id	sid	tid	token	lemma	pos
ejpam-3193	1	1	european	european	PROPN
ejpam-3193	1	2	journal	journal	PROPN
ejpam-3193	1	3	of	of	ADP
ejpam-3193	1	4	pure	pure	ADJ
ejpam-3193	1	5	and	and	CCONJ
ejpam-3193	1	6	applied	apply	VERB
ejpam-3193	1	7	mathematics	mathematic	NOUN
ejpam-3193	1	8	vol	vol	NOUN
ejpam-3193	1	9	.	.	PUNCT
ejpam-3193	2	1	11	11	NUM
ejpam-3193	2	2	,	,	PUNCT
ejpam-3193	2	3	no	no	INTJ
ejpam-3193	2	4	.	.	NOUN
ejpam-3193	2	5	2	2	NUM
ejpam-3193	2	6	,	,	PUNCT
ejpam-3193	2	7	2018	2018	NUM
ejpam-3193	2	8	,	,	PUNCT
ejpam-3193	2	9	449	449	NUM
ejpam-3193	2	10	-	-	SYM
ejpam-3193	2	11	456	456	NUM
ejpam-3193	2	12	issn	issn	PROPN
ejpam-3193	2	13	1307	1307	NUM
ejpam-3193	2	14	-	-	SYM
ejpam-3193	2	15	5543	5543	NUM
ejpam-3193	2	16	–	–	PUNCT
ejpam-3193	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3193	2	18	published	publish	VERB
ejpam-3193	2	19	by	by	ADP
ejpam-3193	2	20	new	new	PROPN
ejpam-3193	2	21	york	york	PROPN
ejpam-3193	2	22	business	business	PROPN
ejpam-3193	2	23	global	global	ADJ
ejpam-3193	2	24	almost	almost	ADV
ejpam-3193	2	25	prime	prime	ADJ
ejpam-3193	2	26	ideal	ideal	NOUN
ejpam-3193	2	27	in	in	ADP
ejpam-3193	2	28	gamma	gamma	PROPN
ejpam-3193	2	29	near	near	ADP
ejpam-3193	2	30	ring	ring	PROPN
ejpam-3193	2	31	waheed	waheed	PROPN
ejpam-3193	2	32	ahmad	ahmad	PROPN
ejpam-3193	2	33	khan1	khan1	PROPN
ejpam-3193	2	34	,	,	PUNCT
ejpam-3193	2	35	adnan	adnan	PROPN
ejpam-3193	2	36	muhammad1	muhammad1	PROPN
ejpam-3193	2	37	,	,	PUNCT
ejpam-3193	2	38	abdelghani	abdelghani	PROPN
ejpam-3193	2	39	taouti2,∗	taouti2,∗	PROPN
ejpam-3193	2	40	,	,	PUNCT
ejpam-3193	2	41	jameel	jameel	PROPN
ejpam-3193	2	42	maki3	maki3	PROPN
ejpam-3193	2	43	1	1	NUM
ejpam-3193	2	44	university	university	NOUN
ejpam-3193	2	45	of	of	ADP
ejpam-3193	2	46	education	education	NOUN
ejpam-3193	2	47	lahore	lahore	NOUN
ejpam-3193	2	48	,	,	PUNCT
ejpam-3193	2	49	attock	attock	PROPN
ejpam-3193	2	50	campus	campus	PROPN
ejpam-3193	2	51	,	,	PUNCT
ejpam-3193	2	52	pakistan	pakistan	PROPN
ejpam-3193	2	53	2	2	NUM
ejpam-3193	2	54	ets	et	NOUN
ejpam-3193	2	55	-	-	PUNCT
ejpam-3193	2	56	maths	math	NOUN
ejpam-3193	2	57	and	and	CCONJ
ejpam-3193	2	58	ns	ns	ADJ
ejpam-3193	2	59	engineering	engineering	NOUN
ejpam-3193	2	60	division	division	NOUN
ejpam-3193	2	61	,	,	PUNCT
ejpam-3193	2	62	hct	hct	PROPN
ejpam-3193	2	63	,	,	PUNCT
ejpam-3193	2	64	university	university	NOUN
ejpam-3193	2	65	city	city	PROPN
ejpam-3193	2	66	p.o	p.o	PROPN
ejpam-3193	2	67	.	.	PROPN
ejpam-3193	2	68	box	box	PROPN
ejpam-3193	2	69	7947	7947	NUM
ejpam-3193	2	70	,	,	PUNCT
ejpam-3193	2	71	sharjah	sharjah	PROPN
ejpam-3193	2	72	,	,	PUNCT
ejpam-3193	2	73	united	united	PROPN
ejpam-3193	2	74	arab	arab	PROPN
ejpam-3193	2	75	emirates	emirates	PROPN
ejpam-3193	2	76	3	3	NUM
ejpam-3193	2	77	department	department	NOUN
ejpam-3193	2	78	of	of	ADP
ejpam-3193	2	79	mathematics	mathematic	NOUN
ejpam-3193	2	80	,	,	PUNCT
ejpam-3193	2	81	r.i.t	r.i.t	NOUN
ejpam-3193	2	82	dubai	dubai	NOUN
ejpam-3193	2	83	.	.	PUNCT
ejpam-3193	3	1	p.	p.	NOUN
ejpam-3193	3	2	o.	o.	PROPN
ejpam-3193	3	3	box	box	PROPN
ejpam-3193	3	4	341055	341055	NUM
ejpam-3193	3	5	,	,	PUNCT
ejpam-3193	3	6	dubai	dubai	PROPN
ejpam-3193	3	7	,	,	PUNCT
ejpam-3193	3	8	united	united	PROPN
ejpam-3193	3	9	arab	arab	PROPN
ejpam-3193	3	10	emirates	emirates	PROPN
ejpam-3193	3	11	abstract	abstract	ADJ
ejpam-3193	3	12	.	.	PUNCT
ejpam-3193	4	1	in	in	ADP
ejpam-3193	4	2	this	this	DET
ejpam-3193	4	3	manuscript	manuscript	NOUN
ejpam-3193	4	4	we	we	PRON
ejpam-3193	4	5	introduce	introduce	VERB
ejpam-3193	4	6	the	the	DET
ejpam-3193	4	7	notion	notion	NOUN
ejpam-3193	4	8	of	of	ADP
ejpam-3193	4	9	almost	almost	ADV
ejpam-3193	4	10	prime	prime	ADJ
ejpam-3193	4	11	ideals	ideal	NOUN
ejpam-3193	4	12	in	in	ADP
ejpam-3193	4	13	γ	γ	NOUN
ejpam-3193	4	14	-	-	PUNCT
ejpam-3193	4	15	near	near	ADP
ejpam-3193	4	16	-	-	PUNCT
ejpam-3193	4	17	rings	ring	NOUN
ejpam-3193	4	18	along	along	ADP
ejpam-3193	4	19	with	with	ADP
ejpam-3193	4	20	few	few	ADJ
ejpam-3193	4	21	of	of	ADP
ejpam-3193	4	22	their	their	PRON
ejpam-3193	4	23	characterizations	characterization	NOUN
ejpam-3193	4	24	.	.	PUNCT
ejpam-3193	5	1	we	we	PRON
ejpam-3193	5	2	also	also	ADV
ejpam-3193	5	3	present	present	VERB
ejpam-3193	5	4	the	the	DET
ejpam-3193	5	5	interesting	interesting	ADJ
ejpam-3193	5	6	relations	relation	NOUN
ejpam-3193	5	7	among	among	ADP
ejpam-3193	5	8	almost	almost	ADV
ejpam-3193	5	9	prime	prime	ADJ
ejpam-3193	5	10	,	,	PUNCT
ejpam-3193	5	11	prime	prime	ADJ
ejpam-3193	5	12	and	and	CCONJ
ejpam-3193	5	13	primary	primary	ADJ
ejpam-3193	5	14	ideal	ideal	NOUN
ejpam-3193	5	15	in	in	ADP
ejpam-3193	5	16	γ	γ	NOUN
ejpam-3193	5	17	-	-	PUNCT
ejpam-3193	5	18	nearrings	nearring	NOUN
ejpam-3193	5	19	.	.	PUNCT
ejpam-3193	6	1	2010	2010	NUM
ejpam-3193	6	2	mathematics	mathematic	NOUN
ejpam-3193	6	3	subject	subject	NOUN
ejpam-3193	6	4	classifications	classification	NOUN
ejpam-3193	6	5	:	:	PUNCT
ejpam-3193	6	6	13a05	13a05	NUM
ejpam-3193	6	7	,	,	PUNCT
ejpam-3193	6	8	13a18	13a18	NUM
ejpam-3193	6	9	,	,	PUNCT
ejpam-3193	6	10	12j20	12j20	NUM
ejpam-3193	6	11	key	key	ADJ
ejpam-3193	6	12	words	word	NOUN
ejpam-3193	6	13	and	and	CCONJ
ejpam-3193	6	14	phrases	phrase	NOUN
ejpam-3193	6	15	:	:	PUNCT
ejpam-3193	6	16	γ	γ	X
ejpam-3193	6	17	-	-	PUNCT
ejpam-3193	6	18	near	near	ADP
ejpam-3193	6	19	-	-	PUNCT
ejpam-3193	6	20	rings	ring	NOUN
ejpam-3193	6	21	,	,	PUNCT
ejpam-3193	6	22	prime	prime	ADJ
ejpam-3193	6	23	ideals	ideal	NOUN
ejpam-3193	6	24	,	,	PUNCT
ejpam-3193	6	25	almost	almost	ADV
ejpam-3193	6	26	prime	prime	ADJ
ejpam-3193	6	27	ideals	ideal	NOUN
ejpam-3193	6	28	.	.	PUNCT
ejpam-3193	7	1	1	1	X
ejpam-3193	7	2	.	.	X
ejpam-3193	7	3	introduction	introduction	NOUN
ejpam-3193	7	4	and	and	CCONJ
ejpam-3193	7	5	preliminaries	preliminary	NOUN
ejpam-3193	7	6	recently	recently	ADV
ejpam-3193	7	7	,	,	PUNCT
ejpam-3193	7	8	the	the	DET
ejpam-3193	7	9	generalization	generalization	NOUN
ejpam-3193	7	10	of	of	ADP
ejpam-3193	7	11	prime	prime	ADJ
ejpam-3193	7	12	ideal	ideal	NOUN
ejpam-3193	7	13	i.e.	i.e.	X
ejpam-3193	7	14	,	,	PUNCT
ejpam-3193	7	15	almost	almost	ADV
ejpam-3193	7	16	prime	prime	ADJ
ejpam-3193	7	17	ideal	ideal	NOUN
ejpam-3193	7	18	in	in	ADP
ejpam-3193	7	19	commutative	commutative	ADJ
ejpam-3193	7	20	rings	ring	NOUN
ejpam-3193	7	21	has	have	AUX
ejpam-3193	7	22	been	be	AUX
ejpam-3193	7	23	introduced	introduce	VERB
ejpam-3193	7	24	and	and	CCONJ
ejpam-3193	7	25	discussed	discuss	VERB
ejpam-3193	7	26	by	by	ADP
ejpam-3193	7	27	srikant	srikant	ADJ
ejpam-3193	7	28	m.	m.	NOUN
ejpam-3193	7	29	bhatwadekar	bhatwadekar	PROPN
ejpam-3193	7	30	and	and	CCONJ
ejpam-3193	7	31	pramod	pramod	PROPN
ejpam-3193	7	32	k.	k.	PROPN
ejpam-3193	7	33	sharma	sharma	PROPN
ejpam-3193	7	34	(	(	PUNCT
ejpam-3193	7	35	see	see	VERB
ejpam-3193	7	36	[	[	X
ejpam-3193	7	37	3	3	NUM
ejpam-3193	7	38	]	]	NUM
ejpam-3193	7	39	)	)	PUNCT
ejpam-3193	7	40	.	.	PUNCT
ejpam-3193	8	1	following	follow	VERB
ejpam-3193	8	2	[	[	X
ejpam-3193	8	3	3	3	NUM
ejpam-3193	8	4	]	]	PUNCT
ejpam-3193	8	5	,	,	PUNCT
ejpam-3193	8	6	an	an	DET
ejpam-3193	8	7	ideal	ideal	ADJ
ejpam-3193	8	8	i	i	PRON
ejpam-3193	8	9	of	of	ADP
ejpam-3193	8	10	a	a	DET
ejpam-3193	8	11	ring	ring	NOUN
ejpam-3193	8	12	r	r	NOUN
ejpam-3193	8	13	is	be	AUX
ejpam-3193	8	14	said	say	VERB
ejpam-3193	8	15	to	to	PART
ejpam-3193	8	16	be	be	AUX
ejpam-3193	8	17	an	an	DET
ejpam-3193	8	18	almost	almost	ADV
ejpam-3193	8	19	prime	prime	ADJ
ejpam-3193	8	20	if	if	SCONJ
ejpam-3193	8	21	for	for	ADP
ejpam-3193	8	22	all	all	DET
ejpam-3193	8	23	a	a	PRON
ejpam-3193	8	24	,	,	PUNCT
ejpam-3193	8	25	b	b	X
ejpam-3193	8	26	∈	∈	NOUN
ejpam-3193	8	27	r	r	NOUN
ejpam-3193	8	28	implies	imply	VERB
ejpam-3193	8	29	ab	ab	PROPN
ejpam-3193	8	30	∈	∈	PROPN
ejpam-3193	9	1	i	i	PRON
ejpam-3193	9	2	−	−	PROPN
ejpam-3193	9	3	i2	i2	PROPN
ejpam-3193	9	4	either	either	CCONJ
ejpam-3193	9	5	a	a	DET
ejpam-3193	9	6	∈	∈	ADJ
ejpam-3193	9	7	i	i	NOUN
ejpam-3193	9	8	or	or	CCONJ
ejpam-3193	9	9	b	b	PROPN
ejpam-3193	9	10	∈	∈	PROPN
ejpam-3193	9	11	i.	i.	NOUN
ejpam-3193	9	12	all	all	ADV
ejpam-3193	9	13	prime	prime	ADJ
ejpam-3193	9	14	and	and	CCONJ
ejpam-3193	9	15	idempotent	idempotent	ADJ
ejpam-3193	9	16	ideals	ideal	NOUN
ejpam-3193	9	17	are	be	AUX
ejpam-3193	9	18	almost	almost	ADV
ejpam-3193	9	19	prime	prime	ADJ
ejpam-3193	9	20	[	[	X
ejpam-3193	9	21	3	3	NUM
ejpam-3193	9	22	]	]	PUNCT
ejpam-3193	9	23	.	.	PUNCT
ejpam-3193	10	1	it	it	PRON
ejpam-3193	10	2	has	have	AUX
ejpam-3193	10	3	been	be	AUX
ejpam-3193	10	4	proved	prove	VERB
ejpam-3193	10	5	that	that	SCONJ
ejpam-3193	10	6	every	every	DET
ejpam-3193	10	7	almost	almost	ADV
ejpam-3193	10	8	prime	prime	ADJ
ejpam-3193	10	9	ideal	ideal	NOUN
ejpam-3193	10	10	in	in	ADP
ejpam-3193	10	11	a	a	DET
ejpam-3193	10	12	noetherian	noetherian	ADJ
ejpam-3193	10	13	domain	domain	NOUN
ejpam-3193	10	14	r	r	NOUN
ejpam-3193	10	15	is	be	AUX
ejpam-3193	10	16	primary	primary	ADJ
ejpam-3193	10	17	[	[	X
ejpam-3193	10	18	3	3	NUM
ejpam-3193	10	19	]	]	PUNCT
ejpam-3193	10	20	.	.	PUNCT
ejpam-3193	11	1	further	far	ADV
ejpam-3193	11	2	to	to	ADP
ejpam-3193	11	3	this	this	PRON
ejpam-3193	11	4	,	,	PUNCT
ejpam-3193	11	5	almost	almost	ADV
ejpam-3193	11	6	primary	primary	ADJ
ejpam-3193	11	7	ideals	ideal	NOUN
ejpam-3193	11	8	in	in	ADP
ejpam-3193	11	9	rings	ring	NOUN
ejpam-3193	11	10	have	have	AUX
ejpam-3193	11	11	been	be	AUX
ejpam-3193	11	12	introduced	introduce	VERB
ejpam-3193	11	13	by	by	ADP
ejpam-3193	11	14	a.	a.	PROPN
ejpam-3193	11	15	k.	k.	PROPN
ejpam-3193	11	16	jabbar	jabbar	PROPN
ejpam-3193	11	17	and	and	CCONJ
ejpam-3193	11	18	c.	c.	PROPN
ejpam-3193	11	19	a.	a.	PROPN
ejpam-3193	11	20	ahmed	ahme	VERB
ejpam-3193	11	21	in	in	ADP
ejpam-3193	11	22	[	[	X
ejpam-3193	11	23	12	12	NUM
ejpam-3193	11	24	]	]	X
ejpam-3193	11	25	,	,	PUNCT
ejpam-3193	11	26	a	a	DET
ejpam-3193	11	27	proper	proper	ADJ
ejpam-3193	11	28	ideal	ideal	NOUN
ejpam-3193	11	29	a	a	PRON
ejpam-3193	11	30	of	of	ADP
ejpam-3193	11	31	a	a	DET
ejpam-3193	11	32	ring	ring	NOUN
ejpam-3193	11	33	r	r	NOUN
ejpam-3193	11	34	is	be	AUX
ejpam-3193	11	35	an	an	DET
ejpam-3193	11	36	almost	almost	ADV
ejpam-3193	11	37	primary	primary	ADJ
ejpam-3193	11	38	ideal	ideal	NOUN
ejpam-3193	11	39	if	if	SCONJ
ejpam-3193	11	40	for	for	ADP
ejpam-3193	11	41	a	a	DET
ejpam-3193	11	42	,	,	PUNCT
ejpam-3193	11	43	b	b	X
ejpam-3193	11	44	∈	∈	NOUN
ejpam-3193	11	45	r	r	NOUN
ejpam-3193	11	46	such	such	ADJ
ejpam-3193	11	47	that	that	SCONJ
ejpam-3193	11	48	ab	ab	PROPN
ejpam-3193	11	49	∈	∈	PROPN
ejpam-3193	11	50	a−a2	a−a2	VERB
ejpam-3193	11	51	,	,	PUNCT
ejpam-3193	11	52	then	then	ADV
ejpam-3193	11	53	a	a	DET
ejpam-3193	11	54	∈	∈	PROPN
ejpam-3193	11	55	a	a	PRON
ejpam-3193	11	56	or	or	CCONJ
ejpam-3193	11	57	b	b	NOUN
ejpam-3193	11	58	∈	∈	PROPN
ejpam-3193	11	59	a	a	PRON
ejpam-3193	11	60	,	,	PUNCT
ejpam-3193	11	61	for	for	ADP
ejpam-3193	11	62	some	some	DET
ejpam-3193	11	63	positive	positive	ADJ
ejpam-3193	11	64	integer	integer	NOUN
ejpam-3193	11	65	n	n	PROPN
ejpam-3193	11	66	[	[	X
ejpam-3193	11	67	12	12	NUM
ejpam-3193	11	68	]	]	PUNCT
ejpam-3193	11	69	.	.	PUNCT
ejpam-3193	12	1	in	in	ADP
ejpam-3193	12	2	[	[	X
ejpam-3193	12	3	12	12	NUM
ejpam-3193	12	4	]	]	PUNCT
ejpam-3193	12	5	,	,	PUNCT
ejpam-3193	12	6	authors	author	NOUN
ejpam-3193	12	7	have	have	AUX
ejpam-3193	12	8	also	also	ADV
ejpam-3193	12	9	discussed	discuss	VERB
ejpam-3193	12	10	several	several	ADJ
ejpam-3193	12	11	characterizations	characterization	NOUN
ejpam-3193	12	12	of	of	ADP
ejpam-3193	12	13	almost	almost	ADV
ejpam-3193	12	14	primary	primary	ADJ
ejpam-3193	12	15	ideals	ideal	NOUN
ejpam-3193	12	16	.	.	PUNCT
ejpam-3193	13	1	it	it	PRON
ejpam-3193	13	2	is	be	AUX
ejpam-3193	13	3	evident	evident	ADJ
ejpam-3193	13	4	that	that	SCONJ
ejpam-3193	13	5	primary	primary	ADJ
ejpam-3193	13	6	ideals	ideal	NOUN
ejpam-3193	13	7	,	,	PUNCT
ejpam-3193	13	8	almost	almost	ADV
ejpam-3193	13	9	prime	prime	ADJ
ejpam-3193	13	10	ideals	ideal	NOUN
ejpam-3193	13	11	and	and	CCONJ
ejpam-3193	13	12	idempotent	idempotent	ADJ
ejpam-3193	13	13	ideals	ideal	NOUN
ejpam-3193	13	14	of	of	ADP
ejpam-3193	13	15	a	a	DET
ejpam-3193	13	16	ring	ring	NOUN
ejpam-3193	13	17	r	r	NOUN
ejpam-3193	13	18	are	be	AUX
ejpam-3193	13	19	almost	almost	ADV
ejpam-3193	13	20	primary	primary	ADJ
ejpam-3193	13	21	ideals	ideal	NOUN
ejpam-3193	13	22	,	,	PUNCT
ejpam-3193	13	23	but	but	CCONJ
ejpam-3193	13	24	the	the	DET
ejpam-3193	13	25	converse	converse	NOUN
ejpam-3193	13	26	is	be	AUX
ejpam-3193	13	27	not	not	PART
ejpam-3193	13	28	true	true	ADJ
ejpam-3193	13	29	in	in	ADP
ejpam-3193	13	30	each	each	DET
ejpam-3193	13	31	case	case	NOUN
ejpam-3193	13	32	.	.	PUNCT
ejpam-3193	14	1	notion	notion	NOUN
ejpam-3193	14	2	of	of	ADP
ejpam-3193	14	3	weakly	weakly	ADJ
ejpam-3193	14	4	prime	prime	ADJ
ejpam-3193	14	5	element	element	NOUN
ejpam-3193	14	6	(	(	PUNCT
ejpam-3193	14	7	author	author	NOUN
ejpam-3193	14	8	called	call	VERB
ejpam-3193	14	9	it	it	PRON
ejpam-3193	14	10	a	a	DET
ejpam-3193	14	11	prime	prime	NOUN
ejpam-3193	14	12	)	)	PUNCT
ejpam-3193	14	13	was	be	AUX
ejpam-3193	14	14	introduced	introduce	VERB
ejpam-3193	14	15	by	by	ADP
ejpam-3193	14	16	steven	steven	PROPN
ejpam-3193	14	17	galovich	galovich	PROPN
ejpam-3193	14	18	while	while	SCONJ
ejpam-3193	14	19	studying	study	VERB
ejpam-3193	14	20	the	the	DET
ejpam-3193	14	21	property	property	NOUN
ejpam-3193	14	22	of	of	ADP
ejpam-3193	14	23	unique	unique	ADJ
ejpam-3193	14	24	factorization	factorization	NOUN
ejpam-3193	14	25	of	of	ADP
ejpam-3193	14	26	rings	ring	NOUN
ejpam-3193	14	27	with	with	ADP
ejpam-3193	14	28	zero	zero	NUM
ejpam-3193	14	29	divisors	divisor	NOUN
ejpam-3193	14	30	[	[	X
ejpam-3193	14	31	10	10	NUM
ejpam-3193	14	32	]	]	PUNCT
ejpam-3193	14	33	.	.	PUNCT
ejpam-3193	15	1	following	follow	VERB
ejpam-3193	15	2	[	[	X
ejpam-3193	15	3	10	10	NUM
ejpam-3193	15	4	]	]	PUNCT
ejpam-3193	15	5	,	,	PUNCT
ejpam-3193	15	6	let	let	VERB
ejpam-3193	15	7	r	r	NOUN
ejpam-3193	15	8	6=	6=	ADP
ejpam-3193	15	9	0	0	NUM
ejpam-3193	15	10	be	be	AUX
ejpam-3193	15	11	in	in	ADP
ejpam-3193	15	12	r	r	NOUN
ejpam-3193	15	13	than	than	SCONJ
ejpam-3193	15	14	r	r	NOUN
ejpam-3193	15	15	is	be	AUX
ejpam-3193	15	16	prime	prime	ADJ
ejpam-3193	16	1	if	if	SCONJ
ejpam-3193	16	2	,	,	PUNCT
ejpam-3193	16	3	whenever	whenever	SCONJ
ejpam-3193	16	4	r	r	NOUN
ejpam-3193	16	5	divides	divide	VERB
ejpam-3193	16	6	ab	ab	PROPN
ejpam-3193	16	7	where	where	SCONJ
ejpam-3193	16	8	ab	ab	PROPN
ejpam-3193	16	9	6=	6=	PROPN
ejpam-3193	16	10	0	0	NUM
ejpam-3193	16	11	,	,	PUNCT
ejpam-3193	16	12	then	then	ADV
ejpam-3193	16	13	r	r	NOUN
ejpam-3193	16	14	divides	divide	VERB
ejpam-3193	16	15	a	a	DET
ejpam-3193	16	16	or	or	CCONJ
ejpam-3193	16	17	r	r	NOUN
ejpam-3193	16	18	divides	divide	NOUN
ejpam-3193	16	19	b.	b.	PROPN
ejpam-3193	16	20	author	author	PROPN
ejpam-3193	16	21	established	establish	VERB
ejpam-3193	16	22	the	the	DET
ejpam-3193	16	23	fundamental	fundamental	ADJ
ejpam-3193	16	24	results	result	NOUN
ejpam-3193	16	25	:	:	PUNCT
ejpam-3193	16	26	(	(	PUNCT
ejpam-3193	16	27	i	i	NOUN
ejpam-3193	16	28	)	)	PUNCT
ejpam-3193	16	29	in	in	ADP
ejpam-3193	16	30	[	[	X
ejpam-3193	16	31	10	10	NUM
ejpam-3193	16	32	]	]	PUNCT
ejpam-3193	16	33	,	,	PUNCT
ejpam-3193	16	34	author	author	NOUN
ejpam-3193	16	35	also	also	ADV
ejpam-3193	16	36	∗corresponding	∗corresponde	VERB
ejpam-3193	16	37	author	author	NOUN
ejpam-3193	16	38	.	.	PUNCT
ejpam-3193	17	1	email	email	NOUN
ejpam-3193	17	2	addresses	address	NOUN
ejpam-3193	17	3	:	:	PUNCT
ejpam-3193	17	4	sirwak2003@yahoo.com	sirwak2003@yahoo.com	X
ejpam-3193	17	5	(	(	PUNCT
ejpam-3193	17	6	w.	w.	PROPN
ejpam-3193	17	7	a.	a.	PROPN
ejpam-3193	17	8	khan	khan	PROPN
ejpam-3193	17	9	)	)	PUNCT
ejpam-3193	17	10	,	,	PUNCT
ejpam-3193	17	11	adnanmuhammad216@gmail.com	adnanmuhammad216@gmail.com	X
ejpam-3193	17	12	(	(	PUNCT
ejpam-3193	17	13	a.	a.	NOUN
ejpam-3193	17	14	muhammad	muhammad	PROPN
ejpam-3193	17	15	)	)	PUNCT
ejpam-3193	17	16	,	,	PUNCT
ejpam-3193	17	17	ganitaouti@yahoo.com.au	ganitaouti@yahoo.com.au	PROPN
ejpam-3193	17	18	(	(	PUNCT
ejpam-3193	17	19	a.	a.	NOUN
ejpam-3193	17	20	taouti	taouti	PROPN
ejpam-3193	17	21	)	)	PUNCT
ejpam-3193	17	22	,	,	PUNCT
ejpam-3193	17	23	jamcad@rit.edu	jamcad@rit.edu	PROPN
ejpam-3193	17	24	(	(	PUNCT
ejpam-3193	17	25	j.	j.	PROPN
ejpam-3193	17	26	maki	maki	PROPN
ejpam-3193	17	27	)	)	PUNCT
ejpam-3193	17	28	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3193	18	1	449	449	NUM
ejpam-3193	18	2	c	c	X
ejpam-3193	18	3	©	©	PROPN
ejpam-3193	18	4	2018	2018	NUM
ejpam-3193	18	5	ejpam	ejpam	VERB
ejpam-3193	18	6	all	all	DET
ejpam-3193	18	7	rights	right	NOUN
ejpam-3193	18	8	reserved	reserve	VERB
ejpam-3193	18	9	.	.	PUNCT
ejpam-3193	19	1	a.	a.	PROPN
ejpam-3193	19	2	taouti	taouti	PROPN
ejpam-3193	19	3	et	et	PROPN
ejpam-3193	19	4	al	al	PROPN
ejpam-3193	19	5	.	.	PUNCT
ejpam-3193	19	6	/	/	SYM
ejpam-3193	19	7	eur	eur	PROPN
ejpam-3193	19	8	.	.	PUNCT
ejpam-3193	20	1	j.	j.	PROPN
ejpam-3193	20	2	pure	pure	PROPN
ejpam-3193	20	3	appl	appl	PROPN
ejpam-3193	20	4	.	.	PROPN
ejpam-3193	20	5	math	math	PROPN
ejpam-3193	20	6	,	,	PUNCT
ejpam-3193	20	7	11	11	NUM
ejpam-3193	20	8	(	(	PUNCT
ejpam-3193	20	9	2	2	NUM
ejpam-3193	20	10	)	)	PUNCT
ejpam-3193	20	11	(	(	PUNCT
ejpam-3193	20	12	2018	2018	NUM
ejpam-3193	20	13	)	)	PUNCT
ejpam-3193	20	14	,	,	PUNCT
ejpam-3193	20	15	449	449	NUM
ejpam-3193	20	16	-	-	SYM
ejpam-3193	20	17	456	456	NUM
ejpam-3193	20	18	450	450	NUM
ejpam-3193	20	19	showed	show	VERB
ejpam-3193	20	20	that	that	SCONJ
ejpam-3193	20	21	every	every	DET
ejpam-3193	20	22	irreducible	irreducible	NOUN
ejpam-3193	20	23	is	be	AUX
ejpam-3193	20	24	a	a	DET
ejpam-3193	20	25	prime	prime	NOUN
ejpam-3193	20	26	,	,	PUNCT
ejpam-3193	20	27	(	(	PUNCT
ejpam-3193	20	28	ii	ii	NOUN
ejpam-3193	20	29	)	)	PUNCT
ejpam-3193	20	30	every	every	PRON
ejpam-3193	20	31	irreducible	irreducible	ADJ
ejpam-3193	20	32	in	in	ADP
ejpam-3193	20	33	r	r	NOUN
ejpam-3193	20	34	is	be	AUX
ejpam-3193	20	35	a	a	DET
ejpam-3193	20	36	zero	zero	NUM
ejpam-3193	20	37	divisor	divisor	NOUN
ejpam-3193	20	38	[	[	X
ejpam-3193	20	39	10	10	NUM
ejpam-3193	20	40	]	]	PUNCT
ejpam-3193	20	41	,	,	PUNCT
ejpam-3193	20	42	(	(	PUNCT
ejpam-3193	20	43	iii	iii	X
ejpam-3193	20	44	)	)	PUNCT
ejpam-3193	20	45	every	every	DET
ejpam-3193	20	46	irreducible	irreducible	ADJ
ejpam-3193	20	47	element	element	NOUN
ejpam-3193	20	48	of	of	ADP
ejpam-3193	20	49	r	r	NOUN
ejpam-3193	20	50	is	be	AUX
ejpam-3193	20	51	nilpotent	nilpotent	ADJ
ejpam-3193	20	52	,	,	PUNCT
ejpam-3193	20	53	and	and	CCONJ
ejpam-3193	20	54	(	(	PUNCT
ejpam-3193	20	55	iv	iv	X
ejpam-3193	20	56	)	)	PUNCT
ejpam-3193	20	57	every	every	DET
ejpam-3193	20	58	nonunit	nonunit	NOUN
ejpam-3193	20	59	in	in	ADP
ejpam-3193	20	60	r	r	NOUN
ejpam-3193	20	61	is	be	AUX
ejpam-3193	20	62	nilpotent	nilpotent	ADJ
ejpam-3193	20	63	.	.	PUNCT
ejpam-3193	21	1	consequently	consequently	ADV
ejpam-3193	21	2	the	the	DET
ejpam-3193	21	3	author	author	NOUN
ejpam-3193	21	4	declared	declare	VERB
ejpam-3193	21	5	the	the	DET
ejpam-3193	21	6	unique	unique	ADJ
ejpam-3193	21	7	maximal	maximal	ADJ
ejpam-3193	21	8	ideal	ideal	NOUN
ejpam-3193	21	9	consists	consist	NOUN
ejpam-3193	21	10	of	of	ADP
ejpam-3193	21	11	nonunit	nonunit	NOUN
ejpam-3193	21	12	elements	element	NOUN
ejpam-3193	21	13	[	[	X
ejpam-3193	21	14	10	10	NUM
ejpam-3193	21	15	]	]	PUNCT
ejpam-3193	21	16	.	.	PUNCT
ejpam-3193	22	1	in	in	ADP
ejpam-3193	22	2	[	[	X
ejpam-3193	22	3	1	1	NUM
ejpam-3193	22	4	]	]	PUNCT
ejpam-3193	22	5	,	,	PUNCT
ejpam-3193	22	6	authors	author	NOUN
ejpam-3193	22	7	declare	declare	VERB
ejpam-3193	22	8	that	that	SCONJ
ejpam-3193	22	9	(	(	PUNCT
ejpam-3193	22	10	which	which	PRON
ejpam-3193	22	11	was	be	AUX
ejpam-3193	22	12	named	name	VERB
ejpam-3193	22	13	prime	prime	ADJ
ejpam-3193	22	14	by	by	ADP
ejpam-3193	22	15	galovich	galovich	NOUN
ejpam-3193	22	16	in	in	ADP
ejpam-3193	22	17	[	[	X
ejpam-3193	22	18	10	10	NUM
ejpam-3193	22	19	]	]	SYM
ejpam-3193	22	20	)	)	PUNCT
ejpam-3193	22	21	a	a	DET
ejpam-3193	22	22	nonzero	nonzero	NOUN
ejpam-3193	22	23	nonunit	nonunit	NOUN
ejpam-3193	22	24	p	p	PROPN
ejpam-3193	22	25	∈	∈	PROPN
ejpam-3193	22	26	r	r	NOUN
ejpam-3193	22	27	is	be	AUX
ejpam-3193	22	28	weakly	weakly	ADV
ejpam-3193	22	29	prime	prime	ADJ
ejpam-3193	22	30	if	if	SCONJ
ejpam-3193	22	31	p|ab	p|ab	PROPN
ejpam-3193	22	32	6=	6=	ADP
ejpam-3193	22	33	0	0	NUM
ejpam-3193	22	34	implies	imply	VERB
ejpam-3193	22	35	p|a	p|a	NOUN
ejpam-3193	22	36	or	or	CCONJ
ejpam-3193	22	37	p|b	p|b	NOUN
ejpam-3193	22	38	.	.	PUNCT
ejpam-3193	23	1	consequently	consequently	ADV
ejpam-3193	23	2	,	,	PUNCT
ejpam-3193	23	3	an	an	DET
ejpam-3193	23	4	ideal	ideal	NOUN
ejpam-3193	23	5	i	i	PRON
ejpam-3193	23	6	of	of	ADP
ejpam-3193	23	7	a	a	DET
ejpam-3193	23	8	commutative	commutative	ADJ
ejpam-3193	23	9	ring	ring	NOUN
ejpam-3193	23	10	r	r	NOUN
ejpam-3193	23	11	is	be	AUX
ejpam-3193	23	12	called	call	VERB
ejpam-3193	23	13	a	a	DET
ejpam-3193	23	14	weakly	weakly	ADJ
ejpam-3193	23	15	prime	prime	NOUN
ejpam-3193	23	16	if	if	SCONJ
ejpam-3193	23	17	0	0	NUM
ejpam-3193	23	18	6=	6=	NUM
ejpam-3193	23	19	ab	ab	PROPN
ejpam-3193	23	20	∈	∈	PROPN
ejpam-3193	24	1	i	i	PRON
ejpam-3193	24	2	implies	imply	VERB
ejpam-3193	24	3	a	a	DET
ejpam-3193	24	4	∈	∈	NOUN
ejpam-3193	24	5	i	i	PRON
ejpam-3193	24	6	or	or	CCONJ
ejpam-3193	24	7	b	b	X
ejpam-3193	24	8	∈	∈	PROPN
ejpam-3193	24	9	i	i	PRON
ejpam-3193	24	10	,	,	PUNCT
ejpam-3193	24	11	and	and	CCONJ
ejpam-3193	24	12	also	also	ADV
ejpam-3193	24	13	p	p	PRON
ejpam-3193	24	14	is	be	AUX
ejpam-3193	24	15	weakly	weakly	ADJ
ejpam-3193	24	16	prime	prime	ADJ
ejpam-3193	24	17	iff	iff	NOUN
ejpam-3193	24	18	(	(	PUNCT
ejpam-3193	24	19	p	p	NOUN
ejpam-3193	24	20	)	)	PUNCT
ejpam-3193	24	21	is	be	AUX
ejpam-3193	24	22	weakly	weakly	ADV
ejpam-3193	24	23	prime	prime	ADJ
ejpam-3193	24	24	[	[	X
ejpam-3193	24	25	1	1	NUM
ejpam-3193	24	26	]	]	PUNCT
ejpam-3193	24	27	.	.	PUNCT
ejpam-3193	25	1	following	follow	VERB
ejpam-3193	25	2	[	[	X
ejpam-3193	25	3	2	2	NUM
ejpam-3193	25	4	]	]	PUNCT
ejpam-3193	25	5	,	,	PUNCT
ejpam-3193	25	6	p	p	NOUN
ejpam-3193	25	7	is	be	AUX
ejpam-3193	25	8	weakly	weakly	ADJ
ejpam-3193	25	9	prime	prime	ADJ
ejpam-3193	25	10	ideal	ideal	NOUN
ejpam-3193	25	11	if	if	SCONJ
ejpam-3193	25	12	and	and	CCONJ
ejpam-3193	25	13	only	only	ADV
ejpam-3193	25	14	if	if	SCONJ
ejpam-3193	25	15	0	0	NUM
ejpam-3193	25	16	6=	6=	NUM
ejpam-3193	25	17	ab	ab	PROPN
ejpam-3193	25	18	⊆	⊆	NUM
ejpam-3193	25	19	p	p	PROPN
ejpam-3193	25	20	,	,	PUNCT
ejpam-3193	25	21	a	a	PRON
ejpam-3193	25	22	and	and	CCONJ
ejpam-3193	25	23	b	b	NOUN
ejpam-3193	25	24	ideals	ideal	NOUN
ejpam-3193	25	25	of	of	ADP
ejpam-3193	25	26	r	r	NOUN
ejpam-3193	25	27	,	,	PUNCT
ejpam-3193	25	28	implies	imply	VERB
ejpam-3193	25	29	a	a	DET
ejpam-3193	25	30	⊆	⊆	NUM
ejpam-3193	25	31	p	p	NOUN
ejpam-3193	25	32	or	or	CCONJ
ejpam-3193	25	33	b	b	NOUN
ejpam-3193	25	34	⊆	⊆	NUM
ejpam-3193	25	35	p	p	NOUN
ejpam-3193	25	36	.	.	PUNCT
ejpam-3193	26	1	further	far	ADV
ejpam-3193	26	2	to	to	ADP
ejpam-3193	26	3	this	this	PRON
ejpam-3193	26	4	,	,	PUNCT
ejpam-3193	26	5	every	every	DET
ejpam-3193	26	6	weakly	weakly	ADJ
ejpam-3193	26	7	prime	prime	ADJ
ejpam-3193	26	8	ideal	ideal	NOUN
ejpam-3193	26	9	is	be	AUX
ejpam-3193	26	10	an	an	DET
ejpam-3193	26	11	almost	almost	ADV
ejpam-3193	26	12	prime	prime	ADJ
ejpam-3193	26	13	ideal	ideal	NOUN
ejpam-3193	26	14	.	.	PUNCT
ejpam-3193	27	1	we	we	PRON
ejpam-3193	27	2	call	call	VERB
ejpam-3193	27	3	an	an	DET
ejpam-3193	27	4	algebraic	algebraic	ADJ
ejpam-3193	27	5	system	system	NOUN
ejpam-3193	27	6	n	n	NOUN
ejpam-3193	27	7	with	with	ADP
ejpam-3193	27	8	two	two	NUM
ejpam-3193	27	9	binary	binary	ADJ
ejpam-3193	27	10	operation	operation	NOUN
ejpam-3193	27	11	”	"	PUNCT
ejpam-3193	28	1	+	+	CCONJ
ejpam-3193	28	2	”	"	PUNCT
ejpam-3193	28	3	and	and	CCONJ
ejpam-3193	28	4	”	"	PUNCT
ejpam-3193	28	5	.	.	PUNCT
ejpam-3193	28	6	”	"	PUNCT
ejpam-3193	29	1	(	(	PUNCT
ejpam-3193	29	2	right	right	NOUN
ejpam-3193	29	3	)	)	PUNCT
ejpam-3193	29	4	near	near	ADP
ejpam-3193	29	5	-	-	PUNCT
ejpam-3193	29	6	ring	ring	NOUN
ejpam-3193	29	7	if	if	SCONJ
ejpam-3193	29	8	it	it	PRON
ejpam-3193	29	9	is	be	AUX
ejpam-3193	29	10	a	a	DET
ejpam-3193	29	11	group	group	NOUN
ejpam-3193	29	12	(	(	PUNCT
ejpam-3193	29	13	not	not	PART
ejpam-3193	29	14	necessarily	necessarily	ADV
ejpam-3193	29	15	abelian	abelian	ADJ
ejpam-3193	29	16	)	)	PUNCT
ejpam-3193	29	17	under	under	ADP
ejpam-3193	29	18	addition	addition	NOUN
ejpam-3193	29	19	,	,	PUNCT
ejpam-3193	29	20	and	and	CCONJ
ejpam-3193	29	21	n	n	PRON
ejpam-3193	29	22	is	be	AUX
ejpam-3193	29	23	associative	associative	ADJ
ejpam-3193	29	24	group	group	NOUN
ejpam-3193	29	25	under	under	ADP
ejpam-3193	29	26	multiplication	multiplication	NOUN
ejpam-3193	29	27	and	and	CCONJ
ejpam-3193	29	28	distribution	distribution	NOUN
ejpam-3193	29	29	of	of	ADP
ejpam-3193	29	30	multiplication	multiplication	NOUN
ejpam-3193	29	31	over	over	ADP
ejpam-3193	29	32	addition	addition	NOUN
ejpam-3193	29	33	on	on	ADP
ejpam-3193	29	34	the	the	DET
ejpam-3193	29	35	right	right	NOUN
ejpam-3193	29	36	holds	hold	NOUN
ejpam-3193	29	37	i.e.	i.e.	ADV
ejpam-3193	29	38	,	,	PUNCT
ejpam-3193	29	39	for	for	ADP
ejpam-3193	29	40	any	any	DET
ejpam-3193	29	41	x	x	NOUN
ejpam-3193	29	42	,	,	PUNCT
ejpam-3193	29	43	y	y	PROPN
ejpam-3193	29	44	,	,	PUNCT
ejpam-3193	29	45	z	z	PROPN
ejpam-3193	29	46	∈	∈	PROPN
ejpam-3193	29	47	n	n	X
ejpam-3193	29	48	,	,	PUNCT
ejpam-3193	29	49	it	it	PRON
ejpam-3193	29	50	satisfies	satisfy	VERB
ejpam-3193	29	51	that	that	SCONJ
ejpam-3193	29	52	(	(	PUNCT
ejpam-3193	29	53	x	x	SYM
ejpam-3193	29	54	+	+	NUM
ejpam-3193	29	55	y)z	y)z	X
ejpam-3193	29	56	=	=	SYM
ejpam-3193	29	57	(	(	PUNCT
ejpam-3193	29	58	xz	xz	PROPN
ejpam-3193	29	59	)	)	PUNCT
ejpam-3193	30	1	+	+	CCONJ
ejpam-3193	30	2	(	(	PUNCT
ejpam-3193	30	3	yz)[15	yz)[15	NOUN
ejpam-3193	30	4	]	]	X
ejpam-3193	30	5	.	.	PUNCT
ejpam-3193	31	1	likewise	likewise	ADV
ejpam-3193	31	2	,	,	PUNCT
ejpam-3193	31	3	a	a	DET
ejpam-3193	31	4	left	left	ADJ
ejpam-3193	31	5	near	near	NOUN
ejpam-3193	31	6	-	-	PUNCT
ejpam-3193	31	7	ring	ring	NOUN
ejpam-3193	31	8	can	can	AUX
ejpam-3193	31	9	be	be	AUX
ejpam-3193	31	10	defined	define	VERB
ejpam-3193	31	11	by	by	ADP
ejpam-3193	31	12	replacing	replace	VERB
ejpam-3193	31	13	the	the	DET
ejpam-3193	31	14	right	right	ADJ
ejpam-3193	31	15	distributive	distributive	ADJ
ejpam-3193	31	16	law	law	NOUN
ejpam-3193	31	17	by	by	ADP
ejpam-3193	31	18	the	the	DET
ejpam-3193	31	19	equivalent	equivalent	NOUN
ejpam-3193	31	20	left	leave	VERB
ejpam-3193	31	21	distributive	distributive	ADJ
ejpam-3193	31	22	law	law	NOUN
ejpam-3193	31	23	.	.	PUNCT
ejpam-3193	32	1	suppose	suppose	VERB
ejpam-3193	32	2	n	n	PRON
ejpam-3193	32	3	is	be	AUX
ejpam-3193	32	4	a	a	DET
ejpam-3193	32	5	left	left	ADJ
ejpam-3193	32	6	near	near	NOUN
ejpam-3193	32	7	-	-	PUNCT
ejpam-3193	32	8	ring	ring	NOUN
ejpam-3193	32	9	with	with	ADP
ejpam-3193	32	10	binary	binary	ADJ
ejpam-3193	32	11	operation	operation	NOUN
ejpam-3193	32	12	”	"	PUNCT
ejpam-3193	33	1	+	+	CCONJ
ejpam-3193	33	2	”	"	PUNCT
ejpam-3193	33	3	and	and	CCONJ
ejpam-3193	33	4	”	"	PUNCT
ejpam-3193	33	5	.	.	PUNCT
ejpam-3193	33	6	”	"	PUNCT
ejpam-3193	34	1	then	then	ADV
ejpam-3193	34	2	a	a	DET
ejpam-3193	34	3	subset	subset	NOUN
ejpam-3193	34	4	i	i	PRON
ejpam-3193	34	5	is	be	AUX
ejpam-3193	34	6	said	say	VERB
ejpam-3193	34	7	to	to	PART
ejpam-3193	34	8	be	be	AUX
ejpam-3193	34	9	an	an	DET
ejpam-3193	34	10	ideal	ideal	NOUN
ejpam-3193	34	11	if	if	SCONJ
ejpam-3193	34	12	(	(	PUNCT
ejpam-3193	34	13	i	i	NOUN
ejpam-3193	34	14	)	)	PUNCT
ejpam-3193	34	15	(	(	PUNCT
ejpam-3193	34	16	i,+	i,+	NUM
ejpam-3193	34	17	)	)	PUNCT
ejpam-3193	34	18	is	be	AUX
ejpam-3193	34	19	a	a	DET
ejpam-3193	34	20	normal	normal	ADJ
ejpam-3193	34	21	subgroup	subgroup	NOUN
ejpam-3193	34	22	of	of	ADP
ejpam-3193	34	23	a	a	DET
ejpam-3193	34	24	(	(	PUNCT
ejpam-3193	34	25	n,+	n,+	NUM
ejpam-3193	34	26	)	)	PUNCT
ejpam-3193	34	27	,	,	PUNCT
ejpam-3193	34	28	(	(	PUNCT
ejpam-3193	34	29	ii	ii	NOUN
ejpam-3193	34	30	)	)	PUNCT
ejpam-3193	34	31	for	for	ADP
ejpam-3193	34	32	each	each	DET
ejpam-3193	34	33	n	n	PRON
ejpam-3193	34	34	∈	∈	PROPN
ejpam-3193	34	35	n	n	NOUN
ejpam-3193	34	36	,	,	PUNCT
ejpam-3193	34	37	i	i	PRON
ejpam-3193	34	38	∈	∈	PROPN
ejpam-3193	35	1	i	i	PRON
ejpam-3193	35	2	,	,	PUNCT
ejpam-3193	35	3	ni	ni	PROPN
ejpam-3193	35	4	∈	∈	PROPN
ejpam-3193	36	1	i	i	PRON
ejpam-3193	36	2	i.e.	i.e.	X
ejpam-3193	36	3	,	,	PUNCT
ejpam-3193	36	4	ni	ni	PROPN
ejpam-3193	36	5	⊆	⊆	NUM
ejpam-3193	36	6	i	i	PROPN
ejpam-3193	36	7	,	,	PUNCT
ejpam-3193	36	8	and	and	CCONJ
ejpam-3193	36	9	(	(	PUNCT
ejpam-3193	36	10	iii	iii	NOUN
ejpam-3193	36	11	)	)	PUNCT
ejpam-3193	36	12	(	(	PUNCT
ejpam-3193	36	13	n1	n1	PROPN
ejpam-3193	36	14	+	+	CCONJ
ejpam-3193	36	15	i)n2	i)n2	PROPN
ejpam-3193	36	16	n1n2	n1n2	NOUN
ejpam-3193	36	17	∈	∈	NOUN
ejpam-3193	36	18	i	i	PRON
ejpam-3193	36	19	for	for	ADP
ejpam-3193	36	20	each	each	DET
ejpam-3193	36	21	n1	n1	NOUN
ejpam-3193	36	22	,	,	PUNCT
ejpam-3193	36	23	n2	n2	NOUN
ejpam-3193	36	24	∈	∈	PROPN
ejpam-3193	36	25	n	n	CCONJ
ejpam-3193	37	1	and	and	CCONJ
ejpam-3193	37	2	i	i	PROPN
ejpam-3193	37	3	∈	∈	PROPN
ejpam-3193	37	4	i.	i.	NOUN
ejpam-3193	37	5	but	but	CCONJ
ejpam-3193	37	6	a.	a.	NOUN
ejpam-3193	37	7	frohlich	frohlich	PROPN
ejpam-3193	38	1	[	[	X
ejpam-3193	38	2	9	9	NUM
ejpam-3193	38	3	]	]	PUNCT
ejpam-3193	38	4	showed	show	VERB
ejpam-3193	38	5	that	that	SCONJ
ejpam-3193	38	6	for	for	ADP
ejpam-3193	38	7	d.g	d.g	PROPN
ejpam-3193	38	8	.	.	PROPN
ejpam-3193	38	9	near	near	NOUN
ejpam-3193	38	10	-	-	PUNCT
ejpam-3193	38	11	rings	ring	NOUN
ejpam-3193	38	12	the	the	DET
ejpam-3193	38	13	third	third	ADJ
ejpam-3193	38	14	condition	condition	NOUN
ejpam-3193	38	15	is	be	AUX
ejpam-3193	38	16	equivalent	equivalent	ADJ
ejpam-3193	38	17	to	to	ADP
ejpam-3193	38	18	in	in	ADP
ejpam-3193	38	19	∈	∈	PROPN
ejpam-3193	38	20	i	i	PRON
ejpam-3193	38	21	i.e.	i.e.	X
ejpam-3193	38	22	,	,	PUNCT
ejpam-3193	38	23	in	in	ADP
ejpam-3193	38	24	⊆	⊆	NUM
ejpam-3193	38	25	i.	i.	NOUN
ejpam-3193	38	26	hence	hence	ADV
ejpam-3193	38	27	a	a	DET
ejpam-3193	38	28	subset	subset	NOUN
ejpam-3193	39	1	i	i	PRON
ejpam-3193	39	2	is	be	AUX
ejpam-3193	39	3	a	a	DET
ejpam-3193	39	4	right	right	NOUN
ejpam-3193	39	5	(	(	PUNCT
ejpam-3193	39	6	left	left	ADJ
ejpam-3193	39	7	)	)	PUNCT
ejpam-3193	39	8	ideal	ideal	NOUN
ejpam-3193	39	9	if	if	SCONJ
ejpam-3193	39	10	i	i	PRON
ejpam-3193	39	11	satisfies	satisfy	VERB
ejpam-3193	39	12	the	the	DET
ejpam-3193	39	13	first	first	ADJ
ejpam-3193	39	14	and	and	CCONJ
ejpam-3193	39	15	third	third	ADJ
ejpam-3193	39	16	(	(	PUNCT
ejpam-3193	39	17	second	second	ADJ
ejpam-3193	39	18	)	)	PUNCT
ejpam-3193	39	19	conditions	condition	NOUN
ejpam-3193	39	20	.	.	PUNCT
ejpam-3193	40	1	a	a	DET
ejpam-3193	40	2	proper	proper	ADJ
ejpam-3193	40	3	ideal	ideal	NOUN
ejpam-3193	40	4	p	p	NOUN
ejpam-3193	40	5	of	of	ADP
ejpam-3193	40	6	a	a	DET
ejpam-3193	40	7	near	near	ADJ
ejpam-3193	40	8	ring	ring	NOUN
ejpam-3193	40	9	n	n	PRON
ejpam-3193	40	10	is	be	AUX
ejpam-3193	40	11	prime	prime	ADJ
ejpam-3193	40	12	if	if	SCONJ
ejpam-3193	40	13	for	for	ADP
ejpam-3193	40	14	ideals	ideal	NOUN
ejpam-3193	40	15	a	a	PRON
ejpam-3193	40	16	and	and	CCONJ
ejpam-3193	40	17	b	b	NOUN
ejpam-3193	40	18	of	of	ADP
ejpam-3193	40	19	n	n	PROPN
ejpam-3193	40	20	,	,	PUNCT
ejpam-3193	40	21	ab	ab	PROPN
ejpam-3193	40	22	⊆	⊆	NUM
ejpam-3193	40	23	p	p	PROPN
ejpam-3193	40	24	implies	imply	VERB
ejpam-3193	40	25	a	a	DET
ejpam-3193	40	26	⊆	⊆	NUM
ejpam-3193	40	27	p	p	NOUN
ejpam-3193	40	28	or	or	CCONJ
ejpam-3193	40	29	b	b	NOUN
ejpam-3193	40	30	⊆	⊆	NUM
ejpam-3193	40	31	p	p	NOUN
ejpam-3193	40	32	.	.	PUNCT
ejpam-3193	41	1	an	an	DET
ejpam-3193	41	2	ideal	ideal	ADJ
ejpam-3193	41	3	p	p	NOUN
ejpam-3193	41	4	of	of	ADP
ejpam-3193	41	5	a	a	DET
ejpam-3193	41	6	near	near	ADJ
ejpam-3193	41	7	-	-	PUNCT
ejpam-3193	41	8	ring	ring	NOUN
ejpam-3193	41	9	n	n	NOUN
ejpam-3193	41	10	is	be	AUX
ejpam-3193	41	11	a	a	DET
ejpam-3193	41	12	completely	completely	ADV
ejpam-3193	41	13	prime	prime	ADJ
ejpam-3193	41	14	(	(	PUNCT
ejpam-3193	41	15	prime	prime	ADJ
ejpam-3193	41	16	ideal	ideal	NOUN
ejpam-3193	41	17	of	of	ADP
ejpam-3193	41	18	type-2	type-2	NUM
ejpam-3193	41	19	)	)	PUNCT
ejpam-3193	41	20	if	if	SCONJ
ejpam-3193	41	21	for	for	ADP
ejpam-3193	41	22	all	all	DET
ejpam-3193	41	23	x	x	NOUN
ejpam-3193	41	24	,	,	PUNCT
ejpam-3193	41	25	y	y	PROPN
ejpam-3193	41	26	∈	∈	PROPN
ejpam-3193	41	27	n	n	AUX
ejpam-3193	41	28	,	,	PUNCT
ejpam-3193	41	29	xy	xy	PROPN
ejpam-3193	41	30	∈	∈	PROPN
ejpam-3193	42	1	p	p	NOUN
ejpam-3193	42	2	implies	imply	VERB
ejpam-3193	42	3	x	x	X
ejpam-3193	42	4	∈	∈	PROPN
ejpam-3193	42	5	p	p	NOUN
ejpam-3193	42	6	or	or	CCONJ
ejpam-3193	42	7	y	y	PROPN
ejpam-3193	42	8	∈	∈	PROPN
ejpam-3193	42	9	p	p	NOUN
ejpam-3193	42	10	.	.	PUNCT
ejpam-3193	43	1	almost	almost	ADV
ejpam-3193	43	2	prime	prime	ADJ
ejpam-3193	43	3	ideals	ideal	NOUN
ejpam-3193	43	4	in	in	ADP
ejpam-3193	43	5	near	near	ADJ
ejpam-3193	43	6	rings	ring	NOUN
ejpam-3193	43	7	have	have	AUX
ejpam-3193	43	8	been	be	AUX
ejpam-3193	43	9	endorsed	endorse	VERB
ejpam-3193	43	10	by	by	ADP
ejpam-3193	43	11	b.	b.	PROPN
ejpam-3193	43	12	elavarasan	elavarasan	PROPN
ejpam-3193	43	13	(	(	PUNCT
ejpam-3193	43	14	see	see	VERB
ejpam-3193	43	15	[	[	X
ejpam-3193	43	16	8	8	NUM
ejpam-3193	43	17	]	]	NUM
ejpam-3193	43	18	)	)	PUNCT
ejpam-3193	43	19	.	.	PUNCT
ejpam-3193	44	1	a	a	DET
ejpam-3193	44	2	proper	proper	ADJ
ejpam-3193	44	3	ideal	ideal	NOUN
ejpam-3193	44	4	p	p	NOUN
ejpam-3193	44	5	of	of	ADP
ejpam-3193	44	6	a	a	DET
ejpam-3193	44	7	near	near	ADJ
ejpam-3193	44	8	ring	ring	NOUN
ejpam-3193	44	9	n	n	PRON
ejpam-3193	44	10	is	be	AUX
ejpam-3193	44	11	said	say	VERB
ejpam-3193	44	12	to	to	PART
ejpam-3193	44	13	be	be	AUX
ejpam-3193	44	14	almost	almost	ADV
ejpam-3193	44	15	prime	prime	ADJ
ejpam-3193	44	16	if	if	SCONJ
ejpam-3193	44	17	for	for	ADP
ejpam-3193	44	18	any	any	DET
ejpam-3193	44	19	ideals	ideal	NOUN
ejpam-3193	44	20	a	a	PRON
ejpam-3193	44	21	and	and	CCONJ
ejpam-3193	44	22	b	b	NOUN
ejpam-3193	44	23	of	of	ADP
ejpam-3193	44	24	n	n	PRON
ejpam-3193	44	25	such	such	ADJ
ejpam-3193	44	26	that	that	SCONJ
ejpam-3193	44	27	ab	ab	PROPN
ejpam-3193	44	28	⊆	⊆	NUM
ejpam-3193	44	29	p	p	NOUN
ejpam-3193	44	30	and	and	CCONJ
ejpam-3193	44	31	ab	ab	PROPN
ejpam-3193	44	32	*	*	PUNCT
ejpam-3193	45	1	p	p	PROPN
ejpam-3193	45	2	2	2	NUM
ejpam-3193	45	3	,	,	PUNCT
ejpam-3193	45	4	we	we	PRON
ejpam-3193	45	5	have	have	VERB
ejpam-3193	45	6	a	a	DET
ejpam-3193	45	7	⊆	⊆	NUM
ejpam-3193	45	8	p	p	NOUN
ejpam-3193	45	9	or	or	CCONJ
ejpam-3193	45	10	b	b	NOUN
ejpam-3193	45	11	⊆	⊆	NUM
ejpam-3193	45	12	p	p	NOUN
ejpam-3193	46	1	[	[	X
ejpam-3193	46	2	8	8	NUM
ejpam-3193	46	3	]	]	PUNCT
ejpam-3193	46	4	.	.	PUNCT
ejpam-3193	47	1	the	the	DET
ejpam-3193	47	2	author	author	NOUN
ejpam-3193	47	3	established	establish	VERB
ejpam-3193	47	4	few	few	ADJ
ejpam-3193	47	5	relationships	relationship	NOUN
ejpam-3193	47	6	between	between	ADP
ejpam-3193	47	7	almost	almost	ADV
ejpam-3193	47	8	prime	prime	ADJ
ejpam-3193	47	9	and	and	CCONJ
ejpam-3193	47	10	prime	prime	ADJ
ejpam-3193	47	11	ideals	ideal	NOUN
ejpam-3193	47	12	[	[	X
ejpam-3193	47	13	8	8	NUM
ejpam-3193	47	14	]	]	PUNCT
ejpam-3193	47	15	.	.	PUNCT
ejpam-3193	48	1	weakly	weakly	ADJ
ejpam-3193	48	2	prime	prime	ADJ
ejpam-3193	48	3	ideals	ideal	NOUN
ejpam-3193	48	4	in	in	ADP
ejpam-3193	48	5	near	near	ADJ
ejpam-3193	48	6	rings	ring	NOUN
ejpam-3193	48	7	have	have	AUX
ejpam-3193	48	8	been	be	AUX
ejpam-3193	48	9	introduced	introduce	VERB
ejpam-3193	48	10	by	by	ADP
ejpam-3193	48	11	p.	p.	PROPN
ejpam-3193	48	12	dheena	dheena	PROPN
ejpam-3193	48	13	and	and	CCONJ
ejpam-3193	48	14	b.	b.	PROPN
ejpam-3193	48	15	elavarasan	elavarasan	PROPN
ejpam-3193	49	1	[	[	X
ejpam-3193	49	2	6	6	NUM
ejpam-3193	49	3	]	]	PUNCT
ejpam-3193	49	4	,	,	PUNCT
ejpam-3193	49	5	a	a	DET
ejpam-3193	49	6	proper	proper	ADJ
ejpam-3193	49	7	ideal	ideal	NOUN
ejpam-3193	49	8	p	p	NOUN
ejpam-3193	49	9	of	of	ADP
ejpam-3193	49	10	near	near	ADJ
ejpam-3193	49	11	ring	ring	NOUN
ejpam-3193	49	12	n	n	PRON
ejpam-3193	49	13	is	be	AUX
ejpam-3193	49	14	said	say	VERB
ejpam-3193	49	15	to	to	PART
ejpam-3193	49	16	be	be	AUX
ejpam-3193	49	17	weakly	weakly	ADV
ejpam-3193	49	18	prime	prime	ADJ
ejpam-3193	49	19	if	if	SCONJ
ejpam-3193	49	20	0	0	NUM
ejpam-3193	49	21	6=	6=	NUM
ejpam-3193	49	22	ab	ab	PROPN
ejpam-3193	49	23	⊆	⊆	NUM
ejpam-3193	49	24	p	p	PROPN
ejpam-3193	49	25	,	,	PUNCT
ejpam-3193	49	26	a	a	PRON
ejpam-3193	49	27	and	and	CCONJ
ejpam-3193	49	28	b	b	NOUN
ejpam-3193	49	29	are	be	AUX
ejpam-3193	49	30	ideals	ideal	NOUN
ejpam-3193	49	31	of	of	ADP
ejpam-3193	49	32	n	n	NUM
ejpam-3193	49	33	,	,	PUNCT
ejpam-3193	49	34	implies	imply	VERB
ejpam-3193	49	35	a	a	DET
ejpam-3193	49	36	⊆	⊆	NUM
ejpam-3193	49	37	p	p	NOUN
ejpam-3193	49	38	or	or	CCONJ
ejpam-3193	49	39	b	b	NOUN
ejpam-3193	49	40	⊆	⊆	NUM
ejpam-3193	49	41	p	p	NOUN
ejpam-3193	49	42	.	.	PUNCT
ejpam-3193	50	1	clearly	clearly	ADV
ejpam-3193	50	2	,	,	PUNCT
ejpam-3193	50	3	every	every	DET
ejpam-3193	50	4	prime	prime	ADJ
ejpam-3193	50	5	ideal	ideal	NOUN
ejpam-3193	50	6	is	be	AUX
ejpam-3193	50	7	weakly	weakly	ADV
ejpam-3193	50	8	prime	prime	ADJ
ejpam-3193	50	9	and	and	CCONJ
ejpam-3193	50	10	{	{	PUNCT
ejpam-3193	50	11	0	0	NUM
ejpam-3193	50	12	}	}	PUNCT
ejpam-3193	50	13	is	be	AUX
ejpam-3193	50	14	always	always	ADV
ejpam-3193	50	15	weakly	weakly	ADJ
ejpam-3193	50	16	prime	prime	ADJ
ejpam-3193	50	17	ideal	ideal	NOUN
ejpam-3193	50	18	of	of	ADP
ejpam-3193	50	19	a	a	DET
ejpam-3193	50	20	near	near	ADJ
ejpam-3193	50	21	ring	ring	NOUN
ejpam-3193	50	22	n	n	NOUN
ejpam-3193	50	23	.	.	PUNCT
ejpam-3193	51	1	also	also	ADV
ejpam-3193	51	2	every	every	DET
ejpam-3193	51	3	prime	prime	ADJ
ejpam-3193	51	4	ideal	ideal	NOUN
ejpam-3193	51	5	is	be	AUX
ejpam-3193	51	6	a	a	DET
ejpam-3193	51	7	weakly	weakly	ADJ
ejpam-3193	51	8	prime	prime	NOUN
ejpam-3193	51	9	,	,	PUNCT
ejpam-3193	51	10	and	and	CCONJ
ejpam-3193	51	11	a	a	DET
ejpam-3193	51	12	weakly	weakly	ADJ
ejpam-3193	51	13	prime	prime	ADJ
ejpam-3193	51	14	ideal	ideal	NOUN
ejpam-3193	51	15	is	be	AUX
ejpam-3193	51	16	an	an	DET
ejpam-3193	51	17	almost	almost	ADV
ejpam-3193	51	18	prime	prime	ADJ
ejpam-3193	51	19	ideal	ideal	NOUN
ejpam-3193	51	20	.	.	PUNCT
ejpam-3193	52	1	an	an	DET
ejpam-3193	52	2	ideal	ideal	ADJ
ejpam-3193	52	3	i	i	PRON
ejpam-3193	52	4	of	of	ADP
ejpam-3193	52	5	a	a	DET
ejpam-3193	52	6	near	near	ADJ
ejpam-3193	52	7	ring	ring	NOUN
ejpam-3193	52	8	n	n	PRON
ejpam-3193	52	9	is	be	AUX
ejpam-3193	52	10	said	say	VERB
ejpam-3193	52	11	to	to	PART
ejpam-3193	52	12	be	be	AUX
ejpam-3193	52	13	a	a	DET
ejpam-3193	52	14	completely	completely	ADV
ejpam-3193	52	15	prime	prime	ADJ
ejpam-3193	52	16	ideal	ideal	NOUN
ejpam-3193	52	17	if	if	SCONJ
ejpam-3193	52	18	x	x	NOUN
ejpam-3193	52	19	,	,	PUNCT
ejpam-3193	52	20	y	y	PROPN
ejpam-3193	52	21	∈	∈	PROPN
ejpam-3193	52	22	n	n	AUX
ejpam-3193	52	23	,	,	PUNCT
ejpam-3193	52	24	xy	xy	PROPN
ejpam-3193	52	25	∈	∈	PROPN
ejpam-3193	53	1	i	i	PRON
ejpam-3193	53	2	implies	imply	VERB
ejpam-3193	53	3	x	x	X
ejpam-3193	53	4	∈	∈	X
ejpam-3193	53	5	i	i	PRON
ejpam-3193	53	6	or	or	CCONJ
ejpam-3193	53	7	y	y	PROPN
ejpam-3193	53	8	∈	∈	PROPN
ejpam-3193	53	9	i	i	PRON
ejpam-3193	54	1	[	[	X
ejpam-3193	54	2	11	11	NUM
ejpam-3193	54	3	]	]	PUNCT
ejpam-3193	54	4	.	.	PUNCT
ejpam-3193	55	1	similarly	similarly	ADV
ejpam-3193	55	2	,	,	PUNCT
ejpam-3193	55	3	an	an	DET
ejpam-3193	55	4	ideal	ideal	NOUN
ejpam-3193	55	5	of	of	ADP
ejpam-3193	55	6	a	a	DET
ejpam-3193	55	7	near	near	ADJ
ejpam-3193	55	8	ring	ring	NOUN
ejpam-3193	55	9	n	n	PRON
ejpam-3193	55	10	is	be	AUX
ejpam-3193	55	11	said	say	VERB
ejpam-3193	55	12	to	to	PART
ejpam-3193	55	13	be	be	AUX
ejpam-3193	55	14	primary	primary	ADJ
ejpam-3193	55	15	ideal	ideal	NOUN
ejpam-3193	55	16	of	of	ADP
ejpam-3193	55	17	n	n	PRON
ejpam-3193	55	18	if	if	SCONJ
ejpam-3193	55	19	x	x	X
ejpam-3193	55	20	,	,	PUNCT
ejpam-3193	55	21	y	y	PROPN
ejpam-3193	55	22	∈	∈	PROPN
ejpam-3193	55	23	n	n	AUX
ejpam-3193	55	24	,	,	PUNCT
ejpam-3193	55	25	xy	xy	PROPN
ejpam-3193	55	26	∈	∈	PROPN
ejpam-3193	56	1	i	i	PRON
ejpam-3193	56	2	implies	imply	VERB
ejpam-3193	56	3	x	x	X
ejpam-3193	56	4	∈	∈	X
ejpam-3193	57	1	i	i	PRON
ejpam-3193	57	2	or	or	CCONJ
ejpam-3193	57	3	ym	ym	INTJ
ejpam-3193	57	4	∈	∈	PROPN
ejpam-3193	57	5	i	i	PRON
ejpam-3193	57	6	for	for	ADP
ejpam-3193	57	7	some	some	DET
ejpam-3193	57	8	m	m	NOUN
ejpam-3193	57	9	∈	∈	PROPN
ejpam-3193	57	10	z.	z.	PROPN
ejpam-3193	58	1	an	an	DET
ejpam-3193	58	2	ideal	ideal	ADJ
ejpam-3193	58	3	i	i	PRON
ejpam-3193	58	4	of	of	ADP
ejpam-3193	58	5	a	a	DET
ejpam-3193	58	6	near	near	ADJ
ejpam-3193	58	7	ring	ring	NOUN
ejpam-3193	58	8	n	n	PRON
ejpam-3193	58	9	is	be	AUX
ejpam-3193	58	10	called	call	VERB
ejpam-3193	58	11	a	a	DET
ejpam-3193	58	12	completely	completely	ADV
ejpam-3193	58	13	semiprime	semiprime	NOUN
ejpam-3193	58	14	ideal	ideal	NOUN
ejpam-3193	58	15	of	of	ADP
ejpam-3193	58	16	a	a	DET
ejpam-3193	58	17	near	near	ADJ
ejpam-3193	58	18	ring	ring	NOUN
ejpam-3193	58	19	n	n	INTJ
ejpam-3193	58	20	if	if	SCONJ
ejpam-3193	58	21	y2	y2	PROPN
ejpam-3193	58	22	∈	∈	PROPN
ejpam-3193	58	23	i	i	PRON
ejpam-3193	58	24	implies	imply	VERB
ejpam-3193	58	25	y	y	PROPN
ejpam-3193	58	26	∈	∈	PROPN
ejpam-3193	58	27	i	i	PRON
ejpam-3193	58	28	for	for	ADP
ejpam-3193	58	29	all	all	DET
ejpam-3193	58	30	y	y	PROPN
ejpam-3193	58	31	∈	∈	PROPN
ejpam-3193	59	1	n	n	CCONJ
ejpam-3193	60	1	[	[	X
ejpam-3193	60	2	11	11	NUM
ejpam-3193	60	3	]	]	PUNCT
ejpam-3193	60	4	.	.	PUNCT
ejpam-3193	61	1	further	far	ADV
ejpam-3193	61	2	to	to	ADP
ejpam-3193	61	3	this	this	PRON
ejpam-3193	61	4	,	,	PUNCT
ejpam-3193	61	5	almost	almost	ADV
ejpam-3193	61	6	prime	prime	ADJ
ejpam-3193	61	7	ideals	ideal	NOUN
ejpam-3193	61	8	in	in	ADP
ejpam-3193	61	9	near	near	ADJ
ejpam-3193	61	10	rings	ring	NOUN
ejpam-3193	61	11	have	have	AUX
ejpam-3193	61	12	been	be	AUX
ejpam-3193	61	13	endorsed	endorse	VERB
ejpam-3193	61	14	by	by	ADP
ejpam-3193	61	15	b.	b.	PROPN
ejpam-3193	61	16	elavarasan	elavarasan	PROPN
ejpam-3193	61	17	(	(	PUNCT
ejpam-3193	61	18	see	see	VERB
ejpam-3193	61	19	[	[	X
ejpam-3193	61	20	8	8	NUM
ejpam-3193	61	21	]	]	NUM
ejpam-3193	61	22	)	)	PUNCT
ejpam-3193	61	23	.	.	PUNCT
ejpam-3193	62	1	a	a	DET
ejpam-3193	62	2	proper	proper	ADJ
ejpam-3193	62	3	ideal	ideal	NOUN
ejpam-3193	62	4	p	p	NOUN
ejpam-3193	62	5	of	of	ADP
ejpam-3193	62	6	a	a	DET
ejpam-3193	62	7	near	near	ADJ
ejpam-3193	62	8	ring	ring	NOUN
ejpam-3193	62	9	n	n	PRON
ejpam-3193	62	10	is	be	AUX
ejpam-3193	62	11	said	say	VERB
ejpam-3193	62	12	to	to	PART
ejpam-3193	62	13	be	be	AUX
ejpam-3193	62	14	almost	almost	ADV
ejpam-3193	62	15	prime	prime	ADJ
ejpam-3193	62	16	if	if	SCONJ
ejpam-3193	62	17	for	for	ADP
ejpam-3193	62	18	any	any	DET
ejpam-3193	62	19	ideals	ideal	NOUN
ejpam-3193	62	20	a	a	PRON
ejpam-3193	62	21	and	and	CCONJ
ejpam-3193	62	22	b	b	NOUN
ejpam-3193	62	23	of	of	ADP
ejpam-3193	62	24	n	n	PRON
ejpam-3193	62	25	such	such	ADJ
ejpam-3193	62	26	that	that	SCONJ
ejpam-3193	62	27	ab	ab	PROPN
ejpam-3193	62	28	⊆	⊆	NUM
ejpam-3193	62	29	p	p	NOUN
ejpam-3193	62	30	and	and	CCONJ
ejpam-3193	62	31	ab	ab	PROPN
ejpam-3193	62	32	*	*	PUNCT
ejpam-3193	63	1	p	p	PROPN
ejpam-3193	63	2	2	2	NUM
ejpam-3193	63	3	,	,	PUNCT
ejpam-3193	63	4	we	we	PRON
ejpam-3193	63	5	have	have	VERB
ejpam-3193	63	6	a	a	DET
ejpam-3193	63	7	⊆	⊆	NUM
ejpam-3193	63	8	p	p	NOUN
ejpam-3193	63	9	or	or	CCONJ
ejpam-3193	63	10	b	b	NOUN
ejpam-3193	63	11	⊆	⊆	NUM
ejpam-3193	63	12	p	p	NOUN
ejpam-3193	64	1	[	[	X
ejpam-3193	64	2	8	8	NUM
ejpam-3193	64	3	]	]	PUNCT
ejpam-3193	64	4	.	.	PUNCT
ejpam-3193	65	1	the	the	DET
ejpam-3193	65	2	author	author	NOUN
ejpam-3193	65	3	established	establish	VERB
ejpam-3193	65	4	few	few	ADJ
ejpam-3193	65	5	relationships	relationship	NOUN
ejpam-3193	65	6	between	between	ADP
ejpam-3193	65	7	almost	almost	ADV
ejpam-3193	65	8	prime	prime	ADJ
ejpam-3193	65	9	and	and	CCONJ
ejpam-3193	65	10	prime	prime	ADJ
ejpam-3193	65	11	ideals	ideal	NOUN
ejpam-3193	65	12	[	[	X
ejpam-3193	65	13	8	8	NUM
ejpam-3193	65	14	]	]	PUNCT
ejpam-3193	65	15	.	.	PUNCT
ejpam-3193	66	1	number	number	NOUN
ejpam-3193	66	2	of	of	ADP
ejpam-3193	66	3	ideals	ideal	NOUN
ejpam-3193	66	4	in	in	ADP
ejpam-3193	66	5	near	near	ADJ
ejpam-3193	66	6	ring	ring	NOUN
ejpam-3193	66	7	have	have	AUX
ejpam-3193	66	8	been	be	AUX
ejpam-3193	66	9	introduced	introduce	VERB
ejpam-3193	66	10	and	and	CCONJ
ejpam-3193	66	11	discussed	discuss	VERB
ejpam-3193	66	12	such	such	ADJ
ejpam-3193	66	13	as	as	ADP
ejpam-3193	66	14	completely	completely	ADV
ejpam-3193	66	15	prime	prime	ADJ
ejpam-3193	66	16	,	,	PUNCT
ejpam-3193	66	17	primary	primary	ADJ
ejpam-3193	66	18	,	,	PUNCT
ejpam-3193	66	19	completely	completely	ADV
ejpam-3193	66	20	primary	primary	ADJ
ejpam-3193	66	21	and	and	CCONJ
ejpam-3193	66	22	so	so	ADV
ejpam-3193	66	23	on	on	ADV
ejpam-3193	66	24	.	.	PUNCT
ejpam-3193	67	1	following	follow	VERB
ejpam-3193	67	2	[	[	X
ejpam-3193	67	3	11	11	NUM
ejpam-3193	67	4	]	]	PUNCT
ejpam-3193	67	5	,	,	PUNCT
ejpam-3193	67	6	an	an	DET
ejpam-3193	67	7	ideal	ideal	NOUN
ejpam-3193	67	8	i	i	PRON
ejpam-3193	67	9	of	of	ADP
ejpam-3193	67	10	a	a	DET
ejpam-3193	67	11	near	near	ADJ
ejpam-3193	67	12	ring	ring	NOUN
ejpam-3193	67	13	n	n	PRON
ejpam-3193	67	14	is	be	AUX
ejpam-3193	67	15	said	say	VERB
ejpam-3193	67	16	to	to	PART
ejpam-3193	67	17	be	be	AUX
ejpam-3193	67	18	a	a	DET
ejpam-3193	67	19	completely	completely	ADV
ejpam-3193	67	20	prime	prime	ADJ
ejpam-3193	67	21	ideal	ideal	NOUN
ejpam-3193	67	22	if	if	SCONJ
ejpam-3193	67	23	x	x	NOUN
ejpam-3193	67	24	,	,	PUNCT
ejpam-3193	67	25	y	y	PROPN
ejpam-3193	67	26	∈	∈	PROPN
ejpam-3193	67	27	n	n	AUX
ejpam-3193	67	28	,	,	PUNCT
ejpam-3193	67	29	xy	xy	PROPN
ejpam-3193	67	30	∈	∈	PROPN
ejpam-3193	68	1	i	i	PRON
ejpam-3193	68	2	implies	imply	VERB
ejpam-3193	68	3	x	x	X
ejpam-3193	68	4	∈	∈	X
ejpam-3193	68	5	i	i	PRON
ejpam-3193	68	6	or	or	CCONJ
ejpam-3193	68	7	y	y	PROPN
ejpam-3193	68	8	∈	∈	PROPN
ejpam-3193	68	9	i	i	PRON
ejpam-3193	69	1	[	[	X
ejpam-3193	69	2	11	11	NUM
ejpam-3193	69	3	]	]	PUNCT
ejpam-3193	69	4	.	.	PUNCT
ejpam-3193	70	1	similarly	similarly	ADV
ejpam-3193	70	2	,	,	PUNCT
ejpam-3193	70	3	an	an	DET
ejpam-3193	70	4	ideal	ideal	NOUN
ejpam-3193	70	5	of	of	ADP
ejpam-3193	70	6	a	a	DET
ejpam-3193	70	7	near	near	ADJ
ejpam-3193	70	8	ring	ring	NOUN
ejpam-3193	70	9	n	n	PRON
ejpam-3193	70	10	is	be	AUX
ejpam-3193	70	11	said	say	VERB
ejpam-3193	70	12	to	to	PART
ejpam-3193	70	13	be	be	AUX
ejpam-3193	70	14	primary	primary	ADJ
ejpam-3193	70	15	ideal	ideal	NOUN
ejpam-3193	70	16	of	of	ADP
ejpam-3193	70	17	n	n	PRON
ejpam-3193	70	18	if	if	SCONJ
ejpam-3193	70	19	x	x	X
ejpam-3193	70	20	,	,	PUNCT
ejpam-3193	70	21	y	y	PROPN
ejpam-3193	70	22	∈	∈	PROPN
ejpam-3193	70	23	n	n	AUX
ejpam-3193	70	24	,	,	PUNCT
ejpam-3193	70	25	xy	xy	PROPN
ejpam-3193	70	26	∈	∈	PROPN
ejpam-3193	71	1	i	i	PRON
ejpam-3193	71	2	implies	imply	VERB
ejpam-3193	71	3	x	x	X
ejpam-3193	71	4	∈	∈	X
ejpam-3193	72	1	i	i	PRON
ejpam-3193	72	2	or	or	CCONJ
ejpam-3193	72	3	ym	ym	INTJ
ejpam-3193	72	4	∈	∈	PROPN
ejpam-3193	72	5	i	i	PRON
ejpam-3193	72	6	for	for	ADP
ejpam-3193	72	7	some	some	DET
ejpam-3193	72	8	m	m	NOUN
ejpam-3193	72	9	∈	∈	PROPN
ejpam-3193	72	10	z.	z.	PROPN
ejpam-3193	73	1	an	an	DET
ejpam-3193	73	2	ideal	ideal	ADJ
ejpam-3193	73	3	i	i	PRON
ejpam-3193	73	4	of	of	ADP
ejpam-3193	73	5	a	a	DET
ejpam-3193	73	6	near	near	ADJ
ejpam-3193	73	7	ring	ring	NOUN
ejpam-3193	73	8	n	n	PRON
ejpam-3193	73	9	is	be	AUX
ejpam-3193	73	10	called	call	VERB
ejpam-3193	73	11	a	a	DET
ejpam-3193	73	12	a.	a.	NOUN
ejpam-3193	73	13	taouti	taouti	NOUN
ejpam-3193	73	14	et	et	PROPN
ejpam-3193	73	15	al	al	PROPN
ejpam-3193	73	16	.	.	PUNCT
ejpam-3193	73	17	/	/	SYM
ejpam-3193	73	18	eur	eur	PROPN
ejpam-3193	73	19	.	.	PUNCT
ejpam-3193	74	1	j.	j.	PROPN
ejpam-3193	74	2	pure	pure	PROPN
ejpam-3193	74	3	appl	appl	PROPN
ejpam-3193	74	4	.	.	PROPN
ejpam-3193	74	5	math	math	PROPN
ejpam-3193	74	6	,	,	PUNCT
ejpam-3193	74	7	11	11	NUM
ejpam-3193	74	8	(	(	PUNCT
ejpam-3193	74	9	2	2	NUM
ejpam-3193	74	10	)	)	PUNCT
ejpam-3193	74	11	(	(	PUNCT
ejpam-3193	74	12	2018	2018	NUM
ejpam-3193	74	13	)	)	PUNCT
ejpam-3193	74	14	,	,	PUNCT
ejpam-3193	74	15	449	449	NUM
ejpam-3193	74	16	-	-	SYM
ejpam-3193	74	17	456	456	NUM
ejpam-3193	74	18	451	451	NUM
ejpam-3193	74	19	completely	completely	ADV
ejpam-3193	74	20	semiprime	semiprime	NOUN
ejpam-3193	74	21	ideal	ideal	NOUN
ejpam-3193	74	22	of	of	ADP
ejpam-3193	74	23	a	a	DET
ejpam-3193	74	24	near	near	ADJ
ejpam-3193	74	25	ring	ring	NOUN
ejpam-3193	74	26	n	n	INTJ
ejpam-3193	74	27	if	if	SCONJ
ejpam-3193	74	28	y2	y2	PROPN
ejpam-3193	74	29	∈	∈	PROPN
ejpam-3193	74	30	i	i	PRON
ejpam-3193	74	31	implies	imply	VERB
ejpam-3193	74	32	y	y	PROPN
ejpam-3193	74	33	∈	∈	PROPN
ejpam-3193	74	34	i	i	PRON
ejpam-3193	74	35	for	for	ADP
ejpam-3193	74	36	all	all	DET
ejpam-3193	74	37	y	y	PROPN
ejpam-3193	74	38	∈	∈	PROPN
ejpam-3193	74	39	n	n	CCONJ
ejpam-3193	75	1	[	[	X
ejpam-3193	75	2	11	11	NUM
ejpam-3193	75	3	]	]	PUNCT
ejpam-3193	75	4	.	.	PUNCT
ejpam-3193	76	1	the	the	DET
ejpam-3193	76	2	ideal	ideal	PROPN
ejpam-3193	76	3	theory	theory	NOUN
ejpam-3193	76	4	is	be	AUX
ejpam-3193	76	5	the	the	DET
ejpam-3193	76	6	most	most	ADV
ejpam-3193	76	7	important	important	ADJ
ejpam-3193	76	8	part	part	NOUN
ejpam-3193	76	9	of	of	ADP
ejpam-3193	76	10	algebra	algebra	NOUN
ejpam-3193	76	11	,	,	PUNCT
ejpam-3193	76	12	different	different	ADJ
ejpam-3193	76	13	types	type	NOUN
ejpam-3193	76	14	of	of	ADP
ejpam-3193	76	15	ideals	ideal	NOUN
ejpam-3193	76	16	in	in	ADP
ejpam-3193	76	17	rings	ring	NOUN
ejpam-3193	76	18	have	have	AUX
ejpam-3193	76	19	been	be	AUX
ejpam-3193	76	20	discussed	discuss	VERB
ejpam-3193	76	21	in	in	ADP
ejpam-3193	76	22	the	the	DET
ejpam-3193	76	23	literature	literature	NOUN
ejpam-3193	76	24	.	.	PUNCT
ejpam-3193	77	1	a	a	DET
ejpam-3193	77	2	right	right	NOUN
ejpam-3193	77	3	(	(	PUNCT
ejpam-3193	77	4	left	left	ADJ
ejpam-3193	77	5	)	)	PUNCT
ejpam-3193	77	6	ideal	ideal	NOUN
ejpam-3193	77	7	of	of	ADP
ejpam-3193	77	8	a	a	DET
ejpam-3193	77	9	γ	γ	NOUN
ejpam-3193	77	10	-	-	PUNCT
ejpam-3193	77	11	ring	ring	NOUN
ejpam-3193	77	12	m	m	NOUN
ejpam-3193	77	13	is	be	AUX
ejpam-3193	77	14	an	an	DET
ejpam-3193	77	15	additive	additive	ADJ
ejpam-3193	77	16	subgroup	subgroup	NOUN
ejpam-3193	77	17	i	i	PROPN
ejpam-3193	77	18	of	of	ADP
ejpam-3193	77	19	m	m	PRON
ejpam-3193	77	20	such	such	ADJ
ejpam-3193	77	21	that	that	DET
ejpam-3193	77	22	iγm	iγm	NOUN
ejpam-3193	77	23	⊆	⊆	NUM
ejpam-3193	77	24	i	i	PROPN
ejpam-3193	77	25	(	(	PUNCT
ejpam-3193	77	26	mγi	mγi	PROPN
ejpam-3193	77	27	⊆	⊆	NUM
ejpam-3193	77	28	i	i	NOUN
ejpam-3193	77	29	)	)	PUNCT
ejpam-3193	77	30	.	.	PUNCT
ejpam-3193	78	1	if	if	SCONJ
ejpam-3193	78	2	i	i	PRON
ejpam-3193	78	3	is	be	AUX
ejpam-3193	78	4	both	both	CCONJ
ejpam-3193	78	5	a	a	DET
ejpam-3193	78	6	right	right	NOUN
ejpam-3193	78	7	and	and	CCONJ
ejpam-3193	78	8	a	a	DET
ejpam-3193	78	9	left	left	ADJ
ejpam-3193	78	10	ideal	ideal	NOUN
ejpam-3193	78	11	,	,	PUNCT
ejpam-3193	78	12	then	then	ADV
ejpam-3193	78	13	we	we	PRON
ejpam-3193	78	14	say	say	VERB
ejpam-3193	78	15	that	that	SCONJ
ejpam-3193	78	16	i	i	PRON
ejpam-3193	78	17	is	be	AUX
ejpam-3193	78	18	an	an	DET
ejpam-3193	78	19	ideal	ideal	NOUN
ejpam-3193	78	20	or	or	CCONJ
ejpam-3193	78	21	a	a	DET
ejpam-3193	78	22	two	two	NUM
ejpam-3193	78	23	-	-	PUNCT
ejpam-3193	78	24	sided	sided	ADJ
ejpam-3193	78	25	ideal	ideal	NOUN
ejpam-3193	78	26	of	of	ADP
ejpam-3193	78	27	m	m	PROPN
ejpam-3193	78	28	.	.	PUNCT
ejpam-3193	79	1	in	in	ADP
ejpam-3193	79	2	rings	ring	NOUN
ejpam-3193	79	3	,	,	PUNCT
ejpam-3193	79	4	an	an	DET
ejpam-3193	79	5	ideal	ideal	NOUN
ejpam-3193	79	6	p	p	NOUN
ejpam-3193	79	7	is	be	AUX
ejpam-3193	79	8	prime	prime	ADJ
ejpam-3193	79	9	ideal	ideal	NOUN
ejpam-3193	79	10	if	if	SCONJ
ejpam-3193	80	1	and	and	CCONJ
ejpam-3193	80	2	only	only	ADV
ejpam-3193	80	3	if	if	SCONJ
ejpam-3193	80	4	a	a	PRON
ejpam-3193	80	5	and	and	CCONJ
ejpam-3193	80	6	b	b	NOUN
ejpam-3193	80	7	are	be	AUX
ejpam-3193	80	8	ideals	ideal	NOUN
ejpam-3193	80	9	in	in	ADP
ejpam-3193	80	10	m	m	PRON
ejpam-3193	80	11	such	such	ADJ
ejpam-3193	80	12	that	that	SCONJ
ejpam-3193	80	13	ab	ab	PROPN
ejpam-3193	80	14	⊆	⊆	NUM
ejpam-3193	80	15	p	p	NOUN
ejpam-3193	80	16	,	,	PUNCT
ejpam-3193	80	17	then	then	ADV
ejpam-3193	80	18	a	a	DET
ejpam-3193	80	19	⊆	⊆	NUM
ejpam-3193	80	20	p	p	NOUN
ejpam-3193	80	21	or	or	CCONJ
ejpam-3193	80	22	b	b	NOUN
ejpam-3193	80	23	⊆	⊆	NUM
ejpam-3193	80	24	p	p	NOUN
ejpam-3193	81	1	[	[	X
ejpam-3193	81	2	13	13	NUM
ejpam-3193	81	3	]	]	PUNCT
ejpam-3193	81	4	.	.	PUNCT
ejpam-3193	82	1	the	the	DET
ejpam-3193	82	2	prime	prime	ADJ
ejpam-3193	82	3	ideals	ideal	NOUN
ejpam-3193	82	4	of	of	ADP
ejpam-3193	82	5	the	the	DET
ejpam-3193	82	6	γn	γn	NOUN
ejpam-3193	82	7	,	,	PUNCT
ejpam-3193	82	8	m	m	VERB
ejpam-3193	82	9	-ring	-re	VERB
ejpam-3193	82	10	mm	mm	PROPN
ejpam-3193	82	11	,	,	PUNCT
ejpam-3193	82	12	n	n	X
ejpam-3193	82	13	are	be	AUX
ejpam-3193	82	14	the	the	DET
ejpam-3193	82	15	sets	set	NOUN
ejpam-3193	82	16	pm	pm	VERB
ejpam-3193	82	17	,	,	PUNCT
ejpam-3193	82	18	n	n	PRON
ejpam-3193	82	19	corresponding	correspond	VERB
ejpam-3193	82	20	to	to	ADP
ejpam-3193	82	21	the	the	DET
ejpam-3193	82	22	prime	prime	ADJ
ejpam-3193	82	23	ideals	ideal	NOUN
ejpam-3193	82	24	p	p	NOUN
ejpam-3193	82	25	of	of	ADP
ejpam-3193	82	26	the	the	DET
ejpam-3193	82	27	γ	γ	NOUN
ejpam-3193	82	28	-	-	PUNCT
ejpam-3193	82	29	ring	ring	NOUN
ejpam-3193	82	30	m	m	NOUN
ejpam-3193	82	31	[	[	X
ejpam-3193	82	32	13	13	NUM
ejpam-3193	82	33	]	]	PUNCT
ejpam-3193	82	34	.	.	PUNCT
ejpam-3193	83	1	if	if	SCONJ
ejpam-3193	83	2	p	p	NOUN
ejpam-3193	83	3	is	be	AUX
ejpam-3193	83	4	an	an	DET
ejpam-3193	83	5	ideal	ideal	NOUN
ejpam-3193	83	6	in	in	ADP
ejpam-3193	83	7	a	a	DET
ejpam-3193	83	8	γ	γ	NOUN
ejpam-3193	83	9	-	-	NOUN
ejpam-3193	83	10	ring	ring	NOUN
ejpam-3193	83	11	m	m	VERB
ejpam-3193	83	12	then	then	ADV
ejpam-3193	83	13	,	,	PUNCT
ejpam-3193	83	14	(	(	PUNCT
ejpam-3193	83	15	i	i	NOUN
ejpam-3193	83	16	)	)	PUNCT
ejpam-3193	83	17	ideal	ideal	NOUN
ejpam-3193	83	18	p	p	NOUN
ejpam-3193	83	19	is	be	AUX
ejpam-3193	83	20	a	a	DET
ejpam-3193	83	21	prime	prime	ADJ
ejpam-3193	83	22	ideal	ideal	NOUN
ejpam-3193	83	23	of	of	ADP
ejpam-3193	83	24	m	m	PRON
ejpam-3193	83	25	,	,	PUNCT
ejpam-3193	83	26	(	(	PUNCT
ejpam-3193	83	27	ii	ii	NOUN
ejpam-3193	83	28	)	)	PUNCT
ejpam-3193	83	29	if	if	SCONJ
ejpam-3193	83	30	a	a	DET
ejpam-3193	83	31	,	,	PUNCT
ejpam-3193	83	32	b	b	NOUN
ejpam-3193	83	33	∈m	∈m	NOUN
ejpam-3193	83	34	and	and	CCONJ
ejpam-3193	83	35	aγmγb	aγmγb	VERB
ejpam-3193	83	36	⊆	⊆	NUM
ejpam-3193	83	37	p	p	NOUN
ejpam-3193	83	38	then	then	ADV
ejpam-3193	83	39	either	either	CCONJ
ejpam-3193	83	40	a	a	DET
ejpam-3193	83	41	∈	∈	PROPN
ejpam-3193	83	42	p	p	NOUN
ejpam-3193	83	43	or	or	CCONJ
ejpam-3193	83	44	b	b	NOUN
ejpam-3193	83	45	∈	∈	PROPN
ejpam-3193	83	46	p	p	X
ejpam-3193	83	47	,	,	PUNCT
ejpam-3193	83	48	(	(	PUNCT
ejpam-3193	83	49	iii	iii	NOUN
ejpam-3193	83	50	)	)	PUNCT
ejpam-3193	83	51	if	if	SCONJ
ejpam-3193	83	52	ideal	ideal	ADJ
ejpam-3193	83	53	generated	generate	VERB
ejpam-3193	83	54	by	by	ADP
ejpam-3193	83	55	<	<	X
ejpam-3193	83	56	a	a	X
ejpam-3193	83	57	>	>	X
ejpam-3193	83	58	and	and	CCONJ
ejpam-3193	83	59	<	<	X
ejpam-3193	83	60	b	b	X
ejpam-3193	83	61	>	>	X
ejpam-3193	83	62	are	be	AUX
ejpam-3193	83	63	called	call	VERB
ejpam-3193	83	64	principal	principal	ADJ
ejpam-3193	83	65	ideals	ideal	NOUN
ejpam-3193	83	66	in	in	ADP
ejpam-3193	83	67	m	m	PROPN
ejpam-3193	83	68	and	and	CCONJ
ejpam-3193	83	69	<	<	X
ejpam-3193	83	70	a	a	X
ejpam-3193	83	71	>	>	X
ejpam-3193	83	72	γ	γ	X
ejpam-3193	83	73	<	<	X
ejpam-3193	83	74	b	b	X
ejpam-3193	83	75	>	>	PUNCT
ejpam-3193	83	76	⊆	⊆	NUM
ejpam-3193	83	77	p	p	NOUN
ejpam-3193	83	78	,	,	PUNCT
ejpam-3193	83	79	then	then	ADV
ejpam-3193	83	80	a	a	DET
ejpam-3193	83	81	∈	∈	PROPN
ejpam-3193	83	82	p	p	NOUN
ejpam-3193	83	83	or	or	CCONJ
ejpam-3193	83	84	b	b	NOUN
ejpam-3193	83	85	∈	∈	PROPN
ejpam-3193	83	86	p	p	NOUN
ejpam-3193	83	87	,	,	PUNCT
ejpam-3193	83	88	(	(	PUNCT
ejpam-3193	83	89	iv	iv	X
ejpam-3193	83	90	)	)	PUNCT
ejpam-3193	83	91	if	if	SCONJ
ejpam-3193	83	92	u	u	NOUN
ejpam-3193	83	93	and	and	CCONJ
ejpam-3193	83	94	v	v	NOUN
ejpam-3193	83	95	are	be	AUX
ejpam-3193	83	96	right	right	ADJ
ejpam-3193	83	97	ideals	ideal	NOUN
ejpam-3193	83	98	in	in	ADP
ejpam-3193	83	99	m	m	PROPN
ejpam-3193	83	100	with	with	ADP
ejpam-3193	83	101	uγv	uγv	ADJ
ejpam-3193	83	102	⊆	⊆	NUM
ejpam-3193	83	103	p	p	NOUN
ejpam-3193	83	104	,	,	PUNCT
ejpam-3193	83	105	then	then	ADV
ejpam-3193	83	106	u	u	NOUN
ejpam-3193	83	107	⊆	⊆	NUM
ejpam-3193	83	108	p	p	NOUN
ejpam-3193	83	109	or	or	CCONJ
ejpam-3193	83	110	v	v	ADP
ejpam-3193	83	111	⊆	⊆	NUM
ejpam-3193	83	112	p	p	NOUN
ejpam-3193	83	113	,	,	PUNCT
ejpam-3193	83	114	(	(	PUNCT
ejpam-3193	83	115	v	v	NOUN
ejpam-3193	83	116	)	)	PUNCT
ejpam-3193	83	117	if	if	SCONJ
ejpam-3193	83	118	u	u	NOUN
ejpam-3193	83	119	and	and	CCONJ
ejpam-3193	83	120	v	v	NOUN
ejpam-3193	83	121	are	be	AUX
ejpam-3193	83	122	left	leave	VERB
ejpam-3193	83	123	ideals	ideal	NOUN
ejpam-3193	83	124	in	in	ADP
ejpam-3193	83	125	m	m	PROPN
ejpam-3193	83	126	with	with	ADP
ejpam-3193	83	127	uγv	uγv	ADJ
ejpam-3193	83	128	⊆	⊆	NUM
ejpam-3193	83	129	p	p	NOUN
ejpam-3193	83	130	,	,	PUNCT
ejpam-3193	83	131	either	either	CCONJ
ejpam-3193	83	132	u	u	PROPN
ejpam-3193	83	133	⊆	⊆	NUM
ejpam-3193	83	134	p	p	NOUN
ejpam-3193	83	135	or	or	CCONJ
ejpam-3193	83	136	v	v	ADP
ejpam-3193	83	137	⊆	⊆	NUM
ejpam-3193	83	138	p	p	NOUN
ejpam-3193	84	1	[	[	X
ejpam-3193	84	2	16	16	NUM
ejpam-3193	84	3	]	]	PUNCT
ejpam-3193	84	4	.	.	PUNCT
ejpam-3193	85	1	γ	γ	X
ejpam-3193	85	2	-	-	PUNCT
ejpam-3193	85	3	near	near	ADJ
ejpam-3193	85	4	rings	ring	NOUN
ejpam-3193	85	5	were	be	AUX
ejpam-3193	85	6	introduced	introduce	VERB
ejpam-3193	85	7	by	by	ADP
ejpam-3193	85	8	satyanarayana	satyanarayana	PROPN
ejpam-3193	85	9	bhavanari	bhavanari	PROPN
ejpam-3193	85	10	(	(	PUNCT
ejpam-3193	85	11	see	see	VERB
ejpam-3193	85	12	[	[	X
ejpam-3193	85	13	14	14	NUM
ejpam-3193	85	14	]	]	PUNCT
ejpam-3193	85	15	,	,	PUNCT
ejpam-3193	85	16	[	[	X
ejpam-3193	85	17	15	15	NUM
ejpam-3193	85	18	]	]	NUM
ejpam-3193	85	19	)	)	PUNCT
ejpam-3193	85	20	.	.	PUNCT
ejpam-3193	86	1	a	a	DET
ejpam-3193	86	2	subset	subset	NOUN
ejpam-3193	86	3	a	a	PRON
ejpam-3193	86	4	of	of	ADP
ejpam-3193	86	5	a	a	DET
ejpam-3193	86	6	γ	γ	NOUN
ejpam-3193	86	7	-	-	PUNCT
ejpam-3193	86	8	near	near	ADP
ejpam-3193	86	9	-	-	PUNCT
ejpam-3193	86	10	ring	ring	NOUN
ejpam-3193	86	11	m	m	NOUN
ejpam-3193	86	12	is	be	AUX
ejpam-3193	86	13	called	call	VERB
ejpam-3193	86	14	a	a	DET
ejpam-3193	86	15	left	left	ADJ
ejpam-3193	86	16	(	(	PUNCT
ejpam-3193	86	17	resp	resp	NOUN
ejpam-3193	86	18	.	.	PUNCT
ejpam-3193	87	1	right	right	ADJ
ejpam-3193	87	2	)	)	PUNCT
ejpam-3193	87	3	ideal	ideal	NOUN
ejpam-3193	87	4	of	of	ADP
ejpam-3193	87	5	m	m	PRON
ejpam-3193	87	6	if	if	SCONJ
ejpam-3193	87	7	(	(	PUNCT
ejpam-3193	87	8	a,+	a,+	NOUN
ejpam-3193	87	9	)	)	PUNCT
ejpam-3193	87	10	is	be	AUX
ejpam-3193	87	11	a	a	DET
ejpam-3193	87	12	normal	normal	ADJ
ejpam-3193	87	13	divisor	divisor	NOUN
ejpam-3193	87	14	of	of	ADP
ejpam-3193	87	15	(	(	PUNCT
ejpam-3193	87	16	m,+	m,+	NOUN
ejpam-3193	87	17	)	)	PUNCT
ejpam-3193	87	18	,	,	PUNCT
ejpam-3193	87	19	uα(x	uα(x	PUNCT
ejpam-3193	88	1	+	+	CCONJ
ejpam-3193	88	2	v	v	X
ejpam-3193	88	3	)	)	PUNCT
ejpam-3193	88	4	−	−	PROPN
ejpam-3193	89	1	uαv	uαv	PROPN
ejpam-3193	89	2	∈	∈	PROPN
ejpam-3193	89	3	a	a	DET
ejpam-3193	89	4	(	(	PUNCT
ejpam-3193	89	5	resp	resp	NOUN
ejpam-3193	89	6	.	.	PUNCT
ejpam-3193	90	1	xαu	xαu	PROPN
ejpam-3193	90	2	∈	∈	PROPN
ejpam-3193	90	3	a	a	PRON
ejpam-3193	90	4	)	)	PUNCT
ejpam-3193	90	5	for	for	ADP
ejpam-3193	90	6	all	all	PRON
ejpam-3193	90	7	x	x	SYM
ejpam-3193	90	8	∈	∈	PROPN
ejpam-3193	90	9	a	a	PRON
ejpam-3193	90	10	,	,	PUNCT
ejpam-3193	90	11	α	α	PROPN
ejpam-3193	90	12	∈	∈	PROPN
ejpam-3193	90	13	γ	γ	X
ejpam-3193	90	14	and	and	CCONJ
ejpam-3193	90	15	u	u	NOUN
ejpam-3193	90	16	,	,	PUNCT
ejpam-3193	90	17	v	v	ADP
ejpam-3193	90	18	∈	∈	NOUN
ejpam-3193	90	19	m	m	NOUN
ejpam-3193	90	20	.	.	PUNCT
ejpam-3193	91	1	an	an	DET
ejpam-3193	91	2	ideal	ideal	ADJ
ejpam-3193	91	3	p	p	NOUN
ejpam-3193	91	4	of	of	ADP
ejpam-3193	91	5	γ	γ	X
ejpam-3193	91	6	-	-	PUNCT
ejpam-3193	91	7	near	near	ADJ
ejpam-3193	91	8	ring	ring	NOUN
ejpam-3193	91	9	(	(	PUNCT
ejpam-3193	91	10	m,+	m,+	PROPN
ejpam-3193	91	11	,	,	PUNCT
ejpam-3193	91	12	(	(	PUNCT
ejpam-3193	91	13	.)γ	.)γ	NUM
ejpam-3193	91	14	)	)	PUNCT
ejpam-3193	91	15	is	be	AUX
ejpam-3193	91	16	called	call	VERB
ejpam-3193	91	17	prime	prime	ADJ
ejpam-3193	91	18	,	,	PUNCT
ejpam-3193	91	19	if	if	SCONJ
ejpam-3193	91	20	for	for	ADP
ejpam-3193	91	21	every	every	DET
ejpam-3193	91	22	two	two	NUM
ejpam-3193	91	23	ideals	ideal	NOUN
ejpam-3193	91	24	i	i	PRON
ejpam-3193	91	25	,	,	PUNCT
ejpam-3193	91	26	j	j	PROPN
ejpam-3193	91	27	of	of	ADP
ejpam-3193	91	28	m	m	PROPN
ejpam-3193	91	29	,	,	PUNCT
ejpam-3193	91	30	iγj	iγj	VERB
ejpam-3193	91	31	⊆	⊆	NUM
ejpam-3193	91	32	p	p	NOUN
ejpam-3193	91	33	implies	imply	VERB
ejpam-3193	91	34	i	i	PRON
ejpam-3193	91	35	⊆	⊆	NUM
ejpam-3193	91	36	p	p	NOUN
ejpam-3193	91	37	or	or	CCONJ
ejpam-3193	91	38	j	j	PROPN
ejpam-3193	91	39	⊆	⊆	NUM
ejpam-3193	91	40	p	p	NOUN
ejpam-3193	91	41	.	.	PUNCT
ejpam-3193	92	1	an	an	DET
ejpam-3193	92	2	ideal	ideal	ADJ
ejpam-3193	92	3	p	p	NOUN
ejpam-3193	92	4	of	of	ADP
ejpam-3193	92	5	a	a	DET
ejpam-3193	92	6	γ	γ	NOUN
ejpam-3193	92	7	-	-	PUNCT
ejpam-3193	92	8	near	near	ADP
ejpam-3193	92	9	-	-	PUNCT
ejpam-3193	92	10	ring	ring	NOUN
ejpam-3193	92	11	n	n	PRON
ejpam-3193	92	12	is	be	AUX
ejpam-3193	92	13	called	call	VERB
ejpam-3193	92	14	a	a	DET
ejpam-3193	92	15	completely	completely	ADV
ejpam-3193	92	16	primary	primary	ADJ
ejpam-3193	92	17	ideal	ideal	NOUN
ejpam-3193	92	18	if	if	SCONJ
ejpam-3193	92	19	for	for	ADP
ejpam-3193	92	20	a	a	DET
ejpam-3193	92	21	,	,	PUNCT
ejpam-3193	92	22	b	b	PROPN
ejpam-3193	92	23	∈	∈	PROPN
ejpam-3193	92	24	n	n	NOUN
ejpam-3193	92	25	and	and	CCONJ
ejpam-3193	92	26	γ	γ	PROPN
ejpam-3193	92	27	∈	∈	PROPN
ejpam-3193	92	28	γ	γ	NOUN
ejpam-3193	92	29	such	such	ADJ
ejpam-3193	92	30	that	that	SCONJ
ejpam-3193	92	31	aγb	aγb	ADV
ejpam-3193	92	32	∈	∈	PROPN
ejpam-3193	92	33	p	p	PROPN
ejpam-3193	92	34	implies	imply	VERB
ejpam-3193	92	35	that	that	SCONJ
ejpam-3193	92	36	a	a	DET
ejpam-3193	92	37	∈	∈	PROPN
ejpam-3193	92	38	p	p	NOUN
ejpam-3193	92	39	or	or	CCONJ
ejpam-3193	92	40	b	b	NOUN
ejpam-3193	92	41	∈	∈	PROPN
ejpam-3193	92	42	p	p	NOUN
ejpam-3193	92	43	,	,	PUNCT
ejpam-3193	92	44	for	for	ADP
ejpam-3193	92	45	some	some	DET
ejpam-3193	92	46	positive	positive	ADJ
ejpam-3193	92	47	integer	integer	NOUN
ejpam-3193	92	48	n	n	PROPN
ejpam-3193	92	49	[	[	X
ejpam-3193	92	50	17	17	NUM
ejpam-3193	92	51	]	]	PUNCT
ejpam-3193	92	52	.	.	PUNCT
ejpam-3193	93	1	if	if	SCONJ
ejpam-3193	93	2	an	an	DET
ejpam-3193	93	3	ideal	ideal	NOUN
ejpam-3193	93	4	i	i	PRON
ejpam-3193	93	5	of	of	ADP
ejpam-3193	93	6	γ	γ	X
ejpam-3193	93	7	-	-	PUNCT
ejpam-3193	93	8	near	near	ADP
ejpam-3193	93	9	-	-	PUNCT
ejpam-3193	93	10	ring	ring	NOUN
ejpam-3193	93	11	m	m	NOUN
ejpam-3193	93	12	is	be	AUX
ejpam-3193	93	13	maximal	maximal	ADJ
ejpam-3193	93	14	,	,	PUNCT
ejpam-3193	93	15	then	then	ADV
ejpam-3193	93	16	it	it	PRON
ejpam-3193	93	17	is	be	AUX
ejpam-3193	93	18	prime	prime	ADJ
ejpam-3193	93	19	or	or	CCONJ
ejpam-3193	93	20	mγm	mγm	NOUN
ejpam-3193	93	21	=	=	SYM
ejpam-3193	93	22	i	i	PRON
ejpam-3193	94	1	[	[	X
ejpam-3193	94	2	7	7	NUM
ejpam-3193	94	3	]	]	PUNCT
ejpam-3193	94	4	.	.	PUNCT
ejpam-3193	95	1	if	if	SCONJ
ejpam-3193	95	2	(	(	PUNCT
ejpam-3193	95	3	m,+	m,+	INTJ
ejpam-3193	95	4	,	,	PUNCT
ejpam-3193	95	5	(	(	PUNCT
ejpam-3193	95	6	.)γ	.)γ	NUM
ejpam-3193	95	7	)	)	PUNCT
ejpam-3193	95	8	is	be	AUX
ejpam-3193	95	9	a	a	DET
ejpam-3193	95	10	γ	γ	NOUN
ejpam-3193	95	11	-	-	PUNCT
ejpam-3193	95	12	near	near	ADP
ejpam-3193	95	13	-	-	PUNCT
ejpam-3193	95	14	ring	ring	NOUN
ejpam-3193	95	15	such	such	ADJ
ejpam-3193	95	16	that	that	PRON
ejpam-3193	95	17	for	for	ADP
ejpam-3193	95	18	any	any	DET
ejpam-3193	95	19	γ	γ	PROPN
ejpam-3193	95	20	∈	∈	PROPN
ejpam-3193	95	21	γ	γ	NOUN
ejpam-3193	95	22	there	there	PRON
ejpam-3193	95	23	is	be	VERB
ejpam-3193	95	24	an	an	DET
ejpam-3193	95	25	element	element	NOUN
ejpam-3193	95	26	which	which	PRON
ejpam-3193	95	27	is	be	AUX
ejpam-3193	95	28	γ	γ	NOUN
ejpam-3193	95	29	-	-	NOUN
ejpam-3193	95	30	unit	unit	NOUN
ejpam-3193	95	31	,	,	PUNCT
ejpam-3193	95	32	then	then	ADV
ejpam-3193	95	33	every	every	DET
ejpam-3193	95	34	maximal	maximal	ADJ
ejpam-3193	95	35	ideal	ideal	NOUN
ejpam-3193	95	36	i	i	PRON
ejpam-3193	95	37	of	of	ADP
ejpam-3193	95	38	m	m	PROPN
ejpam-3193	95	39	is	be	AUX
ejpam-3193	95	40	prime	prime	ADJ
ejpam-3193	95	41	[	[	X
ejpam-3193	95	42	7	7	NUM
ejpam-3193	95	43	]	]	PUNCT
ejpam-3193	95	44	.	.	PUNCT
ejpam-3193	96	1	for	for	ADP
ejpam-3193	96	2	every	every	DET
ejpam-3193	96	3	ideal	ideal	NOUN
ejpam-3193	96	4	i	i	PRON
ejpam-3193	96	5	of	of	ADP
ejpam-3193	96	6	γ	γ	PROPN
ejpam-3193	96	7	-	-	PUNCT
ejpam-3193	96	8	nearring	nearre	VERB
ejpam-3193	96	9	m	m	VERB
ejpam-3193	96	10	exists	exist	VERB
ejpam-3193	96	11	prime	prime	ADJ
ejpam-3193	96	12	minimal	minimal	ADJ
ejpam-3193	96	13	ideal	ideal	NOUN
ejpam-3193	96	14	of	of	ADP
ejpam-3193	96	15	i	i	PRON
ejpam-3193	97	1	[	[	X
ejpam-3193	97	2	7	7	NUM
ejpam-3193	97	3	]	]	PUNCT
ejpam-3193	97	4	.	.	PUNCT
ejpam-3193	98	1	in	in	ADP
ejpam-3193	98	2	this	this	DET
ejpam-3193	98	3	note	note	NOUN
ejpam-3193	98	4	first	first	ADV
ejpam-3193	98	5	we	we	PRON
ejpam-3193	98	6	introduce	introduce	VERB
ejpam-3193	98	7	the	the	DET
ejpam-3193	98	8	notion	notion	NOUN
ejpam-3193	98	9	of	of	ADP
ejpam-3193	98	10	almost	almost	ADV
ejpam-3193	98	11	prime	prime	ADJ
ejpam-3193	98	12	ideals	ideal	NOUN
ejpam-3193	98	13	in	in	ADP
ejpam-3193	98	14	γ	γ	NOUN
ejpam-3193	98	15	-	-	PUNCT
ejpam-3193	98	16	near	near	ADP
ejpam-3193	98	17	-	-	PUNCT
ejpam-3193	98	18	rings	ring	NOUN
ejpam-3193	98	19	along	along	ADP
ejpam-3193	98	20	with	with	ADP
ejpam-3193	98	21	few	few	ADJ
ejpam-3193	98	22	of	of	ADP
ejpam-3193	98	23	their	their	PRON
ejpam-3193	98	24	characterizations	characterization	NOUN
ejpam-3193	98	25	.	.	PUNCT
ejpam-3193	99	1	finally	finally	ADV
ejpam-3193	99	2	,	,	PUNCT
ejpam-3193	99	3	we	we	PRON
ejpam-3193	99	4	present	present	VERB
ejpam-3193	99	5	the	the	DET
ejpam-3193	99	6	interesting	interesting	ADJ
ejpam-3193	99	7	relations	relation	NOUN
ejpam-3193	99	8	of	of	ADP
ejpam-3193	99	9	an	an	DET
ejpam-3193	99	10	almost	almost	ADV
ejpam-3193	99	11	prime	prime	ADJ
ejpam-3193	99	12	with	with	ADP
ejpam-3193	99	13	the	the	DET
ejpam-3193	99	14	prime	prime	ADJ
ejpam-3193	99	15	and	and	CCONJ
ejpam-3193	99	16	primary	primary	ADJ
ejpam-3193	99	17	ideal	ideal	NOUN
ejpam-3193	99	18	in	in	ADP
ejpam-3193	99	19	γ	γ	PROPN
ejpam-3193	99	20	-	-	PUNCT
ejpam-3193	99	21	near	near	ADP
ejpam-3193	99	22	-	-	PUNCT
ejpam-3193	99	23	rings	ring	NOUN
ejpam-3193	99	24	.	.	PUNCT
ejpam-3193	100	1	2	2	X
ejpam-3193	100	2	.	.	X
ejpam-3193	100	3	almost	almost	ADV
ejpam-3193	100	4	prime	prime	ADJ
ejpam-3193	100	5	ideal	ideal	NOUN
ejpam-3193	100	6	in	in	ADP
ejpam-3193	100	7	γ	γ	PROPN
ejpam-3193	100	8	-	-	PUNCT
ejpam-3193	100	9	near	near	ADP
ejpam-3193	100	10	-	-	PUNCT
ejpam-3193	100	11	ring	ring	NOUN
ejpam-3193	100	12	in	in	ADP
ejpam-3193	100	13	this	this	DET
ejpam-3193	100	14	section	section	NOUN
ejpam-3193	100	15	we	we	PRON
ejpam-3193	100	16	introduce	introduce	VERB
ejpam-3193	100	17	almost	almost	ADV
ejpam-3193	100	18	prime	prime	ADJ
ejpam-3193	100	19	ideal	ideal	NOUN
ejpam-3193	100	20	in	in	ADP
ejpam-3193	100	21	γ	γ	PROPN
ejpam-3193	100	22	-	-	PUNCT
ejpam-3193	100	23	near	near	ADP
ejpam-3193	100	24	-	-	PUNCT
ejpam-3193	100	25	rings	ring	NOUN
ejpam-3193	100	26	.	.	PUNCT
ejpam-3193	101	1	furthermore	furthermore	ADV
ejpam-3193	101	2	,	,	PUNCT
ejpam-3193	101	3	we	we	PRON
ejpam-3193	101	4	also	also	ADV
ejpam-3193	101	5	present	present	VERB
ejpam-3193	101	6	its	its	PRON
ejpam-3193	101	7	implications	implication	NOUN
ejpam-3193	101	8	with	with	ADP
ejpam-3193	101	9	the	the	DET
ejpam-3193	101	10	some	some	DET
ejpam-3193	101	11	ideals	ideal	NOUN
ejpam-3193	101	12	,	,	PUNCT
ejpam-3193	101	13	we	we	PRON
ejpam-3193	101	14	start	start	VERB
ejpam-3193	101	15	with	with	ADP
ejpam-3193	101	16	the	the	DET
ejpam-3193	101	17	following	follow	VERB
ejpam-3193	101	18	definition	definition	NOUN
ejpam-3193	101	19	.	.	PUNCT
ejpam-3193	102	1	definition	definition	NOUN
ejpam-3193	102	2	1	1	NUM
ejpam-3193	102	3	.	.	PUNCT
ejpam-3193	103	1	let	let	VERB
ejpam-3193	103	2	m	m	PRON
ejpam-3193	103	3	be	be	AUX
ejpam-3193	103	4	γ	γ	X
ejpam-3193	103	5	-	-	PUNCT
ejpam-3193	103	6	near	near	ADP
ejpam-3193	103	7	-	-	PUNCT
ejpam-3193	103	8	ring	ring	NOUN
ejpam-3193	103	9	and	and	CCONJ
ejpam-3193	103	10	p	p	NOUN
ejpam-3193	103	11	be	be	AUX
ejpam-3193	103	12	a	a	DET
ejpam-3193	103	13	prime	prime	ADJ
ejpam-3193	103	14	ideal	ideal	NOUN
ejpam-3193	103	15	of	of	ADP
ejpam-3193	103	16	m	m	PRON
ejpam-3193	103	17	then	then	ADV
ejpam-3193	103	18	p	p	NOUN
ejpam-3193	103	19	is	be	AUX
ejpam-3193	103	20	almost	almost	ADV
ejpam-3193	103	21	prime	prime	ADJ
ejpam-3193	103	22	ideal	ideal	NOUN
ejpam-3193	103	23	if	if	SCONJ
ejpam-3193	103	24	a	a	DET
ejpam-3193	103	25	,	,	PUNCT
ejpam-3193	103	26	b	b	X
ejpam-3193	103	27	∈	∈	PROPN
ejpam-3193	103	28	r	r	NOUN
ejpam-3193	103	29	,	,	PUNCT
ejpam-3193	103	30	ab	ab	PROPN
ejpam-3193	103	31	∈	∈	PROPN
ejpam-3193	104	1	p	p	X
ejpam-3193	104	2	−	−	PROPN
ejpam-3193	104	3	pγp	pγp	PROPN
ejpam-3193	104	4	,	,	PUNCT
ejpam-3193	104	5	either	either	CCONJ
ejpam-3193	104	6	a	a	DET
ejpam-3193	104	7	∈	∈	PROPN
ejpam-3193	104	8	p	p	NOUN
ejpam-3193	104	9	or	or	CCONJ
ejpam-3193	104	10	b	b	NOUN
ejpam-3193	104	11	∈	∈	PROPN
ejpam-3193	104	12	p	p	NOUN
ejpam-3193	104	13	.	.	PUNCT
ejpam-3193	104	14	example	example	NOUN
ejpam-3193	105	1	1	1	NUM
ejpam-3193	105	2	.	.	PUNCT
ejpam-3193	105	3	suppose	suppose	VERB
ejpam-3193	105	4	z8	z8	NOUN
ejpam-3193	105	5	=	=	SYM
ejpam-3193	105	6	{	{	PUNCT
ejpam-3193	105	7	0	0	NUM
ejpam-3193	105	8	,	,	PUNCT
ejpam-3193	105	9	1	1	NUM
ejpam-3193	105	10	,	,	PUNCT
ejpam-3193	105	11	2	2	NUM
ejpam-3193	105	12	,	,	PUNCT
ejpam-3193	105	13	3	3	NUM
ejpam-3193	105	14	,	,	PUNCT
ejpam-3193	105	15	4	4	NUM
ejpam-3193	105	16	,	,	PUNCT
ejpam-3193	105	17	5	5	NUM
ejpam-3193	105	18	,	,	PUNCT
ejpam-3193	105	19	6	6	NUM
ejpam-3193	105	20	,	,	PUNCT
ejpam-3193	105	21	7	7	NUM
ejpam-3193	105	22	}	}	PUNCT
ejpam-3193	105	23	and	and	CCONJ
ejpam-3193	105	24	γ	γ	X
ejpam-3193	105	25	=	=	SYM
ejpam-3193	105	26	{	{	PUNCT
ejpam-3193	105	27	0	0	NUM
ejpam-3193	105	28	,	,	PUNCT
ejpam-3193	105	29	2	2	NUM
ejpam-3193	105	30	,	,	PUNCT
ejpam-3193	105	31	4	4	NUM
ejpam-3193	105	32	}	}	PUNCT
ejpam-3193	105	33	.	.	PUNCT
ejpam-3193	106	1	let	let	VERB
ejpam-3193	106	2	p	p	NOUN
ejpam-3193	106	3	=	=	NOUN
ejpam-3193	106	4	2z8	2z8	NUM
ejpam-3193	106	5	=	=	SYM
ejpam-3193	106	6	{	{	PUNCT
ejpam-3193	106	7	0	0	NUM
ejpam-3193	106	8	,	,	PUNCT
ejpam-3193	106	9	2	2	NUM
ejpam-3193	106	10	,	,	PUNCT
ejpam-3193	106	11	4	4	NUM
ejpam-3193	106	12	}	}	PUNCT
ejpam-3193	106	13	be	be	AUX
ejpam-3193	106	14	a	a	DET
ejpam-3193	106	15	prime	prime	ADJ
ejpam-3193	106	16	ideal	ideal	NOUN
ejpam-3193	106	17	in	in	ADP
ejpam-3193	106	18	z8	z8	NOUN
ejpam-3193	106	19	and	and	CCONJ
ejpam-3193	106	20	consider	consider	VERB
ejpam-3193	106	21	pγp	pγp	ADJ
ejpam-3193	106	22	=	=	SYM
ejpam-3193	106	23	{	{	PUNCT
ejpam-3193	106	24	0	0	NUM
ejpam-3193	106	25	,	,	PUNCT
ejpam-3193	106	26	6	6	NUM
ejpam-3193	106	27	}	}	PUNCT
ejpam-3193	106	28	,	,	PUNCT
ejpam-3193	106	29	p	p	NOUN
ejpam-3193	106	30	−pγp	−pγp	NOUN
ejpam-3193	106	31	=	=	PUNCT
ejpam-3193	106	32	{	{	PUNCT
ejpam-3193	106	33	2	2	NUM
ejpam-3193	106	34	,	,	PUNCT
ejpam-3193	106	35	4	4	NUM
ejpam-3193	106	36	}	}	PUNCT
ejpam-3193	106	37	.	.	PUNCT
ejpam-3193	107	1	here	here	ADV
ejpam-3193	107	2	2	2	NUM
ejpam-3193	107	3	,	,	PUNCT
ejpam-3193	107	4	3	3	NUM
ejpam-3193	107	5	∈	∈	PROPN
ejpam-3193	107	6	z8	z8	NOUN
ejpam-3193	107	7	and	and	CCONJ
ejpam-3193	107	8	2.2.3	2.2.3	NUM
ejpam-3193	107	9	=	=	SYM
ejpam-3193	107	10	4	4	NUM
ejpam-3193	107	11	∈	∈	NOUN
ejpam-3193	107	12	p	p	NOUN
ejpam-3193	108	1	−	−	NOUN
ejpam-3193	108	2	pγp	pγp	INTJ
ejpam-3193	108	3	where	where	SCONJ
ejpam-3193	108	4	2	2	NUM
ejpam-3193	108	5	∈	∈	NOUN
ejpam-3193	108	6	p	p	NOUN
ejpam-3193	108	7	and	and	CCONJ
ejpam-3193	108	8	3	3	NUM
ejpam-3193	108	9	6∈	6∈	NOUN
ejpam-3193	108	10	p	p	NOUN
ejpam-3193	108	11	.	.	PUNCT
ejpam-3193	109	1	similarly	similarly	ADV
ejpam-3193	109	2	we	we	PRON
ejpam-3193	109	3	can	can	AUX
ejpam-3193	109	4	check	check	VERB
ejpam-3193	109	5	for	for	ADP
ejpam-3193	109	6	other	other	ADJ
ejpam-3193	109	7	elements	element	NOUN
ejpam-3193	109	8	as	as	ADV
ejpam-3193	109	9	well	well	ADV
ejpam-3193	109	10	.	.	PUNCT
ejpam-3193	110	1	hence	hence	ADV
ejpam-3193	110	2	p	p	PROPN
ejpam-3193	110	3	is	be	AUX
ejpam-3193	110	4	an	an	DET
ejpam-3193	110	5	almost	almost	ADV
ejpam-3193	110	6	prime	prime	ADJ
ejpam-3193	110	7	ideal	ideal	NOUN
ejpam-3193	110	8	in	in	ADP
ejpam-3193	110	9	γ	γ	PROPN
ejpam-3193	110	10	-	-	PUNCT
ejpam-3193	110	11	near	near	ADJ
ejpam-3193	110	12	ring	ring	NOUN
ejpam-3193	110	13	.	.	PUNCT
ejpam-3193	110	14	example	example	NOUN
ejpam-3193	111	1	2	2	NUM
ejpam-3193	111	2	.	.	PUNCT
ejpam-3193	111	3	suppose	suppose	VERB
ejpam-3193	111	4	r	r	NOUN
ejpam-3193	111	5	is	be	AUX
ejpam-3193	111	6	a	a	DET
ejpam-3193	111	7	γ	γ	NOUN
ejpam-3193	111	8	-	-	PUNCT
ejpam-3193	111	9	near	near	ADJ
ejpam-3193	111	10	ring	ring	NOUN
ejpam-3193	111	11	of	of	ADP
ejpam-3193	111	12	algebraic	algebraic	ADJ
ejpam-3193	111	13	integers	integer	NOUN
ejpam-3193	111	14	such	such	ADJ
ejpam-3193	111	15	that	that	SCONJ
ejpam-3193	111	16	the	the	DET
ejpam-3193	111	17	integral	integral	ADJ
ejpam-3193	111	18	closure	closure	NOUN
ejpam-3193	111	19	of	of	ADP
ejpam-3193	111	20	z	z	PROPN
ejpam-3193	111	21	in	in	ADP
ejpam-3193	111	22	c.	c.	PROPN
ejpam-3193	111	23	suppose	suppose	VERB
ejpam-3193	111	24	that	that	SCONJ
ejpam-3193	111	25	i	i	PRON
ejpam-3193	111	26	be	be	VERB
ejpam-3193	111	27	a	a	DET
ejpam-3193	111	28	radical	radical	ADJ
ejpam-3193	111	29	ideal	ideal	NOUN
ejpam-3193	111	30	of	of	ADP
ejpam-3193	111	31	r	r	NOUN
ejpam-3193	111	32	say	say	VERB
ejpam-3193	111	33	iγi	iγi	NOUN
ejpam-3193	112	1	=	=	PROPN
ejpam-3193	112	2	i	i	PROPN
ejpam-3193	112	3	,	,	PUNCT
ejpam-3193	113	1	if	if	SCONJ
ejpam-3193	113	2	α	α	PRON
ejpam-3193	113	3	∈	∈	VERB
ejpam-3193	113	4	i	i	PRON
ejpam-3193	113	5	then	then	ADV
ejpam-3193	113	6	β	β	PROPN
ejpam-3193	113	7	∈	∈	NOUN
ejpam-3193	113	8	r	r	NOUN
ejpam-3193	113	9	exist	exist	VERB
ejpam-3193	113	10	such	such	ADJ
ejpam-3193	113	11	that	that	DET
ejpam-3193	113	12	βγβ	βγβ	NOUN
ejpam-3193	114	1	=	=	NOUN
ejpam-3193	114	2	α	α	X
ejpam-3193	114	3	.	.	PUNCT
ejpam-3193	115	1	since	since	SCONJ
ejpam-3193	115	2	βγβ	βγβ	NOUN
ejpam-3193	115	3	=	=	PUNCT
ejpam-3193	115	4	α	α	X
ejpam-3193	115	5	∈	∈	PROPN
ejpam-3193	116	1	i	i	PRON
ejpam-3193	116	2	,	,	PUNCT
ejpam-3193	116	3	β	β	PROPN
ejpam-3193	116	4	∈	∈	PROPN
ejpam-3193	116	5	i	i	PRON
ejpam-3193	116	6	implies	imply	VERB
ejpam-3193	116	7	i	i	PRON
ejpam-3193	116	8	=	=	PUNCT
ejpam-3193	116	9	iγi	iγi	PROPN
ejpam-3193	116	10	.	.	PROPN
ejpam-3193	116	11	example	example	NOUN
ejpam-3193	117	1	3	3	X
ejpam-3193	117	2	.	.	X
ejpam-3193	117	3	consider	consider	VERB
ejpam-3193	117	4	the	the	DET
ejpam-3193	117	5	near	near	ADJ
ejpam-3193	117	6	ring	ring	NOUN
ejpam-3193	117	7	n	n	NOUN
ejpam-3193	117	8	=	=	PUNCT
ejpam-3193	117	9	{	{	PUNCT
ejpam-3193	117	10	0	0	NUM
ejpam-3193	117	11	,	,	PUNCT
ejpam-3193	117	12	1	1	NUM
ejpam-3193	117	13	,	,	PUNCT
ejpam-3193	117	14	2	2	NUM
ejpam-3193	117	15	,	,	PUNCT
ejpam-3193	117	16	3	3	NUM
ejpam-3193	117	17	}	}	PUNCT
ejpam-3193	117	18	and	and	CCONJ
ejpam-3193	117	19	γ	γ	X
ejpam-3193	117	20	=	=	SYM
ejpam-3193	117	21	{	{	PUNCT
ejpam-3193	117	22	0	0	NUM
ejpam-3193	117	23	,	,	PUNCT
ejpam-3193	117	24	2	2	NUM
ejpam-3193	117	25	}	}	PUNCT
ejpam-3193	117	26	such	such	ADJ
ejpam-3193	117	27	that	that	DET
ejpam-3193	117	28	addition	addition	NOUN
ejpam-3193	117	29	and	and	CCONJ
ejpam-3193	117	30	multiplication	multiplication	NOUN
ejpam-3193	117	31	defined	define	VERB
ejpam-3193	117	32	as	as	ADP
ejpam-3193	117	33	follow	follow	NOUN
ejpam-3193	117	34	.	.	PUNCT
ejpam-3193	118	1	a.	a.	NOUN
ejpam-3193	118	2	taouti	taouti	PROPN
ejpam-3193	118	3	et	et	PROPN
ejpam-3193	118	4	al	al	PROPN
ejpam-3193	118	5	.	.	PUNCT
ejpam-3193	118	6	/	/	SYM
ejpam-3193	118	7	eur	eur	PROPN
ejpam-3193	118	8	.	.	PUNCT
ejpam-3193	119	1	j.	j.	PROPN
ejpam-3193	119	2	pure	pure	PROPN
ejpam-3193	119	3	appl	appl	PROPN
ejpam-3193	119	4	.	.	PROPN
ejpam-3193	119	5	math	math	PROPN
ejpam-3193	119	6	,	,	PUNCT
ejpam-3193	119	7	11	11	NUM
ejpam-3193	119	8	(	(	PUNCT
ejpam-3193	119	9	2	2	NUM
ejpam-3193	119	10	)	)	PUNCT
ejpam-3193	119	11	(	(	PUNCT
ejpam-3193	119	12	2018	2018	NUM
ejpam-3193	119	13	)	)	PUNCT
ejpam-3193	119	14	,	,	PUNCT
ejpam-3193	119	15	449	449	NUM
ejpam-3193	119	16	-	-	NOUN
ejpam-3193	119	17	456	456	NUM
ejpam-3193	119	18	452	452	NUM
ejpam-3193	120	1	+	+	CCONJ
ejpam-3193	120	2	0	0	NUM
ejpam-3193	120	3	1	1	NUM
ejpam-3193	120	4	2	2	NUM
ejpam-3193	120	5	3	3	NUM
ejpam-3193	120	6	0	0	NUM
ejpam-3193	120	7	0	0	NUM
ejpam-3193	120	8	1	1	NUM
ejpam-3193	120	9	2	2	NUM
ejpam-3193	120	10	3	3	NUM
ejpam-3193	120	11	1	1	NUM
ejpam-3193	120	12	1	1	NUM
ejpam-3193	120	13	0	0	NUM
ejpam-3193	120	14	3	3	NUM
ejpam-3193	120	15	2	2	NUM
ejpam-3193	120	16	2	2	NUM
ejpam-3193	120	17	2	2	NUM
ejpam-3193	120	18	3	3	NUM
ejpam-3193	120	19	0	0	NUM
ejpam-3193	120	20	1	1	NUM
ejpam-3193	120	21	3	3	NUM
ejpam-3193	120	22	3	3	NUM
ejpam-3193	120	23	2	2	NUM
ejpam-3193	120	24	1	1	NUM
ejpam-3193	120	25	0	0	NUM
ejpam-3193	120	26			NOUN
ejpam-3193	120	27			NOUN
ejpam-3193	120	28	·	·	PUNCT
ejpam-3193	120	29	0	0	NUM
ejpam-3193	120	30	1	1	NUM
ejpam-3193	120	31	2	2	NUM
ejpam-3193	120	32	3	3	NUM
ejpam-3193	120	33	0	0	NUM
ejpam-3193	120	34	0	0	NUM
ejpam-3193	120	35	0	0	NUM
ejpam-3193	120	36	0	0	NUM
ejpam-3193	120	37	0	0	NUM
ejpam-3193	120	38	1	1	NUM
ejpam-3193	120	39	0	0	NUM
ejpam-3193	120	40	1	1	NUM
ejpam-3193	120	41	2	2	NUM
ejpam-3193	120	42	3	3	NUM
ejpam-3193	120	43	2	2	NUM
ejpam-3193	120	44	0	0	NUM
ejpam-3193	120	45	2	2	NUM
ejpam-3193	120	46	0	0	NUM
ejpam-3193	120	47	2	2	NUM
ejpam-3193	120	48	3	3	NUM
ejpam-3193	120	49	0	0	NUM
ejpam-3193	120	50	3	3	NUM
ejpam-3193	120	51	2	2	NUM
ejpam-3193	120	52	1	1	NUM
ejpam-3193	120	53			NOUN
ejpam-3193	120	54	suppose	suppose	VERB
ejpam-3193	120	55	p	p	X
ejpam-3193	120	56	=	=	X
ejpam-3193	120	57	{	{	PUNCT
ejpam-3193	120	58	0	0	NUM
ejpam-3193	120	59	,	,	PUNCT
ejpam-3193	120	60	2	2	NUM
ejpam-3193	120	61	}	}	PUNCT
ejpam-3193	120	62	=	=	NOUN
ejpam-3193	120	63	2n	2n	NUM
ejpam-3193	120	64	be	be	VERB
ejpam-3193	120	65	a	a	DET
ejpam-3193	120	66	prime	prime	ADJ
ejpam-3193	120	67	ideal	ideal	NOUN
ejpam-3193	120	68	of	of	ADP
ejpam-3193	120	69	n	n	NOUN
ejpam-3193	120	70	because	because	SCONJ
ejpam-3193	120	71	for	for	ADP
ejpam-3193	120	72	all	all	DET
ejpam-3193	120	73	a	a	PRON
ejpam-3193	120	74	,	,	PUNCT
ejpam-3193	120	75	b	b	X
ejpam-3193	120	76	∈	∈	PROPN
ejpam-3193	120	77	n	n	NOUN
ejpam-3193	120	78	and	and	CCONJ
ejpam-3193	120	79	aγb	aγb	ADV
ejpam-3193	120	80	∈	∈	PROPN
ejpam-3193	120	81	p	p	NOUN
ejpam-3193	120	82	implies	imply	VERB
ejpam-3193	120	83	a	a	DET
ejpam-3193	120	84	∈	∈	PROPN
ejpam-3193	120	85	p	p	NOUN
ejpam-3193	120	86	or	or	CCONJ
ejpam-3193	120	87	b	b	NOUN
ejpam-3193	120	88	∈	∈	PROPN
ejpam-3193	120	89	p	p	NOUN
ejpam-3193	120	90	.	.	PUNCT
ejpam-3193	121	1	as	as	SCONJ
ejpam-3193	121	2	pγp	pγp	ADV
ejpam-3193	121	3	=	=	SYM
ejpam-3193	121	4	{	{	PUNCT
ejpam-3193	121	5	0	0	NUM
ejpam-3193	121	6	}	}	PUNCT
ejpam-3193	121	7	then	then	ADV
ejpam-3193	121	8	p	p	NOUN
ejpam-3193	121	9	−	−	PROPN
ejpam-3193	121	10	pγp	pγp	NOUN
ejpam-3193	121	11	=	=	PUNCT
ejpam-3193	121	12	{	{	PUNCT
ejpam-3193	121	13	2	2	NUM
ejpam-3193	121	14	}	}	PUNCT
ejpam-3193	121	15	,	,	PUNCT
ejpam-3193	121	16	then	then	ADV
ejpam-3193	121	17	for	for	ADP
ejpam-3193	121	18	all	all	DET
ejpam-3193	121	19	a	a	PRON
ejpam-3193	121	20	,	,	PUNCT
ejpam-3193	121	21	b	b	X
ejpam-3193	121	22	∈	∈	PROPN
ejpam-3193	121	23	n	n	PRON
ejpam-3193	121	24	such	such	ADJ
ejpam-3193	121	25	that	that	SCONJ
ejpam-3193	121	26	aγb	aγb	VERB
ejpam-3193	121	27	∈	∈	PROPN
ejpam-3193	121	28	p	p	NOUN
ejpam-3193	122	1	−	−	PROPN
ejpam-3193	122	2	pγp	pγp	ADJ
ejpam-3193	122	3	either	either	CCONJ
ejpam-3193	122	4	a	a	DET
ejpam-3193	122	5	∈	∈	PROPN
ejpam-3193	122	6	p	p	NOUN
ejpam-3193	122	7	or	or	CCONJ
ejpam-3193	122	8	b	b	NOUN
ejpam-3193	122	9	∈	∈	PROPN
ejpam-3193	122	10	p	p	NOUN
ejpam-3193	122	11	which	which	PRON
ejpam-3193	122	12	is	be	AUX
ejpam-3193	122	13	almost	almost	ADV
ejpam-3193	122	14	prime	prime	ADJ
ejpam-3193	122	15	ideal	ideal	NOUN
ejpam-3193	122	16	.	.	PUNCT
ejpam-3193	123	1	preposition	preposition	NOUN
ejpam-3193	123	2	1	1	NUM
ejpam-3193	123	3	.	.	PUNCT
ejpam-3193	124	1	every	every	DET
ejpam-3193	124	2	prime	prime	ADJ
ejpam-3193	124	3	ideal	ideal	NOUN
ejpam-3193	124	4	in	in	ADP
ejpam-3193	124	5	a	a	DET
ejpam-3193	124	6	γ	γ	X
ejpam-3193	124	7	-	-	PUNCT
ejpam-3193	124	8	near	near	ADJ
ejpam-3193	124	9	ring	ring	NOUN
ejpam-3193	124	10	is	be	AUX
ejpam-3193	124	11	almost	almost	ADV
ejpam-3193	124	12	prime	prime	ADJ
ejpam-3193	124	13	ideal	ideal	NOUN
ejpam-3193	124	14	.	.	PUNCT
ejpam-3193	125	1	proof	proof	NOUN
ejpam-3193	125	2	.	.	PUNCT
ejpam-3193	126	1	suppose	suppose	VERB
ejpam-3193	126	2	p	p	PRON
ejpam-3193	126	3	be	be	AUX
ejpam-3193	126	4	a	a	DET
ejpam-3193	126	5	prime	prime	ADJ
ejpam-3193	126	6	ideal	ideal	NOUN
ejpam-3193	126	7	of	of	ADP
ejpam-3193	126	8	γ	γ	X
ejpam-3193	126	9	-	-	PUNCT
ejpam-3193	126	10	near	near	ADJ
ejpam-3193	126	11	ring	ring	NOUN
ejpam-3193	126	12	but	but	CCONJ
ejpam-3193	126	13	not	not	PART
ejpam-3193	126	14	an	an	DET
ejpam-3193	126	15	almost	almost	ADV
ejpam-3193	126	16	prime	prime	ADJ
ejpam-3193	126	17	.	.	PUNCT
ejpam-3193	127	1	assume	assume	VERB
ejpam-3193	127	2	aγb	aγb	NOUN
ejpam-3193	127	3	∈	∈	PROPN
ejpam-3193	128	1	p	p	NOUN
ejpam-3193	129	1	−	−	PROPN
ejpam-3193	129	2	pγp	pγp	ADV
ejpam-3193	129	3	,	,	PUNCT
ejpam-3193	129	4	implies	imply	VERB
ejpam-3193	129	5	aγb	aγb	NOUN
ejpam-3193	129	6	∈	∈	PROPN
ejpam-3193	129	7	p	p	NOUN
ejpam-3193	129	8	.	.	PUNCT
ejpam-3193	130	1	if	if	SCONJ
ejpam-3193	130	2	aγb	aγb	VERB
ejpam-3193	130	3	6∈	6∈	PROPN
ejpam-3193	130	4	pγp	pγp	PROPN
ejpam-3193	130	5	implies	imply	VERB
ejpam-3193	130	6	a	a	DET
ejpam-3193	130	7	∈	∈	PROPN
ejpam-3193	130	8	p	p	NOUN
ejpam-3193	130	9	or	or	CCONJ
ejpam-3193	130	10	b	b	NOUN
ejpam-3193	130	11	∈	∈	PROPN
ejpam-3193	131	1	p	p	NOUN
ejpam-3193	131	2	then	then	ADV
ejpam-3193	131	3	contradiction	contradiction	NOUN
ejpam-3193	131	4	arise	arise	VERB
ejpam-3193	131	5	to	to	ADP
ejpam-3193	131	6	our	our	PRON
ejpam-3193	131	7	supposition	supposition	NOUN
ejpam-3193	131	8	.	.	PUNCT
ejpam-3193	132	1	hence	hence	ADV
ejpam-3193	132	2	p	p	PRON
ejpam-3193	132	3	must	must	AUX
ejpam-3193	132	4	be	be	AUX
ejpam-3193	132	5	a	a	DET
ejpam-3193	132	6	prime	prime	NOUN
ejpam-3193	132	7	.	.	PUNCT
ejpam-3193	133	1	remark	remark	NOUN
ejpam-3193	133	2	1	1	NUM
ejpam-3193	133	3	.	.	PUNCT
ejpam-3193	134	1	if	if	SCONJ
ejpam-3193	134	2	i	i	PRON
ejpam-3193	134	3	is	be	AUX
ejpam-3193	134	4	a	a	DET
ejpam-3193	134	5	maximal	maximal	ADJ
ejpam-3193	134	6	ideal	ideal	NOUN
ejpam-3193	134	7	of	of	ADP
ejpam-3193	134	8	γ	γ	X
ejpam-3193	134	9	-	-	PUNCT
ejpam-3193	134	10	near	near	ADP
ejpam-3193	134	11	-	-	PUNCT
ejpam-3193	134	12	ring	ring	NOUN
ejpam-3193	134	13	m	m	VERB
ejpam-3193	134	14	then	then	ADV
ejpam-3193	134	15	it	it	PRON
ejpam-3193	134	16	is	be	AUX
ejpam-3193	134	17	prime	prime	ADJ
ejpam-3193	134	18	or	or	CCONJ
ejpam-3193	134	19	mγm	mγm	NOUN
ejpam-3193	134	20	=	=	SYM
ejpam-3193	134	21	i.	i.	NOUN
ejpam-3193	134	22	supporting	support	VERB
ejpam-3193	134	23	the	the	DET
ejpam-3193	134	24	above	above	ADJ
ejpam-3193	134	25	remark	remark	NOUN
ejpam-3193	134	26	1	1	NUM
ejpam-3193	134	27	,	,	PUNCT
ejpam-3193	134	28	we	we	PRON
ejpam-3193	134	29	present	present	VERB
ejpam-3193	134	30	the	the	DET
ejpam-3193	134	31	below	below	ADJ
ejpam-3193	134	32	example	example	NOUN
ejpam-3193	134	33	.	.	PUNCT
ejpam-3193	135	1	example	example	NOUN
ejpam-3193	136	1	4	4	NUM
ejpam-3193	136	2	.	.	PUNCT
ejpam-3193	136	3	let	let	VERB
ejpam-3193	136	4	m	m	VERB
ejpam-3193	136	5	=	=	PUNCT
ejpam-3193	136	6	{	{	PUNCT
ejpam-3193	136	7	0	0	NUM
ejpam-3193	136	8	,	,	PUNCT
ejpam-3193	136	9	1	1	NUM
ejpam-3193	136	10	,	,	PUNCT
ejpam-3193	136	11	2	2	NUM
ejpam-3193	136	12	,	,	PUNCT
ejpam-3193	136	13	3	3	NUM
ejpam-3193	136	14	}	}	PUNCT
ejpam-3193	136	15	is	be	AUX
ejpam-3193	136	16	a	a	DET
ejpam-3193	136	17	γ	γ	NOUN
ejpam-3193	136	18	-	-	PUNCT
ejpam-3193	136	19	near	near	ADP
ejpam-3193	136	20	-	-	PUNCT
ejpam-3193	136	21	ring	ring	NOUN
ejpam-3193	136	22	where	where	SCONJ
ejpam-3193	136	23	γ	γ	X
ejpam-3193	136	24	=	=	SYM
ejpam-3193	136	25	{	{	PUNCT
ejpam-3193	136	26	0	0	NUM
ejpam-3193	136	27	,	,	PUNCT
ejpam-3193	136	28	2	2	NUM
ejpam-3193	136	29	}	}	PUNCT
ejpam-3193	136	30	and	and	CCONJ
ejpam-3193	136	31	ideal	ideal	ADJ
ejpam-3193	136	32	i	i	NOUN
ejpam-3193	136	33	=	=	NOUN
ejpam-3193	136	34	2	2	NUM
ejpam-3193	136	35	m	m	NOUN
ejpam-3193	136	36	=	=	PUNCT
ejpam-3193	136	37	{	{	PUNCT
ejpam-3193	136	38	0	0	NUM
ejpam-3193	136	39	,	,	PUNCT
ejpam-3193	136	40	2	2	NUM
ejpam-3193	136	41	}	}	PUNCT
ejpam-3193	136	42	that	that	PRON
ejpam-3193	136	43	is	be	AUX
ejpam-3193	136	44	maximal	maximal	ADJ
ejpam-3193	136	45	in	in	ADP
ejpam-3193	136	46	m	m	PROPN
ejpam-3193	136	47	.	.	PUNCT
ejpam-3193	137	1	obviously	obviously	ADV
ejpam-3193	137	2	i	i	PRON
ejpam-3193	137	3	is	be	AUX
ejpam-3193	137	4	prime	prime	ADJ
ejpam-3193	137	5	ideal	ideal	NOUN
ejpam-3193	137	6	in	in	ADP
ejpam-3193	137	7	m	m	PRON
ejpam-3193	137	8	also	also	ADV
ejpam-3193	137	9	mγm	mγm	ADJ
ejpam-3193	137	10	=	=	SYM
ejpam-3193	137	11	i.	i.	PROPN
ejpam-3193	137	12	lemma	lemma	PROPN
ejpam-3193	137	13	1	1	X
ejpam-3193	137	14	.	.	PUNCT
ejpam-3193	137	15	suppose	suppose	VERB
ejpam-3193	137	16	n	n	PRON
ejpam-3193	137	17	is	be	AUX
ejpam-3193	137	18	a	a	DET
ejpam-3193	137	19	γ	γ	NOUN
ejpam-3193	137	20	-	-	PUNCT
ejpam-3193	137	21	near	near	ADP
ejpam-3193	137	22	-	-	PUNCT
ejpam-3193	137	23	ring	ring	NOUN
ejpam-3193	137	24	and	and	CCONJ
ejpam-3193	137	25	for	for	ADP
ejpam-3193	137	26	any	any	DET
ejpam-3193	137	27	γ	γ	PROPN
ejpam-3193	137	28	∈	∈	PROPN
ejpam-3193	137	29	γ	γ	NOUN
ejpam-3193	137	30	there	there	PRON
ejpam-3193	137	31	is	be	VERB
ejpam-3193	137	32	an	an	DET
ejpam-3193	137	33	element	element	NOUN
ejpam-3193	137	34	which	which	PRON
ejpam-3193	137	35	is	be	AUX
ejpam-3193	137	36	γ	γ	NOUN
ejpam-3193	137	37	-	-	NOUN
ejpam-3193	137	38	unit	unit	NOUN
ejpam-3193	137	39	then	then	ADV
ejpam-3193	137	40	every	every	DET
ejpam-3193	137	41	maximal	maximal	ADJ
ejpam-3193	137	42	ideal	ideal	NOUN
ejpam-3193	137	43	i	i	PRON
ejpam-3193	137	44	of	of	ADP
ejpam-3193	137	45	m	m	PROPN
ejpam-3193	137	46	is	be	AUX
ejpam-3193	137	47	prime	prime	ADJ
ejpam-3193	137	48	.	.	PUNCT
ejpam-3193	138	1	proof	proof	NOUN
ejpam-3193	138	2	.	.	PUNCT
ejpam-3193	139	1	if	if	SCONJ
ejpam-3193	139	2	for	for	ADP
ejpam-3193	139	3	one	one	NUM
ejpam-3193	139	4	γ	γ	NOUN
ejpam-3193	139	5	∈	∈	NOUN
ejpam-3193	139	6	γ	γ	NOUN
ejpam-3193	139	7	the	the	DET
ejpam-3193	139	8	element	element	NOUN
ejpam-3193	139	9	e	e	NOUN
ejpam-3193	139	10	is	be	AUX
ejpam-3193	139	11	γ	γ	X
ejpam-3193	139	12	-	-	PUNCT
ejpam-3193	139	13	one	one	NUM
ejpam-3193	139	14	of	of	ADP
ejpam-3193	139	15	m	m	PRON
ejpam-3193	139	16	then	then	ADV
ejpam-3193	139	17	mγm	mγm	AUX
ejpam-3193	139	18	=	=	SYM
ejpam-3193	139	19	{	{	PUNCT
ejpam-3193	139	20	m1γm2	m1γm2	PROPN
ejpam-3193	139	21	:	:	PUNCT
ejpam-3193	139	22	m1;m2	m1;m2	PROPN
ejpam-3193	139	23	∈	∈	PROPN
ejpam-3193	139	24	m	m	PROPN
ejpam-3193	139	25	}	}	PUNCT
ejpam-3193	139	26	=	=	PUNCT
ejpam-3193	139	27	m	m	VERB
ejpam-3193	139	28	since	since	SCONJ
ejpam-3193	139	29	for	for	ADP
ejpam-3193	139	30	any	any	DET
ejpam-3193	139	31	m	m	NOUN
ejpam-3193	139	32	∈	∈	NOUN
ejpam-3193	139	33	m	m	NOUN
ejpam-3193	139	34	,	,	PUNCT
ejpam-3193	139	35	m	m	VERB
ejpam-3193	139	36	=	=	ADJ
ejpam-3193	139	37	mγe	mγe	NOUN
ejpam-3193	139	38	.	.	PUNCT
ejpam-3193	140	1	because	because	SCONJ
ejpam-3193	140	2	m	m	PROPN
ejpam-3193	140	3	6=	6=	NUM
ejpam-3193	140	4	i	i	PRON
ejpam-3193	140	5	the	the	DET
ejpam-3193	140	6	equation	equation	NOUN
ejpam-3193	140	7	is	be	AUX
ejpam-3193	140	8	not	not	PART
ejpam-3193	140	9	true	true	ADJ
ejpam-3193	140	10	mγm	mγm	NOUN
ejpam-3193	140	11	=	=	SYM
ejpam-3193	140	12	i.	i.	NOUN
ejpam-3193	140	13	when	when	SCONJ
ejpam-3193	140	14	m	m	VERB
ejpam-3193	140	15	=	=	VERB
ejpam-3193	140	16	i	i	PRON
ejpam-3193	140	17	or	or	CCONJ
ejpam-3193	140	18	m	m	VERB
ejpam-3193	140	19	=	=	NOUN
ejpam-3193	140	20	0	0	NUM
ejpam-3193	140	21	then	then	ADV
ejpam-3193	140	22	equation	equation	NOUN
ejpam-3193	140	23	is	be	AUX
ejpam-3193	140	24	true	true	ADJ
ejpam-3193	140	25	so	so	ADV
ejpam-3193	140	26	m	m	VERB
ejpam-3193	140	27	is	be	AUX
ejpam-3193	140	28	simple	simple	ADJ
ejpam-3193	140	29	and	and	CCONJ
ejpam-3193	140	30	mγm	mγm	NOUN
ejpam-3193	140	31	6=	6=	PROPN
ejpam-3193	140	32	0	0	NUM
ejpam-3193	140	33	,	,	PUNCT
ejpam-3193	140	34	as	as	ADP
ejpam-3193	140	35	a	a	DET
ejpam-3193	140	36	result	result	NOUN
ejpam-3193	140	37	m	m	NOUN
ejpam-3193	140	38	is	be	AUX
ejpam-3193	140	39	prime	prime	ADJ
ejpam-3193	140	40	.	.	PUNCT
ejpam-3193	141	1	preposition	preposition	NOUN
ejpam-3193	141	2	2	2	NUM
ejpam-3193	141	3	.	.	PUNCT
ejpam-3193	141	4	suppose	suppose	VERB
ejpam-3193	141	5	i	i	PRON
ejpam-3193	141	6	be	be	VERB
ejpam-3193	141	7	a	a	DET
ejpam-3193	141	8	p	p	NOUN
ejpam-3193	141	9	-primary	-primary	ADJ
ejpam-3193	141	10	ideal	ideal	NOUN
ejpam-3193	141	11	of	of	ADP
ejpam-3193	141	12	a	a	DET
ejpam-3193	141	13	γ	γ	NOUN
ejpam-3193	141	14	-	-	PUNCT
ejpam-3193	141	15	near	near	ADJ
ejpam-3193	141	16	ring	ring	NOUN
ejpam-3193	141	17	such	such	ADJ
ejpam-3193	141	18	that	that	PRON
ejpam-3193	141	19	pγp	pγp	ADJ
ejpam-3193	141	20	=	=	PUNCT
ejpam-3193	141	21	iγi	iγi	NOUN
ejpam-3193	141	22	implies	imply	VERB
ejpam-3193	141	23	i	i	PRON
ejpam-3193	141	24	is	be	AUX
ejpam-3193	141	25	an	an	DET
ejpam-3193	141	26	almost	almost	ADV
ejpam-3193	141	27	prime	prime	ADJ
ejpam-3193	141	28	.	.	PUNCT
ejpam-3193	142	1	proof	proof	NOUN
ejpam-3193	142	2	.	.	PUNCT
ejpam-3193	143	1	suppose	suppose	VERB
ejpam-3193	143	2	a	a	DET
ejpam-3193	143	3	,	,	PUNCT
ejpam-3193	143	4	b	b	X
ejpam-3193	143	5	∈	∈	PROPN
ejpam-3193	143	6	r	r	NOUN
ejpam-3193	143	7	,	,	PUNCT
ejpam-3193	143	8	aγb	aγb	NOUN
ejpam-3193	143	9	∈	∈	PROPN
ejpam-3193	144	1	i	i	PRON
ejpam-3193	144	2	−	−	VERB
ejpam-3193	144	3	iγi	iγi	PROPN
ejpam-3193	144	4	,	,	PUNCT
ejpam-3193	144	5	a	a	DET
ejpam-3193	144	6	6∈	6∈	NOUN
ejpam-3193	145	1	i	i	PRON
ejpam-3193	145	2	and	and	CCONJ
ejpam-3193	145	3	b	b	PROPN
ejpam-3193	145	4	6∈	6∈	PROPN
ejpam-3193	145	5	i.	i.	NOUN
ejpam-3193	145	6	as	as	ADP
ejpam-3193	145	7	a	a	DET
ejpam-3193	145	8	6∈	6∈	NOUN
ejpam-3193	145	9	i	i	PRON
ejpam-3193	146	1	and	and	CCONJ
ejpam-3193	146	2	i	i	PRON
ejpam-3193	146	3	is	be	AUX
ejpam-3193	146	4	a	a	DET
ejpam-3193	146	5	p	p	ADJ
ejpam-3193	146	6	-primary	-primary	ADJ
ejpam-3193	146	7	ideal	ideal	NOUN
ejpam-3193	146	8	it	it	PRON
ejpam-3193	146	9	implies	imply	VERB
ejpam-3193	146	10	that	that	SCONJ
ejpam-3193	146	11	b	b	PROPN
ejpam-3193	146	12	∈	∈	PROPN
ejpam-3193	146	13	p	p	NOUN
ejpam-3193	146	14	.	.	PUNCT
ejpam-3193	147	1	also	also	ADV
ejpam-3193	147	2	a	a	DET
ejpam-3193	147	3	∈	∈	NOUN
ejpam-3193	147	4	p	p	NOUN
ejpam-3193	147	5	thus	thus	ADV
ejpam-3193	147	6	aγb	aγb	ADP
ejpam-3193	147	7	∈	∈	PROPN
ejpam-3193	147	8	pγp	pγp	NOUN
ejpam-3193	147	9	=	=	SYM
ejpam-3193	147	10	iγi	iγi	NOUN
ejpam-3193	147	11	,	,	PUNCT
ejpam-3193	147	12	which	which	PRON
ejpam-3193	147	13	is	be	AUX
ejpam-3193	147	14	a	a	DET
ejpam-3193	147	15	contradiction	contradiction	NOUN
ejpam-3193	147	16	.	.	PUNCT
ejpam-3193	148	1	lemma	lemma	PROPN
ejpam-3193	148	2	2	2	X
ejpam-3193	148	3	.	.	PUNCT
ejpam-3193	148	4	suppose	suppose	VERB
ejpam-3193	148	5	that	that	SCONJ
ejpam-3193	148	6	r	r	NOUN
ejpam-3193	148	7	be	be	AUX
ejpam-3193	148	8	a	a	DET
ejpam-3193	148	9	near	near	ADJ
ejpam-3193	148	10	integral	integral	ADJ
ejpam-3193	148	11	domain	domain	NOUN
ejpam-3193	148	12	and	and	CCONJ
ejpam-3193	148	13	c	c	AUX
ejpam-3193	148	14	be	be	AUX
ejpam-3193	148	15	a	a	DET
ejpam-3193	148	16	nonzero	nonzero	NOUN
ejpam-3193	148	17	nonunit	nonunit	NOUN
ejpam-3193	148	18	element	element	NOUN
ejpam-3193	148	19	of	of	ADP
ejpam-3193	148	20	r.	r.	PROPN
ejpam-3193	148	21	if	if	SCONJ
ejpam-3193	148	22	element	element	NOUN
ejpam-3193	148	23	c	c	PROPN
ejpam-3193	148	24	is	be	AUX
ejpam-3193	148	25	other	other	ADJ
ejpam-3193	148	26	than	than	ADP
ejpam-3193	148	27	prime	prime	ADJ
ejpam-3193	148	28	element	element	NOUN
ejpam-3193	148	29	then	then	ADV
ejpam-3193	148	30	there	there	PRON
ejpam-3193	148	31	exist	exist	VERB
ejpam-3193	148	32	a	a	DET
ejpam-3193	148	33	6∈	6∈	NOUN
ejpam-3193	148	34	rγc	rγc	NOUN
ejpam-3193	148	35	,	,	PUNCT
ejpam-3193	148	36	b	b	PROPN
ejpam-3193	148	37	6∈	6∈	NOUN
ejpam-3193	148	38	rγc	rγc	NOUN
ejpam-3193	148	39	such	such	ADJ
ejpam-3193	148	40	that	that	SCONJ
ejpam-3193	148	41	aγb	aγb	NOUN
ejpam-3193	148	42	∈	∈	PROPN
ejpam-3193	148	43	rγc	rγc	NOUN
ejpam-3193	148	44	but	but	CCONJ
ejpam-3193	148	45	aγb	aγb	VERB
ejpam-3193	148	46	6∈	6∈	PROPN
ejpam-3193	148	47	rγc2	rγc2	PROPN
ejpam-3193	148	48	.	.	PUNCT
ejpam-3193	149	1	proof	proof	NOUN
ejpam-3193	149	2	.	.	PUNCT
ejpam-3193	150	1	suppose	suppose	VERB
ejpam-3193	150	2	an	an	DET
ejpam-3193	150	3	ideal	ideal	ADJ
ejpam-3193	150	4	rc	rc	PROPN
ejpam-3193	150	5	is	be	AUX
ejpam-3193	150	6	not	not	PART
ejpam-3193	150	7	prime	prime	ADJ
ejpam-3193	150	8	then	then	ADV
ejpam-3193	150	9	there	there	PRON
ejpam-3193	150	10	exist	exist	VERB
ejpam-3193	150	11	a	a	DET
ejpam-3193	150	12	6∈	6∈	NOUN
ejpam-3193	150	13	rγc	rγc	NOUN
ejpam-3193	150	14	,	,	PUNCT
ejpam-3193	150	15	b	b	PROPN
ejpam-3193	150	16	6∈	6∈	NOUN
ejpam-3193	150	17	rγc	rγc	NOUN
ejpam-3193	150	18	such	such	ADJ
ejpam-3193	150	19	that	that	SCONJ
ejpam-3193	150	20	aγb	aγb	NOUN
ejpam-3193	150	21	∈	∈	PROPN
ejpam-3193	150	22	rγc	rγc	NOUN
ejpam-3193	150	23	.	.	PUNCT
ejpam-3193	151	1	if	if	SCONJ
ejpam-3193	151	2	the	the	DET
ejpam-3193	151	3	case	case	NOUN
ejpam-3193	151	4	aγb	aγb	VERB
ejpam-3193	151	5	∈	∈	PROPN
ejpam-3193	151	6	rγc2	rγc2	VERB
ejpam-3193	151	7	then	then	ADV
ejpam-3193	151	8	for	for	ADP
ejpam-3193	151	9	d	d	PROPN
ejpam-3193	151	10	=	=	SYM
ejpam-3193	151	11	(	(	PUNCT
ejpam-3193	151	12	b	b	NOUN
ejpam-3193	151	13	+	+	CCONJ
ejpam-3193	151	14	c)γ	c)γ	X
ejpam-3193	151	15	6∈	6∈	PROPN
ejpam-3193	151	16	rγc	rγc	NOUN
ejpam-3193	151	17	and	and	CCONJ
ejpam-3193	151	18	aγd	aγd	NOUN
ejpam-3193	151	19	∈	∈	PROPN
ejpam-3193	151	20	rγc	rγc	NOUN
ejpam-3193	151	21	.	.	PUNCT
ejpam-3193	152	1	if	if	SCONJ
ejpam-3193	152	2	aγd	aγd	NOUN
ejpam-3193	152	3	∈	∈	PROPN
ejpam-3193	152	4	rγc2	rγc2	PROPN
ejpam-3193	152	5	,	,	PUNCT
ejpam-3193	152	6	implies	imply	VERB
ejpam-3193	152	7	aγc	aγc	PROPN
ejpam-3193	152	8	∈	∈	PROPN
ejpam-3193	152	9	rγc2	rγc2	PROPN
ejpam-3193	152	10	as	as	SCONJ
ejpam-3193	152	11	aγb	aγb	NOUN
ejpam-3193	152	12	∈	∈	PROPN
ejpam-3193	152	13	rγc2	rγc2	PROPN
ejpam-3193	152	14	implies	imply	VERB
ejpam-3193	152	15	a	a	DET
ejpam-3193	152	16	∈	∈	PROPN
ejpam-3193	152	17	rγc	rγc	NOUN
ejpam-3193	152	18	,	,	PUNCT
ejpam-3193	152	19	a	a	DET
ejpam-3193	152	20	contradiction	contradiction	NOUN
ejpam-3193	152	21	to	to	ADP
ejpam-3193	152	22	our	our	PRON
ejpam-3193	152	23	supposition	supposition	NOUN
ejpam-3193	152	24	.	.	PUNCT
ejpam-3193	153	1	hence	hence	ADV
ejpam-3193	153	2	the	the	DET
ejpam-3193	153	3	result	result	NOUN
ejpam-3193	153	4	follows	follow	VERB
ejpam-3193	153	5	.	.	PUNCT
ejpam-3193	154	1	example	example	NOUN
ejpam-3193	154	2	5	5	NUM
ejpam-3193	154	3	.	.	PUNCT
ejpam-3193	155	1	let	let	VERB
ejpam-3193	155	2	z	z	PRON
ejpam-3193	155	3	be	be	AUX
ejpam-3193	155	4	a	a	DET
ejpam-3193	155	5	γ	γ	NOUN
ejpam-3193	155	6	-	-	PUNCT
ejpam-3193	155	7	near	near	ADJ
ejpam-3193	155	8	ring	ring	NOUN
ejpam-3193	155	9	and	and	CCONJ
ejpam-3193	155	10	γ	γ	X
ejpam-3193	155	11	=	=	SYM
ejpam-3193	155	12	{	{	PUNCT
ejpam-3193	155	13	0	0	NUM
ejpam-3193	155	14	,	,	PUNCT
ejpam-3193	155	15	1	1	NUM
ejpam-3193	155	16	,	,	PUNCT
ejpam-3193	155	17	2	2	NUM
ejpam-3193	155	18	,	,	PUNCT
ejpam-3193	155	19	3	3	NUM
ejpam-3193	155	20	}	}	PUNCT
ejpam-3193	155	21	consider	consider	VERB
ejpam-3193	155	22	c	c	NOUN
ejpam-3193	155	23	=	=	SYM
ejpam-3193	155	24	6	6	NUM
ejpam-3193	155	25	be	be	AUX
ejpam-3193	155	26	an	an	DET
ejpam-3193	155	27	non	non	ADJ
ejpam-3193	155	28	prime	prime	ADJ
ejpam-3193	155	29	element	element	NOUN
ejpam-3193	155	30	of	of	ADP
ejpam-3193	155	31	z	z	PROPN
ejpam-3193	155	32	then	then	ADV
ejpam-3193	155	33	zγ6	zγ6	PROPN
ejpam-3193	155	34	is	be	AUX
ejpam-3193	155	35	non	non	ADJ
ejpam-3193	155	36	prime	prime	ADJ
ejpam-3193	155	37	ideal	ideal	NOUN
ejpam-3193	155	38	because	because	SCONJ
ejpam-3193	155	39	3	3	NUM
ejpam-3193	155	40	6∈	6∈	NOUN
ejpam-3193	155	41	zγ6	zγ6	NOUN
ejpam-3193	155	42	and	and	CCONJ
ejpam-3193	155	43	4	4	NUM
ejpam-3193	155	44	6∈	6∈	NOUN
ejpam-3193	155	45	zγ6	zγ6	NOUN
ejpam-3193	155	46	but	but	CCONJ
ejpam-3193	155	47	12	12	NUM
ejpam-3193	155	48	∈	∈	NOUN
ejpam-3193	155	49	zγ6	zγ6	NOUN
ejpam-3193	155	50	and	and	CCONJ
ejpam-3193	155	51	12	12	NUM
ejpam-3193	155	52	6∈	6∈	PROPN
ejpam-3193	155	53	zγ62	zγ62	PROPN
ejpam-3193	155	54	.	.	PUNCT
ejpam-3193	156	1	in	in	ADP
ejpam-3193	156	2	the	the	DET
ejpam-3193	156	3	below	below	ADJ
ejpam-3193	156	4	proposition	proposition	NOUN
ejpam-3193	156	5	,	,	PUNCT
ejpam-3193	156	6	we	we	PRON
ejpam-3193	156	7	reverse	reverse	VERB
ejpam-3193	156	8	the	the	DET
ejpam-3193	156	9	situation	situation	NOUN
ejpam-3193	156	10	occurring	occur	VERB
ejpam-3193	156	11	in	in	ADP
ejpam-3193	156	12	lemma	lemma	PROPN
ejpam-3193	156	13	2	2	NUM
ejpam-3193	156	14	.	.	PUNCT
ejpam-3193	156	15	preposition	preposition	NOUN
ejpam-3193	156	16	3	3	NUM
ejpam-3193	156	17	.	.	PUNCT
ejpam-3193	156	18	suppose	suppose	VERB
ejpam-3193	156	19	that	that	SCONJ
ejpam-3193	156	20	r	r	NOUN
ejpam-3193	156	21	be	be	AUX
ejpam-3193	156	22	γ	γ	X
ejpam-3193	156	23	-	-	PUNCT
ejpam-3193	156	24	near	near	ADJ
ejpam-3193	156	25	integral	integral	ADJ
ejpam-3193	156	26	domain	domain	NOUN
ejpam-3193	156	27	and	and	CCONJ
ejpam-3193	156	28	c	c	AUX
ejpam-3193	156	29	be	be	AUX
ejpam-3193	156	30	a	a	DET
ejpam-3193	156	31	nonzero	nonzero	NOUN
ejpam-3193	156	32	nonunit	nonunit	NOUN
ejpam-3193	156	33	element	element	NOUN
ejpam-3193	156	34	of	of	ADP
ejpam-3193	156	35	r.	r.	PROPN
ejpam-3193	156	36	if	if	SCONJ
ejpam-3193	156	37	c	c	PROPN
ejpam-3193	156	38	is	be	AUX
ejpam-3193	156	39	not	not	PART
ejpam-3193	156	40	a	a	DET
ejpam-3193	156	41	prime	prime	ADJ
ejpam-3193	156	42	element	element	NOUN
ejpam-3193	156	43	then	then	ADV
ejpam-3193	156	44	there	there	PRON
ejpam-3193	156	45	exists	exist	VERB
ejpam-3193	156	46	a	a	DET
ejpam-3193	156	47	∈	∈	ADJ
ejpam-3193	156	48	rγc	rγc	NOUN
ejpam-3193	156	49	and	and	CCONJ
ejpam-3193	156	50	b	b	PROPN
ejpam-3193	156	51	∈	∈	PROPN
ejpam-3193	156	52	rγc	rγc	NOUN
ejpam-3193	156	53	such	such	ADJ
ejpam-3193	157	1	that	that	SCONJ
ejpam-3193	157	2	aγb	aγb	ADP
ejpam-3193	157	3	∈	∈	PROPN
ejpam-3193	157	4	rγc	rγc	NOUN
ejpam-3193	157	5	and	and	CCONJ
ejpam-3193	157	6	aγb	aγb	NOUN
ejpam-3193	157	7	∈	∈	PROPN
ejpam-3193	157	8	rγc2	rγc2	PROPN
ejpam-3193	157	9	.	.	PUNCT
ejpam-3193	158	1	proof	proof	NOUN
ejpam-3193	158	2	.	.	PUNCT
ejpam-3193	159	1	suppose	suppose	VERB
ejpam-3193	159	2	an	an	DET
ejpam-3193	159	3	ideal	ideal	ADJ
ejpam-3193	159	4	rγc	rγc	NOUN
ejpam-3193	159	5	is	be	AUX
ejpam-3193	159	6	not	not	PART
ejpam-3193	159	7	prime	prime	ADJ
ejpam-3193	159	8	and	and	CCONJ
ejpam-3193	159	9	consider	consider	VERB
ejpam-3193	159	10	a	a	DET
ejpam-3193	159	11	∈	∈	PROPN
ejpam-3193	159	12	rγc	rγc	NOUN
ejpam-3193	159	13	,	,	PUNCT
ejpam-3193	159	14	b	b	X
ejpam-3193	159	15	∈	∈	PROPN
ejpam-3193	159	16	rγc	rγc	NOUN
ejpam-3193	159	17	such	such	ADJ
ejpam-3193	159	18	that	that	SCONJ
ejpam-3193	159	19	aγb	aγb	NOUN
ejpam-3193	159	20	∈	∈	PROPN
ejpam-3193	159	21	rγc	rγc	NOUN
ejpam-3193	159	22	.	.	PUNCT
ejpam-3193	160	1	if	if	SCONJ
ejpam-3193	160	2	the	the	DET
ejpam-3193	160	3	case	case	NOUN
ejpam-3193	160	4	,	,	PUNCT
ejpam-3193	160	5	aγb	aγb	VERB
ejpam-3193	160	6	6∈	6∈	PROPN
ejpam-3193	160	7	rγc2	rγc2	VERB
ejpam-3193	160	8	then	then	ADV
ejpam-3193	160	9	for	for	ADP
ejpam-3193	160	10	d	d	PROPN
ejpam-3193	160	11	=	=	SYM
ejpam-3193	160	12	(	(	PUNCT
ejpam-3193	160	13	b+	b+	X
ejpam-3193	160	14	c	c	X
ejpam-3193	160	15	)	)	PUNCT
ejpam-3193	160	16	∈	∈	PROPN
ejpam-3193	160	17	rγc	rγc	NOUN
ejpam-3193	160	18	and	and	CCONJ
ejpam-3193	160	19	aγd	aγd	PROPN
ejpam-3193	160	20	∈	∈	PROPN
ejpam-3193	160	21	rγc	rγc	NOUN
ejpam-3193	160	22	.	.	PUNCT
ejpam-3193	161	1	consider	consider	VERB
ejpam-3193	161	2	aγd	aγd	VERB
ejpam-3193	161	3	6∈	6∈	PROPN
ejpam-3193	161	4	rγc2	rγc2	PROPN
ejpam-3193	161	5	)	)	PUNCT
ejpam-3193	161	6	implies	imply	VERB
ejpam-3193	161	7	ac	ac	PROPN
ejpam-3193	161	8	6∈	6∈	PROPN
ejpam-3193	161	9	rγc2	rγc2	PROPN
ejpam-3193	161	10	and	and	CCONJ
ejpam-3193	161	11	because	because	SCONJ
ejpam-3193	161	12	aγb	aγb	NOUN
ejpam-3193	161	13	6∈	6∈	PROPN
ejpam-3193	161	14	rγc2	rγc2	PROPN
ejpam-3193	161	15	implies	imply	VERB
ejpam-3193	161	16	a	a	DET
ejpam-3193	161	17	6∈	6∈	PROPN
ejpam-3193	161	18	rγc	rγc	NOUN
ejpam-3193	161	19	,	,	PUNCT
ejpam-3193	161	20	a	a	DET
ejpam-3193	161	21	contradiction	contradiction	NOUN
ejpam-3193	161	22	a.	a.	NOUN
ejpam-3193	161	23	taouti	taouti	PROPN
ejpam-3193	162	1	et	et	PROPN
ejpam-3193	162	2	al	al	PROPN
ejpam-3193	162	3	.	.	PUNCT
ejpam-3193	162	4	/	/	SYM
ejpam-3193	162	5	eur	eur	PROPN
ejpam-3193	162	6	.	.	PUNCT
ejpam-3193	163	1	j.	j.	PROPN
ejpam-3193	163	2	pure	pure	PROPN
ejpam-3193	163	3	appl	appl	PROPN
ejpam-3193	163	4	.	.	PROPN
ejpam-3193	163	5	math	math	PROPN
ejpam-3193	163	6	,	,	PUNCT
ejpam-3193	163	7	11	11	NUM
ejpam-3193	163	8	(	(	PUNCT
ejpam-3193	163	9	2	2	NUM
ejpam-3193	163	10	)	)	PUNCT
ejpam-3193	163	11	(	(	PUNCT
ejpam-3193	163	12	2018	2018	NUM
ejpam-3193	163	13	)	)	PUNCT
ejpam-3193	163	14	,	,	PUNCT
ejpam-3193	163	15	449	449	NUM
ejpam-3193	163	16	-	-	SYM
ejpam-3193	163	17	456	456	NUM
ejpam-3193	163	18	453	453	NUM
ejpam-3193	163	19	to	to	ADP
ejpam-3193	163	20	our	our	PRON
ejpam-3193	163	21	hypothesis	hypothesis	NOUN
ejpam-3193	163	22	.	.	PUNCT
ejpam-3193	164	1	hence	hence	ADV
ejpam-3193	164	2	the	the	DET
ejpam-3193	164	3	result	result	NOUN
ejpam-3193	164	4	is	be	AUX
ejpam-3193	164	5	valid	valid	ADJ
ejpam-3193	164	6	.	.	PUNCT
ejpam-3193	165	1	supporting	support	VERB
ejpam-3193	165	2	the	the	DET
ejpam-3193	165	3	above	above	ADJ
ejpam-3193	165	4	lemma3	lemma3	NOUN
ejpam-3193	165	5	we	we	PRON
ejpam-3193	165	6	present	present	VERB
ejpam-3193	165	7	the	the	DET
ejpam-3193	165	8	below	below	ADJ
ejpam-3193	165	9	example	example	NOUN
ejpam-3193	165	10	.	.	PUNCT
ejpam-3193	166	1	example	example	NOUN
ejpam-3193	167	1	6	6	NUM
ejpam-3193	167	2	.	.	PUNCT
ejpam-3193	168	1	let	let	VERB
ejpam-3193	168	2	z8	z8	NOUN
ejpam-3193	168	3	=	=	SYM
ejpam-3193	168	4	{	{	PUNCT
ejpam-3193	168	5	0	0	NUM
ejpam-3193	168	6	,	,	PUNCT
ejpam-3193	168	7	1	1	NUM
ejpam-3193	168	8	,	,	PUNCT
ejpam-3193	168	9	2	2	NUM
ejpam-3193	168	10	,	,	PUNCT
ejpam-3193	168	11	3	3	NUM
ejpam-3193	168	12	,	,	PUNCT
ejpam-3193	168	13	4	4	NUM
ejpam-3193	168	14	,	,	PUNCT
ejpam-3193	168	15	5	5	NUM
ejpam-3193	168	16	,	,	PUNCT
ejpam-3193	168	17	6	6	NUM
ejpam-3193	168	18	,	,	PUNCT
ejpam-3193	168	19	7	7	NUM
ejpam-3193	168	20	}	}	PUNCT
ejpam-3193	168	21	and	and	CCONJ
ejpam-3193	168	22	γ	γ	X
ejpam-3193	168	23	=	=	SYM
ejpam-3193	168	24	{	{	PUNCT
ejpam-3193	168	25	0	0	NUM
ejpam-3193	168	26	,	,	PUNCT
ejpam-3193	168	27	2	2	NUM
ejpam-3193	168	28	,	,	PUNCT
ejpam-3193	168	29	4	4	NUM
ejpam-3193	168	30	}	}	PUNCT
ejpam-3193	168	31	consider	consider	VERB
ejpam-3193	168	32	a	a	DET
ejpam-3193	168	33	non	non	ADJ
ejpam-3193	168	34	-	-	ADJ
ejpam-3193	168	35	prime	prime	ADJ
ejpam-3193	168	36	element	element	NOUN
ejpam-3193	168	37	of	of	ADP
ejpam-3193	168	38	z8	z8	NOUN
ejpam-3193	168	39	i.e.	i.e.	X
ejpam-3193	168	40	,	,	PUNCT
ejpam-3193	168	41	c	c	NOUN
ejpam-3193	168	42	=	=	SYM
ejpam-3193	168	43	6	6	NUM
ejpam-3193	168	44	implies	imply	VERB
ejpam-3193	168	45	6z8	6z8	NUM
ejpam-3193	168	46	=	=	SYM
ejpam-3193	168	47	{	{	PUNCT
ejpam-3193	168	48	0	0	NUM
ejpam-3193	168	49	,	,	PUNCT
ejpam-3193	168	50	2	2	NUM
ejpam-3193	168	51	,	,	PUNCT
ejpam-3193	168	52	4	4	NUM
ejpam-3193	168	53	}	}	PUNCT
ejpam-3193	168	54	.	.	PUNCT
ejpam-3193	169	1	consider	consider	VERB
ejpam-3193	169	2	6	6	NUM
ejpam-3193	169	3	,	,	PUNCT
ejpam-3193	169	4	4	4	NUM
ejpam-3193	169	5	∈	∈	NOUN
ejpam-3193	169	6	6z8	6z8	NUM
ejpam-3193	169	7	such	such	ADJ
ejpam-3193	169	8	that	that	SCONJ
ejpam-3193	169	9	6.2.4	6.2.4	NUM
ejpam-3193	169	10	=	=	SYM
ejpam-3193	169	11	0	0	NUM
ejpam-3193	169	12	∈	∈	NOUN
ejpam-3193	169	13	6z8	6z8	NUM
ejpam-3193	169	14	and	and	CCONJ
ejpam-3193	169	15	c2	c2	PROPN
ejpam-3193	169	16	=	=	PROPN
ejpam-3193	169	17	62	62	NUM
ejpam-3193	169	18	and	and	CCONJ
ejpam-3193	169	19	62z8	62z8	NUM
ejpam-3193	169	20	=	=	SYM
ejpam-3193	169	21	{	{	PUNCT
ejpam-3193	169	22	0	0	NUM
ejpam-3193	169	23	,	,	PUNCT
ejpam-3193	169	24	4	4	NUM
ejpam-3193	169	25	}	}	PUNCT
ejpam-3193	169	26	,	,	PUNCT
ejpam-3193	170	1	hence	hence	ADV
ejpam-3193	170	2	6.2.4	6.2.4	NUM
ejpam-3193	170	3	=	=	SYM
ejpam-3193	170	4	0	0	NUM
ejpam-3193	170	5	∈	∈	PROPN
ejpam-3193	170	6	62z8	62z8	NUM
ejpam-3193	170	7	.	.	PUNCT
ejpam-3193	171	1	further	far	ADV
ejpam-3193	171	2	we	we	PRON
ejpam-3193	171	3	consider	consider	VERB
ejpam-3193	171	4	6.4.4	6.4.4	NUM
ejpam-3193	171	5	=	=	SYM
ejpam-3193	171	6	4	4	NUM
ejpam-3193	171	7	∈	∈	NOUN
ejpam-3193	171	8	62z	62z	NOUN
ejpam-3193	171	9	and	and	CCONJ
ejpam-3193	171	10	take	take	VERB
ejpam-3193	171	11	4	4	NUM
ejpam-3193	171	12	,	,	PUNCT
ejpam-3193	171	13	2	2	NUM
ejpam-3193	171	14	∈	∈	NOUN
ejpam-3193	171	15	6z8	6z8	NUM
ejpam-3193	171	16	then	then	ADV
ejpam-3193	171	17	4.2.2	4.2.2	NUM
ejpam-3193	171	18	=	=	SYM
ejpam-3193	171	19	0	0	NUM
ejpam-3193	171	20	∈	∈	PROPN
ejpam-3193	171	21	6z8	6z8	NUM
ejpam-3193	171	22	,	,	PUNCT
ejpam-3193	171	23	and	and	CCONJ
ejpam-3193	171	24	again	again	ADV
ejpam-3193	171	25	we	we	PRON
ejpam-3193	171	26	get	get	VERB
ejpam-3193	171	27	4.2.2	4.2.2	NUM
ejpam-3193	171	28	=	=	SYM
ejpam-3193	171	29	0	0	PUNCT
ejpam-3193	171	30	∈	∈	PROPN
ejpam-3193	171	31	62z8	62z8	NUM
ejpam-3193	171	32	,	,	PUNCT
ejpam-3193	171	33	similarly	similarly	ADV
ejpam-3193	171	34	4.4.2	4.4.2	NUM
ejpam-3193	171	35	=	=	SYM
ejpam-3193	171	36	0	0	PUNCT
ejpam-3193	171	37	∈	∈	PROPN
ejpam-3193	171	38	6z8	6z8	NUM
ejpam-3193	171	39	and	and	CCONJ
ejpam-3193	171	40	4.4.2	4.4.2	NUM
ejpam-3193	171	41	=	=	SYM
ejpam-3193	171	42	0	0	PUNCT
ejpam-3193	172	1	∈	∈	PROPN
ejpam-3193	172	2	62z8	62z8	NUM
ejpam-3193	172	3	.	.	PUNCT
ejpam-3193	173	1	theorem	theorem	NOUN
ejpam-3193	173	2	1	1	NUM
ejpam-3193	173	3	.	.	PUNCT
ejpam-3193	173	4	suppose	suppose	VERB
ejpam-3193	173	5	n	n	PRON
ejpam-3193	173	6	be	be	AUX
ejpam-3193	173	7	a	a	DET
ejpam-3193	173	8	γ	γ	NOUN
ejpam-3193	173	9	-	-	PUNCT
ejpam-3193	173	10	near	near	ADP
ejpam-3193	173	11	-	-	PUNCT
ejpam-3193	173	12	ring	ring	NOUN
ejpam-3193	173	13	with	with	ADP
ejpam-3193	173	14	identity	identity	NOUN
ejpam-3193	173	15	and	and	CCONJ
ejpam-3193	173	16	p	p	NOUN
ejpam-3193	173	17	be	be	AUX
ejpam-3193	173	18	an	an	DET
ejpam-3193	173	19	almost	almost	ADV
ejpam-3193	173	20	prime	prime	ADJ
ejpam-3193	173	21	ideal	ideal	NOUN
ejpam-3193	173	22	of	of	ADP
ejpam-3193	173	23	n	n	PROPN
ejpam-3193	173	24	.	.	PUNCT
ejpam-3193	174	1	if	if	SCONJ
ejpam-3193	174	2	p	p	NOUN
ejpam-3193	174	3	is	be	AUX
ejpam-3193	174	4	not	not	PART
ejpam-3193	174	5	prime	prime	ADJ
ejpam-3193	174	6	then	then	ADV
ejpam-3193	174	7	pγp	pγp	ADV
ejpam-3193	174	8	=	=	SYM
ejpam-3193	174	9	p	p	NOUN
ejpam-3193	174	10	.	.	PUNCT
ejpam-3193	175	1	proof	proof	NOUN
ejpam-3193	175	2	.	.	PUNCT
ejpam-3193	176	1	let	let	VERB
ejpam-3193	176	2	us	we	PRON
ejpam-3193	176	3	assume	assume	VERB
ejpam-3193	176	4	that	that	SCONJ
ejpam-3193	176	5	p	p	PROPN
ejpam-3193	176	6	⊆	⊆	NUM
ejpam-3193	176	7	pγp	pγp	ADJ
ejpam-3193	176	8	.	.	PUNCT
ejpam-3193	177	1	we	we	PRON
ejpam-3193	177	2	have	have	VERB
ejpam-3193	177	3	to	to	PART
ejpam-3193	177	4	prove	prove	VERB
ejpam-3193	177	5	that	that	SCONJ
ejpam-3193	177	6	p	p	NOUN
ejpam-3193	177	7	is	be	AUX
ejpam-3193	177	8	prime	prime	ADJ
ejpam-3193	177	9	.	.	PUNCT
ejpam-3193	178	1	let	let	VERB
ejpam-3193	178	2	us	we	PRON
ejpam-3193	178	3	suppose	suppose	VERB
ejpam-3193	178	4	that	that	SCONJ
ejpam-3193	178	5	two	two	NUM
ejpam-3193	178	6	ideals	ideal	NOUN
ejpam-3193	178	7	a	a	PRON
ejpam-3193	178	8	and	and	CCONJ
ejpam-3193	178	9	b	b	NOUN
ejpam-3193	178	10	contained	contain	VERB
ejpam-3193	178	11	in	in	ADP
ejpam-3193	178	12	n	n	CCONJ
ejpam-3193	178	13	such	such	ADJ
ejpam-3193	178	14	that	that	SCONJ
ejpam-3193	178	15	aγb	aγb	VERB
ejpam-3193	178	16	⊆	⊆	NUM
ejpam-3193	178	17	p	p	NOUN
ejpam-3193	178	18	.	.	PUNCT
ejpam-3193	179	1	if	if	SCONJ
ejpam-3193	179	2	aγb	aγb	NOUN
ejpam-3193	179	3	*	*	PUNCT
ejpam-3193	179	4	pγp	pγp	INTJ
ejpam-3193	179	5	then	then	ADV
ejpam-3193	179	6	a	a	DET
ejpam-3193	179	7	*	*	PUNCT
ejpam-3193	179	8	p	p	NOUN
ejpam-3193	179	9	or	or	CCONJ
ejpam-3193	179	10	b	b	NOUN
ejpam-3193	180	1	*	*	PUNCT
ejpam-3193	180	2	p	p	NOUN
ejpam-3193	180	3	.	.	PUNCT
ejpam-3193	181	1	we	we	PRON
ejpam-3193	181	2	assume	assume	VERB
ejpam-3193	181	3	that	that	SCONJ
ejpam-3193	181	4	aγb	aγb	VERB
ejpam-3193	181	5	*	*	PUNCT
ejpam-3193	181	6	pγp	pγp	INTJ
ejpam-3193	181	7	.	.	PUNCT
ejpam-3193	182	1	since	since	SCONJ
ejpam-3193	182	2	p	p	NOUN
ejpam-3193	182	3	*	*	PUNCT
ejpam-3193	182	4	pγp	pγp	ADV
ejpam-3193	182	5	as	as	ADP
ejpam-3193	182	6	a	a	DET
ejpam-3193	182	7	result	result	NOUN
ejpam-3193	182	8	p	p	X
ejpam-3193	182	9	∈	∈	PROPN
ejpam-3193	182	10	p	p	NOUN
ejpam-3193	182	11	such	such	ADJ
ejpam-3193	182	12	that	that	SCONJ
ejpam-3193	182	13	<	<	X
ejpam-3193	182	14	p	p	X
ejpam-3193	182	15	>	>	X
ejpam-3193	182	16	*	*	PUNCT
ejpam-3193	182	17	pγp	pγp	INTJ
ejpam-3193	182	18	hence	hence	ADV
ejpam-3193	182	19	(	(	PUNCT
ejpam-3193	182	20	a+	a+	PUNCT
ejpam-3193	182	21	<	<	X
ejpam-3193	182	22	p	p	X
ejpam-3193	182	23	>	>	X
ejpam-3193	182	24	)	)	PUNCT
ejpam-3193	182	25	γ(b	γ(b	PROPN
ejpam-3193	182	26	+	+	NOUN
ejpam-3193	182	27	n	n	CCONJ
ejpam-3193	182	28	)	)	PUNCT
ejpam-3193	182	29	*	*	PUNCT
ejpam-3193	182	30	pγp	pγp	INTJ
ejpam-3193	182	31	.	.	PUNCT
ejpam-3193	183	1	consider	consider	VERB
ejpam-3193	183	2	(	(	PUNCT
ejpam-3193	183	3	a+	a+	PUNCT
ejpam-3193	183	4	<	<	X
ejpam-3193	183	5	p	p	X
ejpam-3193	183	6	>	>	X
ejpam-3193	183	7	)	)	PUNCT
ejpam-3193	183	8	γ(b	γ(b	PROPN
ejpam-3193	183	9	+	+	NOUN
ejpam-3193	183	10	n	n	CCONJ
ejpam-3193	183	11	)	)	PUNCT
ejpam-3193	183	12	*	*	PUNCT
ejpam-3193	184	1	p	p	X
ejpam-3193	184	2	,	,	PUNCT
ejpam-3193	184	3	there	there	PRON
ejpam-3193	184	4	exist	exist	VERB
ejpam-3193	184	5	an	an	DET
ejpam-3193	184	6	element	element	NOUN
ejpam-3193	184	7	a	a	DET
ejpam-3193	184	8	∈	∈	PROPN
ejpam-3193	184	9	a	a	PRON
ejpam-3193	184	10	,	,	PUNCT
ejpam-3193	184	11	b	b	PROPN
ejpam-3193	184	12	∈	∈	PROPN
ejpam-3193	184	13	b	b	PROPN
ejpam-3193	184	14	,	,	PUNCT
ejpam-3193	184	15	p0	p0	PROPN
ejpam-3193	184	16	∈	∈	PROPN
ejpam-3193	184	17	<	<	X
ejpam-3193	184	18	p	p	X
ejpam-3193	184	19	>	>	X
ejpam-3193	184	20	and	and	CCONJ
ejpam-3193	184	21	q0	q0	PROPN
ejpam-3193	184	22	∈	∈	PROPN
ejpam-3193	184	23	n	n	PRON
ejpam-3193	184	24	such	such	ADJ
ejpam-3193	184	25	that	that	SCONJ
ejpam-3193	184	26	(	(	PUNCT
ejpam-3193	184	27	a+p0)γ(b+q0	a+p0)γ(b+q0	NOUN
ejpam-3193	184	28	)	)	PUNCT
ejpam-3193	184	29	6∈	6∈	PROPN
ejpam-3193	184	30	p	p	NOUN
ejpam-3193	184	31	implies	imply	VERB
ejpam-3193	184	32	aγ(b	aγ(b	PUNCT
ejpam-3193	184	33	+	+	CCONJ
ejpam-3193	184	34	q0	q0	ADJ
ejpam-3193	184	35	)	)	PUNCT
ejpam-3193	184	36	6∈	6∈	PROPN
ejpam-3193	184	37	p	p	NOUN
ejpam-3193	184	38	,	,	PUNCT
ejpam-3193	184	39	but	but	CCONJ
ejpam-3193	184	40	aγ(b	aγ(b	X
ejpam-3193	184	41	+	+	CCONJ
ejpam-3193	184	42	q0	q0	ADJ
ejpam-3193	184	43	)	)	PUNCT
ejpam-3193	184	44	=	=	SYM
ejpam-3193	184	45	aγ(b	aγ(b	X
ejpam-3193	184	46	+	+	CCONJ
ejpam-3193	184	47	q0	q0	ADJ
ejpam-3193	184	48	)	)	PUNCT
ejpam-3193	184	49	−	−	PROPN
ejpam-3193	185	1	aγb	aγb	NOUN
ejpam-3193	186	1	+	+	CCONJ
ejpam-3193	186	2	aγb	aγb	NOUN
ejpam-3193	186	3	∈	∈	PROPN
ejpam-3193	186	4	p	p	NOUN
ejpam-3193	186	5	as	as	ADP
ejpam-3193	186	6	aγb	aγb	NOUN
ejpam-3193	186	7	⊆	⊆	NUM
ejpam-3193	186	8	p	p	NOUN
ejpam-3193	186	9	,	,	PUNCT
ejpam-3193	186	10	a	a	DET
ejpam-3193	186	11	contradiction	contradiction	NOUN
ejpam-3193	186	12	.	.	PUNCT
ejpam-3193	187	1	hence	hence	ADV
ejpam-3193	187	2	(	(	PUNCT
ejpam-3193	187	3	a+	a+	PUNCT
ejpam-3193	187	4	<	<	X
ejpam-3193	187	5	p	p	X
ejpam-3193	187	6	>	>	X
ejpam-3193	187	7	)	)	PUNCT
ejpam-3193	187	8	γ(b	γ(b	PROPN
ejpam-3193	187	9	+	+	NOUN
ejpam-3193	187	10	n	n	CCONJ
ejpam-3193	187	11	)	)	PUNCT
ejpam-3193	188	1	⊆	⊆	PROPN
ejpam-3193	188	2	p	p	NOUN
ejpam-3193	188	3	implies	imply	VERB
ejpam-3193	188	4	a	a	DET
ejpam-3193	188	5	⊆	⊆	NUM
ejpam-3193	188	6	p	p	NOUN
ejpam-3193	188	7	.	.	PUNCT
ejpam-3193	189	1	corollary	corollary	ADJ
ejpam-3193	189	2	1	1	NUM
ejpam-3193	189	3	.	.	PUNCT
ejpam-3193	190	1	consider	consider	VERB
ejpam-3193	190	2	n	n	PRON
ejpam-3193	190	3	a	a	DET
ejpam-3193	190	4	γ	γ	X
ejpam-3193	190	5	-	-	PUNCT
ejpam-3193	190	6	near	near	ADP
ejpam-3193	190	7	-	-	PUNCT
ejpam-3193	190	8	ring	ring	NOUN
ejpam-3193	190	9	having	have	VERB
ejpam-3193	190	10	identity	identity	NOUN
ejpam-3193	190	11	and	and	CCONJ
ejpam-3193	190	12	containing	contain	VERB
ejpam-3193	190	13	an	an	DET
ejpam-3193	190	14	ideal	ideal	NOUN
ejpam-3193	190	15	p	p	NOUN
ejpam-3193	190	16	.	.	PUNCT
ejpam-3193	191	1	if	if	SCONJ
ejpam-3193	191	2	pγp	pγp	PROPN
ejpam-3193	191	3	6=	6=	PUNCT
ejpam-3193	192	1	p	p	X
ejpam-3193	192	2	then	then	ADV
ejpam-3193	192	3	p	p	PROPN
ejpam-3193	192	4	is	be	AUX
ejpam-3193	192	5	prime	prime	ADJ
ejpam-3193	192	6	if	if	SCONJ
ejpam-3193	193	1	and	and	CCONJ
ejpam-3193	193	2	only	only	ADV
ejpam-3193	193	3	if	if	SCONJ
ejpam-3193	193	4	p	p	NOUN
ejpam-3193	193	5	is	be	AUX
ejpam-3193	193	6	almost	almost	ADV
ejpam-3193	193	7	prime	prime	ADJ
ejpam-3193	193	8	.	.	PUNCT
ejpam-3193	194	1	proposition	proposition	NOUN
ejpam-3193	194	2	4	4	NUM
ejpam-3193	194	3	.	.	PUNCT
ejpam-3193	195	1	if	if	SCONJ
ejpam-3193	195	2	p	p	PROPN
ejpam-3193	195	3	6=	6=	ADP
ejpam-3193	195	4	0	0	NUM
ejpam-3193	195	5	be	be	AUX
ejpam-3193	195	6	a	a	DET
ejpam-3193	195	7	proper	proper	ADJ
ejpam-3193	195	8	ideal	ideal	NOUN
ejpam-3193	195	9	of	of	ADP
ejpam-3193	195	10	a	a	DET
ejpam-3193	195	11	γ	γ	NOUN
ejpam-3193	195	12	-	-	PUNCT
ejpam-3193	195	13	near	near	ADP
ejpam-3193	195	14	-	-	PUNCT
ejpam-3193	195	15	ring	ring	NOUN
ejpam-3193	195	16	n	n	NOUN
ejpam-3193	195	17	such	such	ADJ
ejpam-3193	195	18	that	that	SCONJ
ejpam-3193	195	19	p	p	NOUN
ejpam-3193	195	20	is	be	AUX
ejpam-3193	195	21	almost	almost	ADV
ejpam-3193	195	22	prime	prime	ADJ
ejpam-3193	195	23	and	and	CCONJ
ejpam-3193	195	24	(	(	PUNCT
ejpam-3193	195	25	pγp	pγp	INTJ
ejpam-3193	195	26	:	:	PUNCT
ejpam-3193	195	27	p	p	X
ejpam-3193	195	28	)	)	PUNCT
ejpam-3193	195	29	⊆	⊆	NUM
ejpam-3193	195	30	p	p	NOUN
ejpam-3193	195	31	then	then	ADV
ejpam-3193	195	32	p	p	NOUN
ejpam-3193	195	33	is	be	AUX
ejpam-3193	195	34	prime	prime	ADJ
ejpam-3193	195	35	.	.	PUNCT
ejpam-3193	196	1	proof	proof	NOUN
ejpam-3193	196	2	.	.	PUNCT
ejpam-3193	197	1	we	we	PRON
ejpam-3193	197	2	suppose	suppose	VERB
ejpam-3193	197	3	that	that	SCONJ
ejpam-3193	197	4	p	p	NOUN
ejpam-3193	197	5	is	be	AUX
ejpam-3193	197	6	not	not	PART
ejpam-3193	197	7	a	a	DET
ejpam-3193	197	8	prime	prime	ADJ
ejpam-3193	197	9	ideal	ideal	NOUN
ejpam-3193	197	10	of	of	ADP
ejpam-3193	197	11	n	n	PROPN
ejpam-3193	197	12	.	.	PUNCT
ejpam-3193	198	1	then	then	ADV
ejpam-3193	198	2	there	there	PRON
ejpam-3193	198	3	exist	exist	VERB
ejpam-3193	198	4	x	x	X
ejpam-3193	198	5	/	/	SYM
ejpam-3193	198	6	pγp	pγp	PROPN
ejpam-3193	198	7	and	and	CCONJ
ejpam-3193	198	8	y	y	PROPN
ejpam-3193	198	9	6∈	6∈	PROPN
ejpam-3193	198	10	p	p	NOUN
ejpam-3193	198	11	such	such	ADJ
ejpam-3193	198	12	that	that	SCONJ
ejpam-3193	198	13	<	<	X
ejpam-3193	198	14	x	x	X
ejpam-3193	198	15	>	>	X
ejpam-3193	198	16	γ	γ	X
ejpam-3193	198	17	<	<	X
ejpam-3193	198	18	y	y	PROPN
ejpam-3193	198	19	>	>	PROPN
ejpam-3193	198	20	⊆	⊆	NUM
ejpam-3193	198	21	p	p	NOUN
ejpam-3193	198	22	.	.	PUNCT
ejpam-3193	199	1	if	if	SCONJ
ejpam-3193	199	2	<	<	X
ejpam-3193	199	3	x	x	X
ejpam-3193	199	4	>	>	X
ejpam-3193	199	5	γ	γ	X
ejpam-3193	199	6	<	<	X
ejpam-3193	199	7	y	y	X
ejpam-3193	199	8	>	>	PUNCT
ejpam-3193	199	9	*	*	PUNCT
ejpam-3193	199	10	pγp	pγp	INTJ
ejpam-3193	199	11	,	,	PUNCT
ejpam-3193	199	12	then	then	ADV
ejpam-3193	199	13	the	the	DET
ejpam-3193	199	14	result	result	NOUN
ejpam-3193	199	15	holds	hold	VERB
ejpam-3193	199	16	.	.	PUNCT
ejpam-3193	200	1	hence	hence	ADV
ejpam-3193	200	2	<	<	X
ejpam-3193	200	3	x	x	X
ejpam-3193	200	4	>	>	X
ejpam-3193	200	5	γ	γ	X
ejpam-3193	200	6	<	<	X
ejpam-3193	200	7	y	y	PROPN
ejpam-3193	200	8	>	>	PROPN
ejpam-3193	200	9	⊆	⊆	NUM
ejpam-3193	200	10	pγp	pγp	ADV
ejpam-3193	200	11	.	.	PUNCT
ejpam-3193	201	1	suppose	suppose	VERB
ejpam-3193	201	2	<	<	X
ejpam-3193	201	3	x	x	X
ejpam-3193	201	4	>	>	X
ejpam-3193	201	5	γ	γ	X
ejpam-3193	201	6	(	(	PUNCT
ejpam-3193	201	7	<	<	X
ejpam-3193	201	8	y	y	X
ejpam-3193	201	9	>	>	X
ejpam-3193	202	1	+	+	PROPN
ejpam-3193	202	2	p	p	X
ejpam-3193	202	3	)	)	PUNCT
ejpam-3193	202	4	⊆	⊆	NUM
ejpam-3193	202	5	p	p	NOUN
ejpam-3193	202	6	.	.	PUNCT
ejpam-3193	203	1	if	if	SCONJ
ejpam-3193	203	2	<	<	X
ejpam-3193	203	3	x	x	X
ejpam-3193	203	4	>	>	X
ejpam-3193	203	5	γ	γ	X
ejpam-3193	203	6	(	(	PUNCT
ejpam-3193	203	7	<	<	X
ejpam-3193	203	8	y	y	X
ejpam-3193	203	9	>	>	X
ejpam-3193	204	1	+	+	PROPN
ejpam-3193	204	2	p	p	NOUN
ejpam-3193	204	3	)	)	PUNCT
ejpam-3193	204	4	*	*	PUNCT
ejpam-3193	205	1	p	p	X
ejpam-3193	205	2	then	then	ADV
ejpam-3193	205	3	we	we	PRON
ejpam-3193	205	4	have	have	VERB
ejpam-3193	205	5	x	x	X
ejpam-3193	205	6	∈	∈	PROPN
ejpam-3193	205	7	p	p	NOUN
ejpam-3193	205	8	or	or	CCONJ
ejpam-3193	205	9	y	y	PROPN
ejpam-3193	205	10	∈	∈	PROPN
ejpam-3193	205	11	p	p	PROPN
ejpam-3193	205	12	,	,	PUNCT
ejpam-3193	205	13	a	a	DET
ejpam-3193	205	14	contradiction	contradiction	NOUN
ejpam-3193	205	15	to	to	ADP
ejpam-3193	205	16	our	our	PRON
ejpam-3193	205	17	assumption	assumption	NOUN
ejpam-3193	205	18	,	,	PUNCT
ejpam-3193	205	19	or	or	CCONJ
ejpam-3193	205	20	else	else	ADV
ejpam-3193	205	21	<	<	X
ejpam-3193	205	22	x	x	X
ejpam-3193	205	23	>	>	X
ejpam-3193	205	24	γ	γ	X
ejpam-3193	205	25	(	(	PUNCT
ejpam-3193	205	26	<	<	X
ejpam-3193	205	27	y	y	X
ejpam-3193	205	28	>	>	X
ejpam-3193	206	1	+	+	PROPN
ejpam-3193	206	2	p	p	X
ejpam-3193	206	3	)	)	PUNCT
ejpam-3193	206	4	⊆	⊆	NUM
ejpam-3193	206	5	pγp	pγp	ADV
ejpam-3193	206	6	.	.	PUNCT
ejpam-3193	207	1	thus	thus	ADV
ejpam-3193	207	2	<	<	X
ejpam-3193	207	3	x	x	X
ejpam-3193	207	4	>	>	X
ejpam-3193	207	5	γp	γp	NOUN
ejpam-3193	207	6	⊆	⊆	NUM
ejpam-3193	207	7	pγp	pγp	NOUN
ejpam-3193	207	8	implies	imply	VERB
ejpam-3193	207	9	x	x	X
ejpam-3193	207	10	∈	∈	PROPN
ejpam-3193	207	11	(	(	PUNCT
ejpam-3193	207	12	pγp	pγp	ADV
ejpam-3193	207	13	:	:	PUNCT
ejpam-3193	207	14	γ	γ	X
ejpam-3193	207	15	:	:	PUNCT
ejpam-3193	207	16	p	p	NOUN
ejpam-3193	207	17	)	)	PUNCT
ejpam-3193	207	18	⊆	⊆	PROPN
ejpam-3193	207	19	p	p	NOUN
ejpam-3193	207	20	.	.	PUNCT
ejpam-3193	208	1	theorem	theorem	NOUN
ejpam-3193	208	2	2	2	NUM
ejpam-3193	208	3	.	.	PUNCT
ejpam-3193	208	4	suppose	suppose	VERB
ejpam-3193	208	5	n	n	PRON
ejpam-3193	208	6	be	be	AUX
ejpam-3193	208	7	a	a	DET
ejpam-3193	208	8	γ	γ	NOUN
ejpam-3193	208	9	-	-	PUNCT
ejpam-3193	208	10	near	near	ADP
ejpam-3193	208	11	-	-	PUNCT
ejpam-3193	208	12	ring	ring	NOUN
ejpam-3193	208	13	and	and	CCONJ
ejpam-3193	208	14	let	let	VERB
ejpam-3193	208	15	p	p	PRON
ejpam-3193	208	16	be	be	AUX
ejpam-3193	208	17	an	an	DET
ejpam-3193	208	18	ideal	ideal	NOUN
ejpam-3193	208	19	of	of	ADP
ejpam-3193	208	20	n	n	PROPN
ejpam-3193	208	21	.	.	PUNCT
ejpam-3193	209	1	then	then	ADV
ejpam-3193	209	2	the	the	DET
ejpam-3193	209	3	following	follow	VERB
ejpam-3193	209	4	statements	statement	NOUN
ejpam-3193	209	5	are	be	AUX
ejpam-3193	209	6	equivalent	equivalent	ADJ
ejpam-3193	209	7	:	:	PUNCT
ejpam-3193	209	8	i	i	NOUN
ejpam-3193	209	9	)	)	PUNCT
ejpam-3193	209	10	if	if	SCONJ
ejpam-3193	209	11	elements	element	NOUN
ejpam-3193	209	12	a	a	DET
ejpam-3193	209	13	,	,	PUNCT
ejpam-3193	209	14	b	b	NOUN
ejpam-3193	209	15	,	,	PUNCT
ejpam-3193	209	16	c	c	PROPN
ejpam-3193	209	17	∈	∈	PROPN
ejpam-3193	209	18	n	n	ADV
ejpam-3193	209	19	with	with	ADP
ejpam-3193	209	20	aγ	aγ	PROPN
ejpam-3193	209	21	(	(	PUNCT
ejpam-3193	209	22	<	<	X
ejpam-3193	209	23	b	b	X
ejpam-3193	209	24	>	>	X
ejpam-3193	210	1	+	+	X
ejpam-3193	210	2	<	<	X
ejpam-3193	210	3	c	c	X
ejpam-3193	210	4	>	>	PUNCT
ejpam-3193	210	5	)	)	PUNCT
ejpam-3193	210	6	∈	∈	PROPN
ejpam-3193	210	7	p	p	NOUN
ejpam-3193	210	8	and	and	CCONJ
ejpam-3193	210	9	aγ	aγ	NOUN
ejpam-3193	210	10	(	(	PUNCT
ejpam-3193	210	11	<	<	X
ejpam-3193	210	12	b	b	X
ejpam-3193	210	13	>	>	X
ejpam-3193	211	1	+	+	X
ejpam-3193	211	2	<	<	X
ejpam-3193	211	3	c	c	X
ejpam-3193	211	4	>	>	PUNCT
ejpam-3193	211	5	)	)	PUNCT
ejpam-3193	212	1	*	*	PUNCT
ejpam-3193	212	2	pγp	pγp	INTJ
ejpam-3193	212	3	then	then	ADV
ejpam-3193	212	4	a	a	DET
ejpam-3193	212	5	∈	∈	PROPN
ejpam-3193	212	6	p	p	NOUN
ejpam-3193	212	7	or	or	CCONJ
ejpam-3193	212	8	b	b	NOUN
ejpam-3193	212	9	,	,	PUNCT
ejpam-3193	212	10	c	c	PROPN
ejpam-3193	212	11	in	in	ADP
ejpam-3193	212	12	p	p	PROPN
ejpam-3193	212	13	.	.	PUNCT
ejpam-3193	213	1	ii	ii	X
ejpam-3193	213	2	)	)	PUNCT
ejpam-3193	213	3	if	if	SCONJ
ejpam-3193	213	4	x	x	SYM
ejpam-3193	213	5	∈	∈	PROPN
ejpam-3193	213	6	n	n	CCONJ
ejpam-3193	213	7	−	−	PROPN
ejpam-3193	213	8	p	p	NOUN
ejpam-3193	213	9	,	,	PUNCT
ejpam-3193	213	10	then	then	ADV
ejpam-3193	213	11	(	(	PUNCT
ejpam-3193	213	12	p	p	X
ejpam-3193	213	13	:	:	PUNCT
ejpam-3193	213	14	γ	γ	X
ejpam-3193	213	15	:	:	PUNCT
ejpam-3193	213	16	<	<	X
ejpam-3193	213	17	x	x	X
ejpam-3193	213	18	>	>	X
ejpam-3193	214	1	+	+	PUNCT
ejpam-3193	214	2	<	<	X
ejpam-3193	214	3	y	y	X
ejpam-3193	214	4	>	>	PUNCT
ejpam-3193	214	5	)	)	PUNCT
ejpam-3193	215	1	=	=	SYM
ejpam-3193	216	1	p	p	NOUN
ejpam-3193	216	2	∪	∪	X
ejpam-3193	216	3	(	(	PUNCT
ejpam-3193	216	4	pγp	pγp	ADV
ejpam-3193	216	5	:	:	PUNCT
ejpam-3193	216	6	γ	γ	X
ejpam-3193	216	7	:	:	PUNCT
ejpam-3193	216	8	<	<	X
ejpam-3193	216	9	x	x	X
ejpam-3193	216	10	>	>	X
ejpam-3193	217	1	+	+	PUNCT
ejpam-3193	217	2	<	<	X
ejpam-3193	217	3	y	y	PROPN
ejpam-3193	217	4	>	>	PUNCT
ejpam-3193	217	5	)	)	PUNCT
ejpam-3193	217	6	for	for	ADP
ejpam-3193	217	7	some	some	DET
ejpam-3193	217	8	y	y	PROPN
ejpam-3193	217	9	∈	∈	PROPN
ejpam-3193	217	10	n	n	X
ejpam-3193	217	11	.	.	PUNCT
ejpam-3193	218	1	iii	iii	X
ejpam-3193	218	2	)	)	PUNCT
ejpam-3193	219	1	if	if	SCONJ
ejpam-3193	219	2	x	x	SYM
ejpam-3193	219	3	∈	∈	NOUN
ejpam-3193	219	4	np	np	INTJ
ejpam-3193	219	5	,	,	PUNCT
ejpam-3193	219	6	then	then	ADV
ejpam-3193	219	7	(	(	PUNCT
ejpam-3193	219	8	p	p	X
ejpam-3193	219	9	:	:	PUNCT
ejpam-3193	219	10	γ	γ	X
ejpam-3193	219	11	:	:	PUNCT
ejpam-3193	219	12	<	<	X
ejpam-3193	219	13	x	x	X
ejpam-3193	219	14	>	>	X
ejpam-3193	220	1	+	+	PUNCT
ejpam-3193	220	2	<	<	X
ejpam-3193	220	3	y	y	X
ejpam-3193	220	4	>	>	PUNCT
ejpam-3193	220	5	)	)	PUNCT
ejpam-3193	221	1	=	=	SYM
ejpam-3193	222	1	p	p	NOUN
ejpam-3193	222	2	or	or	CCONJ
ejpam-3193	222	3	(	(	PUNCT
ejpam-3193	222	4	p	p	X
ejpam-3193	222	5	:	:	PUNCT
ejpam-3193	222	6	γ	γ	X
ejpam-3193	222	7	:	:	PUNCT
ejpam-3193	222	8	<	<	X
ejpam-3193	222	9	x	x	X
ejpam-3193	222	10	>	>	X
ejpam-3193	223	1	+	+	PUNCT
ejpam-3193	223	2	<	<	X
ejpam-3193	223	3	y	y	X
ejpam-3193	223	4	>	>	PUNCT
ejpam-3193	223	5	)	)	PUNCT
ejpam-3193	223	6	=	=	SYM
ejpam-3193	224	1	(	(	PUNCT
ejpam-3193	224	2	pγp	pγp	INTJ
ejpam-3193	224	3	:	:	PUNCT
ejpam-3193	224	4	γ	γ	X
ejpam-3193	224	5	:	:	PUNCT
ejpam-3193	224	6	<	<	X
ejpam-3193	224	7	x	x	X
ejpam-3193	224	8	>	>	X
ejpam-3193	225	1	+	+	PUNCT
ejpam-3193	225	2	<	<	X
ejpam-3193	225	3	y	y	PROPN
ejpam-3193	225	4	>	>	PUNCT
ejpam-3193	225	5	)	)	PUNCT
ejpam-3193	225	6	for	for	ADP
ejpam-3193	225	7	some	some	DET
ejpam-3193	225	8	y	y	PROPN
ejpam-3193	225	9	∈	∈	PROPN
ejpam-3193	225	10	n	n	X
ejpam-3193	225	11	.	.	PUNCT
ejpam-3193	226	1	iv	iv	X
ejpam-3193	226	2	)	)	PUNCT
ejpam-3193	226	3	p	p	NOUN
ejpam-3193	226	4	is	be	AUX
ejpam-3193	226	5	an	an	DET
ejpam-3193	226	6	almost	almost	ADV
ejpam-3193	226	7	prime	prime	ADJ
ejpam-3193	226	8	.	.	PUNCT
ejpam-3193	227	1	proof	proof	NOUN
ejpam-3193	227	2	.	.	PUNCT
ejpam-3193	228	1	(	(	PUNCT
ejpam-3193	228	2	i	i	NOUN
ejpam-3193	228	3	)	)	PUNCT
ejpam-3193	228	4	implies	imply	VERB
ejpam-3193	228	5	(	(	PUNCT
ejpam-3193	228	6	ii	ii	NOUN
ejpam-3193	228	7	)	)	PUNCT
ejpam-3193	228	8	consider	consider	VERB
ejpam-3193	228	9	t	t	NOUN
ejpam-3193	228	10	∈	∈	PROPN
ejpam-3193	228	11	(	(	PUNCT
ejpam-3193	228	12	p	p	X
ejpam-3193	228	13	:	:	PUNCT
ejpam-3193	228	14	γ	γ	X
ejpam-3193	228	15	:	:	PUNCT
ejpam-3193	228	16	<	<	X
ejpam-3193	228	17	x	x	X
ejpam-3193	228	18	>	>	X
ejpam-3193	229	1	+	+	PUNCT
ejpam-3193	229	2	<	<	X
ejpam-3193	229	3	y	y	PROPN
ejpam-3193	229	4	>	>	PUNCT
ejpam-3193	229	5	)	)	PUNCT
ejpam-3193	229	6	for	for	ADP
ejpam-3193	229	7	some	some	DET
ejpam-3193	229	8	x	x	SYM
ejpam-3193	229	9	∈	∈	PROPN
ejpam-3193	229	10	n	n	PRON
ejpam-3193	229	11	−p	−p	NOUN
ejpam-3193	229	12	,	,	PUNCT
ejpam-3193	229	13	γ	γ	PROPN
ejpam-3193	229	14	∈	∈	PROPN
ejpam-3193	229	15	γ	γ	NOUN
ejpam-3193	229	16	and	and	CCONJ
ejpam-3193	229	17	y	y	PROPN
ejpam-3193	229	18	∈	∈	PROPN
ejpam-3193	229	19	n	n	ADV
ejpam-3193	229	20	.	.	PUNCT
ejpam-3193	230	1	after	after	ADP
ejpam-3193	230	2	that	that	DET
ejpam-3193	230	3	tγ	tγ	NOUN
ejpam-3193	230	4	(	(	PUNCT
ejpam-3193	230	5	<	<	X
ejpam-3193	230	6	x	x	X
ejpam-3193	230	7	>	>	X
ejpam-3193	231	1	+	+	PUNCT
ejpam-3193	231	2	<	<	X
ejpam-3193	231	3	y	y	PROPN
ejpam-3193	231	4	>	>	PUNCT
ejpam-3193	231	5	)	)	PUNCT
ejpam-3193	232	1	⊆	⊆	NUM
ejpam-3193	232	2	p	p	NOUN
ejpam-3193	232	3	.	.	PUNCT
ejpam-3193	233	1	if	if	SCONJ
ejpam-3193	233	2	tγ	tγ	PROPN
ejpam-3193	233	3	(	(	PUNCT
ejpam-3193	233	4	<	<	X
ejpam-3193	233	5	x	x	X
ejpam-3193	233	6	>	>	X
ejpam-3193	234	1	+	+	PUNCT
ejpam-3193	234	2	<	<	X
ejpam-3193	234	3	y	y	PROPN
ejpam-3193	234	4	>	>	PUNCT
ejpam-3193	234	5	)	)	PUNCT
ejpam-3193	235	1	⊆	⊆	NUM
ejpam-3193	235	2	pγp	pγp	ADV
ejpam-3193	235	3	subsequently	subsequently	ADV
ejpam-3193	235	4	t2γ(pγp	t2γ(pγp	X
ejpam-3193	235	5	:	:	PUNCT
ejpam-3193	235	6	γ	γ	X
ejpam-3193	235	7	:	:	PUNCT
ejpam-3193	235	8	<	<	X
ejpam-3193	235	9	x	x	X
ejpam-3193	235	10	>	>	X
ejpam-3193	236	1	+	+	PUNCT
ejpam-3193	236	2	<	<	X
ejpam-3193	236	3	y	y	PROPN
ejpam-3193	236	4	>	>	PUNCT
ejpam-3193	236	5	)	)	PUNCT
ejpam-3193	236	6	.	.	PUNCT
ejpam-3193	237	1	if	if	SCONJ
ejpam-3193	237	2	tγ	tγ	PROPN
ejpam-3193	237	3	(	(	PUNCT
ejpam-3193	237	4	<	<	X
ejpam-3193	237	5	x	x	X
ejpam-3193	237	6	>	>	X
ejpam-3193	238	1	+	+	PUNCT
ejpam-3193	238	2	<	<	X
ejpam-3193	238	3	y	y	X
ejpam-3193	238	4	>	>	PUNCT
ejpam-3193	238	5	*	*	PUNCT
ejpam-3193	238	6	pγp	pγp	INTJ
ejpam-3193	238	7	,	,	PUNCT
ejpam-3193	238	8	then	then	ADV
ejpam-3193	238	9	t	t	PROPN
ejpam-3193	238	10	∈	∈	PROPN
ejpam-3193	238	11	p	p	NOUN
ejpam-3193	238	12	by	by	ADP
ejpam-3193	238	13	assumption	assumption	NOUN
ejpam-3193	238	14	.	.	PUNCT
ejpam-3193	239	1	(	(	PUNCT
ejpam-3193	239	2	ii	ii	NOUN
ejpam-3193	239	3	)	)	PUNCT
ejpam-3193	239	4	implies	imply	VERB
ejpam-3193	239	5	(	(	PUNCT
ejpam-3193	239	6	iii	iii	X
ejpam-3193	239	7	)	)	PUNCT
ejpam-3193	239	8	holds	hold	VERB
ejpam-3193	239	9	from	from	ADP
ejpam-3193	239	10	the	the	DET
ejpam-3193	239	11	truth	truth	NOUN
ejpam-3193	239	12	that	that	SCONJ
ejpam-3193	239	13	if	if	SCONJ
ejpam-3193	239	14	union	union	NOUN
ejpam-3193	239	15	of	of	ADP
ejpam-3193	239	16	two	two	NUM
ejpam-3193	239	17	ideal	ideal	NOUN
ejpam-3193	239	18	is	be	AUX
ejpam-3193	239	19	an	an	DET
ejpam-3193	239	20	ideal	ideal	NOUN
ejpam-3193	239	21	then	then	ADV
ejpam-3193	239	22	it	it	PRON
ejpam-3193	239	23	is	be	AUX
ejpam-3193	239	24	equal	equal	ADJ
ejpam-3193	239	25	to	to	ADP
ejpam-3193	239	26	one	one	NUM
ejpam-3193	239	27	of	of	ADP
ejpam-3193	239	28	them.(iii	them.(iii	NOUN
ejpam-3193	239	29	)	)	PUNCT
ejpam-3193	239	30	implies	imply	VERB
ejpam-3193	239	31	(	(	PUNCT
ejpam-3193	239	32	iv	iv	X
ejpam-3193	239	33	)	)	PUNCT
ejpam-3193	239	34	imagine	imagine	VERB
ejpam-3193	239	35	a	a	PRON
ejpam-3193	239	36	and	and	CCONJ
ejpam-3193	239	37	b	b	NOUN
ejpam-3193	239	38	be	be	AUX
ejpam-3193	239	39	ideals	ideal	NOUN
ejpam-3193	239	40	of	of	ADP
ejpam-3193	239	41	n	n	PRON
ejpam-3193	239	42	such	such	ADJ
ejpam-3193	239	43	that	that	SCONJ
ejpam-3193	239	44	aγb	aγb	VERB
ejpam-3193	239	45	⊆	⊆	NUM
ejpam-3193	239	46	p	p	NOUN
ejpam-3193	239	47	.	.	PUNCT
ejpam-3193	240	1	assume	assume	VERB
ejpam-3193	240	2	a	a	DET
ejpam-3193	240	3	*	*	PUNCT
ejpam-3193	240	4	p	p	NOUN
ejpam-3193	240	5	and	and	CCONJ
ejpam-3193	240	6	b	b	NOUN
ejpam-3193	241	1	*	*	PUNCT
ejpam-3193	241	2	p	p	NOUN
ejpam-3193	241	3	implies	imply	VERB
ejpam-3193	241	4	a	a	DET
ejpam-3193	241	5	∈	∈	PROPN
ejpam-3193	241	6	a	a	DET
ejpam-3193	241	7	and	and	CCONJ
ejpam-3193	241	8	b	b	PROPN
ejpam-3193	241	9	∈	∈	NOUN
ejpam-3193	241	10	b	b	NOUN
ejpam-3193	241	11	exist	exist	VERB
ejpam-3193	241	12	with	with	ADP
ejpam-3193	241	13	a	a	DET
ejpam-3193	241	14	,	,	PUNCT
ejpam-3193	241	15	b	b	PROPN
ejpam-3193	241	16	6∈	6∈	PROPN
ejpam-3193	241	17	p	p	X
ejpam-3193	241	18	.	.	PUNCT
ejpam-3193	242	1	now	now	ADV
ejpam-3193	242	2	we	we	PRON
ejpam-3193	242	3	say	say	VERB
ejpam-3193	242	4	that	that	SCONJ
ejpam-3193	242	5	aγb	aγb	VERB
ejpam-3193	242	6	*	*	PUNCT
ejpam-3193	242	7	pγp	pγp	ADJ
ejpam-3193	242	8	and	and	CCONJ
ejpam-3193	242	9	consider	consider	VERB
ejpam-3193	242	10	b1	b1	PROPN
ejpam-3193	242	11	∈	∈	PROPN
ejpam-3193	242	12	b.	b.	PROPN
ejpam-3193	242	13	in	in	ADP
ejpam-3193	242	14	that	that	DET
ejpam-3193	242	15	case	case	NOUN
ejpam-3193	242	16	aγ	aγ	ADV
ejpam-3193	242	17	(	(	PUNCT
ejpam-3193	242	18	<	<	X
ejpam-3193	242	19	b	b	X
ejpam-3193	242	20	>	>	X
ejpam-3193	243	1	+	+	PUNCT
ejpam-3193	243	2	<	<	X
ejpam-3193	243	3	b1	b1	NOUN
ejpam-3193	243	4	>	>	PUNCT
ejpam-3193	243	5	)	)	PUNCT
ejpam-3193	244	1	*	*	PUNCT
ejpam-3193	245	1	p	p	X
ejpam-3193	245	2	which	which	PRON
ejpam-3193	245	3	implies	imply	VERB
ejpam-3193	245	4	a	a	DET
ejpam-3193	245	5	⊆	⊆	NUM
ejpam-3193	245	6	(	(	PUNCT
ejpam-3193	245	7	p	p	X
ejpam-3193	245	8	:	:	PUNCT
ejpam-3193	245	9	γ	γ	X
ejpam-3193	245	10	:	:	PUNCT
ejpam-3193	245	11	<	<	X
ejpam-3193	245	12	b	b	X
ejpam-3193	245	13	>	>	X
ejpam-3193	245	14	+	+	PUNCT
ejpam-3193	245	15	<	<	X
ejpam-3193	245	16	b1	b1	NOUN
ejpam-3193	245	17	>	>	PUNCT
ejpam-3193	245	18	)	)	PUNCT
ejpam-3193	245	19	.	.	PUNCT
ejpam-3193	246	1	then	then	ADV
ejpam-3193	246	2	by	by	ADP
ejpam-3193	246	3	supposition	supposition	NOUN
ejpam-3193	246	4	a	a	DET
ejpam-3193	246	5	⊆	⊆	NUM
ejpam-3193	246	6	(	(	PUNCT
ejpam-3193	246	7	<	<	X
ejpam-3193	246	8	b	b	X
ejpam-3193	246	9	>	>	X
ejpam-3193	246	10	+	+	PUNCT
ejpam-3193	246	11	<	<	X
ejpam-3193	246	12	b1	b1	NOUN
ejpam-3193	246	13	>	>	SYM
ejpam-3193	246	14	)	)	PUNCT
ejpam-3193	246	15	γpγp	γpγp	PROPN
ejpam-3193	246	16	implies	imply	VERB
ejpam-3193	246	17	aγb1	aγb1	PROPN
ejpam-3193	246	18	⊆	⊆	NUM
ejpam-3193	246	19	pγp	pγp	ADV
ejpam-3193	246	20	.	.	PUNCT
ejpam-3193	247	1	consequently	consequently	ADV
ejpam-3193	247	2	ab	ab	PROPN
ejpam-3193	247	3	⊆	⊆	NUM
ejpam-3193	247	4	pγp	pγp	PROPN
ejpam-3193	248	1	and	and	CCONJ
ejpam-3193	248	2	therefore	therefore	ADV
ejpam-3193	248	3	p	p	PRON
ejpam-3193	248	4	is	be	AUX
ejpam-3193	248	5	an	an	DET
ejpam-3193	248	6	almost	almost	ADV
ejpam-3193	248	7	prime	prime	ADJ
ejpam-3193	248	8	ideal	ideal	NOUN
ejpam-3193	248	9	of	of	ADP
ejpam-3193	248	10	n	n	PROPN
ejpam-3193	248	11	.	.	PUNCT
ejpam-3193	249	1	a.	a.	PROPN
ejpam-3193	249	2	taouti	taouti	PROPN
ejpam-3193	249	3	et	et	PROPN
ejpam-3193	249	4	al	al	PROPN
ejpam-3193	249	5	.	.	PUNCT
ejpam-3193	249	6	/	/	SYM
ejpam-3193	249	7	eur	eur	PROPN
ejpam-3193	249	8	.	.	PUNCT
ejpam-3193	250	1	j.	j.	PROPN
ejpam-3193	250	2	pure	pure	PROPN
ejpam-3193	250	3	appl	appl	PROPN
ejpam-3193	250	4	.	.	PROPN
ejpam-3193	250	5	math	math	PROPN
ejpam-3193	250	6	,	,	PUNCT
ejpam-3193	250	7	11	11	NUM
ejpam-3193	250	8	(	(	PUNCT
ejpam-3193	250	9	2	2	NUM
ejpam-3193	250	10	)	)	PUNCT
ejpam-3193	250	11	(	(	PUNCT
ejpam-3193	250	12	2018	2018	NUM
ejpam-3193	250	13	)	)	PUNCT
ejpam-3193	250	14	,	,	PUNCT
ejpam-3193	250	15	449	449	NUM
ejpam-3193	250	16	-	-	SYM
ejpam-3193	250	17	456	456	NUM
ejpam-3193	250	18	454	454	NUM
ejpam-3193	250	19	(	(	PUNCT
ejpam-3193	250	20	iv	iv	NOUN
ejpam-3193	250	21	)	)	PUNCT
ejpam-3193	250	22	implies	imply	VERB
ejpam-3193	250	23	(	(	PUNCT
ejpam-3193	250	24	i	i	NOUN
ejpam-3193	250	25	)	)	PUNCT
ejpam-3193	250	26	is	be	AUX
ejpam-3193	250	27	obvious	obvious	ADJ
ejpam-3193	250	28	.	.	PUNCT
ejpam-3193	251	1	theorem	theorem	NOUN
ejpam-3193	251	2	3	3	X
ejpam-3193	251	3	.	.	PUNCT
ejpam-3193	251	4	suppose	suppose	VERB
ejpam-3193	251	5	n1	n1	PROPN
ejpam-3193	251	6	,	,	PUNCT
ejpam-3193	251	7	n2	n2	ADJ
ejpam-3193	251	8	be	be	AUX
ejpam-3193	251	9	any	any	DET
ejpam-3193	251	10	two	two	NUM
ejpam-3193	251	11	γ	γ	NOUN
ejpam-3193	251	12	-	-	PUNCT
ejpam-3193	251	13	near	near	ADP
ejpam-3193	251	14	-	-	PUNCT
ejpam-3193	251	15	rings	ring	NOUN
ejpam-3193	251	16	with	with	ADP
ejpam-3193	251	17	identity	identity	NOUN
ejpam-3193	251	18	and	and	CCONJ
ejpam-3193	251	19	let	let	VERB
ejpam-3193	251	20	p	p	PRON
ejpam-3193	251	21	be	be	AUX
ejpam-3193	251	22	a	a	DET
ejpam-3193	251	23	proper	proper	ADJ
ejpam-3193	251	24	ideal	ideal	NOUN
ejpam-3193	251	25	of	of	ADP
ejpam-3193	251	26	n1	n1	NOUN
ejpam-3193	251	27	.	.	PUNCT
ejpam-3193	252	1	then	then	ADV
ejpam-3193	252	2	p	p	NOUN
ejpam-3193	252	3	is	be	AUX
ejpam-3193	252	4	almost	almost	ADV
ejpam-3193	252	5	prime	prime	ADJ
ejpam-3193	252	6	if	if	SCONJ
ejpam-3193	253	1	and	and	CCONJ
ejpam-3193	253	2	only	only	ADV
ejpam-3193	253	3	if	if	SCONJ
ejpam-3193	253	4	(	(	PUNCT
ejpam-3193	253	5	p	p	NOUN
ejpam-3193	253	6	×	×	PROPN
ejpam-3193	253	7	n2	n2	NOUN
ejpam-3193	253	8	)	)	PUNCT
ejpam-3193	253	9	is	be	AUX
ejpam-3193	253	10	an	an	DET
ejpam-3193	253	11	almost	almost	ADV
ejpam-3193	253	12	prime	prime	ADJ
ejpam-3193	253	13	ideal	ideal	NOUN
ejpam-3193	253	14	of	of	ADP
ejpam-3193	253	15	n1	n1	ADJ
ejpam-3193	253	16	×n2	×n2	NOUN
ejpam-3193	253	17	.	.	PUNCT
ejpam-3193	254	1	proof	proof	NOUN
ejpam-3193	254	2	.	.	PUNCT
ejpam-3193	255	1	suppose	suppose	VERB
ejpam-3193	255	2	p	p	PRON
ejpam-3193	255	3	be	be	AUX
ejpam-3193	255	4	an	an	DET
ejpam-3193	255	5	almost	almost	ADV
ejpam-3193	255	6	prime	prime	ADJ
ejpam-3193	255	7	ideal	ideal	NOUN
ejpam-3193	255	8	of	of	ADP
ejpam-3193	255	9	n1	n1	NOUN
ejpam-3193	255	10	and	and	CCONJ
ejpam-3193	255	11	consider	consider	VERB
ejpam-3193	255	12	(	(	PUNCT
ejpam-3193	255	13	a1×b1	a1×b1	PROPN
ejpam-3193	255	14	)	)	PUNCT
ejpam-3193	255	15	and	and	CCONJ
ejpam-3193	255	16	(	(	PUNCT
ejpam-3193	255	17	a2×b2	a2×b2	PROPN
ejpam-3193	255	18	)	)	PUNCT
ejpam-3193	255	19	be	be	VERB
ejpam-3193	255	20	ideals	ideal	NOUN
ejpam-3193	255	21	of	of	ADP
ejpam-3193	255	22	n1×n2	n1×n2	NOUN
ejpam-3193	255	23	such	such	ADJ
ejpam-3193	255	24	that	that	SCONJ
ejpam-3193	255	25	(	(	PUNCT
ejpam-3193	255	26	a1×b1)γ(a2×b2	a1×b1)γ(a2×b2	NOUN
ejpam-3193	255	27	)	)	PUNCT
ejpam-3193	255	28	⊆	⊆	NUM
ejpam-3193	255	29	(	(	PUNCT
ejpam-3193	255	30	p	p	NOUN
ejpam-3193	255	31	×n2	×n2	NOUN
ejpam-3193	255	32	)	)	PUNCT
ejpam-3193	255	33	and	and	CCONJ
ejpam-3193	255	34	(	(	PUNCT
ejpam-3193	255	35	a1×b1)γ(a2×b2	a1×b1)γ(a2×b2	NOUN
ejpam-3193	255	36	)	)	PUNCT
ejpam-3193	255	37	*	*	PUNCT
ejpam-3193	256	1	(	(	PUNCT
ejpam-3193	256	2	p	p	NOUN
ejpam-3193	256	3	×n2)γ(p	×n2)γ(p	NOUN
ejpam-3193	256	4	×n2	×n2	PROPN
ejpam-3193	256	5	)	)	PUNCT
ejpam-3193	256	6	.	.	PUNCT
ejpam-3193	257	1	in	in	ADP
ejpam-3193	257	2	this	this	DET
ejpam-3193	257	3	case	case	NOUN
ejpam-3193	257	4	(	(	PUNCT
ejpam-3193	257	5	a1γa2	a1γa2	NOUN
ejpam-3193	257	6	×b1γb2	×b1γb2	NOUN
ejpam-3193	257	7	)	)	PUNCT
ejpam-3193	257	8	⊆	⊆	NUM
ejpam-3193	257	9	(	(	PUNCT
ejpam-3193	257	10	p	p	NOUN
ejpam-3193	257	11	×n2	×n2	NOUN
ejpam-3193	257	12	)	)	PUNCT
ejpam-3193	257	13	and	and	CCONJ
ejpam-3193	257	14	(	(	PUNCT
ejpam-3193	257	15	a1γa2	a1γa2	NOUN
ejpam-3193	257	16	×b1γb2	×b1γb2	NOUN
ejpam-3193	257	17	)	)	PUNCT
ejpam-3193	257	18	*	*	PUNCT
ejpam-3193	258	1	(	(	PUNCT
ejpam-3193	258	2	pγp	pγp	INTJ
ejpam-3193	258	3	×	×	NOUN
ejpam-3193	258	4	nγn),therefore	nγn),therefore	ADJ
ejpam-3193	258	5	a1γa2	a1γa2	NOUN
ejpam-3193	258	6	×	×	NOUN
ejpam-3193	258	7	p	p	NOUN
ejpam-3193	258	8	and	and	CCONJ
ejpam-3193	258	9	a1γa2	a1γa2	NOUN
ejpam-3193	258	10	*	*	PUNCT
ejpam-3193	258	11	pγp	pγp	ADJ
ejpam-3193	258	12	implies	imply	VERB
ejpam-3193	258	13	a1	a1	VERB
ejpam-3193	258	14	⊆	⊆	NUM
ejpam-3193	258	15	p	p	NOUN
ejpam-3193	258	16	or	or	CCONJ
ejpam-3193	258	17	a2	a2	PROPN
ejpam-3193	259	1	⊆	⊆	NUM
ejpam-3193	259	2	p	p	NOUN
ejpam-3193	259	3	.	.	PUNCT
ejpam-3193	260	1	conversely	conversely	ADV
ejpam-3193	260	2	,	,	PUNCT
ejpam-3193	260	3	assume	assume	VERB
ejpam-3193	260	4	that	that	SCONJ
ejpam-3193	260	5	(	(	PUNCT
ejpam-3193	260	6	p	p	NOUN
ejpam-3193	260	7	×n2	×n2	NOUN
ejpam-3193	260	8	)	)	PUNCT
ejpam-3193	260	9	is	be	AUX
ejpam-3193	260	10	an	an	DET
ejpam-3193	260	11	almost	almost	ADV
ejpam-3193	260	12	prime	prime	ADJ
ejpam-3193	260	13	ideal	ideal	NOUN
ejpam-3193	260	14	of	of	ADP
ejpam-3193	260	15	n1×n2	n1×n2	NOUN
ejpam-3193	260	16	and	and	CCONJ
ejpam-3193	260	17	consider	consider	VERB
ejpam-3193	260	18	i	i	PRON
ejpam-3193	260	19	and	and	CCONJ
ejpam-3193	260	20	j	j	PROPN
ejpam-3193	260	21	be	be	VERB
ejpam-3193	260	22	ideals	ideal	NOUN
ejpam-3193	260	23	of	of	ADP
ejpam-3193	260	24	n1	n1	NOUN
ejpam-3193	260	25	such	such	ADJ
ejpam-3193	260	26	that	that	SCONJ
ejpam-3193	260	27	iγj	iγj	VERB
ejpam-3193	260	28	⊆	⊆	NUM
ejpam-3193	260	29	p	p	NOUN
ejpam-3193	260	30	and	and	CCONJ
ejpam-3193	260	31	iγj	iγj	NOUN
ejpam-3193	260	32	*	*	PUNCT
ejpam-3193	260	33	pγp	pγp	INTJ
ejpam-3193	260	34	.	.	PUNCT
ejpam-3193	261	1	then	then	ADV
ejpam-3193	261	2	(	(	PUNCT
ejpam-3193	261	3	i×n2)γ(j×n2	i×n2)γ(j×n2	NOUN
ejpam-3193	261	4	)	)	PUNCT
ejpam-3193	261	5	⊆	⊆	NUM
ejpam-3193	261	6	(	(	PUNCT
ejpam-3193	261	7	p×n2	p×n2	PROPN
ejpam-3193	261	8	)	)	PUNCT
ejpam-3193	261	9	and	and	CCONJ
ejpam-3193	261	10	(	(	PUNCT
ejpam-3193	261	11	i×n2)γ(j×n2	i×n2)γ(j×n2	NOUN
ejpam-3193	261	12	)	)	PUNCT
ejpam-3193	261	13	*	*	PUNCT
ejpam-3193	262	1	(	(	PUNCT
ejpam-3193	262	2	p	p	NOUN
ejpam-3193	262	3	×n2)γ(p	×n2)γ(p	NOUN
ejpam-3193	262	4	×n2	×n2	PROPN
ejpam-3193	262	5	)	)	PUNCT
ejpam-3193	262	6	.	.	PUNCT
ejpam-3193	263	1	by	by	ADP
ejpam-3193	263	2	hypothesis	hypothesis	NOUN
ejpam-3193	263	3	,	,	PUNCT
ejpam-3193	263	4	we	we	PRON
ejpam-3193	263	5	have	have	VERB
ejpam-3193	263	6	(	(	PUNCT
ejpam-3193	263	7	i×n2	i×n2	NOUN
ejpam-3193	263	8	)	)	PUNCT
ejpam-3193	263	9	⊆	⊆	NUM
ejpam-3193	263	10	(	(	PUNCT
ejpam-3193	263	11	p	p	NOUN
ejpam-3193	263	12	×n2	×n2	NOUN
ejpam-3193	263	13	)	)	PUNCT
ejpam-3193	263	14	or	or	CCONJ
ejpam-3193	263	15	(	(	PUNCT
ejpam-3193	263	16	j	j	PROPN
ejpam-3193	263	17	×n2	×n2	PROPN
ejpam-3193	263	18	)	)	PUNCT
ejpam-3193	264	1	⊆	⊆	NUM
ejpam-3193	264	2	(	(	PUNCT
ejpam-3193	264	3	p	p	NOUN
ejpam-3193	264	4	×n2	×n2	NOUN
ejpam-3193	264	5	)	)	PUNCT
ejpam-3193	264	6	.	.	PUNCT
ejpam-3193	265	1	thus	thus	ADV
ejpam-3193	265	2	i	i	PRON
ejpam-3193	265	3	⊆	⊆	NUM
ejpam-3193	265	4	p	p	NOUN
ejpam-3193	265	5	or	or	CCONJ
ejpam-3193	265	6	j	j	PROPN
ejpam-3193	265	7	⊆	⊆	NUM
ejpam-3193	265	8	p	p	NOUN
ejpam-3193	265	9	.	.	PUNCT
ejpam-3193	266	1	lemma	lemma	PROPN
ejpam-3193	267	1	3	3	X
ejpam-3193	267	2	.	.	PUNCT
ejpam-3193	268	1	if	if	SCONJ
ejpam-3193	268	2	c	c	PROPN
ejpam-3193	268	3	6=	6=	ADP
ejpam-3193	268	4	0	0	NUM
ejpam-3193	268	5	is	be	AUX
ejpam-3193	268	6	a	a	DET
ejpam-3193	268	7	nonunit	nonunit	NOUN
ejpam-3193	268	8	element	element	NOUN
ejpam-3193	268	9	in	in	ADP
ejpam-3193	268	10	γ	γ	PROPN
ejpam-3193	268	11	-	-	PUNCT
ejpam-3193	268	12	near	near	ADJ
ejpam-3193	268	13	integral	integral	ADJ
ejpam-3193	268	14	domain	domain	NOUN
ejpam-3193	268	15	r	r	NOUN
ejpam-3193	268	16	then	then	ADV
ejpam-3193	268	17	ideal	ideal	ADJ
ejpam-3193	268	18	rγc	rγc	NOUN
ejpam-3193	268	19	is	be	AUX
ejpam-3193	268	20	prime	prime	ADJ
ejpam-3193	268	21	if	if	SCONJ
ejpam-3193	269	1	and	and	CCONJ
ejpam-3193	269	2	only	only	ADV
ejpam-3193	269	3	if	if	SCONJ
ejpam-3193	269	4	rγc	rγc	NOUN
ejpam-3193	269	5	is	be	AUX
ejpam-3193	269	6	an	an	DET
ejpam-3193	269	7	almost	almost	ADV
ejpam-3193	269	8	prime	prime	ADJ
ejpam-3193	269	9	.	.	PUNCT
ejpam-3193	270	1	proof	proof	NOUN
ejpam-3193	270	2	.	.	PUNCT
ejpam-3193	271	1	let	let	VERB
ejpam-3193	271	2	c	c	NOUN
ejpam-3193	271	3	6=	6=	ADP
ejpam-3193	271	4	0	0	NUM
ejpam-3193	271	5	is	be	AUX
ejpam-3193	271	6	a	a	DET
ejpam-3193	271	7	nonunit	nonunit	NOUN
ejpam-3193	271	8	element	element	NOUN
ejpam-3193	271	9	in	in	ADP
ejpam-3193	271	10	an	an	DET
ejpam-3193	271	11	γ	γ	NOUN
ejpam-3193	271	12	-	-	PUNCT
ejpam-3193	271	13	near	near	ADJ
ejpam-3193	271	14	integral	integral	ADJ
ejpam-3193	271	15	domain	domain	NOUN
ejpam-3193	271	16	r.	r.	NOUN
ejpam-3193	271	17	assume	assume	VERB
ejpam-3193	271	18	that	that	SCONJ
ejpam-3193	271	19	ideal	ideal	ADJ
ejpam-3193	271	20	rγc	rγc	NOUN
ejpam-3193	271	21	is	be	AUX
ejpam-3193	271	22	an	an	DET
ejpam-3193	271	23	almost	almost	ADV
ejpam-3193	271	24	prime	prime	ADJ
ejpam-3193	271	25	we	we	PRON
ejpam-3193	271	26	need	need	VERB
ejpam-3193	271	27	to	to	PART
ejpam-3193	271	28	prove	prove	VERB
ejpam-3193	271	29	that	that	DET
ejpam-3193	271	30	rγc	rγc	NOUN
ejpam-3193	271	31	is	be	AUX
ejpam-3193	271	32	prime	prime	ADJ
ejpam-3193	271	33	.	.	PUNCT
ejpam-3193	272	1	as	as	SCONJ
ejpam-3193	272	2	we	we	PRON
ejpam-3193	272	3	know	know	VERB
ejpam-3193	272	4	that	that	DET
ejpam-3193	272	5	ideal	ideal	ADJ
ejpam-3193	272	6	rγc	rγc	NOUN
ejpam-3193	272	7	is	be	AUX
ejpam-3193	272	8	an	an	DET
ejpam-3193	272	9	almost	almost	ADV
ejpam-3193	272	10	prime	prime	NOUN
ejpam-3193	272	11	for	for	ADP
ejpam-3193	272	12	some	some	PRON
ejpam-3193	272	13	a	a	PRON
ejpam-3193	272	14	,	,	PUNCT
ejpam-3193	272	15	b	b	X
ejpam-3193	272	16	∈	∈	NOUN
ejpam-3193	272	17	r	r	NOUN
ejpam-3193	272	18	and	and	CCONJ
ejpam-3193	272	19	aγb	aγb	NOUN
ejpam-3193	272	20	∈	∈	PROPN
ejpam-3193	272	21	rγc	rγc	NOUN
ejpam-3193	272	22	−	−	PROPN
ejpam-3193	273	1	rγcγrγc	rγcγrγc	NOUN
ejpam-3193	273	2	implies	imply	VERB
ejpam-3193	273	3	either	either	CCONJ
ejpam-3193	273	4	a	a	DET
ejpam-3193	273	5	∈	∈	ADJ
ejpam-3193	273	6	rγc	rγc	NOUN
ejpam-3193	273	7	or	or	CCONJ
ejpam-3193	273	8	b	b	NOUN
ejpam-3193	273	9	∈	∈	PROPN
ejpam-3193	273	10	rγc	rγc	NOUN
ejpam-3193	273	11	where	where	SCONJ
ejpam-3193	273	12	aγb	aγb	NOUN
ejpam-3193	273	13	6∈	6∈	PROPN
ejpam-3193	273	14	rγcγrγc	rγcγrγc	NOUN
ejpam-3193	273	15	implies	imply	VERB
ejpam-3193	273	16	aγb	aγb	NOUN
ejpam-3193	273	17	∈	∈	PROPN
ejpam-3193	273	18	rγc	rγc	NOUN
ejpam-3193	273	19	.	.	PUNCT
ejpam-3193	274	1	hence	hence	ADV
ejpam-3193	274	2	rγc	rγc	PROPN
ejpam-3193	274	3	is	be	AUX
ejpam-3193	274	4	a	a	DET
ejpam-3193	274	5	prime	prime	ADJ
ejpam-3193	274	6	ideal	ideal	NOUN
ejpam-3193	274	7	.	.	PUNCT
ejpam-3193	275	1	conversely	conversely	ADV
ejpam-3193	275	2	,	,	PUNCT
ejpam-3193	275	3	suppose	suppose	VERB
ejpam-3193	275	4	that	that	SCONJ
ejpam-3193	275	5	ideal	ideal	ADJ
ejpam-3193	275	6	rγc	rγc	NOUN
ejpam-3193	275	7	is	be	AUX
ejpam-3193	275	8	prime	prime	ADJ
ejpam-3193	275	9	and	and	CCONJ
ejpam-3193	275	10	we	we	PRON
ejpam-3193	275	11	use	use	VERB
ejpam-3193	275	12	a	a	DET
ejpam-3193	275	13	result	result	NOUN
ejpam-3193	275	14	that	that	SCONJ
ejpam-3193	275	15	every	every	DET
ejpam-3193	275	16	prime	prime	ADJ
ejpam-3193	275	17	ideal	ideal	NOUN
ejpam-3193	275	18	is	be	AUX
ejpam-3193	275	19	almost	almost	ADV
ejpam-3193	275	20	prime	prime	ADJ
ejpam-3193	275	21	then	then	ADV
ejpam-3193	275	22	rγc	rγc	NOUN
ejpam-3193	275	23	is	be	AUX
ejpam-3193	275	24	almost	almost	ADV
ejpam-3193	275	25	prime	prime	ADJ
ejpam-3193	275	26	ideal	ideal	NOUN
ejpam-3193	275	27	which	which	PRON
ejpam-3193	275	28	is	be	AUX
ejpam-3193	275	29	immediate	immediate	ADJ
ejpam-3193	275	30	from	from	ADP
ejpam-3193	275	31	lemma	lemma	PROPN
ejpam-3193	275	32	2	2	NUM
ejpam-3193	275	33	.	.	PUNCT
ejpam-3193	276	1	lemma	lemma	PROPN
ejpam-3193	276	2	4	4	X
ejpam-3193	276	3	.	.	PUNCT
ejpam-3193	276	4	suppose	suppose	VERB
ejpam-3193	276	5	i	i	PRON
ejpam-3193	276	6	be	be	VERB
ejpam-3193	276	7	an	an	DET
ejpam-3193	276	8	almost	almost	ADV
ejpam-3193	276	9	prime	prime	ADJ
ejpam-3193	276	10	ideal	ideal	NOUN
ejpam-3193	276	11	in	in	ADP
ejpam-3193	276	12	a	a	DET
ejpam-3193	276	13	γ	γ	X
ejpam-3193	276	14	-	-	PUNCT
ejpam-3193	276	15	near	near	ADJ
ejpam-3193	276	16	integral	integral	ADJ
ejpam-3193	276	17	domain	domain	NOUN
ejpam-3193	276	18	r.	r.	NOUN
ejpam-3193	276	19	then	then	ADV
ejpam-3193	276	20	the	the	DET
ejpam-3193	276	21	below	below	ADJ
ejpam-3193	276	22	statements	statement	NOUN
ejpam-3193	276	23	hold	hold	VERB
ejpam-3193	276	24	.	.	PUNCT
ejpam-3193	277	1	(	(	PUNCT
ejpam-3193	277	2	i	i	NOUN
ejpam-3193	277	3	)	)	PUNCT
ejpam-3193	277	4	if	if	SCONJ
ejpam-3193	277	5	element	element	NOUN
ejpam-3193	277	6	b	b	PROPN
ejpam-3193	277	7	is	be	AUX
ejpam-3193	277	8	a	a	DET
ejpam-3193	277	9	zero	zero	NUM
ejpam-3193	277	10	divisor	divisor	NOUN
ejpam-3193	277	11	in	in	ADP
ejpam-3193	277	12	r	r	PROPN
ejpam-3193	277	13	/	/	SYM
ejpam-3193	277	14	i	i	PROPN
ejpam-3193	277	15	,	,	PUNCT
ejpam-3193	277	16	in	in	ADP
ejpam-3193	277	17	that	that	DET
ejpam-3193	277	18	case	case	NOUN
ejpam-3193	277	19	bγi	bγi	NOUN
ejpam-3193	277	20	⊆	⊆	NUM
ejpam-3193	277	21	iγi	iγi	NOUN
ejpam-3193	277	22	.	.	PUNCT
ejpam-3193	278	1	(	(	PUNCT
ejpam-3193	278	2	ii	ii	NOUN
ejpam-3193	278	3	)	)	PUNCT
ejpam-3193	278	4	if	if	SCONJ
ejpam-3193	278	5	for	for	ADP
ejpam-3193	278	6	any	any	DET
ejpam-3193	278	7	ideal	ideal	ADJ
ejpam-3193	278	8	j	j	PROPN
ejpam-3193	278	9	of	of	ADP
ejpam-3193	278	10	r	r	NOUN
ejpam-3193	278	11	such	such	ADJ
ejpam-3193	278	12	that	that	SCONJ
ejpam-3193	278	13	i	i	PRON
ejpam-3193	278	14	⊆	⊆	NUM
ejpam-3193	278	15	j	j	NOUN
ejpam-3193	278	16	where	where	SCONJ
ejpam-3193	278	17	j	j	PROPN
ejpam-3193	278	18	consists	consist	VERB
ejpam-3193	278	19	of	of	ADP
ejpam-3193	278	20	zero	zero	NUM
ejpam-3193	278	21	divisors	divisor	NOUN
ejpam-3193	278	22	on	on	ADP
ejpam-3193	278	23	r	r	NOUN
ejpam-3193	278	24	/	/	SYM
ejpam-3193	278	25	i	i	PRON
ejpam-3193	278	26	then	then	ADV
ejpam-3193	278	27	jγi	jγi	VERB
ejpam-3193	278	28	=	=	PUNCT
ejpam-3193	278	29	iγi	iγi	PROPN
ejpam-3193	278	30	.	.	PUNCT
ejpam-3193	279	1	(	(	PUNCT
ejpam-3193	279	2	iii	iii	X
ejpam-3193	279	3	)	)	PUNCT
ejpam-3193	279	4	if	if	SCONJ
ejpam-3193	279	5	i	i	PRON
ejpam-3193	279	6	is	be	AUX
ejpam-3193	279	7	an	an	DET
ejpam-3193	279	8	invertible	invertible	ADJ
ejpam-3193	279	9	ideal	ideal	NOUN
ejpam-3193	279	10	then	then	ADV
ejpam-3193	279	11	i	i	PRON
ejpam-3193	279	12	is	be	AUX
ejpam-3193	279	13	prime	prime	ADJ
ejpam-3193	279	14	.	.	PUNCT
ejpam-3193	280	1	proof	proof	NOUN
ejpam-3193	280	2	.	.	PUNCT
ejpam-3193	281	1	(	(	PUNCT
ejpam-3193	281	2	i	i	NOUN
ejpam-3193	281	3	)	)	PUNCT
ejpam-3193	281	4	let	let	VERB
ejpam-3193	281	5	us	we	PRON
ejpam-3193	281	6	suppose	suppose	VERB
ejpam-3193	281	7	that	that	SCONJ
ejpam-3193	281	8	there	there	PRON
ejpam-3193	281	9	is	be	VERB
ejpam-3193	281	10	an	an	DET
ejpam-3193	281	11	element	element	NOUN
ejpam-3193	281	12	c	c	NOUN
ejpam-3193	281	13	∈	∈	PROPN
ejpam-3193	281	14	i	i	PRON
ejpam-3193	281	15	such	such	ADJ
ejpam-3193	281	16	that	that	DET
ejpam-3193	281	17	bγc	bγc	PROPN
ejpam-3193	281	18	∈	∈	PROPN
ejpam-3193	281	19	i.	i.	NOUN
ejpam-3193	282	1	if	if	SCONJ
ejpam-3193	282	2	b	b	PROPN
ejpam-3193	282	3	∈	∈	PROPN
ejpam-3193	282	4	i	i	PRON
ejpam-3193	282	5	then	then	ADV
ejpam-3193	282	6	obviously	obviously	ADV
ejpam-3193	282	7	bγi	bγi	VERB
ejpam-3193	282	8	⊆	⊆	NUM
ejpam-3193	282	9	iγi	iγi	NOUN
ejpam-3193	282	10	,	,	PUNCT
ejpam-3193	282	11	so	so	ADV
ejpam-3193	282	12	let	let	VERB
ejpam-3193	282	13	b	b	X
ejpam-3193	282	14	∈	∈	PROPN
ejpam-3193	282	15	i.	i.	NOUN
ejpam-3193	282	16	since	since	SCONJ
ejpam-3193	282	17	we	we	PRON
ejpam-3193	282	18	have	have	VERB
ejpam-3193	282	19	b	b	NUM
ejpam-3193	282	20	6∈	6∈	NOUN
ejpam-3193	283	1	i	i	PRON
ejpam-3193	283	2	,	,	PUNCT
ejpam-3193	283	3	c	c	PROPN
ejpam-3193	283	4	6∈	6∈	PROPN
ejpam-3193	283	5	i	i	PRON
ejpam-3193	283	6	and	and	CCONJ
ejpam-3193	283	7	bγc	bγc	PROPN
ejpam-3193	283	8	∈	∈	PROPN
ejpam-3193	283	9	i.	i.	NOUN
ejpam-3193	284	1	furthermore	furthermore	ADV
ejpam-3193	284	2	i	i	PRON
ejpam-3193	284	3	is	be	AUX
ejpam-3193	284	4	an	an	DET
ejpam-3193	284	5	almost	almost	ADV
ejpam-3193	284	6	prime	prime	ADJ
ejpam-3193	284	7	and	and	CCONJ
ejpam-3193	284	8	bγc	bγc	NOUN
ejpam-3193	284	9	∈	∈	PROPN
ejpam-3193	284	10	iγi	iγi	NOUN
ejpam-3193	284	11	.	.	PUNCT
ejpam-3193	285	1	also	also	ADV
ejpam-3193	285	2	,	,	PUNCT
ejpam-3193	285	3	for	for	ADP
ejpam-3193	285	4	any	any	DET
ejpam-3193	285	5	x	x	SYM
ejpam-3193	285	6	∈	∈	PROPN
ejpam-3193	285	7	i	i	PRON
ejpam-3193	285	8	,	,	PUNCT
ejpam-3193	285	9	x	x	PROPN
ejpam-3193	286	1	+	+	CCONJ
ejpam-3193	286	2	c	c	VERB
ejpam-3193	286	3	6∈	6∈	NOUN
ejpam-3193	287	1	i	i	PRON
ejpam-3193	287	2	and	and	CCONJ
ejpam-3193	287	3	bγ(x	bγ(x	PUNCT
ejpam-3193	287	4	+	+	CCONJ
ejpam-3193	287	5	c	c	X
ejpam-3193	287	6	)	)	PUNCT
ejpam-3193	287	7	∈	∈	PROPN
ejpam-3193	287	8	i.	i.	NOUN
ejpam-3193	287	9	thus	thus	ADV
ejpam-3193	287	10	,	,	PUNCT
ejpam-3193	287	11	as	as	SCONJ
ejpam-3193	287	12	i	i	PRON
ejpam-3193	287	13	is	be	AUX
ejpam-3193	287	14	almost	almost	ADV
ejpam-3193	287	15	prime	prime	ADJ
ejpam-3193	287	16	,	,	PUNCT
ejpam-3193	287	17	bγ(x+	bγ(x+	ADV
ejpam-3193	287	18	c	c	NOUN
ejpam-3193	287	19	)	)	PUNCT
ejpam-3193	287	20	∈	∈	PROPN
ejpam-3193	287	21	iγi	iγi	NOUN
ejpam-3193	287	22	.	.	PUNCT
ejpam-3193	288	1	as	as	ADP
ejpam-3193	288	2	a	a	DET
ejpam-3193	288	3	result	result	NOUN
ejpam-3193	288	4	bγc	bγc	PROPN
ejpam-3193	288	5	∈	∈	PROPN
ejpam-3193	288	6	iγi	iγi	NOUN
ejpam-3193	288	7	,	,	PUNCT
ejpam-3193	288	8	bγx	bγx	PROPN
ejpam-3193	288	9	∈	∈	PROPN
ejpam-3193	288	10	iγi	iγi	NOUN
ejpam-3193	288	11	.	.	PUNCT
ejpam-3193	289	1	therefore	therefore	ADV
ejpam-3193	289	2	bγi	bγi	VERB
ejpam-3193	289	3	⊆	⊆	NUM
ejpam-3193	289	4	iγi	iγi	NOUN
ejpam-3193	289	5	.	.	PUNCT
ejpam-3193	290	1	(	(	PUNCT
ejpam-3193	290	2	ii	ii	X
ejpam-3193	290	3	)	)	PUNCT
ejpam-3193	290	4	this	this	PRON
ejpam-3193	290	5	is	be	AUX
ejpam-3193	290	6	obvious	obvious	ADJ
ejpam-3193	290	7	from	from	ADP
ejpam-3193	290	8	(	(	PUNCT
ejpam-3193	290	9	i	i	NOUN
ejpam-3193	290	10	)	)	PUNCT
ejpam-3193	290	11	.	.	PUNCT
ejpam-3193	291	1	(	(	PUNCT
ejpam-3193	291	2	iii	iii	X
ejpam-3193	291	3	)	)	PUNCT
ejpam-3193	291	4	let	let	VERB
ejpam-3193	291	5	xγy	xγy	PROPN
ejpam-3193	291	6	∈	∈	PROPN
ejpam-3193	292	1	i	i	PRON
ejpam-3193	292	2	and	and	CCONJ
ejpam-3193	292	3	x	x	PROPN
ejpam-3193	292	4	∈	∈	PROPN
ejpam-3193	292	5	i.	i.	NOUN
ejpam-3193	292	6	then	then	ADV
ejpam-3193	292	7	from	from	ADP
ejpam-3193	292	8	(	(	PUNCT
ejpam-3193	292	9	i	i	NOUN
ejpam-3193	292	10	)	)	PUNCT
ejpam-3193	292	11	yγi	yγi	NOUN
ejpam-3193	292	12	⊆	⊆	NUM
ejpam-3193	292	13	iγi	iγi	NOUN
ejpam-3193	292	14	.	.	PUNCT
ejpam-3193	293	1	since	since	SCONJ
ejpam-3193	293	2	i	i	PRON
ejpam-3193	293	3	is	be	AUX
ejpam-3193	293	4	invertible	invertible	ADJ
ejpam-3193	293	5	it	it	PRON
ejpam-3193	293	6	is	be	AUX
ejpam-3193	293	7	immediate	immediate	ADJ
ejpam-3193	293	8	that	that	SCONJ
ejpam-3193	293	9	y	y	PROPN
ejpam-3193	293	10	∈	∈	PROPN
ejpam-3193	293	11	i.	i.	NOUN
ejpam-3193	293	12	thus	thus	ADV
ejpam-3193	293	13	i	i	PRON
ejpam-3193	293	14	is	be	AUX
ejpam-3193	293	15	a	a	DET
ejpam-3193	293	16	prime	prime	ADJ
ejpam-3193	293	17	ideal	ideal	NOUN
ejpam-3193	293	18	.	.	PUNCT
ejpam-3193	294	1	lemma	lemma	PROPN
ejpam-3193	294	2	5	5	X
ejpam-3193	294	3	.	.	PUNCT
ejpam-3193	295	1	let	let	VERB
ejpam-3193	295	2	s−1i	s−1i	NOUN
ejpam-3193	295	3	is	be	AUX
ejpam-3193	295	4	an	an	DET
ejpam-3193	295	5	almost	almost	ADV
ejpam-3193	295	6	prime	prime	ADJ
ejpam-3193	295	7	in	in	ADP
ejpam-3193	295	8	the	the	DET
ejpam-3193	295	9	ring	ring	NOUN
ejpam-3193	295	10	s−1r	s−1r	PROPN
ejpam-3193	295	11	,	,	PUNCT
ejpam-3193	295	12	where	where	SCONJ
ejpam-3193	295	13	r	r	NOUN
ejpam-3193	295	14	be	be	AUX
ejpam-3193	295	15	a	a	DET
ejpam-3193	295	16	γ	γ	NOUN
ejpam-3193	295	17	-	-	PUNCT
ejpam-3193	295	18	near	near	ADJ
ejpam-3193	295	19	integral	integral	ADJ
ejpam-3193	295	20	domain	domain	NOUN
ejpam-3193	295	21	.	.	PUNCT
ejpam-3193	296	1	then	then	ADV
ejpam-3193	296	2	i	i	PRON
ejpam-3193	296	3	be	be	VERB
ejpam-3193	296	4	an	an	DET
ejpam-3193	296	5	almost	almost	ADV
ejpam-3193	296	6	prime	prime	ADJ
ejpam-3193	296	7	ideal	ideal	NOUN
ejpam-3193	296	8	in	in	ADP
ejpam-3193	296	9	r	r	NOUN
ejpam-3193	296	10	and	and	CCONJ
ejpam-3193	296	11	s	s	VERB
ejpam-3193	296	12	be	be	AUX
ejpam-3193	296	13	a	a	DET
ejpam-3193	296	14	multiplicatively	multiplicatively	ADV
ejpam-3193	296	15	closed	close	VERB
ejpam-3193	296	16	subset	subset	NOUN
ejpam-3193	296	17	of	of	ADP
ejpam-3193	296	18	r	r	PROPN
ejpam-3193	296	19	disjoint	disjoint	NOUN
ejpam-3193	296	20	from	from	ADP
ejpam-3193	296	21	i.	i.	NOUN
ejpam-3193	296	22	proof	proof	PROPN
ejpam-3193	296	23	.	.	PUNCT
ejpam-3193	297	1	suppose	suppose	VERB
ejpam-3193	297	2	for	for	ADP
ejpam-3193	297	3	x	x	PRON
ejpam-3193	297	4	,	,	PUNCT
ejpam-3193	297	5	y	y	PROPN
ejpam-3193	297	6	∈	∈	PROPN
ejpam-3193	297	7	r	r	NOUN
ejpam-3193	297	8	and	and	CCONJ
ejpam-3193	297	9	s	s	PROPN
ejpam-3193	297	10	,	,	PUNCT
ejpam-3193	297	11	t	t	PROPN
ejpam-3193	297	12	∈	∈	PROPN
ejpam-3193	297	13	s	s	PROPN
ejpam-3193	297	14	,	,	PUNCT
ejpam-3193	297	15	xγy	xγy	PROPN
ejpam-3193	297	16	/	/	SYM
ejpam-3193	297	17	sγt	sγt	PROPN
ejpam-3193	297	18	∈	∈	PROPN
ejpam-3193	297	19	s−1(i	s−1(i	PROPN
ejpam-3193	297	20	−	−	PROPN
ejpam-3193	297	21	iγi	iγi	NOUN
ejpam-3193	297	22	)	)	PUNCT
ejpam-3193	297	23	.	.	PUNCT
ejpam-3193	298	1	then	then	ADV
ejpam-3193	298	2	there	there	PRON
ejpam-3193	298	3	exists	exist	VERB
ejpam-3193	298	4	u	u	NOUN
ejpam-3193	298	5	,	,	PUNCT
ejpam-3193	298	6	w	w	PROPN
ejpam-3193	298	7	∈	∈	PROPN
ejpam-3193	298	8	s	s	VERB
ejpam-3193	298	9	such	such	ADJ
ejpam-3193	298	10	that	that	SCONJ
ejpam-3193	298	11	uγxγy	uγxγy	ADJ
ejpam-3193	298	12	∈	∈	PROPN
ejpam-3193	299	1	i	i	PRON
ejpam-3193	299	2	and	and	CCONJ
ejpam-3193	299	3	wγxγy	wγxγy	PROPN
ejpam-3193	299	4	6∈	6∈	PROPN
ejpam-3193	299	5	iγi	iγi	VERB
ejpam-3193	299	6	.	.	PUNCT
ejpam-3193	300	1	therefore	therefore	ADV
ejpam-3193	300	2	,	,	PUNCT
ejpam-3193	300	3	uγxγy	uγxγy	ADJ
ejpam-3193	300	4	∈	∈	PROPN
ejpam-3193	301	1	i	i	PRON
ejpam-3193	301	2	−	−	VERB
ejpam-3193	301	3	iγi	iγi	VERB
ejpam-3193	301	4	.	.	PUNCT
ejpam-3193	302	1	since	since	SCONJ
ejpam-3193	302	2	i	i	PRON
ejpam-3193	302	3	is	be	AUX
ejpam-3193	302	4	almost	almost	ADV
ejpam-3193	302	5	prime	prime	ADJ
ejpam-3193	303	1	so	so	ADV
ejpam-3193	303	2	uγx	uγx	PROPN
ejpam-3193	303	3	∈	∈	PROPN
ejpam-3193	304	1	i	i	PRON
ejpam-3193	304	2	or	or	CCONJ
ejpam-3193	304	3	y	y	PROPN
ejpam-3193	304	4	∈	∈	PROPN
ejpam-3193	304	5	i.	i.	NOUN
ejpam-3193	304	6	therefore	therefore	ADV
ejpam-3193	304	7	,	,	PUNCT
ejpam-3193	304	8	either	either	CCONJ
ejpam-3193	304	9	x	x	X
ejpam-3193	304	10	/	/	SYM
ejpam-3193	304	11	s	s	PART
ejpam-3193	304	12	∈	∈	NOUN
ejpam-3193	304	13	s−1i	s−1i	NOUN
ejpam-3193	304	14	or	or	CCONJ
ejpam-3193	304	15	y	y	PROPN
ejpam-3193	304	16	/	/	SYM
ejpam-3193	304	17	t	t	NOUN
ejpam-3193	304	18	∈	∈	NOUN
ejpam-3193	304	19	s−1i	s−1i	NOUN
ejpam-3193	304	20	implies	imply	VERB
ejpam-3193	304	21	s−1i	s−1i	NOUN
ejpam-3193	304	22	is	be	AUX
ejpam-3193	304	23	an	an	DET
ejpam-3193	304	24	almost	almost	ADV
ejpam-3193	304	25	prime	prime	ADJ
ejpam-3193	304	26	ideal	ideal	NOUN
ejpam-3193	304	27	.	.	PUNCT
ejpam-3193	305	1	references	reference	NOUN
ejpam-3193	305	2	455	455	NUM
ejpam-3193	305	3	references	reference	NOUN
ejpam-3193	305	4	[	[	X
ejpam-3193	305	5	1	1	NUM
ejpam-3193	305	6	]	]	PUNCT
ejpam-3193	305	7	a.	a.	NOUN
ejpam-3193	305	8	g.	g.	PROPN
ejpam-3193	305	9	agargn	agargn	PROPN
ejpam-3193	305	10	,	,	PUNCT
ejpam-3193	305	11	d.	d.	PROPN
ejpam-3193	305	12	d.	d.	PROPN
ejpam-3193	305	13	anderson	anderson	PROPN
ejpam-3193	305	14	,	,	PUNCT
ejpam-3193	305	15	and	and	CCONJ
ejpam-3193	305	16	s.	s.	PROPN
ejpam-3193	305	17	valdes	valdes	PROPN
ejpam-3193	305	18	-	-	PUNCT
ejpam-3193	305	19	leon	leon	PROPN
ejpam-3193	305	20	,	,	PUNCT
ejpam-3193	305	21	unique	unique	ADJ
ejpam-3193	305	22	factorization	factorization	NOUN
ejpam-3193	305	23	rings	ring	NOUN
ejpam-3193	305	24	with	with	ADP
ejpam-3193	305	25	zero	zero	NUM
ejpam-3193	305	26	divisors	divisor	NOUN
ejpam-3193	305	27	,	,	PUNCT
ejpam-3193	305	28	communication	communication	NOUN
ejpam-3193	305	29	in	in	ADP
ejpam-3193	305	30	algebra	algebra	NOUN
ejpam-3193	305	31	,	,	PUNCT
ejpam-3193	305	32	27	27	NUM
ejpam-3193	305	33	(	(	PUNCT
ejpam-3193	305	34	4	4	NUM
ejpam-3193	305	35	)	)	PUNCT
ejpam-3193	305	36	,	,	PUNCT
ejpam-3193	305	37	1967	1967	NUM
ejpam-3193	305	38	-	-	SYM
ejpam-3193	305	39	1974	1974	NUM
ejpam-3193	305	40	.	.	PUNCT
ejpam-3193	306	1	[	[	X
ejpam-3193	306	2	2	2	X
ejpam-3193	306	3	]	]	X
ejpam-3193	306	4	d.	d.	PROPN
ejpam-3193	306	5	d.	d.	PROPN
ejpam-3193	306	6	anderson	anderson	PROPN
ejpam-3193	306	7	,	,	PUNCT
ejpam-3193	306	8	and	and	CCONJ
ejpam-3193	306	9	e.	e.	PROPN
ejpam-3193	306	10	smith	smith	PROPN
ejpam-3193	306	11	,	,	PUNCT
ejpam-3193	306	12	weakly	weakly	ADJ
ejpam-3193	306	13	prime	prime	ADJ
ejpam-3193	306	14	ideals	ideal	NOUN
ejpam-3193	306	15	,	,	PUNCT
ejpam-3193	306	16	houston	houston	PROPN
ejpam-3193	306	17	journal	journal	PROPN
ejpam-3193	306	18	of	of	ADP
ejpam-3193	306	19	mathematics	mathematics	PROPN
ejpam-3193	306	20	,	,	PUNCT
ejpam-3193	306	21	29(4	29(4	NOUN
ejpam-3193	306	22	)	)	PUNCT
ejpam-3193	306	23	,	,	PUNCT
ejpam-3193	306	24	2003	2003	NUM
ejpam-3193	306	25	,	,	PUNCT
ejpam-3193	306	26	831	831	NUM
ejpam-3193	306	27	-	-	SYM
ejpam-3193	306	28	840	840	NUM
ejpam-3193	306	29	.	.	PUNCT
ejpam-3193	307	1	[	[	X
ejpam-3193	307	2	3	3	X
ejpam-3193	307	3	]	]	PUNCT
ejpam-3193	307	4	s.	s.	PROPN
ejpam-3193	307	5	m.	m.	PROPN
ejpam-3193	307	6	bhatwadekar	bhatwadekar	PROPN
ejpam-3193	307	7	and	and	CCONJ
ejpam-3193	307	8	p.	p.	PROPN
ejpam-3193	307	9	k.	k.	PROPN
ejpam-3193	308	1	sharma	sharma	PROPN
ejpam-3193	308	2	,	,	PUNCT
ejpam-3193	308	3	unique	unique	ADJ
ejpam-3193	308	4	factorization	factorization	NOUN
ejpam-3193	308	5	and	and	CCONJ
ejpam-3193	308	6	birth	birth	NOUN
ejpam-3193	308	7	of	of	ADP
ejpam-3193	308	8	almost	almost	ADV
ejpam-3193	308	9	primes	prime	NOUN
ejpam-3193	308	10	,	,	PUNCT
ejpam-3193	308	11	communication	communication	NOUN
ejpam-3193	308	12	in	in	ADP
ejpam-3193	308	13	algebra	algebra	NOUN
ejpam-3193	308	14	,	,	PUNCT
ejpam-3193	308	15	33	33	NUM
ejpam-3193	308	16	(	(	PUNCT
ejpam-3193	308	17	1	1	NUM
ejpam-3193	308	18	)	)	PUNCT
ejpam-3193	308	19	,	,	PUNCT
ejpam-3193	308	20	43	43	NUM
ejpam-3193	308	21	-	-	SYM
ejpam-3193	308	22	49	49	NUM
ejpam-3193	308	23	,	,	PUNCT
ejpam-3193	308	24	2005	2005	NUM
ejpam-3193	308	25	.	.	PUNCT
ejpam-3193	309	1	[	[	X
ejpam-3193	309	2	4	4	X
ejpam-3193	309	3	]	]	X
ejpam-3193	309	4	s.	s.	PROPN
ejpam-3193	309	5	bhavanari	bhavanari	PROPN
ejpam-3193	309	6	,	,	PUNCT
ejpam-3193	309	7	a	a	DET
ejpam-3193	309	8	note	note	NOUN
ejpam-3193	309	9	on	on	ADP
ejpam-3193	309	10	-near	-near	NOUN
ejpam-3193	309	11	-	-	PUNCT
ejpam-3193	309	12	rings	ring	NOUN
ejpam-3193	309	13	,	,	PUNCT
ejpam-3193	309	14	indian	indian	PROPN
ejpam-3193	309	15	j.	j.	PROPN
ejpam-3193	309	16	math	math	PROPN
ejpam-3193	309	17	.	.	PROPN
ejpam-3193	309	18	,	,	PUNCT
ejpam-3193	309	19	41	41	NUM
ejpam-3193	309	20	,	,	PUNCT
ejpam-3193	309	21	427	427	NUM
ejpam-3193	309	22	–	–	SYM
ejpam-3193	309	23	433	433	NUM
ejpam-3193	309	24	,	,	PUNCT
ejpam-3193	309	25	1999	1999	NUM
ejpam-3193	309	26	.	.	PUNCT
ejpam-3193	310	1	[	[	X
ejpam-3193	310	2	5	5	X
ejpam-3193	310	3	]	]	PUNCT
ejpam-3193	310	4	s.	s.	PROPN
ejpam-3193	310	5	bhavanari	bhavanari	PROPN
ejpam-3193	310	6	,	,	PUNCT
ejpam-3193	310	7	contributions	contribution	NOUN
ejpam-3193	310	8	to	to	ADP
ejpam-3193	310	9	near	near	ADJ
ejpam-3193	310	10	-	-	PUNCT
ejpam-3193	310	11	ring	ring	NOUN
ejpam-3193	310	12	theory	theory	NOUN
ejpam-3193	310	13	,	,	PUNCT
ejpam-3193	310	14	vdm	vdm	PROPN
ejpam-3193	310	15	verlag	verlag	PROPN
ejpam-3193	310	16	dr	dr	PROPN
ejpam-3193	310	17	mullar	mullar	PROPN
ejpam-3193	310	18	,	,	PUNCT
ejpam-3193	310	19	germany	germany	PROPN
ejpam-3193	310	20	,	,	PUNCT
ejpam-3193	310	21	2010	2010	NUM
ejpam-3193	310	22	(	(	PUNCT
ejpam-3193	310	23	isbn	isbn	ADJ
ejpam-3193	310	24	:	:	PUNCT
ejpam-3193	310	25	978	978	NUM
ejpam-3193	310	26	-	-	SYM
ejpam-3193	310	27	3	3	NUM
ejpam-3193	310	28	-	-	PUNCT
ejpam-3193	310	29	639	639	NUM
ejpam-3193	310	30	-	-	PUNCT
ejpam-3193	310	31	22417	22417	NUM
ejpam-3193	310	32	-	-	SYM
ejpam-3193	310	33	7	7	NUM
ejpam-3193	310	34	)	)	PUNCT
ejpam-3193	310	35	.	.	PUNCT
ejpam-3193	311	1	[	[	X
ejpam-3193	311	2	6	6	NUM
ejpam-3193	311	3	]	]	PUNCT
ejpam-3193	311	4	p.	p.	NOUN
ejpam-3193	311	5	dheena	dheena	PROPN
ejpam-3193	311	6	and	and	CCONJ
ejpam-3193	311	7	b.	b.	PROPN
ejpam-3193	311	8	elavarasan	elavarasan	PROPN
ejpam-3193	311	9	,	,	PUNCT
ejpam-3193	311	10	weakly	weakly	ADJ
ejpam-3193	311	11	prime	prime	ADJ
ejpam-3193	311	12	ideals	ideal	NOUN
ejpam-3193	311	13	in	in	ADP
ejpam-3193	311	14	near	near	ADJ
ejpam-3193	311	15	-	-	PUNCT
ejpam-3193	311	16	rings	ring	NOUN
ejpam-3193	311	17	,	,	PUNCT
ejpam-3193	311	18	tamsui	tamsui	PROPN
ejpam-3193	311	19	oxford	oxford	PROPN
ejpam-3193	311	20	journal	journal	NOUN
ejpam-3193	311	21	of	of	ADP
ejpam-3193	311	22	information	information	NOUN
ejpam-3193	311	23	and	and	CCONJ
ejpam-3193	311	24	mathematical	mathematical	ADJ
ejpam-3193	311	25	sciences	science	NOUN
ejpam-3193	311	26	,	,	PUNCT
ejpam-3193	311	27	29	29	NUM
ejpam-3193	311	28	(	(	PUNCT
ejpam-3193	311	29	1	1	NUM
ejpam-3193	311	30	)	)	PUNCT
ejpam-3193	311	31	,	,	PUNCT
ejpam-3193	311	32	55	55	NUM
ejpam-3193	311	33	-	-	SYM
ejpam-3193	311	34	59	59	NUM
ejpam-3193	311	35	,	,	PUNCT
ejpam-3193	311	36	2013	2013	NUM
ejpam-3193	311	37	.	.	PUNCT
ejpam-3193	312	1	[	[	X
ejpam-3193	312	2	7	7	X
ejpam-3193	312	3	]	]	X
ejpam-3193	312	4	e.	e.	PROPN
ejpam-3193	312	5	domi	domi	PROPN
ejpam-3193	312	6	,	,	PUNCT
ejpam-3193	312	7	prime	prime	ADJ
ejpam-3193	312	8	ideals	ideal	NOUN
ejpam-3193	312	9	and	and	CCONJ
ejpam-3193	312	10	bi	bi	ADJ
ejpam-3193	312	11	ideals	ideal	NOUN
ejpam-3193	312	12	in	in	ADP
ejpam-3193	312	13	gamma	gamma	NOUN
ejpam-3193	312	14	near	near	ADP
ejpam-3193	312	15	–	–	PUNCT
ejpam-3193	312	16	rings	ring	NOUN
ejpam-3193	312	17	,	,	PUNCT
ejpam-3193	312	18	1st	1st	ADJ
ejpam-3193	312	19	international	international	ADJ
ejpam-3193	312	20	symposium	symposium	NOUN
ejpam-3193	312	21	on	on	ADP
ejpam-3193	312	22	computing	compute	VERB
ejpam-3193	312	23	in	in	ADP
ejpam-3193	312	24	informatics	informatic	NOUN
ejpam-3193	312	25	and	and	CCONJ
ejpam-3193	312	26	mathematics	mathematics	PROPN
ejpam-3193	312	27	(	(	PUNCT
ejpam-3193	312	28	iscim	iscim	NOUN
ejpam-3193	312	29	2011	2011	NUM
ejpam-3193	312	30	)	)	PUNCT
ejpam-3193	312	31	,	,	PUNCT
ejpam-3193	312	32	in	in	ADP
ejpam-3193	312	33	collabaration	collabaration	NOUN
ejpam-3193	312	34	between	between	ADP
ejpam-3193	312	35	epoka	epoka	NOUN
ejpam-3193	312	36	university	university	NOUN
ejpam-3193	312	37	and	and	CCONJ
ejpam-3193	312	38	”	"	PUNCT
ejpam-3193	312	39	aleksandr	aleksandr	PROPN
ejpam-3193	312	40	moisiu	moisiu	NOUN
ejpam-3193	312	41	”	"	PUNCT
ejpam-3193	312	42	university	university	NOUN
ejpam-3193	312	43	of	of	ADP
ejpam-3193	312	44	durrs	durrs	NOUN
ejpam-3193	312	45	on	on	ADP
ejpam-3193	312	46	june	june	PROPN
ejpam-3193	312	47	2	2	NUM
ejpam-3193	312	48	-	-	SYM
ejpam-3193	312	49	4	4	NUM
ejpam-3193	312	50	2011	2011	NUM
ejpam-3193	312	51	,	,	PUNCT
ejpam-3193	312	52	480	480	NUM
ejpam-3193	312	53	-	-	SYM
ejpam-3193	312	54	485	485	NUM
ejpam-3193	312	55	,	,	PUNCT
ejpam-3193	312	56	tiranadurres	tiranadurre	NOUN
ejpam-3193	312	57	,	,	PUNCT
ejpam-3193	312	58	albania	albania	PROPN
ejpam-3193	312	59	.	.	PUNCT
ejpam-3193	313	1	[	[	X
ejpam-3193	313	2	8	8	NUM
ejpam-3193	313	3	]	]	X
ejpam-3193	313	4	b.	b.	PROPN
ejpam-3193	313	5	elavarasan	elavarasan	PROPN
ejpam-3193	313	6	,	,	PUNCT
ejpam-3193	313	7	generalizations	generalization	NOUN
ejpam-3193	313	8	of	of	ADP
ejpam-3193	313	9	prime	prime	ADJ
ejpam-3193	313	10	ideals	ideal	NOUN
ejpam-3193	313	11	in	in	ADP
ejpam-3193	313	12	near	near	ADJ
ejpam-3193	313	13	-	-	PUNCT
ejpam-3193	313	14	rings	ring	NOUN
ejpam-3193	313	15	,	,	PUNCT
ejpam-3193	313	16	int	int	NOUN
ejpam-3193	313	17	.	.	PUNCT
ejpam-3193	314	1	j.	j.	PROPN
ejpam-3193	314	2	open	open	PROPN
ejpam-3193	314	3	problems	problem	NOUN
ejpam-3193	314	4	compt	compt	VERB
ejpam-3193	314	5	.	.	PUNCT
ejpam-3193	315	1	math	math	NOUN
ejpam-3193	315	2	.	.	PUNCT
ejpam-3193	316	1	,	,	PUNCT
ejpam-3193	316	2	vol	vol	NOUN
ejpam-3193	316	3	.	.	PROPN
ejpam-3193	316	4	4	4	NUM
ejpam-3193	316	5	(	(	PUNCT
ejpam-3193	316	6	4	4	NUM
ejpam-3193	316	7	)	)	PUNCT
ejpam-3193	316	8	,	,	PUNCT
ejpam-3193	316	9	47	47	NUM
ejpam-3193	316	10	-	-	SYM
ejpam-3193	316	11	53	53	NUM
ejpam-3193	316	12	,	,	PUNCT
ejpam-3193	316	13	dec	dec	PROPN
ejpam-3193	316	14	.	.	PROPN
ejpam-3193	316	15	,	,	PUNCT
ejpam-3193	316	16	2011	2011	NUM
ejpam-3193	316	17	.	.	PUNCT
ejpam-3193	317	1	[	[	X
ejpam-3193	317	2	9	9	NUM
ejpam-3193	317	3	]	]	PUNCT
ejpam-3193	317	4	a.	a.	NOUN
ejpam-3193	317	5	frohlich	frohlich	PROPN
ejpam-3193	317	6	,	,	PUNCT
ejpam-3193	317	7	distributively	distributively	ADV
ejpam-3193	317	8	generated	generate	VERB
ejpam-3193	317	9	near	near	ADP
ejpam-3193	317	10	-	-	PUNCT
ejpam-3193	317	11	rings	ring	NOUN
ejpam-3193	317	12	,	,	PUNCT
ejpam-3193	317	13	proc	proc	NOUN
ejpam-3193	317	14	.	.	PUNCT
ejpam-3193	318	1	london	london	PROPN
ejpam-3193	318	2	math	math	PROPN
ejpam-3193	318	3	.	.	PUNCT
ejpam-3193	319	1	soc	soc	PROPN
ejpam-3193	319	2	.	.	PUNCT
ejpam-3193	320	1	3	3	NUM
ejpam-3193	320	2	(	(	PUNCT
ejpam-3193	320	3	8)	8)	NUM
ejpam-3193	320	4	,	,	PUNCT
ejpam-3193	320	5	76108	76108	NUM
ejpam-3193	320	6	,	,	PUNCT
ejpam-3193	320	7	1958	1958	NUM
ejpam-3193	320	8	.	.	PUNCT
ejpam-3193	321	1	[	[	X
ejpam-3193	321	2	10	10	NUM
ejpam-3193	321	3	]	]	X
ejpam-3193	321	4	s.	s.	PROPN
ejpam-3193	321	5	galovich	galovich	PROPN
ejpam-3193	321	6	,	,	PUNCT
ejpam-3193	321	7	unique	unique	ADJ
ejpam-3193	321	8	factorization	factorization	NOUN
ejpam-3193	321	9	rings	ring	NOUN
ejpam-3193	321	10	with	with	ADP
ejpam-3193	321	11	zero	zero	NUM
ejpam-3193	321	12	divisors	divisor	NOUN
ejpam-3193	321	13	,	,	PUNCT
ejpam-3193	321	14	mathematics	mathematics	PROPN
ejpam-3193	321	15	magazine	magazine	NOUN
ejpam-3193	321	16	,	,	PUNCT
ejpam-3193	321	17	51	51	NUM
ejpam-3193	321	18	(	(	PUNCT
ejpam-3193	321	19	5	5	NUM
ejpam-3193	321	20	)	)	PUNCT
ejpam-3193	321	21	,	,	PUNCT
ejpam-3193	321	22	276	276	NUM
ejpam-3193	321	23	-	-	SYM
ejpam-3193	321	24	283	283	NUM
ejpam-3193	321	25	,	,	PUNCT
ejpam-3193	321	26	nov	nov	PROPN
ejpam-3193	321	27	.	.	PROPN
ejpam-3193	321	28	,	,	PUNCT
ejpam-3193	321	29	1978	1978	NUM
ejpam-3193	321	30	.	.	PUNCT
ejpam-3193	322	1	[	[	X
ejpam-3193	322	2	11	11	NUM
ejpam-3193	322	3	]	]	X
ejpam-3193	322	4	n.	n.	PROPN
ejpam-3193	322	5	j.	j.	PROPN
ejpam-3193	322	6	groenewald	groenewald	PROPN
ejpam-3193	322	7	,	,	PUNCT
ejpam-3193	322	8	”	"	PUNCT
ejpam-3193	322	9	the	the	DET
ejpam-3193	322	10	completely	completely	ADV
ejpam-3193	322	11	prime	prime	ADJ
ejpam-3193	322	12	radical	radical	NOUN
ejpam-3193	322	13	in	in	ADP
ejpam-3193	322	14	near	near	ADP
ejpam-3193	322	15	rings	ring	NOUN
ejpam-3193	322	16	”	"	PUNCT
ejpam-3193	322	17	,	,	PUNCT
ejpam-3193	322	18	acta	acta	PROPN
ejpam-3193	322	19	math	math	PROPN
ejpam-3193	322	20	.	.	PUNCT
ejpam-3193	323	1	hung	hung	PROPN
ejpam-3193	323	2	,	,	PUNCT
ejpam-3193	323	3	vol.33	vol.33	PROPN
ejpam-3193	323	4	,	,	PUNCT
ejpam-3193	323	5	301	301	NUM
ejpam-3193	323	6	-	-	SYM
ejpam-3193	323	7	305	305	NUM
ejpam-3193	323	8	,	,	PUNCT
ejpam-3193	323	9	19888	19888	NUM
ejpam-3193	323	10	.	.	PUNCT
ejpam-3193	324	1	[	[	X
ejpam-3193	324	2	12	12	NUM
ejpam-3193	324	3	]	]	PUNCT
ejpam-3193	324	4	a.	a.	NOUN
ejpam-3193	324	5	k.	k.	PROPN
ejpam-3193	324	6	jabbar	jabbar	PROPN
ejpam-3193	324	7	and	and	CCONJ
ejpam-3193	324	8	c.	c.	PROPN
ejpam-3193	324	9	a.	a.	PROPN
ejpam-3193	324	10	ahmed	ahmed	PROPN
ejpam-3193	324	11	,	,	PUNCT
ejpam-3193	324	12	on	on	ADP
ejpam-3193	324	13	almost	almost	ADV
ejpam-3193	324	14	primary	primary	ADJ
ejpam-3193	324	15	ideals	ideal	NOUN
ejpam-3193	324	16	,	,	PUNCT
ejpam-3193	324	17	international	international	ADJ
ejpam-3193	324	18	journal	journal	NOUN
ejpam-3193	324	19	of	of	ADP
ejpam-3193	324	20	algebra	algebra	PROPN
ejpam-3193	324	21	,	,	PUNCT
ejpam-3193	324	22	5	5	NUM
ejpam-3193	324	23	(	(	PUNCT
ejpam-3193	324	24	13	13	NUM
ejpam-3193	324	25	)	)	PUNCT
ejpam-3193	324	26	,	,	PUNCT
ejpam-3193	324	27	627	627	NUM
ejpam-3193	324	28	636	636	NUM
ejpam-3193	324	29	,	,	PUNCT
ejpam-3193	324	30	2011	2011	NUM
ejpam-3193	324	31	.	.	PUNCT
ejpam-3193	325	1	[	[	X
ejpam-3193	325	2	13	13	NUM
ejpam-3193	325	3	]	]	PUNCT
ejpam-3193	325	4	s.	s.	PROPN
ejpam-3193	325	5	kyuno	kyuno	PROPN
ejpam-3193	325	6	,	,	PUNCT
ejpam-3193	325	7	prime	prime	ADJ
ejpam-3193	325	8	ideals	ideal	NOUN
ejpam-3193	325	9	in	in	ADP
ejpam-3193	325	10	gamma	gamma	NOUN
ejpam-3193	325	11	rings	ring	NOUN
ejpam-3193	325	12	,	,	PUNCT
ejpam-3193	325	13	vol	vol	NOUN
ejpam-3193	325	14	.	.	PROPN
ejpam-3193	326	1	98	98	NUM
ejpam-3193	326	2	(	(	PUNCT
ejpam-3193	326	3	2	2	NUM
ejpam-3193	326	4	)	)	PUNCT
ejpam-3193	326	5	,	,	PUNCT
ejpam-3193	326	6	375	375	NUM
ejpam-3193	326	7	-	-	SYM
ejpam-3193	326	8	379	379	NUM
ejpam-3193	326	9	,	,	PUNCT
ejpam-3193	326	10	april	april	PROPN
ejpam-3193	326	11	1982	1982	NUM
ejpam-3193	326	12	.	.	PUNCT
ejpam-3193	327	1	[	[	X
ejpam-3193	327	2	14	14	NUM
ejpam-3193	327	3	]	]	X
ejpam-3193	327	4	n.	n.	PROPN
ejpam-3193	327	5	nobusawa	nobusawa	PROPN
ejpam-3193	327	6	,	,	PUNCT
ejpam-3193	327	7	on	on	ADP
ejpam-3193	327	8	a	a	DET
ejpam-3193	327	9	generalization	generalization	NOUN
ejpam-3193	327	10	of	of	ADP
ejpam-3193	327	11	the	the	DET
ejpam-3193	327	12	ring	ring	NOUN
ejpam-3193	327	13	theory	theory	NOUN
ejpam-3193	327	14	,	,	PUNCT
ejpam-3193	327	15	osaka	osaka	PROPN
ejpam-3193	327	16	j.	j.	PROPN
ejpam-3193	327	17	math	math	PROPN
ejpam-3193	327	18	.	.	PUNCT
ejpam-3193	327	19	,	,	PUNCT
ejpam-3193	327	20	1	1	NUM
ejpam-3193	327	21	,	,	PUNCT
ejpam-3193	327	22	81	81	NUM
ejpam-3193	327	23	-	-	SYM
ejpam-3193	327	24	89	89	NUM
ejpam-3193	327	25	,	,	PUNCT
ejpam-3193	327	26	1964	1964	NUM
ejpam-3193	327	27	.	.	PUNCT
ejpam-3193	328	1	[	[	X
ejpam-3193	328	2	15	15	NUM
ejpam-3193	328	3	]	]	X
ejpam-3193	328	4	g.	g.	PROPN
ejpam-3193	328	5	pilz	pilz	PROPN
ejpam-3193	328	6	,	,	PUNCT
ejpam-3193	328	7	near	near	NOUN
ejpam-3193	328	8	-	-	PUNCT
ejpam-3193	328	9	rings	ring	NOUN
ejpam-3193	328	10	,	,	PUNCT
ejpam-3193	328	11	north	north	NOUN
ejpam-3193	328	12	-	-	PUNCT
ejpam-3193	328	13	holland	holland	PROPN
ejpam-3193	328	14	publishing	publishing	PROPN
ejpam-3193	328	15	co.	co.	PROPN
ejpam-3193	328	16	,	,	PUNCT
ejpam-3193	328	17	amsterdam	amsterdam	PROPN
ejpam-3193	328	18	,	,	PUNCT
ejpam-3193	328	19	second	second	ADJ
ejpam-3193	328	20	edition	edition	NOUN
ejpam-3193	328	21	,	,	PUNCT
ejpam-3193	328	22	1983	1983	NUM
ejpam-3193	328	23	.	.	PUNCT
ejpam-3193	329	1	[	[	X
ejpam-3193	329	2	16	16	NUM
ejpam-3193	329	3	]	]	PUNCT
ejpam-3193	329	4	m.	m.	NOUN
ejpam-3193	329	5	sabur	sabur	PROPN
ejpam-3193	329	6	uddin	uddin	PROPN
ejpam-3193	329	7	and	and	CCONJ
ejpam-3193	329	8	m.	m.	PROPN
ejpam-3193	329	9	shamsul	shamsul	PROPN
ejpam-3193	329	10	islam	islam	PROPN
ejpam-3193	329	11	,	,	PUNCT
ejpam-3193	329	12	”	"	PUNCT
ejpam-3193	329	13	semi	semi	ADJ
ejpam-3193	329	14	-	-	ADJ
ejpam-3193	329	15	prime	prime	ADJ
ejpam-3193	329	16	ideals	ideal	NOUN
ejpam-3193	329	17	of	of	ADP
ejpam-3193	329	18	gamma	gamma	NOUN
ejpam-3193	329	19	rings	ring	NOUN
ejpam-3193	329	20	”	"	PUNCT
ejpam-3193	329	21	,	,	PUNCT
ejpam-3193	329	22	annals	annal	NOUN
ejpam-3193	329	23	of	of	ADP
ejpam-3193	329	24	pure	pure	ADJ
ejpam-3193	329	25	and	and	CCONJ
ejpam-3193	329	26	applied	apply	VERB
ejpam-3193	329	27	mathematics	mathematic	NOUN
ejpam-3193	329	28	vol	vol	NOUN
ejpam-3193	329	29	.	.	PROPN
ejpam-3193	329	30	1	1	NUM
ejpam-3193	329	31	,	,	PUNCT
ejpam-3193	329	32	no	no	INTJ
ejpam-3193	329	33	.	.	NOUN
ejpam-3193	329	34	2	2	NUM
ejpam-3193	329	35	,	,	PUNCT
ejpam-3193	329	36	2012	2012	NUM
ejpam-3193	329	37	,	,	PUNCT
ejpam-3193	329	38	186	186	NUM
ejpam-3193	329	39	-	-	SYM
ejpam-3193	329	40	191	191	NUM
ejpam-3193	329	41	,	,	PUNCT
ejpam-3193	329	42	issn	issn	PROPN
ejpam-3193	329	43	:	:	PUNCT
ejpam-3193	329	44	2279	2279	NUM
ejpam-3193	329	45	-	-	PUNCT
ejpam-3193	329	46	087x	087x	NOUN
ejpam-3193	329	47	(	(	PUNCT
ejpam-3193	329	48	p	p	NOUN
ejpam-3193	329	49	)	)	PUNCT
ejpam-3193	329	50	,	,	PUNCT
ejpam-3193	329	51	2279	2279	NUM
ejpam-3193	329	52	-	-	PUNCT
ejpam-3193	329	53	0888(online	0888(online	NOUN
ejpam-3193	329	54	)	)	PUNCT
ejpam-3193	329	55	published	publish	VERB
ejpam-3193	329	56	on	on	ADP
ejpam-3193	329	57	16	16	NUM
ejpam-3193	329	58	october	october	PROPN
ejpam-3193	329	59	2012	2012	NUM
ejpam-3193	329	60	.	.	PUNCT
ejpam-3193	330	1	references	reference	NOUN
ejpam-3193	330	2	456	456	NUM
ejpam-3193	331	1	[	[	X
ejpam-3193	331	2	17	17	NUM
ejpam-3193	331	3	]	]	PUNCT
ejpam-3193	331	4	p.	p.	NOUN
ejpam-3193	331	5	yiarayong	yiarayong	PROPN
ejpam-3193	331	6	and	and	CCONJ
ejpam-3193	331	7	p.	p.	NOUN
ejpam-3193	331	8	panpho	panpho	NOUN
ejpam-3193	331	9	,	,	PUNCT
ejpam-3193	331	10	some	some	DET
ejpam-3193	331	11	basic	basic	ADJ
ejpam-3193	331	12	properties	property	NOUN
ejpam-3193	331	13	of	of	ADP
ejpam-3193	331	14	weakly	weakly	ADV
ejpam-3193	331	15	completely	completely	ADV
ejpam-3193	331	16	primary	primary	ADJ
ejpam-3193	331	17	ideals	ideal	NOUN
ejpam-3193	331	18	in	in	ADP
ejpam-3193	331	19	-near	-near	NOUN
ejpam-3193	331	20	rings	ring	NOUN
ejpam-3193	331	21	,	,	PUNCT
ejpam-3193	331	22	asian	asian	ADJ
ejpam-3193	331	23	journal	journal	NOUN
ejpam-3193	331	24	of	of	ADP
ejpam-3193	331	25	applied	apply	VERB
ejpam-3193	331	26	sciences	science	NOUN
ejpam-3193	331	27	(	(	PUNCT
ejpam-3193	331	28	issn	issn	PROPN
ejpam-3193	331	29	:	:	PUNCT
ejpam-3193	331	30	2321	2321	NUM
ejpam-3193	331	31	–	–	PUNCT
ejpam-3193	331	32	0893	0893	NUM
ejpam-3193	331	33	)	)	PUNCT
ejpam-3193	331	34	,	,	PUNCT
ejpam-3193	331	35	3(1	3(1	NUM
ejpam-3193	331	36	)	)	PUNCT
ejpam-3193	331	37	,	,	PUNCT
ejpam-3193	331	38	feb	feb	PROPN
ejpam-3193	331	39	.	.	PROPN
ejpam-3193	331	40	,	,	PUNCT
ejpam-3193	331	41	2015	2015	NUM
ejpam-3193	331	42	.	.	PUNCT