id	sid	tid	token	lemma	pos
ejpam-5066	1	1	european	european	PROPN
ejpam-5066	1	2	journal	journal	PROPN
ejpam-5066	1	3	of	of	ADP
ejpam-5066	1	4	pure	pure	ADJ
ejpam-5066	1	5	and	and	CCONJ
ejpam-5066	1	6	applied	apply	VERB
ejpam-5066	1	7	mathematics	mathematic	NOUN
ejpam-5066	1	8	vol	vol	NOUN
ejpam-5066	1	9	.	.	PROPN
ejpam-5066	2	1	17	17	NUM
ejpam-5066	2	2	,	,	PUNCT
ejpam-5066	2	3	no	no	INTJ
ejpam-5066	2	4	.	.	NOUN
ejpam-5066	2	5	2	2	NUM
ejpam-5066	2	6	,	,	PUNCT
ejpam-5066	2	7	2024	2024	NUM
ejpam-5066	2	8	,	,	PUNCT
ejpam-5066	2	9	736	736	NUM
ejpam-5066	2	10	-	-	SYM
ejpam-5066	2	11	752	752	NUM
ejpam-5066	2	12	issn	issn	PROPN
ejpam-5066	2	13	1307	1307	NUM
ejpam-5066	2	14	-	-	SYM
ejpam-5066	2	15	5543	5543	NUM
ejpam-5066	2	16	–	–	PUNCT
ejpam-5066	3	1	ejpam.com	ejpam.com	X
ejpam-5066	3	2	published	publish	VERB
ejpam-5066	3	3	by	by	ADP
ejpam-5066	3	4	new	new	PROPN
ejpam-5066	3	5	york	york	PROPN
ejpam-5066	3	6	business	business	PROPN
ejpam-5066	3	7	global	global	ADJ
ejpam-5066	3	8	extending	extend	VERB
ejpam-5066	3	9	abelian	abelian	ADJ
ejpam-5066	3	10	rings	ring	NOUN
ejpam-5066	3	11	:	:	PUNCT
ejpam-5066	3	12	a	a	DET
ejpam-5066	3	13	generalized	generalized	ADJ
ejpam-5066	3	14	approach	approach	NOUN
ejpam-5066	3	15	muhammad	muhammad	PROPN
ejpam-5066	3	16	saad	saad	PROPN
ejpam-5066	3	17	1	1	NUM
ejpam-5066	3	18	,	,	PUNCT
ejpam-5066	3	19	majed	majed	PROPN
ejpam-5066	3	20	zailaee2,∗	zailaee2,∗	PROPN
ejpam-5066	3	21	1	1	NUM
ejpam-5066	3	22	department	department	PROPN
ejpam-5066	3	23	of	of	ADP
ejpam-5066	3	24	mathematics	mathematic	NOUN
ejpam-5066	3	25	and	and	CCONJ
ejpam-5066	3	26	computer	computer	NOUN
ejpam-5066	3	27	science	science	NOUN
ejpam-5066	3	28	,	,	PUNCT
ejpam-5066	3	29	faculty	faculty	NOUN
ejpam-5066	3	30	of	of	ADP
ejpam-5066	3	31	science	science	NOUN
ejpam-5066	3	32	,	,	PUNCT
ejpam-5066	3	33	alexandria	alexandria	PROPN
ejpam-5066	3	34	university	university	PROPN
ejpam-5066	3	35	,	,	PUNCT
ejpam-5066	3	36	alexandria	alexandria	PROPN
ejpam-5066	3	37	,	,	PUNCT
ejpam-5066	3	38	egypt	egypt	PROPN
ejpam-5066	3	39	2	2	NUM
ejpam-5066	3	40	department	department	NOUN
ejpam-5066	3	41	of	of	ADP
ejpam-5066	3	42	mathematics	mathematic	NOUN
ejpam-5066	3	43	,	,	PUNCT
ejpam-5066	3	44	king	king	PROPN
ejpam-5066	3	45	abdulaziz	abdulaziz	PROPN
ejpam-5066	3	46	university	university	PROPN
ejpam-5066	3	47	,	,	PUNCT
ejpam-5066	3	48	rabigh	rabigh	VERB
ejpam-5066	3	49	,	,	PUNCT
ejpam-5066	3	50	saudi	saudi	PROPN
ejpam-5066	3	51	arabia	arabia	PROPN
ejpam-5066	3	52	abstract	abstract	NOUN
ejpam-5066	3	53	.	.	PUNCT
ejpam-5066	4	1	we	we	PRON
ejpam-5066	4	2	introduce	introduce	VERB
ejpam-5066	4	3	a	a	DET
ejpam-5066	4	4	novel	novel	ADJ
ejpam-5066	4	5	framework	framework	NOUN
ejpam-5066	4	6	for	for	ADP
ejpam-5066	4	7	assessing	assess	VERB
ejpam-5066	4	8	the	the	DET
ejpam-5066	4	9	centrality	centrality	NOUN
ejpam-5066	4	10	of	of	ADP
ejpam-5066	4	11	idempotents	idempotent	NOUN
ejpam-5066	4	12	within	within	ADP
ejpam-5066	4	13	a	a	DET
ejpam-5066	4	14	ring	ring	NOUN
ejpam-5066	4	15	by	by	ADP
ejpam-5066	4	16	presenting	present	VERB
ejpam-5066	4	17	a	a	DET
ejpam-5066	4	18	general	general	ADJ
ejpam-5066	4	19	concept	concept	NOUN
ejpam-5066	4	20	that	that	PRON
ejpam-5066	4	21	assigns	assign	VERB
ejpam-5066	4	22	a	a	DET
ejpam-5066	4	23	degree	degree	NOUN
ejpam-5066	4	24	of	of	ADP
ejpam-5066	4	25	centrality	centrality	NOUN
ejpam-5066	4	26	.	.	PUNCT
ejpam-5066	5	1	this	this	DET
ejpam-5066	5	2	approach	approach	NOUN
ejpam-5066	5	3	aligns	align	VERB
ejpam-5066	5	4	with	with	ADP
ejpam-5066	5	5	the	the	DET
ejpam-5066	5	6	previously	previously	ADV
ejpam-5066	5	7	established	establish	VERB
ejpam-5066	5	8	notions	notion	NOUN
ejpam-5066	5	9	of	of	ADP
ejpam-5066	5	10	semicentral	semicentral	ADJ
ejpam-5066	5	11	and	and	CCONJ
ejpam-5066	5	12	q	q	ADJ
ejpam-5066	5	13	-	-	ADJ
ejpam-5066	5	14	central	central	ADJ
ejpam-5066	5	15	idempotents	idempotent	NOUN
ejpam-5066	5	16	by	by	ADP
ejpam-5066	5	17	birkenmeier	birkenmeier	NOUN
ejpam-5066	5	18	and	and	CCONJ
ejpam-5066	5	19	lam	lam	PROPN
ejpam-5066	5	20	.	.	PUNCT
ejpam-5066	6	1	specifically	specifically	ADV
ejpam-5066	6	2	,	,	PUNCT
ejpam-5066	6	3	we	we	PRON
ejpam-5066	6	4	define	define	VERB
ejpam-5066	6	5	an	an	DET
ejpam-5066	6	6	idempotent	idempotent	ADJ
ejpam-5066	6	7	e	e	NOUN
ejpam-5066	6	8	in	in	ADP
ejpam-5066	6	9	a	a	DET
ejpam-5066	6	10	ring	ring	NOUN
ejpam-5066	6	11	r	r	NOUN
ejpam-5066	6	12	to	to	PART
ejpam-5066	6	13	be	be	AUX
ejpam-5066	6	14	n	n	ADV
ejpam-5066	6	15	-	-	PUNCT
ejpam-5066	6	16	central	central	ADJ
ejpam-5066	6	17	,	,	PUNCT
ejpam-5066	6	18	where	where	SCONJ
ejpam-5066	6	19	n	n	PRON
ejpam-5066	6	20	is	be	AUX
ejpam-5066	6	21	a	a	DET
ejpam-5066	6	22	positive	positive	ADJ
ejpam-5066	6	23	integer	integer	NOUN
ejpam-5066	6	24	,	,	PUNCT
ejpam-5066	6	25	if	if	SCONJ
ejpam-5066	6	26	[	[	X
ejpam-5066	6	27	e	e	NOUN
ejpam-5066	6	28	,	,	PUNCT
ejpam-5066	6	29	r]ne	r]ne	NOUN
ejpam-5066	6	30	=	=	SYM
ejpam-5066	6	31	0	0	NUM
ejpam-5066	6	32	,	,	PUNCT
ejpam-5066	6	33	where	where	SCONJ
ejpam-5066	6	34	[	[	X
ejpam-5066	6	35	x	x	X
ejpam-5066	6	36	,	,	PUNCT
ejpam-5066	6	37	y	y	PROPN
ejpam-5066	6	38	]	]	PUNCT
ejpam-5066	6	39	represents	represent	VERB
ejpam-5066	6	40	the	the	DET
ejpam-5066	6	41	additive	additive	ADJ
ejpam-5066	6	42	commutator	commutator	NOUN
ejpam-5066	6	43	xy−yx	xy−yx	PROPN
ejpam-5066	6	44	.	.	PUNCT
ejpam-5066	7	1	if	if	SCONJ
ejpam-5066	7	2	every	every	DET
ejpam-5066	7	3	idempotent	idempotent	NOUN
ejpam-5066	7	4	in	in	ADP
ejpam-5066	7	5	a	a	DET
ejpam-5066	7	6	ring	ring	NOUN
ejpam-5066	7	7	r	r	NOUN
ejpam-5066	7	8	is	be	AUX
ejpam-5066	7	9	n	n	CCONJ
ejpam-5066	7	10	-	-	PUNCT
ejpam-5066	7	11	central	central	ADJ
ejpam-5066	7	12	,	,	PUNCT
ejpam-5066	7	13	we	we	PRON
ejpam-5066	7	14	refer	refer	VERB
ejpam-5066	7	15	to	to	ADP
ejpam-5066	7	16	r	r	NOUN
ejpam-5066	7	17	as	as	ADP
ejpam-5066	7	18	n	n	NOUN
ejpam-5066	7	19	-	-	PUNCT
ejpam-5066	7	20	abelian	abelian	NOUN
ejpam-5066	7	21	.	.	PUNCT
ejpam-5066	8	1	our	our	PRON
ejpam-5066	8	2	study	study	NOUN
ejpam-5066	8	3	lays	lay	VERB
ejpam-5066	8	4	the	the	DET
ejpam-5066	8	5	groundwork	groundwork	NOUN
ejpam-5066	8	6	by	by	ADP
ejpam-5066	8	7	presenting	present	VERB
ejpam-5066	8	8	foundational	foundational	ADJ
ejpam-5066	8	9	results	result	NOUN
ejpam-5066	8	10	that	that	PRON
ejpam-5066	8	11	support	support	VERB
ejpam-5066	8	12	this	this	DET
ejpam-5066	8	13	concept	concept	NOUN
ejpam-5066	8	14	and	and	CCONJ
ejpam-5066	8	15	examines	examine	VERB
ejpam-5066	8	16	key	key	ADJ
ejpam-5066	8	17	features	feature	NOUN
ejpam-5066	8	18	of	of	ADP
ejpam-5066	8	19	n	n	CCONJ
ejpam-5066	8	20	-	-	PUNCT
ejpam-5066	8	21	central	central	ADJ
ejpam-5066	8	22	idempotents	idempotent	NOUN
ejpam-5066	8	23	essential	essential	ADJ
ejpam-5066	8	24	for	for	ADP
ejpam-5066	8	25	appropriately	appropriately	ADV
ejpam-5066	8	26	categorizing	categorize	VERB
ejpam-5066	8	27	n	n	CCONJ
ejpam-5066	8	28	-	-	PUNCT
ejpam-5066	8	29	abelian	abelian	NOUN
ejpam-5066	8	30	rings	ring	NOUN
ejpam-5066	8	31	among	among	ADP
ejpam-5066	8	32	various	various	ADJ
ejpam-5066	8	33	generalizations	generalization	NOUN
ejpam-5066	8	34	of	of	ADP
ejpam-5066	8	35	abelian	abelian	PROPN
ejpam-5066	8	36	rings	ring	NOUN
ejpam-5066	8	37	introduced	introduce	VERB
ejpam-5066	8	38	in	in	ADP
ejpam-5066	8	39	prior	prior	ADJ
ejpam-5066	8	40	literature	literature	NOUN
ejpam-5066	8	41	.	.	PUNCT
ejpam-5066	9	1	we	we	PRON
ejpam-5066	9	2	provide	provide	VERB
ejpam-5066	9	3	examples	example	NOUN
ejpam-5066	9	4	of	of	ADP
ejpam-5066	9	5	n	n	CCONJ
ejpam-5066	9	6	-	-	PUNCT
ejpam-5066	9	7	central	central	ADJ
ejpam-5066	9	8	idempotents	idempotent	NOUN
ejpam-5066	9	9	that	that	PRON
ejpam-5066	9	10	do	do	AUX
ejpam-5066	9	11	not	not	PART
ejpam-5066	9	12	fall	fall	VERB
ejpam-5066	9	13	under	under	ADP
ejpam-5066	9	14	the	the	DET
ejpam-5066	9	15	categories	category	NOUN
ejpam-5066	9	16	of	of	ADP
ejpam-5066	9	17	semicentral	semicentral	ADJ
ejpam-5066	9	18	or	or	CCONJ
ejpam-5066	9	19	q	q	ADJ
ejpam-5066	9	20	-	-	ADJ
ejpam-5066	9	21	central	central	ADJ
ejpam-5066	9	22	.	.	PUNCT
ejpam-5066	10	1	furthermore	furthermore	ADV
ejpam-5066	10	2	,	,	PUNCT
ejpam-5066	10	3	we	we	PRON
ejpam-5066	10	4	demonstrate	demonstrate	VERB
ejpam-5066	10	5	that	that	SCONJ
ejpam-5066	10	6	the	the	DET
ejpam-5066	10	7	ring	ring	NOUN
ejpam-5066	10	8	of	of	ADP
ejpam-5066	10	9	upper	upper	ADJ
ejpam-5066	10	10	matrices	matrix	NOUN
ejpam-5066	10	11	tn(r	tn(r	PRON
ejpam-5066	10	12	)	)	PUNCT
ejpam-5066	10	13	,	,	PUNCT
ejpam-5066	10	14	where	where	SCONJ
ejpam-5066	10	15	r	r	NOUN
ejpam-5066	10	16	is	be	AUX
ejpam-5066	10	17	abelian	abelian	ADJ
ejpam-5066	10	18	,	,	PUNCT
ejpam-5066	10	19	is	be	AUX
ejpam-5066	10	20	an	an	DET
ejpam-5066	10	21	n	n	CCONJ
ejpam-5066	10	22	-	-	PUNCT
ejpam-5066	10	23	abelian	abelian	NOUN
ejpam-5066	10	24	.	.	PUNCT
ejpam-5066	11	1	we	we	PRON
ejpam-5066	11	2	also	also	ADV
ejpam-5066	11	3	prove	prove	VERB
ejpam-5066	11	4	that	that	SCONJ
ejpam-5066	11	5	a	a	DET
ejpam-5066	11	6	ring	ring	NOUN
ejpam-5066	11	7	where	where	SCONJ
ejpam-5066	11	8	all	all	PRON
ejpam-5066	11	9	of	of	ADP
ejpam-5066	11	10	its	its	PRON
ejpam-5066	11	11	idempotents	idempotent	NOUN
ejpam-5066	11	12	are	be	AUX
ejpam-5066	11	13	n	n	PRON
ejpam-5066	11	14	-	-	PUNCT
ejpam-5066	11	15	central	central	NOUN
ejpam-5066	11	16	is	be	AUX
ejpam-5066	11	17	an	an	DET
ejpam-5066	11	18	exchange	exchange	NOUN
ejpam-5066	11	19	ring	ring	NOUN
ejpam-5066	11	20	if	if	SCONJ
ejpam-5066	11	21	and	and	CCONJ
ejpam-5066	11	22	only	only	ADV
ejpam-5066	11	23	if	if	SCONJ
ejpam-5066	11	24	the	the	DET
ejpam-5066	11	25	ring	ring	NOUN
ejpam-5066	11	26	is	be	AUX
ejpam-5066	11	27	clean	clean	ADJ
ejpam-5066	11	28	.	.	PUNCT
ejpam-5066	12	1	2020	2020	NUM
ejpam-5066	12	2	mathematics	mathematic	NOUN
ejpam-5066	12	3	subject	subject	NOUN
ejpam-5066	12	4	classifications	classification	NOUN
ejpam-5066	12	5	:	:	PUNCT
ejpam-5066	12	6	16u60	16u60	NUM
ejpam-5066	12	7	,	,	PUNCT
ejpam-5066	12	8	16u70	16u70	NUM
ejpam-5066	12	9	,	,	PUNCT
ejpam-5066	12	10	16u80	16u80	NUM
ejpam-5066	12	11	,	,	PUNCT
ejpam-5066	12	12	16e50	16e50	NUM
ejpam-5066	12	13	,	,	PUNCT
ejpam-5066	12	14	16u80	16u80	NUM
ejpam-5066	12	15	key	key	ADJ
ejpam-5066	12	16	words	word	NOUN
ejpam-5066	12	17	and	and	CCONJ
ejpam-5066	12	18	phrases	phrase	NOUN
ejpam-5066	12	19	:	:	PUNCT
ejpam-5066	12	20	idempotent	idempotent	ADJ
ejpam-5066	12	21	,	,	PUNCT
ejpam-5066	12	22	semicentral	semicentral	ADJ
ejpam-5066	12	23	;	;	PUNCT
ejpam-5066	12	24	q	q	ADJ
ejpam-5066	12	25	-	-	ADJ
ejpam-5066	12	26	central	central	ADJ
ejpam-5066	12	27	,	,	PUNCT
ejpam-5066	12	28	n	n	CCONJ
ejpam-5066	12	29	-	-	PUNCT
ejpam-5066	12	30	central	central	ADJ
ejpam-5066	12	31	,	,	PUNCT
ejpam-5066	12	32	n	n	CCONJ
ejpam-5066	12	33	-	-	PUNCT
ejpam-5066	12	34	abelian	abelian	ADJ
ejpam-5066	12	35	1	1	NUM
ejpam-5066	12	36	.	.	PUNCT
ejpam-5066	13	1	introduction	introduction	NOUN
ejpam-5066	13	2	by	by	ADP
ejpam-5066	13	3	the	the	DET
ejpam-5066	13	4	term	term	NOUN
ejpam-5066	13	5	“	"	PUNCT
ejpam-5066	13	6	ring	ring	NOUN
ejpam-5066	13	7	”	"	PUNCT
ejpam-5066	13	8	,	,	PUNCT
ejpam-5066	13	9	we	we	PRON
ejpam-5066	13	10	mean	mean	VERB
ejpam-5066	13	11	an	an	DET
ejpam-5066	13	12	associative	associative	ADJ
ejpam-5066	13	13	ring	ring	NOUN
ejpam-5066	13	14	with	with	ADP
ejpam-5066	13	15	nonzero	nonzero	PROPN
ejpam-5066	13	16	identity	identity	NOUN
ejpam-5066	13	17	.	.	PUNCT
ejpam-5066	14	1	further	far	ADV
ejpam-5066	14	2	,	,	PUNCT
ejpam-5066	14	3	z(r	z(r	NOUN
ejpam-5066	14	4	)	)	PUNCT
ejpam-5066	14	5	,	,	PUNCT
ejpam-5066	14	6	i(r	i(r	PROPN
ejpam-5066	14	7	)	)	PUNCT
ejpam-5066	14	8	,	,	PUNCT
ejpam-5066	14	9	u(r	u(r	PROPN
ejpam-5066	14	10	)	)	PUNCT
ejpam-5066	14	11	,	,	PUNCT
ejpam-5066	14	12	and	and	CCONJ
ejpam-5066	14	13	n	n	CCONJ
ejpam-5066	14	14	(	(	PUNCT
ejpam-5066	14	15	r	r	NOUN
ejpam-5066	14	16	)	)	PUNCT
ejpam-5066	14	17	are	be	AUX
ejpam-5066	14	18	used	use	VERB
ejpam-5066	14	19	for	for	ADP
ejpam-5066	14	20	the	the	DET
ejpam-5066	14	21	set	set	ADJ
ejpam-5066	14	22	central	central	ADJ
ejpam-5066	14	23	elements	element	NOUN
ejpam-5066	14	24	,	,	PUNCT
ejpam-5066	14	25	the	the	DET
ejpam-5066	14	26	set	set	NOUN
ejpam-5066	14	27	of	of	ADP
ejpam-5066	14	28	idempotents	idempotent	NOUN
ejpam-5066	14	29	(	(	PUNCT
ejpam-5066	14	30	that	that	PRON
ejpam-5066	14	31	is	be	AUX
ejpam-5066	14	32	e2	e2	PROPN
ejpam-5066	14	33	=	=	SYM
ejpam-5066	14	34	e	e	PROPN
ejpam-5066	14	35	)	)	PUNCT
ejpam-5066	14	36	,	,	PUNCT
ejpam-5066	14	37	the	the	DET
ejpam-5066	14	38	set	set	NOUN
ejpam-5066	14	39	of	of	ADP
ejpam-5066	14	40	invertible	invertible	ADJ
ejpam-5066	14	41	(	(	PUNCT
ejpam-5066	14	42	unit	unit	NOUN
ejpam-5066	14	43	)	)	PUNCT
ejpam-5066	14	44	elements	element	NOUN
ejpam-5066	14	45	,	,	PUNCT
ejpam-5066	14	46	and	and	CCONJ
ejpam-5066	14	47	the	the	DET
ejpam-5066	14	48	set	set	NOUN
ejpam-5066	14	49	of	of	ADP
ejpam-5066	14	50	nilpotent	nilpotent	ADJ
ejpam-5066	14	51	elements	element	NOUN
ejpam-5066	14	52	of	of	ADP
ejpam-5066	14	53	a	a	DET
ejpam-5066	14	54	ring	ring	NOUN
ejpam-5066	14	55	r.	r.	NOUN
ejpam-5066	14	56	an	an	DET
ejpam-5066	14	57	idempotent	idempotent	ADJ
ejpam-5066	14	58	e	e	NOUN
ejpam-5066	14	59	of	of	ADP
ejpam-5066	14	60	a	a	DET
ejpam-5066	14	61	ring	ring	NOUN
ejpam-5066	14	62	r	r	NOUN
ejpam-5066	14	63	is	be	AUX
ejpam-5066	14	64	called	call	VERB
ejpam-5066	14	65	central	central	ADJ
ejpam-5066	14	66	if	if	SCONJ
ejpam-5066	14	67	e	e	PROPN
ejpam-5066	14	68	∈	∈	PROPN
ejpam-5066	14	69	z(r	z(r	PROPN
ejpam-5066	14	70	)	)	PUNCT
ejpam-5066	14	71	.	.	PUNCT
ejpam-5066	15	1	the	the	DET
ejpam-5066	15	2	set	set	PROPN
ejpam-5066	15	3	b(r	b(r	PROPN
ejpam-5066	15	4	)	)	PUNCT
ejpam-5066	15	5	denotes	denote	VERB
ejpam-5066	15	6	the	the	DET
ejpam-5066	15	7	set	set	NOUN
ejpam-5066	15	8	of	of	ADP
ejpam-5066	15	9	all	all	DET
ejpam-5066	15	10	central	central	ADJ
ejpam-5066	15	11	idempotents	idempotent	NOUN
ejpam-5066	15	12	of	of	ADP
ejpam-5066	15	13	r.	r.	PROPN
ejpam-5066	15	14	a	a	DET
ejpam-5066	15	15	ring	ring	NOUN
ejpam-5066	15	16	r	r	NOUN
ejpam-5066	15	17	is	be	AUX
ejpam-5066	15	18	called	call	VERB
ejpam-5066	15	19	abelian	abelian	ADJ
ejpam-5066	15	20	if	if	SCONJ
ejpam-5066	15	21	all	all	DET
ejpam-5066	15	22	idempotents	idempotent	NOUN
ejpam-5066	15	23	of	of	ADP
ejpam-5066	15	24	r	r	NOUN
ejpam-5066	15	25	are	be	AUX
ejpam-5066	15	26	central	central	ADJ
ejpam-5066	15	27	;	;	PUNCT
ejpam-5066	15	28	that	that	PRON
ejpam-5066	15	29	i(r	i(r	NOUN
ejpam-5066	15	30	)	)	PUNCT
ejpam-5066	15	31	=	=	SYM
ejpam-5066	15	32	b(r	b(r	NOUN
ejpam-5066	15	33	)	)	PUNCT
ejpam-5066	15	34	.	.	PUNCT
ejpam-5066	16	1	throughout	throughout	ADP
ejpam-5066	16	2	this	this	DET
ejpam-5066	16	3	paper	paper	NOUN
ejpam-5066	16	4	,	,	PUNCT
ejpam-5066	16	5	we	we	PRON
ejpam-5066	16	6	will	will	AUX
ejpam-5066	16	7	always	always	ADV
ejpam-5066	16	8	notate	notate	VERB
ejpam-5066	16	9	the	the	DET
ejpam-5066	16	10	ring	ring	NOUN
ejpam-5066	16	11	of	of	ADP
ejpam-5066	16	12	n×n	n×n	PROPN
ejpam-5066	16	13	upper	upper	ADJ
ejpam-5066	16	14	triangular	triangular	NOUN
ejpam-5066	16	15	matrices	matrix	NOUN
ejpam-5066	16	16	over	over	ADP
ejpam-5066	16	17	a	a	DET
ejpam-5066	16	18	ring	ring	NOUN
ejpam-5066	16	19	r	r	NOUN
ejpam-5066	16	20	by	by	ADP
ejpam-5066	16	21	tn(r	tn(r	NUM
ejpam-5066	16	22	)	)	PUNCT
ejpam-5066	16	23	.	.	PUNCT
ejpam-5066	17	1	the	the	DET
ejpam-5066	17	2	concept	concept	NOUN
ejpam-5066	17	3	of	of	ADP
ejpam-5066	17	4	semicentrality	semicentrality	NOUN
ejpam-5066	17	5	of	of	ADP
ejpam-5066	17	6	idempotents	idempotent	NOUN
ejpam-5066	17	7	was	be	AUX
ejpam-5066	17	8	first	first	ADV
ejpam-5066	17	9	introduced	introduce	VERB
ejpam-5066	17	10	by	by	ADP
ejpam-5066	17	11	birkenmeier	birkenmeier	NOUN
ejpam-5066	17	12	in	in	ADP
ejpam-5066	17	13	1983	1983	NUM
ejpam-5066	17	14	[	[	X
ejpam-5066	17	15	1	1	NUM
ejpam-5066	17	16	]	]	PUNCT
ejpam-5066	17	17	as	as	ADP
ejpam-5066	17	18	a	a	DET
ejpam-5066	17	19	form	form	NOUN
ejpam-5066	17	20	of	of	ADP
ejpam-5066	17	21	one	one	NUM
ejpam-5066	17	22	-	-	PUNCT
ejpam-5066	17	23	sided	sided	ADJ
ejpam-5066	17	24	centrality	centrality	NOUN
ejpam-5066	17	25	to	to	PART
ejpam-5066	17	26	generalize	generalize	VERB
ejpam-5066	17	27	some	some	DET
ejpam-5066	17	28	results	result	NOUN
ejpam-5066	17	29	on	on	ADP
ejpam-5066	17	30	von	von	PROPN
ejpam-5066	17	31	neumann	neumann	PROPN
ejpam-5066	17	32	regular	regular	PROPN
ejpam-5066	17	33	rings	ring	NOUN
ejpam-5066	17	34	.	.	PUNCT
ejpam-5066	18	1	the	the	DET
ejpam-5066	18	2	semicentral	semicentral	ADJ
ejpam-5066	18	3	idempotent	idempotent	NOUN
ejpam-5066	18	4	has	have	AUX
ejpam-5066	18	5	since	since	ADV
ejpam-5066	18	6	been	be	AUX
ejpam-5066	18	7	used	use	VERB
ejpam-5066	18	8	in	in	ADP
ejpam-5066	18	9	extensions	extension	NOUN
ejpam-5066	18	10	of	of	ADP
ejpam-5066	18	11	rings	ring	NOUN
ejpam-5066	18	12	and	and	CCONJ
ejpam-5066	18	13	∗corresponding	∗corresponde	VERB
ejpam-5066	18	14	author	author	NOUN
ejpam-5066	18	15	.	.	PUNCT
ejpam-5066	19	1	doi	doi	NOUN
ejpam-5066	19	2	:	:	PUNCT
ejpam-5066	19	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5066	https://doi.org/10.29020/nybg.ejpam.v17i2.5066	VERB
ejpam-5066	19	4	email	email	NOUN
ejpam-5066	19	5	addresses	address	NOUN
ejpam-5066	19	6	:	:	PUNCT
ejpam-5066	19	7	m.saad@alexu.edu.eg	m.saad@alexu.edu.eg	PROPN
ejpam-5066	19	8	(	(	PUNCT
ejpam-5066	19	9	m.	m.	PROPN
ejpam-5066	19	10	saad	saad	PROPN
ejpam-5066	19	11	)	)	PUNCT
ejpam-5066	19	12	,	,	PUNCT
ejpam-5066	19	13	mzailaee@kau.edu.sa	mzailaee@kau.edu.sa	PROPN
ejpam-5066	19	14	(	(	PUNCT
ejpam-5066	19	15	m.	m.	NOUN
ejpam-5066	19	16	zailaee	zailaee	NOUN
ejpam-5066	19	17	)	)	PUNCT
ejpam-5066	19	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5066	20	1	736	736	NUM
ejpam-5066	21	1	©	©	ADP
ejpam-5066	21	2	2024	2024	NUM
ejpam-5066	21	3	ejpam	ejpam	NOUN
ejpam-5066	21	4	all	all	DET
ejpam-5066	21	5	rights	right	NOUN
ejpam-5066	21	6	reserved	reserve	VERB
ejpam-5066	21	7	.	.	PUNCT
ejpam-5066	22	1	m.	m.	NOUN
ejpam-5066	22	2	saad	saad	PROPN
ejpam-5066	22	3	,	,	PUNCT
ejpam-5066	22	4	m.	m.	NOUN
ejpam-5066	22	5	zailaee	zailaee	PROPN
ejpam-5066	22	6	/	/	SYM
ejpam-5066	22	7	eur	eur	PROPN
ejpam-5066	22	8	.	.	PUNCT
ejpam-5066	23	1	j.	j.	PROPN
ejpam-5066	23	2	pure	pure	PROPN
ejpam-5066	23	3	appl	appl	PROPN
ejpam-5066	23	4	.	.	PROPN
ejpam-5066	23	5	math	math	PROPN
ejpam-5066	23	6	,	,	PUNCT
ejpam-5066	23	7	17	17	NUM
ejpam-5066	23	8	(	(	PUNCT
ejpam-5066	23	9	2	2	NUM
ejpam-5066	23	10	)	)	PUNCT
ejpam-5066	23	11	(	(	PUNCT
ejpam-5066	23	12	2024	2024	NUM
ejpam-5066	23	13	)	)	PUNCT
ejpam-5066	23	14	,	,	PUNCT
ejpam-5066	23	15	736	736	NUM
ejpam-5066	23	16	-	-	SYM
ejpam-5066	23	17	752	752	NUM
ejpam-5066	23	18	737	737	NUM
ejpam-5066	23	19	modules	module	NOUN
ejpam-5066	23	20	by	by	ADP
ejpam-5066	23	21	birkenmeier	birkenmeier	NOUN
ejpam-5066	23	22	and	and	CCONJ
ejpam-5066	23	23	others	other	NOUN
ejpam-5066	23	24	.	.	PUNCT
ejpam-5066	24	1	an	an	DET
ejpam-5066	24	2	idempotent	idempotent	ADJ
ejpam-5066	24	3	e	e	NOUN
ejpam-5066	24	4	of	of	ADP
ejpam-5066	24	5	a	a	DET
ejpam-5066	24	6	ring	ring	NOUN
ejpam-5066	24	7	r	r	NOUN
ejpam-5066	24	8	is	be	AUX
ejpam-5066	24	9	considered	consider	VERB
ejpam-5066	24	10	left	left	ADJ
ejpam-5066	24	11	(	(	PUNCT
ejpam-5066	24	12	resp	resp	NOUN
ejpam-5066	24	13	.	.	PUNCT
ejpam-5066	25	1	right	right	ADJ
ejpam-5066	25	2	)	)	PUNCT
ejpam-5066	25	3	semicentral	semicentral	ADJ
ejpam-5066	26	1	if	if	SCONJ
ejpam-5066	26	2	ae	ae	PROPN
ejpam-5066	26	3	=	=	PROPN
ejpam-5066	26	4	eae	eae	PROPN
ejpam-5066	26	5	(	(	PUNCT
ejpam-5066	26	6	resp	resp	PROPN
ejpam-5066	26	7	.	.	PUNCT
ejpam-5066	26	8	,	,	PUNCT
ejpam-5066	26	9	ea	ea	X
ejpam-5066	26	10	=	=	SYM
ejpam-5066	26	11	eae	eae	PROPN
ejpam-5066	26	12	)	)	PUNCT
ejpam-5066	26	13	for	for	ADP
ejpam-5066	26	14	all	all	DET
ejpam-5066	26	15	a	a	DET
ejpam-5066	26	16	∈	∈	PROPN
ejpam-5066	26	17	r.	r.	NOUN
ejpam-5066	26	18	the	the	DET
ejpam-5066	26	19	sets	set	NOUN
ejpam-5066	26	20	of	of	ADP
ejpam-5066	26	21	left	left	ADJ
ejpam-5066	26	22	and	and	CCONJ
ejpam-5066	26	23	right	right	ADJ
ejpam-5066	26	24	semicentral	semicentral	ADJ
ejpam-5066	26	25	idempotents	idempotent	NOUN
ejpam-5066	26	26	are	be	AUX
ejpam-5066	26	27	denoted	denote	VERB
ejpam-5066	26	28	sl(r	sl(r	NOUN
ejpam-5066	26	29	)	)	PUNCT
ejpam-5066	26	30	and	and	CCONJ
ejpam-5066	26	31	sr(r	sr(r	NOUN
ejpam-5066	26	32	)	)	PUNCT
ejpam-5066	26	33	,	,	PUNCT
ejpam-5066	26	34	respectively	respectively	ADV
ejpam-5066	26	35	.	.	PUNCT
ejpam-5066	27	1	those	those	PRON
ejpam-5066	27	2	interested	interested	ADJ
ejpam-5066	27	3	in	in	ADP
ejpam-5066	27	4	delving	delve	VERB
ejpam-5066	27	5	deeper	deeply	ADV
ejpam-5066	27	6	into	into	ADP
ejpam-5066	27	7	semicentral	semicentral	ADJ
ejpam-5066	27	8	idempotents	idempotent	NOUN
ejpam-5066	27	9	and	and	CCONJ
ejpam-5066	27	10	q	q	ADJ
ejpam-5066	27	11	-	-	ADJ
ejpam-5066	27	12	central	central	ADJ
ejpam-5066	27	13	properties	property	NOUN
ejpam-5066	27	14	can	can	AUX
ejpam-5066	27	15	find	find	VERB
ejpam-5066	27	16	valuable	valuable	ADJ
ejpam-5066	27	17	insights	insight	NOUN
ejpam-5066	27	18	in	in	ADP
ejpam-5066	27	19	modern	modern	ADJ
ejpam-5066	27	20	references	reference	NOUN
ejpam-5066	27	21	,	,	PUNCT
ejpam-5066	27	22	such	such	ADJ
ejpam-5066	27	23	as	as	ADP
ejpam-5066	27	24	[	[	X
ejpam-5066	27	25	2	2	NUM
ejpam-5066	27	26	,	,	PUNCT
ejpam-5066	27	27	3	3	NUM
ejpam-5066	27	28	,	,	PUNCT
ejpam-5066	27	29	7	7	NUM
ejpam-5066	27	30	,	,	PUNCT
ejpam-5066	27	31	13	13	NUM
ejpam-5066	27	32	,	,	PUNCT
ejpam-5066	27	33	17	17	NUM
ejpam-5066	27	34	,	,	PUNCT
ejpam-5066	27	35	26	26	NUM
ejpam-5066	27	36	]	]	PUNCT
ejpam-5066	27	37	.	.	PUNCT
ejpam-5066	28	1	an	an	DET
ejpam-5066	28	2	idempotent	idempotent	NOUN
ejpam-5066	28	3	of	of	ADP
ejpam-5066	28	4	a	a	DET
ejpam-5066	28	5	ring	ring	NOUN
ejpam-5066	28	6	r	r	NOUN
ejpam-5066	28	7	is	be	AUX
ejpam-5066	28	8	considered	consider	VERB
ejpam-5066	28	9	central	central	ADJ
ejpam-5066	28	10	if	if	SCONJ
ejpam-5066	28	11	it	it	PRON
ejpam-5066	28	12	is	be	AUX
ejpam-5066	28	13	both	both	PRON
ejpam-5066	28	14	left	left	ADJ
ejpam-5066	28	15	and	and	CCONJ
ejpam-5066	28	16	right	right	ADJ
ejpam-5066	28	17	semicentral	semicentral	ADJ
ejpam-5066	28	18	,	,	PUNCT
ejpam-5066	28	19	and	and	CCONJ
ejpam-5066	28	20	the	the	DET
ejpam-5066	28	21	set	set	NOUN
ejpam-5066	28	22	of	of	ADP
ejpam-5066	28	23	central	central	ADJ
ejpam-5066	28	24	idempotents	idempotent	NOUN
ejpam-5066	28	25	is	be	AUX
ejpam-5066	28	26	denoted	denote	VERB
ejpam-5066	28	27	b(r	b(r	NOUN
ejpam-5066	28	28	)	)	PUNCT
ejpam-5066	28	29	=	=	SYM
ejpam-5066	28	30	sl(r)∩sr(r	sl(r)∩sr(r	PROPN
ejpam-5066	28	31	)	)	PUNCT
ejpam-5066	28	32	.	.	PUNCT
ejpam-5066	29	1	an	an	DET
ejpam-5066	29	2	idempotent	idempotent	NOUN
ejpam-5066	29	3	is	be	AUX
ejpam-5066	29	4	called	call	VERB
ejpam-5066	29	5	semicentral	semicentral	ADJ
ejpam-5066	29	6	if	if	SCONJ
ejpam-5066	29	7	it	it	PRON
ejpam-5066	29	8	is	be	AUX
ejpam-5066	29	9	either	either	CCONJ
ejpam-5066	29	10	left	left	ADJ
ejpam-5066	29	11	or	or	CCONJ
ejpam-5066	29	12	right	right	ADJ
ejpam-5066	29	13	semicentral	semicentral	ADJ
ejpam-5066	29	14	.	.	PUNCT
ejpam-5066	30	1	a	a	DET
ejpam-5066	30	2	ring	ring	NOUN
ejpam-5066	30	3	r	r	NOUN
ejpam-5066	30	4	is	be	AUX
ejpam-5066	30	5	called	call	VERB
ejpam-5066	30	6	semi	semi	ADJ
ejpam-5066	30	7	-	-	ADJ
ejpam-5066	30	8	abelian	abelian	ADJ
ejpam-5066	30	9	if	if	SCONJ
ejpam-5066	30	10	all	all	DET
ejpam-5066	30	11	idempotents	idempotent	NOUN
ejpam-5066	30	12	of	of	ADP
ejpam-5066	30	13	r	r	NOUN
ejpam-5066	30	14	are	be	AUX
ejpam-5066	30	15	semicentral	semicentral	ADJ
ejpam-5066	30	16	.	.	PUNCT
ejpam-5066	31	1	the	the	DET
ejpam-5066	31	2	notion	notion	NOUN
ejpam-5066	31	3	of	of	ADP
ejpam-5066	31	4	idempotent	idempotent	ADJ
ejpam-5066	31	5	centrality	centrality	NOUN
ejpam-5066	31	6	,	,	PUNCT
ejpam-5066	31	7	which	which	PRON
ejpam-5066	31	8	generalizes	generalize	VERB
ejpam-5066	31	9	that	that	SCONJ
ejpam-5066	31	10	of	of	ADP
ejpam-5066	31	11	semicentral	semicentral	ADJ
ejpam-5066	31	12	idempotents	idempotent	NOUN
ejpam-5066	31	13	,	,	PUNCT
ejpam-5066	31	14	was	be	AUX
ejpam-5066	31	15	introduced	introduce	VERB
ejpam-5066	31	16	by	by	ADP
ejpam-5066	31	17	lam	lam	PROPN
ejpam-5066	31	18	in	in	ADP
ejpam-5066	31	19	[	[	X
ejpam-5066	31	20	11	11	NUM
ejpam-5066	31	21	]	]	PUNCT
ejpam-5066	31	22	as	as	ADP
ejpam-5066	31	23	q	q	ADJ
ejpam-5066	31	24	-	-	ADJ
ejpam-5066	31	25	central	central	ADJ
ejpam-5066	31	26	idempotents	idempotent	NOUN
ejpam-5066	31	27	.	.	PUNCT
ejpam-5066	32	1	an	an	DET
ejpam-5066	32	2	idempotent	idempotent	ADJ
ejpam-5066	32	3	e	e	NOUN
ejpam-5066	32	4	of	of	ADP
ejpam-5066	32	5	a	a	DET
ejpam-5066	32	6	ring	ring	NOUN
ejpam-5066	32	7	r	r	NOUN
ejpam-5066	32	8	is	be	AUX
ejpam-5066	32	9	called	call	VERB
ejpam-5066	32	10	q	q	ADJ
ejpam-5066	32	11	-	-	ADJ
ejpam-5066	32	12	central	central	ADJ
ejpam-5066	32	13	if	if	SCONJ
ejpam-5066	32	14	er(1−e)re	er(1−e)re	PROPN
ejpam-5066	32	15	=	=	SYM
ejpam-5066	32	16	0	0	NUM
ejpam-5066	32	17	,	,	PUNCT
ejpam-5066	32	18	and	and	CCONJ
ejpam-5066	32	19	the	the	DET
ejpam-5066	32	20	set	set	NOUN
ejpam-5066	32	21	of	of	ADP
ejpam-5066	32	22	all	all	DET
ejpam-5066	32	23	q	q	ADJ
ejpam-5066	32	24	-	-	ADJ
ejpam-5066	32	25	central	central	ADJ
ejpam-5066	32	26	idempotents	idempotent	NOUN
ejpam-5066	32	27	of	of	ADP
ejpam-5066	32	28	r	r	NOUN
ejpam-5066	32	29	is	be	AUX
ejpam-5066	32	30	denoted	denote	VERB
ejpam-5066	32	31	q	q	NOUN
ejpam-5066	32	32	-	-	PUNCT
ejpam-5066	32	33	idem(r	idem(r	NOUN
ejpam-5066	32	34	)	)	PUNCT
ejpam-5066	32	35	.	.	PUNCT
ejpam-5066	33	1	if	if	SCONJ
ejpam-5066	33	2	every	every	DET
ejpam-5066	33	3	idempotent	idempotent	NOUN
ejpam-5066	33	4	of	of	ADP
ejpam-5066	33	5	a	a	DET
ejpam-5066	33	6	ring	ring	NOUN
ejpam-5066	33	7	r	r	NOUN
ejpam-5066	33	8	is	be	AUX
ejpam-5066	33	9	q	q	ADJ
ejpam-5066	33	10	-	-	ADJ
ejpam-5066	33	11	central	central	ADJ
ejpam-5066	33	12	,	,	PUNCT
ejpam-5066	33	13	then	then	ADV
ejpam-5066	33	14	r	r	NOUN
ejpam-5066	33	15	is	be	AUX
ejpam-5066	33	16	called	call	VERB
ejpam-5066	33	17	q	q	NOUN
ejpam-5066	33	18	-	-	PUNCT
ejpam-5066	33	19	abelian	abelian	ADJ
ejpam-5066	33	20	.	.	PUNCT
ejpam-5066	34	1	this	this	DET
ejpam-5066	34	2	condition	condition	NOUN
ejpam-5066	34	3	has	have	AUX
ejpam-5066	34	4	been	be	AUX
ejpam-5066	34	5	introduced	introduce	VERB
ejpam-5066	34	6	in	in	ADP
ejpam-5066	34	7	several	several	ADJ
ejpam-5066	34	8	works	work	NOUN
ejpam-5066	34	9	,	,	PUNCT
ejpam-5066	34	10	such	such	ADJ
ejpam-5066	34	11	as	as	ADP
ejpam-5066	34	12	[	[	X
ejpam-5066	34	13	22	22	NUM
ejpam-5066	34	14	,	,	PUNCT
ejpam-5066	34	15	24	24	NUM
ejpam-5066	34	16	]	]	PUNCT
ejpam-5066	34	17	,	,	PUNCT
ejpam-5066	34	18	but	but	CCONJ
ejpam-5066	34	19	lam	lam	PROPN
ejpam-5066	34	20	was	be	AUX
ejpam-5066	34	21	the	the	DET
ejpam-5066	34	22	first	first	ADJ
ejpam-5066	34	23	to	to	PART
ejpam-5066	34	24	name	name	VERB
ejpam-5066	34	25	and	and	CCONJ
ejpam-5066	34	26	study	study	VERB
ejpam-5066	34	27	this	this	DET
ejpam-5066	34	28	property	property	NOUN
ejpam-5066	34	29	as	as	ADP
ejpam-5066	34	30	an	an	DET
ejpam-5066	34	31	elemental	elemental	ADJ
ejpam-5066	34	32	property	property	NOUN
ejpam-5066	34	33	.	.	PUNCT
ejpam-5066	35	1	a	a	DET
ejpam-5066	35	2	ring	ring	NOUN
ejpam-5066	35	3	r	r	NOUN
ejpam-5066	35	4	is	be	AUX
ejpam-5066	35	5	called	call	VERB
ejpam-5066	35	6	q	q	NOUN
ejpam-5066	35	7	-	-	PUNCT
ejpam-5066	35	8	abelian	abelian	ADJ
ejpam-5066	35	9	if	if	SCONJ
ejpam-5066	35	10	every	every	DET
ejpam-5066	35	11	element	element	NOUN
ejpam-5066	35	12	of	of	ADP
ejpam-5066	35	13	r	r	NOUN
ejpam-5066	35	14	is	be	AUX
ejpam-5066	35	15	q	q	ADJ
ejpam-5066	35	16	-	-	ADJ
ejpam-5066	35	17	central	central	ADJ
ejpam-5066	35	18	,	,	PUNCT
ejpam-5066	35	19	which	which	PRON
ejpam-5066	35	20	is	be	AUX
ejpam-5066	35	21	referred	refer	VERB
ejpam-5066	35	22	to	to	ADP
ejpam-5066	35	23	as	as	ADP
ejpam-5066	35	24	a	a	DET
ejpam-5066	35	25	quasi	quasi	ADJ
ejpam-5066	35	26	-	-	ADJ
ejpam-5066	35	27	normal	normal	ADJ
ejpam-5066	35	28	ring	ring	NOUN
ejpam-5066	35	29	and	and	CCONJ
ejpam-5066	35	30	defined	define	VERB
ejpam-5066	35	31	in	in	ADP
ejpam-5066	35	32	[	[	X
ejpam-5066	35	33	23	23	NUM
ejpam-5066	35	34	]	]	PUNCT
ejpam-5066	35	35	.	.	PUNCT
ejpam-5066	36	1	it	it	PRON
ejpam-5066	36	2	is	be	AUX
ejpam-5066	36	3	worth	worth	ADJ
ejpam-5066	36	4	noting	note	VERB
ejpam-5066	36	5	that	that	SCONJ
ejpam-5066	36	6	every	every	DET
ejpam-5066	36	7	semicentral	semicentral	ADJ
ejpam-5066	36	8	idempotent	idempotent	NOUN
ejpam-5066	36	9	is	be	AUX
ejpam-5066	36	10	q	q	ADJ
ejpam-5066	36	11	-	-	ADJ
ejpam-5066	36	12	central	central	ADJ
ejpam-5066	36	13	;	;	PUNCT
ejpam-5066	36	14	therefore	therefore	ADV
ejpam-5066	36	15	,	,	PUNCT
ejpam-5066	36	16	every	every	DET
ejpam-5066	36	17	semi	semi	ADJ
ejpam-5066	36	18	-	-	ADJ
ejpam-5066	36	19	abelian	abelian	ADJ
ejpam-5066	36	20	ring	ring	NOUN
ejpam-5066	36	21	is	be	AUX
ejpam-5066	36	22	q	q	ADJ
ejpam-5066	36	23	-	-	PUNCT
ejpam-5066	36	24	abelian	abelian	ADJ
ejpam-5066	36	25	.	.	PUNCT
ejpam-5066	37	1	this	this	DET
ejpam-5066	37	2	paper	paper	NOUN
ejpam-5066	37	3	is	be	AUX
ejpam-5066	37	4	structured	structure	VERB
ejpam-5066	37	5	into	into	ADP
ejpam-5066	37	6	three	three	NUM
ejpam-5066	37	7	primary	primary	ADJ
ejpam-5066	37	8	sections	section	NOUN
ejpam-5066	37	9	.	.	PUNCT
ejpam-5066	38	1	the	the	DET
ejpam-5066	38	2	initial	initial	ADJ
ejpam-5066	38	3	segment	segment	NOUN
ejpam-5066	38	4	is	be	AUX
ejpam-5066	38	5	dedicated	dedicate	VERB
ejpam-5066	38	6	to	to	ADP
ejpam-5066	38	7	a	a	DET
ejpam-5066	38	8	comprehensive	comprehensive	ADJ
ejpam-5066	38	9	elucidation	elucidation	NOUN
ejpam-5066	38	10	of	of	ADP
ejpam-5066	38	11	the	the	DET
ejpam-5066	38	12	concept	concept	NOUN
ejpam-5066	38	13	of	of	ADP
ejpam-5066	38	14	n	n	CCONJ
ejpam-5066	38	15	-	-	PUNCT
ejpam-5066	38	16	central	central	ADJ
ejpam-5066	38	17	idempotents	idempotent	NOUN
ejpam-5066	38	18	.	.	PUNCT
ejpam-5066	39	1	within	within	ADP
ejpam-5066	39	2	this	this	DET
ejpam-5066	39	3	section	section	NOUN
ejpam-5066	39	4	,	,	PUNCT
ejpam-5066	39	5	we	we	PRON
ejpam-5066	39	6	undertake	undertake	VERB
ejpam-5066	39	7	a	a	DET
ejpam-5066	39	8	rigorous	rigorous	ADJ
ejpam-5066	39	9	exploration	exploration	NOUN
ejpam-5066	39	10	to	to	PART
ejpam-5066	39	11	establish	establish	VERB
ejpam-5066	39	12	the	the	DET
ejpam-5066	39	13	existence	existence	NOUN
ejpam-5066	39	14	of	of	ADP
ejpam-5066	39	15	n	n	CCONJ
ejpam-5066	39	16	-	-	PUNCT
ejpam-5066	39	17	central	central	ADJ
ejpam-5066	39	18	idempotents	idempotent	NOUN
ejpam-5066	39	19	distinct	distinct	ADJ
ejpam-5066	39	20	from	from	ADP
ejpam-5066	39	21	semicentral	semicentral	ADJ
ejpam-5066	39	22	and	and	CCONJ
ejpam-5066	39	23	q	q	ADJ
ejpam-5066	39	24	-	-	ADJ
ejpam-5066	39	25	central	central	ADJ
ejpam-5066	39	26	idempotents	idempotent	NOUN
ejpam-5066	39	27	,	,	PUNCT
ejpam-5066	39	28	as	as	SCONJ
ejpam-5066	39	29	substantiated	substantiate	VERB
ejpam-5066	39	30	by	by	ADP
ejpam-5066	39	31	illustrative	illustrative	ADJ
ejpam-5066	39	32	examples	example	NOUN
ejpam-5066	39	33	referenced	reference	VERB
ejpam-5066	39	34	as	as	ADP
ejpam-5066	39	35	1	1	NUM
ejpam-5066	39	36	.	.	PUNCT
ejpam-5066	40	1	furthermore	furthermore	ADV
ejpam-5066	40	2	,	,	PUNCT
ejpam-5066	40	3	proposition	proposition	NOUN
ejpam-5066	40	4	4	4	NUM
ejpam-5066	40	5	rigorously	rigorously	ADV
ejpam-5066	40	6	establishes	establish	VERB
ejpam-5066	40	7	that	that	SCONJ
ejpam-5066	40	8	every	every	DET
ejpam-5066	40	9	n	n	CCONJ
ejpam-5066	40	10	-	-	PUNCT
ejpam-5066	40	11	central	central	ADJ
ejpam-5066	40	12	idempotent	idempotent	NOUN
ejpam-5066	40	13	within	within	ADP
ejpam-5066	40	14	a	a	DET
ejpam-5066	40	15	semiprime	semiprime	NOUN
ejpam-5066	40	16	ring	ring	NOUN
ejpam-5066	40	17	unequivocally	unequivocally	ADV
ejpam-5066	40	18	assumes	assume	VERB
ejpam-5066	40	19	a	a	DET
ejpam-5066	40	20	central	central	ADJ
ejpam-5066	40	21	position	position	NOUN
ejpam-5066	40	22	.	.	PUNCT
ejpam-5066	41	1	the	the	DET
ejpam-5066	41	2	culmination	culmination	NOUN
ejpam-5066	41	3	of	of	ADP
ejpam-5066	41	4	this	this	DET
ejpam-5066	41	5	section	section	NOUN
ejpam-5066	41	6	lies	lie	VERB
ejpam-5066	41	7	in	in	ADP
ejpam-5066	41	8	the	the	DET
ejpam-5066	41	9	proof	proof	NOUN
ejpam-5066	41	10	establishing	establish	VERB
ejpam-5066	41	11	that	that	SCONJ
ejpam-5066	41	12	if	if	SCONJ
ejpam-5066	41	13	idempotents	idempotent	NOUN
ejpam-5066	41	14	e	e	PROPN
ejpam-5066	41	15	and	and	CCONJ
ejpam-5066	41	16	f	f	PROPN
ejpam-5066	41	17	are	be	AUX
ejpam-5066	41	18	conjugate	conjugate	ADJ
ejpam-5066	41	19	,	,	PUNCT
ejpam-5066	41	20	then	then	ADV
ejpam-5066	41	21	e	e	NOUN
ejpam-5066	41	22	assumes	assume	VERB
ejpam-5066	41	23	centrality	centrality	NOUN
ejpam-5066	41	24	if	if	SCONJ
ejpam-5066	41	25	and	and	CCONJ
ejpam-5066	41	26	only	only	ADV
ejpam-5066	41	27	if	if	SCONJ
ejpam-5066	41	28	f	f	PROPN
ejpam-5066	41	29	does	do	VERB
ejpam-5066	41	30	so	so	ADV
ejpam-5066	41	31	.	.	PUNCT
ejpam-5066	42	1	the	the	DET
ejpam-5066	42	2	subsequent	subsequent	ADJ
ejpam-5066	42	3	section	section	NOUN
ejpam-5066	42	4	of	of	ADP
ejpam-5066	42	5	the	the	DET
ejpam-5066	42	6	paper	paper	NOUN
ejpam-5066	42	7	is	be	AUX
ejpam-5066	42	8	dedicated	dedicate	VERB
ejpam-5066	42	9	to	to	ADP
ejpam-5066	42	10	defining	define	VERB
ejpam-5066	42	11	n	n	CCONJ
ejpam-5066	42	12	-	-	PUNCT
ejpam-5066	42	13	abelian	abelian	NOUN
ejpam-5066	42	14	rings	ring	NOUN
ejpam-5066	42	15	.	.	PUNCT
ejpam-5066	43	1	proposition	proposition	NOUN
ejpam-5066	43	2	9	9	NUM
ejpam-5066	43	3	rigorously	rigorously	ADV
ejpam-5066	43	4	demonstrates	demonstrate	VERB
ejpam-5066	43	5	that	that	SCONJ
ejpam-5066	43	6	every	every	DET
ejpam-5066	43	7	n	n	CCONJ
ejpam-5066	43	8	-	-	PUNCT
ejpam-5066	43	9	central	central	ADJ
ejpam-5066	43	10	idempotent	idempotent	NOUN
ejpam-5066	43	11	enjoys	enjoy	VERB
ejpam-5066	43	12	a	a	DET
ejpam-5066	43	13	state	state	NOUN
ejpam-5066	43	14	of	of	ADP
ejpam-5066	43	15	direct	direct	ADJ
ejpam-5066	43	16	finiteness	finiteness	NOUN
ejpam-5066	43	17	.	.	PUNCT
ejpam-5066	44	1	drawing	draw	VERB
ejpam-5066	44	2	from	from	ADP
ejpam-5066	44	3	lam	lam	PROPN
ejpam-5066	44	4	’s	’s	PART
ejpam-5066	44	5	seminal	seminal	ADJ
ejpam-5066	44	6	work	work	NOUN
ejpam-5066	44	7	[	[	X
ejpam-5066	44	8	12	12	NUM
ejpam-5066	44	9	]	]	PUNCT
ejpam-5066	44	10	,	,	PUNCT
ejpam-5066	44	11	where	where	SCONJ
ejpam-5066	44	12	he	he	PRON
ejpam-5066	44	13	established	establish	VERB
ejpam-5066	44	14	the	the	DET
ejpam-5066	44	15	semiabelian	semiabelian	ADJ
ejpam-5066	44	16	nature	nature	NOUN
ejpam-5066	44	17	of	of	ADP
ejpam-5066	44	18	2	2	NUM
ejpam-5066	44	19	×	×	NOUN
ejpam-5066	44	20	2	2	NUM
ejpam-5066	44	21	upper	upper	ADJ
ejpam-5066	44	22	triangular	triangular	NOUN
ejpam-5066	44	23	matrices	matrix	NOUN
ejpam-5066	44	24	denoted	denote	VERB
ejpam-5066	44	25	as	as	ADP
ejpam-5066	44	26	t2(r	t2(r	PROPN
ejpam-5066	44	27	)	)	PUNCT
ejpam-5066	44	28	,	,	PUNCT
ejpam-5066	44	29	we	we	PRON
ejpam-5066	44	30	further	far	ADV
ejpam-5066	44	31	extend	extend	VERB
ejpam-5066	44	32	this	this	DET
ejpam-5066	44	33	understanding	understanding	NOUN
ejpam-5066	44	34	.	.	PUNCT
ejpam-5066	45	1	specifically	specifically	ADV
ejpam-5066	45	2	,	,	PUNCT
ejpam-5066	45	3	theorem	theorem	VERB
ejpam-5066	45	4	4	4	NUM
ejpam-5066	45	5	posited	posit	VERB
ejpam-5066	45	6	within	within	ADP
ejpam-5066	45	7	this	this	DET
ejpam-5066	45	8	section	section	NOUN
ejpam-5066	45	9	firmly	firmly	ADV
ejpam-5066	45	10	establishes	establish	VERB
ejpam-5066	45	11	the	the	DET
ejpam-5066	45	12	n	n	CCONJ
ejpam-5066	45	13	-	-	PUNCT
ejpam-5066	45	14	abelian	abelian	ADJ
ejpam-5066	45	15	property	property	NOUN
ejpam-5066	45	16	for	for	ADP
ejpam-5066	45	17	the	the	DET
ejpam-5066	45	18	set	set	NOUN
ejpam-5066	45	19	tn	tn	NOUN
ejpam-5066	45	20	across	across	ADP
ejpam-5066	45	21	all	all	DET
ejpam-5066	45	22	values	value	NOUN
ejpam-5066	45	23	of	of	ADP
ejpam-5066	45	24	n.	n.	NOUN
ejpam-5066	45	25	2	2	NUM
ejpam-5066	45	26	.	.	NUM
ejpam-5066	46	1	n	n	CCONJ
ejpam-5066	46	2	-	-	PUNCT
ejpam-5066	46	3	central	central	ADJ
ejpam-5066	46	4	idempotents	idempotent	NOUN
ejpam-5066	46	5	this	this	DET
ejpam-5066	46	6	section	section	NOUN
ejpam-5066	46	7	introduces	introduce	VERB
ejpam-5066	46	8	the	the	DET
ejpam-5066	46	9	concept	concept	NOUN
ejpam-5066	46	10	of	of	ADP
ejpam-5066	46	11	n	n	CCONJ
ejpam-5066	46	12	-	-	PUNCT
ejpam-5066	46	13	central	central	ADJ
ejpam-5066	46	14	idempotents	idempotent	NOUN
ejpam-5066	46	15	for	for	ADP
ejpam-5066	46	16	a	a	DET
ejpam-5066	46	17	ring	ring	NOUN
ejpam-5066	46	18	r.	r.	NOUN
ejpam-5066	46	19	the	the	DET
ejpam-5066	46	20	definition	definition	NOUN
ejpam-5066	46	21	of	of	ADP
ejpam-5066	46	22	n	n	CCONJ
ejpam-5066	46	23	-	-	PUNCT
ejpam-5066	46	24	central	central	ADJ
ejpam-5066	46	25	idempotents	idempotent	NOUN
ejpam-5066	46	26	involves	involve	VERB
ejpam-5066	46	27	a	a	DET
ejpam-5066	46	28	recursive	recursive	ADJ
ejpam-5066	46	29	sequence	sequence	NOUN
ejpam-5066	46	30	of	of	ADP
ejpam-5066	46	31	sets	set	NOUN
ejpam-5066	46	32	of	of	ADP
ejpam-5066	46	33	r.	r.	PROPN
ejpam-5066	46	34	let	let	VERB
ejpam-5066	46	35	e	e	PRON
ejpam-5066	46	36	be	be	AUX
ejpam-5066	46	37	an	an	DET
ejpam-5066	46	38	idempotent	idempotent	NOUN
ejpam-5066	46	39	of	of	ADP
ejpam-5066	46	40	r.	r.	NOUN
ejpam-5066	46	41	we	we	PRON
ejpam-5066	46	42	define	define	VERB
ejpam-5066	46	43	a	a	DET
ejpam-5066	46	44	collection	collection	NOUN
ejpam-5066	46	45	of	of	ADP
ejpam-5066	46	46	right	right	ADJ
ejpam-5066	46	47	ideals	ideal	NOUN
ejpam-5066	46	48	[	[	X
ejpam-5066	46	49	e]n	e]n	NOUN
ejpam-5066	46	50	of	of	ADP
ejpam-5066	46	51	r	r	NOUN
ejpam-5066	46	52	,	,	PUNCT
ejpam-5066	46	53	for	for	ADP
ejpam-5066	46	54	n	n	PRON
ejpam-5066	46	55	≥	≥	NOUN
ejpam-5066	46	56	0	0	NUM
ejpam-5066	46	57	,	,	PUNCT
ejpam-5066	46	58	recursively	recursively	ADV
ejpam-5066	46	59	as	as	SCONJ
ejpam-5066	46	60	follows	follow	VERB
ejpam-5066	46	61	:	:	PUNCT
ejpam-5066	46	62	[	[	X
ejpam-5066	46	63	e]0	e]0	X
ejpam-5066	46	64	=	=	SYM
ejpam-5066	46	65	er	er	INTJ
ejpam-5066	46	66	,	,	PUNCT
ejpam-5066	46	67	[	[	X
ejpam-5066	46	68	e]1	e]1	X
ejpam-5066	46	69	=	=	SYM
ejpam-5066	46	70	(	(	PUNCT
ejpam-5066	46	71	1−	1−	NUM
ejpam-5066	46	72	e)rer	e)rer	ADJ
ejpam-5066	46	73	,	,	PUNCT
ejpam-5066	46	74	and	and	CCONJ
ejpam-5066	46	75	for	for	ADP
ejpam-5066	46	76	i	i	PRON
ejpam-5066	46	77	≥	≥	NOUN
ejpam-5066	46	78	2	2	NUM
ejpam-5066	46	79	,	,	PUNCT
ejpam-5066	46	80	[	[	X
ejpam-5066	46	81	e]i	e]i	X
ejpam-5066	46	82	=	=	X
ejpam-5066	46	83	[	[	X
ejpam-5066	46	84	e]i−2(1−	e]i−2(1−	NOUN
ejpam-5066	46	85	e)rer	e)rer	NOUN
ejpam-5066	46	86	.	.	PUNCT
ejpam-5066	47	1	using	use	VERB
ejpam-5066	47	2	the	the	DET
ejpam-5066	47	3	notation	notation	NOUN
ejpam-5066	47	4	introduced	introduce	VERB
ejpam-5066	47	5	earlier	early	ADV
ejpam-5066	47	6	,	,	PUNCT
ejpam-5066	47	7	if	if	SCONJ
ejpam-5066	47	8	there	there	PRON
ejpam-5066	47	9	exists	exist	VERB
ejpam-5066	47	10	some	some	PRON
ejpam-5066	47	11	k	k	ADP
ejpam-5066	48	1	such	such	ADJ
ejpam-5066	48	2	that	that	SCONJ
ejpam-5066	48	3	[	[	X
ejpam-5066	48	4	e]k	e]k	NOUN
ejpam-5066	48	5	=	=	PUNCT
ejpam-5066	48	6	[	[	X
ejpam-5066	48	7	e]k+1	e]k+1	VERB
ejpam-5066	48	8	,	,	PUNCT
ejpam-5066	48	9	then	then	ADV
ejpam-5066	48	10	for	for	ADP
ejpam-5066	48	11	every	every	DET
ejpam-5066	48	12	i	i	PRON
ejpam-5066	48	13	≥	≥	AUX
ejpam-5066	48	14	k	k	NOUN
ejpam-5066	48	15	,	,	PUNCT
ejpam-5066	48	16	we	we	PRON
ejpam-5066	48	17	have	have	VERB
ejpam-5066	48	18	[	[	X
ejpam-5066	48	19	e]i	e]i	NOUN
ejpam-5066	48	20	=	=	SYM
ejpam-5066	48	21	0	0	NUM
ejpam-5066	48	22	.	.	PUNCT
ejpam-5066	49	1	therefore	therefore	ADV
ejpam-5066	49	2	,	,	PUNCT
ejpam-5066	49	3	the	the	DET
ejpam-5066	49	4	sequence	sequence	NOUN
ejpam-5066	49	5	(	(	PUNCT
ejpam-5066	49	6	[	[	X
ejpam-5066	49	7	e]k	e]k	NOUN
ejpam-5066	49	8	)	)	PUNCT
ejpam-5066	49	9	is	be	AUX
ejpam-5066	49	10	eventually	eventually	ADV
ejpam-5066	49	11	-	-	PUNCT
ejpam-5066	49	12	zero	zero	NUM
ejpam-5066	49	13	,	,	PUNCT
ejpam-5066	49	14	as	as	SCONJ
ejpam-5066	49	15	shown	show	VERB
ejpam-5066	49	16	in	in	ADP
ejpam-5066	49	17	example	example	NOUN
ejpam-5066	49	18	1	1	NUM
ejpam-5066	49	19	.	.	PUNCT
ejpam-5066	50	1	on	on	ADP
ejpam-5066	50	2	the	the	DET
ejpam-5066	50	3	other	other	ADJ
ejpam-5066	50	4	hand	hand	NOUN
ejpam-5066	50	5	,	,	PUNCT
ejpam-5066	50	6	if	if	SCONJ
ejpam-5066	50	7	there	there	PRON
ejpam-5066	50	8	exists	exist	VERB
ejpam-5066	50	9	some	some	PRON
ejpam-5066	50	10	k	k	ADP
ejpam-5066	50	11	such	such	ADJ
ejpam-5066	50	12	that	that	SCONJ
ejpam-5066	51	1	[	[	X
ejpam-5066	51	2	e]k	e]k	NOUN
ejpam-5066	51	3	=	=	PUNCT
ejpam-5066	51	4	[	[	X
ejpam-5066	51	5	e]k+2	e]k+2	NOUN
ejpam-5066	51	6	,	,	PUNCT
ejpam-5066	51	7	then	then	ADV
ejpam-5066	51	8	we	we	PRON
ejpam-5066	51	9	m.	m.	NOUN
ejpam-5066	51	10	saad	saad	PROPN
ejpam-5066	51	11	,	,	PUNCT
ejpam-5066	51	12	m.	m.	NOUN
ejpam-5066	51	13	zailaee	zailaee	PROPN
ejpam-5066	51	14	/	/	SYM
ejpam-5066	51	15	eur	eur	PROPN
ejpam-5066	51	16	.	.	PUNCT
ejpam-5066	52	1	j.	j.	PROPN
ejpam-5066	52	2	pure	pure	PROPN
ejpam-5066	52	3	appl	appl	PROPN
ejpam-5066	52	4	.	.	PROPN
ejpam-5066	52	5	math	math	PROPN
ejpam-5066	52	6	,	,	PUNCT
ejpam-5066	52	7	17	17	NUM
ejpam-5066	52	8	(	(	PUNCT
ejpam-5066	52	9	2	2	NUM
ejpam-5066	52	10	)	)	PUNCT
ejpam-5066	52	11	(	(	PUNCT
ejpam-5066	52	12	2024	2024	NUM
ejpam-5066	52	13	)	)	PUNCT
ejpam-5066	52	14	,	,	PUNCT
ejpam-5066	52	15	736	736	NUM
ejpam-5066	52	16	-	-	SYM
ejpam-5066	52	17	752	752	NUM
ejpam-5066	52	18	738	738	NUM
ejpam-5066	52	19	have	have	VERB
ejpam-5066	52	20	[	[	X
ejpam-5066	52	21	e]i+2	e]i+2	X
ejpam-5066	52	22	=	=	PUNCT
ejpam-5066	53	1	[	[	X
ejpam-5066	53	2	e]i	e]i	NOUN
ejpam-5066	53	3	for	for	ADP
ejpam-5066	53	4	every	every	DET
ejpam-5066	53	5	i	i	PRON
ejpam-5066	53	6	≥	≥	NUM
ejpam-5066	53	7	k	k	NOUN
ejpam-5066	53	8	,	,	PUNCT
ejpam-5066	53	9	as	as	ADP
ejpam-5066	53	10	in	in	ADP
ejpam-5066	53	11	example	example	NOUN
ejpam-5066	53	12	2	2	NUM
ejpam-5066	53	13	.	.	X
ejpam-5066	53	14	note	note	VERB
ejpam-5066	53	15	that	that	SCONJ
ejpam-5066	53	16	every	every	DET
ejpam-5066	53	17	two	two	NUM
ejpam-5066	53	18	consecutive	consecutive	ADJ
ejpam-5066	53	19	sets	set	NOUN
ejpam-5066	53	20	in	in	ADP
ejpam-5066	53	21	the	the	DET
ejpam-5066	53	22	sequence	sequence	NOUN
ejpam-5066	53	23	(	(	PUNCT
ejpam-5066	53	24	[	[	X
ejpam-5066	53	25	e]n	e]n	X
ejpam-5066	53	26	)	)	PUNCT
ejpam-5066	53	27	have	have	VERB
ejpam-5066	53	28	zero	zero	NUM
ejpam-5066	53	29	intersection	intersection	NOUN
ejpam-5066	53	30	.	.	PUNCT
ejpam-5066	54	1	we	we	PRON
ejpam-5066	54	2	shall	shall	AUX
ejpam-5066	54	3	refer	refer	VERB
ejpam-5066	54	4	to	to	ADP
ejpam-5066	54	5	[	[	X
ejpam-5066	54	6	e]n	e]n	NOUN
ejpam-5066	54	7	as	as	SCONJ
ejpam-5066	54	8	the	the	DET
ejpam-5066	54	9	right	right	ADJ
ejpam-5066	54	10	centralizer	centralizer	NOUN
ejpam-5066	54	11	of	of	ADP
ejpam-5066	54	12	e	e	PROPN
ejpam-5066	54	13	with	with	ADP
ejpam-5066	54	14	degree	degree	NOUN
ejpam-5066	54	15	n.	n.	NOUN
ejpam-5066	54	16	the	the	DET
ejpam-5066	54	17	additive	additive	ADJ
ejpam-5066	54	18	commutator	commutator	NOUN
ejpam-5066	54	19	of	of	ADP
ejpam-5066	54	20	elements	element	NOUN
ejpam-5066	54	21	x	x	PUNCT
ejpam-5066	54	22	and	and	CCONJ
ejpam-5066	54	23	y	y	PROPN
ejpam-5066	54	24	in	in	ADP
ejpam-5066	54	25	a	a	DET
ejpam-5066	54	26	ringr	ringr	NOUN
ejpam-5066	54	27	is	be	AUX
ejpam-5066	54	28	denoted	denote	VERB
ejpam-5066	54	29	by	by	ADP
ejpam-5066	54	30	[	[	X
ejpam-5066	54	31	x	x	X
ejpam-5066	54	32	,	,	PUNCT
ejpam-5066	54	33	y	y	PROPN
ejpam-5066	54	34	]	]	PUNCT
ejpam-5066	54	35	and	and	CCONJ
ejpam-5066	54	36	defined	define	VERB
ejpam-5066	54	37	as	as	ADP
ejpam-5066	54	38	xy	xy	PROPN
ejpam-5066	54	39	−	−	PROPN
ejpam-5066	54	40	yx	yx	PROPN
ejpam-5066	54	41	.	.	PROPN
ejpam-5066	55	1	for	for	ADP
ejpam-5066	55	2	an	an	DET
ejpam-5066	55	3	idempotent	idempotent	ADJ
ejpam-5066	55	4	e	e	NOUN
ejpam-5066	55	5	and	and	CCONJ
ejpam-5066	55	6	a	a	DET
ejpam-5066	55	7	subset	subset	NOUN
ejpam-5066	55	8	s	s	NOUN
ejpam-5066	55	9	of	of	ADP
ejpam-5066	55	10	a	a	DET
ejpam-5066	55	11	ring	ring	NOUN
ejpam-5066	55	12	r	r	NOUN
ejpam-5066	55	13	,	,	PUNCT
ejpam-5066	55	14	we	we	PRON
ejpam-5066	55	15	use	use	VERB
ejpam-5066	55	16	the	the	DET
ejpam-5066	55	17	notation	notation	NOUN
ejpam-5066	55	18	[	[	X
ejpam-5066	55	19	e	e	NOUN
ejpam-5066	55	20	,	,	PUNCT
ejpam-5066	55	21	s	s	PART
ejpam-5066	55	22	]	]	PUNCT
ejpam-5066	55	23	to	to	PART
ejpam-5066	55	24	represent	represent	VERB
ejpam-5066	55	25	the	the	DET
ejpam-5066	55	26	subset	subset	NOUN
ejpam-5066	55	27	{	{	PUNCT
ejpam-5066	55	28	es	es	ADP
ejpam-5066	55	29	−	−	NOUN
ejpam-5066	55	30	se|s	se|s	NOUN
ejpam-5066	55	31	∈	∈	PROPN
ejpam-5066	55	32	s	s	PART
ejpam-5066	55	33	}	}	PUNCT
ejpam-5066	55	34	of	of	ADP
ejpam-5066	55	35	r.	r.	PROPN
ejpam-5066	55	36	the	the	DET
ejpam-5066	55	37	following	follow	VERB
ejpam-5066	55	38	lemma	lemma	PROPN
ejpam-5066	55	39	provides	provide	VERB
ejpam-5066	55	40	an	an	DET
ejpam-5066	55	41	alternative	alternative	ADJ
ejpam-5066	55	42	definition	definition	NOUN
ejpam-5066	55	43	for	for	ADP
ejpam-5066	55	44	the	the	DET
ejpam-5066	55	45	centralizer	centralizer	NOUN
ejpam-5066	55	46	[	[	X
ejpam-5066	55	47	e]n	e]n	ADV
ejpam-5066	55	48	introduced	introduce	VERB
ejpam-5066	55	49	earlier	early	ADV
ejpam-5066	55	50	,	,	PUNCT
ejpam-5066	55	51	using	use	VERB
ejpam-5066	55	52	additive	additive	ADJ
ejpam-5066	55	53	commutators	commutator	NOUN
ejpam-5066	55	54	.	.	PUNCT
ejpam-5066	56	1	this	this	DET
ejpam-5066	56	2	definition	definition	NOUN
ejpam-5066	56	3	allows	allow	VERB
ejpam-5066	56	4	for	for	SCONJ
ejpam-5066	56	5	the	the	DET
ejpam-5066	56	6	centralizers	centralizer	NOUN
ejpam-5066	56	7	to	to	PART
ejpam-5066	56	8	be	be	AUX
ejpam-5066	56	9	extended	extend	VERB
ejpam-5066	56	10	to	to	ADP
ejpam-5066	56	11	rings	ring	NOUN
ejpam-5066	56	12	without	without	ADP
ejpam-5066	56	13	identity	identity	NOUN
ejpam-5066	56	14	.	.	PUNCT
ejpam-5066	57	1	however	however	ADV
ejpam-5066	57	2	,	,	PUNCT
ejpam-5066	57	3	our	our	PRON
ejpam-5066	57	4	focus	focus	NOUN
ejpam-5066	57	5	in	in	ADP
ejpam-5066	57	6	this	this	DET
ejpam-5066	57	7	research	research	NOUN
ejpam-5066	57	8	is	be	AUX
ejpam-5066	57	9	still	still	ADV
ejpam-5066	57	10	on	on	ADP
ejpam-5066	57	11	unital	unital	ADJ
ejpam-5066	57	12	rings	ring	NOUN
ejpam-5066	57	13	.	.	PUNCT
ejpam-5066	58	1	lemma	lemma	PROPN
ejpam-5066	58	2	1	1	NUM
ejpam-5066	58	3	.	.	PUNCT
ejpam-5066	59	1	for	for	ADP
ejpam-5066	59	2	every	every	DET
ejpam-5066	59	3	idempotent	idempotent	ADJ
ejpam-5066	59	4	e	e	NOUN
ejpam-5066	59	5	of	of	ADP
ejpam-5066	59	6	r	r	NOUN
ejpam-5066	59	7	and	and	CCONJ
ejpam-5066	59	8	non	non	ADJ
ejpam-5066	59	9	-	-	ADJ
ejpam-5066	59	10	negative	negative	ADJ
ejpam-5066	59	11	integer	integer	NOUN
ejpam-5066	59	12	n	n	CCONJ
ejpam-5066	59	13	,	,	PUNCT
ejpam-5066	59	14	we	we	PRON
ejpam-5066	59	15	have	have	VERB
ejpam-5066	59	16	[	[	NOUN
ejpam-5066	59	17	e]n	e]n	NOUN
ejpam-5066	59	18	=	=	SYM
ejpam-5066	60	1	[	[	X
ejpam-5066	60	2	e	e	NOUN
ejpam-5066	60	3	,	,	PUNCT
ejpam-5066	60	4	r]ner	r]ner	NOUN
ejpam-5066	60	5	.	.	PUNCT
ejpam-5066	61	1	proof	proof	NOUN
ejpam-5066	61	2	.	.	PUNCT
ejpam-5066	62	1	we	we	PRON
ejpam-5066	62	2	will	will	AUX
ejpam-5066	62	3	prove	prove	VERB
ejpam-5066	62	4	the	the	DET
ejpam-5066	62	5	relation	relation	NOUN
ejpam-5066	62	6	using	use	VERB
ejpam-5066	62	7	the	the	DET
ejpam-5066	62	8	mathematical	mathematical	ADJ
ejpam-5066	62	9	induction	induction	NOUN
ejpam-5066	62	10	on	on	ADP
ejpam-5066	62	11	n.	n.	PROPN
ejpam-5066	62	12	claim	claim	NOUN
ejpam-5066	62	13	1	1	NUM
ejpam-5066	62	14	.	.	PUNCT
ejpam-5066	63	1	[	[	X
ejpam-5066	63	2	e]1	e]1	X
ejpam-5066	63	3	=	=	PUNCT
ejpam-5066	64	1	[	[	X
ejpam-5066	64	2	e	e	NOUN
ejpam-5066	64	3	,	,	PUNCT
ejpam-5066	64	4	r]er	r]er	PROPN
ejpam-5066	64	5	.	.	PROPN
ejpam-5066	64	6	for	for	ADP
ejpam-5066	64	7	every	every	DET
ejpam-5066	64	8	,	,	PUNCT
ejpam-5066	64	9	r	r	NOUN
ejpam-5066	64	10	,	,	PUNCT
ejpam-5066	64	11	s	s	NOUN
ejpam-5066	64	12	∈	∈	PROPN
ejpam-5066	64	13	r	r	NOUN
ejpam-5066	64	14	,	,	PUNCT
ejpam-5066	64	15	we	we	PRON
ejpam-5066	64	16	have	have	VERB
ejpam-5066	64	17	[	[	X
ejpam-5066	64	18	e	e	NOUN
ejpam-5066	64	19	,	,	PUNCT
ejpam-5066	64	20	r]es	r]es	PRON
ejpam-5066	64	21	=	=	SYM
ejpam-5066	64	22	(	(	PUNCT
ejpam-5066	64	23	er	er	INTJ
ejpam-5066	64	24	−	−	PRON
ejpam-5066	64	25	re)es	re)es	NOUN
ejpam-5066	64	26	=	=	SYM
ejpam-5066	64	27	eres−	eres−	NOUN
ejpam-5066	64	28	res	re	NOUN
ejpam-5066	64	29	=	=	SYM
ejpam-5066	64	30	(	(	PUNCT
ejpam-5066	64	31	1−	1−	NUM
ejpam-5066	64	32	e)(−r)es	e)(−r)es	NOUN
ejpam-5066	64	33	∈	∈	PROPN
ejpam-5066	65	1	[	[	X
ejpam-5066	65	2	e]1	e]1	X
ejpam-5066	65	3	and	and	CCONJ
ejpam-5066	65	4	[	[	X
ejpam-5066	65	5	e	e	NOUN
ejpam-5066	65	6	,	,	PUNCT
ejpam-5066	65	7	r]er	r]er	PROPN
ejpam-5066	65	8	⊆	⊆	NUM
ejpam-5066	65	9	[	[	X
ejpam-5066	65	10	e]1	e]1	NOUN
ejpam-5066	65	11	.	.	PUNCT
ejpam-5066	66	1	conversely	conversely	ADV
ejpam-5066	66	2	,	,	PUNCT
ejpam-5066	66	3	for	for	ADP
ejpam-5066	66	4	every	every	DET
ejpam-5066	66	5	a	a	DET
ejpam-5066	66	6	∈	∈	PROPN
ejpam-5066	66	7	[	[	X
ejpam-5066	66	8	e]1	e]1	NOUN
ejpam-5066	66	9	,	,	PUNCT
ejpam-5066	66	10	there	there	PRON
ejpam-5066	66	11	exist	exist	VERB
ejpam-5066	66	12	x	x	NOUN
ejpam-5066	66	13	,	,	PUNCT
ejpam-5066	66	14	y	y	PROPN
ejpam-5066	66	15	∈	∈	PROPN
ejpam-5066	66	16	r	r	NOUN
ejpam-5066	66	17	such	such	ADJ
ejpam-5066	66	18	that	that	SCONJ
ejpam-5066	66	19	a	a	DET
ejpam-5066	66	20	=	=	X
ejpam-5066	66	21	(	(	PUNCT
ejpam-5066	66	22	1	1	NUM
ejpam-5066	66	23	−	−	PROPN
ejpam-5066	66	24	e)xey	e)xey	NOUN
ejpam-5066	66	25	.	.	PUNCT
ejpam-5066	67	1	hence	hence	ADV
ejpam-5066	67	2	,	,	PUNCT
ejpam-5066	67	3	a	a	DET
ejpam-5066	67	4	=	=	PUNCT
ejpam-5066	67	5	xey	xey	PROPN
ejpam-5066	67	6	−	−	PROPN
ejpam-5066	67	7	exey	exey	NOUN
ejpam-5066	67	8	=	=	SYM
ejpam-5066	67	9	(	(	PUNCT
ejpam-5066	67	10	xe	xe	PROPN
ejpam-5066	67	11	−	−	PROPN
ejpam-5066	67	12	ex)ey	ex)ey	PROPN
ejpam-5066	67	13	=	=	PUNCT
ejpam-5066	68	1	[	[	X
ejpam-5066	68	2	e	e	X
ejpam-5066	68	3	,	,	PUNCT
ejpam-5066	68	4	x]e(−y	x]e(−y	NOUN
ejpam-5066	68	5	)	)	PUNCT
ejpam-5066	68	6	∈	∈	PROPN
ejpam-5066	69	1	[	[	X
ejpam-5066	69	2	e	e	NOUN
ejpam-5066	69	3	,	,	PUNCT
ejpam-5066	69	4	r]er	r]er	PROPN
ejpam-5066	69	5	.	.	NOUN
ejpam-5066	69	6	therefore	therefore	ADV
ejpam-5066	69	7	,	,	PUNCT
ejpam-5066	69	8	[	[	X
ejpam-5066	69	9	e]1	e]1	X
ejpam-5066	69	10	=	=	PUNCT
ejpam-5066	70	1	[	[	X
ejpam-5066	70	2	e	e	NOUN
ejpam-5066	70	3	,	,	PUNCT
ejpam-5066	70	4	r]er	r]er	PROPN
ejpam-5066	70	5	.	.	NOUN
ejpam-5066	70	6	claim	claim	NOUN
ejpam-5066	70	7	2	2	NUM
ejpam-5066	70	8	.	.	PUNCT
ejpam-5066	71	1	e[e	e[e	ADJ
ejpam-5066	71	2	,	,	PUNCT
ejpam-5066	71	3	r	r	NOUN
ejpam-5066	71	4	]	]	PUNCT
ejpam-5066	71	5	=	=	PUNCT
ejpam-5066	72	1	[	[	X
ejpam-5066	72	2	e	e	NOUN
ejpam-5066	72	3	,	,	PUNCT
ejpam-5066	72	4	r](1−	r](1−	NOUN
ejpam-5066	72	5	e	e	NOUN
ejpam-5066	72	6	)	)	PUNCT
ejpam-5066	72	7	and	and	CCONJ
ejpam-5066	72	8	[	[	X
ejpam-5066	72	9	e	e	NOUN
ejpam-5066	72	10	,	,	PUNCT
ejpam-5066	72	11	r]e	r]e	NOUN
ejpam-5066	72	12	=	=	SYM
ejpam-5066	72	13	(	(	PUNCT
ejpam-5066	72	14	1−	1−	NUM
ejpam-5066	72	15	e)[e	e)[e	PROPN
ejpam-5066	72	16	,	,	PUNCT
ejpam-5066	72	17	r	r	NOUN
ejpam-5066	72	18	]	]	PUNCT
ejpam-5066	72	19	.	.	PUNCT
ejpam-5066	73	1	for	for	ADP
ejpam-5066	73	2	every	every	DET
ejpam-5066	73	3	r	r	NOUN
ejpam-5066	73	4	∈	∈	NOUN
ejpam-5066	73	5	r	r	NOUN
ejpam-5066	73	6	,	,	PUNCT
ejpam-5066	73	7	we	we	PRON
ejpam-5066	73	8	have	have	VERB
ejpam-5066	73	9	e[e	e[e	ADJ
ejpam-5066	73	10	,	,	PUNCT
ejpam-5066	73	11	r	r	NOUN
ejpam-5066	73	12	]	]	X
ejpam-5066	73	13	=	=	SYM
ejpam-5066	73	14	e(er	e(er	PROPN
ejpam-5066	73	15	−	−	NUM
ejpam-5066	73	16	re	re	NOUN
ejpam-5066	73	17	)	)	PUNCT
ejpam-5066	74	1	=	=	PUNCT
ejpam-5066	74	2	er	er	INTJ
ejpam-5066	74	3	−	−	NOUN
ejpam-5066	74	4	ere	ere	NOUN
ejpam-5066	75	1	=	=	PUNCT
ejpam-5066	75	2	er(1	er(1	PROPN
ejpam-5066	75	3	−	−	PROPN
ejpam-5066	75	4	e	e	X
ejpam-5066	75	5	)	)	PUNCT
ejpam-5066	75	6	=	=	SYM
ejpam-5066	75	7	er(1	er(1	PROPN
ejpam-5066	75	8	−	−	PROPN
ejpam-5066	75	9	e	e	NOUN
ejpam-5066	75	10	)	)	PUNCT
ejpam-5066	76	1	−	−	PROPN
ejpam-5066	76	2	re(1	re(1	PROPN
ejpam-5066	76	3	−	−	PROPN
ejpam-5066	76	4	e	e	NOUN
ejpam-5066	76	5	)	)	PUNCT
ejpam-5066	76	6	=	=	SYM
ejpam-5066	77	1	(	(	PUNCT
ejpam-5066	77	2	er	er	INTJ
ejpam-5066	77	3	−	−	PROPN
ejpam-5066	77	4	re)(1	re)(1	NOUN
ejpam-5066	77	5	−	−	PROPN
ejpam-5066	77	6	e	e	NOUN
ejpam-5066	77	7	)	)	PUNCT
ejpam-5066	77	8	=	=	PUNCT
ejpam-5066	78	1	[	[	X
ejpam-5066	78	2	e	e	NOUN
ejpam-5066	78	3	,	,	PUNCT
ejpam-5066	78	4	r](1	r](1	VERB
ejpam-5066	78	5	−	−	PROPN
ejpam-5066	78	6	e	e	NOUN
ejpam-5066	78	7	)	)	PUNCT
ejpam-5066	78	8	and	and	CCONJ
ejpam-5066	78	9	e[e	e[e	ADJ
ejpam-5066	78	10	,	,	PUNCT
ejpam-5066	78	11	r	r	NOUN
ejpam-5066	78	12	]	]	PUNCT
ejpam-5066	78	13	=	=	PUNCT
ejpam-5066	79	1	[	[	X
ejpam-5066	79	2	e	e	NOUN
ejpam-5066	79	3	,	,	PUNCT
ejpam-5066	79	4	r](1	r](1	VERB
ejpam-5066	79	5	−	−	PROPN
ejpam-5066	79	6	e	e	NOUN
ejpam-5066	79	7	)	)	PUNCT
ejpam-5066	79	8	.	.	PUNCT
ejpam-5066	80	1	similarly	similarly	ADV
ejpam-5066	80	2	,	,	PUNCT
ejpam-5066	80	3	[	[	X
ejpam-5066	80	4	e	e	NOUN
ejpam-5066	80	5	,	,	PUNCT
ejpam-5066	80	6	r]e	r]e	NOUN
ejpam-5066	80	7	=	=	SYM
ejpam-5066	80	8	(	(	PUNCT
ejpam-5066	80	9	1−	1−	NUM
ejpam-5066	80	10	e)[e	e)[e	PROPN
ejpam-5066	80	11	,	,	PUNCT
ejpam-5066	80	12	r	r	NOUN
ejpam-5066	80	13	]	]	PUNCT
ejpam-5066	80	14	.	.	PUNCT
ejpam-5066	81	1	claim	claim	NOUN
ejpam-5066	81	2	3	3	NUM
ejpam-5066	81	3	.	.	PUNCT
ejpam-5066	82	1	[	[	X
ejpam-5066	82	2	e	e	NOUN
ejpam-5066	82	3	,	,	PUNCT
ejpam-5066	82	4	r]2e	r]2e	NOUN
ejpam-5066	82	5	=	=	SYM
ejpam-5066	82	6	er(1−	er(1−	NOUN
ejpam-5066	83	1	e)re	e)re	PROPN
ejpam-5066	83	2	and	and	CCONJ
ejpam-5066	83	3	[	[	X
ejpam-5066	83	4	e	e	NOUN
ejpam-5066	83	5	,	,	PUNCT
ejpam-5066	83	6	r]2(1−	r]2(1−	ADJ
ejpam-5066	83	7	e	e	NOUN
ejpam-5066	83	8	)	)	PUNCT
ejpam-5066	83	9	=	=	SYM
ejpam-5066	83	10	er(1−	er(1−	NOUN
ejpam-5066	84	1	e)r(1−	e)r(1−	PROPN
ejpam-5066	84	2	e	e	NOUN
ejpam-5066	84	3	)	)	PUNCT
ejpam-5066	84	4	.	.	PUNCT
ejpam-5066	85	1	for	for	ADP
ejpam-5066	85	2	every	every	DET
ejpam-5066	85	3	r	r	NOUN
ejpam-5066	85	4	,	,	PUNCT
ejpam-5066	85	5	s	s	NOUN
ejpam-5066	85	6	∈	∈	PROPN
ejpam-5066	85	7	r	r	NOUN
ejpam-5066	85	8	,	,	PUNCT
ejpam-5066	85	9	we	we	PRON
ejpam-5066	85	10	have	have	VERB
ejpam-5066	85	11	[	[	X
ejpam-5066	85	12	e	e	NOUN
ejpam-5066	85	13	,	,	PUNCT
ejpam-5066	85	14	r][e	r][e	PROPN
ejpam-5066	85	15	,	,	PUNCT
ejpam-5066	85	16	s]e	s]e	PROPN
ejpam-5066	85	17	=	=	PUNCT
ejpam-5066	85	18	(	(	PUNCT
ejpam-5066	85	19	er	er	INTJ
ejpam-5066	85	20	−	−	NOUN
ejpam-5066	85	21	re)(es	re)(es	PUNCT
ejpam-5066	85	22	−	−	NOUN
ejpam-5066	86	1	se)e	se)e	PROPN
ejpam-5066	87	1	=	=	PRON
ejpam-5066	88	1	(	(	PUNCT
ejpam-5066	88	2	er	er	INTJ
ejpam-5066	88	3	−	−	NOUN
ejpam-5066	88	4	re)(ese	re)(ese	NOUN
ejpam-5066	88	5	−	−	PROPN
ejpam-5066	88	6	se	se	X
ejpam-5066	88	7	)	)	PUNCT
ejpam-5066	88	8	=	=	SYM
ejpam-5066	89	1	(	(	PUNCT
ejpam-5066	89	2	er	er	INTJ
ejpam-5066	89	3	−	−	PROPN
ejpam-5066	89	4	re)(1−	re)(1−	PROPN
ejpam-5066	89	5	e)(−se	e)(−se	NOUN
ejpam-5066	89	6	)	)	PUNCT
ejpam-5066	89	7	=	=	SYM
ejpam-5066	90	1	er(1−	er(1−	PROPN
ejpam-5066	90	2	e)(−s)e	e)(−s)e	ADP
ejpam-5066	90	3	∈	∈	NOUN
ejpam-5066	90	4	er(1−	er(1−	NOUN
ejpam-5066	90	5	e)re	e)re	NOUN
ejpam-5066	90	6	.	.	PUNCT
ejpam-5066	91	1	also	also	ADV
ejpam-5066	91	2	,	,	PUNCT
ejpam-5066	91	3	er(1−	er(1−	NOUN
ejpam-5066	91	4	e)se	e)se	PROPN
ejpam-5066	91	5	=	=	SYM
ejpam-5066	91	6	er(1−	er(1−	PROPN
ejpam-5066	92	1	e)se	e)se	PROPN
ejpam-5066	92	2	=	=	PROPN
ejpam-5066	92	3	(	(	PUNCT
ejpam-5066	92	4	er	er	INTJ
ejpam-5066	92	5	−	−	PUNCT
ejpam-5066	92	6	re)(1−	re)(1−	PROPN
ejpam-5066	93	1	e)se	e)se	PROPN
ejpam-5066	93	2	=	=	PUNCT
ejpam-5066	94	1	[	[	X
ejpam-5066	94	2	e	e	NOUN
ejpam-5066	94	3	,	,	PUNCT
ejpam-5066	94	4	r](se−	r](se−	NOUN
ejpam-5066	94	5	ese	ese	NOUN
ejpam-5066	94	6	)	)	PUNCT
ejpam-5066	94	7	=	=	PUNCT
ejpam-5066	95	1	[	[	X
ejpam-5066	95	2	e	e	NOUN
ejpam-5066	95	3	,	,	PUNCT
ejpam-5066	95	4	r](se−	r](se−	NOUN
ejpam-5066	95	5	es)e	es)e	PROPN
ejpam-5066	95	6	=	=	PUNCT
ejpam-5066	96	1	[	[	X
ejpam-5066	96	2	e	e	NOUN
ejpam-5066	96	3	,	,	PUNCT
ejpam-5066	96	4	r][e,−s]e	r][e,−s]e	PROPN
ejpam-5066	96	5	∈	∈	PROPN
ejpam-5066	96	6	er(1−	er(1−	VERB
ejpam-5066	97	1	e)re	e)re	NOUN
ejpam-5066	97	2	.	.	PUNCT
ejpam-5066	98	1	thus	thus	ADV
ejpam-5066	98	2	[	[	X
ejpam-5066	98	3	e	e	NOUN
ejpam-5066	98	4	,	,	PUNCT
ejpam-5066	98	5	r]2e	r]2e	NOUN
ejpam-5066	98	6	=	=	SYM
ejpam-5066	98	7	er(1−	er(1−	NOUN
ejpam-5066	99	1	e)re	e)re	PROPN
ejpam-5066	99	2	and	and	CCONJ
ejpam-5066	99	3	similarly	similarly	ADV
ejpam-5066	99	4	one	one	NUM
ejpam-5066	99	5	can	can	AUX
ejpam-5066	99	6	prove	prove	VERB
ejpam-5066	99	7	that	that	SCONJ
ejpam-5066	99	8	[	[	X
ejpam-5066	99	9	e	e	NOUN
ejpam-5066	99	10	,	,	PUNCT
ejpam-5066	99	11	r]2(1−	r]2(1−	ADJ
ejpam-5066	99	12	e	e	NOUN
ejpam-5066	99	13	)	)	PUNCT
ejpam-5066	99	14	=	=	SYM
ejpam-5066	99	15	er(1−	er(1−	NOUN
ejpam-5066	99	16	e)r(1−	e)r(1−	PROPN
ejpam-5066	99	17	e	e	NOUN
ejpam-5066	99	18	)	)	PUNCT
ejpam-5066	99	19	indeed	indeed	ADV
ejpam-5066	99	20	,	,	PUNCT
ejpam-5066	99	21	[	[	X
ejpam-5066	99	22	e]n	e]n	NOUN
ejpam-5066	99	23	=	=	SYM
ejpam-5066	99	24	[	[	X
ejpam-5066	99	25	e	e	NOUN
ejpam-5066	99	26	,	,	PUNCT
ejpam-5066	99	27	r]ner	r]ner	NOUN
ejpam-5066	99	28	,	,	PUNCT
ejpam-5066	99	29	for	for	ADP
ejpam-5066	99	30	n	n	X
ejpam-5066	99	31	=	=	SYM
ejpam-5066	99	32	0	0	NUM
ejpam-5066	99	33	and	and	CCONJ
ejpam-5066	99	34	n	n	CCONJ
ejpam-5066	99	35	=	=	SYM
ejpam-5066	99	36	1	1	NUM
ejpam-5066	99	37	from	from	ADP
ejpam-5066	99	38	claim	claim	NOUN
ejpam-5066	99	39	1	1	NUM
ejpam-5066	99	40	.	.	X
ejpam-5066	99	41	assume	assume	VERB
ejpam-5066	99	42	that	that	SCONJ
ejpam-5066	100	1	[	[	X
ejpam-5066	100	2	e]k	e]k	X
ejpam-5066	100	3	=	=	PUNCT
ejpam-5066	100	4	[	[	X
ejpam-5066	100	5	e	e	NOUN
ejpam-5066	100	6	,	,	PUNCT
ejpam-5066	100	7	r]ker	r]ker	NOUN
ejpam-5066	100	8	,	,	PUNCT
ejpam-5066	100	9	for	for	ADP
ejpam-5066	100	10	some	some	DET
ejpam-5066	100	11	integer	integer	NOUN
ejpam-5066	100	12	k	k	PROPN
ejpam-5066	100	13	≥	≥	NUM
ejpam-5066	100	14	0	0	NUM
ejpam-5066	100	15	.	.	PUNCT
ejpam-5066	101	1	if	if	SCONJ
ejpam-5066	101	2	k	k	PROPN
ejpam-5066	101	3	is	be	AUX
ejpam-5066	101	4	even	even	ADV
ejpam-5066	101	5	,	,	PUNCT
ejpam-5066	101	6	then	then	ADV
ejpam-5066	101	7	[	[	X
ejpam-5066	101	8	e	e	NOUN
ejpam-5066	101	9	,	,	PUNCT
ejpam-5066	101	10	r]k+2er	r]k+2er	X
ejpam-5066	101	11	=	=	PUNCT
ejpam-5066	102	1	[	[	X
ejpam-5066	102	2	e	e	NOUN
ejpam-5066	102	3	,	,	PUNCT
ejpam-5066	102	4	r]2[e	r]2[e	NOUN
ejpam-5066	102	5	,	,	PUNCT
ejpam-5066	102	6	r]ker	r]ker	X
ejpam-5066	102	7	=	=	PUNCT
ejpam-5066	103	1	[	[	X
ejpam-5066	103	2	e	e	NOUN
ejpam-5066	103	3	,	,	PUNCT
ejpam-5066	103	4	r]2e[e	r]2e[e	NOUN
ejpam-5066	103	5	,	,	PUNCT
ejpam-5066	103	6	r]ker	r]ker	NOUN
ejpam-5066	103	7	=	=	SYM
ejpam-5066	103	8	er(1−	er(1−	NOUN
ejpam-5066	103	9	e)re[e]k	e)re[e]k	VERB
ejpam-5066	103	10	=	=	PUNCT
ejpam-5066	104	1	[	[	X
ejpam-5066	104	2	e]k+2	e]k+2	NOUN
ejpam-5066	104	3	,	,	PUNCT
ejpam-5066	104	4	using	use	VERB
ejpam-5066	104	5	the	the	DET
ejpam-5066	104	6	result	result	NOUN
ejpam-5066	104	7	of	of	ADP
ejpam-5066	104	8	claims	claim	NOUN
ejpam-5066	104	9	2	2	NUM
ejpam-5066	104	10	and	and	CCONJ
ejpam-5066	104	11	3	3	NUM
ejpam-5066	104	12	.	.	PUNCT
ejpam-5066	105	1	if	if	SCONJ
ejpam-5066	105	2	k	k	PROPN
ejpam-5066	105	3	is	be	AUX
ejpam-5066	105	4	odd	odd	ADJ
ejpam-5066	105	5	,	,	PUNCT
ejpam-5066	105	6	then	then	ADV
ejpam-5066	105	7	[	[	X
ejpam-5066	105	8	e	e	NOUN
ejpam-5066	105	9	,	,	PUNCT
ejpam-5066	105	10	r]k+2er	r]k+2er	X
ejpam-5066	105	11	=	=	PUNCT
ejpam-5066	106	1	[	[	X
ejpam-5066	106	2	e	e	NOUN
ejpam-5066	106	3	,	,	PUNCT
ejpam-5066	106	4	r]2[e	r]2[e	NOUN
ejpam-5066	106	5	,	,	PUNCT
ejpam-5066	106	6	r]ker	r]ker	X
ejpam-5066	106	7	=	=	PUNCT
ejpam-5066	107	1	[	[	X
ejpam-5066	107	2	e	e	NOUN
ejpam-5066	107	3	,	,	PUNCT
ejpam-5066	107	4	r]2(1−	r]2(1−	PROPN
ejpam-5066	107	5	e)[e	e)[e	PROPN
ejpam-5066	107	6	,	,	PUNCT
ejpam-5066	107	7	r]ker	r]ker	NOUN
ejpam-5066	107	8	=	=	SYM
ejpam-5066	107	9	(	(	PUNCT
ejpam-5066	107	10	1−	1−	NUM
ejpam-5066	107	11	e)rer(1−	e)rer(1−	ADJ
ejpam-5066	107	12	e)[e]k	e)[e]k	NOUN
ejpam-5066	107	13	=	=	PUNCT
ejpam-5066	107	14	[	[	X
ejpam-5066	107	15	e]k+2	e]k+2	NOUN
ejpam-5066	107	16	.	.	PUNCT
ejpam-5066	108	1	the	the	DET
ejpam-5066	108	2	next	next	ADJ
ejpam-5066	108	3	definition	definition	NOUN
ejpam-5066	108	4	uses	use	VERB
ejpam-5066	108	5	our	our	PRON
ejpam-5066	108	6	notation	notation	NOUN
ejpam-5066	108	7	to	to	PART
ejpam-5066	108	8	give	give	VERB
ejpam-5066	108	9	a	a	DET
ejpam-5066	108	10	generalized	generalized	ADJ
ejpam-5066	108	11	centrality	centrality	NOUN
ejpam-5066	108	12	for	for	ADP
ejpam-5066	108	13	the	the	DET
ejpam-5066	108	14	idempotents	idempotent	NOUN
ejpam-5066	108	15	of	of	ADP
ejpam-5066	108	16	a	a	DET
ejpam-5066	108	17	ring	ring	NOUN
ejpam-5066	108	18	.	.	PUNCT
ejpam-5066	109	1	definition	definition	NOUN
ejpam-5066	109	2	1	1	NUM
ejpam-5066	109	3	.	.	PUNCT
ejpam-5066	110	1	an	an	DET
ejpam-5066	110	2	idempotent	idempotent	ADJ
ejpam-5066	110	3	e	e	NOUN
ejpam-5066	110	4	of	of	ADP
ejpam-5066	110	5	a	a	DET
ejpam-5066	110	6	ring	ring	NOUN
ejpam-5066	110	7	r	r	NOUN
ejpam-5066	110	8	is	be	AUX
ejpam-5066	110	9	said	say	VERB
ejpam-5066	110	10	to	to	PART
ejpam-5066	110	11	be	be	AUX
ejpam-5066	110	12	n	n	ADV
ejpam-5066	110	13	-	-	ADJ
ejpam-5066	110	14	central	central	ADJ
ejpam-5066	110	15	,	,	PUNCT
ejpam-5066	110	16	for	for	ADP
ejpam-5066	110	17	some	some	DET
ejpam-5066	110	18	positive	positive	ADJ
ejpam-5066	110	19	integer	integer	NOUN
ejpam-5066	110	20	n	n	CCONJ
ejpam-5066	110	21	,	,	PUNCT
ejpam-5066	110	22	if	if	SCONJ
ejpam-5066	110	23	[	[	X
ejpam-5066	110	24	e]n	e]n	X
ejpam-5066	110	25	=	=	SYM
ejpam-5066	110	26	0	0	NUM
ejpam-5066	110	27	.	.	PUNCT
ejpam-5066	111	1	moreover	moreover	ADV
ejpam-5066	111	2	,	,	PUNCT
ejpam-5066	111	3	e	e	PROPN
ejpam-5066	111	4	is	be	AUX
ejpam-5066	111	5	called	call	VERB
ejpam-5066	111	6	complementary	complementary	ADJ
ejpam-5066	111	7	n	n	CCONJ
ejpam-5066	111	8	-	-	PUNCT
ejpam-5066	111	9	central	central	ADJ
ejpam-5066	111	10	if	if	SCONJ
ejpam-5066	111	11	1	1	NUM
ejpam-5066	111	12	−	−	NOUN
ejpam-5066	111	13	e	e	NOUN
ejpam-5066	111	14	is	be	AUX
ejpam-5066	111	15	n	n	PRON
ejpam-5066	111	16	-	-	PUNCT
ejpam-5066	111	17	central	central	ADJ
ejpam-5066	111	18	and	and	CCONJ
ejpam-5066	111	19	dual	dual	ADJ
ejpam-5066	111	20	n	n	CCONJ
ejpam-5066	111	21	-	-	PUNCT
ejpam-5066	111	22	central	central	ADJ
ejpam-5066	111	23	if	if	SCONJ
ejpam-5066	111	24	it	it	PRON
ejpam-5066	111	25	is	be	AUX
ejpam-5066	111	26	both	both	PRON
ejpam-5066	111	27	n	n	CCONJ
ejpam-5066	111	28	-	-	PUNCT
ejpam-5066	111	29	central	central	ADJ
ejpam-5066	111	30	and	and	CCONJ
ejpam-5066	111	31	complementary	complementary	ADJ
ejpam-5066	111	32	n	n	CCONJ
ejpam-5066	111	33	-	-	PUNCT
ejpam-5066	111	34	central	central	ADJ
ejpam-5066	111	35	.	.	PUNCT
ejpam-5066	112	1	m.	m.	PROPN
ejpam-5066	112	2	saad	saad	PROPN
ejpam-5066	112	3	,	,	PUNCT
ejpam-5066	112	4	m.	m.	NOUN
ejpam-5066	112	5	zailaee	zailaee	PROPN
ejpam-5066	112	6	/	/	SYM
ejpam-5066	112	7	eur	eur	PROPN
ejpam-5066	112	8	.	.	PUNCT
ejpam-5066	113	1	j.	j.	PROPN
ejpam-5066	113	2	pure	pure	PROPN
ejpam-5066	113	3	appl	appl	PROPN
ejpam-5066	113	4	.	.	PROPN
ejpam-5066	113	5	math	math	PROPN
ejpam-5066	113	6	,	,	PUNCT
ejpam-5066	113	7	17	17	NUM
ejpam-5066	113	8	(	(	PUNCT
ejpam-5066	113	9	2	2	NUM
ejpam-5066	113	10	)	)	PUNCT
ejpam-5066	113	11	(	(	PUNCT
ejpam-5066	113	12	2024	2024	NUM
ejpam-5066	113	13	)	)	PUNCT
ejpam-5066	113	14	,	,	PUNCT
ejpam-5066	113	15	736	736	NUM
ejpam-5066	113	16	-	-	SYM
ejpam-5066	113	17	752	752	NUM
ejpam-5066	113	18	739	739	NUM
ejpam-5066	113	19	the	the	DET
ejpam-5066	113	20	following	follow	VERB
ejpam-5066	113	21	examples	example	NOUN
ejpam-5066	113	22	serve	serve	VERB
ejpam-5066	113	23	to	to	PART
ejpam-5066	113	24	showcase	showcase	VERB
ejpam-5066	113	25	idempotents	idempotent	NOUN
ejpam-5066	113	26	which	which	PRON
ejpam-5066	113	27	possess	possess	VERB
ejpam-5066	113	28	the	the	DET
ejpam-5066	113	29	characteristic	characteristic	NOUN
ejpam-5066	113	30	of	of	ADP
ejpam-5066	113	31	being	be	AUX
ejpam-5066	113	32	n	n	CCONJ
ejpam-5066	113	33	-	-	PUNCT
ejpam-5066	113	34	central	central	ADJ
ejpam-5066	113	35	,	,	PUNCT
ejpam-5066	113	36	yet	yet	CCONJ
ejpam-5066	113	37	they	they	PRON
ejpam-5066	113	38	do	do	AUX
ejpam-5066	113	39	not	not	PART
ejpam-5066	113	40	fall	fall	VERB
ejpam-5066	113	41	within	within	ADP
ejpam-5066	113	42	the	the	DET
ejpam-5066	113	43	categories	category	NOUN
ejpam-5066	113	44	of	of	ADP
ejpam-5066	113	45	semicentral	semicentral	ADJ
ejpam-5066	113	46	or	or	CCONJ
ejpam-5066	113	47	q	q	ADJ
ejpam-5066	113	48	-	-	ADJ
ejpam-5066	113	49	central	central	ADJ
ejpam-5066	113	50	.	.	PUNCT
ejpam-5066	113	51	example	example	NOUN
ejpam-5066	114	1	1	1	NUM
ejpam-5066	114	2	.	.	PUNCT
ejpam-5066	114	3	let	let	VERB
ejpam-5066	114	4	r	r	NOUN
ejpam-5066	114	5	=	=	SYM
ejpam-5066	114	6	t3(f	t3(f	PROPN
ejpam-5066	114	7	)	)	PUNCT
ejpam-5066	114	8	for	for	ADP
ejpam-5066	114	9	some	some	DET
ejpam-5066	114	10	field	field	NOUN
ejpam-5066	114	11	f	f	X
ejpam-5066	114	12	.	.	PUNCT
ejpam-5066	115	1	for	for	ADP
ejpam-5066	115	2	the	the	DET
ejpam-5066	115	3	idempotent	idempotent	ADJ
ejpam-5066	115	4	e	e	NOUN
ejpam-5066	115	5	=	=	SYM
ejpam-5066	115	6	diag(1	diag(1	PROPN
ejpam-5066	115	7	,	,	PUNCT
ejpam-5066	115	8	0	0	NUM
ejpam-5066	115	9	,	,	PUNCT
ejpam-5066	115	10	1	1	NUM
ejpam-5066	115	11	)	)	PUNCT
ejpam-5066	115	12	,	,	PUNCT
ejpam-5066	115	13	we	we	PRON
ejpam-5066	115	14	have	have	VERB
ejpam-5066	115	15	(	(	PUNCT
ejpam-5066	115	16	[	[	X
ejpam-5066	115	17	e]n	e]n	ADJ
ejpam-5066	115	18	)	)	PUNCT
ejpam-5066	115	19	=	=	NOUN
ejpam-5066	115	20			NOUN
ejpam-5066	115	21	f	f	PROPN
ejpam-5066	115	22	f	f	PROPN
ejpam-5066	115	23	f	f	PROPN
ejpam-5066	115	24	0	0	PROPN
ejpam-5066	115	25	0	0	NUM
ejpam-5066	115	26	0	0	NUM
ejpam-5066	115	27	0	0	NUM
ejpam-5066	115	28	0	0	NUM
ejpam-5066	115	29	f	f	NOUN
ejpam-5066	115	30			NOUN
ejpam-5066	115	31	,	,	PUNCT
ejpam-5066	115	32			NOUN
ejpam-5066	115	33	0	0	NUM
ejpam-5066	115	34	0	0	NUM
ejpam-5066	115	35	0	0	NUM
ejpam-5066	115	36	0	0	NUM
ejpam-5066	115	37	0	0	NUM
ejpam-5066	116	1	f	f	NOUN
ejpam-5066	116	2	0	0	NUM
ejpam-5066	116	3	0	0	NUM
ejpam-5066	116	4	0	0	NUM
ejpam-5066	116	5			NOUN
ejpam-5066	116	6	,	,	PUNCT
ejpam-5066	116	7			NOUN
ejpam-5066	116	8	0	0	NUM
ejpam-5066	116	9	0	0	NUM
ejpam-5066	117	1	f	f	NOUN
ejpam-5066	117	2	0	0	NUM
ejpam-5066	117	3	0	0	NUM
ejpam-5066	117	4	0	0	NUM
ejpam-5066	117	5	0	0	NUM
ejpam-5066	117	6	0	0	NUM
ejpam-5066	117	7	0	0	NUM
ejpam-5066	117	8			NOUN
ejpam-5066	117	9	,	,	PUNCT
ejpam-5066	117	10	0	0	NUM
ejpam-5066	117	11	,	,	PUNCT
ejpam-5066	117	12	0	0	NUM
ejpam-5066	117	13	,	,	PUNCT
ejpam-5066	117	14	0	0	NUM
ejpam-5066	117	15	,	,	PUNCT
ejpam-5066	117	16	·	·	PUNCT
ejpam-5066	117	17	·	·	PUNCT
ejpam-5066	117	18	·	·	PUNCT
ejpam-5066	118	1			PROPN
ejpam-5066	118	2	;	;	PUNCT
ejpam-5066	118	3	which	which	PRON
ejpam-5066	118	4	means	mean	VERB
ejpam-5066	118	5	that	that	SCONJ
ejpam-5066	118	6	e	e	NOUN
ejpam-5066	118	7	is	be	AUX
ejpam-5066	118	8	3	3	NUM
ejpam-5066	118	9	-	-	PUNCT
ejpam-5066	118	10	central	central	ADJ
ejpam-5066	118	11	.	.	PUNCT
ejpam-5066	119	1	note	note	VERB
ejpam-5066	119	2	that	that	SCONJ
ejpam-5066	119	3	straightforward	straightforward	ADJ
ejpam-5066	119	4	calculations	calculation	NOUN
ejpam-5066	119	5	can	can	AUX
ejpam-5066	119	6	show	show	VERB
ejpam-5066	119	7	that	that	SCONJ
ejpam-5066	119	8	e	e	NOUN
ejpam-5066	119	9	is	be	AUX
ejpam-5066	119	10	neither	neither	CCONJ
ejpam-5066	119	11	semicentral	semicentral	ADJ
ejpam-5066	119	12	nor	nor	CCONJ
ejpam-5066	119	13	q	q	ADJ
ejpam-5066	119	14	-	-	ADJ
ejpam-5066	119	15	central	central	ADJ
ejpam-5066	119	16	.	.	PUNCT
ejpam-5066	120	1	the	the	DET
ejpam-5066	120	2	example	example	NOUN
ejpam-5066	120	3	below	below	ADV
ejpam-5066	120	4	illustrates	illustrate	VERB
ejpam-5066	120	5	that	that	SCONJ
ejpam-5066	120	6	there	there	PRON
ejpam-5066	120	7	are	be	VERB
ejpam-5066	120	8	idempotents	idempotent	NOUN
ejpam-5066	120	9	that	that	PRON
ejpam-5066	120	10	can	can	AUX
ejpam-5066	120	11	not	not	PART
ejpam-5066	120	12	be	be	AUX
ejpam-5066	120	13	classified	classify	VERB
ejpam-5066	120	14	as	as	ADP
ejpam-5066	120	15	n	n	NOUN
ejpam-5066	120	16	-	-	PUNCT
ejpam-5066	120	17	central	central	ADJ
ejpam-5066	120	18	for	for	ADP
ejpam-5066	120	19	any	any	DET
ejpam-5066	120	20	n.	n.	NOUN
ejpam-5066	120	21	example	example	NOUN
ejpam-5066	121	1	2	2	NUM
ejpam-5066	121	2	.	.	X
ejpam-5066	121	3	in	in	ADP
ejpam-5066	121	4	the	the	DET
ejpam-5066	121	5	ring	ring	NOUN
ejpam-5066	121	6	s	s	X
ejpam-5066	121	7	=	=	SYM
ejpam-5066	121	8	m4(f	m4(f	NOUN
ejpam-5066	121	9	)	)	PUNCT
ejpam-5066	121	10	of	of	ADP
ejpam-5066	121	11	4	4	NUM
ejpam-5066	121	12	×	×	NOUN
ejpam-5066	121	13	4	4	NUM
ejpam-5066	121	14	matrices	matrix	NOUN
ejpam-5066	121	15	over	over	ADP
ejpam-5066	121	16	the	the	DET
ejpam-5066	121	17	filed	file	VERB
ejpam-5066	121	18	f	f	PROPN
ejpam-5066	121	19	,	,	PUNCT
ejpam-5066	121	20	the	the	DET
ejpam-5066	121	21	idempotent	idempotent	NOUN
ejpam-5066	121	22	f	f	NOUN
ejpam-5066	121	23	=	=	NOUN
ejpam-5066	121	24			NOUN
ejpam-5066	121	25	1	1	NUM
ejpam-5066	121	26	0	0	NUM
ejpam-5066	121	27	0	0	NUM
ejpam-5066	121	28	0	0	NUM
ejpam-5066	121	29	0	0	NUM
ejpam-5066	121	30	0	0	NUM
ejpam-5066	121	31	0	0	NUM
ejpam-5066	121	32	0	0	NUM
ejpam-5066	121	33	0	0	NUM
ejpam-5066	121	34	0	0	NUM
ejpam-5066	121	35	1	1	NUM
ejpam-5066	121	36	2	2	NUM
ejpam-5066	121	37	1	1	NUM
ejpam-5066	121	38	2	2	NUM
ejpam-5066	121	39	0	0	NUM
ejpam-5066	121	40	0	0	NUM
ejpam-5066	121	41	1	1	NUM
ejpam-5066	121	42	2	2	NUM
ejpam-5066	121	43	1	1	NUM
ejpam-5066	121	44	2	2	NUM
ejpam-5066	121	45			NOUN
ejpam-5066	121	46	has	have	VERB
ejpam-5066	121	47	the	the	DET
ejpam-5066	121	48	chain	chain	NOUN
ejpam-5066	121	49	(	(	PUNCT
ejpam-5066	121	50	[	[	X
ejpam-5066	121	51	f	f	X
ejpam-5066	121	52	]	]	X
ejpam-5066	121	53	n	n	CCONJ
ejpam-5066	121	54	)	)	PUNCT
ejpam-5066	121	55	=	=	SYM
ejpam-5066	121	56	(	(	PUNCT
ejpam-5066	121	57	a	a	PRON
ejpam-5066	121	58	,	,	PUNCT
ejpam-5066	121	59	b	b	NOUN
ejpam-5066	121	60	,	,	PUNCT
ejpam-5066	121	61	c	c	PROPN
ejpam-5066	121	62	,	,	PUNCT
ejpam-5066	121	63	b	b	PROPN
ejpam-5066	121	64	,	,	PUNCT
ejpam-5066	121	65	c	c	PROPN
ejpam-5066	121	66	,	,	PUNCT
ejpam-5066	121	67	b	b	PROPN
ejpam-5066	121	68	,	,	PUNCT
ejpam-5066	121	69	c	c	NOUN
ejpam-5066	121	70	,	,	PUNCT
ejpam-5066	121	71	·	·	PUNCT
ejpam-5066	121	72	·	·	PUNCT
ejpam-5066	121	73	·	·	PUNCT
ejpam-5066	121	74	)	)	PUNCT
ejpam-5066	121	75	,	,	PUNCT
ejpam-5066	121	76	where	where	SCONJ
ejpam-5066	121	77	a	a	DET
ejpam-5066	121	78	=	=	SYM
ejpam-5066	121	79			NOUN
ejpam-5066	121	80			NOUN
ejpam-5066	121	81	a1	a1	NOUN
ejpam-5066	121	82	a2	a2	PROPN
ejpam-5066	121	83	a3	a3	NOUN
ejpam-5066	121	84	a4	a4	PROPN
ejpam-5066	121	85	0	0	NUM
ejpam-5066	121	86	0	0	NUM
ejpam-5066	121	87	0	0	NUM
ejpam-5066	121	88	0	0	NUM
ejpam-5066	121	89	a5	a5	PROPN
ejpam-5066	121	90	a6	a6	NOUN
ejpam-5066	121	91	a7	a7	PROPN
ejpam-5066	121	92	a8	a8	PROPN
ejpam-5066	121	93	a5	a5	PROPN
ejpam-5066	121	94	a6	a6	PROPN
ejpam-5066	121	95	a7	a7	PROPN
ejpam-5066	121	96	a8	a8	PROPN
ejpam-5066	121	97			PROPN
ejpam-5066	121	98	|	|	ADV
ejpam-5066	121	99	ai	ai	VERB
ejpam-5066	121	100	∈	∈	PROPN
ejpam-5066	121	101	f	f	PROPN
ejpam-5066	121	102			PROPN
ejpam-5066	121	103	,	,	PUNCT
ejpam-5066	121	104	b	b	X
ejpam-5066	121	105	=	=	SYM
ejpam-5066	121	106			X
ejpam-5066	121	107			NOUN
ejpam-5066	121	108	0	0	NUM
ejpam-5066	121	109	0	0	NUM
ejpam-5066	121	110	0	0	SYM
ejpam-5066	121	111	0	0	NUM
ejpam-5066	121	112	a1	a1	NOUN
ejpam-5066	121	113	a2	a2	PROPN
ejpam-5066	121	114	a3	a3	PROPN
ejpam-5066	121	115	a4	a4	PROPN
ejpam-5066	121	116	a5	a5	PROPN
ejpam-5066	121	117	a6	a6	NOUN
ejpam-5066	122	1	a7	a7	PROPN
ejpam-5066	122	2	a8	a8	PROPN
ejpam-5066	122	3	−a5	−a5	PROPN
ejpam-5066	123	1	−a6	−a6	NOUN
ejpam-5066	123	2	−a7	−a7	PROPN
ejpam-5066	123	3	−a8	−a8	PROPN
ejpam-5066	123	4			PROPN
ejpam-5066	123	5	|	|	ADV
ejpam-5066	123	6	ai	ai	VERB
ejpam-5066	123	7	∈	∈	PROPN
ejpam-5066	123	8	f	f	PROPN
ejpam-5066	123	9			PROPN
ejpam-5066	123	10	,	,	PUNCT
ejpam-5066	123	11	c	c	NOUN
ejpam-5066	123	12	=	=	SYM
ejpam-5066	123	13			X
ejpam-5066	123	14			NOUN
ejpam-5066	123	15	a1	a1	NOUN
ejpam-5066	123	16	a2	a2	PROPN
ejpam-5066	123	17	a3	a3	NOUN
ejpam-5066	123	18	a4	a4	PROPN
ejpam-5066	123	19	0	0	NUM
ejpam-5066	123	20	0	0	NUM
ejpam-5066	123	21	0	0	NUM
ejpam-5066	123	22	0	0	NUM
ejpam-5066	123	23	a5	a5	PROPN
ejpam-5066	123	24	a6	a6	NOUN
ejpam-5066	123	25	a7	a7	PROPN
ejpam-5066	124	1	a8	a8	PROPN
ejpam-5066	125	1	−a5	−a5	PROPN
ejpam-5066	126	1	−a6	−a6	NOUN
ejpam-5066	126	2	−a7	−a7	PROPN
ejpam-5066	126	3	−a8	−a8	PROPN
ejpam-5066	126	4			PROPN
ejpam-5066	126	5	|	|	ADV
ejpam-5066	126	6	ai	ai	VERB
ejpam-5066	126	7	∈	∈	PROPN
ejpam-5066	126	8	f	f	PROPN
ejpam-5066	126	9			PROPN
ejpam-5066	126	10	.	.	PUNCT
ejpam-5066	127	1	we	we	PRON
ejpam-5066	127	2	make	make	VERB
ejpam-5066	127	3	cn(r	cn(r	X
ejpam-5066	127	4	)	)	PUNCT
ejpam-5066	127	5	,	,	PUNCT
ejpam-5066	127	6	cn(r	cn(r	X
ejpam-5066	127	7	)	)	PUNCT
ejpam-5066	127	8	,	,	PUNCT
ejpam-5066	127	9	and	and	CCONJ
ejpam-5066	127	10	ĉ(r	ĉ(r	NOUN
ejpam-5066	127	11	)	)	PUNCT
ejpam-5066	127	12	denote	denote	NOUN
ejpam-5066	127	13	,	,	PUNCT
ejpam-5066	127	14	respectively	respectively	ADV
ejpam-5066	127	15	,	,	PUNCT
ejpam-5066	127	16	the	the	DET
ejpam-5066	127	17	sets	set	NOUN
ejpam-5066	127	18	of	of	ADP
ejpam-5066	127	19	n	n	CCONJ
ejpam-5066	127	20	-	-	PUNCT
ejpam-5066	127	21	central	central	ADJ
ejpam-5066	127	22	,	,	PUNCT
ejpam-5066	127	23	complementary	complementary	ADJ
ejpam-5066	127	24	n	n	CCONJ
ejpam-5066	127	25	-	-	PUNCT
ejpam-5066	127	26	central	central	ADJ
ejpam-5066	127	27	,	,	PUNCT
ejpam-5066	127	28	and	and	CCONJ
ejpam-5066	127	29	dual	dual	ADJ
ejpam-5066	127	30	n	n	CCONJ
ejpam-5066	127	31	-	-	PUNCT
ejpam-5066	127	32	central	central	ADJ
ejpam-5066	127	33	idempotents	idempotent	NOUN
ejpam-5066	127	34	of	of	ADP
ejpam-5066	127	35	a	a	DET
ejpam-5066	127	36	ring	ring	NOUN
ejpam-5066	127	37	r.	r.	NOUN
ejpam-5066	127	38	the	the	DET
ejpam-5066	127	39	definitions	definition	NOUN
ejpam-5066	127	40	show	show	VERB
ejpam-5066	127	41	that	that	SCONJ
ejpam-5066	127	42	left	leave	VERB
ejpam-5066	127	43	semicentral	semicentral	ADJ
ejpam-5066	127	44	,	,	PUNCT
ejpam-5066	127	45	right	right	ADJ
ejpam-5066	127	46	semicentral	semicentral	ADJ
ejpam-5066	127	47	,	,	PUNCT
ejpam-5066	127	48	central	central	ADJ
ejpam-5066	127	49	,	,	PUNCT
ejpam-5066	127	50	and	and	CCONJ
ejpam-5066	127	51	quarter	quarter	NOUN
ejpam-5066	127	52	-	-	PUNCT
ejpam-5066	127	53	central	central	ADJ
ejpam-5066	127	54	idempotents	idempotent	NOUN
ejpam-5066	127	55	coincide	coincide	VERB
ejpam-5066	127	56	with	with	ADP
ejpam-5066	127	57	1	1	NUM
ejpam-5066	127	58	-	-	PUNCT
ejpam-5066	127	59	central	central	ADJ
ejpam-5066	127	60	,	,	PUNCT
ejpam-5066	127	61	complement	complement	VERB
ejpam-5066	127	62	1	1	NUM
ejpam-5066	127	63	-	-	PUNCT
ejpam-5066	127	64	central	central	ADJ
ejpam-5066	127	65	,	,	PUNCT
ejpam-5066	127	66	dual	dual	ADJ
ejpam-5066	127	67	1	1	NUM
ejpam-5066	127	68	-	-	ADJ
ejpam-5066	127	69	central	central	ADJ
ejpam-5066	127	70	,	,	PUNCT
ejpam-5066	127	71	and	and	CCONJ
ejpam-5066	127	72	2	2	NUM
ejpam-5066	127	73	-	-	PUNCT
ejpam-5066	127	74	central	central	ADJ
ejpam-5066	127	75	,	,	PUNCT
ejpam-5066	127	76	respectively	respectively	ADV
ejpam-5066	127	77	.	.	PUNCT
ejpam-5066	128	1	in	in	ADP
ejpam-5066	128	2	other	other	ADJ
ejpam-5066	128	3	words	word	NOUN
ejpam-5066	128	4	,	,	PUNCT
ejpam-5066	128	5	sl(r	sl(r	PUNCT
ejpam-5066	128	6	)	)	PUNCT
ejpam-5066	128	7	=	=	SYM
ejpam-5066	128	8	c1(r	c1(r	PROPN
ejpam-5066	128	9	)	)	PUNCT
ejpam-5066	128	10	,	,	PUNCT
ejpam-5066	128	11	sr(r	sr(r	NOUN
ejpam-5066	128	12	)	)	PUNCT
ejpam-5066	128	13	=	=	SYM
ejpam-5066	128	14	cr(r	cr(r	NOUN
ejpam-5066	128	15	)	)	PUNCT
ejpam-5066	128	16	,	,	PUNCT
ejpam-5066	128	17	b(r	b(r	PROPN
ejpam-5066	128	18	)	)	PUNCT
ejpam-5066	128	19	=	=	SYM
ejpam-5066	128	20	ĉ1(r	ĉ1(r	NOUN
ejpam-5066	128	21	)	)	PUNCT
ejpam-5066	128	22	,	,	PUNCT
ejpam-5066	128	23	and	and	CCONJ
ejpam-5066	128	24	q	q	X
ejpam-5066	128	25	-	-	PUNCT
ejpam-5066	128	26	idem(r	idem(r	NOUN
ejpam-5066	128	27	)	)	PUNCT
ejpam-5066	128	28	=	=	SYM
ejpam-5066	129	1	c2(r	c2(r	PROPN
ejpam-5066	129	2	)	)	PUNCT
ejpam-5066	129	3	,	,	PUNCT
ejpam-5066	129	4	of	of	ADP
ejpam-5066	129	5	every	every	DET
ejpam-5066	129	6	ring	ring	NOUN
ejpam-5066	129	7	r.	r.	PROPN
ejpam-5066	129	8	notice	notice	VERB
ejpam-5066	129	9	that	that	SCONJ
ejpam-5066	129	10	every	every	DET
ejpam-5066	129	11	n	n	CCONJ
ejpam-5066	129	12	-	-	PUNCT
ejpam-5066	129	13	central	central	NOUN
ejpam-5066	129	14	is	be	AUX
ejpam-5066	129	15	m	m	NOUN
ejpam-5066	129	16	-	-	ADJ
ejpam-5066	129	17	central	central	ADJ
ejpam-5066	129	18	if	if	SCONJ
ejpam-5066	129	19	n	n	DET
ejpam-5066	129	20	≤	≤	NOUN
ejpam-5066	129	21	m.	m.	NOUN
ejpam-5066	129	22	observe	observe	VERB
ejpam-5066	129	23	that	that	SCONJ
ejpam-5066	129	24	lemma	lemma	PROPN
ejpam-5066	129	25	1	1	NUM
ejpam-5066	129	26	provides	provide	VERB
ejpam-5066	129	27	an	an	DET
ejpam-5066	129	28	alternative	alternative	ADJ
ejpam-5066	129	29	definition	definition	NOUN
ejpam-5066	129	30	for	for	ADP
ejpam-5066	129	31	the	the	DET
ejpam-5066	129	32	n	n	NOUN
ejpam-5066	129	33	-	-	PUNCT
ejpam-5066	129	34	centralizer	centralizer	NOUN
ejpam-5066	129	35	of	of	ADP
ejpam-5066	129	36	an	an	DET
ejpam-5066	129	37	idempotent	idempotent	ADJ
ejpam-5066	129	38	e	e	NOUN
ejpam-5066	129	39	in	in	ADP
ejpam-5066	129	40	a	a	DET
ejpam-5066	129	41	ring	ring	NOUN
ejpam-5066	129	42	r	r	NOUN
ejpam-5066	129	43	,	,	PUNCT
ejpam-5066	129	44	which	which	PRON
ejpam-5066	129	45	is	be	AUX
ejpam-5066	129	46	independent	independent	ADJ
ejpam-5066	129	47	of	of	ADP
ejpam-5066	129	48	the	the	DET
ejpam-5066	129	49	existence	existence	NOUN
ejpam-5066	129	50	of	of	ADP
ejpam-5066	129	51	identity	identity	NOUN
ejpam-5066	129	52	in	in	ADP
ejpam-5066	129	53	r.	r.	PROPN
ejpam-5066	129	54	this	this	DET
ejpam-5066	129	55	definition	definition	NOUN
ejpam-5066	129	56	allows	allow	VERB
ejpam-5066	129	57	us	we	PRON
ejpam-5066	129	58	to	to	PART
ejpam-5066	129	59	extend	extend	VERB
ejpam-5066	129	60	the	the	DET
ejpam-5066	129	61	notion	notion	NOUN
ejpam-5066	129	62	of	of	ADP
ejpam-5066	129	63	n	n	CCONJ
ejpam-5066	129	64	-	-	PUNCT
ejpam-5066	129	65	central	central	ADJ
ejpam-5066	129	66	idempotents	idempotent	NOUN
ejpam-5066	129	67	to	to	ADP
ejpam-5066	129	68	rings	ring	NOUN
ejpam-5066	129	69	without	without	ADP
ejpam-5066	129	70	unity	unity	NOUN
ejpam-5066	129	71	or	or	CCONJ
ejpam-5066	129	72	near	near	NOUN
ejpam-5066	129	73	-	-	PUNCT
ejpam-5066	129	74	rings	ring	NOUN
ejpam-5066	129	75	.	.	PUNCT
ejpam-5066	130	1	nevertheless	nevertheless	ADV
ejpam-5066	130	2	,	,	PUNCT
ejpam-5066	130	3	for	for	ADP
ejpam-5066	130	4	the	the	DET
ejpam-5066	130	5	purpose	purpose	NOUN
ejpam-5066	130	6	of	of	ADP
ejpam-5066	130	7	this	this	DET
ejpam-5066	130	8	paper	paper	NOUN
ejpam-5066	130	9	,	,	PUNCT
ejpam-5066	130	10	we	we	PRON
ejpam-5066	130	11	restrict	restrict	VERB
ejpam-5066	130	12	our	our	PRON
ejpam-5066	130	13	attention	attention	NOUN
ejpam-5066	130	14	to	to	ADP
ejpam-5066	130	15	associative	associative	VERB
ejpam-5066	130	16	and	and	CCONJ
ejpam-5066	130	17	unital	unital	ADJ
ejpam-5066	130	18	rings	ring	NOUN
ejpam-5066	130	19	.	.	PUNCT
ejpam-5066	131	1	m.	m.	PROPN
ejpam-5066	131	2	saad	saad	PROPN
ejpam-5066	131	3	,	,	PUNCT
ejpam-5066	131	4	m.	m.	NOUN
ejpam-5066	131	5	zailaee	zailaee	PROPN
ejpam-5066	131	6	/	/	SYM
ejpam-5066	131	7	eur	eur	PROPN
ejpam-5066	131	8	.	.	PUNCT
ejpam-5066	132	1	j.	j.	PROPN
ejpam-5066	132	2	pure	pure	PROPN
ejpam-5066	132	3	appl	appl	PROPN
ejpam-5066	132	4	.	.	PROPN
ejpam-5066	132	5	math	math	PROPN
ejpam-5066	132	6	,	,	PUNCT
ejpam-5066	132	7	17	17	NUM
ejpam-5066	132	8	(	(	PUNCT
ejpam-5066	132	9	2	2	NUM
ejpam-5066	132	10	)	)	PUNCT
ejpam-5066	132	11	(	(	PUNCT
ejpam-5066	132	12	2024	2024	NUM
ejpam-5066	132	13	)	)	PUNCT
ejpam-5066	132	14	,	,	PUNCT
ejpam-5066	132	15	736	736	NUM
ejpam-5066	132	16	-	-	SYM
ejpam-5066	132	17	752	752	NUM
ejpam-5066	132	18	740	740	NUM
ejpam-5066	132	19	the	the	DET
ejpam-5066	132	20	following	follow	VERB
ejpam-5066	132	21	proposition	proposition	NOUN
ejpam-5066	132	22	presents	present	VERB
ejpam-5066	132	23	several	several	ADJ
ejpam-5066	132	24	equivalent	equivalent	ADJ
ejpam-5066	132	25	characterizations	characterization	NOUN
ejpam-5066	132	26	of	of	ADP
ejpam-5066	132	27	n	n	CCONJ
ejpam-5066	132	28	-	-	PUNCT
ejpam-5066	132	29	central	central	ADJ
ejpam-5066	132	30	idempotents	idempotent	NOUN
ejpam-5066	132	31	.	.	PUNCT
ejpam-5066	133	1	proposition	proposition	NOUN
ejpam-5066	133	2	1	1	NUM
ejpam-5066	133	3	.	.	PUNCT
ejpam-5066	134	1	for	for	ADP
ejpam-5066	134	2	a	a	DET
ejpam-5066	134	3	ring	ring	NOUN
ejpam-5066	134	4	r	r	NOUN
ejpam-5066	134	5	and	and	CCONJ
ejpam-5066	134	6	idempotent	idempotent	NOUN
ejpam-5066	134	7	e	e	NOUN
ejpam-5066	134	8	of	of	ADP
ejpam-5066	134	9	r	r	PROPN
ejpam-5066	134	10	,	,	PUNCT
ejpam-5066	134	11	the	the	DET
ejpam-5066	134	12	following	follow	VERB
ejpam-5066	134	13	statements	statement	NOUN
ejpam-5066	134	14	are	be	AUX
ejpam-5066	134	15	equivalent	equivalent	ADJ
ejpam-5066	134	16	.	.	PUNCT
ejpam-5066	135	1	(	(	PUNCT
ejpam-5066	135	2	i	i	NOUN
ejpam-5066	135	3	)	)	PUNCT
ejpam-5066	135	4	e	e	PROPN
ejpam-5066	135	5	∈	∈	PROPN
ejpam-5066	135	6	cn(r	cn(r	PRON
ejpam-5066	135	7	)	)	PUNCT
ejpam-5066	135	8	.	.	PUNCT
ejpam-5066	136	1	(	(	PUNCT
ejpam-5066	136	2	ii	ii	X
ejpam-5066	136	3	)	)	PUNCT
ejpam-5066	137	1	[	[	X
ejpam-5066	137	2	e]n−1	e]n−1	PROPN
ejpam-5066	137	3	is	be	AUX
ejpam-5066	137	4	an	an	DET
ejpam-5066	137	5	ideal	ideal	NOUN
ejpam-5066	137	6	of	of	ADP
ejpam-5066	137	7	r.	r.	PROPN
ejpam-5066	137	8	(	(	PUNCT
ejpam-5066	137	9	iii	iii	NOUN
ejpam-5066	137	10	)	)	PUNCT
ejpam-5066	138	1	[	[	X
ejpam-5066	138	2	e	e	NOUN
ejpam-5066	138	3	,	,	PUNCT
ejpam-5066	138	4	r]ne	r]ne	NOUN
ejpam-5066	138	5	=	=	SYM
ejpam-5066	138	6	0	0	NUM
ejpam-5066	138	7	.	.	PUNCT
ejpam-5066	138	8	(	(	PUNCT
ejpam-5066	138	9	iv	iv	X
ejpam-5066	138	10	)	)	PUNCT
ejpam-5066	138	11	e[e	e[e	ADJ
ejpam-5066	138	12	,	,	PUNCT
ejpam-5066	138	13	r]n	r]n	NOUN
ejpam-5066	138	14	=	=	NOUN
ejpam-5066	138	15	0	0	PUNCT
ejpam-5066	138	16	if	if	SCONJ
ejpam-5066	138	17	n	n	NOUN
ejpam-5066	138	18	is	be	AUX
ejpam-5066	138	19	even	even	ADV
ejpam-5066	138	20	,	,	PUNCT
ejpam-5066	138	21	(	(	PUNCT
ejpam-5066	138	22	1−	1−	NUM
ejpam-5066	138	23	e)[e	e)[e	NOUN
ejpam-5066	138	24	,	,	PUNCT
ejpam-5066	138	25	r]n	r]n	NOUN
ejpam-5066	138	26	=	=	NOUN
ejpam-5066	138	27	0	0	PUNCT
ejpam-5066	138	28	if	if	SCONJ
ejpam-5066	138	29	n	n	NOUN
ejpam-5066	138	30	is	be	AUX
ejpam-5066	138	31	odd	odd	ADJ
ejpam-5066	138	32	.	.	PUNCT
ejpam-5066	139	1	proof	proof	NOUN
ejpam-5066	139	2	.	.	PUNCT
ejpam-5066	140	1	(	(	PUNCT
ejpam-5066	140	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-5066	140	3	):	):	PUNCT
ejpam-5066	140	4	if	if	SCONJ
ejpam-5066	140	5	n	n	PRON
ejpam-5066	140	6	is	be	AUX
ejpam-5066	140	7	odd	odd	ADJ
ejpam-5066	140	8	,	,	PUNCT
ejpam-5066	140	9	then	then	ADV
ejpam-5066	140	10	(	(	PUNCT
ejpam-5066	140	11	1	1	NUM
ejpam-5066	140	12	−	−	NUM
ejpam-5066	140	13	e)r[e]n−1	e)r[e]n−1	PROPN
ejpam-5066	140	14	=	=	SYM
ejpam-5066	140	15	0	0	NUM
ejpam-5066	140	16	and	and	CCONJ
ejpam-5066	140	17	r[e]n−1	r[e]n−1	ADJ
ejpam-5066	140	18	=	=	PUNCT
ejpam-5066	140	19	er[e]n−1	er[e]n−1	ADJ
ejpam-5066	140	20	=	=	PUNCT
ejpam-5066	140	21	erer[e]n−2	erer[e]n−2	NOUN
ejpam-5066	140	22	⊆	⊆	NUM
ejpam-5066	140	23	er[e]n−2	er[e]n−2	PRON
ejpam-5066	141	1	=	=	PUNCT
ejpam-5066	142	1	[	[	X
ejpam-5066	142	2	e]n−1	e]n−1	ADJ
ejpam-5066	142	3	and	and	CCONJ
ejpam-5066	142	4	[	[	X
ejpam-5066	142	5	e]n−1	e]n−1	PROPN
ejpam-5066	142	6	is	be	AUX
ejpam-5066	142	7	a	a	DET
ejpam-5066	142	8	two	two	NUM
ejpam-5066	142	9	-	-	PUNCT
ejpam-5066	142	10	sided	sided	ADJ
ejpam-5066	142	11	ideal	ideal	NOUN
ejpam-5066	142	12	of	of	ADP
ejpam-5066	142	13	r.	r.	PROPN
ejpam-5066	142	14	similarly	similarly	ADV
ejpam-5066	142	15	,	,	PUNCT
ejpam-5066	142	16	for	for	ADP
ejpam-5066	142	17	the	the	DET
ejpam-5066	142	18	even	even	ADJ
ejpam-5066	142	19	case	case	NOUN
ejpam-5066	142	20	.	.	PUNCT
ejpam-5066	143	1	(	(	PUNCT
ejpam-5066	143	2	ii)⇒(i	ii)⇒(i	NOUN
ejpam-5066	143	3	):	):	PUNCT
ejpam-5066	143	4	if	if	SCONJ
ejpam-5066	143	5	[	[	X
ejpam-5066	143	6	e]n−1	e]n−1	ADJ
ejpam-5066	143	7	is	be	AUX
ejpam-5066	143	8	an	an	DET
ejpam-5066	143	9	ideal	ideal	NOUN
ejpam-5066	143	10	of	of	ADP
ejpam-5066	143	11	r	r	NOUN
ejpam-5066	143	12	and	and	CCONJ
ejpam-5066	143	13	assume	assume	VERB
ejpam-5066	143	14	that	that	SCONJ
ejpam-5066	143	15	n	n	PRON
ejpam-5066	143	16	is	be	AUX
ejpam-5066	143	17	odd	odd	ADJ
ejpam-5066	143	18	.	.	PUNCT
ejpam-5066	144	1	so	so	ADV
ejpam-5066	144	2	that	that	SCONJ
ejpam-5066	144	3	[	[	X
ejpam-5066	144	4	e]n	e]n	NOUN
ejpam-5066	144	5	=	=	SYM
ejpam-5066	144	6	(	(	PUNCT
ejpam-5066	144	7	1	1	NUM
ejpam-5066	144	8	−	−	NUM
ejpam-5066	144	9	e)r[e]n−1	e)r[e]n−1	PROPN
ejpam-5066	144	10	⊆	⊆	NUM
ejpam-5066	144	11	(	(	PUNCT
ejpam-5066	144	12	1−	1−	NUM
ejpam-5066	144	13	e)[e]n−1	e)[e]n−1	X
ejpam-5066	144	14	=	=	SYM
ejpam-5066	144	15	0	0	PUNCT
ejpam-5066	144	16	and	and	CCONJ
ejpam-5066	144	17	e	e	PROPN
ejpam-5066	144	18	is	be	AUX
ejpam-5066	144	19	n	n	CCONJ
ejpam-5066	144	20	-	-	PUNCT
ejpam-5066	144	21	central	central	ADJ
ejpam-5066	144	22	.	.	PUNCT
ejpam-5066	145	1	similarity	similarity	NOUN
ejpam-5066	145	2	,	,	PUNCT
ejpam-5066	145	3	for	for	ADP
ejpam-5066	145	4	the	the	DET
ejpam-5066	145	5	even	even	ADJ
ejpam-5066	145	6	case	case	NOUN
ejpam-5066	145	7	.	.	PUNCT
ejpam-5066	146	1	(	(	PUNCT
ejpam-5066	146	2	i)⇔(iii	i)⇔(iii	NOUN
ejpam-5066	146	3	):	):	PUNCT
ejpam-5066	146	4	it	it	PRON
ejpam-5066	146	5	is	be	AUX
ejpam-5066	146	6	direct	direct	ADJ
ejpam-5066	146	7	by	by	ADP
ejpam-5066	146	8	lemma	lemma	PROPN
ejpam-5066	146	9	1	1	NUM
ejpam-5066	146	10	.	.	PUNCT
ejpam-5066	147	1	(	(	PUNCT
ejpam-5066	147	2	i)⇔(iv	i)⇔(iv	NUM
ejpam-5066	147	3	)	)	PUNCT
ejpam-5066	147	4	is	be	AUX
ejpam-5066	147	5	clear	clear	ADJ
ejpam-5066	147	6	from	from	ADP
ejpam-5066	147	7	claim	claim	NOUN
ejpam-5066	147	8	2	2	NUM
ejpam-5066	147	9	.	.	PUNCT
ejpam-5066	147	10	corollary	corollary	ADJ
ejpam-5066	147	11	1	1	NUM
ejpam-5066	147	12	.	.	PUNCT
ejpam-5066	148	1	for	for	ADP
ejpam-5066	148	2	a	a	DET
ejpam-5066	148	3	ring	ring	NOUN
ejpam-5066	148	4	r	r	NOUN
ejpam-5066	148	5	and	and	CCONJ
ejpam-5066	148	6	idempotent	idempotent	NOUN
ejpam-5066	148	7	e	e	NOUN
ejpam-5066	148	8	of	of	ADP
ejpam-5066	148	9	r	r	PROPN
ejpam-5066	148	10	,	,	PUNCT
ejpam-5066	148	11	e	e	PROPN
ejpam-5066	148	12	∈	∈	PROPN
ejpam-5066	148	13	ĉn(r	ĉn(r	NOUN
ejpam-5066	148	14	)	)	PUNCT
ejpam-5066	148	15	if	if	SCONJ
ejpam-5066	148	16	and	and	CCONJ
ejpam-5066	148	17	only	only	ADV
ejpam-5066	148	18	if	if	SCONJ
ejpam-5066	148	19	[	[	X
ejpam-5066	148	20	e	e	NOUN
ejpam-5066	148	21	,	,	PUNCT
ejpam-5066	148	22	r]n	r]n	NOUN
ejpam-5066	148	23	=	=	NOUN
ejpam-5066	148	24	0	0	X
ejpam-5066	148	25	.	.	PUNCT
ejpam-5066	148	26	corollary	corollary	ADJ
ejpam-5066	148	27	2	2	NUM
ejpam-5066	148	28	(	(	PUNCT
ejpam-5066	148	29	[	[	X
ejpam-5066	148	30	4	4	NUM
ejpam-5066	148	31	]	]	PUNCT
ejpam-5066	148	32	,	,	PUNCT
ejpam-5066	148	33	proposition	proposition	NOUN
ejpam-5066	148	34	1.2.2	1.2.2	NUM
ejpam-5066	148	35	)	)	PUNCT
ejpam-5066	148	36	.	.	PUNCT
ejpam-5066	149	1	for	for	ADP
ejpam-5066	149	2	an	an	DET
ejpam-5066	149	3	idempotent	idempotent	ADJ
ejpam-5066	149	4	e	e	NOUN
ejpam-5066	149	5	of	of	ADP
ejpam-5066	149	6	a	a	DET
ejpam-5066	149	7	ring	ring	NOUN
ejpam-5066	149	8	r	r	NOUN
ejpam-5066	149	9	,	,	PUNCT
ejpam-5066	149	10	the	the	DET
ejpam-5066	149	11	following	follow	VERB
ejpam-5066	149	12	conditions	condition	NOUN
ejpam-5066	149	13	are	be	AUX
ejpam-5066	149	14	equivalent	equivalent	ADJ
ejpam-5066	149	15	:	:	PUNCT
ejpam-5066	149	16	(	(	PUNCT
ejpam-5066	149	17	i	i	NOUN
ejpam-5066	149	18	)	)	PUNCT
ejpam-5066	149	19	e	e	PROPN
ejpam-5066	149	20	∈	∈	PROPN
ejpam-5066	149	21	sl(r	sl(r	NOUN
ejpam-5066	149	22	)	)	PUNCT
ejpam-5066	149	23	;	;	PUNCT
ejpam-5066	149	24	(	(	PUNCT
ejpam-5066	149	25	ii	ii	NOUN
ejpam-5066	149	26	)	)	PUNCT
ejpam-5066	149	27	er	er	INTJ
ejpam-5066	149	28	is	be	AUX
ejpam-5066	149	29	an	an	DET
ejpam-5066	149	30	ideal	ideal	NOUN
ejpam-5066	149	31	of	of	ADP
ejpam-5066	149	32	r	r	NOUN
ejpam-5066	149	33	;	;	PUNCT
ejpam-5066	149	34	(	(	PUNCT
ejpam-5066	149	35	iii	iii	NOUN
ejpam-5066	149	36	)	)	PUNCT
ejpam-5066	149	37	1−	1−	NUM
ejpam-5066	149	38	e	e	NOUN
ejpam-5066	149	39	∈	∈	PROPN
ejpam-5066	149	40	sr(r	sr(r	NOUN
ejpam-5066	149	41	)	)	PUNCT
ejpam-5066	149	42	;	;	PUNCT
ejpam-5066	149	43	(	(	PUNCT
ejpam-5066	149	44	iv	iv	X
ejpam-5066	149	45	)	)	PUNCT
ejpam-5066	149	46	r(1−	r(1−	NOUN
ejpam-5066	149	47	e	e	X
ejpam-5066	149	48	)	)	PUNCT
ejpam-5066	149	49	is	be	AUX
ejpam-5066	149	50	an	an	DET
ejpam-5066	149	51	ideal	ideal	NOUN
ejpam-5066	149	52	of	of	ADP
ejpam-5066	149	53	r	r	NOUN
ejpam-5066	149	54	;	;	PUNCT
ejpam-5066	149	55	(	(	PUNCT
ejpam-5066	149	56	v	v	NOUN
ejpam-5066	149	57	)	)	PUNCT
ejpam-5066	149	58	(	(	PUNCT
ejpam-5066	149	59	1−	1−	NUM
ejpam-5066	149	60	e)re	e)re	PROPN
ejpam-5066	149	61	=	=	SYM
ejpam-5066	149	62	0	0	NUM
ejpam-5066	149	63	;	;	PUNCT
ejpam-5066	149	64	the	the	DET
ejpam-5066	149	65	following	follow	VERB
ejpam-5066	149	66	proposition	proposition	NOUN
ejpam-5066	149	67	provides	provide	VERB
ejpam-5066	149	68	a	a	DET
ejpam-5066	149	69	generalization	generalization	NOUN
ejpam-5066	149	70	of	of	ADP
ejpam-5066	149	71	[	[	X
ejpam-5066	149	72	7	7	NUM
ejpam-5066	149	73	,	,	PUNCT
ejpam-5066	149	74	proposition	proposition	NOUN
ejpam-5066	149	75	2.1	2.1	NUM
ejpam-5066	149	76	.	.	PUNCT
ejpam-5066	149	77	]	]	PUNCT
ejpam-5066	150	1	and	and	CCONJ
ejpam-5066	150	2	[	[	X
ejpam-5066	150	3	11	11	NUM
ejpam-5066	150	4	,	,	PUNCT
ejpam-5066	150	5	proposition	proposition	NOUN
ejpam-5066	150	6	2.1	2.1	NUM
ejpam-5066	150	7	]	]	PUNCT
ejpam-5066	150	8	,	,	PUNCT
ejpam-5066	150	9	which	which	PRON
ejpam-5066	150	10	presents	present	VERB
ejpam-5066	150	11	sufficient	sufficient	ADJ
ejpam-5066	150	12	conditions	condition	NOUN
ejpam-5066	150	13	for	for	ADP
ejpam-5066	150	14	verifying	verify	VERB
ejpam-5066	150	15	the	the	DET
ejpam-5066	150	16	n	n	NOUN
ejpam-5066	150	17	-	-	PUNCT
ejpam-5066	150	18	centrality	centrality	NOUN
ejpam-5066	150	19	of	of	ADP
ejpam-5066	150	20	an	an	DET
ejpam-5066	150	21	idempotent	idempotent	NOUN
ejpam-5066	150	22	.	.	PUNCT
ejpam-5066	151	1	in	in	ADP
ejpam-5066	151	2	this	this	DET
ejpam-5066	151	3	context	context	NOUN
ejpam-5066	151	4	,	,	PUNCT
ejpam-5066	151	5	1+n	1+n	NUM
ejpam-5066	151	6	(	(	PUNCT
ejpam-5066	151	7	r	r	NOUN
ejpam-5066	151	8	)	)	PUNCT
ejpam-5066	151	9	and	and	CCONJ
ejpam-5066	151	10	n2(r	n2(r	NOUN
ejpam-5066	151	11	)	)	PUNCT
ejpam-5066	151	12	respectively	respectively	ADV
ejpam-5066	151	13	denote	denote	VERB
ejpam-5066	151	14	the	the	DET
ejpam-5066	151	15	sets	set	NOUN
ejpam-5066	151	16	of	of	ADP
ejpam-5066	151	17	unipotent	unipotent	ADJ
ejpam-5066	151	18	elements	element	NOUN
ejpam-5066	151	19	of	of	ADP
ejpam-5066	151	20	r	r	NOUN
ejpam-5066	151	21	(	(	PUNCT
ejpam-5066	151	22	i.e.	i.e.	X
ejpam-5066	151	23	,	,	PUNCT
ejpam-5066	151	24	the	the	DET
ejpam-5066	151	25	elements	element	NOUN
ejpam-5066	151	26	of	of	ADP
ejpam-5066	151	27	the	the	DET
ejpam-5066	151	28	form	form	NOUN
ejpam-5066	151	29	1	1	NUM
ejpam-5066	151	30	+	+	CCONJ
ejpam-5066	151	31	a	a	PRON
ejpam-5066	151	32	for	for	ADP
ejpam-5066	151	33	a	a	DET
ejpam-5066	151	34	nilpotent	nilpotent	ADJ
ejpam-5066	151	35	element	element	NOUN
ejpam-5066	151	36	a	a	DET
ejpam-5066	151	37	∈	∈	PROPN
ejpam-5066	151	38	r	r	NOUN
ejpam-5066	151	39	)	)	PUNCT
ejpam-5066	151	40	and	and	CCONJ
ejpam-5066	151	41	the	the	DET
ejpam-5066	151	42	set	set	NOUN
ejpam-5066	151	43	of	of	ADP
ejpam-5066	151	44	all	all	DET
ejpam-5066	151	45	square	square	ADJ
ejpam-5066	151	46	-	-	PUNCT
ejpam-5066	151	47	zero	zero	NUM
ejpam-5066	151	48	elements	element	NOUN
ejpam-5066	151	49	of	of	ADP
ejpam-5066	151	50	r	r	NOUN
ejpam-5066	151	51	(	(	PUNCT
ejpam-5066	151	52	i.e.	i.e.	X
ejpam-5066	151	53	,	,	PUNCT
ejpam-5066	151	54	the	the	DET
ejpam-5066	151	55	nilpotent	nilpotent	ADJ
ejpam-5066	151	56	elements	element	NOUN
ejpam-5066	151	57	of	of	ADP
ejpam-5066	151	58	index	index	NOUN
ejpam-5066	151	59	2	2	NUM
ejpam-5066	151	60	)	)	PUNCT
ejpam-5066	151	61	.	.	PUNCT
ejpam-5066	152	1	note	note	VERB
ejpam-5066	152	2	that	that	SCONJ
ejpam-5066	152	3	although	although	SCONJ
ejpam-5066	152	4	lemma	lemma	PROPN
ejpam-5066	152	5	1	1	NUM
ejpam-5066	152	6	provides	provide	VERB
ejpam-5066	152	7	an	an	DET
ejpam-5066	152	8	equivalent	equivalent	ADJ
ejpam-5066	152	9	definition	definition	NOUN
ejpam-5066	152	10	of	of	ADP
ejpam-5066	152	11	the	the	DET
ejpam-5066	152	12	n	n	CCONJ
ejpam-5066	152	13	-	-	PUNCT
ejpam-5066	152	14	centralizer	centralizer	NOUN
ejpam-5066	152	15	of	of	ADP
ejpam-5066	152	16	an	an	DET
ejpam-5066	152	17	idempotent	idempotent	ADJ
ejpam-5066	152	18	e	e	NOUN
ejpam-5066	152	19	in	in	ADP
ejpam-5066	152	20	a	a	DET
ejpam-5066	152	21	ring	ring	NOUN
ejpam-5066	152	22	r	r	NOUN
ejpam-5066	152	23	that	that	PRON
ejpam-5066	152	24	is	be	AUX
ejpam-5066	152	25	independent	independent	ADJ
ejpam-5066	152	26	of	of	ADP
ejpam-5066	152	27	the	the	DET
ejpam-5066	152	28	identity	identity	NOUN
ejpam-5066	152	29	element	element	NOUN
ejpam-5066	152	30	of	of	ADP
ejpam-5066	152	31	r	r	NOUN
ejpam-5066	152	32	,	,	PUNCT
ejpam-5066	152	33	this	this	DET
ejpam-5066	152	34	paper	paper	NOUN
ejpam-5066	152	35	considers	consider	VERB
ejpam-5066	152	36	only	only	ADV
ejpam-5066	152	37	associative	associative	ADJ
ejpam-5066	152	38	and	and	CCONJ
ejpam-5066	152	39	unital	unital	ADJ
ejpam-5066	152	40	rings	ring	NOUN
ejpam-5066	152	41	.	.	PUNCT
ejpam-5066	153	1	proposition	proposition	NOUN
ejpam-5066	153	2	2	2	NUM
ejpam-5066	153	3	.	.	X
ejpam-5066	153	4	for	for	ADP
ejpam-5066	153	5	an	an	DET
ejpam-5066	153	6	idempotent	idempotent	ADJ
ejpam-5066	153	7	e	e	NOUN
ejpam-5066	153	8	of	of	ADP
ejpam-5066	153	9	a	a	DET
ejpam-5066	153	10	ring	ring	NOUN
ejpam-5066	153	11	r	r	NOUN
ejpam-5066	153	12	the	the	DET
ejpam-5066	153	13	following	following	ADJ
ejpam-5066	153	14	statements	statement	NOUN
ejpam-5066	153	15	are	be	AUX
ejpam-5066	153	16	equivalent	equivalent	ADJ
ejpam-5066	153	17	:	:	PUNCT
ejpam-5066	153	18	(	(	PUNCT
ejpam-5066	153	19	i	i	NOUN
ejpam-5066	153	20	)	)	PUNCT
ejpam-5066	153	21	e	e	PROPN
ejpam-5066	153	22	∈	∈	PROPN
ejpam-5066	153	23	cn(r	cn(r	PRON
ejpam-5066	153	24	)	)	PUNCT
ejpam-5066	153	25	;	;	PUNCT
ejpam-5066	154	1	m.	m.	NOUN
ejpam-5066	154	2	saad	saad	PROPN
ejpam-5066	154	3	,	,	PUNCT
ejpam-5066	154	4	m.	m.	NOUN
ejpam-5066	154	5	zailaee	zailaee	PROPN
ejpam-5066	154	6	/	/	SYM
ejpam-5066	154	7	eur	eur	PROPN
ejpam-5066	154	8	.	.	PUNCT
ejpam-5066	155	1	j.	j.	PROPN
ejpam-5066	155	2	pure	pure	PROPN
ejpam-5066	155	3	appl	appl	PROPN
ejpam-5066	155	4	.	.	PROPN
ejpam-5066	155	5	math	math	PROPN
ejpam-5066	155	6	,	,	PUNCT
ejpam-5066	155	7	17	17	NUM
ejpam-5066	155	8	(	(	PUNCT
ejpam-5066	155	9	2	2	NUM
ejpam-5066	155	10	)	)	PUNCT
ejpam-5066	155	11	(	(	PUNCT
ejpam-5066	155	12	2024	2024	NUM
ejpam-5066	155	13	)	)	PUNCT
ejpam-5066	155	14	,	,	PUNCT
ejpam-5066	155	15	736	736	NUM
ejpam-5066	155	16	-	-	SYM
ejpam-5066	155	17	752	752	NUM
ejpam-5066	155	18	741	741	NUM
ejpam-5066	155	19	(	(	PUNCT
ejpam-5066	155	20	ii	ii	NOUN
ejpam-5066	155	21	)	)	PUNCT
ejpam-5066	156	1	[	[	X
ejpam-5066	156	2	e	e	NOUN
ejpam-5066	156	3	,	,	PUNCT
ejpam-5066	156	4	u(r)]ne	u(r)]ne	NOUN
ejpam-5066	156	5	=	=	SYM
ejpam-5066	156	6	0	0	NUM
ejpam-5066	156	7	;	;	PUNCT
ejpam-5066	156	8	(	(	PUNCT
ejpam-5066	156	9	iii	iii	X
ejpam-5066	156	10	)	)	PUNCT
ejpam-5066	157	1	[	[	X
ejpam-5066	157	2	e	e	NOUN
ejpam-5066	157	3	,	,	PUNCT
ejpam-5066	157	4	1	1	NUM
ejpam-5066	157	5	+	+	NOUN
ejpam-5066	157	6	n	n	NUM
ejpam-5066	157	7	(	(	PUNCT
ejpam-5066	157	8	r)]ne	r)]ne	NOUN
ejpam-5066	157	9	=	=	NOUN
ejpam-5066	157	10	0	0	NUM
ejpam-5066	157	11	;	;	PUNCT
ejpam-5066	157	12	(	(	PUNCT
ejpam-5066	157	13	iv	iv	X
ejpam-5066	157	14	)	)	PUNCT
ejpam-5066	158	1	[	[	X
ejpam-5066	158	2	e	e	X
ejpam-5066	158	3	,	,	PUNCT
ejpam-5066	158	4	n	n	PROPN
ejpam-5066	158	5	(	(	PUNCT
ejpam-5066	158	6	r)]ne	r)]ne	PROPN
ejpam-5066	158	7	=	=	NOUN
ejpam-5066	158	8	0	0	NUM
ejpam-5066	158	9	;	;	PUNCT
ejpam-5066	158	10	(	(	PUNCT
ejpam-5066	158	11	v	v	NOUN
ejpam-5066	158	12	)	)	PUNCT
ejpam-5066	159	1	[	[	X
ejpam-5066	159	2	e	e	X
ejpam-5066	159	3	,	,	PUNCT
ejpam-5066	159	4	n2(r)]ne	n2(r)]ne	NOUN
ejpam-5066	159	5	=	=	NOUN
ejpam-5066	159	6	0	0	NUM
ejpam-5066	159	7	;	;	PUNCT
ejpam-5066	159	8	(	(	PUNCT
ejpam-5066	159	9	vi	vi	NOUN
ejpam-5066	159	10	)	)	PUNCT
ejpam-5066	159	11	[	[	X
ejpam-5066	159	12	e	e	X
ejpam-5066	159	13	,	,	PUNCT
ejpam-5066	159	14	i(r)]ne	i(r)]ne	PROPN
ejpam-5066	159	15	=	=	SYM
ejpam-5066	159	16	0	0	NUM
ejpam-5066	159	17	;	;	PUNCT
ejpam-5066	159	18	proof	proof	NOUN
ejpam-5066	159	19	.	.	PUNCT
ejpam-5066	160	1	(	(	PUNCT
ejpam-5066	160	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-5066	160	3	)	)	PUNCT
ejpam-5066	160	4	is	be	AUX
ejpam-5066	160	5	obvious	obvious	ADJ
ejpam-5066	160	6	from	from	ADP
ejpam-5066	160	7	proposition	proposition	NOUN
ejpam-5066	160	8	1	1	NUM
ejpam-5066	160	9	.	.	PUNCT
ejpam-5066	160	10	(	(	PUNCT
ejpam-5066	160	11	ii)⇒(iii	ii)⇒(iii	NOUN
ejpam-5066	160	12	)	)	PUNCT
ejpam-5066	160	13	and	and	CCONJ
ejpam-5066	160	14	(	(	PUNCT
ejpam-5066	160	15	iv)⇒(v	iv)⇒(v	ADV
ejpam-5066	160	16	)	)	PUNCT
ejpam-5066	160	17	are	be	AUX
ejpam-5066	160	18	obvious	obvious	ADJ
ejpam-5066	160	19	.	.	PUNCT
ejpam-5066	161	1	(	(	PUNCT
ejpam-5066	161	2	iii)⇒(iv	iii)⇒(iv	NOUN
ejpam-5066	161	3	):	):	PUNCT
ejpam-5066	161	4	it	it	PRON
ejpam-5066	161	5	is	be	AUX
ejpam-5066	161	6	direct	direct	ADJ
ejpam-5066	161	7	since	since	SCONJ
ejpam-5066	161	8	[	[	X
ejpam-5066	161	9	a	a	X
ejpam-5066	161	10	,	,	PUNCT
ejpam-5066	161	11	1	1	NUM
ejpam-5066	161	12	+	+	SYM
ejpam-5066	161	13	b	b	X
ejpam-5066	161	14	]	]	X
ejpam-5066	161	15	=	=	PUNCT
ejpam-5066	162	1	[	[	X
ejpam-5066	162	2	a	a	X
ejpam-5066	162	3	,	,	PUNCT
ejpam-5066	162	4	b	b	NOUN
ejpam-5066	162	5	]	]	X
ejpam-5066	162	6	,	,	PUNCT
ejpam-5066	162	7	for	for	ADP
ejpam-5066	162	8	every	every	DET
ejpam-5066	162	9	a	a	PROPN
ejpam-5066	162	10	,	,	PUNCT
ejpam-5066	162	11	b	b	X
ejpam-5066	162	12	∈	∈	PROPN
ejpam-5066	162	13	r	r	NOUN
ejpam-5066	162	14	(	(	PUNCT
ejpam-5066	162	15	v)⇒(vi	v)⇒(vi	NUM
ejpam-5066	162	16	):	):	PUNCT
ejpam-5066	162	17	for	for	ADP
ejpam-5066	162	18	all	all	DET
ejpam-5066	162	19	fi	fi	NOUN
ejpam-5066	162	20	∈	∈	PROPN
ejpam-5066	162	21	i(r	i(r	PROPN
ejpam-5066	162	22	)	)	PUNCT
ejpam-5066	162	23	(	(	PUNCT
ejpam-5066	162	24	for	for	ADP
ejpam-5066	162	25	i	i	PRON
ejpam-5066	162	26	=	=	SYM
ejpam-5066	162	27	1	1	NUM
ejpam-5066	162	28	,	,	PUNCT
ejpam-5066	162	29	·	·	PUNCT
ejpam-5066	162	30	·	·	PUNCT
ejpam-5066	162	31	·	·	PUNCT
ejpam-5066	162	32	,	,	PUNCT
ejpam-5066	162	33	n	n	CCONJ
ejpam-5066	162	34	)	)	PUNCT
ejpam-5066	162	35	,	,	PUNCT
ejpam-5066	162	36	the	the	DET
ejpam-5066	162	37	elements	element	NOUN
ejpam-5066	162	38	efi(1−e	efi(1−e	PROPN
ejpam-5066	162	39	)	)	PUNCT
ejpam-5066	162	40	and	and	CCONJ
ejpam-5066	162	41	(	(	PUNCT
ejpam-5066	162	42	1−e)fie	1−e)fie	NUM
ejpam-5066	162	43	are	be	AUX
ejpam-5066	162	44	square	square	NOUN
ejpam-5066	162	45	-	-	PUNCT
ejpam-5066	162	46	zero	zero	NUM
ejpam-5066	162	47	.	.	PUNCT
ejpam-5066	163	1	but	but	CCONJ
ejpam-5066	163	2	[	[	X
ejpam-5066	163	3	e	e	NOUN
ejpam-5066	163	4	,	,	PUNCT
ejpam-5066	163	5	efi(1−	efi(1−	PROPN
ejpam-5066	163	6	e	e	NOUN
ejpam-5066	163	7	)	)	PUNCT
ejpam-5066	163	8	]	]	PUNCT
ejpam-5066	164	1	=	=	PUNCT
ejpam-5066	164	2	efi(1−	efi(1−	PROPN
ejpam-5066	164	3	e	e	X
ejpam-5066	164	4	)	)	PUNCT
ejpam-5066	164	5	=	=	PUNCT
ejpam-5066	165	1	[	[	X
ejpam-5066	165	2	e	e	NOUN
ejpam-5066	165	3	,	,	PUNCT
ejpam-5066	165	4	fi](1−	fi](1−	NOUN
ejpam-5066	165	5	e	e	NOUN
ejpam-5066	165	6	)	)	PUNCT
ejpam-5066	165	7	and	and	CCONJ
ejpam-5066	165	8	similarity	similarity	NOUN
ejpam-5066	166	1	[	[	X
ejpam-5066	166	2	e	e	NOUN
ejpam-5066	166	3	,	,	PUNCT
ejpam-5066	166	4	(	(	PUNCT
ejpam-5066	166	5	1−	1−	NUM
ejpam-5066	166	6	e)fie	e)fie	NOUN
ejpam-5066	166	7	]	]	X
ejpam-5066	166	8	=	=	SYM
ejpam-5066	166	9	−[e	−[e	PROPN
ejpam-5066	166	10	,	,	PUNCT
ejpam-5066	166	11	fi]e	fi]e	PROPN
ejpam-5066	166	12	.	.	PUNCT
ejpam-5066	167	1	so	so	ADV
ejpam-5066	167	2	that	that	SCONJ
ejpam-5066	167	3	[	[	X
ejpam-5066	167	4	e	e	NOUN
ejpam-5066	167	5	,	,	PUNCT
ejpam-5066	167	6	f1	f1	NOUN
ejpam-5066	167	7	]	]	PUNCT
ejpam-5066	167	8	·	·	PUNCT
ejpam-5066	167	9	·	·	PUNCT
ejpam-5066	167	10	·	·	PUNCT
ejpam-5066	168	1	[	[	X
ejpam-5066	168	2	e	e	NOUN
ejpam-5066	168	3	,	,	PUNCT
ejpam-5066	168	4	fn−2][e	fn−2][e	PROPN
ejpam-5066	168	5	,	,	PUNCT
ejpam-5066	168	6	fn−1][e	fn−1][e	PROPN
ejpam-5066	168	7	,	,	PUNCT
ejpam-5066	168	8	fn]e	fn]e	NOUN
ejpam-5066	168	9	=	=	SYM
ejpam-5066	168	10	[	[	X
ejpam-5066	168	11	e	e	NOUN
ejpam-5066	168	12	,	,	PUNCT
ejpam-5066	168	13	f1	f1	NOUN
ejpam-5066	168	14	]	]	PUNCT
ejpam-5066	168	15	·	·	PUNCT
ejpam-5066	168	16	·	·	PUNCT
ejpam-5066	168	17	·	·	PUNCT
ejpam-5066	169	1	[	[	X
ejpam-5066	169	2	e	e	NOUN
ejpam-5066	169	3	,	,	PUNCT
ejpam-5066	169	4	fn−2][e	fn−2][e	PROPN
ejpam-5066	169	5	,	,	PUNCT
ejpam-5066	169	6	fn−1][e	fn−1][e	PROPN
ejpam-5066	169	7	,	,	PUNCT
ejpam-5066	169	8	fn]e	fn]e	PROPN
ejpam-5066	169	9	2	2	NUM
ejpam-5066	169	10	=	=	SYM
ejpam-5066	169	11	[	[	X
ejpam-5066	169	12	e	e	NOUN
ejpam-5066	169	13	,	,	PUNCT
ejpam-5066	169	14	f1	f1	NOUN
ejpam-5066	169	15	]	]	PUNCT
ejpam-5066	169	16	·	·	PUNCT
ejpam-5066	169	17	·	·	PUNCT
ejpam-5066	169	18	·	·	PUNCT
ejpam-5066	170	1	[	[	X
ejpam-5066	170	2	e	e	NOUN
ejpam-5066	170	3	,	,	PUNCT
ejpam-5066	170	4	fn−2][e	fn−2][e	PROPN
ejpam-5066	170	5	,	,	PUNCT
ejpam-5066	170	6	fn−1](1−	fn−1](1−	PROPN
ejpam-5066	170	7	e)[e	e)[e	PROPN
ejpam-5066	170	8	,	,	PUNCT
ejpam-5066	170	9	fn]e	fn]e	NOUN
ejpam-5066	170	10	=	=	SYM
ejpam-5066	170	11	−[e	−[e	PROPN
ejpam-5066	170	12	,	,	PUNCT
ejpam-5066	170	13	f1	f1	NOUN
ejpam-5066	170	14	]	]	PUNCT
ejpam-5066	170	15	·	·	PUNCT
ejpam-5066	170	16	·	·	PUNCT
ejpam-5066	170	17	·	·	PUNCT
ejpam-5066	171	1	[	[	X
ejpam-5066	171	2	e	e	NOUN
ejpam-5066	171	3	,	,	PUNCT
ejpam-5066	171	4	fn−2][e	fn−2][e	PROPN
ejpam-5066	171	5	,	,	PUNCT
ejpam-5066	171	6	fn−1](1−	fn−1](1−	PROPN
ejpam-5066	171	7	e)[e	e)[e	PROPN
ejpam-5066	171	8	,	,	PUNCT
ejpam-5066	171	9	(	(	PUNCT
ejpam-5066	171	10	1−	1−	NUM
ejpam-5066	171	11	e)fne	e)fne	NOUN
ejpam-5066	171	12	]	]	PUNCT
ejpam-5066	171	13	=	=	SYM
ejpam-5066	171	14	−[e	−[e	PROPN
ejpam-5066	171	15	,	,	PUNCT
ejpam-5066	171	16	f1	f1	NOUN
ejpam-5066	171	17	]	]	PUNCT
ejpam-5066	171	18	·	·	PUNCT
ejpam-5066	171	19	·	·	PUNCT
ejpam-5066	171	20	·	·	PUNCT
ejpam-5066	172	1	[	[	X
ejpam-5066	172	2	e	e	X
ejpam-5066	172	3	,	,	PUNCT
ejpam-5066	172	4	fn−2]e[e	fn−2]e[e	NOUN
ejpam-5066	172	5	,	,	PUNCT
ejpam-5066	172	6	fn−1](1−	fn−1](1−	PROPN
ejpam-5066	172	7	e)[e	e)[e	PROPN
ejpam-5066	172	8	,	,	PUNCT
ejpam-5066	172	9	(	(	PUNCT
ejpam-5066	172	10	1−	1−	NUM
ejpam-5066	172	11	e)fne	e)fne	NOUN
ejpam-5066	172	12	]	]	PUNCT
ejpam-5066	172	13	=	=	SYM
ejpam-5066	172	14	−[e	−[e	PROPN
ejpam-5066	172	15	,	,	PUNCT
ejpam-5066	172	16	f1	f1	NOUN
ejpam-5066	172	17	]	]	PUNCT
ejpam-5066	172	18	·	·	PUNCT
ejpam-5066	172	19	·	·	PUNCT
ejpam-5066	172	20	·	·	PUNCT
ejpam-5066	173	1	[	[	X
ejpam-5066	173	2	e	e	X
ejpam-5066	173	3	,	,	PUNCT
ejpam-5066	173	4	fn−2]e[e	fn−2]e[e	NOUN
ejpam-5066	173	5	,	,	PUNCT
ejpam-5066	173	6	(	(	PUNCT
ejpam-5066	173	7	1−	1−	NUM
ejpam-5066	173	8	e)fn−1e][e	e)fn−1e][e	NOUN
ejpam-5066	173	9	,	,	PUNCT
ejpam-5066	173	10	(	(	PUNCT
ejpam-5066	173	11	1−	1−	NUM
ejpam-5066	173	12	e)fne	e)fne	NOUN
ejpam-5066	173	13	]	]	PUNCT
ejpam-5066	173	14	.	.	PUNCT
ejpam-5066	174	1	continuing	continue	VERB
ejpam-5066	174	2	,	,	PUNCT
ejpam-5066	174	3	we	we	PRON
ejpam-5066	174	4	get	get	VERB
ejpam-5066	174	5	[	[	X
ejpam-5066	174	6	e	e	NOUN
ejpam-5066	174	7	,	,	PUNCT
ejpam-5066	174	8	f1	f1	NOUN
ejpam-5066	174	9	]	]	PUNCT
ejpam-5066	174	10	·	·	PUNCT
ejpam-5066	174	11	·	·	PUNCT
ejpam-5066	174	12	·	·	PUNCT
ejpam-5066	175	1	[	[	X
ejpam-5066	175	2	e	e	NOUN
ejpam-5066	175	3	,	,	PUNCT
ejpam-5066	175	4	fn]e	fn]e	PROPN
ejpam-5066	175	5	=	=	SYM
ejpam-5066	175	6	(	(	PUNCT
ejpam-5066	175	7	−1	−1	NOUN
ejpam-5066	175	8	)	)	PUNCT
ejpam-5066	175	9	n	n	CCONJ
ejpam-5066	175	10	2	2	NUM
ejpam-5066	175	11	e[e	e[e	ADJ
ejpam-5066	175	12	,	,	PUNCT
ejpam-5066	175	13	(	(	PUNCT
ejpam-5066	175	14	1	1	NUM
ejpam-5066	175	15	−	−	NOUN
ejpam-5066	175	16	e)f1e	e)f1e	NOUN
ejpam-5066	175	17	]	]	PUNCT
ejpam-5066	175	18	·	·	PUNCT
ejpam-5066	175	19	·	·	PUNCT
ejpam-5066	175	20	·	·	PUNCT
ejpam-5066	176	1	[	[	X
ejpam-5066	176	2	e	e	X
ejpam-5066	176	3	,	,	PUNCT
ejpam-5066	176	4	(	(	PUNCT
ejpam-5066	176	5	1	1	NUM
ejpam-5066	176	6	−	−	NOUN
ejpam-5066	176	7	e)fne	e)fne	NOUN
ejpam-5066	176	8	]	]	PUNCT
ejpam-5066	176	9	if	if	SCONJ
ejpam-5066	176	10	n	n	PRON
ejpam-5066	176	11	is	be	AUX
ejpam-5066	176	12	even	even	ADV
ejpam-5066	176	13	,	,	PUNCT
ejpam-5066	176	14	and	and	CCONJ
ejpam-5066	176	15	[	[	X
ejpam-5066	176	16	e	e	NOUN
ejpam-5066	176	17	,	,	PUNCT
ejpam-5066	176	18	f1	f1	NOUN
ejpam-5066	176	19	]	]	PUNCT
ejpam-5066	176	20	·	·	PUNCT
ejpam-5066	176	21	·	·	PUNCT
ejpam-5066	176	22	·	·	PUNCT
ejpam-5066	177	1	[	[	X
ejpam-5066	177	2	e	e	NOUN
ejpam-5066	177	3	,	,	PUNCT
ejpam-5066	177	4	fn]e	fn]e	PROPN
ejpam-5066	177	5	=	=	SYM
ejpam-5066	177	6	(	(	PUNCT
ejpam-5066	177	7	−1	−1	NOUN
ejpam-5066	177	8	)	)	PUNCT
ejpam-5066	177	9	n+1	n+1	NUM
ejpam-5066	177	10	2	2	NUM
ejpam-5066	177	11	(	(	PUNCT
ejpam-5066	177	12	1−	1−	NUM
ejpam-5066	177	13	e)[e	e)[e	PROPN
ejpam-5066	177	14	,	,	PUNCT
ejpam-5066	177	15	ef1(1−	ef1(1−	ADJ
ejpam-5066	177	16	e	e	NOUN
ejpam-5066	177	17	)	)	PUNCT
ejpam-5066	177	18	]	]	PUNCT
ejpam-5066	177	19	·	·	PUNCT
ejpam-5066	177	20	·	·	PUNCT
ejpam-5066	177	21	·	·	PUNCT
ejpam-5066	178	1	[	[	X
ejpam-5066	178	2	e	e	NOUN
ejpam-5066	178	3	,	,	PUNCT
ejpam-5066	178	4	(	(	PUNCT
ejpam-5066	178	5	1−	1−	NUM
ejpam-5066	178	6	e)fne	e)fne	NOUN
ejpam-5066	178	7	]	]	PUNCT
ejpam-5066	178	8	if	if	SCONJ
ejpam-5066	178	9	n	n	NOUN
ejpam-5066	178	10	is	be	AUX
ejpam-5066	178	11	odd	odd	ADJ
ejpam-5066	178	12	.	.	PUNCT
ejpam-5066	179	1	using	use	VERB
ejpam-5066	179	2	the	the	DET
ejpam-5066	179	3	claim	claim	NOUN
ejpam-5066	179	4	2	2	NUM
ejpam-5066	179	5	to	to	ADP
ejpam-5066	179	6	e	e	NOUN
ejpam-5066	179	7	and	and	CCONJ
ejpam-5066	179	8	1	1	NUM
ejpam-5066	179	9	−	−	NOUN
ejpam-5066	179	10	e	e	X
ejpam-5066	179	11	in	in	ADP
ejpam-5066	179	12	the	the	DET
ejpam-5066	179	13	left	left	NOUN
ejpam-5066	179	14	of	of	ADP
ejpam-5066	179	15	the	the	DET
ejpam-5066	179	16	previous	previous	ADJ
ejpam-5066	179	17	two	two	NUM
ejpam-5066	179	18	equations	equation	NOUN
ejpam-5066	179	19	to	to	PART
ejpam-5066	179	20	transfer	transfer	VERB
ejpam-5066	179	21	them	they	PRON
ejpam-5066	179	22	to	to	ADP
ejpam-5066	179	23	the	the	DET
ejpam-5066	179	24	right	right	NOUN
ejpam-5066	179	25	as	as	ADP
ejpam-5066	179	26	e	e	NOUN
ejpam-5066	179	27	in	in	ADP
ejpam-5066	179	28	both	both	DET
ejpam-5066	179	29	cases	case	NOUN
ejpam-5066	179	30	,	,	PUNCT
ejpam-5066	179	31	we	we	PRON
ejpam-5066	179	32	get	get	VERB
ejpam-5066	179	33	[	[	X
ejpam-5066	179	34	e	e	NOUN
ejpam-5066	179	35	,	,	PUNCT
ejpam-5066	179	36	i(r)]ne	i(r)]ne	NOUN
ejpam-5066	179	37	⊆	⊆	NUM
ejpam-5066	179	38	[	[	X
ejpam-5066	179	39	e	e	NOUN
ejpam-5066	179	40	,	,	PUNCT
ejpam-5066	179	41	n	n	PROPN
ejpam-5066	179	42	(	(	PUNCT
ejpam-5066	179	43	r)]ne	r)]ne	NOUN
ejpam-5066	179	44	=	=	SYM
ejpam-5066	179	45	0	0	NUM
ejpam-5066	179	46	and	and	CCONJ
ejpam-5066	179	47	the	the	DET
ejpam-5066	179	48	result	result	NOUN
ejpam-5066	179	49	follows	follow	VERB
ejpam-5066	179	50	.	.	PUNCT
ejpam-5066	180	1	(	(	PUNCT
ejpam-5066	180	2	vi)⇒(i	vi)⇒(i	X
ejpam-5066	180	3	):	):	PUNCT
ejpam-5066	180	4	for	for	ADP
ejpam-5066	180	5	every	every	DET
ejpam-5066	180	6	r	r	NOUN
ejpam-5066	180	7	∈	∈	NOUN
ejpam-5066	180	8	r	r	NOUN
ejpam-5066	180	9	,	,	PUNCT
ejpam-5066	180	10	the	the	DET
ejpam-5066	180	11	element	element	NOUN
ejpam-5066	180	12	e+(1−e)re	e+(1−e)re	PROPN
ejpam-5066	180	13	is	be	AUX
ejpam-5066	180	14	idempotent	idempotent	ADJ
ejpam-5066	180	15	and	and	CCONJ
ejpam-5066	180	16	[	[	X
ejpam-5066	180	17	e	e	NOUN
ejpam-5066	180	18	,	,	PUNCT
ejpam-5066	180	19	e+(1−e)re	e+(1−e)re	PROPN
ejpam-5066	180	20	]	]	PUNCT
ejpam-5066	180	21	=	=	SYM
ejpam-5066	180	22	−[e	−[e	PROPN
ejpam-5066	180	23	,	,	PUNCT
ejpam-5066	180	24	r]e	r]e	PROPN
ejpam-5066	180	25	.	.	PUNCT
ejpam-5066	181	1	applying	apply	VERB
ejpam-5066	181	2	the	the	DET
ejpam-5066	181	3	same	same	ADJ
ejpam-5066	181	4	technique	technique	NOUN
ejpam-5066	181	5	of	of	ADP
ejpam-5066	181	6	proving	prove	VERB
ejpam-5066	181	7	(	(	PUNCT
ejpam-5066	181	8	v)⇒(vi	v)⇒(vi	X
ejpam-5066	181	9	)	)	PUNCT
ejpam-5066	181	10	,	,	PUNCT
ejpam-5066	181	11	we	we	PRON
ejpam-5066	181	12	get	get	VERB
ejpam-5066	181	13	[	[	X
ejpam-5066	181	14	e	e	NOUN
ejpam-5066	181	15	,	,	PUNCT
ejpam-5066	181	16	r]ne	r]ne	NOUN
ejpam-5066	181	17	=	=	SYM
ejpam-5066	181	18	0	0	NUM
ejpam-5066	182	1	and	and	CCONJ
ejpam-5066	182	2	e	e	PROPN
ejpam-5066	182	3	is	be	AUX
ejpam-5066	182	4	n	n	CCONJ
ejpam-5066	182	5	-	-	PUNCT
ejpam-5066	182	6	central	central	ADJ
ejpam-5066	182	7	.	.	PUNCT
ejpam-5066	183	1	a	a	DET
ejpam-5066	183	2	ring	ring	NOUN
ejpam-5066	183	3	r	r	NOUN
ejpam-5066	183	4	is	be	AUX
ejpam-5066	183	5	called	call	VERB
ejpam-5066	183	6	2	2	NUM
ejpam-5066	183	7	-	-	PUNCT
ejpam-5066	183	8	primal	primal	ADJ
ejpam-5066	183	9	if	if	SCONJ
ejpam-5066	183	10	n	n	X
ejpam-5066	183	11	(	(	PUNCT
ejpam-5066	183	12	r	r	NOUN
ejpam-5066	183	13	)	)	PUNCT
ejpam-5066	183	14	=	=	SYM
ejpam-5066	183	15	p(r	p(r	PROPN
ejpam-5066	183	16	)	)	PUNCT
ejpam-5066	183	17	where	where	SCONJ
ejpam-5066	183	18	p(r	p(r	PROPN
ejpam-5066	183	19	)	)	PUNCT
ejpam-5066	183	20	the	the	DET
ejpam-5066	183	21	prime	prime	ADJ
ejpam-5066	183	22	radical	radical	NOUN
ejpam-5066	183	23	of	of	ADP
ejpam-5066	183	24	r.	r.	PROPN
ejpam-5066	183	25	hence	hence	ADV
ejpam-5066	183	26	,	,	PUNCT
ejpam-5066	183	27	we	we	PRON
ejpam-5066	183	28	have	have	VERB
ejpam-5066	183	29	the	the	DET
ejpam-5066	183	30	next	next	ADJ
ejpam-5066	183	31	corollary	corollary	NOUN
ejpam-5066	183	32	.	.	PUNCT
ejpam-5066	184	1	corollary	corollary	ADJ
ejpam-5066	184	2	3	3	NUM
ejpam-5066	184	3	.	.	PUNCT
ejpam-5066	185	1	for	for	ADP
ejpam-5066	185	2	a	a	DET
ejpam-5066	185	3	2	2	NUM
ejpam-5066	185	4	-	-	PUNCT
ejpam-5066	185	5	primal	primal	ADJ
ejpam-5066	185	6	ring	ring	NOUN
ejpam-5066	185	7	r	r	NOUN
ejpam-5066	185	8	and	and	CCONJ
ejpam-5066	185	9	e	e	NOUN
ejpam-5066	185	10	∈	∈	PROPN
ejpam-5066	185	11	i(r	i(r	PROPN
ejpam-5066	185	12	)	)	PUNCT
ejpam-5066	185	13	,	,	PUNCT
ejpam-5066	185	14	e	e	PROPN
ejpam-5066	185	15	is	be	AUX
ejpam-5066	185	16	n	n	CCONJ
ejpam-5066	185	17	-	-	PUNCT
ejpam-5066	185	18	central	central	ADJ
ejpam-5066	185	19	if	if	SCONJ
ejpam-5066	186	1	and	and	CCONJ
ejpam-5066	186	2	only	only	ADV
ejpam-5066	186	3	if	if	SCONJ
ejpam-5066	186	4	[	[	X
ejpam-5066	186	5	e	e	NOUN
ejpam-5066	186	6	,	,	PUNCT
ejpam-5066	186	7	p(r)]ne	p(r)]ne	NOUN
ejpam-5066	186	8	=	=	SYM
ejpam-5066	186	9	0	0	NUM
ejpam-5066	186	10	.	.	PUNCT
ejpam-5066	187	1	the	the	DET
ejpam-5066	187	2	argument	argument	NOUN
ejpam-5066	187	3	presented	present	VERB
ejpam-5066	187	4	below	below	ADP
ejpam-5066	187	5	,	,	PUNCT
ejpam-5066	187	6	which	which	PRON
ejpam-5066	187	7	connects	connect	VERB
ejpam-5066	187	8	the	the	DET
ejpam-5066	187	9	n	n	NOUN
ejpam-5066	187	10	-	-	PUNCT
ejpam-5066	187	11	centrality	centrality	NOUN
ejpam-5066	187	12	of	of	ADP
ejpam-5066	187	13	idempotents	idempotent	NOUN
ejpam-5066	187	14	with	with	ADP
ejpam-5066	187	15	consecutive	consecutive	ADJ
ejpam-5066	187	16	degrees	degree	NOUN
ejpam-5066	187	17	by	by	ADP
ejpam-5066	187	18	utilizing	utilize	VERB
ejpam-5066	187	19	the	the	DET
ejpam-5066	187	20	minimality	minimality	NOUN
ejpam-5066	187	21	of	of	ADP
ejpam-5066	187	22	some	some	DET
ejpam-5066	187	23	centralizer	centralizer	NOUN
ejpam-5066	187	24	as	as	ADP
ejpam-5066	187	25	a	a	DET
ejpam-5066	187	26	one	one	NUM
ejpam-5066	187	27	-	-	PUNCT
ejpam-5066	187	28	sided	sided	ADJ
ejpam-5066	187	29	ideal	ideal	NOUN
ejpam-5066	187	30	,	,	PUNCT
ejpam-5066	187	31	is	be	AUX
ejpam-5066	187	32	inspired	inspire	VERB
ejpam-5066	187	33	by	by	ADP
ejpam-5066	187	34	the	the	DET
ejpam-5066	187	35	work	work	NOUN
ejpam-5066	187	36	of	of	ADP
ejpam-5066	187	37	lam	lam	PROPN
ejpam-5066	187	38	in	in	ADP
ejpam-5066	187	39	[	[	X
ejpam-5066	187	40	11	11	NUM
ejpam-5066	187	41	,	,	PUNCT
ejpam-5066	187	42	proposition	proposition	NOUN
ejpam-5066	187	43	2.10	2.10	NUM
ejpam-5066	187	44	]	]	PUNCT
ejpam-5066	187	45	.	.	PUNCT
ejpam-5066	188	1	proposition	proposition	NOUN
ejpam-5066	188	2	3	3	NUM
ejpam-5066	188	3	.	.	PUNCT
ejpam-5066	189	1	if	if	SCONJ
ejpam-5066	189	2	e	e	PROPN
ejpam-5066	189	3	∈	∈	PROPN
ejpam-5066	189	4	cn(r	cn(r	PRON
ejpam-5066	189	5	)	)	PUNCT
ejpam-5066	189	6	(	(	PUNCT
ejpam-5066	189	7	for	for	ADP
ejpam-5066	189	8	n	n	PRON
ejpam-5066	189	9	≥	≥	NOUN
ejpam-5066	189	10	2	2	NUM
ejpam-5066	189	11	)	)	PUNCT
ejpam-5066	189	12	and	and	CCONJ
ejpam-5066	189	13	[	[	X
ejpam-5066	189	14	e]n−2	e]n−2	NOUN
ejpam-5066	189	15	is	be	AUX
ejpam-5066	189	16	a	a	DET
ejpam-5066	189	17	minimal	minimal	ADJ
ejpam-5066	189	18	right	right	ADJ
ejpam-5066	189	19	ideal	ideal	NOUN
ejpam-5066	189	20	in	in	ADP
ejpam-5066	189	21	r	r	NOUN
ejpam-5066	189	22	,	,	PUNCT
ejpam-5066	189	23	then	then	ADV
ejpam-5066	189	24	e	e	PROPN
ejpam-5066	189	25	∈	∈	PROPN
ejpam-5066	189	26	cn−1(r	cn−1(r	NOUN
ejpam-5066	189	27	)	)	PUNCT
ejpam-5066	189	28	∪	∪	ADP
ejpam-5066	189	29	cn−2(r	cn−2(r	NOUN
ejpam-5066	189	30	)	)	PUNCT
ejpam-5066	189	31	.	.	PUNCT
ejpam-5066	190	1	(	(	PUNCT
ejpam-5066	190	2	here	here	ADV
ejpam-5066	190	3	,	,	PUNCT
ejpam-5066	190	4	c0(r	c0(r	NOUN
ejpam-5066	190	5	)	)	PUNCT
ejpam-5066	190	6	=	=	PRON
ejpam-5066	190	7	{	{	PUNCT
ejpam-5066	190	8	0	0	NUM
ejpam-5066	190	9	}	}	PUNCT
ejpam-5066	190	10	)	)	PUNCT
ejpam-5066	191	1	proof	proof	NOUN
ejpam-5066	191	2	.	.	PUNCT
ejpam-5066	192	1	for	for	ADP
ejpam-5066	192	2	e	e	PROPN
ejpam-5066	192	3	∈	∈	PROPN
ejpam-5066	192	4	cn(r	cn(r	PRON
ejpam-5066	192	5	)	)	PUNCT
ejpam-5066	192	6	,	,	PUNCT
ejpam-5066	192	7	we	we	PRON
ejpam-5066	192	8	have	have	VERB
ejpam-5066	192	9	[	[	X
ejpam-5066	192	10	1	1	NUM
ejpam-5066	192	11	−	−	NOUN
ejpam-5066	192	12	e]n−1	e]n−1	ADJ
ejpam-5066	192	13	=	=	PUNCT
ejpam-5066	193	1	[	[	X
ejpam-5066	193	2	e]n−2(1	e]n−2(1	X
ejpam-5066	193	3	−	−	NOUN
ejpam-5066	193	4	e)r	e)r	NOUN
ejpam-5066	193	5	⊆	⊆	NUM
ejpam-5066	194	1	[	[	X
ejpam-5066	194	2	e]n−2	e]n−2	NOUN
ejpam-5066	194	3	.	.	PUNCT
ejpam-5066	195	1	so	so	ADV
ejpam-5066	195	2	either	either	CCONJ
ejpam-5066	195	3	[	[	X
ejpam-5066	195	4	1−	1−	NUM
ejpam-5066	195	5	e]n−1	e]n−1	ADJ
ejpam-5066	195	6	=	=	SYM
ejpam-5066	195	7	0	0	PUNCT
ejpam-5066	195	8	(	(	PUNCT
ejpam-5066	195	9	and	and	CCONJ
ejpam-5066	195	10	e	e	PROPN
ejpam-5066	195	11	∈	∈	PROPN
ejpam-5066	195	12	cn−1(r	cn−1(r	NOUN
ejpam-5066	195	13	)	)	PUNCT
ejpam-5066	195	14	)	)	PUNCT
ejpam-5066	195	15	or	or	CCONJ
ejpam-5066	195	16	[	[	X
ejpam-5066	195	17	1−	1−	NUM
ejpam-5066	195	18	e]n−1	e]n−1	ADJ
ejpam-5066	195	19	=	=	PUNCT
ejpam-5066	196	1	[	[	X
ejpam-5066	196	2	e]n−2	e]n−2	PROPN
ejpam-5066	196	3	,	,	PUNCT
ejpam-5066	196	4	from	from	ADP
ejpam-5066	196	5	the	the	DET
ejpam-5066	196	6	minimality	minimality	NOUN
ejpam-5066	196	7	of	of	ADP
ejpam-5066	196	8	[	[	X
ejpam-5066	196	9	1−	1−	NUM
ejpam-5066	196	10	e]n−1	e]n−1	ADJ
ejpam-5066	196	11	.	.	PUNCT
ejpam-5066	197	1	if	if	SCONJ
ejpam-5066	197	2	[	[	X
ejpam-5066	197	3	1−	1−	NUM
ejpam-5066	197	4	e]n−1	e]n−1	ADJ
ejpam-5066	197	5	=	=	PUNCT
ejpam-5066	198	1	[	[	X
ejpam-5066	198	2	e]n−2	e]n−2	PROPN
ejpam-5066	198	3	,	,	PUNCT
ejpam-5066	198	4	then	then	ADV
ejpam-5066	198	5	0	0	NUM
ejpam-5066	198	6	=	=	SYM
ejpam-5066	199	1	[	[	X
ejpam-5066	199	2	e]n	e]n	X
ejpam-5066	199	3	=	=	SYM
ejpam-5066	200	1	[	[	X
ejpam-5066	200	2	1−	1−	NUM
ejpam-5066	200	3	e]n−1er	e]n−1er	NOUN
ejpam-5066	200	4	=	=	PUNCT
ejpam-5066	201	1	[	[	X
ejpam-5066	201	2	e]n−2er	e]n−2er	NOUN
ejpam-5066	201	3	=	=	PUNCT
ejpam-5066	202	1	[	[	X
ejpam-5066	202	2	e]n−2	e]n−2	NOUN
ejpam-5066	202	3	and	and	CCONJ
ejpam-5066	202	4	e	e	NOUN
ejpam-5066	202	5	∈	∈	PROPN
ejpam-5066	202	6	cn−2(r	cn−2(r	PROPN
ejpam-5066	202	7	)	)	PUNCT
ejpam-5066	202	8	.	.	PUNCT
ejpam-5066	203	1	m.	m.	PROPN
ejpam-5066	203	2	saad	saad	PROPN
ejpam-5066	203	3	,	,	PUNCT
ejpam-5066	203	4	m.	m.	NOUN
ejpam-5066	203	5	zailaee	zailaee	PROPN
ejpam-5066	203	6	/	/	SYM
ejpam-5066	203	7	eur	eur	PROPN
ejpam-5066	203	8	.	.	PUNCT
ejpam-5066	204	1	j.	j.	PROPN
ejpam-5066	204	2	pure	pure	PROPN
ejpam-5066	204	3	appl	appl	PROPN
ejpam-5066	204	4	.	.	PROPN
ejpam-5066	204	5	math	math	PROPN
ejpam-5066	204	6	,	,	PUNCT
ejpam-5066	204	7	17	17	NUM
ejpam-5066	204	8	(	(	PUNCT
ejpam-5066	204	9	2	2	NUM
ejpam-5066	204	10	)	)	PUNCT
ejpam-5066	204	11	(	(	PUNCT
ejpam-5066	204	12	2024	2024	NUM
ejpam-5066	204	13	)	)	PUNCT
ejpam-5066	204	14	,	,	PUNCT
ejpam-5066	204	15	736	736	NUM
ejpam-5066	204	16	-	-	SYM
ejpam-5066	204	17	752	752	NUM
ejpam-5066	204	18	742	742	NUM
ejpam-5066	204	19	corollary	corollary	ADJ
ejpam-5066	204	20	4	4	NUM
ejpam-5066	204	21	(	(	PUNCT
ejpam-5066	204	22	[	[	X
ejpam-5066	204	23	11	11	NUM
ejpam-5066	204	24	]	]	PUNCT
ejpam-5066	204	25	,	,	PUNCT
ejpam-5066	204	26	proposition	proposition	NOUN
ejpam-5066	204	27	2.10	2.10	NUM
ejpam-5066	204	28	)	)	PUNCT
ejpam-5066	204	29	.	.	PUNCT
ejpam-5066	205	1	if	if	SCONJ
ejpam-5066	205	2	e	e	PROPN
ejpam-5066	205	3	∈	∈	PROPN
ejpam-5066	205	4	q	q	NOUN
ejpam-5066	205	5	-	-	PUNCT
ejpam-5066	205	6	idem(r	idem(r	NOUN
ejpam-5066	205	7	)	)	PUNCT
ejpam-5066	205	8	and	and	CCONJ
ejpam-5066	205	9	er	er	INTJ
ejpam-5066	205	10	is	be	AUX
ejpam-5066	205	11	a	a	DET
ejpam-5066	205	12	minimal	minimal	ADJ
ejpam-5066	205	13	right	right	ADJ
ejpam-5066	205	14	ideal	ideal	NOUN
ejpam-5066	205	15	in	in	ADP
ejpam-5066	205	16	r	r	NOUN
ejpam-5066	205	17	,	,	PUNCT
ejpam-5066	205	18	then	then	ADV
ejpam-5066	205	19	e	e	PROPN
ejpam-5066	205	20	is	be	AUX
ejpam-5066	205	21	right	right	ADV
ejpam-5066	205	22	semicentral	semicentral	ADJ
ejpam-5066	205	23	.	.	PUNCT
ejpam-5066	206	1	every	every	DET
ejpam-5066	206	2	idempotent	idempotent	ADJ
ejpam-5066	206	3	element	element	NOUN
ejpam-5066	206	4	that	that	PRON
ejpam-5066	206	5	is	be	AUX
ejpam-5066	206	6	central	central	ADJ
ejpam-5066	206	7	is	be	AUX
ejpam-5066	206	8	n	n	CCONJ
ejpam-5066	206	9	-	-	PUNCT
ejpam-5066	206	10	central	central	ADJ
ejpam-5066	206	11	for	for	ADP
ejpam-5066	206	12	all	all	DET
ejpam-5066	206	13	n.	n.	PROPN
ejpam-5066	206	14	nevertheless	nevertheless	ADV
ejpam-5066	206	15	,	,	PUNCT
ejpam-5066	206	16	there	there	PRON
ejpam-5066	206	17	exist	exist	VERB
ejpam-5066	206	18	n	n	CCONJ
ejpam-5066	206	19	-	-	PUNCT
ejpam-5066	206	20	central	central	ADJ
ejpam-5066	206	21	idempotents	idempotent	NOUN
ejpam-5066	206	22	that	that	PRON
ejpam-5066	206	23	are	be	AUX
ejpam-5066	206	24	not	not	PART
ejpam-5066	206	25	central	central	ADJ
ejpam-5066	206	26	,	,	PUNCT
ejpam-5066	206	27	such	such	ADJ
ejpam-5066	206	28	as	as	ADP
ejpam-5066	206	29	the	the	DET
ejpam-5066	206	30	idempotent	idempotent	ADJ
ejpam-5066	206	31	e	e	NOUN
ejpam-5066	206	32	in	in	ADP
ejpam-5066	206	33	example	example	NOUN
ejpam-5066	207	1	1	1	NUM
ejpam-5066	207	2	,	,	PUNCT
ejpam-5066	207	3	which	which	PRON
ejpam-5066	207	4	is	be	AUX
ejpam-5066	207	5	2	2	NUM
ejpam-5066	207	6	-	-	ADJ
ejpam-5066	207	7	central	central	ADJ
ejpam-5066	207	8	but	but	CCONJ
ejpam-5066	207	9	not	not	PART
ejpam-5066	207	10	central	central	ADJ
ejpam-5066	207	11	.	.	PUNCT
ejpam-5066	208	1	the	the	DET
ejpam-5066	208	2	subsequent	subsequent	ADJ
ejpam-5066	208	3	proposition	proposition	NOUN
ejpam-5066	208	4	provides	provide	VERB
ejpam-5066	208	5	a	a	DET
ejpam-5066	208	6	sufficient	sufficient	ADJ
ejpam-5066	208	7	condition	condition	NOUN
ejpam-5066	208	8	for	for	ADP
ejpam-5066	208	9	a	a	DET
ejpam-5066	208	10	ring	ring	NOUN
ejpam-5066	208	11	r	r	NOUN
ejpam-5066	208	12	to	to	PART
ejpam-5066	208	13	have	have	VERB
ejpam-5066	208	14	the	the	DET
ejpam-5066	208	15	sets	set	NOUN
ejpam-5066	208	16	cn(r	cn(r	NOUN
ejpam-5066	208	17	)	)	PUNCT
ejpam-5066	208	18	coincide	coincide	NOUN
ejpam-5066	208	19	with	with	ADP
ejpam-5066	208	20	b(r	b(r	PROPN
ejpam-5066	208	21	)	)	PUNCT
ejpam-5066	208	22	for	for	ADP
ejpam-5066	208	23	all	all	DET
ejpam-5066	208	24	n.	n.	NOUN
ejpam-5066	208	25	proposition	proposition	NOUN
ejpam-5066	208	26	4	4	NUM
ejpam-5066	208	27	.	.	PUNCT
ejpam-5066	209	1	every	every	DET
ejpam-5066	209	2	n	n	CCONJ
ejpam-5066	209	3	-	-	PUNCT
ejpam-5066	209	4	central	central	ADJ
ejpam-5066	209	5	idempotent	idempotent	NOUN
ejpam-5066	209	6	of	of	ADP
ejpam-5066	209	7	a	a	DET
ejpam-5066	209	8	semiprime	semiprime	NOUN
ejpam-5066	209	9	ring	ring	NOUN
ejpam-5066	209	10	is	be	AUX
ejpam-5066	209	11	central	central	ADJ
ejpam-5066	209	12	,	,	PUNCT
ejpam-5066	209	13	for	for	ADP
ejpam-5066	209	14	every	every	DET
ejpam-5066	209	15	n.	n.	NOUN
ejpam-5066	209	16	proof	proof	NOUN
ejpam-5066	209	17	.	.	PUNCT
ejpam-5066	210	1	let	let	VERB
ejpam-5066	210	2	e	e	PRON
ejpam-5066	210	3	be	be	AUX
ejpam-5066	210	4	an	an	DET
ejpam-5066	210	5	n	n	CCONJ
ejpam-5066	210	6	-	-	PUNCT
ejpam-5066	210	7	central	central	ADJ
ejpam-5066	210	8	idempotent	idempotent	NOUN
ejpam-5066	210	9	of	of	ADP
ejpam-5066	210	10	a	a	DET
ejpam-5066	210	11	semiprime	semiprime	NOUN
ejpam-5066	210	12	ring	ring	NOUN
ejpam-5066	210	13	r	r	NOUN
ejpam-5066	210	14	for	for	ADP
ejpam-5066	210	15	some	some	DET
ejpam-5066	210	16	odd	odd	ADJ
ejpam-5066	210	17	n	n	NOUN
ejpam-5066	210	18	(	(	PUNCT
ejpam-5066	210	19	that	that	PRON
ejpam-5066	210	20	does	do	AUX
ejpam-5066	210	21	not	not	PART
ejpam-5066	210	22	loss	loss	VERB
ejpam-5066	210	23	of	of	ADP
ejpam-5066	210	24	generality	generality	NOUN
ejpam-5066	210	25	of	of	ADP
ejpam-5066	210	26	n	n	CCONJ
ejpam-5066	210	27	)	)	PUNCT
ejpam-5066	210	28	,	,	PUNCT
ejpam-5066	210	29	then	then	ADV
ejpam-5066	210	30	[	[	X
ejpam-5066	210	31	e]n	e]n	X
ejpam-5066	210	32	=	=	SYM
ejpam-5066	210	33	0	0	NUM
ejpam-5066	210	34	and	and	CCONJ
ejpam-5066	210	35	(	(	PUNCT
ejpam-5066	210	36	(	(	PUNCT
ejpam-5066	210	37	1−	1−	NUM
ejpam-5066	210	38	e)rer	e)rer	NOUN
ejpam-5066	210	39	)	)	PUNCT
ejpam-5066	210	40	n+1	n+1	PROPN
ejpam-5066	210	41	2	2	NUM
ejpam-5066	210	42	=	=	SYM
ejpam-5066	210	43	0	0	NUM
ejpam-5066	210	44	.	.	PUNCT
ejpam-5066	211	1	so	so	ADV
ejpam-5066	211	2	,	,	PUNCT
ejpam-5066	211	3	(	(	PUNCT
ejpam-5066	211	4	1−e)re	1−e)re	NUM
ejpam-5066	211	5	=	=	SYM
ejpam-5066	211	6	0	0	NUM
ejpam-5066	211	7	and	and	CCONJ
ejpam-5066	211	8	e	e	PROPN
ejpam-5066	211	9	∈	∈	PROPN
ejpam-5066	211	10	sl(r	sl(r	NOUN
ejpam-5066	211	11	)	)	PUNCT
ejpam-5066	211	12	.	.	PUNCT
ejpam-5066	212	1	also	also	ADV
ejpam-5066	212	2	,	,	PUNCT
ejpam-5066	212	3	rer(1−e)r	rer(1−e)r	ADV
ejpam-5066	212	4	is	be	AUX
ejpam-5066	212	5	a	a	DET
ejpam-5066	212	6	nilpotent	nilpotent	ADJ
ejpam-5066	212	7	ideal	ideal	NOUN
ejpam-5066	212	8	.	.	PUNCT
ejpam-5066	213	1	hence	hence	ADV
ejpam-5066	213	2	er(1−e	er(1−e	NOUN
ejpam-5066	213	3	)	)	PUNCT
ejpam-5066	213	4	=	=	SYM
ejpam-5066	213	5	0	0	NUM
ejpam-5066	213	6	and	and	CCONJ
ejpam-5066	213	7	e	e	PROPN
ejpam-5066	213	8	∈	∈	PROPN
ejpam-5066	213	9	sl(r	sl(r	NOUN
ejpam-5066	213	10	)	)	PUNCT
ejpam-5066	213	11	;	;	PUNCT
ejpam-5066	213	12	that	that	SCONJ
ejpam-5066	213	13	e	e	NOUN
ejpam-5066	213	14	is	be	AUX
ejpam-5066	213	15	central	central	ADJ
ejpam-5066	213	16	.	.	PUNCT
ejpam-5066	214	1	as	as	ADP
ejpam-5066	214	2	per	per	ADP
ejpam-5066	214	3	theorem	theorem	NOUN
ejpam-5066	214	4	[	[	X
ejpam-5066	214	5	10	10	NUM
ejpam-5066	214	6	,	,	PUNCT
ejpam-5066	214	7	example	example	NOUN
ejpam-5066	214	8	10.17	10.17	NUM
ejpam-5066	214	9	]	]	PUNCT
ejpam-5066	214	10	,	,	PUNCT
ejpam-5066	214	11	both	both	PRON
ejpam-5066	214	12	r	r	X
ejpam-5066	214	13	/	/	SYM
ejpam-5066	214	14	p	p	X
ejpam-5066	214	15	(	(	PUNCT
ejpam-5066	214	16	r	r	NOUN
ejpam-5066	214	17	)	)	PUNCT
ejpam-5066	214	18	and	and	CCONJ
ejpam-5066	214	19	r	r	NOUN
ejpam-5066	214	20	/	/	SYM
ejpam-5066	214	21	j(r	j(r	PROPN
ejpam-5066	214	22	)	)	PUNCT
ejpam-5066	214	23	are	be	AUX
ejpam-5066	214	24	semiprime	semiprime	NOUN
ejpam-5066	214	25	rings	ring	NOUN
ejpam-5066	214	26	,	,	PUNCT
ejpam-5066	214	27	where	where	SCONJ
ejpam-5066	214	28	j	j	PROPN
ejpam-5066	214	29	(	(	PUNCT
ejpam-5066	214	30	r	r	NOUN
ejpam-5066	214	31	)	)	PUNCT
ejpam-5066	214	32	denotes	denote	VERB
ejpam-5066	214	33	the	the	DET
ejpam-5066	214	34	jacobson	jacobson	PROPN
ejpam-5066	214	35	radical	radical	PROPN
ejpam-5066	214	36	of	of	ADP
ejpam-5066	214	37	r.	r.	PROPN
ejpam-5066	214	38	therefore	therefore	ADV
ejpam-5066	214	39	,	,	PUNCT
ejpam-5066	214	40	we	we	PRON
ejpam-5066	214	41	can	can	AUX
ejpam-5066	214	42	derive	derive	VERB
ejpam-5066	214	43	the	the	DET
ejpam-5066	214	44	following	follow	VERB
ejpam-5066	214	45	corollary	corollary	NOUN
ejpam-5066	214	46	.	.	PUNCT
ejpam-5066	215	1	corollary	corollary	ADJ
ejpam-5066	215	2	5	5	NUM
ejpam-5066	215	3	.	.	PUNCT
ejpam-5066	216	1	every	every	DET
ejpam-5066	216	2	e	e	PROPN
ejpam-5066	216	3	∈	∈	PROPN
ejpam-5066	216	4	cn(r	cn(r	PRON
ejpam-5066	216	5	)	)	PUNCT
ejpam-5066	216	6	maps	map	VERB
ejpam-5066	216	7	onto	onto	ADP
ejpam-5066	216	8	a	a	DET
ejpam-5066	216	9	central	central	ADJ
ejpam-5066	216	10	idempotent	idempotent	NOUN
ejpam-5066	216	11	in	in	ADP
ejpam-5066	216	12	r	r	PROPN
ejpam-5066	216	13	/	/	SYM
ejpam-5066	216	14	p	p	X
ejpam-5066	216	15	(	(	PUNCT
ejpam-5066	216	16	r	r	NOUN
ejpam-5066	216	17	)	)	PUNCT
ejpam-5066	216	18	and	and	CCONJ
ejpam-5066	216	19	r	r	PROPN
ejpam-5066	216	20	/	/	SYM
ejpam-5066	216	21	j	j	PROPN
ejpam-5066	216	22	(	(	PUNCT
ejpam-5066	216	23	r	r	NOUN
ejpam-5066	216	24	)	)	PUNCT
ejpam-5066	216	25	.	.	PUNCT
ejpam-5066	217	1	therefore	therefore	ADV
ejpam-5066	217	2	,	,	PUNCT
ejpam-5066	217	3	if	if	SCONJ
ejpam-5066	217	4	r	r	NOUN
ejpam-5066	217	5	is	be	AUX
ejpam-5066	217	6	a	a	DET
ejpam-5066	217	7	n	n	CCONJ
ejpam-5066	217	8	-	-	PUNCT
ejpam-5066	217	9	abelian	abelian	NOUN
ejpam-5066	217	10	ring	ring	NOUN
ejpam-5066	217	11	,	,	PUNCT
ejpam-5066	217	12	then	then	ADV
ejpam-5066	217	13	every	every	DET
ejpam-5066	217	14	idempotent	idempotent	NOUN
ejpam-5066	217	15	of	of	ADP
ejpam-5066	217	16	r	r	NOUN
ejpam-5066	217	17	maps	map	NOUN
ejpam-5066	217	18	onto	onto	ADP
ejpam-5066	217	19	a	a	DET
ejpam-5066	217	20	central	central	ADJ
ejpam-5066	217	21	idempotent	idempotent	NOUN
ejpam-5066	217	22	in	in	ADP
ejpam-5066	217	23	r	r	PROPN
ejpam-5066	217	24	/	/	SYM
ejpam-5066	217	25	p(r	p(r	PROPN
ejpam-5066	217	26	)	)	PUNCT
ejpam-5066	217	27	and	and	CCONJ
ejpam-5066	217	28	r	r	NOUN
ejpam-5066	217	29	/	/	SYM
ejpam-5066	217	30	rad	rad	NOUN
ejpam-5066	217	31	(	(	PUNCT
ejpam-5066	217	32	r	r	NOUN
ejpam-5066	217	33	)	)	PUNCT
ejpam-5066	217	34	.	.	PUNCT
ejpam-5066	218	1	indeed	indeed	ADV
ejpam-5066	218	2	,	,	PUNCT
ejpam-5066	218	3	indecomposable	indecomposable	ADJ
ejpam-5066	218	4	rings	ring	NOUN
ejpam-5066	218	5	have	have	VERB
ejpam-5066	218	6	no	no	DET
ejpam-5066	218	7	nontrivial	nontrivial	ADJ
ejpam-5066	218	8	idempotents	idempotent	NOUN
ejpam-5066	218	9	.	.	PUNCT
ejpam-5066	219	1	so	so	ADV
ejpam-5066	219	2	,	,	PUNCT
ejpam-5066	219	3	we	we	PRON
ejpam-5066	219	4	have	have	VERB
ejpam-5066	219	5	the	the	DET
ejpam-5066	219	6	next	next	ADJ
ejpam-5066	219	7	corollary	corollary	ADJ
ejpam-5066	219	8	corollary	corollary	ADJ
ejpam-5066	219	9	6	6	NUM
ejpam-5066	219	10	.	.	PUNCT
ejpam-5066	220	1	if	if	SCONJ
ejpam-5066	220	2	r	r	NOUN
ejpam-5066	220	3	is	be	AUX
ejpam-5066	220	4	an	an	DET
ejpam-5066	220	5	indecomposable	indecomposable	ADJ
ejpam-5066	220	6	semiprime	semiprime	NOUN
ejpam-5066	220	7	ring	ring	NOUN
ejpam-5066	220	8	,	,	PUNCT
ejpam-5066	220	9	then	then	ADV
ejpam-5066	220	10	cn(r	cn(r	PUNCT
ejpam-5066	220	11	)	)	PUNCT
ejpam-5066	220	12	=	=	PUNCT
ejpam-5066	220	13	{	{	PUNCT
ejpam-5066	220	14	0	0	NUM
ejpam-5066	220	15	,	,	PUNCT
ejpam-5066	220	16	1	1	NUM
ejpam-5066	220	17	}	}	PUNCT
ejpam-5066	220	18	,	,	PUNCT
ejpam-5066	220	19	for	for	ADP
ejpam-5066	220	20	every	every	DET
ejpam-5066	220	21	n.	n.	NOUN
ejpam-5066	220	22	here	here	ADV
ejpam-5066	220	23	,	,	PUNCT
ejpam-5066	220	24	we	we	PRON
ejpam-5066	220	25	have	have	VERB
ejpam-5066	220	26	a	a	DET
ejpam-5066	220	27	necessary	necessary	ADJ
ejpam-5066	220	28	and	and	CCONJ
ejpam-5066	220	29	sufficient	sufficient	ADJ
ejpam-5066	220	30	condition	condition	NOUN
ejpam-5066	220	31	making	make	VERB
ejpam-5066	220	32	b(r	b(r	NOUN
ejpam-5066	220	33	)	)	PUNCT
ejpam-5066	220	34	=	=	SYM
ejpam-5066	220	35	cn(r	cn(r	X
ejpam-5066	220	36	)	)	PUNCT
ejpam-5066	220	37	for	for	ADP
ejpam-5066	220	38	some	some	DET
ejpam-5066	220	39	n.	n.	NOUN
ejpam-5066	220	40	proposition	proposition	NOUN
ejpam-5066	220	41	5	5	NUM
ejpam-5066	220	42	.	.	PUNCT
ejpam-5066	221	1	an	an	DET
ejpam-5066	221	2	idempotent	idempotent	ADJ
ejpam-5066	221	3	e	e	NOUN
ejpam-5066	221	4	of	of	ADP
ejpam-5066	221	5	a	a	DET
ejpam-5066	221	6	ring	ring	NOUN
ejpam-5066	221	7	r	r	NOUN
ejpam-5066	221	8	is	be	AUX
ejpam-5066	221	9	central	central	ADJ
ejpam-5066	221	10	if	if	SCONJ
ejpam-5066	221	11	e	e	PROPN
ejpam-5066	221	12	∈	∈	PROPN
ejpam-5066	221	13	cn(r	cn(r	PRON
ejpam-5066	221	14	)	)	PUNCT
ejpam-5066	221	15	,	,	PUNCT
ejpam-5066	221	16	for	for	ADP
ejpam-5066	221	17	some	some	DET
ejpam-5066	221	18	n	n	CCONJ
ejpam-5066	221	19	,	,	PUNCT
ejpam-5066	221	20	and	and	CCONJ
ejpam-5066	221	21	cn(r	cn(r	NUM
ejpam-5066	221	22	)	)	PUNCT
ejpam-5066	221	23	is	be	AUX
ejpam-5066	221	24	commutating	commutate	VERB
ejpam-5066	221	25	.	.	PUNCT
ejpam-5066	222	1	proof	proof	NOUN
ejpam-5066	222	2	.	.	PUNCT
ejpam-5066	223	1	let	let	VERB
ejpam-5066	223	2	e	e	PROPN
ejpam-5066	223	3	∈	∈	PROPN
ejpam-5066	223	4	cn(r	cn(r	PRON
ejpam-5066	223	5	)	)	PUNCT
ejpam-5066	223	6	and	and	CCONJ
ejpam-5066	223	7	define	define	VERB
ejpam-5066	223	8	the	the	DET
ejpam-5066	223	9	idempotent	idempotent	ADJ
ejpam-5066	223	10	element	element	NOUN
ejpam-5066	223	11	f	f	PROPN
ejpam-5066	223	12	=	=	SYM
ejpam-5066	223	13	e+er(1−e	e+er(1−e	PROPN
ejpam-5066	223	14	)	)	PUNCT
ejpam-5066	223	15	,	,	PUNCT
ejpam-5066	223	16	for	for	ADP
ejpam-5066	223	17	arbitrary	arbitrary	ADJ
ejpam-5066	223	18	r	r	NOUN
ejpam-5066	223	19	∈	∈	PROPN
ejpam-5066	223	20	r.	r.	NOUN
ejpam-5066	223	21	but	but	CCONJ
ejpam-5066	223	22	f	f	PROPN
ejpam-5066	223	23	∈	∈	PROPN
ejpam-5066	223	24	er	er	INTJ
ejpam-5066	223	25	and	and	CCONJ
ejpam-5066	223	26	1−	1−	NUM
ejpam-5066	223	27	f	f	X
ejpam-5066	223	28	∈	∈	PROPN
ejpam-5066	223	29	r(1−	r(1−	PROPN
ejpam-5066	223	30	e	e	NOUN
ejpam-5066	223	31	)	)	PUNCT
ejpam-5066	223	32	.	.	PUNCT
ejpam-5066	224	1	hence	hence	ADV
ejpam-5066	224	2	,	,	PUNCT
ejpam-5066	225	1	[	[	X
ejpam-5066	225	2	f	f	X
ejpam-5066	225	3	]	]	X
ejpam-5066	225	4	n	n	CCONJ
ejpam-5066	225	5	⊆	⊆	NUM
ejpam-5066	225	6	[	[	SYM
ejpam-5066	225	7	e]n	e]n	NOUN
ejpam-5066	225	8	=	=	SYM
ejpam-5066	225	9	0	0	NUM
ejpam-5066	225	10	and	and	CCONJ
ejpam-5066	225	11	f	f	PROPN
ejpam-5066	225	12	∈	∈	PROPN
ejpam-5066	225	13	cn(r	cn(r	PRON
ejpam-5066	225	14	)	)	PUNCT
ejpam-5066	225	15	.	.	PUNCT
ejpam-5066	226	1	so	so	ADV
ejpam-5066	226	2	that	that	SCONJ
ejpam-5066	226	3	e	e	PROPN
ejpam-5066	226	4	and	and	CCONJ
ejpam-5066	226	5	f	f	PROPN
ejpam-5066	226	6	are	be	AUX
ejpam-5066	226	7	commutating	commutate	VERB
ejpam-5066	226	8	from	from	ADP
ejpam-5066	226	9	the	the	DET
ejpam-5066	226	10	assumption	assumption	NOUN
ejpam-5066	226	11	and	and	CCONJ
ejpam-5066	226	12	f	f	PROPN
ejpam-5066	226	13	=	=	SYM
ejpam-5066	226	14	ef	ef	PROPN
ejpam-5066	226	15	=	=	SYM
ejpam-5066	226	16	fe	fe	X
ejpam-5066	226	17	=	=	PROPN
ejpam-5066	226	18	e.	e.	PROPN
ejpam-5066	226	19	therefore	therefore	ADV
ejpam-5066	226	20	,	,	PUNCT
ejpam-5066	226	21	er(1−	er(1−	ADJ
ejpam-5066	226	22	e	e	NOUN
ejpam-5066	226	23	)	)	PUNCT
ejpam-5066	226	24	and	and	CCONJ
ejpam-5066	226	25	e	e	PROPN
ejpam-5066	226	26	∈	∈	PROPN
ejpam-5066	226	27	sr(r	sr(r	NOUN
ejpam-5066	226	28	)	)	PUNCT
ejpam-5066	226	29	.	.	PUNCT
ejpam-5066	227	1	similarly	similarly	ADV
ejpam-5066	227	2	,	,	PUNCT
ejpam-5066	227	3	one	one	PRON
ejpam-5066	227	4	can	can	AUX
ejpam-5066	227	5	show	show	VERB
ejpam-5066	227	6	that	that	SCONJ
ejpam-5066	227	7	e	e	PROPN
ejpam-5066	227	8	∈	∈	PROPN
ejpam-5066	227	9	sl(r	sl(r	NOUN
ejpam-5066	227	10	)	)	PUNCT
ejpam-5066	227	11	and	and	CCONJ
ejpam-5066	227	12	hence	hence	ADV
ejpam-5066	227	13	e	e	NOUN
ejpam-5066	227	14	is	be	AUX
ejpam-5066	227	15	central	central	ADJ
ejpam-5066	227	16	.	.	PUNCT
ejpam-5066	228	1	here	here	ADV
ejpam-5066	228	2	,	,	PUNCT
ejpam-5066	228	3	we	we	PRON
ejpam-5066	228	4	give	give	VERB
ejpam-5066	228	5	the	the	DET
ejpam-5066	228	6	same	same	ADJ
ejpam-5066	228	7	result	result	NOUN
ejpam-5066	228	8	of	of	ADP
ejpam-5066	228	9	[	[	X
ejpam-5066	228	10	11	11	NUM
ejpam-5066	228	11	,	,	PUNCT
ejpam-5066	228	12	proposition	proposition	NOUN
ejpam-5066	228	13	2.6	2.6	NUM
ejpam-5066	228	14	]	]	PUNCT
ejpam-5066	228	15	with	with	ADP
ejpam-5066	228	16	a	a	DET
ejpam-5066	228	17	generalized	generalized	ADJ
ejpam-5066	228	18	condition	condition	NOUN
ejpam-5066	228	19	.	.	PUNCT
ejpam-5066	229	1	proposition	proposition	NOUN
ejpam-5066	229	2	6	6	NUM
ejpam-5066	229	3	.	.	PUNCT
ejpam-5066	230	1	let	let	VERB
ejpam-5066	230	2	e	e	PROPN
ejpam-5066	230	3	∈	∈	PROPN
ejpam-5066	230	4	cn(r	cn(r	PRON
ejpam-5066	230	5	)	)	PUNCT
ejpam-5066	230	6	,	,	PUNCT
ejpam-5066	230	7	for	for	ADP
ejpam-5066	230	8	some	some	DET
ejpam-5066	230	9	n	n	CCONJ
ejpam-5066	230	10	,	,	PUNCT
ejpam-5066	230	11	such	such	ADJ
ejpam-5066	230	12	that	that	DET
ejpam-5066	230	13	rer	rer	X
ejpam-5066	230	14	=	=	SYM
ejpam-5066	230	15	r	r	PROPN
ejpam-5066	230	16	,	,	PUNCT
ejpam-5066	230	17	then	then	ADV
ejpam-5066	230	18	e	e	NOUN
ejpam-5066	230	19	=	=	NOUN
ejpam-5066	231	1	1	1	X
ejpam-5066	231	2	.	.	PUNCT
ejpam-5066	231	3	proof	proof	NOUN
ejpam-5066	231	4	.	.	PUNCT
ejpam-5066	232	1	indeed	indeed	ADV
ejpam-5066	232	2	,	,	PUNCT
ejpam-5066	232	3	e	e	PROPN
ejpam-5066	232	4	∈	∈	PROPN
ejpam-5066	232	5	cn(r	cn(r	PRON
ejpam-5066	232	6	)	)	PUNCT
ejpam-5066	232	7	and	and	CCONJ
ejpam-5066	232	8	[	[	X
ejpam-5066	232	9	e]n	e]n	X
ejpam-5066	232	10	=	=	SYM
ejpam-5066	232	11	0	0	NUM
ejpam-5066	232	12	.	.	PUNCT
ejpam-5066	233	1	so	so	ADV
ejpam-5066	233	2	(	(	PUNCT
ejpam-5066	233	3	(	(	PUNCT
ejpam-5066	233	4	1	1	NUM
ejpam-5066	233	5	−	−	NOUN
ejpam-5066	233	6	e)r)m	e)r)m	NOUN
ejpam-5066	233	7	=	=	NOUN
ejpam-5066	233	8	0	0	NUM
ejpam-5066	233	9	where	where	SCONJ
ejpam-5066	233	10	m	m	VERB
ejpam-5066	233	11	=	=	VERB
ejpam-5066	233	12	⌈n2	⌈n2	VERB
ejpam-5066	233	13	⌉.	⌉.	ADV
ejpam-5066	233	14	so	so	SCONJ
ejpam-5066	233	15	that	that	SCONJ
ejpam-5066	233	16	1−	1−	NUM
ejpam-5066	233	17	e	e	X
ejpam-5066	233	18	=	=	SYM
ejpam-5066	233	19	0	0	PUNCT
ejpam-5066	233	20	and	and	CCONJ
ejpam-5066	233	21	e	e	X
ejpam-5066	233	22	=	=	NOUN
ejpam-5066	233	23	1	1	X
ejpam-5066	233	24	.	.	PUNCT
ejpam-5066	234	1	in	in	ADP
ejpam-5066	234	2	the	the	DET
ejpam-5066	234	3	context	context	NOUN
ejpam-5066	234	4	of	of	ADP
ejpam-5066	234	5	a	a	DET
ejpam-5066	234	6	ring	ring	NOUN
ejpam-5066	234	7	r	r	NOUN
ejpam-5066	234	8	,	,	PUNCT
ejpam-5066	234	9	recall	recall	VERB
ejpam-5066	234	10	that	that	SCONJ
ejpam-5066	234	11	two	two	NUM
ejpam-5066	234	12	idempotents	idempotent	NOUN
ejpam-5066	234	13	e	e	NOUN
ejpam-5066	234	14	and	and	CCONJ
ejpam-5066	234	15	f	f	PROPN
ejpam-5066	234	16	are	be	AUX
ejpam-5066	234	17	said	say	VERB
ejpam-5066	234	18	to	to	PART
ejpam-5066	234	19	be	be	AUX
ejpam-5066	234	20	isomorphic	isomorphic	ADJ
ejpam-5066	235	1	if	if	SCONJ
ejpam-5066	235	2	er	er	INTJ
ejpam-5066	235	3	and	and	CCONJ
ejpam-5066	235	4	fr	fr	PROPN
ejpam-5066	235	5	are	be	AUX
ejpam-5066	235	6	isomorphic	isomorphic	ADJ
ejpam-5066	235	7	as	as	ADP
ejpam-5066	235	8	right	right	ADJ
ejpam-5066	235	9	r	r	NOUN
ejpam-5066	235	10	-	-	PUNCT
ejpam-5066	235	11	modules	module	NOUN
ejpam-5066	235	12	.	.	PUNCT
ejpam-5066	236	1	equivalently	equivalently	ADV
ejpam-5066	236	2	,	,	PUNCT
ejpam-5066	236	3	e	e	PROPN
ejpam-5066	236	4	and	and	CCONJ
ejpam-5066	236	5	f	f	PROPN
ejpam-5066	236	6	are	be	AUX
ejpam-5066	236	7	isomorphic	isomorphic	ADJ
ejpam-5066	236	8	if	if	SCONJ
ejpam-5066	236	9	there	there	PRON
ejpam-5066	236	10	exist	exist	VERB
ejpam-5066	236	11	a	a	DET
ejpam-5066	236	12	,	,	PUNCT
ejpam-5066	236	13	b	b	X
ejpam-5066	236	14	∈	∈	NOUN
ejpam-5066	236	15	r	r	NOUN
ejpam-5066	237	1	such	such	ADJ
ejpam-5066	237	2	that	that	DET
ejpam-5066	237	3	e	e	NOUN
ejpam-5066	237	4	=	=	SYM
ejpam-5066	237	5	ab	ab	PROPN
ejpam-5066	237	6	and	and	CCONJ
ejpam-5066	237	7	f	f	PROPN
ejpam-5066	237	8	=	=	SYM
ejpam-5066	237	9	ba	ba	PROPN
ejpam-5066	237	10	.	.	PUNCT
ejpam-5066	238	1	idempotents	idempotents	PROPN
ejpam-5066	238	2	e	e	PROPN
ejpam-5066	238	3	and	and	CCONJ
ejpam-5066	238	4	f	f	PROPN
ejpam-5066	238	5	are	be	AUX
ejpam-5066	238	6	said	say	VERB
ejpam-5066	238	7	to	to	PART
ejpam-5066	238	8	be	be	AUX
ejpam-5066	238	9	m.	m.	NOUN
ejpam-5066	238	10	saad	saad	PROPN
ejpam-5066	238	11	,	,	PUNCT
ejpam-5066	238	12	m.	m.	NOUN
ejpam-5066	238	13	zailaee	zailaee	PROPN
ejpam-5066	238	14	/	/	SYM
ejpam-5066	238	15	eur	eur	PROPN
ejpam-5066	238	16	.	.	PUNCT
ejpam-5066	239	1	j.	j.	PROPN
ejpam-5066	239	2	pure	pure	PROPN
ejpam-5066	239	3	appl	appl	PROPN
ejpam-5066	239	4	.	.	PROPN
ejpam-5066	239	5	math	math	PROPN
ejpam-5066	239	6	,	,	PUNCT
ejpam-5066	239	7	17	17	NUM
ejpam-5066	239	8	(	(	PUNCT
ejpam-5066	239	9	2	2	NUM
ejpam-5066	239	10	)	)	PUNCT
ejpam-5066	239	11	(	(	PUNCT
ejpam-5066	239	12	2024	2024	NUM
ejpam-5066	239	13	)	)	PUNCT
ejpam-5066	239	14	,	,	PUNCT
ejpam-5066	239	15	736	736	NUM
ejpam-5066	239	16	-	-	SYM
ejpam-5066	239	17	752	752	NUM
ejpam-5066	239	18	743	743	NUM
ejpam-5066	239	19	isomorphic	isomorphic	ADJ
ejpam-5066	239	20	complements	complement	NOUN
ejpam-5066	239	21	if	if	SCONJ
ejpam-5066	239	22	1−	1−	NUM
ejpam-5066	239	23	e	e	NOUN
ejpam-5066	239	24	and	and	CCONJ
ejpam-5066	239	25	1−	1−	NUM
ejpam-5066	239	26	f	f	NOUN
ejpam-5066	239	27	are	be	AUX
ejpam-5066	239	28	isomorphic	isomorphic	ADJ
ejpam-5066	239	29	.	.	PUNCT
ejpam-5066	240	1	additionally	additionally	ADV
ejpam-5066	240	2	,	,	PUNCT
ejpam-5066	240	3	e	e	PROPN
ejpam-5066	240	4	and	and	CCONJ
ejpam-5066	240	5	f	f	PROPN
ejpam-5066	240	6	are	be	AUX
ejpam-5066	240	7	said	say	VERB
ejpam-5066	240	8	to	to	PART
ejpam-5066	240	9	be	be	AUX
ejpam-5066	240	10	conjugate	conjugate	ADJ
ejpam-5066	240	11	(	(	PUNCT
ejpam-5066	240	12	resp	resp	NOUN
ejpam-5066	240	13	.	.	PUNCT
ejpam-5066	241	1	right	right	ADJ
ejpam-5066	241	2	associate	associate	NOUN
ejpam-5066	241	3	,	,	PUNCT
ejpam-5066	241	4	left	leave	VERB
ejpam-5066	241	5	associate	associate	NOUN
ejpam-5066	241	6	)	)	PUNCT
ejpam-5066	241	7	if	if	SCONJ
ejpam-5066	241	8	there	there	PRON
ejpam-5066	241	9	exists	exist	VERB
ejpam-5066	241	10	a	a	DET
ejpam-5066	241	11	unit	unit	NOUN
ejpam-5066	241	12	u	u	NOUN
ejpam-5066	241	13	∈	∈	PROPN
ejpam-5066	241	14	r	r	NOUN
ejpam-5066	241	15	such	such	ADJ
ejpam-5066	241	16	that	that	DET
ejpam-5066	241	17	uf	uf	PROPN
ejpam-5066	241	18	=	=	SYM
ejpam-5066	241	19	eu	eu	PROPN
ejpam-5066	241	20	(	(	PUNCT
ejpam-5066	241	21	resp	resp	PROPN
ejpam-5066	241	22	.	.	PUNCT
ejpam-5066	242	1	f	f	X
ejpam-5066	242	2	=	=	SYM
ejpam-5066	242	3	eu	eu	PROPN
ejpam-5066	242	4	,	,	PUNCT
ejpam-5066	242	5	f	f	PROPN
ejpam-5066	242	6	=	=	PUNCT
ejpam-5066	242	7	ue	ue	PROPN
ejpam-5066	242	8	)	)	PUNCT
ejpam-5066	242	9	.	.	PUNCT
ejpam-5066	243	1	the	the	DET
ejpam-5066	243	2	following	follow	VERB
ejpam-5066	243	3	proposition	proposition	NOUN
ejpam-5066	243	4	establishes	establish	VERB
ejpam-5066	243	5	that	that	SCONJ
ejpam-5066	243	6	if	if	SCONJ
ejpam-5066	243	7	two	two	NUM
ejpam-5066	243	8	idempotents	idempotent	NOUN
ejpam-5066	243	9	are	be	AUX
ejpam-5066	243	10	isomorphic	isomorphic	ADJ
ejpam-5066	243	11	,	,	PUNCT
ejpam-5066	243	12	isomorphic	isomorphic	ADJ
ejpam-5066	243	13	complements	complement	NOUN
ejpam-5066	243	14	,	,	PUNCT
ejpam-5066	243	15	or	or	CCONJ
ejpam-5066	243	16	conjugate	conjugate	VERB
ejpam-5066	243	17	,	,	PUNCT
ejpam-5066	243	18	right	right	ADJ
ejpam-5066	243	19	associate	associate	NOUN
ejpam-5066	243	20	,	,	PUNCT
ejpam-5066	243	21	or	or	CCONJ
ejpam-5066	243	22	left	leave	VERB
ejpam-5066	243	23	associate	associate	NOUN
ejpam-5066	243	24	,	,	PUNCT
ejpam-5066	243	25	then	then	ADV
ejpam-5066	243	26	their	their	PRON
ejpam-5066	243	27	n	n	CCONJ
ejpam-5066	243	28	-	-	PUNCT
ejpam-5066	243	29	centrality	centrality	NOUN
ejpam-5066	243	30	for	for	ADP
ejpam-5066	243	31	some	some	DET
ejpam-5066	243	32	n	n	NOUN
ejpam-5066	243	33	is	be	AUX
ejpam-5066	243	34	equivalent	equivalent	ADJ
ejpam-5066	243	35	.	.	PUNCT
ejpam-5066	244	1	proposition	proposition	NOUN
ejpam-5066	244	2	7	7	NUM
ejpam-5066	244	3	.	.	PUNCT
ejpam-5066	245	1	let	let	VERB
ejpam-5066	245	2	r	r	PRON
ejpam-5066	245	3	be	be	AUX
ejpam-5066	245	4	a	a	DET
ejpam-5066	245	5	ring	ring	NOUN
ejpam-5066	245	6	and	and	CCONJ
ejpam-5066	245	7	e	e	NOUN
ejpam-5066	245	8	,	,	PUNCT
ejpam-5066	245	9	f	f	PROPN
ejpam-5066	245	10	∈	∈	PROPN
ejpam-5066	245	11	i(r	i(r	PROPN
ejpam-5066	245	12	)	)	PUNCT
ejpam-5066	245	13	.	.	PUNCT
ejpam-5066	246	1	then	then	ADV
ejpam-5066	246	2	we	we	PRON
ejpam-5066	246	3	have	have	VERB
ejpam-5066	246	4	the	the	DET
ejpam-5066	246	5	following	follow	VERB
ejpam-5066	246	6	identities	identity	NOUN
ejpam-5066	246	7	.	.	PUNCT
ejpam-5066	247	1	(	(	PUNCT
ejpam-5066	247	2	i	i	NOUN
ejpam-5066	247	3	)	)	PUNCT
ejpam-5066	247	4	if	if	SCONJ
ejpam-5066	247	5	e	e	PROPN
ejpam-5066	247	6	and	and	CCONJ
ejpam-5066	247	7	f	f	PROPN
ejpam-5066	247	8	are	be	AUX
ejpam-5066	247	9	conjugate	conjugate	ADJ
ejpam-5066	247	10	,	,	PUNCT
ejpam-5066	247	11	then	then	ADV
ejpam-5066	247	12	e	e	PROPN
ejpam-5066	247	13	∈	∈	PROPN
ejpam-5066	247	14	cn(r	cn(r	PRON
ejpam-5066	247	15	)	)	PUNCT
ejpam-5066	247	16	if	if	SCONJ
ejpam-5066	247	17	and	and	CCONJ
ejpam-5066	247	18	only	only	ADV
ejpam-5066	247	19	if	if	SCONJ
ejpam-5066	247	20	f	f	PROPN
ejpam-5066	247	21	∈	∈	PROPN
ejpam-5066	247	22	cn(r	cn(r	PRON
ejpam-5066	247	23	)	)	PUNCT
ejpam-5066	247	24	.	.	PUNCT
ejpam-5066	248	1	(	(	PUNCT
ejpam-5066	248	2	ii	ii	NOUN
ejpam-5066	248	3	)	)	PUNCT
ejpam-5066	248	4	if	if	SCONJ
ejpam-5066	248	5	e	e	PROPN
ejpam-5066	248	6	and	and	CCONJ
ejpam-5066	248	7	f	f	PROPN
ejpam-5066	248	8	are	be	AUX
ejpam-5066	248	9	right	right	ADJ
ejpam-5066	248	10	associate	associate	NOUN
ejpam-5066	248	11	,	,	PUNCT
ejpam-5066	248	12	then	then	ADV
ejpam-5066	248	13	e	e	PROPN
ejpam-5066	248	14	∈	∈	PROPN
ejpam-5066	248	15	cn(r	cn(r	PRON
ejpam-5066	248	16	)	)	PUNCT
ejpam-5066	248	17	if	if	SCONJ
ejpam-5066	248	18	and	and	CCONJ
ejpam-5066	248	19	only	only	ADV
ejpam-5066	248	20	if	if	SCONJ
ejpam-5066	248	21	f	f	PROPN
ejpam-5066	248	22	∈	∈	PROPN
ejpam-5066	248	23	cn(r	cn(r	PRON
ejpam-5066	248	24	)	)	PUNCT
ejpam-5066	248	25	.	.	PUNCT
ejpam-5066	249	1	(	(	PUNCT
ejpam-5066	249	2	iii	iii	X
ejpam-5066	249	3	)	)	PUNCT
ejpam-5066	249	4	if	if	SCONJ
ejpam-5066	249	5	e	e	PROPN
ejpam-5066	249	6	and	and	CCONJ
ejpam-5066	249	7	f	f	PROPN
ejpam-5066	249	8	are	be	AUX
ejpam-5066	249	9	left	leave	VERB
ejpam-5066	249	10	associate	associate	ADJ
ejpam-5066	249	11	,	,	PUNCT
ejpam-5066	249	12	then	then	ADV
ejpam-5066	249	13	e	e	PROPN
ejpam-5066	249	14	∈	∈	PROPN
ejpam-5066	249	15	cn(r	cn(r	PRON
ejpam-5066	249	16	)	)	PUNCT
ejpam-5066	249	17	if	if	SCONJ
ejpam-5066	249	18	and	and	CCONJ
ejpam-5066	249	19	only	only	ADV
ejpam-5066	249	20	if	if	SCONJ
ejpam-5066	249	21	f	f	PROPN
ejpam-5066	249	22	∈	∈	PROPN
ejpam-5066	249	23	cn(r	cn(r	PRON
ejpam-5066	249	24	)	)	PUNCT
ejpam-5066	249	25	.	.	PUNCT
ejpam-5066	250	1	(	(	PUNCT
ejpam-5066	250	2	iv	iv	X
ejpam-5066	250	3	)	)	PUNCT
ejpam-5066	250	4	if	if	SCONJ
ejpam-5066	250	5	e	e	PROPN
ejpam-5066	250	6	and	and	CCONJ
ejpam-5066	250	7	f	f	PROPN
ejpam-5066	250	8	are	be	AUX
ejpam-5066	250	9	isomorphic	isomorphic	ADJ
ejpam-5066	250	10	and	and	CCONJ
ejpam-5066	250	11	isomorphic	isomorphic	ADJ
ejpam-5066	250	12	complements	complement	NOUN
ejpam-5066	250	13	,	,	PUNCT
ejpam-5066	250	14	then	then	ADV
ejpam-5066	250	15	e	e	PROPN
ejpam-5066	250	16	∈	∈	PROPN
ejpam-5066	250	17	cn(r	cn(r	PRON
ejpam-5066	250	18	)	)	PUNCT
ejpam-5066	250	19	if	if	SCONJ
ejpam-5066	250	20	and	and	CCONJ
ejpam-5066	250	21	only	only	ADV
ejpam-5066	250	22	if	if	SCONJ
ejpam-5066	250	23	f	f	PROPN
ejpam-5066	250	24	∈	∈	PROPN
ejpam-5066	250	25	cn(r	cn(r	PRON
ejpam-5066	250	26	)	)	PUNCT
ejpam-5066	250	27	.	.	PUNCT
ejpam-5066	251	1	proof	proof	NOUN
ejpam-5066	251	2	.	.	PUNCT
ejpam-5066	252	1	(	(	PUNCT
ejpam-5066	252	2	i	i	NOUN
ejpam-5066	252	3	)	)	PUNCT
ejpam-5066	252	4	if	if	SCONJ
ejpam-5066	252	5	e	e	PROPN
ejpam-5066	252	6	and	and	CCONJ
ejpam-5066	252	7	f	f	PROPN
ejpam-5066	252	8	are	be	AUX
ejpam-5066	252	9	conjugate	conjugate	ADJ
ejpam-5066	252	10	,	,	PUNCT
ejpam-5066	252	11	then	then	ADV
ejpam-5066	252	12	f	f	PROPN
ejpam-5066	252	13	=	=	PRON
ejpam-5066	252	14	u−1fu	u−1fu	VERB
ejpam-5066	252	15	for	for	ADP
ejpam-5066	252	16	some	some	DET
ejpam-5066	252	17	unit	unit	NOUN
ejpam-5066	252	18	u	u	NOUN
ejpam-5066	252	19	in	in	ADP
ejpam-5066	252	20	r.	r.	PROPN
ejpam-5066	252	21	by	by	ADP
ejpam-5066	252	22	simple	simple	ADJ
ejpam-5066	252	23	calculations	calculation	NOUN
ejpam-5066	252	24	,	,	PUNCT
ejpam-5066	252	25	one	one	PRON
ejpam-5066	252	26	can	can	AUX
ejpam-5066	252	27	find	find	VERB
ejpam-5066	252	28	that	that	SCONJ
ejpam-5066	252	29	[	[	X
ejpam-5066	252	30	e	e	NOUN
ejpam-5066	252	31	,	,	PUNCT
ejpam-5066	252	32	r	r	NOUN
ejpam-5066	252	33	]	]	X
ejpam-5066	252	34	=	=	SYM
ejpam-5066	252	35	u[f	u[f	NOUN
ejpam-5066	252	36	,	,	PUNCT
ejpam-5066	252	37	u−1ru	u−1ru	NOUN
ejpam-5066	252	38	]	]	PUNCT
ejpam-5066	252	39	and	and	CCONJ
ejpam-5066	252	40	[	[	X
ejpam-5066	252	41	f	f	X
ejpam-5066	252	42	,	,	PUNCT
ejpam-5066	252	43	r	r	X
ejpam-5066	252	44	]	]	X
ejpam-5066	252	45	=	=	SYM
ejpam-5066	252	46	u[f	u[f	NOUN
ejpam-5066	252	47	,	,	PUNCT
ejpam-5066	252	48	uru−1	uru−1	PROPN
ejpam-5066	252	49	]	]	PUNCT
ejpam-5066	252	50	,	,	PUNCT
ejpam-5066	252	51	for	for	ADP
ejpam-5066	252	52	every	every	DET
ejpam-5066	252	53	r	r	NOUN
ejpam-5066	252	54	∈	∈	PROPN
ejpam-5066	252	55	r.	r.	NOUN
ejpam-5066	252	56	by	by	ADP
ejpam-5066	252	57	proposition	proposition	NOUN
ejpam-5066	252	58	1	1	NUM
ejpam-5066	252	59	,	,	PUNCT
ejpam-5066	252	60	the	the	DET
ejpam-5066	252	61	result	result	NOUN
ejpam-5066	252	62	follows	follow	VERB
ejpam-5066	252	63	.	.	PUNCT
ejpam-5066	253	1	(	(	PUNCT
ejpam-5066	253	2	ii	ii	NOUN
ejpam-5066	253	3	)	)	PUNCT
ejpam-5066	253	4	if	if	SCONJ
ejpam-5066	253	5	e	e	PROPN
ejpam-5066	253	6	and	and	CCONJ
ejpam-5066	253	7	f	f	PROPN
ejpam-5066	253	8	are	be	AUX
ejpam-5066	253	9	right	right	ADJ
ejpam-5066	253	10	associate	associate	NOUN
ejpam-5066	253	11	,	,	PUNCT
ejpam-5066	253	12	then	then	ADV
ejpam-5066	253	13	ef	ef	VERB
ejpam-5066	253	14	=	=	SYM
ejpam-5066	253	15	f	f	PROPN
ejpam-5066	253	16	and	and	CCONJ
ejpam-5066	253	17	fe	fe	X
ejpam-5066	253	18	=	=	SYM
ejpam-5066	253	19	e	e	PROPN
ejpam-5066	253	20	,	,	PUNCT
ejpam-5066	253	21	by	by	ADP
ejpam-5066	253	22	[	[	X
ejpam-5066	253	23	5	5	NUM
ejpam-5066	253	24	,	,	PUNCT
ejpam-5066	253	25	lemma	lemma	PROPN
ejpam-5066	253	26	6.2	6.2	NUM
ejpam-5066	253	27	]	]	PUNCT
ejpam-5066	253	28	.	.	PUNCT
ejpam-5066	254	1	so	so	ADV
ejpam-5066	254	2	,	,	PUNCT
ejpam-5066	254	3	(	(	PUNCT
ejpam-5066	254	4	1	1	NUM
ejpam-5066	254	5	−	−	NOUN
ejpam-5066	254	6	e)(1	e)(1	NOUN
ejpam-5066	255	1	−	−	PROPN
ejpam-5066	256	1	f	f	X
ejpam-5066	256	2	)	)	PUNCT
ejpam-5066	256	3	=	=	SYM
ejpam-5066	256	4	1	1	NUM
ejpam-5066	256	5	−	−	NOUN
ejpam-5066	256	6	e	e	NOUN
ejpam-5066	256	7	and	and	CCONJ
ejpam-5066	256	8	(	(	PUNCT
ejpam-5066	256	9	1	1	NUM
ejpam-5066	256	10	−	−	NOUN
ejpam-5066	256	11	f)(1	f)(1	NOUN
ejpam-5066	256	12	−	−	PROPN
ejpam-5066	256	13	e	e	NOUN
ejpam-5066	256	14	)	)	PUNCT
ejpam-5066	256	15	=	=	SYM
ejpam-5066	257	1	1	1	NUM
ejpam-5066	257	2	−	−	PROPN
ejpam-5066	257	3	f	f	NOUN
ejpam-5066	257	4	.	.	PUNCT
ejpam-5066	258	1	therefore	therefore	ADV
ejpam-5066	258	2	,	,	PUNCT
ejpam-5066	258	3	[	[	X
ejpam-5066	258	4	e]n	e]n	NOUN
ejpam-5066	258	5	⊆	⊆	NUM
ejpam-5066	258	6	(	(	PUNCT
ejpam-5066	258	7	1	1	NUM
ejpam-5066	258	8	−	−	PROPN
ejpam-5066	258	9	e)[f	e)[f	PROPN
ejpam-5066	258	10	]	]	PUNCT
ejpam-5066	258	11	n	n	NOUN
ejpam-5066	259	1	and	and	CCONJ
ejpam-5066	259	2	[	[	X
ejpam-5066	259	3	f	f	X
ejpam-5066	259	4	]	]	X
ejpam-5066	259	5	n	n	CCONJ
ejpam-5066	259	6	⊆	⊆	NUM
ejpam-5066	259	7	(	(	PUNCT
ejpam-5066	259	8	1−	1−	NUM
ejpam-5066	259	9	f)[f	f)[f	NOUN
ejpam-5066	259	10	]	]	PUNCT
ejpam-5066	259	11	n	n	CCONJ
ejpam-5066	259	12	,	,	PUNCT
ejpam-5066	259	13	for	for	ADP
ejpam-5066	259	14	every	every	DET
ejpam-5066	259	15	odd	odd	ADJ
ejpam-5066	259	16	n	n	CCONJ
ejpam-5066	259	17	,	,	PUNCT
ejpam-5066	259	18	while	while	SCONJ
ejpam-5066	259	19	[	[	X
ejpam-5066	259	20	e]n	e]n	NOUN
ejpam-5066	259	21	=	=	X
ejpam-5066	260	1	[	[	X
ejpam-5066	260	2	f	f	X
ejpam-5066	260	3	]	]	X
ejpam-5066	260	4	n	n	CCONJ
ejpam-5066	260	5	,	,	PUNCT
ejpam-5066	260	6	for	for	ADP
ejpam-5066	260	7	every	every	DET
ejpam-5066	260	8	even	even	ADV
ejpam-5066	260	9	n.	n.	PROPN
ejpam-5066	260	10	(	(	PUNCT
ejpam-5066	260	11	iii	iii	NOUN
ejpam-5066	260	12	)	)	PUNCT
ejpam-5066	260	13	is	be	AUX
ejpam-5066	260	14	proved	prove	VERB
ejpam-5066	260	15	as	as	ADP
ejpam-5066	260	16	in	in	ADP
ejpam-5066	260	17	(	(	PUNCT
ejpam-5066	260	18	ii	ii	NOUN
ejpam-5066	260	19	)	)	PUNCT
ejpam-5066	260	20	.	.	PUNCT
ejpam-5066	261	1	(	(	PUNCT
ejpam-5066	261	2	iv	iv	X
ejpam-5066	261	3	)	)	PUNCT
ejpam-5066	261	4	is	be	AUX
ejpam-5066	261	5	clear	clear	ADJ
ejpam-5066	261	6	from	from	ADP
ejpam-5066	261	7	(	(	PUNCT
ejpam-5066	261	8	i	i	NOUN
ejpam-5066	261	9	)	)	PUNCT
ejpam-5066	262	1	[	[	X
ejpam-5066	262	2	5	5	NUM
ejpam-5066	262	3	,	,	PUNCT
ejpam-5066	262	4	lemma	lemma	PROPN
ejpam-5066	262	5	6.2	6.2	NUM
ejpam-5066	262	6	]	]	PUNCT
ejpam-5066	262	7	.	.	PUNCT
ejpam-5066	263	1	it	it	PRON
ejpam-5066	263	2	is	be	AUX
ejpam-5066	263	3	important	important	ADJ
ejpam-5066	263	4	to	to	PART
ejpam-5066	263	5	note	note	VERB
ejpam-5066	263	6	that	that	SCONJ
ejpam-5066	263	7	the	the	DET
ejpam-5066	263	8	isomorphism	isomorphism	NOUN
ejpam-5066	263	9	of	of	ADP
ejpam-5066	263	10	two	two	NUM
ejpam-5066	263	11	idempotents	idempotent	NOUN
ejpam-5066	263	12	is	be	AUX
ejpam-5066	263	13	not	not	PART
ejpam-5066	263	14	always	always	ADV
ejpam-5066	263	15	enough	enough	ADJ
ejpam-5066	263	16	to	to	PART
ejpam-5066	263	17	transfer	transfer	VERB
ejpam-5066	263	18	the	the	DET
ejpam-5066	263	19	n	n	NOUN
ejpam-5066	263	20	-	-	PUNCT
ejpam-5066	263	21	centrality	centrality	NOUN
ejpam-5066	263	22	of	of	ADP
ejpam-5066	263	23	one	one	NUM
ejpam-5066	263	24	to	to	ADP
ejpam-5066	263	25	the	the	DET
ejpam-5066	263	26	other	other	ADJ
ejpam-5066	263	27	.	.	PUNCT
ejpam-5066	264	1	for	for	ADP
ejpam-5066	264	2	instance	instance	NOUN
ejpam-5066	264	3	,	,	PUNCT
ejpam-5066	264	4	consider	consider	VERB
ejpam-5066	264	5	a	a	DET
ejpam-5066	264	6	non	non	ADJ
ejpam-5066	264	7	-	-	ADJ
ejpam-5066	264	8	directly	directly	ADV
ejpam-5066	264	9	finite	finite	ADJ
ejpam-5066	264	10	ring	ring	NOUN
ejpam-5066	264	11	r	r	NOUN
ejpam-5066	264	12	(	(	PUNCT
ejpam-5066	264	13	i.e.	i.e.	X
ejpam-5066	264	14	,	,	PUNCT
ejpam-5066	264	15	ab	ab	PROPN
ejpam-5066	264	16	=	=	NOUN
ejpam-5066	264	17	1	1	NUM
ejpam-5066	264	18	does	do	AUX
ejpam-5066	264	19	not	not	PART
ejpam-5066	264	20	imply	imply	VERB
ejpam-5066	264	21	ba	ba	NOUN
ejpam-5066	264	22	=	=	NOUN
ejpam-5066	264	23	1	1	NUM
ejpam-5066	264	24	for	for	ADP
ejpam-5066	264	25	some	some	DET
ejpam-5066	264	26	a	a	PRON
ejpam-5066	264	27	,	,	PUNCT
ejpam-5066	264	28	b	b	X
ejpam-5066	264	29	∈	∈	PROPN
ejpam-5066	264	30	r	r	NOUN
ejpam-5066	264	31	)	)	PUNCT
ejpam-5066	264	32	.	.	PUNCT
ejpam-5066	265	1	such	such	DET
ejpam-5066	265	2	a	a	DET
ejpam-5066	265	3	ring	ring	NOUN
ejpam-5066	265	4	can	can	AUX
ejpam-5066	265	5	have	have	VERB
ejpam-5066	265	6	an	an	DET
ejpam-5066	265	7	n	n	CCONJ
ejpam-5066	265	8	-	-	PUNCT
ejpam-5066	265	9	central	central	ADJ
ejpam-5066	265	10	idempotent	idempotent	NOUN
ejpam-5066	265	11	,	,	PUNCT
ejpam-5066	265	12	for	for	ADP
ejpam-5066	265	13	any	any	DET
ejpam-5066	265	14	n	n	CCONJ
ejpam-5066	265	15	,	,	PUNCT
ejpam-5066	265	16	while	while	SCONJ
ejpam-5066	265	17	its	its	PRON
ejpam-5066	265	18	isomorphic	isomorphic	ADJ
ejpam-5066	265	19	idempotent	idempotent	NOUN
ejpam-5066	265	20	may	may	AUX
ejpam-5066	265	21	not	not	PART
ejpam-5066	265	22	be	be	AUX
ejpam-5066	265	23	n	n	ADV
ejpam-5066	265	24	-	-	ADJ
ejpam-5066	265	25	central	central	ADJ
ejpam-5066	265	26	.	.	PUNCT
ejpam-5066	266	1	to	to	PART
ejpam-5066	266	2	see	see	VERB
ejpam-5066	266	3	why	why	SCONJ
ejpam-5066	266	4	,	,	PUNCT
ejpam-5066	266	5	suppose	suppose	VERB
ejpam-5066	266	6	ab	ab	PROPN
ejpam-5066	266	7	=	=	NOUN
ejpam-5066	266	8	1	1	NUM
ejpam-5066	266	9	for	for	ADP
ejpam-5066	266	10	a	a	DET
ejpam-5066	266	11	,	,	PUNCT
ejpam-5066	266	12	b	b	X
ejpam-5066	266	13	∈	∈	NOUN
ejpam-5066	266	14	r	r	NOUN
ejpam-5066	266	15	with	with	ADP
ejpam-5066	266	16	r	r	NOUN
ejpam-5066	266	17	not	not	PART
ejpam-5066	266	18	directly	directly	ADV
ejpam-5066	266	19	finite	finite	VERB
ejpam-5066	266	20	.	.	PUNCT
ejpam-5066	267	1	then	then	ADV
ejpam-5066	267	2	,	,	PUNCT
ejpam-5066	267	3	ba	ba	PROPN
ejpam-5066	267	4	is	be	AUX
ejpam-5066	267	5	a	a	DET
ejpam-5066	267	6	non	non	ADJ
ejpam-5066	267	7	-	-	ADJ
ejpam-5066	267	8	trivial	trivial	ADJ
ejpam-5066	267	9	idempotent	idempotent	NOUN
ejpam-5066	267	10	,	,	PUNCT
ejpam-5066	267	11	and	and	CCONJ
ejpam-5066	267	12	for	for	ADP
ejpam-5066	267	13	any	any	DET
ejpam-5066	267	14	m	m	NOUN
ejpam-5066	267	15	,	,	PUNCT
ejpam-5066	267	16	(	(	PUNCT
ejpam-5066	267	17	[	[	X
ejpam-5066	267	18	ba	ba	NOUN
ejpam-5066	267	19	,	,	PUNCT
ejpam-5066	267	20	a][ba	a][ba	NOUN
ejpam-5066	267	21	,	,	PUNCT
ejpam-5066	267	22	b])m	b])m	PROPN
ejpam-5066	267	23	=	=	PUNCT
ejpam-5066	267	24	(	(	PUNCT
ejpam-5066	267	25	−1)m(1	−1)m(1	PROPN
ejpam-5066	267	26	−	−	PROPN
ejpam-5066	267	27	ba	ba	NOUN
ejpam-5066	267	28	)	)	PUNCT
ejpam-5066	267	29	̸=	̸=	PROPN
ejpam-5066	267	30	0	0	NUM
ejpam-5066	267	31	.	.	PUNCT
ejpam-5066	268	1	hence	hence	ADV
ejpam-5066	268	2	,	,	PUNCT
ejpam-5066	268	3	ba	ba	PROPN
ejpam-5066	268	4	can	can	AUX
ejpam-5066	268	5	not	not	PART
ejpam-5066	268	6	be	be	AUX
ejpam-5066	268	7	k	k	ADJ
ejpam-5066	268	8	-	-	ADJ
ejpam-5066	268	9	central	central	ADJ
ejpam-5066	268	10	for	for	ADP
ejpam-5066	268	11	any	any	DET
ejpam-5066	268	12	k.	k.	NOUN
ejpam-5066	268	13	3	3	NUM
ejpam-5066	268	14	.	.	NUM
ejpam-5066	268	15	n	n	CCONJ
ejpam-5066	268	16	-	-	PUNCT
ejpam-5066	268	17	abelian	abelian	NOUN
ejpam-5066	268	18	rings	ring	NOUN
ejpam-5066	268	19	this	this	DET
ejpam-5066	268	20	section	section	NOUN
ejpam-5066	268	21	presents	present	VERB
ejpam-5066	268	22	a	a	DET
ejpam-5066	268	23	generalization	generalization	NOUN
ejpam-5066	268	24	of	of	ADP
ejpam-5066	268	25	idempotent	idempotent	ADJ
ejpam-5066	268	26	centrality	centrality	NOUN
ejpam-5066	268	27	,	,	PUNCT
ejpam-5066	268	28	with	with	SCONJ
ejpam-5066	268	29	the	the	DET
ejpam-5066	268	30	introduction	introduction	NOUN
ejpam-5066	268	31	of	of	ADP
ejpam-5066	268	32	a	a	DET
ejpam-5066	268	33	ring	ring	NOUN
ejpam-5066	268	34	that	that	PRON
ejpam-5066	268	35	contains	contain	VERB
ejpam-5066	268	36	only	only	ADV
ejpam-5066	268	37	idempotents	idempotent	NOUN
ejpam-5066	268	38	which	which	PRON
ejpam-5066	268	39	are	be	AUX
ejpam-5066	268	40	n	n	ADV
ejpam-5066	268	41	-	-	PUNCT
ejpam-5066	268	42	central	central	ADJ
ejpam-5066	268	43	for	for	ADP
ejpam-5066	268	44	some	some	DET
ejpam-5066	268	45	n.	n.	NOUN
ejpam-5066	268	46	the	the	DET
ejpam-5066	268	47	following	follow	VERB
ejpam-5066	268	48	definition	definition	NOUN
ejpam-5066	268	49	serves	serve	VERB
ejpam-5066	268	50	as	as	ADP
ejpam-5066	268	51	an	an	DET
ejpam-5066	268	52	introduction	introduction	NOUN
ejpam-5066	268	53	to	to	ADP
ejpam-5066	268	54	this	this	DET
ejpam-5066	268	55	ring	ring	NOUN
ejpam-5066	268	56	.	.	PUNCT
ejpam-5066	269	1	it	it	PRON
ejpam-5066	269	2	states	state	VERB
ejpam-5066	269	3	that	that	SCONJ
ejpam-5066	269	4	the	the	DET
ejpam-5066	269	5	idempotents	idempotent	NOUN
ejpam-5066	269	6	in	in	ADP
ejpam-5066	269	7	a	a	DET
ejpam-5066	269	8	ring	ring	NOUN
ejpam-5066	269	9	r	r	NOUN
ejpam-5066	269	10	are	be	AUX
ejpam-5066	269	11	all	all	PRON
ejpam-5066	269	12	n	n	ADV
ejpam-5066	269	13	-	-	PUNCT
ejpam-5066	269	14	central	central	ADJ
ejpam-5066	269	15	if	if	SCONJ
ejpam-5066	269	16	and	and	CCONJ
ejpam-5066	269	17	only	only	ADV
ejpam-5066	269	18	if	if	SCONJ
ejpam-5066	269	19	they	they	PRON
ejpam-5066	269	20	are	be	AUX
ejpam-5066	269	21	all	all	ADV
ejpam-5066	269	22	complementary	complementary	ADJ
ejpam-5066	269	23	n	n	CCONJ
ejpam-5066	269	24	-	-	PUNCT
ejpam-5066	269	25	central	central	ADJ
ejpam-5066	269	26	if	if	SCONJ
ejpam-5066	269	27	and	and	CCONJ
ejpam-5066	269	28	only	only	ADV
ejpam-5066	269	29	if	if	SCONJ
ejpam-5066	269	30	they	they	PRON
ejpam-5066	269	31	are	be	AUX
ejpam-5066	269	32	all	all	PRON
ejpam-5066	269	33	dual	dual	ADJ
ejpam-5066	269	34	n	n	CCONJ
ejpam-5066	269	35	-	-	PUNCT
ejpam-5066	269	36	central	central	ADJ
ejpam-5066	269	37	.	.	PUNCT
ejpam-5066	270	1	definition	definition	NOUN
ejpam-5066	270	2	2	2	NUM
ejpam-5066	270	3	.	.	PUNCT
ejpam-5066	271	1	a	a	DET
ejpam-5066	271	2	ring	ring	NOUN
ejpam-5066	271	3	r	r	NOUN
ejpam-5066	271	4	is	be	AUX
ejpam-5066	271	5	called	call	VERB
ejpam-5066	271	6	n	n	CCONJ
ejpam-5066	271	7	-	-	PUNCT
ejpam-5066	271	8	abelian	abelian	NOUN
ejpam-5066	271	9	if	if	SCONJ
ejpam-5066	271	10	every	every	DET
ejpam-5066	271	11	idempotent	idempotent	NOUN
ejpam-5066	271	12	of	of	ADP
ejpam-5066	271	13	r	r	NOUN
ejpam-5066	271	14	is	be	AUX
ejpam-5066	271	15	n	n	ADV
ejpam-5066	271	16	-	-	PUNCT
ejpam-5066	271	17	central	central	ADJ
ejpam-5066	271	18	;	;	PUNCT
ejpam-5066	271	19	that	that	DET
ejpam-5066	271	20	i(r	i(r	NOUN
ejpam-5066	271	21	)	)	PUNCT
ejpam-5066	271	22	=	=	SYM
ejpam-5066	271	23	cn(r	cn(r	PRON
ejpam-5066	271	24	)	)	PUNCT
ejpam-5066	271	25	.	.	PUNCT
ejpam-5066	272	1	m.	m.	PROPN
ejpam-5066	272	2	saad	saad	PROPN
ejpam-5066	272	3	,	,	PUNCT
ejpam-5066	272	4	m.	m.	NOUN
ejpam-5066	272	5	zailaee	zailaee	PROPN
ejpam-5066	272	6	/	/	SYM
ejpam-5066	272	7	eur	eur	PROPN
ejpam-5066	272	8	.	.	PUNCT
ejpam-5066	273	1	j.	j.	PROPN
ejpam-5066	273	2	pure	pure	PROPN
ejpam-5066	273	3	appl	appl	PROPN
ejpam-5066	273	4	.	.	PROPN
ejpam-5066	273	5	math	math	PROPN
ejpam-5066	273	6	,	,	PUNCT
ejpam-5066	273	7	17	17	NUM
ejpam-5066	273	8	(	(	PUNCT
ejpam-5066	273	9	2	2	NUM
ejpam-5066	273	10	)	)	PUNCT
ejpam-5066	273	11	(	(	PUNCT
ejpam-5066	273	12	2024	2024	NUM
ejpam-5066	273	13	)	)	PUNCT
ejpam-5066	273	14	,	,	PUNCT
ejpam-5066	273	15	736	736	NUM
ejpam-5066	273	16	-	-	SYM
ejpam-5066	273	17	752	752	NUM
ejpam-5066	273	18	744	744	NUM
ejpam-5066	273	19	based	base	VERB
ejpam-5066	273	20	on	on	ADP
ejpam-5066	273	21	the	the	DET
ejpam-5066	273	22	definition	definition	NOUN
ejpam-5066	273	23	,	,	PUNCT
ejpam-5066	273	24	it	it	PRON
ejpam-5066	273	25	can	can	AUX
ejpam-5066	273	26	be	be	AUX
ejpam-5066	273	27	observed	observe	VERB
ejpam-5066	273	28	that	that	SCONJ
ejpam-5066	273	29	an	an	DET
ejpam-5066	273	30	n	n	CCONJ
ejpam-5066	273	31	-	-	PUNCT
ejpam-5066	273	32	abelian	abelian	NOUN
ejpam-5066	273	33	ring	ring	NOUN
ejpam-5066	273	34	is	be	AUX
ejpam-5066	273	35	also	also	ADV
ejpam-5066	273	36	an	an	DET
ejpam-5066	273	37	m	m	ADJ
ejpam-5066	273	38	-	-	ADJ
ejpam-5066	273	39	abelian	abelian	ADJ
ejpam-5066	273	40	ring	ring	NOUN
ejpam-5066	274	1	if	if	SCONJ
ejpam-5066	274	2	n	n	NOUN
ejpam-5066	274	3	≤	≤	X
ejpam-5066	274	4	m.	m.	NOUN
ejpam-5066	274	5	furthermore	furthermore	ADV
ejpam-5066	274	6	,	,	PUNCT
ejpam-5066	274	7	the	the	DET
ejpam-5066	274	8	classes	class	NOUN
ejpam-5066	274	9	of	of	ADP
ejpam-5066	274	10	abelian	abelian	ADJ
ejpam-5066	274	11	rings	ring	NOUN
ejpam-5066	274	12	and	and	CCONJ
ejpam-5066	274	13	q	q	ADJ
ejpam-5066	274	14	-	-	PUNCT
ejpam-5066	274	15	abelian	abelian	ADJ
ejpam-5066	274	16	rings	ring	NOUN
ejpam-5066	274	17	are	be	AUX
ejpam-5066	274	18	equivalent	equivalent	ADJ
ejpam-5066	274	19	to	to	ADP
ejpam-5066	274	20	the	the	DET
ejpam-5066	274	21	classes	class	NOUN
ejpam-5066	274	22	of	of	ADP
ejpam-5066	274	23	1	1	NUM
ejpam-5066	274	24	-	-	PUNCT
ejpam-5066	274	25	abelian	abelian	NOUN
ejpam-5066	274	26	and	and	CCONJ
ejpam-5066	274	27	2	2	NUM
ejpam-5066	274	28	-	-	PUNCT
ejpam-5066	274	29	abelian	abelian	NOUN
ejpam-5066	274	30	rings	ring	NOUN
ejpam-5066	274	31	,	,	PUNCT
ejpam-5066	274	32	respectively	respectively	ADV
ejpam-5066	274	33	.	.	PUNCT
ejpam-5066	275	1	it	it	PRON
ejpam-5066	275	2	is	be	AUX
ejpam-5066	275	3	worth	worth	ADJ
ejpam-5066	275	4	noting	note	VERB
ejpam-5066	275	5	that	that	SCONJ
ejpam-5066	275	6	every	every	DET
ejpam-5066	275	7	abelian	abelian	ADJ
ejpam-5066	275	8	ring	ring	NOUN
ejpam-5066	275	9	is	be	AUX
ejpam-5066	275	10	n	n	CCONJ
ejpam-5066	275	11	-	-	PUNCT
ejpam-5066	275	12	abelian	abelian	ADJ
ejpam-5066	275	13	for	for	ADP
ejpam-5066	275	14	all	all	DET
ejpam-5066	275	15	values	value	NOUN
ejpam-5066	275	16	of	of	ADP
ejpam-5066	275	17	n.	n.	NOUN
ejpam-5066	275	18	however	however	ADV
ejpam-5066	275	19	,	,	PUNCT
ejpam-5066	275	20	the	the	DET
ejpam-5066	275	21	ring	ring	NOUN
ejpam-5066	275	22	in	in	ADP
ejpam-5066	275	23	example	example	NOUN
ejpam-5066	275	24	2	2	NUM
ejpam-5066	275	25	can	can	AUX
ejpam-5066	275	26	not	not	PART
ejpam-5066	275	27	be	be	AUX
ejpam-5066	275	28	n	n	ADV
ejpam-5066	275	29	-	-	PUNCT
ejpam-5066	275	30	abelian	abelian	ADJ
ejpam-5066	275	31	for	for	ADP
ejpam-5066	275	32	any	any	DET
ejpam-5066	275	33	n.	n.	NOUN
ejpam-5066	275	34	the	the	DET
ejpam-5066	275	35	following	follow	VERB
ejpam-5066	275	36	proposition	proposition	NOUN
ejpam-5066	275	37	provides	provide	VERB
ejpam-5066	275	38	an	an	DET
ejpam-5066	275	39	alternative	alternative	ADJ
ejpam-5066	275	40	definition	definition	NOUN
ejpam-5066	275	41	for	for	ADP
ejpam-5066	275	42	n	n	CCONJ
ejpam-5066	275	43	-	-	PUNCT
ejpam-5066	275	44	abelian	abelian	NOUN
ejpam-5066	275	45	rings	ring	NOUN
ejpam-5066	275	46	.	.	PUNCT
ejpam-5066	276	1	proposition	proposition	NOUN
ejpam-5066	276	2	8	8	NUM
ejpam-5066	276	3	.	.	PUNCT
ejpam-5066	277	1	the	the	DET
ejpam-5066	277	2	following	follow	VERB
ejpam-5066	277	3	conditions	condition	NOUN
ejpam-5066	277	4	are	be	AUX
ejpam-5066	277	5	equivalent	equivalent	ADJ
ejpam-5066	277	6	for	for	ADP
ejpam-5066	277	7	a	a	DET
ejpam-5066	277	8	ring	ring	NOUN
ejpam-5066	277	9	r	r	NOUN
ejpam-5066	277	10	and	and	CCONJ
ejpam-5066	277	11	positive	positive	ADJ
ejpam-5066	277	12	integer	integer	NOUN
ejpam-5066	277	13	n.	n.	NOUN
ejpam-5066	277	14	(	(	PUNCT
ejpam-5066	277	15	i	i	NOUN
ejpam-5066	277	16	)	)	PUNCT
ejpam-5066	277	17	r	r	NOUN
ejpam-5066	277	18	is	be	AUX
ejpam-5066	277	19	n	n	CCONJ
ejpam-5066	277	20	-	-	PUNCT
ejpam-5066	277	21	abelian	abelian	ADJ
ejpam-5066	277	22	.	.	PUNCT
ejpam-5066	278	1	(	(	PUNCT
ejpam-5066	278	2	ii	ii	NOUN
ejpam-5066	278	3	)	)	PUNCT
ejpam-5066	279	1	[	[	X
ejpam-5066	279	2	e	e	NOUN
ejpam-5066	279	3	,	,	PUNCT
ejpam-5066	279	4	r]n	r]n	NOUN
ejpam-5066	279	5	=	=	NOUN
ejpam-5066	279	6	0	0	NUM
ejpam-5066	279	7	,	,	PUNCT
ejpam-5066	279	8	for	for	ADP
ejpam-5066	279	9	every	every	DET
ejpam-5066	279	10	e	e	PROPN
ejpam-5066	279	11	∈	∈	PROPN
ejpam-5066	279	12	i(r	i(r	PROPN
ejpam-5066	279	13	)	)	PUNCT
ejpam-5066	279	14	.	.	PUNCT
ejpam-5066	280	1	(	(	PUNCT
ejpam-5066	280	2	iii	iii	X
ejpam-5066	280	3	)	)	PUNCT
ejpam-5066	281	1	[	[	X
ejpam-5066	281	2	e]n−1	e]n−1	PROPN
ejpam-5066	281	3	is	be	AUX
ejpam-5066	281	4	an	an	DET
ejpam-5066	281	5	ideal	ideal	NOUN
ejpam-5066	281	6	of	of	ADP
ejpam-5066	281	7	r	r	NOUN
ejpam-5066	281	8	,	,	PUNCT
ejpam-5066	281	9	for	for	ADP
ejpam-5066	281	10	every	every	DET
ejpam-5066	281	11	e	e	PROPN
ejpam-5066	281	12	∈	∈	PROPN
ejpam-5066	281	13	i(r	i(r	PROPN
ejpam-5066	281	14	)	)	PUNCT
ejpam-5066	281	15	.	.	PUNCT
ejpam-5066	282	1	proof	proof	NOUN
ejpam-5066	282	2	.	.	PUNCT
ejpam-5066	283	1	the	the	DET
ejpam-5066	283	2	proof	proof	NOUN
ejpam-5066	283	3	is	be	AUX
ejpam-5066	283	4	straightforward	straightforward	ADJ
ejpam-5066	283	5	from	from	ADP
ejpam-5066	283	6	the	the	DET
ejpam-5066	283	7	definition	definition	NOUN
ejpam-5066	283	8	and	and	CCONJ
ejpam-5066	283	9	proposition	proposition	NOUN
ejpam-5066	284	1	1	1	NUM
ejpam-5066	284	2	.	.	PUNCT
ejpam-5066	284	3	recall	recall	NOUN
ejpam-5066	284	4	,	,	PUNCT
ejpam-5066	284	5	an	an	DET
ejpam-5066	284	6	idempotent	idempotent	ADJ
ejpam-5066	284	7	e	e	NOUN
ejpam-5066	284	8	of	of	ADP
ejpam-5066	284	9	a	a	DET
ejpam-5066	284	10	ring	ring	NOUN
ejpam-5066	284	11	r	r	NOUN
ejpam-5066	284	12	is	be	AUX
ejpam-5066	284	13	said	say	VERB
ejpam-5066	284	14	to	to	PART
ejpam-5066	284	15	be	be	AUX
ejpam-5066	284	16	directly	directly	ADV
ejpam-5066	284	17	finite	finite	ADJ
ejpam-5066	284	18	if	if	SCONJ
ejpam-5066	284	19	the	the	DET
ejpam-5066	284	20	ere	ere	NOUN
ejpam-5066	284	21	is	be	AUX
ejpam-5066	284	22	directly	directly	ADV
ejpam-5066	284	23	finite	finite	ADJ
ejpam-5066	284	24	.	.	PUNCT
ejpam-5066	285	1	the	the	DET
ejpam-5066	285	2	next	next	ADJ
ejpam-5066	285	3	proposition	proposition	NOUN
ejpam-5066	285	4	shows	show	VERB
ejpam-5066	285	5	that	that	SCONJ
ejpam-5066	285	6	every	every	DET
ejpam-5066	285	7	n	n	CCONJ
ejpam-5066	285	8	-	-	ADJ
ejpam-5066	285	9	central	central	ADJ
ejpam-5066	285	10	idempotent	idempotent	NOUN
ejpam-5066	285	11	for	for	ADP
ejpam-5066	285	12	any	any	DET
ejpam-5066	285	13	n	n	NOUN
ejpam-5066	285	14	is	be	AUX
ejpam-5066	285	15	directly	directly	ADV
ejpam-5066	285	16	finite	finite	ADJ
ejpam-5066	285	17	,	,	PUNCT
ejpam-5066	285	18	and	and	CCONJ
ejpam-5066	285	19	therefore	therefore	ADV
ejpam-5066	285	20	every	every	DET
ejpam-5066	285	21	n	n	CCONJ
ejpam-5066	285	22	-	-	PUNCT
ejpam-5066	285	23	abelian	abelian	ADJ
ejpam-5066	285	24	ring	ring	NOUN
ejpam-5066	285	25	for	for	ADP
ejpam-5066	285	26	any	any	DET
ejpam-5066	285	27	n	n	NOUN
ejpam-5066	285	28	is	be	AUX
ejpam-5066	285	29	directly	directly	ADV
ejpam-5066	285	30	finite	finite	ADJ
ejpam-5066	285	31	.	.	PUNCT
ejpam-5066	286	1	proposition	proposition	NOUN
ejpam-5066	286	2	9	9	NUM
ejpam-5066	286	3	.	.	PUNCT
ejpam-5066	287	1	if	if	SCONJ
ejpam-5066	287	2	e	e	PROPN
ejpam-5066	287	3	is	be	AUX
ejpam-5066	287	4	an	an	DET
ejpam-5066	287	5	n	n	CCONJ
ejpam-5066	287	6	-	-	PUNCT
ejpam-5066	287	7	central	central	ADJ
ejpam-5066	287	8	idempotent	idempotent	NOUN
ejpam-5066	287	9	of	of	ADP
ejpam-5066	287	10	a	a	DET
ejpam-5066	287	11	ring	ring	NOUN
ejpam-5066	287	12	for	for	ADP
ejpam-5066	287	13	some	some	DET
ejpam-5066	287	14	n	n	CCONJ
ejpam-5066	287	15	,	,	PUNCT
ejpam-5066	287	16	then	then	ADV
ejpam-5066	287	17	e	e	PROPN
ejpam-5066	287	18	is	be	AUX
ejpam-5066	287	19	directly	directly	ADV
ejpam-5066	287	20	finite	finite	ADJ
ejpam-5066	287	21	.	.	PUNCT
ejpam-5066	288	1	proof	proof	NOUN
ejpam-5066	288	2	.	.	PUNCT
ejpam-5066	289	1	let	let	VERB
ejpam-5066	289	2	r	r	PRON
ejpam-5066	289	3	be	be	AUX
ejpam-5066	289	4	a	a	DET
ejpam-5066	289	5	ring	ring	NOUN
ejpam-5066	289	6	e	e	NOUN
ejpam-5066	289	7	an	an	DET
ejpam-5066	289	8	n	n	CCONJ
ejpam-5066	289	9	-	-	PUNCT
ejpam-5066	289	10	central	central	ADJ
ejpam-5066	289	11	idempotent	idempotent	NOUN
ejpam-5066	289	12	of	of	ADP
ejpam-5066	289	13	a	a	DET
ejpam-5066	289	14	ring	ring	NOUN
ejpam-5066	289	15	r	r	NOUN
ejpam-5066	289	16	,	,	PUNCT
ejpam-5066	289	17	for	for	ADP
ejpam-5066	289	18	some	some	DET
ejpam-5066	289	19	n.	n.	NOUN
ejpam-5066	289	20	assume	assume	VERB
ejpam-5066	289	21	that	that	SCONJ
ejpam-5066	289	22	ab	ab	PROPN
ejpam-5066	289	23	=	=	SYM
ejpam-5066	289	24	e	e	PROPN
ejpam-5066	289	25	,	,	PUNCT
ejpam-5066	289	26	for	for	ADP
ejpam-5066	289	27	some	some	DET
ejpam-5066	289	28	a	a	PRON
ejpam-5066	289	29	,	,	PUNCT
ejpam-5066	289	30	b	b	PROPN
ejpam-5066	289	31	∈	∈	PROPN
ejpam-5066	289	32	ere	ere	PROPN
ejpam-5066	289	33	.	.	PUNCT
ejpam-5066	289	34	hence	hence	ADV
ejpam-5066	289	35	,	,	PUNCT
ejpam-5066	289	36	ba	ba	PROPN
ejpam-5066	289	37	is	be	AUX
ejpam-5066	289	38	an	an	DET
ejpam-5066	289	39	idempotent	idempotent	NOUN
ejpam-5066	289	40	in	in	ADP
ejpam-5066	289	41	ere	ere	PROPN
ejpam-5066	289	42	and	and	CCONJ
ejpam-5066	289	43	[	[	X
ejpam-5066	289	44	ba	ba	PROPN
ejpam-5066	289	45	,	,	PUNCT
ejpam-5066	289	46	a][ba	a][ba	NOUN
ejpam-5066	289	47	,	,	PUNCT
ejpam-5066	289	48	b	b	NOUN
ejpam-5066	289	49	]	]	X
ejpam-5066	289	50	=	=	SYM
ejpam-5066	290	1	ba−	ba−	PROPN
ejpam-5066	290	2	e.	e.	PROPN
ejpam-5066	290	3	without	without	ADP
ejpam-5066	290	4	loss	loss	NOUN
ejpam-5066	290	5	of	of	ADP
ejpam-5066	290	6	generality	generality	NOUN
ejpam-5066	290	7	we	we	PRON
ejpam-5066	290	8	assume	assume	VERB
ejpam-5066	290	9	that	that	SCONJ
ejpam-5066	290	10	n	n	PRON
ejpam-5066	290	11	is	be	AUX
ejpam-5066	290	12	even	even	ADV
ejpam-5066	290	13	,	,	PUNCT
ejpam-5066	290	14	then	then	ADV
ejpam-5066	290	15	0	0	NUM
ejpam-5066	290	16	=	=	SYM
ejpam-5066	290	17	(	(	PUNCT
ejpam-5066	290	18	[	[	X
ejpam-5066	290	19	ba	ba	NOUN
ejpam-5066	290	20	,	,	PUNCT
ejpam-5066	290	21	a][ba	a][ba	NOUN
ejpam-5066	290	22	,	,	PUNCT
ejpam-5066	290	23	b	b	NOUN
ejpam-5066	290	24	]	]	X
ejpam-5066	290	25	)	)	PUNCT
ejpam-5066	290	26	n	n	PRON
ejpam-5066	290	27	2	2	NUM
ejpam-5066	290	28	=	=	SYM
ejpam-5066	290	29	(	(	PUNCT
ejpam-5066	290	30	−1	−1	NOUN
ejpam-5066	290	31	)	)	PUNCT
ejpam-5066	290	32	n	n	PRON
ejpam-5066	290	33	2	2	NUM
ejpam-5066	290	34	(	(	PUNCT
ejpam-5066	290	35	e−	e−	PROPN
ejpam-5066	290	36	ba	ba	PROPN
ejpam-5066	290	37	)	)	PUNCT
ejpam-5066	290	38	and	and	CCONJ
ejpam-5066	290	39	ba	ba	PROPN
ejpam-5066	291	1	=	=	SYM
ejpam-5066	291	2	e.	e.	PROPN
ejpam-5066	291	3	thus	thus	ADV
ejpam-5066	291	4	,	,	PUNCT
ejpam-5066	291	5	ere	ere	PROPN
ejpam-5066	291	6	is	be	AUX
ejpam-5066	291	7	differently	differently	ADV
ejpam-5066	291	8	finite	finite	ADJ
ejpam-5066	291	9	.	.	PUNCT
ejpam-5066	292	1	corollary	corollary	ADJ
ejpam-5066	292	2	7	7	NUM
ejpam-5066	292	3	.	.	PUNCT
ejpam-5066	293	1	every	every	DET
ejpam-5066	293	2	n	n	CCONJ
ejpam-5066	293	3	-	-	PUNCT
ejpam-5066	293	4	abelian	abelian	ADJ
ejpam-5066	293	5	ring	ring	NOUN
ejpam-5066	293	6	is	be	AUX
ejpam-5066	293	7	directly	directly	ADV
ejpam-5066	293	8	finite	finite	ADJ
ejpam-5066	293	9	,	,	PUNCT
ejpam-5066	293	10	for	for	ADP
ejpam-5066	293	11	every	every	DET
ejpam-5066	293	12	n.	n.	NOUN
ejpam-5066	293	13	corollary	corollary	ADJ
ejpam-5066	293	14	8	8	NUM
ejpam-5066	293	15	(	(	PUNCT
ejpam-5066	293	16	[	[	X
ejpam-5066	293	17	22],theorem	22],theorem	NUM
ejpam-5066	293	18	2.4	2.4	NUM
ejpam-5066	293	19	)	)	PUNCT
ejpam-5066	293	20	.	.	PUNCT
ejpam-5066	294	1	quasi	quasi	ADJ
ejpam-5066	294	2	-	-	ADJ
ejpam-5066	294	3	normal	normal	ADJ
ejpam-5066	294	4	rings	ring	NOUN
ejpam-5066	294	5	are	be	AUX
ejpam-5066	294	6	directly	directly	ADV
ejpam-5066	294	7	finite	finite	ADJ
ejpam-5066	294	8	.	.	PUNCT
ejpam-5066	295	1	according	accord	VERB
ejpam-5066	295	2	to	to	ADP
ejpam-5066	295	3	to	to	ADP
ejpam-5066	295	4	[	[	X
ejpam-5066	295	5	21	21	NUM
ejpam-5066	295	6	]	]	X
ejpam-5066	295	7	,	,	PUNCT
ejpam-5066	295	8	an	an	DET
ejpam-5066	295	9	element	element	NOUN
ejpam-5066	295	10	a	a	PRON
ejpam-5066	295	11	of	of	ADP
ejpam-5066	295	12	a	a	DET
ejpam-5066	295	13	ring	ring	NOUN
ejpam-5066	295	14	r	r	NOUN
ejpam-5066	295	15	is	be	AUX
ejpam-5066	295	16	called	call	VERB
ejpam-5066	295	17	left	leave	VERB
ejpam-5066	295	18	minimal	minimal	ADJ
ejpam-5066	295	19	if	if	SCONJ
ejpam-5066	295	20	ra	ra	PROPN
ejpam-5066	295	21	is	be	AUX
ejpam-5066	295	22	a	a	DET
ejpam-5066	295	23	minimal	minimal	ADJ
ejpam-5066	295	24	left	left	ADJ
ejpam-5066	295	25	ideal	ideal	NOUN
ejpam-5066	295	26	of	of	ADP
ejpam-5066	295	27	r	r	NOUN
ejpam-5066	295	28	and	and	CCONJ
ejpam-5066	295	29	r	r	NOUN
ejpam-5066	295	30	is	be	AUX
ejpam-5066	295	31	called	call	VERB
ejpam-5066	295	32	left	left	ADJ
ejpam-5066	295	33	min	min	NOUN
ejpam-5066	295	34	-	-	NOUN
ejpam-5066	295	35	abel	abel	NOUN
ejpam-5066	295	36	if	if	SCONJ
ejpam-5066	295	37	each	each	PRON
ejpam-5066	295	38	left	leave	VERB
ejpam-5066	295	39	minimal	minimal	ADJ
ejpam-5066	295	40	idempotent	idempotent	NOUN
ejpam-5066	295	41	left	leave	VERB
ejpam-5066	295	42	semicentral	semicentral	NOUN
ejpam-5066	295	43	.	.	PUNCT
ejpam-5066	296	1	the	the	DET
ejpam-5066	296	2	next	next	ADJ
ejpam-5066	296	3	proposition	proposition	NOUN
ejpam-5066	296	4	states	state	VERB
ejpam-5066	296	5	that	that	SCONJ
ejpam-5066	296	6	a	a	DET
ejpam-5066	296	7	ring	ring	NOUN
ejpam-5066	296	8	r	r	NOUN
ejpam-5066	296	9	is	be	AUX
ejpam-5066	296	10	left	leave	VERB
ejpam-5066	296	11	min	min	NOUN
ejpam-5066	296	12	-	-	NOUN
ejpam-5066	296	13	abel	abel	NOUN
ejpam-5066	296	14	whenever	whenever	SCONJ
ejpam-5066	296	15	it	it	PRON
ejpam-5066	296	16	is	be	AUX
ejpam-5066	296	17	n	n	CCONJ
ejpam-5066	296	18	-	-	PUNCT
ejpam-5066	296	19	abelian	abelian	ADJ
ejpam-5066	296	20	for	for	ADP
ejpam-5066	296	21	some	some	DET
ejpam-5066	296	22	n.	n.	NOUN
ejpam-5066	296	23	proposition	proposition	NOUN
ejpam-5066	296	24	10	10	NUM
ejpam-5066	296	25	.	.	PUNCT
ejpam-5066	297	1	if	if	SCONJ
ejpam-5066	297	2	r	r	NOUN
ejpam-5066	297	3	is	be	AUX
ejpam-5066	297	4	n	n	CCONJ
ejpam-5066	297	5	-	-	PUNCT
ejpam-5066	297	6	abelian	abelian	ADJ
ejpam-5066	297	7	for	for	ADP
ejpam-5066	297	8	some	some	DET
ejpam-5066	297	9	n	n	CCONJ
ejpam-5066	297	10	,	,	PUNCT
ejpam-5066	297	11	then	then	ADV
ejpam-5066	297	12	r	r	NOUN
ejpam-5066	297	13	is	be	AUX
ejpam-5066	297	14	left	leave	VERB
ejpam-5066	297	15	min	min	NOUN
ejpam-5066	297	16	-	-	NOUN
ejpam-5066	297	17	abel	abel	NOUN
ejpam-5066	297	18	.	.	PUNCT
ejpam-5066	298	1	proof	proof	NOUN
ejpam-5066	298	2	.	.	PUNCT
ejpam-5066	299	1	let	let	VERB
ejpam-5066	299	2	e	e	PRON
ejpam-5066	299	3	be	be	AUX
ejpam-5066	299	4	a	a	DET
ejpam-5066	299	5	nonzero	nonzero	NOUN
ejpam-5066	299	6	left	leave	VERB
ejpam-5066	299	7	minimal	minimal	ADJ
ejpam-5066	299	8	idempotent	idempotent	ADJ
ejpam-5066	299	9	element	element	NOUN
ejpam-5066	299	10	of	of	ADP
ejpam-5066	299	11	an	an	DET
ejpam-5066	299	12	n	n	ADV
ejpam-5066	299	13	-	-	PUNCT
ejpam-5066	299	14	abelian	abelian	NOUN
ejpam-5066	299	15	ring	ring	NOUN
ejpam-5066	299	16	r.	r.	PROPN
ejpam-5066	299	17	if	if	SCONJ
ejpam-5066	299	18	e	e	PROPN
ejpam-5066	299	19	is	be	AUX
ejpam-5066	299	20	not	not	PART
ejpam-5066	299	21	left	leave	VERB
ejpam-5066	299	22	semicentral	semicentral	ADJ
ejpam-5066	299	23	,	,	PUNCT
ejpam-5066	299	24	then	then	ADV
ejpam-5066	299	25	(	(	PUNCT
ejpam-5066	299	26	1	1	NUM
ejpam-5066	299	27	−	−	NOUN
ejpam-5066	299	28	e)ae	e)ae	PROPN
ejpam-5066	299	29	̸=	̸=	PROPN
ejpam-5066	299	30	0	0	NUM
ejpam-5066	299	31	for	for	ADP
ejpam-5066	299	32	some	some	PRON
ejpam-5066	299	33	a	a	DET
ejpam-5066	299	34	∈	∈	NOUN
ejpam-5066	299	35	r	r	NOUN
ejpam-5066	299	36	and	and	CCONJ
ejpam-5066	299	37	0	0	NUM
ejpam-5066	299	38	̸=	̸=	PROPN
ejpam-5066	299	39	r(1	r(1	PROPN
ejpam-5066	299	40	−	−	PROPN
ejpam-5066	300	1	e)ae	e)ae	PROPN
ejpam-5066	300	2	⊆	⊆	NUM
ejpam-5066	300	3	re	re	NOUN
ejpam-5066	300	4	.	.	PUNCT
ejpam-5066	301	1	hence	hence	ADV
ejpam-5066	301	2	r(1−e)ae	r(1−e)ae	NUM
ejpam-5066	301	3	=	=	SYM
ejpam-5066	301	4	re	re	VERB
ejpam-5066	301	5	because	because	SCONJ
ejpam-5066	301	6	re	re	VERB
ejpam-5066	301	7	is	be	AUX
ejpam-5066	301	8	minimal	minimal	ADJ
ejpam-5066	301	9	left	leave	VERB
ejpam-5066	301	10	ideal	ideal	NOUN
ejpam-5066	301	11	of	of	ADP
ejpam-5066	301	12	r.	r.	PROPN
ejpam-5066	301	13	so	so	ADV
ejpam-5066	301	14	(	(	PUNCT
ejpam-5066	301	15	re)m	re)m	NOUN
ejpam-5066	301	16	=	=	SYM
ejpam-5066	301	17	(	(	PUNCT
ejpam-5066	301	18	r(1−e)ae)m	r(1−e)ae)m	PROPN
ejpam-5066	301	19	⊆	⊆	NUM
ejpam-5066	301	20	r[e]2m−1	r[e]2m−1	PROPN
ejpam-5066	301	21	,	,	PUNCT
ejpam-5066	301	22	for	for	ADP
ejpam-5066	301	23	every	every	DET
ejpam-5066	301	24	m	m	NOUN
ejpam-5066	301	25	>	>	X
ejpam-5066	301	26	0	0	NUM
ejpam-5066	301	27	.	.	PUNCT
ejpam-5066	302	1	choosing	choose	VERB
ejpam-5066	302	2	m	m	PROPN
ejpam-5066	302	3	≥	≥	NOUN
ejpam-5066	302	4	n+1	n+1	ADV
ejpam-5066	302	5	2	2	NUM
ejpam-5066	302	6	,	,	PUNCT
ejpam-5066	302	7	we	we	PRON
ejpam-5066	302	8	get	get	VERB
ejpam-5066	302	9	e	e	NOUN
ejpam-5066	302	10	=	=	SYM
ejpam-5066	302	11	0	0	NUM
ejpam-5066	302	12	,	,	PUNCT
ejpam-5066	302	13	which	which	PRON
ejpam-5066	302	14	is	be	AUX
ejpam-5066	302	15	a	a	DET
ejpam-5066	302	16	contradiction	contradiction	NOUN
ejpam-5066	302	17	.	.	PUNCT
ejpam-5066	303	1	hence	hence	ADV
ejpam-5066	303	2	(	(	PUNCT
ejpam-5066	303	3	1−	1−	NUM
ejpam-5066	303	4	e)ae	e)ae	X
ejpam-5066	303	5	=	=	PUNCT
ejpam-5066	303	6	0	0	PROPN
ejpam-5066	303	7	for	for	ADP
ejpam-5066	303	8	all	all	DET
ejpam-5066	303	9	a	a	DET
ejpam-5066	303	10	∈	∈	NOUN
ejpam-5066	303	11	r	r	NOUN
ejpam-5066	303	12	and	and	CCONJ
ejpam-5066	303	13	r	r	NOUN
ejpam-5066	303	14	is	be	AUX
ejpam-5066	303	15	left	leave	VERB
ejpam-5066	303	16	min	min	NOUN
ejpam-5066	303	17	-	-	NOUN
ejpam-5066	303	18	abel	abel	NOUN
ejpam-5066	303	19	.	.	PUNCT
ejpam-5066	304	1	corollary	corollary	ADJ
ejpam-5066	304	2	9	9	NUM
ejpam-5066	304	3	(	(	PUNCT
ejpam-5066	304	4	[	[	X
ejpam-5066	304	5	23],theorem	23],theorem	NUM
ejpam-5066	304	6	2.5	2.5	NUM
ejpam-5066	304	7	)	)	PUNCT
ejpam-5066	304	8	.	.	PUNCT
ejpam-5066	305	1	every	every	DET
ejpam-5066	305	2	quasi	quasi	ADJ
ejpam-5066	305	3	-	-	ADJ
ejpam-5066	305	4	normal	normal	ADJ
ejpam-5066	305	5	ring	ring	NOUN
ejpam-5066	305	6	is	be	AUX
ejpam-5066	305	7	left	leave	VERB
ejpam-5066	305	8	min	min	NOUN
ejpam-5066	305	9	-	-	NOUN
ejpam-5066	305	10	abel	abel	NOUN
ejpam-5066	305	11	.	.	PUNCT
ejpam-5066	306	1	m.	m.	PROPN
ejpam-5066	306	2	saad	saad	PROPN
ejpam-5066	306	3	,	,	PUNCT
ejpam-5066	306	4	m.	m.	NOUN
ejpam-5066	306	5	zailaee	zailaee	PROPN
ejpam-5066	306	6	/	/	SYM
ejpam-5066	306	7	eur	eur	PROPN
ejpam-5066	306	8	.	.	PUNCT
ejpam-5066	307	1	j.	j.	PROPN
ejpam-5066	307	2	pure	pure	PROPN
ejpam-5066	307	3	appl	appl	PROPN
ejpam-5066	307	4	.	.	PROPN
ejpam-5066	307	5	math	math	PROPN
ejpam-5066	307	6	,	,	PUNCT
ejpam-5066	307	7	17	17	NUM
ejpam-5066	307	8	(	(	PUNCT
ejpam-5066	307	9	2	2	NUM
ejpam-5066	307	10	)	)	PUNCT
ejpam-5066	307	11	(	(	PUNCT
ejpam-5066	307	12	2024	2024	NUM
ejpam-5066	307	13	)	)	PUNCT
ejpam-5066	307	14	,	,	PUNCT
ejpam-5066	307	15	736	736	NUM
ejpam-5066	307	16	-	-	SYM
ejpam-5066	307	17	752	752	NUM
ejpam-5066	307	18	745	745	NUM
ejpam-5066	307	19	the	the	DET
ejpam-5066	307	20	previous	previous	ADJ
ejpam-5066	307	21	corollary	corollary	NOUN
ejpam-5066	307	22	presented	present	VERB
ejpam-5066	307	23	in	in	ADP
ejpam-5066	307	24	[	[	X
ejpam-5066	307	25	23	23	NUM
ejpam-5066	307	26	]	]	PUNCT
ejpam-5066	307	27	has	have	VERB
ejpam-5066	307	28	a	a	DET
ejpam-5066	307	29	counterexample	counterexample	NOUN
ejpam-5066	307	30	,	,	PUNCT
ejpam-5066	307	31	which	which	PRON
ejpam-5066	307	32	also	also	ADV
ejpam-5066	307	33	serves	serve	VERB
ejpam-5066	307	34	as	as	ADP
ejpam-5066	307	35	a	a	DET
ejpam-5066	307	36	counterexample	counterexample	NOUN
ejpam-5066	307	37	for	for	ADP
ejpam-5066	307	38	proposition	proposition	NOUN
ejpam-5066	307	39	10	10	NUM
ejpam-5066	307	40	.	.	PUNCT
ejpam-5066	308	1	in	in	ADP
ejpam-5066	308	2	accordance	accordance	NOUN
ejpam-5066	308	3	with	with	ADP
ejpam-5066	308	4	[	[	X
ejpam-5066	308	5	25	25	NUM
ejpam-5066	308	6	]	]	PUNCT
ejpam-5066	308	7	,	,	PUNCT
ejpam-5066	308	8	a	a	DET
ejpam-5066	308	9	ring	ring	NOUN
ejpam-5066	308	10	r	r	NOUN
ejpam-5066	308	11	is	be	AUX
ejpam-5066	308	12	referred	refer	VERB
ejpam-5066	308	13	to	to	ADP
ejpam-5066	308	14	as	as	ADV
ejpam-5066	308	15	weakly	weakly	ADV
ejpam-5066	308	16	normal	normal	ADJ
ejpam-5066	308	17	if	if	SCONJ
ejpam-5066	308	18	ae	ae	PROPN
ejpam-5066	308	19	=	=	SYM
ejpam-5066	308	20	0	0	PROPN
ejpam-5066	308	21	implies	imply	VERB
ejpam-5066	308	22	rera	rera	NOUN
ejpam-5066	308	23	is	be	AUX
ejpam-5066	308	24	a	a	DET
ejpam-5066	308	25	nil	nil	ADJ
ejpam-5066	308	26	left	left	ADJ
ejpam-5066	308	27	ideal	ideal	NOUN
ejpam-5066	308	28	of	of	ADP
ejpam-5066	308	29	r	r	NOUN
ejpam-5066	308	30	,	,	PUNCT
ejpam-5066	308	31	where	where	SCONJ
ejpam-5066	308	32	a	a	X
ejpam-5066	308	33	,	,	PUNCT
ejpam-5066	308	34	r	r	NOUN
ejpam-5066	308	35	∈	∈	PROPN
ejpam-5066	308	36	r	r	NOUN
ejpam-5066	308	37	and	and	CCONJ
ejpam-5066	308	38	e	e	NOUN
ejpam-5066	308	39	∈	∈	PROPN
ejpam-5066	308	40	i(r	i(r	PROPN
ejpam-5066	308	41	)	)	PUNCT
ejpam-5066	308	42	.	.	PUNCT
ejpam-5066	309	1	it	it	PRON
ejpam-5066	309	2	is	be	AUX
ejpam-5066	309	3	worth	worth	ADJ
ejpam-5066	309	4	noting	note	VERB
ejpam-5066	309	5	that	that	SCONJ
ejpam-5066	309	6	every	every	DET
ejpam-5066	309	7	quasi	quasi	NOUN
ejpam-5066	309	8	-	-	ADJ
ejpam-5066	309	9	normal	normal	ADJ
ejpam-5066	309	10	(	(	PUNCT
ejpam-5066	309	11	or	or	CCONJ
ejpam-5066	309	12	q	q	NOUN
ejpam-5066	309	13	-	-	PUNCT
ejpam-5066	309	14	abelian	abelian	ADJ
ejpam-5066	309	15	)	)	PUNCT
ejpam-5066	309	16	ring	ring	NOUN
ejpam-5066	309	17	is	be	AUX
ejpam-5066	309	18	weakly	weakly	ADV
ejpam-5066	309	19	normal	normal	ADJ
ejpam-5066	309	20	,	,	PUNCT
ejpam-5066	309	21	as	as	SCONJ
ejpam-5066	309	22	shown	show	VERB
ejpam-5066	309	23	in	in	ADP
ejpam-5066	309	24	[	[	X
ejpam-5066	309	25	25	25	NUM
ejpam-5066	309	26	,	,	PUNCT
ejpam-5066	309	27	corollary	corollary	ADJ
ejpam-5066	309	28	2.3	2.3	NUM
ejpam-5066	309	29	(	(	PUNCT
ejpam-5066	309	30	1	1	NUM
ejpam-5066	309	31	)	)	PUNCT
ejpam-5066	309	32	]	]	PUNCT
ejpam-5066	309	33	.	.	PUNCT
ejpam-5066	310	1	the	the	DET
ejpam-5066	310	2	following	follow	VERB
ejpam-5066	310	3	proposition	proposition	NOUN
ejpam-5066	310	4	extends	extend	VERB
ejpam-5066	310	5	this	this	DET
ejpam-5066	310	6	result	result	NOUN
ejpam-5066	310	7	to	to	ADP
ejpam-5066	310	8	a	a	DET
ejpam-5066	310	9	broader	broad	ADJ
ejpam-5066	310	10	scope	scope	NOUN
ejpam-5066	310	11	.	.	PUNCT
ejpam-5066	311	1	proposition	proposition	NOUN
ejpam-5066	311	2	11	11	NUM
ejpam-5066	311	3	.	.	PUNCT
ejpam-5066	312	1	if	if	SCONJ
ejpam-5066	312	2	r	r	NOUN
ejpam-5066	312	3	is	be	AUX
ejpam-5066	312	4	an	an	DET
ejpam-5066	312	5	n	n	ADV
ejpam-5066	312	6	-	-	PUNCT
ejpam-5066	312	7	abelian	abelian	NOUN
ejpam-5066	312	8	ring	ring	NOUN
ejpam-5066	312	9	for	for	ADP
ejpam-5066	312	10	some	some	DET
ejpam-5066	312	11	n	n	CCONJ
ejpam-5066	312	12	,	,	PUNCT
ejpam-5066	312	13	then	then	ADV
ejpam-5066	312	14	r	r	NOUN
ejpam-5066	312	15	is	be	AUX
ejpam-5066	312	16	weakly	weakly	ADV
ejpam-5066	312	17	normal	normal	ADJ
ejpam-5066	312	18	.	.	PUNCT
ejpam-5066	313	1	proof	proof	NOUN
ejpam-5066	313	2	.	.	PUNCT
ejpam-5066	314	1	it	it	PRON
ejpam-5066	314	2	is	be	AUX
ejpam-5066	314	3	direct	direct	ADJ
ejpam-5066	314	4	by	by	ADP
ejpam-5066	314	5	[	[	X
ejpam-5066	314	6	25	25	NUM
ejpam-5066	314	7	,	,	PUNCT
ejpam-5066	314	8	theorem	theorem	VERB
ejpam-5066	314	9	2.2	2.2	NUM
ejpam-5066	314	10	]	]	PUNCT
ejpam-5066	314	11	.	.	PUNCT
ejpam-5066	315	1	in	in	ADP
ejpam-5066	315	2	[	[	X
ejpam-5066	315	3	8	8	NUM
ejpam-5066	315	4	]	]	PUNCT
ejpam-5066	315	5	,	,	PUNCT
ejpam-5066	315	6	a	a	DET
ejpam-5066	315	7	ring	ring	NOUN
ejpam-5066	315	8	r	r	NOUN
ejpam-5066	315	9	is	be	AUX
ejpam-5066	315	10	called	call	VERB
ejpam-5066	315	11	left	left	ADJ
ejpam-5066	315	12	(	(	PUNCT
ejpam-5066	315	13	resp	resp	NOUN
ejpam-5066	315	14	.	.	PUNCT
ejpam-5066	316	1	right	right	ADJ
ejpam-5066	316	2	)	)	PUNCT
ejpam-5066	316	3	idempotent	idempotent	NOUN
ejpam-5066	316	4	-	-	PUNCT
ejpam-5066	316	5	reflexive	reflexive	ADJ
ejpam-5066	316	6	if	if	SCONJ
ejpam-5066	316	7	are	be	AUX
ejpam-5066	316	8	=	=	SYM
ejpam-5066	316	9	0	0	NUM
ejpam-5066	316	10	(	(	PUNCT
ejpam-5066	316	11	resp	resp	NOUN
ejpam-5066	316	12	.	.	PUNCT
ejpam-5066	317	1	era	era	NOUN
ejpam-5066	317	2	=	=	SYM
ejpam-5066	317	3	0	0	NUM
ejpam-5066	317	4	)	)	PUNCT
ejpam-5066	317	5	implies	imply	VERB
ejpam-5066	317	6	era	era	NOUN
ejpam-5066	317	7	=	=	SYM
ejpam-5066	317	8	0	0	NUM
ejpam-5066	317	9	(	(	PUNCT
ejpam-5066	317	10	resp	resp	NOUN
ejpam-5066	317	11	,	,	PUNCT
ejpam-5066	317	12	are	be	AUX
ejpam-5066	317	13	=	=	NOUN
ejpam-5066	317	14	0	0	NUM
ejpam-5066	317	15	)	)	PUNCT
ejpam-5066	317	16	for	for	ADP
ejpam-5066	317	17	every	every	DET
ejpam-5066	317	18	a	a	DET
ejpam-5066	317	19	∈	∈	PROPN
ejpam-5066	317	20	r	r	NOUN
ejpam-5066	317	21	and	and	CCONJ
ejpam-5066	317	22	e	e	NOUN
ejpam-5066	317	23	∈	∈	PROPN
ejpam-5066	317	24	i(r	i(r	PROPN
ejpam-5066	317	25	)	)	PUNCT
ejpam-5066	317	26	.	.	PUNCT
ejpam-5066	318	1	the	the	DET
ejpam-5066	318	2	following	follow	VERB
ejpam-5066	318	3	proposition	proposition	NOUN
ejpam-5066	318	4	shows	show	VERB
ejpam-5066	318	5	that	that	SCONJ
ejpam-5066	318	6	the	the	DET
ejpam-5066	318	7	sets	set	NOUN
ejpam-5066	318	8	of	of	ADP
ejpam-5066	318	9	n	n	CCONJ
ejpam-5066	318	10	-	-	PUNCT
ejpam-5066	318	11	central	central	ADJ
ejpam-5066	318	12	idempotents	idempotent	NOUN
ejpam-5066	318	13	and	and	CCONJ
ejpam-5066	318	14	complementary	complementary	ADJ
ejpam-5066	318	15	n	n	CCONJ
ejpam-5066	318	16	-	-	PUNCT
ejpam-5066	318	17	central	central	ADJ
ejpam-5066	318	18	idempotents	idempotent	NOUN
ejpam-5066	318	19	of	of	ADP
ejpam-5066	318	20	a	a	DET
ejpam-5066	318	21	ring	ring	NOUN
ejpam-5066	318	22	one	one	NUM
ejpam-5066	318	23	-	-	PUNCT
ejpam-5066	318	24	sided	sided	ADJ
ejpam-5066	318	25	idempotent	idempotent	ADJ
ejpam-5066	318	26	-	-	PUNCT
ejpam-5066	318	27	reflexive	reflexive	ADJ
ejpam-5066	318	28	ring	ring	NOUN
ejpam-5066	318	29	r	r	NOUN
ejpam-5066	318	30	coincide	coincide	NOUN
ejpam-5066	318	31	.	.	PUNCT
ejpam-5066	319	1	proposition	proposition	NOUN
ejpam-5066	319	2	12	12	NUM
ejpam-5066	319	3	.	.	PUNCT
ejpam-5066	320	1	for	for	ADP
ejpam-5066	320	2	every	every	DET
ejpam-5066	320	3	left	left	NOUN
ejpam-5066	320	4	(	(	PUNCT
ejpam-5066	320	5	or	or	CCONJ
ejpam-5066	320	6	right	right	ADJ
ejpam-5066	320	7	)	)	PUNCT
ejpam-5066	320	8	idempotent	idempotent	ADJ
ejpam-5066	320	9	-	-	PUNCT
ejpam-5066	320	10	reflexive	reflexive	ADJ
ejpam-5066	320	11	ring	ring	NOUN
ejpam-5066	320	12	r	r	NOUN
ejpam-5066	320	13	,	,	PUNCT
ejpam-5066	320	14	we	we	PRON
ejpam-5066	320	15	have	have	VERB
ejpam-5066	320	16	cn(r	cn(r	PRON
ejpam-5066	320	17	)	)	PUNCT
ejpam-5066	321	1	=	=	SYM
ejpam-5066	321	2	ĉn(r	ĉn(r	NOUN
ejpam-5066	321	3	)	)	PUNCT
ejpam-5066	321	4	if	if	SCONJ
ejpam-5066	321	5	n	n	NOUN
ejpam-5066	321	6	is	be	AUX
ejpam-5066	321	7	odd	odd	ADJ
ejpam-5066	321	8	and	and	CCONJ
ejpam-5066	321	9	cn(r	cn(r	NUM
ejpam-5066	321	10	)	)	PUNCT
ejpam-5066	321	11	=	=	SYM
ejpam-5066	321	12	ĉn−1(r	ĉn−1(r	NOUN
ejpam-5066	321	13	)	)	PUNCT
ejpam-5066	321	14	if	if	SCONJ
ejpam-5066	321	15	n	n	PRON
ejpam-5066	321	16	is	be	AUX
ejpam-5066	321	17	even	even	ADV
ejpam-5066	321	18	.	.	PUNCT
ejpam-5066	322	1	proof	proof	NOUN
ejpam-5066	322	2	.	.	PUNCT
ejpam-5066	323	1	if	if	SCONJ
ejpam-5066	323	2	e	e	PROPN
ejpam-5066	323	3	∈	∈	PROPN
ejpam-5066	323	4	cn(r	cn(r	PRON
ejpam-5066	323	5	)	)	PUNCT
ejpam-5066	323	6	for	for	ADP
ejpam-5066	323	7	some	some	DET
ejpam-5066	323	8	odd	odd	ADJ
ejpam-5066	323	9	n	n	CCONJ
ejpam-5066	323	10	,	,	PUNCT
ejpam-5066	323	11	then	then	ADV
ejpam-5066	323	12	(	(	PUNCT
ejpam-5066	323	13	(	(	PUNCT
ejpam-5066	323	14	1	1	NUM
ejpam-5066	323	15	−	−	NOUN
ejpam-5066	323	16	e)rer	e)rer	ADJ
ejpam-5066	323	17	)	)	PUNCT
ejpam-5066	323	18	n+1	n+1	PROPN
ejpam-5066	323	19	2	2	NUM
ejpam-5066	323	20	=	=	SYM
ejpam-5066	323	21	0	0	NUM
ejpam-5066	323	22	and	and	CCONJ
ejpam-5066	323	23	therefore	therefore	ADV
ejpam-5066	323	24	(	(	PUNCT
ejpam-5066	323	25	(	(	PUNCT
ejpam-5066	323	26	(	(	PUNCT
ejpam-5066	323	27	1−	1−	NUM
ejpam-5066	323	28	e)rer	e)rer	NOUN
ejpam-5066	323	29	)	)	PUNCT
ejpam-5066	323	30	n−1	n−1	PROPN
ejpam-5066	323	31	2	2	NUM
ejpam-5066	323	32	(	(	PUNCT
ejpam-5066	323	33	1−	1−	NUM
ejpam-5066	323	34	e	e	NOUN
ejpam-5066	323	35	)	)	PUNCT
ejpam-5066	323	36	)	)	PUNCT
ejpam-5066	323	37	re	re	VERB
ejpam-5066	323	38	=	=	NOUN
ejpam-5066	323	39	0	0	X
ejpam-5066	323	40	.	.	PUNCT
ejpam-5066	324	1	ifr	ifr	PROPN
ejpam-5066	324	2	is	be	AUX
ejpam-5066	324	3	left	leave	VERB
ejpam-5066	324	4	idempotent	idempotent	NOUN
ejpam-5066	324	5	-	-	PUNCT
ejpam-5066	324	6	reflexive	reflexive	ADJ
ejpam-5066	324	7	,	,	PUNCT
ejpam-5066	324	8	then	then	ADV
ejpam-5066	324	9	(	(	PUNCT
ejpam-5066	324	10	er(1−e)r	er(1−e)r	ADJ
ejpam-5066	324	11	)	)	PUNCT
ejpam-5066	324	12	n+1	n+1	PROPN
ejpam-5066	324	13	2	2	NUM
ejpam-5066	324	14	=	=	SYM
ejpam-5066	324	15	0	0	NUM
ejpam-5066	324	16	and	and	CCONJ
ejpam-5066	324	17	e	e	PROPN
ejpam-5066	324	18	∈	∈	PROPN
ejpam-5066	324	19	cn(r	cn(r	PRON
ejpam-5066	324	20	)	)	PUNCT
ejpam-5066	324	21	.	.	PUNCT
ejpam-5066	325	1	in	in	ADP
ejpam-5066	325	2	case	case	NOUN
ejpam-5066	325	3	of	of	ADP
ejpam-5066	325	4	n	n	NUM
ejpam-5066	325	5	is	be	AUX
ejpam-5066	325	6	even	even	ADV
ejpam-5066	325	7	,	,	PUNCT
ejpam-5066	325	8	we	we	PRON
ejpam-5066	325	9	have	have	VERB
ejpam-5066	325	10	er((1	er((1	VERB
ejpam-5066	325	11	−	−	PROPN
ejpam-5066	325	12	e)rer	e)rer	ADJ
ejpam-5066	325	13	)	)	PUNCT
ejpam-5066	325	14	n−2	n−2	PROPN
ejpam-5066	325	15	2	2	NUM
ejpam-5066	325	16	=	=	SYM
ejpam-5066	325	17	0	0	NUM
ejpam-5066	325	18	and	and	CCONJ
ejpam-5066	325	19	0	0	NUM
ejpam-5066	326	1	=	=	SYM
ejpam-5066	326	2	(	(	PUNCT
ejpam-5066	326	3	r(1	r(1	PROPN
ejpam-5066	326	4	−	−	PROPN
ejpam-5066	326	5	e)re	e)re	PROPN
ejpam-5066	326	6	)	)	PUNCT
ejpam-5066	326	7	n−2	n−2	PROPN
ejpam-5066	326	8	2	2	NUM
ejpam-5066	326	9	re	re	NOUN
ejpam-5066	326	10	=	=	X
ejpam-5066	326	11	(	(	PUNCT
ejpam-5066	326	12	r(1	r(1	PROPN
ejpam-5066	326	13	−	−	PROPN
ejpam-5066	326	14	e)re	e)re	PROPN
ejpam-5066	326	15	)	)	PUNCT
ejpam-5066	326	16	n−2	n−2	PROPN
ejpam-5066	326	17	2	2	NUM
ejpam-5066	326	18	.	.	PUNCT
ejpam-5066	327	1	therefore	therefore	ADV
ejpam-5066	327	2	,	,	PUNCT
ejpam-5066	327	3	e	e	PROPN
ejpam-5066	327	4	∈	∈	PROPN
ejpam-5066	327	5	cn−1(r	cn−1(r	PROPN
ejpam-5066	327	6	)	)	PUNCT
ejpam-5066	327	7	.	.	PUNCT
ejpam-5066	328	1	but	but	CCONJ
ejpam-5066	328	2	n	n	PRON
ejpam-5066	328	3	−	−	PROPN
ejpam-5066	328	4	1	1	NUM
ejpam-5066	328	5	is	be	AUX
ejpam-5066	328	6	odd	odd	ADJ
ejpam-5066	328	7	and	and	CCONJ
ejpam-5066	328	8	consequently	consequently	ADV
ejpam-5066	328	9	cn(r	cn(r	NUM
ejpam-5066	328	10	)	)	PUNCT
ejpam-5066	328	11	=	=	PUNCT
ejpam-5066	329	1	ĉn−1	ĉn−1	PROPN
ejpam-5066	329	2	,	,	PUNCT
ejpam-5066	329	3	from	from	ADP
ejpam-5066	329	4	the	the	DET
ejpam-5066	329	5	previous	previous	ADJ
ejpam-5066	329	6	result	result	NOUN
ejpam-5066	329	7	.	.	PUNCT
ejpam-5066	330	1	every	every	DET
ejpam-5066	330	2	subring	subre	VERB
ejpam-5066	330	3	s	s	X
ejpam-5066	330	4	(	(	PUNCT
ejpam-5066	330	5	which	which	PRON
ejpam-5066	330	6	is	be	AUX
ejpam-5066	330	7	not	not	PART
ejpam-5066	330	8	necessarily	necessarily	ADV
ejpam-5066	330	9	with	with	ADP
ejpam-5066	330	10	identity	identity	NOUN
ejpam-5066	330	11	)	)	PUNCT
ejpam-5066	330	12	of	of	ADP
ejpam-5066	330	13	an	an	DET
ejpam-5066	330	14	n	n	ADV
ejpam-5066	330	15	-	-	PUNCT
ejpam-5066	330	16	abelian	abelian	NOUN
ejpam-5066	330	17	ring	ring	NOUN
ejpam-5066	330	18	r	r	NOUN
ejpam-5066	330	19	is	be	AUX
ejpam-5066	330	20	also	also	ADV
ejpam-5066	330	21	n	n	CCONJ
ejpam-5066	330	22	-	-	PUNCT
ejpam-5066	330	23	abelian	abelian	ADJ
ejpam-5066	330	24	since	since	SCONJ
ejpam-5066	330	25	for	for	ADP
ejpam-5066	330	26	every	every	DET
ejpam-5066	330	27	e	e	PROPN
ejpam-5066	330	28	∈	∈	PROPN
ejpam-5066	330	29	in(s	in(s	NOUN
ejpam-5066	330	30	)	)	PUNCT
ejpam-5066	330	31	,	,	PUNCT
ejpam-5066	330	32	we	we	PRON
ejpam-5066	330	33	have	have	VERB
ejpam-5066	330	34	e	e	NOUN
ejpam-5066	330	35	∈	∈	PROPN
ejpam-5066	330	36	cn(r	cn(r	PRON
ejpam-5066	330	37	)	)	PUNCT
ejpam-5066	330	38	and	and	CCONJ
ejpam-5066	330	39	[	[	X
ejpam-5066	330	40	e	e	NOUN
ejpam-5066	330	41	,	,	PUNCT
ejpam-5066	330	42	s]ne	s]ne	X
ejpam-5066	330	43	⊆	⊆	NUM
ejpam-5066	330	44	[	[	X
ejpam-5066	330	45	e	e	NOUN
ejpam-5066	330	46	,	,	PUNCT
ejpam-5066	330	47	r]ne	r]ne	NOUN
ejpam-5066	330	48	=	=	SYM
ejpam-5066	330	49	0	0	NUM
ejpam-5066	330	50	and	and	CCONJ
ejpam-5066	330	51	e	e	PROPN
ejpam-5066	330	52	∈	∈	PROPN
ejpam-5066	330	53	cn(s	cn(	NOUN
ejpam-5066	330	54	)	)	PUNCT
ejpam-5066	330	55	.	.	PUNCT
ejpam-5066	331	1	however	however	ADV
ejpam-5066	331	2	,	,	PUNCT
ejpam-5066	331	3	the	the	DET
ejpam-5066	331	4	n	n	CCONJ
ejpam-5066	331	5	-	-	PUNCT
ejpam-5066	331	6	abelianity	abelianity	NOUN
ejpam-5066	331	7	of	of	ADP
ejpam-5066	331	8	a	a	DET
ejpam-5066	331	9	subring	subring	NOUN
ejpam-5066	331	10	is	be	AUX
ejpam-5066	331	11	not	not	PART
ejpam-5066	331	12	necessarily	necessarily	ADV
ejpam-5066	331	13	extended	extend	VERB
ejpam-5066	331	14	to	to	ADP
ejpam-5066	331	15	the	the	DET
ejpam-5066	331	16	ring	ring	NOUN
ejpam-5066	331	17	itself	itself	PRON
ejpam-5066	331	18	.	.	PUNCT
ejpam-5066	332	1	example	example	NOUN
ejpam-5066	333	1	3	3	NUM
ejpam-5066	333	2	.	.	PUNCT
ejpam-5066	334	1	in	in	ADP
ejpam-5066	334	2	the	the	DET
ejpam-5066	334	3	ring	ring	NOUN
ejpam-5066	334	4	r	r	NOUN
ejpam-5066	334	5	of	of	ADP
ejpam-5066	334	6	example	example	NOUN
ejpam-5066	334	7	1	1	NUM
ejpam-5066	334	8	,	,	PUNCT
ejpam-5066	334	9	the	the	DET
ejpam-5066	334	10	element	element	NOUN
ejpam-5066	334	11	e	e	NOUN
ejpam-5066	334	12	=	=	SYM
ejpam-5066	334	13	diag(1	diag(1	PROPN
ejpam-5066	334	14	,	,	PUNCT
ejpam-5066	334	15	0	0	NUM
ejpam-5066	334	16	,	,	PUNCT
ejpam-5066	334	17	1	1	NUM
ejpam-5066	334	18	)	)	PUNCT
ejpam-5066	334	19	is	be	AUX
ejpam-5066	334	20	not	not	PART
ejpam-5066	334	21	1	1	NUM
ejpam-5066	334	22	-	-	PUNCT
ejpam-5066	334	23	central	central	ADJ
ejpam-5066	334	24	in	in	ADP
ejpam-5066	334	25	r.	r.	PROPN
ejpam-5066	334	26	however	however	ADV
ejpam-5066	334	27	,	,	PUNCT
ejpam-5066	334	28	e	e	PROPN
ejpam-5066	334	29	is	be	AUX
ejpam-5066	334	30	an	an	DET
ejpam-5066	334	31	idempotent	idempotent	NOUN
ejpam-5066	334	32	of	of	ADP
ejpam-5066	334	33	subring	subre	VERB
ejpam-5066	334	34	s	s	PART
ejpam-5066	334	35	=	=	SYM
ejpam-5066	334	36			PROPN
ejpam-5066	334	37	f	f	X
ejpam-5066	334	38	0	0	PUNCT
ejpam-5066	334	39	f	f	PROPN
ejpam-5066	334	40	0	0	NUM
ejpam-5066	334	41	0	0	NUM
ejpam-5066	334	42	0	0	NUM
ejpam-5066	334	43	0	0	NUM
ejpam-5066	334	44	0	0	NUM
ejpam-5066	334	45	f	f	NOUN
ejpam-5066	334	46			NOUN
ejpam-5066	334	47	∼=	∼=	NOUN
ejpam-5066	334	48	t2(f	t2(f	NOUN
ejpam-5066	334	49	)	)	PUNCT
ejpam-5066	334	50	and	and	CCONJ
ejpam-5066	334	51	it	it	PRON
ejpam-5066	334	52	is	be	AUX
ejpam-5066	334	53	1	1	NUM
ejpam-5066	334	54	-	-	PUNCT
ejpam-5066	334	55	central	central	ADJ
ejpam-5066	334	56	.	.	PUNCT
ejpam-5066	335	1	also	also	ADV
ejpam-5066	335	2	,	,	PUNCT
ejpam-5066	335	3	f	f	PROPN
ejpam-5066	335	4	is	be	AUX
ejpam-5066	335	5	a	a	DET
ejpam-5066	335	6	subring	subring	NOUN
ejpam-5066	335	7	of	of	ADP
ejpam-5066	335	8	r	r	NOUN
ejpam-5066	335	9	which	which	PRON
ejpam-5066	335	10	is	be	AUX
ejpam-5066	335	11	commutative	commutative	ADJ
ejpam-5066	335	12	and	and	CCONJ
ejpam-5066	335	13	in	in	ADP
ejpam-5066	335	14	particular	particular	ADJ
ejpam-5066	335	15	,	,	PUNCT
ejpam-5066	335	16	is	be	AUX
ejpam-5066	335	17	1	1	NUM
ejpam-5066	335	18	-	-	PUNCT
ejpam-5066	335	19	abelian	abelian	NOUN
ejpam-5066	335	20	while	while	SCONJ
ejpam-5066	335	21	r	r	NOUN
ejpam-5066	335	22	is	be	AUX
ejpam-5066	335	23	not	not	PART
ejpam-5066	335	24	1	1	NUM
ejpam-5066	335	25	-	-	PUNCT
ejpam-5066	335	26	abelian	abelian	NOUN
ejpam-5066	335	27	.	.	PUNCT
ejpam-5066	336	1	proposition	proposition	NOUN
ejpam-5066	336	2	13	13	NUM
ejpam-5066	336	3	.	.	PUNCT
ejpam-5066	337	1	every	every	DET
ejpam-5066	337	2	subdirect	subdirect	ADJ
ejpam-5066	337	3	product	product	NOUN
ejpam-5066	337	4	of	of	ADP
ejpam-5066	337	5	a	a	DET
ejpam-5066	337	6	family	family	NOUN
ejpam-5066	337	7	n	n	CCONJ
ejpam-5066	337	8	-	-	PUNCT
ejpam-5066	337	9	abelian	abelian	NOUN
ejpam-5066	337	10	rings	ring	NOUN
ejpam-5066	337	11	is	be	AUX
ejpam-5066	337	12	also	also	ADV
ejpam-5066	337	13	n	n	CCONJ
ejpam-5066	337	14	-	-	PUNCT
ejpam-5066	337	15	abelian	abelian	ADJ
ejpam-5066	337	16	,	,	PUNCT
ejpam-5066	337	17	for	for	ADP
ejpam-5066	337	18	every	every	DET
ejpam-5066	337	19	n.	n.	NOUN
ejpam-5066	337	20	proof	proof	NOUN
ejpam-5066	337	21	.	.	PUNCT
ejpam-5066	338	1	let	let	VERB
ejpam-5066	338	2	ri	ri	NOUN
ejpam-5066	338	3	=	=	SYM
ejpam-5066	338	4	r	r	X
ejpam-5066	338	5	/	/	SYM
ejpam-5066	338	6	ai	ai	NOUN
ejpam-5066	338	7	,	,	PUNCT
ejpam-5066	338	8	for	for	ADP
ejpam-5066	338	9	some	some	DET
ejpam-5066	338	10	ideals	ideal	NOUN
ejpam-5066	338	11	{	{	PUNCT
ejpam-5066	338	12	ai}i∈λ	ai}i∈λ	NOUN
ejpam-5066	338	13	of	of	ADP
ejpam-5066	338	14	a	a	DET
ejpam-5066	338	15	ring	ring	NOUN
ejpam-5066	338	16	r	r	NOUN
ejpam-5066	338	17	with	with	ADP
ejpam-5066	338	18	⋂	⋂	PROPN
ejpam-5066	338	19	i∈λai	i∈λai	PROPN
ejpam-5066	338	20	=	=	SYM
ejpam-5066	338	21	0	0	X
ejpam-5066	338	22	.	.	PUNCT
ejpam-5066	339	1	for	for	ADP
ejpam-5066	339	2	every	every	DET
ejpam-5066	339	3	e	e	PROPN
ejpam-5066	339	4	∈	∈	PROPN
ejpam-5066	339	5	i(r	i(r	PROPN
ejpam-5066	339	6	)	)	PUNCT
ejpam-5066	339	7	,	,	PUNCT
ejpam-5066	339	8	we	we	PRON
ejpam-5066	339	9	have	have	VERB
ejpam-5066	339	10	ei	ei	NOUN
ejpam-5066	339	11	=	=	PUNCT
ejpam-5066	339	12	e	e	PROPN
ejpam-5066	339	13	+	+	CCONJ
ejpam-5066	339	14	ai	ai	PROPN
ejpam-5066	339	15	∈	∈	PROPN
ejpam-5066	339	16	i(ai	i(ai	NOUN
ejpam-5066	339	17	)	)	PUNCT
ejpam-5066	339	18	where	where	SCONJ
ejpam-5066	339	19	i	i	PRON
ejpam-5066	339	20	∈	∈	PROPN
ejpam-5066	339	21	λ	λ	PROPN
ejpam-5066	339	22	.	.	PUNCT
ejpam-5066	340	1	if	if	SCONJ
ejpam-5066	340	2	each	each	DET
ejpam-5066	340	3	ri	ri	PROPN
ejpam-5066	340	4	is	be	AUX
ejpam-5066	340	5	n	n	CCONJ
ejpam-5066	340	6	-	-	PUNCT
ejpam-5066	340	7	abelian	abelian	ADJ
ejpam-5066	340	8	for	for	ADP
ejpam-5066	340	9	some	some	DET
ejpam-5066	340	10	n	n	NOUN
ejpam-5066	340	11	,	,	PUNCT
ejpam-5066	340	12	then	then	ADV
ejpam-5066	340	13	[	[	X
ejpam-5066	340	14	ei	ei	X
ejpam-5066	340	15	,	,	PUNCT
ejpam-5066	340	16	r]neir	r]neir	PROPN
ejpam-5066	340	17	=	=	NOUN
ejpam-5066	340	18	0	0	NUM
ejpam-5066	340	19	for	for	ADP
ejpam-5066	340	20	every	every	DET
ejpam-5066	340	21	i	i	PROPN
ejpam-5066	340	22	∈	∈	PROPN
ejpam-5066	340	23	λ	λ	PROPN
ejpam-5066	340	24	.	.	PUNCT
ejpam-5066	341	1	so	so	ADV
ejpam-5066	341	2	0	0	X
ejpam-5066	342	1	=	=	SYM
ejpam-5066	343	1	[	[	X
ejpam-5066	343	2	ei	ei	X
ejpam-5066	343	3	,	,	PUNCT
ejpam-5066	343	4	r]neir	r]neir	X
ejpam-5066	343	5	=	=	X
ejpam-5066	344	1	[	[	X
ejpam-5066	344	2	e	e	X
ejpam-5066	344	3	+	+	CCONJ
ejpam-5066	344	4	ai	ai	VERB
ejpam-5066	344	5	,	,	PUNCT
ejpam-5066	344	6	r]neir	r]neir	PRON
ejpam-5066	344	7	⊆	⊆	NUM
ejpam-5066	344	8	[	[	X
ejpam-5066	344	9	e	e	NOUN
ejpam-5066	344	10	,	,	PUNCT
ejpam-5066	344	11	r]n(e+ai)r+[ai	r]n(e+ai)r+[ai	VERB
ejpam-5066	344	12	,	,	PUNCT
ejpam-5066	344	13	r]n(e+ai)r	r]n(e+ai)r	NOUN
ejpam-5066	344	14	⊆	⊆	NUM
ejpam-5066	344	15	[	[	X
ejpam-5066	344	16	e	e	NOUN
ejpam-5066	344	17	,	,	PUNCT
ejpam-5066	344	18	r]ne+[e	r]ne+[e	NOUN
ejpam-5066	344	19	,	,	PUNCT
ejpam-5066	344	20	r]nai+[ai	r]nai+[ai	ADJ
ejpam-5066	344	21	,	,	PUNCT
ejpam-5066	344	22	r]n(e+ai)r	r]n(e+ai)r	PROPN
ejpam-5066	344	23	⊆	⊆	NUM
ejpam-5066	344	24	[	[	X
ejpam-5066	344	25	e	e	NOUN
ejpam-5066	344	26	,	,	PUNCT
ejpam-5066	344	27	r]ne+ai	r]ne+ai	NOUN
ejpam-5066	344	28	and	and	CCONJ
ejpam-5066	344	29	[	[	X
ejpam-5066	344	30	e	e	NOUN
ejpam-5066	344	31	,	,	PUNCT
ejpam-5066	344	32	r]ne	r]ne	NOUN
ejpam-5066	344	33	for	for	ADP
ejpam-5066	344	34	every	every	DET
ejpam-5066	344	35	i	i	PROPN
ejpam-5066	344	36	∈	∈	PROPN
ejpam-5066	344	37	λ	λ	PROPN
ejpam-5066	344	38	.	.	PUNCT
ejpam-5066	345	1	therefore	therefore	ADV
ejpam-5066	345	2	,	,	PUNCT
ejpam-5066	345	3	[	[	X
ejpam-5066	345	4	e	e	NOUN
ejpam-5066	345	5	,	,	PUNCT
ejpam-5066	345	6	r]ne	r]ne	NOUN
ejpam-5066	345	7	∈	∈	PROPN
ejpam-5066	345	8	⋂	⋂	PROPN
ejpam-5066	345	9	i∈λai	i∈λai	PROPN
ejpam-5066	345	10	=	=	PUNCT
ejpam-5066	345	11	0	0	NUM
ejpam-5066	345	12	and	and	CCONJ
ejpam-5066	345	13	e	e	PROPN
ejpam-5066	345	14	is	be	AUX
ejpam-5066	345	15	n	n	CCONJ
ejpam-5066	345	16	-	-	PUNCT
ejpam-5066	345	17	central	central	ADJ
ejpam-5066	345	18	.	.	PUNCT
ejpam-5066	346	1	motivated	motivate	VERB
ejpam-5066	346	2	by	by	ADP
ejpam-5066	346	3	[	[	X
ejpam-5066	346	4	11	11	NUM
ejpam-5066	346	5	,	,	PUNCT
ejpam-5066	346	6	section	section	NOUN
ejpam-5066	346	7	3	3	NUM
ejpam-5066	346	8	]	]	PUNCT
ejpam-5066	346	9	,	,	PUNCT
ejpam-5066	346	10	which	which	PRON
ejpam-5066	346	11	discusses	discuss	VERB
ejpam-5066	346	12	how	how	SCONJ
ejpam-5066	346	13	the	the	DET
ejpam-5066	346	14	abelianity	abelianity	NOUN
ejpam-5066	346	15	and	and	CCONJ
ejpam-5066	346	16	q	q	NOUN
ejpam-5066	346	17	-	-	PUNCT
ejpam-5066	346	18	abelianity	abelianity	NOUN
ejpam-5066	346	19	of	of	ADP
ejpam-5066	346	20	rings	ring	NOUN
ejpam-5066	346	21	can	can	AUX
ejpam-5066	346	22	be	be	AUX
ejpam-5066	346	23	determined	determine	VERB
ejpam-5066	346	24	by	by	ADP
ejpam-5066	346	25	their	their	PRON
ejpam-5066	346	26	upper	upper	ADJ
ejpam-5066	346	27	triangular	triangular	NOUN
ejpam-5066	346	28	matrices	matrix	NOUN
ejpam-5066	346	29	,	,	PUNCT
ejpam-5066	346	30	we	we	PRON
ejpam-5066	346	31	present	present	VERB
ejpam-5066	346	32	a	a	DET
ejpam-5066	346	33	generalization	generalization	NOUN
ejpam-5066	346	34	m.	m.	NOUN
ejpam-5066	346	35	saad	saad	PROPN
ejpam-5066	346	36	,	,	PUNCT
ejpam-5066	346	37	m.	m.	NOUN
ejpam-5066	346	38	zailaee	zailaee	PROPN
ejpam-5066	346	39	/	/	SYM
ejpam-5066	346	40	eur	eur	PROPN
ejpam-5066	346	41	.	.	PUNCT
ejpam-5066	347	1	j.	j.	PROPN
ejpam-5066	347	2	pure	pure	PROPN
ejpam-5066	347	3	appl	appl	PROPN
ejpam-5066	347	4	.	.	PROPN
ejpam-5066	347	5	math	math	PROPN
ejpam-5066	347	6	,	,	PUNCT
ejpam-5066	347	7	17	17	NUM
ejpam-5066	347	8	(	(	PUNCT
ejpam-5066	347	9	2	2	NUM
ejpam-5066	347	10	)	)	PUNCT
ejpam-5066	347	11	(	(	PUNCT
ejpam-5066	347	12	2024	2024	NUM
ejpam-5066	347	13	)	)	PUNCT
ejpam-5066	347	14	,	,	PUNCT
ejpam-5066	347	15	736	736	NUM
ejpam-5066	347	16	-	-	SYM
ejpam-5066	347	17	752	752	NUM
ejpam-5066	347	18	746	746	NUM
ejpam-5066	347	19	of	of	ADP
ejpam-5066	347	20	[	[	X
ejpam-5066	347	21	11	11	NUM
ejpam-5066	347	22	,	,	PUNCT
ejpam-5066	347	23	theorem	theorem	VERB
ejpam-5066	347	24	3.4	3.4	NUM
ejpam-5066	347	25	]	]	PUNCT
ejpam-5066	347	26	.	.	PUNCT
ejpam-5066	348	1	our	our	PRON
ejpam-5066	348	2	approach	approach	NOUN
ejpam-5066	348	3	combines	combine	VERB
ejpam-5066	348	4	the	the	DET
ejpam-5066	348	5	order	order	NOUN
ejpam-5066	348	6	of	of	ADP
ejpam-5066	348	7	the	the	DET
ejpam-5066	348	8	upper	upper	ADJ
ejpam-5066	348	9	triangular	triangular	NOUN
ejpam-5066	348	10	matrix	matrix	NOUN
ejpam-5066	348	11	with	with	ADP
ejpam-5066	348	12	the	the	DET
ejpam-5066	348	13	order	order	NOUN
ejpam-5066	348	14	of	of	ADP
ejpam-5066	348	15	abelianity	abelianity	NOUN
ejpam-5066	348	16	.	.	PUNCT
ejpam-5066	349	1	we	we	PRON
ejpam-5066	349	2	begin	begin	VERB
ejpam-5066	349	3	by	by	ADP
ejpam-5066	349	4	establishing	establish	VERB
ejpam-5066	349	5	the	the	DET
ejpam-5066	349	6	following	follow	VERB
ejpam-5066	349	7	lemma	lemma	PROPN
ejpam-5066	349	8	,	,	PUNCT
ejpam-5066	349	9	which	which	PRON
ejpam-5066	349	10	will	will	AUX
ejpam-5066	349	11	be	be	AUX
ejpam-5066	349	12	used	use	VERB
ejpam-5066	349	13	in	in	ADP
ejpam-5066	349	14	the	the	DET
ejpam-5066	349	15	proof	proof	NOUN
ejpam-5066	349	16	of	of	ADP
ejpam-5066	349	17	our	our	PRON
ejpam-5066	349	18	main	main	ADJ
ejpam-5066	349	19	result	result	NOUN
ejpam-5066	349	20	.	.	PUNCT
ejpam-5066	350	1	lemma	lemma	PROPN
ejpam-5066	350	2	2	2	X
ejpam-5066	350	3	.	.	PUNCT
ejpam-5066	351	1	let	let	VERB
ejpam-5066	351	2	the	the	DET
ejpam-5066	351	3	ring	ring	NOUN
ejpam-5066	351	4	t	t	NOUN
ejpam-5066	352	1	=	=	PUNCT
ejpam-5066	352	2	[	[	PUNCT
ejpam-5066	352	3	r	r	NOUN
ejpam-5066	352	4	m	m	VERB
ejpam-5066	352	5	0	0	NUM
ejpam-5066	352	6	s	s	PART
ejpam-5066	352	7	]	]	PUNCT
ejpam-5066	352	8	,	,	PUNCT
ejpam-5066	352	9	for	for	ADP
ejpam-5066	352	10	any	any	DET
ejpam-5066	352	11	rings	ring	NOUN
ejpam-5066	352	12	r	r	NOUN
ejpam-5066	352	13	,	,	PUNCT
ejpam-5066	352	14	and	and	CCONJ
ejpam-5066	352	15	s	s	NOUN
ejpam-5066	352	16	,	,	PUNCT
ejpam-5066	352	17	and	and	CCONJ
ejpam-5066	352	18	a	a	DET
ejpam-5066	352	19	unital	unital	ADJ
ejpam-5066	352	20	(	(	PUNCT
ejpam-5066	352	21	r	r	NOUN
ejpam-5066	352	22	,	,	PUNCT
ejpam-5066	352	23	s)bimodule	s)bimodule	NOUN
ejpam-5066	352	24	m	m	PRON
ejpam-5066	352	25	,	,	PUNCT
ejpam-5066	352	26	be	be	AUX
ejpam-5066	352	27	n	n	PRON
ejpam-5066	352	28	-	-	PUNCT
ejpam-5066	352	29	abelian	abelian	ADJ
ejpam-5066	352	30	,	,	PUNCT
ejpam-5066	352	31	for	for	ADP
ejpam-5066	352	32	some	some	DET
ejpam-5066	352	33	n.	n.	NOUN
ejpam-5066	352	34	if	if	SCONJ
ejpam-5066	352	35	r	r	NOUN
ejpam-5066	352	36	and	and	CCONJ
ejpam-5066	352	37	s	s	NOUN
ejpam-5066	352	38	are	be	AUX
ejpam-5066	352	39	respectively	respectively	ADV
ejpam-5066	352	40	n1	n1	NOUN
ejpam-5066	352	41	-	-	PUNCT
ejpam-5066	352	42	abelian	abelian	NOUN
ejpam-5066	352	43	and	and	CCONJ
ejpam-5066	352	44	n2abelian	n2abelian	ADJ
ejpam-5066	352	45	rings	ring	NOUN
ejpam-5066	352	46	,	,	PUNCT
ejpam-5066	352	47	then	then	ADV
ejpam-5066	352	48	the	the	DET
ejpam-5066	352	49	ring	ring	NOUN
ejpam-5066	352	50	t	t	PROPN
ejpam-5066	352	51	is	be	AUX
ejpam-5066	352	52	(	(	PUNCT
ejpam-5066	352	53	n1	n1	PROPN
ejpam-5066	352	54	+	+	CCONJ
ejpam-5066	352	55	n2)-abelian	n2)-abelian	ADJ
ejpam-5066	352	56	.	.	PUNCT
ejpam-5066	353	1	while	while	SCONJ
ejpam-5066	353	2	,	,	PUNCT
ejpam-5066	353	3	if	if	SCONJ
ejpam-5066	353	4	t	t	PROPN
ejpam-5066	353	5	is	be	AUX
ejpam-5066	353	6	n	n	CCONJ
ejpam-5066	353	7	-	-	PUNCT
ejpam-5066	353	8	abelian	abelian	ADJ
ejpam-5066	353	9	,	,	PUNCT
ejpam-5066	353	10	then	then	ADV
ejpam-5066	353	11	both	both	DET
ejpam-5066	353	12	r	r	NOUN
ejpam-5066	353	13	and	and	CCONJ
ejpam-5066	353	14	s	s	NOUN
ejpam-5066	353	15	are	be	AUX
ejpam-5066	353	16	(	(	PUNCT
ejpam-5066	353	17	n−	n−	NOUN
ejpam-5066	353	18	1)-abelian	1)-abelian	NOUN
ejpam-5066	353	19	.	.	PUNCT
ejpam-5066	354	1	proof	proof	NOUN
ejpam-5066	354	2	.	.	PUNCT
ejpam-5066	355	1	first	first	ADV
ejpam-5066	355	2	,	,	PUNCT
ejpam-5066	355	3	assume	assume	VERB
ejpam-5066	355	4	that	that	SCONJ
ejpam-5066	355	5	r	r	NOUN
ejpam-5066	355	6	and	and	CCONJ
ejpam-5066	355	7	s	s	NOUN
ejpam-5066	355	8	are	be	AUX
ejpam-5066	355	9	n1	n1	NOUN
ejpam-5066	355	10	-	-	PUNCT
ejpam-5066	355	11	abelian	abelian	NOUN
ejpam-5066	355	12	and	and	CCONJ
ejpam-5066	355	13	n2	n2	ADJ
ejpam-5066	355	14	-	-	PUNCT
ejpam-5066	355	15	abelian	abelian	NOUN
ejpam-5066	355	16	,	,	PUNCT
ejpam-5066	355	17	for	for	ADP
ejpam-5066	355	18	some	some	DET
ejpam-5066	355	19	n1	n1	NOUN
ejpam-5066	355	20	and	and	CCONJ
ejpam-5066	355	21	n2	n2	ADJ
ejpam-5066	355	22	,	,	PUNCT
ejpam-5066	355	23	respectively	respectively	ADV
ejpam-5066	355	24	.	.	PUNCT
ejpam-5066	356	1	for	for	ADP
ejpam-5066	356	2	an	an	DET
ejpam-5066	356	3	arbitrary	arbitrary	ADJ
ejpam-5066	356	4	idempotent	idempotent	NOUN
ejpam-5066	356	5	ε	ε	X
ejpam-5066	356	6	=	=	PUNCT
ejpam-5066	356	7	[	[	PUNCT
ejpam-5066	356	8	e	e	X
ejpam-5066	356	9	s	s	NOUN
ejpam-5066	356	10	0	0	NUM
ejpam-5066	356	11	f	f	X
ejpam-5066	356	12	]	]	PUNCT
ejpam-5066	356	13	of	of	ADP
ejpam-5066	356	14	t	t	PROPN
ejpam-5066	356	15	,	,	PUNCT
ejpam-5066	356	16	e	e	PROPN
ejpam-5066	356	17	,	,	PUNCT
ejpam-5066	356	18	f	f	PROPN
ejpam-5066	356	19	∈	∈	PROPN
ejpam-5066	356	20	i(r	i(r	PROPN
ejpam-5066	356	21	)	)	PUNCT
ejpam-5066	356	22	and	and	CCONJ
ejpam-5066	356	23	es+sf	es+sf	PRON
ejpam-5066	356	24	=	=	SYM
ejpam-5066	356	25	s.	s.	PROPN
ejpam-5066	356	26	indeed	indeed	ADV
ejpam-5066	356	27	,	,	PUNCT
ejpam-5066	356	28	[	[	X
ejpam-5066	356	29	ε	ε	PROPN
ejpam-5066	356	30	,	,	PUNCT
ejpam-5066	356	31	t	t	X
ejpam-5066	356	32	]	]	PUNCT
ejpam-5066	356	33	⊆	⊆	NUM
ejpam-5066	356	34	[	[	PUNCT
ejpam-5066	356	35	[	[	X
ejpam-5066	356	36	e	e	NOUN
ejpam-5066	356	37	,	,	PUNCT
ejpam-5066	356	38	r	r	X
ejpam-5066	356	39	]	]	X
ejpam-5066	356	40	m	m	VERB
ejpam-5066	356	41	0	0	PUNCT
ejpam-5066	357	1	[	[	X
ejpam-5066	357	2	f	f	X
ejpam-5066	357	3	,	,	PUNCT
ejpam-5066	357	4	s	s	X
ejpam-5066	357	5	]	]	X
ejpam-5066	357	6	]	]	PUNCT
ejpam-5066	357	7	and	and	CCONJ
ejpam-5066	357	8	[	[	X
ejpam-5066	357	9	ε	ε	PROPN
ejpam-5066	357	10	,	,	PUNCT
ejpam-5066	357	11	t	t	X
ejpam-5066	357	12	]	]	PUNCT
ejpam-5066	357	13	n1+n2	n1+n2	PROPN
ejpam-5066	357	14	⊆	⊆	NUM
ejpam-5066	357	15	[	[	PUNCT
ejpam-5066	357	16	0	0	NUM
ejpam-5066	357	17	m	m	NOUN
ejpam-5066	357	18	0	0	NUM
ejpam-5066	358	1	[	[	X
ejpam-5066	358	2	f	f	X
ejpam-5066	358	3	,	,	PUNCT
ejpam-5066	358	4	s]n1	s]n1	VERB
ejpam-5066	358	5	]	]	PUNCT
ejpam-5066	358	6	[	[	PUNCT
ejpam-5066	358	7	[	[	X
ejpam-5066	358	8	e	e	NOUN
ejpam-5066	358	9	,	,	PUNCT
ejpam-5066	358	10	r]n2	r]n2	ADP
ejpam-5066	358	11	m	m	PROPN
ejpam-5066	358	12	0	0	NUM
ejpam-5066	358	13	0	0	NUM
ejpam-5066	358	14	]	]	PUNCT
ejpam-5066	359	1	=	=	SYM
ejpam-5066	359	2	0	0	NUM
ejpam-5066	359	3	and	and	CCONJ
ejpam-5066	359	4	ε	ε	PROPN
ejpam-5066	359	5	∈	∈	PROPN
ejpam-5066	359	6	c(n1+n2)(t	c(n1+n2)(t	PROPN
ejpam-5066	359	7	)	)	PUNCT
ejpam-5066	359	8	.	.	PUNCT
ejpam-5066	360	1	since	since	SCONJ
ejpam-5066	360	2	ε	ε	PROPN
ejpam-5066	360	3	∈	∈	PROPN
ejpam-5066	360	4	i(t	i(t	PROPN
ejpam-5066	360	5	)	)	PUNCT
ejpam-5066	360	6	is	be	AUX
ejpam-5066	360	7	arbitrary	arbitrary	ADJ
ejpam-5066	360	8	,	,	PUNCT
ejpam-5066	360	9	this	this	PRON
ejpam-5066	360	10	shows	show	VERB
ejpam-5066	360	11	that	that	SCONJ
ejpam-5066	360	12	t	t	PROPN
ejpam-5066	360	13	is	be	AUX
ejpam-5066	360	14	(	(	PUNCT
ejpam-5066	360	15	n1	n1	PROPN
ejpam-5066	360	16	+	+	X
ejpam-5066	360	17	n2)-abelian	n2)-abelian	ADJ
ejpam-5066	360	18	.	.	PUNCT
ejpam-5066	361	1	secondly	secondly	ADV
ejpam-5066	361	2	,	,	PUNCT
ejpam-5066	361	3	if	if	SCONJ
ejpam-5066	361	4	t	t	PROPN
ejpam-5066	361	5	is	be	AUX
ejpam-5066	361	6	n	n	CCONJ
ejpam-5066	361	7	-	-	PUNCT
ejpam-5066	361	8	abelian	abelian	NOUN
ejpam-5066	361	9	and	and	CCONJ
ejpam-5066	361	10	e2	e2	PROPN
ejpam-5066	361	11	=	=	SYM
ejpam-5066	362	1	e	e	PROPN
ejpam-5066	362	2	,	,	PUNCT
ejpam-5066	362	3	r	r	NOUN
ejpam-5066	362	4	,	,	PUNCT
ejpam-5066	362	5	r1	r1	NOUN
ejpam-5066	362	6	,	,	PUNCT
ejpam-5066	362	7	r2	r2	PROPN
ejpam-5066	362	8	,	,	PUNCT
ejpam-5066	362	9	·	·	PUNCT
ejpam-5066	362	10	·	·	PUNCT
ejpam-5066	362	11	·	·	PUNCT
ejpam-5066	362	12	,	,	PUNCT
ejpam-5066	362	13	rn−1	rn−1	PROPN
ejpam-5066	362	14	∈	∈	PROPN
ejpam-5066	362	15	r	r	NOUN
ejpam-5066	362	16	,	,	PUNCT
ejpam-5066	362	17	define	define	VERB
ejpam-5066	362	18	the	the	DET
ejpam-5066	362	19	elements	element	NOUN
ejpam-5066	362	20	ϵ	ϵ	X
ejpam-5066	363	1	=	=	PUNCT
ejpam-5066	363	2	[	[	PUNCT
ejpam-5066	363	3	e	e	NOUN
ejpam-5066	363	4	0	0	NUM
ejpam-5066	363	5	0	0	NUM
ejpam-5066	363	6	1	1	NUM
ejpam-5066	363	7	]	]	PUNCT
ejpam-5066	363	8	,	,	PUNCT
ejpam-5066	363	9	t	t	PROPN
ejpam-5066	363	10	=	=	PUNCT
ejpam-5066	364	1	[	[	PUNCT
ejpam-5066	364	2	0	0	NUM
ejpam-5066	364	3	1	1	NUM
ejpam-5066	364	4	0	0	NUM
ejpam-5066	364	5	0	0	NUM
ejpam-5066	364	6	]	]	PUNCT
ejpam-5066	364	7	,	,	PUNCT
ejpam-5066	364	8	and	and	CCONJ
ejpam-5066	364	9	tk	tk	PROPN
ejpam-5066	364	10	=	=	PRON
ejpam-5066	364	11	[	[	PUNCT
ejpam-5066	364	12	rk	rk	NOUN
ejpam-5066	364	13	0	0	NUM
ejpam-5066	364	14	0	0	NUM
ejpam-5066	364	15	0	0	NUM
ejpam-5066	364	16	]	]	PUNCT
ejpam-5066	364	17	,	,	PUNCT
ejpam-5066	364	18	for	for	ADP
ejpam-5066	364	19	k	k	PROPN
ejpam-5066	364	20	=	=	SYM
ejpam-5066	364	21	1	1	NUM
ejpam-5066	364	22	,	,	PUNCT
ejpam-5066	364	23	2	2	NUM
ejpam-5066	364	24	,	,	PUNCT
ejpam-5066	364	25	·	·	PUNCT
ejpam-5066	364	26	·	·	PUNCT
ejpam-5066	364	27	·	·	PUNCT
ejpam-5066	364	28	,	,	PUNCT
ejpam-5066	364	29	n	n	CCONJ
ejpam-5066	364	30	−	−	PROPN
ejpam-5066	364	31	1	1	NUM
ejpam-5066	364	32	in	in	ADP
ejpam-5066	364	33	t	t	PROPN
ejpam-5066	364	34	.	.	PUNCT
ejpam-5066	365	1	if	if	SCONJ
ejpam-5066	365	2	t	t	PROPN
ejpam-5066	365	3	is	be	AUX
ejpam-5066	365	4	nabelian	nabelian	NOUN
ejpam-5066	365	5	,	,	PUNCT
ejpam-5066	365	6	then	then	ADV
ejpam-5066	365	7	0	0	NUM
ejpam-5066	366	1	=	=	PUNCT
ejpam-5066	367	1	[	[	X
ejpam-5066	367	2	ϵ	ϵ	X
ejpam-5066	367	3	,	,	PUNCT
ejpam-5066	367	4	t1	t1	PROPN
ejpam-5066	367	5	]	]	X
ejpam-5066	367	6	·	·	PUNCT
ejpam-5066	367	7	·	·	PUNCT
ejpam-5066	367	8	·	·	PUNCT
ejpam-5066	368	1	[	[	X
ejpam-5066	368	2	ϵ	ϵ	X
ejpam-5066	368	3	,	,	PUNCT
ejpam-5066	368	4	tn−1][ϵ	tn−1][ϵ	PROPN
ejpam-5066	368	5	,	,	PUNCT
ejpam-5066	368	6	t	t	PROPN
ejpam-5066	368	7	]	]	X
ejpam-5066	368	8	=	=	PUNCT
ejpam-5066	369	1	[	[	PUNCT
ejpam-5066	369	2	[	[	X
ejpam-5066	369	3	e	e	NOUN
ejpam-5066	369	4	,	,	PUNCT
ejpam-5066	369	5	r1	r1	NOUN
ejpam-5066	369	6	]	]	PUNCT
ejpam-5066	369	7	0	0	NUM
ejpam-5066	369	8	0	0	NUM
ejpam-5066	369	9	0	0	NUM
ejpam-5066	369	10	]	]	PUNCT
ejpam-5066	369	11	·	·	PUNCT
ejpam-5066	369	12	·	·	PUNCT
ejpam-5066	369	13	·	·	PUNCT
ejpam-5066	370	1	[	[	PUNCT
ejpam-5066	370	2	[	[	X
ejpam-5066	370	3	e	e	NOUN
ejpam-5066	370	4	,	,	PUNCT
ejpam-5066	370	5	rn−1	rn−1	PROPN
ejpam-5066	370	6	]	]	PUNCT
ejpam-5066	370	7	0	0	NUM
ejpam-5066	370	8	0	0	NUM
ejpam-5066	370	9	0	0	NUM
ejpam-5066	370	10	]	]	PUNCT
ejpam-5066	371	1	[	[	PUNCT
ejpam-5066	371	2	0	0	NUM
ejpam-5066	371	3	e−	e−	PROPN
ejpam-5066	371	4	1	1	NUM
ejpam-5066	371	5	0	0	NUM
ejpam-5066	371	6	0	0	NUM
ejpam-5066	371	7	]	]	PUNCT
ejpam-5066	371	8	=[	=[	NOUN
ejpam-5066	371	9	0	0	PUNCT
ejpam-5066	372	1	[	[	X
ejpam-5066	372	2	e	e	X
ejpam-5066	372	3	,	,	PUNCT
ejpam-5066	372	4	r1	r1	PROPN
ejpam-5066	372	5	]	]	PUNCT
ejpam-5066	372	6	·	·	PUNCT
ejpam-5066	372	7	·	·	PUNCT
ejpam-5066	372	8	·	·	PUNCT
ejpam-5066	373	1	[	[	X
ejpam-5066	373	2	e	e	X
ejpam-5066	373	3	,	,	PUNCT
ejpam-5066	373	4	rn−1](e−	rn−1](e−	NOUN
ejpam-5066	373	5	1	1	NUM
ejpam-5066	373	6	)	)	PUNCT
ejpam-5066	373	7	0	0	NUM
ejpam-5066	373	8	0	0	NUM
ejpam-5066	373	9	]	]	PUNCT
ejpam-5066	373	10	.	.	PUNCT
ejpam-5066	374	1	so	so	ADV
ejpam-5066	374	2	,	,	PUNCT
ejpam-5066	374	3	[	[	X
ejpam-5066	374	4	e	e	NOUN
ejpam-5066	374	5	,	,	PUNCT
ejpam-5066	374	6	r1	r1	PROPN
ejpam-5066	374	7	]	]	PUNCT
ejpam-5066	374	8	·	·	PUNCT
ejpam-5066	374	9	·	·	PUNCT
ejpam-5066	374	10	·	·	PUNCT
ejpam-5066	375	1	[	[	X
ejpam-5066	375	2	e	e	X
ejpam-5066	375	3	,	,	PUNCT
ejpam-5066	375	4	rn−1](e−	rn−1](e−	NOUN
ejpam-5066	375	5	1	1	NUM
ejpam-5066	375	6	)	)	PUNCT
ejpam-5066	375	7	=	=	SYM
ejpam-5066	375	8	0	0	NUM
ejpam-5066	376	1	and	and	CCONJ
ejpam-5066	376	2	[	[	X
ejpam-5066	376	3	e	e	NOUN
ejpam-5066	376	4	,	,	PUNCT
ejpam-5066	376	5	r]n−1(1−	r]n−1(1−	NOUN
ejpam-5066	376	6	e	e	NOUN
ejpam-5066	376	7	)	)	PUNCT
ejpam-5066	376	8	=	=	SYM
ejpam-5066	376	9	0	0	X
ejpam-5066	376	10	.	.	PUNCT
ejpam-5066	377	1	therefore	therefore	ADV
ejpam-5066	377	2	,	,	PUNCT
ejpam-5066	377	3	e	e	PROPN
ejpam-5066	377	4	∈	∈	PROPN
ejpam-5066	377	5	cn−1(r	cn−1(r	NOUN
ejpam-5066	377	6	)	)	PUNCT
ejpam-5066	377	7	and	and	CCONJ
ejpam-5066	377	8	r	r	NOUN
ejpam-5066	377	9	is	be	AUX
ejpam-5066	377	10	(	(	PUNCT
ejpam-5066	377	11	n	n	CCONJ
ejpam-5066	377	12	−	−	PROPN
ejpam-5066	377	13	1)-abelian	1)-abelian	NUM
ejpam-5066	377	14	since	since	SCONJ
ejpam-5066	377	15	e	e	NOUN
ejpam-5066	377	16	is	be	AUX
ejpam-5066	377	17	arbitrary	arbitrary	ADJ
ejpam-5066	377	18	in	in	ADP
ejpam-5066	377	19	i(r	i(r	PROPN
ejpam-5066	377	20	)	)	PUNCT
ejpam-5066	377	21	.	.	PUNCT
ejpam-5066	378	1	similarly	similarly	ADV
ejpam-5066	378	2	,	,	PUNCT
ejpam-5066	378	3	one	one	PRON
ejpam-5066	378	4	can	can	AUX
ejpam-5066	378	5	verify	verify	VERB
ejpam-5066	378	6	that	that	SCONJ
ejpam-5066	378	7	s	s	VERB
ejpam-5066	378	8	is	be	AUX
ejpam-5066	378	9	also	also	ADV
ejpam-5066	378	10	(	(	PUNCT
ejpam-5066	378	11	n−	n−	NOUN
ejpam-5066	378	12	1)-abelian	1)-abelian	NOUN
ejpam-5066	378	13	.	.	PUNCT
ejpam-5066	379	1	in	in	ADP
ejpam-5066	379	2	example	example	NOUN
ejpam-5066	379	3	4.7	4.7	NUM
ejpam-5066	379	4	of	of	ADP
ejpam-5066	379	5	lam	lam	PROPN
ejpam-5066	379	6	’s	’s	PART
ejpam-5066	379	7	work	work	NOUN
ejpam-5066	380	1	[	[	X
ejpam-5066	380	2	12	12	NUM
ejpam-5066	380	3	]	]	PUNCT
ejpam-5066	380	4	,	,	PUNCT
ejpam-5066	380	5	it	it	PRON
ejpam-5066	380	6	is	be	AUX
ejpam-5066	380	7	demonstrated	demonstrate	VERB
ejpam-5066	380	8	that	that	SCONJ
ejpam-5066	380	9	t3(r	t3(r	VERB
ejpam-5066	380	10	)	)	PUNCT
ejpam-5066	380	11	fails	fail	VERB
ejpam-5066	380	12	to	to	PART
ejpam-5066	380	13	exhibit	exhibit	VERB
ejpam-5066	380	14	semiabelian	semiabelian	ADJ
ejpam-5066	380	15	characteristics	characteristic	NOUN
ejpam-5066	380	16	.	.	PUNCT
ejpam-5066	381	1	expanding	expand	VERB
ejpam-5066	381	2	upon	upon	SCONJ
ejpam-5066	381	3	the	the	DET
ejpam-5066	381	4	findings	finding	NOUN
ejpam-5066	381	5	elucidated	elucidate	VERB
ejpam-5066	381	6	in	in	ADP
ejpam-5066	381	7	lam	lam	PROPN
ejpam-5066	381	8	’s	’s	PART
ejpam-5066	381	9	prior	prior	ADJ
ejpam-5066	381	10	works	work	NOUN
ejpam-5066	381	11	,	,	PUNCT
ejpam-5066	381	12	specifically	specifically	ADV
ejpam-5066	381	13	theorem	theorem	VERB
ejpam-5066	381	14	3.4	3.4	NUM
ejpam-5066	381	15	in	in	ADP
ejpam-5066	381	16	[	[	X
ejpam-5066	381	17	11	11	NUM
ejpam-5066	381	18	]	]	PUNCT
ejpam-5066	381	19	and	and	CCONJ
ejpam-5066	381	20	theorem	theorem	VERB
ejpam-5066	381	21	2.4	2.4	NUM
ejpam-5066	381	22	in	in	ADP
ejpam-5066	381	23	[	[	X
ejpam-5066	381	24	12	12	NUM
ejpam-5066	381	25	]	]	PUNCT
ejpam-5066	381	26	,	,	PUNCT
ejpam-5066	381	27	we	we	PRON
ejpam-5066	381	28	have	have	AUX
ejpam-5066	381	29	undertaken	undertake	VERB
ejpam-5066	381	30	a	a	DET
ejpam-5066	381	31	comprehensive	comprehensive	ADJ
ejpam-5066	381	32	exploration	exploration	NOUN
ejpam-5066	381	33	leading	lead	VERB
ejpam-5066	381	34	to	to	ADP
ejpam-5066	381	35	a	a	DET
ejpam-5066	381	36	noteworthy	noteworthy	ADJ
ejpam-5066	381	37	generalization	generalization	NOUN
ejpam-5066	381	38	,	,	PUNCT
ejpam-5066	381	39	which	which	PRON
ejpam-5066	381	40	we	we	PRON
ejpam-5066	381	41	present	present	VERB
ejpam-5066	381	42	in	in	ADP
ejpam-5066	381	43	the	the	DET
ejpam-5066	381	44	subsequent	subsequent	ADJ
ejpam-5066	381	45	theorem	theorem	NOUN
ejpam-5066	381	46	.	.	PUNCT
ejpam-5066	382	1	our	our	PRON
ejpam-5066	382	2	derived	derived	ADJ
ejpam-5066	382	3	result	result	NOUN
ejpam-5066	382	4	establishes	establish	VERB
ejpam-5066	382	5	a	a	DET
ejpam-5066	382	6	fundamental	fundamental	ADJ
ejpam-5066	382	7	link	link	NOUN
ejpam-5066	382	8	between	between	ADP
ejpam-5066	382	9	the	the	DET
ejpam-5066	382	10	abelian	abelian	ADJ
ejpam-5066	382	11	nature	nature	NOUN
ejpam-5066	382	12	of	of	ADP
ejpam-5066	382	13	a	a	DET
ejpam-5066	382	14	ring	ring	NOUN
ejpam-5066	382	15	and	and	CCONJ
ejpam-5066	382	16	the	the	DET
ejpam-5066	382	17	abelianity	abelianity	NOUN
ejpam-5066	382	18	observed	observe	VERB
ejpam-5066	382	19	within	within	ADP
ejpam-5066	382	20	its	its	PRON
ejpam-5066	382	21	corresponding	corresponding	ADJ
ejpam-5066	382	22	upper	upper	ADJ
ejpam-5066	382	23	triangular	triangular	NOUN
ejpam-5066	382	24	matrix	matrix	NOUN
ejpam-5066	382	25	extension	extension	NOUN
ejpam-5066	382	26	.	.	PUNCT
ejpam-5066	383	1	theorem	theorem	VERB
ejpam-5066	383	2	4	4	NUM
ejpam-5066	383	3	.	.	X
ejpam-5066	383	4	for	for	ADP
ejpam-5066	383	5	a	a	DET
ejpam-5066	383	6	ring	ring	NOUN
ejpam-5066	383	7	r	r	NOUN
ejpam-5066	383	8	,	,	PUNCT
ejpam-5066	383	9	the	the	DET
ejpam-5066	383	10	following	follow	VERB
ejpam-5066	383	11	conditions	condition	NOUN
ejpam-5066	383	12	are	be	AUX
ejpam-5066	383	13	equivalent	equivalent	ADJ
ejpam-5066	383	14	.	.	PUNCT
ejpam-5066	384	1	(	(	PUNCT
ejpam-5066	384	2	i	i	NOUN
ejpam-5066	384	3	)	)	PUNCT
ejpam-5066	384	4	r	r	NOUN
ejpam-5066	384	5	is	be	AUX
ejpam-5066	384	6	abelian	abelian	ADJ
ejpam-5066	384	7	.	.	PUNCT
ejpam-5066	385	1	(	(	PUNCT
ejpam-5066	385	2	ii	ii	NOUN
ejpam-5066	385	3	)	)	PUNCT
ejpam-5066	385	4	tn(r	tn(r	NUM
ejpam-5066	385	5	)	)	PUNCT
ejpam-5066	385	6	is	be	AUX
ejpam-5066	385	7	n	n	CCONJ
ejpam-5066	385	8	-	-	PUNCT
ejpam-5066	385	9	abelian	abelian	ADJ
ejpam-5066	385	10	,	,	PUNCT
ejpam-5066	385	11	for	for	SCONJ
ejpam-5066	385	12	every	every	DET
ejpam-5066	385	13	n.	n.	NOUN
ejpam-5066	385	14	(	(	PUNCT
ejpam-5066	385	15	iii	iii	NOUN
ejpam-5066	385	16	)	)	PUNCT
ejpam-5066	385	17	tn(r	tn(r	NUM
ejpam-5066	385	18	)	)	PUNCT
ejpam-5066	385	19	is	be	AUX
ejpam-5066	385	20	n	n	CCONJ
ejpam-5066	385	21	-	-	PUNCT
ejpam-5066	385	22	abelian	abelian	ADJ
ejpam-5066	385	23	,	,	PUNCT
ejpam-5066	385	24	for	for	ADP
ejpam-5066	385	25	some	some	DET
ejpam-5066	385	26	n.	n.	NOUN
ejpam-5066	385	27	proof	proof	NOUN
ejpam-5066	385	28	.	.	PUNCT
ejpam-5066	386	1	(	(	PUNCT
ejpam-5066	386	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-5066	386	3	):	):	PUNCT
ejpam-5066	386	4	assume	assume	VERB
ejpam-5066	386	5	that	that	SCONJ
ejpam-5066	386	6	r	r	NOUN
ejpam-5066	386	7	is	be	AUX
ejpam-5066	386	8	abelian	abelian	ADJ
ejpam-5066	386	9	.	.	PUNCT
ejpam-5066	387	1	hence	hence	ADV
ejpam-5066	387	2	,	,	PUNCT
ejpam-5066	387	3	t2(r	t2(r	PROPN
ejpam-5066	387	4	)	)	PUNCT
ejpam-5066	387	5	is	be	AUX
ejpam-5066	387	6	2	2	NUM
ejpam-5066	387	7	-	-	PUNCT
ejpam-5066	387	8	abelian	abelian	NOUN
ejpam-5066	387	9	,	,	PUNCT
ejpam-5066	387	10	by	by	ADP
ejpam-5066	387	11	the	the	DET
ejpam-5066	387	12	previous	previous	ADJ
ejpam-5066	387	13	lemma	lemma	PROPN
ejpam-5066	387	14	.	.	PUNCT
ejpam-5066	388	1	also	also	ADV
ejpam-5066	388	2	,	,	PUNCT
ejpam-5066	388	3	t3(r	t3(r	NOUN
ejpam-5066	388	4	)	)	PUNCT
ejpam-5066	388	5	=	=	NOUN
ejpam-5066	388	6	[	[	PUNCT
ejpam-5066	388	7	r	r	NOUN
ejpam-5066	388	8	r2	r2	NOUN
ejpam-5066	388	9	0	0	PUNCT
ejpam-5066	388	10	t2(r	t2(r	PROPN
ejpam-5066	388	11	)	)	PUNCT
ejpam-5066	388	12	]	]	PUNCT
ejpam-5066	388	13	,	,	PUNCT
ejpam-5066	388	14	where	where	SCONJ
ejpam-5066	388	15	r2	r2	PROPN
ejpam-5066	388	16	is	be	AUX
ejpam-5066	388	17	a	a	DET
ejpam-5066	388	18	unital	unital	ADJ
ejpam-5066	388	19	(	(	PUNCT
ejpam-5066	388	20	r	r	NOUN
ejpam-5066	388	21	,	,	PUNCT
ejpam-5066	388	22	t2(r))-bimodule	t2(r))-bimodule	NOUN
ejpam-5066	388	23	.	.	PUNCT
ejpam-5066	389	1	again	again	ADV
ejpam-5066	389	2	,	,	PUNCT
ejpam-5066	389	3	t3(r	t3(r	NOUN
ejpam-5066	389	4	)	)	PUNCT
ejpam-5066	389	5	is	be	AUX
ejpam-5066	389	6	3	3	NUM
ejpam-5066	389	7	-	-	PUNCT
ejpam-5066	389	8	abelian	abelian	NOUN
ejpam-5066	389	9	form	form	NOUN
ejpam-5066	389	10	the	the	DET
ejpam-5066	389	11	previous	previous	ADJ
ejpam-5066	389	12	lemma	lemma	PROPN
ejpam-5066	389	13	.	.	PUNCT
ejpam-5066	390	1	continuing	continue	VERB
ejpam-5066	390	2	,	,	PUNCT
ejpam-5066	390	3	we	we	PRON
ejpam-5066	390	4	have	have	VERB
ejpam-5066	390	5	tn(r	tn(r	ADV
ejpam-5066	390	6	)	)	PUNCT
ejpam-5066	390	7	is	be	AUX
ejpam-5066	390	8	n	n	CCONJ
ejpam-5066	390	9	-	-	PUNCT
ejpam-5066	390	10	abelian	abelian	ADJ
ejpam-5066	390	11	,	,	PUNCT
ejpam-5066	390	12	for	for	ADP
ejpam-5066	390	13	every	every	DET
ejpam-5066	390	14	n.	n.	NOUN
ejpam-5066	390	15	m.	m.	NOUN
ejpam-5066	390	16	saad	saad	PROPN
ejpam-5066	390	17	,	,	PUNCT
ejpam-5066	390	18	m.	m.	NOUN
ejpam-5066	390	19	zailaee	zailaee	PROPN
ejpam-5066	390	20	/	/	SYM
ejpam-5066	390	21	eur	eur	PROPN
ejpam-5066	390	22	.	.	PUNCT
ejpam-5066	391	1	j.	j.	PROPN
ejpam-5066	391	2	pure	pure	PROPN
ejpam-5066	391	3	appl	appl	PROPN
ejpam-5066	391	4	.	.	PROPN
ejpam-5066	391	5	math	math	PROPN
ejpam-5066	391	6	,	,	PUNCT
ejpam-5066	391	7	17	17	NUM
ejpam-5066	391	8	(	(	PUNCT
ejpam-5066	391	9	2	2	NUM
ejpam-5066	391	10	)	)	PUNCT
ejpam-5066	391	11	(	(	PUNCT
ejpam-5066	391	12	2024	2024	NUM
ejpam-5066	391	13	)	)	PUNCT
ejpam-5066	391	14	,	,	PUNCT
ejpam-5066	391	15	736	736	NUM
ejpam-5066	391	16	-	-	SYM
ejpam-5066	391	17	752	752	NUM
ejpam-5066	391	18	747	747	NUM
ejpam-5066	391	19	(	(	PUNCT
ejpam-5066	391	20	ii)⇒(iii	ii)⇒(iii	NOUN
ejpam-5066	391	21	)	)	PUNCT
ejpam-5066	391	22	is	be	AUX
ejpam-5066	391	23	direct	direct	ADJ
ejpam-5066	391	24	.	.	PUNCT
ejpam-5066	392	1	(	(	PUNCT
ejpam-5066	392	2	iii)⇒(i	iii)⇒(i	NOUN
ejpam-5066	392	3	):	):	PUNCT
ejpam-5066	392	4	let	let	AUX
ejpam-5066	392	5	tn(r	tn(r	NUM
ejpam-5066	392	6	)	)	PUNCT
ejpam-5066	392	7	be	be	AUX
ejpam-5066	392	8	n	n	ADV
ejpam-5066	392	9	-	-	PUNCT
ejpam-5066	392	10	abelian	abelian	ADJ
ejpam-5066	392	11	,	,	PUNCT
ejpam-5066	392	12	for	for	ADP
ejpam-5066	392	13	some	some	DET
ejpam-5066	392	14	n.	n.	NOUN
ejpam-5066	392	15	tn(r	tn(r	NOUN
ejpam-5066	392	16	)	)	PUNCT
ejpam-5066	393	1	=	=	PUNCT
ejpam-5066	394	1	[	[	PUNCT
ejpam-5066	394	2	r	r	NOUN
ejpam-5066	394	3	rn−1	rn−1	PROPN
ejpam-5066	394	4	0	0	NUM
ejpam-5066	394	5	tn−1(r	tn−1(r	NOUN
ejpam-5066	394	6	)	)	PUNCT
ejpam-5066	394	7	]	]	PUNCT
ejpam-5066	394	8	.	.	PUNCT
ejpam-5066	395	1	so	so	ADV
ejpam-5066	395	2	,	,	PUNCT
ejpam-5066	395	3	r	r	NOUN
ejpam-5066	395	4	and	and	CCONJ
ejpam-5066	395	5	tn−1(r	tn−1(r	NOUN
ejpam-5066	395	6	)	)	PUNCT
ejpam-5066	395	7	are	be	AUX
ejpam-5066	395	8	(	(	PUNCT
ejpam-5066	395	9	n−	n−	NOUN
ejpam-5066	395	10	1)-abelian	1)-abelian	NUM
ejpam-5066	395	11	.	.	PUNCT
ejpam-5066	396	1	inducting	induct	VERB
ejpam-5066	396	2	on	on	ADP
ejpam-5066	396	3	n	n	CCONJ
ejpam-5066	396	4	gives	give	VERB
ejpam-5066	396	5	r	r	NOUN
ejpam-5066	396	6	is	be	AUX
ejpam-5066	396	7	1	1	NUM
ejpam-5066	396	8	-	-	PUNCT
ejpam-5066	396	9	abelian	abelian	ADJ
ejpam-5066	396	10	,	,	PUNCT
ejpam-5066	396	11	corollary	corollary	NOUN
ejpam-5066	396	12	10	10	NUM
ejpam-5066	396	13	(	(	PUNCT
ejpam-5066	396	14	[	[	X
ejpam-5066	396	15	11	11	NUM
ejpam-5066	396	16	]	]	PUNCT
ejpam-5066	396	17	,	,	PUNCT
ejpam-5066	396	18	theorem	theorem	VERB
ejpam-5066	396	19	3.4	3.4	NUM
ejpam-5066	396	20	)	)	PUNCT
ejpam-5066	396	21	.	.	PUNCT
ejpam-5066	397	1	a	a	DET
ejpam-5066	397	2	ring	ring	NOUN
ejpam-5066	397	3	r	r	NOUN
ejpam-5066	397	4	is	be	AUX
ejpam-5066	397	5	abelian	abelian	ADJ
ejpam-5066	397	6	if	if	SCONJ
ejpam-5066	397	7	and	and	CCONJ
ejpam-5066	397	8	only	only	ADV
ejpam-5066	397	9	if	if	SCONJ
ejpam-5066	397	10	t2(r	t2(r	PROPN
ejpam-5066	397	11	)	)	PUNCT
ejpam-5066	397	12	is	be	AUX
ejpam-5066	397	13	q	q	ADJ
ejpam-5066	397	14	-	-	PUNCT
ejpam-5066	397	15	abelian	abelian	ADJ
ejpam-5066	397	16	.	.	PUNCT
ejpam-5066	398	1	4	4	X
ejpam-5066	398	2	.	.	X
ejpam-5066	398	3	applications	application	NOUN
ejpam-5066	398	4	in	in	ADP
ejpam-5066	398	5	ring	ring	NOUN
ejpam-5066	398	6	theory	theory	NOUN
ejpam-5066	398	7	,	,	PUNCT
ejpam-5066	398	8	several	several	ADJ
ejpam-5066	398	9	types	type	NOUN
ejpam-5066	398	10	of	of	ADP
ejpam-5066	398	11	regularity	regularity	NOUN
ejpam-5066	398	12	are	be	AUX
ejpam-5066	398	13	defined	define	VERB
ejpam-5066	398	14	for	for	ADP
ejpam-5066	398	15	elements	element	NOUN
ejpam-5066	398	16	in	in	ADP
ejpam-5066	398	17	a	a	DET
ejpam-5066	398	18	ring	ring	NOUN
ejpam-5066	398	19	.	.	PUNCT
ejpam-5066	399	1	an	an	DET
ejpam-5066	399	2	element	element	NOUN
ejpam-5066	399	3	a	a	PRON
ejpam-5066	399	4	in	in	ADP
ejpam-5066	399	5	a	a	DET
ejpam-5066	399	6	ring	ring	NOUN
ejpam-5066	399	7	r	r	NOUN
ejpam-5066	399	8	is	be	AUX
ejpam-5066	399	9	called	call	VERB
ejpam-5066	399	10	regular	regular	ADJ
ejpam-5066	399	11	(	(	PUNCT
ejpam-5066	399	12	in	in	ADP
ejpam-5066	399	13	the	the	DET
ejpam-5066	399	14	sense	sense	NOUN
ejpam-5066	399	15	of	of	ADP
ejpam-5066	399	16	von	von	PROPN
ejpam-5066	399	17	neumann	neumann	PROPN
ejpam-5066	399	18	)	)	PUNCT
ejpam-5066	399	19	if	if	SCONJ
ejpam-5066	399	20	there	there	PRON
ejpam-5066	399	21	exists	exist	VERB
ejpam-5066	399	22	an	an	DET
ejpam-5066	399	23	element	element	NOUN
ejpam-5066	399	24	b	b	PROPN
ejpam-5066	399	25	∈	∈	NOUN
ejpam-5066	399	26	r	r	NOUN
ejpam-5066	399	27	such	such	DET
ejpam-5066	399	28	that	that	SCONJ
ejpam-5066	399	29	a	a	DET
ejpam-5066	399	30	=	=	X
ejpam-5066	399	31	aba	aba	PROPN
ejpam-5066	399	32	.	.	PUNCT
ejpam-5066	400	1	here	here	ADV
ejpam-5066	400	2	,	,	PUNCT
ejpam-5066	400	3	the	the	DET
ejpam-5066	400	4	element	element	NOUN
ejpam-5066	400	5	b	b	PROPN
ejpam-5066	400	6	is	be	AUX
ejpam-5066	400	7	called	call	VERB
ejpam-5066	400	8	an	an	DET
ejpam-5066	400	9	inner	inner	ADJ
ejpam-5066	400	10	inverse	inverse	NOUN
ejpam-5066	400	11	of	of	ADP
ejpam-5066	400	12	a	a	PRON
ejpam-5066	400	13	,	,	PUNCT
ejpam-5066	400	14	and	and	CCONJ
ejpam-5066	400	15	i(a	i(a	NUM
ejpam-5066	400	16	)	)	PUNCT
ejpam-5066	400	17	denotes	denote	VERB
ejpam-5066	400	18	the	the	DET
ejpam-5066	400	19	set	set	NOUN
ejpam-5066	400	20	of	of	ADP
ejpam-5066	400	21	all	all	DET
ejpam-5066	400	22	inner	inner	ADJ
ejpam-5066	400	23	inverses	inverse	NOUN
ejpam-5066	400	24	of	of	ADP
ejpam-5066	400	25	a	a	PRON
ejpam-5066	400	26	in	in	ADP
ejpam-5066	400	27	r.	r.	NOUN
ejpam-5066	400	28	we	we	PRON
ejpam-5066	400	29	can	can	AUX
ejpam-5066	400	30	also	also	ADV
ejpam-5066	400	31	define	define	VERB
ejpam-5066	400	32	the	the	DET
ejpam-5066	400	33	left	left	ADJ
ejpam-5066	400	34	regularity	regularity	NOUN
ejpam-5066	400	35	and	and	CCONJ
ejpam-5066	400	36	right	right	ADJ
ejpam-5066	400	37	regularity	regularity	NOUN
ejpam-5066	400	38	of	of	ADP
ejpam-5066	400	39	an	an	DET
ejpam-5066	400	40	element	element	NOUN
ejpam-5066	400	41	a	a	PRON
ejpam-5066	400	42	in	in	ADP
ejpam-5066	400	43	a	a	DET
ejpam-5066	400	44	similar	similar	ADJ
ejpam-5066	400	45	manner	manner	NOUN
ejpam-5066	400	46	.	.	PUNCT
ejpam-5066	401	1	if	if	SCONJ
ejpam-5066	401	2	an	an	DET
ejpam-5066	401	3	element	element	NOUN
ejpam-5066	401	4	is	be	AUX
ejpam-5066	401	5	both	both	PRON
ejpam-5066	401	6	left	left	ADJ
ejpam-5066	401	7	and	and	CCONJ
ejpam-5066	401	8	right	right	ADV
ejpam-5066	401	9	regular	regular	ADV
ejpam-5066	401	10	,	,	PUNCT
ejpam-5066	401	11	it	it	PRON
ejpam-5066	401	12	is	be	AUX
ejpam-5066	401	13	called	call	VERB
ejpam-5066	401	14	strongly	strongly	ADV
ejpam-5066	401	15	regular	regular	ADJ
ejpam-5066	401	16	.	.	PUNCT
ejpam-5066	402	1	a	a	DET
ejpam-5066	402	2	ring	ring	NOUN
ejpam-5066	402	3	r	r	NOUN
ejpam-5066	402	4	is	be	AUX
ejpam-5066	402	5	called	call	VERB
ejpam-5066	402	6	regular	regular	ADJ
ejpam-5066	402	7	(	(	PUNCT
ejpam-5066	402	8	resp	resp	NOUN
ejpam-5066	402	9	.	.	PUNCT
ejpam-5066	403	1	strongly	strongly	ADV
ejpam-5066	403	2	regular	regular	ADJ
ejpam-5066	403	3	)	)	PUNCT
ejpam-5066	403	4	if	if	SCONJ
ejpam-5066	403	5	all	all	PRON
ejpam-5066	403	6	its	its	PRON
ejpam-5066	403	7	elements	element	NOUN
ejpam-5066	403	8	are	be	AUX
ejpam-5066	403	9	regular	regular	ADJ
ejpam-5066	403	10	(	(	PUNCT
ejpam-5066	403	11	resp	resp	NOUN
ejpam-5066	403	12	.	.	PUNCT
ejpam-5066	404	1	strongly	strongly	ADV
ejpam-5066	404	2	regular	regular	ADJ
ejpam-5066	404	3	)	)	PUNCT
ejpam-5066	404	4	.	.	PUNCT
ejpam-5066	405	1	it	it	PRON
ejpam-5066	405	2	is	be	AUX
ejpam-5066	405	3	well	well	ADV
ejpam-5066	405	4	-	-	PUNCT
ejpam-5066	405	5	known	know	VERB
ejpam-5066	405	6	that	that	SCONJ
ejpam-5066	405	7	a	a	DET
ejpam-5066	405	8	ring	ring	NOUN
ejpam-5066	405	9	r	r	NOUN
ejpam-5066	405	10	is	be	AUX
ejpam-5066	405	11	strongly	strongly	ADV
ejpam-5066	405	12	regular	regular	ADJ
ejpam-5066	405	13	if	if	SCONJ
ejpam-5066	406	1	and	and	CCONJ
ejpam-5066	406	2	only	only	ADV
ejpam-5066	406	3	if	if	SCONJ
ejpam-5066	406	4	it	it	PRON
ejpam-5066	406	5	is	be	AUX
ejpam-5066	406	6	regular	regular	ADJ
ejpam-5066	406	7	and	and	CCONJ
ejpam-5066	406	8	abelian	abelian	ADJ
ejpam-5066	406	9	.	.	PUNCT
ejpam-5066	407	1	in	in	ADP
ejpam-5066	407	2	addition	addition	NOUN
ejpam-5066	407	3	to	to	ADP
ejpam-5066	407	4	regularity	regularity	NOUN
ejpam-5066	407	5	,	,	PUNCT
ejpam-5066	407	6	we	we	PRON
ejpam-5066	407	7	can	can	AUX
ejpam-5066	407	8	define	define	VERB
ejpam-5066	407	9	π	π	NOUN
ejpam-5066	407	10	-	-	NOUN
ejpam-5066	407	11	regularity	regularity	NOUN
ejpam-5066	407	12	for	for	ADP
ejpam-5066	407	13	elements	element	NOUN
ejpam-5066	407	14	in	in	ADP
ejpam-5066	407	15	a	a	DET
ejpam-5066	407	16	ring	ring	NOUN
ejpam-5066	407	17	.	.	PUNCT
ejpam-5066	408	1	an	an	DET
ejpam-5066	408	2	element	element	NOUN
ejpam-5066	408	3	a	a	DET
ejpam-5066	408	4	∈	∈	NOUN
ejpam-5066	408	5	r	r	NOUN
ejpam-5066	408	6	is	be	AUX
ejpam-5066	408	7	called	call	VERB
ejpam-5066	408	8	π	π	PROPN
ejpam-5066	408	9	-	-	NOUN
ejpam-5066	408	10	regular	regular	ADJ
ejpam-5066	408	11	if	if	SCONJ
ejpam-5066	408	12	a	a	DET
ejpam-5066	408	13	∈	∈	NOUN
ejpam-5066	408	14	anrn	anrn	NOUN
ejpam-5066	408	15	for	for	ADP
ejpam-5066	408	16	some	some	DET
ejpam-5066	408	17	positive	positive	ADJ
ejpam-5066	408	18	integer	integer	NOUN
ejpam-5066	408	19	n	n	CCONJ
ejpam-5066	408	20	depending	depend	VERB
ejpam-5066	408	21	on	on	ADP
ejpam-5066	408	22	a.	a.	NOUN
ejpam-5066	408	23	if	if	SCONJ
ejpam-5066	408	24	an	an	DET
ejpam-5066	408	25	element	element	NOUN
ejpam-5066	408	26	a	a	PRON
ejpam-5066	408	27	is	be	AUX
ejpam-5066	408	28	both	both	PRON
ejpam-5066	408	29	π	π	NOUN
ejpam-5066	408	30	-	-	ADJ
ejpam-5066	408	31	regular	regular	ADJ
ejpam-5066	408	32	and	and	CCONJ
ejpam-5066	408	33	strongly	strongly	ADV
ejpam-5066	408	34	regular	regular	ADJ
ejpam-5066	408	35	,	,	PUNCT
ejpam-5066	408	36	it	it	PRON
ejpam-5066	408	37	is	be	AUX
ejpam-5066	408	38	called	call	VERB
ejpam-5066	408	39	strongly	strongly	ADV
ejpam-5066	408	40	π	π	NOUN
ejpam-5066	408	41	-	-	NOUN
ejpam-5066	408	42	regular	regular	ADJ
ejpam-5066	408	43	.	.	PUNCT
ejpam-5066	409	1	a	a	DET
ejpam-5066	409	2	ring	ring	NOUN
ejpam-5066	409	3	r	r	NOUN
ejpam-5066	409	4	is	be	AUX
ejpam-5066	409	5	called	call	VERB
ejpam-5066	409	6	π	π	PROPN
ejpam-5066	409	7	-	-	ADJ
ejpam-5066	409	8	regular	regular	ADJ
ejpam-5066	409	9	(	(	PUNCT
ejpam-5066	409	10	resp	resp	NOUN
ejpam-5066	409	11	.	.	PUNCT
ejpam-5066	410	1	strongly	strongly	ADV
ejpam-5066	410	2	π	π	X
ejpam-5066	410	3	-	-	NOUN
ejpam-5066	410	4	regular	regular	ADJ
ejpam-5066	410	5	)	)	PUNCT
ejpam-5066	410	6	if	if	SCONJ
ejpam-5066	410	7	all	all	DET
ejpam-5066	410	8	its	its	PRON
ejpam-5066	410	9	elements	element	NOUN
ejpam-5066	410	10	are	be	AUX
ejpam-5066	410	11	π	π	NOUN
ejpam-5066	410	12	-	-	ADJ
ejpam-5066	410	13	regular	regular	ADJ
ejpam-5066	410	14	(	(	PUNCT
ejpam-5066	410	15	resp	resp	NOUN
ejpam-5066	410	16	.	.	PUNCT
ejpam-5066	411	1	strongly	strongly	ADV
ejpam-5066	411	2	π	π	X
ejpam-5066	411	3	-	-	NOUN
ejpam-5066	411	4	regular	regular	ADJ
ejpam-5066	411	5	)	)	PUNCT
ejpam-5066	411	6	.	.	PUNCT
ejpam-5066	412	1	it	it	PRON
ejpam-5066	412	2	is	be	AUX
ejpam-5066	412	3	worth	worth	ADJ
ejpam-5066	412	4	noting	note	VERB
ejpam-5066	412	5	that	that	SCONJ
ejpam-5066	412	6	a	a	DET
ejpam-5066	412	7	ring	ring	NOUN
ejpam-5066	412	8	r	r	NOUN
ejpam-5066	412	9	is	be	AUX
ejpam-5066	412	10	strongly	strongly	ADV
ejpam-5066	412	11	π	π	NOUN
ejpam-5066	412	12	-	-	NOUN
ejpam-5066	412	13	regular	regular	ADJ
ejpam-5066	412	14	if	if	SCONJ
ejpam-5066	412	15	and	and	CCONJ
ejpam-5066	412	16	only	only	ADV
ejpam-5066	412	17	if	if	SCONJ
ejpam-5066	412	18	it	it	PRON
ejpam-5066	412	19	is	be	AUX
ejpam-5066	412	20	abelian	abelian	ADJ
ejpam-5066	412	21	and	and	CCONJ
ejpam-5066	412	22	π	π	NOUN
ejpam-5066	412	23	-	-	NOUN
ejpam-5066	412	24	regular	regular	ADJ
ejpam-5066	412	25	.	.	PUNCT
ejpam-5066	413	1	previous	previous	ADJ
ejpam-5066	413	2	research	research	NOUN
ejpam-5066	413	3	by	by	ADP
ejpam-5066	413	4	wei	wei	PROPN
ejpam-5066	413	5	and	and	CCONJ
ejpam-5066	413	6	li	li	PROPN
ejpam-5066	413	7	in	in	ADP
ejpam-5066	413	8	[	[	X
ejpam-5066	413	9	23	23	NUM
ejpam-5066	413	10	]	]	PUNCT
ejpam-5066	413	11	showed	show	VERB
ejpam-5066	413	12	that	that	SCONJ
ejpam-5066	413	13	for	for	ADP
ejpam-5066	413	14	a	a	DET
ejpam-5066	413	15	regular	regular	ADJ
ejpam-5066	413	16	(	(	PUNCT
ejpam-5066	413	17	resp	resp	NOUN
ejpam-5066	413	18	.	.	PUNCT
ejpam-5066	414	1	π	π	X
ejpam-5066	414	2	-	-	ADJ
ejpam-5066	414	3	regular	regular	ADJ
ejpam-5066	414	4	)	)	PUNCT
ejpam-5066	414	5	ring	ring	NOUN
ejpam-5066	414	6	,	,	PUNCT
ejpam-5066	414	7	being	be	AUX
ejpam-5066	414	8	quasi	quasi	ADJ
ejpam-5066	414	9	-	-	NOUN
ejpam-5066	414	10	abelian	abelian	ADJ
ejpam-5066	414	11	(	(	PUNCT
ejpam-5066	414	12	2	2	NUM
ejpam-5066	414	13	-	-	PUNCT
ejpam-5066	414	14	abelian	abelian	NOUN
ejpam-5066	414	15	)	)	PUNCT
ejpam-5066	414	16	is	be	AUX
ejpam-5066	414	17	enough	enough	ADJ
ejpam-5066	414	18	to	to	PART
ejpam-5066	414	19	become	become	VERB
ejpam-5066	414	20	strongly	strongly	ADV
ejpam-5066	414	21	regular	regular	ADJ
ejpam-5066	414	22	(	(	PUNCT
ejpam-5066	414	23	strongly	strongly	ADV
ejpam-5066	414	24	πregular	πregular	ADJ
ejpam-5066	414	25	)	)	PUNCT
ejpam-5066	414	26	.	.	PUNCT
ejpam-5066	415	1	moreover	moreover	ADV
ejpam-5066	415	2	,	,	PUNCT
ejpam-5066	415	3	lam	lam	PROPN
ejpam-5066	415	4	in	in	ADP
ejpam-5066	415	5	[	[	PUNCT
ejpam-5066	415	6	10	10	NUM
ejpam-5066	415	7	]	]	PUNCT
ejpam-5066	415	8	showed	show	VERB
ejpam-5066	415	9	that	that	SCONJ
ejpam-5066	415	10	if	if	SCONJ
ejpam-5066	415	11	the	the	DET
ejpam-5066	415	12	idempotent	idempotent	ADJ
ejpam-5066	415	13	ba	ba	NOUN
ejpam-5066	415	14	of	of	ADP
ejpam-5066	415	15	a	a	DET
ejpam-5066	415	16	regular	regular	ADJ
ejpam-5066	415	17	element	element	NOUN
ejpam-5066	415	18	a	a	DET
ejpam-5066	415	19	=	=	X
ejpam-5066	415	20	aba	aba	PROPN
ejpam-5066	415	21	is	be	AUX
ejpam-5066	415	22	q	q	ADJ
ejpam-5066	415	23	-	-	ADJ
ejpam-5066	415	24	central	central	ADJ
ejpam-5066	415	25	(	(	PUNCT
ejpam-5066	415	26	2	2	NUM
ejpam-5066	415	27	-	-	NUM
ejpam-5066	415	28	central	central	ADJ
ejpam-5066	415	29	)	)	PUNCT
ejpam-5066	415	30	,	,	PUNCT
ejpam-5066	415	31	then	then	ADV
ejpam-5066	415	32	a	a	PRON
ejpam-5066	415	33	is	be	AUX
ejpam-5066	415	34	strongly	strongly	ADV
ejpam-5066	415	35	regular	regular	ADJ
ejpam-5066	415	36	.	.	PUNCT
ejpam-5066	416	1	in	in	ADP
ejpam-5066	416	2	the	the	DET
ejpam-5066	416	3	following	following	NOUN
ejpam-5066	416	4	,	,	PUNCT
ejpam-5066	416	5	we	we	PRON
ejpam-5066	416	6	present	present	VERB
ejpam-5066	416	7	a	a	DET
ejpam-5066	416	8	weak	weak	ADJ
ejpam-5066	416	9	condition	condition	NOUN
ejpam-5066	416	10	that	that	PRON
ejpam-5066	416	11	guarantees	guarantee	VERB
ejpam-5066	416	12	a	a	DET
ejpam-5066	416	13	regular	regular	ADJ
ejpam-5066	416	14	element	element	NOUN
ejpam-5066	416	15	to	to	PART
ejpam-5066	416	16	be	be	AUX
ejpam-5066	416	17	strongly	strongly	ADV
ejpam-5066	416	18	regular	regular	ADJ
ejpam-5066	416	19	.	.	PUNCT
ejpam-5066	417	1	theorem	theorem	NOUN
ejpam-5066	417	2	5	5	NUM
ejpam-5066	417	3	.	.	PUNCT
ejpam-5066	418	1	let	let	VERB
ejpam-5066	418	2	a	a	PRON
ejpam-5066	418	3	be	be	AUX
ejpam-5066	418	4	a	a	DET
ejpam-5066	418	5	regular	regular	ADJ
ejpam-5066	418	6	element	element	NOUN
ejpam-5066	418	7	of	of	ADP
ejpam-5066	418	8	a	a	DET
ejpam-5066	418	9	ring	ring	NOUN
ejpam-5066	418	10	r.	r.	PROPN
ejpam-5066	418	11	then	then	ADV
ejpam-5066	418	12	the	the	DET
ejpam-5066	418	13	following	follow	VERB
ejpam-5066	418	14	statements	statement	NOUN
ejpam-5066	418	15	are	be	AUX
ejpam-5066	418	16	satisfied	satisfied	ADJ
ejpam-5066	418	17	.	.	PUNCT
ejpam-5066	419	1	(	(	PUNCT
ejpam-5066	419	2	i	i	NOUN
ejpam-5066	419	3	)	)	PUNCT
ejpam-5066	419	4	if	if	SCONJ
ejpam-5066	419	5	ab	ab	PROPN
ejpam-5066	419	6	is	be	AUX
ejpam-5066	419	7	n	n	ADV
ejpam-5066	419	8	-	-	PUNCT
ejpam-5066	419	9	central	central	ADJ
ejpam-5066	419	10	for	for	ADP
ejpam-5066	419	11	some	some	DET
ejpam-5066	419	12	n	n	NOUN
ejpam-5066	419	13	and	and	CCONJ
ejpam-5066	419	14	b	b	X
ejpam-5066	419	15	∈	∈	PROPN
ejpam-5066	419	16	i(a	i(a	PROPN
ejpam-5066	419	17	)	)	PUNCT
ejpam-5066	419	18	,	,	PUNCT
ejpam-5066	419	19	then	then	ADV
ejpam-5066	419	20	ac	ac	PROPN
ejpam-5066	419	21	is	be	AUX
ejpam-5066	419	22	n	n	ADV
ejpam-5066	419	23	-	-	PUNCT
ejpam-5066	419	24	central	central	ADJ
ejpam-5066	419	25	for	for	ADP
ejpam-5066	419	26	every	every	DET
ejpam-5066	419	27	c	c	PROPN
ejpam-5066	419	28	∈	∈	PROPN
ejpam-5066	419	29	i(a	i(a	PROPN
ejpam-5066	419	30	)	)	PUNCT
ejpam-5066	419	31	(	(	PUNCT
ejpam-5066	419	32	ii	ii	NOUN
ejpam-5066	419	33	)	)	PUNCT
ejpam-5066	419	34	if	if	SCONJ
ejpam-5066	419	35	ba	ba	PROPN
ejpam-5066	419	36	is	be	AUX
ejpam-5066	419	37	n	n	ADV
ejpam-5066	419	38	-	-	PUNCT
ejpam-5066	419	39	central	central	ADJ
ejpam-5066	419	40	for	for	ADP
ejpam-5066	419	41	some	some	DET
ejpam-5066	419	42	n	n	NOUN
ejpam-5066	419	43	and	and	CCONJ
ejpam-5066	419	44	b	b	X
ejpam-5066	419	45	∈	∈	PROPN
ejpam-5066	419	46	i(a	i(a	PROPN
ejpam-5066	419	47	)	)	PUNCT
ejpam-5066	419	48	,	,	PUNCT
ejpam-5066	419	49	then	then	ADV
ejpam-5066	419	50	ca	can	AUX
ejpam-5066	419	51	is	be	AUX
ejpam-5066	419	52	n	n	ADV
ejpam-5066	419	53	-	-	PUNCT
ejpam-5066	419	54	central	central	ADJ
ejpam-5066	419	55	for	for	ADP
ejpam-5066	419	56	every	every	DET
ejpam-5066	419	57	c	c	PROPN
ejpam-5066	419	58	∈	∈	PROPN
ejpam-5066	419	59	i(a	i(a	PROPN
ejpam-5066	419	60	)	)	PUNCT
ejpam-5066	419	61	(	(	PUNCT
ejpam-5066	419	62	iii	iii	X
ejpam-5066	419	63	)	)	PUNCT
ejpam-5066	419	64	if	if	SCONJ
ejpam-5066	419	65	ab	ab	PROPN
ejpam-5066	419	66	is	be	AUX
ejpam-5066	419	67	n	n	ADV
ejpam-5066	419	68	-	-	PUNCT
ejpam-5066	419	69	central	central	ADJ
ejpam-5066	419	70	for	for	ADP
ejpam-5066	419	71	some	some	DET
ejpam-5066	419	72	n	n	NOUN
ejpam-5066	419	73	and	and	CCONJ
ejpam-5066	419	74	b	b	X
ejpam-5066	419	75	∈	∈	PROPN
ejpam-5066	419	76	i(a	i(a	PROPN
ejpam-5066	419	77	)	)	PUNCT
ejpam-5066	419	78	,	,	PUNCT
ejpam-5066	419	79	then	then	ADV
ejpam-5066	419	80	a	a	PRON
ejpam-5066	419	81	is	be	AUX
ejpam-5066	419	82	right	right	ADV
ejpam-5066	419	83	regular	regular	ADJ
ejpam-5066	419	84	(	(	PUNCT
ejpam-5066	419	85	iv	iv	X
ejpam-5066	419	86	)	)	PUNCT
ejpam-5066	419	87	if	if	SCONJ
ejpam-5066	419	88	ba	ba	PROPN
ejpam-5066	419	89	is	be	AUX
ejpam-5066	419	90	n	n	ADV
ejpam-5066	419	91	-	-	PUNCT
ejpam-5066	419	92	central	central	ADJ
ejpam-5066	419	93	for	for	ADP
ejpam-5066	419	94	some	some	DET
ejpam-5066	419	95	n	n	NOUN
ejpam-5066	419	96	and	and	CCONJ
ejpam-5066	419	97	b	b	X
ejpam-5066	419	98	∈	∈	PROPN
ejpam-5066	419	99	i(a	i(a	PROPN
ejpam-5066	419	100	)	)	PUNCT
ejpam-5066	419	101	,	,	PUNCT
ejpam-5066	419	102	then	then	ADV
ejpam-5066	419	103	a	a	PRON
ejpam-5066	419	104	is	be	AUX
ejpam-5066	419	105	left	leave	VERB
ejpam-5066	419	106	regular	regular	ADJ
ejpam-5066	419	107	(	(	PUNCT
ejpam-5066	419	108	v	v	NOUN
ejpam-5066	419	109	)	)	PUNCT
ejpam-5066	419	110	if	if	SCONJ
ejpam-5066	419	111	both	both	PRON
ejpam-5066	419	112	ab1	ab1	ADV
ejpam-5066	419	113	and	and	CCONJ
ejpam-5066	419	114	b2a	b2a	PROPN
ejpam-5066	419	115	are	be	AUX
ejpam-5066	419	116	n	n	ADV
ejpam-5066	419	117	-	-	PUNCT
ejpam-5066	419	118	central	central	ADJ
ejpam-5066	419	119	for	for	ADP
ejpam-5066	419	120	some	some	DET
ejpam-5066	419	121	n	n	NOUN
ejpam-5066	419	122	and	and	CCONJ
ejpam-5066	419	123	b1	b1	NOUN
ejpam-5066	419	124	,	,	PUNCT
ejpam-5066	419	125	b2	b2	NOUN
ejpam-5066	419	126	∈	∈	PROPN
ejpam-5066	419	127	i(a	i(a	PROPN
ejpam-5066	419	128	)	)	PUNCT
ejpam-5066	419	129	,	,	PUNCT
ejpam-5066	419	130	then	then	ADV
ejpam-5066	419	131	a	a	PRON
ejpam-5066	419	132	is	be	AUX
ejpam-5066	419	133	strongly	strongly	ADV
ejpam-5066	419	134	regular	regular	ADJ
ejpam-5066	419	135	.	.	PUNCT
ejpam-5066	420	1	proof	proof	NOUN
ejpam-5066	420	2	.	.	PUNCT
ejpam-5066	421	1	(	(	PUNCT
ejpam-5066	421	2	i	i	NOUN
ejpam-5066	421	3	)	)	PUNCT
ejpam-5066	421	4	easy	easy	ADJ
ejpam-5066	421	5	calculations	calculation	NOUN
ejpam-5066	421	6	show	show	VERB
ejpam-5066	421	7	that	that	SCONJ
ejpam-5066	421	8	the	the	DET
ejpam-5066	421	9	idempotents	idempotent	NOUN
ejpam-5066	421	10	of	of	ADP
ejpam-5066	421	11	ai(a	ai(a	NOUN
ejpam-5066	421	12	)	)	PUNCT
ejpam-5066	421	13	are	be	AUX
ejpam-5066	421	14	isomorphic	isomorphic	ADJ
ejpam-5066	421	15	and	and	CCONJ
ejpam-5066	421	16	isomorphic	isomorphic	ADJ
ejpam-5066	421	17	complements	complement	NOUN
ejpam-5066	421	18	.	.	PUNCT
ejpam-5066	422	1	so	so	ADV
ejpam-5066	422	2	,	,	PUNCT
ejpam-5066	422	3	therefore	therefore	ADV
ejpam-5066	422	4	,	,	PUNCT
ejpam-5066	422	5	ac	ac	PROPN
ejpam-5066	422	6	is	be	AUX
ejpam-5066	422	7	n	n	ADV
ejpam-5066	422	8	-	-	PUNCT
ejpam-5066	422	9	central	central	ADJ
ejpam-5066	422	10	for	for	ADP
ejpam-5066	422	11	every	every	DET
ejpam-5066	422	12	c	c	PROPN
ejpam-5066	422	13	∈	∈	PROPN
ejpam-5066	422	14	i(a	i(a	PROPN
ejpam-5066	422	15	)	)	PUNCT
ejpam-5066	422	16	.	.	PUNCT
ejpam-5066	423	1	m.	m.	PROPN
ejpam-5066	423	2	saad	saad	PROPN
ejpam-5066	423	3	,	,	PUNCT
ejpam-5066	423	4	m.	m.	NOUN
ejpam-5066	423	5	zailaee	zailaee	PROPN
ejpam-5066	423	6	/	/	SYM
ejpam-5066	423	7	eur	eur	PROPN
ejpam-5066	423	8	.	.	PUNCT
ejpam-5066	424	1	j.	j.	PROPN
ejpam-5066	424	2	pure	pure	PROPN
ejpam-5066	424	3	appl	appl	PROPN
ejpam-5066	424	4	.	.	PROPN
ejpam-5066	424	5	math	math	PROPN
ejpam-5066	424	6	,	,	PUNCT
ejpam-5066	424	7	17	17	NUM
ejpam-5066	424	8	(	(	PUNCT
ejpam-5066	424	9	2	2	NUM
ejpam-5066	424	10	)	)	PUNCT
ejpam-5066	424	11	(	(	PUNCT
ejpam-5066	424	12	2024	2024	NUM
ejpam-5066	424	13	)	)	PUNCT
ejpam-5066	424	14	,	,	PUNCT
ejpam-5066	424	15	736	736	NUM
ejpam-5066	424	16	-	-	SYM
ejpam-5066	424	17	752	752	NUM
ejpam-5066	424	18	748	748	NUM
ejpam-5066	424	19	(	(	PUNCT
ejpam-5066	424	20	ii	ii	NOUN
ejpam-5066	424	21	)	)	PUNCT
ejpam-5066	424	22	similarly	similarly	ADV
ejpam-5066	424	23	as	as	ADP
ejpam-5066	424	24	(	(	PUNCT
ejpam-5066	424	25	i	i	NOUN
ejpam-5066	424	26	)	)	PUNCT
ejpam-5066	424	27	.	.	PUNCT
ejpam-5066	425	1	(	(	PUNCT
ejpam-5066	425	2	iii	iii	X
ejpam-5066	425	3	)	)	PUNCT
ejpam-5066	425	4	write	write	NOUN
ejpam-5066	425	5	e	e	NOUN
ejpam-5066	425	6	=	=	PROPN
ejpam-5066	425	7	ab	ab	PROPN
ejpam-5066	425	8	and	and	CCONJ
ejpam-5066	425	9	consider	consider	VERB
ejpam-5066	425	10	n	n	PRON
ejpam-5066	425	11	is	be	AUX
ejpam-5066	425	12	even	even	ADV
ejpam-5066	425	13	which	which	PRON
ejpam-5066	425	14	does	do	AUX
ejpam-5066	425	15	not	not	PART
ejpam-5066	425	16	lose	lose	VERB
ejpam-5066	425	17	the	the	DET
ejpam-5066	425	18	geniality	geniality	NOUN
ejpam-5066	425	19	of	of	ADP
ejpam-5066	425	20	n.	n.	NOUN
ejpam-5066	425	21	write	write	PROPN
ejpam-5066	425	22	n	n	PROPN
ejpam-5066	425	23	=	=	SYM
ejpam-5066	425	24	2k	2k	NUM
ejpam-5066	425	25	,	,	PUNCT
ejpam-5066	425	26	for	for	ADP
ejpam-5066	425	27	integer	integer	PROPN
ejpam-5066	425	28	k	k	PROPN
ejpam-5066	425	29	≥	≥	NUM
ejpam-5066	425	30	1	1	NUM
ejpam-5066	425	31	and	and	CCONJ
ejpam-5066	425	32	er((1	er((1	NUM
ejpam-5066	425	33	−	−	PROPN
ejpam-5066	425	34	e)rer)k	e)rer)k	PROPN
ejpam-5066	425	35	=	=	SYM
ejpam-5066	425	36	0	0	NUM
ejpam-5066	425	37	.	.	PUNCT
ejpam-5066	426	1	so	so	ADV
ejpam-5066	426	2	,	,	PUNCT
ejpam-5066	426	3	0	0	NUM
ejpam-5066	426	4	=	=	SYM
ejpam-5066	426	5	e(a(1	e(a(1	NUM
ejpam-5066	426	6	−	−	PROPN
ejpam-5066	426	7	e)be)k	e)be)k	NOUN
ejpam-5066	426	8	=	=	PUNCT
ejpam-5066	426	9	e(abe−	e(abe−	PRON
ejpam-5066	426	10	aebe)k	aebe)k	PUNCT
ejpam-5066	426	11	=	=	SYM
ejpam-5066	426	12	e(e−	e(e−	PROPN
ejpam-5066	426	13	aebe)k	aebe)k	X
ejpam-5066	426	14	=	=	SYM
ejpam-5066	426	15	(	(	PUNCT
ejpam-5066	426	16	e−	e−	PROPN
ejpam-5066	426	17	aebe)k	aebe)k	PROPN
ejpam-5066	426	18	.	.	PUNCT
ejpam-5066	427	1	expanding	expand	VERB
ejpam-5066	427	2	and	and	CCONJ
ejpam-5066	427	3	using	use	VERB
ejpam-5066	427	4	the	the	DET
ejpam-5066	427	5	fact	fact	NOUN
ejpam-5066	427	6	ea	ea	NOUN
ejpam-5066	427	7	=	=	SYM
ejpam-5066	427	8	a	a	PROPN
ejpam-5066	427	9	,	,	PUNCT
ejpam-5066	427	10	we	we	PRON
ejpam-5066	427	11	get	get	VERB
ejpam-5066	427	12	e	e	PRON
ejpam-5066	427	13	∈	∈	NOUN
ejpam-5066	427	14	aere	aere	ADV
ejpam-5066	427	15	.	.	PUNCT
ejpam-5066	428	1	so	so	ADV
ejpam-5066	428	2	that	that	SCONJ
ejpam-5066	428	3	a	a	DET
ejpam-5066	428	4	=	=	SYM
ejpam-5066	428	5	ea	ea	PROPN
ejpam-5066	428	6	∈	∈	PROPN
ejpam-5066	428	7	aera	aera	NOUN
ejpam-5066	428	8	=	=	PUNCT
ejpam-5066	428	9	a2bra	a2bra	PROPN
ejpam-5066	428	10	⊆	⊆	NUM
ejpam-5066	428	11	a2r	a2r	PROPN
ejpam-5066	428	12	and	and	CCONJ
ejpam-5066	428	13	a	a	PRON
ejpam-5066	428	14	is	be	AUX
ejpam-5066	428	15	left	leave	VERB
ejpam-5066	428	16	regular	regular	ADV
ejpam-5066	428	17	.	.	PUNCT
ejpam-5066	429	1	(	(	PUNCT
ejpam-5066	429	2	iv	iv	X
ejpam-5066	429	3	)	)	PUNCT
ejpam-5066	429	4	similar	similar	ADJ
ejpam-5066	429	5	as	as	ADP
ejpam-5066	429	6	(	(	PUNCT
ejpam-5066	429	7	iii	iii	NOUN
ejpam-5066	429	8	)	)	PUNCT
ejpam-5066	429	9	.	.	PUNCT
ejpam-5066	430	1	(	(	PUNCT
ejpam-5066	430	2	iii	iii	X
ejpam-5066	430	3	)	)	PUNCT
ejpam-5066	430	4	let	let	VERB
ejpam-5066	430	5	a	a	PRON
ejpam-5066	430	6	be	be	AUX
ejpam-5066	430	7	a	a	DET
ejpam-5066	430	8	regular	regular	ADJ
ejpam-5066	430	9	element	element	NOUN
ejpam-5066	430	10	a	a	PRON
ejpam-5066	430	11	of	of	ADP
ejpam-5066	430	12	r	r	NOUN
ejpam-5066	430	13	with	with	ADP
ejpam-5066	430	14	a	a	DET
ejpam-5066	430	15	=	=	SYM
ejpam-5066	430	16	aba	aba	PROPN
ejpam-5066	430	17	,	,	PUNCT
ejpam-5066	430	18	for	for	ADP
ejpam-5066	430	19	some	some	DET
ejpam-5066	430	20	b	b	PROPN
ejpam-5066	430	21	∈	∈	PROPN
ejpam-5066	430	22	r.	r.	NOUN
ejpam-5066	430	23	we	we	PRON
ejpam-5066	430	24	prove	prove	VERB
ejpam-5066	430	25	the	the	DET
ejpam-5066	430	26	case	case	NOUN
ejpam-5066	430	27	of	of	ADP
ejpam-5066	430	28	the	the	DET
ejpam-5066	430	29	idempotent	idempotent	NOUN
ejpam-5066	430	30	e	e	NOUN
ejpam-5066	430	31	=	=	SYM
ejpam-5066	430	32	ba	ba	PROPN
ejpam-5066	430	33	is	be	AUX
ejpam-5066	430	34	n	n	ADV
ejpam-5066	430	35	-	-	PUNCT
ejpam-5066	430	36	central	central	ADJ
ejpam-5066	430	37	for	for	ADP
ejpam-5066	430	38	some	some	DET
ejpam-5066	430	39	n.	n.	NOUN
ejpam-5066	430	40	the	the	DET
ejpam-5066	430	41	case	case	NOUN
ejpam-5066	430	42	of	of	ADP
ejpam-5066	430	43	n	n	NOUN
ejpam-5066	430	44	=	=	SYM
ejpam-5066	430	45	2	2	NUM
ejpam-5066	430	46	has	have	AUX
ejpam-5066	430	47	been	be	AUX
ejpam-5066	430	48	shown	show	VERB
ejpam-5066	430	49	in	in	ADP
ejpam-5066	430	50	[	[	X
ejpam-5066	430	51	11	11	NUM
ejpam-5066	430	52	,	,	PUNCT
ejpam-5066	430	53	theorem	theorem	ADJ
ejpam-5066	430	54	c.	c.	PROPN
ejpam-5066	430	55	]	]	PUNCT
ejpam-5066	430	56	and	and	CCONJ
ejpam-5066	430	57	considering	consider	VERB
ejpam-5066	430	58	n	n	X
ejpam-5066	430	59	is	be	AUX
ejpam-5066	430	60	even	even	ADV
ejpam-5066	430	61	does	do	AUX
ejpam-5066	430	62	not	not	PART
ejpam-5066	430	63	lose	lose	VERB
ejpam-5066	430	64	the	the	DET
ejpam-5066	430	65	geniality	geniality	NOUN
ejpam-5066	430	66	of	of	ADP
ejpam-5066	430	67	n.	n.	NOUN
ejpam-5066	430	68	write	write	PROPN
ejpam-5066	430	69	n	n	PROPN
ejpam-5066	430	70	=	=	SYM
ejpam-5066	430	71	2k+	2k+	NUM
ejpam-5066	430	72	2	2	NUM
ejpam-5066	430	73	,	,	PUNCT
ejpam-5066	430	74	for	for	ADP
ejpam-5066	430	75	integer	integer	PROPN
ejpam-5066	430	76	k	k	PROPN
ejpam-5066	430	77	≥	≥	NUM
ejpam-5066	430	78	0	0	NUM
ejpam-5066	431	1	and	and	CCONJ
ejpam-5066	431	2	(	(	PUNCT
ejpam-5066	431	3	(	(	PUNCT
ejpam-5066	431	4	1−e)rer)k	1−e)rer)k	NUM
ejpam-5066	431	5	=	=	NOUN
ejpam-5066	431	6	0	0	NUM
ejpam-5066	431	7	from	from	ADP
ejpam-5066	431	8	the	the	DET
ejpam-5066	431	9	n	n	NOUN
ejpam-5066	431	10	-	-	PUNCT
ejpam-5066	431	11	centrality	centrality	NOUN
ejpam-5066	431	12	of	of	ADP
ejpam-5066	431	13	e.	e.	PROPN
ejpam-5066	432	1	so	so	ADV
ejpam-5066	432	2	,	,	PUNCT
ejpam-5066	432	3	er((1−e)rer)k	er((1−e)rer)k	PROPN
ejpam-5066	432	4	=	=	SYM
ejpam-5066	432	5	0	0	PUNCT
ejpam-5066	433	1	and	and	CCONJ
ejpam-5066	433	2	(	(	PUNCT
ejpam-5066	433	3	eb(1	eb(1	NOUN
ejpam-5066	433	4	−	−	NOUN
ejpam-5066	433	5	e)a)k	e)a)k	NOUN
ejpam-5066	433	6	=	=	SYM
ejpam-5066	433	7	0	0	NUM
ejpam-5066	433	8	,	,	PUNCT
ejpam-5066	433	9	since	since	SCONJ
ejpam-5066	433	10	ae	ae	PROPN
ejpam-5066	433	11	=	=	NOUN
ejpam-5066	433	12	a.	a.	PROPN
ejpam-5066	433	13	but	but	CCONJ
ejpam-5066	433	14	eb(1	eb(1	PROPN
ejpam-5066	433	15	−	−	PROPN
ejpam-5066	433	16	e)a	e)a	PUNCT
ejpam-5066	433	17	=	=	PUNCT
ejpam-5066	433	18	eba	eba	NOUN
ejpam-5066	433	19	−	−	NOUN
ejpam-5066	433	20	ebea	ebea	PROPN
ejpam-5066	433	21	=	=	SYM
ejpam-5066	433	22	e	e	NOUN
ejpam-5066	433	23	−	−	NOUN
ejpam-5066	433	24	ebea	ebea	NOUN
ejpam-5066	433	25	.	.	PUNCT
ejpam-5066	434	1	hence	hence	ADV
ejpam-5066	434	2	,	,	PUNCT
ejpam-5066	434	3	(	(	PUNCT
ejpam-5066	434	4	e	e	NOUN
ejpam-5066	434	5	−	−	PROPN
ejpam-5066	434	6	ebea)k	ebea)k	NOUN
ejpam-5066	434	7	=	=	SYM
ejpam-5066	434	8	0	0	NUM
ejpam-5066	434	9	and	and	CCONJ
ejpam-5066	434	10	the	the	DET
ejpam-5066	434	11	expanding	expand	VERB
ejpam-5066	434	12	gives	give	VERB
ejpam-5066	434	13	e	e	PROPN
ejpam-5066	434	14	∈	∈	PROPN
ejpam-5066	434	15	erea	erea	NOUN
ejpam-5066	434	16	.	.	PUNCT
ejpam-5066	435	1	so	so	ADV
ejpam-5066	435	2	that	that	SCONJ
ejpam-5066	435	3	a	a	DET
ejpam-5066	435	4	=	=	SYM
ejpam-5066	435	5	ae	ae	PROPN
ejpam-5066	435	6	∈	∈	PROPN
ejpam-5066	435	7	e	e	PROPN
ejpam-5066	435	8	∈	∈	PROPN
ejpam-5066	435	9	aerea	aerea	X
ejpam-5066	435	10	⊆	⊆	NUM
ejpam-5066	435	11	ra2	ra2	PROPN
ejpam-5066	435	12	and	and	CCONJ
ejpam-5066	435	13	a	a	PRON
ejpam-5066	435	14	is	be	AUX
ejpam-5066	435	15	right	right	ADV
ejpam-5066	435	16	regular	regular	ADJ
ejpam-5066	435	17	.	.	PUNCT
ejpam-5066	436	1	(	(	PUNCT
ejpam-5066	436	2	iv	iv	X
ejpam-5066	436	3	)	)	PUNCT
ejpam-5066	436	4	can	can	AUX
ejpam-5066	436	5	be	be	AUX
ejpam-5066	436	6	proved	prove	VERB
ejpam-5066	436	7	similarly	similarly	ADV
ejpam-5066	436	8	as	as	ADP
ejpam-5066	436	9	in	in	ADP
ejpam-5066	436	10	(	(	PUNCT
ejpam-5066	436	11	iii	iii	NOUN
ejpam-5066	436	12	)	)	PUNCT
ejpam-5066	436	13	.	.	PUNCT
ejpam-5066	437	1	(	(	PUNCT
ejpam-5066	437	2	v	v	NOUN
ejpam-5066	437	3	)	)	PUNCT
ejpam-5066	437	4	is	be	AUX
ejpam-5066	437	5	direct	direct	ADJ
ejpam-5066	437	6	from	from	ADP
ejpam-5066	437	7	(	(	PUNCT
ejpam-5066	437	8	iii	iii	NOUN
ejpam-5066	437	9	)	)	PUNCT
ejpam-5066	437	10	and	and	CCONJ
ejpam-5066	437	11	(	(	PUNCT
ejpam-5066	437	12	iv	iv	X
ejpam-5066	437	13	)	)	PUNCT
ejpam-5066	437	14	.	.	PUNCT
ejpam-5066	438	1	corollary	corollary	ADJ
ejpam-5066	438	2	11	11	NUM
ejpam-5066	438	3	.	.	PUNCT
ejpam-5066	439	1	a	a	DET
ejpam-5066	439	2	ring	ring	NOUN
ejpam-5066	439	3	r	r	NOUN
ejpam-5066	439	4	is	be	AUX
ejpam-5066	439	5	strongly	strongly	ADV
ejpam-5066	439	6	regular	regular	ADJ
ejpam-5066	439	7	if	if	SCONJ
ejpam-5066	440	1	and	and	CCONJ
ejpam-5066	440	2	only	only	ADV
ejpam-5066	440	3	if	if	SCONJ
ejpam-5066	440	4	r	r	NOUN
ejpam-5066	440	5	is	be	AUX
ejpam-5066	440	6	regular	regular	ADJ
ejpam-5066	440	7	and	and	CCONJ
ejpam-5066	440	8	n	n	CCONJ
ejpam-5066	440	9	-	-	PUNCT
ejpam-5066	440	10	abelian	abelian	NOUN
ejpam-5066	440	11	for	for	ADP
ejpam-5066	440	12	some	some	DET
ejpam-5066	440	13	n.	n.	NOUN
ejpam-5066	440	14	corollary	corollary	NOUN
ejpam-5066	440	15	12	12	NUM
ejpam-5066	440	16	.	.	PUNCT
ejpam-5066	441	1	let	let	VERB
ejpam-5066	441	2	r	r	PRON
ejpam-5066	441	3	be	be	AUX
ejpam-5066	441	4	a	a	DET
ejpam-5066	441	5	regular	regular	ADJ
ejpam-5066	441	6	ring	ring	NOUN
ejpam-5066	441	7	.	.	PUNCT
ejpam-5066	442	1	then	then	ADV
ejpam-5066	442	2	the	the	DET
ejpam-5066	442	3	following	follow	VERB
ejpam-5066	442	4	statements	statement	NOUN
ejpam-5066	442	5	are	be	AUX
ejpam-5066	442	6	equivalent	equivalent	ADJ
ejpam-5066	442	7	:	:	PUNCT
ejpam-5066	442	8	(	(	PUNCT
ejpam-5066	442	9	i	i	NOUN
ejpam-5066	442	10	)	)	PUNCT
ejpam-5066	442	11	r	r	NOUN
ejpam-5066	442	12	is	be	AUX
ejpam-5066	442	13	reduced	reduce	VERB
ejpam-5066	442	14	.	.	PUNCT
ejpam-5066	443	1	(	(	PUNCT
ejpam-5066	443	2	ii	ii	X
ejpam-5066	443	3	)	)	PUNCT
ejpam-5066	443	4	r	r	NOUN
ejpam-5066	443	5	is	be	AUX
ejpam-5066	443	6	abelian	abelian	ADJ
ejpam-5066	443	7	.	.	PUNCT
ejpam-5066	444	1	(	(	PUNCT
ejpam-5066	444	2	iii	iii	X
ejpam-5066	444	3	)	)	PUNCT
ejpam-5066	444	4	r	r	NOUN
ejpam-5066	444	5	is	be	AUX
ejpam-5066	444	6	n	n	CCONJ
ejpam-5066	444	7	-	-	PUNCT
ejpam-5066	444	8	abelian	abelian	NOUN
ejpam-5066	444	9	for	for	ADP
ejpam-5066	444	10	every	every	DET
ejpam-5066	444	11	n.	n.	NOUN
ejpam-5066	444	12	(	(	PUNCT
ejpam-5066	444	13	iv	iv	X
ejpam-5066	444	14	)	)	PUNCT
ejpam-5066	444	15	r	r	NOUN
ejpam-5066	444	16	is	be	AUX
ejpam-5066	444	17	n	n	CCONJ
ejpam-5066	444	18	-	-	PUNCT
ejpam-5066	444	19	abelian	abelian	ADJ
ejpam-5066	444	20	for	for	ADP
ejpam-5066	444	21	some	some	DET
ejpam-5066	444	22	n.	n.	NOUN
ejpam-5066	444	23	proposition	proposition	NOUN
ejpam-5066	444	24	14	14	NUM
ejpam-5066	444	25	.	.	PUNCT
ejpam-5066	445	1	let	let	VERB
ejpam-5066	445	2	r	r	PRON
ejpam-5066	445	3	be	be	AUX
ejpam-5066	445	4	an	an	DET
ejpam-5066	445	5	n	n	CCONJ
ejpam-5066	445	6	-	-	PUNCT
ejpam-5066	445	7	abelian	abelian	ADJ
ejpam-5066	445	8	ring	ring	NOUN
ejpam-5066	445	9	,	,	PUNCT
ejpam-5066	445	10	for	for	ADP
ejpam-5066	445	11	some	some	DET
ejpam-5066	445	12	n.	n.	NOUN
ejpam-5066	445	13	if	if	SCONJ
ejpam-5066	445	14	r	r	NOUN
ejpam-5066	445	15	is	be	AUX
ejpam-5066	445	16	π	π	NOUN
ejpam-5066	445	17	-	-	NOUN
ejpam-5066	445	18	regular	regular	ADJ
ejpam-5066	445	19	,	,	PUNCT
ejpam-5066	445	20	then	then	ADV
ejpam-5066	445	21	r	r	NOUN
ejpam-5066	445	22	is	be	AUX
ejpam-5066	445	23	abelian	abelian	NOUN
ejpam-5066	445	24	(	(	PUNCT
ejpam-5066	445	25	consequently	consequently	ADV
ejpam-5066	445	26	r	r	NOUN
ejpam-5066	445	27	is	be	AUX
ejpam-5066	445	28	strongly	strongly	ADV
ejpam-5066	445	29	π	π	NOUN
ejpam-5066	445	30	-	-	NOUN
ejpam-5066	445	31	regular	regular	ADJ
ejpam-5066	445	32	)	)	PUNCT
ejpam-5066	445	33	.	.	PUNCT
ejpam-5066	446	1	proof	proof	NOUN
ejpam-5066	446	2	.	.	PUNCT
ejpam-5066	447	1	straightforward	straightforward	ADJ
ejpam-5066	447	2	.	.	PUNCT
ejpam-5066	448	1	by	by	ADP
ejpam-5066	448	2	the	the	DET
ejpam-5066	448	3	previous	previous	ADJ
ejpam-5066	448	4	proposition	proposition	NOUN
ejpam-5066	448	5	,	,	PUNCT
ejpam-5066	448	6	we	we	PRON
ejpam-5066	448	7	can	can	AUX
ejpam-5066	448	8	get	get	VERB
ejpam-5066	448	9	the	the	DET
ejpam-5066	448	10	results	result	NOUN
ejpam-5066	448	11	[	[	X
ejpam-5066	448	12	23	23	NUM
ejpam-5066	448	13	,	,	PUNCT
ejpam-5066	448	14	theorems	theorem	VERB
ejpam-5066	448	15	3.8	3.8	NUM
ejpam-5066	448	16	and	and	CCONJ
ejpam-5066	448	17	3.10	3.10	NUM
ejpam-5066	448	18	]	]	PUNCT
ejpam-5066	448	19	using	use	VERB
ejpam-5066	448	20	the	the	DET
ejpam-5066	448	21	n	n	NOUN
ejpam-5066	448	22	-	-	PUNCT
ejpam-5066	448	23	centrality	centrality	NOUN
ejpam-5066	448	24	with	with	ADP
ejpam-5066	448	25	any	any	DET
ejpam-5066	448	26	degree	degree	NOUN
ejpam-5066	448	27	.	.	PUNCT
ejpam-5066	449	1	in	in	ADP
ejpam-5066	449	2	[	[	X
ejpam-5066	449	3	20	20	NUM
ejpam-5066	449	4	]	]	PUNCT
ejpam-5066	449	5	,	,	PUNCT
ejpam-5066	449	6	warfield	warfield	PROPN
ejpam-5066	449	7	called	call	VERB
ejpam-5066	449	8	a	a	DET
ejpam-5066	449	9	ring	ring	NOUN
ejpam-5066	449	10	r	r	NOUN
ejpam-5066	449	11	an	an	DET
ejpam-5066	449	12	exchange	exchange	NOUN
ejpam-5066	449	13	ring	ring	NOUN
ejpam-5066	449	14	if	if	SCONJ
ejpam-5066	449	15	rr	rr	PROPN
ejpam-5066	449	16	has	have	VERB
ejpam-5066	449	17	the	the	DET
ejpam-5066	449	18	finite	finite	ADJ
ejpam-5066	449	19	exchange	exchange	NOUN
ejpam-5066	449	20	property	property	NOUN
ejpam-5066	449	21	.	.	PUNCT
ejpam-5066	450	1	an	an	DET
ejpam-5066	450	2	equivalent	equivalent	ADJ
ejpam-5066	450	3	idempotent	idempotent	ADJ
ejpam-5066	450	4	-	-	PUNCT
ejpam-5066	450	5	wise	wise	ADJ
ejpam-5066	450	6	definition	definition	NOUN
ejpam-5066	450	7	of	of	ADP
ejpam-5066	450	8	exchange	exchange	NOUN
ejpam-5066	450	9	rings	ring	NOUN
ejpam-5066	450	10	was	be	AUX
ejpam-5066	450	11	introduced	introduce	VERB
ejpam-5066	450	12	in	in	ADP
ejpam-5066	450	13	[	[	X
ejpam-5066	450	14	6	6	NUM
ejpam-5066	450	15	,	,	PUNCT
ejpam-5066	450	16	14	14	NUM
ejpam-5066	450	17	]	]	PUNCT
ejpam-5066	450	18	;	;	PUNCT
ejpam-5066	450	19	that	that	SCONJ
ejpam-5066	450	20	a	a	DET
ejpam-5066	450	21	ring	ring	NOUN
ejpam-5066	450	22	r	r	NOUN
ejpam-5066	450	23	is	be	AUX
ejpam-5066	450	24	exchange	exchange	NOUN
ejpam-5066	450	25	if	if	SCONJ
ejpam-5066	450	26	and	and	CCONJ
ejpam-5066	450	27	only	only	ADV
ejpam-5066	450	28	if	if	SCONJ
ejpam-5066	450	29	for	for	ADP
ejpam-5066	450	30	every	every	DET
ejpam-5066	450	31	a	a	DET
ejpam-5066	450	32	∈	∈	PROPN
ejpam-5066	450	33	r	r	NOUN
ejpam-5066	450	34	,	,	PUNCT
ejpam-5066	450	35	there	there	PRON
ejpam-5066	450	36	exists	exist	VERB
ejpam-5066	450	37	an	an	DET
ejpam-5066	450	38	idempotent	idempotent	ADJ
ejpam-5066	450	39	e	e	NOUN
ejpam-5066	450	40	of	of	ADP
ejpam-5066	450	41	r	r	NOUN
ejpam-5066	450	42	such	such	ADJ
ejpam-5066	450	43	that	that	SCONJ
ejpam-5066	450	44	e	e	PROPN
ejpam-5066	450	45	∈	∈	PROPN
ejpam-5066	450	46	ar	ar	PROPN
ejpam-5066	450	47	and	and	CCONJ
ejpam-5066	450	48	1−	1−	NUM
ejpam-5066	450	49	e	e	X
ejpam-5066	450	50	∈	∈	PROPN
ejpam-5066	450	51	(	(	PUNCT
ejpam-5066	450	52	1−	1−	NUM
ejpam-5066	450	53	a)r	a)r	NOUN
ejpam-5066	450	54	.	.	PUNCT
ejpam-5066	451	1	according	accord	VERB
ejpam-5066	451	2	to	to	ADP
ejpam-5066	451	3	[	[	X
ejpam-5066	451	4	14	14	NUM
ejpam-5066	451	5	]	]	PUNCT
ejpam-5066	451	6	,	,	PUNCT
ejpam-5066	451	7	a	a	DET
ejpam-5066	451	8	ring	ring	NOUN
ejpam-5066	451	9	r	r	NOUN
ejpam-5066	451	10	is	be	AUX
ejpam-5066	451	11	said	say	VERB
ejpam-5066	451	12	to	to	PART
ejpam-5066	451	13	be	be	AUX
ejpam-5066	451	14	clean	clean	ADJ
ejpam-5066	451	15	if	if	SCONJ
ejpam-5066	451	16	every	every	DET
ejpam-5066	451	17	element	element	NOUN
ejpam-5066	451	18	a	a	DET
ejpam-5066	451	19	∈	∈	NOUN
ejpam-5066	451	20	r	r	NOUN
ejpam-5066	451	21	can	can	AUX
ejpam-5066	451	22	be	be	AUX
ejpam-5066	451	23	written	write	VERB
ejpam-5066	451	24	as	as	ADP
ejpam-5066	451	25	a	a	DET
ejpam-5066	451	26	sum	sum	NOUN
ejpam-5066	451	27	of	of	ADP
ejpam-5066	451	28	a	a	DET
ejpam-5066	451	29	unit	unit	NOUN
ejpam-5066	451	30	and	and	CCONJ
ejpam-5066	451	31	an	an	DET
ejpam-5066	451	32	idempotent	idempotent	NOUN
ejpam-5066	451	33	.	.	PUNCT
ejpam-5066	452	1	the	the	DET
ejpam-5066	452	2	next	next	ADJ
ejpam-5066	452	3	theorem	theorem	NOUN
ejpam-5066	452	4	gives	give	VERB
ejpam-5066	452	5	an	an	DET
ejpam-5066	452	6	equivalent	equivalent	ADJ
ejpam-5066	452	7	definition	definition	NOUN
ejpam-5066	452	8	of	of	ADP
ejpam-5066	452	9	n	n	CCONJ
ejpam-5066	452	10	-	-	PUNCT
ejpam-5066	452	11	abelian	abelian	PROPN
ejpam-5066	452	12	exchange	exchange	NOUN
ejpam-5066	452	13	rings	ring	NOUN
ejpam-5066	452	14	,	,	PUNCT
ejpam-5066	452	15	drawing	draw	VERB
ejpam-5066	452	16	inspiration	inspiration	NOUN
ejpam-5066	452	17	from	from	ADP
ejpam-5066	452	18	[	[	X
ejpam-5066	452	19	11	11	NUM
ejpam-5066	452	20	,	,	PUNCT
ejpam-5066	452	21	theorem	theorem	VERB
ejpam-5066	452	22	5.10	5.10	NUM
ejpam-5066	452	23	.	.	PUNCT
ejpam-5066	452	24	]	]	PUNCT
ejpam-5066	453	1	and	and	CCONJ
ejpam-5066	453	2	extending	extend	VERB
ejpam-5066	453	3	it	it	PRON
ejpam-5066	453	4	to	to	ADP
ejpam-5066	453	5	a	a	DET
ejpam-5066	453	6	more	more	ADV
ejpam-5066	453	7	general	general	ADJ
ejpam-5066	453	8	context	context	NOUN
ejpam-5066	453	9	.	.	PUNCT
ejpam-5066	454	1	theorem	theorem	VERB
ejpam-5066	454	2	6	6	NUM
ejpam-5066	454	3	.	.	PUNCT
ejpam-5066	455	1	the	the	DET
ejpam-5066	455	2	following	follow	VERB
ejpam-5066	455	3	statements	statement	NOUN
ejpam-5066	455	4	are	be	AUX
ejpam-5066	455	5	equivalent	equivalent	ADJ
ejpam-5066	455	6	for	for	ADP
ejpam-5066	455	7	any	any	DET
ejpam-5066	455	8	ring	ring	NOUN
ejpam-5066	455	9	r	r	NOUN
ejpam-5066	455	10	and	and	CCONJ
ejpam-5066	455	11	positive	positive	ADJ
ejpam-5066	455	12	integer	integer	NOUN
ejpam-5066	455	13	n	n	CCONJ
ejpam-5066	455	14	:	:	PUNCT
ejpam-5066	455	15	m.	m.	NOUN
ejpam-5066	455	16	saad	saad	PROPN
ejpam-5066	455	17	,	,	PUNCT
ejpam-5066	455	18	m.	m.	NOUN
ejpam-5066	455	19	zailaee	zailaee	PROPN
ejpam-5066	455	20	/	/	SYM
ejpam-5066	455	21	eur	eur	PROPN
ejpam-5066	455	22	.	.	PUNCT
ejpam-5066	456	1	j.	j.	PROPN
ejpam-5066	456	2	pure	pure	PROPN
ejpam-5066	456	3	appl	appl	PROPN
ejpam-5066	456	4	.	.	PROPN
ejpam-5066	456	5	math	math	PROPN
ejpam-5066	456	6	,	,	PUNCT
ejpam-5066	456	7	17	17	NUM
ejpam-5066	456	8	(	(	PUNCT
ejpam-5066	456	9	2	2	NUM
ejpam-5066	456	10	)	)	PUNCT
ejpam-5066	456	11	(	(	PUNCT
ejpam-5066	456	12	2024	2024	NUM
ejpam-5066	456	13	)	)	PUNCT
ejpam-5066	456	14	,	,	PUNCT
ejpam-5066	456	15	736	736	NUM
ejpam-5066	456	16	-	-	SYM
ejpam-5066	456	17	752	752	NUM
ejpam-5066	456	18	749	749	NUM
ejpam-5066	456	19	(	(	PUNCT
ejpam-5066	456	20	i	i	NOUN
ejpam-5066	456	21	)	)	PUNCT
ejpam-5066	456	22	r	r	NOUN
ejpam-5066	456	23	is	be	AUX
ejpam-5066	456	24	a	a	DET
ejpam-5066	456	25	n	n	CCONJ
ejpam-5066	456	26	-	-	PUNCT
ejpam-5066	456	27	abelian	abelian	ADJ
ejpam-5066	456	28	exchange	exchange	NOUN
ejpam-5066	456	29	ring	ring	NOUN
ejpam-5066	456	30	.	.	PUNCT
ejpam-5066	457	1	(	(	PUNCT
ejpam-5066	457	2	ii	ii	NOUN
ejpam-5066	457	3	)	)	PUNCT
ejpam-5066	457	4	r	r	NOUN
ejpam-5066	457	5	is	be	AUX
ejpam-5066	457	6	a	a	DET
ejpam-5066	457	7	n	n	CCONJ
ejpam-5066	457	8	-	-	PUNCT
ejpam-5066	457	9	abelian	abelian	ADJ
ejpam-5066	457	10	clean	clean	ADJ
ejpam-5066	457	11	ring	ring	NOUN
ejpam-5066	457	12	.	.	PUNCT
ejpam-5066	458	1	(	(	PUNCT
ejpam-5066	458	2	iii	iii	X
ejpam-5066	458	3	)	)	PUNCT
ejpam-5066	458	4	every	every	DET
ejpam-5066	458	5	element	element	NOUN
ejpam-5066	458	6	in	in	ADP
ejpam-5066	458	7	r	r	NOUN
ejpam-5066	458	8	is	be	AUX
ejpam-5066	458	9	the	the	DET
ejpam-5066	458	10	sum	sum	NOUN
ejpam-5066	458	11	of	of	ADP
ejpam-5066	458	12	a	a	DET
ejpam-5066	458	13	unit	unit	NOUN
ejpam-5066	458	14	and	and	CCONJ
ejpam-5066	458	15	an	an	DET
ejpam-5066	458	16	n	n	CCONJ
ejpam-5066	458	17	-	-	PUNCT
ejpam-5066	458	18	central	central	ADJ
ejpam-5066	458	19	idempotent	idempotent	NOUN
ejpam-5066	458	20	.	.	PUNCT
ejpam-5066	459	1	(	(	PUNCT
ejpam-5066	459	2	iv	iv	X
ejpam-5066	459	3	)	)	PUNCT
ejpam-5066	459	4	for	for	ADP
ejpam-5066	459	5	any	any	DET
ejpam-5066	459	6	a	a	DET
ejpam-5066	459	7	∈	∈	NOUN
ejpam-5066	459	8	r	r	NOUN
ejpam-5066	459	9	,	,	PUNCT
ejpam-5066	459	10	there	there	PRON
ejpam-5066	459	11	exists	exist	VERB
ejpam-5066	459	12	e	e	PROPN
ejpam-5066	459	13	∈	∈	PROPN
ejpam-5066	459	14	cn(r	cn(r	PRON
ejpam-5066	459	15	)	)	PUNCT
ejpam-5066	459	16	such	such	ADJ
ejpam-5066	459	17	that	that	SCONJ
ejpam-5066	459	18	e	e	PROPN
ejpam-5066	459	19	∈	∈	PROPN
ejpam-5066	459	20	ar	ar	PROPN
ejpam-5066	459	21	and	and	CCONJ
ejpam-5066	459	22	1−	1−	NUM
ejpam-5066	459	23	e	e	X
ejpam-5066	459	24	∈	∈	PROPN
ejpam-5066	459	25	(	(	PUNCT
ejpam-5066	459	26	1−	1−	NUM
ejpam-5066	459	27	a)r	a)r	NOUN
ejpam-5066	459	28	.	.	PUNCT
ejpam-5066	460	1	proof	proof	NOUN
ejpam-5066	460	2	.	.	PUNCT
ejpam-5066	461	1	(	(	PUNCT
ejpam-5066	461	2	i)⇒	i)⇒	PROPN
ejpam-5066	461	3	(	(	PUNCT
ejpam-5066	461	4	ii	ii	PROPN
ejpam-5066	461	5	):	):	PUNCT
ejpam-5066	461	6	by	by	ADP
ejpam-5066	461	7	utilizing	utilize	VERB
ejpam-5066	461	8	the	the	DET
ejpam-5066	461	9	exchange	exchange	NOUN
ejpam-5066	461	10	property	property	NOUN
ejpam-5066	461	11	of	of	ADP
ejpam-5066	461	12	r	r	NOUN
ejpam-5066	461	13	,	,	PUNCT
ejpam-5066	461	14	any	any	DET
ejpam-5066	461	15	idempotent	idempotent	ADJ
ejpam-5066	461	16	e	e	X
ejpam-5066	461	17	∈	∈	PROPN
ejpam-5066	461	18	r	r	PROPN
ejpam-5066	461	19	/	/	SYM
ejpam-5066	461	20	j	j	PROPN
ejpam-5066	461	21	(	(	PUNCT
ejpam-5066	461	22	r	r	NOUN
ejpam-5066	461	23	)	)	PUNCT
ejpam-5066	461	24	can	can	AUX
ejpam-5066	461	25	be	be	AUX
ejpam-5066	461	26	raised	raise	VERB
ejpam-5066	461	27	to	to	ADP
ejpam-5066	461	28	an	an	DET
ejpam-5066	461	29	idempotent	idempotent	NOUN
ejpam-5066	461	30	in	in	ADP
ejpam-5066	461	31	r.	r.	NOUN
ejpam-5066	461	32	using	use	VERB
ejpam-5066	461	33	corollary	corollary	ADJ
ejpam-5066	461	34	5	5	NUM
ejpam-5066	461	35	leads	lead	NOUN
ejpam-5066	461	36	to	to	ADP
ejpam-5066	461	37	the	the	DET
ejpam-5066	461	38	conclusion	conclusion	NOUN
ejpam-5066	461	39	that	that	SCONJ
ejpam-5066	461	40	e	e	NOUN
ejpam-5066	461	41	must	must	AUX
ejpam-5066	461	42	be	be	AUX
ejpam-5066	461	43	central	central	ADJ
ejpam-5066	461	44	in	in	ADP
ejpam-5066	461	45	r	r	PROPN
ejpam-5066	461	46	/	/	SYM
ejpam-5066	461	47	j	j	PROPN
ejpam-5066	461	48	(	(	PUNCT
ejpam-5066	461	49	r	r	NOUN
ejpam-5066	461	50	)	)	PUNCT
ejpam-5066	461	51	.	.	PUNCT
ejpam-5066	462	1	consequently	consequently	ADV
ejpam-5066	462	2	,	,	PUNCT
ejpam-5066	462	3	rj	rj	PROPN
ejpam-5066	462	4	(	(	PUNCT
ejpam-5066	462	5	r	r	NOUN
ejpam-5066	462	6	)	)	PUNCT
ejpam-5066	462	7	is	be	AUX
ejpam-5066	462	8	demonstrated	demonstrate	VERB
ejpam-5066	462	9	to	to	PART
ejpam-5066	462	10	be	be	AUX
ejpam-5066	462	11	an	an	DET
ejpam-5066	462	12	abelian	abelian	ADJ
ejpam-5066	462	13	exchange	exchange	NOUN
ejpam-5066	462	14	ring	ring	NOUN
ejpam-5066	462	15	.	.	PUNCT
ejpam-5066	463	1	by	by	ADP
ejpam-5066	463	2	[	[	X
ejpam-5066	463	3	14	14	NUM
ejpam-5066	463	4	,	,	PUNCT
ejpam-5066	463	5	proposition	proposition	NOUN
ejpam-5066	463	6	1.8	1.8	NUM
ejpam-5066	463	7	]	]	PUNCT
ejpam-5066	463	8	,	,	PUNCT
ejpam-5066	463	9	r	r	NOUN
ejpam-5066	463	10	/	/	SYM
ejpam-5066	463	11	j(r	j(r	PROPN
ejpam-5066	463	12	)	)	PUNCT
ejpam-5066	463	13	is	be	AUX
ejpam-5066	463	14	a	a	DET
ejpam-5066	463	15	clean	clean	ADJ
ejpam-5066	463	16	ring	ring	NOUN
ejpam-5066	463	17	and	and	CCONJ
ejpam-5066	463	18	we	we	PRON
ejpam-5066	463	19	can	can	AUX
ejpam-5066	463	20	find	find	VERB
ejpam-5066	463	21	e	e	NOUN
ejpam-5066	463	22	∈	∈	PROPN
ejpam-5066	463	23	i(r	i(r	PROPN
ejpam-5066	463	24	)	)	PUNCT
ejpam-5066	463	25	such	such	ADJ
ejpam-5066	463	26	that	that	SCONJ
ejpam-5066	463	27	a−	a−	PROPN
ejpam-5066	463	28	e	e	NOUN
ejpam-5066	463	29	is	be	AUX
ejpam-5066	463	30	a	a	DET
ejpam-5066	463	31	unit	unit	NOUN
ejpam-5066	463	32	in	in	ADP
ejpam-5066	463	33	r	r	PROPN
ejpam-5066	463	34	/	/	SYM
ejpam-5066	463	35	j	j	PROPN
ejpam-5066	463	36	(	(	PUNCT
ejpam-5066	463	37	r	r	NOUN
ejpam-5066	463	38	)	)	PUNCT
ejpam-5066	463	39	,	,	PUNCT
ejpam-5066	463	40	for	for	ADP
ejpam-5066	463	41	every	every	DET
ejpam-5066	463	42	a	a	DET
ejpam-5066	463	43	∈	∈	PROPN
ejpam-5066	463	44	r.	r.	PROPN
ejpam-5066	463	45	so	so	ADV
ejpam-5066	463	46	,	,	PUNCT
ejpam-5066	463	47	a	a	DET
ejpam-5066	463	48	-	-	PUNCT
ejpam-5066	463	49	e	e	NOUN
ejpam-5066	463	50	must	must	AUX
ejpam-5066	463	51	also	also	ADV
ejpam-5066	463	52	be	be	AUX
ejpam-5066	463	53	a	a	DET
ejpam-5066	463	54	unit	unit	NOUN
ejpam-5066	463	55	in	in	ADP
ejpam-5066	463	56	r	r	NOUN
ejpam-5066	463	57	since	since	SCONJ
ejpam-5066	463	58	e	e	PROPN
ejpam-5066	463	59	∈	∈	PROPN
ejpam-5066	463	60	cn(r	cn(r	PRON
ejpam-5066	463	61	)	)	PUNCT
ejpam-5066	463	62	is	be	AUX
ejpam-5066	463	63	a	a	DET
ejpam-5066	463	64	n	n	CCONJ
ejpam-5066	463	65	-	-	PUNCT
ejpam-5066	463	66	central	central	ADJ
ejpam-5066	463	67	idempotent	idempotent	NOUN
ejpam-5066	463	68	according	accord	VERB
ejpam-5066	463	69	to	to	ADP
ejpam-5066	463	70	the	the	DET
ejpam-5066	463	71	given	give	VERB
ejpam-5066	463	72	assumption	assumption	NOUN
ejpam-5066	463	73	.	.	PUNCT
ejpam-5066	464	1	(	(	PUNCT
ejpam-5066	464	2	ii)⇒	ii)⇒	X
ejpam-5066	464	3	(	(	PUNCT
ejpam-5066	464	4	iii	iii	NOUN
ejpam-5066	464	5	)	)	PUNCT
ejpam-5066	464	6	is	be	AUX
ejpam-5066	464	7	direct	direct	ADJ
ejpam-5066	464	8	from	from	ADP
ejpam-5066	464	9	the	the	DET
ejpam-5066	464	10	definitions	definition	NOUN
ejpam-5066	464	11	.	.	PUNCT
ejpam-5066	465	1	(	(	PUNCT
ejpam-5066	465	2	iii)⇒	iii)⇒	PROPN
ejpam-5066	465	3	(	(	PUNCT
ejpam-5066	465	4	iv	iv	X
ejpam-5066	465	5	):	):	PUNCT
ejpam-5066	465	6	if	if	SCONJ
ejpam-5066	465	7	a	a	DET
ejpam-5066	465	8	∈	∈	PROPN
ejpam-5066	465	9	r	r	NOUN
ejpam-5066	465	10	,	,	PUNCT
ejpam-5066	465	11	then	then	ADV
ejpam-5066	465	12	1	1	NUM
ejpam-5066	465	13	−	−	NOUN
ejpam-5066	465	14	a	a	DET
ejpam-5066	465	15	=	=	SYM
ejpam-5066	465	16	e	e	X
ejpam-5066	465	17	+	+	CCONJ
ejpam-5066	465	18	u	u	NOUN
ejpam-5066	465	19	,	,	PUNCT
ejpam-5066	465	20	for	for	ADP
ejpam-5066	465	21	some	some	DET
ejpam-5066	465	22	e	e	NOUN
ejpam-5066	465	23	∈	∈	PROPN
ejpam-5066	465	24	cn(r	cn(r	PRON
ejpam-5066	465	25	)	)	PUNCT
ejpam-5066	465	26	and	and	CCONJ
ejpam-5066	465	27	u	u	PROPN
ejpam-5066	465	28	∈	∈	PROPN
ejpam-5066	465	29	u(r	u(r	PROPN
ejpam-5066	465	30	)	)	PUNCT
ejpam-5066	465	31	for	for	ADP
ejpam-5066	465	32	the	the	DET
ejpam-5066	465	33	assumption	assumption	NOUN
ejpam-5066	465	34	.	.	PUNCT
ejpam-5066	466	1	define	define	VERB
ejpam-5066	466	2	the	the	DET
ejpam-5066	466	3	idempotent	idempotent	NOUN
ejpam-5066	466	4	f	f	NOUN
ejpam-5066	466	5	=	=	SYM
ejpam-5066	466	6	ueu−1	ueu−1	NOUN
ejpam-5066	466	7	which	which	PRON
ejpam-5066	466	8	is	be	AUX
ejpam-5066	466	9	conjugate	conjugate	ADJ
ejpam-5066	466	10	to	to	ADP
ejpam-5066	466	11	e	e	NOUN
ejpam-5066	466	12	and	and	CCONJ
ejpam-5066	466	13	consequently	consequently	ADV
ejpam-5066	466	14	f	f	PROPN
ejpam-5066	466	15	∈	∈	PROPN
ejpam-5066	466	16	cn(r	cn(r	PRON
ejpam-5066	466	17	)	)	PUNCT
ejpam-5066	466	18	,	,	PUNCT
ejpam-5066	466	19	by	by	ADP
ejpam-5066	466	20	proposition	proposition	NOUN
ejpam-5066	466	21	7	7	NUM
ejpam-5066	466	22	.	.	PUNCT
ejpam-5066	467	1	now	now	ADV
ejpam-5066	467	2	,	,	PUNCT
ejpam-5066	467	3	f	f	PROPN
ejpam-5066	468	1	=	=	PUNCT
ejpam-5066	468	2	ueu−1	ueu−1	PROPN
ejpam-5066	468	3	=	=	SYM
ejpam-5066	468	4	(	(	PUNCT
ejpam-5066	468	5	1	1	NUM
ejpam-5066	468	6	−	−	NOUN
ejpam-5066	468	7	a	a	DET
ejpam-5066	468	8	−	−	NOUN
ejpam-5066	468	9	e)eu−1	e)eu−1	NOUN
ejpam-5066	468	10	=	=	PUNCT
ejpam-5066	468	11	−aeu−1	−aeu−1	PROPN
ejpam-5066	468	12	∈	∈	PROPN
ejpam-5066	468	13	ar	ar	PROPN
ejpam-5066	468	14	and	and	CCONJ
ejpam-5066	468	15	1−	1−	NUM
ejpam-5066	468	16	f	f	NOUN
ejpam-5066	468	17	=	=	SYM
ejpam-5066	468	18	1−ueu−1	1−ueu−1	PROPN
ejpam-5066	469	1	=	=	PUNCT
ejpam-5066	469	2	uu−1−ueu−1	uu−1−ueu−1	ADJ
ejpam-5066	469	3	=	=	PUNCT
ejpam-5066	469	4	u(1−	u(1−	ADJ
ejpam-5066	469	5	e)u−1	e)u−1	NOUN
ejpam-5066	469	6	=	=	SYM
ejpam-5066	469	7	u(1−	u(1−	ADJ
ejpam-5066	469	8	e)u−1	e)u−1	NOUN
ejpam-5066	469	9	=	=	SYM
ejpam-5066	469	10	(	(	PUNCT
ejpam-5066	469	11	1−a−	1−a−	NUM
ejpam-5066	469	12	e)(1−	e)(1−	ADJ
ejpam-5066	469	13	e)u−1	e)u−1	NOUN
ejpam-5066	469	14	=	=	SYM
ejpam-5066	469	15	(	(	PUNCT
ejpam-5066	469	16	1−	1−	NUM
ejpam-5066	469	17	a)(1−	a)(1−	PROPN
ejpam-5066	469	18	e)u−1	e)u−1	PROPN
ejpam-5066	469	19	∈	∈	PROPN
ejpam-5066	469	20	(	(	PUNCT
ejpam-5066	469	21	1−	1−	NUM
ejpam-5066	469	22	a)r	a)r	NUM
ejpam-5066	469	23	;	;	PUNCT
ejpam-5066	469	24	it	it	PRON
ejpam-5066	469	25	follows	follow	VERB
ejpam-5066	469	26	.	.	PUNCT
ejpam-5066	470	1	(	(	PUNCT
ejpam-5066	470	2	iv)⇒	iv)⇒	X
ejpam-5066	470	3	(	(	PUNCT
ejpam-5066	470	4	i	i	NOUN
ejpam-5066	470	5	):	):	PUNCT
ejpam-5066	470	6	the	the	DET
ejpam-5066	470	7	condition	condition	NOUN
ejpam-5066	470	8	shows	show	VERB
ejpam-5066	470	9	that	that	SCONJ
ejpam-5066	470	10	r	r	NOUN
ejpam-5066	470	11	is	be	AUX
ejpam-5066	470	12	exchange	exchange	NOUN
ejpam-5066	470	13	and	and	CCONJ
ejpam-5066	470	14	it	it	PRON
ejpam-5066	470	15	is	be	AUX
ejpam-5066	470	16	enough	enough	ADJ
ejpam-5066	470	17	to	to	PART
ejpam-5066	470	18	show	show	VERB
ejpam-5066	470	19	that	that	SCONJ
ejpam-5066	470	20	r	r	NOUN
ejpam-5066	470	21	is	be	AUX
ejpam-5066	470	22	n	n	CCONJ
ejpam-5066	470	23	-	-	PUNCT
ejpam-5066	470	24	abelian	abelian	ADJ
ejpam-5066	470	25	.	.	PUNCT
ejpam-5066	471	1	for	for	ADP
ejpam-5066	471	2	every	every	DET
ejpam-5066	471	3	e	e	PROPN
ejpam-5066	471	4	∈	∈	PROPN
ejpam-5066	471	5	i(r	i(r	PROPN
ejpam-5066	471	6	)	)	PUNCT
ejpam-5066	471	7	,	,	PUNCT
ejpam-5066	471	8	the	the	DET
ejpam-5066	471	9	assumption	assumption	NOUN
ejpam-5066	471	10	yields	yield	VERB
ejpam-5066	471	11	that	that	SCONJ
ejpam-5066	471	12	there	there	PRON
ejpam-5066	471	13	exist	exist	VERB
ejpam-5066	471	14	f	f	PROPN
ejpam-5066	471	15	∈	∈	PROPN
ejpam-5066	471	16	cn(r	cn(r	PRON
ejpam-5066	471	17	)	)	PUNCT
ejpam-5066	471	18	such	such	ADJ
ejpam-5066	471	19	that	that	SCONJ
ejpam-5066	471	20	f	f	PROPN
ejpam-5066	471	21	∈	∈	PROPN
ejpam-5066	472	1	er	er	INTJ
ejpam-5066	472	2	and	and	CCONJ
ejpam-5066	472	3	1−	1−	NUM
ejpam-5066	472	4	f	f	X
ejpam-5066	472	5	∈	∈	PROPN
ejpam-5066	472	6	(	(	PUNCT
ejpam-5066	472	7	1−	1−	NUM
ejpam-5066	472	8	e)r	e)r	X
ejpam-5066	472	9	.	.	PUNCT
ejpam-5066	473	1	so	so	ADV
ejpam-5066	473	2	f	f	PROPN
ejpam-5066	473	3	=	=	SYM
ejpam-5066	473	4	ef	ef	PROPN
ejpam-5066	473	5	and	and	CCONJ
ejpam-5066	473	6	1−	1−	NUM
ejpam-5066	473	7	f	f	X
ejpam-5066	474	1	=	=	SYM
ejpam-5066	474	2	(	(	PUNCT
ejpam-5066	474	3	1−	1−	NUM
ejpam-5066	474	4	e)(1−	e)(1−	PROPN
ejpam-5066	474	5	f	f	NOUN
ejpam-5066	474	6	)	)	PUNCT
ejpam-5066	474	7	=	=	SYM
ejpam-5066	474	8	1−	1−	NUM
ejpam-5066	474	9	e−	e−	PROPN
ejpam-5066	474	10	f	f	PROPN
ejpam-5066	475	1	+	+	CCONJ
ejpam-5066	475	2	ef	ef	X
ejpam-5066	475	3	=	=	SYM
ejpam-5066	475	4	1−.	1−.	PROPN
ejpam-5066	476	1	hence	hence	ADV
ejpam-5066	476	2	,	,	PUNCT
ejpam-5066	476	3	e	e	X
ejpam-5066	476	4	=	=	PUNCT
ejpam-5066	476	5	f	f	PROPN
ejpam-5066	476	6	∈∈	∈∈	NOUN
ejpam-5066	476	7	cn(r	cn(r	NUM
ejpam-5066	476	8	)	)	PUNCT
ejpam-5066	476	9	and	and	CCONJ
ejpam-5066	476	10	r	r	NOUN
ejpam-5066	476	11	is	be	AUX
ejpam-5066	476	12	n	n	CCONJ
ejpam-5066	476	13	-	-	PUNCT
ejpam-5066	476	14	abelian	abelian	NOUN
ejpam-5066	476	15	.	.	PUNCT
ejpam-5066	477	1	vaserstein	vaserstein	PROPN
ejpam-5066	478	1	[	[	X
ejpam-5066	478	2	19	19	NUM
ejpam-5066	478	3	]	]	PUNCT
ejpam-5066	478	4	defines	define	VERB
ejpam-5066	478	5	a	a	DET
ejpam-5066	478	6	ring	ring	NOUN
ejpam-5066	478	7	r	r	NOUN
ejpam-5066	478	8	to	to	PART
ejpam-5066	478	9	have	have	VERB
ejpam-5066	478	10	stable	stable	ADJ
ejpam-5066	478	11	range	range	NOUN
ejpam-5066	478	12	1	1	NUM
ejpam-5066	478	13	if	if	SCONJ
ejpam-5066	478	14	for	for	ADP
ejpam-5066	478	15	any	any	DET
ejpam-5066	478	16	a	a	NOUN
ejpam-5066	478	17	,	,	PUNCT
ejpam-5066	478	18	b	b	X
ejpam-5066	478	19	∈	∈	NOUN
ejpam-5066	478	20	r	r	NOUN
ejpam-5066	478	21	with	with	ADP
ejpam-5066	478	22	ar+br	ar+br	X
ejpam-5066	478	23	=	=	SYM
ejpam-5066	478	24	r	r	NOUN
ejpam-5066	478	25	,	,	PUNCT
ejpam-5066	478	26	there	there	PRON
ejpam-5066	478	27	exists	exist	VERB
ejpam-5066	478	28	y	y	PROPN
ejpam-5066	478	29	∈	∈	PROPN
ejpam-5066	478	30	r	r	NOUN
ejpam-5066	478	31	such	such	ADJ
ejpam-5066	478	32	that	that	PRON
ejpam-5066	478	33	a+	a+	PUNCT
ejpam-5066	478	34	by	by	SCONJ
ejpam-5066	478	35	is	be	AUX
ejpam-5066	478	36	right	right	ADV
ejpam-5066	478	37	invertible	invertible	ADJ
ejpam-5066	478	38	.	.	PUNCT
ejpam-5066	479	1	r	r	NOUN
ejpam-5066	479	2	has	have	VERB
ejpam-5066	479	3	stable	stable	ADJ
ejpam-5066	479	4	range	range	NOUN
ejpam-5066	479	5	1	1	NUM
ejpam-5066	479	6	if	if	SCONJ
ejpam-5066	479	7	and	and	CCONJ
ejpam-5066	479	8	only	only	ADV
ejpam-5066	479	9	if	if	SCONJ
ejpam-5066	479	10	r	r	NOUN
ejpam-5066	479	11	/	/	SYM
ejpam-5066	479	12	j	j	PROPN
ejpam-5066	479	13	(	(	PUNCT
ejpam-5066	479	14	r	r	NOUN
ejpam-5066	479	15	)	)	PUNCT
ejpam-5066	479	16	has	have	VERB
ejpam-5066	479	17	stable	stable	ADJ
ejpam-5066	479	18	range	range	NOUN
ejpam-5066	479	19	1	1	NUM
ejpam-5066	479	20	.	.	PUNCT
ejpam-5066	480	1	the	the	DET
ejpam-5066	480	2	next	next	ADJ
ejpam-5066	480	3	corollary	corollary	NOUN
ejpam-5066	480	4	demonstrates	demonstrate	VERB
ejpam-5066	480	5	that	that	SCONJ
ejpam-5066	480	6	exchange	exchange	NOUN
ejpam-5066	480	7	rings	ring	NOUN
ejpam-5066	480	8	with	with	ADP
ejpam-5066	480	9	n	n	CCONJ
ejpam-5066	480	10	-	-	PUNCT
ejpam-5066	480	11	central	central	ADJ
ejpam-5066	480	12	idempotents	idempotent	NOUN
ejpam-5066	480	13	,	,	PUNCT
ejpam-5066	480	14	for	for	ADP
ejpam-5066	480	15	some	some	DET
ejpam-5066	480	16	n	n	NOUN
ejpam-5066	480	17	have	have	VERB
ejpam-5066	480	18	stable	stable	ADJ
ejpam-5066	480	19	range	range	NOUN
ejpam-5066	480	20	1	1	NUM
ejpam-5066	480	21	.	.	PUNCT
ejpam-5066	480	22	corollary	corollary	ADJ
ejpam-5066	480	23	13	13	NUM
ejpam-5066	480	24	.	.	PUNCT
ejpam-5066	481	1	every	every	DET
ejpam-5066	481	2	n	n	NUM
ejpam-5066	481	3	-	-	PUNCT
ejpam-5066	481	4	abelain	abelain	NOUN
ejpam-5066	481	5	exchange	exchange	NOUN
ejpam-5066	481	6	rings	ring	NOUN
ejpam-5066	481	7	,	,	PUNCT
ejpam-5066	481	8	for	for	ADP
ejpam-5066	481	9	some	some	DET
ejpam-5066	481	10	n	n	CCONJ
ejpam-5066	481	11	,	,	PUNCT
ejpam-5066	481	12	has	have	VERB
ejpam-5066	481	13	stable	stable	ADJ
ejpam-5066	481	14	range	range	NOUN
ejpam-5066	481	15	1	1	NUM
ejpam-5066	481	16	.	.	PUNCT
ejpam-5066	482	1	proof	proof	NOUN
ejpam-5066	482	2	.	.	PUNCT
ejpam-5066	483	1	corollary	corollary	ADJ
ejpam-5066	483	2	5	5	NUM
ejpam-5066	483	3	and	and	CCONJ
ejpam-5066	483	4	[	[	X
ejpam-5066	483	5	27	27	NUM
ejpam-5066	483	6	,	,	PUNCT
ejpam-5066	483	7	theorem	theorem	VERB
ejpam-5066	483	8	6	6	NUM
ejpam-5066	483	9	]	]	PUNCT
ejpam-5066	483	10	jointly	jointly	ADV
ejpam-5066	483	11	yield	yield	VERB
ejpam-5066	483	12	the	the	DET
ejpam-5066	483	13	result	result	NOUN
ejpam-5066	483	14	.	.	PUNCT
ejpam-5066	484	1	the	the	DET
ejpam-5066	484	2	inherent	inherent	ADJ
ejpam-5066	484	3	generality	generality	NOUN
ejpam-5066	484	4	of	of	ADP
ejpam-5066	484	5	hyperrings	hyperring	NOUN
ejpam-5066	484	6	as	as	ADP
ejpam-5066	484	7	an	an	DET
ejpam-5066	484	8	extension	extension	NOUN
ejpam-5066	484	9	of	of	ADP
ejpam-5066	484	10	rings	ring	NOUN
ejpam-5066	484	11	prompts	prompt	VERB
ejpam-5066	484	12	a	a	DET
ejpam-5066	484	13	pertinent	pertinent	ADJ
ejpam-5066	484	14	consideration	consideration	NOUN
ejpam-5066	484	15	:	:	PUNCT
ejpam-5066	484	16	the	the	DET
ejpam-5066	484	17	exploration	exploration	NOUN
ejpam-5066	484	18	of	of	ADP
ejpam-5066	484	19	the	the	DET
ejpam-5066	484	20	concepts	concept	NOUN
ejpam-5066	484	21	of	of	ADP
ejpam-5066	484	22	n	n	CCONJ
ejpam-5066	484	23	-	-	PUNCT
ejpam-5066	484	24	central	central	ADJ
ejpam-5066	484	25	idempotents	idempotent	NOUN
ejpam-5066	484	26	and	and	CCONJ
ejpam-5066	484	27	n	n	CCONJ
ejpam-5066	484	28	-	-	PUNCT
ejpam-5066	484	29	abelian	abelian	NOUN
ejpam-5066	484	30	rings	ring	NOUN
ejpam-5066	484	31	within	within	ADP
ejpam-5066	484	32	the	the	DET
ejpam-5066	484	33	domain	domain	NOUN
ejpam-5066	484	34	of	of	ADP
ejpam-5066	484	35	hyperrings	hyperring	NOUN
ejpam-5066	484	36	.	.	PUNCT
ejpam-5066	485	1	this	this	DET
ejpam-5066	485	2	avenue	avenue	NOUN
ejpam-5066	485	3	of	of	ADP
ejpam-5066	485	4	inquiry	inquiry	NOUN
ejpam-5066	485	5	holds	hold	VERB
ejpam-5066	485	6	promise	promise	NOUN
ejpam-5066	485	7	in	in	ADP
ejpam-5066	485	8	further	far	ADV
ejpam-5066	485	9	elucidating	elucidate	VERB
ejpam-5066	485	10	the	the	DET
ejpam-5066	485	11	structural	structural	ADJ
ejpam-5066	485	12	properties	property	NOUN
ejpam-5066	485	13	and	and	CCONJ
ejpam-5066	485	14	algebraic	algebraic	ADJ
ejpam-5066	485	15	characteristics	characteristic	NOUN
ejpam-5066	485	16	inherent	inherent	ADJ
ejpam-5066	485	17	in	in	ADP
ejpam-5066	485	18	hyperring	hyperre	VERB
ejpam-5066	485	19	theory	theory	NOUN
ejpam-5066	485	20	.	.	PUNCT
ejpam-5066	486	1	for	for	ADP
ejpam-5066	486	2	readers	reader	NOUN
ejpam-5066	486	3	intrigued	intrigue	VERB
ejpam-5066	486	4	by	by	ADP
ejpam-5066	486	5	the	the	DET
ejpam-5066	486	6	realm	realm	NOUN
ejpam-5066	486	7	of	of	ADP
ejpam-5066	486	8	hyperrings	hyperring	NOUN
ejpam-5066	486	9	and	and	CCONJ
ejpam-5066	486	10	desiring	desire	VERB
ejpam-5066	486	11	a	a	DET
ejpam-5066	486	12	deeper	deep	ADJ
ejpam-5066	486	13	understanding	understanding	NOUN
ejpam-5066	486	14	,	,	PUNCT
ejpam-5066	486	15	an	an	DET
ejpam-5066	486	16	extensive	extensive	ADJ
ejpam-5066	486	17	exploration	exploration	NOUN
ejpam-5066	486	18	can	can	AUX
ejpam-5066	486	19	be	be	AUX
ejpam-5066	486	20	found	find	VERB
ejpam-5066	486	21	in	in	ADP
ejpam-5066	486	22	the	the	DET
ejpam-5066	486	23	following	follow	VERB
ejpam-5066	486	24	scholarly	scholarly	ADJ
ejpam-5066	486	25	sources	source	NOUN
ejpam-5066	486	26	:	:	PUNCT
ejpam-5066	487	1	[	[	X
ejpam-5066	487	2	9	9	NUM
ejpam-5066	487	3	,	,	PUNCT
ejpam-5066	487	4	15	15	NUM
ejpam-5066	487	5	,	,	PUNCT
ejpam-5066	487	6	16	16	NUM
ejpam-5066	487	7	,	,	PUNCT
ejpam-5066	487	8	18	18	NUM
ejpam-5066	487	9	]	]	PUNCT
ejpam-5066	487	10	.	.	PUNCT
ejpam-5066	488	1	these	these	DET
ejpam-5066	488	2	references	reference	NOUN
ejpam-5066	488	3	offer	offer	VERB
ejpam-5066	488	4	comprehensive	comprehensive	ADJ
ejpam-5066	488	5	insights	insight	NOUN
ejpam-5066	488	6	into	into	ADP
ejpam-5066	488	7	hyperrings	hyperring	NOUN
ejpam-5066	488	8	,	,	PUNCT
ejpam-5066	488	9	serving	serve	VERB
ejpam-5066	488	10	as	as	ADP
ejpam-5066	488	11	valuable	valuable	ADJ
ejpam-5066	488	12	resources	resource	NOUN
ejpam-5066	488	13	for	for	ADP
ejpam-5066	488	14	those	those	PRON
ejpam-5066	488	15	engaged	engage	VERB
ejpam-5066	488	16	in	in	ADP
ejpam-5066	488	17	advanced	advanced	ADJ
ejpam-5066	488	18	studies	study	NOUN
ejpam-5066	488	19	or	or	CCONJ
ejpam-5066	488	20	research	research	NOUN
ejpam-5066	488	21	endeavors	endeavor	NOUN
ejpam-5066	488	22	within	within	ADP
ejpam-5066	488	23	this	this	DET
ejpam-5066	488	24	domain	domain	NOUN
ejpam-5066	488	25	.	.	PUNCT
ejpam-5066	489	1	references	reference	NOUN
ejpam-5066	489	2	750	750	NUM
ejpam-5066	489	3	5	5	NUM
ejpam-5066	489	4	.	.	PUNCT
ejpam-5066	489	5	conclusion	conclusion	NOUN
ejpam-5066	489	6	within	within	ADP
ejpam-5066	489	7	this	this	DET
ejpam-5066	489	8	paper	paper	NOUN
ejpam-5066	489	9	,	,	PUNCT
ejpam-5066	489	10	we	we	PRON
ejpam-5066	489	11	present	present	VERB
ejpam-5066	489	12	a	a	DET
ejpam-5066	489	13	series	series	NOUN
ejpam-5066	489	14	of	of	ADP
ejpam-5066	489	15	comprehensive	comprehensive	ADJ
ejpam-5066	489	16	findings	finding	NOUN
ejpam-5066	489	17	.	.	PUNCT
ejpam-5066	490	1	initially	initially	ADV
ejpam-5066	490	2	,	,	PUNCT
ejpam-5066	490	3	we	we	PRON
ejpam-5066	490	4	establish	establish	VERB
ejpam-5066	490	5	that	that	SCONJ
ejpam-5066	490	6	every	every	DET
ejpam-5066	490	7	n	n	CCONJ
ejpam-5066	490	8	-	-	PUNCT
ejpam-5066	490	9	central	central	ADJ
ejpam-5066	490	10	idempotent	idempotent	NOUN
ejpam-5066	490	11	within	within	ADP
ejpam-5066	490	12	a	a	DET
ejpam-5066	490	13	semiprime	semiprime	NOUN
ejpam-5066	490	14	ring	ring	NOUN
ejpam-5066	490	15	is	be	AUX
ejpam-5066	490	16	central	central	ADJ
ejpam-5066	490	17	.	.	PUNCT
ejpam-5066	491	1	additionally	additionally	ADV
ejpam-5066	491	2	,	,	PUNCT
ejpam-5066	491	3	we	we	PRON
ejpam-5066	491	4	demonstrate	demonstrate	VERB
ejpam-5066	491	5	that	that	SCONJ
ejpam-5066	491	6	a	a	DET
ejpam-5066	491	7	full	full	ADJ
ejpam-5066	491	8	idempotent	idempotent	NOUN
ejpam-5066	491	9	e	e	NOUN
ejpam-5066	491	10	is	be	AUX
ejpam-5066	491	11	an	an	DET
ejpam-5066	491	12	n	n	CCONJ
ejpam-5066	491	13	-	-	PUNCT
ejpam-5066	491	14	central	central	ADJ
ejpam-5066	491	15	idempotent	idempotent	NOUN
ejpam-5066	491	16	if	if	SCONJ
ejpam-5066	491	17	and	and	CCONJ
ejpam-5066	491	18	only	only	ADV
ejpam-5066	491	19	if	if	SCONJ
ejpam-5066	491	20	e	e	PRON
ejpam-5066	491	21	equals	equal	VERB
ejpam-5066	491	22	zero	zero	NUM
ejpam-5066	491	23	.	.	PUNCT
ejpam-5066	492	1	moreover	moreover	ADV
ejpam-5066	492	2	,	,	PUNCT
ejpam-5066	492	3	we	we	PRON
ejpam-5066	492	4	showcase	showcase	VERB
ejpam-5066	492	5	that	that	SCONJ
ejpam-5066	492	6	if	if	SCONJ
ejpam-5066	492	7	e	e	PROPN
ejpam-5066	492	8	and	and	CCONJ
ejpam-5066	492	9	f	f	PROPN
ejpam-5066	492	10	represent	represent	VERB
ejpam-5066	492	11	isomorphic	isomorphic	ADJ
ejpam-5066	492	12	idempotents	idempotent	NOUN
ejpam-5066	492	13	with	with	ADP
ejpam-5066	492	14	isomorphic	isomorphic	ADJ
ejpam-5066	492	15	complements	complement	NOUN
ejpam-5066	492	16	,	,	PUNCT
ejpam-5066	492	17	then	then	ADV
ejpam-5066	492	18	e	e	X
ejpam-5066	492	19	being	be	AUX
ejpam-5066	492	20	n	n	CCONJ
ejpam-5066	492	21	-	-	PUNCT
ejpam-5066	492	22	central	central	NOUN
ejpam-5066	492	23	is	be	AUX
ejpam-5066	492	24	equivalent	equivalent	ADJ
ejpam-5066	492	25	to	to	ADP
ejpam-5066	492	26	f	f	PROPN
ejpam-5066	492	27	being	be	AUX
ejpam-5066	492	28	n	n	CCONJ
ejpam-5066	492	29	-	-	PUNCT
ejpam-5066	492	30	central	central	ADJ
ejpam-5066	492	31	.	.	PUNCT
ejpam-5066	493	1	furthermore	furthermore	ADV
ejpam-5066	493	2	,	,	PUNCT
ejpam-5066	493	3	we	we	PRON
ejpam-5066	493	4	provide	provide	VERB
ejpam-5066	493	5	proof	proof	NOUN
ejpam-5066	493	6	indicating	indicate	VERB
ejpam-5066	493	7	that	that	SCONJ
ejpam-5066	493	8	if	if	SCONJ
ejpam-5066	493	9	e	e	PROPN
ejpam-5066	493	10	denotes	denote	VERB
ejpam-5066	493	11	an	an	DET
ejpam-5066	493	12	n	n	CCONJ
ejpam-5066	493	13	-	-	PUNCT
ejpam-5066	493	14	central	central	ADJ
ejpam-5066	493	15	idempotent	idempotent	NOUN
ejpam-5066	493	16	within	within	ADP
ejpam-5066	493	17	a	a	DET
ejpam-5066	493	18	ring	ring	NOUN
ejpam-5066	493	19	r	r	NOUN
ejpam-5066	493	20	,	,	PUNCT
ejpam-5066	493	21	then	then	ADV
ejpam-5066	493	22	the	the	DET
ejpam-5066	493	23	left	left	ADJ
ejpam-5066	493	24	ideal	ideal	NOUN
ejpam-5066	493	25	er	er	INTJ
ejpam-5066	493	26	is	be	AUX
ejpam-5066	493	27	directly	directly	ADV
ejpam-5066	493	28	finite	finite	ADJ
ejpam-5066	493	29	.	.	PUNCT
ejpam-5066	494	1	additionally	additionally	ADV
ejpam-5066	494	2	,	,	PUNCT
ejpam-5066	494	3	we	we	PRON
ejpam-5066	494	4	illustrate	illustrate	VERB
ejpam-5066	494	5	that	that	SCONJ
ejpam-5066	494	6	the	the	DET
ejpam-5066	494	7	condition	condition	NOUN
ejpam-5066	494	8	mandating	mandate	VERB
ejpam-5066	494	9	all	all	DET
ejpam-5066	494	10	idempotents	idempotent	NOUN
ejpam-5066	494	11	to	to	PART
ejpam-5066	494	12	be	be	AUX
ejpam-5066	494	13	n	n	PRON
ejpam-5066	494	14	-	-	PUNCT
ejpam-5066	494	15	central	central	ADJ
ejpam-5066	494	16	extends	extend	VERB
ejpam-5066	494	17	to	to	ADP
ejpam-5066	494	18	upper	upper	ADJ
ejpam-5066	494	19	triangular	triangular	NOUN
ejpam-5066	494	20	matrix	matrix	NOUN
ejpam-5066	494	21	rings	ring	NOUN
ejpam-5066	494	22	,	,	PUNCT
ejpam-5066	494	23	thereby	thereby	ADV
ejpam-5066	494	24	compelling	compel	VERB
ejpam-5066	494	25	a	a	DET
ejpam-5066	494	26	von	von	PROPN
ejpam-5066	494	27	neumann	neumann	PROPN
ejpam-5066	494	28	ring	ring	PROPN
ejpam-5066	494	29	to	to	PART
ejpam-5066	494	30	attain	attain	VERB
ejpam-5066	494	31	the	the	DET
ejpam-5066	494	32	status	status	NOUN
ejpam-5066	494	33	of	of	ADP
ejpam-5066	494	34	strong	strong	ADJ
ejpam-5066	494	35	regularity	regularity	NOUN
ejpam-5066	494	36	.	.	PUNCT
ejpam-5066	495	1	lastly	lastly	ADV
ejpam-5066	495	2	,	,	PUNCT
ejpam-5066	495	3	we	we	PRON
ejpam-5066	495	4	conclude	conclude	VERB
ejpam-5066	495	5	our	our	PRON
ejpam-5066	495	6	findings	finding	NOUN
ejpam-5066	495	7	by	by	ADP
ejpam-5066	495	8	demonstrating	demonstrate	VERB
ejpam-5066	495	9	that	that	SCONJ
ejpam-5066	495	10	a	a	DET
ejpam-5066	495	11	ring	ring	NOUN
ejpam-5066	495	12	wherein	wherein	SCONJ
ejpam-5066	495	13	all	all	DET
ejpam-5066	495	14	idempotents	idempotent	NOUN
ejpam-5066	495	15	are	be	AUX
ejpam-5066	495	16	n	n	PRON
ejpam-5066	495	17	-	-	PUNCT
ejpam-5066	495	18	central	central	NOUN
ejpam-5066	495	19	achieves	achieve	VERB
ejpam-5066	495	20	the	the	DET
ejpam-5066	495	21	status	status	NOUN
ejpam-5066	495	22	of	of	ADP
ejpam-5066	495	23	an	an	DET
ejpam-5066	495	24	exchange	exchange	NOUN
ejpam-5066	495	25	ring	ring	NOUN
ejpam-5066	495	26	if	if	SCONJ
ejpam-5066	495	27	and	and	CCONJ
ejpam-5066	495	28	only	only	ADV
ejpam-5066	495	29	the	the	DET
ejpam-5066	495	30	ring	ring	NOUN
ejpam-5066	495	31	is	be	AUX
ejpam-5066	495	32	clean	clean	ADJ
ejpam-5066	495	33	.	.	PUNCT
ejpam-5066	496	1	acknowledgements	acknowledgement	NOUN
ejpam-5066	496	2	the	the	DET
ejpam-5066	496	3	authors	author	NOUN
ejpam-5066	496	4	would	would	AUX
ejpam-5066	496	5	like	like	VERB
ejpam-5066	496	6	to	to	PART
ejpam-5066	496	7	express	express	VERB
ejpam-5066	496	8	their	their	PRON
ejpam-5066	496	9	sincere	sincere	ADJ
ejpam-5066	496	10	gratitude	gratitude	NOUN
ejpam-5066	496	11	to	to	ADP
ejpam-5066	496	12	the	the	DET
ejpam-5066	496	13	anonymous	anonymous	ADJ
ejpam-5066	496	14	referees	referee	NOUN
ejpam-5066	496	15	for	for	ADP
ejpam-5066	496	16	their	their	PRON
ejpam-5066	496	17	invaluable	invaluable	ADJ
ejpam-5066	496	18	feedback	feedback	NOUN
ejpam-5066	496	19	and	and	CCONJ
ejpam-5066	496	20	expert	expert	ADJ
ejpam-5066	496	21	evaluation	evaluation	NOUN
ejpam-5066	496	22	of	of	ADP
ejpam-5066	496	23	our	our	PRON
ejpam-5066	496	24	manuscript	manuscript	NOUN
ejpam-5066	496	25	,	,	PUNCT
ejpam-5066	496	26	which	which	PRON
ejpam-5066	496	27	greatly	greatly	ADV
ejpam-5066	496	28	contributed	contribute	VERB
ejpam-5066	496	29	to	to	ADP
ejpam-5066	496	30	its	its	PRON
ejpam-5066	496	31	improvement	improvement	NOUN
ejpam-5066	496	32	references	reference	NOUN
ejpam-5066	496	33	[	[	X
ejpam-5066	496	34	1	1	NUM
ejpam-5066	496	35	]	]	PUNCT
ejpam-5066	496	36	g.	g.	PROPN
ejpam-5066	496	37	f.	f.	PROPN
ejpam-5066	496	38	birkenmeier	birkenmeier	PROPN
ejpam-5066	496	39	.	.	PUNCT
ejpam-5066	497	1	idempotents	idempotent	NOUN
ejpam-5066	497	2	and	and	CCONJ
ejpam-5066	497	3	completely	completely	ADV
ejpam-5066	497	4	semiprime	semiprime	NOUN
ejpam-5066	497	5	ideals	ideal	NOUN
ejpam-5066	497	6	.	.	PUNCT
ejpam-5066	498	1	commun	commun	PROPN
ejpam-5066	498	2	.	.	PUNCT
ejpam-5066	499	1	algebra	algebra	PROPN
ejpam-5066	499	2	,	,	PUNCT
ejpam-5066	499	3	11:567–58	11:567–58	NUM
ejpam-5066	499	4	,	,	PUNCT
ejpam-5066	499	5	1983	1983	NUM
ejpam-5066	499	6	.	.	PUNCT
ejpam-5066	500	1	[	[	X
ejpam-5066	500	2	2	2	X
ejpam-5066	500	3	]	]	X
ejpam-5066	500	4	g.	g.	PROPN
ejpam-5066	500	5	f.	f.	PROPN
ejpam-5066	500	6	birkenmeier	birkenmeier	PROPN
ejpam-5066	500	7	,	,	PUNCT
ejpam-5066	500	8	h.	h.	PROPN
ejpam-5066	500	9	heatherly	heatherly	PROPN
ejpam-5066	500	10	,	,	PUNCT
ejpam-5066	500	11	j.	j.	PROPN
ejpam-5066	500	12	y.	y.	PROPN
ejpam-5066	500	13	kim	kim	PROPN
ejpam-5066	500	14	,	,	PUNCT
ejpam-5066	500	15	and	and	CCONJ
ejpam-5066	500	16	j.	j.	PROPN
ejpam-5066	500	17	k.	k.	PROPN
ejpam-5066	500	18	park	park	PROPN
ejpam-5066	500	19	.	.	PUNCT
ejpam-5066	501	1	algebras	algebras	PROPN
ejpam-5066	501	2	generated	generate	VERB
ejpam-5066	501	3	by	by	ADP
ejpam-5066	501	4	semicentral	semicentral	ADJ
ejpam-5066	501	5	idempotents	idempotent	NOUN
ejpam-5066	501	6	.	.	PUNCT
ejpam-5066	502	1	acta	acta	PROPN
ejpam-5066	502	2	math	math	PROPN
ejpam-5066	502	3	.	.	PUNCT
ejpam-5066	503	1	hungar	hungar	PROPN
ejpam-5066	503	2	.	.	PUNCT
ejpam-5066	503	3	,	,	PUNCT
ejpam-5066	503	4	95(1	95(1	NOUN
ejpam-5066	503	5	-	-	SYM
ejpam-5066	503	6	2):101–114	2):101–114	NUM
ejpam-5066	503	7	,	,	PUNCT
ejpam-5066	503	8	2002	2002	NUM
ejpam-5066	503	9	.	.	PUNCT
ejpam-5066	504	1	[	[	X
ejpam-5066	504	2	3	3	X
ejpam-5066	504	3	]	]	X
ejpam-5066	504	4	g.	g.	PROPN
ejpam-5066	504	5	f.	f.	PROPN
ejpam-5066	504	6	birkenmeier	birkenmeier	PROPN
ejpam-5066	504	7	,	,	PUNCT
ejpam-5066	504	8	j.	j.	PROPN
ejpam-5066	504	9	y.	y.	PROPN
ejpam-5066	504	10	kim	kim	PROPN
ejpam-5066	504	11	,	,	PUNCT
ejpam-5066	504	12	and	and	CCONJ
ejpam-5066	504	13	j.	j.	PROPN
ejpam-5066	504	14	k.	k.	PROPN
ejpam-5066	504	15	park	park	PROPN
ejpam-5066	504	16	.	.	PUNCT
ejpam-5066	505	1	semicentral	semicentral	ADJ
ejpam-5066	505	2	reduced	reduce	VERB
ejpam-5066	505	3	algebras	algebra	NOUN
ejpam-5066	505	4	.	.	PUNCT
ejpam-5066	506	1	in	in	ADP
ejpam-5066	506	2	international	international	ADJ
ejpam-5066	506	3	symposium	symposium	NOUN
ejpam-5066	506	4	on	on	ADP
ejpam-5066	506	5	ring	ring	NOUN
ejpam-5066	506	6	theory	theory	NOUN
ejpam-5066	506	7	,	,	PUNCT
ejpam-5066	506	8	pages	page	VERB
ejpam-5066	506	9	67–84	67–84	PROPN
ejpam-5066	506	10	.	.	PUNCT
ejpam-5066	507	1	springer	springer	NOUN
ejpam-5066	507	2	,	,	PUNCT
ejpam-5066	507	3	2001	2001	NUM
ejpam-5066	507	4	.	.	PUNCT
ejpam-5066	508	1	[	[	X
ejpam-5066	508	2	4	4	X
ejpam-5066	508	3	]	]	X
ejpam-5066	508	4	g.	g.	PROPN
ejpam-5066	508	5	f.	f.	PROPN
ejpam-5066	508	6	birkenmeier	birkenmeier	PROPN
ejpam-5066	508	7	,	,	PUNCT
ejpam-5066	508	8	j.	j.	PROPN
ejpam-5066	508	9	k.	k.	PROPN
ejpam-5066	508	10	park	park	PROPN
ejpam-5066	508	11	,	,	PUNCT
ejpam-5066	508	12	and	and	CCONJ
ejpam-5066	508	13	s.	s.	PROPN
ejpam-5066	508	14	t.	t.	PROPN
ejpam-5066	508	15	rizvi	rizvi	PROPN
ejpam-5066	508	16	.	.	PUNCT
ejpam-5066	509	1	extensions	extension	NOUN
ejpam-5066	509	2	of	of	ADP
ejpam-5066	509	3	rings	ring	NOUN
ejpam-5066	509	4	and	and	CCONJ
ejpam-5066	509	5	modules	module	NOUN
ejpam-5066	509	6	.	.	PUNCT
ejpam-5066	510	1	springer	springer	NOUN
ejpam-5066	510	2	,	,	PUNCT
ejpam-5066	510	3	new	new	PROPN
ejpam-5066	510	4	york	york	PROPN
ejpam-5066	510	5	heidelberg	heidelberg	PROPN
ejpam-5066	510	6	dordrecht	dordrecht	PROPN
ejpam-5066	510	7	london	london	PROPN
ejpam-5066	510	8	,	,	PUNCT
ejpam-5066	510	9	2013	2013	NUM
ejpam-5066	510	10	.	.	PUNCT
ejpam-5066	511	1	[	[	X
ejpam-5066	511	2	5	5	NUM
ejpam-5066	511	3	]	]	PUNCT
ejpam-5066	511	4	a.	a.	NOUN
ejpam-5066	511	5	j.	j.	PROPN
ejpam-5066	511	6	diesl	diesl	PROPN
ejpam-5066	511	7	,	,	PUNCT
ejpam-5066	511	8	s.	s.	PROPN
ejpam-5066	511	9	j.	j.	PROPN
ejpam-5066	511	10	dittmer	dittmer	PROPN
ejpam-5066	511	11	,	,	PUNCT
ejpam-5066	511	12	and	and	CCONJ
ejpam-5066	511	13	p.	p.	NOUN
ejpam-5066	511	14	p	p	PROPN
ejpam-5066	512	1	nielsen	nielsen	PROPN
ejpam-5066	512	2	.	.	PUNCT
ejpam-5066	513	1	idempotent	idempotent	ADJ
ejpam-5066	513	2	lifting	lifting	NOUN
ejpam-5066	513	3	and	and	CCONJ
ejpam-5066	513	4	ring	ring	NOUN
ejpam-5066	513	5	extensions	extension	NOUN
ejpam-5066	513	6	.	.	PUNCT
ejpam-5066	514	1	j.	j.	PROPN
ejpam-5066	514	2	algebra	algebra	PROPN
ejpam-5066	514	3	appl	appl	PROPN
ejpam-5066	514	4	.	.	PROPN
ejpam-5066	514	5	,	,	PUNCT
ejpam-5066	514	6	15(06):1650112	15(06):1650112	NUM
ejpam-5066	514	7	,	,	PUNCT
ejpam-5066	514	8	2016	2016	NUM
ejpam-5066	514	9	.	.	PUNCT
ejpam-5066	515	1	[	[	X
ejpam-5066	515	2	6	6	NUM
ejpam-5066	515	3	]	]	PUNCT
ejpam-5066	515	4	k.	k.	PROPN
ejpam-5066	515	5	r.	r.	PROPN
ejpam-5066	515	6	goodearl	goodearl	PROPN
ejpam-5066	515	7	and	and	CCONJ
ejpam-5066	515	8	r.	r.	PROPN
ejpam-5066	515	9	b.	b.	PROPN
ejpam-5066	515	10	warfield	warfield	PROPN
ejpam-5066	515	11	jr	jr	PROPN
ejpam-5066	515	12	.	.	PROPN
ejpam-5066	515	13	algebras	algebras	PROPN
ejpam-5066	515	14	over	over	ADP
ejpam-5066	515	15	zero	zero	NUM
ejpam-5066	515	16	-	-	PUNCT
ejpam-5066	515	17	dimensional	dimensional	ADJ
ejpam-5066	515	18	rings	ring	NOUN
ejpam-5066	515	19	.	.	PUNCT
ejpam-5066	516	1	math	math	NOUN
ejpam-5066	516	2	.	.	PUNCT
ejpam-5066	517	1	annal	annal	PROPN
ejpam-5066	517	2	.	.	PUNCT
ejpam-5066	517	3	,	,	PUNCT
ejpam-5066	518	1	223(2):157–168	223(2):157–168	PROPN
ejpam-5066	518	2	,	,	PUNCT
ejpam-5066	518	3	1976	1976	NUM
ejpam-5066	518	4	.	.	PUNCT
ejpam-5066	519	1	[	[	X
ejpam-5066	519	2	7	7	X
ejpam-5066	519	3	]	]	X
ejpam-5066	519	4	j.	j.	PROPN
ejpam-5066	519	5	han	han	PROPN
ejpam-5066	519	6	,	,	PUNCT
ejpam-5066	519	7	y.	y.	PROPN
ejpam-5066	519	8	lee	lee	PROPN
ejpam-5066	519	9	,	,	PUNCT
ejpam-5066	519	10	and	and	CCONJ
ejpam-5066	519	11	s.	s.	PROPN
ejpam-5066	519	12	park	park	PROPN
ejpam-5066	519	13	.	.	PUNCT
ejpam-5066	520	1	semicentral	semicentral	ADJ
ejpam-5066	520	2	idempotents	idempotent	NOUN
ejpam-5066	520	3	in	in	ADP
ejpam-5066	520	4	a	a	DET
ejpam-5066	520	5	ring	ring	NOUN
ejpam-5066	520	6	.	.	PUNCT
ejpam-5066	521	1	j.	j.	PROPN
ejpam-5066	521	2	korean	korean	PROPN
ejpam-5066	521	3	math	math	PROPN
ejpam-5066	521	4	.	.	PUNCT
ejpam-5066	522	1	soc	soc	PROPN
ejpam-5066	522	2	.	.	PUNCT
ejpam-5066	522	3	,	,	PUNCT
ejpam-5066	523	1	51(3):463–472	51(3):463–472	NUM
ejpam-5066	523	2	,	,	PUNCT
ejpam-5066	523	3	2014	2014	NUM
ejpam-5066	523	4	.	.	PUNCT
ejpam-5066	524	1	[	[	X
ejpam-5066	524	2	8	8	X
ejpam-5066	524	3	]	]	PUNCT
ejpam-5066	524	4	j.	j.	PROPN
ejpam-5066	524	5	y.	y.	PROPN
ejpam-5066	524	6	kim	kim	PROPN
ejpam-5066	524	7	and	and	CCONJ
ejpam-5066	524	8	j.	j.	PROPN
ejpam-5066	524	9	u.	u.	PROPN
ejpam-5066	524	10	baik	baik	PROPN
ejpam-5066	524	11	.	.	PUNCT
ejpam-5066	525	1	on	on	ADP
ejpam-5066	525	2	idempotent	idempotent	ADJ
ejpam-5066	525	3	reflexive	reflexive	ADJ
ejpam-5066	525	4	rings	ring	NOUN
ejpam-5066	525	5	.	.	PUNCT
ejpam-5066	526	1	kyungpook	kyungpook	PROPN
ejpam-5066	526	2	math	math	PROPN
ejpam-5066	526	3	.	.	PUNCT
ejpam-5066	527	1	j.	j.	PROPN
ejpam-5066	527	2	,	,	PUNCT
ejpam-5066	527	3	46:597–601	46:597–601	PROPN
ejpam-5066	527	4	,	,	PUNCT
ejpam-5066	527	5	2006	2006	NUM
ejpam-5066	527	6	.	.	PUNCT
ejpam-5066	528	1	references	reference	NOUN
ejpam-5066	528	2	751	751	NUM
ejpam-5066	529	1	[	[	X
ejpam-5066	529	2	9	9	NUM
ejpam-5066	529	3	]	]	PUNCT
ejpam-5066	529	4	z.	z.	PROPN
ejpam-5066	529	5	kou	kou	PROPN
ejpam-5066	529	6	,	,	PUNCT
ejpam-5066	529	7	s.	s.	PROPN
ejpam-5066	529	8	kosari	kosari	PROPN
ejpam-5066	529	9	,	,	PUNCT
ejpam-5066	529	10	m.	m.	NOUN
ejpam-5066	529	11	monemrad	monemrad	PROPN
ejpam-5066	529	12	,	,	PUNCT
ejpam-5066	529	13	m.	m.	NOUN
ejpam-5066	529	14	akhoundi	akhoundi	PROPN
ejpam-5066	529	15	,	,	PUNCT
ejpam-5066	529	16	and	and	CCONJ
ejpam-5066	529	17	s.	s.	PROPN
ejpam-5066	529	18	omidi	omidi	PROPN
ejpam-5066	529	19	.	.	PUNCT
ejpam-5066	530	1	a	a	DET
ejpam-5066	530	2	note	note	NOUN
ejpam-5066	530	3	on	on	ADP
ejpam-5066	530	4	the	the	DET
ejpam-5066	530	5	connection	connection	NOUN
ejpam-5066	530	6	between	between	ADP
ejpam-5066	530	7	ordered	order	VERB
ejpam-5066	530	8	semihyperrings	semihyperring	NOUN
ejpam-5066	530	9	.	.	PUNCT
ejpam-5066	531	1	symmetry	symmetry	PROPN
ejpam-5066	531	2	,	,	PUNCT
ejpam-5066	531	3	13(11):2035	13(11):2035	NUM
ejpam-5066	531	4	,	,	PUNCT
ejpam-5066	531	5	2021	2021	NUM
ejpam-5066	531	6	.	.	PUNCT
ejpam-5066	532	1	[	[	X
ejpam-5066	532	2	10	10	NUM
ejpam-5066	532	3	]	]	PUNCT
ejpam-5066	532	4	t.	t.	PROPN
ejpam-5066	532	5	y.	y.	PROPN
ejpam-5066	532	6	lam	lam	PROPN
ejpam-5066	532	7	.	.	PUNCT
ejpam-5066	533	1	a	a	DET
ejpam-5066	533	2	first	first	ADJ
ejpam-5066	533	3	course	course	NOUN
ejpam-5066	533	4	in	in	ADP
ejpam-5066	533	5	noncommutative	noncommutative	ADJ
ejpam-5066	533	6	rings	ring	NOUN
ejpam-5066	533	7	,	,	PUNCT
ejpam-5066	533	8	volume	volume	NOUN
ejpam-5066	533	9	131	131	NUM
ejpam-5066	533	10	.	.	PUNCT
ejpam-5066	534	1	springer	springer	PROPN
ejpam-5066	534	2	science	science	PROPN
ejpam-5066	534	3	&	&	CCONJ
ejpam-5066	534	4	business	business	NOUN
ejpam-5066	534	5	media	medium	NOUN
ejpam-5066	534	6	,	,	PUNCT
ejpam-5066	534	7	new	new	PROPN
ejpam-5066	534	8	york	york	PROPN
ejpam-5066	534	9	heidelberg	heidelberg	PROPN
ejpam-5066	534	10	dordrecht	dordrecht	PROPN
ejpam-5066	534	11	london	london	PROPN
ejpam-5066	534	12	,	,	PUNCT
ejpam-5066	534	13	2013	2013	NUM
ejpam-5066	534	14	.	.	PUNCT
ejpam-5066	535	1	[	[	X
ejpam-5066	535	2	11	11	NUM
ejpam-5066	535	3	]	]	PUNCT
ejpam-5066	535	4	t.	t.	PROPN
ejpam-5066	535	5	y.	y.	PROPN
ejpam-5066	535	6	lam	lam	PROPN
ejpam-5066	535	7	.	.	PUNCT
ejpam-5066	536	1	an	an	DET
ejpam-5066	536	2	introduction	introduction	NOUN
ejpam-5066	536	3	to	to	ADP
ejpam-5066	536	4	q	q	ADJ
ejpam-5066	536	5	-	-	ADJ
ejpam-5066	536	6	central	central	ADJ
ejpam-5066	536	7	idempotents	idempotent	NOUN
ejpam-5066	536	8	and	and	CCONJ
ejpam-5066	536	9	q	q	ADJ
ejpam-5066	536	10	-	-	PUNCT
ejpam-5066	536	11	abelian	abelian	ADJ
ejpam-5066	536	12	rings	ring	NOUN
ejpam-5066	536	13	.	.	PUNCT
ejpam-5066	537	1	commun	commun	PROPN
ejpam-5066	537	2	.	.	PUNCT
ejpam-5066	538	1	algebra	algebra	PROPN
ejpam-5066	538	2	,	,	PUNCT
ejpam-5066	538	3	pages	page	NOUN
ejpam-5066	538	4	1–18	1–18	NUM
ejpam-5066	538	5	,	,	PUNCT
ejpam-5066	538	6	2022	2022	NUM
ejpam-5066	538	7	.	.	PUNCT
ejpam-5066	539	1	[	[	X
ejpam-5066	539	2	12	12	NUM
ejpam-5066	539	3	]	]	PUNCT
ejpam-5066	539	4	t.	t.	PROPN
ejpam-5066	539	5	y.	y.	PROPN
ejpam-5066	539	6	lam	lam	PROPN
ejpam-5066	539	7	.	.	PUNCT
ejpam-5066	540	1	on	on	ADP
ejpam-5066	540	2	some	some	DET
ejpam-5066	540	3	generalizations	generalization	NOUN
ejpam-5066	540	4	of	of	ADP
ejpam-5066	540	5	abelian	abelian	ADJ
ejpam-5066	540	6	rings	ring	NOUN
ejpam-5066	540	7	.	.	PUNCT
ejpam-5066	541	1	j.	j.	PROPN
ejpam-5066	541	2	algebra	algebra	PROPN
ejpam-5066	541	3	appl	appl	PROPN
ejpam-5066	541	4	.	.	PROPN
ejpam-5066	541	5	,	,	PUNCT
ejpam-5066	541	6	page	page	NOUN
ejpam-5066	541	7	2550146	2550146	NUM
ejpam-5066	541	8	,	,	PUNCT
ejpam-5066	541	9	2023	2023	NUM
ejpam-5066	541	10	.	.	PUNCT
ejpam-5066	542	1	[	[	X
ejpam-5066	542	2	13	13	NUM
ejpam-5066	542	3	]	]	X
ejpam-5066	542	4	c.	c.	NOUN
ejpam-5066	542	5	lomp	lomp	PROPN
ejpam-5066	542	6	and	and	CCONJ
ejpam-5066	542	7	j.	j.	PROPN
ejpam-5066	542	8	matczuk	matczuk	PROPN
ejpam-5066	542	9	.	.	PUNCT
ejpam-5066	543	1	a	a	DET
ejpam-5066	543	2	note	note	NOUN
ejpam-5066	543	3	on	on	ADP
ejpam-5066	543	4	semicentral	semicentral	ADJ
ejpam-5066	543	5	idempotents	idempotent	NOUN
ejpam-5066	543	6	.	.	PUNCT
ejpam-5066	544	1	commun	commun	PROPN
ejpam-5066	544	2	.	.	PUNCT
ejpam-5066	545	1	algebra	algebra	PROPN
ejpam-5066	545	2	,	,	PUNCT
ejpam-5066	545	3	45(6):2735–2737	45(6):2735–2737	PROPN
ejpam-5066	545	4	,	,	PUNCT
ejpam-5066	545	5	2017	2017	NUM
ejpam-5066	545	6	.	.	PUNCT
ejpam-5066	546	1	[	[	X
ejpam-5066	546	2	14	14	NUM
ejpam-5066	546	3	]	]	X
ejpam-5066	546	4	w.	w.	PROPN
ejpam-5066	546	5	k.	k.	PROPN
ejpam-5066	546	6	nicholson	nicholson	PROPN
ejpam-5066	546	7	.	.	PUNCT
ejpam-5066	547	1	lifting	lift	VERB
ejpam-5066	547	2	idempotents	idempotent	NOUN
ejpam-5066	547	3	and	and	CCONJ
ejpam-5066	547	4	exchange	exchange	NOUN
ejpam-5066	547	5	rings	ring	NOUN
ejpam-5066	547	6	.	.	PUNCT
ejpam-5066	548	1	trans	trans	PROPN
ejpam-5066	548	2	.	.	PROPN
ejpam-5066	549	1	am	be	AUX
ejpam-5066	549	2	.	.	PUNCT
ejpam-5066	550	1	math	math	NOUN
ejpam-5066	550	2	.	.	PUNCT
ejpam-5066	551	1	soc	soc	PROPN
ejpam-5066	551	2	.	.	PROPN
ejpam-5066	551	3	,	,	PUNCT
ejpam-5066	551	4	229:269–278	229:269–278	NUM
ejpam-5066	551	5	,	,	PUNCT
ejpam-5066	551	6	1977	1977	NUM
ejpam-5066	551	7	.	.	PUNCT
ejpam-5066	552	1	[	[	X
ejpam-5066	552	2	15	15	NUM
ejpam-5066	552	3	]	]	X
ejpam-5066	552	4	y.	y.	PROPN
ejpam-5066	552	5	rao	rao	PROPN
ejpam-5066	552	6	,	,	PUNCT
ejpam-5066	552	7	s.	s.	PROPN
ejpam-5066	552	8	kosari	kosari	PROPN
ejpam-5066	552	9	,	,	PUNCT
ejpam-5066	552	10	z.	z.	PROPN
ejpam-5066	552	11	shao	shao	PROPN
ejpam-5066	552	12	,	,	PUNCT
ejpam-5066	552	13	m.	m.	NOUN
ejpam-5066	552	14	akhoundi	akhoundi	PROPN
ejpam-5066	552	15	,	,	PUNCT
ejpam-5066	552	16	and	and	CCONJ
ejpam-5066	552	17	s.	s.	PROPN
ejpam-5066	552	18	omidi	omidi	PROPN
ejpam-5066	552	19	.	.	PUNCT
ejpam-5066	553	1	a	a	DET
ejpam-5066	553	2	study	study	NOUN
ejpam-5066	553	3	on	on	ADP
ejpam-5066	553	4	-hyperideals	-hyperideal	NOUN
ejpam-5066	553	5	and	and	CCONJ
ejpam-5066	553	6	-hyperfilters	-hyperfilter	NOUN
ejpam-5066	553	7	in	in	ADP
ejpam-5066	553	8	ordered	order	VERB
ejpam-5066	553	9	-semihypergroups	-semihypergroup	NOUN
ejpam-5066	553	10	.	.	PUNCT
ejpam-5066	554	1	discrete	discrete	ADJ
ejpam-5066	554	2	dynamics	dynamic	NOUN
ejpam-5066	554	3	in	in	ADP
ejpam-5066	554	4	nature	nature	NOUN
ejpam-5066	554	5	and	and	CCONJ
ejpam-5066	554	6	society	society	NOUN
ejpam-5066	554	7	,	,	PUNCT
ejpam-5066	554	8	2021:6683910	2021:6683910	NUM
ejpam-5066	554	9	,	,	PUNCT
ejpam-5066	554	10	2021	2021	NUM
ejpam-5066	554	11	.	.	PUNCT
ejpam-5066	555	1	[	[	X
ejpam-5066	555	2	16	16	NUM
ejpam-5066	555	3	]	]	X
ejpam-5066	555	4	y.	y.	PROPN
ejpam-5066	555	5	rao	rao	PROPN
ejpam-5066	555	6	,	,	PUNCT
ejpam-5066	555	7	s.	s.	PROPN
ejpam-5066	555	8	kosari	kosari	PROPN
ejpam-5066	555	9	,	,	PUNCT
ejpam-5066	555	10	z.	z.	PROPN
ejpam-5066	555	11	shao	shao	PROPN
ejpam-5066	555	12	,	,	PUNCT
ejpam-5066	555	13	and	and	CCONJ
ejpam-5066	555	14	s.	s.	PROPN
ejpam-5066	555	15	omidi	omidi	PROPN
ejpam-5066	555	16	.	.	PUNCT
ejpam-5066	556	1	some	some	DET
ejpam-5066	556	2	properties	property	NOUN
ejpam-5066	556	3	of	of	ADP
ejpam-5066	556	4	derivations	derivation	NOUN
ejpam-5066	556	5	and	and	CCONJ
ejpam-5066	556	6	m	m	NOUN
ejpam-5066	556	7	-	-	PUNCT
ejpam-5066	556	8	khyperideals	khyperideal	NOUN
ejpam-5066	556	9	in	in	ADP
ejpam-5066	556	10	ordered	order	VERB
ejpam-5066	556	11	semihyperrings	semihyperring	NOUN
ejpam-5066	556	12	.	.	PUNCT
ejpam-5066	557	1	politehn	politehn	PROPN
ejpam-5066	557	2	.	.	PUNCT
ejpam-5066	558	1	univ	univ	PROPN
ejpam-5066	558	2	.	.	PUNCT
ejpam-5066	559	1	bucharest	bucharest	PROPN
ejpam-5066	559	2	sci	sci	PROPN
ejpam-5066	559	3	.	.	PUNCT
ejpam-5066	559	4	bull	bull	PROPN
ejpam-5066	559	5	.	.	PUNCT
ejpam-5066	560	1	ser	ser	PROPN
ejpam-5066	560	2	.	.	PUNCT
ejpam-5066	561	1	a	a	DET
ejpam-5066	561	2	appl	appl	PROPN
ejpam-5066	561	3	.	.	PUNCT
ejpam-5066	561	4	math	math	NOUN
ejpam-5066	561	5	.	.	PUNCT
ejpam-5066	562	1	phys	phy	NOUN
ejpam-5066	562	2	,	,	PUNCT
ejpam-5066	562	3	83(3):87–96	83(3):87–96	NUM
ejpam-5066	562	4	,	,	PUNCT
ejpam-5066	562	5	2021	2021	NUM
ejpam-5066	562	6	.	.	PUNCT
ejpam-5066	563	1	[	[	X
ejpam-5066	563	2	17	17	NUM
ejpam-5066	563	3	]	]	PUNCT
ejpam-5066	563	4	m.	m.	NOUN
ejpam-5066	563	5	saad	saad	PROPN
ejpam-5066	563	6	.	.	PUNCT
ejpam-5066	563	7	rings	ring	NOUN
ejpam-5066	563	8	in	in	ADP
ejpam-5066	563	9	which	which	PRON
ejpam-5066	563	10	every	every	DET
ejpam-5066	563	11	semicentral	semicentral	ADJ
ejpam-5066	563	12	idempotent	idempotent	NOUN
ejpam-5066	563	13	is	be	AUX
ejpam-5066	563	14	central	central	ADJ
ejpam-5066	563	15	.	.	PUNCT
ejpam-5066	564	1	korean	korean	PROPN
ejpam-5066	564	2	j.	j.	PROPN
ejpam-5066	564	3	math	math	PROPN
ejpam-5066	564	4	.	.	PUNCT
ejpam-5066	564	5	,	,	PUNCT
ejpam-5066	565	1	31(4):405–417	31(4):405–417	NUM
ejpam-5066	565	2	,	,	PUNCT
ejpam-5066	565	3	2023	2023	NUM
ejpam-5066	565	4	.	.	PUNCT
ejpam-5066	566	1	[	[	X
ejpam-5066	566	2	18	18	NUM
ejpam-5066	566	3	]	]	PUNCT
ejpam-5066	566	4	z.	z.	PROPN
ejpam-5066	566	5	shao	shao	PROPN
ejpam-5066	566	6	,	,	PUNCT
ejpam-5066	566	7	x.	x.	PROPN
ejpam-5066	566	8	chen	chen	PROPN
ejpam-5066	566	9	,	,	PUNCT
ejpam-5066	566	10	s.	s.	PROPN
ejpam-5066	566	11	kosari	kosari	PROPN
ejpam-5066	566	12	,	,	PUNCT
ejpam-5066	566	13	and	and	CCONJ
ejpam-5066	566	14	s.	s.	PROPN
ejpam-5066	566	15	omidi	omidi	PROPN
ejpam-5066	566	16	.	.	PUNCT
ejpam-5066	567	1	on	on	ADP
ejpam-5066	567	2	some	some	DET
ejpam-5066	567	3	properties	property	NOUN
ejpam-5066	567	4	of	of	ADP
ejpam-5066	567	5	right	right	ADJ
ejpam-5066	567	6	pure	pure	ADJ
ejpam-5066	567	7	(	(	PUNCT
ejpam-5066	567	8	biquasi-	biquasi-	NUM
ejpam-5066	567	9	)	)	PUNCT
ejpam-5066	567	10	hyperideals	hyperideal	NOUN
ejpam-5066	567	11	in	in	ADP
ejpam-5066	567	12	ordered	order	VERB
ejpam-5066	567	13	semihyperrings	semihyperring	NOUN
ejpam-5066	567	14	.	.	PUNCT
ejpam-5066	568	1	politehn	politehn	PROPN
ejpam-5066	568	2	.	.	PUNCT
ejpam-5066	569	1	univ	univ	PROPN
ejpam-5066	569	2	.	.	PUNCT
ejpam-5066	570	1	bucharest	bucharest	PROPN
ejpam-5066	570	2	sci	sci	PROPN
ejpam-5066	570	3	.	.	PUNCT
ejpam-5066	570	4	bull	bull	PROPN
ejpam-5066	570	5	.	.	PUNCT
ejpam-5066	571	1	ser	ser	PROPN
ejpam-5066	571	2	.	.	PUNCT
ejpam-5066	572	1	a	a	DET
ejpam-5066	572	2	appl	appl	PROPN
ejpam-5066	572	3	.	.	PUNCT
ejpam-5066	572	4	math	math	NOUN
ejpam-5066	572	5	.	.	PUNCT
ejpam-5066	573	1	phys	phy	NOUN
ejpam-5066	573	2	,	,	PUNCT
ejpam-5066	573	3	83(4):95–104	83(4):95–104	NUM
ejpam-5066	573	4	,	,	PUNCT
ejpam-5066	573	5	2021	2021	NUM
ejpam-5066	573	6	.	.	PUNCT
ejpam-5066	574	1	[	[	X
ejpam-5066	574	2	19	19	NUM
ejpam-5066	574	3	]	]	PUNCT
ejpam-5066	574	4	l.	l.	PROPN
ejpam-5066	574	5	n.	n.	PROPN
ejpam-5066	574	6	vaserstein	vaserstein	PROPN
ejpam-5066	574	7	.	.	PUNCT
ejpam-5066	575	1	bass	bass	PROPN
ejpam-5066	575	2	’s	’s	PART
ejpam-5066	575	3	first	first	ADJ
ejpam-5066	575	4	stable	stable	ADJ
ejpam-5066	575	5	range	range	NOUN
ejpam-5066	575	6	condition	condition	NOUN
ejpam-5066	575	7	.	.	PUNCT
ejpam-5066	576	1	j.	j.	PROPN
ejpam-5066	576	2	pure	pure	PROPN
ejpam-5066	576	3	appl	appl	PROPN
ejpam-5066	576	4	.	.	PUNCT
ejpam-5066	577	1	algebra	algebra	PROPN
ejpam-5066	577	2	,	,	PUNCT
ejpam-5066	577	3	34(23):319–330	34(23):319–330	NUM
ejpam-5066	577	4	,	,	PUNCT
ejpam-5066	577	5	1984	1984	NUM
ejpam-5066	577	6	.	.	PUNCT
ejpam-5066	578	1	[	[	X
ejpam-5066	578	2	20	20	NUM
ejpam-5066	578	3	]	]	PUNCT
ejpam-5066	578	4	r.	r.	PROPN
ejpam-5066	578	5	b.	b.	PROPN
ejpam-5066	578	6	warfield	warfield	PROPN
ejpam-5066	578	7	jr	jr	PROPN
ejpam-5066	578	8	.	.	PROPN
ejpam-5066	578	9	exchange	exchange	NOUN
ejpam-5066	578	10	rings	ring	NOUN
ejpam-5066	578	11	and	and	CCONJ
ejpam-5066	578	12	decompositions	decomposition	NOUN
ejpam-5066	578	13	of	of	ADP
ejpam-5066	578	14	modules	module	NOUN
ejpam-5066	578	15	.	.	PUNCT
ejpam-5066	579	1	math	math	NOUN
ejpam-5066	579	2	.	.	PUNCT
ejpam-5066	580	1	annal	annal	PROPN
ejpam-5066	580	2	.	.	PROPN
ejpam-5066	580	3	,	,	PUNCT
ejpam-5066	580	4	199(1):31–36	199(1):31–36	NUM
ejpam-5066	580	5	,	,	PUNCT
ejpam-5066	580	6	1972	1972	NUM
ejpam-5066	580	7	.	.	PUNCT
ejpam-5066	581	1	[	[	X
ejpam-5066	581	2	21	21	NUM
ejpam-5066	581	3	]	]	X
ejpam-5066	581	4	j.	j.	PROPN
ejpam-5066	581	5	wei	wei	PROPN
ejpam-5066	581	6	.	.	PUNCT
ejpam-5066	582	1	certain	certain	ADJ
ejpam-5066	582	2	rings	ring	NOUN
ejpam-5066	582	3	whose	whose	DET
ejpam-5066	582	4	simple	simple	ADJ
ejpam-5066	582	5	singular	singular	ADJ
ejpam-5066	582	6	modules	module	NOUN
ejpam-5066	582	7	are	be	AUX
ejpam-5066	582	8	nil	nil	ADJ
ejpam-5066	582	9	-	-	PUNCT
ejpam-5066	582	10	injective	injective	ADJ
ejpam-5066	582	11	.	.	PUNCT
ejpam-5066	583	1	turk	turk	PROPN
ejpam-5066	583	2	.	.	PUNCT
ejpam-5066	584	1	j.	j.	PROPN
ejpam-5066	584	2	math	math	PROPN
ejpam-5066	584	3	.	.	PUNCT
ejpam-5066	584	4	,	,	PUNCT
ejpam-5066	584	5	32(4):393–408	32(4):393–408	NUM
ejpam-5066	584	6	,	,	PUNCT
ejpam-5066	584	7	2008	2008	NUM
ejpam-5066	584	8	.	.	PUNCT
ejpam-5066	585	1	[	[	X
ejpam-5066	585	2	22	22	NUM
ejpam-5066	585	3	]	]	PUNCT
ejpam-5066	585	4	j.	j.	PROPN
ejpam-5066	585	5	wei	wei	PROPN
ejpam-5066	585	6	and	and	CCONJ
ejpam-5066	585	7	j.	j.	PROPN
ejpam-5066	585	8	chen	chen	PROPN
ejpam-5066	585	9	.	.	PUNCT
ejpam-5066	586	1	nilinjective	nilinjective	ADJ
ejpam-5066	586	2	rings	ring	NOUN
ejpam-5066	586	3	.	.	PUNCT
ejpam-5066	587	1	inter	inter	PROPN
ejpam-5066	587	2	.	.	PUNCT
ejpam-5066	588	1	electron	electron	PROPN
ejpam-5066	588	2	.	.	PUNCT
ejpam-5066	589	1	j.	j.	PROPN
ejpam-5066	589	2	algebra	algebra	PROPN
ejpam-5066	589	3	,	,	PUNCT
ejpam-5066	589	4	2(2):1–21	2(2):1–21	NUM
ejpam-5066	589	5	,	,	PUNCT
ejpam-5066	589	6	2007	2007	NUM
ejpam-5066	589	7	.	.	PUNCT
ejpam-5066	590	1	[	[	X
ejpam-5066	590	2	23	23	NUM
ejpam-5066	590	3	]	]	PUNCT
ejpam-5066	590	4	j.	j.	PROPN
ejpam-5066	590	5	wei	wei	PROPN
ejpam-5066	590	6	and	and	CCONJ
ejpam-5066	590	7	l.	l.	PROPN
ejpam-5066	590	8	li	li	PROPN
ejpam-5066	590	9	.	.	PUNCT
ejpam-5066	590	10	quasi	quasi	ADJ
ejpam-5066	590	11	-	-	ADJ
ejpam-5066	590	12	normal	normal	ADJ
ejpam-5066	590	13	rings	ring	NOUN
ejpam-5066	590	14	.	.	PUNCT
ejpam-5066	591	1	commun	commun	PROPN
ejpam-5066	591	2	.	.	PUNCT
ejpam-5066	592	1	algebra	algebra	PROPN
ejpam-5066	592	2	,	,	PUNCT
ejpam-5066	592	3	38(5):1855–1868	38(5):1855–1868	NUM
ejpam-5066	592	4	,	,	PUNCT
ejpam-5066	592	5	2010	2010	NUM
ejpam-5066	592	6	.	.	PUNCT
ejpam-5066	593	1	[	[	X
ejpam-5066	593	2	24	24	NUM
ejpam-5066	593	3	]	]	PUNCT
ejpam-5066	593	4	j.	j.	PROPN
ejpam-5066	593	5	wei	wei	PROPN
ejpam-5066	593	6	and	and	CCONJ
ejpam-5066	593	7	l	l	PROPN
ejpam-5066	593	8	li	li	PROPN
ejpam-5066	593	9	.	.	PROPN
ejpam-5066	593	10	nilpotent	nilpotent	ADJ
ejpam-5066	593	11	elements	element	NOUN
ejpam-5066	593	12	and	and	CCONJ
ejpam-5066	593	13	reduced	reduced	ADJ
ejpam-5066	593	14	rings	ring	NOUN
ejpam-5066	593	15	.	.	PUNCT
ejpam-5066	594	1	turk	turk	PROPN
ejpam-5066	594	2	.	.	PUNCT
ejpam-5066	595	1	j.	j.	PROPN
ejpam-5066	595	2	math	math	PROPN
ejpam-5066	595	3	.	.	PUNCT
ejpam-5066	595	4	,	,	PUNCT
ejpam-5066	595	5	35(2):341–353	35(2):341–353	PROPN
ejpam-5066	595	6	,	,	PUNCT
ejpam-5066	595	7	2011	2011	NUM
ejpam-5066	595	8	.	.	PUNCT
ejpam-5066	596	1	references	reference	NOUN
ejpam-5066	596	2	752	752	NUM
ejpam-5066	597	1	[	[	X
ejpam-5066	597	2	25	25	NUM
ejpam-5066	597	3	]	]	PUNCT
ejpam-5066	597	4	j.	j.	PROPN
ejpam-5066	597	5	wei	wei	PROPN
ejpam-5066	597	6	and	and	CCONJ
ejpam-5066	597	7	l.	l.	PROPN
ejpam-5066	597	8	li	li	PROPN
ejpam-5066	597	9	.	.	PROPN
ejpam-5066	598	1	weakly	weakly	ADJ
ejpam-5066	598	2	normal	normal	ADJ
ejpam-5066	598	3	rings	ring	NOUN
ejpam-5066	598	4	.	.	PUNCT
ejpam-5066	599	1	turk	turk	PROPN
ejpam-5066	599	2	.	.	PUNCT
ejpam-5066	600	1	j.	j.	PROPN
ejpam-5066	600	2	math	math	PROPN
ejpam-5066	600	3	.	.	PUNCT
ejpam-5066	600	4	,	,	PUNCT
ejpam-5066	600	5	36(1):47–57	36(1):47–57	NUM
ejpam-5066	600	6	,	,	PUNCT
ejpam-5066	600	7	2012	2012	NUM
ejpam-5066	600	8	.	.	PUNCT
ejpam-5066	601	1	[	[	X
ejpam-5066	601	2	26	26	NUM
ejpam-5066	601	3	]	]	PUNCT
ejpam-5066	601	4	j.	j.	PROPN
ejpam-5066	601	5	wei	wei	PROPN
ejpam-5066	601	6	and	and	CCONJ
ejpam-5066	601	7	n.	n.	PROPN
ejpam-5066	601	8	li	li	PROPN
ejpam-5066	601	9	.	.	PUNCT
ejpam-5066	602	1	some	some	DET
ejpam-5066	602	2	notes	note	NOUN
ejpam-5066	602	3	on	on	ADP
ejpam-5066	602	4	semiabelian	semiabelian	ADJ
ejpam-5066	602	5	rings	ring	NOUN
ejpam-5066	602	6	.	.	PUNCT
ejpam-5066	603	1	j.	j.	PROPN
ejpam-5066	603	2	math	math	PROPN
ejpam-5066	603	3	.	.	PUNCT
ejpam-5066	604	1	math	math	NOUN
ejpam-5066	604	2	.	.	PUNCT
ejpam-5066	605	1	sci	sci	PROPN
ejpam-5066	605	2	.	.	PROPN
ejpam-5066	605	3	,	,	PUNCT
ejpam-5066	605	4	2011	2011	NUM
ejpam-5066	605	5	,	,	PUNCT
ejpam-5066	605	6	2011	2011	NUM
ejpam-5066	605	7	.	.	PUNCT
ejpam-5066	606	1	[	[	X
ejpam-5066	606	2	27	27	NUM
ejpam-5066	606	3	]	]	X
ejpam-5066	606	4	h	h	PROPN
ejpam-5066	606	5	-	-	PUNCT
ejpam-5066	606	6	p.	p.	NOUN
ejpam-5066	606	7	yu	yu	PROPN
ejpam-5066	606	8	.	.	PROPN
ejpam-5066	606	9	stable	stable	ADJ
ejpam-5066	606	10	range	range	NOUN
ejpam-5066	606	11	one	one	NUM
ejpam-5066	606	12	for	for	ADP
ejpam-5066	606	13	exchange	exchange	NOUN
ejpam-5066	606	14	rings	ring	NOUN
ejpam-5066	606	15	.	.	PUNCT
ejpam-5066	607	1	j.	j.	PROPN
ejpam-5066	607	2	pure	pure	PROPN
ejpam-5066	607	3	appl	appl	PROPN
ejpam-5066	607	4	.	.	PUNCT
ejpam-5066	608	1	algebra	algebra	PROPN
ejpam-5066	608	2	,	,	PUNCT
ejpam-5066	608	3	98(1):105–109	98(1):105–109	NUM
ejpam-5066	608	4	,	,	PUNCT
ejpam-5066	608	5	1995	1995	NUM
ejpam-5066	608	6	.	.	PUNCT