam-2272	457	20	is	be	AUX
ejpam-2272	457	21	,	,	PUNCT
ejpam-2272	457	22	either	either	CCONJ
ejpam-2272	457	23	ab	ab	PROPN
ejpam-2272	457	24	∈	∈	PROPN
ejpam-2272	458	1	i	i	PRON
ejpam-2272	458	2	or	or	CCONJ
ejpam-2272	458	3	bmcm	bmcm	PROPN
ejpam-2272	458	4	∈	∈	PROPN
ejpam-2272	458	5	i	i	PRON
ejpam-2272	458	6	or	or	CCONJ
ejpam-2272	458	7	ancn	ancn	PROPN
ejpam-2272	458	8	∈	∈	PROPN
ejpam-2272	458	9	i	i	PRON
ejpam-2272	458	10	for	for	ADP
ejpam-2272	458	11	some	some	DET
ejpam-2272	458	12	positive	positive	ADJ
ejpam-2272	458	13	integers	integer	NOUN
ejpam-2272	458	14	m	m	VERB
ejpam-2272	458	15	,	,	PUNCT
ejpam-2272	458	16	n.	n.	NOUN
ejpam-2272	458	17	this	this	PRON
ejpam-2272	458	18	shows	show	VERB
ejpam-2272	458	19	that	that	SCONJ
ejpam-2272	458	20	i	i	PRON
ejpam-2272	458	21	is	be	AUX
ejpam-2272	458	22	a	a	DET
ejpam-2272	458	23	2	2	NUM
ejpam-2272	458	24	-	-	PUNCT
ejpam-2272	458	25	absorbing	absorbing	ADJ
ejpam-2272	458	26	primary	primary	ADJ
ejpam-2272	458	27	ideal	ideal	NOUN
ejpam-2272	458	28	of	of	ADP
ejpam-2272	458	29	s1	s1	PROPN
ejpam-2272	458	30	.	.	PUNCT
ejpam-2272	459	1	theorem	theorem	NOUN
ejpam-2272	459	2	12	12	NUM
ejpam-2272	459	3	.	.	PUNCT
ejpam-2272	460	1	let	let	AUX
ejpam-2272	460	2	(	(	PUNCT
ejpam-2272	460	3	s	s	X
ejpam-2272	460	4	,	,	PUNCT
ejpam-2272	460	5	m	m	VERB
ejpam-2272	460	6	)	)	PUNCT
ejpam-2272	460	7	be	be	VERB
ejpam-2272	460	8	a	a	DET
ejpam-2272	460	9	local	local	ADJ
ejpam-2272	460	10	semiring	semiring	NOUN
ejpam-2272	460	11	with	with	ADP
ejpam-2272	460	12	m3	m3	PROPN
ejpam-2272	460	13	=	=	SYM
ejpam-2272	460	14	0	0	PROPN
ejpam-2272	460	15	.	.	PUNCT
ejpam-2272	461	1	then	then	ADV
ejpam-2272	461	2	every	every	DET
ejpam-2272	461	3	proper	proper	ADJ
ejpam-2272	461	4	subtractive	subtractive	NOUN
ejpam-2272	461	5	ideal	ideal	NOUN
ejpam-2272	461	6	of	of	ADP
ejpam-2272	461	7	s	s	PROPN
ejpam-2272	461	8	is	be	AUX
ejpam-2272	461	9	a	a	DET
ejpam-2272	461	10	weakly	weakly	ADJ
ejpam-2272	461	11	2	2	NUM
ejpam-2272	461	12	-	-	PUNCT
ejpam-2272	461	13	absorbing	absorbing	ADJ
ejpam-2272	461	14	primary	primary	ADJ
ejpam-2272	461	15	ideal	ideal	NOUN
ejpam-2272	461	16	of	of	ADP
ejpam-2272	461	17	s.	s.	PROPN
ejpam-2272	461	18	proof	proof	PROPN
ejpam-2272	461	19	.	.	PUNCT
ejpam-2272	462	1	proof	proof	NOUN
ejpam-2272	462	2	is	be	AUX
ejpam-2272	462	3	analogous	analogous	ADJ
ejpam-2272	462	4	to	to	ADP
ejpam-2272	462	5	the	the	DET
ejpam-2272	462	6	proof	proof	NOUN
ejpam-2272	462	7	of	of	ADP
ejpam-2272	462	8	[	[	X
ejpam-2272	462	9	10	10	NUM
ejpam-2272	462	10	,	,	PUNCT
ejpam-2272	462	11	theorem	theorem	VERB
ejpam-2272	462	12	2.8	2.8	NUM
ejpam-2272	462	13	]	]	PUNCT
ejpam-2272	462	14	.	.	PUNCT
ejpam-2272	463	1	references	reference	NOUN
ejpam-2272	463	2	194	194	NUM
ejpam-2272	463	3	theorem	theorem	NOUN
ejpam-2272	463	4	13	13	NUM
ejpam-2272	463	5	.	.	PUNCT
ejpam-2272	464	1	let	let	VERB
ejpam-2272	464	2	s	s	PRON
ejpam-2272	464	3	be	be	AUX
ejpam-2272	464	4	a	a	DET
ejpam-2272	464	5	semiring	semiring	NOUN
ejpam-2272	464	6	,	,	PUNCT
ejpam-2272	464	7	i	i	PRON
ejpam-2272	464	8	a	a	DET
ejpam-2272	464	9	q	q	NOUN
ejpam-2272	464	10	-	-	PUNCT
ejpam-2272	464	11	ideal	ideal	NOUN
ejpam-2272	464	12	of	of	ADP
ejpam-2272	464	13	s	s	PRON
ejpam-2272	464	14	and	and	CCONJ
ejpam-2272	464	15	p	p	X
ejpam-2272	464	16	a	a	DET
ejpam-2272	464	17	subtractive	subtractive	NOUN
ejpam-2272	464	18	ideal	ideal	NOUN
ejpam-2272	464	19	of	of	ADP
ejpam-2272	464	20	s	s	PRON
ejpam-2272	464	21	such	such	ADJ
ejpam-2272	464	22	that	that	SCONJ
ejpam-2272	464	23	i	i	PRON
ejpam-2272	464	24	⊆	⊆	NUM
ejpam-2272	465	1	p.	p.	NOUN
ejpam-2272	465	2	then	then	ADV
ejpam-2272	465	3	(	(	PUNCT
ejpam-2272	465	4	1	1	X
ejpam-2272	465	5	)	)	PUNCT
ejpam-2272	465	6	if	if	SCONJ
ejpam-2272	465	7	p	p	NOUN
ejpam-2272	465	8	is	be	AUX
ejpam-2272	465	9	a	a	DET
ejpam-2272	465	10	weakly	weakly	ADJ
ejpam-2272	465	11	2	2	NUM
ejpam-2272	465	12	-	-	PUNCT
ejpam-2272	465	13	absorbing	absorbing	ADJ
ejpam-2272	465	14	primary	primary	ADJ
ejpam-2272	465	15	ideal	ideal	NOUN
ejpam-2272	465	16	of	of	ADP
ejpam-2272	465	17	s	s	PROPN
ejpam-2272	465	18	,	,	PUNCT
ejpam-2272	465	19	then	then	ADV
ejpam-2272	465	20	p	p	X
ejpam-2272	465	21	/	/	SYM
ejpam-2272	465	22	i(q∩p	i(q∩p	NOUN
ejpam-2272	465	23	)	)	PUNCT
ejpam-2272	465	24	is	be	AUX
ejpam-2272	465	25	a	a	DET
ejpam-2272	465	26	weakly	weakly	ADJ
ejpam-2272	465	27	2	2	NUM
ejpam-2272	465	28	-	-	PUNCT
ejpam-2272	465	29	absorbing	absorbing	ADJ
ejpam-2272	465	30	primary	primary	ADJ
ejpam-2272	465	31	ideal	ideal	NOUN
ejpam-2272	465	32	of	of	ADP
ejpam-2272	465	33	s	s	NOUN
ejpam-2272	465	34	/	/	SYM
ejpam-2272	465	35	i(q	i(q	NOUN
ejpam-2272	465	36	)	)	PUNCT
ejpam-2272	465	37	.	.	PUNCT
ejpam-2272	466	1	(	(	PUNCT
ejpam-2272	466	2	2	2	X
ejpam-2272	466	3	)	)	PUNCT
ejpam-2272	466	4	if	if	SCONJ
ejpam-2272	466	5	i	i	PRON
ejpam-2272	466	6	and	and	CCONJ
ejpam-2272	466	7	p	p	X
ejpam-2272	466	8	/	/	SYM
ejpam-2272	466	9	i(q∩p	i(q∩p	NOUN
ejpam-2272	466	10	)	)	PUNCT
ejpam-2272	466	11	are	be	AUX
ejpam-2272	466	12	weakly	weakly	ADJ
ejpam-2272	466	13	2	2	NUM
ejpam-2272	466	14	-	-	PUNCT
ejpam-2272	466	15	absorbing	absorbing	ADJ
ejpam-2272	466	16	primary	primary	ADJ
ejpam-2272	466	17	ideals	ideal	NOUN
ejpam-2272	466	18	of	of	ADP
ejpam-2272	466	19	s	s	PRON
ejpam-2272	466	20	and	and	CCONJ
ejpam-2272	466	21	s	s	NOUN
ejpam-2272	466	22	/	/	SYM
ejpam-2272	466	23	i(q	i(q	NOUN
ejpam-2272	466	24	)	)	PUNCT
ejpam-2272	466	25	respectively	respectively	ADV
ejpam-2272	466	26	,	,	PUNCT
ejpam-2272	466	27	then	then	ADV
ejpam-2272	466	28	p	p	PROPN
ejpam-2272	466	29	is	be	AUX
ejpam-2272	466	30	a	a	DET
ejpam-2272	466	31	weakly	weakly	ADJ
ejpam-2272	466	32	2	2	NUM
ejpam-2272	466	33	-	-	PUNCT
ejpam-2272	466	34	absorbing	absorbing	ADJ
ejpam-2272	466	35	primary	primary	ADJ
ejpam-2272	466	36	ideal	ideal	NOUN
ejpam-2272	466	37	of	of	ADP
ejpam-2272	466	38	s.	s.	PROPN
ejpam-2272	466	39	proof	proof	PROPN
ejpam-2272	466	40	.	.	PUNCT
ejpam-2272	467	1	(	(	PUNCT
ejpam-2272	467	2	1	1	X
ejpam-2272	467	3	)	)	PUNCT
ejpam-2272	467	4	if	if	SCONJ
ejpam-2272	467	5	(	(	PUNCT
ejpam-2272	467	6	q1	q1	PROPN
ejpam-2272	467	7	+	+	CCONJ
ejpam-2272	467	8	i)⊙	i)⊙	PROPN
ejpam-2272	467	9	(	(	PUNCT
ejpam-2272	467	10	q2	q2	NOUN
ejpam-2272	467	11	+	+	CCONJ
ejpam-2272	467	12	i)⊙	i)⊙	PROPN
ejpam-2272	467	13	(	(	PUNCT
ejpam-2272	467	14	q3	q3	PROPN
ejpam-2272	467	15	+	+	PROPN
ejpam-2272	467	16	i	i	PROPN
ejpam-2272	467	17	)	)	PUNCT
ejpam-2272	467	18	≠	≠	PROPN
ejpam-2272	467	19	0	0	NUM
ejpam-2272	467	20	in	in	ADP
ejpam-2272	467	21	s	s	NOUN
ejpam-2272	467	22	/	/	SYM
ejpam-2272	467	23	iq	iq	NOUN
ejpam-2272	467	24	then	then	ADV
ejpam-2272	467	25	q1q2q3	q1q2q3	PRON
ejpam-2272	467	26	≠	≠	PROPN
ejpam-2272	467	27	0	0	NUM
ejpam-2272	467	28	in	in	ADP
ejpam-2272	467	29	s	s	PROPN
ejpam-2272	467	30	,	,	PUNCT
ejpam-2272	467	31	then	then	ADV
ejpam-2272	467	32	the	the	DET
ejpam-2272	467	33	proof	proof	NOUN
ejpam-2272	467	34	follows	follow	VERB
ejpam-2272	467	35	from	from	ADP
ejpam-2272	467	36	theorem	theorem	ADJ
ejpam-2272	467	37	9	9	NUM
ejpam-2272	467	38	.	.	PUNCT
ejpam-2272	468	1	(	(	PUNCT
ejpam-2272	468	2	2	2	X
ejpam-2272	468	3	)	)	PUNCT
ejpam-2272	468	4	let	let	VERB
ejpam-2272	468	5	a	a	DET
ejpam-2272	468	6	,	,	PUNCT
ejpam-2272	468	7	b	b	NOUN
ejpam-2272	468	8	,	,	PUNCT
ejpam-2272	468	9	c	c	PROPN
ejpam-2272	468	10	∈	∈	PROPN
ejpam-2272	468	11	s	s	AUX
ejpam-2272	468	12	be	be	AUX
ejpam-2272	469	1	such	such	ADJ
ejpam-2272	469	2	that	that	SCONJ
ejpam-2272	469	3	0	0	NUM
ejpam-2272	469	4	≠	≠	PROPN
ejpam-2272	469	5	abc	abc	PROPN
ejpam-2272	469	6	∈	∈	PROPN
ejpam-2272	469	7	p.	p.	NOUN
ejpam-2272	469	8	if	if	SCONJ
ejpam-2272	469	9	abc	abc	PROPN
ejpam-2272	469	10	∈	∈	PROPN
ejpam-2272	470	1	i	i	PRON
ejpam-2272	470	2	,	,	PUNCT
ejpam-2272	470	3	then	then	ADV
ejpam-2272	470	4	either	either	CCONJ
ejpam-2272	470	5	ab	ab	PROPN
ejpam-2272	470	6	∈	∈	PROPN
ejpam-2272	470	7	i	i	PRON
ejpam-2272	470	8	⊆	⊆	PROPN
ejpam-2272	470	9	p	p	NOUN
ejpam-2272	470	10	or	or	CCONJ
ejpam-2272	470	11	bc	bc	PROPN
ejpam-2272	470	12	∈	∈	PROPN
ejpam-2272	470	13	i	i	NOUN
ejpam-2272	470	14	⊆	⊆	NUM
ejpam-2272	470	15	√	√	VERB
ejpam-2272	470	16	i	i	PRON
ejpam-2272	470	17	⊆	⊆	NUM
ejpam-2272	470	18	√	√	ADP
ejpam-2272	470	19	p	p	NOUN
ejpam-2272	470	20	or	or	CCONJ
ejpam-2272	470	21	ca	ca	AUX
ejpam-2272	470	22	∈	∈	PROPN
ejpam-2272	470	23	i	i	PRON
ejpam-2272	470	24	⊆	⊆	NUM
ejpam-2272	470	25	√	√	VERB
ejpam-2272	470	26	i	i	PRON
ejpam-2272	470	27	⊆	⊆	NUM
ejpam-2272	470	28	√	√	ADP
ejpam-2272	470	29	p	p	X
ejpam-2272	470	30	,	,	PUNCT
ejpam-2272	470	31	since	since	SCONJ
ejpam-2272	470	32	i	i	PRON
ejpam-2272	470	33	is	be	AUX
ejpam-2272	470	34	a	a	DET
ejpam-2272	470	35	weakly	weakly	ADJ
ejpam-2272	470	36	2	2	NUM
ejpam-2272	470	37	-	-	PUNCT
ejpam-2272	470	38	absorbing	absorbing	ADJ
ejpam-2272	470	39	primary	primary	ADJ
ejpam-2272	470	40	ideal	ideal	NOUN
ejpam-2272	470	41	of	of	ADP
ejpam-2272	470	42	s.	s.	PROPN
ejpam-2272	470	43	so	so	ADV
ejpam-2272	470	44	,	,	PUNCT
ejpam-2272	470	45	assume	assume	VERB
ejpam-2272	470	46	that	that	SCONJ
ejpam-2272	470	47	abc	abc	PROPN
ejpam-2272	470	48	∉	∉	PROPN
ejpam-2272	470	49	i	i	PRON
ejpam-2272	470	50	.	.	PUNCT
ejpam-2272	471	1	then	then	ADV
ejpam-2272	471	2	there	there	PRON
ejpam-2272	471	3	are	be	VERB
ejpam-2272	471	4	elements	element	NOUN
ejpam-2272	471	5	q1	q1	PROPN
ejpam-2272	471	6	,	,	PUNCT
ejpam-2272	471	7	q2	q2	PROPN
ejpam-2272	471	8	,	,	PUNCT
ejpam-2272	471	9	q3	q3	PROPN
ejpam-2272	471	10	∈	∈	PROPN
ejpam-2272	471	11	q	q	NOUN
ejpam-2272	471	12	such	such	ADJ
ejpam-2272	471	13	that	that	SCONJ
ejpam-2272	471	14	a	a	DET
ejpam-2272	471	15	∈	∈	PROPN
ejpam-2272	471	16	q1	q1	NOUN
ejpam-2272	471	17	+	+	CCONJ
ejpam-2272	471	18	i	i	PROPN
ejpam-2272	471	19	,	,	PUNCT
ejpam-2272	471	20	b	b	PROPN
ejpam-2272	471	21	∈	∈	PROPN
ejpam-2272	471	22	q2	q2	NOUN
ejpam-2272	471	23	+	+	CCONJ
ejpam-2272	471	24	i	i	PRON
ejpam-2272	471	25	,	,	PUNCT
ejpam-2272	471	26	c	c	PROPN
ejpam-2272	471	27	∈	∈	PROPN
ejpam-2272	471	28	q3	q3	NOUN
ejpam-2272	472	1	+	+	CCONJ
ejpam-2272	472	2	i	i	PRON
ejpam-2272	472	3	.	.	PUNCT
ejpam-2272	473	1	therefore	therefore	ADV
ejpam-2272	473	2	,	,	PUNCT
ejpam-2272	473	3	for	for	ADP
ejpam-2272	473	4	some	some	DET
ejpam-2272	473	5	i1	i1	PROPN
ejpam-2272	473	6	,	,	PUNCT
ejpam-2272	473	7	i2	i2	PROPN
ejpam-2272	473	8	,	,	PUNCT
ejpam-2272	473	9	i3	i3	NOUN
ejpam-2272	473	10	∈	∈	PROPN
ejpam-2272	473	11	i	i	PRON
ejpam-2272	473	12	,	,	PUNCT
ejpam-2272	473	13	a	a	DET
ejpam-2272	473	14	=	=	PROPN
ejpam-2272	473	15	q1	q1	PROPN
ejpam-2272	473	16	+	+	CCONJ
ejpam-2272	473	17	i1	i1	PROPN
ejpam-2272	473	18	,	,	PUNCT
ejpam-2272	473	19	b	b	PROPN
ejpam-2272	473	20	=	=	SYM
ejpam-2272	473	21	q2	q2	PROPN
ejpam-2272	473	22	+	+	CCONJ
ejpam-2272	473	23	i2	i2	PROPN
ejpam-2272	473	24	,	,	PUNCT
ejpam-2272	473	25	c	c	NOUN
ejpam-2272	473	26	=	=	SYM
ejpam-2272	473	27	q3	q3	PROPN
ejpam-2272	473	28	+	+	NUM
ejpam-2272	473	29	i3	i3	NOUN
ejpam-2272	473	30	.	.	PUNCT
ejpam-2272	474	1	as	as	SCONJ
ejpam-2272	474	2	abc	abc	PROPN
ejpam-2272	474	3	=	=	PUNCT
ejpam-2272	474	4	q1q2q3	q1q2q3	PROPN
ejpam-2272	474	5	+	+	CCONJ
ejpam-2272	474	6	q1q2i3	q1q2i3	VERB
ejpam-2272	474	7	+	+	CCONJ
ejpam-2272	474	8	q1q3i2	q1q3i2	ADJ
ejpam-2272	474	9	+	+	CCONJ
ejpam-2272	474	10	q1i2i3	q1i2i3	ADJ
ejpam-2272	474	11	+	+	CCONJ
ejpam-2272	474	12	q2q3i1	q2q3i1	ADJ
ejpam-2272	474	13	+	+	CCONJ
ejpam-2272	474	14	q2i1i3	q2i1i3	NOUN
ejpam-2272	474	15	+	+	CCONJ
ejpam-2272	474	16	q3i1i2	q3i1i2	NOUN
ejpam-2272	474	17	+	+	CCONJ
ejpam-2272	474	18	i1i2i3	i1i2i3	PROPN
ejpam-2272	474	19	∈	∈	PROPN
ejpam-2272	474	20	p	p	NOUN
ejpam-2272	474	21	and	and	CCONJ
ejpam-2272	474	22	since	since	SCONJ
ejpam-2272	474	23	p	p	NOUN
ejpam-2272	474	24	is	be	AUX
ejpam-2272	474	25	subtractive	subtractive	NOUN
ejpam-2272	474	26	,	,	PUNCT
ejpam-2272	474	27	we	we	PRON
ejpam-2272	474	28	have	have	VERB
ejpam-2272	474	29	q1q2q3	q1q2q3	PROPN
ejpam-2272	474	30	∈	∈	PROPN
ejpam-2272	474	31	p.	p.	NOUN
ejpam-2272	474	32	consider	consider	VERB
ejpam-2272	474	33	,	,	PUNCT
ejpam-2272	474	34	(	(	PUNCT
ejpam-2272	474	35	q1	q1	PROPN
ejpam-2272	474	36	+	+	CCONJ
ejpam-2272	474	37	i)⊙	i)⊙	PROPN
ejpam-2272	474	38	(	(	PUNCT
ejpam-2272	474	39	q2	q2	NOUN
ejpam-2272	474	40	+	+	CCONJ
ejpam-2272	474	41	i)⊙	i)⊙	PROPN
ejpam-2272	474	42	(	(	PUNCT
ejpam-2272	474	43	q3	q3	PROPN
ejpam-2272	474	44	+	+	PROPN
ejpam-2272	474	45	i	i	NOUN
ejpam-2272	474	46	)	)	PUNCT
ejpam-2272	475	1	=	=	SYM
ejpam-2272	475	2	q4	q4	PROPN
ejpam-2272	475	3	+	+	CCONJ
ejpam-2272	475	4	i	i	PRON
ejpam-2272	475	5	where	where	SCONJ
ejpam-2272	475	6	q4	q4	PROPN
ejpam-2272	475	7	is	be	AUX
ejpam-2272	475	8	the	the	DET
ejpam-2272	475	9	unique	unique	ADJ
ejpam-2272	475	10	element	element	NOUN
ejpam-2272	475	11	such	such	ADJ
ejpam-2272	475	12	that	that	SCONJ
ejpam-2272	475	13	q1q2q3	q1q2q3	NOUN
ejpam-2272	475	14	+	+	PROPN
ejpam-2272	475	15	i	i	PROPN
ejpam-2272	475	16	⊆	⊆	NUM
ejpam-2272	475	17	q4	q4	PROPN
ejpam-2272	475	18	+	+	CCONJ
ejpam-2272	475	19	i	i	PRON
ejpam-2272	475	20	.	.	PUNCT
ejpam-2272	476	1	since	since	SCONJ
ejpam-2272	476	2	p	p	NOUN
ejpam-2272	476	3	is	be	AUX
ejpam-2272	476	4	subtractive	subtractive	NOUN
ejpam-2272	476	5	,	,	PUNCT
ejpam-2272	476	6	we	we	PRON
ejpam-2272	476	7	have	have	VERB
ejpam-2272	476	8	q4	q4	PROPN
ejpam-2272	476	9	∈	∈	PROPN
ejpam-2272	476	10	p	p	PROPN
ejpam-2272	476	11	∩q	∩q	PROPN
ejpam-2272	476	12	,	,	PUNCT
ejpam-2272	476	13	hence	hence	ADV
ejpam-2272	476	14	q1q2q3	q1q2q3	NOUN
ejpam-2272	477	1	+	+	PROPN
ejpam-2272	477	2	i	i	PROPN
ejpam-2272	477	3	⊆	⊆	NUM
ejpam-2272	477	4	q4	q4	PROPN
ejpam-2272	478	1	+	+	CCONJ
ejpam-2272	478	2	i	i	PROPN
ejpam-2272	478	3	∈	∈	PROPN
ejpam-2272	478	4	p	p	X
ejpam-2272	478	5	/	/	SYM
ejpam-2272	478	6	iq∩p	iq∩p	PROPN
ejpam-2272	478	7	,	,	PUNCT
ejpam-2272	478	8	that	that	ADV
ejpam-2272	478	9	is	is	ADV
ejpam-2272	478	10	,	,	PUNCT
ejpam-2272	478	11	(	(	PUNCT
ejpam-2272	478	12	q1	q1	X
ejpam-2272	478	13	+	+	X
ejpam-2272	478	14	i)⊙(q2	i)⊙(q2	NUM
ejpam-2272	478	15	+	+	SYM
ejpam-2272	478	16	i)⊙(q3	i)⊙(q3	NUM
ejpam-2272	478	17	+	+	ADJ
ejpam-2272	478	18	i	i	NOUN
ejpam-2272	478	19	)	)	PUNCT
ejpam-2272	478	20	∈	∈	PROPN
ejpam-2272	478	21	p	p	X
ejpam-2272	478	22	/	/	SYM
ejpam-2272	478	23	iq∩p	iq∩p	PROPN
ejpam-2272	478	24	.	.	PUNCT
ejpam-2272	479	1	let	let	VERB
ejpam-2272	479	2	q	q	PROPN
ejpam-2272	479	3	∈	∈	PROPN
ejpam-2272	479	4	q	q	AUX
ejpam-2272	479	5	be	be	AUX
ejpam-2272	479	6	the	the	DET
ejpam-2272	479	7	unique	unique	ADJ
ejpam-2272	479	8	element	element	NOUN
ejpam-2272	479	9	such	such	ADJ
ejpam-2272	479	10	that	that	SCONJ
ejpam-2272	479	11	q+	q+	ADV
ejpam-2272	479	12	i	i	PRON
ejpam-2272	479	13	is	be	AUX
ejpam-2272	479	14	the	the	DET
ejpam-2272	479	15	zero	zero	NUM
ejpam-2272	479	16	element	element	NOUN
ejpam-2272	479	17	in	in	ADP
ejpam-2272	479	18	s	s	PROPN
ejpam-2272	479	19	/	/	SYM
ejpam-2272	479	20	iq	iq	NOUN
ejpam-2272	479	21	.	.	PUNCT
ejpam-2272	480	1	if	if	SCONJ
ejpam-2272	480	2	(	(	PUNCT
ejpam-2272	480	3	q1	q1	NOUN
ejpam-2272	480	4	+	+	X
ejpam-2272	480	5	i)⊙(q2	i)⊙(q2	NUM
ejpam-2272	480	6	+	+	SYM
ejpam-2272	480	7	i)⊙(q3	i)⊙(q3	NUM
ejpam-2272	480	8	+	+	NUM
ejpam-2272	480	9	i	i	NOUN
ejpam-2272	480	10	)	)	PUNCT
ejpam-2272	480	11	=	=	PUNCT
ejpam-2272	480	12	0s	0	NOUN
ejpam-2272	480	13	/	/	SYM
ejpam-2272	480	14	iq	iq	NOUN
ejpam-2272	480	15	=	=	PUNCT
ejpam-2272	481	1	q+	q+	NOUN
ejpam-2272	481	2	i	i	PRON
ejpam-2272	481	3	,	,	PUNCT
ejpam-2272	481	4	then	then	ADV
ejpam-2272	481	5	there	there	PRON
ejpam-2272	481	6	exit	exit	NOUN
ejpam-2272	481	7	r	r	NOUN
ejpam-2272	481	8	,	,	PUNCT
ejpam-2272	481	9	s	s	PART
ejpam-2272	481	10	∈	∈	NOUN
ejpam-2272	481	11	i	i	PRON
ejpam-2272	481	12	such	such	ADJ
ejpam-2272	481	13	that	that	SCONJ
ejpam-2272	481	14	q1q2q3	q1q2q3	NOUN
ejpam-2272	481	15	+	+	NOUN
ejpam-2272	481	16	r	r	NOUN
ejpam-2272	481	17	=	=	PUNCT
ejpam-2272	481	18	q	q	NOUN
ejpam-2272	482	1	+	+	NUM
ejpam-2272	482	2	s	s	VERB
ejpam-2272	482	3	∈	∈	NOUN
ejpam-2272	483	1	i	i	PRON
ejpam-2272	483	2	.	.	PUNCT
ejpam-2272	484	1	therefore	therefore	ADV
ejpam-2272	484	2	,	,	PUNCT
ejpam-2272	484	3	q1q2q3	q1q2q3	PROPN
ejpam-2272	484	4	∈	∈	VERB
ejpam-2272	484	5	i	i	PRON
ejpam-2272	484	6	,	,	PUNCT
ejpam-2272	484	7	since	since	SCONJ
ejpam-2272	484	8	i	i	PRON
ejpam-2272	484	9	is	be	AUX
ejpam-2272	484	10	a	a	DET
ejpam-2272	484	11	qideal	qideal	NOUN
ejpam-2272	484	12	of	of	ADP
ejpam-2272	484	13	s	s	PROPN
ejpam-2272	484	14	,	,	PUNCT
ejpam-2272	484	15	it	it	PRON
ejpam-2272	484	16	is	be	AUX
ejpam-2272	484	17	subtractive	subtractive	NOUN
ejpam-2272	484	18	by	by	ADP
ejpam-2272	484	19	[	[	X
ejpam-2272	484	20	cor	cor	X
ejpam-2272	484	21	.	.	PROPN
ejpam-2272	484	22	8.23	8.23	NUM
ejpam-2272	484	23	,	,	PUNCT
ejpam-2272	484	24	11	11	NUM
ejpam-2272	484	25	]	]	PUNCT
ejpam-2272	484	26	.	.	PUNCT
ejpam-2272	485	1	this	this	PRON
ejpam-2272	485	2	gives	give	VERB
ejpam-2272	485	3	abc	abc	PROPN
ejpam-2272	485	4	∈	∈	PROPN
ejpam-2272	485	5	i	i	PRON
ejpam-2272	485	6	,	,	PUNCT
ejpam-2272	485	7	a	a	DET
ejpam-2272	485	8	contradiction	contradiction	NOUN
ejpam-2272	485	9	.	.	PUNCT
ejpam-2272	486	1	hence	hence	ADV
ejpam-2272	486	2	,	,	PUNCT
ejpam-2272	486	3	0s	0s	NOUN
ejpam-2272	486	4	/	/	SYM
ejpam-2272	486	5	iq	iq	PROPN
ejpam-2272	486	6	≠	≠	PROPN
ejpam-2272	486	7	(	(	PUNCT
ejpam-2272	486	8	q1	q1	PROPN
ejpam-2272	486	9	+	+	CCONJ
ejpam-2272	486	10	i	i	PROPN
ejpam-2272	486	11	)	)	PUNCT
ejpam-2272	486	12	⊙	⊙	PROPN
ejpam-2272	486	13	(	(	PUNCT
ejpam-2272	486	14	q2	q2	PROPN
ejpam-2272	486	15	+	+	CCONJ
ejpam-2272	486	16	i	i	PROPN
ejpam-2272	486	17	)	)	PUNCT
ejpam-2272	486	18	⊙	⊙	PROPN
ejpam-2272	486	19	(	(	PUNCT
ejpam-2272	486	20	q3	q3	PROPN
ejpam-2272	486	21	+	+	CCONJ
ejpam-2272	486	22	i	i	NOUN
ejpam-2272	486	23	)	)	PUNCT
ejpam-2272	486	24	∈	∈	PROPN
ejpam-2272	486	25	p	p	X
ejpam-2272	486	26	/	/	SYM
ejpam-2272	486	27	iq∩p	iq∩p	PROPN
ejpam-2272	486	28	.	.	PUNCT
ejpam-2272	487	1	this	this	PRON
ejpam-2272	487	2	gives	give	VERB
ejpam-2272	487	3	either	either	PRON
ejpam-2272	487	4	(	(	PUNCT
ejpam-2272	487	5	q1	q1	PROPN
ejpam-2272	487	6	+	+	CCONJ
ejpam-2272	487	7	i	i	PROPN
ejpam-2272	487	8	)	)	PUNCT
ejpam-2272	487	9	⊙	⊙	PROPN
ejpam-2272	487	10	(	(	PUNCT
ejpam-2272	487	11	q2	q2	PROPN
ejpam-2272	487	12	+	+	CCONJ
ejpam-2272	488	1	i	i	PROPN
ejpam-2272	488	2	)	)	PUNCT
ejpam-2272	488	3	∈	∈	PROPN
ejpam-2272	488	4	p	p	PROPN
ejpam-2272	488	5	/	/	SYM
ejpam-2272	488	6	iq∩p	iq∩p	PROPN
ejpam-2272	488	7	or	or	CCONJ
ejpam-2272	488	8	(	(	PUNCT
ejpam-2272	488	9	ql	ql	ADP
ejpam-2272	488	10	2	2	NUM
ejpam-2272	488	11	+	+	NUM
ejpam-2272	488	12	i	i	PROPN
ejpam-2272	488	13	)	)	PUNCT
ejpam-2272	488	14	⊙	⊙	NOUN
ejpam-2272	488	15	(	(	PUNCT
ejpam-2272	488	16	ql	ql	ADP
ejpam-2272	488	17	3	3	NUM
ejpam-2272	488	18	+	+	NUM
ejpam-2272	489	1	i	i	NOUN
ejpam-2272	489	2	)	)	PUNCT
ejpam-2272	489	3	∈	∈	PROPN
ejpam-2272	489	4	p	p	PROPN
ejpam-2272	489	5	/	/	SYM
ejpam-2272	489	6	iq∩p	iq∩p	PROPN
ejpam-2272	489	7	or	or	CCONJ
ejpam-2272	489	8	(	(	PUNCT
ejpam-2272	489	9	qt	qt	NOUN
ejpam-2272	489	10	3	3	NUM
ejpam-2272	489	11	+	+	CCONJ
ejpam-2272	489	12	i	i	PROPN
ejpam-2272	489	13	)	)	PUNCT
ejpam-2272	489	14	⊙	⊙	PROPN
ejpam-2272	489	15	(	(	PUNCT
ejpam-2272	489	16	qt	qt	PROPN
ejpam-2272	489	17	1	1	NUM
ejpam-2272	489	18	+	+	CCONJ
ejpam-2272	489	19	i	i	NOUN
ejpam-2272	489	20	)	)	PUNCT
ejpam-2272	489	21	∈	∈	PROPN
ejpam-2272	489	22	p	p	X
ejpam-2272	489	23	/	/	SYM
ejpam-2272	489	24	iq∩p	iq∩p	PROPN
ejpam-2272	489	25	for	for	ADP
ejpam-2272	489	26	some	some	DET
ejpam-2272	489	27	positive	positive	ADJ
ejpam-2272	489	28	integers	integer	NOUN
ejpam-2272	489	29	l	l	NOUN
ejpam-2272	489	30	,	,	PUNCT
ejpam-2272	489	31	t	t	PROPN
ejpam-2272	489	32	since	since	SCONJ
ejpam-2272	489	33	p	p	PROPN
ejpam-2272	489	34	/	/	SYM
ejpam-2272	489	35	iq∩p	iq∩p	PROPN
ejpam-2272	489	36	is	be	AUX
ejpam-2272	489	37	a	a	DET
ejpam-2272	489	38	weakly	weakly	ADJ
ejpam-2272	489	39	2	2	NUM
ejpam-2272	489	40	-	-	PUNCT
ejpam-2272	489	41	absorbing	absorbing	ADJ
ejpam-2272	489	42	primary	primary	ADJ
ejpam-2272	489	43	ideal	ideal	NOUN
ejpam-2272	489	44	of	of	ADP
ejpam-2272	489	45	s	s	PROPN
ejpam-2272	489	46	/	/	SYM
ejpam-2272	489	47	iq	iq	NOUN
ejpam-2272	489	48	.	.	PUNCT
ejpam-2272	490	1	thus	thus	ADV
ejpam-2272	490	2	,	,	PUNCT
ejpam-2272	490	3	either	either	CCONJ
ejpam-2272	490	4	ab	ab	PROPN
ejpam-2272	490	5	∈	∈	PROPN
ejpam-2272	490	6	p	p	PROPN
ejpam-2272	490	7	or	or	CCONJ
ejpam-2272	490	8	(	(	PUNCT
ejpam-2272	490	9	bc)l	bc)l	PROPN
ejpam-2272	490	10	∈	∈	PROPN
ejpam-2272	490	11	p	p	PROPN
ejpam-2272	490	12	or	or	CCONJ
ejpam-2272	490	13	(	(	PUNCT
ejpam-2272	490	14	ca)t	ca)t	ADP
ejpam-2272	490	15	∈	∈	PROPN
ejpam-2272	490	16	p	p	NOUN
ejpam-2272	490	17	for	for	ADP
ejpam-2272	490	18	some	some	DET
ejpam-2272	490	19	positive	positive	ADJ
ejpam-2272	490	20	integers	integer	NOUN
ejpam-2272	490	21	l	l	NOUN
ejpam-2272	490	22	,	,	PUNCT
ejpam-2272	490	23	t.	t.	PROPN
ejpam-2272	490	24	hence	hence	ADV
ejpam-2272	490	25	,	,	PUNCT
ejpam-2272	490	26	p	p	PROPN
ejpam-2272	490	27	is	be	AUX
ejpam-2272	490	28	a	a	DET
ejpam-2272	490	29	weakly	weakly	ADJ
ejpam-2272	490	30	2	2	NUM
ejpam-2272	490	31	-	-	PUNCT
ejpam-2272	490	32	absorbing	absorbing	ADJ
ejpam-2272	490	33	primary	primary	ADJ
ejpam-2272	490	34	ideal	ideal	NOUN
ejpam-2272	490	35	of	of	ADP
ejpam-2272	490	36	s.	s.	PROPN
ejpam-2272	490	37	references	reference	NOUN
ejpam-2272	490	38	[	[	X
ejpam-2272	490	39	1	1	NUM
ejpam-2272	490	40	]	]	X
ejpam-2272	490	41	p.j	p.j	PROPN
ejpam-2272	490	42	.	.	PROPN
ejpam-2272	490	43	allen	allen	PROPN
ejpam-2272	490	44	.	.	PUNCT
ejpam-2272	491	1	a	a	DET
ejpam-2272	491	2	fundamental	fundamental	ADJ
ejpam-2272	491	3	theorem	theorem	NOUN
ejpam-2272	491	4	of	of	ADP
ejpam-2272	491	5	homomorphism	homomorphism	PROPN
ejpam-2272	491	6	for	for	ADP
ejpam-2272	491	7	semirings	semiring	NOUN
ejpam-2272	491	8	,	,	PUNCT
ejpam-2272	491	9	proceedings	proceeding	NOUN
ejpam-2272	491	10	of	of	ADP
ejpam-2272	491	11	the	the	DET
ejpam-2272	491	12	american	american	PROPN
ejpam-2272	491	13	mathematical	mathematical	PROPN
ejpam-2272	491	14	society	society	NOUN
ejpam-2272	491	15	,	,	PUNCT
ejpam-2272	491	16	21	21	NUM
ejpam-2272	491	17	,	,	PUNCT
ejpam-2272	491	18	412–416	412–416	NUM
ejpam-2272	491	19	.	.	NOUN
ejpam-2272	491	20	1969	1969	NUM
ejpam-2272	491	21	.	.	PUNCT
ejpam-2272	492	1	[	[	X
ejpam-2272	492	2	2	2	NUM
ejpam-2272	492	3	]	]	X
ejpam-2272	492	4	d.d	d.d	PROPN
ejpam-2272	492	5	.	.	PROPN
ejpam-2272	492	6	anderson	anderson	PROPN
ejpam-2272	492	7	and	and	CCONJ
ejpam-2272	492	8	e	e	PROPN
ejpam-2272	492	9	smith	smith	PROPN
ejpam-2272	492	10	.	.	PUNCT
ejpam-2272	493	1	weakly	weakly	ADJ
ejpam-2272	493	2	prime	prime	ADJ
ejpam-2272	493	3	ideals	ideal	NOUN
ejpam-2272	493	4	,	,	PUNCT
ejpam-2272	493	5	houston	houston	PROPN
ejpam-2272	493	6	journal	journal	PROPN
ejpam-2272	493	7	of	of	ADP
ejpam-2272	493	8	mathematics	mathematic	NOUN
ejpam-2272	493	9	,	,	PUNCT
ejpam-2272	493	10	29	29	NUM
ejpam-2272	493	11	,	,	PUNCT
ejpam-2272	493	12	831–840	831–840	NUM
ejpam-2272	493	13	.	.	PUNCT
ejpam-2272	493	14	2003	2003	NUM
ejpam-2272	493	15	.	.	PUNCT
ejpam-2272	494	1	[	[	X
ejpam-2272	494	2	3	3	X
ejpam-2272	494	3	]	]	X
ejpam-2272	494	4	r.e	r.e	PROPN
ejpam-2272	494	5	.	.	PROPN
ejpam-2272	494	6	atani	atani	PROPN
ejpam-2272	494	7	and	and	CCONJ
ejpam-2272	494	8	s.e	s.e	PROPN
ejpam-2272	494	9	.	.	PROPN
ejpam-2272	494	10	atani	atani	PROPN
ejpam-2272	494	11	.	.	PUNCT
ejpam-2272	494	12	ideal	ideal	ADJ
ejpam-2272	494	13	theory	theory	NOUN
ejpam-2272	494	14	in	in	ADP
ejpam-2272	494	15	commutative	commutative	ADJ
ejpam-2272	494	16	semirings	semiring	NOUN
ejpam-2272	494	17	,	,	PUNCT
ejpam-2272	494	18	buletinul	buletinul	NOUN
ejpam-2272	494	19	academiei	academiei	PROPN
ejpam-2272	494	20	de	de	X
ejpam-2272	494	21	stiinte	stiinte	PROPN
ejpam-2272	494	22	,	,	PUNCT
ejpam-2272	494	23	a	a	DET
ejpam-2272	494	24	republich	republich	PROPN
ejpam-2272	494	25	moldova	moldova	PROPN
ejpam-2272	494	26	,	,	PUNCT
ejpam-2272	494	27	matematica	matematica	PROPN
ejpam-2272	494	28	,	,	PUNCT
ejpam-2272	494	29	57(2	57(2	NUM
ejpam-2272	494	30	)	)	PUNCT
ejpam-2272	494	31	,	,	PUNCT
ejpam-2272	494	32	14–23	14–23	PROPN
ejpam-2272	494	33	.	.	NOUN
ejpam-2272	494	34	2008	2008	NUM
ejpam-2272	494	35	.	.	PUNCT
ejpam-2272	495	1	[	[	X
ejpam-2272	495	2	4	4	NUM
ejpam-2272	495	3	]	]	X
ejpam-2272	495	4	r.e	r.e	PROPN
ejpam-2272	495	5	.	.	PROPN
ejpam-2272	495	6	atani	atani	PROPN
ejpam-2272	495	7	and	and	CCONJ
ejpam-2272	495	8	s.e	s.e	PROPN
ejpam-2272	495	9	.	.	PROPN
ejpam-2272	495	10	atani	atani	PROPN
ejpam-2272	495	11	.	.	PUNCT
ejpam-2272	496	1	spectra	spectra	NOUN
ejpam-2272	496	2	of	of	ADP
ejpam-2272	496	3	semimodules	semimodule	NOUN
ejpam-2272	496	4	,	,	PUNCT
ejpam-2272	496	5	buletinul	buletinul	NOUN
ejpam-2272	496	6	academiei	academiei	PROPN
ejpam-2272	496	7	de	de	X
ejpam-2272	496	8	stiinte	stiinte	PROPN
ejpam-2272	496	9	,	,	PUNCT
ejpam-2272	496	10	a	a	DET
ejpam-2272	496	11	republich	republich	PROPN
ejpam-2272	496	12	moldova	moldova	PROPN
ejpam-2272	496	13	,	,	PUNCT
ejpam-2272	496	14	matematica	matematica	PROPN
ejpam-2272	496	15	,	,	PUNCT
ejpam-2272	496	16	3(67	3(67	NUM
ejpam-2272	496	17	)	)	PUNCT
ejpam-2272	496	18	,	,	PUNCT
ejpam-2272	496	19	15	15	NUM
ejpam-2272	496	20	-	-	SYM
ejpam-2272	496	21	28	28	NUM
ejpam-2272	496	22	.	.	PUNCT
ejpam-2272	497	1	2011	2011	NUM
ejpam-2272	497	2	.	.	PUNCT
ejpam-2272	498	1	[	[	X
ejpam-2272	498	2	5	5	NUM
ejpam-2272	498	3	]	]	X
ejpam-2272	498	4	s.e	s.e	PROPN
ejpam-2272	498	5	.	.	PROPN
ejpam-2272	498	6	atani	atani	PROPN
ejpam-2272	498	7	and	and	CCONJ
ejpam-2272	498	8	f.	f.	PROPN
ejpam-2272	498	9	farzalipour	farzalipour	PROPN
ejpam-2272	498	10	.	.	PUNCT
ejpam-2272	499	1	on	on	ADP
ejpam-2272	499	2	weakly	weakly	ADJ
ejpam-2272	499	3	primary	primary	ADJ
ejpam-2272	499	4	ideals	ideal	NOUN
ejpam-2272	499	5	,	,	PUNCT
ejpam-2272	499	6	georgian	georgian	PROPN
ejpam-2272	499	7	mathematical	mathematical	ADJ
ejpam-2272	499	8	journal	journal	NOUN
ejpam-2272	499	9	,	,	PUNCT
ejpam-2272	499	10	12(3	12(3	NUM
ejpam-2272	499	11	)	)	PUNCT
ejpam-2272	499	12	,	,	PUNCT
ejpam-2272	499	13	423–429	423–429	NUM
ejpam-2272	499	14	.	.	NOUN
ejpam-2272	499	15	2005	2005	NUM
ejpam-2272	499	16	.	.	PUNCT
ejpam-2272	500	1	references	reference	NOUN
ejpam-2272	500	2	195	195	NUM
ejpam-2272	500	3	[	[	SYM
ejpam-2272	500	4	6	6	NUM
ejpam-2272	500	5	]	]	X
ejpam-2272	500	6	s.e	s.e	PROPN
ejpam-2272	500	7	.	.	PROPN
ejpam-2272	500	8	atani	atani	PROPN
ejpam-2272	500	9	.	.	PUNCT
ejpam-2272	501	1	on	on	ADP
ejpam-2272	501	2	k−weakly	k−weakly	ADV
ejpam-2272	501	3	primary	primary	ADJ
ejpam-2272	501	4	ideals	ideal	NOUN
ejpam-2272	501	5	over	over	ADP
ejpam-2272	501	6	semirings	semiring	NOUN
ejpam-2272	501	7	,	,	PUNCT
ejpam-2272	501	8	sarajevo	sarajevo	PROPN
ejpam-2272	501	9	journal	journal	PROPN
ejpam-2272	501	10	of	of	ADP
ejpam-2272	501	11	mathematics	mathematic	NOUN
ejpam-2272	501	12	,	,	PUNCT
ejpam-2272	501	13	15(3	15(3	NUM
ejpam-2272	501	14	)	)	PUNCT
ejpam-2272	501	15	,	,	PUNCT
ejpam-2272	501	16	9–13	9–13	NOUN
ejpam-2272	501	17	.	.	PUNCT
ejpam-2272	501	18	2007	2007	NUM
ejpam-2272	501	19	.	.	PUNCT
ejpam-2272	502	1	[	[	X
ejpam-2272	502	2	7	7	NUM
ejpam-2272	502	3	]	]	PUNCT
ejpam-2272	502	4	a.	a.	NOUN
ejpam-2272	502	5	badawi	badawi	PROPN
ejpam-2272	502	6	.	.	PUNCT
ejpam-2272	503	1	on	on	ADP
ejpam-2272	503	2	2	2	NUM
ejpam-2272	503	3	-	-	PUNCT
ejpam-2272	503	4	absorbing	absorbing	ADJ
ejpam-2272	503	5	ideals	ideal	NOUN
ejpam-2272	503	6	of	of	ADP
ejpam-2272	503	7	commutative	commutative	ADJ
ejpam-2272	503	8	rings	ring	NOUN
ejpam-2272	503	9	,	,	PUNCT
ejpam-2272	503	10	bulletin	bulletin	NOUN
ejpam-2272	503	11	of	of	ADP
ejpam-2272	503	12	the	the	DET
ejpam-2272	503	13	australian	australian	ADJ
ejpam-2272	503	14	mathematical	mathematical	ADJ
ejpam-2272	503	15	society	society	NOUN
ejpam-2272	503	16	,	,	PUNCT
ejpam-2272	503	17	75	75	NUM
ejpam-2272	503	18	,	,	PUNCT
ejpam-2272	503	19	417–429	417–429	NUM
ejpam-2272	503	20	.	.	PUNCT
ejpam-2272	503	21	2007	2007	NUM
ejpam-2272	503	22	.	.	PUNCT
ejpam-2272	504	1	[	[	X
ejpam-2272	504	2	8	8	NUM
ejpam-2272	504	3	]	]	PUNCT
ejpam-2272	504	4	a.	a.	NOUN
ejpam-2272	504	5	badawi	badawi	PROPN
ejpam-2272	504	6	and	and	CCONJ
ejpam-2272	504	7	a.y	a.y	PROPN
ejpam-2272	504	8	.	.	PROPN
ejpam-2272	504	9	darani	darani	PROPN
ejpam-2272	504	10	.	.	PUNCT
ejpam-2272	505	1	on	on	ADP
ejpam-2272	505	2	weakly	weakly	ADJ
ejpam-2272	505	3	2	2	NUM
ejpam-2272	505	4	-	-	PUNCT
ejpam-2272	505	5	absorbing	absorbing	ADJ
ejpam-2272	505	6	ideals	ideal	NOUN
ejpam-2272	505	7	of	of	ADP
ejpam-2272	505	8	commutative	commutative	ADJ
ejpam-2272	505	9	rings	ring	NOUN
ejpam-2272	505	10	,	,	PUNCT
ejpam-2272	505	11	houston	houston	PROPN
ejpam-2272	505	12	journal	journal	PROPN
ejpam-2272	505	13	of	of	ADP
ejpam-2272	505	14	mathematics	mathematic	NOUN
ejpam-2272	505	15	,	,	PUNCT
ejpam-2272	505	16	39(2	39(2	NUM
ejpam-2272	505	17	)	)	PUNCT
ejpam-2272	505	18	,	,	PUNCT
ejpam-2272	505	19	441–452	441–452	NUM
ejpam-2272	505	20	.	.	NOUN
ejpam-2272	505	21	2013	2013	NUM
ejpam-2272	505	22	.	.	PUNCT
ejpam-2272	506	1	[	[	X
ejpam-2272	506	2	9	9	NUM
ejpam-2272	506	3	]	]	PUNCT
ejpam-2272	506	4	a.	a.	NOUN
ejpam-2272	506	5	badawi	badawi	PROPN
ejpam-2272	506	6	,	,	PUNCT
ejpam-2272	506	7	u.	u.	PROPN
ejpam-2272	506	8	tekir	tekir	PROPN
ejpam-2272	506	9	,	,	PUNCT
ejpam-2272	506	10	and	and	CCONJ
ejpam-2272	506	11	e.	e.	PROPN
ejpam-2272	506	12	yetkin	yetkin	PROPN
ejpam-2272	506	13	.	.	PUNCT
ejpam-2272	507	1	on	on	ADP
ejpam-2272	507	2	2	2	NUM
ejpam-2272	507	3	-	-	PUNCT
ejpam-2272	507	4	absorbing	absorbing	ADJ
ejpam-2272	507	5	primary	primary	ADJ
ejpam-2272	507	6	ideals	ideal	NOUN
ejpam-2272	507	7	in	in	ADP
ejpam-2272	507	8	commutative	commutative	ADJ
ejpam-2272	507	9	rings	ring	NOUN
ejpam-2272	507	10	,	,	PUNCT
ejpam-2272	507	11	bulletin	bulletin	NOUN
ejpam-2272	507	12	of	of	ADP
ejpam-2272	507	13	korean	korean	PROPN
ejpam-2272	507	14	mathematical	mathematical	ADJ
ejpam-2272	507	15	society	society	NOUN
ejpam-2272	507	16	,	,	PUNCT
ejpam-2272	507	17	51(4	51(4	NUM
ejpam-2272	507	18	)	)	PUNCT
ejpam-2272	507	19	,	,	PUNCT
ejpam-2272	507	20	1163–1173	1163–1173	NUM
ejpam-2272	507	21	.	.	PUNCT
ejpam-2272	507	22	2014	2014	NUM
ejpam-2272	507	23	.	.	PUNCT
ejpam-2272	508	1	[	[	X
ejpam-2272	508	2	10	10	NUM
ejpam-2272	508	3	]	]	X
ejpam-2272	508	4	a.y	a.y	PROPN
ejpam-2272	508	5	.	.	PROPN
ejpam-2272	508	6	darani	darani	PROPN
ejpam-2272	508	7	.	.	PUNCT
ejpam-2272	509	1	on	on	ADP
ejpam-2272	509	2	2	2	NUM
ejpam-2272	509	3	-	-	PUNCT
ejpam-2272	509	4	absorbing	absorbing	ADJ
ejpam-2272	509	5	and	and	CCONJ
ejpam-2272	509	6	weakly	weakly	ADJ
ejpam-2272	509	7	2	2	NUM
ejpam-2272	509	8	-	-	PUNCT
ejpam-2272	509	9	absorbing	absorbing	ADJ
ejpam-2272	509	10	ideals	ideal	NOUN
ejpam-2272	509	11	of	of	ADP
ejpam-2272	509	12	commutative	commutative	ADJ
ejpam-2272	509	13	semirings	semiring	NOUN
ejpam-2272	509	14	,	,	PUNCT
ejpam-2272	509	15	kyungpook	kyungpook	PROPN
ejpam-2272	509	16	mathematical	mathematical	ADJ
ejpam-2272	509	17	journal	journal	NOUN
ejpam-2272	509	18	,	,	PUNCT
ejpam-2272	509	19	52(1	52(1	NOUN
ejpam-2272	509	20	)	)	PUNCT
ejpam-2272	509	21	,	,	PUNCT
ejpam-2272	509	22	(	(	PUNCT
ejpam-2272	509	23	2012	2012	NUM
ejpam-2272	509	24	)	)	PUNCT
ejpam-2272	509	25	,	,	PUNCT
ejpam-2272	509	26	91–97	91–97	NUM
ejpam-2272	509	27	.	.	PUNCT
ejpam-2272	509	28	2012	2012	NUM
ejpam-2272	509	29	.	.	PUNCT
ejpam-2272	510	1	[	[	X
ejpam-2272	510	2	11	11	NUM
ejpam-2272	510	3	]	]	X
ejpam-2272	510	4	j.s	j.s	PROPN
ejpam-2272	510	5	.	.	PUNCT
ejpam-2272	510	6	golan	golan	PROPN
ejpam-2272	510	7	.	.	PUNCT
ejpam-2272	510	8	semirings	semiring	NOUN
ejpam-2272	510	9	and	and	CCONJ
ejpam-2272	510	10	their	their	PRON
ejpam-2272	510	11	applications	application	NOUN
ejpam-2272	510	12	,	,	PUNCT
ejpam-2272	510	13	kluwer	kluwer	NOUN
ejpam-2272	510	14	academic	academic	ADJ
ejpam-2272	510	15	publishers	publisher	NOUN
ejpam-2272	510	16	,	,	PUNCT
ejpam-2272	510	17	dordrecht	dordrecht	PROPN
ejpam-2272	510	18	,	,	PUNCT
ejpam-2272	510	19	1999	1999	NUM
ejpam-2272	510	20	.	.	PUNCT
ejpam-2272	511	1	[	[	X
ejpam-2272	511	2	12	12	NUM
ejpam-2272	511	3	]	]	X
ejpam-2272	511	4	r.y	r.y	PROPN
ejpam-2272	511	5	.	.	PROPN
ejpam-2272	511	6	sharp	sharp	PROPN
ejpam-2272	511	7	.	.	PUNCT
ejpam-2272	512	1	steps	step	NOUN
ejpam-2272	512	2	in	in	ADP
ejpam-2272	512	3	commutative	commutative	ADJ
ejpam-2272	512	4	algebra	algebra	NOUN
ejpam-2272	512	5	,	,	PUNCT
ejpam-2272	512	6	second	second	ADJ
ejpam-2272	512	7	edition	edition	NOUN
ejpam-2272	512	8	,	,	PUNCT
ejpam-2272	512	9	cambridge	cambridge	PROPN
ejpam-2272	512	10	university	university	PROPN
ejpam-2272	512	11	press	press	PROPN
ejpam-2272	512	12	,	,	PUNCT
ejpam-2272	512	13	cambridge	cambridge	PROPN
ejpam-2272	512	14	,	,	PUNCT
ejpam-2272	512	15	2000	2000	NUM
ejpam-2272	512	16	.	.	PUNCT
ejpam-2272	513	1	[	[	X
ejpam-2272	513	2	13	13	NUM
ejpam-2272	513	3	]	]	SYM
ejpam-2272	513	4	h.s	h.s	PROPN
ejpam-2272	513	5	.	.	PROPN
ejpam-2272	513	6	vandiver	vandiver	PROPN
ejpam-2272	513	7	.	.	PUNCT
ejpam-2272	514	1	note	note	NOUN
ejpam-2272	514	2	on	on	ADP
ejpam-2272	514	3	a	a	DET
ejpam-2272	514	4	simple	simple	ADJ
ejpam-2272	514	5	type	type	NOUN
ejpam-2272	514	6	of	of	ADP
ejpam-2272	514	7	algebra	algebra	NOUN
ejpam-2272	514	8	in	in	ADP
ejpam-2272	514	9	which	which	PRON
ejpam-2272	514	10	the	the	DET
ejpam-2272	514	11	cancellation	cancellation	NOUN
ejpam-2272	514	12	law	law	NOUN
ejpam-2272	514	13	of	of	ADP
ejpam-2272	514	14	addition	addition	NOUN
ejpam-2272	514	15	does	do	AUX
ejpam-2272	514	16	not	not	PART
ejpam-2272	514	17	hold	hold	VERB
ejpam-2272	514	18	,	,	PUNCT
ejpam-2272	514	19	bulletin	bulletin	NOUN
ejpam-2272	514	20	of	of	ADP
ejpam-2272	514	21	the	the	DET
ejpam-2272	514	22	american	american	PROPN
ejpam-2272	514	23	mathematical	mathematical	PROPN
ejpam-2272	514	24	society	society	NOUN
ejpam-2272	514	25	,	,	PUNCT
ejpam-2272	514	26	40(3	40(3	NUM
ejpam-2272	514	27	)	)	PUNCT
ejpam-2272	514	28	,	,	PUNCT
ejpam-2272	514	29	916–920	916–920	NUM
ejpam-2272	514	30	.	.	NOUN
ejpam-2272	514	31	1934	1934	NUM
ejpam-2272	514	32	.	.	PUNCT