id	sid	tid	token	lemma	pos
ejpam-3397	1	1	european	european	PROPN
ejpam-3397	1	2	journal	journal	PROPN
ejpam-3397	1	3	of	of	ADP
ejpam-3397	1	4	pure	pure	ADJ
ejpam-3397	1	5	and	and	CCONJ
ejpam-3397	1	6	applied	apply	VERB
ejpam-3397	1	7	mathematics	mathematic	NOUN
ejpam-3397	1	8	vol	vol	NOUN
ejpam-3397	1	9	.	.	PROPN
ejpam-3397	2	1	12	12	NUM
ejpam-3397	2	2	,	,	PUNCT
ejpam-3397	2	3	no	no	INTJ
ejpam-3397	2	4	.	.	NOUN
ejpam-3397	2	5	2	2	NUM
ejpam-3397	2	6	,	,	PUNCT
ejpam-3397	2	7	2019	2019	NUM
ejpam-3397	2	8	,	,	PUNCT
ejpam-3397	2	9	544	544	NUM
ejpam-3397	2	10	-	-	SYM
ejpam-3397	2	11	552	552	NUM
ejpam-3397	2	12	issn	issn	PROPN
ejpam-3397	2	13	1307	1307	NUM
ejpam-3397	2	14	-	-	SYM
ejpam-3397	2	15	5543	5543	NUM
ejpam-3397	2	16	–	–	PUNCT
ejpam-3397	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3397	2	18	published	publish	VERB
ejpam-3397	2	19	by	by	ADP
ejpam-3397	2	20	new	new	PROPN
ejpam-3397	2	21	york	york	PROPN
ejpam-3397	2	22	business	business	PROPN
ejpam-3397	2	23	global	global	ADJ
ejpam-3397	2	24	weakly	weakly	ADJ
ejpam-3397	2	25	prime	prime	ADJ
ejpam-3397	2	26	and	and	CCONJ
ejpam-3397	2	27	weakly	weakly	ADJ
ejpam-3397	2	28	primary	primary	ADJ
ejpam-3397	2	29	ideals	ideal	NOUN
ejpam-3397	2	30	in	in	ADP
ejpam-3397	2	31	gamma	gamma	NOUN
ejpam-3397	2	32	seminearrings	seminearring	NOUN
ejpam-3397	2	33	waheed	waheed	PROPN
ejpam-3397	2	34	ahmad	ahmad	PROPN
ejpam-3397	2	35	khan1	khan1	PROPN
ejpam-3397	2	36	,	,	PUNCT
ejpam-3397	2	37	abdelghani	abdelghani	PROPN
ejpam-3397	2	38	taouti2,∗	taouti2,∗	PROPN
ejpam-3397	2	39	,	,	PUNCT
ejpam-3397	2	40	seema	seema	PROPN
ejpam-3397	2	41	karkain2	karkain2	PROPN
ejpam-3397	2	42	,	,	PUNCT
ejpam-3397	2	43	azar	azar	PROPN
ejpam-3397	2	44	salami2	salami2	PROPN
ejpam-3397	2	45	,	,	PUNCT
ejpam-3397	2	46	waqar	waqar	VERB
ejpam-3397	3	1	arif1	arif1	PROPN
ejpam-3397	3	2	1	1	NUM
ejpam-3397	3	3	department	department	NOUN
ejpam-3397	3	4	of	of	ADP
ejpam-3397	3	5	mathematics	mathematic	NOUN
ejpam-3397	3	6	,	,	PUNCT
ejpam-3397	3	7	university	university	NOUN
ejpam-3397	3	8	of	of	ADP
ejpam-3397	3	9	education	education	NOUN
ejpam-3397	3	10	,	,	PUNCT
ejpam-3397	3	11	attock	attock	PROPN
ejpam-3397	3	12	campus	campus	PROPN
ejpam-3397	3	13	,	,	PUNCT
ejpam-3397	3	14	pakistan	pakistan	PROPN
ejpam-3397	3	15	2	2	NUM
ejpam-3397	3	16	ets	et	NOUN
ejpam-3397	3	17	-	-	PUNCT
ejpam-3397	3	18	maths	math	NOUN
ejpam-3397	3	19	and	and	CCONJ
ejpam-3397	3	20	ns	ns	NUM
ejpam-3397	3	21	engineering	engineering	NOUN
ejpam-3397	3	22	division	division	NOUN
ejpam-3397	3	23	hct	hct	PROPN
ejpam-3397	3	24	,	,	PUNCT
ejpam-3397	3	25	university	university	NOUN
ejpam-3397	3	26	city	city	NOUN
ejpam-3397	3	27	,	,	PUNCT
ejpam-3397	3	28	p.	p.	PROPN
ejpam-3397	3	29	o.	o.	PROPN
ejpam-3397	3	30	box	box	PROPN
ejpam-3397	3	31	:	:	PUNCT
ejpam-3397	3	32	7947	7947	NUM
ejpam-3397	3	33	united	united	PROPN
ejpam-3397	3	34	arab	arab	PROPN
ejpam-3397	3	35	emirates	emirates	PROPN
ejpam-3397	3	36	abstract	abstract	ADV
ejpam-3397	3	37	.	.	PUNCT
ejpam-3397	4	1	we	we	PRON
ejpam-3397	4	2	introduce	introduce	VERB
ejpam-3397	4	3	and	and	CCONJ
ejpam-3397	4	4	discuss	discuss	VERB
ejpam-3397	4	5	the	the	DET
ejpam-3397	4	6	weakly	weakly	ADJ
ejpam-3397	4	7	prime	prime	ADJ
ejpam-3397	4	8	and	and	CCONJ
ejpam-3397	4	9	weakly	weakly	ADJ
ejpam-3397	4	10	primary	primary	ADJ
ejpam-3397	4	11	ideals	ideal	NOUN
ejpam-3397	4	12	of	of	ADP
ejpam-3397	4	13	a	a	DET
ejpam-3397	4	14	gamma	gamma	NOUN
ejpam-3397	4	15	seminearrings	seminearring	NOUN
ejpam-3397	4	16	with	with	ADP
ejpam-3397	4	17	illustrative	illustrative	ADJ
ejpam-3397	4	18	examples	example	NOUN
ejpam-3397	4	19	.	.	PUNCT
ejpam-3397	5	1	we	we	PRON
ejpam-3397	5	2	also	also	ADV
ejpam-3397	5	3	present	present	VERB
ejpam-3397	5	4	few	few	ADJ
ejpam-3397	5	5	of	of	ADP
ejpam-3397	5	6	characterizations	characterization	NOUN
ejpam-3397	5	7	of	of	ADP
ejpam-3397	5	8	these	these	DET
ejpam-3397	5	9	ideals	ideal	NOUN
ejpam-3397	5	10	.	.	PUNCT
ejpam-3397	6	1	2010	2010	NUM
ejpam-3397	6	2	mathematics	mathematic	NOUN
ejpam-3397	6	3	subject	subject	NOUN
ejpam-3397	6	4	classifications	classification	NOUN
ejpam-3397	6	5	:	:	PUNCT
ejpam-3397	6	6	16y30	16y30	NUM
ejpam-3397	6	7	,	,	PUNCT
ejpam-3397	6	8	16y60	16y60	NUM
ejpam-3397	6	9	key	key	ADJ
ejpam-3397	6	10	words	word	NOUN
ejpam-3397	6	11	and	and	CCONJ
ejpam-3397	6	12	phrases	phrase	NOUN
ejpam-3397	6	13	:	:	PUNCT
ejpam-3397	6	14	gamma	gamma	PROPN
ejpam-3397	6	15	seminearrrings	seminearrring	NOUN
ejpam-3397	6	16	,	,	PUNCT
ejpam-3397	6	17	prime	prime	ADJ
ejpam-3397	6	18	ideals	ideal	NOUN
ejpam-3397	6	19	,	,	PUNCT
ejpam-3397	6	20	primary	primary	ADJ
ejpam-3397	6	21	ideals	ideal	NOUN
ejpam-3397	6	22	.	.	PUNCT
ejpam-3397	7	1	1	1	X
ejpam-3397	7	2	.	.	X
ejpam-3397	7	3	introduction	introduction	NOUN
ejpam-3397	7	4	and	and	CCONJ
ejpam-3397	7	5	preliminaries	preliminary	NOUN
ejpam-3397	7	6	the	the	DET
ejpam-3397	7	7	concept	concept	NOUN
ejpam-3397	7	8	of	of	ADP
ejpam-3397	7	9	seminearring	seminearring	NOUN
ejpam-3397	7	10	was	be	AUX
ejpam-3397	7	11	introduced	introduce	VERB
ejpam-3397	7	12	by	by	ADP
ejpam-3397	7	13	w.	w.	PROPN
ejpam-3397	7	14	g.	g.	PROPN
ejpam-3397	7	15	van	van	PROPN
ejpam-3397	7	16	hoorn	hoorn	PROPN
ejpam-3397	7	17	et	et	PROPN
ejpam-3397	7	18	al	al	PROPN
ejpam-3397	7	19	.	.	PUNCT
ejpam-3397	8	1	in	in	ADP
ejpam-3397	8	2	[	[	X
ejpam-3397	8	3	1	1	NUM
ejpam-3397	8	4	]	]	PUNCT
ejpam-3397	8	5	.	.	PUNCT
ejpam-3397	9	1	seminearfields	seminearfield	NOUN
ejpam-3397	9	2	have	have	AUX
ejpam-3397	9	3	been	be	AUX
ejpam-3397	9	4	introduced	introduce	VERB
ejpam-3397	9	5	in	in	ADP
ejpam-3397	9	6	[	[	X
ejpam-3397	9	7	5	5	NUM
ejpam-3397	9	8	]	]	PUNCT
ejpam-3397	9	9	.	.	PUNCT
ejpam-3397	10	1	as	as	ADP
ejpam-3397	10	2	a	a	DET
ejpam-3397	10	3	generalization	generalization	NOUN
ejpam-3397	10	4	of	of	ADP
ejpam-3397	10	5	seminearrings	seminearring	NOUN
ejpam-3397	10	6	that	that	PRON
ejpam-3397	10	7	is	be	AUX
ejpam-3397	10	8	γseminear	γseminear	ADJ
ejpam-3397	10	9	-	-	PUNCT
ejpam-3397	10	10	rings	ring	NOUN
ejpam-3397	10	11	were	be	AUX
ejpam-3397	10	12	introduced	introduce	VERB
ejpam-3397	10	13	in	in	ADP
ejpam-3397	10	14	[	[	X
ejpam-3397	10	15	2	2	NUM
ejpam-3397	10	16	]	]	PUNCT
ejpam-3397	10	17	.	.	PUNCT
ejpam-3397	11	1	subsequently	subsequently	ADV
ejpam-3397	11	2	,	,	PUNCT
ejpam-3397	11	3	prime	prime	ADJ
ejpam-3397	11	4	and	and	CCONJ
ejpam-3397	11	5	semiprime	semiprime	NOUN
ejpam-3397	11	6	ideals	ideal	NOUN
ejpam-3397	11	7	in	in	ADP
ejpam-3397	11	8	gamma	gamma	NOUN
ejpam-3397	11	9	seminearrings	seminearring	NOUN
ejpam-3397	11	10	have	have	AUX
ejpam-3397	11	11	been	be	AUX
ejpam-3397	11	12	explored	explore	VERB
ejpam-3397	11	13	in	in	ADP
ejpam-3397	11	14	[	[	X
ejpam-3397	11	15	3	3	NUM
ejpam-3397	11	16	]	]	PUNCT
ejpam-3397	11	17	.	.	PUNCT
ejpam-3397	12	1	in	in	ADP
ejpam-3397	12	2	a	a	DET
ejpam-3397	12	3	sequel	sequel	NOUN
ejpam-3397	12	4	,	,	PUNCT
ejpam-3397	12	5	we	we	PRON
ejpam-3397	12	6	introduce	introduce	VERB
ejpam-3397	12	7	the	the	DET
ejpam-3397	12	8	notion	notion	NOUN
ejpam-3397	12	9	of	of	ADP
ejpam-3397	12	10	weakly	weakly	ADJ
ejpam-3397	12	11	prime	prime	ADJ
ejpam-3397	12	12	and	and	CCONJ
ejpam-3397	12	13	weakly	weakly	ADJ
ejpam-3397	12	14	primary	primary	ADJ
ejpam-3397	12	15	ideals	ideal	NOUN
ejpam-3397	12	16	γ	γ	X
ejpam-3397	12	17	-	-	PUNCT
ejpam-3397	12	18	seminearring	seminearring	NOUN
ejpam-3397	12	19	and	and	CCONJ
ejpam-3397	12	20	few	few	ADJ
ejpam-3397	12	21	of	of	ADP
ejpam-3397	12	22	their	their	PRON
ejpam-3397	12	23	characterizations	characterization	NOUN
ejpam-3397	12	24	.	.	PUNCT
ejpam-3397	13	1	we	we	PRON
ejpam-3397	13	2	recall	recall	VERB
ejpam-3397	13	3	some	some	DET
ejpam-3397	13	4	useful	useful	ADJ
ejpam-3397	13	5	concepts	concept	NOUN
ejpam-3397	13	6	for	for	ADP
ejpam-3397	13	7	the	the	DET
ejpam-3397	13	8	sake	sake	NOUN
ejpam-3397	13	9	of	of	ADP
ejpam-3397	13	10	completeness	completeness	NOUN
ejpam-3397	13	11	.	.	PUNCT
ejpam-3397	14	1	a	a	DET
ejpam-3397	14	2	nonempty	nonempty	ADV
ejpam-3397	14	3	set	set	VERB
ejpam-3397	14	4	r	r	NOUN
ejpam-3397	14	5	with	with	ADP
ejpam-3397	14	6	two	two	NUM
ejpam-3397	14	7	binary	binary	ADJ
ejpam-3397	14	8	operations	operation	NOUN
ejpam-3397	14	9	”	"	PUNCT
ejpam-3397	14	10	+	+	CCONJ
ejpam-3397	14	11	”	"	PUNCT
ejpam-3397	14	12	(	(	PUNCT
ejpam-3397	14	13	addition	addition	NOUN
ejpam-3397	14	14	)	)	PUNCT
ejpam-3397	14	15	and	and	CCONJ
ejpam-3397	14	16	”	"	PUNCT
ejpam-3397	14	17	.	.	PUNCT
ejpam-3397	14	18	”	"	PUNCT
ejpam-3397	15	1	(	(	PUNCT
ejpam-3397	15	2	multiplication	multiplication	NOUN
ejpam-3397	15	3	)	)	PUNCT
ejpam-3397	15	4	is	be	AUX
ejpam-3397	15	5	called	call	VERB
ejpam-3397	15	6	a	a	DET
ejpam-3397	15	7	seminearring	seminearring	NOUN
ejpam-3397	15	8	if	if	SCONJ
ejpam-3397	15	9	it	it	PRON
ejpam-3397	15	10	satisfies	satisfy	VERB
ejpam-3397	15	11	(	(	PUNCT
ejpam-3397	15	12	i	i	NOUN
ejpam-3397	15	13	)	)	PUNCT
ejpam-3397	15	14	(	(	PUNCT
ejpam-3397	15	15	r	r	NOUN
ejpam-3397	15	16	,	,	PUNCT
ejpam-3397	15	17	+	+	NOUN
ejpam-3397	15	18	)	)	PUNCT
ejpam-3397	15	19	and	and	CCONJ
ejpam-3397	15	20	(	(	PUNCT
ejpam-3397	15	21	r	r	NOUN
ejpam-3397	15	22	,	,	PUNCT
ejpam-3397	15	23	.	.	PUNCT
ejpam-3397	15	24	)	)	PUNCT
ejpam-3397	16	1	are	be	AUX
ejpam-3397	16	2	semigroups	semigroup	NOUN
ejpam-3397	16	3	;	;	PUNCT
ejpam-3397	16	4	(	(	PUNCT
ejpam-3397	16	5	ii	ii	NOUN
ejpam-3397	16	6	)	)	PUNCT
ejpam-3397	16	7	(	(	PUNCT
ejpam-3397	16	8	x+	x+	PROPN
ejpam-3397	16	9	y).z	y).z	PROPN
ejpam-3397	16	10	=	=	SYM
ejpam-3397	16	11	x.z+	x.z+	NOUN
ejpam-3397	16	12	y.z	y.z	PROPN
ejpam-3397	16	13	for	for	ADP
ejpam-3397	16	14	all	all	DET
ejpam-3397	16	15	x	x	NOUN
ejpam-3397	16	16	,	,	PUNCT
ejpam-3397	16	17	y	y	PROPN
ejpam-3397	16	18	,	,	PUNCT
ejpam-3397	16	19	z	z	PROPN
ejpam-3397	16	20	∈	∈	PROPN
ejpam-3397	16	21	r.	r.	PROPN
ejpam-3397	16	22	in	in	ADP
ejpam-3397	16	23	2005	2005	NUM
ejpam-3397	16	24	,	,	PUNCT
ejpam-3397	16	25	krishna	krishna	PROPN
ejpam-3397	16	26	&	&	CCONJ
ejpam-3397	16	27	chatterjee	chatterjee	PROPN
ejpam-3397	17	1	[	[	X
ejpam-3397	17	2	4	4	NUM
ejpam-3397	17	3	]	]	PUNCT
ejpam-3397	17	4	,	,	PUNCT
ejpam-3397	17	5	introduced	introduce	VERB
ejpam-3397	17	6	the	the	DET
ejpam-3397	17	7	condition	condition	NOUN
ejpam-3397	17	8	of	of	ADP
ejpam-3397	17	9	minimality	minimality	NOUN
ejpam-3397	17	10	of	of	ADP
ejpam-3397	17	11	generalized	generalized	ADJ
ejpam-3397	17	12	linear	linear	ADJ
ejpam-3397	17	13	sequential	sequential	ADJ
ejpam-3397	17	14	machines	machine	NOUN
ejpam-3397	17	15	using	use	VERB
ejpam-3397	17	16	the	the	DET
ejpam-3397	17	17	theory	theory	NOUN
ejpam-3397	17	18	of	of	ADP
ejpam-3397	17	19	near	near	ADJ
ejpam-3397	17	20	-	-	PUNCT
ejpam-3397	17	21	semirings	semiring	NOUN
ejpam-3397	17	22	.	.	PUNCT
ejpam-3397	18	1	near	near	ADJ
ejpam-3397	18	2	-	-	PUNCT
ejpam-3397	18	3	semirings	semiring	NOUN
ejpam-3397	18	4	have	have	AUX
ejpam-3397	18	5	proven	prove	VERB
ejpam-3397	18	6	to	to	PART
ejpam-3397	18	7	be	be	AUX
ejpam-3397	18	8	useful	useful	ADJ
ejpam-3397	18	9	in	in	ADP
ejpam-3397	18	10	studying	study	VERB
ejpam-3397	18	11	automata	automata	NOUN
ejpam-3397	18	12	and	and	CCONJ
ejpam-3397	18	13	formal	formal	ADJ
ejpam-3397	18	14	languages	language	NOUN
ejpam-3397	18	15	.	.	PUNCT
ejpam-3397	19	1	following	follow	VERB
ejpam-3397	19	2	[	[	X
ejpam-3397	19	3	3	3	NUM
ejpam-3397	19	4	]	]	PUNCT
ejpam-3397	19	5	,	,	PUNCT
ejpam-3397	19	6	γ	γ	X
ejpam-3397	19	7	-	-	PUNCT
ejpam-3397	19	8	seminearring	seminearring	NOUN
ejpam-3397	19	9	is	be	AUX
ejpam-3397	19	10	a	a	DET
ejpam-3397	19	11	triple	triple	ADJ
ejpam-3397	19	12	(	(	PUNCT
ejpam-3397	19	13	r	r	NOUN
ejpam-3397	19	14	,	,	PUNCT
ejpam-3397	19	15	+	+	ADJ
ejpam-3397	19	16	,	,	PUNCT
ejpam-3397	19	17	γ	γ	NOUN
ejpam-3397	19	18	)	)	PUNCT
ejpam-3397	19	19	where	where	SCONJ
ejpam-3397	19	20	,	,	PUNCT
ejpam-3397	19	21	(	(	PUNCT
ejpam-3397	19	22	i	i	NOUN
ejpam-3397	19	23	)	)	PUNCT
ejpam-3397	19	24	γ	γ	PROPN
ejpam-3397	19	25	is	be	AUX
ejpam-3397	19	26	a	a	DET
ejpam-3397	19	27	non	non	ADJ
ejpam-3397	19	28	-	-	ADJ
ejpam-3397	19	29	empty	empty	ADJ
ejpam-3397	19	30	set	set	NOUN
ejpam-3397	19	31	of	of	ADP
ejpam-3397	19	32	binary	binary	ADJ
ejpam-3397	19	33	operators	operator	NOUN
ejpam-3397	19	34	on	on	ADP
ejpam-3397	19	35	r	r	NOUN
ejpam-3397	19	36	such	such	ADJ
ejpam-3397	19	37	that	that	PRON
ejpam-3397	19	38	for	for	SCONJ
ejpam-3397	19	39	each	each	DET
ejpam-3397	19	40	α	α	PROPN
ejpam-3397	19	41	∈	∈	PROPN
ejpam-3397	19	42	γ	γ	X
ejpam-3397	19	43	,	,	PUNCT
ejpam-3397	19	44	(	(	PUNCT
ejpam-3397	19	45	r	r	NOUN
ejpam-3397	19	46	,	,	PUNCT
ejpam-3397	19	47	+	+	NOUN
ejpam-3397	19	48	,	,	PUNCT
ejpam-3397	19	49	.	.	PUNCT
ejpam-3397	19	50	)	)	PUNCT
ejpam-3397	19	51	is	be	AUX
ejpam-3397	19	52	a	a	DET
ejpam-3397	19	53	seminearring	seminearring	NOUN
ejpam-3397	19	54	,	,	PUNCT
ejpam-3397	19	55	(	(	PUNCT
ejpam-3397	19	56	ii	ii	NOUN
ejpam-3397	19	57	)	)	PUNCT
ejpam-3397	19	58	xα(yβz	xα(yβz	PROPN
ejpam-3397	19	59	)	)	PUNCT
ejpam-3397	20	1	=	=	SYM
ejpam-3397	20	2	(	(	PUNCT
ejpam-3397	20	3	xαy)βz	xαy)βz	VERB
ejpam-3397	20	4	for	for	ADP
ejpam-3397	20	5	all	all	DET
ejpam-3397	20	6	x	x	NOUN
ejpam-3397	20	7	,	,	PUNCT
ejpam-3397	20	8	y	y	PROPN
ejpam-3397	20	9	,	,	PUNCT
ejpam-3397	20	10	z	z	NOUN
ejpam-3397	20	11	∈	∈	PROPN
ejpam-3397	20	12	r	r	NOUN
ejpam-3397	20	13	and	and	CCONJ
ejpam-3397	20	14	α	α	NOUN
ejpam-3397	20	15	,	,	PUNCT
ejpam-3397	20	16	β	β	PROPN
ejpam-3397	20	17	∈	∈	PROPN
ejpam-3397	20	18	γ	γ	PROPN
ejpam-3397	20	19	.	.	PROPN
ejpam-3397	20	20	similarly	similarly	ADV
ejpam-3397	20	21	,	,	PUNCT
ejpam-3397	20	22	let	let	VERB
ejpam-3397	20	23	r	r	PRON
ejpam-3397	20	24	be	be	AUX
ejpam-3397	20	25	a	a	DET
ejpam-3397	20	26	γ	γ	NOUN
ejpam-3397	20	27	-	-	PUNCT
ejpam-3397	20	28	seminearring	seminearring	NOUN
ejpam-3397	20	29	,	,	PUNCT
ejpam-3397	20	30	a	a	DET
ejpam-3397	20	31	subsemigroup	subsemigroup	NOUN
ejpam-3397	20	32	a	a	PRON
ejpam-3397	20	33	of	of	ADP
ejpam-3397	20	34	(	(	PUNCT
ejpam-3397	20	35	r,+	r,+	NUM
ejpam-3397	20	36	)	)	PUNCT
ejpam-3397	20	37	is	be	AUX
ejpam-3397	20	38	called	call	VERB
ejpam-3397	20	39	a	a	DET
ejpam-3397	20	40	left	left	ADJ
ejpam-3397	20	41	(	(	PUNCT
ejpam-3397	20	42	resp	resp	NOUN
ejpam-3397	20	43	.	.	PUNCT
ejpam-3397	20	44	,	,	PUNCT
ejpam-3397	20	45	right	right	ADJ
ejpam-3397	20	46	)	)	PUNCT
ejpam-3397	20	47	ideal	ideal	NOUN
ejpam-3397	20	48	of	of	ADP
ejpam-3397	20	49	r	r	NOUN
ejpam-3397	20	50	if	if	SCONJ
ejpam-3397	20	51	rγa	rγa	ADJ
ejpam-3397	20	52	⊆	⊆	NUM
ejpam-3397	20	53	a	a	DET
ejpam-3397	20	54	(	(	PUNCT
ejpam-3397	20	55	resp	resp	NOUN
ejpam-3397	20	56	.	.	PUNCT
ejpam-3397	20	57	,	,	PUNCT
ejpam-3397	20	58	aγr	aγr	ADJ
ejpam-3397	20	59	⊆	⊆	NUM
ejpam-3397	20	60	a	a	PRON
ejpam-3397	20	61	)	)	PUNCT
ejpam-3397	20	62	.	.	PUNCT
ejpam-3397	21	1	a	a	DET
ejpam-3397	21	2	left	left	ADJ
ejpam-3397	21	3	and	and	CCONJ
ejpam-3397	21	4	right	right	ADJ
ejpam-3397	21	5	ideal	ideal	NOUN
ejpam-3397	21	6	is	be	AUX
ejpam-3397	21	7	called	call	VERB
ejpam-3397	21	8	an	an	DET
ejpam-3397	21	9	ideal	ideal	NOUN
ejpam-3397	21	10	.	.	PUNCT
ejpam-3397	22	1	let	let	VERB
ejpam-3397	22	2	∗corresponding	∗corresponde	VERB
ejpam-3397	22	3	author	author	NOUN
ejpam-3397	22	4	.	.	PUNCT
ejpam-3397	23	1	doi	doi	NOUN
ejpam-3397	23	2	:	:	PUNCT
ejpam-3397	23	3	https://doi.org/10.29020/nybg.ejpam.v12i2.3397	https://doi.org/10.29020/nybg.ejpam.v12i2.3397	ADP
ejpam-3397	23	4	email	email	NOUN
ejpam-3397	23	5	addresses	address	NOUN
ejpam-3397	23	6	:	:	PUNCT
ejpam-3397	23	7	sirwak2003@	sirwak2003@	X
ejpam-3397	23	8	yahoo.com	yahoo.com	X
ejpam-3397	23	9	(	(	PUNCT
ejpam-3397	23	10	w.	w.	PROPN
ejpam-3397	23	11	a.	a.	PROPN
ejpam-3397	23	12	khan	khan	PROPN
ejpam-3397	23	13	)	)	PUNCT
ejpam-3397	23	14	,	,	PUNCT
ejpam-3397	23	15	ganitaouti@yahoo.com.au	ganitaouti@yahoo.com.au	PROPN
ejpam-3397	23	16	(	(	PUNCT
ejpam-3397	23	17	a.	a.	NOUN
ejpam-3397	23	18	taouti	taouti	PROPN
ejpam-3397	23	19	)	)	PUNCT
ejpam-3397	23	20	,	,	PUNCT
ejpam-3397	23	21	skarkain@hct.ac.ae	skarkain@hct.ac.ae	ADJ
ejpam-3397	23	22	(	(	PUNCT
ejpam-3397	23	23	s.	s.	PROPN
ejpam-3397	23	24	karkain	karkain	PROPN
ejpam-3397	23	25	)	)	PUNCT
ejpam-3397	23	26	,	,	PUNCT
ejpam-3397	23	27	asalami@hct.ac.ae	asalami@hct.ac.ae	NOUN
ejpam-3397	23	28	(	(	PUNCT
ejpam-3397	23	29	a.	a.	NOUN
ejpam-3397	23	30	salami	salami	PROPN
ejpam-3397	23	31	)	)	PUNCT
ejpam-3397	23	32	,	,	PUNCT
ejpam-3397	23	33	waqarvicky6699@gmail.com	waqarvicky6699@gmail.com	X
ejpam-3397	24	1	(	(	PUNCT
ejpam-3397	24	2	w.	w.	PROPN
ejpam-3397	24	3	arif	arif	PROPN
ejpam-3397	24	4	)	)	PUNCT
ejpam-3397	24	5	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3397	25	1	544	544	NUM
ejpam-3397	25	2	c	c	NOUN
ejpam-3397	25	3	©	©	PROPN
ejpam-3397	25	4	2019	2019	NUM
ejpam-3397	25	5	ejpam	ejpam	NOUN
ejpam-3397	25	6	all	all	DET
ejpam-3397	25	7	rights	right	NOUN
ejpam-3397	25	8	reserved	reserve	VERB
ejpam-3397	25	9	.	.	PUNCT
ejpam-3397	26	1	abdelghani	abdelghani	PROPN
ejpam-3397	26	2	taouti	taouti	PROPN
ejpam-3397	26	3	et	et	PROPN
ejpam-3397	26	4	al	al	PROPN
ejpam-3397	26	5	.	.	PUNCT
ejpam-3397	26	6	/	/	SYM
ejpam-3397	26	7	eur	eur	PROPN
ejpam-3397	26	8	.	.	PUNCT
ejpam-3397	27	1	j.	j.	PROPN
ejpam-3397	27	2	pure	pure	PROPN
ejpam-3397	27	3	appl	appl	PROPN
ejpam-3397	27	4	.	.	PROPN
ejpam-3397	27	5	math	math	PROPN
ejpam-3397	27	6	,	,	PUNCT
ejpam-3397	27	7	12	12	NUM
ejpam-3397	27	8	(	(	PUNCT
ejpam-3397	27	9	2	2	NUM
ejpam-3397	27	10	)	)	PUNCT
ejpam-3397	27	11	(	(	PUNCT
ejpam-3397	27	12	2019	2019	NUM
ejpam-3397	27	13	)	)	PUNCT
ejpam-3397	27	14	,	,	PUNCT
ejpam-3397	27	15	544	544	NUM
ejpam-3397	27	16	-	-	SYM
ejpam-3397	27	17	552	552	NUM
ejpam-3397	27	18	545	545	NUM
ejpam-3397	27	19	r	r	NOUN
ejpam-3397	27	20	be	be	VERB
ejpam-3397	27	21	a	a	DET
ejpam-3397	27	22	γ	γ	NOUN
ejpam-3397	27	23	-	-	PUNCT
ejpam-3397	27	24	seminearring	seminearring	NOUN
ejpam-3397	27	25	and	and	CCONJ
ejpam-3397	27	26	i	i	PRON
ejpam-3397	27	27	,	,	PUNCT
ejpam-3397	27	28	j	j	PROPN
ejpam-3397	27	29	⊆	⊆	NUM
ejpam-3397	27	30	r.	r.	PROPN
ejpam-3397	27	31	we	we	PRON
ejpam-3397	27	32	denote	denote	VERB
ejpam-3397	27	33	it	it	PRON
ejpam-3397	27	34	by	by	ADP
ejpam-3397	27	35	iγj	iγj	NOUN
ejpam-3397	27	36	=	=	PUNCT
ejpam-3397	27	37	{	{	PUNCT
ejpam-3397	27	38	aαb	aαb	NOUN
ejpam-3397	27	39	|	|	ADV
ejpam-3397	27	40	a	a	NOUN
ejpam-3397	27	41	,	,	PUNCT
ejpam-3397	27	42	b	b	X
ejpam-3397	27	43	∈	∈	PROPN
ejpam-3397	27	44	r	r	NOUN
ejpam-3397	27	45	and	and	CCONJ
ejpam-3397	27	46	α	α	NOUN
ejpam-3397	27	47	∈	∈	PROPN
ejpam-3397	27	48	γ	γ	X
ejpam-3397	27	49	}	}	PUNCT
ejpam-3397	27	50	.	.	PUNCT
ejpam-3397	28	1	a	a	DET
ejpam-3397	28	2	mapping	mapping	NOUN
ejpam-3397	28	3	f	f	NOUN
ejpam-3397	28	4	:	:	PUNCT
ejpam-3397	28	5	r	r	X
ejpam-3397	28	6	→	→	SYM
ejpam-3397	28	7	r′	r′	NOUN
ejpam-3397	28	8	between	between	ADP
ejpam-3397	28	9	two	two	NUM
ejpam-3397	28	10	gamma	gamma	NOUN
ejpam-3397	28	11	seminearrings	seminearring	NOUN
ejpam-3397	28	12	is	be	AUX
ejpam-3397	28	13	called	call	VERB
ejpam-3397	28	14	a	a	DET
ejpam-3397	28	15	γ	γ	NOUN
ejpam-3397	28	16	-	-	PUNCT
ejpam-3397	28	17	seminearring	seminearre	VERB
ejpam-3397	28	18	homomorphism	homomorphism	NOUN
ejpam-3397	28	19	(	(	PUNCT
ejpam-3397	28	20	γ	γ	NOUN
ejpam-3397	28	21	-	-	PUNCT
ejpam-3397	28	22	homomorphism	homomorphism	NOUN
ejpam-3397	28	23	)	)	PUNCT
ejpam-3397	28	24	,	,	PUNCT
ejpam-3397	28	25	if	if	SCONJ
ejpam-3397	28	26	f(x	f(x	PROPN
ejpam-3397	28	27	+	+	CCONJ
ejpam-3397	28	28	y	y	X
ejpam-3397	28	29	)	)	PUNCT
ejpam-3397	28	30	=	=	SYM
ejpam-3397	28	31	f(x	f(x	PROPN
ejpam-3397	28	32	)	)	PUNCT
ejpam-3397	29	1	+	+	SYM
ejpam-3397	29	2	f(y	f(y	NOUN
ejpam-3397	29	3	)	)	PUNCT
ejpam-3397	29	4	and	and	CCONJ
ejpam-3397	29	5	f(xγy	f(xγy	NOUN
ejpam-3397	29	6	)	)	PUNCT
ejpam-3397	29	7	=	=	SYM
ejpam-3397	29	8	f(x)γf(y	f(x)γf(y	PROPN
ejpam-3397	29	9	)	)	PUNCT
ejpam-3397	29	10	for	for	ADP
ejpam-3397	29	11	all	all	DET
ejpam-3397	29	12	x	x	NOUN
ejpam-3397	29	13	,	,	PUNCT
ejpam-3397	29	14	y	y	PROPN
ejpam-3397	29	15	∈	∈	PROPN
ejpam-3397	29	16	r	r	NOUN
ejpam-3397	29	17	and	and	CCONJ
ejpam-3397	29	18	γ	γ	PROPN
ejpam-3397	29	19	∈	∈	PROPN
ejpam-3397	29	20	γ	γ	X
ejpam-3397	29	21	.	.	PUNCT
ejpam-3397	30	1	let	let	AUX
ejpam-3397	30	2	r	r	NOUN
ejpam-3397	30	3	and	and	CCONJ
ejpam-3397	30	4	r′	r′	PROPN
ejpam-3397	30	5	be	be	AUX
ejpam-3397	30	6	a	a	DET
ejpam-3397	30	7	γ	γ	NOUN
ejpam-3397	30	8	-	-	PUNCT
ejpam-3397	30	9	seminearrings	seminearring	NOUN
ejpam-3397	30	10	and	and	CCONJ
ejpam-3397	30	11	f	f	NOUN
ejpam-3397	30	12	:	:	PUNCT
ejpam-3397	30	13	r	r	NOUN
ejpam-3397	30	14	→	→	SYM
ejpam-3397	30	15	r′	r′	X
ejpam-3397	30	16	be	be	AUX
ejpam-3397	30	17	a	a	DET
ejpam-3397	30	18	γ	γ	NOUN
ejpam-3397	30	19	-	-	PUNCT
ejpam-3397	30	20	seminearring	seminearre	VERB
ejpam-3397	30	21	homomorphism	homomorphism	NOUN
ejpam-3397	30	22	.	.	PUNCT
ejpam-3397	31	1	then	then	ADV
ejpam-3397	31	2	,	,	PUNCT
ejpam-3397	31	3	(	(	PUNCT
ejpam-3397	31	4	i	i	NOUN
ejpam-3397	31	5	)	)	PUNCT
ejpam-3397	31	6	f(i1γi2	f(i1γi2	NOUN
ejpam-3397	31	7	)	)	PUNCT
ejpam-3397	31	8	=	=	SYM
ejpam-3397	31	9	f(i1)γf(i2	f(i1)γf(i2	NOUN
ejpam-3397	31	10	)	)	PUNCT
ejpam-3397	31	11	for	for	ADP
ejpam-3397	31	12	all	all	DET
ejpam-3397	31	13	i1	i1	PROPN
ejpam-3397	31	14	,	,	PUNCT
ejpam-3397	31	15	i2	i2	PROPN
ejpam-3397	31	16	∈	∈	PROPN
ejpam-3397	31	17	r	r	PROPN
ejpam-3397	31	18	,	,	PUNCT
ejpam-3397	31	19	(	(	PUNCT
ejpam-3397	31	20	ii	ii	NOUN
ejpam-3397	31	21	)	)	PUNCT
ejpam-3397	31	22	f−1(j1)γf−1(j2	f−1(j1)γf−1(j2	NOUN
ejpam-3397	31	23	)	)	PUNCT
ejpam-3397	31	24	=	=	SYM
ejpam-3397	31	25	f−1(j1γj2	f−1(j1γj2	PROPN
ejpam-3397	31	26	)	)	PUNCT
ejpam-3397	31	27	for	for	ADP
ejpam-3397	31	28	all	all	DET
ejpam-3397	31	29	j1	j1	PROPN
ejpam-3397	31	30	,	,	PUNCT
ejpam-3397	31	31	j2	j2	PROPN
ejpam-3397	31	32	∈	∈	PROPN
ejpam-3397	31	33	r′.	r′.	PROPN
ejpam-3397	31	34	2	2	NUM
ejpam-3397	31	35	.	.	PUNCT
ejpam-3397	31	36	weakly	weakly	ADJ
ejpam-3397	31	37	prime	prime	ADJ
ejpam-3397	31	38	and	and	CCONJ
ejpam-3397	31	39	weakly	weakly	ADJ
ejpam-3397	31	40	primary	primary	ADJ
ejpam-3397	31	41	ideals	ideal	NOUN
ejpam-3397	31	42	in	in	ADP
ejpam-3397	31	43	γ	γ	NOUN
ejpam-3397	31	44	-	-	NOUN
ejpam-3397	31	45	seminearrings	seminearring	NOUN
ejpam-3397	31	46	in	in	ADP
ejpam-3397	31	47	this	this	DET
ejpam-3397	31	48	section	section	NOUN
ejpam-3397	31	49	we	we	PRON
ejpam-3397	31	50	introduce	introduce	VERB
ejpam-3397	31	51	the	the	DET
ejpam-3397	31	52	notion	notion	NOUN
ejpam-3397	31	53	of	of	ADP
ejpam-3397	31	54	weakly	weakly	ADJ
ejpam-3397	31	55	prime	prime	ADJ
ejpam-3397	31	56	and	and	CCONJ
ejpam-3397	31	57	weakly	weakly	ADJ
ejpam-3397	31	58	primary	primary	ADJ
ejpam-3397	31	59	ideals	ideal	NOUN
ejpam-3397	31	60	in	in	ADP
ejpam-3397	31	61	γ	γ	NOUN
ejpam-3397	31	62	-	-	NOUN
ejpam-3397	31	63	seminearrings	seminearring	NOUN
ejpam-3397	31	64	.	.	PUNCT
ejpam-3397	32	1	by	by	ADP
ejpam-3397	32	2	an	an	DET
ejpam-3397	32	3	ideal	ideal	NOUN
ejpam-3397	32	4	we	we	PRON
ejpam-3397	32	5	mean	mean	VERB
ejpam-3397	32	6	two	two	NUM
ejpam-3397	32	7	-	-	PUNCT
ejpam-3397	32	8	sided	sided	ADJ
ejpam-3397	32	9	ideal	ideal	NOUN
ejpam-3397	32	10	unless	unless	SCONJ
ejpam-3397	32	11	otherwise	otherwise	ADV
ejpam-3397	32	12	stated	state	VERB
ejpam-3397	32	13	.	.	PUNCT
ejpam-3397	33	1	we	we	PRON
ejpam-3397	33	2	begin	begin	VERB
ejpam-3397	33	3	with	with	ADP
ejpam-3397	33	4	the	the	DET
ejpam-3397	33	5	following	follow	VERB
ejpam-3397	33	6	definition	definition	NOUN
ejpam-3397	33	7	.	.	PUNCT
ejpam-3397	34	1	definition	definition	NOUN
ejpam-3397	34	2	1	1	NUM
ejpam-3397	34	3	.	.	PUNCT
ejpam-3397	35	1	let	let	VERB
ejpam-3397	35	2	r	r	PRON
ejpam-3397	35	3	be	be	AUX
ejpam-3397	35	4	a	a	DET
ejpam-3397	35	5	γ	γ	NOUN
ejpam-3397	35	6	-	-	PUNCT
ejpam-3397	35	7	seminearring	seminearring	NOUN
ejpam-3397	35	8	.	.	PUNCT
ejpam-3397	36	1	a	a	DET
ejpam-3397	36	2	proper	proper	ADJ
ejpam-3397	36	3	ideal	ideal	NOUN
ejpam-3397	36	4	p	p	NOUN
ejpam-3397	36	5	of	of	ADP
ejpam-3397	36	6	r	r	NOUN
ejpam-3397	36	7	is	be	AUX
ejpam-3397	36	8	called	call	VERB
ejpam-3397	36	9	weakly	weakly	ADJ
ejpam-3397	36	10	prime	prime	NOUN
ejpam-3397	36	11	if	if	SCONJ
ejpam-3397	36	12	for	for	ADP
ejpam-3397	36	13	ideals	ideal	NOUN
ejpam-3397	36	14	i	i	PRON
ejpam-3397	36	15	and	and	CCONJ
ejpam-3397	36	16	j	j	PROPN
ejpam-3397	36	17	,	,	PUNCT
ejpam-3397	36	18	0	0	NUM
ejpam-3397	36	19	6=	6=	NUM
ejpam-3397	36	20	iγj	iγj	VERB
ejpam-3397	36	21	⊆	⊆	NUM
ejpam-3397	36	22	p	p	NOUN
ejpam-3397	36	23	implies	imply	VERB
ejpam-3397	36	24	i	i	PRON
ejpam-3397	36	25	⊆	⊆	NUM
ejpam-3397	36	26	p	p	NOUN
ejpam-3397	36	27	or	or	CCONJ
ejpam-3397	36	28	j	j	PROPN
ejpam-3397	36	29	⊆	⊆	NUM
ejpam-3397	36	30	p.	p.	NOUN
ejpam-3397	36	31	proposition	proposition	NOUN
ejpam-3397	37	1	1	1	X
ejpam-3397	37	2	.	.	PUNCT
ejpam-3397	38	1	let	let	VERB
ejpam-3397	38	2	p	p	PRON
ejpam-3397	38	3	be	be	AUX
ejpam-3397	38	4	a	a	DET
ejpam-3397	38	5	proper	proper	ADJ
ejpam-3397	38	6	ideal	ideal	NOUN
ejpam-3397	38	7	of	of	ADP
ejpam-3397	38	8	a	a	DET
ejpam-3397	38	9	γ	γ	NOUN
ejpam-3397	38	10	-	-	PUNCT
ejpam-3397	38	11	seminearring	seminearre	VERB
ejpam-3397	38	12	r.	r.	NOUN
ejpam-3397	38	13	the	the	DET
ejpam-3397	38	14	following	follow	VERB
ejpam-3397	38	15	statements	statement	NOUN
ejpam-3397	38	16	are	be	AUX
ejpam-3397	38	17	equivalent	equivalent	ADJ
ejpam-3397	38	18	.	.	PUNCT
ejpam-3397	39	1	(	(	PUNCT
ejpam-3397	39	2	i	i	NOUN
ejpam-3397	39	3	)	)	PUNCT
ejpam-3397	39	4	p	p	NOUN
ejpam-3397	39	5	is	be	AUX
ejpam-3397	39	6	weakly	weakly	ADV
ejpam-3397	39	7	prime	prime	ADJ
ejpam-3397	39	8	.	.	PUNCT
ejpam-3397	40	1	(	(	PUNCT
ejpam-3397	40	2	ii	ii	NOUN
ejpam-3397	40	3	)	)	PUNCT
ejpam-3397	40	4	for	for	ADP
ejpam-3397	40	5	ideals	ideal	NOUN
ejpam-3397	41	1	i	i	PRON
ejpam-3397	41	2	and	and	CCONJ
ejpam-3397	41	3	j	j	PROPN
ejpam-3397	41	4	of	of	ADP
ejpam-3397	41	5	r	r	PROPN
ejpam-3397	41	6	,	,	PUNCT
ejpam-3397	41	7	0	0	NUM
ejpam-3397	41	8	6=	6=	NUM
ejpam-3397	41	9	(	(	PUNCT
ejpam-3397	41	10	iγj	iγj	PROPN
ejpam-3397	41	11	)	)	PUNCT
ejpam-3397	41	12	⊆	⊆	NUM
ejpam-3397	41	13	p	p	NOUN
ejpam-3397	41	14	implies	imply	VERB
ejpam-3397	41	15	i	i	PRON
ejpam-3397	41	16	⊆	⊆	NUM
ejpam-3397	41	17	p	p	NOUN
ejpam-3397	41	18	or	or	CCONJ
ejpam-3397	41	19	j	j	PROPN
ejpam-3397	41	20	⊆	⊆	NUM
ejpam-3397	41	21	p	p	NOUN
ejpam-3397	41	22	.	.	PUNCT
ejpam-3397	42	1	(	(	PUNCT
ejpam-3397	42	2	iii	iii	NOUN
ejpam-3397	42	3	)	)	PUNCT
ejpam-3397	42	4	for	for	ADP
ejpam-3397	42	5	elements	element	NOUN
ejpam-3397	43	1	i	i	PRON
ejpam-3397	43	2	and	and	CCONJ
ejpam-3397	43	3	j	j	PROPN
ejpam-3397	43	4	in	in	ADP
ejpam-3397	43	5	r	r	PROPN
ejpam-3397	43	6	,	,	PUNCT
ejpam-3397	43	7	i	i	PRON
ejpam-3397	43	8	/∈	/∈	PUNCT
ejpam-3397	44	1	p	p	NOUN
ejpam-3397	44	2	and	and	CCONJ
ejpam-3397	44	3	j	j	PROPN
ejpam-3397	44	4	/∈	/∈	PUNCT
ejpam-3397	45	1	p	p	PROPN
ejpam-3397	45	2	implies	imply	VERB
ejpam-3397	45	3	0	0	NUM
ejpam-3397	45	4	6=	6=	NUM
ejpam-3397	45	5	(	(	PUNCT
ejpam-3397	45	6	i)γ(j	i)γ(j	PROPN
ejpam-3397	45	7	)	)	PUNCT
ejpam-3397	45	8	*	*	PUNCT
ejpam-3397	46	1	p.	p.	NOUN
ejpam-3397	46	2	proof	proof	NOUN
ejpam-3397	46	3	.	.	PUNCT
ejpam-3397	47	1	following	follow	VERB
ejpam-3397	47	2	definition1	definition1	NOUN
ejpam-3397	47	3	,	,	PUNCT
ejpam-3397	47	4	clearly	clearly	ADV
ejpam-3397	47	5	(	(	PUNCT
ejpam-3397	47	6	i	i	NOUN
ejpam-3397	47	7	)	)	PUNCT
ejpam-3397	47	8	and	and	CCONJ
ejpam-3397	47	9	(	(	PUNCT
ejpam-3397	47	10	ii	ii	NOUN
ejpam-3397	47	11	)	)	PUNCT
ejpam-3397	47	12	are	be	AUX
ejpam-3397	47	13	equivalent	equivalent	ADJ
ejpam-3397	47	14	.	.	PUNCT
ejpam-3397	48	1	now	now	ADV
ejpam-3397	48	2	,	,	PUNCT
ejpam-3397	48	3	(	(	PUNCT
ejpam-3397	48	4	i	i	NOUN
ejpam-3397	48	5	)	)	PUNCT
ejpam-3397	48	6	=	=	NOUN
ejpam-3397	48	7	⇒	⇒	NOUN
ejpam-3397	48	8	(	(	PUNCT
ejpam-3397	48	9	iii	iii	X
ejpam-3397	48	10	)	)	PUNCT
ejpam-3397	48	11	let	let	VERB
ejpam-3397	48	12	p	p	PRON
ejpam-3397	48	13	be	be	AUX
ejpam-3397	48	14	a	a	DET
ejpam-3397	48	15	weakly	weakly	ADJ
ejpam-3397	48	16	prime	prime	NOUN
ejpam-3397	48	17	,	,	PUNCT
ejpam-3397	49	1	i	i	PRON
ejpam-3397	49	2	/∈	/∈	PUNCT
ejpam-3397	50	1	p	p	NOUN
ejpam-3397	50	2	and	and	CCONJ
ejpam-3397	50	3	j	j	PROPN
ejpam-3397	50	4	/∈	/∈	PUNCT
ejpam-3397	51	1	p	p	X
ejpam-3397	51	2	.	.	PUNCT
ejpam-3397	52	1	assume	assume	VERB
ejpam-3397	52	2	0	0	NUM
ejpam-3397	53	1	6=	6=	NUM
ejpam-3397	53	2	(	(	PUNCT
ejpam-3397	53	3	i	i	NOUN
ejpam-3397	53	4	)	)	PUNCT
ejpam-3397	53	5	γ(j	γ(j	NOUN
ejpam-3397	53	6	)	)	PUNCT
ejpam-3397	53	7	⊆	⊆	NUM
ejpam-3397	53	8	p	p	NOUN
ejpam-3397	53	9	⇒	⇒	NOUN
ejpam-3397	53	10	(	(	PUNCT
ejpam-3397	53	11	i	i	NOUN
ejpam-3397	53	12	)	)	PUNCT
ejpam-3397	54	1	⊆	⊆	NUM
ejpam-3397	54	2	p	p	NOUN
ejpam-3397	54	3	or	or	CCONJ
ejpam-3397	54	4	(	(	PUNCT
ejpam-3397	54	5	j	j	PROPN
ejpam-3397	54	6	)	)	PUNCT
ejpam-3397	54	7	⊆	⊆	NUM
ejpam-3397	54	8	p	p	NOUN
ejpam-3397	54	9	.	.	PUNCT
ejpam-3397	55	1	hence	hence	ADV
ejpam-3397	55	2	,	,	PUNCT
ejpam-3397	55	3	i	i	PRON
ejpam-3397	55	4	∈	∈	VERB
ejpam-3397	55	5	p	p	NOUN
ejpam-3397	55	6	or	or	CCONJ
ejpam-3397	55	7	j	j	PROPN
ejpam-3397	55	8	∈	∈	PROPN
ejpam-3397	55	9	p	p	PROPN
ejpam-3397	55	10	,	,	PUNCT
ejpam-3397	55	11	a	a	DET
ejpam-3397	55	12	contradiction	contradiction	NOUN
ejpam-3397	55	13	.	.	PUNCT
ejpam-3397	56	1	thus	thus	ADV
ejpam-3397	56	2	,	,	PUNCT
ejpam-3397	56	3	0	0	NUM
ejpam-3397	56	4	6=	6=	NUM
ejpam-3397	56	5	(	(	PUNCT
ejpam-3397	56	6	i)γ(j	i)γ(j	PROPN
ejpam-3397	56	7	)	)	PUNCT
ejpam-3397	56	8	*	*	PUNCT
ejpam-3397	57	1	p	p	X
ejpam-3397	57	2	.	.	PUNCT
ejpam-3397	58	1	(	(	PUNCT
ejpam-3397	58	2	iii	iii	NOUN
ejpam-3397	58	3	)	)	PUNCT
ejpam-3397	58	4	=	=	NOUN
ejpam-3397	58	5	⇒	⇒	NOUN
ejpam-3397	58	6	(	(	PUNCT
ejpam-3397	58	7	i	i	NOUN
ejpam-3397	58	8	)	)	PUNCT
ejpam-3397	58	9	assume	assume	VERB
ejpam-3397	58	10	that	that	SCONJ
ejpam-3397	58	11	i	i	PRON
ejpam-3397	58	12	*	*	PUNCT
ejpam-3397	58	13	p	p	PROPN
ejpam-3397	58	14	and	and	CCONJ
ejpam-3397	58	15	j	j	PROPN
ejpam-3397	59	1	*	*	PUNCT
ejpam-3397	59	2	p	p	X
ejpam-3397	59	3	.	.	PUNCT
ejpam-3397	60	1	then	then	ADV
ejpam-3397	60	2	there	there	PRON
ejpam-3397	60	3	exists	exist	VERB
ejpam-3397	60	4	i	i	PRON
ejpam-3397	60	5	∈	∈	PROPN
ejpam-3397	60	6	i\p	i\p	PROPN
ejpam-3397	60	7	and	and	CCONJ
ejpam-3397	60	8	j	j	PROPN
ejpam-3397	60	9	∈	∈	PROPN
ejpam-3397	60	10	j\p	j\p	PROPN
ejpam-3397	60	11	.	.	PUNCT
ejpam-3397	61	1	hence	hence	ADV
ejpam-3397	61	2	,	,	PUNCT
ejpam-3397	61	3	0	0	PROPN
ejpam-3397	61	4	6=	6=	NUM
ejpam-3397	61	5	(	(	PUNCT
ejpam-3397	61	6	i)γ(j	i)γ(j	PROPN
ejpam-3397	61	7	)	)	PUNCT
ejpam-3397	61	8	⊆	⊆	NUM
ejpam-3397	61	9	0	0	NUM
ejpam-3397	61	10	6=	6=	ADP
ejpam-3397	61	11	iγj	iγj	NOUN
ejpam-3397	61	12	but	but	CCONJ
ejpam-3397	61	13	0	0	NUM
ejpam-3397	61	14	6=	6=	NUM
ejpam-3397	61	15	(	(	PUNCT
ejpam-3397	61	16	i)γ(j	i)γ(j	PROPN
ejpam-3397	61	17	)	)	PUNCT
ejpam-3397	61	18	*	*	PUNCT
ejpam-3397	62	1	p	p	NOUN
ejpam-3397	62	2	by	by	ADP
ejpam-3397	62	3	(	(	PUNCT
ejpam-3397	62	4	iii	iii	NOUN
ejpam-3397	62	5	)	)	PUNCT
ejpam-3397	62	6	.	.	PUNCT
ejpam-3397	63	1	thus	thus	ADV
ejpam-3397	63	2	,	,	PUNCT
ejpam-3397	63	3	0	0	PUNCT
ejpam-3397	63	4	6=	6=	NUM
ejpam-3397	63	5	iγj	iγj	NOUN
ejpam-3397	63	6	*	*	PUNCT
ejpam-3397	63	7	p.	p.	NOUN
ejpam-3397	63	8	example	example	NOUN
ejpam-3397	64	1	1	1	X
ejpam-3397	64	2	.	.	PUNCT
ejpam-3397	65	1	let	let	AUX
ejpam-3397	65	2	r	r	NOUN
ejpam-3397	65	3	=	=	SYM
ejpam-3397	65	4	{	{	PUNCT
ejpam-3397	65	5	0	0	NUM
ejpam-3397	65	6	,	,	PUNCT
ejpam-3397	65	7	1	1	NUM
ejpam-3397	65	8	,	,	PUNCT
ejpam-3397	65	9	e	e	NOUN
ejpam-3397	65	10	,	,	PUNCT
ejpam-3397	65	11	a	a	DET
ejpam-3397	65	12	,	,	PUNCT
ejpam-3397	65	13	b	b	NOUN
ejpam-3397	65	14	,	,	PUNCT
ejpam-3397	65	15	c	c	AUX
ejpam-3397	65	16	}	}	PUNCT
ejpam-3397	65	17	be	be	AUX
ejpam-3397	65	18	a	a	DET
ejpam-3397	65	19	γ	γ	NOUN
ejpam-3397	65	20	-	-	PUNCT
ejpam-3397	65	21	seminearring	seminearring	NOUN
ejpam-3397	65	22	with	with	ADP
ejpam-3397	65	23	γ	γ	X
ejpam-3397	65	24	=	=	SYM
ejpam-3397	65	25	{	{	PUNCT
ejpam-3397	65	26	α	α	NOUN
ejpam-3397	65	27	,	,	PUNCT
ejpam-3397	65	28	1	1	NUM
ejpam-3397	65	29	}	}	PUNCT
ejpam-3397	65	30	.	.	PUNCT
ejpam-3397	66	1	+	+	CCONJ
ejpam-3397	66	2	0	0	NUM
ejpam-3397	66	3	1	1	NUM
ejpam-3397	66	4	e	e	NOUN
ejpam-3397	66	5	a	a	DET
ejpam-3397	66	6	b	b	X
ejpam-3397	66	7	c	c	NOUN
ejpam-3397	66	8	0	0	NUM
ejpam-3397	66	9	0	0	NUM
ejpam-3397	66	10	1	1	NUM
ejpam-3397	66	11	e	e	NOUN
ejpam-3397	66	12	a	a	DET
ejpam-3397	66	13	a	a	DET
ejpam-3397	66	14	c	c	NOUN
ejpam-3397	66	15	1	1	NUM
ejpam-3397	66	16	1	1	NUM
ejpam-3397	66	17	1	1	NUM
ejpam-3397	66	18	1	1	NUM
ejpam-3397	66	19	1	1	NUM
ejpam-3397	66	20	1	1	NUM
ejpam-3397	66	21	1	1	NUM
ejpam-3397	66	22	e	e	NOUN
ejpam-3397	66	23	e	e	X
ejpam-3397	66	24	1	1	NUM
ejpam-3397	66	25	e	e	SYM
ejpam-3397	66	26	1	1	NUM
ejpam-3397	66	27	1	1	NUM
ejpam-3397	66	28	e	e	NOUN
ejpam-3397	66	29	a	a	DET
ejpam-3397	66	30	a	a	DET
ejpam-3397	66	31	1	1	NUM
ejpam-3397	66	32	1	1	NUM
ejpam-3397	66	33	a	a	DET
ejpam-3397	66	34	a	a	DET
ejpam-3397	66	35	a	a	DET
ejpam-3397	66	36	b	b	NOUN
ejpam-3397	66	37	a	a	DET
ejpam-3397	66	38	1	1	NUM
ejpam-3397	66	39	1	1	NUM
ejpam-3397	66	40	a	a	PRON
ejpam-3397	66	41	a	a	DET
ejpam-3397	66	42	a	a	DET
ejpam-3397	66	43	c	c	NOUN
ejpam-3397	66	44	c	c	NOUN
ejpam-3397	66	45	1	1	NUM
ejpam-3397	66	46	e	e	NOUN
ejpam-3397	66	47	a	a	DET
ejpam-3397	66	48	a	a	DET
ejpam-3397	66	49	c	c	PROPN
ejpam-3397	66	50	abdelghani	abdelghani	PROPN
ejpam-3397	66	51	taouti	taouti	PROPN
ejpam-3397	66	52	et	et	PROPN
ejpam-3397	66	53	al	al	PROPN
ejpam-3397	66	54	.	.	PUNCT
ejpam-3397	66	55	/	/	SYM
ejpam-3397	66	56	eur	eur	PROPN
ejpam-3397	66	57	.	.	PUNCT
ejpam-3397	67	1	j.	j.	PROPN
ejpam-3397	67	2	pure	pure	PROPN
ejpam-3397	67	3	appl	appl	PROPN
ejpam-3397	67	4	.	.	PROPN
ejpam-3397	67	5	math	math	PROPN
ejpam-3397	67	6	,	,	PUNCT
ejpam-3397	67	7	12	12	NUM
ejpam-3397	67	8	(	(	PUNCT
ejpam-3397	67	9	2	2	NUM
ejpam-3397	67	10	)	)	PUNCT
ejpam-3397	67	11	(	(	PUNCT
ejpam-3397	67	12	2019	2019	NUM
ejpam-3397	67	13	)	)	PUNCT
ejpam-3397	67	14	,	,	PUNCT
ejpam-3397	67	15	544	544	NUM
ejpam-3397	67	16	-	-	SYM
ejpam-3397	67	17	552	552	NUM
ejpam-3397	67	18	546	546	NUM
ejpam-3397	67	19	α	α	NOUN
ejpam-3397	67	20	0	0	NUM
ejpam-3397	67	21	1	1	NUM
ejpam-3397	67	22	e	e	NOUN
ejpam-3397	67	23	a	a	DET
ejpam-3397	67	24	b	b	X
ejpam-3397	67	25	c	c	NOUN
ejpam-3397	67	26	0	0	NUM
ejpam-3397	67	27	0	0	NUM
ejpam-3397	67	28	0	0	NUM
ejpam-3397	67	29	0	0	NUM
ejpam-3397	67	30	0	0	NUM
ejpam-3397	67	31	0	0	NUM
ejpam-3397	67	32	0	0	NUM
ejpam-3397	67	33	1	1	NUM
ejpam-3397	67	34	0	0	NUM
ejpam-3397	67	35	1	1	NUM
ejpam-3397	67	36	e	e	NOUN
ejpam-3397	67	37	a	a	DET
ejpam-3397	67	38	b	b	NOUN
ejpam-3397	67	39	c	c	NOUN
ejpam-3397	67	40	e	e	X
ejpam-3397	67	41	0	0	NUM
ejpam-3397	67	42	e	e	X
ejpam-3397	67	43	e	e	X
ejpam-3397	67	44	0	0	PROPN
ejpam-3397	68	1	c	c	NOUN
ejpam-3397	68	2	c	c	PROPN
ejpam-3397	68	3	a	a	DET
ejpam-3397	68	4	0	0	NUM
ejpam-3397	68	5	a	a	DET
ejpam-3397	68	6	0	0	NUM
ejpam-3397	68	7	a	a	DET
ejpam-3397	68	8	a	a	DET
ejpam-3397	68	9	0	0	NUM
ejpam-3397	68	10	b	b	NOUN
ejpam-3397	68	11	0	0	NUM
ejpam-3397	68	12	b	b	NOUN
ejpam-3397	68	13	0	0	NUM
ejpam-3397	68	14	a	a	DET
ejpam-3397	68	15	a	a	DET
ejpam-3397	68	16	0	0	NUM
ejpam-3397	68	17	c	c	NOUN
ejpam-3397	68	18	0	0	PUNCT
ejpam-3397	69	1	c	c	NOUN
ejpam-3397	69	2	0	0	NUM
ejpam-3397	69	3	0	0	NUM
ejpam-3397	69	4	0	0	NUM
ejpam-3397	69	5	0	0	NUM
ejpam-3397	70	1	p	p	NOUN
ejpam-3397	70	2	=	=	X
ejpam-3397	70	3	{	{	PUNCT
ejpam-3397	70	4	0	0	NUM
ejpam-3397	70	5	,	,	PUNCT
ejpam-3397	70	6	a	a	DET
ejpam-3397	70	7	,	,	PUNCT
ejpam-3397	70	8	b	b	NOUN
ejpam-3397	70	9	}	}	PUNCT
ejpam-3397	70	10	of	of	ADP
ejpam-3397	70	11	a	a	DET
ejpam-3397	70	12	seminearring	seminearring	NOUN
ejpam-3397	70	13	r	r	NOUN
ejpam-3397	70	14	is	be	AUX
ejpam-3397	70	15	a	a	DET
ejpam-3397	70	16	weakly	weakly	ADJ
ejpam-3397	70	17	prime	prime	ADJ
ejpam-3397	70	18	ideal	ideal	NOUN
ejpam-3397	70	19	but	but	CCONJ
ejpam-3397	70	20	not	not	PART
ejpam-3397	70	21	a	a	DET
ejpam-3397	70	22	prime	prime	ADJ
ejpam-3397	70	23	ideal	ideal	NOUN
ejpam-3397	70	24	,	,	PUNCT
ejpam-3397	70	25	since	since	SCONJ
ejpam-3397	70	26	cαc	cαc	NOUN
ejpam-3397	70	27	=	=	SYM
ejpam-3397	70	28	0	0	NUM
ejpam-3397	70	29	and	and	CCONJ
ejpam-3397	70	30	c	c	NOUN
ejpam-3397	70	31	/∈	/∈	PUNCT
ejpam-3397	71	1	p	p	X
ejpam-3397	71	2	.	.	PUNCT
ejpam-3397	72	1	on	on	ADP
ejpam-3397	72	2	the	the	DET
ejpam-3397	72	3	other	other	ADJ
ejpam-3397	72	4	hand	hand	NOUN
ejpam-3397	72	5	,	,	PUNCT
ejpam-3397	72	6	consider	consider	VERB
ejpam-3397	72	7	a	a	DET
ejpam-3397	72	8	prime	prime	ADJ
ejpam-3397	72	9	ideal	ideal	NOUN
ejpam-3397	72	10	q	q	NOUN
ejpam-3397	73	1	=	=	PUNCT
ejpam-3397	73	2	{	{	PUNCT
ejpam-3397	73	3	0	0	NUM
ejpam-3397	73	4	,	,	PUNCT
ejpam-3397	73	5	e	e	NOUN
ejpam-3397	73	6	,	,	PUNCT
ejpam-3397	73	7	c	c	NOUN
ejpam-3397	73	8	}	}	PUNCT
ejpam-3397	73	9	of	of	ADP
ejpam-3397	73	10	r.	r.	PROPN
ejpam-3397	73	11	it	it	PRON
ejpam-3397	73	12	is	be	AUX
ejpam-3397	73	13	easy	easy	ADJ
ejpam-3397	73	14	to	to	PART
ejpam-3397	73	15	show	show	VERB
ejpam-3397	73	16	that	that	PRON
ejpam-3397	73	17	q	q	NOUN
ejpam-3397	73	18	is	be	AUX
ejpam-3397	73	19	a	a	DET
ejpam-3397	73	20	weakly	weakly	ADJ
ejpam-3397	73	21	prime	prime	ADJ
ejpam-3397	73	22	ideal	ideal	NOUN
ejpam-3397	73	23	.	.	PUNCT
ejpam-3397	74	1	for	for	ADP
ejpam-3397	74	2	this	this	PRON
ejpam-3397	74	3	,	,	PUNCT
ejpam-3397	74	4	let	let	VERB
ejpam-3397	74	5	i	i	PRON
ejpam-3397	74	6	and	and	CCONJ
ejpam-3397	74	7	j	j	PROPN
ejpam-3397	74	8	are	be	AUX
ejpam-3397	74	9	ideals	ideal	NOUN
ejpam-3397	74	10	of	of	ADP
ejpam-3397	74	11	r	r	NOUN
ejpam-3397	74	12	,	,	PUNCT
ejpam-3397	74	13	where	where	SCONJ
ejpam-3397	74	14	i	i	PRON
ejpam-3397	74	15	=	=	PUNCT
ejpam-3397	74	16	{	{	PUNCT
ejpam-3397	74	17	0	0	NUM
ejpam-3397	74	18	,	,	PUNCT
ejpam-3397	74	19	e	e	NOUN
ejpam-3397	74	20	,	,	PUNCT
ejpam-3397	74	21	c	c	NOUN
ejpam-3397	74	22	}	}	PUNCT
ejpam-3397	74	23	and	and	CCONJ
ejpam-3397	74	24	j	j	PROPN
ejpam-3397	74	25	=	=	PUNCT
ejpam-3397	74	26	{	{	PUNCT
ejpam-3397	74	27	0	0	NUM
ejpam-3397	74	28	,	,	PUNCT
ejpam-3397	74	29	a	a	DET
ejpam-3397	74	30	,	,	PUNCT
ejpam-3397	74	31	b	b	NOUN
ejpam-3397	74	32	}	}	PUNCT
ejpam-3397	74	33	.	.	PUNCT
ejpam-3397	75	1	then	then	ADV
ejpam-3397	75	2	,	,	PUNCT
ejpam-3397	75	3	0	0	NUM
ejpam-3397	75	4	6=	6=	NUM
ejpam-3397	75	5	iγj	iγj	VERB
ejpam-3397	75	6	⊆	⊆	NUM
ejpam-3397	75	7	q	q	NOUN
ejpam-3397	75	8	=	=	NOUN
ejpam-3397	75	9	⇒	⇒	NOUN
ejpam-3397	75	10	i	i	PRON
ejpam-3397	75	11	⊆	⊆	NUM
ejpam-3397	75	12	q	q	NOUN
ejpam-3397	75	13	=	=	NOUN
ejpam-3397	75	14	⇒	⇒	NOUN
ejpam-3397	75	15	q	q	PUNCT
ejpam-3397	75	16	is	be	AUX
ejpam-3397	75	17	a	a	DET
ejpam-3397	75	18	weakly	weakly	ADJ
ejpam-3397	75	19	prime	prime	ADJ
ejpam-3397	75	20	ideal	ideal	NOUN
ejpam-3397	75	21	.	.	PUNCT
ejpam-3397	76	1	hence	hence	ADV
ejpam-3397	76	2	every	every	DET
ejpam-3397	76	3	prime	prime	ADJ
ejpam-3397	76	4	ideal	ideal	NOUN
ejpam-3397	76	5	of	of	ADP
ejpam-3397	76	6	a	a	DET
ejpam-3397	76	7	gamma	gamma	NOUN
ejpam-3397	76	8	seminearring	seminearring	NOUN
ejpam-3397	76	9	is	be	AUX
ejpam-3397	76	10	a	a	DET
ejpam-3397	76	11	weakly	weakly	ADJ
ejpam-3397	76	12	prime	prime	ADJ
ejpam-3397	76	13	ideal	ideal	NOUN
ejpam-3397	76	14	.	.	PUNCT
ejpam-3397	77	1	example	example	NOUN
ejpam-3397	78	1	2	2	NUM
ejpam-3397	78	2	.	.	PUNCT
ejpam-3397	78	3	let	let	VERB
ejpam-3397	78	4	r	r	NOUN
ejpam-3397	78	5	=	=	SYM
ejpam-3397	78	6	{	{	PUNCT
ejpam-3397	78	7	0	0	NUM
ejpam-3397	78	8	,	,	PUNCT
ejpam-3397	78	9	1	1	NUM
ejpam-3397	78	10	,	,	PUNCT
ejpam-3397	78	11	e	e	NOUN
ejpam-3397	78	12	,	,	PUNCT
ejpam-3397	78	13	a	a	DET
ejpam-3397	78	14	,	,	PUNCT
ejpam-3397	78	15	b	b	NOUN
ejpam-3397	78	16	,	,	PUNCT
ejpam-3397	78	17	c	c	AUX
ejpam-3397	78	18	}	}	PUNCT
ejpam-3397	78	19	be	be	AUX
ejpam-3397	78	20	a	a	DET
ejpam-3397	78	21	γ	γ	NOUN
ejpam-3397	78	22	-	-	PUNCT
ejpam-3397	78	23	seminearring	seminearring	NOUN
ejpam-3397	78	24	with	with	ADP
ejpam-3397	78	25	γ	γ	X
ejpam-3397	78	26	=	=	SYM
ejpam-3397	78	27	{	{	PUNCT
ejpam-3397	78	28	α	α	NOUN
ejpam-3397	78	29	,	,	PUNCT
ejpam-3397	78	30	1	1	NUM
ejpam-3397	78	31	}	}	PUNCT
ejpam-3397	78	32	as	as	SCONJ
ejpam-3397	78	33	defined	define	VERB
ejpam-3397	78	34	in	in	ADP
ejpam-3397	78	35	example1	example1	PROPN
ejpam-3397	78	36	.	.	PUNCT
ejpam-3397	79	1	let	let	VERB
ejpam-3397	79	2	s	s	PRON
ejpam-3397	79	3	=	=	X
ejpam-3397	79	4	{	{	PUNCT
ejpam-3397	79	5	0	0	NUM
ejpam-3397	79	6	,	,	PUNCT
ejpam-3397	79	7	1	1	NUM
ejpam-3397	79	8	,	,	PUNCT
ejpam-3397	79	9	e	e	NOUN
ejpam-3397	79	10	,	,	PUNCT
ejpam-3397	79	11	a	a	PRON
ejpam-3397	79	12	,	,	PUNCT
ejpam-3397	79	13	c	c	NOUN
ejpam-3397	79	14	}	}	PUNCT
ejpam-3397	79	15	⊆	⊆	NUM
ejpam-3397	79	16	r	r	NOUN
ejpam-3397	79	17	be	be	VERB
ejpam-3397	79	18	a	a	DET
ejpam-3397	79	19	γ	γ	NOUN
ejpam-3397	79	20	-	-	PUNCT
ejpam-3397	79	21	sub	sub	NOUN
ejpam-3397	79	22	-	-	ADJ
ejpam-3397	79	23	seminearring	seminearring	NOUN
ejpam-3397	79	24	of	of	ADP
ejpam-3397	79	25	r	r	NOUN
ejpam-3397	79	26	with	with	ADP
ejpam-3397	79	27	γ	γ	X
ejpam-3397	79	28	=	=	SYM
ejpam-3397	79	29	{	{	PUNCT
ejpam-3397	79	30	1	1	NUM
ejpam-3397	79	31	,	,	PUNCT
ejpam-3397	79	32	α}.clearly	α}.clearly	ADV
ejpam-3397	79	33	,	,	PUNCT
ejpam-3397	79	34	in	in	ADP
ejpam-3397	79	35	s	s	PRON
ejpam-3397	79	36	the	the	DET
ejpam-3397	79	37	ideals	ideal	NOUN
ejpam-3397	80	1	i	i	PRON
ejpam-3397	80	2	=	=	PUNCT
ejpam-3397	80	3	{	{	PUNCT
ejpam-3397	80	4	0	0	NUM
ejpam-3397	80	5	,	,	PUNCT
ejpam-3397	80	6	a	a	PRON
ejpam-3397	80	7	}	}	PUNCT
ejpam-3397	80	8	and	and	CCONJ
ejpam-3397	80	9	j	j	PROPN
ejpam-3397	80	10	=	=	PUNCT
ejpam-3397	80	11	{	{	PUNCT
ejpam-3397	80	12	0	0	NUM
ejpam-3397	80	13	,	,	PUNCT
ejpam-3397	80	14	c	c	NOUN
ejpam-3397	80	15	}	}	PUNCT
ejpam-3397	80	16	are	be	AUX
ejpam-3397	80	17	weakly	weakly	ADJ
ejpam-3397	80	18	prime	prime	ADJ
ejpam-3397	80	19	ideals	ideal	NOUN
ejpam-3397	80	20	but	but	CCONJ
ejpam-3397	80	21	not	not	PART
ejpam-3397	80	22	prime	prime	ADJ
ejpam-3397	80	23	.	.	PUNCT
ejpam-3397	81	1	proposition	proposition	NOUN
ejpam-3397	81	2	2	2	NUM
ejpam-3397	81	3	.	.	PUNCT
ejpam-3397	82	1	let	let	VERB
ejpam-3397	82	2	p	p	PRON
ejpam-3397	82	3	be	be	AUX
ejpam-3397	82	4	a	a	DET
ejpam-3397	82	5	proper	proper	ADJ
ejpam-3397	82	6	ideal	ideal	NOUN
ejpam-3397	82	7	of	of	ADP
ejpam-3397	82	8	a	a	DET
ejpam-3397	82	9	γ	γ	NOUN
ejpam-3397	82	10	-	-	PUNCT
ejpam-3397	82	11	seminearring	seminearre	VERB
ejpam-3397	82	12	r	r	NOUN
ejpam-3397	82	13	and	and	CCONJ
ejpam-3397	82	14	{	{	PUNCT
ejpam-3397	82	15	0	0	PROPN
ejpam-3397	82	16	6=	6=	ADP
ejpam-3397	82	17	aαrβb	aαrβb	PROPN
ejpam-3397	82	18	:	:	PUNCT
ejpam-3397	82	19	r	r	NOUN
ejpam-3397	82	20	∈	∈	PROPN
ejpam-3397	82	21	r	r	NOUN
ejpam-3397	82	22	,	,	PUNCT
ejpam-3397	82	23	α	α	NOUN
ejpam-3397	82	24	,	,	PUNCT
ejpam-3397	82	25	β	β	PROPN
ejpam-3397	82	26	∈	∈	PROPN
ejpam-3397	82	27	γ	γ	X
ejpam-3397	82	28	}	}	PUNCT
ejpam-3397	82	29	⊆	⊆	NUM
ejpam-3397	82	30	p	p	NOUN
ejpam-3397	82	31	if	if	SCONJ
ejpam-3397	82	32	and	and	CCONJ
ejpam-3397	82	33	only	only	ADV
ejpam-3397	82	34	if	if	SCONJ
ejpam-3397	82	35	a	a	DET
ejpam-3397	82	36	∈	∈	PROPN
ejpam-3397	82	37	p	p	NOUN
ejpam-3397	82	38	or	or	CCONJ
ejpam-3397	82	39	b	b	NOUN
ejpam-3397	82	40	∈	∈	PROPN
ejpam-3397	82	41	p	p	NOUN
ejpam-3397	82	42	,	,	PUNCT
ejpam-3397	82	43	then	then	ADV
ejpam-3397	82	44	p	p	NOUN
ejpam-3397	82	45	is	be	AUX
ejpam-3397	82	46	a	a	DET
ejpam-3397	82	47	weakly	weakly	ADJ
ejpam-3397	82	48	prime	prime	ADJ
ejpam-3397	82	49	ideal	ideal	NOUN
ejpam-3397	82	50	.	.	PUNCT
ejpam-3397	83	1	proof	proof	NOUN
ejpam-3397	83	2	.	.	PUNCT
ejpam-3397	84	1	let	let	VERB
ejpam-3397	84	2	i	i	PRON
ejpam-3397	84	3	and	and	CCONJ
ejpam-3397	84	4	j	j	PROPN
ejpam-3397	84	5	are	be	AUX
ejpam-3397	84	6	ideals	ideal	NOUN
ejpam-3397	84	7	of	of	ADP
ejpam-3397	84	8	r	r	NOUN
ejpam-3397	84	9	with	with	ADP
ejpam-3397	84	10	0	0	NUM
ejpam-3397	84	11	6=	6=	NUM
ejpam-3397	84	12	iγj	iγj	NOUN
ejpam-3397	84	13	⊆	⊆	NUM
ejpam-3397	84	14	p	p	NOUN
ejpam-3397	84	15	.	.	PUNCT
ejpam-3397	85	1	let	let	VERB
ejpam-3397	86	1	i	i	PRON
ejpam-3397	86	2	*	*	VERB
ejpam-3397	86	3	p	p	X
ejpam-3397	86	4	,	,	PUNCT
ejpam-3397	86	5	and	and	CCONJ
ejpam-3397	86	6	for	for	ADP
ejpam-3397	86	7	a	a	DET
ejpam-3397	86	8	∈	∈	PROPN
ejpam-3397	86	9	i\p	i\p	NOUN
ejpam-3397	86	10	,	,	PUNCT
ejpam-3397	86	11	b	b	PROPN
ejpam-3397	86	12	∈	∈	PROPN
ejpam-3397	86	13	j	j	PROPN
ejpam-3397	86	14	,	,	PUNCT
ejpam-3397	86	15	we	we	PRON
ejpam-3397	86	16	have	have	VERB
ejpam-3397	86	17	{	{	PUNCT
ejpam-3397	86	18	0	0	NUM
ejpam-3397	86	19	6=	6=	NUM
ejpam-3397	86	20	aαrβb	aαrβb	PROPN
ejpam-3397	86	21	:	:	PUNCT
ejpam-3397	86	22	r	r	NOUN
ejpam-3397	86	23	∈	∈	PROPN
ejpam-3397	86	24	r	r	NOUN
ejpam-3397	86	25	,	,	PUNCT
ejpam-3397	86	26	α	α	NOUN
ejpam-3397	86	27	,	,	PUNCT
ejpam-3397	86	28	β	β	PROPN
ejpam-3397	86	29	∈	∈	PROPN
ejpam-3397	86	30	γ	γ	PROPN
ejpam-3397	86	31	}	}	PUNCT
ejpam-3397	86	32	⊆	⊆	NUM
ejpam-3397	86	33	iγj	iγj	NOUN
ejpam-3397	86	34	6=	6=	ADP
ejpam-3397	86	35	0	0	NUM
ejpam-3397	86	36	⊆	⊆	NUM
ejpam-3397	86	37	p.	p.	NOUN
ejpam-3397	86	38	since	since	SCONJ
ejpam-3397	86	39	a	a	DET
ejpam-3397	86	40	/∈	/∈	PUNCT
ejpam-3397	86	41	p	p	NOUN
ejpam-3397	86	42	and	and	CCONJ
ejpam-3397	86	43	b	b	NOUN
ejpam-3397	86	44	∈	∈	PROPN
ejpam-3397	86	45	p	p	PROPN
ejpam-3397	86	46	⇒	⇒	NOUN
ejpam-3397	86	47	j	j	PROPN
ejpam-3397	86	48	⊆	⊆	NUM
ejpam-3397	86	49	p	p	NOUN
ejpam-3397	86	50	.	.	PUNCT
ejpam-3397	87	1	hence	hence	ADV
ejpam-3397	87	2	,	,	PUNCT
ejpam-3397	87	3	p	p	PROPN
ejpam-3397	87	4	is	be	AUX
ejpam-3397	87	5	a	a	DET
ejpam-3397	87	6	weakly	weakly	ADJ
ejpam-3397	87	7	prime	prime	ADJ
ejpam-3397	87	8	ideal	ideal	NOUN
ejpam-3397	87	9	.	.	PUNCT
ejpam-3397	88	1	proposition	proposition	NOUN
ejpam-3397	88	2	3	3	NUM
ejpam-3397	88	3	.	.	PUNCT
ejpam-3397	88	4	intersection	intersection	NOUN
ejpam-3397	88	5	of	of	ADP
ejpam-3397	88	6	finite	finite	ADJ
ejpam-3397	88	7	numbers	number	NOUN
ejpam-3397	88	8	of	of	ADP
ejpam-3397	88	9	weakly	weakly	ADJ
ejpam-3397	88	10	prime	prime	ADJ
ejpam-3397	88	11	ideals	ideal	NOUN
ejpam-3397	88	12	of	of	ADP
ejpam-3397	88	13	a	a	DET
ejpam-3397	88	14	γ	γ	X
ejpam-3397	88	15	-	-	PUNCT
ejpam-3397	88	16	seminearring	seminearre	VERB
ejpam-3397	88	17	r	r	NOUN
ejpam-3397	88	18	which	which	PRON
ejpam-3397	88	19	are	be	AUX
ejpam-3397	88	20	totally	totally	ADV
ejpam-3397	88	21	ordered	order	VERB
ejpam-3397	88	22	by	by	ADP
ejpam-3397	88	23	inclusion	inclusion	NOUN
ejpam-3397	88	24	is	be	AUX
ejpam-3397	88	25	a	a	DET
ejpam-3397	88	26	weakly	weakly	ADJ
ejpam-3397	88	27	prime	prime	ADJ
ejpam-3397	88	28	ideal	ideal	NOUN
ejpam-3397	88	29	.	.	PUNCT
ejpam-3397	89	1	proof	proof	NOUN
ejpam-3397	89	2	.	.	PUNCT
ejpam-3397	90	1	let	let	VERB
ejpam-3397	90	2	{	{	PUNCT
ejpam-3397	90	3	pα}α∈λ	pα}α∈λ	VERB
ejpam-3397	90	4	be	be	AUX
ejpam-3397	90	5	the	the	DET
ejpam-3397	90	6	family	family	NOUN
ejpam-3397	90	7	of	of	ADP
ejpam-3397	90	8	weakly	weakly	ADJ
ejpam-3397	90	9	prime	prime	ADJ
ejpam-3397	90	10	ideals	ideal	NOUN
ejpam-3397	90	11	which	which	PRON
ejpam-3397	90	12	are	be	AUX
ejpam-3397	90	13	totally	totally	ADV
ejpam-3397	90	14	ordered	order	VERB
ejpam-3397	90	15	by	by	ADP
ejpam-3397	90	16	inclusion	inclusion	NOUN
ejpam-3397	90	17	.	.	PUNCT
ejpam-3397	91	1	suppose	suppose	VERB
ejpam-3397	91	2	i	i	PRON
ejpam-3397	91	3	and	and	CCONJ
ejpam-3397	91	4	j	j	PROPN
ejpam-3397	91	5	be	be	VERB
ejpam-3397	91	6	ideals	ideal	NOUN
ejpam-3397	91	7	of	of	ADP
ejpam-3397	91	8	r.	r.	PROPN
ejpam-3397	91	9	if	if	SCONJ
ejpam-3397	91	10	0	0	NUM
ejpam-3397	91	11	6=	6=	NUM
ejpam-3397	91	12	iγj	iγj	NOUN
ejpam-3397	91	13	⊆	⊆	NUM
ejpam-3397	91	14	∩α∈λpα	∩α∈λpα	NOUN
ejpam-3397	91	15	,	,	PUNCT
ejpam-3397	91	16	then	then	ADV
ejpam-3397	91	17	0	0	NUM
ejpam-3397	91	18	6=	6=	NUM
ejpam-3397	91	19	iγj	iγj	NOUN
ejpam-3397	91	20	⊆	⊆	NUM
ejpam-3397	91	21	pα	pα	NOUN
ejpam-3397	91	22	,	,	PUNCT
ejpam-3397	91	23	for	for	ADP
ejpam-3397	91	24	all	all	DET
ejpam-3397	91	25	α	α	DET
ejpam-3397	91	26	∈	∈	PROPN
ejpam-3397	91	27	λ	λ	PROPN
ejpam-3397	91	28	.	.	PROPN
ejpam-3397	91	29	suppose	suppose	VERB
ejpam-3397	91	30	that	that	SCONJ
ejpam-3397	91	31	there	there	PRON
ejpam-3397	91	32	exists	exist	VERB
ejpam-3397	91	33	α	α	PRON
ejpam-3397	91	34	∈	∈	PROPN
ejpam-3397	91	35	λ	λ	NOUN
ejpam-3397	91	36	such	such	ADJ
ejpam-3397	91	37	that	that	SCONJ
ejpam-3397	91	38	i	i	PRON
ejpam-3397	91	39	*	*	VERB
ejpam-3397	91	40	pα	pα	INTJ
ejpam-3397	91	41	.	.	PUNCT
ejpam-3397	92	1	then	then	ADV
ejpam-3397	92	2	,	,	PUNCT
ejpam-3397	92	3	j	j	PROPN
ejpam-3397	92	4	⊆	⊆	NUM
ejpam-3397	92	5	pα	pα	NOUN
ejpam-3397	92	6	and	and	CCONJ
ejpam-3397	92	7	hence	hence	ADV
ejpam-3397	92	8	j	j	PROPN
ejpam-3397	92	9	⊆	⊆	NUM
ejpam-3397	92	10	pβ	pβ	ADV
ejpam-3397	92	11	for	for	ADP
ejpam-3397	92	12	all	all	DET
ejpam-3397	92	13	β	β	NOUN
ejpam-3397	92	14	≥	≥	NUM
ejpam-3397	92	15	α	α	X
ejpam-3397	92	16	.	.	PUNCT
ejpam-3397	93	1	we	we	PRON
ejpam-3397	93	2	assume	assume	VERB
ejpam-3397	93	3	that	that	SCONJ
ejpam-3397	93	4	there	there	PRON
ejpam-3397	93	5	exist	exist	VERB
ejpam-3397	93	6	γ	γ	NOUN
ejpam-3397	93	7	<	<	X
ejpam-3397	93	8	α	α	PRON
ejpam-3397	93	9	such	such	ADJ
ejpam-3397	93	10	that	that	SCONJ
ejpam-3397	93	11	j	j	PROPN
ejpam-3397	93	12	⊆	⊆	NUM
ejpam-3397	93	13	pγ	pγ	NOUN
ejpam-3397	93	14	.then	.then	ADP
ejpam-3397	93	15	,	,	PUNCT
ejpam-3397	93	16	i	i	PRON
ejpam-3397	93	17	⊆	⊆	NUM
ejpam-3397	93	18	pγ	pγ	VERB
ejpam-3397	94	1	and	and	CCONJ
ejpam-3397	94	2	hence	hence	ADV
ejpam-3397	94	3	i	i	PRON
ejpam-3397	94	4	⊆	⊆	NUM
ejpam-3397	94	5	pα	pα	NOUN
ejpam-3397	94	6	,	,	PUNCT
ejpam-3397	94	7	which	which	PRON
ejpam-3397	94	8	is	be	AUX
ejpam-3397	94	9	impossible	impossible	ADJ
ejpam-3397	94	10	.	.	PUNCT
ejpam-3397	95	1	hence	hence	ADV
ejpam-3397	95	2	,	,	PUNCT
ejpam-3397	95	3	j	j	PROPN
ejpam-3397	95	4	⊆	⊆	NUM
ejpam-3397	95	5	pβ	pβ	NOUN
ejpam-3397	95	6	for	for	ADP
ejpam-3397	95	7	any	any	DET
ejpam-3397	95	8	β	β	X
ejpam-3397	95	9	∈	∈	PROPN
ejpam-3397	95	10	λ	λ	PROPN
ejpam-3397	95	11	.	.	PUNCT
ejpam-3397	96	1	thus	thus	ADV
ejpam-3397	96	2	,	,	PUNCT
ejpam-3397	96	3	∩α∈λpα	∩α∈λpα	ADV
ejpam-3397	96	4	is	be	AUX
ejpam-3397	96	5	a	a	DET
ejpam-3397	96	6	weakly	weakly	ADJ
ejpam-3397	96	7	prime	prime	ADJ
ejpam-3397	96	8	ideal	ideal	NOUN
ejpam-3397	96	9	of	of	ADP
ejpam-3397	96	10	a	a	DET
ejpam-3397	96	11	γ	γ	NOUN
ejpam-3397	96	12	-	-	PUNCT
ejpam-3397	96	13	seminearring	seminearre	VERB
ejpam-3397	96	14	r.	r.	PROPN
ejpam-3397	96	15	below	below	ADP
ejpam-3397	96	16	we	we	PRON
ejpam-3397	96	17	provide	provide	VERB
ejpam-3397	96	18	an	an	DET
ejpam-3397	96	19	illustrative	illustrative	ADJ
ejpam-3397	96	20	example	example	NOUN
ejpam-3397	96	21	.	.	PUNCT
ejpam-3397	97	1	example	example	NOUN
ejpam-3397	98	1	3	3	X
ejpam-3397	98	2	.	.	PUNCT
ejpam-3397	98	3	let	let	VERB
ejpam-3397	98	4	s	s	VERB
ejpam-3397	98	5	=	=	X
ejpam-3397	98	6	{	{	PUNCT
ejpam-3397	98	7	0	0	NUM
ejpam-3397	98	8	,	,	PUNCT
ejpam-3397	98	9	1	1	NUM
ejpam-3397	98	10	,	,	PUNCT
ejpam-3397	98	11	e	e	NOUN
ejpam-3397	98	12	,	,	PUNCT
ejpam-3397	98	13	a	a	PRON
ejpam-3397	98	14	,	,	PUNCT
ejpam-3397	98	15	c	c	NOUN
ejpam-3397	98	16	}	}	PUNCT
ejpam-3397	98	17	with	with	ADP
ejpam-3397	98	18	γ	γ	X
ejpam-3397	98	19	=	=	SYM
ejpam-3397	98	20	{	{	PUNCT
ejpam-3397	98	21	1	1	NUM
ejpam-3397	98	22	,	,	PUNCT
ejpam-3397	98	23	α	α	NOUN
ejpam-3397	98	24	}	}	PUNCT
ejpam-3397	98	25	be	be	AUX
ejpam-3397	98	26	a	a	DET
ejpam-3397	98	27	γ	γ	NOUN
ejpam-3397	98	28	-	-	PUNCT
ejpam-3397	98	29	seminearring	seminearring	NOUN
ejpam-3397	98	30	defined	define	VERB
ejpam-3397	98	31	in	in	ADP
ejpam-3397	98	32	the	the	DET
ejpam-3397	98	33	tables	table	NOUN
ejpam-3397	98	34	given	give	VERB
ejpam-3397	98	35	below	below	ADV
ejpam-3397	98	36	.	.	PUNCT
ejpam-3397	99	1	+	+	CCONJ
ejpam-3397	99	2	0	0	NUM
ejpam-3397	99	3	1	1	NUM
ejpam-3397	99	4	e	e	NOUN
ejpam-3397	99	5	a	a	PROPN
ejpam-3397	99	6	c	c	NOUN
ejpam-3397	99	7	0	0	NUM
ejpam-3397	99	8	0	0	NUM
ejpam-3397	99	9	1	1	NUM
ejpam-3397	99	10	e	e	NOUN
ejpam-3397	99	11	a	a	PROPN
ejpam-3397	99	12	c	c	NOUN
ejpam-3397	99	13	1	1	NUM
ejpam-3397	99	14	1	1	NUM
ejpam-3397	99	15	1	1	NUM
ejpam-3397	99	16	1	1	NUM
ejpam-3397	99	17	1	1	NUM
ejpam-3397	99	18	1	1	NUM
ejpam-3397	99	19	e	e	NOUN
ejpam-3397	99	20	e	e	X
ejpam-3397	99	21	1	1	NUM
ejpam-3397	99	22	e	e	SYM
ejpam-3397	99	23	1	1	NUM
ejpam-3397	99	24	e	e	NOUN
ejpam-3397	99	25	a	a	DET
ejpam-3397	99	26	a	a	DET
ejpam-3397	99	27	1	1	NUM
ejpam-3397	99	28	1	1	NUM
ejpam-3397	99	29	a	a	PRON
ejpam-3397	99	30	a	a	DET
ejpam-3397	99	31	c	c	NOUN
ejpam-3397	99	32	c	c	NOUN
ejpam-3397	99	33	1	1	NUM
ejpam-3397	99	34	e	e	NOUN
ejpam-3397	99	35	a	a	PROPN
ejpam-3397	99	36	c	c	NOUN
ejpam-3397	99	37	α	α	NOUN
ejpam-3397	99	38	0	0	NUM
ejpam-3397	99	39	1	1	NUM
ejpam-3397	99	40	e	e	NOUN
ejpam-3397	99	41	a	a	PROPN
ejpam-3397	99	42	c	c	NOUN
ejpam-3397	99	43	0	0	NUM
ejpam-3397	99	44	0	0	NUM
ejpam-3397	99	45	0	0	NUM
ejpam-3397	99	46	0	0	NUM
ejpam-3397	99	47	0	0	NUM
ejpam-3397	99	48	0	0	NUM
ejpam-3397	99	49	1	1	NUM
ejpam-3397	99	50	0	0	NUM
ejpam-3397	99	51	1	1	NUM
ejpam-3397	99	52	e	e	NOUN
ejpam-3397	99	53	a	a	DET
ejpam-3397	99	54	c	c	NOUN
ejpam-3397	99	55	e	e	NOUN
ejpam-3397	99	56	0	0	NUM
ejpam-3397	99	57	e	e	X
ejpam-3397	99	58	e	e	X
ejpam-3397	99	59	0	0	PUNCT
ejpam-3397	99	60	c	c	PROPN
ejpam-3397	99	61	a	a	DET
ejpam-3397	99	62	0	0	NUM
ejpam-3397	99	63	a	a	DET
ejpam-3397	99	64	0	0	NUM
ejpam-3397	99	65	a	a	DET
ejpam-3397	99	66	0	0	NUM
ejpam-3397	99	67	c	c	NOUN
ejpam-3397	99	68	0	0	PUNCT
ejpam-3397	100	1	c	c	NOUN
ejpam-3397	100	2	0	0	NUM
ejpam-3397	100	3	0	0	SYM
ejpam-3397	100	4	0	0	NUM
ejpam-3397	100	5	abdelghani	abdelghani	PROPN
ejpam-3397	100	6	taouti	taouti	NOUN
ejpam-3397	100	7	et	et	PROPN
ejpam-3397	100	8	al	al	PROPN
ejpam-3397	100	9	.	.	PUNCT
ejpam-3397	100	10	/	/	SYM
ejpam-3397	100	11	eur	eur	PROPN
ejpam-3397	100	12	.	.	PUNCT
ejpam-3397	101	1	j.	j.	PROPN
ejpam-3397	101	2	pure	pure	PROPN
ejpam-3397	101	3	appl	appl	PROPN
ejpam-3397	101	4	.	.	PROPN
ejpam-3397	101	5	math	math	PROPN
ejpam-3397	101	6	,	,	PUNCT
ejpam-3397	101	7	12	12	NUM
ejpam-3397	101	8	(	(	PUNCT
ejpam-3397	101	9	2	2	NUM
ejpam-3397	101	10	)	)	PUNCT
ejpam-3397	101	11	(	(	PUNCT
ejpam-3397	101	12	2019	2019	NUM
ejpam-3397	101	13	)	)	PUNCT
ejpam-3397	101	14	,	,	PUNCT
ejpam-3397	101	15	544	544	NUM
ejpam-3397	101	16	-	-	SYM
ejpam-3397	101	17	552	552	NUM
ejpam-3397	101	18	547	547	NUM
ejpam-3397	101	19	consider	consider	VERB
ejpam-3397	101	20	p1	p1	NOUN
ejpam-3397	101	21	=	=	SYM
ejpam-3397	101	22	{	{	PUNCT
ejpam-3397	101	23	0	0	NUM
ejpam-3397	101	24	,	,	PUNCT
ejpam-3397	101	25	c	c	NOUN
ejpam-3397	101	26	}	}	PUNCT
ejpam-3397	101	27	and	and	CCONJ
ejpam-3397	101	28	p2	p2	PROPN
ejpam-3397	101	29	=	=	SYM
ejpam-3397	101	30	{	{	PUNCT
ejpam-3397	101	31	0	0	NUM
ejpam-3397	101	32	,	,	PUNCT
ejpam-3397	101	33	e	e	NOUN
ejpam-3397	101	34	,	,	PUNCT
ejpam-3397	101	35	c	c	NOUN
ejpam-3397	101	36	}	}	PUNCT
ejpam-3397	101	37	are	be	AUX
ejpam-3397	101	38	the	the	DET
ejpam-3397	101	39	weakly	weakly	ADJ
ejpam-3397	101	40	prime	prime	ADJ
ejpam-3397	101	41	ideals	ideal	NOUN
ejpam-3397	101	42	of	of	ADP
ejpam-3397	101	43	r	r	NOUN
ejpam-3397	101	44	and	and	CCONJ
ejpam-3397	101	45	are	be	AUX
ejpam-3397	101	46	totally	totally	ADV
ejpam-3397	101	47	ordered	order	VERB
ejpam-3397	101	48	by	by	ADP
ejpam-3397	101	49	inclusion	inclusion	NOUN
ejpam-3397	101	50	as	as	ADV
ejpam-3397	101	51	well	well	ADV
ejpam-3397	101	52	.	.	PUNCT
ejpam-3397	102	1	since	since	SCONJ
ejpam-3397	102	2	,	,	PUNCT
ejpam-3397	102	3	p1	p1	NOUN
ejpam-3397	102	4	∩	∩	ADJ
ejpam-3397	102	5	p2	p2	PROPN
ejpam-3397	102	6	=	=	SYM
ejpam-3397	102	7	p1	p1	PROPN
ejpam-3397	102	8	,	,	PUNCT
ejpam-3397	102	9	which	which	PRON
ejpam-3397	102	10	is	be	AUX
ejpam-3397	102	11	a	a	DET
ejpam-3397	102	12	weakly	weakly	ADJ
ejpam-3397	102	13	prime	prime	ADJ
ejpam-3397	102	14	ideal	ideal	NOUN
ejpam-3397	102	15	of	of	ADP
ejpam-3397	102	16	r.	r.	PROPN
ejpam-3397	102	17	hence	hence	ADV
ejpam-3397	102	18	,	,	PUNCT
ejpam-3397	102	19	∩α∈apα	∩α∈apα	PROPN
ejpam-3397	102	20	is	be	AUX
ejpam-3397	102	21	a	a	DET
ejpam-3397	102	22	weakly	weakly	ADJ
ejpam-3397	102	23	prime	prime	ADJ
ejpam-3397	102	24	ideal	ideal	NOUN
ejpam-3397	102	25	.	.	PUNCT
ejpam-3397	103	1	proposition	proposition	NOUN
ejpam-3397	103	2	4	4	NUM
ejpam-3397	103	3	.	.	PUNCT
ejpam-3397	104	1	let	let	VERB
ejpam-3397	104	2	i	i	PRON
ejpam-3397	104	3	be	be	AUX
ejpam-3397	104	4	an	an	DET
ejpam-3397	104	5	ideal	ideal	NOUN
ejpam-3397	104	6	of	of	ADP
ejpam-3397	104	7	a	a	DET
ejpam-3397	104	8	γ	γ	NOUN
ejpam-3397	104	9	-	-	PUNCT
ejpam-3397	104	10	seminearring	seminearre	VERB
ejpam-3397	104	11	r	r	NOUN
ejpam-3397	104	12	with	with	ADP
ejpam-3397	104	13	r	r	NOUN
ejpam-3397	105	1	+	+	CCONJ
ejpam-3397	105	2	i	i	PROPN
ejpam-3397	105	3	⊆	⊆	NUM
ejpam-3397	105	4	i	i	PRON
ejpam-3397	105	5	and	and	CCONJ
ejpam-3397	105	6	i	i	PRON
ejpam-3397	106	1	+	+	NOUN
ejpam-3397	106	2	r	r	NOUN
ejpam-3397	106	3	⊆	⊆	NUM
ejpam-3397	106	4	i.	i.	NOUN
ejpam-3397	106	5	let	let	VERB
ejpam-3397	106	6	p	p	PRON
ejpam-3397	106	7	be	be	AUX
ejpam-3397	106	8	a	a	DET
ejpam-3397	106	9	proper	proper	ADJ
ejpam-3397	106	10	ideal	ideal	NOUN
ejpam-3397	106	11	of	of	ADP
ejpam-3397	106	12	r	r	NOUN
ejpam-3397	106	13	containing	contain	VERB
ejpam-3397	106	14	i	i	PRON
ejpam-3397	106	15	and	and	CCONJ
ejpam-3397	106	16	ψ	ψ	X
ejpam-3397	106	17	:	:	PUNCT
ejpam-3397	106	18	r→	r→	PROPN
ejpam-3397	106	19	r	r	X
ejpam-3397	106	20	/	/	SYM
ejpam-3397	106	21	i	i	PRON
ejpam-3397	106	22	be	be	VERB
ejpam-3397	106	23	the	the	DET
ejpam-3397	106	24	canonical	canonical	ADJ
ejpam-3397	106	25	epimorphism	epimorphism	NOUN
ejpam-3397	106	26	.	.	PUNCT
ejpam-3397	107	1	then	then	ADV
ejpam-3397	107	2	,	,	PUNCT
ejpam-3397	107	3	p	p	PRON
ejpam-3397	107	4	is	be	AUX
ejpam-3397	107	5	a	a	DET
ejpam-3397	107	6	weakly	weakly	ADJ
ejpam-3397	107	7	prime	prime	ADJ
ejpam-3397	107	8	ideal	ideal	NOUN
ejpam-3397	107	9	if	if	SCONJ
ejpam-3397	107	10	and	and	CCONJ
ejpam-3397	107	11	only	only	ADV
ejpam-3397	107	12	if	if	SCONJ
ejpam-3397	107	13	ψ(p	ψ(p	NOUN
ejpam-3397	107	14	)	)	PUNCT
ejpam-3397	107	15	is	be	AUX
ejpam-3397	107	16	a	a	DET
ejpam-3397	107	17	weakly	weakly	ADJ
ejpam-3397	107	18	prime	prime	NOUN
ejpam-3397	107	19	.	.	PUNCT
ejpam-3397	108	1	proof	proof	NOUN
ejpam-3397	108	2	.	.	PUNCT
ejpam-3397	109	1	let	let	VERB
ejpam-3397	109	2	p	p	PRON
ejpam-3397	109	3	be	be	AUX
ejpam-3397	109	4	a	a	DET
ejpam-3397	109	5	weakly	weakly	ADJ
ejpam-3397	109	6	prime	prime	ADJ
ejpam-3397	109	7	ideal	ideal	NOUN
ejpam-3397	109	8	of	of	ADP
ejpam-3397	109	9	r.	r.	PROPN
ejpam-3397	109	10	suppose	suppose	VERB
ejpam-3397	109	11	j1	j1	PROPN
ejpam-3397	109	12	and	and	CCONJ
ejpam-3397	109	13	j2	j2	PROPN
ejpam-3397	109	14	are	be	AUX
ejpam-3397	109	15	ideals	ideal	NOUN
ejpam-3397	109	16	in	in	ADP
ejpam-3397	109	17	r	r	NOUN
ejpam-3397	109	18	/	/	SYM
ejpam-3397	109	19	i	i	PRON
ejpam-3397	109	20	such	such	ADJ
ejpam-3397	109	21	that	that	DET
ejpam-3397	109	22	0	0	NUM
ejpam-3397	109	23	6=	6=	NUM
ejpam-3397	109	24	j1γj2	j1γj2	NOUN
ejpam-3397	109	25	⊆	⊆	NUM
ejpam-3397	109	26	ψ(p	ψ(p	PROPN
ejpam-3397	109	27	)	)	PUNCT
ejpam-3397	109	28	.	.	PUNCT
ejpam-3397	110	1	assume	assume	VERB
ejpam-3397	110	2	that	that	SCONJ
ejpam-3397	110	3	ψ−1(j1	ψ−1(j1	AUX
ejpam-3397	110	4	)	)	PUNCT
ejpam-3397	110	5	=	=	SYM
ejpam-3397	110	6	i1	i1	PROPN
ejpam-3397	110	7	and	and	CCONJ
ejpam-3397	110	8	ψ−1(j2	ψ−1(j2	NOUN
ejpam-3397	110	9	)	)	PUNCT
ejpam-3397	110	10	=	=	SYM
ejpam-3397	110	11	i2	i2	PROPN
ejpam-3397	110	12	.	.	PUNCT
ejpam-3397	111	1	then	then	ADV
ejpam-3397	111	2	,	,	PUNCT
ejpam-3397	111	3	0	0	NUM
ejpam-3397	111	4	6=	6=	ADP
ejpam-3397	111	5	i1γi2	i1γi2	NOUN
ejpam-3397	111	6	=	=	SYM
ejpam-3397	111	7	0	0	PUNCT
ejpam-3397	111	8	6=	6=	NUM
ejpam-3397	111	9	ψ−1(j1)γψ−1(j2	ψ−1(j1)γψ−1(j2	NUM
ejpam-3397	111	10	)	)	PUNCT
ejpam-3397	112	1	⊆	⊆	NUM
ejpam-3397	112	2	0	0	PUNCT
ejpam-3397	112	3	6=	6=	SYM
ejpam-3397	112	4	ψ−1(j1γj2	ψ−1(j1γj2	PROPN
ejpam-3397	112	5	)	)	PUNCT
ejpam-3397	112	6	⊆	⊆	NUM
ejpam-3397	112	7	0	0	NUM
ejpam-3397	112	8	6=	6=	NUM
ejpam-3397	112	9	ψ−1(π(p	ψ−1(π(p	NOUN
ejpam-3397	112	10	)	)	PUNCT
ejpam-3397	112	11	)	)	PUNCT
ejpam-3397	113	1	=	=	PUNCT
ejpam-3397	114	1	p	p	NOUN
ejpam-3397	114	2	.	.	PUNCT
ejpam-3397	115	1	since	since	SCONJ
ejpam-3397	115	2	p	p	NOUN
ejpam-3397	115	3	is	be	AUX
ejpam-3397	115	4	a	a	DET
ejpam-3397	115	5	weakly	weakly	ADJ
ejpam-3397	115	6	prime	prime	ADJ
ejpam-3397	115	7	ideal	ideal	NOUN
ejpam-3397	115	8	,	,	PUNCT
ejpam-3397	115	9	it	it	PRON
ejpam-3397	115	10	implies	imply	VERB
ejpam-3397	115	11	i1	i1	PROPN
ejpam-3397	115	12	⊆	⊆	NUM
ejpam-3397	115	13	p	p	NOUN
ejpam-3397	115	14	or	or	CCONJ
ejpam-3397	115	15	i2	i2	PROPN
ejpam-3397	115	16	⊆	⊆	NUM
ejpam-3397	115	17	p	p	NOUN
ejpam-3397	115	18	.	.	PUNCT
ejpam-3397	116	1	hence	hence	ADV
ejpam-3397	116	2	,	,	PUNCT
ejpam-3397	116	3	j1	j1	PROPN
ejpam-3397	116	4	=	=	PUNCT
ejpam-3397	116	5	ψ(ψ−1(j1	ψ(ψ−1(j1	X
ejpam-3397	116	6	)	)	PUNCT
ejpam-3397	116	7	)	)	PUNCT
ejpam-3397	117	1	=	=	SYM
ejpam-3397	117	2	ψ(i1	ψ(i1	NUM
ejpam-3397	117	3	)	)	PUNCT
ejpam-3397	117	4	⊆	⊆	NUM
ejpam-3397	117	5	ψ(p	ψ(p	PROPN
ejpam-3397	117	6	)	)	PUNCT
ejpam-3397	117	7	or	or	CCONJ
ejpam-3397	117	8	j2	j2	PROPN
ejpam-3397	117	9	=	=	SYM
ejpam-3397	117	10	ψ(ψ−1(j2	ψ(ψ−1(j2	NOUN
ejpam-3397	117	11	)	)	PUNCT
ejpam-3397	117	12	)	)	PUNCT
ejpam-3397	118	1	=	=	PUNCT
ejpam-3397	118	2	ψ(j2	ψ(j2	NOUN
ejpam-3397	118	3	)	)	PUNCT
ejpam-3397	118	4	⊆	⊆	NUM
ejpam-3397	118	5	ψ(p	ψ(p	PROPN
ejpam-3397	118	6	)	)	PUNCT
ejpam-3397	118	7	.	.	PUNCT
ejpam-3397	119	1	hence	hence	ADV
ejpam-3397	119	2	,	,	PUNCT
ejpam-3397	119	3	ψ(p	ψ(p	PROPN
ejpam-3397	119	4	)	)	PUNCT
ejpam-3397	119	5	is	be	AUX
ejpam-3397	119	6	a	a	DET
ejpam-3397	119	7	weakly	weakly	ADJ
ejpam-3397	119	8	prime	prime	NOUN
ejpam-3397	119	9	.	.	PUNCT
ejpam-3397	120	1	conversely	conversely	ADV
ejpam-3397	120	2	,	,	PUNCT
ejpam-3397	120	3	suppose	suppose	VERB
ejpam-3397	120	4	ψ(p	ψ(p	NOUN
ejpam-3397	120	5	)	)	PUNCT
ejpam-3397	120	6	be	be	AUX
ejpam-3397	120	7	a	a	DET
ejpam-3397	120	8	weakly	weakly	ADJ
ejpam-3397	120	9	prime	prime	ADJ
ejpam-3397	120	10	ideal	ideal	NOUN
ejpam-3397	120	11	and	and	CCONJ
ejpam-3397	120	12	let	let	VERB
ejpam-3397	120	13	i1	i1	PROPN
ejpam-3397	120	14	,	,	PUNCT
ejpam-3397	120	15	i2	i2	PROPN
ejpam-3397	120	16	are	be	AUX
ejpam-3397	120	17	ideals	ideal	NOUN
ejpam-3397	120	18	of	of	ADP
ejpam-3397	120	19	r	r	NOUN
ejpam-3397	121	1	such	such	ADJ
ejpam-3397	121	2	that	that	DET
ejpam-3397	121	3	0	0	NUM
ejpam-3397	122	1	6=	6=	NUM
ejpam-3397	122	2	i1γi2	i1γi2	NOUN
ejpam-3397	122	3	⊆	⊆	NUM
ejpam-3397	122	4	p	p	NOUN
ejpam-3397	122	5	.	.	PUNCT
ejpam-3397	123	1	then	then	ADV
ejpam-3397	123	2	,	,	PUNCT
ejpam-3397	123	3	0	0	NUM
ejpam-3397	123	4	6=	6=	NUM
ejpam-3397	123	5	ψ(i1)γψ(i2	ψ(i1)γψ(i2	NOUN
ejpam-3397	123	6	)	)	PUNCT
ejpam-3397	123	7	=	=	PUNCT
ejpam-3397	123	8	0	0	PUNCT
ejpam-3397	123	9	6=	6=	NUM
ejpam-3397	123	10	ψ(i1γi2	ψ(i1γi2	NOUN
ejpam-3397	123	11	)	)	PUNCT
ejpam-3397	123	12	⊆	⊆	NUM
ejpam-3397	123	13	ψ(p	ψ(p	PROPN
ejpam-3397	123	14	)	)	PUNCT
ejpam-3397	123	15	.since	.since	NOUN
ejpam-3397	123	16	ψ(p	ψ(p	PROPN
ejpam-3397	123	17	)	)	PUNCT
ejpam-3397	123	18	is	be	AUX
ejpam-3397	123	19	a	a	DET
ejpam-3397	123	20	weakly	weakly	ADJ
ejpam-3397	123	21	prime	prime	ADJ
ejpam-3397	123	22	ideal	ideal	NOUN
ejpam-3397	123	23	,	,	PUNCT
ejpam-3397	123	24	it	it	PRON
ejpam-3397	123	25	implies	imply	VERB
ejpam-3397	123	26	that	that	SCONJ
ejpam-3397	123	27	ψ(i1	ψ(i1	NUM
ejpam-3397	123	28	)	)	PUNCT
ejpam-3397	123	29	⊆	⊆	NUM
ejpam-3397	123	30	ψ(p	ψ(p	PROPN
ejpam-3397	123	31	)	)	PUNCT
ejpam-3397	123	32	or	or	CCONJ
ejpam-3397	123	33	ψ(i2	ψ(i2	NOUN
ejpam-3397	123	34	)	)	PUNCT
ejpam-3397	123	35	⊆	⊆	NUM
ejpam-3397	123	36	ψ(p	ψ(p	PROPN
ejpam-3397	123	37	)	)	PUNCT
ejpam-3397	123	38	.	.	PUNCT
ejpam-3397	124	1	thus	thus	ADV
ejpam-3397	124	2	,	,	PUNCT
ejpam-3397	124	3	i1	i1	PROPN
ejpam-3397	124	4	⊆	⊆	NUM
ejpam-3397	124	5	p	p	NOUN
ejpam-3397	124	6	or	or	CCONJ
ejpam-3397	124	7	i2	i2	PROPN
ejpam-3397	124	8	⊆	⊆	NUM
ejpam-3397	124	9	p	p	NOUN
ejpam-3397	124	10	,	,	PUNCT
ejpam-3397	124	11	and	and	CCONJ
ejpam-3397	124	12	hence	hence	ADV
ejpam-3397	124	13	p	p	PRON
ejpam-3397	124	14	is	be	AUX
ejpam-3397	124	15	a	a	DET
ejpam-3397	124	16	weakly	weakly	ADJ
ejpam-3397	124	17	prime	prime	ADJ
ejpam-3397	124	18	ideal	ideal	NOUN
ejpam-3397	124	19	of	of	ADP
ejpam-3397	124	20	a	a	DET
ejpam-3397	124	21	γ	γ	NOUN
ejpam-3397	124	22	-	-	PUNCT
ejpam-3397	124	23	seminearring	seminearre	VERB
ejpam-3397	124	24	r.	r.	PROPN
ejpam-3397	124	25	definition	definition	NOUN
ejpam-3397	124	26	2	2	NUM
ejpam-3397	124	27	.	.	PUNCT
ejpam-3397	125	1	let	let	VERB
ejpam-3397	125	2	r	r	PRON
ejpam-3397	125	3	be	be	AUX
ejpam-3397	125	4	a	a	DET
ejpam-3397	125	5	γ	γ	NOUN
ejpam-3397	125	6	-	-	PUNCT
ejpam-3397	125	7	seminearring	seminearring	NOUN
ejpam-3397	125	8	and	and	CCONJ
ejpam-3397	125	9	m	m	AUX
ejpam-3397	125	10	be	be	AUX
ejpam-3397	125	11	a	a	DET
ejpam-3397	125	12	non	non	ADJ
ejpam-3397	125	13	-	-	ADJ
ejpam-3397	125	14	empty	empty	ADJ
ejpam-3397	125	15	subset	subset	NOUN
ejpam-3397	125	16	of	of	ADP
ejpam-3397	125	17	r.	r.	PROPN
ejpam-3397	125	18	we	we	PRON
ejpam-3397	125	19	call	call	VERB
ejpam-3397	125	20	m	m	VERB
ejpam-3397	125	21	an	an	DET
ejpam-3397	125	22	m	m	NOUN
ejpam-3397	125	23	-	-	NOUN
ejpam-3397	125	24	system	system	NOUN
ejpam-3397	125	25	if	if	SCONJ
ejpam-3397	125	26	for	for	ADP
ejpam-3397	125	27	a	a	PRON
ejpam-3397	125	28	,	,	PUNCT
ejpam-3397	125	29	b	b	PROPN
ejpam-3397	125	30	∈	∈	ADV
ejpam-3397	125	31	m	m	NOUN
ejpam-3397	125	32	,	,	PUNCT
ejpam-3397	125	33	there	there	PRON
ejpam-3397	125	34	exist	exist	VERB
ejpam-3397	125	35	a1	a1	NOUN
ejpam-3397	125	36	∈	∈	NOUN
ejpam-3397	125	37	(	(	PUNCT
ejpam-3397	125	38	a	a	NOUN
ejpam-3397	125	39	)	)	PUNCT
ejpam-3397	125	40	,	,	PUNCT
ejpam-3397	125	41	b1	b1	NOUN
ejpam-3397	125	42	∈	∈	PROPN
ejpam-3397	125	43	(	(	PUNCT
ejpam-3397	125	44	b	b	NOUN
ejpam-3397	125	45	)	)	PUNCT
ejpam-3397	125	46	and	and	CCONJ
ejpam-3397	125	47	α	α	PRON
ejpam-3397	125	48	∈	∈	PROPN
ejpam-3397	125	49	γ	γ	NOUN
ejpam-3397	125	50	such	such	ADJ
ejpam-3397	125	51	that	that	DET
ejpam-3397	125	52	0	0	NUM
ejpam-3397	125	53	6=	6=	NUM
ejpam-3397	125	54	a1αb1	a1αb1	PROPN
ejpam-3397	125	55	∈m	∈m	NOUN
ejpam-3397	125	56	.	.	PUNCT
ejpam-3397	126	1	proposition	proposition	NOUN
ejpam-3397	126	2	5	5	NUM
ejpam-3397	126	3	.	.	PUNCT
ejpam-3397	127	1	let	let	VERB
ejpam-3397	127	2	p	p	PRON
ejpam-3397	127	3	be	be	AUX
ejpam-3397	127	4	a	a	DET
ejpam-3397	127	5	proper	proper	ADJ
ejpam-3397	127	6	ideal	ideal	NOUN
ejpam-3397	127	7	of	of	ADP
ejpam-3397	127	8	a	a	DET
ejpam-3397	127	9	γ	γ	NOUN
ejpam-3397	127	10	-	-	PUNCT
ejpam-3397	127	11	seminearring	seminearre	VERB
ejpam-3397	127	12	r.	r.	PROPN
ejpam-3397	127	13	then	then	ADV
ejpam-3397	127	14	,	,	PUNCT
ejpam-3397	127	15	p	p	PRON
ejpam-3397	127	16	is	be	AUX
ejpam-3397	127	17	a	a	DET
ejpam-3397	127	18	weakly	weakly	ADJ
ejpam-3397	127	19	prime	prime	ADJ
ejpam-3397	127	20	ideal	ideal	NOUN
ejpam-3397	127	21	if	if	SCONJ
ejpam-3397	127	22	and	and	CCONJ
ejpam-3397	127	23	only	only	ADV
ejpam-3397	127	24	if	if	SCONJ
ejpam-3397	127	25	r\p	r\p	NOUN
ejpam-3397	127	26	is	be	AUX
ejpam-3397	127	27	m	m	NOUN
ejpam-3397	127	28	-	-	NOUN
ejpam-3397	127	29	system	system	NOUN
ejpam-3397	127	30	.	.	PUNCT
ejpam-3397	128	1	proof	proof	NOUN
ejpam-3397	128	2	.	.	PUNCT
ejpam-3397	129	1	let	let	VERB
ejpam-3397	129	2	p	p	PRON
ejpam-3397	129	3	be	be	AUX
ejpam-3397	129	4	a	a	DET
ejpam-3397	129	5	weakly	weakly	ADJ
ejpam-3397	129	6	prime	prime	ADJ
ejpam-3397	129	7	ideal	ideal	NOUN
ejpam-3397	129	8	of	of	ADP
ejpam-3397	129	9	a	a	DET
ejpam-3397	129	10	γ	γ	NOUN
ejpam-3397	129	11	-	-	PUNCT
ejpam-3397	129	12	seminearring	seminearre	VERB
ejpam-3397	129	13	r.	r.	PROPN
ejpam-3397	129	14	consider	consider	VERB
ejpam-3397	129	15	a	a	DET
ejpam-3397	129	16	,	,	PUNCT
ejpam-3397	129	17	b	b	PROPN
ejpam-3397	129	18	∈	∈	PROPN
ejpam-3397	129	19	r\p	r\p	NOUN
ejpam-3397	129	20	and	and	CCONJ
ejpam-3397	129	21	0	0	NUM
ejpam-3397	129	22	6=	6=	NUM
ejpam-3397	129	23	(	(	PUNCT
ejpam-3397	129	24	a)γ(b	a)γ(b	X
ejpam-3397	129	25	)	)	PUNCT
ejpam-3397	129	26	*	*	PUNCT
ejpam-3397	130	1	p	p	NOUN
ejpam-3397	130	2	.	.	PUNCT
ejpam-3397	131	1	let	let	VERB
ejpam-3397	131	2	a1	a1	NOUN
ejpam-3397	131	3	∈	∈	PROPN
ejpam-3397	131	4	(	(	PUNCT
ejpam-3397	131	5	a	a	NOUN
ejpam-3397	131	6	)	)	PUNCT
ejpam-3397	131	7	,	,	PUNCT
ejpam-3397	131	8	b1	b1	NOUN
ejpam-3397	131	9	∈	∈	PROPN
ejpam-3397	131	10	(	(	PUNCT
ejpam-3397	131	11	b	b	NOUN
ejpam-3397	131	12	)	)	PUNCT
ejpam-3397	131	13	and	and	CCONJ
ejpam-3397	131	14	α	α	PRON
ejpam-3397	131	15	∈	∈	PROPN
ejpam-3397	131	16	γ	γ	NOUN
ejpam-3397	131	17	such	such	ADJ
ejpam-3397	131	18	that	that	PRON
ejpam-3397	131	19	0	0	NUM
ejpam-3397	131	20	6=	6=	ADP
ejpam-3397	131	21	a1αb1	a1αb1	PROPN
ejpam-3397	131	22	/∈	/∈	PUNCT
ejpam-3397	132	1	p	p	X
ejpam-3397	132	2	,	,	PUNCT
ejpam-3397	132	3	i.e.	i.e.	X
ejpam-3397	132	4	,	,	PUNCT
ejpam-3397	132	5	a1αb1	a1αb1	PROPN
ejpam-3397	132	6	∈	∈	PROPN
ejpam-3397	132	7	r\p	r\p	NOUN
ejpam-3397	132	8	.	.	PUNCT
ejpam-3397	133	1	thus	thus	ADV
ejpam-3397	133	2	,	,	PUNCT
ejpam-3397	133	3	r\p	r\p	NOUN
ejpam-3397	133	4	is	be	AUX
ejpam-3397	133	5	an	an	DET
ejpam-3397	133	6	m	m	NOUN
ejpam-3397	133	7	-	-	NOUN
ejpam-3397	133	8	system	system	NOUN
ejpam-3397	133	9	.	.	PUNCT
ejpam-3397	134	1	conversely	conversely	ADV
ejpam-3397	134	2	,	,	PUNCT
ejpam-3397	134	3	suppose	suppose	VERB
ejpam-3397	134	4	r\p	r\p	NOUN
ejpam-3397	134	5	is	be	AUX
ejpam-3397	134	6	an	an	DET
ejpam-3397	134	7	m	m	NOUN
ejpam-3397	134	8	-	-	NOUN
ejpam-3397	134	9	system	system	NOUN
ejpam-3397	134	10	and	and	CCONJ
ejpam-3397	134	11	let	let	VERB
ejpam-3397	134	12	a	a	DET
ejpam-3397	134	13	,	,	PUNCT
ejpam-3397	134	14	b	b	PROPN
ejpam-3397	134	15	∈	∈	PROPN
ejpam-3397	134	16	r\p	r\p	NOUN
ejpam-3397	134	17	.	.	PUNCT
ejpam-3397	135	1	then	then	ADV
ejpam-3397	135	2	,	,	PUNCT
ejpam-3397	135	3	there	there	PRON
ejpam-3397	135	4	exist	exist	VERB
ejpam-3397	135	5	a1	a1	NOUN
ejpam-3397	135	6	∈	∈	NOUN
ejpam-3397	135	7	(	(	PUNCT
ejpam-3397	135	8	a	a	NOUN
ejpam-3397	135	9	)	)	PUNCT
ejpam-3397	135	10	,	,	PUNCT
ejpam-3397	135	11	b1	b1	NOUN
ejpam-3397	135	12	∈	∈	PROPN
ejpam-3397	135	13	(	(	PUNCT
ejpam-3397	135	14	b	b	NOUN
ejpam-3397	135	15	)	)	PUNCT
ejpam-3397	135	16	and	and	CCONJ
ejpam-3397	135	17	α	α	PRON
ejpam-3397	135	18	∈	∈	PROPN
ejpam-3397	135	19	γ	γ	NOUN
ejpam-3397	135	20	such	such	ADJ
ejpam-3397	135	21	that	that	SCONJ
ejpam-3397	135	22	a1αb1	a1αb1	PROPN
ejpam-3397	135	23	∈	∈	PROPN
ejpam-3397	135	24	r\p	r\p	NOUN
ejpam-3397	135	25	.	.	PUNCT
ejpam-3397	136	1	thus	thus	ADV
ejpam-3397	136	2	,	,	PUNCT
ejpam-3397	136	3	0	0	NUM
ejpam-3397	136	4	6=	6=	NUM
ejpam-3397	136	5	(	(	PUNCT
ejpam-3397	136	6	a)γ(b	a)γ(b	X
ejpam-3397	136	7	)	)	PUNCT
ejpam-3397	136	8	*	*	PUNCT
ejpam-3397	137	1	p	p	NOUN
ejpam-3397	137	2	and	and	CCONJ
ejpam-3397	137	3	hence	hence	ADV
ejpam-3397	137	4	p	p	PRON
ejpam-3397	137	5	is	be	AUX
ejpam-3397	137	6	a	a	DET
ejpam-3397	137	7	weakly	weakly	ADJ
ejpam-3397	137	8	prime	prime	ADJ
ejpam-3397	137	9	ideal	ideal	NOUN
ejpam-3397	137	10	of	of	ADP
ejpam-3397	137	11	a	a	DET
ejpam-3397	137	12	γ	γ	NOUN
ejpam-3397	137	13	-	-	PUNCT
ejpam-3397	137	14	seminearring	seminearre	VERB
ejpam-3397	137	15	r.	r.	PROPN
ejpam-3397	137	16	definition	definition	NOUN
ejpam-3397	137	17	3	3	NUM
ejpam-3397	137	18	.	.	PUNCT
ejpam-3397	138	1	a	a	DET
ejpam-3397	138	2	subset	subset	NOUN
ejpam-3397	138	3	a	a	PRON
ejpam-3397	138	4	of	of	ADP
ejpam-3397	138	5	a	a	DET
ejpam-3397	138	6	γ	γ	NOUN
ejpam-3397	138	7	-	-	PUNCT
ejpam-3397	138	8	seminearring	seminearre	VERB
ejpam-3397	138	9	r	r	NOUN
ejpam-3397	138	10	is	be	AUX
ejpam-3397	138	11	a	a	DET
ejpam-3397	138	12	subtractive	subtractive	NOUN
ejpam-3397	138	13	,	,	PUNCT
ejpam-3397	138	14	if	if	SCONJ
ejpam-3397	138	15	a	a	DET
ejpam-3397	138	16	∈	∈	PROPN
ejpam-3397	138	17	a	a	PRON
ejpam-3397	138	18	and	and	CCONJ
ejpam-3397	138	19	a+	a+	PRON
ejpam-3397	138	20	b	b	X
ejpam-3397	138	21	∈	∈	PROPN
ejpam-3397	138	22	a	a	DET
ejpam-3397	138	23	implies	imply	VERB
ejpam-3397	138	24	b	b	X
ejpam-3397	138	25	∈	∈	NOUN
ejpam-3397	138	26	a.	a.	NOUN
ejpam-3397	138	27	proposition	proposition	NOUN
ejpam-3397	138	28	6	6	NUM
ejpam-3397	138	29	.	.	PUNCT
ejpam-3397	139	1	let	let	VERB
ejpam-3397	139	2	r	r	PRON
ejpam-3397	139	3	be	be	AUX
ejpam-3397	139	4	a	a	DET
ejpam-3397	139	5	γ	γ	NOUN
ejpam-3397	139	6	-	-	PUNCT
ejpam-3397	139	7	seminearring	seminearring	NOUN
ejpam-3397	139	8	whose	whose	DET
ejpam-3397	139	9	all	all	DET
ejpam-3397	139	10	ideals	ideal	NOUN
ejpam-3397	139	11	are	be	AUX
ejpam-3397	139	12	subtractive	subtractive	NOUN
ejpam-3397	139	13	,	,	PUNCT
ejpam-3397	139	14	and	and	CCONJ
ejpam-3397	139	15	let	let	VERB
ejpam-3397	139	16	p	p	PRON
ejpam-3397	139	17	be	be	AUX
ejpam-3397	139	18	a	a	DET
ejpam-3397	139	19	proper	proper	ADJ
ejpam-3397	139	20	ideal	ideal	NOUN
ejpam-3397	139	21	of	of	ADP
ejpam-3397	139	22	r.	r.	PROPN
ejpam-3397	139	23	then	then	ADV
ejpam-3397	139	24	,	,	PUNCT
ejpam-3397	139	25	p	p	PRON
ejpam-3397	139	26	is	be	AUX
ejpam-3397	139	27	a	a	DET
ejpam-3397	139	28	weakly	weakly	ADJ
ejpam-3397	139	29	prime	prime	NOUN
ejpam-3397	139	30	if	if	SCONJ
ejpam-3397	139	31	and	and	CCONJ
ejpam-3397	139	32	only	only	ADV
ejpam-3397	139	33	if	if	SCONJ
ejpam-3397	139	34	for	for	ADP
ejpam-3397	139	35	any	any	DET
ejpam-3397	139	36	ideals	ideal	NOUN
ejpam-3397	139	37	i	i	PRON
ejpam-3397	139	38	,	,	PUNCT
ejpam-3397	139	39	j	j	PROPN
ejpam-3397	139	40	of	of	ADP
ejpam-3397	139	41	r	r	PROPN
ejpam-3397	139	42	,	,	PUNCT
ejpam-3397	139	43	p	p	X
ejpam-3397	139	44	⊂	⊂	PROPN
ejpam-3397	139	45	i	i	PROPN
ejpam-3397	139	46	and	and	CCONJ
ejpam-3397	139	47	p	p	PROPN
ejpam-3397	139	48	⊂	⊂	PROPN
ejpam-3397	139	49	j	j	PROPN
ejpam-3397	139	50	implies	imply	VERB
ejpam-3397	139	51	0	0	NUM
ejpam-3397	139	52	6=	6=	NUM
ejpam-3397	139	53	iγj	iγj	NOUN
ejpam-3397	139	54	*	*	PUNCT
ejpam-3397	139	55	p	p	X
ejpam-3397	139	56	.	.	PUNCT
ejpam-3397	140	1	proof	proof	NOUN
ejpam-3397	140	2	.	.	PUNCT
ejpam-3397	141	1	suppose	suppose	VERB
ejpam-3397	141	2	for	for	ADP
ejpam-3397	141	3	any	any	DET
ejpam-3397	141	4	ideals	ideal	NOUN
ejpam-3397	141	5	i	i	PRON
ejpam-3397	141	6	,	,	PUNCT
ejpam-3397	141	7	j	j	PROPN
ejpam-3397	141	8	of	of	ADP
ejpam-3397	141	9	r	r	PROPN
ejpam-3397	141	10	,	,	PUNCT
ejpam-3397	141	11	p	p	X
ejpam-3397	141	12	⊂	⊂	PROPN
ejpam-3397	141	13	i	i	PROPN
ejpam-3397	141	14	and	and	CCONJ
ejpam-3397	141	15	p	p	PROPN
ejpam-3397	141	16	⊂	⊂	PROPN
ejpam-3397	141	17	j	j	PROPN
ejpam-3397	141	18	implies	imply	VERB
ejpam-3397	141	19	0	0	NUM
ejpam-3397	141	20	6=	6=	NUM
ejpam-3397	141	21	iγj	iγj	NOUN
ejpam-3397	141	22	*	*	PUNCT
ejpam-3397	141	23	p	p	X
ejpam-3397	141	24	.	.	PUNCT
ejpam-3397	142	1	let	let	VERB
ejpam-3397	142	2	us	we	PRON
ejpam-3397	142	3	suppose	suppose	VERB
ejpam-3397	142	4	that	that	SCONJ
ejpam-3397	142	5	i	i	PRON
ejpam-3397	142	6	*	*	PUNCT
ejpam-3397	142	7	p	p	PROPN
ejpam-3397	142	8	and	and	CCONJ
ejpam-3397	142	9	j	j	PROPN
ejpam-3397	142	10	*	*	PUNCT
ejpam-3397	142	11	p	p	X
ejpam-3397	142	12	.	.	PUNCT
ejpam-3397	143	1	then	then	ADV
ejpam-3397	143	2	there	there	PRON
ejpam-3397	143	3	exist	exist	VERB
ejpam-3397	143	4	i	i	PRON
ejpam-3397	143	5	∈	∈	PROPN
ejpam-3397	143	6	i\p	i\p	PROPN
ejpam-3397	143	7	and	and	CCONJ
ejpam-3397	143	8	j	j	PROPN
ejpam-3397	143	9	∈	∈	PROPN
ejpam-3397	143	10	j\p	j\p	PROPN
ejpam-3397	143	11	and	and	CCONJ
ejpam-3397	143	12	hence	hence	ADV
ejpam-3397	143	13	p	p	X
ejpam-3397	144	1	⊂	⊂	PROPN
ejpam-3397	144	2	p	p	X
ejpam-3397	145	1	+	+	PROPN
ejpam-3397	145	2	(	(	PUNCT
ejpam-3397	145	3	i	i	NOUN
ejpam-3397	145	4	)	)	PUNCT
ejpam-3397	145	5	.	.	PUNCT
ejpam-3397	146	1	by	by	ADP
ejpam-3397	146	2	hypothesis	hypothesis	NOUN
ejpam-3397	146	3	,	,	PUNCT
ejpam-3397	146	4	0	0	PUNCT
ejpam-3397	146	5	6=	6=	NUM
ejpam-3397	146	6	(	(	PUNCT
ejpam-3397	146	7	p	p	X
ejpam-3397	146	8	+	+	X
ejpam-3397	146	9	(	(	PUNCT
ejpam-3397	146	10	i))γ(p	i))γ(p	NOUN
ejpam-3397	146	11	+	+	CCONJ
ejpam-3397	146	12	(	(	PUNCT
ejpam-3397	146	13	j	j	NOUN
ejpam-3397	146	14	)	)	PUNCT
ejpam-3397	146	15	)	)	PUNCT
ejpam-3397	147	1	*	*	PUNCT
ejpam-3397	148	1	p	p	NOUN
ejpam-3397	149	1	and	and	CCONJ
ejpam-3397	149	2	so	so	ADV
ejpam-3397	149	3	there	there	PRON
ejpam-3397	149	4	exist	exist	VERB
ejpam-3397	149	5	i′	i′	NOUN
ejpam-3397	149	6	∈	∈	NOUN
ejpam-3397	149	7	(	(	PUNCT
ejpam-3397	149	8	i	i	NOUN
ejpam-3397	149	9	)	)	PUNCT
ejpam-3397	149	10	,	,	PUNCT
ejpam-3397	149	11	j′	j′	PROPN
ejpam-3397	149	12	∈	∈	PROPN
ejpam-3397	149	13	(	(	PUNCT
ejpam-3397	149	14	j	j	NOUN
ejpam-3397	149	15	)	)	PUNCT
ejpam-3397	149	16	,	,	PUNCT
ejpam-3397	149	17	p	p	X
ejpam-3397	149	18	,	,	PUNCT
ejpam-3397	149	19	p′	p′	NOUN
ejpam-3397	149	20	∈	∈	PROPN
ejpam-3397	149	21	p	p	NOUN
ejpam-3397	149	22	and	and	CCONJ
ejpam-3397	149	23	α	α	NOUN
ejpam-3397	149	24	∈	∈	PROPN
ejpam-3397	149	25	γ	γ	NOUN
ejpam-3397	150	1	such	such	ADJ
ejpam-3397	150	2	that	that	DET
ejpam-3397	150	3	0	0	NUM
ejpam-3397	150	4	6=	6=	NUM
ejpam-3397	150	5	(	(	PUNCT
ejpam-3397	150	6	p+i′)α(p′+j′	p+i′)α(p′+j′	PROPN
ejpam-3397	150	7	)	)	PUNCT
ejpam-3397	150	8	/∈	/∈	PUNCT
ejpam-3397	151	1	p	p	NOUN
ejpam-3397	151	2	.	.	PUNCT
ejpam-3397	152	1	since	since	SCONJ
ejpam-3397	152	2	,	,	PUNCT
ejpam-3397	152	3	0	0	NUM
ejpam-3397	152	4	6=	6=	ADP
ejpam-3397	152	5	pα(p′+j′	pα(p′+j′	PROPN
ejpam-3397	152	6	)	)	PUNCT
ejpam-3397	152	7	∈	∈	PROPN
ejpam-3397	153	1	p	p	NOUN
ejpam-3397	153	2	,	,	PUNCT
ejpam-3397	153	3	0	0	NUM
ejpam-3397	153	4	6=	6=	NUM
ejpam-3397	153	5	i′α(p′	i′α(p′	NOUN
ejpam-3397	153	6	+	+	CCONJ
ejpam-3397	153	7	j′	j′	NOUN
ejpam-3397	153	8	)	)	PUNCT
ejpam-3397	153	9	/∈	/∈	PUNCT
ejpam-3397	154	1	p	p	NOUN
ejpam-3397	155	1	and	and	CCONJ
ejpam-3397	155	2	p	p	NOUN
ejpam-3397	155	3	is	be	AUX
ejpam-3397	155	4	an	an	DET
ejpam-3397	155	5	ideal	ideal	NOUN
ejpam-3397	155	6	,	,	PUNCT
ejpam-3397	155	7	then	then	ADV
ejpam-3397	155	8	i′	i′	VERB
ejpam-3397	155	9	/∈	/∈	PUNCT
ejpam-3397	156	1	p	p	NOUN
ejpam-3397	156	2	and	and	CCONJ
ejpam-3397	156	3	p′	p′	NOUN
ejpam-3397	156	4	+	+	CCONJ
ejpam-3397	156	5	j′	j′	PROPN
ejpam-3397	156	6	/∈	/∈	PUNCT
ejpam-3397	157	1	p	p	X
ejpam-3397	157	2	.	.	PUNCT
ejpam-3397	158	1	thus	thus	ADV
ejpam-3397	158	2	,	,	PUNCT
ejpam-3397	158	3	i′	i′	NOUN
ejpam-3397	158	4	/∈	/∈	PUNCT
ejpam-3397	159	1	p	p	NOUN
ejpam-3397	159	2	and	and	CCONJ
ejpam-3397	159	3	j′	j′	PROPN
ejpam-3397	159	4	/∈	/∈	PUNCT
ejpam-3397	160	1	p	p	NOUN
ejpam-3397	160	2	because	because	SCONJ
ejpam-3397	160	3	p	p	PROPN
ejpam-3397	160	4	is	be	AUX
ejpam-3397	160	5	subtractive	subtractive	NOUN
ejpam-3397	160	6	.	.	PUNCT
ejpam-3397	161	1	it	it	PRON
ejpam-3397	161	2	implies	imply	VERB
ejpam-3397	161	3	0	0	NUM
ejpam-3397	162	1	6=	6=	X
ejpam-3397	162	2	(	(	PUNCT
ejpam-3397	162	3	i′)γ(j′	i′)γ(j′	PROPN
ejpam-3397	162	4	)	)	PUNCT
ejpam-3397	162	5	*	*	PUNCT
ejpam-3397	163	1	p	p	NOUN
ejpam-3397	163	2	.	.	PUNCT
ejpam-3397	164	1	but	but	CCONJ
ejpam-3397	164	2	0	0	NUM
ejpam-3397	165	1	6=	6=	NUM
ejpam-3397	165	2	(	(	PUNCT
ejpam-3397	165	3	i′)γ(j′	i′)γ(j′	PROPN
ejpam-3397	165	4	)	)	PUNCT
ejpam-3397	165	5	⊆	⊆	NUM
ejpam-3397	165	6	0	0	NUM
ejpam-3397	165	7	6=	6=	NUM
ejpam-3397	165	8	iγj	iγj	PROPN
ejpam-3397	165	9	⇒	⇒	NOUN
ejpam-3397	165	10	0	0	PUNCT
ejpam-3397	166	1	6=	6=	NUM
ejpam-3397	166	2	iγj	iγj	NOUN
ejpam-3397	166	3	*	*	PUNCT
ejpam-3397	166	4	p	p	X
ejpam-3397	166	5	.	.	PUNCT
ejpam-3397	167	1	hence	hence	ADV
ejpam-3397	167	2	,	,	PUNCT
ejpam-3397	167	3	p	p	PROPN
ejpam-3397	167	4	is	be	AUX
ejpam-3397	167	5	a	a	DET
ejpam-3397	167	6	weakly	weakly	ADJ
ejpam-3397	167	7	prime	prime	ADJ
ejpam-3397	167	8	ideal	ideal	NOUN
ejpam-3397	167	9	.	.	PUNCT
ejpam-3397	168	1	the	the	DET
ejpam-3397	168	2	converse	converse	NOUN
ejpam-3397	168	3	is	be	AUX
ejpam-3397	168	4	obvious	obvious	ADJ
ejpam-3397	168	5	by	by	ADP
ejpam-3397	168	6	the	the	DET
ejpam-3397	168	7	definition	definition	NOUN
ejpam-3397	168	8	of	of	ADP
ejpam-3397	168	9	a	a	DET
ejpam-3397	168	10	weakly	weakly	ADJ
ejpam-3397	168	11	prime	prime	ADJ
ejpam-3397	168	12	ideal	ideal	NOUN
ejpam-3397	168	13	of	of	ADP
ejpam-3397	168	14	a	a	DET
ejpam-3397	168	15	γ	γ	NOUN
ejpam-3397	168	16	-	-	PUNCT
ejpam-3397	168	17	seminearring	seminearring	NOUN
ejpam-3397	168	18	.	.	PUNCT
ejpam-3397	169	1	abdelghani	abdelghani	PROPN
ejpam-3397	169	2	taouti	taouti	PROPN
ejpam-3397	169	3	et	et	PROPN
ejpam-3397	169	4	al	al	PROPN
ejpam-3397	169	5	.	.	PUNCT
ejpam-3397	169	6	/	/	SYM
ejpam-3397	169	7	eur	eur	PROPN
ejpam-3397	169	8	.	.	PUNCT
ejpam-3397	170	1	j.	j.	PROPN
ejpam-3397	170	2	pure	pure	PROPN
ejpam-3397	170	3	appl	appl	PROPN
ejpam-3397	170	4	.	.	PROPN
ejpam-3397	170	5	math	math	PROPN
ejpam-3397	170	6	,	,	PUNCT
ejpam-3397	170	7	12	12	NUM
ejpam-3397	170	8	(	(	PUNCT
ejpam-3397	170	9	2	2	NUM
ejpam-3397	170	10	)	)	PUNCT
ejpam-3397	170	11	(	(	PUNCT
ejpam-3397	170	12	2019	2019	NUM
ejpam-3397	170	13	)	)	PUNCT
ejpam-3397	170	14	,	,	PUNCT
ejpam-3397	170	15	544	544	NUM
ejpam-3397	170	16	-	-	SYM
ejpam-3397	170	17	552	552	NUM
ejpam-3397	170	18	548	548	NUM
ejpam-3397	170	19	theorem	theorem	NOUN
ejpam-3397	170	20	1	1	NUM
ejpam-3397	170	21	.	.	PUNCT
ejpam-3397	171	1	let	let	VERB
ejpam-3397	171	2	m	m	PRON
ejpam-3397	171	3	be	be	AUX
ejpam-3397	171	4	an	an	DET
ejpam-3397	171	5	m	m	NOUN
ejpam-3397	171	6	-	-	NOUN
ejpam-3397	171	7	system	system	NOUN
ejpam-3397	171	8	of	of	ADP
ejpam-3397	171	9	a	a	DET
ejpam-3397	171	10	γ	γ	X
ejpam-3397	171	11	-	-	PUNCT
ejpam-3397	171	12	seminearring	seminearre	VERB
ejpam-3397	171	13	r	r	NOUN
ejpam-3397	171	14	whose	whose	DET
ejpam-3397	171	15	each	each	DET
ejpam-3397	171	16	ideal	ideal	NOUN
ejpam-3397	171	17	is	be	AUX
ejpam-3397	171	18	a	a	DET
ejpam-3397	171	19	subtractive	subtractive	NOUN
ejpam-3397	171	20	.	.	PUNCT
ejpam-3397	172	1	let	let	VERB
ejpam-3397	172	2	i	i	PRON
ejpam-3397	172	3	be	be	AUX
ejpam-3397	172	4	an	an	DET
ejpam-3397	172	5	ideal	ideal	NOUN
ejpam-3397	172	6	with	with	ADP
ejpam-3397	172	7	i	i	PRON
ejpam-3397	172	8	∩m	∩m	PROPN
ejpam-3397	173	1	=	=	PUNCT
ejpam-3397	173	2	∅.	∅.	VERB
ejpam-3397	173	3	then	then	ADV
ejpam-3397	173	4	,	,	PUNCT
ejpam-3397	173	5	there	there	PRON
ejpam-3397	173	6	exists	exist	VERB
ejpam-3397	173	7	a	a	DET
ejpam-3397	173	8	weakly	weakly	ADJ
ejpam-3397	173	9	prime	prime	ADJ
ejpam-3397	173	10	ideal	ideal	NOUN
ejpam-3397	173	11	p	p	NOUN
ejpam-3397	173	12	such	such	ADJ
ejpam-3397	173	13	that	that	SCONJ
ejpam-3397	173	14	i	i	PRON
ejpam-3397	173	15	⊆	⊆	NUM
ejpam-3397	173	16	p	p	NOUN
ejpam-3397	173	17	and	and	CCONJ
ejpam-3397	173	18	p	p	NOUN
ejpam-3397	173	19	∩m	∩m	PROPN
ejpam-3397	173	20	=	=	PUNCT
ejpam-3397	173	21	∅.	∅.	NOUN
ejpam-3397	173	22	proof	proof	NOUN
ejpam-3397	173	23	.	.	PUNCT
ejpam-3397	174	1	let	let	VERB
ejpam-3397	174	2	=	=	PRON
ejpam-3397	175	1	=	=	PRON
ejpam-3397	175	2	{	{	PUNCT
ejpam-3397	175	3	j	j	NOUN
ejpam-3397	175	4	:	:	PUNCT
ejpam-3397	175	5	j	j	PROPN
ejpam-3397	175	6	is	be	AUX
ejpam-3397	175	7	an	an	DET
ejpam-3397	175	8	ideal	ideal	NOUN
ejpam-3397	175	9	of	of	ADP
ejpam-3397	175	10	r	r	NOUN
ejpam-3397	175	11	,	,	PUNCT
ejpam-3397	175	12	i	i	PROPN
ejpam-3397	175	13	⊆	⊆	NUM
ejpam-3397	175	14	j	j	PROPN
ejpam-3397	175	15	and	and	CCONJ
ejpam-3397	175	16	j	j	PROPN
ejpam-3397	175	17	∩m	∩m	PROPN
ejpam-3397	175	18	6=	6=	ADP
ejpam-3397	175	19	∅	∅	NOUN
ejpam-3397	175	20	}	}	PUNCT
ejpam-3397	175	21	.	.	PUNCT
ejpam-3397	176	1	then	then	ADV
ejpam-3397	176	2	,	,	PUNCT
ejpam-3397	176	3	=	=	SYM
ejpam-3397	176	4	6=	6=	NOUN
ejpam-3397	176	5	∅	∅	NOUN
ejpam-3397	176	6	and	and	CCONJ
ejpam-3397	176	7	let	let	VERB
ejpam-3397	176	8	{	{	PUNCT
ejpam-3397	176	9	jα}α∈a	jα}α∈a	VERB
ejpam-3397	176	10	be	be	AUX
ejpam-3397	176	11	a	a	DET
ejpam-3397	176	12	chain	chain	NOUN
ejpam-3397	176	13	in	in	ADP
ejpam-3397	176	14	i	i	PRON
ejpam-3397	176	15	which	which	PRON
ejpam-3397	176	16	is	be	AUX
ejpam-3397	176	17	ordered	order	VERB
ejpam-3397	176	18	under	under	ADP
ejpam-3397	176	19	set	set	VERB
ejpam-3397	176	20	inclusion	inclusion	NOUN
ejpam-3397	176	21	.	.	PUNCT
ejpam-3397	177	1	then	then	ADV
ejpam-3397	177	2	,	,	PUNCT
ejpam-3397	177	3	i	i	PRON
ejpam-3397	177	4	⊆	⊆	NUM
ejpam-3397	177	5	∩α∈λjα	∩α∈λjα	ADP
ejpam-3397	177	6	and	and	CCONJ
ejpam-3397	177	7	(	(	PUNCT
ejpam-3397	177	8	∪α∈λjα	∪α∈λjα	PROPN
ejpam-3397	177	9	)	)	PUNCT
ejpam-3397	177	10	∩m	∩m	PROPN
ejpam-3397	178	1	=	=	PUNCT
ejpam-3397	178	2	∪α∈λ(jα	∪α∈λ(jα	PROPN
ejpam-3397	178	3	∩m	∩m	PROPN
ejpam-3397	178	4	)	)	PUNCT
ejpam-3397	178	5	6=	6=	ADP
ejpam-3397	178	6	∅.	∅.	ADP
ejpam-3397	178	7	thus	thus	ADV
ejpam-3397	178	8	,	,	PUNCT
ejpam-3397	178	9	∪α∈λjα	∪α∈λjα	PROPN
ejpam-3397	178	10	∈	∈	PROPN
ejpam-3397	178	11	i.	i.	NOUN
ejpam-3397	178	12	by	by	ADP
ejpam-3397	178	13	zorn	zorn	PROPN
ejpam-3397	178	14	’s	’s	PART
ejpam-3397	178	15	lemma	lemma	PROPN
ejpam-3397	178	16	,	,	PUNCT
ejpam-3397	178	17	=	=	PRON
ejpam-3397	178	18	has	have	VERB
ejpam-3397	178	19	a	a	DET
ejpam-3397	178	20	maximal	maximal	ADJ
ejpam-3397	178	21	element	element	NOUN
ejpam-3397	178	22	say	say	VERB
ejpam-3397	178	23	p	p	NOUN
ejpam-3397	178	24	.	.	PUNCT
ejpam-3397	179	1	we	we	PRON
ejpam-3397	179	2	also	also	ADV
ejpam-3397	179	3	claim	claim	VERB
ejpam-3397	179	4	that	that	SCONJ
ejpam-3397	179	5	p	p	NOUN
ejpam-3397	179	6	is	be	AUX
ejpam-3397	179	7	a	a	DET
ejpam-3397	179	8	weakly	weakly	ADJ
ejpam-3397	179	9	prime	prime	ADJ
ejpam-3397	179	10	ideal	ideal	NOUN
ejpam-3397	179	11	.	.	PUNCT
ejpam-3397	180	1	if	if	SCONJ
ejpam-3397	180	2	p	p	PROPN
ejpam-3397	180	3	⊂	⊂	PROPN
ejpam-3397	180	4	k1	k1	PROPN
ejpam-3397	180	5	and	and	CCONJ
ejpam-3397	180	6	p	p	PROPN
ejpam-3397	180	7	⊂	⊂	PROPN
ejpam-3397	180	8	k2	k2	PROPN
ejpam-3397	180	9	,	,	PUNCT
ejpam-3397	180	10	then	then	ADV
ejpam-3397	180	11	there	there	PRON
ejpam-3397	180	12	exist	exist	VERB
ejpam-3397	180	13	k1	k1	PROPN
ejpam-3397	180	14	∈	∈	PROPN
ejpam-3397	180	15	k1∩m	k1∩m	NOUN
ejpam-3397	180	16	,	,	PUNCT
ejpam-3397	180	17	k2	k2	PROPN
ejpam-3397	180	18	∈	∈	PROPN
ejpam-3397	180	19	k2∩m	k2∩m	X
ejpam-3397	180	20	and	and	CCONJ
ejpam-3397	180	21	α	α	PROPN
ejpam-3397	180	22	∈	∈	PROPN
ejpam-3397	180	23	γ	γ	NOUN
ejpam-3397	181	1	such	such	ADJ
ejpam-3397	181	2	that	that	DET
ejpam-3397	181	3	0	0	NUM
ejpam-3397	181	4	6=	6=	NUM
ejpam-3397	181	5	(	(	PUNCT
ejpam-3397	181	6	k1)α(k2	k1)α(k2	PROPN
ejpam-3397	181	7	)	)	PUNCT
ejpam-3397	181	8	⊆	⊆	NUM
ejpam-3397	181	9	0	0	NUM
ejpam-3397	181	10	6=	6=	ADP
ejpam-3397	181	11	k1γk2	k1γk2	NOUN
ejpam-3397	181	12	and	and	CCONJ
ejpam-3397	181	13	there	there	PRON
ejpam-3397	181	14	exist	exist	VERB
ejpam-3397	181	15	k′1	k′1	X
ejpam-3397	181	16	∈	∈	PROPN
ejpam-3397	181	17	(	(	PUNCT
ejpam-3397	181	18	k1	k1	NOUN
ejpam-3397	181	19	)	)	PUNCT
ejpam-3397	181	20	and	and	CCONJ
ejpam-3397	181	21	k′2	k′2	PROPN
ejpam-3397	181	22	∈	∈	PROPN
ejpam-3397	181	23	(	(	PUNCT
ejpam-3397	181	24	k2	k2	NOUN
ejpam-3397	181	25	)	)	PUNCT
ejpam-3397	182	1	such	such	ADJ
ejpam-3397	182	2	that	that	DET
ejpam-3397	182	3	0	0	NUM
ejpam-3397	182	4	6=	6=	NUM
ejpam-3397	182	5	k′1αk	k′1αk	PROPN
ejpam-3397	182	6	′	′	NOUN
ejpam-3397	182	7	2	2	NUM
ejpam-3397	182	8	∈m	∈m	NOUN
ejpam-3397	182	9	.	.	PUNCT
ejpam-3397	183	1	thus	thus	ADV
ejpam-3397	183	2	,	,	PUNCT
ejpam-3397	183	3	0	0	PROPN
ejpam-3397	183	4	6=	6=	NUM
ejpam-3397	183	5	k′1αk	k′1αk	PROPN
ejpam-3397	183	6	′	′	NOUN
ejpam-3397	183	7	2	2	NUM
ejpam-3397	183	8	∈	∈	NOUN
ejpam-3397	183	9	0	0	PUNCT
ejpam-3397	183	10	6=	6=	NUM
ejpam-3397	183	11	k1γk2	k1γk2	NOUN
ejpam-3397	183	12	∩m	∩m	PROPN
ejpam-3397	183	13	.	.	PUNCT
ejpam-3397	184	1	since	since	SCONJ
ejpam-3397	184	2	p	p	PRON
ejpam-3397	184	3	∩m	∩m	NOUN
ejpam-3397	184	4	=	=	PUNCT
ejpam-3397	184	5	∅	∅	NOUN
ejpam-3397	184	6	,	,	PUNCT
ejpam-3397	184	7	(	(	PUNCT
ejpam-3397	184	8	k1γk2	k1γk2	NOUN
ejpam-3397	184	9	)	)	PUNCT
ejpam-3397	184	10	*	*	PUNCT
ejpam-3397	185	1	p	p	NOUN
ejpam-3397	185	2	.	.	PUNCT
ejpam-3397	186	1	hence	hence	ADV
ejpam-3397	186	2	,	,	PUNCT
ejpam-3397	186	3	p	p	PROPN
ejpam-3397	186	4	is	be	AUX
ejpam-3397	186	5	a	a	DET
ejpam-3397	186	6	weakly	weakly	ADJ
ejpam-3397	186	7	prime	prime	ADJ
ejpam-3397	186	8	ideal	ideal	NOUN
ejpam-3397	186	9	.	.	PUNCT
ejpam-3397	187	1	now	now	ADV
ejpam-3397	187	2	we	we	PRON
ejpam-3397	187	3	present	present	VERB
ejpam-3397	187	4	few	few	ADJ
ejpam-3397	187	5	results	result	NOUN
ejpam-3397	187	6	about	about	ADP
ejpam-3397	187	7	such	such	DET
ejpam-3397	187	8	a	a	DET
ejpam-3397	187	9	γ	γ	X
ejpam-3397	187	10	-	-	PUNCT
ejpam-3397	187	11	seminearring	seminearre	VERB
ejpam-3397	187	12	r	r	NOUN
ejpam-3397	187	13	in	in	ADP
ejpam-3397	187	14	which	which	PRON
ejpam-3397	187	15	each	each	DET
ejpam-3397	187	16	ideal	ideal	NOUN
ejpam-3397	187	17	is	be	AUX
ejpam-3397	187	18	weakly	weakly	ADV
ejpam-3397	187	19	prime	prime	ADJ
ejpam-3397	187	20	.	.	PUNCT
ejpam-3397	188	1	proposition	proposition	NOUN
ejpam-3397	188	2	7	7	NUM
ejpam-3397	188	3	.	.	PUNCT
ejpam-3397	189	1	every	every	DET
ejpam-3397	189	2	ideal	ideal	NOUN
ejpam-3397	189	3	of	of	ADP
ejpam-3397	189	4	a	a	DET
ejpam-3397	189	5	γ	γ	NOUN
ejpam-3397	189	6	-	-	PUNCT
ejpam-3397	189	7	seminearring	seminearre	VERB
ejpam-3397	189	8	r	r	NOUN
ejpam-3397	189	9	is	be	AUX
ejpam-3397	189	10	a	a	DET
ejpam-3397	189	11	weakly	weakly	ADJ
ejpam-3397	189	12	prime	prime	NOUN
ejpam-3397	189	13	if	if	SCONJ
ejpam-3397	189	14	and	and	CCONJ
ejpam-3397	189	15	only	only	ADV
ejpam-3397	189	16	if	if	SCONJ
ejpam-3397	189	17	for	for	ADP
ejpam-3397	189	18	any	any	DET
ejpam-3397	189	19	ideals	ideal	NOUN
ejpam-3397	190	1	i	i	PRON
ejpam-3397	190	2	,	,	PUNCT
ejpam-3397	190	3	j	j	PROPN
ejpam-3397	190	4	,	,	PUNCT
ejpam-3397	190	5	k	k	PROPN
ejpam-3397	190	6	of	of	ADP
ejpam-3397	190	7	r	r	PROPN
ejpam-3397	190	8	,	,	PUNCT
ejpam-3397	190	9	iγj	iγj	NOUN
ejpam-3397	190	10	=	=	PUNCT
ejpam-3397	191	1	i	i	PROPN
ejpam-3397	191	2	,	,	PUNCT
ejpam-3397	191	3	iγj	iγj	PROPN
ejpam-3397	191	4	=	=	SYM
ejpam-3397	191	5	j	j	PROPN
ejpam-3397	191	6	,	,	PUNCT
ejpam-3397	191	7	iγj	iγj	PROPN
ejpam-3397	191	8	=	=	PUNCT
ejpam-3397	192	1	k	k	PROPN
ejpam-3397	192	2	where	where	SCONJ
ejpam-3397	192	3	k	k	PROPN
ejpam-3397	192	4	is	be	AUX
ejpam-3397	192	5	the	the	DET
ejpam-3397	192	6	ideal	ideal	NOUN
ejpam-3397	192	7	contained	contain	VERB
ejpam-3397	192	8	in	in	ADP
ejpam-3397	192	9	both	both	CCONJ
ejpam-3397	192	10	i	i	PROPN
ejpam-3397	192	11	and	and	CCONJ
ejpam-3397	192	12	j	j	PROPN
ejpam-3397	192	13	,	,	PUNCT
ejpam-3397	192	14	or	or	CCONJ
ejpam-3397	192	15	iγj	iγj	X
ejpam-3397	192	16	=	=	NOUN
ejpam-3397	192	17	0	0	X
ejpam-3397	192	18	.	.	PUNCT
ejpam-3397	192	19	proof	proof	NOUN
ejpam-3397	192	20	.	.	PUNCT
ejpam-3397	193	1	suppose	suppose	VERB
ejpam-3397	193	2	that	that	SCONJ
ejpam-3397	193	3	every	every	DET
ejpam-3397	193	4	ideal	ideal	NOUN
ejpam-3397	193	5	of	of	ADP
ejpam-3397	193	6	r	r	NOUN
ejpam-3397	193	7	is	be	AUX
ejpam-3397	193	8	a	a	DET
ejpam-3397	193	9	weakly	weakly	ADJ
ejpam-3397	193	10	prime	prime	NOUN
ejpam-3397	193	11	.	.	PUNCT
ejpam-3397	194	1	let	let	VERB
ejpam-3397	194	2	i	i	PRON
ejpam-3397	194	3	,	,	PUNCT
ejpam-3397	194	4	j	j	PROPN
ejpam-3397	194	5	are	be	AUX
ejpam-3397	194	6	ideals	ideal	NOUN
ejpam-3397	194	7	of	of	ADP
ejpam-3397	194	8	a	a	DET
ejpam-3397	194	9	γ	γ	NOUN
ejpam-3397	194	10	-	-	PUNCT
ejpam-3397	194	11	seminearring	seminearre	VERB
ejpam-3397	194	12	r.	r.	NOUN
ejpam-3397	194	13	if	if	SCONJ
ejpam-3397	194	14	iγj	iγj	PROPN
ejpam-3397	194	15	6=	6=	ADP
ejpam-3397	194	16	r	r	NOUN
ejpam-3397	194	17	,	,	PUNCT
ejpam-3397	194	18	then	then	ADV
ejpam-3397	194	19	iγj	iγj	PROPN
ejpam-3397	194	20	is	be	AUX
ejpam-3397	194	21	a	a	DET
ejpam-3397	194	22	weakly	weakly	ADJ
ejpam-3397	194	23	prime	prime	NOUN
ejpam-3397	194	24	.	.	PUNCT
ejpam-3397	195	1	if	if	SCONJ
ejpam-3397	195	2	0	0	NUM
ejpam-3397	195	3	6=	6=	NUM
ejpam-3397	195	4	iγj	iγj	NOUN
ejpam-3397	195	5	⊆	⊆	NUM
ejpam-3397	195	6	iγj	iγj	NOUN
ejpam-3397	195	7	,	,	PUNCT
ejpam-3397	195	8	then	then	ADV
ejpam-3397	195	9	we	we	PRON
ejpam-3397	195	10	have	have	VERB
ejpam-3397	195	11	i	i	PRON
ejpam-3397	195	12	⊆	⊆	NUM
ejpam-3397	195	13	iγj	iγj	NOUN
ejpam-3397	195	14	or	or	CCONJ
ejpam-3397	195	15	j	j	PROPN
ejpam-3397	195	16	⊆	⊆	NUM
ejpam-3397	195	17	iγj	iγj	NOUN
ejpam-3397	195	18	i.e.	i.e.	X
ejpam-3397	195	19	,	,	PUNCT
ejpam-3397	195	20	i	i	PRON
ejpam-3397	195	21	=	=	VERB
ejpam-3397	195	22	iγj	iγj	NOUN
ejpam-3397	195	23	or	or	CCONJ
ejpam-3397	195	24	j	j	NOUN
ejpam-3397	195	25	=	=	SYM
ejpam-3397	195	26	iγj	iγj	NOUN
ejpam-3397	195	27	.	.	PUNCT
ejpam-3397	196	1	if	if	SCONJ
ejpam-3397	196	2	iγj	iγj	PRON
ejpam-3397	196	3	=	=	PUNCT
ejpam-3397	197	1	k	k	PROPN
ejpam-3397	197	2	then	then	ADV
ejpam-3397	197	3	clearly	clearly	ADV
ejpam-3397	197	4	k	k	PROPN
ejpam-3397	197	5	=	=	PUNCT
ejpam-3397	198	1	i	i	PRON
ejpam-3397	198	2	∩j	∩j	PROPN
ejpam-3397	198	3	is	be	AUX
ejpam-3397	198	4	a	a	DET
ejpam-3397	198	5	weakly	weakly	ADJ
ejpam-3397	198	6	prime	prime	ADJ
ejpam-3397	198	7	ideal	ideal	NOUN
ejpam-3397	198	8	then	then	ADV
ejpam-3397	198	9	by	by	ADP
ejpam-3397	198	10	proposition3	proposition3	PROPN
ejpam-3397	198	11	,	,	PUNCT
ejpam-3397	199	1	k	k	PROPN
ejpam-3397	199	2	⊂	⊂	PROPN
ejpam-3397	199	3	i	i	PROPN
ejpam-3397	199	4	and	and	CCONJ
ejpam-3397	199	5	k	k	PROPN
ejpam-3397	199	6	⊂	⊂	PROPN
ejpam-3397	199	7	j	j	PROPN
ejpam-3397	199	8	.	.	PUNCT
ejpam-3397	200	1	finally	finally	ADV
ejpam-3397	200	2	,	,	PUNCT
ejpam-3397	200	3	if	if	SCONJ
ejpam-3397	200	4	iγj	iγj	NOUN
ejpam-3397	200	5	=	=	SYM
ejpam-3397	200	6	r	r	NOUN
ejpam-3397	200	7	,	,	PUNCT
ejpam-3397	200	8	then	then	ADV
ejpam-3397	200	9	we	we	PRON
ejpam-3397	200	10	have	have	VERB
ejpam-3397	200	11	i	i	NOUN
ejpam-3397	200	12	=	=	PUNCT
ejpam-3397	200	13	j	j	PROPN
ejpam-3397	200	14	=	=	SYM
ejpam-3397	200	15	r	r	NOUN
ejpam-3397	200	16	and	and	CCONJ
ejpam-3397	200	17	hence	hence	ADV
ejpam-3397	200	18	rγr	rγr	NOUN
ejpam-3397	200	19	=	=	SYM
ejpam-3397	200	20	r.	r.	PROPN
ejpam-3397	200	21	conversely	conversely	ADV
ejpam-3397	200	22	,	,	PUNCT
ejpam-3397	200	23	let	let	VERB
ejpam-3397	200	24	l	l	NOUN
ejpam-3397	200	25	be	be	AUX
ejpam-3397	200	26	any	any	DET
ejpam-3397	200	27	proper	proper	ADJ
ejpam-3397	200	28	ideal	ideal	NOUN
ejpam-3397	200	29	of	of	ADP
ejpam-3397	200	30	r	r	NOUN
ejpam-3397	200	31	and	and	CCONJ
ejpam-3397	200	32	suppose	suppose	VERB
ejpam-3397	200	33	that	that	SCONJ
ejpam-3397	200	34	0	0	NUM
ejpam-3397	200	35	6=	6=	NUM
ejpam-3397	200	36	iγj	iγj	VERB
ejpam-3397	200	37	⊆	⊆	NUM
ejpam-3397	200	38	l	l	NOUN
ejpam-3397	200	39	for	for	ADP
ejpam-3397	200	40	ideals	ideal	NOUN
ejpam-3397	200	41	i	i	PRON
ejpam-3397	200	42	and	and	CCONJ
ejpam-3397	200	43	j	j	PROPN
ejpam-3397	200	44	of	of	ADP
ejpam-3397	200	45	r.	r.	PROPN
ejpam-3397	200	46	then	then	ADV
ejpam-3397	200	47	,	,	PUNCT
ejpam-3397	200	48	we	we	PRON
ejpam-3397	200	49	have	have	VERB
ejpam-3397	200	50	either	either	CCONJ
ejpam-3397	201	1	i	i	PRON
ejpam-3397	201	2	=	=	PRON
ejpam-3397	201	3	iγj	iγj	VERB
ejpam-3397	201	4	⊆	⊆	NUM
ejpam-3397	201	5	l	l	NOUN
ejpam-3397	201	6	or	or	CCONJ
ejpam-3397	201	7	j	j	NOUN
ejpam-3397	201	8	=	=	PRON
ejpam-3397	201	9	iγj	iγj	VERB
ejpam-3397	201	10	⊆	⊆	NUM
ejpam-3397	201	11	l.	l.	NOUN
ejpam-3397	202	1	and	and	CCONJ
ejpam-3397	202	2	if	if	SCONJ
ejpam-3397	202	3	k	k	PROPN
ejpam-3397	202	4	=	=	PRON
ejpam-3397	202	5	iγj	iγj	VERB
ejpam-3397	202	6	⊆	⊆	NUM
ejpam-3397	202	7	l	l	NOUN
ejpam-3397	202	8	,	,	PUNCT
ejpam-3397	202	9	where	where	SCONJ
ejpam-3397	202	10	k	k	PROPN
ejpam-3397	202	11	⊂	⊂	PROPN
ejpam-3397	202	12	i	i	PRON
ejpam-3397	202	13	∩	∩	VERB
ejpam-3397	202	14	j	j	PROPN
ejpam-3397	202	15	and	and	CCONJ
ejpam-3397	202	16	hence	hence	ADV
ejpam-3397	202	17	k	k	PROPN
ejpam-3397	202	18	∩	∩	PROPN
ejpam-3397	202	19	i	i	PRON
ejpam-3397	202	20	⊆	⊆	NUM
ejpam-3397	202	21	i	i	PROPN
ejpam-3397	202	22	and	and	CCONJ
ejpam-3397	202	23	k	k	PROPN
ejpam-3397	202	24	∩	∩	PROPN
ejpam-3397	202	25	j	j	PROPN
ejpam-3397	202	26	⊆	⊆	NUM
ejpam-3397	202	27	l.	l.	PROPN
ejpam-3397	202	28	example	example	NOUN
ejpam-3397	202	29	4	4	NUM
ejpam-3397	202	30	.	.	PUNCT
ejpam-3397	202	31	refer	refer	VERB
ejpam-3397	202	32	to	to	ADP
ejpam-3397	202	33	the	the	DET
ejpam-3397	202	34	γ	γ	X
ejpam-3397	202	35	-	-	PUNCT
ejpam-3397	202	36	seminearring	seminearre	VERB
ejpam-3397	202	37	s	s	PRON
ejpam-3397	202	38	defined	define	VERB
ejpam-3397	202	39	by	by	ADP
ejpam-3397	202	40	tables	table	NOUN
ejpam-3397	202	41	in	in	ADP
ejpam-3397	202	42	example3	example3	PROPN
ejpam-3397	202	43	.	.	PUNCT
ejpam-3397	203	1	clearly	clearly	ADV
ejpam-3397	203	2	,	,	PUNCT
ejpam-3397	203	3	s	s	VERB
ejpam-3397	203	4	has	have	VERB
ejpam-3397	203	5	four	four	NUM
ejpam-3397	203	6	ideals	ideal	NOUN
ejpam-3397	203	7	,	,	PUNCT
ejpam-3397	203	8	i	i	PRON
ejpam-3397	203	9	=	=	PUNCT
ejpam-3397	203	10	{	{	PUNCT
ejpam-3397	203	11	0	0	NUM
ejpam-3397	203	12	,	,	PUNCT
ejpam-3397	203	13	a	a	PRON
ejpam-3397	203	14	}	}	PUNCT
ejpam-3397	203	15	,	,	PUNCT
ejpam-3397	204	1	j	j	PROPN
ejpam-3397	204	2	=	=	PUNCT
ejpam-3397	204	3	{	{	PUNCT
ejpam-3397	204	4	0	0	NUM
ejpam-3397	204	5	,	,	PUNCT
ejpam-3397	204	6	c	c	NOUN
ejpam-3397	204	7	}	}	PUNCT
ejpam-3397	204	8	,	,	PUNCT
ejpam-3397	204	9	k	k	PROPN
ejpam-3397	204	10	=	=	PUNCT
ejpam-3397	204	11	{	{	PUNCT
ejpam-3397	204	12	0	0	NUM
ejpam-3397	204	13	,	,	PUNCT
ejpam-3397	204	14	e	e	NOUN
ejpam-3397	204	15	,	,	PUNCT
ejpam-3397	204	16	c	c	NOUN
ejpam-3397	204	17	}	}	PUNCT
ejpam-3397	204	18	and	and	CCONJ
ejpam-3397	204	19	l	l	NOUN
ejpam-3397	204	20	=	=	PUNCT
ejpam-3397	204	21	{	{	PUNCT
ejpam-3397	204	22	0	0	NUM
ejpam-3397	204	23	,	,	PUNCT
ejpam-3397	204	24	a	a	DET
ejpam-3397	204	25	,	,	PUNCT
ejpam-3397	204	26	c	c	NOUN
ejpam-3397	204	27	}	}	PUNCT
ejpam-3397	204	28	.	.	PUNCT
ejpam-3397	205	1	now	now	ADV
ejpam-3397	205	2	,	,	PUNCT
ejpam-3397	205	3	iγj	iγj	PROPN
ejpam-3397	205	4	=	=	PUNCT
ejpam-3397	205	5	{	{	PUNCT
ejpam-3397	205	6	0	0	NUM
ejpam-3397	205	7	}	}	PUNCT
ejpam-3397	205	8	,	,	PUNCT
ejpam-3397	205	9	iγk	iγk	VERB
ejpam-3397	205	10	=	=	SYM
ejpam-3397	205	11	{	{	PUNCT
ejpam-3397	205	12	0	0	NUM
ejpam-3397	205	13	}	}	PUNCT
ejpam-3397	205	14	,	,	PUNCT
ejpam-3397	205	15	iγl	iγl	NOUN
ejpam-3397	206	1	=	=	SYM
ejpam-3397	206	2	i	i	PROPN
ejpam-3397	206	3	,	,	PUNCT
ejpam-3397	206	4	jγi	jγi	NOUN
ejpam-3397	206	5	=	=	SYM
ejpam-3397	206	6	{	{	PUNCT
ejpam-3397	206	7	0	0	NUM
ejpam-3397	206	8	}	}	PUNCT
ejpam-3397	206	9	,	,	PUNCT
ejpam-3397	206	10	jγk	jγk	NOUN
ejpam-3397	206	11	=	=	SYM
ejpam-3397	206	12	{	{	PUNCT
ejpam-3397	206	13	0	0	NUM
ejpam-3397	206	14	}	}	PUNCT
ejpam-3397	206	15	,	,	PUNCT
ejpam-3397	206	16	jγl	jγl	NOUN
ejpam-3397	206	17	=	=	PUNCT
ejpam-3397	206	18	{	{	PUNCT
ejpam-3397	206	19	0	0	NUM
ejpam-3397	206	20	}	}	PUNCT
ejpam-3397	206	21	,	,	PUNCT
ejpam-3397	206	22	kγi	kγi	NOUN
ejpam-3397	206	23	=	=	SYM
ejpam-3397	206	24	{	{	PUNCT
ejpam-3397	206	25	0	0	NUM
ejpam-3397	206	26	}	}	PUNCT
ejpam-3397	206	27	,	,	PUNCT
ejpam-3397	206	28	kγj	kγj	PROPN
ejpam-3397	206	29	=	=	SYM
ejpam-3397	206	30	j	j	PROPN
ejpam-3397	206	31	,	,	PUNCT
ejpam-3397	206	32	kγl	kγl	PROPN
ejpam-3397	206	33	=	=	SYM
ejpam-3397	206	34	j	j	PROPN
ejpam-3397	206	35	where	where	SCONJ
ejpam-3397	206	36	j	j	PROPN
ejpam-3397	206	37	⊂	⊂	PROPN
ejpam-3397	206	38	k	k	PROPN
ejpam-3397	206	39	and	and	CCONJ
ejpam-3397	206	40	j	j	PROPN
ejpam-3397	206	41	⊂	⊂	PROPN
ejpam-3397	206	42	l	l	PROPN
ejpam-3397	206	43	,	,	PUNCT
ejpam-3397	206	44	lγi	lγi	NOUN
ejpam-3397	206	45	=	=	SYM
ejpam-3397	206	46	i	i	PROPN
ejpam-3397	206	47	,	,	PUNCT
ejpam-3397	206	48	lγj	lγj	PROPN
ejpam-3397	206	49	=	=	X
ejpam-3397	206	50	{	{	PUNCT
ejpam-3397	206	51	0	0	NUM
ejpam-3397	206	52	}	}	PUNCT
ejpam-3397	206	53	,	,	PUNCT
ejpam-3397	206	54	lγk	lγk	NOUN
ejpam-3397	206	55	=	=	PUNCT
ejpam-3397	206	56	{	{	PUNCT
ejpam-3397	206	57	0	0	NUM
ejpam-3397	206	58	}	}	PUNCT
ejpam-3397	206	59	.	.	PUNCT
ejpam-3397	207	1	hence	hence	ADV
ejpam-3397	207	2	,	,	PUNCT
ejpam-3397	207	3	we	we	PRON
ejpam-3397	207	4	can	can	AUX
ejpam-3397	207	5	check	check	VERB
ejpam-3397	207	6	easily	easily	ADV
ejpam-3397	207	7	that	that	SCONJ
ejpam-3397	207	8	every	every	DET
ejpam-3397	207	9	ideal	ideal	NOUN
ejpam-3397	207	10	of	of	ADP
ejpam-3397	207	11	s	s	PROPN
ejpam-3397	207	12	is	be	AUX
ejpam-3397	207	13	a	a	DET
ejpam-3397	207	14	weakly	weakly	ADJ
ejpam-3397	207	15	prime	prime	ADJ
ejpam-3397	207	16	ideal	ideal	NOUN
ejpam-3397	207	17	.	.	PUNCT
ejpam-3397	208	1	corollary	corollary	ADJ
ejpam-3397	208	2	1	1	NUM
ejpam-3397	208	3	.	.	PUNCT
ejpam-3397	209	1	let	let	VERB
ejpam-3397	209	2	r	r	PRON
ejpam-3397	209	3	be	be	AUX
ejpam-3397	209	4	a	a	DET
ejpam-3397	209	5	γ	γ	NOUN
ejpam-3397	209	6	-	-	PUNCT
ejpam-3397	209	7	seminearring	seminearring	NOUN
ejpam-3397	209	8	in	in	ADP
ejpam-3397	209	9	which	which	PRON
ejpam-3397	209	10	every	every	DET
ejpam-3397	209	11	ideal	ideal	NOUN
ejpam-3397	209	12	of	of	ADP
ejpam-3397	209	13	r	r	NOUN
ejpam-3397	209	14	is	be	AUX
ejpam-3397	209	15	a	a	DET
ejpam-3397	209	16	weakly	weakly	ADJ
ejpam-3397	209	17	prime	prime	NOUN
ejpam-3397	209	18	.	.	PUNCT
ejpam-3397	210	1	then	then	ADV
ejpam-3397	210	2	for	for	ADP
ejpam-3397	210	3	any	any	DET
ejpam-3397	210	4	ideal	ideal	NOUN
ejpam-3397	210	5	i	i	PRON
ejpam-3397	210	6	of	of	ADP
ejpam-3397	210	7	r	r	PROPN
ejpam-3397	210	8	,	,	PUNCT
ejpam-3397	210	9	either	either	CCONJ
ejpam-3397	210	10	iγi	iγi	NOUN
ejpam-3397	210	11	=	=	PROPN
ejpam-3397	210	12	i2	i2	PROPN
ejpam-3397	210	13	=	=	PUNCT
ejpam-3397	210	14	i	i	PROPN
ejpam-3397	210	15	or	or	CCONJ
ejpam-3397	210	16	iγi	iγi	PROPN
ejpam-3397	210	17	=	=	PROPN
ejpam-3397	210	18	i2	i2	PROPN
ejpam-3397	210	19	=	=	SYM
ejpam-3397	210	20	0	0	PROPN
ejpam-3397	210	21	.	.	NOUN
ejpam-3397	210	22	example	example	NOUN
ejpam-3397	210	23	5	5	NUM
ejpam-3397	210	24	.	.	X
ejpam-3397	210	25	refer	refer	VERB
ejpam-3397	210	26	to	to	ADP
ejpam-3397	210	27	example4	example4	NOUN
ejpam-3397	210	28	,	,	PUNCT
ejpam-3397	210	29	since	since	SCONJ
ejpam-3397	210	30	i	i	PRON
ejpam-3397	210	31	=	=	PUNCT
ejpam-3397	210	32	{	{	PUNCT
ejpam-3397	210	33	0	0	NUM
ejpam-3397	210	34	,	,	PUNCT
ejpam-3397	210	35	a	a	PRON
ejpam-3397	210	36	}	}	PUNCT
ejpam-3397	210	37	be	be	AUX
ejpam-3397	210	38	the	the	DET
ejpam-3397	210	39	weakly	weakly	ADJ
ejpam-3397	210	40	prime	prime	ADJ
ejpam-3397	210	41	ideal	ideal	NOUN
ejpam-3397	210	42	of	of	ADP
ejpam-3397	210	43	s	s	PRON
ejpam-3397	210	44	and	and	CCONJ
ejpam-3397	210	45	hence	hence	ADV
ejpam-3397	210	46	iγi	iγi	NOUN
ejpam-3397	210	47	=	=	PROPN
ejpam-3397	210	48	i2	i2	PROPN
ejpam-3397	210	49	=	=	PROPN
ejpam-3397	210	50	i.	i.	PROPN
ejpam-3397	210	51	also	also	ADV
ejpam-3397	210	52	,	,	PUNCT
ejpam-3397	210	53	for	for	ADP
ejpam-3397	210	54	another	another	DET
ejpam-3397	210	55	weakly	weakly	ADJ
ejpam-3397	210	56	prime	prime	ADJ
ejpam-3397	210	57	ideal	ideal	NOUN
ejpam-3397	210	58	j	j	PROPN
ejpam-3397	211	1	=	=	PUNCT
ejpam-3397	211	2	{	{	PUNCT
ejpam-3397	211	3	0	0	NUM
ejpam-3397	211	4	,	,	PUNCT
ejpam-3397	211	5	c	c	NOUN
ejpam-3397	211	6	}	}	PUNCT
ejpam-3397	211	7	of	of	ADP
ejpam-3397	211	8	s	s	PRON
ejpam-3397	211	9	we	we	PRON
ejpam-3397	211	10	have	have	VERB
ejpam-3397	211	11	jγj	jγj	NOUN
ejpam-3397	211	12	=	=	PROPN
ejpam-3397	211	13	j2	j2	PROPN
ejpam-3397	211	14	=	=	SYM
ejpam-3397	211	15	0	0	PROPN
ejpam-3397	211	16	.	.	PUNCT
ejpam-3397	212	1	in	in	ADP
ejpam-3397	212	2	the	the	DET
ejpam-3397	212	3	above	above	ADJ
ejpam-3397	212	4	example	example	NOUN
ejpam-3397	212	5	the	the	DET
ejpam-3397	212	6	ideal	ideal	NOUN
ejpam-3397	212	7	k	k	PROPN
ejpam-3397	213	1	=	=	PUNCT
ejpam-3397	213	2	{	{	PUNCT
ejpam-3397	213	3	0	0	NUM
ejpam-3397	213	4	,	,	PUNCT
ejpam-3397	213	5	e	e	NOUN
ejpam-3397	213	6	,	,	PUNCT
ejpam-3397	213	7	c	c	X
ejpam-3397	213	8	}	}	PUNCT
ejpam-3397	213	9	we	we	PRON
ejpam-3397	213	10	have	have	VERB
ejpam-3397	213	11	kγk	kγk	NOUN
ejpam-3397	213	12	=	=	SYM
ejpam-3397	213	13	k2	k2	NOUN
ejpam-3397	213	14	=	=	PUNCT
ejpam-3397	213	15	{	{	PUNCT
ejpam-3397	213	16	0	0	NUM
ejpam-3397	213	17	,	,	PUNCT
ejpam-3397	213	18	e	e	NOUN
ejpam-3397	213	19	}	}	PUNCT
ejpam-3397	213	20	which	which	PRON
ejpam-3397	213	21	is	be	AUX
ejpam-3397	213	22	a	a	DET
ejpam-3397	213	23	subset	subset	NOUN
ejpam-3397	213	24	of	of	ADP
ejpam-3397	213	25	γ	γ	NOUN
ejpam-3397	213	26	-	-	PUNCT
ejpam-3397	213	27	seminearring	seminearring	NOUN
ejpam-3397	213	28	but	but	CCONJ
ejpam-3397	213	29	not	not	PART
ejpam-3397	213	30	an	an	DET
ejpam-3397	213	31	ideal	ideal	NOUN
ejpam-3397	213	32	.	.	PUNCT
ejpam-3397	214	1	and	and	CCONJ
ejpam-3397	214	2	for	for	ADP
ejpam-3397	214	3	the	the	DET
ejpam-3397	214	4	ideal	ideal	ADJ
ejpam-3397	214	5	l	l	NOUN
ejpam-3397	214	6	=	=	PUNCT
ejpam-3397	214	7	{	{	PUNCT
ejpam-3397	214	8	0	0	NUM
ejpam-3397	214	9	,	,	PUNCT
ejpam-3397	214	10	a	a	PRON
ejpam-3397	214	11	,	,	PUNCT
ejpam-3397	214	12	c	c	NOUN
ejpam-3397	214	13	}	}	PUNCT
ejpam-3397	214	14	we	we	PRON
ejpam-3397	214	15	have	have	VERB
ejpam-3397	214	16	lγl	lγl	ADJ
ejpam-3397	214	17	=	=	SYM
ejpam-3397	214	18	l2	l2	NOUN
ejpam-3397	214	19	=	=	SYM
ejpam-3397	214	20	{	{	PUNCT
ejpam-3397	214	21	0	0	NUM
ejpam-3397	214	22	,	,	PUNCT
ejpam-3397	214	23	a}which	a}which	PROPN
ejpam-3397	214	24	is	be	AUX
ejpam-3397	214	25	a	a	DET
ejpam-3397	214	26	weakly	weakly	ADJ
ejpam-3397	214	27	prime	prime	ADJ
ejpam-3397	214	28	ideal	ideal	NOUN
ejpam-3397	214	29	of	of	ADP
ejpam-3397	214	30	s.	s.	PROPN
ejpam-3397	214	31	)	)	PUNCT
ejpam-3397	214	32	proposition	proposition	NOUN
ejpam-3397	214	33	8	8	NUM
ejpam-3397	214	34	.	.	PUNCT
ejpam-3397	214	35	suppose	suppose	VERB
ejpam-3397	214	36	that	that	SCONJ
ejpam-3397	214	37	every	every	DET
ejpam-3397	214	38	ideal	ideal	NOUN
ejpam-3397	214	39	of	of	ADP
ejpam-3397	214	40	a	a	DET
ejpam-3397	214	41	γ	γ	NOUN
ejpam-3397	214	42	-	-	PUNCT
ejpam-3397	214	43	seminearring	seminearre	VERB
ejpam-3397	214	44	r	r	NOUN
ejpam-3397	214	45	is	be	AUX
ejpam-3397	214	46	a	a	DET
ejpam-3397	214	47	weakly	weakly	ADJ
ejpam-3397	214	48	prime	prime	NOUN
ejpam-3397	214	49	.	.	PUNCT
ejpam-3397	215	1	if	if	SCONJ
ejpam-3397	215	2	m1	m1	PROPN
ejpam-3397	215	3	and	and	CCONJ
ejpam-3397	215	4	m2	m2	PROPN
ejpam-3397	215	5	are	be	AUX
ejpam-3397	215	6	two	two	NUM
ejpam-3397	215	7	maximal	maximal	ADJ
ejpam-3397	215	8	ideals	ideal	NOUN
ejpam-3397	215	9	of	of	ADP
ejpam-3397	215	10	r	r	NOUN
ejpam-3397	215	11	then	then	ADV
ejpam-3397	215	12	m1γm2	m1γm2	VERB
ejpam-3397	215	13	=	=	SYM
ejpam-3397	215	14	0	0	NUM
ejpam-3397	215	15	or	or	CCONJ
ejpam-3397	215	16	m1γm2	m1γm2	PROPN
ejpam-3397	215	17	=	=	SYM
ejpam-3397	216	1	n	n	PROPN
ejpam-3397	216	2	=	=	SYM
ejpam-3397	216	3	m1	m1	PROPN
ejpam-3397	216	4	∩m2	∩m2	PROPN
ejpam-3397	216	5	.	.	PUNCT
ejpam-3397	216	6	abdelghani	abdelghani	PROPN
ejpam-3397	216	7	taouti	taouti	PROPN
ejpam-3397	216	8	et	et	PROPN
ejpam-3397	216	9	al	al	PROPN
ejpam-3397	216	10	.	.	PUNCT
ejpam-3397	216	11	/	/	SYM
ejpam-3397	216	12	eur	eur	PROPN
ejpam-3397	216	13	.	.	PUNCT
ejpam-3397	217	1	j.	j.	PROPN
ejpam-3397	217	2	pure	pure	PROPN
ejpam-3397	217	3	appl	appl	PROPN
ejpam-3397	217	4	.	.	PROPN
ejpam-3397	217	5	math	math	PROPN
ejpam-3397	217	6	,	,	PUNCT
ejpam-3397	217	7	12	12	NUM
ejpam-3397	217	8	(	(	PUNCT
ejpam-3397	217	9	2	2	NUM
ejpam-3397	217	10	)	)	PUNCT
ejpam-3397	217	11	(	(	PUNCT
ejpam-3397	217	12	2019	2019	NUM
ejpam-3397	217	13	)	)	PUNCT
ejpam-3397	217	14	,	,	PUNCT
ejpam-3397	217	15	544	544	NUM
ejpam-3397	217	16	-	-	SYM
ejpam-3397	217	17	552	552	NUM
ejpam-3397	217	18	549	549	NUM
ejpam-3397	217	19	proof	proof	NOUN
ejpam-3397	217	20	.	.	PUNCT
ejpam-3397	218	1	suppose	suppose	VERB
ejpam-3397	218	2	every	every	DET
ejpam-3397	218	3	ideal	ideal	NOUN
ejpam-3397	218	4	of	of	ADP
ejpam-3397	218	5	a	a	DET
ejpam-3397	218	6	γ	γ	NOUN
ejpam-3397	218	7	-	-	PUNCT
ejpam-3397	218	8	seminearring	seminearre	VERB
ejpam-3397	218	9	r	r	NOUN
ejpam-3397	218	10	is	be	AUX
ejpam-3397	218	11	a	a	DET
ejpam-3397	218	12	weakly	weakly	ADJ
ejpam-3397	218	13	prime	prime	ADJ
ejpam-3397	218	14	ideal	ideal	NOUN
ejpam-3397	218	15	.	.	PUNCT
ejpam-3397	219	1	let	let	VERB
ejpam-3397	219	2	m1	m1	PROPN
ejpam-3397	219	3	and	and	CCONJ
ejpam-3397	219	4	m2	m2	PROPN
ejpam-3397	219	5	be	be	VERB
ejpam-3397	219	6	the	the	DET
ejpam-3397	219	7	two	two	NUM
ejpam-3397	219	8	distinct	distinct	ADJ
ejpam-3397	219	9	maximal	maximal	ADJ
ejpam-3397	219	10	ideals	ideal	NOUN
ejpam-3397	219	11	.	.	PUNCT
ejpam-3397	220	1	since	since	ADV
ejpam-3397	220	2	,	,	PUNCT
ejpam-3397	220	3	m1	m1	PROPN
ejpam-3397	220	4	∩m2	∩m2	PROPN
ejpam-3397	220	5	is	be	AUX
ejpam-3397	220	6	a	a	DET
ejpam-3397	220	7	weakly	weakly	ADJ
ejpam-3397	220	8	prime	prime	NOUN
ejpam-3397	220	9	and	and	CCONJ
ejpam-3397	220	10	hence	hence	ADV
ejpam-3397	220	11	m1γm2	m1γm2	VERB
ejpam-3397	220	12	⊆m1∩m2	⊆m1∩m2	PROPN
ejpam-3397	220	13	,	,	PUNCT
ejpam-3397	220	14	we	we	PRON
ejpam-3397	220	15	must	must	AUX
ejpam-3397	220	16	have	have	AUX
ejpam-3397	220	17	m1γm2	m1γm2	VERB
ejpam-3397	220	18	=	=	SYM
ejpam-3397	220	19	0	0	NUM
ejpam-3397	221	1	and	and	CCONJ
ejpam-3397	221	2	similarlym2γm1	similarlym2γm1	ADV
ejpam-3397	221	3	=	=	SYM
ejpam-3397	221	4	0	0	NUM
ejpam-3397	221	5	,	,	PUNCT
ejpam-3397	221	6	or	or	CCONJ
ejpam-3397	221	7	m2γm1	m2γm1	NOUN
ejpam-3397	221	8	=	=	SYM
ejpam-3397	221	9	n	n	NOUN
ejpam-3397	221	10	,	,	PUNCT
ejpam-3397	221	11	being	be	AUX
ejpam-3397	221	12	every	every	DET
ejpam-3397	221	13	ideal	ideal	NOUN
ejpam-3397	221	14	a	a	DET
ejpam-3397	221	15	weakly	weakly	ADJ
ejpam-3397	221	16	prime	prime	ADJ
ejpam-3397	221	17	ideal	ideal	NOUN
ejpam-3397	221	18	of	of	ADP
ejpam-3397	221	19	r	r	NOUN
ejpam-3397	221	20	,	,	PUNCT
ejpam-3397	221	21	the	the	DET
ejpam-3397	221	22	result	result	NOUN
ejpam-3397	221	23	follows	follow	VERB
ejpam-3397	221	24	from	from	ADP
ejpam-3397	221	25	proposition3	proposition3	PROPN
ejpam-3397	221	26	.	.	PUNCT
ejpam-3397	222	1	example	example	NOUN
ejpam-3397	222	2	6	6	NUM
ejpam-3397	222	3	.	.	PUNCT
ejpam-3397	222	4	refer	refer	VERB
ejpam-3397	222	5	to	to	ADP
ejpam-3397	222	6	the	the	DET
ejpam-3397	222	7	γ	γ	X
ejpam-3397	222	8	-	-	PUNCT
ejpam-3397	222	9	seminearring	seminearre	VERB
ejpam-3397	222	10	s	s	PRON
ejpam-3397	222	11	defined	define	VERB
ejpam-3397	222	12	in	in	ADP
ejpam-3397	222	13	tables	table	NOUN
ejpam-3397	222	14	of	of	ADP
ejpam-3397	222	15	an	an	DET
ejpam-3397	222	16	example3	example3	NOUN
ejpam-3397	222	17	.	.	PUNCT
ejpam-3397	223	1	let	let	VERB
ejpam-3397	223	2	i	i	PRON
ejpam-3397	223	3	=	=	PUNCT
ejpam-3397	223	4	{	{	PUNCT
ejpam-3397	223	5	0	0	NUM
ejpam-3397	223	6	,	,	PUNCT
ejpam-3397	223	7	a	a	PRON
ejpam-3397	223	8	,	,	PUNCT
ejpam-3397	223	9	c	c	NOUN
ejpam-3397	223	10	}	}	PUNCT
ejpam-3397	223	11	and	and	CCONJ
ejpam-3397	223	12	j	j	PROPN
ejpam-3397	223	13	=	=	PUNCT
ejpam-3397	223	14	{	{	PUNCT
ejpam-3397	223	15	0	0	NUM
ejpam-3397	223	16	,	,	PUNCT
ejpam-3397	223	17	e	e	NOUN
ejpam-3397	223	18	,	,	PUNCT
ejpam-3397	223	19	c	c	AUX
ejpam-3397	223	20	}	}	PUNCT
ejpam-3397	223	21	be	be	AUX
ejpam-3397	223	22	the	the	DET
ejpam-3397	223	23	two	two	NUM
ejpam-3397	223	24	maximal	maximal	ADJ
ejpam-3397	223	25	ideals	ideal	NOUN
ejpam-3397	223	26	of	of	ADP
ejpam-3397	223	27	s.	s.	PROPN
ejpam-3397	223	28	clearly	clearly	ADV
ejpam-3397	223	29	iγj	iγj	VERB
ejpam-3397	223	30	=	=	PUNCT
ejpam-3397	223	31	0	0	NUM
ejpam-3397	223	32	and	and	CCONJ
ejpam-3397	223	33	jγi	jγi	NOUN
ejpam-3397	223	34	=	=	SYM
ejpam-3397	223	35	{	{	PUNCT
ejpam-3397	223	36	0	0	NUM
ejpam-3397	223	37	,	,	PUNCT
ejpam-3397	223	38	c	c	NOUN
ejpam-3397	223	39	}	}	PUNCT
ejpam-3397	223	40	=	=	PUNCT
ejpam-3397	223	41	i	i	PRON
ejpam-3397	223	42	∩	∩	NOUN
ejpam-3397	223	43	j	j	PROPN
ejpam-3397	223	44	=	=	PUNCT
ejpam-3397	223	45	{	{	PUNCT
ejpam-3397	223	46	0	0	NUM
ejpam-3397	223	47	,	,	PUNCT
ejpam-3397	223	48	c	c	NOUN
ejpam-3397	223	49	}	}	PUNCT
ejpam-3397	223	50	.	.	PUNCT
ejpam-3397	224	1	corollary	corollary	ADJ
ejpam-3397	224	2	2	2	NUM
ejpam-3397	224	3	.	.	PUNCT
ejpam-3397	225	1	let	let	VERB
ejpam-3397	225	2	every	every	DET
ejpam-3397	225	3	ideal	ideal	NOUN
ejpam-3397	225	4	of	of	ADP
ejpam-3397	225	5	a	a	DET
ejpam-3397	225	6	γ	γ	PROPN
ejpam-3397	225	7	-	-	PUNCT
ejpam-3397	225	8	seminear	seminear	NOUN
ejpam-3397	225	9	-	-	PUNCT
ejpam-3397	225	10	ring	ring	NOUN
ejpam-3397	225	11	r	r	NOUN
ejpam-3397	225	12	is	be	AUX
ejpam-3397	225	13	a	a	DET
ejpam-3397	225	14	weakly	weakly	ADJ
ejpam-3397	225	15	prime	prime	NOUN
ejpam-3397	225	16	.	.	PUNCT
ejpam-3397	226	1	then	then	ADV
ejpam-3397	226	2	,	,	PUNCT
ejpam-3397	226	3	every	every	DET
ejpam-3397	226	4	nonzero	nonzero	NOUN
ejpam-3397	226	5	ideal	ideal	NOUN
ejpam-3397	226	6	of	of	ADP
ejpam-3397	226	7	r	r	NOUN
ejpam-3397	226	8	/	/	SYM
ejpam-3397	226	9	n(r	n(r	NOUN
ejpam-3397	226	10	)	)	PUNCT
ejpam-3397	226	11	is	be	AUX
ejpam-3397	226	12	prime	prime	ADJ
ejpam-3397	226	13	.	.	PUNCT
ejpam-3397	227	1	corollary	corollary	ADJ
ejpam-3397	227	2	3	3	NUM
ejpam-3397	227	3	.	.	PUNCT
ejpam-3397	227	4	suppose	suppose	VERB
ejpam-3397	227	5	that	that	SCONJ
ejpam-3397	227	6	every	every	DET
ejpam-3397	227	7	ideal	ideal	NOUN
ejpam-3397	227	8	of	of	ADP
ejpam-3397	227	9	a	a	DET
ejpam-3397	227	10	γ	γ	PROPN
ejpam-3397	227	11	-	-	PUNCT
ejpam-3397	227	12	seminear	seminear	NOUN
ejpam-3397	227	13	-	-	PUNCT
ejpam-3397	227	14	ring	ring	NOUN
ejpam-3397	227	15	r	r	NOUN
ejpam-3397	227	16	is	be	AUX
ejpam-3397	227	17	a	a	DET
ejpam-3397	227	18	weakly	weakly	ADJ
ejpam-3397	227	19	prime	prime	NOUN
ejpam-3397	227	20	.	.	PUNCT
ejpam-3397	228	1	then	then	ADV
ejpam-3397	228	2	(	(	PUNCT
ejpam-3397	228	3	n(r))γ(n(r	n(r))γ(n(r	NOUN
ejpam-3397	228	4	)	)	PUNCT
ejpam-3397	228	5	)	)	PUNCT
ejpam-3397	229	1	=	=	SYM
ejpam-3397	229	2	0	0	NUM
ejpam-3397	230	1	and	and	CCONJ
ejpam-3397	230	2	every	every	DET
ejpam-3397	230	3	prime	prime	ADJ
ejpam-3397	230	4	ideal	ideal	NOUN
ejpam-3397	230	5	p	p	X
ejpam-3397	230	6	(	(	PUNCT
ejpam-3397	230	7	r	r	NOUN
ejpam-3397	230	8	)	)	PUNCT
ejpam-3397	230	9	contains	contain	VERB
ejpam-3397	230	10	n(r	n(r	NOUN
ejpam-3397	230	11	)	)	PUNCT
ejpam-3397	230	12	.	.	PUNCT
ejpam-3397	231	1	there	there	PRON
ejpam-3397	231	2	are	be	VERB
ejpam-3397	231	3	three	three	NUM
ejpam-3397	231	4	possibilities	possibility	NOUN
ejpam-3397	231	5	.	.	PUNCT
ejpam-3397	232	1	(	(	PUNCT
ejpam-3397	232	2	a	a	X
ejpam-3397	232	3	)	)	PUNCT
ejpam-3397	232	4	n(r	n(r	NOUN
ejpam-3397	232	5	)	)	PUNCT
ejpam-3397	233	1	=	=	SYM
ejpam-3397	233	2	r.	r.	X
ejpam-3397	233	3	(	(	PUNCT
ejpam-3397	233	4	b	b	NOUN
ejpam-3397	233	5	)	)	PUNCT
ejpam-3397	233	6	n(r	n(r	NOUN
ejpam-3397	233	7	)	)	PUNCT
ejpam-3397	234	1	=	=	SYM
ejpam-3397	234	2	p	p	X
ejpam-3397	234	3	(	(	PUNCT
ejpam-3397	234	4	r	r	NOUN
ejpam-3397	234	5	)	)	PUNCT
ejpam-3397	234	6	is	be	AUX
ejpam-3397	234	7	the	the	DET
ejpam-3397	234	8	smallest	small	ADJ
ejpam-3397	234	9	prime	prime	ADJ
ejpam-3397	234	10	ideal	ideal	NOUN
ejpam-3397	234	11	and	and	CCONJ
ejpam-3397	234	12	all	all	DET
ejpam-3397	234	13	other	other	ADJ
ejpam-3397	234	14	prime	prime	ADJ
ejpam-3397	234	15	ideals	ideal	NOUN
ejpam-3397	234	16	are	be	AUX
ejpam-3397	234	17	idempotent	idempotent	ADJ
ejpam-3397	234	18	and	and	CCONJ
ejpam-3397	234	19	are	be	AUX
ejpam-3397	234	20	linearly	linearly	ADV
ejpam-3397	234	21	ordered	order	VERB
ejpam-3397	234	22	.	.	PUNCT
ejpam-3397	235	1	if	if	SCONJ
ejpam-3397	235	2	n(r	n(r	NUM
ejpam-3397	235	3	)	)	PUNCT
ejpam-3397	235	4	6=	6=	ADP
ejpam-3397	235	5	0	0	NUM
ejpam-3397	235	6	,	,	PUNCT
ejpam-3397	235	7	then	then	ADV
ejpam-3397	235	8	it	it	PRON
ejpam-3397	235	9	is	be	AUX
ejpam-3397	235	10	the	the	DET
ejpam-3397	235	11	only	only	ADJ
ejpam-3397	235	12	non	non	ADJ
ejpam-3397	235	13	-	-	ADJ
ejpam-3397	235	14	idempotent	idempotent	ADJ
ejpam-3397	235	15	prime	prime	ADJ
ejpam-3397	235	16	ideal	ideal	NOUN
ejpam-3397	235	17	.	.	PUNCT
ejpam-3397	236	1	(	(	PUNCT
ejpam-3397	236	2	c	c	X
ejpam-3397	236	3	)	)	PUNCT
ejpam-3397	236	4	n(r	n(r	NOUN
ejpam-3397	236	5	)	)	PUNCT
ejpam-3397	237	1	=	=	SYM
ejpam-3397	237	2	p	p	X
ejpam-3397	237	3	(	(	PUNCT
ejpam-3397	237	4	r	r	NOUN
ejpam-3397	237	5	)	)	PUNCT
ejpam-3397	237	6	is	be	AUX
ejpam-3397	237	7	not	not	PART
ejpam-3397	237	8	a	a	DET
ejpam-3397	237	9	prime	prime	ADJ
ejpam-3397	237	10	ideal	ideal	NOUN
ejpam-3397	237	11	.	.	PUNCT
ejpam-3397	238	1	and	and	CCONJ
ejpam-3397	238	2	in	in	ADP
ejpam-3397	238	3	such	such	ADJ
ejpam-3397	238	4	case	case	NOUN
ejpam-3397	238	5	there	there	PRON
ejpam-3397	238	6	exist	exist	VERB
ejpam-3397	238	7	two	two	NUM
ejpam-3397	238	8	nonzero	nonzero	NOUN
ejpam-3397	238	9	minimal	minimal	ADJ
ejpam-3397	238	10	prime	prime	ADJ
ejpam-3397	238	11	ideals	ideal	NOUN
ejpam-3397	238	12	j1	j1	PROPN
ejpam-3397	238	13	and	and	CCONJ
ejpam-3397	238	14	j2	j2	PROPN
ejpam-3397	238	15	with	with	ADP
ejpam-3397	238	16	n(r	n(r	NOUN
ejpam-3397	238	17	)	)	PUNCT
ejpam-3397	239	1	=	=	SYM
ejpam-3397	239	2	j1	j1	PROPN
ejpam-3397	239	3	∩	∩	ADJ
ejpam-3397	239	4	j2	j2	NOUN
ejpam-3397	239	5	and	and	CCONJ
ejpam-3397	239	6	j1γj2	j1γj2	NOUN
ejpam-3397	239	7	=	=	PUNCT
ejpam-3397	239	8	{	{	PUNCT
ejpam-3397	239	9	0	0	NUM
ejpam-3397	239	10	}	}	PUNCT
ejpam-3397	239	11	or	or	CCONJ
ejpam-3397	239	12	(	(	PUNCT
ejpam-3397	239	13	d	d	NOUN
ejpam-3397	239	14	)	)	PUNCT
ejpam-3397	239	15	,	,	PUNCT
ejpam-3397	239	16	j2γj1	j2γj1	NOUN
ejpam-3397	239	17	=	=	PUNCT
ejpam-3397	239	18	{	{	PUNCT
ejpam-3397	239	19	0	0	NUM
ejpam-3397	239	20	}	}	PUNCT
ejpam-3397	239	21	or	or	CCONJ
ejpam-3397	239	22	(	(	PUNCT
ejpam-3397	239	23	c	c	NOUN
ejpam-3397	239	24	)	)	PUNCT
ejpam-3397	239	25	.	.	PUNCT
ejpam-3397	240	1	all	all	DET
ejpam-3397	240	2	other	other	ADJ
ejpam-3397	240	3	ideals	ideal	NOUN
ejpam-3397	240	4	containing	contain	VERB
ejpam-3397	240	5	n(r	n(r	NOUN
ejpam-3397	240	6	)	)	PUNCT
ejpam-3397	240	7	also	also	ADV
ejpam-3397	240	8	contain	contain	VERB
ejpam-3397	240	9	j1	j1	PROPN
ejpam-3397	240	10	+	+	CCONJ
ejpam-3397	240	11	j2	j2	PROPN
ejpam-3397	240	12	and	and	CCONJ
ejpam-3397	240	13	they	they	PRON
ejpam-3397	240	14	are	be	AUX
ejpam-3397	240	15	linearly	linearly	ADV
ejpam-3397	240	16	ordered	order	VERB
ejpam-3397	240	17	.	.	PUNCT
ejpam-3397	241	1	we	we	PRON
ejpam-3397	241	2	elaborate	elaborate	VERB
ejpam-3397	241	3	the	the	DET
ejpam-3397	241	4	above	above	ADJ
ejpam-3397	241	5	proposition	proposition	NOUN
ejpam-3397	241	6	in	in	ADP
ejpam-3397	241	7	the	the	DET
ejpam-3397	241	8	below	below	ADJ
ejpam-3397	241	9	example	example	NOUN
ejpam-3397	241	10	.	.	PUNCT
ejpam-3397	242	1	example	example	NOUN
ejpam-3397	243	1	7	7	NUM
ejpam-3397	243	2	.	.	X
ejpam-3397	243	3	refer	refer	VERB
ejpam-3397	243	4	to	to	ADP
ejpam-3397	243	5	the	the	DET
ejpam-3397	243	6	γ	γ	X
ejpam-3397	243	7	-	-	PUNCT
ejpam-3397	243	8	seminearring	seminearre	VERB
ejpam-3397	243	9	s	s	PRON
ejpam-3397	243	10	defined	define	VERB
ejpam-3397	243	11	in	in	ADP
ejpam-3397	243	12	tables	table	NOUN
ejpam-3397	243	13	of	of	ADP
ejpam-3397	243	14	an	an	DET
ejpam-3397	243	15	example3	example3	NOUN
ejpam-3397	243	16	.	.	PUNCT
ejpam-3397	244	1	in	in	ADP
ejpam-3397	244	2	s	s	PROPN
ejpam-3397	244	3	,	,	PUNCT
ejpam-3397	244	4	n(s	n(s	PROPN
ejpam-3397	244	5	)	)	PUNCT
ejpam-3397	244	6	=	=	PRON
ejpam-3397	244	7	{	{	PUNCT
ejpam-3397	244	8	0	0	NUM
ejpam-3397	244	9	,	,	PUNCT
ejpam-3397	244	10	c	c	NOUN
ejpam-3397	244	11	}	}	PUNCT
ejpam-3397	244	12	and	and	CCONJ
ejpam-3397	244	13	we	we	PRON
ejpam-3397	244	14	have	have	VERB
ejpam-3397	244	15	(	(	PUNCT
ejpam-3397	244	16	n(s))2	n(s))2	NOUN
ejpam-3397	244	17	=	=	SYM
ejpam-3397	244	18	(	(	PUNCT
ejpam-3397	244	19	n(r	n(r	NOUN
ejpam-3397	244	20	)	)	PUNCT
ejpam-3397	244	21	)	)	PUNCT
ejpam-3397	245	1	γ(n(r	γ(n(r	NOUN
ejpam-3397	245	2	)	)	PUNCT
ejpam-3397	245	3	)	)	PUNCT
ejpam-3397	246	1	=	=	PUNCT
ejpam-3397	246	2	{	{	PUNCT
ejpam-3397	246	3	0	0	NUM
ejpam-3397	246	4	,	,	PUNCT
ejpam-3397	246	5	c}2	c}2	VERB
ejpam-3397	246	6	=	=	SYM
ejpam-3397	246	7	{	{	PUNCT
ejpam-3397	246	8	0	0	NUM
ejpam-3397	246	9	}	}	PUNCT
ejpam-3397	246	10	.	.	PUNCT
ejpam-3397	247	1	as	as	ADP
ejpam-3397	247	2	,	,	PUNCT
ejpam-3397	247	3	n(s	n(s	PROPN
ejpam-3397	247	4	)	)	PUNCT
ejpam-3397	247	5	=	=	SYM
ejpam-3397	247	6	p	p	X
ejpam-3397	247	7	(	(	PUNCT
ejpam-3397	247	8	s	s	NOUN
ejpam-3397	247	9	)	)	PUNCT
ejpam-3397	247	10	is	be	AUX
ejpam-3397	247	11	not	not	PART
ejpam-3397	247	12	a	a	DET
ejpam-3397	247	13	prime	prime	ADJ
ejpam-3397	247	14	ideal	ideal	NOUN
ejpam-3397	247	15	and	and	CCONJ
ejpam-3397	247	16	possibility	possibility	NOUN
ejpam-3397	247	17	(	(	PUNCT
ejpam-3397	247	18	c	c	NOUN
ejpam-3397	247	19	)	)	PUNCT
ejpam-3397	247	20	of	of	ADP
ejpam-3397	247	21	the	the	DET
ejpam-3397	247	22	above	above	ADJ
ejpam-3397	247	23	corollary3	corollary3	NOUN
ejpam-3397	247	24	is	be	AUX
ejpam-3397	247	25	valid	valid	ADJ
ejpam-3397	247	26	for	for	ADP
ejpam-3397	247	27	this	this	PRON
ejpam-3397	247	28	i.e.	i.e.	X
ejpam-3397	247	29	,	,	PUNCT
ejpam-3397	247	30	there	there	PRON
ejpam-3397	247	31	exist	exist	VERB
ejpam-3397	247	32	two	two	NUM
ejpam-3397	247	33	nonzero	nonzero	NOUN
ejpam-3397	247	34	minimal	minimal	ADJ
ejpam-3397	247	35	prime	prime	ADJ
ejpam-3397	247	36	ideals	ideal	NOUN
ejpam-3397	247	37	j1	j1	PROPN
ejpam-3397	247	38	and	and	CCONJ
ejpam-3397	247	39	j2	j2	PROPN
ejpam-3397	247	40	with	with	ADP
ejpam-3397	247	41	n(r	n(r	NOUN
ejpam-3397	247	42	)	)	PUNCT
ejpam-3397	247	43	=	=	SYM
ejpam-3397	247	44	j1	j1	PROPN
ejpam-3397	247	45	∩	∩	ADJ
ejpam-3397	247	46	j2	j2	NOUN
ejpam-3397	247	47	and	and	CCONJ
ejpam-3397	247	48	j1γj2	j1γj2	NOUN
ejpam-3397	247	49	=	=	PUNCT
ejpam-3397	247	50	{	{	PUNCT
ejpam-3397	247	51	0	0	NUM
ejpam-3397	247	52	}	}	PUNCT
ejpam-3397	247	53	and	and	CCONJ
ejpam-3397	247	54	j2γj1	j2γj1	NOUN
ejpam-3397	247	55	=	=	PUNCT
ejpam-3397	247	56	(	(	PUNCT
ejpam-3397	247	57	c	c	X
ejpam-3397	247	58	)	)	PUNCT
ejpam-3397	247	59	=	=	SYM
ejpam-3397	247	60	{	{	PUNCT
ejpam-3397	247	61	0	0	NUM
ejpam-3397	247	62	,	,	PUNCT
ejpam-3397	247	63	c	c	NOUN
ejpam-3397	247	64	}	}	PUNCT
ejpam-3397	247	65	.	.	PUNCT
ejpam-3397	248	1	all	all	DET
ejpam-3397	248	2	other	other	ADJ
ejpam-3397	248	3	ideals	ideal	NOUN
ejpam-3397	248	4	containing	contain	VERB
ejpam-3397	248	5	n(r	n(r	NOUN
ejpam-3397	248	6	)	)	PUNCT
ejpam-3397	248	7	also	also	ADV
ejpam-3397	248	8	contain	contain	VERB
ejpam-3397	248	9	j1	j1	PROPN
ejpam-3397	248	10	+	+	CCONJ
ejpam-3397	248	11	j2	j2	PROPN
ejpam-3397	248	12	and	and	CCONJ
ejpam-3397	248	13	are	be	AUX
ejpam-3397	248	14	linearly	linearly	ADV
ejpam-3397	248	15	ordered	order	VERB
ejpam-3397	248	16	.	.	PUNCT
ejpam-3397	249	1	let	let	VERB
ejpam-3397	249	2	j1	j1	PROPN
ejpam-3397	249	3	=	=	PUNCT
ejpam-3397	249	4	{	{	PUNCT
ejpam-3397	249	5	0	0	NUM
ejpam-3397	249	6	,	,	PUNCT
ejpam-3397	249	7	a	a	PRON
ejpam-3397	249	8	,	,	PUNCT
ejpam-3397	249	9	c	c	NOUN
ejpam-3397	249	10	}	}	PUNCT
ejpam-3397	249	11	and	and	CCONJ
ejpam-3397	249	12	j2	j2	PROPN
ejpam-3397	249	13	=	=	SYM
ejpam-3397	249	14	{	{	PUNCT
ejpam-3397	249	15	0	0	NUM
ejpam-3397	249	16	,	,	PUNCT
ejpam-3397	249	17	e	e	NOUN
ejpam-3397	249	18	,	,	PUNCT
ejpam-3397	249	19	c	c	AUX
ejpam-3397	249	20	}	}	PUNCT
ejpam-3397	249	21	be	be	AUX
ejpam-3397	249	22	the	the	DET
ejpam-3397	249	23	minimal	minimal	ADJ
ejpam-3397	249	24	prime	prime	ADJ
ejpam-3397	249	25	ideals	ideal	NOUN
ejpam-3397	249	26	of	of	ADP
ejpam-3397	249	27	s.	s.	PROPN
ejpam-3397	249	28	we	we	PRON
ejpam-3397	249	29	have	have	VERB
ejpam-3397	249	30	n(s	n(s	NUM
ejpam-3397	249	31	)	)	PUNCT
ejpam-3397	249	32	=	=	PRON
ejpam-3397	249	33	{	{	PUNCT
ejpam-3397	249	34	0	0	NUM
ejpam-3397	249	35	,	,	PUNCT
ejpam-3397	249	36	c	c	NOUN
ejpam-3397	249	37	}	}	PUNCT
ejpam-3397	249	38	=	=	SYM
ejpam-3397	249	39	j1	j1	PROPN
ejpam-3397	249	40	∩	∩	ADJ
ejpam-3397	249	41	j2	j2	PROPN
ejpam-3397	249	42	and	and	CCONJ
ejpam-3397	249	43	j1j2	j1j2	PROPN
ejpam-3397	249	44	=	=	PUNCT
ejpam-3397	249	45	j2j1	j2j1	PROPN
ejpam-3397	249	46	=	=	SYM
ejpam-3397	249	47	0	0	PROPN
ejpam-3397	249	48	.	.	PUNCT
ejpam-3397	250	1	beside	beside	ADP
ejpam-3397	250	2	these	these	DET
ejpam-3397	250	3	two	two	NUM
ejpam-3397	250	4	ideals	ideal	NOUN
ejpam-3397	250	5	another	another	DET
ejpam-3397	250	6	ideal	ideal	NOUN
ejpam-3397	250	7	of	of	ADP
ejpam-3397	250	8	s	s	PRON
ejpam-3397	250	9	is	be	AUX
ejpam-3397	250	10	s	s	PRON
ejpam-3397	250	11	itself	itself	PRON
ejpam-3397	250	12	and	and	CCONJ
ejpam-3397	250	13	clearly	clearly	ADV
ejpam-3397	250	14	it	it	PRON
ejpam-3397	250	15	contains	contain	VERB
ejpam-3397	250	16	n(s	n(s	PROPN
ejpam-3397	250	17	)	)	PUNCT
ejpam-3397	250	18	and	and	CCONJ
ejpam-3397	250	19	also	also	ADV
ejpam-3397	250	20	j1	j1	PROPN
ejpam-3397	250	21	+	+	CCONJ
ejpam-3397	250	22	j2	j2	PROPN
ejpam-3397	250	23	where	where	SCONJ
ejpam-3397	250	24	j1	j1	PROPN
ejpam-3397	250	25	+	+	CCONJ
ejpam-3397	250	26	j2	j2	PROPN
ejpam-3397	250	27	=	=	PROPN
ejpam-3397	250	28	s.	s.	PROPN
ejpam-3397	250	29	example	example	NOUN
ejpam-3397	250	30	8	8	NUM
ejpam-3397	250	31	.	.	PUNCT
ejpam-3397	251	1	let	let	VERB
ejpam-3397	251	2	t	t	NOUN
ejpam-3397	251	3	=	=	SYM
ejpam-3397	251	4	{	{	PUNCT
ejpam-3397	251	5	0	0	NUM
ejpam-3397	251	6	,	,	PUNCT
ejpam-3397	251	7	a	a	PRON
ejpam-3397	251	8	,	,	PUNCT
ejpam-3397	251	9	b	b	X
ejpam-3397	251	10	}	}	PUNCT
ejpam-3397	251	11	be	be	AUX
ejpam-3397	251	12	a	a	DET
ejpam-3397	251	13	right	right	ADJ
ejpam-3397	251	14	seminearring	seminearring	NOUN
ejpam-3397	251	15	under	under	ADP
ejpam-3397	251	16	the	the	DET
ejpam-3397	251	17	operations	operation	NOUN
ejpam-3397	251	18	defined	define	VERB
ejpam-3397	251	19	in	in	ADP
ejpam-3397	251	20	given	give	VERB
ejpam-3397	251	21	below	below	ADP
ejpam-3397	251	22	tables	table	NOUN
ejpam-3397	251	23	.	.	PUNCT
ejpam-3397	252	1	+	+	CCONJ
ejpam-3397	252	2	0	0	NUM
ejpam-3397	252	3	a	a	DET
ejpam-3397	252	4	b	b	NOUN
ejpam-3397	252	5	0	0	NUM
ejpam-3397	252	6	0	0	NUM
ejpam-3397	253	1	a	a	DET
ejpam-3397	253	2	b	b	NOUN
ejpam-3397	253	3	a	a	PRON
ejpam-3397	253	4	a	a	PRON
ejpam-3397	253	5	a	a	DET
ejpam-3397	253	6	a	a	DET
ejpam-3397	253	7	b	b	PROPN
ejpam-3397	253	8	b	b	PROPN
ejpam-3397	253	9	b	b	PROPN
ejpam-3397	253	10	b	b	PROPN
ejpam-3397	253	11	.	.	PROPN
ejpam-3397	253	12	0	0	PUNCT
ejpam-3397	254	1	a	a	DET
ejpam-3397	254	2	b	b	NOUN
ejpam-3397	254	3	0	0	NUM
ejpam-3397	254	4	0	0	NUM
ejpam-3397	254	5	0	0	NUM
ejpam-3397	254	6	0	0	NUM
ejpam-3397	254	7	a	a	DET
ejpam-3397	254	8	0	0	NUM
ejpam-3397	254	9	a	a	DET
ejpam-3397	254	10	a	a	DET
ejpam-3397	254	11	b	b	NOUN
ejpam-3397	254	12	0	0	NUM
ejpam-3397	254	13	a	a	DET
ejpam-3397	254	14	b	b	NOUN
ejpam-3397	254	15	here	here	ADV
ejpam-3397	254	16	n(t	n(t	PROPN
ejpam-3397	254	17	)	)	PUNCT
ejpam-3397	255	1	=	=	PUNCT
ejpam-3397	256	1	p	p	X
ejpam-3397	256	2	(	(	PUNCT
ejpam-3397	256	3	t	t	PROPN
ejpam-3397	256	4	)	)	PUNCT
ejpam-3397	256	5	=	=	PUNCT
ejpam-3397	256	6	{	{	PUNCT
ejpam-3397	256	7	0	0	NUM
ejpam-3397	256	8	}	}	PUNCT
ejpam-3397	256	9	and	and	CCONJ
ejpam-3397	256	10	it	it	PRON
ejpam-3397	256	11	is	be	AUX
ejpam-3397	256	12	the	the	DET
ejpam-3397	256	13	smallest	small	ADJ
ejpam-3397	256	14	prime	prime	ADJ
ejpam-3397	256	15	ideal	ideal	NOUN
ejpam-3397	256	16	.	.	PUNCT
ejpam-3397	257	1	possibility	possibility	NOUN
ejpam-3397	257	2	(	(	PUNCT
ejpam-3397	257	3	b	b	NOUN
ejpam-3397	257	4	)	)	PUNCT
ejpam-3397	257	5	of	of	ADP
ejpam-3397	257	6	above	above	ADP
ejpam-3397	257	7	corollary3	corollary3	NOUN
ejpam-3397	257	8	is	be	AUX
ejpam-3397	257	9	valid	valid	ADJ
ejpam-3397	257	10	for	for	ADP
ejpam-3397	257	11	this	this	DET
ejpam-3397	257	12	seminearring	seminearring	NOUN
ejpam-3397	257	13	.	.	PUNCT
ejpam-3397	258	1	definition	definition	NOUN
ejpam-3397	258	2	4	4	NUM
ejpam-3397	258	3	.	.	PUNCT
ejpam-3397	259	1	let	let	VERB
ejpam-3397	259	2	r	r	PRON
ejpam-3397	259	3	be	be	AUX
ejpam-3397	259	4	a	a	DET
ejpam-3397	259	5	γ	γ	NOUN
ejpam-3397	259	6	-	-	PUNCT
ejpam-3397	259	7	seminearring	seminearring	NOUN
ejpam-3397	259	8	under	under	ADP
ejpam-3397	259	9	the	the	DET
ejpam-3397	259	10	mapping	mapping	NOUN
ejpam-3397	259	11	from	from	ADP
ejpam-3397	259	12	r	r	PROPN
ejpam-3397	259	13	×	×	PROPN
ejpam-3397	259	14	γ	γ	X
ejpam-3397	259	15	×	×	NOUN
ejpam-3397	259	16	r	r	NOUN
ejpam-3397	259	17	into	into	ADP
ejpam-3397	259	18	r	r	NOUN
ejpam-3397	259	19	,	,	PUNCT
ejpam-3397	259	20	say	say	VERB
ejpam-3397	259	21	f	f	NOUN
ejpam-3397	259	22	,	,	PUNCT
ejpam-3397	259	23	and	and	CCONJ
ejpam-3397	259	24	d	d	PRON
ejpam-3397	259	25	be	be	VERB
ejpam-3397	259	26	the	the	DET
ejpam-3397	259	27	set	set	NOUN
ejpam-3397	259	28	of	of	ADP
ejpam-3397	259	29	all	all	DET
ejpam-3397	259	30	destributive	destributive	ADJ
ejpam-3397	259	31	elements	element	NOUN
ejpam-3397	259	32	of	of	ADP
ejpam-3397	259	33	r	r	NOUN
ejpam-3397	259	34	,	,	PUNCT
ejpam-3397	259	35	i.e.	i.e.	X
ejpam-3397	259	36	,	,	PUNCT
ejpam-3397	259	37	d	d	X
ejpam-3397	259	38	=	=	SYM
ejpam-3397	259	39	{	{	PUNCT
ejpam-3397	259	40	d	d	X
ejpam-3397	259	41	∈	∈	PROPN
ejpam-3397	259	42	r	r	NOUN
ejpam-3397	259	43	|	|	NOUN
ejpam-3397	259	44	dα(a	dα(a	NOUN
ejpam-3397	260	1	+	+	CCONJ
ejpam-3397	260	2	b	b	X
ejpam-3397	260	3	)	)	PUNCT
ejpam-3397	260	4	=	=	SYM
ejpam-3397	260	5	abdelghani	abdelghani	PROPN
ejpam-3397	260	6	taouti	taouti	NOUN
ejpam-3397	260	7	et	et	PROPN
ejpam-3397	260	8	al	al	PROPN
ejpam-3397	260	9	.	.	PUNCT
ejpam-3397	260	10	/	/	SYM
ejpam-3397	260	11	eur	eur	PROPN
ejpam-3397	260	12	.	.	PUNCT
ejpam-3397	261	1	j.	j.	PROPN
ejpam-3397	261	2	pure	pure	PROPN
ejpam-3397	261	3	appl	appl	PROPN
ejpam-3397	261	4	.	.	PROPN
ejpam-3397	261	5	math	math	PROPN
ejpam-3397	261	6	,	,	PUNCT
ejpam-3397	261	7	12	12	NUM
ejpam-3397	261	8	(	(	PUNCT
ejpam-3397	261	9	2	2	NUM
ejpam-3397	261	10	)	)	PUNCT
ejpam-3397	261	11	(	(	PUNCT
ejpam-3397	261	12	2019	2019	NUM
ejpam-3397	261	13	)	)	PUNCT
ejpam-3397	261	14	,	,	PUNCT
ejpam-3397	261	15	544	544	NUM
ejpam-3397	261	16	-	-	SYM
ejpam-3397	261	17	552	552	NUM
ejpam-3397	261	18	550	550	NUM
ejpam-3397	261	19	dαa	dαa	NOUN
ejpam-3397	261	20	+	+	X
ejpam-3397	261	21	dαb	dαb	ADJ
ejpam-3397	261	22	for	for	ADP
ejpam-3397	261	23	all	all	DET
ejpam-3397	261	24	a	a	DET
ejpam-3397	261	25	,	,	PUNCT
ejpam-3397	261	26	b	b	X
ejpam-3397	261	27	∈	∈	PROPN
ejpam-3397	261	28	r	r	NOUN
ejpam-3397	261	29	and	and	CCONJ
ejpam-3397	261	30	α	α	NOUN
ejpam-3397	261	31	∈	∈	PROPN
ejpam-3397	261	32	γ	γ	X
ejpam-3397	261	33	}	}	PUNCT
ejpam-3397	261	34	.	.	PUNCT
ejpam-3397	262	1	then	then	ADV
ejpam-3397	262	2	r	r	NOUN
ejpam-3397	262	3	is	be	AUX
ejpam-3397	262	4	called	call	VERB
ejpam-3397	262	5	distributively	distributively	ADV
ejpam-3397	262	6	generated	generate	VERB
ejpam-3397	262	7	(	(	PUNCT
ejpam-3397	262	8	in	in	ADP
ejpam-3397	262	9	short	short	ADJ
ejpam-3397	262	10	,	,	PUNCT
ejpam-3397	262	11	d.g	d.g	PROPN
ejpam-3397	262	12	.	.	PUNCT
ejpam-3397	262	13	)	)	PUNCT
ejpam-3397	263	1	if	if	SCONJ
ejpam-3397	263	2	the	the	DET
ejpam-3397	263	3	set	set	NOUN
ejpam-3397	263	4	d	d	NOUN
ejpam-3397	263	5	is	be	AUX
ejpam-3397	263	6	non	non	X
ejpam-3397	263	7	empty	empty	ADJ
ejpam-3397	263	8	subset	subset	NOUN
ejpam-3397	263	9	of	of	ADP
ejpam-3397	263	10	r	r	NOUN
ejpam-3397	263	11	which	which	PRON
ejpam-3397	263	12	fd×γ×d	fd×γ×d	ADV
ejpam-3397	263	13	:	:	PUNCT
ejpam-3397	263	14	d	d	X
ejpam-3397	263	15	×	×	NOUN
ejpam-3397	263	16	γ×d	γ×d	PUNCT
ejpam-3397	263	17	→	→	SYM
ejpam-3397	263	18	d	d	PROPN
ejpam-3397	263	19	and	and	CCONJ
ejpam-3397	263	20	(	(	PUNCT
ejpam-3397	263	21	<	<	X
ejpam-3397	263	22	d,+	d,+	X
ejpam-3397	263	23	>	>	PUNCT
ejpam-3397	263	24	)	)	PUNCT
ejpam-3397	263	25	=	=	SYM
ejpam-3397	263	26	(	(	PUNCT
ejpam-3397	263	27	r,+	r,+	NUM
ejpam-3397	263	28	)	)	PUNCT
ejpam-3397	263	29	where	where	SCONJ
ejpam-3397	263	30	<	<	X
ejpam-3397	263	31	d	d	X
ejpam-3397	263	32	>	>	X
ejpam-3397	263	33	=	=	SYM
ejpam-3397	263	34	{	{	PUNCT
ejpam-3397	263	35	m∑	m∑	INTJ
ejpam-3397	263	36	i=1	i=1	PROPN
ejpam-3397	263	37	αidi	αidi	NUM
ejpam-3397	263	38	|	|	ADV
ejpam-3397	263	39	m	m	PROPN
ejpam-3397	263	40	,	,	PUNCT
ejpam-3397	263	41	αi	αi	PROPN
ejpam-3397	263	42	∈	∈	PROPN
ejpam-3397	263	43	n	n	NOUN
ejpam-3397	263	44	and	and	CCONJ
ejpam-3397	263	45	di	di	NOUN
ejpam-3397	263	46	∈	∈	PROPN
ejpam-3397	263	47	d	d	PROPN
ejpam-3397	263	48	for	for	ADP
ejpam-3397	263	49	all	all	DET
ejpam-3397	263	50	i	i	PRON
ejpam-3397	263	51	}	}	PUNCT
ejpam-3397	263	52	.	.	PUNCT
ejpam-3397	264	1	in	in	ADP
ejpam-3397	264	2	fact	fact	NOUN
ejpam-3397	264	3	,	,	PUNCT
ejpam-3397	264	4	<	<	X
ejpam-3397	264	5	d	d	X
ejpam-3397	264	6	>	>	X
ejpam-3397	264	7	=	=	X
ejpam-3397	264	8	{	{	PUNCT
ejpam-3397	264	9	∑n	∑n	PROPN
ejpam-3397	264	10	i=1	i=1	PROPN
ejpam-3397	264	11	di	di	NOUN
ejpam-3397	264	12	|	|	ADV
ejpam-3397	264	13	n	n	CCONJ
ejpam-3397	264	14	∈	∈	PROPN
ejpam-3397	264	15	n	n	NOUN
ejpam-3397	264	16	and	and	CCONJ
ejpam-3397	264	17	di	di	NOUN
ejpam-3397	264	18	∈	∈	PROPN
ejpam-3397	264	19	d	d	NOUN
ejpam-3397	264	20	}	}	PUNCT
ejpam-3397	264	21	where	where	SCONJ
ejpam-3397	264	22	all	all	DET
ejpam-3397	264	23	d’is	d’is	PROPN
ejpam-3397	264	24	in	in	ADP
ejpam-3397	264	25	∑	∑	PROPN
ejpam-3397	264	26	di	di	NOUN
ejpam-3397	264	27	may	may	AUX
ejpam-3397	264	28	not	not	PART
ejpam-3397	264	29	be	be	AUX
ejpam-3397	264	30	distinct	distinct	ADJ
ejpam-3397	264	31	.	.	PUNCT
ejpam-3397	265	1	in	in	ADP
ejpam-3397	265	2	addition	addition	NOUN
ejpam-3397	265	3	,	,	PUNCT
ejpam-3397	265	4	(	(	PUNCT
ejpam-3397	265	5	<	<	X
ejpam-3397	265	6	d,+	d,+	X
ejpam-3397	265	7	>	>	PUNCT
ejpam-3397	265	8	)	)	PUNCT
ejpam-3397	266	1	=	=	SYM
ejpam-3397	266	2	(	(	PUNCT
ejpam-3397	266	3	r,+	r,+	PRON
ejpam-3397	266	4	)	)	PUNCT
ejpam-3397	266	5	means	mean	VERB
ejpam-3397	266	6	that	that	SCONJ
ejpam-3397	266	7	every	every	DET
ejpam-3397	266	8	element	element	NOUN
ejpam-3397	266	9	in	in	ADP
ejpam-3397	266	10	r	r	NOUN
ejpam-3397	266	11	can	can	AUX
ejpam-3397	266	12	be	be	AUX
ejpam-3397	266	13	written	write	VERB
ejpam-3397	266	14	as	as	ADP
ejpam-3397	266	15	a	a	DET
ejpam-3397	266	16	finite	finite	ADJ
ejpam-3397	266	17	sum	sum	NOUN
ejpam-3397	266	18	of	of	ADP
ejpam-3397	266	19	desrtributive	desrtributive	ADJ
ejpam-3397	266	20	elements	element	NOUN
ejpam-3397	266	21	.	.	PUNCT
ejpam-3397	267	1	example	example	NOUN
ejpam-3397	267	2	9	9	NUM
ejpam-3397	267	3	.	.	X
ejpam-3397	267	4	refer	refer	VERB
ejpam-3397	267	5	to	to	ADP
ejpam-3397	267	6	the	the	DET
ejpam-3397	267	7	γ	γ	X
ejpam-3397	267	8	-	-	PUNCT
ejpam-3397	267	9	seminearring	seminearre	VERB
ejpam-3397	267	10	s	s	AUX
ejpam-3397	267	11	defined	define	VERB
ejpam-3397	267	12	in	in	ADP
ejpam-3397	267	13	tables	table	NOUN
ejpam-3397	267	14	of	of	ADP
ejpam-3397	267	15	an	an	DET
ejpam-3397	267	16	example3	example3	NOUN
ejpam-3397	267	17	.	.	PUNCT
ejpam-3397	268	1	let	let	VERB
ejpam-3397	268	2	d	d	NOUN
ejpam-3397	268	3	=	=	PUNCT
ejpam-3397	268	4	{	{	PUNCT
ejpam-3397	268	5	0	0	NUM
ejpam-3397	268	6	,	,	PUNCT
ejpam-3397	268	7	1	1	NUM
ejpam-3397	268	8	}	}	PUNCT
ejpam-3397	268	9	,	,	PUNCT
ejpam-3397	268	10	where	where	SCONJ
ejpam-3397	268	11	all	all	DET
ejpam-3397	268	12	elements	element	NOUN
ejpam-3397	268	13	of	of	ADP
ejpam-3397	268	14	d	d	NOUN
ejpam-3397	268	15	are	be	AUX
ejpam-3397	268	16	distributive	distributive	ADJ
ejpam-3397	268	17	elements	element	NOUN
ejpam-3397	268	18	of	of	ADP
ejpam-3397	268	19	r	r	NOUN
ejpam-3397	268	20	i.e.	i.e.	X
ejpam-3397	268	21	,	,	PUNCT
ejpam-3397	268	22	d	d	NOUN
ejpam-3397	268	23	=	=	SYM
ejpam-3397	268	24	{	{	PUNCT
ejpam-3397	268	25	d	d	X
ejpam-3397	268	26	∈	∈	PROPN
ejpam-3397	268	27	r	r	NOUN
ejpam-3397	268	28	|	|	ADV
ejpam-3397	268	29	dα(a+	dα(a+	PROPN
ejpam-3397	268	30	b	b	NOUN
ejpam-3397	268	31	)	)	PUNCT
ejpam-3397	268	32	=	=	VERB
ejpam-3397	268	33	dαa	dαa	NOUN
ejpam-3397	268	34	+	+	X
ejpam-3397	268	35	dαb	dαb	ADJ
ejpam-3397	268	36	for	for	ADP
ejpam-3397	268	37	all	all	DET
ejpam-3397	268	38	a	a	DET
ejpam-3397	268	39	,	,	PUNCT
ejpam-3397	268	40	b	b	X
ejpam-3397	268	41	∈	∈	PROPN
ejpam-3397	268	42	r	r	NOUN
ejpam-3397	268	43	and	and	CCONJ
ejpam-3397	268	44	α	α	NOUN
ejpam-3397	268	45	∈	∈	PROPN
ejpam-3397	268	46	γ	γ	X
ejpam-3397	268	47	}	}	PUNCT
ejpam-3397	268	48	.	.	PUNCT
ejpam-3397	269	1	s	s	PART
ejpam-3397	269	2	is	be	AUX
ejpam-3397	269	3	called	call	VERB
ejpam-3397	269	4	distributively	distributively	ADV
ejpam-3397	269	5	generated	generate	VERB
ejpam-3397	269	6	because	because	SCONJ
ejpam-3397	269	7	the	the	DET
ejpam-3397	269	8	set	set	NOUN
ejpam-3397	269	9	d	d	X
ejpam-3397	269	10	=	=	SYM
ejpam-3397	269	11	{	{	PUNCT
ejpam-3397	269	12	0	0	NUM
ejpam-3397	269	13	,	,	PUNCT
ejpam-3397	269	14	1	1	NUM
ejpam-3397	269	15	}	}	PUNCT
ejpam-3397	269	16	is	be	AUX
ejpam-3397	269	17	a	a	DET
ejpam-3397	269	18	nonempty	nonempty	ADJ
ejpam-3397	269	19	subset	subset	NOUN
ejpam-3397	269	20	of	of	ADP
ejpam-3397	269	21	r	r	NOUN
ejpam-3397	269	22	which	which	PRON
ejpam-3397	269	23	satisfies	satisfy	VERB
ejpam-3397	269	24	fd×γ×d	fd×γ×d	ADV
ejpam-3397	270	1	:	:	PUNCT
ejpam-3397	270	2	d	d	X
ejpam-3397	270	3	×	×	NOUN
ejpam-3397	270	4	γ×d	γ×d	PUNCT
ejpam-3397	270	5	→	→	SYM
ejpam-3397	270	6	d	d	PROPN
ejpam-3397	270	7	and	and	CCONJ
ejpam-3397	270	8	(	(	PUNCT
ejpam-3397	270	9	<	<	X
ejpam-3397	270	10	d	d	X
ejpam-3397	270	11	>	>	X
ejpam-3397	270	12	,	,	PUNCT
ejpam-3397	270	13	+	+	NOUN
ejpam-3397	270	14	)	)	PUNCT
ejpam-3397	270	15	=	=	SYM
ejpam-3397	270	16	(	(	PUNCT
ejpam-3397	270	17	r	r	NOUN
ejpam-3397	270	18	,	,	PUNCT
ejpam-3397	270	19	+	+	NOUN
ejpam-3397	270	20	)	)	PUNCT
ejpam-3397	270	21	.	.	PUNCT
ejpam-3397	271	1	theorem	theorem	NOUN
ejpam-3397	271	2	2	2	NUM
ejpam-3397	271	3	.	.	PUNCT
ejpam-3397	272	1	let	let	VERB
ejpam-3397	272	2	r	r	PRON
ejpam-3397	272	3	be	be	AUX
ejpam-3397	272	4	a	a	DET
ejpam-3397	272	5	distributively	distributively	ADV
ejpam-3397	272	6	generated	generate	VERB
ejpam-3397	272	7	γ	γ	X
ejpam-3397	272	8	-	-	PUNCT
ejpam-3397	272	9	seminearring	seminearring	NOUN
ejpam-3397	272	10	.	.	PUNCT
ejpam-3397	273	1	(	(	PUNCT
ejpam-3397	273	2	1	1	X
ejpam-3397	273	3	)	)	PUNCT
ejpam-3397	273	4	if	if	SCONJ
ejpam-3397	273	5	a	a	PRON
ejpam-3397	273	6	is	be	AUX
ejpam-3397	273	7	weakly	weakly	ADJ
ejpam-3397	273	8	prime	prime	ADJ
ejpam-3397	273	9	ideal	ideal	NOUN
ejpam-3397	273	10	of	of	ADP
ejpam-3397	273	11	r	r	NOUN
ejpam-3397	273	12	and	and	CCONJ
ejpam-3397	273	13	b	b	NOUN
ejpam-3397	273	14	is	be	AUX
ejpam-3397	273	15	a	a	DET
ejpam-3397	273	16	nonempty	nonempty	ADJ
ejpam-3397	273	17	subset	subset	NOUN
ejpam-3397	273	18	of	of	ADP
ejpam-3397	273	19	r.	r.	PROPN
ejpam-3397	273	20	then	then	ADV
ejpam-3397	273	21	,	,	PUNCT
ejpam-3397	273	22	aγb	aγb	NOUN
ejpam-3397	273	23	is	be	VERB
ejpam-3397	273	24	a	a	DET
ejpam-3397	273	25	weakly	weakly	ADJ
ejpam-3397	273	26	prime	prime	ADJ
ejpam-3397	273	27	ideal	ideal	NOUN
ejpam-3397	273	28	of	of	ADP
ejpam-3397	273	29	r.	r.	PROPN
ejpam-3397	273	30	(	(	PUNCT
ejpam-3397	273	31	2	2	NUM
ejpam-3397	273	32	)	)	PUNCT
ejpam-3397	273	33	if	if	SCONJ
ejpam-3397	273	34	a	a	PRON
ejpam-3397	273	35	and	and	CCONJ
ejpam-3397	273	36	b	b	NOUN
ejpam-3397	273	37	are	be	AUX
ejpam-3397	273	38	weakly	weakly	ADJ
ejpam-3397	273	39	prime	prime	ADJ
ejpam-3397	273	40	ideals	ideal	NOUN
ejpam-3397	273	41	of	of	ADP
ejpam-3397	273	42	r	r	NOUN
ejpam-3397	273	43	,	,	PUNCT
ejpam-3397	273	44	then	then	ADV
ejpam-3397	273	45	aγb	aγb	PRON
ejpam-3397	273	46	is	be	VERB
ejpam-3397	273	47	an	an	DET
ejpam-3397	273	48	ideal	ideal	NOUN
ejpam-3397	273	49	of	of	ADP
ejpam-3397	273	50	r.	r.	PROPN
ejpam-3397	273	51	example	example	PROPN
ejpam-3397	273	52	10	10	NUM
ejpam-3397	273	53	.	.	PUNCT
ejpam-3397	274	1	refer	refer	VERB
ejpam-3397	274	2	to	to	ADP
ejpam-3397	274	3	the	the	DET
ejpam-3397	274	4	γ	γ	X
ejpam-3397	274	5	-	-	PUNCT
ejpam-3397	274	6	seminearring	seminearre	VERB
ejpam-3397	274	7	s	s	AUX
ejpam-3397	274	8	defined	define	VERB
ejpam-3397	274	9	in	in	ADP
ejpam-3397	274	10	tables	table	NOUN
ejpam-3397	274	11	of	of	ADP
ejpam-3397	274	12	an	an	DET
ejpam-3397	274	13	example3	example3	NOUN
ejpam-3397	274	14	.	.	PUNCT
ejpam-3397	275	1	let	let	VERB
ejpam-3397	275	2	a	a	PRON
ejpam-3397	275	3	=	=	PUNCT
ejpam-3397	275	4	{	{	PUNCT
ejpam-3397	275	5	0	0	NUM
ejpam-3397	275	6	,	,	PUNCT
ejpam-3397	275	7	a	a	PRON
ejpam-3397	275	8	}	}	PUNCT
ejpam-3397	275	9	be	be	AUX
ejpam-3397	275	10	a	a	DET
ejpam-3397	275	11	weakly	weakly	ADJ
ejpam-3397	275	12	prime	prime	ADJ
ejpam-3397	275	13	ideal	ideal	NOUN
ejpam-3397	275	14	of	of	ADP
ejpam-3397	275	15	r	r	NOUN
ejpam-3397	275	16	and	and	CCONJ
ejpam-3397	275	17	b	b	NOUN
ejpam-3397	275	18	=	=	PUNCT
ejpam-3397	275	19	{	{	PUNCT
ejpam-3397	275	20	1	1	NUM
ejpam-3397	275	21	,	,	PUNCT
ejpam-3397	275	22	e	e	AUX
ejpam-3397	275	23	}	}	PUNCT
ejpam-3397	275	24	be	be	AUX
ejpam-3397	275	25	a	a	DET
ejpam-3397	275	26	nonempty	nonempty	ADJ
ejpam-3397	275	27	subset	subset	NOUN
ejpam-3397	275	28	of	of	ADP
ejpam-3397	275	29	r.	r.	PROPN
ejpam-3397	275	30	clearly	clearly	ADV
ejpam-3397	275	31	aγb	aγb	VERB
ejpam-3397	275	32	=	=	SYM
ejpam-3397	275	33	{	{	PUNCT
ejpam-3397	275	34	0	0	NUM
ejpam-3397	275	35	,	,	PUNCT
ejpam-3397	275	36	a	a	PRON
ejpam-3397	275	37	}	}	PUNCT
ejpam-3397	275	38	is	be	AUX
ejpam-3397	275	39	a	a	DET
ejpam-3397	275	40	weakly	weakly	ADJ
ejpam-3397	275	41	prime	prime	ADJ
ejpam-3397	275	42	ideal	ideal	NOUN
ejpam-3397	275	43	of	of	ADP
ejpam-3397	275	44	s.	s.	PROPN
ejpam-3397	275	45	let	let	VERB
ejpam-3397	275	46	c	c	NOUN
ejpam-3397	275	47	=	=	PRON
ejpam-3397	275	48	{	{	PUNCT
ejpam-3397	275	49	0	0	NUM
ejpam-3397	275	50	,	,	PUNCT
ejpam-3397	275	51	c	c	AUX
ejpam-3397	275	52	}	}	PUNCT
ejpam-3397	275	53	be	be	AUX
ejpam-3397	275	54	another	another	DET
ejpam-3397	275	55	weakly	weakly	ADJ
ejpam-3397	275	56	prime	prime	ADJ
ejpam-3397	275	57	ideal	ideal	NOUN
ejpam-3397	275	58	.	.	PUNCT
ejpam-3397	276	1	also	also	ADV
ejpam-3397	276	2	aγc	aγc	VERB
ejpam-3397	276	3	=	=	PUNCT
ejpam-3397	276	4	{	{	PUNCT
ejpam-3397	276	5	0	0	NUM
ejpam-3397	276	6	}	}	PUNCT
ejpam-3397	276	7	and	and	CCONJ
ejpam-3397	276	8	it	it	PRON
ejpam-3397	276	9	is	be	AUX
ejpam-3397	276	10	a	a	DET
ejpam-3397	276	11	minimal	minimal	ADJ
ejpam-3397	276	12	prime	prime	ADJ
ejpam-3397	276	13	ideal	ideal	NOUN
ejpam-3397	276	14	of	of	ADP
ejpam-3397	276	15	s.	s.	PROPN
ejpam-3397	276	16	weakly	weakly	ADJ
ejpam-3397	276	17	primary	primary	ADJ
ejpam-3397	276	18	ideals	ideal	NOUN
ejpam-3397	276	19	definition	definition	NOUN
ejpam-3397	276	20	5	5	NUM
ejpam-3397	276	21	.	.	PUNCT
ejpam-3397	277	1	let	let	VERB
ejpam-3397	277	2	r	r	PRON
ejpam-3397	277	3	be	be	AUX
ejpam-3397	277	4	a	a	DET
ejpam-3397	277	5	γ	γ	NOUN
ejpam-3397	277	6	-	-	PUNCT
ejpam-3397	277	7	seminearring	seminearring	NOUN
ejpam-3397	277	8	.	.	PUNCT
ejpam-3397	278	1	a	a	DET
ejpam-3397	278	2	proper	proper	ADJ
ejpam-3397	278	3	ideal	ideal	NOUN
ejpam-3397	278	4	p	p	NOUN
ejpam-3397	278	5	of	of	ADP
ejpam-3397	278	6	r	r	NOUN
ejpam-3397	278	7	is	be	AUX
ejpam-3397	278	8	said	say	VERB
ejpam-3397	278	9	to	to	PART
ejpam-3397	278	10	be	be	AUX
ejpam-3397	278	11	a	a	DET
ejpam-3397	278	12	weakly	weakly	ADJ
ejpam-3397	278	13	primary	primary	ADJ
ejpam-3397	278	14	ideal	ideal	NOUN
ejpam-3397	278	15	if	if	SCONJ
ejpam-3397	278	16	0	0	NUM
ejpam-3397	278	17	6=	6=	NUM
ejpam-3397	278	18	pγq	pγq	NOUN
ejpam-3397	278	19	∈	∈	PROPN
ejpam-3397	278	20	p	p	NOUN
ejpam-3397	278	21	implies	imply	VERB
ejpam-3397	278	22	p	p	X
ejpam-3397	278	23	∈	∈	PROPN
ejpam-3397	278	24	p	p	NOUN
ejpam-3397	278	25	or	or	CCONJ
ejpam-3397	278	26	qn	qn	NOUN
ejpam-3397	278	27	∈	∈	PROPN
ejpam-3397	278	28	p	p	NOUN
ejpam-3397	278	29	.	.	PUNCT
ejpam-3397	279	1	+	+	CCONJ
ejpam-3397	279	2	0	0	NUM
ejpam-3397	279	3	1	1	NUM
ejpam-3397	279	4	e	e	NOUN
ejpam-3397	279	5	a	a	DET
ejpam-3397	279	6	b	b	X
ejpam-3397	279	7	c	c	NOUN
ejpam-3397	279	8	0	0	NUM
ejpam-3397	279	9	0	0	NUM
ejpam-3397	279	10	1	1	NUM
ejpam-3397	279	11	e	e	NOUN
ejpam-3397	279	12	a	a	DET
ejpam-3397	279	13	a	a	DET
ejpam-3397	279	14	c	c	NOUN
ejpam-3397	279	15	1	1	NUM
ejpam-3397	279	16	1	1	NUM
ejpam-3397	279	17	1	1	NUM
ejpam-3397	279	18	1	1	NUM
ejpam-3397	279	19	1	1	NUM
ejpam-3397	279	20	1	1	NUM
ejpam-3397	279	21	1	1	NUM
ejpam-3397	279	22	e	e	NOUN
ejpam-3397	279	23	e	e	X
ejpam-3397	279	24	1	1	NUM
ejpam-3397	279	25	e	e	SYM
ejpam-3397	279	26	1	1	NUM
ejpam-3397	279	27	1	1	NUM
ejpam-3397	279	28	e	e	NOUN
ejpam-3397	279	29	a	a	DET
ejpam-3397	279	30	a	a	DET
ejpam-3397	279	31	1	1	NUM
ejpam-3397	279	32	1	1	NUM
ejpam-3397	279	33	a	a	DET
ejpam-3397	279	34	a	a	DET
ejpam-3397	279	35	a	a	DET
ejpam-3397	279	36	b	b	NOUN
ejpam-3397	279	37	a	a	DET
ejpam-3397	279	38	1	1	NUM
ejpam-3397	279	39	1	1	NUM
ejpam-3397	279	40	a	a	PRON
ejpam-3397	279	41	a	a	DET
ejpam-3397	279	42	a	a	DET
ejpam-3397	279	43	c	c	NOUN
ejpam-3397	279	44	c	c	NOUN
ejpam-3397	279	45	1	1	NUM
ejpam-3397	279	46	e	e	NOUN
ejpam-3397	279	47	a	a	PRON
ejpam-3397	279	48	a	a	DET
ejpam-3397	279	49	c	c	NOUN
ejpam-3397	279	50	α	α	NOUN
ejpam-3397	279	51	0	0	NUM
ejpam-3397	279	52	1	1	NUM
ejpam-3397	279	53	e	e	NOUN
ejpam-3397	279	54	a	a	DET
ejpam-3397	279	55	b	b	X
ejpam-3397	279	56	c	c	NOUN
ejpam-3397	279	57	0	0	NUM
ejpam-3397	279	58	0	0	NUM
ejpam-3397	279	59	0	0	NUM
ejpam-3397	279	60	0	0	NUM
ejpam-3397	279	61	0	0	NUM
ejpam-3397	279	62	0	0	NUM
ejpam-3397	279	63	0	0	NUM
ejpam-3397	279	64	1	1	NUM
ejpam-3397	279	65	0	0	NUM
ejpam-3397	279	66	1	1	NUM
ejpam-3397	279	67	e	e	NOUN
ejpam-3397	279	68	a	a	DET
ejpam-3397	279	69	b	b	NOUN
ejpam-3397	279	70	c	c	NOUN
ejpam-3397	279	71	e	e	X
ejpam-3397	279	72	0	0	NUM
ejpam-3397	279	73	e	e	X
ejpam-3397	279	74	e	e	X
ejpam-3397	279	75	0	0	PROPN
ejpam-3397	279	76	c	c	NOUN
ejpam-3397	279	77	c	c	PROPN
ejpam-3397	279	78	a	a	DET
ejpam-3397	279	79	0	0	NUM
ejpam-3397	279	80	a	a	DET
ejpam-3397	279	81	0	0	NUM
ejpam-3397	280	1	a	a	DET
ejpam-3397	280	2	a	a	DET
ejpam-3397	280	3	0	0	NUM
ejpam-3397	280	4	b	b	NOUN
ejpam-3397	280	5	0	0	NUM
ejpam-3397	280	6	b	b	NOUN
ejpam-3397	280	7	0	0	NUM
ejpam-3397	280	8	a	a	DET
ejpam-3397	280	9	a	a	DET
ejpam-3397	280	10	0	0	NUM
ejpam-3397	280	11	c	c	NOUN
ejpam-3397	280	12	0	0	PUNCT
ejpam-3397	281	1	c	c	NOUN
ejpam-3397	281	2	0	0	NUM
ejpam-3397	281	3	0	0	NUM
ejpam-3397	281	4	0	0	SYM
ejpam-3397	281	5	0	0	NUM
ejpam-3397	281	6	abdelghani	abdelghani	PROPN
ejpam-3397	281	7	taouti	taouti	NOUN
ejpam-3397	281	8	et	et	PROPN
ejpam-3397	281	9	al	al	PROPN
ejpam-3397	281	10	.	.	PUNCT
ejpam-3397	281	11	/	/	SYM
ejpam-3397	281	12	eur	eur	PROPN
ejpam-3397	281	13	.	.	PUNCT
ejpam-3397	282	1	j.	j.	PROPN
ejpam-3397	282	2	pure	pure	PROPN
ejpam-3397	282	3	appl	appl	PROPN
ejpam-3397	282	4	.	.	PROPN
ejpam-3397	282	5	math	math	PROPN
ejpam-3397	282	6	,	,	PUNCT
ejpam-3397	282	7	12	12	NUM
ejpam-3397	282	8	(	(	PUNCT
ejpam-3397	282	9	2	2	NUM
ejpam-3397	282	10	)	)	PUNCT
ejpam-3397	282	11	(	(	PUNCT
ejpam-3397	282	12	2019	2019	NUM
ejpam-3397	282	13	)	)	PUNCT
ejpam-3397	282	14	,	,	PUNCT
ejpam-3397	282	15	544	544	NUM
ejpam-3397	282	16	-	-	SYM
ejpam-3397	282	17	552	552	NUM
ejpam-3397	282	18	551	551	NUM
ejpam-3397	282	19	example	example	NOUN
ejpam-3397	282	20	11	11	NUM
ejpam-3397	282	21	.	.	PUNCT
ejpam-3397	283	1	let	let	VERB
ejpam-3397	283	2	r	r	NOUN
ejpam-3397	283	3	=	=	SYM
ejpam-3397	283	4	{	{	PUNCT
ejpam-3397	283	5	0	0	NUM
ejpam-3397	283	6	,	,	PUNCT
ejpam-3397	283	7	1	1	NUM
ejpam-3397	283	8	,	,	PUNCT
ejpam-3397	283	9	e	e	NOUN
ejpam-3397	283	10	,	,	PUNCT
ejpam-3397	283	11	a	a	DET
ejpam-3397	283	12	,	,	PUNCT
ejpam-3397	283	13	b	b	NOUN
ejpam-3397	283	14	,	,	PUNCT
ejpam-3397	283	15	c	c	AUX
ejpam-3397	283	16	}	}	PUNCT
ejpam-3397	283	17	be	be	AUX
ejpam-3397	283	18	a	a	DET
ejpam-3397	283	19	γ	γ	NOUN
ejpam-3397	283	20	-	-	PUNCT
ejpam-3397	283	21	seminearring	seminearring	NOUN
ejpam-3397	283	22	with	with	ADP
ejpam-3397	283	23	γ	γ	X
ejpam-3397	283	24	=	=	SYM
ejpam-3397	283	25	{	{	PUNCT
ejpam-3397	283	26	α	α	NOUN
ejpam-3397	283	27	,	,	PUNCT
ejpam-3397	283	28	1	1	NUM
ejpam-3397	283	29	}	}	PUNCT
ejpam-3397	283	30	defined	define	VERB
ejpam-3397	283	31	in	in	ADP
ejpam-3397	283	32	example1	example1	PROPN
ejpam-3397	283	33	.	.	PUNCT
ejpam-3397	284	1	here	here	ADV
ejpam-3397	284	2	i	i	PRON
ejpam-3397	284	3	=	=	PUNCT
ejpam-3397	284	4	{	{	PUNCT
ejpam-3397	284	5	0	0	NUM
ejpam-3397	284	6	,	,	PUNCT
ejpam-3397	284	7	a	a	PRON
ejpam-3397	284	8	}	}	PUNCT
ejpam-3397	284	9	,	,	PUNCT
ejpam-3397	284	10	j	j	PROPN
ejpam-3397	284	11	=	=	PUNCT
ejpam-3397	284	12	{	{	PUNCT
ejpam-3397	284	13	0	0	NUM
ejpam-3397	284	14	,	,	PUNCT
ejpam-3397	284	15	a	a	PRON
ejpam-3397	284	16	,	,	PUNCT
ejpam-3397	284	17	c	c	NOUN
ejpam-3397	284	18	}	}	PUNCT
ejpam-3397	284	19	are	be	AUX
ejpam-3397	284	20	weakly	weakly	ADV
ejpam-3397	284	21	primary	primary	ADJ
ejpam-3397	284	22	ideals	ideal	NOUN
ejpam-3397	284	23	but	but	CCONJ
ejpam-3397	284	24	not	not	PART
ejpam-3397	284	25	a	a	DET
ejpam-3397	284	26	weakly	weakly	ADJ
ejpam-3397	284	27	prime	prime	NOUN
ejpam-3397	284	28	.	.	PUNCT
ejpam-3397	285	1	clearly	clearly	ADV
ejpam-3397	285	2	,	,	PUNCT
ejpam-3397	285	3	bαb	bαb	PROPN
ejpam-3397	285	4	=	=	PUNCT
ejpam-3397	285	5	a	a	DET
ejpam-3397	285	6	∈	∈	NOUN
ejpam-3397	286	1	i	i	PRON
ejpam-3397	286	2	but	but	CCONJ
ejpam-3397	286	3	b2	b2	NOUN
ejpam-3397	286	4	=	=	PUNCT
ejpam-3397	286	5	a	a	DET
ejpam-3397	286	6	∈	∈	PROPN
ejpam-3397	286	7	i.	i.	NOUN
ejpam-3397	286	8	similarly	similarly	ADV
ejpam-3397	286	9	,	,	PUNCT
ejpam-3397	286	10	j	j	PROPN
ejpam-3397	286	11	is	be	AUX
ejpam-3397	286	12	also	also	ADV
ejpam-3397	286	13	a	a	DET
ejpam-3397	286	14	weakly	weakly	ADJ
ejpam-3397	286	15	primary	primary	NOUN
ejpam-3397	286	16	but	but	CCONJ
ejpam-3397	286	17	not	not	PART
ejpam-3397	286	18	a	a	DET
ejpam-3397	286	19	weakly	weakly	ADJ
ejpam-3397	286	20	prime	prime	ADJ
ejpam-3397	286	21	ideal	ideal	NOUN
ejpam-3397	286	22	.	.	PUNCT
ejpam-3397	287	1	neither	neither	CCONJ
ejpam-3397	287	2	prime	prime	NOUN
ejpam-3397	287	3	because	because	SCONJ
ejpam-3397	287	4	in	in	ADP
ejpam-3397	287	5	j	j	PROPN
ejpam-3397	287	6	,	,	PUNCT
ejpam-3397	287	7	as	as	ADP
ejpam-3397	287	8	e.b	e.b	PROPN
ejpam-3397	287	9	=	=	SYM
ejpam-3397	287	10	c	c	PROPN
ejpam-3397	287	11	∈	∈	PROPN
ejpam-3397	287	12	j.	j.	PROPN
ejpam-3397	287	13	clearly	clearly	ADV
ejpam-3397	287	14	,	,	PUNCT
ejpam-3397	287	15	e	e	PROPN
ejpam-3397	287	16	,	,	PUNCT
ejpam-3397	287	17	b	b	PROPN
ejpam-3397	287	18	/∈	/∈	PROPN
ejpam-3397	288	1	j	j	PROPN
ejpam-3397	288	2	but	but	CCONJ
ejpam-3397	288	3	b2	b2	NOUN
ejpam-3397	288	4	=	=	PUNCT
ejpam-3397	288	5	a	a	DET
ejpam-3397	288	6	∈	∈	NOUN
ejpam-3397	288	7	j.	j.	NOUN
ejpam-3397	288	8	proposition	proposition	NOUN
ejpam-3397	288	9	9	9	NUM
ejpam-3397	288	10	.	.	PUNCT
ejpam-3397	289	1	every	every	DET
ejpam-3397	289	2	weakly	weakly	ADJ
ejpam-3397	289	3	prime	prime	ADJ
ejpam-3397	289	4	ideal	ideal	NOUN
ejpam-3397	289	5	is	be	AUX
ejpam-3397	289	6	a	a	DET
ejpam-3397	289	7	weakly	weakly	ADJ
ejpam-3397	289	8	primary	primary	ADJ
ejpam-3397	289	9	ideal	ideal	NOUN
ejpam-3397	289	10	but	but	CCONJ
ejpam-3397	289	11	converse	converse	NOUN
ejpam-3397	289	12	is	be	AUX
ejpam-3397	289	13	not	not	PART
ejpam-3397	289	14	true	true	ADJ
ejpam-3397	289	15	.	.	PUNCT
ejpam-3397	290	1	example	example	NOUN
ejpam-3397	290	2	12	12	NUM
ejpam-3397	290	3	.	.	PUNCT
ejpam-3397	291	1	let	let	VERB
ejpam-3397	291	2	r	r	NOUN
ejpam-3397	291	3	=	=	SYM
ejpam-3397	291	4	{	{	PUNCT
ejpam-3397	291	5	0	0	NUM
ejpam-3397	291	6	,	,	PUNCT
ejpam-3397	291	7	1	1	NUM
ejpam-3397	291	8	,	,	PUNCT
ejpam-3397	291	9	e	e	NOUN
ejpam-3397	291	10	,	,	PUNCT
ejpam-3397	291	11	a	a	DET
ejpam-3397	291	12	,	,	PUNCT
ejpam-3397	291	13	b	b	NOUN
ejpam-3397	291	14	,	,	PUNCT
ejpam-3397	291	15	c	c	AUX
ejpam-3397	291	16	}	}	PUNCT
ejpam-3397	291	17	be	be	AUX
ejpam-3397	291	18	a	a	DET
ejpam-3397	291	19	γ	γ	NOUN
ejpam-3397	291	20	-	-	PUNCT
ejpam-3397	291	21	seminearring	seminearring	NOUN
ejpam-3397	291	22	with	with	ADP
ejpam-3397	291	23	γ	γ	X
ejpam-3397	291	24	=	=	SYM
ejpam-3397	291	25	{	{	PUNCT
ejpam-3397	291	26	α	α	NOUN
ejpam-3397	291	27	,	,	PUNCT
ejpam-3397	291	28	1	1	NUM
ejpam-3397	291	29	}	}	PUNCT
ejpam-3397	291	30	.	.	PUNCT
ejpam-3397	292	1	in	in	ADP
ejpam-3397	292	2	r	r	NOUN
ejpam-3397	292	3	the	the	DET
ejpam-3397	292	4	ideal	ideal	NOUN
ejpam-3397	292	5	i	i	X
ejpam-3397	292	6	=	=	PUNCT
ejpam-3397	292	7	{	{	PUNCT
ejpam-3397	292	8	0	0	NUM
ejpam-3397	292	9	,	,	PUNCT
ejpam-3397	292	10	a	a	DET
ejpam-3397	292	11	,	,	PUNCT
ejpam-3397	292	12	b	b	X
ejpam-3397	292	13	}	}	PUNCT
ejpam-3397	292	14	is	be	AUX
ejpam-3397	292	15	weakly	weakly	ADV
ejpam-3397	292	16	prime	prime	ADJ
ejpam-3397	292	17	and	and	CCONJ
ejpam-3397	292	18	also	also	ADV
ejpam-3397	292	19	by	by	ADP
ejpam-3397	292	20	above	above	ADP
ejpam-3397	292	21	proposition	proposition	NOUN
ejpam-3397	292	22	it	it	PRON
ejpam-3397	292	23	is	be	AUX
ejpam-3397	292	24	weakly	weakly	ADV
ejpam-3397	292	25	primary	primary	ADJ
ejpam-3397	293	1	but	but	CCONJ
ejpam-3397	293	2	it	it	PRON
ejpam-3397	293	3	is	be	AUX
ejpam-3397	293	4	not	not	PART
ejpam-3397	293	5	prime	prime	ADJ
ejpam-3397	293	6	b	b	PROPN
ejpam-3397	293	7	/	/	SYM
ejpam-3397	293	8	c	c	NOUN
ejpam-3397	293	9	cαc	cαc	NOUN
ejpam-3397	294	1	=	=	SYM
ejpam-3397	294	2	0	0	NUM
ejpam-3397	294	3	and	and	CCONJ
ejpam-3397	294	4	c	c	PROPN
ejpam-3397	294	5	/∈	/∈	PUNCT
ejpam-3397	295	1	i.	i.	PROPN
ejpam-3397	295	2	another	another	DET
ejpam-3397	295	3	ideal	ideal	NOUN
ejpam-3397	295	4	j	j	PROPN
ejpam-3397	295	5	=	=	PUNCT
ejpam-3397	295	6	{	{	PUNCT
ejpam-3397	295	7	0	0	NUM
ejpam-3397	295	8	,	,	PUNCT
ejpam-3397	295	9	a	a	PRON
ejpam-3397	295	10	}	}	PUNCT
ejpam-3397	295	11	is	be	AUX
ejpam-3397	295	12	weakly	weakly	ADV
ejpam-3397	295	13	primary	primary	ADJ
ejpam-3397	295	14	but	but	CCONJ
ejpam-3397	295	15	not	not	PART
ejpam-3397	295	16	weakly	weakly	ADV
ejpam-3397	295	17	prime	prime	ADJ
ejpam-3397	295	18	neither	neither	CCONJ
ejpam-3397	295	19	prime	prime	ADJ
ejpam-3397	295	20	b	b	PROPN
ejpam-3397	295	21	/	/	SYM
ejpam-3397	295	22	c	c	NOUN
ejpam-3397	295	23	bαb	bαb	NOUN
ejpam-3397	296	1	=	=	NOUN
ejpam-3397	296	2	a	a	DET
ejpam-3397	296	3	∈	∈	PROPN
ejpam-3397	296	4	i.	i.	NOUN
ejpam-3397	296	5	clearly	clearly	ADV
ejpam-3397	296	6	,	,	PUNCT
ejpam-3397	296	7	b	b	X
ejpam-3397	296	8	/∈	/∈	PUNCT
ejpam-3397	297	1	i	i	PRON
ejpam-3397	297	2	but	but	CCONJ
ejpam-3397	297	3	b2	b2	NOUN
ejpam-3397	297	4	=	=	PUNCT
ejpam-3397	297	5	a	a	DET
ejpam-3397	297	6	∈	∈	PROPN
ejpam-3397	297	7	i.	i.	NOUN
ejpam-3397	297	8	proposition	proposition	NOUN
ejpam-3397	297	9	10	10	NUM
ejpam-3397	297	10	.	.	PUNCT
ejpam-3397	298	1	intersection	intersection	NOUN
ejpam-3397	298	2	of	of	ADP
ejpam-3397	298	3	finite	finite	ADJ
ejpam-3397	298	4	numbers	number	NOUN
ejpam-3397	298	5	of	of	ADP
ejpam-3397	298	6	weakly	weakly	ADJ
ejpam-3397	298	7	primary	primary	ADJ
ejpam-3397	298	8	ideals	ideal	NOUN
ejpam-3397	298	9	of	of	ADP
ejpam-3397	298	10	a	a	DET
ejpam-3397	298	11	γ	γ	X
ejpam-3397	298	12	-	-	PUNCT
ejpam-3397	298	13	seminearring	seminearre	VERB
ejpam-3397	298	14	r	r	NOUN
ejpam-3397	298	15	which	which	PRON
ejpam-3397	298	16	are	be	AUX
ejpam-3397	298	17	totally	totally	ADV
ejpam-3397	298	18	ordered	order	VERB
ejpam-3397	298	19	by	by	ADP
ejpam-3397	298	20	inclusion	inclusion	NOUN
ejpam-3397	298	21	is	be	AUX
ejpam-3397	298	22	a	a	DET
ejpam-3397	298	23	weakly	weakly	ADJ
ejpam-3397	298	24	primary	primary	ADJ
ejpam-3397	298	25	ideal	ideal	NOUN
ejpam-3397	298	26	.	.	PUNCT
ejpam-3397	299	1	proof	proof	NOUN
ejpam-3397	299	2	.	.	PUNCT
ejpam-3397	300	1	let	let	VERB
ejpam-3397	300	2	{	{	PUNCT
ejpam-3397	300	3	pα}α∈λ	pα}α∈λ	VERB
ejpam-3397	300	4	be	be	AUX
ejpam-3397	300	5	the	the	DET
ejpam-3397	300	6	family	family	NOUN
ejpam-3397	300	7	of	of	ADP
ejpam-3397	300	8	weakly	weakly	ADJ
ejpam-3397	300	9	primary	primary	ADJ
ejpam-3397	300	10	ideals	ideal	NOUN
ejpam-3397	300	11	which	which	PRON
ejpam-3397	300	12	are	be	AUX
ejpam-3397	300	13	totally	totally	ADV
ejpam-3397	300	14	ordered	order	VERB
ejpam-3397	300	15	by	by	ADP
ejpam-3397	300	16	inclusion	inclusion	NOUN
ejpam-3397	300	17	.	.	PUNCT
ejpam-3397	301	1	suppose	suppose	VERB
ejpam-3397	301	2	i	i	PRON
ejpam-3397	301	3	and	and	CCONJ
ejpam-3397	301	4	j	j	PROPN
ejpam-3397	301	5	be	be	VERB
ejpam-3397	301	6	ideals	ideal	NOUN
ejpam-3397	301	7	of	of	ADP
ejpam-3397	301	8	r.	r.	PROPN
ejpam-3397	301	9	if	if	SCONJ
ejpam-3397	301	10	0	0	NUM
ejpam-3397	301	11	6=	6=	NUM
ejpam-3397	301	12	iγj	iγj	NOUN
ejpam-3397	301	13	⊆	⊆	NUM
ejpam-3397	301	14	∩α∈λpα	∩α∈λpα	NOUN
ejpam-3397	301	15	,	,	PUNCT
ejpam-3397	301	16	then	then	ADV
ejpam-3397	301	17	0	0	NUM
ejpam-3397	301	18	6=	6=	NUM
ejpam-3397	301	19	iγj	iγj	NOUN
ejpam-3397	301	20	⊆	⊆	NUM
ejpam-3397	301	21	pα	pα	NOUN
ejpam-3397	301	22	,	,	PUNCT
ejpam-3397	301	23	for	for	ADP
ejpam-3397	301	24	all	all	DET
ejpam-3397	301	25	α	α	DET
ejpam-3397	301	26	∈	∈	PROPN
ejpam-3397	301	27	λ	λ	PROPN
ejpam-3397	301	28	.	.	PROPN
ejpam-3397	301	29	suppose	suppose	VERB
ejpam-3397	301	30	that	that	SCONJ
ejpam-3397	301	31	there	there	PRON
ejpam-3397	301	32	exists	exist	VERB
ejpam-3397	301	33	α	α	PRON
ejpam-3397	301	34	∈	∈	PROPN
ejpam-3397	301	35	λ	λ	NOUN
ejpam-3397	301	36	such	such	ADJ
ejpam-3397	301	37	that	that	SCONJ
ejpam-3397	301	38	i	i	PRON
ejpam-3397	301	39	*	*	VERB
ejpam-3397	301	40	pα	pα	INTJ
ejpam-3397	301	41	.	.	PUNCT
ejpam-3397	302	1	then	then	ADV
ejpam-3397	302	2	,	,	PUNCT
ejpam-3397	302	3	jn	jn	PROPN
ejpam-3397	302	4	⊆	⊆	NUM
ejpam-3397	302	5	pα	pα	VERB
ejpam-3397	302	6	and	and	CCONJ
ejpam-3397	302	7	hence	hence	ADV
ejpam-3397	302	8	jn	jn	PROPN
ejpam-3397	302	9	⊆	⊆	NUM
ejpam-3397	302	10	pβ	pβ	ADV
ejpam-3397	302	11	for	for	ADP
ejpam-3397	302	12	all	all	DET
ejpam-3397	302	13	β	β	NOUN
ejpam-3397	302	14	≥	≥	NUM
ejpam-3397	302	15	α	α	X
ejpam-3397	302	16	.	.	PUNCT
ejpam-3397	303	1	we	we	PRON
ejpam-3397	303	2	assume	assume	VERB
ejpam-3397	303	3	that	that	SCONJ
ejpam-3397	303	4	there	there	PRON
ejpam-3397	303	5	exist	exist	VERB
ejpam-3397	303	6	γ	γ	NOUN
ejpam-3397	303	7	<	<	X
ejpam-3397	303	8	α	α	PRON
ejpam-3397	303	9	such	such	ADJ
ejpam-3397	303	10	that	that	SCONJ
ejpam-3397	303	11	jn	jn	PROPN
ejpam-3397	303	12	⊆	⊆	NUM
ejpam-3397	303	13	pγ	pγ	NOUN
ejpam-3397	303	14	.then	.then	ADP
ejpam-3397	303	15	,	,	PUNCT
ejpam-3397	303	16	i	i	PRON
ejpam-3397	303	17	⊆	⊆	NUM
ejpam-3397	303	18	pγ	pγ	VERB
ejpam-3397	304	1	and	and	CCONJ
ejpam-3397	304	2	hence	hence	ADV
ejpam-3397	304	3	i	i	PRON
ejpam-3397	304	4	⊆	⊆	NUM
ejpam-3397	304	5	pα	pα	NOUN
ejpam-3397	304	6	,	,	PUNCT
ejpam-3397	304	7	which	which	PRON
ejpam-3397	304	8	is	be	AUX
ejpam-3397	304	9	impossible	impossible	ADJ
ejpam-3397	304	10	.	.	PUNCT
ejpam-3397	305	1	hence	hence	ADV
ejpam-3397	305	2	,	,	PUNCT
ejpam-3397	305	3	jn	jn	PROPN
ejpam-3397	305	4	⊆	⊆	NUM
ejpam-3397	305	5	pβ	pβ	ADV
ejpam-3397	305	6	for	for	ADP
ejpam-3397	305	7	any	any	DET
ejpam-3397	305	8	β	β	X
ejpam-3397	305	9	∈	∈	PROPN
ejpam-3397	305	10	λ	λ	PROPN
ejpam-3397	305	11	.	.	PUNCT
ejpam-3397	306	1	thus	thus	ADV
ejpam-3397	306	2	,	,	PUNCT
ejpam-3397	306	3	∩α∈λpα	∩α∈λpα	ADV
ejpam-3397	306	4	is	be	AUX
ejpam-3397	306	5	a	a	DET
ejpam-3397	306	6	weakly	weakly	ADJ
ejpam-3397	306	7	primary	primary	ADJ
ejpam-3397	306	8	ideal	ideal	NOUN
ejpam-3397	306	9	of	of	ADP
ejpam-3397	306	10	a	a	DET
ejpam-3397	306	11	γ	γ	NOUN
ejpam-3397	306	12	-	-	PUNCT
ejpam-3397	306	13	seminearring	seminearre	VERB
ejpam-3397	306	14	r.	r.	PROPN
ejpam-3397	306	15	example	example	NOUN
ejpam-3397	306	16	13	13	NUM
ejpam-3397	306	17	.	.	PUNCT
ejpam-3397	307	1	let	let	VERB
ejpam-3397	307	2	r	r	NOUN
ejpam-3397	307	3	=	=	SYM
ejpam-3397	307	4	{	{	PUNCT
ejpam-3397	307	5	0	0	NUM
ejpam-3397	307	6	,	,	PUNCT
ejpam-3397	307	7	1	1	NUM
ejpam-3397	307	8	,	,	PUNCT
ejpam-3397	307	9	e	e	NOUN
ejpam-3397	307	10	,	,	PUNCT
ejpam-3397	307	11	a	a	DET
ejpam-3397	307	12	,	,	PUNCT
ejpam-3397	307	13	b	b	NOUN
ejpam-3397	307	14	,	,	PUNCT
ejpam-3397	307	15	c	c	AUX
ejpam-3397	307	16	}	}	PUNCT
ejpam-3397	307	17	be	be	AUX
ejpam-3397	307	18	a	a	DET
ejpam-3397	307	19	γ	γ	NOUN
ejpam-3397	307	20	-	-	PUNCT
ejpam-3397	307	21	seminearring	seminearring	NOUN
ejpam-3397	307	22	with	with	ADP
ejpam-3397	307	23	γ	γ	X
ejpam-3397	307	24	=	=	SYM
ejpam-3397	307	25	{	{	PUNCT
ejpam-3397	307	26	α	α	NOUN
ejpam-3397	307	27	,	,	PUNCT
ejpam-3397	307	28	1	1	NUM
ejpam-3397	307	29	}	}	PUNCT
ejpam-3397	307	30	.	.	PUNCT
ejpam-3397	308	1	here	here	ADV
ejpam-3397	308	2	i	i	PRON
ejpam-3397	308	3	=	=	PUNCT
ejpam-3397	308	4	{	{	PUNCT
ejpam-3397	308	5	0	0	NUM
ejpam-3397	308	6	,	,	PUNCT
ejpam-3397	308	7	a	a	PRON
ejpam-3397	308	8	}	}	PUNCT
ejpam-3397	308	9	,	,	PUNCT
ejpam-3397	308	10	j	j	PROPN
ejpam-3397	308	11	=	=	PUNCT
ejpam-3397	308	12	{	{	PUNCT
ejpam-3397	308	13	0	0	NUM
ejpam-3397	308	14	,	,	PUNCT
ejpam-3397	308	15	a	a	PRON
ejpam-3397	308	16	,	,	PUNCT
ejpam-3397	308	17	c	c	NOUN
ejpam-3397	308	18	}	}	PUNCT
ejpam-3397	308	19	are	be	AUX
ejpam-3397	308	20	weakly	weakly	ADV
ejpam-3397	308	21	primary	primary	ADJ
ejpam-3397	308	22	ideals	ideal	NOUN
ejpam-3397	308	23	but	but	CCONJ
ejpam-3397	308	24	not	not	PART
ejpam-3397	308	25	weakly	weakly	ADV
ejpam-3397	308	26	prime	prime	ADJ
ejpam-3397	308	27	and	and	CCONJ
ejpam-3397	308	28	k	k	NOUN
ejpam-3397	308	29	=	=	PUNCT
ejpam-3397	308	30	{	{	PUNCT
ejpam-3397	308	31	0	0	NUM
ejpam-3397	308	32	,	,	PUNCT
ejpam-3397	308	33	a	a	DET
ejpam-3397	308	34	,	,	PUNCT
ejpam-3397	308	35	b	b	NOUN
ejpam-3397	308	36	,	,	PUNCT
ejpam-3397	308	37	c	c	NOUN
ejpam-3397	308	38	}	}	PUNCT
ejpam-3397	308	39	is	be	AUX
ejpam-3397	308	40	prime	prime	ADJ
ejpam-3397	308	41	and	and	CCONJ
ejpam-3397	308	42	hence	hence	ADV
ejpam-3397	308	43	weakly	weakly	ADV
ejpam-3397	308	44	primary	primary	ADJ
ejpam-3397	308	45	because	because	SCONJ
ejpam-3397	308	46	every	every	DET
ejpam-3397	308	47	prime	prime	ADJ
ejpam-3397	308	48	ideal	ideal	NOUN
ejpam-3397	308	49	is	be	AUX
ejpam-3397	308	50	weakly	weakly	ADV
ejpam-3397	308	51	primary	primary	ADJ
ejpam-3397	308	52	.	.	PUNCT
ejpam-3397	309	1	clearly	clearly	ADV
ejpam-3397	309	2	,	,	PUNCT
ejpam-3397	309	3	these	these	DET
ejpam-3397	309	4	ideals	ideal	NOUN
ejpam-3397	309	5	are	be	AUX
ejpam-3397	309	6	totally	totally	ADV
ejpam-3397	309	7	ordered	order	VERB
ejpam-3397	309	8	by	by	ADP
ejpam-3397	309	9	inclusion	inclusion	NOUN
ejpam-3397	309	10	i.e.	i.e.	X
ejpam-3397	309	11	i	i	PROPN
ejpam-3397	309	12	⊆	⊆	NUM
ejpam-3397	309	13	j	j	PROPN
ejpam-3397	309	14	⊆	⊆	NUM
ejpam-3397	309	15	k.	k.	PROPN
ejpam-3397	309	16	since	since	ADV
ejpam-3397	309	17	,	,	PUNCT
ejpam-3397	309	18	i	i	PRON
ejpam-3397	309	19	∩j	∩j	VERB
ejpam-3397	309	20	∩k	∩k	NOUN
ejpam-3397	310	1	=	=	PUNCT
ejpam-3397	310	2	i	i	PRON
ejpam-3397	310	3	=	=	PUNCT
ejpam-3397	310	4	{	{	PUNCT
ejpam-3397	310	5	0	0	NUM
ejpam-3397	310	6	,	,	PUNCT
ejpam-3397	310	7	a	a	PRON
ejpam-3397	310	8	}	}	PUNCT
ejpam-3397	310	9	,	,	PUNCT
ejpam-3397	310	10	which	which	PRON
ejpam-3397	310	11	is	be	AUX
ejpam-3397	310	12	also	also	ADV
ejpam-3397	310	13	a	a	DET
ejpam-3397	310	14	primary	primary	ADJ
ejpam-3397	310	15	ideal	ideal	NOUN
ejpam-3397	310	16	b	b	PROPN
ejpam-3397	310	17	/	/	SYM
ejpam-3397	310	18	c	c	PROPN
ejpam-3397	310	19	b.b	b.b	PROPN
ejpam-3397	310	20	=	=	DET
ejpam-3397	310	21	a	a	DET
ejpam-3397	310	22	∈	∈	PROPN
ejpam-3397	310	23	i.	i.	NOUN
ejpam-3397	310	24	clearly	clearly	ADV
ejpam-3397	310	25	,	,	PUNCT
ejpam-3397	310	26	b	b	X
ejpam-3397	310	27	/∈	/∈	PUNCT
ejpam-3397	311	1	i	i	PRON
ejpam-3397	311	2	but	but	CCONJ
ejpam-3397	311	3	b2	b2	NOUN
ejpam-3397	311	4	=	=	PUNCT
ejpam-3397	311	5	a	a	DET
ejpam-3397	311	6	∈	∈	PROPN
ejpam-3397	311	7	i.	i.	NOUN
ejpam-3397	311	8	proposition	proposition	NOUN
ejpam-3397	311	9	11	11	NUM
ejpam-3397	311	10	.	.	PUNCT
ejpam-3397	312	1	every	every	DET
ejpam-3397	312	2	ideal	ideal	NOUN
ejpam-3397	312	3	of	of	ADP
ejpam-3397	312	4	a	a	DET
ejpam-3397	312	5	γ	γ	NOUN
ejpam-3397	312	6	-	-	PUNCT
ejpam-3397	312	7	seminearring	seminearre	VERB
ejpam-3397	312	8	r	r	NOUN
ejpam-3397	312	9	is	be	AUX
ejpam-3397	312	10	a	a	DET
ejpam-3397	312	11	weakly	weakly	ADJ
ejpam-3397	312	12	primary	primary	NOUN
ejpam-3397	312	13	if	if	SCONJ
ejpam-3397	312	14	and	and	CCONJ
ejpam-3397	312	15	only	only	ADV
ejpam-3397	312	16	if	if	SCONJ
ejpam-3397	312	17	for	for	ADP
ejpam-3397	312	18	any	any	DET
ejpam-3397	312	19	ideals	ideal	NOUN
ejpam-3397	312	20	i	i	PRON
ejpam-3397	312	21	,	,	PUNCT
ejpam-3397	312	22	j	j	PROPN
ejpam-3397	312	23	,	,	PUNCT
ejpam-3397	312	24	k	k	PROPN
ejpam-3397	312	25	of	of	ADP
ejpam-3397	312	26	r	r	PROPN
ejpam-3397	312	27	,	,	PUNCT
ejpam-3397	312	28	iγj	iγj	NOUN
ejpam-3397	313	1	=	=	PUNCT
ejpam-3397	313	2	i	i	PROPN
ejpam-3397	313	3	,	,	PUNCT
ejpam-3397	313	4	iγj	iγj	PROPN
ejpam-3397	313	5	=	=	SYM
ejpam-3397	313	6	j	j	PROPN
ejpam-3397	313	7	,	,	PUNCT
ejpam-3397	313	8	iγj	iγj	PROPN
ejpam-3397	313	9	=	=	PUNCT
ejpam-3397	314	1	k	k	PROPN
ejpam-3397	314	2	where	where	SCONJ
ejpam-3397	314	3	k	k	PROPN
ejpam-3397	314	4	is	be	AUX
ejpam-3397	314	5	the	the	DET
ejpam-3397	314	6	ideal	ideal	NOUN
ejpam-3397	314	7	contained	contain	VERB
ejpam-3397	314	8	in	in	ADP
ejpam-3397	314	9	both	both	CCONJ
ejpam-3397	314	10	i	i	PROPN
ejpam-3397	314	11	and	and	CCONJ
ejpam-3397	314	12	j	j	PROPN
ejpam-3397	314	13	or	or	CCONJ
ejpam-3397	314	14	either	either	CCONJ
ejpam-3397	314	15	in	in	ADP
ejpam-3397	314	16	i	i	PRON
ejpam-3397	314	17	or	or	CCONJ
ejpam-3397	314	18	in	in	ADP
ejpam-3397	314	19	j	j	PROPN
ejpam-3397	314	20	i.e.k	i.e.k	VERB
ejpam-3397	314	21	⊆	⊆	NUM
ejpam-3397	314	22	i	i	PROPN
ejpam-3397	314	23	,	,	PUNCT
ejpam-3397	314	24	j	j	PROPN
ejpam-3397	314	25	or	or	CCONJ
ejpam-3397	314	26	k	k	PROPN
ejpam-3397	314	27	⊆	⊆	NUM
ejpam-3397	314	28	i	i	PROPN
ejpam-3397	314	29	or	or	CCONJ
ejpam-3397	314	30	k	k	PROPN
ejpam-3397	314	31	⊆	⊆	NUM
ejpam-3397	314	32	j	j	PROPN
ejpam-3397	314	33	,	,	PUNCT
ejpam-3397	314	34	or	or	CCONJ
ejpam-3397	314	35	iγj	iγj	X
ejpam-3397	314	36	=	=	SYM
ejpam-3397	314	37	0	0	X
ejpam-3397	314	38	.	.	PUNCT
ejpam-3397	315	1	proof	proof	NOUN
ejpam-3397	315	2	.	.	PUNCT
ejpam-3397	316	1	suppose	suppose	VERB
ejpam-3397	316	2	that	that	SCONJ
ejpam-3397	316	3	every	every	DET
ejpam-3397	316	4	ideal	ideal	NOUN
ejpam-3397	316	5	of	of	ADP
ejpam-3397	316	6	r	r	NOUN
ejpam-3397	316	7	is	be	AUX
ejpam-3397	316	8	a	a	DET
ejpam-3397	316	9	weakly	weakly	ADJ
ejpam-3397	316	10	prime	prime	NOUN
ejpam-3397	316	11	.	.	PUNCT
ejpam-3397	317	1	let	let	VERB
ejpam-3397	317	2	i	i	PRON
ejpam-3397	317	3	,	,	PUNCT
ejpam-3397	317	4	j	j	PROPN
ejpam-3397	317	5	are	be	AUX
ejpam-3397	317	6	ideals	ideal	NOUN
ejpam-3397	317	7	of	of	ADP
ejpam-3397	317	8	a	a	DET
ejpam-3397	317	9	γ	γ	NOUN
ejpam-3397	317	10	-	-	PUNCT
ejpam-3397	317	11	seminearring	seminearre	VERB
ejpam-3397	317	12	r.	r.	NOUN
ejpam-3397	317	13	if	if	SCONJ
ejpam-3397	317	14	iγj	iγj	PROPN
ejpam-3397	317	15	6=	6=	ADP
ejpam-3397	317	16	r	r	NOUN
ejpam-3397	317	17	,	,	PUNCT
ejpam-3397	317	18	then	then	ADV
ejpam-3397	317	19	iγj	iγj	PROPN
ejpam-3397	317	20	is	be	AUX
ejpam-3397	317	21	a	a	DET
ejpam-3397	317	22	weakly	weakly	ADJ
ejpam-3397	317	23	prime	prime	NOUN
ejpam-3397	317	24	.	.	PUNCT
ejpam-3397	318	1	if	if	SCONJ
ejpam-3397	318	2	0	0	NUM
ejpam-3397	318	3	6=	6=	NUM
ejpam-3397	318	4	iγj	iγj	NOUN
ejpam-3397	318	5	⊆	⊆	NUM
ejpam-3397	318	6	iγj	iγj	NOUN
ejpam-3397	318	7	,	,	PUNCT
ejpam-3397	318	8	then	then	ADV
ejpam-3397	318	9	we	we	PRON
ejpam-3397	318	10	have	have	VERB
ejpam-3397	318	11	i	i	PRON
ejpam-3397	318	12	⊆	⊆	NUM
ejpam-3397	318	13	iγj	iγj	NOUN
ejpam-3397	318	14	or	or	CCONJ
ejpam-3397	318	15	jn	jn	PROPN
ejpam-3397	318	16	⊆	⊆	NUM
ejpam-3397	318	17	iγj	iγj	NOUN
ejpam-3397	318	18	i.e.	i.e.	X
ejpam-3397	318	19	,	,	PUNCT
ejpam-3397	318	20	i	i	PRON
ejpam-3397	318	21	=	=	X
ejpam-3397	318	22	iγj	iγj	NOUN
ejpam-3397	318	23	or	or	CCONJ
ejpam-3397	318	24	jn	jn	PROPN
ejpam-3397	318	25	=	=	PROPN
ejpam-3397	318	26	iγj	iγj	PROPN
ejpam-3397	318	27	.	.	PUNCT
ejpam-3397	319	1	if	if	SCONJ
ejpam-3397	319	2	iγj	iγj	PRON
ejpam-3397	319	3	=	=	PUNCT
ejpam-3397	320	1	k	k	PROPN
ejpam-3397	320	2	then	then	ADV
ejpam-3397	320	3	clearly	clearly	ADV
ejpam-3397	320	4	k	k	PROPN
ejpam-3397	320	5	=	=	PUNCT
ejpam-3397	321	1	i	i	PRON
ejpam-3397	321	2	∩j	∩j	PROPN
ejpam-3397	321	3	is	be	AUX
ejpam-3397	321	4	a	a	DET
ejpam-3397	321	5	weakly	weakly	ADJ
ejpam-3397	321	6	primary	primary	ADJ
ejpam-3397	321	7	ideal	ideal	NOUN
ejpam-3397	321	8	then	then	ADV
ejpam-3397	321	9	,	,	PUNCT
ejpam-3397	322	1	k	k	PROPN
ejpam-3397	322	2	⊂	⊂	PROPN
ejpam-3397	322	3	i	i	PROPN
ejpam-3397	322	4	and	and	CCONJ
ejpam-3397	322	5	k	k	PROPN
ejpam-3397	322	6	⊂	⊂	PROPN
ejpam-3397	322	7	jn	jn	PROPN
ejpam-3397	322	8	.	.	PUNCT
ejpam-3397	323	1	finally	finally	ADV
ejpam-3397	323	2	,	,	PUNCT
ejpam-3397	323	3	if	if	SCONJ
ejpam-3397	323	4	iγj	iγj	NOUN
ejpam-3397	323	5	=	=	SYM
ejpam-3397	323	6	r	r	NOUN
ejpam-3397	323	7	,	,	PUNCT
ejpam-3397	323	8	then	then	ADV
ejpam-3397	323	9	we	we	PRON
ejpam-3397	323	10	have	have	VERB
ejpam-3397	323	11	i	i	NOUN
ejpam-3397	323	12	=	=	PUNCT
ejpam-3397	323	13	j	j	PROPN
ejpam-3397	323	14	=	=	SYM
ejpam-3397	323	15	r	r	NOUN
ejpam-3397	323	16	and	and	CCONJ
ejpam-3397	323	17	hence	hence	ADV
ejpam-3397	323	18	rγr	rγr	NOUN
ejpam-3397	323	19	=	=	SYM
ejpam-3397	323	20	r.	r.	PROPN
ejpam-3397	323	21	conversely	conversely	ADV
ejpam-3397	323	22	,	,	PUNCT
ejpam-3397	323	23	let	let	VERB
ejpam-3397	323	24	l	l	NOUN
ejpam-3397	323	25	be	be	AUX
ejpam-3397	323	26	any	any	DET
ejpam-3397	323	27	proper	proper	ADJ
ejpam-3397	323	28	ideal	ideal	NOUN
ejpam-3397	323	29	of	of	ADP
ejpam-3397	323	30	r	r	NOUN
ejpam-3397	323	31	and	and	CCONJ
ejpam-3397	323	32	suppose	suppose	VERB
ejpam-3397	323	33	that	that	SCONJ
ejpam-3397	323	34	0	0	NUM
ejpam-3397	323	35	6=	6=	NUM
ejpam-3397	323	36	iγj	iγj	VERB
ejpam-3397	323	37	⊆	⊆	NUM
ejpam-3397	323	38	l	l	NOUN
ejpam-3397	323	39	for	for	ADP
ejpam-3397	323	40	ideals	ideal	NOUN
ejpam-3397	323	41	i	i	PRON
ejpam-3397	323	42	and	and	CCONJ
ejpam-3397	323	43	j	j	PROPN
ejpam-3397	323	44	of	of	ADP
ejpam-3397	323	45	r.	r.	PROPN
ejpam-3397	323	46	then	then	ADV
ejpam-3397	323	47	,	,	PUNCT
ejpam-3397	323	48	we	we	PRON
ejpam-3397	323	49	have	have	VERB
ejpam-3397	323	50	either	either	CCONJ
ejpam-3397	324	1	i	i	PRON
ejpam-3397	324	2	=	=	PRON
ejpam-3397	324	3	iγj	iγj	VERB
ejpam-3397	324	4	⊆	⊆	NUM
ejpam-3397	324	5	l	l	NOUN
ejpam-3397	324	6	or	or	CCONJ
ejpam-3397	324	7	jn	jn	PROPN
ejpam-3397	324	8	=	=	PRON
ejpam-3397	324	9	iγj	iγj	VERB
ejpam-3397	324	10	⊆	⊆	NUM
ejpam-3397	324	11	l.	l.	NOUN
ejpam-3397	325	1	and	and	CCONJ
ejpam-3397	325	2	if	if	SCONJ
ejpam-3397	325	3	k	k	PROPN
ejpam-3397	325	4	=	=	PRON
ejpam-3397	325	5	iγj	iγj	VERB
ejpam-3397	325	6	⊆	⊆	NUM
ejpam-3397	325	7	l	l	NOUN
ejpam-3397	325	8	,	,	PUNCT
ejpam-3397	325	9	where	where	SCONJ
ejpam-3397	325	10	k	k	PROPN
ejpam-3397	325	11	⊂	⊂	PROPN
ejpam-3397	325	12	i	i	PRON
ejpam-3397	325	13	∩	∩	VERB
ejpam-3397	325	14	j	j	PROPN
ejpam-3397	325	15	and	and	CCONJ
ejpam-3397	325	16	hence	hence	ADV
ejpam-3397	325	17	k	k	PROPN
ejpam-3397	325	18	∩	∩	PROPN
ejpam-3397	325	19	i	i	PRON
ejpam-3397	325	20	⊆	⊆	NUM
ejpam-3397	325	21	i	i	PROPN
ejpam-3397	325	22	and	and	CCONJ
ejpam-3397	325	23	k	k	PROPN
ejpam-3397	325	24	∩	∩	PROPN
ejpam-3397	325	25	j	j	PROPN
ejpam-3397	325	26	⊆	⊆	NUM
ejpam-3397	325	27	l.	l.	PROPN
ejpam-3397	325	28	example	example	NOUN
ejpam-3397	325	29	14	14	NUM
ejpam-3397	325	30	.	.	PUNCT
ejpam-3397	326	1	let	let	VERB
ejpam-3397	326	2	r	r	NOUN
ejpam-3397	326	3	=	=	SYM
ejpam-3397	326	4	{	{	PUNCT
ejpam-3397	326	5	0	0	NUM
ejpam-3397	326	6	,	,	PUNCT
ejpam-3397	326	7	1	1	NUM
ejpam-3397	326	8	,	,	PUNCT
ejpam-3397	326	9	e	e	NOUN
ejpam-3397	326	10	,	,	PUNCT
ejpam-3397	326	11	a	a	DET
ejpam-3397	326	12	,	,	PUNCT
ejpam-3397	326	13	b	b	NOUN
ejpam-3397	326	14	,	,	PUNCT
ejpam-3397	326	15	c	c	AUX
ejpam-3397	326	16	}	}	PUNCT
ejpam-3397	326	17	be	be	AUX
ejpam-3397	326	18	a	a	DET
ejpam-3397	326	19	γ	γ	NOUN
ejpam-3397	326	20	-	-	PUNCT
ejpam-3397	326	21	seminearring	seminearring	NOUN
ejpam-3397	326	22	with	with	ADP
ejpam-3397	326	23	γ	γ	X
ejpam-3397	326	24	=	=	SYM
ejpam-3397	326	25	{	{	PUNCT
ejpam-3397	326	26	1	1	NUM
ejpam-3397	326	27	,	,	PUNCT
ejpam-3397	326	28	α	α	NOUN
ejpam-3397	326	29	}	}	PUNCT
ejpam-3397	326	30	.	.	PUNCT
ejpam-3397	327	1	as	as	SCONJ
ejpam-3397	327	2	r	r	NOUN
ejpam-3397	327	3	has	have	VERB
ejpam-3397	327	4	six	six	NUM
ejpam-3397	327	5	different	different	ADJ
ejpam-3397	327	6	ideals	ideal	NOUN
ejpam-3397	327	7	i.e.	i.e.	X
ejpam-3397	327	8	i	i	X
ejpam-3397	327	9	=	=	PUNCT
ejpam-3397	327	10	{	{	PUNCT
ejpam-3397	327	11	0	0	NUM
ejpam-3397	327	12	,	,	PUNCT
ejpam-3397	327	13	a	a	PRON
ejpam-3397	327	14	}	}	PUNCT
ejpam-3397	327	15	,	,	PUNCT
ejpam-3397	327	16	j	j	PROPN
ejpam-3397	327	17	=	=	PUNCT
ejpam-3397	327	18	{	{	PUNCT
ejpam-3397	327	19	0	0	NUM
ejpam-3397	327	20	,	,	PUNCT
ejpam-3397	327	21	c	c	NOUN
ejpam-3397	327	22	}	}	PUNCT
ejpam-3397	327	23	,	,	PUNCT
ejpam-3397	327	24	k	k	PROPN
ejpam-3397	327	25	=	=	PUNCT
ejpam-3397	327	26	{	{	PUNCT
ejpam-3397	327	27	0	0	NUM
ejpam-3397	327	28	,	,	PUNCT
ejpam-3397	327	29	e	e	NOUN
ejpam-3397	327	30	,	,	PUNCT
ejpam-3397	327	31	c	c	NOUN
ejpam-3397	327	32	}	}	PUNCT
ejpam-3397	327	33	,	,	PUNCT
ejpam-3397	327	34	l	l	NOUN
ejpam-3397	327	35	=	=	SYM
ejpam-3397	327	36	{	{	PUNCT
ejpam-3397	327	37	0	0	NUM
ejpam-3397	327	38	,	,	PUNCT
ejpam-3397	327	39	a	a	DET
ejpam-3397	327	40	,	,	PUNCT
ejpam-3397	327	41	c	c	NOUN
ejpam-3397	327	42	}	}	PUNCT
ejpam-3397	327	43	,	,	PUNCT
ejpam-3397	327	44	m	m	VERB
ejpam-3397	327	45	=	=	PUNCT
ejpam-3397	327	46	{	{	PUNCT
ejpam-3397	327	47	0	0	NUM
ejpam-3397	327	48	,	,	PUNCT
ejpam-3397	327	49	a	a	DET
ejpam-3397	327	50	,	,	PUNCT
ejpam-3397	327	51	b	b	NOUN
ejpam-3397	327	52	}	}	PUNCT
ejpam-3397	327	53	,	,	PUNCT
ejpam-3397	327	54	and	and	CCONJ
ejpam-3397	327	55	references	reference	NOUN
ejpam-3397	327	56	552	552	NUM
ejpam-3397	327	57	n	n	NOUN
ejpam-3397	327	58	=	=	PRON
ejpam-3397	327	59	{	{	PUNCT
ejpam-3397	327	60	0	0	NUM
ejpam-3397	327	61	,	,	PUNCT
ejpam-3397	327	62	a	a	DET
ejpam-3397	327	63	,	,	PUNCT
ejpam-3397	327	64	b	b	NOUN
ejpam-3397	327	65	,	,	PUNCT
ejpam-3397	327	66	c	c	NOUN
ejpam-3397	327	67	}	}	PUNCT
ejpam-3397	327	68	.	.	PUNCT
ejpam-3397	328	1	now	now	ADV
ejpam-3397	328	2	,	,	PUNCT
ejpam-3397	328	3	iγi	iγi	PROPN
ejpam-3397	328	4	=	=	VERB
ejpam-3397	328	5	i	i	PROPN
ejpam-3397	328	6	,	,	PUNCT
ejpam-3397	328	7	iγj	iγj	PROPN
ejpam-3397	328	8	=	=	PUNCT
ejpam-3397	328	9	{	{	PUNCT
ejpam-3397	328	10	0	0	NUM
ejpam-3397	328	11	}	}	PUNCT
ejpam-3397	328	12	,	,	PUNCT
ejpam-3397	328	13	iγk	iγk	VERB
ejpam-3397	328	14	=	=	SYM
ejpam-3397	328	15	{	{	PUNCT
ejpam-3397	328	16	0	0	NUM
ejpam-3397	328	17	}	}	PUNCT
ejpam-3397	328	18	,	,	PUNCT
ejpam-3397	328	19	iγl	iγl	NOUN
ejpam-3397	329	1	=	=	SYM
ejpam-3397	329	2	i	i	PROPN
ejpam-3397	329	3	,	,	PUNCT
ejpam-3397	329	4	iγm	iγm	PROPN
ejpam-3397	330	1	=	=	SYM
ejpam-3397	330	2	i	i	PROPN
ejpam-3397	330	3	,	,	PUNCT
ejpam-3397	330	4	iγn	iγn	PROPN
ejpam-3397	330	5	=	=	SYM
ejpam-3397	330	6	i	i	PROPN
ejpam-3397	330	7	,	,	PUNCT
ejpam-3397	330	8	jγi	jγi	NOUN
ejpam-3397	330	9	=	=	SYM
ejpam-3397	330	10	{	{	PUNCT
ejpam-3397	330	11	0	0	NUM
ejpam-3397	330	12	}	}	PUNCT
ejpam-3397	330	13	,	,	PUNCT
ejpam-3397	330	14	jγj	jγj	NOUN
ejpam-3397	330	15	=	=	PUNCT
ejpam-3397	330	16	{	{	PUNCT
ejpam-3397	330	17	0	0	NUM
ejpam-3397	330	18	}	}	PUNCT
ejpam-3397	330	19	,	,	PUNCT
ejpam-3397	330	20	jγk	jγk	NOUN
ejpam-3397	330	21	=	=	SYM
ejpam-3397	330	22	{	{	PUNCT
ejpam-3397	330	23	0	0	NUM
ejpam-3397	330	24	}	}	PUNCT
ejpam-3397	330	25	,	,	PUNCT
ejpam-3397	330	26	jγl	jγl	NOUN
ejpam-3397	330	27	=	=	PUNCT
ejpam-3397	330	28	{	{	PUNCT
ejpam-3397	330	29	0	0	NUM
ejpam-3397	330	30	}	}	PUNCT
ejpam-3397	330	31	,	,	PUNCT
ejpam-3397	330	32	jγm	jγm	ADV
ejpam-3397	330	33	=	=	PUNCT
ejpam-3397	330	34	{	{	PUNCT
ejpam-3397	330	35	0	0	NUM
ejpam-3397	330	36	}	}	PUNCT
ejpam-3397	330	37	,	,	PUNCT
ejpam-3397	330	38	jγn	jγn	PROPN
ejpam-3397	330	39	=	=	PUNCT
ejpam-3397	330	40	{	{	PUNCT
ejpam-3397	330	41	0	0	NUM
ejpam-3397	330	42	}	}	PUNCT
ejpam-3397	330	43	,	,	PUNCT
ejpam-3397	330	44	kγi	kγi	NOUN
ejpam-3397	330	45	=	=	SYM
ejpam-3397	330	46	{	{	PUNCT
ejpam-3397	330	47	0	0	NUM
ejpam-3397	330	48	}	}	PUNCT
ejpam-3397	330	49	,	,	PUNCT
ejpam-3397	330	50	kγj	kγj	PROPN
ejpam-3397	330	51	=	=	SYM
ejpam-3397	330	52	j	j	PROPN
ejpam-3397	330	53	,	,	PUNCT
ejpam-3397	330	54	kγk	kγk	NOUN
ejpam-3397	330	55	=	=	SYM
ejpam-3397	330	56	k	k	NOUN
ejpam-3397	330	57	,	,	PUNCT
ejpam-3397	330	58	kγl	kγl	PROPN
ejpam-3397	330	59	=	=	SYM
ejpam-3397	330	60	j	j	PROPN
ejpam-3397	330	61	,	,	PUNCT
ejpam-3397	330	62	kγm	kγm	PROPN
ejpam-3397	331	1	=	=	SYM
ejpam-3397	331	2	j	j	PROPN
ejpam-3397	331	3	where	where	SCONJ
ejpam-3397	331	4	j	j	PROPN
ejpam-3397	331	5	⊆	⊆	NUM
ejpam-3397	331	6	k	k	PROPN
ejpam-3397	331	7	,	,	PUNCT
ejpam-3397	331	8	kγn	kγn	PROPN
ejpam-3397	331	9	=	=	SYM
ejpam-3397	331	10	j	j	PROPN
ejpam-3397	331	11	,	,	PUNCT
ejpam-3397	331	12	where	where	SCONJ
ejpam-3397	331	13	j	j	PROPN
ejpam-3397	331	14	⊆	⊆	NUM
ejpam-3397	331	15	k	k	PROPN
ejpam-3397	331	16	and	and	CCONJ
ejpam-3397	331	17	j	j	PROPN
ejpam-3397	331	18	⊆	⊆	NUM
ejpam-3397	331	19	n	n	CCONJ
ejpam-3397	331	20	,	,	PUNCT
ejpam-3397	331	21	lγi	lγi	NOUN
ejpam-3397	331	22	=	=	SYM
ejpam-3397	331	23	i	i	PROPN
ejpam-3397	331	24	,	,	PUNCT
ejpam-3397	331	25	lγj	lγj	PROPN
ejpam-3397	331	26	=	=	X
ejpam-3397	331	27	{	{	PUNCT
ejpam-3397	331	28	0	0	NUM
ejpam-3397	331	29	}	}	PUNCT
ejpam-3397	331	30	,	,	PUNCT
ejpam-3397	331	31	lγk	lγk	NOUN
ejpam-3397	331	32	=	=	PUNCT
ejpam-3397	331	33	{	{	PUNCT
ejpam-3397	331	34	0	0	NUM
ejpam-3397	331	35	}	}	PUNCT
ejpam-3397	331	36	,	,	PUNCT
ejpam-3397	331	37	lγl	lγl	PROPN
ejpam-3397	332	1	=	=	SYM
ejpam-3397	332	2	i	i	PROPN
ejpam-3397	332	3	,	,	PUNCT
ejpam-3397	332	4	where	where	SCONJ
ejpam-3397	332	5	i	i	PRON
ejpam-3397	332	6	⊆	⊆	NUM
ejpam-3397	332	7	l	l	NOUN
ejpam-3397	332	8	,	,	PUNCT
ejpam-3397	332	9	lγm	lγm	NOUN
ejpam-3397	332	10	=	=	PUNCT
ejpam-3397	333	1	i	i	PRON
ejpam-3397	333	2	where	where	SCONJ
ejpam-3397	333	3	i	i	PRON
ejpam-3397	333	4	⊆	⊆	NUM
ejpam-3397	333	5	l	l	NOUN
ejpam-3397	333	6	and	and	CCONJ
ejpam-3397	333	7	i	i	PRON
ejpam-3397	333	8	⊆	⊆	NUM
ejpam-3397	333	9	m	m	NOUN
ejpam-3397	333	10	,	,	PUNCT
ejpam-3397	333	11	lγn	lγn	NOUN
ejpam-3397	333	12	=	=	SYM
ejpam-3397	333	13	i	i	PROPN
ejpam-3397	333	14	,	,	PUNCT
ejpam-3397	333	15	where	where	SCONJ
ejpam-3397	333	16	i	i	PRON
ejpam-3397	333	17	⊆	⊆	NUM
ejpam-3397	333	18	l	l	NOUN
ejpam-3397	333	19	and	and	CCONJ
ejpam-3397	333	20	i	i	PRON
ejpam-3397	333	21	⊆	⊆	NUM
ejpam-3397	333	22	n	n	CCONJ
ejpam-3397	333	23	,	,	PUNCT
ejpam-3397	333	24	mγi	mγi	X
ejpam-3397	333	25	=	=	SYM
ejpam-3397	333	26	i	i	PROPN
ejpam-3397	333	27	,	,	PUNCT
ejpam-3397	333	28	mγj	mγj	NOUN
ejpam-3397	333	29	=	=	SYM
ejpam-3397	333	30	{	{	PUNCT
ejpam-3397	333	31	0	0	NUM
ejpam-3397	333	32	}	}	PUNCT
ejpam-3397	333	33	,	,	PUNCT
ejpam-3397	333	34	mγk	mγk	PROPN
ejpam-3397	333	35	=	=	SYM
ejpam-3397	333	36	{	{	PUNCT
ejpam-3397	333	37	0	0	NUM
ejpam-3397	333	38	}	}	PUNCT
ejpam-3397	333	39	,	,	PUNCT
ejpam-3397	333	40	mγl	mγl	NOUN
ejpam-3397	333	41	=	=	VERB
ejpam-3397	334	1	i	i	PRON
ejpam-3397	334	2	where	where	SCONJ
ejpam-3397	334	3	i	i	PRON
ejpam-3397	334	4	⊆	⊆	NUM
ejpam-3397	334	5	m	m	VERB
ejpam-3397	334	6	and	and	CCONJ
ejpam-3397	334	7	i	i	PRON
ejpam-3397	334	8	⊆	⊆	NUM
ejpam-3397	334	9	l	l	NOUN
ejpam-3397	334	10	,	,	PUNCT
ejpam-3397	334	11	mγm	mγm	NOUN
ejpam-3397	334	12	=	=	SYM
ejpam-3397	334	13	i	i	PROPN
ejpam-3397	334	14	,	,	PUNCT
ejpam-3397	334	15	where	where	SCONJ
ejpam-3397	334	16	i	i	PRON
ejpam-3397	334	17	⊆	⊆	NUM
ejpam-3397	334	18	m	m	NOUN
ejpam-3397	334	19	,	,	PUNCT
ejpam-3397	334	20	mγn	mγn	NOUN
ejpam-3397	334	21	=	=	SYM
ejpam-3397	334	22	i	i	PROPN
ejpam-3397	334	23	,	,	PUNCT
ejpam-3397	335	1	where	where	SCONJ
ejpam-3397	335	2	i	i	PRON
ejpam-3397	335	3	⊆	⊆	NUM
ejpam-3397	335	4	m	m	VERB
ejpam-3397	335	5	and	and	CCONJ
ejpam-3397	335	6	i	i	PRON
ejpam-3397	335	7	⊆	⊆	NUM
ejpam-3397	335	8	n.	n.	NOUN
ejpam-3397	335	9	hence	hence	ADV
ejpam-3397	335	10	,	,	PUNCT
ejpam-3397	335	11	we	we	PRON
ejpam-3397	335	12	can	can	AUX
ejpam-3397	335	13	easily	easily	ADV
ejpam-3397	335	14	check	check	VERB
ejpam-3397	335	15	that	that	SCONJ
ejpam-3397	335	16	every	every	DET
ejpam-3397	335	17	ideal	ideal	NOUN
ejpam-3397	335	18	of	of	ADP
ejpam-3397	335	19	r	r	NOUN
ejpam-3397	335	20	is	be	AUX
ejpam-3397	335	21	weakly	weakly	ADV
ejpam-3397	335	22	primary	primary	ADJ
ejpam-3397	335	23	.	.	PUNCT
ejpam-3397	336	1	proposition	proposition	NOUN
ejpam-3397	336	2	12	12	NUM
ejpam-3397	336	3	.	.	PUNCT
ejpam-3397	336	4	suppose	suppose	VERB
ejpam-3397	336	5	that	that	SCONJ
ejpam-3397	336	6	every	every	DET
ejpam-3397	336	7	ideal	ideal	NOUN
ejpam-3397	336	8	of	of	ADP
ejpam-3397	336	9	a	a	DET
ejpam-3397	336	10	γ	γ	NOUN
ejpam-3397	336	11	-	-	PUNCT
ejpam-3397	336	12	seminearring	seminearre	VERB
ejpam-3397	336	13	r	r	NOUN
ejpam-3397	336	14	is	be	AUX
ejpam-3397	336	15	a	a	DET
ejpam-3397	336	16	weakly	weakly	ADJ
ejpam-3397	336	17	primary	primary	NOUN
ejpam-3397	336	18	.	.	PUNCT
ejpam-3397	337	1	if	if	SCONJ
ejpam-3397	337	2	m1	m1	PROPN
ejpam-3397	337	3	and	and	CCONJ
ejpam-3397	337	4	m2	m2	PROPN
ejpam-3397	337	5	are	be	AUX
ejpam-3397	337	6	two	two	NUM
ejpam-3397	337	7	maximal	maximal	ADJ
ejpam-3397	337	8	ideals	ideal	NOUN
ejpam-3397	337	9	of	of	ADP
ejpam-3397	337	10	r	r	NOUN
ejpam-3397	337	11	then	then	ADV
ejpam-3397	337	12	either	either	CCONJ
ejpam-3397	337	13	m1γm2	m1γm2	ADV
ejpam-3397	337	14	=	=	SYM
ejpam-3397	337	15	0	0	NUM
ejpam-3397	337	16	or	or	CCONJ
ejpam-3397	337	17	m1γm2	m1γm2	PROPN
ejpam-3397	337	18	=	=	SYM
ejpam-3397	337	19	n	n	PROPN
ejpam-3397	337	20	=	=	SYM
ejpam-3397	337	21	m1	m1	PROPN
ejpam-3397	337	22	∩m2	∩m2	PROPN
ejpam-3397	337	23	.	.	PROPN
ejpam-3397	337	24	example	example	NOUN
ejpam-3397	338	1	15	15	NUM
ejpam-3397	338	2	.	.	PUNCT
ejpam-3397	339	1	let	let	VERB
ejpam-3397	339	2	r	r	NOUN
ejpam-3397	339	3	=	=	SYM
ejpam-3397	339	4	{	{	PUNCT
ejpam-3397	339	5	0	0	NUM
ejpam-3397	339	6	,	,	PUNCT
ejpam-3397	339	7	1	1	NUM
ejpam-3397	339	8	,	,	PUNCT
ejpam-3397	339	9	e	e	NOUN
ejpam-3397	339	10	,	,	PUNCT
ejpam-3397	339	11	a	a	DET
ejpam-3397	339	12	,	,	PUNCT
ejpam-3397	339	13	b	b	NOUN
ejpam-3397	339	14	,	,	PUNCT
ejpam-3397	339	15	c	c	AUX
ejpam-3397	339	16	}	}	PUNCT
ejpam-3397	339	17	be	be	AUX
ejpam-3397	339	18	a	a	DET
ejpam-3397	339	19	γseminearring	γseminearring	NOUN
ejpam-3397	339	20	with	with	ADP
ejpam-3397	339	21	γ	γ	X
ejpam-3397	339	22	=	=	SYM
ejpam-3397	339	23	{	{	PUNCT
ejpam-3397	339	24	1	1	NUM
ejpam-3397	339	25	,	,	PUNCT
ejpam-3397	339	26	α	α	NOUN
ejpam-3397	339	27	}	}	PUNCT
ejpam-3397	339	28	.	.	PUNCT
ejpam-3397	340	1	let	let	VERB
ejpam-3397	340	2	i	i	PRON
ejpam-3397	340	3	=	=	PUNCT
ejpam-3397	340	4	{	{	PUNCT
ejpam-3397	340	5	0	0	NUM
ejpam-3397	340	6	,	,	PUNCT
ejpam-3397	340	7	a	a	DET
ejpam-3397	340	8	,	,	PUNCT
ejpam-3397	340	9	b	b	NOUN
ejpam-3397	340	10	,	,	PUNCT
ejpam-3397	340	11	c	c	NOUN
ejpam-3397	340	12	}	}	PUNCT
ejpam-3397	340	13	and	and	CCONJ
ejpam-3397	340	14	j	j	PROPN
ejpam-3397	340	15	=	=	PUNCT
ejpam-3397	340	16	{	{	PUNCT
ejpam-3397	340	17	0	0	NUM
ejpam-3397	340	18	,	,	PUNCT
ejpam-3397	340	19	e	e	NOUN
ejpam-3397	340	20	,	,	PUNCT
ejpam-3397	340	21	c	c	AUX
ejpam-3397	340	22	}	}	PUNCT
ejpam-3397	340	23	be	be	AUX
ejpam-3397	340	24	the	the	DET
ejpam-3397	340	25	two	two	NUM
ejpam-3397	340	26	maximal	maximal	ADJ
ejpam-3397	340	27	ideals	ideal	NOUN
ejpam-3397	340	28	of	of	ADP
ejpam-3397	340	29	r.	r.	PROPN
ejpam-3397	340	30	clearly	clearly	ADV
ejpam-3397	340	31	iγj	iγj	VERB
ejpam-3397	340	32	=	=	PUNCT
ejpam-3397	340	33	{	{	PUNCT
ejpam-3397	340	34	0	0	NUM
ejpam-3397	340	35	}	}	PUNCT
ejpam-3397	340	36	and	and	CCONJ
ejpam-3397	340	37	jγi	jγi	NUM
ejpam-3397	340	38	=	=	SYM
ejpam-3397	340	39	{	{	PUNCT
ejpam-3397	340	40	0	0	NUM
ejpam-3397	340	41	,	,	PUNCT
ejpam-3397	340	42	c	c	NOUN
ejpam-3397	340	43	}	}	PUNCT
ejpam-3397	340	44	=	=	SYM
ejpam-3397	341	1	i	i	PROPN
ejpam-3397	341	2	∩	∩	PROPN
ejpam-3397	341	3	j.	j.	PROPN
ejpam-3397	341	4	references	reference	NOUN
ejpam-3397	341	5	[	[	X
ejpam-3397	341	6	1	1	NUM
ejpam-3397	341	7	]	]	PUNCT
ejpam-3397	341	8	w.	w.	PROPN
ejpam-3397	341	9	g.	g.	PROPN
ejpam-3397	341	10	van	van	PROPN
ejpam-3397	341	11	hoorn	hoorn	PROPN
ejpam-3397	341	12	,	,	PUNCT
ejpam-3397	341	13	and	and	CCONJ
ejpam-3397	341	14	b.	b.	PROPN
ejpam-3397	341	15	van	van	PROPN
ejpam-3397	341	16	rootselaar	rootselaar	NOUN
ejpam-3397	341	17	,	,	PUNCT
ejpam-3397	341	18	fundamental	fundamental	ADJ
ejpam-3397	341	19	notions	notion	NOUN
ejpam-3397	341	20	in	in	ADP
ejpam-3397	341	21	the	the	DET
ejpam-3397	341	22	theory	theory	NOUN
ejpam-3397	341	23	of	of	ADP
ejpam-3397	341	24	seminearrings	seminearring	NOUN
ejpam-3397	341	25	,	,	PUNCT
ejpam-3397	341	26	compositio	compositio	NOUN
ejpam-3397	341	27	math	math	NOUN
ejpam-3397	341	28	.	.	PUNCT
ejpam-3397	342	1	18	18	NUM
ejpam-3397	342	2	(	(	PUNCT
ejpam-3397	342	3	1967	1967	NUM
ejpam-3397	342	4	)	)	PUNCT
ejpam-3397	342	5	,	,	PUNCT
ejpam-3397	342	6	65	65	NUM
ejpam-3397	342	7	-	-	SYM
ejpam-3397	342	8	78	78	NUM
ejpam-3397	342	9	.	.	PUNCT
ejpam-3397	343	1	[	[	X
ejpam-3397	343	2	2	2	X
ejpam-3397	343	3	]	]	X
ejpam-3397	343	4	y.	y.	PROPN
ejpam-3397	343	5	b.	b.	PROPN
ejpam-3397	343	6	jun	jun	PROPN
ejpam-3397	343	7	and	and	CCONJ
ejpam-3397	343	8	k.	k.	PROPN
ejpam-3397	343	9	h.	h.	PROPN
ejpam-3397	343	10	kim	kim	PROPN
ejpam-3397	343	11	,	,	PUNCT
ejpam-3397	343	12	on	on	ADP
ejpam-3397	343	13	structures	structure	NOUN
ejpam-3397	343	14	of	of	ADP
ejpam-3397	343	15	gamma	gamma	NOUN
ejpam-3397	343	16	-	-	PUNCT
ejpam-3397	343	17	seminear	seminear	NOUN
ejpam-3397	343	18	-	-	PUNCT
ejpam-3397	343	19	rings	ring	NOUN
ejpam-3397	343	20	,	,	PUNCT
ejpam-3397	343	21	(	(	PUNCT
ejpam-3397	343	22	submitted	submit	VERB
ejpam-3397	343	23	)	)	PUNCT
ejpam-3397	344	1	[	[	X
ejpam-3397	344	2	3	3	X
ejpam-3397	344	3	]	]	PUNCT
ejpam-3397	344	4	k.	k.	PROPN
ejpam-3397	344	5	h.	h.	PROPN
ejpam-3397	344	6	kim	kim	PROPN
ejpam-3397	344	7	,	,	PUNCT
ejpam-3397	344	8	on	on	ADP
ejpam-3397	344	9	prime	prime	ADJ
ejpam-3397	344	10	and	and	CCONJ
ejpam-3397	344	11	semiprime	semiprime	NOUN
ejpam-3397	344	12	ideals	ideal	NOUN
ejpam-3397	344	13	in	in	ADP
ejpam-3397	344	14	gamma	gamma	NOUN
ejpam-3397	344	15	-	-	PUNCT
ejpam-3397	344	16	seminearrings	seminearring	NOUN
ejpam-3397	344	17	,	,	PUNCT
ejpam-3397	344	18	sci	sci	PROPN
ejpam-3397	344	19	.	.	PROPN
ejpam-3397	344	20	math	math	PROPN
ejpam-3397	344	21	.	.	PUNCT
ejpam-3397	345	1	jap	jap	PROPN
ejpam-3397	345	2	.	.	PROPN
ejpam-3397	346	1	online	online	PROPN
ejpam-3397	346	2	,	,	PUNCT
ejpam-3397	346	3	4	4	NUM
ejpam-3397	346	4	,	,	PUNCT
ejpam-3397	346	5	(	(	PUNCT
ejpam-3397	346	6	2001	2001	NUM
ejpam-3397	346	7	)	)	PUNCT
ejpam-3397	346	8	,	,	PUNCT
ejpam-3397	346	9	885	885	NUM
ejpam-3397	346	10	-	-	SYM
ejpam-3397	346	11	889	889	NUM
ejpam-3397	346	12	.	.	PUNCT
ejpam-3397	347	1	[	[	X
ejpam-3397	347	2	4	4	X
ejpam-3397	347	3	]	]	PUNCT
ejpam-3397	347	4	k.	k.	PROPN
ejpam-3397	348	1	v.	v.	PROPN
ejpam-3397	348	2	krishna	krishna	PROPN
ejpam-3397	348	3	and	and	CCONJ
ejpam-3397	348	4	n.	n.	PROPN
ejpam-3397	348	5	chatterjee	chatterjee	PROPN
ejpam-3397	348	6	,	,	PUNCT
ejpam-3397	348	7	a	a	DET
ejpam-3397	348	8	necessary	necessary	ADJ
ejpam-3397	348	9	condition	condition	NOUN
ejpam-3397	348	10	to	to	PART
ejpam-3397	348	11	test	test	VERB
ejpam-3397	348	12	the	the	DET
ejpam-3397	348	13	minimality	minimality	NOUN
ejpam-3397	348	14	of	of	ADP
ejpam-3397	348	15	generalized	generalized	ADJ
ejpam-3397	348	16	linear	linear	ADJ
ejpam-3397	348	17	sequential	sequential	ADJ
ejpam-3397	348	18	machines	machine	NOUN
ejpam-3397	348	19	using	use	VERB
ejpam-3397	348	20	the	the	DET
ejpam-3397	348	21	theory	theory	NOUN
ejpam-3397	348	22	of	of	ADP
ejpam-3397	348	23	near	near	ADJ
ejpam-3397	348	24	-	-	PUNCT
ejpam-3397	348	25	semirings	semiring	NOUN
ejpam-3397	348	26	,	,	PUNCT
ejpam-3397	348	27	algebra	algebra	NOUN
ejpam-3397	348	28	and	and	CCONJ
ejpam-3397	348	29	discrete	discrete	ADJ
ejpam-3397	348	30	mathematics	mathematic	NOUN
ejpam-3397	348	31	.	.	PUNCT
ejpam-3397	349	1	3	3	NUM
ejpam-3397	349	2	(	(	PUNCT
ejpam-3397	349	3	2005	2005	NUM
ejpam-3397	349	4	)	)	PUNCT
ejpam-3397	349	5	,	,	PUNCT
ejpam-3397	349	6	30	30	NUM
ejpam-3397	349	7	–	–	SYM
ejpam-3397	349	8	45	45	NUM
ejpam-3397	349	9	.	.	PUNCT
ejpam-3397	350	1	[	[	X
ejpam-3397	350	2	5	5	X
ejpam-3397	350	3	]	]	PUNCT
ejpam-3397	350	4	h.	h.	PROPN
ejpam-3397	350	5	j.	j.	PROPN
ejpam-3397	350	6	weinert	weinert	PROPN
ejpam-3397	350	7	,	,	PUNCT
ejpam-3397	350	8	seminear	seminear	NOUN
ejpam-3397	350	9	-	-	PUNCT
ejpam-3397	350	10	rings	ring	NOUN
ejpam-3397	350	11	,	,	PUNCT
ejpam-3397	350	12	seminearfieds	seminearfied	NOUN
ejpam-3397	350	13	and	and	CCONJ
ejpam-3397	350	14	their	their	PRON
ejpam-3397	350	15	semigroup	semigroup	ADJ
ejpam-3397	350	16	theoretic	theoretic	ADJ
ejpam-3397	350	17	background	background	NOUN
ejpam-3397	350	18	,	,	PUNCT
ejpam-3397	350	19	semigroup	semigroup	PROPN
ejpam-3397	350	20	forum	forum	PROPN
ejpam-3397	350	21	24	24	NUM
ejpam-3397	350	22	(	(	PUNCT
ejpam-3397	350	23	1982	1982	NUM
ejpam-3397	350	24	)	)	PUNCT
ejpam-3397	350	25	,	,	PUNCT
ejpam-3397	350	26	235	235	NUM
ejpam-3397	350	27	-	-	SYM
ejpam-3397	350	28	254	254	NUM
ejpam-3397	350	29	.	.	PUNCT