id	sid	tid	token	lemma	pos
ejpam-6576	1	1	european	european	PROPN
ejpam-6576	1	2	journal	journal	PROPN
ejpam-6576	1	3	of	of	ADP
ejpam-6576	1	4	pure	pure	ADJ
ejpam-6576	1	5	and	and	CCONJ
ejpam-6576	1	6	applied	applied	ADJ
ejpam-6576	1	7	mathematics	mathematic	NOUN
ejpam-6576	1	8	2025	2025	NUM
ejpam-6576	1	9	,	,	PUNCT
ejpam-6576	1	10	vol	vol	NOUN
ejpam-6576	1	11	.	.	PROPN
ejpam-6576	1	12	18	18	NUM
ejpam-6576	1	13	,	,	PUNCT
ejpam-6576	1	14	issue	issue	NOUN
ejpam-6576	1	15	3	3	NUM
ejpam-6576	1	16	,	,	PUNCT
ejpam-6576	1	17	article	article	NOUN
ejpam-6576	1	18	number	number	NOUN
ejpam-6576	1	19	6576	6576	NUM
ejpam-6576	1	20	issn	issn	PROPN
ejpam-6576	1	21	1307	1307	NUM
ejpam-6576	1	22	-	-	SYM
ejpam-6576	1	23	5543	5543	NUM
ejpam-6576	1	24	–	–	PUNCT
ejpam-6576	1	25	ejpam.com	ejpam.com	X
ejpam-6576	1	26	published	publish	VERB
ejpam-6576	1	27	by	by	ADP
ejpam-6576	1	28	new	new	PROPN
ejpam-6576	1	29	york	york	PROPN
ejpam-6576	1	30	business	business	PROPN
ejpam-6576	1	31	global	global	PROPN
ejpam-6576	1	32	a	a	DET
ejpam-6576	1	33	kurosh	kurosh	ADV
ejpam-6576	1	34	-	-	PUNCT
ejpam-6576	1	35	amitsur	amitsur	VERB
ejpam-6576	1	36	completely	completely	ADV
ejpam-6576	1	37	prime	prime	ADJ
ejpam-6576	1	38	radical	radical	NOUN
ejpam-6576	1	39	for	for	ADP
ejpam-6576	1	40	near	near	ADJ
ejpam-6576	1	41	-	-	PUNCT
ejpam-6576	1	42	rings	ring	NOUN
ejpam-6576	1	43	kilaru	kilaru	PROPN
ejpam-6576	1	44	j.	j.	PROPN
ejpam-6576	1	45	lakshminarayana1,∗	lakshminarayana1,∗	PROPN
ejpam-6576	1	46	,	,	PUNCT
ejpam-6576	1	47	v.b.v.n	v.b.v.n	PROPN
ejpam-6576	1	48	.	.	PROPN
ejpam-6576	1	49	prasad1	prasad1	PROPN
ejpam-6576	1	50	,	,	PUNCT
ejpam-6576	1	51	srinivasa	srinivasa	PROPN
ejpam-6576	1	52	rao	rao	PROPN
ejpam-6576	1	53	ravi2	ravi2	PROPN
ejpam-6576	1	54	,	,	PUNCT
ejpam-6576	1	55	siva	siva	PROPN
ejpam-6576	1	56	prasad	prasad	PROPN
ejpam-6576	1	57	korrapati2	korrapati2	PROPN
ejpam-6576	2	1	a.v	a.v	PROPN
ejpam-6576	2	2	.	.	PUNCT
ejpam-6576	2	3	ramakrishna3	ramakrishna3	NOUN
ejpam-6576	2	4	1	1	NUM
ejpam-6576	2	5	department	department	NOUN
ejpam-6576	2	6	of	of	ADP
ejpam-6576	2	7	engineering	engineering	NOUN
ejpam-6576	2	8	mathematics	mathematics	PROPN
ejpam-6576	2	9	koneru	koneru	PROPN
ejpam-6576	2	10	lakshmaiah	lakshmaiah	PROPN
ejpam-6576	2	11	education	education	PROPN
ejpam-6576	2	12	foundation	foundation	PROPN
ejpam-6576	2	13	,	,	PUNCT
ejpam-6576	2	14	vaddeswaram-522502	vaddeswaram-522502	NOUN
ejpam-6576	2	15	,	,	PUNCT
ejpam-6576	2	16	guntur	guntur	PROPN
ejpam-6576	2	17	(	(	PUNCT
ejpam-6576	2	18	dist	dist	NOUN
ejpam-6576	2	19	.	.	PUNCT
ejpam-6576	2	20	)	)	PUNCT
ejpam-6576	3	1	,	,	PUNCT
ejpam-6576	3	2	andhra	andhra	PROPN
ejpam-6576	3	3	pradesh	pradesh	PROPN
ejpam-6576	3	4	,	,	PUNCT
ejpam-6576	3	5	india	india	PROPN
ejpam-6576	3	6	2	2	NUM
ejpam-6576	3	7	department	department	NOUN
ejpam-6576	3	8	of	of	ADP
ejpam-6576	3	9	mathematics	mathematics	PROPN
ejpam-6576	3	10	university	university	PROPN
ejpam-6576	3	11	college	college	PROPN
ejpam-6576	3	12	of	of	ADP
ejpam-6576	3	13	sciences	sciences	PROPN
ejpam-6576	3	14	,	,	PUNCT
ejpam-6576	3	15	acharya	acharya	PROPN
ejpam-6576	3	16	nagarjuna	nagarjuna	PROPN
ejpam-6576	3	17	university	university	PROPN
ejpam-6576	3	18	,	,	PUNCT
ejpam-6576	4	1	nagarjuna	nagarjuna	PROPN
ejpam-6576	4	2	nagar-522510	nagar-522510	PROPN
ejpam-6576	4	3	guntur	guntur	PROPN
ejpam-6576	4	4	(	(	PUNCT
ejpam-6576	4	5	dist	dist	NOUN
ejpam-6576	4	6	.	.	PUNCT
ejpam-6576	4	7	)	)	PUNCT
ejpam-6576	4	8	,	,	PUNCT
ejpam-6576	4	9	andhra	andhra	PROPN
ejpam-6576	4	10	pradesh	pradesh	PROPN
ejpam-6576	4	11	,	,	PUNCT
ejpam-6576	4	12	india	india	PROPN
ejpam-6576	4	13	3	3	PROPN
ejpam-6576	4	14	department	department	PROPN
ejpam-6576	4	15	of	of	ADP
ejpam-6576	4	16	mathematics	mathematics	PROPN
ejpam-6576	4	17	r.v.r	r.v.r	PROPN
ejpam-6576	4	18	and	and	CCONJ
ejpam-6576	4	19	j.c	j.c	PROPN
ejpam-6576	4	20	college	college	PROPN
ejpam-6576	4	21	of	of	ADP
ejpam-6576	4	22	engineering	engineering	NOUN
ejpam-6576	4	23	,	,	PUNCT
ejpam-6576	4	24	chowdavaram-522019	chowdavaram-522019	NOUN
ejpam-6576	4	25	,	,	PUNCT
ejpam-6576	4	26	guntur	guntur	PROPN
ejpam-6576	4	27	(	(	PUNCT
ejpam-6576	4	28	dist.),andhra	dist.),andhra	PROPN
ejpam-6576	4	29	pradesh	pradesh	PROPN
ejpam-6576	4	30	,	,	PUNCT
ejpam-6576	4	31	india	india	PROPN
ejpam-6576	4	32	abstract	abstract	NOUN
ejpam-6576	4	33	.	.	PUNCT
ejpam-6576	5	1	two	two	NUM
ejpam-6576	5	2	generalizations	generalization	NOUN
ejpam-6576	5	3	of	of	ADP
ejpam-6576	5	4	the	the	DET
ejpam-6576	5	5	completely	completely	ADV
ejpam-6576	5	6	prime	prime	ADJ
ejpam-6576	5	7	radical	radical	NOUN
ejpam-6576	5	8	of	of	ADP
ejpam-6576	5	9	rings	ring	NOUN
ejpam-6576	5	10	to	to	ADP
ejpam-6576	5	11	near	near	ADJ
ejpam-6576	5	12	-	-	PUNCT
ejpam-6576	5	13	rings	ring	NOUN
ejpam-6576	5	14	,	,	PUNCT
ejpam-6576	5	15	namely	namely	ADV
ejpam-6576	5	16	the	the	DET
ejpam-6576	5	17	completely	completely	ADV
ejpam-6576	5	18	prime	prime	ADJ
ejpam-6576	5	19	radical	radical	NOUN
ejpam-6576	5	20	of	of	ADP
ejpam-6576	5	21	near	near	ADJ
ejpam-6576	5	22	-	-	PUNCT
ejpam-6576	5	23	rings	ring	NOUN
ejpam-6576	5	24	and	and	CCONJ
ejpam-6576	5	25	the	the	DET
ejpam-6576	5	26	completely	completely	ADV
ejpam-6576	5	27	equiprime	equiprime	NOUN
ejpam-6576	5	28	radical	radical	ADJ
ejpam-6576	5	29	of	of	ADP
ejpam-6576	5	30	near	near	ADJ
ejpam-6576	5	31	-	-	PUNCT
ejpam-6576	5	32	rings	ring	NOUN
ejpam-6576	5	33	were	be	AUX
ejpam-6576	5	34	introduced	introduce	VERB
ejpam-6576	5	35	and	and	CCONJ
ejpam-6576	5	36	studied	study	VERB
ejpam-6576	5	37	.	.	PUNCT
ejpam-6576	6	1	first	first	ADJ
ejpam-6576	6	2	one	one	NUM
ejpam-6576	6	3	is	be	AUX
ejpam-6576	6	4	not	not	PART
ejpam-6576	6	5	a	a	DET
ejpam-6576	6	6	kurosh	kurosh	ADV
ejpam-6576	6	7	-	-	PUNCT
ejpam-6576	6	8	amitsur	amitsur	NOUN
ejpam-6576	6	9	radical	radical	NOUN
ejpam-6576	6	10	but	but	CCONJ
ejpam-6576	6	11	the	the	DET
ejpam-6576	6	12	second	second	ADJ
ejpam-6576	6	13	one	one	NOUN
ejpam-6576	6	14	is	be	AUX
ejpam-6576	6	15	a	a	DET
ejpam-6576	6	16	special	special	ADJ
ejpam-6576	6	17	radical	radical	NOUN
ejpam-6576	6	18	in	in	ADP
ejpam-6576	6	19	near	near	ADJ
ejpam-6576	6	20	-	-	PUNCT
ejpam-6576	6	21	rings	ring	NOUN
ejpam-6576	6	22	.	.	PUNCT
ejpam-6576	7	1	in	in	ADP
ejpam-6576	7	2	this	this	DET
ejpam-6576	7	3	article	article	NOUN
ejpam-6576	7	4	another	another	DET
ejpam-6576	7	5	generalization	generalization	NOUN
ejpam-6576	7	6	of	of	ADP
ejpam-6576	7	7	the	the	DET
ejpam-6576	7	8	completely	completely	ADV
ejpam-6576	7	9	prime	prime	ADJ
ejpam-6576	7	10	radical	radical	NOUN
ejpam-6576	7	11	of	of	ADP
ejpam-6576	7	12	rings	ring	NOUN
ejpam-6576	7	13	is	be	AUX
ejpam-6576	7	14	introduced	introduce	VERB
ejpam-6576	7	15	in	in	ADP
ejpam-6576	7	16	nearrings	nearring	NOUN
ejpam-6576	7	17	using	use	VERB
ejpam-6576	7	18	right	right	ADJ
ejpam-6576	7	19	modules	module	NOUN
ejpam-6576	7	20	of	of	ADP
ejpam-6576	7	21	near	near	ADJ
ejpam-6576	7	22	-	-	PUNCT
ejpam-6576	7	23	rings	ring	NOUN
ejpam-6576	7	24	.	.	PUNCT
ejpam-6576	8	1	for	for	ADP
ejpam-6576	8	2	this	this	DET
ejpam-6576	8	3	completely	completely	ADV
ejpam-6576	8	4	prime	prime	ADJ
ejpam-6576	8	5	right	right	NOUN
ejpam-6576	8	6	n	n	PRON
ejpam-6576	8	7	-groups	-group	NOUN
ejpam-6576	8	8	of	of	ADP
ejpam-6576	8	9	type	type	NOUN
ejpam-6576	8	10	-	-	PUNCT
ejpam-6576	8	11	r(1	r(1	PROPN
ejpam-6576	8	12	)	)	PUNCT
ejpam-6576	8	13	are	be	AUX
ejpam-6576	8	14	introduced	introduce	VERB
ejpam-6576	8	15	in	in	ADP
ejpam-6576	8	16	near	near	ADJ
ejpam-6576	8	17	-	-	PUNCT
ejpam-6576	8	18	rings	ring	NOUN
ejpam-6576	8	19	,	,	PUNCT
ejpam-6576	8	20	n	n	X
ejpam-6576	8	21	is	be	AUX
ejpam-6576	8	22	a	a	DET
ejpam-6576	8	23	near	near	ADJ
ejpam-6576	8	24	-	-	PUNCT
ejpam-6576	8	25	ring	ring	NOUN
ejpam-6576	8	26	.	.	PUNCT
ejpam-6576	9	1	making	make	VERB
ejpam-6576	9	2	use	use	NOUN
ejpam-6576	9	3	of	of	ADP
ejpam-6576	9	4	these	these	DET
ejpam-6576	9	5	right	right	ADJ
ejpam-6576	10	1	n	n	PRON
ejpam-6576	10	2	-groups	-group	NOUN
ejpam-6576	10	3	of	of	ADP
ejpam-6576	10	4	type	type	NOUN
ejpam-6576	10	5	-	-	PUNCT
ejpam-6576	10	6	r(1	r(1	PROPN
ejpam-6576	10	7	)	)	PUNCT
ejpam-6576	10	8	,	,	PUNCT
ejpam-6576	10	9	the	the	DET
ejpam-6576	10	10	completely	completely	ADV
ejpam-6576	10	11	prime	prime	ADJ
ejpam-6576	10	12	radical	radical	NOUN
ejpam-6576	10	13	of	of	ADP
ejpam-6576	10	14	near	near	ADJ
ejpam-6576	10	15	-	-	PUNCT
ejpam-6576	10	16	rings	ring	NOUN
ejpam-6576	10	17	of	of	ADP
ejpam-6576	10	18	type	type	NOUN
ejpam-6576	10	19	-	-	PUNCT
ejpam-6576	10	20	r(1	r(1	PROPN
ejpam-6576	10	21	)	)	PUNCT
ejpam-6576	10	22	is	be	AUX
ejpam-6576	10	23	introduced	introduce	VERB
ejpam-6576	10	24	.	.	PUNCT
ejpam-6576	11	1	it	it	PRON
ejpam-6576	11	2	is	be	AUX
ejpam-6576	11	3	observed	observe	VERB
ejpam-6576	11	4	that	that	SCONJ
ejpam-6576	11	5	the	the	DET
ejpam-6576	11	6	completely	completely	ADV
ejpam-6576	11	7	prime	prime	ADJ
ejpam-6576	11	8	radical	radical	NOUN
ejpam-6576	11	9	of	of	ADP
ejpam-6576	11	10	type	type	NOUN
ejpam-6576	11	11	-	-	PUNCT
ejpam-6576	11	12	r(1	r(1	PROPN
ejpam-6576	11	13	)	)	PUNCT
ejpam-6576	11	14	is	be	AUX
ejpam-6576	11	15	a	a	DET
ejpam-6576	11	16	kurosh	kurosh	ADV
ejpam-6576	11	17	-	-	PUNCT
ejpam-6576	11	18	amitsur	amitsur	NOUN
ejpam-6576	11	19	radical	radical	NOUN
ejpam-6576	11	20	.	.	PUNCT
ejpam-6576	12	1	2020	2020	NUM
ejpam-6576	12	2	mathematics	mathematic	NOUN
ejpam-6576	12	3	subject	subject	NOUN
ejpam-6576	12	4	classifications	classification	NOUN
ejpam-6576	12	5	:	:	PUNCT
ejpam-6576	12	6	16y30	16y30	NUM
ejpam-6576	12	7	key	key	ADJ
ejpam-6576	12	8	words	word	NOUN
ejpam-6576	12	9	and	and	CCONJ
ejpam-6576	12	10	phrases	phrase	NOUN
ejpam-6576	12	11	:	:	PUNCT
ejpam-6576	12	12	near	near	ADJ
ejpam-6576	12	13	-	-	PUNCT
ejpam-6576	12	14	ring	ring	NOUN
ejpam-6576	12	15	,	,	PUNCT
ejpam-6576	12	16	right	right	ADJ
ejpam-6576	12	17	n	n	PRON
ejpam-6576	12	18	-group	-group	NOUN
ejpam-6576	12	19	,	,	PUNCT
ejpam-6576	12	20	rightn	rightn	PROPN
ejpam-6576	12	21	-groups	-group	NOUN
ejpam-6576	12	22	of	of	ADP
ejpam-6576	12	23	type	type	NOUN
ejpam-6576	12	24	r(1	r(1	PROPN
ejpam-6576	12	25	)	)	PUNCT
ejpam-6576	12	26	,	,	PUNCT
ejpam-6576	12	27	completely	completely	ADV
ejpam-6576	12	28	prime	prime	ADJ
ejpam-6576	12	29	radical	radical	ADJ
ejpam-6576	12	30	of	of	ADP
ejpam-6576	12	31	type	type	NOUN
ejpam-6576	12	32	r(1	r(1	PROPN
ejpam-6576	12	33	)	)	PUNCT
ejpam-6576	12	34	1	1	NUM
ejpam-6576	12	35	.	.	PUNCT
ejpam-6576	13	1	introduction	introduction	NOUN
ejpam-6576	13	2	in	in	ADP
ejpam-6576	13	3	this	this	DET
ejpam-6576	13	4	article	article	NOUN
ejpam-6576	13	5	,	,	PUNCT
ejpam-6576	13	6	we	we	PRON
ejpam-6576	13	7	consider	consider	VERB
ejpam-6576	13	8	right	right	ADJ
ejpam-6576	13	9	zero	zero	NUM
ejpam-6576	13	10	-	-	PUNCT
ejpam-6576	13	11	symmetric	symmetric	ADJ
ejpam-6576	13	12	near	near	ADJ
ejpam-6576	13	13	-	-	PUNCT
ejpam-6576	13	14	rings	ring	NOUN
ejpam-6576	13	15	.	.	PUNCT
ejpam-6576	14	1	the	the	DET
ejpam-6576	14	2	concept	concept	NOUN
ejpam-6576	14	3	of	of	ADP
ejpam-6576	14	4	the	the	DET
ejpam-6576	14	5	completely	completely	ADV
ejpam-6576	14	6	prime	prime	ADJ
ejpam-6576	14	7	radical	radical	ADJ
ejpam-6576	14	8	,	,	PUNCT
ejpam-6576	14	9	originally	originally	ADV
ejpam-6576	14	10	developed	develop	VERB
ejpam-6576	14	11	for	for	ADP
ejpam-6576	14	12	rings	ring	NOUN
ejpam-6576	14	13	,	,	PUNCT
ejpam-6576	14	14	was	be	AUX
ejpam-6576	14	15	extended	extend	VERB
ejpam-6576	14	16	to	to	ADP
ejpam-6576	14	17	near	near	ADJ
ejpam-6576	14	18	-	-	PUNCT
ejpam-6576	14	19	rings	ring	NOUN
ejpam-6576	14	20	by	by	ADP
ejpam-6576	14	21	n.	n.	PROPN
ejpam-6576	14	22	j.	j.	PROPN
ejpam-6576	14	23	groenewald	groenewald	PROPN
ejpam-6576	15	1	[	[	X
ejpam-6576	15	2	1	1	NUM
ejpam-6576	15	3	]	]	PUNCT
ejpam-6576	15	4	.	.	PUNCT
ejpam-6576	16	1	he	he	PRON
ejpam-6576	16	2	demonstrated	demonstrate	VERB
ejpam-6576	16	3	that	that	SCONJ
ejpam-6576	16	4	,	,	PUNCT
ejpam-6576	16	5	analogous	analogous	ADJ
ejpam-6576	16	6	to	to	ADP
ejpam-6576	16	7	the	the	DET
ejpam-6576	16	8	prime	prime	ADJ
ejpam-6576	16	9	radical	radical	NOUN
ejpam-6576	16	10	in	in	ADP
ejpam-6576	16	11	near	near	ADJ
ejpam-6576	16	12	-	-	PUNCT
ejpam-6576	16	13	rings	ring	NOUN
ejpam-6576	16	14	,	,	PUNCT
ejpam-6576	16	15	the	the	DET
ejpam-6576	16	16	completely	completely	ADV
ejpam-6576	16	17	prime	prime	ADJ
ejpam-6576	16	18	radical	radical	NOUN
ejpam-6576	16	19	of	of	ADP
ejpam-6576	16	20	near	near	ADJ
ejpam-6576	16	21	-	-	PUNCT
ejpam-6576	16	22	rings	ring	NOUN
ejpam-6576	16	23	does	do	AUX
ejpam-6576	16	24	not	not	PART
ejpam-6576	16	25	satisfy	satisfy	VERB
ejpam-6576	16	26	the	the	DET
ejpam-6576	16	27	properties	property	NOUN
ejpam-6576	16	28	of	of	ADP
ejpam-6576	16	29	a	a	DET
ejpam-6576	16	30	kurosh	kurosh	ADV
ejpam-6576	16	31	-	-	PUNCT
ejpam-6576	16	32	amitsur	amitsur	NOUN
ejpam-6576	16	33	radical	radical	NOUN
ejpam-6576	16	34	.	.	PUNCT
ejpam-6576	17	1	based	base	VERB
ejpam-6576	17	2	on	on	ADP
ejpam-6576	17	3	the	the	DET
ejpam-6576	17	4	notion	notion	NOUN
ejpam-6576	17	5	of	of	ADP
ejpam-6576	17	6	equiprime	equiprime	ADJ
ejpam-6576	17	7	ideals	ideal	NOUN
ejpam-6576	17	8	in	in	ADP
ejpam-6576	17	9	near	near	ADJ
ejpam-6576	17	10	-	-	PUNCT
ejpam-6576	17	11	rings	ring	NOUN
ejpam-6576	17	12	,	,	PUNCT
ejpam-6576	17	13	a	a	DET
ejpam-6576	17	14	further	further	ADJ
ejpam-6576	17	15	generalization	generalization	NOUN
ejpam-6576	17	16	of	of	ADP
ejpam-6576	17	17	∗corresponding	∗corresponde	VERB
ejpam-6576	17	18	author	author	NOUN
ejpam-6576	17	19	.	.	PUNCT
ejpam-6576	18	1	doi	doi	NOUN
ejpam-6576	18	2	:	:	PUNCT
ejpam-6576	18	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6576	https://doi.org/10.29020/nybg.ejpam.v18i3.6576	PROPN
ejpam-6576	18	4	email	email	NOUN
ejpam-6576	18	5	addresses	address	VERB
ejpam-6576	18	6	:	:	PUNCT
ejpam-6576	18	7	2002511005@kluniversity.in	2002511005@kluniversity.in	NUM
ejpam-6576	18	8	(	(	PUNCT
ejpam-6576	18	9	kilaru	kilaru	PROPN
ejpam-6576	18	10	j.	j.	PROPN
ejpam-6576	18	11	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	18	12	)	)	PUNCT
ejpam-6576	18	13	,	,	PUNCT
ejpam-6576	18	14	vbvnprasad@kluniversity.in	vbvnprasad@kluniversity.in	PROPN
ejpam-6576	18	15	(	(	PUNCT
ejpam-6576	18	16	v.b.v.n	v.b.v.n	PROPN
ejpam-6576	18	17	.	.	PROPN
ejpam-6576	18	18	prasad	prasad	PROPN
ejpam-6576	18	19	)	)	PUNCT
ejpam-6576	18	20	,	,	PUNCT
ejpam-6576	18	21	dr	dr	PROPN
ejpam-6576	18	22	rsrao@yahoo.com	rsrao@yahoo.com	PROPN
ejpam-6576	18	23	(	(	PUNCT
ejpam-6576	18	24	s.	s.	PROPN
ejpam-6576	18	25	rao	rao	PROPN
ejpam-6576	18	26	ravi	ravi	PROPN
ejpam-6576	18	27	)	)	PUNCT
ejpam-6576	18	28	,	,	PUNCT
ejpam-6576	18	29	siva235prasad@yahoo.co.in	siva235prasad@yahoo.co.in	INTJ
ejpam-6576	18	30	(	(	PUNCT
ejpam-6576	18	31	s.	s.	PROPN
ejpam-6576	18	32	p.	p.	PROPN
ejpam-6576	18	33	korrapati	korrapati	PROPN
ejpam-6576	18	34	)	)	PUNCT
ejpam-6576	18	35	,	,	PUNCT
ejpam-6576	18	36	amathi7@gmail.com	amathi7@gmail.com	X
ejpam-6576	18	37	(	(	PUNCT
ejpam-6576	18	38	a.	a.	PROPN
ejpam-6576	18	39	v.	v.	PROPN
ejpam-6576	18	40	ramakrishna	ramakrishna	PROPN
ejpam-6576	18	41	)	)	PUNCT
ejpam-6576	18	42	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6576	18	43	1	1	NUM
ejpam-6576	18	44	copyright	copyright	NOUN
ejpam-6576	18	45	:	:	PUNCT
ejpam-6576	19	1	©	©	PROPN
ejpam-6576	19	2	2025	2025	NUM
ejpam-6576	19	3	the	the	DET
ejpam-6576	19	4	author(s	author(s	NOUN
ejpam-6576	19	5	)	)	PUNCT
ejpam-6576	19	6	.	.	PUNCT
ejpam-6576	20	1	(	(	PUNCT
ejpam-6576	20	2	cc	cc	NOUN
ejpam-6576	20	3	by	by	ADP
ejpam-6576	20	4	-	-	PUNCT
ejpam-6576	20	5	nc	nc	PROPN
ejpam-6576	20	6	4.0	4.0	NUM
ejpam-6576	20	7	)	)	PUNCT
ejpam-6576	20	8	k.	k.	PROPN
ejpam-6576	21	1	j.	j.	PROPN
ejpam-6576	21	2	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	21	3	et	et	PROPN
ejpam-6576	21	4	al	al	PROPN
ejpam-6576	21	5	.	.	PUNCT
ejpam-6576	21	6	/	/	SYM
ejpam-6576	21	7	eur	eur	PROPN
ejpam-6576	21	8	.	.	PUNCT
ejpam-6576	22	1	j.	j.	PROPN
ejpam-6576	22	2	pure	pure	PROPN
ejpam-6576	22	3	appl	appl	PROPN
ejpam-6576	22	4	.	.	PROPN
ejpam-6576	22	5	math	math	PROPN
ejpam-6576	22	6	,	,	PUNCT
ejpam-6576	22	7	18	18	NUM
ejpam-6576	22	8	(	(	PUNCT
ejpam-6576	22	9	3	3	NUM
ejpam-6576	22	10	)	)	PUNCT
ejpam-6576	22	11	(	(	PUNCT
ejpam-6576	22	12	2025	2025	NUM
ejpam-6576	22	13	)	)	PUNCT
ejpam-6576	22	14	,	,	PUNCT
ejpam-6576	22	15	6576	6576	NUM
ejpam-6576	22	16	2	2	NUM
ejpam-6576	22	17	of	of	ADP
ejpam-6576	22	18	7	7	NUM
ejpam-6576	22	19	completely	completely	ADV
ejpam-6576	22	20	prime	prime	ADJ
ejpam-6576	22	21	radical	radical	ADJ
ejpam-6576	22	22	of	of	ADP
ejpam-6576	22	23	rings	ring	NOUN
ejpam-6576	22	24	called	call	VERB
ejpam-6576	22	25	the	the	DET
ejpam-6576	22	26	completely	completely	ADV
ejpam-6576	22	27	equiprime	equiprime	ADJ
ejpam-6576	22	28	radical	radical	ADJ
ejpam-6576	22	29	was	be	AUX
ejpam-6576	22	30	introduced	introduce	VERB
ejpam-6576	22	31	in	in	ADP
ejpam-6576	22	32	[	[	X
ejpam-6576	22	33	2	2	NUM
ejpam-6576	22	34	]	]	PUNCT
ejpam-6576	22	35	.	.	PUNCT
ejpam-6576	23	1	it	it	PRON
ejpam-6576	23	2	was	be	AUX
ejpam-6576	23	3	shown	show	VERB
ejpam-6576	23	4	that	that	SCONJ
ejpam-6576	23	5	this	this	DET
ejpam-6576	23	6	radical	radical	ADJ
ejpam-6576	23	7	forms	form	NOUN
ejpam-6576	23	8	a	a	DET
ejpam-6576	23	9	special	special	ADJ
ejpam-6576	23	10	class	class	NOUN
ejpam-6576	23	11	of	of	ADP
ejpam-6576	23	12	radicals	radical	NOUN
ejpam-6576	23	13	within	within	ADP
ejpam-6576	23	14	the	the	DET
ejpam-6576	23	15	framework	framework	NOUN
ejpam-6576	23	16	of	of	ADP
ejpam-6576	23	17	near	near	ADJ
ejpam-6576	23	18	-	-	PUNCT
ejpam-6576	23	19	rings	ring	NOUN
ejpam-6576	23	20	.	.	PUNCT
ejpam-6576	24	1	completely	completely	ADV
ejpam-6576	24	2	prime	prime	ADJ
ejpam-6576	24	3	modules	module	NOUN
ejpam-6576	24	4	of	of	ADP
ejpam-6576	24	5	rings	ring	NOUN
ejpam-6576	24	6	were	be	AUX
ejpam-6576	24	7	introduced	introduce	VERB
ejpam-6576	24	8	in	in	ADP
ejpam-6576	24	9	[	[	X
ejpam-6576	24	10	3	3	NUM
ejpam-6576	24	11	]	]	PUNCT
ejpam-6576	24	12	.	.	PUNCT
ejpam-6576	25	1	in	in	ADP
ejpam-6576	25	2	[	[	X
ejpam-6576	25	3	4	4	NUM
ejpam-6576	25	4	]	]	PUNCT
ejpam-6576	25	5	,	,	PUNCT
ejpam-6576	25	6	completely	completely	ADV
ejpam-6576	25	7	prime	prime	ADJ
ejpam-6576	25	8	modules	module	NOUN
ejpam-6576	25	9	of	of	ADP
ejpam-6576	25	10	rings	ring	NOUN
ejpam-6576	25	11	were	be	AUX
ejpam-6576	25	12	extended	extend	VERB
ejpam-6576	25	13	to	to	ADP
ejpam-6576	25	14	(	(	PUNCT
ejpam-6576	25	15	left	left	ADJ
ejpam-6576	25	16	)	)	PUNCT
ejpam-6576	25	17	n	n	PRON
ejpam-6576	25	18	-groups	-groups	PROPN
ejpam-6576	25	19	,	,	PUNCT
ejpam-6576	25	20	n	n	PROPN
ejpam-6576	25	21	is	be	AUX
ejpam-6576	25	22	a	a	DET
ejpam-6576	25	23	near	near	ADJ
ejpam-6576	25	24	-	-	PUNCT
ejpam-6576	25	25	ring	ring	NOUN
ejpam-6576	25	26	and	and	CCONJ
ejpam-6576	25	27	the	the	DET
ejpam-6576	25	28	corresponding	corresponding	ADJ
ejpam-6576	25	29	radical	radical	NOUN
ejpam-6576	25	30	is	be	AUX
ejpam-6576	25	31	the	the	DET
ejpam-6576	25	32	completely	completely	ADV
ejpam-6576	25	33	prime	prime	ADJ
ejpam-6576	25	34	radical	radical	NOUN
ejpam-6576	25	35	of	of	ADP
ejpam-6576	25	36	near	near	ADJ
ejpam-6576	25	37	-	-	PUNCT
ejpam-6576	25	38	ring	ring	NOUN
ejpam-6576	25	39	which	which	PRON
ejpam-6576	25	40	is	be	AUX
ejpam-6576	25	41	not	not	PART
ejpam-6576	25	42	a	a	DET
ejpam-6576	25	43	kurosh	kurosh	ADV
ejpam-6576	25	44	-	-	PUNCT
ejpam-6576	25	45	amitsur	amitsur	NOUN
ejpam-6576	25	46	radical	radical	NOUN
ejpam-6576	25	47	of	of	ADP
ejpam-6576	25	48	near	near	ADJ
ejpam-6576	25	49	-	-	PUNCT
ejpam-6576	25	50	rings	ring	NOUN
ejpam-6576	25	51	.	.	PUNCT
ejpam-6576	26	1	right	right	ADJ
ejpam-6576	26	2	module	module	NOUN
ejpam-6576	26	3	theoretic	theoretic	ADJ
ejpam-6576	26	4	characterization	characterization	NOUN
ejpam-6576	26	5	of	of	ADP
ejpam-6576	26	6	radicals	radical	NOUN
ejpam-6576	26	7	of	of	ADP
ejpam-6576	26	8	near	near	ADJ
ejpam-6576	26	9	-	-	PUNCT
ejpam-6576	26	10	rings	ring	NOUN
ejpam-6576	26	11	was	be	AUX
ejpam-6576	26	12	studied	study	VERB
ejpam-6576	26	13	in	in	ADP
ejpam-6576	26	14	[	[	X
ejpam-6576	26	15	5	5	NUM
ejpam-6576	26	16	]	]	PUNCT
ejpam-6576	26	17	.	.	PUNCT
ejpam-6576	27	1	prime	prime	ADJ
ejpam-6576	27	2	right	right	PROPN
ejpam-6576	27	3	n	n	PRON
ejpam-6576	27	4	-groups	-group	NOUN
ejpam-6576	27	5	were	be	AUX
ejpam-6576	27	6	introduced	introduce	VERB
ejpam-6576	27	7	and	and	CCONJ
ejpam-6576	27	8	their	their	PRON
ejpam-6576	27	9	correspoding	correspode	VERB
ejpam-6576	27	10	radicals	radical	NOUN
ejpam-6576	27	11	were	be	AUX
ejpam-6576	27	12	studied	study	VERB
ejpam-6576	27	13	in	in	ADP
ejpam-6576	27	14	[	[	X
ejpam-6576	27	15	6	6	NUM
ejpam-6576	27	16	]	]	PUNCT
ejpam-6576	27	17	,	,	PUNCT
ejpam-6576	28	1	[	[	X
ejpam-6576	28	2	7	7	X
ejpam-6576	28	3	]	]	PUNCT
ejpam-6576	28	4	and	and	CCONJ
ejpam-6576	28	5	[	[	X
ejpam-6576	28	6	8	8	NUM
ejpam-6576	28	7	]	]	PUNCT
ejpam-6576	28	8	.	.	PUNCT
ejpam-6576	29	1	in	in	ADP
ejpam-6576	29	2	this	this	DET
ejpam-6576	29	3	article	article	NOUN
ejpam-6576	29	4	,	,	PUNCT
ejpam-6576	29	5	the	the	DET
ejpam-6576	29	6	concept	concept	NOUN
ejpam-6576	29	7	of	of	ADP
ejpam-6576	29	8	completely	completely	ADV
ejpam-6576	29	9	prime	prime	ADJ
ejpam-6576	29	10	module	module	NOUN
ejpam-6576	29	11	is	be	AUX
ejpam-6576	29	12	extended	extend	VERB
ejpam-6576	29	13	to	to	ADP
ejpam-6576	29	14	(	(	PUNCT
ejpam-6576	29	15	right	right	ADJ
ejpam-6576	29	16	)	)	PUNCT
ejpam-6576	30	1	n	n	PRON
ejpam-6576	30	2	-groups	-group	NOUN
ejpam-6576	30	3	which	which	PRON
ejpam-6576	30	4	leads	lead	VERB
ejpam-6576	30	5	to	to	ADP
ejpam-6576	30	6	another	another	DET
ejpam-6576	30	7	completely	completely	ADV
ejpam-6576	30	8	prime	prime	ADJ
ejpam-6576	30	9	radical	radical	NOUN
ejpam-6576	30	10	of	of	ADP
ejpam-6576	30	11	near	near	ADJ
ejpam-6576	30	12	-	-	PUNCT
ejpam-6576	30	13	rings	ring	NOUN
ejpam-6576	30	14	and	and	CCONJ
ejpam-6576	30	15	is	be	AUX
ejpam-6576	30	16	a	a	DET
ejpam-6576	30	17	kurosh	kurosh	ADV
ejpam-6576	30	18	-	-	PUNCT
ejpam-6576	30	19	amitsur	amitsur	NOUN
ejpam-6576	30	20	radical	radical	NOUN
ejpam-6576	30	21	of	of	ADP
ejpam-6576	30	22	near	near	ADJ
ejpam-6576	30	23	-	-	PUNCT
ejpam-6576	30	24	rings	ring	NOUN
ejpam-6576	30	25	.	.	PUNCT
ejpam-6576	31	1	a	a	DET
ejpam-6576	31	2	group	group	NOUN
ejpam-6576	31	3	(	(	PUNCT
ejpam-6576	31	4	t,+	t,+	NOUN
ejpam-6576	31	5	)	)	PUNCT
ejpam-6576	31	6	is	be	AUX
ejpam-6576	31	7	a	a	DET
ejpam-6576	31	8	right	right	ADJ
ejpam-6576	31	9	n	n	ADP
ejpam-6576	31	10	-group	-group	NOUN
ejpam-6576	31	11	if	if	SCONJ
ejpam-6576	31	12	there	there	PRON
ejpam-6576	31	13	is	be	VERB
ejpam-6576	31	14	a	a	DET
ejpam-6576	31	15	mapping	mapping	NOUN
ejpam-6576	31	16	(	(	PUNCT
ejpam-6576	31	17	t	t	PROPN
ejpam-6576	31	18	,	,	PUNCT
ejpam-6576	31	19	a	a	NOUN
ejpam-6576	31	20	)	)	PUNCT
ejpam-6576	31	21	→	→	SYM
ejpam-6576	31	22	ta	ta	PROPN
ejpam-6576	31	23	of	of	ADP
ejpam-6576	31	24	t	t	PROPN
ejpam-6576	31	25	×n	×n	PROPN
ejpam-6576	31	26	into	into	ADP
ejpam-6576	31	27	t	t	PROPN
ejpam-6576	31	28	such	such	ADJ
ejpam-6576	31	29	that	that	PRON
ejpam-6576	31	30	:	:	PUNCT
ejpam-6576	31	31	(	(	PUNCT
ejpam-6576	31	32	i	i	NOUN
ejpam-6576	31	33	)	)	PUNCT
ejpam-6576	32	1	t(ab	t(ab	ADP
ejpam-6576	32	2	)	)	PUNCT
ejpam-6576	32	3	=	=	SYM
ejpam-6576	32	4	(	(	PUNCT
ejpam-6576	32	5	ta)b	ta)b	PROPN
ejpam-6576	32	6	;	;	PUNCT
ejpam-6576	32	7	(	(	PUNCT
ejpam-6576	32	8	ii	ii	NOUN
ejpam-6576	32	9	)	)	PUNCT
ejpam-6576	32	10	t(a+	t(a+	NOUN
ejpam-6576	32	11	b	b	X
ejpam-6576	32	12	)	)	PUNCT
ejpam-6576	32	13	=	=	NOUN
ejpam-6576	32	14	ta+	ta+	NOUN
ejpam-6576	32	15	tb	tb	NOUN
ejpam-6576	32	16	for	for	ADP
ejpam-6576	32	17	all	all	DET
ejpam-6576	32	18	t	t	NOUN
ejpam-6576	32	19	∈	∈	PROPN
ejpam-6576	32	20	t	t	PROPN
ejpam-6576	32	21	,	,	PUNCT
ejpam-6576	32	22	a	a	PRON
ejpam-6576	32	23	,	,	PUNCT
ejpam-6576	32	24	b	b	PROPN
ejpam-6576	32	25	∈	∈	PROPN
ejpam-6576	32	26	n	n	ADV
ejpam-6576	32	27	.	.	PUNCT
ejpam-6576	33	1	i	i	PRON
ejpam-6576	33	2	is	be	AUX
ejpam-6576	33	3	a	a	DET
ejpam-6576	33	4	right	right	ADJ
ejpam-6576	33	5	n	n	ADP
ejpam-6576	33	6	-group	-group	NOUN
ejpam-6576	33	7	for	for	ADP
ejpam-6576	33	8	any	any	DET
ejpam-6576	33	9	right	right	ADJ
ejpam-6576	33	10	ideal	ideal	NOUN
ejpam-6576	33	11	i	i	PRON
ejpam-6576	33	12	of	of	ADP
ejpam-6576	33	13	n	n	CCONJ
ejpam-6576	33	14	under	under	ADP
ejpam-6576	33	15	the	the	DET
ejpam-6576	33	16	multiplication	multiplication	NOUN
ejpam-6576	33	17	in	in	ADP
ejpam-6576	33	18	n	n	PROPN
ejpam-6576	33	19	.	.	PUNCT
ejpam-6576	34	1	moreover	moreover	ADV
ejpam-6576	34	2	,	,	PUNCT
ejpam-6576	34	3	for	for	ADP
ejpam-6576	34	4	a	a	DET
ejpam-6576	34	5	right	right	ADJ
ejpam-6576	34	6	ideal	ideal	NOUN
ejpam-6576	34	7	i	i	PRON
ejpam-6576	34	8	of	of	ADP
ejpam-6576	34	9	n	n	PROPN
ejpam-6576	34	10	,	,	PUNCT
ejpam-6576	34	11	n	n	CCONJ
ejpam-6576	34	12	/	/	SYM
ejpam-6576	34	13	i	i	PRON
ejpam-6576	34	14	is	be	AUX
ejpam-6576	34	15	a	a	DET
ejpam-6576	34	16	right	right	NOUN
ejpam-6576	34	17	n	n	ADP
ejpam-6576	34	18	-group	-group	NOUN
ejpam-6576	34	19	under	under	ADP
ejpam-6576	34	20	(	(	PUNCT
ejpam-6576	34	21	a+	a+	X
ejpam-6576	34	22	i)b	i)b	ADJ
ejpam-6576	34	23	=	=	PROPN
ejpam-6576	34	24	ab+	ab+	PROPN
ejpam-6576	34	25	i	i	PRON
ejpam-6576	34	26	,	,	PUNCT
ejpam-6576	34	27	a	a	PRON
ejpam-6576	34	28	,	,	PUNCT
ejpam-6576	34	29	b	b	PROPN
ejpam-6576	34	30	∈	∈	PROPN
ejpam-6576	34	31	n	n	ADV
ejpam-6576	34	32	.	.	PUNCT
ejpam-6576	35	1	a	a	DET
ejpam-6576	35	2	subgroup	subgroup	NOUN
ejpam-6576	35	3	(	(	PUNCT
ejpam-6576	35	4	normal	normal	ADJ
ejpam-6576	35	5	subgroup	subgroup	NOUN
ejpam-6576	35	6	)	)	PUNCT
ejpam-6576	36	1	d	d	NOUN
ejpam-6576	36	2	of	of	ADP
ejpam-6576	36	3	the	the	DET
ejpam-6576	36	4	right	right	ADJ
ejpam-6576	36	5	n	n	NUM
ejpam-6576	36	6	-group	-group	NOUN
ejpam-6576	36	7	t	t	PROPN
ejpam-6576	36	8	is	be	AUX
ejpam-6576	36	9	a	a	DET
ejpam-6576	36	10	right	right	NOUN
ejpam-6576	36	11	n	n	PRON
ejpam-6576	36	12	-subgroup	-subgroup	NOUN
ejpam-6576	36	13	(	(	PUNCT
ejpam-6576	36	14	ideal	ideal	NOUN
ejpam-6576	36	15	)	)	PUNCT
ejpam-6576	36	16	of	of	ADP
ejpam-6576	36	17	t	t	PROPN
ejpam-6576	36	18	if	if	SCONJ
ejpam-6576	36	19	da	da	PROPN
ejpam-6576	36	20	∈	∈	PROPN
ejpam-6576	36	21	d	d	NOUN
ejpam-6576	36	22	for	for	ADP
ejpam-6576	36	23	all	all	DET
ejpam-6576	36	24	d	d	PROPN
ejpam-6576	36	25	∈	∈	PROPN
ejpam-6576	36	26	d	d	NOUN
ejpam-6576	36	27	,	,	PUNCT
ejpam-6576	36	28	a	a	DET
ejpam-6576	36	29	∈	∈	PROPN
ejpam-6576	36	30	n	n	NOUN
ejpam-6576	36	31	.	.	PUNCT
ejpam-6576	37	1	t	t	PROPN
ejpam-6576	37	2	∈	∈	PROPN
ejpam-6576	37	3	t	t	PROPN
ejpam-6576	37	4	is	be	AUX
ejpam-6576	37	5	called	call	VERB
ejpam-6576	37	6	a	a	DET
ejpam-6576	37	7	distributive	distributive	ADJ
ejpam-6576	37	8	element	element	NOUN
ejpam-6576	37	9	of	of	ADP
ejpam-6576	37	10	the	the	DET
ejpam-6576	37	11	right	right	ADJ
ejpam-6576	37	12	n	n	NUM
ejpam-6576	37	13	-group	-group	NOUN
ejpam-6576	37	14	t	t	NOUN
ejpam-6576	37	15	if	if	SCONJ
ejpam-6576	37	16	t(a	t(a	PROPN
ejpam-6576	37	17	+	+	NOUN
ejpam-6576	37	18	b	b	X
ejpam-6576	37	19	)	)	PUNCT
ejpam-6576	37	20	=	=	SYM
ejpam-6576	37	21	ta	ta	X
ejpam-6576	37	22	+	+	CCONJ
ejpam-6576	37	23	tb	tb	X
ejpam-6576	37	24	for	for	ADP
ejpam-6576	37	25	all	all	DET
ejpam-6576	37	26	a	a	DET
ejpam-6576	37	27	,	,	PUNCT
ejpam-6576	37	28	b	b	X
ejpam-6576	37	29	∈	∈	PROPN
ejpam-6576	37	30	n	n	NOUN
ejpam-6576	37	31	.	.	PUNCT
ejpam-6576	38	1	2	2	X
ejpam-6576	38	2	.	.	X
ejpam-6576	38	3	completely	completely	ADV
ejpam-6576	38	4	prime	prime	ADJ
ejpam-6576	38	5	right	right	ADJ
ejpam-6576	38	6	n	n	CCONJ
ejpam-6576	38	7	-	-	PUNCT
ejpam-6576	38	8	groups	group	NOUN
ejpam-6576	38	9	of	of	ADP
ejpam-6576	38	10	type	type	NOUN
ejpam-6576	38	11	r(1	r(1	PROPN
ejpam-6576	38	12	)	)	PUNCT
ejpam-6576	38	13	unless	unless	SCONJ
ejpam-6576	38	14	or	or	CCONJ
ejpam-6576	38	15	otherwise	otherwise	ADV
ejpam-6576	38	16	specified	specify	VERB
ejpam-6576	38	17	near	near	ADJ
ejpam-6576	38	18	-	-	PUNCT
ejpam-6576	38	19	rings	ring	NOUN
ejpam-6576	38	20	considered	consider	VERB
ejpam-6576	38	21	are	be	AUX
ejpam-6576	38	22	zero	zero	NUM
ejpam-6576	38	23	-	-	PUNCT
ejpam-6576	38	24	symmetric	symmetric	ADJ
ejpam-6576	38	25	right	right	ADJ
ejpam-6576	38	26	near	near	NOUN
ejpam-6576	38	27	-	-	PUNCT
ejpam-6576	38	28	rings	ring	NOUN
ejpam-6576	38	29	.	.	PUNCT
ejpam-6576	39	1	definition	definition	NOUN
ejpam-6576	39	2	1	1	NUM
ejpam-6576	39	3	.	.	PUNCT
ejpam-6576	40	1	a	a	DET
ejpam-6576	40	2	right	right	ADJ
ejpam-6576	40	3	n	n	ADP
ejpam-6576	40	4	-group	-group	NOUN
ejpam-6576	40	5	g	g	NOUN
ejpam-6576	40	6	with	with	ADP
ejpam-6576	40	7	gn	gn	PROPN
ejpam-6576	40	8	̸=	̸=	PROPN
ejpam-6576	40	9	{	{	PUNCT
ejpam-6576	40	10	0	0	NUM
ejpam-6576	40	11	}	}	PUNCT
ejpam-6576	40	12	and	and	CCONJ
ejpam-6576	40	13	g0	g0	NOUN
ejpam-6576	40	14	=	=	SYM
ejpam-6576	40	15	{	{	PUNCT
ejpam-6576	40	16	0	0	NUM
ejpam-6576	40	17	}	}	PUNCT
ejpam-6576	40	18	is	be	AUX
ejpam-6576	40	19	called	call	VERB
ejpam-6576	40	20	a	a	DET
ejpam-6576	40	21	completely	completely	ADV
ejpam-6576	40	22	prime	prime	ADJ
ejpam-6576	40	23	n	n	PRON
ejpam-6576	40	24	-group	-group	NOUN
ejpam-6576	40	25	of	of	ADP
ejpam-6576	40	26	type	type	NOUN
ejpam-6576	40	27	r(1	r(1	PROPN
ejpam-6576	40	28	)	)	PUNCT
ejpam-6576	41	1	if	if	SCONJ
ejpam-6576	41	2	(	(	PUNCT
ejpam-6576	41	3	i	i	NOUN
ejpam-6576	41	4	)	)	PUNCT
ejpam-6576	41	5	every	every	DET
ejpam-6576	41	6	non	non	ADJ
ejpam-6576	41	7	-	-	ADJ
ejpam-6576	41	8	zero	zero	NUM
ejpam-6576	41	9	right	right	NOUN
ejpam-6576	41	10	n	n	PRON
ejpam-6576	41	11	-subgroup	-subgroup	NOUN
ejpam-6576	41	12	of	of	ADP
ejpam-6576	41	13	g	g	PROPN
ejpam-6576	41	14	contains	contain	VERB
ejpam-6576	41	15	a	a	DET
ejpam-6576	41	16	distributive	distributive	ADJ
ejpam-6576	41	17	element	element	NOUN
ejpam-6576	41	18	g0(̸=	g0(̸=	PROPN
ejpam-6576	41	19	0	0	NUM
ejpam-6576	41	20	)	)	PUNCT
ejpam-6576	41	21	and	and	CCONJ
ejpam-6576	41	22	;	;	PUNCT
ejpam-6576	41	23	(	(	PUNCT
ejpam-6576	41	24	ii	ii	NOUN
ejpam-6576	41	25	)	)	PUNCT
ejpam-6576	41	26	gr	gr	NOUN
ejpam-6576	41	27	=	=	SYM
ejpam-6576	41	28	0	0	NUM
ejpam-6576	41	29	implies	imply	VERB
ejpam-6576	41	30	either	either	CCONJ
ejpam-6576	41	31	g	g	PROPN
ejpam-6576	41	32	=	=	SYM
ejpam-6576	41	33	0	0	NUM
ejpam-6576	41	34	or	or	CCONJ
ejpam-6576	41	35	gr	gr	NOUN
ejpam-6576	41	36	=	=	SYM
ejpam-6576	41	37	{	{	PUNCT
ejpam-6576	41	38	0	0	NUM
ejpam-6576	41	39	}	}	PUNCT
ejpam-6576	41	40	for	for	ADP
ejpam-6576	41	41	all	all	PRON
ejpam-6576	41	42	g	g	PROPN
ejpam-6576	41	43	∈	∈	PROPN
ejpam-6576	41	44	g	g	NOUN
ejpam-6576	41	45	,	,	PUNCT
ejpam-6576	41	46	r	r	PROPN
ejpam-6576	41	47	∈	∈	PROPN
ejpam-6576	41	48	r.	r.	NOUN
ejpam-6576	41	49	remark	remark	NOUN
ejpam-6576	41	50	1	1	NUM
ejpam-6576	41	51	.	.	PUNCT
ejpam-6576	42	1	let	let	VERB
ejpam-6576	42	2	r	r	PRON
ejpam-6576	42	3	be	be	AUX
ejpam-6576	42	4	a	a	DET
ejpam-6576	42	5	ring	ring	NOUN
ejpam-6576	42	6	and	and	CCONJ
ejpam-6576	42	7	m	m	AUX
ejpam-6576	42	8	be	be	AUX
ejpam-6576	42	9	a	a	DET
ejpam-6576	42	10	completely	completely	ADV
ejpam-6576	42	11	prime	prime	ADJ
ejpam-6576	42	12	(	(	PUNCT
ejpam-6576	42	13	right	right	ADJ
ejpam-6576	42	14	)	)	PUNCT
ejpam-6576	42	15	r	r	NOUN
ejpam-6576	42	16	-	-	PUNCT
ejpam-6576	42	17	module	module	NOUN
ejpam-6576	42	18	.	.	PUNCT
ejpam-6576	43	1	then	then	ADV
ejpam-6576	43	2	m	m	PROPN
ejpam-6576	43	3	is	be	AUX
ejpam-6576	43	4	also	also	ADV
ejpam-6576	43	5	a	a	DET
ejpam-6576	43	6	completely	completely	ADV
ejpam-6576	43	7	prime	prime	ADJ
ejpam-6576	43	8	right	right	ADJ
ejpam-6576	43	9	r	r	NOUN
ejpam-6576	43	10	-	-	NOUN
ejpam-6576	43	11	group	group	NOUN
ejpam-6576	43	12	of	of	ADP
ejpam-6576	43	13	type	type	NOUN
ejpam-6576	43	14	r(1	r(1	PROPN
ejpam-6576	43	15	)	)	PUNCT
ejpam-6576	43	16	,	,	PUNCT
ejpam-6576	43	17	when	when	SCONJ
ejpam-6576	43	18	r	r	NOUN
ejpam-6576	43	19	is	be	AUX
ejpam-6576	43	20	considered	consider	VERB
ejpam-6576	43	21	as	as	ADP
ejpam-6576	43	22	a	a	DET
ejpam-6576	43	23	near	near	ADV
ejpam-6576	43	24	-	-	PUNCT
ejpam-6576	43	25	ring	ring	NOUN
ejpam-6576	43	26	.	.	PUNCT
ejpam-6576	43	27	example	example	NOUN
ejpam-6576	44	1	1	1	NUM
ejpam-6576	44	2	.	.	PUNCT
ejpam-6576	45	1	let	let	VERB
ejpam-6576	45	2	n	n	PRON
ejpam-6576	45	3	be	be	AUX
ejpam-6576	45	4	a	a	DET
ejpam-6576	45	5	near	near	ADJ
ejpam-6576	45	6	-	-	PUNCT
ejpam-6576	45	7	field	field	NOUN
ejpam-6576	45	8	.	.	PUNCT
ejpam-6576	46	1	it	it	PRON
ejpam-6576	46	2	is	be	AUX
ejpam-6576	46	3	clear	clear	ADJ
ejpam-6576	46	4	that	that	SCONJ
ejpam-6576	46	5	n	n	X
ejpam-6576	46	6	is	be	AUX
ejpam-6576	46	7	a	a	DET
ejpam-6576	46	8	completely	completely	ADV
ejpam-6576	46	9	prime	prime	ADJ
ejpam-6576	46	10	right	right	NOUN
ejpam-6576	46	11	n	n	PRON
ejpam-6576	46	12	-group	-group	NOUN
ejpam-6576	46	13	of	of	ADP
ejpam-6576	46	14	type	type	NOUN
ejpam-6576	46	15	r(1	r(1	PROPN
ejpam-6576	46	16	)	)	PUNCT
ejpam-6576	46	17	.	.	PUNCT
ejpam-6576	47	1	the	the	DET
ejpam-6576	47	2	following	follow	VERB
ejpam-6576	47	3	proposition	proposition	NOUN
ejpam-6576	47	4	is	be	AUX
ejpam-6576	47	5	obvious	obvious	ADJ
ejpam-6576	47	6	in	in	ADP
ejpam-6576	47	7	view	view	NOUN
ejpam-6576	47	8	of	of	ADP
ejpam-6576	47	9	the	the	DET
ejpam-6576	47	10	above	above	ADJ
ejpam-6576	47	11	definition	definition	NOUN
ejpam-6576	47	12	.	.	PUNCT
ejpam-6576	48	1	k.	k.	PROPN
ejpam-6576	49	1	j.	j.	PROPN
ejpam-6576	49	2	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	49	3	et	et	PROPN
ejpam-6576	50	1	al	al	PROPN
ejpam-6576	50	2	.	.	PUNCT
ejpam-6576	50	3	/	/	SYM
ejpam-6576	50	4	eur	eur	PROPN
ejpam-6576	50	5	.	.	PUNCT
ejpam-6576	51	1	j.	j.	PROPN
ejpam-6576	51	2	pure	pure	PROPN
ejpam-6576	51	3	appl	appl	PROPN
ejpam-6576	51	4	.	.	PROPN
ejpam-6576	51	5	math	math	PROPN
ejpam-6576	51	6	,	,	PUNCT
ejpam-6576	51	7	18	18	NUM
ejpam-6576	51	8	(	(	PUNCT
ejpam-6576	51	9	3	3	NUM
ejpam-6576	51	10	)	)	PUNCT
ejpam-6576	51	11	(	(	PUNCT
ejpam-6576	51	12	2025	2025	NUM
ejpam-6576	51	13	)	)	PUNCT
ejpam-6576	51	14	,	,	PUNCT
ejpam-6576	51	15	6576	6576	NUM
ejpam-6576	51	16	3	3	NUM
ejpam-6576	51	17	of	of	ADP
ejpam-6576	51	18	7	7	NUM
ejpam-6576	51	19	proposition	proposition	NOUN
ejpam-6576	51	20	1	1	NUM
ejpam-6576	51	21	.	.	PUNCT
ejpam-6576	52	1	let	let	VERB
ejpam-6576	52	2	g	g	PRON
ejpam-6576	52	3	be	be	AUX
ejpam-6576	52	4	right	right	ADJ
ejpam-6576	52	5	n	n	DET
ejpam-6576	52	6	-group	-group	NOUN
ejpam-6576	52	7	with	with	ADP
ejpam-6576	52	8	gn	gn	PROPN
ejpam-6576	52	9	̸=	̸=	PROPN
ejpam-6576	52	10	{	{	PUNCT
ejpam-6576	52	11	0	0	NUM
ejpam-6576	52	12	}	}	PUNCT
ejpam-6576	52	13	and	and	CCONJ
ejpam-6576	52	14	each	each	DET
ejpam-6576	52	15	non	non	ADJ
ejpam-6576	52	16	-	-	ADJ
ejpam-6576	52	17	zero	zero	NUM
ejpam-6576	52	18	n	n	PRON
ejpam-6576	52	19	-subgroup	-subgroup	NOUN
ejpam-6576	52	20	of	of	ADP
ejpam-6576	52	21	g	g	PROPN
ejpam-6576	52	22	has	have	VERB
ejpam-6576	52	23	a	a	DET
ejpam-6576	52	24	non	non	ADJ
ejpam-6576	52	25	-	-	ADJ
ejpam-6576	52	26	zero	zero	NUM
ejpam-6576	52	27	distributive	distributive	ADJ
ejpam-6576	52	28	element	element	NOUN
ejpam-6576	52	29	.	.	PUNCT
ejpam-6576	53	1	then	then	ADV
ejpam-6576	53	2	g	g	PROPN
ejpam-6576	53	3	is	be	AUX
ejpam-6576	53	4	a	a	DET
ejpam-6576	53	5	completely	completely	ADV
ejpam-6576	53	6	prime	prime	ADJ
ejpam-6576	53	7	n	n	DET
ejpam-6576	53	8	-group	-group	NOUN
ejpam-6576	53	9	of	of	ADP
ejpam-6576	53	10	type	type	NOUN
ejpam-6576	53	11	r(1	r(1	PROPN
ejpam-6576	53	12	)	)	PUNCT
ejpam-6576	54	1	if	if	SCONJ
ejpam-6576	54	2	and	and	CCONJ
ejpam-6576	54	3	only	only	ADV
ejpam-6576	54	4	if	if	SCONJ
ejpam-6576	54	5	(	(	PUNCT
ejpam-6576	54	6	g	g	NOUN
ejpam-6576	54	7	:	:	PUNCT
ejpam-6576	54	8	0	0	NUM
ejpam-6576	54	9	)	)	PUNCT
ejpam-6576	54	10	=	=	SYM
ejpam-6576	54	11	(	(	PUNCT
ejpam-6576	54	12	g	g	NOUN
ejpam-6576	54	13	:	:	PUNCT
ejpam-6576	54	14	0	0	NUM
ejpam-6576	54	15	)	)	PUNCT
ejpam-6576	54	16	for	for	ADP
ejpam-6576	54	17	all	all	PRON
ejpam-6576	54	18	0	0	NUM
ejpam-6576	55	1	̸=	̸=	PROPN
ejpam-6576	55	2	g	g	PROPN
ejpam-6576	55	3	∈	∈	PROPN
ejpam-6576	55	4	g.	g.	NOUN
ejpam-6576	55	5	proof	proof	NOUN
ejpam-6576	55	6	.	.	PUNCT
ejpam-6576	56	1	suppose	suppose	VERB
ejpam-6576	56	2	that	that	SCONJ
ejpam-6576	56	3	g	g	PROPN
ejpam-6576	56	4	completely	completely	ADV
ejpam-6576	56	5	prime	prime	ADJ
ejpam-6576	56	6	right	right	ADJ
ejpam-6576	56	7	n	n	PRON
ejpam-6576	56	8	-group	-group	NOUN
ejpam-6576	56	9	of	of	ADP
ejpam-6576	56	10	type	type	NOUN
ejpam-6576	56	11	r(1	r(1	PROPN
ejpam-6576	56	12	)	)	PUNCT
ejpam-6576	56	13	.	.	PUNCT
ejpam-6576	57	1	let	let	VERB
ejpam-6576	57	2	0	0	NUM
ejpam-6576	57	3	̸=	̸=	PROPN
ejpam-6576	57	4	g	g	ADP
ejpam-6576	57	5	∈	∈	PROPN
ejpam-6576	57	6	g	g	PROPN
ejpam-6576	57	7	and	and	CCONJ
ejpam-6576	57	8	r	r	NOUN
ejpam-6576	57	9	∈	∈	PROPN
ejpam-6576	57	10	n	n	NOUN
ejpam-6576	57	11	and	and	CCONJ
ejpam-6576	57	12	gr	gr	NOUN
ejpam-6576	57	13	=	=	NOUN
ejpam-6576	57	14	0	0	NUM
ejpam-6576	57	15	.	.	PUNCT
ejpam-6576	58	1	by	by	ADP
ejpam-6576	58	2	definition	definition	NOUN
ejpam-6576	58	3	of	of	ADP
ejpam-6576	58	4	completely	completely	ADV
ejpam-6576	58	5	prime	prime	ADJ
ejpam-6576	58	6	right	right	NOUN
ejpam-6576	58	7	n	n	PRON
ejpam-6576	58	8	-group	-group	NOUN
ejpam-6576	58	9	of	of	ADP
ejpam-6576	58	10	type	type	NOUN
ejpam-6576	58	11	r(1	r(1	PROPN
ejpam-6576	58	12	)	)	PUNCT
ejpam-6576	58	13	,	,	PUNCT
ejpam-6576	58	14	gr	gr	NOUN
ejpam-6576	58	15	=	=	SYM
ejpam-6576	58	16	{	{	PUNCT
ejpam-6576	58	17	0	0	NUM
ejpam-6576	58	18	}	}	PUNCT
ejpam-6576	58	19	.	.	PUNCT
ejpam-6576	59	1	so	so	ADV
ejpam-6576	59	2	(	(	PUNCT
ejpam-6576	59	3	g	g	NOUN
ejpam-6576	59	4	:	:	PUNCT
ejpam-6576	59	5	0	0	NUM
ejpam-6576	59	6	)	)	PUNCT
ejpam-6576	59	7	⊆	⊆	NUM
ejpam-6576	59	8	(	(	PUNCT
ejpam-6576	59	9	g	g	NOUN
ejpam-6576	59	10	:	:	PUNCT
ejpam-6576	59	11	0	0	NUM
ejpam-6576	59	12	)	)	PUNCT
ejpam-6576	59	13	.	.	PUNCT
ejpam-6576	60	1	since	since	SCONJ
ejpam-6576	60	2	(	(	PUNCT
ejpam-6576	60	3	g	g	NOUN
ejpam-6576	60	4	:	:	PUNCT
ejpam-6576	60	5	0	0	NUM
ejpam-6576	60	6	)	)	PUNCT
ejpam-6576	60	7	⊆	⊆	NUM
ejpam-6576	60	8	(	(	PUNCT
ejpam-6576	60	9	g	g	NOUN
ejpam-6576	60	10	:	:	PUNCT
ejpam-6576	60	11	0	0	NUM
ejpam-6576	60	12	)	)	PUNCT
ejpam-6576	60	13	,	,	PUNCT
ejpam-6576	60	14	we	we	PRON
ejpam-6576	60	15	have	have	VERB
ejpam-6576	60	16	(	(	PUNCT
ejpam-6576	60	17	g	g	NOUN
ejpam-6576	60	18	:	:	PUNCT
ejpam-6576	60	19	0	0	NUM
ejpam-6576	60	20	)	)	PUNCT
ejpam-6576	60	21	=	=	SYM
ejpam-6576	60	22	(	(	PUNCT
ejpam-6576	60	23	g	g	NOUN
ejpam-6576	60	24	:	:	PUNCT
ejpam-6576	60	25	0	0	NUM
ejpam-6576	60	26	)	)	PUNCT
ejpam-6576	60	27	.	.	PUNCT
ejpam-6576	61	1	conversely	conversely	ADV
ejpam-6576	61	2	suppose	suppose	VERB
ejpam-6576	61	3	that	that	SCONJ
ejpam-6576	61	4	(	(	PUNCT
ejpam-6576	61	5	g	g	NOUN
ejpam-6576	61	6	:	:	PUNCT
ejpam-6576	61	7	0	0	NUM
ejpam-6576	61	8	)	)	PUNCT
ejpam-6576	61	9	=	=	SYM
ejpam-6576	61	10	(	(	PUNCT
ejpam-6576	61	11	g	g	NOUN
ejpam-6576	61	12	:	:	PUNCT
ejpam-6576	61	13	0	0	NUM
ejpam-6576	61	14	)	)	PUNCT
ejpam-6576	61	15	for	for	ADP
ejpam-6576	61	16	all	all	PRON
ejpam-6576	61	17	0	0	NUM
ejpam-6576	61	18	̸=	̸=	PROPN
ejpam-6576	61	19	g	g	PROPN
ejpam-6576	61	20	∈	∈	PROPN
ejpam-6576	61	21	g.	g.	NOUN
ejpam-6576	61	22	let	let	VERB
ejpam-6576	61	23	0	0	NUM
ejpam-6576	61	24	̸=	̸=	PROPN
ejpam-6576	61	25	g	g	PROPN
ejpam-6576	61	26	∈	∈	PROPN
ejpam-6576	61	27	g	g	PROPN
ejpam-6576	61	28	,	,	PUNCT
ejpam-6576	61	29	r	r	NOUN
ejpam-6576	61	30	∈	∈	PROPN
ejpam-6576	61	31	n	n	NOUN
ejpam-6576	61	32	and	and	CCONJ
ejpam-6576	61	33	gr	gr	NOUN
ejpam-6576	61	34	=	=	NOUN
ejpam-6576	61	35	0	0	NUM
ejpam-6576	61	36	.	.	PUNCT
ejpam-6576	62	1	by	by	ADP
ejpam-6576	62	2	assumption	assumption	NOUN
ejpam-6576	62	3	,	,	PUNCT
ejpam-6576	62	4	gr	gr	NOUN
ejpam-6576	62	5	=	=	SYM
ejpam-6576	62	6	{	{	PUNCT
ejpam-6576	62	7	0	0	NUM
ejpam-6576	62	8	}	}	PUNCT
ejpam-6576	62	9	.	.	PUNCT
ejpam-6576	63	1	so	so	ADV
ejpam-6576	63	2	g	g	PROPN
ejpam-6576	63	3	completely	completely	ADV
ejpam-6576	63	4	prime	prime	ADJ
ejpam-6576	63	5	right	right	ADJ
ejpam-6576	63	6	n	n	PRON
ejpam-6576	63	7	-group	-group	NOUN
ejpam-6576	63	8	of	of	ADP
ejpam-6576	63	9	type	type	NOUN
ejpam-6576	63	10	r(1	r(1	PROPN
ejpam-6576	63	11	)	)	PUNCT
ejpam-6576	63	12	.	.	PUNCT
ejpam-6576	64	1	proposition	proposition	NOUN
ejpam-6576	64	2	2	2	NUM
ejpam-6576	64	3	.	.	PUNCT
ejpam-6576	65	1	let	let	VERB
ejpam-6576	65	2	g	g	PRON
ejpam-6576	65	3	be	be	AUX
ejpam-6576	65	4	a	a	DET
ejpam-6576	65	5	completely	completely	ADV
ejpam-6576	65	6	prime	prime	ADJ
ejpam-6576	65	7	right	right	NOUN
ejpam-6576	65	8	n	n	PRON
ejpam-6576	65	9	-group	-group	NOUN
ejpam-6576	65	10	of	of	ADP
ejpam-6576	65	11	type	type	NOUN
ejpam-6576	65	12	r(1	r(1	PROPN
ejpam-6576	65	13	)	)	PUNCT
ejpam-6576	66	1	and	and	CCONJ
ejpam-6576	66	2	h	h	NOUN
ejpam-6576	66	3	be	be	VERB
ejpam-6576	66	4	n	n	PRON
ejpam-6576	66	5	subgroup	subgroup	NOUN
ejpam-6576	66	6	of	of	ADP
ejpam-6576	66	7	g.	g.	PROPN
ejpam-6576	67	1	then	then	ADV
ejpam-6576	67	2	h	h	PROPN
ejpam-6576	67	3	is	be	AUX
ejpam-6576	67	4	a	a	DET
ejpam-6576	67	5	completely	completely	ADV
ejpam-6576	67	6	prime	prime	ADJ
ejpam-6576	67	7	right	right	NOUN
ejpam-6576	67	8	n	n	PRON
ejpam-6576	67	9	-group	-group	NOUN
ejpam-6576	67	10	of	of	ADP
ejpam-6576	67	11	type	type	NOUN
ejpam-6576	67	12	r(1	r(1	PROPN
ejpam-6576	67	13	)	)	PUNCT
ejpam-6576	67	14	.	.	PUNCT
ejpam-6576	68	1	proof	proof	NOUN
ejpam-6576	68	2	.	.	PUNCT
ejpam-6576	69	1	this	this	PRON
ejpam-6576	69	2	follows	follow	VERB
ejpam-6576	69	3	from	from	ADP
ejpam-6576	69	4	the	the	DET
ejpam-6576	69	5	definition	definition	NOUN
ejpam-6576	69	6	of	of	ADP
ejpam-6576	69	7	completely	completely	ADV
ejpam-6576	69	8	prime	prime	ADJ
ejpam-6576	69	9	right	right	NOUN
ejpam-6576	69	10	n	n	NUM
ejpam-6576	69	11	-group	-group	NOUN
ejpam-6576	69	12	.	.	PUNCT
ejpam-6576	70	1	proposition	proposition	NOUN
ejpam-6576	70	2	3	3	X
ejpam-6576	70	3	.	.	PUNCT
ejpam-6576	71	1	let	let	VERB
ejpam-6576	71	2	g	g	PRON
ejpam-6576	71	3	be	be	AUX
ejpam-6576	71	4	a	a	DET
ejpam-6576	71	5	completely	completely	ADV
ejpam-6576	71	6	prime	prime	ADJ
ejpam-6576	71	7	right	right	NOUN
ejpam-6576	71	8	n	n	PRON
ejpam-6576	71	9	-group	-group	NOUN
ejpam-6576	71	10	of	of	ADP
ejpam-6576	71	11	type	type	NOUN
ejpam-6576	71	12	r(1	r(1	PROPN
ejpam-6576	71	13	)	)	PUNCT
ejpam-6576	71	14	.	.	PUNCT
ejpam-6576	72	1	if	if	SCONJ
ejpam-6576	72	2	n	n	PRON
ejpam-6576	72	3	is	be	AUX
ejpam-6576	72	4	a	a	DET
ejpam-6576	72	5	ring	ring	NOUN
ejpam-6576	72	6	,	,	PUNCT
ejpam-6576	72	7	then	then	ADV
ejpam-6576	72	8	g	g	PROPN
ejpam-6576	72	9	is	be	AUX
ejpam-6576	72	10	a	a	DET
ejpam-6576	72	11	completely	completely	ADV
ejpam-6576	72	12	prime	prime	ADJ
ejpam-6576	72	13	ring	ring	NOUN
ejpam-6576	72	14	n	n	PRON
ejpam-6576	72	15	-module	-module	NOUN
ejpam-6576	72	16	.	.	PUNCT
ejpam-6576	73	1	proof	proof	NOUN
ejpam-6576	73	2	.	.	PUNCT
ejpam-6576	74	1	obvious	obvious	ADJ
ejpam-6576	74	2	from	from	ADP
ejpam-6576	74	3	the	the	DET
ejpam-6576	74	4	definition	definition	NOUN
ejpam-6576	74	5	of	of	ADP
ejpam-6576	74	6	the	the	DET
ejpam-6576	74	7	completely	completely	ADV
ejpam-6576	74	8	prime	prime	ADJ
ejpam-6576	74	9	right	right	NOUN
ejpam-6576	74	10	n	n	NUM
ejpam-6576	74	11	-group	-group	NOUN
ejpam-6576	74	12	.	.	PUNCT
ejpam-6576	75	1	proposition	proposition	NOUN
ejpam-6576	75	2	4	4	NUM
ejpam-6576	75	3	.	.	PUNCT
ejpam-6576	76	1	let	let	VERB
ejpam-6576	76	2	g	g	PRON
ejpam-6576	76	3	be	be	AUX
ejpam-6576	76	4	a	a	DET
ejpam-6576	76	5	completely	completely	ADV
ejpam-6576	76	6	prime	prime	ADJ
ejpam-6576	76	7	right	right	NOUN
ejpam-6576	76	8	n	n	PRON
ejpam-6576	76	9	-group	-group	NOUN
ejpam-6576	76	10	of	of	ADP
ejpam-6576	76	11	type	type	NOUN
ejpam-6576	76	12	r(1	r(1	PROPN
ejpam-6576	76	13	)	)	PUNCT
ejpam-6576	77	1	and	and	CCONJ
ejpam-6576	77	2	i	i	PRON
ejpam-6576	77	3	be	be	VERB
ejpam-6576	77	4	an	an	DET
ejpam-6576	77	5	ideal	ideal	NOUN
ejpam-6576	77	6	of	of	ADP
ejpam-6576	77	7	n	n	NOUN
ejpam-6576	77	8	and	and	CCONJ
ejpam-6576	77	9	gi	gi	VERB
ejpam-6576	77	10	=	=	PUNCT
ejpam-6576	78	1	{	{	PUNCT
ejpam-6576	78	2	0}.then	0}.then	ADV
ejpam-6576	78	3	g	g	PROPN
ejpam-6576	78	4	has	have	VERB
ejpam-6576	78	5	a	a	DET
ejpam-6576	78	6	right	right	ADJ
ejpam-6576	78	7	n	n	CCONJ
ejpam-6576	78	8	-subgroup	-subgroup	NOUN
ejpam-6576	78	9	h	h	NOUN
ejpam-6576	78	10	which	which	PRON
ejpam-6576	78	11	is	be	AUX
ejpam-6576	78	12	a	a	DET
ejpam-6576	78	13	completely	completely	ADV
ejpam-6576	78	14	prime	prime	ADJ
ejpam-6576	78	15	n	n	CCONJ
ejpam-6576	78	16	/	/	SYM
ejpam-6576	78	17	i	i	PROPN
ejpam-6576	78	18	-	-	PUNCT
ejpam-6576	78	19	group	group	NOUN
ejpam-6576	78	20	of	of	ADP
ejpam-6576	78	21	type	type	NOUN
ejpam-6576	78	22	r(1	r(1	PROPN
ejpam-6576	78	23	)	)	PUNCT
ejpam-6576	78	24	.	.	PUNCT
ejpam-6576	79	1	proof	proof	NOUN
ejpam-6576	79	2	.	.	PUNCT
ejpam-6576	80	1	let	let	VERB
ejpam-6576	80	2	g0	g0	PROPN
ejpam-6576	80	3	be	be	AUX
ejpam-6576	80	4	a	a	DET
ejpam-6576	80	5	distributive	distributive	ADJ
ejpam-6576	80	6	element	element	NOUN
ejpam-6576	80	7	of	of	ADP
ejpam-6576	80	8	the	the	DET
ejpam-6576	80	9	right	right	ADJ
ejpam-6576	80	10	n	n	NOUN
ejpam-6576	80	11	-group	-group	NOUN
ejpam-6576	80	12	g.	g.	NOUN
ejpam-6576	80	13	consider	consider	VERB
ejpam-6576	80	14	g0n	g0n	NOUN
ejpam-6576	80	15	:	:	PUNCT
ejpam-6576	80	16	=	=	SYM
ejpam-6576	80	17	{	{	PUNCT
ejpam-6576	80	18	g0a	g0a	ADJ
ejpam-6576	80	19	|	|	ADV
ejpam-6576	80	20	a	a	DET
ejpam-6576	80	21	∈	∈	PROPN
ejpam-6576	80	22	n	n	CCONJ
ejpam-6576	80	23	}	}	PUNCT
ejpam-6576	80	24	.	.	PUNCT
ejpam-6576	81	1	for	for	ADP
ejpam-6576	81	2	g1	g1	NOUN
ejpam-6576	81	3	,	,	PUNCT
ejpam-6576	81	4	g2	g2	PROPN
ejpam-6576	81	5	∈	∈	PROPN
ejpam-6576	81	6	g0n	g0n	NOUN
ejpam-6576	81	7	,	,	PUNCT
ejpam-6576	81	8	g1	g1	PROPN
ejpam-6576	81	9	=	=	SYM
ejpam-6576	81	10	g0b	g0b	PROPN
ejpam-6576	81	11	and	and	CCONJ
ejpam-6576	81	12	g2	g2	PROPN
ejpam-6576	81	13	=	=	PUNCT
ejpam-6576	81	14	g0c	g0c	VERB
ejpam-6576	81	15	for	for	ADP
ejpam-6576	81	16	some	some	DET
ejpam-6576	81	17	b	b	NOUN
ejpam-6576	81	18	,	,	PUNCT
ejpam-6576	81	19	c	c	PROPN
ejpam-6576	81	20	∈	∈	PROPN
ejpam-6576	81	21	n	n	ADV
ejpam-6576	81	22	.	.	PUNCT
ejpam-6576	82	1	now	now	ADV
ejpam-6576	82	2	g1	g1	VERB
ejpam-6576	82	3	−	−	PROPN
ejpam-6576	82	4	g2	g2	PROPN
ejpam-6576	82	5	=	=	PUNCT
ejpam-6576	82	6	g0(b−	g0(b−	NOUN
ejpam-6576	82	7	c	c	NOUN
ejpam-6576	82	8	)	)	PUNCT
ejpam-6576	82	9	∈	∈	PROPN
ejpam-6576	82	10	g0n	g0n	NOUN
ejpam-6576	82	11	.	.	PUNCT
ejpam-6576	83	1	so	so	ADV
ejpam-6576	83	2	g0n	g0n	NOUN
ejpam-6576	83	3	is	be	AUX
ejpam-6576	83	4	a	a	DET
ejpam-6576	83	5	subgroup	subgroup	NOUN
ejpam-6576	83	6	of	of	ADP
ejpam-6576	83	7	(	(	PUNCT
ejpam-6576	83	8	g,+	g,+	PROPN
ejpam-6576	83	9	)	)	PUNCT
ejpam-6576	83	10	.	.	PUNCT
ejpam-6576	84	1	since	since	SCONJ
ejpam-6576	84	2	(	(	PUNCT
ejpam-6576	84	3	g0n)n	g0n)n	PROPN
ejpam-6576	84	4	⊆	⊆	NUM
ejpam-6576	84	5	g0(nn	g0(nn	PROPN
ejpam-6576	84	6	)	)	PUNCT
ejpam-6576	84	7	⊆	⊆	NUM
ejpam-6576	84	8	g0n	g0n	NOUN
ejpam-6576	84	9	,	,	PUNCT
ejpam-6576	84	10	g0n	g0n	NOUN
ejpam-6576	84	11	is	be	AUX
ejpam-6576	84	12	a	a	DET
ejpam-6576	84	13	right	right	ADJ
ejpam-6576	84	14	n	n	PRON
ejpam-6576	84	15	-subgroup	-subgroup	NOUN
ejpam-6576	84	16	of	of	ADP
ejpam-6576	84	17	g.	g.	PROPN
ejpam-6576	84	18	we	we	PRON
ejpam-6576	84	19	claim	claim	VERB
ejpam-6576	84	20	that	that	SCONJ
ejpam-6576	84	21	g0n	g0n	NOUN
ejpam-6576	84	22	is	be	AUX
ejpam-6576	84	23	a	a	DET
ejpam-6576	84	24	right	right	ADJ
ejpam-6576	84	25	n	n	CCONJ
ejpam-6576	84	26	/	/	SYM
ejpam-6576	84	27	i	i	PROPN
ejpam-6576	84	28	-	-	PUNCT
ejpam-6576	84	29	group	group	NOUN
ejpam-6576	84	30	,	,	PUNCT
ejpam-6576	84	31	under	under	ADP
ejpam-6576	84	32	g(x+	g(x+	ADV
ejpam-6576	84	33	i	i	NOUN
ejpam-6576	84	34	)	)	PUNCT
ejpam-6576	84	35	=	=	SYM
ejpam-6576	84	36	gx	gx	PROPN
ejpam-6576	84	37	,	,	PUNCT
ejpam-6576	84	38	g	g	PROPN
ejpam-6576	84	39	∈	∈	PROPN
ejpam-6576	84	40	g0n	g0n	NOUN
ejpam-6576	84	41	and	and	CCONJ
ejpam-6576	84	42	x+	x+	NUM
ejpam-6576	84	43	i	i	PROPN
ejpam-6576	84	44	∈	∈	PROPN
ejpam-6576	84	45	n	n	CCONJ
ejpam-6576	84	46	/	/	SYM
ejpam-6576	84	47	i.	i.	NOUN
ejpam-6576	84	48	let	let	VERB
ejpam-6576	84	49	g	g	NOUN
ejpam-6576	84	50	:	:	PUNCT
ejpam-6576	84	51	=	=	SYM
ejpam-6576	84	52	g0a	g0a	PROPN
ejpam-6576	84	53	∈	∈	PROPN
ejpam-6576	84	54	g0n	g0n	NOUN
ejpam-6576	84	55	and	and	CCONJ
ejpam-6576	84	56	x+	x+	X
ejpam-6576	84	57	i	i	INTJ
ejpam-6576	84	58	,	,	PUNCT
ejpam-6576	84	59	y+	y+	PROPN
ejpam-6576	84	60	i	i	NOUN
ejpam-6576	84	61	∈	∈	PROPN
ejpam-6576	84	62	n	n	CCONJ
ejpam-6576	84	63	/	/	SYM
ejpam-6576	84	64	i	i	PRON
ejpam-6576	84	65	and	and	CCONJ
ejpam-6576	84	66	x+	x+	ADJ
ejpam-6576	85	1	i	i	NOUN
ejpam-6576	85	2	=	=	PUNCT
ejpam-6576	85	3	y+	y+	NUM
ejpam-6576	85	4	i.	i.	NOUN
ejpam-6576	85	5	we	we	PRON
ejpam-6576	85	6	have	have	VERB
ejpam-6576	85	7	x	x	X
ejpam-6576	85	8	−	−	PROPN
ejpam-6576	85	9	y	y	PROPN
ejpam-6576	85	10	∈	∈	PROPN
ejpam-6576	85	11	i.	i.	PROPN
ejpam-6576	85	12	gx	gx	PROPN
ejpam-6576	85	13	=	=	PRON
ejpam-6576	85	14	(	(	PUNCT
ejpam-6576	85	15	g0a)x	g0a)x	X
ejpam-6576	85	16	=	=	SYM
ejpam-6576	85	17	g0(ax	g0(ax	PROPN
ejpam-6576	85	18	)	)	PUNCT
ejpam-6576	85	19	=	=	SYM
ejpam-6576	86	1	g0(a((x	g0(a((x	VERB
ejpam-6576	86	2	−	−	PROPN
ejpam-6576	86	3	y	y	PROPN
ejpam-6576	86	4	)	)	PUNCT
ejpam-6576	87	1	+	+	NOUN
ejpam-6576	87	2	y	y	NOUN
ejpam-6576	87	3	)	)	PUNCT
ejpam-6576	87	4	)	)	PUNCT
ejpam-6576	88	1	−	−	PROPN
ejpam-6576	88	2	g0(ay	g0(ay	PROPN
ejpam-6576	88	3	)	)	PUNCT
ejpam-6576	88	4	+	+	PUNCT
ejpam-6576	89	1	g0(ay	g0(ay	PROPN
ejpam-6576	89	2	)	)	PUNCT
ejpam-6576	89	3	=	=	PUNCT
ejpam-6576	89	4	g0(a((x	g0(a((x	VERB
ejpam-6576	89	5	−	−	PROPN
ejpam-6576	89	6	y	y	PROPN
ejpam-6576	89	7	)	)	PUNCT
ejpam-6576	90	1	+	+	NOUN
ejpam-6576	90	2	y	y	X
ejpam-6576	90	3	)	)	PUNCT
ejpam-6576	90	4	−	−	ADP
ejpam-6576	90	5	ay	ay	NOUN
ejpam-6576	90	6	)	)	PUNCT
ejpam-6576	90	7	+	+	NUM
ejpam-6576	91	1	g0(ay	g0(ay	PROPN
ejpam-6576	91	2	)	)	PUNCT
ejpam-6576	91	3	=	=	SYM
ejpam-6576	91	4	0	0	PUNCT
ejpam-6576	92	1	+	+	CCONJ
ejpam-6576	92	2	(	(	PUNCT
ejpam-6576	92	3	g0a)y	g0a)y	X
ejpam-6576	92	4	)	)	PUNCT
ejpam-6576	92	5	=	=	SYM
ejpam-6576	92	6	gy	gy	PROPN
ejpam-6576	92	7	.	.	PUNCT
ejpam-6576	93	1	so	so	ADV
ejpam-6576	93	2	the	the	DET
ejpam-6576	93	3	operation	operation	NOUN
ejpam-6576	93	4	is	be	AUX
ejpam-6576	93	5	well	well	ADV
ejpam-6576	93	6	defined	define	VERB
ejpam-6576	93	7	and	and	CCONJ
ejpam-6576	93	8	g0n	g0n	NOUN
ejpam-6576	93	9	is	be	AUX
ejpam-6576	93	10	a	a	DET
ejpam-6576	93	11	right	right	ADJ
ejpam-6576	93	12	n	n	CCONJ
ejpam-6576	93	13	/	/	SYM
ejpam-6576	93	14	i	i	PROPN
ejpam-6576	93	15	-	-	NOUN
ejpam-6576	93	16	group	group	NOUN
ejpam-6576	93	17	.	.	PUNCT
ejpam-6576	94	1	it	it	PRON
ejpam-6576	94	2	is	be	AUX
ejpam-6576	94	3	clear	clear	ADJ
ejpam-6576	94	4	that	that	SCONJ
ejpam-6576	94	5	g	g	PROPN
ejpam-6576	94	6	′	′	NUM
ejpam-6576	94	7	∈	∈	PROPN
ejpam-6576	94	8	g0n	g0n	NOUN
ejpam-6576	94	9	is	be	AUX
ejpam-6576	94	10	a	a	DET
ejpam-6576	94	11	distributive	distributive	ADJ
ejpam-6576	94	12	element	element	NOUN
ejpam-6576	94	13	of	of	ADP
ejpam-6576	94	14	the	the	DET
ejpam-6576	94	15	right	right	NOUN
ejpam-6576	94	16	n	n	CCONJ
ejpam-6576	94	17	/	/	SYM
ejpam-6576	94	18	i	i	PROPN
ejpam-6576	94	19	-	-	PUNCT
ejpam-6576	94	20	group	group	NOUN
ejpam-6576	94	21	g0n	g0n	PUNCT
ejpam-6576	94	22	if	if	SCONJ
ejpam-6576	94	23	and	and	CCONJ
ejpam-6576	94	24	only	only	ADV
ejpam-6576	94	25	if	if	SCONJ
ejpam-6576	94	26	g	g	PROPN
ejpam-6576	94	27	′	′	NOUN
ejpam-6576	94	28	is	be	AUX
ejpam-6576	94	29	a	a	DET
ejpam-6576	94	30	distributive	distributive	ADJ
ejpam-6576	94	31	element	element	NOUN
ejpam-6576	94	32	of	of	ADP
ejpam-6576	94	33	the	the	DET
ejpam-6576	94	34	right	right	ADJ
ejpam-6576	94	35	n	n	NUM
ejpam-6576	94	36	-group	-group	NOUN
ejpam-6576	94	37	g.	g.	NOUN
ejpam-6576	94	38	since	since	SCONJ
ejpam-6576	94	39	every	every	DET
ejpam-6576	94	40	right	right	NOUN
ejpam-6576	94	41	n	n	CCONJ
ejpam-6576	94	42	/	/	SYM
ejpam-6576	94	43	i	i	PROPN
ejpam-6576	94	44	-	-	PUNCT
ejpam-6576	94	45	subgroup	subgroup	NOUN
ejpam-6576	94	46	of	of	ADP
ejpam-6576	94	47	g0n	g0n	NOUN
ejpam-6576	94	48	is	be	AUX
ejpam-6576	94	49	an	an	DET
ejpam-6576	94	50	n	n	NUM
ejpam-6576	94	51	subgroup	subgroup	NOUN
ejpam-6576	94	52	of	of	ADP
ejpam-6576	94	53	g	g	PROPN
ejpam-6576	94	54	,	,	PUNCT
ejpam-6576	94	55	every	every	DET
ejpam-6576	94	56	non	non	ADJ
ejpam-6576	94	57	-	-	ADJ
ejpam-6576	94	58	zero	zero	NUM
ejpam-6576	94	59	n	n	CCONJ
ejpam-6576	94	60	/	/	SYM
ejpam-6576	94	61	isubgroup	isubgroup	NOUN
ejpam-6576	94	62	of	of	ADP
ejpam-6576	94	63	g0n	g0n	PROPN
ejpam-6576	94	64	contains	contain	VERB
ejpam-6576	94	65	a	a	DET
ejpam-6576	94	66	non	non	ADJ
ejpam-6576	94	67	-	-	ADJ
ejpam-6576	94	68	zero	zero	NUM
ejpam-6576	94	69	distributive	distributive	ADJ
ejpam-6576	94	70	element	element	NOUN
ejpam-6576	94	71	.	.	PUNCT
ejpam-6576	95	1	let	let	VERB
ejpam-6576	95	2	0	0	NUM
ejpam-6576	95	3	̸=	̸=	PROPN
ejpam-6576	95	4	g3	g3	PROPN
ejpam-6576	95	5	∈	∈	PROPN
ejpam-6576	95	6	g0n	g0n	NOUN
ejpam-6576	95	7	.	.	PUNCT
ejpam-6576	96	1	we	we	PRON
ejpam-6576	96	2	have	have	VERB
ejpam-6576	96	3	(	(	PUNCT
ejpam-6576	97	1	g3	g3	NOUN
ejpam-6576	97	2	:	:	PUNCT
ejpam-6576	97	3	0)n	0)n	X
ejpam-6576	97	4	/	/	SYM
ejpam-6576	97	5	i	i	PRON
ejpam-6576	97	6	=	=	PUNCT
ejpam-6576	97	7	{	{	PUNCT
ejpam-6576	97	8	n	n	PROPN
ejpam-6576	97	9	+	+	CCONJ
ejpam-6576	97	10	i	i	PROPN
ejpam-6576	97	11	∈	∈	PROPN
ejpam-6576	97	12	n	n	CCONJ
ejpam-6576	97	13	/	/	SYM
ejpam-6576	97	14	i	i	PRON
ejpam-6576	97	15	|	|	ADV
ejpam-6576	97	16	g3(n	g3(n	VERB
ejpam-6576	98	1	+	+	CCONJ
ejpam-6576	99	1	i	i	NOUN
ejpam-6576	99	2	)	)	PUNCT
ejpam-6576	100	1	=	=	PUNCT
ejpam-6576	100	2	0	0	X
ejpam-6576	100	3	}	}	PUNCT
ejpam-6576	100	4	=	=	SYM
ejpam-6576	100	5	{	{	PUNCT
ejpam-6576	100	6	n	n	PROPN
ejpam-6576	100	7	+	+	CCONJ
ejpam-6576	100	8	i	i	PROPN
ejpam-6576	100	9	∈	∈	PROPN
ejpam-6576	100	10	n	n	CCONJ
ejpam-6576	100	11	/	/	SYM
ejpam-6576	100	12	i	i	PRON
ejpam-6576	100	13	|	|	ADV
ejpam-6576	100	14	g3n	g3n	PROPN
ejpam-6576	100	15	=	=	SYM
ejpam-6576	100	16	0	0	NUM
ejpam-6576	100	17	}	}	PUNCT
ejpam-6576	100	18	=	=	SYM
ejpam-6576	100	19	(	(	PUNCT
ejpam-6576	100	20	g3	g3	NOUN
ejpam-6576	100	21	:	:	PUNCT
ejpam-6576	100	22	0)n	0)n	X
ejpam-6576	100	23	/	/	SYM
ejpam-6576	100	24	i	i	PRON
ejpam-6576	100	25	=	=	PUNCT
ejpam-6576	100	26	(	(	PUNCT
ejpam-6576	100	27	g0n	g0n	NOUN
ejpam-6576	100	28	:	:	PUNCT
ejpam-6576	100	29	0)n	0)n	X
ejpam-6576	100	30	/	/	SYM
ejpam-6576	101	1	i	i	PRON
ejpam-6576	101	2	=	=	PUNCT
ejpam-6576	101	3	(	(	PUNCT
ejpam-6576	101	4	g0n	g0n	NOUN
ejpam-6576	101	5	:	:	PUNCT
ejpam-6576	101	6	0)n	0)n	X
ejpam-6576	101	7	/	/	SYM
ejpam-6576	102	1	i	i	PRON
ejpam-6576	102	2	.	.	PUNCT
ejpam-6576	103	1	hence	hence	ADV
ejpam-6576	103	2	g0n	g0n	NOUN
ejpam-6576	103	3	is	be	AUX
ejpam-6576	103	4	a	a	DET
ejpam-6576	103	5	completely	completely	ADV
ejpam-6576	103	6	prime	prime	ADJ
ejpam-6576	103	7	right	right	NOUN
ejpam-6576	103	8	n	n	CCONJ
ejpam-6576	103	9	/	/	SYM
ejpam-6576	103	10	i	i	PROPN
ejpam-6576	103	11	-	-	PUNCT
ejpam-6576	103	12	group	group	NOUN
ejpam-6576	103	13	of	of	ADP
ejpam-6576	103	14	type	type	NOUN
ejpam-6576	103	15	r(1	r(1	PROPN
ejpam-6576	103	16	)	)	PUNCT
ejpam-6576	103	17	.	.	PUNCT
ejpam-6576	104	1	proposition	proposition	NOUN
ejpam-6576	104	2	5	5	NUM
ejpam-6576	104	3	.	.	PUNCT
ejpam-6576	105	1	let	let	VERB
ejpam-6576	105	2	n	n	PRON
ejpam-6576	105	3	be	be	AUX
ejpam-6576	105	4	a	a	DET
ejpam-6576	105	5	near	near	ADJ
ejpam-6576	105	6	-	-	PUNCT
ejpam-6576	105	7	ring	ring	NOUN
ejpam-6576	106	1	and	and	CCONJ
ejpam-6576	106	2	i	i	PRON
ejpam-6576	106	3	is	be	AUX
ejpam-6576	106	4	an	an	DET
ejpam-6576	106	5	ideal	ideal	NOUN
ejpam-6576	106	6	of	of	ADP
ejpam-6576	106	7	n	n	PROPN
ejpam-6576	106	8	and	and	CCONJ
ejpam-6576	106	9	g	g	PROPN
ejpam-6576	106	10	be	be	AUX
ejpam-6576	106	11	completely	completely	ADV
ejpam-6576	106	12	prime	prime	ADJ
ejpam-6576	106	13	right	right	ADJ
ejpam-6576	107	1	n	n	CCONJ
ejpam-6576	107	2	/	/	SYM
ejpam-6576	107	3	i	i	PROPN
ejpam-6576	107	4	-	-	PUNCT
ejpam-6576	107	5	group	group	NOUN
ejpam-6576	107	6	type	type	NOUN
ejpam-6576	107	7	r(1	r(1	PROPN
ejpam-6576	107	8	)	)	PUNCT
ejpam-6576	107	9	.	.	PUNCT
ejpam-6576	108	1	then	then	ADV
ejpam-6576	108	2	g	g	PROPN
ejpam-6576	108	3	is	be	AUX
ejpam-6576	108	4	a	a	DET
ejpam-6576	108	5	completely	completely	ADV
ejpam-6576	108	6	prime	prime	ADJ
ejpam-6576	108	7	right	right	NOUN
ejpam-6576	108	8	n	n	PRON
ejpam-6576	108	9	-group	-group	NOUN
ejpam-6576	108	10	of	of	ADP
ejpam-6576	108	11	type	type	NOUN
ejpam-6576	108	12	r(1	r(1	PROPN
ejpam-6576	108	13	)	)	PUNCT
ejpam-6576	108	14	.	.	PUNCT
ejpam-6576	109	1	proof	proof	NOUN
ejpam-6576	109	2	.	.	PUNCT
ejpam-6576	110	1	n	n	PRON
ejpam-6576	110	2	is	be	AUX
ejpam-6576	110	3	a	a	DET
ejpam-6576	110	4	near	near	ADJ
ejpam-6576	110	5	-	-	PUNCT
ejpam-6576	110	6	ring	ring	NOUN
ejpam-6576	110	7	and	and	CCONJ
ejpam-6576	110	8	i	i	PRON
ejpam-6576	110	9	is	be	AUX
ejpam-6576	110	10	an	an	DET
ejpam-6576	110	11	ideal	ideal	NOUN
ejpam-6576	110	12	of	of	ADP
ejpam-6576	110	13	n	n	NUM
ejpam-6576	110	14	and	and	CCONJ
ejpam-6576	110	15	g	g	PROPN
ejpam-6576	110	16	is	be	AUX
ejpam-6576	110	17	a	a	DET
ejpam-6576	110	18	completely	completely	ADV
ejpam-6576	110	19	prime	prime	ADJ
ejpam-6576	110	20	right	right	NOUN
ejpam-6576	110	21	n	n	CCONJ
ejpam-6576	110	22	/	/	SYM
ejpam-6576	110	23	i	i	PROPN
ejpam-6576	110	24	-	-	PUNCT
ejpam-6576	110	25	group	group	NOUN
ejpam-6576	110	26	type	type	NOUN
ejpam-6576	110	27	r(1	r(1	PROPN
ejpam-6576	110	28	)	)	PUNCT
ejpam-6576	110	29	.	.	PUNCT
ejpam-6576	111	1	define	define	VERB
ejpam-6576	111	2	gx	gx	PROPN
ejpam-6576	111	3	:	:	PUNCT
ejpam-6576	111	4	=	=	SYM
ejpam-6576	111	5	g(x	g(x	PROPN
ejpam-6576	111	6	+	+	CCONJ
ejpam-6576	111	7	i	i	NOUN
ejpam-6576	111	8	)	)	PUNCT
ejpam-6576	111	9	for	for	ADP
ejpam-6576	111	10	all	all	PRON
ejpam-6576	111	11	g	g	PROPN
ejpam-6576	111	12	∈	∈	PROPN
ejpam-6576	111	13	g	g	NOUN
ejpam-6576	111	14	,	,	PUNCT
ejpam-6576	111	15	x	x	SYM
ejpam-6576	111	16	∈	∈	PROPN
ejpam-6576	111	17	n	n	ADV
ejpam-6576	111	18	.	.	PUNCT
ejpam-6576	112	1	this	this	PRON
ejpam-6576	112	2	makes	make	VERB
ejpam-6576	112	3	g	g	PROPN
ejpam-6576	112	4	a	a	DET
ejpam-6576	112	5	right	right	NOUN
ejpam-6576	112	6	n	n	NUM
ejpam-6576	112	7	-group	-group	NOUN
ejpam-6576	112	8	.	.	PUNCT
ejpam-6576	113	1	let	let	VERB
ejpam-6576	113	2	h	h	PRON
ejpam-6576	113	3	be	be	AUX
ejpam-6576	113	4	a	a	DET
ejpam-6576	113	5	non	non	ADJ
ejpam-6576	113	6	-	-	ADJ
ejpam-6576	113	7	zero	zero	NUM
ejpam-6576	113	8	right	right	NOUN
ejpam-6576	113	9	n	n	PRON
ejpam-6576	113	10	-subgroup	-subgroup	NOUN
ejpam-6576	113	11	of	of	ADP
ejpam-6576	113	12	g.	g.	PROPN
ejpam-6576	114	1	it	it	PRON
ejpam-6576	114	2	is	be	AUX
ejpam-6576	114	3	clear	clear	ADJ
ejpam-6576	114	4	that	that	SCONJ
ejpam-6576	114	5	h	h	NOUN
ejpam-6576	114	6	is	be	AUX
ejpam-6576	114	7	also	also	ADV
ejpam-6576	114	8	a	a	DET
ejpam-6576	114	9	non	non	ADJ
ejpam-6576	114	10	-	-	ADJ
ejpam-6576	114	11	zero	zero	NUM
ejpam-6576	114	12	k.	k.	NOUN
ejpam-6576	115	1	j.	j.	PROPN
ejpam-6576	115	2	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	115	3	et	et	PROPN
ejpam-6576	115	4	al	al	PROPN
ejpam-6576	115	5	.	.	PUNCT
ejpam-6576	115	6	/	/	SYM
ejpam-6576	115	7	eur	eur	PROPN
ejpam-6576	115	8	.	.	PUNCT
ejpam-6576	116	1	j.	j.	PROPN
ejpam-6576	116	2	pure	pure	PROPN
ejpam-6576	116	3	appl	appl	PROPN
ejpam-6576	116	4	.	.	PROPN
ejpam-6576	116	5	math	math	PROPN
ejpam-6576	116	6	,	,	PUNCT
ejpam-6576	116	7	18	18	NUM
ejpam-6576	116	8	(	(	PUNCT
ejpam-6576	116	9	3	3	NUM
ejpam-6576	116	10	)	)	PUNCT
ejpam-6576	116	11	(	(	PUNCT
ejpam-6576	116	12	2025	2025	NUM
ejpam-6576	116	13	)	)	PUNCT
ejpam-6576	116	14	,	,	PUNCT
ejpam-6576	116	15	6576	6576	NUM
ejpam-6576	116	16	4	4	NUM
ejpam-6576	116	17	of	of	ADP
ejpam-6576	116	18	7	7	NUM
ejpam-6576	116	19	right	right	ADJ
ejpam-6576	116	20	n	n	CCONJ
ejpam-6576	116	21	/	/	SYM
ejpam-6576	116	22	i	i	PROPN
ejpam-6576	116	23	-	-	PUNCT
ejpam-6576	116	24	subgroup	subgroup	NOUN
ejpam-6576	116	25	of	of	ADP
ejpam-6576	116	26	g	g	PROPN
ejpam-6576	116	27	and	and	CCONJ
ejpam-6576	116	28	it	it	PRON
ejpam-6576	116	29	contains	contain	VERB
ejpam-6576	116	30	a	a	DET
ejpam-6576	116	31	non	non	ADJ
ejpam-6576	116	32	-	-	ADJ
ejpam-6576	116	33	zero	zero	NUM
ejpam-6576	116	34	distributive	distributive	ADJ
ejpam-6576	116	35	element	element	NOUN
ejpam-6576	116	36	h0	h0	NOUN
ejpam-6576	116	37	.	.	PUNCT
ejpam-6576	117	1	we	we	PRON
ejpam-6576	117	2	have	have	VERB
ejpam-6576	117	3	h0(x	h0(x	X
ejpam-6576	117	4	+	+	NOUN
ejpam-6576	117	5	y	y	NOUN
ejpam-6576	117	6	)	)	PUNCT
ejpam-6576	118	1	=	=	PUNCT
ejpam-6576	118	2	h0(x	h0(x	PROPN
ejpam-6576	119	1	+	+	NUM
ejpam-6576	119	2	y	y	PROPN
ejpam-6576	120	1	+	+	CCONJ
ejpam-6576	120	2	i	i	NOUN
ejpam-6576	120	3	)	)	PUNCT
ejpam-6576	121	1	=	=	VERB
ejpam-6576	121	2	h0((x	h0((x	VERB
ejpam-6576	121	3	+	+	CCONJ
ejpam-6576	121	4	i	i	NOUN
ejpam-6576	121	5	)	)	PUNCT
ejpam-6576	122	1	+	+	CCONJ
ejpam-6576	122	2	(	(	PUNCT
ejpam-6576	122	3	y	y	PROPN
ejpam-6576	122	4	+	+	CCONJ
ejpam-6576	122	5	i	i	NOUN
ejpam-6576	122	6	)	)	PUNCT
ejpam-6576	122	7	)	)	PUNCT
ejpam-6576	123	1	=	=	PUNCT
ejpam-6576	123	2	h0(x	h0(x	PROPN
ejpam-6576	124	1	+	+	NUM
ejpam-6576	124	2	i	i	NOUN
ejpam-6576	124	3	)	)	PUNCT
ejpam-6576	125	1	+	+	CCONJ
ejpam-6576	125	2	h0(y	h0(y	X
ejpam-6576	126	1	+	+	CCONJ
ejpam-6576	126	2	i	i	NOUN
ejpam-6576	126	3	)	)	PUNCT
ejpam-6576	127	1	=	=	X
ejpam-6576	127	2	h0x	h0x	NOUN
ejpam-6576	127	3	+	+	CCONJ
ejpam-6576	127	4	h0y	h0y	NOUN
ejpam-6576	127	5	.	.	PUNCT
ejpam-6576	128	1	so	so	ADV
ejpam-6576	128	2	h0	h0	PROPN
ejpam-6576	128	3	is	be	AUX
ejpam-6576	128	4	also	also	ADV
ejpam-6576	128	5	a	a	DET
ejpam-6576	128	6	non	non	ADJ
ejpam-6576	128	7	-	-	ADJ
ejpam-6576	128	8	zero	zero	NUM
ejpam-6576	128	9	distributive	distributive	ADJ
ejpam-6576	128	10	element	element	NOUN
ejpam-6576	128	11	of	of	ADP
ejpam-6576	128	12	the	the	DET
ejpam-6576	128	13	right	right	NOUN
ejpam-6576	128	14	n	n	PROPN
ejpam-6576	128	15	subgroup	subgroup	NOUN
ejpam-6576	128	16	h	h	PROPN
ejpam-6576	128	17	of	of	ADP
ejpam-6576	128	18	g.	g.	PROPN
ejpam-6576	128	19	let	let	VERB
ejpam-6576	128	20	0	0	NUM
ejpam-6576	128	21	̸=	̸=	PROPN
ejpam-6576	128	22	g1	g1	NOUN
ejpam-6576	128	23	,	,	PUNCT
ejpam-6576	128	24	0	0	NUM
ejpam-6576	129	1	̸=	̸=	PROPN
ejpam-6576	129	2	g2	g2	PROPN
ejpam-6576	129	3	∈	∈	PROPN
ejpam-6576	129	4	g.	g.	PROPN
ejpam-6576	130	1	now	now	ADV
ejpam-6576	130	2	(	(	PUNCT
ejpam-6576	130	3	g1	g1	PROPN
ejpam-6576	130	4	:	:	PUNCT
ejpam-6576	130	5	0)n	0)n	X
ejpam-6576	130	6	=	=	PUNCT
ejpam-6576	130	7	{	{	PUNCT
ejpam-6576	130	8	x	x	SYM
ejpam-6576	130	9	∈	∈	PROPN
ejpam-6576	130	10	n	n	CCONJ
ejpam-6576	130	11	|	|	ADV
ejpam-6576	130	12	g1x	g1x	NOUN
ejpam-6576	130	13	=	=	NOUN
ejpam-6576	130	14	0	0	X
ejpam-6576	130	15	}	}	PUNCT
ejpam-6576	130	16	=	=	PRON
ejpam-6576	130	17	{	{	PUNCT
ejpam-6576	130	18	x	x	SYM
ejpam-6576	130	19	∈	∈	PROPN
ejpam-6576	130	20	n	n	CCONJ
ejpam-6576	131	1	|	|	ADV
ejpam-6576	131	2	g1(x+	g1(x+	PROPN
ejpam-6576	131	3	i	i	PRON
ejpam-6576	131	4	)	)	PUNCT
ejpam-6576	132	1	=	=	PUNCT
ejpam-6576	132	2	0	0	X
ejpam-6576	132	3	}	}	PUNCT
ejpam-6576	132	4	=	=	PRON
ejpam-6576	132	5	{	{	PUNCT
ejpam-6576	132	6	x	x	SYM
ejpam-6576	132	7	∈	∈	PROPN
ejpam-6576	132	8	n	n	CCONJ
ejpam-6576	133	1	|	|	ADV
ejpam-6576	133	2	g2(x+	g2(x+	PROPN
ejpam-6576	133	3	i	i	NOUN
ejpam-6576	133	4	)	)	PUNCT
ejpam-6576	133	5	=	=	PUNCT
ejpam-6576	134	1	0	0	X
ejpam-6576	134	2	}	}	PUNCT
ejpam-6576	134	3	=	=	PRON
ejpam-6576	134	4	{	{	PUNCT
ejpam-6576	134	5	x	x	SYM
ejpam-6576	134	6	∈	∈	PROPN
ejpam-6576	134	7	n	n	CCONJ
ejpam-6576	134	8	|	|	ADV
ejpam-6576	134	9	g2x	g2x	NOUN
ejpam-6576	134	10	=	=	NOUN
ejpam-6576	134	11	0	0	NUM
ejpam-6576	134	12	}	}	PUNCT
ejpam-6576	134	13	=	=	SYM
ejpam-6576	134	14	(	(	PUNCT
ejpam-6576	134	15	g2	g2	PROPN
ejpam-6576	134	16	:	:	PUNCT
ejpam-6576	134	17	0)n	0)n	X
ejpam-6576	134	18	.	.	PUNCT
ejpam-6576	135	1	hence	hence	ADV
ejpam-6576	135	2	g	g	PROPN
ejpam-6576	135	3	is	be	AUX
ejpam-6576	135	4	a	a	DET
ejpam-6576	135	5	completely	completely	ADV
ejpam-6576	135	6	prime	prime	ADJ
ejpam-6576	135	7	right	right	NOUN
ejpam-6576	135	8	n	n	PRON
ejpam-6576	135	9	-group	-group	NOUN
ejpam-6576	135	10	of	of	ADP
ejpam-6576	135	11	typer(1	typer(1	PROPN
ejpam-6576	135	12	)	)	PUNCT
ejpam-6576	135	13	.	.	PUNCT
ejpam-6576	136	1	proposition	proposition	NOUN
ejpam-6576	136	2	6	6	NUM
ejpam-6576	136	3	.	.	PUNCT
ejpam-6576	137	1	let	let	VERB
ejpam-6576	137	2	g	g	PRON
ejpam-6576	137	3	be	be	AUX
ejpam-6576	137	4	a	a	DET
ejpam-6576	137	5	completely	completely	ADV
ejpam-6576	137	6	prime	prime	ADJ
ejpam-6576	137	7	right	right	NOUN
ejpam-6576	137	8	n	n	PRON
ejpam-6576	137	9	-group	-group	NOUN
ejpam-6576	137	10	of	of	ADP
ejpam-6576	137	11	type	type	NOUN
ejpam-6576	137	12	1	1	NUM
ejpam-6576	137	13	and	and	CCONJ
ejpam-6576	137	14	(	(	PUNCT
ejpam-6576	137	15	g	g	NOUN
ejpam-6576	137	16	:	:	PUNCT
ejpam-6576	137	17	0	0	NUM
ejpam-6576	137	18	)	)	PUNCT
ejpam-6576	137	19	=	=	PRON
ejpam-6576	138	1	{	{	PUNCT
ejpam-6576	138	2	x	x	SYM
ejpam-6576	138	3	∈	∈	PROPN
ejpam-6576	138	4	n	n	CCONJ
ejpam-6576	138	5	|	|	ADV
ejpam-6576	138	6	gx	gx	PROPN
ejpam-6576	138	7	=	=	NOUN
ejpam-6576	138	8	0	0	PROPN
ejpam-6576	138	9	for	for	ADP
ejpam-6576	138	10	all	all	PRON
ejpam-6576	138	11	g	g	PROPN
ejpam-6576	138	12	∈	∈	PRON
ejpam-6576	138	13	g	g	AUX
ejpam-6576	138	14	}	}	PUNCT
ejpam-6576	138	15	be	be	AUX
ejpam-6576	138	16	the	the	DET
ejpam-6576	138	17	annihilator	annihilator	NOUN
ejpam-6576	138	18	of	of	ADP
ejpam-6576	138	19	g.	g.	PROPN
ejpam-6576	138	20	then	then	ADV
ejpam-6576	138	21	there	there	PRON
ejpam-6576	138	22	is	be	VERB
ejpam-6576	138	23	a	a	DET
ejpam-6576	138	24	largest	large	ADJ
ejpam-6576	138	25	ideal	ideal	NOUN
ejpam-6576	138	26	of	of	ADP
ejpam-6576	138	27	n	n	NUM
ejpam-6576	138	28	contained	contain	VERB
ejpam-6576	138	29	in	in	ADP
ejpam-6576	138	30	(	(	PUNCT
ejpam-6576	138	31	g	g	NOUN
ejpam-6576	138	32	:	:	PUNCT
ejpam-6576	138	33	0	0	NUM
ejpam-6576	138	34	)	)	PUNCT
ejpam-6576	138	35	.	.	PUNCT
ejpam-6576	139	1	proof	proof	NOUN
ejpam-6576	139	2	.	.	PUNCT
ejpam-6576	140	1	we	we	PRON
ejpam-6576	140	2	have	have	VERB
ejpam-6576	140	3	0	0	NUM
ejpam-6576	140	4	∈	∈	NOUN
ejpam-6576	140	5	(	(	PUNCT
ejpam-6576	140	6	g	g	NOUN
ejpam-6576	140	7	:	:	PUNCT
ejpam-6576	140	8	0	0	NUM
ejpam-6576	140	9	)	)	PUNCT
ejpam-6576	140	10	.	.	PUNCT
ejpam-6576	141	1	let	let	VERB
ejpam-6576	141	2	i	i	PRON
ejpam-6576	141	3	,	,	PUNCT
ejpam-6576	141	4	j	j	PROPN
ejpam-6576	141	5	be	be	VERB
ejpam-6576	141	6	ideals	ideal	NOUN
ejpam-6576	141	7	of	of	ADP
ejpam-6576	141	8	n	n	NUM
ejpam-6576	141	9	contained	contain	VERB
ejpam-6576	141	10	in	in	ADP
ejpam-6576	141	11	(	(	PUNCT
ejpam-6576	141	12	g	g	NOUN
ejpam-6576	141	13	:	:	PUNCT
ejpam-6576	141	14	0	0	NUM
ejpam-6576	141	15	)	)	PUNCT
ejpam-6576	141	16	.	.	PUNCT
ejpam-6576	142	1	by	by	ADP
ejpam-6576	142	2	definition	definition	NOUN
ejpam-6576	142	3	,	,	PUNCT
ejpam-6576	142	4	g	g	PROPN
ejpam-6576	142	5	has	have	VERB
ejpam-6576	142	6	a	a	DET
ejpam-6576	142	7	non	non	ADJ
ejpam-6576	142	8	zero	zero	NUM
ejpam-6576	142	9	distributive	distributive	ADJ
ejpam-6576	142	10	element	element	NOUN
ejpam-6576	142	11	g0	g0	NOUN
ejpam-6576	142	12	.	.	PUNCT
ejpam-6576	143	1	we	we	PRON
ejpam-6576	143	2	have	have	VERB
ejpam-6576	143	3	(	(	PUNCT
ejpam-6576	143	4	g0	g0	NOUN
ejpam-6576	143	5	:	:	PUNCT
ejpam-6576	143	6	0	0	NUM
ejpam-6576	143	7	)	)	PUNCT
ejpam-6576	143	8	=	=	SYM
ejpam-6576	144	1	(	(	PUNCT
ejpam-6576	144	2	g	g	NOUN
ejpam-6576	144	3	:	:	PUNCT
ejpam-6576	144	4	0	0	NUM
ejpam-6576	144	5	)	)	PUNCT
ejpam-6576	144	6	and	and	CCONJ
ejpam-6576	144	7	g0(y	g0(y	PROPN
ejpam-6576	145	1	+	+	CCONJ
ejpam-6576	145	2	z	z	NOUN
ejpam-6576	145	3	)	)	PUNCT
ejpam-6576	145	4	=	=	PUNCT
ejpam-6576	145	5	g0y	g0y	PROPN
ejpam-6576	145	6	+	+	CCONJ
ejpam-6576	145	7	g0z	g0z	PROPN
ejpam-6576	145	8	=	=	PUNCT
ejpam-6576	145	9	0	0	PUNCT
ejpam-6576	146	1	+	+	CCONJ
ejpam-6576	146	2	0	0	NUM
ejpam-6576	146	3	=	=	SYM
ejpam-6576	146	4	0	0	NUM
ejpam-6576	146	5	for	for	ADP
ejpam-6576	146	6	all	all	DET
ejpam-6576	146	7	y	y	PROPN
ejpam-6576	146	8	∈	∈	PROPN
ejpam-6576	147	1	i	i	PRON
ejpam-6576	147	2	and	and	CCONJ
ejpam-6576	147	3	z	z	PROPN
ejpam-6576	147	4	∈	∈	PROPN
ejpam-6576	147	5	j	j	PROPN
ejpam-6576	147	6	.	.	PUNCT
ejpam-6576	148	1	so	so	ADV
ejpam-6576	148	2	g0(i	g0(i	PROPN
ejpam-6576	148	3	+	+	NUM
ejpam-6576	148	4	j	j	NOUN
ejpam-6576	148	5	)	)	PUNCT
ejpam-6576	149	1	=	=	SYM
ejpam-6576	149	2	0	0	X
ejpam-6576	149	3	.	.	PUNCT
ejpam-6576	150	1	therefore	therefore	ADV
ejpam-6576	150	2	i	i	PRON
ejpam-6576	150	3	+	+	NUM
ejpam-6576	150	4	j	j	PROPN
ejpam-6576	150	5	⊆	⊆	NUM
ejpam-6576	150	6	(	(	PUNCT
ejpam-6576	150	7	g0	g0	NOUN
ejpam-6576	150	8	:	:	PUNCT
ejpam-6576	150	9	0	0	NUM
ejpam-6576	150	10	)	)	PUNCT
ejpam-6576	150	11	=	=	SYM
ejpam-6576	150	12	(	(	PUNCT
ejpam-6576	150	13	g	g	NOUN
ejpam-6576	150	14	:	:	PUNCT
ejpam-6576	150	15	0	0	NUM
ejpam-6576	150	16	)	)	PUNCT
ejpam-6576	150	17	.	.	PUNCT
ejpam-6576	151	1	hence	hence	ADV
ejpam-6576	151	2	there	there	PRON
ejpam-6576	151	3	is	be	VERB
ejpam-6576	151	4	a	a	DET
ejpam-6576	151	5	largest	large	ADJ
ejpam-6576	151	6	ideal	ideal	NOUN
ejpam-6576	151	7	of	of	ADP
ejpam-6576	151	8	n	n	NUM
ejpam-6576	151	9	contained	contain	VERB
ejpam-6576	151	10	in	in	ADP
ejpam-6576	151	11	(	(	PUNCT
ejpam-6576	151	12	g	g	NOUN
ejpam-6576	151	13	:	:	PUNCT
ejpam-6576	151	14	0	0	NUM
ejpam-6576	151	15	)	)	PUNCT
ejpam-6576	151	16	.	.	PUNCT
ejpam-6576	152	1	(	(	PUNCT
ejpam-6576	152	2	g	g	NOUN
ejpam-6576	152	3	:	:	PUNCT
ejpam-6576	152	4	0	0	NUM
ejpam-6576	152	5	)	)	PUNCT
ejpam-6576	152	6	denotes	denote	VERB
ejpam-6576	152	7	the	the	DET
ejpam-6576	152	8	largest	large	ADJ
ejpam-6576	152	9	ideal	ideal	NOUN
ejpam-6576	152	10	of	of	ADP
ejpam-6576	152	11	n	n	NUM
ejpam-6576	152	12	contained	contain	VERB
ejpam-6576	152	13	in	in	ADP
ejpam-6576	152	14	(	(	PUNCT
ejpam-6576	152	15	g	g	NOUN
ejpam-6576	152	16	:	:	PUNCT
ejpam-6576	152	17	0	0	NUM
ejpam-6576	152	18	)	)	PUNCT
ejpam-6576	152	19	.	.	PUNCT
ejpam-6576	153	1	definition	definition	NOUN
ejpam-6576	153	2	2	2	NUM
ejpam-6576	153	3	.	.	PUNCT
ejpam-6576	154	1	let	let	VERB
ejpam-6576	154	2	g	g	PRON
ejpam-6576	154	3	be	be	AUX
ejpam-6576	154	4	a	a	DET
ejpam-6576	154	5	completely	completely	ADV
ejpam-6576	154	6	prime	prime	ADJ
ejpam-6576	154	7	right	right	NOUN
ejpam-6576	154	8	n	n	PRON
ejpam-6576	154	9	-group	-group	NOUN
ejpam-6576	154	10	of	of	ADP
ejpam-6576	154	11	type	type	NOUN
ejpam-6576	154	12	1	1	NUM
ejpam-6576	154	13	and	and	CCONJ
ejpam-6576	154	14	p	p	X
ejpam-6576	154	15	:	:	PUNCT
ejpam-6576	154	16	=	=	SYM
ejpam-6576	154	17	(	(	PUNCT
ejpam-6576	154	18	g	g	NOUN
ejpam-6576	154	19	:	:	PUNCT
ejpam-6576	154	20	0	0	NUM
ejpam-6576	154	21	)	)	PUNCT
ejpam-6576	154	22	.	.	PUNCT
ejpam-6576	155	1	then	then	ADV
ejpam-6576	155	2	p	p	PROPN
ejpam-6576	155	3	is	be	AUX
ejpam-6576	155	4	called	call	VERB
ejpam-6576	155	5	a	a	DET
ejpam-6576	155	6	completely	completely	ADV
ejpam-6576	155	7	prime	prime	ADJ
ejpam-6576	155	8	ideal	ideal	NOUN
ejpam-6576	155	9	of	of	ADP
ejpam-6576	155	10	n	n	PROPN
ejpam-6576	155	11	of	of	ADP
ejpam-6576	155	12	type	type	NOUN
ejpam-6576	155	13	r(1	r(1	PROPN
ejpam-6576	155	14	)	)	PUNCT
ejpam-6576	155	15	.	.	PUNCT
ejpam-6576	156	1	definition	definition	NOUN
ejpam-6576	156	2	3	3	NUM
ejpam-6576	156	3	.	.	PUNCT
ejpam-6576	157	1	a	a	DET
ejpam-6576	157	2	near	near	ADV
ejpam-6576	157	3	-	-	PUNCT
ejpam-6576	157	4	ring	ring	NOUN
ejpam-6576	157	5	n	n	NOUN
ejpam-6576	157	6	is	be	AUX
ejpam-6576	157	7	called	call	VERB
ejpam-6576	157	8	a	a	DET
ejpam-6576	157	9	completely	completely	ADV
ejpam-6576	157	10	prime	prime	ADJ
ejpam-6576	157	11	near	near	ADP
ejpam-6576	157	12	-	-	PUNCT
ejpam-6576	157	13	ring	ring	NOUN
ejpam-6576	157	14	of	of	ADP
ejpam-6576	157	15	type	type	NOUN
ejpam-6576	157	16	r(1	r(1	PROPN
ejpam-6576	157	17	)	)	PUNCT
ejpam-6576	157	18	if	if	SCONJ
ejpam-6576	157	19	it	it	PRON
ejpam-6576	157	20	’s	’	VERB
ejpam-6576	157	21	zero	zero	NUM
ejpam-6576	157	22	ideal	ideal	NOUN
ejpam-6576	157	23	is	be	AUX
ejpam-6576	157	24	a	a	DET
ejpam-6576	157	25	completely	completely	ADV
ejpam-6576	157	26	prime	prime	ADJ
ejpam-6576	157	27	ideal	ideal	NOUN
ejpam-6576	157	28	of	of	ADP
ejpam-6576	157	29	type	type	NOUN
ejpam-6576	157	30	r(1	r(1	PROPN
ejpam-6576	157	31	)	)	PUNCT
ejpam-6576	157	32	.	.	PUNCT
ejpam-6576	158	1	corollary	corollary	ADJ
ejpam-6576	158	2	1	1	NUM
ejpam-6576	158	3	.	.	PUNCT
ejpam-6576	159	1	let	let	VERB
ejpam-6576	159	2	n	n	PRON
ejpam-6576	159	3	be	be	AUX
ejpam-6576	159	4	a	a	DET
ejpam-6576	159	5	near	near	ADJ
ejpam-6576	159	6	-	-	PUNCT
ejpam-6576	159	7	ring	ring	NOUN
ejpam-6576	159	8	and	and	CCONJ
ejpam-6576	159	9	p	p	NOUN
ejpam-6576	159	10	be	be	AUX
ejpam-6576	159	11	a	a	DET
ejpam-6576	159	12	completely	completely	ADV
ejpam-6576	159	13	prime	prime	ADJ
ejpam-6576	159	14	ideal	ideal	NOUN
ejpam-6576	159	15	of	of	ADP
ejpam-6576	159	16	n	n	PROPN
ejpam-6576	159	17	of	of	ADP
ejpam-6576	159	18	type	type	NOUN
ejpam-6576	159	19	r(1	r(1	PROPN
ejpam-6576	159	20	)	)	PUNCT
ejpam-6576	159	21	.	.	PUNCT
ejpam-6576	160	1	then	then	ADV
ejpam-6576	160	2	n	n	CCONJ
ejpam-6576	160	3	/	/	SYM
ejpam-6576	160	4	p	p	X
ejpam-6576	160	5	completely	completely	ADV
ejpam-6576	160	6	prime	prime	ADJ
ejpam-6576	160	7	near	near	ADP
ejpam-6576	160	8	-	-	PUNCT
ejpam-6576	160	9	ring	ring	NOUN
ejpam-6576	160	10	of	of	ADP
ejpam-6576	160	11	type	type	NOUN
ejpam-6576	160	12	r(1	r(1	PROPN
ejpam-6576	160	13	)	)	PUNCT
ejpam-6576	160	14	.	.	PUNCT
ejpam-6576	161	1	proof	proof	NOUN
ejpam-6576	161	2	.	.	PUNCT
ejpam-6576	162	1	since	since	SCONJ
ejpam-6576	162	2	p	p	NOUN
ejpam-6576	162	3	is	be	AUX
ejpam-6576	162	4	a	a	DET
ejpam-6576	162	5	completely	completely	ADV
ejpam-6576	162	6	prime	prime	ADJ
ejpam-6576	162	7	ideal	ideal	NOUN
ejpam-6576	162	8	of	of	ADP
ejpam-6576	162	9	n	n	PROPN
ejpam-6576	162	10	of	of	ADP
ejpam-6576	162	11	type	type	NOUN
ejpam-6576	162	12	r(1	r(1	PROPN
ejpam-6576	162	13	)	)	PUNCT
ejpam-6576	162	14	,	,	PUNCT
ejpam-6576	162	15	there	there	PRON
ejpam-6576	162	16	is	be	VERB
ejpam-6576	162	17	a	a	DET
ejpam-6576	162	18	completely	completely	ADV
ejpam-6576	162	19	prime	prime	ADJ
ejpam-6576	162	20	right	right	ADJ
ejpam-6576	162	21	n	n	NUM
ejpam-6576	162	22	-group	-group	NOUN
ejpam-6576	162	23	g	g	NOUN
ejpam-6576	162	24	of	of	ADP
ejpam-6576	162	25	type	type	NOUN
ejpam-6576	162	26	r(1	r(1	PROPN
ejpam-6576	162	27	)	)	PUNCT
ejpam-6576	162	28	and	and	CCONJ
ejpam-6576	162	29	p	p	NOUN
ejpam-6576	162	30	is	be	AUX
ejpam-6576	162	31	the	the	DET
ejpam-6576	162	32	largest	large	ADJ
ejpam-6576	162	33	ideal	ideal	NOUN
ejpam-6576	162	34	of	of	ADP
ejpam-6576	162	35	n	n	NUM
ejpam-6576	162	36	contained	contain	VERB
ejpam-6576	162	37	in	in	ADP
ejpam-6576	162	38	(	(	PUNCT
ejpam-6576	162	39	g	g	NOUN
ejpam-6576	162	40	:	:	PUNCT
ejpam-6576	162	41	0	0	NUM
ejpam-6576	162	42	)	)	PUNCT
ejpam-6576	162	43	.	.	PUNCT
ejpam-6576	163	1	by	by	ADP
ejpam-6576	163	2	propostion	propostion	NOUN
ejpam-6576	163	3	4	4	NUM
ejpam-6576	163	4	,	,	PUNCT
ejpam-6576	163	5	there	there	PRON
ejpam-6576	163	6	is	be	VERB
ejpam-6576	163	7	a	a	DET
ejpam-6576	163	8	right	right	ADJ
ejpam-6576	163	9	n	n	CCONJ
ejpam-6576	163	10	-subgroup	-subgroup	NOUN
ejpam-6576	163	11	h	h	NOUN
ejpam-6576	163	12	of	of	ADP
ejpam-6576	163	13	g	g	PROPN
ejpam-6576	163	14	which	which	PRON
ejpam-6576	163	15	is	be	AUX
ejpam-6576	163	16	a	a	DET
ejpam-6576	163	17	completely	completely	ADV
ejpam-6576	163	18	prime	prime	ADJ
ejpam-6576	163	19	n	n	CCONJ
ejpam-6576	163	20	/	/	SYM
ejpam-6576	163	21	p	p	NOUN
ejpam-6576	163	22	group	group	NOUN
ejpam-6576	163	23	of	of	ADP
ejpam-6576	163	24	type	type	NOUN
ejpam-6576	163	25	r(1	r(1	PROPN
ejpam-6576	163	26	)	)	PUNCT
ejpam-6576	163	27	,	,	PUNCT
ejpam-6576	163	28	where	where	SCONJ
ejpam-6576	163	29	h(x	h(x	PROPN
ejpam-6576	163	30	+	+	CCONJ
ejpam-6576	163	31	p	p	NOUN
ejpam-6576	163	32	)	)	PUNCT
ejpam-6576	163	33	:	:	PUNCT
ejpam-6576	163	34	=	=	NOUN
ejpam-6576	163	35	hx	hx	PROPN
ejpam-6576	163	36	for	for	ADP
ejpam-6576	163	37	all	all	DET
ejpam-6576	163	38	h	h	NOUN
ejpam-6576	163	39	∈	∈	PROPN
ejpam-6576	163	40	h	h	NOUN
ejpam-6576	163	41	,	,	PUNCT
ejpam-6576	163	42	x	x	SYM
ejpam-6576	163	43	∈	∈	PROPN
ejpam-6576	163	44	n	n	NOUN
ejpam-6576	164	1	and	and	CCONJ
ejpam-6576	164	2	(	(	PUNCT
ejpam-6576	164	3	h	h	NOUN
ejpam-6576	164	4	:	:	PUNCT
ejpam-6576	164	5	0)n	0)n	PROPN
ejpam-6576	164	6	/	/	SYM
ejpam-6576	164	7	p	p	NOUN
ejpam-6576	165	1	=	=	X
ejpam-6576	166	1	(	(	PUNCT
ejpam-6576	166	2	g	g	NOUN
ejpam-6576	166	3	:	:	PUNCT
ejpam-6576	166	4	0)n	0)n	PROPN
ejpam-6576	166	5	/	/	SYM
ejpam-6576	166	6	p	p	X
ejpam-6576	166	7	.	.	PUNCT
ejpam-6576	167	1	therefore	therefore	ADV
ejpam-6576	167	2	the	the	DET
ejpam-6576	167	3	zero	zero	NUM
ejpam-6576	167	4	ideal	ideal	NOUN
ejpam-6576	167	5	,	,	PUNCT
ejpam-6576	167	6	(	(	PUNCT
ejpam-6576	167	7	0	0	NUM
ejpam-6576	167	8	)	)	PUNCT
ejpam-6576	167	9	,	,	PUNCT
ejpam-6576	167	10	is	be	AUX
ejpam-6576	167	11	the	the	DET
ejpam-6576	167	12	largest	large	ADJ
ejpam-6576	167	13	ideal	ideal	NOUN
ejpam-6576	167	14	of	of	ADP
ejpam-6576	167	15	n	n	CCONJ
ejpam-6576	167	16	/	/	SYM
ejpam-6576	167	17	p	p	NOUN
ejpam-6576	167	18	contained	contain	VERB
ejpam-6576	167	19	in	in	ADP
ejpam-6576	167	20	(	(	PUNCT
ejpam-6576	167	21	h	h	NOUN
ejpam-6576	167	22	:	:	PUNCT
ejpam-6576	167	23	0)n	0)n	PROPN
ejpam-6576	167	24	/	/	SYM
ejpam-6576	167	25	p	p	NOUN
ejpam-6576	167	26	.	.	PUNCT
ejpam-6576	168	1	so	so	ADV
ejpam-6576	168	2	(	(	PUNCT
ejpam-6576	168	3	0	0	X
ejpam-6576	168	4	)	)	PUNCT
ejpam-6576	168	5	is	be	AUX
ejpam-6576	168	6	a	a	DET
ejpam-6576	168	7	completely	completely	ADV
ejpam-6576	168	8	prime	prime	ADJ
ejpam-6576	168	9	ideal	ideal	NOUN
ejpam-6576	168	10	of	of	ADP
ejpam-6576	168	11	n	n	CCONJ
ejpam-6576	168	12	/	/	SYM
ejpam-6576	168	13	p	p	NOUN
ejpam-6576	168	14	of	of	ADP
ejpam-6576	168	15	type	type	NOUN
ejpam-6576	168	16	r(1	r(1	PROPN
ejpam-6576	168	17	)	)	PUNCT
ejpam-6576	168	18	.	.	PUNCT
ejpam-6576	169	1	hence	hence	ADV
ejpam-6576	169	2	n	n	CCONJ
ejpam-6576	169	3	/	/	SYM
ejpam-6576	169	4	p	p	PRON
ejpam-6576	169	5	is	be	AUX
ejpam-6576	169	6	a	a	DET
ejpam-6576	169	7	completely	completely	ADV
ejpam-6576	169	8	prime	prime	ADJ
ejpam-6576	169	9	near	near	ADP
ejpam-6576	169	10	-	-	PUNCT
ejpam-6576	169	11	ring	ring	NOUN
ejpam-6576	169	12	of	of	ADP
ejpam-6576	169	13	type	type	NOUN
ejpam-6576	169	14	r(1	r(1	PROPN
ejpam-6576	169	15	)	)	PUNCT
ejpam-6576	169	16	.	.	PUNCT
ejpam-6576	170	1	corollary	corollary	ADJ
ejpam-6576	170	2	2	2	NUM
ejpam-6576	170	3	.	.	PUNCT
ejpam-6576	171	1	let	let	VERB
ejpam-6576	171	2	n	n	PRON
ejpam-6576	171	3	be	be	AUX
ejpam-6576	171	4	a	a	DET
ejpam-6576	171	5	near	near	ADJ
ejpam-6576	171	6	-	-	PUNCT
ejpam-6576	171	7	ring	ring	NOUN
ejpam-6576	171	8	and	and	CCONJ
ejpam-6576	171	9	p	p	NOUN
ejpam-6576	171	10	be	be	AUX
ejpam-6576	171	11	an	an	DET
ejpam-6576	171	12	ideal	ideal	NOUN
ejpam-6576	171	13	of	of	ADP
ejpam-6576	171	14	n	n	PROPN
ejpam-6576	171	15	and	and	CCONJ
ejpam-6576	171	16	n	n	CCONJ
ejpam-6576	171	17	/	/	SYM
ejpam-6576	171	18	p	p	X
ejpam-6576	171	19	be	be	AUX
ejpam-6576	171	20	a	a	DET
ejpam-6576	171	21	completely	completely	ADV
ejpam-6576	171	22	prime	prime	ADJ
ejpam-6576	171	23	near	near	ADP
ejpam-6576	171	24	ring	ring	NOUN
ejpam-6576	171	25	of	of	ADP
ejpam-6576	171	26	type	type	NOUN
ejpam-6576	171	27	r(1	r(1	PROPN
ejpam-6576	171	28	)	)	PUNCT
ejpam-6576	171	29	.	.	PUNCT
ejpam-6576	172	1	then	then	ADV
ejpam-6576	172	2	p	p	PROPN
ejpam-6576	172	3	is	be	AUX
ejpam-6576	172	4	a	a	DET
ejpam-6576	172	5	completely	completely	ADV
ejpam-6576	172	6	prime	prime	ADJ
ejpam-6576	172	7	ideal	ideal	NOUN
ejpam-6576	172	8	of	of	ADP
ejpam-6576	172	9	n	n	PROPN
ejpam-6576	172	10	of	of	ADP
ejpam-6576	172	11	type	type	NOUN
ejpam-6576	172	12	r(1	r(1	PROPN
ejpam-6576	172	13	)	)	PUNCT
ejpam-6576	172	14	.	.	PUNCT
ejpam-6576	173	1	proof	proof	NOUN
ejpam-6576	173	2	.	.	PUNCT
ejpam-6576	174	1	p	p	NOUN
ejpam-6576	174	2	is	be	AUX
ejpam-6576	174	3	an	an	DET
ejpam-6576	174	4	ideal	ideal	NOUN
ejpam-6576	174	5	of	of	ADP
ejpam-6576	174	6	a	a	DET
ejpam-6576	174	7	near	near	ADJ
ejpam-6576	174	8	-	-	PUNCT
ejpam-6576	174	9	ring	ring	NOUN
ejpam-6576	174	10	n	n	NOUN
ejpam-6576	174	11	and	and	CCONJ
ejpam-6576	174	12	n	n	CCONJ
ejpam-6576	174	13	/	/	SYM
ejpam-6576	174	14	p	p	PRON
ejpam-6576	174	15	is	be	AUX
ejpam-6576	174	16	a	a	DET
ejpam-6576	174	17	completely	completely	ADV
ejpam-6576	174	18	prime	prime	ADJ
ejpam-6576	174	19	near	near	ADP
ejpam-6576	174	20	-	-	PUNCT
ejpam-6576	174	21	ring	ring	NOUN
ejpam-6576	174	22	of	of	ADP
ejpam-6576	174	23	type	type	NOUN
ejpam-6576	174	24	r(1	r(1	PROPN
ejpam-6576	174	25	)	)	PUNCT
ejpam-6576	174	26	.	.	PUNCT
ejpam-6576	175	1	so	so	ADV
ejpam-6576	175	2	the	the	DET
ejpam-6576	175	3	zero	zero	NUM
ejpam-6576	175	4	ideal	ideal	NOUN
ejpam-6576	175	5	,	,	PUNCT
ejpam-6576	175	6	(	(	PUNCT
ejpam-6576	175	7	0	0	NUM
ejpam-6576	175	8	)	)	PUNCT
ejpam-6576	175	9	,	,	PUNCT
ejpam-6576	175	10	is	be	AUX
ejpam-6576	175	11	a	a	DET
ejpam-6576	175	12	completely	completely	ADV
ejpam-6576	175	13	prime	prime	ADJ
ejpam-6576	175	14	ideal	ideal	NOUN
ejpam-6576	175	15	of	of	ADP
ejpam-6576	175	16	n	n	CCONJ
ejpam-6576	175	17	/	/	SYM
ejpam-6576	175	18	p	p	NOUN
ejpam-6576	175	19	of	of	ADP
ejpam-6576	175	20	type	type	NOUN
ejpam-6576	175	21	r(1).therefore	r(1).therefore	ADV
ejpam-6576	175	22	there	there	PRON
ejpam-6576	175	23	is	be	VERB
ejpam-6576	175	24	a	a	DET
ejpam-6576	175	25	completely	completely	ADV
ejpam-6576	175	26	prime	prime	ADJ
ejpam-6576	175	27	right	right	NOUN
ejpam-6576	175	28	n	n	CCONJ
ejpam-6576	175	29	/	/	SYM
ejpam-6576	175	30	p	p	PROPN
ejpam-6576	175	31	group	group	NOUN
ejpam-6576	175	32	g	g	NOUN
ejpam-6576	175	33	of	of	ADP
ejpam-6576	175	34	type	type	NOUN
ejpam-6576	175	35	r(1	r(1	PROPN
ejpam-6576	175	36	)	)	PUNCT
ejpam-6576	175	37	such	such	ADJ
ejpam-6576	175	38	that	that	SCONJ
ejpam-6576	175	39	the	the	DET
ejpam-6576	175	40	zero	zero	NUM
ejpam-6576	175	41	ideal	ideal	NOUN
ejpam-6576	175	42	of	of	ADP
ejpam-6576	175	43	n	n	CCONJ
ejpam-6576	175	44	/	/	SYM
ejpam-6576	175	45	p	p	NOUN
ejpam-6576	175	46	is	be	AUX
ejpam-6576	175	47	the	the	DET
ejpam-6576	175	48	largest	large	ADJ
ejpam-6576	175	49	ideal	ideal	NOUN
ejpam-6576	175	50	of	of	ADP
ejpam-6576	175	51	n	n	CCONJ
ejpam-6576	175	52	/	/	SYM
ejpam-6576	175	53	p	p	NOUN
ejpam-6576	175	54	contained	contain	VERB
ejpam-6576	175	55	in	in	ADP
ejpam-6576	175	56	(	(	PUNCT
ejpam-6576	175	57	g	g	NOUN
ejpam-6576	175	58	:	:	PUNCT
ejpam-6576	175	59	0)n	0)n	PROPN
ejpam-6576	175	60	/	/	SYM
ejpam-6576	175	61	p	p	NOUN
ejpam-6576	175	62	.	.	PUNCT
ejpam-6576	176	1	by	by	ADP
ejpam-6576	176	2	propostion	propostion	NOUN
ejpam-6576	176	3	5	5	NUM
ejpam-6576	176	4	,	,	PUNCT
ejpam-6576	176	5	g	g	PROPN
ejpam-6576	176	6	is	be	AUX
ejpam-6576	176	7	also	also	ADV
ejpam-6576	176	8	a	a	DET
ejpam-6576	176	9	completely	completely	ADV
ejpam-6576	176	10	prime	prime	ADJ
ejpam-6576	176	11	right	right	NOUN
ejpam-6576	176	12	n	n	PRON
ejpam-6576	176	13	-group	-group	NOUN
ejpam-6576	176	14	of	of	ADP
ejpam-6576	176	15	type	type	NOUN
ejpam-6576	176	16	r(1	r(1	PROPN
ejpam-6576	176	17	)	)	PUNCT
ejpam-6576	176	18	,	,	PUNCT
ejpam-6576	176	19	where	where	SCONJ
ejpam-6576	176	20	gx	gx	PROPN
ejpam-6576	176	21	:	:	PUNCT
ejpam-6576	176	22	=	=	SYM
ejpam-6576	176	23	g(x	g(x	PROPN
ejpam-6576	176	24	+	+	CCONJ
ejpam-6576	176	25	i	i	NOUN
ejpam-6576	176	26	)	)	PUNCT
ejpam-6576	176	27	for	for	ADP
ejpam-6576	176	28	all	all	PRON
ejpam-6576	176	29	g	g	PROPN
ejpam-6576	176	30	∈	∈	PROPN
ejpam-6576	176	31	g	g	NOUN
ejpam-6576	176	32	and	and	CCONJ
ejpam-6576	176	33	x	x	SYM
ejpam-6576	176	34	∈	∈	PROPN
ejpam-6576	176	35	n	n	NOUN
ejpam-6576	176	36	and	and	CCONJ
ejpam-6576	176	37	(	(	PUNCT
ejpam-6576	176	38	g	g	NOUN
ejpam-6576	176	39	:	:	PUNCT
ejpam-6576	176	40	0)n	0)n	PROPN
ejpam-6576	176	41	/	/	SYM
ejpam-6576	176	42	p	p	NOUN
ejpam-6576	176	43	=	=	X
ejpam-6576	176	44	(	(	PUNCT
ejpam-6576	176	45	g	g	NOUN
ejpam-6576	176	46	:	:	PUNCT
ejpam-6576	176	47	0)n	0)n	PROPN
ejpam-6576	176	48	/	/	SYM
ejpam-6576	176	49	p	p	X
ejpam-6576	176	50	.	.	PUNCT
ejpam-6576	177	1	therefore	therefore	ADV
ejpam-6576	177	2	p	p	PROPN
ejpam-6576	177	3	is	be	AUX
ejpam-6576	177	4	the	the	DET
ejpam-6576	177	5	largest	large	ADJ
ejpam-6576	177	6	ideal	ideal	NOUN
ejpam-6576	177	7	of	of	ADP
ejpam-6576	177	8	n	n	NUM
ejpam-6576	177	9	contained	contain	VERB
ejpam-6576	177	10	in	in	ADP
ejpam-6576	177	11	(	(	PUNCT
ejpam-6576	177	12	g	g	NOUN
ejpam-6576	177	13	:	:	PUNCT
ejpam-6576	177	14	0)n	0)n	X
ejpam-6576	177	15	.	.	PUNCT
ejpam-6576	178	1	hence	hence	ADV
ejpam-6576	178	2	p	p	NOUN
ejpam-6576	178	3	is	be	AUX
ejpam-6576	178	4	a	a	DET
ejpam-6576	178	5	completely	completely	ADV
ejpam-6576	178	6	prime	prime	ADJ
ejpam-6576	178	7	ideal	ideal	NOUN
ejpam-6576	178	8	of	of	ADP
ejpam-6576	178	9	n	n	PROPN
ejpam-6576	178	10	of	of	ADP
ejpam-6576	178	11	type	type	NOUN
ejpam-6576	178	12	r(1	r(1	PROPN
ejpam-6576	178	13	)	)	PUNCT
ejpam-6576	178	14	.	.	PUNCT
ejpam-6576	179	1	k.	k.	PROPN
ejpam-6576	180	1	j.	j.	PROPN
ejpam-6576	180	2	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	180	3	et	et	PROPN
ejpam-6576	181	1	al	al	PROPN
ejpam-6576	181	2	.	.	PUNCT
ejpam-6576	181	3	/	/	SYM
ejpam-6576	181	4	eur	eur	PROPN
ejpam-6576	181	5	.	.	PUNCT
ejpam-6576	182	1	j.	j.	PROPN
ejpam-6576	182	2	pure	pure	PROPN
ejpam-6576	182	3	appl	appl	PROPN
ejpam-6576	182	4	.	.	PROPN
ejpam-6576	182	5	math	math	PROPN
ejpam-6576	182	6	,	,	PUNCT
ejpam-6576	182	7	18	18	NUM
ejpam-6576	182	8	(	(	PUNCT
ejpam-6576	182	9	3	3	NUM
ejpam-6576	182	10	)	)	PUNCT
ejpam-6576	182	11	(	(	PUNCT
ejpam-6576	182	12	2025	2025	NUM
ejpam-6576	182	13	)	)	PUNCT
ejpam-6576	182	14	,	,	PUNCT
ejpam-6576	182	15	6576	6576	NUM
ejpam-6576	182	16	5	5	NUM
ejpam-6576	182	17	of	of	ADP
ejpam-6576	182	18	7	7	NUM
ejpam-6576	182	19	definition	definition	NOUN
ejpam-6576	182	20	4	4	NUM
ejpam-6576	182	21	.	.	PUNCT
ejpam-6576	183	1	let	let	VERB
ejpam-6576	183	2	n	n	PRON
ejpam-6576	183	3	be	be	AUX
ejpam-6576	183	4	nearring	nearre	VERB
ejpam-6576	183	5	.	.	PUNCT
ejpam-6576	184	1	define	define	VERB
ejpam-6576	184	2	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	184	3	)	)	PUNCT
ejpam-6576	184	4	:	:	PUNCT
ejpam-6576	185	1	=	=	PUNCT
ejpam-6576	185	2	∩{i	∩{i	PRON
ejpam-6576	186	1	|	|	ADV
ejpam-6576	186	2	i	i	PRON
ejpam-6576	186	3	is	be	AUX
ejpam-6576	186	4	a	a	DET
ejpam-6576	186	5	completely	completely	ADV
ejpam-6576	186	6	prime	prime	ADJ
ejpam-6576	186	7	ideal	ideal	NOUN
ejpam-6576	186	8	of	of	ADP
ejpam-6576	186	9	n	n	PROPN
ejpam-6576	186	10	of	of	ADP
ejpam-6576	186	11	type	type	NOUN
ejpam-6576	186	12	r(1	r(1	PROPN
ejpam-6576	186	13	)	)	PUNCT
ejpam-6576	186	14	}	}	PUNCT
ejpam-6576	186	15	and	and	CCONJ
ejpam-6576	186	16	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	186	17	)	)	PUNCT
ejpam-6576	186	18	:	:	PUNCT
ejpam-6576	187	1	=	=	SYM
ejpam-6576	187	2	n	n	CCONJ
ejpam-6576	187	3	if	if	SCONJ
ejpam-6576	187	4	n	n	PRON
ejpam-6576	187	5	has	have	VERB
ejpam-6576	187	6	no	no	DET
ejpam-6576	187	7	completely	completely	ADV
ejpam-6576	187	8	prime	prime	ADJ
ejpam-6576	187	9	ideals	ideal	NOUN
ejpam-6576	187	10	of	of	ADP
ejpam-6576	187	11	type	type	NOUN
ejpam-6576	187	12	r(1	r(1	PROPN
ejpam-6576	187	13	)	)	PUNCT
ejpam-6576	187	14	.	.	PUNCT
ejpam-6576	188	1	pcr(1	pcr(1	PROPN
ejpam-6576	188	2	)	)	PUNCT
ejpam-6576	189	1	is	be	AUX
ejpam-6576	189	2	the	the	DET
ejpam-6576	189	3	completely	completely	ADV
ejpam-6576	189	4	prime	prime	ADJ
ejpam-6576	189	5	radical	radical	NOUN
ejpam-6576	189	6	of	of	ADP
ejpam-6576	189	7	type	type	NOUN
ejpam-6576	189	8	r(1	r(1	PROPN
ejpam-6576	189	9	)	)	PUNCT
ejpam-6576	189	10	.	.	PUNCT
ejpam-6576	190	1	proposition	proposition	NOUN
ejpam-6576	190	2	7	7	NUM
ejpam-6576	190	3	.	.	PUNCT
ejpam-6576	190	4	pcr(1	pcr(1	PROPN
ejpam-6576	190	5	)	)	PUNCT
ejpam-6576	190	6	is	be	AUX
ejpam-6576	190	7	a	a	DET
ejpam-6576	190	8	h	h	ADJ
ejpam-6576	190	9	-	-	PUNCT
ejpam-6576	190	10	radical	radical	ADJ
ejpam-6576	190	11	corresponding	corresponding	NOUN
ejpam-6576	190	12	to	to	ADP
ejpam-6576	190	13	the	the	DET
ejpam-6576	190	14	class	class	NOUN
ejpam-6576	190	15	of	of	ADP
ejpam-6576	190	16	all	all	DET
ejpam-6576	190	17	completely	completely	ADV
ejpam-6576	190	18	prime	prime	ADJ
ejpam-6576	190	19	zero	zero	NUM
ejpam-6576	190	20	-	-	PUNCT
ejpam-6576	190	21	symmetric	symmetric	NOUN
ejpam-6576	190	22	near	near	ADP
ejpam-6576	190	23	rings	ring	NOUN
ejpam-6576	190	24	of	of	ADP
ejpam-6576	190	25	type	type	NOUN
ejpam-6576	190	26	r(1	r(1	PROPN
ejpam-6576	190	27	)	)	PUNCT
ejpam-6576	190	28	.	.	PUNCT
ejpam-6576	191	1	proof	proof	NOUN
ejpam-6576	191	2	.	.	PUNCT
ejpam-6576	192	1	let	let	VERB
ejpam-6576	192	2	mcr(1	mcr(1	NOUN
ejpam-6576	192	3	)	)	PUNCT
ejpam-6576	193	1	:	:	PUNCT
ejpam-6576	193	2	=	=	SYM
ejpam-6576	193	3	{	{	PUNCT
ejpam-6576	193	4	n	n	CCONJ
ejpam-6576	193	5	|	|	ADV
ejpam-6576	193	6	n	n	ADV
ejpam-6576	193	7	is	be	AUX
ejpam-6576	193	8	completely	completely	ADV
ejpam-6576	193	9	prime	prime	ADJ
ejpam-6576	193	10	zero	zero	NUM
ejpam-6576	193	11	-	-	PUNCT
ejpam-6576	193	12	symmetric	symmetric	NOUN
ejpam-6576	193	13	near	near	ADP
ejpam-6576	193	14	rings	ring	NOUN
ejpam-6576	193	15	of	of	ADP
ejpam-6576	193	16	type	type	NOUN
ejpam-6576	193	17	r(1	r(1	PROPN
ejpam-6576	193	18	)	)	PUNCT
ejpam-6576	193	19	}	}	PUNCT
ejpam-6576	193	20	.	.	PUNCT
ejpam-6576	194	1	for	for	ADP
ejpam-6576	194	2	a	a	DET
ejpam-6576	194	3	near	near	ADJ
ejpam-6576	194	4	-	-	PUNCT
ejpam-6576	194	5	ring	ring	NOUN
ejpam-6576	194	6	n	n	NOUN
ejpam-6576	194	7	,	,	PUNCT
ejpam-6576	194	8	we	we	PRON
ejpam-6576	194	9	have	have	VERB
ejpam-6576	194	10	(	(	PUNCT
ejpam-6576	194	11	n)mcr(1	n)mcr(1	X
ejpam-6576	194	12	)	)	PUNCT
ejpam-6576	194	13	:	:	PUNCT
ejpam-6576	195	1	=	=	PUNCT
ejpam-6576	195	2	∩{j	∩{j	PUNCT
ejpam-6576	195	3	|	|	ADV
ejpam-6576	195	4	j	j	PROPN
ejpam-6576	195	5	is	be	AUX
ejpam-6576	195	6	an	an	DET
ejpam-6576	195	7	ideal	ideal	NOUN
ejpam-6576	195	8	of	of	ADP
ejpam-6576	195	9	n	n	PROPN
ejpam-6576	195	10	and	and	CCONJ
ejpam-6576	195	11	n	n	CCONJ
ejpam-6576	195	12	/	/	SYM
ejpam-6576	195	13	j	j	PROPN
ejpam-6576	195	14	∈	∈	PROPN
ejpam-6576	195	15	mcr(1	mcr(1	NOUN
ejpam-6576	195	16	)	)	PUNCT
ejpam-6576	195	17	}	}	PUNCT
ejpam-6576	195	18	.	.	PUNCT
ejpam-6576	196	1	now	now	ADV
ejpam-6576	196	2	q	q	X
ejpam-6576	196	3	defined	define	VERB
ejpam-6576	196	4	by	by	ADP
ejpam-6576	196	5	q(n	q(n	PROPN
ejpam-6576	196	6	)	)	PUNCT
ejpam-6576	196	7	:	:	PUNCT
ejpam-6576	196	8	=	=	SYM
ejpam-6576	196	9	(	(	PUNCT
ejpam-6576	196	10	n)mcr(1	n)mcr(1	NOUN
ejpam-6576	196	11	)	)	PUNCT
ejpam-6576	196	12	,	,	PUNCT
ejpam-6576	196	13	is	be	AUX
ejpam-6576	196	14	the	the	DET
ejpam-6576	196	15	h	h	ADJ
ejpam-6576	196	16	-	-	PUNCT
ejpam-6576	196	17	radical	radical	ADJ
ejpam-6576	196	18	corresponding	corresponding	NOUN
ejpam-6576	196	19	to	to	ADP
ejpam-6576	196	20	the	the	DET
ejpam-6576	196	21	class	class	NOUN
ejpam-6576	196	22	of	of	ADP
ejpam-6576	196	23	all	all	DET
ejpam-6576	196	24	completely	completely	ADV
ejpam-6576	196	25	prime	prime	ADJ
ejpam-6576	196	26	zero	zero	NUM
ejpam-6576	196	27	-	-	PUNCT
ejpam-6576	196	28	symmetric	symmetric	NOUN
ejpam-6576	196	29	near	near	ADP
ejpam-6576	196	30	rings	ring	NOUN
ejpam-6576	196	31	of	of	ADP
ejpam-6576	196	32	type	type	NOUN
ejpam-6576	196	33	r(1	r(1	PROPN
ejpam-6576	196	34	)	)	PUNCT
ejpam-6576	196	35	.	.	PUNCT
ejpam-6576	197	1	from	from	ADP
ejpam-6576	197	2	the	the	DET
ejpam-6576	197	3	corollaries	corollary	NOUN
ejpam-6576	197	4	1	1	NUM
ejpam-6576	197	5	and	and	CCONJ
ejpam-6576	197	6	2	2	NUM
ejpam-6576	197	7	,	,	PUNCT
ejpam-6576	197	8	pcr(1	pcr(1	X
ejpam-6576	197	9	)	)	PUNCT
ejpam-6576	198	1	=	=	PUNCT
ejpam-6576	198	2	q.	q.	NOUN
ejpam-6576	198	3	hence	hence	ADV
ejpam-6576	198	4	pcr(1	pcr(1	PROPN
ejpam-6576	198	5	)	)	PUNCT
ejpam-6576	198	6	is	be	AUX
ejpam-6576	198	7	the	the	DET
ejpam-6576	198	8	h	h	PROPN
ejpam-6576	198	9	radical	radical	PROPN
ejpam-6576	198	10	corresponding	corresponding	NOUN
ejpam-6576	198	11	to	to	ADP
ejpam-6576	198	12	the	the	DET
ejpam-6576	198	13	class	class	NOUN
ejpam-6576	198	14	of	of	ADP
ejpam-6576	198	15	all	all	DET
ejpam-6576	198	16	completely	completely	ADV
ejpam-6576	198	17	prime	prime	ADJ
ejpam-6576	198	18	zero	zero	NUM
ejpam-6576	198	19	-	-	PUNCT
ejpam-6576	198	20	symmetric	symmetric	NOUN
ejpam-6576	198	21	near	near	ADP
ejpam-6576	198	22	rings	ring	NOUN
ejpam-6576	198	23	of	of	ADP
ejpam-6576	198	24	type	type	NOUN
ejpam-6576	198	25	r(1	r(1	PROPN
ejpam-6576	198	26	)	)	PUNCT
ejpam-6576	198	27	.	.	PUNCT
ejpam-6576	199	1	lemma	lemma	PROPN
ejpam-6576	199	2	1	1	X
ejpam-6576	199	3	.	.	PUNCT
ejpam-6576	200	1	let	let	VERB
ejpam-6576	200	2	g	g	PRON
ejpam-6576	200	3	be	be	AUX
ejpam-6576	200	4	a	a	DET
ejpam-6576	200	5	completely	completely	ADV
ejpam-6576	200	6	prime	prime	ADJ
ejpam-6576	200	7	right	right	NOUN
ejpam-6576	200	8	n	n	PRON
ejpam-6576	200	9	-group	-group	NOUN
ejpam-6576	200	10	of	of	ADP
ejpam-6576	200	11	type	type	NOUN
ejpam-6576	200	12	(	(	PUNCT
ejpam-6576	200	13	1	1	NUM
ejpam-6576	200	14	)	)	PUNCT
ejpam-6576	201	1	and	and	CCONJ
ejpam-6576	201	2	i	i	PRON
ejpam-6576	201	3	be	be	VERB
ejpam-6576	201	4	an	an	DET
ejpam-6576	201	5	ideal	ideal	NOUN
ejpam-6576	201	6	of	of	ADP
ejpam-6576	201	7	n	n	PRON
ejpam-6576	201	8	and	and	CCONJ
ejpam-6576	201	9	gi	gi	VERB
ejpam-6576	201	10	̸=	̸=	PROPN
ejpam-6576	201	11	{	{	PUNCT
ejpam-6576	201	12	0	0	NUM
ejpam-6576	201	13	}	}	PUNCT
ejpam-6576	201	14	.	.	PUNCT
ejpam-6576	202	1	then	then	ADV
ejpam-6576	202	2	g	g	PROPN
ejpam-6576	202	3	is	be	AUX
ejpam-6576	202	4	a	a	DET
ejpam-6576	202	5	completely	completely	ADV
ejpam-6576	202	6	prime	prime	ADJ
ejpam-6576	202	7	right	right	INTJ
ejpam-6576	203	1	i	i	PROPN
ejpam-6576	203	2	-	-	PUNCT
ejpam-6576	203	3	group	group	NOUN
ejpam-6576	203	4	of	of	ADP
ejpam-6576	203	5	type	type	NOUN
ejpam-6576	203	6	1	1	NUM
ejpam-6576	203	7	and	and	CCONJ
ejpam-6576	203	8	(	(	PUNCT
ejpam-6576	203	9	g	g	NOUN
ejpam-6576	203	10	:	:	PUNCT
ejpam-6576	203	11	0)i	0)i	ADJ
ejpam-6576	203	12	⊇	⊇	NOUN
ejpam-6576	203	13	(	(	PUNCT
ejpam-6576	203	14	g	g	NOUN
ejpam-6576	203	15	:	:	PUNCT
ejpam-6576	203	16	0)n	0)n	PROPN
ejpam-6576	203	17	∩	∩	ADJ
ejpam-6576	203	18	i.	i.	NOUN
ejpam-6576	203	19	proof	proof	NOUN
ejpam-6576	203	20	.	.	PUNCT
ejpam-6576	204	1	since	since	SCONJ
ejpam-6576	204	2	g	g	PROPN
ejpam-6576	204	3	is	be	AUX
ejpam-6576	204	4	a	a	DET
ejpam-6576	204	5	right	right	ADJ
ejpam-6576	204	6	n	n	ADP
ejpam-6576	204	7	-group	-group	NOUN
ejpam-6576	204	8	and	and	CCONJ
ejpam-6576	204	9	gi	gi	VERB
ejpam-6576	204	10	̸=	̸=	PROPN
ejpam-6576	204	11	{	{	PUNCT
ejpam-6576	204	12	0	0	NUM
ejpam-6576	204	13	}	}	PUNCT
ejpam-6576	204	14	,	,	PUNCT
ejpam-6576	204	15	g	g	PROPN
ejpam-6576	204	16	is	be	AUX
ejpam-6576	204	17	also	also	ADV
ejpam-6576	204	18	a	a	DET
ejpam-6576	204	19	non	non	ADJ
ejpam-6576	204	20	-	-	ADJ
ejpam-6576	204	21	trivial	trivial	ADJ
ejpam-6576	204	22	i	i	NOUN
ejpam-6576	204	23	-	-	NOUN
ejpam-6576	204	24	group	group	NOUN
ejpam-6576	204	25	under	under	ADP
ejpam-6576	204	26	restriction	restriction	NOUN
ejpam-6576	204	27	.	.	PUNCT
ejpam-6576	205	1	let	let	VERB
ejpam-6576	205	2	h	h	PRON
ejpam-6576	205	3	be	be	AUX
ejpam-6576	205	4	a	a	DET
ejpam-6576	205	5	non	non	ADJ
ejpam-6576	205	6	-	-	ADJ
ejpam-6576	205	7	zero	zero	NUM
ejpam-6576	205	8	right	right	NOUN
ejpam-6576	205	9	i	i	PROPN
ejpam-6576	205	10	-	-	PUNCT
ejpam-6576	205	11	subgroup	subgroup	NOUN
ejpam-6576	205	12	of	of	ADP
ejpam-6576	205	13	g	g	PROPN
ejpam-6576	205	14	and	and	CCONJ
ejpam-6576	205	15	0	0	NUM
ejpam-6576	205	16	̸=	̸=	PROPN
ejpam-6576	205	17	h	h	NOUN
ejpam-6576	205	18	∈	∈	PROPN
ejpam-6576	205	19	h.	h.	NOUN
ejpam-6576	205	20	now	now	ADV
ejpam-6576	205	21	hi	hi	INTJ
ejpam-6576	205	22	̸=	̸=	PROPN
ejpam-6576	205	23	{	{	PUNCT
ejpam-6576	205	24	0	0	NUM
ejpam-6576	205	25	}	}	PUNCT
ejpam-6576	205	26	as	as	ADP
ejpam-6576	205	27	gi	gi	PROPN
ejpam-6576	205	28	̸=	̸=	PROPN
ejpam-6576	205	29	{	{	PUNCT
ejpam-6576	205	30	0	0	NUM
ejpam-6576	205	31	}	}	PUNCT
ejpam-6576	205	32	.	.	PUNCT
ejpam-6576	206	1	let	let	VERB
ejpam-6576	206	2	t	t	NOUN
ejpam-6576	206	3	be	be	AUX
ejpam-6576	206	4	the	the	DET
ejpam-6576	206	5	subgroup	subgroup	NOUN
ejpam-6576	206	6	of	of	ADP
ejpam-6576	206	7	g	g	PROPN
ejpam-6576	206	8	generated	generate	VERB
ejpam-6576	206	9	by	by	ADP
ejpam-6576	206	10	hi	hi	INTJ
ejpam-6576	206	11	:	:	PUNCT
ejpam-6576	206	12	=	=	PRON
ejpam-6576	206	13	{	{	PUNCT
ejpam-6576	206	14	ha	ha	INTJ
ejpam-6576	206	15	|	|	ADV
ejpam-6576	206	16	a	a	DET
ejpam-6576	206	17	∈	∈	NOUN
ejpam-6576	206	18	i	i	X
ejpam-6576	206	19	}	}	PUNCT
ejpam-6576	206	20	.	.	PUNCT
ejpam-6576	207	1	t	t	PROPN
ejpam-6576	207	2	is	be	AUX
ejpam-6576	207	3	a	a	DET
ejpam-6576	207	4	right	right	ADJ
ejpam-6576	207	5	i	i	PROPN
ejpam-6576	207	6	-	-	PUNCT
ejpam-6576	207	7	subgroup	subgroup	NOUN
ejpam-6576	207	8	of	of	ADP
ejpam-6576	207	9	g	g	PROPN
ejpam-6576	207	10	as	as	ADV
ejpam-6576	207	11	well	well	ADV
ejpam-6576	207	12	as	as	ADP
ejpam-6576	207	13	right	right	ADJ
ejpam-6576	207	14	n	n	PRON
ejpam-6576	207	15	-subgroup	-subgroup	NOUN
ejpam-6576	207	16	of	of	ADP
ejpam-6576	207	17	g.	g.	PROPN
ejpam-6576	207	18	being	be	AUX
ejpam-6576	207	19	right	right	ADJ
ejpam-6576	207	20	n	n	DET
ejpam-6576	207	21	-subgroup	-subgroup	NOUN
ejpam-6576	207	22	of	of	ADP
ejpam-6576	207	23	g	g	PROPN
ejpam-6576	207	24	,	,	PUNCT
ejpam-6576	207	25	t	t	PROPN
ejpam-6576	207	26	has	have	VERB
ejpam-6576	207	27	a	a	DET
ejpam-6576	207	28	distributive	distributive	ADJ
ejpam-6576	207	29	element	element	NOUN
ejpam-6576	207	30	t0	t0	NOUN
ejpam-6576	207	31	.	.	PUNCT
ejpam-6576	208	1	so	so	ADV
ejpam-6576	208	2	t0(x+	t0(x+	ADV
ejpam-6576	208	3	y	y	NOUN
ejpam-6576	208	4	)	)	PUNCT
ejpam-6576	209	1	=	=	PUNCT
ejpam-6576	210	1	t0x+	t0x+	PRON
ejpam-6576	210	2	t0y	t0y	ADJ
ejpam-6576	210	3	for	for	ADP
ejpam-6576	210	4	all	all	DET
ejpam-6576	210	5	x	x	NOUN
ejpam-6576	210	6	,	,	PUNCT
ejpam-6576	210	7	y	y	PROPN
ejpam-6576	210	8	∈	∈	PROPN
ejpam-6576	210	9	n	n	ADV
ejpam-6576	210	10	.	.	PUNCT
ejpam-6576	211	1	hence	hence	ADV
ejpam-6576	211	2	t0	t0	PROPN
ejpam-6576	211	3	is	be	AUX
ejpam-6576	211	4	a	a	DET
ejpam-6576	211	5	distributive	distributive	ADJ
ejpam-6576	211	6	element	element	NOUN
ejpam-6576	211	7	of	of	ADP
ejpam-6576	211	8	the	the	DET
ejpam-6576	211	9	right	right	ADJ
ejpam-6576	212	1	i	i	PROPN
ejpam-6576	212	2	-	-	PUNCT
ejpam-6576	212	3	group	group	NOUN
ejpam-6576	212	4	t	t	PROPN
ejpam-6576	212	5	⊆	⊆	NUM
ejpam-6576	212	6	h.	h.	NOUN
ejpam-6576	212	7	for	for	ADP
ejpam-6576	212	8	0	0	NUM
ejpam-6576	212	9	̸=	̸=	PROPN
ejpam-6576	212	10	g1	g1	NOUN
ejpam-6576	212	11	,	,	PUNCT
ejpam-6576	212	12	0	0	NUM
ejpam-6576	212	13	̸=	̸=	PROPN
ejpam-6576	212	14	g2	g2	PROPN
ejpam-6576	212	15	∈	∈	PROPN
ejpam-6576	212	16	g	g	PROPN
ejpam-6576	212	17	,	,	PUNCT
ejpam-6576	212	18	(	(	PUNCT
ejpam-6576	212	19	g1	g1	PROPN
ejpam-6576	212	20	:	:	PUNCT
ejpam-6576	212	21	0)i	0)i	ADJ
ejpam-6576	212	22	=	=	SYM
ejpam-6576	212	23	(	(	PUNCT
ejpam-6576	212	24	g1	g1	PROPN
ejpam-6576	212	25	:	:	PUNCT
ejpam-6576	212	26	0)n	0)n	NOUN
ejpam-6576	212	27	∩	∩	ADJ
ejpam-6576	212	28	i	i	PRON
ejpam-6576	212	29	=	=	SYM
ejpam-6576	212	30	(	(	PUNCT
ejpam-6576	212	31	g2	g2	PROPN
ejpam-6576	212	32	:	:	PUNCT
ejpam-6576	212	33	0)n	0)n	NOUN
ejpam-6576	212	34	∩	∩	ADJ
ejpam-6576	212	35	i	i	PRON
ejpam-6576	212	36	=	=	SYM
ejpam-6576	212	37	(	(	PUNCT
ejpam-6576	212	38	g2	g2	PROPN
ejpam-6576	212	39	:	:	PUNCT
ejpam-6576	212	40	0)i	0)i	ADJ
ejpam-6576	212	41	.	.	PUNCT
ejpam-6576	213	1	also	also	ADV
ejpam-6576	213	2	(	(	PUNCT
ejpam-6576	213	3	g	g	NOUN
ejpam-6576	213	4	:	:	PUNCT
ejpam-6576	213	5	0)i	0)i	ADJ
ejpam-6576	213	6	=	=	SYM
ejpam-6576	213	7	(	(	PUNCT
ejpam-6576	213	8	g1	g1	PROPN
ejpam-6576	213	9	:	:	PUNCT
ejpam-6576	213	10	0)i	0)i	ADJ
ejpam-6576	213	11	=	=	SYM
ejpam-6576	213	12	(	(	PUNCT
ejpam-6576	213	13	g1	g1	PROPN
ejpam-6576	213	14	:	:	PUNCT
ejpam-6576	213	15	0)n	0)n	NOUN
ejpam-6576	213	16	∩	∩	ADJ
ejpam-6576	213	17	i	i	PRON
ejpam-6576	213	18	=	=	PUNCT
ejpam-6576	213	19	(	(	PUNCT
ejpam-6576	213	20	g	g	NOUN
ejpam-6576	213	21	:	:	PUNCT
ejpam-6576	213	22	0)n	0)n	PROPN
ejpam-6576	213	23	∩	∩	PROPN
ejpam-6576	213	24	i.	i.	PROPN
ejpam-6576	214	1	therefore	therefore	ADV
ejpam-6576	214	2	(	(	PUNCT
ejpam-6576	214	3	g	g	NOUN
ejpam-6576	214	4	:	:	PUNCT
ejpam-6576	214	5	0)i	0)i	ADJ
ejpam-6576	214	6	⊇	⊇	NOUN
ejpam-6576	214	7	(	(	PUNCT
ejpam-6576	214	8	g	g	NOUN
ejpam-6576	214	9	:	:	PUNCT
ejpam-6576	214	10	0)n	0)n	PROPN
ejpam-6576	214	11	∩	∩	PROPN
ejpam-6576	214	12	i.	i.	PROPN
ejpam-6576	214	13	lemma	lemma	PROPN
ejpam-6576	214	14	2	2	X
ejpam-6576	214	15	.	.	PUNCT
ejpam-6576	215	1	let	let	VERB
ejpam-6576	215	2	g	g	PRON
ejpam-6576	215	3	be	be	AUX
ejpam-6576	215	4	a	a	DET
ejpam-6576	215	5	completely	completely	ADV
ejpam-6576	215	6	prime	prime	ADJ
ejpam-6576	215	7	right	right	INTJ
ejpam-6576	216	1	i	i	PROPN
ejpam-6576	216	2	-	-	PUNCT
ejpam-6576	216	3	group	group	NOUN
ejpam-6576	216	4	of	of	ADP
ejpam-6576	216	5	type	type	NOUN
ejpam-6576	216	6	1	1	NUM
ejpam-6576	216	7	and	and	CCONJ
ejpam-6576	216	8	i	i	PRON
ejpam-6576	216	9	,	,	PUNCT
ejpam-6576	216	10	an	an	DET
ejpam-6576	216	11	ideal	ideal	NOUN
ejpam-6576	216	12	of	of	ADP
ejpam-6576	216	13	a	a	DET
ejpam-6576	216	14	near	near	ADJ
ejpam-6576	216	15	-	-	PUNCT
ejpam-6576	216	16	ring	ring	NOUN
ejpam-6576	216	17	n	n	NOUN
ejpam-6576	216	18	.	.	PUNCT
ejpam-6576	217	1	then	then	ADV
ejpam-6576	217	2	h	h	NOUN
ejpam-6576	217	3	:	:	PUNCT
ejpam-6576	217	4	=	=	PUNCT
ejpam-6576	217	5	g0i	g0i	PROPN
ejpam-6576	217	6	is	be	AUX
ejpam-6576	217	7	a	a	DET
ejpam-6576	217	8	completely	completely	ADV
ejpam-6576	217	9	prime	prime	ADJ
ejpam-6576	217	10	right	right	NOUN
ejpam-6576	218	1	n	n	PRON
ejpam-6576	218	2	-group	-group	NOUN
ejpam-6576	218	3	of	of	ADP
ejpam-6576	218	4	type	type	NOUN
ejpam-6576	218	5	r(1	r(1	PROPN
ejpam-6576	218	6	)	)	PUNCT
ejpam-6576	218	7	and	and	CCONJ
ejpam-6576	218	8	(	(	PUNCT
ejpam-6576	218	9	g	g	NOUN
ejpam-6576	218	10	:	:	PUNCT
ejpam-6576	218	11	0)i	0)i	ADJ
ejpam-6576	218	12	=	=	SYM
ejpam-6576	218	13	(	(	PUNCT
ejpam-6576	218	14	h	h	NOUN
ejpam-6576	218	15	:	:	PUNCT
ejpam-6576	218	16	0)i	0)i	ADJ
ejpam-6576	218	17	⊇	⊇	NOUN
ejpam-6576	218	18	(	(	PUNCT
ejpam-6576	218	19	h	h	NOUN
ejpam-6576	218	20	:	:	PUNCT
ejpam-6576	218	21	0)n	0)n	PROPN
ejpam-6576	218	22	∩	∩	PROPN
ejpam-6576	218	23	i	i	PRON
ejpam-6576	218	24	,	,	PUNCT
ejpam-6576	218	25	g0	g0	PROPN
ejpam-6576	218	26	is	be	AUX
ejpam-6576	218	27	a	a	DET
ejpam-6576	218	28	distributive	distributive	ADJ
ejpam-6576	218	29	element	element	NOUN
ejpam-6576	218	30	of	of	ADP
ejpam-6576	218	31	the	the	DET
ejpam-6576	218	32	right	right	ADJ
ejpam-6576	218	33	i	i	PROPN
ejpam-6576	218	34	-	-	PUNCT
ejpam-6576	218	35	group	group	NOUN
ejpam-6576	218	36	g.	g.	NOUN
ejpam-6576	218	37	proof	proof	NOUN
ejpam-6576	218	38	.	.	PUNCT
ejpam-6576	219	1	it	it	PRON
ejpam-6576	219	2	is	be	AUX
ejpam-6576	219	3	clear	clear	ADJ
ejpam-6576	219	4	that	that	SCONJ
ejpam-6576	219	5	h	h	NOUN
ejpam-6576	219	6	=	=	PUNCT
ejpam-6576	219	7	g0i	g0i	PROPN
ejpam-6576	219	8	:	:	PUNCT
ejpam-6576	219	9	=	=	SYM
ejpam-6576	219	10	{	{	PUNCT
ejpam-6576	219	11	g0a	g0a	ADJ
ejpam-6576	219	12	|	|	ADV
ejpam-6576	219	13	a	a	DET
ejpam-6576	219	14	∈	∈	NOUN
ejpam-6576	220	1	i	i	X
ejpam-6576	220	2	}	}	PUNCT
ejpam-6576	220	3	is	be	AUX
ejpam-6576	220	4	a	a	DET
ejpam-6576	220	5	subgroup	subgroup	NOUN
ejpam-6576	220	6	of	of	ADP
ejpam-6576	220	7	(	(	PUNCT
ejpam-6576	220	8	g,+	g,+	PROPN
ejpam-6576	220	9	)	)	PUNCT
ejpam-6576	220	10	and	and	CCONJ
ejpam-6576	220	11	hence	hence	ADV
ejpam-6576	220	12	h	h	NOUN
ejpam-6576	220	13	is	be	AUX
ejpam-6576	220	14	a	a	DET
ejpam-6576	220	15	right	right	ADJ
ejpam-6576	221	1	i	i	PROPN
ejpam-6576	221	2	-	-	PUNCT
ejpam-6576	221	3	subgroup	subgroup	NOUN
ejpam-6576	221	4	of	of	ADP
ejpam-6576	221	5	g.	g.	PROPN
ejpam-6576	221	6	for	for	ADP
ejpam-6576	221	7	g0a	g0a	PROPN
ejpam-6576	221	8	∈	∈	PROPN
ejpam-6576	221	9	h	h	NOUN
ejpam-6576	221	10	define	define	VERB
ejpam-6576	221	11	(	(	PUNCT
ejpam-6576	221	12	g0a)x	g0a)x	X
ejpam-6576	221	13	:	:	PUNCT
ejpam-6576	221	14	=	=	SYM
ejpam-6576	221	15	g0(ax	g0(ax	PROPN
ejpam-6576	221	16	)	)	PUNCT
ejpam-6576	221	17	for	for	ADP
ejpam-6576	221	18	all	all	DET
ejpam-6576	221	19	x	x	SYM
ejpam-6576	221	20	∈	∈	NOUN
ejpam-6576	221	21	n	n	NOUN
ejpam-6576	221	22	.	.	PUNCT
ejpam-6576	222	1	we	we	PRON
ejpam-6576	222	2	claim	claim	VERB
ejpam-6576	222	3	that	that	SCONJ
ejpam-6576	222	4	this	this	DET
ejpam-6576	222	5	operation	operation	NOUN
ejpam-6576	222	6	is	be	AUX
ejpam-6576	222	7	well	well	ADV
ejpam-6576	222	8	defined	define	VERB
ejpam-6576	222	9	.	.	PUNCT
ejpam-6576	223	1	let	let	VERB
ejpam-6576	223	2	g0a	g0a	PRON
ejpam-6576	223	3	=	=	SYM
ejpam-6576	223	4	g0b	g0b	PROPN
ejpam-6576	223	5	,	,	PUNCT
ejpam-6576	223	6	a	a	PRON
ejpam-6576	223	7	,	,	PUNCT
ejpam-6576	223	8	b	b	X
ejpam-6576	223	9	∈	∈	NOUN
ejpam-6576	224	1	i	i	PRON
ejpam-6576	224	2	and	and	CCONJ
ejpam-6576	224	3	x	x	SYM
ejpam-6576	224	4	∈	∈	PROPN
ejpam-6576	224	5	n	n	NOUN
ejpam-6576	224	6	and	and	CCONJ
ejpam-6576	224	7	c	c	PROPN
ejpam-6576	224	8	∈	∈	PROPN
ejpam-6576	224	9	i.	i.	NOUN
ejpam-6576	225	1	[	[	X
ejpam-6576	225	2	g0(ax)−g0(bx)]c	g0(ax)−g0(bx)]c	NOUN
ejpam-6576	225	3	=	=	SYM
ejpam-6576	225	4	g0(ax)c−g0(bx)c	g0(ax)c−g0(bx)c	NOUN
ejpam-6576	225	5	=	=	SYM
ejpam-6576	225	6	(	(	PUNCT
ejpam-6576	225	7	g0a)(xc)−(g0b)(xc	g0a)(xc)−(g0b)(xc	NOUN
ejpam-6576	225	8	)	)	PUNCT
ejpam-6576	225	9	=	=	SYM
ejpam-6576	226	1	(	(	PUNCT
ejpam-6576	226	2	g0a−g0b)(xc	g0a−g0b)(xc	ADJ
ejpam-6576	226	3	)	)	PUNCT
ejpam-6576	226	4	=	=	SYM
ejpam-6576	226	5	0(xc	0(xc	NUM
ejpam-6576	226	6	)	)	PUNCT
ejpam-6576	226	7	=	=	SYM
ejpam-6576	227	1	0	0	X
ejpam-6576	227	2	.	.	PUNCT
ejpam-6576	228	1	if	if	SCONJ
ejpam-6576	228	2	g0(ax	g0(ax	PROPN
ejpam-6576	228	3	)	)	PUNCT
ejpam-6576	228	4	−	−	PROPN
ejpam-6576	228	5	g0(bx	g0(bx	PROPN
ejpam-6576	228	6	)	)	PUNCT
ejpam-6576	228	7	̸=	̸=	PROPN
ejpam-6576	228	8	0	0	NUM
ejpam-6576	228	9	then	then	ADV
ejpam-6576	228	10	gi	gi	VERB
ejpam-6576	228	11	=	=	SYM
ejpam-6576	228	12	0	0	PROPN
ejpam-6576	228	13	,	,	PUNCT
ejpam-6576	228	14	a	a	DET
ejpam-6576	228	15	contraction	contraction	NOUN
ejpam-6576	228	16	.	.	PUNCT
ejpam-6576	229	1	so	so	ADV
ejpam-6576	229	2	the	the	DET
ejpam-6576	229	3	operation	operation	NOUN
ejpam-6576	229	4	is	be	AUX
ejpam-6576	229	5	well	well	ADV
ejpam-6576	229	6	defined	define	VERB
ejpam-6576	229	7	and	and	CCONJ
ejpam-6576	229	8	h	h	NOUN
ejpam-6576	229	9	is	be	AUX
ejpam-6576	229	10	a	a	DET
ejpam-6576	229	11	right	right	NOUN
ejpam-6576	229	12	n	n	NUM
ejpam-6576	229	13	-group	-group	NOUN
ejpam-6576	229	14	with	with	ADP
ejpam-6576	229	15	hn	hn	PROPN
ejpam-6576	229	16	̸=	̸=	PROPN
ejpam-6576	229	17	(	(	PUNCT
ejpam-6576	229	18	0	0	NUM
ejpam-6576	229	19	)	)	PUNCT
ejpam-6576	229	20	.	.	PUNCT
ejpam-6576	230	1	let	let	VERB
ejpam-6576	230	2	△	△	X
ejpam-6576	230	3	be	be	AUX
ejpam-6576	230	4	a	a	DET
ejpam-6576	230	5	non	non	ADJ
ejpam-6576	230	6	zero	zero	NUM
ejpam-6576	230	7	right	right	NOUN
ejpam-6576	230	8	n	n	PRON
ejpam-6576	230	9	-subgroup	-subgroup	NOUN
ejpam-6576	230	10	of	of	ADP
ejpam-6576	230	11	h.	h.	PROPN
ejpam-6576	230	12	it	it	PRON
ejpam-6576	230	13	is	be	AUX
ejpam-6576	230	14	clear	clear	ADJ
ejpam-6576	230	15	that	that	SCONJ
ejpam-6576	230	16	△	△	PROPN
ejpam-6576	230	17	is	be	AUX
ejpam-6576	230	18	also	also	ADV
ejpam-6576	230	19	a	a	DET
ejpam-6576	230	20	right	right	ADJ
ejpam-6576	231	1	i	i	PROPN
ejpam-6576	231	2	-	-	PUNCT
ejpam-6576	231	3	subgroup	subgroup	NOUN
ejpam-6576	231	4	of	of	ADP
ejpam-6576	231	5	h	h	NOUN
ejpam-6576	232	1	and	and	CCONJ
ejpam-6576	232	2	hence	hence	ADV
ejpam-6576	232	3	there	there	PRON
ejpam-6576	232	4	is	be	VERB
ejpam-6576	232	5	a	a	DET
ejpam-6576	232	6	h0	h0	PROPN
ejpam-6576	232	7	∈	∈	PROPN
ejpam-6576	232	8	△	△	X
ejpam-6576	232	9	such	such	ADJ
ejpam-6576	232	10	that	that	SCONJ
ejpam-6576	232	11	h0(a	h0(a	PROPN
ejpam-6576	232	12	+	+	NUM
ejpam-6576	232	13	b	b	NOUN
ejpam-6576	232	14	)	)	PUNCT
ejpam-6576	232	15	=	=	NOUN
ejpam-6576	233	1	h0a	h0a	NOUN
ejpam-6576	233	2	+	+	CCONJ
ejpam-6576	233	3	h0b	h0b	NOUN
ejpam-6576	233	4	for	for	ADP
ejpam-6576	233	5	all	all	DET
ejpam-6576	233	6	a	a	DET
ejpam-6576	233	7	,	,	PUNCT
ejpam-6576	233	8	b	b	X
ejpam-6576	233	9	∈	∈	PROPN
ejpam-6576	233	10	i.	i.	NOUN
ejpam-6576	233	11	now	now	ADV
ejpam-6576	233	12	h0	h0	PROPN
ejpam-6576	233	13	:	:	PUNCT
ejpam-6576	233	14	=	=	PUNCT
ejpam-6576	233	15	g0d	g0d	PROPN
ejpam-6576	233	16	for	for	ADP
ejpam-6576	233	17	some	some	DET
ejpam-6576	233	18	d	d	PROPN
ejpam-6576	233	19	∈	∈	PROPN
ejpam-6576	233	20	i.	i.	NOUN
ejpam-6576	233	21	let	let	VERB
ejpam-6576	233	22	x	x	PRON
ejpam-6576	233	23	,	,	PUNCT
ejpam-6576	233	24	y	y	PROPN
ejpam-6576	233	25	∈	∈	PROPN
ejpam-6576	233	26	n	n	ADV
ejpam-6576	233	27	.	.	PUNCT
ejpam-6576	234	1	[	[	X
ejpam-6576	234	2	h0(x	h0(x	X
ejpam-6576	234	3	+	+	NUM
ejpam-6576	234	4	y	y	NOUN
ejpam-6576	234	5	)	)	PUNCT
ejpam-6576	234	6	−	−	PROPN
ejpam-6576	234	7	(	(	PUNCT
ejpam-6576	234	8	h0x	h0x	NOUN
ejpam-6576	234	9	+	+	CCONJ
ejpam-6576	234	10	h0y)]c	h0y)]c	NOUN
ejpam-6576	234	11	=	=	SYM
ejpam-6576	234	12	(	(	PUNCT
ejpam-6576	234	13	g0d)(x	g0d)(x	NOUN
ejpam-6576	234	14	+	+	CCONJ
ejpam-6576	234	15	y)c	y)c	NOUN
ejpam-6576	234	16	−	−	PROPN
ejpam-6576	234	17	(	(	PUNCT
ejpam-6576	234	18	(	(	PUNCT
ejpam-6576	234	19	g0d)xc	g0d)xc	X
ejpam-6576	234	20	+	+	CCONJ
ejpam-6576	234	21	(	(	PUNCT
ejpam-6576	234	22	g0d)yc	g0d)yc	NOUN
ejpam-6576	234	23	)	)	PUNCT
ejpam-6576	234	24	=	=	PUNCT
ejpam-6576	234	25	(	(	PUNCT
ejpam-6576	234	26	g0d)(xc	g0d)(xc	NOUN
ejpam-6576	234	27	+	+	CCONJ
ejpam-6576	234	28	yc	yc	NOUN
ejpam-6576	234	29	)	)	PUNCT
ejpam-6576	235	1	−	−	PROPN
ejpam-6576	235	2	(	(	PUNCT
ejpam-6576	235	3	(	(	PUNCT
ejpam-6576	235	4	g0d)(xc	g0d)(xc	NOUN
ejpam-6576	235	5	)	)	PUNCT
ejpam-6576	235	6	+	+	CCONJ
ejpam-6576	235	7	(	(	PUNCT
ejpam-6576	235	8	g0d)(yc	g0d)(yc	NOUN
ejpam-6576	235	9	)	)	PUNCT
ejpam-6576	235	10	)	)	PUNCT
ejpam-6576	236	1	=	=	SYM
ejpam-6576	236	2	h0(xc	h0(xc	NUM
ejpam-6576	236	3	)	)	PUNCT
ejpam-6576	236	4	+	+	CCONJ
ejpam-6576	236	5	h0(yc)−	h0(yc)−	NOUN
ejpam-6576	236	6	(	(	PUNCT
ejpam-6576	236	7	h0(xc	h0(xc	NUM
ejpam-6576	236	8	)	)	PUNCT
ejpam-6576	236	9	+	+	CCONJ
ejpam-6576	236	10	h0(yc	h0(yc	NUM
ejpam-6576	236	11	)	)	PUNCT
ejpam-6576	236	12	)	)	PUNCT
ejpam-6576	237	1	=	=	SYM
ejpam-6576	237	2	0	0	NUM
ejpam-6576	238	1	for	for	ADP
ejpam-6576	238	2	all	all	DET
ejpam-6576	238	3	c	c	PROPN
ejpam-6576	238	4	∈	∈	PROPN
ejpam-6576	238	5	i.	i.	NOUN
ejpam-6576	238	6	therefore	therefore	ADV
ejpam-6576	238	7	h0(x	h0(x	PROPN
ejpam-6576	238	8	+	+	NOUN
ejpam-6576	238	9	y	y	NOUN
ejpam-6576	238	10	)	)	PUNCT
ejpam-6576	239	1	=	=	SYM
ejpam-6576	239	2	h0x	h0x	NOUN
ejpam-6576	239	3	+	+	CCONJ
ejpam-6576	239	4	h0y	h0y	NOUN
ejpam-6576	239	5	and	and	CCONJ
ejpam-6576	239	6	hence	hence	ADV
ejpam-6576	239	7	a	a	DET
ejpam-6576	239	8	distributive	distributive	ADJ
ejpam-6576	239	9	element	element	NOUN
ejpam-6576	239	10	of	of	ADP
ejpam-6576	239	11	the	the	DET
ejpam-6576	239	12	right	right	ADJ
ejpam-6576	239	13	n	n	PRON
ejpam-6576	239	14	-subgroup	-subgroup	NOUN
ejpam-6576	239	15	△	△	PROPN
ejpam-6576	239	16	.	.	PUNCT
ejpam-6576	240	1	let	let	VERB
ejpam-6576	240	2	0	0	NUM
ejpam-6576	240	3	̸=	̸=	PROPN
ejpam-6576	240	4	g0a	g0a	X
ejpam-6576	240	5	,	,	PUNCT
ejpam-6576	240	6	0	0	NUM
ejpam-6576	241	1	̸=	̸=	PROPN
ejpam-6576	241	2	g0b	g0b	PROPN
ejpam-6576	241	3	∈	∈	PROPN
ejpam-6576	241	4	h	h	NOUN
ejpam-6576	241	5	,	,	PUNCT
ejpam-6576	241	6	a	a	PRON
ejpam-6576	241	7	,	,	PUNCT
ejpam-6576	241	8	b	b	PROPN
ejpam-6576	241	9	∈	∈	PROPN
ejpam-6576	241	10	i.	i.	NOUN
ejpam-6576	241	11	(	(	PUNCT
ejpam-6576	241	12	g0a	g0a	X
ejpam-6576	241	13	:	:	PUNCT
ejpam-6576	241	14	0)n	0)n	X
ejpam-6576	241	15	=	=	PUNCT
ejpam-6576	241	16	(	(	PUNCT
ejpam-6576	241	17	g0a	g0a	X
ejpam-6576	241	18	:	:	PUNCT
ejpam-6576	241	19	0)i	0)i	ADJ
ejpam-6576	241	20	∩n	∩n	NOUN
ejpam-6576	241	21	=	=	PRON
ejpam-6576	241	22	(	(	PUNCT
ejpam-6576	241	23	g0b	g0b	PROPN
ejpam-6576	241	24	:	:	PUNCT
ejpam-6576	241	25	0)i	0)i	ADJ
ejpam-6576	241	26	∩n	∩n	NOUN
ejpam-6576	241	27	=	=	PRON
ejpam-6576	241	28	(	(	PUNCT
ejpam-6576	241	29	g0b	g0b	PROPN
ejpam-6576	241	30	:	:	PUNCT
ejpam-6576	241	31	0)n	0)n	X
ejpam-6576	241	32	.	.	PUNCT
ejpam-6576	242	1	therefore	therefore	ADV
ejpam-6576	242	2	h	h	PROPN
ejpam-6576	242	3	is	be	AUX
ejpam-6576	242	4	a	a	DET
ejpam-6576	242	5	completely	completely	ADV
ejpam-6576	242	6	prime	prime	ADJ
ejpam-6576	242	7	n	n	PRON
ejpam-6576	242	8	-group	-group	NOUN
ejpam-6576	242	9	of	of	ADP
ejpam-6576	242	10	type	type	NOUN
ejpam-6576	242	11	r(1	r(1	PROPN
ejpam-6576	242	12	)	)	PUNCT
ejpam-6576	242	13	.	.	PUNCT
ejpam-6576	243	1	we	we	PRON
ejpam-6576	243	2	have	have	VERB
ejpam-6576	243	3	(	(	PUNCT
ejpam-6576	243	4	h	h	NOUN
ejpam-6576	243	5	:	:	PUNCT
ejpam-6576	243	6	0)i	0)i	X
ejpam-6576	244	1	=	=	PUNCT
ejpam-6576	245	1	(	(	PUNCT
ejpam-6576	245	2	h	h	NOUN
ejpam-6576	245	3	:	:	PUNCT
ejpam-6576	245	4	0)n	0)n	X
ejpam-6576	245	5	∩i	∩i	PROPN
ejpam-6576	245	6	.	.	PUNCT
ejpam-6576	246	1	so	so	ADV
ejpam-6576	246	2	(	(	PUNCT
ejpam-6576	246	3	g	g	NOUN
ejpam-6576	246	4	:	:	PUNCT
ejpam-6576	246	5	0)i	0)i	ADJ
ejpam-6576	246	6	=	=	SYM
ejpam-6576	246	7	(	(	PUNCT
ejpam-6576	246	8	h	h	NOUN
ejpam-6576	246	9	:	:	PUNCT
ejpam-6576	246	10	0)i	0)i	ADJ
ejpam-6576	246	11	⊇	⊇	NOUN
ejpam-6576	246	12	(	(	PUNCT
ejpam-6576	246	13	h	h	NOUN
ejpam-6576	246	14	:	:	PUNCT
ejpam-6576	246	15	0)n	0)n	PROPN
ejpam-6576	247	1	∩	∩	PROPN
ejpam-6576	247	2	i.	i.	PROPN
ejpam-6576	247	3	k.	k.	PROPN
ejpam-6576	248	1	j.	j.	PROPN
ejpam-6576	248	2	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	248	3	et	et	PROPN
ejpam-6576	249	1	al	al	PROPN
ejpam-6576	249	2	.	.	PUNCT
ejpam-6576	249	3	/	/	SYM
ejpam-6576	249	4	eur	eur	PROPN
ejpam-6576	249	5	.	.	PUNCT
ejpam-6576	250	1	j.	j.	PROPN
ejpam-6576	250	2	pure	pure	PROPN
ejpam-6576	250	3	appl	appl	PROPN
ejpam-6576	250	4	.	.	PROPN
ejpam-6576	250	5	math	math	PROPN
ejpam-6576	250	6	,	,	PUNCT
ejpam-6576	250	7	18	18	NUM
ejpam-6576	250	8	(	(	PUNCT
ejpam-6576	250	9	3	3	NUM
ejpam-6576	250	10	)	)	PUNCT
ejpam-6576	250	11	(	(	PUNCT
ejpam-6576	250	12	2025	2025	NUM
ejpam-6576	250	13	)	)	PUNCT
ejpam-6576	250	14	,	,	PUNCT
ejpam-6576	250	15	6576	6576	NUM
ejpam-6576	250	16	6	6	NUM
ejpam-6576	250	17	of	of	ADP
ejpam-6576	250	18	7	7	NUM
ejpam-6576	250	19	theorem	theorem	ADJ
ejpam-6576	250	20	1	1	NUM
ejpam-6576	250	21	.	.	PUNCT
ejpam-6576	250	22	pcr(1	pcr(1	PROPN
ejpam-6576	250	23	)	)	PUNCT
ejpam-6576	250	24	is	be	AUX
ejpam-6576	250	25	a	a	DET
ejpam-6576	250	26	complete	complete	ADJ
ejpam-6576	250	27	h	h	NOUN
ejpam-6576	250	28	-	-	PUNCT
ejpam-6576	250	29	radical	radical	ADJ
ejpam-6576	250	30	.	.	PUNCT
ejpam-6576	251	1	proof	proof	NOUN
ejpam-6576	251	2	.	.	PUNCT
ejpam-6576	252	1	suppose	suppose	VERB
ejpam-6576	252	2	that	that	SCONJ
ejpam-6576	252	3	pcr(1)(j	pcr(1)(j	NOUN
ejpam-6576	252	4	)	)	PUNCT
ejpam-6576	253	1	=	=	SYM
ejpam-6576	253	2	j	j	PROPN
ejpam-6576	253	3	,	,	PUNCT
ejpam-6576	253	4	j	j	PROPN
ejpam-6576	253	5	is	be	AUX
ejpam-6576	253	6	an	an	DET
ejpam-6576	253	7	ideal	ideal	NOUN
ejpam-6576	253	8	of	of	ADP
ejpam-6576	253	9	n	n	PROPN
ejpam-6576	253	10	.	.	PUNCT
ejpam-6576	254	1	we	we	PRON
ejpam-6576	254	2	claim	claim	VERB
ejpam-6576	254	3	that	that	SCONJ
ejpam-6576	254	4	j	j	PROPN
ejpam-6576	254	5	⊆	⊆	NUM
ejpam-6576	254	6	pcr(1)(n	pcr(1)(n	NOUN
ejpam-6576	254	7	)	)	PUNCT
ejpam-6576	254	8	.	.	PUNCT
ejpam-6576	255	1	on	on	ADP
ejpam-6576	255	2	the	the	DET
ejpam-6576	255	3	contrary	contrary	NOUN
ejpam-6576	255	4	suppose	suppose	VERB
ejpam-6576	255	5	that	that	SCONJ
ejpam-6576	255	6	j	j	PROPN
ejpam-6576	255	7	̸⊆	̸⊆	PROPN
ejpam-6576	255	8	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	255	9	)	)	PUNCT
ejpam-6576	255	10	.	.	PUNCT
ejpam-6576	256	1	so	so	ADV
ejpam-6576	256	2	there	there	PRON
ejpam-6576	256	3	is	be	VERB
ejpam-6576	256	4	completely	completely	ADV
ejpam-6576	256	5	prime	prime	ADJ
ejpam-6576	257	1	right	right	ADJ
ejpam-6576	257	2	n	n	NUM
ejpam-6576	257	3	-group	-group	NOUN
ejpam-6576	257	4	g	g	NOUN
ejpam-6576	257	5	of	of	ADP
ejpam-6576	257	6	type	type	NOUN
ejpam-6576	257	7	r(1	r(1	PROPN
ejpam-6576	257	8	)	)	PUNCT
ejpam-6576	257	9	such	such	ADJ
ejpam-6576	257	10	that	that	PRON
ejpam-6576	257	11	gj	gj	NOUN
ejpam-6576	257	12	̸=	̸=	PROPN
ejpam-6576	257	13	{	{	PUNCT
ejpam-6576	257	14	0	0	NUM
ejpam-6576	257	15	}	}	PUNCT
ejpam-6576	257	16	.	.	PUNCT
ejpam-6576	258	1	by	by	ADP
ejpam-6576	258	2	lemma	lemma	PROPN
ejpam-6576	258	3	1	1	NUM
ejpam-6576	258	4	,	,	PUNCT
ejpam-6576	258	5	g	g	PROPN
ejpam-6576	258	6	is	be	AUX
ejpam-6576	258	7	a	a	DET
ejpam-6576	258	8	completely	completely	ADV
ejpam-6576	258	9	prime	prime	ADJ
ejpam-6576	258	10	right	right	ADJ
ejpam-6576	258	11	j	j	PROPN
ejpam-6576	258	12	-	-	NOUN
ejpam-6576	258	13	group	group	NOUN
ejpam-6576	258	14	of	of	ADP
ejpam-6576	258	15	type	type	NOUN
ejpam-6576	258	16	r(1	r(1	PROPN
ejpam-6576	258	17	)	)	PUNCT
ejpam-6576	258	18	.	.	PUNCT
ejpam-6576	259	1	this	this	PRON
ejpam-6576	259	2	is	be	AUX
ejpam-6576	259	3	a	a	DET
ejpam-6576	259	4	contradiction	contradiction	NOUN
ejpam-6576	259	5	to	to	PART
ejpam-6576	259	6	pcr(1)(j	pcr(1)(j	VERB
ejpam-6576	259	7	)	)	PUNCT
ejpam-6576	260	1	=	=	SYM
ejpam-6576	260	2	j	j	PROPN
ejpam-6576	260	3	.	.	PUNCT
ejpam-6576	261	1	therefore	therefore	ADV
ejpam-6576	261	2	j	j	PROPN
ejpam-6576	261	3	⊆	⊆	NUM
ejpam-6576	261	4	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	261	5	)	)	PUNCT
ejpam-6576	261	6	.	.	PUNCT
ejpam-6576	262	1	hence	hence	ADV
ejpam-6576	262	2	pcr(1	pcr(1	PROPN
ejpam-6576	262	3	)	)	PUNCT
ejpam-6576	262	4	is	be	AUX
ejpam-6576	262	5	a	a	DET
ejpam-6576	262	6	complete	complete	ADJ
ejpam-6576	262	7	h	h	NOUN
ejpam-6576	262	8	-	-	PUNCT
ejpam-6576	262	9	radical	radical	ADJ
ejpam-6576	262	10	.	.	PUNCT
ejpam-6576	263	1	theorem	theorem	NOUN
ejpam-6576	263	2	2	2	NUM
ejpam-6576	263	3	.	.	PUNCT
ejpam-6576	264	1	the	the	DET
ejpam-6576	264	2	h	h	NOUN
ejpam-6576	264	3	-	-	PUNCT
ejpam-6576	264	4	radical	radical	ADJ
ejpam-6576	264	5	pcr(1	pcr(1	NOUN
ejpam-6576	264	6	)	)	PUNCT
ejpam-6576	264	7	is	be	AUX
ejpam-6576	264	8	r	r	NOUN
ejpam-6576	264	9	-	-	NOUN
ejpam-6576	264	10	hereditary	hereditary	ADJ
ejpam-6576	264	11	.	.	PUNCT
ejpam-6576	265	1	proof	proof	NOUN
ejpam-6576	265	2	.	.	PUNCT
ejpam-6576	266	1	if	if	SCONJ
ejpam-6576	266	2	j	j	PROPN
ejpam-6576	266	3	is	be	AUX
ejpam-6576	266	4	an	an	DET
ejpam-6576	266	5	ideal	ideal	NOUN
ejpam-6576	266	6	of	of	ADP
ejpam-6576	266	7	a	a	DET
ejpam-6576	266	8	near	near	ADJ
ejpam-6576	266	9	-	-	PUNCT
ejpam-6576	266	10	ring	ring	NOUN
ejpam-6576	266	11	n	n	NOUN
ejpam-6576	266	12	and	and	CCONJ
ejpam-6576	266	13	g	g	PROPN
ejpam-6576	266	14	is	be	AUX
ejpam-6576	266	15	a	a	DET
ejpam-6576	266	16	completely	completely	ADV
ejpam-6576	266	17	prime	prime	ADJ
ejpam-6576	266	18	right	right	ADJ
ejpam-6576	266	19	j	j	PROPN
ejpam-6576	266	20	-	-	NOUN
ejpam-6576	266	21	group	group	NOUN
ejpam-6576	266	22	of	of	ADP
ejpam-6576	266	23	type	type	NOUN
ejpam-6576	266	24	1	1	NUM
ejpam-6576	266	25	then	then	ADV
ejpam-6576	266	26	by	by	ADP
ejpam-6576	266	27	lemma	lemma	PROPN
ejpam-6576	266	28	2	2	NUM
ejpam-6576	266	29	,	,	PUNCT
ejpam-6576	266	30	there	there	PRON
ejpam-6576	266	31	is	be	VERB
ejpam-6576	266	32	a	a	DET
ejpam-6576	266	33	right	right	ADJ
ejpam-6576	266	34	j	j	PROPN
ejpam-6576	266	35	-	-	PROPN
ejpam-6576	266	36	subgroup	subgroup	PROPN
ejpam-6576	266	37	h	h	NOUN
ejpam-6576	266	38	of	of	ADP
ejpam-6576	266	39	g	g	PROPN
ejpam-6576	266	40	such	such	ADJ
ejpam-6576	266	41	that	that	PRON
ejpam-6576	266	42	h	h	NOUN
ejpam-6576	266	43	is	be	AUX
ejpam-6576	266	44	a	a	DET
ejpam-6576	266	45	completely	completely	ADV
ejpam-6576	266	46	prime	prime	ADJ
ejpam-6576	266	47	right	right	NOUN
ejpam-6576	266	48	n	n	PRON
ejpam-6576	266	49	-group	-group	NOUN
ejpam-6576	266	50	of	of	ADP
ejpam-6576	266	51	type	type	NOUN
ejpam-6576	266	52	r(1	r(1	PROPN
ejpam-6576	266	53	)	)	PUNCT
ejpam-6576	266	54	and	and	CCONJ
ejpam-6576	266	55	(	(	PUNCT
ejpam-6576	266	56	g	g	NOUN
ejpam-6576	266	57	:	:	PUNCT
ejpam-6576	266	58	0)j	0)j	PROPN
ejpam-6576	266	59	=	=	SYM
ejpam-6576	266	60	(	(	PUNCT
ejpam-6576	266	61	h	h	NOUN
ejpam-6576	266	62	:	:	PUNCT
ejpam-6576	267	1	0)j	0)j	PROPN
ejpam-6576	267	2	⊇	⊇	NOUN
ejpam-6576	267	3	(	(	PUNCT
ejpam-6576	267	4	h	h	NOUN
ejpam-6576	267	5	:	:	PUNCT
ejpam-6576	267	6	0)n	0)n	PROPN
ejpam-6576	267	7	∩	∩	PROPN
ejpam-6576	267	8	j	j	PROPN
ejpam-6576	267	9	.	.	PUNCT
ejpam-6576	267	10	therefore	therefore	ADV
ejpam-6576	267	11	pcr(1)(j	pcr(1)(j	PROPN
ejpam-6576	267	12	)	)	PUNCT
ejpam-6576	267	13	⊇	⊇	PROPN
ejpam-6576	267	14	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	267	15	)	)	PUNCT
ejpam-6576	267	16	∩	∩	PROPN
ejpam-6576	267	17	j	j	PROPN
ejpam-6576	267	18	.	.	PUNCT
ejpam-6576	268	1	hence	hence	ADV
ejpam-6576	268	2	pcr(1	pcr(1	PROPN
ejpam-6576	268	3	)	)	PUNCT
ejpam-6576	268	4	is	be	AUX
ejpam-6576	268	5	r	r	NOUN
ejpam-6576	268	6	-	-	NOUN
ejpam-6576	268	7	hereditary	hereditary	ADJ
ejpam-6576	268	8	.	.	PUNCT
ejpam-6576	269	1	theorem	theorem	NOUN
ejpam-6576	269	2	3	3	NUM
ejpam-6576	269	3	.	.	PUNCT
ejpam-6576	269	4	pcr(1	pcr(1	PROPN
ejpam-6576	269	5	)	)	PUNCT
ejpam-6576	270	1	is	be	AUX
ejpam-6576	270	2	an	an	DET
ejpam-6576	270	3	idempotent	idempotent	ADJ
ejpam-6576	270	4	h	h	ADJ
ejpam-6576	270	5	-	-	PUNCT
ejpam-6576	270	6	radical	radical	ADJ
ejpam-6576	270	7	.	.	PUNCT
ejpam-6576	271	1	proof	proof	NOUN
ejpam-6576	271	2	.	.	PUNCT
ejpam-6576	272	1	let	let	VERB
ejpam-6576	272	2	n	n	PRON
ejpam-6576	272	3	be	be	AUX
ejpam-6576	272	4	a	a	DET
ejpam-6576	272	5	near	near	ADJ
ejpam-6576	272	6	-	-	PUNCT
ejpam-6576	272	7	ring	ring	NOUN
ejpam-6576	272	8	and	and	CCONJ
ejpam-6576	272	9	j	j	NOUN
ejpam-6576	272	10	:	:	PUNCT
ejpam-6576	272	11	=	=	NOUN
ejpam-6576	272	12	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	272	13	)	)	PUNCT
ejpam-6576	272	14	.	.	PUNCT
ejpam-6576	273	1	from	from	ADP
ejpam-6576	273	2	theorem	theorem	ADJ
ejpam-6576	273	3	2	2	NUM
ejpam-6576	273	4	,	,	PUNCT
ejpam-6576	273	5	pcr(1)(j	pcr(1)(j	ADJ
ejpam-6576	273	6	)	)	PUNCT
ejpam-6576	273	7	⊇	⊇	PROPN
ejpam-6576	273	8	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	273	9	)	)	PUNCT
ejpam-6576	273	10	∩	∩	PROPN
ejpam-6576	273	11	j	j	PROPN
ejpam-6576	273	12	.	.	PUNCT
ejpam-6576	274	1	taking	take	VERB
ejpam-6576	274	2	pcr(1)(n	pcr(1)(n	NOUN
ejpam-6576	274	3	)	)	PUNCT
ejpam-6576	274	4	for	for	ADP
ejpam-6576	274	5	j	j	PROPN
ejpam-6576	274	6	in	in	ADP
ejpam-6576	274	7	the	the	DET
ejpam-6576	274	8	above	above	ADJ
ejpam-6576	274	9	inclusion	inclusion	NOUN
ejpam-6576	274	10	,	,	PUNCT
ejpam-6576	274	11	we	we	PRON
ejpam-6576	274	12	have	have	VERB
ejpam-6576	274	13	pcr(1)(pcr(1)(n	pcr(1)(pcr(1)(n	NUM
ejpam-6576	274	14	)	)	PUNCT
ejpam-6576	274	15	)	)	PUNCT
ejpam-6576	275	1	=	=	SYM
ejpam-6576	275	2	pcr(1)(n	pcr(1)(n	PROPN
ejpam-6576	275	3	)	)	PUNCT
ejpam-6576	275	4	.	.	PUNCT
ejpam-6576	276	1	therefore	therefore	ADV
ejpam-6576	276	2	pcr(1	pcr(1	PROPN
ejpam-6576	276	3	)	)	PUNCT
ejpam-6576	276	4	is	be	AUX
ejpam-6576	276	5	an	an	DET
ejpam-6576	276	6	idempotent	idempotent	ADJ
ejpam-6576	276	7	h	h	NOUN
ejpam-6576	276	8	-	-	PUNCT
ejpam-6576	276	9	radical	radical	ADJ
ejpam-6576	276	10	.	.	PUNCT
ejpam-6576	277	1	from	from	ADP
ejpam-6576	277	2	proposition	proposition	NOUN
ejpam-6576	277	3	7	7	NUM
ejpam-6576	277	4	and	and	CCONJ
ejpam-6576	277	5	theorems	theorem	NOUN
ejpam-6576	277	6	3	3	NUM
ejpam-6576	277	7	and	and	CCONJ
ejpam-6576	277	8	1	1	NUM
ejpam-6576	277	9	,	,	PUNCT
ejpam-6576	277	10	we	we	PRON
ejpam-6576	277	11	have	have	VERB
ejpam-6576	277	12	the	the	DET
ejpam-6576	277	13	following	following	NOUN
ejpam-6576	277	14	:	:	PUNCT
ejpam-6576	277	15	theorem	theorem	NOUN
ejpam-6576	277	16	4	4	NUM
ejpam-6576	277	17	.	.	PUNCT
ejpam-6576	278	1	the	the	DET
ejpam-6576	278	2	h	h	NOUN
ejpam-6576	278	3	-	-	PUNCT
ejpam-6576	278	4	radical	radical	ADJ
ejpam-6576	278	5	pcr(1	pcr(1	NOUN
ejpam-6576	278	6	)	)	PUNCT
ejpam-6576	278	7	is	be	AUX
ejpam-6576	278	8	a	a	DET
ejpam-6576	278	9	kurosh	kurosh	ADV
ejpam-6576	278	10	-	-	PUNCT
ejpam-6576	278	11	amitsur	amitsur	NOUN
ejpam-6576	278	12	radical	radical	NOUN
ejpam-6576	278	13	in	in	ADP
ejpam-6576	278	14	the	the	DET
ejpam-6576	278	15	class	class	NOUN
ejpam-6576	278	16	of	of	ADP
ejpam-6576	278	17	all	all	DET
ejpam-6576	278	18	zerosymmetric	zerosymmetric	ADJ
ejpam-6576	278	19	near	near	NOUN
ejpam-6576	278	20	-	-	PUNCT
ejpam-6576	278	21	rings	ring	NOUN
ejpam-6576	278	22	.	.	PUNCT
ejpam-6576	279	1	proof	proof	NOUN
ejpam-6576	279	2	.	.	PUNCT
ejpam-6576	280	1	since	since	SCONJ
ejpam-6576	280	2	an	an	DET
ejpam-6576	280	3	idempotent	idempotent	NOUN
ejpam-6576	280	4	,	,	PUNCT
ejpam-6576	280	5	complete	complete	ADJ
ejpam-6576	280	6	h	h	NOUN
ejpam-6576	280	7	-	-	PUNCT
ejpam-6576	280	8	radical	radical	ADJ
ejpam-6576	280	9	is	be	AUX
ejpam-6576	280	10	a	a	DET
ejpam-6576	280	11	kurosh	kurosh	ADV
ejpam-6576	280	12	-	-	PUNCT
ejpam-6576	280	13	amitsur	amitsur	NOUN
ejpam-6576	280	14	radical	radical	NOUN
ejpam-6576	280	15	,	,	PUNCT
ejpam-6576	280	16	from	from	ADP
ejpam-6576	280	17	proposition	proposition	NOUN
ejpam-6576	280	18	7	7	NUM
ejpam-6576	280	19	and	and	CCONJ
ejpam-6576	280	20	theorems	theorem	NOUN
ejpam-6576	280	21	1	1	NUM
ejpam-6576	280	22	and	and	CCONJ
ejpam-6576	280	23	3	3	NUM
ejpam-6576	280	24	,	,	PUNCT
ejpam-6576	280	25	pcr(1	pcr(1	PROPN
ejpam-6576	280	26	)	)	PUNCT
ejpam-6576	280	27	is	be	AUX
ejpam-6576	280	28	a	a	DET
ejpam-6576	280	29	kurosh	kurosh	ADV
ejpam-6576	280	30	-	-	PUNCT
ejpam-6576	280	31	amitsur	amitsur	NOUN
ejpam-6576	280	32	radical	radical	NOUN
ejpam-6576	280	33	.	.	PUNCT
ejpam-6576	281	1	3	3	X
ejpam-6576	281	2	.	.	X
ejpam-6576	281	3	conclusions	conclusion	NOUN
ejpam-6576	281	4	some	some	DET
ejpam-6576	281	5	important	important	ADJ
ejpam-6576	281	6	kurosh	kurosh	NOUN
ejpam-6576	281	7	-	-	PUNCT
ejpam-6576	281	8	amitsur	amitsur	ADJ
ejpam-6576	281	9	prime	prime	ADJ
ejpam-6576	281	10	radicals	radical	NOUN
ejpam-6576	281	11	of	of	ADP
ejpam-6576	281	12	rings	ring	NOUN
ejpam-6576	281	13	fails	fail	VERB
ejpam-6576	281	14	to	to	PART
ejpam-6576	281	15	be	be	AUX
ejpam-6576	281	16	kurosh	kurosh	ADV
ejpam-6576	281	17	-	-	PUNCT
ejpam-6576	281	18	amitsur	amitsur	NOUN
ejpam-6576	281	19	radicals	radical	NOUN
ejpam-6576	281	20	,	,	PUNCT
ejpam-6576	281	21	when	when	SCONJ
ejpam-6576	281	22	they	they	PRON
ejpam-6576	281	23	are	be	AUX
ejpam-6576	281	24	generalized	generalize	VERB
ejpam-6576	281	25	to	to	ADP
ejpam-6576	281	26	near	near	ADJ
ejpam-6576	281	27	-	-	PUNCT
ejpam-6576	281	28	rings	ring	NOUN
ejpam-6576	281	29	using	use	VERB
ejpam-6576	281	30	left	left	ADJ
ejpam-6576	281	31	n	n	CCONJ
ejpam-6576	281	32	-	-	PUNCT
ejpam-6576	281	33	groups	group	NOUN
ejpam-6576	281	34	.	.	PUNCT
ejpam-6576	282	1	to	to	PART
ejpam-6576	282	2	see	see	VERB
ejpam-6576	282	3	whether	whether	SCONJ
ejpam-6576	282	4	the	the	DET
ejpam-6576	282	5	situation	situation	NOUN
ejpam-6576	282	6	will	will	AUX
ejpam-6576	282	7	be	be	AUX
ejpam-6576	282	8	different	different	ADJ
ejpam-6576	282	9	with	with	ADP
ejpam-6576	282	10	right	right	ADJ
ejpam-6576	283	1	n	n	DET
ejpam-6576	283	2	-groups	-group	NOUN
ejpam-6576	283	3	was	be	AUX
ejpam-6576	283	4	studied	study	VERB
ejpam-6576	283	5	in	in	ADP
ejpam-6576	283	6	[	[	X
ejpam-6576	283	7	6	6	NUM
ejpam-6576	283	8	]	]	PUNCT
ejpam-6576	283	9	,	,	PUNCT
ejpam-6576	283	10	[	[	X
ejpam-6576	283	11	5	5	NUM
ejpam-6576	283	12	]	]	PUNCT
ejpam-6576	283	13	and	and	CCONJ
ejpam-6576	283	14	[	[	X
ejpam-6576	283	15	8	8	NUM
ejpam-6576	283	16	]	]	PUNCT
ejpam-6576	283	17	.	.	PUNCT
ejpam-6576	284	1	as	as	SCONJ
ejpam-6576	284	2	proved	prove	VERB
ejpam-6576	284	3	in	in	ADP
ejpam-6576	284	4	[	[	X
ejpam-6576	284	5	6	6	NUM
ejpam-6576	284	6	]	]	PUNCT
ejpam-6576	284	7	,	,	PUNCT
ejpam-6576	284	8	right	right	ADJ
ejpam-6576	284	9	n	n	PRON
ejpam-6576	284	10	-groups	-group	NOUN
ejpam-6576	284	11	play	play	VERB
ejpam-6576	284	12	significant	significant	ADJ
ejpam-6576	284	13	role	role	NOUN
ejpam-6576	284	14	in	in	ADP
ejpam-6576	284	15	generalizing	generalize	VERB
ejpam-6576	284	16	the	the	DET
ejpam-6576	284	17	prime	prime	ADJ
ejpam-6576	284	18	radicals	radical	NOUN
ejpam-6576	284	19	of	of	ADP
ejpam-6576	284	20	rings	ring	NOUN
ejpam-6576	284	21	to	to	ADP
ejpam-6576	284	22	near	near	ADJ
ejpam-6576	284	23	-	-	PUNCT
ejpam-6576	284	24	rings	ring	NOUN
ejpam-6576	284	25	.	.	PUNCT
ejpam-6576	285	1	this	this	DET
ejpam-6576	285	2	fact	fact	NOUN
ejpam-6576	285	3	was	be	AUX
ejpam-6576	285	4	completely	completely	ADV
ejpam-6576	285	5	established	establish	VERB
ejpam-6576	285	6	in	in	ADP
ejpam-6576	285	7	[	[	X
ejpam-6576	285	8	5	5	NUM
ejpam-6576	285	9	]	]	PUNCT
ejpam-6576	285	10	in	in	ADP
ejpam-6576	285	11	general	general	ADJ
ejpam-6576	285	12	.	.	PUNCT
ejpam-6576	286	1	in	in	ADP
ejpam-6576	286	2	[	[	X
ejpam-6576	286	3	8	8	NUM
ejpam-6576	286	4	]	]	PUNCT
ejpam-6576	286	5	,	,	PUNCT
ejpam-6576	286	6	the	the	DET
ejpam-6576	286	7	prime	prime	ADJ
ejpam-6576	286	8	radical	radical	NOUN
ejpam-6576	286	9	of	of	ADP
ejpam-6576	286	10	rings	ring	NOUN
ejpam-6576	286	11	was	be	AUX
ejpam-6576	286	12	generalized	generalize	VERB
ejpam-6576	286	13	to	to	ADP
ejpam-6576	286	14	near	near	ADJ
ejpam-6576	286	15	-	-	PUNCT
ejpam-6576	286	16	rings	ring	NOUN
ejpam-6576	286	17	using	use	VERB
ejpam-6576	286	18	right	right	ADJ
ejpam-6576	286	19	n	n	NUM
ejpam-6576	286	20	-groups	-group	NOUN
ejpam-6576	286	21	which	which	PRON
ejpam-6576	286	22	is	be	AUX
ejpam-6576	286	23	also	also	ADV
ejpam-6576	286	24	a	a	DET
ejpam-6576	286	25	kurosh	kurosh	ADV
ejpam-6576	286	26	-	-	PUNCT
ejpam-6576	286	27	amitsur	amitsur	NOUN
ejpam-6576	286	28	radical	radical	NOUN
ejpam-6576	286	29	of	of	ADP
ejpam-6576	286	30	near	near	ADJ
ejpam-6576	286	31	-	-	PUNCT
ejpam-6576	286	32	rings	ring	NOUN
ejpam-6576	286	33	.	.	PUNCT
ejpam-6576	287	1	in	in	ADP
ejpam-6576	287	2	this	this	DET
ejpam-6576	287	3	article	article	NOUN
ejpam-6576	287	4	,	,	PUNCT
ejpam-6576	287	5	the	the	DET
ejpam-6576	287	6	completely	completely	ADV
ejpam-6576	287	7	prime	prime	ADJ
ejpam-6576	287	8	radical	radical	NOUN
ejpam-6576	287	9	of	of	ADP
ejpam-6576	287	10	rings	ring	NOUN
ejpam-6576	287	11	which	which	PRON
ejpam-6576	287	12	is	be	AUX
ejpam-6576	287	13	a	a	DET
ejpam-6576	287	14	kurosh	kurosh	ADV
ejpam-6576	287	15	-	-	PUNCT
ejpam-6576	287	16	amitsur	amitsur	NOUN
ejpam-6576	287	17	radical	radical	NOUN
ejpam-6576	287	18	of	of	ADP
ejpam-6576	287	19	rings	ring	NOUN
ejpam-6576	287	20	is	be	AUX
ejpam-6576	287	21	generalized	generalize	VERB
ejpam-6576	287	22	to	to	ADP
ejpam-6576	287	23	near	near	ADJ
ejpam-6576	287	24	-	-	PUNCT
ejpam-6576	287	25	rings	ring	NOUN
ejpam-6576	287	26	using	use	VERB
ejpam-6576	287	27	right	right	ADJ
ejpam-6576	287	28	n	n	NUM
ejpam-6576	287	29	-groups	-group	NOUN
ejpam-6576	287	30	which	which	PRON
ejpam-6576	287	31	is	be	AUX
ejpam-6576	287	32	also	also	ADV
ejpam-6576	287	33	a	a	DET
ejpam-6576	287	34	kurosh	kurosh	ADV
ejpam-6576	287	35	-	-	PUNCT
ejpam-6576	287	36	amitsur	amitsur	NOUN
ejpam-6576	287	37	radical	radical	NOUN
ejpam-6576	287	38	of	of	ADP
ejpam-6576	287	39	near	near	ADJ
ejpam-6576	287	40	-	-	PUNCT
ejpam-6576	287	41	rings	ring	NOUN
ejpam-6576	287	42	.	.	PUNCT
ejpam-6576	288	1	k.	k.	PROPN
ejpam-6576	289	1	j.	j.	PROPN
ejpam-6576	289	2	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	289	3	et	et	PROPN
ejpam-6576	290	1	al	al	PROPN
ejpam-6576	290	2	.	.	PUNCT
ejpam-6576	290	3	/	/	SYM
ejpam-6576	290	4	eur	eur	PROPN
ejpam-6576	290	5	.	.	PUNCT
ejpam-6576	291	1	j.	j.	PROPN
ejpam-6576	291	2	pure	pure	PROPN
ejpam-6576	291	3	appl	appl	PROPN
ejpam-6576	291	4	.	.	PROPN
ejpam-6576	291	5	math	math	PROPN
ejpam-6576	291	6	,	,	PUNCT
ejpam-6576	291	7	18	18	NUM
ejpam-6576	291	8	(	(	PUNCT
ejpam-6576	291	9	3	3	NUM
ejpam-6576	291	10	)	)	PUNCT
ejpam-6576	291	11	(	(	PUNCT
ejpam-6576	291	12	2025	2025	NUM
ejpam-6576	291	13	)	)	PUNCT
ejpam-6576	291	14	,	,	PUNCT
ejpam-6576	291	15	6576	6576	NUM
ejpam-6576	291	16	7	7	NUM
ejpam-6576	291	17	of	of	ADP
ejpam-6576	291	18	7	7	NUM
ejpam-6576	291	19	references	reference	NOUN
ejpam-6576	291	20	[	[	X
ejpam-6576	291	21	1	1	NUM
ejpam-6576	291	22	]	]	X
ejpam-6576	291	23	n.	n.	PROPN
ejpam-6576	291	24	j.	j.	PROPN
ejpam-6576	291	25	groenewald	groenewald	PROPN
ejpam-6576	291	26	.	.	PUNCT
ejpam-6576	292	1	note	note	NOUN
ejpam-6576	292	2	on	on	ADP
ejpam-6576	292	3	the	the	DET
ejpam-6576	292	4	completely	completely	ADV
ejpam-6576	292	5	prime	prime	ADJ
ejpam-6576	292	6	radical	radical	ADJ
ejpam-6576	292	7	in	in	ADP
ejpam-6576	292	8	near	near	ADJ
ejpam-6576	292	9	-	-	PUNCT
ejpam-6576	292	10	rings	ring	NOUN
ejpam-6576	292	11	.	.	PUNCT
ejpam-6576	293	1	near	near	ADP
ejpam-6576	293	2	-	-	PUNCT
ejpam-6576	293	3	rings	ring	NOUN
ejpam-6576	293	4	and	and	CCONJ
ejpam-6576	293	5	near	near	NOUN
ejpam-6576	293	6	-	-	PUNCT
ejpam-6576	293	7	fields	field	NOUN
ejpam-6576	293	8	,	,	PUNCT
ejpam-6576	293	9	north	north	NOUN
ejpam-6576	293	10	-	-	PUNCT
ejpam-6576	293	11	holland	holland	PROPN
ejpam-6576	293	12	mathematics	mathematics	PROPN
ejpam-6576	293	13	studies	study	NOUN
ejpam-6576	293	14	,	,	PUNCT
ejpam-6576	293	15	137:97–100	137:97–100	NUM
ejpam-6576	293	16	,	,	PUNCT
ejpam-6576	293	17	1987	1987	NUM
ejpam-6576	293	18	.	.	PUNCT
ejpam-6576	294	1	[	[	X
ejpam-6576	294	2	2	2	X
ejpam-6576	294	3	]	]	X
ejpam-6576	294	4	g.	g.	PROPN
ejpam-6576	294	5	l.	l.	PROPN
ejpam-6576	294	6	booth	booth	PROPN
ejpam-6576	294	7	and	and	CCONJ
ejpam-6576	294	8	n.	n.	PROPN
ejpam-6576	294	9	j.	j.	PROPN
ejpam-6576	294	10	groenewald	groenewald	PROPN
ejpam-6576	294	11	.	.	PUNCT
ejpam-6576	295	1	equiprime	equiprime	PROPN
ejpam-6576	295	2	left	leave	VERB
ejpam-6576	295	3	ideals	ideal	NOUN
ejpam-6576	295	4	and	and	CCONJ
ejpam-6576	295	5	equiprime	equiprime	NOUN
ejpam-6576	295	6	n	n	CCONJ
ejpam-6576	295	7	-	-	PUNCT
ejpam-6576	295	8	groups	group	NOUN
ejpam-6576	295	9	of	of	ADP
ejpam-6576	295	10	a	a	DET
ejpam-6576	295	11	near	near	ADV
ejpam-6576	295	12	-	-	PUNCT
ejpam-6576	295	13	ring	ring	NOUN
ejpam-6576	295	14	.	.	PUNCT
ejpam-6576	296	1	contributions	contribution	NOUN
ejpam-6576	296	2	to	to	ADP
ejpam-6576	296	3	general	general	ADJ
ejpam-6576	296	4	algebra	algebra	NOUN
ejpam-6576	296	5	,	,	PUNCT
ejpam-6576	296	6	8:25–38	8:25–38	NUM
ejpam-6576	296	7	,	,	PUNCT
ejpam-6576	296	8	1992	1992	NUM
ejpam-6576	296	9	.	.	PUNCT
ejpam-6576	297	1	[	[	X
ejpam-6576	297	2	3	3	X
ejpam-6576	297	3	]	]	X
ejpam-6576	297	4	n.	n.	PROPN
ejpam-6576	297	5	j.	j.	PROPN
ejpam-6576	297	6	groenewald	groenewald	PROPN
ejpam-6576	297	7	.	.	PUNCT
ejpam-6576	298	1	completely	completely	ADV
ejpam-6576	298	2	prime	prime	ADJ
ejpam-6576	298	3	submodules	submodule	NOUN
ejpam-6576	298	4	.	.	PUNCT
ejpam-6576	299	1	international	international	ADJ
ejpam-6576	299	2	electronic	electronic	ADJ
ejpam-6576	299	3	journal	journal	NOUN
ejpam-6576	299	4	of	of	ADP
ejpam-6576	299	5	algebra	algebra	PROPN
ejpam-6576	299	6	,	,	PUNCT
ejpam-6576	299	7	137:1–14	137:1–14	NUM
ejpam-6576	299	8	,	,	PUNCT
ejpam-6576	299	9	2013	2013	NUM
ejpam-6576	299	10	.	.	PUNCT
ejpam-6576	300	1	[	[	X
ejpam-6576	300	2	4	4	X
ejpam-6576	300	3	]	]	PUNCT
ejpam-6576	300	4	s.	s.	PROPN
ejpam-6576	300	5	juglal	juglal	PROPN
ejpam-6576	300	6	and	and	CCONJ
ejpam-6576	300	7	n.j	n.j	PROPN
ejpam-6576	300	8	.	.	PROPN
ejpam-6576	300	9	groenewaldy	groenewaldy	PROPN
ejpam-6576	300	10	.	.	PUNCT
ejpam-6576	301	1	different	different	ADJ
ejpam-6576	301	2	prime	prime	ADJ
ejpam-6576	301	3	r	r	NOUN
ejpam-6576	301	4	-	-	PUNCT
ejpam-6576	301	5	ideals	ideal	NOUN
ejpam-6576	301	6	.	.	PUNCT
ejpam-6576	302	1	algebra	algebra	NOUN
ejpam-6576	302	2	colloquium	colloquium	NOUN
ejpam-6576	302	3	,	,	PUNCT
ejpam-6576	302	4	17(1):887–904	17(1):887–904	NUM
ejpam-6576	302	5	,	,	PUNCT
ejpam-6576	302	6	2010	2010	NUM
ejpam-6576	302	7	.	.	PUNCT
ejpam-6576	303	1	[	[	X
ejpam-6576	303	2	5	5	NUM
ejpam-6576	303	3	]	]	PUNCT
ejpam-6576	303	4	r.	r.	PROPN
ejpam-6576	303	5	srinivasa	srinivasa	PROPN
ejpam-6576	303	6	rao	rao	PROPN
ejpam-6576	303	7	and	and	CCONJ
ejpam-6576	303	8	s.	s.	PROPN
ejpam-6576	303	9	veldsman	veldsman	PROPN
ejpam-6576	303	10	.	.	PUNCT
ejpam-6576	304	1	right	right	ADJ
ejpam-6576	304	2	representations	representation	NOUN
ejpam-6576	304	3	of	of	ADP
ejpam-6576	304	4	right	right	ADJ
ejpam-6576	304	5	near	near	ADJ
ejpam-6576	304	6	-	-	PUNCT
ejpam-6576	304	7	ring	ring	NOUN
ejpam-6576	304	8	radicals	radical	NOUN
ejpam-6576	304	9	.	.	PUNCT
ejpam-6576	305	1	afrika	afrika	PROPN
ejpam-6576	305	2	matematika	matematika	PROPN
ejpam-6576	305	3	,	,	PUNCT
ejpam-6576	305	4	30(1	30(1	PROPN
ejpam-6576	305	5	-	-	SYM
ejpam-6576	305	6	2):1333–1339	2):1333–1339	NUM
ejpam-6576	305	7	,	,	PUNCT
ejpam-6576	305	8	2019	2019	NUM
ejpam-6576	305	9	.	.	PUNCT
ejpam-6576	306	1	[	[	X
ejpam-6576	306	2	6	6	NUM
ejpam-6576	306	3	]	]	PUNCT
ejpam-6576	306	4	r.	r.	PROPN
ejpam-6576	306	5	srinivasa	srinivasa	PROPN
ejpam-6576	306	6	rao	rao	PROPN
ejpam-6576	306	7	kilaru	kilaru	PROPN
ejpam-6576	306	8	j.	j.	PROPN
ejpam-6576	306	9	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	306	10	,	,	PUNCT
ejpam-6576	306	11	v.	v.	PROPN
ejpam-6576	306	12	b.	b.	PROPN
ejpam-6576	307	1	v.	v.	ADP
ejpam-6576	307	2	n.	n.	PROPN
ejpam-6576	307	3	prsad	prsad	PROPN
ejpam-6576	307	4	and	and	CCONJ
ejpam-6576	307	5	a.	a.	NOUN
ejpam-6576	307	6	v.	v.	PROPN
ejpam-6576	307	7	ramakrishna	ramakrishna	PROPN
ejpam-6576	307	8	.	.	PUNCT
ejpam-6576	308	1	a	a	DET
ejpam-6576	308	2	module	module	NOUN
ejpam-6576	308	3	theoretic	theoretic	ADJ
ejpam-6576	308	4	characterization	characterization	NOUN
ejpam-6576	308	5	of	of	ADP
ejpam-6576	308	6	the	the	DET
ejpam-6576	308	7	prime	prime	ADJ
ejpam-6576	308	8	radical	radical	NOUN
ejpam-6576	308	9	of	of	ADP
ejpam-6576	308	10	near	near	ADJ
ejpam-6576	308	11	-	-	PUNCT
ejpam-6576	308	12	rings	ring	NOUN
ejpam-6576	308	13	.	.	PUNCT
ejpam-6576	309	1	beitr	beitr	NOUN
ejpam-6576	309	2	.	.	PUNCT
ejpam-6576	310	1	algebra	algebra	PROPN
ejpam-6576	310	2	geom	geom	PROPN
ejpam-6576	310	3	,	,	PUNCT
ejpam-6576	310	4	59(1):51–60	59(1):51–60	NUM
ejpam-6576	310	5	,	,	PUNCT
ejpam-6576	310	6	2018	2018	NUM
ejpam-6576	310	7	.	.	PUNCT
ejpam-6576	311	1	[	[	X
ejpam-6576	311	2	7	7	X
ejpam-6576	311	3	]	]	PUNCT
ejpam-6576	311	4	k.	k.	PROPN
ejpam-6576	311	5	siva	siva	PROPN
ejpam-6576	311	6	prasad	prasad	PROPN
ejpam-6576	311	7	r.	r.	PROPN
ejpam-6576	311	8	srinivasa	srinivasa	PROPN
ejpam-6576	311	9	rao	rao	PROPN
ejpam-6576	311	10	,	,	PUNCT
ejpam-6576	311	11	k.	k.	PROPN
ejpam-6576	311	12	naga	naga	PROPN
ejpam-6576	311	13	koteswara	koteswara	PROPN
ejpam-6576	311	14	rao	rao	PROPN
ejpam-6576	311	15	and	and	CCONJ
ejpam-6576	311	16	k	k	PROPN
ejpam-6576	311	17	jaya	jaya	PROPN
ejpam-6576	311	18	lakshmi	lakshmi	PROPN
ejpam-6576	311	19	narayana	narayana	PROPN
ejpam-6576	311	20	.	.	PUNCT
ejpam-6576	312	1	a	a	DET
ejpam-6576	312	2	non	non	ADJ
ejpam-6576	312	3	-	-	ADJ
ejpam-6576	312	4	ideal	ideal	ADJ
ejpam-6576	312	5	-	-	PUNCT
ejpam-6576	312	6	hereditary	hereditary	ADJ
ejpam-6576	312	7	kurosh	kurosh	NOUN
ejpam-6576	312	8	-	-	PUNCT
ejpam-6576	312	9	amitsur	amitsur	ADJ
ejpam-6576	312	10	prime	prime	PROPN
ejpam-6576	312	11	radical	radical	NOUN
ejpam-6576	312	12	for	for	ADP
ejpam-6576	312	13	near	near	ADJ
ejpam-6576	312	14	-	-	PUNCT
ejpam-6576	312	15	rings	ring	NOUN
ejpam-6576	312	16	.	.	PUNCT
ejpam-6576	313	1	afrika	afrika	PROPN
ejpam-6576	313	2	matematika	matematika	PROPN
ejpam-6576	313	3	,	,	PUNCT
ejpam-6576	313	4	32:1333–1339	32:1333–1339	NUM
ejpam-6576	313	5	,	,	PUNCT
ejpam-6576	313	6	2021	2021	NUM
ejpam-6576	313	7	.	.	PUNCT
ejpam-6576	314	1	[	[	X
ejpam-6576	314	2	8	8	NUM
ejpam-6576	314	3	]	]	PUNCT
ejpam-6576	314	4	r.	r.	PROPN
ejpam-6576	314	5	srinivasa	srinivasa	PROPN
ejpam-6576	314	6	rao	rao	PROPN
ejpam-6576	314	7	kilaru	kilaru	PROPN
ejpam-6576	314	8	j.	j.	PROPN
ejpam-6576	314	9	lakshminarayana	lakshminarayana	PROPN
ejpam-6576	314	10	,	,	PUNCT
ejpam-6576	314	11	v.	v.	PROPN
ejpam-6576	314	12	b.	b.	PROPN
ejpam-6576	315	1	v.	v.	ADP
ejpam-6576	315	2	n.	n.	PROPN
ejpam-6576	315	3	prsad	prsad	PROPN
ejpam-6576	315	4	and	and	CCONJ
ejpam-6576	315	5	a.	a.	NOUN
ejpam-6576	315	6	v.	v.	PROPN
ejpam-6576	315	7	ramakrishna	ramakrishna	PROPN
ejpam-6576	315	8	.	.	PUNCT
ejpam-6576	316	1	on	on	ADP
ejpam-6576	316	2	the	the	DET
ejpam-6576	316	3	prime	prime	ADJ
ejpam-6576	316	4	radicals	radical	NOUN
ejpam-6576	316	5	of	of	ADP
ejpam-6576	316	6	near	near	ADJ
ejpam-6576	316	7	-	-	PUNCT
ejpam-6576	316	8	rings	ring	NOUN
ejpam-6576	316	9	which	which	PRON
ejpam-6576	316	10	is	be	AUX
ejpam-6576	316	11	kurosh	kurosh	ADV
ejpam-6576	316	12	-	-	PUNCT
ejpam-6576	316	13	amitsur	amitsur	NOUN
ejpam-6576	316	14	.	.	PUNCT
ejpam-6576	317	1	european	european	PROPN
ejpam-6576	317	2	journal	journal	PROPN
ejpam-6576	317	3	of	of	ADP
ejpam-6576	317	4	pure	pure	ADJ
ejpam-6576	317	5	and	and	CCONJ
ejpam-6576	317	6	applied	applied	ADJ
ejpam-6576	317	7	mathematics	mathematic	NOUN
ejpam-6576	317	8	,	,	PUNCT
ejpam-6576	317	9	17(2):1206–1212	17(2):1206–1212	NUM
ejpam-6576	317	10	,	,	PUNCT
ejpam-6576	317	11	2024	2024	NUM
ejpam-6576	317	12	.	.	PUNCT