id	sid	tid	token	lemma	pos
ejpam-4034	1	1	european	european	PROPN
ejpam-4034	1	2	journal	journal	PROPN
ejpam-4034	1	3	of	of	ADP
ejpam-4034	1	4	pure	pure	ADJ
ejpam-4034	1	5	and	and	CCONJ
ejpam-4034	1	6	applied	apply	VERB
ejpam-4034	1	7	mathematics	mathematic	NOUN
ejpam-4034	1	8	vol	vol	NOUN
ejpam-4034	1	9	.	.	PUNCT
ejpam-4034	2	1	14	14	NUM
ejpam-4034	2	2	,	,	PUNCT
ejpam-4034	2	3	no	no	INTJ
ejpam-4034	2	4	.	.	NOUN
ejpam-4034	2	5	3	3	NUM
ejpam-4034	2	6	,	,	PUNCT
ejpam-4034	2	7	2021	2021	NUM
ejpam-4034	2	8	,	,	PUNCT
ejpam-4034	2	9	989	989	NUM
ejpam-4034	2	10	-	-	PUNCT
ejpam-4034	2	11	1001	1001	NUM
ejpam-4034	2	12	issn	issn	PROPN
ejpam-4034	2	13	1307	1307	NUM
ejpam-4034	2	14	-	-	SYM
ejpam-4034	2	15	5543	5543	NUM
ejpam-4034	2	16	–	–	PUNCT
ejpam-4034	3	1	ejpam.com	ejpam.com	X
ejpam-4034	3	2	published	publish	VERB
ejpam-4034	3	3	by	by	ADP
ejpam-4034	3	4	new	new	PROPN
ejpam-4034	3	5	york	york	PROPN
ejpam-4034	3	6	business	business	PROPN
ejpam-4034	3	7	global	global	PROPN
ejpam-4034	3	8	on	on	ADP
ejpam-4034	3	9	gamma	gamma	PROPN
ejpam-4034	3	10	la	la	PROPN
ejpam-4034	3	11	-	-	PUNCT
ejpam-4034	3	12	rings	ring	NOUN
ejpam-4034	3	13	and	and	CCONJ
ejpam-4034	3	14	gamma	gamma	PROPN
ejpam-4034	3	15	la	la	PROPN
ejpam-4034	3	16	-	-	PUNCT
ejpam-4034	3	17	semirings	semirings	PROPN
ejpam-4034	3	18	waheed	waheed	PROPN
ejpam-4034	3	19	ahmad	ahmad	PROPN
ejpam-4034	3	20	khan1	khan1	PROPN
ejpam-4034	3	21	,	,	PUNCT
ejpam-4034	3	22	,	,	PUNCT
ejpam-4034	3	23	abdelghani	abdelghani	PROPN
ejpam-4034	3	24	taouti2,∗	taouti2,∗	PROPN
ejpam-4034	3	25	,	,	PUNCT
ejpam-4034	3	26	azar	azar	PROPN
ejpam-4034	3	27	salami2	salami2	PROPN
ejpam-4034	3	28	,	,	PUNCT
ejpam-4034	3	29	zahid	zahid	PROPN
ejpam-4034	3	30	hussain1	hussain1	NOUN
ejpam-4034	3	31	1	1	NUM
ejpam-4034	3	32	department	department	NOUN
ejpam-4034	3	33	of	of	ADP
ejpam-4034	3	34	mathematics	mathematic	NOUN
ejpam-4034	3	35	,	,	PUNCT
ejpam-4034	3	36	university	university	NOUN
ejpam-4034	3	37	of	of	ADP
ejpam-4034	3	38	education	education	NOUN
ejpam-4034	3	39	lahore	lahore	PROPN
ejpam-4034	3	40	attock	attock	PROPN
ejpam-4034	3	41	campus	campus	PROPN
ejpam-4034	3	42	,	,	PUNCT
ejpam-4034	3	43	pakistan	pakistan	PROPN
ejpam-4034	3	44	2	2	NUM
ejpam-4034	3	45	ets	et	NOUN
ejpam-4034	3	46	-	-	PUNCT
ejpam-4034	3	47	maths	math	NOUN
ejpam-4034	3	48	and	and	CCONJ
ejpam-4034	3	49	ns	ns	ADJ
ejpam-4034	3	50	engineering	engineering	NOUN
ejpam-4034	3	51	division	division	NOUN
ejpam-4034	3	52	,	,	PUNCT
ejpam-4034	3	53	hct	hct	PROPN
ejpam-4034	3	54	,	,	PUNCT
ejpam-4034	3	55	university	university	NOUN
ejpam-4034	3	56	city	city	NOUN
ejpam-4034	3	57	p.	p.	PROPN
ejpam-4034	3	58	o.	o.	PROPN
ejpam-4034	3	59	box	box	PROPN
ejpam-4034	3	60	7947	7947	NUM
ejpam-4034	3	61	,	,	PUNCT
ejpam-4034	3	62	sharjah	sharjah	PROPN
ejpam-4034	3	63	,	,	PUNCT
ejpam-4034	3	64	united	united	PROPN
ejpam-4034	3	65	arab	arab	PROPN
ejpam-4034	3	66	emirates	emirates	PROPN
ejpam-4034	3	67	abstract	abstract	ADJ
ejpam-4034	3	68	.	.	PUNCT
ejpam-4034	4	1	in	in	ADP
ejpam-4034	4	2	this	this	DET
ejpam-4034	4	3	note	note	NOUN
ejpam-4034	4	4	,	,	PUNCT
ejpam-4034	4	5	first	first	ADV
ejpam-4034	4	6	we	we	PRON
ejpam-4034	4	7	add	add	VERB
ejpam-4034	4	8	some	some	DET
ejpam-4034	4	9	new	new	ADJ
ejpam-4034	4	10	results	result	NOUN
ejpam-4034	4	11	in	in	ADP
ejpam-4034	4	12	gamma	gamma	PROPN
ejpam-4034	4	13	la	la	PROPN
ejpam-4034	4	14	-	-	PUNCT
ejpam-4034	4	15	rings	ring	NOUN
ejpam-4034	4	16	and	and	CCONJ
ejpam-4034	4	17	then	then	ADV
ejpam-4034	4	18	we	we	PRON
ejpam-4034	4	19	initiate	initiate	VERB
ejpam-4034	4	20	the	the	DET
ejpam-4034	4	21	notion	notion	NOUN
ejpam-4034	4	22	of	of	ADP
ejpam-4034	4	23	γ	γ	PROPN
ejpam-4034	4	24	-	-	PUNCT
ejpam-4034	4	25	la	la	ADJ
ejpam-4034	4	26	-	-	PUNCT
ejpam-4034	4	27	semirings	semiring	NOUN
ejpam-4034	4	28	.	.	PUNCT
ejpam-4034	5	1	moreover	moreover	ADV
ejpam-4034	5	2	,	,	PUNCT
ejpam-4034	5	3	we	we	PRON
ejpam-4034	5	4	introduce	introduce	VERB
ejpam-4034	5	5	and	and	CCONJ
ejpam-4034	5	6	discuss	discuss	VERB
ejpam-4034	5	7	the	the	DET
ejpam-4034	5	8	terms	term	NOUN
ejpam-4034	5	9	left	leave	VERB
ejpam-4034	5	10	ideals	ideal	NOUN
ejpam-4034	5	11	,	,	PUNCT
ejpam-4034	5	12	right	right	ADJ
ejpam-4034	5	13	ideals	ideal	NOUN
ejpam-4034	5	14	,	,	PUNCT
ejpam-4034	5	15	bi	bi	NOUN
ejpam-4034	5	16	-	-	ADJ
ejpam-4034	5	17	ideal	ideal	ADJ
ejpam-4034	5	18	,	,	PUNCT
ejpam-4034	5	19	quasi	quasi	NOUN
ejpam-4034	5	20	ideals	ideal	NOUN
ejpam-4034	5	21	,	,	PUNCT
ejpam-4034	5	22	almost	almost	ADV
ejpam-4034	5	23	prime	prime	ADJ
ejpam-4034	5	24	and	and	CCONJ
ejpam-4034	5	25	weakly	weakly	ADJ
ejpam-4034	5	26	almost	almost	ADV
ejpam-4034	5	27	prime	prime	ADJ
ejpam-4034	5	28	ideals	ideal	NOUN
ejpam-4034	5	29	of	of	ADP
ejpam-4034	5	30	a	a	DET
ejpam-4034	5	31	γ	γ	X
ejpam-4034	5	32	-	-	PUNCT
ejpam-4034	5	33	la	la	ADV
ejpam-4034	5	34	-	-	PUNCT
ejpam-4034	5	35	semiring	semiring	NOUN
ejpam-4034	5	36	and	and	CCONJ
ejpam-4034	5	37	their	their	PRON
ejpam-4034	5	38	characterizations	characterization	NOUN
ejpam-4034	5	39	.	.	PUNCT
ejpam-4034	6	1	2020	2020	NUM
ejpam-4034	6	2	mathematics	mathematic	NOUN
ejpam-4034	6	3	subject	subject	NOUN
ejpam-4034	6	4	classifications	classification	NOUN
ejpam-4034	6	5	:	:	PUNCT
ejpam-4034	6	6	17d20	17d20	NUM
ejpam-4034	6	7	,	,	PUNCT
ejpam-4034	6	8	16y60	16y60	NUM
ejpam-4034	6	9	key	key	ADJ
ejpam-4034	6	10	words	word	NOUN
ejpam-4034	6	11	and	and	CCONJ
ejpam-4034	6	12	phrases	phrase	NOUN
ejpam-4034	6	13	:	:	PUNCT
ejpam-4034	6	14	gamma	gamma	PROPN
ejpam-4034	6	15	la	la	PROPN
ejpam-4034	6	16	-	-	PUNCT
ejpam-4034	6	17	rings	ring	NOUN
ejpam-4034	6	18	,	,	PUNCT
ejpam-4034	6	19	gamma	gamma	PROPN
ejpam-4034	6	20	la	la	PROPN
ejpam-4034	6	21	-	-	PUNCT
ejpam-4034	6	22	semirings	semiring	NOUN
ejpam-4034	6	23	,	,	PUNCT
ejpam-4034	6	24	quasi	quasi	ADJ
ejpam-4034	6	25	ideals	ideal	NOUN
ejpam-4034	6	26	of	of	ADP
ejpam-4034	6	27	a	a	DET
ejpam-4034	6	28	gammala	gammala	NOUN
ejpam-4034	6	29	-	-	PUNCT
ejpam-4034	6	30	semiring	semiring	NOUN
ejpam-4034	6	31	,	,	PUNCT
ejpam-4034	6	32	weakly	weakly	ADJ
ejpam-4034	6	33	almost	almost	ADV
ejpam-4034	6	34	prime	prime	ADJ
ejpam-4034	6	35	ideals	ideal	NOUN
ejpam-4034	6	36	of	of	ADP
ejpam-4034	6	37	a	a	DET
ejpam-4034	6	38	gamma	gamma	NOUN
ejpam-4034	6	39	-	-	PUNCT
ejpam-4034	6	40	la	la	NOUN
ejpam-4034	6	41	-	-	PUNCT
ejpam-4034	6	42	semiring	semire	VERB
ejpam-4034	6	43	1	1	NUM
ejpam-4034	6	44	.	.	PUNCT
ejpam-4034	7	1	introduction	introduction	NOUN
ejpam-4034	7	2	γ	γ	PROPN
ejpam-4034	7	3	-ring	-ring	PROPN
ejpam-4034	7	4	was	be	AUX
ejpam-4034	7	5	introduced	introduce	VERB
ejpam-4034	7	6	by	by	ADP
ejpam-4034	7	7	n.	n.	PROPN
ejpam-4034	7	8	nobusawa	nobusawa	PROPN
ejpam-4034	7	9	in	in	ADP
ejpam-4034	7	10	[	[	X
ejpam-4034	7	11	13	13	NUM
ejpam-4034	7	12	]	]	PUNCT
ejpam-4034	7	13	as	as	ADP
ejpam-4034	7	14	a	a	DET
ejpam-4034	7	15	generalization	generalization	NOUN
ejpam-4034	7	16	of	of	ADP
ejpam-4034	7	17	classical	classical	ADJ
ejpam-4034	7	18	rings	ring	NOUN
ejpam-4034	7	19	.	.	PUNCT
ejpam-4034	8	1	γ	γ	PROPN
ejpam-4034	8	2	-rings	-ring	NOUN
ejpam-4034	8	3	have	have	AUX
ejpam-4034	8	4	also	also	ADV
ejpam-4034	8	5	viewed	view	VERB
ejpam-4034	8	6	as	as	ADP
ejpam-4034	8	7	the	the	DET
ejpam-4034	8	8	connection	connection	NOUN
ejpam-4034	8	9	with	with	ADP
ejpam-4034	8	10	the	the	DET
ejpam-4034	8	11	abelian	abelian	ADJ
ejpam-4034	8	12	additive	additive	ADJ
ejpam-4034	8	13	groups	group	NOUN
ejpam-4034	8	14	of	of	ADP
ejpam-4034	8	15	all	all	DET
ejpam-4034	8	16	linear	linear	ADJ
ejpam-4034	8	17	mappings	mapping	NOUN
ejpam-4034	8	18	between	between	ADP
ejpam-4034	8	19	two	two	NUM
ejpam-4034	8	20	finite	finite	ADJ
ejpam-4034	8	21	dimensional	dimensional	ADJ
ejpam-4034	8	22	spaces	space	NOUN
ejpam-4034	8	23	over	over	ADP
ejpam-4034	8	24	a	a	DET
ejpam-4034	8	25	field	field	NOUN
ejpam-4034	8	26	.	.	PUNCT
ejpam-4034	9	1	classical	classical	ADJ
ejpam-4034	9	2	example	example	NOUN
ejpam-4034	9	3	of	of	ADP
ejpam-4034	9	4	γ	γ	X
ejpam-4034	9	5	-ring	-ring	PROPN
ejpam-4034	9	6	presented	present	VERB
ejpam-4034	9	7	by	by	ADP
ejpam-4034	9	8	nobusawa	nobusawa	PROPN
ejpam-4034	9	9	was	be	AUX
ejpam-4034	9	10	by	by	ADP
ejpam-4034	9	11	taking	take	VERB
ejpam-4034	9	12	an	an	DET
ejpam-4034	9	13	additive	additive	ADJ
ejpam-4034	9	14	group	group	NOUN
ejpam-4034	9	15	m	m	PROPN
ejpam-4034	9	16	consisting	consist	VERB
ejpam-4034	9	17	of	of	ADP
ejpam-4034	9	18	homomorphisms	homomorphism	NOUN
ejpam-4034	9	19	of	of	ADP
ejpam-4034	9	20	a	a	DET
ejpam-4034	9	21	module	module	NOUN
ejpam-4034	9	22	a	a	PRON
ejpam-4034	9	23	to	to	ADP
ejpam-4034	9	24	a	a	DET
ejpam-4034	9	25	module	module	NOUN
ejpam-4034	9	26	b	b	NOUN
ejpam-4034	9	27	and	and	CCONJ
ejpam-4034	9	28	an	an	DET
ejpam-4034	9	29	additive	additive	ADJ
ejpam-4034	9	30	group	group	NOUN
ejpam-4034	9	31	γ	γ	NOUN
ejpam-4034	9	32	consisting	consist	VERB
ejpam-4034	9	33	of	of	ADP
ejpam-4034	9	34	homomorphisms	homomorphism	NOUN
ejpam-4034	9	35	of	of	ADP
ejpam-4034	9	36	b	b	PROPN
ejpam-4034	9	37	to	to	ADP
ejpam-4034	9	38	a	a	PRON
ejpam-4034	9	39	,	,	PUNCT
ejpam-4034	9	40	and	and	CCONJ
ejpam-4034	9	41	aαb	aαb	VERB
ejpam-4034	9	42	the	the	DET
ejpam-4034	9	43	usual	usual	ADJ
ejpam-4034	9	44	composite	composite	ADJ
ejpam-4034	9	45	map	map	NOUN
ejpam-4034	9	46	,	,	PUNCT
ejpam-4034	9	47	where	where	SCONJ
ejpam-4034	9	48	a	a	PRON
ejpam-4034	9	49	,	,	PUNCT
ejpam-4034	9	50	b	b	X
ejpam-4034	9	51	∈	∈	NOUN
ejpam-4034	9	52	m	m	VERB
ejpam-4034	9	53	and	and	CCONJ
ejpam-4034	9	54	α	α	PRON
ejpam-4034	9	55	∈	∈	PROPN
ejpam-4034	9	56	γ	γ	X
ejpam-4034	9	57	.	.	PROPN
ejpam-4034	9	58	barnes	barnes	PROPN
ejpam-4034	9	59	introduced	introduce	VERB
ejpam-4034	9	60	radical	radical	ADJ
ejpam-4034	9	61	theory	theory	NOUN
ejpam-4034	9	62	of	of	ADP
ejpam-4034	9	63	γ	γ	NOUN
ejpam-4034	9	64	-	-	PUNCT
ejpam-4034	9	65	rings	ring	NOUN
ejpam-4034	9	66	in	in	ADP
ejpam-4034	9	67	[	[	X
ejpam-4034	9	68	1	1	NUM
ejpam-4034	9	69	]	]	PUNCT
ejpam-4034	9	70	.	.	PUNCT
ejpam-4034	10	1	afterwards	afterwards	ADV
ejpam-4034	10	2	,	,	PUNCT
ejpam-4034	10	3	numbers	number	NOUN
ejpam-4034	10	4	of	of	ADP
ejpam-4034	10	5	researchers	researcher	NOUN
ejpam-4034	10	6	have	have	AUX
ejpam-4034	10	7	been	be	AUX
ejpam-4034	10	8	published	publish	VERB
ejpam-4034	10	9	their	their	PRON
ejpam-4034	10	10	research	research	NOUN
ejpam-4034	10	11	articles	article	NOUN
ejpam-4034	10	12	on	on	ADP
ejpam-4034	10	13	γ	γ	NOUN
ejpam-4034	10	14	-	-	PUNCT
ejpam-4034	10	15	rings	ring	NOUN
ejpam-4034	10	16	.	.	PUNCT
ejpam-4034	11	1	similarly	similarly	ADV
ejpam-4034	11	2	,	,	PUNCT
ejpam-4034	11	3	γ	γ	NOUN
ejpam-4034	11	4	-	-	PUNCT
ejpam-4034	11	5	nearrings	nearring	NOUN
ejpam-4034	11	6	were	be	AUX
ejpam-4034	11	7	introduced	introduce	VERB
ejpam-4034	11	8	by	by	ADP
ejpam-4034	11	9	satyanarayana	satyanarayana	PROPN
ejpam-4034	11	10	in	in	ADP
ejpam-4034	11	11	[	[	X
ejpam-4034	11	12	17	17	NUM
ejpam-4034	11	13	]	]	PUNCT
ejpam-4034	11	14	.	.	PUNCT
ejpam-4034	12	1	booth	booth	NOUN
ejpam-4034	12	2	et	et	PROPN
ejpam-4034	12	3	al	al	PROPN
ejpam-4034	12	4	.	.	PROPN
ejpam-4034	12	5	provided	provide	VERB
ejpam-4034	12	6	different	different	ADJ
ejpam-4034	12	7	ways	way	NOUN
ejpam-4034	12	8	to	to	PART
ejpam-4034	12	9	construct	construct	VERB
ejpam-4034	12	10	equiprime	equiprime	PROPN
ejpam-4034	12	11	γ	γ	NOUN
ejpam-4034	12	12	-	-	PUNCT
ejpam-4034	12	13	nearrings	nearring	NOUN
ejpam-4034	12	14	[	[	X
ejpam-4034	12	15	2	2	NUM
ejpam-4034	12	16	]	]	PUNCT
ejpam-4034	12	17	.	.	PUNCT
ejpam-4034	13	1	γsemirings	γsemiring	NOUN
ejpam-4034	13	2	were	be	AUX
ejpam-4034	13	3	introduced	introduce	VERB
ejpam-4034	13	4	by	by	ADP
ejpam-4034	13	5	rao	rao	NOUN
ejpam-4034	13	6	in	in	ADP
ejpam-4034	13	7	[	[	X
ejpam-4034	13	8	14	14	NUM
ejpam-4034	13	9	]	]	PUNCT
ejpam-4034	13	10	.	.	PUNCT
ejpam-4034	14	1	prime	prime	ADJ
ejpam-4034	14	2	and	and	CCONJ
ejpam-4034	14	3	semi	semi	ADJ
ejpam-4034	14	4	-	-	ADJ
ejpam-4034	14	5	prime	prime	ADJ
ejpam-4034	14	6	ideals	ideal	NOUN
ejpam-4034	14	7	of	of	ADP
ejpam-4034	14	8	of	of	ADP
ejpam-4034	14	9	γ	γ	NOUN
ejpam-4034	14	10	-	-	PUNCT
ejpam-4034	14	11	semirings	semiring	NOUN
ejpam-4034	14	12	were	be	AUX
ejpam-4034	14	13	discussed	discuss	VERB
ejpam-4034	14	14	in	in	ADP
ejpam-4034	14	15	[	[	X
ejpam-4034	14	16	5	5	NUM
ejpam-4034	14	17	,	,	PUNCT
ejpam-4034	14	18	6	6	NUM
ejpam-4034	14	19	]	]	PUNCT
ejpam-4034	14	20	.	.	PUNCT
ejpam-4034	15	1	moreover	moreover	ADV
ejpam-4034	15	2	,	,	PUNCT
ejpam-4034	15	3	quasi	quasi	NOUN
ejpam-4034	15	4	-	-	NOUN
ejpam-4034	15	5	ideals	ideal	NOUN
ejpam-4034	15	6	in	in	ADP
ejpam-4034	15	7	γ	γ	NOUN
ejpam-4034	15	8	-	-	PUNCT
ejpam-4034	15	9	semiring	semiring	NOUN
ejpam-4034	15	10	were	be	AUX
ejpam-4034	15	11	discussed	discuss	VERB
ejpam-4034	15	12	in	in	ADP
ejpam-4034	15	13	[	[	X
ejpam-4034	15	14	7	7	NUM
ejpam-4034	15	15	,	,	PUNCT
ejpam-4034	15	16	8	8	NUM
ejpam-4034	15	17	]	]	PUNCT
ejpam-4034	15	18	.	.	PUNCT
ejpam-4034	16	1	the	the	DET
ejpam-4034	16	2	properties	property	NOUN
ejpam-4034	16	3	of	of	ADP
ejpam-4034	16	4	ideals	ideal	NOUN
ejpam-4034	16	5	,	,	PUNCT
ejpam-4034	16	6	prime	prime	ADJ
ejpam-4034	16	7	ideals	ideal	NOUN
ejpam-4034	16	8	,	,	PUNCT
ejpam-4034	16	9	semi	semi	ADJ
ejpam-4034	16	10	-	-	ADJ
ejpam-4034	16	11	prime	prime	ADJ
ejpam-4034	16	12	ideals	ideal	NOUN
ejpam-4034	16	13	and	and	CCONJ
ejpam-4034	16	14	their	their	PRON
ejpam-4034	16	15	generalization	generalization	NOUN
ejpam-4034	16	16	plays	play	VERB
ejpam-4034	16	17	a	a	DET
ejpam-4034	16	18	key	key	ADJ
ejpam-4034	16	19	role	role	NOUN
ejpam-4034	16	20	in	in	ADP
ejpam-4034	16	21	structure	structure	NOUN
ejpam-4034	16	22	of	of	ADP
ejpam-4034	16	23	γ	γ	NOUN
ejpam-4034	16	24	-	-	NOUN
ejpam-4034	16	25	semirings	semiring	NOUN
ejpam-4034	16	26	.	.	PUNCT
ejpam-4034	17	1	however	however	ADV
ejpam-4034	17	2	,	,	PUNCT
ejpam-4034	17	3	the	the	DET
ejpam-4034	17	4	properties	property	NOUN
ejpam-4034	17	5	of	of	ADP
ejpam-4034	17	6	an	an	DET
ejpam-4034	17	7	ideal	ideal	NOUN
ejpam-4034	17	8	in	in	ADP
ejpam-4034	17	9	semirings	semiring	NOUN
ejpam-4034	17	10	and	and	CCONJ
ejpam-4034	17	11	γ	γ	NOUN
ejpam-4034	17	12	-	-	PUNCT
ejpam-4034	17	13	semirings	semiring	NOUN
ejpam-4034	17	14	are	be	AUX
ejpam-4034	17	15	slightly	slightly	ADV
ejpam-4034	17	16	changed	change	VERB
ejpam-4034	17	17	from	from	ADP
ejpam-4034	17	18	the	the	DET
ejpam-4034	17	19	properties	property	NOUN
ejpam-4034	17	20	of	of	ADP
ejpam-4034	17	21	the	the	DET
ejpam-4034	17	22	usual	usual	ADJ
ejpam-4034	17	23	ring	ring	NOUN
ejpam-4034	17	24	ideals	ideal	NOUN
ejpam-4034	17	25	.	.	PUNCT
ejpam-4034	18	1	theory	theory	NOUN
ejpam-4034	18	2	of	of	ADP
ejpam-4034	18	3	ideals	ideal	NOUN
ejpam-4034	18	4	in	in	ADP
ejpam-4034	18	5	an	an	DET
ejpam-4034	18	6	ordered	order	VERB
ejpam-4034	18	7	γ	γ	NOUN
ejpam-4034	18	8	-	-	PUNCT
ejpam-4034	18	9	semiring	semiring	NOUN
ejpam-4034	18	10	have	have	AUX
ejpam-4034	18	11	been	be	AUX
ejpam-4034	18	12	introduced	introduce	VERB
ejpam-4034	18	13	in	in	ADP
ejpam-4034	18	14	[	[	X
ejpam-4034	18	15	15	15	NUM
ejpam-4034	18	16	]	]	PUNCT
ejpam-4034	18	17	.	.	PUNCT
ejpam-4034	19	1	similarly	similarly	ADV
ejpam-4034	19	2	,	,	PUNCT
ejpam-4034	19	3	weakly	weakly	ADV
ejpam-4034	19	4	prime	prime	ADJ
ejpam-4034	19	5	and	and	CCONJ
ejpam-4034	19	6	weakly	weakly	ADJ
ejpam-4034	19	7	primary	primary	ADJ
ejpam-4034	19	8	ideals	ideal	NOUN
ejpam-4034	19	9	in	in	ADP
ejpam-4034	19	10	gamma	gamma	NOUN
ejpam-4034	19	11	seminearrings	seminearring	NOUN
ejpam-4034	19	12	have	have	AUX
ejpam-4034	19	13	been	be	AUX
ejpam-4034	19	14	introduced	introduce	VERB
ejpam-4034	19	15	in	in	ADP
ejpam-4034	19	16	[	[	X
ejpam-4034	19	17	9	9	NUM
ejpam-4034	19	18	]	]	PUNCT
ejpam-4034	19	19	.	.	PUNCT
ejpam-4034	20	1	∗corresponding	∗corresponde	VERB
ejpam-4034	20	2	author	author	NOUN
ejpam-4034	20	3	.	.	PUNCT
ejpam-4034	21	1	doi	doi	NOUN
ejpam-4034	21	2	:	:	PUNCT
ejpam-4034	21	3	https://doi.org/10.29020/nybg.ejpam.v14i3.4034	https://doi.org/10.29020/nybg.ejpam.v14i3.4034	NOUN
ejpam-4034	21	4	email	email	NOUN
ejpam-4034	21	5	addresses	address	NOUN
ejpam-4034	21	6	:	:	PUNCT
ejpam-4034	21	7	sirwak2003@yahoo.com	sirwak2003@yahoo.com	X
ejpam-4034	21	8	(	(	PUNCT
ejpam-4034	21	9	w.	w.	PROPN
ejpam-4034	21	10	a.	a.	PROPN
ejpam-4034	21	11	khan	khan	PROPN
ejpam-4034	21	12	)	)	PUNCT
ejpam-4034	21	13	,	,	PUNCT
ejpam-4034	21	14	ganitaouti@yahoo.com.au	ganitaouti@yahoo.com.au	PROPN
ejpam-4034	21	15	(	(	PUNCT
ejpam-4034	21	16	ab	ab	PROPN
ejpam-4034	21	17	.	.	PUNCT
ejpam-4034	21	18	taouti	taouti	PROPN
ejpam-4034	21	19	)	)	PUNCT
ejpam-4034	21	20	,	,	PUNCT
ejpam-4034	21	21	asalami@hct.ac.ae	asalami@hct.ac.ae	NOUN
ejpam-4034	21	22	(	(	PUNCT
ejpam-4034	21	23	a.	a.	NOUN
ejpam-4034	21	24	salami	salami	PROPN
ejpam-4034	21	25	)	)	PUNCT
ejpam-4034	21	26	,	,	PUNCT
ejpam-4034	21	27	zahidbsc449@gmail.com	zahidbsc449@gmail.com	NUM
ejpam-4034	21	28	(	(	PUNCT
ejpam-4034	21	29	z.	z.	PROPN
ejpam-4034	21	30	hussain	hussain	PROPN
ejpam-4034	21	31	)	)	PUNCT
ejpam-4034	21	32	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4034	22	1	989	989	NUM
ejpam-4034	22	2	c	c	X
ejpam-4034	22	3	©	©	PROPN
ejpam-4034	22	4	2021	2021	NUM
ejpam-4034	22	5	ejpam	ejpam	VERB
ejpam-4034	22	6	all	all	DET
ejpam-4034	22	7	rights	right	NOUN
ejpam-4034	22	8	reserved	reserve	VERB
ejpam-4034	22	9	.	.	PUNCT
ejpam-4034	23	1	w.	w.	PROPN
ejpam-4034	23	2	a.	a.	PROPN
ejpam-4034	23	3	khan	khan	PROPN
ejpam-4034	23	4	et	et	PROPN
ejpam-4034	23	5	al	al	PROPN
ejpam-4034	23	6	.	.	PUNCT
ejpam-4034	23	7	/	/	SYM
ejpam-4034	23	8	eur	eur	PROPN
ejpam-4034	23	9	.	.	PUNCT
ejpam-4034	24	1	j.	j.	PROPN
ejpam-4034	24	2	pure	pure	PROPN
ejpam-4034	24	3	appl	appl	PROPN
ejpam-4034	24	4	.	.	PROPN
ejpam-4034	24	5	math	math	PROPN
ejpam-4034	24	6	,	,	PUNCT
ejpam-4034	24	7	14	14	NUM
ejpam-4034	24	8	(	(	PUNCT
ejpam-4034	24	9	3	3	NUM
ejpam-4034	24	10	)	)	PUNCT
ejpam-4034	24	11	(	(	PUNCT
ejpam-4034	24	12	2021	2021	NUM
ejpam-4034	24	13	)	)	PUNCT
ejpam-4034	24	14	,	,	PUNCT
ejpam-4034	24	15	989	989	NUM
ejpam-4034	24	16	-	-	SYM
ejpam-4034	24	17	1001	1001	NUM
ejpam-4034	24	18	990	990	NUM
ejpam-4034	24	19	a	a	DET
ejpam-4034	24	20	groupoid	groupoid	NOUN
ejpam-4034	24	21	which	which	PRON
ejpam-4034	24	22	satisfies	satisfy	VERB
ejpam-4034	24	23	the	the	DET
ejpam-4034	24	24	left	left	ADJ
ejpam-4034	24	25	invertive	invertive	ADJ
ejpam-4034	24	26	law	law	NOUN
ejpam-4034	24	27	i.e.	i.e.	X
ejpam-4034	24	28	,	,	PUNCT
ejpam-4034	24	29	(	(	PUNCT
ejpam-4034	24	30	xy)z	xy)z	SYM
ejpam-4034	24	31	=	=	SYM
ejpam-4034	24	32	(	(	PUNCT
ejpam-4034	24	33	zy)x	zy)x	PROPN
ejpam-4034	24	34	is	be	AUX
ejpam-4034	24	35	said	say	VERB
ejpam-4034	24	36	to	to	PART
ejpam-4034	24	37	be	be	AUX
ejpam-4034	24	38	an	an	DET
ejpam-4034	24	39	lagroupoid	lagroupoid	NOUN
ejpam-4034	24	40	.	.	PUNCT
ejpam-4034	25	1	a	a	DET
ejpam-4034	25	2	groupoid	groupoid	NOUN
ejpam-4034	25	3	satisfying	satisfy	VERB
ejpam-4034	25	4	the	the	DET
ejpam-4034	25	5	medial	medial	ADJ
ejpam-4034	25	6	law	law	NOUN
ejpam-4034	25	7	i.e.	i.e.	X
ejpam-4034	25	8	,	,	PUNCT
ejpam-4034	25	9	(	(	PUNCT
ejpam-4034	25	10	xy)(zt	xy)(zt	X
ejpam-4034	25	11	)	)	PUNCT
ejpam-4034	25	12	=	=	SYM
ejpam-4034	25	13	(	(	PUNCT
ejpam-4034	25	14	xz)(yt	xz)(yt	PROPN
ejpam-4034	25	15	)	)	PUNCT
ejpam-4034	25	16	holds	hold	VERB
ejpam-4034	25	17	by	by	ADP
ejpam-4034	25	18	groupoid	groupoid	PROPN
ejpam-4034	25	19	is	be	AUX
ejpam-4034	25	20	called	call	VERB
ejpam-4034	25	21	medial	medial	ADJ
ejpam-4034	25	22	[	[	X
ejpam-4034	25	23	3	3	NUM
ejpam-4034	25	24	]	]	PUNCT
ejpam-4034	25	25	,	,	PUNCT
ejpam-4034	25	26	whereas	whereas	SCONJ
ejpam-4034	25	27	a	a	DET
ejpam-4034	25	28	groupoid	groupoid	NOUN
ejpam-4034	25	29	which	which	PRON
ejpam-4034	25	30	satisfying	satisfy	VERB
ejpam-4034	25	31	the	the	DET
ejpam-4034	25	32	paramedial	paramedial	ADJ
ejpam-4034	25	33	law	law	NOUN
ejpam-4034	25	34	i.e.	i.e.	X
ejpam-4034	25	35	,	,	PUNCT
ejpam-4034	25	36	(	(	PUNCT
ejpam-4034	25	37	st)(uv	st)(uv	NOUN
ejpam-4034	25	38	)	)	PUNCT
ejpam-4034	25	39	=	=	SYM
ejpam-4034	25	40	(	(	PUNCT
ejpam-4034	25	41	vt)(us	vt)(us	X
ejpam-4034	25	42	)	)	PUNCT
ejpam-4034	25	43	is	be	AUX
ejpam-4034	25	44	a	a	DET
ejpam-4034	25	45	paramedial	paramedial	NOUN
ejpam-4034	25	46	.	.	PUNCT
ejpam-4034	26	1	la	la	ADJ
ejpam-4034	26	2	-	-	PUNCT
ejpam-4034	26	3	groupoid	groupoid	PROPN
ejpam-4034	26	4	s	s	VERB
ejpam-4034	26	5	always	always	ADV
ejpam-4034	26	6	obeys	obey	VERB
ejpam-4034	26	7	medial	medial	ADJ
ejpam-4034	26	8	law	law	NOUN
ejpam-4034	26	9	,	,	PUNCT
ejpam-4034	26	10	whereas	whereas	SCONJ
ejpam-4034	26	11	paramedial	paramedial	ADJ
ejpam-4034	26	12	law	law	NOUN
ejpam-4034	26	13	holds	hold	VERB
ejpam-4034	26	14	only	only	ADV
ejpam-4034	26	15	by	by	ADP
ejpam-4034	26	16	la	la	ADJ
ejpam-4034	26	17	-	-	NOUN
ejpam-4034	26	18	groupoid	groupoid	PROPN
ejpam-4034	26	19	s	s	PROPN
ejpam-4034	26	20	with	with	ADP
ejpam-4034	26	21	left	left	ADJ
ejpam-4034	26	22	identity	identity	NOUN
ejpam-4034	26	23	e	e	NOUN
ejpam-4034	27	1	[	[	X
ejpam-4034	27	2	3	3	NUM
ejpam-4034	27	3	]	]	PUNCT
ejpam-4034	27	4	.	.	PUNCT
ejpam-4034	28	1	la	la	ADJ
ejpam-4034	28	2	-	-	PUNCT
ejpam-4034	28	3	groupoid	groupoid	PROPN
ejpam-4034	28	4	s	s	AUX
ejpam-4034	28	5	having	have	VERB
ejpam-4034	28	6	e	e	NOUN
ejpam-4034	28	7	as	as	ADP
ejpam-4034	28	8	a	a	DET
ejpam-4034	28	9	left	left	ADJ
ejpam-4034	28	10	identity	identity	NOUN
ejpam-4034	28	11	holds	hold	VERB
ejpam-4034	28	12	p(qr	p(qr	NOUN
ejpam-4034	28	13	)	)	PUNCT
ejpam-4034	28	14	=	=	PUNCT
ejpam-4034	29	1	q(pr	q(pr	NOUN
ejpam-4034	29	2	)	)	PUNCT
ejpam-4034	30	1	[	[	X
ejpam-4034	30	2	12	12	NUM
ejpam-4034	30	3	]	]	PUNCT
ejpam-4034	30	4	,	,	PUNCT
ejpam-4034	30	5	a	a	DET
ejpam-4034	30	6	∈	∈	NOUN
ejpam-4034	30	7	s	s	PART
ejpam-4034	30	8	is	be	AUX
ejpam-4034	30	9	left	leave	VERB
ejpam-4034	30	10	(	(	PUNCT
ejpam-4034	30	11	right	right	ADJ
ejpam-4034	30	12	)	)	PUNCT
ejpam-4034	30	13	cancellative	cancellative	ADJ
ejpam-4034	30	14	if	if	SCONJ
ejpam-4034	30	15	al	al	PROPN
ejpam-4034	30	16	=	=	PRON
ejpam-4034	30	17	am	be	AUX
ejpam-4034	30	18	⇒	⇒	NOUN
ejpam-4034	30	19	l	l	PROPN
ejpam-4034	31	1	=	=	PUNCT
ejpam-4034	31	2	m	m	PROPN
ejpam-4034	31	3	(	(	PUNCT
ejpam-4034	31	4	la	la	X
ejpam-4034	31	5	=	=	SYM
ejpam-4034	31	6	ma	ma	PROPN
ejpam-4034	31	7	⇒	⇒	PROPN
ejpam-4034	31	8	l	l	PROPN
ejpam-4034	32	1	=	=	PUNCT
ejpam-4034	32	2	m	m	NOUN
ejpam-4034	32	3	)	)	PUNCT
ejpam-4034	32	4	∀	∀	PUNCT
ejpam-4034	33	1	l	l	NOUN
ejpam-4034	33	2	,	,	PUNCT
ejpam-4034	33	3	m	m	VERB
ejpam-4034	33	4	∈	∈	PROPN
ejpam-4034	33	5	s.	s.	PROPN
ejpam-4034	33	6	if	if	SCONJ
ejpam-4034	33	7	every	every	DET
ejpam-4034	33	8	element	element	NOUN
ejpam-4034	33	9	is	be	AUX
ejpam-4034	33	10	left	leave	VERB
ejpam-4034	33	11	and	and	CCONJ
ejpam-4034	33	12	right	right	ADV
ejpam-4034	33	13	cancellative	cancellative	ADJ
ejpam-4034	33	14	then	then	ADV
ejpam-4034	33	15	s	s	VERB
ejpam-4034	33	16	is	be	AUX
ejpam-4034	33	17	cancellative	cancellative	ADJ
ejpam-4034	33	18	and	and	CCONJ
ejpam-4034	33	19	x	x	SYM
ejpam-4034	33	20	∈	∈	NOUN
ejpam-4034	33	21	s	s	VERB
ejpam-4034	33	22	is	be	AUX
ejpam-4034	33	23	cancellative	cancellative	ADJ
ejpam-4034	33	24	if	if	SCONJ
ejpam-4034	33	25	x	x	PRON
ejpam-4034	33	26	is	be	AUX
ejpam-4034	33	27	left	leave	VERB
ejpam-4034	33	28	and	and	CCONJ
ejpam-4034	33	29	right	right	ADV
ejpam-4034	33	30	cancellative	cancellative	ADJ
ejpam-4034	33	31	.	.	PUNCT
ejpam-4034	34	1	the	the	DET
ejpam-4034	34	2	notion	notion	NOUN
ejpam-4034	34	3	la	la	PROPN
ejpam-4034	34	4	-	-	NOUN
ejpam-4034	34	5	groupoid	groupoid	PROPN
ejpam-4034	34	6	to	to	ADP
ejpam-4034	34	7	la	la	PROPN
ejpam-4034	34	8	-	-	PUNCT
ejpam-4034	34	9	group	group	NOUN
ejpam-4034	34	10	was	be	AUX
ejpam-4034	34	11	extended	extend	VERB
ejpam-4034	34	12	by	by	ADP
ejpam-4034	34	13	kamran	kamran	PROPN
ejpam-4034	35	1	[	[	X
ejpam-4034	35	2	16	16	NUM
ejpam-4034	35	3	]	]	PUNCT
ejpam-4034	35	4	.	.	PUNCT
ejpam-4034	36	1	similarly	similarly	ADV
ejpam-4034	36	2	,	,	PUNCT
ejpam-4034	36	3	if	if	SCONJ
ejpam-4034	36	4	e	e	PROPN
ejpam-4034	36	5	is	be	AUX
ejpam-4034	36	6	left	leave	VERB
ejpam-4034	36	7	identity	identity	NOUN
ejpam-4034	36	8	in	in	ADP
ejpam-4034	36	9	la	la	ADJ
ejpam-4034	36	10	-	-	NOUN
ejpam-4034	36	11	groupoid	groupoid	PROPN
ejpam-4034	36	12	(	(	PUNCT
ejpam-4034	36	13	i	i	NOUN
ejpam-4034	36	14	-	-	PUNCT
ejpam-4034	36	15	e	e	VERB
ejpam-4034	36	16	em	em	NOUN
ejpam-4034	36	17	=	=	PUNCT
ejpam-4034	36	18	m	m	VERB
ejpam-4034	36	19	∀	∀	VERB
ejpam-4034	36	20	m	m	VERB
ejpam-4034	36	21	∈	∈	NOUN
ejpam-4034	36	22	s	s	NOUN
ejpam-4034	36	23	)	)	PUNCT
ejpam-4034	36	24	and	and	CCONJ
ejpam-4034	36	25	∀	∀	NUM
ejpam-4034	37	1	m	m	VERB
ejpam-4034	37	2	∈	∈	NOUN
ejpam-4034	37	3	s	s	PART
ejpam-4034	37	4	∃	∃	NOUN
ejpam-4034	37	5	m−1	m−1	PROPN
ejpam-4034	37	6	∈	∈	PROPN
ejpam-4034	37	7	s	s	VERB
ejpam-4034	37	8	such	such	ADJ
ejpam-4034	37	9	that	that	SCONJ
ejpam-4034	37	10	m−1	m−1	PROPN
ejpam-4034	37	11	m	m	PROPN
ejpam-4034	37	12	=	=	NOUN
ejpam-4034	37	13	mm−1	mm−1	PROPN
ejpam-4034	37	14	=	=	SYM
ejpam-4034	37	15	e	e	NOUN
ejpam-4034	37	16	,	,	PUNCT
ejpam-4034	37	17	then	then	ADV
ejpam-4034	37	18	s	s	VERB
ejpam-4034	37	19	is	be	AUX
ejpam-4034	37	20	called	call	VERB
ejpam-4034	37	21	la	la	NOUN
ejpam-4034	37	22	-	-	NOUN
ejpam-4034	37	23	group	group	NOUN
ejpam-4034	37	24	.	.	PUNCT
ejpam-4034	38	1	la	la	ADJ
ejpam-4034	38	2	-	-	PUNCT
ejpam-4034	38	3	semirings	semiring	NOUN
ejpam-4034	38	4	are	be	AUX
ejpam-4034	38	5	developed	develop	VERB
ejpam-4034	38	6	by	by	ADP
ejpam-4034	38	7	the	the	DET
ejpam-4034	38	8	concepts	concept	NOUN
ejpam-4034	38	9	of	of	ADP
ejpam-4034	38	10	lasemigroup	lasemigroup	NOUN
ejpam-4034	38	11	[	[	X
ejpam-4034	38	12	10	10	NUM
ejpam-4034	38	13	,	,	PUNCT
ejpam-4034	38	14	11	11	NUM
ejpam-4034	38	15	]	]	PUNCT
ejpam-4034	38	16	.	.	PUNCT
ejpam-4034	39	1	la	la	ADJ
ejpam-4034	39	2	-	-	PUNCT
ejpam-4034	39	3	semiring	semiring	NOUN
ejpam-4034	39	4	and	and	CCONJ
ejpam-4034	39	5	certain	certain	ADJ
ejpam-4034	39	6	results	result	NOUN
ejpam-4034	39	7	on	on	ADP
ejpam-4034	39	8	la	la	ADJ
ejpam-4034	39	9	-	-	PUNCT
ejpam-4034	39	10	semirings	semiring	NOUN
ejpam-4034	39	11	having	have	VERB
ejpam-4034	39	12	two	two	NUM
ejpam-4034	39	13	variables	variable	NOUN
ejpam-4034	39	14	are	be	AUX
ejpam-4034	39	15	described	describe	VERB
ejpam-4034	39	16	in	in	ADP
ejpam-4034	39	17	[	[	X
ejpam-4034	39	18	4	4	NUM
ejpam-4034	39	19	]	]	PUNCT
ejpam-4034	39	20	.	.	PUNCT
ejpam-4034	40	1	a	a	DET
ejpam-4034	40	2	nonempty	nonempty	ADV
ejpam-4034	40	3	set	set	VERB
ejpam-4034	40	4	r	r	NOUN
ejpam-4034	40	5	with	with	ADP
ejpam-4034	40	6	two	two	NUM
ejpam-4034	40	7	binary	binary	ADJ
ejpam-4034	40	8	operation	operation	NOUN
ejpam-4034	40	9	"	"	PUNCT
ejpam-4034	40	10	.	.	PUNCT
ejpam-4034	40	11	"	"	PUNCT
ejpam-4034	40	12	and	and	CCONJ
ejpam-4034	40	13	"	"	PUNCT
ejpam-4034	40	14	+	+	ADJ
ejpam-4034	40	15	"	"	PUNCT
ejpam-4034	40	16	such	such	ADJ
ejpam-4034	40	17	that	that	SCONJ
ejpam-4034	40	18	(	(	PUNCT
ejpam-4034	40	19	i	i	NOUN
ejpam-4034	40	20	)	)	PUNCT
ejpam-4034	40	21	(	(	PUNCT
ejpam-4034	40	22	r,+	r,+	NUM
ejpam-4034	40	23	)	)	PUNCT
ejpam-4034	40	24	is	be	AUX
ejpam-4034	40	25	la	la	ADJ
ejpam-4034	40	26	-	-	PUNCT
ejpam-4034	40	27	group	group	NOUN
ejpam-4034	40	28	(	(	PUNCT
ejpam-4034	40	29	ii	ii	NOUN
ejpam-4034	40	30	)	)	PUNCT
ejpam-4034	40	31	(	(	PUNCT
ejpam-4034	40	32	r	r	NOUN
ejpam-4034	40	33	,	,	PUNCT
ejpam-4034	40	34	·	·	PUNCT
ejpam-4034	40	35	)	)	PUNCT
ejpam-4034	40	36	is	be	AUX
ejpam-4034	40	37	la	la	ADJ
ejpam-4034	40	38	-	-	NOUN
ejpam-4034	40	39	groupoid	groupoid	NOUN
ejpam-4034	40	40	,	,	PUNCT
ejpam-4034	40	41	and	and	CCONJ
ejpam-4034	40	42	nonassociative	nonassociative	ADJ
ejpam-4034	40	43	structure	structure	NOUN
ejpam-4034	40	44	w.r.t	w.r.t	VERB
ejpam-4034	40	45	′+′	′+′	PROPN
ejpam-4034	40	46	and	and	CCONJ
ejpam-4034	40	47	′·′	′·′	PROPN
ejpam-4034	40	48	satisfying	satisfying	NOUN
ejpam-4034	40	49	left	leave	VERB
ejpam-4034	40	50	and	and	CCONJ
ejpam-4034	40	51	right	right	ADJ
ejpam-4034	40	52	distributive	distributive	ADJ
ejpam-4034	40	53	laws	law	NOUN
ejpam-4034	40	54	is	be	AUX
ejpam-4034	40	55	called	call	VERB
ejpam-4034	40	56	la	la	ADJ
ejpam-4034	40	57	-	-	PUNCT
ejpam-4034	40	58	ring	ring	NOUN
ejpam-4034	40	59	[	[	X
ejpam-4034	40	60	20	20	NUM
ejpam-4034	40	61	]	]	PUNCT
ejpam-4034	40	62	.	.	PUNCT
ejpam-4034	41	1	la	la	ADJ
ejpam-4034	41	2	-	-	PUNCT
ejpam-4034	41	3	ring	ring	NOUN
ejpam-4034	41	4	was	be	AUX
ejpam-4034	41	5	further	far	ADV
ejpam-4034	41	6	elaborated	elaborate	VERB
ejpam-4034	41	7	in	in	ADP
ejpam-4034	41	8	[	[	X
ejpam-4034	41	9	18	18	NUM
ejpam-4034	41	10	]	]	PUNCT
ejpam-4034	41	11	.	.	PUNCT
ejpam-4034	42	1	every	every	DET
ejpam-4034	42	2	x	x	X
ejpam-4034	42	3	6=	6=	ADP
ejpam-4034	42	4	0	0	NUM
ejpam-4034	42	5	element	element	NOUN
ejpam-4034	42	6	of	of	ADP
ejpam-4034	42	7	left	left	ADJ
ejpam-4034	42	8	almost	almost	ADV
ejpam-4034	42	9	ring	ring	NOUN
ejpam-4034	42	10	r	r	NOUN
ejpam-4034	42	11	has	have	VERB
ejpam-4034	42	12	multiplicative	multiplicative	ADJ
ejpam-4034	42	13	inverse	inverse	NOUN
ejpam-4034	42	14	x−1	x−1	PROPN
ejpam-4034	42	15	and	and	CCONJ
ejpam-4034	42	16	having	having	AUX
ejpam-4034	42	17	left	leave	VERB
ejpam-4034	42	18	identity	identity	NOUN
ejpam-4034	42	19	e	e	NOUN
ejpam-4034	42	20	then	then	ADV
ejpam-4034	42	21	la	la	ADJ
ejpam-4034	42	22	-	-	PUNCT
ejpam-4034	42	23	ring	ring	NOUN
ejpam-4034	42	24	r	r	NOUN
ejpam-4034	42	25	is	be	AUX
ejpam-4034	42	26	called	call	VERB
ejpam-4034	42	27	la	la	ADJ
ejpam-4034	42	28	-	-	PUNCT
ejpam-4034	42	29	field	field	NOUN
ejpam-4034	42	30	.	.	PUNCT
ejpam-4034	43	1	la	la	ADJ
ejpam-4034	43	2	-	-	PUNCT
ejpam-4034	43	3	ring	ring	NOUN
ejpam-4034	43	4	<	<	X
ejpam-4034	43	5	r	r	NOUN
ejpam-4034	43	6	,	,	PUNCT
ejpam-4034	43	7	⊕	⊕	PROPN
ejpam-4034	43	8	,	,	PUNCT
ejpam-4034	43	9	.	.	PUNCT
ejpam-4034	44	1	>	>	X
ejpam-4034	44	2	can	can	AUX
ejpam-4034	44	3	be	be	AUX
ejpam-4034	44	4	obtain	obtain	VERB
ejpam-4034	44	5	by	by	ADP
ejpam-4034	44	6	defining	define	VERB
ejpam-4034	44	7	p	p	PROPN
ejpam-4034	44	8	⊕	⊕	PROPN
ejpam-4034	44	9	q	q	PROPN
ejpam-4034	45	1	=	=	PUNCT
ejpam-4034	45	2	q	q	NOUN
ejpam-4034	45	3	−	−	PROPN
ejpam-4034	45	4	p	p	PROPN
ejpam-4034	45	5	and	and	CCONJ
ejpam-4034	45	6	pq	pq	NOUN
ejpam-4034	45	7	,	,	PUNCT
ejpam-4034	45	8	for	for	ADP
ejpam-4034	45	9	p	p	PRON
ejpam-4034	45	10	,	,	PUNCT
ejpam-4034	45	11	q	q	ADJ
ejpam-4034	45	12	,	,	PUNCT
ejpam-4034	45	13	r	r	NOUN
ejpam-4034	45	14	∈	∈	PROPN
ejpam-4034	45	15	r	r	NOUN
ejpam-4034	45	16	,	,	PUNCT
ejpam-4034	45	17	is	be	AUX
ejpam-4034	45	18	similar	similar	ADJ
ejpam-4034	45	19	as	as	ADP
ejpam-4034	45	20	in	in	ADP
ejpam-4034	45	21	the	the	DET
ejpam-4034	45	22	ring	ring	NOUN
ejpam-4034	45	23	.	.	PUNCT
ejpam-4034	46	1	the	the	DET
ejpam-4034	46	2	addition	addition	NOUN
ejpam-4034	46	3	in	in	ADP
ejpam-4034	46	4	la	la	ADJ
ejpam-4034	46	5	-	-	PUNCT
ejpam-4034	46	6	ring	ring	NOUN
ejpam-4034	46	7	can	can	AUX
ejpam-4034	46	8	not	not	PART
ejpam-4034	46	9	assume	assume	VERB
ejpam-4034	46	10	to	to	PART
ejpam-4034	46	11	be	be	AUX
ejpam-4034	46	12	commutative	commutative	ADJ
ejpam-4034	46	13	.	.	PUNCT
ejpam-4034	47	1	if	if	SCONJ
ejpam-4034	47	2	for	for	ADP
ejpam-4034	47	3	p	p	NOUN
ejpam-4034	47	4	,	,	PUNCT
ejpam-4034	47	5	q	q	PUNCT
ejpam-4034	47	6	∈	∈	PROPN
ejpam-4034	47	7	r	r	NOUN
ejpam-4034	47	8	,	,	PUNCT
ejpam-4034	47	9	pq	pq	NOUN
ejpam-4034	47	10	=	=	SYM
ejpam-4034	47	11	0	0	PROPN
ejpam-4034	47	12	implies	imply	VERB
ejpam-4034	47	13	p	p	X
ejpam-4034	47	14	=	=	SYM
ejpam-4034	47	15	0	0	NUM
ejpam-4034	47	16	or	or	CCONJ
ejpam-4034	47	17	q	q	ADJ
ejpam-4034	47	18	=	=	SYM
ejpam-4034	47	19	0	0	PROPN
ejpam-4034	47	20	then	then	ADV
ejpam-4034	47	21	la	la	ADJ
ejpam-4034	47	22	-	-	PUNCT
ejpam-4034	47	23	ring	ring	NOUN
ejpam-4034	47	24	r	r	NOUN
ejpam-4034	47	25	is	be	AUX
ejpam-4034	47	26	called	call	VERB
ejpam-4034	47	27	la	la	ADJ
ejpam-4034	47	28	-	-	ADJ
ejpam-4034	47	29	integral	integral	ADJ
ejpam-4034	47	30	domain	domain	NOUN
ejpam-4034	47	31	.	.	PUNCT
ejpam-4034	48	1	if	if	SCONJ
ejpam-4034	48	2	∅	∅	NOUN
ejpam-4034	48	3	6=	6=	ADP
ejpam-4034	48	4	s	s	ADP
ejpam-4034	48	5	⊆	⊆	NUM
ejpam-4034	48	6	r	r	NOUN
ejpam-4034	48	7	and	and	CCONJ
ejpam-4034	48	8	s	s	NOUN
ejpam-4034	48	9	is	be	AUX
ejpam-4034	48	10	la	la	ADJ
ejpam-4034	48	11	-	-	NOUN
ejpam-4034	48	12	ring	ring	NOUN
ejpam-4034	48	13	under	under	ADP
ejpam-4034	48	14	binary	binary	ADJ
ejpam-4034	48	15	operation	operation	NOUN
ejpam-4034	48	16	defined	define	VERB
ejpam-4034	48	17	in	in	ADP
ejpam-4034	48	18	r	r	NOUN
ejpam-4034	48	19	,	,	PUNCT
ejpam-4034	48	20	then	then	ADV
ejpam-4034	48	21	s	s	VERB
ejpam-4034	48	22	is	be	AUX
ejpam-4034	48	23	la	la	ADJ
ejpam-4034	48	24	-	-	PUNCT
ejpam-4034	48	25	subring	subre	VERB
ejpam-4034	48	26	.	.	PUNCT
ejpam-4034	49	1	if	if	SCONJ
ejpam-4034	49	2	rs	rs	PROPN
ejpam-4034	49	3	⊆	⊆	NUM
ejpam-4034	49	4	s	s	NOUN
ejpam-4034	49	5	,	,	PUNCT
ejpam-4034	49	6	then	then	ADV
ejpam-4034	49	7	s	s	VERB
ejpam-4034	49	8	is	be	AUX
ejpam-4034	49	9	left	leave	VERB
ejpam-4034	49	10	ideal	ideal	NOUN
ejpam-4034	49	11	of	of	ADP
ejpam-4034	49	12	r.	r.	PROPN
ejpam-4034	49	13	similarly	similarly	ADV
ejpam-4034	49	14	we	we	PRON
ejpam-4034	49	15	can	can	AUX
ejpam-4034	49	16	define	define	VERB
ejpam-4034	49	17	right	right	ADJ
ejpam-4034	49	18	and	and	CCONJ
ejpam-4034	49	19	two	two	NUM
ejpam-4034	49	20	-	-	PUNCT
ejpam-4034	49	21	sided	sided	ADJ
ejpam-4034	49	22	ideals	ideal	NOUN
ejpam-4034	49	23	.	.	PUNCT
ejpam-4034	50	1	if	if	SCONJ
ejpam-4034	50	2	pq	pq	PROPN
ejpam-4034	50	3	∈	∈	PROPN
ejpam-4034	50	4	a	a	DET
ejpam-4034	50	5	=	=	NOUN
ejpam-4034	50	6	⇒	⇒	VERB
ejpam-4034	50	7	p	p	PROPN
ejpam-4034	50	8	∈	∈	PROPN
ejpam-4034	50	9	a	a	PRON
ejpam-4034	50	10	or	or	CCONJ
ejpam-4034	50	11	q	q	NOUN
ejpam-4034	50	12	∈	∈	PROPN
ejpam-4034	50	13	a	a	DET
ejpam-4034	50	14	then	then	ADV
ejpam-4034	50	15	ideal	ideal	ADJ
ejpam-4034	50	16	a	a	PRON
ejpam-4034	50	17	of	of	ADP
ejpam-4034	50	18	r	r	NOUN
ejpam-4034	50	19	is	be	AUX
ejpam-4034	50	20	called	call	VERB
ejpam-4034	50	21	prime	prime	ADJ
ejpam-4034	50	22	.	.	PUNCT
ejpam-4034	51	1	left	leave	VERB
ejpam-4034	51	2	primary	primary	ADJ
ejpam-4034	51	3	and	and	CCONJ
ejpam-4034	51	4	weakly	weakly	ADJ
ejpam-4034	51	5	left	left	ADJ
ejpam-4034	51	6	primary	primary	ADJ
ejpam-4034	51	7	ideals	ideal	NOUN
ejpam-4034	51	8	in	in	ADP
ejpam-4034	51	9	γ	γ	PROPN
ejpam-4034	51	10	-	-	PUNCT
ejpam-4034	51	11	la	la	NOUN
ejpam-4034	51	12	-	-	PUNCT
ejpam-4034	51	13	rings	ring	NOUN
ejpam-4034	51	14	and	and	CCONJ
ejpam-4034	51	15	their	their	PRON
ejpam-4034	51	16	characterizations	characterization	NOUN
ejpam-4034	51	17	are	be	AUX
ejpam-4034	51	18	presented	present	VERB
ejpam-4034	51	19	in	in	ADP
ejpam-4034	51	20	[	[	X
ejpam-4034	51	21	19	19	NUM
ejpam-4034	51	22	]	]	PUNCT
ejpam-4034	51	23	.	.	PUNCT
ejpam-4034	52	1	it	it	PRON
ejpam-4034	52	2	is	be	AUX
ejpam-4034	52	3	well	well	ADV
ejpam-4034	52	4	known	know	VERB
ejpam-4034	52	5	that	that	SCONJ
ejpam-4034	52	6	an	an	DET
ejpam-4034	52	7	ideal	ideal	NOUN
ejpam-4034	52	8	i	i	PRON
ejpam-4034	52	9	of	of	ADP
ejpam-4034	52	10	a	a	DET
ejpam-4034	52	11	semiring	semiring	NOUN
ejpam-4034	52	12	r	r	NOUN
ejpam-4034	52	13	is	be	AUX
ejpam-4034	52	14	called	call	VERB
ejpam-4034	52	15	subtractive	subtractive	NOUN
ejpam-4034	52	16	,	,	PUNCT
ejpam-4034	52	17	if	if	SCONJ
ejpam-4034	52	18	whenever	whenever	SCONJ
ejpam-4034	52	19	a	a	DET
ejpam-4034	52	20	,	,	PUNCT
ejpam-4034	52	21	a+b	a+b	NUM
ejpam-4034	52	22	∈	∈	PROPN
ejpam-4034	52	23	i	i	PRON
ejpam-4034	52	24	,	,	PUNCT
ejpam-4034	52	25	br	br	PROPN
ejpam-4034	52	26	,	,	PUNCT
ejpam-4034	52	27	we	we	PRON
ejpam-4034	52	28	have	have	VERB
ejpam-4034	52	29	b	b	PROPN
ejpam-4034	52	30	∈	∈	PROPN
ejpam-4034	52	31	i.	i.	NOUN
ejpam-4034	52	32	similarly	similarly	ADV
ejpam-4034	52	33	,	,	PUNCT
ejpam-4034	52	34	a	a	DET
ejpam-4034	52	35	left	left	ADJ
ejpam-4034	52	36	k	k	NOUN
ejpam-4034	52	37	-	-	NOUN
ejpam-4034	52	38	ideal	ideal	ADJ
ejpam-4034	52	39	i	i	PRON
ejpam-4034	52	40	of	of	ADP
ejpam-4034	52	41	a	a	DET
ejpam-4034	52	42	semiring	semiring	NOUN
ejpam-4034	52	43	s	s	X
ejpam-4034	52	44	is	be	AUX
ejpam-4034	52	45	a	a	DET
ejpam-4034	52	46	left	left	ADJ
ejpam-4034	52	47	ideal	ideal	NOUN
ejpam-4034	52	48	such	such	ADJ
ejpam-4034	52	49	that	that	SCONJ
ejpam-4034	52	50	if	if	SCONJ
ejpam-4034	52	51	a	a	DET
ejpam-4034	52	52	∈	∈	NOUN
ejpam-4034	52	53	and	and	CCONJ
ejpam-4034	52	54	x	x	PART
ejpam-4034	52	55	∈	∈	NOUN
ejpam-4034	52	56	s	s	X
ejpam-4034	52	57	and	and	CCONJ
ejpam-4034	52	58	if	if	SCONJ
ejpam-4034	52	59	either	either	ADV
ejpam-4034	52	60	a+	a+	PUNCT
ejpam-4034	52	61	x	x	SYM
ejpam-4034	52	62	∈	∈	PROPN
ejpam-4034	52	63	i.	i.	NOUN
ejpam-4034	52	64	or	or	CCONJ
ejpam-4034	52	65	x+	x+	X
ejpam-4034	52	66	a	a	DET
ejpam-4034	52	67	∈	∈	PROPN
ejpam-4034	52	68	i	i	PRON
ejpam-4034	52	69	,	,	PUNCT
ejpam-4034	52	70	then	then	ADV
ejpam-4034	52	71	x	x	PART
ejpam-4034	52	72	∈	∈	PROPN
ejpam-4034	52	73	i.	i.	NOUN
ejpam-4034	52	74	in	in	ADP
ejpam-4034	52	75	this	this	DET
ejpam-4034	52	76	note	note	NOUN
ejpam-4034	52	77	,	,	PUNCT
ejpam-4034	52	78	first	first	ADV
ejpam-4034	52	79	we	we	PRON
ejpam-4034	52	80	add	add	VERB
ejpam-4034	52	81	few	few	ADJ
ejpam-4034	52	82	new	new	ADJ
ejpam-4034	52	83	theorems	theorem	NOUN
ejpam-4034	52	84	and	and	CCONJ
ejpam-4034	52	85	examples	example	NOUN
ejpam-4034	52	86	in	in	ADP
ejpam-4034	52	87	the	the	DET
ejpam-4034	52	88	theory	theory	NOUN
ejpam-4034	52	89	of	of	ADP
ejpam-4034	52	90	γ	γ	PROPN
ejpam-4034	52	91	-	-	PUNCT
ejpam-4034	52	92	la	la	NOUN
ejpam-4034	52	93	-	-	PUNCT
ejpam-4034	52	94	rings	ring	NOUN
ejpam-4034	52	95	and	and	CCONJ
ejpam-4034	52	96	then	then	ADV
ejpam-4034	52	97	we	we	PRON
ejpam-4034	52	98	introduce	introduce	VERB
ejpam-4034	52	99	the	the	DET
ejpam-4034	52	100	notion	notion	NOUN
ejpam-4034	52	101	of	of	ADP
ejpam-4034	52	102	γ	γ	PROPN
ejpam-4034	52	103	-	-	PUNCT
ejpam-4034	52	104	la	la	ADJ
ejpam-4034	52	105	-	-	PUNCT
ejpam-4034	52	106	semirings	semiring	NOUN
ejpam-4034	52	107	.	.	PUNCT
ejpam-4034	53	1	in	in	ADP
ejpam-4034	53	2	due	due	ADJ
ejpam-4034	53	3	course	course	NOUN
ejpam-4034	53	4	,	,	PUNCT
ejpam-4034	53	5	we	we	PRON
ejpam-4034	53	6	describe	describe	VERB
ejpam-4034	53	7	c	c	NOUN
ejpam-4034	53	8	-	-	PUNCT
ejpam-4034	53	9	prime	prime	ADJ
ejpam-4034	53	10	,	,	PUNCT
ejpam-4034	53	11	3	3	NUM
ejpam-4034	53	12	-	-	PUNCT
ejpam-4034	53	13	prime	prime	ADJ
ejpam-4034	53	14	ideals	ideal	NOUN
ejpam-4034	53	15	and	and	CCONJ
ejpam-4034	53	16	their	their	PRON
ejpam-4034	53	17	relationships	relationship	NOUN
ejpam-4034	53	18	among	among	ADP
ejpam-4034	53	19	themselves	themselves	PRON
ejpam-4034	53	20	in	in	ADP
ejpam-4034	53	21	γ	γ	PROPN
ejpam-4034	53	22	-	-	PUNCT
ejpam-4034	53	23	la	la	ADJ
ejpam-4034	53	24	-	-	PUNCT
ejpam-4034	53	25	ring	ring	NOUN
ejpam-4034	53	26	and	and	CCONJ
ejpam-4034	53	27	γ	γ	NOUN
ejpam-4034	53	28	-	-	PUNCT
ejpam-4034	53	29	la	la	ADJ
ejpam-4034	53	30	-	-	PUNCT
ejpam-4034	53	31	semirings	semiring	NOUN
ejpam-4034	53	32	.	.	PUNCT
ejpam-4034	54	1	finally	finally	ADV
ejpam-4034	54	2	,	,	PUNCT
ejpam-4034	54	3	we	we	PRON
ejpam-4034	54	4	discuss	discuss	VERB
ejpam-4034	54	5	left	left	ADJ
ejpam-4034	54	6	ideals	ideal	NOUN
ejpam-4034	54	7	,	,	PUNCT
ejpam-4034	54	8	right	right	ADJ
ejpam-4034	54	9	ideals	ideal	NOUN
ejpam-4034	54	10	,	,	PUNCT
ejpam-4034	54	11	and	and	CCONJ
ejpam-4034	54	12	some	some	DET
ejpam-4034	54	13	results	result	NOUN
ejpam-4034	54	14	on	on	ADP
ejpam-4034	54	15	bi	bi	ADJ
ejpam-4034	54	16	-	-	ADJ
ejpam-4034	54	17	ideal	ideal	ADJ
ejpam-4034	54	18	,	,	PUNCT
ejpam-4034	54	19	quasi	quasi	NOUN
ejpam-4034	54	20	ideals	ideal	NOUN
ejpam-4034	54	21	,	,	PUNCT
ejpam-4034	54	22	almost	almost	ADV
ejpam-4034	54	23	prime	prime	ADJ
ejpam-4034	54	24	and	and	CCONJ
ejpam-4034	54	25	weakly	weakly	ADJ
ejpam-4034	54	26	almost	almost	ADV
ejpam-4034	54	27	prime	prime	ADJ
ejpam-4034	54	28	ideals	ideal	NOUN
ejpam-4034	54	29	in	in	ADP
ejpam-4034	54	30	γ	γ	PROPN
ejpam-4034	54	31	-	-	PUNCT
ejpam-4034	54	32	la	la	ADJ
ejpam-4034	54	33	-	-	PUNCT
ejpam-4034	54	34	semiring	semiring	NOUN
ejpam-4034	54	35	.	.	PUNCT
ejpam-4034	55	1	2	2	X
ejpam-4034	55	2	.	.	X
ejpam-4034	55	3	main	main	ADJ
ejpam-4034	55	4	results	result	NOUN
ejpam-4034	55	5	and	and	CCONJ
ejpam-4034	55	6	discussions	discussion	NOUN
ejpam-4034	55	7	2.1	2.1	NUM
ejpam-4034	55	8	.	.	PUNCT
ejpam-4034	56	1	some	some	DET
ejpam-4034	56	2	applications	application	NOUN
ejpam-4034	56	3	of	of	ADP
ejpam-4034	56	4	prime	prime	ADJ
ejpam-4034	56	5	ideals	ideal	NOUN
ejpam-4034	56	6	in	in	ADP
ejpam-4034	56	7	γ	γ	PROPN
ejpam-4034	56	8	-	-	PUNCT
ejpam-4034	56	9	la	la	ADJ
ejpam-4034	56	10	-	-	PUNCT
ejpam-4034	56	11	ring	ring	NOUN
ejpam-4034	56	12	in	in	ADP
ejpam-4034	56	13	this	this	DET
ejpam-4034	56	14	section	section	NOUN
ejpam-4034	56	15	,	,	PUNCT
ejpam-4034	56	16	we	we	PRON
ejpam-4034	56	17	introduce	introduce	VERB
ejpam-4034	56	18	different	different	ADJ
ejpam-4034	56	19	types	type	NOUN
ejpam-4034	56	20	of	of	ADP
ejpam-4034	56	21	prime	prime	ADJ
ejpam-4034	56	22	ideals	ideal	NOUN
ejpam-4034	56	23	in	in	ADP
ejpam-4034	56	24	γ	γ	PROPN
ejpam-4034	56	25	-	-	PUNCT
ejpam-4034	56	26	la	la	NOUN
ejpam-4034	56	27	-	-	PUNCT
ejpam-4034	56	28	rings	ring	NOUN
ejpam-4034	56	29	along	along	ADP
ejpam-4034	56	30	with	with	ADP
ejpam-4034	56	31	their	their	PRON
ejpam-4034	56	32	applications	application	NOUN
ejpam-4034	56	33	.	.	PUNCT
ejpam-4034	57	1	we	we	PRON
ejpam-4034	57	2	begin	begin	VERB
ejpam-4034	57	3	by	by	ADP
ejpam-4034	57	4	recalling	recall	VERB
ejpam-4034	57	5	definition	definition	NOUN
ejpam-4034	57	6	of	of	ADP
ejpam-4034	57	7	γ	γ	PROPN
ejpam-4034	57	8	-	-	PUNCT
ejpam-4034	57	9	la	la	ADJ
ejpam-4034	57	10	-	-	PUNCT
ejpam-4034	57	11	ring	ring	NOUN
ejpam-4034	57	12	and	and	CCONJ
ejpam-4034	57	13	then	then	ADV
ejpam-4034	57	14	we	we	PRON
ejpam-4034	57	15	add	add	VERB
ejpam-4034	57	16	few	few	ADJ
ejpam-4034	57	17	new	new	ADJ
ejpam-4034	57	18	results	result	NOUN
ejpam-4034	57	19	and	and	CCONJ
ejpam-4034	57	20	examples	example	NOUN
ejpam-4034	57	21	in	in	ADP
ejpam-4034	57	22	the	the	DET
ejpam-4034	57	23	theory	theory	NOUN
ejpam-4034	57	24	of	of	ADP
ejpam-4034	57	25	γ	γ	PROPN
ejpam-4034	57	26	-	-	PUNCT
ejpam-4034	57	27	la	la	ADJ
ejpam-4034	57	28	-	-	PUNCT
ejpam-4034	57	29	ring	ring	NOUN
ejpam-4034	57	30	.	.	PUNCT
ejpam-4034	58	1	definition	definition	NOUN
ejpam-4034	58	2	1	1	NUM
ejpam-4034	58	3	.	.	PUNCT
ejpam-4034	59	1	[	[	X
ejpam-4034	59	2	19	19	NUM
ejpam-4034	59	3	]	]	X
ejpam-4034	59	4	let	let	VERB
ejpam-4034	59	5	(	(	PUNCT
ejpam-4034	59	6	r,+	r,+	NUM
ejpam-4034	59	7	)	)	PUNCT
ejpam-4034	59	8	and	and	CCONJ
ejpam-4034	59	9	(	(	PUNCT
ejpam-4034	59	10	γ	γ	X
ejpam-4034	59	11	,	,	PUNCT
ejpam-4034	59	12	+	+	PROPN
ejpam-4034	59	13	)	)	PUNCT
ejpam-4034	59	14	be	be	VERB
ejpam-4034	59	15	the	the	DET
ejpam-4034	59	16	two	two	NUM
ejpam-4034	59	17	la	la	ADJ
ejpam-4034	59	18	-	-	PUNCT
ejpam-4034	59	19	groups	group	NOUN
ejpam-4034	59	20	and	and	CCONJ
ejpam-4034	59	21	there	there	PRON
ejpam-4034	59	22	exists	exist	VERB
ejpam-4034	59	23	a	a	DET
ejpam-4034	59	24	mapping	mapping	NOUN
ejpam-4034	59	25	r×γ	r×γ	PROPN
ejpam-4034	59	26	×r→	×r→	ADV
ejpam-4034	59	27	r	r	NOUN
ejpam-4034	59	28	by	by	ADP
ejpam-4034	59	29	(	(	PUNCT
ejpam-4034	59	30	a	a	DET
ejpam-4034	59	31	,	,	PUNCT
ejpam-4034	59	32	α	α	NOUN
ejpam-4034	59	33	,	,	PUNCT
ejpam-4034	59	34	b)→	b)→	VERB
ejpam-4034	59	35	aαb	aαb	NOUN
ejpam-4034	59	36	,	,	PUNCT
ejpam-4034	59	37	for	for	ADP
ejpam-4034	59	38	all	all	DET
ejpam-4034	59	39	a	a	PRON
ejpam-4034	59	40	,	,	PUNCT
ejpam-4034	59	41	b	b	X
ejpam-4034	59	42	∈	∈	PROPN
ejpam-4034	59	43	r	r	NOUN
ejpam-4034	59	44	and	and	CCONJ
ejpam-4034	59	45	α	α	NOUN
ejpam-4034	59	46	∈	∈	NOUN
ejpam-4034	59	47	γ	γ	NOUN
ejpam-4034	59	48	is	be	AUX
ejpam-4034	59	49	called	call	VERB
ejpam-4034	59	50	a	a	DET
ejpam-4034	59	51	gamma	gamma	NOUN
ejpam-4034	59	52	la	la	PROPN
ejpam-4034	59	53	-	-	PUNCT
ejpam-4034	59	54	ring	ring	NOUN
ejpam-4034	59	55	,	,	PUNCT
ejpam-4034	59	56	if	if	SCONJ
ejpam-4034	59	57	it	it	PRON
ejpam-4034	59	58	satifies	satifie	VERB
ejpam-4034	59	59	the	the	DET
ejpam-4034	59	60	following	follow	VERB
ejpam-4034	59	61	conditions	condition	NOUN
ejpam-4034	59	62	.	.	PUNCT
ejpam-4034	60	1	w.	w.	PROPN
ejpam-4034	60	2	a.	a.	PROPN
ejpam-4034	60	3	khan	khan	PROPN
ejpam-4034	60	4	et	et	PROPN
ejpam-4034	60	5	al	al	PROPN
ejpam-4034	60	6	.	.	PUNCT
ejpam-4034	60	7	/	/	SYM
ejpam-4034	60	8	eur	eur	PROPN
ejpam-4034	60	9	.	.	PUNCT
ejpam-4034	61	1	j.	j.	PROPN
ejpam-4034	61	2	pure	pure	PROPN
ejpam-4034	61	3	appl	appl	PROPN
ejpam-4034	61	4	.	.	PROPN
ejpam-4034	61	5	math	math	PROPN
ejpam-4034	61	6	,	,	PUNCT
ejpam-4034	61	7	14	14	NUM
ejpam-4034	61	8	(	(	PUNCT
ejpam-4034	61	9	3	3	NUM
ejpam-4034	61	10	)	)	PUNCT
ejpam-4034	61	11	(	(	PUNCT
ejpam-4034	61	12	2021	2021	NUM
ejpam-4034	61	13	)	)	PUNCT
ejpam-4034	61	14	,	,	PUNCT
ejpam-4034	61	15	989	989	NUM
ejpam-4034	61	16	-	-	SYM
ejpam-4034	61	17	1001	1001	NUM
ejpam-4034	61	18	991	991	NUM
ejpam-4034	61	19	1	1	NUM
ejpam-4034	61	20	.	.	PUNCT
ejpam-4034	62	1	aα(b+	aα(b+	PRON
ejpam-4034	63	1	c	c	X
ejpam-4034	63	2	)	)	PUNCT
ejpam-4034	63	3	=	=	NOUN
ejpam-4034	63	4	aαb+	aαb+	NOUN
ejpam-4034	63	5	aαc	aαc	VERB
ejpam-4034	63	6	2	2	NUM
ejpam-4034	63	7	.	.	PUNCT
ejpam-4034	64	1	(	(	PUNCT
ejpam-4034	64	2	a+	a+	PUNCT
ejpam-4034	64	3	b)αc	b)αc	PROPN
ejpam-4034	64	4	=	=	SYM
ejpam-4034	64	5	aαc+	aαc+	PROPN
ejpam-4034	64	6	bαc	bαc	ADJ
ejpam-4034	64	7	3	3	NUM
ejpam-4034	64	8	.	.	PUNCT
ejpam-4034	64	9	a(α+	a(α+	X
ejpam-4034	64	10	β)b	β)b	X
ejpam-4034	65	1	=	=	PUNCT
ejpam-4034	65	2	aαb+	aαb+	NOUN
ejpam-4034	65	3	aβb	aβb	VERB
ejpam-4034	65	4	4	4	NUM
ejpam-4034	65	5	.	.	PUNCT
ejpam-4034	66	1	(	(	PUNCT
ejpam-4034	66	2	aαb)βc	aαb)βc	NOUN
ejpam-4034	66	3	=	=	SYM
ejpam-4034	66	4	(	(	PUNCT
ejpam-4034	66	5	cαb)βa,∀	cαb)βa,∀	PROPN
ejpam-4034	66	6	a	a	PROPN
ejpam-4034	66	7	,	,	PUNCT
ejpam-4034	66	8	.b	.b	PROPN
ejpam-4034	66	9	,	,	PUNCT
ejpam-4034	66	10	c	c	PROPN
ejpam-4034	66	11	∈	∈	PROPN
ejpam-4034	66	12	r	r	NOUN
ejpam-4034	66	13	,	,	PUNCT
ejpam-4034	66	14	α	α	PROPN
ejpam-4034	66	15	,	,	PUNCT
ejpam-4034	66	16	β	β	PROPN
ejpam-4034	66	17	∈	∈	PROPN
ejpam-4034	66	18	γ	γ	PROPN
ejpam-4034	66	19	example	example	NOUN
ejpam-4034	66	20	1	1	NUM
ejpam-4034	66	21	.	.	PUNCT
ejpam-4034	67	1	let	let	AUX
ejpam-4034	67	2	r	r	NOUN
ejpam-4034	67	3	=	=	SYM
ejpam-4034	67	4	{	{	PUNCT
ejpam-4034	67	5	a1	a1	PROPN
ejpam-4034	67	6	,	,	PUNCT
ejpam-4034	67	7	a2	a2	PROPN
ejpam-4034	67	8	,	,	PUNCT
ejpam-4034	67	9	a3	a3	NOUN
ejpam-4034	67	10	,	,	PUNCT
ejpam-4034	67	11	a4	a4	PROPN
ejpam-4034	67	12	,	,	PUNCT
ejpam-4034	67	13	a5	a5	NOUN
ejpam-4034	67	14	,	,	PUNCT
ejpam-4034	67	15	a6	a6	NOUN
ejpam-4034	67	16	,	,	PUNCT
ejpam-4034	67	17	a7	a7	PROPN
ejpam-4034	67	18	,	,	PUNCT
ejpam-4034	67	19	a8	a8	PROPN
ejpam-4034	67	20	}	}	PUNCT
ejpam-4034	67	21	be	be	VERB
ejpam-4034	67	22	a	a	DET
ejpam-4034	67	23	set	set	NOUN
ejpam-4034	67	24	with	with	ADP
ejpam-4034	67	25	two	two	NUM
ejpam-4034	67	26	binary	binary	ADJ
ejpam-4034	67	27	operations	operation	NOUN
ejpam-4034	67	28	”	"	PUNCT
ejpam-4034	67	29	+	+	PROPN
ejpam-4034	67	30	”	"	PUNCT
ejpam-4034	67	31	and	and	CCONJ
ejpam-4034	67	32	”	"	PUNCT
ejpam-4034	67	33	.	.	PUNCT
ejpam-4034	67	34	”	"	PUNCT
ejpam-4034	68	1	given	give	VERB
ejpam-4034	68	2	in	in	ADP
ejpam-4034	68	3	the	the	DET
ejpam-4034	68	4	tables	table	NOUN
ejpam-4034	68	5	set	set	VERB
ejpam-4034	68	6	1	1	NUM
ejpam-4034	68	7	be	be	AUX
ejpam-4034	68	8	the	the	DET
ejpam-4034	68	9	la	la	NOUN
ejpam-4034	68	10	-	-	PUNCT
ejpam-4034	68	11	ring	ring	NOUN
ejpam-4034	68	12	and	and	CCONJ
ejpam-4034	68	13	γ	γ	X
ejpam-4034	68	14	=	=	SYM
ejpam-4034	68	15	{	{	PUNCT
ejpam-4034	68	16	s1	s1	PROPN
ejpam-4034	68	17	,	,	PUNCT
ejpam-4034	68	18	s2	s2	PROPN
ejpam-4034	68	19	,	,	PUNCT
ejpam-4034	68	20	s3	s3	PROPN
ejpam-4034	68	21	}	}	PUNCT
ejpam-4034	68	22	with	with	SCONJ
ejpam-4034	68	23	binary	binary	PROPN
ejpam-4034	68	24	operation⊕	operation⊕	PROPN
ejpam-4034	68	25	is	be	AUX
ejpam-4034	68	26	la	la	ADJ
ejpam-4034	68	27	-	-	NOUN
ejpam-4034	68	28	group	group	NOUN
ejpam-4034	68	29	.	.	PUNCT
ejpam-4034	69	1	tables	table	NOUN
ejpam-4034	69	2	set	set	VERB
ejpam-4034	69	3	1	1	NUM
ejpam-4034	69	4	+	+	CCONJ
ejpam-4034	69	5	a1	a1	NOUN
ejpam-4034	69	6	a2	a2	PROPN
ejpam-4034	69	7	a3	a3	PROPN
ejpam-4034	69	8	a4	a4	PROPN
ejpam-4034	69	9	a5	a5	PROPN
ejpam-4034	69	10	a6	a6	NOUN
ejpam-4034	69	11	a7	a7	PROPN
ejpam-4034	69	12	a8	a8	PROPN
ejpam-4034	69	13	a1	a1	NOUN
ejpam-4034	69	14	a1	a1	NOUN
ejpam-4034	69	15	a2	a2	PROPN
ejpam-4034	69	16	a3	a3	PROPN
ejpam-4034	69	17	a4	a4	PROPN
ejpam-4034	69	18	a5	a5	PROPN
ejpam-4034	69	19	a6	a6	PROPN
ejpam-4034	69	20	a7	a7	PROPN
ejpam-4034	69	21	a8	a8	PROPN
ejpam-4034	69	22	a2	a2	PROPN
ejpam-4034	69	23	a3	a3	NOUN
ejpam-4034	69	24	a1	a1	NOUN
ejpam-4034	69	25	a4	a4	PROPN
ejpam-4034	69	26	a2	a2	PROPN
ejpam-4034	69	27	a7	a7	PROPN
ejpam-4034	69	28	a5	a5	PROPN
ejpam-4034	69	29	a8	a8	PROPN
ejpam-4034	69	30	a6	a6	PROPN
ejpam-4034	69	31	a3	a3	PROPN
ejpam-4034	69	32	a2	a2	PROPN
ejpam-4034	69	33	a4	a4	NOUN
ejpam-4034	69	34	a1	a1	NOUN
ejpam-4034	69	35	a3	a3	NOUN
ejpam-4034	69	36	a6	a6	PROPN
ejpam-4034	69	37	a8	a8	PROPN
ejpam-4034	69	38	a5	a5	PROPN
ejpam-4034	69	39	a7	a7	PROPN
ejpam-4034	69	40	a4	a4	PROPN
ejpam-4034	69	41	a4	a4	NOUN
ejpam-4034	69	42	a3	a3	NOUN
ejpam-4034	69	43	a2	a2	PROPN
ejpam-4034	69	44	a1	a1	NOUN
ejpam-4034	69	45	a8	a8	PROPN
ejpam-4034	69	46	a7	a7	PROPN
ejpam-4034	69	47	a6	a6	PROPN
ejpam-4034	69	48	a5	a5	PROPN
ejpam-4034	69	49	a5	a5	PROPN
ejpam-4034	69	50	a5	a5	PROPN
ejpam-4034	69	51	a6	a6	PROPN
ejpam-4034	69	52	a7	a7	PROPN
ejpam-4034	69	53	a8	a8	PROPN
ejpam-4034	69	54	a1	a1	PROPN
ejpam-4034	69	55	a2	a2	PROPN
ejpam-4034	69	56	a3	a3	PROPN
ejpam-4034	69	57	a4	a4	PROPN
ejpam-4034	69	58	a6	a6	PROPN
ejpam-4034	69	59	a7	a7	PROPN
ejpam-4034	69	60	a5	a5	PROPN
ejpam-4034	69	61	a8	a8	PROPN
ejpam-4034	69	62	a6	a6	PROPN
ejpam-4034	69	63	a3	a3	NOUN
ejpam-4034	69	64	a1	a1	NOUN
ejpam-4034	69	65	a4	a4	PROPN
ejpam-4034	69	66	a2	a2	PROPN
ejpam-4034	69	67	a7	a7	PROPN
ejpam-4034	69	68	a6	a6	PROPN
ejpam-4034	69	69	a8	a8	PROPN
ejpam-4034	69	70	a5	a5	PROPN
ejpam-4034	69	71	a7	a7	PROPN
ejpam-4034	69	72	a2	a2	PROPN
ejpam-4034	69	73	a4	a4	NOUN
ejpam-4034	69	74	a1	a1	NOUN
ejpam-4034	69	75	a3	a3	NOUN
ejpam-4034	69	76	a8	a8	PROPN
ejpam-4034	69	77	a8	a8	PROPN
ejpam-4034	69	78	a7	a7	PROPN
ejpam-4034	69	79	a6	a6	PROPN
ejpam-4034	69	80	a5	a5	PROPN
ejpam-4034	69	81	a4	a4	PROPN
ejpam-4034	69	82	a3	a3	NOUN
ejpam-4034	69	83	a2	a2	PROPN
ejpam-4034	69	84	a1	a1	PROPN
ejpam-4034	69	85	·	·	PUNCT
ejpam-4034	69	86	a1	a1	NOUN
ejpam-4034	69	87	a2	a2	PROPN
ejpam-4034	69	88	a3	a3	PROPN
ejpam-4034	69	89	a4	a4	PROPN
ejpam-4034	69	90	a5	a5	PROPN
ejpam-4034	69	91	a6	a6	NOUN
ejpam-4034	69	92	a7	a7	PROPN
ejpam-4034	69	93	a8	a8	PROPN
ejpam-4034	69	94	a1	a1	NOUN
ejpam-4034	69	95	a1	a1	NOUN
ejpam-4034	70	1	a1	a1	NOUN
ejpam-4034	71	1	a1	a1	NOUN
ejpam-4034	71	2	a1	a1	NOUN
ejpam-4034	71	3	a1	a1	NOUN
ejpam-4034	71	4	a1	a1	NOUN
ejpam-4034	71	5	a1	a1	NOUN
ejpam-4034	71	6	a1	a1	NOUN
ejpam-4034	71	7	a2	a2	PROPN
ejpam-4034	71	8	a1	a1	PROPN
ejpam-4034	71	9	a5	a5	PROPN
ejpam-4034	71	10	a5	a5	PROPN
ejpam-4034	71	11	a1	a1	NOUN
ejpam-4034	71	12	a1	a1	NOUN
ejpam-4034	71	13	a5	a5	PROPN
ejpam-4034	71	14	a5	a5	PROPN
ejpam-4034	71	15	a1	a1	NOUN
ejpam-4034	71	16	a3	a3	NOUN
ejpam-4034	71	17	a1	a1	NOUN
ejpam-4034	71	18	a5	a5	PROPN
ejpam-4034	71	19	a5	a5	PROPN
ejpam-4034	71	20	a1	a1	NOUN
ejpam-4034	71	21	a1	a1	NOUN
ejpam-4034	71	22	a5	a5	PROPN
ejpam-4034	71	23	a5	a5	PROPN
ejpam-4034	71	24	a1	a1	NOUN
ejpam-4034	71	25	a4	a4	NOUN
ejpam-4034	71	26	a1	a1	NOUN
ejpam-4034	71	27	a1	a1	NOUN
ejpam-4034	71	28	a1	a1	NOUN
ejpam-4034	71	29	a1	a1	NOUN
ejpam-4034	71	30	a1	a1	NOUN
ejpam-4034	71	31	a1	a1	NOUN
ejpam-4034	71	32	a1	a1	NOUN
ejpam-4034	71	33	a1	a1	NOUN
ejpam-4034	71	34	a5	a5	NOUN
ejpam-4034	71	35	a1	a1	NOUN
ejpam-4034	71	36	a4	a4	NOUN
ejpam-4034	71	37	a4	a4	NOUN
ejpam-4034	71	38	a1	a1	NOUN
ejpam-4034	71	39	a1	a1	NOUN
ejpam-4034	71	40	a4	a4	NOUN
ejpam-4034	71	41	a4	a4	NOUN
ejpam-4034	71	42	a1	a1	NOUN
ejpam-4034	71	43	a6	a6	NOUN
ejpam-4034	71	44	a1	a1	NOUN
ejpam-4034	71	45	a8	a8	PROPN
ejpam-4034	71	46	a8	a8	PROPN
ejpam-4034	71	47	a1	a1	NOUN
ejpam-4034	71	48	a1	a1	NOUN
ejpam-4034	71	49	a8	a8	PROPN
ejpam-4034	71	50	a8	a8	PROPN
ejpam-4034	71	51	a1	a1	NOUN
ejpam-4034	71	52	a7	a7	PROPN
ejpam-4034	71	53	a1	a1	NOUN
ejpam-4034	71	54	a8	a8	PROPN
ejpam-4034	71	55	a8	a8	PROPN
ejpam-4034	71	56	a1	a1	NOUN
ejpam-4034	71	57	a1	a1	NOUN
ejpam-4034	71	58	a8	a8	PROPN
ejpam-4034	71	59	a8	a8	PROPN
ejpam-4034	71	60	a1	a1	NOUN
ejpam-4034	71	61	a8	a8	PROPN
ejpam-4034	71	62	a1	a1	NOUN
ejpam-4034	71	63	a4	a4	NOUN
ejpam-4034	71	64	a4	a4	NOUN
ejpam-4034	71	65	a1	a1	NOUN
ejpam-4034	71	66	a1	a1	NOUN
ejpam-4034	71	67	a4	a4	NOUN
ejpam-4034	71	68	a4	a4	NOUN
ejpam-4034	71	69	a1	a1	NOUN
ejpam-4034	71	70	⊕	⊕	PROPN
ejpam-4034	71	71	s1	s1	PROPN
ejpam-4034	71	72	s2	s2	PROPN
ejpam-4034	71	73	s3	s3	PROPN
ejpam-4034	71	74	s1	s1	PROPN
ejpam-4034	71	75	s1	s1	PROPN
ejpam-4034	71	76	s2	s2	PROPN
ejpam-4034	71	77	s3	s3	PROPN
ejpam-4034	71	78	s2	s2	PROPN
ejpam-4034	71	79	s3	s3	PROPN
ejpam-4034	71	80	s1	s1	PROPN
ejpam-4034	71	81	s2	s2	PROPN
ejpam-4034	71	82	s3	s3	PROPN
ejpam-4034	71	83	s2	s2	PROPN
ejpam-4034	71	84	s3	s3	PROPN
ejpam-4034	71	85	s1	s1	PROPN
ejpam-4034	71	86	then	then	ADV
ejpam-4034	71	87	,	,	PUNCT
ejpam-4034	71	88	clearly	clearly	ADV
ejpam-4034	71	89	r	r	NOUN
ejpam-4034	71	90	is	be	AUX
ejpam-4034	71	91	γ	γ	PRON
ejpam-4034	71	92	-la	-la	NOUN
ejpam-4034	71	93	-	-	PUNCT
ejpam-4034	71	94	ring	ring	NOUN
ejpam-4034	71	95	under	under	ADP
ejpam-4034	71	96	operation	operation	NOUN
ejpam-4034	71	97	xγy	xγy	PROPN
ejpam-4034	72	1	=	=	PUNCT
ejpam-4034	72	2	xy	xy	PROPN
ejpam-4034	72	3	where	where	SCONJ
ejpam-4034	72	4	x	x	X
ejpam-4034	72	5	,	,	PUNCT
ejpam-4034	72	6	y	y	PROPN
ejpam-4034	72	7	∈	∈	PROPN
ejpam-4034	72	8	r	r	NOUN
ejpam-4034	72	9	andγ	andγ	NOUN
ejpam-4034	72	10	∈	∈	PROPN
ejpam-4034	72	11	γ	γ	X
ejpam-4034	72	12	.	.	PUNCT
ejpam-4034	72	13	example	example	NOUN
ejpam-4034	73	1	2	2	NUM
ejpam-4034	73	2	.	.	PUNCT
ejpam-4034	73	3	let	let	VERB
ejpam-4034	73	4	r	r	NOUN
ejpam-4034	73	5	=	=	SYM
ejpam-4034	73	6	{	{	PUNCT
ejpam-4034	73	7	k	k	NOUN
ejpam-4034	73	8	,	,	PUNCT
ejpam-4034	73	9	l	l	NOUN
ejpam-4034	73	10	,	,	PUNCT
ejpam-4034	73	11	m	m	PROPN
ejpam-4034	73	12	,	,	PUNCT
ejpam-4034	73	13	n	n	CCONJ
ejpam-4034	73	14	,	,	PUNCT
ejpam-4034	73	15	o	o	NOUN
ejpam-4034	73	16	,	,	PUNCT
ejpam-4034	73	17	p	p	X
ejpam-4034	73	18	,	,	PUNCT
ejpam-4034	73	19	q	q	ADJ
ejpam-4034	73	20	,	,	PUNCT
ejpam-4034	73	21	r	r	NOUN
ejpam-4034	73	22	}	}	PUNCT
ejpam-4034	73	23	with	with	ADP
ejpam-4034	73	24	two	two	NUM
ejpam-4034	73	25	binary	binary	ADJ
ejpam-4034	73	26	operations	operation	NOUN
ejpam-4034	73	27	”	"	PUNCT
ejpam-4034	73	28	+	+	CCONJ
ejpam-4034	73	29	”	"	PUNCT
ejpam-4034	73	30	and	and	CCONJ
ejpam-4034	73	31	”	"	PUNCT
ejpam-4034	73	32	.	.	PUNCT
ejpam-4034	73	33	”	"	PUNCT
ejpam-4034	74	1	given	give	VERB
ejpam-4034	74	2	in	in	ADP
ejpam-4034	74	3	tables	table	NOUN
ejpam-4034	74	4	set	set	VERB
ejpam-4034	74	5	2	2	NUM
ejpam-4034	74	6	be	be	AUX
ejpam-4034	74	7	the	the	DET
ejpam-4034	74	8	la	la	NOUN
ejpam-4034	74	9	-	-	PUNCT
ejpam-4034	74	10	ring	ring	NOUN
ejpam-4034	74	11	and	and	CCONJ
ejpam-4034	74	12	γ	γ	X
ejpam-4034	74	13	=	=	X
ejpam-4034	74	14	{	{	PUNCT
ejpam-4034	74	15	s	s	PROPN
ejpam-4034	74	16	,	,	PUNCT
ejpam-4034	74	17	t	t	PROPN
ejpam-4034	74	18	,	,	PUNCT
ejpam-4034	74	19	u	u	NOUN
ejpam-4034	74	20	,	,	PUNCT
ejpam-4034	74	21	v	v	NOUN
ejpam-4034	74	22	,	,	PUNCT
ejpam-4034	74	23	w	w	NOUN
ejpam-4034	74	24	}	}	PUNCT
ejpam-4034	74	25	with	with	SCONJ
ejpam-4034	74	26	binary	binary	ADJ
ejpam-4034	74	27	operation	operation	PROPN
ejpam-4034	74	28	⊕	⊕	PROPN
ejpam-4034	74	29	is	be	AUX
ejpam-4034	74	30	la	la	ADJ
ejpam-4034	74	31	-	-	NOUN
ejpam-4034	74	32	group	group	NOUN
ejpam-4034	74	33	.	.	PUNCT
ejpam-4034	75	1	tables	table	NOUN
ejpam-4034	75	2	set	set	VERB
ejpam-4034	75	3	2	2	NUM
ejpam-4034	76	1	+	+	CCONJ
ejpam-4034	76	2	k	k	PROPN
ejpam-4034	76	3	l	l	NOUN
ejpam-4034	76	4	m	m	VERB
ejpam-4034	76	5	n	n	ADV
ejpam-4034	76	6	o	o	NOUN
ejpam-4034	76	7	p	p	X
ejpam-4034	76	8	q	q	NOUN
ejpam-4034	77	1	r	r	NOUN
ejpam-4034	77	2	k	k	NOUN
ejpam-4034	77	3	m	m	VERB
ejpam-4034	77	4	k	k	NOUN
ejpam-4034	77	5	n	n	PRON
ejpam-4034	77	6	l	l	NOUN
ejpam-4034	77	7	q	q	NOUN
ejpam-4034	77	8	r	r	NOUN
ejpam-4034	77	9	o	o	X
ejpam-4034	77	10	p	p	X
ejpam-4034	77	11	l	l	NOUN
ejpam-4034	77	12	n	n	CCONJ
ejpam-4034	77	13	m	m	NOUN
ejpam-4034	77	14	l	l	NOUN
ejpam-4034	78	1	k	k	NOUN
ejpam-4034	78	2	r	r	NOUN
ejpam-4034	78	3	q	q	NOUN
ejpam-4034	78	4	p	p	X
ejpam-4034	78	5	o	o	NOUN
ejpam-4034	78	6	m	m	VERB
ejpam-4034	78	7	k	k	X
ejpam-4034	78	8	l	l	NOUN
ejpam-4034	78	9	m	m	VERB
ejpam-4034	78	10	n	n	ADV
ejpam-4034	78	11	o	o	NOUN
ejpam-4034	78	12	p	p	X
ejpam-4034	78	13	q	q	NOUN
ejpam-4034	78	14	r	r	NOUN
ejpam-4034	78	15	n	n	ADP
ejpam-4034	78	16	l	l	NOUN
ejpam-4034	78	17	n	n	CCONJ
ejpam-4034	78	18	k	k	NOUN
ejpam-4034	78	19	m	m	VERB
ejpam-4034	78	20	p	p	X
ejpam-4034	78	21	o	o	X
ejpam-4034	78	22	r	r	NOUN
ejpam-4034	78	23	q	q	NOUN
ejpam-4034	78	24	o	o	NOUN
ejpam-4034	78	25	q	q	X
ejpam-4034	78	26	r	r	NOUN
ejpam-4034	78	27	o	o	X
ejpam-4034	79	1	p	p	NOUN
ejpam-4034	79	2	m	m	PROPN
ejpam-4034	79	3	k	k	NOUN
ejpam-4034	79	4	n	n	PRON
ejpam-4034	79	5	l	l	NOUN
ejpam-4034	79	6	p	p	NOUN
ejpam-4034	79	7	r	r	NOUN
ejpam-4034	79	8	q	q	X
ejpam-4034	79	9	p	p	X
ejpam-4034	79	10	o	o	NOUN
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ejpam-4034	79	13	l	l	NOUN
ejpam-4034	80	1	k	k	NOUN
ejpam-4034	80	2	q	q	PUNCT
ejpam-4034	81	1	o	o	X
ejpam-4034	81	2	p	p	X
ejpam-4034	81	3	q	q	NOUN
ejpam-4034	81	4	r	r	NOUN
ejpam-4034	81	5	k	k	PROPN
ejpam-4034	81	6	l	l	NOUN
ejpam-4034	81	7	m	m	VERB
ejpam-4034	81	8	n	n	PRON
ejpam-4034	82	1	r	r	NOUN
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ejpam-4034	82	3	o	o	NOUN
ejpam-4034	82	4	r	r	NOUN
ejpam-4034	82	5	q	q	PROPN
ejpam-4034	82	6	l	l	NOUN
ejpam-4034	83	1	n	n	CCONJ
ejpam-4034	83	2	k	k	PROPN
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ejpam-4034	83	4	·	·	PUNCT
ejpam-4034	83	5	k	k	X
ejpam-4034	83	6	l	l	NOUN
ejpam-4034	83	7	m	m	VERB
ejpam-4034	83	8	n	n	ADV
ejpam-4034	83	9	o	o	NOUN
ejpam-4034	83	10	p	p	X
ejpam-4034	83	11	q	q	NOUN
ejpam-4034	83	12	r	r	NOUN
ejpam-4034	84	1	k	k	PROPN
ejpam-4034	84	2	k	k	PROPN
ejpam-4034	85	1	k	k	PROPN
ejpam-4034	86	1	k	k	PROPN
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ejpam-4034	89	1	k	k	PROPN
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ejpam-4034	92	1	l	l	PUNCT
ejpam-4034	93	1	k	k	NOUN
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ejpam-4034	93	3	k	k	NOUN
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ejpam-4034	93	5	k	k	PROPN
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ejpam-4034	93	7	m	m	VERB
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ejpam-4034	93	9	m	m	VERB
ejpam-4034	93	10	k	k	NOUN
ejpam-4034	93	11	o	o	X
ejpam-4034	94	1	k	k	X
ejpam-4034	94	2	o	o	X
ejpam-4034	95	1	k	k	PROPN
ejpam-4034	96	1	k	k	X
ejpam-4034	96	2	o	o	X
ejpam-4034	97	1	o	o	X
ejpam-4034	97	2	n	n	CCONJ
ejpam-4034	98	1	k	k	NOUN
ejpam-4034	98	2	p	p	X
ejpam-4034	98	3	k	k	PROPN
ejpam-4034	99	1	p	p	X
ejpam-4034	99	2	k	k	PROPN
ejpam-4034	100	1	k	k	PROPN
ejpam-4034	101	1	p	p	X
ejpam-4034	102	1	p	p	X
ejpam-4034	102	2	o	o	X
ejpam-4034	103	1	k	k	PROPN
ejpam-4034	103	2	k	k	PROPN
ejpam-4034	104	1	k	k	PROPN
ejpam-4034	105	1	k	k	PROPN
ejpam-4034	106	1	k	k	PROPN
ejpam-4034	107	1	k	k	PROPN
ejpam-4034	108	1	k	k	PROPN
ejpam-4034	109	1	k	k	PROPN
ejpam-4034	110	1	p	p	X
ejpam-4034	111	1	k	k	X
ejpam-4034	111	2	o	o	X
ejpam-4034	112	1	k	k	X
ejpam-4034	113	1	o	o	X
ejpam-4034	113	2	k	k	PROPN
ejpam-4034	114	1	k	k	X
ejpam-4034	115	1	o	o	X
ejpam-4034	115	2	o	o	X
ejpam-4034	115	3	q	q	X
ejpam-4034	116	1	k	k	NOUN
ejpam-4034	116	2	m	m	VERB
ejpam-4034	116	3	k	k	NOUN
ejpam-4034	116	4	m	m	VERB
ejpam-4034	116	5	k	k	X
ejpam-4034	117	1	k	k	X
ejpam-4034	117	2	m	m	VERB
ejpam-4034	117	3	m	m	VERB
ejpam-4034	117	4	r	r	NOUN
ejpam-4034	117	5	k	k	NOUN
ejpam-4034	118	1	p	p	X
ejpam-4034	118	2	k	k	PROPN
ejpam-4034	119	1	p	p	X
ejpam-4034	119	2	k	k	PROPN
ejpam-4034	120	1	k	k	PROPN
ejpam-4034	120	2	p	p	PROPN
ejpam-4034	120	3	p	p	PROPN
ejpam-4034	120	4	w.	w.	PROPN
ejpam-4034	120	5	a.	a.	PROPN
ejpam-4034	120	6	khan	khan	PROPN
ejpam-4034	120	7	et	et	PROPN
ejpam-4034	120	8	al	al	PROPN
ejpam-4034	120	9	.	.	PUNCT
ejpam-4034	120	10	/	/	SYM
ejpam-4034	120	11	eur	eur	PROPN
ejpam-4034	120	12	.	.	PUNCT
ejpam-4034	121	1	j.	j.	PROPN
ejpam-4034	121	2	pure	pure	PROPN
ejpam-4034	121	3	appl	appl	PROPN
ejpam-4034	121	4	.	.	PROPN
ejpam-4034	121	5	math	math	PROPN
ejpam-4034	121	6	,	,	PUNCT
ejpam-4034	121	7	14	14	NUM
ejpam-4034	121	8	(	(	PUNCT
ejpam-4034	121	9	3	3	NUM
ejpam-4034	121	10	)	)	PUNCT
ejpam-4034	121	11	(	(	PUNCT
ejpam-4034	121	12	2021	2021	NUM
ejpam-4034	121	13	)	)	PUNCT
ejpam-4034	121	14	,	,	PUNCT
ejpam-4034	121	15	989	989	NUM
ejpam-4034	121	16	-	-	SYM
ejpam-4034	121	17	1001	1001	NUM
ejpam-4034	121	18	992⊕	992⊕	NUM
ejpam-4034	121	19	s	s	NOUN
ejpam-4034	121	20	t	t	NOUN
ejpam-4034	121	21	u	u	NOUN
ejpam-4034	121	22	v	v	ADP
ejpam-4034	121	23	w	w	PROPN
ejpam-4034	121	24	s	s	PROPN
ejpam-4034	121	25	s	s	PROPN
ejpam-4034	121	26	t	t	X
ejpam-4034	121	27	u	u	NOUN
ejpam-4034	121	28	v	v	PROPN
ejpam-4034	121	29	w	w	PROPN
ejpam-4034	121	30	t	t	PROPN
ejpam-4034	122	1	w	w	PROPN
ejpam-4034	122	2	s	s	PROPN
ejpam-4034	122	3	t	t	PROPN
ejpam-4034	122	4	u	u	NOUN
ejpam-4034	122	5	v	v	ADP
ejpam-4034	122	6	u	u	NOUN
ejpam-4034	122	7	v	v	ADP
ejpam-4034	122	8	w	w	PROPN
ejpam-4034	122	9	s	s	PROPN
ejpam-4034	122	10	t	t	NOUN
ejpam-4034	122	11	u	u	NOUN
ejpam-4034	122	12	v	v	ADP
ejpam-4034	122	13	u	u	NOUN
ejpam-4034	122	14	v	v	ADP
ejpam-4034	122	15	w	w	PROPN
ejpam-4034	122	16	s	s	PROPN
ejpam-4034	122	17	t	t	PROPN
ejpam-4034	122	18	w	w	PROPN
ejpam-4034	122	19	t	t	PROPN
ejpam-4034	122	20	u	u	PROPN
ejpam-4034	122	21	v	v	ADP
ejpam-4034	122	22	w	w	NOUN
ejpam-4034	122	23	s	s	X
ejpam-4034	122	24	then	then	ADV
ejpam-4034	122	25	r	r	NOUN
ejpam-4034	122	26	is	be	AUX
ejpam-4034	122	27	γ	γ	PRON
ejpam-4034	122	28	-la	-la	NOUN
ejpam-4034	122	29	-	-	PUNCT
ejpam-4034	122	30	ring	ring	NOUN
ejpam-4034	122	31	under	under	ADP
ejpam-4034	122	32	operation	operation	NOUN
ejpam-4034	122	33	xγy	xγy	NOUN
ejpam-4034	123	1	=	=	SYM
ejpam-4034	123	2	x+	x+	NUM
ejpam-4034	123	3	γ	γ	X
ejpam-4034	123	4	+	+	X
ejpam-4034	123	5	y	y	PROPN
ejpam-4034	123	6	where	where	SCONJ
ejpam-4034	123	7	x	x	X
ejpam-4034	123	8	,	,	PUNCT
ejpam-4034	123	9	y	y	PROPN
ejpam-4034	123	10	∈	∈	PROPN
ejpam-4034	123	11	r	r	NOUN
ejpam-4034	123	12	and	and	CCONJ
ejpam-4034	123	13	γ	γ	PROPN
ejpam-4034	123	14	∈	∈	PROPN
ejpam-4034	123	15	γ	γ	X
ejpam-4034	123	16	definition	definition	NOUN
ejpam-4034	123	17	2	2	NUM
ejpam-4034	123	18	.	.	PUNCT
ejpam-4034	124	1	an	an	DET
ejpam-4034	124	2	ideal	ideal	NOUN
ejpam-4034	124	3	i	i	PRON
ejpam-4034	124	4	is	be	AUX
ejpam-4034	124	5	said	say	VERB
ejpam-4034	124	6	to	to	PART
ejpam-4034	124	7	be	be	AUX
ejpam-4034	124	8	a	a	DET
ejpam-4034	124	9	c	c	NOUN
ejpam-4034	124	10	-	-	PUNCT
ejpam-4034	124	11	prime	prime	ADJ
ejpam-4034	124	12	ideal	ideal	NOUN
ejpam-4034	124	13	of	of	ADP
ejpam-4034	124	14	a	a	DET
ejpam-4034	124	15	γ	γ	X
ejpam-4034	124	16	-	-	PUNCT
ejpam-4034	124	17	la	la	ADJ
ejpam-4034	124	18	-	-	PUNCT
ejpam-4034	124	19	ring	ring	NOUN
ejpam-4034	124	20	r	r	NOUN
ejpam-4034	124	21	,	,	PUNCT
ejpam-4034	124	22	if	if	SCONJ
ejpam-4034	124	23	x	x	NOUN
ejpam-4034	124	24	,	,	PUNCT
ejpam-4034	124	25	y	y	PROPN
ejpam-4034	124	26	∈	∈	PROPN
ejpam-4034	124	27	r	r	PROPN
ejpam-4034	124	28	,	,	PUNCT
ejpam-4034	124	29	γ	γ	PROPN
ejpam-4034	124	30	∈	∈	PROPN
ejpam-4034	124	31	γ	γ	NOUN
ejpam-4034	124	32	and	and	CCONJ
ejpam-4034	124	33	xγy	xγy	NOUN
ejpam-4034	124	34	∈	∈	PROPN
ejpam-4034	124	35	i	i	PRON
ejpam-4034	124	36	=	=	VERB
ejpam-4034	124	37	⇒	⇒	VERB
ejpam-4034	124	38	x	x	PUNCT
ejpam-4034	124	39	∈	∈	PROPN
ejpam-4034	124	40	i	i	PRON
ejpam-4034	124	41	or	or	CCONJ
ejpam-4034	124	42	y	y	PROPN
ejpam-4034	124	43	∈	∈	PROPN
ejpam-4034	124	44	i.	i.	NOUN
ejpam-4034	124	45	definition	definition	NOUN
ejpam-4034	124	46	3	3	NUM
ejpam-4034	124	47	.	.	PUNCT
ejpam-4034	125	1	an	an	DET
ejpam-4034	125	2	ideal	ideal	NOUN
ejpam-4034	125	3	i	i	PRON
ejpam-4034	125	4	is	be	AUX
ejpam-4034	125	5	said	say	VERB
ejpam-4034	125	6	to	to	PART
ejpam-4034	125	7	be	be	AUX
ejpam-4034	125	8	a	a	DET
ejpam-4034	125	9	3	3	NUM
ejpam-4034	125	10	-	-	PUNCT
ejpam-4034	125	11	prime	prime	ADJ
ejpam-4034	125	12	ideal	ideal	NOUN
ejpam-4034	125	13	of	of	ADP
ejpam-4034	125	14	a	a	DET
ejpam-4034	125	15	γ	γ	X
ejpam-4034	125	16	-	-	PUNCT
ejpam-4034	125	17	la	la	ADJ
ejpam-4034	125	18	-	-	PUNCT
ejpam-4034	125	19	ring	ring	NOUN
ejpam-4034	125	20	r	r	NOUN
ejpam-4034	125	21	if	if	SCONJ
ejpam-4034	125	22	x	x	PROPN
ejpam-4034	125	23	,	,	PUNCT
ejpam-4034	125	24	y	y	PROPN
ejpam-4034	125	25	∈	∈	PROPN
ejpam-4034	125	26	r	r	PROPN
ejpam-4034	125	27	,	,	PUNCT
ejpam-4034	125	28	β	β	X
ejpam-4034	125	29	∈	∈	NOUN
ejpam-4034	125	30	γ	γ	NOUN
ejpam-4034	125	31	and	and	CCONJ
ejpam-4034	125	32	xβsβy	xβsβy	PROPN
ejpam-4034	125	33	∈	∈	PROPN
ejpam-4034	125	34	i	i	PRON
ejpam-4034	125	35	for	for	ADP
ejpam-4034	125	36	all	all	DET
ejpam-4034	125	37	s	s	PART
ejpam-4034	125	38	∈	∈	NOUN
ejpam-4034	125	39	r	r	NOUN
ejpam-4034	125	40	implies	imply	VERB
ejpam-4034	125	41	x	x	X
ejpam-4034	125	42	∈	∈	NOUN
ejpam-4034	126	1	i	i	PRON
ejpam-4034	126	2	or	or	CCONJ
ejpam-4034	126	3	y	y	PROPN
ejpam-4034	126	4	∈	∈	PROPN
ejpam-4034	126	5	i.	i.	NOUN
ejpam-4034	126	6	we	we	PRON
ejpam-4034	126	7	present	present	VERB
ejpam-4034	126	8	few	few	ADJ
ejpam-4034	126	9	relationships	relationship	NOUN
ejpam-4034	126	10	among	among	ADP
ejpam-4034	126	11	c	c	NOUN
ejpam-4034	126	12	-	-	PUNCT
ejpam-4034	126	13	prime	prime	ADJ
ejpam-4034	126	14	,	,	PUNCT
ejpam-4034	126	15	3	3	NUM
ejpam-4034	126	16	-	-	PUNCT
ejpam-4034	126	17	prime	prime	ADJ
ejpam-4034	126	18	and	and	CCONJ
ejpam-4034	126	19	prime	prime	ADJ
ejpam-4034	126	20	ideals	ideal	NOUN
ejpam-4034	126	21	.	.	PUNCT
ejpam-4034	127	1	lemma	lemma	PROPN
ejpam-4034	127	2	1	1	NUM
ejpam-4034	127	3	.	.	PUNCT
ejpam-4034	128	1	in	in	ADP
ejpam-4034	128	2	a	a	DET
ejpam-4034	128	3	γ	γ	X
ejpam-4034	128	4	-	-	PUNCT
ejpam-4034	128	5	la	la	ADJ
ejpam-4034	128	6	-	-	PUNCT
ejpam-4034	128	7	ring	ring	NOUN
ejpam-4034	128	8	r	r	NOUN
ejpam-4034	128	9	,	,	PUNCT
ejpam-4034	128	10	every	every	DET
ejpam-4034	128	11	c	c	NOUN
ejpam-4034	128	12	-	-	PUNCT
ejpam-4034	128	13	prime	prime	ADJ
ejpam-4034	128	14	ideal	ideal	NOUN
ejpam-4034	128	15	is	be	AUX
ejpam-4034	128	16	a	a	DET
ejpam-4034	128	17	3	3	NUM
ejpam-4034	128	18	-	-	PUNCT
ejpam-4034	128	19	prime	prime	ADJ
ejpam-4034	128	20	ideal	ideal	NOUN
ejpam-4034	128	21	.	.	PUNCT
ejpam-4034	129	1	proof	proof	NOUN
ejpam-4034	129	2	.	.	PUNCT
ejpam-4034	130	1	let	let	VERB
ejpam-4034	130	2	i	i	PRON
ejpam-4034	130	3	be	be	AUX
ejpam-4034	130	4	a	a	DET
ejpam-4034	130	5	c	c	NOUN
ejpam-4034	130	6	-	-	PUNCT
ejpam-4034	130	7	prime	prime	ADJ
ejpam-4034	130	8	ideal	ideal	NOUN
ejpam-4034	130	9	of	of	ADP
ejpam-4034	130	10	γ	γ	X
ejpam-4034	130	11	-	-	PUNCT
ejpam-4034	130	12	la	la	ADJ
ejpam-4034	130	13	-	-	PUNCT
ejpam-4034	130	14	ring	ring	NOUN
ejpam-4034	130	15	r.	r.	PROPN
ejpam-4034	130	16	let	let	VERB
ejpam-4034	130	17	x	x	PRON
ejpam-4034	130	18	,	,	PUNCT
ejpam-4034	130	19	y	y	PROPN
ejpam-4034	130	20	∈	∈	PROPN
ejpam-4034	130	21	r	r	PROPN
ejpam-4034	130	22	,	,	PUNCT
ejpam-4034	130	23	β	β	X
ejpam-4034	130	24	∈	∈	NOUN
ejpam-4034	130	25	γ	γ	NOUN
ejpam-4034	130	26	and	and	CCONJ
ejpam-4034	130	27	xβsβy	xβsβy	PROPN
ejpam-4034	130	28	∈	∈	PROPN
ejpam-4034	130	29	i	i	PRON
ejpam-4034	130	30	for	for	ADP
ejpam-4034	130	31	all	all	DET
ejpam-4034	130	32	s	s	PROPN
ejpam-4034	130	33	∈	∈	PROPN
ejpam-4034	130	34	r.	r.	NOUN
ejpam-4034	130	35	as	as	SCONJ
ejpam-4034	130	36	i	i	PRON
ejpam-4034	130	37	is	be	AUX
ejpam-4034	130	38	c	c	NOUN
ejpam-4034	130	39	-	-	PUNCT
ejpam-4034	130	40	prime	prime	ADJ
ejpam-4034	130	41	ideal	ideal	NOUN
ejpam-4034	131	1	so	so	SCONJ
ejpam-4034	131	2	x	x	SYM
ejpam-4034	131	3	∈	∈	PROPN
ejpam-4034	131	4	i	i	PRON
ejpam-4034	131	5	or	or	CCONJ
ejpam-4034	131	6	y	y	PROPN
ejpam-4034	131	7	∈	∈	PROPN
ejpam-4034	131	8	i.	i.	NOUN
ejpam-4034	132	1	so	so	ADV
ejpam-4034	132	2	i	i	PRON
ejpam-4034	132	3	is	be	AUX
ejpam-4034	132	4	a	a	DET
ejpam-4034	132	5	3	3	NUM
ejpam-4034	132	6	-	-	PUNCT
ejpam-4034	132	7	prime	prime	ADJ
ejpam-4034	132	8	ideal	ideal	NOUN
ejpam-4034	132	9	of	of	ADP
ejpam-4034	132	10	r.	r.	PROPN
ejpam-4034	132	11	lemma	lemma	PROPN
ejpam-4034	133	1	2	2	X
ejpam-4034	133	2	.	.	PUNCT
ejpam-4034	133	3	in	in	ADP
ejpam-4034	133	4	a	a	DET
ejpam-4034	133	5	γ	γ	X
ejpam-4034	133	6	-	-	PUNCT
ejpam-4034	133	7	la	la	ADJ
ejpam-4034	133	8	-	-	PUNCT
ejpam-4034	133	9	ring	ring	NOUN
ejpam-4034	133	10	r	r	NOUN
ejpam-4034	133	11	,	,	PUNCT
ejpam-4034	133	12	every	every	DET
ejpam-4034	133	13	3	3	NUM
ejpam-4034	133	14	-	-	PUNCT
ejpam-4034	133	15	prime	prime	ADJ
ejpam-4034	133	16	ideal	ideal	NOUN
ejpam-4034	133	17	is	be	AUX
ejpam-4034	133	18	a	a	DET
ejpam-4034	133	19	prime	prime	ADJ
ejpam-4034	133	20	ideal	ideal	NOUN
ejpam-4034	133	21	.	.	PUNCT
ejpam-4034	134	1	proof	proof	NOUN
ejpam-4034	134	2	.	.	PUNCT
ejpam-4034	135	1	let	let	VERB
ejpam-4034	135	2	i	i	PRON
ejpam-4034	135	3	be	be	AUX
ejpam-4034	135	4	a	a	DET
ejpam-4034	135	5	3	3	NUM
ejpam-4034	135	6	-	-	PUNCT
ejpam-4034	135	7	prime	prime	ADJ
ejpam-4034	135	8	ideal	ideal	NOUN
ejpam-4034	135	9	of	of	ADP
ejpam-4034	135	10	γ	γ	X
ejpam-4034	135	11	-	-	PUNCT
ejpam-4034	135	12	la	la	ADJ
ejpam-4034	135	13	-	-	PUNCT
ejpam-4034	135	14	ring	ring	NOUN
ejpam-4034	135	15	r.	r.	NOUN
ejpam-4034	135	16	let	let	VERB
ejpam-4034	135	17	y	y	PROPN
ejpam-4034	135	18	∈	∈	PROPN
ejpam-4034	136	1	i	i	PRON
ejpam-4034	136	2	,	,	PUNCT
ejpam-4034	136	3	γ	γ	PROPN
ejpam-4034	136	4	∈	∈	PROPN
ejpam-4034	136	5	γ	γ	NOUN
ejpam-4034	136	6	and	and	CCONJ
ejpam-4034	136	7	xγy	xγy	PROPN
ejpam-4034	136	8	∈	∈	PROPN
ejpam-4034	136	9	i.	i.	NOUN
ejpam-4034	136	10	as	as	SCONJ
ejpam-4034	136	11	i	i	PRON
ejpam-4034	136	12	is	be	AUX
ejpam-4034	136	13	3	3	NUM
ejpam-4034	136	14	-	-	PUNCT
ejpam-4034	136	15	prime	prime	ADJ
ejpam-4034	136	16	ideal	ideal	NOUN
ejpam-4034	137	1	so	so	SCONJ
ejpam-4034	137	2	x	x	SYM
ejpam-4034	137	3	∈	∈	PROPN
ejpam-4034	137	4	i	i	PRON
ejpam-4034	137	5	or	or	CCONJ
ejpam-4034	137	6	y	y	PROPN
ejpam-4034	137	7	∈	∈	PROPN
ejpam-4034	137	8	i.	i.	NOUN
ejpam-4034	137	9	clearly	clearly	ADV
ejpam-4034	137	10	,	,	PUNCT
ejpam-4034	137	11	i	i	PRON
ejpam-4034	137	12	is	be	AUX
ejpam-4034	137	13	prime	prime	ADJ
ejpam-4034	137	14	ideal	ideal	NOUN
ejpam-4034	137	15	of	of	ADP
ejpam-4034	137	16	r.	r.	PROPN
ejpam-4034	137	17	lemma	lemma	PROPN
ejpam-4034	138	1	3	3	X
ejpam-4034	138	2	.	.	PUNCT
ejpam-4034	139	1	every	every	DET
ejpam-4034	139	2	c	c	NOUN
ejpam-4034	139	3	-	-	PUNCT
ejpam-4034	139	4	prime	prime	ADJ
ejpam-4034	139	5	ideal	ideal	NOUN
ejpam-4034	139	6	in	in	ADP
ejpam-4034	139	7	γ	γ	PROPN
ejpam-4034	139	8	-	-	PUNCT
ejpam-4034	139	9	la	la	ADJ
ejpam-4034	139	10	-	-	PUNCT
ejpam-4034	139	11	ring	ring	NOUN
ejpam-4034	139	12	is	be	AUX
ejpam-4034	139	13	a	a	DET
ejpam-4034	139	14	prime	prime	ADJ
ejpam-4034	139	15	ideal	ideal	NOUN
ejpam-4034	139	16	.	.	PUNCT
ejpam-4034	140	1	proof	proof	NOUN
ejpam-4034	140	2	.	.	PUNCT
ejpam-4034	141	1	let	let	VERB
ejpam-4034	141	2	i	i	PRON
ejpam-4034	141	3	be	be	AUX
ejpam-4034	141	4	a	a	DET
ejpam-4034	141	5	c	c	NOUN
ejpam-4034	141	6	-	-	PUNCT
ejpam-4034	141	7	prime	prime	ADJ
ejpam-4034	141	8	ideal	ideal	NOUN
ejpam-4034	141	9	of	of	ADP
ejpam-4034	141	10	γ	γ	X
ejpam-4034	141	11	-	-	PUNCT
ejpam-4034	141	12	la	la	ADJ
ejpam-4034	141	13	-	-	PUNCT
ejpam-4034	141	14	ring	ring	NOUN
ejpam-4034	141	15	r	r	NOUN
ejpam-4034	141	16	,	,	PUNCT
ejpam-4034	141	17	and	and	CCONJ
ejpam-4034	141	18	let	let	VERB
ejpam-4034	141	19	y	y	PROPN
ejpam-4034	141	20	∈	∈	PROPN
ejpam-4034	141	21	i	i	PRON
ejpam-4034	141	22	,	,	PUNCT
ejpam-4034	141	23	γ	γ	PROPN
ejpam-4034	141	24	∈	∈	PROPN
ejpam-4034	141	25	γ	γ	NOUN
ejpam-4034	141	26	and	and	CCONJ
ejpam-4034	141	27	xγy	xγy	PROPN
ejpam-4034	141	28	∈	∈	PROPN
ejpam-4034	141	29	i.	i.	NOUN
ejpam-4034	141	30	since	since	SCONJ
ejpam-4034	141	31	i	i	PRON
ejpam-4034	141	32	is	be	AUX
ejpam-4034	141	33	c	c	NOUN
ejpam-4034	141	34	-	-	PUNCT
ejpam-4034	141	35	prime	prime	ADJ
ejpam-4034	141	36	ideal	ideal	NOUN
ejpam-4034	141	37	,	,	PUNCT
ejpam-4034	141	38	x	x	SYM
ejpam-4034	141	39	∈	∈	NOUN
ejpam-4034	142	1	i	i	PRON
ejpam-4034	142	2	or	or	CCONJ
ejpam-4034	142	3	y	y	PROPN
ejpam-4034	142	4	∈	∈	PROPN
ejpam-4034	142	5	i.	i.	NOUN
ejpam-4034	143	1	then	then	ADV
ejpam-4034	143	2	i	i	PRON
ejpam-4034	143	3	is	be	AUX
ejpam-4034	143	4	a	a	DET
ejpam-4034	143	5	prime	prime	ADJ
ejpam-4034	143	6	ideal	ideal	NOUN
ejpam-4034	143	7	of	of	ADP
ejpam-4034	143	8	r.	r.	PROPN
ejpam-4034	143	9	theorem	theorem	PROPN
ejpam-4034	143	10	1	1	X
ejpam-4034	143	11	.	.	PUNCT
ejpam-4034	144	1	let	let	VERB
ejpam-4034	144	2	r	r	PRON
ejpam-4034	144	3	be	be	AUX
ejpam-4034	144	4	a	a	DET
ejpam-4034	144	5	γ	γ	NOUN
ejpam-4034	144	6	-la	-la	NOUN
ejpam-4034	144	7	-	-	PUNCT
ejpam-4034	144	8	ring	ring	NOUN
ejpam-4034	144	9	.	.	PUNCT
ejpam-4034	145	1	then	then	ADV
ejpam-4034	145	2	,	,	PUNCT
ejpam-4034	145	3	i	i	PRON
ejpam-4034	145	4	is	be	AUX
ejpam-4034	145	5	a	a	DET
ejpam-4034	145	6	3	3	NUM
ejpam-4034	145	7	-	-	PUNCT
ejpam-4034	145	8	prime	prime	ADJ
ejpam-4034	145	9	ideal	ideal	NOUN
ejpam-4034	145	10	in	in	ADP
ejpam-4034	145	11	r	r	PROPN
ejpam-4034	145	12	iff	iff	PROPN
ejpam-4034	145	13	r	r	NOUN
ejpam-4034	145	14	/	/	SYM
ejpam-4034	145	15	i	i	PRON
ejpam-4034	145	16	is	be	AUX
ejpam-4034	145	17	a	a	DET
ejpam-4034	145	18	γ	γ	X
ejpam-4034	145	19	-laintegral	-laintegral	ADJ
ejpam-4034	145	20	domain	domain	NOUN
ejpam-4034	145	21	.	.	PUNCT
ejpam-4034	146	1	proof	proof	NOUN
ejpam-4034	146	2	.	.	PUNCT
ejpam-4034	147	1	(=	(=	NOUN
ejpam-4034	147	2	⇒)let	⇒)let	VERB
ejpam-4034	147	3	i	i	PRON
ejpam-4034	147	4	be	be	VERB
ejpam-4034	147	5	a	a	DET
ejpam-4034	147	6	3	3	NUM
ejpam-4034	147	7	-	-	PUNCT
ejpam-4034	147	8	prime	prime	ADJ
ejpam-4034	147	9	ideal	ideal	NOUN
ejpam-4034	147	10	of	of	ADP
ejpam-4034	147	11	r	r	NOUN
ejpam-4034	147	12	,	,	PUNCT
ejpam-4034	147	13	then	then	ADV
ejpam-4034	147	14	by	by	ADP
ejpam-4034	147	15	lemma	lemma	PROPN
ejpam-4034	147	16	2	2	NUM
ejpam-4034	147	17	,	,	PUNCT
ejpam-4034	147	18	i	i	PRON
ejpam-4034	147	19	is	be	AUX
ejpam-4034	147	20	a	a	DET
ejpam-4034	147	21	prime	prime	ADJ
ejpam-4034	147	22	ideal	ideal	NOUN
ejpam-4034	147	23	.	.	PUNCT
ejpam-4034	148	1	thus	thus	ADV
ejpam-4034	148	2	,	,	PUNCT
ejpam-4034	148	3	r	r	X
ejpam-4034	148	4	/	/	SYM
ejpam-4034	148	5	i	i	PRON
ejpam-4034	148	6	is	be	AUX
ejpam-4034	148	7	a	a	DET
ejpam-4034	148	8	γ	γ	NOUN
ejpam-4034	148	9	-la	-la	ADV
ejpam-4034	148	10	-	-	PUNCT
ejpam-4034	148	11	integral	integral	ADJ
ejpam-4034	148	12	domain	domain	NOUN
ejpam-4034	148	13	.	.	PUNCT
ejpam-4034	149	1	(	(	PUNCT
ejpam-4034	149	2	⇐	⇐	ADJ
ejpam-4034	149	3	=)	=)	PROPN
ejpam-4034	149	4	suppose	suppose	VERB
ejpam-4034	149	5	that	that	SCONJ
ejpam-4034	149	6	r	r	NOUN
ejpam-4034	149	7	/	/	SYM
ejpam-4034	149	8	i	i	PRON
ejpam-4034	149	9	is	be	AUX
ejpam-4034	149	10	a	a	DET
ejpam-4034	149	11	γ	γ	X
ejpam-4034	149	12	-	-	PUNCT
ejpam-4034	149	13	la	la	ADJ
ejpam-4034	149	14	-	-	ADJ
ejpam-4034	149	15	integral	integral	ADJ
ejpam-4034	149	16	domain	domain	NOUN
ejpam-4034	149	17	with	with	ADP
ejpam-4034	149	18	xβsβy	xβsβy	PROPN
ejpam-4034	149	19	∈	∈	PROPN
ejpam-4034	150	1	i	i	PRON
ejpam-4034	150	2	for	for	ADP
ejpam-4034	150	3	all	all	DET
ejpam-4034	150	4	s	s	PROPN
ejpam-4034	150	5	∈	∈	PROPN
ejpam-4034	150	6	r.	r.	NOUN
ejpam-4034	150	7	then	then	ADV
ejpam-4034	150	8	i	i	PRON
ejpam-4034	150	9	+	+	CCONJ
ejpam-4034	150	10	arb	arb	X
ejpam-4034	150	11	=	=	PUNCT
ejpam-4034	150	12	i	i	PROPN
ejpam-4034	150	13	,	,	PUNCT
ejpam-4034	150	14	so	so	CCONJ
ejpam-4034	150	15	(	(	PUNCT
ejpam-4034	150	16	i	i	PRON
ejpam-4034	150	17	+	+	CCONJ
ejpam-4034	150	18	a	a	X
ejpam-4034	150	19	)	)	PUNCT
ejpam-4034	150	20	(	(	PUNCT
ejpam-4034	150	21	i	i	PRON
ejpam-4034	150	22	+	+	NUM
ejpam-4034	150	23	b	b	X
ejpam-4034	150	24	)	)	PUNCT
ejpam-4034	150	25	=	=	SYM
ejpam-4034	151	1	i	i	PROPN
ejpam-4034	151	2	,	,	PUNCT
ejpam-4034	151	3	where	where	SCONJ
ejpam-4034	151	4	a	a	X
ejpam-4034	151	5	,	,	PUNCT
ejpam-4034	151	6	b	b	PROPN
ejpam-4034	151	7	∈	∈	PROPN
ejpam-4034	151	8	r.	r.	NOUN
ejpam-4034	151	9	since	since	SCONJ
ejpam-4034	151	10	r	r	PROPN
ejpam-4034	151	11	/	/	SYM
ejpam-4034	151	12	i	i	PRON
ejpam-4034	151	13	is	be	AUX
ejpam-4034	151	14	γ	γ	NOUN
ejpam-4034	151	15	-la	-la	ADJ
ejpam-4034	151	16	-	-	PUNCT
ejpam-4034	151	17	integral	integral	ADJ
ejpam-4034	151	18	domain	domain	NOUN
ejpam-4034	151	19	,	,	PUNCT
ejpam-4034	151	20	we	we	PRON
ejpam-4034	151	21	have	have	VERB
ejpam-4034	151	22	i	i	PRON
ejpam-4034	151	23	+	+	CCONJ
ejpam-4034	152	1	a	a	DET
ejpam-4034	152	2	=	=	X
ejpam-4034	152	3	i	i	PRON
ejpam-4034	152	4	or	or	CCONJ
ejpam-4034	152	5	i	i	PRON
ejpam-4034	152	6	+	+	CCONJ
ejpam-4034	152	7	b	b	X
ejpam-4034	152	8	=	=	SYM
ejpam-4034	152	9	i	i	PROPN
ejpam-4034	152	10	,	,	PUNCT
ejpam-4034	152	11	then	then	ADV
ejpam-4034	152	12	ai	ai	VERB
ejpam-4034	152	13	or	or	CCONJ
ejpam-4034	152	14	bi	bi	NOUN
ejpam-4034	152	15	is	be	AUX
ejpam-4034	152	16	in	in	ADP
ejpam-4034	152	17	i.	i.	NOUN
ejpam-4034	153	1	hence	hence	ADV
ejpam-4034	153	2	i	i	PRON
ejpam-4034	153	3	is	be	AUX
ejpam-4034	153	4	a	a	DET
ejpam-4034	153	5	3	3	NUM
ejpam-4034	153	6	-	-	PUNCT
ejpam-4034	153	7	prime	prime	ADJ
ejpam-4034	153	8	ideal	ideal	NOUN
ejpam-4034	153	9	of	of	ADP
ejpam-4034	153	10	r.	r.	PROPN
ejpam-4034	153	11	theorem	theorem	PROPN
ejpam-4034	153	12	2	2	X
ejpam-4034	153	13	.	.	PUNCT
ejpam-4034	154	1	let	let	VERB
ejpam-4034	154	2	i	i	PRON
ejpam-4034	154	3	be	be	AUX
ejpam-4034	154	4	an	an	DET
ejpam-4034	154	5	ideal	ideal	NOUN
ejpam-4034	154	6	of	of	ADP
ejpam-4034	154	7	a	a	DET
ejpam-4034	154	8	γ	γ	NOUN
ejpam-4034	154	9	-la	-la	NOUN
ejpam-4034	154	10	-	-	PUNCT
ejpam-4034	154	11	ring	ring	NOUN
ejpam-4034	154	12	r.	r.	PROPN
ejpam-4034	154	13	then	then	ADV
ejpam-4034	154	14	,	,	PUNCT
ejpam-4034	154	15	i	i	PRON
ejpam-4034	154	16	is	be	AUX
ejpam-4034	154	17	a	a	DET
ejpam-4034	154	18	c	c	NOUN
ejpam-4034	154	19	-	-	PUNCT
ejpam-4034	154	20	prime	prime	ADJ
ejpam-4034	154	21	ideal	ideal	NOUN
ejpam-4034	154	22	in	in	ADP
ejpam-4034	154	23	r	r	PROPN
ejpam-4034	154	24	iff	iff	PROPN
ejpam-4034	154	25	r	r	NOUN
ejpam-4034	154	26	/	/	SYM
ejpam-4034	154	27	i	i	PRON
ejpam-4034	154	28	is	be	AUX
ejpam-4034	154	29	a	a	DET
ejpam-4034	154	30	γ	γ	NOUN
ejpam-4034	154	31	-la	-la	ADV
ejpam-4034	154	32	-	-	PUNCT
ejpam-4034	154	33	integral	integral	ADJ
ejpam-4034	154	34	domain	domain	NOUN
ejpam-4034	154	35	.	.	PUNCT
ejpam-4034	155	1	w.	w.	PROPN
ejpam-4034	155	2	a.	a.	PROPN
ejpam-4034	155	3	khan	khan	PROPN
ejpam-4034	155	4	et	et	PROPN
ejpam-4034	155	5	al	al	PROPN
ejpam-4034	155	6	.	.	PUNCT
ejpam-4034	155	7	/	/	SYM
ejpam-4034	155	8	eur	eur	PROPN
ejpam-4034	155	9	.	.	PUNCT
ejpam-4034	156	1	j.	j.	PROPN
ejpam-4034	156	2	pure	pure	PROPN
ejpam-4034	156	3	appl	appl	PROPN
ejpam-4034	156	4	.	.	PROPN
ejpam-4034	156	5	math	math	PROPN
ejpam-4034	156	6	,	,	PUNCT
ejpam-4034	156	7	14	14	NUM
ejpam-4034	156	8	(	(	PUNCT
ejpam-4034	156	9	3	3	NUM
ejpam-4034	156	10	)	)	PUNCT
ejpam-4034	156	11	(	(	PUNCT
ejpam-4034	156	12	2021	2021	NUM
ejpam-4034	156	13	)	)	PUNCT
ejpam-4034	156	14	,	,	PUNCT
ejpam-4034	156	15	989	989	NUM
ejpam-4034	156	16	-	-	SYM
ejpam-4034	156	17	1001	1001	NUM
ejpam-4034	156	18	993	993	NUM
ejpam-4034	156	19	proof	proof	NOUN
ejpam-4034	156	20	.	.	PUNCT
ejpam-4034	157	1	(	(	PUNCT
ejpam-4034	157	2	⇒	⇒	NOUN
ejpam-4034	157	3	)	)	PUNCT
ejpam-4034	157	4	let	let	VERB
ejpam-4034	157	5	i	i	PRON
ejpam-4034	157	6	be	be	AUX
ejpam-4034	157	7	a	a	DET
ejpam-4034	157	8	c	c	NOUN
ejpam-4034	157	9	-	-	PUNCT
ejpam-4034	157	10	prime	prime	ADJ
ejpam-4034	157	11	ideal	ideal	NOUN
ejpam-4034	157	12	in	in	ADP
ejpam-4034	157	13	r.	r.	PROPN
ejpam-4034	157	14	then	then	ADV
ejpam-4034	157	15	i	i	PRON
ejpam-4034	157	16	is	be	AUX
ejpam-4034	157	17	a	a	DET
ejpam-4034	157	18	prime	prime	ADJ
ejpam-4034	157	19	ideal	ideal	NOUN
ejpam-4034	157	20	by	by	ADP
ejpam-4034	157	21	lemma	lemma	PROPN
ejpam-4034	157	22	3	3	NUM
ejpam-4034	157	23	.	.	PUNCT
ejpam-4034	158	1	thus	thus	ADV
ejpam-4034	158	2	r	r	X
ejpam-4034	158	3	/	/	SYM
ejpam-4034	158	4	i	i	PRON
ejpam-4034	158	5	is	be	AUX
ejpam-4034	158	6	a	a	DET
ejpam-4034	158	7	γ	γ	NOUN
ejpam-4034	158	8	-la	-la	ADV
ejpam-4034	158	9	-	-	PUNCT
ejpam-4034	158	10	integral	integral	ADJ
ejpam-4034	158	11	domain	domain	NOUN
ejpam-4034	158	12	.	.	PUNCT
ejpam-4034	159	1	(	(	PUNCT
ejpam-4034	159	2	⇐	⇐	ADJ
ejpam-4034	159	3	)	)	PUNCT
ejpam-4034	159	4	assume	assume	VERB
ejpam-4034	159	5	that	that	SCONJ
ejpam-4034	159	6	r	r	NOUN
ejpam-4034	159	7	/	/	SYM
ejpam-4034	159	8	i	i	PRON
ejpam-4034	159	9	is	be	AUX
ejpam-4034	159	10	a	a	DET
ejpam-4034	159	11	γ	γ	X
ejpam-4034	159	12	-	-	PUNCT
ejpam-4034	159	13	la	la	ADJ
ejpam-4034	159	14	-	-	ADJ
ejpam-4034	159	15	integral	integral	ADJ
ejpam-4034	159	16	domain	domain	NOUN
ejpam-4034	159	17	with	with	ADP
ejpam-4034	159	18	xβsβy	xβsβy	PROPN
ejpam-4034	159	19	∈	∈	PROPN
ejpam-4034	160	1	i	i	PRON
ejpam-4034	160	2	for	for	ADP
ejpam-4034	160	3	all	all	DET
ejpam-4034	160	4	s	s	PROPN
ejpam-4034	160	5	∈	∈	PROPN
ejpam-4034	160	6	r.	r.	NOUN
ejpam-4034	160	7	then	then	ADV
ejpam-4034	160	8	,	,	PUNCT
ejpam-4034	160	9	i+arb	i+arb	VERB
ejpam-4034	160	10	=	=	NOUN
ejpam-4034	161	1	i	i	PRON
ejpam-4034	161	2	so	so	ADV
ejpam-4034	161	3	(	(	PUNCT
ejpam-4034	161	4	i	i	PRON
ejpam-4034	161	5	+	+	CCONJ
ejpam-4034	161	6	a	a	X
ejpam-4034	161	7	)	)	PUNCT
ejpam-4034	161	8	(	(	PUNCT
ejpam-4034	161	9	i+b	i+b	NUM
ejpam-4034	161	10	)	)	PUNCT
ejpam-4034	162	1	=	=	SYM
ejpam-4034	162	2	i	i	PROPN
ejpam-4034	162	3	,	,	PUNCT
ejpam-4034	162	4	a	a	PRON
ejpam-4034	162	5	,	,	PUNCT
ejpam-4034	162	6	b	b	PROPN
ejpam-4034	162	7	∈	∈	PROPN
ejpam-4034	162	8	r.	r.	NOUN
ejpam-4034	162	9	since	since	SCONJ
ejpam-4034	162	10	r	r	PROPN
ejpam-4034	162	11	/	/	SYM
ejpam-4034	162	12	i	i	PRON
ejpam-4034	162	13	is	be	AUX
ejpam-4034	162	14	a	a	DET
ejpam-4034	162	15	γ	γ	NOUN
ejpam-4034	162	16	-la	-la	ADV
ejpam-4034	162	17	-	-	PUNCT
ejpam-4034	162	18	integral	integral	ADJ
ejpam-4034	162	19	domain	domain	NOUN
ejpam-4034	162	20	,	,	PUNCT
ejpam-4034	162	21	i+a	i+a	PUNCT
ejpam-4034	162	22	=	=	SYM
ejpam-4034	163	1	i	i	PRON
ejpam-4034	163	2	or	or	CCONJ
ejpam-4034	163	3	i	i	PRON
ejpam-4034	163	4	+	+	CCONJ
ejpam-4034	163	5	b	b	X
ejpam-4034	163	6	=	=	SYM
ejpam-4034	163	7	i	i	PROPN
ejpam-4034	163	8	,	,	PUNCT
ejpam-4034	163	9	then	then	ADV
ejpam-4034	163	10	ai	ai	VERB
ejpam-4034	163	11	or	or	CCONJ
ejpam-4034	163	12	bi	bi	NOUN
ejpam-4034	163	13	is	be	AUX
ejpam-4034	163	14	in	in	ADP
ejpam-4034	163	15	i.	i.	NOUN
ejpam-4034	164	1	thus	thus	ADV
ejpam-4034	164	2	i	i	PRON
ejpam-4034	164	3	is	be	AUX
ejpam-4034	164	4	a	a	DET
ejpam-4034	164	5	c	c	NOUN
ejpam-4034	164	6	-	-	PUNCT
ejpam-4034	164	7	prime	prime	ADJ
ejpam-4034	164	8	ideal	ideal	NOUN
ejpam-4034	164	9	of	of	ADP
ejpam-4034	164	10	r.	r.	PROPN
ejpam-4034	164	11	here	here	ADV
ejpam-4034	164	12	we	we	PRON
ejpam-4034	164	13	introduce	introduce	VERB
ejpam-4034	164	14	the	the	DET
ejpam-4034	164	15	notion	notion	NOUN
ejpam-4034	164	16	of	of	ADP
ejpam-4034	164	17	weakly	weakly	ADJ
ejpam-4034	164	18	prime	prime	ADJ
ejpam-4034	164	19	ideal	ideal	NOUN
ejpam-4034	164	20	in	in	ADP
ejpam-4034	164	21	γ	γ	PROPN
ejpam-4034	164	22	-	-	PUNCT
ejpam-4034	164	23	la	la	ADJ
ejpam-4034	164	24	-	-	PUNCT
ejpam-4034	164	25	rings	ring	NOUN
ejpam-4034	164	26	.	.	PUNCT
ejpam-4034	165	1	definition	definition	NOUN
ejpam-4034	165	2	4	4	NUM
ejpam-4034	165	3	.	.	PUNCT
ejpam-4034	166	1	a	a	DET
ejpam-4034	166	2	proper	proper	ADJ
ejpam-4034	166	3	ideal	ideal	NOUN
ejpam-4034	166	4	i	i	PRON
ejpam-4034	166	5	is	be	AUX
ejpam-4034	166	6	said	say	VERB
ejpam-4034	166	7	to	to	PART
ejpam-4034	166	8	be	be	AUX
ejpam-4034	166	9	a	a	DET
ejpam-4034	166	10	weakly	weakly	ADJ
ejpam-4034	166	11	prime	prime	ADJ
ejpam-4034	166	12	ideal	ideal	NOUN
ejpam-4034	166	13	of	of	ADP
ejpam-4034	166	14	a	a	DET
ejpam-4034	166	15	γ	γ	NOUN
ejpam-4034	166	16	-la	-la	NOUN
ejpam-4034	166	17	-	-	PUNCT
ejpam-4034	166	18	ring	ring	NOUN
ejpam-4034	166	19	r	r	NOUN
ejpam-4034	166	20	if	if	SCONJ
ejpam-4034	166	21	0	0	NUM
ejpam-4034	166	22	6=	6=	NUM
ejpam-4034	166	23	aγb	aγb	NOUN
ejpam-4034	166	24	⊆	⊆	NUM
ejpam-4034	166	25	i	i	PRON
ejpam-4034	166	26	implies	imply	VERB
ejpam-4034	166	27	either	either	CCONJ
ejpam-4034	166	28	a	a	DET
ejpam-4034	166	29	⊆	⊆	NUM
ejpam-4034	166	30	i	i	NOUN
ejpam-4034	166	31	or	or	CCONJ
ejpam-4034	166	32	b	b	NOUN
ejpam-4034	166	33	⊆	⊆	NUM
ejpam-4034	166	34	i	i	PRON
ejpam-4034	166	35	for	for	ADP
ejpam-4034	166	36	any	any	DET
ejpam-4034	166	37	ideals	ideal	NOUN
ejpam-4034	166	38	a	a	PRON
ejpam-4034	166	39	and	and	CCONJ
ejpam-4034	166	40	b	b	PROPN
ejpam-4034	166	41	of	of	ADP
ejpam-4034	166	42	r.	r.	PROPN
ejpam-4034	166	43	remark	remark	PROPN
ejpam-4034	166	44	1	1	NUM
ejpam-4034	166	45	.	.	PUNCT
ejpam-4034	167	1	obviously	obviously	ADV
ejpam-4034	167	2	every	every	DET
ejpam-4034	167	3	prime	prime	ADJ
ejpam-4034	167	4	ideal	ideal	NOUN
ejpam-4034	167	5	is	be	AUX
ejpam-4034	167	6	weakly	weakly	ADV
ejpam-4034	167	7	prime	prime	ADJ
ejpam-4034	167	8	and	and	CCONJ
ejpam-4034	167	9	{	{	PUNCT
ejpam-4034	167	10	0	0	NUM
ejpam-4034	167	11	}	}	PUNCT
ejpam-4034	167	12	is	be	AUX
ejpam-4034	167	13	always	always	ADV
ejpam-4034	167	14	weakly	weakly	ADJ
ejpam-4034	167	15	prime	prime	ADJ
ejpam-4034	167	16	ideal	ideal	NOUN
ejpam-4034	167	17	.	.	PUNCT
ejpam-4034	168	1	theorem	theorem	NOUN
ejpam-4034	168	2	3	3	X
ejpam-4034	168	3	.	.	PUNCT
ejpam-4034	169	1	let	let	VERB
ejpam-4034	169	2	i	i	PRON
ejpam-4034	169	3	be	be	AUX
ejpam-4034	169	4	a	a	DET
ejpam-4034	169	5	weakly	weakly	ADJ
ejpam-4034	169	6	prime	prime	ADJ
ejpam-4034	169	7	ideal	ideal	NOUN
ejpam-4034	169	8	of	of	ADP
ejpam-4034	169	9	γ	γ	X
ejpam-4034	169	10	-	-	PUNCT
ejpam-4034	169	11	la	la	ADJ
ejpam-4034	169	12	-	-	PUNCT
ejpam-4034	169	13	ring	ring	NOUN
ejpam-4034	169	14	which	which	PRON
ejpam-4034	169	15	is	be	AUX
ejpam-4034	169	16	not	not	PART
ejpam-4034	169	17	prime	prime	ADJ
ejpam-4034	169	18	.	.	PUNCT
ejpam-4034	170	1	then	then	ADV
ejpam-4034	170	2	i	i	PRON
ejpam-4034	170	3	=	=	NOUN
ejpam-4034	170	4	0	0	X
ejpam-4034	170	5	.	.	PUNCT
ejpam-4034	171	1	proof	proof	NOUN
ejpam-4034	171	2	.	.	PUNCT
ejpam-4034	172	1	since	since	SCONJ
ejpam-4034	172	2	i	i	PRON
ejpam-4034	172	3	is	be	AUX
ejpam-4034	172	4	a	a	DET
ejpam-4034	172	5	weakly	weakly	ADJ
ejpam-4034	172	6	prime	prime	NOUN
ejpam-4034	172	7	(	(	PUNCT
ejpam-4034	172	8	but	but	CCONJ
ejpam-4034	172	9	not	not	PART
ejpam-4034	172	10	prime	prime	ADJ
ejpam-4034	172	11	)	)	PUNCT
ejpam-4034	172	12	,	,	PUNCT
ejpam-4034	172	13	there	there	PRON
ejpam-4034	172	14	exist	exist	VERB
ejpam-4034	172	15	ideals	ideal	NOUN
ejpam-4034	172	16	a	a	DET
ejpam-4034	172	17	6⊆	6⊆	NUM
ejpam-4034	172	18	i	i	PRON
ejpam-4034	172	19	and	and	CCONJ
ejpam-4034	172	20	b	b	NOUN
ejpam-4034	172	21	6⊆	6⊆	NUM
ejpam-4034	172	22	i	i	NOUN
ejpam-4034	172	23	but	but	CCONJ
ejpam-4034	172	24	0	0	X
ejpam-4034	172	25	=	=	NOUN
ejpam-4034	172	26	aγb	aγb	NOUN
ejpam-4034	172	27	⊆	⊆	NUM
ejpam-4034	172	28	i.	i.	NOUN
ejpam-4034	172	29	since	since	SCONJ
ejpam-4034	172	30	i	i	PRON
ejpam-4034	172	31	⊆	⊆	SYM
ejpam-4034	172	32	a	a	PRON
ejpam-4034	172	33	+	+	X
ejpam-4034	173	1	i	i	NOUN
ejpam-4034	174	1	and	and	CCONJ
ejpam-4034	174	2	b	b	NOUN
ejpam-4034	174	3	⊆	⊆	NUM
ejpam-4034	174	4	b	b	NOUN
ejpam-4034	174	5	+	+	NUM
ejpam-4034	174	6	i.	i.	NOUN
ejpam-4034	175	1	but	but	CCONJ
ejpam-4034	175	2	,	,	PUNCT
ejpam-4034	175	3	if	if	SCONJ
ejpam-4034	175	4	i2	i2	PROPN
ejpam-4034	175	5	6=	6=	ADP
ejpam-4034	175	6	0	0	NUM
ejpam-4034	175	7	,	,	PUNCT
ejpam-4034	175	8	by	by	ADP
ejpam-4034	175	9	distributive	distributive	ADJ
ejpam-4034	175	10	laws	law	NOUN
ejpam-4034	175	11	“	"	PUNCT
ejpam-4034	175	12	.	.	PUNCT
ejpam-4034	175	13	”	"	PUNCT
ejpam-4034	176	1	over	over	ADP
ejpam-4034	176	2	”	"	PUNCT
ejpam-4034	176	3	+	+	CCONJ
ejpam-4034	176	4	”	"	PUNCT
ejpam-4034	176	5	of	of	ADP
ejpam-4034	176	6	γ	γ	PROPN
ejpam-4034	176	7	-	-	PUNCT
ejpam-4034	176	8	la	la	ADJ
ejpam-4034	176	9	-	-	PUNCT
ejpam-4034	176	10	ring	ring	NOUN
ejpam-4034	176	11	,	,	PUNCT
ejpam-4034	176	12	we	we	PRON
ejpam-4034	176	13	have	have	VERB
ejpam-4034	176	14	0	0	NUM
ejpam-4034	176	15	6=	6=	NUM
ejpam-4034	176	16	i2	i2	PROPN
ejpam-4034	176	17	=	=	PUNCT
ejpam-4034	176	18	iγi	iγi	PROPN
ejpam-4034	176	19	⊆	⊆	NUM
ejpam-4034	176	20	(	(	PUNCT
ejpam-4034	176	21	a+	a+	PUNCT
ejpam-4034	176	22	i)γ(b	i)γ(b	PROPN
ejpam-4034	177	1	+	+	CCONJ
ejpam-4034	177	2	i	i	NOUN
ejpam-4034	177	3	)	)	PUNCT
ejpam-4034	178	1	=	=	PUNCT
ejpam-4034	179	1	[	[	X
ejpam-4034	179	2	(	(	PUNCT
ejpam-4034	179	3	a+	a+	X
ejpam-4034	179	4	i)γb	i)γb	NOUN
ejpam-4034	179	5	]	]	X
ejpam-4034	179	6	+	+	CCONJ
ejpam-4034	180	1	[	[	X
ejpam-4034	180	2	(	(	PUNCT
ejpam-4034	180	3	a+	a+	X
ejpam-4034	180	4	i)γi	i)γi	PROPN
ejpam-4034	180	5	]	]	X
ejpam-4034	180	6	=	=	PUNCT
ejpam-4034	181	1	aγb	aγb	NOUN
ejpam-4034	181	2	+	+	CCONJ
ejpam-4034	181	3	iγb	iγb	VERB
ejpam-4034	181	4	+	+	ADV
ejpam-4034	181	5	aγi	aγi	ADJ
ejpam-4034	181	6	+	+	CCONJ
ejpam-4034	181	7	iγi	iγi	PROPN
ejpam-4034	181	8	⊆	⊆	NUM
ejpam-4034	181	9	i.	i.	NOUN
ejpam-4034	181	10	which	which	PRON
ejpam-4034	181	11	implies	imply	VERB
ejpam-4034	181	12	(	(	PUNCT
ejpam-4034	181	13	a	a	DET
ejpam-4034	181	14	+	+	NOUN
ejpam-4034	181	15	i	i	NOUN
ejpam-4034	181	16	)	)	PUNCT
ejpam-4034	182	1	⊆	⊆	NUM
ejpam-4034	183	1	i	i	PRON
ejpam-4034	183	2	and(b	and(b	PROPN
ejpam-4034	183	3	+	+	CCONJ
ejpam-4034	183	4	i	i	NOUN
ejpam-4034	183	5	)	)	PUNCT
ejpam-4034	184	1	⊆	⊆	NUM
ejpam-4034	184	2	i	i	PRON
ejpam-4034	184	3	,	,	PUNCT
ejpam-4034	184	4	since	since	SCONJ
ejpam-4034	184	5	i	i	PRON
ejpam-4034	184	6	is	be	AUX
ejpam-4034	184	7	a	a	DET
ejpam-4034	184	8	weakly	weakly	ADJ
ejpam-4034	184	9	prime	prime	NOUN
ejpam-4034	184	10	i.e.	i.e.	X
ejpam-4034	184	11	,	,	PUNCT
ejpam-4034	184	12	a	a	DET
ejpam-4034	184	13	⊆	⊆	NUM
ejpam-4034	184	14	i	i	NOUN
ejpam-4034	184	15	or	or	CCONJ
ejpam-4034	184	16	b	b	NOUN
ejpam-4034	184	17	+	+	CCONJ
ejpam-4034	184	18	i	i	PROPN
ejpam-4034	184	19	⊆	⊆	NUM
ejpam-4034	184	20	i	i	PRON
ejpam-4034	184	21	,	,	PUNCT
ejpam-4034	184	22	a	a	DET
ejpam-4034	184	23	contradiction	contradiction	NOUN
ejpam-4034	184	24	.	.	PUNCT
ejpam-4034	185	1	hence	hence	ADV
ejpam-4034	185	2	,	,	PUNCT
ejpam-4034	185	3	i2	i2	PROPN
ejpam-4034	185	4	=	=	SYM
ejpam-4034	185	5	0	0	X
ejpam-4034	185	6	.	.	PUNCT
ejpam-4034	185	7	remark	remark	PROPN
ejpam-4034	185	8	2	2	NUM
ejpam-4034	185	9	.	.	PUNCT
ejpam-4034	186	1	it	it	PRON
ejpam-4034	186	2	is	be	AUX
ejpam-4034	186	3	clear	clear	ADJ
ejpam-4034	186	4	that	that	SCONJ
ejpam-4034	186	5	if	if	SCONJ
ejpam-4034	186	6	r2	r2	PROPN
ejpam-4034	186	7	=	=	SYM
ejpam-4034	186	8	rγr	rγr	NOUN
ejpam-4034	187	1	=	=	NOUN
ejpam-4034	187	2	0	0	PUNCT
ejpam-4034	187	3	then	then	ADV
ejpam-4034	187	4	every	every	DET
ejpam-4034	187	5	ideal	ideal	NOUN
ejpam-4034	187	6	of	of	ADP
ejpam-4034	187	7	gamma	gamma	NOUN
ejpam-4034	187	8	left	leave	VERB
ejpam-4034	187	9	almost	almost	ADV
ejpam-4034	187	10	ring	ring	NOUN
ejpam-4034	187	11	is	be	AUX
ejpam-4034	187	12	a	a	DET
ejpam-4034	187	13	weakly	weakly	ADJ
ejpam-4034	187	14	prime	prime	NOUN
ejpam-4034	187	15	.	.	PUNCT
ejpam-4034	188	1	theorem	theorem	ADJ
ejpam-4034	188	2	4	4	NUM
ejpam-4034	188	3	.	.	PUNCT
ejpam-4034	189	1	in	in	ADP
ejpam-4034	189	2	gamma	gamma	NOUN
ejpam-4034	189	3	left	leave	VERB
ejpam-4034	189	4	almost	almost	ADV
ejpam-4034	189	5	ring	ring	NOUN
ejpam-4034	189	6	r	r	NOUN
ejpam-4034	189	7	,	,	PUNCT
ejpam-4034	189	8	every	every	DET
ejpam-4034	189	9	ideal	ideal	NOUN
ejpam-4034	189	10	is	be	AUX
ejpam-4034	189	11	a	a	DET
ejpam-4034	189	12	weakly	weakly	ADJ
ejpam-4034	189	13	prime	prime	ADJ
ejpam-4034	189	14	iff	iff	PROPN
ejpam-4034	189	15	aγb	aγb	NOUN
ejpam-4034	189	16	=	=	SYM
ejpam-4034	189	17	a	a	PROPN
ejpam-4034	189	18	,	,	PUNCT
ejpam-4034	189	19	aγb	aγb	NOUN
ejpam-4034	189	20	=	=	SYM
ejpam-4034	189	21	b	b	PROPN
ejpam-4034	189	22	or	or	CCONJ
ejpam-4034	189	23	aγb	aγb	ADV
ejpam-4034	189	24	=	=	SYM
ejpam-4034	189	25	0	0	NUM
ejpam-4034	189	26	,	,	PUNCT
ejpam-4034	189	27	for	for	ADP
ejpam-4034	189	28	any	any	DET
ejpam-4034	189	29	ideals	ideal	NOUN
ejpam-4034	189	30	a	a	DET
ejpam-4034	189	31	,	,	PUNCT
ejpam-4034	189	32	b	b	PROPN
ejpam-4034	189	33	of	of	ADP
ejpam-4034	189	34	r.	r.	PROPN
ejpam-4034	189	35	proof	proof	PROPN
ejpam-4034	189	36	.	.	PUNCT
ejpam-4034	190	1	assume	assume	VERB
ejpam-4034	190	2	that	that	SCONJ
ejpam-4034	190	3	every	every	DET
ejpam-4034	190	4	ideal	ideal	NOUN
ejpam-4034	190	5	in	in	ADP
ejpam-4034	190	6	r	r	NOUN
ejpam-4034	190	7	is	be	AUX
ejpam-4034	190	8	a	a	DET
ejpam-4034	190	9	weakly	weakly	ADJ
ejpam-4034	190	10	prime	prime	ADJ
ejpam-4034	190	11	ideal	ideal	NOUN
ejpam-4034	190	12	.	.	PUNCT
ejpam-4034	191	1	let	let	VERB
ejpam-4034	191	2	a	a	PRON
ejpam-4034	191	3	,	,	PUNCT
ejpam-4034	191	4	b	b	NOUN
ejpam-4034	191	5	be	be	AUX
ejpam-4034	191	6	weakly	weakly	ADJ
ejpam-4034	191	7	prime	prime	ADJ
ejpam-4034	191	8	ideals	ideal	NOUN
ejpam-4034	191	9	of	of	ADP
ejpam-4034	191	10	r	r	NOUN
ejpam-4034	191	11	,	,	PUNCT
ejpam-4034	191	12	then	then	ADV
ejpam-4034	191	13	aγb	aγb	PRON
ejpam-4034	191	14	is	be	VERB
ejpam-4034	191	15	a	a	DET
ejpam-4034	191	16	left	left	ADJ
ejpam-4034	191	17	ideal	ideal	NOUN
ejpam-4034	191	18	of	of	ADP
ejpam-4034	191	19	r	r	NOUN
ejpam-4034	191	20	provided	provide	VERB
ejpam-4034	191	21	that	that	SCONJ
ejpam-4034	191	22	ab	ab	PROPN
ejpam-4034	191	23	6	6	NUM
ejpam-4034	191	24	=	=	SYM
ejpam-4034	191	25	r	r	NOUN
ejpam-4034	191	26	,	,	PUNCT
ejpam-4034	191	27	then	then	ADV
ejpam-4034	191	28	by	by	ADP
ejpam-4034	191	29	hypothesis	hypothesis	NOUN
ejpam-4034	191	30	,	,	PUNCT
ejpam-4034	191	31	aγb	aγb	PRON
ejpam-4034	191	32	is	be	VERB
ejpam-4034	191	33	weakly	weakly	ADV
ejpam-4034	191	34	prime	prime	ADJ
ejpam-4034	191	35	.	.	PUNCT
ejpam-4034	192	1	we	we	PRON
ejpam-4034	192	2	consider	consider	VERB
ejpam-4034	192	3	two	two	NUM
ejpam-4034	192	4	situations	situation	NOUN
ejpam-4034	192	5	,	,	PUNCT
ejpam-4034	192	6	that	that	PRON
ejpam-4034	192	7	is	be	AUX
ejpam-4034	192	8	aγb	aγb	ADV
ejpam-4034	192	9	=	=	SYM
ejpam-4034	192	10	0	0	NUM
ejpam-4034	192	11	or	or	CCONJ
ejpam-4034	192	12	aγb	aγb	X
ejpam-4034	192	13	6=	6=	ADP
ejpam-4034	192	14	0	0	NUM
ejpam-4034	192	15	.	.	PUNCT
ejpam-4034	193	1	if	if	SCONJ
ejpam-4034	193	2	0	0	NUM
ejpam-4034	193	3	6=	6=	NUM
ejpam-4034	193	4	aγb	aγb	NOUN
ejpam-4034	193	5	⊆	⊆	NUM
ejpam-4034	193	6	ab	ab	PROPN
ejpam-4034	193	7	,	,	PUNCT
ejpam-4034	193	8	then	then	ADV
ejpam-4034	193	9	by	by	ADP
ejpam-4034	193	10	definition	definition	NOUN
ejpam-4034	193	11	4	4	NUM
ejpam-4034	193	12	we	we	PRON
ejpam-4034	193	13	have	have	VERB
ejpam-4034	193	14	a	a	DET
ejpam-4034	193	15	⊆	⊆	NUM
ejpam-4034	193	16	aγb	aγb	NOUN
ejpam-4034	193	17	or	or	CCONJ
ejpam-4034	193	18	b	b	NOUN
ejpam-4034	193	19	⊆	⊆	NUM
ejpam-4034	193	20	aγb	aγb	NOUN
ejpam-4034	193	21	.	.	PUNCT
ejpam-4034	194	1	since	since	SCONJ
ejpam-4034	194	2	a	a	PRON
ejpam-4034	194	3	and	and	CCONJ
ejpam-4034	194	4	b	b	NOUN
ejpam-4034	194	5	are	be	AUX
ejpam-4034	194	6	ideals	ideal	NOUN
ejpam-4034	194	7	of	of	ADP
ejpam-4034	194	8	r	r	NOUN
ejpam-4034	194	9	,	,	PUNCT
ejpam-4034	194	10	we	we	PRON
ejpam-4034	194	11	have	have	VERB
ejpam-4034	194	12	aγb	aγb	NOUN
ejpam-4034	194	13	⊆	⊆	NUM
ejpam-4034	194	14	a	a	PRON
ejpam-4034	194	15	and	and	CCONJ
ejpam-4034	194	16	aγb	aγb	ADV
ejpam-4034	194	17	⊆	⊆	PROPN
ejpam-4034	194	18	b.	b.	PROPN
ejpam-4034	194	19	therefore	therefore	ADV
ejpam-4034	194	20	,	,	PUNCT
ejpam-4034	194	21	a	a	DET
ejpam-4034	194	22	=	=	NOUN
ejpam-4034	194	23	aγb	aγb	NOUN
ejpam-4034	194	24	or	or	CCONJ
ejpam-4034	194	25	b	b	NOUN
ejpam-4034	194	26	=	=	NOUN
ejpam-4034	194	27	aγb	aγb	NOUN
ejpam-4034	194	28	.	.	PUNCT
ejpam-4034	195	1	if	if	SCONJ
ejpam-4034	195	2	aγb	aγb	VERB
ejpam-4034	195	3	=	=	NOUN
ejpam-4034	195	4	r	r	NOUN
ejpam-4034	195	5	then	then	ADV
ejpam-4034	195	6	a	a	DET
ejpam-4034	195	7	=	=	SYM
ejpam-4034	195	8	b	b	NOUN
ejpam-4034	195	9	=	=	SYM
ejpam-4034	195	10	r	r	NOUN
ejpam-4034	195	11	,	,	PUNCT
ejpam-4034	195	12	whence	whence	NOUN
ejpam-4034	195	13	r2	r2	PROPN
ejpam-4034	195	14	=	=	PUNCT
ejpam-4034	195	15	r.	r.	PROPN
ejpam-4034	195	16	conversely	conversely	ADV
ejpam-4034	195	17	,	,	PUNCT
ejpam-4034	195	18	for	for	ADP
ejpam-4034	195	19	proper	proper	ADJ
ejpam-4034	195	20	ideal	ideal	NOUN
ejpam-4034	195	21	i	i	PRON
ejpam-4034	195	22	of	of	ADP
ejpam-4034	195	23	r	r	NOUN
ejpam-4034	195	24	and	and	CCONJ
ejpam-4034	195	25	ideals	ideal	NOUN
ejpam-4034	195	26	a	a	PRON
ejpam-4034	195	27	and	and	CCONJ
ejpam-4034	195	28	b	b	NOUN
ejpam-4034	195	29	,	,	PUNCT
ejpam-4034	195	30	suppose	suppose	VERB
ejpam-4034	195	31	that	that	SCONJ
ejpam-4034	195	32	0	0	NUM
ejpam-4034	195	33	6=	6=	NUM
ejpam-4034	195	34	aγb	aγb	NOUN
ejpam-4034	195	35	⊆	⊆	NUM
ejpam-4034	195	36	i.	i.	NOUN
ejpam-4034	195	37	then	then	ADV
ejpam-4034	195	38	either	either	CCONJ
ejpam-4034	195	39	a	a	DET
ejpam-4034	195	40	=	=	PUNCT
ejpam-4034	195	41	aγb	aγb	NOUN
ejpam-4034	195	42	⊆	⊆	NUM
ejpam-4034	195	43	i	i	PROPN
ejpam-4034	195	44	or	or	CCONJ
ejpam-4034	195	45	b=	b=	NOUN
ejpam-4034	195	46	aγb	aγb	PART
ejpam-4034	195	47	⊆	⊆	NUM
ejpam-4034	195	48	i.	i.	NOUN
ejpam-4034	195	49	corollary	corollary	NOUN
ejpam-4034	195	50	1	1	NUM
ejpam-4034	195	51	.	.	PUNCT
ejpam-4034	196	1	if	if	SCONJ
ejpam-4034	196	2	every	every	DET
ejpam-4034	196	3	ideal	ideal	NOUN
ejpam-4034	196	4	of	of	ADP
ejpam-4034	196	5	gamma	gamma	NOUN
ejpam-4034	196	6	left	leave	VERB
ejpam-4034	196	7	almost	almost	ADV
ejpam-4034	196	8	ring	ring	NOUN
ejpam-4034	196	9	is	be	AUX
ejpam-4034	196	10	weakly	weakly	ADV
ejpam-4034	196	11	prime	prime	ADJ
ejpam-4034	196	12	.	.	PUNCT
ejpam-4034	197	1	then	then	ADV
ejpam-4034	197	2	,	,	PUNCT
ejpam-4034	197	3	either	either	CCONJ
ejpam-4034	197	4	a2	a2	PROPN
ejpam-4034	197	5	=	=	PUNCT
ejpam-4034	197	6	a	a	PRON
ejpam-4034	197	7	or	or	CCONJ
ejpam-4034	197	8	a2	a2	PROPN
ejpam-4034	197	9	=	=	SYM
ejpam-4034	197	10	0	0	NUM
ejpam-4034	198	1	for	for	ADP
ejpam-4034	198	2	ideal	ideal	ADJ
ejpam-4034	198	3	a.	a.	PROPN
ejpam-4034	198	4	w.	w.	PROPN
ejpam-4034	198	5	a.	a.	PROPN
ejpam-4034	198	6	khan	khan	PROPN
ejpam-4034	198	7	et	et	PROPN
ejpam-4034	198	8	al	al	PROPN
ejpam-4034	198	9	.	.	PUNCT
ejpam-4034	198	10	/	/	SYM
ejpam-4034	198	11	eur	eur	PROPN
ejpam-4034	198	12	.	.	PUNCT
ejpam-4034	199	1	j.	j.	PROPN
ejpam-4034	199	2	pure	pure	PROPN
ejpam-4034	199	3	appl	appl	PROPN
ejpam-4034	199	4	.	.	PROPN
ejpam-4034	199	5	math	math	PROPN
ejpam-4034	199	6	,	,	PUNCT
ejpam-4034	199	7	14	14	NUM
ejpam-4034	199	8	(	(	PUNCT
ejpam-4034	199	9	3	3	NUM
ejpam-4034	199	10	)	)	PUNCT
ejpam-4034	199	11	(	(	PUNCT
ejpam-4034	199	12	2021	2021	NUM
ejpam-4034	199	13	)	)	PUNCT
ejpam-4034	199	14	,	,	PUNCT
ejpam-4034	199	15	989	989	NUM
ejpam-4034	199	16	-	-	SYM
ejpam-4034	199	17	1001	1001	NUM
ejpam-4034	199	18	994	994	NUM
ejpam-4034	199	19	theorem	theorem	NOUN
ejpam-4034	199	20	5	5	NUM
ejpam-4034	199	21	.	.	PUNCT
ejpam-4034	199	22	in	in	ADP
ejpam-4034	199	23	γ	γ	PROPN
ejpam-4034	199	24	-la	-la	NOUN
ejpam-4034	199	25	-	-	PUNCT
ejpam-4034	199	26	ring	ring	NOUN
ejpam-4034	199	27	every	every	DET
ejpam-4034	199	28	c	c	NOUN
ejpam-4034	199	29	-	-	PUNCT
ejpam-4034	199	30	prime	prime	ADJ
ejpam-4034	199	31	ideal	ideal	NOUN
ejpam-4034	199	32	is	be	AUX
ejpam-4034	199	33	a	a	DET
ejpam-4034	199	34	weakly	weakly	ADJ
ejpam-4034	199	35	prime	prime	ADJ
ejpam-4034	199	36	ideal	ideal	NOUN
ejpam-4034	199	37	.	.	PUNCT
ejpam-4034	200	1	proof	proof	NOUN
ejpam-4034	200	2	.	.	PUNCT
ejpam-4034	201	1	let	let	VERB
ejpam-4034	201	2	i	i	PRON
ejpam-4034	201	3	be	be	AUX
ejpam-4034	201	4	c	c	NOUN
ejpam-4034	201	5	-	-	PUNCT
ejpam-4034	201	6	prime	prime	ADJ
ejpam-4034	201	7	ideal	ideal	NOUN
ejpam-4034	201	8	of	of	ADP
ejpam-4034	201	9	γ	γ	PROPN
ejpam-4034	201	10	-la	-la	NOUN
ejpam-4034	201	11	-	-	PUNCT
ejpam-4034	201	12	ring	ring	NOUN
ejpam-4034	201	13	r	r	NOUN
ejpam-4034	201	14	,	,	PUNCT
ejpam-4034	201	15	by	by	ADP
ejpam-4034	201	16	lemma	lemma	PROPN
ejpam-4034	201	17	1	1	NUM
ejpam-4034	201	18	,	,	PUNCT
ejpam-4034	201	19	i	i	PRON
ejpam-4034	201	20	is	be	AUX
ejpam-4034	201	21	a	a	DET
ejpam-4034	201	22	prime	prime	ADJ
ejpam-4034	201	23	ideal	ideal	NOUN
ejpam-4034	201	24	.	.	PUNCT
ejpam-4034	202	1	so	so	ADV
ejpam-4034	202	2	i	i	PRON
ejpam-4034	202	3	is	be	AUX
ejpam-4034	202	4	a	a	DET
ejpam-4034	202	5	weakly	weakly	ADJ
ejpam-4034	202	6	prime	prime	ADJ
ejpam-4034	202	7	ideal	ideal	NOUN
ejpam-4034	202	8	of	of	ADP
ejpam-4034	202	9	r	r	NOUN
ejpam-4034	202	10	because	because	SCONJ
ejpam-4034	202	11	every	every	DET
ejpam-4034	202	12	prime	prime	ADJ
ejpam-4034	202	13	ideal	ideal	NOUN
ejpam-4034	202	14	is	be	AUX
ejpam-4034	202	15	weakly	weakly	ADJ
ejpam-4034	202	16	prime	prime	ADJ
ejpam-4034	202	17	ideal	ideal	NOUN
ejpam-4034	202	18	.	.	PUNCT
ejpam-4034	203	1	theorem	theorem	VERB
ejpam-4034	203	2	6	6	NUM
ejpam-4034	203	3	.	.	PUNCT
ejpam-4034	204	1	every	every	DET
ejpam-4034	204	2	3	3	NUM
ejpam-4034	204	3	-	-	PUNCT
ejpam-4034	204	4	prime	prime	ADJ
ejpam-4034	204	5	ideal	ideal	NOUN
ejpam-4034	204	6	in	in	ADP
ejpam-4034	204	7	γ	γ	PROPN
ejpam-4034	204	8	-la	-la	NOUN
ejpam-4034	204	9	-	-	PUNCT
ejpam-4034	204	10	ring	ring	NOUN
ejpam-4034	204	11	is	be	AUX
ejpam-4034	204	12	a	a	DET
ejpam-4034	204	13	weakly	weakly	ADJ
ejpam-4034	204	14	prime	prime	ADJ
ejpam-4034	204	15	ideal	ideal	NOUN
ejpam-4034	204	16	.	.	PUNCT
ejpam-4034	205	1	proof	proof	NOUN
ejpam-4034	205	2	.	.	PUNCT
ejpam-4034	206	1	let	let	VERB
ejpam-4034	206	2	i	i	PRON
ejpam-4034	206	3	is	be	AUX
ejpam-4034	206	4	3	3	NUM
ejpam-4034	206	5	-	-	PUNCT
ejpam-4034	206	6	prime	prime	ADJ
ejpam-4034	206	7	ideal	ideal	NOUN
ejpam-4034	206	8	of	of	ADP
ejpam-4034	206	9	γ	γ	PROPN
ejpam-4034	206	10	-la	-la	NOUN
ejpam-4034	206	11	-	-	PUNCT
ejpam-4034	206	12	ring	ring	NOUN
ejpam-4034	206	13	r	r	NOUN
ejpam-4034	206	14	,	,	PUNCT
ejpam-4034	206	15	by	by	ADP
ejpam-4034	206	16	lemma	lemma	PROPN
ejpam-4034	206	17	2	2	NUM
ejpam-4034	206	18	,	,	PUNCT
ejpam-4034	206	19	we	we	PRON
ejpam-4034	206	20	have	have	VERB
ejpam-4034	206	21	i	i	PRON
ejpam-4034	206	22	is	be	AUX
ejpam-4034	206	23	a	a	DET
ejpam-4034	206	24	prime	prime	ADJ
ejpam-4034	206	25	ideal	ideal	NOUN
ejpam-4034	206	26	.	.	PUNCT
ejpam-4034	207	1	since	since	SCONJ
ejpam-4034	207	2	every	every	DET
ejpam-4034	207	3	prime	prime	ADJ
ejpam-4034	207	4	ideal	ideal	NOUN
ejpam-4034	207	5	is	be	AUX
ejpam-4034	207	6	a	a	DET
ejpam-4034	207	7	weakly	weakly	ADJ
ejpam-4034	207	8	prime	prime	ADJ
ejpam-4034	207	9	ideal	ideal	NOUN
ejpam-4034	207	10	,	,	PUNCT
ejpam-4034	207	11	we	we	PRON
ejpam-4034	207	12	have	have	VERB
ejpam-4034	207	13	i	i	PRON
ejpam-4034	207	14	a	a	DET
ejpam-4034	207	15	weakly	weakly	ADJ
ejpam-4034	207	16	prime	prime	ADJ
ejpam-4034	207	17	ideal	ideal	NOUN
ejpam-4034	207	18	of	of	ADP
ejpam-4034	207	19	r.	r.	PROPN
ejpam-4034	207	20	3	3	NUM
ejpam-4034	207	21	.	.	PUNCT
ejpam-4034	208	1	γ	γ	PROPN
ejpam-4034	208	2	-	-	PUNCT
ejpam-4034	208	3	la	la	ADJ
ejpam-4034	208	4	-	-	PUNCT
ejpam-4034	208	5	semirings	semiring	NOUN
ejpam-4034	208	6	in	in	ADP
ejpam-4034	208	7	this	this	DET
ejpam-4034	208	8	section	section	NOUN
ejpam-4034	208	9	,	,	PUNCT
ejpam-4034	208	10	we	we	PRON
ejpam-4034	208	11	introduce	introduce	VERB
ejpam-4034	208	12	the	the	DET
ejpam-4034	208	13	notion	notion	NOUN
ejpam-4034	208	14	of	of	ADP
ejpam-4034	208	15	γ	γ	PROPN
ejpam-4034	208	16	-	-	PUNCT
ejpam-4034	208	17	la	la	ADJ
ejpam-4034	208	18	-	-	PUNCT
ejpam-4034	208	19	semiring	semiring	NOUN
ejpam-4034	208	20	.	.	PUNCT
ejpam-4034	209	1	furthermore	furthermore	ADV
ejpam-4034	209	2	,	,	PUNCT
ejpam-4034	209	3	we	we	PRON
ejpam-4034	209	4	introduce	introduce	VERB
ejpam-4034	209	5	prime	prime	ADJ
ejpam-4034	209	6	,	,	PUNCT
ejpam-4034	209	7	weakly	weakly	ADJ
ejpam-4034	209	8	prime	prime	NOUN
ejpam-4034	209	9	,	,	PUNCT
ejpam-4034	209	10	subtracted	subtract	VERB
ejpam-4034	209	11	ideals	ideal	NOUN
ejpam-4034	209	12	and	and	CCONJ
ejpam-4034	209	13	nilpotent	nilpotent	ADJ
ejpam-4034	209	14	elements	element	NOUN
ejpam-4034	209	15	in	in	ADP
ejpam-4034	209	16	γ	γ	PROPN
ejpam-4034	209	17	-la	-la	NOUN
ejpam-4034	209	18	-	-	PUNCT
ejpam-4034	209	19	semiring	semiring	NOUN
ejpam-4034	209	20	along	along	ADV
ejpam-4034	209	21	with	with	ADP
ejpam-4034	209	22	some	some	DET
ejpam-4034	209	23	interesting	interesting	ADJ
ejpam-4034	209	24	results	result	NOUN
ejpam-4034	209	25	.	.	PUNCT
ejpam-4034	210	1	definition	definition	NOUN
ejpam-4034	210	2	5	5	NUM
ejpam-4034	210	3	.	.	PUNCT
ejpam-4034	211	1	let	let	VERB
ejpam-4034	211	2	(	(	PUNCT
ejpam-4034	211	3	s,+	s,+	NUM
ejpam-4034	211	4	)	)	PUNCT
ejpam-4034	211	5	and	and	CCONJ
ejpam-4034	211	6	(	(	PUNCT
ejpam-4034	211	7	γ	γ	X
ejpam-4034	211	8	,	,	PUNCT
ejpam-4034	211	9	+	+	PROPN
ejpam-4034	211	10	)	)	PUNCT
ejpam-4034	211	11	be	be	VERB
ejpam-4034	211	12	the	the	DET
ejpam-4034	211	13	two	two	NUM
ejpam-4034	211	14	la	la	ADJ
ejpam-4034	211	15	-	-	PUNCT
ejpam-4034	211	16	monoids	monoids	PROPN
ejpam-4034	211	17	.	.	PUNCT
ejpam-4034	212	1	then	then	ADV
ejpam-4034	212	2	s	s	VERB
ejpam-4034	212	3	is	be	AUX
ejpam-4034	212	4	said	say	VERB
ejpam-4034	212	5	to	to	PART
ejpam-4034	212	6	be	be	AUX
ejpam-4034	212	7	a	a	DET
ejpam-4034	212	8	gamma	gamma	NOUN
ejpam-4034	212	9	la	la	PROPN
ejpam-4034	212	10	-	-	PUNCT
ejpam-4034	212	11	semiring	semire	VERB
ejpam-4034	212	12	(	(	PUNCT
ejpam-4034	212	13	or	or	CCONJ
ejpam-4034	212	14	γ	γ	PRON
ejpam-4034	212	15	-la	-la	NOUN
ejpam-4034	212	16	-	-	PUNCT
ejpam-4034	212	17	semiring	semiring	NOUN
ejpam-4034	212	18	)	)	PUNCT
ejpam-4034	212	19	if	if	SCONJ
ejpam-4034	212	20	there	there	PRON
ejpam-4034	212	21	exists	exist	VERB
ejpam-4034	212	22	a	a	DET
ejpam-4034	212	23	mapping	mapping	NOUN
ejpam-4034	212	24	s×γ	s×γ	PUNCT
ejpam-4034	212	25	×s	×s	ADV
ejpam-4034	212	26	→	→	SYM
ejpam-4034	212	27	s	s	VERB
ejpam-4034	212	28	written	write	VERB
ejpam-4034	212	29	(	(	PUNCT
ejpam-4034	212	30	x	x	X
ejpam-4034	212	31	,	,	PUNCT
ejpam-4034	212	32	γ	γ	PROPN
ejpam-4034	212	33	,	,	PUNCT
ejpam-4034	212	34	y	y	PROPN
ejpam-4034	212	35	)	)	PUNCT
ejpam-4034	212	36	by	by	ADP
ejpam-4034	212	37	xγy	xγy	PROPN
ejpam-4034	212	38	such	such	ADJ
ejpam-4034	212	39	that	that	SCONJ
ejpam-4034	212	40	the	the	DET
ejpam-4034	212	41	following	follow	VERB
ejpam-4034	212	42	axioms	axiom	NOUN
ejpam-4034	212	43	hold	hold	VERB
ejpam-4034	212	44	1	1	NUM
ejpam-4034	212	45	.	.	NOUN
ejpam-4034	212	46	xγ(y	xγ(y	PUNCT
ejpam-4034	213	1	+	+	CCONJ
ejpam-4034	214	1	z	z	X
ejpam-4034	214	2	)	)	PUNCT
ejpam-4034	214	3	=	=	SYM
ejpam-4034	214	4	xγy	xγy	PROPN
ejpam-4034	215	1	+	+	CCONJ
ejpam-4034	215	2	xγz	xγz	PROPN
ejpam-4034	215	3	and	and	CCONJ
ejpam-4034	215	4	(	(	PUNCT
ejpam-4034	215	5	x+	x+	X
ejpam-4034	215	6	y)γz	y)γz	NOUN
ejpam-4034	215	7	=	=	SYM
ejpam-4034	215	8	xγz	xγz	PROPN
ejpam-4034	216	1	+	+	CCONJ
ejpam-4034	216	2	yγz	yγz	PROPN
ejpam-4034	216	3	2	2	NUM
ejpam-4034	216	4	.	.	PUNCT
ejpam-4034	216	5	x(γ	x(γ	PROPN
ejpam-4034	217	1	+	+	CCONJ
ejpam-4034	217	2	β)y	β)y	NOUN
ejpam-4034	218	1	=	=	PUNCT
ejpam-4034	218	2	xγy	xγy	NOUN
ejpam-4034	219	1	+	+	CCONJ
ejpam-4034	219	2	xβy	xβy	PROPN
ejpam-4034	219	3	3	3	X
ejpam-4034	219	4	.	.	PUNCT
ejpam-4034	219	5	(	(	PUNCT
ejpam-4034	219	6	xγy)βz	xγy)βz	X
ejpam-4034	219	7	=	=	SYM
ejpam-4034	219	8	(	(	PUNCT
ejpam-4034	219	9	zγy)βx	zγy)βx	NOUN
ejpam-4034	219	10	for	for	ADP
ejpam-4034	219	11	all	all	DET
ejpam-4034	219	12	x	x	NOUN
ejpam-4034	219	13	,	,	PUNCT
ejpam-4034	219	14	y	y	PROPN
ejpam-4034	219	15	,	,	PUNCT
ejpam-4034	219	16	z	z	PROPN
ejpam-4034	219	17	∈	∈	PROPN
ejpam-4034	219	18	s	s	PROPN
ejpam-4034	219	19	,	,	PUNCT
ejpam-4034	219	20	γ	γ	X
ejpam-4034	219	21	,	,	PUNCT
ejpam-4034	219	22	β	β	X
ejpam-4034	219	23	∈	∈	PROPN
ejpam-4034	219	24	γ	γ	X
ejpam-4034	219	25	.	.	PUNCT
ejpam-4034	220	1	in	in	ADP
ejpam-4034	220	2	this	this	DET
ejpam-4034	220	3	case	case	NOUN
ejpam-4034	220	4	we	we	PRON
ejpam-4034	220	5	denote	denote	VERB
ejpam-4034	220	6	γ	γ	PROPN
ejpam-4034	220	7	-la	-la	NOUN
ejpam-4034	220	8	-	-	PUNCT
ejpam-4034	220	9	semiring	semiring	NOUN
ejpam-4034	220	10	by	by	ADP
ejpam-4034	220	11	(	(	PUNCT
ejpam-4034	220	12	s	s	PROPN
ejpam-4034	220	13	,	,	PUNCT
ejpam-4034	220	14	γ	γ	NOUN
ejpam-4034	220	15	)	)	PUNCT
ejpam-4034	220	16	.	.	PUNCT
ejpam-4034	221	1	example	example	NOUN
ejpam-4034	222	1	3	3	X
ejpam-4034	222	2	.	.	PUNCT
ejpam-4034	222	3	let	let	VERB
ejpam-4034	222	4	s	s	VERB
ejpam-4034	222	5	=	=	X
ejpam-4034	222	6	{	{	PUNCT
ejpam-4034	222	7	a	a	PRON
ejpam-4034	222	8	,	,	PUNCT
ejpam-4034	222	9	b	b	NOUN
ejpam-4034	222	10	,	,	PUNCT
ejpam-4034	222	11	c	c	NOUN
ejpam-4034	222	12	,	,	PUNCT
ejpam-4034	222	13	d	d	NOUN
ejpam-4034	222	14	,	,	PUNCT
ejpam-4034	222	15	e	e	NOUN
ejpam-4034	222	16	}	}	PUNCT
ejpam-4034	222	17	with	with	ADP
ejpam-4034	222	18	two	two	NUM
ejpam-4034	222	19	binary	binary	ADJ
ejpam-4034	222	20	operations	operation	NOUN
ejpam-4034	222	21	”	"	PUNCT
ejpam-4034	223	1	+	+	CCONJ
ejpam-4034	223	2	”	"	PUNCT
ejpam-4034	223	3	and	and	CCONJ
ejpam-4034	223	4	”	"	PUNCT
ejpam-4034	223	5	.	.	PUNCT
ejpam-4034	223	6	”	"	PUNCT
ejpam-4034	224	1	given	give	VERB
ejpam-4034	224	2	in	in	ADP
ejpam-4034	224	3	the	the	DET
ejpam-4034	224	4	tables	table	NOUN
ejpam-4034	224	5	set	set	VERB
ejpam-4034	224	6	3	3	NUM
ejpam-4034	224	7	be	be	AUX
ejpam-4034	224	8	the	the	DET
ejpam-4034	224	9	la	la	NOUN
ejpam-4034	224	10	-	-	PUNCT
ejpam-4034	224	11	semiring	semiring	NOUN
ejpam-4034	224	12	and	and	CCONJ
ejpam-4034	224	13	γ	γ	X
ejpam-4034	224	14	=	=	SYM
ejpam-4034	224	15	{	{	PUNCT
ejpam-4034	224	16	p	p	X
ejpam-4034	224	17	,	,	PUNCT
ejpam-4034	224	18	q	q	ADJ
ejpam-4034	224	19	,	,	PUNCT
ejpam-4034	224	20	r	r	NOUN
ejpam-4034	224	21	,	,	PUNCT
ejpam-4034	224	22	s	s	PART
ejpam-4034	224	23	}	}	PUNCT
ejpam-4034	224	24	with	with	ADP
ejpam-4034	224	25	binary	binary	ADJ
ejpam-4034	224	26	operation	operation	NOUN
ejpam-4034	224	27	”	"	PUNCT
ejpam-4034	224	28	⊕	⊕	PROPN
ejpam-4034	224	29	”	"	PUNCT
ejpam-4034	224	30	is	be	AUX
ejpam-4034	224	31	la	la	PROPN
ejpam-4034	224	32	-	-	PUNCT
ejpam-4034	224	33	monoid	monoid	NOUN
ejpam-4034	224	34	.	.	PUNCT
ejpam-4034	225	1	tables	table	NOUN
ejpam-4034	225	2	set	set	VERB
ejpam-4034	225	3	3	3	NUM
ejpam-4034	225	4	+	+	CCONJ
ejpam-4034	225	5	a	a	DET
ejpam-4034	225	6	b	b	NOUN
ejpam-4034	225	7	c	c	NOUN
ejpam-4034	225	8	d	d	X
ejpam-4034	225	9	e	e	PROPN
ejpam-4034	225	10	a	a	PRON
ejpam-4034	225	11	s	s	X
ejpam-4034	225	12	t	t	X
ejpam-4034	225	13	u	u	NOUN
ejpam-4034	225	14	v	v	PROPN
ejpam-4034	225	15	w	w	PROPN
ejpam-4034	225	16	b	b	PROPN
ejpam-4034	225	17	a	a	DET
ejpam-4034	225	18	a	a	PROPN
ejpam-4034	225	19	d	d	NOUN
ejpam-4034	225	20	a	a	DET
ejpam-4034	225	21	b	b	NOUN
ejpam-4034	225	22	c	c	NOUN
ejpam-4034	225	23	a	a	DET
ejpam-4034	225	24	b	b	PROPN
ejpam-4034	225	25	b	b	PROPN
ejpam-4034	225	26	d	d	X
ejpam-4034	225	27	e	e	PROPN
ejpam-4034	225	28	d	d	X
ejpam-4034	225	29	a	a	DET
ejpam-4034	225	30	a	a	DET
ejpam-4034	225	31	b	b	NOUN
ejpam-4034	225	32	a	a	PRON
ejpam-4034	225	33	d	d	X
ejpam-4034	225	34	e	e	PROPN
ejpam-4034	225	35	a	a	PROPN
ejpam-4034	225	36	d	d	X
ejpam-4034	225	37	e	e	X
ejpam-4034	225	38	b	b	PROPN
ejpam-4034	225	39	c	c	X
ejpam-4034	225	40	·	·	PUNCT
ejpam-4034	225	41	a	a	DET
ejpam-4034	225	42	b	b	X
ejpam-4034	225	43	c	c	NOUN
ejpam-4034	225	44	d	d	PROPN
ejpam-4034	225	45	e	e	PROPN
ejpam-4034	225	46	a	a	PRON
ejpam-4034	225	47	a	a	DET
ejpam-4034	225	48	b	b	NOUN
ejpam-4034	225	49	c	c	NOUN
ejpam-4034	225	50	d	d	PROPN
ejpam-4034	225	51	e	e	PROPN
ejpam-4034	225	52	b	b	PROPN
ejpam-4034	225	53	e	e	PROPN
ejpam-4034	225	54	a	a	PRON
ejpam-4034	225	55	b	b	NOUN
ejpam-4034	225	56	c	c	NOUN
ejpam-4034	225	57	d	d	NOUN
ejpam-4034	225	58	c	c	PROPN
ejpam-4034	226	1	d	d	X
ejpam-4034	226	2	e	e	PROPN
ejpam-4034	226	3	a	a	DET
ejpam-4034	226	4	b	b	NOUN
ejpam-4034	226	5	c	c	NOUN
ejpam-4034	226	6	d	d	NOUN
ejpam-4034	226	7	c	c	PROPN
ejpam-4034	226	8	d	d	X
ejpam-4034	226	9	e	e	PROPN
ejpam-4034	226	10	a	a	DET
ejpam-4034	226	11	b	b	PROPN
ejpam-4034	226	12	e	e	X
ejpam-4034	226	13	b	b	PROPN
ejpam-4034	226	14	c	c	PROPN
ejpam-4034	226	15	d	d	X
ejpam-4034	226	16	e	e	PROPN
ejpam-4034	226	17	a	a	PROPN
ejpam-4034	226	18	⊕	⊕	PROPN
ejpam-4034	226	19	p	p	NOUN
ejpam-4034	226	20	q	q	NOUN
ejpam-4034	226	21	r	r	NOUN
ejpam-4034	226	22	s	s	X
ejpam-4034	226	23	p	p	NOUN
ejpam-4034	226	24	p	p	NOUN
ejpam-4034	226	25	p	p	PROPN
ejpam-4034	226	26	p	p	X
ejpam-4034	226	27	p	p	X
ejpam-4034	226	28	q	q	X
ejpam-4034	226	29	p	p	X
ejpam-4034	226	30	q	q	NOUN
ejpam-4034	226	31	r	r	NOUN
ejpam-4034	226	32	s	s	NOUN
ejpam-4034	226	33	r	r	NOUN
ejpam-4034	226	34	p	p	X
ejpam-4034	226	35	s	s	PART
ejpam-4034	226	36	q	q	NOUN
ejpam-4034	226	37	r	r	NOUN
ejpam-4034	226	38	s	s	X
ejpam-4034	226	39	p	p	NOUN
ejpam-4034	226	40	r	r	NOUN
ejpam-4034	226	41	s	s	PART
ejpam-4034	226	42	q	q	PUNCT
ejpam-4034	226	43	then	then	ADV
ejpam-4034	226	44	,	,	PUNCT
ejpam-4034	226	45	s	s	X
ejpam-4034	226	46	is	be	AUX
ejpam-4034	226	47	laγ	laγ	ADJ
ejpam-4034	226	48	-semiring	-semiring	NOUN
ejpam-4034	226	49	under	under	ADP
ejpam-4034	226	50	operations	operation	NOUN
ejpam-4034	226	51	,	,	PUNCT
ejpam-4034	226	52	(	(	PUNCT
ejpam-4034	226	53	xγy)βz	xγy)βz	X
ejpam-4034	226	54	=	=	SYM
ejpam-4034	226	55	(	(	PUNCT
ejpam-4034	226	56	zγy)βx	zγy)βx	NOUN
ejpam-4034	226	57	and	and	CCONJ
ejpam-4034	226	58	xγy	xγy	NUM
ejpam-4034	227	1	=	=	PUNCT
ejpam-4034	227	2	xy	xy	PROPN
ejpam-4034	227	3	where	where	SCONJ
ejpam-4034	227	4	x	x	X
ejpam-4034	227	5	,	,	PUNCT
ejpam-4034	227	6	y	y	PROPN
ejpam-4034	227	7	∈	∈	PROPN
ejpam-4034	227	8	s	s	X
ejpam-4034	227	9	and	and	CCONJ
ejpam-4034	227	10	γ	γ	PROPN
ejpam-4034	227	11	∈	∈	PROPN
ejpam-4034	227	12	γ	γ	X
ejpam-4034	227	13	.	.	PUNCT
ejpam-4034	228	1	definition	definition	NOUN
ejpam-4034	228	2	6	6	NUM
ejpam-4034	228	3	.	.	PUNCT
ejpam-4034	229	1	let	let	VERB
ejpam-4034	229	2	i	i	PRON
ejpam-4034	229	3	be	be	AUX
ejpam-4034	229	4	a	a	DET
ejpam-4034	229	5	proper	proper	ADJ
ejpam-4034	229	6	ideal	ideal	NOUN
ejpam-4034	229	7	of	of	ADP
ejpam-4034	229	8	gamma	gamma	PROPN
ejpam-4034	229	9	la	la	PROPN
ejpam-4034	229	10	-	-	PUNCT
ejpam-4034	229	11	semiring	semire	VERB
ejpam-4034	229	12	s	s	NOUN
ejpam-4034	229	13	and	and	CCONJ
ejpam-4034	229	14	ab	ab	PROPN
ejpam-4034	229	15	⊆	⊆	NUM
ejpam-4034	229	16	i	i	PRON
ejpam-4034	229	17	such	such	ADJ
ejpam-4034	229	18	that	that	SCONJ
ejpam-4034	229	19	a	a	DET
ejpam-4034	229	20	⊆	⊆	NUM
ejpam-4034	229	21	i	i	NOUN
ejpam-4034	229	22	or	or	CCONJ
ejpam-4034	229	23	b	b	NOUN
ejpam-4034	229	24	⊆	⊆	NUM
ejpam-4034	229	25	i	i	PRON
ejpam-4034	229	26	for	for	ADP
ejpam-4034	229	27	any	any	DET
ejpam-4034	229	28	ideals	ideal	NOUN
ejpam-4034	229	29	a	a	PRON
ejpam-4034	229	30	,	,	PUNCT
ejpam-4034	229	31	b	b	PROPN
ejpam-4034	229	32	of	of	ADP
ejpam-4034	229	33	s	s	PROPN
ejpam-4034	229	34	,	,	PUNCT
ejpam-4034	229	35	then	then	ADV
ejpam-4034	229	36	proper	proper	ADJ
ejpam-4034	229	37	ideal	ideal	NOUN
ejpam-4034	229	38	i	i	PRON
ejpam-4034	229	39	of	of	ADP
ejpam-4034	229	40	a	a	DET
ejpam-4034	229	41	gamma	gamma	NOUN
ejpam-4034	229	42	la	la	PROPN
ejpam-4034	229	43	-	-	PUNCT
ejpam-4034	229	44	semiring	semire	VERB
ejpam-4034	229	45	s	s	X
ejpam-4034	229	46	is	be	AUX
ejpam-4034	229	47	a	a	DET
ejpam-4034	229	48	prime	prime	ADJ
ejpam-4034	229	49	ideal	ideal	NOUN
ejpam-4034	229	50	.	.	PUNCT
ejpam-4034	230	1	definition	definition	NOUN
ejpam-4034	230	2	7	7	NUM
ejpam-4034	230	3	.	.	PUNCT
ejpam-4034	231	1	if	if	SCONJ
ejpam-4034	231	2	i	i	PRON
ejpam-4034	231	3	is	be	AUX
ejpam-4034	231	4	a	a	DET
ejpam-4034	231	5	proper	proper	ADJ
ejpam-4034	231	6	ideal	ideal	NOUN
ejpam-4034	231	7	of	of	ADP
ejpam-4034	231	8	γ	γ	PROPN
ejpam-4034	231	9	-la	-la	NOUN
ejpam-4034	231	10	-	-	PUNCT
ejpam-4034	231	11	semiring	semire	VERB
ejpam-4034	231	12	s	s	X
ejpam-4034	231	13	and	and	CCONJ
ejpam-4034	231	14	{	{	PUNCT
ejpam-4034	231	15	0	0	NUM
ejpam-4034	231	16	}	}	PUNCT
ejpam-4034	231	17	6=	6=	ADP
ejpam-4034	231	18	aγb	aγb	NOUN
ejpam-4034	231	19	⊆	⊆	NUM
ejpam-4034	231	20	i	i	PRON
ejpam-4034	231	21	such	such	ADJ
ejpam-4034	231	22	that	that	SCONJ
ejpam-4034	231	23	a	a	DET
ejpam-4034	231	24	⊆	⊆	NUM
ejpam-4034	231	25	i	i	NOUN
ejpam-4034	231	26	or	or	CCONJ
ejpam-4034	231	27	b	b	NOUN
ejpam-4034	231	28	⊆	⊆	NUM
ejpam-4034	231	29	i	i	PRON
ejpam-4034	231	30	for	for	ADP
ejpam-4034	231	31	any	any	DET
ejpam-4034	231	32	ideals	ideal	NOUN
ejpam-4034	231	33	a	a	PRON
ejpam-4034	231	34	,	,	PUNCT
ejpam-4034	231	35	b	b	PROPN
ejpam-4034	231	36	of	of	ADP
ejpam-4034	231	37	s	s	PROPN
ejpam-4034	231	38	,	,	PUNCT
ejpam-4034	231	39	then	then	ADV
ejpam-4034	231	40	i	i	PRON
ejpam-4034	231	41	is	be	AUX
ejpam-4034	231	42	called	call	VERB
ejpam-4034	231	43	weakly	weakly	ADJ
ejpam-4034	231	44	prime	prime	ADJ
ejpam-4034	231	45	ideal	ideal	NOUN
ejpam-4034	231	46	of	of	ADP
ejpam-4034	231	47	gamma	gamma	PROPN
ejpam-4034	231	48	la	la	PROPN
ejpam-4034	231	49	-	-	PUNCT
ejpam-4034	231	50	semiring	semire	VERB
ejpam-4034	231	51	s.	s.	PROPN
ejpam-4034	231	52	w.	w.	PROPN
ejpam-4034	231	53	a.	a.	PROPN
ejpam-4034	231	54	khan	khan	PROPN
ejpam-4034	231	55	et	et	PROPN
ejpam-4034	231	56	al	al	PROPN
ejpam-4034	231	57	.	.	PUNCT
ejpam-4034	231	58	/	/	SYM
ejpam-4034	231	59	eur	eur	PROPN
ejpam-4034	231	60	.	.	PUNCT
ejpam-4034	232	1	j.	j.	PROPN
ejpam-4034	232	2	pure	pure	PROPN
ejpam-4034	232	3	appl	appl	PROPN
ejpam-4034	232	4	.	.	PROPN
ejpam-4034	232	5	math	math	PROPN
ejpam-4034	232	6	,	,	PUNCT
ejpam-4034	232	7	14	14	NUM
ejpam-4034	232	8	(	(	PUNCT
ejpam-4034	232	9	3	3	NUM
ejpam-4034	232	10	)	)	PUNCT
ejpam-4034	232	11	(	(	PUNCT
ejpam-4034	232	12	2021	2021	NUM
ejpam-4034	232	13	)	)	PUNCT
ejpam-4034	232	14	,	,	PUNCT
ejpam-4034	232	15	989	989	NUM
ejpam-4034	232	16	-	-	SYM
ejpam-4034	232	17	1001	1001	NUM
ejpam-4034	232	18	995	995	NUM
ejpam-4034	232	19	definition	definition	NOUN
ejpam-4034	232	20	8	8	NUM
ejpam-4034	232	21	.	.	PUNCT
ejpam-4034	233	1	if	if	SCONJ
ejpam-4034	233	2	sn	sn	PROPN
ejpam-4034	233	3	=	=	NOUN
ejpam-4034	233	4	0	0	NUM
ejpam-4034	233	5	for	for	ADP
ejpam-4034	233	6	s	s	PROPN
ejpam-4034	233	7	∈	∈	PROPN
ejpam-4034	233	8	s	s	X
ejpam-4034	233	9	and	and	CCONJ
ejpam-4034	233	10	positive	positive	ADJ
ejpam-4034	233	11	integer	integer	NOUN
ejpam-4034	233	12	n	n	CCONJ
ejpam-4034	233	13	(	(	PUNCT
ejpam-4034	233	14	depending	depend	VERB
ejpam-4034	233	15	on	on	ADP
ejpam-4034	233	16	s	s	NOUN
ejpam-4034	233	17	)	)	PUNCT
ejpam-4034	233	18	,	,	PUNCT
ejpam-4034	233	19	then	then	ADV
ejpam-4034	233	20	the	the	DET
ejpam-4034	233	21	element	element	NOUN
ejpam-4034	233	22	s	s	VERB
ejpam-4034	233	23	in	in	ADP
ejpam-4034	233	24	a	a	DET
ejpam-4034	233	25	γ	γ	NOUN
ejpam-4034	233	26	-la	-la	ADV
ejpam-4034	233	27	-	-	PUNCT
ejpam-4034	233	28	semiring	semire	VERB
ejpam-4034	233	29	s	s	NOUN
ejpam-4034	233	30	is	be	AUX
ejpam-4034	233	31	nilpotent	nilpotent	ADJ
ejpam-4034	233	32	.	.	PUNCT
ejpam-4034	234	1	the	the	DET
ejpam-4034	234	2	set	set	NOUN
ejpam-4034	234	3	of	of	ADP
ejpam-4034	234	4	all	all	DET
ejpam-4034	234	5	nilpotent	nilpotent	ADJ
ejpam-4034	234	6	element	element	NOUN
ejpam-4034	234	7	of	of	ADP
ejpam-4034	234	8	s	s	PROPN
ejpam-4034	234	9	is	be	AUX
ejpam-4034	234	10	denoted	denote	VERB
ejpam-4034	234	11	by	by	ADP
ejpam-4034	234	12	nil	nil	ADJ
ejpam-4034	234	13	s.	s.	PROPN
ejpam-4034	234	14	definition	definition	NOUN
ejpam-4034	234	15	9	9	NUM
ejpam-4034	234	16	.	.	PUNCT
ejpam-4034	235	1	if	if	SCONJ
ejpam-4034	235	2	in	in	ADP
ejpam-4034	235	3	=	=	NOUN
ejpam-4034	235	4	0	0	NUM
ejpam-4034	235	5	for	for	ADP
ejpam-4034	235	6	positive	positive	ADJ
ejpam-4034	235	7	integer	integer	NOUN
ejpam-4034	235	8	n	n	CCONJ
ejpam-4034	235	9	(	(	PUNCT
ejpam-4034	235	10	depending	depend	VERB
ejpam-4034	235	11	on	on	ADP
ejpam-4034	235	12	i	i	PRON
ejpam-4034	235	13	)	)	PUNCT
ejpam-4034	235	14	,	,	PUNCT
ejpam-4034	235	15	then	then	ADV
ejpam-4034	235	16	ideal	ideal	VERB
ejpam-4034	235	17	i	i	PRON
ejpam-4034	235	18	in	in	ADP
ejpam-4034	235	19	a	a	DET
ejpam-4034	235	20	γ	γ	X
ejpam-4034	235	21	-lasemiring	-lasemiring	NOUN
ejpam-4034	235	22	s	s	PART
ejpam-4034	235	23	is	be	AUX
ejpam-4034	235	24	nilpotent	nilpotent	ADJ
ejpam-4034	235	25	.	.	PUNCT
ejpam-4034	236	1	theorem	theorem	VERB
ejpam-4034	236	2	7	7	NUM
ejpam-4034	236	3	.	.	PUNCT
ejpam-4034	237	1	let	let	VERB
ejpam-4034	237	2	a	a	PRON
ejpam-4034	237	3	be	be	AUX
ejpam-4034	237	4	a	a	DET
ejpam-4034	237	5	subtractive	subtractive	NOUN
ejpam-4034	237	6	ideal	ideal	NOUN
ejpam-4034	237	7	in	in	ADP
ejpam-4034	237	8	a	a	DET
ejpam-4034	237	9	γ	γ	NOUN
ejpam-4034	237	10	-la	-la	ADV
ejpam-4034	237	11	-	-	PUNCT
ejpam-4034	237	12	semiring	semire	VERB
ejpam-4034	237	13	s	s	NOUN
ejpam-4034	237	14	with	with	ADP
ejpam-4034	237	15	1	1	NUM
ejpam-4034	237	16	6=	6=	SYM
ejpam-4034	237	17	0	0	NUM
ejpam-4034	237	18	.	.	PUNCT
ejpam-4034	238	1	then	then	ADV
ejpam-4034	238	2	the	the	DET
ejpam-4034	238	3	followings	following	NOUN
ejpam-4034	238	4	are	be	AUX
ejpam-4034	238	5	equivalent	equivalent	ADJ
ejpam-4034	238	6	.	.	PUNCT
ejpam-4034	239	1	(	(	PUNCT
ejpam-4034	239	2	i	i	NOUN
ejpam-4034	239	3	)	)	PUNCT
ejpam-4034	239	4	a	a	PRON
ejpam-4034	239	5	is	be	AUX
ejpam-4034	239	6	a	a	DET
ejpam-4034	239	7	weakly	weakly	ADJ
ejpam-4034	239	8	prime	prime	ADJ
ejpam-4034	239	9	ideal	ideal	NOUN
ejpam-4034	239	10	.	.	PUNCT
ejpam-4034	240	1	(	(	PUNCT
ejpam-4034	240	2	ii	ii	NOUN
ejpam-4034	240	3	)	)	PUNCT
ejpam-4034	240	4	if	if	SCONJ
ejpam-4034	240	5	{	{	PUNCT
ejpam-4034	240	6	0	0	NUM
ejpam-4034	240	7	}	}	PUNCT
ejpam-4034	240	8	6=	6=	NUM
ejpam-4034	240	9	xγy	xγy	NOUN
ejpam-4034	241	1	⊆	⊆	NUM
ejpam-4034	241	2	a	a	DET
ejpam-4034	241	3	for	for	ADP
ejpam-4034	241	4	right	right	ADJ
ejpam-4034	241	5	(	(	PUNCT
ejpam-4034	241	6	left	left	ADJ
ejpam-4034	241	7	)	)	PUNCT
ejpam-4034	241	8	ideals	ideal	NOUN
ejpam-4034	241	9	x	x	X
ejpam-4034	241	10	,	,	PUNCT
ejpam-4034	241	11	y	y	PROPN
ejpam-4034	241	12	of	of	ADP
ejpam-4034	241	13	s	s	PROPN
ejpam-4034	241	14	,	,	PUNCT
ejpam-4034	241	15	then	then	ADV
ejpam-4034	241	16	x	x	X
ejpam-4034	241	17	⊆	⊆	SYM
ejpam-4034	241	18	a	a	DET
ejpam-4034	241	19	or	or	CCONJ
ejpam-4034	241	20	y	y	PROPN
ejpam-4034	241	21	⊆	⊆	NUM
ejpam-4034	241	22	a.	a.	NOUN
ejpam-4034	241	23	(	(	PUNCT
ejpam-4034	241	24	iii	iii	NOUN
ejpam-4034	241	25	)	)	PUNCT
ejpam-4034	241	26	if	if	SCONJ
ejpam-4034	241	27	x	x	X
ejpam-4034	241	28	,	,	PUNCT
ejpam-4034	241	29	y	y	PROPN
ejpam-4034	241	30	∈	∈	PROPN
ejpam-4034	241	31	s	s	VERB
ejpam-4034	241	32	such	such	ADJ
ejpam-4034	241	33	that	that	SCONJ
ejpam-4034	241	34	{	{	PUNCT
ejpam-4034	241	35	0	0	NUM
ejpam-4034	241	36	}	}	PUNCT
ejpam-4034	241	37	6=	6=	NUM
ejpam-4034	241	38	xγsγy	xγsγy	NOUN
ejpam-4034	241	39	⊆	⊆	NUM
ejpam-4034	241	40	a	a	PRON
ejpam-4034	241	41	,	,	PUNCT
ejpam-4034	241	42	then	then	ADV
ejpam-4034	241	43	x	x	SYM
ejpam-4034	241	44	∈	∈	PROPN
ejpam-4034	241	45	a	a	DET
ejpam-4034	241	46	or	or	CCONJ
ejpam-4034	241	47	y	y	PROPN
ejpam-4034	241	48	∈	∈	PROPN
ejpam-4034	241	49	a.	a.	NOUN
ejpam-4034	241	50	proof	proof	NOUN
ejpam-4034	241	51	.	.	PUNCT
ejpam-4034	242	1	(	(	PUNCT
ejpam-4034	242	2	i)⇒	i)⇒	PROPN
ejpam-4034	242	3	(	(	PUNCT
ejpam-4034	242	4	ii	ii	NOUN
ejpam-4034	242	5	)	)	PUNCT
ejpam-4034	242	6	let	let	VERB
ejpam-4034	242	7	a	a	PRON
ejpam-4034	242	8	be	be	AUX
ejpam-4034	242	9	a	a	DET
ejpam-4034	242	10	weakly	weakly	ADJ
ejpam-4034	242	11	prime	prime	ADJ
ejpam-4034	242	12	ideal	ideal	NOUN
ejpam-4034	242	13	of	of	ADP
ejpam-4034	242	14	s	s	PRON
ejpam-4034	242	15	and	and	CCONJ
ejpam-4034	242	16	x	x	X
ejpam-4034	242	17	,	,	PUNCT
ejpam-4034	242	18	y	y	PROPN
ejpam-4034	242	19	are	be	AUX
ejpam-4034	242	20	two	two	NUM
ejpam-4034	242	21	right	right	ADJ
ejpam-4034	242	22	(	(	PUNCT
ejpam-4034	242	23	left	left	ADJ
ejpam-4034	242	24	)	)	PUNCT
ejpam-4034	242	25	ideals	ideal	NOUN
ejpam-4034	242	26	of	of	ADP
ejpam-4034	242	27	s	s	PRON
ejpam-4034	243	1	such	such	ADJ
ejpam-4034	243	2	that	that	SCONJ
ejpam-4034	243	3	{	{	PUNCT
ejpam-4034	243	4	0	0	NUM
ejpam-4034	243	5	}	}	PUNCT
ejpam-4034	243	6	6=	6=	NUM
ejpam-4034	243	7	xγy	xγy	NOUN
ejpam-4034	244	1	⊆	⊆	NUM
ejpam-4034	244	2	a.	a.	NOUN
ejpam-4034	244	3	let	let	VERB
ejpam-4034	244	4	the	the	DET
ejpam-4034	244	5	ideals	ideal	NOUN
ejpam-4034	244	6	generated	generate	VERB
ejpam-4034	244	7	by	by	ADP
ejpam-4034	244	8	x	x	PROPN
ejpam-4034	244	9	,	,	PUNCT
ejpam-4034	244	10	y	y	PROPN
ejpam-4034	244	11	are	be	AUX
ejpam-4034	244	12	<	<	X
ejpam-4034	244	13	x	x	X
ejpam-4034	244	14	>	>	X
ejpam-4034	244	15	,	,	PUNCT
ejpam-4034	244	16	<	<	X
ejpam-4034	244	17	y	y	X
ejpam-4034	244	18	>	>	X
ejpam-4034	244	19	,	,	PUNCT
ejpam-4034	244	20	respectively	respectively	ADV
ejpam-4034	244	21	.	.	PUNCT
ejpam-4034	245	1	then	then	ADV
ejpam-4034	245	2	{	{	PUNCT
ejpam-4034	245	3	0	0	NUM
ejpam-4034	245	4	}	}	PUNCT
ejpam-4034	245	5	6=	6=	ADP
ejpam-4034	245	6	<	<	X
ejpam-4034	245	7	x	x	X
ejpam-4034	245	8	>	>	X
ejpam-4034	245	9	γ	γ	X
ejpam-4034	245	10	<	<	X
ejpam-4034	245	11	y	y	PROPN
ejpam-4034	245	12	>	>	PROPN
ejpam-4034	245	13	⊆	⊆	NUM
ejpam-4034	245	14	a	a	DET
ejpam-4034	245	15	implies	imply	VERB
ejpam-4034	245	16	<	<	X
ejpam-4034	245	17	x	x	X
ejpam-4034	245	18	>	>	PUNCT
ejpam-4034	245	19	⊆	⊆	NUM
ejpam-4034	245	20	a	a	PRON
ejpam-4034	245	21	or	or	CCONJ
ejpam-4034	245	22	<	<	X
ejpam-4034	245	23	y	y	PROPN
ejpam-4034	245	24	>	>	PROPN
ejpam-4034	245	25	⊆	⊆	NUM
ejpam-4034	245	26	a	a	PRON
ejpam-4034	245	27	and	and	CCONJ
ejpam-4034	245	28	x	x	SYM
ejpam-4034	245	29	⊆	⊆	SYM
ejpam-4034	245	30	<	<	X
ejpam-4034	245	31	x	x	X
ejpam-4034	245	32	>	>	X
ejpam-4034	245	33	⊆	⊆	NUM
ejpam-4034	245	34	a	a	PRON
ejpam-4034	245	35	or	or	CCONJ
ejpam-4034	245	36	y	y	PROPN
ejpam-4034	245	37	⊆<y	⊆<y	PROPN
ejpam-4034	245	38	>	>	PUNCT
ejpam-4034	245	39	⊆	⊆	NUM
ejpam-4034	245	40	a.	a.	NOUN
ejpam-4034	245	41	therefore	therefore	ADV
ejpam-4034	245	42	,	,	PUNCT
ejpam-4034	245	43	x	x	PUNCT
ejpam-4034	245	44	⊆	⊆	SYM
ejpam-4034	245	45	a	a	PRON
ejpam-4034	245	46	or	or	CCONJ
ejpam-4034	245	47	y	y	PROPN
ejpam-4034	245	48	⊆	⊆	NUM
ejpam-4034	245	49	a.	a.	NOUN
ejpam-4034	245	50	(	(	PUNCT
ejpam-4034	245	51	ii)⇒	ii)⇒	PROPN
ejpam-4034	245	52	(	(	PUNCT
ejpam-4034	245	53	iii	iii	NOUN
ejpam-4034	245	54	)	)	PUNCT
ejpam-4034	245	55	let	let	VERB
ejpam-4034	245	56	{	{	PUNCT
ejpam-4034	245	57	0	0	NUM
ejpam-4034	245	58	}	}	PUNCT
ejpam-4034	245	59	6=	6=	NUM
ejpam-4034	245	60	xγsγy	xγsγy	NOUN
ejpam-4034	245	61	⊆	⊆	NUM
ejpam-4034	245	62	a.	a.	NOUN
ejpam-4034	245	63	since	since	SCONJ
ejpam-4034	245	64	s	s	PROPN
ejpam-4034	245	65	has	have	VERB
ejpam-4034	245	66	an	an	DET
ejpam-4034	245	67	identity	identity	NOUN
ejpam-4034	245	68	,	,	PUNCT
ejpam-4034	245	69	therefore	therefore	ADV
ejpam-4034	245	70	{	{	PUNCT
ejpam-4034	245	71	0	0	NUM
ejpam-4034	245	72	}	}	PUNCT
ejpam-4034	245	73	6=	6=	NUM
ejpam-4034	245	74	(	(	PUNCT
ejpam-4034	245	75	xγs)(yγs	xγs)(yγs	PROPN
ejpam-4034	245	76	)	)	PUNCT
ejpam-4034	246	1	⊆	⊆	NUM
ejpam-4034	246	2	a	a	DET
ejpam-4034	246	3	implies	implie	NOUN
ejpam-4034	246	4	x	x	X
ejpam-4034	246	5	∈	∈	NOUN
ejpam-4034	246	6	xγs	xγs	VERB
ejpam-4034	247	1	⊆	⊆	NUM
ejpam-4034	247	2	a	a	PRON
ejpam-4034	247	3	or	or	CCONJ
ejpam-4034	247	4	y	y	PROPN
ejpam-4034	247	5	∈	∈	PROPN
ejpam-4034	247	6	yγs	yγs	NOUN
ejpam-4034	247	7	⊆	⊆	NUM
ejpam-4034	247	8	a.	a.	NOUN
ejpam-4034	247	9	(	(	PUNCT
ejpam-4034	247	10	iii	iii	NOUN
ejpam-4034	247	11	)	)	PUNCT
ejpam-4034	247	12	⇒	⇒	NOUN
ejpam-4034	247	13	(	(	PUNCT
ejpam-4034	247	14	i	i	NOUN
ejpam-4034	247	15	)	)	PUNCT
ejpam-4034	247	16	suppose	suppose	VERB
ejpam-4034	247	17	that	that	SCONJ
ejpam-4034	247	18	xγy	xγy	PROPN
ejpam-4034	247	19	⊆	⊆	NUM
ejpam-4034	247	20	a	a	PRON
ejpam-4034	247	21	,	,	PUNCT
ejpam-4034	247	22	for	for	ADP
ejpam-4034	247	23	ideals	ideal	NOUN
ejpam-4034	247	24	x	x	PUNCT
ejpam-4034	247	25	and	and	CCONJ
ejpam-4034	247	26	y	y	PROPN
ejpam-4034	247	27	of	of	ADP
ejpam-4034	247	28	s	s	PROPN
ejpam-4034	247	29	,	,	PUNCT
ejpam-4034	247	30	where	where	SCONJ
ejpam-4034	247	31	x	x	PUNCT
ejpam-4034	247	32	6⊆	6⊆	PROPN
ejpam-4034	247	33	a	a	PRON
ejpam-4034	247	34	and	and	CCONJ
ejpam-4034	247	35	y	y	PROPN
ejpam-4034	247	36	6⊆	6⊆	PROPN
ejpam-4034	247	37	a.	a.	NOUN
ejpam-4034	247	38	let	let	VERB
ejpam-4034	247	39	x	x	PROPN
ejpam-4034	247	40	∈	∈	PROPN
ejpam-4034	247	41	x\a	x\a	PROPN
ejpam-4034	247	42	,	,	PUNCT
ejpam-4034	247	43	y	y	PROPN
ejpam-4034	247	44	∈	∈	PROPN
ejpam-4034	247	45	y	y	PROPN
ejpam-4034	247	46	\a	\a	PROPN
ejpam-4034	247	47	.	.	PUNCT
ejpam-4034	248	1	also	also	ADV
ejpam-4034	248	2	let	let	VERB
ejpam-4034	248	3	x′	x′	PROPN
ejpam-4034	248	4	∈	∈	PROPN
ejpam-4034	248	5	xna	xna	PROPN
ejpam-4034	248	6	,	,	PUNCT
ejpam-4034	248	7	y′	y′	NOUN
ejpam-4034	248	8	∈	∈	PROPN
ejpam-4034	248	9	y	y	NOUN
ejpam-4034	248	10	na	na	PART
ejpam-4034	248	11	be	be	AUX
ejpam-4034	248	12	chosen	choose	VERB
ejpam-4034	248	13	arbitrary	arbitrary	ADJ
ejpam-4034	248	14	.	.	PUNCT
ejpam-4034	249	1	since	since	SCONJ
ejpam-4034	249	2	x	x	PROPN
ejpam-4034	249	3	+	+	NUM
ejpam-4034	249	4	x′	x′	NUM
ejpam-4034	249	5	,	,	PUNCT
ejpam-4034	249	6	y	y	PROPN
ejpam-4034	249	7	+	+	NUM
ejpam-4034	249	8	y′	y′	VERB
ejpam-4034	249	9	6∈	6∈	NOUN
ejpam-4034	249	10	a	a	PRON
ejpam-4034	249	11	,	,	PUNCT
ejpam-4034	249	12	we	we	PRON
ejpam-4034	249	13	must	must	AUX
ejpam-4034	249	14	have	have	VERB
ejpam-4034	249	15	{	{	PUNCT
ejpam-4034	249	16	0	0	NUM
ejpam-4034	249	17	}	}	PUNCT
ejpam-4034	249	18	=	=	SYM
ejpam-4034	249	19	(	(	PUNCT
ejpam-4034	249	20	x	x	SYM
ejpam-4034	250	1	+	+	NUM
ejpam-4034	250	2	x′)γsγ	x′)γsγ	NOUN
ejpam-4034	250	3	(	(	PUNCT
ejpam-4034	250	4	y	y	NOUN
ejpam-4034	250	5	+	+	NUM
ejpam-4034	250	6	y′	y′	NUM
ejpam-4034	250	7	)	)	PUNCT
ejpam-4034	250	8	.	.	PUNCT
ejpam-4034	251	1	now	now	ADV
ejpam-4034	251	2	if	if	SCONJ
ejpam-4034	251	3	we	we	PRON
ejpam-4034	251	4	are	be	AUX
ejpam-4034	251	5	letting	let	VERB
ejpam-4034	251	6	x′	x′	PUNCT
ejpam-4034	251	7	=	=	SYM
ejpam-4034	251	8	0	0	NUM
ejpam-4034	251	9	or	or	CCONJ
ejpam-4034	251	10	y′	y′	NUM
ejpam-4034	251	11	=	=	SYM
ejpam-4034	251	12	0	0	NUM
ejpam-4034	251	13	or	or	CCONJ
ejpam-4034	251	14	x′	x′	PROPN
ejpam-4034	252	1	=	=	SYM
ejpam-4034	252	2	0	0	NUM
ejpam-4034	252	3	and	and	CCONJ
ejpam-4034	252	4	y′	y′	NUM
ejpam-4034	252	5	=	=	SYM
ejpam-4034	252	6	0	0	PUNCT
ejpam-4034	253	1	and	and	CCONJ
ejpam-4034	253	2	considering	consider	VERB
ejpam-4034	253	3	all	all	DET
ejpam-4034	253	4	combinations	combination	NOUN
ejpam-4034	253	5	we	we	PRON
ejpam-4034	253	6	get	get	VERB
ejpam-4034	253	7	0	0	PUNCT
ejpam-4034	253	8	=	=	SYM
ejpam-4034	253	9	xγy	xγy	NOUN
ejpam-4034	254	1	=	=	PUNCT
ejpam-4034	254	2	x′γy	x′γy	X
ejpam-4034	255	1	=	=	PUNCT
ejpam-4034	255	2	xγy′	xγy′	NOUN
ejpam-4034	255	3	=	=	PUNCT
ejpam-4034	255	4	x′γy′	x′γy′	NOUN
ejpam-4034	255	5	and	and	CCONJ
ejpam-4034	255	6	hence	hence	ADV
ejpam-4034	255	7	xγy	xγy	PROPN
ejpam-4034	256	1	=	=	PUNCT
ejpam-4034	256	2	{	{	PUNCT
ejpam-4034	256	3	0	0	NUM
ejpam-4034	256	4	}	}	PUNCT
ejpam-4034	256	5	.	.	PUNCT
ejpam-4034	257	1	proposition	proposition	NOUN
ejpam-4034	257	2	1	1	NUM
ejpam-4034	257	3	.	.	PUNCT
ejpam-4034	258	1	every	every	DET
ejpam-4034	258	2	ideal	ideal	NOUN
ejpam-4034	258	3	of	of	ADP
ejpam-4034	258	4	a	a	DET
ejpam-4034	258	5	gamma	gamma	NOUN
ejpam-4034	258	6	la	la	PROPN
ejpam-4034	258	7	-	-	PUNCT
ejpam-4034	258	8	semiring	semire	VERB
ejpam-4034	258	9	s	s	X
ejpam-4034	258	10	is	be	AUX
ejpam-4034	258	11	weakly	weakly	ADJ
ejpam-4034	258	12	prime	prime	ADJ
ejpam-4034	258	13	iff	iff	NOUN
ejpam-4034	258	14	we	we	PRON
ejpam-4034	258	15	have	have	VERB
ejpam-4034	258	16	xγy	xγy	NOUN
ejpam-4034	259	1	=	=	SYM
ejpam-4034	259	2	x	x	NOUN
ejpam-4034	259	3	,	,	PUNCT
ejpam-4034	259	4	xγy	xγy	PROPN
ejpam-4034	260	1	=	=	SYM
ejpam-4034	260	2	y	y	PROPN
ejpam-4034	260	3	,	,	PUNCT
ejpam-4034	260	4	or	or	CCONJ
ejpam-4034	260	5	xγy	xγy	NUM
ejpam-4034	261	1	=	=	SYM
ejpam-4034	261	2	0	0	PROPN
ejpam-4034	261	3	,	,	PUNCT
ejpam-4034	261	4	for	for	ADP
ejpam-4034	261	5	any	any	DET
ejpam-4034	261	6	ideals	ideal	NOUN
ejpam-4034	261	7	x	x	PRON
ejpam-4034	261	8	,	,	PUNCT
ejpam-4034	261	9	y	y	PROPN
ejpam-4034	261	10	in	in	ADP
ejpam-4034	261	11	s.	s.	PROPN
ejpam-4034	261	12	proof	proof	PROPN
ejpam-4034	261	13	.	.	PUNCT
ejpam-4034	262	1	assume	assume	VERB
ejpam-4034	262	2	x	x	PUNCT
ejpam-4034	262	3	and	and	CCONJ
ejpam-4034	262	4	y	y	PROPN
ejpam-4034	262	5	are	be	AUX
ejpam-4034	262	6	the	the	DET
ejpam-4034	262	7	weakly	weakly	ADJ
ejpam-4034	262	8	prime	prime	ADJ
ejpam-4034	262	9	ideals	ideal	NOUN
ejpam-4034	262	10	of	of	ADP
ejpam-4034	262	11	s.	s.	PROPN
ejpam-4034	262	12	suppose	suppose	VERB
ejpam-4034	262	13	xγy	xγy	PROPN
ejpam-4034	263	1	6=	6=	NUM
ejpam-4034	263	2	s.	s.	PROPN
ejpam-4034	263	3	then	then	ADV
ejpam-4034	263	4	xγy	xγy	PROPN
ejpam-4034	263	5	is	be	AUX
ejpam-4034	263	6	a	a	DET
ejpam-4034	263	7	weakly	weakly	ADJ
ejpam-4034	263	8	prime	prime	NOUN
ejpam-4034	263	9	.	.	PUNCT
ejpam-4034	264	1	if	if	SCONJ
ejpam-4034	264	2	{	{	PUNCT
ejpam-4034	264	3	0	0	NUM
ejpam-4034	264	4	}	}	PUNCT
ejpam-4034	264	5	6=	6=	NUM
ejpam-4034	264	6	xγy	xγy	NOUN
ejpam-4034	264	7	⊆	⊆	NUM
ejpam-4034	264	8	xγy	xγy	NOUN
ejpam-4034	264	9	,	,	PUNCT
ejpam-4034	264	10	then	then	ADV
ejpam-4034	264	11	we	we	PRON
ejpam-4034	264	12	have	have	VERB
ejpam-4034	264	13	x	x	NUM
ejpam-4034	264	14	⊆	⊆	NUM
ejpam-4034	264	15	xγy	xγy	NOUN
ejpam-4034	264	16	or	or	CCONJ
ejpam-4034	264	17	y	y	PROPN
ejpam-4034	264	18	⊆	⊆	NUM
ejpam-4034	264	19	xγy	xγy	PROPN
ejpam-4034	264	20	(	(	PUNCT
ejpam-4034	264	21	since	since	SCONJ
ejpam-4034	264	22	xγy	xγy	PROPN
ejpam-4034	264	23	is	be	AUX
ejpam-4034	264	24	weakly	weakly	ADJ
ejpam-4034	264	25	prime	prime	ADJ
ejpam-4034	264	26	ideal	ideal	NOUN
ejpam-4034	264	27	of	of	ADP
ejpam-4034	264	28	s	s	NOUN
ejpam-4034	264	29	)	)	PUNCT
ejpam-4034	264	30	,	,	PUNCT
ejpam-4034	264	31	that	that	ADV
ejpam-4034	264	32	is	is	ADV
ejpam-4034	264	33	,	,	PUNCT
ejpam-4034	264	34	x	x	PUNCT
ejpam-4034	264	35	=	=	PUNCT
ejpam-4034	264	36	xγy	xγy	PROPN
ejpam-4034	264	37	or	or	CCONJ
ejpam-4034	264	38	y	y	PROPN
ejpam-4034	264	39	=	=	NOUN
ejpam-4034	264	40	xγy	xγy	PROPN
ejpam-4034	264	41	.	.	PUNCT
ejpam-4034	265	1	if	if	SCONJ
ejpam-4034	265	2	xγy	xγy	PROPN
ejpam-4034	265	3	=	=	SYM
ejpam-4034	265	4	s	s	VERB
ejpam-4034	265	5	then	then	ADV
ejpam-4034	265	6	we	we	PRON
ejpam-4034	265	7	have	have	VERB
ejpam-4034	265	8	x	x	NOUN
ejpam-4034	265	9	=	=	PUNCT
ejpam-4034	265	10	y	y	PROPN
ejpam-4034	265	11	=	=	SYM
ejpam-4034	265	12	s	s	PROPN
ejpam-4034	265	13	,	,	PUNCT
ejpam-4034	265	14	whence	whence	NOUN
ejpam-4034	265	15	sas	sas	PROPN
ejpam-4034	265	16	=	=	PUNCT
ejpam-4034	265	17	s.	s.	PROPN
ejpam-4034	265	18	conversely	conversely	ADV
ejpam-4034	265	19	,	,	PUNCT
ejpam-4034	265	20	let	let	VERB
ejpam-4034	265	21	a	a	PRON
ejpam-4034	265	22	be	be	AUX
ejpam-4034	265	23	any	any	DET
ejpam-4034	265	24	proper	proper	ADJ
ejpam-4034	265	25	ideal	ideal	NOUN
ejpam-4034	265	26	of	of	ADP
ejpam-4034	265	27	s	s	PRON
ejpam-4034	265	28	and	and	CCONJ
ejpam-4034	265	29	let	let	VERB
ejpam-4034	265	30	{	{	PUNCT
ejpam-4034	265	31	0	0	NUM
ejpam-4034	265	32	}	}	PUNCT
ejpam-4034	265	33	6=	6=	NUM
ejpam-4034	265	34	xγy	xγy	NOUN
ejpam-4034	266	1	⊆	⊆	NUM
ejpam-4034	266	2	a	a	PRON
ejpam-4034	266	3	for	for	ADP
ejpam-4034	266	4	ideals	ideal	NOUN
ejpam-4034	266	5	x	x	PUNCT
ejpam-4034	266	6	and	and	CCONJ
ejpam-4034	266	7	y	y	PROPN
ejpam-4034	266	8	of	of	ADP
ejpam-4034	266	9	s.	s.	PROPN
ejpam-4034	266	10	then	then	ADV
ejpam-4034	266	11	,	,	PUNCT
ejpam-4034	266	12	we	we	PRON
ejpam-4034	266	13	have	have	VERB
ejpam-4034	266	14	either	either	CCONJ
ejpam-4034	266	15	x	x	PUNCT
ejpam-4034	266	16	=	=	PUNCT
ejpam-4034	266	17	xγy	xγy	PROPN
ejpam-4034	267	1	⊆	⊆	NUM
ejpam-4034	267	2	a	a	PRON
ejpam-4034	267	3	or	or	CCONJ
ejpam-4034	267	4	y	y	NOUN
ejpam-4034	267	5	=	=	PUNCT
ejpam-4034	267	6	xγy	xγy	PROPN
ejpam-4034	268	1	⊆	⊆	NUM
ejpam-4034	268	2	a.	a.	NOUN
ejpam-4034	268	3	on	on	ADP
ejpam-4034	268	4	the	the	DET
ejpam-4034	268	5	basis	basis	NOUN
ejpam-4034	268	6	of	of	ADP
ejpam-4034	268	7	above	above	ADJ
ejpam-4034	268	8	proposition	proposition	NOUN
ejpam-4034	268	9	we	we	PRON
ejpam-4034	268	10	can	can	AUX
ejpam-4034	268	11	easily	easily	ADV
ejpam-4034	268	12	prove	prove	VERB
ejpam-4034	268	13	the	the	DET
ejpam-4034	268	14	following	follow	VERB
ejpam-4034	268	15	results	result	NOUN
ejpam-4034	268	16	.	.	PUNCT
ejpam-4034	269	1	remark	remark	VERB
ejpam-4034	269	2	3	3	NUM
ejpam-4034	269	3	.	.	PUNCT
ejpam-4034	270	1	if	if	SCONJ
ejpam-4034	270	2	every	every	DET
ejpam-4034	270	3	ideal	ideal	NOUN
ejpam-4034	270	4	of	of	ADP
ejpam-4034	270	5	γ	γ	PROPN
ejpam-4034	270	6	-la	-la	NOUN
ejpam-4034	270	7	-	-	PUNCT
ejpam-4034	270	8	semiring	semiring	NOUN
ejpam-4034	270	9	s	s	X
ejpam-4034	270	10	is	be	AUX
ejpam-4034	270	11	a	a	DET
ejpam-4034	270	12	weakly	weakly	ADJ
ejpam-4034	270	13	prime	prime	NOUN
ejpam-4034	270	14	,	,	PUNCT
ejpam-4034	270	15	then	then	ADV
ejpam-4034	270	16	we	we	PRON
ejpam-4034	270	17	have	have	VERB
ejpam-4034	270	18	either	either	CCONJ
ejpam-4034	270	19	x2	x2	PROPN
ejpam-4034	271	1	=	=	PUNCT
ejpam-4034	272	1	x	x	X
ejpam-4034	273	1	or	or	CCONJ
ejpam-4034	273	2	x2	x2	ADJ
ejpam-4034	273	3	=	=	SYM
ejpam-4034	273	4	0	0	NUM
ejpam-4034	273	5	,	,	PUNCT
ejpam-4034	273	6	for	for	ADP
ejpam-4034	273	7	any	any	DET
ejpam-4034	273	8	ideal	ideal	NOUN
ejpam-4034	273	9	x	x	PROPN
ejpam-4034	273	10	of	of	ADP
ejpam-4034	273	11	s.	s.	PROPN
ejpam-4034	273	12	lemma	lemma	PROPN
ejpam-4034	273	13	4	4	X
ejpam-4034	273	14	.	.	PUNCT
ejpam-4034	273	15	let	let	VERB
ejpam-4034	273	16	p	p	PRON
ejpam-4034	273	17	be	be	AUX
ejpam-4034	273	18	a	a	DET
ejpam-4034	273	19	subtractive	subtractive	NOUN
ejpam-4034	273	20	and	and	CCONJ
ejpam-4034	273	21	weakly	weakly	ADJ
ejpam-4034	273	22	prime	prime	ADJ
ejpam-4034	273	23	ideal	ideal	NOUN
ejpam-4034	273	24	but	but	CCONJ
ejpam-4034	273	25	not	not	PART
ejpam-4034	273	26	a	a	DET
ejpam-4034	273	27	prime	prime	ADJ
ejpam-4034	273	28	ideal	ideal	NOUN
ejpam-4034	273	29	of	of	ADP
ejpam-4034	273	30	γ	γ	PROPN
ejpam-4034	273	31	-lasemiring	-lasemiring	PROPN
ejpam-4034	273	32	s.	s.	PROPN
ejpam-4034	273	33	let	let	VERB
ejpam-4034	273	34	xγy	xγy	PROPN
ejpam-4034	274	1	=	=	SYM
ejpam-4034	274	2	0	0	PROPN
ejpam-4034	274	3	,	,	PUNCT
ejpam-4034	274	4	for	for	ADP
ejpam-4034	274	5	some	some	DET
ejpam-4034	274	6	x	x	NOUN
ejpam-4034	274	7	,	,	PUNCT
ejpam-4034	274	8	y	y	PROPN
ejpam-4034	274	9	/∈	/∈	PUNCT
ejpam-4034	275	1	p	p	NOUN
ejpam-4034	275	2	,	,	PUNCT
ejpam-4034	275	3	then	then	ADV
ejpam-4034	275	4	we	we	PRON
ejpam-4034	275	5	have	have	VERB
ejpam-4034	275	6	xγp	xγp	NOUN
ejpam-4034	276	1	=	=	NOUN
ejpam-4034	276	2	pγy	pγy	NOUN
ejpam-4034	276	3	=	=	SYM
ejpam-4034	276	4	{	{	PUNCT
ejpam-4034	276	5	0	0	NUM
ejpam-4034	276	6	}	}	PUNCT
ejpam-4034	276	7	.	.	PUNCT
ejpam-4034	277	1	proof	proof	NOUN
ejpam-4034	277	2	.	.	PUNCT
ejpam-4034	278	1	suppose	suppose	VERB
ejpam-4034	278	2	xγp1	xγp1	PROPN
ejpam-4034	278	3	6=	6=	ADP
ejpam-4034	278	4	0	0	NUM
ejpam-4034	278	5	,	,	PUNCT
ejpam-4034	278	6	for	for	ADP
ejpam-4034	278	7	some	some	DET
ejpam-4034	278	8	p1	p1	PROPN
ejpam-4034	278	9	∈	∈	PROPN
ejpam-4034	278	10	p	p	NOUN
ejpam-4034	278	11	and	and	CCONJ
ejpam-4034	278	12	γ	γ	X
ejpam-4034	278	13	∈	∈	PROPN
ejpam-4034	278	14	γ	γ	X
ejpam-4034	278	15	.	.	PUNCT
ejpam-4034	279	1	then	then	ADV
ejpam-4034	279	2	0	0	NUM
ejpam-4034	279	3	6=	6=	NUM
ejpam-4034	279	4	xγ(y+p1	xγ(y+p1	NOUN
ejpam-4034	279	5	)	)	PUNCT
ejpam-4034	279	6	∈	∈	PROPN
ejpam-4034	279	7	p	p	NOUN
ejpam-4034	279	8	.	.	PUNCT
ejpam-4034	280	1	since	since	SCONJ
ejpam-4034	280	2	p	p	NOUN
ejpam-4034	280	3	is	be	AUX
ejpam-4034	280	4	a	a	DET
ejpam-4034	280	5	weakly	weakly	ADJ
ejpam-4034	280	6	prime	prime	ADJ
ejpam-4034	280	7	ideal	ideal	NOUN
ejpam-4034	280	8	of	of	ADP
ejpam-4034	280	9	s	s	PROPN
ejpam-4034	280	10	,	,	PUNCT
ejpam-4034	280	11	therefore	therefore	ADV
ejpam-4034	280	12	y	y	PROPN
ejpam-4034	280	13	+	+	PROPN
ejpam-4034	280	14	p1	p1	PROPN
ejpam-4034	280	15	∈	∈	PROPN
ejpam-4034	280	16	p	p	NOUN
ejpam-4034	280	17	or	or	CCONJ
ejpam-4034	280	18	x	x	SYM
ejpam-4034	280	19	∈	∈	PROPN
ejpam-4034	280	20	p	p	NOUN
ejpam-4034	280	21	,	,	PUNCT
ejpam-4034	280	22	that	that	ADV
ejpam-4034	280	23	is	is	ADV
ejpam-4034	280	24	,	,	PUNCT
ejpam-4034	280	25	x	x	PUNCT
ejpam-4034	280	26	∈	∈	PROPN
ejpam-4034	280	27	p	p	NOUN
ejpam-4034	280	28	or	or	CCONJ
ejpam-4034	280	29	y	y	PROPN
ejpam-4034	280	30	∈	∈	PROPN
ejpam-4034	280	31	p	p	PROPN
ejpam-4034	280	32	,	,	PUNCT
ejpam-4034	280	33	a	a	DET
ejpam-4034	280	34	contradiction	contradiction	NOUN
ejpam-4034	280	35	.	.	PUNCT
ejpam-4034	281	1	therefore	therefore	ADV
ejpam-4034	281	2	xγp	xγp	NOUN
ejpam-4034	282	1	=	=	PUNCT
ejpam-4034	282	2	{	{	PUNCT
ejpam-4034	282	3	0	0	NUM
ejpam-4034	282	4	}	}	PUNCT
ejpam-4034	282	5	.	.	PUNCT
ejpam-4034	283	1	similarly	similarly	ADV
ejpam-4034	283	2	,	,	PUNCT
ejpam-4034	283	3	we	we	PRON
ejpam-4034	283	4	can	can	AUX
ejpam-4034	283	5	show	show	VERB
ejpam-4034	283	6	that	that	DET
ejpam-4034	283	7	pγy	pγy	NOUN
ejpam-4034	283	8	=	=	PUNCT
ejpam-4034	283	9	{	{	PUNCT
ejpam-4034	283	10	0	0	NUM
ejpam-4034	283	11	}	}	PUNCT
ejpam-4034	283	12	.	.	PUNCT
ejpam-4034	284	1	w.	w.	PROPN
ejpam-4034	284	2	a.	a.	PROPN
ejpam-4034	284	3	khan	khan	PROPN
ejpam-4034	284	4	et	et	PROPN
ejpam-4034	284	5	al	al	PROPN
ejpam-4034	284	6	.	.	PUNCT
ejpam-4034	284	7	/	/	SYM
ejpam-4034	284	8	eur	eur	PROPN
ejpam-4034	284	9	.	.	PUNCT
ejpam-4034	285	1	j.	j.	PROPN
ejpam-4034	285	2	pure	pure	PROPN
ejpam-4034	285	3	appl	appl	PROPN
ejpam-4034	285	4	.	.	PROPN
ejpam-4034	285	5	math	math	PROPN
ejpam-4034	285	6	,	,	PUNCT
ejpam-4034	285	7	14	14	NUM
ejpam-4034	285	8	(	(	PUNCT
ejpam-4034	285	9	3	3	NUM
ejpam-4034	285	10	)	)	PUNCT
ejpam-4034	285	11	(	(	PUNCT
ejpam-4034	285	12	2021	2021	NUM
ejpam-4034	285	13	)	)	PUNCT
ejpam-4034	285	14	,	,	PUNCT
ejpam-4034	285	15	989	989	NUM
ejpam-4034	285	16	-	-	SYM
ejpam-4034	285	17	1001	1001	NUM
ejpam-4034	285	18	996	996	NUM
ejpam-4034	285	19	theorem	theorem	NOUN
ejpam-4034	285	20	8	8	NUM
ejpam-4034	285	21	.	.	PUNCT
ejpam-4034	286	1	let	let	VERB
ejpam-4034	286	2	p	p	PRON
ejpam-4034	286	3	be	be	AUX
ejpam-4034	286	4	a	a	DET
ejpam-4034	286	5	subtractive	subtractive	NOUN
ejpam-4034	286	6	ideal	ideal	NOUN
ejpam-4034	286	7	of	of	ADP
ejpam-4034	286	8	a	a	DET
ejpam-4034	286	9	γ	γ	NOUN
ejpam-4034	286	10	-la	-la	ADV
ejpam-4034	286	11	-	-	PUNCT
ejpam-4034	286	12	semiring	semire	VERB
ejpam-4034	286	13	s.	s.	PROPN
ejpam-4034	286	14	if	if	SCONJ
ejpam-4034	286	15	p	p	NOUN
ejpam-4034	286	16	is	be	AUX
ejpam-4034	286	17	weakly	weakly	ADV
ejpam-4034	286	18	prime	prime	ADJ
ejpam-4034	286	19	but	but	CCONJ
ejpam-4034	286	20	not	not	PART
ejpam-4034	286	21	a	a	DET
ejpam-4034	286	22	prime	prime	NOUN
ejpam-4034	286	23	,	,	PUNCT
ejpam-4034	286	24	then	then	ADV
ejpam-4034	286	25	p	p	X
ejpam-4034	286	26	2	2	NUM
ejpam-4034	286	27	=	=	SYM
ejpam-4034	286	28	{	{	PUNCT
ejpam-4034	286	29	0	0	NUM
ejpam-4034	286	30	}	}	PUNCT
ejpam-4034	286	31	.	.	PUNCT
ejpam-4034	287	1	proof	proof	NOUN
ejpam-4034	287	2	.	.	PUNCT
ejpam-4034	288	1	suppose	suppose	VERB
ejpam-4034	288	2	p1γp2	p1γp2	ADJ
ejpam-4034	288	3	6=	6=	NUM
ejpam-4034	288	4	0	0	NUM
ejpam-4034	288	5	,	,	PUNCT
ejpam-4034	288	6	for	for	ADP
ejpam-4034	288	7	some	some	DET
ejpam-4034	288	8	p1	p1	NOUN
ejpam-4034	288	9	,	,	PUNCT
ejpam-4034	288	10	p2	p2	PROPN
ejpam-4034	288	11	∈	∈	PROPN
ejpam-4034	288	12	p	p	NOUN
ejpam-4034	288	13	and	and	CCONJ
ejpam-4034	288	14	γ	γ	PROPN
ejpam-4034	288	15	∈	∈	PROPN
ejpam-4034	288	16	γ	γ	NOUN
ejpam-4034	288	17	and	and	CCONJ
ejpam-4034	288	18	xγy	xγy	NUM
ejpam-4034	289	1	=	=	SYM
ejpam-4034	289	2	0	0	PROPN
ejpam-4034	289	3	,	,	PUNCT
ejpam-4034	289	4	for	for	ADP
ejpam-4034	289	5	some	some	DET
ejpam-4034	289	6	x	x	NOUN
ejpam-4034	289	7	,	,	PUNCT
ejpam-4034	289	8	y	y	PROPN
ejpam-4034	289	9	/∈	/∈	PUNCT
ejpam-4034	290	1	p	p	NOUN
ejpam-4034	290	2	,	,	PUNCT
ejpam-4034	290	3	where	where	SCONJ
ejpam-4034	290	4	p	p	NOUN
ejpam-4034	290	5	is	be	AUX
ejpam-4034	290	6	not	not	PART
ejpam-4034	290	7	a	a	DET
ejpam-4034	290	8	prime	prime	ADJ
ejpam-4034	290	9	ideal	ideal	NOUN
ejpam-4034	290	10	of	of	ADP
ejpam-4034	290	11	s.	s.	PROPN
ejpam-4034	290	12	then	then	ADV
ejpam-4034	290	13	by	by	ADP
ejpam-4034	290	14	lemma	lemma	PROPN
ejpam-4034	290	15	4	4	NUM
ejpam-4034	290	16	we	we	PRON
ejpam-4034	290	17	have	have	VERB
ejpam-4034	290	18	(	(	PUNCT
ejpam-4034	290	19	x+p1)γ(y+p2	x+p1)γ(y+p2	X
ejpam-4034	290	20	)	)	PUNCT
ejpam-4034	290	21	=	=	PUNCT
ejpam-4034	290	22	p1γp2	p1γp2	ADJ
ejpam-4034	290	23	6=	6=	NUM
ejpam-4034	290	24	0	0	NUM
ejpam-4034	290	25	.	.	PUNCT
ejpam-4034	291	1	hence	hence	ADV
ejpam-4034	291	2	either	either	CCONJ
ejpam-4034	291	3	(	(	PUNCT
ejpam-4034	291	4	x+	x+	ADJ
ejpam-4034	291	5	p1	p1	NOUN
ejpam-4034	291	6	)	)	PUNCT
ejpam-4034	291	7	∈	∈	PROPN
ejpam-4034	291	8	p	p	NOUN
ejpam-4034	291	9	or	or	CCONJ
ejpam-4034	291	10	(	(	PUNCT
ejpam-4034	291	11	y	y	NOUN
ejpam-4034	291	12	+	+	CCONJ
ejpam-4034	291	13	p2	p2	PROPN
ejpam-4034	291	14	)	)	PUNCT
ejpam-4034	291	15	∈	∈	PROPN
ejpam-4034	291	16	p	p	NOUN
ejpam-4034	291	17	,	,	PUNCT
ejpam-4034	291	18	and	and	CCONJ
ejpam-4034	291	19	thus	thus	ADV
ejpam-4034	291	20	either	either	CCONJ
ejpam-4034	291	21	x	x	SYM
ejpam-4034	291	22	∈	∈	PROPN
ejpam-4034	291	23	p	p	NOUN
ejpam-4034	291	24	or	or	CCONJ
ejpam-4034	291	25	y	y	PROPN
ejpam-4034	291	26	∈	∈	PROPN
ejpam-4034	291	27	p	p	PROPN
ejpam-4034	291	28	,	,	PUNCT
ejpam-4034	291	29	a	a	DET
ejpam-4034	291	30	contradiction	contradiction	NOUN
ejpam-4034	291	31	.	.	PUNCT
ejpam-4034	292	1	hence	hence	ADV
ejpam-4034	292	2	p	p	X
ejpam-4034	292	3	2	2	NUM
ejpam-4034	292	4	=	=	SYM
ejpam-4034	292	5	{	{	PUNCT
ejpam-4034	292	6	0	0	NUM
ejpam-4034	292	7	}	}	PUNCT
ejpam-4034	292	8	.	.	PUNCT
ejpam-4034	293	1	3.1	3.1	NUM
ejpam-4034	293	2	.	.	PUNCT
ejpam-4034	293	3	ideals	ideal	NOUN
ejpam-4034	293	4	in	in	ADP
ejpam-4034	293	5	γ	γ	PROPN
ejpam-4034	293	6	-	-	PUNCT
ejpam-4034	293	7	la	la	NOUN
ejpam-4034	293	8	-	-	PUNCT
ejpam-4034	293	9	semiring	semiring	NOUN
ejpam-4034	293	10	in	in	ADP
ejpam-4034	293	11	this	this	DET
ejpam-4034	293	12	section	section	NOUN
ejpam-4034	293	13	,	,	PUNCT
ejpam-4034	293	14	we	we	PRON
ejpam-4034	293	15	introduce	introduce	VERB
ejpam-4034	293	16	the	the	DET
ejpam-4034	293	17	left	left	ADJ
ejpam-4034	293	18	and	and	CCONJ
ejpam-4034	293	19	right	right	ADJ
ejpam-4034	293	20	ideals	ideal	NOUN
ejpam-4034	293	21	of	of	ADP
ejpam-4034	293	22	γ	γ	PROPN
ejpam-4034	293	23	-la	-la	ADV
ejpam-4034	293	24	-	-	PUNCT
ejpam-4034	293	25	semiring	semiring	NOUN
ejpam-4034	293	26	and	and	CCONJ
ejpam-4034	293	27	present	present	VERB
ejpam-4034	293	28	some	some	DET
ejpam-4034	293	29	results	result	NOUN
ejpam-4034	293	30	on	on	ADP
ejpam-4034	293	31	bi	bi	ADJ
ejpam-4034	293	32	-	-	ADJ
ejpam-4034	293	33	ideal	ideal	ADJ
ejpam-4034	293	34	and	and	CCONJ
ejpam-4034	293	35	quasi	quasi	ADJ
ejpam-4034	293	36	ideal	ideal	NOUN
ejpam-4034	293	37	in	in	ADP
ejpam-4034	293	38	γ	γ	PROPN
ejpam-4034	293	39	-la	-la	NOUN
ejpam-4034	293	40	-	-	PUNCT
ejpam-4034	293	41	semiring	semiring	NOUN
ejpam-4034	293	42	.	.	PUNCT
ejpam-4034	294	1	lemma	lemma	PROPN
ejpam-4034	294	2	5	5	X
ejpam-4034	294	3	.	.	PUNCT
ejpam-4034	295	1	let	let	VERB
ejpam-4034	295	2	s	s	PRON
ejpam-4034	295	3	be	be	AUX
ejpam-4034	295	4	a	a	DET
ejpam-4034	295	5	gamma	gamma	NOUN
ejpam-4034	295	6	la	la	PROPN
ejpam-4034	295	7	-	-	PUNCT
ejpam-4034	295	8	semiring	semiring	NOUN
ejpam-4034	295	9	with	with	ADP
ejpam-4034	295	10	identity	identity	NOUN
ejpam-4034	295	11	.	.	PUNCT
ejpam-4034	296	1	then	then	ADV
ejpam-4034	296	2	aγb	aγb	ADV
ejpam-4034	296	3	=	=	PRON
ejpam-4034	296	4	aβb	aβb	VERB
ejpam-4034	296	5	,	,	PUNCT
ejpam-4034	296	6	for	for	ADP
ejpam-4034	296	7	all	all	DET
ejpam-4034	296	8	a	a	DET
ejpam-4034	296	9	,	,	PUNCT
ejpam-4034	296	10	b	b	X
ejpam-4034	296	11	∈	∈	PROPN
ejpam-4034	296	12	s	s	X
ejpam-4034	296	13	and	and	CCONJ
ejpam-4034	296	14	γ	γ	X
ejpam-4034	296	15	,	,	PUNCT
ejpam-4034	296	16	β	β	PROPN
ejpam-4034	296	17	∈	∈	PROPN
ejpam-4034	296	18	γ	γ	X
ejpam-4034	296	19	.	.	PUNCT
ejpam-4034	297	1	proof	proof	NOUN
ejpam-4034	297	2	.	.	PUNCT
ejpam-4034	298	1	let	let	VERB
ejpam-4034	298	2	s	s	PRON
ejpam-4034	298	3	be	be	AUX
ejpam-4034	298	4	a	a	DET
ejpam-4034	298	5	γ	γ	X
ejpam-4034	298	6	-la	-la	ADV
ejpam-4034	298	7	-	-	PUNCT
ejpam-4034	298	8	semiring	semiring	NOUN
ejpam-4034	298	9	and	and	CCONJ
ejpam-4034	298	10	e	e	NOUN
ejpam-4034	298	11	be	be	AUX
ejpam-4034	298	12	the	the	DET
ejpam-4034	298	13	identity	identity	NOUN
ejpam-4034	298	14	of	of	ADP
ejpam-4034	298	15	s.	s.	PROPN
ejpam-4034	298	16	let	let	VERB
ejpam-4034	298	17	x	x	PRON
ejpam-4034	298	18	,	,	PUNCT
ejpam-4034	298	19	y	y	PROPN
ejpam-4034	298	20	∈	∈	PROPN
ejpam-4034	298	21	s	s	X
ejpam-4034	298	22	and	and	CCONJ
ejpam-4034	298	23	γ	γ	X
ejpam-4034	298	24	,	,	PUNCT
ejpam-4034	298	25	β	β	PROPN
ejpam-4034	298	26	∈	∈	PROPN
ejpam-4034	298	27	γ	γ	X
ejpam-4034	298	28	.	.	PUNCT
ejpam-4034	299	1	then	then	ADV
ejpam-4034	299	2	,	,	PUNCT
ejpam-4034	299	3	we	we	PRON
ejpam-4034	299	4	have	have	VERB
ejpam-4034	299	5	xγy	xγy	NOUN
ejpam-4034	299	6	=	=	SYM
ejpam-4034	299	7	xγ(eβy	xγ(eβy	ADJ
ejpam-4034	299	8	)	)	PUNCT
ejpam-4034	299	9	=	=	SYM
ejpam-4034	299	10	eγ(xβy	eγ(xβy	NOUN
ejpam-4034	299	11	)	)	PUNCT
ejpam-4034	299	12	=	=	PUNCT
ejpam-4034	300	1	xβy	xβy	PROPN
ejpam-4034	300	2	lemma	lemma	PROPN
ejpam-4034	300	3	6	6	X
ejpam-4034	300	4	.	.	PUNCT
ejpam-4034	301	1	let	let	VERB
ejpam-4034	301	2	s	s	PRON
ejpam-4034	301	3	be	be	AUX
ejpam-4034	301	4	a	a	DET
ejpam-4034	301	5	gamma	gamma	NOUN
ejpam-4034	301	6	la	la	ADP
ejpam-4034	301	7	semiring	semire	VERB
ejpam-4034	301	8	with	with	ADP
ejpam-4034	301	9	identity	identity	NOUN
ejpam-4034	301	10	and	and	CCONJ
ejpam-4034	301	11	x	x	SYM
ejpam-4034	301	12	∈	∈	PROPN
ejpam-4034	301	13	s.	s.	PROPN
ejpam-4034	301	14	if	if	SCONJ
ejpam-4034	301	15	x	x	PRON
ejpam-4034	301	16	is	be	AUX
ejpam-4034	301	17	a	a	DET
ejpam-4034	301	18	left	left	ADJ
ejpam-4034	301	19	ideal	ideal	NOUN
ejpam-4034	301	20	of	of	ADP
ejpam-4034	301	21	s	s	PRON
ejpam-4034	301	22	then	then	ADV
ejpam-4034	301	23	xγx	xγx	ADJ
ejpam-4034	301	24	is	be	AUX
ejpam-4034	301	25	a	a	DET
ejpam-4034	301	26	left	left	ADJ
ejpam-4034	301	27	ideal	ideal	NOUN
ejpam-4034	301	28	ins	in	NOUN
ejpam-4034	301	29	,	,	PUNCT
ejpam-4034	301	30	whereγ	whereγ	PROPN
ejpam-4034	301	31	∈	∈	PROPN
ejpam-4034	301	32	γ	γ	X
ejpam-4034	301	33	.	.	PUNCT
ejpam-4034	302	1	proof	proof	NOUN
ejpam-4034	302	2	.	.	PUNCT
ejpam-4034	303	1	if	if	SCONJ
ejpam-4034	303	2	s	s	PROPN
ejpam-4034	303	3	is	be	AUX
ejpam-4034	303	4	gamma	gamma	PROPN
ejpam-4034	303	5	la	la	PROPN
ejpam-4034	303	6	-	-	PUNCT
ejpam-4034	303	7	semiring	semiring	NOUN
ejpam-4034	303	8	having	having	AUX
ejpam-4034	303	9	left	leave	VERB
ejpam-4034	303	10	identity	identity	NOUN
ejpam-4034	303	11	and	and	CCONJ
ejpam-4034	303	12	let	let	VERB
ejpam-4034	303	13	x	x	PROPN
ejpam-4034	303	14	∈	∈	PROPN
ejpam-4034	303	15	s.	s.	PROPN
ejpam-4034	303	16	now	now	ADV
ejpam-4034	303	17	consider	consider	VERB
ejpam-4034	303	18	sγx+	sγx+	NOUN
ejpam-4034	303	19	rγx	rγx	VERB
ejpam-4034	303	20	=	=	PUNCT
ejpam-4034	303	21	(	(	PUNCT
ejpam-4034	303	22	s+	s+	ADV
ejpam-4034	303	23	r)γx	r)γx	PROPN
ejpam-4034	303	24	∈	∈	PROPN
ejpam-4034	303	25	xγx	xγx	NOUN
ejpam-4034	303	26	.	.	PUNCT
ejpam-4034	304	1	and	and	CCONJ
ejpam-4034	304	2	sγ	sγ	PROPN
ejpam-4034	304	3	(	(	PUNCT
ejpam-4034	304	4	xγx	xγx	PROPN
ejpam-4034	304	5	)	)	PUNCT
ejpam-4034	304	6	⊆	⊆	NUM
ejpam-4034	304	7	(	(	PUNCT
ejpam-4034	304	8	sγx)γx	sγx)γx	NOUN
ejpam-4034	304	9	⊆	⊆	NUM
ejpam-4034	304	10	xγx	xγx	NOUN
ejpam-4034	304	11	for	for	ADP
ejpam-4034	304	12	all	all	DET
ejpam-4034	304	13	r	r	NOUN
ejpam-4034	304	14	,	,	PUNCT
ejpam-4034	304	15	s	s	NOUN
ejpam-4034	304	16	∈	∈	PROPN
ejpam-4034	304	17	x	x	X
ejpam-4034	304	18	and	and	CCONJ
ejpam-4034	304	19	γ	γ	PROPN
ejpam-4034	304	20	∈	∈	PROPN
ejpam-4034	304	21	γ	γ	NOUN
ejpam-4034	304	22	.	.	PUNCT
ejpam-4034	305	1	hence	hence	ADV
ejpam-4034	305	2	xγx	xγx	ADJ
ejpam-4034	305	3	is	be	AUX
ejpam-4034	305	4	a	a	DET
ejpam-4034	305	5	left	left	ADJ
ejpam-4034	305	6	ideal	ideal	ADJ
ejpam-4034	305	7	ins	in	NOUN
ejpam-4034	305	8	.	.	PUNCT
ejpam-4034	306	1	corollary	corollary	ADJ
ejpam-4034	306	2	2	2	NUM
ejpam-4034	306	3	.	.	PUNCT
ejpam-4034	307	1	let	let	VERB
ejpam-4034	307	2	s	s	PRON
ejpam-4034	307	3	be	be	AUX
ejpam-4034	307	4	a	a	DET
ejpam-4034	307	5	gamma	gamma	NOUN
ejpam-4034	307	6	la	la	PROPN
ejpam-4034	307	7	-	-	PUNCT
ejpam-4034	307	8	semiring	semiring	NOUN
ejpam-4034	307	9	with	with	ADP
ejpam-4034	307	10	identity	identity	NOUN
ejpam-4034	307	11	and	and	CCONJ
ejpam-4034	307	12	x	x	SYM
ejpam-4034	307	13	∈	∈	PROPN
ejpam-4034	307	14	s.	s.	PROPN
ejpam-4034	307	15	if	if	SCONJ
ejpam-4034	307	16	x	x	PRON
ejpam-4034	307	17	is	be	AUX
ejpam-4034	307	18	a	a	DET
ejpam-4034	307	19	right	right	ADJ
ejpam-4034	307	20	ideal	ideal	NOUN
ejpam-4034	307	21	of	of	ADP
ejpam-4034	307	22	s	s	PROPN
ejpam-4034	307	23	,	,	PUNCT
ejpam-4034	307	24	then	then	ADV
ejpam-4034	307	25	xγx	xγx	NOUN
ejpam-4034	307	26	is	be	AUX
ejpam-4034	307	27	a	a	DET
ejpam-4034	307	28	right	right	ADJ
ejpam-4034	307	29	ideal	ideal	NOUN
ejpam-4034	307	30	in	in	ADP
ejpam-4034	307	31	s	s	PROPN
ejpam-4034	307	32	,	,	PUNCT
ejpam-4034	307	33	where	where	SCONJ
ejpam-4034	307	34	γ	γ	X
ejpam-4034	307	35	∈	∈	PROPN
ejpam-4034	307	36	γ	γ	X
ejpam-4034	307	37	.	.	PUNCT
ejpam-4034	308	1	proof	proof	NOUN
ejpam-4034	308	2	.	.	PUNCT
ejpam-4034	309	1	it	it	PRON
ejpam-4034	309	2	is	be	AUX
ejpam-4034	309	3	similar	similar	ADJ
ejpam-4034	309	4	to	to	ADP
ejpam-4034	309	5	the	the	DET
ejpam-4034	309	6	proof	proof	NOUN
ejpam-4034	309	7	of	of	ADP
ejpam-4034	309	8	lemma	lemma	PROPN
ejpam-4034	309	9	6	6	NUM
ejpam-4034	309	10	.	.	PUNCT
ejpam-4034	310	1	lemma	lemma	PROPN
ejpam-4034	310	2	7	7	X
ejpam-4034	310	3	.	.	PUNCT
ejpam-4034	311	1	let	let	VERB
ejpam-4034	311	2	s	s	PRON
ejpam-4034	311	3	be	be	AUX
ejpam-4034	311	4	a	a	DET
ejpam-4034	311	5	gamma	gamma	NOUN
ejpam-4034	311	6	la	la	PROPN
ejpam-4034	311	7	-	-	PUNCT
ejpam-4034	311	8	semiring	semiring	NOUN
ejpam-4034	311	9	with	with	ADP
ejpam-4034	311	10	identity	identity	NOUN
ejpam-4034	311	11	and	and	CCONJ
ejpam-4034	311	12	x	x	NOUN
ejpam-4034	311	13	,	,	PUNCT
ejpam-4034	311	14	y	y	PROPN
ejpam-4034	311	15	be	be	VERB
ejpam-4034	311	16	the	the	DET
ejpam-4034	311	17	left	left	ADJ
ejpam-4034	311	18	ideals	ideal	NOUN
ejpam-4034	311	19	of	of	ADP
ejpam-4034	311	20	s.	s.	PROPN
ejpam-4034	311	21	then	then	ADV
ejpam-4034	311	22	,	,	PUNCT
ejpam-4034	311	23	for	for	ADP
ejpam-4034	311	24	each	each	DET
ejpam-4034	311	25	left	leave	VERB
ejpam-4034	311	26	ideal	ideal	NOUN
ejpam-4034	311	27	y	y	PROPN
ejpam-4034	311	28	of	of	ADP
ejpam-4034	311	29	s	s	PROPN
ejpam-4034	311	30	,	,	PUNCT
ejpam-4034	311	31	(	(	PUNCT
ejpam-4034	311	32	x	x	X
ejpam-4034	311	33	:	:	PUNCT
ejpam-4034	311	34	γ	γ	X
ejpam-4034	311	35	:	:	PUNCT
ejpam-4034	311	36	y	y	PROPN
ejpam-4034	311	37	)	)	PUNCT
ejpam-4034	311	38	is	be	AUX
ejpam-4034	311	39	a	a	DET
ejpam-4034	311	40	left	left	ADJ
ejpam-4034	311	41	ideal	ideal	NOUN
ejpam-4034	311	42	in	in	ADP
ejpam-4034	311	43	s	s	PROPN
ejpam-4034	311	44	,	,	PUNCT
ejpam-4034	311	45	where	where	SCONJ
ejpam-4034	311	46	(	(	PUNCT
ejpam-4034	311	47	x	x	X
ejpam-4034	311	48	:	:	PUNCT
ejpam-4034	311	49	γ	γ	X
ejpam-4034	311	50	:	:	PUNCT
ejpam-4034	311	51	y	y	PROPN
ejpam-4034	311	52	)	)	PUNCT
ejpam-4034	312	1	=	=	PRON
ejpam-4034	313	1	{	{	PUNCT
ejpam-4034	313	2	x	x	PUNCT
ejpam-4034	313	3	∈	∈	PROPN
ejpam-4034	313	4	s	s	PART
ejpam-4034	313	5	:	:	PUNCT
ejpam-4034	313	6	xγy	xγy	PROPN
ejpam-4034	314	1	⊆	⊆	NUM
ejpam-4034	314	2	x	x	SYM
ejpam-4034	314	3	}	}	PUNCT
ejpam-4034	314	4	.	.	PUNCT
ejpam-4034	315	1	w.	w.	PROPN
ejpam-4034	315	2	a.	a.	PROPN
ejpam-4034	315	3	khan	khan	PROPN
ejpam-4034	315	4	et	et	PROPN
ejpam-4034	315	5	al	al	PROPN
ejpam-4034	315	6	.	.	PUNCT
ejpam-4034	315	7	/	/	SYM
ejpam-4034	315	8	eur	eur	PROPN
ejpam-4034	315	9	.	.	PUNCT
ejpam-4034	316	1	j.	j.	PROPN
ejpam-4034	316	2	pure	pure	PROPN
ejpam-4034	316	3	appl	appl	PROPN
ejpam-4034	316	4	.	.	PROPN
ejpam-4034	316	5	math	math	PROPN
ejpam-4034	316	6	,	,	PUNCT
ejpam-4034	316	7	14	14	NUM
ejpam-4034	316	8	(	(	PUNCT
ejpam-4034	316	9	3	3	NUM
ejpam-4034	316	10	)	)	PUNCT
ejpam-4034	316	11	(	(	PUNCT
ejpam-4034	316	12	2021	2021	NUM
ejpam-4034	316	13	)	)	PUNCT
ejpam-4034	316	14	,	,	PUNCT
ejpam-4034	316	15	989	989	NUM
ejpam-4034	316	16	-	-	SYM
ejpam-4034	316	17	1001	1001	NUM
ejpam-4034	316	18	997	997	NUM
ejpam-4034	316	19	proof	proof	NOUN
ejpam-4034	316	20	.	.	PUNCT
ejpam-4034	316	21	suppose	suppose	VERB
ejpam-4034	316	22	that	that	SCONJ
ejpam-4034	316	23	s	s	VERB
ejpam-4034	316	24	is	be	AUX
ejpam-4034	316	25	a	a	DET
ejpam-4034	316	26	gamma	gamma	NOUN
ejpam-4034	316	27	la	la	PROPN
ejpam-4034	316	28	-	-	PUNCT
ejpam-4034	316	29	semiring	semiring	NOUN
ejpam-4034	316	30	with	with	ADP
ejpam-4034	316	31	left	left	ADJ
ejpam-4034	316	32	identity	identity	NOUN
ejpam-4034	316	33	.	.	PUNCT
ejpam-4034	317	1	let	let	VERB
ejpam-4034	317	2	s	s	PRON
ejpam-4034	317	3	∈	∈	NOUN
ejpam-4034	317	4	s	s	PART
ejpam-4034	317	5	and	and	CCONJ
ejpam-4034	317	6	let	let	VERB
ejpam-4034	317	7	x	x	PRON
ejpam-4034	317	8	,	,	PUNCT
ejpam-4034	317	9	y	y	PROPN
ejpam-4034	317	10	∈	∈	PROPN
ejpam-4034	317	11	(	(	PUNCT
ejpam-4034	317	12	x	x	X
ejpam-4034	317	13	:	:	PUNCT
ejpam-4034	317	14	γ	γ	X
ejpam-4034	317	15	:	:	PUNCT
ejpam-4034	317	16	y	y	PROPN
ejpam-4034	317	17	)	)	PUNCT
ejpam-4034	317	18	.	.	PUNCT
ejpam-4034	318	1	then	then	ADV
ejpam-4034	318	2	xγy	xγy	NOUN
ejpam-4034	319	1	⊆	⊆	NUM
ejpam-4034	319	2	x	x	PUNCT
ejpam-4034	319	3	and	and	CCONJ
ejpam-4034	319	4	yγy	yγy	PROPN
ejpam-4034	319	5	⊆	⊆	NUM
ejpam-4034	319	6	x	x	PUNCT
ejpam-4034	320	1	so	so	SCONJ
ejpam-4034	320	2	that	that	SCONJ
ejpam-4034	320	3	(	(	PUNCT
ejpam-4034	320	4	x+	x+	X
ejpam-4034	320	5	y)γy	y)γy	PROPN
ejpam-4034	320	6	=	=	PUNCT
ejpam-4034	320	7	xγy	xγy	PROPN
ejpam-4034	321	1	+	+	CCONJ
ejpam-4034	321	2	yγy	yγy	PROPN
ejpam-4034	321	3	⊆	⊆	NUM
ejpam-4034	321	4	x	x	PUNCT
ejpam-4034	321	5	+	+	NOUN
ejpam-4034	321	6	x	x	X
ejpam-4034	321	7	=	=	PUNCT
ejpam-4034	321	8	x.	x.	NOUN
ejpam-4034	321	9	and	and	CCONJ
ejpam-4034	321	10	(	(	PUNCT
ejpam-4034	321	11	sγx)γy	sγx)γy	NOUN
ejpam-4034	321	12	=	=	PUNCT
ejpam-4034	321	13	sγ(xγy	sγ(xγy	ADJ
ejpam-4034	321	14	)	)	PUNCT
ejpam-4034	322	1	⊆	⊆	NUM
ejpam-4034	322	2	sγx	sγx	NOUN
ejpam-4034	322	3	⊆	⊆	NUM
ejpam-4034	322	4	x	x	PUNCT
ejpam-4034	322	5	for	for	ADP
ejpam-4034	322	6	all	all	DET
ejpam-4034	322	7	γ	γ	PROPN
ejpam-4034	322	8	∈	∈	PROPN
ejpam-4034	322	9	γ	γ	X
ejpam-4034	322	10	.	.	PUNCT
ejpam-4034	323	1	hence	hence	ADV
ejpam-4034	323	2	x	x	PUNCT
ejpam-4034	324	1	+	+	CCONJ
ejpam-4034	324	2	y	y	PROPN
ejpam-4034	324	3	∈	∈	PROPN
ejpam-4034	324	4	(	(	PUNCT
ejpam-4034	324	5	x	x	X
ejpam-4034	324	6	:	:	PUNCT
ejpam-4034	324	7	γ	γ	X
ejpam-4034	324	8	:	:	PUNCT
ejpam-4034	324	9	y	y	PROPN
ejpam-4034	324	10	)	)	PUNCT
ejpam-4034	324	11	and	and	CCONJ
ejpam-4034	324	12	sγ	sγ	INTJ
ejpam-4034	324	13	(	(	PUNCT
ejpam-4034	324	14	x	x	X
ejpam-4034	324	15	:	:	PUNCT
ejpam-4034	324	16	γ	γ	X
ejpam-4034	324	17	:	:	PUNCT
ejpam-4034	324	18	y	y	PROPN
ejpam-4034	324	19	)	)	PUNCT
ejpam-4034	324	20	⊆	⊆	X
ejpam-4034	324	21	(	(	PUNCT
ejpam-4034	324	22	x	x	X
ejpam-4034	324	23	:	:	PUNCT
ejpam-4034	324	24	γ	γ	X
ejpam-4034	324	25	:	:	PUNCT
ejpam-4034	324	26	y	y	PROPN
ejpam-4034	324	27	)	)	PUNCT
ejpam-4034	324	28	.	.	PUNCT
ejpam-4034	325	1	thus	thus	ADV
ejpam-4034	325	2	(	(	PUNCT
ejpam-4034	325	3	x	x	X
ejpam-4034	325	4	:	:	PUNCT
ejpam-4034	325	5	γ	γ	X
ejpam-4034	325	6	:	:	PUNCT
ejpam-4034	325	7	y	y	PROPN
ejpam-4034	325	8	)	)	PUNCT
ejpam-4034	325	9	is	be	AUX
ejpam-4034	325	10	a	a	DET
ejpam-4034	325	11	left	left	ADJ
ejpam-4034	325	12	ideal	ideal	NOUN
ejpam-4034	325	13	in	in	ADP
ejpam-4034	325	14	s.	s.	PROPN
ejpam-4034	325	15	corollary	corollary	PROPN
ejpam-4034	325	16	3	3	X
ejpam-4034	325	17	.	.	PUNCT
ejpam-4034	326	1	let	let	VERB
ejpam-4034	326	2	s	s	PRON
ejpam-4034	326	3	be	be	AUX
ejpam-4034	326	4	a	a	DET
ejpam-4034	326	5	gamma	gamma	NOUN
ejpam-4034	326	6	la	la	PROPN
ejpam-4034	326	7	-	-	PUNCT
ejpam-4034	326	8	semiring	semiring	NOUN
ejpam-4034	326	9	with	with	ADP
ejpam-4034	326	10	identity	identity	NOUN
ejpam-4034	326	11	and	and	CCONJ
ejpam-4034	326	12	x	x	PART
ejpam-4034	326	13	be	be	AUX
ejpam-4034	326	14	a	a	DET
ejpam-4034	326	15	left	left	ADJ
ejpam-4034	326	16	ideal	ideal	NOUN
ejpam-4034	326	17	of	of	ADP
ejpam-4034	326	18	s.	s.	PROPN
ejpam-4034	326	19	then	then	ADV
ejpam-4034	326	20	,	,	PUNCT
ejpam-4034	326	21	(	(	PUNCT
ejpam-4034	326	22	x	x	X
ejpam-4034	326	23	:	:	PUNCT
ejpam-4034	326	24	γ	γ	X
ejpam-4034	326	25	:	:	PUNCT
ejpam-4034	326	26	r	r	X
ejpam-4034	326	27	)	)	PUNCT
ejpam-4034	326	28	is	be	AUX
ejpam-4034	326	29	a	a	DET
ejpam-4034	326	30	left	left	ADJ
ejpam-4034	326	31	ideal	ideal	NOUN
ejpam-4034	326	32	in	in	ADP
ejpam-4034	326	33	s	s	PROPN
ejpam-4034	326	34	,	,	PUNCT
ejpam-4034	327	1	where	where	SCONJ
ejpam-4034	327	2	(	(	PUNCT
ejpam-4034	327	3	x	x	X
ejpam-4034	327	4	:	:	PUNCT
ejpam-4034	327	5	γ	γ	X
ejpam-4034	327	6	:	:	PUNCT
ejpam-4034	327	7	r	r	X
ejpam-4034	327	8	)	)	PUNCT
ejpam-4034	327	9	=	=	SYM
ejpam-4034	327	10	{	{	PUNCT
ejpam-4034	327	11	x	x	PUNCT
ejpam-4034	327	12	∈	∈	PROPN
ejpam-4034	327	13	s	s	PART
ejpam-4034	327	14	:	:	PUNCT
ejpam-4034	327	15	xγr	xγr	PROPN
ejpam-4034	327	16	∈	∈	PROPN
ejpam-4034	327	17	x	x	X
ejpam-4034	327	18	}	}	PUNCT
ejpam-4034	327	19	.	.	PUNCT
ejpam-4034	328	1	proof	proof	NOUN
ejpam-4034	328	2	.	.	PUNCT
ejpam-4034	329	1	this	this	PRON
ejpam-4034	329	2	follows	follow	VERB
ejpam-4034	329	3	from	from	ADP
ejpam-4034	329	4	lemma	lemma	PROPN
ejpam-4034	329	5	7	7	NUM
ejpam-4034	329	6	remark	remark	NOUN
ejpam-4034	329	7	4	4	NUM
ejpam-4034	329	8	.	.	PUNCT
ejpam-4034	330	1	let	let	VERB
ejpam-4034	330	2	x	x	PRON
ejpam-4034	330	3	,	,	PUNCT
ejpam-4034	330	4	y	y	PROPN
ejpam-4034	330	5	and	and	CCONJ
ejpam-4034	330	6	z	z	NOUN
ejpam-4034	330	7	be	be	AUX
ejpam-4034	330	8	the	the	DET
ejpam-4034	330	9	left	left	ADJ
ejpam-4034	330	10	ideals	ideal	NOUN
ejpam-4034	330	11	of	of	ADP
ejpam-4034	330	12	a	a	DET
ejpam-4034	330	13	gamma	gamma	NOUN
ejpam-4034	330	14	la	la	PROPN
ejpam-4034	330	15	-	-	PUNCT
ejpam-4034	330	16	semiring	semire	VERB
ejpam-4034	330	17	s.	s.	PROPN
ejpam-4034	330	18	then	then	ADV
ejpam-4034	330	19	(	(	PUNCT
ejpam-4034	330	20	x	x	X
ejpam-4034	330	21	:	:	PUNCT
ejpam-4034	330	22	γ	γ	X
ejpam-4034	330	23	:	:	PUNCT
ejpam-4034	330	24	z	z	X
ejpam-4034	330	25	)	)	PUNCT
ejpam-4034	330	26	⊆	⊆	NUM
ejpam-4034	330	27	(	(	PUNCT
ejpam-4034	330	28	x	x	X
ejpam-4034	330	29	:	:	PUNCT
ejpam-4034	330	30	γ	γ	X
ejpam-4034	330	31	:	:	PUNCT
ejpam-4034	330	32	y	y	PROPN
ejpam-4034	330	33	)	)	PUNCT
ejpam-4034	330	34	,	,	PUNCT
ejpam-4034	330	35	where	where	SCONJ
ejpam-4034	330	36	y	y	PROPN
ejpam-4034	330	37	⊆	⊆	NUM
ejpam-4034	330	38	z.	z.	PROPN
ejpam-4034	330	39	theorem	theorem	VERB
ejpam-4034	330	40	9	9	NUM
ejpam-4034	330	41	.	.	PUNCT
ejpam-4034	331	1	let	let	VERB
ejpam-4034	331	2	s	s	PRON
ejpam-4034	331	3	be	be	AUX
ejpam-4034	331	4	a	a	DET
ejpam-4034	331	5	γ	γ	X
ejpam-4034	331	6	-	-	PUNCT
ejpam-4034	331	7	la	la	NOUN
ejpam-4034	331	8	-	-	PUNCT
ejpam-4034	331	9	semiring	semiring	NOUN
ejpam-4034	331	10	with	with	ADP
ejpam-4034	331	11	identity	identity	NOUN
ejpam-4034	331	12	.	.	PUNCT
ejpam-4034	332	1	then	then	ADV
ejpam-4034	332	2	,	,	PUNCT
ejpam-4034	332	3	(	(	PUNCT
ejpam-4034	332	4	x	x	X
ejpam-4034	332	5	:	:	PUNCT
ejpam-4034	332	6	γ	γ	X
ejpam-4034	332	7	:	:	PUNCT
ejpam-4034	332	8	y	y	PROPN
ejpam-4034	332	9	)	)	PUNCT
ejpam-4034	332	10	is	be	AUX
ejpam-4034	332	11	a	a	DET
ejpam-4034	332	12	quasi	quasi	NOUN
ejpam-4034	332	13	-	-	NOUN
ejpam-4034	332	14	ideal	ideal	ADJ
ejpam-4034	332	15	in	in	ADP
ejpam-4034	332	16	s	s	PRON
ejpam-4034	332	17	if	if	SCONJ
ejpam-4034	332	18	x	x	PRON
ejpam-4034	332	19	is	be	AUX
ejpam-4034	332	20	quasi	quasi	ADJ
ejpam-4034	332	21	-	-	NOUN
ejpam-4034	332	22	ideal	ideal	ADJ
ejpam-4034	332	23	of	of	ADP
ejpam-4034	332	24	s.	s.	PROPN
ejpam-4034	332	25	proof	proof	PROPN
ejpam-4034	332	26	.	.	PUNCT
ejpam-4034	333	1	assume	assume	VERB
ejpam-4034	333	2	that	that	SCONJ
ejpam-4034	333	3	x	x	PRON
ejpam-4034	333	4	is	be	AUX
ejpam-4034	333	5	a	a	DET
ejpam-4034	333	6	quasi	quasi	NOUN
ejpam-4034	333	7	-	-	NOUN
ejpam-4034	333	8	ideal	ideal	NOUN
ejpam-4034	333	9	of	of	ADP
ejpam-4034	333	10	s	s	PROPN
ejpam-4034	333	11	,	,	PUNCT
ejpam-4034	333	12	then	then	ADV
ejpam-4034	333	13	by	by	ADP
ejpam-4034	333	14	lemma	lemma	PROPN
ejpam-4034	333	15	7	7	NUM
ejpam-4034	333	16	,	,	PUNCT
ejpam-4034	333	17	we	we	PRON
ejpam-4034	333	18	have	have	AUX
ejpam-4034	333	19	(	(	PUNCT
ejpam-4034	333	20	x	x	X
ejpam-4034	333	21	:	:	PUNCT
ejpam-4034	333	22	γ	γ	X
ejpam-4034	333	23	:	:	PUNCT
ejpam-4034	333	24	y	y	PROPN
ejpam-4034	333	25	)	)	PUNCT
ejpam-4034	333	26	is	be	AUX
ejpam-4034	333	27	a	a	DET
ejpam-4034	333	28	left	left	ADJ
ejpam-4034	333	29	ideal	ideal	NOUN
ejpam-4034	333	30	in	in	ADP
ejpam-4034	333	31	s.	s.	PROPN
ejpam-4034	333	32	then	then	ADV
ejpam-4034	333	33	,	,	PUNCT
ejpam-4034	333	34	(	(	PUNCT
ejpam-4034	333	35	sγ	sγ	INTJ
ejpam-4034	333	36	(	(	PUNCT
ejpam-4034	333	37	x	x	X
ejpam-4034	333	38	:	:	PUNCT
ejpam-4034	333	39	γ	γ	X
ejpam-4034	333	40	:	:	PUNCT
ejpam-4034	333	41	y	y	PROPN
ejpam-4034	333	42	)	)	PUNCT
ejpam-4034	333	43	)	)	PUNCT
ejpam-4034	333	44	∩	∩	NOUN
ejpam-4034	333	45	(	(	PUNCT
ejpam-4034	333	46	(	(	PUNCT
ejpam-4034	333	47	x	x	X
ejpam-4034	333	48	:	:	PUNCT
ejpam-4034	333	49	γ	γ	X
ejpam-4034	333	50	:	:	PUNCT
ejpam-4034	333	51	y	y	NOUN
ejpam-4034	333	52	)	)	PUNCT
ejpam-4034	333	53	γs	γs	VERB
ejpam-4034	333	54	)	)	PUNCT
ejpam-4034	333	55	⊆	⊆	NUM
ejpam-4034	333	56	(	(	PUNCT
ejpam-4034	333	57	x	x	X
ejpam-4034	333	58	:	:	PUNCT
ejpam-4034	333	59	γ	γ	X
ejpam-4034	333	60	:	:	PUNCT
ejpam-4034	333	61	y	y	PROPN
ejpam-4034	333	62	)	)	PUNCT
ejpam-4034	333	63	∩	∩	NOUN
ejpam-4034	333	64	(	(	PUNCT
ejpam-4034	333	65	x	x	X
ejpam-4034	333	66	:	:	PUNCT
ejpam-4034	333	67	γ	γ	X
ejpam-4034	333	68	:	:	PUNCT
ejpam-4034	333	69	y	y	PROPN
ejpam-4034	333	70	)	)	PUNCT
ejpam-4034	333	71	⊆	⊆	X
ejpam-4034	333	72	(	(	PUNCT
ejpam-4034	333	73	x	x	X
ejpam-4034	333	74	:	:	PUNCT
ejpam-4034	333	75	γ	γ	X
ejpam-4034	333	76	:	:	PUNCT
ejpam-4034	333	77	y	y	PROPN
ejpam-4034	333	78	)	)	PUNCT
ejpam-4034	333	79	.	.	PUNCT
ejpam-4034	334	1	hence	hence	ADV
ejpam-4034	334	2	(	(	PUNCT
ejpam-4034	334	3	x	x	X
ejpam-4034	334	4	:	:	PUNCT
ejpam-4034	334	5	γ	γ	X
ejpam-4034	334	6	:	:	PUNCT
ejpam-4034	334	7	y	y	PROPN
ejpam-4034	334	8	)	)	PUNCT
ejpam-4034	334	9	is	be	AUX
ejpam-4034	334	10	a	a	DET
ejpam-4034	334	11	quasi	quasi	NOUN
ejpam-4034	334	12	-	-	NOUN
ejpam-4034	334	13	ideal	ideal	ADJ
ejpam-4034	334	14	in	in	ADP
ejpam-4034	334	15	s.	s.	PROPN
ejpam-4034	334	16	theorem	theorem	VERB
ejpam-4034	334	17	10	10	NUM
ejpam-4034	334	18	.	.	PUNCT
ejpam-4034	335	1	let	let	VERB
ejpam-4034	335	2	s	s	PRON
ejpam-4034	335	3	be	be	AUX
ejpam-4034	335	4	a	a	DET
ejpam-4034	335	5	gamma	gamma	NOUN
ejpam-4034	335	6	la	la	PROPN
ejpam-4034	335	7	-	-	PUNCT
ejpam-4034	335	8	semiring	semiring	NOUN
ejpam-4034	335	9	with	with	ADP
ejpam-4034	335	10	identity	identity	NOUN
ejpam-4034	335	11	.	.	PUNCT
ejpam-4034	336	1	then	then	ADV
ejpam-4034	336	2	(	(	PUNCT
ejpam-4034	336	3	x	x	X
ejpam-4034	336	4	:	:	PUNCT
ejpam-4034	336	5	γ	γ	X
ejpam-4034	336	6	:	:	PUNCT
ejpam-4034	336	7	y	y	PROPN
ejpam-4034	336	8	)	)	PUNCT
ejpam-4034	336	9	is	be	AUX
ejpam-4034	336	10	a	a	DET
ejpam-4034	336	11	left	left	ADJ
ejpam-4034	336	12	k	k	NOUN
ejpam-4034	336	13	-	-	NOUN
ejpam-4034	336	14	ideal	ideal	NOUN
ejpam-4034	336	15	in	in	ADP
ejpam-4034	336	16	s	s	PROPN
ejpam-4034	336	17	,	,	PUNCT
ejpam-4034	336	18	if	if	SCONJ
ejpam-4034	336	19	x	x	PRON
ejpam-4034	336	20	be	be	AUX
ejpam-4034	336	21	a	a	DET
ejpam-4034	336	22	left	left	ADJ
ejpam-4034	336	23	k	k	NOUN
ejpam-4034	336	24	-	-	NOUN
ejpam-4034	336	25	ideal	ideal	NOUN
ejpam-4034	336	26	of	of	ADP
ejpam-4034	336	27	s.	s.	PROPN
ejpam-4034	336	28	proof	proof	PROPN
ejpam-4034	336	29	.	.	PUNCT
ejpam-4034	337	1	assume	assume	VERB
ejpam-4034	337	2	that	that	SCONJ
ejpam-4034	337	3	x	x	PRON
ejpam-4034	337	4	is	be	AUX
ejpam-4034	337	5	a	a	DET
ejpam-4034	337	6	left	left	ADJ
ejpam-4034	337	7	k	k	NOUN
ejpam-4034	337	8	-	-	NOUN
ejpam-4034	337	9	ideal	ideal	NOUN
ejpam-4034	337	10	of	of	ADP
ejpam-4034	337	11	s	s	PRON
ejpam-4034	337	12	then	then	ADV
ejpam-4034	337	13	by	by	ADP
ejpam-4034	337	14	lemma	lemma	PROPN
ejpam-4034	337	15	7	7	NUM
ejpam-4034	337	16	,	,	PUNCT
ejpam-4034	337	17	(	(	PUNCT
ejpam-4034	337	18	x	x	X
ejpam-4034	337	19	:	:	PUNCT
ejpam-4034	337	20	γ	γ	X
ejpam-4034	337	21	:	:	PUNCT
ejpam-4034	337	22	y	y	PROPN
ejpam-4034	337	23	)	)	PUNCT
ejpam-4034	337	24	is	be	AUX
ejpam-4034	337	25	a	a	DET
ejpam-4034	337	26	left	left	ADJ
ejpam-4034	337	27	ideal	ideal	NOUN
ejpam-4034	337	28	in	in	ADP
ejpam-4034	337	29	s.	s.	PROPN
ejpam-4034	337	30	similarly	similarly	ADV
ejpam-4034	337	31	,	,	PUNCT
ejpam-4034	337	32	if	if	SCONJ
ejpam-4034	337	33	x	x	X
ejpam-4034	337	34	,	,	PUNCT
ejpam-4034	337	35	x	x	PROPN
ejpam-4034	338	1	+	+	NUM
ejpam-4034	338	2	t	t	X
ejpam-4034	338	3	∈	∈	PROPN
ejpam-4034	338	4	(	(	PUNCT
ejpam-4034	338	5	x	x	X
ejpam-4034	338	6	:	:	PUNCT
ejpam-4034	338	7	γ	γ	X
ejpam-4034	338	8	:	:	PUNCT
ejpam-4034	338	9	y	y	PROPN
ejpam-4034	338	10	)	)	PUNCT
ejpam-4034	338	11	then	then	ADV
ejpam-4034	338	12	xγy	xγy	NOUN
ejpam-4034	339	1	⊆	⊆	NUM
ejpam-4034	339	2	x	x	PUNCT
ejpam-4034	339	3	and	and	CCONJ
ejpam-4034	339	4	(	(	PUNCT
ejpam-4034	339	5	x	x	X
ejpam-4034	339	6	+	+	NUM
ejpam-4034	339	7	t)γy	t)γy	PROPN
ejpam-4034	339	8	⊆	⊆	NUM
ejpam-4034	339	9	x	x	SYM
ejpam-4034	339	10	that	that	PRON
ejpam-4034	339	11	is	be	AUX
ejpam-4034	339	12	xγy	xγy	PROPN
ejpam-4034	340	1	⊆	⊆	NUM
ejpam-4034	340	2	x	x	PUNCT
ejpam-4034	340	3	and	and	CCONJ
ejpam-4034	340	4	xγy	xγy	NOUN
ejpam-4034	341	1	+	+	CCONJ
ejpam-4034	341	2	tγy	tγy	PROPN
ejpam-4034	341	3	⊆	⊆	NUM
ejpam-4034	341	4	x.	x.	NOUN
ejpam-4034	341	5	then	then	ADV
ejpam-4034	341	6	,	,	PUNCT
ejpam-4034	341	7	we	we	PRON
ejpam-4034	341	8	get	get	VERB
ejpam-4034	341	9	tγy	tγy	NOUN
ejpam-4034	341	10	⊆	⊆	NUM
ejpam-4034	341	11	x.	x.	NOUN
ejpam-4034	341	12	hence	hence	ADV
ejpam-4034	341	13	(	(	PUNCT
ejpam-4034	341	14	x	x	X
ejpam-4034	341	15	:	:	PUNCT
ejpam-4034	341	16	γ	γ	X
ejpam-4034	341	17	:	:	PUNCT
ejpam-4034	341	18	y	y	PROPN
ejpam-4034	341	19	)	)	PUNCT
ejpam-4034	341	20	is	be	AUX
ejpam-4034	341	21	a	a	DET
ejpam-4034	341	22	left	left	ADJ
ejpam-4034	341	23	k	k	NOUN
ejpam-4034	341	24	-	-	NOUN
ejpam-4034	341	25	ideal	ideal	NOUN
ejpam-4034	341	26	in	in	ADP
ejpam-4034	341	27	s.	s.	PROPN
ejpam-4034	341	28	w.	w.	PROPN
ejpam-4034	341	29	a.	a.	PROPN
ejpam-4034	341	30	khan	khan	PROPN
ejpam-4034	341	31	et	et	PROPN
ejpam-4034	341	32	al	al	PROPN
ejpam-4034	341	33	.	.	PUNCT
ejpam-4034	341	34	/	/	SYM
ejpam-4034	341	35	eur	eur	PROPN
ejpam-4034	341	36	.	.	PUNCT
ejpam-4034	342	1	j.	j.	PROPN
ejpam-4034	342	2	pure	pure	PROPN
ejpam-4034	342	3	appl	appl	PROPN
ejpam-4034	342	4	.	.	PROPN
ejpam-4034	342	5	math	math	PROPN
ejpam-4034	342	6	,	,	PUNCT
ejpam-4034	342	7	14	14	NUM
ejpam-4034	342	8	(	(	PUNCT
ejpam-4034	342	9	3	3	NUM
ejpam-4034	342	10	)	)	PUNCT
ejpam-4034	342	11	(	(	PUNCT
ejpam-4034	342	12	2021	2021	NUM
ejpam-4034	342	13	)	)	PUNCT
ejpam-4034	342	14	,	,	PUNCT
ejpam-4034	342	15	989	989	NUM
ejpam-4034	342	16	-	-	SYM
ejpam-4034	342	17	1001	1001	NUM
ejpam-4034	342	18	998	998	NUM
ejpam-4034	342	19	3.2	3.2	NUM
ejpam-4034	342	20	.	.	PUNCT
ejpam-4034	343	1	almost	almost	ADV
ejpam-4034	343	2	prime	prime	ADJ
ejpam-4034	343	3	ideals	ideal	NOUN
ejpam-4034	343	4	in	in	ADP
ejpam-4034	343	5	γ	γ	PROPN
ejpam-4034	343	6	-	-	PUNCT
ejpam-4034	343	7	la	la	NOUN
ejpam-4034	343	8	-	-	PUNCT
ejpam-4034	343	9	semiring	semiring	NOUN
ejpam-4034	343	10	in	in	ADP
ejpam-4034	343	11	this	this	DET
ejpam-4034	343	12	section	section	NOUN
ejpam-4034	343	13	,	,	PUNCT
ejpam-4034	343	14	we	we	PRON
ejpam-4034	343	15	initiate	initiate	VERB
ejpam-4034	343	16	the	the	DET
ejpam-4034	343	17	term	term	NOUN
ejpam-4034	343	18	almost	almost	ADV
ejpam-4034	343	19	prime	prime	ADJ
ejpam-4034	343	20	and	and	CCONJ
ejpam-4034	343	21	weakly	weakly	ADJ
ejpam-4034	343	22	almost	almost	ADV
ejpam-4034	343	23	prime	prime	ADJ
ejpam-4034	343	24	ideals	ideal	NOUN
ejpam-4034	343	25	in	in	ADP
ejpam-4034	343	26	γ	γ	X
ejpam-4034	343	27	-lasemiring	-lasemiring	NOUN
ejpam-4034	343	28	.	.	PUNCT
ejpam-4034	344	1	our	our	PRON
ejpam-4034	344	2	starting	starting	NOUN
ejpam-4034	344	3	point	point	NOUN
ejpam-4034	344	4	is	be	AUX
ejpam-4034	344	5	the	the	DET
ejpam-4034	344	6	following	follow	VERB
ejpam-4034	344	7	definition	definition	NOUN
ejpam-4034	344	8	.	.	PUNCT
ejpam-4034	345	1	definition	definition	NOUN
ejpam-4034	345	2	10	10	NUM
ejpam-4034	345	3	.	.	PUNCT
ejpam-4034	346	1	a	a	DET
ejpam-4034	346	2	left	left	ADJ
ejpam-4034	346	3	ideal	ideal	NOUN
ejpam-4034	346	4	p	p	NOUN
ejpam-4034	346	5	is	be	AUX
ejpam-4034	346	6	called	call	VERB
ejpam-4034	346	7	almost	almost	ADV
ejpam-4034	346	8	-	-	PUNCT
ejpam-4034	346	9	prime	prime	NOUN
ejpam-4034	346	10	if	if	SCONJ
ejpam-4034	346	11	xγy	xγy	PROPN
ejpam-4034	346	12	⊆	⊆	NUM
ejpam-4034	346	13	p	p	NOUN
ejpam-4034	346	14	implies	imply	VERB
ejpam-4034	346	15	that	that	SCONJ
ejpam-4034	346	16	x	x	PROPN
ejpam-4034	346	17	⊆	⊆	NUM
ejpam-4034	346	18	p	p	NOUN
ejpam-4034	346	19	or	or	CCONJ
ejpam-4034	346	20	y	y	PROPN
ejpam-4034	347	1	⊆	⊆	NUM
ejpam-4034	347	2	p	p	NOUN
ejpam-4034	347	3	,	,	PUNCT
ejpam-4034	347	4	where	where	SCONJ
ejpam-4034	347	5	x	x	PUNCT
ejpam-4034	347	6	and	and	CCONJ
ejpam-4034	347	7	y	y	PROPN
ejpam-4034	347	8	are	be	AUX
ejpam-4034	347	9	respectively	respectively	ADV
ejpam-4034	347	10	left	leave	VERB
ejpam-4034	347	11	and	and	CCONJ
ejpam-4034	347	12	right	right	ADJ
ejpam-4034	347	13	ideal	ideal	NOUN
ejpam-4034	347	14	of	of	ADP
ejpam-4034	347	15	s.	s.	PROPN
ejpam-4034	347	16	remark	remark	PROPN
ejpam-4034	347	17	5	5	NUM
ejpam-4034	347	18	.	.	PUNCT
ejpam-4034	348	1	it	it	PRON
ejpam-4034	348	2	is	be	AUX
ejpam-4034	348	3	easy	easy	ADJ
ejpam-4034	348	4	to	to	PART
ejpam-4034	348	5	see	see	VERB
ejpam-4034	348	6	that	that	SCONJ
ejpam-4034	348	7	every	every	DET
ejpam-4034	348	8	almost	almost	ADV
ejpam-4034	348	9	-	-	PUNCT
ejpam-4034	348	10	prime	prime	ADJ
ejpam-4034	348	11	left	leave	VERB
ejpam-4034	348	12	ideal	ideal	NOUN
ejpam-4034	348	13	is	be	AUX
ejpam-4034	348	14	prime	prime	ADJ
ejpam-4034	348	15	.	.	PUNCT
ejpam-4034	349	1	definition	definition	NOUN
ejpam-4034	349	2	11	11	NUM
ejpam-4034	349	3	.	.	PUNCT
ejpam-4034	350	1	a	a	DET
ejpam-4034	350	2	left	left	ADJ
ejpam-4034	350	3	ideal	ideal	NOUN
ejpam-4034	350	4	p	p	NOUN
ejpam-4034	350	5	is	be	AUX
ejpam-4034	350	6	called	call	VERB
ejpam-4034	350	7	weakly	weakly	ADJ
ejpam-4034	350	8	almost	almost	ADV
ejpam-4034	350	9	-	-	PUNCT
ejpam-4034	350	10	prime	prime	ADJ
ejpam-4034	350	11	if	if	SCONJ
ejpam-4034	350	12	{	{	PUNCT
ejpam-4034	350	13	0	0	NUM
ejpam-4034	350	14	}	}	PUNCT
ejpam-4034	350	15	6=	6=	NUM
ejpam-4034	350	16	xγy	xγy	NOUN
ejpam-4034	351	1	⊆	⊆	NUM
ejpam-4034	351	2	p	p	NOUN
ejpam-4034	351	3	implies	imply	VERB
ejpam-4034	351	4	x	x	SYM
ejpam-4034	351	5	⊆	⊆	NUM
ejpam-4034	351	6	p	p	NOUN
ejpam-4034	351	7	or	or	CCONJ
ejpam-4034	351	8	y	y	PROPN
ejpam-4034	351	9	⊆	⊆	NUM
ejpam-4034	351	10	p	p	NOUN
ejpam-4034	351	11	,	,	PUNCT
ejpam-4034	351	12	where	where	SCONJ
ejpam-4034	351	13	x	x	PUNCT
ejpam-4034	351	14	and	and	CCONJ
ejpam-4034	351	15	y	y	PROPN
ejpam-4034	351	16	are	be	AUX
ejpam-4034	351	17	respectively	respectively	ADV
ejpam-4034	351	18	left	leave	VERB
ejpam-4034	351	19	and	and	CCONJ
ejpam-4034	351	20	right	right	ADJ
ejpam-4034	351	21	ideal	ideal	NOUN
ejpam-4034	351	22	of	of	ADP
ejpam-4034	351	23	s.	s.	PROPN
ejpam-4034	351	24	remark	remark	PROPN
ejpam-4034	351	25	6	6	NUM
ejpam-4034	351	26	.	.	PUNCT
ejpam-4034	352	1	it	it	PRON
ejpam-4034	352	2	is	be	AUX
ejpam-4034	352	3	easy	easy	ADJ
ejpam-4034	352	4	to	to	PART
ejpam-4034	352	5	see	see	VERB
ejpam-4034	352	6	that	that	SCONJ
ejpam-4034	352	7	every	every	DET
ejpam-4034	352	8	almost	almost	ADV
ejpam-4034	352	9	-	-	PUNCT
ejpam-4034	352	10	prime	prime	ADJ
ejpam-4034	352	11	left	leave	VERB
ejpam-4034	352	12	ideal	ideal	NOUN
ejpam-4034	352	13	is	be	AUX
ejpam-4034	352	14	weakly	weakly	ADJ
ejpam-4034	352	15	almost	almost	ADV
ejpam-4034	352	16	-	-	PUNCT
ejpam-4034	352	17	prime	prime	NOUN
ejpam-4034	352	18	.	.	PUNCT
ejpam-4034	353	1	lemma	lemma	PROPN
ejpam-4034	353	2	8	8	NUM
ejpam-4034	353	3	.	.	PUNCT
ejpam-4034	354	1	let	let	VERB
ejpam-4034	354	2	p	p	PRON
ejpam-4034	354	3	be	be	AUX
ejpam-4034	354	4	the	the	DET
ejpam-4034	354	5	ideal	ideal	NOUN
ejpam-4034	354	6	of	of	ADP
ejpam-4034	354	7	a	a	DET
ejpam-4034	354	8	γ	γ	NOUN
ejpam-4034	354	9	-la	-la	ADV
ejpam-4034	354	10	-	-	PUNCT
ejpam-4034	354	11	semiring	semire	VERB
ejpam-4034	354	12	s	s	NOUN
ejpam-4034	354	13	with	with	ADP
ejpam-4034	354	14	identity	identity	NOUN
ejpam-4034	354	15	.	.	PUNCT
ejpam-4034	355	1	then	then	ADV
ejpam-4034	355	2	p	p	NOUN
ejpam-4034	355	3	is	be	AUX
ejpam-4034	355	4	an	an	DET
ejpam-4034	355	5	almostprime	almostprime	NOUN
ejpam-4034	355	6	left	leave	VERB
ejpam-4034	355	7	ideal	ideal	NOUN
ejpam-4034	355	8	of	of	ADP
ejpam-4034	355	9	s	s	PRON
ejpam-4034	355	10	if	if	SCONJ
ejpam-4034	355	11	xγ	xγ	PROPN
ejpam-4034	355	12	(	(	PUNCT
ejpam-4034	355	13	sγy	sγy	NOUN
ejpam-4034	355	14	)	)	PUNCT
ejpam-4034	355	15	⊆	⊆	NUM
ejpam-4034	355	16	p	p	NOUN
ejpam-4034	355	17	implies	imply	VERB
ejpam-4034	355	18	x	x	X
ejpam-4034	355	19	∈	∈	PROPN
ejpam-4034	355	20	p	p	NOUN
ejpam-4034	355	21	or	or	CCONJ
ejpam-4034	355	22	y	y	PROPN
ejpam-4034	355	23	∈	∈	PROPN
ejpam-4034	355	24	p	p	NOUN
ejpam-4034	355	25	.	.	PUNCT
ejpam-4034	356	1	proof	proof	NOUN
ejpam-4034	356	2	.	.	PUNCT
ejpam-4034	357	1	let	let	VERB
ejpam-4034	357	2	p	p	PRON
ejpam-4034	357	3	be	be	AUX
ejpam-4034	357	4	an	an	DET
ejpam-4034	357	5	almost	almost	ADV
ejpam-4034	357	6	-	-	PUNCT
ejpam-4034	357	7	prime	prime	ADJ
ejpam-4034	357	8	left	left	ADJ
ejpam-4034	357	9	ideal	ideal	NOUN
ejpam-4034	357	10	of	of	ADP
ejpam-4034	357	11	a	a	DET
ejpam-4034	357	12	γ	γ	NOUN
ejpam-4034	357	13	-la	-la	ADV
ejpam-4034	357	14	-	-	PUNCT
ejpam-4034	357	15	semiring	semire	VERB
ejpam-4034	357	16	s	s	NOUN
ejpam-4034	357	17	with	with	ADP
ejpam-4034	357	18	identity	identity	NOUN
ejpam-4034	357	19	.	.	PUNCT
ejpam-4034	358	1	now	now	ADV
ejpam-4034	358	2	suppose	suppose	VERB
ejpam-4034	358	3	that	that	SCONJ
ejpam-4034	358	4	xγ	xγ	PROPN
ejpam-4034	358	5	(	(	PUNCT
ejpam-4034	358	6	sγy	sγy	NOUN
ejpam-4034	358	7	)	)	PUNCT
ejpam-4034	358	8	⊆	⊆	NUM
ejpam-4034	358	9	p	p	NOUN
ejpam-4034	358	10	.	.	PUNCT
ejpam-4034	359	1	then	then	ADV
ejpam-4034	359	2	by	by	ADP
ejpam-4034	359	3	hypothesis	hypothesis	NOUN
ejpam-4034	359	4	,	,	PUNCT
ejpam-4034	359	5	we	we	PRON
ejpam-4034	359	6	have	have	VERB
ejpam-4034	359	7	(	(	PUNCT
ejpam-4034	359	8	sγx)γ	sγx)γ	PROPN
ejpam-4034	359	9	(	(	PUNCT
ejpam-4034	359	10	yγs	yγs	PROPN
ejpam-4034	359	11	)	)	PUNCT
ejpam-4034	359	12	⊆	⊆	NUM
ejpam-4034	359	13	(	(	PUNCT
ejpam-4034	359	14	sγx)γsγ	sγx)γsγ	PROPN
ejpam-4034	359	15	(	(	PUNCT
ejpam-4034	359	16	yγs	yγs	PROPN
ejpam-4034	359	17	)	)	PUNCT
ejpam-4034	359	18	=	=	PUNCT
ejpam-4034	360	1	(	(	PUNCT
ejpam-4034	360	2	xγs)γsγ	xγs)γsγ	X
ejpam-4034	360	3	(	(	PUNCT
ejpam-4034	360	4	sγy	sγy	NOUN
ejpam-4034	360	5	)	)	PUNCT
ejpam-4034	360	6	=	=	SYM
ejpam-4034	360	7	(	(	PUNCT
ejpam-4034	360	8	sγs)γxγ	sγs)γxγ	ADJ
ejpam-4034	360	9	(	(	PUNCT
ejpam-4034	360	10	sγy	sγy	NOUN
ejpam-4034	360	11	)	)	PUNCT
ejpam-4034	360	12	⊆	⊆	NUM
ejpam-4034	360	13	(	(	PUNCT
ejpam-4034	360	14	sγs)γp	sγs)γp	NOUN
ejpam-4034	360	15	=	=	SYM
ejpam-4034	360	16	(	(	PUNCT
ejpam-4034	360	17	pγs)γs	pγs)γ	NOUN
ejpam-4034	360	18	⊆	⊆	NUM
ejpam-4034	360	19	pγs	pγs	NOUN
ejpam-4034	360	20	⊆	⊆	NUM
ejpam-4034	360	21	p	p	NOUN
ejpam-4034	360	22	which	which	PRON
ejpam-4034	360	23	implies	imply	VERB
ejpam-4034	360	24	(	(	PUNCT
ejpam-4034	360	25	sγx)γ	sγx)γ	PROPN
ejpam-4034	360	26	(	(	PUNCT
ejpam-4034	360	27	yγs	yγs	PROPN
ejpam-4034	360	28	)	)	PUNCT
ejpam-4034	360	29	⊆	⊆	NUM
ejpam-4034	360	30	p	p	NOUN
ejpam-4034	360	31	.	.	PUNCT
ejpam-4034	361	1	then	then	ADV
ejpam-4034	361	2	,	,	PUNCT
ejpam-4034	361	3	x	x	PUNCT
ejpam-4034	361	4	=	=	PUNCT
ejpam-4034	361	5	eγx	eγx	PROPN
ejpam-4034	361	6	∈	∈	NOUN
ejpam-4034	361	7	sγx	sγx	VERB
ejpam-4034	361	8	⊆	⊆	NUM
ejpam-4034	361	9	p	p	NOUN
ejpam-4034	361	10	or	or	CCONJ
ejpam-4034	361	11	y	y	PROPN
ejpam-4034	361	12	=	=	PUNCT
ejpam-4034	361	13	yγe	yγe	VERB
ejpam-4034	361	14	∈	∈	PROPN
ejpam-4034	361	15	yγs	yγs	NOUN
ejpam-4034	361	16	⊆	⊆	NUM
ejpam-4034	361	17	p	p	NOUN
ejpam-4034	361	18	.	.	PUNCT
ejpam-4034	362	1	hence	hence	ADV
ejpam-4034	362	2	x	x	PUNCT
ejpam-4034	362	3	∈	∈	PROPN
ejpam-4034	362	4	p	p	NOUN
ejpam-4034	362	5	ory	ory	PROPN
ejpam-4034	362	6	∈	∈	PROPN
ejpam-4034	362	7	p	p	NOUN
ejpam-4034	362	8	.	.	PUNCT
ejpam-4034	363	1	corollary	corollary	ADJ
ejpam-4034	363	2	4	4	NUM
ejpam-4034	363	3	.	.	PUNCT
ejpam-4034	364	1	let	let	VERB
ejpam-4034	364	2	p	p	PRON
ejpam-4034	364	3	be	be	AUX
ejpam-4034	364	4	an	an	DET
ejpam-4034	364	5	almost	almost	ADV
ejpam-4034	364	6	-	-	PUNCT
ejpam-4034	364	7	prime	prime	ADJ
ejpam-4034	364	8	left	left	ADJ
ejpam-4034	364	9	ideal	ideal	NOUN
ejpam-4034	364	10	of	of	ADP
ejpam-4034	364	11	a	a	DET
ejpam-4034	364	12	γ	γ	NOUN
ejpam-4034	364	13	-la	-la	ADV
ejpam-4034	364	14	-	-	PUNCT
ejpam-4034	364	15	semiring	semire	VERB
ejpam-4034	364	16	s	s	NOUN
ejpam-4034	364	17	with	with	ADP
ejpam-4034	364	18	identity	identity	NOUN
ejpam-4034	364	19	.	.	PUNCT
ejpam-4034	365	1	then	then	ADV
ejpam-4034	365	2	p	p	NOUN
ejpam-4034	365	3	is	be	AUX
ejpam-4034	365	4	a	a	DET
ejpam-4034	365	5	weakly	weakly	ADJ
ejpam-4034	365	6	almost	almost	ADV
ejpam-4034	365	7	-	-	PUNCT
ejpam-4034	365	8	prime	prime	ADJ
ejpam-4034	365	9	left	left	ADJ
ejpam-4034	365	10	ideal	ideal	NOUN
ejpam-4034	365	11	of	of	ADP
ejpam-4034	365	12	s	s	PRON
ejpam-4034	365	13	if	if	SCONJ
ejpam-4034	365	14	{	{	PUNCT
ejpam-4034	365	15	0	0	NUM
ejpam-4034	365	16	}	}	PUNCT
ejpam-4034	365	17	6=	6=	NUM
ejpam-4034	365	18	xγ	xγ	NOUN
ejpam-4034	365	19	(	(	PUNCT
ejpam-4034	365	20	sγy	sγy	NOUN
ejpam-4034	365	21	)	)	PUNCT
ejpam-4034	365	22	⊆	⊆	NUM
ejpam-4034	365	23	p	p	NOUN
ejpam-4034	365	24	,	,	PUNCT
ejpam-4034	365	25	then	then	ADV
ejpam-4034	365	26	x	x	SYM
ejpam-4034	365	27	∈	∈	PROPN
ejpam-4034	365	28	p	p	NOUN
ejpam-4034	365	29	or	or	CCONJ
ejpam-4034	365	30	y	y	PROPN
ejpam-4034	365	31	∈	∈	PROPN
ejpam-4034	365	32	p	p	NOUN
ejpam-4034	365	33	.	.	PUNCT
ejpam-4034	366	1	proof	proof	NOUN
ejpam-4034	366	2	.	.	PUNCT
ejpam-4034	367	1	this	this	PRON
ejpam-4034	367	2	follows	follow	VERB
ejpam-4034	367	3	from	from	ADP
ejpam-4034	367	4	lemma	lemma	PROPN
ejpam-4034	367	5	8	8	NUM
ejpam-4034	367	6	theorem	theorem	NOUN
ejpam-4034	367	7	11	11	NUM
ejpam-4034	367	8	.	.	PUNCT
ejpam-4034	368	1	let	let	VERB
ejpam-4034	368	2	s	s	PRON
ejpam-4034	368	3	be	be	AUX
ejpam-4034	368	4	a	a	DET
ejpam-4034	368	5	gamma	gamma	NOUN
ejpam-4034	368	6	la	la	PROPN
ejpam-4034	368	7	-	-	PUNCT
ejpam-4034	368	8	semiring	semiring	NOUN
ejpam-4034	368	9	with	with	ADP
ejpam-4034	368	10	identity	identity	NOUN
ejpam-4034	368	11	and	and	CCONJ
ejpam-4034	368	12	x	x	NOUN
ejpam-4034	368	13	,	,	PUNCT
ejpam-4034	368	14	y	y	PROPN
ejpam-4034	368	15	∈	∈	PROPN
ejpam-4034	368	16	s	s	X
ejpam-4034	368	17	and	and	CCONJ
ejpam-4034	368	18	γ	γ	PROPN
ejpam-4034	368	19	∈	∈	PROPN
ejpam-4034	368	20	γ	γ	X
ejpam-4034	368	21	.	.	PUNCT
ejpam-4034	369	1	then	then	ADV
ejpam-4034	369	2	a	a	DET
ejpam-4034	369	3	left	left	ADJ
ejpam-4034	369	4	ideal	ideal	NOUN
ejpam-4034	369	5	p	p	NOUN
ejpam-4034	369	6	of	of	ADP
ejpam-4034	369	7	s	s	PRON
ejpam-4034	369	8	is	be	AUX
ejpam-4034	370	1	almost	almost	ADV
ejpam-4034	370	2	-	-	PUNCT
ejpam-4034	370	3	prime	prime	NOUN
ejpam-4034	370	4	iff	iff	PROPN
ejpam-4034	370	5	xγy	xγy	PROPN
ejpam-4034	370	6	∈	∈	PROPN
ejpam-4034	371	1	p	p	PROPN
ejpam-4034	371	2	implies	imply	VERB
ejpam-4034	371	3	x	x	PUNCT
ejpam-4034	371	4	∈	∈	PROPN
ejpam-4034	371	5	p	p	NOUN
ejpam-4034	371	6	ory	ory	PROPN
ejpam-4034	371	7	∈	∈	PROPN
ejpam-4034	371	8	p	p	NOUN
ejpam-4034	371	9	.	.	PUNCT
ejpam-4034	372	1	proof	proof	NOUN
ejpam-4034	372	2	.	.	PUNCT
ejpam-4034	373	1	let	let	VERB
ejpam-4034	373	2	p	p	PRON
ejpam-4034	373	3	be	be	AUX
ejpam-4034	373	4	a	a	DET
ejpam-4034	373	5	left	left	ADJ
ejpam-4034	373	6	ideal	ideal	NOUN
ejpam-4034	373	7	of	of	ADP
ejpam-4034	373	8	a	a	DET
ejpam-4034	373	9	γ	γ	NOUN
ejpam-4034	373	10	-la	-la	ADV
ejpam-4034	373	11	-	-	PUNCT
ejpam-4034	373	12	semiring	semiring	NOUN
ejpam-4034	373	13	with	with	ADP
ejpam-4034	373	14	identity	identity	NOUN
ejpam-4034	373	15	.	.	PUNCT
ejpam-4034	374	1	now	now	ADV
ejpam-4034	374	2	suppose	suppose	VERB
ejpam-4034	374	3	that	that	SCONJ
ejpam-4034	374	4	xγy	xγy	PROPN
ejpam-4034	374	5	∈	∈	PROPN
ejpam-4034	374	6	p	p	NOUN
ejpam-4034	374	7	,	,	PUNCT
ejpam-4034	374	8	where	where	SCONJ
ejpam-4034	374	9	x	x	X
ejpam-4034	374	10	,	,	PUNCT
ejpam-4034	374	11	y	y	PROPN
ejpam-4034	374	12	∈	∈	PROPN
ejpam-4034	374	13	s	s	X
ejpam-4034	374	14	and	and	CCONJ
ejpam-4034	374	15	γ	γ	PROPN
ejpam-4034	374	16	∈	∈	PROPN
ejpam-4034	374	17	γ	γ	X
ejpam-4034	374	18	.	.	PUNCT
ejpam-4034	375	1	then	then	ADV
ejpam-4034	375	2	by	by	ADP
ejpam-4034	375	3	hypothesis	hypothesis	NOUN
ejpam-4034	375	4	,	,	PUNCT
ejpam-4034	375	5	we	we	PRON
ejpam-4034	375	6	get	get	VERB
ejpam-4034	375	7	;	;	PUNCT
ejpam-4034	375	8	(	(	PUNCT
ejpam-4034	375	9	sγx)γ(yγs	sγx)γ(yγs	X
ejpam-4034	375	10	)	)	PUNCT
ejpam-4034	375	11	⊆	⊆	NUM
ejpam-4034	375	12	sγ	sγ	X
ejpam-4034	375	13	(	(	PUNCT
ejpam-4034	375	14	(	(	PUNCT
ejpam-4034	375	15	xγy)γs	xγy)γs	X
ejpam-4034	375	16	)	)	PUNCT
ejpam-4034	375	17	w.	w.	PROPN
ejpam-4034	375	18	a.	a.	PROPN
ejpam-4034	375	19	khan	khan	PROPN
ejpam-4034	375	20	et	et	PROPN
ejpam-4034	375	21	al	al	PROPN
ejpam-4034	375	22	.	.	PUNCT
ejpam-4034	375	23	/	/	SYM
ejpam-4034	375	24	eur	eur	PROPN
ejpam-4034	375	25	.	.	PUNCT
ejpam-4034	376	1	j.	j.	PROPN
ejpam-4034	376	2	pure	pure	PROPN
ejpam-4034	376	3	appl	appl	PROPN
ejpam-4034	376	4	.	.	PROPN
ejpam-4034	376	5	math	math	PROPN
ejpam-4034	376	6	,	,	PUNCT
ejpam-4034	376	7	14	14	NUM
ejpam-4034	376	8	(	(	PUNCT
ejpam-4034	376	9	3	3	NUM
ejpam-4034	376	10	)	)	PUNCT
ejpam-4034	376	11	(	(	PUNCT
ejpam-4034	376	12	2021	2021	NUM
ejpam-4034	376	13	)	)	PUNCT
ejpam-4034	376	14	,	,	PUNCT
ejpam-4034	376	15	989	989	NUM
ejpam-4034	376	16	-	-	SYM
ejpam-4034	376	17	1001	1001	NUM
ejpam-4034	376	18	999	999	NUM
ejpam-4034	376	19	⊆	⊆	NUM
ejpam-4034	376	20	sγ	sγ	NOUN
ejpam-4034	376	21	(	(	PUNCT
ejpam-4034	376	22	pγs	pγs	NOUN
ejpam-4034	376	23	)	)	PUNCT
ejpam-4034	376	24	⊆	⊆	NUM
ejpam-4034	376	25	sγp	sγp	NOUN
ejpam-4034	376	26	⊆	⊆	NUM
ejpam-4034	376	27	p.	p.	NOUN
ejpam-4034	376	28	so	so	ADV
ejpam-4034	376	29	by	by	ADP
ejpam-4034	376	30	the	the	DET
ejpam-4034	376	31	definition	definition	NOUN
ejpam-4034	376	32	of	of	ADP
ejpam-4034	376	33	almost	almost	ADV
ejpam-4034	376	34	-	-	PUNCT
ejpam-4034	376	35	prime	prime	NOUN
ejpam-4034	376	36	,	,	PUNCT
ejpam-4034	376	37	we	we	PRON
ejpam-4034	376	38	have	have	VERB
ejpam-4034	376	39	x	x	X
ejpam-4034	376	40	∈	∈	PROPN
ejpam-4034	376	41	p	p	NOUN
ejpam-4034	376	42	or	or	CCONJ
ejpam-4034	376	43	y	y	PROPN
ejpam-4034	376	44	∈	∈	PROPN
ejpam-4034	376	45	p	p	NOUN
ejpam-4034	376	46	.	.	PUNCT
ejpam-4034	377	1	conversely	conversely	ADV
ejpam-4034	377	2	,	,	PUNCT
ejpam-4034	377	3	assume	assume	VERB
ejpam-4034	377	4	that	that	SCONJ
ejpam-4034	377	5	if	if	SCONJ
ejpam-4034	377	6	xγy	xγy	PROPN
ejpam-4034	377	7	∈	∈	PROPN
ejpam-4034	377	8	p	p	PROPN
ejpam-4034	377	9	implies	imply	VERB
ejpam-4034	377	10	x	x	X
ejpam-4034	377	11	∈	∈	PROPN
ejpam-4034	377	12	p	p	NOUN
ejpam-4034	377	13	or	or	CCONJ
ejpam-4034	377	14	y	y	PROPN
ejpam-4034	377	15	∈	∈	PROPN
ejpam-4034	377	16	p	p	NOUN
ejpam-4034	377	17	and	and	CCONJ
ejpam-4034	377	18	x	x	NOUN
ejpam-4034	377	19	is	be	AUX
ejpam-4034	377	20	left	leave	VERB
ejpam-4034	377	21	ideal	ideal	NOUN
ejpam-4034	377	22	of	of	ADP
ejpam-4034	377	23	s.	s.	PROPN
ejpam-4034	377	24	let	let	VERB
ejpam-4034	377	25	xγy	xγy	PROPN
ejpam-4034	378	1	⊆	⊆	NUM
ejpam-4034	378	2	p	p	NOUN
ejpam-4034	378	3	,	,	PUNCT
ejpam-4034	378	4	where	where	SCONJ
ejpam-4034	378	5	y	y	PROPN
ejpam-4034	378	6	is	be	AUX
ejpam-4034	378	7	right	right	ADV
ejpam-4034	378	8	ideal	ideal	NOUN
ejpam-4034	378	9	of	of	ADP
ejpam-4034	378	10	s	s	PRON
ejpam-4034	378	11	such	such	ADJ
ejpam-4034	378	12	that	that	SCONJ
ejpam-4034	378	13	y	y	PROPN
ejpam-4034	378	14	⊆	⊆	NUM
ejpam-4034	378	15	s−p	s−p	PROPN
ejpam-4034	378	16	.	.	PUNCT
ejpam-4034	379	1	then	then	ADV
ejpam-4034	379	2	there	there	PRON
ejpam-4034	379	3	exists	exist	VERB
ejpam-4034	379	4	y	y	PROPN
ejpam-4034	379	5	∈	∈	PROPN
ejpam-4034	379	6	y	y	PROPN
ejpam-4034	379	7	such	such	ADJ
ejpam-4034	379	8	that	that	SCONJ
ejpam-4034	379	9	y	y	PROPN
ejpam-4034	379	10	/∈	/∈	PUNCT
ejpam-4034	380	1	p	p	INTJ
ejpam-4034	380	2	.	.	PUNCT
ejpam-4034	381	1	now	now	ADV
ejpam-4034	381	2	we	we	PRON
ejpam-4034	381	3	get	get	VERB
ejpam-4034	381	4	xγy	xγy	PROPN
ejpam-4034	381	5	∈	∈	PROPN
ejpam-4034	381	6	p	p	NOUN
ejpam-4034	381	7	.	.	PUNCT
ejpam-4034	382	1	so	so	ADV
ejpam-4034	382	2	by	by	ADP
ejpam-4034	382	3	hypothesis	hypothesis	NOUN
ejpam-4034	382	4	,	,	PUNCT
ejpam-4034	382	5	x	x	X
ejpam-4034	382	6	∈	∈	PROPN
ejpam-4034	382	7	p	p	NOUN
ejpam-4034	382	8	,	,	PUNCT
ejpam-4034	382	9	for	for	ADP
ejpam-4034	382	10	all	all	DET
ejpam-4034	382	11	x	x	SYM
ejpam-4034	382	12	∈	∈	ADJ
ejpam-4034	382	13	x	x	PUNCT
ejpam-4034	383	1	=	=	NOUN
ejpam-4034	383	2	⇒	⇒	NOUN
ejpam-4034	383	3	x	x	NOUN
ejpam-4034	383	4	⊆	⊆	NUM
ejpam-4034	383	5	p	p	NOUN
ejpam-4034	383	6	.	.	PUNCT
ejpam-4034	384	1	so	so	ADV
ejpam-4034	384	2	p	p	PROPN
ejpam-4034	384	3	is	be	AUX
ejpam-4034	384	4	almost	almost	ADV
ejpam-4034	384	5	-	-	PUNCT
ejpam-4034	384	6	prime	prime	ADJ
ejpam-4034	384	7	left	leave	VERB
ejpam-4034	384	8	ideal	ideal	NOUN
ejpam-4034	384	9	in	in	ADP
ejpam-4034	384	10	s.	s.	PROPN
ejpam-4034	384	11	corollary	corollary	PROPN
ejpam-4034	384	12	5	5	NUM
ejpam-4034	384	13	.	.	PUNCT
ejpam-4034	385	1	let	let	VERB
ejpam-4034	385	2	s	s	PRON
ejpam-4034	385	3	be	be	AUX
ejpam-4034	385	4	a	a	DET
ejpam-4034	385	5	gamma	gamma	NOUN
ejpam-4034	385	6	la	la	PROPN
ejpam-4034	385	7	-	-	PUNCT
ejpam-4034	385	8	semiring	semire	VERB
ejpam-4034	385	9	having	have	VERB
ejpam-4034	385	10	identity	identity	NOUN
ejpam-4034	385	11	and	and	CCONJ
ejpam-4034	385	12	let	let	VERB
ejpam-4034	385	13	x	x	PRON
ejpam-4034	385	14	,	,	PUNCT
ejpam-4034	385	15	y	y	PROPN
ejpam-4034	385	16	∈	∈	PROPN
ejpam-4034	385	17	s	s	PROPN
ejpam-4034	385	18	,	,	PUNCT
ejpam-4034	385	19	γ	γ	PROPN
ejpam-4034	385	20	∈	∈	PROPN
ejpam-4034	385	21	γ	γ	X
ejpam-4034	385	22	.	.	PUNCT
ejpam-4034	386	1	then	then	ADV
ejpam-4034	386	2	a	a	DET
ejpam-4034	386	3	left	left	ADJ
ejpam-4034	386	4	ideal	ideal	NOUN
ejpam-4034	386	5	p	p	NOUN
ejpam-4034	386	6	of	of	ADP
ejpam-4034	386	7	s	s	PRON
ejpam-4034	386	8	is	be	AUX
ejpam-4034	386	9	weakly	weakly	ADJ
ejpam-4034	386	10	almost	almost	ADV
ejpam-4034	386	11	-	-	PUNCT
ejpam-4034	386	12	prime	prime	NOUN
ejpam-4034	386	13	iff	iff	PROPN
ejpam-4034	386	14	0	0	NUM
ejpam-4034	386	15	6=	6=	NUM
ejpam-4034	386	16	xγy	xγy	PROPN
ejpam-4034	387	1	∈	∈	PROPN
ejpam-4034	387	2	p	p	NOUN
ejpam-4034	387	3	implies	imply	VERB
ejpam-4034	387	4	x	x	X
ejpam-4034	387	5	∈	∈	PROPN
ejpam-4034	387	6	p	p	NOUN
ejpam-4034	387	7	or	or	CCONJ
ejpam-4034	387	8	y	y	PROPN
ejpam-4034	387	9	∈	∈	PROPN
ejpam-4034	387	10	p	p	NOUN
ejpam-4034	387	11	.	.	PUNCT
ejpam-4034	388	1	proof	proof	NOUN
ejpam-4034	388	2	.	.	PUNCT
ejpam-4034	389	1	this	this	PRON
ejpam-4034	389	2	follows	follow	VERB
ejpam-4034	389	3	from	from	ADP
ejpam-4034	389	4	theorem	theorem	ADJ
ejpam-4034	389	5	11	11	NUM
ejpam-4034	389	6	theorem	theorem	NOUN
ejpam-4034	389	7	12	12	NUM
ejpam-4034	389	8	.	.	PUNCT
ejpam-4034	390	1	let	let	VERB
ejpam-4034	390	2	s	s	PRON
ejpam-4034	390	3	be	be	AUX
ejpam-4034	390	4	a	a	DET
ejpam-4034	390	5	gamma	gamma	NOUN
ejpam-4034	390	6	la	la	PROPN
ejpam-4034	390	7	-	-	PUNCT
ejpam-4034	390	8	semiring	semiring	NOUN
ejpam-4034	390	9	having	having	AUX
ejpam-4034	390	10	left	leave	VERB
ejpam-4034	390	11	identity	identity	NOUN
ejpam-4034	390	12	and	and	CCONJ
ejpam-4034	390	13	x	x	PART
ejpam-4034	390	14	be	be	AUX
ejpam-4034	390	15	an	an	DET
ejpam-4034	390	16	almostprime	almostprime	NOUN
ejpam-4034	390	17	left	leave	VERB
ejpam-4034	390	18	ideal	ideal	NOUN
ejpam-4034	390	19	of	of	ADP
ejpam-4034	390	20	s.	s.	PROPN
ejpam-4034	390	21	then	then	ADV
ejpam-4034	390	22	(	(	PUNCT
ejpam-4034	390	23	x	x	X
ejpam-4034	390	24	:	:	PUNCT
ejpam-4034	390	25	γ	γ	X
ejpam-4034	390	26	:	:	PUNCT
ejpam-4034	390	27	y	y	PROPN
ejpam-4034	390	28	)	)	PUNCT
ejpam-4034	390	29	is	be	AUX
ejpam-4034	390	30	an	an	DET
ejpam-4034	390	31	almost	almost	ADV
ejpam-4034	390	32	-	-	PUNCT
ejpam-4034	390	33	prime	prime	ADJ
ejpam-4034	390	34	left	leave	VERB
ejpam-4034	390	35	ideal	ideal	NOUN
ejpam-4034	390	36	in	in	ADP
ejpam-4034	390	37	s	s	PROPN
ejpam-4034	390	38	,	,	PUNCT
ejpam-4034	390	39	where	where	SCONJ
ejpam-4034	390	40	y	y	PROPN
ejpam-4034	390	41	⊆	⊆	NUM
ejpam-4034	390	42	s−x	s−x	NOUN
ejpam-4034	390	43	.	.	PUNCT
ejpam-4034	391	1	proof	proof	NOUN
ejpam-4034	391	2	.	.	PUNCT
ejpam-4034	392	1	assume	assume	VERB
ejpam-4034	392	2	that	that	SCONJ
ejpam-4034	392	3	x	x	PRON
ejpam-4034	392	4	is	be	AUX
ejpam-4034	392	5	a	a	DET
ejpam-4034	392	6	almost	almost	ADV
ejpam-4034	392	7	-	-	PUNCT
ejpam-4034	392	8	prime	prime	ADJ
ejpam-4034	392	9	left	left	ADJ
ejpam-4034	392	10	ideal	ideal	NOUN
ejpam-4034	392	11	of	of	ADP
ejpam-4034	392	12	s.	s.	PROPN
ejpam-4034	392	13	by	by	ADP
ejpam-4034	392	14	lemma	lemma	PROPN
ejpam-4034	392	15	7	7	NUM
ejpam-4034	392	16	,	,	PUNCT
ejpam-4034	392	17	we	we	PRON
ejpam-4034	392	18	have	have	VERB
ejpam-4034	392	19	(	(	PUNCT
ejpam-4034	392	20	x	x	X
ejpam-4034	392	21	:	:	PUNCT
ejpam-4034	392	22	γ	γ	X
ejpam-4034	392	23	:	:	PUNCT
ejpam-4034	392	24	y	y	PROPN
ejpam-4034	392	25	)	)	PUNCT
ejpam-4034	392	26	,	,	PUNCT
ejpam-4034	392	27	a	a	DET
ejpam-4034	392	28	left	left	ADJ
ejpam-4034	392	29	ideal	ideal	NOUN
ejpam-4034	392	30	in	in	ADP
ejpam-4034	392	31	s.	s.	PROPN
ejpam-4034	392	32	let	let	VERB
ejpam-4034	392	33	xγy	xγy	PROPN
ejpam-4034	392	34	∈	∈	PROPN
ejpam-4034	392	35	(	(	PUNCT
ejpam-4034	392	36	x	x	X
ejpam-4034	392	37	:	:	PUNCT
ejpam-4034	392	38	γ	γ	X
ejpam-4034	392	39	:	:	PUNCT
ejpam-4034	392	40	y	y	PROPN
ejpam-4034	392	41	)	)	PUNCT
ejpam-4034	392	42	,	,	PUNCT
ejpam-4034	392	43	where	where	SCONJ
ejpam-4034	392	44	x	x	X
ejpam-4034	392	45	,	,	PUNCT
ejpam-4034	392	46	y	y	PROPN
ejpam-4034	392	47	∈	∈	PROPN
ejpam-4034	392	48	s	s	X
ejpam-4034	392	49	and	and	CCONJ
ejpam-4034	392	50	γ	γ	PROPN
ejpam-4034	392	51	∈	∈	PROPN
ejpam-4034	392	52	γ	γ	X
ejpam-4034	392	53	.	.	PUNCT
ejpam-4034	392	54	suppose	suppose	VERB
ejpam-4034	392	55	that	that	SCONJ
ejpam-4034	392	56	y	y	PROPN
ejpam-4034	392	57	/∈	/∈	PUNCT
ejpam-4034	393	1	(	(	PUNCT
ejpam-4034	393	2	x	x	X
ejpam-4034	393	3	:	:	PUNCT
ejpam-4034	393	4	γ	γ	X
ejpam-4034	393	5	:	:	PUNCT
ejpam-4034	393	6	y	y	PROPN
ejpam-4034	393	7	)	)	PUNCT
ejpam-4034	393	8	.	.	PUNCT
ejpam-4034	394	1	since	since	SCONJ
ejpam-4034	394	2	xγy	xγy	PROPN
ejpam-4034	394	3	∈	∈	PROPN
ejpam-4034	394	4	(	(	PUNCT
ejpam-4034	394	5	x	x	X
ejpam-4034	394	6	:	:	PUNCT
ejpam-4034	394	7	γ	γ	X
ejpam-4034	394	8	:	:	PUNCT
ejpam-4034	394	9	y	y	PROPN
ejpam-4034	394	10	)	)	PUNCT
ejpam-4034	394	11	,	,	PUNCT
ejpam-4034	394	12	we	we	PRON
ejpam-4034	394	13	have	have	VERB
ejpam-4034	394	14	(	(	PUNCT
ejpam-4034	394	15	xγy)γy	xγy)γy	PROPN
ejpam-4034	394	16	⊆	⊆	NUM
ejpam-4034	394	17	x.	x.	NOUN
ejpam-4034	394	18	so	so	ADV
ejpam-4034	394	19	by	by	ADP
ejpam-4034	394	20	hypothesis	hypothesis	NOUN
ejpam-4034	394	21	(	(	PUNCT
ejpam-4034	394	22	sγx)γ(yγy	sγx)γ(yγy	PROPN
ejpam-4034	394	23	)	)	PUNCT
ejpam-4034	395	1	=	=	NOUN
ejpam-4034	395	2	sγ	sγ	X
ejpam-4034	395	3	(	(	PUNCT
ejpam-4034	395	4	(	(	PUNCT
ejpam-4034	395	5	xγy)γy	xγy)γy	PROPN
ejpam-4034	395	6	)	)	PUNCT
ejpam-4034	396	1	⊆	⊆	NUM
ejpam-4034	396	2	sγx	sγx	NOUN
ejpam-4034	396	3	⊆	⊆	NUM
ejpam-4034	396	4	x.	x.	NOUN
ejpam-4034	396	5	then	then	ADV
ejpam-4034	396	6	,	,	PUNCT
ejpam-4034	396	7	following	follow	VERB
ejpam-4034	396	8	the	the	DET
ejpam-4034	396	9	definition	definition	NOUN
ejpam-4034	396	10	of	of	ADP
ejpam-4034	396	11	almost	almost	ADV
ejpam-4034	396	12	-	-	PUNCT
ejpam-4034	396	13	prime	prime	NOUN
ejpam-4034	396	14	,	,	PUNCT
ejpam-4034	396	15	we	we	PRON
ejpam-4034	396	16	have	have	VERB
ejpam-4034	396	17	x	x	NOUN
ejpam-4034	396	18	=	=	PUNCT
ejpam-4034	396	19	eγx	eγx	PROPN
ejpam-4034	396	20	∈	∈	NOUN
ejpam-4034	396	21	sγx	sγx	VERB
ejpam-4034	396	22	⊆	⊆	NUM
ejpam-4034	396	23	x	x	PUNCT
ejpam-4034	396	24	or	or	CCONJ
ejpam-4034	396	25	yγy	yγy	PROPN
ejpam-4034	396	26	⊆	⊆	NUM
ejpam-4034	396	27	x	x	NOUN
ejpam-4034	396	28	implies	imply	VERB
ejpam-4034	396	29	that	that	SCONJ
ejpam-4034	396	30	xγs	xγs	PROPN
ejpam-4034	396	31	⊆	⊆	NUM
ejpam-4034	396	32	xγs	xγs	NOUN
ejpam-4034	396	33	⊆	⊆	NUM
ejpam-4034	396	34	x.	x.	NOUN
ejpam-4034	396	35	hence	hence	ADV
ejpam-4034	396	36	(	(	PUNCT
ejpam-4034	396	37	x	x	X
ejpam-4034	396	38	:	:	PUNCT
ejpam-4034	396	39	γ	γ	X
ejpam-4034	396	40	:	:	PUNCT
ejpam-4034	396	41	y	y	PROPN
ejpam-4034	396	42	)	)	PUNCT
ejpam-4034	396	43	is	be	AUX
ejpam-4034	396	44	an	an	DET
ejpam-4034	396	45	almost	almost	ADV
ejpam-4034	396	46	-	-	PUNCT
ejpam-4034	396	47	prime	prime	ADJ
ejpam-4034	396	48	left	leave	VERB
ejpam-4034	396	49	ideal	ideal	NOUN
ejpam-4034	396	50	in	in	ADP
ejpam-4034	396	51	s.	s.	PROPN
ejpam-4034	396	52	corollary	corollary	PROPN
ejpam-4034	396	53	6	6	NUM
ejpam-4034	396	54	.	.	PUNCT
ejpam-4034	397	1	let	let	VERB
ejpam-4034	397	2	s	s	PRON
ejpam-4034	397	3	be	be	AUX
ejpam-4034	397	4	a	a	DET
ejpam-4034	397	5	gamma	gamma	NOUN
ejpam-4034	397	6	la	la	PROPN
ejpam-4034	397	7	-	-	PUNCT
ejpam-4034	397	8	semiring	semiring	NOUN
ejpam-4034	397	9	having	having	AUX
ejpam-4034	397	10	left	leave	VERB
ejpam-4034	397	11	identity	identity	NOUN
ejpam-4034	397	12	and	and	CCONJ
ejpam-4034	397	13	let	let	VERB
ejpam-4034	397	14	x	x	PRON
ejpam-4034	397	15	be	be	AUX
ejpam-4034	397	16	an	an	DET
ejpam-4034	397	17	ideal	ideal	NOUN
ejpam-4034	397	18	of	of	ADP
ejpam-4034	397	19	s.	s.	PROPN
ejpam-4034	397	20	if	if	SCONJ
ejpam-4034	397	21	x	x	PRON
ejpam-4034	397	22	is	be	AUX
ejpam-4034	397	23	a	a	DET
ejpam-4034	397	24	weakly	weakly	ADJ
ejpam-4034	397	25	almost	almost	ADV
ejpam-4034	397	26	-	-	PUNCT
ejpam-4034	397	27	prime	prime	ADJ
ejpam-4034	397	28	left	left	ADJ
ejpam-4034	397	29	ideal	ideal	NOUN
ejpam-4034	397	30	of	of	ADP
ejpam-4034	397	31	s	s	PROPN
ejpam-4034	397	32	,	,	PUNCT
ejpam-4034	397	33	then	then	ADV
ejpam-4034	397	34	(	(	PUNCT
ejpam-4034	397	35	x	x	X
ejpam-4034	397	36	:	:	PUNCT
ejpam-4034	397	37	γ	γ	X
ejpam-4034	397	38	:	:	PUNCT
ejpam-4034	397	39	y	y	PROPN
ejpam-4034	397	40	)	)	PUNCT
ejpam-4034	397	41	,	,	PUNCT
ejpam-4034	397	42	is	be	AUX
ejpam-4034	397	43	a	a	DET
ejpam-4034	397	44	weakly	weakly	ADJ
ejpam-4034	397	45	almost	almost	ADV
ejpam-4034	397	46	-	-	PUNCT
ejpam-4034	397	47	prime	prime	ADJ
ejpam-4034	397	48	left	leave	VERB
ejpam-4034	397	49	ideal	ideal	NOUN
ejpam-4034	397	50	in	in	ADP
ejpam-4034	397	51	s	s	PROPN
ejpam-4034	397	52	,	,	PUNCT
ejpam-4034	398	1	where	where	SCONJ
ejpam-4034	398	2	y	y	PROPN
ejpam-4034	398	3	⊆	⊆	NUM
ejpam-4034	398	4	s	s	NOUN
ejpam-4034	398	5	−x	−x	NOUN
ejpam-4034	398	6	.	.	PUNCT
ejpam-4034	399	1	proof	proof	NOUN
ejpam-4034	399	2	.	.	PUNCT
ejpam-4034	400	1	this	this	PRON
ejpam-4034	400	2	follows	follow	VERB
ejpam-4034	400	3	from	from	ADP
ejpam-4034	400	4	theorem	theorem	ADJ
ejpam-4034	400	5	12	12	NUM
ejpam-4034	400	6	corollary	corollary	ADJ
ejpam-4034	400	7	7	7	NUM
ejpam-4034	400	8	.	.	PUNCT
ejpam-4034	401	1	let	let	VERB
ejpam-4034	401	2	s	s	PRON
ejpam-4034	401	3	be	be	AUX
ejpam-4034	401	4	a	a	DET
ejpam-4034	401	5	gamma	gamma	NOUN
ejpam-4034	401	6	la	la	PROPN
ejpam-4034	401	7	-	-	PUNCT
ejpam-4034	401	8	semiring	semiring	NOUN
ejpam-4034	401	9	having	having	AUX
ejpam-4034	401	10	left	leave	VERB
ejpam-4034	401	11	identity	identity	NOUN
ejpam-4034	401	12	and	and	CCONJ
ejpam-4034	401	13	let	let	VERB
ejpam-4034	401	14	x	x	PRON
ejpam-4034	401	15	be	be	AUX
ejpam-4034	401	16	an	an	DET
ejpam-4034	401	17	ideal	ideal	NOUN
ejpam-4034	401	18	of	of	ADP
ejpam-4034	401	19	s.	s.	PROPN
ejpam-4034	401	20	if	if	SCONJ
ejpam-4034	401	21	x	x	PRON
ejpam-4034	401	22	is	be	AUX
ejpam-4034	401	23	an	an	DET
ejpam-4034	401	24	almost	almost	ADV
ejpam-4034	401	25	-	-	PUNCT
ejpam-4034	401	26	prime	prime	ADJ
ejpam-4034	401	27	left	left	ADJ
ejpam-4034	401	28	ideal	ideal	NOUN
ejpam-4034	401	29	of	of	ADP
ejpam-4034	401	30	s	s	PROPN
ejpam-4034	401	31	,	,	PUNCT
ejpam-4034	401	32	then	then	ADV
ejpam-4034	401	33	(	(	PUNCT
ejpam-4034	401	34	x	x	X
ejpam-4034	401	35	:	:	PUNCT
ejpam-4034	401	36	γ	γ	X
ejpam-4034	401	37	:	:	PUNCT
ejpam-4034	401	38	s	s	X
ejpam-4034	401	39	)	)	PUNCT
ejpam-4034	401	40	is	be	AUX
ejpam-4034	401	41	an	an	DET
ejpam-4034	401	42	almost	almost	ADV
ejpam-4034	401	43	-	-	PUNCT
ejpam-4034	401	44	prime	prime	ADJ
ejpam-4034	401	45	left	leave	VERB
ejpam-4034	401	46	ideal	ideal	NOUN
ejpam-4034	401	47	in	in	ADP
ejpam-4034	401	48	s	s	PROPN
ejpam-4034	401	49	,	,	PUNCT
ejpam-4034	401	50	where	where	SCONJ
ejpam-4034	401	51	s	s	VERB
ejpam-4034	401	52	∈	∈	PROPN
ejpam-4034	401	53	s	s	VERB
ejpam-4034	401	54	−x	−x	NOUN
ejpam-4034	401	55	and	and	CCONJ
ejpam-4034	401	56	γ	γ	PROPN
ejpam-4034	401	57	∈	∈	PROPN
ejpam-4034	401	58	γ	γ	X
ejpam-4034	401	59	.	.	PUNCT
ejpam-4034	402	1	proof	proof	NOUN
ejpam-4034	402	2	.	.	PUNCT
ejpam-4034	403	1	this	this	PRON
ejpam-4034	403	2	follows	follow	VERB
ejpam-4034	403	3	from	from	ADP
ejpam-4034	403	4	theorem	theorem	ADJ
ejpam-4034	403	5	12	12	NUM
ejpam-4034	403	6	corollary	corollary	ADJ
ejpam-4034	403	7	8	8	NUM
ejpam-4034	403	8	.	.	PUNCT
ejpam-4034	404	1	let	let	VERB
ejpam-4034	404	2	s	s	PRON
ejpam-4034	404	3	be	be	AUX
ejpam-4034	404	4	a	a	DET
ejpam-4034	404	5	gamma	gamma	NOUN
ejpam-4034	404	6	la	la	PROPN
ejpam-4034	404	7	-	-	PUNCT
ejpam-4034	404	8	semiring	semiring	NOUN
ejpam-4034	404	9	with	with	ADP
ejpam-4034	404	10	left	left	ADJ
ejpam-4034	404	11	identity	identity	NOUN
ejpam-4034	404	12	and	and	CCONJ
ejpam-4034	404	13	let	let	VERB
ejpam-4034	404	14	x	x	PRON
ejpam-4034	404	15	be	be	AUX
ejpam-4034	404	16	an	an	DET
ejpam-4034	404	17	ideal	ideal	NOUN
ejpam-4034	404	18	of	of	ADP
ejpam-4034	404	19	s.	s.	PROPN
ejpam-4034	404	20	if	if	SCONJ
ejpam-4034	404	21	x	x	PRON
ejpam-4034	404	22	is	be	AUX
ejpam-4034	404	23	a	a	DET
ejpam-4034	404	24	weakly	weakly	ADJ
ejpam-4034	404	25	almost	almost	ADV
ejpam-4034	404	26	-	-	PUNCT
ejpam-4034	404	27	prime	prime	ADJ
ejpam-4034	404	28	left	left	ADJ
ejpam-4034	404	29	ideal	ideal	NOUN
ejpam-4034	404	30	of	of	ADP
ejpam-4034	404	31	s	s	PROPN
ejpam-4034	404	32	,	,	PUNCT
ejpam-4034	404	33	then	then	ADV
ejpam-4034	404	34	(	(	PUNCT
ejpam-4034	404	35	x	x	X
ejpam-4034	404	36	:	:	PUNCT
ejpam-4034	404	37	γ	γ	X
ejpam-4034	404	38	:	:	PUNCT
ejpam-4034	404	39	s	s	X
ejpam-4034	404	40	)	)	PUNCT
ejpam-4034	404	41	,	,	PUNCT
ejpam-4034	404	42	is	be	AUX
ejpam-4034	404	43	a	a	DET
ejpam-4034	404	44	weakly	weakly	ADJ
ejpam-4034	404	45	almost	almost	ADV
ejpam-4034	404	46	-	-	PUNCT
ejpam-4034	404	47	prime	prime	ADJ
ejpam-4034	404	48	left	leave	VERB
ejpam-4034	404	49	ideal	ideal	NOUN
ejpam-4034	404	50	in	in	ADP
ejpam-4034	404	51	s	s	PROPN
ejpam-4034	404	52	,	,	PUNCT
ejpam-4034	404	53	where	where	SCONJ
ejpam-4034	404	54	s	s	VERB
ejpam-4034	404	55	∈	∈	PROPN
ejpam-4034	404	56	s	s	VERB
ejpam-4034	404	57	−x	−x	NOUN
ejpam-4034	404	58	and	and	CCONJ
ejpam-4034	404	59	γ	γ	PROPN
ejpam-4034	404	60	∈	∈	PROPN
ejpam-4034	404	61	γ	γ	X
ejpam-4034	404	62	.	.	PUNCT
ejpam-4034	405	1	proof	proof	NOUN
ejpam-4034	405	2	.	.	PUNCT
ejpam-4034	406	1	this	this	PRON
ejpam-4034	406	2	follows	follow	VERB
ejpam-4034	406	3	from	from	ADP
ejpam-4034	406	4	corollary	corollary	ADJ
ejpam-4034	406	5	5	5	NUM
ejpam-4034	406	6	references	reference	NOUN
ejpam-4034	406	7	1000	1000	NUM
ejpam-4034	406	8	4	4	NUM
ejpam-4034	406	9	.	.	PUNCT
ejpam-4034	406	10	conclusion	conclusion	NOUN
ejpam-4034	406	11	in	in	ADP
ejpam-4034	406	12	this	this	DET
ejpam-4034	406	13	manuscript	manuscript	NOUN
ejpam-4034	406	14	,	,	PUNCT
ejpam-4034	406	15	firstly	firstly	ADV
ejpam-4034	406	16	we	we	PRON
ejpam-4034	406	17	have	have	AUX
ejpam-4034	406	18	added	add	VERB
ejpam-4034	406	19	some	some	DET
ejpam-4034	406	20	fresh	fresh	ADJ
ejpam-4034	406	21	examples	example	NOUN
ejpam-4034	406	22	along	along	ADP
ejpam-4034	406	23	with	with	ADP
ejpam-4034	406	24	new	new	ADJ
ejpam-4034	406	25	results	result	NOUN
ejpam-4034	406	26	in	in	ADP
ejpam-4034	406	27	γ	γ	PROPN
ejpam-4034	406	28	-la	-la	NOUN
ejpam-4034	406	29	-	-	PUNCT
ejpam-4034	406	30	rings	ring	NOUN
ejpam-4034	406	31	.	.	PUNCT
ejpam-4034	407	1	next	next	ADV
ejpam-4034	407	2	,	,	PUNCT
ejpam-4034	407	3	we	we	PRON
ejpam-4034	407	4	introduced	introduce	VERB
ejpam-4034	407	5	the	the	DET
ejpam-4034	407	6	notion	notion	NOUN
ejpam-4034	407	7	of	of	ADP
ejpam-4034	407	8	γ	γ	PROPN
ejpam-4034	407	9	-la	-la	NOUN
ejpam-4034	407	10	-	-	PUNCT
ejpam-4034	407	11	semirings	semiring	NOUN
ejpam-4034	407	12	and	and	CCONJ
ejpam-4034	407	13	discussed	discuss	VERB
ejpam-4034	407	14	different	different	ADJ
ejpam-4034	407	15	types	type	NOUN
ejpam-4034	407	16	of	of	ADP
ejpam-4034	407	17	ideals	ideal	NOUN
ejpam-4034	407	18	in	in	ADP
ejpam-4034	407	19	γ	γ	PROPN
ejpam-4034	407	20	-la	-la	NOUN
ejpam-4034	407	21	-	-	PUNCT
ejpam-4034	407	22	semirings	semiring	NOUN
ejpam-4034	407	23	.	.	PUNCT
ejpam-4034	408	1	we	we	PRON
ejpam-4034	408	2	examined	examine	VERB
ejpam-4034	408	3	that	that	SCONJ
ejpam-4034	408	4	almost	almost	ADV
ejpam-4034	408	5	all	all	DET
ejpam-4034	408	6	the	the	DET
ejpam-4034	408	7	results	result	NOUN
ejpam-4034	408	8	of	of	ADP
ejpam-4034	408	9	la	la	NOUN
ejpam-4034	408	10	-	-	PUNCT
ejpam-4034	408	11	rings	ring	NOUN
ejpam-4034	408	12	and	and	CCONJ
ejpam-4034	408	13	la	la	ADJ
ejpam-4034	408	14	-	-	PUNCT
ejpam-4034	408	15	semirings	semiring	NOUN
ejpam-4034	408	16	are	be	AUX
ejpam-4034	408	17	valid	valid	ADJ
ejpam-4034	408	18	in	in	ADP
ejpam-4034	408	19	case	case	NOUN
ejpam-4034	408	20	of	of	ADP
ejpam-4034	408	21	γ	γ	PROPN
ejpam-4034	408	22	-la	-la	NOUN
ejpam-4034	408	23	-	-	PUNCT
ejpam-4034	408	24	rings	ring	NOUN
ejpam-4034	408	25	and	and	CCONJ
ejpam-4034	408	26	γ	γ	PROPN
ejpam-4034	408	27	-la	-la	NOUN
ejpam-4034	408	28	-	-	PUNCT
ejpam-4034	408	29	semirings	semiring	NOUN
ejpam-4034	408	30	.	.	PUNCT
ejpam-4034	409	1	one	one	PRON
ejpam-4034	409	2	could	could	AUX
ejpam-4034	409	3	extend	extend	VERB
ejpam-4034	409	4	this	this	DET
ejpam-4034	409	5	work	work	NOUN
ejpam-4034	409	6	by	by	ADP
ejpam-4034	409	7	shifting	shift	VERB
ejpam-4034	409	8	our	our	PRON
ejpam-4034	409	9	results	result	NOUN
ejpam-4034	409	10	towards	towards	ADP
ejpam-4034	409	11	the	the	DET
ejpam-4034	409	12	theory	theory	NOUN
ejpam-4034	409	13	of	of	ADP
ejpam-4034	409	14	γ	γ	PROPN
ejpam-4034	409	15	-la	-la	NOUN
ejpam-4034	409	16	-	-	PUNCT
ejpam-4034	409	17	nearrings	nearring	NOUN
ejpam-4034	409	18	,	,	PUNCT
ejpam-4034	409	19	γ	γ	NOUN
ejpam-4034	409	20	-la	-la	NOUN
ejpam-4034	409	21	-	-	PUNCT
ejpam-4034	409	22	hemirings	hemiring	NOUN
ejpam-4034	409	23	etc	etc	X
ejpam-4034	409	24	.	.	X
ejpam-4034	409	25	acknowledgements	acknowledgement	VERB
ejpam-4034	409	26	some	some	DET
ejpam-4034	409	27	results	result	NOUN
ejpam-4034	409	28	presented	present	VERB
ejpam-4034	409	29	in	in	ADP
ejpam-4034	409	30	this	this	DET
ejpam-4034	409	31	article	article	NOUN
ejpam-4034	409	32	are	be	AUX
ejpam-4034	409	33	extracted	extract	VERB
ejpam-4034	409	34	from	from	ADP
ejpam-4034	409	35	ms	ms	ADJ
ejpam-4034	409	36	thesis	thesis	NOUN
ejpam-4034	409	37	of	of	ADP
ejpam-4034	409	38	fourth	fourth	ADJ
ejpam-4034	409	39	author	author	NOUN
ejpam-4034	409	40	(	(	PUNCT
ejpam-4034	409	41	zahid	zahid	PROPN
ejpam-4034	409	42	hussain	hussain	PROPN
ejpam-4034	409	43	)	)	PUNCT
ejpam-4034	409	44	which	which	PRON
ejpam-4034	409	45	was	be	AUX
ejpam-4034	409	46	written	write	VERB
ejpam-4034	409	47	under	under	ADP
ejpam-4034	409	48	the	the	DET
ejpam-4034	409	49	supervision	supervision	NOUN
ejpam-4034	409	50	of	of	ADP
ejpam-4034	409	51	the	the	DET
ejpam-4034	409	52	first	first	ADJ
ejpam-4034	409	53	author	author	NOUN
ejpam-4034	409	54	(	(	PUNCT
ejpam-4034	409	55	waheed	waheed	PROPN
ejpam-4034	409	56	ahmad	ahmad	PROPN
ejpam-4034	409	57	khan	khan	PROPN
ejpam-4034	409	58	)	)	PUNCT
ejpam-4034	409	59	and	and	CCONJ
ejpam-4034	409	60	was	be	AUX
ejpam-4034	409	61	submitted	submit	VERB
ejpam-4034	409	62	to	to	ADP
ejpam-4034	409	63	higher	high	ADJ
ejpam-4034	409	64	education	education	NOUN
ejpam-4034	409	65	of	of	ADP
ejpam-4034	409	66	pakistan	pakistan	PROPN
ejpam-4034	409	67	.	.	PUNCT
ejpam-4034	410	1	references	reference	NOUN
ejpam-4034	410	2	[	[	X
ejpam-4034	410	3	1	1	NUM
ejpam-4034	410	4	]	]	PUNCT
ejpam-4034	410	5	w.	w.	PROPN
ejpam-4034	410	6	barnes	barnes	PROPN
ejpam-4034	410	7	.	.	PUNCT
ejpam-4034	411	1	on	on	ADP
ejpam-4034	411	2	the	the	DET
ejpam-4034	411	3	gamma	gamma	NOUN
ejpam-4034	411	4	-	-	PUNCT
ejpam-4034	411	5	rings	ring	NOUN
ejpam-4034	411	6	of	of	ADP
ejpam-4034	411	7	nobusawa	nobusawa	PROPN
ejpam-4034	411	8	.	.	PUNCT
ejpam-4034	411	9	pacific	pacific	PROPN
ejpam-4034	411	10	journal	journal	PROPN
ejpam-4034	411	11	of	of	ADP
ejpam-4034	411	12	mathematics	mathematic	NOUN
ejpam-4034	411	13	,	,	PUNCT
ejpam-4034	411	14	18(3):411–422	18(3):411–422	NUM
ejpam-4034	411	15	,	,	PUNCT
ejpam-4034	411	16	1966	1966	NUM
ejpam-4034	411	17	.	.	PUNCT
ejpam-4034	412	1	[	[	X
ejpam-4034	412	2	2	2	X
ejpam-4034	412	3	]	]	X
ejpam-4034	412	4	g.	g.	NOUN
ejpam-4034	412	5	booth	booth	PROPN
ejpam-4034	412	6	and	and	CCONJ
ejpam-4034	412	7	n.	n.	PROPN
ejpam-4034	412	8	groenewald	groenewald	PROPN
ejpam-4034	412	9	.	.	PUNCT
ejpam-4034	413	1	equiprime	equiprime	PROPN
ejpam-4034	413	2	gamma	gamma	NOUN
ejpam-4034	413	3	-	-	PUNCT
ejpam-4034	413	4	near	near	ADP
ejpam-4034	413	5	-	-	PUNCT
ejpam-4034	413	6	rings	ring	NOUN
ejpam-4034	413	7	.	.	PUNCT
ejpam-4034	414	1	quaestiones	quaestione	NOUN
ejpam-4034	414	2	mathematicae	mathematicae	PROPN
ejpam-4034	414	3	,	,	PUNCT
ejpam-4034	414	4	14:411–417	14:411–417	NUM
ejpam-4034	414	5	,	,	PUNCT
ejpam-4034	414	6	1991	1991	NUM
ejpam-4034	414	7	.	.	PUNCT
ejpam-4034	415	1	[	[	X
ejpam-4034	415	2	3	3	X
ejpam-4034	415	3	]	]	X
ejpam-4034	415	4	j.	j.	PROPN
ejpam-4034	415	5	r.	r.	PROPN
ejpam-4034	415	6	cho	cho	PROPN
ejpam-4034	415	7	,	,	PUNCT
ejpam-4034	415	8	j.	j.	PROPN
ejpam-4034	415	9	jezek	jezek	PROPN
ejpam-4034	415	10	pusan	pusan	PROPN
ejpam-4034	415	11	,	,	PUNCT
ejpam-4034	415	12	and	and	CCONJ
ejpam-4034	415	13	t.	t.	PROPN
ejpam-4034	415	14	kepka	kepka	NOUN
ejpam-4034	415	15	.	.	PUNCT
ejpam-4034	416	1	praha	praha	PROPN
ejpam-4034	416	2	,	,	PUNCT
ejpam-4034	416	3	paramedial	paramedial	ADJ
ejpam-4034	416	4	groupoids	groupoid	NOUN
ejpam-4034	416	5	.	.	PUNCT
ejpam-4034	417	1	czechoslovak	czechoslovak	ADJ
ejpam-4034	417	2	mathematical	mathematical	PROPN
ejpam-4034	417	3	journal	journal	PROPN
ejpam-4034	417	4	,	,	PUNCT
ejpam-4034	417	5	49:124	49:124	NUM
ejpam-4034	417	6	,	,	PUNCT
ejpam-4034	417	7	1996	1996	NUM
ejpam-4034	417	8	.	.	PUNCT
ejpam-4034	418	1	[	[	X
ejpam-4034	418	2	4	4	X
ejpam-4034	418	3	]	]	X
ejpam-4034	418	4	d.	d.	PROPN
ejpam-4034	418	5	m.	m.	PROPN
ejpam-4034	418	6	devi	devi	PROPN
ejpam-4034	418	7	and	and	CCONJ
ejpam-4034	418	8	g.	g.	PROPN
ejpam-4034	418	9	s.	s.	PROPN
ejpam-4034	418	10	latha	latha	PROPN
ejpam-4034	418	11	.	.	PUNCT
ejpam-4034	419	1	la	la	ADJ
ejpam-4034	419	2	–	–	PUNCT
ejpam-4034	419	3	semirings	semiring	NOUN
ejpam-4034	419	4	satisfying	satisfy	VERB
ejpam-4034	419	5	the	the	DET
ejpam-4034	419	6	identity	identity	NOUN
ejpam-4034	419	7	ab=	ab=	NOUN
ejpam-4034	419	8	a+	a+	PUNCT
ejpam-4034	419	9	b+	b+	NUM
ejpam-4034	419	10	1	1	X
ejpam-4034	419	11	.	.	PUNCT
ejpam-4034	419	12	international	international	ADJ
ejpam-4034	419	13	journal	journal	NOUN
ejpam-4034	419	14	of	of	ADP
ejpam-4034	419	15	innovative	innovative	ADJ
ejpam-4034	419	16	science	science	NOUN
ejpam-4034	419	17	,	,	PUNCT
ejpam-4034	419	18	engineering	engineering	NOUN
ejpam-4034	419	19	&	&	CCONJ
ejpam-4034	419	20	technology	technology	NOUN
ejpam-4034	419	21	,	,	PUNCT
ejpam-4034	419	22	2:378–389	2:378–389	NUM
ejpam-4034	419	23	,	,	PUNCT
ejpam-4034	419	24	2015	2015	NUM
ejpam-4034	419	25	.	.	PUNCT
ejpam-4034	420	1	[	[	X
ejpam-4034	420	2	5	5	X
ejpam-4034	420	3	]	]	PUNCT
ejpam-4034	420	4	t.	t.	PROPN
ejpam-4034	420	5	k.	k.	PROPN
ejpam-4034	420	6	dutta	dutta	PROPN
ejpam-4034	420	7	and	and	CCONJ
ejpam-4034	420	8	s.	s.	PROPN
ejpam-4034	420	9	k.	k.	PROPN
ejpam-4034	420	10	sardar	sardar	PROPN
ejpam-4034	420	11	.	.	PUNCT
ejpam-4034	421	1	on	on	ADP
ejpam-4034	421	2	prime	prime	ADJ
ejpam-4034	421	3	ideals	ideal	NOUN
ejpam-4034	421	4	and	and	CCONJ
ejpam-4034	421	5	prime	prime	ADJ
ejpam-4034	421	6	radicals	radical	NOUN
ejpam-4034	421	7	of	of	ADP
ejpam-4034	421	8	a	a	DET
ejpam-4034	421	9	gammasemirings	gammasemiring	NOUN
ejpam-4034	421	10	.	.	PUNCT
ejpam-4034	422	1	an	an	DET
ejpam-4034	422	2	.	.	NOUN
ejpam-4034	422	3	stiint	stiint	PROPN
ejpam-4034	422	4	.	.	PUNCT
ejpam-4034	423	1	univ	univ	PROPN
ejpam-4034	423	2	.	.	PUNCT
ejpam-4034	424	1	al	al	PROPN
ejpam-4034	424	2	.	.	PROPN
ejpam-4034	424	3	i.	i.	PROPN
ejpam-4034	424	4	cuza	cuza	PROPN
ejpam-4034	424	5	iasi	iasi	PROPN
ejpam-4034	424	6	,	,	PUNCT
ejpam-4034	424	7	mat.(ns	mat.(ns	NOUN
ejpam-4034	424	8	)	)	PUNCT
ejpam-4034	424	9	,	,	PUNCT
ejpam-4034	424	10	46:319–329	46:319–329	PROPN
ejpam-4034	424	11	,	,	PUNCT
ejpam-4034	424	12	2000	2000	NUM
ejpam-4034	424	13	.	.	PUNCT
ejpam-4034	425	1	[	[	X
ejpam-4034	425	2	6	6	NUM
ejpam-4034	425	3	]	]	PUNCT
ejpam-4034	425	4	t.	t.	PROPN
ejpam-4034	425	5	k.	k.	PROPN
ejpam-4034	425	6	dutta	dutta	PROPN
ejpam-4034	425	7	and	and	CCONJ
ejpam-4034	425	8	s.	s.	PROPN
ejpam-4034	425	9	k.	k.	PROPN
ejpam-4034	425	10	sardar	sardar	PROPN
ejpam-4034	425	11	.	.	PUNCT
ejpam-4034	426	1	semiprime	semiprime	NOUN
ejpam-4034	426	2	ideals	ideal	NOUN
ejpam-4034	426	3	and	and	CCONJ
ejpam-4034	426	4	irreducible	irreducible	ADJ
ejpam-4034	426	5	ideals	ideal	NOUN
ejpam-4034	426	6	of	of	ADP
ejpam-4034	426	7	r	r	NOUN
ejpam-4034	426	8	-	-	PUNCT
ejpam-4034	426	9	semirings	semiring	NOUN
ejpam-4034	426	10	.	.	PUNCT
ejpam-4034	427	1	novi	novi	PROPN
ejpam-4034	427	2	sad	sad	PROPN
ejpam-4034	427	3	j.	j.	PROPN
ejpam-4034	427	4	math	math	PROPN
ejpam-4034	427	5	,	,	PUNCT
ejpam-4034	427	6	30(1):97–108	30(1):97–108	NUM
ejpam-4034	427	7	,	,	PUNCT
ejpam-4034	427	8	2000	2000	NUM
ejpam-4034	427	9	.	.	PUNCT
ejpam-4034	428	1	[	[	X
ejpam-4034	428	2	7	7	X
ejpam-4034	428	3	]	]	X
ejpam-4034	428	4	r.	r.	PROPN
ejpam-4034	428	5	d.	d.	PROPN
ejpam-4034	428	6	jagatap	jagatap	PROPN
ejpam-4034	428	7	and	and	CCONJ
ejpam-4034	428	8	y.	y.	PROPN
ejpam-4034	428	9	s.	s.	PROPN
ejpam-4034	428	10	pawar	pawar	PROPN
ejpam-4034	428	11	.	.	PUNCT
ejpam-4034	429	1	quasi	quasi	ADJ
ejpam-4034	429	2	-	-	NOUN
ejpam-4034	429	3	ideals	ideal	NOUN
ejpam-4034	429	4	and	and	CCONJ
ejpam-4034	429	5	minimal	minimal	ADJ
ejpam-4034	429	6	quasi	quasi	NOUN
ejpam-4034	429	7	-	-	NOUN
ejpam-4034	429	8	ideals	ideal	NOUN
ejpam-4034	429	9	in	in	ADP
ejpam-4034	429	10	gammasemirings	gammasemiring	NOUN
ejpam-4034	429	11	.	.	PUNCT
ejpam-4034	430	1	novi	novi	PROPN
ejpam-4034	430	2	sad	sad	PROPN
ejpam-4034	430	3	j.	j.	PROPN
ejpam-4034	430	4	math	math	PROPN
ejpam-4034	430	5	,	,	PUNCT
ejpam-4034	430	6	39(2):79–87	39(2):79–87	NUM
ejpam-4034	430	7	,	,	PUNCT
ejpam-4034	430	8	2009	2009	NUM
ejpam-4034	430	9	.	.	PUNCT
ejpam-4034	431	1	[	[	X
ejpam-4034	431	2	8	8	NUM
ejpam-4034	431	3	]	]	X
ejpam-4034	431	4	r.	r.	PROPN
ejpam-4034	431	5	d.	d.	PROPN
ejpam-4034	431	6	jagatap	jagatap	PROPN
ejpam-4034	431	7	and	and	CCONJ
ejpam-4034	431	8	y.	y.	PROPN
ejpam-4034	431	9	s.	s.	PROPN
ejpam-4034	431	10	pawar	pawar	PROPN
ejpam-4034	431	11	.	.	PUNCT
ejpam-4034	432	1	quasi	quasi	ADJ
ejpam-4034	432	2	-	-	NOUN
ejpam-4034	432	3	ideals	ideal	NOUN
ejpam-4034	432	4	in	in	ADP
ejpam-4034	432	5	regular	regular	ADJ
ejpam-4034	432	6	gamma	gamma	NOUN
ejpam-4034	432	7	-	-	PUNCT
ejpam-4034	432	8	semirings	semiring	NOUN
ejpam-4034	432	9	.	.	PUNCT
ejpam-4034	433	1	bulletin	bulletin	NOUN
ejpam-4034	433	2	of	of	ADP
ejpam-4034	433	3	kerala	kerala	PROPN
ejpam-4034	433	4	mathematics	mathematics	PROPN
ejpam-4034	433	5	association	association	PROPN
ejpam-4034	433	6	,	,	PUNCT
ejpam-4034	433	7	7(2):51–61	7(2):51–61	NUM
ejpam-4034	433	8	,	,	PUNCT
ejpam-4034	433	9	2010	2010	NUM
ejpam-4034	433	10	.	.	PUNCT
ejpam-4034	434	1	[	[	X
ejpam-4034	434	2	9	9	NUM
ejpam-4034	434	3	]	]	PUNCT
ejpam-4034	434	4	waheed	waheed	PROPN
ejpam-4034	434	5	ahmad	ahmad	PROPN
ejpam-4034	434	6	khan	khan	PROPN
ejpam-4034	434	7	,	,	PUNCT
ejpam-4034	434	8	abdelghani	abdelghani	PROPN
ejpam-4034	434	9	taouti	taouti	PROPN
ejpam-4034	434	10	,	,	PUNCT
ejpam-4034	434	11	seema	seema	PROPN
ejpam-4034	434	12	karkain	karkain	PROPN
ejpam-4034	434	13	,	,	PUNCT
ejpam-4034	434	14	azar	azar	PROPN
ejpam-4034	434	15	salami	salami	NOUN
ejpam-4034	434	16	,	,	PUNCT
ejpam-4034	434	17	and	and	CCONJ
ejpam-4034	434	18	waqar	waqar	PROPN
ejpam-4034	434	19	arif	arif	PROPN
ejpam-4034	434	20	.	.	PUNCT
ejpam-4034	435	1	weakly	weakly	ADJ
ejpam-4034	435	2	prime	prime	ADJ
ejpam-4034	435	3	and	and	CCONJ
ejpam-4034	435	4	weakly	weakly	ADJ
ejpam-4034	435	5	primary	primary	ADJ
ejpam-4034	435	6	ideals	ideal	NOUN
ejpam-4034	435	7	in	in	ADP
ejpam-4034	435	8	gamma	gamma	NOUN
ejpam-4034	435	9	seminearrings	seminearring	NOUN
ejpam-4034	435	10	.	.	PUNCT
ejpam-4034	436	1	european	european	ADJ
ejpam-4034	436	2	journal	journal	PROPN
ejpam-4034	436	3	of	of	ADP
ejpam-4034	436	4	pure	pure	ADJ
ejpam-4034	436	5	and	and	CCONJ
ejpam-4034	436	6	applied	applied	ADJ
ejpam-4034	436	7	mathematics	mathematic	NOUN
ejpam-4034	436	8	,	,	PUNCT
ejpam-4034	436	9	12(2):544–552	12(2):544–552	NUM
ejpam-4034	436	10	,	,	PUNCT
ejpam-4034	436	11	2019	2019	NUM
ejpam-4034	436	12	.	.	PUNCT
ejpam-4034	437	1	references	reference	NOUN
ejpam-4034	437	2	1001	1001	NUM
ejpam-4034	438	1	[	[	X
ejpam-4034	438	2	10	10	NUM
ejpam-4034	438	3	]	]	X
ejpam-4034	438	4	q.	q.	PROPN
ejpam-4034	438	5	mushtaq	mushtaq	PROPN
ejpam-4034	438	6	and	and	CCONJ
ejpam-4034	438	7	m.	m.	PROPN
ejpam-4034	438	8	khan	khan	PROPN
ejpam-4034	438	9	.	.	PUNCT
ejpam-4034	439	1	ideals	ideal	NOUN
ejpam-4034	439	2	in	in	ADP
ejpam-4034	439	3	la	la	NOUN
ejpam-4034	439	4	-	-	PUNCT
ejpam-4034	439	5	semigroups	semigroup	NOUN
ejpam-4034	439	6	.	.	PUNCT
ejpam-4034	440	1	in	in	ADP
ejpam-4034	440	2	proc	proc	NOUN
ejpam-4034	440	3	.	.	PUNCT
ejpam-4034	441	1	of	of	ADP
ejpam-4034	441	2	4th	4th	ADJ
ejpam-4034	441	3	international	international	ADJ
ejpam-4034	441	4	pure	pure	ADJ
ejpam-4034	441	5	mathematics	mathematic	NOUN
ejpam-4034	441	6	conference	conference	NOUN
ejpam-4034	441	7	,	,	PUNCT
ejpam-4034	441	8	2003	2003	NUM
ejpam-4034	442	1	.	.	PUNCT
ejpam-4034	443	1	[	[	X
ejpam-4034	443	2	11	11	NUM
ejpam-4034	443	3	]	]	X
ejpam-4034	443	4	q.	q.	PROPN
ejpam-4034	443	5	mushtaq	mushtaq	PROPN
ejpam-4034	443	6	and	and	CCONJ
ejpam-4034	443	7	s.	s.	PROPN
ejpam-4034	443	8	m.	m.	PROPN
ejpam-4034	443	9	yusuf	yusuf	PROPN
ejpam-4034	443	10	.	.	PUNCT
ejpam-4034	444	1	on	on	ADP
ejpam-4034	444	2	la	la	ADJ
ejpam-4034	444	3	-	-	PUNCT
ejpam-4034	444	4	semi	semi	ADJ
ejpam-4034	444	5	groups	group	NOUN
ejpam-4034	444	6	.	.	PUNCT
ejpam-4034	445	1	the	the	DET
ejpam-4034	445	2	alig	alig	PROPN
ejpam-4034	445	3	.	.	PUNCT
ejpam-4034	446	1	bull.math	bull.math	NOUN
ejpam-4034	446	2	.	.	PUNCT
ejpam-4034	446	3	,	,	PUNCT
ejpam-4034	446	4	8:65–70	8:65–70	PROPN
ejpam-4034	446	5	,	,	PUNCT
ejpam-4034	446	6	1978	1978	NUM
ejpam-4034	446	7	.	.	PUNCT
ejpam-4034	447	1	[	[	X
ejpam-4034	447	2	12	12	NUM
ejpam-4034	447	3	]	]	X
ejpam-4034	447	4	q.	q.	PROPN
ejpam-4034	447	5	mushtaq	mushtaq	PROPN
ejpam-4034	447	6	and	and	CCONJ
ejpam-4034	447	7	s.	s.	PROPN
ejpam-4034	447	8	m.	m.	PROPN
ejpam-4034	447	9	yusuf	yusuf	PROPN
ejpam-4034	447	10	.	.	PUNCT
ejpam-4034	448	1	on	on	ADP
ejpam-4034	448	2	locally	locally	ADV
ejpam-4034	448	3	associative	associative	ADJ
ejpam-4034	448	4	la	la	NOUN
ejpam-4034	448	5	-	-	PUNCT
ejpam-4034	448	6	semigroups	semigroup	NOUN
ejpam-4034	448	7	.	.	PUNCT
ejpam-4034	449	1	j.	j.	PROPN
ejpam-4034	449	2	nat	nat	PROPN
ejpam-4034	449	3	.	.	PUNCT
ejpam-4034	450	1	sci	sci	PROPN
ejpam-4034	450	2	.	.	PROPN
ejpam-4034	450	3	math	math	PROPN
ejpam-4034	450	4	,	,	PUNCT
ejpam-4034	450	5	19(1):57–62	19(1):57–62	NUM
ejpam-4034	450	6	,	,	PUNCT
ejpam-4034	450	7	1979	1979	NUM
ejpam-4034	450	8	.	.	PUNCT
ejpam-4034	451	1	[	[	X
ejpam-4034	451	2	13	13	NUM
ejpam-4034	451	3	]	]	X
ejpam-4034	451	4	n.	n.	PROPN
ejpam-4034	451	5	nobusawa	nobusawa	PROPN
ejpam-4034	451	6	.	.	PUNCT
ejpam-4034	452	1	on	on	ADP
ejpam-4034	452	2	a	a	DET
ejpam-4034	452	3	generalization	generalization	NOUN
ejpam-4034	452	4	of	of	ADP
ejpam-4034	452	5	the	the	DET
ejpam-4034	452	6	ring	ring	NOUN
ejpam-4034	452	7	theory	theory	NOUN
ejpam-4034	452	8	.	.	PUNCT
ejpam-4034	453	1	osaka	osaka	PROPN
ejpam-4034	453	2	journal	journal	PROPN
ejpam-4034	453	3	of	of	ADP
ejpam-4034	453	4	mathematics	mathematic	NOUN
ejpam-4034	453	5	,	,	PUNCT
ejpam-4034	453	6	1(1):81–89	1(1):81–89	NUM
ejpam-4034	453	7	,	,	PUNCT
ejpam-4034	453	8	1964	1964	NUM
ejpam-4034	453	9	.	.	PUNCT
ejpam-4034	454	1	[	[	X
ejpam-4034	454	2	14	14	NUM
ejpam-4034	454	3	]	]	PUNCT
ejpam-4034	454	4	m.	m.	NOUN
ejpam-4034	454	5	m.	m.	PROPN
ejpam-4034	454	6	k.	k.	PROPN
ejpam-4034	454	7	rao	rao	PROPN
ejpam-4034	454	8	.	.	PUNCT
ejpam-4034	455	1	gamma	gamma	PROPN
ejpam-4034	455	2	-	-	PUNCT
ejpam-4034	455	3	semirings	semirings	PROPN
ejpam-4034	455	4	-	-	PUNCT
ejpam-4034	455	5	i.	i.	PROPN
ejpam-4034	455	6	southeast	southeast	PROPN
ejpam-4034	455	7	asian	asian	PROPN
ejpam-4034	455	8	bull	bull	PROPN
ejpam-4034	455	9	.	.	PUNCT
ejpam-4034	456	1	math	math	NOUN
ejpam-4034	456	2	,	,	PUNCT
ejpam-4034	456	3	19(1):49–54	19(1):49–54	NUM
ejpam-4034	456	4	,	,	PUNCT
ejpam-4034	456	5	1995	1995	NUM
ejpam-4034	456	6	.	.	PUNCT
ejpam-4034	457	1	[	[	X
ejpam-4034	457	2	15	15	NUM
ejpam-4034	457	3	]	]	X
ejpam-4034	457	4	m.	m.	NOUN
ejpam-4034	457	5	m.	m.	PROPN
ejpam-4034	457	6	k.	k.	PROPN
ejpam-4034	457	7	rao	rao	PROPN
ejpam-4034	457	8	.	.	PUNCT
ejpam-4034	458	1	ideals	ideal	NOUN
ejpam-4034	458	2	in	in	ADP
ejpam-4034	458	3	ordered	order	VERB
ejpam-4034	458	4	gamma	gamma	NOUN
ejpam-4034	458	5	-	-	PUNCT
ejpam-4034	458	6	semirings	semiring	NOUN
ejpam-4034	458	7	.	.	PUNCT
ejpam-4034	459	1	discussiones	discussione	NOUN
ejpam-4034	459	2	mathematicaegeneral	mathematicaegeneral	ADJ
ejpam-4034	459	3	algebra	algebra	NOUN
ejpam-4034	459	4	and	and	CCONJ
ejpam-4034	459	5	applications	application	NOUN
ejpam-4034	459	6	,	,	PUNCT
ejpam-4034	459	7	38(1):47–68	38(1):47–68	NUM
ejpam-4034	459	8	,	,	PUNCT
ejpam-4034	459	9	2018	2018	NUM
ejpam-4034	459	10	.	.	PUNCT
ejpam-4034	460	1	[	[	X
ejpam-4034	460	2	16	16	NUM
ejpam-4034	460	3	]	]	PUNCT
ejpam-4034	460	4	m.	m.	NOUN
ejpam-4034	460	5	sarwar	sarwar	PROPN
ejpam-4034	460	6	.	.	PUNCT
ejpam-4034	461	1	conditions	condition	NOUN
ejpam-4034	461	2	for	for	ADP
ejpam-4034	461	3	la	la	NOUN
ejpam-4034	461	4	-	-	PUNCT
ejpam-4034	461	5	semigroups	semigroup	NOUN
ejpam-4034	461	6	to	to	PART
ejpam-4034	461	7	resemble	resemble	VERB
ejpam-4034	461	8	associative	associative	ADJ
ejpam-4034	461	9	structures	structure	NOUN
ejpam-4034	461	10	.	.	PUNCT
ejpam-4034	462	1	phd	phd	NOUN
ejpam-4034	462	2	thesis	thesis	NOUN
ejpam-4034	462	3	,	,	PUNCT
ejpam-4034	462	4	quaid	quaid	PROPN
ejpam-4034	462	5	-	-	PUNCT
ejpam-4034	462	6	i	i	PROPN
ejpam-4034	462	7	-	-	PUNCT
ejpam-4034	462	8	azam	azam	PROPN
ejpam-4034	462	9	university	university	PROPN
ejpam-4034	462	10	islamabad	islamabad	NOUN
ejpam-4034	462	11	,	,	PUNCT
ejpam-4034	462	12	1993	1993	NUM
ejpam-4034	462	13	.	.	PUNCT
ejpam-4034	463	1	[	[	X
ejpam-4034	463	2	17	17	NUM
ejpam-4034	463	3	]	]	X
ejpam-4034	463	4	b.	b.	PROPN
ejpam-4034	463	5	satyanarayana	satyanarayana	PROPN
ejpam-4034	463	6	.	.	PUNCT
ejpam-4034	464	1	a	a	DET
ejpam-4034	464	2	note	note	NOUN
ejpam-4034	464	3	on	on	ADP
ejpam-4034	464	4	gamma	gamma	NOUN
ejpam-4034	464	5	-	-	PUNCT
ejpam-4034	464	6	near	near	ADP
ejpam-4034	464	7	-	-	PUNCT
ejpam-4034	464	8	rings	ring	NOUN
ejpam-4034	464	9	.	.	PUNCT
ejpam-4034	465	1	indian	indian	PROPN
ejpam-4034	465	2	j.	j.	PROPN
ejpam-4034	465	3	math	math	PROPN
ejpam-4034	465	4	,	,	PUNCT
ejpam-4034	465	5	41(3):427–433	41(3):427–433	PROPN
ejpam-4034	465	6	,	,	PUNCT
ejpam-4034	465	7	1999	1999	NUM
ejpam-4034	465	8	.	.	PUNCT
ejpam-4034	466	1	[	[	X
ejpam-4034	466	2	18	18	NUM
ejpam-4034	466	3	]	]	PUNCT
ejpam-4034	466	4	t	t	NOUN
ejpam-4034	466	5	shah	shah	NOUN
ejpam-4034	466	6	and	and	CCONJ
ejpam-4034	466	7	i	i	PROPN
ejpam-4034	466	8	rehman	rehman	PROPN
ejpam-4034	466	9	.	.	PUNCT
ejpam-4034	467	1	on	on	ADP
ejpam-4034	467	2	la	la	PROPN
ejpam-4034	467	3	-	-	PUNCT
ejpam-4034	467	4	rings	ring	NOUN
ejpam-4034	467	5	of	of	ADP
ejpam-4034	467	6	finitely	finitely	ADJ
ejpam-4034	467	7	non	non	ADJ
ejpam-4034	467	8	-	-	ADJ
ejpam-4034	467	9	zero	zero	NUM
ejpam-4034	467	10	functions	function	NOUN
ejpam-4034	467	11	.	.	PUNCT
ejpam-4034	468	1	int	int	NOUN
ejpam-4034	468	2	.	.	PUNCT
ejpam-4034	469	1	j.	j.	PROPN
ejpam-4034	469	2	contemp	contemp	PROPN
ejpam-4034	469	3	.	.	PUNCT
ejpam-4034	470	1	math	math	NOUN
ejpam-4034	470	2	.	.	PUNCT
ejpam-4034	471	1	sciences	science	NOUN
ejpam-4034	471	2	,	,	PUNCT
ejpam-4034	471	3	5(5):209–222	5(5):209–222	NUM
ejpam-4034	471	4	,	,	PUNCT
ejpam-4034	471	5	2010	2010	NUM
ejpam-4034	471	6	.	.	PUNCT
ejpam-4034	472	1	[	[	X
ejpam-4034	472	2	19	19	NUM
ejpam-4034	472	3	]	]	PUNCT
ejpam-4034	472	4	p.	p.	NOUN
ejpam-4034	472	5	yiarayong	yiarayong	PROPN
ejpam-4034	472	6	.	.	PUNCT
ejpam-4034	473	1	on	on	ADP
ejpam-4034	473	2	left	left	ADJ
ejpam-4034	473	3	primary	primary	ADJ
ejpam-4034	473	4	and	and	CCONJ
ejpam-4034	473	5	weakly	weakly	ADJ
ejpam-4034	473	6	left	left	ADJ
ejpam-4034	473	7	primary	primary	ADJ
ejpam-4034	473	8	ideals	ideal	NOUN
ejpam-4034	473	9	in	in	ADP
ejpam-4034	473	10	gamma	gamma	NOUN
ejpam-4034	473	11	-	-	PUNCT
ejpam-4034	473	12	la	la	NOUN
ejpam-4034	473	13	-	-	PUNCT
ejpam-4034	473	14	rings	ring	NOUN
ejpam-4034	473	15	.	.	PUNCT
ejpam-4034	474	1	gazi	gazi	PROPN
ejpam-4034	474	2	university	university	PROPN
ejpam-4034	474	3	journal	journal	PROPN
ejpam-4034	474	4	of	of	ADP
ejpam-4034	474	5	science	science	NOUN
ejpam-4034	474	6	,	,	PUNCT
ejpam-4034	474	7	29(1):143–148	29(1):143–148	PROPN
ejpam-4034	474	8	,	,	PUNCT
ejpam-4034	474	9	2016	2016	NUM
ejpam-4034	474	10	.	.	PUNCT
ejpam-4034	475	1	[	[	X
ejpam-4034	475	2	20	20	NUM
ejpam-4034	475	3	]	]	PUNCT
ejpam-4034	475	4	s.	s.	PROPN
ejpam-4034	475	5	m.	m.	PROPN
ejpam-4034	475	6	yusuf	yusuf	PROPN
ejpam-4034	475	7	.	.	PUNCT
ejpam-4034	476	1	on	on	ADP
ejpam-4034	476	2	left	left	ADJ
ejpam-4034	476	3	almost	almost	ADV
ejpam-4034	476	4	ring	ring	NOUN
ejpam-4034	476	5	.	.	PUNCT
ejpam-4034	477	1	in	in	ADP
ejpam-4034	477	2	proc	proc	PROPN
ejpam-4034	477	3	.	.	PUNCT
ejpam-4034	478	1	of	of	ADP
ejpam-4034	478	2	7th	7th	ADJ
ejpam-4034	478	3	international	international	ADJ
ejpam-4034	478	4	pure	pure	ADJ
ejpam-4034	478	5	math	math	NOUN
ejpam-4034	478	6	.	.	PUNCT
ejpam-4034	479	1	conference	conference	NOUN
ejpam-4034	479	2	,	,	PUNCT
ejpam-4034	479	3	islamabad	islamabad	PROPN
ejpam-4034	479	4	,	,	PUNCT
ejpam-4034	479	5	2006	2006	NUM
ejpam-4034	479	6	.	.	PUNCT