id	sid	tid	token	lemma	pos
ejpam-1802	1	1	european	european	PROPN
ejpam-1802	1	2	journal	journal	PROPN
ejpam-1802	1	3	of	of	ADP
ejpam-1802	1	4	pure	pure	ADJ
ejpam-1802	1	5	and	and	CCONJ
ejpam-1802	1	6	applied	apply	VERB
ejpam-1802	1	7	mathematics	mathematic	NOUN
ejpam-1802	1	8	vol	vol	NOUN
ejpam-1802	1	9	.	.	PUNCT
ejpam-1802	2	1	7	7	NUM
ejpam-1802	2	2	,	,	PUNCT
ejpam-1802	2	3	no	no	INTJ
ejpam-1802	2	4	.	.	NOUN
ejpam-1802	2	5	1	1	NUM
ejpam-1802	2	6	,	,	PUNCT
ejpam-1802	2	7	2014	2014	NUM
ejpam-1802	2	8	,	,	PUNCT
ejpam-1802	2	9	109	109	NUM
ejpam-1802	2	10	-	-	SYM
ejpam-1802	2	11	113	113	NUM
ejpam-1802	2	12	issn	issn	PROPN
ejpam-1802	2	13	1307	1307	NUM
ejpam-1802	2	14	-	-	SYM
ejpam-1802	2	15	5543	5543	NUM
ejpam-1802	2	16	–	–	PUNCT
ejpam-1802	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1802	2	18	on	on	ADP
ejpam-1802	2	19	distinguishing	distinguish	VERB
ejpam-1802	2	20	local	local	ADJ
ejpam-1802	2	21	finite	finite	NOUN
ejpam-1802	2	22	rings	ring	NOUN
ejpam-1802	2	23	from	from	ADP
ejpam-1802	2	24	finite	finite	ADJ
ejpam-1802	2	25	rings	ring	NOUN
ejpam-1802	2	26	only	only	ADV
ejpam-1802	2	27	by	by	ADP
ejpam-1802	2	28	counting	count	VERB
ejpam-1802	2	29	elements	element	NOUN
ejpam-1802	2	30	and	and	CCONJ
ejpam-1802	2	31	zero	zero	NUM
ejpam-1802	2	32	divisors	divisor	NOUN
ejpam-1802	2	33	marcos	marcos	PROPN
ejpam-1802	2	34	j.	j.	PROPN
ejpam-1802	2	35	gonzález	gonzález	PROPN
ejpam-1802	2	36	departamento	departamento	PROPN
ejpam-1802	2	37	de	de	PROPN
ejpam-1802	2	38	matemáticas	matemáticas	PROPN
ejpam-1802	2	39	,	,	PUNCT
ejpam-1802	2	40	universidad	universidad	PROPN
ejpam-1802	2	41	simón	simón	PROPN
ejpam-1802	2	42	bolívar	bolívar	PROPN
ejpam-1802	2	43	,	,	PUNCT
ejpam-1802	2	44	caracas	caracas	PROPN
ejpam-1802	2	45	1080	1080	NUM
ejpam-1802	2	46	-	-	PUNCT
ejpam-1802	2	47	a	a	PRON
ejpam-1802	2	48	,	,	PUNCT
ejpam-1802	2	49	venezuela	venezuela	PROPN
ejpam-1802	2	50	abstract	abstract	NOUN
ejpam-1802	2	51	.	.	PUNCT
ejpam-1802	3	1	the	the	DET
ejpam-1802	3	2	purpose	purpose	NOUN
ejpam-1802	3	3	of	of	ADP
ejpam-1802	3	4	this	this	DET
ejpam-1802	3	5	short	short	ADJ
ejpam-1802	3	6	communication	communication	NOUN
ejpam-1802	3	7	is	be	AUX
ejpam-1802	3	8	to	to	PART
ejpam-1802	3	9	prove	prove	VERB
ejpam-1802	3	10	the	the	DET
ejpam-1802	3	11	following	following	NOUN
ejpam-1802	3	12	:	:	PUNCT
ejpam-1802	3	13	let	let	VERB
ejpam-1802	3	14	r	r	PRON
ejpam-1802	3	15	be	be	AUX
ejpam-1802	3	16	a	a	DET
ejpam-1802	3	17	finite	finite	ADJ
ejpam-1802	3	18	associative	associative	ADJ
ejpam-1802	3	19	ring	ring	NOUN
ejpam-1802	3	20	with	with	ADP
ejpam-1802	3	21	unit	unit	NOUN
ejpam-1802	3	22	.	.	PUNCT
ejpam-1802	4	1	then	then	ADV
ejpam-1802	4	2	r	r	NOUN
ejpam-1802	4	3	is	be	AUX
ejpam-1802	4	4	local	local	ADJ
ejpam-1802	4	5	if	if	SCONJ
ejpam-1802	4	6	and	and	CCONJ
ejpam-1802	4	7	only	only	ADV
ejpam-1802	4	8	if	if	SCONJ
ejpam-1802	4	9	|r|=	|r|=	PRON
ejpam-1802	4	10	pn	pn	VERB
ejpam-1802	4	11	and	and	CCONJ
ejpam-1802	4	12	|z(r)|=	|z(r)|=	VERB
ejpam-1802	4	13	pm	pm	NOUN
ejpam-1802	4	14	for	for	ADP
ejpam-1802	4	15	some	some	DET
ejpam-1802	4	16	prime	prime	ADJ
ejpam-1802	4	17	number	number	NOUN
ejpam-1802	4	18	p	p	NOUN
ejpam-1802	4	19	and	and	CCONJ
ejpam-1802	4	20	integers	integer	NOUN
ejpam-1802	4	21	1	1	NUM
ejpam-1802	4	22	≤	≤	NUM
ejpam-1802	4	23	m	m	VERB
ejpam-1802	4	24	<	<	X
ejpam-1802	4	25	n.	n.	NOUN
ejpam-1802	4	26	for	for	ADP
ejpam-1802	4	27	the	the	DET
ejpam-1802	4	28	commutative	commutative	ADJ
ejpam-1802	4	29	case	case	NOUN
ejpam-1802	4	30	,	,	PUNCT
ejpam-1802	4	31	this	this	PRON
ejpam-1802	4	32	have	have	AUX
ejpam-1802	4	33	been	be	AUX
ejpam-1802	4	34	recently	recently	ADV
ejpam-1802	4	35	discovered	discover	VERB
ejpam-1802	4	36	by	by	ADP
ejpam-1802	4	37	behboodi	behboodi	NOUN
ejpam-1802	4	38	and	and	CCONJ
ejpam-1802	4	39	beyranvand	beyranvand	NOUN
ejpam-1802	5	1	[	[	X
ejpam-1802	5	2	1	1	NUM
ejpam-1802	5	3	,	,	PUNCT
ejpam-1802	5	4	theorem	theorem	VERB
ejpam-1802	5	5	3	3	NUM
ejpam-1802	5	6	]	]	PUNCT
ejpam-1802	5	7	.	.	PUNCT
ejpam-1802	6	1	we	we	PRON
ejpam-1802	6	2	will	will	AUX
ejpam-1802	6	3	also	also	ADV
ejpam-1802	6	4	present	present	VERB
ejpam-1802	6	5	yet	yet	ADV
ejpam-1802	6	6	another	another	DET
ejpam-1802	6	7	proof	proof	NOUN
ejpam-1802	6	8	for	for	ADP
ejpam-1802	6	9	the	the	DET
ejpam-1802	6	10	commutative	commutative	ADJ
ejpam-1802	6	11	case	case	NOUN
ejpam-1802	6	12	.	.	PUNCT
ejpam-1802	7	1	2010	2010	NUM
ejpam-1802	7	2	mathematics	mathematic	NOUN
ejpam-1802	7	3	subject	subject	NOUN
ejpam-1802	7	4	classifications	classification	NOUN
ejpam-1802	7	5	:	:	PUNCT
ejpam-1802	7	6	16b99	16b99	NUM
ejpam-1802	7	7	,	,	PUNCT
ejpam-1802	7	8	13a99	13a99	NUM
ejpam-1802	7	9	,	,	PUNCT
ejpam-1802	7	10	68r10	68r10	NUM
ejpam-1802	7	11	key	key	ADJ
ejpam-1802	7	12	words	word	NOUN
ejpam-1802	7	13	and	and	CCONJ
ejpam-1802	7	14	phrases	phrase	NOUN
ejpam-1802	7	15	:	:	PUNCT
ejpam-1802	7	16	finite	finite	PROPN
ejpam-1802	7	17	ring	ring	NOUN
ejpam-1802	7	18	,	,	PUNCT
ejpam-1802	7	19	zero	zero	NUM
ejpam-1802	7	20	-	-	PUNCT
ejpam-1802	7	21	divisor	divisor	NOUN
ejpam-1802	7	22	,	,	PUNCT
ejpam-1802	7	23	local	local	ADJ
ejpam-1802	7	24	rings	ring	NOUN
ejpam-1802	7	25	1	1	NUM
ejpam-1802	7	26	.	.	PUNCT
ejpam-1802	7	27	introduction	introduction	NOUN
ejpam-1802	7	28	the	the	DET
ejpam-1802	7	29	purpose	purpose	NOUN
ejpam-1802	7	30	of	of	ADP
ejpam-1802	7	31	this	this	DET
ejpam-1802	7	32	paper	paper	NOUN
ejpam-1802	7	33	is	be	AUX
ejpam-1802	7	34	to	to	PART
ejpam-1802	7	35	prove	prove	VERB
ejpam-1802	7	36	the	the	DET
ejpam-1802	7	37	following	following	NOUN
ejpam-1802	7	38	:	:	PUNCT
ejpam-1802	7	39	theorem	theorem	NOUN
ejpam-1802	7	40	1	1	X
ejpam-1802	7	41	.	.	PUNCT
ejpam-1802	8	1	let	let	VERB
ejpam-1802	8	2	r	r	PRON
ejpam-1802	8	3	be	be	AUX
ejpam-1802	8	4	a	a	DET
ejpam-1802	8	5	finite	finite	ADJ
ejpam-1802	8	6	associative	associative	ADJ
ejpam-1802	8	7	ring	ring	NOUN
ejpam-1802	8	8	with	with	ADP
ejpam-1802	8	9	unit	unit	NOUN
ejpam-1802	8	10	.	.	PUNCT
ejpam-1802	9	1	then	then	ADV
ejpam-1802	9	2	r	r	NOUN
ejpam-1802	9	3	is	be	AUX
ejpam-1802	9	4	local	local	ADJ
ejpam-1802	9	5	if	if	SCONJ
ejpam-1802	9	6	and	and	CCONJ
ejpam-1802	9	7	only	only	ADV
ejpam-1802	9	8	if	if	SCONJ
ejpam-1802	9	9	|r|	|r|	NOUN
ejpam-1802	9	10	=	=	SYM
ejpam-1802	9	11	pm	pm	NOUN
ejpam-1802	9	12	and	and	CCONJ
ejpam-1802	9	13	|z(r)|=	|z(r)|=	VERB
ejpam-1802	9	14	pn	pn	NOUN
ejpam-1802	9	15	for	for	ADP
ejpam-1802	9	16	some	some	DET
ejpam-1802	9	17	prime	prime	ADJ
ejpam-1802	9	18	number	number	NOUN
ejpam-1802	9	19	p	p	NOUN
ejpam-1802	9	20	and	and	CCONJ
ejpam-1802	9	21	integers	integer	NOUN
ejpam-1802	9	22	1≤	1≤	INTJ
ejpam-1802	9	23	n	n	CCONJ
ejpam-1802	9	24	<	<	X
ejpam-1802	9	25	m.	m.	NOUN
ejpam-1802	9	26	this	this	PRON
ejpam-1802	9	27	allows	allow	VERB
ejpam-1802	9	28	us	we	PRON
ejpam-1802	9	29	to	to	PART
ejpam-1802	9	30	recognize	recognize	VERB
ejpam-1802	9	31	local	local	ADJ
ejpam-1802	9	32	rings	ring	NOUN
ejpam-1802	9	33	out	out	ADP
ejpam-1802	9	34	of	of	ADP
ejpam-1802	9	35	finite	finite	ADJ
ejpam-1802	9	36	rings	ring	NOUN
ejpam-1802	9	37	only	only	ADV
ejpam-1802	9	38	by	by	ADP
ejpam-1802	9	39	counting	count	VERB
ejpam-1802	9	40	the	the	DET
ejpam-1802	9	41	number	number	NOUN
ejpam-1802	9	42	of	of	ADP
ejpam-1802	9	43	elements	element	NOUN
ejpam-1802	9	44	in	in	ADP
ejpam-1802	9	45	the	the	DET
ejpam-1802	9	46	ring	ring	NOUN
ejpam-1802	9	47	and	and	CCONJ
ejpam-1802	9	48	the	the	DET
ejpam-1802	9	49	number	number	NOUN
ejpam-1802	9	50	of	of	ADP
ejpam-1802	9	51	zero	zero	NUM
ejpam-1802	9	52	divisors	divisor	NOUN
ejpam-1802	9	53	.	.	PUNCT
ejpam-1802	10	1	this	this	DET
ejpam-1802	10	2	result	result	NOUN
ejpam-1802	10	3	is	be	AUX
ejpam-1802	10	4	inspired	inspire	VERB
ejpam-1802	10	5	by	by	ADP
ejpam-1802	10	6	the	the	DET
ejpam-1802	10	7	corresponding	corresponding	ADJ
ejpam-1802	10	8	result	result	NOUN
ejpam-1802	10	9	behboodi	behboodi	NOUN
ejpam-1802	10	10	and	and	CCONJ
ejpam-1802	10	11	beyranvand	beyranvand	NOUN
ejpam-1802	11	1	[	[	X
ejpam-1802	11	2	1	1	NUM
ejpam-1802	11	3	,	,	PUNCT
ejpam-1802	11	4	theorem	theorem	ADJ
ejpam-1802	11	5	3	3	NUM
ejpam-1802	11	6	,	,	PUNCT
ejpam-1802	11	7	pp	pp	ADJ
ejpam-1802	11	8	.	.	PUNCT
ejpam-1802	12	1	306–307	306–307	NUM
ejpam-1802	12	2	]	]	PUNCT
ejpam-1802	12	3	where	where	SCONJ
ejpam-1802	12	4	r	r	NOUN
ejpam-1802	12	5	is	be	AUX
ejpam-1802	12	6	assumed	assume	VERB
ejpam-1802	12	7	to	to	PART
ejpam-1802	12	8	be	be	AUX
ejpam-1802	12	9	commutative	commutative	ADJ
ejpam-1802	12	10	.	.	PUNCT
ejpam-1802	13	1	more	more	ADV
ejpam-1802	13	2	precisely	precisely	ADV
ejpam-1802	13	3	,	,	PUNCT
ejpam-1802	13	4	they	they	PRON
ejpam-1802	13	5	proved	prove	VERB
ejpam-1802	13	6	the	the	DET
ejpam-1802	13	7	following	following	NOUN
ejpam-1802	13	8	:	:	PUNCT
ejpam-1802	13	9	theorem	theorem	NOUN
ejpam-1802	13	10	2	2	NUM
ejpam-1802	13	11	.	.	PUNCT
ejpam-1802	14	1	let	let	VERB
ejpam-1802	14	2	r	r	PRON
ejpam-1802	14	3	be	be	AUX
ejpam-1802	14	4	a	a	DET
ejpam-1802	14	5	finite	finite	ADJ
ejpam-1802	14	6	commutative	commutative	ADJ
ejpam-1802	14	7	ring	ring	NOUN
ejpam-1802	14	8	with	with	ADP
ejpam-1802	14	9	unit	unit	NOUN
ejpam-1802	14	10	.	.	PUNCT
ejpam-1802	15	1	then	then	ADV
ejpam-1802	15	2	r	r	NOUN
ejpam-1802	15	3	is	be	AUX
ejpam-1802	15	4	local	local	ADJ
ejpam-1802	15	5	if	if	SCONJ
ejpam-1802	15	6	and	and	CCONJ
ejpam-1802	15	7	only	only	ADV
ejpam-1802	15	8	if	if	SCONJ
ejpam-1802	15	9	|r|=	|r|=	PRON
ejpam-1802	15	10	pm	pm	VERB
ejpam-1802	15	11	and	and	CCONJ
ejpam-1802	15	12	|z(r)|=	|z(r)|=	VERB
ejpam-1802	15	13	pn	pn	NOUN
ejpam-1802	15	14	for	for	ADP
ejpam-1802	15	15	some	some	DET
ejpam-1802	15	16	prime	prime	ADJ
ejpam-1802	15	17	number	number	NOUN
ejpam-1802	15	18	p	p	NOUN
ejpam-1802	15	19	and	and	CCONJ
ejpam-1802	15	20	integers	integer	NOUN
ejpam-1802	15	21	1≤	1≤	INTJ
ejpam-1802	15	22	n	n	CCONJ
ejpam-1802	15	23	<	<	X
ejpam-1802	15	24	m.	m.	NOUN
ejpam-1802	15	25	we	we	PRON
ejpam-1802	15	26	will	will	AUX
ejpam-1802	15	27	also	also	ADV
ejpam-1802	15	28	provide	provide	VERB
ejpam-1802	15	29	another	another	DET
ejpam-1802	15	30	proof	proof	NOUN
ejpam-1802	15	31	for	for	ADP
ejpam-1802	15	32	this	this	DET
ejpam-1802	15	33	theorem	theorem	NOUN
ejpam-1802	15	34	which	which	PRON
ejpam-1802	15	35	differs	differ	VERB
ejpam-1802	15	36	both	both	PRON
ejpam-1802	15	37	from	from	ADP
ejpam-1802	15	38	the	the	DET
ejpam-1802	15	39	proof	proof	NOUN
ejpam-1802	15	40	of	of	ADP
ejpam-1802	15	41	theorem	theorem	NOUN
ejpam-1802	15	42	1	1	NUM
ejpam-1802	15	43	and	and	CCONJ
ejpam-1802	15	44	the	the	DET
ejpam-1802	15	45	original	original	ADJ
ejpam-1802	15	46	proof	proof	NOUN
ejpam-1802	15	47	presented	present	VERB
ejpam-1802	15	48	in	in	ADP
ejpam-1802	15	49	[	[	X
ejpam-1802	15	50	1	1	NUM
ejpam-1802	15	51	]	]	PUNCT
ejpam-1802	15	52	.	.	PUNCT
ejpam-1802	16	1	email	email	NOUN
ejpam-1802	16	2	address	address	NOUN
ejpam-1802	16	3	:	:	PUNCT
ejpam-1802	16	4	mago@usb.ve	mago@usb.ve	PROPN
ejpam-1802	16	5	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1802	16	6	109	109	NUM
ejpam-1802	17	1	c	c	X
ejpam-1802	17	2	©	©	PROPN
ejpam-1802	17	3	2014	2014	NUM
ejpam-1802	17	4	ejpam	ejpam	NOUN
ejpam-1802	17	5	all	all	DET
ejpam-1802	17	6	rights	right	NOUN
ejpam-1802	17	7	reserved	reserve	VERB
ejpam-1802	17	8	.	.	PUNCT
ejpam-1802	18	1	m.	m.	NOUN
ejpam-1802	18	2	gonzález	gonzález	PROPN
ejpam-1802	18	3	/	/	SYM
ejpam-1802	18	4	eur	eur	PROPN
ejpam-1802	18	5	.	.	PUNCT
ejpam-1802	19	1	j.	j.	PROPN
ejpam-1802	19	2	pure	pure	PROPN
ejpam-1802	19	3	appl	appl	PROPN
ejpam-1802	19	4	.	.	PROPN
ejpam-1802	19	5	math	math	PROPN
ejpam-1802	19	6	,	,	PUNCT
ejpam-1802	19	7	7	7	NUM
ejpam-1802	19	8	(	(	PUNCT
ejpam-1802	19	9	2014	2014	NUM
ejpam-1802	19	10	)	)	PUNCT
ejpam-1802	19	11	,	,	PUNCT
ejpam-1802	19	12	109	109	NUM
ejpam-1802	19	13	-	-	SYM
ejpam-1802	19	14	113	113	NUM
ejpam-1802	19	15	110	110	NUM
ejpam-1802	19	16	2	2	NUM
ejpam-1802	19	17	.	.	PUNCT
ejpam-1802	19	18	preliminaries	preliminary	NOUN
ejpam-1802	19	19	almost	almost	ADV
ejpam-1802	19	20	all	all	DET
ejpam-1802	19	21	the	the	DET
ejpam-1802	19	22	facts	fact	NOUN
ejpam-1802	19	23	we	we	PRON
ejpam-1802	19	24	require	require	VERB
ejpam-1802	19	25	for	for	SCONJ
ejpam-1802	19	26	the	the	DET
ejpam-1802	19	27	both	both	DET
ejpam-1802	19	28	proofs	proof	NOUN
ejpam-1802	19	29	can	can	AUX
ejpam-1802	19	30	be	be	AUX
ejpam-1802	19	31	found	find	VERB
ejpam-1802	19	32	in	in	ADP
ejpam-1802	19	33	mcdonald	mcdonald	PROPN
ejpam-1802	19	34	’s	’s	PART
ejpam-1802	19	35	monograph	monograph	NOUN
ejpam-1802	20	1	[	[	X
ejpam-1802	20	2	4	4	NUM
ejpam-1802	20	3	]	]	PUNCT
ejpam-1802	20	4	.	.	PUNCT
ejpam-1802	21	1	we	we	PRON
ejpam-1802	21	2	use	use	VERB
ejpam-1802	21	3	a	a	DET
ejpam-1802	21	4	different	different	ADJ
ejpam-1802	21	5	notation	notation	NOUN
ejpam-1802	21	6	though	though	ADV
ejpam-1802	21	7	,	,	PUNCT
ejpam-1802	21	8	similar	similar	ADJ
ejpam-1802	21	9	to	to	ADP
ejpam-1802	21	10	the	the	DET
ejpam-1802	21	11	one	one	NOUN
ejpam-1802	21	12	used	use	VERB
ejpam-1802	21	13	in	in	ADP
ejpam-1802	21	14	[	[	X
ejpam-1802	21	15	1	1	NUM
ejpam-1802	21	16	]	]	PUNCT
ejpam-1802	21	17	.	.	PUNCT
ejpam-1802	22	1	the	the	DET
ejpam-1802	22	2	letter	letter	NOUN
ejpam-1802	22	3	r	r	NOUN
ejpam-1802	22	4	always	always	ADV
ejpam-1802	22	5	denotes	denote	VERB
ejpam-1802	22	6	a	a	DET
ejpam-1802	22	7	finite	finite	ADJ
ejpam-1802	22	8	associative	associative	ADJ
ejpam-1802	22	9	ring	ring	NOUN
ejpam-1802	22	10	with	with	ADP
ejpam-1802	22	11	identity	identity	NOUN
ejpam-1802	22	12	.	.	PUNCT
ejpam-1802	23	1	an	an	DET
ejpam-1802	23	2	element	element	NOUN
ejpam-1802	23	3	r	r	NOUN
ejpam-1802	23	4	∈	∈	NOUN
ejpam-1802	23	5	r	r	NOUN
ejpam-1802	23	6	is	be	AUX
ejpam-1802	23	7	said	say	VERB
ejpam-1802	23	8	to	to	PART
ejpam-1802	23	9	be	be	AUX
ejpam-1802	23	10	invertible	invertible	ADJ
ejpam-1802	23	11	is	be	AUX
ejpam-1802	23	12	there	there	PRON
ejpam-1802	23	13	exists	exist	VERB
ejpam-1802	23	14	s	s	PROPN
ejpam-1802	23	15	∈	∈	PROPN
ejpam-1802	23	16	r	r	NOUN
ejpam-1802	23	17	such	such	ADJ
ejpam-1802	23	18	that	that	SCONJ
ejpam-1802	23	19	rs=	rs=	PROPN
ejpam-1802	23	20	sr=	sr=	NOUN
ejpam-1802	23	21	1	1	NUM
ejpam-1802	23	22	.	.	PUNCT
ejpam-1802	24	1	the	the	DET
ejpam-1802	24	2	set	set	NOUN
ejpam-1802	24	3	of	of	ADP
ejpam-1802	24	4	all	all	DET
ejpam-1802	24	5	invertible	invertible	ADJ
ejpam-1802	24	6	elements	element	NOUN
ejpam-1802	24	7	of	of	ADP
ejpam-1802	24	8	r	r	NOUN
ejpam-1802	24	9	forms	form	NOUN
ejpam-1802	24	10	a	a	DET
ejpam-1802	24	11	group	group	NOUN
ejpam-1802	24	12	denoted	denote	VERB
ejpam-1802	24	13	by	by	ADP
ejpam-1802	24	14	r×.	r×.	ADP
ejpam-1802	24	15	an	an	DET
ejpam-1802	24	16	element	element	NOUN
ejpam-1802	24	17	r	r	NOUN
ejpam-1802	24	18	∈	∈	NOUN
ejpam-1802	24	19	r	r	NOUN
ejpam-1802	24	20	is	be	AUX
ejpam-1802	24	21	said	say	VERB
ejpam-1802	24	22	to	to	PART
ejpam-1802	24	23	be	be	AUX
ejpam-1802	24	24	a	a	DET
ejpam-1802	24	25	left	left	ADJ
ejpam-1802	24	26	(	(	PUNCT
ejpam-1802	24	27	right	right	ADJ
ejpam-1802	24	28	,	,	PUNCT
ejpam-1802	24	29	resp	resp	NOUN
ejpam-1802	24	30	.	.	PUNCT
ejpam-1802	24	31	)	)	PUNCT
ejpam-1802	25	1	zero	zero	NUM
ejpam-1802	25	2	divisor	divisor	NOUN
ejpam-1802	25	3	if	if	SCONJ
ejpam-1802	25	4	there	there	PRON
ejpam-1802	25	5	exists	exist	VERB
ejpam-1802	25	6	s	s	PROPN
ejpam-1802	25	7	∈	∈	PROPN
ejpam-1802	25	8	r	r	NOUN
ejpam-1802	25	9	such	such	ADJ
ejpam-1802	25	10	that	that	DET
ejpam-1802	25	11	sr	sr	PROPN
ejpam-1802	25	12	=	=	SYM
ejpam-1802	25	13	0	0	PROPN
ejpam-1802	26	1	(	(	PUNCT
ejpam-1802	26	2	rs	rs	NOUN
ejpam-1802	26	3	=	=	SYM
ejpam-1802	26	4	0	0	NUM
ejpam-1802	26	5	,	,	PUNCT
ejpam-1802	26	6	resp	resp	NOUN
ejpam-1802	26	7	.	.	PUNCT
ejpam-1802	26	8	)	)	PUNCT
ejpam-1802	27	1	and	and	CCONJ
ejpam-1802	27	2	r	r	NOUN
ejpam-1802	27	3	is	be	AUX
ejpam-1802	27	4	said	say	VERB
ejpam-1802	27	5	to	to	PART
ejpam-1802	27	6	be	be	AUX
ejpam-1802	27	7	a	a	DET
ejpam-1802	27	8	two	two	NUM
ejpam-1802	27	9	-	-	PUNCT
ejpam-1802	27	10	sided	sided	ADJ
ejpam-1802	27	11	zero	zero	NUM
ejpam-1802	27	12	divisor	divisor	NOUN
ejpam-1802	27	13	if	if	SCONJ
ejpam-1802	27	14	it	it	PRON
ejpam-1802	27	15	is	be	AUX
ejpam-1802	27	16	both	both	CCONJ
ejpam-1802	27	17	a	a	DET
ejpam-1802	27	18	left	left	NOUN
ejpam-1802	27	19	and	and	CCONJ
ejpam-1802	27	20	a	a	DET
ejpam-1802	27	21	right	right	ADJ
ejpam-1802	27	22	zero	zero	NUM
ejpam-1802	27	23	divisor	divisor	NOUN
ejpam-1802	27	24	.	.	PUNCT
ejpam-1802	28	1	the	the	DET
ejpam-1802	28	2	set	set	NOUN
ejpam-1802	28	3	of	of	ADP
ejpam-1802	28	4	all	all	DET
ejpam-1802	28	5	zero	zero	NUM
ejpam-1802	28	6	divisors	divisor	NOUN
ejpam-1802	28	7	is	be	AUX
ejpam-1802	28	8	denoted	denote	VERB
ejpam-1802	28	9	by	by	ADP
ejpam-1802	28	10	z(r	z(r	NOUN
ejpam-1802	28	11	)	)	PUNCT
ejpam-1802	28	12	.	.	PUNCT
ejpam-1802	29	1	for	for	ADP
ejpam-1802	29	2	r	r	NOUN
ejpam-1802	29	3	a	a	DET
ejpam-1802	29	4	finite	finite	NOUN
ejpam-1802	29	5	ring	ring	NOUN
ejpam-1802	29	6	with	with	ADP
ejpam-1802	29	7	identity	identity	NOUN
ejpam-1802	29	8	,	,	PUNCT
ejpam-1802	29	9	any	any	DET
ejpam-1802	29	10	zero	zero	NUM
ejpam-1802	29	11	divisor	divisor	NOUN
ejpam-1802	29	12	is	be	AUX
ejpam-1802	29	13	a	a	DET
ejpam-1802	29	14	two	two	NUM
ejpam-1802	29	15	-	-	PUNCT
ejpam-1802	29	16	sided	sided	ADJ
ejpam-1802	29	17	zero	zero	NUM
ejpam-1802	29	18	divisor	divisor	NOUN
ejpam-1802	29	19	and	and	CCONJ
ejpam-1802	29	20	every	every	DET
ejpam-1802	29	21	element	element	NOUN
ejpam-1802	29	22	that	that	PRON
ejpam-1802	29	23	is	be	AUX
ejpam-1802	29	24	not	not	PART
ejpam-1802	29	25	a	a	DET
ejpam-1802	29	26	zero	zero	NUM
ejpam-1802	29	27	divisor	divisor	NOUN
ejpam-1802	29	28	is	be	AUX
ejpam-1802	29	29	invertible	invertible	ADJ
ejpam-1802	29	30	(	(	PUNCT
ejpam-1802	29	31	see	see	VERB
ejpam-1802	29	32	ganesan	ganesan	PROPN
ejpam-1802	30	1	[	[	X
ejpam-1802	30	2	2	2	NUM
ejpam-1802	30	3	,	,	PUNCT
ejpam-1802	30	4	theorem	theorem	ADJ
ejpam-1802	30	5	5	5	NUM
ejpam-1802	30	6	,	,	PUNCT
ejpam-1802	30	7	p.245	p.245	NOUN
ejpam-1802	30	8	]	]	PUNCT
ejpam-1802	30	9	)	)	PUNCT
ejpam-1802	30	10	.	.	PUNCT
ejpam-1802	31	1	consequently	consequently	ADV
ejpam-1802	31	2	,	,	PUNCT
ejpam-1802	31	3	(	(	PUNCT
ejpam-1802	31	4	i	i	NOUN
ejpam-1802	31	5	)	)	PUNCT
ejpam-1802	31	6	|r×|=	|r×|=	NOUN
ejpam-1802	31	7	|r|−	|r|−	PROPN
ejpam-1802	31	8	|z(r)|	|z(r)|	PROPN
ejpam-1802	31	9	,	,	PUNCT
ejpam-1802	31	10	and	and	CCONJ
ejpam-1802	31	11	(	(	PUNCT
ejpam-1802	31	12	ii	ii	NOUN
ejpam-1802	31	13	)	)	PUNCT
ejpam-1802	31	14	if	if	SCONJ
ejpam-1802	31	15	r=	r=	ADJ
ejpam-1802	31	16	r1⊕	r1⊕	X
ejpam-1802	31	17	·	·	PUNCT
ejpam-1802	31	18	·	·	PUNCT
ejpam-1802	31	19	·	·	PUNCT
ejpam-1802	32	1	⊕rt	⊕rt	NOUN
ejpam-1802	32	2	is	be	AUX
ejpam-1802	32	3	a	a	DET
ejpam-1802	32	4	direct	direct	ADJ
ejpam-1802	32	5	sum	sum	NOUN
ejpam-1802	32	6	of	of	ADP
ejpam-1802	32	7	rings	ring	NOUN
ejpam-1802	32	8	,	,	PUNCT
ejpam-1802	32	9	then	then	ADV
ejpam-1802	32	10	r×	r×	NOUN
ejpam-1802	32	11	=	=	SYM
ejpam-1802	32	12	r×	r×	NOUN
ejpam-1802	32	13	1	1	NUM
ejpam-1802	32	14	⊕	⊕	PROPN
ejpam-1802	32	15	·	·	PUNCT
ejpam-1802	32	16	·	·	PUNCT
ejpam-1802	32	17	·	·	PUNCT
ejpam-1802	33	1	⊕r×t	⊕r×t	X
ejpam-1802	33	2	.	.	PUNCT
ejpam-1802	34	1	the	the	DET
ejpam-1802	34	2	symbol	symbol	NOUN
ejpam-1802	34	3	j(r	j(r	PROPN
ejpam-1802	34	4	)	)	PUNCT
ejpam-1802	34	5	denotes	denote	VERB
ejpam-1802	34	6	the	the	DET
ejpam-1802	34	7	jacobson	jacobson	PROPN
ejpam-1802	34	8	radical	radical	PROPN
ejpam-1802	34	9	,	,	PUNCT
ejpam-1802	34	10	or	or	CCONJ
ejpam-1802	34	11	briefly	briefly	ADV
ejpam-1802	34	12	the	the	DET
ejpam-1802	34	13	radical	radical	ADJ
ejpam-1802	34	14	,	,	PUNCT
ejpam-1802	34	15	of	of	ADP
ejpam-1802	34	16	r	r	NOUN
ejpam-1802	34	17	,	,	PUNCT
ejpam-1802	34	18	which	which	PRON
ejpam-1802	34	19	can	can	AUX
ejpam-1802	34	20	be	be	AUX
ejpam-1802	34	21	equivalently	equivalently	ADV
ejpam-1802	34	22	defined	define	VERB
ejpam-1802	34	23	as	as	ADP
ejpam-1802	34	24	:	:	PUNCT
ejpam-1802	34	25	(	(	PUNCT
ejpam-1802	34	26	a	a	X
ejpam-1802	34	27	)	)	PUNCT
ejpam-1802	34	28	the	the	DET
ejpam-1802	34	29	intersection	intersection	NOUN
ejpam-1802	34	30	of	of	ADP
ejpam-1802	34	31	all	all	DET
ejpam-1802	34	32	left	leave	VERB
ejpam-1802	34	33	maximal	maximal	ADJ
ejpam-1802	34	34	ideals	ideal	NOUN
ejpam-1802	34	35	of	of	ADP
ejpam-1802	34	36	r	r	NOUN
ejpam-1802	34	37	,	,	PUNCT
ejpam-1802	34	38	or	or	CCONJ
ejpam-1802	34	39	(	(	PUNCT
ejpam-1802	34	40	b	b	X
ejpam-1802	34	41	)	)	PUNCT
ejpam-1802	34	42	the	the	DET
ejpam-1802	34	43	intersection	intersection	NOUN
ejpam-1802	34	44	of	of	ADP
ejpam-1802	34	45	all	all	DET
ejpam-1802	34	46	right	right	ADJ
ejpam-1802	34	47	maximal	maximal	ADJ
ejpam-1802	34	48	ideals	ideal	NOUN
ejpam-1802	34	49	of	of	ADP
ejpam-1802	34	50	r	r	NOUN
ejpam-1802	34	51	,	,	PUNCT
ejpam-1802	34	52	or	or	CCONJ
ejpam-1802	34	53	(	(	PUNCT
ejpam-1802	34	54	c	c	X
ejpam-1802	34	55	)	)	PUNCT
ejpam-1802	34	56	the	the	DET
ejpam-1802	34	57	set	set	NOUN
ejpam-1802	34	58	of	of	ADP
ejpam-1802	34	59	all	all	DET
ejpam-1802	34	60	elements	element	NOUN
ejpam-1802	34	61	r	r	NOUN
ejpam-1802	34	62	∈	∈	NOUN
ejpam-1802	34	63	r	r	NOUN
ejpam-1802	34	64	such	such	ADJ
ejpam-1802	34	65	that	that	DET
ejpam-1802	34	66	1	1	NUM
ejpam-1802	34	67	+	+	NUM
ejpam-1802	34	68	sr	sr	PROPN
ejpam-1802	34	69	is	be	AUX
ejpam-1802	34	70	invertible	invertible	ADJ
ejpam-1802	34	71	for	for	ADP
ejpam-1802	34	72	every	every	DET
ejpam-1802	34	73	s	s	PROPN
ejpam-1802	34	74	∈	∈	PROPN
ejpam-1802	34	75	r.	r.	PROPN
ejpam-1802	34	76	consequently	consequently	ADV
ejpam-1802	34	77	,	,	PUNCT
ejpam-1802	34	78	1	1	NUM
ejpam-1802	34	79	+	+	NUM
ejpam-1802	34	80	j(r	j(r	NOUN
ejpam-1802	34	81	)	)	PUNCT
ejpam-1802	34	82	is	be	AUX
ejpam-1802	34	83	a	a	DET
ejpam-1802	34	84	subgroup	subgroup	NOUN
ejpam-1802	34	85	of	of	ADP
ejpam-1802	34	86	the	the	DET
ejpam-1802	34	87	group	group	NOUN
ejpam-1802	34	88	r×	r×	PROPN
ejpam-1802	34	89	of	of	ADP
ejpam-1802	34	90	invertible	invertible	ADJ
ejpam-1802	34	91	element	element	NOUN
ejpam-1802	34	92	in	in	ADP
ejpam-1802	34	93	r.	r.	PROPN
ejpam-1802	34	94	a	a	DET
ejpam-1802	34	95	finite	finite	PROPN
ejpam-1802	34	96	ring	ring	NOUN
ejpam-1802	34	97	r	r	NOUN
ejpam-1802	34	98	is	be	AUX
ejpam-1802	34	99	semi	semi	ADJ
ejpam-1802	34	100	-	-	ADJ
ejpam-1802	34	101	simple	simple	ADJ
ejpam-1802	34	102	if	if	SCONJ
ejpam-1802	34	103	and	and	CCONJ
ejpam-1802	34	104	only	only	ADV
ejpam-1802	34	105	if	if	SCONJ
ejpam-1802	34	106	j(r	j(r	NOUN
ejpam-1802	34	107	)	)	PUNCT
ejpam-1802	34	108	=	=	PRON
ejpam-1802	35	1	{	{	PUNCT
ejpam-1802	35	2	0	0	NUM
ejpam-1802	35	3	}	}	PUNCT
ejpam-1802	35	4	.	.	PUNCT
ejpam-1802	36	1	for	for	ADP
ejpam-1802	36	2	any	any	DET
ejpam-1802	36	3	finite	finite	NOUN
ejpam-1802	36	4	ring	ring	NOUN
ejpam-1802	36	5	r	r	NOUN
ejpam-1802	36	6	,	,	PUNCT
ejpam-1802	36	7	the	the	DET
ejpam-1802	36	8	quotient	quotient	NOUN
ejpam-1802	36	9	ring	ring	NOUN
ejpam-1802	36	10	r	r	PROPN
ejpam-1802	36	11	/	/	SYM
ejpam-1802	36	12	j(r	j(r	PROPN
ejpam-1802	36	13	)	)	PUNCT
ejpam-1802	36	14	is	be	AUX
ejpam-1802	36	15	semi	semi	ADJ
ejpam-1802	36	16	-	-	ADJ
ejpam-1802	36	17	simple	simple	ADJ
ejpam-1802	36	18	.	.	PUNCT
ejpam-1802	37	1	a	a	DET
ejpam-1802	37	2	finite	finite	NOUN
ejpam-1802	37	3	ring	ring	NOUN
ejpam-1802	37	4	r	r	NOUN
ejpam-1802	37	5	is	be	AUX
ejpam-1802	37	6	local	local	ADJ
ejpam-1802	37	7	if	if	SCONJ
ejpam-1802	37	8	and	and	CCONJ
ejpam-1802	37	9	only	only	ADV
ejpam-1802	37	10	if	if	SCONJ
ejpam-1802	37	11	r	r	NOUN
ejpam-1802	37	12	/	/	SYM
ejpam-1802	37	13	j(r	j(r	PROPN
ejpam-1802	37	14	)	)	PUNCT
ejpam-1802	37	15	is	be	AUX
ejpam-1802	37	16	a	a	DET
ejpam-1802	37	17	finite	finite	ADJ
ejpam-1802	37	18	field	field	NOUN
ejpam-1802	37	19	,	,	PUNCT
ejpam-1802	37	20	called	call	VERB
ejpam-1802	37	21	the	the	DET
ejpam-1802	37	22	residue	residue	NOUN
ejpam-1802	37	23	field	field	NOUN
ejpam-1802	37	24	.	.	PUNCT
ejpam-1802	38	1	notice	notice	VERB
ejpam-1802	38	2	that	that	SCONJ
ejpam-1802	38	3	a	a	DET
ejpam-1802	38	4	semi	semi	ADJ
ejpam-1802	38	5	-	-	ADJ
ejpam-1802	38	6	simple	simple	ADJ
ejpam-1802	38	7	ring	ring	NOUN
ejpam-1802	38	8	is	be	AUX
ejpam-1802	38	9	local	local	ADJ
ejpam-1802	38	10	if	if	SCONJ
ejpam-1802	38	11	and	and	CCONJ
ejpam-1802	38	12	only	only	ADV
ejpam-1802	38	13	if	if	SCONJ
ejpam-1802	38	14	it	it	PRON
ejpam-1802	38	15	is	be	AUX
ejpam-1802	38	16	a	a	DET
ejpam-1802	38	17	field	field	NOUN
ejpam-1802	38	18	.	.	PUNCT
ejpam-1802	39	1	finite	finite	PROPN
ejpam-1802	39	2	fields	field	NOUN
ejpam-1802	39	3	are	be	AUX
ejpam-1802	39	4	denoted	denote	VERB
ejpam-1802	39	5	by	by	ADP
ejpam-1802	39	6	fq	fq	PROPN
ejpam-1802	39	7	,	,	PUNCT
ejpam-1802	39	8	where	where	SCONJ
ejpam-1802	39	9	q=	q=	ADV
ejpam-1802	39	10	|fq|	|fq|	PROPN
ejpam-1802	39	11	is	be	AUX
ejpam-1802	39	12	a	a	DET
ejpam-1802	39	13	power	power	NOUN
ejpam-1802	39	14	of	of	ADP
ejpam-1802	39	15	some	some	DET
ejpam-1802	39	16	prime	prime	NOUN
ejpam-1802	39	17	.	.	PUNCT
ejpam-1802	40	1	if	if	SCONJ
ejpam-1802	40	2	fq	fq	PROPN
ejpam-1802	40	3	is	be	AUX
ejpam-1802	40	4	the	the	DET
ejpam-1802	40	5	residue	residue	NOUN
ejpam-1802	40	6	field	field	NOUN
ejpam-1802	40	7	of	of	ADP
ejpam-1802	40	8	some	some	DET
ejpam-1802	40	9	finite	finite	ADJ
ejpam-1802	40	10	local	local	ADJ
ejpam-1802	40	11	ring	ring	NOUN
ejpam-1802	40	12	having	have	VERB
ejpam-1802	40	13	pn	pn	PROPN
ejpam-1802	40	14	elements	element	NOUN
ejpam-1802	40	15	,	,	PUNCT
ejpam-1802	40	16	then	then	ADV
ejpam-1802	40	17	q=	q=	ADV
ejpam-1802	40	18	pr	pr	NOUN
ejpam-1802	40	19	,	,	PUNCT
ejpam-1802	40	20	for	for	ADP
ejpam-1802	40	21	some	some	DET
ejpam-1802	40	22	1≤	1≤	NUM
ejpam-1802	40	23	r≤	r≤	PROPN
ejpam-1802	40	24	n.	n.	NOUN
ejpam-1802	40	25	in	in	ADP
ejpam-1802	40	26	1969	1969	NUM
ejpam-1802	40	27	,	,	PUNCT
ejpam-1802	40	28	raghavendra	raghavendra	PROPN
ejpam-1802	41	1	[	[	X
ejpam-1802	41	2	5	5	NUM
ejpam-1802	41	3	]	]	PUNCT
ejpam-1802	41	4	proved	prove	VERB
ejpam-1802	41	5	the	the	DET
ejpam-1802	41	6	following	following	NOUN
ejpam-1802	41	7	:	:	PUNCT
ejpam-1802	41	8	proposition	proposition	NOUN
ejpam-1802	41	9	1	1	NUM
ejpam-1802	41	10	.	.	PUNCT
ejpam-1802	42	1	let	let	VERB
ejpam-1802	42	2	r	r	PRON
ejpam-1802	42	3	be	be	AUX
ejpam-1802	42	4	a	a	DET
ejpam-1802	42	5	finite	finite	ADJ
ejpam-1802	42	6	ring	ring	NOUN
ejpam-1802	42	7	with	with	ADP
ejpam-1802	42	8	multiplicative	multiplicative	ADJ
ejpam-1802	42	9	identity	identity	NOUN
ejpam-1802	42	10	1	1	NUM
ejpam-1802	42	11	6=	6=	SYM
ejpam-1802	42	12	0	0	NUM
ejpam-1802	42	13	and	and	CCONJ
ejpam-1802	42	14	suppose	suppose	VERB
ejpam-1802	42	15	that	that	SCONJ
ejpam-1802	42	16	the	the	DET
ejpam-1802	42	17	set	set	NOUN
ejpam-1802	42	18	of	of	ADP
ejpam-1802	42	19	zero	zero	NUM
ejpam-1802	42	20	divisors	divisor	NOUN
ejpam-1802	42	21	z(r	z(r	NOUN
ejpam-1802	42	22	)	)	PUNCT
ejpam-1802	42	23	forms	form	VERB
ejpam-1802	42	24	an	an	DET
ejpam-1802	42	25	additive	additive	ADJ
ejpam-1802	42	26	group	group	NOUN
ejpam-1802	42	27	.	.	PUNCT
ejpam-1802	43	1	then	then	ADV
ejpam-1802	43	2	(	(	PUNCT
ejpam-1802	43	3	i	i	NOUN
ejpam-1802	43	4	)	)	PUNCT
ejpam-1802	43	5	z(r	z(r	NOUN
ejpam-1802	43	6	)	)	PUNCT
ejpam-1802	43	7	is	be	AUX
ejpam-1802	43	8	the	the	DET
ejpam-1802	43	9	jacobson	jacobson	PROPN
ejpam-1802	43	10	radical	radical	PROPN
ejpam-1802	43	11	of	of	ADP
ejpam-1802	43	12	r	r	PROPN
ejpam-1802	43	13	;	;	PUNCT
ejpam-1802	43	14	(	(	PUNCT
ejpam-1802	43	15	ii	ii	NOUN
ejpam-1802	43	16	)	)	PUNCT
ejpam-1802	43	17	|r|=	|r|=	NOUN
ejpam-1802	43	18	pnr	pnr	NOUN
ejpam-1802	43	19	,	,	PUNCT
ejpam-1802	43	20	and	and	CCONJ
ejpam-1802	43	21	|j|=	|j|=	ADJ
ejpam-1802	43	22	p(n−1)r	p(n−1)r	NOUN
ejpam-1802	43	23	for	for	ADP
ejpam-1802	43	24	some	some	DET
ejpam-1802	43	25	prime	prime	ADJ
ejpam-1802	43	26	number	number	NOUN
ejpam-1802	43	27	p	p	NOUN
ejpam-1802	43	28	,	,	PUNCT
ejpam-1802	43	29	and	and	CCONJ
ejpam-1802	43	30	some	some	DET
ejpam-1802	43	31	positive	positive	ADJ
ejpam-1802	43	32	integers	integer	NOUN
ejpam-1802	43	33	n	n	CCONJ
ejpam-1802	43	34	,	,	PUNCT
ejpam-1802	43	35	r	r	NOUN
ejpam-1802	43	36	;	;	PUNCT
ejpam-1802	43	37	(	(	PUNCT
ejpam-1802	43	38	iii	iii	X
ejpam-1802	43	39	)	)	PUNCT
ejpam-1802	43	40	z(r)n	z(r)n	NOUN
ejpam-1802	43	41	=	=	SYM
ejpam-1802	43	42	{	{	PUNCT
ejpam-1802	43	43	0	0	NUM
ejpam-1802	43	44	}	}	PUNCT
ejpam-1802	43	45	;	;	PUNCT
ejpam-1802	43	46	(	(	PUNCT
ejpam-1802	43	47	iv	iv	X
ejpam-1802	43	48	)	)	PUNCT
ejpam-1802	43	49	the	the	DET
ejpam-1802	43	50	characteristic	characteristic	NOUN
ejpam-1802	43	51	of	of	ADP
ejpam-1802	43	52	the	the	DET
ejpam-1802	43	53	ring	ring	NOUN
ejpam-1802	43	54	r	r	NOUN
ejpam-1802	43	55	is	be	AUX
ejpam-1802	43	56	pk	pk	NOUN
ejpam-1802	43	57	for	for	ADP
ejpam-1802	43	58	some	some	DET
ejpam-1802	43	59	integer	integer	NOUN
ejpam-1802	43	60	k	k	NOUN
ejpam-1802	43	61	with	with	ADP
ejpam-1802	43	62	1	1	NUM
ejpam-1802	43	63	≤	≤	NUM
ejpam-1802	43	64	k	k	NOUN
ejpam-1802	43	65	≤	≤	PROPN
ejpam-1802	43	66	n	n	CCONJ
ejpam-1802	43	67	,	,	PUNCT
ejpam-1802	43	68	where	where	SCONJ
ejpam-1802	43	69	p	p	NOUN
ejpam-1802	43	70	is	be	AUX
ejpam-1802	43	71	the	the	DET
ejpam-1802	43	72	same	same	ADJ
ejpam-1802	43	73	prime	prime	ADJ
ejpam-1802	43	74	number	number	NOUN
ejpam-1802	43	75	as	as	ADP
ejpam-1802	43	76	in	in	ADP
ejpam-1802	43	77	(	(	PUNCT
ejpam-1802	43	78	ii	ii	NOUN
ejpam-1802	43	79	)	)	PUNCT
ejpam-1802	43	80	;	;	PUNCT
ejpam-1802	43	81	and	and	CCONJ
ejpam-1802	43	82	(	(	PUNCT
ejpam-1802	43	83	v	v	NOUN
ejpam-1802	43	84	)	)	PUNCT
ejpam-1802	43	85	if	if	SCONJ
ejpam-1802	43	86	the	the	DET
ejpam-1802	43	87	characteristic	characteristic	NOUN
ejpam-1802	43	88	of	of	ADP
ejpam-1802	43	89	the	the	DET
ejpam-1802	43	90	ring	ring	NOUN
ejpam-1802	43	91	r	r	NOUN
ejpam-1802	43	92	is	be	AUX
ejpam-1802	43	93	pn	pn	INTJ
ejpam-1802	43	94	(	(	PUNCT
ejpam-1802	43	95	i.e.	i.e.	X
ejpam-1802	43	96	,	,	PUNCT
ejpam-1802	43	97	the	the	DET
ejpam-1802	43	98	largest	large	ADJ
ejpam-1802	43	99	possible	possible	ADJ
ejpam-1802	43	100	choice	choice	NOUN
ejpam-1802	43	101	of	of	ADP
ejpam-1802	43	102	pk	pk	NOUN
ejpam-1802	43	103	in	in	X
ejpam-1802	43	104	(	(	PUNCT
ejpam-1802	43	105	iv	iv	NOUN
ejpam-1802	43	106	)	)	PUNCT
ejpam-1802	43	107	)	)	PUNCT
ejpam-1802	43	108	,	,	PUNCT
ejpam-1802	43	109	then	then	ADV
ejpam-1802	43	110	r	r	NOUN
ejpam-1802	43	111	is	be	AUX
ejpam-1802	43	112	commutative	commutative	ADJ
ejpam-1802	43	113	.	.	PUNCT
ejpam-1802	44	1	we	we	PRON
ejpam-1802	44	2	also	also	ADV
ejpam-1802	44	3	refer	refer	VERB
ejpam-1802	44	4	to	to	ADP
ejpam-1802	44	5	gilmer	gilmer	PROPN
ejpam-1802	44	6	[	[	X
ejpam-1802	44	7	3]where	3]where	NUM
ejpam-1802	44	8	alternative	alternative	ADJ
ejpam-1802	44	9	proofs	proof	NOUN
ejpam-1802	44	10	for	for	ADP
ejpam-1802	44	11	some	some	PRON
ejpam-1802	44	12	of	of	ADP
ejpam-1802	44	13	these	these	DET
ejpam-1802	44	14	results	result	NOUN
ejpam-1802	44	15	can	can	AUX
ejpam-1802	44	16	be	be	AUX
ejpam-1802	44	17	found	find	VERB
ejpam-1802	44	18	,	,	PUNCT
ejpam-1802	44	19	as	as	ADV
ejpam-1802	44	20	well	well	ADV
ejpam-1802	44	21	as	as	ADP
ejpam-1802	44	22	examples	example	NOUN
ejpam-1802	44	23	showing	show	VERB
ejpam-1802	44	24	that	that	SCONJ
ejpam-1802	44	25	all	all	DET
ejpam-1802	44	26	possible	possible	ADJ
ejpam-1802	44	27	values	value	NOUN
ejpam-1802	44	28	of	of	ADP
ejpam-1802	44	29	n	n	CCONJ
ejpam-1802	44	30	,	,	PUNCT
ejpam-1802	44	31	r	r	NOUN
ejpam-1802	44	32	and	and	CCONJ
ejpam-1802	44	33	k	k	PROPN
ejpam-1802	44	34	in	in	ADP
ejpam-1802	44	35	the	the	DET
ejpam-1802	44	36	above	above	ADJ
ejpam-1802	44	37	proposition	proposition	NOUN
ejpam-1802	44	38	already	already	ADV
ejpam-1802	44	39	appear	appear	VERB
ejpam-1802	44	40	in	in	ADP
ejpam-1802	44	41	the	the	DET
ejpam-1802	44	42	commutative	commutative	ADJ
ejpam-1802	44	43	case	case	NOUN
ejpam-1802	44	44	.	.	PUNCT
ejpam-1802	45	1	m.	m.	NOUN
ejpam-1802	45	2	gonzález	gonzález	PROPN
ejpam-1802	45	3	/	/	SYM
ejpam-1802	45	4	eur	eur	PROPN
ejpam-1802	45	5	.	.	PUNCT
ejpam-1802	46	1	j.	j.	PROPN
ejpam-1802	46	2	pure	pure	PROPN
ejpam-1802	46	3	appl	appl	PROPN
ejpam-1802	46	4	.	.	PROPN
ejpam-1802	46	5	math	math	PROPN
ejpam-1802	46	6	,	,	PUNCT
ejpam-1802	46	7	7	7	NUM
ejpam-1802	46	8	(	(	PUNCT
ejpam-1802	46	9	2014	2014	NUM
ejpam-1802	46	10	)	)	PUNCT
ejpam-1802	46	11	,	,	PUNCT
ejpam-1802	46	12	109	109	NUM
ejpam-1802	46	13	-	-	SYM
ejpam-1802	46	14	113	113	NUM
ejpam-1802	46	15	111	111	NUM
ejpam-1802	46	16	3	3	NUM
ejpam-1802	46	17	.	.	PUNCT
ejpam-1802	47	1	proof	proof	NOUN
ejpam-1802	47	2	of	of	ADP
ejpam-1802	47	3	theorem	theorem	NOUN
ejpam-1802	47	4	1	1	NUM
ejpam-1802	47	5	in	in	ADP
ejpam-1802	47	6	this	this	DET
ejpam-1802	47	7	section	section	NOUN
ejpam-1802	47	8	,	,	PUNCT
ejpam-1802	47	9	we	we	PRON
ejpam-1802	47	10	shall	shall	AUX
ejpam-1802	47	11	provide	provide	VERB
ejpam-1802	47	12	a	a	DET
ejpam-1802	47	13	proof	proof	NOUN
ejpam-1802	47	14	for	for	ADP
ejpam-1802	47	15	theorem	theorem	NOUN
ejpam-1802	47	16	1	1	NUM
ejpam-1802	47	17	.	.	PUNCT
ejpam-1802	48	1	this	this	PRON
ejpam-1802	48	2	also	also	ADV
ejpam-1802	48	3	provides	provide	VERB
ejpam-1802	48	4	a	a	DET
ejpam-1802	48	5	new	new	ADJ
ejpam-1802	48	6	proof	proof	NOUN
ejpam-1802	48	7	for	for	ADP
ejpam-1802	48	8	theorem	theorem	NOUN
ejpam-1802	48	9	2	2	X
ejpam-1802	48	10	.	.	PUNCT
ejpam-1802	49	1	our	our	PRON
ejpam-1802	49	2	idea	idea	NOUN
ejpam-1802	49	3	is	be	AUX
ejpam-1802	49	4	to	to	PART
ejpam-1802	49	5	reduce	reduce	VERB
ejpam-1802	49	6	the	the	DET
ejpam-1802	49	7	statement	statement	NOUN
ejpam-1802	49	8	to	to	ADP
ejpam-1802	49	9	the	the	DET
ejpam-1802	49	10	case	case	NOUN
ejpam-1802	49	11	in	in	ADP
ejpam-1802	49	12	which	which	PRON
ejpam-1802	49	13	r	r	NOUN
ejpam-1802	49	14	is	be	AUX
ejpam-1802	49	15	semi	semi	ADJ
ejpam-1802	49	16	-	-	ADJ
ejpam-1802	49	17	simple	simple	ADJ
ejpam-1802	49	18	.	.	PUNCT
ejpam-1802	50	1	it	it	PRON
ejpam-1802	50	2	is	be	AUX
ejpam-1802	50	3	natural	natural	ADJ
ejpam-1802	50	4	to	to	PART
ejpam-1802	50	5	replace	replace	VERB
ejpam-1802	50	6	r	r	NOUN
ejpam-1802	50	7	by	by	ADP
ejpam-1802	50	8	r	r	NOUN
ejpam-1802	50	9	/	/	SYM
ejpam-1802	50	10	j(r	j(r	PROPN
ejpam-1802	50	11	)	)	PUNCT
ejpam-1802	50	12	and	and	CCONJ
ejpam-1802	50	13	the	the	DET
ejpam-1802	50	14	next	next	ADJ
ejpam-1802	50	15	lemma	lemma	PROPN
ejpam-1802	50	16	tells	tell	VERB
ejpam-1802	50	17	us	we	PRON
ejpam-1802	50	18	that	that	SCONJ
ejpam-1802	50	19	,	,	PUNCT
ejpam-1802	50	20	if	if	SCONJ
ejpam-1802	50	21	we	we	PRON
ejpam-1802	50	22	do	do	VERB
ejpam-1802	50	23	so	so	ADV
ejpam-1802	50	24	,	,	PUNCT
ejpam-1802	50	25	the	the	DET
ejpam-1802	50	26	hypotheses	hypothesis	NOUN
ejpam-1802	50	27	still	still	ADV
ejpam-1802	50	28	hold	hold	VERB
ejpam-1802	50	29	for	for	ADP
ejpam-1802	50	30	r	r	NOUN
ejpam-1802	50	31	/	/	SYM
ejpam-1802	50	32	j(r	j(r	PROPN
ejpam-1802	50	33	)	)	PUNCT
ejpam-1802	50	34	.	.	PUNCT
ejpam-1802	51	1	lemma	lemma	PROPN
ejpam-1802	51	2	1	1	X
ejpam-1802	51	3	.	.	PUNCT
ejpam-1802	52	1	let	let	VERB
ejpam-1802	52	2	p	p	PRON
ejpam-1802	52	3	be	be	AUX
ejpam-1802	52	4	a	a	DET
ejpam-1802	52	5	prime	prime	ADJ
ejpam-1802	52	6	number	number	NOUN
ejpam-1802	52	7	,	,	PUNCT
ejpam-1802	52	8	and	and	CCONJ
ejpam-1802	52	9	let	let	VERB
ejpam-1802	52	10	r	r	PRON
ejpam-1802	52	11	be	be	AUX
ejpam-1802	52	12	a	a	DET
ejpam-1802	52	13	finite	finite	ADJ
ejpam-1802	52	14	ring	ring	NOUN
ejpam-1802	52	15	with	with	ADP
ejpam-1802	52	16	identity	identity	NOUN
ejpam-1802	52	17	such	such	ADJ
ejpam-1802	52	18	that	that	DET
ejpam-1802	52	19	|r|	|r|	NOUN
ejpam-1802	52	20	=	=	NOUN
ejpam-1802	52	21	pm	pm	NOUN
ejpam-1802	52	22	and	and	CCONJ
ejpam-1802	52	23	|z(r)|=	|z(r)|=	VERB
ejpam-1802	52	24	pn	pn	PROPN
ejpam-1802	52	25	,	,	PUNCT
ejpam-1802	52	26	for	for	ADP
ejpam-1802	52	27	some	some	DET
ejpam-1802	52	28	0≤	0≤	ADJ
ejpam-1802	52	29	n	n	CCONJ
ejpam-1802	52	30	<	<	X
ejpam-1802	52	31	m	m	PROPN
ejpam-1802	52	32	,	,	PUNCT
ejpam-1802	52	33	and	and	CCONJ
ejpam-1802	52	34	let	let	VERB
ejpam-1802	52	35	s	s	PRON
ejpam-1802	52	36	:	:	PUNCT
ejpam-1802	52	37	=	=	SYM
ejpam-1802	52	38	r	r	X
ejpam-1802	52	39	/	/	SYM
ejpam-1802	52	40	j(r	j(r	PROPN
ejpam-1802	52	41	)	)	PUNCT
ejpam-1802	52	42	.	.	PUNCT
ejpam-1802	53	1	then	then	ADV
ejpam-1802	53	2	,	,	PUNCT
ejpam-1802	53	3	|s|=	|s|=	DET
ejpam-1802	53	4	pm−k	pm−k	VERB
ejpam-1802	53	5	and	and	CCONJ
ejpam-1802	53	6	|z(s)|=	|z(s)|=	PROPN
ejpam-1802	53	7	pn−k	pn−k	NOUN
ejpam-1802	53	8	,	,	PUNCT
ejpam-1802	53	9	where	where	SCONJ
ejpam-1802	53	10	0≤	0≤	ADJ
ejpam-1802	53	11	k≤	k≤	NOUN
ejpam-1802	53	12	n	n	PART
ejpam-1802	53	13	is	be	AUX
ejpam-1802	53	14	the	the	DET
ejpam-1802	53	15	integer	integer	NOUN
ejpam-1802	53	16	determined	determine	VERB
ejpam-1802	53	17	by	by	ADP
ejpam-1802	53	18	|j(r)|=	|j(r)|=	PROPN
ejpam-1802	53	19	pk	pk	PROPN
ejpam-1802	53	20	.	.	PUNCT
ejpam-1802	53	21	let	let	VERB
ejpam-1802	53	22	π	π	NOUN
ejpam-1802	53	23	:	:	PUNCT
ejpam-1802	53	24	r	r	NOUN
ejpam-1802	53	25	→	→	SYM
ejpam-1802	53	26	s	s	NOUN
ejpam-1802	53	27	=	=	SYM
ejpam-1802	53	28	r	r	NOUN
ejpam-1802	53	29	/	/	SYM
ejpam-1802	53	30	j(r	j(r	PROPN
ejpam-1802	53	31	)	)	PUNCT
ejpam-1802	53	32	be	be	VERB
ejpam-1802	53	33	the	the	DET
ejpam-1802	53	34	quotient	quotient	NOUN
ejpam-1802	53	35	map	map	NOUN
ejpam-1802	53	36	.	.	PUNCT
ejpam-1802	54	1	then	then	ADV
ejpam-1802	54	2	the	the	DET
ejpam-1802	54	3	restriction	restriction	NOUN
ejpam-1802	54	4	π|r×	π|r×	PROPN
ejpam-1802	54	5	:	:	PUNCT
ejpam-1802	54	6	r×	r×	NOUN
ejpam-1802	54	7	→	→	SYM
ejpam-1802	54	8	s×	s×	PROPN
ejpam-1802	54	9	is	be	AUX
ejpam-1802	54	10	a	a	DET
ejpam-1802	54	11	surjective	surjective	ADJ
ejpam-1802	54	12	group	group	NOUN
ejpam-1802	54	13	homomorphism	homomorphism	NOUN
ejpam-1802	54	14	.	.	PUNCT
ejpam-1802	55	1	since	since	SCONJ
ejpam-1802	55	2	ker	ker	PROPN
ejpam-1802	55	3	π|r×	π|r×	PROPN
ejpam-1802	55	4	=	=	PUNCT
ejpam-1802	55	5	1+j(r	1+j(r	NUM
ejpam-1802	55	6	)	)	PUNCT
ejpam-1802	55	7	,	,	PUNCT
ejpam-1802	55	8	we	we	PRON
ejpam-1802	55	9	have	have	VERB
ejpam-1802	55	10	that	that	DET
ejpam-1802	55	11	r×/(1+j(r	r×/(1+j(r	PROPN
ejpam-1802	55	12	)	)	PUNCT
ejpam-1802	55	13	)	)	PUNCT
ejpam-1802	56	1	∼=	∼=	ADV
ejpam-1802	56	2	s×.	s×.	NOUN
ejpam-1802	56	3	consequently	consequently	ADV
ejpam-1802	56	4	,	,	PUNCT
ejpam-1802	56	5	|s×|=	|s×|=	NOUN
ejpam-1802	56	6	|r×|	|r×|	NOUN
ejpam-1802	56	7	|1	|1	X
ejpam-1802	56	8	+	+	NOUN
ejpam-1802	56	9	j(r)|	j(r)|	NOUN
ejpam-1802	56	10	=	=	SYM
ejpam-1802	56	11	|r|−	|r|−	ADJ
ejpam-1802	56	12	|z(r)|	|z(r)|	PROPN
ejpam-1802	56	13	|j(r)|	|j(r)|	PROPN
ejpam-1802	56	14	=	=	PUNCT
ejpam-1802	56	15	|r|	|r|	PROPN
ejpam-1802	56	16	|j(r)|	|j(r)|	PROPN
ejpam-1802	56	17	−	−	PROPN
ejpam-1802	56	18	|z(r)|	|z(r)|	PROPN
ejpam-1802	56	19	|j(r)|	|j(r)|	PROPN
ejpam-1802	56	20	=	=	PROPN
ejpam-1802	56	21	pm−k−	pm−k−	PROPN
ejpam-1802	56	22	pn−k	pn−k	NOUN
ejpam-1802	56	23	,	,	PUNCT
ejpam-1802	56	24	which	which	PRON
ejpam-1802	56	25	implies	imply	VERB
ejpam-1802	56	26	that	that	SCONJ
ejpam-1802	56	27	|z(s)|=	|z(s)|=	PROPN
ejpam-1802	56	28	|s|−	|s|−	NOUN
ejpam-1802	56	29	|s×|=	|s×|=	NOUN
ejpam-1802	56	30	pn−k	pn−k	NOUN
ejpam-1802	56	31	.	.	PUNCT
ejpam-1802	57	1	this	this	PRON
ejpam-1802	57	2	concludes	conclude	VERB
ejpam-1802	57	3	the	the	DET
ejpam-1802	57	4	proof	proof	NOUN
ejpam-1802	57	5	of	of	ADP
ejpam-1802	57	6	the	the	DET
ejpam-1802	57	7	lemma	lemma	PROPN
ejpam-1802	57	8	.	.	PUNCT
ejpam-1802	58	1	now	now	ADV
ejpam-1802	58	2	we	we	PRON
ejpam-1802	58	3	can	can	AUX
ejpam-1802	58	4	prove	prove	VERB
ejpam-1802	58	5	theorem	theorem	ADJ
ejpam-1802	58	6	1	1	NUM
ejpam-1802	58	7	.	.	PUNCT
ejpam-1802	59	1	the	the	DET
ejpam-1802	59	2	forward	forward	ADJ
ejpam-1802	59	3	direction	direction	NOUN
ejpam-1802	59	4	follows	follow	VERB
ejpam-1802	59	5	from	from	ADP
ejpam-1802	59	6	proposition	proposition	NOUN
ejpam-1802	59	7	1	1	NUM
ejpam-1802	59	8	,	,	PUNCT
ejpam-1802	59	9	(	(	PUNCT
ejpam-1802	59	10	ii	ii	NOUN
ejpam-1802	59	11	)	)	PUNCT
ejpam-1802	59	12	.	.	PUNCT
ejpam-1802	60	1	for	for	ADP
ejpam-1802	60	2	the	the	DET
ejpam-1802	60	3	converse	converse	NOUN
ejpam-1802	60	4	,	,	PUNCT
ejpam-1802	60	5	let	let	VERB
ejpam-1802	60	6	r	r	PRON
ejpam-1802	60	7	be	be	AUX
ejpam-1802	60	8	such	such	ADJ
ejpam-1802	60	9	that	that	SCONJ
ejpam-1802	60	10	|r|=	|r|=	PRON
ejpam-1802	60	11	pm	pm	NOUN
ejpam-1802	60	12	and	and	CCONJ
ejpam-1802	60	13	|z(r)|=	|z(r)|=	VERB
ejpam-1802	60	14	pn	pn	AUX
ejpam-1802	60	15	.	.	PROPN
ejpam-1802	60	16	suppose	suppose	VERB
ejpam-1802	60	17	first	first	ADV
ejpam-1802	60	18	that	that	SCONJ
ejpam-1802	60	19	j(r	j(r	NOUN
ejpam-1802	60	20	)	)	PUNCT
ejpam-1802	61	1	=	=	PRON
ejpam-1802	61	2	{	{	PUNCT
ejpam-1802	61	3	0	0	NUM
ejpam-1802	61	4	}	}	PUNCT
ejpam-1802	61	5	.	.	PUNCT
ejpam-1802	62	1	in	in	ADP
ejpam-1802	62	2	this	this	DET
ejpam-1802	62	3	case	case	NOUN
ejpam-1802	62	4	,	,	PUNCT
ejpam-1802	62	5	r	r	NOUN
ejpam-1802	62	6	is	be	AUX
ejpam-1802	62	7	semi	semi	ADJ
ejpam-1802	62	8	-	-	ADJ
ejpam-1802	62	9	simple	simple	ADJ
ejpam-1802	62	10	and	and	CCONJ
ejpam-1802	62	11	the	the	DET
ejpam-1802	62	12	wedderburn	wedderburn	PROPN
ejpam-1802	62	13	-	-	PUNCT
ejpam-1802	62	14	artin	artin	NOUN
ejpam-1802	62	15	theorem	theorem	NOUN
ejpam-1802	62	16	(	(	PUNCT
ejpam-1802	62	17	cf	cf	NOUN
ejpam-1802	62	18	.	.	PUNCT
ejpam-1802	63	1	[	[	X
ejpam-1802	63	2	4	4	NUM
ejpam-1802	63	3	,	,	PUNCT
ejpam-1802	63	4	theorem	theorem	VERB
ejpam-1802	63	5	viii.4	viii.4	PROPN
ejpam-1802	63	6	,	,	PUNCT
ejpam-1802	63	7	pp.128	pp.128	NOUN
ejpam-1802	63	8	–	–	PUNCT
ejpam-1802	63	9	130	130	NUM
ejpam-1802	63	10	]	]	PUNCT
ejpam-1802	63	11	)	)	PUNCT
ejpam-1802	63	12	asserts	assert	VERB
ejpam-1802	63	13	that	that	SCONJ
ejpam-1802	63	14	r	r	NOUN
ejpam-1802	63	15	is	be	AUX
ejpam-1802	63	16	a	a	DET
ejpam-1802	63	17	direct	direct	ADJ
ejpam-1802	63	18	sum	sum	NOUN
ejpam-1802	63	19	of	of	ADP
ejpam-1802	63	20	full	full	ADJ
ejpam-1802	63	21	matrix	matrix	NOUN
ejpam-1802	63	22	rings	ring	NOUN
ejpam-1802	63	23	over	over	ADP
ejpam-1802	63	24	fields	field	NOUN
ejpam-1802	63	25	.	.	PUNCT
ejpam-1802	64	1	more	more	ADV
ejpam-1802	64	2	precisely	precisely	ADV
ejpam-1802	64	3	,	,	PUNCT
ejpam-1802	64	4	there	there	PRON
ejpam-1802	64	5	exist	exist	VERB
ejpam-1802	64	6	finite	finite	ADJ
ejpam-1802	64	7	fields	field	NOUN
ejpam-1802	64	8	fq1	fq1	ADV
ejpam-1802	64	9	,	,	PUNCT
ejpam-1802	64	10	.	.	PUNCT
ejpam-1802	64	11	.	.	PUNCT
ejpam-1802	65	1	.	.	PUNCT
ejpam-1802	66	1	,	,	PUNCT
ejpam-1802	66	2	fqt	fqt	VERB
ejpam-1802	66	3	having	have	VERB
ejpam-1802	66	4	qs	qs	NOUN
ejpam-1802	66	5	=	=	SYM
ejpam-1802	66	6	p	p	X
ejpam-1802	66	7	ds	ds	ADJ
ejpam-1802	66	8	elements	element	NOUN
ejpam-1802	66	9	,	,	PUNCT
ejpam-1802	66	10	for	for	ADP
ejpam-1802	66	11	every	every	DET
ejpam-1802	66	12	s=	s=	NOUN
ejpam-1802	66	13	1	1	NUM
ejpam-1802	66	14	,	,	PUNCT
ejpam-1802	66	15	.	.	PUNCT
ejpam-1802	66	16	.	.	PUNCT
ejpam-1802	67	1	.	.	PUNCT
ejpam-1802	68	1	,	,	PUNCT
ejpam-1802	68	2	t	t	NOUN
ejpam-1802	68	3	and	and	CCONJ
ejpam-1802	68	4	positive	positive	ADJ
ejpam-1802	68	5	integers	integer	NOUN
ejpam-1802	68	6	n1	n1	NOUN
ejpam-1802	68	7	,	,	PUNCT
ejpam-1802	68	8	.	.	PUNCT
ejpam-1802	68	9	.	.	PUNCT
ejpam-1802	69	1	.	.	PUNCT
ejpam-1802	70	1	,	,	PUNCT
ejpam-1802	70	2	nt	not	PART
ejpam-1802	70	3	,	,	PUNCT
ejpam-1802	70	4	such	such	ADJ
ejpam-1802	70	5	that	that	SCONJ
ejpam-1802	70	6	r	r	PROPN
ejpam-1802	70	7	=	=	SYM
ejpam-1802	70	8	mn1	mn1	X
ejpam-1802	70	9	�	�	PROPN
ejpam-1802	70	10	fq1	fq1	CCONJ
ejpam-1802	70	11	�	�	PROPN
ejpam-1802	70	12	⊕	⊕	PROPN
ejpam-1802	70	13	·	·	PUNCT
ejpam-1802	70	14	·	·	PUNCT
ejpam-1802	70	15	·	·	PUNCT
ejpam-1802	70	16	⊕mnt	⊕mnt	NUM
ejpam-1802	70	17	�	�	PROPN
ejpam-1802	70	18	fqt	fqt	PROPN
ejpam-1802	70	19	�	�	PROPN
ejpam-1802	70	20	=	=	SYM
ejpam-1802	70	21	rn1,q1	rn1,q1	NOUN
ejpam-1802	70	22	⊕	⊕	PROPN
ejpam-1802	70	23	·	·	PUNCT
ejpam-1802	70	24	·	·	PUNCT
ejpam-1802	70	25	·	·	PUNCT
ejpam-1802	71	1	⊕rnt	⊕rnt	NUM
ejpam-1802	71	2	,	,	PUNCT
ejpam-1802	71	3	qs	qs	INTJ
ejpam-1802	71	4	,	,	PUNCT
ejpam-1802	71	5	(	(	PUNCT
ejpam-1802	71	6	1	1	X
ejpam-1802	71	7	)	)	PUNCT
ejpam-1802	71	8	where	where	SCONJ
ejpam-1802	71	9	rn	rn	NOUN
ejpam-1802	71	10	,	,	PUNCT
ejpam-1802	71	11	q	q	NOUN
ejpam-1802	71	12	:	:	PUNCT
ejpam-1802	71	13	=	=	SYM
ejpam-1802	71	14	mn(fq	mn(fq	PROPN
ejpam-1802	71	15	)	)	PUNCT
ejpam-1802	71	16	.	.	PUNCT
ejpam-1802	72	1	now	now	ADV
ejpam-1802	72	2	clearly	clearly	ADV
ejpam-1802	72	3	|mn(fq)|	|mn(fq)|	VERB
ejpam-1802	72	4	=	=	PRON
ejpam-1802	73	1	qn	qn	PROPN
ejpam-1802	73	2	2	2	NUM
ejpam-1802	73	3	and	and	CCONJ
ejpam-1802	73	4	the	the	DET
ejpam-1802	73	5	group	group	NOUN
ejpam-1802	73	6	of	of	ADP
ejpam-1802	73	7	invertible	invertible	ADJ
ejpam-1802	73	8	elements	element	NOUN
ejpam-1802	73	9	in	in	ADP
ejpam-1802	73	10	mn(fq	mn(fq	PROPN
ejpam-1802	73	11	)	)	PUNCT
ejpam-1802	73	12	is	be	AUX
ejpam-1802	73	13	the	the	DET
ejpam-1802	73	14	general	general	ADJ
ejpam-1802	73	15	linear	linear	PROPN
ejpam-1802	73	16	group	group	NOUN
ejpam-1802	73	17	gln(fq	gln(fq	PROPN
ejpam-1802	73	18	)	)	PUNCT
ejpam-1802	73	19	which	which	PRON
ejpam-1802	73	20	has	have	AUX
ejpam-1802	73	21	(	(	PUNCT
ejpam-1802	73	22	qn−	qn−	PUNCT
ejpam-1802	73	23	1)(qn−q	1)(qn−q	NUM
ejpam-1802	73	24	)	)	PUNCT
ejpam-1802	73	25	.	.	PUNCT
ejpam-1802	73	26	.	.	PUNCT
ejpam-1802	73	27	.	.	PUNCT
ejpam-1802	74	1	(	(	PUNCT
ejpam-1802	74	2	qn−qn−1	qn−qn−1	PROPN
ejpam-1802	74	3	)	)	PUNCT
ejpam-1802	74	4	=	=	PUNCT
ejpam-1802	74	5	qn(n−1)/2	qn(n−1)/2	PROPN
ejpam-1802	74	6	n∏	n∏	PROPN
ejpam-1802	74	7	k=1	k=1	PROPN
ejpam-1802	74	8	(	(	PUNCT
ejpam-1802	74	9	qk−	qk−	NUM
ejpam-1802	74	10	1	1	NUM
ejpam-1802	74	11	)	)	PUNCT
ejpam-1802	74	12	elements	element	NOUN
ejpam-1802	74	13	(	(	PUNCT
ejpam-1802	74	14	cf	cf	NOUN
ejpam-1802	74	15	.	.	PUNCT
ejpam-1802	75	1	[	[	X
ejpam-1802	75	2	4	4	NUM
ejpam-1802	75	3	,	,	PUNCT
ejpam-1802	75	4	theorem	theorem	ADJ
ejpam-1802	75	5	viii.19	viii.19	NOUN
ejpam-1802	75	6	,	,	PUNCT
ejpam-1802	75	7	p.156	p.156	NOUN
ejpam-1802	75	8	]	]	X
ejpam-1802	75	9	)	)	PUNCT
ejpam-1802	75	10	.	.	PUNCT
ejpam-1802	76	1	since	since	SCONJ
ejpam-1802	76	2	|r|	|r|	NOUN
ejpam-1802	76	3	=	=	SYM
ejpam-1802	76	4	pm	pm	PROPN
ejpam-1802	76	5	,	,	PUNCT
ejpam-1802	76	6	we	we	PRON
ejpam-1802	76	7	have	have	VERB
ejpam-1802	76	8	that	that	PRON
ejpam-1802	76	9	qs	qs	PROPN
ejpam-1802	76	10	=	=	SYM
ejpam-1802	76	11	p	p	X
ejpam-1802	76	12	ds	ds	NOUN
ejpam-1802	76	13	,	,	PUNCT
ejpam-1802	76	14	for	for	ADP
ejpam-1802	76	15	every	every	DET
ejpam-1802	76	16	s=	s=	NOUN
ejpam-1802	76	17	1	1	NUM
ejpam-1802	76	18	,	,	PUNCT
ejpam-1802	76	19	.	.	PUNCT
ejpam-1802	76	20	.	.	PUNCT
ejpam-1802	77	1	.	.	PUNCT
ejpam-1802	78	1	,	,	PUNCT
ejpam-1802	78	2	t	t	PROPN
ejpam-1802	78	3	(	(	PUNCT
ejpam-1802	78	4	that	that	PRON
ejpam-1802	78	5	is	is	ADV
ejpam-1802	78	6	,	,	PUNCT
ejpam-1802	78	7	all	all	DET
ejpam-1802	78	8	the	the	DET
ejpam-1802	78	9	qs	qs	NOUN
ejpam-1802	78	10	’s	’s	PART
ejpam-1802	78	11	are	be	AUX
ejpam-1802	78	12	powers	power	NOUN
ejpam-1802	78	13	of	of	ADP
ejpam-1802	78	14	the	the	DET
ejpam-1802	78	15	same	same	ADJ
ejpam-1802	78	16	prime	prime	NOUN
ejpam-1802	78	17	)	)	PUNCT
ejpam-1802	78	18	and	and	CCONJ
ejpam-1802	78	19	the	the	DET
ejpam-1802	78	20	numbers	number	NOUN
ejpam-1802	78	21	ds	ds	INTJ
ejpam-1802	78	22	satisfy	satisfy	VERB
ejpam-1802	78	23	the	the	DET
ejpam-1802	78	24	equation	equation	NOUN
ejpam-1802	78	25	m=	m=	PRON
ejpam-1802	78	26	∑t	∑t	PROPN
ejpam-1802	78	27	s=1n	s=1n	PROPN
ejpam-1802	78	28	2	2	NUM
ejpam-1802	78	29	sds	sds	NOUN
ejpam-1802	78	30	.	.	PUNCT
ejpam-1802	79	1	since	since	SCONJ
ejpam-1802	79	2	r×	r×	NOUN
ejpam-1802	79	3	=	=	SYM
ejpam-1802	79	4	r×n1,q1	r×n1,q1	PROPN
ejpam-1802	79	5	⊕	⊕	PROPN
ejpam-1802	79	6	·	·	PUNCT
ejpam-1802	79	7	·	·	PUNCT
ejpam-1802	79	8	·	·	PUNCT
ejpam-1802	79	9	⊕r×nt	⊕r×nt	NUM
ejpam-1802	79	10	,	,	PUNCT
ejpam-1802	79	11	qt	qt	NOUN
ejpam-1802	79	12	,	,	PUNCT
ejpam-1802	79	13	we	we	PRON
ejpam-1802	79	14	have	have	VERB
ejpam-1802	79	15	that	that	DET
ejpam-1802	79	16	|z(r)|=|r|−	|z(r)|=|r|−	ADJ
ejpam-1802	79	17	|r×|=	|r×|=	NOUN
ejpam-1802	79	18	|r|−	|r|−	NOUN
ejpam-1802	79	19	|r×n1,q1	|r×n1,q1	VERB
ejpam-1802	79	20	|×	|×	ADV
ejpam-1802	79	21	·	·	PUNCT
ejpam-1802	79	22	·	·	PUNCT
ejpam-1802	79	23	·	·	PUNCT
ejpam-1802	80	1	×	×	NOUN
ejpam-1802	80	2	|r×nt	|r×nt	NOUN
ejpam-1802	80	3	,	,	PUNCT
ejpam-1802	80	4	qt	qt	NOUN
ejpam-1802	80	5	|	|	ADV
ejpam-1802	80	6	=	=	NOUN
ejpam-1802	80	7	pm−	pm−	NUM
ejpam-1802	80	8	t∏	t∏	PROPN
ejpam-1802	80	9	s=1	s=1	X
ejpam-1802	80	10	q	q	PROPN
ejpam-1802	80	11	ns(ns−1)/2	ns(ns−1)/2	PROPN
ejpam-1802	80	12	s	s	PROPN
ejpam-1802	80	13	ns∏	ns∏	NOUN
ejpam-1802	80	14	k=1	k=1	PUNCT
ejpam-1802	80	15	(	(	PUNCT
ejpam-1802	80	16	qks	qks	INTJ
ejpam-1802	80	17	−	−	NOUN
ejpam-1802	80	18	1	1	NUM
ejpam-1802	80	19	)	)	PUNCT
ejpam-1802	80	20	!	!	PUNCT
ejpam-1802	81	1	=	=	PUNCT
ejpam-1802	82	1	t∏	t∏	PROPN
ejpam-1802	82	2	s=1	s=1	X
ejpam-1802	83	1	q	q	PROPN
ejpam-1802	83	2	n2	n2	PROPN
ejpam-1802	83	3	s	s	PROPN
ejpam-1802	83	4	s	s	PART
ejpam-1802	83	5	−	−	PROPN
ejpam-1802	83	6	t∏	t∏	PROPN
ejpam-1802	83	7	s=1	s=1	X
ejpam-1802	83	8	q	q	PROPN
ejpam-1802	83	9	ns(ns−1)/2	ns(ns−1)/2	PROPN
ejpam-1802	83	10	s	s	X
ejpam-1802	83	11	!	!	PUNCT
ejpam-1802	84	1	t∏	t∏	PROPN
ejpam-1802	84	2	s=1	s=1	X
ejpam-1802	85	1	ns∏	ns∏	NOUN
ejpam-1802	85	2	k=1	k=1	PUNCT
ejpam-1802	86	1	(	(	PUNCT
ejpam-1802	86	2	qks	qks	INTJ
ejpam-1802	86	3	−	−	NOUN
ejpam-1802	86	4	1	1	NUM
ejpam-1802	86	5	)	)	PUNCT
ejpam-1802	86	6	!	!	PUNCT
ejpam-1802	87	1	m.	m.	NOUN
ejpam-1802	87	2	gonzález	gonzález	PROPN
ejpam-1802	87	3	/	/	SYM
ejpam-1802	87	4	eur	eur	PROPN
ejpam-1802	87	5	.	.	PUNCT
ejpam-1802	88	1	j.	j.	PROPN
ejpam-1802	88	2	pure	pure	PROPN
ejpam-1802	88	3	appl	appl	PROPN
ejpam-1802	88	4	.	.	PROPN
ejpam-1802	88	5	math	math	PROPN
ejpam-1802	88	6	,	,	PUNCT
ejpam-1802	88	7	7	7	NUM
ejpam-1802	88	8	(	(	PUNCT
ejpam-1802	88	9	2014	2014	NUM
ejpam-1802	88	10	)	)	PUNCT
ejpam-1802	88	11	,	,	PUNCT
ejpam-1802	89	1	109	109	NUM
ejpam-1802	89	2	-	-	SYM
ejpam-1802	89	3	113	113	NUM
ejpam-1802	89	4	112	112	NUM
ejpam-1802	89	5	=	=	SYM
ejpam-1802	89	6	t∏	t∏	PROPN
ejpam-1802	89	7	s=1	s=1	X
ejpam-1802	89	8	q	q	PROPN
ejpam-1802	90	1	ns(ns−1)/2	ns(ns−1)/2	PROPN
ejpam-1802	90	2	s	s	X
ejpam-1802	90	3	t∏	t∏	PROPN
ejpam-1802	90	4	s=1	s=1	X
ejpam-1802	90	5	q	q	X
ejpam-1802	90	6	ns(ns+1)/2	ns(ns+1)/2	PROPN
ejpam-1802	90	7	s	s	PART
ejpam-1802	90	8	−	−	PROPN
ejpam-1802	91	1	t∏	t∏	PROPN
ejpam-1802	91	2	s=1	s=1	X
ejpam-1802	92	1	ns∏	ns∏	NOUN
ejpam-1802	92	2	k=1	k=1	PUNCT
ejpam-1802	93	1	(	(	PUNCT
ejpam-1802	93	2	qks	qks	INTJ
ejpam-1802	93	3	−	−	NOUN
ejpam-1802	93	4	1	1	NUM
ejpam-1802	93	5	)	)	PUNCT
ejpam-1802	93	6	!	!	PUNCT
ejpam-1802	94	1	,	,	PUNCT
ejpam-1802	94	2	and	and	CCONJ
ejpam-1802	94	3	hence	hence	ADV
ejpam-1802	94	4	|z(r)|	|z(r)|	PROPN
ejpam-1802	94	5	is	be	AUX
ejpam-1802	94	6	a	a	DET
ejpam-1802	94	7	power	power	NOUN
ejpam-1802	94	8	of	of	ADP
ejpam-1802	94	9	p	p	NOUN
ejpam-1802	94	10	if	if	SCONJ
ejpam-1802	95	1	and	and	CCONJ
ejpam-1802	95	2	only	only	ADV
ejpam-1802	95	3	if	if	SCONJ
ejpam-1802	95	4	∆	∆	PROPN
ejpam-1802	95	5	:	:	PUNCT
ejpam-1802	95	6	=	=	PUNCT
ejpam-1802	95	7	∏t	∏t	PROPN
ejpam-1802	95	8	s=1q	s=1q	PROPN
ejpam-1802	95	9	ns(ns+1)/2	ns(ns+1)/2	PROPN
ejpam-1802	95	10	s	s	PART
ejpam-1802	95	11	−	−	PROPN
ejpam-1802	95	12	∏t	∏t	NOUN
ejpam-1802	95	13	s=1	s=1	X
ejpam-1802	95	14	∏ns	∏ns	X
ejpam-1802	95	15	k=1(q	k=1(q	X
ejpam-1802	95	16	k	k	PROPN
ejpam-1802	95	17	s	s	PROPN
ejpam-1802	95	18	−	−	PROPN
ejpam-1802	95	19	1	1	NUM
ejpam-1802	95	20	)	)	PUNCT
ejpam-1802	95	21	is	be	AUX
ejpam-1802	95	22	a	a	DET
ejpam-1802	95	23	power	power	NOUN
ejpam-1802	95	24	of	of	ADP
ejpam-1802	95	25	p.	p.	NOUN
ejpam-1802	95	26	since	since	SCONJ
ejpam-1802	95	27	∆	∆	PROPN
ejpam-1802	95	28	≡	≡	PROPN
ejpam-1802	95	29	±1	±1	VERB
ejpam-1802	95	30	mod	mod	PROPN
ejpam-1802	95	31	p	p	X
ejpam-1802	95	32	,	,	PUNCT
ejpam-1802	95	33	and	and	CCONJ
ejpam-1802	95	34	∆	∆	PROPN
ejpam-1802	95	35	6=	6=	ADP
ejpam-1802	95	36	1	1	NUM
ejpam-1802	95	37	unless	unless	SCONJ
ejpam-1802	95	38	n1	n1	NOUN
ejpam-1802	95	39	=	=	SYM
ejpam-1802	95	40	·	·	PUNCT
ejpam-1802	95	41	·	·	PUNCT
ejpam-1802	95	42	·	·	PUNCT
ejpam-1802	96	1	=	=	PUNCT
ejpam-1802	96	2	nt	not	PART
ejpam-1802	96	3	=	=	SYM
ejpam-1802	96	4	1	1	NUM
ejpam-1802	96	5	,	,	PUNCT
ejpam-1802	96	6	and	and	CCONJ
ejpam-1802	96	7	t	t	X
ejpam-1802	96	8	=	=	SYM
ejpam-1802	96	9	1	1	NUM
ejpam-1802	96	10	,	,	PUNCT
ejpam-1802	96	11	we	we	PRON
ejpam-1802	96	12	conclude	conclude	VERB
ejpam-1802	96	13	that	that	SCONJ
ejpam-1802	96	14	|z(r)|	|z(r)|	PROPN
ejpam-1802	96	15	is	be	AUX
ejpam-1802	96	16	a	a	DET
ejpam-1802	96	17	power	power	NOUN
ejpam-1802	96	18	of	of	ADP
ejpam-1802	96	19	p	p	NOUN
ejpam-1802	96	20	if	if	SCONJ
ejpam-1802	96	21	and	and	CCONJ
ejpam-1802	96	22	only	only	ADV
ejpam-1802	96	23	if	if	SCONJ
ejpam-1802	96	24	r	r	NOUN
ejpam-1802	96	25	=	=	PUNCT
ejpam-1802	96	26	fq1	fq1	NOUN
ejpam-1802	96	27	.	.	PUNCT
ejpam-1802	97	1	for	for	ADP
ejpam-1802	97	2	the	the	DET
ejpam-1802	97	3	general	general	ADJ
ejpam-1802	97	4	case	case	NOUN
ejpam-1802	97	5	,	,	PUNCT
ejpam-1802	97	6	we	we	PRON
ejpam-1802	97	7	have	have	AUX
ejpam-1802	97	8	that	that	DET
ejpam-1802	97	9	s	s	NOUN
ejpam-1802	97	10	=	=	SYM
ejpam-1802	97	11	r	r	NOUN
ejpam-1802	97	12	/	/	SYM
ejpam-1802	97	13	j(r	j(r	PROPN
ejpam-1802	97	14	)	)	PUNCT
ejpam-1802	97	15	is	be	AUX
ejpam-1802	97	16	semi	semi	ADJ
ejpam-1802	97	17	-	-	ADJ
ejpam-1802	97	18	simple	simple	ADJ
ejpam-1802	97	19	and	and	CCONJ
ejpam-1802	97	20	,	,	PUNCT
ejpam-1802	97	21	according	accord	VERB
ejpam-1802	97	22	to	to	ADP
ejpam-1802	97	23	lemma	lemma	PROPN
ejpam-1802	97	24	1	1	NUM
ejpam-1802	97	25	,	,	PUNCT
ejpam-1802	97	26	both	both	PRON
ejpam-1802	97	27	|s|	|s|	PROPN
ejpam-1802	97	28	and	and	CCONJ
ejpam-1802	97	29	z(s	z(s	PROPN
ejpam-1802	97	30	)	)	PUNCT
ejpam-1802	97	31	are	be	AUX
ejpam-1802	97	32	powers	power	NOUN
ejpam-1802	97	33	of	of	ADP
ejpam-1802	97	34	p.	p.	NOUN
ejpam-1802	97	35	consequently	consequently	ADV
ejpam-1802	97	36	,	,	PUNCT
ejpam-1802	97	37	by	by	ADP
ejpam-1802	97	38	applying	apply	VERB
ejpam-1802	97	39	the	the	DET
ejpam-1802	97	40	first	first	ADJ
ejpam-1802	97	41	part	part	NOUN
ejpam-1802	97	42	of	of	ADP
ejpam-1802	97	43	the	the	DET
ejpam-1802	97	44	proof	proof	NOUN
ejpam-1802	97	45	,	,	PUNCT
ejpam-1802	97	46	s	s	PART
ejpam-1802	97	47	is	be	AUX
ejpam-1802	97	48	a	a	DET
ejpam-1802	97	49	field	field	NOUN
ejpam-1802	97	50	and	and	CCONJ
ejpam-1802	97	51	hence	hence	ADV
ejpam-1802	97	52	r	r	NOUN
ejpam-1802	97	53	is	be	AUX
ejpam-1802	97	54	a	a	DET
ejpam-1802	97	55	local	local	ADJ
ejpam-1802	97	56	ring	ring	NOUN
ejpam-1802	97	57	.	.	PUNCT
ejpam-1802	98	1	4	4	X
ejpam-1802	98	2	.	.	X
ejpam-1802	98	3	new	new	ADJ
ejpam-1802	98	4	proof	proof	NOUN
ejpam-1802	98	5	of	of	ADP
ejpam-1802	98	6	theorem	theorem	NOUN
ejpam-1802	98	7	2	2	NUM
ejpam-1802	98	8	in	in	ADP
ejpam-1802	98	9	this	this	DET
ejpam-1802	98	10	section	section	NOUN
ejpam-1802	98	11	,	,	PUNCT
ejpam-1802	98	12	we	we	PRON
ejpam-1802	98	13	shall	shall	AUX
ejpam-1802	98	14	provide	provide	VERB
ejpam-1802	98	15	a	a	DET
ejpam-1802	98	16	new	new	ADJ
ejpam-1802	98	17	proof	proof	NOUN
ejpam-1802	98	18	for	for	ADP
ejpam-1802	98	19	theorem	theorem	NOUN
ejpam-1802	98	20	2	2	NUM
ejpam-1802	98	21	based	base	VERB
ejpam-1802	98	22	on	on	ADP
ejpam-1802	98	23	the	the	DET
ejpam-1802	98	24	fact	fact	NOUN
ejpam-1802	98	25	that	that	SCONJ
ejpam-1802	98	26	,	,	PUNCT
ejpam-1802	98	27	for	for	ADP
ejpam-1802	98	28	the	the	DET
ejpam-1802	98	29	case	case	NOUN
ejpam-1802	98	30	in	in	ADP
ejpam-1802	98	31	which	which	PRON
ejpam-1802	98	32	r	r	NOUN
ejpam-1802	98	33	is	be	AUX
ejpam-1802	98	34	a	a	DET
ejpam-1802	98	35	finite	finite	ADJ
ejpam-1802	98	36	commutative	commutative	ADJ
ejpam-1802	98	37	ring	ring	NOUN
ejpam-1802	98	38	,	,	PUNCT
ejpam-1802	98	39	the	the	DET
ejpam-1802	98	40	jacobson	jacobson	PROPN
ejpam-1802	98	41	radical	radical	PROPN
ejpam-1802	98	42	j(r	j(r	PROPN
ejpam-1802	98	43	)	)	PUNCT
ejpam-1802	98	44	coincides	coincide	VERB
ejpam-1802	98	45	with	with	ADP
ejpam-1802	98	46	the	the	DET
ejpam-1802	98	47	ideal	ideal	NOUN
ejpam-1802	98	48	of	of	ADP
ejpam-1802	98	49	all	all	DET
ejpam-1802	98	50	the	the	DET
ejpam-1802	98	51	nilpotent	nilpotent	ADJ
ejpam-1802	98	52	elements	element	NOUN
ejpam-1802	98	53	of	of	ADP
ejpam-1802	98	54	the	the	DET
ejpam-1802	98	55	ring	ring	NOUN
ejpam-1802	98	56	r.	r.	PROPN
ejpam-1802	98	57	let	let	VERB
ejpam-1802	98	58	r	r	NOUN
ejpam-1802	98	59	be	be	AUX
ejpam-1802	98	60	finite	finite	ADJ
ejpam-1802	98	61	commutative	commutative	ADJ
ejpam-1802	98	62	ring	ring	NOUN
ejpam-1802	98	63	with	with	ADP
ejpam-1802	98	64	identity	identity	NOUN
ejpam-1802	98	65	of	of	ADP
ejpam-1802	98	66	order	order	NOUN
ejpam-1802	98	67	|r|=	|r|=	PRON
ejpam-1802	98	68	pm	pm	VERB
ejpam-1802	98	69	and	and	CCONJ
ejpam-1802	98	70	having	have	VERB
ejpam-1802	98	71	|z(r)|	|z(r)|	PROPN
ejpam-1802	98	72	=	=	SYM
ejpam-1802	98	73	pn	pn	PROPN
ejpam-1802	98	74	zero	zero	NUM
ejpam-1802	98	75	divisors	divisor	NOUN
ejpam-1802	98	76	.	.	PUNCT
ejpam-1802	99	1	we	we	PRON
ejpam-1802	99	2	want	want	VERB
ejpam-1802	99	3	to	to	PART
ejpam-1802	99	4	prove	prove	VERB
ejpam-1802	99	5	that	that	SCONJ
ejpam-1802	99	6	r	r	NOUN
ejpam-1802	99	7	is	be	AUX
ejpam-1802	99	8	local	local	ADJ
ejpam-1802	99	9	,	,	PUNCT
ejpam-1802	99	10	for	for	ADP
ejpam-1802	99	11	which	which	PRON
ejpam-1802	99	12	it	it	PRON
ejpam-1802	99	13	is	be	AUX
ejpam-1802	99	14	enough	enough	ADJ
ejpam-1802	99	15	to	to	PART
ejpam-1802	99	16	prove	prove	VERB
ejpam-1802	99	17	that	that	SCONJ
ejpam-1802	99	18	z(r	z(r	NOUN
ejpam-1802	99	19	)	)	PUNCT
ejpam-1802	99	20	=	=	SYM
ejpam-1802	99	21	j(r	j(r	PROPN
ejpam-1802	99	22	)	)	PUNCT
ejpam-1802	99	23	.	.	PUNCT
ejpam-1802	100	1	moreover	moreover	ADV
ejpam-1802	100	2	,	,	PUNCT
ejpam-1802	100	3	since	since	SCONJ
ejpam-1802	100	4	j(r	j(r	PROPN
ejpam-1802	100	5	)	)	PUNCT
ejpam-1802	100	6	⊂	⊂	PROPN
ejpam-1802	100	7	z(r	z(r	NOUN
ejpam-1802	100	8	)	)	PUNCT
ejpam-1802	100	9	,	,	PUNCT
ejpam-1802	100	10	it	it	PRON
ejpam-1802	100	11	is	be	AUX
ejpam-1802	100	12	enough	enough	ADJ
ejpam-1802	100	13	to	to	PART
ejpam-1802	100	14	prove	prove	VERB
ejpam-1802	100	15	that	that	SCONJ
ejpam-1802	100	16	|j(r)|	|j(r)|	PROPN
ejpam-1802	100	17	=	=	SYM
ejpam-1802	100	18	|z(r)|	|z(r)|	PROPN
ejpam-1802	100	19	.	.	PUNCT
ejpam-1802	101	1	since	since	SCONJ
ejpam-1802	101	2	|r×|	|r×|	NOUN
ejpam-1802	101	3	=	=	SYM
ejpam-1802	101	4	|r|−	|r|−	ADJ
ejpam-1802	101	5	|z(r)|	|z(r)|	PROPN
ejpam-1802	101	6	=	=	SYM
ejpam-1802	101	7	(	(	PUNCT
ejpam-1802	101	8	pm−n	pm−n	NOUN
ejpam-1802	101	9	−	−	PROPN
ejpam-1802	101	10	1)pn	1)pn	PROPN
ejpam-1802	101	11	,	,	PUNCT
ejpam-1802	101	12	and	and	CCONJ
ejpam-1802	101	13	the	the	DET
ejpam-1802	101	14	integers	integer	NOUN
ejpam-1802	101	15	pm−n	pm−n	VERB
ejpam-1802	101	16	−	−	PROPN
ejpam-1802	101	17	1	1	NUM
ejpam-1802	101	18	and	and	CCONJ
ejpam-1802	101	19	pn	pn	PROPN
ejpam-1802	101	20	are	be	AUX
ejpam-1802	101	21	relatively	relatively	ADV
ejpam-1802	101	22	prime	prime	ADJ
ejpam-1802	101	23	,	,	PUNCT
ejpam-1802	101	24	there	there	PRON
ejpam-1802	101	25	exists	exist	VERB
ejpam-1802	101	26	a	a	DET
ejpam-1802	101	27	sylow	sylow	NOUN
ejpam-1802	101	28	p	p	NOUN
ejpam-1802	101	29	-	-	PUNCT
ejpam-1802	101	30	subgroup	subgroup	NOUN
ejpam-1802	101	31	of	of	ADP
ejpam-1802	101	32	|r×|	|r×|	NOUN
ejpam-1802	101	33	,	,	PUNCT
ejpam-1802	101	34	say	say	VERB
ejpam-1802	101	35	g	g	NOUN
ejpam-1802	101	36	,	,	PUNCT
ejpam-1802	101	37	having	have	VERB
ejpam-1802	101	38	order	order	NOUN
ejpam-1802	101	39	pn	pn	X
ejpam-1802	101	40	(	(	PUNCT
ejpam-1802	101	41	cf	cf	NOUN
ejpam-1802	101	42	.	.	PUNCT
ejpam-1802	102	1	wielandt	wielandt	PROPN
ejpam-1802	103	1	[	[	X
ejpam-1802	103	2	6	6	NUM
ejpam-1802	103	3	]	]	NUM
ejpam-1802	103	4	)	)	PUNCT
ejpam-1802	103	5	.	.	PUNCT
ejpam-1802	104	1	we	we	PRON
ejpam-1802	104	2	claim	claim	VERB
ejpam-1802	104	3	that	that	SCONJ
ejpam-1802	104	4	g	g	PROPN
ejpam-1802	104	5	⊂	⊂	PROPN
ejpam-1802	104	6	1	1	NUM
ejpam-1802	104	7	+	+	NUM
ejpam-1802	104	8	j(r	j(r	NOUN
ejpam-1802	104	9	)	)	PUNCT
ejpam-1802	104	10	.	.	PUNCT
ejpam-1802	105	1	indeed	indeed	ADV
ejpam-1802	105	2	,	,	PUNCT
ejpam-1802	105	3	the	the	DET
ejpam-1802	105	4	characteristic	characteristic	NOUN
ejpam-1802	105	5	of	of	ADP
ejpam-1802	105	6	r	r	NOUN
ejpam-1802	105	7	is	be	AUX
ejpam-1802	105	8	a	a	DET
ejpam-1802	105	9	power	power	NOUN
ejpam-1802	105	10	of	of	ADP
ejpam-1802	105	11	p	p	NOUN
ejpam-1802	105	12	,	,	PUNCT
ejpam-1802	105	13	and	and	CCONJ
ejpam-1802	105	14	in	in	ADP
ejpam-1802	105	15	particular	particular	ADJ
ejpam-1802	105	16	p	p	NOUN
ejpam-1802	105	17	is	be	AUX
ejpam-1802	105	18	nilpotent	nilpotent	ADJ
ejpam-1802	105	19	,	,	PUNCT
ejpam-1802	105	20	so	so	SCONJ
ejpam-1802	105	21	p	p	PROPN
ejpam-1802	105	22	∈	∈	PROPN
ejpam-1802	105	23	j(r	j(r	PROPN
ejpam-1802	105	24	)	)	PUNCT
ejpam-1802	105	25	.	.	PUNCT
ejpam-1802	106	1	we	we	PRON
ejpam-1802	106	2	have	have	VERB
ejpam-1802	106	3	that	that	PRON
ejpam-1802	106	4	(	(	PUNCT
ejpam-1802	106	5	a+	a+	PUNCT
ejpam-1802	106	6	b)p	b)p	NOUN
ejpam-1802	106	7	=	=	SYM
ejpam-1802	106	8	ap	ap	PROPN
ejpam-1802	106	9	+	+	NUM
ejpam-1802	106	10	bp	bp	PROPN
ejpam-1802	107	1	+	+	CCONJ
ejpam-1802	107	2	c	c	X
ejpam-1802	107	3	,	,	PUNCT
ejpam-1802	107	4	where	where	SCONJ
ejpam-1802	107	5	c	c	PROPN
ejpam-1802	107	6	is	be	AUX
ejpam-1802	107	7	nilpotent	nilpotent	ADJ
ejpam-1802	107	8	,	,	PUNCT
ejpam-1802	107	9	whence	whence	NOUN
ejpam-1802	107	10	c	c	PROPN
ejpam-1802	107	11	∈	∈	PROPN
ejpam-1802	107	12	j(r	j(r	PROPN
ejpam-1802	107	13	)	)	PUNCT
ejpam-1802	107	14	.	.	PUNCT
ejpam-1802	108	1	let	let	VERB
ejpam-1802	108	2	g	g	PROPN
ejpam-1802	108	3	∈	∈	PROPN
ejpam-1802	108	4	g	g	PROPN
ejpam-1802	108	5	be	be	AUX
ejpam-1802	108	6	an	an	DET
ejpam-1802	108	7	arbitrary	arbitrary	ADJ
ejpam-1802	108	8	element	element	NOUN
ejpam-1802	108	9	.	.	PUNCT
ejpam-1802	109	1	then	then	ADV
ejpam-1802	109	2	gp	gp	PROPN
ejpam-1802	110	1	n	n	PROPN
ejpam-1802	110	2	=	=	SYM
ejpam-1802	110	3	1	1	NUM
ejpam-1802	110	4	(	(	PUNCT
ejpam-1802	110	5	because	because	SCONJ
ejpam-1802	110	6	pn	pn	PROPN
ejpam-1802	110	7	is	be	AUX
ejpam-1802	110	8	the	the	DET
ejpam-1802	110	9	order	order	NOUN
ejpam-1802	110	10	of	of	ADP
ejpam-1802	110	11	g	g	NOUN
ejpam-1802	110	12	)	)	PUNCT
ejpam-1802	110	13	.	.	PUNCT
ejpam-1802	111	1	hence	hence	ADV
ejpam-1802	111	2	we	we	PRON
ejpam-1802	111	3	can	can	AUX
ejpam-1802	111	4	conclude	conclude	VERB
ejpam-1802	111	5	that	that	PRON
ejpam-1802	111	6	(	(	PUNCT
ejpam-1802	111	7	g−	g−	PROPN
ejpam-1802	111	8	1)p	1)p	NUM
ejpam-1802	111	9	n	n	NOUN
ejpam-1802	111	10	=	=	NOUN
ejpam-1802	111	11	gp	gp	NOUN
ejpam-1802	111	12	n	n	CCONJ
ejpam-1802	111	13	−	−	PROPN
ejpam-1802	111	14	1	1	NUM
ejpam-1802	111	15	+	+	NUM
ejpam-1802	111	16	d	d	NOUN
ejpam-1802	111	17	=	=	SYM
ejpam-1802	111	18	d	d	PROPN
ejpam-1802	111	19	∈	∈	PROPN
ejpam-1802	111	20	j(r	j(r	PROPN
ejpam-1802	111	21	)	)	PUNCT
ejpam-1802	111	22	(	(	PUNCT
ejpam-1802	111	23	even	even	ADV
ejpam-1802	111	24	for	for	ADP
ejpam-1802	111	25	the	the	DET
ejpam-1802	111	26	case	case	NOUN
ejpam-1802	111	27	p	p	X
ejpam-1802	111	28	=	=	SYM
ejpam-1802	111	29	2	2	NUM
ejpam-1802	111	30	because	because	SCONJ
ejpam-1802	111	31	1	1	NUM
ejpam-1802	111	32	=	=	SYM
ejpam-1802	111	33	−1	−1	NOUN
ejpam-1802	111	34	+	+	X
ejpam-1802	111	35	p	p	NOUN
ejpam-1802	111	36	)	)	PUNCT
ejpam-1802	111	37	and	and	CCONJ
ejpam-1802	111	38	so	so	ADV
ejpam-1802	111	39	g−	g−	ADJ
ejpam-1802	111	40	1	1	NUM
ejpam-1802	111	41	∈	∈	PROPN
ejpam-1802	111	42	j(r	j(r	PROPN
ejpam-1802	111	43	)	)	PUNCT
ejpam-1802	111	44	(	(	PUNCT
ejpam-1802	111	45	because	because	SCONJ
ejpam-1802	111	46	,	,	PUNCT
ejpam-1802	111	47	if	if	SCONJ
ejpam-1802	111	48	some	some	DET
ejpam-1802	111	49	power	power	NOUN
ejpam-1802	111	50	of	of	ADP
ejpam-1802	111	51	a	a	DET
ejpam-1802	111	52	∈	∈	NOUN
ejpam-1802	111	53	r	r	NOUN
ejpam-1802	111	54	belongs	belong	VERB
ejpam-1802	111	55	to	to	ADP
ejpam-1802	111	56	j(r	j(r	PROPN
ejpam-1802	111	57	)	)	PUNCT
ejpam-1802	111	58	,	,	PUNCT
ejpam-1802	111	59	then	then	ADV
ejpam-1802	111	60	a	a	DET
ejpam-1802	111	61	∈	∈	PROPN
ejpam-1802	111	62	j(r	j(r	PROPN
ejpam-1802	111	63	)	)	PUNCT
ejpam-1802	111	64	)	)	PUNCT
ejpam-1802	111	65	.	.	PUNCT
ejpam-1802	112	1	we	we	PRON
ejpam-1802	112	2	have	have	AUX
ejpam-1802	112	3	proved	prove	VERB
ejpam-1802	112	4	that	that	SCONJ
ejpam-1802	112	5	g	g	PROPN
ejpam-1802	112	6	∈	∈	PROPN
ejpam-1802	112	7	1	1	NUM
ejpam-1802	112	8	+	+	NUM
ejpam-1802	112	9	j(r	j(r	NOUN
ejpam-1802	112	10	)	)	PUNCT
ejpam-1802	112	11	,	,	PUNCT
ejpam-1802	112	12	for	for	ADP
ejpam-1802	112	13	every	every	DET
ejpam-1802	112	14	g	g	PROPN
ejpam-1802	112	15	∈	∈	PROPN
ejpam-1802	112	16	g	g	NOUN
ejpam-1802	112	17	,	,	PUNCT
ejpam-1802	112	18	that	that	ADV
ejpam-1802	112	19	is	is	ADV
ejpam-1802	112	20	,	,	PUNCT
ejpam-1802	112	21	g	g	PROPN
ejpam-1802	112	22	⊂	⊂	PROPN
ejpam-1802	112	23	1	1	NUM
ejpam-1802	112	24	+	+	NUM
ejpam-1802	112	25	j(r	j(r	NOUN
ejpam-1802	112	26	)	)	PUNCT
ejpam-1802	112	27	,	,	PUNCT
ejpam-1802	112	28	as	as	SCONJ
ejpam-1802	112	29	claimed	claim	VERB
ejpam-1802	112	30	.	.	PUNCT
ejpam-1802	113	1	finally	finally	ADV
ejpam-1802	113	2	,	,	PUNCT
ejpam-1802	113	3	pn	pn	PROPN
ejpam-1802	113	4	=	=	PUNCT
ejpam-1802	113	5	|g|	|g|	PROPN
ejpam-1802	113	6	≤	≤	NOUN
ejpam-1802	113	7	|1	|1	X
ejpam-1802	113	8	+	+	NOUN
ejpam-1802	113	9	j(r)|	j(r)|	NOUN
ejpam-1802	113	10	=	=	SYM
ejpam-1802	113	11	|j(r)|	|j(r)|	PROPN
ejpam-1802	113	12	≤	≤	PROPN
ejpam-1802	113	13	|z(r)|	|z(r)|	PROPN
ejpam-1802	113	14	=	=	SYM
ejpam-1802	113	15	pn	pn	PROPN
ejpam-1802	113	16	and	and	CCONJ
ejpam-1802	113	17	hence	hence	ADV
ejpam-1802	113	18	|j(r)|	|j(r)|	PROPN
ejpam-1802	113	19	=	=	SYM
ejpam-1802	113	20	pn	pn	PROPN
ejpam-1802	113	21	=	=	PUNCT
ejpam-1802	113	22	|z(r)|	|z(r)|	PROPN
ejpam-1802	113	23	,	,	PUNCT
ejpam-1802	113	24	whence	whence	NOUN
ejpam-1802	113	25	j(r	j(r	PROPN
ejpam-1802	113	26	)	)	PUNCT
ejpam-1802	113	27	=	=	PUNCT
ejpam-1802	114	1	z(r	z(r	NOUN
ejpam-1802	114	2	)	)	PUNCT
ejpam-1802	114	3	.	.	PUNCT
ejpam-1802	115	1	5	5	X
ejpam-1802	115	2	.	.	X
ejpam-1802	115	3	final	final	ADJ
ejpam-1802	115	4	remarks	remark	NOUN
ejpam-1802	115	5	concerning	concern	VERB
ejpam-1802	115	6	the	the	DET
ejpam-1802	115	7	proof	proof	NOUN
ejpam-1802	115	8	of	of	ADP
ejpam-1802	115	9	theorem	theorem	NOUN
ejpam-1802	115	10	1	1	NUM
ejpam-1802	115	11	,	,	PUNCT
ejpam-1802	115	12	the	the	DET
ejpam-1802	115	13	present	present	ADJ
ejpam-1802	115	14	version	version	NOUN
ejpam-1802	115	15	of	of	ADP
ejpam-1802	115	16	wedderburn	wedderburn	PROPN
ejpam-1802	115	17	-	-	PUNCT
ejpam-1802	115	18	artin	artin	NOUN
ejpam-1802	115	19	theorem	theorem	NOUN
ejpam-1802	115	20	given	give	VERB
ejpam-1802	115	21	by	by	ADP
ejpam-1802	115	22	[	[	X
ejpam-1802	115	23	4	4	NUM
ejpam-1802	115	24	,	,	PUNCT
ejpam-1802	115	25	theorem	theorem	VERB
ejpam-1802	115	26	viii.4	viii.4	NOUN
ejpam-1802	115	27	,	,	PUNCT
ejpam-1802	115	28	pp.128–130	pp.128–130	NOUN
ejpam-1802	115	29	]	]	PUNCT
ejpam-1802	115	30	,	,	PUNCT
ejpam-1802	115	31	asserts	assert	VERB
ejpam-1802	115	32	that	that	SCONJ
ejpam-1802	115	33	there	there	PRON
ejpam-1802	115	34	exist	exist	VERB
ejpam-1802	115	35	finite	finite	ADJ
ejpam-1802	115	36	fields	field	NOUN
ejpam-1802	115	37	fq1	fq1	ADV
ejpam-1802	115	38	,	,	PUNCT
ejpam-1802	115	39	.	.	PUNCT
ejpam-1802	115	40	.	.	PUNCT
ejpam-1802	116	1	.	.	PUNCT
ejpam-1802	117	1	,	,	PUNCT
ejpam-1802	117	2	fqt	fqt	NOUN
ejpam-1802	117	3	and	and	CCONJ
ejpam-1802	117	4	some	some	DET
ejpam-1802	117	5	positive	positive	ADJ
ejpam-1802	117	6	integers	integer	NOUN
ejpam-1802	117	7	n1	n1	NOUN
ejpam-1802	117	8	,	,	PUNCT
ejpam-1802	117	9	.	.	PUNCT
ejpam-1802	117	10	.	.	PUNCT
ejpam-1802	117	11	.	.	PUNCT
ejpam-1802	118	1	,	,	PUNCT
ejpam-1802	118	2	nt	not	PART
ejpam-1802	118	3	such	such	ADJ
ejpam-1802	118	4	that	that	SCONJ
ejpam-1802	118	5	r	r	PROPN
ejpam-1802	118	6	=	=	SYM
ejpam-1802	118	7	mn1	mn1	X
ejpam-1802	118	8	�	�	PROPN
ejpam-1802	118	9	fq1	fq1	CCONJ
ejpam-1802	118	10	�	�	PROPN
ejpam-1802	118	11	⊕	⊕	PROPN
ejpam-1802	118	12	·	·	PUNCT
ejpam-1802	118	13	·	·	PUNCT
ejpam-1802	118	14	·	·	PUNCT
ejpam-1802	118	15	⊕mnt	⊕mnt	NUM
ejpam-1802	118	16	�	�	PROPN
ejpam-1802	118	17	fqt	fqt	PROPN
ejpam-1802	118	18	�	�	PROPN
ejpam-1802	118	19	.	.	PUNCT
ejpam-1802	119	1	(	(	PUNCT
ejpam-1802	119	2	2	2	X
ejpam-1802	119	3	)	)	PUNCT
ejpam-1802	119	4	the	the	DET
ejpam-1802	119	5	fact	fact	NOUN
ejpam-1802	119	6	that	that	SCONJ
ejpam-1802	119	7	fqs	fqs	PROPN
ejpam-1802	119	8	has	have	AUX
ejpam-1802	119	9	qs	qs	NOUN
ejpam-1802	119	10	=	=	SYM
ejpam-1802	119	11	p	p	X
ejpam-1802	119	12	ds	ds	ADJ
ejpam-1802	119	13	elements	element	NOUN
ejpam-1802	119	14	,	,	PUNCT
ejpam-1802	119	15	for	for	ADP
ejpam-1802	119	16	every	every	DET
ejpam-1802	119	17	s=	s=	NOUN
ejpam-1802	119	18	1	1	NUM
ejpam-1802	119	19	,	,	PUNCT
ejpam-1802	119	20	.	.	PUNCT
ejpam-1802	119	21	.	.	PUNCT
ejpam-1802	120	1	.	.	PUNCT
ejpam-1802	121	1	,	,	PUNCT
ejpam-1802	121	2	t	t	PROPN
ejpam-1802	121	3	,	,	PUNCT
ejpam-1802	121	4	is	be	AUX
ejpam-1802	121	5	a	a	DET
ejpam-1802	121	6	consequence	consequence	NOUN
ejpam-1802	121	7	of	of	ADP
ejpam-1802	121	8	two	two	NUM
ejpam-1802	121	9	facts	fact	NOUN
ejpam-1802	121	10	:	:	PUNCT
ejpam-1802	121	11	(	(	PUNCT
ejpam-1802	121	12	i	i	NOUN
ejpam-1802	121	13	)	)	PUNCT
ejpam-1802	121	14	�	�	PROPN
ejpam-1802	121	15	�	�	PROPN
ejpam-1802	121	16	�	�	PROPN
ejpam-1802	121	17	mns	mns	PROPN
ejpam-1802	121	18	�	�	PROPN
ejpam-1802	121	19	fqs	fqs	PROPN
ejpam-1802	121	20	�	�	PROPN
ejpam-1802	121	21	�	�	PROPN
ejpam-1802	121	22	�	�	PROPN
ejpam-1802	121	23	�	�	PROPN
ejpam-1802	121	24	divides	divide	VERB
ejpam-1802	121	25	|r|	|r|	NOUN
ejpam-1802	121	26	for	for	ADP
ejpam-1802	121	27	every	every	DET
ejpam-1802	121	28	s=	s=	NOUN
ejpam-1802	121	29	1	1	NUM
ejpam-1802	121	30	,	,	PUNCT
ejpam-1802	121	31	.	.	PUNCT
ejpam-1802	121	32	.	.	PUNCT
ejpam-1802	122	1	.	.	PUNCT
ejpam-1802	123	1	,	,	PUNCT
ejpam-1802	123	2	t	t	PROPN
ejpam-1802	123	3	,	,	PUNCT
ejpam-1802	123	4	and	and	CCONJ
ejpam-1802	123	5	(	(	PUNCT
ejpam-1802	123	6	ii	ii	NOUN
ejpam-1802	123	7	)	)	PUNCT
ejpam-1802	123	8	the	the	DET
ejpam-1802	123	9	assumption	assumption	NOUN
ejpam-1802	123	10	that	that	SCONJ
ejpam-1802	123	11	|r|=	|r|=	PRON
ejpam-1802	123	12	pm	pm	VERB
ejpam-1802	123	13	.	.	PUNCT
ejpam-1802	124	1	references	reference	NOUN
ejpam-1802	124	2	113	113	NUM
ejpam-1802	124	3	finally	finally	ADV
ejpam-1802	124	4	,	,	PUNCT
ejpam-1802	124	5	we	we	PRON
ejpam-1802	124	6	have	have	VERB
ejpam-1802	124	7	that	that	DET
ejpam-1802	124	8	pm	pm	NOUN
ejpam-1802	124	9	=	=	SYM
ejpam-1802	125	1	|r|=	|r|=	NUM
ejpam-1802	125	2	�	�	PROPN
ejpam-1802	125	3	�	�	PROPN
ejpam-1802	125	4	�	�	PROPN
ejpam-1802	125	5	mn1	mn1	PROPN
ejpam-1802	125	6	�	�	PROPN
ejpam-1802	125	7	fq1	fq1	CCONJ
ejpam-1802	125	8	�	�	PROPN
ejpam-1802	125	9	�	�	PROPN
ejpam-1802	125	10	�	�	PROPN
ejpam-1802	125	11	�	�	PROPN
ejpam-1802	125	12	×	×	NOUN
ejpam-1802	125	13	·	·	PUNCT
ejpam-1802	125	14	·	·	PUNCT
ejpam-1802	125	15	·	·	PUNCT
ejpam-1802	125	16	×	×	PROPN
ejpam-1802	125	17	�	�	PROPN
ejpam-1802	125	18	�	�	PROPN
ejpam-1802	125	19	�	�	PROPN
ejpam-1802	125	20	mnt	mnt	VERB
ejpam-1802	125	21	�	�	PROPN
ejpam-1802	125	22	fqt	fqt	PROPN
ejpam-1802	125	23	�	�	PROPN
ejpam-1802	125	24	�	�	PROPN
ejpam-1802	125	25	�	�	PROPN
ejpam-1802	125	26	�	�	PROPN
ejpam-1802	125	27	=	=	SYM
ejpam-1802	125	28	t∏	t∏	PROPN
ejpam-1802	125	29	s=1	s=1	X
ejpam-1802	125	30	pn	pn	PROPN
ejpam-1802	125	31	2	2	NUM
ejpam-1802	125	32	sds	sds	NOUN
ejpam-1802	126	1	and	and	CCONJ
ejpam-1802	126	2	hence	hence	ADV
ejpam-1802	126	3	we	we	PRON
ejpam-1802	126	4	obtain	obtain	VERB
ejpam-1802	126	5	the	the	DET
ejpam-1802	126	6	condition	condition	NOUN
ejpam-1802	126	7	m=	m=	PRON
ejpam-1802	126	8	∑t	∑t	PROPN
ejpam-1802	126	9	s=1n	s=1n	PROPN
ejpam-1802	126	10	2	2	NUM
ejpam-1802	126	11	sds	sds	NOUN
ejpam-1802	126	12	.	.	PUNCT
ejpam-1802	127	1	acknowledgements	acknowledgement	VERB
ejpam-1802	127	2	the	the	DET
ejpam-1802	127	3	author	author	NOUN
ejpam-1802	127	4	is	be	AUX
ejpam-1802	127	5	indebted	indebte	VERB
ejpam-1802	127	6	to	to	ADP
ejpam-1802	127	7	professor	professor	PROPN
ejpam-1802	127	8	marek	marek	PROPN
ejpam-1802	127	9	wójtowicz	wójtowicz	NOUN
ejpam-1802	127	10	for	for	ADP
ejpam-1802	127	11	making	make	VERB
ejpam-1802	127	12	many	many	ADJ
ejpam-1802	127	13	important	important	ADJ
ejpam-1802	127	14	observations	observation	NOUN
ejpam-1802	127	15	regarding	regard	VERB
ejpam-1802	127	16	the	the	DET
ejpam-1802	127	17	composition	composition	NOUN
ejpam-1802	127	18	of	of	ADP
ejpam-1802	127	19	this	this	DET
ejpam-1802	127	20	paper	paper	NOUN
ejpam-1802	127	21	.	.	PUNCT
ejpam-1802	128	1	references	reference	NOUN
ejpam-1802	128	2	[	[	X
ejpam-1802	128	3	1	1	NUM
ejpam-1802	128	4	]	]	PUNCT
ejpam-1802	128	5	m.	m.	NOUN
ejpam-1802	128	6	behboodi	behboodi	NOUN
ejpam-1802	128	7	and	and	CCONJ
ejpam-1802	128	8	r.	r.	PROPN
ejpam-1802	128	9	beyranvand	beyranvand	PROPN
ejpam-1802	128	10	.	.	PUNCT
ejpam-1802	129	1	on	on	ADP
ejpam-1802	129	2	the	the	DET
ejpam-1802	129	3	structure	structure	NOUN
ejpam-1802	129	4	of	of	ADP
ejpam-1802	129	5	commutative	commutative	ADJ
ejpam-1802	129	6	rings	ring	NOUN
ejpam-1802	129	7	with	with	ADP
ejpam-1802	129	8	pk1	pk1	PROPN
ejpam-1802	129	9	1	1	NUM
ejpam-1802	129	10	·	·	PUNCT
ejpam-1802	129	11	·	·	PUNCT
ejpam-1802	129	12	·	·	PUNCT
ejpam-1802	129	13	pknn	pknn	NOUN
ejpam-1802	129	14	(	(	PUNCT
ejpam-1802	129	15	1	1	NUM
ejpam-1802	129	16	≤	≤	NUM
ejpam-1802	129	17	k	k	X
ejpam-1802	129	18	≤	≤	NUM
ejpam-1802	129	19	7	7	NUM
ejpam-1802	129	20	)	)	PUNCT
ejpam-1802	129	21	zero	zero	NUM
ejpam-1802	129	22	divisors	divisor	NOUN
ejpam-1802	129	23	.	.	PUNCT
ejpam-1802	130	1	european	european	ADJ
ejpam-1802	130	2	journal	journal	PROPN
ejpam-1802	130	3	of	of	ADP
ejpam-1802	130	4	pure	pure	ADJ
ejpam-1802	130	5	and	and	CCONJ
ejpam-1802	130	6	applied	applied	ADJ
ejpam-1802	130	7	mathematics	mathematic	NOUN
ejpam-1802	130	8	,	,	PUNCT
ejpam-1802	130	9	3(2):303	3(2):303	NUM
ejpam-1802	130	10	–	–	PUNCT
ejpam-1802	130	11	316	316	NUM
ejpam-1802	130	12	,	,	PUNCT
ejpam-1802	130	13	2010	2010	NUM
ejpam-1802	130	14	.	.	PUNCT
ejpam-1802	131	1	[	[	X
ejpam-1802	131	2	2	2	NUM
ejpam-1802	131	3	]	]	X
ejpam-1802	131	4	n.	n.	PROPN
ejpam-1802	131	5	ganesan	ganesan	PROPN
ejpam-1802	131	6	.	.	PUNCT
ejpam-1802	131	7	properties	property	NOUN
ejpam-1802	131	8	of	of	ADP
ejpam-1802	131	9	rings	ring	NOUN
ejpam-1802	131	10	with	with	ADP
ejpam-1802	131	11	a	a	DET
ejpam-1802	131	12	finite	finite	ADJ
ejpam-1802	131	13	number	number	NOUN
ejpam-1802	131	14	of	of	ADP
ejpam-1802	131	15	zero	zero	NUM
ejpam-1802	131	16	-	-	PUNCT
ejpam-1802	131	17	divisors	divisors	PROPN
ejpam-1802	131	18	ii	ii	PROPN
ejpam-1802	131	19	.	.	PROPN
ejpam-1802	131	20	mathematische	mathematische	PROPN
ejpam-1802	131	21	annale	annale	PROPN
ejpam-1802	131	22	,	,	PUNCT
ejpam-1802	131	23	161:241–246	161:241–246	NUM
ejpam-1802	131	24	,	,	PUNCT
ejpam-1802	131	25	1965	1965	NUM
ejpam-1802	131	26	.	.	PUNCT
ejpam-1802	132	1	[	[	X
ejpam-1802	132	2	3	3	X
ejpam-1802	132	3	]	]	X
ejpam-1802	132	4	r.	r.	PROPN
ejpam-1802	132	5	gilmer	gilmer	PROPN
ejpam-1802	132	6	.	.	PUNCT
ejpam-1802	133	1	zero	zero	NUM
ejpam-1802	133	2	-	-	PUNCT
ejpam-1802	133	3	divisors	divisor	NOUN
ejpam-1802	133	4	in	in	ADP
ejpam-1802	133	5	commutative	commutative	ADJ
ejpam-1802	133	6	rings	ring	NOUN
ejpam-1802	133	7	.	.	PUNCT
ejpam-1802	134	1	the	the	DET
ejpam-1802	134	2	american	american	PROPN
ejpam-1802	134	3	mathematical	mathematical	PROPN
ejpam-1802	134	4	monthly	monthly	ADJ
ejpam-1802	134	5	,	,	PUNCT
ejpam-1802	134	6	93(5):382–387	93(5):382–387	NOUN
ejpam-1802	134	7	,	,	PUNCT
ejpam-1802	134	8	1986	1986	NUM
ejpam-1802	134	9	.	.	PUNCT
ejpam-1802	135	1	[	[	X
ejpam-1802	135	2	4	4	X
ejpam-1802	135	3	]	]	PUNCT
ejpam-1802	135	4	b.	b.	PROPN
ejpam-1802	135	5	r.	r.	PROPN
ejpam-1802	135	6	mcdonald	mcdonald	PROPN
ejpam-1802	135	7	.	.	PUNCT
ejpam-1802	136	1	finite	finite	PROPN
ejpam-1802	136	2	rings	ring	NOUN
ejpam-1802	136	3	with	with	ADP
ejpam-1802	136	4	identity	identity	NOUN
ejpam-1802	136	5	.	.	PUNCT
ejpam-1802	137	1	marcel	marcel	PROPN
ejpam-1802	137	2	dekker	dekker	PROPN
ejpam-1802	137	3	,	,	PUNCT
ejpam-1802	137	4	new	new	PROPN
ejpam-1802	137	5	york	york	PROPN
ejpam-1802	137	6	,	,	PUNCT
ejpam-1802	137	7	1974	1974	NUM
ejpam-1802	137	8	.	.	PUNCT
ejpam-1802	138	1	[	[	X
ejpam-1802	138	2	5	5	NUM
ejpam-1802	138	3	]	]	PUNCT
ejpam-1802	138	4	r.	r.	NOUN
ejpam-1802	138	5	raghavendran	raghavendran	PROPN
ejpam-1802	138	6	.	.	PUNCT
ejpam-1802	139	1	finite	finite	PROPN
ejpam-1802	139	2	associative	associative	PROPN
ejpam-1802	139	3	rings	ring	NOUN
ejpam-1802	139	4	.	.	PUNCT
ejpam-1802	140	1	compositio	compositio	PROPN
ejpam-1802	140	2	mathematica	mathematica	PROPN
ejpam-1802	140	3	,	,	PUNCT
ejpam-1802	140	4	21:195–229	21:195–229	PROPN
ejpam-1802	140	5	,	,	PUNCT
ejpam-1802	140	6	1969	1969	NUM
ejpam-1802	140	7	.	.	PUNCT
ejpam-1802	141	1	[	[	X
ejpam-1802	141	2	6	6	NUM
ejpam-1802	141	3	]	]	PUNCT
ejpam-1802	141	4	h.	h.	NOUN
ejpam-1802	141	5	wielandt	wielandt	PROPN
ejpam-1802	141	6	.	.	PUNCT
ejpam-1802	142	1	ein	ein	PROPN
ejpam-1802	142	2	beweis	beweis	PROPN
ejpam-1802	142	3	für	für	PROPN
ejpam-1802	142	4	die	die	VERB
ejpam-1802	142	5	existenz	existenz	PROPN
ejpam-1802	142	6	der	der	PROPN
ejpam-1802	142	7	sylowgruppen	sylowgruppen	VERB
ejpam-1802	142	8	.	.	PUNCT
ejpam-1802	143	1	archiv	archiv	PROPN
ejpam-1802	143	2	der	der	PROPN
ejpam-1802	143	3	mathematik	mathematik	PROPN
ejpam-1802	143	4	,	,	PUNCT
ejpam-1802	143	5	10(1):401–402	10(1):401–402	PROPN
ejpam-1802	143	6	,	,	PUNCT
ejpam-1802	143	7	1959	1959	NUM
ejpam-1802	143	8	.	.	PUNCT