id	sid	tid	token	lemma	pos
ejpam-609	1	1	8_609_bhat.dvi	8_609_bhat.dvi	NUM
ejpam-609	1	2	european	european	ADJ
ejpam-609	1	3	journal	journal	NOUN
ejpam-609	1	4	of	of	ADP
ejpam-609	1	5	pure	pure	ADJ
ejpam-609	1	6	and	and	CCONJ
ejpam-609	1	7	applied	apply	VERB
ejpam-609	1	8	mathematics	mathematic	NOUN
ejpam-609	1	9	vol	vol	NOUN
ejpam-609	1	10	.	.	PUNCT
ejpam-609	2	1	3	3	NUM
ejpam-609	2	2	,	,	PUNCT
ejpam-609	2	3	no	no	INTJ
ejpam-609	2	4	.	.	NOUN
ejpam-609	2	5	4	4	NUM
ejpam-609	2	6	,	,	PUNCT
ejpam-609	2	7	2010	2010	NUM
ejpam-609	2	8	,	,	PUNCT
ejpam-609	2	9	695	695	NUM
ejpam-609	2	10	-	-	SYM
ejpam-609	2	11	703	703	NUM
ejpam-609	2	12	issn	issn	PROPN
ejpam-609	2	13	1307	1307	NUM
ejpam-609	2	14	-	-	SYM
ejpam-609	2	15	5543	5543	NUM
ejpam-609	2	16	–	–	PUNCT
ejpam-609	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-609	2	18	ore	ore	NOUN
ejpam-609	2	19	extensions	extension	NOUN
ejpam-609	2	20	over	over	ADP
ejpam-609	2	21	weak	weak	ADJ
ejpam-609	2	22	σ	σ	ADJ
ejpam-609	2	23	-	-	ADJ
ejpam-609	2	24	rigid	rigid	ADJ
ejpam-609	2	25	rings	ring	NOUN
ejpam-609	2	26	and	and	CCONJ
ejpam-609	2	27	σ(∗)-rings	σ(∗)-rings	PROPN
ejpam-609	2	28	v.	v.	PROPN
ejpam-609	2	29	k.	k.	PROPN
ejpam-609	3	1	bhat	bhat	PROPN
ejpam-609	3	2	school	school	NOUN
ejpam-609	3	3	of	of	ADP
ejpam-609	3	4	mathematics	mathematics	PROPN
ejpam-609	3	5	,	,	PUNCT
ejpam-609	3	6	smvd	smvd	PROPN
ejpam-609	3	7	university	university	PROPN
ejpam-609	3	8	,	,	PUNCT
ejpam-609	3	9	p.o	p.o	PROPN
ejpam-609	3	10	.	.	PROPN
ejpam-609	3	11	smvd	smvd	PROPN
ejpam-609	3	12	university	university	PROPN
ejpam-609	3	13	,	,	PUNCT
ejpam-609	3	14	katra	katra	PROPN
ejpam-609	3	15	,	,	PUNCT
ejpam-609	3	16	j	j	PROPN
ejpam-609	3	17	and	and	CCONJ
ejpam-609	3	18	k	k	PROPN
ejpam-609	3	19	,	,	PUNCT
ejpam-609	3	20	india-182320	india-182320	VERB
ejpam-609	3	21	abstract	abstract	ADJ
ejpam-609	3	22	.	.	PUNCT
ejpam-609	4	1	let	let	VERB
ejpam-609	4	2	r	r	PRON
ejpam-609	4	3	be	be	AUX
ejpam-609	4	4	a	a	DET
ejpam-609	4	5	ring	ring	NOUN
ejpam-609	4	6	and	and	CCONJ
ejpam-609	4	7	σ	σ	NOUN
ejpam-609	4	8	an	an	DET
ejpam-609	4	9	endomorphism	endomorphism	NOUN
ejpam-609	4	10	of	of	ADP
ejpam-609	4	11	a	a	DET
ejpam-609	4	12	ring	ring	PROPN
ejpam-609	4	13	r.	r.	PROPN
ejpam-609	4	14	recall	recall	NOUN
ejpam-609	4	15	that	that	SCONJ
ejpam-609	4	16	r	r	NOUN
ejpam-609	4	17	is	be	AUX
ejpam-609	4	18	said	say	VERB
ejpam-609	4	19	to	to	PART
ejpam-609	4	20	be	be	AUX
ejpam-609	4	21	a	a	DET
ejpam-609	4	22	σ(∗)-ring	σ(∗)-re	VERB
ejpam-609	4	23	if	if	SCONJ
ejpam-609	4	24	aσ(a	aσ(a	NUM
ejpam-609	4	25	)	)	PUNCT
ejpam-609	4	26	∈	∈	PROPN
ejpam-609	4	27	p(r	p(r	PROPN
ejpam-609	4	28	)	)	PUNCT
ejpam-609	4	29	implies	imply	VERB
ejpam-609	4	30	a	a	DET
ejpam-609	4	31	∈	∈	PROPN
ejpam-609	4	32	p(r	p(r	PROPN
ejpam-609	4	33	)	)	PUNCT
ejpam-609	4	34	for	for	ADP
ejpam-609	4	35	a	a	DET
ejpam-609	4	36	∈	∈	PROPN
ejpam-609	4	37	r	r	NOUN
ejpam-609	4	38	,	,	PUNCT
ejpam-609	4	39	where	where	SCONJ
ejpam-609	4	40	p(r	p(r	NOUN
ejpam-609	4	41	)	)	PUNCT
ejpam-609	4	42	is	be	AUX
ejpam-609	4	43	the	the	DET
ejpam-609	4	44	prime	prime	ADJ
ejpam-609	4	45	radical	radical	NOUN
ejpam-609	4	46	of	of	ADP
ejpam-609	4	47	r.	r.	PROPN
ejpam-609	4	48	we	we	PRON
ejpam-609	4	49	also	also	ADV
ejpam-609	4	50	recall	recall	VERB
ejpam-609	4	51	that	that	SCONJ
ejpam-609	4	52	r	r	NOUN
ejpam-609	4	53	is	be	AUX
ejpam-609	4	54	said	say	VERB
ejpam-609	4	55	to	to	PART
ejpam-609	4	56	be	be	AUX
ejpam-609	4	57	a	a	DET
ejpam-609	4	58	weak	weak	ADJ
ejpam-609	4	59	σ	σ	ADJ
ejpam-609	4	60	-	-	ADJ
ejpam-609	4	61	rigid	rigid	ADJ
ejpam-609	4	62	ring	ring	NOUN
ejpam-609	4	63	if	if	SCONJ
ejpam-609	4	64	aσ(a	aσ(a	NUM
ejpam-609	4	65	)	)	PUNCT
ejpam-609	4	66	∈	∈	PROPN
ejpam-609	4	67	n(r	n(r	NOUN
ejpam-609	4	68	)	)	PUNCT
ejpam-609	5	1	if	if	SCONJ
ejpam-609	5	2	and	and	CCONJ
ejpam-609	5	3	only	only	ADV
ejpam-609	5	4	if	if	SCONJ
ejpam-609	5	5	a	a	DET
ejpam-609	5	6	∈	∈	PROPN
ejpam-609	5	7	n(r	n(r	NOUN
ejpam-609	5	8	)	)	PUNCT
ejpam-609	5	9	for	for	ADP
ejpam-609	5	10	a	a	DET
ejpam-609	5	11	∈	∈	PROPN
ejpam-609	5	12	r	r	NOUN
ejpam-609	5	13	,	,	PUNCT
ejpam-609	5	14	where	where	SCONJ
ejpam-609	5	15	n(r	n(r	NOUN
ejpam-609	5	16	)	)	PUNCT
ejpam-609	5	17	is	be	AUX
ejpam-609	5	18	the	the	DET
ejpam-609	5	19	set	set	NOUN
ejpam-609	5	20	of	of	ADP
ejpam-609	5	21	nilpotent	nilpotent	ADJ
ejpam-609	5	22	elements	element	NOUN
ejpam-609	5	23	of	of	ADP
ejpam-609	5	24	r.	r.	PROPN
ejpam-609	5	25	in	in	ADP
ejpam-609	5	26	this	this	DET
ejpam-609	5	27	paper	paper	NOUN
ejpam-609	5	28	we	we	PRON
ejpam-609	5	29	give	give	VERB
ejpam-609	5	30	a	a	DET
ejpam-609	5	31	relation	relation	NOUN
ejpam-609	5	32	between	between	ADP
ejpam-609	5	33	a	a	DET
ejpam-609	5	34	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	5	35	and	and	CCONJ
ejpam-609	5	36	a	a	DET
ejpam-609	5	37	weak	weak	ADJ
ejpam-609	5	38	σ	σ	ADJ
ejpam-609	5	39	-	-	ADJ
ejpam-609	5	40	rigid	rigid	ADJ
ejpam-609	5	41	ring	ring	NOUN
ejpam-609	5	42	.	.	PUNCT
ejpam-609	6	1	we	we	PRON
ejpam-609	6	2	also	also	ADV
ejpam-609	6	3	give	give	VERB
ejpam-609	6	4	a	a	DET
ejpam-609	6	5	necessary	necessary	ADJ
ejpam-609	6	6	and	and	CCONJ
ejpam-609	6	7	sufficient	sufficient	ADJ
ejpam-609	6	8	condition	condition	NOUN
ejpam-609	6	9	for	for	ADP
ejpam-609	6	10	a	a	DET
ejpam-609	6	11	noetherian	noetherian	ADJ
ejpam-609	6	12	ring	ring	NOUN
ejpam-609	6	13	to	to	PART
ejpam-609	6	14	be	be	AUX
ejpam-609	6	15	a	a	DET
ejpam-609	6	16	weak	weak	ADJ
ejpam-609	6	17	σ	σ	ADJ
ejpam-609	6	18	-	-	ADJ
ejpam-609	6	19	rigid	rigid	ADJ
ejpam-609	6	20	ring	ring	NOUN
ejpam-609	6	21	.	.	PUNCT
ejpam-609	7	1	let	let	VERB
ejpam-609	7	2	σ	σ	NOUN
ejpam-609	7	3	be	be	AUX
ejpam-609	7	4	an	an	DET
ejpam-609	7	5	endomorphism	endomorphism	NOUN
ejpam-609	7	6	of	of	ADP
ejpam-609	7	7	a	a	DET
ejpam-609	7	8	ring	ring	NOUN
ejpam-609	7	9	r	r	NOUN
ejpam-609	7	10	and	and	CCONJ
ejpam-609	7	11	δ	δ	PROPN
ejpam-609	7	12	a	a	DET
ejpam-609	7	13	σ	σ	NOUN
ejpam-609	7	14	-	-	PUNCT
ejpam-609	7	15	derivation	derivation	NOUN
ejpam-609	7	16	of	of	ADP
ejpam-609	7	17	r	r	NOUN
ejpam-609	7	18	such	such	ADJ
ejpam-609	7	19	that	that	DET
ejpam-609	7	20	σ(δ(a	σ(δ(a	NOUN
ejpam-609	7	21	)	)	PUNCT
ejpam-609	7	22	)	)	PUNCT
ejpam-609	8	1	=	=	PUNCT
ejpam-609	8	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	8	3	)	)	PUNCT
ejpam-609	8	4	)	)	PUNCT
ejpam-609	8	5	for	for	ADP
ejpam-609	8	6	all	all	DET
ejpam-609	8	7	a	a	DET
ejpam-609	8	8	∈	∈	PROPN
ejpam-609	8	9	r.	r.	NOUN
ejpam-609	8	10	then	then	ADV
ejpam-609	8	11	σ	σ	PROPN
ejpam-609	8	12	can	can	AUX
ejpam-609	8	13	be	be	AUX
ejpam-609	8	14	extended	extend	VERB
ejpam-609	8	15	to	to	ADP
ejpam-609	8	16	an	an	DET
ejpam-609	8	17	endomorphism	endomorphism	NOUN
ejpam-609	8	18	(	(	PUNCT
ejpam-609	8	19	say	say	INTJ
ejpam-609	8	20	σ	σ	NOUN
ejpam-609	8	21	)	)	PUNCT
ejpam-609	8	22	of	of	ADP
ejpam-609	8	23	r[x;σ	r[x;σ	NOUN
ejpam-609	8	24	,	,	PUNCT
ejpam-609	8	25	δ	δ	PROPN
ejpam-609	8	26	]	]	PUNCT
ejpam-609	8	27	and	and	CCONJ
ejpam-609	8	28	δ	δ	PROPN
ejpam-609	8	29	can	can	AUX
ejpam-609	8	30	be	be	AUX
ejpam-609	8	31	extended	extend	VERB
ejpam-609	8	32	to	to	ADP
ejpam-609	8	33	a	a	DET
ejpam-609	8	34	σ	σ	NOUN
ejpam-609	8	35	-	-	PUNCT
ejpam-609	8	36	derivation	derivation	NOUN
ejpam-609	8	37	(	(	PUNCT
ejpam-609	8	38	say	say	VERB
ejpam-609	8	39	δ	δ	PROPN
ejpam-609	8	40	)	)	PUNCT
ejpam-609	8	41	of	of	ADP
ejpam-609	8	42	r[x;σ	r[x;σ	NOUN
ejpam-609	8	43	,	,	PUNCT
ejpam-609	8	44	δ	δ	NOUN
ejpam-609	8	45	]	]	PUNCT
ejpam-609	8	46	.	.	PUNCT
ejpam-609	9	1	with	with	ADP
ejpam-609	9	2	this	this	PRON
ejpam-609	9	3	we	we	PRON
ejpam-609	9	4	show	show	VERB
ejpam-609	9	5	that	that	SCONJ
ejpam-609	9	6	if	if	SCONJ
ejpam-609	9	7	r	r	NOUN
ejpam-609	9	8	is	be	AUX
ejpam-609	9	9	a	a	DET
ejpam-609	9	10	2	2	NUM
ejpam-609	9	11	-	-	PUNCT
ejpam-609	9	12	primal	primal	ADJ
ejpam-609	9	13	commutative	commutative	ADJ
ejpam-609	9	14	noetherian	noetherian	ADJ
ejpam-609	9	15	ring	ring	NOUN
ejpam-609	9	16	which	which	PRON
ejpam-609	9	17	is	be	AUX
ejpam-609	9	18	also	also	ADV
ejpam-609	9	19	an	an	DET
ejpam-609	9	20	algebra	algebra	NOUN
ejpam-609	9	21	over	over	ADP
ejpam-609	9	22	q	q	PROPN
ejpam-609	9	23	(	(	PUNCT
ejpam-609	9	24	where	where	SCONJ
ejpam-609	9	25	q	q	NOUN
ejpam-609	9	26	is	be	AUX
ejpam-609	9	27	the	the	DET
ejpam-609	9	28	field	field	NOUN
ejpam-609	9	29	of	of	ADP
ejpam-609	9	30	rational	rational	ADJ
ejpam-609	9	31	numbers	number	NOUN
ejpam-609	9	32	)	)	PUNCT
ejpam-609	9	33	,	,	PUNCT
ejpam-609	9	34	σ	σ	PROPN
ejpam-609	9	35	is	be	AUX
ejpam-609	9	36	an	an	DET
ejpam-609	9	37	automorphism	automorphism	NOUN
ejpam-609	9	38	of	of	ADP
ejpam-609	9	39	r	r	NOUN
ejpam-609	9	40	and	and	CCONJ
ejpam-609	9	41	δ	δ	PROPN
ejpam-609	9	42	a	a	DET
ejpam-609	9	43	σ	σ	NOUN
ejpam-609	9	44	-	-	PUNCT
ejpam-609	9	45	derivation	derivation	NOUN
ejpam-609	9	46	of	of	ADP
ejpam-609	9	47	r	r	NOUN
ejpam-609	9	48	such	such	ADJ
ejpam-609	9	49	that	that	DET
ejpam-609	9	50	σ(δ(a	σ(δ(a	NOUN
ejpam-609	9	51	)	)	PUNCT
ejpam-609	9	52	)	)	PUNCT
ejpam-609	10	1	=	=	PUNCT
ejpam-609	10	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	10	3	)	)	PUNCT
ejpam-609	10	4	)	)	PUNCT
ejpam-609	10	5	for	for	ADP
ejpam-609	10	6	all	all	DET
ejpam-609	10	7	a	a	DET
ejpam-609	10	8	∈	∈	NOUN
ejpam-609	10	9	r	r	NOUN
ejpam-609	10	10	,	,	PUNCT
ejpam-609	10	11	then	then	ADV
ejpam-609	10	12	r	r	NOUN
ejpam-609	10	13	is	be	AUX
ejpam-609	10	14	a	a	DET
ejpam-609	10	15	weak	weak	ADJ
ejpam-609	10	16	σ	σ	ADJ
ejpam-609	10	17	-	-	ADJ
ejpam-609	10	18	rigid	rigid	ADJ
ejpam-609	10	19	ring	ring	NOUN
ejpam-609	10	20	implies	imply	VERB
ejpam-609	10	21	that	that	SCONJ
ejpam-609	10	22	r[x;σ	r[x;σ	NOUN
ejpam-609	10	23	,	,	PUNCT
ejpam-609	10	24	δ	δ	PROPN
ejpam-609	10	25	]	]	PUNCT
ejpam-609	10	26	is	be	AUX
ejpam-609	10	27	a	a	DET
ejpam-609	10	28	weak	weak	ADJ
ejpam-609	10	29	σ	σ	ADJ
ejpam-609	10	30	-	-	ADJ
ejpam-609	10	31	rigid	rigid	ADJ
ejpam-609	10	32	ring	ring	NOUN
ejpam-609	10	33	.	.	PUNCT
ejpam-609	11	1	2000	2000	NUM
ejpam-609	11	2	mathematics	mathematic	NOUN
ejpam-609	11	3	subject	subject	NOUN
ejpam-609	11	4	classifications	classification	NOUN
ejpam-609	11	5	:	:	PUNCT
ejpam-609	11	6	16s36	16s36	NUM
ejpam-609	11	7	,	,	PUNCT
ejpam-609	11	8	16p40	16p40	NUM
ejpam-609	11	9	,	,	PUNCT
ejpam-609	11	10	16p50	16p50	NUM
ejpam-609	11	11	,	,	PUNCT
ejpam-609	11	12	16u20,16w25	16u20,16w25	NUM
ejpam-609	11	13	key	key	ADJ
ejpam-609	11	14	words	word	NOUN
ejpam-609	11	15	and	and	CCONJ
ejpam-609	11	16	phrases	phrase	NOUN
ejpam-609	11	17	:	:	PUNCT
ejpam-609	11	18	automorphism	automorphism	NOUN
ejpam-609	11	19	,	,	PUNCT
ejpam-609	11	20	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	11	21	,	,	PUNCT
ejpam-609	11	22	weak	weak	ADJ
ejpam-609	11	23	σ	σ	VERB
ejpam-609	11	24	-	-	ADJ
ejpam-609	11	25	rigid	rigid	ADJ
ejpam-609	11	26	ring	ring	NOUN
ejpam-609	11	27	,	,	PUNCT
ejpam-609	11	28	2	2	NUM
ejpam-609	11	29	-	-	PUNCT
ejpam-609	11	30	primal	primal	ADJ
ejpam-609	11	31	ring	ring	NOUN
ejpam-609	11	32	1	1	NUM
ejpam-609	11	33	.	.	PUNCT
ejpam-609	12	1	introduction	introduction	NOUN
ejpam-609	12	2	throughout	throughout	ADP
ejpam-609	12	3	this	this	DET
ejpam-609	12	4	paper	paper	NOUN
ejpam-609	12	5	r	r	NOUN
ejpam-609	12	6	will	will	AUX
ejpam-609	12	7	denote	denote	VERB
ejpam-609	12	8	an	an	DET
ejpam-609	12	9	associative	associative	ADJ
ejpam-609	12	10	ring	ring	NOUN
ejpam-609	12	11	with	with	ADP
ejpam-609	12	12	identity	identity	NOUN
ejpam-609	12	13	1	1	NUM
ejpam-609	12	14	6=	6=	ADP
ejpam-609	12	15	0	0	NUM
ejpam-609	12	16	.	.	PUNCT
ejpam-609	13	1	let	let	AUX
ejpam-609	13	2	now	now	ADV
ejpam-609	13	3	σ	σ	NOUN
ejpam-609	13	4	be	be	AUX
ejpam-609	13	5	an	an	DET
ejpam-609	13	6	endomorphism	endomorphism	NOUN
ejpam-609	13	7	of	of	ADP
ejpam-609	13	8	a	a	DET
ejpam-609	13	9	ring	ring	NOUN
ejpam-609	13	10	r.	r.	NOUN
ejpam-609	13	11	the	the	DET
ejpam-609	13	12	field	field	NOUN
ejpam-609	13	13	of	of	ADP
ejpam-609	13	14	complex	complex	ADJ
ejpam-609	13	15	numbers	number	NOUN
ejpam-609	13	16	is	be	AUX
ejpam-609	13	17	denoted	denote	VERB
ejpam-609	13	18	by	by	ADP
ejpam-609	13	19	c	c	PROPN
ejpam-609	13	20	,	,	PUNCT
ejpam-609	13	21	the	the	DET
ejpam-609	13	22	field	field	NOUN
ejpam-609	13	23	of	of	ADP
ejpam-609	13	24	rational	rational	ADJ
ejpam-609	13	25	numbers	number	NOUN
ejpam-609	13	26	is	be	AUX
ejpam-609	13	27	denoted	denote	VERB
ejpam-609	13	28	by	by	ADP
ejpam-609	13	29	q	q	PROPN
ejpam-609	13	30	,	,	PUNCT
ejpam-609	13	31	the	the	DET
ejpam-609	13	32	ring	ring	NOUN
ejpam-609	13	33	of	of	ADP
ejpam-609	13	34	integers	integer	NOUN
ejpam-609	13	35	is	be	AUX
ejpam-609	13	36	denoted	denote	VERB
ejpam-609	13	37	by	by	ADP
ejpam-609	13	38	z	z	PROPN
ejpam-609	13	39	,	,	PUNCT
ejpam-609	13	40	and	and	CCONJ
ejpam-609	13	41	the	the	DET
ejpam-609	13	42	set	set	NOUN
ejpam-609	13	43	of	of	ADP
ejpam-609	13	44	positive	positive	ADJ
ejpam-609	13	45	integers	integer	NOUN
ejpam-609	13	46	is	be	AUX
ejpam-609	13	47	denoted	denote	VERB
ejpam-609	13	48	by	by	ADP
ejpam-609	13	49	n.	n.	NOUN
ejpam-609	13	50	the	the	DET
ejpam-609	13	51	set	set	NOUN
ejpam-609	13	52	of	of	ADP
ejpam-609	13	53	prime	prime	ADJ
ejpam-609	13	54	ideals	ideal	NOUN
ejpam-609	13	55	of	of	ADP
ejpam-609	13	56	r	r	NOUN
ejpam-609	13	57	is	be	AUX
ejpam-609	13	58	denoted	denote	VERB
ejpam-609	13	59	by	by	ADP
ejpam-609	13	60	spec(r	spec(r	PROPN
ejpam-609	13	61	)	)	PUNCT
ejpam-609	13	62	.	.	PUNCT
ejpam-609	14	1	the	the	DET
ejpam-609	14	2	set	set	NOUN
ejpam-609	14	3	of	of	ADP
ejpam-609	14	4	minimal	minimal	ADJ
ejpam-609	14	5	prime	prime	ADJ
ejpam-609	14	6	ideals	ideal	NOUN
ejpam-609	14	7	of	of	ADP
ejpam-609	14	8	r	r	NOUN
ejpam-609	14	9	is	be	AUX
ejpam-609	14	10	denoted	denote	VERB
ejpam-609	14	11	by	by	ADP
ejpam-609	14	12	min.spec(r	min.spec(r	PROPN
ejpam-609	14	13	)	)	PUNCT
ejpam-609	14	14	.	.	PUNCT
ejpam-609	15	1	the	the	DET
ejpam-609	15	2	prime	prime	ADJ
ejpam-609	15	3	radical	radical	ADJ
ejpam-609	15	4	and	and	CCONJ
ejpam-609	15	5	the	the	DET
ejpam-609	15	6	set	set	NOUN
ejpam-609	15	7	of	of	ADP
ejpam-609	15	8	nilpotent	nilpotent	ADJ
ejpam-609	15	9	elements	element	NOUN
ejpam-609	15	10	of	of	ADP
ejpam-609	15	11	r	r	NOUN
ejpam-609	15	12	are	be	AUX
ejpam-609	15	13	denoted	denote	VERB
ejpam-609	15	14	by	by	ADP
ejpam-609	15	15	p(r	p(r	PROPN
ejpam-609	15	16	)	)	PUNCT
ejpam-609	15	17	and	and	CCONJ
ejpam-609	15	18	n(r	n(r	NOUN
ejpam-609	15	19	)	)	PUNCT
ejpam-609	15	20	respectively	respectively	ADV
ejpam-609	15	21	.	.	PUNCT
ejpam-609	16	1	we	we	PRON
ejpam-609	16	2	note	note	VERB
ejpam-609	16	3	that	that	SCONJ
ejpam-609	16	4	for	for	ADP
ejpam-609	16	5	a	a	DET
ejpam-609	16	6	commutative	commutative	ADJ
ejpam-609	16	7	ring	ring	NOUN
ejpam-609	16	8	p(r	p(r	PROPN
ejpam-609	16	9	)	)	PUNCT
ejpam-609	16	10	and	and	CCONJ
ejpam-609	16	11	n(r	n(r	NUM
ejpam-609	16	12	)	)	PUNCT
ejpam-609	16	13	.	.	PUNCT
ejpam-609	17	1	now	now	ADV
ejpam-609	17	2	let	let	VERB
ejpam-609	17	3	r	r	NOUN
ejpam-609	17	4	be	be	AUX
ejpam-609	17	5	a	a	DET
ejpam-609	17	6	ring	ring	NOUN
ejpam-609	17	7	and	and	CCONJ
ejpam-609	17	8	σ	σ	NOUN
ejpam-609	17	9	an	an	DET
ejpam-609	17	10	endomorphism	endomorphism	NOUN
ejpam-609	17	11	of	of	ADP
ejpam-609	17	12	r	r	NOUN
ejpam-609	17	13	and	and	CCONJ
ejpam-609	17	14	δ	δ	PROPN
ejpam-609	17	15	is	be	AUX
ejpam-609	17	16	a	a	DET
ejpam-609	17	17	σ	σ	NOUN
ejpam-609	17	18	-	-	PUNCT
ejpam-609	17	19	derivation	derivation	NOUN
ejpam-609	17	20	of	of	ADP
ejpam-609	17	21	r.	r.	PROPN
ejpam-609	17	22	recall	recall	PROPN
ejpam-609	17	23	that	that	SCONJ
ejpam-609	17	24	the	the	DET
ejpam-609	17	25	skew	skew	ADJ
ejpam-609	17	26	polynomial	polynomial	ADJ
ejpam-609	17	27	ring	ring	NOUN
ejpam-609	17	28	r[x	r[x	NOUN
ejpam-609	17	29	;	;	PUNCT
ejpam-609	17	30	σ	σ	PROPN
ejpam-609	17	31	,	,	PUNCT
ejpam-609	17	32	δ	δ	PROPN
ejpam-609	17	33	]	]	PUNCT
ejpam-609	17	34	is	be	AUX
ejpam-609	17	35	the	the	DET
ejpam-609	17	36	set	set	NOUN
ejpam-609	17	37	of	of	ADP
ejpam-609	17	38	polynomials	polynomial	NOUN
ejpam-609	17	39	{	{	PUNCT
ejpam-609	17	40	n	n	CCONJ
ejpam-609	17	41	∑	∑	ADP
ejpam-609	17	42	i=0	i=0	PROPN
ejpam-609	17	43	x	x	SYM
ejpam-609	17	44	iai	iai	PROPN
ejpam-609	17	45	,	,	PUNCT
ejpam-609	17	46	ai	ai	VERB
ejpam-609	17	47	∈	∈	PROPN
ejpam-609	17	48	r	r	NOUN
ejpam-609	17	49	,	,	PUNCT
ejpam-609	17	50	n	n	PRON
ejpam-609	17	51	∈	∈	PROPN
ejpam-609	17	52	n	n	CCONJ
ejpam-609	17	53	}	}	PUNCT
ejpam-609	17	54	email	email	NOUN
ejpam-609	17	55	address	address	NOUN
ejpam-609	17	56	:	:	PUNCT
ejpam-609	17	57	vijaykumarbhat2000	vijaykumarbhat2000	PROPN
ejpam-609	17	58	�	�	NOUN
ejpam-609	17	59	yahoo	yahoo	PROPN
ejpam-609	17	60	.	.	PUNCT
ejpam-609	18	1	om	om	PROPN
ejpam-609	18	2	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-609	19	1	695	695	NUM
ejpam-609	19	2	c	c	NOUN
ejpam-609	19	3	©	©	PROPN
ejpam-609	19	4	2010	2010	NUM
ejpam-609	19	5	ejpam	ejpam	NOUN
ejpam-609	19	6	all	all	DET
ejpam-609	19	7	rights	right	NOUN
ejpam-609	19	8	reserved	reserve	VERB
ejpam-609	19	9	.	.	PUNCT
ejpam-609	20	1	v.	v.	ADP
ejpam-609	20	2	bhat	bhat	PROPN
ejpam-609	20	3	/	/	SYM
ejpam-609	20	4	eur	eur	PROPN
ejpam-609	20	5	.	.	PUNCT
ejpam-609	21	1	j.	j.	PROPN
ejpam-609	21	2	pure	pure	PROPN
ejpam-609	21	3	appl	appl	PROPN
ejpam-609	21	4	.	.	PROPN
ejpam-609	21	5	math	math	PROPN
ejpam-609	21	6	,	,	PUNCT
ejpam-609	21	7	3	3	NUM
ejpam-609	21	8	(	(	PUNCT
ejpam-609	21	9	2010	2010	NUM
ejpam-609	21	10	)	)	PUNCT
ejpam-609	21	11	,	,	PUNCT
ejpam-609	21	12	695	695	NUM
ejpam-609	21	13	-	-	SYM
ejpam-609	21	14	703	703	NUM
ejpam-609	21	15	696	696	NUM
ejpam-609	21	16	with	with	ADP
ejpam-609	21	17	usual	usual	ADJ
ejpam-609	21	18	addition	addition	NOUN
ejpam-609	21	19	of	of	ADP
ejpam-609	21	20	polynomials	polynomial	NOUN
ejpam-609	21	21	and	and	CCONJ
ejpam-609	21	22	multiplication	multiplication	NOUN
ejpam-609	21	23	subject	subject	ADJ
ejpam-609	21	24	to	to	ADP
ejpam-609	21	25	the	the	DET
ejpam-609	21	26	relation	relation	NOUN
ejpam-609	21	27	ax	ax	NOUN
ejpam-609	21	28	=	=	PUNCT
ejpam-609	21	29	xσ(a	xσ(a	PUNCT
ejpam-609	21	30	)	)	PUNCT
ejpam-609	22	1	+	+	CCONJ
ejpam-609	22	2	δ(a	δ(a	PROPN
ejpam-609	22	3	)	)	PUNCT
ejpam-609	22	4	for	for	ADP
ejpam-609	22	5	all	all	DET
ejpam-609	22	6	a	a	DET
ejpam-609	22	7	∈	∈	PROPN
ejpam-609	22	8	r.	r.	NOUN
ejpam-609	22	9	we	we	PRON
ejpam-609	22	10	take	take	VERB
ejpam-609	22	11	any	any	DET
ejpam-609	22	12	f	f	NOUN
ejpam-609	22	13	(	(	PUNCT
ejpam-609	22	14	x	x	X
ejpam-609	22	15	)	)	PUNCT
ejpam-609	22	16	∈	∈	PROPN
ejpam-609	22	17	r[x	r[x	NOUN
ejpam-609	22	18	;	;	PUNCT
ejpam-609	22	19	σ	σ	PROPN
ejpam-609	22	20	,	,	PUNCT
ejpam-609	22	21	δ	δ	PROPN
ejpam-609	22	22	]	]	PUNCT
ejpam-609	22	23	to	to	PART
ejpam-609	22	24	be	be	AUX
ejpam-609	22	25	of	of	ADP
ejpam-609	22	26	the	the	DET
ejpam-609	22	27	form	form	NOUN
ejpam-609	22	28	f	f	X
ejpam-609	22	29	(	(	PUNCT
ejpam-609	22	30	x	x	X
ejpam-609	22	31	)	)	PUNCT
ejpam-609	22	32	=	=	SYM
ejpam-609	23	1	∑n	∑n	PROPN
ejpam-609	23	2	i=0	i=0	PROPN
ejpam-609	23	3	x	x	SYM
ejpam-609	23	4	iai	iai	PROPN
ejpam-609	23	5	,	,	PUNCT
ejpam-609	23	6	ai	ai	VERB
ejpam-609	23	7	∈	∈	NOUN
ejpam-609	23	8	r	r	NOUN
ejpam-609	23	9	as	as	ADP
ejpam-609	23	10	in	in	ADP
ejpam-609	23	11	mcconnell	mcconnell	PROPN
ejpam-609	23	12	and	and	CCONJ
ejpam-609	23	13	robson	robson	NOUN
ejpam-609	23	14	[	[	X
ejpam-609	23	15	14	14	NUM
ejpam-609	23	16	]	]	PUNCT
ejpam-609	23	17	.	.	PUNCT
ejpam-609	24	1	we	we	PRON
ejpam-609	24	2	denote	denote	VERB
ejpam-609	24	3	r[x	r[x	NOUN
ejpam-609	24	4	;	;	PUNCT
ejpam-609	24	5	σ	σ	PROPN
ejpam-609	24	6	,	,	PUNCT
ejpam-609	24	7	δ	δ	PROPN
ejpam-609	24	8	]	]	PUNCT
ejpam-609	24	9	by	by	ADP
ejpam-609	24	10	o(r	o(r	NOUN
ejpam-609	24	11	)	)	PUNCT
ejpam-609	24	12	.	.	PUNCT
ejpam-609	25	1	in	in	ADP
ejpam-609	25	2	case	case	NOUN
ejpam-609	25	3	σ	σ	PROPN
ejpam-609	25	4	is	be	AUX
ejpam-609	25	5	the	the	DET
ejpam-609	25	6	identity	identity	NOUN
ejpam-609	25	7	map	map	NOUN
ejpam-609	25	8	,	,	PUNCT
ejpam-609	25	9	we	we	PRON
ejpam-609	25	10	denote	denote	VERB
ejpam-609	25	11	the	the	DET
ejpam-609	25	12	differential	differential	ADJ
ejpam-609	25	13	operator	operator	NOUN
ejpam-609	25	14	ring	ring	NOUN
ejpam-609	25	15	r[x	r[x	NOUN
ejpam-609	25	16	;	;	PUNCT
ejpam-609	25	17	δ	δ	PROPN
ejpam-609	25	18	]	]	PUNCT
ejpam-609	25	19	by	by	ADP
ejpam-609	25	20	d(r	d(r	NOUN
ejpam-609	25	21	)	)	PUNCT
ejpam-609	25	22	and	and	CCONJ
ejpam-609	25	23	in	in	ADP
ejpam-609	25	24	case	case	NOUN
ejpam-609	25	25	δ	δ	PROPN
ejpam-609	25	26	is	be	AUX
ejpam-609	25	27	the	the	DET
ejpam-609	25	28	zero	zero	NUM
ejpam-609	25	29	map	map	NOUN
ejpam-609	25	30	,	,	PUNCT
ejpam-609	25	31	we	we	PRON
ejpam-609	25	32	denote	denote	VERB
ejpam-609	25	33	r[x	r[x	NOUN
ejpam-609	25	34	;	;	PUNCT
ejpam-609	25	35	σ	σ	X
ejpam-609	25	36	]	]	PUNCT
ejpam-609	25	37	by	by	ADP
ejpam-609	25	38	s(r	s(r	PROPN
ejpam-609	25	39	)	)	PUNCT
ejpam-609	25	40	.	.	PUNCT
ejpam-609	26	1	for	for	ADP
ejpam-609	26	2	more	more	ADJ
ejpam-609	26	3	details	detail	NOUN
ejpam-609	26	4	on	on	ADP
ejpam-609	26	5	ore	ore	NOUN
ejpam-609	26	6	extensions	extension	NOUN
ejpam-609	26	7	(	(	PUNCT
ejpam-609	26	8	the	the	DET
ejpam-609	26	9	skew	skew	ADJ
ejpam-609	26	10	polynomial	polynomial	ADJ
ejpam-609	26	11	rings	ring	NOUN
ejpam-609	26	12	)	)	PUNCT
ejpam-609	26	13	,	,	PUNCT
ejpam-609	26	14	we	we	PRON
ejpam-609	26	15	refer	refer	VERB
ejpam-609	26	16	the	the	DET
ejpam-609	26	17	reader	reader	NOUN
ejpam-609	26	18	to	to	ADP
ejpam-609	26	19	chapter	chapter	NOUN
ejpam-609	26	20	(	(	PUNCT
ejpam-609	26	21	1	1	NUM
ejpam-609	26	22	)	)	PUNCT
ejpam-609	26	23	of	of	ADP
ejpam-609	26	24	mcconnell	mcconnell	PROPN
ejpam-609	26	25	and	and	CCONJ
ejpam-609	26	26	robson	robson	NOUN
ejpam-609	26	27	[	[	X
ejpam-609	26	28	14	14	NUM
ejpam-609	26	29	]	]	PUNCT
ejpam-609	26	30	.	.	PUNCT
ejpam-609	27	1	ore	ore	NOUN
ejpam-609	27	2	-	-	PUNCT
ejpam-609	27	3	extensions	extension	NOUN
ejpam-609	27	4	including	include	VERB
ejpam-609	27	5	skew	skew	ADJ
ejpam-609	27	6	-	-	PUNCT
ejpam-609	27	7	polynomial	polynomial	ADJ
ejpam-609	27	8	rings	ring	NOUN
ejpam-609	27	9	and	and	CCONJ
ejpam-609	27	10	differential	differential	NOUN
ejpam-609	27	11	operator	operator	NOUN
ejpam-609	27	12	rings	ring	NOUN
ejpam-609	27	13	have	have	AUX
ejpam-609	27	14	been	be	AUX
ejpam-609	27	15	of	of	ADP
ejpam-609	27	16	interest	interest	NOUN
ejpam-609	27	17	to	to	ADP
ejpam-609	27	18	many	many	ADJ
ejpam-609	27	19	authors	author	NOUN
ejpam-609	27	20	.	.	PUNCT
ejpam-609	28	1	for	for	ADP
ejpam-609	28	2	example	example	NOUN
ejpam-609	28	3	[	[	X
ejpam-609	28	4	1	1	NUM
ejpam-609	28	5	,	,	PUNCT
ejpam-609	28	6	3	3	NUM
ejpam-609	28	7	,	,	PUNCT
ejpam-609	28	8	6	6	NUM
ejpam-609	28	9	,	,	PUNCT
ejpam-609	28	10	8	8	NUM
ejpam-609	28	11	,	,	PUNCT
ejpam-609	28	12	11	11	NUM
ejpam-609	28	13	,	,	PUNCT
ejpam-609	28	14	12	12	NUM
ejpam-609	28	15	,	,	PUNCT
ejpam-609	28	16	15	15	NUM
ejpam-609	28	17	]	]	PUNCT
ejpam-609	28	18	.	.	PUNCT
ejpam-609	29	1	the	the	DET
ejpam-609	29	2	classical	classical	ADJ
ejpam-609	29	3	study	study	NOUN
ejpam-609	29	4	of	of	ADP
ejpam-609	29	5	any	any	DET
ejpam-609	29	6	commutative	commutative	ADJ
ejpam-609	29	7	noetherian	noetherian	ADJ
ejpam-609	29	8	ring	ring	NOUN
ejpam-609	29	9	is	be	AUX
ejpam-609	29	10	done	do	VERB
ejpam-609	29	11	by	by	ADP
ejpam-609	29	12	studying	study	VERB
ejpam-609	29	13	its	its	PRON
ejpam-609	29	14	primary	primary	ADJ
ejpam-609	29	15	decomposition	decomposition	NOUN
ejpam-609	29	16	and	and	CCONJ
ejpam-609	29	17	this	this	PRON
ejpam-609	29	18	forms	form	VERB
ejpam-609	29	19	the	the	DET
ejpam-609	29	20	fundamental	fundamental	ADJ
ejpam-609	29	21	edifice	edifice	NOUN
ejpam-609	29	22	on	on	ADP
ejpam-609	29	23	which	which	PRON
ejpam-609	29	24	any	any	DET
ejpam-609	29	25	such	such	ADJ
ejpam-609	29	26	ring	ring	NOUN
ejpam-609	29	27	is	be	AUX
ejpam-609	29	28	studied	study	VERB
ejpam-609	29	29	.	.	PUNCT
ejpam-609	30	1	further	far	ADV
ejpam-609	30	2	there	there	PRON
ejpam-609	30	3	are	be	VERB
ejpam-609	30	4	other	other	ADJ
ejpam-609	30	5	structural	structural	ADJ
ejpam-609	30	6	properties	property	NOUN
ejpam-609	30	7	of	of	ADP
ejpam-609	30	8	rings	ring	NOUN
ejpam-609	30	9	,	,	PUNCT
ejpam-609	30	10	for	for	ADP
ejpam-609	30	11	example	example	NOUN
ejpam-609	30	12	the	the	DET
ejpam-609	30	13	existence	existence	NOUN
ejpam-609	30	14	of	of	ADP
ejpam-609	30	15	quotient	quotient	NOUN
ejpam-609	30	16	rings	ring	NOUN
ejpam-609	30	17	or	or	CCONJ
ejpam-609	30	18	more	more	ADJ
ejpam-609	30	19	particularly	particularly	ADV
ejpam-609	30	20	the	the	DET
ejpam-609	30	21	existence	existence	NOUN
ejpam-609	30	22	of	of	ADP
ejpam-609	30	23	artinian	artinian	ADJ
ejpam-609	30	24	quotient	quotient	NOUN
ejpam-609	30	25	rings	ring	NOUN
ejpam-609	30	26	etc	etc	X
ejpam-609	30	27	.	.	X
ejpam-609	30	28	which	which	PRON
ejpam-609	30	29	can	can	AUX
ejpam-609	30	30	be	be	AUX
ejpam-609	30	31	nicely	nicely	ADV
ejpam-609	30	32	tied	tie	VERB
ejpam-609	30	33	to	to	ADP
ejpam-609	30	34	primary	primary	ADJ
ejpam-609	30	35	decomposition	decomposition	NOUN
ejpam-609	30	36	of	of	ADP
ejpam-609	30	37	a	a	DET
ejpam-609	30	38	noetherian	noetherian	ADJ
ejpam-609	30	39	ring	ring	NOUN
ejpam-609	30	40	.	.	PUNCT
ejpam-609	31	1	the	the	DET
ejpam-609	31	2	notion	notion	NOUN
ejpam-609	31	3	of	of	ADP
ejpam-609	31	4	the	the	DET
ejpam-609	31	5	quotient	quotient	NOUN
ejpam-609	31	6	ring	ring	NOUN
ejpam-609	31	7	of	of	ADP
ejpam-609	31	8	a	a	DET
ejpam-609	31	9	ring	ring	NOUN
ejpam-609	31	10	appears	appear	VERB
ejpam-609	31	11	in	in	ADP
ejpam-609	31	12	chapter	chapter	NOUN
ejpam-609	31	13	(	(	PUNCT
ejpam-609	31	14	9	9	NUM
ejpam-609	31	15	)	)	PUNCT
ejpam-609	31	16	of	of	ADP
ejpam-609	31	17	[	[	X
ejpam-609	31	18	8	8	NUM
ejpam-609	31	19	]	]	PUNCT
ejpam-609	31	20	.	.	PUNCT
ejpam-609	32	1	in	in	ADP
ejpam-609	32	2	[	[	X
ejpam-609	32	3	6	6	NUM
ejpam-609	32	4	]	]	PUNCT
ejpam-609	32	5	it	it	PRON
ejpam-609	32	6	is	be	AUX
ejpam-609	32	7	shown	show	VERB
ejpam-609	32	8	that	that	SCONJ
ejpam-609	32	9	if	if	SCONJ
ejpam-609	32	10	r	r	NOUN
ejpam-609	32	11	is	be	AUX
ejpam-609	32	12	embeddable	embeddable	ADJ
ejpam-609	32	13	in	in	ADP
ejpam-609	32	14	a	a	DET
ejpam-609	32	15	right	right	ADJ
ejpam-609	32	16	artinian	artinian	ADJ
ejpam-609	32	17	ring	ring	NOUN
ejpam-609	32	18	and	and	CCONJ
ejpam-609	32	19	if	if	SCONJ
ejpam-609	32	20	characteristic	characteristic	ADJ
ejpam-609	32	21	of	of	ADP
ejpam-609	32	22	r	r	NOUN
ejpam-609	32	23	is	be	AUX
ejpam-609	32	24	zero	zero	NUM
ejpam-609	32	25	,	,	PUNCT
ejpam-609	32	26	then	then	ADV
ejpam-609	32	27	the	the	DET
ejpam-609	32	28	differential	differential	ADJ
ejpam-609	32	29	operator	operator	NOUN
ejpam-609	32	30	ring	ring	NOUN
ejpam-609	32	31	r[x	r[x	NOUN
ejpam-609	32	32	;	;	PUNCT
ejpam-609	32	33	δ	δ	PROPN
ejpam-609	32	34	]	]	PUNCT
ejpam-609	32	35	embeds	embed	VERB
ejpam-609	32	36	in	in	ADP
ejpam-609	32	37	a	a	DET
ejpam-609	32	38	right	right	ADJ
ejpam-609	32	39	artinian	artinian	ADJ
ejpam-609	32	40	ring	ring	NOUN
ejpam-609	32	41	.	.	PUNCT
ejpam-609	33	1	it	it	PRON
ejpam-609	33	2	is	be	AUX
ejpam-609	33	3	also	also	ADV
ejpam-609	33	4	shown	show	VERB
ejpam-609	33	5	in	in	ADP
ejpam-609	33	6	[	[	X
ejpam-609	33	7	6	6	NUM
ejpam-609	33	8	]	]	PUNCT
ejpam-609	33	9	that	that	SCONJ
ejpam-609	33	10	if	if	SCONJ
ejpam-609	33	11	r	r	NOUN
ejpam-609	33	12	is	be	AUX
ejpam-609	33	13	a	a	DET
ejpam-609	33	14	commutative	commutative	ADJ
ejpam-609	33	15	noetherian	noetherian	ADJ
ejpam-609	33	16	ring	ring	NOUN
ejpam-609	33	17	and	and	CCONJ
ejpam-609	33	18	σ	σ	PROPN
ejpam-609	33	19	is	be	AUX
ejpam-609	33	20	an	an	DET
ejpam-609	33	21	automorphism	automorphism	NOUN
ejpam-609	33	22	of	of	ADP
ejpam-609	33	23	r	r	NOUN
ejpam-609	33	24	,	,	PUNCT
ejpam-609	33	25	then	then	ADV
ejpam-609	33	26	the	the	DET
ejpam-609	33	27	skew	skew	ADJ
ejpam-609	33	28	-	-	PUNCT
ejpam-609	33	29	polynomial	polynomial	ADJ
ejpam-609	33	30	ring	ring	NOUN
ejpam-609	33	31	r[x	r[x	NOUN
ejpam-609	33	32	;	;	PUNCT
ejpam-609	33	33	σ	σ	PROPN
ejpam-609	33	34	]	]	PUNCT
ejpam-609	33	35	embeds	embed	VERB
ejpam-609	33	36	in	in	ADP
ejpam-609	33	37	an	an	DET
ejpam-609	33	38	artinian	artinian	ADJ
ejpam-609	33	39	ring	ring	NOUN
ejpam-609	33	40	.	.	PUNCT
ejpam-609	34	1	a	a	DET
ejpam-609	34	2	non	non	X
ejpam-609	34	3	commutative	commutative	ADJ
ejpam-609	34	4	analogue	analogue	NOUN
ejpam-609	34	5	of	of	ADP
ejpam-609	34	6	associated	associate	VERB
ejpam-609	34	7	prime	prime	ADJ
ejpam-609	34	8	ideals	ideal	NOUN
ejpam-609	34	9	of	of	ADP
ejpam-609	34	10	a	a	DET
ejpam-609	34	11	noetherian	noetherian	ADJ
ejpam-609	34	12	ring	ring	NOUN
ejpam-609	34	13	has	have	AUX
ejpam-609	34	14	also	also	ADV
ejpam-609	34	15	been	be	AUX
ejpam-609	34	16	also	also	ADV
ejpam-609	34	17	discussed	discuss	VERB
ejpam-609	34	18	.	.	PUNCT
ejpam-609	35	1	we	we	PRON
ejpam-609	35	2	would	would	AUX
ejpam-609	35	3	like	like	VERB
ejpam-609	35	4	to	to	PART
ejpam-609	35	5	note	note	VERB
ejpam-609	35	6	that	that	SCONJ
ejpam-609	35	7	a	a	DET
ejpam-609	35	8	considerable	considerable	ADJ
ejpam-609	35	9	work	work	NOUN
ejpam-609	35	10	has	have	AUX
ejpam-609	35	11	been	be	AUX
ejpam-609	35	12	done	do	VERB
ejpam-609	35	13	in	in	ADP
ejpam-609	35	14	the	the	DET
ejpam-609	35	15	investigation	investigation	NOUN
ejpam-609	35	16	of	of	ADP
ejpam-609	35	17	prime	prime	ADJ
ejpam-609	35	18	ideals	ideal	NOUN
ejpam-609	35	19	(	(	PUNCT
ejpam-609	35	20	in	in	ADP
ejpam-609	35	21	particular	particular	ADJ
ejpam-609	35	22	minimal	minimal	ADJ
ejpam-609	35	23	prime	prime	ADJ
ejpam-609	35	24	ideals	ideal	NOUN
ejpam-609	35	25	and	and	CCONJ
ejpam-609	35	26	associated	associate	VERB
ejpam-609	35	27	prime	prime	ADJ
ejpam-609	35	28	ideals	ideal	NOUN
ejpam-609	35	29	)	)	PUNCT
ejpam-609	35	30	of	of	ADP
ejpam-609	35	31	skew	skew	ADJ
ejpam-609	35	32	polynomial	polynomial	ADJ
ejpam-609	35	33	rings	ring	NOUN
ejpam-609	35	34	(	(	PUNCT
ejpam-609	35	35	k.	k.	PROPN
ejpam-609	35	36	r.	r.	PROPN
ejpam-609	35	37	goodearl	goodearl	PROPN
ejpam-609	35	38	and	and	CCONJ
ejpam-609	35	39	e.	e.	PROPN
ejpam-609	35	40	s.	s.	PROPN
ejpam-609	35	41	letzter	letzter	PROPN
ejpam-609	36	1	[	[	X
ejpam-609	36	2	9	9	NUM
ejpam-609	36	3	]	]	PUNCT
ejpam-609	36	4	,	,	PUNCT
ejpam-609	36	5	c.	c.	PROPN
ejpam-609	36	6	faith	faith	NOUN
ejpam-609	36	7	[	[	X
ejpam-609	36	8	7	7	NUM
ejpam-609	36	9	]	]	PUNCT
ejpam-609	36	10	,	,	PUNCT
ejpam-609	36	11	s.	s.	PROPN
ejpam-609	36	12	annin	annin	PROPN
ejpam-609	37	1	[	[	X
ejpam-609	37	2	1	1	NUM
ejpam-609	37	3	]	]	PUNCT
ejpam-609	37	4	,	,	PUNCT
ejpam-609	37	5	leroy	leroy	PROPN
ejpam-609	37	6	and	and	CCONJ
ejpam-609	37	7	matczuk	matczuk	ADJ
ejpam-609	37	8	[	[	X
ejpam-609	37	9	12	12	NUM
ejpam-609	37	10	]	]	PUNCT
ejpam-609	37	11	,	,	PUNCT
ejpam-609	37	12	nordstrom	nordstrom	NOUN
ejpam-609	37	13	[	[	X
ejpam-609	37	14	15	15	NUM
ejpam-609	37	15	]	]	PUNCT
ejpam-609	37	16	)	)	PUNCT
ejpam-609	37	17	and	and	CCONJ
ejpam-609	37	18	bhat	bhat	X
ejpam-609	38	1	[	[	X
ejpam-609	38	2	3	3	NUM
ejpam-609	38	3	]	]	PUNCT
ejpam-609	38	4	.	.	PUNCT
ejpam-609	39	1	another	another	DET
ejpam-609	39	2	related	related	ADJ
ejpam-609	39	3	area	area	NOUN
ejpam-609	39	4	of	of	ADP
ejpam-609	39	5	interest	interest	NOUN
ejpam-609	39	6	since	since	SCONJ
ejpam-609	39	7	recent	recent	ADJ
ejpam-609	39	8	past	past	NOUN
ejpam-609	39	9	has	have	AUX
ejpam-609	39	10	been	be	AUX
ejpam-609	39	11	the	the	DET
ejpam-609	39	12	study	study	NOUN
ejpam-609	39	13	of	of	ADP
ejpam-609	39	14	2	2	NUM
ejpam-609	39	15	-	-	PUNCT
ejpam-609	39	16	primal	primal	ADJ
ejpam-609	39	17	rings	ring	NOUN
ejpam-609	39	18	.	.	PUNCT
ejpam-609	40	1	this	this	PRON
ejpam-609	40	2	involves	involve	VERB
ejpam-609	40	3	the	the	DET
ejpam-609	40	4	notions	notion	NOUN
ejpam-609	40	5	of	of	ADP
ejpam-609	40	6	prime	prime	ADJ
ejpam-609	40	7	radical	radical	ADJ
ejpam-609	40	8	and	and	CCONJ
ejpam-609	40	9	the	the	DET
ejpam-609	40	10	set	set	NOUN
ejpam-609	40	11	of	of	ADP
ejpam-609	40	12	nilpotent	nilpotent	ADJ
ejpam-609	40	13	elements	element	NOUN
ejpam-609	40	14	of	of	ADP
ejpam-609	40	15	a	a	DET
ejpam-609	40	16	ring	ring	NOUN
ejpam-609	40	17	.	.	PUNCT
ejpam-609	41	1	further	far	ADV
ejpam-609	41	2	more	more	ADV
ejpam-609	41	3	the	the	DET
ejpam-609	41	4	concept	concept	NOUN
ejpam-609	41	5	of	of	ADP
ejpam-609	41	6	completely	completely	ADV
ejpam-609	41	7	prime	prime	ADJ
ejpam-609	41	8	ideals	ideal	NOUN
ejpam-609	41	9	and	and	CCONJ
ejpam-609	41	10	the	the	DET
ejpam-609	41	11	completely	completely	ADV
ejpam-609	41	12	semiprime	semiprime	NOUN
ejpam-609	41	13	ideals	ideal	NOUN
ejpam-609	41	14	are	be	AUX
ejpam-609	41	15	also	also	ADV
ejpam-609	41	16	studied	study	VERB
ejpam-609	41	17	in	in	ADP
ejpam-609	41	18	this	this	DET
ejpam-609	41	19	area	area	NOUN
ejpam-609	41	20	.	.	PUNCT
ejpam-609	42	1	recall	recall	VERB
ejpam-609	42	2	that	that	SCONJ
ejpam-609	42	3	a	a	DET
ejpam-609	42	4	ring	ring	NOUN
ejpam-609	42	5	r	r	NOUN
ejpam-609	42	6	is	be	AUX
ejpam-609	42	7	2	2	NUM
ejpam-609	42	8	-	-	PUNCT
ejpam-609	42	9	primal	primal	ADJ
ejpam-609	42	10	if	if	SCONJ
ejpam-609	42	11	n(r	n(r	NUM
ejpam-609	42	12	)	)	PUNCT
ejpam-609	42	13	=	=	SYM
ejpam-609	42	14	p(r	p(r	PROPN
ejpam-609	42	15	)	)	PUNCT
ejpam-609	42	16	;	;	PUNCT
ejpam-609	42	17	i.e.	i.e.	X
ejpam-609	42	18	if	if	SCONJ
ejpam-609	42	19	the	the	DET
ejpam-609	42	20	prime	prime	ADJ
ejpam-609	42	21	radical	radical	NOUN
ejpam-609	42	22	is	be	AUX
ejpam-609	42	23	a	a	DET
ejpam-609	42	24	completely	completely	ADV
ejpam-609	42	25	semiprime	semiprime	NOUN
ejpam-609	42	26	ideal	ideal	NOUN
ejpam-609	42	27	.	.	PUNCT
ejpam-609	43	1	an	an	DET
ejpam-609	43	2	ideal	ideal	ADJ
ejpam-609	43	3	i	i	PRON
ejpam-609	43	4	of	of	ADP
ejpam-609	43	5	a	a	DET
ejpam-609	43	6	ring	ring	NOUN
ejpam-609	43	7	r	r	NOUN
ejpam-609	43	8	is	be	AUX
ejpam-609	43	9	called	call	VERB
ejpam-609	43	10	completely	completely	ADV
ejpam-609	43	11	semiprime	semiprime	NOUN
ejpam-609	43	12	if	if	SCONJ
ejpam-609	43	13	a2	a2	PROPN
ejpam-609	43	14	∈	∈	PROPN
ejpam-609	43	15	i	i	PRON
ejpam-609	43	16	implies	imply	VERB
ejpam-609	43	17	a	a	DET
ejpam-609	43	18	∈	∈	NOUN
ejpam-609	43	19	i	i	PRON
ejpam-609	43	20	for	for	ADP
ejpam-609	43	21	a	a	DET
ejpam-609	43	22	∈	∈	PROPN
ejpam-609	43	23	r.	r.	NOUN
ejpam-609	43	24	we	we	PRON
ejpam-609	43	25	also	also	ADV
ejpam-609	43	26	note	note	VERB
ejpam-609	43	27	that	that	SCONJ
ejpam-609	43	28	a	a	DET
ejpam-609	43	29	reduced	reduced	NOUN
ejpam-609	43	30	is	be	AUX
ejpam-609	43	31	2	2	NUM
ejpam-609	43	32	-	-	PUNCT
ejpam-609	43	33	primal	primal	ADJ
ejpam-609	43	34	and	and	CCONJ
ejpam-609	43	35	a	a	DET
ejpam-609	43	36	commutative	commutative	ADJ
ejpam-609	43	37	ring	ring	NOUN
ejpam-609	43	38	is	be	AUX
ejpam-609	43	39	also	also	ADV
ejpam-609	43	40	2	2	NUM
ejpam-609	43	41	-	-	PUNCT
ejpam-609	43	42	primal	primal	ADJ
ejpam-609	43	43	.	.	PUNCT
ejpam-609	44	1	2	2	NUM
ejpam-609	44	2	-	-	PUNCT
ejpam-609	44	3	primal	primal	ADJ
ejpam-609	44	4	rings	ring	NOUN
ejpam-609	44	5	have	have	AUX
ejpam-609	44	6	been	be	AUX
ejpam-609	44	7	studied	study	VERB
ejpam-609	44	8	in	in	ADP
ejpam-609	44	9	recent	recent	ADJ
ejpam-609	44	10	years	year	NOUN
ejpam-609	44	11	and	and	CCONJ
ejpam-609	44	12	are	be	AUX
ejpam-609	44	13	being	be	AUX
ejpam-609	44	14	treated	treat	VERB
ejpam-609	44	15	by	by	ADP
ejpam-609	44	16	authors	author	NOUN
ejpam-609	44	17	for	for	ADP
ejpam-609	44	18	different	different	ADJ
ejpam-609	44	19	structures	structure	NOUN
ejpam-609	44	20	.	.	PUNCT
ejpam-609	45	1	in	in	ADP
ejpam-609	45	2	[	[	X
ejpam-609	45	3	13	13	NUM
ejpam-609	45	4	]	]	PUNCT
ejpam-609	45	5	,	,	PUNCT
ejpam-609	45	6	marks	mark	NOUN
ejpam-609	45	7	discusses	discuss	VERB
ejpam-609	45	8	the	the	DET
ejpam-609	45	9	2	2	NUM
ejpam-609	45	10	-	-	PUNCT
ejpam-609	45	11	primal	primal	ADJ
ejpam-609	45	12	property	property	NOUN
ejpam-609	45	13	of	of	ADP
ejpam-609	45	14	r[x	r[x	NOUN
ejpam-609	45	15	;	;	PUNCT
ejpam-609	45	16	σ	σ	PROPN
ejpam-609	45	17	,	,	PUNCT
ejpam-609	45	18	δ	δ	PROPN
ejpam-609	45	19	]	]	X
ejpam-609	45	20	,	,	PUNCT
ejpam-609	45	21	where	where	SCONJ
ejpam-609	45	22	r	r	NOUN
ejpam-609	45	23	is	be	AUX
ejpam-609	45	24	a	a	DET
ejpam-609	45	25	local	local	ADJ
ejpam-609	45	26	ring	ring	NOUN
ejpam-609	45	27	,	,	PUNCT
ejpam-609	45	28	σ	σ	VERB
ejpam-609	45	29	an	an	DET
ejpam-609	45	30	automorphism	automorphism	NOUN
ejpam-609	45	31	of	of	ADP
ejpam-609	45	32	r	r	NOUN
ejpam-609	45	33	and	and	CCONJ
ejpam-609	45	34	δ	δ	PROPN
ejpam-609	45	35	a	a	DET
ejpam-609	45	36	σ	σ	NOUN
ejpam-609	45	37	-	-	PUNCT
ejpam-609	45	38	derivation	derivation	NOUN
ejpam-609	45	39	of	of	ADP
ejpam-609	45	40	r.	r.	PROPN
ejpam-609	45	41	in	in	ADP
ejpam-609	45	42	marks	mark	NOUN
ejpam-609	45	43	[	[	X
ejpam-609	45	44	13	13	NUM
ejpam-609	45	45	]	]	PUNCT
ejpam-609	45	46	,	,	PUNCT
ejpam-609	45	47	it	it	PRON
ejpam-609	45	48	has	have	AUX
ejpam-609	45	49	been	be	AUX
ejpam-609	45	50	shown	show	VERB
ejpam-609	45	51	that	that	SCONJ
ejpam-609	45	52	for	for	ADP
ejpam-609	45	53	a	a	DET
ejpam-609	45	54	local	local	ADJ
ejpam-609	45	55	ring	ring	NOUN
ejpam-609	45	56	r	r	NOUN
ejpam-609	45	57	with	with	ADP
ejpam-609	45	58	a	a	DET
ejpam-609	45	59	nilpotent	nilpotent	ADJ
ejpam-609	45	60	maximal	maximal	ADJ
ejpam-609	45	61	ideal	ideal	NOUN
ejpam-609	45	62	,	,	PUNCT
ejpam-609	45	63	the	the	DET
ejpam-609	45	64	ore	ore	NOUN
ejpam-609	45	65	extension	extension	NOUN
ejpam-609	45	66	r[x	r[x	NOUN
ejpam-609	45	67	;	;	PUNCT
ejpam-609	45	68	σ	σ	PROPN
ejpam-609	45	69	,	,	PUNCT
ejpam-609	45	70	δ	δ	PROPN
ejpam-609	45	71	]	]	PUNCT
ejpam-609	45	72	will	will	AUX
ejpam-609	45	73	or	or	CCONJ
ejpam-609	45	74	will	will	AUX
ejpam-609	45	75	not	not	PART
ejpam-609	45	76	be	be	AUX
ejpam-609	45	77	2	2	NUM
ejpam-609	45	78	-	-	NOUN
ejpam-609	45	79	primal	primal	ADJ
ejpam-609	45	80	depending	depend	VERB
ejpam-609	45	81	on	on	ADP
ejpam-609	45	82	the	the	DET
ejpam-609	45	83	δ	δ	NOUN
ejpam-609	45	84	-	-	NOUN
ejpam-609	45	85	stability	stability	NOUN
ejpam-609	45	86	of	of	ADP
ejpam-609	45	87	the	the	DET
ejpam-609	45	88	maximal	maximal	ADJ
ejpam-609	45	89	ideal	ideal	NOUN
ejpam-609	45	90	of	of	ADP
ejpam-609	45	91	r.	r.	PROPN
ejpam-609	45	92	in	in	ADP
ejpam-609	45	93	the	the	DET
ejpam-609	45	94	case	case	NOUN
ejpam-609	45	95	where	where	SCONJ
ejpam-609	45	96	r[x	r[x	NOUN
ejpam-609	45	97	;	;	PUNCT
ejpam-609	45	98	σ	σ	PROPN
ejpam-609	45	99	,	,	PUNCT
ejpam-609	45	100	δ	δ	PROPN
ejpam-609	45	101	]	]	PUNCT
ejpam-609	45	102	is	be	AUX
ejpam-609	45	103	2	2	NUM
ejpam-609	45	104	-	-	PUNCT
ejpam-609	45	105	primal	primal	ADJ
ejpam-609	45	106	,	,	PUNCT
ejpam-609	45	107	it	it	PRON
ejpam-609	45	108	will	will	AUX
ejpam-609	45	109	satisfy	satisfy	VERB
ejpam-609	45	110	an	an	DET
ejpam-609	45	111	even	even	ADV
ejpam-609	45	112	stronger	strong	ADJ
ejpam-609	45	113	condition	condition	NOUN
ejpam-609	45	114	;	;	PUNCT
ejpam-609	45	115	in	in	ADP
ejpam-609	45	116	the	the	DET
ejpam-609	45	117	case	case	NOUN
ejpam-609	45	118	where	where	SCONJ
ejpam-609	45	119	r[x	r[x	NOUN
ejpam-609	45	120	;	;	PUNCT
ejpam-609	45	121	σ	σ	PROPN
ejpam-609	45	122	,	,	PUNCT
ejpam-609	45	123	δ	δ	PROPN
ejpam-609	45	124	]	]	PUNCT
ejpam-609	45	125	is	be	AUX
ejpam-609	45	126	not	not	PART
ejpam-609	45	127	2	2	NUM
ejpam-609	45	128	-	-	PUNCT
ejpam-609	45	129	primal	primal	ADJ
ejpam-609	45	130	,	,	PUNCT
ejpam-609	45	131	it	it	PRON
ejpam-609	45	132	will	will	AUX
ejpam-609	45	133	fail	fail	VERB
ejpam-609	45	134	to	to	PART
ejpam-609	45	135	satisfy	satisfy	VERB
ejpam-609	45	136	an	an	DET
ejpam-609	45	137	even	even	ADV
ejpam-609	45	138	weaker	weak	ADJ
ejpam-609	45	139	condition	condition	NOUN
ejpam-609	45	140	.	.	PUNCT
ejpam-609	46	1	krempa	krempa	NOUN
ejpam-609	46	2	in	in	ADP
ejpam-609	46	3	[	[	X
ejpam-609	46	4	10	10	NUM
ejpam-609	46	5	]	]	PUNCT
ejpam-609	46	6	introduced	introduce	VERB
ejpam-609	46	7	σ	σ	PROPN
ejpam-609	46	8	-	-	ADJ
ejpam-609	46	9	rigid	rigid	ADJ
ejpam-609	46	10	rings	ring	NOUN
ejpam-609	46	11	;	;	PUNCT
ejpam-609	46	12	kwak	kwak	PROPN
ejpam-609	46	13	in	in	ADP
ejpam-609	46	14	[	[	X
ejpam-609	46	15	11	11	NUM
ejpam-609	46	16	]	]	PUNCT
ejpam-609	46	17	introduced	introduce	VERB
ejpam-609	46	18	σ(∗)-rings	σ(∗)-ring	NOUN
ejpam-609	46	19	and	and	CCONJ
ejpam-609	46	20	established	establish	VERB
ejpam-609	46	21	a	a	DET
ejpam-609	46	22	relation	relation	NOUN
ejpam-609	46	23	between	between	ADP
ejpam-609	46	24	a	a	DET
ejpam-609	46	25	2	2	NUM
ejpam-609	46	26	-	-	PUNCT
ejpam-609	46	27	primal	primal	ADJ
ejpam-609	46	28	ring	ring	NOUN
ejpam-609	46	29	and	and	CCONJ
ejpam-609	46	30	a	a	DET
ejpam-609	46	31	σ(∗)-ring	σ(∗)-ring	PROPN
ejpam-609	46	32	.	.	PUNCT
ejpam-609	46	33	ouyang	ouyang	PROPN
ejpam-609	46	34	in	in	ADP
ejpam-609	46	35	[	[	X
ejpam-609	46	36	16	16	NUM
ejpam-609	46	37	]	]	PUNCT
ejpam-609	46	38	introduced	introduce	VERB
ejpam-609	46	39	weak	weak	ADJ
ejpam-609	46	40	σ	σ	ADJ
ejpam-609	46	41	-	-	ADJ
ejpam-609	46	42	rigid	rigid	ADJ
ejpam-609	46	43	rings	ring	NOUN
ejpam-609	46	44	,	,	PUNCT
ejpam-609	46	45	where	where	SCONJ
ejpam-609	46	46	σ	σ	PROPN
ejpam-609	46	47	is	be	AUX
ejpam-609	46	48	an	an	DET
ejpam-609	46	49	endomorphism	endomorphism	NOUN
ejpam-609	46	50	of	of	ADP
ejpam-609	46	51	ring	ring	PROPN
ejpam-609	46	52	r.	r.	PROPN
ejpam-609	46	53	these	these	DET
ejpam-609	46	54	rings	ring	NOUN
ejpam-609	46	55	are	be	AUX
ejpam-609	46	56	related	relate	VERB
ejpam-609	46	57	to	to	ADP
ejpam-609	46	58	2	2	NUM
ejpam-609	46	59	-	-	PUNCT
ejpam-609	46	60	primal	primal	ADJ
ejpam-609	46	61	rings	ring	NOUN
ejpam-609	46	62	.	.	PUNCT
ejpam-609	47	1	in	in	ADP
ejpam-609	47	2	this	this	DET
ejpam-609	47	3	paper	paper	NOUN
ejpam-609	47	4	we	we	PRON
ejpam-609	47	5	study	study	VERB
ejpam-609	47	6	these	these	DET
ejpam-609	47	7	rings	ring	NOUN
ejpam-609	47	8	and	and	CCONJ
ejpam-609	47	9	find	find	VERB
ejpam-609	47	10	a	a	DET
ejpam-609	47	11	relation	relation	NOUN
ejpam-609	47	12	between	between	ADP
ejpam-609	47	13	these	these	DET
ejpam-609	47	14	rings	ring	NOUN
ejpam-609	47	15	.	.	PUNCT
ejpam-609	48	1	towards	towards	ADP
ejpam-609	48	2	this	this	PRON
ejpam-609	48	3	v.	v.	ADP
ejpam-609	48	4	bhat	bhat	PROPN
ejpam-609	48	5	/	/	SYM
ejpam-609	48	6	eur	eur	PROPN
ejpam-609	48	7	.	.	PUNCT
ejpam-609	49	1	j.	j.	PROPN
ejpam-609	49	2	pure	pure	PROPN
ejpam-609	49	3	appl	appl	PROPN
ejpam-609	49	4	.	.	PROPN
ejpam-609	49	5	math	math	PROPN
ejpam-609	49	6	,	,	PUNCT
ejpam-609	49	7	3	3	NUM
ejpam-609	49	8	(	(	PUNCT
ejpam-609	49	9	2010	2010	NUM
ejpam-609	49	10	)	)	PUNCT
ejpam-609	49	11	,	,	PUNCT
ejpam-609	49	12	695	695	NUM
ejpam-609	49	13	-	-	SYM
ejpam-609	49	14	703	703	NUM
ejpam-609	49	15	697	697	NUM
ejpam-609	49	16	we	we	PRON
ejpam-609	49	17	prove	prove	VERB
ejpam-609	49	18	the	the	DET
ejpam-609	49	19	following	following	NOUN
ejpam-609	49	20	:	:	PUNCT
ejpam-609	49	21	theorem	theorem	NOUN
ejpam-609	49	22	1	1	X
ejpam-609	49	23	.	.	PUNCT
ejpam-609	50	1	let	let	VERB
ejpam-609	50	2	r	r	PRON
ejpam-609	50	3	be	be	AUX
ejpam-609	50	4	a	a	DET
ejpam-609	50	5	noetherian	noetherian	ADJ
ejpam-609	50	6	ring	ring	NOUN
ejpam-609	50	7	.	.	PUNCT
ejpam-609	51	1	let	let	VERB
ejpam-609	51	2	σ	σ	NOUN
ejpam-609	51	3	be	be	AUX
ejpam-609	51	4	an	an	DET
ejpam-609	51	5	automorphism	automorphism	NOUN
ejpam-609	51	6	of	of	ADP
ejpam-609	51	7	r	r	NOUN
ejpam-609	51	8	such	such	ADJ
ejpam-609	51	9	that	that	SCONJ
ejpam-609	51	10	r	r	NOUN
ejpam-609	51	11	is	be	AUX
ejpam-609	51	12	a	a	DET
ejpam-609	51	13	σ(∗)ring	σ(∗)ring	NOUN
ejpam-609	51	14	.	.	PUNCT
ejpam-609	52	1	then	then	ADV
ejpam-609	52	2	r	r	NOUN
ejpam-609	52	3	is	be	AUX
ejpam-609	52	4	a	a	DET
ejpam-609	52	5	weak	weak	ADJ
ejpam-609	52	6	σ	σ	ADJ
ejpam-609	52	7	-	-	ADJ
ejpam-609	52	8	rigid	rigid	ADJ
ejpam-609	52	9	ring	ring	NOUN
ejpam-609	52	10	.	.	PUNCT
ejpam-609	53	1	conversely	conversely	ADV
ejpam-609	53	2	a	a	DET
ejpam-609	53	3	2	2	NUM
ejpam-609	53	4	-	-	PUNCT
ejpam-609	53	5	primal	primal	ADJ
ejpam-609	53	6	weak	weak	ADJ
ejpam-609	53	7	σ	σ	ADJ
ejpam-609	53	8	-	-	ADJ
ejpam-609	53	9	rigid	rigid	ADJ
ejpam-609	53	10	ring	ring	NOUN
ejpam-609	53	11	is	be	AUX
ejpam-609	53	12	a	a	DET
ejpam-609	53	13	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	53	14	.	.	PUNCT
ejpam-609	54	1	(	(	PUNCT
ejpam-609	54	2	this	this	PRON
ejpam-609	54	3	is	be	AUX
ejpam-609	54	4	proved	prove	VERB
ejpam-609	54	5	in	in	ADP
ejpam-609	54	6	theorem	theorem	NOUN
ejpam-609	54	7	(	(	PUNCT
ejpam-609	54	8	5	5	NUM
ejpam-609	54	9	)	)	PUNCT
ejpam-609	54	10	)	)	PUNCT
ejpam-609	54	11	.	.	PUNCT
ejpam-609	55	1	we	we	PRON
ejpam-609	55	2	also	also	ADV
ejpam-609	55	3	discuss	discuss	VERB
ejpam-609	55	4	skew	skew	ADJ
ejpam-609	55	5	polynomial	polynomial	ADJ
ejpam-609	55	6	rings	ring	NOUN
ejpam-609	55	7	over	over	ADP
ejpam-609	55	8	weak	weak	ADJ
ejpam-609	55	9	σ	σ	ADJ
ejpam-609	55	10	-	-	ADJ
ejpam-609	55	11	rigid	rigid	ADJ
ejpam-609	55	12	rings	ring	NOUN
ejpam-609	55	13	.	.	PUNCT
ejpam-609	56	1	towards	towards	ADP
ejpam-609	56	2	this	this	PRON
ejpam-609	56	3	we	we	PRON
ejpam-609	56	4	have	have	VERB
ejpam-609	56	5	the	the	DET
ejpam-609	56	6	following	following	NOUN
ejpam-609	56	7	:	:	PUNCT
ejpam-609	56	8	let	let	VERB
ejpam-609	56	9	σ	σ	NOUN
ejpam-609	56	10	be	be	AUX
ejpam-609	56	11	an	an	DET
ejpam-609	56	12	endomorphism	endomorphism	NOUN
ejpam-609	56	13	of	of	ADP
ejpam-609	56	14	a	a	DET
ejpam-609	56	15	ring	ring	NOUN
ejpam-609	56	16	r	r	NOUN
ejpam-609	56	17	and	and	CCONJ
ejpam-609	56	18	δ	δ	PROPN
ejpam-609	56	19	a	a	DET
ejpam-609	56	20	σ	σ	NOUN
ejpam-609	56	21	-	-	PUNCT
ejpam-609	56	22	derivation	derivation	NOUN
ejpam-609	56	23	of	of	ADP
ejpam-609	56	24	r	r	NOUN
ejpam-609	56	25	such	such	ADJ
ejpam-609	56	26	that	that	DET
ejpam-609	56	27	σ(δ(a	σ(δ(a	NOUN
ejpam-609	56	28	)	)	PUNCT
ejpam-609	56	29	)	)	PUNCT
ejpam-609	57	1	=	=	PUNCT
ejpam-609	57	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	57	3	)	)	PUNCT
ejpam-609	57	4	)	)	PUNCT
ejpam-609	57	5	for	for	ADP
ejpam-609	57	6	all	all	DET
ejpam-609	57	7	a	a	DET
ejpam-609	57	8	∈	∈	PROPN
ejpam-609	57	9	r.	r.	NOUN
ejpam-609	57	10	then	then	ADV
ejpam-609	57	11	σ	σ	PROPN
ejpam-609	57	12	can	can	AUX
ejpam-609	57	13	be	be	AUX
ejpam-609	57	14	extended	extend	VERB
ejpam-609	57	15	to	to	ADP
ejpam-609	57	16	an	an	DET
ejpam-609	57	17	endomorphism	endomorphism	NOUN
ejpam-609	57	18	(	(	PUNCT
ejpam-609	57	19	say	say	INTJ
ejpam-609	57	20	σ	σ	NOUN
ejpam-609	57	21	)	)	PUNCT
ejpam-609	57	22	of	of	ADP
ejpam-609	57	23	r[x	r[x	NOUN
ejpam-609	57	24	;	;	PUNCT
ejpam-609	57	25	σ	σ	PROPN
ejpam-609	57	26	,	,	PUNCT
ejpam-609	57	27	δ	δ	PROPN
ejpam-609	57	28	]	]	PUNCT
ejpam-609	57	29	by	by	ADP
ejpam-609	57	30	σ	σ	PROPN
ejpam-609	57	31	(	(	PUNCT
ejpam-609	57	32	∑m	∑m	PROPN
ejpam-609	57	33	i=0	i=0	PROPN
ejpam-609	57	34	x	x	X
ejpam-609	57	35	iai	iai	ADJ
ejpam-609	57	36	)	)	PUNCT
ejpam-609	57	37	=	=	VERB
ejpam-609	57	38	∑m	∑m	PROPN
ejpam-609	57	39	i=0	i=0	PROPN
ejpam-609	57	40	x	x	SYM
ejpam-609	57	41	iσ(ai	iσ(ai	PROPN
ejpam-609	57	42	)	)	PUNCT
ejpam-609	57	43	.	.	PUNCT
ejpam-609	58	1	also	also	ADV
ejpam-609	58	2	δ	δ	PROPN
ejpam-609	58	3	can	can	AUX
ejpam-609	58	4	be	be	AUX
ejpam-609	58	5	extended	extend	VERB
ejpam-609	58	6	to	to	ADP
ejpam-609	58	7	a	a	DET
ejpam-609	58	8	σ	σ	NOUN
ejpam-609	58	9	-	-	PUNCT
ejpam-609	58	10	derivation	derivation	NOUN
ejpam-609	58	11	(	(	PUNCT
ejpam-609	58	12	say	say	VERB
ejpam-609	58	13	δ	δ	PROPN
ejpam-609	58	14	)	)	PUNCT
ejpam-609	58	15	of	of	ADP
ejpam-609	58	16	r[x	r[x	NOUN
ejpam-609	58	17	;	;	PUNCT
ejpam-609	58	18	σ	σ	PROPN
ejpam-609	58	19	,	,	PUNCT
ejpam-609	58	20	δ	δ	PROPN
ejpam-609	58	21	]	]	PUNCT
ejpam-609	58	22	by	by	ADP
ejpam-609	58	23	δ	δ	PROPN
ejpam-609	58	24	(	(	PUNCT
ejpam-609	58	25	∑m	∑m	PROPN
ejpam-609	58	26	i=0	i=0	PROPN
ejpam-609	58	27	x	x	X
ejpam-609	58	28	iai	iai	ADJ
ejpam-609	58	29	)	)	PUNCT
ejpam-609	59	1	=	=	PUNCT
ejpam-609	59	2	∑m	∑m	PROPN
ejpam-609	59	3	i=0	i=0	PROPN
ejpam-609	59	4	x	x	SYM
ejpam-609	59	5	iδ(ai	iδ(ai	PROPN
ejpam-609	59	6	)	)	PUNCT
ejpam-609	59	7	.	.	PUNCT
ejpam-609	60	1	we	we	PRON
ejpam-609	60	2	note	note	VERB
ejpam-609	60	3	that	that	SCONJ
ejpam-609	60	4	if	if	SCONJ
ejpam-609	60	5	σ(δ(a	σ(δ(a	PROPN
ejpam-609	60	6	)	)	PUNCT
ejpam-609	60	7	)	)	PUNCT
ejpam-609	60	8	6=	6=	X
ejpam-609	61	1	δ(σ(a	δ(σ(a	NOUN
ejpam-609	61	2	)	)	PUNCT
ejpam-609	61	3	)	)	PUNCT
ejpam-609	61	4	for	for	ADP
ejpam-609	61	5	all	all	DET
ejpam-609	61	6	a	a	DET
ejpam-609	61	7	∈	∈	NOUN
ejpam-609	61	8	r	r	NOUN
ejpam-609	61	9	,	,	PUNCT
ejpam-609	61	10	then	then	ADV
ejpam-609	61	11	the	the	DET
ejpam-609	61	12	above	above	ADJ
ejpam-609	61	13	does	do	AUX
ejpam-609	61	14	not	not	PART
ejpam-609	61	15	hold	hold	VERB
ejpam-609	61	16	.	.	PUNCT
ejpam-609	62	1	for	for	ADP
ejpam-609	62	2	example	example	NOUN
ejpam-609	62	3	let	let	VERB
ejpam-609	62	4	f	f	PROPN
ejpam-609	62	5	(	(	PUNCT
ejpam-609	62	6	x	x	NOUN
ejpam-609	62	7	)	)	PUNCT
ejpam-609	62	8	=	=	SYM
ejpam-609	62	9	xa	xa	PROPN
ejpam-609	62	10	and	and	CCONJ
ejpam-609	62	11	g(x	g(x	NOUN
ejpam-609	62	12	)	)	PUNCT
ejpam-609	63	1	=	=	PUNCT
ejpam-609	63	2	x	x	SYM
ejpam-609	63	3	b	b	PROPN
ejpam-609	63	4	,	,	PUNCT
ejpam-609	63	5	a	a	PRON
ejpam-609	63	6	,	,	PUNCT
ejpam-609	63	7	b	b	PROPN
ejpam-609	63	8	∈	∈	PROPN
ejpam-609	63	9	r.	r.	PROPN
ejpam-609	63	10	then	then	ADV
ejpam-609	63	11	δ	δ	PROPN
ejpam-609	63	12	(	(	PUNCT
ejpam-609	63	13	f	f	PROPN
ejpam-609	63	14	(	(	PUNCT
ejpam-609	63	15	x)g(x	x)g(x	ADJ
ejpam-609	63	16	)	)	PUNCT
ejpam-609	63	17	)	)	PUNCT
ejpam-609	64	1	=	=	SYM
ejpam-609	64	2	x2{δ(σ(a))σ(b	x2{δ(σ(a))σ(b	NUM
ejpam-609	64	3	)	)	PUNCT
ejpam-609	65	1	+	+	NOUN
ejpam-609	65	2	σ(a)δ(b)}+	σ(a)δ(b)}+	ADJ
ejpam-609	65	3	x{δ2(a)σ(b	x{δ2(a)σ(b	NOUN
ejpam-609	65	4	)	)	PUNCT
ejpam-609	65	5	+	+	NUM
ejpam-609	65	6	δ(a)σ(b	δ(a)σ(b	NOUN
ejpam-609	65	7	)	)	PUNCT
ejpam-609	65	8	}	}	PUNCT
ejpam-609	65	9	,	,	PUNCT
ejpam-609	65	10	but	but	CCONJ
ejpam-609	65	11	δ	δ	PROPN
ejpam-609	65	12	(	(	PUNCT
ejpam-609	65	13	f	f	PROPN
ejpam-609	65	14	(	(	PUNCT
ejpam-609	65	15	x))σ(g(x))+	x))σ(g(x))+	PROPN
ejpam-609	65	16	f	f	PROPN
ejpam-609	65	17	(	(	PUNCT
ejpam-609	65	18	x)δ(g(x	x)δ(g(x	NUM
ejpam-609	65	19	)	)	PUNCT
ejpam-609	65	20	)	)	PUNCT
ejpam-609	66	1	=	=	SYM
ejpam-609	66	2	x2{σ(δ(a))σ(b)+σ(a)δ(b)}+	x2{σ(δ(a))σ(b)+σ(a)δ(b)}+	PROPN
ejpam-609	66	3	x{δ2(a)σ(b)+δ(a)σ(b	x{δ2(a)σ(b)+δ(a)σ(b	PROPN
ejpam-609	66	4	)	)	PUNCT
ejpam-609	66	5	}	}	PUNCT
ejpam-609	66	6	.	.	PUNCT
ejpam-609	67	1	with	with	ADP
ejpam-609	67	2	this	this	PRON
ejpam-609	67	3	we	we	PRON
ejpam-609	67	4	prove	prove	VERB
ejpam-609	67	5	the	the	DET
ejpam-609	67	6	following	following	NOUN
ejpam-609	67	7	:	:	PUNCT
ejpam-609	67	8	theorem	theorem	NOUN
ejpam-609	67	9	2	2	NUM
ejpam-609	67	10	.	.	PUNCT
ejpam-609	68	1	let	let	VERB
ejpam-609	68	2	r	r	PRON
ejpam-609	68	3	be	be	AUX
ejpam-609	68	4	a	a	DET
ejpam-609	68	5	commutative	commutative	ADJ
ejpam-609	68	6	noetherian	noetherian	ADJ
ejpam-609	68	7	ring	ring	NOUN
ejpam-609	68	8	which	which	PRON
ejpam-609	68	9	is	be	AUX
ejpam-609	68	10	also	also	ADV
ejpam-609	68	11	an	an	DET
ejpam-609	68	12	algebra	algebra	NOUN
ejpam-609	68	13	over	over	ADP
ejpam-609	68	14	q.	q.	PROPN
ejpam-609	68	15	let	let	VERB
ejpam-609	68	16	σ	σ	NOUN
ejpam-609	68	17	be	be	AUX
ejpam-609	68	18	an	an	DET
ejpam-609	68	19	automorphism	automorphism	NOUN
ejpam-609	68	20	of	of	ADP
ejpam-609	68	21	r	r	NOUN
ejpam-609	68	22	and	and	CCONJ
ejpam-609	68	23	δ	δ	PROPN
ejpam-609	69	1	a	a	DET
ejpam-609	69	2	σ	σ	NOUN
ejpam-609	69	3	-	-	PUNCT
ejpam-609	69	4	derivation	derivation	NOUN
ejpam-609	69	5	of	of	ADP
ejpam-609	69	6	r	r	NOUN
ejpam-609	69	7	such	such	ADJ
ejpam-609	69	8	that	that	DET
ejpam-609	69	9	σ(δ(a	σ(δ(a	NOUN
ejpam-609	69	10	)	)	PUNCT
ejpam-609	69	11	)	)	PUNCT
ejpam-609	70	1	=	=	PUNCT
ejpam-609	70	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	70	3	)	)	PUNCT
ejpam-609	70	4	)	)	PUNCT
ejpam-609	70	5	for	for	ADP
ejpam-609	70	6	all	all	DET
ejpam-609	70	7	a	a	DET
ejpam-609	70	8	∈	∈	PROPN
ejpam-609	70	9	r.	r.	NOUN
ejpam-609	70	10	then	then	ADV
ejpam-609	70	11	r	r	NOUN
ejpam-609	70	12	is	be	AUX
ejpam-609	70	13	a	a	DET
ejpam-609	70	14	weak	weak	ADJ
ejpam-609	70	15	σ	σ	ADJ
ejpam-609	70	16	-	-	ADJ
ejpam-609	70	17	rigid	rigid	ADJ
ejpam-609	70	18	ring	ring	NOUN
ejpam-609	70	19	if	if	SCONJ
ejpam-609	70	20	and	and	CCONJ
ejpam-609	70	21	only	only	ADV
ejpam-609	70	22	if	if	SCONJ
ejpam-609	70	23	o(r	o(r	PRON
ejpam-609	70	24	)	)	PUNCT
ejpam-609	70	25	=	=	SYM
ejpam-609	70	26	r[x	r[x	NOUN
ejpam-609	70	27	;	;	PUNCT
ejpam-609	70	28	σ	σ	PROPN
ejpam-609	70	29	,	,	PUNCT
ejpam-609	70	30	δ	δ	PROPN
ejpam-609	70	31	]	]	PUNCT
ejpam-609	70	32	is	be	AUX
ejpam-609	70	33	a	a	DET
ejpam-609	70	34	weak	weak	ADJ
ejpam-609	70	35	σ	σ	ADJ
ejpam-609	70	36	-	-	ADJ
ejpam-609	70	37	rigid	rigid	ADJ
ejpam-609	70	38	ring	ring	NOUN
ejpam-609	70	39	.	.	PUNCT
ejpam-609	71	1	(	(	PUNCT
ejpam-609	71	2	this	this	PRON
ejpam-609	71	3	is	be	AUX
ejpam-609	71	4	proved	prove	VERB
ejpam-609	71	5	in	in	ADP
ejpam-609	71	6	theorem	theorem	NOUN
ejpam-609	71	7	(	(	PUNCT
ejpam-609	71	8	7	7	NUM
ejpam-609	71	9	)	)	PUNCT
ejpam-609	71	10	)	)	PUNCT
ejpam-609	71	11	.	.	PUNCT
ejpam-609	72	1	2	2	X
ejpam-609	72	2	.	.	X
ejpam-609	72	3	preliminaries	preliminary	NOUN
ejpam-609	72	4	2.1	2.1	NUM
ejpam-609	72	5	.	.	PUNCT
ejpam-609	73	1	general	general	ADJ
ejpam-609	73	2	we	we	PRON
ejpam-609	73	3	begin	begin	VERB
ejpam-609	73	4	with	with	ADP
ejpam-609	73	5	the	the	DET
ejpam-609	73	6	following	follow	VERB
ejpam-609	73	7	definitions	definition	NOUN
ejpam-609	73	8	:	:	PUNCT
ejpam-609	73	9	definition	definition	NOUN
ejpam-609	73	10	1	1	NUM
ejpam-609	73	11	(	(	PUNCT
ejpam-609	73	12	krempa[10	krempa[10	X
ejpam-609	73	13	]	]	X
ejpam-609	73	14	)	)	PUNCT
ejpam-609	73	15	.	.	PUNCT
ejpam-609	74	1	let	let	VERB
ejpam-609	74	2	r	r	PRON
ejpam-609	74	3	be	be	AUX
ejpam-609	74	4	a	a	DET
ejpam-609	74	5	ring	ring	NOUN
ejpam-609	74	6	and	and	CCONJ
ejpam-609	74	7	σ	σ	NOUN
ejpam-609	74	8	an	an	DET
ejpam-609	74	9	endomorphism	endomorphism	NOUN
ejpam-609	74	10	of	of	ADP
ejpam-609	74	11	r.	r.	PROPN
ejpam-609	74	12	then	then	ADV
ejpam-609	74	13	σ	σ	PROPN
ejpam-609	74	14	is	be	AUX
ejpam-609	74	15	said	say	VERB
ejpam-609	74	16	to	to	PART
ejpam-609	74	17	be	be	AUX
ejpam-609	74	18	a	a	DET
ejpam-609	74	19	rigid	rigid	ADJ
ejpam-609	74	20	endomorphism	endomorphism	NOUN
ejpam-609	74	21	if	if	SCONJ
ejpam-609	74	22	aσ(a	aσ(a	VERB
ejpam-609	74	23	)	)	PUNCT
ejpam-609	74	24	=	=	SYM
ejpam-609	74	25	0	0	NUM
ejpam-609	74	26	implies	imply	VERB
ejpam-609	74	27	that	that	SCONJ
ejpam-609	74	28	a	a	DET
ejpam-609	74	29	=	=	SYM
ejpam-609	74	30	0	0	NUM
ejpam-609	74	31	,	,	PUNCT
ejpam-609	74	32	for	for	ADP
ejpam-609	74	33	a	a	DET
ejpam-609	74	34	∈	∈	PROPN
ejpam-609	74	35	r.	r.	NOUN
ejpam-609	74	36	the	the	DET
ejpam-609	74	37	ring	ring	NOUN
ejpam-609	74	38	r	r	NOUN
ejpam-609	74	39	is	be	AUX
ejpam-609	74	40	said	say	VERB
ejpam-609	74	41	to	to	PART
ejpam-609	74	42	be	be	AUX
ejpam-609	74	43	a	a	DET
ejpam-609	74	44	σ	σ	ADJ
ejpam-609	74	45	-	-	ADJ
ejpam-609	74	46	rigid	rigid	ADJ
ejpam-609	74	47	ring	ring	NOUN
ejpam-609	74	48	if	if	SCONJ
ejpam-609	74	49	there	there	PRON
ejpam-609	74	50	exists	exist	VERB
ejpam-609	74	51	a	a	DET
ejpam-609	74	52	σ	σ	PROPN
ejpam-609	74	53	-	-	ADJ
ejpam-609	74	54	rigid	rigid	ADJ
ejpam-609	74	55	endomorphism	endomorphism	PROPN
ejpam-609	74	56	r.	r.	PROPN
ejpam-609	74	57	for	for	ADP
ejpam-609	74	58	example	example	NOUN
ejpam-609	74	59	let	let	VERB
ejpam-609	74	60	r=	r=	ADJ
ejpam-609	74	61	c	c	NOUN
ejpam-609	74	62	,	,	PUNCT
ejpam-609	74	63	and	and	CCONJ
ejpam-609	74	64	σ	σ	NOUN
ejpam-609	74	65	:	:	PUNCT
ejpam-609	74	66	c→	c→	X
ejpam-609	74	67	c	c	X
ejpam-609	74	68	be	be	AUX
ejpam-609	74	69	the	the	DET
ejpam-609	74	70	map	map	NOUN
ejpam-609	74	71	defined	define	VERB
ejpam-609	74	72	by	by	ADP
ejpam-609	74	73	σ(a+	σ(a+	PROPN
ejpam-609	74	74	i	i	PROPN
ejpam-609	74	75	b	b	NOUN
ejpam-609	74	76	)	)	PUNCT
ejpam-609	74	77	=	=	SYM
ejpam-609	74	78	a−	a−	PROPN
ejpam-609	74	79	i	i	NOUN
ejpam-609	74	80	b	b	PROPN
ejpam-609	74	81	,	,	PUNCT
ejpam-609	74	82	a	a	PRON
ejpam-609	74	83	,	,	PUNCT
ejpam-609	74	84	b	b	PROPN
ejpam-609	74	85	∈	∈	PROPN
ejpam-609	74	86	r.	r.	NOUN
ejpam-609	74	87	then	then	ADV
ejpam-609	74	88	it	it	PRON
ejpam-609	74	89	can	can	AUX
ejpam-609	74	90	be	be	AUX
ejpam-609	74	91	seen	see	VERB
ejpam-609	74	92	that	that	SCONJ
ejpam-609	74	93	σ	σ	PROPN
ejpam-609	74	94	is	be	AUX
ejpam-609	74	95	a	a	DET
ejpam-609	74	96	rigid	rigid	ADJ
ejpam-609	74	97	endomorphism	endomorphism	NOUN
ejpam-609	74	98	of	of	ADP
ejpam-609	74	99	r.	r.	PROPN
ejpam-609	74	100	in	in	ADP
ejpam-609	74	101	theorem	theorem	PROPN
ejpam-609	74	102	3.3	3.3	NUM
ejpam-609	74	103	of	of	ADP
ejpam-609	74	104	[	[	X
ejpam-609	74	105	10	10	NUM
ejpam-609	74	106	]	]	PUNCT
ejpam-609	74	107	,	,	PUNCT
ejpam-609	74	108	krempa	krempa	PROPN
ejpam-609	74	109	has	have	AUX
ejpam-609	74	110	proved	prove	VERB
ejpam-609	74	111	the	the	DET
ejpam-609	74	112	following	following	NOUN
ejpam-609	74	113	:	:	PUNCT
ejpam-609	74	114	let	let	VERB
ejpam-609	74	115	r	r	PRON
ejpam-609	74	116	be	be	AUX
ejpam-609	74	117	a	a	DET
ejpam-609	74	118	ring	ring	NOUN
ejpam-609	74	119	,	,	PUNCT
ejpam-609	74	120	let	let	VERB
ejpam-609	74	121	σ	σ	NOUN
ejpam-609	74	122	be	be	AUX
ejpam-609	74	123	an	an	DET
ejpam-609	74	124	endomorphism	endomorphism	NOUN
ejpam-609	74	125	and	and	CCONJ
ejpam-609	74	126	δ	δ	PROPN
ejpam-609	74	127	a	a	DET
ejpam-609	74	128	σ	σ	NOUN
ejpam-609	74	129	-	-	PUNCT
ejpam-609	74	130	derivation	derivation	NOUN
ejpam-609	74	131	of	of	ADP
ejpam-609	74	132	r.	r.	PROPN
ejpam-609	74	133	if	if	SCONJ
ejpam-609	74	134	σ	σ	PROPN
ejpam-609	74	135	is	be	AUX
ejpam-609	74	136	a	a	DET
ejpam-609	74	137	monomorphism	monomorphism	NOUN
ejpam-609	74	138	,	,	PUNCT
ejpam-609	74	139	then	then	ADV
ejpam-609	74	140	the	the	DET
ejpam-609	74	141	skew	skew	ADJ
ejpam-609	74	142	polynomial	polynomial	ADJ
ejpam-609	74	143	ring	ring	NOUN
ejpam-609	74	144	r[x	r[x	NOUN
ejpam-609	74	145	;	;	PUNCT
ejpam-609	74	146	σ	σ	PROPN
ejpam-609	74	147	,	,	PUNCT
ejpam-609	74	148	δ	δ	PROPN
ejpam-609	74	149	]	]	PUNCT
ejpam-609	74	150	is	be	AUX
ejpam-609	74	151	reduced	reduce	VERB
ejpam-609	74	152	if	if	SCONJ
ejpam-609	74	153	and	and	CCONJ
ejpam-609	74	154	only	only	ADV
ejpam-609	74	155	if	if	SCONJ
ejpam-609	74	156	r	r	NOUN
ejpam-609	74	157	is	be	AUX
ejpam-609	74	158	reduced	reduce	VERB
ejpam-609	74	159	and	and	CCONJ
ejpam-609	74	160	σ	σ	PROPN
ejpam-609	74	161	is	be	AUX
ejpam-609	74	162	rigid	rigid	ADJ
ejpam-609	74	163	.	.	PUNCT
ejpam-609	75	1	under	under	ADP
ejpam-609	75	2	this	this	DET
ejpam-609	75	3	conditions	condition	NOUN
ejpam-609	75	4	any	any	DET
ejpam-609	75	5	minimal	minimal	ADJ
ejpam-609	75	6	prime	prime	ADJ
ejpam-609	75	7	ideal	ideal	NOUN
ejpam-609	75	8	(	(	PUNCT
ejpam-609	75	9	annihilator	annihilator	NOUN
ejpam-609	75	10	)	)	PUNCT
ejpam-609	75	11	of	of	ADP
ejpam-609	75	12	r[x	r[x	NOUN
ejpam-609	75	13	;	;	PUNCT
ejpam-609	76	1	σ;δ	σ;δ	PROPN
ejpam-609	76	2	]	]	X
ejpam-609	76	3	is	be	AUX
ejpam-609	76	4	of	of	ADP
ejpam-609	76	5	the	the	DET
ejpam-609	76	6	form	form	NOUN
ejpam-609	76	7	p[x	p[x	ADV
ejpam-609	76	8	;	;	PUNCT
ejpam-609	76	9	σ;δ	σ;δ	PROPN
ejpam-609	76	10	]	]	PUNCT
ejpam-609	76	11	where	where	SCONJ
ejpam-609	76	12	p	p	NOUN
ejpam-609	76	13	is	be	AUX
ejpam-609	76	14	a	a	DET
ejpam-609	76	15	minimal	minimal	ADJ
ejpam-609	76	16	prime	prime	ADJ
ejpam-609	76	17	ideal	ideal	NOUN
ejpam-609	76	18	(	(	PUNCT
ejpam-609	76	19	annihilator	annihilator	PROPN
ejpam-609	76	20	)	)	PUNCT
ejpam-609	76	21	in	in	ADP
ejpam-609	76	22	r.	r.	PROPN
ejpam-609	76	23	v.	v.	PROPN
ejpam-609	76	24	bhat	bhat	PROPN
ejpam-609	76	25	/	/	SYM
ejpam-609	76	26	eur	eur	PROPN
ejpam-609	76	27	.	.	PUNCT
ejpam-609	77	1	j.	j.	PROPN
ejpam-609	77	2	pure	pure	PROPN
ejpam-609	77	3	appl	appl	PROPN
ejpam-609	77	4	.	.	PROPN
ejpam-609	77	5	math	math	PROPN
ejpam-609	77	6	,	,	PUNCT
ejpam-609	77	7	3	3	NUM
ejpam-609	77	8	(	(	PUNCT
ejpam-609	77	9	2010	2010	NUM
ejpam-609	77	10	)	)	PUNCT
ejpam-609	77	11	,	,	PUNCT
ejpam-609	77	12	695	695	NUM
ejpam-609	77	13	-	-	SYM
ejpam-609	77	14	703	703	NUM
ejpam-609	77	15	698	698	NUM
ejpam-609	77	16	definition	definition	NOUN
ejpam-609	77	17	2	2	NUM
ejpam-609	77	18	(	(	PUNCT
ejpam-609	77	19	kwak	kwak	PROPN
ejpam-609	77	20	[	[	X
ejpam-609	77	21	11	11	NUM
ejpam-609	77	22	]	]	PUNCT
ejpam-609	77	23	)	)	PUNCT
ejpam-609	77	24	.	.	PUNCT
ejpam-609	78	1	let	let	VERB
ejpam-609	78	2	r	r	PRON
ejpam-609	78	3	be	be	AUX
ejpam-609	78	4	a	a	DET
ejpam-609	78	5	ring	ring	NOUN
ejpam-609	78	6	and	and	CCONJ
ejpam-609	78	7	σ	σ	NOUN
ejpam-609	78	8	an	an	DET
ejpam-609	78	9	endomorphism	endomorphism	NOUN
ejpam-609	78	10	of	of	ADP
ejpam-609	78	11	r.	r.	PROPN
ejpam-609	78	12	then	then	ADV
ejpam-609	78	13	r	r	NOUN
ejpam-609	78	14	is	be	AUX
ejpam-609	78	15	said	say	VERB
ejpam-609	78	16	to	to	PART
ejpam-609	78	17	be	be	AUX
ejpam-609	78	18	a	a	DET
ejpam-609	78	19	σ(∗)-ring	σ(∗)-re	VERB
ejpam-609	78	20	if	if	SCONJ
ejpam-609	78	21	aσ(a	aσ(a	NUM
ejpam-609	78	22	)	)	PUNCT
ejpam-609	78	23	∈	∈	PROPN
ejpam-609	78	24	p(r	p(r	PROPN
ejpam-609	78	25	)	)	PUNCT
ejpam-609	78	26	implies	imply	VERB
ejpam-609	78	27	a	a	DET
ejpam-609	78	28	∈	∈	PROPN
ejpam-609	78	29	p(r	p(r	PROPN
ejpam-609	78	30	)	)	PUNCT
ejpam-609	78	31	for	for	ADP
ejpam-609	78	32	a	a	DET
ejpam-609	78	33	∈	∈	PROPN
ejpam-609	78	34	r.	r.	PROPN
ejpam-609	78	35	example	example	NOUN
ejpam-609	78	36	1	1	NUM
ejpam-609	78	37	(	(	PUNCT
ejpam-609	78	38	example	example	NOUN
ejpam-609	78	39	2	2	NUM
ejpam-609	78	40	of	of	ADP
ejpam-609	78	41	[	[	X
ejpam-609	78	42	11	11	NUM
ejpam-609	78	43	]	]	NUM
ejpam-609	78	44	)	)	PUNCT
ejpam-609	78	45	.	.	PUNCT
ejpam-609	79	1	let	let	VERB
ejpam-609	79	2	r	r	NOUN
ejpam-609	79	3	=	=	SYM
ejpam-609	79	4	�	�	PROPN
ejpam-609	79	5	f	f	PROPN
ejpam-609	79	6	f	f	PROPN
ejpam-609	79	7	0	0	PROPN
ejpam-609	79	8	f	f	PROPN
ejpam-609	79	9	�	�	PROPN
ejpam-609	79	10	,	,	PUNCT
ejpam-609	79	11	where	where	SCONJ
ejpam-609	79	12	f	f	PROPN
ejpam-609	79	13	is	be	AUX
ejpam-609	79	14	a	a	DET
ejpam-609	79	15	field	field	NOUN
ejpam-609	79	16	.	.	PUNCT
ejpam-609	80	1	then	then	ADV
ejpam-609	80	2	p(r	p(r	PROPN
ejpam-609	80	3	)	)	PUNCT
ejpam-609	80	4	=	=	SYM
ejpam-609	80	5	�	�	PROPN
ejpam-609	80	6	0	0	NUM
ejpam-609	80	7	f	f	PROPN
ejpam-609	80	8	0	0	SYM
ejpam-609	80	9	0	0	NUM
ejpam-609	80	10	�	�	PROPN
ejpam-609	80	11	.	.	PUNCT
ejpam-609	81	1	let	let	VERB
ejpam-609	81	2	σ	σ	NOUN
ejpam-609	81	3	:	:	PUNCT
ejpam-609	81	4	r	r	NOUN
ejpam-609	81	5	→	→	SYM
ejpam-609	81	6	r	r	NOUN
ejpam-609	81	7	be	be	AUX
ejpam-609	81	8	defined	define	VERB
ejpam-609	81	9	by	by	ADP
ejpam-609	81	10	σ	σ	PROPN
ejpam-609	81	11	�	�	PROPN
ejpam-609	81	12	�	�	PROPN
ejpam-609	81	13	a	a	DET
ejpam-609	81	14	b	b	PROPN
ejpam-609	81	15	0	0	NUM
ejpam-609	81	16	c	c	PROPN
ejpam-609	81	17	�	�	PROPN
ejpam-609	81	18	�	�	PROPN
ejpam-609	81	19	=	=	SYM
ejpam-609	81	20	�	�	PROPN
ejpam-609	81	21	a	a	DET
ejpam-609	81	22	0	0	NUM
ejpam-609	81	23	0	0	NUM
ejpam-609	81	24	c	c	PROPN
ejpam-609	81	25	�	�	PROPN
ejpam-609	81	26	.	.	PUNCT
ejpam-609	82	1	then	then	ADV
ejpam-609	82	2	it	it	PRON
ejpam-609	82	3	can	can	AUX
ejpam-609	82	4	be	be	AUX
ejpam-609	82	5	seen	see	VERB
ejpam-609	82	6	that	that	SCONJ
ejpam-609	82	7	σ	σ	PROPN
ejpam-609	82	8	is	be	AUX
ejpam-609	82	9	an	an	DET
ejpam-609	82	10	endomorphism	endomorphism	NOUN
ejpam-609	82	11	of	of	ADP
ejpam-609	82	12	r	r	NOUN
ejpam-609	82	13	and	and	CCONJ
ejpam-609	82	14	r	r	NOUN
ejpam-609	82	15	is	be	AUX
ejpam-609	82	16	a	a	DET
ejpam-609	82	17	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	82	18	.	.	PUNCT
ejpam-609	83	1	we	we	PRON
ejpam-609	83	2	note	note	VERB
ejpam-609	83	3	that	that	SCONJ
ejpam-609	83	4	the	the	DET
ejpam-609	83	5	above	above	ADJ
ejpam-609	83	6	ring	ring	NOUN
ejpam-609	83	7	is	be	AUX
ejpam-609	83	8	not	not	PART
ejpam-609	83	9	σ	σ	NOUN
ejpam-609	83	10	-	-	ADJ
ejpam-609	83	11	rigid	rigid	ADJ
ejpam-609	83	12	.	.	PUNCT
ejpam-609	84	1	for	for	ADP
ejpam-609	84	2	let	let	VERB
ejpam-609	84	3	0	0	NUM
ejpam-609	84	4	6=	6=	ADP
ejpam-609	84	5	a	a	DET
ejpam-609	84	6	∈	∈	PROPN
ejpam-609	84	7	f	f	X
ejpam-609	84	8	.	.	PUNCT
ejpam-609	85	1	then	then	ADV
ejpam-609	85	2	�	�	PROPN
ejpam-609	85	3	0	0	NUM
ejpam-609	85	4	a	a	DET
ejpam-609	85	5	0	0	NUM
ejpam-609	85	6	0	0	NUM
ejpam-609	85	7	�	�	PROPN
ejpam-609	85	8	σ	σ	X
ejpam-609	85	9	�	�	PROPN
ejpam-609	85	10	0	0	NUM
ejpam-609	85	11	a	a	DET
ejpam-609	85	12	0	0	NUM
ejpam-609	85	13	0	0	NUM
ejpam-609	85	14	�	�	PROPN
ejpam-609	85	15	=	=	SYM
ejpam-609	85	16	�	�	PROPN
ejpam-609	85	17	0	0	NUM
ejpam-609	85	18	0	0	NUM
ejpam-609	85	19	0	0	SYM
ejpam-609	85	20	0	0	NUM
ejpam-609	85	21	�	�	PROPN
ejpam-609	85	22	,	,	PUNCT
ejpam-609	85	23	but	but	CCONJ
ejpam-609	85	24	�	�	PROPN
ejpam-609	85	25	0	0	NUM
ejpam-609	85	26	a	a	DET
ejpam-609	85	27	0	0	NUM
ejpam-609	85	28	0	0	NUM
ejpam-609	85	29	�	�	PROPN
ejpam-609	85	30	6=	6=	SYM
ejpam-609	85	31	�	�	PROPN
ejpam-609	85	32	0	0	NUM
ejpam-609	85	33	0	0	NUM
ejpam-609	85	34	0	0	SYM
ejpam-609	85	35	0	0	NUM
ejpam-609	85	36	�	�	PROPN
ejpam-609	85	37	.	.	PUNCT
ejpam-609	86	1	the	the	DET
ejpam-609	86	2	ring	ring	NOUN
ejpam-609	86	3	in	in	ADP
ejpam-609	86	4	example	example	NOUN
ejpam-609	86	5	(	(	PUNCT
ejpam-609	86	6	1	1	X
ejpam-609	86	7	)	)	PUNCT
ejpam-609	86	8	is	be	AUX
ejpam-609	86	9	also	also	ADV
ejpam-609	86	10	2	2	NUM
ejpam-609	86	11	-	-	PUNCT
ejpam-609	86	12	primal	primal	ADJ
ejpam-609	86	13	.	.	PUNCT
ejpam-609	87	1	we	we	PRON
ejpam-609	87	2	also	also	ADV
ejpam-609	87	3	note	note	VERB
ejpam-609	87	4	that	that	SCONJ
ejpam-609	87	5	a	a	DET
ejpam-609	87	6	ring	ring	NOUN
ejpam-609	87	7	r	r	NOUN
ejpam-609	87	8	is	be	AUX
ejpam-609	87	9	a	a	DET
ejpam-609	87	10	i(∗)-ring	i(∗)-ring	ADJ
ejpam-609	87	11	if	if	SCONJ
ejpam-609	88	1	and	and	CCONJ
ejpam-609	88	2	only	only	ADV
ejpam-609	88	3	if	if	SCONJ
ejpam-609	88	4	r	r	NOUN
ejpam-609	88	5	is	be	AUX
ejpam-609	88	6	a	a	DET
ejpam-609	88	7	2	2	NUM
ejpam-609	88	8	-	-	PUNCT
ejpam-609	88	9	primal	primal	ADJ
ejpam-609	88	10	ring	ring	NOUN
ejpam-609	88	11	,	,	PUNCT
ejpam-609	88	12	where	where	SCONJ
ejpam-609	88	13	i	i	PRON
ejpam-609	88	14	is	be	AUX
ejpam-609	88	15	the	the	DET
ejpam-609	88	16	identity	identity	NOUN
ejpam-609	88	17	map	map	NOUN
ejpam-609	88	18	on	on	ADP
ejpam-609	88	19	r.	r.	PROPN
ejpam-609	88	20	in	in	ADP
ejpam-609	88	21	[	[	X
ejpam-609	88	22	11	11	NUM
ejpam-609	88	23	]	]	PUNCT
ejpam-609	88	24	,	,	PUNCT
ejpam-609	88	25	the	the	DET
ejpam-609	88	26	2	2	NUM
ejpam-609	88	27	-	-	PUNCT
ejpam-609	88	28	primal	primal	ADJ
ejpam-609	88	29	property	property	NOUN
ejpam-609	88	30	has	have	AUX
ejpam-609	88	31	also	also	ADV
ejpam-609	88	32	been	be	AUX
ejpam-609	88	33	extended	extend	VERB
ejpam-609	88	34	to	to	ADP
ejpam-609	88	35	the	the	DET
ejpam-609	88	36	skew	skew	ADJ
ejpam-609	88	37	-	-	PUNCT
ejpam-609	88	38	polynomial	polynomial	ADJ
ejpam-609	88	39	ring	ring	NOUN
ejpam-609	88	40	r[x	r[x	NOUN
ejpam-609	88	41	;	;	PUNCT
ejpam-609	88	42	σ	σ	PROPN
ejpam-609	88	43	]	]	PUNCT
ejpam-609	88	44	.	.	PUNCT
ejpam-609	89	1	kwak	kwak	PROPN
ejpam-609	89	2	in	in	ADP
ejpam-609	89	3	[	[	X
ejpam-609	89	4	11	11	NUM
ejpam-609	89	5	]	]	PUNCT
ejpam-609	89	6	also	also	ADV
ejpam-609	89	7	establishes	establish	VERB
ejpam-609	89	8	a	a	DET
ejpam-609	89	9	relation	relation	NOUN
ejpam-609	89	10	between	between	ADP
ejpam-609	89	11	a	a	DET
ejpam-609	89	12	2	2	NUM
ejpam-609	89	13	-	-	PUNCT
ejpam-609	89	14	primal	primal	ADJ
ejpam-609	89	15	ring	ring	NOUN
ejpam-609	89	16	and	and	CCONJ
ejpam-609	89	17	a	a	DET
ejpam-609	89	18	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	89	19	.	.	PUNCT
ejpam-609	90	1	it	it	PRON
ejpam-609	90	2	has	have	AUX
ejpam-609	90	3	been	be	AUX
ejpam-609	90	4	proved	prove	VERB
ejpam-609	90	5	in	in	ADP
ejpam-609	90	6	theorem	theorem	NOUN
ejpam-609	90	7	5	5	NUM
ejpam-609	90	8	of	of	ADP
ejpam-609	90	9	[	[	X
ejpam-609	90	10	11	11	NUM
ejpam-609	90	11	]	]	PUNCT
ejpam-609	90	12	that	that	SCONJ
ejpam-609	90	13	if	if	SCONJ
ejpam-609	90	14	r	r	NOUN
ejpam-609	90	15	is	be	AUX
ejpam-609	90	16	a	a	DET
ejpam-609	90	17	2	2	NUM
ejpam-609	90	18	-	-	PUNCT
ejpam-609	90	19	primal	primal	ADJ
ejpam-609	90	20	ring	ring	NOUN
ejpam-609	90	21	and	and	CCONJ
ejpam-609	90	22	σ	σ	PROPN
ejpam-609	90	23	is	be	AUX
ejpam-609	90	24	an	an	DET
ejpam-609	90	25	automorphism	automorphism	NOUN
ejpam-609	90	26	of	of	ADP
ejpam-609	90	27	r	r	NOUN
ejpam-609	90	28	,	,	PUNCT
ejpam-609	90	29	then	then	ADV
ejpam-609	90	30	r	r	NOUN
ejpam-609	90	31	is	be	AUX
ejpam-609	90	32	a	a	DET
ejpam-609	90	33	σ(∗)-ring	σ(∗)-re	VERB
ejpam-609	90	34	if	if	SCONJ
ejpam-609	90	35	and	and	CCONJ
ejpam-609	90	36	only	only	ADV
ejpam-609	90	37	if	if	SCONJ
ejpam-609	90	38	σ(p	σ(p	PROPN
ejpam-609	90	39	)	)	PUNCT
ejpam-609	90	40	=	=	SYM
ejpam-609	91	1	p	p	NOUN
ejpam-609	91	2	for	for	ADP
ejpam-609	91	3	all	all	DET
ejpam-609	91	4	p	p	PROPN
ejpam-609	91	5	∈	∈	PROPN
ejpam-609	91	6	min.spec(r	min.spec(r	NOUN
ejpam-609	91	7	)	)	PUNCT
ejpam-609	91	8	.	.	PUNCT
ejpam-609	92	1	in	in	ADP
ejpam-609	92	2	theorem	theorem	NOUN
ejpam-609	92	3	12	12	NUM
ejpam-609	92	4	of	of	ADP
ejpam-609	92	5	[	[	X
ejpam-609	92	6	11	11	NUM
ejpam-609	92	7	]	]	PUNCT
ejpam-609	92	8	it	it	PRON
ejpam-609	92	9	has	have	AUX
ejpam-609	92	10	been	be	AUX
ejpam-609	92	11	proved	prove	VERB
ejpam-609	92	12	that	that	SCONJ
ejpam-609	92	13	if	if	SCONJ
ejpam-609	92	14	r	r	NOUN
ejpam-609	92	15	is	be	AUX
ejpam-609	92	16	a	a	DET
ejpam-609	92	17	σ(∗)-ring	σ(∗)-re	VERB
ejpam-609	92	18	with	with	ADP
ejpam-609	92	19	σ(p(r	σ(p(r	PROPN
ejpam-609	92	20	)	)	PUNCT
ejpam-609	92	21	)	)	PUNCT
ejpam-609	93	1	=	=	SYM
ejpam-609	93	2	p(r	p(r	PROPN
ejpam-609	93	3	)	)	PUNCT
ejpam-609	93	4	,	,	PUNCT
ejpam-609	93	5	then	then	ADV
ejpam-609	93	6	r[x	r[x	NOUN
ejpam-609	93	7	;	;	PUNCT
ejpam-609	93	8	σ	σ	PROPN
ejpam-609	93	9	]	]	X
ejpam-609	93	10	is	be	AUX
ejpam-609	93	11	2	2	NUM
ejpam-609	93	12	-	-	PUNCT
ejpam-609	93	13	primal	primal	ADJ
ejpam-609	93	14	if	if	SCONJ
ejpam-609	93	15	and	and	CCONJ
ejpam-609	93	16	only	only	ADV
ejpam-609	93	17	if	if	SCONJ
ejpam-609	93	18	p(r)[x	p(r)[x	NOUN
ejpam-609	93	19	;	;	PUNCT
ejpam-609	93	20	σ	σ	X
ejpam-609	93	21	]	]	X
ejpam-609	93	22	=	=	PUNCT
ejpam-609	93	23	p(r[x	p(r[x	NOUN
ejpam-609	93	24	;	;	PUNCT
ejpam-609	93	25	σ	σ	PROPN
ejpam-609	93	26	]	]	NOUN
ejpam-609	93	27	)	)	PUNCT
ejpam-609	93	28	.	.	PUNCT
ejpam-609	94	1	we	we	PRON
ejpam-609	94	2	now	now	ADV
ejpam-609	94	3	give	give	VERB
ejpam-609	94	4	an	an	DET
ejpam-609	94	5	example	example	NOUN
ejpam-609	94	6	of	of	ADP
ejpam-609	94	7	a	a	DET
ejpam-609	94	8	ring	ring	NOUN
ejpam-609	94	9	r	r	NOUN
ejpam-609	94	10	,	,	PUNCT
ejpam-609	94	11	and	and	CCONJ
ejpam-609	94	12	an	an	DET
ejpam-609	94	13	endomorphism	endomorphism	PROPN
ejpam-609	94	14	σ	σ	NOUN
ejpam-609	94	15	of	of	ADP
ejpam-609	94	16	r	r	NOUN
ejpam-609	94	17	such	such	ADJ
ejpam-609	94	18	that	that	SCONJ
ejpam-609	94	19	r	r	NOUN
ejpam-609	94	20	is	be	AUX
ejpam-609	94	21	not	not	PART
ejpam-609	94	22	a	a	DET
ejpam-609	94	23	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	94	24	,	,	PUNCT
ejpam-609	94	25	however	however	ADV
ejpam-609	94	26	r	r	NOUN
ejpam-609	94	27	is	be	AUX
ejpam-609	94	28	2	2	NUM
ejpam-609	94	29	-	-	PUNCT
ejpam-609	94	30	primal	primal	ADJ
ejpam-609	94	31	.	.	PUNCT
ejpam-609	95	1	example	example	NOUN
ejpam-609	95	2	2	2	NUM
ejpam-609	95	3	(	(	PUNCT
ejpam-609	95	4	example	example	NOUN
ejpam-609	95	5	4	4	NUM
ejpam-609	95	6	of	of	ADP
ejpam-609	95	7	[	[	X
ejpam-609	95	8	11	11	NUM
ejpam-609	95	9	]	]	NUM
ejpam-609	95	10	)	)	PUNCT
ejpam-609	95	11	.	.	PUNCT
ejpam-609	96	1	let	let	VERB
ejpam-609	96	2	r=	r=	PROPN
ejpam-609	96	3	f[x	f[x	PROPN
ejpam-609	96	4	]	]	PUNCT
ejpam-609	96	5	be	be	VERB
ejpam-609	96	6	the	the	DET
ejpam-609	96	7	polynomial	polynomial	ADJ
ejpam-609	96	8	ring	ring	NOUN
ejpam-609	96	9	over	over	ADP
ejpam-609	96	10	a	a	DET
ejpam-609	96	11	field	field	NOUN
ejpam-609	97	1	f.	f.	NOUN
ejpam-609	97	2	then	then	ADV
ejpam-609	97	3	r	r	NOUN
ejpam-609	97	4	is	be	AUX
ejpam-609	97	5	2	2	NUM
ejpam-609	97	6	-	-	NOUN
ejpam-609	97	7	primal	primal	ADJ
ejpam-609	97	8	with	with	ADP
ejpam-609	97	9	p(r	p(r	NOUN
ejpam-609	97	10	)	)	PUNCT
ejpam-609	97	11	=	=	SYM
ejpam-609	98	1	0	0	X
ejpam-609	98	2	.	.	PUNCT
ejpam-609	99	1	let	let	VERB
ejpam-609	99	2	σ	σ	NOUN
ejpam-609	99	3	:	:	PUNCT
ejpam-609	99	4	r→	r→	PROPN
ejpam-609	99	5	r	r	NOUN
ejpam-609	99	6	be	be	AUX
ejpam-609	99	7	an	an	DET
ejpam-609	99	8	endomorphism	endomorphism	NOUN
ejpam-609	99	9	defined	define	VERB
ejpam-609	99	10	by	by	ADP
ejpam-609	99	11	σ	σ	PROPN
ejpam-609	99	12	(	(	PUNCT
ejpam-609	99	13	f	f	PROPN
ejpam-609	99	14	(	(	PUNCT
ejpam-609	99	15	x	x	NOUN
ejpam-609	99	16	)	)	PUNCT
ejpam-609	99	17	)	)	PUNCT
ejpam-609	100	1	=	=	SYM
ejpam-609	100	2	f	f	PROPN
ejpam-609	100	3	(	(	PUNCT
ejpam-609	100	4	0	0	NUM
ejpam-609	100	5	)	)	PUNCT
ejpam-609	100	6	.	.	PUNCT
ejpam-609	101	1	then	then	ADV
ejpam-609	101	2	r	r	NOUN
ejpam-609	101	3	is	be	AUX
ejpam-609	101	4	not	not	PART
ejpam-609	101	5	a	a	DET
ejpam-609	101	6	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	101	7	.	.	PUNCT
ejpam-609	102	1	for	for	ADP
ejpam-609	102	2	example	example	NOUN
ejpam-609	102	3	consider	consider	VERB
ejpam-609	102	4	f	f	PROPN
ejpam-609	102	5	(	(	PUNCT
ejpam-609	102	6	x	x	NOUN
ejpam-609	102	7	)	)	PUNCT
ejpam-609	102	8	=	=	SYM
ejpam-609	102	9	xa	xa	PROPN
ejpam-609	102	10	,	,	PUNCT
ejpam-609	102	11	a	a	PRON
ejpam-609	102	12	6=	6=	NUM
ejpam-609	102	13	0	0	NUM
ejpam-609	102	14	.	.	PUNCT
ejpam-609	103	1	let	let	VERB
ejpam-609	103	2	r	r	PRON
ejpam-609	103	3	be	be	AUX
ejpam-609	103	4	a	a	DET
ejpam-609	103	5	ring	ring	NOUN
ejpam-609	103	6	and	and	CCONJ
ejpam-609	103	7	σ	σ	NOUN
ejpam-609	103	8	an	an	DET
ejpam-609	103	9	automorphism	automorphism	NOUN
ejpam-609	103	10	of	of	ADP
ejpam-609	103	11	r.	r.	PROPN
ejpam-609	103	12	we	we	PRON
ejpam-609	103	13	now	now	ADV
ejpam-609	103	14	give	give	VERB
ejpam-609	103	15	a	a	DET
ejpam-609	103	16	necessary	necessary	ADJ
ejpam-609	103	17	and	and	CCONJ
ejpam-609	103	18	sufficient	sufficient	ADJ
ejpam-609	103	19	condition	condition	NOUN
ejpam-609	103	20	for	for	SCONJ
ejpam-609	103	21	r	r	NOUN
ejpam-609	103	22	to	to	PART
ejpam-609	103	23	be	be	AUX
ejpam-609	103	24	a	a	DET
ejpam-609	103	25	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	103	26	in	in	ADP
ejpam-609	103	27	the	the	DET
ejpam-609	103	28	following	following	NOUN
ejpam-609	103	29	theorem	theorem	NOUN
ejpam-609	103	30	:	:	PUNCT
ejpam-609	103	31	proposition	proposition	NOUN
ejpam-609	103	32	1	1	NUM
ejpam-609	103	33	.	.	PUNCT
ejpam-609	104	1	let	let	VERB
ejpam-609	104	2	r	r	PRON
ejpam-609	104	3	be	be	AUX
ejpam-609	104	4	a	a	DET
ejpam-609	104	5	ring	ring	NOUN
ejpam-609	104	6	and	and	CCONJ
ejpam-609	104	7	σ	σ	NOUN
ejpam-609	104	8	an	an	DET
ejpam-609	104	9	automorphism	automorphism	NOUN
ejpam-609	104	10	of	of	ADP
ejpam-609	104	11	r.	r.	PROPN
ejpam-609	104	12	then	then	ADV
ejpam-609	104	13	r	r	NOUN
ejpam-609	104	14	is	be	AUX
ejpam-609	104	15	a	a	DET
ejpam-609	104	16	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	104	17	implies	implie	NOUN
ejpam-609	104	18	that	that	SCONJ
ejpam-609	104	19	p(r	p(r	PROPN
ejpam-609	104	20	)	)	PUNCT
ejpam-609	104	21	is	be	AUX
ejpam-609	104	22	completely	completely	ADV
ejpam-609	104	23	semiprime	semiprime	NOUN
ejpam-609	104	24	.	.	PUNCT
ejpam-609	105	1	proof	proof	NOUN
ejpam-609	105	2	.	.	PUNCT
ejpam-609	106	1	proposition	proposition	NOUN
ejpam-609	106	2	(	(	PUNCT
ejpam-609	106	3	2.2	2.2	NUM
ejpam-609	106	4	)	)	PUNCT
ejpam-609	106	5	of	of	ADP
ejpam-609	106	6	bhat	bhat	PROPN
ejpam-609	106	7	and	and	CCONJ
ejpam-609	106	8	neetu	neetu	NOUN
ejpam-609	107	1	[	[	X
ejpam-609	107	2	4	4	NUM
ejpam-609	107	3	]	]	PUNCT
ejpam-609	107	4	.	.	PUNCT
ejpam-609	108	1	proposition	proposition	NOUN
ejpam-609	108	2	2	2	NUM
ejpam-609	108	3	.	.	PUNCT
ejpam-609	109	1	let	let	VERB
ejpam-609	109	2	r	r	PRON
ejpam-609	109	3	be	be	AUX
ejpam-609	109	4	a	a	DET
ejpam-609	109	5	noetharian	noetharian	ADJ
ejpam-609	109	6	ring	ring	NOUN
ejpam-609	109	7	and	and	CCONJ
ejpam-609	109	8	σ	σ	NOUN
ejpam-609	109	9	an	an	DET
ejpam-609	109	10	automorphism	automorphism	NOUN
ejpam-609	109	11	of	of	ADP
ejpam-609	109	12	r.	r.	PROPN
ejpam-609	109	13	then	then	ADV
ejpam-609	109	14	r	r	NOUN
ejpam-609	109	15	is	be	AUX
ejpam-609	109	16	a	a	DET
ejpam-609	109	17	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	109	18	implies	implie	NOUN
ejpam-609	109	19	that	that	SCONJ
ejpam-609	109	20	r	r	NOUN
ejpam-609	109	21	is	be	AUX
ejpam-609	109	22	2	2	NUM
ejpam-609	109	23	-	-	PUNCT
ejpam-609	109	24	primal	primal	ADJ
ejpam-609	109	25	.	.	PUNCT
ejpam-609	110	1	proof	proof	NOUN
ejpam-609	110	2	.	.	PUNCT
ejpam-609	111	1	by	by	ADP
ejpam-609	111	2	proposition	proposition	NOUN
ejpam-609	111	3	(	(	PUNCT
ejpam-609	111	4	1	1	X
ejpam-609	111	5	)	)	PUNCT
ejpam-609	111	6	p(r	p(r	PROPN
ejpam-609	111	7	)	)	PUNCT
ejpam-609	111	8	is	be	AUX
ejpam-609	111	9	completely	completely	ADV
ejpam-609	111	10	semiprime	semiprime	NOUN
ejpam-609	111	11	.	.	PUNCT
ejpam-609	112	1	therefore	therefore	ADV
ejpam-609	112	2	,	,	PUNCT
ejpam-609	112	3	r	r	NOUN
ejpam-609	112	4	is	be	AUX
ejpam-609	112	5	2	2	NUM
ejpam-609	112	6	-	-	PUNCT
ejpam-609	112	7	primal	primal	ADJ
ejpam-609	112	8	.	.	PUNCT
ejpam-609	113	1	recall	recall	VERB
ejpam-609	113	2	that	that	SCONJ
ejpam-609	113	3	a	a	DET
ejpam-609	113	4	completely	completely	ADV
ejpam-609	113	5	prime	prime	ADJ
ejpam-609	113	6	ideal	ideal	NOUN
ejpam-609	113	7	in	in	ADP
ejpam-609	113	8	a	a	DET
ejpam-609	113	9	ring	ring	NOUN
ejpam-609	113	10	r	r	NOUN
ejpam-609	113	11	is	be	AUX
ejpam-609	113	12	any	any	DET
ejpam-609	113	13	(	(	PUNCT
ejpam-609	113	14	prime	prime	ADJ
ejpam-609	113	15	)	)	PUNCT
ejpam-609	113	16	ideal	ideal	NOUN
ejpam-609	113	17	such	such	ADJ
ejpam-609	113	18	that	that	SCONJ
ejpam-609	113	19	r	r	NOUN
ejpam-609	113	20	/	/	SYM
ejpam-609	113	21	p	p	NOUN
ejpam-609	113	22	is	be	AUX
ejpam-609	113	23	a	a	DET
ejpam-609	113	24	domain	domain	NOUN
ejpam-609	113	25	(	(	PUNCT
ejpam-609	113	26	chapter	chapter	NOUN
ejpam-609	113	27	9	9	NUM
ejpam-609	113	28	of	of	ADP
ejpam-609	113	29	goodearl	goodearl	PROPN
ejpam-609	113	30	and	and	CCONJ
ejpam-609	113	31	warfield	warfield	VERB
ejpam-609	114	1	[	[	X
ejpam-609	114	2	8	8	NUM
ejpam-609	114	3	]	]	PUNCT
ejpam-609	114	4	)	)	PUNCT
ejpam-609	114	5	.	.	PUNCT
ejpam-609	115	1	by	by	ADP
ejpam-609	115	2	definition	definition	NOUN
ejpam-609	115	3	we	we	PRON
ejpam-609	115	4	note	note	VERB
ejpam-609	115	5	that	that	SCONJ
ejpam-609	115	6	every	every	DET
ejpam-609	115	7	completely	completely	ADV
ejpam-609	115	8	prime	prime	ADJ
ejpam-609	115	9	ideal	ideal	NOUN
ejpam-609	115	10	of	of	ADP
ejpam-609	115	11	a	a	DET
ejpam-609	115	12	ring	ring	NOUN
ejpam-609	115	13	r	r	NOUN
ejpam-609	115	14	is	be	AUX
ejpam-609	115	15	a	a	DET
ejpam-609	115	16	prime	prime	ADJ
ejpam-609	115	17	ideal	ideal	NOUN
ejpam-609	115	18	,	,	PUNCT
ejpam-609	115	19	but	but	CCONJ
ejpam-609	115	20	the	the	DET
ejpam-609	115	21	converse	converse	NOUN
ejpam-609	115	22	need	need	AUX
ejpam-609	115	23	not	not	PART
ejpam-609	115	24	be	be	AUX
ejpam-609	115	25	true	true	ADJ
ejpam-609	115	26	.	.	PUNCT
ejpam-609	116	1	v.	v.	ADP
ejpam-609	116	2	bhat	bhat	PROPN
ejpam-609	116	3	/	/	SYM
ejpam-609	116	4	eur	eur	PROPN
ejpam-609	116	5	.	.	PUNCT
ejpam-609	117	1	j.	j.	PROPN
ejpam-609	117	2	pure	pure	PROPN
ejpam-609	117	3	appl	appl	PROPN
ejpam-609	117	4	.	.	PROPN
ejpam-609	117	5	math	math	PROPN
ejpam-609	117	6	,	,	PUNCT
ejpam-609	117	7	3	3	NUM
ejpam-609	117	8	(	(	PUNCT
ejpam-609	117	9	2010	2010	NUM
ejpam-609	117	10	)	)	PUNCT
ejpam-609	117	11	,	,	PUNCT
ejpam-609	117	12	695	695	NUM
ejpam-609	117	13	-	-	SYM
ejpam-609	117	14	703	703	NUM
ejpam-609	117	15	699	699	NUM
ejpam-609	117	16	example	example	NOUN
ejpam-609	117	17	3	3	NUM
ejpam-609	117	18	.	.	PUNCT
ejpam-609	118	1	let	let	VERB
ejpam-609	118	2	r=	r=	PROPN
ejpam-609	118	3	�	�	PROPN
ejpam-609	118	4	z	z	NOUN
ejpam-609	118	5	z	z	PROPN
ejpam-609	118	6	z	z	NOUN
ejpam-609	118	7	z	z	NOUN
ejpam-609	118	8	�	�	PROPN
ejpam-609	118	9	=	=	SYM
ejpam-609	118	10	m2(z	m2(z	PROPN
ejpam-609	118	11	)	)	PUNCT
ejpam-609	118	12	.	.	PUNCT
ejpam-609	119	1	if	if	SCONJ
ejpam-609	119	2	p	p	NOUN
ejpam-609	119	3	is	be	AUX
ejpam-609	119	4	a	a	DET
ejpam-609	119	5	prime	prime	ADJ
ejpam-609	119	6	number	number	NOUN
ejpam-609	119	7	,	,	PUNCT
ejpam-609	119	8	then	then	ADV
ejpam-609	119	9	the	the	DET
ejpam-609	119	10	ideal	ideal	NOUN
ejpam-609	119	11	p	p	X
ejpam-609	119	12	=	=	SYM
ejpam-609	119	13	m2(pz	m2(pz	PROPN
ejpam-609	119	14	)	)	PUNCT
ejpam-609	119	15	is	be	AUX
ejpam-609	119	16	a	a	DET
ejpam-609	119	17	prime	prime	ADJ
ejpam-609	119	18	ideal	ideal	NOUN
ejpam-609	119	19	of	of	ADP
ejpam-609	119	20	r	r	NOUN
ejpam-609	119	21	,	,	PUNCT
ejpam-609	119	22	but	but	CCONJ
ejpam-609	119	23	is	be	AUX
ejpam-609	119	24	not	not	PART
ejpam-609	119	25	completely	completely	ADV
ejpam-609	119	26	prime	prime	ADJ
ejpam-609	119	27	,	,	PUNCT
ejpam-609	119	28	since	since	SCONJ
ejpam-609	119	29	for	for	ADP
ejpam-609	119	30	a	a	DET
ejpam-609	119	31	=	=	SYM
ejpam-609	119	32	�	�	PROPN
ejpam-609	119	33	1	1	NUM
ejpam-609	119	34	0	0	NUM
ejpam-609	119	35	0	0	NUM
ejpam-609	119	36	0	0	NUM
ejpam-609	119	37	�	�	PROPN
ejpam-609	119	38	and	and	CCONJ
ejpam-609	119	39	b	b	NOUN
ejpam-609	119	40	=	=	SYM
ejpam-609	119	41	�	�	PROPN
ejpam-609	119	42	0	0	NUM
ejpam-609	119	43	0	0	NUM
ejpam-609	119	44	0	0	NUM
ejpam-609	119	45	1	1	NUM
ejpam-609	119	46	�	�	PROPN
ejpam-609	119	47	,	,	PUNCT
ejpam-609	119	48	we	we	PRON
ejpam-609	119	49	have	have	VERB
ejpam-609	119	50	ab	ab	PROPN
ejpam-609	119	51	∈	∈	PROPN
ejpam-609	119	52	p	p	NOUN
ejpam-609	119	53	,	,	PUNCT
ejpam-609	119	54	even	even	ADV
ejpam-609	119	55	though	though	SCONJ
ejpam-609	119	56	a	a	DET
ejpam-609	119	57	/∈	/∈	SYM
ejpam-609	119	58	p	p	NOUN
ejpam-609	119	59	and	and	CCONJ
ejpam-609	119	60	b	b	PROPN
ejpam-609	119	61	/∈	/∈	PUNCT
ejpam-609	120	1	p.	p.	NOUN
ejpam-609	120	2	a	a	DET
ejpam-609	120	3	necessary	necessary	ADJ
ejpam-609	120	4	and	and	CCONJ
ejpam-609	120	5	sufficient	sufficient	ADJ
ejpam-609	120	6	condition	condition	NOUN
ejpam-609	120	7	for	for	ADP
ejpam-609	120	8	a	a	DET
ejpam-609	120	9	noetherian	noetherian	ADJ
ejpam-609	120	10	ring	ring	NOUN
ejpam-609	120	11	r	r	NOUN
ejpam-609	120	12	to	to	PART
ejpam-609	120	13	be	be	AUX
ejpam-609	120	14	a	a	DET
ejpam-609	120	15	σ(∗)-ring	σ(∗)-re	VERB
ejpam-609	120	16	(	(	PUNCT
ejpam-609	120	17	where	where	SCONJ
ejpam-609	120	18	σ	σ	PROPN
ejpam-609	120	19	is	be	AUX
ejpam-609	120	20	an	an	DET
ejpam-609	120	21	automorphism	automorphism	NOUN
ejpam-609	120	22	of	of	ADP
ejpam-609	120	23	r	r	NOUN
ejpam-609	120	24	)	)	PUNCT
ejpam-609	120	25	has	have	AUX
ejpam-609	120	26	been	be	AUX
ejpam-609	120	27	given	give	VERB
ejpam-609	120	28	in	in	ADP
ejpam-609	120	29	theorem	theorem	NOUN
ejpam-609	120	30	(	(	PUNCT
ejpam-609	120	31	2.4	2.4	NUM
ejpam-609	120	32	)	)	PUNCT
ejpam-609	120	33	of	of	ADP
ejpam-609	120	34	[	[	X
ejpam-609	120	35	4	4	NUM
ejpam-609	120	36	]	]	PUNCT
ejpam-609	120	37	:	:	PUNCT
ejpam-609	120	38	theorem	theorem	NOUN
ejpam-609	120	39	3	3	X
ejpam-609	120	40	.	.	PUNCT
ejpam-609	121	1	let	let	VERB
ejpam-609	121	2	r	r	PRON
ejpam-609	121	3	be	be	AUX
ejpam-609	121	4	a	a	DET
ejpam-609	121	5	noetherian	noetherian	ADJ
ejpam-609	121	6	ring	ring	NOUN
ejpam-609	121	7	.	.	PUNCT
ejpam-609	122	1	let	let	VERB
ejpam-609	122	2	σ	σ	NOUN
ejpam-609	122	3	be	be	AUX
ejpam-609	122	4	an	an	DET
ejpam-609	122	5	automorphism	automorphism	NOUN
ejpam-609	122	6	of	of	ADP
ejpam-609	122	7	r.	r.	PROPN
ejpam-609	122	8	then	then	ADV
ejpam-609	122	9	r	r	NOUN
ejpam-609	122	10	is	be	AUX
ejpam-609	122	11	a	a	DET
ejpam-609	122	12	σ(∗)-ring	σ(∗)-re	VERB
ejpam-609	122	13	if	if	SCONJ
ejpam-609	122	14	and	and	CCONJ
ejpam-609	122	15	only	only	ADV
ejpam-609	122	16	if	if	SCONJ
ejpam-609	122	17	for	for	ADP
ejpam-609	122	18	each	each	DET
ejpam-609	122	19	minimal	minimal	ADJ
ejpam-609	122	20	prime	prime	ADJ
ejpam-609	122	21	u	u	NOUN
ejpam-609	122	22	of	of	ADP
ejpam-609	122	23	r	r	NOUN
ejpam-609	122	24	,	,	PUNCT
ejpam-609	122	25	σ(u	σ(u	NOUN
ejpam-609	122	26	)	)	PUNCT
ejpam-609	122	27	=	=	SYM
ejpam-609	122	28	u	u	NOUN
ejpam-609	122	29	and	and	CCONJ
ejpam-609	122	30	u	u	NOUN
ejpam-609	122	31	is	be	AUX
ejpam-609	122	32	completely	completely	ADV
ejpam-609	122	33	prime	prime	ADJ
ejpam-609	122	34	ideal	ideal	NOUN
ejpam-609	122	35	of	of	ADP
ejpam-609	122	36	r.	r.	PROPN
ejpam-609	122	37	proof	proof	NOUN
ejpam-609	122	38	.	.	PUNCT
ejpam-609	123	1	see	see	VERB
ejpam-609	123	2	theorem	theorem	NOUN
ejpam-609	123	3	(	(	PUNCT
ejpam-609	123	4	2.4	2.4	NUM
ejpam-609	123	5	)	)	PUNCT
ejpam-609	123	6	of	of	ADP
ejpam-609	123	7	[	[	X
ejpam-609	123	8	4	4	NUM
ejpam-609	123	9	]	]	PUNCT
ejpam-609	123	10	.	.	PUNCT
ejpam-609	124	1	proposition	proposition	NOUN
ejpam-609	124	2	3	3	X
ejpam-609	124	3	.	.	PUNCT
ejpam-609	125	1	let	let	VERB
ejpam-609	125	2	r	r	PRON
ejpam-609	125	3	be	be	AUX
ejpam-609	125	4	a	a	DET
ejpam-609	125	5	noetherian	noetherian	ADJ
ejpam-609	125	6	ring	ring	NOUN
ejpam-609	125	7	which	which	PRON
ejpam-609	125	8	is	be	AUX
ejpam-609	125	9	also	also	ADV
ejpam-609	125	10	an	an	DET
ejpam-609	125	11	algebra	algebra	NOUN
ejpam-609	125	12	over	over	ADP
ejpam-609	125	13	q.	q.	PROPN
ejpam-609	125	14	let	let	VERB
ejpam-609	125	15	σ	σ	NOUN
ejpam-609	125	16	be	be	AUX
ejpam-609	125	17	an	an	DET
ejpam-609	125	18	automorphism	automorphism	NOUN
ejpam-609	125	19	of	of	ADP
ejpam-609	125	20	r	r	NOUN
ejpam-609	125	21	such	such	ADJ
ejpam-609	125	22	that	that	SCONJ
ejpam-609	125	23	r	r	NOUN
ejpam-609	125	24	is	be	AUX
ejpam-609	125	25	a	a	DET
ejpam-609	125	26	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	125	27	and	and	CCONJ
ejpam-609	125	28	δ	δ	PROPN
ejpam-609	125	29	a	a	DET
ejpam-609	125	30	σ	σ	NOUN
ejpam-609	125	31	-	-	PUNCT
ejpam-609	125	32	derivation	derivation	NOUN
ejpam-609	125	33	of	of	ADP
ejpam-609	125	34	r.	r.	PROPN
ejpam-609	125	35	then	then	ADV
ejpam-609	125	36	δ(u	δ(u	PROPN
ejpam-609	125	37	)	)	PUNCT
ejpam-609	125	38	⊆	⊆	NUM
ejpam-609	125	39	u	u	NOUN
ejpam-609	125	40	for	for	ADP
ejpam-609	125	41	all	all	DET
ejpam-609	125	42	u	u	PROPN
ejpam-609	125	43	∈	∈	PROPN
ejpam-609	125	44	min.spec(r	min.spec(r	PROPN
ejpam-609	125	45	)	)	PUNCT
ejpam-609	125	46	.	.	PUNCT
ejpam-609	126	1	proof	proof	NOUN
ejpam-609	126	2	.	.	PUNCT
ejpam-609	127	1	we	we	PRON
ejpam-609	127	2	note	note	VERB
ejpam-609	127	3	that	that	SCONJ
ejpam-609	127	4	proposition	proposition	NOUN
ejpam-609	127	5	(	(	PUNCT
ejpam-609	127	6	1	1	NUM
ejpam-609	127	7	)	)	PUNCT
ejpam-609	127	8	implies	imply	VERB
ejpam-609	127	9	that	that	SCONJ
ejpam-609	127	10	p(r	p(r	PROPN
ejpam-609	127	11	)	)	PUNCT
ejpam-609	127	12	is	be	AUX
ejpam-609	127	13	completely	completely	ADV
ejpam-609	127	14	semiprime	semiprime	ADJ
ejpam-609	127	15	.	.	PUNCT
ejpam-609	128	1	let	let	VERB
ejpam-609	128	2	u	u	PRON
ejpam-609	128	3	∈	∈	PROPN
ejpam-609	128	4	min.spec(r	min.spec(r	PROPN
ejpam-609	128	5	)	)	PUNCT
ejpam-609	128	6	.	.	PUNCT
ejpam-609	129	1	then	then	ADV
ejpam-609	129	2	theorem	theorem	ADJ
ejpam-609	129	3	(	(	PUNCT
ejpam-609	129	4	3	3	NUM
ejpam-609	129	5	)	)	PUNCT
ejpam-609	129	6	implies	imply	VERB
ejpam-609	129	7	that	that	SCONJ
ejpam-609	129	8	σ(u	σ(u	NOUN
ejpam-609	129	9	)	)	PUNCT
ejpam-609	129	10	=	=	SYM
ejpam-609	129	11	u	u	NOUN
ejpam-609	129	12	.	.	PUNCT
ejpam-609	130	1	let	let	VERB
ejpam-609	130	2	now	now	ADV
ejpam-609	130	3	t	t	PROPN
ejpam-609	131	1	=	=	PUNCT
ejpam-609	131	2	{	{	PUNCT
ejpam-609	131	3	a	a	DET
ejpam-609	131	4	∈	∈	X
ejpam-609	131	5	u	u	NOUN
ejpam-609	131	6	|	|	ADV
ejpam-609	131	7	such	such	ADJ
ejpam-609	131	8	that	that	SCONJ
ejpam-609	131	9	δk(a	δk(a	X
ejpam-609	131	10	)	)	PUNCT
ejpam-609	131	11	∈	∈	PROPN
ejpam-609	131	12	u	u	NOUN
ejpam-609	131	13	for	for	ADP
ejpam-609	131	14	all	all	DET
ejpam-609	131	15	integers	integer	NOUN
ejpam-609	131	16	k	k	X
ejpam-609	131	17	≥	≥	NUM
ejpam-609	131	18	1	1	NUM
ejpam-609	131	19	}	}	PUNCT
ejpam-609	131	20	.	.	PUNCT
ejpam-609	132	1	first	first	ADV
ejpam-609	132	2	of	of	ADP
ejpam-609	132	3	all	all	PRON
ejpam-609	132	4	,	,	PUNCT
ejpam-609	132	5	we	we	PRON
ejpam-609	132	6	will	will	AUX
ejpam-609	132	7	show	show	VERB
ejpam-609	132	8	that	that	SCONJ
ejpam-609	132	9	t	t	PROPN
ejpam-609	132	10	is	be	AUX
ejpam-609	132	11	an	an	DET
ejpam-609	132	12	ideal	ideal	NOUN
ejpam-609	132	13	of	of	ADP
ejpam-609	132	14	r.	r.	PROPN
ejpam-609	132	15	let	let	VERB
ejpam-609	132	16	a	a	DET
ejpam-609	132	17	,	,	PUNCT
ejpam-609	132	18	b	b	PROPN
ejpam-609	132	19	∈	∈	PROPN
ejpam-609	132	20	t	t	NOUN
ejpam-609	132	21	.	.	PUNCT
ejpam-609	133	1	then	then	ADV
ejpam-609	133	2	δk(a	δk(a	PUNCT
ejpam-609	133	3	)	)	PUNCT
ejpam-609	133	4	∈	∈	PROPN
ejpam-609	133	5	u	u	NOUN
ejpam-609	133	6	and	and	CCONJ
ejpam-609	133	7	δk(b	δk(b	NUM
ejpam-609	133	8	)	)	PUNCT
ejpam-609	133	9	∈	∈	PROPN
ejpam-609	133	10	u	u	NOUN
ejpam-609	133	11	for	for	ADP
ejpam-609	133	12	all	all	DET
ejpam-609	133	13	integers	integer	NOUN
ejpam-609	133	14	k	k	X
ejpam-609	133	15	≥	≥	NUM
ejpam-609	133	16	1	1	NUM
ejpam-609	133	17	}	}	PUNCT
ejpam-609	133	18	.	.	PUNCT
ejpam-609	134	1	now	now	ADV
ejpam-609	134	2	δk(a	δk(a	VERB
ejpam-609	134	3	−	−	PROPN
ejpam-609	134	4	b	b	X
ejpam-609	134	5	)	)	PUNCT
ejpam-609	134	6	=	=	SYM
ejpam-609	134	7	δk(a)−	δk(a)−	NOUN
ejpam-609	134	8	δk(b	δk(b	NOUN
ejpam-609	134	9	)	)	PUNCT
ejpam-609	134	10	∈	∈	PROPN
ejpam-609	134	11	u	u	NOUN
ejpam-609	134	12	for	for	ADP
ejpam-609	134	13	all	all	DET
ejpam-609	134	14	k	k	PROPN
ejpam-609	134	15	≥	≥	NUM
ejpam-609	134	16	1	1	NUM
ejpam-609	134	17	}	}	PUNCT
ejpam-609	134	18	.	.	PUNCT
ejpam-609	135	1	therefore	therefore	ADV
ejpam-609	135	2	a	a	DET
ejpam-609	135	3	−	−	PROPN
ejpam-609	135	4	b	b	PROPN
ejpam-609	135	5	∈	∈	PROPN
ejpam-609	135	6	t	t	PROPN
ejpam-609	135	7	.	.	PUNCT
ejpam-609	136	1	therefore	therefore	ADV
ejpam-609	136	2	t	t	PROPN
ejpam-609	136	3	is	be	AUX
ejpam-609	136	4	a	a	DET
ejpam-609	136	5	δ	δ	NOUN
ejpam-609	136	6	-	-	PUNCT
ejpam-609	136	7	invariant	invariant	ADJ
ejpam-609	136	8	ideal	ideal	NOUN
ejpam-609	136	9	of	of	ADP
ejpam-609	136	10	r.	r.	PROPN
ejpam-609	136	11	we	we	PRON
ejpam-609	136	12	will	will	AUX
ejpam-609	136	13	now	now	ADV
ejpam-609	136	14	show	show	VERB
ejpam-609	136	15	that	that	SCONJ
ejpam-609	136	16	t	t	PROPN
ejpam-609	136	17	∈	∈	PROPN
ejpam-609	136	18	spec(r	spec(r	PROPN
ejpam-609	136	19	)	)	PUNCT
ejpam-609	136	20	.	.	PUNCT
ejpam-609	137	1	suppose	suppose	VERB
ejpam-609	138	1	t	t	PROPN
ejpam-609	138	2	/∈	/∈	PUNCT
ejpam-609	138	3	spec(r	spec(r	PROPN
ejpam-609	138	4	)	)	PUNCT
ejpam-609	138	5	.	.	PUNCT
ejpam-609	139	1	let	let	VERB
ejpam-609	139	2	a	a	DET
ejpam-609	139	3	/∈	/∈	NOUN
ejpam-609	139	4	t	t	NOUN
ejpam-609	139	5	,	,	PUNCT
ejpam-609	139	6	b	b	PROPN
ejpam-609	139	7	/∈	/∈	PROPN
ejpam-609	139	8	t	t	PROPN
ejpam-609	139	9	be	be	VERB
ejpam-609	139	10	such	such	ADJ
ejpam-609	139	11	that	that	SCONJ
ejpam-609	139	12	arb	arb	PROPN
ejpam-609	139	13	⊆	⊆	NUM
ejpam-609	139	14	t	t	NOUN
ejpam-609	139	15	.	.	PUNCT
ejpam-609	140	1	let	let	VERB
ejpam-609	140	2	t	t	PROPN
ejpam-609	140	3	,	,	PUNCT
ejpam-609	140	4	s	s	AUX
ejpam-609	140	5	be	be	AUX
ejpam-609	140	6	least	least	ADJ
ejpam-609	140	7	such	such	ADJ
ejpam-609	140	8	that	that	PRON
ejpam-609	140	9	δt(a	δt(a	NOUN
ejpam-609	140	10	)	)	PUNCT
ejpam-609	140	11	/∈	/∈	PUNCT
ejpam-609	141	1	u	u	NOUN
ejpam-609	141	2	and	and	CCONJ
ejpam-609	141	3	δs(b	δs(b	ADV
ejpam-609	141	4	)	)	PUNCT
ejpam-609	141	5	/∈	/∈	PUNCT
ejpam-609	142	1	u	u	INTJ
ejpam-609	142	2	.	.	PUNCT
ejpam-609	143	1	now	now	ADV
ejpam-609	143	2	there	there	PRON
ejpam-609	143	3	exists	exist	VERB
ejpam-609	143	4	c	c	NOUN
ejpam-609	143	5	∈	∈	PROPN
ejpam-609	143	6	r	r	NOUN
ejpam-609	143	7	such	such	ADJ
ejpam-609	143	8	that	that	DET
ejpam-609	143	9	δt(a)cσt	δt(a)cσt	NOUN
ejpam-609	143	10	(	(	PUNCT
ejpam-609	143	11	δs(b	δs(b	NUM
ejpam-609	143	12	)	)	PUNCT
ejpam-609	143	13	)	)	PUNCT
ejpam-609	143	14	/∈	/∈	PUNCT
ejpam-609	144	1	u	u	INTJ
ejpam-609	144	2	.	.	PUNCT
ejpam-609	145	1	let	let	VERB
ejpam-609	145	2	d	d	NOUN
ejpam-609	145	3	=	=	SYM
ejpam-609	145	4	σ−t(c	σ−t(c	NOUN
ejpam-609	145	5	)	)	PUNCT
ejpam-609	145	6	.	.	PUNCT
ejpam-609	146	1	now	now	ADV
ejpam-609	146	2	δt+s(ad	δt+s(ad	PROPN
ejpam-609	146	3	b	b	X
ejpam-609	146	4	)	)	PUNCT
ejpam-609	146	5	∈	∈	PROPN
ejpam-609	146	6	u	u	NOUN
ejpam-609	146	7	as	as	ADP
ejpam-609	146	8	arb	arb	PROPN
ejpam-609	146	9	⊆	⊆	NUM
ejpam-609	146	10	t	t	NOUN
ejpam-609	146	11	.	.	PUNCT
ejpam-609	147	1	this	this	PRON
ejpam-609	147	2	implies	imply	VERB
ejpam-609	147	3	on	on	ADP
ejpam-609	147	4	simplification	simplification	NOUN
ejpam-609	147	5	that	that	PRON
ejpam-609	147	6	δt(a)σt(d)σt(δs(b))+u	δt(a)σt(d)σt(δs(b))+u	VERB
ejpam-609	147	7	∈	∈	PROPN
ejpam-609	147	8	u	u	NOUN
ejpam-609	147	9	,	,	PUNCT
ejpam-609	147	10	where	where	SCONJ
ejpam-609	147	11	u	u	NOUN
ejpam-609	147	12	is	be	AUX
ejpam-609	147	13	sum	sum	NOUN
ejpam-609	147	14	of	of	ADP
ejpam-609	147	15	terms	term	NOUN
ejpam-609	147	16	involving	involve	VERB
ejpam-609	147	17	δl(a	δl(a	NOUN
ejpam-609	147	18	)	)	PUNCT
ejpam-609	147	19	or	or	CCONJ
ejpam-609	147	20	δm(b	δm(b	NOUN
ejpam-609	147	21	)	)	PUNCT
ejpam-609	147	22	,	,	PUNCT
ejpam-609	147	23	where	where	SCONJ
ejpam-609	147	24	l	l	PROPN
ejpam-609	147	25	<	<	X
ejpam-609	147	26	t	t	PROPN
ejpam-609	147	27	and	and	CCONJ
ejpam-609	147	28	m	m	PROPN
ejpam-609	147	29	<	<	X
ejpam-609	147	30	s.	s.	PROPN
ejpam-609	147	31	therefore	therefore	ADV
ejpam-609	147	32	by	by	ADP
ejpam-609	147	33	assumption	assumption	NOUN
ejpam-609	147	34	u	u	PROPN
ejpam-609	147	35	∈	∈	PROPN
ejpam-609	147	36	u	u	NOUN
ejpam-609	147	37	which	which	PRON
ejpam-609	147	38	implies	imply	VERB
ejpam-609	147	39	that	that	SCONJ
ejpam-609	147	40	δt(a)σt(d)σt(δs(b	δt(a)σt(d)σt(δs(b	ADV
ejpam-609	147	41	)	)	PUNCT
ejpam-609	147	42	)	)	PUNCT
ejpam-609	148	1	∈	∈	PROPN
ejpam-609	148	2	u	u	NOUN
ejpam-609	148	3	.	.	PUNCT
ejpam-609	149	1	this	this	PRON
ejpam-609	149	2	is	be	AUX
ejpam-609	149	3	a	a	DET
ejpam-609	149	4	contradiction	contradiction	NOUN
ejpam-609	149	5	.	.	PUNCT
ejpam-609	150	1	therefore	therefore	ADV
ejpam-609	150	2	,	,	PUNCT
ejpam-609	150	3	our	our	PRON
ejpam-609	150	4	supposition	supposition	NOUN
ejpam-609	150	5	must	must	AUX
ejpam-609	150	6	be	be	AUX
ejpam-609	150	7	wrong	wrong	ADJ
ejpam-609	150	8	.	.	PUNCT
ejpam-609	151	1	hence	hence	ADV
ejpam-609	151	2	t	t	PROPN
ejpam-609	151	3	∈	∈	PROPN
ejpam-609	151	4	spec(r	spec(r	PROPN
ejpam-609	151	5	)	)	PUNCT
ejpam-609	151	6	.	.	PUNCT
ejpam-609	152	1	now	now	ADV
ejpam-609	152	2	t	t	VERB
ejpam-609	152	3	⊆	⊆	NUM
ejpam-609	152	4	u	u	NOUN
ejpam-609	152	5	,	,	PUNCT
ejpam-609	152	6	so	so	ADV
ejpam-609	152	7	t	t	PROPN
ejpam-609	152	8	=	=	SYM
ejpam-609	152	9	u	u	PROPN
ejpam-609	152	10	as	as	ADP
ejpam-609	152	11	u	u	NOUN
ejpam-609	152	12	∈	∈	PROPN
ejpam-609	152	13	min.spec(r	min.spec(r	PROPN
ejpam-609	152	14	)	)	PUNCT
ejpam-609	152	15	.	.	PUNCT
ejpam-609	153	1	hence	hence	ADV
ejpam-609	153	2	δ(u)⊆	δ(u)⊆	PROPN
ejpam-609	153	3	u	u	PROPN
ejpam-609	153	4	.	.	PUNCT
ejpam-609	154	1	theorem	theorem	ADJ
ejpam-609	154	2	4	4	NUM
ejpam-609	154	3	.	.	PUNCT
ejpam-609	155	1	let	let	VERB
ejpam-609	155	2	r	r	PRON
ejpam-609	155	3	be	be	AUX
ejpam-609	155	4	a	a	DET
ejpam-609	155	5	ring	ring	NOUN
ejpam-609	155	6	.	.	PUNCT
ejpam-609	156	1	let	let	VERB
ejpam-609	156	2	σ	σ	NOUN
ejpam-609	156	3	be	be	AUX
ejpam-609	156	4	an	an	DET
ejpam-609	156	5	automorphism	automorphism	NOUN
ejpam-609	156	6	of	of	ADP
ejpam-609	156	7	r	r	NOUN
ejpam-609	156	8	and	and	CCONJ
ejpam-609	156	9	δ	δ	PROPN
ejpam-609	156	10	be	be	AUX
ejpam-609	156	11	a	a	DET
ejpam-609	156	12	σ	σ	NOUN
ejpam-609	156	13	-	-	PUNCT
ejpam-609	156	14	derivation	derivation	NOUN
ejpam-609	156	15	of	of	ADP
ejpam-609	156	16	r.	r.	PROPN
ejpam-609	156	17	then	then	ADV
ejpam-609	156	18	:	:	PUNCT
ejpam-609	156	19	1	1	X
ejpam-609	156	20	.	.	X
ejpam-609	156	21	for	for	ADP
ejpam-609	156	22	any	any	DET
ejpam-609	156	23	completely	completely	ADV
ejpam-609	156	24	prime	prime	ADJ
ejpam-609	156	25	ideal	ideal	NOUN
ejpam-609	156	26	p	p	NOUN
ejpam-609	156	27	of	of	ADP
ejpam-609	156	28	r	r	NOUN
ejpam-609	156	29	with	with	ADP
ejpam-609	156	30	δ(p)⊆	δ(p)⊆	PROPN
ejpam-609	156	31	p	p	NOUN
ejpam-609	156	32	and	and	CCONJ
ejpam-609	156	33	σ(p	σ(p	PROPN
ejpam-609	156	34	)	)	PUNCT
ejpam-609	156	35	=	=	SYM
ejpam-609	156	36	p	p	PROPN
ejpam-609	156	37	,	,	PUNCT
ejpam-609	156	38	o(p	o(p	PROPN
ejpam-609	156	39	)	)	PUNCT
ejpam-609	156	40	=	=	PUNCT
ejpam-609	156	41	p[x	p[x	X
ejpam-609	156	42	;	;	PUNCT
ejpam-609	156	43	σ	σ	PROPN
ejpam-609	156	44	,	,	PUNCT
ejpam-609	156	45	δ	δ	PROPN
ejpam-609	156	46	]	]	PUNCT
ejpam-609	156	47	is	be	AUX
ejpam-609	156	48	a	a	DET
ejpam-609	156	49	completely	completely	ADV
ejpam-609	156	50	prime	prime	ADJ
ejpam-609	156	51	ideal	ideal	NOUN
ejpam-609	156	52	of	of	ADP
ejpam-609	156	53	o(r	o(r	PROPN
ejpam-609	156	54	)	)	PUNCT
ejpam-609	156	55	.	.	PUNCT
ejpam-609	157	1	2	2	X
ejpam-609	157	2	.	.	X
ejpam-609	157	3	for	for	ADP
ejpam-609	157	4	any	any	DET
ejpam-609	157	5	completely	completely	ADV
ejpam-609	157	6	prime	prime	ADJ
ejpam-609	157	7	ideal	ideal	ADJ
ejpam-609	157	8	u	u	NOUN
ejpam-609	157	9	of	of	ADP
ejpam-609	157	10	o(r	o(r	PROPN
ejpam-609	157	11	)	)	PUNCT
ejpam-609	157	12	,	,	PUNCT
ejpam-609	157	13	u	u	NOUN
ejpam-609	157	14	∩	∩	NOUN
ejpam-609	157	15	r	r	NOUN
ejpam-609	157	16	is	be	AUX
ejpam-609	157	17	a	a	DET
ejpam-609	157	18	completely	completely	ADV
ejpam-609	157	19	prime	prime	ADJ
ejpam-609	157	20	ideal	ideal	NOUN
ejpam-609	157	21	of	of	ADP
ejpam-609	157	22	r.	r.	PROPN
ejpam-609	157	23	proof	proof	PROPN
ejpam-609	157	24	.	.	PUNCT
ejpam-609	158	1	see	see	VERB
ejpam-609	158	2	proposition	proposition	NOUN
ejpam-609	158	3	(	(	PUNCT
ejpam-609	158	4	2.2	2.2	NUM
ejpam-609	158	5	)	)	PUNCT
ejpam-609	158	6	of	of	ADP
ejpam-609	158	7	bhat	bhat	PROPN
ejpam-609	159	1	[	[	X
ejpam-609	159	2	2	2	NUM
ejpam-609	159	3	]	]	PUNCT
ejpam-609	159	4	.	.	PUNCT
ejpam-609	160	1	proposition	proposition	NOUN
ejpam-609	160	2	4	4	NUM
ejpam-609	160	3	.	.	PUNCT
ejpam-609	161	1	let	let	VERB
ejpam-609	161	2	r	r	PRON
ejpam-609	161	3	be	be	AUX
ejpam-609	161	4	a	a	DET
ejpam-609	161	5	noetherian	noetherian	ADJ
ejpam-609	161	6	ring	ring	NOUN
ejpam-609	161	7	which	which	PRON
ejpam-609	161	8	is	be	AUX
ejpam-609	161	9	also	also	ADV
ejpam-609	161	10	an	an	DET
ejpam-609	161	11	algebra	algebra	NOUN
ejpam-609	161	12	overq	overq	ADJ
ejpam-609	161	13	.	.	PUNCT
ejpam-609	162	1	let	let	VERB
ejpam-609	162	2	σ	σ	NOUN
ejpam-609	162	3	be	be	AUX
ejpam-609	162	4	an	an	DET
ejpam-609	162	5	automorphism	automorphism	NOUN
ejpam-609	162	6	of	of	ADP
ejpam-609	162	7	r	r	NOUN
ejpam-609	162	8	such	such	ADJ
ejpam-609	162	9	that	that	SCONJ
ejpam-609	162	10	r	r	NOUN
ejpam-609	162	11	is	be	AUX
ejpam-609	162	12	a	a	DET
ejpam-609	162	13	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	162	14	.	.	PUNCT
ejpam-609	163	1	then	then	ADV
ejpam-609	163	2	u	u	PROPN
ejpam-609	163	3	∈	∈	PROPN
ejpam-609	163	4	min.spec(r	min.spec(r	PROPN
ejpam-609	163	5	)	)	PUNCT
ejpam-609	163	6	implies	imply	VERB
ejpam-609	163	7	that	that	SCONJ
ejpam-609	163	8	uo(r	uo(r	PUNCT
ejpam-609	163	9	)	)	PUNCT
ejpam-609	163	10	=	=	SYM
ejpam-609	163	11	u[x	u[x	PRON
ejpam-609	163	12	;	;	PUNCT
ejpam-609	163	13	σ	σ	PROPN
ejpam-609	163	14	,	,	PUNCT
ejpam-609	163	15	δ	δ	PROPN
ejpam-609	163	16	]	]	PUNCT
ejpam-609	163	17	is	be	AUX
ejpam-609	163	18	a	a	DET
ejpam-609	163	19	completely	completely	ADV
ejpam-609	163	20	prime	prime	ADJ
ejpam-609	163	21	ideal	ideal	NOUN
ejpam-609	163	22	of	of	ADP
ejpam-609	163	23	o(r	o(r	NOUN
ejpam-609	163	24	)	)	PUNCT
ejpam-609	163	25	=	=	SYM
ejpam-609	163	26	r[x	r[x	NOUN
ejpam-609	163	27	;	;	PUNCT
ejpam-609	163	28	σ	σ	PROPN
ejpam-609	163	29	,	,	PUNCT
ejpam-609	163	30	δ	δ	PROPN
ejpam-609	163	31	]	]	PUNCT
ejpam-609	163	32	.	.	PUNCT
ejpam-609	164	1	v.	v.	ADP
ejpam-609	164	2	bhat	bhat	PROPN
ejpam-609	164	3	/	/	SYM
ejpam-609	164	4	eur	eur	PROPN
ejpam-609	164	5	.	.	PUNCT
ejpam-609	165	1	j.	j.	PROPN
ejpam-609	165	2	pure	pure	PROPN
ejpam-609	165	3	appl	appl	PROPN
ejpam-609	165	4	.	.	PROPN
ejpam-609	165	5	math	math	PROPN
ejpam-609	165	6	,	,	PUNCT
ejpam-609	165	7	3	3	NUM
ejpam-609	165	8	(	(	PUNCT
ejpam-609	165	9	2010	2010	NUM
ejpam-609	165	10	)	)	PUNCT
ejpam-609	165	11	,	,	PUNCT
ejpam-609	165	12	695	695	NUM
ejpam-609	165	13	-	-	SYM
ejpam-609	165	14	703	703	NUM
ejpam-609	165	15	700	700	NUM
ejpam-609	165	16	proof	proof	NOUN
ejpam-609	165	17	.	.	PUNCT
ejpam-609	166	1	proposition	proposition	NOUN
ejpam-609	166	2	(	(	PUNCT
ejpam-609	166	3	1	1	NUM
ejpam-609	166	4	)	)	PUNCT
ejpam-609	166	5	implies	imply	VERB
ejpam-609	166	6	that	that	SCONJ
ejpam-609	166	7	p(r	p(r	PROPN
ejpam-609	166	8	)	)	PUNCT
ejpam-609	166	9	is	be	AUX
ejpam-609	166	10	completely	completely	ADV
ejpam-609	166	11	semiprime	semiprime	NOUN
ejpam-609	166	12	ideal	ideal	NOUN
ejpam-609	166	13	of	of	ADP
ejpam-609	166	14	r.	r.	PROPN
ejpam-609	166	15	let	let	VERB
ejpam-609	166	16	u	u	PRON
ejpam-609	166	17	∈	∈	PROPN
ejpam-609	166	18	min.spec(r	min.spec(r	PROPN
ejpam-609	166	19	)	)	PUNCT
ejpam-609	166	20	.	.	PUNCT
ejpam-609	167	1	then	then	ADV
ejpam-609	167	2	theorem	theorem	ADJ
ejpam-609	167	3	(	(	PUNCT
ejpam-609	167	4	3	3	NUM
ejpam-609	167	5	)	)	PUNCT
ejpam-609	167	6	implies	imply	VERB
ejpam-609	167	7	that	that	SCONJ
ejpam-609	167	8	σ(u	σ(u	NOUN
ejpam-609	167	9	)	)	PUNCT
ejpam-609	167	10	=	=	SYM
ejpam-609	167	11	u	u	NOUN
ejpam-609	167	12	and	and	CCONJ
ejpam-609	167	13	u	u	NOUN
ejpam-609	167	14	is	be	AUX
ejpam-609	167	15	completely	completely	ADV
ejpam-609	167	16	prime	prime	ADJ
ejpam-609	167	17	.	.	PUNCT
ejpam-609	168	1	also	also	ADV
ejpam-609	168	2	by	by	ADP
ejpam-609	168	3	proposition	proposition	NOUN
ejpam-609	168	4	(	(	PUNCT
ejpam-609	168	5	3	3	NUM
ejpam-609	168	6	)	)	PUNCT
ejpam-609	168	7	δ(u)⊆	δ(u)⊆	PROPN
ejpam-609	168	8	u	u	NOUN
ejpam-609	168	9	.	.	PUNCT
ejpam-609	169	1	now	now	ADV
ejpam-609	169	2	theorem	theorem	ADJ
ejpam-609	169	3	(	(	PUNCT
ejpam-609	169	4	4	4	NUM
ejpam-609	169	5	)	)	PUNCT
ejpam-609	169	6	implies	imply	VERB
ejpam-609	169	7	that	that	SCONJ
ejpam-609	169	8	uo(r	uo(r	PUNCT
ejpam-609	169	9	)	)	PUNCT
ejpam-609	169	10	=	=	SYM
ejpam-609	169	11	u[x	u[x	PRON
ejpam-609	169	12	;	;	PUNCT
ejpam-609	169	13	σ	σ	PROPN
ejpam-609	169	14	,	,	PUNCT
ejpam-609	169	15	δ	δ	PROPN
ejpam-609	169	16	]	]	PUNCT
ejpam-609	169	17	is	be	AUX
ejpam-609	169	18	a	a	DET
ejpam-609	169	19	completely	completely	ADV
ejpam-609	169	20	prime	prime	ADJ
ejpam-609	169	21	ideal	ideal	NOUN
ejpam-609	169	22	of	of	ADP
ejpam-609	169	23	o(r	o(r	NOUN
ejpam-609	169	24	)	)	PUNCT
ejpam-609	169	25	=	=	SYM
ejpam-609	169	26	r[x	r[x	NOUN
ejpam-609	169	27	;	;	PUNCT
ejpam-609	169	28	σ	σ	PROPN
ejpam-609	169	29	,	,	PUNCT
ejpam-609	169	30	δ	δ	PROPN
ejpam-609	169	31	]	]	PUNCT
ejpam-609	169	32	.	.	PUNCT
ejpam-609	170	1	3	3	X
ejpam-609	170	2	.	.	X
ejpam-609	170	3	skew	skew	ADJ
ejpam-609	170	4	polynomial	polynomial	ADJ
ejpam-609	170	5	rings	ring	NOUN
ejpam-609	170	6	over	over	ADP
ejpam-609	170	7	weak	weak	ADJ
ejpam-609	170	8	σ	σ	ADJ
ejpam-609	170	9	-	-	ADJ
ejpam-609	170	10	rigid	rigid	ADJ
ejpam-609	170	11	rings	ring	NOUN
ejpam-609	170	12	definition	definition	NOUN
ejpam-609	170	13	3	3	NUM
ejpam-609	170	14	(	(	PUNCT
ejpam-609	170	15	ouyang	ouyang	X
ejpam-609	171	1	[	[	X
ejpam-609	171	2	16	16	NUM
ejpam-609	171	3	]	]	PUNCT
ejpam-609	171	4	)	)	PUNCT
ejpam-609	171	5	.	.	PUNCT
ejpam-609	172	1	let	let	VERB
ejpam-609	172	2	r	r	PRON
ejpam-609	172	3	be	be	AUX
ejpam-609	172	4	a	a	DET
ejpam-609	172	5	ring	ring	NOUN
ejpam-609	172	6	.	.	PUNCT
ejpam-609	173	1	then	then	ADV
ejpam-609	173	2	r	r	NOUN
ejpam-609	173	3	is	be	AUX
ejpam-609	173	4	said	say	VERB
ejpam-609	173	5	to	to	PART
ejpam-609	173	6	be	be	AUX
ejpam-609	173	7	a	a	DET
ejpam-609	173	8	weak	weak	ADJ
ejpam-609	173	9	σ	σ	ADJ
ejpam-609	173	10	-	-	ADJ
ejpam-609	173	11	rigid	rigid	ADJ
ejpam-609	173	12	ring	ring	NOUN
ejpam-609	173	13	if	if	SCONJ
ejpam-609	173	14	aσ(a	aσ(a	NUM
ejpam-609	173	15	)	)	PUNCT
ejpam-609	173	16	∈	∈	PROPN
ejpam-609	173	17	n(r	n(r	NOUN
ejpam-609	173	18	)	)	PUNCT
ejpam-609	174	1	if	if	SCONJ
ejpam-609	174	2	and	and	CCONJ
ejpam-609	174	3	only	only	ADV
ejpam-609	174	4	if	if	SCONJ
ejpam-609	174	5	a	a	DET
ejpam-609	174	6	∈	∈	PROPN
ejpam-609	174	7	n(r	n(r	NOUN
ejpam-609	174	8	)	)	PUNCT
ejpam-609	174	9	for	for	ADP
ejpam-609	174	10	a	a	DET
ejpam-609	174	11	∈	∈	PROPN
ejpam-609	174	12	r.	r.	PROPN
ejpam-609	174	13	example	example	NOUN
ejpam-609	174	14	4	4	NUM
ejpam-609	174	15	(	(	PUNCT
ejpam-609	174	16	example	example	NOUN
ejpam-609	174	17	(	(	PUNCT
ejpam-609	174	18	2.1	2.1	NUM
ejpam-609	174	19	)	)	PUNCT
ejpam-609	174	20	of	of	ADP
ejpam-609	174	21	ouyang	ouyang	PROPN
ejpam-609	174	22	[	[	X
ejpam-609	174	23	16	16	NUM
ejpam-609	174	24	]	]	PUNCT
ejpam-609	174	25	)	)	PUNCT
ejpam-609	174	26	.	.	PUNCT
ejpam-609	175	1	let	let	VERB
ejpam-609	175	2	σ	σ	NOUN
ejpam-609	175	3	be	be	AUX
ejpam-609	175	4	an	an	DET
ejpam-609	175	5	endomorphism	endomorphism	NOUN
ejpam-609	175	6	of	of	ADP
ejpam-609	175	7	a	a	DET
ejpam-609	175	8	ring	ring	NOUN
ejpam-609	175	9	r	r	NOUN
ejpam-609	175	10	such	such	ADJ
ejpam-609	175	11	that	that	SCONJ
ejpam-609	175	12	r	r	NOUN
ejpam-609	175	13	is	be	AUX
ejpam-609	175	14	a	a	DET
ejpam-609	175	15	σ	σ	PROPN
ejpam-609	175	16	-	-	ADJ
ejpam-609	175	17	rigid	rigid	ADJ
ejpam-609	175	18	ring	ring	NOUN
ejpam-609	175	19	.	.	PUNCT
ejpam-609	176	1	let	let	VERB
ejpam-609	176	2	a=	a=	ADV
ejpam-609	176	3	n	n	PART
ejpam-609	176	4			VERB
ejpam-609	176	5			NOUN
ejpam-609	176	6			NOUN
ejpam-609	176	7	a	a	DET
ejpam-609	176	8	b	b	NOUN
ejpam-609	176	9	c	c	NOUN
ejpam-609	176	10	0	0	PUNCT
ejpam-609	176	11	a	a	DET
ejpam-609	176	12	d	d	NOUN
ejpam-609	176	13	0	0	NUM
ejpam-609	176	14	0	0	NUM
ejpam-609	177	1	a	a	DET
ejpam-609	177	2			NOUN
ejpam-609	177	3			NOUN
ejpam-609	177	4			PUNCT
ejpam-609	178	1	|	|	ADV
ejpam-609	178	2	a	a	DET
ejpam-609	178	3	,	,	PUNCT
ejpam-609	178	4	b	b	NOUN
ejpam-609	178	5	,	,	PUNCT
ejpam-609	178	6	c	c	NOUN
ejpam-609	178	7	,	,	PUNCT
ejpam-609	178	8	d	d	PROPN
ejpam-609	178	9	∈	∈	PROPN
ejpam-609	178	10	r	r	NOUN
ejpam-609	178	11	o	o	AUX
ejpam-609	178	12	be	be	AUX
ejpam-609	178	13	a	a	DET
ejpam-609	178	14	subring	subring	NOUN
ejpam-609	178	15	of	of	ADP
ejpam-609	178	16	t3(r	t3(r	NOUN
ejpam-609	178	17	)	)	PUNCT
ejpam-609	178	18	,	,	PUNCT
ejpam-609	178	19	the	the	DET
ejpam-609	178	20	ring	ring	NOUN
ejpam-609	178	21	of	of	ADP
ejpam-609	178	22	upper	upper	ADJ
ejpam-609	178	23	triangular	triangular	NOUN
ejpam-609	178	24	matrices	matrix	NOUN
ejpam-609	178	25	over	over	ADP
ejpam-609	178	26	r.	r.	PROPN
ejpam-609	178	27	now	now	ADV
ejpam-609	178	28	σ	σ	PROPN
ejpam-609	178	29	can	can	AUX
ejpam-609	178	30	be	be	AUX
ejpam-609	178	31	extended	extend	VERB
ejpam-609	178	32	to	to	ADP
ejpam-609	178	33	an	an	DET
ejpam-609	178	34	endomorphism	endomorphism	PROPN
ejpam-609	178	35	σ	σ	NOUN
ejpam-609	178	36	of	of	ADP
ejpam-609	178	37	a	a	PRON
ejpam-609	178	38	by	by	ADP
ejpam-609	178	39	σ((ai	σ((ai	PROPN
ejpam-609	178	40	j	j	PROPN
ejpam-609	178	41	)	)	PUNCT
ejpam-609	178	42	)	)	PUNCT
ejpam-609	179	1	=	=	PRON
ejpam-609	179	2	(	(	PUNCT
ejpam-609	179	3	σ(ai	σ(ai	PROPN
ejpam-609	179	4	j	j	PROPN
ejpam-609	179	5	)	)	PUNCT
ejpam-609	179	6	)	)	PUNCT
ejpam-609	179	7	.	.	PUNCT
ejpam-609	180	1	the	the	DET
ejpam-609	180	2	it	it	PRON
ejpam-609	180	3	can	can	AUX
ejpam-609	180	4	be	be	AUX
ejpam-609	180	5	seen	see	VERB
ejpam-609	180	6	that	that	SCONJ
ejpam-609	180	7	a	a	PRON
ejpam-609	180	8	is	be	AUX
ejpam-609	180	9	a	a	DET
ejpam-609	180	10	weak	weak	ADJ
ejpam-609	180	11	σ	σ	ADJ
ejpam-609	180	12	-	-	ADJ
ejpam-609	180	13	rigid	rigid	ADJ
ejpam-609	180	14	ring	ring	NOUN
ejpam-609	180	15	.	.	PUNCT
ejpam-609	181	1	ouyang	ouyang	PROPN
ejpam-609	181	2	has	have	AUX
ejpam-609	181	3	proved	prove	VERB
ejpam-609	181	4	in	in	ADP
ejpam-609	181	5	[	[	X
ejpam-609	181	6	16	16	NUM
ejpam-609	181	7	]	]	PUNCT
ejpam-609	181	8	that	that	SCONJ
ejpam-609	181	9	if	if	SCONJ
ejpam-609	181	10	σ	σ	PROPN
ejpam-609	181	11	is	be	AUX
ejpam-609	181	12	an	an	DET
ejpam-609	181	13	endomorphism	endomorphism	NOUN
ejpam-609	181	14	of	of	ADP
ejpam-609	181	15	a	a	DET
ejpam-609	181	16	ring	ring	NOUN
ejpam-609	181	17	r	r	NOUN
ejpam-609	181	18	,	,	PUNCT
ejpam-609	181	19	then	then	ADV
ejpam-609	181	20	r	r	NOUN
ejpam-609	181	21	is	be	AUX
ejpam-609	181	22	σ	σ	NOUN
ejpam-609	181	23	-	-	ADJ
ejpam-609	181	24	rigid	rigid	ADJ
ejpam-609	181	25	if	if	SCONJ
ejpam-609	181	26	and	and	CCONJ
ejpam-609	181	27	only	only	ADV
ejpam-609	181	28	if	if	SCONJ
ejpam-609	181	29	r	r	NOUN
ejpam-609	181	30	is	be	AUX
ejpam-609	181	31	weak	weak	ADJ
ejpam-609	181	32	σ	σ	NOUN
ejpam-609	181	33	-	-	ADJ
ejpam-609	181	34	rigid	rigid	ADJ
ejpam-609	181	35	and	and	CCONJ
ejpam-609	181	36	reduced	reduce	VERB
ejpam-609	181	37	.	.	PUNCT
ejpam-609	182	1	we	we	PRON
ejpam-609	182	2	now	now	ADV
ejpam-609	182	3	give	give	VERB
ejpam-609	182	4	a	a	DET
ejpam-609	182	5	relation	relation	NOUN
ejpam-609	182	6	between	between	ADP
ejpam-609	182	7	a	a	DET
ejpam-609	182	8	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	182	9	and	and	CCONJ
ejpam-609	182	10	a	a	DET
ejpam-609	182	11	weak	weak	ADJ
ejpam-609	182	12	σ	σ	ADJ
ejpam-609	182	13	-	-	ADJ
ejpam-609	182	14	rigid	rigid	ADJ
ejpam-609	182	15	ring	ring	NOUN
ejpam-609	182	16	in	in	ADP
ejpam-609	182	17	the	the	DET
ejpam-609	182	18	following	following	NOUN
ejpam-609	182	19	theorem	theorem	NOUN
ejpam-609	182	20	:	:	PUNCT
ejpam-609	182	21	theorem	theorem	NOUN
ejpam-609	182	22	5	5	NUM
ejpam-609	182	23	.	.	PUNCT
ejpam-609	183	1	let	let	VERB
ejpam-609	183	2	r	r	PRON
ejpam-609	183	3	be	be	AUX
ejpam-609	183	4	a	a	DET
ejpam-609	183	5	noetherian	noetherian	ADJ
ejpam-609	183	6	ring	ring	NOUN
ejpam-609	183	7	.	.	PUNCT
ejpam-609	184	1	let	let	VERB
ejpam-609	184	2	σ	σ	NOUN
ejpam-609	184	3	be	be	AUX
ejpam-609	184	4	an	an	DET
ejpam-609	184	5	automorphism	automorphism	NOUN
ejpam-609	184	6	of	of	ADP
ejpam-609	184	7	r	r	NOUN
ejpam-609	184	8	such	such	ADJ
ejpam-609	184	9	that	that	SCONJ
ejpam-609	184	10	r	r	NOUN
ejpam-609	184	11	is	be	AUX
ejpam-609	184	12	a	a	DET
ejpam-609	184	13	σ(∗)ring	σ(∗)ring	NOUN
ejpam-609	184	14	.	.	PUNCT
ejpam-609	185	1	then	then	ADV
ejpam-609	185	2	r	r	NOUN
ejpam-609	185	3	is	be	AUX
ejpam-609	185	4	a	a	DET
ejpam-609	185	5	weak	weak	ADJ
ejpam-609	185	6	σ	σ	ADJ
ejpam-609	185	7	-	-	ADJ
ejpam-609	185	8	rigid	rigid	ADJ
ejpam-609	185	9	ring	ring	NOUN
ejpam-609	185	10	.	.	PUNCT
ejpam-609	186	1	conversely	conversely	ADV
ejpam-609	186	2	a	a	DET
ejpam-609	186	3	2	2	NUM
ejpam-609	186	4	-	-	PUNCT
ejpam-609	186	5	primal	primal	ADJ
ejpam-609	186	6	weak	weak	ADJ
ejpam-609	186	7	σ	σ	ADJ
ejpam-609	186	8	-	-	ADJ
ejpam-609	186	9	rigid	rigid	ADJ
ejpam-609	186	10	ring	ring	NOUN
ejpam-609	186	11	is	be	AUX
ejpam-609	186	12	a	a	DET
ejpam-609	186	13	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	186	14	.	.	PUNCT
ejpam-609	187	1	proof	proof	NOUN
ejpam-609	187	2	.	.	PUNCT
ejpam-609	188	1	let	let	VERB
ejpam-609	188	2	σ	σ	NOUN
ejpam-609	188	3	be	be	AUX
ejpam-609	188	4	an	an	DET
ejpam-609	188	5	automorphism	automorphism	NOUN
ejpam-609	188	6	of	of	ADP
ejpam-609	188	7	r	r	NOUN
ejpam-609	188	8	such	such	ADJ
ejpam-609	188	9	that	that	SCONJ
ejpam-609	188	10	r	r	NOUN
ejpam-609	188	11	is	be	AUX
ejpam-609	188	12	a	a	DET
ejpam-609	188	13	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	188	14	.	.	PUNCT
ejpam-609	189	1	now	now	ADV
ejpam-609	189	2	proposition	proposition	NOUN
ejpam-609	189	3	(	(	PUNCT
ejpam-609	189	4	2	2	NUM
ejpam-609	189	5	)	)	PUNCT
ejpam-609	189	6	implies	imply	VERB
ejpam-609	189	7	that	that	SCONJ
ejpam-609	189	8	r	r	NOUN
ejpam-609	189	9	is	be	AUX
ejpam-609	189	10	2	2	NUM
ejpam-609	189	11	-	-	PUNCT
ejpam-609	189	12	primal	primal	ADJ
ejpam-609	189	13	,	,	PUNCT
ejpam-609	189	14	i.e.	i.e.	X
ejpam-609	189	15	n(r	n(r	NOUN
ejpam-609	189	16	)	)	PUNCT
ejpam-609	189	17	=	=	SYM
ejpam-609	189	18	p(r	p(r	PROPN
ejpam-609	189	19	)	)	PUNCT
ejpam-609	189	20	.	.	PUNCT
ejpam-609	190	1	thus	thus	ADV
ejpam-609	190	2	aσ(a	aσ(a	X
ejpam-609	190	3	)	)	PUNCT
ejpam-609	190	4	∈	∈	PROPN
ejpam-609	190	5	n(r	n(r	NOUN
ejpam-609	190	6	)	)	PUNCT
ejpam-609	190	7	=	=	SYM
ejpam-609	190	8	p(r	p(r	NOUN
ejpam-609	190	9	)	)	PUNCT
ejpam-609	190	10	implies	imply	VERB
ejpam-609	190	11	that	that	SCONJ
ejpam-609	190	12	a	a	DET
ejpam-609	190	13	∈	∈	PROPN
ejpam-609	190	14	p(r	p(r	NOUN
ejpam-609	190	15	)	)	PUNCT
ejpam-609	190	16	=	=	SYM
ejpam-609	190	17	n(r	n(r	NOUN
ejpam-609	190	18	)	)	PUNCT
ejpam-609	190	19	.	.	PUNCT
ejpam-609	191	1	hence	hence	ADV
ejpam-609	191	2	r	r	NOUN
ejpam-609	191	3	is	be	AUX
ejpam-609	191	4	weak	weak	ADJ
ejpam-609	191	5	σ	σ	ADJ
ejpam-609	191	6	-	-	ADJ
ejpam-609	191	7	rigid	rigid	ADJ
ejpam-609	191	8	ring	ring	NOUN
ejpam-609	191	9	.	.	PUNCT
ejpam-609	192	1	conversely	conversely	ADV
ejpam-609	192	2	let	let	VERB
ejpam-609	192	3	r	r	NOUN
ejpam-609	192	4	be	be	AUX
ejpam-609	192	5	2	2	NUM
ejpam-609	192	6	-	-	PUNCT
ejpam-609	192	7	primal	primal	ADJ
ejpam-609	192	8	weak	weak	ADJ
ejpam-609	192	9	σ	σ	ADJ
ejpam-609	192	10	-	-	ADJ
ejpam-609	192	11	rigid	rigid	ADJ
ejpam-609	192	12	ring	ring	NOUN
ejpam-609	192	13	.	.	PUNCT
ejpam-609	193	1	then	then	ADV
ejpam-609	193	2	n(r	n(r	PRON
ejpam-609	193	3	)	)	PUNCT
ejpam-609	193	4	=	=	SYM
ejpam-609	193	5	p(r	p(r	PROPN
ejpam-609	193	6	)	)	PUNCT
ejpam-609	193	7	and	and	CCONJ
ejpam-609	193	8	aσ(a	aσ(a	NUM
ejpam-609	193	9	)	)	PUNCT
ejpam-609	193	10	∈	∈	PROPN
ejpam-609	193	11	n(r	n(r	NOUN
ejpam-609	193	12	)	)	PUNCT
ejpam-609	193	13	implies	imply	VERB
ejpam-609	193	14	that	that	SCONJ
ejpam-609	193	15	a	a	DET
ejpam-609	193	16	∈	∈	PROPN
ejpam-609	193	17	n(r	n(r	NOUN
ejpam-609	193	18	)	)	PUNCT
ejpam-609	193	19	.	.	PUNCT
ejpam-609	194	1	therefore	therefore	ADV
ejpam-609	194	2	,	,	PUNCT
ejpam-609	194	3	aσ(a	aσ(a	X
ejpam-609	194	4	)	)	PUNCT
ejpam-609	194	5	∈	∈	PROPN
ejpam-609	194	6	p(r	p(r	PROPN
ejpam-609	194	7	)	)	PUNCT
ejpam-609	194	8	implies	imply	VERB
ejpam-609	194	9	that	that	SCONJ
ejpam-609	194	10	a	a	DET
ejpam-609	194	11	∈	∈	PROPN
ejpam-609	194	12	p(r	p(r	PROPN
ejpam-609	194	13	)	)	PUNCT
ejpam-609	194	14	.	.	PUNCT
ejpam-609	195	1	hence	hence	ADV
ejpam-609	195	2	r	r	NOUN
ejpam-609	195	3	is	be	AUX
ejpam-609	195	4	a	a	DET
ejpam-609	195	5	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	195	6	.	.	PUNCT
ejpam-609	196	1	let	let	VERB
ejpam-609	196	2	r	r	PRON
ejpam-609	196	3	be	be	AUX
ejpam-609	196	4	a	a	DET
ejpam-609	196	5	noetherian	noetherian	ADJ
ejpam-609	196	6	ring	ring	NOUN
ejpam-609	196	7	and	and	CCONJ
ejpam-609	196	8	σ	σ	NOUN
ejpam-609	196	9	an	an	DET
ejpam-609	196	10	automorphism	automorphism	NOUN
ejpam-609	196	11	of	of	ADP
ejpam-609	196	12	r.	r.	PROPN
ejpam-609	196	13	we	we	PRON
ejpam-609	196	14	now	now	ADV
ejpam-609	196	15	give	give	VERB
ejpam-609	196	16	a	a	DET
ejpam-609	196	17	characterization	characterization	NOUN
ejpam-609	196	18	for	for	SCONJ
ejpam-609	196	19	r	r	NOUN
ejpam-609	196	20	to	to	PART
ejpam-609	196	21	be	be	AUX
ejpam-609	196	22	a	a	DET
ejpam-609	196	23	weak	weak	ADJ
ejpam-609	196	24	σ	σ	ADJ
ejpam-609	196	25	-	-	ADJ
ejpam-609	196	26	rigid	rigid	ADJ
ejpam-609	196	27	ring	ring	NOUN
ejpam-609	196	28	.	.	PUNCT
ejpam-609	197	1	(	(	PUNCT
ejpam-609	197	2	an	an	DET
ejpam-609	197	3	analog	analog	NOUN
ejpam-609	197	4	of	of	ADP
ejpam-609	197	5	proposition	proposition	NOUN
ejpam-609	197	6	(	(	PUNCT
ejpam-609	197	7	1	1	NUM
ejpam-609	197	8	)	)	PUNCT
ejpam-609	197	9	for	for	ADP
ejpam-609	197	10	weak	weak	ADJ
ejpam-609	197	11	σ	σ	ADJ
ejpam-609	197	12	-	-	ADJ
ejpam-609	197	13	rigid	rigid	ADJ
ejpam-609	197	14	rings	ring	NOUN
ejpam-609	197	15	)	)	PUNCT
ejpam-609	197	16	theorem	theorem	VERB
ejpam-609	197	17	6	6	NUM
ejpam-609	197	18	.	.	PUNCT
ejpam-609	198	1	let	let	VERB
ejpam-609	198	2	r	r	PRON
ejpam-609	198	3	be	be	AUX
ejpam-609	198	4	a	a	DET
ejpam-609	198	5	commutative	commutative	ADJ
ejpam-609	198	6	noetherian	noetherian	ADJ
ejpam-609	198	7	ring	ring	NOUN
ejpam-609	198	8	.	.	PUNCT
ejpam-609	199	1	let	let	VERB
ejpam-609	199	2	σ	σ	NOUN
ejpam-609	199	3	be	be	AUX
ejpam-609	199	4	an	an	DET
ejpam-609	199	5	automorphism	automorphism	NOUN
ejpam-609	199	6	of	of	ADP
ejpam-609	199	7	r.	r.	PROPN
ejpam-609	199	8	then	then	ADV
ejpam-609	199	9	r	r	NOUN
ejpam-609	199	10	is	be	AUX
ejpam-609	199	11	a	a	DET
ejpam-609	199	12	weak	weak	ADJ
ejpam-609	199	13	σ	σ	ADJ
ejpam-609	199	14	-	-	ADJ
ejpam-609	199	15	rigid	rigid	ADJ
ejpam-609	199	16	ring	ring	NOUN
ejpam-609	199	17	implies	imply	VERB
ejpam-609	199	18	that	that	SCONJ
ejpam-609	199	19	n(r	n(r	NOUN
ejpam-609	199	20	)	)	PUNCT
ejpam-609	199	21	is	be	AUX
ejpam-609	199	22	completely	completely	ADV
ejpam-609	199	23	semiprime	semiprime	NOUN
ejpam-609	199	24	.	.	PUNCT
ejpam-609	200	1	proof	proof	NOUN
ejpam-609	200	2	.	.	PUNCT
ejpam-609	201	1	first	first	ADV
ejpam-609	201	2	of	of	ADP
ejpam-609	201	3	all	all	PRON
ejpam-609	201	4	we	we	PRON
ejpam-609	201	5	show	show	VERB
ejpam-609	201	6	that	that	SCONJ
ejpam-609	201	7	σ(n(r	σ(n(r	NOUN
ejpam-609	201	8	)	)	PUNCT
ejpam-609	201	9	)	)	PUNCT
ejpam-609	202	1	=	=	SYM
ejpam-609	202	2	n(r	n(r	NOUN
ejpam-609	202	3	)	)	PUNCT
ejpam-609	202	4	.	.	PUNCT
ejpam-609	203	1	we	we	PRON
ejpam-609	203	2	have	have	VERB
ejpam-609	203	3	σ(n(r	σ(n(r	PROPN
ejpam-609	203	4	)	)	PUNCT
ejpam-609	203	5	)	)	PUNCT
ejpam-609	204	1	⊆	⊆	NUM
ejpam-609	204	2	n(r	n(r	NOUN
ejpam-609	204	3	)	)	PUNCT
ejpam-609	204	4	as	as	ADP
ejpam-609	204	5	σ(n(r	σ(n(r	PROPN
ejpam-609	204	6	)	)	PUNCT
ejpam-609	204	7	)	)	PUNCT
ejpam-609	204	8	is	be	AUX
ejpam-609	204	9	a	a	DET
ejpam-609	204	10	nilpotent	nilpotent	ADJ
ejpam-609	204	11	ideal	ideal	NOUN
ejpam-609	204	12	of	of	ADP
ejpam-609	204	13	r.	r.	PROPN
ejpam-609	204	14	now	now	ADV
ejpam-609	204	15	for	for	ADP
ejpam-609	204	16	any	any	DET
ejpam-609	204	17	n	n	PRON
ejpam-609	204	18	∈	∈	PROPN
ejpam-609	204	19	n(r	n(r	NOUN
ejpam-609	204	20	)	)	PUNCT
ejpam-609	204	21	,	,	PUNCT
ejpam-609	204	22	there	there	PRON
ejpam-609	204	23	exists	exist	VERB
ejpam-609	204	24	a	a	DET
ejpam-609	204	25	∈	∈	NOUN
ejpam-609	204	26	r	r	NOUN
ejpam-609	204	27	such	such	ADJ
ejpam-609	204	28	that	that	SCONJ
ejpam-609	204	29	n	n	NOUN
ejpam-609	204	30	=	=	SYM
ejpam-609	204	31	σ(a	σ(a	PROPN
ejpam-609	204	32	)	)	PUNCT
ejpam-609	204	33	.	.	PUNCT
ejpam-609	205	1	so	so	ADV
ejpam-609	205	2	i	i	PRON
ejpam-609	205	3	=	=	SYM
ejpam-609	205	4	σ−1(n(r	σ−1(n(r	NOUN
ejpam-609	205	5	)	)	PUNCT
ejpam-609	205	6	)	)	PUNCT
ejpam-609	206	1	=	=	PRON
ejpam-609	206	2	{	{	PUNCT
ejpam-609	206	3	a	a	DET
ejpam-609	206	4	∈	∈	NOUN
ejpam-609	206	5	r	r	NOUN
ejpam-609	206	6	such	such	ADJ
ejpam-609	206	7	that	that	DET
ejpam-609	206	8	σ(a	σ(a	PROPN
ejpam-609	206	9	)	)	PUNCT
ejpam-609	206	10	=	=	SYM
ejpam-609	206	11	n	n	CCONJ
ejpam-609	206	12	∈	∈	PROPN
ejpam-609	206	13	n(r	n(r	NOUN
ejpam-609	206	14	)	)	PUNCT
ejpam-609	206	15	}	}	PUNCT
ejpam-609	206	16	is	be	AUX
ejpam-609	206	17	an	an	DET
ejpam-609	206	18	ideal	ideal	NOUN
ejpam-609	206	19	of	of	ADP
ejpam-609	206	20	r.	r.	PROPN
ejpam-609	206	21	now	now	ADV
ejpam-609	206	22	i	i	PRON
ejpam-609	206	23	is	be	AUX
ejpam-609	206	24	nilpotent	nilpotent	ADJ
ejpam-609	206	25	,	,	PUNCT
ejpam-609	206	26	therefore	therefore	ADV
ejpam-609	206	27	i	i	PROPN
ejpam-609	206	28	⊆	⊆	NUM
ejpam-609	206	29	n(r	n(r	NUM
ejpam-609	206	30	)	)	PUNCT
ejpam-609	206	31	,	,	PUNCT
ejpam-609	206	32	which	which	PRON
ejpam-609	206	33	implies	imply	VERB
ejpam-609	206	34	that	that	SCONJ
ejpam-609	206	35	n(r)⊆	n(r)⊆	PROPN
ejpam-609	206	36	σ(n(r	σ(n(r	PROPN
ejpam-609	206	37	)	)	PUNCT
ejpam-609	206	38	)	)	PUNCT
ejpam-609	206	39	.	.	PUNCT
ejpam-609	207	1	hence	hence	ADV
ejpam-609	207	2	σ(n(r	σ(n(r	NUM
ejpam-609	207	3	)	)	PUNCT
ejpam-609	207	4	)	)	PUNCT
ejpam-609	208	1	=	=	SYM
ejpam-609	208	2	n(r	n(r	NOUN
ejpam-609	208	3	)	)	PUNCT
ejpam-609	208	4	.	.	PUNCT
ejpam-609	209	1	now	now	ADV
ejpam-609	209	2	let	let	VERB
ejpam-609	209	3	r	r	NOUN
ejpam-609	209	4	be	be	AUX
ejpam-609	209	5	a	a	DET
ejpam-609	209	6	weak	weak	ADJ
ejpam-609	209	7	σ	σ	ADJ
ejpam-609	209	8	-	-	ADJ
ejpam-609	209	9	rigid	rigid	ADJ
ejpam-609	209	10	ring	ring	NOUN
ejpam-609	209	11	.	.	PUNCT
ejpam-609	210	1	we	we	PRON
ejpam-609	210	2	will	will	AUX
ejpam-609	210	3	show	show	VERB
ejpam-609	210	4	that	that	SCONJ
ejpam-609	210	5	n(r	n(r	NOUN
ejpam-609	210	6	)	)	PUNCT
ejpam-609	210	7	is	be	AUX
ejpam-609	210	8	completely	completely	ADV
ejpam-609	210	9	semiprime	semiprime	ADJ
ejpam-609	210	10	.	.	PUNCT
ejpam-609	211	1	let	let	VERB
ejpam-609	211	2	a	a	DET
ejpam-609	211	3	∈	∈	NOUN
ejpam-609	211	4	r	r	NOUN
ejpam-609	211	5	be	be	VERB
ejpam-609	211	6	such	such	ADJ
ejpam-609	211	7	that	that	SCONJ
ejpam-609	211	8	a2	a2	PROPN
ejpam-609	211	9	∈	∈	PROPN
ejpam-609	211	10	n(r	n(r	NOUN
ejpam-609	211	11	)	)	PUNCT
ejpam-609	211	12	.	.	PUNCT
ejpam-609	212	1	then	then	ADV
ejpam-609	212	2	aσ(a)σ(aσ(a	aσ(a)σ(aσ(a	X
ejpam-609	212	3	)	)	PUNCT
ejpam-609	212	4	)	)	PUNCT
ejpam-609	213	1	=	=	SYM
ejpam-609	213	2	aσ(a)σ(a)σ2(a	aσ(a)σ(a)σ2(a	NOUN
ejpam-609	213	3	)	)	PUNCT
ejpam-609	213	4	∈	∈	PROPN
ejpam-609	213	5	σ(n(r	σ(n(r	PROPN
ejpam-609	213	6	)	)	PUNCT
ejpam-609	213	7	)	)	PUNCT
ejpam-609	214	1	=	=	SYM
ejpam-609	214	2	n(r	n(r	NOUN
ejpam-609	214	3	)	)	PUNCT
ejpam-609	214	4	.	.	PUNCT
ejpam-609	215	1	therefore	therefore	ADV
ejpam-609	215	2	aσ(a	aσ(a	X
ejpam-609	215	3	)	)	PUNCT
ejpam-609	215	4	∈	∈	PROPN
ejpam-609	215	5	n(r	n(r	NOUN
ejpam-609	215	6	)	)	PUNCT
ejpam-609	215	7	and	and	CCONJ
ejpam-609	215	8	hence	hence	ADV
ejpam-609	215	9	a	a	DET
ejpam-609	215	10	∈	∈	NOUN
ejpam-609	215	11	n(r	n(r	NOUN
ejpam-609	215	12	)	)	PUNCT
ejpam-609	215	13	.	.	PUNCT
ejpam-609	216	1	so	so	ADV
ejpam-609	216	2	n(r	n(r	NOUN
ejpam-609	216	3	)	)	PUNCT
ejpam-609	216	4	is	be	AUX
ejpam-609	216	5	completely	completely	ADV
ejpam-609	216	6	semiprime	semiprime	ADJ
ejpam-609	216	7	.	.	PUNCT
ejpam-609	217	1	v.	v.	ADP
ejpam-609	217	2	bhat	bhat	PROPN
ejpam-609	217	3	/	/	SYM
ejpam-609	217	4	eur	eur	PROPN
ejpam-609	217	5	.	.	PUNCT
ejpam-609	218	1	j.	j.	PROPN
ejpam-609	218	2	pure	pure	PROPN
ejpam-609	218	3	appl	appl	PROPN
ejpam-609	218	4	.	.	PROPN
ejpam-609	218	5	math	math	PROPN
ejpam-609	218	6	,	,	PUNCT
ejpam-609	218	7	3	3	NUM
ejpam-609	218	8	(	(	PUNCT
ejpam-609	218	9	2010	2010	NUM
ejpam-609	218	10	)	)	PUNCT
ejpam-609	218	11	,	,	PUNCT
ejpam-609	218	12	695	695	NUM
ejpam-609	218	13	-	-	SYM
ejpam-609	218	14	703	703	NUM
ejpam-609	218	15	701	701	NUM
ejpam-609	218	16	corollary	corollary	ADJ
ejpam-609	218	17	1	1	NUM
ejpam-609	218	18	.	.	PUNCT
ejpam-609	219	1	let	let	VERB
ejpam-609	219	2	r	r	PRON
ejpam-609	219	3	be	be	AUX
ejpam-609	219	4	a	a	DET
ejpam-609	219	5	commutative	commutative	ADJ
ejpam-609	219	6	noetherian	noetherian	ADJ
ejpam-609	219	7	ring	ring	NOUN
ejpam-609	219	8	.	.	PUNCT
ejpam-609	220	1	let	let	VERB
ejpam-609	220	2	σ	σ	NOUN
ejpam-609	220	3	be	be	AUX
ejpam-609	220	4	an	an	DET
ejpam-609	220	5	automorphism	automorphism	NOUN
ejpam-609	220	6	of	of	ADP
ejpam-609	220	7	r.	r.	PROPN
ejpam-609	220	8	then	then	ADV
ejpam-609	220	9	r	r	NOUN
ejpam-609	220	10	is	be	AUX
ejpam-609	220	11	a	a	DET
ejpam-609	220	12	2	2	NUM
ejpam-609	220	13	-	-	PUNCT
ejpam-609	220	14	primal	primal	ADJ
ejpam-609	220	15	weak	weak	ADJ
ejpam-609	220	16	σ	σ	ADJ
ejpam-609	220	17	-	-	ADJ
ejpam-609	220	18	rigid	rigid	ADJ
ejpam-609	220	19	ring	ring	NOUN
ejpam-609	220	20	if	if	SCONJ
ejpam-609	220	21	and	and	CCONJ
ejpam-609	220	22	only	only	ADV
ejpam-609	220	23	if	if	SCONJ
ejpam-609	220	24	for	for	ADP
ejpam-609	220	25	each	each	DET
ejpam-609	220	26	minimal	minimal	ADJ
ejpam-609	220	27	prime	prime	ADJ
ejpam-609	220	28	u	u	NOUN
ejpam-609	220	29	of	of	ADP
ejpam-609	220	30	r	r	NOUN
ejpam-609	220	31	,	,	PUNCT
ejpam-609	220	32	σ(u	σ(u	NOUN
ejpam-609	220	33	)	)	PUNCT
ejpam-609	220	34	=	=	SYM
ejpam-609	220	35	u	u	NOUN
ejpam-609	220	36	and	and	CCONJ
ejpam-609	220	37	u	u	NOUN
ejpam-609	220	38	is	be	AUX
ejpam-609	220	39	completely	completely	ADV
ejpam-609	220	40	prime	prime	ADJ
ejpam-609	220	41	ideal	ideal	NOUN
ejpam-609	220	42	of	of	ADP
ejpam-609	220	43	r.	r.	PROPN
ejpam-609	220	44	proof	proof	PROPN
ejpam-609	220	45	.	.	PUNCT
ejpam-609	221	1	combine	combine	PROPN
ejpam-609	221	2	theorem	theorem	ADJ
ejpam-609	221	3	(	(	PUNCT
ejpam-609	221	4	3	3	NUM
ejpam-609	221	5	)	)	PUNCT
ejpam-609	221	6	and	and	CCONJ
ejpam-609	221	7	theorem	theorem	VERB
ejpam-609	221	8	(	(	PUNCT
ejpam-609	221	9	6	6	NUM
ejpam-609	221	10	)	)	PUNCT
ejpam-609	221	11	.	.	PUNCT
ejpam-609	222	1	proposition	proposition	NOUN
ejpam-609	222	2	5	5	NUM
ejpam-609	222	3	.	.	PUNCT
ejpam-609	223	1	let	let	VERB
ejpam-609	223	2	r	r	PRON
ejpam-609	223	3	be	be	AUX
ejpam-609	223	4	a	a	DET
ejpam-609	223	5	commutative	commutative	ADJ
ejpam-609	223	6	noetherian	noetherian	ADJ
ejpam-609	223	7	ring	ring	NOUN
ejpam-609	223	8	.	.	PUNCT
ejpam-609	224	1	let	let	VERB
ejpam-609	224	2	σ	σ	NOUN
ejpam-609	224	3	be	be	AUX
ejpam-609	224	4	an	an	DET
ejpam-609	224	5	automorphism	automorphism	NOUN
ejpam-609	224	6	of	of	ADP
ejpam-609	224	7	r	r	NOUN
ejpam-609	224	8	such	such	ADJ
ejpam-609	224	9	that	that	SCONJ
ejpam-609	224	10	r	r	NOUN
ejpam-609	224	11	is	be	AUX
ejpam-609	224	12	a	a	DET
ejpam-609	224	13	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	224	14	.	.	PUNCT
ejpam-609	225	1	then	then	ADV
ejpam-609	225	2	o(n(r	o(n(r	NOUN
ejpam-609	225	3	)	)	PUNCT
ejpam-609	225	4	)	)	PUNCT
ejpam-609	226	1	=	=	PUNCT
ejpam-609	226	2	n(o(r	n(o(r	ADJ
ejpam-609	226	3	)	)	PUNCT
ejpam-609	226	4	)	)	PUNCT
ejpam-609	226	5	.	.	PUNCT
ejpam-609	227	1	proof	proof	NOUN
ejpam-609	227	2	.	.	PUNCT
ejpam-609	228	1	proposition	proposition	NOUN
ejpam-609	228	2	(	(	PUNCT
ejpam-609	228	3	2	2	NUM
ejpam-609	228	4	)	)	PUNCT
ejpam-609	228	5	implies	imply	VERB
ejpam-609	228	6	that	that	SCONJ
ejpam-609	228	7	r	r	NOUN
ejpam-609	228	8	is	be	AUX
ejpam-609	228	9	2	2	NUM
ejpam-609	228	10	-	-	PUNCT
ejpam-609	228	11	primal	primal	ADJ
ejpam-609	228	12	.	.	PUNCT
ejpam-609	229	1	now	now	ADV
ejpam-609	229	2	it	it	PRON
ejpam-609	229	3	is	be	AUX
ejpam-609	229	4	easy	easy	ADJ
ejpam-609	229	5	to	to	PART
ejpam-609	229	6	see	see	VERB
ejpam-609	229	7	that	that	DET
ejpam-609	229	8	o(n(r	o(n(r	NOUN
ejpam-609	229	9	)	)	PUNCT
ejpam-609	229	10	)	)	PUNCT
ejpam-609	230	1	⊆	⊆	NUM
ejpam-609	230	2	n(o(r	n(o(r	NUM
ejpam-609	230	3	)	)	PUNCT
ejpam-609	230	4	)	)	PUNCT
ejpam-609	230	5	.	.	PUNCT
ejpam-609	231	1	we	we	PRON
ejpam-609	231	2	will	will	AUX
ejpam-609	231	3	show	show	VERB
ejpam-609	231	4	that	that	SCONJ
ejpam-609	231	5	n(o(r))⊆	n(o(r))⊆	NOUN
ejpam-609	231	6	o(n(r	o(n(r	NOUN
ejpam-609	231	7	)	)	PUNCT
ejpam-609	231	8	)	)	PUNCT
ejpam-609	231	9	.	.	PUNCT
ejpam-609	232	1	let	let	VERB
ejpam-609	232	2	f	f	PROPN
ejpam-609	232	3	=	=	SYM
ejpam-609	232	4	∑m	∑m	PROPN
ejpam-609	232	5	i=0	i=0	PROPN
ejpam-609	232	6	x	x	PRON
ejpam-609	232	7	iai	iai	PROPN
ejpam-609	232	8	∈	∈	PROPN
ejpam-609	232	9	n(o(r	n(o(r	PROPN
ejpam-609	232	10	)	)	PUNCT
ejpam-609	232	11	)	)	PUNCT
ejpam-609	232	12	.	.	PUNCT
ejpam-609	233	1	then	then	ADV
ejpam-609	233	2	(	(	PUNCT
ejpam-609	233	3	f	f	X
ejpam-609	233	4	)	)	PUNCT
ejpam-609	233	5	(	(	PUNCT
ejpam-609	233	6	o(r))⊆	o(r))⊆	ADP
ejpam-609	233	7	n(o(r	n(o(r	NUM
ejpam-609	233	8	)	)	PUNCT
ejpam-609	233	9	)	)	PUNCT
ejpam-609	233	10	,	,	PUNCT
ejpam-609	233	11	and	and	CCONJ
ejpam-609	233	12	(	(	PUNCT
ejpam-609	233	13	f	f	PROPN
ejpam-609	233	14	)	)	PUNCT
ejpam-609	233	15	(	(	PUNCT
ejpam-609	233	16	r)⊆	r)⊆	X
ejpam-609	233	17	n(o(r	n(o(r	NUM
ejpam-609	233	18	)	)	PUNCT
ejpam-609	233	19	)	)	PUNCT
ejpam-609	233	20	.	.	PUNCT
ejpam-609	234	1	let	let	VERB
ejpam-609	234	2	(	(	PUNCT
ejpam-609	234	3	(	(	PUNCT
ejpam-609	234	4	f	f	X
ejpam-609	234	5	)	)	PUNCT
ejpam-609	234	6	(	(	PUNCT
ejpam-609	234	7	r))k	r))k	NOUN
ejpam-609	234	8	=	=	SYM
ejpam-609	234	9	0	0	PROPN
ejpam-609	234	10	,	,	PUNCT
ejpam-609	234	11	k	k	X
ejpam-609	234	12	>	>	X
ejpam-609	234	13	0	0	X
ejpam-609	234	14	.	.	PUNCT
ejpam-609	235	1	then	then	ADV
ejpam-609	235	2	equating	equate	VERB
ejpam-609	235	3	leading	lead	VERB
ejpam-609	235	4	term	term	NOUN
ejpam-609	235	5	to	to	ADP
ejpam-609	235	6	zero	zero	NUM
ejpam-609	235	7	,	,	PUNCT
ejpam-609	235	8	we	we	PRON
ejpam-609	235	9	get	get	VERB
ejpam-609	235	10	(	(	PUNCT
ejpam-609	235	11	xmamr)k	xmamr)k	PUNCT
ejpam-609	235	12	=	=	SYM
ejpam-609	236	1	0	0	X
ejpam-609	236	2	.	.	PUNCT
ejpam-609	237	1	after	after	ADP
ejpam-609	237	2	simplification	simplification	NOUN
ejpam-609	237	3	equating	equate	VERB
ejpam-609	237	4	leading	lead	VERB
ejpam-609	237	5	term	term	NOUN
ejpam-609	237	6	to	to	ADP
ejpam-609	237	7	zero	zero	NUM
ejpam-609	237	8	,	,	PUNCT
ejpam-609	237	9	we	we	PRON
ejpam-609	237	10	get	get	VERB
ejpam-609	237	11	x	x	NOUN
ejpam-609	237	12	kmσ(k−1)m(amr).σ(k−2)m(amr).σ(k−3)m(amr	kmσ(k−1)m(amr).σ(k−2)m(amr).σ(k−3)m(amr	NOUN
ejpam-609	237	13	)	)	PUNCT
ejpam-609	237	14	.	.	PUNCT
ejpam-609	237	15	.	.	PUNCT
ejpam-609	237	16	.	.	PUNCT
ejpam-609	238	1	amr=	amr=	PROPN
ejpam-609	238	2	0	0	NUM
ejpam-609	238	3	.	.	PUNCT
ejpam-609	239	1	therefore	therefore	ADV
ejpam-609	239	2	,	,	PUNCT
ejpam-609	239	3	σ(k−1)m(amr).σ(k−2)m(amr).σ(k−3)m(amr)	σ(k−1)m(amr).σ(k−2)m(amr).σ(k−3)m(amr)	NOUN
ejpam-609	239	4	...	...	PUNCT
ejpam-609	239	5	amr=	amr=	VERB
ejpam-609	239	6	0⊆	0⊆	NOUN
ejpam-609	239	7	p	p	X
ejpam-609	239	8	,	,	PUNCT
ejpam-609	239	9	for	for	ADP
ejpam-609	239	10	all	all	PRON
ejpam-609	239	11	p	p	PROPN
ejpam-609	239	12	∈	∈	PROPN
ejpam-609	239	13	min.spec(r	min.spec(r	NOUN
ejpam-609	239	14	)	)	PUNCT
ejpam-609	239	15	.	.	PUNCT
ejpam-609	240	1	this	this	PRON
ejpam-609	240	2	implies	imply	VERB
ejpam-609	240	3	that	that	SCONJ
ejpam-609	240	4	σ(k−	σ(k−	PROPN
ejpam-609	240	5	j)m(amr	j)m(amr	PROPN
ejpam-609	240	6	)	)	PUNCT
ejpam-609	240	7	⊆	⊆	NUM
ejpam-609	240	8	p	p	NOUN
ejpam-609	240	9	,	,	PUNCT
ejpam-609	240	10	for	for	ADP
ejpam-609	240	11	some	some	DET
ejpam-609	240	12	j	j	NOUN
ejpam-609	240	13	,	,	PUNCT
ejpam-609	240	14	1	1	NUM
ejpam-609	240	15	≤	≤	NUM
ejpam-609	240	16	j	j	PROPN
ejpam-609	240	17	≤	≤	PROPN
ejpam-609	240	18	k.	k.	PROPN
ejpam-609	240	19	therefore	therefore	ADV
ejpam-609	240	20	,	,	PUNCT
ejpam-609	240	21	amr	amr	PROPN
ejpam-609	240	22	⊆	⊆	NUM
ejpam-609	240	23	σ−(k−	σ−(k−	ADJ
ejpam-609	240	24	j)m(p	j)m(p	NOUN
ejpam-609	240	25	)	)	PUNCT
ejpam-609	240	26	.	.	PUNCT
ejpam-609	241	1	but	but	CCONJ
ejpam-609	241	2	σ−(k−	σ−(k−	PROPN
ejpam-609	241	3	j)m(p	j)m(p	NOUN
ejpam-609	241	4	)	)	PUNCT
ejpam-609	242	1	=	=	PUNCT
ejpam-609	242	2	p	p	NOUN
ejpam-609	242	3	by	by	ADP
ejpam-609	242	4	theorem	theorem	NOUN
ejpam-609	242	5	(	(	PUNCT
ejpam-609	242	6	3	3	NUM
ejpam-609	242	7	)	)	PUNCT
ejpam-609	242	8	.	.	PUNCT
ejpam-609	243	1	so	so	ADV
ejpam-609	243	2	we	we	PRON
ejpam-609	243	3	have	have	AUX
ejpam-609	243	4	amr	amr	NOUN
ejpam-609	243	5	⊆	⊆	NUM
ejpam-609	243	6	p	p	NOUN
ejpam-609	243	7	,	,	PUNCT
ejpam-609	243	8	for	for	ADP
ejpam-609	243	9	all	all	DET
ejpam-609	243	10	p	p	PROPN
ejpam-609	243	11	∈	∈	PROPN
ejpam-609	243	12	min.spec(r	min.spec(r	PROPN
ejpam-609	243	13	)	)	PUNCT
ejpam-609	243	14	.	.	PUNCT
ejpam-609	244	1	therefore	therefore	ADV
ejpam-609	244	2	,	,	PUNCT
ejpam-609	244	3	am	be	AUX
ejpam-609	244	4	∈	∈	PROPN
ejpam-609	244	5	p(r	p(r	PROPN
ejpam-609	244	6	)	)	PUNCT
ejpam-609	244	7	,	,	PUNCT
ejpam-609	244	8	and	and	CCONJ
ejpam-609	244	9	r	r	NOUN
ejpam-609	244	10	being	be	AUX
ejpam-609	244	11	2	2	NUM
ejpam-609	244	12	-	-	PUNCT
ejpam-609	244	13	primal	primal	ADJ
ejpam-609	244	14	implies	imply	VERB
ejpam-609	244	15	that	that	PRON
ejpam-609	244	16	am	be	AUX
ejpam-609	244	17	∈	∈	NOUN
ejpam-609	244	18	n(r	n(r	NOUN
ejpam-609	244	19	)	)	PUNCT
ejpam-609	244	20	.	.	PUNCT
ejpam-609	245	1	now	now	ADV
ejpam-609	245	2	xmam	xmam	PROPN
ejpam-609	245	3	∈	∈	PROPN
ejpam-609	245	4	o(n(r	o(n(r	NOUN
ejpam-609	245	5	)	)	PUNCT
ejpam-609	245	6	)	)	PUNCT
ejpam-609	246	1	⊆	⊆	NUM
ejpam-609	246	2	n(o(r	n(o(r	NUM
ejpam-609	246	3	)	)	PUNCT
ejpam-609	246	4	)	)	PUNCT
ejpam-609	246	5	implies	imply	VERB
ejpam-609	246	6	that	that	SCONJ
ejpam-609	246	7	∑m−1	∑m−1	PRON
ejpam-609	246	8	i=0	i=0	PROPN
ejpam-609	246	9	x	x	SYM
ejpam-609	246	10	iai	iai	PROPN
ejpam-609	246	11	∈	∈	PROPN
ejpam-609	246	12	n(o(r	n(o(r	PROPN
ejpam-609	246	13	)	)	PUNCT
ejpam-609	246	14	)	)	PUNCT
ejpam-609	246	15	,	,	PUNCT
ejpam-609	246	16	and	and	CCONJ
ejpam-609	246	17	with	with	ADP
ejpam-609	246	18	the	the	DET
ejpam-609	246	19	same	same	ADJ
ejpam-609	246	20	process	process	NOUN
ejpam-609	246	21	,	,	PUNCT
ejpam-609	246	22	in	in	ADP
ejpam-609	246	23	a	a	DET
ejpam-609	246	24	finite	finite	ADJ
ejpam-609	246	25	number	number	NOUN
ejpam-609	246	26	of	of	ADP
ejpam-609	246	27	steps	step	NOUN
ejpam-609	246	28	,	,	PUNCT
ejpam-609	246	29	it	it	PRON
ejpam-609	246	30	can	can	AUX
ejpam-609	246	31	be	be	AUX
ejpam-609	246	32	seen	see	VERB
ejpam-609	246	33	that	that	SCONJ
ejpam-609	246	34	ai	ai	VERB
ejpam-609	246	35	∈	∈	PROPN
ejpam-609	246	36	p(r	p(r	PROPN
ejpam-609	246	37	)	)	PUNCT
ejpam-609	246	38	=	=	SYM
ejpam-609	246	39	n(r	n(r	NOUN
ejpam-609	246	40	)	)	PUNCT
ejpam-609	246	41	,	,	PUNCT
ejpam-609	246	42	0	0	NUM
ejpam-609	246	43	≤	≤	NUM
ejpam-609	246	44	i	i	PRON
ejpam-609	246	45	≤	≤	ADJ
ejpam-609	246	46	m−	m−	PROPN
ejpam-609	246	47	1	1	NUM
ejpam-609	246	48	.	.	PUNCT
ejpam-609	247	1	therefore	therefore	ADV
ejpam-609	247	2	,	,	PUNCT
ejpam-609	247	3	f	f	PROPN
ejpam-609	247	4	∈	∈	PROPN
ejpam-609	247	5	o(n(r	o(n(r	NOUN
ejpam-609	247	6	)	)	PUNCT
ejpam-609	247	7	)	)	PUNCT
ejpam-609	247	8	.	.	PUNCT
ejpam-609	248	1	hence	hence	ADV
ejpam-609	248	2	n(o(r))⊆	n(o(r))⊆	ADV
ejpam-609	248	3	o(n(r	o(n(r	NOUN
ejpam-609	248	4	)	)	PUNCT
ejpam-609	248	5	)	)	PUNCT
ejpam-609	249	1	and	and	CCONJ
ejpam-609	249	2	the	the	DET
ejpam-609	249	3	result	result	NOUN
ejpam-609	249	4	follows	follow	VERB
ejpam-609	249	5	.	.	PUNCT
ejpam-609	250	1	as	as	SCONJ
ejpam-609	250	2	mentioned	mention	VERB
ejpam-609	250	3	earlier	early	ADV
ejpam-609	250	4	,	,	PUNCT
ejpam-609	250	5	we	we	PRON
ejpam-609	250	6	note	note	VERB
ejpam-609	250	7	that	that	SCONJ
ejpam-609	250	8	if	if	SCONJ
ejpam-609	250	9	σ	σ	PROPN
ejpam-609	250	10	is	be	AUX
ejpam-609	250	11	an	an	DET
ejpam-609	250	12	endomorphism	endomorphism	NOUN
ejpam-609	250	13	of	of	ADP
ejpam-609	250	14	a	a	DET
ejpam-609	250	15	ring	ring	NOUN
ejpam-609	250	16	r	r	NOUN
ejpam-609	250	17	and	and	CCONJ
ejpam-609	250	18	δ	δ	PROPN
ejpam-609	250	19	a	a	DET
ejpam-609	250	20	σderivation	σderivation	NOUN
ejpam-609	250	21	of	of	ADP
ejpam-609	250	22	r	r	NOUN
ejpam-609	250	23	such	such	ADJ
ejpam-609	250	24	that	that	DET
ejpam-609	250	25	σ(δ(a	σ(δ(a	NOUN
ejpam-609	250	26	)	)	PUNCT
ejpam-609	250	27	)	)	PUNCT
ejpam-609	251	1	=	=	PUNCT
ejpam-609	251	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	251	3	)	)	PUNCT
ejpam-609	251	4	)	)	PUNCT
ejpam-609	251	5	for	for	ADP
ejpam-609	251	6	all	all	DET
ejpam-609	251	7	a	a	DET
ejpam-609	251	8	∈	∈	PROPN
ejpam-609	251	9	r.	r.	NOUN
ejpam-609	251	10	then	then	ADV
ejpam-609	251	11	σ	σ	PROPN
ejpam-609	251	12	can	can	AUX
ejpam-609	251	13	be	be	AUX
ejpam-609	251	14	extended	extend	VERB
ejpam-609	251	15	to	to	ADP
ejpam-609	251	16	an	an	DET
ejpam-609	251	17	endomorphism	endomorphism	NOUN
ejpam-609	251	18	(	(	PUNCT
ejpam-609	251	19	say	say	INTJ
ejpam-609	251	20	σ	σ	NOUN
ejpam-609	251	21	)	)	PUNCT
ejpam-609	251	22	of	of	ADP
ejpam-609	251	23	r[x	r[x	NOUN
ejpam-609	251	24	;	;	PUNCT
ejpam-609	251	25	σ	σ	PROPN
ejpam-609	251	26	,	,	PUNCT
ejpam-609	251	27	δ	δ	PROPN
ejpam-609	251	28	]	]	PUNCT
ejpam-609	251	29	by	by	ADP
ejpam-609	251	30	σ	σ	PROPN
ejpam-609	251	31	(	(	PUNCT
ejpam-609	251	32	∑m	∑m	PROPN
ejpam-609	251	33	i=0	i=0	PROPN
ejpam-609	251	34	x	x	X
ejpam-609	251	35	iai	iai	ADJ
ejpam-609	251	36	)	)	PUNCT
ejpam-609	251	37	=	=	VERB
ejpam-609	251	38	∑m	∑m	PROPN
ejpam-609	251	39	i=0	i=0	PROPN
ejpam-609	251	40	x	x	SYM
ejpam-609	251	41	iσ(ai	iσ(ai	PROPN
ejpam-609	251	42	)	)	PUNCT
ejpam-609	251	43	.	.	PUNCT
ejpam-609	252	1	also	also	ADV
ejpam-609	252	2	δ	δ	PROPN
ejpam-609	252	3	can	can	AUX
ejpam-609	252	4	be	be	AUX
ejpam-609	252	5	extended	extend	VERB
ejpam-609	252	6	to	to	ADP
ejpam-609	252	7	a	a	DET
ejpam-609	252	8	σ	σ	NOUN
ejpam-609	252	9	derivation	derivation	NOUN
ejpam-609	252	10	(	(	PUNCT
ejpam-609	252	11	say	say	VERB
ejpam-609	252	12	δ	δ	PROPN
ejpam-609	252	13	)	)	PUNCT
ejpam-609	252	14	of	of	ADP
ejpam-609	252	15	r[x	r[x	NOUN
ejpam-609	252	16	;	;	PUNCT
ejpam-609	252	17	σ	σ	PROPN
ejpam-609	252	18	,	,	PUNCT
ejpam-609	252	19	δ	δ	PROPN
ejpam-609	252	20	]	]	PUNCT
ejpam-609	252	21	by	by	ADP
ejpam-609	252	22	δ	δ	PROPN
ejpam-609	252	23	(	(	PUNCT
ejpam-609	252	24	∑m	∑m	PROPN
ejpam-609	252	25	i=0	i=0	PROPN
ejpam-609	252	26	x	x	X
ejpam-609	252	27	iai	iai	ADJ
ejpam-609	252	28	)	)	PUNCT
ejpam-609	253	1	=	=	PUNCT
ejpam-609	253	2	∑m	∑m	PROPN
ejpam-609	253	3	i=0	i=0	PROPN
ejpam-609	253	4	x	x	SYM
ejpam-609	253	5	iδ(ai	iδ(ai	PROPN
ejpam-609	253	6	)	)	PUNCT
ejpam-609	253	7	.	.	PUNCT
ejpam-609	254	1	we	we	PRON
ejpam-609	254	2	now	now	ADV
ejpam-609	254	3	prove	prove	VERB
ejpam-609	254	4	the	the	DET
ejpam-609	254	5	following	following	NOUN
ejpam-609	254	6	:	:	PUNCT
ejpam-609	254	7	theorem	theorem	NOUN
ejpam-609	254	8	7	7	NUM
ejpam-609	254	9	.	.	PUNCT
ejpam-609	255	1	let	let	VERB
ejpam-609	255	2	r	r	PRON
ejpam-609	255	3	be	be	AUX
ejpam-609	255	4	a	a	DET
ejpam-609	255	5	2	2	NUM
ejpam-609	255	6	-	-	PUNCT
ejpam-609	255	7	primal	primal	ADJ
ejpam-609	255	8	commutative	commutative	ADJ
ejpam-609	255	9	noetherian	noetherian	ADJ
ejpam-609	255	10	ring	ring	NOUN
ejpam-609	255	11	.	.	PUNCT
ejpam-609	256	1	let	let	VERB
ejpam-609	256	2	σ	σ	NOUN
ejpam-609	256	3	be	be	AUX
ejpam-609	256	4	an	an	DET
ejpam-609	256	5	automorphism	automorphism	NOUN
ejpam-609	256	6	of	of	ADP
ejpam-609	256	7	r	r	NOUN
ejpam-609	256	8	and	and	CCONJ
ejpam-609	256	9	δ	δ	PROPN
ejpam-609	256	10	a	a	DET
ejpam-609	256	11	σ	σ	NOUN
ejpam-609	256	12	-	-	PUNCT
ejpam-609	256	13	derivation	derivation	NOUN
ejpam-609	256	14	of	of	ADP
ejpam-609	256	15	r	r	NOUN
ejpam-609	256	16	such	such	ADJ
ejpam-609	256	17	that	that	DET
ejpam-609	256	18	σ(δ(a	σ(δ(a	NOUN
ejpam-609	256	19	)	)	PUNCT
ejpam-609	256	20	)	)	PUNCT
ejpam-609	257	1	=	=	PUNCT
ejpam-609	257	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	257	3	)	)	PUNCT
ejpam-609	257	4	)	)	PUNCT
ejpam-609	257	5	for	for	ADP
ejpam-609	257	6	all	all	DET
ejpam-609	257	7	a	a	DET
ejpam-609	257	8	∈	∈	PROPN
ejpam-609	257	9	r.	r.	NOUN
ejpam-609	257	10	then	then	ADV
ejpam-609	257	11	r	r	NOUN
ejpam-609	257	12	is	be	AUX
ejpam-609	257	13	a	a	DET
ejpam-609	257	14	weak	weak	ADJ
ejpam-609	257	15	σ	σ	ADJ
ejpam-609	257	16	-	-	ADJ
ejpam-609	257	17	rigid	rigid	ADJ
ejpam-609	257	18	ring	ring	NOUN
ejpam-609	257	19	implies	imply	VERB
ejpam-609	257	20	that	that	SCONJ
ejpam-609	257	21	o(r	o(r	NOUN
ejpam-609	257	22	)	)	PUNCT
ejpam-609	257	23	=	=	SYM
ejpam-609	257	24	r[x	r[x	NOUN
ejpam-609	257	25	;	;	PUNCT
ejpam-609	257	26	σ	σ	PROPN
ejpam-609	257	27	,	,	PUNCT
ejpam-609	257	28	δ	δ	PROPN
ejpam-609	257	29	]	]	PUNCT
ejpam-609	257	30	is	be	AUX
ejpam-609	257	31	a	a	DET
ejpam-609	257	32	weak	weak	ADJ
ejpam-609	257	33	σ	σ	ADJ
ejpam-609	257	34	-	-	ADJ
ejpam-609	257	35	rigid	rigid	ADJ
ejpam-609	257	36	ring	ring	NOUN
ejpam-609	257	37	.	.	PUNCT
ejpam-609	258	1	proof	proof	NOUN
ejpam-609	258	2	.	.	PUNCT
ejpam-609	259	1	let	let	VERB
ejpam-609	259	2	r	r	PRON
ejpam-609	259	3	be	be	AUX
ejpam-609	259	4	a	a	DET
ejpam-609	259	5	weak	weak	ADJ
ejpam-609	259	6	σ	σ	ADJ
ejpam-609	259	7	-	-	ADJ
ejpam-609	259	8	rigid	rigid	ADJ
ejpam-609	259	9	ring	ring	NOUN
ejpam-609	259	10	.	.	PUNCT
ejpam-609	260	1	then	then	ADV
ejpam-609	260	2	theorem	theorem	ADJ
ejpam-609	260	3	(	(	PUNCT
ejpam-609	260	4	5	5	NUM
ejpam-609	260	5	)	)	PUNCT
ejpam-609	260	6	implies	imply	VERB
ejpam-609	260	7	that	that	SCONJ
ejpam-609	260	8	r	r	NOUN
ejpam-609	260	9	is	be	AUX
ejpam-609	260	10	a	a	DET
ejpam-609	260	11	σ(∗)-ring	σ(∗)-ring	NOUN
ejpam-609	260	12	.	.	PUNCT
ejpam-609	261	1	also	also	ADV
ejpam-609	261	2	proposition	proposition	NOUN
ejpam-609	261	3	(	(	PUNCT
ejpam-609	261	4	5	5	NUM
ejpam-609	261	5	)	)	PUNCT
ejpam-609	261	6	implies	imply	VERB
ejpam-609	261	7	that	that	SCONJ
ejpam-609	261	8	o(n(r	o(n(r	NOUN
ejpam-609	261	9	)	)	PUNCT
ejpam-609	261	10	)	)	PUNCT
ejpam-609	262	1	=	=	PUNCT
ejpam-609	262	2	n(o(r	n(o(r	ADJ
ejpam-609	262	3	)	)	PUNCT
ejpam-609	262	4	)	)	PUNCT
ejpam-609	262	5	.	.	PUNCT
ejpam-609	263	1	we	we	PRON
ejpam-609	263	2	show	show	VERB
ejpam-609	263	3	that	that	DET
ejpam-609	263	4	r[x	r[x	NOUN
ejpam-609	263	5	;	;	PUNCT
ejpam-609	263	6	σ	σ	PROPN
ejpam-609	263	7	,	,	PUNCT
ejpam-609	263	8	δ	δ	PROPN
ejpam-609	263	9	]	]	PUNCT
ejpam-609	263	10	is	be	AUX
ejpam-609	263	11	a	a	DET
ejpam-609	263	12	weak	weak	ADJ
ejpam-609	263	13	σ	σ	ADJ
ejpam-609	263	14	-	-	ADJ
ejpam-609	263	15	rigid	rigid	ADJ
ejpam-609	263	16	ring	ring	NOUN
ejpam-609	263	17	.	.	PUNCT
ejpam-609	264	1	let	let	VERB
ejpam-609	264	2	f	f	PROPN
ejpam-609	264	3	∈	∈	PROPN
ejpam-609	264	4	o(r	o(r	PROPN
ejpam-609	264	5	)	)	PUNCT
ejpam-609	264	6	(	(	PUNCT
ejpam-609	264	7	say	say	VERB
ejpam-609	264	8	f	f	PROPN
ejpam-609	264	9	=	=	SYM
ejpam-609	264	10	∑m	∑m	PROPN
ejpam-609	264	11	i=0	i=0	PROPN
ejpam-609	264	12	x	x	SYM
ejpam-609	264	13	iai	iai	NOUN
ejpam-609	264	14	)	)	PUNCT
ejpam-609	264	15	be	be	VERB
ejpam-609	264	16	such	such	ADJ
ejpam-609	264	17	that	that	SCONJ
ejpam-609	264	18	f	f	PROPN
ejpam-609	264	19	σ	σ	PROPN
ejpam-609	264	20	(	(	PUNCT
ejpam-609	264	21	f	f	PROPN
ejpam-609	264	22	)	)	PUNCT
ejpam-609	264	23	∈	∈	PROPN
ejpam-609	264	24	n(o(r	n(o(r	PROPN
ejpam-609	264	25	)	)	PUNCT
ejpam-609	264	26	)	)	PUNCT
ejpam-609	264	27	.	.	PUNCT
ejpam-609	265	1	we	we	PRON
ejpam-609	265	2	use	use	VERB
ejpam-609	265	3	induction	induction	NOUN
ejpam-609	265	4	on	on	ADP
ejpam-609	265	5	m	m	NOUN
ejpam-609	265	6	to	to	PART
ejpam-609	265	7	prove	prove	VERB
ejpam-609	265	8	the	the	DET
ejpam-609	265	9	result	result	NOUN
ejpam-609	265	10	.	.	PUNCT
ejpam-609	266	1	for	for	ADP
ejpam-609	266	2	m	m	PROPN
ejpam-609	266	3	=	=	SYM
ejpam-609	266	4	1	1	NUM
ejpam-609	266	5	,	,	PUNCT
ejpam-609	266	6	f	f	PROPN
ejpam-609	266	7	=	=	PUNCT
ejpam-609	266	8	xa1	xa1	PROPN
ejpam-609	266	9	+	+	NUM
ejpam-609	266	10	a0	a0	PROPN
ejpam-609	266	11	.	.	PUNCT
ejpam-609	267	1	now	now	ADV
ejpam-609	267	2	f	f	PROPN
ejpam-609	267	3	σ	σ	PROPN
ejpam-609	267	4	(	(	PUNCT
ejpam-609	267	5	f	f	PROPN
ejpam-609	267	6	)	)	PUNCT
ejpam-609	267	7	∈	∈	PROPN
ejpam-609	267	8	n(o(r	n(o(r	PROPN
ejpam-609	267	9	)	)	PUNCT
ejpam-609	267	10	)	)	PUNCT
ejpam-609	267	11	implies	imply	VERB
ejpam-609	267	12	that	that	SCONJ
ejpam-609	267	13	(	(	PUNCT
ejpam-609	267	14	xa1	xa1	PROPN
ejpam-609	267	15	+	+	PROPN
ejpam-609	267	16	a0)(xσ(a1	a0)(xσ(a1	X
ejpam-609	267	17	)	)	PUNCT
ejpam-609	268	1	+	+	NOUN
ejpam-609	268	2	σ(a0	σ(a0	X
ejpam-609	268	3	)	)	PUNCT
ejpam-609	268	4	)	)	PUNCT
ejpam-609	269	1	∈	∈	PROPN
ejpam-609	269	2	n(o(r	n(o(r	PROPN
ejpam-609	269	3	)	)	PUNCT
ejpam-609	269	4	)	)	PUNCT
ejpam-609	270	1	=	=	SYM
ejpam-609	270	2	o(n(r	o(n(r	NOUN
ejpam-609	270	3	)	)	PUNCT
ejpam-609	270	4	)	)	PUNCT
ejpam-609	270	5	,	,	PUNCT
ejpam-609	270	6	i.e.	i.e.	X
ejpam-609	270	7	x2σ2(a1	x2σ2(a1	X
ejpam-609	270	8	)	)	PUNCT
ejpam-609	270	9	+	+	NUM
ejpam-609	270	10	xδ(a1)σ(a1	xδ(a1)σ(a1	NOUN
ejpam-609	270	11	)	)	PUNCT
ejpam-609	271	1	+	+	CCONJ
ejpam-609	271	2	xσ(a0)σ(a1	xσ(a0)σ(a1	X
ejpam-609	271	3	)	)	PUNCT
ejpam-609	271	4	references	reference	NOUN
ejpam-609	271	5	702	702	NUM
ejpam-609	271	6	+	+	NOUN
ejpam-609	271	7	δ(a0)σ(a1	δ(a0)σ(a1	ADJ
ejpam-609	271	8	)	)	PUNCT
ejpam-609	271	9	+	+	CCONJ
ejpam-609	271	10	xa1σ(a0	xa1σ(a0	X
ejpam-609	271	11	)	)	PUNCT
ejpam-609	271	12	+	+	CCONJ
ejpam-609	271	13	a0σ(a0	a0σ(a0	X
ejpam-609	271	14	)	)	PUNCT
ejpam-609	271	15	∈	∈	PROPN
ejpam-609	271	16	o(n(r	o(n(r	NOUN
ejpam-609	271	17	)	)	PUNCT
ejpam-609	271	18	)	)	PUNCT
ejpam-609	272	1	(	(	PUNCT
ejpam-609	272	2	1	1	X
ejpam-609	272	3	)	)	PUNCT
ejpam-609	272	4	therefore	therefore	ADV
ejpam-609	272	5	,	,	PUNCT
ejpam-609	272	6	σ2(a1	σ2(a1	ADJ
ejpam-609	272	7	)	)	PUNCT
ejpam-609	272	8	∈	∈	PROPN
ejpam-609	272	9	n(r	n(r	NOUN
ejpam-609	272	10	)	)	PUNCT
ejpam-609	272	11	.	.	PUNCT
ejpam-609	273	1	now	now	ADV
ejpam-609	273	2	σ(n(r	σ(n(r	NUM
ejpam-609	273	3	)	)	PUNCT
ejpam-609	273	4	)	)	PUNCT
ejpam-609	274	1	=	=	SYM
ejpam-609	274	2	n(r	n(r	NOUN
ejpam-609	274	3	)	)	PUNCT
ejpam-609	274	4	implies	imply	VERB
ejpam-609	274	5	that	that	SCONJ
ejpam-609	274	6	σi(a1	σi(a1	NOUN
ejpam-609	274	7	)	)	PUNCT
ejpam-609	274	8	∈	∈	NOUN
ejpam-609	274	9	n(r	n(r	NOUN
ejpam-609	274	10	)	)	PUNCT
ejpam-609	274	11	for	for	ADP
ejpam-609	274	12	all	all	PRON
ejpam-609	274	13	i	i	PRON
ejpam-609	274	14	≥	≥	VERB
ejpam-609	274	15	1	1	NUM
ejpam-609	274	16	.	.	PUNCT
ejpam-609	275	1	so	so	ADV
ejpam-609	275	2	(	(	PUNCT
ejpam-609	275	3	1	1	X
ejpam-609	275	4	)	)	PUNCT
ejpam-609	275	5	implies	imply	VERB
ejpam-609	275	6	that	that	SCONJ
ejpam-609	275	7	a0σ(a0	a0σ(a0	VERB
ejpam-609	275	8	)	)	PUNCT
ejpam-609	275	9	∈	∈	PROPN
ejpam-609	275	10	n(r	n(r	NOUN
ejpam-609	275	11	)	)	PUNCT
ejpam-609	275	12	,	,	PUNCT
ejpam-609	275	13	and	and	CCONJ
ejpam-609	275	14	r	r	NOUN
ejpam-609	275	15	being	be	AUX
ejpam-609	275	16	a	a	DET
ejpam-609	275	17	weak	weak	ADJ
ejpam-609	275	18	σ	σ	ADJ
ejpam-609	275	19	-	-	ADJ
ejpam-609	275	20	rigid	rigid	ADJ
ejpam-609	275	21	ring	ring	NOUN
ejpam-609	275	22	implies	imply	VERB
ejpam-609	275	23	that	that	SCONJ
ejpam-609	275	24	a0	a0	PROPN
ejpam-609	275	25	∈	∈	PROPN
ejpam-609	275	26	n(r	n(r	PROPN
ejpam-609	275	27	)	)	PUNCT
ejpam-609	275	28	.	.	PUNCT
ejpam-609	276	1	therefore	therefore	ADV
ejpam-609	276	2	,	,	PUNCT
ejpam-609	276	3	f	f	PROPN
ejpam-609	276	4	∈	∈	PROPN
ejpam-609	276	5	o(n(r	o(n(r	NOUN
ejpam-609	276	6	)	)	PUNCT
ejpam-609	276	7	)	)	PUNCT
ejpam-609	277	1	=	=	PUNCT
ejpam-609	277	2	n(o(r	n(o(r	ADJ
ejpam-609	277	3	)	)	PUNCT
ejpam-609	277	4	)	)	PUNCT
ejpam-609	277	5	.	.	PUNCT
ejpam-609	278	1	suppose	suppose	VERB
ejpam-609	278	2	the	the	DET
ejpam-609	278	3	result	result	NOUN
ejpam-609	278	4	is	be	AUX
ejpam-609	278	5	true	true	ADJ
ejpam-609	278	6	for	for	ADP
ejpam-609	278	7	m	m	PROPN
ejpam-609	279	1	=	=	PUNCT
ejpam-609	279	2	k.	k.	NOUN
ejpam-609	279	3	we	we	PRON
ejpam-609	279	4	prove	prove	VERB
ejpam-609	279	5	for	for	ADP
ejpam-609	279	6	m	m	PROPN
ejpam-609	280	1	=	=	SYM
ejpam-609	280	2	k	k	PROPN
ejpam-609	281	1	+	+	NOUN
ejpam-609	281	2	1	1	X
ejpam-609	281	3	.	.	PUNCT
ejpam-609	281	4	now	now	ADV
ejpam-609	281	5	f	f	PROPN
ejpam-609	281	6	σ	σ	PROPN
ejpam-609	281	7	(	(	PUNCT
ejpam-609	281	8	f	f	PROPN
ejpam-609	281	9	)	)	PUNCT
ejpam-609	281	10	∈	∈	PROPN
ejpam-609	281	11	n(o(r	n(o(r	PROPN
ejpam-609	281	12	)	)	PUNCT
ejpam-609	281	13	)	)	PUNCT
ejpam-609	281	14	implies	imply	VERB
ejpam-609	281	15	that	that	SCONJ
ejpam-609	281	16	(	(	PUNCT
ejpam-609	281	17	x	x	X
ejpam-609	281	18	k+1ak+1	k+1ak+1	NOUN
ejpam-609	281	19	+	+	NUM
ejpam-609	281	20	...	...	PUNCT
ejpam-609	282	1	+	+	CCONJ
ejpam-609	282	2	a0)(x	a0)(x	NUM
ejpam-609	282	3	k+1σ(ak+1	k+1σ(ak+1	NOUN
ejpam-609	282	4	)	)	PUNCT
ejpam-609	283	1	+	+	CCONJ
ejpam-609	283	2	...	...	PUNCT
ejpam-609	284	1	+	+	ADJ
ejpam-609	284	2	σ(a0	σ(a0	ADJ
ejpam-609	284	3	)	)	PUNCT
ejpam-609	284	4	)	)	PUNCT
ejpam-609	285	1	∈	∈	PROPN
ejpam-609	285	2	n(o(r	n(o(r	PROPN
ejpam-609	285	3	)	)	PUNCT
ejpam-609	285	4	)	)	PUNCT
ejpam-609	286	1	=	=	SYM
ejpam-609	286	2	o(n(r	o(n(r	NOUN
ejpam-609	286	3	)	)	PUNCT
ejpam-609	286	4	)	)	PUNCT
ejpam-609	286	5	,	,	PUNCT
ejpam-609	286	6	i.e.	i.e.	X
ejpam-609	286	7	x2k+2σk+2(ak+1	x2k+2σk+2(ak+1	X
ejpam-609	286	8	)	)	PUNCT
ejpam-609	286	9	+	+	CCONJ
ejpam-609	286	10	x2k+1(σk(ak+1)σ(ak	x2k+1(σk(ak+1)σ(ak	X
ejpam-609	286	11	)	)	PUNCT
ejpam-609	287	1	+	+	NOUN
ejpam-609	287	2	σ	σ	X
ejpam-609	287	3	k+1(ak)σ(ak+1))+	k+1(ak)σ(ak+1))+	NOUN
ejpam-609	287	4	gσ(g	gσ(g	NOUN
ejpam-609	287	5	)	)	PUNCT
ejpam-609	287	6	∈	∈	PROPN
ejpam-609	287	7	o(n(r	o(n(r	NOUN
ejpam-609	287	8	)	)	PUNCT
ejpam-609	287	9	)	)	PUNCT
ejpam-609	287	10	,	,	PUNCT
ejpam-609	287	11	where	where	SCONJ
ejpam-609	287	12	g	g	PROPN
ejpam-609	287	13	=	=	PROPN
ejpam-609	287	14	∑k	∑k	PROPN
ejpam-609	287	15	i=0	i=0	PROPN
ejpam-609	287	16	x	x	SYM
ejpam-609	287	17	iai	iai	PROPN
ejpam-609	287	18	.	.	PUNCT
ejpam-609	288	1	therefore	therefore	ADV
ejpam-609	288	2	,	,	PUNCT
ejpam-609	288	3	σk+2(ak+1	σk+2(ak+1	X
ejpam-609	288	4	)	)	PUNCT
ejpam-609	288	5	∈	∈	PROPN
ejpam-609	288	6	n(r	n(r	NOUN
ejpam-609	288	7	)	)	PUNCT
ejpam-609	288	8	implies	imply	VERB
ejpam-609	288	9	that	that	SCONJ
ejpam-609	288	10	ak+1	ak+1	VERB
ejpam-609	288	11	∈	∈	PROPN
ejpam-609	288	12	n(r	n(r	NOUN
ejpam-609	288	13	)	)	PUNCT
ejpam-609	288	14	.	.	PUNCT
ejpam-609	289	1	also	also	ADV
ejpam-609	289	2	σk(ak+1)σ(ak	σk(ak+1)σ(ak	PROPN
ejpam-609	289	3	)	)	PUNCT
ejpam-609	289	4	+	+	PROPN
ejpam-609	289	5	σ	σ	PROPN
ejpam-609	289	6	k+1(ak)σ(ak+1	k+1(ak)σ(ak+1	NOUN
ejpam-609	289	7	)	)	PUNCT
ejpam-609	289	8	∈	∈	PROPN
ejpam-609	289	9	n(r	n(r	NOUN
ejpam-609	289	10	)	)	PUNCT
ejpam-609	289	11	implies	imply	VERB
ejpam-609	289	12	that	that	SCONJ
ejpam-609	289	13	gσ(g	gσ(g	X
ejpam-609	289	14	)	)	PUNCT
ejpam-609	289	15	∈	∈	PROPN
ejpam-609	289	16	n(o(r	n(o(r	PROPN
ejpam-609	289	17	)	)	PUNCT
ejpam-609	289	18	)	)	PUNCT
ejpam-609	289	19	,	,	PUNCT
ejpam-609	289	20	but	but	CCONJ
ejpam-609	289	21	degree	degree	NOUN
ejpam-609	289	22	of	of	ADP
ejpam-609	289	23	g	g	PROPN
ejpam-609	289	24	is	be	AUX
ejpam-609	289	25	k	k	PROPN
ejpam-609	289	26	,	,	PUNCT
ejpam-609	289	27	therefore	therefore	ADV
ejpam-609	289	28	,	,	PUNCT
ejpam-609	289	29	by	by	ADP
ejpam-609	289	30	induction	induction	NOUN
ejpam-609	289	31	hypothesis	hypothesis	NOUN
ejpam-609	289	32	,	,	PUNCT
ejpam-609	289	33	the	the	DET
ejpam-609	289	34	result	result	NOUN
ejpam-609	289	35	is	be	AUX
ejpam-609	289	36	true	true	ADJ
ejpam-609	289	37	for	for	ADP
ejpam-609	289	38	all	all	DET
ejpam-609	289	39	m.	m.	NOUN
ejpam-609	289	40	question	question	NOUN
ejpam-609	289	41	:	:	PUNCT
ejpam-609	289	42	let	let	VERB
ejpam-609	289	43	r	r	PRON
ejpam-609	289	44	be	be	AUX
ejpam-609	289	45	a	a	DET
ejpam-609	289	46	commutative	commutative	ADJ
ejpam-609	289	47	noetherian	noetherian	ADJ
ejpam-609	289	48	ring	ring	NOUN
ejpam-609	289	49	.	.	PUNCT
ejpam-609	290	1	let	let	VERB
ejpam-609	290	2	σ	σ	NOUN
ejpam-609	290	3	be	be	AUX
ejpam-609	290	4	an	an	DET
ejpam-609	290	5	automorphism	automorphism	NOUN
ejpam-609	290	6	of	of	ADP
ejpam-609	290	7	r	r	NOUN
ejpam-609	290	8	and	and	CCONJ
ejpam-609	290	9	δ	δ	PROPN
ejpam-609	290	10	a	a	DET
ejpam-609	290	11	σ	σ	NOUN
ejpam-609	290	12	-	-	PUNCT
ejpam-609	290	13	derivation	derivation	NOUN
ejpam-609	290	14	of	of	ADP
ejpam-609	290	15	r	r	NOUN
ejpam-609	290	16	such	such	ADJ
ejpam-609	290	17	that	that	DET
ejpam-609	290	18	σ(δ(a	σ(δ(a	NOUN
ejpam-609	290	19	)	)	PUNCT
ejpam-609	290	20	)	)	PUNCT
ejpam-609	291	1	=	=	PUNCT
ejpam-609	291	2	δ(σ(a	δ(σ(a	NOUN
ejpam-609	291	3	)	)	PUNCT
ejpam-609	291	4	)	)	PUNCT
ejpam-609	291	5	for	for	ADP
ejpam-609	291	6	all	all	DET
ejpam-609	291	7	a	a	DET
ejpam-609	291	8	∈	∈	PROPN
ejpam-609	291	9	r.	r.	NOUN
ejpam-609	291	10	let	let	VERB
ejpam-609	291	11	r	r	PRON
ejpam-609	291	12	be	be	AUX
ejpam-609	291	13	a	a	DET
ejpam-609	291	14	weak	weak	ADJ
ejpam-609	291	15	σ	σ	ADJ
ejpam-609	291	16	-	-	ADJ
ejpam-609	291	17	rigid	rigid	ADJ
ejpam-609	291	18	ring	ring	NOUN
ejpam-609	291	19	.	.	PUNCT
ejpam-609	292	1	is	be	AUX
ejpam-609	292	2	o(r	o(r	PROPN
ejpam-609	292	3	)	)	PUNCT
ejpam-609	293	1	=	=	SYM
ejpam-609	293	2	r[x	r[x	NOUN
ejpam-609	293	3	;	;	PUNCT
ejpam-609	293	4	σ	σ	PROPN
ejpam-609	293	5	,	,	PUNCT
ejpam-609	293	6	δ	δ	PROPN
ejpam-609	293	7	]	]	X
ejpam-609	293	8	a	a	DET
ejpam-609	293	9	weak	weak	ADJ
ejpam-609	293	10	σ	σ	ADJ
ejpam-609	293	11	-	-	ADJ
ejpam-609	293	12	rigid	rigid	ADJ
ejpam-609	293	13	ring	ring	NOUN
ejpam-609	293	14	?	?	PUNCT
ejpam-609	294	1	references	reference	NOUN
ejpam-609	294	2	[	[	X
ejpam-609	294	3	1	1	X
ejpam-609	294	4	]	]	PUNCT
ejpam-609	294	5	s.	s.	PROPN
ejpam-609	294	6	annin	annin	PROPN
ejpam-609	294	7	,	,	PUNCT
ejpam-609	294	8	associated	associate	VERB
ejpam-609	294	9	primes	prime	NOUN
ejpam-609	294	10	over	over	ADP
ejpam-609	294	11	skew	skew	ADJ
ejpam-609	294	12	polynomial	polynomial	ADJ
ejpam-609	294	13	rings	ring	NOUN
ejpam-609	294	14	,	,	PUNCT
ejpam-609	294	15	comm	comm	NOUN
ejpam-609	294	16	.	.	PUNCT
ejpam-609	295	1	algebra	algebra	NOUN
ejpam-609	295	2	,	,	PUNCT
ejpam-609	295	3	vol	vol	NOUN
ejpam-609	295	4	.	.	PUNCT
ejpam-609	295	5	30(5	30(5	NUM
ejpam-609	295	6	)	)	PUNCT
ejpam-609	295	7	,	,	PUNCT
ejpam-609	295	8	2511	2511	NUM
ejpam-609	295	9	-	-	SYM
ejpam-609	295	10	2528	2528	NUM
ejpam-609	295	11	.	.	PUNCT
ejpam-609	296	1	mr1940490	mr1940490	PROPN
ejpam-609	296	2	(	(	PUNCT
ejpam-609	296	3	2003k:16037	2003k:16037	NUM
ejpam-609	296	4	)	)	PUNCT
ejpam-609	296	5	.	.	PUNCT
ejpam-609	297	1	2002	2002	NUM
ejpam-609	298	1	[	[	X
ejpam-609	298	2	2	2	X
ejpam-609	298	3	]	]	PUNCT
ejpam-609	298	4	v.	v.	PROPN
ejpam-609	298	5	k.	k.	PROPN
ejpam-609	298	6	bhat	bhat	PROPN
ejpam-609	298	7	,	,	PUNCT
ejpam-609	298	8	on	on	ADP
ejpam-609	298	9	2	2	NUM
ejpam-609	298	10	-	-	PUNCT
ejpam-609	298	11	primal	primal	ADJ
ejpam-609	298	12	ore	ore	NOUN
ejpam-609	298	13	extensions	extension	NOUN
ejpam-609	298	14	,	,	PUNCT
ejpam-609	298	15	ukrainian	ukrainian	ADJ
ejpam-609	298	16	math	math	NOUN
ejpam-609	298	17	.	.	PUNCT
ejpam-609	299	1	bulletin	bulletin	NOUN
ejpam-609	299	2	,	,	PUNCT
ejpam-609	299	3	vol	vol	NOUN
ejpam-609	299	4	.	.	PROPN
ejpam-609	299	5	4	4	NUM
ejpam-609	299	6	,	,	PUNCT
ejpam-609	299	7	173	173	NUM
ejpam-609	299	8	-	-	SYM
ejpam-609	299	9	179	179	NUM
ejpam-609	299	10	.	.	PUNCT
ejpam-609	300	1	mr2503886	mr2503886	PROPN
ejpam-609	300	2	(	(	PUNCT
ejpam-609	300	3	2010b:16082	2010b:16082	NUM
ejpam-609	300	4	)	)	PUNCT
ejpam-609	300	5	.	.	PUNCT
ejpam-609	301	1	2007	2007	NUM
ejpam-609	301	2	.	.	PUNCT
ejpam-609	302	1	[	[	X
ejpam-609	302	2	3	3	X
ejpam-609	302	3	]	]	PUNCT
ejpam-609	302	4	v.	v.	PROPN
ejpam-609	302	5	k.	k.	PROPN
ejpam-609	302	6	bhat	bhat	PROPN
ejpam-609	302	7	,	,	PUNCT
ejpam-609	302	8	associated	associate	VERB
ejpam-609	302	9	prime	prime	ADJ
ejpam-609	302	10	ideals	ideal	NOUN
ejpam-609	302	11	of	of	ADP
ejpam-609	302	12	skew	skew	ADJ
ejpam-609	302	13	polynomial	polynomial	ADJ
ejpam-609	302	14	rings	ring	NOUN
ejpam-609	302	15	,	,	PUNCT
ejpam-609	302	16	beiträge	beiträge	ADJ
ejpam-609	302	17	algebra	algebra	PROPN
ejpam-609	302	18	geom	geom	PROPN
ejpam-609	302	19	.	.	PUNCT
ejpam-609	302	20	,	,	PUNCT
ejpam-609	302	21	vol	vol	NOUN
ejpam-609	302	22	.	.	PUNCT
ejpam-609	302	23	49(1	49(1	NUM
ejpam-609	302	24	)	)	PUNCT
ejpam-609	302	25	,	,	PUNCT
ejpam-609	302	26	277	277	NUM
ejpam-609	302	27	-	-	SYM
ejpam-609	302	28	283	283	NUM
ejpam-609	302	29	.	.	PUNCT
ejpam-609	303	1	mr2410584	mr2410584	NOUN
ejpam-609	303	2	(	(	PUNCT
ejpam-609	303	3	2009e:16046	2009e:16046	NUM
ejpam-609	303	4	)	)	PUNCT
ejpam-609	303	5	.	.	PUNCT
ejpam-609	304	1	2008	2008	NUM
ejpam-609	304	2	.	.	PUNCT
ejpam-609	305	1	[	[	X
ejpam-609	305	2	4	4	X
ejpam-609	305	3	]	]	PUNCT
ejpam-609	305	4	v.	v.	PROPN
ejpam-609	305	5	k.	k.	PROPN
ejpam-609	305	6	bhat	bhat	PROPN
ejpam-609	305	7	and	and	CCONJ
ejpam-609	305	8	neetu	neetu	PROPN
ejpam-609	305	9	kumari	kumari	PROPN
ejpam-609	305	10	,	,	PUNCT
ejpam-609	305	11	transparency	transparency	NOUN
ejpam-609	305	12	of	of	ADP
ejpam-609	305	13	σ(∗)-rings	σ(∗)-ring	NOUN
ejpam-609	305	14	and	and	CCONJ
ejpam-609	305	15	their	their	PRON
ejpam-609	305	16	extensions	extension	NOUN
ejpam-609	305	17	,	,	PUNCT
ejpam-609	305	18	int	int	NOUN
ejpam-609	305	19	.	.	PUNCT
ejpam-609	306	1	j.	j.	PROPN
ejpam-609	306	2	algebra	algebra	PROPN
ejpam-609	306	3	,	,	PUNCT
ejpam-609	306	4	vol	vol	NOUN
ejpam-609	306	5	.	.	PUNCT
ejpam-609	306	6	2(19	2(19	NUM
ejpam-609	306	7	)	)	PUNCT
ejpam-609	306	8	,	,	PUNCT
ejpam-609	306	9	919	919	NUM
ejpam-609	306	10	-	-	SYM
ejpam-609	306	11	924	924	NUM
ejpam-609	306	12	.	.	PUNCT
ejpam-609	307	1	mr2481211	mr2481211	PROPN
ejpam-609	307	2	(	(	PUNCT
ejpam-609	307	3	2010d:16032	2010d:16032	NUM
ejpam-609	307	4	)	)	PUNCT
ejpam-609	307	5	.	.	PUNCT
ejpam-609	307	6	2008	2008	NUM
ejpam-609	307	7	.	.	PUNCT
ejpam-609	308	1	[	[	X
ejpam-609	308	2	5	5	X
ejpam-609	308	3	]	]	PUNCT
ejpam-609	308	4	v.	v.	PROPN
ejpam-609	308	5	k.	k.	PROPN
ejpam-609	308	6	bhat	bhat	PROPN
ejpam-609	308	7	and	and	CCONJ
ejpam-609	308	8	ravi	ravi	PROPN
ejpam-609	308	9	raina	raina	PROPN
ejpam-609	308	10	,	,	PUNCT
ejpam-609	308	11	ore	ore	NOUN
ejpam-609	308	12	extensions	extension	NOUN
ejpam-609	308	13	over	over	ADP
ejpam-609	308	14	2	2	NUM
ejpam-609	308	15	-	-	PUNCT
ejpam-609	308	16	primal	primal	ADJ
ejpam-609	308	17	rings	ring	NOUN
ejpam-609	308	18	,	,	PUNCT
ejpam-609	308	19	vietnam	vietnam	PROPN
ejpam-609	308	20	j.	j.	PROPN
ejpam-609	308	21	math	math	PROPN
ejpam-609	308	22	.	.	PUNCT
ejpam-609	308	23	,	,	PUNCT
ejpam-609	308	24	vol	vol	NOUN
ejpam-609	308	25	.	.	PROPN
ejpam-609	308	26	36(4	36(4	NUM
ejpam-609	308	27	)	)	PUNCT
ejpam-609	308	28	,	,	PUNCT
ejpam-609	308	29	455	455	NUM
ejpam-609	308	30	-	-	SYM
ejpam-609	308	31	461	461	NUM
ejpam-609	308	32	.	.	PUNCT
ejpam-609	309	1	mr2522617	mr2522617	PROPN
ejpam-609	309	2	(	(	PUNCT
ejpam-609	309	3	2010h:16064	2010h:16064	NUM
ejpam-609	309	4	)	)	PUNCT
ejpam-609	309	5	.	.	PUNCT
ejpam-609	310	1	2008	2008	NUM
ejpam-609	310	2	.	.	PUNCT
ejpam-609	311	1	[	[	X
ejpam-609	311	2	6	6	NUM
ejpam-609	311	3	]	]	PUNCT
ejpam-609	311	4	w.	w.	PROPN
ejpam-609	311	5	d.	d.	PROPN
ejpam-609	311	6	blair	blair	PROPN
ejpam-609	311	7	,	,	PUNCT
ejpam-609	311	8	l.w	l.w	PROPN
ejpam-609	311	9	.	.	PROPN
ejpam-609	311	10	small	small	ADJ
ejpam-609	311	11	,	,	PUNCT
ejpam-609	311	12	embedding	embed	VERB
ejpam-609	311	13	differential	differential	NOUN
ejpam-609	311	14	and	and	CCONJ
ejpam-609	311	15	skew	skew	ADJ
ejpam-609	311	16	polynomial	polynomial	ADJ
ejpam-609	311	17	rings	ring	NOUN
ejpam-609	311	18	into	into	ADP
ejpam-609	311	19	artinian	artinian	ADJ
ejpam-609	311	20	rings	ring	NOUN
ejpam-609	311	21	,	,	PUNCT
ejpam-609	311	22	proc	proc	NOUN
ejpam-609	311	23	.	.	PUNCT
ejpam-609	311	24	amer	amer	PROPN
ejpam-609	311	25	.	.	PUNCT
ejpam-609	311	26	math	math	PROPN
ejpam-609	311	27	.	.	PUNCT
ejpam-609	312	1	soc	soc	PROPN
ejpam-609	312	2	.	.	PUNCT
ejpam-609	313	1	,	,	PUNCT
ejpam-609	313	2	vol	vol	NOUN
ejpam-609	313	3	.	.	PROPN
ejpam-609	314	1	109	109	NUM
ejpam-609	314	2	(	(	PUNCT
ejpam-609	314	3	4	4	NUM
ejpam-609	314	4	)	)	PUNCT
ejpam-609	314	5	,	,	PUNCT
ejpam-609	314	6	881	881	NUM
ejpam-609	314	7	-	-	SYM
ejpam-609	314	8	886	886	NUM
ejpam-609	314	9	.	.	PUNCT
ejpam-609	315	1	mr1025276	mr1025276	NOUN
ejpam-609	315	2	(	(	PUNCT
ejpam-609	315	3	90k:16003	90k:16003	NUM
ejpam-609	315	4	)	)	PUNCT
ejpam-609	315	5	.	.	PUNCT
ejpam-609	316	1	1990	1990	NUM
ejpam-609	316	2	.	.	PUNCT
ejpam-609	317	1	[	[	X
ejpam-609	317	2	7	7	X
ejpam-609	317	3	]	]	X
ejpam-609	317	4	c.	c.	NOUN
ejpam-609	317	5	faith	faith	NOUN
ejpam-609	317	6	,	,	PUNCT
ejpam-609	317	7	associated	associate	VERB
ejpam-609	317	8	primes	prime	NOUN
ejpam-609	317	9	in	in	ADP
ejpam-609	317	10	commutative	commutative	ADJ
ejpam-609	317	11	polynomial	polynomial	ADJ
ejpam-609	317	12	rings	ring	NOUN
ejpam-609	317	13	,	,	PUNCT
ejpam-609	317	14	comm	comm	NOUN
ejpam-609	317	15	.	.	PUNCT
ejpam-609	318	1	algebra	algebra	NOUN
ejpam-609	318	2	,	,	PUNCT
ejpam-609	318	3	vol	vol	NOUN
ejpam-609	318	4	.	.	PROPN
ejpam-609	318	5	28	28	NUM
ejpam-609	318	6	,	,	PUNCT
ejpam-609	318	7	3983	3983	NUM
ejpam-609	318	8	-	-	SYM
ejpam-609	318	9	3986	3986	NUM
ejpam-609	318	10	.	.	PUNCT
ejpam-609	319	1	mr1767601	mr1767601	PROPN
ejpam-609	319	2	(	(	PUNCT
ejpam-609	319	3	2001a:13038	2001a:13038	NUM
ejpam-609	319	4	)	)	PUNCT
ejpam-609	319	5	.	.	PUNCT
ejpam-609	320	1	2000	2000	NUM
ejpam-609	320	2	.	.	PUNCT
ejpam-609	321	1	[	[	X
ejpam-609	321	2	8	8	X
ejpam-609	321	3	]	]	PUNCT
ejpam-609	321	4	k.	k.	PROPN
ejpam-609	321	5	r.	r.	PROPN
ejpam-609	321	6	goodearl	goodearl	PROPN
ejpam-609	321	7	,	,	PUNCT
ejpam-609	321	8	r.b	r.b	PROPN
ejpam-609	321	9	.	.	PROPN
ejpam-609	321	10	warfield	warfield	PROPN
ejpam-609	321	11	,	,	PUNCT
ejpam-609	321	12	an	an	DET
ejpam-609	321	13	introduction	introduction	NOUN
ejpam-609	321	14	to	to	ADP
ejpam-609	321	15	non	non	ADJ
ejpam-609	321	16	-	-	ADJ
ejpam-609	321	17	commutative	commutative	ADJ
ejpam-609	321	18	noetherian	noetherian	ADJ
ejpam-609	321	19	rings	ring	NOUN
ejpam-609	321	20	,	,	PUNCT
ejpam-609	321	21	camb	camb	PROPN
ejpam-609	321	22	.	.	PUNCT
ejpam-609	322	1	uni	uni	PROPN
ejpam-609	322	2	.	.	PUNCT
ejpam-609	322	3	press	press	PROPN
ejpam-609	322	4	.	.	PUNCT
ejpam-609	323	1	mr1020298	mr1020298	NOUN
ejpam-609	323	2	(	(	PUNCT
ejpam-609	323	3	91c:16001	91c:16001	NUM
ejpam-609	323	4	)	)	PUNCT
ejpam-609	323	5	.	.	PUNCT
ejpam-609	324	1	1989	1989	NUM
ejpam-609	324	2	.	.	PUNCT
ejpam-609	325	1	references	reference	NOUN
ejpam-609	325	2	703	703	NUM
ejpam-609	326	1	[	[	X
ejpam-609	326	2	9	9	NUM
ejpam-609	326	3	]	]	PUNCT
ejpam-609	326	4	k.	k.	PROPN
ejpam-609	326	5	r.	r.	PROPN
ejpam-609	326	6	goodearl	goodearl	PROPN
ejpam-609	326	7	and	and	CCONJ
ejpam-609	326	8	e.	e.	PROPN
ejpam-609	326	9	s.	s.	PROPN
ejpam-609	326	10	letzter	letzter	PROPN
ejpam-609	326	11	,	,	PUNCT
ejpam-609	326	12	prime	prime	ADJ
ejpam-609	326	13	ideals	ideal	NOUN
ejpam-609	326	14	in	in	ADP
ejpam-609	326	15	skew	skew	ADJ
ejpam-609	326	16	and	and	CCONJ
ejpam-609	326	17	q	q	ADJ
ejpam-609	326	18	-	-	PUNCT
ejpam-609	326	19	skew	skew	ADJ
ejpam-609	326	20	polynomial	polynomial	ADJ
ejpam-609	326	21	rings	ring	NOUN
ejpam-609	326	22	,	,	PUNCT
ejpam-609	326	23	memoirs	memoir	NOUN
ejpam-609	326	24	of	of	ADP
ejpam-609	326	25	amer	amer	PROPN
ejpam-609	326	26	.	.	PUNCT
ejpam-609	326	27	math	math	PROPN
ejpam-609	326	28	.	.	PUNCT
ejpam-609	327	1	soc	soc	PROPN
ejpam-609	327	2	.	.	PUNCT
ejpam-609	327	3	,	,	PUNCT
ejpam-609	327	4	521	521	NUM
ejpam-609	327	5	.	.	PUNCT
ejpam-609	327	6	mr1197519	mr1197519	ADJ
ejpam-609	327	7	(	(	PUNCT
ejpam-609	327	8	94j:16051	94j:16051	NUM
ejpam-609	327	9	)	)	PUNCT
ejpam-609	327	10	.	.	PUNCT
ejpam-609	328	1	1994	1994	NUM
ejpam-609	328	2	.	.	PUNCT
ejpam-609	329	1	[	[	X
ejpam-609	329	2	10	10	NUM
ejpam-609	329	3	]	]	X
ejpam-609	329	4	j.	j.	PROPN
ejpam-609	329	5	krempa	krempa	PROPN
ejpam-609	329	6	,	,	PUNCT
ejpam-609	329	7	some	some	DET
ejpam-609	329	8	examples	example	NOUN
ejpam-609	329	9	of	of	ADP
ejpam-609	329	10	reduced	reduce	VERB
ejpam-609	329	11	rings	ring	NOUN
ejpam-609	329	12	,	,	PUNCT
ejpam-609	329	13	algebra	algebra	PROPN
ejpam-609	329	14	colloq	colloq	PROPN
ejpam-609	329	15	.	.	PUNCT
ejpam-609	329	16	,	,	PUNCT
ejpam-609	329	17	vol	vol	NOUN
ejpam-609	329	18	.	.	PUNCT
ejpam-609	329	19	3(4	3(4	NUM
ejpam-609	329	20	)	)	PUNCT
ejpam-609	329	21	,	,	PUNCT
ejpam-609	329	22	289	289	NUM
ejpam-609	329	23	-	-	SYM
ejpam-609	329	24	300	300	NUM
ejpam-609	329	25	.	.	PUNCT
ejpam-609	330	1	mr1422968	mr1422968	PROPN
ejpam-609	330	2	(	(	PUNCT
ejpam-609	330	3	98e:16027	98e:16027	NUM
ejpam-609	330	4	)	)	PUNCT
ejpam-609	330	5	.	.	PUNCT
ejpam-609	330	6	1996	1996	NUM
ejpam-609	330	7	.	.	PUNCT
ejpam-609	331	1	[	[	X
ejpam-609	331	2	11	11	NUM
ejpam-609	331	3	]	]	PUNCT
ejpam-609	331	4	t.	t.	PROPN
ejpam-609	331	5	k.	k.	PROPN
ejpam-609	331	6	kwak	kwak	PROPN
ejpam-609	331	7	,	,	PUNCT
ejpam-609	331	8	prime	prime	ADJ
ejpam-609	331	9	radicals	radical	NOUN
ejpam-609	331	10	of	of	ADP
ejpam-609	331	11	skew	skew	ADJ
ejpam-609	331	12	-	-	PUNCT
ejpam-609	331	13	polynomial	polynomial	ADJ
ejpam-609	331	14	rings	ring	NOUN
ejpam-609	331	15	,	,	PUNCT
ejpam-609	331	16	int	int	NOUN
ejpam-609	331	17	.	.	PUNCT
ejpam-609	332	1	j.	j.	PROPN
ejpam-609	332	2	math	math	PROPN
ejpam-609	332	3	.	.	PUNCT
ejpam-609	333	1	sci	sci	PROPN
ejpam-609	333	2	.	.	PROPN
ejpam-609	333	3	,	,	PUNCT
ejpam-609	333	4	vol	vol	NOUN
ejpam-609	333	5	.	.	PROPN
ejpam-609	333	6	2(2	2(2	NUM
ejpam-609	333	7	)	)	PUNCT
ejpam-609	333	8	,	,	PUNCT
ejpam-609	333	9	219227	219227	NUM
ejpam-609	333	10	.	.	PUNCT
ejpam-609	334	1	mr2061508	mr2061508	PROPN
ejpam-609	334	2	(	(	PUNCT
ejpam-609	334	3	2006a:16035	2006a:16035	NUM
ejpam-609	334	4	)	)	PUNCT
ejpam-609	334	5	.	.	PUNCT
ejpam-609	335	1	2003	2003	NUM
ejpam-609	335	2	.	.	PUNCT
ejpam-609	336	1	[	[	X
ejpam-609	336	2	12	12	NUM
ejpam-609	336	3	]	]	PUNCT
ejpam-609	336	4	a.	a.	PROPN
ejpam-609	336	5	leroy	leroy	PROPN
ejpam-609	336	6	and	and	CCONJ
ejpam-609	336	7	j.	j.	PROPN
ejpam-609	336	8	matczuk	matczuk	PROPN
ejpam-609	336	9	,	,	PUNCT
ejpam-609	336	10	on	on	ADP
ejpam-609	336	11	induced	induced	ADJ
ejpam-609	336	12	modules	module	NOUN
ejpam-609	336	13	over	over	ADP
ejpam-609	336	14	ore	ore	NOUN
ejpam-609	336	15	extensions	extension	NOUN
ejpam-609	336	16	,	,	PUNCT
ejpam-609	336	17	comm	comm	NOUN
ejpam-609	336	18	.	.	PUNCT
ejpam-609	337	1	algebra	algebra	NOUN
ejpam-609	337	2	,	,	PUNCT
ejpam-609	337	3	vol	vol	NOUN
ejpam-609	337	4	.	.	PUNCT
ejpam-609	337	5	32(7	32(7	NOUN
ejpam-609	337	6	)	)	PUNCT
ejpam-609	337	7	,	,	PUNCT
ejpam-609	337	8	2743	2743	NUM
ejpam-609	337	9	-	-	SYM
ejpam-609	337	10	2766	2766	NUM
ejpam-609	337	11	.	.	PUNCT
ejpam-609	338	1	mr2099932	mr2099932	PROPN
ejpam-609	338	2	(	(	PUNCT
ejpam-609	338	3	2005g:16051	2005g:16051	NUM
ejpam-609	338	4	)	)	PUNCT
ejpam-609	338	5	.	.	PUNCT
ejpam-609	339	1	2004	2004	NUM
ejpam-609	339	2	.	.	PUNCT
ejpam-609	340	1	[	[	X
ejpam-609	340	2	13	13	NUM
ejpam-609	340	3	]	]	X
ejpam-609	340	4	g.	g.	NOUN
ejpam-609	340	5	marks	marks	PROPN
ejpam-609	340	6	,	,	PUNCT
ejpam-609	340	7	on	on	ADP
ejpam-609	340	8	2	2	NUM
ejpam-609	340	9	-	-	PUNCT
ejpam-609	340	10	primal	primal	ADJ
ejpam-609	340	11	ore	ore	NOUN
ejpam-609	340	12	extensions	extension	NOUN
ejpam-609	340	13	,	,	PUNCT
ejpam-609	340	14	comm	comm	NOUN
ejpam-609	340	15	.	.	PUNCT
ejpam-609	341	1	algebra	algebra	NOUN
ejpam-609	341	2	,	,	PUNCT
ejpam-609	341	3	vol	vol	NOUN
ejpam-609	341	4	.	.	PROPN
ejpam-609	341	5	29	29	NUM
ejpam-609	341	6	(	(	PUNCT
ejpam-609	341	7	5	5	NUM
ejpam-609	341	8	)	)	PUNCT
ejpam-609	341	9	,	,	PUNCT
ejpam-609	341	10	2113	2113	NUM
ejpam-609	341	11	-	-	SYM
ejpam-609	341	12	2123	2123	NUM
ejpam-609	341	13	.	.	PUNCT
ejpam-609	342	1	mr1837966	mr1837966	NOUN
ejpam-609	342	2	(	(	PUNCT
ejpam-609	342	3	2002e:16042	2002e:16042	NUM
ejpam-609	342	4	)	)	PUNCT
ejpam-609	342	5	.	.	PUNCT
ejpam-609	343	1	2001	2001	NUM
ejpam-609	343	2	.	.	PUNCT
ejpam-609	344	1	[	[	X
ejpam-609	344	2	14	14	NUM
ejpam-609	344	3	]	]	X
ejpam-609	344	4	j.	j.	PROPN
ejpam-609	344	5	c.	c.	PROPN
ejpam-609	344	6	mcconnell	mcconnell	PROPN
ejpam-609	344	7	and	and	CCONJ
ejpam-609	344	8	j.	j.	PROPN
ejpam-609	344	9	c.	c.	PROPN
ejpam-609	344	10	robson	robson	PROPN
ejpam-609	344	11	,	,	PUNCT
ejpam-609	344	12	noncommutative	noncommutative	ADJ
ejpam-609	344	13	noetherian	noetherian	ADJ
ejpam-609	344	14	rings	ring	NOUN
ejpam-609	344	15	,	,	PUNCT
ejpam-609	344	16	wiley	wiley	NOUN
ejpam-609	344	17	1987	1987	NUM
ejpam-609	344	18	;	;	PUNCT
ejpam-609	344	19	revised	revise	VERB
ejpam-609	344	20	edition	edition	NOUN
ejpam-609	344	21	:	:	PUNCT
ejpam-609	344	22	american	american	PROPN
ejpam-609	344	23	mathematical	mathematical	ADJ
ejpam-609	344	24	society	society	NOUN
ejpam-609	344	25	mr1811901	mr1811901	NOUN
ejpam-609	344	26	(	(	PUNCT
ejpam-609	344	27	2001i:16039	2001i:16039	NUM
ejpam-609	344	28	)	)	PUNCT
ejpam-609	344	29	,	,	PUNCT
ejpam-609	344	30	zbl	zbl	PROPN
ejpam-609	344	31	0980.16019	0980.16019	NUM
ejpam-609	344	32	.	.	PUNCT
ejpam-609	345	1	2001	2001	NUM
ejpam-609	345	2	.	.	PUNCT
ejpam-609	346	1	[	[	X
ejpam-609	346	2	15	15	NUM
ejpam-609	346	3	]	]	X
ejpam-609	346	4	h.	h.	PROPN
ejpam-609	346	5	e.	e.	PROPN
ejpam-609	346	6	nordstorm	nordstorm	PROPN
ejpam-609	346	7	,	,	PUNCT
ejpam-609	346	8	associated	associate	VERB
ejpam-609	346	9	primes	prime	NOUN
ejpam-609	346	10	over	over	ADP
ejpam-609	346	11	ore	ore	NOUN
ejpam-609	346	12	extensions	extension	NOUN
ejpam-609	346	13	,	,	PUNCT
ejpam-609	346	14	j.	j.	PROPN
ejpam-609	346	15	algebra	algebra	PROPN
ejpam-609	346	16	,	,	PUNCT
ejpam-609	346	17	vol	vol	NOUN
ejpam-609	346	18	.	.	PUNCT
ejpam-609	346	19	286(1	286(1	NUM
ejpam-609	346	20	)	)	PUNCT
ejpam-609	346	21	,	,	PUNCT
ejpam-609	346	22	69	69	NUM
ejpam-609	346	23	-	-	SYM
ejpam-609	346	24	75	75	NUM
ejpam-609	346	25	.	.	PUNCT
ejpam-609	347	1	mr2124809	mr2124809	PROPN
ejpam-609	347	2	(	(	PUNCT
ejpam-609	347	3	2006c:16049	2006c:16049	NUM
ejpam-609	347	4	)	)	PUNCT
ejpam-609	347	5	.	.	PUNCT
ejpam-609	348	1	2005	2005	NUM
ejpam-609	348	2	.	.	PUNCT
ejpam-609	349	1	[	[	X
ejpam-609	349	2	16	16	NUM
ejpam-609	349	3	]	]	PUNCT
ejpam-609	349	4	l.	l.	PROPN
ejpam-609	349	5	ouyang	ouyang	PROPN
ejpam-609	349	6	,	,	PUNCT
ejpam-609	349	7	extensions	extension	NOUN
ejpam-609	349	8	of	of	ADP
ejpam-609	349	9	generalized	generalized	ADJ
ejpam-609	349	10	α	α	ADJ
ejpam-609	349	11	-	-	ADJ
ejpam-609	349	12	rigid	rigid	ADJ
ejpam-609	349	13	rings	ring	NOUN
ejpam-609	349	14	,	,	PUNCT
ejpam-609	349	15	int	int	NOUN
ejpam-609	349	16	.	.	PUNCT
ejpam-609	349	17	electron	electron	PROPN
ejpam-609	349	18	.	.	PUNCT
ejpam-609	350	1	j.	j.	PROPN
ejpam-609	350	2	algebra	algebra	PROPN
ejpam-609	350	3	,	,	PUNCT
ejpam-609	350	4	vol	vol	NOUN
ejpam-609	350	5	.	.	PROPN
ejpam-609	350	6	3	3	NUM
ejpam-609	350	7	,	,	PUNCT
ejpam-609	350	8	103116	103116	NUM
ejpam-609	350	9	.	.	PUNCT
ejpam-609	351	1	mr2369408	mr2369408	NOUN
ejpam-609	351	2	(	(	PUNCT
ejpam-609	351	3	2008m:16054	2008m:16054	NUM
ejpam-609	351	4	)	)	PUNCT
ejpam-609	351	5	.	.	PUNCT
ejpam-609	352	1	2008	2008	NUM
ejpam-609	352	2	.	.	PUNCT