id	sid	tid	token	lemma	pos
ejpam-6011	1	1	european	european	PROPN
ejpam-6011	1	2	journal	journal	PROPN
ejpam-6011	1	3	of	of	ADP
ejpam-6011	1	4	pure	pure	ADJ
ejpam-6011	1	5	and	and	CCONJ
ejpam-6011	1	6	applied	applied	ADJ
ejpam-6011	1	7	mathematics	mathematic	NOUN
ejpam-6011	1	8	2025	2025	NUM
ejpam-6011	1	9	,	,	PUNCT
ejpam-6011	1	10	vol	vol	NOUN
ejpam-6011	1	11	.	.	PROPN
ejpam-6011	1	12	18	18	NUM
ejpam-6011	1	13	,	,	PUNCT
ejpam-6011	1	14	issue	issue	NOUN
ejpam-6011	1	15	2	2	NUM
ejpam-6011	1	16	,	,	PUNCT
ejpam-6011	1	17	article	article	NOUN
ejpam-6011	1	18	number	number	NOUN
ejpam-6011	1	19	6011	6011	NUM
ejpam-6011	1	20	issn	issn	VERB
ejpam-6011	1	21	1307	1307	NUM
ejpam-6011	1	22	-	-	SYM
ejpam-6011	1	23	5543	5543	NUM
ejpam-6011	1	24	–	–	PUNCT
ejpam-6011	1	25	ejpam.com	ejpam.com	X
ejpam-6011	1	26	published	publish	VERB
ejpam-6011	1	27	by	by	ADP
ejpam-6011	1	28	new	new	PROPN
ejpam-6011	1	29	york	york	PROPN
ejpam-6011	1	30	business	business	PROPN
ejpam-6011	1	31	global	global	PROPN
ejpam-6011	1	32	nj	nj	PROPN
ejpam-6011	1	33	-	-	PUNCT
ejpam-6011	1	34	abelian	abelian	ADJ
ejpam-6011	1	35	rings	ring	NOUN
ejpam-6011	1	36	:	:	PUNCT
ejpam-6011	1	37	an	an	DET
ejpam-6011	1	38	abelian	abelian	ADJ
ejpam-6011	1	39	-	-	PUNCT
ejpam-6011	1	40	like	like	ADJ
ejpam-6011	1	41	approach	approach	NOUN
ejpam-6011	1	42	muhammad	muhammad	PROPN
ejpam-6011	1	43	saad1	saad1	PROPN
ejpam-6011	1	44	,	,	PUNCT
ejpam-6011	1	45	samia	samia	PROPN
ejpam-6011	1	46	m.	m.	PROPN
ejpam-6011	1	47	abdelwahab2,3,∗	abdelwahab2,3,∗	PROPN
ejpam-6011	1	48	1	1	NUM
ejpam-6011	1	49	department	department	NOUN
ejpam-6011	1	50	of	of	ADP
ejpam-6011	1	51	mathematics	mathematic	NOUN
ejpam-6011	1	52	and	and	CCONJ
ejpam-6011	1	53	computer	computer	NOUN
ejpam-6011	1	54	science	science	NOUN
ejpam-6011	1	55	,	,	PUNCT
ejpam-6011	1	56	faculty	faculty	NOUN
ejpam-6011	1	57	of	of	ADP
ejpam-6011	1	58	science	science	NOUN
ejpam-6011	1	59	,	,	PUNCT
ejpam-6011	1	60	alexandria	alexandria	PROPN
ejpam-6011	1	61	university	university	PROPN
ejpam-6011	1	62	,	,	PUNCT
ejpam-6011	1	63	21511	21511	NUM
ejpam-6011	1	64	alexandria	alexandria	PROPN
ejpam-6011	1	65	,	,	PUNCT
ejpam-6011	1	66	egypt	egypt	PROPN
ejpam-6011	1	67	2	2	NUM
ejpam-6011	1	68	department	department	NOUN
ejpam-6011	1	69	of	of	ADP
ejpam-6011	1	70	mathematics	mathematic	NOUN
ejpam-6011	1	71	,	,	PUNCT
ejpam-6011	1	72	faculty	faculty	NOUN
ejpam-6011	1	73	of	of	ADP
ejpam-6011	1	74	science	science	NOUN
ejpam-6011	1	75	,	,	PUNCT
ejpam-6011	1	76	helwan	helwan	PROPN
ejpam-6011	1	77	university	university	PROPN
ejpam-6011	1	78	,	,	PUNCT
ejpam-6011	1	79	ain	ain	PROPN
ejpam-6011	1	80	helwan	helwan	PROPN
ejpam-6011	1	81	,	,	PUNCT
ejpam-6011	1	82	11790	11790	NUM
ejpam-6011	1	83	helwan	helwan	NOUN
ejpam-6011	1	84	,	,	PUNCT
ejpam-6011	1	85	egypt	egypt	PROPN
ejpam-6011	1	86	3department	3department	NUM
ejpam-6011	1	87	of	of	ADP
ejpam-6011	1	88	mathematics	mathematic	NOUN
ejpam-6011	1	89	,	,	PUNCT
ejpam-6011	1	90	college	college	NOUN
ejpam-6011	1	91	of	of	ADP
ejpam-6011	1	92	science	science	NOUN
ejpam-6011	1	93	,	,	PUNCT
ejpam-6011	1	94	qassim	qassim	PROPN
ejpam-6011	1	95	university	university	PROPN
ejpam-6011	1	96	,	,	PUNCT
ejpam-6011	1	97	p.o	p.o	PROPN
ejpam-6011	1	98	.	.	PROPN
ejpam-6011	1	99	box	box	PROPN
ejpam-6011	1	100	6644	6644	NUM
ejpam-6011	1	101	,	,	PUNCT
ejpam-6011	1	102	51452	51452	NUM
ejpam-6011	1	103	buraidah	buraidah	NOUN
ejpam-6011	1	104	,	,	PUNCT
ejpam-6011	1	105	saudi	saudi	PROPN
ejpam-6011	1	106	arabia	arabia	PROPN
ejpam-6011	1	107	abstract	abstract	NOUN
ejpam-6011	1	108	.	.	PUNCT
ejpam-6011	2	1	this	this	DET
ejpam-6011	2	2	article	article	NOUN
ejpam-6011	2	3	extends	extend	VERB
ejpam-6011	2	4	the	the	DET
ejpam-6011	2	5	concept	concept	NOUN
ejpam-6011	2	6	of	of	ADP
ejpam-6011	2	7	nj	nj	PROPN
ejpam-6011	2	8	-	-	PUNCT
ejpam-6011	2	9	semicommutative	semicommutative	NOUN
ejpam-6011	2	10	rings	ring	NOUN
ejpam-6011	2	11	to	to	PART
ejpam-6011	2	12	introduce	introduce	VERB
ejpam-6011	2	13	the	the	DET
ejpam-6011	2	14	broader	broad	ADJ
ejpam-6011	2	15	class	class	NOUN
ejpam-6011	2	16	of	of	ADP
ejpam-6011	2	17	nj	nj	PROPN
ejpam-6011	2	18	-	-	PUNCT
ejpam-6011	2	19	abelian	abelian	ADJ
ejpam-6011	2	20	rings	ring	NOUN
ejpam-6011	2	21	,	,	PUNCT
ejpam-6011	2	22	which	which	PRON
ejpam-6011	2	23	are	be	AUX
ejpam-6011	2	24	defined	define	VERB
ejpam-6011	2	25	by	by	ADP
ejpam-6011	2	26	properties	property	NOUN
ejpam-6011	2	27	involving	involve	VERB
ejpam-6011	2	28	nilpotent	nilpotent	ADJ
ejpam-6011	2	29	elements	element	NOUN
ejpam-6011	2	30	and	and	CCONJ
ejpam-6011	2	31	the	the	DET
ejpam-6011	2	32	jacobson	jacobson	PROPN
ejpam-6011	2	33	radical	radical	PROPN
ejpam-6011	2	34	.	.	PUNCT
ejpam-6011	3	1	we	we	PRON
ejpam-6011	3	2	investigate	investigate	VERB
ejpam-6011	3	3	the	the	DET
ejpam-6011	3	4	unique	unique	ADJ
ejpam-6011	3	5	algebraic	algebraic	ADJ
ejpam-6011	3	6	properties	property	NOUN
ejpam-6011	3	7	of	of	ADP
ejpam-6011	3	8	nj	nj	PROPN
ejpam-6011	3	9	-	-	PUNCT
ejpam-6011	3	10	abelian	abelian	ADJ
ejpam-6011	3	11	rings	ring	NOUN
ejpam-6011	3	12	and	and	CCONJ
ejpam-6011	3	13	analyze	analyze	VERB
ejpam-6011	3	14	their	their	PRON
ejpam-6011	3	15	relationships	relationship	NOUN
ejpam-6011	3	16	with	with	ADP
ejpam-6011	3	17	various	various	ADJ
ejpam-6011	3	18	types	type	NOUN
ejpam-6011	3	19	of	of	ADP
ejpam-6011	3	20	rings	ring	NOUN
ejpam-6011	3	21	,	,	PUNCT
ejpam-6011	3	22	including	include	VERB
ejpam-6011	3	23	abelian	abelian	NOUN
ejpam-6011	3	24	,	,	PUNCT
ejpam-6011	3	25	reduced	reduce	VERB
ejpam-6011	3	26	,	,	PUNCT
ejpam-6011	3	27	j	j	PROPN
ejpam-6011	3	28	-	-	PUNCT
ejpam-6011	3	29	clean	clean	ADJ
ejpam-6011	3	30	,	,	PUNCT
ejpam-6011	3	31	local	local	ADJ
ejpam-6011	3	32	,	,	PUNCT
ejpam-6011	3	33	and	and	CCONJ
ejpam-6011	3	34	dedekind	dedekind	ADJ
ejpam-6011	3	35	-	-	PUNCT
ejpam-6011	3	36	finite	finite	NOUN
ejpam-6011	3	37	rings	ring	NOUN
ejpam-6011	3	38	.	.	PUNCT
ejpam-6011	4	1	in	in	ADP
ejpam-6011	4	2	particular	particular	ADJ
ejpam-6011	4	3	,	,	PUNCT
ejpam-6011	4	4	we	we	PRON
ejpam-6011	4	5	show	show	VERB
ejpam-6011	4	6	that	that	SCONJ
ejpam-6011	4	7	every	every	DET
ejpam-6011	4	8	nj	nj	ADJ
ejpam-6011	4	9	-	-	PUNCT
ejpam-6011	4	10	semicommutative	semicommutative	NOUN
ejpam-6011	4	11	ring	ring	NOUN
ejpam-6011	4	12	is	be	AUX
ejpam-6011	4	13	nj	nj	NOUN
ejpam-6011	4	14	-	-	PUNCT
ejpam-6011	4	15	abelian	abelian	ADJ
ejpam-6011	4	16	,	,	PUNCT
ejpam-6011	4	17	much	much	ADV
ejpam-6011	4	18	like	like	SCONJ
ejpam-6011	4	19	every	every	DET
ejpam-6011	4	20	semicommutative	semicommutative	NOUN
ejpam-6011	4	21	ring	ring	NOUN
ejpam-6011	4	22	is	be	AUX
ejpam-6011	4	23	abelian	abelian	ADJ
ejpam-6011	4	24	.	.	PUNCT
ejpam-6011	5	1	2020	2020	NUM
ejpam-6011	5	2	mathematics	mathematic	NOUN
ejpam-6011	5	3	subject	subject	NOUN
ejpam-6011	5	4	classifications	classification	NOUN
ejpam-6011	5	5	:	:	PUNCT
ejpam-6011	5	6	16u80	16u80	NUM
ejpam-6011	5	7	,	,	PUNCT
ejpam-6011	5	8	16u99	16u99	NUM
ejpam-6011	5	9	,	,	PUNCT
ejpam-6011	5	10	16u40	16u40	NUM
ejpam-6011	5	11	key	key	ADJ
ejpam-6011	5	12	words	word	NOUN
ejpam-6011	5	13	and	and	CCONJ
ejpam-6011	5	14	phrases	phrase	NOUN
ejpam-6011	5	15	:	:	PUNCT
ejpam-6011	5	16	nj	nj	NOUN
ejpam-6011	5	17	-	-	PUNCT
ejpam-6011	5	18	abelian	abelian	PROPN
ejpam-6011	5	19	,	,	PUNCT
ejpam-6011	5	20	j	j	NOUN
ejpam-6011	5	21	-	-	PUNCT
ejpam-6011	5	22	abelian	abelian	PROPN
ejpam-6011	5	23	,	,	PUNCT
ejpam-6011	5	24	nj	nj	NOUN
ejpam-6011	5	25	-	-	PUNCT
ejpam-6011	5	26	semicommutative	semicommutative	NOUN
ejpam-6011	5	27	,	,	PUNCT
ejpam-6011	5	28	j	j	NOUN
ejpam-6011	5	29	-	-	PUNCT
ejpam-6011	5	30	reduced	reduce	VERB
ejpam-6011	5	31	1	1	NUM
ejpam-6011	5	32	.	.	PUNCT
ejpam-6011	5	33	introduction	introduction	NOUN
ejpam-6011	5	34	though	though	SCONJ
ejpam-6011	5	35	r	r	NOUN
ejpam-6011	5	36	is	be	AUX
ejpam-6011	5	37	an	an	DET
ejpam-6011	5	38	associative	associative	ADJ
ejpam-6011	5	39	ring	ring	NOUN
ejpam-6011	5	40	with	with	ADP
ejpam-6011	5	41	an	an	DET
ejpam-6011	5	42	identity	identity	NOUN
ejpam-6011	5	43	,	,	PUNCT
ejpam-6011	5	44	j(r	j(r	PROPN
ejpam-6011	5	45	)	)	PUNCT
ejpam-6011	5	46	is	be	AUX
ejpam-6011	5	47	the	the	DET
ejpam-6011	5	48	jacobson	jacobson	PROPN
ejpam-6011	5	49	radical	radical	PROPN
ejpam-6011	5	50	of	of	ADP
ejpam-6011	5	51	r	r	NOUN
ejpam-6011	5	52	,	,	PUNCT
ejpam-6011	5	53	and	and	CCONJ
ejpam-6011	5	54	n(r	n(r	NOUN
ejpam-6011	5	55	)	)	PUNCT
ejpam-6011	5	56	is	be	AUX
ejpam-6011	5	57	the	the	DET
ejpam-6011	5	58	set	set	NOUN
ejpam-6011	5	59	of	of	ADP
ejpam-6011	5	60	nilpotent	nilpotent	ADJ
ejpam-6011	5	61	elements	element	NOUN
ejpam-6011	5	62	of	of	ADP
ejpam-6011	5	63	r.	r.	PROPN
ejpam-6011	5	64	a	a	DET
ejpam-6011	5	65	ring	ring	NOUN
ejpam-6011	5	66	r	r	NOUN
ejpam-6011	5	67	is	be	AUX
ejpam-6011	5	68	called	call	VERB
ejpam-6011	5	69	semiprimative	semiprimative	ADJ
ejpam-6011	5	70	if	if	SCONJ
ejpam-6011	5	71	j(r	j(r	NOUN
ejpam-6011	5	72	)	)	PUNCT
ejpam-6011	6	1	=	=	SYM
ejpam-6011	6	2	0	0	PUNCT
ejpam-6011	7	1	and	and	CCONJ
ejpam-6011	7	2	reduced	reduce	VERB
ejpam-6011	7	3	if	if	SCONJ
ejpam-6011	7	4	n(r	n(r	PRON
ejpam-6011	7	5	)	)	PUNCT
ejpam-6011	7	6	=	=	SYM
ejpam-6011	8	1	0	0	X
ejpam-6011	8	2	.	.	PUNCT
ejpam-6011	9	1	an	an	DET
ejpam-6011	9	2	idempotent	idempotent	ADJ
ejpam-6011	9	3	e	e	NOUN
ejpam-6011	9	4	of	of	ADP
ejpam-6011	9	5	a	a	DET
ejpam-6011	9	6	ring	ring	NOUN
ejpam-6011	9	7	r	r	NOUN
ejpam-6011	9	8	is	be	AUX
ejpam-6011	9	9	said	say	VERB
ejpam-6011	9	10	to	to	PART
ejpam-6011	9	11	be	be	AUX
ejpam-6011	9	12	left	leave	VERB
ejpam-6011	9	13	(	(	PUNCT
ejpam-6011	9	14	resp	resp	NOUN
ejpam-6011	9	15	.	.	PUNCT
ejpam-6011	10	1	right	right	ADJ
ejpam-6011	10	2	)	)	PUNCT
ejpam-6011	10	3	semicentral	semicentral	ADJ
ejpam-6011	11	1	if	if	SCONJ
ejpam-6011	11	2	(	(	PUNCT
ejpam-6011	11	3	1−e)re	1−e)re	NUM
ejpam-6011	11	4	=	=	SYM
ejpam-6011	11	5	0	0	NUM
ejpam-6011	11	6	(	(	PUNCT
ejpam-6011	11	7	resp	resp	NOUN
ejpam-6011	11	8	.	.	PUNCT
ejpam-6011	11	9	er(1−e	er(1−e	X
ejpam-6011	11	10	)	)	PUNCT
ejpam-6011	11	11	=	=	NOUN
ejpam-6011	11	12	0	0	NUM
ejpam-6011	11	13	)	)	PUNCT
ejpam-6011	11	14	.	.	PUNCT
ejpam-6011	12	1	if	if	SCONJ
ejpam-6011	12	2	an	an	DET
ejpam-6011	12	3	idempotent	idempotent	NOUN
ejpam-6011	12	4	e	e	NOUN
ejpam-6011	12	5	is	be	AUX
ejpam-6011	12	6	both	both	PRON
ejpam-6011	12	7	left	left	ADJ
ejpam-6011	12	8	and	and	CCONJ
ejpam-6011	12	9	right	right	ADJ
ejpam-6011	12	10	semicentral	semicentral	NOUN
ejpam-6011	12	11	,	,	PUNCT
ejpam-6011	12	12	then	then	ADV
ejpam-6011	12	13	it	it	PRON
ejpam-6011	12	14	is	be	AUX
ejpam-6011	12	15	central	central	ADJ
ejpam-6011	12	16	.	.	PUNCT
ejpam-6011	13	1	a	a	DET
ejpam-6011	13	2	ring	ring	NOUN
ejpam-6011	13	3	r	r	NOUN
ejpam-6011	13	4	is	be	AUX
ejpam-6011	13	5	abelian	abelian	ADJ
ejpam-6011	13	6	if	if	SCONJ
ejpam-6011	13	7	all	all	PRON
ejpam-6011	13	8	of	of	ADP
ejpam-6011	13	9	its	its	PRON
ejpam-6011	13	10	idempotents	idempotent	NOUN
ejpam-6011	13	11	are	be	AUX
ejpam-6011	13	12	central	central	ADJ
ejpam-6011	13	13	.	.	PUNCT
ejpam-6011	14	1	a	a	DET
ejpam-6011	14	2	ring	ring	NOUN
ejpam-6011	14	3	r	r	NOUN
ejpam-6011	14	4	is	be	AUX
ejpam-6011	14	5	called	call	VERB
ejpam-6011	14	6	j	j	NOUN
ejpam-6011	14	7	-	-	NOUN
ejpam-6011	14	8	abelian	abelian	ADJ
ejpam-6011	14	9	if	if	SCONJ
ejpam-6011	14	10	ae−	ae−	NUM
ejpam-6011	14	11	ea	ea	NUM
ejpam-6011	14	12	∈	∈	PROPN
ejpam-6011	14	13	j(r	j(r	PROPN
ejpam-6011	14	14	)	)	PUNCT
ejpam-6011	14	15	for	for	ADP
ejpam-6011	14	16	all	all	DET
ejpam-6011	14	17	e2	e2	PROPN
ejpam-6011	14	18	=	=	SYM
ejpam-6011	14	19	e	e	NOUN
ejpam-6011	14	20	,	,	PUNCT
ejpam-6011	14	21	a	a	DET
ejpam-6011	14	22	∈	∈	PROPN
ejpam-6011	14	23	r.	r.	PROPN
ejpam-6011	14	24	abelian	abelian	PROPN
ejpam-6011	14	25	rings	ring	NOUN
ejpam-6011	14	26	are	be	AUX
ejpam-6011	14	27	easily	easily	ADV
ejpam-6011	14	28	shown	show	VERB
ejpam-6011	14	29	to	to	PART
ejpam-6011	14	30	be	be	AUX
ejpam-6011	14	31	j	j	NOUN
ejpam-6011	14	32	-	-	NOUN
ejpam-6011	14	33	abelian	abelian	ADJ
ejpam-6011	14	34	,	,	PUNCT
ejpam-6011	14	35	but	but	CCONJ
ejpam-6011	14	36	the	the	DET
ejpam-6011	14	37	converse	converse	NOUN
ejpam-6011	14	38	is	be	AUX
ejpam-6011	14	39	not	not	PART
ejpam-6011	14	40	true	true	ADJ
ejpam-6011	14	41	in	in	ADP
ejpam-6011	14	42	general	general	ADJ
ejpam-6011	14	43	(	(	PUNCT
ejpam-6011	14	44	see	see	VERB
ejpam-6011	14	45	[	[	X
ejpam-6011	14	46	1	1	NUM
ejpam-6011	14	47	,	,	PUNCT
ejpam-6011	14	48	2	2	NUM
ejpam-6011	14	49	]	]	NUM
ejpam-6011	14	50	)	)	PUNCT
ejpam-6011	14	51	.	.	PUNCT
ejpam-6011	15	1	recall	recall	VERB
ejpam-6011	16	1	[	[	X
ejpam-6011	16	2	3	3	NUM
ejpam-6011	16	3	]	]	PUNCT
ejpam-6011	16	4	,	,	PUNCT
ejpam-6011	16	5	a	a	DET
ejpam-6011	16	6	ring	ring	NOUN
ejpam-6011	16	7	r	r	NOUN
ejpam-6011	16	8	is	be	AUX
ejpam-6011	16	9	said	say	VERB
ejpam-6011	16	10	to	to	PART
ejpam-6011	16	11	be	be	AUX
ejpam-6011	16	12	semicommutative	semicommutative	ADJ
ejpam-6011	16	13	if	if	SCONJ
ejpam-6011	16	14	ab	ab	PROPN
ejpam-6011	16	15	=	=	NOUN
ejpam-6011	16	16	0	0	NUM
ejpam-6011	16	17	implies	imply	VERB
ejpam-6011	16	18	arb	arb	NOUN
ejpam-6011	16	19	=	=	SYM
ejpam-6011	16	20	0	0	NUM
ejpam-6011	16	21	for	for	ADP
ejpam-6011	16	22	any	any	DET
ejpam-6011	16	23	a	a	NOUN
ejpam-6011	16	24	,	,	PUNCT
ejpam-6011	16	25	b	b	PROPN
ejpam-6011	16	26	∈	∈	PROPN
ejpam-6011	16	27	r.	r.	NOUN
ejpam-6011	16	28	the	the	DET
ejpam-6011	16	29	concept	concept	NOUN
ejpam-6011	16	30	of	of	ADP
ejpam-6011	16	31	semicommutative	semicommutative	NOUN
ejpam-6011	16	32	rings	ring	NOUN
ejpam-6011	16	33	has	have	AUX
ejpam-6011	16	34	been	be	AUX
ejpam-6011	16	35	introduced	introduce	VERB
ejpam-6011	16	36	in	in	ADP
ejpam-6011	16	37	other	other	ADJ
ejpam-6011	16	38	terms	term	NOUN
ejpam-6011	16	39	in	in	ADP
ejpam-6011	16	40	[	[	X
ejpam-6011	16	41	4–6	4–6	X
ejpam-6011	16	42	]	]	X
ejpam-6011	16	43	.	.	PUNCT
ejpam-6011	17	1	in	in	ADP
ejpam-6011	17	2	[	[	X
ejpam-6011	17	3	7	7	NUM
ejpam-6011	17	4	]	]	PUNCT
ejpam-6011	17	5	,	,	PUNCT
ejpam-6011	17	6	subba	subba	NOUN
ejpam-6011	17	7	and	and	CCONJ
ejpam-6011	17	8	subedi	subedi	PROPN
ejpam-6011	17	9	investigated	investigate	VERB
ejpam-6011	17	10	a	a	DET
ejpam-6011	17	11	new	new	ADJ
ejpam-6011	17	12	class	class	NOUN
ejpam-6011	17	13	of	of	ADP
ejpam-6011	17	14	rings	ring	NOUN
ejpam-6011	17	15	called	call	VERB
ejpam-6011	17	16	nj	nj	PROPN
ejpam-6011	17	17	-	-	PUNCT
ejpam-6011	17	18	semicomutative	semicomutative	PROPN
ejpam-6011	17	19	.	.	PUNCT
ejpam-6011	18	1	these	these	DET
ejpam-6011	18	2	rings	ring	NOUN
ejpam-6011	18	3	generalize	generalize	VERB
ejpam-6011	18	4	the	the	DET
ejpam-6011	18	5	notion	notion	NOUN
ejpam-6011	18	6	of	of	ADP
ejpam-6011	18	7	semicommutative	semicommutative	NOUN
ejpam-6011	18	8	rings	ring	NOUN
ejpam-6011	18	9	by	by	ADP
ejpam-6011	18	10	exploring	explore	VERB
ejpam-6011	18	11	the	the	DET
ejpam-6011	18	12	relationship	relationship	NOUN
ejpam-6011	18	13	between	between	ADP
ejpam-6011	18	14	nilpotent	nilpotent	ADJ
ejpam-6011	18	15	elements	element	NOUN
ejpam-6011	18	16	and	and	CCONJ
ejpam-6011	18	17	the	the	DET
ejpam-6011	18	18	jacobson	jacobson	PROPN
ejpam-6011	18	19	radical	radical	PROPN
ejpam-6011	18	20	.	.	PUNCT
ejpam-6011	19	1	a	a	DET
ejpam-6011	19	2	ring	ring	NOUN
ejpam-6011	19	3	r	r	NOUN
ejpam-6011	19	4	is	be	AUX
ejpam-6011	19	5	said	say	VERB
ejpam-6011	19	6	to	to	PART
ejpam-6011	19	7	be	be	AUX
ejpam-6011	19	8	njsemicommutative	njsemicommutative	ADJ
ejpam-6011	19	9	if	if	SCONJ
ejpam-6011	19	10	arb	arb	PROPN
ejpam-6011	19	11	⊆	⊆	NUM
ejpam-6011	19	12	j(r	j(r	NOUN
ejpam-6011	19	13	)	)	PUNCT
ejpam-6011	19	14	wherever	wherever	SCONJ
ejpam-6011	19	15	ab	ab	PROPN
ejpam-6011	19	16	∈	∈	PROPN
ejpam-6011	19	17	n(r	n(r	NOUN
ejpam-6011	19	18	)	)	PUNCT
ejpam-6011	19	19	for	for	ADP
ejpam-6011	19	20	all	all	DET
ejpam-6011	19	21	elements	element	NOUN
ejpam-6011	19	22	a	a	PRON
ejpam-6011	19	23	,	,	PUNCT
ejpam-6011	19	24	b	b	PROPN
ejpam-6011	19	25	∈	∈	PROPN
ejpam-6011	19	26	r.	r.	NOUN
ejpam-6011	19	27	this	this	DET
ejpam-6011	19	28	∗corresponding	∗corresponde	VERB
ejpam-6011	19	29	author	author	NOUN
ejpam-6011	19	30	.	.	PUNCT
ejpam-6011	20	1	doi	doi	NOUN
ejpam-6011	20	2	:	:	PUNCT
ejpam-6011	20	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6011	https://doi.org/10.29020/nybg.ejpam.v18i2.6011	ADJ
ejpam-6011	20	4	email	email	NOUN
ejpam-6011	20	5	addresses	address	VERB
ejpam-6011	20	6	:	:	PUNCT
ejpam-6011	20	7	m.saad@alexu.edu.eg	m.saad@alexu.edu.eg	PROPN
ejpam-6011	20	8	(	(	PUNCT
ejpam-6011	20	9	m.	m.	PROPN
ejpam-6011	20	10	saad	saad	PROPN
ejpam-6011	20	11	)	)	PUNCT
ejpam-6011	20	12	,	,	PUNCT
ejpam-6011	20	13	sam.mahmoud@qu.edu.sa	sam.mahmoud@qu.edu.sa	PROPN
ejpam-6011	20	14	(	(	PUNCT
ejpam-6011	20	15	s.	s.	PROPN
ejpam-6011	20	16	m.	m.	PROPN
ejpam-6011	20	17	abdelwahab	abdelwahab	PROPN
ejpam-6011	20	18	)	)	PUNCT
ejpam-6011	20	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6011	21	1	1	1	NUM
ejpam-6011	21	2	copyright	copyright	NOUN
ejpam-6011	21	3	:	:	PUNCT
ejpam-6011	21	4	©	©	PROPN
ejpam-6011	21	5	2025	2025	NUM
ejpam-6011	21	6	the	the	DET
ejpam-6011	21	7	author(s	author(s	NOUN
ejpam-6011	21	8	)	)	PUNCT
ejpam-6011	21	9	.	.	PUNCT
ejpam-6011	22	1	(	(	PUNCT
ejpam-6011	22	2	cc	cc	NOUN
ejpam-6011	22	3	by	by	ADP
ejpam-6011	22	4	-	-	PUNCT
ejpam-6011	22	5	nc	nc	PROPN
ejpam-6011	22	6	4.0	4.0	NUM
ejpam-6011	22	7	)	)	PUNCT
ejpam-6011	22	8	m.	m.	NOUN
ejpam-6011	22	9	saad	saad	PROPN
ejpam-6011	22	10	,	,	PUNCT
ejpam-6011	22	11	s.	s.	PROPN
ejpam-6011	22	12	m.	m.	PROPN
ejpam-6011	22	13	abdelwahab	abdelwahab	PROPN
ejpam-6011	22	14	/	/	SYM
ejpam-6011	22	15	eur	eur	PROPN
ejpam-6011	22	16	.	.	PUNCT
ejpam-6011	23	1	j.	j.	PROPN
ejpam-6011	23	2	pure	pure	PROPN
ejpam-6011	23	3	appl	appl	PROPN
ejpam-6011	23	4	.	.	PROPN
ejpam-6011	23	5	math	math	PROPN
ejpam-6011	23	6	,	,	PUNCT
ejpam-6011	23	7	18	18	NUM
ejpam-6011	23	8	(	(	PUNCT
ejpam-6011	23	9	2	2	NUM
ejpam-6011	23	10	)	)	PUNCT
ejpam-6011	23	11	(	(	PUNCT
ejpam-6011	23	12	2025	2025	NUM
ejpam-6011	23	13	)	)	PUNCT
ejpam-6011	23	14	,	,	PUNCT
ejpam-6011	23	15	6011	6011	NUM
ejpam-6011	23	16	2	2	NUM
ejpam-6011	23	17	of	of	ADP
ejpam-6011	23	18	13	13	NUM
ejpam-6011	23	19	allows	allow	VERB
ejpam-6011	23	20	for	for	ADP
ejpam-6011	23	21	a	a	DET
ejpam-6011	23	22	deeper	deep	ADJ
ejpam-6011	23	23	understanding	understanding	NOUN
ejpam-6011	23	24	of	of	ADP
ejpam-6011	23	25	the	the	DET
ejpam-6011	23	26	structure	structure	NOUN
ejpam-6011	23	27	of	of	ADP
ejpam-6011	23	28	these	these	DET
ejpam-6011	23	29	rings	ring	NOUN
ejpam-6011	23	30	,	,	PUNCT
ejpam-6011	23	31	particularly	particularly	ADV
ejpam-6011	23	32	how	how	SCONJ
ejpam-6011	23	33	the	the	DET
ejpam-6011	23	34	properties	property	NOUN
ejpam-6011	23	35	of	of	ADP
ejpam-6011	23	36	nilpotent	nilpotent	ADJ
ejpam-6011	23	37	elements	element	NOUN
ejpam-6011	23	38	influence	influence	VERB
ejpam-6011	23	39	the	the	DET
ejpam-6011	23	40	behavior	behavior	NOUN
ejpam-6011	23	41	of	of	ADP
ejpam-6011	23	42	the	the	DET
ejpam-6011	23	43	ring	ring	NOUN
ejpam-6011	23	44	as	as	ADP
ejpam-6011	23	45	a	a	DET
ejpam-6011	23	46	whole	whole	NOUN
ejpam-6011	23	47	.	.	PUNCT
ejpam-6011	24	1	in	in	ADP
ejpam-6011	24	2	the	the	DET
ejpam-6011	24	3	same	same	ADJ
ejpam-6011	24	4	context	context	NOUN
ejpam-6011	24	5	,	,	PUNCT
ejpam-6011	24	6	we	we	PRON
ejpam-6011	24	7	define	define	VERB
ejpam-6011	24	8	the	the	DET
ejpam-6011	24	9	nj	nj	PROPN
ejpam-6011	24	10	-	-	PUNCT
ejpam-6011	24	11	abelian	abelian	ADJ
ejpam-6011	24	12	property	property	NOUN
ejpam-6011	24	13	.	.	PUNCT
ejpam-6011	25	1	by	by	ADP
ejpam-6011	25	2	introducing	introduce	VERB
ejpam-6011	25	3	the	the	DET
ejpam-6011	25	4	concept	concept	NOUN
ejpam-6011	25	5	of	of	ADP
ejpam-6011	25	6	nj	nj	PROPN
ejpam-6011	25	7	-	-	PUNCT
ejpam-6011	25	8	abelian	abelian	ADJ
ejpam-6011	25	9	rings	ring	NOUN
ejpam-6011	25	10	,	,	PUNCT
ejpam-6011	25	11	we	we	PRON
ejpam-6011	25	12	aim	aim	VERB
ejpam-6011	25	13	to	to	PART
ejpam-6011	25	14	define	define	VERB
ejpam-6011	25	15	a	a	DET
ejpam-6011	25	16	class	class	NOUN
ejpam-6011	25	17	of	of	ADP
ejpam-6011	25	18	rings	ring	NOUN
ejpam-6011	25	19	that	that	PRON
ejpam-6011	25	20	incorporates	incorporate	VERB
ejpam-6011	25	21	both	both	CCONJ
ejpam-6011	25	22	nilpotent	nilpotent	ADJ
ejpam-6011	25	23	elements	element	NOUN
ejpam-6011	25	24	and	and	CCONJ
ejpam-6011	25	25	the	the	DET
ejpam-6011	25	26	jacobson	jacobson	PROPN
ejpam-6011	25	27	radical	radical	PROPN
ejpam-6011	25	28	,	,	PUNCT
ejpam-6011	25	29	potentially	potentially	ADV
ejpam-6011	25	30	exploring	explore	VERB
ejpam-6011	25	31	the	the	DET
ejpam-6011	25	32	interactions	interaction	NOUN
ejpam-6011	25	33	between	between	ADP
ejpam-6011	25	34	these	these	DET
ejpam-6011	25	35	two	two	NUM
ejpam-6011	25	36	sets	set	NOUN
ejpam-6011	25	37	and	and	CCONJ
ejpam-6011	25	38	the	the	DET
ejpam-6011	25	39	properties	property	NOUN
ejpam-6011	25	40	that	that	PRON
ejpam-6011	25	41	result	result	VERB
ejpam-6011	25	42	from	from	ADP
ejpam-6011	25	43	them	they	PRON
ejpam-6011	25	44	.	.	PUNCT
ejpam-6011	26	1	investigating	investigate	VERB
ejpam-6011	26	2	the	the	DET
ejpam-6011	26	3	structure	structure	NOUN
ejpam-6011	26	4	of	of	ADP
ejpam-6011	26	5	nj	nj	PROPN
ejpam-6011	26	6	-	-	PUNCT
ejpam-6011	26	7	abelian	abelian	ADJ
ejpam-6011	26	8	rings	ring	NOUN
ejpam-6011	26	9	may	may	AUX
ejpam-6011	26	10	reveal	reveal	VERB
ejpam-6011	26	11	intriguing	intriguing	ADJ
ejpam-6011	26	12	characteristics	characteristic	NOUN
ejpam-6011	26	13	and	and	CCONJ
ejpam-6011	26	14	properties	property	NOUN
ejpam-6011	26	15	that	that	PRON
ejpam-6011	26	16	either	either	CCONJ
ejpam-6011	26	17	align	align	VERB
ejpam-6011	26	18	with	with	ADP
ejpam-6011	26	19	or	or	CCONJ
ejpam-6011	26	20	contrast	contrast	VERB
ejpam-6011	26	21	with	with	ADP
ejpam-6011	26	22	those	those	PRON
ejpam-6011	26	23	found	find	VERB
ejpam-6011	26	24	in	in	ADP
ejpam-6011	26	25	nj	nj	ADJ
ejpam-6011	26	26	-	-	PUNCT
ejpam-6011	26	27	semicommutative	semicommutative	NOUN
ejpam-6011	26	28	rings	ring	NOUN
ejpam-6011	26	29	.	.	PUNCT
ejpam-6011	27	1	you	you	PRON
ejpam-6011	27	2	might	might	AUX
ejpam-6011	27	3	explore	explore	VERB
ejpam-6011	27	4	questions	question	NOUN
ejpam-6011	27	5	such	such	ADJ
ejpam-6011	27	6	as	as	ADP
ejpam-6011	27	7	•	•	NOUN
ejpam-6011	27	8	how	how	SCONJ
ejpam-6011	27	9	do	do	AUX
ejpam-6011	27	10	nj	nj	NOUN
ejpam-6011	27	11	-	-	PUNCT
ejpam-6011	27	12	abelian	abelian	ADJ
ejpam-6011	27	13	rings	ring	NOUN
ejpam-6011	27	14	relate	relate	VERB
ejpam-6011	27	15	to	to	ADP
ejpam-6011	27	16	traditional	traditional	ADJ
ejpam-6011	27	17	abelian	abelian	ADJ
ejpam-6011	27	18	rings	ring	NOUN
ejpam-6011	27	19	?	?	PUNCT
ejpam-6011	28	1	•	•	NUM
ejpam-6011	28	2	what	what	DET
ejpam-6011	28	3	conditions	condition	NOUN
ejpam-6011	28	4	define	define	VERB
ejpam-6011	28	5	a	a	DET
ejpam-6011	28	6	ring	ring	NOUN
ejpam-6011	28	7	as	as	ADP
ejpam-6011	28	8	nj	nj	NOUN
ejpam-6011	28	9	-	-	PUNCT
ejpam-6011	28	10	abelian	abelian	ADJ
ejpam-6011	28	11	?	?	PUNCT
ejpam-6011	29	1	•	•	NUM
ejpam-6011	29	2	are	be	AUX
ejpam-6011	29	3	there	there	PRON
ejpam-6011	29	4	any	any	DET
ejpam-6011	29	5	specific	specific	ADJ
ejpam-6011	29	6	examples	example	NOUN
ejpam-6011	29	7	or	or	CCONJ
ejpam-6011	29	8	counterexamples	counterexample	NOUN
ejpam-6011	29	9	that	that	PRON
ejpam-6011	29	10	illustrate	illustrate	VERB
ejpam-6011	29	11	the	the	DET
ejpam-6011	29	12	behavior	behavior	NOUN
ejpam-6011	29	13	of	of	ADP
ejpam-6011	29	14	these	these	DET
ejpam-6011	29	15	rings	ring	NOUN
ejpam-6011	29	16	?	?	PUNCT
ejpam-6011	30	1	in	in	ADP
ejpam-6011	30	2	this	this	DET
ejpam-6011	30	3	article	article	NOUN
ejpam-6011	30	4	,	,	PUNCT
ejpam-6011	30	5	the	the	DET
ejpam-6011	30	6	following	follow	VERB
ejpam-6011	30	7	notations	notation	NOUN
ejpam-6011	30	8	are	be	AUX
ejpam-6011	30	9	used	use	VERB
ejpam-6011	30	10	for	for	ADP
ejpam-6011	30	11	a	a	DET
ejpam-6011	30	12	ring	ring	NOUN
ejpam-6011	30	13	r	r	NOUN
ejpam-6011	30	14	:	:	PUNCT
ejpam-6011	30	15	i(r	i(r	PROPN
ejpam-6011	30	16	)	)	PUNCT
ejpam-6011	30	17	for	for	ADP
ejpam-6011	30	18	the	the	DET
ejpam-6011	30	19	set	set	NOUN
ejpam-6011	30	20	of	of	ADP
ejpam-6011	30	21	idempotents	idempotent	NOUN
ejpam-6011	30	22	of	of	ADP
ejpam-6011	30	23	r	r	NOUN
ejpam-6011	30	24	,	,	PUNCT
ejpam-6011	30	25	n2(r	n2(r	PROPN
ejpam-6011	30	26	)	)	PUNCT
ejpam-6011	30	27	for	for	ADP
ejpam-6011	30	28	the	the	DET
ejpam-6011	30	29	set	set	NOUN
ejpam-6011	30	30	of	of	ADP
ejpam-6011	30	31	all	all	DET
ejpam-6011	30	32	square	square	ADJ
ejpam-6011	30	33	-	-	PUNCT
ejpam-6011	30	34	zero	zero	NUM
ejpam-6011	30	35	elements	element	NOUN
ejpam-6011	30	36	of	of	ADP
ejpam-6011	30	37	r	r	NOUN
ejpam-6011	30	38	(	(	PUNCT
ejpam-6011	30	39	i.e.	i.e.	X
ejpam-6011	30	40	,	,	PUNCT
ejpam-6011	30	41	the	the	DET
ejpam-6011	30	42	nilpotent	nilpotent	ADJ
ejpam-6011	30	43	elements	element	NOUN
ejpam-6011	30	44	of	of	ADP
ejpam-6011	30	45	index	index	NOUN
ejpam-6011	30	46	2	2	NUM
ejpam-6011	30	47	or	or	CCONJ
ejpam-6011	30	48	1	1	NUM
ejpam-6011	30	49	)	)	PUNCT
ejpam-6011	30	50	,	,	PUNCT
ejpam-6011	30	51	u(r	u(r	PROPN
ejpam-6011	30	52	)	)	PUNCT
ejpam-6011	30	53	for	for	ADP
ejpam-6011	30	54	the	the	DET
ejpam-6011	30	55	set	set	NOUN
ejpam-6011	30	56	of	of	ADP
ejpam-6011	30	57	units	unit	NOUN
ejpam-6011	30	58	of	of	ADP
ejpam-6011	30	59	r	r	NOUN
ejpam-6011	30	60	,	,	PUNCT
ejpam-6011	30	61	and	and	CCONJ
ejpam-6011	30	62	mn(r	mn(r	NUM
ejpam-6011	30	63	)	)	PUNCT
ejpam-6011	30	64	(	(	PUNCT
ejpam-6011	30	65	resp	resp	NOUN
ejpam-6011	30	66	.	.	PUNCT
ejpam-6011	31	1	t	t	PROPN
ejpam-6011	31	2	n(r	n(r	NUM
ejpam-6011	31	3	)	)	PUNCT
ejpam-6011	31	4	)	)	PUNCT
ejpam-6011	32	1	for	for	ADP
ejpam-6011	32	2	the	the	DET
ejpam-6011	32	3	ring	ring	NOUN
ejpam-6011	32	4	of	of	ADP
ejpam-6011	32	5	all	all	DET
ejpam-6011	32	6	matrices	matrix	NOUN
ejpam-6011	32	7	(	(	PUNCT
ejpam-6011	32	8	resp	resp	NOUN
ejpam-6011	32	9	.	.	PUNCT
ejpam-6011	33	1	upper	upper	ADJ
ejpam-6011	33	2	triangular	triangular	NOUN
ejpam-6011	33	3	matrices	matrix	NOUN
ejpam-6011	33	4	)	)	PUNCT
ejpam-6011	33	5	over	over	ADP
ejpam-6011	33	6	r.	r.	PROPN
ejpam-6011	33	7	2	2	NUM
ejpam-6011	33	8	.	.	PUNCT
ejpam-6011	33	9	basic	basic	ADJ
ejpam-6011	33	10	results	result	NOUN
ejpam-6011	33	11	we	we	PRON
ejpam-6011	33	12	will	will	AUX
ejpam-6011	33	13	talk	talk	VERB
ejpam-6011	33	14	about	about	ADP
ejpam-6011	33	15	the	the	DET
ejpam-6011	33	16	nj	nj	PROPN
ejpam-6011	33	17	-	-	PUNCT
ejpam-6011	33	18	abelian	abelian	ADJ
ejpam-6011	33	19	concept	concept	NOUN
ejpam-6011	33	20	,	,	PUNCT
ejpam-6011	33	21	come	come	VERB
ejpam-6011	33	22	up	up	ADP
ejpam-6011	33	23	with	with	ADP
ejpam-6011	33	24	some	some	DET
ejpam-6011	33	25	basic	basic	ADJ
ejpam-6011	33	26	results	result	NOUN
ejpam-6011	33	27	,	,	PUNCT
ejpam-6011	33	28	and	and	CCONJ
ejpam-6011	33	29	show	show	VERB
ejpam-6011	33	30	how	how	SCONJ
ejpam-6011	33	31	it	it	PRON
ejpam-6011	33	32	is	be	AUX
ejpam-6011	33	33	related	relate	VERB
ejpam-6011	33	34	to	to	ADP
ejpam-6011	33	35	other	other	ADJ
ejpam-6011	33	36	concepts	concept	NOUN
ejpam-6011	33	37	like	like	ADP
ejpam-6011	33	38	the	the	DET
ejpam-6011	33	39	j	j	PROPN
ejpam-6011	33	40	-	-	PUNCT
ejpam-6011	33	41	abelian	abelian	PROPN
ejpam-6011	33	42	,	,	PUNCT
ejpam-6011	33	43	nj	nj	NOUN
ejpam-6011	33	44	-	-	PUNCT
ejpam-6011	33	45	semicommutative	semicommutative	NOUN
ejpam-6011	33	46	,	,	PUNCT
ejpam-6011	33	47	and	and	CCONJ
ejpam-6011	33	48	j	j	X
ejpam-6011	33	49	-	-	PUNCT
ejpam-6011	33	50	reduced	reduce	VERB
ejpam-6011	33	51	conditions	condition	NOUN
ejpam-6011	33	52	.	.	PUNCT
ejpam-6011	34	1	definition	definition	NOUN
ejpam-6011	34	2	1	1	NUM
ejpam-6011	34	3	.	.	PUNCT
ejpam-6011	35	1	a	a	PRON
ejpam-6011	35	2	ring	ring	NOUN
ejpam-6011	35	3	r	r	NOUN
ejpam-6011	35	4	(	(	PUNCT
ejpam-6011	35	5	not	not	PART
ejpam-6011	35	6	necessarily	necessarily	ADV
ejpam-6011	35	7	with	with	ADP
ejpam-6011	35	8	identity	identity	NOUN
ejpam-6011	35	9	)	)	PUNCT
ejpam-6011	35	10	is	be	AUX
ejpam-6011	35	11	called	call	VERB
ejpam-6011	35	12	nj	nj	PROPN
ejpam-6011	35	13	-	-	PUNCT
ejpam-6011	35	14	abelian	abelian	ADJ
ejpam-6011	35	15	if	if	SCONJ
ejpam-6011	35	16	e2	e2	PROPN
ejpam-6011	35	17	=	=	SYM
ejpam-6011	35	18	e	e	PROPN
ejpam-6011	35	19	,	,	PUNCT
ejpam-6011	35	20	a	a	DET
ejpam-6011	35	21	∈	∈	PROPN
ejpam-6011	35	22	r	r	NOUN
ejpam-6011	35	23	,	,	PUNCT
ejpam-6011	35	24	and	and	CCONJ
ejpam-6011	35	25	ae	ae	PROPN
ejpam-6011	35	26	∈	∈	PROPN
ejpam-6011	35	27	n(r	n(r	PROPN
ejpam-6011	35	28	)	)	PUNCT
ejpam-6011	35	29	,	,	PUNCT
ejpam-6011	35	30	then	then	ADV
ejpam-6011	35	31	are	be	AUX
ejpam-6011	35	32	⊆	⊆	NUM
ejpam-6011	35	33	j(r	j(r	NOUN
ejpam-6011	35	34	)	)	PUNCT
ejpam-6011	35	35	.	.	PUNCT
ejpam-6011	36	1	the	the	DET
ejpam-6011	36	2	next	next	ADJ
ejpam-6011	36	3	proposition	proposition	NOUN
ejpam-6011	36	4	gives	give	VERB
ejpam-6011	36	5	an	an	DET
ejpam-6011	36	6	equivalent	equivalent	ADJ
ejpam-6011	36	7	condition	condition	NOUN
ejpam-6011	36	8	for	for	ADP
ejpam-6011	36	9	j	j	PROPN
ejpam-6011	36	10	-	-	PUNCT
ejpam-6011	36	11	abelianity	abelianity	NOUN
ejpam-6011	36	12	of	of	ADP
ejpam-6011	36	13	rings	ring	NOUN
ejpam-6011	36	14	,	,	PUNCT
ejpam-6011	36	15	which	which	PRON
ejpam-6011	36	16	will	will	AUX
ejpam-6011	36	17	be	be	AUX
ejpam-6011	36	18	used	use	VERB
ejpam-6011	36	19	to	to	PART
ejpam-6011	36	20	show	show	VERB
ejpam-6011	36	21	that	that	SCONJ
ejpam-6011	36	22	every	every	DET
ejpam-6011	36	23	nj	nj	PROPN
ejpam-6011	36	24	-	-	PUNCT
ejpam-6011	36	25	abelian	abelian	ADJ
ejpam-6011	36	26	ring	ring	NOUN
ejpam-6011	36	27	is	be	AUX
ejpam-6011	36	28	j	j	NOUN
ejpam-6011	36	29	-	-	PUNCT
ejpam-6011	36	30	abelian	abelian	PROPN
ejpam-6011	36	31	.	.	PUNCT
ejpam-6011	37	1	proposition	proposition	NOUN
ejpam-6011	37	2	1	1	NUM
ejpam-6011	37	3	.	.	PUNCT
ejpam-6011	38	1	a	a	DET
ejpam-6011	38	2	ring	ring	NOUN
ejpam-6011	38	3	r	r	NOUN
ejpam-6011	38	4	is	be	AUX
ejpam-6011	38	5	j	j	NOUN
ejpam-6011	38	6	-	-	NOUN
ejpam-6011	38	7	abelian	abelian	ADJ
ejpam-6011	39	1	if	if	SCONJ
ejpam-6011	39	2	and	and	CCONJ
ejpam-6011	39	3	only	only	ADV
ejpam-6011	39	4	if	if	SCONJ
ejpam-6011	39	5	ef	ef	PROPN
ejpam-6011	39	6	∈	∈	PROPN
ejpam-6011	39	7	n(r	n(r	NOUN
ejpam-6011	39	8	)	)	PUNCT
ejpam-6011	39	9	for	for	ADP
ejpam-6011	39	10	any	any	DET
ejpam-6011	39	11	e	e	NOUN
ejpam-6011	39	12	,	,	PUNCT
ejpam-6011	39	13	f	f	PROPN
ejpam-6011	39	14	∈	∈	PROPN
ejpam-6011	39	15	i(r	i(r	PROPN
ejpam-6011	39	16	)	)	PUNCT
ejpam-6011	39	17	implies	imply	VERB
ejpam-6011	39	18	erf	erf	NOUN
ejpam-6011	39	19	⊆	⊆	NUM
ejpam-6011	39	20	j(r	j(r	NOUN
ejpam-6011	39	21	)	)	PUNCT
ejpam-6011	39	22	.	.	PUNCT
ejpam-6011	40	1	proof	proof	NOUN
ejpam-6011	40	2	.	.	PUNCT
ejpam-6011	41	1	(	(	PUNCT
ejpam-6011	41	2	⇐	⇐	NOUN
ejpam-6011	41	3	)	)	PUNCT
ejpam-6011	41	4	let	let	VERB
ejpam-6011	41	5	e	e	PRON
ejpam-6011	41	6	be	be	AUX
ejpam-6011	41	7	an	an	DET
ejpam-6011	41	8	idempotent	idempotent	NOUN
ejpam-6011	41	9	of	of	ADP
ejpam-6011	41	10	r.	r.	PROPN
ejpam-6011	41	11	then	then	ADV
ejpam-6011	42	1	e(1	e(1	PROPN
ejpam-6011	42	2	−	−	PROPN
ejpam-6011	42	3	e	e	X
ejpam-6011	42	4	)	)	PUNCT
ejpam-6011	42	5	=	=	SYM
ejpam-6011	42	6	0	0	NUM
ejpam-6011	42	7	∈	∈	PROPN
ejpam-6011	42	8	n(r	n(r	NOUN
ejpam-6011	42	9	)	)	PUNCT
ejpam-6011	42	10	,	,	PUNCT
ejpam-6011	42	11	and	and	CCONJ
ejpam-6011	42	12	hence	hence	ADV
ejpam-6011	42	13	er(1	er(1	PROPN
ejpam-6011	42	14	−	−	PROPN
ejpam-6011	42	15	e	e	NOUN
ejpam-6011	42	16	)	)	PUNCT
ejpam-6011	42	17	⊆	⊆	NUM
ejpam-6011	42	18	j(r	j(r	NOUN
ejpam-6011	42	19	)	)	PUNCT
ejpam-6011	42	20	.	.	PUNCT
ejpam-6011	43	1	similarly	similarly	ADV
ejpam-6011	43	2	,	,	PUNCT
ejpam-6011	43	3	er(1	er(1	PROPN
ejpam-6011	43	4	−	−	PROPN
ejpam-6011	43	5	e	e	NOUN
ejpam-6011	43	6	)	)	PUNCT
ejpam-6011	43	7	⊆	⊆	NUM
ejpam-6011	43	8	j(r	j(r	NOUN
ejpam-6011	43	9	)	)	PUNCT
ejpam-6011	43	10	.	.	PUNCT
ejpam-6011	44	1	so	so	ADV
ejpam-6011	44	2	,	,	PUNCT
ejpam-6011	44	3	for	for	ADP
ejpam-6011	44	4	every	every	DET
ejpam-6011	44	5	r	r	NOUN
ejpam-6011	44	6	∈	∈	NOUN
ejpam-6011	44	7	r	r	NOUN
ejpam-6011	44	8	,	,	PUNCT
ejpam-6011	44	9	we	we	PRON
ejpam-6011	44	10	have	have	VERB
ejpam-6011	44	11	er	er	INTJ
ejpam-6011	44	12	−	−	NOUN
ejpam-6011	44	13	re	re	NOUN
ejpam-6011	44	14	=	=	NOUN
ejpam-6011	44	15	er(1−	er(1−	NOUN
ejpam-6011	44	16	e	e	X
ejpam-6011	44	17	)	)	PUNCT
ejpam-6011	45	1	+	+	CCONJ
ejpam-6011	45	2	(	(	PUNCT
ejpam-6011	45	3	1−	1−	NUM
ejpam-6011	45	4	e)(−r)e	e)(−r)e	NUM
ejpam-6011	45	5	∈	∈	NOUN
ejpam-6011	45	6	er(1−	er(1−	ADJ
ejpam-6011	45	7	e	e	X
ejpam-6011	45	8	)	)	PUNCT
ejpam-6011	45	9	+	+	CCONJ
ejpam-6011	45	10	(	(	PUNCT
ejpam-6011	45	11	1−	1−	NUM
ejpam-6011	45	12	e)re	e)re	PROPN
ejpam-6011	45	13	⊆	⊆	NUM
ejpam-6011	45	14	j(r	j(r	NOUN
ejpam-6011	45	15	)	)	PUNCT
ejpam-6011	45	16	and	and	CCONJ
ejpam-6011	45	17	r	r	NOUN
ejpam-6011	45	18	is	be	AUX
ejpam-6011	45	19	j	j	NOUN
ejpam-6011	45	20	-	-	PUNCT
ejpam-6011	45	21	abelian	abelian	ADJ
ejpam-6011	45	22	.	.	PUNCT
ejpam-6011	46	1	(	(	PUNCT
ejpam-6011	46	2	⇒	⇒	PROPN
ejpam-6011	46	3	)	)	PUNCT
ejpam-6011	46	4	let	let	VERB
ejpam-6011	46	5	ef	ef	VERB
ejpam-6011	46	6	∈	∈	PROPN
ejpam-6011	46	7	n(r	n(r	NOUN
ejpam-6011	46	8	)	)	PUNCT
ejpam-6011	46	9	for	for	ADP
ejpam-6011	46	10	some	some	DET
ejpam-6011	46	11	idempotents	idempotent	NOUN
ejpam-6011	46	12	e	e	NOUN
ejpam-6011	46	13	and	and	CCONJ
ejpam-6011	46	14	f	f	PROPN
ejpam-6011	46	15	of	of	ADP
ejpam-6011	46	16	r	r	NOUN
ejpam-6011	46	17	of	of	ADP
ejpam-6011	46	18	nilpotency	nilpotency	NOUN
ejpam-6011	46	19	index	index	NOUN
ejpam-6011	46	20	n.	n.	NOUN
ejpam-6011	46	21	define	define	VERB
ejpam-6011	46	22	the	the	DET
ejpam-6011	46	23	idempotent	idempotent	NOUN
ejpam-6011	46	24	g	g	PROPN
ejpam-6011	46	25	=	=	SYM
ejpam-6011	46	26	1−f+(1−f)erf	1−f+(1−f)erf	NUM
ejpam-6011	46	27	for	for	ADP
ejpam-6011	46	28	arbitrary	arbitrary	ADJ
ejpam-6011	46	29	r	r	NOUN
ejpam-6011	46	30	in	in	ADP
ejpam-6011	46	31	r.	r.	PROPN
ejpam-6011	46	32	so	so	ADV
ejpam-6011	46	33	,	,	PUNCT
ejpam-6011	46	34	(	(	PUNCT
ejpam-6011	46	35	1−f)erf	1−f)erf	NUM
ejpam-6011	46	36	=	=	SYM
ejpam-6011	46	37	fg−gf	fg−gf	PROPN
ejpam-6011	46	38	∈	∈	PROPN
ejpam-6011	46	39	j(r	j(r	PROPN
ejpam-6011	46	40	)	)	PUNCT
ejpam-6011	46	41	and	and	CCONJ
ejpam-6011	46	42	erf−eferf	erf−eferf	NOUN
ejpam-6011	46	43	=	=	NOUN
ejpam-6011	46	44	e(erf−ferf	e(erf−ferf	PROPN
ejpam-6011	46	45	)	)	PUNCT
ejpam-6011	47	1	=	=	X
ejpam-6011	47	2	e(1−f)erf	e(1−f)erf	NOUN
ejpam-6011	47	3	∈	∈	PROPN
ejpam-6011	47	4	j(r	j(r	PROPN
ejpam-6011	47	5	)	)	PUNCT
ejpam-6011	47	6	.	.	PUNCT
ejpam-6011	48	1	applying	apply	VERB
ejpam-6011	48	2	the	the	DET
ejpam-6011	48	3	j	j	NOUN
ejpam-6011	48	4	-	-	PUNCT
ejpam-6011	48	5	abelian	abelian	ADJ
ejpam-6011	48	6	condition	condition	NOUN
ejpam-6011	48	7	,	,	PUNCT
ejpam-6011	48	8	we	we	PRON
ejpam-6011	48	9	get	get	VERB
ejpam-6011	48	10	fer	fer	PROPN
ejpam-6011	48	11	−	−	PROPN
ejpam-6011	48	12	fefer	fefer	PROPN
ejpam-6011	48	13	∈	∈	PROPN
ejpam-6011	48	14	j(r	j(r	PROPN
ejpam-6011	48	15	)	)	PUNCT
ejpam-6011	48	16	.	.	PUNCT
ejpam-6011	49	1	beginning	begin	VERB
ejpam-6011	49	2	with	with	ADP
ejpam-6011	49	3	the	the	DET
ejpam-6011	49	4	inclusion	inclusion	NOUN
ejpam-6011	49	5	and	and	CCONJ
ejpam-6011	49	6	sequentially	sequentially	ADV
ejpam-6011	49	7	multiplying	multiply	VERB
ejpam-6011	49	8	the	the	DET
ejpam-6011	49	9	element	element	NOUN
ejpam-6011	49	10	by	by	ADP
ejpam-6011	49	11	ef	ef	PROPN
ejpam-6011	49	12	from	from	ADP
ejpam-6011	49	13	left	leave	VERB
ejpam-6011	49	14	to	to	ADP
ejpam-6011	49	15	right	right	NOUN
ejpam-6011	49	16	,	,	PUNCT
ejpam-6011	49	17	we	we	PRON
ejpam-6011	49	18	obtain	obtain	VERB
ejpam-6011	49	19	(	(	PUNCT
ejpam-6011	49	20	fer	fer	PROPN
ejpam-6011	49	21	−	−	PROPN
ejpam-6011	49	22	(	(	PUNCT
ejpam-6011	49	23	fe)2r	fe)2r	NOUN
ejpam-6011	49	24	)	)	PUNCT
ejpam-6011	50	1	+	+	CCONJ
ejpam-6011	50	2	(	(	PUNCT
ejpam-6011	50	3	(	(	PUNCT
ejpam-6011	50	4	fe)2r	fe)2r	INTJ
ejpam-6011	50	5	−	−	X
ejpam-6011	50	6	(	(	PUNCT
ejpam-6011	50	7	fe)3r	fe)3r	NOUN
ejpam-6011	50	8	)	)	PUNCT
ejpam-6011	51	1	+	+	CCONJ
ejpam-6011	51	2	(	(	PUNCT
ejpam-6011	51	3	(	(	PUNCT
ejpam-6011	51	4	fe)3r	fe)3r	NOUN
ejpam-6011	51	5	−	−	X
ejpam-6011	51	6	(	(	PUNCT
ejpam-6011	51	7	fe)4r	fe)4r	INTJ
ejpam-6011	51	8	)	)	PUNCT
ejpam-6011	51	9	+	+	X
ejpam-6011	51	10	·	·	PUNCT
ejpam-6011	51	11	·	·	PUNCT
ejpam-6011	51	12	·	·	PUNCT
ejpam-6011	51	13	(	(	PUNCT
ejpam-6011	51	14	(	(	PUNCT
ejpam-6011	51	15	fe)n−1r	fe)n−1r	PROPN
ejpam-6011	51	16	−	−	PROPN
ejpam-6011	51	17	(	(	PUNCT
ejpam-6011	51	18	fe)nr	fe)nr	X
ejpam-6011	51	19	)	)	PUNCT
ejpam-6011	51	20	∈	∈	PROPN
ejpam-6011	51	21	j(r	j(r	PROPN
ejpam-6011	51	22	)	)	PUNCT
ejpam-6011	51	23	m.	m.	NOUN
ejpam-6011	51	24	saad	saad	PROPN
ejpam-6011	51	25	,	,	PUNCT
ejpam-6011	51	26	s.	s.	PROPN
ejpam-6011	51	27	m.	m.	PROPN
ejpam-6011	51	28	abdelwahab	abdelwahab	PROPN
ejpam-6011	51	29	/	/	SYM
ejpam-6011	51	30	eur	eur	PROPN
ejpam-6011	51	31	.	.	PUNCT
ejpam-6011	52	1	j.	j.	PROPN
ejpam-6011	52	2	pure	pure	PROPN
ejpam-6011	52	3	appl	appl	PROPN
ejpam-6011	52	4	.	.	PROPN
ejpam-6011	52	5	math	math	PROPN
ejpam-6011	52	6	,	,	PUNCT
ejpam-6011	52	7	18	18	NUM
ejpam-6011	52	8	(	(	PUNCT
ejpam-6011	52	9	2	2	NUM
ejpam-6011	52	10	)	)	PUNCT
ejpam-6011	52	11	(	(	PUNCT
ejpam-6011	52	12	2025	2025	NUM
ejpam-6011	52	13	)	)	PUNCT
ejpam-6011	52	14	,	,	PUNCT
ejpam-6011	52	15	6011	6011	NUM
ejpam-6011	52	16	3	3	NUM
ejpam-6011	52	17	of	of	ADP
ejpam-6011	52	18	13	13	NUM
ejpam-6011	52	19	and	and	CCONJ
ejpam-6011	52	20	fer	fer	PROPN
ejpam-6011	52	21	∈	∈	PROPN
ejpam-6011	52	22	j(r	j(r	PROPN
ejpam-6011	52	23	)	)	PUNCT
ejpam-6011	52	24	for	for	ADP
ejpam-6011	52	25	every	every	DET
ejpam-6011	52	26	r	r	NOUN
ejpam-6011	52	27	∈	∈	PROPN
ejpam-6011	52	28	r.	r.	NOUN
ejpam-6011	52	29	from	from	ADP
ejpam-6011	52	30	the	the	DET
ejpam-6011	52	31	j	j	PROPN
ejpam-6011	52	32	-	-	PUNCT
ejpam-6011	52	33	abelianity	abelianity	NOUN
ejpam-6011	52	34	of	of	ADP
ejpam-6011	52	35	r	r	NOUN
ejpam-6011	52	36	,	,	PUNCT
ejpam-6011	52	37	we	we	PRON
ejpam-6011	52	38	get	get	VERB
ejpam-6011	52	39	erf	erf	NOUN
ejpam-6011	52	40	⊆	⊆	NUM
ejpam-6011	52	41	j(r	j(r	NOUN
ejpam-6011	52	42	)	)	PUNCT
ejpam-6011	52	43	.	.	PUNCT
ejpam-6011	53	1	corollary	corollary	ADJ
ejpam-6011	53	2	1	1	NUM
ejpam-6011	53	3	.	.	PUNCT
ejpam-6011	54	1	every	every	DET
ejpam-6011	54	2	nj	nj	PROPN
ejpam-6011	54	3	-	-	PUNCT
ejpam-6011	54	4	abelian	abelian	ADJ
ejpam-6011	54	5	ring	ring	NOUN
ejpam-6011	54	6	r	r	NOUN
ejpam-6011	54	7	is	be	AUX
ejpam-6011	54	8	j	j	NOUN
ejpam-6011	54	9	-	-	PUNCT
ejpam-6011	54	10	abelian	abelian	PROPN
ejpam-6011	54	11	.	.	PUNCT
ejpam-6011	55	1	here	here	ADV
ejpam-6011	55	2	is	be	AUX
ejpam-6011	55	3	a	a	DET
ejpam-6011	55	4	j	j	NOUN
ejpam-6011	55	5	-	-	PUNCT
ejpam-6011	55	6	abelian	abelian	ADJ
ejpam-6011	55	7	ring	ring	NOUN
ejpam-6011	55	8	that	that	PRON
ejpam-6011	55	9	is	be	AUX
ejpam-6011	55	10	not	not	PART
ejpam-6011	55	11	nj	nj	NOUN
ejpam-6011	55	12	-	-	PUNCT
ejpam-6011	55	13	abelian	abelian	PROPN
ejpam-6011	55	14	.	.	PUNCT
ejpam-6011	55	15	example	example	NOUN
ejpam-6011	56	1	1	1	NUM
ejpam-6011	56	2	.	.	PUNCT
ejpam-6011	57	1	let	let	VERB
ejpam-6011	57	2	r	r	NOUN
ejpam-6011	57	3	=	=	SYM
ejpam-6011	57	4	z4[x	z4[x	X
ejpam-6011	57	5	]	]	PUNCT
ejpam-6011	57	6	and	and	CCONJ
ejpam-6011	57	7	s	s	X
ejpam-6011	57	8	=	=	X
ejpam-6011	57	9	r/⟨x⟩.	r/⟨x⟩.	NOUN
ejpam-6011	57	10	so	so	ADV
ejpam-6011	57	11	,	,	PUNCT
ejpam-6011	57	12	j(s	j(s	NOUN
ejpam-6011	57	13	)	)	PUNCT
ejpam-6011	58	1	=	=	SYM
ejpam-6011	58	2	⟨2x⟩	⟨2x⟩	NUM
ejpam-6011	58	3	and	and	CCONJ
ejpam-6011	58	4	n(s	n(s	PROPN
ejpam-6011	58	5	)	)	PUNCT
ejpam-6011	58	6	=	=	SYM
ejpam-6011	59	1	⟨2	⟨2	PROPN
ejpam-6011	59	2	,	,	PUNCT
ejpam-6011	59	3	x⟩.	x⟩.	VERB
ejpam-6011	59	4	we	we	PRON
ejpam-6011	59	5	have	have	VERB
ejpam-6011	59	6	1x	1x	NUM
ejpam-6011	59	7	∈	∈	PROPN
ejpam-6011	59	8	n(s	n(s	PROPN
ejpam-6011	59	9	)	)	PUNCT
ejpam-6011	59	10	while	while	SCONJ
ejpam-6011	59	11	1sx	1sx	ADJ
ejpam-6011	59	12	⊈	⊈	PROPN
ejpam-6011	59	13	j(s	j(s	NOUN
ejpam-6011	59	14	)	)	PUNCT
ejpam-6011	59	15	.	.	PUNCT
ejpam-6011	60	1	thus	thus	ADV
ejpam-6011	60	2	,	,	PUNCT
ejpam-6011	60	3	s	s	VERB
ejpam-6011	60	4	is	be	AUX
ejpam-6011	60	5	not	not	PART
ejpam-6011	60	6	nj	nj	NOUN
ejpam-6011	60	7	-	-	NOUN
ejpam-6011	60	8	abelian	abelian	ADJ
ejpam-6011	60	9	even	even	ADV
ejpam-6011	60	10	though	though	SCONJ
ejpam-6011	60	11	s	s	NOUN
ejpam-6011	60	12	is	be	AUX
ejpam-6011	60	13	j	j	NOUN
ejpam-6011	60	14	-	-	PUNCT
ejpam-6011	60	15	abelian	abelian	PROPN
ejpam-6011	60	16	.	.	PUNCT
ejpam-6011	61	1	the	the	DET
ejpam-6011	61	2	next	next	ADJ
ejpam-6011	61	3	proposition	proposition	NOUN
ejpam-6011	61	4	shows	show	VERB
ejpam-6011	61	5	that	that	SCONJ
ejpam-6011	61	6	the	the	DET
ejpam-6011	61	7	definition	definition	NOUN
ejpam-6011	61	8	of	of	ADP
ejpam-6011	61	9	the	the	DET
ejpam-6011	61	10	nj	nj	PROPN
ejpam-6011	61	11	-	-	PUNCT
ejpam-6011	61	12	abelian	abelian	ADJ
ejpam-6011	61	13	property	property	NOUN
ejpam-6011	61	14	is	be	AUX
ejpam-6011	61	15	left	leave	VERB
ejpam-6011	61	16	-	-	PUNCT
ejpam-6011	61	17	right	right	NOUN
ejpam-6011	61	18	symmetric	symmetric	NOUN
ejpam-6011	61	19	.	.	PUNCT
ejpam-6011	62	1	proposition	proposition	NOUN
ejpam-6011	62	2	2	2	NUM
ejpam-6011	62	3	.	.	PUNCT
ejpam-6011	63	1	any	any	DET
ejpam-6011	63	2	ring	ring	NOUN
ejpam-6011	63	3	r	r	NOUN
ejpam-6011	63	4	can	can	AUX
ejpam-6011	63	5	satisfy	satisfy	VERB
ejpam-6011	63	6	the	the	DET
ejpam-6011	63	7	following	follow	VERB
ejpam-6011	63	8	equivalent	equivalent	ADJ
ejpam-6011	63	9	conditions	condition	NOUN
ejpam-6011	63	10	:	:	PUNCT
ejpam-6011	63	11	(	(	PUNCT
ejpam-6011	63	12	i	i	NOUN
ejpam-6011	63	13	)	)	PUNCT
ejpam-6011	63	14	r	r	NOUN
ejpam-6011	63	15	is	be	AUX
ejpam-6011	63	16	nj	nj	NOUN
ejpam-6011	63	17	-	-	PUNCT
ejpam-6011	63	18	abelian	abelian	ADJ
ejpam-6011	63	19	;	;	PUNCT
ejpam-6011	63	20	(	(	PUNCT
ejpam-6011	63	21	ii	ii	NOUN
ejpam-6011	63	22	)	)	PUNCT
ejpam-6011	63	23	ae	ae	PROPN
ejpam-6011	63	24	∈	∈	PROPN
ejpam-6011	63	25	n(r	n(r	NOUN
ejpam-6011	63	26	)	)	PUNCT
ejpam-6011	63	27	implies	imply	VERB
ejpam-6011	63	28	then	then	ADV
ejpam-6011	63	29	era	era	VERB
ejpam-6011	63	30	⊆	⊆	NUM
ejpam-6011	63	31	j(r	j(r	NOUN
ejpam-6011	63	32	)	)	PUNCT
ejpam-6011	63	33	,	,	PUNCT
ejpam-6011	63	34	where	where	SCONJ
ejpam-6011	63	35	e2	e2	PROPN
ejpam-6011	63	36	=	=	SYM
ejpam-6011	63	37	e	e	PROPN
ejpam-6011	63	38	,	,	PUNCT
ejpam-6011	63	39	a	a	DET
ejpam-6011	63	40	∈	∈	NOUN
ejpam-6011	63	41	r	r	NOUN
ejpam-6011	63	42	;	;	PUNCT
ejpam-6011	63	43	(	(	PUNCT
ejpam-6011	63	44	iii	iii	NOUN
ejpam-6011	63	45	)	)	PUNCT
ejpam-6011	63	46	ea	ea	NOUN
ejpam-6011	63	47	∈	∈	PROPN
ejpam-6011	63	48	n(r	n(r	NOUN
ejpam-6011	63	49	)	)	PUNCT
ejpam-6011	63	50	implies	imply	VERB
ejpam-6011	63	51	then	then	ADV
ejpam-6011	63	52	era	era	VERB
ejpam-6011	63	53	⊆	⊆	NUM
ejpam-6011	63	54	j(r	j(r	NOUN
ejpam-6011	63	55	)	)	PUNCT
ejpam-6011	63	56	,	,	PUNCT
ejpam-6011	63	57	where	where	SCONJ
ejpam-6011	63	58	e2	e2	PROPN
ejpam-6011	63	59	=	=	SYM
ejpam-6011	63	60	e	e	PROPN
ejpam-6011	63	61	,	,	PUNCT
ejpam-6011	63	62	a	a	DET
ejpam-6011	63	63	∈	∈	NOUN
ejpam-6011	63	64	r	r	NOUN
ejpam-6011	63	65	;	;	PUNCT
ejpam-6011	63	66	(	(	PUNCT
ejpam-6011	63	67	iv	iv	X
ejpam-6011	63	68	)	)	PUNCT
ejpam-6011	63	69	ea	ea	NOUN
ejpam-6011	63	70	∈	∈	PROPN
ejpam-6011	63	71	n(r	n(r	NOUN
ejpam-6011	63	72	)	)	PUNCT
ejpam-6011	63	73	implies	imply	VERB
ejpam-6011	63	74	then	then	ADV
ejpam-6011	63	75	are	be	AUX
ejpam-6011	63	76	⊆	⊆	NUM
ejpam-6011	63	77	j(r	j(r	NOUN
ejpam-6011	63	78	)	)	PUNCT
ejpam-6011	63	79	,	,	PUNCT
ejpam-6011	63	80	where	where	SCONJ
ejpam-6011	63	81	e2	e2	PROPN
ejpam-6011	63	82	=	=	SYM
ejpam-6011	63	83	e	e	PROPN
ejpam-6011	63	84	,	,	PUNCT
ejpam-6011	63	85	a	a	DET
ejpam-6011	63	86	∈	∈	PROPN
ejpam-6011	63	87	r.	r.	NOUN
ejpam-6011	63	88	proof	proof	NOUN
ejpam-6011	63	89	.	.	PUNCT
ejpam-6011	64	1	(	(	PUNCT
ejpam-6011	64	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-6011	64	3	):	):	PUNCT
ejpam-6011	64	4	if	if	SCONJ
ejpam-6011	64	5	ae	ae	PROPN
ejpam-6011	64	6	∈	∈	PROPN
ejpam-6011	64	7	n(r	n(r	PROPN
ejpam-6011	64	8	)	)	PUNCT
ejpam-6011	64	9	,	,	PUNCT
ejpam-6011	64	10	then	then	ADV
ejpam-6011	64	11	are	be	AUX
ejpam-6011	64	12	⊆	⊆	NUM
ejpam-6011	64	13	j(r	j(r	NOUN
ejpam-6011	64	14	)	)	PUNCT
ejpam-6011	64	15	from	from	ADP
ejpam-6011	64	16	the	the	DET
ejpam-6011	64	17	nj	nj	PROPN
ejpam-6011	64	18	-	-	PUNCT
ejpam-6011	64	19	abelianity	abelianity	NOUN
ejpam-6011	64	20	of	of	ADP
ejpam-6011	64	21	r.	r.	PROPN
ejpam-6011	64	22	for	for	ADP
ejpam-6011	64	23	every	every	DET
ejpam-6011	64	24	r	r	NOUN
ejpam-6011	64	25	∈	∈	NOUN
ejpam-6011	64	26	r	r	NOUN
ejpam-6011	64	27	,	,	PUNCT
ejpam-6011	64	28	we	we	PRON
ejpam-6011	64	29	have	have	VERB
ejpam-6011	64	30	era	era	NOUN
ejpam-6011	64	31	=	=	SYM
ejpam-6011	64	32	e(ra	e(ra	PROPN
ejpam-6011	64	33	)	)	PUNCT
ejpam-6011	64	34	−	−	PROPN
ejpam-6011	65	1	(	(	PUNCT
ejpam-6011	65	2	ra)e	ra)e	PROPN
ejpam-6011	65	3	+	+	NUM
ejpam-6011	65	4	rae	rae	PROPN
ejpam-6011	65	5	∈	∈	PROPN
ejpam-6011	65	6	j(r	j(r	PROPN
ejpam-6011	65	7	)	)	PUNCT
ejpam-6011	65	8	,	,	PUNCT
ejpam-6011	65	9	since	since	SCONJ
ejpam-6011	65	10	r	r	NOUN
ejpam-6011	65	11	is	be	AUX
ejpam-6011	65	12	j	j	NOUN
ejpam-6011	65	13	-	-	NOUN
ejpam-6011	65	14	abelian	abelian	ADJ
ejpam-6011	65	15	from	from	ADP
ejpam-6011	65	16	proposition	proposition	NOUN
ejpam-6011	65	17	1	1	NUM
ejpam-6011	65	18	,	,	PUNCT
ejpam-6011	65	19	and	and	CCONJ
ejpam-6011	65	20	hence	hence	ADV
ejpam-6011	65	21	era	era	VERB
ejpam-6011	65	22	⊆	⊆	NUM
ejpam-6011	65	23	j(r	j(r	NOUN
ejpam-6011	65	24	)	)	PUNCT
ejpam-6011	65	25	.	.	PUNCT
ejpam-6011	66	1	(	(	PUNCT
ejpam-6011	66	2	ii)⇒(iii	ii)⇒(iii	X
ejpam-6011	66	3	)	)	PUNCT
ejpam-6011	66	4	is	be	AUX
ejpam-6011	66	5	direct	direct	ADJ
ejpam-6011	66	6	since	since	SCONJ
ejpam-6011	66	7	ae	ae	PROPN
ejpam-6011	66	8	is	be	AUX
ejpam-6011	66	9	nilpotent	nilpotent	ADJ
ejpam-6011	66	10	if	if	SCONJ
ejpam-6011	66	11	and	and	CCONJ
ejpam-6011	66	12	only	only	ADV
ejpam-6011	66	13	if	if	SCONJ
ejpam-6011	66	14	ea	ea	PROPN
ejpam-6011	66	15	is	be	AUX
ejpam-6011	66	16	.	.	PUNCT
ejpam-6011	67	1	(	(	PUNCT
ejpam-6011	67	2	iii)⇒(iv	iii)⇒(iv	NOUN
ejpam-6011	67	3	):	):	PUNCT
ejpam-6011	67	4	as	as	ADP
ejpam-6011	67	5	in	in	ADP
ejpam-6011	67	6	proposition	proposition	NOUN
ejpam-6011	67	7	1	1	NUM
ejpam-6011	67	8	,	,	PUNCT
ejpam-6011	67	9	one	one	PRON
ejpam-6011	67	10	can	can	AUX
ejpam-6011	67	11	prove	prove	VERB
ejpam-6011	67	12	that	that	SCONJ
ejpam-6011	67	13	r	r	NOUN
ejpam-6011	67	14	is	be	AUX
ejpam-6011	67	15	j	j	NOUN
ejpam-6011	67	16	-	-	PUNCT
ejpam-6011	67	17	abelian	abelian	PROPN
ejpam-6011	67	18	.	.	PUNCT
ejpam-6011	68	1	now	now	ADV
ejpam-6011	68	2	,	,	PUNCT
ejpam-6011	68	3	are	be	AUX
ejpam-6011	68	4	=	=	PUNCT
ejpam-6011	68	5	(	(	PUNCT
ejpam-6011	68	6	ar)e−e(ar)+ear	ar)e−e(ar)+ear	PROPN
ejpam-6011	68	7	∈	∈	PROPN
ejpam-6011	68	8	j(r	j(r	PROPN
ejpam-6011	68	9	)	)	PUNCT
ejpam-6011	68	10	,	,	PUNCT
ejpam-6011	68	11	for	for	ADP
ejpam-6011	68	12	every	every	DET
ejpam-6011	68	13	e2	e2	PROPN
ejpam-6011	68	14	=	=	SYM
ejpam-6011	68	15	e	e	NOUN
ejpam-6011	68	16	,	,	PUNCT
ejpam-6011	68	17	r	r	NOUN
ejpam-6011	68	18	∈	∈	NOUN
ejpam-6011	68	19	r	r	NOUN
ejpam-6011	68	20	by	by	ADP
ejpam-6011	68	21	the	the	DET
ejpam-6011	68	22	hypotheses	hypothesis	NOUN
ejpam-6011	68	23	;	;	PUNCT
ejpam-6011	68	24	that	that	PRON
ejpam-6011	68	25	is	is	ADV
ejpam-6011	68	26	,	,	PUNCT
ejpam-6011	68	27	are	be	AUX
ejpam-6011	68	28	⊆	⊆	NUM
ejpam-6011	68	29	j(r	j(r	NOUN
ejpam-6011	68	30	)	)	PUNCT
ejpam-6011	68	31	.	.	PUNCT
ejpam-6011	69	1	(	(	PUNCT
ejpam-6011	69	2	iv)⇒(i	iv)⇒(i	X
ejpam-6011	69	3	)	)	PUNCT
ejpam-6011	69	4	is	be	AUX
ejpam-6011	69	5	direct	direct	ADJ
ejpam-6011	69	6	again	again	ADV
ejpam-6011	69	7	.	.	PUNCT
ejpam-6011	70	1	now	now	ADV
ejpam-6011	70	2	,	,	PUNCT
ejpam-6011	70	3	we	we	PRON
ejpam-6011	70	4	aim	aim	VERB
ejpam-6011	70	5	to	to	PART
ejpam-6011	70	6	get	get	VERB
ejpam-6011	70	7	a	a	DET
ejpam-6011	70	8	sufficient	sufficient	ADJ
ejpam-6011	70	9	condition	condition	NOUN
ejpam-6011	70	10	for	for	ADP
ejpam-6011	70	11	the	the	DET
ejpam-6011	70	12	j	j	PROPN
ejpam-6011	70	13	-	-	PUNCT
ejpam-6011	70	14	abelian	abelian	ADJ
ejpam-6011	70	15	ring	ring	NOUN
ejpam-6011	70	16	to	to	PART
ejpam-6011	70	17	be	be	AUX
ejpam-6011	70	18	nj	nj	NOUN
ejpam-6011	70	19	-	-	PUNCT
ejpam-6011	70	20	abelian	abelian	ADJ
ejpam-6011	70	21	.	.	PUNCT
ejpam-6011	71	1	according	accord	VERB
ejpam-6011	71	2	to	to	ADP
ejpam-6011	71	3	[	[	X
ejpam-6011	71	4	8	8	NUM
ejpam-6011	71	5	]	]	PUNCT
ejpam-6011	71	6	,	,	PUNCT
ejpam-6011	71	7	a	a	DET
ejpam-6011	71	8	ring	ring	NOUN
ejpam-6011	71	9	r	r	NOUN
ejpam-6011	71	10	is	be	AUX
ejpam-6011	71	11	called	call	VERB
ejpam-6011	71	12	j	j	NOUN
ejpam-6011	71	13	-	-	PUNCT
ejpam-6011	71	14	reduced	reduce	VERB
ejpam-6011	71	15	if	if	SCONJ
ejpam-6011	71	16	n(r	n(r	NOUN
ejpam-6011	71	17	)	)	PUNCT
ejpam-6011	71	18	⊆	⊆	NUM
ejpam-6011	71	19	j(r	j(r	NOUN
ejpam-6011	71	20	)	)	PUNCT
ejpam-6011	71	21	.	.	PUNCT
ejpam-6011	72	1	in	in	ADP
ejpam-6011	72	2	[	[	X
ejpam-6011	72	3	9	9	NUM
ejpam-6011	72	4	]	]	PUNCT
ejpam-6011	72	5	,	,	PUNCT
ejpam-6011	72	6	a	a	DET
ejpam-6011	72	7	j	j	NOUN
ejpam-6011	72	8	-	-	PUNCT
ejpam-6011	72	9	reduced	reduce	VERB
ejpam-6011	72	10	ring	ring	NOUN
ejpam-6011	72	11	is	be	AUX
ejpam-6011	72	12	called	call	VERB
ejpam-6011	72	13	an	an	DET
ejpam-6011	72	14	nj	nj	PROPN
ejpam-6011	72	15	ring	ring	NOUN
ejpam-6011	72	16	.	.	PUNCT
ejpam-6011	73	1	obviously	obviously	ADV
ejpam-6011	73	2	,	,	PUNCT
ejpam-6011	73	3	every	every	DET
ejpam-6011	73	4	nj	nj	PROPN
ejpam-6011	73	5	-	-	PUNCT
ejpam-6011	73	6	abelian	abelian	ADJ
ejpam-6011	73	7	ring	ring	NOUN
ejpam-6011	73	8	r	r	NOUN
ejpam-6011	73	9	is	be	AUX
ejpam-6011	73	10	j	j	NOUN
ejpam-6011	73	11	-	-	PUNCT
ejpam-6011	73	12	reduced	reduce	VERB
ejpam-6011	73	13	since	since	SCONJ
ejpam-6011	73	14	all	all	DET
ejpam-6011	73	15	rings	ring	NOUN
ejpam-6011	73	16	are	be	AUX
ejpam-6011	73	17	with	with	ADP
ejpam-6011	73	18	identity	identity	NOUN
ejpam-6011	73	19	.	.	PUNCT
ejpam-6011	74	1	the	the	DET
ejpam-6011	74	2	converse	converse	NOUN
ejpam-6011	74	3	is	be	AUX
ejpam-6011	74	4	not	not	PART
ejpam-6011	74	5	necessarily	necessarily	ADV
ejpam-6011	74	6	true	true	ADJ
ejpam-6011	74	7	,	,	PUNCT
ejpam-6011	74	8	as	as	SCONJ
ejpam-6011	74	9	the	the	DET
ejpam-6011	74	10	ring	ring	NOUN
ejpam-6011	74	11	s	s	PRON
ejpam-6011	74	12	in	in	ADP
ejpam-6011	74	13	example	example	NOUN
ejpam-6011	74	14	1	1	X
ejpam-6011	74	15	.	.	X
ejpam-6011	75	1	remind	remind	VERB
ejpam-6011	75	2	that	that	SCONJ
ejpam-6011	75	3	every	every	DET
ejpam-6011	75	4	reduced	reduce	VERB
ejpam-6011	75	5	ring	ring	NOUN
ejpam-6011	75	6	is	be	AUX
ejpam-6011	75	7	abelian	abelian	ADJ
ejpam-6011	75	8	.	.	PUNCT
ejpam-6011	76	1	hence	hence	ADV
ejpam-6011	76	2	,	,	PUNCT
ejpam-6011	76	3	every	every	DET
ejpam-6011	76	4	reduced	reduce	VERB
ejpam-6011	76	5	ring	ring	NOUN
ejpam-6011	76	6	is	be	AUX
ejpam-6011	76	7	nj	nj	NOUN
ejpam-6011	76	8	-	-	PUNCT
ejpam-6011	76	9	abelian	abelian	ADJ
ejpam-6011	76	10	.	.	PUNCT
ejpam-6011	77	1	the	the	DET
ejpam-6011	77	2	next	next	ADJ
ejpam-6011	77	3	proposition	proposition	NOUN
ejpam-6011	77	4	shows	show	VERB
ejpam-6011	77	5	that	that	SCONJ
ejpam-6011	77	6	j	j	PROPN
ejpam-6011	77	7	-	-	NOUN
ejpam-6011	77	8	reducedness	reducedness	PROPN
ejpam-6011	77	9	is	be	AUX
ejpam-6011	77	10	a	a	DET
ejpam-6011	77	11	sufficient	sufficient	ADJ
ejpam-6011	77	12	and	and	CCONJ
ejpam-6011	77	13	necessary	necessary	ADJ
ejpam-6011	77	14	condition	condition	NOUN
ejpam-6011	77	15	for	for	ADP
ejpam-6011	77	16	the	the	DET
ejpam-6011	77	17	j	j	PROPN
ejpam-6011	77	18	-	-	PUNCT
ejpam-6011	77	19	abelian	abelian	ADJ
ejpam-6011	77	20	ring	ring	NOUN
ejpam-6011	77	21	to	to	PART
ejpam-6011	77	22	become	become	VERB
ejpam-6011	77	23	nj	nj	NOUN
ejpam-6011	77	24	-	-	PUNCT
ejpam-6011	77	25	abelian	abelian	NOUN
ejpam-6011	77	26	.	.	PUNCT
ejpam-6011	78	1	proposition	proposition	NOUN
ejpam-6011	78	2	3	3	NUM
ejpam-6011	78	3	.	.	PUNCT
ejpam-6011	79	1	a	a	DET
ejpam-6011	79	2	ring	ring	NOUN
ejpam-6011	79	3	r	r	NOUN
ejpam-6011	79	4	is	be	AUX
ejpam-6011	79	5	nj	nj	NOUN
ejpam-6011	79	6	-	-	PUNCT
ejpam-6011	79	7	abelian	abelian	ADJ
ejpam-6011	80	1	if	if	SCONJ
ejpam-6011	80	2	and	and	CCONJ
ejpam-6011	80	3	only	only	ADV
ejpam-6011	80	4	if	if	SCONJ
ejpam-6011	80	5	it	it	PRON
ejpam-6011	80	6	is	be	AUX
ejpam-6011	80	7	j	j	NOUN
ejpam-6011	80	8	-	-	PUNCT
ejpam-6011	80	9	reduced	reduce	VERB
ejpam-6011	80	10	and	and	CCONJ
ejpam-6011	80	11	j	j	NOUN
ejpam-6011	80	12	-	-	PUNCT
ejpam-6011	80	13	abelian	abelian	PROPN
ejpam-6011	80	14	.	.	PUNCT
ejpam-6011	81	1	proof	proof	NOUN
ejpam-6011	81	2	.	.	PUNCT
ejpam-6011	82	1	the	the	DET
ejpam-6011	82	2	necessity	necessity	NOUN
ejpam-6011	82	3	is	be	AUX
ejpam-6011	82	4	obvious	obvious	ADJ
ejpam-6011	82	5	.	.	PUNCT
ejpam-6011	83	1	for	for	ADP
ejpam-6011	83	2	sufficiency	sufficiency	NOUN
ejpam-6011	83	3	,	,	PUNCT
ejpam-6011	83	4	assume	assume	VERB
ejpam-6011	83	5	that	that	SCONJ
ejpam-6011	83	6	r	r	NOUN
ejpam-6011	83	7	is	be	AUX
ejpam-6011	83	8	a	a	DET
ejpam-6011	83	9	j	j	NOUN
ejpam-6011	83	10	-	-	PUNCT
ejpam-6011	83	11	reduced	reduce	VERB
ejpam-6011	83	12	and	and	CCONJ
ejpam-6011	83	13	j	j	NOUN
ejpam-6011	83	14	-	-	PUNCT
ejpam-6011	83	15	abelian	abelian	ADJ
ejpam-6011	83	16	ring	ring	NOUN
ejpam-6011	83	17	.	.	PUNCT
ejpam-6011	84	1	let	let	VERB
ejpam-6011	84	2	ae	ae	PROPN
ejpam-6011	84	3	∈	∈	PROPN
ejpam-6011	84	4	n(r	n(r	NOUN
ejpam-6011	84	5	)	)	PUNCT
ejpam-6011	84	6	for	for	ADP
ejpam-6011	84	7	some	some	DET
ejpam-6011	84	8	e2	e2	NOUN
ejpam-6011	84	9	=	=	SYM
ejpam-6011	84	10	e	e	PROPN
ejpam-6011	84	11	,	,	PUNCT
ejpam-6011	84	12	a	a	DET
ejpam-6011	84	13	∈	∈	PROPN
ejpam-6011	84	14	r.	r.	PROPN
ejpam-6011	85	1	so	so	ADV
ejpam-6011	85	2	,	,	PUNCT
ejpam-6011	85	3	ea	ea	PROPN
ejpam-6011	85	4	∈	∈	PROPN
ejpam-6011	85	5	j(r	j(r	PROPN
ejpam-6011	85	6	)	)	PUNCT
ejpam-6011	85	7	from	from	ADP
ejpam-6011	85	8	the	the	DET
ejpam-6011	85	9	jreducedness	jreducedness	NOUN
ejpam-6011	85	10	of	of	ADP
ejpam-6011	85	11	r.	r.	PROPN
ejpam-6011	85	12	for	for	ADP
ejpam-6011	85	13	every	every	DET
ejpam-6011	85	14	r	r	NOUN
ejpam-6011	85	15	∈	∈	NOUN
ejpam-6011	85	16	r	r	NOUN
ejpam-6011	85	17	,	,	PUNCT
ejpam-6011	85	18	we	we	PRON
ejpam-6011	85	19	have	have	VERB
ejpam-6011	85	20	era	era	NOUN
ejpam-6011	85	21	=	=	SYM
ejpam-6011	85	22	(	(	PUNCT
ejpam-6011	85	23	e(ra)−	e(ra)−	X
ejpam-6011	85	24	(	(	PUNCT
ejpam-6011	85	25	ra)e	ra)e	NOUN
ejpam-6011	85	26	)	)	PUNCT
ejpam-6011	86	1	+	+	CCONJ
ejpam-6011	86	2	r(ae	r(ae	NOUN
ejpam-6011	86	3	)	)	PUNCT
ejpam-6011	86	4	∈	∈	PROPN
ejpam-6011	86	5	j(r	j(r	PROPN
ejpam-6011	86	6	)	)	PUNCT
ejpam-6011	86	7	since	since	SCONJ
ejpam-6011	86	8	r	r	NOUN
ejpam-6011	86	9	is	be	AUX
ejpam-6011	86	10	j	j	NOUN
ejpam-6011	86	11	-	-	PUNCT
ejpam-6011	86	12	abelian	abelian	ADJ
ejpam-6011	86	13	.	.	PUNCT
ejpam-6011	87	1	thus	thus	ADV
ejpam-6011	87	2	era	era	VERB
ejpam-6011	87	3	⊆	⊆	NUM
ejpam-6011	87	4	j(r	j(r	NOUN
ejpam-6011	87	5	)	)	PUNCT
ejpam-6011	87	6	and	and	CCONJ
ejpam-6011	87	7	r	r	NOUN
ejpam-6011	87	8	is	be	AUX
ejpam-6011	87	9	nj	nj	NOUN
ejpam-6011	87	10	-	-	PUNCT
ejpam-6011	87	11	abelian	abelian	ADJ
ejpam-6011	87	12	.	.	PUNCT
ejpam-6011	88	1	nevertheless	nevertheless	ADV
ejpam-6011	88	2	,	,	PUNCT
ejpam-6011	88	3	it	it	PRON
ejpam-6011	88	4	’s	’	VERB
ejpam-6011	88	5	worth	worth	ADJ
ejpam-6011	88	6	noting	note	VERB
ejpam-6011	88	7	that	that	SCONJ
ejpam-6011	88	8	nj	nj	PROPN
ejpam-6011	88	9	-	-	PUNCT
ejpam-6011	88	10	abelian	abelian	ADJ
ejpam-6011	88	11	and	and	CCONJ
ejpam-6011	88	12	abelian	abelian	ADJ
ejpam-6011	88	13	properties	property	NOUN
ejpam-6011	88	14	are	be	AUX
ejpam-6011	88	15	distinct	distinct	ADJ
ejpam-6011	88	16	.	.	PUNCT
ejpam-6011	89	1	the	the	DET
ejpam-6011	89	2	next	next	ADJ
ejpam-6011	89	3	examples	example	NOUN
ejpam-6011	89	4	show	show	VERB
ejpam-6011	89	5	that	that	SCONJ
ejpam-6011	89	6	the	the	DET
ejpam-6011	89	7	classes	class	NOUN
ejpam-6011	89	8	of	of	ADP
ejpam-6011	89	9	nj	nj	PROPN
ejpam-6011	89	10	-	-	PUNCT
ejpam-6011	89	11	abelian	abelian	ADJ
ejpam-6011	89	12	rings	ring	NOUN
ejpam-6011	89	13	and	and	CCONJ
ejpam-6011	89	14	abelian	abelian	NOUN
ejpam-6011	89	15	rings	ring	NOUN
ejpam-6011	89	16	are	be	AUX
ejpam-6011	89	17	independent	independent	ADJ
ejpam-6011	89	18	.	.	PUNCT
ejpam-6011	90	1	m.	m.	PROPN
ejpam-6011	90	2	saad	saad	PROPN
ejpam-6011	90	3	,	,	PUNCT
ejpam-6011	90	4	s.	s.	PROPN
ejpam-6011	90	5	m.	m.	PROPN
ejpam-6011	90	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	90	7	/	/	SYM
ejpam-6011	90	8	eur	eur	PROPN
ejpam-6011	90	9	.	.	PUNCT
ejpam-6011	91	1	j.	j.	PROPN
ejpam-6011	91	2	pure	pure	PROPN
ejpam-6011	91	3	appl	appl	PROPN
ejpam-6011	91	4	.	.	PROPN
ejpam-6011	91	5	math	math	PROPN
ejpam-6011	91	6	,	,	PUNCT
ejpam-6011	91	7	18	18	NUM
ejpam-6011	91	8	(	(	PUNCT
ejpam-6011	91	9	2	2	NUM
ejpam-6011	91	10	)	)	PUNCT
ejpam-6011	91	11	(	(	PUNCT
ejpam-6011	91	12	2025	2025	NUM
ejpam-6011	91	13	)	)	PUNCT
ejpam-6011	91	14	,	,	PUNCT
ejpam-6011	91	15	6011	6011	NUM
ejpam-6011	91	16	4	4	NUM
ejpam-6011	91	17	of	of	ADP
ejpam-6011	91	18	13	13	NUM
ejpam-6011	91	19	example	example	NOUN
ejpam-6011	91	20	2	2	NUM
ejpam-6011	91	21	.	.	X
ejpam-6011	91	22	consider	consider	VERB
ejpam-6011	91	23	the	the	DET
ejpam-6011	91	24	ring	ring	NOUN
ejpam-6011	91	25	r	r	NOUN
ejpam-6011	91	26	=	=	PUNCT
ejpam-6011	91	27	{	{	PUNCT
ejpam-6011	91	28	[	[	PUNCT
ejpam-6011	91	29	a	a	PRON
ejpam-6011	91	30	b	b	NOUN
ejpam-6011	91	31	c	c	NOUN
ejpam-6011	91	32	d	d	X
ejpam-6011	91	33	]	]	X
ejpam-6011	92	1	|	|	ADV
ejpam-6011	92	2	a	a	DET
ejpam-6011	92	3	≡	≡	PROPN
ejpam-6011	92	4	d	d	X
ejpam-6011	92	5	mod	mod	PROPN
ejpam-6011	92	6	2	2	NUM
ejpam-6011	92	7	,	,	PUNCT
ejpam-6011	92	8	b	b	PROPN
ejpam-6011	92	9	≡	≡	PROPN
ejpam-6011	92	10	c	c	PROPN
ejpam-6011	92	11	mod	mod	PROPN
ejpam-6011	92	12	2	2	NUM
ejpam-6011	92	13	,	,	PUNCT
ejpam-6011	92	14	a	a	DET
ejpam-6011	92	15	,	,	PUNCT
ejpam-6011	92	16	b	b	NOUN
ejpam-6011	92	17	,	,	PUNCT
ejpam-6011	92	18	c	c	NOUN
ejpam-6011	92	19	,	,	PUNCT
ejpam-6011	92	20	d	d	PROPN
ejpam-6011	92	21	∈	∈	PROPN
ejpam-6011	92	22	z	z	NOUN
ejpam-6011	92	23	}	}	PUNCT
ejpam-6011	92	24	.	.	PUNCT
ejpam-6011	93	1	r	r	NOUN
ejpam-6011	93	2	has	have	VERB
ejpam-6011	93	3	no	no	DET
ejpam-6011	93	4	nontrivial	nontrivial	ADJ
ejpam-6011	93	5	idempotents	idempotent	NOUN
ejpam-6011	93	6	;	;	PUNCT
ejpam-6011	93	7	consequently	consequently	ADV
ejpam-6011	93	8	,	,	PUNCT
ejpam-6011	93	9	r	r	NOUN
ejpam-6011	93	10	is	be	AUX
ejpam-6011	93	11	abelian	abelian	ADJ
ejpam-6011	93	12	.	.	PUNCT
ejpam-6011	94	1	the	the	DET
ejpam-6011	94	2	element	element	NOUN
ejpam-6011	94	3	α	α	NOUN
ejpam-6011	94	4	=	=	PUNCT
ejpam-6011	95	1	[	[	PUNCT
ejpam-6011	95	2	0	0	NUM
ejpam-6011	95	3	2	2	NUM
ejpam-6011	95	4	0	0	NUM
ejpam-6011	95	5	0	0	NUM
ejpam-6011	95	6	]	]	PUNCT
ejpam-6011	95	7	in	in	ADP
ejpam-6011	95	8	r	r	NOUN
ejpam-6011	95	9	satisfies	satisfie	NOUN
ejpam-6011	95	10	α1	α1	PROPN
ejpam-6011	95	11	∈	∈	PROPN
ejpam-6011	95	12	n(r	n(r	NOUN
ejpam-6011	95	13	)	)	PUNCT
ejpam-6011	95	14	while	while	SCONJ
ejpam-6011	95	15	αr1	αr1	PRON
ejpam-6011	95	16	=	=	PUNCT
ejpam-6011	95	17	[	[	PUNCT
ejpam-6011	95	18	2z	2z	NUM
ejpam-6011	95	19	2z	2z	NUM
ejpam-6011	95	20	0	0	NUM
ejpam-6011	95	21	0	0	NUM
ejpam-6011	95	22	]	]	PUNCT
ejpam-6011	96	1	⊈	⊈	PROPN
ejpam-6011	96	2	j(r	j(r	NOUN
ejpam-6011	96	3	)	)	PUNCT
ejpam-6011	96	4	=	=	PUNCT
ejpam-6011	97	1	0	0	X
ejpam-6011	97	2	.	.	PUNCT
ejpam-6011	98	1	therefore	therefore	ADV
ejpam-6011	98	2	,	,	PUNCT
ejpam-6011	98	3	r	r	NOUN
ejpam-6011	98	4	is	be	AUX
ejpam-6011	98	5	not	not	PART
ejpam-6011	98	6	nj	nj	NOUN
ejpam-6011	98	7	-	-	PUNCT
ejpam-6011	98	8	abelian	abelian	PROPN
ejpam-6011	98	9	.	.	PUNCT
ejpam-6011	98	10	example	example	NOUN
ejpam-6011	99	1	3	3	NUM
ejpam-6011	99	2	.	.	X
ejpam-6011	99	3	for	for	ADP
ejpam-6011	99	4	any	any	DET
ejpam-6011	99	5	field	field	NOUN
ejpam-6011	99	6	f	f	NOUN
ejpam-6011	99	7	,	,	PUNCT
ejpam-6011	99	8	the	the	DET
ejpam-6011	99	9	ring	ring	NOUN
ejpam-6011	99	10	t	t	PROPN
ejpam-6011	99	11	2(f	2(f	NUM
ejpam-6011	99	12	)	)	PUNCT
ejpam-6011	99	13	is	be	AUX
ejpam-6011	99	14	not	not	PART
ejpam-6011	99	15	abelian	abelian	ADJ
ejpam-6011	99	16	,	,	PUNCT
ejpam-6011	99	17	where	where	SCONJ
ejpam-6011	99	18	the	the	DET
ejpam-6011	99	19	set	set	NOUN
ejpam-6011	99	20	of	of	ADP
ejpam-6011	99	21	nontrivial	nontrivial	ADJ
ejpam-6011	99	22	idempotents	idempotent	NOUN
ejpam-6011	99	23	of	of	ADP
ejpam-6011	99	24	r	r	NOUN
ejpam-6011	99	25	is	be	AUX
ejpam-6011	99	26	{	{	PUNCT
ejpam-6011	99	27	[	[	PUNCT
ejpam-6011	99	28	1	1	NUM
ejpam-6011	99	29	a	a	DET
ejpam-6011	99	30	0	0	NUM
ejpam-6011	99	31	0	0	NUM
ejpam-6011	99	32	]	]	PUNCT
ejpam-6011	99	33	,	,	PUNCT
ejpam-6011	99	34	[	[	PUNCT
ejpam-6011	99	35	0	0	NUM
ejpam-6011	99	36	a	a	DET
ejpam-6011	99	37	0	0	NUM
ejpam-6011	99	38	1	1	NUM
ejpam-6011	99	39	]	]	PUNCT
ejpam-6011	99	40	|	|	ADV
ejpam-6011	99	41	a	a	DET
ejpam-6011	99	42	∈	∈	NOUN
ejpam-6011	99	43	f	f	X
ejpam-6011	99	44	}	}	PUNCT
ejpam-6011	99	45	is	be	AUX
ejpam-6011	99	46	not	not	PART
ejpam-6011	99	47	central	central	ADJ
ejpam-6011	99	48	.	.	PUNCT
ejpam-6011	100	1	for	for	ADP
ejpam-6011	100	2	the	the	DET
ejpam-6011	100	3	idempotents	idempotent	NOUN
ejpam-6011	100	4	of	of	ADP
ejpam-6011	100	5	the	the	DET
ejpam-6011	100	6	form	form	NOUN
ejpam-6011	100	7	[	[	PUNCT
ejpam-6011	100	8	0	0	NUM
ejpam-6011	100	9	a	a	DET
ejpam-6011	100	10	0	0	NUM
ejpam-6011	100	11	1	1	NUM
ejpam-6011	100	12	]	]	PUNCT
ejpam-6011	100	13	,	,	PUNCT
ejpam-6011	100	14	if	if	SCONJ
ejpam-6011	100	15	αe	αe	PROPN
ejpam-6011	100	16	∈	∈	PROPN
ejpam-6011	100	17	n(r	n(r	NOUN
ejpam-6011	100	18	)	)	PUNCT
ejpam-6011	100	19	for	for	ADP
ejpam-6011	100	20	some	some	DET
ejpam-6011	100	21	α	α	NOUN
ejpam-6011	100	22	∈	∈	NOUN
ejpam-6011	100	23	r	r	NOUN
ejpam-6011	100	24	,	,	PUNCT
ejpam-6011	100	25	then	then	ADV
ejpam-6011	100	26	α	α	NOUN
ejpam-6011	100	27	=	=	PUNCT
ejpam-6011	101	1	[	[	PUNCT
ejpam-6011	101	2	b	b	X
ejpam-6011	101	3	c	c	NOUN
ejpam-6011	101	4	0	0	NUM
ejpam-6011	101	5	0	0	NUM
ejpam-6011	101	6	]	]	PUNCT
ejpam-6011	101	7	for	for	ADP
ejpam-6011	101	8	arbitrary	arbitrary	ADJ
ejpam-6011	101	9	b	b	NOUN
ejpam-6011	101	10	,	,	PUNCT
ejpam-6011	101	11	c	c	PROPN
ejpam-6011	101	12	∈	∈	PROPN
ejpam-6011	101	13	f	f	X
ejpam-6011	101	14	.	.	PUNCT
ejpam-6011	102	1	so	so	ADV
ejpam-6011	102	2	,	,	PUNCT
ejpam-6011	102	3	αre	αre	NUM
ejpam-6011	102	4	∈	∈	PROPN
ejpam-6011	102	5	[	[	PUNCT
ejpam-6011	102	6	0	0	NUM
ejpam-6011	102	7	f	f	NOUN
ejpam-6011	102	8	0	0	NUM
ejpam-6011	102	9	0	0	NUM
ejpam-6011	102	10	]	]	PUNCT
ejpam-6011	102	11	=	=	PUNCT
ejpam-6011	102	12	j(r	j(r	PROPN
ejpam-6011	102	13	)	)	PUNCT
ejpam-6011	102	14	.	.	PUNCT
ejpam-6011	103	1	therefore	therefore	ADV
ejpam-6011	103	2	,	,	PUNCT
ejpam-6011	103	3	r	r	NOUN
ejpam-6011	103	4	is	be	AUX
ejpam-6011	103	5	nj	nj	NOUN
ejpam-6011	103	6	-	-	PUNCT
ejpam-6011	103	7	abelian	abelian	ADJ
ejpam-6011	103	8	.	.	PUNCT
ejpam-6011	104	1	the	the	DET
ejpam-6011	104	2	two	two	NUM
ejpam-6011	104	3	examples	example	NOUN
ejpam-6011	104	4	above	above	ADV
ejpam-6011	104	5	demonstrate	demonstrate	VERB
ejpam-6011	104	6	that	that	SCONJ
ejpam-6011	104	7	the	the	DET
ejpam-6011	104	8	abelian	abelian	NOUN
ejpam-6011	104	9	and	and	CCONJ
ejpam-6011	104	10	nj	nj	PROPN
ejpam-6011	104	11	-	-	PUNCT
ejpam-6011	104	12	abelian	abelian	ADJ
ejpam-6011	104	13	properties	property	NOUN
ejpam-6011	104	14	of	of	ADP
ejpam-6011	104	15	rings	ring	NOUN
ejpam-6011	104	16	are	be	AUX
ejpam-6011	104	17	independent	independent	ADJ
ejpam-6011	104	18	of	of	ADP
ejpam-6011	104	19	each	each	DET
ejpam-6011	104	20	other	other	ADJ
ejpam-6011	104	21	.	.	PUNCT
ejpam-6011	105	1	however	however	ADV
ejpam-6011	105	2	,	,	PUNCT
ejpam-6011	105	3	this	this	PRON
ejpam-6011	105	4	does	do	AUX
ejpam-6011	105	5	not	not	PART
ejpam-6011	105	6	imply	imply	VERB
ejpam-6011	105	7	that	that	SCONJ
ejpam-6011	105	8	the	the	DET
ejpam-6011	105	9	abelian	abelian	NOUN
ejpam-6011	105	10	and	and	CCONJ
ejpam-6011	105	11	nj	nj	PROPN
ejpam-6011	105	12	-	-	PUNCT
ejpam-6011	105	13	abelian	abelian	ADJ
ejpam-6011	105	14	classes	class	NOUN
ejpam-6011	105	15	are	be	AUX
ejpam-6011	105	16	completely	completely	ADV
ejpam-6011	105	17	disjoint	disjoint	ADJ
ejpam-6011	105	18	.	.	PUNCT
ejpam-6011	106	1	for	for	ADP
ejpam-6011	106	2	instance	instance	NOUN
ejpam-6011	106	3	,	,	PUNCT
ejpam-6011	106	4	the	the	DET
ejpam-6011	106	5	ring	ring	NOUN
ejpam-6011	106	6	z	z	PROPN
ejpam-6011	106	7	is	be	AUX
ejpam-6011	106	8	both	both	CCONJ
ejpam-6011	106	9	abelian	abelian	ADJ
ejpam-6011	106	10	and	and	CCONJ
ejpam-6011	106	11	nj	nj	PROPN
ejpam-6011	106	12	-	-	PUNCT
ejpam-6011	106	13	abelian	abelian	PROPN
ejpam-6011	106	14	.	.	PUNCT
ejpam-6011	107	1	additionally	additionally	ADV
ejpam-6011	107	2	,	,	PUNCT
ejpam-6011	107	3	there	there	PRON
ejpam-6011	107	4	exists	exist	VERB
ejpam-6011	107	5	a	a	DET
ejpam-6011	107	6	ring	ring	NOUN
ejpam-6011	107	7	that	that	PRON
ejpam-6011	107	8	is	be	AUX
ejpam-6011	107	9	neither	neither	CCONJ
ejpam-6011	107	10	abelian	abelian	ADJ
ejpam-6011	107	11	nor	nor	CCONJ
ejpam-6011	107	12	nj	nj	NOUN
ejpam-6011	107	13	-	-	PUNCT
ejpam-6011	107	14	abelian	abelian	NOUN
ejpam-6011	107	15	(	(	PUNCT
ejpam-6011	107	16	see	see	VERB
ejpam-6011	107	17	the	the	DET
ejpam-6011	107	18	next	next	ADJ
ejpam-6011	107	19	example	example	NOUN
ejpam-6011	107	20	)	)	PUNCT
ejpam-6011	107	21	.	.	PUNCT
ejpam-6011	108	1	example	example	NOUN
ejpam-6011	109	1	4	4	X
ejpam-6011	109	2	.	.	PUNCT
ejpam-6011	109	3	consider	consider	VERB
ejpam-6011	109	4	the	the	DET
ejpam-6011	109	5	ring	ring	NOUN
ejpam-6011	109	6	r	r	NOUN
ejpam-6011	109	7	=	=	PUNCT
ejpam-6011	109	8	f	f	PROPN
ejpam-6011	110	1	+	+	PROPN
ejpam-6011	110	2	f	f	PROPN
ejpam-6011	110	3	j	j	PROPN
ejpam-6011	110	4	,	,	PUNCT
ejpam-6011	110	5	where	where	SCONJ
ejpam-6011	110	6	f	f	PROPN
ejpam-6011	110	7	is	be	AUX
ejpam-6011	110	8	a	a	DET
ejpam-6011	110	9	field	field	NOUN
ejpam-6011	110	10	,	,	PUNCT
ejpam-6011	110	11	with	with	ADP
ejpam-6011	110	12	2−1	2−1	NUM
ejpam-6011	110	13	∈	∈	PROPN
ejpam-6011	110	14	f	f	X
ejpam-6011	110	15	,	,	PUNCT
ejpam-6011	110	16	and	and	CCONJ
ejpam-6011	110	17	j2	j2	PROPN
ejpam-6011	110	18	=	=	SYM
ejpam-6011	110	19	1	1	X
ejpam-6011	110	20	.	.	PUNCT
ejpam-6011	111	1	r	r	NOUN
ejpam-6011	111	2	is	be	AUX
ejpam-6011	111	3	a	a	DET
ejpam-6011	111	4	commutative	commutative	ADJ
ejpam-6011	111	5	reduced	reduce	VERB
ejpam-6011	111	6	ring	ring	NOUN
ejpam-6011	111	7	,	,	PUNCT
ejpam-6011	111	8	and	and	CCONJ
ejpam-6011	111	9	its	its	PRON
ejpam-6011	111	10	set	set	NOUN
ejpam-6011	111	11	of	of	ADP
ejpam-6011	111	12	idempotents	idempotent	NOUN
ejpam-6011	111	13	is	be	AUX
ejpam-6011	111	14	{	{	PUNCT
ejpam-6011	111	15	0	0	NUM
ejpam-6011	111	16	,	,	PUNCT
ejpam-6011	111	17	1	1	NUM
ejpam-6011	111	18	,	,	PUNCT
ejpam-6011	111	19	12(1	12(1	NUM
ejpam-6011	111	20	+	+	CCONJ
ejpam-6011	111	21	j	j	PROPN
ejpam-6011	111	22	)	)	PUNCT
ejpam-6011	111	23	,	,	PUNCT
ejpam-6011	111	24	12(1	12(1	X
ejpam-6011	111	25	−	−	PROPN
ejpam-6011	111	26	j	j	PROPN
ejpam-6011	111	27	)	)	PUNCT
ejpam-6011	111	28	}	}	PUNCT
ejpam-6011	111	29	.	.	PUNCT
ejpam-6011	112	1	define	define	VERB
ejpam-6011	112	2	the	the	DET
ejpam-6011	112	3	automorphism	automorphism	NOUN
ejpam-6011	112	4	σ	σ	NOUN
ejpam-6011	112	5	:	:	PUNCT
ejpam-6011	112	6	r	r	NOUN
ejpam-6011	112	7	→	→	SYM
ejpam-6011	112	8	r	r	NOUN
ejpam-6011	112	9	as	as	ADP
ejpam-6011	112	10	σ(a	σ(a	PROPN
ejpam-6011	112	11	+	+	CCONJ
ejpam-6011	112	12	bj	bj	NOUN
ejpam-6011	112	13	)	)	PUNCT
ejpam-6011	112	14	=	=	PUNCT
ejpam-6011	112	15	a	a	DET
ejpam-6011	112	16	−	−	NOUN
ejpam-6011	112	17	bj	bj	NOUN
ejpam-6011	112	18	,	,	PUNCT
ejpam-6011	112	19	for	for	ADP
ejpam-6011	112	20	every	every	DET
ejpam-6011	112	21	a	a	PROPN
ejpam-6011	112	22	,	,	PUNCT
ejpam-6011	112	23	b	b	PROPN
ejpam-6011	112	24	∈	∈	PROPN
ejpam-6011	112	25	f	f	X
ejpam-6011	112	26	.	.	PUNCT
ejpam-6011	113	1	let	let	AUX
ejpam-6011	113	2	s	s	PRON
ejpam-6011	113	3	=	=	VERB
ejpam-6011	113	4	r[x;σ	r[x;σ	NOUN
ejpam-6011	113	5	]	]	PUNCT
ejpam-6011	113	6	be	be	VERB
ejpam-6011	113	7	the	the	DET
ejpam-6011	113	8	skew	skew	ADJ
ejpam-6011	113	9	polynomial	polynomial	ADJ
ejpam-6011	113	10	ring	ring	NOUN
ejpam-6011	113	11	with	with	ADP
ejpam-6011	113	12	an	an	DET
ejpam-6011	113	13	indeterminate	indeterminate	ADJ
ejpam-6011	113	14	x	x	NOUN
ejpam-6011	113	15	over	over	ADP
ejpam-6011	113	16	r.	r.	PROPN
ejpam-6011	113	17	the	the	DET
ejpam-6011	113	18	element	element	NOUN
ejpam-6011	113	19	α	α	PROPN
ejpam-6011	114	1	=	=	PUNCT
ejpam-6011	115	1	(	(	PUNCT
ejpam-6011	115	2	1	1	NUM
ejpam-6011	115	3	+	+	NUM
ejpam-6011	115	4	j)x	j)x	X
ejpam-6011	115	5	in	in	ADP
ejpam-6011	115	6	s	s	PROPN
ejpam-6011	115	7	satisfies	satisfie	NOUN
ejpam-6011	115	8	α1	α1	PROPN
ejpam-6011	115	9	∈	∈	PROPN
ejpam-6011	115	10	n(s	n(s	PROPN
ejpam-6011	115	11	)	)	PUNCT
ejpam-6011	115	12	while	while	SCONJ
ejpam-6011	115	13	αs1	αs1	PROPN
ejpam-6011	115	14	̸=	̸=	PROPN
ejpam-6011	115	15	0	0	NUM
ejpam-6011	115	16	.	.	PUNCT
ejpam-6011	116	1	therefore	therefore	ADV
ejpam-6011	116	2	,	,	PUNCT
ejpam-6011	116	3	s	s	VERB
ejpam-6011	116	4	is	be	AUX
ejpam-6011	116	5	not	not	PART
ejpam-6011	116	6	nj	nj	NOUN
ejpam-6011	116	7	-	-	NOUN
ejpam-6011	116	8	abelian	abelian	ADJ
ejpam-6011	116	9	since	since	SCONJ
ejpam-6011	116	10	r	r	NOUN
ejpam-6011	116	11	is	be	AUX
ejpam-6011	116	12	semiprimitive	semiprimitive	ADJ
ejpam-6011	116	13	.	.	PUNCT
ejpam-6011	117	1	also	also	ADV
ejpam-6011	117	2	,	,	PUNCT
ejpam-6011	117	3	the	the	DET
ejpam-6011	117	4	nontrivial	nontrivial	ADJ
ejpam-6011	117	5	idempotents	idempotent	NOUN
ejpam-6011	117	6	of	of	ADP
ejpam-6011	117	7	s	s	PRON
ejpam-6011	117	8	have	have	VERB
ejpam-6011	117	9	the	the	DET
ejpam-6011	117	10	form	form	NOUN
ejpam-6011	117	11	e+(1±	e+(1±	ADJ
ejpam-6011	117	12	j)fx	j)fx	PROPN
ejpam-6011	117	13	,	,	PUNCT
ejpam-6011	117	14	where	where	SCONJ
ejpam-6011	117	15	e2	e2	PROPN
ejpam-6011	117	16	=	=	SYM
ejpam-6011	117	17	e	e	PROPN
ejpam-6011	117	18	,	,	PUNCT
ejpam-6011	117	19	f	f	PROPN
ejpam-6011	117	20	∈	∈	PROPN
ejpam-6011	117	21	r.	r.	PROPN
ejpam-6011	118	1	so	so	ADV
ejpam-6011	118	2	r	r	NOUN
ejpam-6011	118	3	is	be	AUX
ejpam-6011	118	4	not	not	PART
ejpam-6011	118	5	abelian	abelian	ADJ
ejpam-6011	118	6	.	.	PUNCT
ejpam-6011	119	1	the	the	DET
ejpam-6011	119	2	next	next	ADJ
ejpam-6011	119	3	proposition	proposition	NOUN
ejpam-6011	119	4	gives	give	VERB
ejpam-6011	119	5	a	a	DET
ejpam-6011	119	6	sufficient	sufficient	ADJ
ejpam-6011	119	7	condition	condition	NOUN
ejpam-6011	119	8	to	to	PART
ejpam-6011	119	9	make	make	VERB
ejpam-6011	119	10	an	an	DET
ejpam-6011	119	11	nj	nj	ADJ
ejpam-6011	119	12	-	-	PUNCT
ejpam-6011	119	13	abelian	abelian	ADJ
ejpam-6011	119	14	ring	ring	NOUN
ejpam-6011	119	15	abelian	abelian	PROPN
ejpam-6011	119	16	.	.	PUNCT
ejpam-6011	120	1	proposition	proposition	NOUN
ejpam-6011	120	2	4	4	NUM
ejpam-6011	120	3	.	.	PUNCT
ejpam-6011	121	1	if	if	SCONJ
ejpam-6011	121	2	r	r	NOUN
ejpam-6011	121	3	is	be	AUX
ejpam-6011	121	4	a	a	DET
ejpam-6011	121	5	semiprimitive	semiprimitive	ADJ
ejpam-6011	121	6	nj	nj	ADJ
ejpam-6011	121	7	-	-	PUNCT
ejpam-6011	121	8	abelian	abelian	ADJ
ejpam-6011	121	9	ring	ring	NOUN
ejpam-6011	121	10	,	,	PUNCT
ejpam-6011	121	11	then	then	ADV
ejpam-6011	121	12	r	r	NOUN
ejpam-6011	121	13	is	be	AUX
ejpam-6011	121	14	abelian	abelian	ADJ
ejpam-6011	121	15	.	.	PUNCT
ejpam-6011	122	1	proof	proof	NOUN
ejpam-6011	122	2	.	.	PUNCT
ejpam-6011	123	1	for	for	ADP
ejpam-6011	123	2	every	every	DET
ejpam-6011	123	3	idempotent	idempotent	ADJ
ejpam-6011	123	4	e	e	NOUN
ejpam-6011	123	5	of	of	ADP
ejpam-6011	123	6	a	a	DET
ejpam-6011	123	7	ring	ring	NOUN
ejpam-6011	123	8	r	r	NOUN
ejpam-6011	123	9	,	,	PUNCT
ejpam-6011	123	10	we	we	PRON
ejpam-6011	123	11	have	have	VERB
ejpam-6011	123	12	0	0	NUM
ejpam-6011	123	13	=	=	SYM
ejpam-6011	123	14	e(1	e(1	PROPN
ejpam-6011	123	15	−	−	PROPN
ejpam-6011	123	16	e	e	X
ejpam-6011	123	17	)	)	PUNCT
ejpam-6011	123	18	∈	∈	PROPN
ejpam-6011	123	19	n(r	n(r	NOUN
ejpam-6011	123	20	)	)	PUNCT
ejpam-6011	123	21	.	.	PUNCT
ejpam-6011	124	1	so	so	ADV
ejpam-6011	124	2	,	,	PUNCT
ejpam-6011	124	3	er(1−	er(1−	ADJ
ejpam-6011	124	4	e	e	NOUN
ejpam-6011	124	5	)	)	PUNCT
ejpam-6011	124	6	⊆	⊆	NUM
ejpam-6011	124	7	j(r	j(r	NOUN
ejpam-6011	124	8	)	)	PUNCT
ejpam-6011	124	9	=	=	PUNCT
ejpam-6011	125	1	0	0	X
ejpam-6011	125	2	.	.	PUNCT
ejpam-6011	126	1	hence	hence	ADV
ejpam-6011	126	2	,	,	PUNCT
ejpam-6011	126	3	e	e	X
ejpam-6011	126	4	is	be	AUX
ejpam-6011	126	5	right	right	ADV
ejpam-6011	126	6	semicentral	semicentral	ADJ
ejpam-6011	126	7	.	.	PUNCT
ejpam-6011	127	1	similarly	similarly	ADV
ejpam-6011	127	2	,	,	PUNCT
ejpam-6011	127	3	we	we	PRON
ejpam-6011	127	4	demonstrate	demonstrate	VERB
ejpam-6011	127	5	that	that	SCONJ
ejpam-6011	127	6	e	e	NOUN
ejpam-6011	127	7	is	be	AUX
ejpam-6011	127	8	also	also	ADV
ejpam-6011	127	9	left	leave	VERB
ejpam-6011	127	10	semicentral	semicentral	ADJ
ejpam-6011	127	11	.	.	PUNCT
ejpam-6011	128	1	therefore	therefore	ADV
ejpam-6011	128	2	,	,	PUNCT
ejpam-6011	128	3	e	e	NOUN
ejpam-6011	128	4	is	be	AUX
ejpam-6011	128	5	central	central	ADJ
ejpam-6011	128	6	,	,	PUNCT
ejpam-6011	128	7	and	and	CCONJ
ejpam-6011	128	8	hence	hence	ADV
ejpam-6011	128	9	r	r	NOUN
ejpam-6011	128	10	is	be	AUX
ejpam-6011	128	11	abelian	abelian	ADJ
ejpam-6011	128	12	.	.	PUNCT
ejpam-6011	129	1	according	accord	VERB
ejpam-6011	129	2	to	to	ADP
ejpam-6011	129	3	the	the	DET
ejpam-6011	129	4	definitions	definition	NOUN
ejpam-6011	129	5	,	,	PUNCT
ejpam-6011	129	6	every	every	DET
ejpam-6011	129	7	nj	nj	ADJ
ejpam-6011	129	8	-	-	PUNCT
ejpam-6011	129	9	semicommutative	semicommutative	NOUN
ejpam-6011	129	10	ring	ring	NOUN
ejpam-6011	129	11	is	be	AUX
ejpam-6011	129	12	nj	nj	NOUN
ejpam-6011	129	13	-	-	PUNCT
ejpam-6011	129	14	abelian	abelian	NOUN
ejpam-6011	129	15	.	.	PUNCT
ejpam-6011	130	1	however	however	ADV
ejpam-6011	130	2	,	,	PUNCT
ejpam-6011	130	3	the	the	DET
ejpam-6011	130	4	converse	converse	NOUN
ejpam-6011	130	5	does	do	AUX
ejpam-6011	130	6	not	not	PART
ejpam-6011	130	7	necessarily	necessarily	ADV
ejpam-6011	130	8	hold	hold	VERB
ejpam-6011	130	9	,	,	PUNCT
ejpam-6011	130	10	as	as	SCONJ
ejpam-6011	130	11	illustrated	illustrate	VERB
ejpam-6011	130	12	by	by	ADP
ejpam-6011	130	13	the	the	DET
ejpam-6011	130	14	following	follow	VERB
ejpam-6011	130	15	example	example	NOUN
ejpam-6011	130	16	.	.	PUNCT
ejpam-6011	131	1	example	example	NOUN
ejpam-6011	132	1	5	5	NUM
ejpam-6011	132	2	.	.	PUNCT
ejpam-6011	132	3	let	let	VERB
ejpam-6011	132	4	k	k	PRON
ejpam-6011	132	5	be	be	AUX
ejpam-6011	132	6	a	a	DET
ejpam-6011	132	7	countable	countable	ADJ
ejpam-6011	132	8	field	field	NOUN
ejpam-6011	132	9	.	.	PUNCT
ejpam-6011	133	1	by	by	ADP
ejpam-6011	133	2	[	[	X
ejpam-6011	133	3	10	10	NUM
ejpam-6011	133	4	,	,	PUNCT
ejpam-6011	133	5	lemma	lemma	PROPN
ejpam-6011	133	6	3.7	3.7	NUM
ejpam-6011	133	7	]	]	PUNCT
ejpam-6011	133	8	,	,	PUNCT
ejpam-6011	133	9	there	there	PRON
ejpam-6011	133	10	exists	exist	VERB
ejpam-6011	133	11	a	a	DET
ejpam-6011	133	12	nonzero	nonzero	NOUN
ejpam-6011	133	13	nil	nil	NOUN
ejpam-6011	133	14	algebra	algebra	NOUN
ejpam-6011	133	15	a	a	PRON
ejpam-6011	133	16	over	over	ADP
ejpam-6011	133	17	k	k	NOUN
ejpam-6011	133	18	such	such	ADJ
ejpam-6011	133	19	that	that	DET
ejpam-6011	133	20	a[x	a[x	NOUN
ejpam-6011	133	21	]	]	PUNCT
ejpam-6011	133	22	has	have	VERB
ejpam-6011	133	23	a	a	DET
ejpam-6011	133	24	zero	zero	NUM
ejpam-6011	133	25	upper	upper	ADJ
ejpam-6011	133	26	nil	nil	NOUN
ejpam-6011	133	27	radical	radical	NOUN
ejpam-6011	133	28	.	.	PUNCT
ejpam-6011	134	1	the	the	DET
ejpam-6011	134	2	ring	ring	NOUN
ejpam-6011	134	3	r	r	NOUN
ejpam-6011	134	4	=	=	PUNCT
ejpam-6011	134	5	(	(	PUNCT
ejpam-6011	134	6	k	k	PROPN
ejpam-6011	134	7	+	+	CCONJ
ejpam-6011	134	8	a)[x	a)[x	PROPN
ejpam-6011	134	9	]	]	X
ejpam-6011	134	10	is	be	AUX
ejpam-6011	134	11	not	not	PART
ejpam-6011	134	12	nj	nj	NOUN
ejpam-6011	134	13	-	-	PUNCT
ejpam-6011	134	14	semicommutative	semicommutative	NOUN
ejpam-6011	134	15	,	,	PUNCT
ejpam-6011	134	16	as	as	SCONJ
ejpam-6011	134	17	shown	show	VERB
ejpam-6011	134	18	in	in	ADP
ejpam-6011	134	19	[	[	X
ejpam-6011	134	20	7	7	NUM
ejpam-6011	134	21	,	,	PUNCT
ejpam-6011	134	22	example	example	NOUN
ejpam-6011	134	23	4	4	NUM
ejpam-6011	134	24	]	]	PUNCT
ejpam-6011	134	25	.	.	PUNCT
ejpam-6011	135	1	however	however	ADV
ejpam-6011	135	2	,	,	PUNCT
ejpam-6011	135	3	r	r	NOUN
ejpam-6011	135	4	is	be	AUX
ejpam-6011	135	5	j	j	NOUN
ejpam-6011	135	6	-	-	PUNCT
ejpam-6011	135	7	reduced	reduce	VERB
ejpam-6011	135	8	,	,	PUNCT
ejpam-6011	135	9	with	with	ADP
ejpam-6011	135	10	n(r	n(r	NOUN
ejpam-6011	135	11	)	)	PUNCT
ejpam-6011	135	12	=	=	SYM
ejpam-6011	135	13	j(r	j(r	PROPN
ejpam-6011	135	14	)	)	PUNCT
ejpam-6011	135	15	=	=	SYM
ejpam-6011	135	16	a[x	a[x	NOUN
ejpam-6011	135	17	]	]	PUNCT
ejpam-6011	135	18	,	,	PUNCT
ejpam-6011	135	19	and	and	CCONJ
ejpam-6011	135	20	has	have	VERB
ejpam-6011	135	21	only	only	ADV
ejpam-6011	135	22	trivial	trivial	ADJ
ejpam-6011	135	23	idempotents	idempotent	NOUN
ejpam-6011	135	24	.	.	PUNCT
ejpam-6011	136	1	therefore	therefore	ADV
ejpam-6011	136	2	,	,	PUNCT
ejpam-6011	136	3	r	r	NOUN
ejpam-6011	136	4	is	be	AUX
ejpam-6011	136	5	nj	nj	NOUN
ejpam-6011	136	6	-	-	PUNCT
ejpam-6011	136	7	abelian	abelian	NOUN
ejpam-6011	136	8	.	.	PUNCT
ejpam-6011	137	1	m.	m.	PROPN
ejpam-6011	137	2	saad	saad	PROPN
ejpam-6011	137	3	,	,	PUNCT
ejpam-6011	137	4	s.	s.	PROPN
ejpam-6011	137	5	m.	m.	PROPN
ejpam-6011	137	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	137	7	/	/	SYM
ejpam-6011	137	8	eur	eur	PROPN
ejpam-6011	137	9	.	.	PUNCT
ejpam-6011	138	1	j.	j.	PROPN
ejpam-6011	138	2	pure	pure	PROPN
ejpam-6011	138	3	appl	appl	PROPN
ejpam-6011	138	4	.	.	PROPN
ejpam-6011	138	5	math	math	PROPN
ejpam-6011	138	6	,	,	PUNCT
ejpam-6011	138	7	18	18	NUM
ejpam-6011	138	8	(	(	PUNCT
ejpam-6011	138	9	2	2	NUM
ejpam-6011	138	10	)	)	PUNCT
ejpam-6011	138	11	(	(	PUNCT
ejpam-6011	138	12	2025	2025	NUM
ejpam-6011	138	13	)	)	PUNCT
ejpam-6011	138	14	,	,	PUNCT
ejpam-6011	138	15	6011	6011	NUM
ejpam-6011	138	16	5	5	NUM
ejpam-6011	138	17	of	of	ADP
ejpam-6011	138	18	13	13	NUM
ejpam-6011	138	19	remind	remind	VERB
ejpam-6011	138	20	that	that	SCONJ
ejpam-6011	138	21	a	a	DET
ejpam-6011	138	22	ring	ring	NOUN
ejpam-6011	138	23	r	r	NOUN
ejpam-6011	138	24	is	be	AUX
ejpam-6011	138	25	said	say	VERB
ejpam-6011	138	26	to	to	PART
ejpam-6011	138	27	be	be	AUX
ejpam-6011	138	28	dedekind	dedekind	NOUN
ejpam-6011	138	29	-	-	PUNCT
ejpam-6011	138	30	finite	finite	ADJ
ejpam-6011	138	31	if	if	SCONJ
ejpam-6011	138	32	ab	ab	PROPN
ejpam-6011	138	33	=	=	SYM
ejpam-6011	138	34	1	1	NUM
ejpam-6011	138	35	implies	imply	VERB
ejpam-6011	138	36	ba	ba	PROPN
ejpam-6011	138	37	=	=	NOUN
ejpam-6011	138	38	1	1	NUM
ejpam-6011	138	39	for	for	ADP
ejpam-6011	138	40	every	every	DET
ejpam-6011	138	41	a	a	PROPN
ejpam-6011	138	42	,	,	PUNCT
ejpam-6011	138	43	b	b	PROPN
ejpam-6011	138	44	∈	∈	PROPN
ejpam-6011	138	45	r.	r.	NOUN
ejpam-6011	138	46	the	the	DET
ejpam-6011	138	47	next	next	ADJ
ejpam-6011	138	48	proposition	proposition	NOUN
ejpam-6011	138	49	shows	show	VERB
ejpam-6011	138	50	that	that	SCONJ
ejpam-6011	138	51	every	every	DET
ejpam-6011	138	52	nj	nj	PROPN
ejpam-6011	138	53	-	-	PUNCT
ejpam-6011	138	54	abelian	abelian	ADJ
ejpam-6011	138	55	ring	ring	NOUN
ejpam-6011	138	56	is	be	AUX
ejpam-6011	138	57	dedekind	dedekind	NOUN
ejpam-6011	138	58	-	-	PUNCT
ejpam-6011	138	59	finite	finite	ADJ
ejpam-6011	138	60	.	.	PUNCT
ejpam-6011	139	1	proposition	proposition	NOUN
ejpam-6011	139	2	5	5	NUM
ejpam-6011	139	3	.	.	PUNCT
ejpam-6011	140	1	every	every	DET
ejpam-6011	140	2	nj	nj	PROPN
ejpam-6011	140	3	-	-	PUNCT
ejpam-6011	140	4	abelian	abelian	ADJ
ejpam-6011	140	5	ring	ring	NOUN
ejpam-6011	140	6	is	be	AUX
ejpam-6011	140	7	dedekind	dedekind	NOUN
ejpam-6011	140	8	-	-	PUNCT
ejpam-6011	140	9	finite	finite	ADJ
ejpam-6011	140	10	.	.	PUNCT
ejpam-6011	141	1	proof	proof	NOUN
ejpam-6011	141	2	.	.	PUNCT
ejpam-6011	142	1	let	let	VERB
ejpam-6011	142	2	r	r	PRON
ejpam-6011	142	3	be	be	AUX
ejpam-6011	142	4	an	an	DET
ejpam-6011	142	5	nj	nj	ADJ
ejpam-6011	142	6	-	-	PUNCT
ejpam-6011	142	7	abelian	abelian	ADJ
ejpam-6011	142	8	ring	ring	NOUN
ejpam-6011	142	9	with	with	ADP
ejpam-6011	142	10	ab	ab	PROPN
ejpam-6011	142	11	=	=	NOUN
ejpam-6011	142	12	1	1	NUM
ejpam-6011	142	13	for	for	ADP
ejpam-6011	142	14	some	some	DET
ejpam-6011	142	15	a	a	PRON
ejpam-6011	142	16	,	,	PUNCT
ejpam-6011	142	17	b	b	PROPN
ejpam-6011	142	18	∈	∈	PROPN
ejpam-6011	142	19	r.	r.	NOUN
ejpam-6011	143	1	so	so	SCONJ
ejpam-6011	143	2	that	that	SCONJ
ejpam-6011	143	3	(	(	PUNCT
ejpam-6011	143	4	1−	1−	NUM
ejpam-6011	143	5	ba)2	ba)2	PROPN
ejpam-6011	143	6	=	=	SYM
ejpam-6011	143	7	1−	1−	NUM
ejpam-6011	143	8	2ba+	2ba+	NUM
ejpam-6011	143	9	baba	baba	NOUN
ejpam-6011	143	10	=	=	PUNCT
ejpam-6011	143	11	1−	1−	NUM
ejpam-6011	143	12	ba	ba	PROPN
ejpam-6011	143	13	and	and	CCONJ
ejpam-6011	143	14	1−	1−	NUM
ejpam-6011	143	15	ba	ba	PROPN
ejpam-6011	143	16	is	be	AUX
ejpam-6011	143	17	an	an	DET
ejpam-6011	143	18	idempotent	idempotent	NOUN
ejpam-6011	143	19	of	of	ADP
ejpam-6011	143	20	r.	r.	PROPN
ejpam-6011	143	21	we	we	PRON
ejpam-6011	143	22	have	have	VERB
ejpam-6011	143	23	(	(	PUNCT
ejpam-6011	143	24	1−	1−	NUM
ejpam-6011	143	25	ba)b	ba)b	NOUN
ejpam-6011	143	26	=	=	SYM
ejpam-6011	143	27	0	0	NUM
ejpam-6011	143	28	∈	∈	PROPN
ejpam-6011	143	29	n(r	n(r	NOUN
ejpam-6011	143	30	)	)	PUNCT
ejpam-6011	143	31	and	and	CCONJ
ejpam-6011	143	32	consequently	consequently	ADV
ejpam-6011	143	33	1−	1−	NUM
ejpam-6011	143	34	ba	ba	NOUN
ejpam-6011	143	35	=	=	PRON
ejpam-6011	143	36	(	(	PUNCT
ejpam-6011	143	37	1−	1−	NUM
ejpam-6011	143	38	ba)ab	ba)ab	SYM
ejpam-6011	143	39	∈	∈	PROPN
ejpam-6011	143	40	(	(	PUNCT
ejpam-6011	143	41	1−	1−	NUM
ejpam-6011	143	42	ba)rb	ba)rb	NUM
ejpam-6011	143	43	⊆	⊆	NUM
ejpam-6011	143	44	j(r	j(r	NOUN
ejpam-6011	143	45	)	)	PUNCT
ejpam-6011	143	46	,	,	PUNCT
ejpam-6011	143	47	from	from	ADP
ejpam-6011	143	48	the	the	DET
ejpam-6011	143	49	nj	nj	PROPN
ejpam-6011	143	50	-	-	PUNCT
ejpam-6011	143	51	abelianity	abelianity	NOUN
ejpam-6011	143	52	of	of	ADP
ejpam-6011	143	53	r.	r.	PROPN
ejpam-6011	143	54	therefore	therefore	ADV
ejpam-6011	143	55	,	,	PUNCT
ejpam-6011	143	56	we	we	PRON
ejpam-6011	143	57	have	have	VERB
ejpam-6011	143	58	1−	1−	NUM
ejpam-6011	143	59	ba	ba	PROPN
ejpam-6011	144	1	=	=	NOUN
ejpam-6011	144	2	0	0	PROPN
ejpam-6011	144	3	and	and	CCONJ
ejpam-6011	144	4	ba	ba	PROPN
ejpam-6011	144	5	=	=	SYM
ejpam-6011	144	6	1	1	NUM
ejpam-6011	144	7	,	,	PUNCT
ejpam-6011	144	8	indicating	indicate	VERB
ejpam-6011	144	9	that	that	SCONJ
ejpam-6011	144	10	r	r	NOUN
ejpam-6011	144	11	is	be	AUX
ejpam-6011	144	12	dedekind	dedekind	NOUN
ejpam-6011	144	13	-	-	PUNCT
ejpam-6011	144	14	finite	finite	ADJ
ejpam-6011	144	15	.	.	PUNCT
ejpam-6011	145	1	in	in	ADP
ejpam-6011	145	2	[	[	X
ejpam-6011	145	3	11	11	NUM
ejpam-6011	145	4	]	]	PUNCT
ejpam-6011	145	5	,	,	PUNCT
ejpam-6011	145	6	an	an	DET
ejpam-6011	145	7	element	element	NOUN
ejpam-6011	145	8	a	a	PRON
ejpam-6011	145	9	of	of	ADP
ejpam-6011	145	10	a	a	DET
ejpam-6011	145	11	ring	ring	NOUN
ejpam-6011	145	12	r	r	NOUN
ejpam-6011	145	13	is	be	AUX
ejpam-6011	145	14	called	call	VERB
ejpam-6011	145	15	left	leave	VERB
ejpam-6011	145	16	minimal	minimal	ADJ
ejpam-6011	145	17	if	if	SCONJ
ejpam-6011	145	18	ra	ra	PROPN
ejpam-6011	145	19	is	be	AUX
ejpam-6011	145	20	a	a	DET
ejpam-6011	145	21	minimal	minimal	ADJ
ejpam-6011	145	22	left	leave	VERB
ejpam-6011	145	23	ideal	ideal	NOUN
ejpam-6011	145	24	of	of	ADP
ejpam-6011	145	25	r.	r.	PROPN
ejpam-6011	145	26	write	write	PROPN
ejpam-6011	145	27	mel(r	mel(r	NOUN
ejpam-6011	145	28	)	)	PUNCT
ejpam-6011	145	29	to	to	PART
ejpam-6011	145	30	denote	denote	VERB
ejpam-6011	145	31	the	the	DET
ejpam-6011	145	32	set	set	NOUN
ejpam-6011	145	33	of	of	ADP
ejpam-6011	145	34	all	all	PRON
ejpam-6011	145	35	left	leave	VERB
ejpam-6011	145	36	minimal	minimal	ADJ
ejpam-6011	145	37	idempotents	idempotent	NOUN
ejpam-6011	145	38	of	of	ADP
ejpam-6011	145	39	r.	r.	PROPN
ejpam-6011	145	40	a	a	DET
ejpam-6011	145	41	ring	ring	NOUN
ejpam-6011	145	42	r	r	NOUN
ejpam-6011	145	43	is	be	AUX
ejpam-6011	145	44	called	call	VERB
ejpam-6011	145	45	left	left	ADJ
ejpam-6011	145	46	min	min	NOUN
ejpam-6011	145	47	-	-	NOUN
ejpam-6011	145	48	abel	abel	NOUN
ejpam-6011	145	49	if	if	SCONJ
ejpam-6011	145	50	each	each	PRON
ejpam-6011	145	51	left	leave	VERB
ejpam-6011	145	52	minimal	minimal	ADJ
ejpam-6011	145	53	idempotent	idempotent	NOUN
ejpam-6011	145	54	is	be	AUX
ejpam-6011	145	55	left	leave	VERB
ejpam-6011	145	56	semicentral	semicentral	ADJ
ejpam-6011	145	57	.	.	PUNCT
ejpam-6011	146	1	the	the	DET
ejpam-6011	146	2	next	next	ADJ
ejpam-6011	146	3	proposition	proposition	NOUN
ejpam-6011	146	4	states	state	VERB
ejpam-6011	146	5	that	that	SCONJ
ejpam-6011	146	6	a	a	DET
ejpam-6011	146	7	ring	ring	NOUN
ejpam-6011	146	8	r	r	NOUN
ejpam-6011	146	9	is	be	AUX
ejpam-6011	146	10	left	leave	VERB
ejpam-6011	146	11	min	min	NOUN
ejpam-6011	146	12	-	-	NOUN
ejpam-6011	146	13	abel	abel	NOUN
ejpam-6011	146	14	whenever	whenever	SCONJ
ejpam-6011	146	15	it	it	PRON
ejpam-6011	146	16	is	be	AUX
ejpam-6011	146	17	nj	nj	NOUN
ejpam-6011	146	18	-	-	PUNCT
ejpam-6011	146	19	abelian	abelian	NOUN
ejpam-6011	146	20	.	.	PUNCT
ejpam-6011	147	1	proposition	proposition	NOUN
ejpam-6011	147	2	6	6	NUM
ejpam-6011	147	3	.	.	PUNCT
ejpam-6011	148	1	every	every	DET
ejpam-6011	148	2	nj	nj	PROPN
ejpam-6011	148	3	-	-	PUNCT
ejpam-6011	148	4	abelian	abelian	ADJ
ejpam-6011	148	5	ring	ring	NOUN
ejpam-6011	148	6	is	be	AUX
ejpam-6011	148	7	left	leave	VERB
ejpam-6011	148	8	-	-	PUNCT
ejpam-6011	148	9	min	min	NOUN
ejpam-6011	148	10	abel	abel	NOUN
ejpam-6011	148	11	.	.	PUNCT
ejpam-6011	149	1	proof	proof	NOUN
ejpam-6011	149	2	.	.	PUNCT
ejpam-6011	150	1	assume	assume	VERB
ejpam-6011	150	2	e	e	X
ejpam-6011	150	3	∈	∈	PROPN
ejpam-6011	150	4	mel(r	mel(r	PROPN
ejpam-6011	150	5	)	)	PUNCT
ejpam-6011	150	6	and	and	CCONJ
ejpam-6011	150	7	r	r	PROPN
ejpam-6011	150	8	∈	∈	PROPN
ejpam-6011	150	9	r.	r.	NOUN
ejpam-6011	150	10	define	define	VERB
ejpam-6011	150	11	the	the	DET
ejpam-6011	150	12	element	element	NOUN
ejpam-6011	150	13	x	x	PUNCT
ejpam-6011	151	1	=	=	SYM
ejpam-6011	151	2	re	re	X
ejpam-6011	151	3	−	−	PROPN
ejpam-6011	151	4	ere	ere	NOUN
ejpam-6011	151	5	;	;	PUNCT
ejpam-6011	151	6	hence	hence	ADV
ejpam-6011	151	7	,	,	PUNCT
ejpam-6011	151	8	rx	rx	VERB
ejpam-6011	151	9	⊆	⊆	NUM
ejpam-6011	151	10	r.	r.	NOUN
ejpam-6011	151	11	but	but	CCONJ
ejpam-6011	151	12	re	re	NOUN
ejpam-6011	151	13	is	be	AUX
ejpam-6011	151	14	a	a	DET
ejpam-6011	151	15	minimal	minimal	ADJ
ejpam-6011	151	16	left	leave	VERB
ejpam-6011	151	17	ideal	ideal	NOUN
ejpam-6011	151	18	of	of	ADP
ejpam-6011	151	19	r.	r.	PROPN
ejpam-6011	152	1	so	so	ADV
ejpam-6011	152	2	rx	rx	VERB
ejpam-6011	152	3	=	=	SYM
ejpam-6011	152	4	re	re	NOUN
ejpam-6011	152	5	or	or	CCONJ
ejpam-6011	152	6	x	x	SYM
ejpam-6011	152	7	=	=	SYM
ejpam-6011	152	8	0	0	X
ejpam-6011	152	9	.	.	PUNCT
ejpam-6011	153	1	indeed	indeed	ADV
ejpam-6011	153	2	,	,	PUNCT
ejpam-6011	153	3	xe	xe	PROPN
ejpam-6011	153	4	∈	∈	PROPN
ejpam-6011	153	5	n(r	n(r	NOUN
ejpam-6011	153	6	)	)	PUNCT
ejpam-6011	153	7	and	and	CCONJ
ejpam-6011	153	8	xre	xre	PROPN
ejpam-6011	153	9	⊆	⊆	NUM
ejpam-6011	153	10	j(r	j(r	NOUN
ejpam-6011	153	11	)	)	PUNCT
ejpam-6011	153	12	,	,	PUNCT
ejpam-6011	153	13	from	from	ADP
ejpam-6011	153	14	the	the	DET
ejpam-6011	153	15	nj	nj	PROPN
ejpam-6011	153	16	-	-	PUNCT
ejpam-6011	153	17	abelianity	abelianity	NOUN
ejpam-6011	153	18	of	of	ADP
ejpam-6011	153	19	r.	r.	PROPN
ejpam-6011	153	20	if	if	SCONJ
ejpam-6011	153	21	rx	rx	VERB
ejpam-6011	153	22	=	=	SYM
ejpam-6011	153	23	re	re	NOUN
ejpam-6011	153	24	,	,	PUNCT
ejpam-6011	153	25	then	then	ADV
ejpam-6011	153	26	e	e	PROPN
ejpam-6011	153	27	=	=	PROPN
ejpam-6011	153	28	e2	e2	PROPN
ejpam-6011	153	29	∈	∈	PROPN
ejpam-6011	153	30	rxre	rxre	NOUN
ejpam-6011	154	1	⊆	⊆	NUM
ejpam-6011	154	2	j(r	j(r	PROPN
ejpam-6011	154	3	)	)	PUNCT
ejpam-6011	154	4	and	and	CCONJ
ejpam-6011	154	5	e	e	X
ejpam-6011	154	6	=	=	SYM
ejpam-6011	154	7	0	0	NUM
ejpam-6011	155	1	;	;	PUNCT
ejpam-6011	155	2	it	it	PRON
ejpam-6011	155	3	is	be	AUX
ejpam-6011	155	4	a	a	DET
ejpam-6011	155	5	contradiction	contradiction	NOUN
ejpam-6011	155	6	.	.	PUNCT
ejpam-6011	156	1	so	so	ADV
ejpam-6011	156	2	,	,	PUNCT
ejpam-6011	156	3	x	x	PUNCT
ejpam-6011	156	4	=	=	SYM
ejpam-6011	156	5	0	0	NUM
ejpam-6011	156	6	and	and	CCONJ
ejpam-6011	156	7	(	(	PUNCT
ejpam-6011	156	8	1−	1−	NUM
ejpam-6011	156	9	e)re	e)re	PROPN
ejpam-6011	156	10	=	=	SYM
ejpam-6011	156	11	0	0	X
ejpam-6011	156	12	.	.	PUNCT
ejpam-6011	157	1	thus	thus	ADV
ejpam-6011	157	2	e	e	NOUN
ejpam-6011	157	3	is	be	AUX
ejpam-6011	157	4	a	a	DET
ejpam-6011	157	5	left	left	ADJ
ejpam-6011	157	6	semicentral	semicentral	ADJ
ejpam-6011	157	7	idempotent	idempotent	NOUN
ejpam-6011	157	8	of	of	ADP
ejpam-6011	157	9	r	r	NOUN
ejpam-6011	157	10	,	,	PUNCT
ejpam-6011	157	11	and	and	CCONJ
ejpam-6011	157	12	hence	hence	ADV
ejpam-6011	157	13	r	r	NOUN
ejpam-6011	157	14	left	leave	VERB
ejpam-6011	157	15	-	-	PUNCT
ejpam-6011	157	16	min	min	NOUN
ejpam-6011	157	17	abel	abel	PROPN
ejpam-6011	157	18	.	.	PUNCT
ejpam-6011	158	1	remind	remind	VERB
ejpam-6011	158	2	that	that	SCONJ
ejpam-6011	158	3	a	a	DET
ejpam-6011	158	4	ring	ring	NOUN
ejpam-6011	158	5	r	r	NOUN
ejpam-6011	158	6	is	be	AUX
ejpam-6011	158	7	called	call	VERB
ejpam-6011	158	8	j	j	NOUN
ejpam-6011	158	9	-	-	NOUN
ejpam-6011	158	10	clean	clean	ADJ
ejpam-6011	158	11	if	if	SCONJ
ejpam-6011	158	12	for	for	ADP
ejpam-6011	158	13	every	every	DET
ejpam-6011	158	14	a	a	DET
ejpam-6011	158	15	∈	∈	PROPN
ejpam-6011	158	16	r	r	NOUN
ejpam-6011	158	17	,	,	PUNCT
ejpam-6011	158	18	there	there	PRON
ejpam-6011	158	19	exists	exist	VERB
ejpam-6011	158	20	an	an	DET
ejpam-6011	158	21	idempotent	idempotent	ADJ
ejpam-6011	158	22	e	e	NOUN
ejpam-6011	158	23	∈	∈	NOUN
ejpam-6011	158	24	r	r	NOUN
ejpam-6011	158	25	and	and	CCONJ
ejpam-6011	158	26	b	b	PROPN
ejpam-6011	158	27	∈	∈	PROPN
ejpam-6011	158	28	j(r	j(r	PROPN
ejpam-6011	158	29	)	)	PUNCT
ejpam-6011	159	1	such	such	ADJ
ejpam-6011	159	2	that	that	SCONJ
ejpam-6011	159	3	a	a	DET
ejpam-6011	159	4	=	=	PUNCT
ejpam-6011	159	5	e+	e+	PUNCT
ejpam-6011	159	6	b	b	NOUN
ejpam-6011	159	7	;	;	PUNCT
ejpam-6011	159	8	that	that	PRON
ejpam-6011	159	9	is	is	ADV
ejpam-6011	159	10	,	,	PUNCT
ejpam-6011	159	11	r	r	NOUN
ejpam-6011	159	12	=	=	PUNCT
ejpam-6011	159	13	i(r	i(r	PROPN
ejpam-6011	159	14	)	)	PUNCT
ejpam-6011	159	15	+	+	NUM
ejpam-6011	159	16	j(r	j(r	NOUN
ejpam-6011	159	17	)	)	PUNCT
ejpam-6011	159	18	.	.	PUNCT
ejpam-6011	160	1	we	we	PRON
ejpam-6011	160	2	show	show	VERB
ejpam-6011	160	3	that	that	SCONJ
ejpam-6011	160	4	every	every	DET
ejpam-6011	160	5	j	j	PROPN
ejpam-6011	160	6	-	-	PUNCT
ejpam-6011	160	7	clean	clean	ADJ
ejpam-6011	160	8	ring	ring	NOUN
ejpam-6011	160	9	is	be	AUX
ejpam-6011	160	10	nj	nj	NOUN
ejpam-6011	160	11	-	-	PUNCT
ejpam-6011	160	12	abelian	abelian	ADJ
ejpam-6011	160	13	in	in	ADP
ejpam-6011	160	14	the	the	DET
ejpam-6011	160	15	following	follow	VERB
ejpam-6011	160	16	proposition	proposition	NOUN
ejpam-6011	160	17	.	.	PUNCT
ejpam-6011	161	1	proposition	proposition	NOUN
ejpam-6011	161	2	7	7	NUM
ejpam-6011	161	3	.	.	PUNCT
ejpam-6011	162	1	every	every	DET
ejpam-6011	162	2	j	j	PROPN
ejpam-6011	162	3	-	-	ADJ
ejpam-6011	162	4	clean	clean	ADJ
ejpam-6011	162	5	ring	ring	NOUN
ejpam-6011	162	6	r	r	NOUN
ejpam-6011	162	7	is	be	AUX
ejpam-6011	162	8	nj	nj	NOUN
ejpam-6011	162	9	-	-	PUNCT
ejpam-6011	162	10	abelian	abelian	ADJ
ejpam-6011	162	11	.	.	PUNCT
ejpam-6011	163	1	proof	proof	NOUN
ejpam-6011	163	2	.	.	PUNCT
ejpam-6011	164	1	let	let	VERB
ejpam-6011	164	2	r	r	PRON
ejpam-6011	164	3	be	be	AUX
ejpam-6011	164	4	a	a	DET
ejpam-6011	164	5	j	j	PROPN
ejpam-6011	164	6	-	-	ADJ
ejpam-6011	164	7	clean	clean	ADJ
ejpam-6011	164	8	ring	ring	NOUN
ejpam-6011	164	9	and	and	CCONJ
ejpam-6011	164	10	a	a	DET
ejpam-6011	164	11	be	be	AUX
ejpam-6011	164	12	a	a	DET
ejpam-6011	164	13	nilpotent	nilpotent	ADJ
ejpam-6011	164	14	element	element	NOUN
ejpam-6011	164	15	of	of	ADP
ejpam-6011	164	16	r	r	NOUN
ejpam-6011	164	17	with	with	ADP
ejpam-6011	164	18	index	index	NOUN
ejpam-6011	164	19	of	of	ADP
ejpam-6011	164	20	nilpotency	nilpotency	NOUN
ejpam-6011	164	21	n.	n.	PROPN
ejpam-6011	165	1	so	so	ADV
ejpam-6011	165	2	,	,	PUNCT
ejpam-6011	165	3	a	a	DET
ejpam-6011	165	4	=	=	PUNCT
ejpam-6011	165	5	e+	e+	PUNCT
ejpam-6011	165	6	b	b	NOUN
ejpam-6011	165	7	for	for	ADP
ejpam-6011	165	8	some	some	DET
ejpam-6011	165	9	e	e	PROPN
ejpam-6011	165	10	∈	∈	PROPN
ejpam-6011	165	11	i(r	i(r	PROPN
ejpam-6011	165	12	)	)	PUNCT
ejpam-6011	165	13	and	and	CCONJ
ejpam-6011	165	14	0	0	NUM
ejpam-6011	166	1	=	=	SYM
ejpam-6011	166	2	(	(	PUNCT
ejpam-6011	166	3	e+	e+	X
ejpam-6011	166	4	b)n	b)n	X
ejpam-6011	166	5	∈	∈	PROPN
ejpam-6011	166	6	e+	e+	ADJ
ejpam-6011	166	7	j(r	j(r	NOUN
ejpam-6011	166	8	)	)	PUNCT
ejpam-6011	166	9	and	and	CCONJ
ejpam-6011	166	10	e	e	PROPN
ejpam-6011	166	11	∈	∈	PROPN
ejpam-6011	166	12	j(r	j(r	PROPN
ejpam-6011	166	13	)	)	PUNCT
ejpam-6011	166	14	.	.	PUNCT
ejpam-6011	167	1	therefore	therefore	ADV
ejpam-6011	167	2	,	,	PUNCT
ejpam-6011	167	3	e	e	X
ejpam-6011	167	4	=	=	SYM
ejpam-6011	167	5	0	0	NUM
ejpam-6011	167	6	and	and	CCONJ
ejpam-6011	167	7	a	a	DET
ejpam-6011	167	8	∈	∈	PROPN
ejpam-6011	167	9	j(r	j(r	PROPN
ejpam-6011	167	10	)	)	PUNCT
ejpam-6011	167	11	;	;	PUNCT
ejpam-6011	167	12	that	that	SCONJ
ejpam-6011	167	13	r	r	NOUN
ejpam-6011	167	14	is	be	AUX
ejpam-6011	167	15	j	j	NOUN
ejpam-6011	167	16	-	-	PUNCT
ejpam-6011	167	17	reduced	reduce	VERB
ejpam-6011	167	18	.	.	PUNCT
ejpam-6011	168	1	hence	hence	ADV
ejpam-6011	168	2	,	,	PUNCT
ejpam-6011	168	3	r	r	NOUN
ejpam-6011	168	4	is	be	AUX
ejpam-6011	168	5	nj	nj	NOUN
ejpam-6011	168	6	-	-	PUNCT
ejpam-6011	168	7	abelian	abelian	NOUN
ejpam-6011	168	8	based	base	VERB
ejpam-6011	168	9	on	on	ADP
ejpam-6011	168	10	the	the	DET
ejpam-6011	168	11	results	result	NOUN
ejpam-6011	168	12	of	of	ADP
ejpam-6011	168	13	[	[	X
ejpam-6011	168	14	1	1	NUM
ejpam-6011	168	15	,	,	PUNCT
ejpam-6011	168	16	lemma	lemma	PROPN
ejpam-6011	168	17	2.4	2.4	NUM
ejpam-6011	168	18	.	.	PUNCT
ejpam-6011	168	19	]	]	PUNCT
ejpam-6011	168	20	and	and	CCONJ
ejpam-6011	168	21	proposition	proposition	NOUN
ejpam-6011	168	22	3	3	NUM
ejpam-6011	168	23	.	.	PUNCT
ejpam-6011	168	24	from	from	ADP
ejpam-6011	168	25	the	the	DET
ejpam-6011	168	26	proposition	proposition	NOUN
ejpam-6011	168	27	above	above	ADV
ejpam-6011	168	28	,	,	PUNCT
ejpam-6011	168	29	we	we	PRON
ejpam-6011	168	30	can	can	AUX
ejpam-6011	168	31	get	get	VERB
ejpam-6011	168	32	some	some	DET
ejpam-6011	168	33	corollaries	corollary	NOUN
ejpam-6011	168	34	that	that	PRON
ejpam-6011	168	35	have	have	AUX
ejpam-6011	168	36	been	be	AUX
ejpam-6011	168	37	proved	prove	VERB
ejpam-6011	168	38	before	before	ADV
ejpam-6011	168	39	in	in	ADP
ejpam-6011	168	40	another	another	DET
ejpam-6011	168	41	work	work	NOUN
ejpam-6011	168	42	.	.	PUNCT
ejpam-6011	169	1	corollary	corollary	ADJ
ejpam-6011	169	2	2	2	NUM
ejpam-6011	169	3	(	(	PUNCT
ejpam-6011	169	4	[	[	X
ejpam-6011	169	5	1	1	NUM
ejpam-6011	169	6	]	]	PUNCT
ejpam-6011	169	7	,	,	PUNCT
ejpam-6011	169	8	lemma	lemma	PROPN
ejpam-6011	169	9	2.4	2.4	NUM
ejpam-6011	169	10	)	)	PUNCT
ejpam-6011	169	11	.	.	PUNCT
ejpam-6011	170	1	every	every	DET
ejpam-6011	170	2	j	j	PROPN
ejpam-6011	170	3	-	-	ADJ
ejpam-6011	170	4	clean	clean	ADJ
ejpam-6011	170	5	ring	ring	NOUN
ejpam-6011	170	6	is	be	AUX
ejpam-6011	170	7	j	j	NOUN
ejpam-6011	170	8	-	-	PUNCT
ejpam-6011	170	9	abelian	abelian	PROPN
ejpam-6011	170	10	.	.	PUNCT
ejpam-6011	171	1	by	by	ADP
ejpam-6011	171	2	proposition	proposition	NOUN
ejpam-6011	171	3	5	5	NUM
ejpam-6011	171	4	,	,	PUNCT
ejpam-6011	171	5	we	we	PRON
ejpam-6011	171	6	also	also	ADV
ejpam-6011	171	7	have	have	VERB
ejpam-6011	171	8	the	the	DET
ejpam-6011	171	9	next	next	ADJ
ejpam-6011	171	10	corollary	corollary	NOUN
ejpam-6011	171	11	.	.	PUNCT
ejpam-6011	172	1	corollary	corollary	ADJ
ejpam-6011	172	2	3	3	NUM
ejpam-6011	172	3	(	(	PUNCT
ejpam-6011	172	4	[	[	X
ejpam-6011	172	5	1	1	NUM
ejpam-6011	172	6	]	]	PUNCT
ejpam-6011	172	7	,	,	PUNCT
ejpam-6011	172	8	theorem	theorem	VERB
ejpam-6011	172	9	2.10	2.10	NUM
ejpam-6011	172	10	)	)	PUNCT
ejpam-6011	172	11	.	.	PUNCT
ejpam-6011	173	1	every	every	DET
ejpam-6011	173	2	j	j	PROPN
ejpam-6011	173	3	-	-	ADJ
ejpam-6011	173	4	clean	clean	ADJ
ejpam-6011	173	5	ring	ring	NOUN
ejpam-6011	173	6	is	be	AUX
ejpam-6011	173	7	dedekind	dedekind	NOUN
ejpam-6011	173	8	-	-	PUNCT
ejpam-6011	173	9	finite	finite	VERB
ejpam-6011	173	10	.	.	PUNCT
ejpam-6011	174	1	recall	recall	NOUN
ejpam-6011	174	2	from	from	ADP
ejpam-6011	174	3	[	[	X
ejpam-6011	174	4	12	12	NUM
ejpam-6011	174	5	]	]	PUNCT
ejpam-6011	174	6	that	that	SCONJ
ejpam-6011	174	7	the	the	DET
ejpam-6011	174	8	set	set	NOUN
ejpam-6011	174	9	of	of	ADP
ejpam-6011	174	10	all	all	DET
ejpam-6011	174	11	elements	element	NOUN
ejpam-6011	174	12	of	of	ADP
ejpam-6011	174	13	r	r	NOUN
ejpam-6011	174	14	that	that	PRON
ejpam-6011	174	15	are	be	AUX
ejpam-6011	174	16	nilpotent	nilpotent	ADJ
ejpam-6011	174	17	in	in	ADP
ejpam-6011	174	18	r	r	PROPN
ejpam-6011	174	19	/	/	SYM
ejpam-6011	174	20	j(r	j(r	PROPN
ejpam-6011	174	21	)	)	PUNCT
ejpam-6011	174	22	is	be	AUX
ejpam-6011	174	23	denoted	denote	VERB
ejpam-6011	174	24	by	by	ADP
ejpam-6011	174	25	j#(r	j#(r	PROPN
ejpam-6011	174	26	)	)	PUNCT
ejpam-6011	174	27	;	;	PUNCT
ejpam-6011	174	28	that	that	PRON
ejpam-6011	174	29	is	be	AUX
ejpam-6011	174	30	,	,	PUNCT
ejpam-6011	174	31	j#(r	j#(r	PROPN
ejpam-6011	174	32	)	)	PUNCT
ejpam-6011	174	33	=	=	PUNCT
ejpam-6011	175	1	{	{	PUNCT
ejpam-6011	175	2	a	a	DET
ejpam-6011	175	3	∈	∈	NOUN
ejpam-6011	175	4	r	r	NOUN
ejpam-6011	175	5	|	|	ADV
ejpam-6011	175	6	an	an	DET
ejpam-6011	175	7	∈	∈	PROPN
ejpam-6011	175	8	j(r	j(r	PROPN
ejpam-6011	175	9	)	)	PUNCT
ejpam-6011	175	10	}	}	PUNCT
ejpam-6011	175	11	.	.	PUNCT
ejpam-6011	176	1	it	it	PRON
ejpam-6011	176	2	is	be	AUX
ejpam-6011	176	3	obvious	obvious	ADJ
ejpam-6011	176	4	that	that	SCONJ
ejpam-6011	176	5	both	both	DET
ejpam-6011	176	6	j(r	j(r	NOUN
ejpam-6011	176	7	)	)	PUNCT
ejpam-6011	176	8	and	and	CCONJ
ejpam-6011	176	9	n(r	n(r	NOUN
ejpam-6011	176	10	)	)	PUNCT
ejpam-6011	176	11	are	be	AUX
ejpam-6011	176	12	contained	contain	VERB
ejpam-6011	176	13	in	in	ADP
ejpam-6011	176	14	j#(r	j#(r	NOUN
ejpam-6011	176	15	)	)	PUNCT
ejpam-6011	176	16	.	.	PUNCT
ejpam-6011	177	1	in	in	ADP
ejpam-6011	177	2	[	[	X
ejpam-6011	177	3	13	13	NUM
ejpam-6011	177	4	]	]	PUNCT
ejpam-6011	177	5	,	,	PUNCT
ejpam-6011	177	6	a	a	DET
ejpam-6011	177	7	ring	ring	NOUN
ejpam-6011	177	8	r	r	NOUN
ejpam-6011	177	9	is	be	AUX
ejpam-6011	177	10	called	call	VERB
ejpam-6011	177	11	feckly	feckly	ADV
ejpam-6011	177	12	reduced	reduce	VERB
ejpam-6011	177	13	if	if	SCONJ
ejpam-6011	177	14	r	r	NOUN
ejpam-6011	177	15	/	/	SYM
ejpam-6011	177	16	j(r	j(r	PROPN
ejpam-6011	177	17	)	)	PUNCT
ejpam-6011	177	18	is	be	AUX
ejpam-6011	177	19	a	a	DET
ejpam-6011	177	20	reduced	reduce	VERB
ejpam-6011	177	21	ring	ring	NOUN
ejpam-6011	177	22	.	.	PUNCT
ejpam-6011	178	1	the	the	DET
ejpam-6011	178	2	following	follow	VERB
ejpam-6011	178	3	proposition	proposition	NOUN
ejpam-6011	178	4	provides	provide	VERB
ejpam-6011	178	5	a	a	DET
ejpam-6011	178	6	more	more	ADV
ejpam-6011	178	7	general	general	ADJ
ejpam-6011	178	8	result	result	NOUN
ejpam-6011	178	9	of	of	ADP
ejpam-6011	178	10	[	[	X
ejpam-6011	178	11	1	1	NUM
ejpam-6011	178	12	,	,	PUNCT
ejpam-6011	178	13	proposition	proposition	NOUN
ejpam-6011	178	14	2.6	2.6	NUM
ejpam-6011	178	15	]	]	PUNCT
ejpam-6011	178	16	under	under	ADP
ejpam-6011	178	17	the	the	DET
ejpam-6011	178	18	same	same	ADJ
ejpam-6011	178	19	assumptions	assumption	NOUN
ejpam-6011	178	20	and	and	CCONJ
ejpam-6011	178	21	shows	show	VERB
ejpam-6011	178	22	that	that	SCONJ
ejpam-6011	178	23	every	every	DET
ejpam-6011	178	24	feckly	feckly	ADV
ejpam-6011	178	25	reduced	reduced	ADJ
ejpam-6011	178	26	ring	ring	NOUN
ejpam-6011	178	27	is	be	AUX
ejpam-6011	178	28	nj	nj	NOUN
ejpam-6011	178	29	-	-	PUNCT
ejpam-6011	178	30	abelian	abelian	NOUN
ejpam-6011	178	31	.	.	PUNCT
ejpam-6011	179	1	m.	m.	PROPN
ejpam-6011	179	2	saad	saad	PROPN
ejpam-6011	179	3	,	,	PUNCT
ejpam-6011	179	4	s.	s.	PROPN
ejpam-6011	179	5	m.	m.	PROPN
ejpam-6011	179	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	179	7	/	/	SYM
ejpam-6011	179	8	eur	eur	PROPN
ejpam-6011	179	9	.	.	PUNCT
ejpam-6011	180	1	j.	j.	PROPN
ejpam-6011	180	2	pure	pure	PROPN
ejpam-6011	180	3	appl	appl	PROPN
ejpam-6011	180	4	.	.	PROPN
ejpam-6011	180	5	math	math	PROPN
ejpam-6011	180	6	,	,	PUNCT
ejpam-6011	180	7	18	18	NUM
ejpam-6011	180	8	(	(	PUNCT
ejpam-6011	180	9	2	2	NUM
ejpam-6011	180	10	)	)	PUNCT
ejpam-6011	180	11	(	(	PUNCT
ejpam-6011	180	12	2025	2025	NUM
ejpam-6011	180	13	)	)	PUNCT
ejpam-6011	180	14	,	,	PUNCT
ejpam-6011	180	15	6011	6011	NUM
ejpam-6011	180	16	6	6	NUM
ejpam-6011	180	17	of	of	ADP
ejpam-6011	180	18	13	13	NUM
ejpam-6011	180	19	proposition	proposition	NOUN
ejpam-6011	180	20	8	8	NUM
ejpam-6011	180	21	.	.	PUNCT
ejpam-6011	181	1	if	if	SCONJ
ejpam-6011	181	2	r	r	NOUN
ejpam-6011	181	3	is	be	AUX
ejpam-6011	181	4	a	a	DET
ejpam-6011	181	5	feckly	feckly	ADV
ejpam-6011	181	6	reduced	reduce	VERB
ejpam-6011	181	7	ring	ring	NOUN
ejpam-6011	181	8	,	,	PUNCT
ejpam-6011	181	9	then	then	ADV
ejpam-6011	181	10	r	r	NOUN
ejpam-6011	181	11	is	be	AUX
ejpam-6011	181	12	nj	nj	NOUN
ejpam-6011	181	13	-	-	PUNCT
ejpam-6011	181	14	abelian	abelian	ADJ
ejpam-6011	181	15	.	.	PUNCT
ejpam-6011	182	1	proof	proof	NOUN
ejpam-6011	182	2	.	.	PUNCT
ejpam-6011	183	1	from	from	ADP
ejpam-6011	183	2	[	[	X
ejpam-6011	183	3	13	13	NUM
ejpam-6011	183	4	,	,	PUNCT
ejpam-6011	183	5	proposition	proposition	NOUN
ejpam-6011	183	6	2.6	2.6	NUM
ejpam-6011	183	7	]	]	PUNCT
ejpam-6011	183	8	,	,	PUNCT
ejpam-6011	183	9	r	r	NOUN
ejpam-6011	183	10	satisfies	satisfie	NOUN
ejpam-6011	183	11	j#(r	j#(r	PROPN
ejpam-6011	183	12	)	)	PUNCT
ejpam-6011	183	13	.	.	PUNCT
ejpam-6011	184	1	now	now	ADV
ejpam-6011	184	2	,	,	PUNCT
ejpam-6011	184	3	let	let	VERB
ejpam-6011	184	4	ea	ea	PRON
ejpam-6011	184	5	∈	∈	VERB
ejpam-6011	184	6	n(r	n(r	NOUN
ejpam-6011	184	7	)	)	PUNCT
ejpam-6011	184	8	where	where	SCONJ
ejpam-6011	184	9	e2	e2	PROPN
ejpam-6011	184	10	=	=	SYM
ejpam-6011	184	11	e	e	PROPN
ejpam-6011	184	12	,	,	PUNCT
ejpam-6011	184	13	a	a	DET
ejpam-6011	184	14	∈	∈	PROPN
ejpam-6011	184	15	r.	r.	NOUN
ejpam-6011	184	16	so	so	ADV
ejpam-6011	184	17	,	,	PUNCT
ejpam-6011	184	18	for	for	ADP
ejpam-6011	184	19	every	every	DET
ejpam-6011	184	20	r	r	NOUN
ejpam-6011	184	21	∈	∈	NOUN
ejpam-6011	184	22	r	r	NOUN
ejpam-6011	184	23	,	,	PUNCT
ejpam-6011	184	24	we	we	PRON
ejpam-6011	184	25	have	have	VERB
ejpam-6011	184	26	(	(	PUNCT
ejpam-6011	184	27	e	e	X
ejpam-6011	184	28	−	−	PROPN
ejpam-6011	184	29	1)are	1)are	NUM
ejpam-6011	184	30	is	be	AUX
ejpam-6011	184	31	nilpotent	nilpotent	ADJ
ejpam-6011	184	32	,	,	PUNCT
ejpam-6011	184	33	and	and	CCONJ
ejpam-6011	184	34	consequently	consequently	ADV
ejpam-6011	184	35	ea	ea	NUM
ejpam-6011	184	36	,	,	PUNCT
ejpam-6011	184	37	(	(	PUNCT
ejpam-6011	184	38	e	e	X
ejpam-6011	184	39	−	−	PROPN
ejpam-6011	184	40	1)are	1)are	NUM
ejpam-6011	184	41	∈	∈	PROPN
ejpam-6011	184	42	j#(r	j#(r	PROPN
ejpam-6011	184	43	)	)	PUNCT
ejpam-6011	184	44	=	=	SYM
ejpam-6011	185	1	j(r	j(r	PROPN
ejpam-6011	185	2	)	)	PUNCT
ejpam-6011	185	3	.	.	PUNCT
ejpam-6011	186	1	so	so	ADV
ejpam-6011	186	2	,	,	PUNCT
ejpam-6011	186	3	are	be	AUX
ejpam-6011	186	4	=	=	PUNCT
ejpam-6011	186	5	(	(	PUNCT
ejpam-6011	186	6	e	e	X
ejpam-6011	186	7	−	−	PROPN
ejpam-6011	186	8	1)are	1)are	NUM
ejpam-6011	186	9	+	+	CCONJ
ejpam-6011	186	10	eare	eare	NOUN
ejpam-6011	186	11	∈	∈	PROPN
ejpam-6011	186	12	j(r	j(r	PROPN
ejpam-6011	186	13	)	)	PUNCT
ejpam-6011	186	14	,	,	PUNCT
ejpam-6011	186	15	for	for	ADP
ejpam-6011	186	16	every	every	DET
ejpam-6011	186	17	r	r	NOUN
ejpam-6011	186	18	∈	∈	NOUN
ejpam-6011	186	19	r	r	NOUN
ejpam-6011	186	20	,	,	PUNCT
ejpam-6011	186	21	thus	thus	ADV
ejpam-6011	186	22	,	,	PUNCT
ejpam-6011	186	23	are	be	AUX
ejpam-6011	186	24	⊆	⊆	NUM
ejpam-6011	186	25	j(r	j(r	NOUN
ejpam-6011	186	26	)	)	PUNCT
ejpam-6011	186	27	,	,	PUNCT
ejpam-6011	186	28	and	and	CCONJ
ejpam-6011	186	29	hence	hence	ADV
ejpam-6011	186	30	r	r	NOUN
ejpam-6011	186	31	is	be	AUX
ejpam-6011	186	32	nj	nj	NOUN
ejpam-6011	186	33	-	-	PUNCT
ejpam-6011	186	34	abelian	abelian	ADJ
ejpam-6011	186	35	.	.	PUNCT
ejpam-6011	187	1	remind	remind	VERB
ejpam-6011	187	2	that	that	SCONJ
ejpam-6011	187	3	a	a	DET
ejpam-6011	187	4	ring	ring	NOUN
ejpam-6011	187	5	r	r	NOUN
ejpam-6011	187	6	is	be	AUX
ejpam-6011	187	7	said	say	VERB
ejpam-6011	187	8	to	to	PART
ejpam-6011	187	9	be	be	AUX
ejpam-6011	187	10	local	local	ADJ
ejpam-6011	187	11	if	if	SCONJ
ejpam-6011	187	12	it	it	PRON
ejpam-6011	187	13	has	have	VERB
ejpam-6011	187	14	only	only	ADV
ejpam-6011	187	15	one	one	NUM
ejpam-6011	187	16	maximal	maximal	ADJ
ejpam-6011	187	17	left	left	NOUN
ejpam-6011	187	18	(	(	PUNCT
ejpam-6011	187	19	or	or	CCONJ
ejpam-6011	187	20	right	right	ADJ
ejpam-6011	187	21	)	)	PUNCT
ejpam-6011	187	22	ideal	ideal	NOUN
ejpam-6011	187	23	;	;	PUNCT
ejpam-6011	187	24	equivalently	equivalently	ADV
ejpam-6011	187	25	,	,	PUNCT
ejpam-6011	187	26	r	r	NOUN
ejpam-6011	187	27	/	/	SYM
ejpam-6011	187	28	j(r	j(r	PROPN
ejpam-6011	187	29	)	)	PUNCT
ejpam-6011	187	30	is	be	AUX
ejpam-6011	187	31	a	a	DET
ejpam-6011	187	32	division	division	NOUN
ejpam-6011	187	33	ring	ring	NOUN
ejpam-6011	187	34	.	.	PUNCT
ejpam-6011	188	1	from	from	ADP
ejpam-6011	188	2	proposition	proposition	NOUN
ejpam-6011	188	3	8	8	NUM
ejpam-6011	188	4	and	and	CCONJ
ejpam-6011	188	5	[	[	X
ejpam-6011	188	6	12	12	NUM
ejpam-6011	188	7	,	,	PUNCT
ejpam-6011	188	8	lemma	lemma	PROPN
ejpam-6011	188	9	1	1	NUM
ejpam-6011	188	10	]	]	PUNCT
ejpam-6011	188	11	,	,	PUNCT
ejpam-6011	188	12	we	we	PRON
ejpam-6011	188	13	get	get	VERB
ejpam-6011	188	14	directly	directly	ADV
ejpam-6011	188	15	the	the	DET
ejpam-6011	188	16	following	follow	VERB
ejpam-6011	188	17	result	result	NOUN
ejpam-6011	188	18	.	.	PUNCT
ejpam-6011	189	1	corollary	corollary	ADJ
ejpam-6011	189	2	4	4	NUM
ejpam-6011	189	3	.	.	PUNCT
ejpam-6011	190	1	every	every	DET
ejpam-6011	190	2	local	local	ADJ
ejpam-6011	190	3	ring	ring	NOUN
ejpam-6011	190	4	is	be	AUX
ejpam-6011	190	5	nj	nj	NOUN
ejpam-6011	190	6	-	-	PUNCT
ejpam-6011	190	7	abelian	abelian	ADJ
ejpam-6011	190	8	.	.	PUNCT
ejpam-6011	191	1	here	here	ADV
ejpam-6011	191	2	are	be	AUX
ejpam-6011	191	3	interesting	interesting	ADJ
ejpam-6011	191	4	nontrivial	nontrivial	ADJ
ejpam-6011	191	5	implications	implication	NOUN
ejpam-6011	191	6	in	in	ADP
ejpam-6011	191	7	the	the	DET
ejpam-6011	191	8	class	class	NOUN
ejpam-6011	191	9	of	of	ADP
ejpam-6011	191	10	rings	ring	NOUN
ejpam-6011	191	11	with	with	ADP
ejpam-6011	191	12	respect	respect	NOUN
ejpam-6011	191	13	to	to	ADP
ejpam-6011	191	14	the	the	DET
ejpam-6011	191	15	class	class	NOUN
ejpam-6011	191	16	of	of	ADP
ejpam-6011	191	17	nj	nj	PROPN
ejpam-6011	191	18	-	-	PUNCT
ejpam-6011	191	19	abelian	abelian	ADJ
ejpam-6011	191	20	rings	ring	NOUN
ejpam-6011	191	21	.	.	PUNCT
ejpam-6011	192	1	semicommutative	semicommutative	VERB
ejpam-6011	192	2	reduced	reduce	VERB
ejpam-6011	192	3	abelian	abelian	PROPN
ejpam-6011	192	4	j	j	PROPN
ejpam-6011	192	5	-	-	PUNCT
ejpam-6011	192	6	clean	clean	ADJ
ejpam-6011	192	7	nj	nj	PROPN
ejpam-6011	192	8	-	-	PUNCT
ejpam-6011	192	9	semicommutative	semicommutative	NOUN
ejpam-6011	192	10	nj	nj	PROPN
ejpam-6011	192	11	-	-	PUNCT
ejpam-6011	192	12	abelian	abelian	ADJ
ejpam-6011	192	13	j	j	PROPN
ejpam-6011	192	14	-	-	PUNCT
ejpam-6011	192	15	abelian	abelian	ADJ
ejpam-6011	192	16	feckly	feckly	ADV
ejpam-6011	192	17	reduced	reduce	VERB
ejpam-6011	192	18	j	j	PROPN
ejpam-6011	192	19	-	-	PUNCT
ejpam-6011	192	20	reduced	reduce	VERB
ejpam-6011	192	21	dedekind	dedekind	NOUN
ejpam-6011	192	22	-	-	PUNCT
ejpam-6011	192	23	finite	finite	ADJ
ejpam-6011	192	24	3	3	NUM
ejpam-6011	192	25	.	.	X
ejpam-6011	193	1	extending	extend	VERB
ejpam-6011	193	2	of	of	ADP
ejpam-6011	193	3	nj	nj	PROPN
ejpam-6011	193	4	-	-	PUNCT
ejpam-6011	193	5	abelianity	abelianity	NOUN
ejpam-6011	193	6	note	note	NOUN
ejpam-6011	193	7	that	that	SCONJ
ejpam-6011	193	8	the	the	DET
ejpam-6011	193	9	class	class	NOUN
ejpam-6011	193	10	of	of	ADP
ejpam-6011	193	11	nj	nj	PROPN
ejpam-6011	193	12	-	-	PUNCT
ejpam-6011	193	13	abelian	abelian	ADJ
ejpam-6011	193	14	rings	ring	NOUN
ejpam-6011	193	15	is	be	AUX
ejpam-6011	193	16	closed	close	VERB
ejpam-6011	193	17	under	under	ADP
ejpam-6011	193	18	direct	direct	ADJ
ejpam-6011	193	19	products	product	NOUN
ejpam-6011	193	20	but	but	CCONJ
ejpam-6011	193	21	not	not	PART
ejpam-6011	193	22	under	under	ADP
ejpam-6011	193	23	closed	closed	ADJ
ejpam-6011	193	24	subrings	subring	NOUN
ejpam-6011	193	25	.	.	PUNCT
ejpam-6011	194	1	remember	remember	VERB
ejpam-6011	194	2	that	that	SCONJ
ejpam-6011	194	3	the	the	DET
ejpam-6011	194	4	nagata	nagata	PROPN
ejpam-6011	194	5	extension	extension	NOUN
ejpam-6011	194	6	of	of	ADP
ejpam-6011	194	7	a	a	DET
ejpam-6011	194	8	commutative	commutative	ADJ
ejpam-6011	194	9	ring	ring	NOUN
ejpam-6011	194	10	r	r	NOUN
ejpam-6011	194	11	by	by	ADP
ejpam-6011	194	12	an	an	DET
ejpam-6011	194	13	r	r	NOUN
ejpam-6011	194	14	-	-	PUNCT
ejpam-6011	194	15	module	module	NOUN
ejpam-6011	194	16	m	m	NOUN
ejpam-6011	194	17	and	and	CCONJ
ejpam-6011	194	18	an	an	DET
ejpam-6011	194	19	endomorphism	endomorphism	PROPN
ejpam-6011	194	20	σ	σ	NOUN
ejpam-6011	194	21	of	of	ADP
ejpam-6011	194	22	r	r	NOUN
ejpam-6011	194	23	is	be	AUX
ejpam-6011	194	24	the	the	DET
ejpam-6011	194	25	ring	ring	NOUN
ejpam-6011	194	26	of	of	ADP
ejpam-6011	194	27	direct	direct	ADJ
ejpam-6011	194	28	sum	sum	NOUN
ejpam-6011	194	29	of	of	ADP
ejpam-6011	194	30	the	the	DET
ejpam-6011	194	31	abelian	abelian	ADJ
ejpam-6011	194	32	groups	group	NOUN
ejpam-6011	194	33	r	r	NOUN
ejpam-6011	194	34	and	and	CCONJ
ejpam-6011	194	35	m	m	VERB
ejpam-6011	194	36	,	,	PUNCT
ejpam-6011	194	37	with	with	ADP
ejpam-6011	194	38	componentwise	componentwise	NOUN
ejpam-6011	194	39	addition	addition	NOUN
ejpam-6011	194	40	and	and	CCONJ
ejpam-6011	194	41	multiplication	multiplication	NOUN
ejpam-6011	194	42	defined	define	VERB
ejpam-6011	194	43	as	as	ADP
ejpam-6011	194	44	(	(	PUNCT
ejpam-6011	194	45	r1,m1)(r2,m2	r1,m1)(r2,m2	NOUN
ejpam-6011	194	46	)	)	PUNCT
ejpam-6011	194	47	=	=	PRON
ejpam-6011	195	1	(	(	PUNCT
ejpam-6011	195	2	r1r2	r1r2	ADJ
ejpam-6011	195	3	,	,	PUNCT
ejpam-6011	195	4	σ(r1)m2	σ(r1)m2	NOUN
ejpam-6011	195	5	+	+	CCONJ
ejpam-6011	195	6	m1r2	m1r2	VERB
ejpam-6011	195	7	)	)	PUNCT
ejpam-6011	195	8	for	for	ADP
ejpam-6011	195	9	all	all	DET
ejpam-6011	195	10	r1	r1	NOUN
ejpam-6011	195	11	,	,	PUNCT
ejpam-6011	195	12	r2	r2	PROPN
ejpam-6011	195	13	∈	∈	PROPN
ejpam-6011	195	14	r	r	NOUN
ejpam-6011	195	15	and	and	CCONJ
ejpam-6011	195	16	m1,m2	m1,m2	PROPN
ejpam-6011	195	17	∈	∈	PROPN
ejpam-6011	195	18	m	m	VERB
ejpam-6011	195	19	.	.	PUNCT
ejpam-6011	196	1	the	the	DET
ejpam-6011	196	2	next	next	ADJ
ejpam-6011	196	3	examples	example	NOUN
ejpam-6011	196	4	give	give	VERB
ejpam-6011	196	5	an	an	DET
ejpam-6011	196	6	nj	nj	ADJ
ejpam-6011	196	7	-	-	PUNCT
ejpam-6011	196	8	abelian	abelian	ADJ
ejpam-6011	196	9	subring	subring	NOUN
ejpam-6011	196	10	of	of	ADP
ejpam-6011	196	11	a	a	DET
ejpam-6011	196	12	ring	ring	NOUN
ejpam-6011	196	13	that	that	PRON
ejpam-6011	196	14	is	be	AUX
ejpam-6011	196	15	not	not	PART
ejpam-6011	196	16	nj	nj	PROPN
ejpam-6011	196	17	.	.	PUNCT
ejpam-6011	197	1	example	example	NOUN
ejpam-6011	198	1	6	6	NUM
ejpam-6011	198	2	.	.	PUNCT
ejpam-6011	199	1	let	let	VERB
ejpam-6011	199	2	r	r	NOUN
ejpam-6011	199	3	=	=	PUNCT
ejpam-6011	199	4	z⊕z	z⊕z	PROPN
ejpam-6011	199	5	and	and	CCONJ
ejpam-6011	199	6	σ	σ	PROPN
ejpam-6011	199	7	be	be	AUX
ejpam-6011	199	8	an	an	DET
ejpam-6011	199	9	automorphism	automorphism	NOUN
ejpam-6011	199	10	on	on	ADP
ejpam-6011	199	11	r	r	NOUN
ejpam-6011	199	12	defined	define	VERB
ejpam-6011	199	13	as	as	ADP
ejpam-6011	199	14	σ(a	σ(a	PROPN
ejpam-6011	199	15	,	,	PUNCT
ejpam-6011	199	16	b	b	NOUN
ejpam-6011	199	17	)	)	PUNCT
ejpam-6011	200	1	=	=	SYM
ejpam-6011	200	2	(	(	PUNCT
ejpam-6011	200	3	b	b	NOUN
ejpam-6011	200	4	,	,	PUNCT
ejpam-6011	200	5	a	a	PRON
ejpam-6011	200	6	)	)	PUNCT
ejpam-6011	200	7	for	for	ADP
ejpam-6011	200	8	every	every	DET
ejpam-6011	200	9	(	(	PUNCT
ejpam-6011	200	10	a	a	PRON
ejpam-6011	200	11	,	,	PUNCT
ejpam-6011	200	12	b	b	NOUN
ejpam-6011	200	13	)	)	PUNCT
ejpam-6011	200	14	∈	∈	PROPN
ejpam-6011	200	15	r.	r.	NOUN
ejpam-6011	200	16	the	the	DET
ejpam-6011	200	17	nagata	nagata	PROPN
ejpam-6011	200	18	extension	extension	NOUN
ejpam-6011	200	19	of	of	ADP
ejpam-6011	200	20	r	r	NOUN
ejpam-6011	200	21	by	by	ADP
ejpam-6011	200	22	s	s	NOUN
ejpam-6011	200	23	and	and	CCONJ
ejpam-6011	200	24	σ	σ	PROPN
ejpam-6011	200	25	,	,	PUNCT
ejpam-6011	200	26	denoted	denote	VERB
ejpam-6011	200	27	as	as	ADP
ejpam-6011	200	28	s	s	PROPN
ejpam-6011	200	29	,	,	PUNCT
ejpam-6011	200	30	is	be	AUX
ejpam-6011	200	31	semiprimitive	semiprimitive	ADJ
ejpam-6011	200	32	since	since	SCONJ
ejpam-6011	200	33	r	r	NOUN
ejpam-6011	200	34	is	be	AUX
ejpam-6011	200	35	reduced	reduce	VERB
ejpam-6011	200	36	.	.	PUNCT
ejpam-6011	201	1	moreover	moreover	ADV
ejpam-6011	201	2	,	,	PUNCT
ejpam-6011	201	3	the	the	DET
ejpam-6011	201	4	idempotent	idempotent	NOUN
ejpam-6011	201	5	(	(	PUNCT
ejpam-6011	201	6	(	(	PUNCT
ejpam-6011	201	7	1	1	NUM
ejpam-6011	201	8	,	,	PUNCT
ejpam-6011	201	9	0	0	NUM
ejpam-6011	201	10	)	)	PUNCT
ejpam-6011	201	11	,	,	PUNCT
ejpam-6011	201	12	(	(	PUNCT
ejpam-6011	201	13	0	0	NUM
ejpam-6011	201	14	,	,	PUNCT
ejpam-6011	201	15	1	1	NUM
ejpam-6011	201	16	)	)	PUNCT
ejpam-6011	201	17	)	)	PUNCT
ejpam-6011	201	18	and	and	CCONJ
ejpam-6011	201	19	element	element	NOUN
ejpam-6011	201	20	(	(	PUNCT
ejpam-6011	201	21	(	(	PUNCT
ejpam-6011	201	22	0	0	NUM
ejpam-6011	201	23	,	,	PUNCT
ejpam-6011	201	24	1	1	NUM
ejpam-6011	201	25	)	)	PUNCT
ejpam-6011	201	26	,	,	PUNCT
ejpam-6011	201	27	(	(	PUNCT
ejpam-6011	201	28	0	0	NUM
ejpam-6011	201	29	,	,	PUNCT
ejpam-6011	201	30	1	1	NUM
ejpam-6011	201	31	)	)	PUNCT
ejpam-6011	201	32	)	)	PUNCT
ejpam-6011	201	33	of	of	ADP
ejpam-6011	201	34	s	s	PRON
ejpam-6011	201	35	satisfy	satisfy	NOUN
ejpam-6011	201	36	ea	ea	ADP
ejpam-6011	201	37	∈	∈	PROPN
ejpam-6011	201	38	n(s	n(s	PROPN
ejpam-6011	201	39	)	)	PUNCT
ejpam-6011	201	40	while	while	SCONJ
ejpam-6011	201	41	esa	esa	PROPN
ejpam-6011	201	42	=	=	SYM
ejpam-6011	201	43	(	(	PUNCT
ejpam-6011	201	44	(	(	PUNCT
ejpam-6011	201	45	0	0	NUM
ejpam-6011	201	46	,	,	PUNCT
ejpam-6011	201	47	0	0	NUM
ejpam-6011	201	48	)	)	PUNCT
ejpam-6011	201	49	,	,	PUNCT
ejpam-6011	201	50	(	(	PUNCT
ejpam-6011	201	51	0,z	0,z	NOUN
ejpam-6011	201	52	)	)	PUNCT
ejpam-6011	201	53	)	)	PUNCT
ejpam-6011	201	54	⊈	⊈	PROPN
ejpam-6011	201	55	j(s	j(s	NOUN
ejpam-6011	201	56	)	)	PUNCT
ejpam-6011	202	1	=	=	PUNCT
ejpam-6011	202	2	0	0	X
ejpam-6011	202	3	.	.	PUNCT
ejpam-6011	203	1	therefore	therefore	ADV
ejpam-6011	203	2	,	,	PUNCT
ejpam-6011	203	3	s	s	VERB
ejpam-6011	203	4	is	be	AUX
ejpam-6011	203	5	not	not	PART
ejpam-6011	203	6	nj	nj	NOUN
ejpam-6011	203	7	-	-	PUNCT
ejpam-6011	203	8	abelian	abelian	ADJ
ejpam-6011	203	9	,	,	PUNCT
ejpam-6011	203	10	while	while	SCONJ
ejpam-6011	203	11	r	r	NOUN
ejpam-6011	203	12	is	be	AUX
ejpam-6011	203	13	an	an	DET
ejpam-6011	203	14	nj	nj	ADJ
ejpam-6011	203	15	-	-	PUNCT
ejpam-6011	203	16	abelian	abelian	ADJ
ejpam-6011	203	17	subring	subring	NOUN
ejpam-6011	203	18	of	of	ADP
ejpam-6011	203	19	s.	s.	PROPN
ejpam-6011	203	20	here	here	ADV
ejpam-6011	203	21	is	be	AUX
ejpam-6011	203	22	an	an	DET
ejpam-6011	203	23	nj	nj	ADJ
ejpam-6011	203	24	-	-	PUNCT
ejpam-6011	203	25	abelian	abelian	ADJ
ejpam-6011	203	26	ring	ring	NOUN
ejpam-6011	203	27	that	that	PRON
ejpam-6011	203	28	has	have	VERB
ejpam-6011	203	29	a	a	DET
ejpam-6011	203	30	non	non	ADJ
ejpam-6011	203	31	-	-	ADJ
ejpam-6011	203	32	nj	nj	ADJ
ejpam-6011	203	33	subring	subring	NOUN
ejpam-6011	203	34	.	.	PUNCT
ejpam-6011	203	35	example	example	NOUN
ejpam-6011	204	1	7	7	NUM
ejpam-6011	204	2	.	.	PUNCT
ejpam-6011	204	3	from	from	ADP
ejpam-6011	204	4	[	[	X
ejpam-6011	204	5	14	14	NUM
ejpam-6011	204	6	,	,	PUNCT
ejpam-6011	204	7	example	example	NOUN
ejpam-6011	204	8	4.8	4.8	NUM
ejpam-6011	204	9	]	]	PUNCT
ejpam-6011	204	10	,	,	PUNCT
ejpam-6011	204	11	let	let	VERB
ejpam-6011	204	12	r	r	NOUN
ejpam-6011	204	13	=	=	SYM
ejpam-6011	204	14	f	f	PROPN
ejpam-6011	204	15	⟨x	⟨x	NUM
ejpam-6011	204	16	,	,	PUNCT
ejpam-6011	204	17	y⟩	y⟩	NOUN
ejpam-6011	204	18	be	be	AUX
ejpam-6011	204	19	a	a	DET
ejpam-6011	204	20	free	free	ADJ
ejpam-6011	204	21	algebra	algebra	NOUN
ejpam-6011	204	22	over	over	ADP
ejpam-6011	204	23	a	a	DET
ejpam-6011	204	24	field	field	NOUN
ejpam-6011	204	25	f	f	NOUN
ejpam-6011	204	26	generated	generate	VERB
ejpam-6011	204	27	by	by	ADP
ejpam-6011	204	28	the	the	DET
ejpam-6011	204	29	noncommuting	noncommuting	NOUN
ejpam-6011	204	30	indeterminates	indeterminate	VERB
ejpam-6011	204	31	x	x	X
ejpam-6011	204	32	and	and	CCONJ
ejpam-6011	204	33	y.	y.	PROPN
ejpam-6011	204	34	consider	consider	VERB
ejpam-6011	204	35	the	the	DET
ejpam-6011	204	36	subring	subring	NOUN
ejpam-6011	204	37	s	s	PART
ejpam-6011	204	38	=	=	ADJ
ejpam-6011	204	39	r/⟨y2⟩	r/⟨y2⟩	NOUN
ejpam-6011	204	40	of	of	ADP
ejpam-6011	204	41	r.	r.	PROPN
ejpam-6011	204	42	according	accord	VERB
ejpam-6011	204	43	to	to	ADP
ejpam-6011	204	44	[	[	X
ejpam-6011	204	45	9	9	NUM
ejpam-6011	204	46	]	]	PUNCT
ejpam-6011	204	47	,	,	PUNCT
ejpam-6011	204	48	s	s	VERB
ejpam-6011	204	49	is	be	AUX
ejpam-6011	204	50	not	not	PART
ejpam-6011	204	51	j	j	NOUN
ejpam-6011	204	52	-	-	PUNCT
ejpam-6011	204	53	reduced	reduce	VERB
ejpam-6011	204	54	,	,	PUNCT
ejpam-6011	204	55	and	and	CCONJ
ejpam-6011	204	56	consequently	consequently	ADV
ejpam-6011	204	57	it	it	PRON
ejpam-6011	204	58	is	be	AUX
ejpam-6011	204	59	not	not	PART
ejpam-6011	204	60	nj	nj	NOUN
ejpam-6011	204	61	-	-	PUNCT
ejpam-6011	204	62	reduced	reduce	VERB
ejpam-6011	204	63	since	since	SCONJ
ejpam-6011	204	64	s	s	PROPN
ejpam-6011	204	65	has	have	VERB
ejpam-6011	204	66	identity	identity	NOUN
ejpam-6011	204	67	.	.	PUNCT
ejpam-6011	205	1	however	however	ADV
ejpam-6011	205	2	,	,	PUNCT
ejpam-6011	205	3	r	r	NOUN
ejpam-6011	205	4	is	be	AUX
ejpam-6011	205	5	an	an	DET
ejpam-6011	205	6	nj	nj	ADJ
ejpam-6011	205	7	-	-	PUNCT
ejpam-6011	205	8	abelian	abelian	ADJ
ejpam-6011	205	9	ring	ring	NOUN
ejpam-6011	205	10	.	.	PUNCT
ejpam-6011	206	1	now	now	ADV
ejpam-6011	206	2	,	,	PUNCT
ejpam-6011	206	3	we	we	PRON
ejpam-6011	206	4	give	give	VERB
ejpam-6011	206	5	some	some	DET
ejpam-6011	206	6	results	result	NOUN
ejpam-6011	206	7	for	for	ADP
ejpam-6011	206	8	subrings	subring	NOUN
ejpam-6011	206	9	that	that	PRON
ejpam-6011	206	10	are	be	AUX
ejpam-6011	206	11	nj	nj	NOUN
ejpam-6011	206	12	-	-	PUNCT
ejpam-6011	206	13	abelian	abelian	ADJ
ejpam-6011	206	14	due	due	ADP
ejpam-6011	206	15	to	to	ADP
ejpam-6011	206	16	the	the	DET
ejpam-6011	206	17	nj	nj	PROPN
ejpam-6011	206	18	-	-	PUNCT
ejpam-6011	206	19	abelianity	abelianity	NOUN
ejpam-6011	206	20	of	of	ADP
ejpam-6011	206	21	their	their	PRON
ejpam-6011	206	22	rings	ring	NOUN
ejpam-6011	206	23	.	.	PUNCT
ejpam-6011	207	1	m.	m.	PROPN
ejpam-6011	207	2	saad	saad	PROPN
ejpam-6011	207	3	,	,	PUNCT
ejpam-6011	207	4	s.	s.	PROPN
ejpam-6011	207	5	m.	m.	PROPN
ejpam-6011	207	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	207	7	/	/	SYM
ejpam-6011	207	8	eur	eur	PROPN
ejpam-6011	207	9	.	.	PUNCT
ejpam-6011	208	1	j.	j.	PROPN
ejpam-6011	208	2	pure	pure	PROPN
ejpam-6011	208	3	appl	appl	PROPN
ejpam-6011	208	4	.	.	PROPN
ejpam-6011	208	5	math	math	PROPN
ejpam-6011	208	6	,	,	PUNCT
ejpam-6011	208	7	18	18	NUM
ejpam-6011	208	8	(	(	PUNCT
ejpam-6011	208	9	2	2	NUM
ejpam-6011	208	10	)	)	PUNCT
ejpam-6011	208	11	(	(	PUNCT
ejpam-6011	208	12	2025	2025	NUM
ejpam-6011	208	13	)	)	PUNCT
ejpam-6011	208	14	,	,	PUNCT
ejpam-6011	208	15	6011	6011	NUM
ejpam-6011	208	16	7	7	NUM
ejpam-6011	208	17	of	of	ADP
ejpam-6011	208	18	13	13	NUM
ejpam-6011	208	19	proposition	proposition	NOUN
ejpam-6011	208	20	9	9	NUM
ejpam-6011	208	21	.	.	PUNCT
ejpam-6011	209	1	let	let	VERB
ejpam-6011	209	2	{	{	PUNCT
ejpam-6011	209	3	ri}i∈λ	ri}i∈λ	INTJ
ejpam-6011	209	4	be	be	AUX
ejpam-6011	209	5	a	a	DET
ejpam-6011	209	6	class	class	NOUN
ejpam-6011	209	7	of	of	ADP
ejpam-6011	209	8	rings	ring	NOUN
ejpam-6011	209	9	for	for	ADP
ejpam-6011	209	10	some	some	DET
ejpam-6011	209	11	index	index	NOUN
ejpam-6011	209	12	set	set	VERB
ejpam-6011	209	13	λ	λ	PROPN
ejpam-6011	209	14	.	.	PUNCT
ejpam-6011	210	1	then	then	ADV
ejpam-6011	210	2	∏	∏	PROPN
ejpam-6011	210	3	i∈λri	i∈λri	PROPN
ejpam-6011	210	4	is	be	AUX
ejpam-6011	210	5	nj	nj	NOUN
ejpam-6011	210	6	-	-	PUNCT
ejpam-6011	210	7	abelian	abelian	ADJ
ejpam-6011	211	1	if	if	SCONJ
ejpam-6011	211	2	and	and	CCONJ
ejpam-6011	211	3	only	only	ADV
ejpam-6011	211	4	if	if	SCONJ
ejpam-6011	211	5	ri	ri	PROPN
ejpam-6011	211	6	is	be	AUX
ejpam-6011	211	7	nj	nj	NOUN
ejpam-6011	211	8	-	-	PUNCT
ejpam-6011	211	9	abelian	abelian	ADJ
ejpam-6011	211	10	for	for	ADP
ejpam-6011	211	11	every	every	DET
ejpam-6011	211	12	i	i	PROPN
ejpam-6011	211	13	∈	∈	PROPN
ejpam-6011	211	14	λ	λ	PROPN
ejpam-6011	211	15	.	.	PUNCT
ejpam-6011	211	16	proof	proof	NOUN
ejpam-6011	211	17	.	.	PUNCT
ejpam-6011	212	1	the	the	DET
ejpam-6011	212	2	proof	proof	NOUN
ejpam-6011	212	3	is	be	AUX
ejpam-6011	212	4	routine	routine	ADJ
ejpam-6011	212	5	.	.	PUNCT
ejpam-6011	213	1	corollary	corollary	ADJ
ejpam-6011	213	2	5	5	NUM
ejpam-6011	213	3	.	.	PUNCT
ejpam-6011	214	1	let	let	VERB
ejpam-6011	214	2	r	r	PRON
ejpam-6011	214	3	be	be	AUX
ejpam-6011	214	4	a	a	DET
ejpam-6011	214	5	ring	ring	NOUN
ejpam-6011	214	6	and	and	CCONJ
ejpam-6011	214	7	e	e	NOUN
ejpam-6011	214	8	be	be	AUX
ejpam-6011	214	9	a	a	DET
ejpam-6011	214	10	central	central	ADJ
ejpam-6011	214	11	idempotent	idempotent	NOUN
ejpam-6011	214	12	of	of	ADP
ejpam-6011	214	13	r.	r.	PROPN
ejpam-6011	214	14	then	then	ADV
ejpam-6011	214	15	er	er	INTJ
ejpam-6011	214	16	and	and	CCONJ
ejpam-6011	214	17	(	(	PUNCT
ejpam-6011	214	18	1−	1−	NUM
ejpam-6011	214	19	e)r	e)r	ADV
ejpam-6011	214	20	are	be	AUX
ejpam-6011	214	21	nj	nj	NOUN
ejpam-6011	214	22	-	-	PUNCT
ejpam-6011	214	23	abelian	abelian	ADJ
ejpam-6011	215	1	if	if	SCONJ
ejpam-6011	215	2	and	and	CCONJ
ejpam-6011	215	3	only	only	ADV
ejpam-6011	215	4	if	if	SCONJ
ejpam-6011	215	5	r	r	NOUN
ejpam-6011	215	6	is	be	AUX
ejpam-6011	215	7	nj	nj	NOUN
ejpam-6011	215	8	-	-	PUNCT
ejpam-6011	215	9	abelian	abelian	NOUN
ejpam-6011	215	10	.	.	PUNCT
ejpam-6011	216	1	proposition	proposition	NOUN
ejpam-6011	216	2	10	10	NUM
ejpam-6011	216	3	.	.	PUNCT
ejpam-6011	217	1	a	a	DET
ejpam-6011	217	2	ring	ring	NOUN
ejpam-6011	217	3	r	r	NOUN
ejpam-6011	217	4	is	be	AUX
ejpam-6011	217	5	nj	nj	NOUN
ejpam-6011	217	6	-	-	PUNCT
ejpam-6011	217	7	abelian	abelian	ADJ
ejpam-6011	218	1	if	if	SCONJ
ejpam-6011	218	2	and	and	CCONJ
ejpam-6011	218	3	only	only	ADV
ejpam-6011	218	4	if	if	SCONJ
ejpam-6011	218	5	every	every	DET
ejpam-6011	218	6	corner	corner	NOUN
ejpam-6011	218	7	of	of	ADP
ejpam-6011	218	8	r	r	NOUN
ejpam-6011	218	9	is	be	AUX
ejpam-6011	218	10	nj	nj	NOUN
ejpam-6011	218	11	-	-	PUNCT
ejpam-6011	218	12	abelian	abelian	ADJ
ejpam-6011	218	13	.	.	PUNCT
ejpam-6011	219	1	proof	proof	NOUN
ejpam-6011	219	2	.	.	PUNCT
ejpam-6011	220	1	the	the	DET
ejpam-6011	220	2	sufficiency	sufficiency	NOUN
ejpam-6011	220	3	is	be	AUX
ejpam-6011	220	4	trivial	trivial	ADJ
ejpam-6011	220	5	.	.	PUNCT
ejpam-6011	221	1	for	for	ADP
ejpam-6011	221	2	the	the	DET
ejpam-6011	221	3	necessity	necessity	NOUN
ejpam-6011	221	4	,	,	PUNCT
ejpam-6011	221	5	let	let	VERB
ejpam-6011	221	6	f2	f2	NOUN
ejpam-6011	221	7	=	=	SYM
ejpam-6011	221	8	f	f	PROPN
ejpam-6011	221	9	,	,	PUNCT
ejpam-6011	221	10	a	a	DET
ejpam-6011	221	11	∈	∈	PROPN
ejpam-6011	221	12	ere	ere	PROPN
ejpam-6011	221	13	∈	∈	PROPN
ejpam-6011	221	14	r	r	NOUN
ejpam-6011	221	15	,	,	PUNCT
ejpam-6011	221	16	for	for	ADP
ejpam-6011	221	17	some	some	DET
ejpam-6011	221	18	idempotent	idempotent	ADJ
ejpam-6011	221	19	e	e	NOUN
ejpam-6011	221	20	of	of	ADP
ejpam-6011	221	21	r	r	NOUN
ejpam-6011	221	22	such	such	ADJ
ejpam-6011	221	23	that	that	SCONJ
ejpam-6011	221	24	af	af	PROPN
ejpam-6011	221	25	∈	∈	PROPN
ejpam-6011	221	26	n(ere	n(ere	ADV
ejpam-6011	221	27	)	)	PUNCT
ejpam-6011	221	28	⊆	⊆	NUM
ejpam-6011	221	29	n(r	n(r	NUM
ejpam-6011	221	30	)	)	PUNCT
ejpam-6011	221	31	.	.	PUNCT
ejpam-6011	222	1	so	so	ADV
ejpam-6011	222	2	,	,	PUNCT
ejpam-6011	222	3	a(ere)f	a(ere)f	NOUN
ejpam-6011	222	4	=	=	SYM
ejpam-6011	222	5	arf	arf	PROPN
ejpam-6011	222	6	∈	∈	PROPN
ejpam-6011	222	7	j(r	j(r	PROPN
ejpam-6011	222	8	)	)	PUNCT
ejpam-6011	222	9	since	since	SCONJ
ejpam-6011	222	10	r	r	NOUN
ejpam-6011	222	11	is	be	AUX
ejpam-6011	222	12	nj	nj	NOUN
ejpam-6011	222	13	-	-	PUNCT
ejpam-6011	222	14	abelian	abelian	ADJ
ejpam-6011	222	15	.	.	PUNCT
ejpam-6011	223	1	but	but	CCONJ
ejpam-6011	223	2	j(ere	j(ere	X
ejpam-6011	223	3	)	)	PUNCT
ejpam-6011	224	1	=	=	PUNCT
ejpam-6011	224	2	ej(r)e	ej(r)e	PROPN
ejpam-6011	224	3	.	.	PUNCT
ejpam-6011	225	1	so	so	ADV
ejpam-6011	225	2	,	,	PUNCT
ejpam-6011	225	3	a(ere)f	a(ere)f	NOUN
ejpam-6011	225	4	=	=	SYM
ejpam-6011	225	5	earfe	earfe	NOUN
ejpam-6011	225	6	⊆	⊆	NUM
ejpam-6011	225	7	ej(r)e	ej(r)e	X
ejpam-6011	225	8	=	=	SYM
ejpam-6011	225	9	j(ere	j(ere	X
ejpam-6011	225	10	)	)	PUNCT
ejpam-6011	225	11	and	and	CCONJ
ejpam-6011	225	12	ere	ere	PROPN
ejpam-6011	225	13	is	be	AUX
ejpam-6011	225	14	nj	nj	NOUN
ejpam-6011	225	15	-	-	PUNCT
ejpam-6011	225	16	abelian	abelian	ADJ
ejpam-6011	225	17	.	.	PUNCT
ejpam-6011	226	1	the	the	DET
ejpam-6011	226	2	next	next	ADJ
ejpam-6011	226	3	example	example	NOUN
ejpam-6011	226	4	shows	show	VERB
ejpam-6011	226	5	that	that	SCONJ
ejpam-6011	226	6	even	even	ADV
ejpam-6011	226	7	if	if	SCONJ
ejpam-6011	226	8	every	every	DET
ejpam-6011	226	9	corner	corner	NOUN
ejpam-6011	226	10	ere	ere	NOUN
ejpam-6011	226	11	of	of	ADP
ejpam-6011	226	12	a	a	DET
ejpam-6011	226	13	ring	ring	NOUN
ejpam-6011	226	14	r	r	NOUN
ejpam-6011	226	15	is	be	AUX
ejpam-6011	226	16	nj	nj	NOUN
ejpam-6011	226	17	-	-	PUNCT
ejpam-6011	226	18	abelian	abelian	ADJ
ejpam-6011	226	19	for	for	ADP
ejpam-6011	226	20	all	all	DET
ejpam-6011	226	21	nonidentity	nonidentity	NOUN
ejpam-6011	226	22	idempotents	idempotent	NOUN
ejpam-6011	226	23	e	e	NOUN
ejpam-6011	226	24	,	,	PUNCT
ejpam-6011	226	25	r	r	NOUN
ejpam-6011	226	26	is	be	AUX
ejpam-6011	226	27	not	not	PART
ejpam-6011	226	28	necessarily	necessarily	ADV
ejpam-6011	226	29	nj	nj	NOUN
ejpam-6011	226	30	-	-	PUNCT
ejpam-6011	226	31	abelian	abelian	PROPN
ejpam-6011	226	32	.	.	PUNCT
ejpam-6011	227	1	example	example	NOUN
ejpam-6011	227	2	8	8	NUM
ejpam-6011	227	3	.	.	PUNCT
ejpam-6011	228	1	let	let	VERB
ejpam-6011	228	2	r	r	NOUN
ejpam-6011	228	3	=	=	SYM
ejpam-6011	228	4	m2(z	m2(z	X
ejpam-6011	228	5	)	)	PUNCT
ejpam-6011	228	6	be	be	VERB
ejpam-6011	228	7	the	the	DET
ejpam-6011	228	8	ring	ring	NOUN
ejpam-6011	228	9	of	of	ADP
ejpam-6011	228	10	all	all	DET
ejpam-6011	228	11	2×	2×	NUM
ejpam-6011	228	12	2	2	NUM
ejpam-6011	228	13	matrices	matrix	NOUN
ejpam-6011	228	14	over	over	ADP
ejpam-6011	228	15	the	the	DET
ejpam-6011	228	16	ring	ring	NOUN
ejpam-6011	228	17	of	of	ADP
ejpam-6011	228	18	integers	integer	NOUN
ejpam-6011	228	19	z.	z.	PROPN
ejpam-6011	229	1	every	every	DET
ejpam-6011	229	2	nontrivial	nontrivial	ADJ
ejpam-6011	229	3	idempotent	idempotent	ADJ
ejpam-6011	229	4	e	e	NOUN
ejpam-6011	229	5	of	of	ADP
ejpam-6011	229	6	r	r	NOUN
ejpam-6011	229	7	satisfies	satisfie	NOUN
ejpam-6011	229	8	ere	ere	X
ejpam-6011	230	1	=	=	SYM
ejpam-6011	230	2	z	z	PROPN
ejpam-6011	230	3	,	,	PUNCT
ejpam-6011	230	4	which	which	PRON
ejpam-6011	230	5	is	be	AUX
ejpam-6011	230	6	nj	nj	NOUN
ejpam-6011	230	7	-	-	PUNCT
ejpam-6011	230	8	abelian	abelian	ADJ
ejpam-6011	230	9	,	,	PUNCT
ejpam-6011	230	10	but	but	CCONJ
ejpam-6011	230	11	r	r	NOUN
ejpam-6011	230	12	is	be	AUX
ejpam-6011	230	13	not	not	PART
ejpam-6011	230	14	nj	nj	NOUN
ejpam-6011	230	15	-	-	NOUN
ejpam-6011	230	16	abelian	abelian	ADJ
ejpam-6011	230	17	since	since	SCONJ
ejpam-6011	230	18	the	the	DET
ejpam-6011	230	19	elements	element	NOUN
ejpam-6011	231	1	α	α	X
ejpam-6011	232	1	=	=	PUNCT
ejpam-6011	233	1	[	[	PUNCT
ejpam-6011	233	2	0	0	NUM
ejpam-6011	233	3	1	1	NUM
ejpam-6011	233	4	0	0	NUM
ejpam-6011	233	5	0	0	NUM
ejpam-6011	233	6	]	]	PUNCT
ejpam-6011	233	7	and	and	CCONJ
ejpam-6011	233	8	ϵ2	ϵ2	NOUN
ejpam-6011	233	9	=	=	PUNCT
ejpam-6011	233	10	ϵ	ϵ	X
ejpam-6011	233	11	=	=	PUNCT
ejpam-6011	234	1	[	[	PUNCT
ejpam-6011	234	2	0	0	NUM
ejpam-6011	234	3	1	1	NUM
ejpam-6011	234	4	0	0	NUM
ejpam-6011	234	5	1	1	NUM
ejpam-6011	234	6	]	]	PUNCT
ejpam-6011	234	7	satisfy	satisfy	VERB
ejpam-6011	234	8	αϵ	αϵ	PROPN
ejpam-6011	234	9	∈	∈	PROPN
ejpam-6011	234	10	n(r	n(r	NOUN
ejpam-6011	234	11	)	)	PUNCT
ejpam-6011	234	12	but	but	CCONJ
ejpam-6011	234	13	αrϵ	αrϵ	PRON
ejpam-6011	234	14	=	=	PUNCT
ejpam-6011	234	15	[	[	PUNCT
ejpam-6011	234	16	0	0	NUM
ejpam-6011	234	17	z	z	NOUN
ejpam-6011	234	18	0	0	NUM
ejpam-6011	234	19	0	0	NUM
ejpam-6011	234	20	]	]	PUNCT
ejpam-6011	234	21	.	.	PUNCT
ejpam-6011	235	1	the	the	DET
ejpam-6011	235	2	following	follow	VERB
ejpam-6011	235	3	proposition	proposition	NOUN
ejpam-6011	235	4	shows	show	VERB
ejpam-6011	235	5	that	that	SCONJ
ejpam-6011	235	6	if	if	SCONJ
ejpam-6011	235	7	a	a	DET
ejpam-6011	235	8	subring	subring	NOUN
ejpam-6011	235	9	of	of	ADP
ejpam-6011	235	10	an	an	DET
ejpam-6011	235	11	nj	nj	PROPN
ejpam-6011	235	12	-	-	PUNCT
ejpam-6011	235	13	abelian	abelian	ADJ
ejpam-6011	235	14	ring	ring	NOUN
ejpam-6011	235	15	is	be	AUX
ejpam-6011	235	16	an	an	DET
ejpam-6011	235	17	ideal	ideal	NOUN
ejpam-6011	235	18	,	,	PUNCT
ejpam-6011	235	19	then	then	ADV
ejpam-6011	235	20	it	it	PRON
ejpam-6011	235	21	is	be	AUX
ejpam-6011	235	22	also	also	ADV
ejpam-6011	235	23	nj	nj	NOUN
ejpam-6011	235	24	-	-	PUNCT
ejpam-6011	235	25	abelian	abelian	ADJ
ejpam-6011	235	26	.	.	PUNCT
ejpam-6011	236	1	proposition	proposition	NOUN
ejpam-6011	236	2	11	11	NUM
ejpam-6011	236	3	.	.	PUNCT
ejpam-6011	237	1	every	every	DET
ejpam-6011	237	2	ideal	ideal	NOUN
ejpam-6011	237	3	of	of	ADP
ejpam-6011	237	4	an	an	DET
ejpam-6011	237	5	nj	nj	PROPN
ejpam-6011	237	6	-	-	PUNCT
ejpam-6011	237	7	abelian	abelian	ADJ
ejpam-6011	237	8	ring	ring	NOUN
ejpam-6011	237	9	is	be	AUX
ejpam-6011	237	10	nj	nj	NOUN
ejpam-6011	237	11	-	-	PUNCT
ejpam-6011	237	12	abelian	abelian	NOUN
ejpam-6011	237	13	(	(	PUNCT
ejpam-6011	237	14	as	as	ADP
ejpam-6011	237	15	a	a	DET
ejpam-6011	237	16	ring	ring	NOUN
ejpam-6011	237	17	without	without	ADP
ejpam-6011	237	18	identity	identity	NOUN
ejpam-6011	237	19	)	)	PUNCT
ejpam-6011	237	20	.	.	PUNCT
ejpam-6011	238	1	proof	proof	NOUN
ejpam-6011	238	2	.	.	PUNCT
ejpam-6011	239	1	let	let	VERB
ejpam-6011	239	2	r	r	PRON
ejpam-6011	239	3	be	be	AUX
ejpam-6011	239	4	an	an	DET
ejpam-6011	239	5	nj	nj	ADJ
ejpam-6011	239	6	-	-	PUNCT
ejpam-6011	239	7	abelian	abelian	ADJ
ejpam-6011	239	8	ring	ring	NOUN
ejpam-6011	239	9	and	and	CCONJ
ejpam-6011	239	10	i	i	PRON
ejpam-6011	239	11	be	be	VERB
ejpam-6011	239	12	an	an	DET
ejpam-6011	239	13	ideal	ideal	NOUN
ejpam-6011	239	14	of	of	ADP
ejpam-6011	239	15	r.	r.	PROPN
ejpam-6011	239	16	assume	assume	VERB
ejpam-6011	239	17	that	that	SCONJ
ejpam-6011	239	18	ea	ea	PROPN
ejpam-6011	239	19	∈	∈	PROPN
ejpam-6011	239	20	n(i	n(i	PROPN
ejpam-6011	239	21	)	)	PUNCT
ejpam-6011	239	22	where	where	SCONJ
ejpam-6011	239	23	e2	e2	PROPN
ejpam-6011	239	24	,	,	PUNCT
ejpam-6011	239	25	a	a	DET
ejpam-6011	239	26	∈	∈	PROPN
ejpam-6011	239	27	i.	i.	NOUN
ejpam-6011	239	28	but	but	CCONJ
ejpam-6011	239	29	n(i	n(i	PROPN
ejpam-6011	239	30	)	)	PUNCT
ejpam-6011	239	31	⊆	⊆	NUM
ejpam-6011	239	32	n(r	n(r	NUM
ejpam-6011	239	33	)	)	PUNCT
ejpam-6011	239	34	and	and	CCONJ
ejpam-6011	239	35	r	r	NOUN
ejpam-6011	239	36	is	be	AUX
ejpam-6011	239	37	nj	nj	NOUN
ejpam-6011	239	38	-	-	PUNCT
ejpam-6011	239	39	abelian	abelian	ADJ
ejpam-6011	239	40	;	;	PUNCT
ejpam-6011	239	41	hence	hence	ADV
ejpam-6011	239	42	,	,	PUNCT
ejpam-6011	239	43	are	be	AUX
ejpam-6011	239	44	⊆	⊆	NUM
ejpam-6011	239	45	j(r	j(r	NOUN
ejpam-6011	239	46	)	)	PUNCT
ejpam-6011	239	47	.	.	PUNCT
ejpam-6011	240	1	so	so	ADV
ejpam-6011	240	2	,	,	PUNCT
ejpam-6011	240	3	aie	aie	PROPN
ejpam-6011	240	4	⊆	⊆	NUM
ejpam-6011	240	5	are	be	AUX
ejpam-6011	240	6	⊆	⊆	NUM
ejpam-6011	240	7	i	i	PROPN
ejpam-6011	240	8	∩	∩	ADJ
ejpam-6011	240	9	j(r	j(r	NOUN
ejpam-6011	240	10	)	)	PUNCT
ejpam-6011	241	1	=	=	SYM
ejpam-6011	241	2	j(r	j(r	PROPN
ejpam-6011	241	3	)	)	PUNCT
ejpam-6011	241	4	and	and	CCONJ
ejpam-6011	241	5	i	i	PRON
ejpam-6011	241	6	is	be	AUX
ejpam-6011	241	7	nj	nj	NOUN
ejpam-6011	241	8	-	-	PUNCT
ejpam-6011	241	9	abelian	abelian	NOUN
ejpam-6011	241	10	.	.	PUNCT
ejpam-6011	242	1	proposition	proposition	NOUN
ejpam-6011	242	2	12	12	NUM
ejpam-6011	242	3	.	.	PUNCT
ejpam-6011	243	1	let	let	VERB
ejpam-6011	243	2	r	r	PRON
ejpam-6011	243	3	be	be	AUX
ejpam-6011	243	4	a	a	DET
ejpam-6011	243	5	ring	ring	NOUN
ejpam-6011	243	6	such	such	ADJ
ejpam-6011	243	7	that	that	DET
ejpam-6011	243	8	r[x	r[x	NOUN
ejpam-6011	243	9	]	]	PUNCT
ejpam-6011	243	10	is	be	AUX
ejpam-6011	243	11	nj	nj	NOUN
ejpam-6011	243	12	-	-	PUNCT
ejpam-6011	243	13	abelian	abelian	PROPN
ejpam-6011	243	14	.	.	PUNCT
ejpam-6011	244	1	then	then	ADV
ejpam-6011	244	2	r	r	NOUN
ejpam-6011	244	3	is	be	AUX
ejpam-6011	244	4	an	an	DET
ejpam-6011	244	5	nj	nj	ADJ
ejpam-6011	244	6	-	-	PUNCT
ejpam-6011	244	7	abelian	abelian	ADJ
ejpam-6011	244	8	ring	ring	NOUN
ejpam-6011	244	9	.	.	PUNCT
ejpam-6011	245	1	proof	proof	NOUN
ejpam-6011	245	2	.	.	PUNCT
ejpam-6011	246	1	let	let	VERB
ejpam-6011	246	2	e2	e2	PROPN
ejpam-6011	246	3	=	=	SYM
ejpam-6011	246	4	e	e	PROPN
ejpam-6011	246	5	,	,	PUNCT
ejpam-6011	246	6	a	a	DET
ejpam-6011	246	7	∈	∈	NOUN
ejpam-6011	246	8	r	r	NOUN
ejpam-6011	246	9	such	such	ADJ
ejpam-6011	246	10	that	that	DET
ejpam-6011	246	11	ea	ea	PROPN
ejpam-6011	246	12	∈	∈	PROPN
ejpam-6011	246	13	n(r	n(r	NOUN
ejpam-6011	246	14	)	)	PUNCT
ejpam-6011	246	15	⊆	⊆	NUM
ejpam-6011	246	16	n(r[x	n(r[x	NOUN
ejpam-6011	246	17	]	]	PUNCT
ejpam-6011	246	18	)	)	PUNCT
ejpam-6011	246	19	.	.	PUNCT
ejpam-6011	247	1	from	from	ADP
ejpam-6011	247	2	the	the	DET
ejpam-6011	247	3	nj	nj	PROPN
ejpam-6011	247	4	-	-	PUNCT
ejpam-6011	247	5	abelianity	abelianity	NOUN
ejpam-6011	247	6	of	of	ADP
ejpam-6011	247	7	r[x	r[x	NOUN
ejpam-6011	247	8	]	]	PUNCT
ejpam-6011	247	9	,	,	PUNCT
ejpam-6011	247	10	we	we	PRON
ejpam-6011	247	11	have	have	VERB
ejpam-6011	247	12	era	era	NOUN
ejpam-6011	247	13	⊆	⊆	NUM
ejpam-6011	247	14	er[x]a	er[x]a	PROPN
ejpam-6011	247	15	∈	∈	PROPN
ejpam-6011	247	16	j(r[x	j(r[x	NOUN
ejpam-6011	247	17	]	]	PUNCT
ejpam-6011	247	18	)	)	PUNCT
ejpam-6011	247	19	,	,	PUNCT
ejpam-6011	247	20	and	and	CCONJ
ejpam-6011	247	21	1−	1−	NUM
ejpam-6011	247	22	eras	era	NOUN
ejpam-6011	247	23	is	be	AUX
ejpam-6011	247	24	invertible	invertible	ADJ
ejpam-6011	247	25	in	in	ADP
ejpam-6011	247	26	r[x	r[x	NOUN
ejpam-6011	247	27	]	]	PUNCT
ejpam-6011	247	28	for	for	ADP
ejpam-6011	247	29	all	all	DET
ejpam-6011	247	30	r	r	NOUN
ejpam-6011	247	31	,	,	PUNCT
ejpam-6011	247	32	s	s	PART
ejpam-6011	247	33	∈	∈	PROPN
ejpam-6011	247	34	r.	r.	NOUN
ejpam-6011	247	35	but	but	CCONJ
ejpam-6011	247	36	1−	1−	NUM
ejpam-6011	247	37	eras	era	NOUN
ejpam-6011	247	38	∈	∈	PROPN
ejpam-6011	247	39	r	r	NOUN
ejpam-6011	247	40	,	,	PUNCT
ejpam-6011	247	41	and	and	CCONJ
ejpam-6011	247	42	hence	hence	ADV
ejpam-6011	247	43	1−	1−	NUM
ejpam-6011	247	44	eras	era	NOUN
ejpam-6011	247	45	∈	∈	PROPN
ejpam-6011	247	46	u(r	u(r	PROPN
ejpam-6011	247	47	)	)	PUNCT
ejpam-6011	247	48	for	for	ADP
ejpam-6011	247	49	all	all	DET
ejpam-6011	247	50	r	r	NOUN
ejpam-6011	247	51	,	,	PUNCT
ejpam-6011	247	52	s	s	PART
ejpam-6011	247	53	∈	∈	PROPN
ejpam-6011	247	54	r.	r.	NOUN
ejpam-6011	247	55	thus	thus	ADV
ejpam-6011	247	56	r	r	NOUN
ejpam-6011	247	57	is	be	AUX
ejpam-6011	247	58	nj	nj	NOUN
ejpam-6011	247	59	-	-	PUNCT
ejpam-6011	247	60	abelian	abelian	ADJ
ejpam-6011	247	61	.	.	PUNCT
ejpam-6011	248	1	it	it	PRON
ejpam-6011	248	2	is	be	AUX
ejpam-6011	248	3	natural	natural	ADJ
ejpam-6011	248	4	to	to	PART
ejpam-6011	248	5	conjecture	conjecture	VERB
ejpam-6011	248	6	that	that	SCONJ
ejpam-6011	248	7	r	r	NOUN
ejpam-6011	248	8	is	be	AUX
ejpam-6011	248	9	an	an	DET
ejpam-6011	248	10	nj	nj	ADJ
ejpam-6011	248	11	-	-	PUNCT
ejpam-6011	248	12	abelian	abelian	ADJ
ejpam-6011	248	13	ring	ring	NOUN
ejpam-6011	248	14	if	if	SCONJ
ejpam-6011	248	15	for	for	ADP
ejpam-6011	248	16	any	any	DET
ejpam-6011	248	17	nonzero	nonzero	ADJ
ejpam-6011	248	18	proper	proper	ADJ
ejpam-6011	248	19	ideal	ideal	NOUN
ejpam-6011	248	20	i	i	PRON
ejpam-6011	248	21	of	of	ADP
ejpam-6011	248	22	r	r	NOUN
ejpam-6011	248	23	,	,	PUNCT
ejpam-6011	248	24	r	r	NOUN
ejpam-6011	248	25	/	/	SYM
ejpam-6011	248	26	i	i	PRON
ejpam-6011	248	27	and	and	CCONJ
ejpam-6011	248	28	i	i	PRON
ejpam-6011	248	29	are	be	AUX
ejpam-6011	248	30	both	both	PRON
ejpam-6011	248	31	nj	nj	PROPN
ejpam-6011	248	32	-	-	PUNCT
ejpam-6011	248	33	abelian	abelian	ADJ
ejpam-6011	248	34	rings	ring	NOUN
ejpam-6011	248	35	,	,	PUNCT
ejpam-6011	248	36	where	where	SCONJ
ejpam-6011	248	37	i	i	PRON
ejpam-6011	248	38	is	be	AUX
ejpam-6011	248	39	considered	consider	VERB
ejpam-6011	248	40	a	a	DET
ejpam-6011	248	41	ring	ring	NOUN
ejpam-6011	248	42	without	without	ADP
ejpam-6011	248	43	identity	identity	NOUN
ejpam-6011	248	44	.	.	PUNCT
ejpam-6011	249	1	however	however	ADV
ejpam-6011	249	2	,	,	PUNCT
ejpam-6011	249	3	the	the	DET
ejpam-6011	249	4	following	follow	VERB
ejpam-6011	249	5	example	example	NOUN
ejpam-6011	249	6	provides	provide	VERB
ejpam-6011	249	7	a	a	DET
ejpam-6011	249	8	negative	negative	ADJ
ejpam-6011	249	9	answer	answer	NOUN
ejpam-6011	249	10	to	to	ADP
ejpam-6011	249	11	this	this	DET
ejpam-6011	249	12	conjecture	conjecture	NOUN
ejpam-6011	249	13	.	.	PUNCT
ejpam-6011	250	1	m.	m.	NOUN
ejpam-6011	250	2	saad	saad	PROPN
ejpam-6011	250	3	,	,	PUNCT
ejpam-6011	250	4	s.	s.	PROPN
ejpam-6011	250	5	m.	m.	PROPN
ejpam-6011	250	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	250	7	/	/	SYM
ejpam-6011	250	8	eur	eur	PROPN
ejpam-6011	250	9	.	.	PUNCT
ejpam-6011	251	1	j.	j.	PROPN
ejpam-6011	251	2	pure	pure	PROPN
ejpam-6011	251	3	appl	appl	PROPN
ejpam-6011	251	4	.	.	PROPN
ejpam-6011	251	5	math	math	PROPN
ejpam-6011	251	6	,	,	PUNCT
ejpam-6011	251	7	18	18	NUM
ejpam-6011	251	8	(	(	PUNCT
ejpam-6011	251	9	2	2	NUM
ejpam-6011	251	10	)	)	PUNCT
ejpam-6011	251	11	(	(	PUNCT
ejpam-6011	251	12	2025	2025	NUM
ejpam-6011	251	13	)	)	PUNCT
ejpam-6011	251	14	,	,	PUNCT
ejpam-6011	251	15	6011	6011	NUM
ejpam-6011	251	16	8	8	NUM
ejpam-6011	251	17	of	of	ADP
ejpam-6011	251	18	13	13	NUM
ejpam-6011	251	19	example	example	NOUN
ejpam-6011	251	20	9	9	NUM
ejpam-6011	251	21	.	.	X
ejpam-6011	252	1	for	for	ADP
ejpam-6011	252	2	a	a	DET
ejpam-6011	252	3	ring	ring	NOUN
ejpam-6011	252	4	r	r	NOUN
ejpam-6011	252	5	,	,	PUNCT
ejpam-6011	252	6	let	let	VERB
ejpam-6011	252	7	s	s	PRON
ejpam-6011	252	8	=	=	ADJ
ejpam-6011	252	9	m2(r	m2(r	ADJ
ejpam-6011	252	10	)	)	PUNCT
ejpam-6011	252	11	,	,	PUNCT
ejpam-6011	252	12	and	and	CCONJ
ejpam-6011	252	13	i	i	PRON
ejpam-6011	252	14	is	be	AUX
ejpam-6011	252	15	the	the	DET
ejpam-6011	252	16	ideal	ideal	NOUN
ejpam-6011	252	17	generated	generate	VERB
ejpam-6011	252	18	by	by	ADP
ejpam-6011	252	19	the	the	DET
ejpam-6011	252	20	commutators	commutator	NOUN
ejpam-6011	252	21	of	of	ADP
ejpam-6011	252	22	s.	s.	PROPN
ejpam-6011	252	23	then	then	ADV
ejpam-6011	252	24	s	s	VERB
ejpam-6011	252	25	/	/	SYM
ejpam-6011	252	26	i	i	PRON
ejpam-6011	252	27	is	be	AUX
ejpam-6011	252	28	commutative	commutative	ADJ
ejpam-6011	252	29	,	,	PUNCT
ejpam-6011	252	30	and	and	CCONJ
ejpam-6011	252	31	therefore	therefore	ADV
ejpam-6011	252	32	it	it	PRON
ejpam-6011	252	33	is	be	AUX
ejpam-6011	252	34	nj	nj	NOUN
ejpam-6011	252	35	-	-	PUNCT
ejpam-6011	252	36	abelian	abelian	ADJ
ejpam-6011	252	37	.	.	PUNCT
ejpam-6011	253	1	the	the	DET
ejpam-6011	253	2	elements	element	NOUN
ejpam-6011	253	3	e	e	X
ejpam-6011	253	4	=	=	PUNCT
ejpam-6011	253	5	[	[	PUNCT
ejpam-6011	253	6	1	1	NUM
ejpam-6011	253	7	1	1	NUM
ejpam-6011	253	8	0	0	NUM
ejpam-6011	253	9	0	0	NUM
ejpam-6011	253	10	]	]	PUNCT
ejpam-6011	253	11	and	and	CCONJ
ejpam-6011	253	12	a	a	DET
ejpam-6011	253	13	=	=	X
ejpam-6011	253	14	[	[	PUNCT
ejpam-6011	253	15	1	1	NUM
ejpam-6011	253	16	1	1	NUM
ejpam-6011	253	17	−1	−1	NOUN
ejpam-6011	253	18	0	0	NUM
ejpam-6011	253	19	]	]	PUNCT
ejpam-6011	253	20	of	of	ADP
ejpam-6011	253	21	s	s	PROPN
ejpam-6011	253	22	satisfy	satisfy	PROPN
ejpam-6011	253	23	e2	e2	PROPN
ejpam-6011	253	24	=	=	SYM
ejpam-6011	253	25	e	e	PROPN
ejpam-6011	253	26	and	and	CCONJ
ejpam-6011	253	27	ea	ea	NOUN
ejpam-6011	253	28	∈	∈	PROPN
ejpam-6011	253	29	n(r	n(r	NOUN
ejpam-6011	253	30	)	)	PUNCT
ejpam-6011	253	31	.	.	PUNCT
ejpam-6011	254	1	however	however	ADV
ejpam-6011	254	2	,	,	PUNCT
ejpam-6011	254	3	esa	esa	PROPN
ejpam-6011	254	4	=	=	PRON
ejpam-6011	254	5	[	[	PUNCT
ejpam-6011	254	6	r	r	NOUN
ejpam-6011	254	7	r	r	NOUN
ejpam-6011	254	8	0	0	NUM
ejpam-6011	254	9	0	0	NUM
ejpam-6011	254	10	]	]	PUNCT
ejpam-6011	254	11	⊈	⊈	PROPN
ejpam-6011	254	12	j(r	j(r	NOUN
ejpam-6011	254	13	)	)	PUNCT
ejpam-6011	254	14	.	.	PUNCT
ejpam-6011	255	1	thus	thus	ADV
ejpam-6011	255	2	,	,	PUNCT
ejpam-6011	255	3	s	s	VERB
ejpam-6011	255	4	is	be	AUX
ejpam-6011	255	5	not	not	PART
ejpam-6011	255	6	nj	nj	NOUN
ejpam-6011	255	7	-	-	PUNCT
ejpam-6011	255	8	abelian	abelian	ADJ
ejpam-6011	255	9	.	.	PUNCT
ejpam-6011	256	1	however	however	ADV
ejpam-6011	256	2	,	,	PUNCT
ejpam-6011	256	3	if	if	SCONJ
ejpam-6011	256	4	we	we	PRON
ejpam-6011	256	5	take	take	VERB
ejpam-6011	256	6	stronger	strong	ADJ
ejpam-6011	256	7	independent	independent	ADJ
ejpam-6011	256	8	conditions	condition	NOUN
ejpam-6011	256	9	,	,	PUNCT
ejpam-6011	256	10	such	such	ADJ
ejpam-6011	256	11	as	as	ADP
ejpam-6011	256	12	“	"	PUNCT
ejpam-6011	256	13	i	i	PRON
ejpam-6011	256	14	is	be	AUX
ejpam-6011	256	15	nil	nil	ADJ
ejpam-6011	256	16	”	"	PUNCT
ejpam-6011	256	17	and	and	CCONJ
ejpam-6011	256	18	“	"	PUNCT
ejpam-6011	256	19	i	i	PROPN
ejpam-6011	256	20	=	=	SYM
ejpam-6011	256	21	j(r	j(r	PROPN
ejpam-6011	256	22	)	)	PUNCT
ejpam-6011	256	23	”	"	PUNCT
ejpam-6011	256	24	,	,	PUNCT
ejpam-6011	256	25	then	then	ADV
ejpam-6011	256	26	we	we	PRON
ejpam-6011	256	27	may	may	AUX
ejpam-6011	256	28	have	have	VERB
ejpam-6011	256	29	an	an	DET
ejpam-6011	256	30	affirmative	affirmative	ADJ
ejpam-6011	256	31	answer	answer	NOUN
ejpam-6011	256	32	,	,	PUNCT
ejpam-6011	256	33	as	as	ADP
ejpam-6011	256	34	in	in	ADP
ejpam-6011	256	35	the	the	DET
ejpam-6011	256	36	following	following	NOUN
ejpam-6011	256	37	.	.	PUNCT
ejpam-6011	257	1	proposition	proposition	NOUN
ejpam-6011	257	2	13	13	NUM
ejpam-6011	257	3	.	.	PUNCT
ejpam-6011	258	1	for	for	ADP
ejpam-6011	258	2	a	a	DET
ejpam-6011	258	3	ring	ring	NOUN
ejpam-6011	258	4	r	r	NOUN
ejpam-6011	258	5	,	,	PUNCT
ejpam-6011	258	6	if	if	SCONJ
ejpam-6011	258	7	r	r	NOUN
ejpam-6011	258	8	/	/	SYM
ejpam-6011	258	9	j(r	j(r	PROPN
ejpam-6011	258	10	)	)	PUNCT
ejpam-6011	258	11	is	be	AUX
ejpam-6011	258	12	nj	nj	NOUN
ejpam-6011	258	13	-	-	PUNCT
ejpam-6011	258	14	abelian	abelian	ADJ
ejpam-6011	258	15	,	,	PUNCT
ejpam-6011	258	16	then	then	ADV
ejpam-6011	258	17	r	r	NOUN
ejpam-6011	258	18	is	be	AUX
ejpam-6011	258	19	nj	nj	NOUN
ejpam-6011	258	20	-	-	PUNCT
ejpam-6011	258	21	abelian	abelian	ADJ
ejpam-6011	258	22	.	.	PUNCT
ejpam-6011	259	1	proof	proof	NOUN
ejpam-6011	259	2	.	.	PUNCT
ejpam-6011	260	1	if	if	SCONJ
ejpam-6011	260	2	e	e	X
ejpam-6011	260	3	,	,	PUNCT
ejpam-6011	260	4	f	f	PROPN
ejpam-6011	260	5	∈	∈	PROPN
ejpam-6011	260	6	i(r	i(r	PROPN
ejpam-6011	260	7	)	)	PUNCT
ejpam-6011	260	8	and	and	CCONJ
ejpam-6011	260	9	ef	ef	ADP
ejpam-6011	260	10	∈	∈	PROPN
ejpam-6011	260	11	n(r	n(r	NOUN
ejpam-6011	260	12	)	)	PUNCT
ejpam-6011	260	13	,	,	PUNCT
ejpam-6011	260	14	then	then	ADV
ejpam-6011	260	15	ēf̄	ēf̄	PROPN
ejpam-6011	260	16	∈	∈	PROPN
ejpam-6011	260	17	n(r	n(r	NOUN
ejpam-6011	260	18	/	/	SYM
ejpam-6011	260	19	j(r	j(r	PROPN
ejpam-6011	260	20	)	)	PUNCT
ejpam-6011	260	21	)	)	PUNCT
ejpam-6011	260	22	.	.	PUNCT
ejpam-6011	261	1	but	but	CCONJ
ejpam-6011	261	2	r	r	X
ejpam-6011	261	3	/	/	SYM
ejpam-6011	261	4	j(r	j(r	PROPN
ejpam-6011	261	5	)	)	PUNCT
ejpam-6011	261	6	is	be	AUX
ejpam-6011	261	7	njabelian	njabelian	ADJ
ejpam-6011	261	8	and	and	CCONJ
ejpam-6011	261	9	ēr̄f̄	ēr̄f̄	PUNCT
ejpam-6011	261	10	∈	∈	PROPN
ejpam-6011	261	11	n(r	n(r	NOUN
ejpam-6011	261	12	/	/	SYM
ejpam-6011	261	13	j(r	j(r	PROPN
ejpam-6011	261	14	)	)	PUNCT
ejpam-6011	261	15	)	)	PUNCT
ejpam-6011	262	1	=	=	PUNCT
ejpam-6011	262	2	0	0	NUM
ejpam-6011	262	3	,	,	PUNCT
ejpam-6011	262	4	for	for	ADP
ejpam-6011	262	5	every	every	DET
ejpam-6011	262	6	r	r	NOUN
ejpam-6011	262	7	∈	∈	PROPN
ejpam-6011	262	8	r.	r.	PROPN
ejpam-6011	262	9	thus	thus	ADV
ejpam-6011	262	10	,	,	PUNCT
ejpam-6011	262	11	arb	arb	PROPN
ejpam-6011	262	12	⊆	⊆	NUM
ejpam-6011	262	13	j(r	j(r	NOUN
ejpam-6011	262	14	)	)	PUNCT
ejpam-6011	262	15	,	,	PUNCT
ejpam-6011	262	16	and	and	CCONJ
ejpam-6011	262	17	r	r	NOUN
ejpam-6011	262	18	is	be	AUX
ejpam-6011	262	19	nj	nj	NOUN
ejpam-6011	262	20	-	-	PUNCT
ejpam-6011	262	21	abelian	abelian	PROPN
ejpam-6011	262	22	.	.	PUNCT
ejpam-6011	263	1	since	since	SCONJ
ejpam-6011	263	2	every	every	DET
ejpam-6011	263	3	reduced	reduce	VERB
ejpam-6011	263	4	ring	ring	NOUN
ejpam-6011	263	5	is	be	AUX
ejpam-6011	263	6	nj	nj	NOUN
ejpam-6011	263	7	-	-	PUNCT
ejpam-6011	263	8	abelian	abelian	ADJ
ejpam-6011	263	9	,	,	PUNCT
ejpam-6011	263	10	we	we	PRON
ejpam-6011	263	11	have	have	VERB
ejpam-6011	263	12	the	the	DET
ejpam-6011	263	13	following	follow	VERB
ejpam-6011	263	14	corollary	corollary	NOUN
ejpam-6011	263	15	.	.	PUNCT
ejpam-6011	264	1	corollary	corollary	ADJ
ejpam-6011	264	2	6	6	NUM
ejpam-6011	264	3	.	.	PUNCT
ejpam-6011	265	1	if	if	SCONJ
ejpam-6011	265	2	r	r	NOUN
ejpam-6011	265	3	/	/	SYM
ejpam-6011	265	4	j(r	j(r	PROPN
ejpam-6011	265	5	)	)	PUNCT
ejpam-6011	265	6	is	be	AUX
ejpam-6011	265	7	reduced	reduce	VERB
ejpam-6011	265	8	,	,	PUNCT
ejpam-6011	265	9	then	then	ADV
ejpam-6011	265	10	r	r	NOUN
ejpam-6011	265	11	is	be	AUX
ejpam-6011	265	12	nj	nj	NOUN
ejpam-6011	265	13	-	-	PUNCT
ejpam-6011	265	14	abelian	abelian	ADJ
ejpam-6011	265	15	.	.	PUNCT
ejpam-6011	266	1	the	the	DET
ejpam-6011	266	2	converse	converse	NOUN
ejpam-6011	266	3	of	of	ADP
ejpam-6011	266	4	the	the	DET
ejpam-6011	266	5	previous	previous	ADJ
ejpam-6011	266	6	proposition	proposition	NOUN
ejpam-6011	266	7	is	be	AUX
ejpam-6011	266	8	not	not	PART
ejpam-6011	266	9	necessarily	necessarily	ADV
ejpam-6011	266	10	true	true	ADJ
ejpam-6011	266	11	,	,	PUNCT
ejpam-6011	266	12	as	as	SCONJ
ejpam-6011	266	13	shown	show	VERB
ejpam-6011	266	14	in	in	ADP
ejpam-6011	266	15	the	the	DET
ejpam-6011	266	16	next	next	ADJ
ejpam-6011	266	17	example	example	NOUN
ejpam-6011	266	18	.	.	PUNCT
ejpam-6011	267	1	example	example	NOUN
ejpam-6011	268	1	10	10	NUM
ejpam-6011	268	2	.	.	PUNCT
ejpam-6011	269	1	let	let	VERB
ejpam-6011	269	2	s	s	PRON
ejpam-6011	269	3	be	be	AUX
ejpam-6011	269	4	the	the	DET
ejpam-6011	269	5	localization	localization	NOUN
ejpam-6011	269	6	of	of	ADP
ejpam-6011	269	7	z	z	NOUN
ejpam-6011	269	8	at	at	ADP
ejpam-6011	269	9	3z	3z	NUM
ejpam-6011	269	10	and	and	CCONJ
ejpam-6011	269	11	r	r	NOUN
ejpam-6011	269	12	the	the	DET
ejpam-6011	269	13	set	set	NOUN
ejpam-6011	269	14	of	of	ADP
ejpam-6011	269	15	quaternions	quaternion	NOUN
ejpam-6011	269	16	over	over	ADP
ejpam-6011	269	17	the	the	DET
ejpam-6011	269	18	ring	ring	NOUN
ejpam-6011	269	19	s.	s.	PROPN
ejpam-6011	269	20	according	accord	VERB
ejpam-6011	269	21	to	to	ADP
ejpam-6011	269	22	[	[	X
ejpam-6011	269	23	7	7	NUM
ejpam-6011	269	24	]	]	PUNCT
ejpam-6011	269	25	,	,	PUNCT
ejpam-6011	269	26	j(r	j(r	PROPN
ejpam-6011	269	27	)	)	PUNCT
ejpam-6011	270	1	=	=	SYM
ejpam-6011	270	2	3r	3r	NUM
ejpam-6011	270	3	and	and	CCONJ
ejpam-6011	270	4	r	r	NOUN
ejpam-6011	270	5	/	/	SYM
ejpam-6011	270	6	j(r	j(r	PROPN
ejpam-6011	270	7	)	)	PUNCT
ejpam-6011	270	8	=	=	SYM
ejpam-6011	271	1	m2(z3	m2(z3	NUM
ejpam-6011	271	2	)	)	PUNCT
ejpam-6011	271	3	.	.	PUNCT
ejpam-6011	272	1	also	also	ADV
ejpam-6011	272	2	,	,	PUNCT
ejpam-6011	272	3	r	r	NOUN
ejpam-6011	272	4	is	be	AUX
ejpam-6011	272	5	an	an	DET
ejpam-6011	272	6	njsemicommutative	njsemicommutative	ADJ
ejpam-6011	272	7	ring	ring	NOUN
ejpam-6011	272	8	and	and	CCONJ
ejpam-6011	272	9	consequently	consequently	ADV
ejpam-6011	272	10	nj	nj	PROPN
ejpam-6011	272	11	-	-	PUNCT
ejpam-6011	272	12	abelian	abelian	PROPN
ejpam-6011	272	13	.	.	PUNCT
ejpam-6011	273	1	on	on	ADP
ejpam-6011	273	2	the	the	DET
ejpam-6011	273	3	other	other	ADJ
ejpam-6011	273	4	hand	hand	NOUN
ejpam-6011	273	5	,	,	PUNCT
ejpam-6011	273	6	the	the	DET
ejpam-6011	273	7	idempotents	idempotent	NOUN
ejpam-6011	273	8	e	e	X
ejpam-6011	273	9	=	=	PUNCT
ejpam-6011	273	10	[	[	PUNCT
ejpam-6011	273	11	2	2	NUM
ejpam-6011	273	12	2	2	NUM
ejpam-6011	273	13	2	2	NUM
ejpam-6011	273	14	2	2	NUM
ejpam-6011	273	15	]	]	PUNCT
ejpam-6011	273	16	and	and	CCONJ
ejpam-6011	273	17	f	f	X
ejpam-6011	273	18	=	=	PUNCT
ejpam-6011	274	1	[	[	PUNCT
ejpam-6011	274	2	0	0	NUM
ejpam-6011	274	3	2	2	NUM
ejpam-6011	274	4	0	0	NUM
ejpam-6011	274	5	1	1	NUM
ejpam-6011	274	6	]	]	PUNCT
ejpam-6011	274	7	of	of	ADP
ejpam-6011	274	8	r	r	PROPN
ejpam-6011	274	9	/	/	SYM
ejpam-6011	274	10	j(r	j(r	PROPN
ejpam-6011	274	11	)	)	PUNCT
ejpam-6011	274	12	satisfy	satisfy	VERB
ejpam-6011	274	13	ef	ef	ADP
ejpam-6011	274	14	∈	∈	PROPN
ejpam-6011	274	15	n(r	n(r	NOUN
ejpam-6011	274	16	/	/	SYM
ejpam-6011	274	17	j(r	j(r	PROPN
ejpam-6011	274	18	)	)	PUNCT
ejpam-6011	274	19	)	)	PUNCT
ejpam-6011	275	1	while	while	SCONJ
ejpam-6011	275	2	e	e	X
ejpam-6011	275	3	[	[	PUNCT
ejpam-6011	275	4	1	1	NUM
ejpam-6011	275	5	1	1	NUM
ejpam-6011	275	6	2	2	NUM
ejpam-6011	275	7	1	1	NUM
ejpam-6011	275	8	]	]	PUNCT
ejpam-6011	275	9	f	f	PROPN
ejpam-6011	275	10	=[	=[	NOUN
ejpam-6011	275	11	0	0	NUM
ejpam-6011	275	12	1	1	NUM
ejpam-6011	275	13	0	0	NUM
ejpam-6011	275	14	1	1	NUM
ejpam-6011	275	15	]	]	PUNCT
ejpam-6011	275	16	̸∈	̸∈	PROPN
ejpam-6011	275	17	j(r	j(r	PROPN
ejpam-6011	275	18	/	/	SYM
ejpam-6011	275	19	j(r	j(r	PROPN
ejpam-6011	275	20	)	)	PUNCT
ejpam-6011	275	21	)	)	PUNCT
ejpam-6011	275	22	.	.	PUNCT
ejpam-6011	276	1	thus	thus	ADV
ejpam-6011	276	2	,	,	PUNCT
ejpam-6011	276	3	r	r	NOUN
ejpam-6011	276	4	/	/	SYM
ejpam-6011	276	5	j(r	j(r	PROPN
ejpam-6011	276	6	)	)	PUNCT
ejpam-6011	276	7	is	be	AUX
ejpam-6011	276	8	not	not	PART
ejpam-6011	276	9	nj	nj	NOUN
ejpam-6011	276	10	-	-	PUNCT
ejpam-6011	276	11	abelian	abelian	ADJ
ejpam-6011	276	12	.	.	PUNCT
ejpam-6011	277	1	proposition	proposition	NOUN
ejpam-6011	277	2	14	14	NUM
ejpam-6011	277	3	.	.	PUNCT
ejpam-6011	278	1	let	let	VERB
ejpam-6011	278	2	i	i	PRON
ejpam-6011	278	3	be	be	AUX
ejpam-6011	278	4	a	a	DET
ejpam-6011	278	5	nil	nil	ADJ
ejpam-6011	278	6	ideal	ideal	NOUN
ejpam-6011	278	7	of	of	ADP
ejpam-6011	278	8	r	r	NOUN
ejpam-6011	278	9	such	such	ADJ
ejpam-6011	278	10	that	that	PRON
ejpam-6011	278	11	r	r	NOUN
ejpam-6011	278	12	/	/	SYM
ejpam-6011	278	13	i	i	PRON
ejpam-6011	278	14	is	be	AUX
ejpam-6011	278	15	an	an	DET
ejpam-6011	278	16	nj	nj	ADJ
ejpam-6011	278	17	-	-	PUNCT
ejpam-6011	278	18	abelian	abelian	ADJ
ejpam-6011	278	19	ring	ring	NOUN
ejpam-6011	278	20	.	.	PUNCT
ejpam-6011	279	1	then	then	ADV
ejpam-6011	279	2	r	r	NOUN
ejpam-6011	279	3	is	be	AUX
ejpam-6011	279	4	nj	nj	NOUN
ejpam-6011	279	5	-	-	PUNCT
ejpam-6011	279	6	abelian	abelian	ADJ
ejpam-6011	279	7	.	.	PUNCT
ejpam-6011	280	1	proof	proof	NOUN
ejpam-6011	280	2	.	.	PUNCT
ejpam-6011	281	1	suppose	suppose	VERB
ejpam-6011	281	2	that	that	SCONJ
ejpam-6011	281	3	r	r	NOUN
ejpam-6011	281	4	/	/	SYM
ejpam-6011	281	5	i	i	PROPN
ejpam-6011	281	6	is	be	AUX
ejpam-6011	281	7	nj	nj	NOUN
ejpam-6011	281	8	-	-	PUNCT
ejpam-6011	281	9	abelian	abelian	ADJ
ejpam-6011	281	10	and	and	CCONJ
ejpam-6011	281	11	e	e	NOUN
ejpam-6011	281	12	=	=	PROPN
ejpam-6011	281	13	e2	e2	PROPN
ejpam-6011	281	14	,	,	PUNCT
ejpam-6011	281	15	a	a	DET
ejpam-6011	281	16	∈	∈	NOUN
ejpam-6011	281	17	r	r	NOUN
ejpam-6011	281	18	such	such	ADJ
ejpam-6011	281	19	that	that	SCONJ
ejpam-6011	281	20	ae	ae	PROPN
ejpam-6011	281	21	∈	∈	PROPN
ejpam-6011	281	22	n(r	n(r	PROPN
ejpam-6011	281	23	)	)	PUNCT
ejpam-6011	281	24	.	.	PUNCT
ejpam-6011	282	1	then	then	ADV
ejpam-6011	282	2	ae	ae	PROPN
ejpam-6011	282	3	∈	∈	PROPN
ejpam-6011	282	4	n(r	n(r	PROPN
ejpam-6011	282	5	/	/	SYM
ejpam-6011	282	6	i	i	PROPN
ejpam-6011	282	7	)	)	PUNCT
ejpam-6011	282	8	and	and	CCONJ
ejpam-6011	282	9	consequently	consequently	ADV
ejpam-6011	282	10	a(r	a(r	NOUN
ejpam-6011	282	11	/	/	SYM
ejpam-6011	282	12	i)e	i)e	ADJ
ejpam-6011	282	13	⊆	⊆	NUM
ejpam-6011	282	14	j(r	j(r	PROPN
ejpam-6011	282	15	/	/	SYM
ejpam-6011	282	16	i	i	PROPN
ejpam-6011	282	17	)	)	PUNCT
ejpam-6011	282	18	.	.	PUNCT
ejpam-6011	283	1	therefore	therefore	ADV
ejpam-6011	283	2	,	,	PUNCT
ejpam-6011	283	3	1−	1−	NUM
ejpam-6011	283	4	ares	are	NOUN
ejpam-6011	283	5	∈	∈	PROPN
ejpam-6011	283	6	u(r	u(r	PROPN
ejpam-6011	283	7	/	/	SYM
ejpam-6011	283	8	i	i	PROPN
ejpam-6011	283	9	)	)	PUNCT
ejpam-6011	283	10	,	,	PUNCT
ejpam-6011	283	11	for	for	ADP
ejpam-6011	283	12	every	every	DET
ejpam-6011	283	13	r	r	NOUN
ejpam-6011	283	14	,	,	PUNCT
ejpam-6011	283	15	s	s	PART
ejpam-6011	283	16	∈	∈	PROPN
ejpam-6011	283	17	r.	r.	NOUN
ejpam-6011	283	18	it	it	PRON
ejpam-6011	283	19	means	mean	VERB
ejpam-6011	283	20	that	that	SCONJ
ejpam-6011	283	21	1	1	NUM
ejpam-6011	283	22	−	−	NOUN
ejpam-6011	283	23	(	(	PUNCT
ejpam-6011	283	24	1	1	NUM
ejpam-6011	283	25	−	−	PROPN
ejpam-6011	283	26	ares)x	ares)x	PROPN
ejpam-6011	283	27	∈	∈	PROPN
ejpam-6011	283	28	i	i	NOUN
ejpam-6011	283	29	⊆	⊆	NUM
ejpam-6011	283	30	n(r	n(r	NUM
ejpam-6011	283	31	)	)	PUNCT
ejpam-6011	283	32	,	,	PUNCT
ejpam-6011	283	33	for	for	ADP
ejpam-6011	283	34	some	some	DET
ejpam-6011	283	35	x	x	SYM
ejpam-6011	283	36	∈	∈	PROPN
ejpam-6011	283	37	r.	r.	NOUN
ejpam-6011	283	38	thus	thus	ADV
ejpam-6011	283	39	(	(	PUNCT
ejpam-6011	283	40	1	1	NUM
ejpam-6011	283	41	−	−	PROPN
ejpam-6011	283	42	ares)x	ares)x	PROPN
ejpam-6011	283	43	is	be	AUX
ejpam-6011	283	44	a	a	DET
ejpam-6011	283	45	unit	unit	NOUN
ejpam-6011	283	46	in	in	ADP
ejpam-6011	283	47	r	r	NOUN
ejpam-6011	283	48	,	,	PUNCT
ejpam-6011	283	49	and	and	CCONJ
ejpam-6011	283	50	hence	hence	ADV
ejpam-6011	283	51	1	1	NUM
ejpam-6011	283	52	−	−	NOUN
ejpam-6011	283	53	ares	are	NOUN
ejpam-6011	283	54	has	have	VERB
ejpam-6011	283	55	a	a	DET
ejpam-6011	283	56	right	right	ADJ
ejpam-6011	283	57	inverse	inverse	NOUN
ejpam-6011	283	58	for	for	ADP
ejpam-6011	283	59	every	every	DET
ejpam-6011	283	60	r	r	NOUN
ejpam-6011	283	61	,	,	PUNCT
ejpam-6011	283	62	s	s	PROPN
ejpam-6011	283	63	∈	∈	PROPN
ejpam-6011	283	64	r.	r.	PROPN
ejpam-6011	283	65	therefore	therefore	ADV
ejpam-6011	283	66	,	,	PUNCT
ejpam-6011	283	67	are	be	AUX
ejpam-6011	283	68	⊆	⊆	NUM
ejpam-6011	283	69	j(r	j(r	NOUN
ejpam-6011	283	70	)	)	PUNCT
ejpam-6011	283	71	and	and	CCONJ
ejpam-6011	283	72	r	r	NOUN
ejpam-6011	283	73	are	be	AUX
ejpam-6011	283	74	nj	nj	NOUN
ejpam-6011	283	75	-	-	PUNCT
ejpam-6011	283	76	abelian	abelian	ADJ
ejpam-6011	283	77	.	.	PUNCT
ejpam-6011	284	1	4	4	X
ejpam-6011	284	2	.	.	NOUN
ejpam-6011	284	3	matrix	matrix	NOUN
ejpam-6011	284	4	extensions	extension	NOUN
ejpam-6011	284	5	of	of	ADP
ejpam-6011	284	6	nj	nj	PROPN
ejpam-6011	284	7	-	-	PUNCT
ejpam-6011	284	8	abelian	abelian	ADJ
ejpam-6011	284	9	rings	ring	NOUN
ejpam-6011	284	10	in	in	ADP
ejpam-6011	284	11	this	this	DET
ejpam-6011	284	12	section	section	NOUN
ejpam-6011	284	13	,	,	PUNCT
ejpam-6011	284	14	we	we	PRON
ejpam-6011	284	15	study	study	VERB
ejpam-6011	284	16	the	the	DET
ejpam-6011	284	17	nj	nj	PROPN
ejpam-6011	284	18	-	-	PUNCT
ejpam-6011	284	19	abelian	abelian	ADJ
ejpam-6011	284	20	property	property	NOUN
ejpam-6011	284	21	for	for	ADP
ejpam-6011	284	22	some	some	DET
ejpam-6011	284	23	ring	ring	NOUN
ejpam-6011	284	24	extensions	extension	NOUN
ejpam-6011	284	25	and	and	CCONJ
ejpam-6011	284	26	their	their	PRON
ejpam-6011	284	27	subrings	subring	NOUN
ejpam-6011	284	28	.	.	PUNCT
ejpam-6011	285	1	first	first	ADV
ejpam-6011	285	2	,	,	PUNCT
ejpam-6011	285	3	we	we	PRON
ejpam-6011	285	4	show	show	VERB
ejpam-6011	285	5	that	that	SCONJ
ejpam-6011	285	6	the	the	DET
ejpam-6011	285	7	matrix	matrix	NOUN
ejpam-6011	285	8	ring	ring	NOUN
ejpam-6011	285	9	mn(r	mn(r	NOUN
ejpam-6011	285	10	)	)	PUNCT
ejpam-6011	285	11	over	over	ADP
ejpam-6011	285	12	a	a	DET
ejpam-6011	285	13	ring	ring	NOUN
ejpam-6011	285	14	r	r	NOUN
ejpam-6011	285	15	is	be	AUX
ejpam-6011	285	16	not	not	PART
ejpam-6011	285	17	nj	nj	NOUN
ejpam-6011	285	18	-	-	NOUN
ejpam-6011	285	19	abelian	abelian	ADJ
ejpam-6011	285	20	for	for	ADP
ejpam-6011	285	21	any	any	DET
ejpam-6011	285	22	ring	ring	NOUN
ejpam-6011	285	23	r	r	NOUN
ejpam-6011	285	24	and	and	CCONJ
ejpam-6011	285	25	n	n	PRON
ejpam-6011	285	26	≥	≥	NOUN
ejpam-6011	285	27	2	2	NUM
ejpam-6011	285	28	.	.	PUNCT
ejpam-6011	285	29	proposition	proposition	NOUN
ejpam-6011	285	30	15	15	NUM
ejpam-6011	285	31	.	.	PUNCT
ejpam-6011	286	1	for	for	ADP
ejpam-6011	286	2	any	any	DET
ejpam-6011	286	3	ring	ring	NOUN
ejpam-6011	286	4	r	r	NOUN
ejpam-6011	286	5	and	and	CCONJ
ejpam-6011	286	6	integer	integer	PROPN
ejpam-6011	286	7	n	n	PRON
ejpam-6011	286	8	≥	≥	NUM
ejpam-6011	286	9	2	2	NUM
ejpam-6011	286	10	,	,	PUNCT
ejpam-6011	286	11	mn(r	mn(r	NUM
ejpam-6011	286	12	)	)	PUNCT
ejpam-6011	286	13	is	be	AUX
ejpam-6011	286	14	not	not	PART
ejpam-6011	286	15	nj	nj	NOUN
ejpam-6011	286	16	-	-	PUNCT
ejpam-6011	286	17	abelian	abelian	ADJ
ejpam-6011	286	18	.	.	PUNCT
ejpam-6011	287	1	m.	m.	PROPN
ejpam-6011	287	2	saad	saad	PROPN
ejpam-6011	287	3	,	,	PUNCT
ejpam-6011	287	4	s.	s.	PROPN
ejpam-6011	287	5	m.	m.	PROPN
ejpam-6011	287	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	287	7	/	/	SYM
ejpam-6011	287	8	eur	eur	PROPN
ejpam-6011	287	9	.	.	PUNCT
ejpam-6011	288	1	j.	j.	PROPN
ejpam-6011	288	2	pure	pure	PROPN
ejpam-6011	288	3	appl	appl	PROPN
ejpam-6011	288	4	.	.	PROPN
ejpam-6011	288	5	math	math	PROPN
ejpam-6011	288	6	,	,	PUNCT
ejpam-6011	288	7	18	18	NUM
ejpam-6011	288	8	(	(	PUNCT
ejpam-6011	288	9	2	2	NUM
ejpam-6011	288	10	)	)	PUNCT
ejpam-6011	288	11	(	(	PUNCT
ejpam-6011	288	12	2025	2025	NUM
ejpam-6011	288	13	)	)	PUNCT
ejpam-6011	288	14	,	,	PUNCT
ejpam-6011	288	15	6011	6011	NUM
ejpam-6011	288	16	9	9	NUM
ejpam-6011	288	17	of	of	ADP
ejpam-6011	288	18	13	13	NUM
ejpam-6011	288	19	proof	proof	NOUN
ejpam-6011	288	20	.	.	PUNCT
ejpam-6011	289	1	as	as	SCONJ
ejpam-6011	289	2	shown	show	VERB
ejpam-6011	289	3	in	in	ADP
ejpam-6011	289	4	example	example	NOUN
ejpam-6011	289	5	9	9	NUM
ejpam-6011	289	6	,	,	PUNCT
ejpam-6011	289	7	m2(r	m2(r	NOUN
ejpam-6011	289	8	)	)	PUNCT
ejpam-6011	289	9	is	be	AUX
ejpam-6011	289	10	not	not	PART
ejpam-6011	289	11	nj	nj	NOUN
ejpam-6011	289	12	-	-	NOUN
ejpam-6011	289	13	abelian	abelian	ADJ
ejpam-6011	289	14	for	for	ADP
ejpam-6011	289	15	any	any	DET
ejpam-6011	289	16	ring	ring	NOUN
ejpam-6011	289	17	r.	r.	PROPN
ejpam-6011	289	18	from	from	ADP
ejpam-6011	289	19	proposition	proposition	NOUN
ejpam-6011	289	20	10	10	NUM
ejpam-6011	289	21	,	,	PUNCT
ejpam-6011	289	22	every	every	DET
ejpam-6011	289	23	corner	corner	NOUN
ejpam-6011	289	24	of	of	ADP
ejpam-6011	289	25	an	an	DET
ejpam-6011	289	26	nj	nj	PROPN
ejpam-6011	289	27	-	-	PUNCT
ejpam-6011	289	28	abelian	abelian	ADJ
ejpam-6011	289	29	ring	ring	NOUN
ejpam-6011	289	30	is	be	AUX
ejpam-6011	289	31	nj	nj	NOUN
ejpam-6011	289	32	-	-	PUNCT
ejpam-6011	289	33	abelian	abelian	NOUN
ejpam-6011	289	34	.	.	PUNCT
ejpam-6011	290	1	by	by	ADP
ejpam-6011	290	2	induction	induction	NOUN
ejpam-6011	290	3	,	,	PUNCT
ejpam-6011	290	4	mn(r	mn(r	NUM
ejpam-6011	290	5	)	)	PUNCT
ejpam-6011	290	6	is	be	AUX
ejpam-6011	290	7	not	not	PART
ejpam-6011	290	8	nj	nj	NOUN
ejpam-6011	290	9	-	-	NOUN
ejpam-6011	290	10	abelian	abelian	ADJ
ejpam-6011	290	11	for	for	ADP
ejpam-6011	290	12	every	every	DET
ejpam-6011	290	13	n	n	PRON
ejpam-6011	290	14	≥	≥	NOUN
ejpam-6011	290	15	2	2	NUM
ejpam-6011	290	16	since	since	SCONJ
ejpam-6011	290	17	mn(r	mn(r	NOUN
ejpam-6011	290	18	)	)	PUNCT
ejpam-6011	290	19	is	be	AUX
ejpam-6011	290	20	a	a	DET
ejpam-6011	290	21	corner	corner	NOUN
ejpam-6011	290	22	of	of	ADP
ejpam-6011	290	23	mn+1(r	mn+1(r	PROPN
ejpam-6011	290	24	)	)	PUNCT
ejpam-6011	290	25	for	for	ADP
ejpam-6011	290	26	every	every	DET
ejpam-6011	290	27	n	n	PRON
ejpam-6011	290	28	≥	≥	NOUN
ejpam-6011	290	29	1	1	NUM
ejpam-6011	290	30	.	.	PUNCT
ejpam-6011	291	1	while	while	SCONJ
ejpam-6011	291	2	it	it	PRON
ejpam-6011	291	3	is	be	AUX
ejpam-6011	291	4	impossible	impossible	ADJ
ejpam-6011	291	5	to	to	PART
ejpam-6011	291	6	get	get	VERB
ejpam-6011	291	7	a	a	DET
ejpam-6011	291	8	matrix	matrix	NOUN
ejpam-6011	291	9	ring	ring	NOUN
ejpam-6011	291	10	satisfying	satisfy	VERB
ejpam-6011	291	11	the	the	DET
ejpam-6011	291	12	nj	nj	PROPN
ejpam-6011	291	13	-	-	PUNCT
ejpam-6011	291	14	abelian	abelian	ADJ
ejpam-6011	291	15	condition	condition	NOUN
ejpam-6011	291	16	,	,	PUNCT
ejpam-6011	291	17	as	as	SCONJ
ejpam-6011	291	18	shown	show	VERB
ejpam-6011	291	19	in	in	ADP
ejpam-6011	291	20	the	the	DET
ejpam-6011	291	21	example	example	NOUN
ejpam-6011	291	22	,	,	PUNCT
ejpam-6011	291	23	we	we	PRON
ejpam-6011	291	24	will	will	AUX
ejpam-6011	291	25	explore	explore	VERB
ejpam-6011	291	26	the	the	DET
ejpam-6011	291	27	extent	extent	NOUN
ejpam-6011	291	28	to	to	PART
ejpam-6011	291	29	which	which	PRON
ejpam-6011	291	30	this	this	DET
ejpam-6011	291	31	property	property	NOUN
ejpam-6011	291	32	holds	hold	VERB
ejpam-6011	291	33	in	in	ADP
ejpam-6011	291	34	some	some	PRON
ejpam-6011	291	35	of	of	ADP
ejpam-6011	291	36	its	its	PRON
ejpam-6011	291	37	subrings	subring	NOUN
ejpam-6011	291	38	or	or	CCONJ
ejpam-6011	291	39	certain	certain	ADJ
ejpam-6011	291	40	matrix	matrix	NOUN
ejpam-6011	291	41	contexts	contexts	NOUN
ejpam-6011	291	42	.	.	PUNCT
ejpam-6011	292	1	for	for	ADP
ejpam-6011	292	2	rings	ring	NOUN
ejpam-6011	292	3	r	r	NOUN
ejpam-6011	292	4	and	and	CCONJ
ejpam-6011	292	5	s	s	NOUN
ejpam-6011	292	6	,	,	PUNCT
ejpam-6011	292	7	letm	letm	NOUN
ejpam-6011	292	8	and	and	CCONJ
ejpam-6011	292	9	n	n	ADV
ejpam-6011	292	10	be	be	AUX
ejpam-6011	292	11	(	(	PUNCT
ejpam-6011	292	12	r	r	NOUN
ejpam-6011	292	13	,	,	PUNCT
ejpam-6011	292	14	s)-bimodule	s)-bimodule	NOUN
ejpam-6011	292	15	and	and	CCONJ
ejpam-6011	292	16	(	(	PUNCT
ejpam-6011	292	17	s	s	X
ejpam-6011	292	18	,	,	PUNCT
ejpam-6011	292	19	r)-bimodule	r)-bimodule	NOUN
ejpam-6011	292	20	,	,	PUNCT
ejpam-6011	292	21	respectively	respectively	ADV
ejpam-6011	292	22	.	.	PUNCT
ejpam-6011	293	1	the	the	DET
ejpam-6011	293	2	set	set	NOUN
ejpam-6011	293	3	of	of	ADP
ejpam-6011	293	4	all	all	DET
ejpam-6011	293	5	matrices	matrix	NOUN
ejpam-6011	293	6	of	of	ADP
ejpam-6011	293	7	the	the	DET
ejpam-6011	293	8	form	form	NOUN
ejpam-6011	293	9	[	[	PUNCT
ejpam-6011	293	10	r	r	NOUN
ejpam-6011	293	11	m	m	VERB
ejpam-6011	293	12	n	n	NOUN
ejpam-6011	293	13	s	s	X
ejpam-6011	293	14	]	]	PUNCT
ejpam-6011	293	15	,	,	PUNCT
ejpam-6011	293	16	where	where	SCONJ
ejpam-6011	293	17	r	r	NOUN
ejpam-6011	293	18	∈	∈	PROPN
ejpam-6011	293	19	r	r	NOUN
ejpam-6011	293	20	,	,	PUNCT
ejpam-6011	293	21	s	s	PART
ejpam-6011	293	22	∈	∈	PROPN
ejpam-6011	293	23	s	s	NOUN
ejpam-6011	293	24	,	,	PUNCT
ejpam-6011	293	25	m	m	VERB
ejpam-6011	293	26	∈	∈	ADJ
ejpam-6011	293	27	m	m	NOUN
ejpam-6011	293	28	,	,	PUNCT
ejpam-6011	293	29	and	and	CCONJ
ejpam-6011	293	30	n	n	PRON
ejpam-6011	293	31	∈	∈	PROPN
ejpam-6011	293	32	n	n	X
ejpam-6011	293	33	.	.	PUNCT
ejpam-6011	294	1	using	use	VERB
ejpam-6011	294	2	the	the	DET
ejpam-6011	294	3	standard	standard	ADJ
ejpam-6011	294	4	matrix	matrix	NOUN
ejpam-6011	294	5	addition	addition	NOUN
ejpam-6011	294	6	,	,	PUNCT
ejpam-6011	294	7	we	we	PRON
ejpam-6011	294	8	can	can	AUX
ejpam-6011	294	9	identify	identify	VERB
ejpam-6011	294	10	this	this	PRON
ejpam-6011	294	11	as	as	ADP
ejpam-6011	294	12	an	an	DET
ejpam-6011	294	13	abelian	abelian	ADJ
ejpam-6011	294	14	group	group	NOUN
ejpam-6011	294	15	.	.	PUNCT
ejpam-6011	295	1	to	to	PART
ejpam-6011	295	2	define	define	VERB
ejpam-6011	295	3	a	a	DET
ejpam-6011	295	4	matrix	matrix	NOUN
ejpam-6011	295	5	multiplication	multiplication	NOUN
ejpam-6011	295	6	for	for	ADP
ejpam-6011	295	7	these	these	DET
ejpam-6011	295	8	elements	element	NOUN
ejpam-6011	295	9	,	,	PUNCT
ejpam-6011	295	10	we	we	PRON
ejpam-6011	295	11	need	need	VERB
ejpam-6011	295	12	to	to	PART
ejpam-6011	295	13	define	define	VERB
ejpam-6011	295	14	the	the	DET
ejpam-6011	295	15	products	product	NOUN
ejpam-6011	295	16	of	of	ADP
ejpam-6011	295	17	mn	mn	PROPN
ejpam-6011	295	18	and	and	CCONJ
ejpam-6011	295	19	nm	nm	VERB
ejpam-6011	295	20	in	in	ADP
ejpam-6011	295	21	r	r	NOUN
ejpam-6011	295	22	and	and	CCONJ
ejpam-6011	295	23	s	s	NOUN
ejpam-6011	295	24	,	,	PUNCT
ejpam-6011	295	25	respectively	respectively	ADV
ejpam-6011	295	26	,	,	PUNCT
ejpam-6011	295	27	for	for	ADP
ejpam-6011	295	28	every	every	DET
ejpam-6011	295	29	m	m	NOUN
ejpam-6011	295	30	∈	∈	NOUN
ejpam-6011	295	31	m	m	NOUN
ejpam-6011	295	32	and	and	CCONJ
ejpam-6011	295	33	n	n	PRON
ejpam-6011	295	34	∈	∈	PROPN
ejpam-6011	295	35	n	n	NOUN
ejpam-6011	295	36	.	.	PUNCT
ejpam-6011	296	1	we	we	PRON
ejpam-6011	296	2	assume	assume	VERB
ejpam-6011	296	3	that	that	SCONJ
ejpam-6011	296	4	there	there	PRON
ejpam-6011	296	5	are	be	VERB
ejpam-6011	296	6	two	two	NUM
ejpam-6011	296	7	bimodule	bimodule	NOUN
ejpam-6011	296	8	homomorphisms	homomorphism	NOUN
ejpam-6011	296	9	ϕ	ϕ	NOUN
ejpam-6011	296	10	:	:	PUNCT
ejpam-6011	296	11	(	(	PUNCT
ejpam-6011	296	12	m	m	NOUN
ejpam-6011	296	13	,	,	PUNCT
ejpam-6011	296	14	n	n	CCONJ
ejpam-6011	296	15	)	)	PUNCT
ejpam-6011	296	16	→	→	SYM
ejpam-6011	296	17	r	r	NOUN
ejpam-6011	296	18	and	and	CCONJ
ejpam-6011	296	19	ψ(n	ψ(n	PROPN
ejpam-6011	296	20	,	,	PUNCT
ejpam-6011	296	21	m	m	NOUN
ejpam-6011	296	22	)	)	PUNCT
ejpam-6011	296	23	→	→	PUNCT
ejpam-6011	296	24	s.	s.	PROPN
ejpam-6011	296	25	simplify	simplify	VERB
ejpam-6011	296	26	mn	mn	PROPN
ejpam-6011	297	1	=	=	SYM
ejpam-6011	297	2	ϕ(m	ϕ(m	PROPN
ejpam-6011	297	3	,	,	PUNCT
ejpam-6011	297	4	n	n	CCONJ
ejpam-6011	297	5	)	)	PUNCT
ejpam-6011	297	6	and	and	CCONJ
ejpam-6011	297	7	nm	nm	NOUN
ejpam-6011	297	8	=	=	SYM
ejpam-6011	297	9	ψ(n	ψ(n	PROPN
ejpam-6011	297	10	,	,	PUNCT
ejpam-6011	297	11	m	m	NOUN
ejpam-6011	297	12	)	)	PUNCT
ejpam-6011	297	13	for	for	ADP
ejpam-6011	297	14	all	all	DET
ejpam-6011	297	15	m	m	NOUN
ejpam-6011	297	16	∈	∈	NOUN
ejpam-6011	297	17	m	m	NOUN
ejpam-6011	297	18	and	and	CCONJ
ejpam-6011	297	19	n	n	PRON
ejpam-6011	297	20	∈	∈	PROPN
ejpam-6011	297	21	n	n	NOUN
ejpam-6011	297	22	.	.	PUNCT
ejpam-6011	298	1	these	these	DET
ejpam-6011	298	2	maps	map	NOUN
ejpam-6011	298	3	satisfy	satisfy	VERB
ejpam-6011	298	4	the	the	DET
ejpam-6011	298	5	associativity	associativity	NOUN
ejpam-6011	298	6	conditions	condition	NOUN
ejpam-6011	298	7	that	that	PRON
ejpam-6011	298	8	are	be	AUX
ejpam-6011	298	9	required	require	VERB
ejpam-6011	298	10	to	to	PART
ejpam-6011	298	11	make	make	VERB
ejpam-6011	298	12	the	the	DET
ejpam-6011	298	13	set	set	NOUN
ejpam-6011	298	14	with	with	ADP
ejpam-6011	298	15	usual	usual	ADJ
ejpam-6011	298	16	matrix	matrix	NOUN
ejpam-6011	298	17	addition	addition	NOUN
ejpam-6011	298	18	and	and	CCONJ
ejpam-6011	298	19	context	context	NOUN
ejpam-6011	298	20	multiplication	multiplication	NOUN
ejpam-6011	298	21	an	an	DET
ejpam-6011	298	22	associative	associative	ADJ
ejpam-6011	298	23	ring	ring	NOUN
ejpam-6011	298	24	with	with	ADP
ejpam-6011	298	25	identity	identity	NOUN
ejpam-6011	298	26	,	,	PUNCT
ejpam-6011	298	27	notated	notate	VERB
ejpam-6011	298	28	by	by	ADP
ejpam-6011	298	29	[	[	PUNCT
ejpam-6011	298	30	r	r	NOUN
ejpam-6011	298	31	m	m	VERB
ejpam-6011	298	32	n	n	NOUN
ejpam-6011	298	33	s	s	X
ejpam-6011	298	34	]	]	PUNCT
ejpam-6011	298	35	.	.	PUNCT
ejpam-6011	299	1	this	this	DET
ejpam-6011	299	2	ring	ring	NOUN
ejpam-6011	299	3	is	be	AUX
ejpam-6011	299	4	called	call	VERB
ejpam-6011	299	5	the	the	DET
ejpam-6011	299	6	morita	morita	PROPN
ejpam-6011	299	7	context	context	PROPN
ejpam-6011	299	8	(	(	PUNCT
ejpam-6011	299	9	r	r	NOUN
ejpam-6011	299	10	,	,	PUNCT
ejpam-6011	299	11	m	m	PROPN
ejpam-6011	299	12	,	,	PUNCT
ejpam-6011	299	13	n	n	CCONJ
ejpam-6011	299	14	,	,	PUNCT
ejpam-6011	299	15	s	s	PROPN
ejpam-6011	299	16	,	,	PUNCT
ejpam-6011	299	17	ϕ	ϕ	NOUN
ejpam-6011	299	18	,	,	PUNCT
ejpam-6011	299	19	ψ	ψ	NOUN
ejpam-6011	299	20	)	)	PUNCT
ejpam-6011	299	21	,	,	PUNCT
ejpam-6011	299	22	or	or	CCONJ
ejpam-6011	299	23	a	a	DET
ejpam-6011	299	24	formal	formal	ADJ
ejpam-6011	299	25	matrix	matrix	NOUN
ejpam-6011	299	26	ring	ring	NOUN
ejpam-6011	299	27	(	(	PUNCT
ejpam-6011	299	28	of	of	ADP
ejpam-6011	299	29	order	order	NOUN
ejpam-6011	299	30	2	2	NUM
ejpam-6011	299	31	)	)	PUNCT
ejpam-6011	299	32	,	,	PUNCT
ejpam-6011	299	33	or	or	CCONJ
ejpam-6011	299	34	a	a	DET
ejpam-6011	299	35	ring	ring	NOUN
ejpam-6011	299	36	of	of	ADP
ejpam-6011	299	37	generalized	generalized	ADJ
ejpam-6011	299	38	matrices	matrix	NOUN
ejpam-6011	299	39	.	.	PUNCT
ejpam-6011	300	1	the	the	DET
ejpam-6011	300	2	readers	reader	NOUN
ejpam-6011	300	3	are	be	AUX
ejpam-6011	300	4	referred	refer	VERB
ejpam-6011	300	5	to	to	ADP
ejpam-6011	300	6	[	[	X
ejpam-6011	300	7	15–19	15–19	NUM
ejpam-6011	300	8	]	]	PUNCT
ejpam-6011	300	9	as	as	ADV
ejpam-6011	300	10	well	well	ADV
ejpam-6011	300	11	as	as	ADP
ejpam-6011	300	12	the	the	DET
ejpam-6011	300	13	references	reference	NOUN
ejpam-6011	300	14	there	there	ADV
ejpam-6011	300	15	for	for	ADP
ejpam-6011	300	16	detailed	detailed	ADJ
ejpam-6011	300	17	information	information	NOUN
ejpam-6011	300	18	on	on	ADP
ejpam-6011	300	19	the	the	DET
ejpam-6011	300	20	study	study	NOUN
ejpam-6011	300	21	in	in	ADP
ejpam-6011	300	22	morita	morita	PROPN
ejpam-6011	300	23	contexts	contexts	PROPN
ejpam-6011	300	24	.	.	PUNCT
ejpam-6011	301	1	recall	recall	VERB
ejpam-6011	302	1	[	[	X
ejpam-6011	302	2	20	20	NUM
ejpam-6011	302	3	]	]	PUNCT
ejpam-6011	302	4	,	,	PUNCT
ejpam-6011	302	5	a	a	DET
ejpam-6011	302	6	morita	morita	PROPN
ejpam-6011	302	7	context	context	PROPN
ejpam-6011	302	8	t	t	PROPN
ejpam-6011	302	9	=	=	PUNCT
ejpam-6011	303	1	[	[	PUNCT
ejpam-6011	303	2	r	r	NOUN
ejpam-6011	303	3	m	m	VERB
ejpam-6011	303	4	n	n	NUM
ejpam-6011	303	5	s	s	X
ejpam-6011	303	6	]	]	X
ejpam-6011	303	7	is	be	AUX
ejpam-6011	303	8	called	call	VERB
ejpam-6011	303	9	trivial	trivial	ADJ
ejpam-6011	303	10	if	if	SCONJ
ejpam-6011	303	11	mn	mn	PROPN
ejpam-6011	303	12	=	=	PROPN
ejpam-6011	303	13	0	0	PROPN
ejpam-6011	303	14	and	and	CCONJ
ejpam-6011	303	15	nm	nm	ADJ
ejpam-6011	304	1	=	=	NOUN
ejpam-6011	304	2	0	0	X
ejpam-6011	304	3	.	.	PUNCT
ejpam-6011	305	1	the	the	DET
ejpam-6011	305	2	following	follow	VERB
ejpam-6011	305	3	lemma	lemma	PROPN
ejpam-6011	305	4	describes	describe	VERB
ejpam-6011	305	5	the	the	DET
ejpam-6011	305	6	idempotents	idempotent	NOUN
ejpam-6011	305	7	,	,	PUNCT
ejpam-6011	305	8	nilpotent	nilpotent	ADJ
ejpam-6011	305	9	elements	element	NOUN
ejpam-6011	305	10	,	,	PUNCT
ejpam-6011	305	11	and	and	CCONJ
ejpam-6011	305	12	jacobson	jacobson	PROPN
ejpam-6011	305	13	radicals	radical	NOUN
ejpam-6011	305	14	in	in	ADP
ejpam-6011	305	15	a	a	DET
ejpam-6011	305	16	trivial	trivial	ADJ
ejpam-6011	305	17	morita	morita	NOUN
ejpam-6011	305	18	context	context	NOUN
ejpam-6011	305	19	.	.	PUNCT
ejpam-6011	306	1	lemma	lemma	PROPN
ejpam-6011	306	2	1	1	NUM
ejpam-6011	306	3	.	.	PUNCT
ejpam-6011	307	1	for	for	ADP
ejpam-6011	307	2	two	two	NUM
ejpam-6011	307	3	rings	ring	NOUN
ejpam-6011	307	4	,	,	PUNCT
ejpam-6011	307	5	r	r	NOUN
ejpam-6011	307	6	and	and	CCONJ
ejpam-6011	307	7	s	s	PROPN
ejpam-6011	307	8	,	,	PUNCT
ejpam-6011	307	9	let	let	VERB
ejpam-6011	307	10	t	t	NOUN
ejpam-6011	307	11	=	=	PUNCT
ejpam-6011	308	1	[	[	PUNCT
ejpam-6011	308	2	r	r	NOUN
ejpam-6011	308	3	m	m	VERB
ejpam-6011	308	4	n	n	NUM
ejpam-6011	308	5	s	s	X
ejpam-6011	308	6	]	]	X
ejpam-6011	308	7	be	be	AUX
ejpam-6011	308	8	a	a	DET
ejpam-6011	308	9	trivial	trivial	ADJ
ejpam-6011	308	10	morita	morita	NOUN
ejpam-6011	308	11	context	context	NOUN
ejpam-6011	308	12	.	.	PUNCT
ejpam-6011	309	1	then	then	ADV
ejpam-6011	309	2	we	we	PRON
ejpam-6011	309	3	have	have	VERB
ejpam-6011	309	4	the	the	DET
ejpam-6011	309	5	following	following	ADJ
ejpam-6011	309	6	descriptions	description	NOUN
ejpam-6011	309	7	:	:	PUNCT
ejpam-6011	309	8	(	(	PUNCT
ejpam-6011	309	9	i	i	NOUN
ejpam-6011	309	10	)	)	PUNCT
ejpam-6011	309	11	i(t	i(t	NOUN
ejpam-6011	309	12	)	)	PUNCT
ejpam-6011	310	1	⊆	⊆	NUM
ejpam-6011	310	2	[	[	PUNCT
ejpam-6011	310	3	i(r	i(r	PROPN
ejpam-6011	310	4	)	)	PUNCT
ejpam-6011	310	5	m	m	VERB
ejpam-6011	310	6	n	n	PRON
ejpam-6011	310	7	i(s	i(s	NOUN
ejpam-6011	310	8	)	)	PUNCT
ejpam-6011	310	9	]	]	PUNCT
ejpam-6011	310	10	;	;	PUNCT
ejpam-6011	310	11	(	(	PUNCT
ejpam-6011	310	12	ii	ii	NOUN
ejpam-6011	310	13	)	)	PUNCT
ejpam-6011	310	14	n(t	n(t	PROPN
ejpam-6011	310	15	)	)	PUNCT
ejpam-6011	310	16	=	=	PUNCT
ejpam-6011	310	17	[	[	PUNCT
ejpam-6011	310	18	n(r	n(r	NOUN
ejpam-6011	310	19	)	)	PUNCT
ejpam-6011	310	20	m	m	PROPN
ejpam-6011	310	21	n	n	PRON
ejpam-6011	310	22	n(s	n(s	NOUN
ejpam-6011	310	23	)	)	PUNCT
ejpam-6011	310	24	]	]	PUNCT
ejpam-6011	310	25	;	;	PUNCT
ejpam-6011	310	26	(	(	PUNCT
ejpam-6011	310	27	iii	iii	X
ejpam-6011	310	28	)	)	PUNCT
ejpam-6011	310	29	j(t	j(t	PROPN
ejpam-6011	310	30	)	)	PUNCT
ejpam-6011	311	1	=	=	PUNCT
ejpam-6011	311	2	[	[	PUNCT
ejpam-6011	311	3	j(r	j(r	PROPN
ejpam-6011	311	4	)	)	PUNCT
ejpam-6011	311	5	m	m	PROPN
ejpam-6011	311	6	n	n	PRON
ejpam-6011	311	7	j(s	j(s	NOUN
ejpam-6011	311	8	)	)	PUNCT
ejpam-6011	311	9	]	]	PUNCT
ejpam-6011	311	10	.	.	PUNCT
ejpam-6011	312	1	proof	proof	NOUN
ejpam-6011	312	2	.	.	PUNCT
ejpam-6011	313	1	the	the	DET
ejpam-6011	313	2	proofs	proof	NOUN
ejpam-6011	313	3	of	of	ADP
ejpam-6011	313	4	(	(	PUNCT
ejpam-6011	313	5	i	i	NOUN
ejpam-6011	313	6	)	)	PUNCT
ejpam-6011	313	7	and	and	CCONJ
ejpam-6011	313	8	(	(	PUNCT
ejpam-6011	313	9	ii	ii	NOUN
ejpam-6011	313	10	)	)	PUNCT
ejpam-6011	313	11	are	be	AUX
ejpam-6011	313	12	straightforward	straightforward	ADJ
ejpam-6011	313	13	,	,	PUNCT
ejpam-6011	313	14	while	while	SCONJ
ejpam-6011	313	15	(	(	PUNCT
ejpam-6011	313	16	iii	iii	X
ejpam-6011	313	17	)	)	PUNCT
ejpam-6011	313	18	is	be	AUX
ejpam-6011	313	19	obtained	obtain	VERB
ejpam-6011	313	20	directly	directly	ADV
ejpam-6011	313	21	from	from	ADP
ejpam-6011	313	22	[	[	X
ejpam-6011	313	23	21	21	NUM
ejpam-6011	313	24	,	,	PUNCT
ejpam-6011	313	25	lemma	lemma	PROPN
ejpam-6011	313	26	3.1	3.1	NUM
ejpam-6011	313	27	]	]	PUNCT
ejpam-6011	313	28	.	.	PUNCT
ejpam-6011	314	1	proposition	proposition	NOUN
ejpam-6011	314	2	16	16	NUM
ejpam-6011	314	3	.	.	PUNCT
ejpam-6011	314	4	suppose	suppose	VERB
ejpam-6011	314	5	that	that	SCONJ
ejpam-6011	314	6	t	t	NOUN
ejpam-6011	314	7	=	=	PUNCT
ejpam-6011	315	1	[	[	PUNCT
ejpam-6011	315	2	r	r	NOUN
ejpam-6011	315	3	m	m	VERB
ejpam-6011	315	4	n	n	PRON
ejpam-6011	315	5	r	r	NOUN
ejpam-6011	315	6	]	]	PUNCT
ejpam-6011	315	7	is	be	AUX
ejpam-6011	315	8	a	a	DET
ejpam-6011	315	9	trivial	trivial	ADJ
ejpam-6011	315	10	morita	morita	NOUN
ejpam-6011	315	11	context	context	NOUN
ejpam-6011	315	12	.	.	PUNCT
ejpam-6011	316	1	then	then	ADV
ejpam-6011	316	2	r	r	NOUN
ejpam-6011	316	3	is	be	AUX
ejpam-6011	316	4	nj	nj	NOUN
ejpam-6011	316	5	-	-	PUNCT
ejpam-6011	316	6	abelian	abelian	ADJ
ejpam-6011	317	1	if	if	SCONJ
ejpam-6011	317	2	and	and	CCONJ
ejpam-6011	317	3	only	only	ADV
ejpam-6011	317	4	if	if	SCONJ
ejpam-6011	317	5	m	m	VERB
ejpam-6011	317	6	and	and	CCONJ
ejpam-6011	317	7	n	n	PROPN
ejpam-6011	317	8	are	be	AUX
ejpam-6011	317	9	nj	nj	NOUN
ejpam-6011	317	10	-	-	PUNCT
ejpam-6011	317	11	abelian	abelian	NOUN
ejpam-6011	317	12	.	.	PUNCT
ejpam-6011	318	1	m.	m.	PROPN
ejpam-6011	318	2	saad	saad	PROPN
ejpam-6011	318	3	,	,	PUNCT
ejpam-6011	318	4	s.	s.	PROPN
ejpam-6011	318	5	m.	m.	PROPN
ejpam-6011	318	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	318	7	/	/	SYM
ejpam-6011	318	8	eur	eur	PROPN
ejpam-6011	318	9	.	.	PUNCT
ejpam-6011	319	1	j.	j.	PROPN
ejpam-6011	319	2	pure	pure	PROPN
ejpam-6011	319	3	appl	appl	PROPN
ejpam-6011	319	4	.	.	PROPN
ejpam-6011	319	5	math	math	PROPN
ejpam-6011	319	6	,	,	PUNCT
ejpam-6011	319	7	18	18	NUM
ejpam-6011	319	8	(	(	PUNCT
ejpam-6011	319	9	2	2	NUM
ejpam-6011	319	10	)	)	PUNCT
ejpam-6011	319	11	(	(	PUNCT
ejpam-6011	319	12	2025	2025	NUM
ejpam-6011	319	13	)	)	PUNCT
ejpam-6011	319	14	,	,	PUNCT
ejpam-6011	319	15	6011	6011	NUM
ejpam-6011	319	16	10	10	NUM
ejpam-6011	319	17	of	of	ADP
ejpam-6011	319	18	13	13	NUM
ejpam-6011	319	19	proof	proof	NOUN
ejpam-6011	319	20	.	.	PUNCT
ejpam-6011	320	1	the	the	DET
ejpam-6011	320	2	sufficiency	sufficiency	NOUN
ejpam-6011	320	3	is	be	AUX
ejpam-6011	320	4	straightforward	straightforward	ADJ
ejpam-6011	320	5	from	from	ADP
ejpam-6011	320	6	proposition	proposition	NOUN
ejpam-6011	320	7	10	10	NUM
ejpam-6011	320	8	.	.	PUNCT
ejpam-6011	321	1	for	for	ADP
ejpam-6011	321	2	the	the	DET
ejpam-6011	321	3	necessity	necessity	NOUN
ejpam-6011	321	4	,	,	PUNCT
ejpam-6011	321	5	assume	assume	VERB
ejpam-6011	321	6	that	that	SCONJ
ejpam-6011	321	7	both	both	DET
ejpam-6011	321	8	m	m	VERB
ejpam-6011	321	9	and	and	CCONJ
ejpam-6011	321	10	n	n	PROPN
ejpam-6011	321	11	are	be	AUX
ejpam-6011	321	12	nj	nj	NOUN
ejpam-6011	321	13	-	-	PUNCT
ejpam-6011	321	14	abelian	abelian	ADJ
ejpam-6011	321	15	.	.	PUNCT
ejpam-6011	322	1	let	let	VERB
ejpam-6011	322	2	ϵ	ϵ	X
ejpam-6011	323	1	=	=	PUNCT
ejpam-6011	324	1	[	[	PUNCT
ejpam-6011	324	2	e	e	X
ejpam-6011	324	3	m	m	VERB
ejpam-6011	324	4	n	n	PRON
ejpam-6011	324	5	f	f	X
ejpam-6011	324	6	]	]	PUNCT
ejpam-6011	324	7	be	be	AUX
ejpam-6011	324	8	an	an	DET
ejpam-6011	324	9	idempotent	idempotent	NOUN
ejpam-6011	324	10	of	of	ADP
ejpam-6011	324	11	t	t	PROPN
ejpam-6011	324	12	and	and	CCONJ
ejpam-6011	324	13	α	α	NOUN
ejpam-6011	324	14	=	=	PUNCT
ejpam-6011	325	1	[	[	PUNCT
ejpam-6011	325	2	a	a	X
ejpam-6011	325	3	x	x	X
ejpam-6011	325	4	y	y	PROPN
ejpam-6011	325	5	b	b	PROPN
ejpam-6011	325	6	]	]	PUNCT
ejpam-6011	325	7	be	be	AUX
ejpam-6011	325	8	an	an	DET
ejpam-6011	325	9	arbitrary	arbitrary	ADJ
ejpam-6011	325	10	element	element	NOUN
ejpam-6011	325	11	of	of	ADP
ejpam-6011	325	12	t	t	PROPN
ejpam-6011	325	13	.	.	PUNCT
ejpam-6011	326	1	so	so	ADV
ejpam-6011	326	2	,	,	PUNCT
ejpam-6011	326	3	e	e	PROPN
ejpam-6011	326	4	∈	∈	PROPN
ejpam-6011	326	5	i(r	i(r	PROPN
ejpam-6011	326	6	)	)	PUNCT
ejpam-6011	326	7	and	and	CCONJ
ejpam-6011	326	8	f	f	PROPN
ejpam-6011	326	9	∈	∈	PROPN
ejpam-6011	326	10	i(s	i(s	PROPN
ejpam-6011	326	11	)	)	PUNCT
ejpam-6011	326	12	,	,	PUNCT
ejpam-6011	326	13	from	from	ADP
ejpam-6011	326	14	the	the	DET
ejpam-6011	326	15	previous	previous	ADJ
ejpam-6011	326	16	lemma	lemma	PROPN
ejpam-6011	326	17	.	.	PUNCT
ejpam-6011	327	1	if	if	SCONJ
ejpam-6011	327	2	ϵα	ϵα	PROPN
ejpam-6011	327	3	∈	∈	PROPN
ejpam-6011	327	4	n(t	n(t	PROPN
ejpam-6011	327	5	)	)	PUNCT
ejpam-6011	327	6	,	,	PUNCT
ejpam-6011	327	7	then	then	ADV
ejpam-6011	327	8	ea	ea	NOUN
ejpam-6011	327	9	∈	∈	PROPN
ejpam-6011	327	10	n(r	n(r	NOUN
ejpam-6011	327	11	)	)	PUNCT
ejpam-6011	327	12	and	and	CCONJ
ejpam-6011	327	13	fb	fb	INTJ
ejpam-6011	327	14	∈	∈	PROPN
ejpam-6011	327	15	n(s	n(s	PROPN
ejpam-6011	327	16	)	)	PUNCT
ejpam-6011	327	17	.	.	PUNCT
ejpam-6011	328	1	so	so	ADV
ejpam-6011	328	2	,	,	PUNCT
ejpam-6011	328	3	era	era	NOUN
ejpam-6011	328	4	⊆	⊆	NUM
ejpam-6011	328	5	j(r	j(r	NOUN
ejpam-6011	328	6	)	)	PUNCT
ejpam-6011	328	7	and	and	CCONJ
ejpam-6011	328	8	frb	frb	PROPN
ejpam-6011	328	9	⊆	⊆	NUM
ejpam-6011	328	10	j(s	j(s	PROPN
ejpam-6011	328	11	)	)	PUNCT
ejpam-6011	328	12	from	from	ADP
ejpam-6011	328	13	the	the	DET
ejpam-6011	328	14	nj	nj	PROPN
ejpam-6011	328	15	-	-	PUNCT
ejpam-6011	328	16	abelianity	abelianity	NOUN
ejpam-6011	328	17	of	of	ADP
ejpam-6011	328	18	r	r	NOUN
ejpam-6011	328	19	and	and	CCONJ
ejpam-6011	328	20	s.	s.	PROPN
ejpam-6011	328	21	now	now	ADV
ejpam-6011	328	22	,	,	PUNCT
ejpam-6011	328	23	ϵtα	ϵtα	NOUN
ejpam-6011	328	24	=	=	PUNCT
ejpam-6011	328	25	[	[	PUNCT
ejpam-6011	328	26	era	era	NOUN
ejpam-6011	328	27	erx	erx	NOUN
ejpam-6011	328	28	fny	fny	NOUN
ejpam-6011	328	29	fsb	fsb	VERB
ejpam-6011	328	30	]	]	PUNCT
ejpam-6011	328	31	⊆	⊆	NUM
ejpam-6011	328	32	[	[	PUNCT
ejpam-6011	328	33	j(r	j(r	PROPN
ejpam-6011	328	34	)	)	PUNCT
ejpam-6011	328	35	m	m	PROPN
ejpam-6011	328	36	n	n	PRON
ejpam-6011	328	37	j(s	j(s	NOUN
ejpam-6011	328	38	)	)	PUNCT
ejpam-6011	328	39	]	]	PUNCT
ejpam-6011	329	1	=	=	SYM
ejpam-6011	329	2	j(t	j(t	PROPN
ejpam-6011	329	3	)	)	PUNCT
ejpam-6011	329	4	and	and	CCONJ
ejpam-6011	329	5	t	t	PROPN
ejpam-6011	329	6	is	be	AUX
ejpam-6011	329	7	an	an	DET
ejpam-6011	329	8	nj	nj	ADJ
ejpam-6011	329	9	-	-	PUNCT
ejpam-6011	329	10	abelian	abelian	ADJ
ejpam-6011	329	11	ring	ring	NOUN
ejpam-6011	329	12	.	.	PUNCT
ejpam-6011	330	1	notice	notice	VERB
ejpam-6011	330	2	that	that	SCONJ
ejpam-6011	330	3	formal	formal	ADJ
ejpam-6011	330	4	triangular	triangular	NOUN
ejpam-6011	330	5	matrix	matrix	NOUN
ejpam-6011	330	6	rings	ring	NOUN
ejpam-6011	330	7	are	be	AUX
ejpam-6011	330	8	obvious	obvious	ADJ
ejpam-6011	330	9	examples	example	NOUN
ejpam-6011	330	10	of	of	ADP
ejpam-6011	330	11	trivial	trivial	ADJ
ejpam-6011	330	12	morita	morita	NOUN
ejpam-6011	330	13	contexts	contexts	NOUN
ejpam-6011	330	14	.	.	PUNCT
ejpam-6011	331	1	so	so	ADV
ejpam-6011	331	2	,	,	PUNCT
ejpam-6011	331	3	we	we	PRON
ejpam-6011	331	4	have	have	VERB
ejpam-6011	331	5	the	the	DET
ejpam-6011	331	6	following	follow	VERB
ejpam-6011	331	7	corollary	corollary	NOUN
ejpam-6011	331	8	.	.	PUNCT
ejpam-6011	332	1	corollary	corollary	ADJ
ejpam-6011	332	2	7	7	NUM
ejpam-6011	332	3	.	.	PUNCT
ejpam-6011	333	1	let	let	VERB
ejpam-6011	333	2	m	m	PRON
ejpam-6011	333	3	represent	represent	VERB
ejpam-6011	333	4	an	an	DET
ejpam-6011	333	5	(	(	PUNCT
ejpam-6011	333	6	r	r	NOUN
ejpam-6011	333	7	,	,	PUNCT
ejpam-6011	333	8	s)-bimodule	s)-bimodule	NOUN
ejpam-6011	333	9	for	for	ADP
ejpam-6011	333	10	the	the	DET
ejpam-6011	333	11	rings	ring	NOUN
ejpam-6011	333	12	r	r	NOUN
ejpam-6011	333	13	and	and	CCONJ
ejpam-6011	333	14	s.	s.	PROPN
ejpam-6011	334	1	then	then	ADV
ejpam-6011	334	2	t	t	PROPN
ejpam-6011	334	3	=[	=[	NOUN
ejpam-6011	334	4	r	r	NOUN
ejpam-6011	334	5	m	m	VERB
ejpam-6011	334	6	0	0	NUM
ejpam-6011	334	7	s	s	PART
ejpam-6011	334	8	]	]	X
ejpam-6011	334	9	is	be	AUX
ejpam-6011	334	10	nj	nj	NOUN
ejpam-6011	334	11	-	-	PUNCT
ejpam-6011	334	12	abelian	abelian	ADJ
ejpam-6011	335	1	if	if	SCONJ
ejpam-6011	335	2	and	and	CCONJ
ejpam-6011	335	3	only	only	ADV
ejpam-6011	335	4	if	if	SCONJ
ejpam-6011	335	5	both	both	DET
ejpam-6011	335	6	r	r	NOUN
ejpam-6011	335	7	and	and	CCONJ
ejpam-6011	335	8	s	s	NOUN
ejpam-6011	335	9	are	be	AUX
ejpam-6011	335	10	nj	nj	NOUN
ejpam-6011	335	11	-	-	PUNCT
ejpam-6011	335	12	abelian	abelian	ADJ
ejpam-6011	335	13	.	.	PUNCT
ejpam-6011	336	1	in	in	ADP
ejpam-6011	336	2	[	[	X
ejpam-6011	336	3	22	22	NUM
ejpam-6011	336	4	]	]	PUNCT
ejpam-6011	336	5	,	,	PUNCT
ejpam-6011	336	6	if	if	SCONJ
ejpam-6011	336	7	r	r	NOUN
ejpam-6011	336	8	is	be	AUX
ejpam-6011	336	9	a	a	DET
ejpam-6011	336	10	ring	ring	NOUN
ejpam-6011	336	11	and	and	CCONJ
ejpam-6011	336	12	m	m	NOUN
ejpam-6011	336	13	is	be	AUX
ejpam-6011	336	14	an	an	DET
ejpam-6011	336	15	(	(	PUNCT
ejpam-6011	336	16	r	r	NOUN
ejpam-6011	336	17	,	,	PUNCT
ejpam-6011	336	18	r)-bimodule	r)-bimodule	NOUN
ejpam-6011	336	19	,	,	PUNCT
ejpam-6011	336	20	then	then	ADV
ejpam-6011	336	21	the	the	DET
ejpam-6011	336	22	direct	direct	ADJ
ejpam-6011	336	23	sum	sum	NOUN
ejpam-6011	336	24	of	of	ADP
ejpam-6011	336	25	abelian	abelian	ADJ
ejpam-6011	336	26	groups	group	NOUN
ejpam-6011	336	27	r	r	NOUN
ejpam-6011	336	28	and	and	CCONJ
ejpam-6011	336	29	m	m	VERB
ejpam-6011	336	30	with	with	ADP
ejpam-6011	336	31	standard	standard	ADJ
ejpam-6011	336	32	addition	addition	NOUN
ejpam-6011	336	33	and	and	CCONJ
ejpam-6011	336	34	multiplication	multiplication	NOUN
ejpam-6011	336	35	defined	define	VERB
ejpam-6011	336	36	as	as	ADP
ejpam-6011	336	37	(	(	PUNCT
ejpam-6011	336	38	r1,m1)(r2,m2	r1,m1)(r2,m2	NOUN
ejpam-6011	336	39	)	)	PUNCT
ejpam-6011	337	1	=	=	PRON
ejpam-6011	337	2	(	(	PUNCT
ejpam-6011	337	3	r1r2	r1r2	ADJ
ejpam-6011	337	4	,	,	PUNCT
ejpam-6011	337	5	r1m2	r1m2	X
ejpam-6011	337	6	+	+	X
ejpam-6011	337	7	m1r2	m1r2	VERB
ejpam-6011	337	8	)	)	PUNCT
ejpam-6011	337	9	,	,	PUNCT
ejpam-6011	337	10	for	for	ADP
ejpam-6011	337	11	all	all	DET
ejpam-6011	337	12	r1	r1	NOUN
ejpam-6011	337	13	,	,	PUNCT
ejpam-6011	337	14	r2	r2	PROPN
ejpam-6011	337	15	∈	∈	PROPN
ejpam-6011	337	16	r	r	NOUN
ejpam-6011	337	17	,	,	PUNCT
ejpam-6011	337	18	m1,m2	m1,m2	PROPN
ejpam-6011	337	19	∈	∈	PROPN
ejpam-6011	337	20	m	m	PROPN
ejpam-6011	337	21	,	,	PUNCT
ejpam-6011	337	22	is	be	AUX
ejpam-6011	337	23	a	a	DET
ejpam-6011	337	24	ring	ring	NOUN
ejpam-6011	337	25	with	with	ADP
ejpam-6011	337	26	the	the	DET
ejpam-6011	337	27	identity	identity	NOUN
ejpam-6011	337	28	(	(	PUNCT
ejpam-6011	337	29	1	1	NUM
ejpam-6011	337	30	,	,	PUNCT
ejpam-6011	337	31	0	0	NUM
ejpam-6011	337	32	)	)	PUNCT
ejpam-6011	337	33	.	.	PUNCT
ejpam-6011	338	1	this	this	DET
ejpam-6011	338	2	ring	ring	NOUN
ejpam-6011	338	3	is	be	AUX
ejpam-6011	338	4	the	the	DET
ejpam-6011	338	5	trivial	trivial	ADJ
ejpam-6011	338	6	extension	extension	NOUN
ejpam-6011	338	7	of	of	ADP
ejpam-6011	338	8	r	r	NOUN
ejpam-6011	338	9	by	by	ADP
ejpam-6011	338	10	m	m	PROPN
ejpam-6011	338	11	,	,	PUNCT
ejpam-6011	338	12	denoted	denote	VERB
ejpam-6011	338	13	by	by	ADP
ejpam-6011	338	14	t	t	PROPN
ejpam-6011	338	15	(	(	PUNCT
ejpam-6011	338	16	r	r	NOUN
ejpam-6011	338	17	,	,	PUNCT
ejpam-6011	338	18	m	m	NOUN
ejpam-6011	338	19	)	)	PUNCT
ejpam-6011	338	20	.	.	PUNCT
ejpam-6011	339	1	notice	notice	VERB
ejpam-6011	339	2	that	that	SCONJ
ejpam-6011	339	3	the	the	DET
ejpam-6011	339	4	trivial	trivial	ADJ
ejpam-6011	339	5	extension	extension	NOUN
ejpam-6011	339	6	of	of	ADP
ejpam-6011	339	7	r	r	NOUN
ejpam-6011	339	8	by	by	ADP
ejpam-6011	339	9	m	m	NOUN
ejpam-6011	339	10	is	be	AUX
ejpam-6011	339	11	that	that	SCONJ
ejpam-6011	339	12	the	the	DET
ejpam-6011	339	13	formal	formal	ADJ
ejpam-6011	339	14	triangular	triangular	NOUN
ejpam-6011	339	15	matrix	matrix	NOUN
ejpam-6011	339	16	ring	ring	NOUN
ejpam-6011	339	17	[	[	PUNCT
ejpam-6011	339	18	r	r	NOUN
ejpam-6011	339	19	m	m	VERB
ejpam-6011	339	20	0	0	NUM
ejpam-6011	339	21	r	r	NOUN
ejpam-6011	339	22	]	]	PUNCT
ejpam-6011	339	23	.	.	PUNCT
ejpam-6011	340	1	so	so	ADV
ejpam-6011	340	2	,	,	PUNCT
ejpam-6011	340	3	we	we	PRON
ejpam-6011	340	4	have	have	VERB
ejpam-6011	340	5	the	the	DET
ejpam-6011	340	6	following	follow	VERB
ejpam-6011	340	7	corollary	corollary	NOUN
ejpam-6011	340	8	.	.	PUNCT
ejpam-6011	341	1	corollary	corollary	ADJ
ejpam-6011	341	2	8	8	NUM
ejpam-6011	341	3	.	.	PUNCT
ejpam-6011	341	4	letm	letm	PROPN
ejpam-6011	341	5	represent	represent	VERB
ejpam-6011	341	6	an	an	DET
ejpam-6011	341	7	(	(	PUNCT
ejpam-6011	341	8	r	r	NOUN
ejpam-6011	341	9	,	,	PUNCT
ejpam-6011	341	10	r)-bimodule	r)-bimodule	NOUN
ejpam-6011	341	11	associated	associate	VERB
ejpam-6011	341	12	with	with	ADP
ejpam-6011	341	13	a	a	DET
ejpam-6011	341	14	ring	ring	NOUN
ejpam-6011	341	15	r.	r.	PROPN
ejpam-6011	341	16	then	then	ADV
ejpam-6011	341	17	t	t	PROPN
ejpam-6011	341	18	(	(	PUNCT
ejpam-6011	341	19	r	r	NOUN
ejpam-6011	341	20	,	,	PUNCT
ejpam-6011	341	21	m	m	NOUN
ejpam-6011	341	22	)	)	PUNCT
ejpam-6011	341	23	is	be	AUX
ejpam-6011	341	24	nj	nj	NOUN
ejpam-6011	341	25	-	-	PUNCT
ejpam-6011	341	26	abelian	abelian	ADJ
ejpam-6011	342	1	if	if	SCONJ
ejpam-6011	342	2	and	and	CCONJ
ejpam-6011	342	3	only	only	ADV
ejpam-6011	342	4	if	if	SCONJ
ejpam-6011	342	5	r	r	NOUN
ejpam-6011	342	6	is	be	AUX
ejpam-6011	342	7	nj	nj	NOUN
ejpam-6011	342	8	-	-	PUNCT
ejpam-6011	342	9	abelian	abelian	NOUN
ejpam-6011	342	10	.	.	PUNCT
ejpam-6011	343	1	proposition	proposition	NOUN
ejpam-6011	343	2	17	17	NUM
ejpam-6011	343	3	.	.	PUNCT
ejpam-6011	344	1	for	for	ADP
ejpam-6011	344	2	any	any	DET
ejpam-6011	344	3	ring	ring	NOUN
ejpam-6011	344	4	r	r	NOUN
ejpam-6011	344	5	,	,	PUNCT
ejpam-6011	344	6	the	the	DET
ejpam-6011	344	7	following	follow	VERB
ejpam-6011	344	8	conditions	condition	NOUN
ejpam-6011	344	9	are	be	AUX
ejpam-6011	344	10	equivalent	equivalent	ADJ
ejpam-6011	344	11	.	.	PUNCT
ejpam-6011	345	1	(	(	PUNCT
ejpam-6011	345	2	i	i	NOUN
ejpam-6011	345	3	)	)	PUNCT
ejpam-6011	345	4	r	r	NOUN
ejpam-6011	345	5	is	be	AUX
ejpam-6011	345	6	nj	nj	NOUN
ejpam-6011	345	7	-	-	PUNCT
ejpam-6011	345	8	abelian	abelian	ADJ
ejpam-6011	345	9	;	;	PUNCT
ejpam-6011	345	10	(	(	PUNCT
ejpam-6011	345	11	ii	ii	NOUN
ejpam-6011	345	12	)	)	PUNCT
ejpam-6011	345	13	t	t	PROPN
ejpam-6011	345	14	n(r	n(r	PROPN
ejpam-6011	345	15	)	)	PUNCT
ejpam-6011	345	16	is	be	AUX
ejpam-6011	345	17	nj	nj	NOUN
ejpam-6011	345	18	-	-	PUNCT
ejpam-6011	345	19	abelian	abelian	ADJ
ejpam-6011	345	20	for	for	ADP
ejpam-6011	345	21	some	some	DET
ejpam-6011	345	22	n	n	NOUN
ejpam-6011	345	23	>	>	X
ejpam-6011	345	24	1	1	NUM
ejpam-6011	345	25	;	;	PUNCT
ejpam-6011	345	26	(	(	PUNCT
ejpam-6011	345	27	iii	iii	X
ejpam-6011	345	28	)	)	PUNCT
ejpam-6011	345	29	t	t	NOUN
ejpam-6011	345	30	n(r	n(r	NOUN
ejpam-6011	345	31	)	)	PUNCT
ejpam-6011	346	1	is	be	AUX
ejpam-6011	346	2	nj	nj	NOUN
ejpam-6011	346	3	-	-	PUNCT
ejpam-6011	346	4	abelian	abelian	ADJ
ejpam-6011	346	5	for	for	ADP
ejpam-6011	346	6	every	every	DET
ejpam-6011	346	7	n	n	NOUN
ejpam-6011	346	8	>	>	SYM
ejpam-6011	346	9	1	1	NUM
ejpam-6011	346	10	.	.	PUNCT
ejpam-6011	346	11	proof	proof	NOUN
ejpam-6011	346	12	.	.	PUNCT
ejpam-6011	347	1	(	(	PUNCT
ejpam-6011	347	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-6011	347	3	):	):	PUNCT
ejpam-6011	347	4	if	if	SCONJ
ejpam-6011	347	5	r	r	NOUN
ejpam-6011	347	6	is	be	AUX
ejpam-6011	347	7	an	an	DET
ejpam-6011	347	8	nj	nj	NOUN
ejpam-6011	347	9	-	-	PUNCT
ejpam-6011	347	10	abelian	abelian	NOUN
ejpam-6011	347	11	,	,	PUNCT
ejpam-6011	347	12	then	then	ADV
ejpam-6011	347	13	t	t	PROPN
ejpam-6011	347	14	2(r	2(r	NUM
ejpam-6011	347	15	)	)	PUNCT
ejpam-6011	347	16	is	be	AUX
ejpam-6011	347	17	a	a	DET
ejpam-6011	347	18	formal	formal	ADJ
ejpam-6011	347	19	triangular	triangular	NOUN
ejpam-6011	347	20	matrix	matrix	NOUN
ejpam-6011	347	21	,	,	PUNCT
ejpam-6011	347	22	and	and	CCONJ
ejpam-6011	347	23	consequently	consequently	ADV
ejpam-6011	347	24	it	it	PRON
ejpam-6011	347	25	is	be	AUX
ejpam-6011	347	26	nj	nj	NOUN
ejpam-6011	347	27	-	-	PUNCT
ejpam-6011	347	28	abelian	abelian	ADJ
ejpam-6011	347	29	from	from	ADP
ejpam-6011	347	30	corollary	corollary	ADJ
ejpam-6011	347	31	7	7	NUM
ejpam-6011	347	32	.	.	PUNCT
ejpam-6011	348	1	(	(	PUNCT
ejpam-6011	348	2	ii)⇒(iii	ii)⇒(iii	NOUN
ejpam-6011	348	3	):	):	PUNCT
ejpam-6011	348	4	applying	apply	VERB
ejpam-6011	348	5	the	the	DET
ejpam-6011	348	6	mathematical	mathematical	ADJ
ejpam-6011	348	7	induction	induction	NOUN
ejpam-6011	348	8	on	on	ADP
ejpam-6011	348	9	n	n	CCONJ
ejpam-6011	348	10	,	,	PUNCT
ejpam-6011	348	11	assume	assume	VERB
ejpam-6011	348	12	that	that	SCONJ
ejpam-6011	348	13	t	t	PROPN
ejpam-6011	348	14	n(r	n(r	NOUN
ejpam-6011	348	15	)	)	PUNCT
ejpam-6011	348	16	is	be	AUX
ejpam-6011	348	17	nj	nj	NOUN
ejpam-6011	348	18	-	-	PUNCT
ejpam-6011	348	19	abelian	abelian	ADJ
ejpam-6011	348	20	for	for	ADP
ejpam-6011	348	21	some	some	DET
ejpam-6011	348	22	n	n	NOUN
ejpam-6011	348	23	>	>	X
ejpam-6011	348	24	1	1	NUM
ejpam-6011	348	25	;	;	PUNCT
ejpam-6011	348	26	then	then	ADV
ejpam-6011	348	27	r	r	NOUN
ejpam-6011	348	28	is	be	AUX
ejpam-6011	348	29	nj	nj	NOUN
ejpam-6011	348	30	-	-	PUNCT
ejpam-6011	348	31	abelian	abelian	ADJ
ejpam-6011	348	32	by	by	ADP
ejpam-6011	348	33	proposition	proposition	NOUN
ejpam-6011	348	34	10	10	NUM
ejpam-6011	348	35	.	.	PUNCT
ejpam-6011	349	1	notice	notice	VERB
ejpam-6011	349	2	that	that	SCONJ
ejpam-6011	349	3	t	t	PROPN
ejpam-6011	349	4	n+1(r	n+1(r	PROPN
ejpam-6011	349	5	)	)	PUNCT
ejpam-6011	349	6	is	be	AUX
ejpam-6011	349	7	the	the	DET
ejpam-6011	349	8	formal	formal	ADJ
ejpam-6011	349	9	triangular	triangular	NOUN
ejpam-6011	349	10	matrix	matrix	NOUN
ejpam-6011	349	11	[	[	PUNCT
ejpam-6011	349	12	t	t	NOUN
ejpam-6011	349	13	n(r	n(r	NUM
ejpam-6011	349	14	)	)	PUNCT
ejpam-6011	349	15	rn	rn	NOUN
ejpam-6011	349	16	0	0	NUM
ejpam-6011	349	17	r	r	NOUN
ejpam-6011	349	18	]	]	PUNCT
ejpam-6011	349	19	,	,	PUNCT
ejpam-6011	349	20	where	where	SCONJ
ejpam-6011	349	21	rn	rn	PROPN
ejpam-6011	349	22	is	be	AUX
ejpam-6011	349	23	the	the	DET
ejpam-6011	349	24	(	(	PUNCT
ejpam-6011	349	25	t	t	NOUN
ejpam-6011	349	26	n(r	n(r	NUM
ejpam-6011	349	27	)	)	PUNCT
ejpam-6011	349	28	,	,	PUNCT
ejpam-6011	349	29	r)-bimodule	r)-bimodule	NOUN
ejpam-6011	349	30	of	of	ADP
ejpam-6011	349	31	n	n	CCONJ
ejpam-6011	349	32	-	-	PUNCT
ejpam-6011	349	33	by-1	by-1	NOUN
ejpam-6011	349	34	matrices	matrix	NOUN
ejpam-6011	349	35	over	over	ADP
ejpam-6011	349	36	r.	r.	PROPN
ejpam-6011	349	37	therefore	therefore	ADV
ejpam-6011	349	38	t	t	PROPN
ejpam-6011	349	39	n+1(r	n+1(r	PROPN
ejpam-6011	349	40	)	)	PUNCT
ejpam-6011	349	41	is	be	AUX
ejpam-6011	349	42	nj	nj	NOUN
ejpam-6011	349	43	-	-	PUNCT
ejpam-6011	349	44	abelian	abelian	ADJ
ejpam-6011	349	45	by	by	ADP
ejpam-6011	349	46	corollary	corollary	ADJ
ejpam-6011	349	47	7	7	NUM
ejpam-6011	349	48	.	.	PUNCT
ejpam-6011	349	49	m.	m.	NOUN
ejpam-6011	349	50	saad	saad	PROPN
ejpam-6011	349	51	,	,	PUNCT
ejpam-6011	349	52	s.	s.	PROPN
ejpam-6011	349	53	m.	m.	PROPN
ejpam-6011	349	54	abdelwahab	abdelwahab	PROPN
ejpam-6011	349	55	/	/	SYM
ejpam-6011	349	56	eur	eur	PROPN
ejpam-6011	349	57	.	.	PUNCT
ejpam-6011	350	1	j.	j.	PROPN
ejpam-6011	350	2	pure	pure	PROPN
ejpam-6011	350	3	appl	appl	PROPN
ejpam-6011	350	4	.	.	PROPN
ejpam-6011	350	5	math	math	PROPN
ejpam-6011	350	6	,	,	PUNCT
ejpam-6011	350	7	18	18	NUM
ejpam-6011	350	8	(	(	PUNCT
ejpam-6011	350	9	2	2	NUM
ejpam-6011	350	10	)	)	PUNCT
ejpam-6011	350	11	(	(	PUNCT
ejpam-6011	350	12	2025	2025	NUM
ejpam-6011	350	13	)	)	PUNCT
ejpam-6011	350	14	,	,	PUNCT
ejpam-6011	350	15	6011	6011	NUM
ejpam-6011	350	16	11	11	NUM
ejpam-6011	350	17	of	of	ADP
ejpam-6011	350	18	13	13	NUM
ejpam-6011	350	19	for	for	ADP
ejpam-6011	350	20	a	a	DET
ejpam-6011	350	21	ring	ring	NOUN
ejpam-6011	350	22	r	r	NOUN
ejpam-6011	350	23	,	,	PUNCT
ejpam-6011	350	24	define	define	VERB
ejpam-6011	350	25	a	a	DET
ejpam-6011	350	26	subring	subre	VERB
ejpam-6011	350	27	sn(r	sn(r	NOUN
ejpam-6011	350	28	)	)	PUNCT
ejpam-6011	350	29	of	of	ADP
ejpam-6011	350	30	t	t	PROPN
ejpam-6011	350	31	n(r	n(r	NOUN
ejpam-6011	350	32	)	)	PUNCT
ejpam-6011	350	33	as	as	ADP
ejpam-6011	350	34	sn(r	sn(r	PRON
ejpam-6011	350	35	)	)	PUNCT
ejpam-6011	350	36	=	=	SYM
ejpam-6011	350	37			PROPN
ejpam-6011	350	38			PROPN
ejpam-6011	350	39	a	a	DET
ejpam-6011	350	40	a12	a12	NOUN
ejpam-6011	350	41	a13	a13	PROPN
ejpam-6011	350	42	a14	a14	PROPN
ejpam-6011	350	43	·	·	PUNCT
ejpam-6011	350	44	·	·	PUNCT
ejpam-6011	350	45	·	·	PUNCT
ejpam-6011	351	1	a1n	a1n	ADP
ejpam-6011	351	2	0	0	NUM
ejpam-6011	351	3	a	a	DET
ejpam-6011	351	4	a23	a23	PROPN
ejpam-6011	351	5	a24	a24	PROPN
ejpam-6011	351	6	·	·	PUNCT
ejpam-6011	351	7	·	·	PUNCT
ejpam-6011	351	8	·	·	PUNCT
ejpam-6011	351	9	a2n	a2n	PUNCT
ejpam-6011	351	10	0	0	NUM
ejpam-6011	351	11	0	0	NUM
ejpam-6011	351	12	a	a	DET
ejpam-6011	351	13	a34	a34	NOUN
ejpam-6011	351	14	·	·	PUNCT
ejpam-6011	351	15	·	·	PUNCT
ejpam-6011	351	16	·	·	PUNCT
ejpam-6011	351	17	a3n	a3n	NOUN
ejpam-6011	351	18	0	0	NUM
ejpam-6011	351	19	0	0	NUM
ejpam-6011	351	20	0	0	NUM
ejpam-6011	351	21	a	a	PRON
ejpam-6011	351	22	·	·	PUNCT
ejpam-6011	351	23	·	·	PUNCT
ejpam-6011	351	24	·	·	PUNCT
ejpam-6011	351	25	a4n	a4n	PUNCT
ejpam-6011	351	26	...	...	PUNCT
ejpam-6011	351	27	...	...	PUNCT
ejpam-6011	351	28	...	...	PUNCT
ejpam-6011	351	29	...	...	PUNCT
ejpam-6011	351	30	.	.	PUNCT
ejpam-6011	351	31	.	.	PUNCT
ejpam-6011	351	32	.	.	PUNCT
ejpam-6011	352	1	...	...	PUNCT
ejpam-6011	353	1	0	0	NUM
ejpam-6011	353	2	0	0	NUM
ejpam-6011	353	3	0	0	NUM
ejpam-6011	353	4	0	0	NUM
ejpam-6011	353	5	·	·	PUNCT
ejpam-6011	353	6	·	·	PUNCT
ejpam-6011	353	7	·	·	PUNCT
ejpam-6011	354	1	a	a	DET
ejpam-6011	354	2			NUM
ejpam-6011	354	3	|a	|a	NOUN
ejpam-6011	354	4	,	,	PUNCT
ejpam-6011	354	5	aij	aij	PROPN
ejpam-6011	354	6	∈	∈	PROPN
ejpam-6011	354	7	r	r	NOUN
ejpam-6011	354	8			NOUN
ejpam-6011	354	9	,	,	PUNCT
ejpam-6011	354	10	where	where	SCONJ
ejpam-6011	354	11	n	n	PRON
ejpam-6011	354	12	≥	≥	NOUN
ejpam-6011	354	13	2	2	NUM
ejpam-6011	354	14	is	be	AUX
ejpam-6011	354	15	a	a	DET
ejpam-6011	354	16	positive	positive	ADJ
ejpam-6011	354	17	integer	integer	NOUN
ejpam-6011	354	18	.	.	PUNCT
ejpam-6011	355	1	also	also	ADV
ejpam-6011	355	2	,	,	PUNCT
ejpam-6011	355	3	we	we	PRON
ejpam-6011	355	4	have	have	VERB
ejpam-6011	355	5	a	a	DET
ejpam-6011	355	6	subring	subring	NOUN
ejpam-6011	355	7	of	of	ADP
ejpam-6011	355	8	sn(r	sn(r	NOUN
ejpam-6011	355	9	)	)	PUNCT
ejpam-6011	355	10	defined	define	VERB
ejpam-6011	355	11	below	below	ADP
ejpam-6011	355	12	:	:	PUNCT
ejpam-6011	355	13	v	v	ADP
ejpam-6011	355	14	n(r	n(r	NOUN
ejpam-6011	355	15	)	)	PUNCT
ejpam-6011	355	16	=	=	SYM
ejpam-6011	355	17			PROPN
ejpam-6011	355	18			ADJ
ejpam-6011	355	19	a1	a1	PROPN
ejpam-6011	355	20	a2	a2	PROPN
ejpam-6011	355	21	a3	a3	PROPN
ejpam-6011	355	22	a4	a4	PROPN
ejpam-6011	355	23	·	·	PUNCT
ejpam-6011	355	24	·	·	PUNCT
ejpam-6011	355	25	·	·	PUNCT
ejpam-6011	356	1	an	an	DET
ejpam-6011	356	2	0	0	NUM
ejpam-6011	356	3	a1	a1	NOUN
ejpam-6011	356	4	a2	a2	PROPN
ejpam-6011	356	5	a3	a3	NOUN
ejpam-6011	356	6	·	·	PUNCT
ejpam-6011	356	7	·	·	PUNCT
ejpam-6011	356	8	·	·	PUNCT
ejpam-6011	357	1	an−1	an−1	ADJ
ejpam-6011	357	2	0	0	NUM
ejpam-6011	357	3	0	0	NUM
ejpam-6011	357	4	a1	a1	NOUN
ejpam-6011	357	5	a2	a2	PROPN
ejpam-6011	357	6	·	·	PUNCT
ejpam-6011	357	7	·	·	PUNCT
ejpam-6011	357	8	·	·	PUNCT
ejpam-6011	358	1	an−2	an−2	NOUN
ejpam-6011	358	2	0	0	NUM
ejpam-6011	358	3	0	0	SYM
ejpam-6011	358	4	0	0	NUM
ejpam-6011	358	5	a1	a1	NOUN
ejpam-6011	358	6	·	·	PUNCT
ejpam-6011	358	7	·	·	PUNCT
ejpam-6011	358	8	·	·	PUNCT
ejpam-6011	359	1	an−3	an−3	INTJ
ejpam-6011	359	2	...	...	PUNCT
ejpam-6011	359	3	...	...	PUNCT
ejpam-6011	359	4	...	...	PUNCT
ejpam-6011	359	5	...	...	PUNCT
ejpam-6011	359	6	.	.	PUNCT
ejpam-6011	359	7	.	.	PUNCT
ejpam-6011	359	8	.	.	PUNCT
ejpam-6011	360	1	...	...	PUNCT
ejpam-6011	361	1	0	0	NUM
ejpam-6011	361	2	0	0	NUM
ejpam-6011	361	3	0	0	NUM
ejpam-6011	361	4	0	0	NUM
ejpam-6011	361	5	·	·	PUNCT
ejpam-6011	361	6	·	·	PUNCT
ejpam-6011	361	7	·	·	PUNCT
ejpam-6011	361	8	a1	a1	NOUN
ejpam-6011	361	9			NOUN
ejpam-6011	361	10	|ai	|ai	NUM
ejpam-6011	361	11	∈	∈	NOUN
ejpam-6011	361	12	r	r	NOUN
ejpam-6011	361	13			NOUN
ejpam-6011	361	14	,	,	PUNCT
ejpam-6011	361	15	where	where	SCONJ
ejpam-6011	361	16	n	n	PRON
ejpam-6011	361	17	≥	≥	NOUN
ejpam-6011	361	18	2	2	NUM
ejpam-6011	361	19	is	be	AUX
ejpam-6011	361	20	a	a	DET
ejpam-6011	361	21	positive	positive	ADJ
ejpam-6011	361	22	integer	integer	NOUN
ejpam-6011	361	23	.	.	PUNCT
ejpam-6011	362	1	the	the	DET
ejpam-6011	362	2	next	next	ADJ
ejpam-6011	362	3	proposition	proposition	NOUN
ejpam-6011	362	4	shows	show	VERB
ejpam-6011	362	5	that	that	SCONJ
ejpam-6011	362	6	the	the	DET
ejpam-6011	362	7	nj	nj	PROPN
ejpam-6011	362	8	-	-	PUNCT
ejpam-6011	362	9	abelianity	abelianity	NOUN
ejpam-6011	362	10	of	of	ADP
ejpam-6011	362	11	a	a	DET
ejpam-6011	362	12	ring	ring	NOUN
ejpam-6011	362	13	r	r	NOUN
ejpam-6011	362	14	,	,	PUNCT
ejpam-6011	362	15	sn(r	sn(r	PRON
ejpam-6011	362	16	)	)	PUNCT
ejpam-6011	362	17	,	,	PUNCT
ejpam-6011	362	18	and	and	CCONJ
ejpam-6011	362	19	vn(r	vn(r	NUM
ejpam-6011	362	20	)	)	PUNCT
ejpam-6011	362	21	are	be	AUX
ejpam-6011	362	22	equivalent	equivalent	ADJ
ejpam-6011	362	23	for	for	ADP
ejpam-6011	362	24	every	every	DET
ejpam-6011	362	25	n.	n.	NOUN
ejpam-6011	362	26	proposition	proposition	NOUN
ejpam-6011	362	27	18	18	NUM
ejpam-6011	362	28	.	.	PUNCT
ejpam-6011	363	1	for	for	ADP
ejpam-6011	363	2	any	any	DET
ejpam-6011	363	3	ring	ring	NOUN
ejpam-6011	363	4	r	r	NOUN
ejpam-6011	363	5	,	,	PUNCT
ejpam-6011	363	6	the	the	DET
ejpam-6011	363	7	following	follow	VERB
ejpam-6011	363	8	conditions	condition	NOUN
ejpam-6011	363	9	are	be	AUX
ejpam-6011	363	10	equivalent	equivalent	ADJ
ejpam-6011	363	11	.	.	PUNCT
ejpam-6011	364	1	(	(	PUNCT
ejpam-6011	364	2	i	i	NOUN
ejpam-6011	364	3	)	)	PUNCT
ejpam-6011	364	4	r	r	NOUN
ejpam-6011	364	5	is	be	AUX
ejpam-6011	364	6	nj	nj	NOUN
ejpam-6011	364	7	-	-	PUNCT
ejpam-6011	364	8	abelian	abelian	ADJ
ejpam-6011	364	9	;	;	PUNCT
ejpam-6011	364	10	(	(	PUNCT
ejpam-6011	364	11	ii	ii	NOUN
ejpam-6011	364	12	)	)	PUNCT
ejpam-6011	364	13	sn(r	sn(r	PRON
ejpam-6011	364	14	)	)	PUNCT
ejpam-6011	364	15	is	be	AUX
ejpam-6011	364	16	nj	nj	NOUN
ejpam-6011	364	17	-	-	PUNCT
ejpam-6011	364	18	abelian	abelian	ADJ
ejpam-6011	364	19	for	for	ADP
ejpam-6011	364	20	some	some	DET
ejpam-6011	364	21	n	n	NOUN
ejpam-6011	364	22	>	>	X
ejpam-6011	364	23	1	1	NUM
ejpam-6011	364	24	;	;	PUNCT
ejpam-6011	364	25	(	(	PUNCT
ejpam-6011	364	26	iii	iii	NOUN
ejpam-6011	364	27	)	)	PUNCT
ejpam-6011	364	28	sn(r	sn(r	PRON
ejpam-6011	364	29	)	)	PUNCT
ejpam-6011	365	1	is	be	AUX
ejpam-6011	365	2	nj	nj	NOUN
ejpam-6011	365	3	-	-	PUNCT
ejpam-6011	365	4	abelian	abelian	ADJ
ejpam-6011	365	5	for	for	ADP
ejpam-6011	365	6	every	every	DET
ejpam-6011	365	7	n	n	NOUN
ejpam-6011	365	8	>	>	SYM
ejpam-6011	365	9	1	1	NUM
ejpam-6011	365	10	;	;	PUNCT
ejpam-6011	365	11	(	(	PUNCT
ejpam-6011	365	12	iv	iv	X
ejpam-6011	365	13	)	)	PUNCT
ejpam-6011	365	14	v	v	ADP
ejpam-6011	365	15	n(r	n(r	NOUN
ejpam-6011	365	16	)	)	PUNCT
ejpam-6011	365	17	is	be	AUX
ejpam-6011	365	18	nj	nj	NOUN
ejpam-6011	365	19	-	-	PUNCT
ejpam-6011	365	20	abelian	abelian	ADJ
ejpam-6011	365	21	for	for	ADP
ejpam-6011	365	22	some	some	DET
ejpam-6011	365	23	n	n	NOUN
ejpam-6011	365	24	>	>	X
ejpam-6011	365	25	1	1	NUM
ejpam-6011	365	26	;	;	PUNCT
ejpam-6011	365	27	(	(	PUNCT
ejpam-6011	365	28	v	v	NOUN
ejpam-6011	365	29	)	)	PUNCT
ejpam-6011	365	30	v	v	ADP
ejpam-6011	365	31	n(r	n(r	NOUN
ejpam-6011	365	32	)	)	PUNCT
ejpam-6011	365	33	is	be	AUX
ejpam-6011	365	34	nj	nj	NOUN
ejpam-6011	365	35	-	-	PUNCT
ejpam-6011	365	36	abelian	abelian	ADJ
ejpam-6011	365	37	for	for	ADP
ejpam-6011	365	38	every	every	DET
ejpam-6011	365	39	n	n	NOUN
ejpam-6011	365	40	>	>	SYM
ejpam-6011	365	41	1	1	NUM
ejpam-6011	365	42	.	.	PUNCT
ejpam-6011	365	43	proof	proof	NOUN
ejpam-6011	365	44	.	.	PUNCT
ejpam-6011	366	1	(	(	PUNCT
ejpam-6011	366	2	i)⇒(iii	i)⇒(iii	NOUN
ejpam-6011	366	3	):	):	PUNCT
ejpam-6011	366	4	for	for	ADP
ejpam-6011	366	5	every	every	DET
ejpam-6011	366	6	n	n	PRON
ejpam-6011	366	7	≥	≥	NOUN
ejpam-6011	366	8	2	2	NUM
ejpam-6011	366	9	,	,	PUNCT
ejpam-6011	366	10	consider	consider	VERB
ejpam-6011	366	11	the	the	DET
ejpam-6011	366	12	ideal	ideal	ADJ
ejpam-6011	366	13	in(r	in(r	PRON
ejpam-6011	366	14	)	)	PUNCT
ejpam-6011	366	15	of	of	ADP
ejpam-6011	366	16	sn(r	sn(r	PRON
ejpam-6011	366	17	)	)	PUNCT
ejpam-6011	366	18	consisting	consist	VERB
ejpam-6011	366	19	of	of	ADP
ejpam-6011	366	20	all	all	DET
ejpam-6011	366	21	elements	element	NOUN
ejpam-6011	366	22	of	of	ADP
ejpam-6011	366	23	sn(r	sn(r	NOUN
ejpam-6011	366	24	)	)	PUNCT
ejpam-6011	366	25	with	with	ADP
ejpam-6011	366	26	zero	zero	NUM
ejpam-6011	366	27	diagonal	diagonal	ADJ
ejpam-6011	366	28	entries	entry	NOUN
ejpam-6011	366	29	.	.	PUNCT
ejpam-6011	367	1	notice	notice	VERB
ejpam-6011	367	2	that	that	SCONJ
ejpam-6011	367	3	i	i	PRON
ejpam-6011	367	4	is	be	AUX
ejpam-6011	367	5	a	a	DET
ejpam-6011	367	6	nil	nil	ADJ
ejpam-6011	367	7	ideal	ideal	NOUN
ejpam-6011	367	8	and	and	CCONJ
ejpam-6011	367	9	r	r	NOUN
ejpam-6011	367	10	∼=	∼=	PROPN
ejpam-6011	367	11	sn(r)/in(r	sn(r)/in(r	NOUN
ejpam-6011	367	12	)	)	PUNCT
ejpam-6011	367	13	.	.	PUNCT
ejpam-6011	368	1	thus	thus	ADV
ejpam-6011	368	2	,	,	PUNCT
ejpam-6011	368	3	sn(r	sn(r	PRON
ejpam-6011	368	4	)	)	PUNCT
ejpam-6011	368	5	is	be	AUX
ejpam-6011	368	6	nj	nj	NOUN
ejpam-6011	368	7	-	-	PUNCT
ejpam-6011	368	8	abelian	abelian	ADJ
ejpam-6011	368	9	from	from	ADP
ejpam-6011	368	10	proposition	proposition	NOUN
ejpam-6011	368	11	14	14	NUM
ejpam-6011	368	12	.	.	PUNCT
ejpam-6011	369	1	(	(	PUNCT
ejpam-6011	369	2	iii)⇒(i	iii)⇒(i	X
ejpam-6011	369	3	)	)	PUNCT
ejpam-6011	369	4	is	be	AUX
ejpam-6011	369	5	direct	direct	ADJ
ejpam-6011	369	6	from	from	ADP
ejpam-6011	369	7	proposition	proposition	NOUN
ejpam-6011	369	8	10	10	NUM
ejpam-6011	369	9	.	.	PUNCT
ejpam-6011	370	1	(	(	PUNCT
ejpam-6011	370	2	i)⇔(v	i)⇔(v	NOUN
ejpam-6011	370	3	)	)	PUNCT
ejpam-6011	370	4	may	may	AUX
ejpam-6011	370	5	be	be	AUX
ejpam-6011	370	6	proved	prove	VERB
ejpam-6011	370	7	by	by	ADP
ejpam-6011	370	8	the	the	DET
ejpam-6011	370	9	same	same	ADJ
ejpam-6011	370	10	technique	technique	NOUN
ejpam-6011	370	11	of	of	ADP
ejpam-6011	370	12	proving	proving	NOUN
ejpam-6011	370	13	(	(	PUNCT
ejpam-6011	370	14	i)⇔(iii	i)⇔(iii	NOUN
ejpam-6011	370	15	)	)	PUNCT
ejpam-6011	370	16	.	.	PUNCT
ejpam-6011	371	1	(	(	PUNCT
ejpam-6011	371	2	i)⇔(ii)⇔(iv	i)⇔(ii)⇔(iv	NOUN
ejpam-6011	371	3	)	)	PUNCT
ejpam-6011	371	4	is	be	AUX
ejpam-6011	371	5	obtained	obtain	VERB
ejpam-6011	371	6	directly	directly	ADV
ejpam-6011	371	7	from	from	ADP
ejpam-6011	371	8	corollary	corollary	ADJ
ejpam-6011	371	9	8	8	NUM
ejpam-6011	371	10	since	since	SCONJ
ejpam-6011	371	11	t	t	PROPN
ejpam-6011	371	12	(	(	PUNCT
ejpam-6011	371	13	r	r	NOUN
ejpam-6011	371	14	,	,	PUNCT
ejpam-6011	371	15	r	r	NOUN
ejpam-6011	371	16	)	)	PUNCT
ejpam-6011	371	17	=	=	SYM
ejpam-6011	371	18	s2(r	s2(r	X
ejpam-6011	371	19	)	)	PUNCT
ejpam-6011	371	20	=	=	SYM
ejpam-6011	371	21	v	v	ADP
ejpam-6011	371	22	2(r	2(r	NUM
ejpam-6011	371	23	)	)	PUNCT
ejpam-6011	371	24	.	.	PUNCT
ejpam-6011	372	1	corollary	corollary	ADJ
ejpam-6011	372	2	9	9	NUM
ejpam-6011	372	3	.	.	PUNCT
ejpam-6011	373	1	a	a	DET
ejpam-6011	373	2	ring	ring	NOUN
ejpam-6011	373	3	r	r	NOUN
ejpam-6011	373	4	is	be	AUX
ejpam-6011	373	5	nj	nj	NOUN
ejpam-6011	373	6	-	-	PUNCT
ejpam-6011	373	7	abelian	abelian	ADJ
ejpam-6011	374	1	if	if	SCONJ
ejpam-6011	374	2	and	and	CCONJ
ejpam-6011	374	3	only	only	ADV
ejpam-6011	374	4	if	if	SCONJ
ejpam-6011	374	5	r[x]/⟨xn⟩	r[x]/⟨xn⟩	PRON
ejpam-6011	374	6	is	be	AUX
ejpam-6011	374	7	nj	nj	NOUN
ejpam-6011	374	8	-	-	PUNCT
ejpam-6011	374	9	abelian	abelian	ADJ
ejpam-6011	374	10	for	for	ADP
ejpam-6011	374	11	any	any	DET
ejpam-6011	374	12	positive	positive	ADJ
ejpam-6011	374	13	integer	integer	NOUN
ejpam-6011	374	14	n.	n.	PROPN
ejpam-6011	374	15	m.	m.	PROPN
ejpam-6011	374	16	saad	saad	PROPN
ejpam-6011	374	17	,	,	PUNCT
ejpam-6011	374	18	s.	s.	PROPN
ejpam-6011	374	19	m.	m.	PROPN
ejpam-6011	374	20	abdelwahab	abdelwahab	PROPN
ejpam-6011	374	21	/	/	SYM
ejpam-6011	374	22	eur	eur	PROPN
ejpam-6011	374	23	.	.	PUNCT
ejpam-6011	375	1	j.	j.	PROPN
ejpam-6011	375	2	pure	pure	PROPN
ejpam-6011	375	3	appl	appl	PROPN
ejpam-6011	375	4	.	.	PROPN
ejpam-6011	375	5	math	math	PROPN
ejpam-6011	375	6	,	,	PUNCT
ejpam-6011	375	7	18	18	NUM
ejpam-6011	375	8	(	(	PUNCT
ejpam-6011	375	9	2	2	NUM
ejpam-6011	375	10	)	)	PUNCT
ejpam-6011	375	11	(	(	PUNCT
ejpam-6011	375	12	2025	2025	NUM
ejpam-6011	375	13	)	)	PUNCT
ejpam-6011	375	14	,	,	PUNCT
ejpam-6011	375	15	6011	6011	NUM
ejpam-6011	375	16	12	12	NUM
ejpam-6011	375	17	of	of	ADP
ejpam-6011	375	18	13	13	NUM
ejpam-6011	375	19	conclusion	conclusion	NOUN
ejpam-6011	375	20	in	in	ADP
ejpam-6011	375	21	this	this	DET
ejpam-6011	375	22	article	article	NOUN
ejpam-6011	375	23	,	,	PUNCT
ejpam-6011	375	24	we	we	PRON
ejpam-6011	375	25	introduced	introduce	VERB
ejpam-6011	375	26	and	and	CCONJ
ejpam-6011	375	27	studied	study	VERB
ejpam-6011	375	28	a	a	DET
ejpam-6011	375	29	new	new	ADJ
ejpam-6011	375	30	class	class	NOUN
ejpam-6011	375	31	of	of	ADP
ejpam-6011	375	32	rings	ring	NOUN
ejpam-6011	375	33	,	,	PUNCT
ejpam-6011	375	34	called	call	VERB
ejpam-6011	375	35	nj	nj	PROPN
ejpam-6011	375	36	-	-	PUNCT
ejpam-6011	375	37	abelian	abelian	NOUN
ejpam-6011	375	38	rings	ring	NOUN
ejpam-6011	375	39	,	,	PUNCT
ejpam-6011	375	40	defined	define	VERB
ejpam-6011	375	41	by	by	ADP
ejpam-6011	375	42	a	a	DET
ejpam-6011	375	43	condition	condition	NOUN
ejpam-6011	375	44	that	that	PRON
ejpam-6011	375	45	connects	connect	VERB
ejpam-6011	375	46	idempotents	idempotent	NOUN
ejpam-6011	375	47	and	and	CCONJ
ejpam-6011	375	48	nilpotent	nilpotent	ADJ
ejpam-6011	375	49	elements	element	NOUN
ejpam-6011	375	50	through	through	ADP
ejpam-6011	375	51	the	the	DET
ejpam-6011	375	52	jacobson	jacobson	PROPN
ejpam-6011	375	53	radical	radical	PROPN
ejpam-6011	375	54	.	.	PUNCT
ejpam-6011	376	1	we	we	PRON
ejpam-6011	376	2	showed	show	VERB
ejpam-6011	376	3	that	that	SCONJ
ejpam-6011	376	4	this	this	DET
ejpam-6011	376	5	class	class	NOUN
ejpam-6011	376	6	properly	properly	ADV
ejpam-6011	376	7	extends	extend	VERB
ejpam-6011	376	8	nj	nj	PROPN
ejpam-6011	376	9	-	-	PUNCT
ejpam-6011	376	10	semicommutative	semicommutative	NOUN
ejpam-6011	376	11	rings	ring	NOUN
ejpam-6011	376	12	,	,	PUNCT
ejpam-6011	376	13	analogous	analogous	ADJ
ejpam-6011	376	14	to	to	ADP
ejpam-6011	376	15	how	how	SCONJ
ejpam-6011	376	16	abelian	abelian	ADJ
ejpam-6011	376	17	rings	ring	NOUN
ejpam-6011	376	18	extend	extend	VERB
ejpam-6011	376	19	semicommutative	semicommutative	NOUN
ejpam-6011	376	20	rings	ring	NOUN
ejpam-6011	376	21	.	.	PUNCT
ejpam-6011	377	1	we	we	PRON
ejpam-6011	377	2	proved	prove	VERB
ejpam-6011	377	3	that	that	SCONJ
ejpam-6011	377	4	every	every	DET
ejpam-6011	377	5	nj	nj	PROPN
ejpam-6011	377	6	-	-	PUNCT
ejpam-6011	377	7	abelian	abelian	ADJ
ejpam-6011	377	8	ring	ring	NOUN
ejpam-6011	377	9	is	be	AUX
ejpam-6011	377	10	j	j	NOUN
ejpam-6011	377	11	-	-	NOUN
ejpam-6011	377	12	abelian	abelian	PROPN
ejpam-6011	377	13	and	and	CCONJ
ejpam-6011	377	14	that	that	SCONJ
ejpam-6011	377	15	the	the	DET
ejpam-6011	377	16	nj	nj	PROPN
ejpam-6011	377	17	-	-	PUNCT
ejpam-6011	377	18	abelian	abelian	ADJ
ejpam-6011	377	19	condition	condition	NOUN
ejpam-6011	377	20	is	be	AUX
ejpam-6011	377	21	symmetric	symmetric	ADJ
ejpam-6011	377	22	with	with	ADP
ejpam-6011	377	23	respect	respect	NOUN
ejpam-6011	377	24	to	to	ADP
ejpam-6011	377	25	left	left	ADJ
ejpam-6011	377	26	and	and	CCONJ
ejpam-6011	377	27	right	right	ADJ
ejpam-6011	377	28	multiplication	multiplication	NOUN
ejpam-6011	377	29	.	.	PUNCT
ejpam-6011	378	1	several	several	ADJ
ejpam-6011	378	2	equivalent	equivalent	ADJ
ejpam-6011	378	3	characterizations	characterization	NOUN
ejpam-6011	378	4	were	be	AUX
ejpam-6011	378	5	established	establish	VERB
ejpam-6011	378	6	,	,	PUNCT
ejpam-6011	378	7	notably	notably	ADV
ejpam-6011	378	8	that	that	SCONJ
ejpam-6011	378	9	a	a	DET
ejpam-6011	378	10	ring	ring	NOUN
ejpam-6011	378	11	is	be	AUX
ejpam-6011	378	12	nj	nj	NOUN
ejpam-6011	378	13	-	-	PUNCT
ejpam-6011	378	14	abelian	abelian	ADJ
ejpam-6011	379	1	if	if	SCONJ
ejpam-6011	379	2	and	and	CCONJ
ejpam-6011	379	3	only	only	ADV
ejpam-6011	379	4	if	if	SCONJ
ejpam-6011	379	5	it	it	PRON
ejpam-6011	379	6	is	be	AUX
ejpam-6011	379	7	both	both	PRON
ejpam-6011	379	8	j	j	NOUN
ejpam-6011	379	9	-	-	PUNCT
ejpam-6011	379	10	reduced	reduce	VERB
ejpam-6011	379	11	and	and	CCONJ
ejpam-6011	379	12	j	j	NOUN
ejpam-6011	379	13	-	-	PUNCT
ejpam-6011	379	14	abelian	abelian	PROPN
ejpam-6011	379	15	.	.	PUNCT
ejpam-6011	380	1	moreover	moreover	ADV
ejpam-6011	380	2	,	,	PUNCT
ejpam-6011	380	3	we	we	PRON
ejpam-6011	380	4	demonstrated	demonstrate	VERB
ejpam-6011	380	5	that	that	SCONJ
ejpam-6011	380	6	many	many	ADJ
ejpam-6011	380	7	classical	classical	ADJ
ejpam-6011	380	8	ring	ring	NOUN
ejpam-6011	380	9	classes	class	NOUN
ejpam-6011	380	10	,	,	PUNCT
ejpam-6011	380	11	such	such	ADJ
ejpam-6011	380	12	as	as	ADP
ejpam-6011	380	13	jclean	jclean	PROPN
ejpam-6011	380	14	rings	ring	NOUN
ejpam-6011	380	15	,	,	PUNCT
ejpam-6011	380	16	local	local	ADJ
ejpam-6011	380	17	rings	ring	NOUN
ejpam-6011	380	18	,	,	PUNCT
ejpam-6011	380	19	and	and	CCONJ
ejpam-6011	380	20	feckly	feckly	ADV
ejpam-6011	380	21	reduced	reduced	ADJ
ejpam-6011	380	22	rings	ring	NOUN
ejpam-6011	380	23	,	,	PUNCT
ejpam-6011	380	24	are	be	AUX
ejpam-6011	380	25	nj	nj	NOUN
ejpam-6011	380	26	-	-	PUNCT
ejpam-6011	380	27	abelian	abelian	ADJ
ejpam-6011	380	28	,	,	PUNCT
ejpam-6011	380	29	and	and	CCONJ
ejpam-6011	380	30	we	we	PRON
ejpam-6011	380	31	clarified	clarify	VERB
ejpam-6011	380	32	their	their	PRON
ejpam-6011	380	33	implications	implication	NOUN
ejpam-6011	380	34	with	with	ADP
ejpam-6011	380	35	suitable	suitable	ADJ
ejpam-6011	380	36	counterexamples	counterexample	NOUN
ejpam-6011	380	37	.	.	PUNCT
ejpam-6011	381	1	in	in	ADP
ejpam-6011	381	2	addition	addition	NOUN
ejpam-6011	381	3	,	,	PUNCT
ejpam-6011	381	4	we	we	PRON
ejpam-6011	381	5	examined	examine	VERB
ejpam-6011	381	6	the	the	DET
ejpam-6011	381	7	behavior	behavior	NOUN
ejpam-6011	381	8	of	of	ADP
ejpam-6011	381	9	the	the	DET
ejpam-6011	381	10	nj	nj	PROPN
ejpam-6011	381	11	-	-	PUNCT
ejpam-6011	381	12	abelian	abelian	ADJ
ejpam-6011	381	13	property	property	NOUN
ejpam-6011	381	14	under	under	ADP
ejpam-6011	381	15	various	various	ADJ
ejpam-6011	381	16	ring	ring	NOUN
ejpam-6011	381	17	constructions	construction	NOUN
ejpam-6011	381	18	and	and	CCONJ
ejpam-6011	381	19	extensions	extension	NOUN
ejpam-6011	381	20	.	.	PUNCT
ejpam-6011	382	1	we	we	PRON
ejpam-6011	382	2	showed	show	VERB
ejpam-6011	382	3	that	that	SCONJ
ejpam-6011	382	4	nj	nj	PROPN
ejpam-6011	382	5	-	-	PUNCT
ejpam-6011	382	6	abelianity	abelianity	NOUN
ejpam-6011	382	7	is	be	AUX
ejpam-6011	382	8	not	not	PART
ejpam-6011	382	9	generally	generally	ADV
ejpam-6011	382	10	preserved	preserve	VERB
ejpam-6011	382	11	in	in	ADP
ejpam-6011	382	12	full	full	ADJ
ejpam-6011	382	13	matrix	matrix	NOUN
ejpam-6011	382	14	rings	ring	NOUN
ejpam-6011	382	15	or	or	CCONJ
ejpam-6011	382	16	certain	certain	ADJ
ejpam-6011	382	17	direct	direct	ADJ
ejpam-6011	382	18	sums	sum	NOUN
ejpam-6011	382	19	and	and	CCONJ
ejpam-6011	382	20	ideals	ideal	NOUN
ejpam-6011	382	21	,	,	PUNCT
ejpam-6011	382	22	whereas	whereas	SCONJ
ejpam-6011	382	23	it	it	PRON
ejpam-6011	382	24	can	can	AUX
ejpam-6011	382	25	hold	hold	VERB
ejpam-6011	382	26	in	in	ADP
ejpam-6011	382	27	specific	specific	ADJ
ejpam-6011	382	28	subrings	subring	NOUN
ejpam-6011	382	29	like	like	ADP
ejpam-6011	382	30	upper	upper	ADJ
ejpam-6011	382	31	triangular	triangular	NOUN
ejpam-6011	382	32	matrix	matrix	NOUN
ejpam-6011	382	33	rings	ring	NOUN
ejpam-6011	382	34	and	and	CCONJ
ejpam-6011	382	35	some	some	PRON
ejpam-6011	382	36	of	of	ADP
ejpam-6011	382	37	their	their	PRON
ejpam-6011	382	38	substructures	substructure	NOUN
ejpam-6011	382	39	.	.	PUNCT
ejpam-6011	383	1	the	the	DET
ejpam-6011	383	2	results	result	NOUN
ejpam-6011	383	3	presented	present	VERB
ejpam-6011	383	4	here	here	ADV
ejpam-6011	383	5	contribute	contribute	VERB
ejpam-6011	383	6	to	to	ADP
ejpam-6011	383	7	a	a	DET
ejpam-6011	383	8	deeper	deep	ADJ
ejpam-6011	383	9	understanding	understanding	NOUN
ejpam-6011	383	10	of	of	ADP
ejpam-6011	383	11	the	the	DET
ejpam-6011	383	12	interaction	interaction	NOUN
ejpam-6011	383	13	between	between	ADP
ejpam-6011	383	14	nilpotent	nilpotent	ADJ
ejpam-6011	383	15	elements	element	NOUN
ejpam-6011	383	16	,	,	PUNCT
ejpam-6011	383	17	idempotents	idempotent	NOUN
ejpam-6011	383	18	,	,	PUNCT
ejpam-6011	383	19	and	and	CCONJ
ejpam-6011	383	20	the	the	DET
ejpam-6011	383	21	jacobson	jacobson	PROPN
ejpam-6011	383	22	radical	radical	PROPN
ejpam-6011	383	23	.	.	PUNCT
ejpam-6011	384	1	they	they	PRON
ejpam-6011	384	2	open	open	VERB
ejpam-6011	384	3	possible	possible	ADJ
ejpam-6011	384	4	avenues	avenue	NOUN
ejpam-6011	384	5	for	for	ADP
ejpam-6011	384	6	further	further	ADJ
ejpam-6011	384	7	exploration	exploration	NOUN
ejpam-6011	384	8	,	,	PUNCT
ejpam-6011	384	9	such	such	ADJ
ejpam-6011	384	10	as	as	ADP
ejpam-6011	384	11	the	the	DET
ejpam-6011	384	12	behavior	behavior	NOUN
ejpam-6011	384	13	of	of	ADP
ejpam-6011	384	14	nj	nj	PROPN
ejpam-6011	384	15	-	-	PUNCT
ejpam-6011	384	16	abelianity	abelianity	NOUN
ejpam-6011	384	17	in	in	ADP
ejpam-6011	384	18	skew	skew	ADJ
ejpam-6011	384	19	polynomial	polynomial	ADJ
ejpam-6011	384	20	rings	ring	NOUN
ejpam-6011	384	21	,	,	PUNCT
ejpam-6011	384	22	endomorphism	endomorphism	PROPN
ejpam-6011	384	23	rings	ring	NOUN
ejpam-6011	384	24	,	,	PUNCT
ejpam-6011	384	25	or	or	CCONJ
ejpam-6011	384	26	more	more	ADV
ejpam-6011	384	27	general	general	ADJ
ejpam-6011	384	28	morita	morita	PROPN
ejpam-6011	384	29	contexts	contexts	PROPN
ejpam-6011	384	30	.	.	PUNCT
ejpam-6011	385	1	acknowledgements	acknowledgement	NOUN
ejpam-6011	385	2	the	the	DET
ejpam-6011	385	3	authors	author	NOUN
ejpam-6011	385	4	would	would	AUX
ejpam-6011	385	5	like	like	VERB
ejpam-6011	385	6	to	to	PART
ejpam-6011	385	7	thank	thank	VERB
ejpam-6011	385	8	the	the	DET
ejpam-6011	385	9	anonymous	anonymous	ADJ
ejpam-6011	385	10	reviewers	reviewer	NOUN
ejpam-6011	385	11	for	for	ADP
ejpam-6011	385	12	their	their	PRON
ejpam-6011	385	13	valuable	valuable	ADJ
ejpam-6011	385	14	comments	comment	NOUN
ejpam-6011	385	15	and	and	CCONJ
ejpam-6011	385	16	suggestions	suggestion	NOUN
ejpam-6011	385	17	,	,	PUNCT
ejpam-6011	385	18	which	which	PRON
ejpam-6011	385	19	helped	help	VERB
ejpam-6011	385	20	improve	improve	VERB
ejpam-6011	385	21	the	the	DET
ejpam-6011	385	22	clarity	clarity	NOUN
ejpam-6011	385	23	and	and	CCONJ
ejpam-6011	385	24	quality	quality	NOUN
ejpam-6011	385	25	of	of	ADP
ejpam-6011	385	26	the	the	DET
ejpam-6011	385	27	manuscript	manuscript	NOUN
ejpam-6011	385	28	.	.	PUNCT
ejpam-6011	386	1	references	reference	NOUN
ejpam-6011	386	2	[	[	X
ejpam-6011	386	3	1	1	NUM
ejpam-6011	386	4	]	]	X
ejpam-6011	386	5	sait	sait	X
ejpam-6011	386	6	halicioglu	halicioglu	PROPN
ejpam-6011	386	7	,	,	PUNCT
ejpam-6011	386	8	abdullah	abdullah	PROPN
ejpam-6011	386	9	harmanci	harmanci	PROPN
ejpam-6011	386	10	,	,	PUNCT
ejpam-6011	386	11	and	and	CCONJ
ejpam-6011	386	12	burcu	burcu	PROPN
ejpam-6011	386	13	ungor	ungor	PROPN
ejpam-6011	386	14	.	.	PUNCT
ejpam-6011	387	1	a	a	DET
ejpam-6011	387	2	class	class	NOUN
ejpam-6011	387	3	of	of	ADP
ejpam-6011	387	4	abelian	abelian	ADJ
ejpam-6011	387	5	rings	ring	NOUN
ejpam-6011	387	6	.	.	PUNCT
ejpam-6011	388	1	bolet́ın	bolet́ın	PROPN
ejpam-6011	388	2	de	de	X
ejpam-6011	388	3	matemáticas	matemáticas	PROPN
ejpam-6011	388	4	,	,	PUNCT
ejpam-6011	388	5	25(1):27–37	25(1):27–37	NUM
ejpam-6011	388	6	,	,	PUNCT
ejpam-6011	388	7	2018	2018	NUM
ejpam-6011	388	8	.	.	PUNCT
ejpam-6011	389	1	[	[	X
ejpam-6011	389	2	2	2	X
ejpam-6011	389	3	]	]	PUNCT
ejpam-6011	389	4	t.	t.	PROPN
ejpam-6011	389	5	y.	y.	PROPN
ejpam-6011	389	6	lam	lam	PROPN
ejpam-6011	389	7	.	.	PUNCT
ejpam-6011	390	1	on	on	ADP
ejpam-6011	390	2	some	some	DET
ejpam-6011	390	3	generalizations	generalization	NOUN
ejpam-6011	390	4	of	of	ADP
ejpam-6011	390	5	abelian	abelian	ADJ
ejpam-6011	390	6	rings	ring	NOUN
ejpam-6011	390	7	.	.	PUNCT
ejpam-6011	391	1	journal	journal	PROPN
ejpam-6011	391	2	of	of	ADP
ejpam-6011	391	3	algebra	algebra	PROPN
ejpam-6011	391	4	and	and	CCONJ
ejpam-6011	391	5	its	its	PRON
ejpam-6011	391	6	applications	application	NOUN
ejpam-6011	391	7	,	,	PUNCT
ejpam-6011	391	8	page	page	NOUN
ejpam-6011	391	9	2550146	2550146	NUM
ejpam-6011	391	10	,	,	PUNCT
ejpam-6011	391	11	2023	2023	NUM
ejpam-6011	391	12	.	.	PUNCT
ejpam-6011	392	1	[	[	X
ejpam-6011	392	2	3	3	X
ejpam-6011	392	3	]	]	X
ejpam-6011	392	4	l.	l.	PROPN
ejpam-6011	392	5	motais	motais	PROPN
ejpam-6011	392	6	de	de	PROPN
ejpam-6011	392	7	narbonne	narbonne	PROPN
ejpam-6011	392	8	.	.	PUNCT
ejpam-6011	393	1	anneaux	anneaux	PROPN
ejpam-6011	393	2	semi	semi	NOUN
ejpam-6011	393	3	-	-	NOUN
ejpam-6011	393	4	commutatifs	commutatifs	ADJ
ejpam-6011	393	5	et	et	PROPN
ejpam-6011	393	6	unis	unis	NOUN
ejpam-6011	393	7	riels	riel	NOUN
ejpam-6011	393	8	anneaux	anneaux	ADV
ejpam-6011	393	9	do	do	AUX
ejpam-6011	393	10	nt	not	PART
ejpam-6011	393	11	les	les	VERB
ejpam-6011	393	12	i	i	PROPN
ejpam-6011	393	13	d	d	PROPN
ejpam-6011	393	14	aux	aux	PROPN
ejpam-6011	393	15	principaux	principaux	PROPN
ejpam-6011	393	16	sont	sont	PROPN
ejpam-6011	393	17	idempotents	idempotent	NOUN
ejpam-6011	393	18	.	.	PUNCT
ejpam-6011	394	1	in	in	ADP
ejpam-6011	394	2	proceedings	proceeding	NOUN
ejpam-6011	394	3	of	of	ADP
ejpam-6011	394	4	the	the	DET
ejpam-6011	394	5	106th	106th	ADJ
ejpam-6011	394	6	national	national	PROPN
ejpam-6011	394	7	congress	congress	PROPN
ejpam-6011	394	8	of	of	ADP
ejpam-6011	394	9	learned	learn	VERB
ejpam-6011	394	10	societies	society	NOUN
ejpam-6011	394	11	(	(	PUNCT
ejpam-6011	394	12	perpignan	perpignan	NOUN
ejpam-6011	394	13	,	,	PUNCT
ejpam-6011	394	14	1981	1981	NUM
ejpam-6011	394	15	)	)	PUNCT
ejpam-6011	394	16	,	,	PUNCT
ejpam-6011	394	17	bib	bib	NOUN
ejpam-6011	394	18	.	.	PUNCT
ejpam-6011	395	1	nat	nat	PROPN
ejpam-6011	395	2	.	.	PROPN
ejpam-6011	395	3	,	,	PUNCT
ejpam-6011	395	4	paris	paris	PROPN
ejpam-6011	395	5	,	,	PUNCT
ejpam-6011	395	6	pages	page	NOUN
ejpam-6011	395	7	71–73	71–73	NUM
ejpam-6011	395	8	,	,	PUNCT
ejpam-6011	395	9	1982	1982	NUM
ejpam-6011	395	10	.	.	PUNCT
ejpam-6011	396	1	[	[	X
ejpam-6011	396	2	4	4	NUM
ejpam-6011	396	3	]	]	X
ejpam-6011	396	4	gooyong	gooyong	PROPN
ejpam-6011	396	5	shin	shin	PROPN
ejpam-6011	396	6	.	.	PUNCT
ejpam-6011	397	1	prime	prime	ADJ
ejpam-6011	397	2	ideals	ideal	NOUN
ejpam-6011	397	3	and	and	CCONJ
ejpam-6011	397	4	sheaf	sheaf	NOUN
ejpam-6011	397	5	representation	representation	NOUN
ejpam-6011	397	6	of	of	ADP
ejpam-6011	397	7	a	a	DET
ejpam-6011	397	8	pseudo	pseudo	NOUN
ejpam-6011	397	9	symmetric	symmetric	ADJ
ejpam-6011	397	10	ring	ring	NOUN
ejpam-6011	397	11	.	.	PUNCT
ejpam-6011	398	1	t.	t.	PROPN
ejpam-6011	398	2	am	am	PROPN
ejpam-6011	398	3	.	.	PUNCT
ejpam-6011	399	1	math	math	NOUN
ejpam-6011	399	2	.	.	PUNCT
ejpam-6011	400	1	soc	soc	PROPN
ejpam-6011	400	2	.	.	PUNCT
ejpam-6011	400	3	,	,	PUNCT
ejpam-6011	400	4	184:43–60	184:43–60	NUM
ejpam-6011	400	5	,	,	PUNCT
ejpam-6011	400	6	1973	1973	NUM
ejpam-6011	400	7	.	.	PUNCT
ejpam-6011	401	1	[	[	X
ejpam-6011	401	2	5	5	NUM
ejpam-6011	401	3	]	]	PUNCT
ejpam-6011	401	4	jebrel	jebrel	PROPN
ejpam-6011	401	5	m.	m.	PROPN
ejpam-6011	401	6	habeb	habeb	PROPN
ejpam-6011	401	7	.	.	PUNCT
ejpam-6011	402	1	a	a	DET
ejpam-6011	402	2	note	note	NOUN
ejpam-6011	402	3	on	on	ADP
ejpam-6011	402	4	zero	zero	NUM
ejpam-6011	402	5	commutative	commutative	ADJ
ejpam-6011	402	6	and	and	CCONJ
ejpam-6011	402	7	duo	duo	NOUN
ejpam-6011	402	8	rings	ring	NOUN
ejpam-6011	402	9	.	.	PUNCT
ejpam-6011	403	1	mathematical	mathematical	ADJ
ejpam-6011	403	2	journal	journal	PROPN
ejpam-6011	403	3	of	of	ADP
ejpam-6011	403	4	okayama	okayama	PROPN
ejpam-6011	403	5	university	university	PROPN
ejpam-6011	403	6	,	,	PUNCT
ejpam-6011	403	7	32:73–76	32:73–76	PROPN
ejpam-6011	403	8	,	,	PUNCT
ejpam-6011	403	9	1990	1990	NUM
ejpam-6011	403	10	.	.	PUNCT
ejpam-6011	404	1	[	[	X
ejpam-6011	404	2	6	6	X
ejpam-6011	404	3	]	]	PUNCT
ejpam-6011	404	4	howard	howard	PROPN
ejpam-6011	404	5	e.	e.	PROPN
ejpam-6011	404	6	bell	bell	PROPN
ejpam-6011	404	7	.	.	PUNCT
ejpam-6011	405	1	near	near	ADP
ejpam-6011	405	2	-	-	PUNCT
ejpam-6011	405	3	rings	ring	NOUN
ejpam-6011	405	4	in	in	ADP
ejpam-6011	405	5	which	which	PRON
ejpam-6011	405	6	each	each	DET
ejpam-6011	405	7	element	element	NOUN
ejpam-6011	405	8	is	be	AUX
ejpam-6011	405	9	a	a	DET
ejpam-6011	405	10	power	power	NOUN
ejpam-6011	405	11	of	of	ADP
ejpam-6011	405	12	itself	itself	PRON
ejpam-6011	405	13	.	.	PUNCT
ejpam-6011	406	1	bulletin	bulletin	NOUN
ejpam-6011	406	2	of	of	ADP
ejpam-6011	406	3	the	the	DET
ejpam-6011	406	4	australian	australian	ADJ
ejpam-6011	406	5	mathematical	mathematical	ADJ
ejpam-6011	406	6	society	society	NOUN
ejpam-6011	406	7	,	,	PUNCT
ejpam-6011	406	8	2:363–368	2:363–368	NUM
ejpam-6011	406	9	,	,	PUNCT
ejpam-6011	406	10	1970	1970	NUM
ejpam-6011	406	11	.	.	PUNCT
ejpam-6011	407	1	[	[	X
ejpam-6011	407	2	7	7	X
ejpam-6011	407	3	]	]	X
ejpam-6011	407	4	sanjiv	sanjiv	PROPN
ejpam-6011	407	5	subba	subba	PROPN
ejpam-6011	407	6	and	and	CCONJ
ejpam-6011	407	7	tikaram	tikaram	PROPN
ejpam-6011	407	8	subedi	subedi	PROPN
ejpam-6011	407	9	.	.	PUNCT
ejpam-6011	408	1	nj	nj	PROPN
ejpam-6011	408	2	-	-	PUNCT
ejpam-6011	408	3	semicommutative	semicommutative	NOUN
ejpam-6011	408	4	rings	ring	NOUN
ejpam-6011	408	5	.	.	PUNCT
ejpam-6011	409	1	miskolc	miskolc	ADJ
ejpam-6011	409	2	mathematical	mathematical	ADJ
ejpam-6011	409	3	notes	note	NOUN
ejpam-6011	409	4	,	,	PUNCT
ejpam-6011	409	5	24(3):1569–1579	24(3):1569–1579	NUM
ejpam-6011	409	6	,	,	PUNCT
ejpam-6011	409	7	2023	2023	NUM
ejpam-6011	409	8	.	.	PUNCT
ejpam-6011	410	1	m.	m.	NOUN
ejpam-6011	410	2	saad	saad	PROPN
ejpam-6011	410	3	,	,	PUNCT
ejpam-6011	410	4	s.	s.	PROPN
ejpam-6011	410	5	m.	m.	PROPN
ejpam-6011	410	6	abdelwahab	abdelwahab	PROPN
ejpam-6011	410	7	/	/	SYM
ejpam-6011	410	8	eur	eur	PROPN
ejpam-6011	410	9	.	.	PUNCT
ejpam-6011	411	1	j.	j.	PROPN
ejpam-6011	411	2	pure	pure	PROPN
ejpam-6011	411	3	appl	appl	PROPN
ejpam-6011	411	4	.	.	PROPN
ejpam-6011	411	5	math	math	PROPN
ejpam-6011	411	6	,	,	PUNCT
ejpam-6011	411	7	18	18	NUM
ejpam-6011	411	8	(	(	PUNCT
ejpam-6011	411	9	2	2	NUM
ejpam-6011	411	10	)	)	PUNCT
ejpam-6011	411	11	(	(	PUNCT
ejpam-6011	411	12	2025	2025	NUM
ejpam-6011	411	13	)	)	PUNCT
ejpam-6011	411	14	,	,	PUNCT
ejpam-6011	411	15	6011	6011	NUM
ejpam-6011	411	16	13	13	NUM
ejpam-6011	411	17	of	of	ADP
ejpam-6011	411	18	13	13	NUM
ejpam-6011	412	1	[	[	SYM
ejpam-6011	412	2	8	8	NUM
ejpam-6011	412	3	]	]	X
ejpam-6011	412	4	h	h	NOUN
ejpam-6011	412	5	chen	chen	PROPN
ejpam-6011	412	6	,	,	PUNCT
ejpam-6011	412	7	o	o	PROPN
ejpam-6011	412	8	gurgun	gurgun	PROPN
ejpam-6011	412	9	,	,	PUNCT
ejpam-6011	412	10	s	s	VERB
ejpam-6011	412	11	halicioglu	halicioglu	NOUN
ejpam-6011	412	12	,	,	PUNCT
ejpam-6011	412	13	and	and	CCONJ
ejpam-6011	412	14	a	a	DET
ejpam-6011	412	15	harmanci	harmanci	NOUN
ejpam-6011	412	16	.	.	PUNCT
ejpam-6011	413	1	rings	ring	NOUN
ejpam-6011	413	2	in	in	ADP
ejpam-6011	413	3	which	which	PRON
ejpam-6011	413	4	nilpotents	nilpotent	NOUN
ejpam-6011	413	5	belong	belong	VERB
ejpam-6011	413	6	to	to	ADP
ejpam-6011	413	7	jacobson	jacobson	PROPN
ejpam-6011	413	8	radical	radical	PROPN
ejpam-6011	413	9	.	.	PUNCT
ejpam-6011	414	1	an	an	DET
ejpam-6011	414	2	.	.	NOUN
ejpam-6011	414	3	stiint	stiint	PROPN
ejpam-6011	414	4	.	.	PUNCT
ejpam-6011	415	1	univ	univ	PROPN
ejpam-6011	415	2	.	.	PUNCT
ejpam-6011	416	1	al	al	PROPN
ejpam-6011	416	2	.	.	PROPN
ejpam-6011	416	3	i.	i.	PROPN
ejpam-6011	416	4	cuza	cuza	PROPN
ejpam-6011	416	5	iasi	iasi	PROPN
ejpam-6011	416	6	.	.	PUNCT
ejpam-6011	417	1	mat.(ns	mat.(ns	NOUN
ejpam-6011	417	2	)	)	PUNCT
ejpam-6011	418	1	,	,	PUNCT
ejpam-6011	418	2	62(2):595–606	62(2):595–606	PROPN
ejpam-6011	418	3	,	,	PUNCT
ejpam-6011	418	4	2016	2016	NUM
ejpam-6011	418	5	.	.	PUNCT
ejpam-6011	419	1	[	[	X
ejpam-6011	419	2	9	9	NUM
ejpam-6011	419	3	]	]	X
ejpam-6011	419	4	chang	chang	PROPN
ejpam-6011	419	5	ik	ik	PROPN
ejpam-6011	419	6	lee	lee	PROPN
ejpam-6011	419	7	and	and	CCONJ
ejpam-6011	419	8	soo	soo	PROPN
ejpam-6011	419	9	yong	yong	PROPN
ejpam-6011	419	10	park	park	PROPN
ejpam-6011	419	11	.	.	PUNCT
ejpam-6011	420	1	when	when	SCONJ
ejpam-6011	420	2	nilpotents	nilpotent	NOUN
ejpam-6011	420	3	are	be	AUX
ejpam-6011	420	4	contained	contain	VERB
ejpam-6011	420	5	in	in	ADP
ejpam-6011	420	6	jacobson	jacobson	PROPN
ejpam-6011	420	7	radicals	radicals	PROPN
ejpam-6011	420	8	.	.	PUNCT
ejpam-6011	421	1	journal	journal	NOUN
ejpam-6011	421	2	of	of	ADP
ejpam-6011	421	3	the	the	DET
ejpam-6011	421	4	korean	korean	PROPN
ejpam-6011	421	5	mathematical	mathematical	ADJ
ejpam-6011	421	6	society	society	NOUN
ejpam-6011	421	7	,	,	PUNCT
ejpam-6011	421	8	55(5):1193–1205	55(5):1193–1205	NUM
ejpam-6011	421	9	,	,	PUNCT
ejpam-6011	421	10	2018	2018	NUM
ejpam-6011	421	11	.	.	PUNCT
ejpam-6011	422	1	[	[	X
ejpam-6011	422	2	10	10	NUM
ejpam-6011	422	3	]	]	PUNCT
ejpam-6011	422	4	weixing	weixe	VERB
ejpam-6011	422	5	chen	chen	PROPN
ejpam-6011	422	6	.	.	PUNCT
ejpam-6011	423	1	on	on	ADP
ejpam-6011	423	2	linearly	linearly	ADV
ejpam-6011	423	3	weak	weak	ADJ
ejpam-6011	423	4	armendariz	armendariz	ADJ
ejpam-6011	423	5	rings	ring	NOUN
ejpam-6011	423	6	.	.	PUNCT
ejpam-6011	424	1	journal	journal	PROPN
ejpam-6011	424	2	of	of	ADP
ejpam-6011	424	3	pure	pure	ADJ
ejpam-6011	424	4	and	and	CCONJ
ejpam-6011	424	5	applied	applied	ADJ
ejpam-6011	424	6	algebra	algebra	NOUN
ejpam-6011	424	7	,	,	PUNCT
ejpam-6011	424	8	219(4):1122–1130	219(4):1122–1130	NUM
ejpam-6011	424	9	,	,	PUNCT
ejpam-6011	424	10	2015	2015	NUM
ejpam-6011	424	11	.	.	PUNCT
ejpam-6011	425	1	[	[	X
ejpam-6011	425	2	11	11	NUM
ejpam-6011	425	3	]	]	X
ejpam-6011	425	4	junchao	junchao	PROPN
ejpam-6011	425	5	wei	wei	PROPN
ejpam-6011	425	6	.	.	PUNCT
ejpam-6011	426	1	certain	certain	ADJ
ejpam-6011	426	2	rings	ring	NOUN
ejpam-6011	426	3	whose	whose	DET
ejpam-6011	426	4	simple	simple	ADJ
ejpam-6011	426	5	singular	singular	ADJ
ejpam-6011	426	6	modules	module	NOUN
ejpam-6011	426	7	are	be	AUX
ejpam-6011	426	8	nil	nil	ADJ
ejpam-6011	426	9	-	-	PUNCT
ejpam-6011	426	10	injective	injective	ADJ
ejpam-6011	426	11	.	.	PUNCT
ejpam-6011	427	1	turkish	turkish	ADJ
ejpam-6011	427	2	journal	journal	NOUN
ejpam-6011	427	3	of	of	ADP
ejpam-6011	427	4	mathematics	mathematic	NOUN
ejpam-6011	427	5	,	,	PUNCT
ejpam-6011	427	6	32(4):393–408	32(4):393–408	NUM
ejpam-6011	427	7	,	,	PUNCT
ejpam-6011	427	8	2008	2008	NUM
ejpam-6011	427	9	.	.	PUNCT
ejpam-6011	428	1	[	[	X
ejpam-6011	428	2	12	12	NUM
ejpam-6011	428	3	]	]	X
ejpam-6011	428	4	orhan	orhan	PROPN
ejpam-6011	428	5	gurgun	gurgun	PROPN
ejpam-6011	428	6	,	,	PUNCT
ejpam-6011	428	7	sait	sait	PROPN
ejpam-6011	428	8	halicioglu	halicioglu	PROPN
ejpam-6011	428	9	ungor	ungor	PROPN
ejpam-6011	428	10	,	,	PUNCT
ejpam-6011	428	11	et	et	PROPN
ejpam-6011	428	12	al	al	PROPN
ejpam-6011	428	13	.	.	PUNCT
ejpam-6011	429	1	a	a	DET
ejpam-6011	429	2	subclass	subclass	NOUN
ejpam-6011	429	3	of	of	ADP
ejpam-6011	429	4	strongly	strongly	ADV
ejpam-6011	429	5	clean	clean	ADJ
ejpam-6011	429	6	rings	ring	NOUN
ejpam-6011	429	7	.	.	PUNCT
ejpam-6011	430	1	communications	communication	NOUN
ejpam-6011	430	2	in	in	ADP
ejpam-6011	430	3	mathematics	mathematic	NOUN
ejpam-6011	430	4	,	,	PUNCT
ejpam-6011	430	5	23(1):13–31	23(1):13–31	NUM
ejpam-6011	430	6	,	,	PUNCT
ejpam-6011	430	7	2015	2015	NUM
ejpam-6011	430	8	.	.	PUNCT
ejpam-6011	431	1	[	[	X
ejpam-6011	431	2	13	13	NUM
ejpam-6011	431	3	]	]	PUNCT
ejpam-6011	431	4	burcu	burcu	NOUN
ejpam-6011	431	5	ungor	ungor	PROPN
ejpam-6011	431	6	,	,	PUNCT
ejpam-6011	431	7	orhan	orhan	PROPN
ejpam-6011	431	8	gurgun	gurgun	PROPN
ejpam-6011	431	9	,	,	PUNCT
ejpam-6011	431	10	sait	sait	PROPN
ejpam-6011	431	11	halicioglu	halicioglu	PROPN
ejpam-6011	431	12	,	,	PUNCT
ejpam-6011	431	13	and	and	CCONJ
ejpam-6011	431	14	abdullah	abdullah	PROPN
ejpam-6011	431	15	harmanci	harmanci	PROPN
ejpam-6011	431	16	.	.	PUNCT
ejpam-6011	432	1	feckly	feckly	ADV
ejpam-6011	432	2	reduced	reduce	VERB
ejpam-6011	432	3	rings	ring	NOUN
ejpam-6011	432	4	.	.	PUNCT
ejpam-6011	433	1	hacettepe	hacettepe	PROPN
ejpam-6011	433	2	journal	journal	PROPN
ejpam-6011	433	3	of	of	ADP
ejpam-6011	433	4	mathematics	mathematic	NOUN
ejpam-6011	433	5	and	and	CCONJ
ejpam-6011	433	6	statistics	statistic	NOUN
ejpam-6011	433	7	,	,	PUNCT
ejpam-6011	433	8	44(2):375–384	44(2):375–384	PROPN
ejpam-6011	433	9	,	,	PUNCT
ejpam-6011	433	10	2015	2015	NUM
ejpam-6011	433	11	.	.	PUNCT
ejpam-6011	434	1	[	[	X
ejpam-6011	434	2	14	14	NUM
ejpam-6011	434	3	]	]	X
ejpam-6011	434	4	ramon	ramon	PROPN
ejpam-6011	434	5	antoine	antoine	PROPN
ejpam-6011	434	6	.	.	PUNCT
ejpam-6011	435	1	nilpotent	nilpotent	ADJ
ejpam-6011	435	2	elements	element	NOUN
ejpam-6011	435	3	and	and	CCONJ
ejpam-6011	435	4	armendariz	armendariz	ADJ
ejpam-6011	435	5	rings	ring	NOUN
ejpam-6011	435	6	.	.	PUNCT
ejpam-6011	436	1	journal	journal	PROPN
ejpam-6011	436	2	of	of	ADP
ejpam-6011	436	3	algebra	algebra	PROPN
ejpam-6011	436	4	,	,	PUNCT
ejpam-6011	436	5	319(8):3128–3140	319(8):3128–3140	NUM
ejpam-6011	436	6	,	,	PUNCT
ejpam-6011	436	7	2008	2008	NUM
ejpam-6011	436	8	.	.	PUNCT
ejpam-6011	437	1	[	[	X
ejpam-6011	437	2	15	15	NUM
ejpam-6011	437	3	]	]	X
ejpam-6011	437	4	philippe	philippe	PROPN
ejpam-6011	437	5	loustaunau	loustaunau	PROPN
ejpam-6011	437	6	and	and	CCONJ
ejpam-6011	437	7	jay	jay	PROPN
ejpam-6011	437	8	shapiro	shapiro	PROPN
ejpam-6011	437	9	.	.	PUNCT
ejpam-6011	437	10	morita	morita	PROPN
ejpam-6011	437	11	contexts	contexts	PROPN
ejpam-6011	437	12	.	.	PUNCT
ejpam-6011	438	1	in	in	ADP
ejpam-6011	438	2	non	non	ADJ
ejpam-6011	438	3	-	-	ADJ
ejpam-6011	438	4	commutative	commutative	ADJ
ejpam-6011	438	5	ring	ring	NOUN
ejpam-6011	438	6	theory	theory	NOUN
ejpam-6011	438	7	:	:	PUNCT
ejpam-6011	438	8	proceedings	proceeding	NOUN
ejpam-6011	438	9	of	of	ADP
ejpam-6011	438	10	a	a	DET
ejpam-6011	438	11	conference	conference	NOUN
ejpam-6011	438	12	held	hold	VERB
ejpam-6011	438	13	in	in	ADP
ejpam-6011	438	14	athens	athens	PROPN
ejpam-6011	438	15	,	,	PUNCT
ejpam-6011	438	16	ohio	ohio	PROPN
ejpam-6011	438	17	sept	sept	PROPN
ejpam-6011	438	18	.	.	PROPN
ejpam-6011	438	19	29–30	29–30	NUM
ejpam-6011	438	20	,	,	PUNCT
ejpam-6011	438	21	1989	1989	NUM
ejpam-6011	438	22	,	,	PUNCT
ejpam-6011	438	23	pages	page	NOUN
ejpam-6011	438	24	80–92	80–92	PROPN
ejpam-6011	438	25	.	.	PUNCT
ejpam-6011	438	26	springer	springer	NOUN
ejpam-6011	438	27	,	,	PUNCT
ejpam-6011	438	28	2006	2006	NUM
ejpam-6011	438	29	.	.	PUNCT
ejpam-6011	439	1	[	[	X
ejpam-6011	439	2	16	16	NUM
ejpam-6011	439	3	]	]	X
ejpam-6011	439	4	kiiti	kiiti	PROPN
ejpam-6011	439	5	morita	morita	PROPN
ejpam-6011	439	6	.	.	PUNCT
ejpam-6011	440	1	duality	duality	NOUN
ejpam-6011	440	2	for	for	ADP
ejpam-6011	440	3	modules	module	NOUN
ejpam-6011	440	4	and	and	CCONJ
ejpam-6011	440	5	its	its	PRON
ejpam-6011	440	6	applications	application	NOUN
ejpam-6011	440	7	to	to	ADP
ejpam-6011	440	8	the	the	DET
ejpam-6011	440	9	theory	theory	NOUN
ejpam-6011	440	10	of	of	ADP
ejpam-6011	440	11	rings	ring	NOUN
ejpam-6011	440	12	with	with	ADP
ejpam-6011	440	13	minimum	minimum	ADJ
ejpam-6011	440	14	condition	condition	NOUN
ejpam-6011	440	15	.	.	PUNCT
ejpam-6011	441	1	science	science	NOUN
ejpam-6011	441	2	reports	report	NOUN
ejpam-6011	441	3	of	of	ADP
ejpam-6011	441	4	the	the	DET
ejpam-6011	441	5	tokyo	tokyo	PROPN
ejpam-6011	441	6	kyoiku	kyoiku	PROPN
ejpam-6011	441	7	daigaku	daigaku	PROPN
ejpam-6011	441	8	,	,	PUNCT
ejpam-6011	441	9	section	section	NOUN
ejpam-6011	441	10	a	a	PRON
ejpam-6011	441	11	,	,	PUNCT
ejpam-6011	441	12	6(150):83	6(150):83	NUM
ejpam-6011	441	13	–	–	PUNCT
ejpam-6011	441	14	142	142	NUM
ejpam-6011	441	15	,	,	PUNCT
ejpam-6011	441	16	1958	1958	NUM
ejpam-6011	441	17	.	.	PUNCT
ejpam-6011	442	1	[	[	X
ejpam-6011	442	2	17	17	NUM
ejpam-6011	442	3	]	]	X
ejpam-6011	442	4	shimshon	shimshon	PROPN
ejpam-6011	442	5	a.	a.	NOUN
ejpam-6011	442	6	amitsur	amitsur	PROPN
ejpam-6011	442	7	.	.	PUNCT
ejpam-6011	443	1	rings	ring	NOUN
ejpam-6011	443	2	of	of	ADP
ejpam-6011	443	3	quotients	quotient	NOUN
ejpam-6011	443	4	and	and	CCONJ
ejpam-6011	443	5	morita	morita	PROPN
ejpam-6011	443	6	contexts	contexts	PROPN
ejpam-6011	443	7	.	.	PUNCT
ejpam-6011	444	1	journal	journal	NOUN
ejpam-6011	444	2	of	of	ADP
ejpam-6011	444	3	algebra	algebra	PROPN
ejpam-6011	444	4	,	,	PUNCT
ejpam-6011	444	5	17(2):273–298	17(2):273–298	NUM
ejpam-6011	444	6	,	,	PUNCT
ejpam-6011	444	7	1971	1971	NUM
ejpam-6011	444	8	.	.	PUNCT
ejpam-6011	445	1	[	[	X
ejpam-6011	445	2	18	18	NUM
ejpam-6011	445	3	]	]	X
ejpam-6011	445	4	piotr	piotr	PROPN
ejpam-6011	445	5	krylov	krylov	PROPN
ejpam-6011	445	6	and	and	CCONJ
ejpam-6011	445	7	askar	askar	PROPN
ejpam-6011	445	8	tuganbaev	tuganbaev	NOUN
ejpam-6011	445	9	.	.	PUNCT
ejpam-6011	446	1	formal	formal	ADJ
ejpam-6011	446	2	matrices	matrix	NOUN
ejpam-6011	446	3	,	,	PUNCT
ejpam-6011	446	4	volume	volume	NOUN
ejpam-6011	446	5	23	23	NUM
ejpam-6011	446	6	.	.	PUNCT
ejpam-6011	446	7	springer	springer	NOUN
ejpam-6011	446	8	,	,	PUNCT
ejpam-6011	446	9	2017	2017	NUM
ejpam-6011	446	10	.	.	PUNCT
ejpam-6011	447	1	[	[	X
ejpam-6011	447	2	19	19	NUM
ejpam-6011	447	3	]	]	X
ejpam-6011	447	4	muhammad	muhammad	PROPN
ejpam-6011	447	5	saad	saad	PROPN
ejpam-6011	447	6	,	,	PUNCT
ejpam-6011	447	7	usama	usama	PROPN
ejpam-6011	447	8	a	a	DET
ejpam-6011	447	9	aburawash	aburawash	NOUN
ejpam-6011	447	10	,	,	PUNCT
ejpam-6011	447	11	ahmed	ahmed	PROPN
ejpam-6011	447	12	ma	ma	PROPN
ejpam-6011	447	13	el	el	PROPN
ejpam-6011	447	14	-	-	PUNCT
ejpam-6011	447	15	sayed	say	VERB
ejpam-6011	447	16	,	,	PUNCT
ejpam-6011	447	17	and	and	CCONJ
ejpam-6011	447	18	nour	nour	PROPN
ejpam-6011	447	19	nabil	nabil	PROPN
ejpam-6011	447	20	.	.	PUNCT
ejpam-6011	448	1	an	an	DET
ejpam-6011	448	2	introduction	introduction	NOUN
ejpam-6011	448	3	to	to	ADP
ejpam-6011	448	4	i	i	PROPN
ejpam-6011	448	5	-	-	PUNCT
ejpam-6011	448	6	commutative	commutative	ADJ
ejpam-6011	448	7	rings	ring	NOUN
ejpam-6011	448	8	.	.	PUNCT
ejpam-6011	449	1	mathematics	mathematic	NOUN
ejpam-6011	449	2	,	,	PUNCT
ejpam-6011	449	3	13(2):253	13(2):253	NUM
ejpam-6011	449	4	,	,	PUNCT
ejpam-6011	449	5	2025	2025	NUM
ejpam-6011	449	6	.	.	PUNCT
ejpam-6011	450	1	[	[	X
ejpam-6011	450	2	20	20	NUM
ejpam-6011	450	3	]	]	SYM
ejpam-6011	450	4	müller	müller	PROPN
ejpam-6011	450	5	marianne	marianne	PROPN
ejpam-6011	450	6	.	.	PUNCT
ejpam-6011	450	7	rings	ring	NOUN
ejpam-6011	450	8	of	of	ADP
ejpam-6011	450	9	quotients	quotient	NOUN
ejpam-6011	450	10	of	of	ADP
ejpam-6011	450	11	generalized	generalized	ADJ
ejpam-6011	450	12	matrix	matrix	NOUN
ejpam-6011	450	13	rings	ring	NOUN
ejpam-6011	450	14	.	.	PUNCT
ejpam-6011	451	1	communications	communication	NOUN
ejpam-6011	451	2	in	in	ADP
ejpam-6011	451	3	algebra	algebra	NOUN
ejpam-6011	451	4	,	,	PUNCT
ejpam-6011	451	5	15(10):1991–2015	15(10):1991–2015	NUM
ejpam-6011	451	6	,	,	PUNCT
ejpam-6011	451	7	1987	1987	NUM
ejpam-6011	451	8	.	.	PUNCT
ejpam-6011	452	1	[	[	X
ejpam-6011	452	2	21	21	NUM
ejpam-6011	452	3	]	]	X
ejpam-6011	452	4	gaohua	gaohua	PROPN
ejpam-6011	452	5	tang	tang	PROPN
ejpam-6011	452	6	,	,	PUNCT
ejpam-6011	452	7	chunna	chunna	NOUN
ejpam-6011	452	8	li	li	PROPN
ejpam-6011	452	9	,	,	PUNCT
ejpam-6011	452	10	and	and	CCONJ
ejpam-6011	452	11	yiqiang	yiqiang	PROPN
ejpam-6011	452	12	zhou	zhou	PROPN
ejpam-6011	452	13	.	.	PUNCT
ejpam-6011	453	1	study	study	PROPN
ejpam-6011	453	2	of	of	ADP
ejpam-6011	453	3	morita	morita	PROPN
ejpam-6011	453	4	contexts	contexts	PROPN
ejpam-6011	453	5	.	.	PUNCT
ejpam-6011	454	1	communications	communication	NOUN
ejpam-6011	454	2	in	in	ADP
ejpam-6011	454	3	algebra	algebra	NOUN
ejpam-6011	454	4	,	,	PUNCT
ejpam-6011	454	5	42(4):1668–1681	42(4):1668–1681	NUM
ejpam-6011	454	6	,	,	PUNCT
ejpam-6011	454	7	2014	2014	NUM
ejpam-6011	454	8	.	.	PUNCT
ejpam-6011	455	1	[	[	X
ejpam-6011	455	2	22	22	NUM
ejpam-6011	455	3	]	]	X
ejpam-6011	455	4	robert	robert	PROPN
ejpam-6011	455	5	m	m	PROPN
ejpam-6011	455	6	fossum	fossum	PROPN
ejpam-6011	455	7	,	,	PUNCT
ejpam-6011	455	8	phillip	phillip	VERB
ejpam-6011	455	9	a	a	DET
ejpam-6011	455	10	griffith	griffith	PROPN
ejpam-6011	455	11	,	,	PUNCT
ejpam-6011	455	12	and	and	CCONJ
ejpam-6011	455	13	idun	idun	PROPN
ejpam-6011	455	14	reiten	reiten	VERB
ejpam-6011	455	15	.	.	PUNCT
ejpam-6011	456	1	trivial	trivial	ADJ
ejpam-6011	456	2	extensions	extension	NOUN
ejpam-6011	456	3	of	of	ADP
ejpam-6011	456	4	abelian	abelian	ADJ
ejpam-6011	456	5	categories	category	NOUN
ejpam-6011	456	6	:	:	PUNCT
ejpam-6011	456	7	homological	homological	ADJ
ejpam-6011	456	8	algebra	algebra	NOUN
ejpam-6011	456	9	of	of	ADP
ejpam-6011	456	10	trivial	trivial	ADJ
ejpam-6011	456	11	extensions	extension	NOUN
ejpam-6011	456	12	of	of	ADP
ejpam-6011	456	13	abelian	abelian	ADJ
ejpam-6011	456	14	catergories	catergorie	NOUN
ejpam-6011	456	15	with	with	ADP
ejpam-6011	456	16	applications	application	NOUN
ejpam-6011	456	17	to	to	PART
ejpam-6011	456	18	ring	ring	NOUN
ejpam-6011	456	19	theory	theory	NOUN
ejpam-6011	456	20	,	,	PUNCT
ejpam-6011	456	21	volume	volume	NOUN
ejpam-6011	456	22	456	456	NUM
ejpam-6011	456	23	.	.	PUNCT
ejpam-6011	457	1	springer	springer	NOUN
ejpam-6011	457	2	,	,	PUNCT
ejpam-6011	457	3	2006	2006	NUM
ejpam-6011	457	4	.	.	PUNCT