id	sid	tid	token	lemma	pos
ejpam-4797	1	1	european	european	PROPN
ejpam-4797	1	2	journal	journal	PROPN
ejpam-4797	1	3	of	of	ADP
ejpam-4797	1	4	pure	pure	ADJ
ejpam-4797	1	5	and	and	CCONJ
ejpam-4797	1	6	applied	apply	VERB
ejpam-4797	1	7	mathematics	mathematic	NOUN
ejpam-4797	1	8	vol	vol	NOUN
ejpam-4797	1	9	.	.	PUNCT
ejpam-4797	2	1	16	16	NUM
ejpam-4797	2	2	,	,	PUNCT
ejpam-4797	2	3	no	no	INTJ
ejpam-4797	2	4	.	.	NOUN
ejpam-4797	2	5	3	3	NUM
ejpam-4797	2	6	,	,	PUNCT
ejpam-4797	2	7	2023	2023	NUM
ejpam-4797	2	8	,	,	PUNCT
ejpam-4797	2	9	1675	1675	NUM
ejpam-4797	2	10	-	-	SYM
ejpam-4797	2	11	1684	1684	NUM
ejpam-4797	2	12	issn	issn	PROPN
ejpam-4797	2	13	1307	1307	NUM
ejpam-4797	2	14	-	-	SYM
ejpam-4797	2	15	5543	5543	NUM
ejpam-4797	2	16	–	–	PUNCT
ejpam-4797	3	1	ejpam.com	ejpam.com	X
ejpam-4797	3	2	published	publish	VERB
ejpam-4797	3	3	by	by	ADP
ejpam-4797	3	4	new	new	PROPN
ejpam-4797	3	5	york	york	PROPN
ejpam-4797	3	6	business	business	PROPN
ejpam-4797	3	7	global	global	ADJ
ejpam-4797	3	8	strongly	strongly	ADV
ejpam-4797	3	9	2	2	NUM
ejpam-4797	3	10	-	-	PUNCT
ejpam-4797	3	11	nil	nil	NOUN
ejpam-4797	3	12	clean	clean	ADJ
ejpam-4797	3	13	rings	ring	NOUN
ejpam-4797	3	14	with	with	ADP
ejpam-4797	3	15	units	unit	NOUN
ejpam-4797	3	16	of	of	ADP
ejpam-4797	3	17	order	order	NOUN
ejpam-4797	3	18	two	two	NUM
ejpam-4797	3	19	renas	rena	NOUN
ejpam-4797	3	20	t.	t.	PROPN
ejpam-4797	3	21	m.salim1	m.salim1	PROPN
ejpam-4797	3	22	,	,	PUNCT
ejpam-4797	3	23	nazar	nazar	PROPN
ejpam-4797	3	24	h.	h.	PROPN
ejpam-4797	3	25	shuker2,∗	shuker2,∗	PROPN
ejpam-4797	3	26	1	1	NUM
ejpam-4797	3	27	department	department	NOUN
ejpam-4797	3	28	of	of	ADP
ejpam-4797	3	29	mathematics	mathematic	NOUN
ejpam-4797	3	30	,	,	PUNCT
ejpam-4797	3	31	university	university	NOUN
ejpam-4797	3	32	of	of	ADP
ejpam-4797	3	33	zakho	zakho	PROPN
ejpam-4797	3	34	,	,	PUNCT
ejpam-4797	3	35	faculty	faculty	NOUN
ejpam-4797	3	36	of	of	ADP
ejpam-4797	3	37	science	science	NOUN
ejpam-4797	3	38	,	,	PUNCT
ejpam-4797	3	39	zakho	zakho	PROPN
ejpam-4797	3	40	,	,	PUNCT
ejpam-4797	3	41	iraq	iraq	PROPN
ejpam-4797	3	42	2	2	NUM
ejpam-4797	3	43	department	department	NOUN
ejpam-4797	3	44	of	of	ADP
ejpam-4797	3	45	math	math	NOUN
ejpam-4797	3	46	.	.	PUNCT
ejpam-4797	4	1	college	college	NOUN
ejpam-4797	4	2	of	of	ADP
ejpam-4797	4	3	computer	computer	NOUN
ejpam-4797	4	4	science	science	NOUN
ejpam-4797	4	5	and	and	CCONJ
ejpam-4797	4	6	math	math	NOUN
ejpam-4797	4	7	.	.	PUNCT
ejpam-4797	5	1	mosul	mosul	PROPN
ejpam-4797	5	2	university	university	PROPN
ejpam-4797	5	3	,	,	PUNCT
ejpam-4797	5	4	mosul	mosul	PROPN
ejpam-4797	5	5	,	,	PUNCT
ejpam-4797	5	6	iraq	iraq	PROPN
ejpam-4797	5	7	abstract	abstract	NOUN
ejpam-4797	5	8	.	.	PUNCT
ejpam-4797	6	1	a	a	DET
ejpam-4797	6	2	ring	ring	NOUN
ejpam-4797	6	3	r	r	NOUN
ejpam-4797	6	4	is	be	AUX
ejpam-4797	6	5	considered	consider	VERB
ejpam-4797	6	6	a	a	DET
ejpam-4797	6	7	strongly	strongly	ADV
ejpam-4797	6	8	2	2	NUM
ejpam-4797	6	9	-	-	PUNCT
ejpam-4797	6	10	nil	nil	NOUN
ejpam-4797	6	11	clean	clean	ADJ
ejpam-4797	6	12	ring	ring	NOUN
ejpam-4797	6	13	,	,	PUNCT
ejpam-4797	6	14	or	or	CCONJ
ejpam-4797	6	15	(	(	PUNCT
ejpam-4797	6	16	strongly	strongly	ADV
ejpam-4797	6	17	2	2	NUM
ejpam-4797	6	18	-	-	PUNCT
ejpam-4797	6	19	nc	nc	NOUN
ejpam-4797	6	20	ring	ring	NOUN
ejpam-4797	6	21	for	for	ADP
ejpam-4797	6	22	short	short	ADJ
ejpam-4797	6	23	)	)	PUNCT
ejpam-4797	6	24	,	,	PUNCT
ejpam-4797	6	25	if	if	SCONJ
ejpam-4797	6	26	each	each	DET
ejpam-4797	6	27	element	element	NOUN
ejpam-4797	6	28	in	in	ADP
ejpam-4797	6	29	r	r	NOUN
ejpam-4797	6	30	can	can	AUX
ejpam-4797	6	31	be	be	AUX
ejpam-4797	6	32	expressed	express	VERB
ejpam-4797	6	33	as	as	ADP
ejpam-4797	6	34	the	the	DET
ejpam-4797	6	35	sum	sum	NOUN
ejpam-4797	6	36	of	of	ADP
ejpam-4797	6	37	a	a	DET
ejpam-4797	6	38	nilpotent	nilpotent	NOUN
ejpam-4797	6	39	and	and	CCONJ
ejpam-4797	6	40	two	two	NUM
ejpam-4797	6	41	idempotents	idempotent	NOUN
ejpam-4797	6	42	that	that	PRON
ejpam-4797	6	43	commute	commute	VERB
ejpam-4797	6	44	with	with	ADP
ejpam-4797	6	45	each	each	DET
ejpam-4797	6	46	other	other	ADJ
ejpam-4797	6	47	.	.	PUNCT
ejpam-4797	7	1	in	in	ADP
ejpam-4797	7	2	this	this	DET
ejpam-4797	7	3	paper	paper	NOUN
ejpam-4797	7	4	,	,	PUNCT
ejpam-4797	7	5	further	further	ADJ
ejpam-4797	7	6	properties	property	NOUN
ejpam-4797	7	7	of	of	ADP
ejpam-4797	7	8	strongly	strongly	ADV
ejpam-4797	7	9	2	2	NUM
ejpam-4797	7	10	-	-	PUNCT
ejpam-4797	7	11	nc	nc	PROPN
ejpam-4797	7	12	rings	ring	NOUN
ejpam-4797	7	13	are	be	AUX
ejpam-4797	7	14	given	give	VERB
ejpam-4797	7	15	.	.	PUNCT
ejpam-4797	8	1	furthermore	furthermore	ADV
ejpam-4797	8	2	,	,	PUNCT
ejpam-4797	8	3	we	we	PRON
ejpam-4797	8	4	introduce	introduce	VERB
ejpam-4797	8	5	and	and	CCONJ
ejpam-4797	8	6	explore	explore	VERB
ejpam-4797	8	7	a	a	DET
ejpam-4797	8	8	special	special	ADJ
ejpam-4797	8	9	type	type	NOUN
ejpam-4797	8	10	of	of	ADP
ejpam-4797	8	11	strongly	strongly	ADV
ejpam-4797	8	12	2	2	NUM
ejpam-4797	8	13	-	-	PUNCT
ejpam-4797	8	14	nc	nc	NOUN
ejpam-4797	8	15	ring	ring	NOUN
ejpam-4797	8	16	where	where	SCONJ
ejpam-4797	8	17	every	every	DET
ejpam-4797	8	18	unit	unit	NOUN
ejpam-4797	8	19	is	be	AUX
ejpam-4797	8	20	of	of	ADP
ejpam-4797	8	21	order	order	NOUN
ejpam-4797	8	22	2	2	NUM
ejpam-4797	8	23	,	,	PUNCT
ejpam-4797	8	24	which	which	PRON
ejpam-4797	8	25	we	we	PRON
ejpam-4797	8	26	refer	refer	VERB
ejpam-4797	8	27	to	to	ADP
ejpam-4797	8	28	as	as	ADP
ejpam-4797	8	29	a	a	DET
ejpam-4797	8	30	strongly	strongly	ADV
ejpam-4797	8	31	2	2	NUM
ejpam-4797	8	32	-	-	PUNCT
ejpam-4797	8	33	nc	nc	PROPN
ejpam-4797	8	34	rings	ring	NOUN
ejpam-4797	8	35	with	with	ADP
ejpam-4797	8	36	u(r	u(r	NOUN
ejpam-4797	8	37	)	)	PUNCT
ejpam-4797	8	38	=	=	SYM
ejpam-4797	9	1	2	2	X
ejpam-4797	9	2	.	.	X
ejpam-4797	10	1	it	it	PRON
ejpam-4797	10	2	was	be	AUX
ejpam-4797	10	3	proved	prove	VERB
ejpam-4797	10	4	that	that	SCONJ
ejpam-4797	10	5	the	the	DET
ejpam-4797	10	6	jacobson	jacobson	PROPN
ejpam-4797	10	7	radical	radical	ADJ
ejpam-4797	10	8	over	over	ADP
ejpam-4797	10	9	a	a	DET
ejpam-4797	10	10	strongly	strongly	ADV
ejpam-4797	10	11	2	2	NUM
ejpam-4797	10	12	-	-	PUNCT
ejpam-4797	10	13	nc	nc	PROPN
ejpam-4797	10	14	ring	ring	NOUN
ejpam-4797	10	15	is	be	AUX
ejpam-4797	10	16	a	a	DET
ejpam-4797	10	17	nil	nil	ADJ
ejpam-4797	10	18	ideal	ideal	NOUN
ejpam-4797	10	19	,	,	PUNCT
ejpam-4797	10	20	here	here	ADV
ejpam-4797	10	21	,	,	PUNCT
ejpam-4797	10	22	we	we	PRON
ejpam-4797	10	23	demonstrated	demonstrate	VERB
ejpam-4797	10	24	that	that	SCONJ
ejpam-4797	10	25	the	the	DET
ejpam-4797	10	26	jacobson	jacobson	PROPN
ejpam-4797	10	27	radical	radical	PROPN
ejpam-4797	10	28	over	over	ADP
ejpam-4797	10	29	strongly	strongly	ADV
ejpam-4797	10	30	2	2	NUM
ejpam-4797	10	31	-	-	PUNCT
ejpam-4797	10	32	nc	nc	NOUN
ejpam-4797	10	33	ring	ring	NOUN
ejpam-4797	10	34	with	with	ADP
ejpam-4797	10	35	u(r	u(r	NOUN
ejpam-4797	10	36	)	)	PUNCT
ejpam-4797	11	1	=	=	SYM
ejpam-4797	11	2	2	2	NUM
ejpam-4797	11	3	is	be	AUX
ejpam-4797	11	4	a	a	DET
ejpam-4797	11	5	nil	nil	ADJ
ejpam-4797	11	6	ideal	ideal	NOUN
ejpam-4797	11	7	of	of	ADP
ejpam-4797	11	8	characteristic	characteristic	ADJ
ejpam-4797	11	9	4	4	NUM
ejpam-4797	11	10	.	.	PUNCT
ejpam-4797	12	1	we	we	PRON
ejpam-4797	12	2	compare	compare	VERB
ejpam-4797	12	3	this	this	DET
ejpam-4797	12	4	ring	ring	NOUN
ejpam-4797	12	5	with	with	ADP
ejpam-4797	12	6	other	other	ADJ
ejpam-4797	12	7	rings	ring	NOUN
ejpam-4797	12	8	,	,	PUNCT
ejpam-4797	12	9	since	since	SCONJ
ejpam-4797	12	10	every	every	DET
ejpam-4797	12	11	snc	snc	NOUN
ejpam-4797	12	12	ring	ring	NOUN
ejpam-4797	12	13	is	be	AUX
ejpam-4797	12	14	strongly	strongly	ADV
ejpam-4797	12	15	2	2	NUM
ejpam-4797	12	16	-	-	PUNCT
ejpam-4797	12	17	nc	nc	NOUN
ejpam-4797	12	18	,	,	PUNCT
ejpam-4797	12	19	but	but	CCONJ
ejpam-4797	12	20	not	not	PART
ejpam-4797	12	21	every	every	DET
ejpam-4797	12	22	unit	unit	NOUN
ejpam-4797	12	23	of	of	ADP
ejpam-4797	12	24	order	order	NOUN
ejpam-4797	12	25	2	2	NUM
ejpam-4797	12	26	,	,	PUNCT
ejpam-4797	12	27	and	and	CCONJ
ejpam-4797	12	28	if	if	SCONJ
ejpam-4797	12	29	r	r	NOUN
ejpam-4797	12	30	is	be	AUX
ejpam-4797	12	31	a	a	DET
ejpam-4797	12	32	strongly	strongly	ADV
ejpam-4797	12	33	2	2	NUM
ejpam-4797	12	34	-	-	PUNCT
ejpam-4797	12	35	nc	nc	NOUN
ejpam-4797	12	36	with	with	ADP
ejpam-4797	12	37	u(r	u(r	NOUN
ejpam-4797	12	38	)	)	PUNCT
ejpam-4797	13	1	=	=	SYM
ejpam-4797	13	2	2	2	NUM
ejpam-4797	13	3	,	,	PUNCT
ejpam-4797	13	4	then	then	ADV
ejpam-4797	13	5	r	r	NOUN
ejpam-4797	13	6	need	need	AUX
ejpam-4797	13	7	not	not	PART
ejpam-4797	13	8	be	be	AUX
ejpam-4797	13	9	snc	snc	PROPN
ejpam-4797	13	10	ring	ring	NOUN
ejpam-4797	13	11	.	.	PUNCT
ejpam-4797	14	1	in	in	ADP
ejpam-4797	14	2	order	order	NOUN
ejpam-4797	14	3	to	to	PART
ejpam-4797	14	4	get	get	VERB
ejpam-4797	14	5	nil(r	nil(r	NOUN
ejpam-4797	14	6	)	)	PUNCT
ejpam-4797	14	7	=	=	SYM
ejpam-4797	14	8	0	0	NUM
ejpam-4797	14	9	,	,	PUNCT
ejpam-4797	14	10	we	we	PRON
ejpam-4797	14	11	added	add	VERB
ejpam-4797	14	12	one	one	NUM
ejpam-4797	14	13	more	more	ADJ
ejpam-4797	14	14	condition	condition	NOUN
ejpam-4797	14	15	involving	involve	VERB
ejpam-4797	14	16	this	this	DET
ejpam-4797	14	17	ring	ring	NOUN
ejpam-4797	14	18	.	.	PUNCT
ejpam-4797	15	1	2020	2020	NUM
ejpam-4797	15	2	mathematics	mathematic	NOUN
ejpam-4797	15	3	subject	subject	NOUN
ejpam-4797	15	4	classifications	classification	NOUN
ejpam-4797	15	5	:	:	PUNCT
ejpam-4797	15	6	05c69	05c69	X
ejpam-4797	15	7	key	key	ADJ
ejpam-4797	15	8	words	word	NOUN
ejpam-4797	15	9	and	and	CCONJ
ejpam-4797	15	10	phrases	phrase	NOUN
ejpam-4797	15	11	:	:	PUNCT
ejpam-4797	15	12	clean	clean	ADJ
ejpam-4797	15	13	,	,	PUNCT
ejpam-4797	15	14	nil	nil	ADJ
ejpam-4797	15	15	clean	clean	ADJ
ejpam-4797	15	16	,	,	PUNCT
ejpam-4797	15	17	strongly	strongly	ADV
ejpam-4797	15	18	2	2	NUM
ejpam-4797	15	19	-	-	PUNCT
ejpam-4797	15	20	nil	nil	NOUN
ejpam-4797	15	21	clean	clean	ADJ
ejpam-4797	15	22	,	,	PUNCT
ejpam-4797	15	23	tripotent	tripotent	ADJ
ejpam-4797	15	24	1	1	NUM
ejpam-4797	15	25	.	.	PUNCT
ejpam-4797	15	26	introduction	introduction	NOUN
ejpam-4797	15	27	in	in	ADP
ejpam-4797	15	28	[	[	X
ejpam-4797	15	29	1	1	NUM
ejpam-4797	15	30	]	]	X
ejpam-4797	15	31	w.k	w.k	PROPN
ejpam-4797	15	32	.	.	PROPN
ejpam-4797	15	33	nicholson	nicholson	PROPN
ejpam-4797	15	34	defined	define	VERB
ejpam-4797	15	35	a	a	DET
ejpam-4797	15	36	clean	clean	ADJ
ejpam-4797	15	37	ring	ring	NOUN
ejpam-4797	15	38	as	as	ADP
ejpam-4797	15	39	having	have	VERB
ejpam-4797	15	40	an	an	DET
ejpam-4797	15	41	σ	σ	NOUN
ejpam-4797	15	42	=	=	PROPN
ejpam-4797	15	43	σ2	σ2	PROPN
ejpam-4797	15	44	and	and	CCONJ
ejpam-4797	15	45	a	a	DET
ejpam-4797	15	46	unit	unit	NOUN
ejpam-4797	15	47	u	u	NOUN
ejpam-4797	15	48	with	with	ADP
ejpam-4797	15	49	a	a	DET
ejpam-4797	15	50	=	=	SYM
ejpam-4797	15	51	σ	σ	PROPN
ejpam-4797	15	52	+	+	CCONJ
ejpam-4797	15	53	u.	u.	NOUN
ejpam-4797	15	54	in	in	ADP
ejpam-4797	15	55	[	[	X
ejpam-4797	15	56	2	2	NUM
ejpam-4797	15	57	]	]	PUNCT
ejpam-4797	15	58	,	,	PUNCT
ejpam-4797	15	59	an	an	DET
ejpam-4797	15	60	element	element	NOUN
ejpam-4797	15	61	a	a	DET
ejpam-4797	15	62	∈	∈	NOUN
ejpam-4797	15	63	r	r	NOUN
ejpam-4797	15	64	is	be	AUX
ejpam-4797	15	65	said	say	VERB
ejpam-4797	15	66	to	to	PART
ejpam-4797	15	67	be	be	AUX
ejpam-4797	15	68	strongly	strongly	ADV
ejpam-4797	15	69	clean	clean	ADJ
ejpam-4797	15	70	if	if	SCONJ
ejpam-4797	15	71	a	a	DET
ejpam-4797	15	72	=	=	X
ejpam-4797	15	73	σ	σ	PROPN
ejpam-4797	15	74	+	+	NOUN
ejpam-4797	15	75	u	u	NOUN
ejpam-4797	15	76	with	with	ADP
ejpam-4797	15	77	u	u	NOUN
ejpam-4797	15	78	∈	∈	PROPN
ejpam-4797	15	79	u(r),σ	u(r),σ	PROPN
ejpam-4797	15	80	∈	∈	PROPN
ejpam-4797	15	81	id(r	id(r	NOUN
ejpam-4797	15	82	)	)	PUNCT
ejpam-4797	15	83	and	and	CCONJ
ejpam-4797	15	84	uς	uς	PRON
ejpam-4797	15	85	=	=	SYM
ejpam-4797	16	1	σu	σu	INTJ
ejpam-4797	16	2	.	.	PUNCT
ejpam-4797	17	1	while	while	SCONJ
ejpam-4797	17	2	the	the	DET
ejpam-4797	17	3	ring	ring	NOUN
ejpam-4797	17	4	r	r	NOUN
ejpam-4797	17	5	is	be	AUX
ejpam-4797	17	6	strongly	strongly	ADV
ejpam-4797	17	7	clean	clean	ADJ
ejpam-4797	17	8	if	if	SCONJ
ejpam-4797	17	9	every	every	DET
ejpam-4797	17	10	element	element	NOUN
ejpam-4797	17	11	of	of	ADP
ejpam-4797	17	12	r	r	NOUN
ejpam-4797	17	13	is	be	AUX
ejpam-4797	17	14	strongly	strongly	ADV
ejpam-4797	17	15	clean	clean	ADJ
ejpam-4797	17	16	.	.	PUNCT
ejpam-4797	18	1	clearly	clearly	ADV
ejpam-4797	18	2	,	,	PUNCT
ejpam-4797	18	3	z9	z9	PROPN
ejpam-4797	18	4	is	be	AUX
ejpam-4797	18	5	a	a	DET
ejpam-4797	18	6	strongly	strongly	ADV
ejpam-4797	18	7	clean	clean	ADJ
ejpam-4797	18	8	ring	ring	NOUN
ejpam-4797	18	9	.	.	PUNCT
ejpam-4797	19	1	a	a	DET
ejpam-4797	19	2	nil	nil	ADJ
ejpam-4797	19	3	-	-	PUNCT
ejpam-4797	19	4	clean	clean	ADJ
ejpam-4797	19	5	ring	ring	NOUN
ejpam-4797	19	6	is	be	AUX
ejpam-4797	19	7	defined	define	VERB
ejpam-4797	19	8	as	as	ADP
ejpam-4797	19	9	a	a	DET
ejpam-4797	19	10	ring	ring	NOUN
ejpam-4797	19	11	with	with	ADP
ejpam-4797	19	12	each	each	DET
ejpam-4797	19	13	element	element	NOUN
ejpam-4797	19	14	is	be	AUX
ejpam-4797	19	15	the	the	DET
ejpam-4797	19	16	sum	sum	NOUN
ejpam-4797	19	17	of	of	ADP
ejpam-4797	19	18	an	an	DET
ejpam-4797	19	19	idempotent	idempotent	NOUN
ejpam-4797	19	20	and	and	CCONJ
ejpam-4797	19	21	a	a	DET
ejpam-4797	19	22	nilpotent	nilpotent	NOUN
ejpam-4797	19	23	was	be	AUX
ejpam-4797	19	24	first	first	ADV
ejpam-4797	19	25	proposed	propose	VERB
ejpam-4797	19	26	by	by	ADP
ejpam-4797	19	27	diesl	diesl	PROPN
ejpam-4797	19	28	in	in	ADP
ejpam-4797	19	29	[	[	X
ejpam-4797	19	30	3	3	NUM
ejpam-4797	19	31	]	]	PUNCT
ejpam-4797	19	32	,	,	PUNCT
ejpam-4797	19	33	r	r	NOUN
ejpam-4797	19	34	is	be	AUX
ejpam-4797	19	35	considered	consider	VERB
ejpam-4797	19	36	a	a	DET
ejpam-4797	19	37	strongly	strongly	ADV
ejpam-4797	19	38	nil	nil	ADJ
ejpam-4797	19	39	clean	clean	ADJ
ejpam-4797	19	40	(	(	PUNCT
ejpam-4797	19	41	snc	snc	NOUN
ejpam-4797	19	42	for	for	ADP
ejpam-4797	19	43	short	short	ADJ
ejpam-4797	19	44	)	)	PUNCT
ejpam-4797	19	45	if	if	SCONJ
ejpam-4797	19	46	the	the	DET
ejpam-4797	19	47	idempotent	idempotent	NOUN
ejpam-4797	19	48	and	and	CCONJ
ejpam-4797	19	49	nilpotent	nilpotent	ADJ
ejpam-4797	19	50	commute	commute	NOUN
ejpam-4797	19	51	[	[	X
ejpam-4797	19	52	4	4	NUM
ejpam-4797	19	53	]	]	PUNCT
ejpam-4797	19	54	.	.	PUNCT
ejpam-4797	20	1	the	the	DET
ejpam-4797	20	2	structure	structure	NOUN
ejpam-4797	20	3	of	of	ADP
ejpam-4797	20	4	snc	snc	PROPN
ejpam-4797	20	5	rings	ring	NOUN
ejpam-4797	20	6	and	and	CCONJ
ejpam-4797	20	7	related	related	ADJ
ejpam-4797	20	8	topics	topic	NOUN
ejpam-4797	20	9	was	be	AUX
ejpam-4797	20	10	given	give	VERB
ejpam-4797	20	11	for	for	ADP
ejpam-4797	20	12	example	example	NOUN
ejpam-4797	20	13	in	in	ADP
ejpam-4797	20	14	[	[	X
ejpam-4797	20	15	5	5	NUM
ejpam-4797	20	16	]	]	PUNCT
ejpam-4797	20	17	and	and	CCONJ
ejpam-4797	20	18	[	[	X
ejpam-4797	20	19	6	6	NUM
ejpam-4797	20	20	]	]	PUNCT
ejpam-4797	20	21	.	.	PUNCT
ejpam-4797	21	1	clearly	clearly	ADV
ejpam-4797	21	2	,	,	PUNCT
ejpam-4797	21	3	z8	z8	NOUN
ejpam-4797	21	4	is	be	AUX
ejpam-4797	21	5	an	an	DET
ejpam-4797	21	6	snc	snc	PROPN
ejpam-4797	21	7	ring	ring	NOUN
ejpam-4797	21	8	.	.	PUNCT
ejpam-4797	22	1	a	a	DET
ejpam-4797	22	2	strongly	strongly	ADV
ejpam-4797	22	3	2	2	NUM
ejpam-4797	22	4	-	-	PUNCT
ejpam-4797	22	5	nc	nc	PROPN
ejpam-4797	22	6	ring	ring	NOUN
ejpam-4797	22	7	was	be	AUX
ejpam-4797	22	8	defined	define	VERB
ejpam-4797	22	9	by	by	ADP
ejpam-4797	22	10	chen	chen	PROPN
ejpam-4797	22	11	and	and	CCONJ
ejpam-4797	22	12	sheibani	sheibani	NOUN
ejpam-4797	22	13	in	in	ADP
ejpam-4797	22	14	[	[	X
ejpam-4797	22	15	7	7	NUM
ejpam-4797	22	16	]	]	PUNCT
ejpam-4797	22	17	as	as	ADP
ejpam-4797	22	18	a	a	DET
ejpam-4797	22	19	ring	ring	NOUN
ejpam-4797	22	20	r	r	NOUN
ejpam-4797	22	21	with	with	ADP
ejpam-4797	22	22	each	each	DET
ejpam-4797	22	23	element	element	NOUN
ejpam-4797	22	24	is	be	AUX
ejpam-4797	22	25	a	a	DET
ejpam-4797	22	26	sum	sum	NOUN
ejpam-4797	22	27	of	of	ADP
ejpam-4797	22	28	two	two	NUM
ejpam-4797	22	29	idempotents	idempotent	NOUN
ejpam-4797	22	30	and	and	CCONJ
ejpam-4797	22	31	a	a	DET
ejpam-4797	22	32	nilpotent	nilpotent	NOUN
ejpam-4797	22	33	that	that	PRON
ejpam-4797	22	34	commute	commute	VERB
ejpam-4797	22	35	with	with	ADP
ejpam-4797	22	36	each	each	DET
ejpam-4797	22	37	other	other	ADJ
ejpam-4797	22	38	.	.	PUNCT
ejpam-4797	23	1	many	many	ADJ
ejpam-4797	23	2	authors	author	NOUN
ejpam-4797	23	3	have	have	AUX
ejpam-4797	23	4	been	be	AUX
ejpam-4797	23	5	working	work	VERB
ejpam-4797	23	6	on	on	ADP
ejpam-4797	23	7	these	these	DET
ejpam-4797	23	8	topics	topic	NOUN
ejpam-4797	23	9	see	see	VERB
ejpam-4797	23	10	for	for	ADP
ejpam-4797	23	11	example	example	NOUN
ejpam-4797	23	12	[	[	X
ejpam-4797	23	13	8	8	X
ejpam-4797	23	14	]	]	X
ejpam-4797	23	15	a	a	DET
ejpam-4797	23	16	ring	ring	NOUN
ejpam-4797	23	17	r	r	NOUN
ejpam-4797	23	18	is	be	AUX
ejpam-4797	23	19	called	call	VERB
ejpam-4797	23	20	strongly	strongly	ADV
ejpam-4797	23	21	∗corresponding	∗corresponde	VERB
ejpam-4797	23	22	author	author	NOUN
ejpam-4797	23	23	.	.	PUNCT
ejpam-4797	24	1	doi	doi	NOUN
ejpam-4797	24	2	:	:	PUNCT
ejpam-4797	24	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4797	https://doi.org/10.29020/nybg.ejpam.v16i3.4797	NUM
ejpam-4797	24	4	email	email	NOUN
ejpam-4797	24	5	addresses	address	NOUN
ejpam-4797	24	6	:	:	PUNCT
ejpam-4797	24	7	renas.salim@uoz.edu.krd	renas.salim@uoz.edu.krd	NOUN
ejpam-4797	24	8	(	(	PUNCT
ejpam-4797	24	9	r.	r.	PROPN
ejpam-4797	24	10	t.	t.	PROPN
ejpam-4797	24	11	m.salim	m.salim	PROPN
ejpam-4797	24	12	)	)	PUNCT
ejpam-4797	24	13	,	,	PUNCT
ejpam-4797	24	14	nazarh	nazarh	NOUN
ejpam-4797	24	15	2013@yahoo.com	2013@yahoo.com	X
ejpam-4797	24	16	(	(	PUNCT
ejpam-4797	24	17	n.	n.	PROPN
ejpam-4797	24	18	h.	h.	PROPN
ejpam-4797	24	19	shuker	shuker	PROPN
ejpam-4797	24	20	)	)	PUNCT
ejpam-4797	24	21	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4797	24	22	1675	1675	NUM
ejpam-4797	25	1	©	©	PROPN
ejpam-4797	25	2	2023	2023	NUM
ejpam-4797	25	3	ejpam	ejpam	NOUN
ejpam-4797	25	4	all	all	DET
ejpam-4797	25	5	rights	right	NOUN
ejpam-4797	25	6	reserved	reserve	VERB
ejpam-4797	25	7	.	.	PUNCT
ejpam-4797	26	1	r.	r.	PROPN
ejpam-4797	26	2	t.	t.	PROPN
ejpam-4797	26	3	m.salim	m.salim	PROPN
ejpam-4797	26	4	,	,	PUNCT
ejpam-4797	26	5	n.	n.	PROPN
ejpam-4797	26	6	h.	h.	PROPN
ejpam-4797	26	7	shuker	shuker	PROPN
ejpam-4797	26	8	/	/	SYM
ejpam-4797	26	9	eur	eur	PROPN
ejpam-4797	26	10	.	.	PUNCT
ejpam-4797	27	1	j.	j.	PROPN
ejpam-4797	27	2	pure	pure	PROPN
ejpam-4797	27	3	appl	appl	PROPN
ejpam-4797	27	4	.	.	PROPN
ejpam-4797	27	5	math	math	PROPN
ejpam-4797	27	6	,	,	PUNCT
ejpam-4797	27	7	16	16	NUM
ejpam-4797	27	8	(	(	PUNCT
ejpam-4797	27	9	3	3	NUM
ejpam-4797	27	10	)	)	PUNCT
ejpam-4797	27	11	(	(	PUNCT
ejpam-4797	27	12	2023	2023	NUM
ejpam-4797	27	13	)	)	PUNCT
ejpam-4797	27	14	,	,	PUNCT
ejpam-4797	27	15	1675	1675	NUM
ejpam-4797	27	16	-	-	SYM
ejpam-4797	27	17	1684	1684	NUM
ejpam-4797	27	18	1676	1676	NUM
ejpam-4797	27	19	2	2	NUM
ejpam-4797	27	20	-	-	PUNCT
ejpam-4797	27	21	nil-*-clean	nil-*-clean	ADJ
ejpam-4797	27	22	if	if	SCONJ
ejpam-4797	27	23	every	every	DET
ejpam-4797	27	24	element	element	NOUN
ejpam-4797	27	25	in	in	ADP
ejpam-4797	27	26	r	r	NOUN
ejpam-4797	27	27	is	be	AUX
ejpam-4797	27	28	the	the	DET
ejpam-4797	27	29	sum	sum	NOUN
ejpam-4797	27	30	of	of	ADP
ejpam-4797	27	31	two	two	NUM
ejpam-4797	27	32	projections	projection	NOUN
ejpam-4797	27	33	and	and	CCONJ
ejpam-4797	27	34	a	a	DET
ejpam-4797	27	35	nilpotent	nilpotent	NOUN
ejpam-4797	27	36	that	that	SCONJ
ejpam-4797	27	37	commute	commute	NOUN
ejpam-4797	27	38	,	,	PUNCT
ejpam-4797	27	39	[	[	X
ejpam-4797	27	40	9	9	NUM
ejpam-4797	27	41	]	]	PUNCT
ejpam-4797	27	42	if	if	SCONJ
ejpam-4797	27	43	every	every	DET
ejpam-4797	27	44	element	element	NOUN
ejpam-4797	27	45	in	in	ADP
ejpam-4797	27	46	r	r	NOUN
ejpam-4797	27	47	is	be	AUX
ejpam-4797	27	48	the	the	DET
ejpam-4797	27	49	sum	sum	NOUN
ejpam-4797	27	50	of	of	ADP
ejpam-4797	27	51	an	an	DET
ejpam-4797	27	52	idempotent	idempotent	NOUN
ejpam-4797	27	53	and	and	CCONJ
ejpam-4797	27	54	two	two	NUM
ejpam-4797	27	55	nilpotents	nilpotent	NOUN
ejpam-4797	27	56	,	,	PUNCT
ejpam-4797	27	57	then	then	ADV
ejpam-4797	27	58	r	r	NOUN
ejpam-4797	27	59	is	be	AUX
ejpam-4797	27	60	called	call	VERB
ejpam-4797	27	61	2	2	NUM
ejpam-4797	27	62	-	-	PUNCT
ejpam-4797	27	63	nil	nil	ADV
ejpam-4797	27	64	-	-	PUNCT
ejpam-4797	27	65	clean	clean	ADJ
ejpam-4797	27	66	and	and	CCONJ
ejpam-4797	27	67	[	[	X
ejpam-4797	27	68	10	10	NUM
ejpam-4797	27	69	]	]	X
ejpam-4797	27	70	a	a	DET
ejpam-4797	27	71	ring	ring	NOUN
ejpam-4797	27	72	r	r	NOUN
ejpam-4797	27	73	is	be	AUX
ejpam-4797	27	74	defined	define	VERB
ejpam-4797	27	75	to	to	PART
ejpam-4797	27	76	be	be	AUX
ejpam-4797	27	77	2	2	NUM
ejpam-4797	27	78	-	-	PUNCT
ejpam-4797	27	79	nil	nil	NOUN
ejpam-4797	27	80	-	-	PUNCT
ejpam-4797	27	81	good	good	ADJ
ejpam-4797	27	82	if	if	SCONJ
ejpam-4797	27	83	every	every	DET
ejpam-4797	27	84	element	element	NOUN
ejpam-4797	27	85	in	in	ADP
ejpam-4797	27	86	r	r	NOUN
ejpam-4797	27	87	is	be	AUX
ejpam-4797	27	88	the	the	DET
ejpam-4797	27	89	sum	sum	NOUN
ejpam-4797	27	90	of	of	ADP
ejpam-4797	27	91	two	two	NUM
ejpam-4797	27	92	units	unit	NOUN
ejpam-4797	27	93	and	and	CCONJ
ejpam-4797	27	94	a	a	DET
ejpam-4797	27	95	nilpotent	nilpotent	NOUN
ejpam-4797	27	96	.	.	PUNCT
ejpam-4797	28	1	the	the	DET
ejpam-4797	28	2	purpose	purpose	NOUN
ejpam-4797	28	3	of	of	ADP
ejpam-4797	28	4	this	this	DET
ejpam-4797	28	5	paper	paper	NOUN
ejpam-4797	28	6	is	be	AUX
ejpam-4797	28	7	to	to	PART
ejpam-4797	28	8	present	present	VERB
ejpam-4797	28	9	new	new	ADJ
ejpam-4797	28	10	properties	property	NOUN
ejpam-4797	28	11	of	of	ADP
ejpam-4797	28	12	strongly	strongly	ADV
ejpam-4797	28	13	2	2	NUM
ejpam-4797	28	14	-	-	PUNCT
ejpam-4797	28	15	nc	nc	NOUN
ejpam-4797	28	16	rings	ring	NOUN
ejpam-4797	28	17	,	,	PUNCT
ejpam-4797	28	18	and	and	CCONJ
ejpam-4797	28	19	their	their	PRON
ejpam-4797	28	20	connection	connection	NOUN
ejpam-4797	28	21	with	with	ADP
ejpam-4797	28	22	other	other	ADJ
ejpam-4797	28	23	related	related	ADJ
ejpam-4797	28	24	rings	ring	NOUN
ejpam-4797	28	25	.	.	PUNCT
ejpam-4797	29	1	we	we	PRON
ejpam-4797	29	2	prove	prove	VERB
ejpam-4797	29	3	that	that	SCONJ
ejpam-4797	29	4	if	if	SCONJ
ejpam-4797	29	5	r	r	NOUN
ejpam-4797	29	6	is	be	AUX
ejpam-4797	29	7	a	a	DET
ejpam-4797	29	8	strongly	strongly	ADV
ejpam-4797	29	9	2	2	NUM
ejpam-4797	29	10	-	-	PUNCT
ejpam-4797	29	11	nc	nc	NOUN
ejpam-4797	29	12	ring	ring	NOUN
ejpam-4797	29	13	,	,	PUNCT
ejpam-4797	29	14	with	with	ADP
ejpam-4797	29	15	n2	n2	ADJ
ejpam-4797	29	16	+	+	CCONJ
ejpam-4797	29	17	2n	2n	NUM
ejpam-4797	29	18	=	=	SYM
ejpam-4797	29	19	0	0	NUM
ejpam-4797	29	20	for	for	ADP
ejpam-4797	29	21	every	every	DET
ejpam-4797	29	22	n	n	PRON
ejpam-4797	29	23	∈	∈	PROPN
ejpam-4797	29	24	nil(r	nil(r	PROPN
ejpam-4797	29	25	)	)	PUNCT
ejpam-4797	29	26	.	.	PUNCT
ejpam-4797	30	1	then	then	ADV
ejpam-4797	30	2	r	r	NOUN
ejpam-4797	30	3	is	be	AUX
ejpam-4797	30	4	of	of	ADP
ejpam-4797	30	5	characteristic	characteristic	ADJ
ejpam-4797	30	6	48	48	NUM
ejpam-4797	30	7	with	with	ADP
ejpam-4797	30	8	every	every	DET
ejpam-4797	30	9	unit	unit	NOUN
ejpam-4797	30	10	is	be	AUX
ejpam-4797	30	11	of	of	ADP
ejpam-4797	30	12	order	order	NOUN
ejpam-4797	30	13	4	4	NUM
ejpam-4797	30	14	.	.	PUNCT
ejpam-4797	31	1	additionally	additionally	ADV
ejpam-4797	31	2	,	,	PUNCT
ejpam-4797	31	3	we	we	PRON
ejpam-4797	31	4	introduce	introduce	VERB
ejpam-4797	31	5	and	and	CCONJ
ejpam-4797	31	6	investigate	investigate	VERB
ejpam-4797	31	7	a	a	DET
ejpam-4797	31	8	strongly	strongly	ADV
ejpam-4797	31	9	2	2	NUM
ejpam-4797	31	10	-	-	PUNCT
ejpam-4797	31	11	nc	nc	PROPN
ejpam-4797	31	12	rings	ring	NOUN
ejpam-4797	31	13	with	with	ADP
ejpam-4797	31	14	u(r	u(r	NOUN
ejpam-4797	31	15	)	)	PUNCT
ejpam-4797	32	1	=	=	SYM
ejpam-4797	32	2	2	2	NUM
ejpam-4797	32	3	,	,	PUNCT
ejpam-4797	32	4	providing	provide	VERB
ejpam-4797	32	5	their	their	PRON
ejpam-4797	32	6	fundamental	fundamental	ADJ
ejpam-4797	32	7	properties	property	NOUN
ejpam-4797	32	8	and	and	CCONJ
ejpam-4797	32	9	their	their	PRON
ejpam-4797	32	10	connection	connection	NOUN
ejpam-4797	32	11	with	with	ADP
ejpam-4797	32	12	tripotent	tripotent	ADJ
ejpam-4797	32	13	rings	ring	NOUN
ejpam-4797	32	14	and	and	CCONJ
ejpam-4797	32	15	other	other	ADJ
ejpam-4797	32	16	related	related	ADJ
ejpam-4797	32	17	rings	ring	NOUN
ejpam-4797	32	18	.	.	PUNCT
ejpam-4797	33	1	among	among	ADP
ejpam-4797	33	2	other	other	ADJ
ejpam-4797	33	3	results	result	NOUN
ejpam-4797	33	4	we	we	PRON
ejpam-4797	33	5	prove	prove	VERB
ejpam-4797	33	6	that	that	SCONJ
ejpam-4797	33	7	:	:	PUNCT
ejpam-4797	33	8	if	if	SCONJ
ejpam-4797	33	9	r	r	NOUN
ejpam-4797	33	10	is	be	AUX
ejpam-4797	33	11	a	a	DET
ejpam-4797	33	12	strongly	strongly	ADV
ejpam-4797	33	13	2	2	NUM
ejpam-4797	33	14	-	-	PUNCT
ejpam-4797	33	15	nc	nc	NOUN
ejpam-4797	33	16	ring	ring	NOUN
ejpam-4797	33	17	with	with	ADP
ejpam-4797	33	18	2	2	NUM
ejpam-4797	33	19	∈	∈	PROPN
ejpam-4797	33	20	u(r	u(r	NOUN
ejpam-4797	33	21	)	)	PUNCT
ejpam-4797	33	22	.	.	PUNCT
ejpam-4797	34	1	then	then	ADV
ejpam-4797	34	2	24	24	NUM
ejpam-4797	34	3	=	=	SYM
ejpam-4797	34	4	0	0	NUM
ejpam-4797	34	5	,	,	PUNCT
ejpam-4797	34	6	and	and	CCONJ
ejpam-4797	34	7	the	the	DET
ejpam-4797	34	8	jacobson	jacobson	PROPN
ejpam-4797	34	9	radical	radical	PROPN
ejpam-4797	34	10	over	over	ADP
ejpam-4797	34	11	a	a	DET
ejpam-4797	34	12	strongly	strongly	ADV
ejpam-4797	34	13	2	2	NUM
ejpam-4797	34	14	-	-	PUNCT
ejpam-4797	34	15	nc	nc	PROPN
ejpam-4797	34	16	ring	ring	NOUN
ejpam-4797	34	17	is	be	AUX
ejpam-4797	34	18	a	a	DET
ejpam-4797	34	19	nil	nil	ADJ
ejpam-4797	34	20	ideal	ideal	NOUN
ejpam-4797	34	21	of	of	ADP
ejpam-4797	34	22	characteristic	characteristic	ADJ
ejpam-4797	34	23	4	4	NUM
ejpam-4797	34	24	.	.	PUNCT
ejpam-4797	35	1	in	in	ADP
ejpam-4797	35	2	addition	addition	NOUN
ejpam-4797	35	3	,	,	PUNCT
ejpam-4797	35	4	we	we	PRON
ejpam-4797	35	5	show	show	VERB
ejpam-4797	35	6	that	that	SCONJ
ejpam-4797	35	7	if	if	SCONJ
ejpam-4797	35	8	r	r	NOUN
ejpam-4797	35	9	is	be	AUX
ejpam-4797	35	10	a	a	DET
ejpam-4797	35	11	strongly	strongly	ADV
ejpam-4797	35	12	2	2	NUM
ejpam-4797	35	13	-	-	PUNCT
ejpam-4797	35	14	nc	nc	NOUN
ejpam-4797	35	15	ring	ring	NOUN
ejpam-4797	35	16	with	with	ADP
ejpam-4797	35	17	u(r	u(r	NOUN
ejpam-4797	35	18	)	)	PUNCT
ejpam-4797	35	19	=	=	SYM
ejpam-4797	35	20	2	2	NUM
ejpam-4797	35	21	and	and	CCONJ
ejpam-4797	35	22	2	2	NUM
ejpam-4797	35	23	∈	∈	PROPN
ejpam-4797	35	24	u(r	u(r	NOUN
ejpam-4797	35	25	)	)	PUNCT
ejpam-4797	35	26	,	,	PUNCT
ejpam-4797	35	27	then	then	ADV
ejpam-4797	35	28	nil(r	nil(r	NUM
ejpam-4797	35	29	)	)	PUNCT
ejpam-4797	35	30	=	=	NOUN
ejpam-4797	35	31	0	0	X
ejpam-4797	35	32	.	.	PUNCT
ejpam-4797	36	1	in	in	ADP
ejpam-4797	36	2	this	this	DET
ejpam-4797	36	3	paper	paper	NOUN
ejpam-4797	36	4	,	,	PUNCT
ejpam-4797	36	5	we	we	PRON
ejpam-4797	36	6	define	define	VERB
ejpam-4797	36	7	r	r	NOUN
ejpam-4797	36	8	as	as	ADP
ejpam-4797	36	9	an	an	DET
ejpam-4797	36	10	associative	associative	ADJ
ejpam-4797	36	11	ring	ring	NOUN
ejpam-4797	36	12	containing	contain	VERB
ejpam-4797	36	13	an	an	DET
ejpam-4797	36	14	identity	identity	NOUN
ejpam-4797	36	15	element	element	NOUN
ejpam-4797	36	16	.	.	PUNCT
ejpam-4797	37	1	finally	finally	ADV
ejpam-4797	37	2	,	,	PUNCT
ejpam-4797	37	3	it	it	PRON
ejpam-4797	37	4	is	be	AUX
ejpam-4797	37	5	worth	worth	ADJ
ejpam-4797	37	6	mentioning	mention	VERB
ejpam-4797	37	7	that	that	SCONJ
ejpam-4797	37	8	ring	ring	NOUN
ejpam-4797	37	9	theory	theory	NOUN
ejpam-4797	37	10	has	have	VERB
ejpam-4797	37	11	several	several	ADJ
ejpam-4797	37	12	applications	application	NOUN
ejpam-4797	37	13	in	in	ADP
ejpam-4797	37	14	many	many	ADJ
ejpam-4797	37	15	field	field	NOUN
ejpam-4797	37	16	,	,	PUNCT
ejpam-4797	37	17	see	see	VERB
ejpam-4797	37	18	for	for	ADP
ejpam-4797	37	19	example	example	NOUN
ejpam-4797	38	1	[	[	X
ejpam-4797	38	2	11	11	NUM
ejpam-4797	38	3	]	]	PUNCT
ejpam-4797	38	4	,	,	PUNCT
ejpam-4797	38	5	[	[	X
ejpam-4797	38	6	12	12	NUM
ejpam-4797	38	7	]	]	PUNCT
ejpam-4797	38	8	and	and	CCONJ
ejpam-4797	38	9	[	[	X
ejpam-4797	38	10	13	13	NUM
ejpam-4797	38	11	]	]	PUNCT
ejpam-4797	38	12	.	.	PUNCT
ejpam-4797	39	1	to	to	PART
ejpam-4797	39	2	represent	represent	VERB
ejpam-4797	39	3	the	the	DET
ejpam-4797	39	4	set	set	NOUN
ejpam-4797	39	5	of	of	ADP
ejpam-4797	39	6	units	unit	NOUN
ejpam-4797	39	7	,	,	PUNCT
ejpam-4797	39	8	idempotents	idempotent	NOUN
ejpam-4797	39	9	and	and	CCONJ
ejpam-4797	39	10	nilpotents	nilpotent	NOUN
ejpam-4797	39	11	in	in	ADP
ejpam-4797	39	12	r	r	NOUN
ejpam-4797	39	13	,	,	PUNCT
ejpam-4797	39	14	we	we	PRON
ejpam-4797	39	15	will	will	AUX
ejpam-4797	39	16	use	use	VERB
ejpam-4797	39	17	the	the	DET
ejpam-4797	39	18	symbols	symbol	NOUN
ejpam-4797	39	19	u(r	u(r	PROPN
ejpam-4797	39	20	)	)	PUNCT
ejpam-4797	39	21	,	,	PUNCT
ejpam-4797	39	22	id(r	id(r	NOUN
ejpam-4797	39	23	)	)	PUNCT
ejpam-4797	39	24	and	and	CCONJ
ejpam-4797	39	25	nil(r	nil(r	NUM
ejpam-4797	39	26	)	)	PUNCT
ejpam-4797	39	27	,	,	PUNCT
ejpam-4797	39	28	respectively	respectively	ADV
ejpam-4797	39	29	.	.	PUNCT
ejpam-4797	40	1	additionally	additionally	ADV
ejpam-4797	40	2	,	,	PUNCT
ejpam-4797	40	3	we	we	PRON
ejpam-4797	40	4	will	will	AUX
ejpam-4797	40	5	use	use	VERB
ejpam-4797	40	6	j(r	j(r	PROPN
ejpam-4797	40	7	)	)	PUNCT
ejpam-4797	40	8	to	to	PART
ejpam-4797	40	9	denote	denote	VERB
ejpam-4797	40	10	the	the	DET
ejpam-4797	40	11	jacobson	jacobson	PROPN
ejpam-4797	40	12	radical	radical	PROPN
ejpam-4797	40	13	and	and	CCONJ
ejpam-4797	40	14	zn	zn	PROPN
ejpam-4797	40	15	for	for	ADP
ejpam-4797	40	16	the	the	DET
ejpam-4797	40	17	ring	ring	NOUN
ejpam-4797	40	18	of	of	ADP
ejpam-4797	40	19	integers	integer	NOUN
ejpam-4797	40	20	modulo	modulo	PROPN
ejpam-4797	40	21	n.	n.	PROPN
ejpam-4797	40	22	recall	recall	PROPN
ejpam-4797	40	23	that	that	PRON
ejpam-4797	40	24	:	:	PUNCT
ejpam-4797	40	25	definition	definition	NOUN
ejpam-4797	40	26	1	1	NUM
ejpam-4797	40	27	.	.	PUNCT
ejpam-4797	41	1	[	[	X
ejpam-4797	41	2	14	14	NUM
ejpam-4797	41	3	]	]	PUNCT
ejpam-4797	41	4	.	.	PUNCT
ejpam-4797	42	1	a	a	DET
ejpam-4797	42	2	ring	ring	NOUN
ejpam-4797	42	3	r	r	NOUN
ejpam-4797	42	4	is	be	AUX
ejpam-4797	42	5	considered	consider	VERB
ejpam-4797	42	6	to	to	PART
ejpam-4797	42	7	be	be	AUX
ejpam-4797	42	8	n	n	X
ejpam-4797	42	9	-	-	PUNCT
ejpam-4797	42	10	good	good	ADJ
ejpam-4797	42	11	if	if	SCONJ
ejpam-4797	42	12	each	each	DET
ejpam-4797	42	13	element	element	NOUN
ejpam-4797	42	14	is	be	AUX
ejpam-4797	42	15	a	a	DET
ejpam-4797	42	16	sum	sum	NOUN
ejpam-4797	42	17	of	of	ADP
ejpam-4797	42	18	n	n	DET
ejpam-4797	42	19	units	unit	NOUN
ejpam-4797	42	20	.	.	PUNCT
ejpam-4797	43	1	definition	definition	NOUN
ejpam-4797	43	2	2	2	NUM
ejpam-4797	43	3	.	.	PUNCT
ejpam-4797	44	1	[	[	X
ejpam-4797	44	2	15	15	NUM
ejpam-4797	44	3	]	]	PUNCT
ejpam-4797	44	4	.	.	PUNCT
ejpam-4797	45	1	if	if	SCONJ
ejpam-4797	45	2	t	t	PROPN
ejpam-4797	45	3	=	=	SYM
ejpam-4797	45	4	t3	t3	PROPN
ejpam-4797	45	5	is	be	AUX
ejpam-4797	45	6	referred	refer	VERB
ejpam-4797	45	7	to	to	ADP
ejpam-4797	45	8	as	as	ADP
ejpam-4797	45	9	a	a	DET
ejpam-4797	45	10	tripotent	tripotent	NOUN
ejpam-4797	45	11	.	.	PUNCT
ejpam-4797	46	1	r	r	NOUN
ejpam-4797	46	2	is	be	AUX
ejpam-4797	46	3	called	call	VERB
ejpam-4797	46	4	a	a	DET
ejpam-4797	46	5	tripotent	tripotent	ADJ
ejpam-4797	46	6	ring	ring	NOUN
ejpam-4797	46	7	if	if	SCONJ
ejpam-4797	46	8	every	every	DET
ejpam-4797	46	9	element	element	NOUN
ejpam-4797	46	10	of	of	ADP
ejpam-4797	46	11	r	r	NOUN
ejpam-4797	46	12	is	be	AUX
ejpam-4797	46	13	tripotent	tripotent	ADJ
ejpam-4797	46	14	.	.	PUNCT
ejpam-4797	47	1	clearly	clearly	ADV
ejpam-4797	47	2	,	,	PUNCT
ejpam-4797	47	3	z6	z6	PROPN
ejpam-4797	47	4	is	be	AUX
ejpam-4797	47	5	a	a	DET
ejpam-4797	47	6	tripotent	tripotent	ADJ
ejpam-4797	47	7	ring	ring	NOUN
ejpam-4797	47	8	.	.	PUNCT
ejpam-4797	48	1	definition	definition	NOUN
ejpam-4797	48	2	3	3	NUM
ejpam-4797	48	3	.	.	PUNCT
ejpam-4797	49	1	for	for	ADP
ejpam-4797	49	2	any	any	DET
ejpam-4797	49	3	a	a	DET
ejpam-4797	49	4	∈	∈	NOUN
ejpam-4797	49	5	r	r	NOUN
ejpam-4797	49	6	,	,	PUNCT
ejpam-4797	49	7	we	we	PRON
ejpam-4797	49	8	define	define	VERB
ejpam-4797	49	9	ann(a	ann(a	PROPN
ejpam-4797	49	10	)	)	PUNCT
ejpam-4797	49	11	=	=	PRON
ejpam-4797	49	12	{	{	PUNCT
ejpam-4797	49	13	b	b	X
ejpam-4797	49	14	∈	∈	PROPN
ejpam-4797	49	15	r	r	NOUN
ejpam-4797	49	16	:	:	PUNCT
ejpam-4797	49	17	ab	ab	PROPN
ejpam-4797	49	18	=	=	PUNCT
ejpam-4797	49	19	ba	ba	PROPN
ejpam-4797	49	20	=	=	NOUN
ejpam-4797	49	21	0	0	NUM
ejpam-4797	49	22	}	}	PUNCT
ejpam-4797	49	23	.	.	PUNCT
ejpam-4797	50	1	theorem	theorem	NOUN
ejpam-4797	50	2	1	1	NUM
ejpam-4797	50	3	.	.	PUNCT
ejpam-4797	51	1	[	[	X
ejpam-4797	51	2	7	7	NUM
ejpam-4797	51	3	]	]	PUNCT
ejpam-4797	51	4	.	.	PUNCT
ejpam-4797	52	1	let	let	VERB
ejpam-4797	52	2	r	r	PRON
ejpam-4797	52	3	be	be	AUX
ejpam-4797	52	4	a	a	DET
ejpam-4797	52	5	ring	ring	NOUN
ejpam-4797	52	6	.	.	PUNCT
ejpam-4797	53	1	then	then	ADV
ejpam-4797	53	2	the	the	DET
ejpam-4797	53	3	following	follow	VERB
ejpam-4797	53	4	are	be	AUX
ejpam-4797	53	5	equivalent	equivalent	ADJ
ejpam-4797	53	6	:	:	PUNCT
ejpam-4797	54	1	1	1	X
ejpam-4797	54	2	.	.	X
ejpam-4797	54	3	r	r	NOUN
ejpam-4797	54	4	is	be	AUX
ejpam-4797	54	5	strongly	strongly	ADV
ejpam-4797	54	6	2	2	NUM
ejpam-4797	54	7	-	-	PUNCT
ejpam-4797	54	8	nc	nc	NOUN
ejpam-4797	54	9	.	.	PROPN
ejpam-4797	54	10	2	2	NUM
ejpam-4797	54	11	.	.	X
ejpam-4797	55	1	for	for	ADP
ejpam-4797	55	2	all	all	DET
ejpam-4797	55	3	a	a	DET
ejpam-4797	55	4	∈	∈	NOUN
ejpam-4797	55	5	r	r	NOUN
ejpam-4797	55	6	,	,	PUNCT
ejpam-4797	55	7	a−	a−	PROPN
ejpam-4797	55	8	a3	a3	NOUN
ejpam-4797	55	9	∈	∈	PROPN
ejpam-4797	55	10	nil(r	nil(r	PROPN
ejpam-4797	55	11	)	)	PUNCT
ejpam-4797	55	12	.	.	PUNCT
ejpam-4797	56	1	3	3	X
ejpam-4797	56	2	.	.	X
ejpam-4797	56	3	for	for	ADP
ejpam-4797	56	4	all	all	DET
ejpam-4797	56	5	a	a	DET
ejpam-4797	56	6	∈	∈	PROPN
ejpam-4797	56	7	r	r	NOUN
ejpam-4797	56	8	,	,	PUNCT
ejpam-4797	56	9	a2	a2	PROPN
ejpam-4797	56	10	is	be	AUX
ejpam-4797	56	11	snc	snc	PROPN
ejpam-4797	56	12	element	element	NOUN
ejpam-4797	56	13	.	.	PUNCT
ejpam-4797	57	1	theorem	theorem	VERB
ejpam-4797	57	2	2	2	NUM
ejpam-4797	57	3	.	.	PUNCT
ejpam-4797	58	1	[	[	X
ejpam-4797	58	2	7	7	NUM
ejpam-4797	58	3	]	]	PUNCT
ejpam-4797	58	4	.	.	PUNCT
ejpam-4797	59	1	a	a	DET
ejpam-4797	59	2	ring	ring	NOUN
ejpam-4797	59	3	r	r	NOUN
ejpam-4797	59	4	is	be	AUX
ejpam-4797	59	5	strongly	strongly	ADV
ejpam-4797	59	6	2	2	NUM
ejpam-4797	59	7	-	-	PUNCT
ejpam-4797	59	8	nc	nc	NOUN
ejpam-4797	59	9	if	if	SCONJ
ejpam-4797	59	10	and	and	CCONJ
ejpam-4797	59	11	only	only	ADV
ejpam-4797	59	12	if	if	SCONJ
ejpam-4797	59	13	1	1	NUM
ejpam-4797	59	14	.	.	PUNCT
ejpam-4797	59	15	j(r	j(r	NOUN
ejpam-4797	59	16	)	)	PUNCT
ejpam-4797	60	1	is	be	AUX
ejpam-4797	60	2	nil	nil	ADJ
ejpam-4797	60	3	.	.	NOUN
ejpam-4797	61	1	2	2	X
ejpam-4797	61	2	.	.	X
ejpam-4797	61	3	r	r	X
ejpam-4797	61	4	/	/	SYM
ejpam-4797	61	5	j(r	j(r	PROPN
ejpam-4797	61	6	)	)	PUNCT
ejpam-4797	61	7	is	be	AUX
ejpam-4797	61	8	tripotent	tripotent	ADJ
ejpam-4797	61	9	.	.	PUNCT
ejpam-4797	62	1	theorem	theorem	NOUN
ejpam-4797	62	2	3	3	NUM
ejpam-4797	62	3	.	.	PUNCT
ejpam-4797	63	1	[	[	X
ejpam-4797	63	2	16	16	NUM
ejpam-4797	63	3	]	]	X
ejpam-4797	63	4	the	the	DET
ejpam-4797	63	5	following	follow	VERB
ejpam-4797	63	6	are	be	AUX
ejpam-4797	63	7	equivalent	equivalent	ADJ
ejpam-4797	63	8	for	for	ADP
ejpam-4797	63	9	a	a	DET
ejpam-4797	63	10	ring	ring	NOUN
ejpam-4797	63	11	r	r	NOUN
ejpam-4797	63	12	:	:	PUNCT
ejpam-4797	63	13	1	1	NUM
ejpam-4797	63	14	.	.	X
ejpam-4797	64	1	every	every	DET
ejpam-4797	64	2	element	element	NOUN
ejpam-4797	64	3	of	of	ADP
ejpam-4797	64	4	r	r	NOUN
ejpam-4797	64	5	is	be	AUX
ejpam-4797	64	6	a	a	DET
ejpam-4797	64	7	sum	sum	NOUN
ejpam-4797	64	8	of	of	ADP
ejpam-4797	64	9	a	a	DET
ejpam-4797	64	10	nilpotent	nilpotent	NOUN
ejpam-4797	64	11	and	and	CCONJ
ejpam-4797	64	12	two	two	NUM
ejpam-4797	64	13	tripotents	tripotent	NOUN
ejpam-4797	64	14	that	that	PRON
ejpam-4797	64	15	commute	commute	VERB
ejpam-4797	64	16	with	with	ADP
ejpam-4797	64	17	one	one	NUM
ejpam-4797	64	18	another	another	DET
ejpam-4797	64	19	.	.	PUNCT
ejpam-4797	65	1	2	2	X
ejpam-4797	65	2	.	.	X
ejpam-4797	65	3	a5	a5	NOUN
ejpam-4797	65	4	−	−	PROPN
ejpam-4797	65	5	a	a	PRON
ejpam-4797	65	6	is	be	AUX
ejpam-4797	65	7	nilpotent	nilpotent	ADJ
ejpam-4797	65	8	for	for	ADP
ejpam-4797	65	9	all	all	DET
ejpam-4797	65	10	a	a	DET
ejpam-4797	65	11	∈	∈	PROPN
ejpam-4797	65	12	r.	r.	PROPN
ejpam-4797	65	13	r.	r.	PROPN
ejpam-4797	65	14	t.	t.	PROPN
ejpam-4797	65	15	m.salim	m.salim	PROPN
ejpam-4797	65	16	,	,	PUNCT
ejpam-4797	65	17	n.	n.	PROPN
ejpam-4797	65	18	h.	h.	PROPN
ejpam-4797	65	19	shuker	shuker	PROPN
ejpam-4797	65	20	/	/	SYM
ejpam-4797	65	21	eur	eur	PROPN
ejpam-4797	65	22	.	.	PUNCT
ejpam-4797	66	1	j.	j.	PROPN
ejpam-4797	66	2	pure	pure	PROPN
ejpam-4797	66	3	appl	appl	PROPN
ejpam-4797	66	4	.	.	PROPN
ejpam-4797	66	5	math	math	PROPN
ejpam-4797	66	6	,	,	PUNCT
ejpam-4797	66	7	16	16	NUM
ejpam-4797	66	8	(	(	PUNCT
ejpam-4797	66	9	3	3	NUM
ejpam-4797	66	10	)	)	PUNCT
ejpam-4797	66	11	(	(	PUNCT
ejpam-4797	66	12	2023	2023	NUM
ejpam-4797	66	13	)	)	PUNCT
ejpam-4797	66	14	,	,	PUNCT
ejpam-4797	66	15	1675	1675	NUM
ejpam-4797	66	16	-	-	SYM
ejpam-4797	66	17	1684	1684	NUM
ejpam-4797	66	18	1677	1677	NUM
ejpam-4797	66	19	2	2	NUM
ejpam-4797	66	20	.	.	PUNCT
ejpam-4797	66	21	fundamental	fundamental	ADJ
ejpam-4797	66	22	properties	property	NOUN
ejpam-4797	66	23	of	of	ADP
ejpam-4797	66	24	strongly	strongly	ADV
ejpam-4797	66	25	2	2	NUM
ejpam-4797	66	26	-	-	PUNCT
ejpam-4797	66	27	nc	nc	PROPN
ejpam-4797	66	28	rings	ring	NOUN
ejpam-4797	66	29	this	this	DET
ejpam-4797	66	30	section	section	NOUN
ejpam-4797	66	31	presents	present	VERB
ejpam-4797	66	32	new	new	ADJ
ejpam-4797	66	33	properties	property	NOUN
ejpam-4797	66	34	of	of	ADP
ejpam-4797	66	35	strongly	strongly	ADV
ejpam-4797	66	36	2	2	NUM
ejpam-4797	66	37	-	-	PUNCT
ejpam-4797	66	38	nc	nc	NOUN
ejpam-4797	66	39	rings	ring	NOUN
ejpam-4797	66	40	,	,	PUNCT
ejpam-4797	66	41	and	and	CCONJ
ejpam-4797	66	42	we	we	PRON
ejpam-4797	66	43	provide	provide	VERB
ejpam-4797	66	44	a	a	DET
ejpam-4797	66	45	condition	condition	NOUN
ejpam-4797	66	46	for	for	ADP
ejpam-4797	66	47	strongly	strongly	ADV
ejpam-4797	66	48	2	2	NUM
ejpam-4797	66	49	-	-	PUNCT
ejpam-4797	66	50	nc	nc	PROPN
ejpam-4797	66	51	rings	ring	NOUN
ejpam-4797	66	52	to	to	PART
ejpam-4797	66	53	be	be	AUX
ejpam-4797	66	54	tripotent	tripotent	ADJ
ejpam-4797	66	55	rings	ring	NOUN
ejpam-4797	66	56	.	.	PUNCT
ejpam-4797	66	57	example	example	NOUN
ejpam-4797	67	1	1	1	NUM
ejpam-4797	67	2	.	.	X
ejpam-4797	67	3	consider	consider	VERB
ejpam-4797	67	4	the	the	DET
ejpam-4797	67	5	ring	ring	NOUN
ejpam-4797	67	6	z18	z18	NOUN
ejpam-4797	67	7	.	.	PUNCT
ejpam-4797	68	1	note	note	VERB
ejpam-4797	68	2	that	that	SCONJ
ejpam-4797	68	3	:	:	PUNCT
ejpam-4797	68	4	nil(z18	nil(z18	NUM
ejpam-4797	68	5	)	)	PUNCT
ejpam-4797	68	6	=	=	NOUN
ejpam-4797	68	7	{	{	PUNCT
ejpam-4797	68	8	0	0	NUM
ejpam-4797	68	9	,	,	PUNCT
ejpam-4797	68	10	6	6	NUM
ejpam-4797	68	11	,	,	PUNCT
ejpam-4797	68	12	12	12	NUM
ejpam-4797	68	13	}	}	PUNCT
ejpam-4797	68	14	,	,	PUNCT
ejpam-4797	68	15	and	and	CCONJ
ejpam-4797	68	16	id(z18	id(z18	PROPN
ejpam-4797	68	17	)	)	PUNCT
ejpam-4797	68	18	=	=	PRON
ejpam-4797	68	19	{	{	PUNCT
ejpam-4797	68	20	0	0	NUM
ejpam-4797	68	21	,	,	PUNCT
ejpam-4797	68	22	1	1	NUM
ejpam-4797	68	23	,	,	PUNCT
ejpam-4797	68	24	9	9	NUM
ejpam-4797	68	25	,	,	PUNCT
ejpam-4797	68	26	10	10	NUM
ejpam-4797	68	27	}	}	PUNCT
ejpam-4797	68	28	.	.	PUNCT
ejpam-4797	69	1	by	by	ADP
ejpam-4797	69	2	direct	direct	ADJ
ejpam-4797	69	3	calculation	calculation	NOUN
ejpam-4797	69	4	,	,	PUNCT
ejpam-4797	69	5	we	we	PRON
ejpam-4797	69	6	may	may	AUX
ejpam-4797	69	7	find	find	VERB
ejpam-4797	69	8	that	that	SCONJ
ejpam-4797	69	9	z18	z18	NOUN
ejpam-4797	69	10	is	be	AUX
ejpam-4797	69	11	a	a	DET
ejpam-4797	69	12	strongly	strongly	ADV
ejpam-4797	69	13	2	2	NUM
ejpam-4797	69	14	-	-	PUNCT
ejpam-4797	69	15	nc	nc	PROPN
ejpam-4797	69	16	.	.	PROPN
ejpam-4797	69	17	chen	chen	PROPN
ejpam-4797	69	18	and	and	CCONJ
ejpam-4797	69	19	sheibani	sheibani	NOUN
ejpam-4797	69	20	in	in	ADP
ejpam-4797	69	21	[	[	X
ejpam-4797	69	22	7	7	NUM
ejpam-4797	69	23	]	]	PUNCT
ejpam-4797	69	24	proved	prove	VERB
ejpam-4797	69	25	that	that	SCONJ
ejpam-4797	69	26	:	:	PUNCT
ejpam-4797	69	27	lemma	lemma	PROPN
ejpam-4797	69	28	1	1	X
ejpam-4797	69	29	.	.	PUNCT
ejpam-4797	70	1	the	the	DET
ejpam-4797	70	2	following	follow	VERB
ejpam-4797	70	3	two	two	NUM
ejpam-4797	70	4	issues	issue	NOUN
ejpam-4797	70	5	are	be	AUX
ejpam-4797	70	6	equivalent	equivalent	ADJ
ejpam-4797	70	7	:	:	PUNCT
ejpam-4797	70	8	1	1	X
ejpam-4797	70	9	.	.	X
ejpam-4797	70	10	r	r	NOUN
ejpam-4797	70	11	is	be	AUX
ejpam-4797	70	12	a	a	DET
ejpam-4797	70	13	strongly	strongly	ADV
ejpam-4797	70	14	2	2	NUM
ejpam-4797	70	15	-	-	PUNCT
ejpam-4797	70	16	nc	nc	NOUN
ejpam-4797	70	17	ring	ring	NOUN
ejpam-4797	70	18	.	.	PUNCT
ejpam-4797	71	1	2	2	X
ejpam-4797	71	2	.	.	X
ejpam-4797	71	3	a	a	DET
ejpam-4797	71	4	=	=	PROPN
ejpam-4797	71	5	σ1	σ1	PROPN
ejpam-4797	71	6	−	−	PROPN
ejpam-4797	71	7	σ2	σ2	PROPN
ejpam-4797	71	8	+	+	CCONJ
ejpam-4797	71	9	n	n	CCONJ
ejpam-4797	71	10	,	,	PUNCT
ejpam-4797	71	11	for	for	ADP
ejpam-4797	71	12	each	each	DET
ejpam-4797	71	13	a	a	DET
ejpam-4797	71	14	∈	∈	PROPN
ejpam-4797	71	15	r	r	NOUN
ejpam-4797	71	16	,	,	PUNCT
ejpam-4797	71	17	and	and	CCONJ
ejpam-4797	71	18	some	some	DET
ejpam-4797	71	19	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	71	20	∈	∈	PROPN
ejpam-4797	71	21	id(r	id(r	NOUN
ejpam-4797	71	22	)	)	PUNCT
ejpam-4797	71	23	,	,	PUNCT
ejpam-4797	71	24	n	n	PROPN
ejpam-4797	71	25	∈	∈	PROPN
ejpam-4797	71	26	nil(r	nil(r	PROPN
ejpam-4797	71	27	)	)	PUNCT
ejpam-4797	71	28	,	,	PUNCT
ejpam-4797	71	29	that	that	DET
ejpam-4797	71	30	commute	commute	NOUN
ejpam-4797	71	31	.	.	PUNCT
ejpam-4797	72	1	next	next	ADV
ejpam-4797	72	2	,	,	PUNCT
ejpam-4797	72	3	we	we	PRON
ejpam-4797	72	4	shall	shall	AUX
ejpam-4797	72	5	record	record	VERB
ejpam-4797	72	6	the	the	DET
ejpam-4797	72	7	following	follow	VERB
ejpam-4797	72	8	two	two	NUM
ejpam-4797	72	9	lemmas	lemma	NOUN
ejpam-4797	72	10	,	,	PUNCT
ejpam-4797	72	11	that	that	PRON
ejpam-4797	72	12	will	will	AUX
ejpam-4797	72	13	be	be	AUX
ejpam-4797	72	14	used	use	VERB
ejpam-4797	72	15	extensively	extensively	ADV
ejpam-4797	72	16	throughout	throughout	ADP
ejpam-4797	72	17	our	our	PRON
ejpam-4797	72	18	current	current	ADJ
ejpam-4797	72	19	work	work	NOUN
ejpam-4797	72	20	.	.	PUNCT
ejpam-4797	73	1	lemma	lemma	PROPN
ejpam-4797	73	2	2	2	NUM
ejpam-4797	73	3	.	.	PUNCT
ejpam-4797	74	1	[	[	X
ejpam-4797	74	2	17	17	NUM
ejpam-4797	74	3	]	]	PUNCT
ejpam-4797	74	4	.	.	PUNCT
ejpam-4797	75	1	if	if	SCONJ
ejpam-4797	75	2	u	u	PROPN
ejpam-4797	75	3	∈	∈	PROPN
ejpam-4797	75	4	u(r	u(r	PROPN
ejpam-4797	75	5	)	)	PUNCT
ejpam-4797	75	6	and	and	CCONJ
ejpam-4797	75	7	n	n	PRON
ejpam-4797	75	8	∈	∈	PROPN
ejpam-4797	75	9	nil(r	nil(r	PROPN
ejpam-4797	75	10	)	)	PUNCT
ejpam-4797	75	11	,	,	PUNCT
ejpam-4797	75	12	and	and	CCONJ
ejpam-4797	75	13	if	if	SCONJ
ejpam-4797	75	14	un	un	PROPN
ejpam-4797	75	15	=	=	PROPN
ejpam-4797	75	16	nu	nu	PROPN
ejpam-4797	75	17	,	,	PUNCT
ejpam-4797	75	18	then	then	ADV
ejpam-4797	75	19	1	1	NUM
ejpam-4797	75	20	+	+	CCONJ
ejpam-4797	75	21	n	n	NUM
ejpam-4797	75	22	and	and	CCONJ
ejpam-4797	75	23	u+	u+	NUM
ejpam-4797	75	24	n	n	CCONJ
ejpam-4797	75	25	are	be	AUX
ejpam-4797	75	26	units	unit	NOUN
ejpam-4797	75	27	.	.	PUNCT
ejpam-4797	76	1	lemma	lemma	PROPN
ejpam-4797	76	2	3	3	X
ejpam-4797	76	3	.	.	PUNCT
ejpam-4797	76	4	suppose	suppose	VERB
ejpam-4797	76	5	that	that	SCONJ
ejpam-4797	76	6	σ1	σ1	PROPN
ejpam-4797	76	7	and	and	CCONJ
ejpam-4797	76	8	σ2	σ2	PROPN
ejpam-4797	76	9	are	be	AUX
ejpam-4797	76	10	two	two	NUM
ejpam-4797	76	11	commuting	commuting	NOUN
ejpam-4797	76	12	idempotents	idempotent	NOUN
ejpam-4797	76	13	.	.	PUNCT
ejpam-4797	77	1	then	then	ADV
ejpam-4797	77	2	:	:	PUNCT
ejpam-4797	77	3	1	1	X
ejpam-4797	77	4	.	.	PUNCT
ejpam-4797	77	5	(	(	PUNCT
ejpam-4797	77	6	σ1	σ1	PROPN
ejpam-4797	77	7	−	−	PROPN
ejpam-4797	77	8	σ2	σ2	PROPN
ejpam-4797	77	9	)	)	PUNCT
ejpam-4797	77	10	2	2	NUM
ejpam-4797	77	11	is	be	AUX
ejpam-4797	77	12	an	an	DET
ejpam-4797	77	13	idempotent	idempotent	NOUN
ejpam-4797	77	14	.	.	PUNCT
ejpam-4797	78	1	2	2	X
ejpam-4797	78	2	.	.	X
ejpam-4797	78	3	(	(	PUNCT
ejpam-4797	78	4	σ1	σ1	PROPN
ejpam-4797	78	5	−	−	PROPN
ejpam-4797	78	6	σ2	σ2	PROPN
ejpam-4797	78	7	)	)	PUNCT
ejpam-4797	78	8	3	3	NUM
ejpam-4797	78	9	is	be	AUX
ejpam-4797	78	10	tripotent	tripotent	ADJ
ejpam-4797	78	11	.	.	PUNCT
ejpam-4797	79	1	3	3	X
ejpam-4797	79	2	.	.	X
ejpam-4797	79	3	(	(	PUNCT
ejpam-4797	79	4	σ1	σ1	PROPN
ejpam-4797	79	5	−	−	PROPN
ejpam-4797	79	6	σ2	σ2	PROPN
ejpam-4797	79	7	)	)	PUNCT
ejpam-4797	79	8	2	2	NUM
ejpam-4797	80	1	+	+	CCONJ
ejpam-4797	80	2	(	(	PUNCT
ejpam-4797	80	3	σ1	σ1	PROPN
ejpam-4797	80	4	−	−	PROPN
ejpam-4797	80	5	σ2)−	σ2)−	ADJ
ejpam-4797	80	6	1	1	NUM
ejpam-4797	80	7	is	be	AUX
ejpam-4797	80	8	a	a	DET
ejpam-4797	80	9	unit	unit	NOUN
ejpam-4797	80	10	of	of	ADP
ejpam-4797	80	11	order	order	NOUN
ejpam-4797	80	12	2	2	NUM
ejpam-4797	80	13	.	.	NOUN
ejpam-4797	80	14	4	4	NUM
ejpam-4797	80	15	.	.	X
ejpam-4797	80	16	2(σ1	2(σ1	NUM
ejpam-4797	80	17	−	−	PROPN
ejpam-4797	80	18	σ2	σ2	NOUN
ejpam-4797	80	19	)	)	PUNCT
ejpam-4797	80	20	2	2	NUM
ejpam-4797	80	21	−	−	NOUN
ejpam-4797	80	22	1	1	NUM
ejpam-4797	80	23	is	be	AUX
ejpam-4797	80	24	a	a	DET
ejpam-4797	80	25	unit	unit	NOUN
ejpam-4797	80	26	of	of	ADP
ejpam-4797	80	27	order	order	NOUN
ejpam-4797	80	28	2	2	NUM
ejpam-4797	80	29	.	.	PUNCT
ejpam-4797	80	30	proof	proof	NOUN
ejpam-4797	80	31	.	.	PUNCT
ejpam-4797	81	1	1	1	X
ejpam-4797	81	2	.	.	X
ejpam-4797	81	3	(	(	PUNCT
ejpam-4797	81	4	σ1	σ1	PROPN
ejpam-4797	81	5	−	−	PROPN
ejpam-4797	81	6	σ2	σ2	PROPN
ejpam-4797	81	7	)	)	PUNCT
ejpam-4797	81	8	4	4	NUM
ejpam-4797	81	9	=	=	SYM
ejpam-4797	81	10	σ4	σ4	NOUN
ejpam-4797	81	11	1	1	NUM
ejpam-4797	81	12	−	−	NOUN
ejpam-4797	81	13	4σ3	4σ3	NUM
ejpam-4797	81	14	1σ2	1σ2	NUM
ejpam-4797	81	15	+	+	CCONJ
ejpam-4797	81	16	6σ2	6σ2	NUM
ejpam-4797	81	17	1σ	1σ	NUM
ejpam-4797	81	18	2	2	NUM
ejpam-4797	81	19	2	2	NUM
ejpam-4797	81	20	−	−	NUM
ejpam-4797	81	21	4σ1σ	4σ1σ	NUM
ejpam-4797	81	22	3	3	NUM
ejpam-4797	81	23	2	2	NUM
ejpam-4797	81	24	+	+	NOUN
ejpam-4797	81	25	σ4	σ4	NOUN
ejpam-4797	81	26	2	2	NUM
ejpam-4797	81	27	=	=	SYM
ejpam-4797	81	28	σ1	σ1	NOUN
ejpam-4797	81	29	−	−	PROPN
ejpam-4797	81	30	4σ1σ2	4σ1σ2	NUM
ejpam-4797	81	31	+	+	CCONJ
ejpam-4797	81	32	6σ1σ2	6σ1σ2	NUM
ejpam-4797	81	33	−	−	NUM
ejpam-4797	81	34	4σ1σ2	4σ1σ2	NUM
ejpam-4797	81	35	+	+	ADJ
ejpam-4797	81	36	σ2	σ2	NOUN
ejpam-4797	81	37	=	=	SYM
ejpam-4797	81	38	(	(	PUNCT
ejpam-4797	81	39	σ1	σ1	PROPN
ejpam-4797	81	40	−	−	PROPN
ejpam-4797	81	41	σ2	σ2	PROPN
ejpam-4797	81	42	)	)	PUNCT
ejpam-4797	81	43	2	2	NUM
ejpam-4797	81	44	.	.	NOUN
ejpam-4797	81	45	2	2	NUM
ejpam-4797	81	46	.	.	PUNCT
ejpam-4797	82	1	(	(	PUNCT
ejpam-4797	82	2	σ1	σ1	PROPN
ejpam-4797	82	3	−	−	PROPN
ejpam-4797	82	4	σ2	σ2	PROPN
ejpam-4797	82	5	)	)	PUNCT
ejpam-4797	82	6	3	3	NUM
ejpam-4797	82	7	=	=	SYM
ejpam-4797	82	8	σ3	σ3	NOUN
ejpam-4797	82	9	1	1	NUM
ejpam-4797	82	10	−	−	PROPN
ejpam-4797	82	11	3σ2	3σ2	NUM
ejpam-4797	82	12	1σ2	1σ2	NUM
ejpam-4797	82	13	+	+	CCONJ
ejpam-4797	82	14	3σ1σ	3σ1σ	NUM
ejpam-4797	82	15	2	2	NUM
ejpam-4797	82	16	2	2	NUM
ejpam-4797	82	17	−	−	NOUN
ejpam-4797	82	18	σ3	σ3	NOUN
ejpam-4797	82	19	2	2	NUM
ejpam-4797	82	20	=	=	SYM
ejpam-4797	82	21	σ1	σ1	PROPN
ejpam-4797	82	22	−	−	PROPN
ejpam-4797	82	23	3σ1σ2	3σ1σ2	NUM
ejpam-4797	82	24	+	+	CCONJ
ejpam-4797	82	25	3σ1σ2	3σ1σ2	NUM
ejpam-4797	82	26	−	−	PROPN
ejpam-4797	82	27	σ2	σ2	PROPN
ejpam-4797	82	28	=	=	SYM
ejpam-4797	82	29	(	(	PUNCT
ejpam-4797	82	30	σ1	σ1	PROPN
ejpam-4797	82	31	−	−	PROPN
ejpam-4797	82	32	σ2	σ2	PROPN
ejpam-4797	82	33	)	)	PUNCT
ejpam-4797	82	34	.	.	PUNCT
ejpam-4797	83	1	3	3	X
ejpam-4797	83	2	.	.	X
ejpam-4797	83	3	(	(	PUNCT
ejpam-4797	83	4	(	(	PUNCT
ejpam-4797	83	5	σ1	σ1	PROPN
ejpam-4797	83	6	−	−	PROPN
ejpam-4797	83	7	σ2	σ2	PROPN
ejpam-4797	83	8	)	)	PUNCT
ejpam-4797	83	9	2	2	NUM
ejpam-4797	84	1	+	+	CCONJ
ejpam-4797	84	2	(	(	PUNCT
ejpam-4797	84	3	σ1	σ1	PROPN
ejpam-4797	84	4	−	−	PROPN
ejpam-4797	84	5	σ2)−	σ2)−	ADJ
ejpam-4797	84	6	1)((σ1	1)((σ1	NUM
ejpam-4797	84	7	−	−	PROPN
ejpam-4797	84	8	σ2	σ2	PROPN
ejpam-4797	84	9	)	)	PUNCT
ejpam-4797	84	10	2	2	NUM
ejpam-4797	85	1	+	+	CCONJ
ejpam-4797	85	2	(	(	PUNCT
ejpam-4797	85	3	σ1	σ1	PROPN
ejpam-4797	85	4	−	−	PROPN
ejpam-4797	85	5	σ2)−	σ2)−	ADJ
ejpam-4797	85	6	1	1	NUM
ejpam-4797	85	7	)	)	PUNCT
ejpam-4797	85	8	=	=	SYM
ejpam-4797	85	9	(	(	PUNCT
ejpam-4797	85	10	σ1	σ1	PROPN
ejpam-4797	85	11	−	−	PROPN
ejpam-4797	85	12	σ2	σ2	PROPN
ejpam-4797	85	13	)	)	PUNCT
ejpam-4797	85	14	4	4	NUM
ejpam-4797	85	15	+	+	CCONJ
ejpam-4797	85	16	(	(	PUNCT
ejpam-4797	85	17	σ1	σ1	PROPN
ejpam-4797	85	18	−	−	PROPN
ejpam-4797	85	19	σ2	σ2	PROPN
ejpam-4797	85	20	)	)	PUNCT
ejpam-4797	85	21	3	3	NUM
ejpam-4797	85	22	−	−	PROPN
ejpam-4797	85	23	(	(	PUNCT
ejpam-4797	85	24	σ1	σ1	PROPN
ejpam-4797	85	25	−	−	PROPN
ejpam-4797	85	26	σ2	σ2	PROPN
ejpam-4797	85	27	)	)	PUNCT
ejpam-4797	85	28	2	2	NUM
ejpam-4797	86	1	+	+	CCONJ
ejpam-4797	86	2	(	(	PUNCT
ejpam-4797	86	3	σ1	σ1	PROPN
ejpam-4797	86	4	−	−	PROPN
ejpam-4797	86	5	σ2	σ2	PROPN
ejpam-4797	86	6	)	)	PUNCT
ejpam-4797	86	7	3	3	NUM
ejpam-4797	86	8	+	+	CCONJ
ejpam-4797	86	9	(	(	PUNCT
ejpam-4797	86	10	σ1	σ1	PROPN
ejpam-4797	86	11	−	−	PROPN
ejpam-4797	86	12	σ2	σ2	PROPN
ejpam-4797	86	13	)	)	PUNCT
ejpam-4797	86	14	2	2	NUM
ejpam-4797	86	15	−	−	PROPN
ejpam-4797	86	16	(	(	PUNCT
ejpam-4797	86	17	σ1	σ1	PROPN
ejpam-4797	86	18	−	−	PROPN
ejpam-4797	86	19	σ2)−	σ2)−	PROPN
ejpam-4797	86	20	(	(	PUNCT
ejpam-4797	86	21	σ1	σ1	PROPN
ejpam-4797	86	22	−	−	PROPN
ejpam-4797	86	23	σ2	σ2	PROPN
ejpam-4797	86	24	)	)	PUNCT
ejpam-4797	86	25	2	2	NUM
ejpam-4797	86	26	−	−	PROPN
ejpam-4797	87	1	(	(	PUNCT
ejpam-4797	87	2	σ1	σ1	PROPN
ejpam-4797	87	3	−	−	PROPN
ejpam-4797	87	4	σ2	σ2	PROPN
ejpam-4797	87	5	)	)	PUNCT
ejpam-4797	88	1	+	+	CCONJ
ejpam-4797	88	2	1	1	NUM
ejpam-4797	88	3	=	=	SYM
ejpam-4797	88	4	(	(	PUNCT
ejpam-4797	88	5	σ1	σ1	PROPN
ejpam-4797	88	6	−	−	PROPN
ejpam-4797	88	7	σ2	σ2	PROPN
ejpam-4797	88	8	)	)	PUNCT
ejpam-4797	88	9	2	2	NUM
ejpam-4797	89	1	+	+	CCONJ
ejpam-4797	89	2	(	(	PUNCT
ejpam-4797	89	3	σ1	σ1	PROPN
ejpam-4797	89	4	−	−	PROPN
ejpam-4797	89	5	σ2)−	σ2)−	PROPN
ejpam-4797	89	6	(	(	PUNCT
ejpam-4797	89	7	σ1	σ1	PROPN
ejpam-4797	89	8	−	−	PROPN
ejpam-4797	89	9	σ2	σ2	PROPN
ejpam-4797	89	10	)	)	PUNCT
ejpam-4797	89	11	2	2	NUM
ejpam-4797	90	1	+	+	CCONJ
ejpam-4797	90	2	(	(	PUNCT
ejpam-4797	90	3	σ1	σ1	PROPN
ejpam-4797	90	4	−	−	PROPN
ejpam-4797	90	5	σ2	σ2	PROPN
ejpam-4797	90	6	)	)	PUNCT
ejpam-4797	91	1	+	+	CCONJ
ejpam-4797	91	2	(	(	PUNCT
ejpam-4797	91	3	σ1	σ1	PROPN
ejpam-4797	91	4	−	−	PROPN
ejpam-4797	91	5	σ2	σ2	PROPN
ejpam-4797	91	6	)	)	PUNCT
ejpam-4797	91	7	2	2	NUM
ejpam-4797	91	8	−	−	PROPN
ejpam-4797	91	9	(	(	PUNCT
ejpam-4797	91	10	σ1	σ1	PROPN
ejpam-4797	91	11	−	−	PROPN
ejpam-4797	91	12	σ2)−	σ2)−	PROPN
ejpam-4797	91	13	(	(	PUNCT
ejpam-4797	91	14	σ1	σ1	PROPN
ejpam-4797	91	15	−	−	PROPN
ejpam-4797	91	16	σ2	σ2	PROPN
ejpam-4797	91	17	)	)	PUNCT
ejpam-4797	91	18	2	2	NUM
ejpam-4797	91	19	−	−	PROPN
ejpam-4797	91	20	(	(	PUNCT
ejpam-4797	91	21	σ1	σ1	PROPN
ejpam-4797	91	22	−	−	PROPN
ejpam-4797	91	23	σ2	σ2	PROPN
ejpam-4797	91	24	)	)	PUNCT
ejpam-4797	91	25	+	+	CCONJ
ejpam-4797	91	26	1	1	NUM
ejpam-4797	91	27	=	=	SYM
ejpam-4797	91	28	1	1	NUM
ejpam-4797	91	29	.	.	PUNCT
ejpam-4797	91	30	r.	r.	PROPN
ejpam-4797	91	31	t.	t.	PROPN
ejpam-4797	91	32	m.salim	m.salim	PROPN
ejpam-4797	91	33	,	,	PUNCT
ejpam-4797	92	1	n.	n.	PROPN
ejpam-4797	92	2	h.	h.	PROPN
ejpam-4797	92	3	shuker	shuker	PROPN
ejpam-4797	92	4	/	/	SYM
ejpam-4797	92	5	eur	eur	PROPN
ejpam-4797	92	6	.	.	PUNCT
ejpam-4797	93	1	j.	j.	PROPN
ejpam-4797	93	2	pure	pure	PROPN
ejpam-4797	93	3	appl	appl	PROPN
ejpam-4797	93	4	.	.	PROPN
ejpam-4797	93	5	math	math	PROPN
ejpam-4797	93	6	,	,	PUNCT
ejpam-4797	93	7	16	16	NUM
ejpam-4797	93	8	(	(	PUNCT
ejpam-4797	93	9	3	3	NUM
ejpam-4797	93	10	)	)	PUNCT
ejpam-4797	93	11	(	(	PUNCT
ejpam-4797	93	12	2023	2023	NUM
ejpam-4797	93	13	)	)	PUNCT
ejpam-4797	93	14	,	,	PUNCT
ejpam-4797	93	15	1675	1675	NUM
ejpam-4797	93	16	-	-	SYM
ejpam-4797	93	17	1684	1684	NUM
ejpam-4797	93	18	1678	1678	NUM
ejpam-4797	93	19	4	4	NUM
ejpam-4797	93	20	.	.	PUNCT
ejpam-4797	94	1	(	(	PUNCT
ejpam-4797	94	2	2(σ1	2(σ1	NUM
ejpam-4797	94	3	−	−	NOUN
ejpam-4797	94	4	σ2	σ2	NOUN
ejpam-4797	94	5	)	)	PUNCT
ejpam-4797	94	6	2	2	NUM
ejpam-4797	94	7	−	−	PROPN
ejpam-4797	94	8	1)(2(σ1	1)(2(σ1	NUM
ejpam-4797	94	9	−	−	PROPN
ejpam-4797	94	10	σ2	σ2	PROPN
ejpam-4797	94	11	)	)	PUNCT
ejpam-4797	94	12	2	2	NUM
ejpam-4797	94	13	−	−	NOUN
ejpam-4797	94	14	1	1	NUM
ejpam-4797	94	15	)	)	PUNCT
ejpam-4797	94	16	=	=	SYM
ejpam-4797	94	17	4(σ1	4(σ1	NUM
ejpam-4797	94	18	−	−	PROPN
ejpam-4797	94	19	σ2	σ2	NOUN
ejpam-4797	94	20	)	)	PUNCT
ejpam-4797	94	21	4	4	NUM
ejpam-4797	94	22	−	−	PROPN
ejpam-4797	94	23	2(σ1	2(σ1	NUM
ejpam-4797	94	24	−	−	NOUN
ejpam-4797	94	25	σ2	σ2	NOUN
ejpam-4797	94	26	)	)	PUNCT
ejpam-4797	94	27	2	2	NUM
ejpam-4797	94	28	−	−	PROPN
ejpam-4797	94	29	2(σ1	2(σ1	NUM
ejpam-4797	94	30	−	−	NOUN
ejpam-4797	94	31	σ2	σ2	NOUN
ejpam-4797	94	32	)	)	PUNCT
ejpam-4797	94	33	2	2	NUM
ejpam-4797	95	1	+	+	SYM
ejpam-4797	95	2	1	1	NUM
ejpam-4797	95	3	=	=	SYM
ejpam-4797	95	4	4(σ1	4(σ1	NUM
ejpam-4797	95	5	−	−	NOUN
ejpam-4797	95	6	σ2	σ2	NOUN
ejpam-4797	95	7	)	)	PUNCT
ejpam-4797	95	8	2	2	NUM
ejpam-4797	95	9	−	−	PROPN
ejpam-4797	95	10	2(σ1	2(σ1	NUM
ejpam-4797	95	11	−	−	NOUN
ejpam-4797	95	12	σ2	σ2	NOUN
ejpam-4797	95	13	)	)	PUNCT
ejpam-4797	95	14	2	2	NUM
ejpam-4797	95	15	−	−	PROPN
ejpam-4797	95	16	2(σ1	2(σ1	NUM
ejpam-4797	95	17	−	−	NOUN
ejpam-4797	95	18	σ2	σ2	NOUN
ejpam-4797	95	19	)	)	PUNCT
ejpam-4797	95	20	2	2	NUM
ejpam-4797	96	1	+	+	SYM
ejpam-4797	96	2	1	1	NUM
ejpam-4797	96	3	=	=	SYM
ejpam-4797	96	4	1	1	NUM
ejpam-4797	96	5	.	.	PUNCT
ejpam-4797	97	1	next	next	ADV
ejpam-4797	97	2	,	,	PUNCT
ejpam-4797	97	3	we	we	PRON
ejpam-4797	97	4	shall	shall	AUX
ejpam-4797	97	5	give	give	VERB
ejpam-4797	97	6	the	the	DET
ejpam-4797	97	7	following	follow	VERB
ejpam-4797	97	8	results	result	NOUN
ejpam-4797	97	9	.	.	PUNCT
ejpam-4797	98	1	proposition	proposition	NOUN
ejpam-4797	98	2	1	1	NUM
ejpam-4797	98	3	.	.	PUNCT
ejpam-4797	99	1	let	let	VERB
ejpam-4797	99	2	r	r	PRON
ejpam-4797	99	3	be	be	AUX
ejpam-4797	99	4	a	a	DET
ejpam-4797	99	5	strongly	strongly	ADV
ejpam-4797	99	6	2	2	NUM
ejpam-4797	99	7	-	-	PUNCT
ejpam-4797	99	8	nc	nc	NOUN
ejpam-4797	99	9	ring	ring	NOUN
ejpam-4797	99	10	,	,	PUNCT
ejpam-4797	99	11	then	then	ADV
ejpam-4797	99	12	for	for	ADP
ejpam-4797	99	13	any	any	DET
ejpam-4797	99	14	a	a	DET
ejpam-4797	99	15	∈	∈	NOUN
ejpam-4797	99	16	r	r	NOUN
ejpam-4797	99	17	we	we	PRON
ejpam-4797	99	18	have	have	VERB
ejpam-4797	99	19	:	:	PUNCT
ejpam-4797	99	20	1	1	X
ejpam-4797	99	21	.	.	X
ejpam-4797	99	22	a2	a2	PROPN
ejpam-4797	99	23	is	be	AUX
ejpam-4797	99	24	an	an	DET
ejpam-4797	99	25	snc	snc	NOUN
ejpam-4797	99	26	.	.	PROPN
ejpam-4797	99	27	2	2	X
ejpam-4797	99	28	.	.	X
ejpam-4797	99	29	a2	a2	PROPN
ejpam-4797	99	30	is	be	AUX
ejpam-4797	99	31	the	the	DET
ejpam-4797	99	32	sum	sum	NOUN
ejpam-4797	99	33	of	of	ADP
ejpam-4797	99	34	a	a	DET
ejpam-4797	99	35	tripotent	tripotent	NOUN
ejpam-4797	99	36	and	and	CCONJ
ejpam-4797	99	37	a	a	DET
ejpam-4797	99	38	nilpotent	nilpotent	NOUN
ejpam-4797	99	39	that	that	DET
ejpam-4797	99	40	commute	commute	NOUN
ejpam-4797	99	41	.	.	PUNCT
ejpam-4797	100	1	3	3	X
ejpam-4797	100	2	.	.	X
ejpam-4797	100	3	a2	a2	PROPN
ejpam-4797	100	4	is	be	AUX
ejpam-4797	100	5	the	the	DET
ejpam-4797	100	6	sum	sum	NOUN
ejpam-4797	100	7	of	of	ADP
ejpam-4797	100	8	an	an	DET
ejpam-4797	100	9	idempotent	idempotent	NOUN
ejpam-4797	100	10	,	,	PUNCT
ejpam-4797	100	11	a	a	DET
ejpam-4797	100	12	unit	unit	NOUN
ejpam-4797	100	13	of	of	ADP
ejpam-4797	100	14	order	order	NOUN
ejpam-4797	100	15	2	2	NUM
ejpam-4797	100	16	,	,	PUNCT
ejpam-4797	100	17	and	and	CCONJ
ejpam-4797	100	18	a	a	DET
ejpam-4797	100	19	nilpotent	nilpotent	NOUN
ejpam-4797	100	20	that	that	SCONJ
ejpam-4797	100	21	commutes	commute	VERB
ejpam-4797	100	22	.	.	PUNCT
ejpam-4797	101	1	proof	proof	NOUN
ejpam-4797	101	2	.	.	PUNCT
ejpam-4797	102	1	1	1	X
ejpam-4797	102	2	.	.	PUNCT
ejpam-4797	102	3	given	give	VERB
ejpam-4797	102	4	a	a	DET
ejpam-4797	102	5	∈	∈	ADJ
ejpam-4797	102	6	r	r	NOUN
ejpam-4797	102	7	,	,	PUNCT
ejpam-4797	102	8	there	there	ADV
ejpam-4797	102	9	existing	exist	VERB
ejpam-4797	102	10	some	some	DET
ejpam-4797	102	11	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	102	12	∈	∈	PROPN
ejpam-4797	102	13	id(r	id(r	NOUN
ejpam-4797	102	14	)	)	PUNCT
ejpam-4797	102	15	and	and	CCONJ
ejpam-4797	102	16	n	n	PRON
ejpam-4797	102	17	∈	∈	PROPN
ejpam-4797	102	18	nil(r	nil(r	PROPN
ejpam-4797	102	19	)	)	PUNCT
ejpam-4797	102	20	,	,	PUNCT
ejpam-4797	102	21	that	that	PRON
ejpam-4797	102	22	commute	commute	VERB
ejpam-4797	102	23	with	with	ADP
ejpam-4797	102	24	one	one	NUM
ejpam-4797	102	25	another	another	DET
ejpam-4797	102	26	,	,	PUNCT
ejpam-4797	102	27	such	such	ADJ
ejpam-4797	102	28	that	that	SCONJ
ejpam-4797	102	29	a	a	DET
ejpam-4797	102	30	=	=	NOUN
ejpam-4797	102	31	σ1−σ2+n	σ1−σ2+n	NOUN
ejpam-4797	102	32	.	.	PUNCT
ejpam-4797	103	1	thus	thus	ADV
ejpam-4797	103	2	,	,	PUNCT
ejpam-4797	103	3	a2	a2	PROPN
ejpam-4797	103	4	=	=	PUNCT
ejpam-4797	103	5	(	(	PUNCT
ejpam-4797	103	6	σ1−σ2	σ1−σ2	NOUN
ejpam-4797	103	7	)	)	PUNCT
ejpam-4797	103	8	2	2	NUM
ejpam-4797	103	9	+	+	NOUN
ejpam-4797	103	10	2(σ1−σ2)n+n2	2(σ1−σ2)n+n2	NUM
ejpam-4797	103	11	.	.	PUNCT
ejpam-4797	104	1	but	but	CCONJ
ejpam-4797	104	2	2(σ1	2(σ1	NUM
ejpam-4797	104	3	−	−	PROPN
ejpam-4797	104	4	σ2)n	σ2)n	NOUN
ejpam-4797	104	5	+	+	CCONJ
ejpam-4797	104	6	n2	n2	ADJ
ejpam-4797	104	7	=	=	SYM
ejpam-4797	104	8	n1	n1	PROPN
ejpam-4797	104	9	∈	∈	PROPN
ejpam-4797	104	10	nil(r	nil(r	NOUN
ejpam-4797	104	11	)	)	PUNCT
ejpam-4797	104	12	,	,	PUNCT
ejpam-4797	104	13	so	so	ADV
ejpam-4797	104	14	a2	a2	PROPN
ejpam-4797	104	15	=	=	SYM
ejpam-4797	104	16	(	(	PUNCT
ejpam-4797	104	17	σ1	σ1	PROPN
ejpam-4797	104	18	−	−	PROPN
ejpam-4797	104	19	σ2	σ2	PROPN
ejpam-4797	104	20	)	)	PUNCT
ejpam-4797	104	21	2	2	NUM
ejpam-4797	105	1	+	+	SYM
ejpam-4797	105	2	n1	n1	NOUN
ejpam-4797	105	3	.	.	PUNCT
ejpam-4797	106	1	according	accord	VERB
ejpam-4797	106	2	to	to	ADP
ejpam-4797	106	3	lemma	lemma	PROPN
ejpam-4797	106	4	3(1	3(1	NUM
ejpam-4797	106	5	)	)	PUNCT
ejpam-4797	106	6	(	(	PUNCT
ejpam-4797	106	7	σ1	σ1	PROPN
ejpam-4797	106	8	−	−	PROPN
ejpam-4797	106	9	σ2	σ2	PROPN
ejpam-4797	106	10	)	)	PUNCT
ejpam-4797	106	11	2	2	NUM
ejpam-4797	106	12	is	be	AUX
ejpam-4797	106	13	an	an	DET
ejpam-4797	106	14	idempotent	idempotent	NOUN
ejpam-4797	106	15	.	.	PUNCT
ejpam-4797	107	1	yielding	yield	VERB
ejpam-4797	107	2	a2	a2	PROPN
ejpam-4797	107	3	is	be	AUX
ejpam-4797	107	4	an	an	DET
ejpam-4797	107	5	snc	snc	NOUN
ejpam-4797	107	6	element	element	NOUN
ejpam-4797	107	7	.	.	PUNCT
ejpam-4797	108	1	2	2	X
ejpam-4797	108	2	.	.	NUM
ejpam-4797	108	3	follows	follow	VERB
ejpam-4797	108	4	from	from	ADP
ejpam-4797	108	5	lemma	lemma	PROPN
ejpam-4797	108	6	3(2	3(2	NUM
ejpam-4797	108	7	)	)	PUNCT
ejpam-4797	108	8	.	.	PUNCT
ejpam-4797	109	1	3	3	X
ejpam-4797	109	2	.	.	X
ejpam-4797	109	3	by	by	ADP
ejpam-4797	109	4	(	(	PUNCT
ejpam-4797	109	5	1	1	X
ejpam-4797	109	6	)	)	PUNCT
ejpam-4797	109	7	a2	a2	PROPN
ejpam-4797	109	8	is	be	AUX
ejpam-4797	109	9	a	a	DET
ejpam-4797	109	10	snc	snc	NOUN
ejpam-4797	109	11	element	element	NOUN
ejpam-4797	109	12	,	,	PUNCT
ejpam-4797	109	13	then	then	ADV
ejpam-4797	110	1	a2	a2	PROPN
ejpam-4797	110	2	=	=	PROPN
ejpam-4797	110	3	σ	σ	PROPN
ejpam-4797	110	4	+	+	NOUN
ejpam-4797	110	5	n	n	NUM
ejpam-4797	110	6	where	where	SCONJ
ejpam-4797	110	7	σ	σ	PROPN
ejpam-4797	110	8	∈	∈	PROPN
ejpam-4797	110	9	id(r	id(r	NOUN
ejpam-4797	110	10	)	)	PUNCT
ejpam-4797	110	11	,	,	PUNCT
ejpam-4797	110	12	n	n	PROPN
ejpam-4797	110	13	∈	∈	PROPN
ejpam-4797	110	14	nil(r	nil(r	PROPN
ejpam-4797	110	15	)	)	PUNCT
ejpam-4797	110	16	that	that	DET
ejpam-4797	110	17	commute	commute	NOUN
ejpam-4797	110	18	,	,	PUNCT
ejpam-4797	110	19	we	we	PRON
ejpam-4797	110	20	may	may	AUX
ejpam-4797	110	21	write	write	VERB
ejpam-4797	110	22	a2	a2	PROPN
ejpam-4797	110	23	=	=	PUNCT
ejpam-4797	110	24	(	(	PUNCT
ejpam-4797	110	25	1	1	NUM
ejpam-4797	110	26	−	−	PROPN
ejpam-4797	110	27	σ	σ	PROPN
ejpam-4797	110	28	)	)	PUNCT
ejpam-4797	111	1	+	+	CCONJ
ejpam-4797	111	2	(	(	PUNCT
ejpam-4797	111	3	2σ	2σ	X
ejpam-4797	111	4	−	−	NOUN
ejpam-4797	111	5	1	1	X
ejpam-4797	111	6	)	)	PUNCT
ejpam-4797	111	7	+	+	CCONJ
ejpam-4797	111	8	n.	n.	NOUN
ejpam-4797	111	9	clearly	clearly	ADV
ejpam-4797	111	10	,	,	PUNCT
ejpam-4797	111	11	(	(	PUNCT
ejpam-4797	111	12	1	1	NUM
ejpam-4797	111	13	−	−	PROPN
ejpam-4797	111	14	σ)2	σ)2	NOUN
ejpam-4797	111	15	=	=	SYM
ejpam-4797	111	16	1−σ	1−σ	NUM
ejpam-4797	111	17	,	,	PUNCT
ejpam-4797	111	18	(	(	PUNCT
ejpam-4797	111	19	2σ−	2σ−	PROPN
ejpam-4797	111	20	1)2	1)2	NUM
ejpam-4797	111	21	=	=	SYM
ejpam-4797	111	22	1	1	X
ejpam-4797	111	23	.	.	PUNCT
ejpam-4797	111	24	thus	thus	ADV
ejpam-4797	111	25	,	,	PUNCT
ejpam-4797	111	26	a2	a2	PROPN
ejpam-4797	111	27	is	be	AUX
ejpam-4797	111	28	the	the	DET
ejpam-4797	111	29	sum	sum	NOUN
ejpam-4797	111	30	of	of	ADP
ejpam-4797	111	31	an	an	DET
ejpam-4797	111	32	idempotent	idempotent	NOUN
ejpam-4797	111	33	,	,	PUNCT
ejpam-4797	111	34	a	a	DET
ejpam-4797	111	35	unit	unit	NOUN
ejpam-4797	111	36	of	of	ADP
ejpam-4797	111	37	order	order	NOUN
ejpam-4797	111	38	2	2	NUM
ejpam-4797	111	39	,	,	PUNCT
ejpam-4797	111	40	and	and	CCONJ
ejpam-4797	111	41	a	a	DET
ejpam-4797	111	42	nilpotent	nilpotent	NOUN
ejpam-4797	111	43	.	.	PUNCT
ejpam-4797	112	1	proposition	proposition	NOUN
ejpam-4797	112	2	2	2	NUM
ejpam-4797	112	3	.	.	PUNCT
ejpam-4797	112	4	suppose	suppose	VERB
ejpam-4797	112	5	r	r	NOUN
ejpam-4797	112	6	is	be	AUX
ejpam-4797	112	7	a	a	DET
ejpam-4797	112	8	ring	ring	NOUN
ejpam-4797	112	9	,	,	PUNCT
ejpam-4797	112	10	and	and	CCONJ
ejpam-4797	112	11	let	let	VERB
ejpam-4797	112	12	a	a	DET
ejpam-4797	112	13	∈	∈	PROPN
ejpam-4797	112	14	r.	r.	NOUN
ejpam-4797	112	15	then	then	ADV
ejpam-4797	112	16	:	:	PUNCT
ejpam-4797	112	17	1	1	X
ejpam-4797	112	18	.	.	X
ejpam-4797	112	19	if	if	SCONJ
ejpam-4797	112	20	a2	a2	PROPN
ejpam-4797	112	21	is	be	AUX
ejpam-4797	112	22	a	a	DET
ejpam-4797	112	23	strongly	strongly	ADV
ejpam-4797	112	24	2	2	NUM
ejpam-4797	112	25	-	-	PUNCT
ejpam-4797	112	26	nc	nc	NOUN
ejpam-4797	112	27	,	,	PUNCT
ejpam-4797	112	28	then	then	ADV
ejpam-4797	112	29	a	a	PRON
ejpam-4797	112	30	and	and	CCONJ
ejpam-4797	112	31	−a	−a	NOUN
ejpam-4797	112	32	are	be	AUX
ejpam-4797	112	33	strongly	strongly	ADV
ejpam-4797	112	34	clean	clean	ADJ
ejpam-4797	112	35	.	.	PUNCT
ejpam-4797	113	1	2	2	X
ejpam-4797	113	2	.	.	X
ejpam-4797	113	3	if	if	SCONJ
ejpam-4797	113	4	a2	a2	PROPN
ejpam-4797	113	5	is	be	AUX
ejpam-4797	113	6	a	a	DET
ejpam-4797	113	7	strongly	strongly	ADV
ejpam-4797	113	8	2	2	NUM
ejpam-4797	113	9	-	-	PUNCT
ejpam-4797	113	10	nc	nc	NOUN
ejpam-4797	113	11	,	,	PUNCT
ejpam-4797	113	12	then	then	ADV
ejpam-4797	113	13	a	a	PRON
ejpam-4797	113	14	is	be	AUX
ejpam-4797	113	15	the	the	DET
ejpam-4797	113	16	sum	sum	NOUN
ejpam-4797	113	17	of	of	ADP
ejpam-4797	113	18	two	two	NUM
ejpam-4797	113	19	tripotents	tripotent	NOUN
ejpam-4797	113	20	and	and	CCONJ
ejpam-4797	113	21	a	a	DET
ejpam-4797	113	22	nilpotent	nilpotent	ADJ
ejpam-4797	113	23	commute	commute	NOUN
ejpam-4797	113	24	one	one	NOUN
ejpam-4797	113	25	another	another	DET
ejpam-4797	113	26	.	.	PUNCT
ejpam-4797	114	1	proof	proof	NOUN
ejpam-4797	114	2	.	.	PUNCT
ejpam-4797	115	1	1	1	X
ejpam-4797	115	2	.	.	X
ejpam-4797	115	3	take	take	VERB
ejpam-4797	115	4	a2	a2	PROPN
ejpam-4797	115	5	=	=	SYM
ejpam-4797	115	6	σ1	σ1	PROPN
ejpam-4797	115	7	−	−	PROPN
ejpam-4797	115	8	σ2	σ2	PROPN
ejpam-4797	115	9	+	+	CCONJ
ejpam-4797	115	10	n	n	CCONJ
ejpam-4797	115	11	,	,	PUNCT
ejpam-4797	115	12	by	by	ADP
ejpam-4797	115	13	proposition	proposition	NOUN
ejpam-4797	115	14	1(1	1(1	NUM
ejpam-4797	115	15	)	)	PUNCT
ejpam-4797	115	16	,	,	PUNCT
ejpam-4797	115	17	a4	a4	NOUN
ejpam-4797	115	18	is	be	AUX
ejpam-4797	115	19	a	a	DET
ejpam-4797	115	20	snc	snc	NOUN
ejpam-4797	115	21	element	element	NOUN
ejpam-4797	115	22	,	,	PUNCT
ejpam-4797	115	23	so	so	ADV
ejpam-4797	115	24	a4	a4	PROPN
ejpam-4797	115	25	=	=	SYM
ejpam-4797	115	26	σ	σ	PROPN
ejpam-4797	115	27	+	+	NOUN
ejpam-4797	115	28	n	n	NUM
ejpam-4797	115	29	where	where	SCONJ
ejpam-4797	115	30	σ	σ	PROPN
ejpam-4797	115	31	∈	∈	PROPN
ejpam-4797	115	32	id(r	id(r	NOUN
ejpam-4797	115	33	)	)	PUNCT
ejpam-4797	115	34	,	,	PUNCT
ejpam-4797	115	35	n	n	PROPN
ejpam-4797	115	36	∈	∈	PROPN
ejpam-4797	115	37	nil(r	nil(r	PROPN
ejpam-4797	115	38	)	)	PUNCT
ejpam-4797	115	39	that	that	DET
ejpam-4797	115	40	commute	commute	NOUN
ejpam-4797	115	41	.	.	PUNCT
ejpam-4797	116	1	write	write	NOUN
ejpam-4797	116	2	a4	a4	NOUN
ejpam-4797	116	3	=	=	SYM
ejpam-4797	116	4	(	(	PUNCT
ejpam-4797	116	5	1	1	NUM
ejpam-4797	116	6	−	−	PROPN
ejpam-4797	116	7	σ	σ	PROPN
ejpam-4797	116	8	)	)	PUNCT
ejpam-4797	117	1	+	+	CCONJ
ejpam-4797	117	2	(	(	PUNCT
ejpam-4797	117	3	2σ	2σ	X
ejpam-4797	117	4	−	−	NOUN
ejpam-4797	117	5	1	1	X
ejpam-4797	117	6	)	)	PUNCT
ejpam-4797	117	7	+	+	CCONJ
ejpam-4797	118	1	n.	n.	NOUN
ejpam-4797	118	2	but	but	CCONJ
ejpam-4797	118	3	(	(	PUNCT
ejpam-4797	118	4	2σ	2σ	NUM
ejpam-4797	118	5	−	−	PROPN
ejpam-4797	119	1	1)2	1)2	NUM
ejpam-4797	119	2	=	=	SYM
ejpam-4797	119	3	1	1	NUM
ejpam-4797	119	4	,	,	PUNCT
ejpam-4797	119	5	then	then	ADV
ejpam-4797	119	6	(	(	PUNCT
ejpam-4797	119	7	2σ	2σ	NOUN
ejpam-4797	119	8	−	−	NOUN
ejpam-4797	119	9	1	1	X
ejpam-4797	119	10	)	)	PUNCT
ejpam-4797	119	11	+	+	NUM
ejpam-4797	119	12	n	n	NOUN
ejpam-4797	119	13	=	=	SYM
ejpam-4797	119	14	u1	u1	PROPN
ejpam-4797	119	15	∈	∈	PROPN
ejpam-4797	119	16	u(r	u(r	PROPN
ejpam-4797	119	17	)	)	PUNCT
ejpam-4797	119	18	.	.	PUNCT
ejpam-4797	120	1	so	so	ADV
ejpam-4797	120	2	a4	a4	NOUN
ejpam-4797	120	3	=	=	SYM
ejpam-4797	120	4	(	(	PUNCT
ejpam-4797	120	5	1	1	NUM
ejpam-4797	120	6	−	−	PROPN
ejpam-4797	120	7	σ	σ	PROPN
ejpam-4797	120	8	)	)	PUNCT
ejpam-4797	120	9	+	+	NUM
ejpam-4797	120	10	u1	u1	NOUN
ejpam-4797	120	11	,	,	PUNCT
ejpam-4797	120	12	implies	imply	VERB
ejpam-4797	120	13	a4	a4	NOUN
ejpam-4797	120	14	−	−	PROPN
ejpam-4797	120	15	(	(	PUNCT
ejpam-4797	120	16	1	1	NUM
ejpam-4797	120	17	−	−	PROPN
ejpam-4797	120	18	σ	σ	PROPN
ejpam-4797	120	19	)	)	PUNCT
ejpam-4797	120	20	=	=	SYM
ejpam-4797	120	21	u1	u1	NOUN
ejpam-4797	120	22	,	,	PUNCT
ejpam-4797	120	23	but	but	CCONJ
ejpam-4797	120	24	(	(	PUNCT
ejpam-4797	120	25	1	1	NUM
ejpam-4797	120	26	−	−	NOUN
ejpam-4797	120	27	σ)4	σ)4	NOUN
ejpam-4797	120	28	=	=	NOUN
ejpam-4797	120	29	1	1	NUM
ejpam-4797	120	30	−	−	PROPN
ejpam-4797	120	31	σ	σ	PROPN
ejpam-4797	120	32	,	,	PUNCT
ejpam-4797	120	33	yields	yield	NOUN
ejpam-4797	120	34	(	(	PUNCT
ejpam-4797	120	35	a2	a2	PROPN
ejpam-4797	120	36	−	−	PROPN
ejpam-4797	121	1	(	(	PUNCT
ejpam-4797	121	2	1	1	NUM
ejpam-4797	121	3	−	−	PROPN
ejpam-4797	121	4	σ))(a2	σ))(a2	NOUN
ejpam-4797	122	1	+	+	CCONJ
ejpam-4797	122	2	(	(	PUNCT
ejpam-4797	122	3	1	1	NUM
ejpam-4797	122	4	−	−	PROPN
ejpam-4797	122	5	σ	σ	NOUN
ejpam-4797	122	6	)	)	PUNCT
ejpam-4797	122	7	)	)	PUNCT
ejpam-4797	123	1	=	=	SYM
ejpam-4797	123	2	u1	u1	NOUN
ejpam-4797	123	3	.	.	PUNCT
ejpam-4797	124	1	and	and	CCONJ
ejpam-4797	124	2	hence	hence	ADV
ejpam-4797	124	3	,	,	PUNCT
ejpam-4797	124	4	(	(	PUNCT
ejpam-4797	124	5	a−	a−	PROPN
ejpam-4797	124	6	(	(	PUNCT
ejpam-4797	124	7	1−σ))(a+	1−σ))(a+	PROPN
ejpam-4797	124	8	(	(	PUNCT
ejpam-4797	124	9	1−σ))(a2	1−σ))(a2	NOUN
ejpam-4797	124	10	+	+	CCONJ
ejpam-4797	124	11	(	(	PUNCT
ejpam-4797	124	12	1−σ	1−σ	NUM
ejpam-4797	124	13	)	)	PUNCT
ejpam-4797	124	14	)	)	PUNCT
ejpam-4797	124	15	=	=	SYM
ejpam-4797	124	16	u1	u1	NOUN
ejpam-4797	124	17	.	.	PUNCT
ejpam-4797	125	1	thus	thus	ADV
ejpam-4797	125	2	,	,	PUNCT
ejpam-4797	125	3	a−	a−	PROPN
ejpam-4797	125	4	(	(	PUNCT
ejpam-4797	125	5	1−σ	1−σ	NUM
ejpam-4797	125	6	)	)	PUNCT
ejpam-4797	125	7	∈	∈	PROPN
ejpam-4797	125	8	u(r	u(r	NOUN
ejpam-4797	125	9	)	)	PUNCT
ejpam-4797	125	10	and	and	CCONJ
ejpam-4797	125	11	−a−	−a−	NOUN
ejpam-4797	125	12	(	(	PUNCT
ejpam-4797	125	13	1−	1−	NUM
ejpam-4797	125	14	σ	σ	NOUN
ejpam-4797	125	15	)	)	PUNCT
ejpam-4797	125	16	∈	∈	PROPN
ejpam-4797	125	17	u(r	u(r	NOUN
ejpam-4797	125	18	)	)	PUNCT
ejpam-4797	125	19	.	.	PUNCT
ejpam-4797	126	1	2	2	X
ejpam-4797	126	2	.	.	X
ejpam-4797	126	3	let	let	VERB
ejpam-4797	126	4	a	a	PRON
ejpam-4797	126	5	in	in	ADP
ejpam-4797	126	6	r.	r.	NOUN
ejpam-4797	126	7	applying	applying	NOUN
ejpam-4797	126	8	theorem	theorem	NOUN
ejpam-4797	126	9	1	1	NUM
ejpam-4797	126	10	,	,	PUNCT
ejpam-4797	126	11	(	(	PUNCT
ejpam-4797	126	12	a2)3	a2)3	NUM
ejpam-4797	126	13	−	−	PROPN
ejpam-4797	126	14	a2	a2	PROPN
ejpam-4797	126	15	∈	∈	PROPN
ejpam-4797	126	16	nil(r	nil(r	PROPN
ejpam-4797	126	17	)	)	PUNCT
ejpam-4797	126	18	.	.	PUNCT
ejpam-4797	127	1	hence	hence	ADV
ejpam-4797	127	2	a(a5	a(a5	ADV
ejpam-4797	127	3	−	−	ADP
ejpam-4797	127	4	a	a	DET
ejpam-4797	127	5	)	)	PUNCT
ejpam-4797	127	6	∈	∈	PROPN
ejpam-4797	127	7	nil(r	nil(r	PROPN
ejpam-4797	127	8	)	)	PUNCT
ejpam-4797	127	9	,	,	PUNCT
ejpam-4797	127	10	so	so	CCONJ
ejpam-4797	127	11	(	(	PUNCT
ejpam-4797	127	12	a4	a4	NOUN
ejpam-4797	127	13	−	−	ADP
ejpam-4797	127	14	1)a(a5	1)a(a5	ADV
ejpam-4797	127	15	−	−	PROPN
ejpam-4797	127	16	a	a	X
ejpam-4797	127	17	)	)	PUNCT
ejpam-4797	127	18	=	=	SYM
ejpam-4797	127	19	(	(	PUNCT
ejpam-4797	127	20	a5	a5	PROPN
ejpam-4797	127	21	−	−	PROPN
ejpam-4797	127	22	a)2	a)2	PROPN
ejpam-4797	127	23	∈	∈	PROPN
ejpam-4797	127	24	nil(r	nil(r	PROPN
ejpam-4797	127	25	)	)	PUNCT
ejpam-4797	127	26	.	.	PUNCT
ejpam-4797	128	1	using	use	VERB
ejpam-4797	128	2	theorem	theorem	NOUN
ejpam-4797	128	3	3	3	NUM
ejpam-4797	128	4	,	,	PUNCT
ejpam-4797	128	5	a	a	PRON
ejpam-4797	128	6	is	be	AUX
ejpam-4797	128	7	a	a	DET
ejpam-4797	128	8	sum	sum	NOUN
ejpam-4797	128	9	of	of	ADP
ejpam-4797	128	10	two	two	NUM
ejpam-4797	128	11	tripotents	tripotent	NOUN
ejpam-4797	128	12	and	and	CCONJ
ejpam-4797	128	13	a	a	DET
ejpam-4797	128	14	nilpotent	nilpotent	NOUN
ejpam-4797	128	15	that	that	DET
ejpam-4797	128	16	commute	commute	NOUN
ejpam-4797	128	17	.	.	PUNCT
ejpam-4797	129	1	r.	r.	PROPN
ejpam-4797	129	2	t.	t.	PROPN
ejpam-4797	129	3	m.salim	m.salim	PROPN
ejpam-4797	129	4	,	,	PUNCT
ejpam-4797	129	5	n.	n.	PROPN
ejpam-4797	129	6	h.	h.	PROPN
ejpam-4797	129	7	shuker	shuker	PROPN
ejpam-4797	129	8	/	/	SYM
ejpam-4797	129	9	eur	eur	PROPN
ejpam-4797	129	10	.	.	PUNCT
ejpam-4797	130	1	j.	j.	PROPN
ejpam-4797	130	2	pure	pure	PROPN
ejpam-4797	130	3	appl	appl	PROPN
ejpam-4797	130	4	.	.	PROPN
ejpam-4797	130	5	math	math	PROPN
ejpam-4797	130	6	,	,	PUNCT
ejpam-4797	130	7	16	16	NUM
ejpam-4797	130	8	(	(	PUNCT
ejpam-4797	130	9	3	3	NUM
ejpam-4797	130	10	)	)	PUNCT
ejpam-4797	130	11	(	(	PUNCT
ejpam-4797	130	12	2023	2023	NUM
ejpam-4797	130	13	)	)	PUNCT
ejpam-4797	130	14	,	,	PUNCT
ejpam-4797	130	15	1675	1675	NUM
ejpam-4797	130	16	-	-	SYM
ejpam-4797	130	17	1684	1684	NUM
ejpam-4797	130	18	1679	1679	NUM
ejpam-4797	130	19	proposition	proposition	NOUN
ejpam-4797	130	20	3	3	NUM
ejpam-4797	130	21	.	.	PUNCT
ejpam-4797	130	22	suppose	suppose	VERB
ejpam-4797	130	23	r	r	NOUN
ejpam-4797	130	24	is	be	AUX
ejpam-4797	130	25	a	a	DET
ejpam-4797	130	26	strongly	strongly	ADV
ejpam-4797	130	27	2	2	NUM
ejpam-4797	130	28	-	-	PUNCT
ejpam-4797	130	29	nc	nc	NOUN
ejpam-4797	130	30	ring	ring	NOUN
ejpam-4797	130	31	,	,	PUNCT
ejpam-4797	130	32	and	and	CCONJ
ejpam-4797	130	33	a	a	DET
ejpam-4797	130	34	=	=	PROPN
ejpam-4797	130	35	σ1	σ1	PROPN
ejpam-4797	130	36	−	−	PROPN
ejpam-4797	130	37	σ2	σ2	PROPN
ejpam-4797	130	38	+	+	CCONJ
ejpam-4797	130	39	n	n	PROPN
ejpam-4797	130	40	for	for	ADP
ejpam-4797	130	41	any	any	DET
ejpam-4797	130	42	a	a	DET
ejpam-4797	130	43	∈	∈	PROPN
ejpam-4797	130	44	r.	r.	NOUN
ejpam-4797	130	45	then	then	ADV
ejpam-4797	130	46	:	:	PUNCT
ejpam-4797	131	1	1	1	X
ejpam-4797	131	2	.	.	X
ejpam-4797	131	3	ann(a	ann(a	NOUN
ejpam-4797	131	4	)	)	PUNCT
ejpam-4797	131	5	∩	∩	NOUN
ejpam-4797	131	6	(	(	PUNCT
ejpam-4797	131	7	σ1	σ1	NOUN
ejpam-4797	131	8	−	−	PROPN
ejpam-4797	131	9	σ2)r	σ2)r	NOUN
ejpam-4797	131	10	=	=	PUNCT
ejpam-4797	131	11	0	0	NUM
ejpam-4797	131	12	.	.	NOUN
ejpam-4797	132	1	2	2	NUM
ejpam-4797	132	2	.	.	X
ejpam-4797	133	1	if	if	SCONJ
ejpam-4797	133	2	2	2	NUM
ejpam-4797	133	3	∈	∈	PROPN
ejpam-4797	133	4	u(r	u(r	NOUN
ejpam-4797	133	5	)	)	PUNCT
ejpam-4797	133	6	,	,	PUNCT
ejpam-4797	133	7	then	then	ADV
ejpam-4797	133	8	a	a	PRON
ejpam-4797	133	9	is	be	AUX
ejpam-4797	133	10	3	3	NUM
ejpam-4797	133	11	-	-	PUNCT
ejpam-4797	133	12	good	good	ADJ
ejpam-4797	133	13	element	element	NOUN
ejpam-4797	133	14	.	.	PUNCT
ejpam-4797	134	1	3	3	X
ejpam-4797	134	2	.	.	X
ejpam-4797	135	1	if	if	SCONJ
ejpam-4797	135	2	a	a	DET
ejpam-4797	135	3	∈	∈	PROPN
ejpam-4797	135	4	u(r	u(r	NOUN
ejpam-4797	135	5	)	)	PUNCT
ejpam-4797	135	6	,	,	PUNCT
ejpam-4797	135	7	then	then	ADV
ejpam-4797	135	8	(	(	PUNCT
ejpam-4797	135	9	σ1	σ1	PROPN
ejpam-4797	135	10	−	−	PROPN
ejpam-4797	135	11	σ2	σ2	PROPN
ejpam-4797	135	12	)	)	PUNCT
ejpam-4797	135	13	2	2	NUM
ejpam-4797	135	14	=	=	SYM
ejpam-4797	135	15	1	1	NUM
ejpam-4797	135	16	.	.	NOUN
ejpam-4797	135	17	4	4	NUM
ejpam-4797	135	18	.	.	X
ejpam-4797	136	1	if	if	SCONJ
ejpam-4797	136	2	a	a	PRON
ejpam-4797	136	3	is	be	AUX
ejpam-4797	136	4	a	a	DET
ejpam-4797	136	5	non	non	ADJ
ejpam-4797	136	6	-	-	ADJ
ejpam-4797	136	7	zero	zero	NUM
ejpam-4797	136	8	divisor	divisor	NOUN
ejpam-4797	136	9	,	,	PUNCT
ejpam-4797	136	10	then	then	ADV
ejpam-4797	136	11	a	a	DET
ejpam-4797	136	12	∈	∈	PROPN
ejpam-4797	136	13	u(r	u(r	NOUN
ejpam-4797	136	14	)	)	PUNCT
ejpam-4797	136	15	.	.	PUNCT
ejpam-4797	137	1	proof	proof	NOUN
ejpam-4797	137	2	.	.	PUNCT
ejpam-4797	138	1	1	1	X
ejpam-4797	138	2	.	.	X
ejpam-4797	138	3	let	let	VERB
ejpam-4797	138	4	c	c	PROPN
ejpam-4797	138	5	∈	∈	PROPN
ejpam-4797	138	6	ann(a)∩(σ1−σ2)r	ann(a)∩(σ1−σ2)r	PROPN
ejpam-4797	138	7	.	.	PUNCT
ejpam-4797	139	1	then	then	ADV
ejpam-4797	139	2	ac	ac	PROPN
ejpam-4797	140	1	=	=	PUNCT
ejpam-4797	140	2	ca	ca	NOUN
ejpam-4797	140	3	=	=	SYM
ejpam-4797	140	4	0	0	NUM
ejpam-4797	140	5	and	and	CCONJ
ejpam-4797	140	6	c	c	NOUN
ejpam-4797	141	1	=	=	SYM
ejpam-4797	141	2	(	(	PUNCT
ejpam-4797	141	3	σ1−σ2)r	σ1−σ2)r	PROPN
ejpam-4797	141	4	,	,	PUNCT
ejpam-4797	141	5	for	for	ADP
ejpam-4797	141	6	some	some	DET
ejpam-4797	141	7	r	r	NOUN
ejpam-4797	141	8	∈	∈	PROPN
ejpam-4797	141	9	r.	r.	NOUN
ejpam-4797	141	10	hence	hence	ADV
ejpam-4797	141	11	a(σ1−σ2)r	a(σ1−σ2)r	PROPN
ejpam-4797	142	1	=	=	PUNCT
ejpam-4797	142	2	0	0	NUM
ejpam-4797	142	3	,	,	PUNCT
ejpam-4797	142	4	so	so	ADV
ejpam-4797	142	5	(	(	PUNCT
ejpam-4797	142	6	σ1−σ2+n)(σ1−σ2)r	σ1−σ2+n)(σ1−σ2)r	NOUN
ejpam-4797	142	7	=	=	SYM
ejpam-4797	142	8	0	0	NUM
ejpam-4797	142	9	,	,	PUNCT
ejpam-4797	142	10	(	(	PUNCT
ejpam-4797	142	11	σ1−σ2	σ1−σ2	NOUN
ejpam-4797	142	12	)	)	PUNCT
ejpam-4797	142	13	2+n(σ1−σ2	2+n(σ1−σ2	NUM
ejpam-4797	142	14	)	)	PUNCT
ejpam-4797	142	15	=	=	SYM
ejpam-4797	143	1	0	0	X
ejpam-4797	143	2	.	.	X
ejpam-4797	143	3	applying	apply	VERB
ejpam-4797	143	4	lemma	lemma	PROPN
ejpam-4797	143	5	3	3	NUM
ejpam-4797	143	6	,	,	PUNCT
ejpam-4797	143	7	we	we	PRON
ejpam-4797	143	8	get	get	VERB
ejpam-4797	143	9	(	(	PUNCT
ejpam-4797	143	10	(	(	PUNCT
ejpam-4797	143	11	σ1	σ1	PROPN
ejpam-4797	143	12	−σ2	−σ2	NOUN
ejpam-4797	143	13	)	)	PUNCT
ejpam-4797	143	14	2	2	NUM
ejpam-4797	143	15	+	+	NUM
ejpam-4797	143	16	n(σ1	n(σ1	NOUN
ejpam-4797	143	17	−σ2	−σ2	ADJ
ejpam-4797	143	18	)	)	PUNCT
ejpam-4797	143	19	3)r	3)r	NUM
ejpam-4797	144	1	=	=	SYM
ejpam-4797	144	2	0	0	PROPN
ejpam-4797	144	3	,	,	PUNCT
ejpam-4797	144	4	(	(	PUNCT
ejpam-4797	144	5	σ1	σ1	NOUN
ejpam-4797	144	6	−σ2	−σ2	PROPN
ejpam-4797	144	7	)	)	PUNCT
ejpam-4797	144	8	2(1	2(1	NUM
ejpam-4797	144	9	+	+	NUM
ejpam-4797	144	10	n(σ1	n(σ1	NOUN
ejpam-4797	144	11	−	−	NOUN
ejpam-4797	144	12	σ2))r	σ2))r	PRON
ejpam-4797	144	13	=	=	NOUN
ejpam-4797	144	14	0	0	PROPN
ejpam-4797	144	15	.	.	PUNCT
ejpam-4797	145	1	but	but	CCONJ
ejpam-4797	145	2	1	1	NUM
ejpam-4797	145	3	+	+	NUM
ejpam-4797	145	4	n(σ1	n(σ1	NOUN
ejpam-4797	145	5	−	−	PROPN
ejpam-4797	145	6	σ2	σ2	PROPN
ejpam-4797	145	7	)	)	PUNCT
ejpam-4797	145	8	∈	∈	PROPN
ejpam-4797	145	9	u(r	u(r	PROPN
ejpam-4797	145	10	)	)	PUNCT
ejpam-4797	145	11	,	,	PUNCT
ejpam-4797	145	12	say	say	VERB
ejpam-4797	145	13	u	u	NOUN
ejpam-4797	145	14	,	,	PUNCT
ejpam-4797	145	15	then	then	ADV
ejpam-4797	145	16	we	we	PRON
ejpam-4797	145	17	have	have	VERB
ejpam-4797	145	18	(	(	PUNCT
ejpam-4797	145	19	σ1	σ1	PROPN
ejpam-4797	145	20	−	−	PROPN
ejpam-4797	145	21	σ2	σ2	PROPN
ejpam-4797	145	22	)	)	PUNCT
ejpam-4797	145	23	2ur	2ur	NOUN
ejpam-4797	145	24	=	=	SYM
ejpam-4797	145	25	0	0	NUM
ejpam-4797	145	26	,	,	PUNCT
ejpam-4797	145	27	so	so	CCONJ
ejpam-4797	145	28	(	(	PUNCT
ejpam-4797	145	29	σ1	σ1	PROPN
ejpam-4797	145	30	−	−	PROPN
ejpam-4797	145	31	σ2	σ2	PROPN
ejpam-4797	145	32	)	)	PUNCT
ejpam-4797	145	33	2r	2r	NUM
ejpam-4797	145	34	=	=	SYM
ejpam-4797	146	1	0	0	X
ejpam-4797	146	2	.	.	PUNCT
ejpam-4797	146	3	multiply	multiply	VERB
ejpam-4797	146	4	by	by	ADP
ejpam-4797	146	5	(	(	PUNCT
ejpam-4797	146	6	σ1	σ1	PROPN
ejpam-4797	146	7	−	−	PROPN
ejpam-4797	146	8	σ2	σ2	PROPN
ejpam-4797	146	9	)	)	PUNCT
ejpam-4797	146	10	,	,	PUNCT
ejpam-4797	146	11	we	we	PRON
ejpam-4797	146	12	have	have	VERB
ejpam-4797	146	13	(	(	PUNCT
ejpam-4797	146	14	σ1	σ1	NOUN
ejpam-4797	146	15	−	−	PROPN
ejpam-4797	146	16	σ2)r	σ2)r	NOUN
ejpam-4797	147	1	=	=	PUNCT
ejpam-4797	147	2	c	c	NOUN
ejpam-4797	147	3	=	=	SYM
ejpam-4797	147	4	0	0	X
ejpam-4797	147	5	.	.	PUNCT
ejpam-4797	148	1	therefore	therefore	ADV
ejpam-4797	148	2	,	,	PUNCT
ejpam-4797	148	3	ann(a	ann(a	PROPN
ejpam-4797	148	4	)	)	PUNCT
ejpam-4797	148	5	∩	∩	NOUN
ejpam-4797	148	6	(	(	PUNCT
ejpam-4797	148	7	σ1	σ1	NOUN
ejpam-4797	148	8	−	−	PROPN
ejpam-4797	148	9	σ2)r	σ2)r	NOUN
ejpam-4797	148	10	=	=	PUNCT
ejpam-4797	148	11	0	0	NUM
ejpam-4797	148	12	.	.	NOUN
ejpam-4797	149	1	2	2	X
ejpam-4797	149	2	.	.	X
ejpam-4797	149	3	we	we	PRON
ejpam-4797	149	4	may	may	AUX
ejpam-4797	149	5	write	write	VERB
ejpam-4797	149	6	a	a	DET
ejpam-4797	149	7	=	=	PROPN
ejpam-4797	149	8	σ1	σ1	NOUN
ejpam-4797	149	9	+	+	NOUN
ejpam-4797	149	10	1+σ2	1+σ2	NUM
ejpam-4797	149	11	+	+	ADJ
ejpam-4797	149	12	1+n−2	1+n−2	NUM
ejpam-4797	149	13	.	.	PUNCT
ejpam-4797	150	1	consider	consider	VERB
ejpam-4797	150	2	(	(	PUNCT
ejpam-4797	150	3	σ1	σ1	PROPN
ejpam-4797	150	4	+	+	NOUN
ejpam-4797	150	5	1)(2−σ1	1)(2−σ1	PROPN
ejpam-4797	150	6	)	)	PUNCT
ejpam-4797	150	7	=	=	PUNCT
ejpam-4797	151	1	2σ1−σ1	2σ1−σ1	VERB
ejpam-4797	151	2	+	+	PROPN
ejpam-4797	151	3	2−	2−	NUM
ejpam-4797	151	4	σ1	σ1	NOUN
ejpam-4797	151	5	=	=	SYM
ejpam-4797	152	1	2	2	X
ejpam-4797	152	2	.	.	PUNCT
ejpam-4797	153	1	since	since	SCONJ
ejpam-4797	153	2	2	2	NUM
ejpam-4797	153	3	∈	∈	PROPN
ejpam-4797	153	4	u(r	u(r	NOUN
ejpam-4797	153	5	)	)	PUNCT
ejpam-4797	153	6	,	,	PUNCT
ejpam-4797	153	7	then	then	ADV
ejpam-4797	153	8	σ1	σ1	NOUN
ejpam-4797	153	9	+	+	CCONJ
ejpam-4797	153	10	1	1	NUM
ejpam-4797	153	11	=	=	SYM
ejpam-4797	153	12	u1	u1	NOUN
ejpam-4797	153	13	∈	∈	PROPN
ejpam-4797	153	14	u(r	u(r	PROPN
ejpam-4797	153	15	)	)	PUNCT
ejpam-4797	153	16	.	.	PUNCT
ejpam-4797	154	1	similarly	similarly	ADV
ejpam-4797	154	2	σ2	σ2	PROPN
ejpam-4797	154	3	+	+	CCONJ
ejpam-4797	154	4	1	1	NUM
ejpam-4797	154	5	=	=	SYM
ejpam-4797	154	6	u2	u2	PROPN
ejpam-4797	154	7	∈	∈	PROPN
ejpam-4797	154	8	u(r	u(r	PROPN
ejpam-4797	154	9	)	)	PUNCT
ejpam-4797	154	10	.	.	PUNCT
ejpam-4797	155	1	furthermore	furthermore	ADV
ejpam-4797	155	2	,	,	PUNCT
ejpam-4797	155	3	n−	n−	NOUN
ejpam-4797	155	4	2	2	NUM
ejpam-4797	155	5	∈	∈	PROPN
ejpam-4797	155	6	u(r	u(r	NOUN
ejpam-4797	155	7	)	)	PUNCT
ejpam-4797	155	8	,	,	PUNCT
ejpam-4797	155	9	say	say	VERB
ejpam-4797	155	10	u3	u3	PROPN
ejpam-4797	155	11	.	.	PUNCT
ejpam-4797	156	1	thus	thus	ADV
ejpam-4797	156	2	,	,	PUNCT
ejpam-4797	156	3	a	a	DET
ejpam-4797	156	4	=	=	NOUN
ejpam-4797	156	5	u1	u1	NOUN
ejpam-4797	156	6	+	+	CCONJ
ejpam-4797	156	7	u2	u2	PROPN
ejpam-4797	156	8	+	+	CCONJ
ejpam-4797	156	9	u3	u3	NOUN
ejpam-4797	156	10	.	.	PUNCT
ejpam-4797	157	1	3	3	X
ejpam-4797	157	2	.	.	X
ejpam-4797	157	3	let	let	VERB
ejpam-4797	157	4	a	a	DET
ejpam-4797	157	5	=	=	X
ejpam-4797	157	6	(	(	PUNCT
ejpam-4797	157	7	σ1−σ2)+n	σ1−σ2)+n	PROPN
ejpam-4797	157	8	,	,	PUNCT
ejpam-4797	157	9	and	and	CCONJ
ejpam-4797	157	10	let	let	VERB
ejpam-4797	157	11	a	a	DET
ejpam-4797	157	12	∈	∈	PROPN
ejpam-4797	157	13	u(r	u(r	NOUN
ejpam-4797	157	14	)	)	PUNCT
ejpam-4797	157	15	.	.	PUNCT
ejpam-4797	158	1	then	then	ADV
ejpam-4797	158	2	a−n	a−n	PROPN
ejpam-4797	158	3	=	=	PUNCT
ejpam-4797	158	4	(	(	PUNCT
ejpam-4797	158	5	σ1−σ2	σ1−σ2	NOUN
ejpam-4797	158	6	)	)	PUNCT
ejpam-4797	158	7	∈	∈	PROPN
ejpam-4797	158	8	u(r	u(r	NOUN
ejpam-4797	158	9	)	)	PUNCT
ejpam-4797	158	10	.	.	PUNCT
ejpam-4797	159	1	applying	apply	VERB
ejpam-4797	159	2	lemma	lemma	PROPN
ejpam-4797	159	3	3(2	3(2	NUM
ejpam-4797	159	4	)	)	PUNCT
ejpam-4797	159	5	,	,	PUNCT
ejpam-4797	159	6	then	then	ADV
ejpam-4797	159	7	(	(	PUNCT
ejpam-4797	159	8	σ1	σ1	PROPN
ejpam-4797	159	9	−	−	PROPN
ejpam-4797	159	10	σ2	σ2	PROPN
ejpam-4797	159	11	)	)	PUNCT
ejpam-4797	159	12	=	=	SYM
ejpam-4797	159	13	(	(	PUNCT
ejpam-4797	159	14	σ1	σ1	PROPN
ejpam-4797	159	15	−	−	PROPN
ejpam-4797	159	16	σ2	σ2	PROPN
ejpam-4797	159	17	)	)	PUNCT
ejpam-4797	159	18	3	3	NUM
ejpam-4797	159	19	.	.	PUNCT
ejpam-4797	160	1	thus	thus	ADV
ejpam-4797	160	2	(	(	PUNCT
ejpam-4797	160	3	σ1	σ1	PROPN
ejpam-4797	160	4	−	−	PROPN
ejpam-4797	160	5	σ2	σ2	PROPN
ejpam-4797	160	6	)	)	PUNCT
ejpam-4797	160	7	2	2	NUM
ejpam-4797	160	8	=	=	SYM
ejpam-4797	160	9	1	1	NUM
ejpam-4797	160	10	.	.	NOUN
ejpam-4797	160	11	4	4	X
ejpam-4797	160	12	.	.	X
ejpam-4797	161	1	let	let	VERB
ejpam-4797	161	2	a	a	PRON
ejpam-4797	161	3	be	be	AUX
ejpam-4797	161	4	a	a	DET
ejpam-4797	161	5	non	non	ADJ
ejpam-4797	161	6	-	-	ADJ
ejpam-4797	161	7	zero	zero	NUM
ejpam-4797	161	8	divisor	divisor	NOUN
ejpam-4797	161	9	element	element	NOUN
ejpam-4797	161	10	.	.	PUNCT
ejpam-4797	162	1	applying	apply	VERB
ejpam-4797	162	2	theorem	theorem	ADJ
ejpam-4797	162	3	1	1	NUM
ejpam-4797	162	4	,	,	PUNCT
ejpam-4797	162	5	a3	a3	VERB
ejpam-4797	162	6	−	−	PROPN
ejpam-4797	162	7	a	a	DET
ejpam-4797	162	8	∈	∈	PROPN
ejpam-4797	162	9	nil(r	nil(r	NOUN
ejpam-4797	162	10	)	)	PUNCT
ejpam-4797	162	11	,	,	PUNCT
ejpam-4797	162	12	this	this	PRON
ejpam-4797	162	13	gives	give	VERB
ejpam-4797	162	14	a(a2	a(a2	NOUN
ejpam-4797	162	15	−	−	NUM
ejpam-4797	162	16	1	1	NUM
ejpam-4797	162	17	)	)	PUNCT
ejpam-4797	162	18	∈	∈	PROPN
ejpam-4797	162	19	nil(r	nil(r	PROPN
ejpam-4797	162	20	)	)	PUNCT
ejpam-4797	162	21	,	,	PUNCT
ejpam-4797	162	22	thus	thus	ADV
ejpam-4797	162	23	,	,	PUNCT
ejpam-4797	162	24	ar(a2	ar(a2	PRON
ejpam-4797	162	25	−	−	PROPN
ejpam-4797	162	26	1)r	1)r	X
ejpam-4797	162	27	=	=	SYM
ejpam-4797	162	28	0	0	NUM
ejpam-4797	162	29	,	,	PUNCT
ejpam-4797	162	30	for	for	ADP
ejpam-4797	162	31	some	some	DET
ejpam-4797	162	32	positive	positive	ADJ
ejpam-4797	162	33	integer	integer	NOUN
ejpam-4797	162	34	r.	r.	PROPN
ejpam-4797	162	35	since	since	SCONJ
ejpam-4797	162	36	ar	ar	PROPN
ejpam-4797	162	37	is	be	AUX
ejpam-4797	162	38	a	a	DET
ejpam-4797	162	39	non	non	ADJ
ejpam-4797	162	40	-	-	ADJ
ejpam-4797	162	41	zero	zero	NUM
ejpam-4797	162	42	divisor	divisor	NOUN
ejpam-4797	162	43	,	,	PUNCT
ejpam-4797	162	44	then	then	ADV
ejpam-4797	162	45	(	(	PUNCT
ejpam-4797	162	46	a2	a2	PROPN
ejpam-4797	162	47	−	−	PROPN
ejpam-4797	162	48	1)r	1)r	NUM
ejpam-4797	162	49	=	=	SYM
ejpam-4797	162	50	0	0	NUM
ejpam-4797	162	51	,	,	PUNCT
ejpam-4797	162	52	so	so	ADV
ejpam-4797	162	53	a2	a2	PROPN
ejpam-4797	162	54	−	−	PROPN
ejpam-4797	162	55	1	1	NUM
ejpam-4797	162	56	=	=	SYM
ejpam-4797	162	57	n1	n1	PROPN
ejpam-4797	162	58	∈	∈	PROPN
ejpam-4797	162	59	nil(r	nil(r	NOUN
ejpam-4797	162	60	)	)	PUNCT
ejpam-4797	162	61	,	,	PUNCT
ejpam-4797	162	62	implies	imply	VERB
ejpam-4797	162	63	a2	a2	PROPN
ejpam-4797	162	64	=	=	SYM
ejpam-4797	162	65	1	1	NUM
ejpam-4797	162	66	+	+	CCONJ
ejpam-4797	162	67	n1	n1	PROPN
ejpam-4797	162	68	∈	∈	PROPN
ejpam-4797	162	69	u(r	u(r	NOUN
ejpam-4797	162	70	)	)	PUNCT
ejpam-4797	162	71	,	,	PUNCT
ejpam-4797	162	72	then	then	ADV
ejpam-4797	162	73	a	a	DET
ejpam-4797	162	74	∈	∈	PROPN
ejpam-4797	162	75	u(r	u(r	NOUN
ejpam-4797	162	76	)	)	PUNCT
ejpam-4797	162	77	.	.	PUNCT
ejpam-4797	163	1	it	it	PRON
ejpam-4797	163	2	was	be	AUX
ejpam-4797	163	3	proved	prove	VERB
ejpam-4797	163	4	in	in	ADP
ejpam-4797	163	5	[	[	X
ejpam-4797	163	6	18	18	NUM
ejpam-4797	163	7	]	]	PUNCT
ejpam-4797	163	8	,	,	PUNCT
ejpam-4797	163	9	that	that	PRON
ejpam-4797	163	10	.	.	PUNCT
ejpam-4797	164	1	proposition	proposition	NOUN
ejpam-4797	164	2	4	4	NUM
ejpam-4797	164	3	.	.	PUNCT
ejpam-4797	165	1	[	[	X
ejpam-4797	165	2	18	18	NUM
ejpam-4797	165	3	,	,	PUNCT
ejpam-4797	165	4	proposition	proposition	NOUN
ejpam-4797	165	5	1	1	NUM
ejpam-4797	165	6	]	]	PUNCT
ejpam-4797	165	7	.	.	PUNCT
ejpam-4797	166	1	assume	assume	VERB
ejpam-4797	166	2	r	r	NOUN
ejpam-4797	166	3	is	be	AUX
ejpam-4797	166	4	a	a	DET
ejpam-4797	166	5	nil	nil	ADJ
ejpam-4797	166	6	clean	clean	ADJ
ejpam-4797	166	7	ring	ring	NOUN
ejpam-4797	166	8	with	with	ADP
ejpam-4797	166	9	every	every	DET
ejpam-4797	166	10	nilpotent	nilpotent	NOUN
ejpam-4797	166	11	is	be	AUX
ejpam-4797	166	12	the	the	DET
ejpam-4797	166	13	difference	difference	NOUN
ejpam-4797	166	14	between	between	ADP
ejpam-4797	166	15	two	two	NUM
ejpam-4797	166	16	commuting	commuting	NOUN
ejpam-4797	166	17	idempotents	idempotent	NOUN
ejpam-4797	166	18	,	,	PUNCT
ejpam-4797	166	19	then	then	ADV
ejpam-4797	166	20	r	r	NOUN
ejpam-4797	166	21	is	be	AUX
ejpam-4797	166	22	a	a	DET
ejpam-4797	166	23	boolean	boolean	ADJ
ejpam-4797	166	24	ring	ring	NOUN
ejpam-4797	166	25	.	.	PUNCT
ejpam-4797	167	1	we	we	PRON
ejpam-4797	167	2	here	here	ADV
ejpam-4797	167	3	extend	extend	VERB
ejpam-4797	167	4	this	this	DET
ejpam-4797	167	5	result	result	NOUN
ejpam-4797	167	6	as	as	SCONJ
ejpam-4797	167	7	follows	follow	VERB
ejpam-4797	167	8	:	:	PUNCT
ejpam-4797	167	9	theorem	theorem	NOUN
ejpam-4797	167	10	4	4	NUM
ejpam-4797	167	11	.	.	PUNCT
ejpam-4797	167	12	suppose	suppose	VERB
ejpam-4797	167	13	r	r	NOUN
ejpam-4797	167	14	is	be	AUX
ejpam-4797	167	15	a	a	DET
ejpam-4797	167	16	strongly	strongly	ADV
ejpam-4797	167	17	2	2	NUM
ejpam-4797	167	18	-	-	PUNCT
ejpam-4797	167	19	nc	nc	NOUN
ejpam-4797	167	20	ring	ring	NOUN
ejpam-4797	167	21	,	,	PUNCT
ejpam-4797	167	22	with	with	SCONJ
ejpam-4797	167	23	any	any	DET
ejpam-4797	167	24	nilpotent	nilpotent	NOUN
ejpam-4797	167	25	is	be	AUX
ejpam-4797	167	26	the	the	DET
ejpam-4797	167	27	difference	difference	NOUN
ejpam-4797	167	28	between	between	ADP
ejpam-4797	167	29	two	two	NUM
ejpam-4797	167	30	commuting	commuting	NOUN
ejpam-4797	167	31	idempotents	idempotent	NOUN
ejpam-4797	167	32	.	.	PUNCT
ejpam-4797	168	1	then	then	ADV
ejpam-4797	168	2	r	r	NOUN
ejpam-4797	168	3	is	be	AUX
ejpam-4797	168	4	a	a	DET
ejpam-4797	168	5	tripotent	tripotent	ADJ
ejpam-4797	168	6	ring	ring	NOUN
ejpam-4797	168	7	.	.	PUNCT
ejpam-4797	169	1	proof	proof	NOUN
ejpam-4797	169	2	.	.	PUNCT
ejpam-4797	170	1	let	let	VERB
ejpam-4797	170	2	a	a	PRON
ejpam-4797	170	3	in	in	ADP
ejpam-4797	170	4	r	r	NOUN
ejpam-4797	170	5	,	,	PUNCT
ejpam-4797	170	6	then	then	ADV
ejpam-4797	170	7	a	a	DET
ejpam-4797	170	8	=	=	PUNCT
ejpam-4797	170	9	σ1−σ2+n	σ1−σ2+n	NOUN
ejpam-4797	170	10	for	for	ADP
ejpam-4797	170	11	some	some	DET
ejpam-4797	170	12	existing	exist	VERB
ejpam-4797	170	13	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	170	14	∈	∈	PROPN
ejpam-4797	170	15	id(r	id(r	NOUN
ejpam-4797	170	16	)	)	PUNCT
ejpam-4797	170	17	,	,	PUNCT
ejpam-4797	170	18	n	n	PROPN
ejpam-4797	170	19	∈	∈	PROPN
ejpam-4797	170	20	nil(r	nil(r	PROPN
ejpam-4797	170	21	)	)	PUNCT
ejpam-4797	170	22	,	,	PUNCT
ejpam-4797	170	23	that	that	DET
ejpam-4797	170	24	commute	commute	NOUN
ejpam-4797	170	25	which	which	PRON
ejpam-4797	170	26	each	each	DET
ejpam-4797	170	27	other	other	ADJ
ejpam-4797	170	28	.	.	PUNCT
ejpam-4797	171	1	then	then	ADV
ejpam-4797	171	2	n	n	PROPN
ejpam-4797	171	3	=	=	SYM
ejpam-4797	171	4	σ3	σ3	PROPN
ejpam-4797	171	5	−σ4	−σ4	NOUN
ejpam-4797	171	6	for	for	ADP
ejpam-4797	171	7	some	some	DET
ejpam-4797	171	8	σ3,σ4	σ3,σ4	PROPN
ejpam-4797	171	9	∈	∈	PROPN
ejpam-4797	171	10	id(r	id(r	NOUN
ejpam-4797	171	11	)	)	PUNCT
ejpam-4797	171	12	and	and	CCONJ
ejpam-4797	171	13	σ3σ4	σ3σ4	X
ejpam-4797	171	14	=	=	SYM
ejpam-4797	171	15	σ4σ3	σ4σ3	X
ejpam-4797	171	16	.	.	PUNCT
ejpam-4797	172	1	so	so	ADV
ejpam-4797	172	2	n+σ4	n+σ4	NOUN
ejpam-4797	172	3	=	=	PROPN
ejpam-4797	172	4	σ3	σ3	PROPN
ejpam-4797	172	5	,	,	PUNCT
ejpam-4797	172	6	this	this	PRON
ejpam-4797	172	7	implies	imply	VERB
ejpam-4797	172	8	(	(	PUNCT
ejpam-4797	172	9	n+σ4	n+σ4	NOUN
ejpam-4797	172	10	)	)	PUNCT
ejpam-4797	172	11	2	2	NUM
ejpam-4797	173	1	=	=	SYM
ejpam-4797	174	1	(	(	PUNCT
ejpam-4797	175	1	n+σ4	n+σ4	NOUN
ejpam-4797	175	2	)	)	PUNCT
ejpam-4797	175	3	,	,	PUNCT
ejpam-4797	175	4	then	then	ADV
ejpam-4797	175	5	n2	n2	ADJ
ejpam-4797	175	6	+2nς4	+2nς4	PROPN
ejpam-4797	176	1	+	+	PROPN
ejpam-4797	176	2	σ2	σ2	PROPN
ejpam-4797	176	3	4	4	NUM
ejpam-4797	176	4	=	=	SYM
ejpam-4797	176	5	n+σ4	n+σ4	NOUN
ejpam-4797	176	6	,	,	PUNCT
ejpam-4797	176	7	this	this	PRON
ejpam-4797	176	8	gives	give	VERB
ejpam-4797	176	9	n2	n2	NOUN
ejpam-4797	176	10	+	+	CCONJ
ejpam-4797	176	11	2nς4	2nς4	NUM
ejpam-4797	176	12	−	−	NOUN
ejpam-4797	177	1	n	n	NOUN
ejpam-4797	177	2	=	=	SYM
ejpam-4797	177	3	0	0	NUM
ejpam-4797	177	4	,	,	PUNCT
ejpam-4797	177	5	so	so	ADV
ejpam-4797	177	6	n2	n2	ADJ
ejpam-4797	177	7	+	+	CCONJ
ejpam-4797	177	8	n(2σ4	n(2σ4	NUM
ejpam-4797	177	9	−	−	PROPN
ejpam-4797	177	10	1	1	NUM
ejpam-4797	177	11	)	)	PUNCT
ejpam-4797	177	12	=	=	SYM
ejpam-4797	178	1	0	0	NUM
ejpam-4797	178	2	,	,	PUNCT
ejpam-4797	178	3	but	but	CCONJ
ejpam-4797	178	4	(	(	PUNCT
ejpam-4797	178	5	2σ4	2σ4	NUM
ejpam-4797	178	6	−	−	PROPN
ejpam-4797	178	7	1)2	1)2	NUM
ejpam-4797	178	8	=	=	SYM
ejpam-4797	178	9	1	1	NUM
ejpam-4797	178	10	,	,	PUNCT
ejpam-4797	178	11	then	then	ADV
ejpam-4797	178	12	we	we	PRON
ejpam-4797	178	13	have	have	VERB
ejpam-4797	178	14	n	n	NOUN
ejpam-4797	178	15	=	=	PUNCT
ejpam-4797	178	16	−n2(2σ4	−n2(2σ4	PROPN
ejpam-4797	178	17	−	−	PROPN
ejpam-4797	178	18	1)−1	1)−1	NUM
ejpam-4797	178	19	.	.	PUNCT
ejpam-4797	179	1	as	as	SCONJ
ejpam-4797	179	2	n	n	PRON
ejpam-4797	179	3	is	be	AUX
ejpam-4797	179	4	nilpotent	nilpotent	ADJ
ejpam-4797	179	5	,	,	PUNCT
ejpam-4797	179	6	then	then	ADV
ejpam-4797	179	7	n	n	CCONJ
ejpam-4797	179	8	=	=	SYM
ejpam-4797	179	9	0	0	NUM
ejpam-4797	179	10	.	.	PUNCT
ejpam-4797	180	1	thus	thus	ADV
ejpam-4797	180	2	,	,	PUNCT
ejpam-4797	180	3	a	a	DET
ejpam-4797	180	4	=	=	PROPN
ejpam-4797	180	5	σ1	σ1	PROPN
ejpam-4797	180	6	−	−	PROPN
ejpam-4797	180	7	σ2	σ2	PROPN
ejpam-4797	180	8	.	.	PUNCT
ejpam-4797	180	9	applying	apply	VERB
ejpam-4797	180	10	lemma	lemma	PROPN
ejpam-4797	180	11	3(2	3(2	NUM
ejpam-4797	180	12	)	)	PUNCT
ejpam-4797	180	13	,	,	PUNCT
ejpam-4797	180	14	(	(	PUNCT
ejpam-4797	180	15	σ1	σ1	PROPN
ejpam-4797	180	16	−	−	PROPN
ejpam-4797	180	17	σ2	σ2	PROPN
ejpam-4797	180	18	)	)	PUNCT
ejpam-4797	180	19	3	3	NUM
ejpam-4797	180	20	=	=	SYM
ejpam-4797	180	21	σ1	σ1	PROPN
ejpam-4797	180	22	−	−	PROPN
ejpam-4797	180	23	σ2	σ2	PROPN
ejpam-4797	180	24	.	.	PUNCT
ejpam-4797	181	1	hence	hence	ADV
ejpam-4797	181	2	,	,	PUNCT
ejpam-4797	181	3	a	a	DET
ejpam-4797	181	4	=	=	NOUN
ejpam-4797	181	5	a3	a3	NOUN
ejpam-4797	181	6	therefore	therefore	ADV
ejpam-4797	181	7	,	,	PUNCT
ejpam-4797	181	8	r	r	NOUN
ejpam-4797	181	9	is	be	AUX
ejpam-4797	181	10	a	a	DET
ejpam-4797	181	11	tripotent	tripotent	ADJ
ejpam-4797	181	12	ring	ring	NOUN
ejpam-4797	181	13	.	.	PUNCT
ejpam-4797	182	1	r.	r.	PROPN
ejpam-4797	182	2	t.	t.	PROPN
ejpam-4797	182	3	m.salim	m.salim	PROPN
ejpam-4797	182	4	,	,	PUNCT
ejpam-4797	182	5	n.	n.	PROPN
ejpam-4797	182	6	h.	h.	PROPN
ejpam-4797	182	7	shuker	shuker	PROPN
ejpam-4797	182	8	/	/	SYM
ejpam-4797	182	9	eur	eur	PROPN
ejpam-4797	182	10	.	.	PUNCT
ejpam-4797	183	1	j.	j.	PROPN
ejpam-4797	183	2	pure	pure	PROPN
ejpam-4797	183	3	appl	appl	PROPN
ejpam-4797	183	4	.	.	PROPN
ejpam-4797	183	5	math	math	PROPN
ejpam-4797	183	6	,	,	PUNCT
ejpam-4797	183	7	16	16	NUM
ejpam-4797	183	8	(	(	PUNCT
ejpam-4797	183	9	3	3	NUM
ejpam-4797	183	10	)	)	PUNCT
ejpam-4797	183	11	(	(	PUNCT
ejpam-4797	183	12	2023	2023	NUM
ejpam-4797	183	13	)	)	PUNCT
ejpam-4797	183	14	,	,	PUNCT
ejpam-4797	183	15	1675	1675	NUM
ejpam-4797	183	16	-	-	SYM
ejpam-4797	183	17	1684	1684	NUM
ejpam-4797	183	18	1680	1680	NUM
ejpam-4797	183	19	3	3	NUM
ejpam-4797	183	20	.	.	PUNCT
ejpam-4797	184	1	strongly	strongly	ADV
ejpam-4797	184	2	2	2	NUM
ejpam-4797	184	3	-	-	PUNCT
ejpam-4797	184	4	nc	nc	PROPN
ejpam-4797	184	5	rings	ring	NOUN
ejpam-4797	184	6	with	with	ADP
ejpam-4797	184	7	units	unit	NOUN
ejpam-4797	184	8	of	of	ADP
ejpam-4797	184	9	order	order	NOUN
ejpam-4797	184	10	two	two	NUM
ejpam-4797	184	11	in	in	ADP
ejpam-4797	184	12	this	this	DET
ejpam-4797	184	13	section	section	NOUN
ejpam-4797	184	14	,	,	PUNCT
ejpam-4797	184	15	we	we	PRON
ejpam-4797	184	16	introduce	introduce	VERB
ejpam-4797	184	17	and	and	CCONJ
ejpam-4797	184	18	investigate	investigate	VERB
ejpam-4797	184	19	a	a	DET
ejpam-4797	184	20	strongly	strongly	ADV
ejpam-4797	184	21	2	2	NUM
ejpam-4797	184	22	-	-	PUNCT
ejpam-4797	184	23	nc	nc	PROPN
ejpam-4797	184	24	rings	ring	NOUN
ejpam-4797	184	25	with	with	ADP
ejpam-4797	184	26	every	every	DET
ejpam-4797	184	27	unit	unit	NOUN
ejpam-4797	184	28	is	be	AUX
ejpam-4797	184	29	of	of	ADP
ejpam-4797	184	30	order	order	NOUN
ejpam-4797	184	31	2	2	NUM
ejpam-4797	184	32	,	,	PUNCT
ejpam-4797	184	33	we	we	PRON
ejpam-4797	184	34	refer	refer	VERB
ejpam-4797	184	35	to	to	ADP
ejpam-4797	184	36	this	this	DET
ejpam-4797	184	37	type	type	NOUN
ejpam-4797	184	38	of	of	ADP
ejpam-4797	184	39	ring	ring	NOUN
ejpam-4797	184	40	as	as	ADP
ejpam-4797	184	41	strongly	strongly	ADV
ejpam-4797	184	42	2	2	NUM
ejpam-4797	184	43	-	-	PUNCT
ejpam-4797	184	44	nc	nc	PROPN
ejpam-4797	184	45	rings	ring	NOUN
ejpam-4797	184	46	with	with	ADP
ejpam-4797	184	47	u(r	u(r	NOUN
ejpam-4797	184	48	)	)	PUNCT
ejpam-4797	185	1	=	=	SYM
ejpam-4797	185	2	2	2	X
ejpam-4797	185	3	.	.	X
ejpam-4797	185	4	definition	definition	NOUN
ejpam-4797	185	5	4	4	NUM
ejpam-4797	185	6	.	.	PUNCT
ejpam-4797	186	1	a	a	DET
ejpam-4797	186	2	ring	ring	NOUN
ejpam-4797	186	3	r	r	NOUN
ejpam-4797	186	4	is	be	AUX
ejpam-4797	186	5	called	call	VERB
ejpam-4797	186	6	strongly	strongly	ADV
ejpam-4797	186	7	2	2	NUM
ejpam-4797	186	8	-	-	PUNCT
ejpam-4797	186	9	nc	nc	NOUN
ejpam-4797	186	10	with	with	ADP
ejpam-4797	186	11	u(r	u(r	NOUN
ejpam-4797	186	12	)	)	PUNCT
ejpam-4797	187	1	=	=	SYM
ejpam-4797	187	2	2	2	NUM
ejpam-4797	187	3	if	if	SCONJ
ejpam-4797	187	4	for	for	ADP
ejpam-4797	187	5	every	every	DET
ejpam-4797	187	6	a	a	DET
ejpam-4797	187	7	∈	∈	PROPN
ejpam-4797	187	8	r	r	NOUN
ejpam-4797	187	9	,	,	PUNCT
ejpam-4797	187	10	existing	exist	VERB
ejpam-4797	187	11	two	two	NUM
ejpam-4797	187	12	idempotents	idempotent	NOUN
ejpam-4797	187	13	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	187	14	and	and	CCONJ
ejpam-4797	187	15	a	a	DET
ejpam-4797	187	16	nilpotent	nilpotent	ADJ
ejpam-4797	187	17	n	n	CCONJ
ejpam-4797	187	18	,	,	PUNCT
ejpam-4797	187	19	that	that	DET
ejpam-4797	187	20	commute	commute	NOUN
ejpam-4797	187	21	and	and	CCONJ
ejpam-4797	187	22	every	every	DET
ejpam-4797	187	23	unit	unit	NOUN
ejpam-4797	187	24	is	be	AUX
ejpam-4797	187	25	of	of	ADP
ejpam-4797	187	26	order	order	NOUN
ejpam-4797	187	27	2	2	NUM
ejpam-4797	187	28	,	,	PUNCT
ejpam-4797	188	1	such	such	ADJ
ejpam-4797	188	2	that	that	SCONJ
ejpam-4797	188	3	a	a	DET
ejpam-4797	188	4	=	=	X
ejpam-4797	188	5	σ1	σ1	PROPN
ejpam-4797	188	6	+	+	PROPN
ejpam-4797	188	7	σ2	σ2	PROPN
ejpam-4797	188	8	+	+	CCONJ
ejpam-4797	188	9	n.	n.	PROPN
ejpam-4797	188	10	example	example	NOUN
ejpam-4797	188	11	2	2	X
ejpam-4797	188	12	.	.	PUNCT
ejpam-4797	188	13	the	the	DET
ejpam-4797	188	14	rings	ring	NOUN
ejpam-4797	188	15	z4	z4	PROPN
ejpam-4797	188	16	,	,	PUNCT
ejpam-4797	188	17	z6	z6	PROPN
ejpam-4797	188	18	,	,	PUNCT
ejpam-4797	188	19	z8	z8	PROPN
ejpam-4797	188	20	,	,	PUNCT
ejpam-4797	188	21	z12	z12	PROPN
ejpam-4797	188	22	,	,	PUNCT
ejpam-4797	188	23	z24	z24	PROPN
ejpam-4797	188	24	are	be	AUX
ejpam-4797	188	25	all	all	ADV
ejpam-4797	188	26	strongly	strongly	ADV
ejpam-4797	188	27	2	2	NUM
ejpam-4797	188	28	-	-	PUNCT
ejpam-4797	188	29	nc	nc	NOUN
ejpam-4797	188	30	with	with	ADP
ejpam-4797	188	31	u(r	u(r	NOUN
ejpam-4797	188	32	)	)	PUNCT
ejpam-4797	189	1	=	=	SYM
ejpam-4797	189	2	2	2	NUM
ejpam-4797	189	3	,	,	PUNCT
ejpam-4797	189	4	while	while	SCONJ
ejpam-4797	189	5	the	the	DET
ejpam-4797	189	6	ring	ring	NOUN
ejpam-4797	189	7	z9	z9	PROPN
ejpam-4797	189	8	is	be	AUX
ejpam-4797	189	9	not	not	PART
ejpam-4797	189	10	strongly	strongly	ADV
ejpam-4797	189	11	2	2	NUM
ejpam-4797	189	12	-	-	PUNCT
ejpam-4797	189	13	nc	nc	NOUN
ejpam-4797	189	14	with	with	ADP
ejpam-4797	189	15	u(r	u(r	NOUN
ejpam-4797	189	16	)	)	PUNCT
ejpam-4797	189	17	=	=	SYM
ejpam-4797	190	1	2	2	X
ejpam-4797	190	2	.	.	X
ejpam-4797	190	3	we	we	PRON
ejpam-4797	190	4	start	start	VERB
ejpam-4797	190	5	this	this	DET
ejpam-4797	190	6	section	section	NOUN
ejpam-4797	190	7	with	with	ADP
ejpam-4797	190	8	some	some	DET
ejpam-4797	190	9	fundamental	fundamental	ADJ
ejpam-4797	190	10	properties	property	NOUN
ejpam-4797	190	11	of	of	ADP
ejpam-4797	190	12	a	a	DET
ejpam-4797	190	13	strongly	strongly	ADV
ejpam-4797	190	14	2	2	NUM
ejpam-4797	190	15	-	-	PUNCT
ejpam-4797	190	16	nc	nc	NOUN
ejpam-4797	190	17	ring	ring	NOUN
ejpam-4797	190	18	with	with	ADP
ejpam-4797	190	19	u(r	u(r	NOUN
ejpam-4797	190	20	)	)	PUNCT
ejpam-4797	191	1	=	=	SYM
ejpam-4797	191	2	2	2	X
ejpam-4797	191	3	.	.	X
ejpam-4797	191	4	proposition	proposition	NOUN
ejpam-4797	191	5	5	5	NUM
ejpam-4797	191	6	.	.	PUNCT
ejpam-4797	191	7	homomorphic	homomorphic	ADJ
ejpam-4797	191	8	images	image	NOUN
ejpam-4797	191	9	of	of	ADP
ejpam-4797	191	10	strongly	strongly	ADV
ejpam-4797	191	11	2	2	NUM
ejpam-4797	191	12	-	-	PUNCT
ejpam-4797	191	13	nc	nc	NOUN
ejpam-4797	191	14	ring	ring	NOUN
ejpam-4797	191	15	with	with	ADP
ejpam-4797	191	16	u(r	u(r	NOUN
ejpam-4797	191	17	)	)	PUNCT
ejpam-4797	192	1	=	=	SYM
ejpam-4797	192	2	2	2	NUM
ejpam-4797	192	3	is	be	AUX
ejpam-4797	192	4	again	again	ADV
ejpam-4797	192	5	strongly	strongly	ADV
ejpam-4797	192	6	2	2	NUM
ejpam-4797	192	7	-	-	PUNCT
ejpam-4797	192	8	nc	nc	NOUN
ejpam-4797	192	9	ring	ring	NOUN
ejpam-4797	192	10	with	with	ADP
ejpam-4797	192	11	every	every	DET
ejpam-4797	192	12	unit	unit	NOUN
ejpam-4797	192	13	is	be	AUX
ejpam-4797	192	14	of	of	ADP
ejpam-4797	192	15	order	order	NOUN
ejpam-4797	192	16	2	2	NUM
ejpam-4797	192	17	.	.	PUNCT
ejpam-4797	193	1	proof	proof	NOUN
ejpam-4797	193	2	.	.	PUNCT
ejpam-4797	194	1	let	let	VERB
ejpam-4797	194	2	f	f	NOUN
ejpam-4797	194	3	:	:	PUNCT
ejpam-4797	194	4	r	r	X
ejpam-4797	194	5	→	→	SYM
ejpam-4797	194	6	r′	r′	X
ejpam-4797	194	7	be	be	AUX
ejpam-4797	194	8	a	a	DET
ejpam-4797	194	9	homomorphism	homomorphism	NOUN
ejpam-4797	194	10	from	from	ADP
ejpam-4797	194	11	a	a	DET
ejpam-4797	194	12	strongly	strongly	ADV
ejpam-4797	194	13	2	2	NUM
ejpam-4797	194	14	-	-	PUNCT
ejpam-4797	194	15	nc	nc	NOUN
ejpam-4797	194	16	ring	ring	NOUN
ejpam-4797	194	17	r	r	NOUN
ejpam-4797	194	18	with	with	ADP
ejpam-4797	194	19	u(r	u(r	NOUN
ejpam-4797	194	20	)	)	PUNCT
ejpam-4797	194	21	=	=	SYM
ejpam-4797	194	22	2	2	NUM
ejpam-4797	194	23	onto	onto	ADP
ejpam-4797	194	24	r′.	r′.	NOUN
ejpam-4797	194	25	then	then	ADV
ejpam-4797	194	26	for	for	ADP
ejpam-4797	194	27	any	any	DET
ejpam-4797	194	28	b	b	PROPN
ejpam-4797	194	29	∈	∈	PROPN
ejpam-4797	194	30	r′	r′	PROPN
ejpam-4797	194	31	,	,	PUNCT
ejpam-4797	194	32	there	there	PRON
ejpam-4797	194	33	exists	exist	VERB
ejpam-4797	194	34	a	a	DET
ejpam-4797	194	35	∈	∈	PROPN
ejpam-4797	194	36	r	r	NOUN
ejpam-4797	194	37	,	,	PUNCT
ejpam-4797	194	38	such	such	ADJ
ejpam-4797	194	39	that	that	DET
ejpam-4797	194	40	b	b	X
ejpam-4797	194	41	=	=	SYM
ejpam-4797	194	42	f(a	f(a	PROPN
ejpam-4797	194	43	)	)	PUNCT
ejpam-4797	194	44	,	,	PUNCT
ejpam-4797	194	45	a	a	DET
ejpam-4797	194	46	=	=	X
ejpam-4797	194	47	σ1	σ1	PROPN
ejpam-4797	194	48	+	+	PROPN
ejpam-4797	194	49	σ2	σ2	PROPN
ejpam-4797	194	50	+	+	NOUN
ejpam-4797	194	51	n	n	NOUN
ejpam-4797	194	52	and	and	CCONJ
ejpam-4797	194	53	u(r	u(r	ADJ
ejpam-4797	194	54	)	)	PUNCT
ejpam-4797	195	1	=	=	SYM
ejpam-4797	195	2	2	2	NUM
ejpam-4797	195	3	,	,	PUNCT
ejpam-4797	195	4	where	where	SCONJ
ejpam-4797	195	5	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	195	6	∈	∈	PROPN
ejpam-4797	195	7	id(r	id(r	NOUN
ejpam-4797	195	8	)	)	PUNCT
ejpam-4797	195	9	,	,	PUNCT
ejpam-4797	195	10	n	n	PROPN
ejpam-4797	195	11	∈	∈	PROPN
ejpam-4797	195	12	nil(r	nil(r	PROPN
ejpam-4797	195	13	)	)	PUNCT
ejpam-4797	195	14	that	that	PRON
ejpam-4797	195	15	commute	commute	NOUN
ejpam-4797	195	16	of	of	ADP
ejpam-4797	195	17	with	with	ADP
ejpam-4797	195	18	one	one	NUM
ejpam-4797	195	19	another	another	DET
ejpam-4797	195	20	.	.	PUNCT
ejpam-4797	196	1	now	now	ADV
ejpam-4797	196	2	,	,	PUNCT
ejpam-4797	196	3	b	b	X
ejpam-4797	196	4	=	=	SYM
ejpam-4797	196	5	f(a	f(a	NOUN
ejpam-4797	196	6	)	)	PUNCT
ejpam-4797	196	7	=	=	SYM
ejpam-4797	197	1	f(σ1	f(σ1	NOUN
ejpam-4797	197	2	+	+	CCONJ
ejpam-4797	197	3	σ2	σ2	NOUN
ejpam-4797	197	4	+	+	CCONJ
ejpam-4797	197	5	n	n	CCONJ
ejpam-4797	197	6	)	)	PUNCT
ejpam-4797	197	7	=	=	SYM
ejpam-4797	197	8	f(σ1	f(σ1	NOUN
ejpam-4797	197	9	)	)	PUNCT
ejpam-4797	197	10	+	+	NUM
ejpam-4797	197	11	f(σ2	f(σ2	NOUN
ejpam-4797	197	12	)	)	PUNCT
ejpam-4797	197	13	+	+	NUM
ejpam-4797	197	14	f(n	f(n	PROPN
ejpam-4797	197	15	)	)	PUNCT
ejpam-4797	197	16	.	.	PUNCT
ejpam-4797	198	1	clearly	clearly	ADV
ejpam-4797	198	2	,	,	PUNCT
ejpam-4797	198	3	f(σ1	f(σ1	NOUN
ejpam-4797	198	4	)	)	PUNCT
ejpam-4797	198	5	,	,	PUNCT
ejpam-4797	198	6	f(σ2	f(σ2	NOUN
ejpam-4797	198	7	)	)	PUNCT
ejpam-4797	198	8	∈	∈	PROPN
ejpam-4797	198	9	id(r′	id(r′	PROPN
ejpam-4797	198	10	)	)	PUNCT
ejpam-4797	198	11	and	and	CCONJ
ejpam-4797	198	12	f(n	f(n	PROPN
ejpam-4797	198	13	)	)	PUNCT
ejpam-4797	198	14	∈	∈	PROPN
ejpam-4797	198	15	nil(r′	nil(r′	PROPN
ejpam-4797	198	16	)	)	PUNCT
ejpam-4797	198	17	.	.	PUNCT
ejpam-4797	199	1	on	on	ADP
ejpam-4797	199	2	the	the	DET
ejpam-4797	199	3	other	other	ADJ
ejpam-4797	199	4	hand	hand	NOUN
ejpam-4797	199	5	for	for	ADP
ejpam-4797	199	6	any	any	DET
ejpam-4797	199	7	u	u	NOUN
ejpam-4797	199	8	∈	∈	PROPN
ejpam-4797	199	9	(	(	PUNCT
ejpam-4797	199	10	r	r	NOUN
ejpam-4797	199	11	)	)	PUNCT
ejpam-4797	199	12	,	,	PUNCT
ejpam-4797	199	13	where	where	SCONJ
ejpam-4797	199	14	u	u	NOUN
ejpam-4797	199	15	is	be	AUX
ejpam-4797	199	16	a	a	DET
ejpam-4797	199	17	unit	unit	NOUN
ejpam-4797	199	18	,	,	PUNCT
ejpam-4797	199	19	(	(	PUNCT
ejpam-4797	199	20	f(u))2	f(u))2	PROPN
ejpam-4797	199	21	=	=	SYM
ejpam-4797	199	22	f(u2	f(u2	NOUN
ejpam-4797	199	23	)	)	PUNCT
ejpam-4797	199	24	=	=	PUNCT
ejpam-4797	199	25	f(1	f(1	PROPN
ejpam-4797	199	26	)	)	PUNCT
ejpam-4797	199	27	,	,	PUNCT
ejpam-4797	199	28	this	this	PRON
ejpam-4797	199	29	shows	show	VERB
ejpam-4797	199	30	that	that	SCONJ
ejpam-4797	199	31	f(u	f(u	PROPN
ejpam-4797	199	32	)	)	PUNCT
ejpam-4797	199	33	is	be	AUX
ejpam-4797	199	34	a	a	DET
ejpam-4797	199	35	unit	unit	NOUN
ejpam-4797	199	36	of	of	ADP
ejpam-4797	199	37	order	order	NOUN
ejpam-4797	199	38	2	2	X
ejpam-4797	199	39	.	.	PUNCT
ejpam-4797	199	40	therefore	therefore	ADV
ejpam-4797	199	41	r′	r′	PROPN
ejpam-4797	199	42	is	be	AUX
ejpam-4797	199	43	a	a	DET
ejpam-4797	199	44	strongly	strongly	ADV
ejpam-4797	199	45	2	2	NUM
ejpam-4797	199	46	-	-	PUNCT
ejpam-4797	199	47	nc	nc	NOUN
ejpam-4797	199	48	ring	ring	NOUN
ejpam-4797	199	49	with	with	ADP
ejpam-4797	199	50	u(r′	u(r′	NUM
ejpam-4797	199	51	)	)	PUNCT
ejpam-4797	199	52	=	=	SYM
ejpam-4797	199	53	2	2	X
ejpam-4797	199	54	.	.	X
ejpam-4797	199	55	proposition	proposition	NOUN
ejpam-4797	199	56	6	6	NUM
ejpam-4797	199	57	.	.	PUNCT
ejpam-4797	200	1	if	if	SCONJ
ejpam-4797	200	2	r	r	NOUN
ejpam-4797	200	3	is	be	AUX
ejpam-4797	200	4	a	a	DET
ejpam-4797	200	5	strongly	strongly	ADV
ejpam-4797	200	6	2	2	NUM
ejpam-4797	200	7	-	-	PUNCT
ejpam-4797	200	8	nc	nc	NOUN
ejpam-4797	200	9	ring	ring	NOUN
ejpam-4797	200	10	with	with	ADP
ejpam-4797	200	11	u(r	u(r	NOUN
ejpam-4797	200	12	)	)	PUNCT
ejpam-4797	201	1	=	=	SYM
ejpam-4797	201	2	2	2	X
ejpam-4797	201	3	.	.	X
ejpam-4797	201	4	then	then	ADV
ejpam-4797	201	5	24	24	NUM
ejpam-4797	201	6	=	=	SYM
ejpam-4797	201	7	0	0	NUM
ejpam-4797	201	8	.	.	PUNCT
ejpam-4797	202	1	proof	proof	NOUN
ejpam-4797	202	2	.	.	PUNCT
ejpam-4797	203	1	assume	assume	VERB
ejpam-4797	203	2	that	that	SCONJ
ejpam-4797	203	3	a	a	PRON
ejpam-4797	203	4	in	in	ADP
ejpam-4797	203	5	r	r	NOUN
ejpam-4797	203	6	,	,	PUNCT
ejpam-4797	203	7	then	then	ADV
ejpam-4797	203	8	existing	exist	VERB
ejpam-4797	203	9	two	two	NUM
ejpam-4797	203	10	idempotents	idempotent	NOUN
ejpam-4797	203	11	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	203	12	and	and	CCONJ
ejpam-4797	203	13	a	a	DET
ejpam-4797	203	14	nilpotent	nilpotent	NOUN
ejpam-4797	203	15	n	n	PRON
ejpam-4797	203	16	that	that	DET
ejpam-4797	203	17	commute	commute	VERB
ejpam-4797	203	18	with	with	ADP
ejpam-4797	203	19	one	one	NUM
ejpam-4797	203	20	another	another	DET
ejpam-4797	203	21	,	,	PUNCT
ejpam-4797	203	22	such	such	ADJ
ejpam-4797	203	23	that	that	SCONJ
ejpam-4797	203	24	a	a	DET
ejpam-4797	203	25	=	=	PROPN
ejpam-4797	203	26	σ1	σ1	PROPN
ejpam-4797	203	27	−	−	PROPN
ejpam-4797	203	28	σ2	σ2	PROPN
ejpam-4797	203	29	+	+	CCONJ
ejpam-4797	203	30	n.	n.	NOUN
ejpam-4797	203	31	by	by	ADP
ejpam-4797	203	32	theorem	theorem	NOUN
ejpam-4797	203	33	1	1	NUM
ejpam-4797	203	34	,	,	PUNCT
ejpam-4797	203	35	a3	a3	VERB
ejpam-4797	203	36	−	−	PROPN
ejpam-4797	203	37	a	a	DET
ejpam-4797	203	38	∈	∈	PROPN
ejpam-4797	203	39	nil(r	nil(r	NOUN
ejpam-4797	203	40	)	)	PUNCT
ejpam-4797	203	41	,	,	PUNCT
ejpam-4797	203	42	this	this	PRON
ejpam-4797	203	43	gives	give	VERB
ejpam-4797	203	44	23	23	NUM
ejpam-4797	203	45	−	−	NUM
ejpam-4797	203	46	2	2	NUM
ejpam-4797	203	47	=	=	SYM
ejpam-4797	203	48	6	6	NUM
ejpam-4797	203	49	∈	∈	NOUN
ejpam-4797	203	50	nil(r	nil(r	NOUN
ejpam-4797	203	51	)	)	PUNCT
ejpam-4797	203	52	.	.	PUNCT
ejpam-4797	204	1	since	since	SCONJ
ejpam-4797	204	2	every	every	DET
ejpam-4797	204	3	unit	unit	NOUN
ejpam-4797	204	4	is	be	AUX
ejpam-4797	204	5	of	of	ADP
ejpam-4797	204	6	order	order	NOUN
ejpam-4797	204	7	2	2	NUM
ejpam-4797	204	8	,	,	PUNCT
ejpam-4797	204	9	and	and	CCONJ
ejpam-4797	204	10	since	since	SCONJ
ejpam-4797	204	11	6	6	NUM
ejpam-4797	204	12	is	be	AUX
ejpam-4797	204	13	nilpotent	nilpotent	ADJ
ejpam-4797	204	14	,	,	PUNCT
ejpam-4797	204	15	then	then	ADV
ejpam-4797	204	16	6−	6−	NUM
ejpam-4797	204	17	1	1	NUM
ejpam-4797	204	18	=	=	SYM
ejpam-4797	204	19	5	5	NUM
ejpam-4797	204	20	∈	∈	PROPN
ejpam-4797	204	21	u(r	u(r	NOUN
ejpam-4797	204	22	)	)	PUNCT
ejpam-4797	204	23	.	.	PUNCT
ejpam-4797	205	1	this	this	PRON
ejpam-4797	205	2	gives	give	VERB
ejpam-4797	205	3	52	52	NUM
ejpam-4797	205	4	=	=	SYM
ejpam-4797	205	5	1	1	NUM
ejpam-4797	205	6	,	,	PUNCT
ejpam-4797	205	7	so	so	ADV
ejpam-4797	205	8	24	24	NUM
ejpam-4797	205	9	=	=	SYM
ejpam-4797	205	10	0	0	PROPN
ejpam-4797	205	11	.	.	NOUN
ejpam-4797	205	12	example	example	NOUN
ejpam-4797	206	1	3	3	X
ejpam-4797	206	2	.	.	X
ejpam-4797	206	3	consider	consider	VERB
ejpam-4797	206	4	the	the	DET
ejpam-4797	206	5	ring	ring	NOUN
ejpam-4797	206	6	z24	z24	PROPN
ejpam-4797	206	7	.	.	PUNCT
ejpam-4797	207	1	clearly	clearly	ADV
ejpam-4797	207	2	,	,	PUNCT
ejpam-4797	207	3	z24	z24	PROPN
ejpam-4797	207	4	is	be	AUX
ejpam-4797	207	5	a	a	DET
ejpam-4797	207	6	strongly	strongly	ADV
ejpam-4797	207	7	2	2	NUM
ejpam-4797	207	8	-	-	PUNCT
ejpam-4797	207	9	nc	nc	NOUN
ejpam-4797	207	10	,	,	PUNCT
ejpam-4797	207	11	with	with	ADP
ejpam-4797	207	12	u(z24	u(z24	NOUN
ejpam-4797	207	13	)	)	PUNCT
ejpam-4797	207	14	=	=	SYM
ejpam-4797	207	15	{	{	PUNCT
ejpam-4797	207	16	1	1	NUM
ejpam-4797	207	17	,	,	PUNCT
ejpam-4797	207	18	5	5	NUM
ejpam-4797	207	19	,	,	PUNCT
ejpam-4797	207	20	7	7	NUM
ejpam-4797	207	21	,	,	PUNCT
ejpam-4797	207	22	11	11	NUM
ejpam-4797	207	23	,	,	PUNCT
ejpam-4797	207	24	13	13	NUM
ejpam-4797	207	25	,	,	PUNCT
ejpam-4797	207	26	17	17	NUM
ejpam-4797	207	27	,	,	PUNCT
ejpam-4797	207	28	19	19	NUM
ejpam-4797	207	29	,	,	PUNCT
ejpam-4797	207	30	23	23	NUM
ejpam-4797	207	31	}	}	PUNCT
ejpam-4797	207	32	.	.	PUNCT
ejpam-4797	208	1	observe	observe	VERB
ejpam-4797	208	2	that	that	SCONJ
ejpam-4797	208	3	12	12	NUM
ejpam-4797	208	4	=	=	SYM
ejpam-4797	208	5	52	52	NUM
ejpam-4797	208	6	=	=	SYM
ejpam-4797	208	7	72	72	NUM
ejpam-4797	208	8	=	=	SYM
ejpam-4797	208	9	112	112	NUM
ejpam-4797	208	10	=	=	SYM
ejpam-4797	208	11	132	132	NUM
ejpam-4797	208	12	=	=	SYM
ejpam-4797	208	13	172	172	NUM
ejpam-4797	208	14	=	=	SYM
ejpam-4797	208	15	192	192	NUM
ejpam-4797	208	16	=	=	SYM
ejpam-4797	208	17	232	232	NUM
ejpam-4797	208	18	=	=	SYM
ejpam-4797	208	19	1	1	X
ejpam-4797	208	20	.	.	X
ejpam-4797	208	21	observe	observe	VERB
ejpam-4797	208	22	that	that	SCONJ
ejpam-4797	208	23	every	every	DET
ejpam-4797	208	24	snc	snc	NOUN
ejpam-4797	208	25	ring	ring	NOUN
ejpam-4797	208	26	is	be	AUX
ejpam-4797	208	27	strongly	strongly	ADV
ejpam-4797	208	28	2	2	NUM
ejpam-4797	208	29	-	-	PUNCT
ejpam-4797	208	30	nc	nc	NOUN
ejpam-4797	208	31	,	,	PUNCT
ejpam-4797	208	32	but	but	CCONJ
ejpam-4797	208	33	not	not	PART
ejpam-4797	208	34	every	every	DET
ejpam-4797	208	35	unit	unit	NOUN
ejpam-4797	208	36	of	of	ADP
ejpam-4797	208	37	order	order	NOUN
ejpam-4797	208	38	2	2	NUM
ejpam-4797	208	39	.	.	NOUN
ejpam-4797	208	40	example	example	NOUN
ejpam-4797	209	1	4	4	NUM
ejpam-4797	209	2	.	.	PUNCT
ejpam-4797	210	1	the	the	DET
ejpam-4797	210	2	ring	ring	NOUN
ejpam-4797	210	3	z16	z16	PROPN
ejpam-4797	210	4	is	be	AUX
ejpam-4797	210	5	an	an	DET
ejpam-4797	210	6	snc	snc	NOUN
ejpam-4797	210	7	which	which	PRON
ejpam-4797	210	8	is	be	AUX
ejpam-4797	210	9	strongly	strongly	ADV
ejpam-4797	210	10	2	2	NUM
ejpam-4797	210	11	-	-	PUNCT
ejpam-4797	210	12	nc	nc	NOUN
ejpam-4797	210	13	,	,	PUNCT
ejpam-4797	210	14	but	but	CCONJ
ejpam-4797	210	15	z16	z16	NOUN
ejpam-4797	210	16	is	be	AUX
ejpam-4797	210	17	not	not	PART
ejpam-4797	210	18	strongly	strongly	ADV
ejpam-4797	210	19	2	2	NUM
ejpam-4797	210	20	-	-	PUNCT
ejpam-4797	210	21	nc	nc	NOUN
ejpam-4797	210	22	ring	ring	NOUN
ejpam-4797	210	23	with	with	ADP
ejpam-4797	210	24	u(r	u(r	NOUN
ejpam-4797	210	25	)	)	PUNCT
ejpam-4797	211	1	=	=	SYM
ejpam-4797	211	2	2	2	NUM
ejpam-4797	211	3	,	,	PUNCT
ejpam-4797	211	4	since	since	SCONJ
ejpam-4797	211	5	the	the	DET
ejpam-4797	211	6	units	unit	NOUN
ejpam-4797	211	7	3	3	NUM
ejpam-4797	211	8	,	,	PUNCT
ejpam-4797	211	9	5	5	NUM
ejpam-4797	211	10	,	,	PUNCT
ejpam-4797	211	11	11	11	NUM
ejpam-4797	211	12	,	,	PUNCT
ejpam-4797	211	13	13	13	NUM
ejpam-4797	211	14	are	be	AUX
ejpam-4797	211	15	not	not	PART
ejpam-4797	211	16	of	of	ADP
ejpam-4797	211	17	order	order	NOUN
ejpam-4797	211	18	2	2	X
ejpam-4797	211	19	.	.	X
ejpam-4797	211	20	note	note	VERB
ejpam-4797	211	21	that	that	SCONJ
ejpam-4797	211	22	:	:	PUNCT
ejpam-4797	211	23	if	if	SCONJ
ejpam-4797	211	24	r	r	NOUN
ejpam-4797	211	25	is	be	AUX
ejpam-4797	211	26	a	a	DET
ejpam-4797	211	27	strongly	strongly	ADV
ejpam-4797	211	28	2	2	NUM
ejpam-4797	211	29	-	-	PUNCT
ejpam-4797	211	30	nc	nc	NOUN
ejpam-4797	211	31	with	with	ADP
ejpam-4797	211	32	u(r	u(r	NOUN
ejpam-4797	211	33	)	)	PUNCT
ejpam-4797	212	1	=	=	SYM
ejpam-4797	212	2	2	2	NUM
ejpam-4797	212	3	,	,	PUNCT
ejpam-4797	212	4	then	then	ADV
ejpam-4797	212	5	r	r	NOUN
ejpam-4797	212	6	need	need	AUX
ejpam-4797	212	7	not	not	PART
ejpam-4797	212	8	to	to	PART
ejpam-4797	212	9	be	be	AUX
ejpam-4797	212	10	snc	snc	PROPN
ejpam-4797	212	11	ring	ring	NOUN
ejpam-4797	212	12	.	.	PUNCT
ejpam-4797	212	13	example	example	NOUN
ejpam-4797	213	1	5	5	NUM
ejpam-4797	213	2	.	.	PUNCT
ejpam-4797	214	1	in	in	ADP
ejpam-4797	214	2	the	the	DET
ejpam-4797	214	3	ring	ring	NOUN
ejpam-4797	214	4	z12	z12	PROPN
ejpam-4797	214	5	.	.	PUNCT
ejpam-4797	215	1	then	then	ADV
ejpam-4797	215	2	u(z12	u(z12	NOUN
ejpam-4797	215	3	)	)	PUNCT
ejpam-4797	215	4	=	=	SYM
ejpam-4797	215	5	{	{	PUNCT
ejpam-4797	215	6	1	1	NUM
ejpam-4797	215	7	,	,	PUNCT
ejpam-4797	215	8	5	5	NUM
ejpam-4797	215	9	,	,	PUNCT
ejpam-4797	215	10	7	7	NUM
ejpam-4797	215	11	,	,	PUNCT
ejpam-4797	215	12	11	11	NUM
ejpam-4797	215	13	}	}	PUNCT
ejpam-4797	215	14	and	and	CCONJ
ejpam-4797	215	15	12	12	NUM
ejpam-4797	215	16	=	=	SYM
ejpam-4797	215	17	52	52	NUM
ejpam-4797	215	18	=	=	SYM
ejpam-4797	215	19	72	72	NUM
ejpam-4797	215	20	=	=	SYM
ejpam-4797	215	21	112	112	NUM
ejpam-4797	215	22	=	=	SYM
ejpam-4797	215	23	1	1	X
ejpam-4797	215	24	.	.	PUNCT
ejpam-4797	215	25	clearly	clearly	ADV
ejpam-4797	215	26	,	,	PUNCT
ejpam-4797	215	27	z12	z12	PROPN
ejpam-4797	215	28	is	be	AUX
ejpam-4797	215	29	a	a	DET
ejpam-4797	215	30	strongly	strongly	ADV
ejpam-4797	215	31	2	2	NUM
ejpam-4797	215	32	-	-	PUNCT
ejpam-4797	215	33	nc	nc	NOUN
ejpam-4797	215	34	with	with	ADP
ejpam-4797	215	35	u(z12	u(z12	NOUN
ejpam-4797	215	36	)	)	PUNCT
ejpam-4797	215	37	=	=	SYM
ejpam-4797	215	38	2	2	NUM
ejpam-4797	215	39	,	,	PUNCT
ejpam-4797	215	40	but	but	CCONJ
ejpam-4797	215	41	(	(	PUNCT
ejpam-4797	215	42	z12	z12	NOUN
ejpam-4797	215	43	)	)	PUNCT
ejpam-4797	215	44	is	be	AUX
ejpam-4797	215	45	not	not	PART
ejpam-4797	215	46	snc	snc	PROPN
ejpam-4797	215	47	ring	ring	NOUN
ejpam-4797	215	48	.	.	PUNCT
ejpam-4797	216	1	since	since	SCONJ
ejpam-4797	216	2	2	2	NUM
ejpam-4797	216	3	is	be	AUX
ejpam-4797	216	4	not	not	PART
ejpam-4797	216	5	snc	snc	NOUN
ejpam-4797	216	6	element	element	NOUN
ejpam-4797	216	7	.	.	PUNCT
ejpam-4797	217	1	r.	r.	PROPN
ejpam-4797	217	2	t.	t.	PROPN
ejpam-4797	217	3	m.salim	m.salim	PROPN
ejpam-4797	217	4	,	,	PUNCT
ejpam-4797	217	5	n.	n.	PROPN
ejpam-4797	217	6	h.	h.	PROPN
ejpam-4797	217	7	shuker	shuker	PROPN
ejpam-4797	217	8	/	/	SYM
ejpam-4797	217	9	eur	eur	PROPN
ejpam-4797	217	10	.	.	PUNCT
ejpam-4797	218	1	j.	j.	PROPN
ejpam-4797	218	2	pure	pure	PROPN
ejpam-4797	218	3	appl	appl	PROPN
ejpam-4797	218	4	.	.	PROPN
ejpam-4797	218	5	math	math	PROPN
ejpam-4797	218	6	,	,	PUNCT
ejpam-4797	218	7	16	16	NUM
ejpam-4797	218	8	(	(	PUNCT
ejpam-4797	218	9	3	3	NUM
ejpam-4797	218	10	)	)	PUNCT
ejpam-4797	218	11	(	(	PUNCT
ejpam-4797	218	12	2023	2023	NUM
ejpam-4797	218	13	)	)	PUNCT
ejpam-4797	218	14	,	,	PUNCT
ejpam-4797	218	15	1675	1675	NUM
ejpam-4797	218	16	-	-	SYM
ejpam-4797	218	17	1684	1684	NUM
ejpam-4797	218	18	1681	1681	NUM
ejpam-4797	218	19	proposition	proposition	NOUN
ejpam-4797	218	20	7	7	NUM
ejpam-4797	218	21	.	.	PUNCT
ejpam-4797	219	1	if	if	SCONJ
ejpam-4797	219	2	a	a	DET
ejpam-4797	219	3	ring	ring	NOUN
ejpam-4797	219	4	r	r	NOUN
ejpam-4797	219	5	is	be	AUX
ejpam-4797	219	6	a	a	DET
ejpam-4797	219	7	strongly	strongly	ADV
ejpam-4797	219	8	2	2	NUM
ejpam-4797	219	9	-	-	PUNCT
ejpam-4797	219	10	nc	nc	NOUN
ejpam-4797	219	11	ring	ring	NOUN
ejpam-4797	219	12	with	with	ADP
ejpam-4797	219	13	u(r	u(r	NOUN
ejpam-4797	219	14	)	)	PUNCT
ejpam-4797	220	1	=	=	SYM
ejpam-4797	220	2	2	2	NUM
ejpam-4797	220	3	,	,	PUNCT
ejpam-4797	220	4	for	for	ADP
ejpam-4797	220	5	which	which	PRON
ejpam-4797	220	6	3	3	NUM
ejpam-4797	220	7	∈	∈	PROPN
ejpam-4797	220	8	u(r	u(r	NOUN
ejpam-4797	220	9	)	)	PUNCT
ejpam-4797	220	10	,	,	PUNCT
ejpam-4797	220	11	then	then	ADV
ejpam-4797	220	12	r	r	NOUN
ejpam-4797	220	13	is	be	AUX
ejpam-4797	220	14	snc	snc	PROPN
ejpam-4797	220	15	ring	ring	NOUN
ejpam-4797	220	16	of	of	ADP
ejpam-4797	220	17	characteristic	characteristic	ADJ
ejpam-4797	220	18	8	8	NUM
ejpam-4797	220	19	.	.	PUNCT
ejpam-4797	220	20	proof	proof	NOUN
ejpam-4797	220	21	.	.	PUNCT
ejpam-4797	221	1	assume	assume	VERB
ejpam-4797	221	2	r	r	NOUN
ejpam-4797	221	3	is	be	AUX
ejpam-4797	221	4	a	a	DET
ejpam-4797	221	5	strongly	strongly	ADV
ejpam-4797	221	6	2	2	NUM
ejpam-4797	221	7	-	-	PUNCT
ejpam-4797	221	8	nc	nc	NOUN
ejpam-4797	221	9	ring	ring	NOUN
ejpam-4797	221	10	with	with	ADP
ejpam-4797	221	11	u(r	u(r	NOUN
ejpam-4797	221	12	)	)	PUNCT
ejpam-4797	222	1	=	=	SYM
ejpam-4797	222	2	2	2	X
ejpam-4797	222	3	.	.	PUNCT
ejpam-4797	222	4	then	then	ADV
ejpam-4797	222	5	by	by	ADP
ejpam-4797	222	6	proposition	proposition	NOUN
ejpam-4797	222	7	1(1	1(1	NUM
ejpam-4797	222	8	)	)	PUNCT
ejpam-4797	222	9	,	,	PUNCT
ejpam-4797	222	10	a2	a2	PROPN
ejpam-4797	222	11	is	be	AUX
ejpam-4797	222	12	a	a	DET
ejpam-4797	222	13	snc	snc	NOUN
ejpam-4797	222	14	element	element	NOUN
ejpam-4797	222	15	for	for	ADP
ejpam-4797	222	16	every	every	PRON
ejpam-4797	222	17	a	a	PRON
ejpam-4797	222	18	∈	∈	PROPN
ejpam-4797	222	19	r.	r.	NOUN
ejpam-4797	222	20	applying	apply	VERB
ejpam-4797	222	21	proposition	proposition	NOUN
ejpam-4797	222	22	2(1	2(1	NUM
ejpam-4797	222	23	)	)	PUNCT
ejpam-4797	222	24	,	,	PUNCT
ejpam-4797	222	25	a	a	PRON
ejpam-4797	222	26	is	be	AUX
ejpam-4797	222	27	strongly	strongly	ADV
ejpam-4797	222	28	clean	clean	ADJ
ejpam-4797	222	29	.	.	PUNCT
ejpam-4797	223	1	then	then	ADV
ejpam-4797	223	2	a	a	PRON
ejpam-4797	223	3	may	may	AUX
ejpam-4797	223	4	be	be	AUX
ejpam-4797	223	5	written	write	VERB
ejpam-4797	223	6	a	a	DET
ejpam-4797	223	7	−	−	PROPN
ejpam-4797	223	8	1	1	NUM
ejpam-4797	223	9	=	=	SYM
ejpam-4797	223	10	σ	σ	PROPN
ejpam-4797	223	11	+	+	NUM
ejpam-4797	223	12	u	u	NOUN
ejpam-4797	223	13	,	,	PUNCT
ejpam-4797	223	14	where	where	SCONJ
ejpam-4797	223	15	σ	σ	PROPN
ejpam-4797	223	16	∈	∈	PROPN
ejpam-4797	223	17	id(r	id(r	NOUN
ejpam-4797	223	18	)	)	PUNCT
ejpam-4797	223	19	and	and	CCONJ
ejpam-4797	223	20	u2	u2	NOUN
ejpam-4797	223	21	=	=	NOUN
ejpam-4797	223	22	1	1	NUM
ejpam-4797	223	23	.	.	PUNCT
ejpam-4797	224	1	then	then	ADV
ejpam-4797	224	2	a	a	DET
ejpam-4797	224	3	=	=	SYM
ejpam-4797	224	4	σ	σ	PROPN
ejpam-4797	224	5	+	+	NUM
ejpam-4797	224	6	u	u	NOUN
ejpam-4797	224	7	+	+	NOUN
ejpam-4797	224	8	1	1	NUM
ejpam-4797	224	9	.	.	PUNCT
ejpam-4797	225	1	since	since	SCONJ
ejpam-4797	225	2	3	3	NUM
ejpam-4797	225	3	∈	∈	PROPN
ejpam-4797	225	4	u(r	u(r	NOUN
ejpam-4797	225	5	)	)	PUNCT
ejpam-4797	225	6	,	,	PUNCT
ejpam-4797	225	7	then	then	ADV
ejpam-4797	225	8	32	32	NUM
ejpam-4797	225	9	=	=	SYM
ejpam-4797	225	10	1	1	NUM
ejpam-4797	225	11	,	,	PUNCT
ejpam-4797	225	12	gives	give	VERB
ejpam-4797	225	13	8	8	NUM
ejpam-4797	225	14	=	=	SYM
ejpam-4797	225	15	0	0	NUM
ejpam-4797	225	16	thus	thus	ADV
ejpam-4797	225	17	,	,	PUNCT
ejpam-4797	225	18	2	2	NUM
ejpam-4797	225	19	∈	∈	NOUN
ejpam-4797	225	20	nil(r	nil(r	NOUN
ejpam-4797	225	21	)	)	PUNCT
ejpam-4797	225	22	.	.	PUNCT
ejpam-4797	226	1	so	so	ADV
ejpam-4797	226	2	(	(	PUNCT
ejpam-4797	226	3	u	u	NOUN
ejpam-4797	226	4	+	+	X
ejpam-4797	226	5	1)2	1)2	NUM
ejpam-4797	226	6	=	=	SYM
ejpam-4797	226	7	u2	u2	NOUN
ejpam-4797	226	8	+	+	CCONJ
ejpam-4797	226	9	2u	2u	NOUN
ejpam-4797	226	10	+	+	CCONJ
ejpam-4797	226	11	1	1	NUM
ejpam-4797	226	12	=	=	SYM
ejpam-4797	226	13	2(u+	2(u+	NUM
ejpam-4797	226	14	1	1	NUM
ejpam-4797	226	15	)	)	PUNCT
ejpam-4797	226	16	∈	∈	PROPN
ejpam-4797	226	17	nil(r	nil(r	PROPN
ejpam-4797	226	18	)	)	PUNCT
ejpam-4797	226	19	.	.	PUNCT
ejpam-4797	227	1	thus	thus	ADV
ejpam-4797	227	2	,	,	PUNCT
ejpam-4797	227	3	u+	u+	NUM
ejpam-4797	227	4	1	1	NUM
ejpam-4797	227	5	∈	∈	NOUN
ejpam-4797	227	6	nil(r	nil(r	NOUN
ejpam-4797	227	7	)	)	PUNCT
ejpam-4797	227	8	.	.	PUNCT
ejpam-4797	228	1	therefore	therefore	ADV
ejpam-4797	228	2	r	r	NOUN
ejpam-4797	228	3	is	be	AUX
ejpam-4797	228	4	an	an	DET
ejpam-4797	228	5	snc	snc	PROPN
ejpam-4797	228	6	ring	ring	NOUN
ejpam-4797	228	7	.	.	PUNCT
ejpam-4797	228	8	example	example	NOUN
ejpam-4797	229	1	6	6	NUM
ejpam-4797	229	2	.	.	PUNCT
ejpam-4797	229	3	consider	consider	VERB
ejpam-4797	229	4	the	the	DET
ejpam-4797	229	5	ring	ring	NOUN
ejpam-4797	229	6	z8	z8	PROPN
ejpam-4797	229	7	.	.	PUNCT
ejpam-4797	230	1	then	then	ADV
ejpam-4797	230	2	u(z8	u(z8	NOUN
ejpam-4797	230	3	)	)	PUNCT
ejpam-4797	230	4	=	=	PRON
ejpam-4797	230	5	{	{	PUNCT
ejpam-4797	230	6	1	1	NUM
ejpam-4797	230	7	,	,	PUNCT
ejpam-4797	230	8	3	3	NUM
ejpam-4797	230	9	,	,	PUNCT
ejpam-4797	230	10	5	5	NUM
ejpam-4797	230	11	,	,	PUNCT
ejpam-4797	230	12	7	7	NUM
ejpam-4797	230	13	}	}	PUNCT
ejpam-4797	230	14	.	.	PUNCT
ejpam-4797	231	1	so	so	ADV
ejpam-4797	231	2	12	12	NUM
ejpam-4797	231	3	=	=	SYM
ejpam-4797	231	4	32	32	NUM
ejpam-4797	231	5	=	=	SYM
ejpam-4797	231	6	52	52	NUM
ejpam-4797	231	7	=	=	SYM
ejpam-4797	231	8	72	72	NUM
ejpam-4797	231	9	=	=	SYM
ejpam-4797	231	10	1	1	X
ejpam-4797	231	11	.	.	PUNCT
ejpam-4797	231	12	clearly	clearly	ADV
ejpam-4797	231	13	,	,	PUNCT
ejpam-4797	231	14	z8	z8	NOUN
ejpam-4797	231	15	is	be	AUX
ejpam-4797	231	16	a	a	DET
ejpam-4797	231	17	strongly	strongly	ADV
ejpam-4797	231	18	2	2	NUM
ejpam-4797	231	19	-	-	PUNCT
ejpam-4797	231	20	nc	nc	NOUN
ejpam-4797	231	21	with	with	ADP
ejpam-4797	231	22	u(z8	u(z8	NOUN
ejpam-4797	231	23	)	)	PUNCT
ejpam-4797	231	24	=	=	SYM
ejpam-4797	231	25	2	2	X
ejpam-4797	231	26	.	.	X
ejpam-4797	231	27	observe	observe	VERB
ejpam-4797	231	28	that	that	SCONJ
ejpam-4797	231	29	3	3	NUM
ejpam-4797	231	30	∈	∈	PROPN
ejpam-4797	231	31	u(z8	u(z8	NOUN
ejpam-4797	231	32	)	)	PUNCT
ejpam-4797	231	33	.	.	PUNCT
ejpam-4797	232	1	then	then	ADV
ejpam-4797	232	2	z8	z8	NOUN
ejpam-4797	232	3	is	be	AUX
ejpam-4797	232	4	an	an	DET
ejpam-4797	232	5	snc	snc	PROPN
ejpam-4797	232	6	ring	ring	NOUN
ejpam-4797	232	7	.	.	PUNCT
ejpam-4797	233	1	it	it	PRON
ejpam-4797	233	2	was	be	AUX
ejpam-4797	233	3	proved	prove	VERB
ejpam-4797	233	4	in	in	ADP
ejpam-4797	233	5	theorem	theorem	NOUN
ejpam-4797	233	6	2	2	NUM
ejpam-4797	233	7	,	,	PUNCT
ejpam-4797	233	8	if	if	SCONJ
ejpam-4797	233	9	a	a	DET
ejpam-4797	233	10	ring	ring	NOUN
ejpam-4797	233	11	r	r	NOUN
ejpam-4797	233	12	is	be	AUX
ejpam-4797	233	13	a	a	DET
ejpam-4797	233	14	strongly	strongly	ADV
ejpam-4797	233	15	2	2	NUM
ejpam-4797	233	16	-	-	PUNCT
ejpam-4797	233	17	nc	nc	NOUN
ejpam-4797	233	18	,	,	PUNCT
ejpam-4797	233	19	then	then	ADV
ejpam-4797	233	20	j(r	j(r	PROPN
ejpam-4797	233	21	)	)	PUNCT
ejpam-4797	233	22	is	be	AUX
ejpam-4797	233	23	nil	nil	ADJ
ejpam-4797	233	24	.	.	PUNCT
ejpam-4797	234	1	in	in	ADP
ejpam-4797	234	2	the	the	DET
ejpam-4797	234	3	next	next	ADJ
ejpam-4797	234	4	result	result	NOUN
ejpam-4797	234	5	,	,	PUNCT
ejpam-4797	234	6	we	we	PRON
ejpam-4797	234	7	consider	consider	VERB
ejpam-4797	234	8	j(r	j(r	NOUN
ejpam-4797	234	9	)	)	PUNCT
ejpam-4797	234	10	over	over	ADP
ejpam-4797	234	11	a	a	DET
ejpam-4797	234	12	strongly	strongly	ADV
ejpam-4797	234	13	2	2	NUM
ejpam-4797	234	14	-	-	PUNCT
ejpam-4797	234	15	nc	nc	NOUN
ejpam-4797	234	16	ring	ring	NOUN
ejpam-4797	234	17	with	with	ADP
ejpam-4797	234	18	u(r	u(r	NOUN
ejpam-4797	234	19	)	)	PUNCT
ejpam-4797	235	1	=	=	SYM
ejpam-4797	235	2	2	2	X
ejpam-4797	235	3	.	.	X
ejpam-4797	235	4	theorem	theorem	NOUN
ejpam-4797	235	5	5	5	NUM
ejpam-4797	235	6	.	.	PUNCT
ejpam-4797	236	1	if	if	SCONJ
ejpam-4797	236	2	r	r	NOUN
ejpam-4797	236	3	is	be	AUX
ejpam-4797	236	4	a	a	DET
ejpam-4797	236	5	strongly	strongly	ADV
ejpam-4797	236	6	2	2	NUM
ejpam-4797	236	7	-	-	PUNCT
ejpam-4797	236	8	nc	nc	NOUN
ejpam-4797	236	9	ring	ring	NOUN
ejpam-4797	236	10	with	with	ADP
ejpam-4797	236	11	u(r	u(r	NOUN
ejpam-4797	236	12	)	)	PUNCT
ejpam-4797	237	1	=	=	SYM
ejpam-4797	237	2	2	2	NUM
ejpam-4797	237	3	,	,	PUNCT
ejpam-4797	237	4	then	then	ADV
ejpam-4797	237	5	j(r	j(r	PROPN
ejpam-4797	237	6	)	)	PUNCT
ejpam-4797	237	7	is	be	AUX
ejpam-4797	237	8	nil	nil	NOUN
ejpam-4797	237	9	of	of	ADP
ejpam-4797	237	10	characteristic	characteristic	ADJ
ejpam-4797	237	11	4	4	NUM
ejpam-4797	237	12	.	.	PUNCT
ejpam-4797	238	1	proof	proof	NOUN
ejpam-4797	238	2	.	.	PUNCT
ejpam-4797	239	1	given	give	VERB
ejpam-4797	239	2	a	a	DET
ejpam-4797	239	3	∈	∈	PROPN
ejpam-4797	239	4	j(r	j(r	PROPN
ejpam-4797	239	5	)	)	PUNCT
ejpam-4797	239	6	,	,	PUNCT
ejpam-4797	239	7	then	then	ADV
ejpam-4797	239	8	a	a	DET
ejpam-4797	239	9	=	=	X
ejpam-4797	239	10	σ1	σ1	X
ejpam-4797	239	11	−σ2	−σ2	PROPN
ejpam-4797	239	12	+	+	PROPN
ejpam-4797	239	13	n	n	CCONJ
ejpam-4797	239	14	,	,	PUNCT
ejpam-4797	239	15	where	where	SCONJ
ejpam-4797	239	16	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	239	17	∈	∈	PROPN
ejpam-4797	239	18	id(r	id(r	NOUN
ejpam-4797	239	19	)	)	PUNCT
ejpam-4797	239	20	and	and	CCONJ
ejpam-4797	239	21	n	n	PRON
ejpam-4797	239	22	∈	∈	PROPN
ejpam-4797	239	23	nil(r	nil(r	PROPN
ejpam-4797	239	24	)	)	PUNCT
ejpam-4797	239	25	,	,	PUNCT
ejpam-4797	239	26	that	that	PRON
ejpam-4797	239	27	commute	commute	VERB
ejpam-4797	239	28	with	with	ADP
ejpam-4797	239	29	one	one	NUM
ejpam-4797	239	30	another	another	DET
ejpam-4797	239	31	.	.	PUNCT
ejpam-4797	240	1	write	write	VERB
ejpam-4797	240	2	a	a	DET
ejpam-4797	240	3	=	=	SYM
ejpam-4797	240	4	1−(σ1−σ2	1−(σ1−σ2	NUM
ejpam-4797	240	5	)	)	PUNCT
ejpam-4797	240	6	2+(σ1−σ2	2+(σ1−σ2	NUM
ejpam-4797	240	7	)	)	PUNCT
ejpam-4797	240	8	2+(σ1−σ2)−1+n	2+(σ1−σ2)−1+n	X
ejpam-4797	240	9	.	.	PUNCT
ejpam-4797	241	1	according	accord	VERB
ejpam-4797	241	2	to	to	ADP
ejpam-4797	241	3	lemma	lemma	PROPN
ejpam-4797	241	4	3(3	3(3	NUM
ejpam-4797	241	5	)	)	PUNCT
ejpam-4797	241	6	,	,	PUNCT
ejpam-4797	241	7	(	(	PUNCT
ejpam-4797	241	8	σ1	σ1	PROPN
ejpam-4797	241	9	−	−	PROPN
ejpam-4797	241	10	σ2	σ2	PROPN
ejpam-4797	241	11	)	)	PUNCT
ejpam-4797	241	12	2	2	NUM
ejpam-4797	242	1	+	+	CCONJ
ejpam-4797	242	2	(	(	PUNCT
ejpam-4797	242	3	σ1	σ1	PROPN
ejpam-4797	242	4	−	−	PROPN
ejpam-4797	242	5	σ2	σ2	PROPN
ejpam-4797	242	6	)	)	PUNCT
ejpam-4797	242	7	−	−	PROPN
ejpam-4797	242	8	1	1	NUM
ejpam-4797	242	9	=	=	NOUN
ejpam-4797	242	10	u1	u1	NOUN
ejpam-4797	242	11	is	be	AUX
ejpam-4797	242	12	a	a	DET
ejpam-4797	242	13	unit	unit	NOUN
ejpam-4797	242	14	of	of	ADP
ejpam-4797	242	15	order	order	NOUN
ejpam-4797	242	16	2	2	NUM
ejpam-4797	242	17	,	,	PUNCT
ejpam-4797	242	18	then	then	ADV
ejpam-4797	242	19	a	a	DET
ejpam-4797	242	20	=	=	SYM
ejpam-4797	242	21	1−(σ1−σ2	1−(σ1−σ2	NUM
ejpam-4797	242	22	)	)	PUNCT
ejpam-4797	242	23	2+u1+n	2+u1+n	NUM
ejpam-4797	242	24	implies	imply	VERB
ejpam-4797	242	25	a	a	DET
ejpam-4797	242	26	=	=	SYM
ejpam-4797	242	27	1−(σ1−σ2	1−(σ1−σ2	NUM
ejpam-4797	242	28	)	)	PUNCT
ejpam-4797	242	29	2+u2	2+u2	NUM
ejpam-4797	242	30	,	,	PUNCT
ejpam-4797	242	31	where	where	SCONJ
ejpam-4797	242	32	u2	u2	PROPN
ejpam-4797	242	33	=	=	SYM
ejpam-4797	242	34	u1+n	u1+n	PROPN
ejpam-4797	242	35	.	.	PUNCT
ejpam-4797	243	1	since	since	SCONJ
ejpam-4797	243	2	a	a	DET
ejpam-4797	243	3	∈	∈	PROPN
ejpam-4797	243	4	j(r	j(r	PROPN
ejpam-4797	243	5	)	)	PUNCT
ejpam-4797	243	6	,	,	PUNCT
ejpam-4797	243	7	so	so	SCONJ
ejpam-4797	243	8	a	a	DET
ejpam-4797	243	9	−	−	PROPN
ejpam-4797	243	10	u2	u2	PROPN
ejpam-4797	243	11	∈	∈	PROPN
ejpam-4797	243	12	u(r	u(r	PROPN
ejpam-4797	243	13	)	)	PUNCT
ejpam-4797	243	14	,	,	PUNCT
ejpam-4797	243	15	applying	apply	VERB
ejpam-4797	243	16	to	to	ADP
ejpam-4797	243	17	proposition	proposition	NOUN
ejpam-4797	243	18	3(3	3(3	NUM
ejpam-4797	243	19	)	)	PUNCT
ejpam-4797	243	20	,	,	PUNCT
ejpam-4797	243	21	we	we	PRON
ejpam-4797	243	22	conclude	conclude	VERB
ejpam-4797	243	23	that	that	SCONJ
ejpam-4797	243	24	1	1	NUM
ejpam-4797	243	25	−	−	PROPN
ejpam-4797	243	26	(	(	PUNCT
ejpam-4797	243	27	σ1	σ1	PROPN
ejpam-4797	243	28	−	−	PROPN
ejpam-4797	243	29	σ2	σ2	PROPN
ejpam-4797	243	30	)	)	PUNCT
ejpam-4797	243	31	2	2	NUM
ejpam-4797	243	32	=	=	SYM
ejpam-4797	243	33	1	1	NUM
ejpam-4797	243	34	,	,	PUNCT
ejpam-4797	243	35	gives	give	VERB
ejpam-4797	243	36	(	(	PUNCT
ejpam-4797	243	37	σ1	σ1	PROPN
ejpam-4797	243	38	−	−	PROPN
ejpam-4797	243	39	σ2	σ2	PROPN
ejpam-4797	243	40	)	)	PUNCT
ejpam-4797	243	41	2	2	NUM
ejpam-4797	243	42	=	=	SYM
ejpam-4797	243	43	0	0	NUM
ejpam-4797	243	44	,	,	PUNCT
ejpam-4797	243	45	whence	whence	SCONJ
ejpam-4797	243	46	it	it	PRON
ejpam-4797	243	47	follows	follow	VERB
ejpam-4797	243	48	that	that	SCONJ
ejpam-4797	243	49	a	a	DET
ejpam-4797	243	50	=	=	SYM
ejpam-4797	243	51	1	1	NUM
ejpam-4797	243	52	+	+	CCONJ
ejpam-4797	243	53	u2	u2	NOUN
ejpam-4797	243	54	,	,	PUNCT
ejpam-4797	243	55	with	with	ADP
ejpam-4797	243	56	u22	u22	PROPN
ejpam-4797	243	57	=	=	SYM
ejpam-4797	243	58	1	1	X
ejpam-4797	243	59	.	.	PUNCT
ejpam-4797	243	60	now	now	ADV
ejpam-4797	243	61	consider	consider	VERB
ejpam-4797	243	62	a2	a2	PROPN
ejpam-4797	243	63	=	=	PUNCT
ejpam-4797	243	64	(	(	PUNCT
ejpam-4797	243	65	1	1	NUM
ejpam-4797	243	66	+	+	CCONJ
ejpam-4797	243	67	u2	u2	NOUN
ejpam-4797	243	68	)	)	PUNCT
ejpam-4797	244	1	2	2	NUM
ejpam-4797	244	2	=	=	SYM
ejpam-4797	244	3	2(1	2(1	NUM
ejpam-4797	244	4	+	+	NUM
ejpam-4797	244	5	u2	u2	NOUN
ejpam-4797	244	6	)	)	PUNCT
ejpam-4797	244	7	=	=	SYM
ejpam-4797	244	8	2a	2a	NUM
ejpam-4797	244	9	,	,	PUNCT
ejpam-4797	244	10	and	and	CCONJ
ejpam-4797	244	11	a3	a3	NOUN
ejpam-4797	244	12	=	=	SYM
ejpam-4797	244	13	22(1	22(1	PROPN
ejpam-4797	245	1	+	+	CCONJ
ejpam-4797	245	2	u2	u2	NOUN
ejpam-4797	245	3	)	)	PUNCT
ejpam-4797	245	4	=	=	SYM
ejpam-4797	245	5	4a	4a	NOUN
ejpam-4797	245	6	.	.	PUNCT
ejpam-4797	246	1	choose	choose	VERB
ejpam-4797	246	2	a	a	DET
ejpam-4797	246	3	=	=	SYM
ejpam-4797	246	4	2b	2b	NOUN
ejpam-4797	246	5	,	,	PUNCT
ejpam-4797	246	6	then	then	ADV
ejpam-4797	246	7	(	(	PUNCT
ejpam-4797	246	8	2b)3	2b)3	NUM
ejpam-4797	246	9	=	=	SYM
ejpam-4797	246	10	4(2b	4(2b	NUM
ejpam-4797	246	11	)	)	PUNCT
ejpam-4797	246	12	,	,	PUNCT
ejpam-4797	246	13	so	so	ADV
ejpam-4797	246	14	8b3	8b3	NUM
ejpam-4797	246	15	=	=	SYM
ejpam-4797	246	16	8b	8b	NUM
ejpam-4797	246	17	,	,	PUNCT
ejpam-4797	246	18	implies	imply	VERB
ejpam-4797	246	19	8b(1	8b(1	NUM
ejpam-4797	246	20	−	−	ADP
ejpam-4797	246	21	b2	b2	NOUN
ejpam-4797	246	22	)	)	PUNCT
ejpam-4797	246	23	=	=	SYM
ejpam-4797	246	24	0	0	NUM
ejpam-4797	246	25	,	,	PUNCT
ejpam-4797	246	26	but	but	CCONJ
ejpam-4797	246	27	b	b	X
ejpam-4797	246	28	∈	∈	PROPN
ejpam-4797	246	29	j(r	j(r	PROPN
ejpam-4797	246	30	)	)	PUNCT
ejpam-4797	246	31	,	,	PUNCT
ejpam-4797	246	32	gives	give	VERB
ejpam-4797	246	33	1	1	NUM
ejpam-4797	246	34	−	−	NOUN
ejpam-4797	246	35	b2	b2	NOUN
ejpam-4797	246	36	∈	∈	PROPN
ejpam-4797	246	37	u(r	u(r	PROPN
ejpam-4797	246	38	)	)	PUNCT
ejpam-4797	246	39	,	,	PUNCT
ejpam-4797	246	40	gives	give	VERB
ejpam-4797	246	41	8b	8b	PROPN
ejpam-4797	246	42	=	=	SYM
ejpam-4797	246	43	0	0	NUM
ejpam-4797	246	44	.	.	PUNCT
ejpam-4797	247	1	thus	thus	ADV
ejpam-4797	247	2	4a	4a	NUM
ejpam-4797	247	3	=	=	SYM
ejpam-4797	247	4	a3	a3	NOUN
ejpam-4797	247	5	=	=	SYM
ejpam-4797	247	6	0	0	X
ejpam-4797	247	7	.	.	NOUN
ejpam-4797	247	8	example	example	NOUN
ejpam-4797	247	9	7	7	X
ejpam-4797	247	10	.	.	PUNCT
ejpam-4797	247	11	consider	consider	VERB
ejpam-4797	247	12	the	the	DET
ejpam-4797	247	13	ring	ring	NOUN
ejpam-4797	247	14	z24	z24	PROPN
ejpam-4797	247	15	.	.	PUNCT
ejpam-4797	248	1	then	then	ADV
ejpam-4797	248	2	z24	z24	PROPN
ejpam-4797	248	3	is	be	AUX
ejpam-4797	248	4	a	a	DET
ejpam-4797	248	5	strongly	strongly	ADV
ejpam-4797	248	6	2	2	NUM
ejpam-4797	248	7	-	-	PUNCT
ejpam-4797	248	8	nc	nc	NOUN
ejpam-4797	248	9	with	with	ADP
ejpam-4797	248	10	u(z24	u(z24	NOUN
ejpam-4797	248	11	)	)	PUNCT
ejpam-4797	249	1	=	=	SYM
ejpam-4797	249	2	2	2	X
ejpam-4797	249	3	.	.	PUNCT
ejpam-4797	249	4	now	now	ADV
ejpam-4797	249	5	j(z24	j(z24	ADV
ejpam-4797	249	6	)	)	PUNCT
ejpam-4797	250	1	=	=	PRON
ejpam-4797	250	2	{	{	PUNCT
ejpam-4797	250	3	0	0	NUM
ejpam-4797	250	4	,	,	PUNCT
ejpam-4797	250	5	6	6	NUM
ejpam-4797	250	6	,	,	PUNCT
ejpam-4797	250	7	12	12	NUM
ejpam-4797	250	8	,	,	PUNCT
ejpam-4797	250	9	18	18	NUM
ejpam-4797	250	10	}	}	PUNCT
ejpam-4797	250	11	.	.	PUNCT
ejpam-4797	251	1	so	so	ADV
ejpam-4797	251	2	j(z24	j(z24	NOUN
ejpam-4797	251	3	)	)	PUNCT
ejpam-4797	251	4	is	be	AUX
ejpam-4797	251	5	a	a	DET
ejpam-4797	251	6	nil	nil	ADJ
ejpam-4797	251	7	ideal	ideal	NOUN
ejpam-4797	251	8	of	of	ADP
ejpam-4797	251	9	characteristic	characteristic	ADJ
ejpam-4797	251	10	4	4	NUM
ejpam-4797	251	11	.	.	PUNCT
ejpam-4797	251	12	corollary	corollary	ADJ
ejpam-4797	251	13	1	1	NUM
ejpam-4797	251	14	.	.	PUNCT
ejpam-4797	252	1	if	if	SCONJ
ejpam-4797	252	2	r	r	NOUN
ejpam-4797	252	3	is	be	AUX
ejpam-4797	252	4	a	a	DET
ejpam-4797	252	5	strongly	strongly	ADV
ejpam-4797	252	6	2	2	NUM
ejpam-4797	252	7	-	-	PUNCT
ejpam-4797	252	8	nc	nc	NOUN
ejpam-4797	252	9	ring	ring	NOUN
ejpam-4797	252	10	with	with	ADP
ejpam-4797	252	11	u(r	u(r	NOUN
ejpam-4797	252	12	)	)	PUNCT
ejpam-4797	252	13	=	=	SYM
ejpam-4797	252	14	2	2	NUM
ejpam-4797	252	15	and	and	CCONJ
ejpam-4797	252	16	if	if	SCONJ
ejpam-4797	252	17	2	2	NUM
ejpam-4797	252	18	∈	∈	PROPN
ejpam-4797	252	19	u(r	u(r	NOUN
ejpam-4797	252	20	)	)	PUNCT
ejpam-4797	252	21	,	,	PUNCT
ejpam-4797	252	22	then	then	ADV
ejpam-4797	252	23	j(r	j(r	PROPN
ejpam-4797	252	24	)	)	PUNCT
ejpam-4797	253	1	=	=	SYM
ejpam-4797	253	2	0	0	X
ejpam-4797	253	3	.	.	PUNCT
ejpam-4797	254	1	proof	proof	NOUN
ejpam-4797	254	2	.	.	PUNCT
ejpam-4797	255	1	let	let	VERB
ejpam-4797	255	2	a	a	DET
ejpam-4797	255	3	∈	∈	PROPN
ejpam-4797	255	4	j(r	j(r	PROPN
ejpam-4797	255	5	)	)	PUNCT
ejpam-4797	255	6	,	,	PUNCT
ejpam-4797	255	7	then	then	ADV
ejpam-4797	255	8	by	by	ADP
ejpam-4797	255	9	theorem	theorem	NOUN
ejpam-4797	255	10	5	5	NUM
ejpam-4797	255	11	,	,	PUNCT
ejpam-4797	255	12	4a	4a	NOUN
ejpam-4797	255	13	=	=	SYM
ejpam-4797	255	14	0	0	NUM
ejpam-4797	255	15	,	,	PUNCT
ejpam-4797	255	16	since	since	SCONJ
ejpam-4797	255	17	2	2	NUM
ejpam-4797	255	18	∈	∈	PROPN
ejpam-4797	255	19	u(r	u(r	NOUN
ejpam-4797	255	20	)	)	PUNCT
ejpam-4797	255	21	,	,	PUNCT
ejpam-4797	255	22	then	then	ADV
ejpam-4797	255	23	a	a	DET
ejpam-4797	255	24	=	=	ADJ
ejpam-4797	255	25	0	0	X
ejpam-4797	255	26	.	.	PUNCT
ejpam-4797	255	27	proposition	proposition	NOUN
ejpam-4797	255	28	8	8	NUM
ejpam-4797	255	29	.	.	PUNCT
ejpam-4797	256	1	if	if	SCONJ
ejpam-4797	256	2	r	r	NOUN
ejpam-4797	256	3	is	be	AUX
ejpam-4797	256	4	a	a	DET
ejpam-4797	256	5	strongly	strongly	ADV
ejpam-4797	256	6	2	2	NUM
ejpam-4797	256	7	-	-	PUNCT
ejpam-4797	256	8	nc	nc	NOUN
ejpam-4797	256	9	ring	ring	NOUN
ejpam-4797	256	10	with	with	ADP
ejpam-4797	256	11	u(r	u(r	NOUN
ejpam-4797	256	12	)	)	PUNCT
ejpam-4797	257	1	=	=	SYM
ejpam-4797	257	2	2	2	NUM
ejpam-4797	257	3	,	,	PUNCT
ejpam-4797	257	4	and	and	CCONJ
ejpam-4797	257	5	if	if	SCONJ
ejpam-4797	257	6	2	2	NUM
ejpam-4797	257	7	∈	∈	PROPN
ejpam-4797	257	8	u(r	u(r	NOUN
ejpam-4797	257	9	)	)	PUNCT
ejpam-4797	257	10	,	,	PUNCT
ejpam-4797	257	11	then	then	ADV
ejpam-4797	257	12	nil(r	nil(r	NUM
ejpam-4797	257	13	)	)	PUNCT
ejpam-4797	257	14	=	=	SYM
ejpam-4797	257	15	0	0	X
ejpam-4797	257	16	.	.	PUNCT
ejpam-4797	258	1	proof	proof	NOUN
ejpam-4797	258	2	.	.	PUNCT
ejpam-4797	259	1	given	give	VERB
ejpam-4797	259	2	a	a	DET
ejpam-4797	259	3	∈	∈	ADJ
ejpam-4797	259	4	r	r	NOUN
ejpam-4797	259	5	,	,	PUNCT
ejpam-4797	259	6	then	then	ADV
ejpam-4797	259	7	a	a	DET
ejpam-4797	259	8	−	−	PROPN
ejpam-4797	259	9	1	1	NUM
ejpam-4797	259	10	=	=	SYM
ejpam-4797	259	11	σ1	σ1	PROPN
ejpam-4797	259	12	−	−	PROPN
ejpam-4797	259	13	σ2	σ2	PROPN
ejpam-4797	259	14	+	+	CCONJ
ejpam-4797	259	15	n	n	CCONJ
ejpam-4797	259	16	,	,	PUNCT
ejpam-4797	259	17	so	so	ADV
ejpam-4797	259	18	a	a	DET
ejpam-4797	259	19	=	=	X
ejpam-4797	259	20	σ1	σ1	PROPN
ejpam-4797	259	21	−	−	PROPN
ejpam-4797	259	22	σ2	σ2	PROPN
ejpam-4797	259	23	+	+	CCONJ
ejpam-4797	259	24	n	n	PROPN
ejpam-4797	259	25	+	+	NUM
ejpam-4797	259	26	1	1	NUM
ejpam-4797	259	27	,	,	PUNCT
ejpam-4797	259	28	but	but	CCONJ
ejpam-4797	259	29	n+1	n+1	PROPN
ejpam-4797	259	30	∈	∈	PROPN
ejpam-4797	259	31	u(r	u(r	PROPN
ejpam-4797	259	32	)	)	PUNCT
ejpam-4797	259	33	,	,	PUNCT
ejpam-4797	259	34	say	say	VERB
ejpam-4797	259	35	u	u	NOUN
ejpam-4797	259	36	,	,	PUNCT
ejpam-4797	259	37	then	then	ADV
ejpam-4797	259	38	a	a	DET
ejpam-4797	259	39	=	=	PUNCT
ejpam-4797	259	40	σ1−σ2+u	σ1−σ2+u	NOUN
ejpam-4797	259	41	.	.	PUNCT
ejpam-4797	260	1	let	let	VERB
ejpam-4797	260	2	n	n	PRON
ejpam-4797	260	3	∈	∈	PROPN
ejpam-4797	260	4	nil(r	nil(r	PROPN
ejpam-4797	260	5	)	)	PUNCT
ejpam-4797	260	6	,	,	PUNCT
ejpam-4797	260	7	then	then	ADV
ejpam-4797	260	8	n	n	PROPN
ejpam-4797	260	9	=	=	SYM
ejpam-4797	260	10	σ1−σ2+u	σ1−σ2+u	ADJ
ejpam-4797	260	11	,	,	PUNCT
ejpam-4797	260	12	implies	imply	VERB
ejpam-4797	260	13	σ1	σ1	PROPN
ejpam-4797	260	14	−	−	PROPN
ejpam-4797	260	15	σ2	σ2	PROPN
ejpam-4797	260	16	=	=	PUNCT
ejpam-4797	260	17	n	n	CCONJ
ejpam-4797	260	18	−	−	PROPN
ejpam-4797	260	19	u	u	NOUN
ejpam-4797	260	20	,	,	PUNCT
ejpam-4797	260	21	since	since	SCONJ
ejpam-4797	260	22	n	n	NUM
ejpam-4797	260	23	−	−	PROPN
ejpam-4797	260	24	u	u	PROPN
ejpam-4797	260	25	∈	∈	PROPN
ejpam-4797	260	26	u(r	u(r	PROPN
ejpam-4797	260	27	)	)	PUNCT
ejpam-4797	260	28	,	,	PUNCT
ejpam-4797	260	29	according	accord	VERB
ejpam-4797	260	30	to	to	ADP
ejpam-4797	260	31	proposition	proposition	NOUN
ejpam-4797	260	32	3(3	3(3	NUM
ejpam-4797	260	33	)	)	PUNCT
ejpam-4797	260	34	,	,	PUNCT
ejpam-4797	260	35	(	(	PUNCT
ejpam-4797	260	36	σ1	σ1	PROPN
ejpam-4797	260	37	−	−	PROPN
ejpam-4797	260	38	σ2	σ2	PROPN
ejpam-4797	260	39	)	)	PUNCT
ejpam-4797	260	40	2	2	NUM
ejpam-4797	260	41	=	=	SYM
ejpam-4797	260	42	1	1	X
ejpam-4797	260	43	.	.	PUNCT
ejpam-4797	261	1	furthermore	furthermore	ADV
ejpam-4797	261	2	,	,	PUNCT
ejpam-4797	261	3	n2	n2	NOUN
ejpam-4797	261	4	=	=	SYM
ejpam-4797	261	5	(	(	PUNCT
ejpam-4797	261	6	σ1−σ2	σ1−σ2	NOUN
ejpam-4797	261	7	)	)	PUNCT
ejpam-4797	261	8	2	2	NUM
ejpam-4797	261	9	+	+	NUM
ejpam-4797	261	10	2(σ1−σ2)u+u2	2(σ1−σ2)u+u2	ADJ
ejpam-4797	261	11	=	=	SYM
ejpam-4797	261	12	1	1	NUM
ejpam-4797	261	13	+	+	NUM
ejpam-4797	261	14	2(σ1−σ2)u+1	2(σ1−σ2)u+1	NUM
ejpam-4797	261	15	=	=	SYM
ejpam-4797	261	16	2(1+(σ1−σ2)u	2(1+(σ1−σ2)u	NOUN
ejpam-4797	261	17	)	)	PUNCT
ejpam-4797	261	18	.	.	PUNCT
ejpam-4797	262	1	observe	observe	VERB
ejpam-4797	262	2	that	that	SCONJ
ejpam-4797	262	3	nu	nu	PROPN
ejpam-4797	262	4	=	=	SYM
ejpam-4797	262	5	(	(	PUNCT
ejpam-4797	262	6	σ1	σ1	PROPN
ejpam-4797	262	7	−	−	PROPN
ejpam-4797	262	8	σ2)u	σ2)u	PROPN
ejpam-4797	262	9	+	+	NOUN
ejpam-4797	262	10	1	1	X
ejpam-4797	262	11	.	.	PUNCT
ejpam-4797	262	12	thus	thus	ADV
ejpam-4797	262	13	,	,	PUNCT
ejpam-4797	262	14	n2	n2	ADJ
ejpam-4797	262	15	=	=	SYM
ejpam-4797	262	16	2nu	2nu	NOUN
ejpam-4797	262	17	,	,	PUNCT
ejpam-4797	262	18	so	so	ADV
ejpam-4797	262	19	n(n	n(n	PROPN
ejpam-4797	262	20	−	−	NOUN
ejpam-4797	262	21	2u	2u	NOUN
ejpam-4797	262	22	)	)	PUNCT
ejpam-4797	262	23	=	=	SYM
ejpam-4797	263	1	0	0	X
ejpam-4797	263	2	.	.	PUNCT
ejpam-4797	264	1	since	since	SCONJ
ejpam-4797	264	2	2	2	NUM
ejpam-4797	264	3	∈	∈	PROPN
ejpam-4797	264	4	u(r	u(r	NOUN
ejpam-4797	264	5	)	)	PUNCT
ejpam-4797	264	6	,	,	PUNCT
ejpam-4797	264	7	by	by	ADP
ejpam-4797	264	8	assumption	assumption	NOUN
ejpam-4797	264	9	then	then	ADV
ejpam-4797	264	10	n−	n−	PROPN
ejpam-4797	264	11	2u	2u	PROPN
ejpam-4797	264	12	∈	∈	PROPN
ejpam-4797	264	13	u(r	u(r	PROPN
ejpam-4797	264	14	)	)	PUNCT
ejpam-4797	264	15	.	.	PUNCT
ejpam-4797	265	1	whence	whence	ADP
ejpam-4797	265	2	it	it	PRON
ejpam-4797	265	3	follows	follow	VERB
ejpam-4797	265	4	that	that	PRON
ejpam-4797	265	5	n	n	NOUN
ejpam-4797	265	6	=	=	SYM
ejpam-4797	265	7	0	0	NUM
ejpam-4797	265	8	.	.	PUNCT
ejpam-4797	266	1	next	next	ADV
ejpam-4797	266	2	,	,	PUNCT
ejpam-4797	266	3	we	we	PRON
ejpam-4797	266	4	shall	shall	AUX
ejpam-4797	266	5	explore	explore	VERB
ejpam-4797	266	6	the	the	DET
ejpam-4797	266	7	relationship	relationship	NOUN
ejpam-4797	266	8	between	between	ADP
ejpam-4797	266	9	strongly	strongly	ADV
ejpam-4797	266	10	2	2	NUM
ejpam-4797	266	11	-	-	PUNCT
ejpam-4797	266	12	nc	nc	NOUN
ejpam-4797	266	13	ring	ring	NOUN
ejpam-4797	266	14	with	with	ADP
ejpam-4797	266	15	u(r	u(r	NOUN
ejpam-4797	266	16	)	)	PUNCT
ejpam-4797	267	1	=	=	SYM
ejpam-4797	267	2	2	2	NUM
ejpam-4797	267	3	and	and	CCONJ
ejpam-4797	267	4	a	a	DET
ejpam-4797	267	5	tripotent	tripotent	ADJ
ejpam-4797	267	6	ring	ring	NOUN
ejpam-4797	267	7	.	.	PUNCT
ejpam-4797	268	1	r.	r.	PROPN
ejpam-4797	268	2	t.	t.	PROPN
ejpam-4797	268	3	m.salim	m.salim	PROPN
ejpam-4797	268	4	,	,	PUNCT
ejpam-4797	268	5	n.	n.	PROPN
ejpam-4797	268	6	h.	h.	PROPN
ejpam-4797	268	7	shuker	shuker	PROPN
ejpam-4797	268	8	/	/	SYM
ejpam-4797	268	9	eur	eur	PROPN
ejpam-4797	268	10	.	.	PUNCT
ejpam-4797	269	1	j.	j.	PROPN
ejpam-4797	269	2	pure	pure	PROPN
ejpam-4797	269	3	appl	appl	PROPN
ejpam-4797	269	4	.	.	PROPN
ejpam-4797	269	5	math	math	PROPN
ejpam-4797	269	6	,	,	PUNCT
ejpam-4797	269	7	16	16	NUM
ejpam-4797	269	8	(	(	PUNCT
ejpam-4797	269	9	3	3	NUM
ejpam-4797	269	10	)	)	PUNCT
ejpam-4797	269	11	(	(	PUNCT
ejpam-4797	269	12	2023	2023	NUM
ejpam-4797	269	13	)	)	PUNCT
ejpam-4797	269	14	,	,	PUNCT
ejpam-4797	269	15	1675	1675	NUM
ejpam-4797	269	16	-	-	SYM
ejpam-4797	269	17	1684	1684	NUM
ejpam-4797	269	18	1682	1682	NUM
ejpam-4797	269	19	theorem	theorem	VERB
ejpam-4797	269	20	6	6	NUM
ejpam-4797	269	21	.	.	PUNCT
ejpam-4797	270	1	a	a	DET
ejpam-4797	270	2	ring	ring	NOUN
ejpam-4797	270	3	r	r	NOUN
ejpam-4797	270	4	with	with	ADP
ejpam-4797	270	5	2	2	NUM
ejpam-4797	270	6	∈	∈	PROPN
ejpam-4797	270	7	u(r	u(r	NOUN
ejpam-4797	270	8	)	)	PUNCT
ejpam-4797	270	9	is	be	AUX
ejpam-4797	270	10	strongly	strongly	ADV
ejpam-4797	270	11	2	2	NUM
ejpam-4797	270	12	-	-	PUNCT
ejpam-4797	270	13	nc	nc	NOUN
ejpam-4797	270	14	with	with	ADP
ejpam-4797	270	15	u(r	u(r	NOUN
ejpam-4797	270	16	)	)	PUNCT
ejpam-4797	271	1	=	=	SYM
ejpam-4797	271	2	2	2	NUM
ejpam-4797	271	3	if	if	SCONJ
ejpam-4797	271	4	and	and	CCONJ
ejpam-4797	271	5	only	only	ADV
ejpam-4797	271	6	if	if	SCONJ
ejpam-4797	271	7	r	r	NOUN
ejpam-4797	271	8	is	be	AUX
ejpam-4797	271	9	a	a	DET
ejpam-4797	271	10	tripotent	tripotent	NOUN
ejpam-4797	271	11	.	.	PUNCT
ejpam-4797	272	1	proof	proof	NOUN
ejpam-4797	272	2	.	.	PUNCT
ejpam-4797	273	1	let	let	VERB
ejpam-4797	273	2	r	r	PRON
ejpam-4797	273	3	be	be	AUX
ejpam-4797	273	4	a	a	DET
ejpam-4797	273	5	strongly	strongly	ADV
ejpam-4797	273	6	2	2	NUM
ejpam-4797	273	7	-	-	PUNCT
ejpam-4797	273	8	nc	nc	NOUN
ejpam-4797	273	9	ring	ring	NOUN
ejpam-4797	273	10	with	with	ADP
ejpam-4797	273	11	u(r	u(r	NOUN
ejpam-4797	273	12	)	)	PUNCT
ejpam-4797	274	1	=	=	SYM
ejpam-4797	274	2	2	2	NUM
ejpam-4797	274	3	,	,	PUNCT
ejpam-4797	274	4	and	and	CCONJ
ejpam-4797	274	5	let	let	VERB
ejpam-4797	274	6	a	a	DET
ejpam-4797	274	7	∈	∈	ADJ
ejpam-4797	274	8	r	r	NOUN
ejpam-4797	274	9	,	,	PUNCT
ejpam-4797	274	10	then	then	ADV
ejpam-4797	274	11	a	a	DET
ejpam-4797	274	12	=	=	X
ejpam-4797	274	13	σ1	σ1	X
ejpam-4797	274	14	−σ2	−σ2	PROPN
ejpam-4797	274	15	+	+	PROPN
ejpam-4797	274	16	n	n	CCONJ
ejpam-4797	274	17	,	,	PUNCT
ejpam-4797	274	18	where	where	SCONJ
ejpam-4797	274	19	σ1,σ2	σ1,σ2	PROPN
ejpam-4797	274	20	∈	∈	PROPN
ejpam-4797	274	21	id(r	id(r	NOUN
ejpam-4797	274	22	)	)	PUNCT
ejpam-4797	274	23	,	,	PUNCT
ejpam-4797	274	24	n	n	PROPN
ejpam-4797	274	25	∈	∈	PROPN
ejpam-4797	274	26	nil(r	nil(r	PROPN
ejpam-4797	274	27	)	)	PUNCT
ejpam-4797	274	28	,	,	PUNCT
ejpam-4797	274	29	that	that	PRON
ejpam-4797	274	30	commute	commute	VERB
ejpam-4797	274	31	with	with	ADP
ejpam-4797	274	32	one	one	NUM
ejpam-4797	274	33	another	another	DET
ejpam-4797	274	34	.	.	PUNCT
ejpam-4797	275	1	according	accord	VERB
ejpam-4797	275	2	to	to	ADP
ejpam-4797	275	3	proposition	proposition	NOUN
ejpam-4797	275	4	8	8	NUM
ejpam-4797	275	5	,	,	PUNCT
ejpam-4797	275	6	n	n	NOUN
ejpam-4797	275	7	=	=	SYM
ejpam-4797	275	8	0	0	NUM
ejpam-4797	275	9	.	.	PUNCT
ejpam-4797	276	1	thus	thus	ADV
ejpam-4797	276	2	,	,	PUNCT
ejpam-4797	276	3	a	a	DET
ejpam-4797	276	4	=	=	X
ejpam-4797	276	5	σ1	σ1	PROPN
ejpam-4797	276	6	−	−	PROPN
ejpam-4797	276	7	σ2	σ2	PROPN
ejpam-4797	276	8	=	=	SYM
ejpam-4797	276	9	(	(	PUNCT
ejpam-4797	276	10	σ1	σ1	PROPN
ejpam-4797	276	11	−	−	PROPN
ejpam-4797	276	12	σ2	σ2	PROPN
ejpam-4797	276	13	)	)	PUNCT
ejpam-4797	276	14	3	3	NUM
ejpam-4797	276	15	=	=	NOUN
ejpam-4797	276	16	a3	a3	NOUN
ejpam-4797	276	17	.	.	PUNCT
ejpam-4797	277	1	conversely	conversely	ADV
ejpam-4797	277	2	,	,	PUNCT
ejpam-4797	277	3	assume	assume	VERB
ejpam-4797	277	4	that	that	SCONJ
ejpam-4797	277	5	r	r	NOUN
ejpam-4797	277	6	is	be	AUX
ejpam-4797	277	7	a	a	DET
ejpam-4797	277	8	tripotent	tripotent	ADJ
ejpam-4797	277	9	ring	ring	NOUN
ejpam-4797	277	10	,	,	PUNCT
ejpam-4797	277	11	and	and	CCONJ
ejpam-4797	277	12	t	t	X
ejpam-4797	277	13	=	=	SYM
ejpam-4797	277	14	t3	t3	PROPN
ejpam-4797	277	15	∈	∈	PROPN
ejpam-4797	277	16	r	r	NOUN
ejpam-4797	277	17	,	,	PUNCT
ejpam-4797	277	18	since	since	SCONJ
ejpam-4797	277	19	2	2	NUM
ejpam-4797	277	20	∈	∈	PROPN
ejpam-4797	277	21	u(r	u(r	NOUN
ejpam-4797	277	22	)	)	PUNCT
ejpam-4797	277	23	,	,	PUNCT
ejpam-4797	277	24	then	then	ADV
ejpam-4797	277	25	t	t	PROPN
ejpam-4797	277	26	may	may	AUX
ejpam-4797	277	27	be	be	AUX
ejpam-4797	277	28	written	write	VERB
ejpam-4797	277	29	as	as	ADP
ejpam-4797	277	30	t	t	PROPN
ejpam-4797	277	31	=	=	SYM
ejpam-4797	277	32	t2+t	t2+t	PROPN
ejpam-4797	277	33	2	2	NUM
ejpam-4797	277	34	−	−	NOUN
ejpam-4797	277	35	t2−t	t2−t	PROPN
ejpam-4797	277	36	2	2	NUM
ejpam-4797	277	37	.	.	PUNCT
ejpam-4797	278	1	note	note	VERB
ejpam-4797	278	2	that	that	SCONJ
ejpam-4797	278	3	:	:	PUNCT
ejpam-4797	278	4	(	(	PUNCT
ejpam-4797	278	5	t	t	NOUN
ejpam-4797	278	6	2+t	2+t	NUM
ejpam-4797	278	7	2	2	NUM
ejpam-4797	278	8	)	)	SYM
ejpam-4797	278	9	2	2	NUM
ejpam-4797	278	10	=	=	SYM
ejpam-4797	278	11	t2	t2	NOUN
ejpam-4797	278	12	+	+	NOUN
ejpam-4797	278	13	2t+t2	2t+t2	PROPN
ejpam-4797	278	14	4	4	NUM
ejpam-4797	278	15	=	=	SYM
ejpam-4797	278	16	t2+t	t2+t	X
ejpam-4797	278	17	2	2	NUM
ejpam-4797	278	18	,	,	PUNCT
ejpam-4797	278	19	and	and	CCONJ
ejpam-4797	278	20	(	(	PUNCT
ejpam-4797	278	21	t	t	PROPN
ejpam-4797	278	22	2−t	2−t	NUM
ejpam-4797	278	23	2	2	NUM
ejpam-4797	278	24	)	)	PUNCT
ejpam-4797	278	25	2	2	NUM
ejpam-4797	278	26	=	=	NOUN
ejpam-4797	278	27	t2−2t+t2	t2−2t+t2	NOUN
ejpam-4797	278	28	4	4	NUM
ejpam-4797	278	29	=	=	SYM
ejpam-4797	278	30	t2−t	t2−t	VERB
ejpam-4797	278	31	2	2	NUM
ejpam-4797	278	32	,	,	PUNCT
ejpam-4797	278	33	so	so	CCONJ
ejpam-4797	278	34	(	(	PUNCT
ejpam-4797	278	35	t	t	PROPN
ejpam-4797	278	36	2+t	2+t	NUM
ejpam-4797	278	37	2	2	NUM
ejpam-4797	278	38	)	)	PUNCT
ejpam-4797	278	39	,	,	PUNCT
ejpam-4797	278	40	(	(	PUNCT
ejpam-4797	278	41	t	t	NOUN
ejpam-4797	278	42	2−t	2−t	NUM
ejpam-4797	278	43	2	2	NUM
ejpam-4797	278	44	)	)	PUNCT
ejpam-4797	278	45	∈	∈	PROPN
ejpam-4797	278	46	id(r	id(r	NOUN
ejpam-4797	278	47	)	)	PUNCT
ejpam-4797	278	48	.	.	PUNCT
ejpam-4797	279	1	observe	observe	VERB
ejpam-4797	279	2	that	that	SCONJ
ejpam-4797	279	3	for	for	ADP
ejpam-4797	279	4	any	any	DET
ejpam-4797	279	5	unit	unit	NOUN
ejpam-4797	279	6	u	u	NOUN
ejpam-4797	279	7	,	,	PUNCT
ejpam-4797	279	8	u3	u3	NOUN
ejpam-4797	279	9	=	=	SYM
ejpam-4797	279	10	u	u	PROPN
ejpam-4797	279	11	thus	thus	ADV
ejpam-4797	279	12	,	,	PUNCT
ejpam-4797	279	13	u2	u2	PROPN
ejpam-4797	279	14	=	=	NOUN
ejpam-4797	279	15	1	1	NUM
ejpam-4797	279	16	.	.	PUNCT
ejpam-4797	279	17	therefore	therefore	ADV
ejpam-4797	279	18	,	,	PUNCT
ejpam-4797	279	19	r	r	NOUN
ejpam-4797	279	20	is	be	AUX
ejpam-4797	279	21	a	a	DET
ejpam-4797	279	22	strongly	strongly	ADV
ejpam-4797	279	23	2	2	NUM
ejpam-4797	279	24	-	-	PUNCT
ejpam-4797	279	25	nc	nc	NOUN
ejpam-4797	279	26	ring	ring	NOUN
ejpam-4797	279	27	with	with	ADP
ejpam-4797	279	28	u(r	u(r	NOUN
ejpam-4797	279	29	)	)	PUNCT
ejpam-4797	279	30	=	=	SYM
ejpam-4797	280	1	2	2	X
ejpam-4797	280	2	.	.	PUNCT
ejpam-4797	280	3	to	to	PART
ejpam-4797	280	4	end	end	VERB
ejpam-4797	280	5	this	this	DET
ejpam-4797	280	6	section	section	NOUN
ejpam-4797	280	7	,	,	PUNCT
ejpam-4797	280	8	we	we	PRON
ejpam-4797	280	9	consider	consider	VERB
ejpam-4797	280	10	a	a	DET
ejpam-4797	280	11	strongly	strongly	ADV
ejpam-4797	280	12	2	2	NUM
ejpam-4797	280	13	-	-	PUNCT
ejpam-4797	280	14	nc	nc	NOUN
ejpam-4797	280	15	ring	ring	NOUN
ejpam-4797	280	16	,	,	PUNCT
ejpam-4797	280	17	with	with	SCONJ
ejpam-4797	280	18	every	every	DET
ejpam-4797	280	19	unit	unit	NOUN
ejpam-4797	280	20	is	be	AUX
ejpam-4797	280	21	of	of	ADP
ejpam-4797	280	22	order	order	NOUN
ejpam-4797	280	23	4	4	NUM
ejpam-4797	280	24	.	.	PUNCT
ejpam-4797	280	25	proposition	proposition	NOUN
ejpam-4797	280	26	9	9	NUM
ejpam-4797	280	27	.	.	PUNCT
ejpam-4797	281	1	suppose	suppose	VERB
ejpam-4797	281	2	r	r	NOUN
ejpam-4797	281	3	is	be	AUX
ejpam-4797	281	4	a	a	DET
ejpam-4797	281	5	strongly	strongly	ADV
ejpam-4797	281	6	2	2	NUM
ejpam-4797	281	7	-	-	PUNCT
ejpam-4797	281	8	nc	nc	NOUN
ejpam-4797	281	9	ring	ring	NOUN
ejpam-4797	281	10	,	,	PUNCT
ejpam-4797	281	11	and	and	CCONJ
ejpam-4797	281	12	if	if	SCONJ
ejpam-4797	281	13	n2	n2	ADJ
ejpam-4797	281	14	+	+	PROPN
ejpam-4797	281	15	2n	2n	X
ejpam-4797	281	16	=	=	SYM
ejpam-4797	281	17	0	0	NUM
ejpam-4797	281	18	for	for	ADP
ejpam-4797	281	19	every	every	DET
ejpam-4797	281	20	nilpotent	nilpotent	ADJ
ejpam-4797	281	21	n.	n.	NOUN
ejpam-4797	281	22	then	then	ADV
ejpam-4797	281	23	every	every	DET
ejpam-4797	281	24	unit	unit	NOUN
ejpam-4797	281	25	of	of	ADP
ejpam-4797	281	26	r	r	NOUN
ejpam-4797	281	27	is	be	AUX
ejpam-4797	281	28	of	of	ADP
ejpam-4797	281	29	order	order	NOUN
ejpam-4797	281	30	4	4	NUM
ejpam-4797	281	31	,	,	PUNCT
ejpam-4797	281	32	and	and	CCONJ
ejpam-4797	281	33	48	48	NUM
ejpam-4797	281	34	=	=	SYM
ejpam-4797	281	35	0	0	NUM
ejpam-4797	281	36	.	.	PUNCT
ejpam-4797	282	1	proof	proof	NOUN
ejpam-4797	282	2	.	.	PUNCT
ejpam-4797	283	1	given	give	VERB
ejpam-4797	283	2	a	a	DET
ejpam-4797	283	3	∈	∈	ADJ
ejpam-4797	283	4	r	r	NOUN
ejpam-4797	283	5	,	,	PUNCT
ejpam-4797	283	6	then	then	ADV
ejpam-4797	283	7	by	by	ADP
ejpam-4797	283	8	proposition	proposition	NOUN
ejpam-4797	283	9	1(1	1(1	NUM
ejpam-4797	283	10	)	)	PUNCT
ejpam-4797	283	11	,	,	PUNCT
ejpam-4797	283	12	a2	a2	PROPN
ejpam-4797	283	13	is	be	AUX
ejpam-4797	283	14	an	an	DET
ejpam-4797	283	15	snc	snc	NOUN
ejpam-4797	283	16	element	element	NOUN
ejpam-4797	283	17	.	.	PUNCT
ejpam-4797	284	1	write	write	PROPN
ejpam-4797	284	2	a2	a2	PROPN
ejpam-4797	284	3	=	=	PUNCT
ejpam-4797	284	4	σ+n	σ+n	PROPN
ejpam-4797	284	5	,	,	PUNCT
ejpam-4797	284	6	where	where	SCONJ
ejpam-4797	284	7	σ	σ	PROPN
ejpam-4797	284	8	∈	∈	PROPN
ejpam-4797	284	9	id(r	id(r	NOUN
ejpam-4797	284	10	)	)	PUNCT
ejpam-4797	284	11	,	,	PUNCT
ejpam-4797	284	12	n	n	PROPN
ejpam-4797	284	13	∈	∈	PROPN
ejpam-4797	284	14	nil(r	nil(r	PROPN
ejpam-4797	284	15	)	)	PUNCT
ejpam-4797	284	16	and	and	CCONJ
ejpam-4797	284	17	σn	σn	NOUN
ejpam-4797	284	18	=	=	SYM
ejpam-4797	284	19	nς	nς	PROPN
ejpam-4797	284	20	.	.	PUNCT
ejpam-4797	285	1	let	let	VERB
ejpam-4797	285	2	u	u	PRON
ejpam-4797	285	3	∈	∈	PROPN
ejpam-4797	285	4	u(r	u(r	PROPN
ejpam-4797	285	5	)	)	PUNCT
ejpam-4797	285	6	,	,	PUNCT
ejpam-4797	285	7	then	then	ADV
ejpam-4797	285	8	u2	u2	PROPN
ejpam-4797	285	9	=	=	PROPN
ejpam-4797	285	10	σ	σ	PROPN
ejpam-4797	285	11	+	+	CCONJ
ejpam-4797	285	12	n	n	CCONJ
ejpam-4797	285	13	,	,	PUNCT
ejpam-4797	285	14	implies	imply	VERB
ejpam-4797	285	15	σ	σ	NOUN
ejpam-4797	285	16	=	=	PUNCT
ejpam-4797	285	17	u2−n	u2−n	PROPN
ejpam-4797	285	18	=	=	PUNCT
ejpam-4797	285	19	v	v	PROPN
ejpam-4797	285	20	∈	∈	PROPN
ejpam-4797	285	21	u(r	u(r	NOUN
ejpam-4797	285	22	)	)	PUNCT
ejpam-4797	285	23	.	.	PUNCT
ejpam-4797	286	1	thus	thus	ADV
ejpam-4797	286	2	,	,	PUNCT
ejpam-4797	286	3	σ	σ	PROPN
ejpam-4797	286	4	=	=	SYM
ejpam-4797	286	5	1	1	X
ejpam-4797	286	6	.	.	PUNCT
ejpam-4797	286	7	hence	hence	ADV
ejpam-4797	286	8	u2	u2	NOUN
ejpam-4797	286	9	=	=	SYM
ejpam-4797	286	10	1+n	1+n	NUM
ejpam-4797	286	11	,	,	PUNCT
ejpam-4797	286	12	implies	imply	VERB
ejpam-4797	286	13	u4	u4	PROPN
ejpam-4797	286	14	=	=	SYM
ejpam-4797	286	15	(	(	PUNCT
ejpam-4797	286	16	1+n)2	1+n)2	NUM
ejpam-4797	286	17	=	=	SYM
ejpam-4797	286	18	1	1	NUM
ejpam-4797	286	19	+	+	NOUN
ejpam-4797	286	20	2n+n2	2n+n2	NUM
ejpam-4797	286	21	.	.	PUNCT
ejpam-4797	287	1	by	by	ADP
ejpam-4797	287	2	assumption	assumption	NOUN
ejpam-4797	287	3	n2	n2	NOUN
ejpam-4797	287	4	+	+	CCONJ
ejpam-4797	287	5	2n	2n	NUM
ejpam-4797	287	6	=	=	SYM
ejpam-4797	287	7	0	0	NUM
ejpam-4797	287	8	,	,	PUNCT
ejpam-4797	287	9	then	then	ADV
ejpam-4797	287	10	u4	u4	PROPN
ejpam-4797	287	11	=	=	PROPN
ejpam-4797	287	12	1	1	PROPN
ejpam-4797	287	13	.	.	PUNCT
ejpam-4797	288	1	on	on	ADP
ejpam-4797	288	2	the	the	DET
ejpam-4797	288	3	other	other	ADJ
ejpam-4797	288	4	hand	hand	NOUN
ejpam-4797	288	5	6	6	NUM
ejpam-4797	288	6	∈	∈	PROPN
ejpam-4797	288	7	nil(r	nil(r	NOUN
ejpam-4797	288	8	)	)	PUNCT
ejpam-4797	288	9	theorem	theorem	NOUN
ejpam-4797	288	10	1	1	NUM
ejpam-4797	288	11	.	.	PUNCT
ejpam-4797	289	1	thus	thus	ADV
ejpam-4797	289	2	,	,	PUNCT
ejpam-4797	289	3	62	62	NUM
ejpam-4797	289	4	+	+	NOUN
ejpam-4797	289	5	2(6	2(6	NUM
ejpam-4797	289	6	)	)	PUNCT
ejpam-4797	289	7	=	=	SYM
ejpam-4797	289	8	0	0	NUM
ejpam-4797	289	9	,	,	PUNCT
ejpam-4797	289	10	gives	give	VERB
ejpam-4797	289	11	48	48	NUM
ejpam-4797	289	12	=	=	SYM
ejpam-4797	289	13	0	0	PROPN
ejpam-4797	289	14	.	.	NOUN
ejpam-4797	289	15	example	example	NOUN
ejpam-4797	289	16	8	8	NUM
ejpam-4797	289	17	.	.	PUNCT
ejpam-4797	290	1	in	in	ADP
ejpam-4797	290	2	the	the	DET
ejpam-4797	290	3	ring	ring	NOUN
ejpam-4797	290	4	z48	z48	PROPN
ejpam-4797	290	5	.	.	PUNCT
ejpam-4797	291	1	then	then	ADV
ejpam-4797	291	2	u(z48	u(z48	VERB
ejpam-4797	291	3	)	)	PUNCT
ejpam-4797	292	1	=	=	PRON
ejpam-4797	292	2	{	{	PUNCT
ejpam-4797	292	3	1	1	NUM
ejpam-4797	292	4	,	,	PUNCT
ejpam-4797	292	5	5	5	NUM
ejpam-4797	292	6	,	,	PUNCT
ejpam-4797	292	7	7	7	NUM
ejpam-4797	292	8	,	,	PUNCT
ejpam-4797	292	9	11	11	NUM
ejpam-4797	292	10	,	,	PUNCT
ejpam-4797	292	11	13	13	NUM
ejpam-4797	292	12	,	,	PUNCT
ejpam-4797	292	13	17	17	NUM
ejpam-4797	292	14	,	,	PUNCT
ejpam-4797	292	15	19	19	NUM
ejpam-4797	292	16	,	,	PUNCT
ejpam-4797	292	17	23	23	NUM
ejpam-4797	292	18	,	,	PUNCT
ejpam-4797	292	19	25	25	NUM
ejpam-4797	292	20	,	,	PUNCT
ejpam-4797	292	21	29	29	NUM
ejpam-4797	292	22	,	,	PUNCT
ejpam-4797	292	23	31	31	NUM
ejpam-4797	292	24	,	,	PUNCT
ejpam-4797	292	25	35	35	NUM
ejpam-4797	292	26	,	,	PUNCT
ejpam-4797	292	27	37	37	NUM
ejpam-4797	292	28	,	,	PUNCT
ejpam-4797	292	29	41	41	NUM
ejpam-4797	292	30	,	,	PUNCT
ejpam-4797	292	31	43	43	NUM
ejpam-4797	292	32	,	,	PUNCT
ejpam-4797	292	33	47	47	NUM
ejpam-4797	292	34	}	}	PUNCT
ejpam-4797	292	35	,	,	PUNCT
ejpam-4797	292	36	nil(z48	nil(z48	X
ejpam-4797	292	37	)	)	PUNCT
ejpam-4797	292	38	=	=	PUNCT
ejpam-4797	292	39	{	{	PUNCT
ejpam-4797	292	40	0	0	NUM
ejpam-4797	292	41	,	,	PUNCT
ejpam-4797	292	42	6	6	NUM
ejpam-4797	292	43	,	,	PUNCT
ejpam-4797	292	44	12	12	NUM
ejpam-4797	292	45	,	,	PUNCT
ejpam-4797	292	46	18	18	NUM
ejpam-4797	292	47	,	,	PUNCT
ejpam-4797	292	48	24	24	NUM
ejpam-4797	292	49	,	,	PUNCT
ejpam-4797	292	50	30	30	NUM
ejpam-4797	292	51	,	,	PUNCT
ejpam-4797	292	52	36	36	NUM
ejpam-4797	292	53	,	,	PUNCT
ejpam-4797	292	54	42	42	NUM
ejpam-4797	292	55	}	}	PUNCT
ejpam-4797	292	56	,	,	PUNCT
ejpam-4797	292	57	id(z48	id(z48	PROPN
ejpam-4797	292	58	)	)	PUNCT
ejpam-4797	292	59	=	=	PUNCT
ejpam-4797	292	60	{	{	PUNCT
ejpam-4797	292	61	0	0	NUM
ejpam-4797	292	62	,	,	PUNCT
ejpam-4797	292	63	1	1	NUM
ejpam-4797	292	64	,	,	PUNCT
ejpam-4797	292	65	16	16	NUM
ejpam-4797	292	66	,	,	PUNCT
ejpam-4797	292	67	33	33	NUM
ejpam-4797	292	68	}	}	PUNCT
ejpam-4797	292	69	.	.	PUNCT
ejpam-4797	293	1	by	by	ADP
ejpam-4797	293	2	direct	direct	ADJ
ejpam-4797	293	3	calculation	calculation	NOUN
ejpam-4797	293	4	,	,	PUNCT
ejpam-4797	293	5	one	one	NUM
ejpam-4797	293	6	easily	easily	ADV
ejpam-4797	293	7	check	check	VERB
ejpam-4797	293	8	that	that	SCONJ
ejpam-4797	293	9	z48	z48	PROPN
ejpam-4797	293	10	is	be	AUX
ejpam-4797	293	11	a	a	DET
ejpam-4797	293	12	strongly	strongly	ADV
ejpam-4797	293	13	2	2	NUM
ejpam-4797	293	14	-	-	PUNCT
ejpam-4797	293	15	nc	nc	NOUN
ejpam-4797	293	16	ring	ring	NOUN
ejpam-4797	293	17	,	,	PUNCT
ejpam-4797	293	18	with	with	SCONJ
ejpam-4797	293	19	every	every	DET
ejpam-4797	293	20	unit	unit	NOUN
ejpam-4797	293	21	is	be	AUX
ejpam-4797	293	22	of	of	ADP
ejpam-4797	293	23	order	order	NOUN
ejpam-4797	293	24	4	4	NUM
ejpam-4797	293	25	.	.	NOUN
ejpam-4797	293	26	4	4	NUM
ejpam-4797	293	27	.	.	X
ejpam-4797	293	28	conclusion	conclusion	NOUN
ejpam-4797	293	29	in	in	ADP
ejpam-4797	293	30	this	this	DET
ejpam-4797	293	31	article	article	NOUN
ejpam-4797	293	32	,	,	PUNCT
ejpam-4797	293	33	new	new	ADJ
ejpam-4797	293	34	properties	property	NOUN
ejpam-4797	293	35	of	of	ADP
ejpam-4797	293	36	a	a	DET
ejpam-4797	293	37	strongly	strongly	ADV
ejpam-4797	293	38	2	2	NUM
ejpam-4797	293	39	-	-	PUNCT
ejpam-4797	293	40	nc	nc	PROPN
ejpam-4797	293	41	rings	ring	NOUN
ejpam-4797	293	42	are	be	AUX
ejpam-4797	293	43	given	give	VERB
ejpam-4797	293	44	.	.	PUNCT
ejpam-4797	294	1	additionally	additionally	ADV
ejpam-4797	294	2	,	,	PUNCT
ejpam-4797	294	3	we	we	PRON
ejpam-4797	294	4	added	add	VERB
ejpam-4797	294	5	certain	certain	ADJ
ejpam-4797	294	6	conditions	condition	NOUN
ejpam-4797	294	7	for	for	ADP
ejpam-4797	294	8	strongly	strongly	ADV
ejpam-4797	294	9	2	2	NUM
ejpam-4797	294	10	-	-	PUNCT
ejpam-4797	294	11	nc	nc	NOUN
ejpam-4797	294	12	ring	ring	NOUN
ejpam-4797	294	13	with	with	ADP
ejpam-4797	294	14	each	each	DET
ejpam-4797	294	15	unit	unit	NOUN
ejpam-4797	294	16	must	must	AUX
ejpam-4797	294	17	be	be	AUX
ejpam-4797	294	18	present	present	ADJ
ejpam-4797	294	19	of	of	ADP
ejpam-4797	294	20	order	order	NOUN
ejpam-4797	294	21	four	four	NUM
ejpam-4797	294	22	.	.	PUNCT
ejpam-4797	295	1	we	we	PRON
ejpam-4797	295	2	also	also	ADV
ejpam-4797	295	3	introduce	introduce	VERB
ejpam-4797	295	4	and	and	CCONJ
ejpam-4797	295	5	investigated	investigate	VERB
ejpam-4797	295	6	a	a	DET
ejpam-4797	295	7	strongly	strongly	ADV
ejpam-4797	295	8	2	2	NUM
ejpam-4797	295	9	-	-	PUNCT
ejpam-4797	295	10	nc	nc	NOUN
ejpam-4797	295	11	ring	ring	NOUN
ejpam-4797	295	12	with	with	ADP
ejpam-4797	295	13	every	every	DET
ejpam-4797	295	14	unit	unit	NOUN
ejpam-4797	295	15	of	of	ADP
ejpam-4797	295	16	order	order	NOUN
ejpam-4797	295	17	two	two	NUM
ejpam-4797	295	18	.	.	PUNCT
ejpam-4797	296	1	we	we	PRON
ejpam-4797	296	2	discuss	discuss	VERB
ejpam-4797	296	3	some	some	PRON
ejpam-4797	296	4	of	of	ADP
ejpam-4797	296	5	the	the	DET
ejpam-4797	296	6	fundamental	fundamental	ADJ
ejpam-4797	296	7	properties	property	NOUN
ejpam-4797	296	8	and	and	CCONJ
ejpam-4797	296	9	present	present	VERB
ejpam-4797	296	10	several	several	ADJ
ejpam-4797	296	11	examples	example	NOUN
ejpam-4797	296	12	.	.	PUNCT
ejpam-4797	297	1	it	it	PRON
ejpam-4797	297	2	was	be	AUX
ejpam-4797	297	3	proved	prove	VERB
ejpam-4797	297	4	that	that	SCONJ
ejpam-4797	297	5	the	the	DET
ejpam-4797	297	6	jacobson	jacobson	PROPN
ejpam-4797	297	7	radical	radical	ADJ
ejpam-4797	297	8	over	over	ADP
ejpam-4797	297	9	a	a	DET
ejpam-4797	297	10	strongly	strongly	ADV
ejpam-4797	297	11	2	2	NUM
ejpam-4797	297	12	-	-	PUNCT
ejpam-4797	297	13	nc	nc	PROPN
ejpam-4797	297	14	ring	ring	NOUN
ejpam-4797	297	15	is	be	AUX
ejpam-4797	297	16	a	a	DET
ejpam-4797	297	17	nil	nil	ADJ
ejpam-4797	297	18	ideal	ideal	NOUN
ejpam-4797	297	19	,	,	PUNCT
ejpam-4797	297	20	here	here	ADV
ejpam-4797	297	21	,	,	PUNCT
ejpam-4797	297	22	we	we	PRON
ejpam-4797	297	23	demonstrated	demonstrate	VERB
ejpam-4797	297	24	that	that	SCONJ
ejpam-4797	297	25	the	the	DET
ejpam-4797	297	26	jacobson	jacobson	PROPN
ejpam-4797	297	27	radical	radical	PROPN
ejpam-4797	297	28	over	over	ADP
ejpam-4797	297	29	strongly	strongly	ADV
ejpam-4797	297	30	2	2	NUM
ejpam-4797	297	31	-	-	PUNCT
ejpam-4797	297	32	nc	nc	NOUN
ejpam-4797	297	33	ring	ring	NOUN
ejpam-4797	297	34	with	with	ADP
ejpam-4797	297	35	u(r	u(r	NOUN
ejpam-4797	297	36	)	)	PUNCT
ejpam-4797	298	1	=	=	SYM
ejpam-4797	298	2	2	2	NUM
ejpam-4797	298	3	is	be	AUX
ejpam-4797	298	4	a	a	DET
ejpam-4797	298	5	nil	nil	ADJ
ejpam-4797	298	6	ideal	ideal	NOUN
ejpam-4797	298	7	of	of	ADP
ejpam-4797	298	8	characteristic	characteristic	ADJ
ejpam-4797	298	9	4	4	NUM
ejpam-4797	298	10	.	.	PUNCT
ejpam-4797	299	1	in	in	ADP
ejpam-4797	299	2	order	order	NOUN
ejpam-4797	299	3	to	to	PART
ejpam-4797	299	4	getnil(r	getnil(r	VERB
ejpam-4797	299	5	)	)	PUNCT
ejpam-4797	299	6	=	=	SYM
ejpam-4797	299	7	0	0	NUM
ejpam-4797	299	8	,	,	PUNCT
ejpam-4797	299	9	we	we	PRON
ejpam-4797	299	10	added	add	VERB
ejpam-4797	299	11	one	one	NUM
ejpam-4797	299	12	more	more	ADJ
ejpam-4797	299	13	condition	condition	NOUN
ejpam-4797	299	14	involving	involve	VERB
ejpam-4797	299	15	this	this	DET
ejpam-4797	299	16	ring	ring	NOUN
ejpam-4797	299	17	.	.	PUNCT
ejpam-4797	300	1	the	the	DET
ejpam-4797	300	2	relationships	relationship	NOUN
ejpam-4797	300	3	between	between	ADP
ejpam-4797	300	4	these	these	DET
ejpam-4797	300	5	rings	ring	NOUN
ejpam-4797	300	6	,	,	PUNCT
ejpam-4797	300	7	tripotent	tripotent	NOUN
ejpam-4797	300	8	rings	ring	NOUN
ejpam-4797	300	9	,	,	PUNCT
ejpam-4797	300	10	and	and	CCONJ
ejpam-4797	300	11	other	other	ADJ
ejpam-4797	300	12	related	related	ADJ
ejpam-4797	300	13	rings	ring	NOUN
ejpam-4797	300	14	are	be	AUX
ejpam-4797	300	15	given	give	VERB
ejpam-4797	300	16	.	.	PUNCT
ejpam-4797	301	1	future	future	ADJ
ejpam-4797	301	2	goals	goal	NOUN
ejpam-4797	301	3	include	include	VERB
ejpam-4797	301	4	obtaining	obtain	VERB
ejpam-4797	301	5	a	a	DET
ejpam-4797	301	6	deeper	deep	ADJ
ejpam-4797	301	7	outcome	outcome	NOUN
ejpam-4797	301	8	on	on	ADP
ejpam-4797	301	9	issues	issue	NOUN
ejpam-4797	301	10	raised	raise	VERB
ejpam-4797	301	11	in	in	ADP
ejpam-4797	301	12	this	this	DET
ejpam-4797	301	13	article	article	NOUN
ejpam-4797	301	14	,	,	PUNCT
ejpam-4797	301	15	such	such	ADJ
ejpam-4797	301	16	as	as	ADP
ejpam-4797	301	17	1	1	NUM
ejpam-4797	301	18	.	.	PUNCT
ejpam-4797	302	1	the	the	DET
ejpam-4797	302	2	snc	snc	PROPN
ejpam-4797	302	3	ring	ring	NOUN
ejpam-4797	302	4	with	with	ADP
ejpam-4797	302	5	u(r	u(r	NOUN
ejpam-4797	302	6	)	)	PUNCT
ejpam-4797	302	7	=	=	SYM
ejpam-4797	303	1	2	2	NUM
ejpam-4797	303	2	,	,	PUNCT
ejpam-4797	303	3	3	3	NUM
ejpam-4797	303	4	or	or	CCONJ
ejpam-4797	303	5	4	4	NUM
ejpam-4797	303	6	.	.	PUNCT
ejpam-4797	303	7	references	reference	NOUN
ejpam-4797	303	8	1683	1683	NUM
ejpam-4797	303	9	2	2	NUM
ejpam-4797	303	10	.	.	PUNCT
ejpam-4797	304	1	the	the	DET
ejpam-4797	304	2	strongly	strongly	ADV
ejpam-4797	304	3	2	2	NUM
ejpam-4797	304	4	-	-	PUNCT
ejpam-4797	304	5	nc	nc	NOUN
ejpam-4797	304	6	ring	ring	NOUN
ejpam-4797	304	7	with	with	ADP
ejpam-4797	304	8	u(r	u(r	NOUN
ejpam-4797	304	9	)	)	PUNCT
ejpam-4797	305	1	=	=	SYM
ejpam-4797	305	2	3	3	NUM
ejpam-4797	305	3	or	or	CCONJ
ejpam-4797	305	4	4	4	NUM
ejpam-4797	305	5	.	.	NOUN
ejpam-4797	305	6	3	3	NUM
ejpam-4797	305	7	.	.	X
ejpam-4797	306	1	the	the	DET
ejpam-4797	306	2	divisor	divisor	NOUN
ejpam-4797	306	3	graph	graph	NOUN
ejpam-4797	306	4	of	of	ADP
ejpam-4797	306	5	a	a	DET
ejpam-4797	306	6	strongly	strongly	ADV
ejpam-4797	306	7	2	2	NUM
ejpam-4797	306	8	-	-	PUNCT
ejpam-4797	306	9	nc	nc	NOUN
ejpam-4797	306	10	ring	ring	NOUN
ejpam-4797	306	11	with	with	ADP
ejpam-4797	306	12	u(r	u(r	NOUN
ejpam-4797	306	13	)	)	PUNCT
ejpam-4797	307	1	=	=	SYM
ejpam-4797	307	2	2	2	X
ejpam-4797	307	3	.	.	NUM
ejpam-4797	307	4	references	reference	NOUN
ejpam-4797	307	5	[	[	X
ejpam-4797	307	6	1	1	NUM
ejpam-4797	307	7	]	]	PUNCT
ejpam-4797	307	8	w.	w.	PROPN
ejpam-4797	307	9	k.	k.	PROPN
ejpam-4797	307	10	nicholson	nicholson	PROPN
ejpam-4797	307	11	,	,	PUNCT
ejpam-4797	307	12	“	"	PUNCT
ejpam-4797	307	13	lifting	lift	VERB
ejpam-4797	307	14	idempotents	idempotent	NOUN
ejpam-4797	307	15	and	and	CCONJ
ejpam-4797	307	16	exchange	exchange	NOUN
ejpam-4797	307	17	rings	ring	NOUN
ejpam-4797	307	18	,	,	PUNCT
ejpam-4797	307	19	”	"	PUNCT
ejpam-4797	307	20	transactions	transaction	NOUN
ejpam-4797	307	21	of	of	ADP
ejpam-4797	307	22	the	the	DET
ejpam-4797	307	23	american	american	PROPN
ejpam-4797	307	24	mathematical	mathematical	PROPN
ejpam-4797	307	25	society	society	NOUN
ejpam-4797	307	26	,	,	PUNCT
ejpam-4797	307	27	vol	vol	NOUN
ejpam-4797	307	28	.	.	PROPN
ejpam-4797	307	29	229	229	NUM
ejpam-4797	307	30	,	,	PUNCT
ejpam-4797	307	31	pp	pp	ADJ
ejpam-4797	307	32	.	.	PUNCT
ejpam-4797	308	1	269–278	269–278	NUM
ejpam-4797	308	2	,	,	PUNCT
ejpam-4797	308	3	1977	1977	NUM
ejpam-4797	308	4	.	.	PUNCT
ejpam-4797	309	1	[	[	X
ejpam-4797	309	2	2	2	X
ejpam-4797	309	3	]	]	PUNCT
ejpam-4797	309	4	w.	w.	PROPN
ejpam-4797	309	5	k.	k.	PROPN
ejpam-4797	309	6	nicholson	nicholson	PROPN
ejpam-4797	309	7	,	,	PUNCT
ejpam-4797	309	8	“	"	PUNCT
ejpam-4797	309	9	strongly	strongly	ADV
ejpam-4797	309	10	clean	clean	ADJ
ejpam-4797	309	11	rings	ring	NOUN
ejpam-4797	309	12	and	and	CCONJ
ejpam-4797	309	13	fitting	fitting	PROPN
ejpam-4797	309	14	’s	’s	PART
ejpam-4797	309	15	lemma	lemma	PROPN
ejpam-4797	309	16	,	,	PUNCT
ejpam-4797	309	17	”	"	PUNCT
ejpam-4797	309	18	communications	communication	NOUN
ejpam-4797	309	19	in	in	ADP
ejpam-4797	309	20	algebra	algebra	NOUN
ejpam-4797	309	21	,	,	PUNCT
ejpam-4797	309	22	vol	vol	NOUN
ejpam-4797	309	23	.	.	PROPN
ejpam-4797	309	24	27	27	NUM
ejpam-4797	309	25	,	,	PUNCT
ejpam-4797	309	26	no	no	INTJ
ejpam-4797	309	27	.	.	NOUN
ejpam-4797	309	28	8	8	NUM
ejpam-4797	309	29	,	,	PUNCT
ejpam-4797	309	30	pp	pp	ADJ
ejpam-4797	309	31	.	.	PUNCT
ejpam-4797	309	32	3583–3592	3583–3592	NUM
ejpam-4797	309	33	,	,	PUNCT
ejpam-4797	309	34	1999	1999	NUM
ejpam-4797	309	35	.	.	PUNCT
ejpam-4797	310	1	[	[	X
ejpam-4797	310	2	3	3	NUM
ejpam-4797	310	3	]	]	PUNCT
ejpam-4797	310	4	a.	a.	NOUN
ejpam-4797	310	5	j.	j.	PROPN
ejpam-4797	310	6	diesl	diesl	PROPN
ejpam-4797	310	7	,	,	PUNCT
ejpam-4797	310	8	“	"	PUNCT
ejpam-4797	310	9	nil	nil	ADJ
ejpam-4797	310	10	clean	clean	ADJ
ejpam-4797	310	11	rings	ring	NOUN
ejpam-4797	310	12	,	,	PUNCT
ejpam-4797	310	13	”	"	PUNCT
ejpam-4797	310	14	journal	journal	NOUN
ejpam-4797	310	15	of	of	ADP
ejpam-4797	310	16	algebra	algebra	PROPN
ejpam-4797	310	17	,	,	PUNCT
ejpam-4797	310	18	vol	vol	NOUN
ejpam-4797	310	19	.	.	NOUN
ejpam-4797	310	20	383	383	NUM
ejpam-4797	310	21	,	,	PUNCT
ejpam-4797	310	22	pp	pp	ADJ
ejpam-4797	310	23	.	.	PUNCT
ejpam-4797	311	1	197–211	197–211	NUM
ejpam-4797	311	2	,	,	PUNCT
ejpam-4797	311	3	2013	2013	NUM
ejpam-4797	311	4	.	.	PUNCT
ejpam-4797	312	1	[	[	X
ejpam-4797	312	2	4	4	X
ejpam-4797	312	3	]	]	PUNCT
ejpam-4797	312	4	m.	m.	NOUN
ejpam-4797	312	5	t.	t.	PROPN
ejpam-4797	312	6	koşan	koşan	PROPN
ejpam-4797	312	7	and	and	CCONJ
ejpam-4797	312	8	y.	y.	PROPN
ejpam-4797	312	9	zhou	zhou	PROPN
ejpam-4797	312	10	,	,	PUNCT
ejpam-4797	312	11	“	"	PUNCT
ejpam-4797	312	12	on	on	ADP
ejpam-4797	312	13	weakly	weakly	ADJ
ejpam-4797	312	14	nil	nil	ADJ
ejpam-4797	312	15	-	-	PUNCT
ejpam-4797	312	16	clean	clean	ADJ
ejpam-4797	312	17	rings	ring	NOUN
ejpam-4797	312	18	,	,	PUNCT
ejpam-4797	312	19	”	"	PUNCT
ejpam-4797	312	20	frontiers	frontier	NOUN
ejpam-4797	312	21	of	of	ADP
ejpam-4797	312	22	mathematics	mathematic	NOUN
ejpam-4797	312	23	in	in	ADP
ejpam-4797	312	24	china	china	PROPN
ejpam-4797	312	25	,	,	PUNCT
ejpam-4797	312	26	vol	vol	NOUN
ejpam-4797	312	27	.	.	PROPN
ejpam-4797	312	28	11	11	NUM
ejpam-4797	312	29	,	,	PUNCT
ejpam-4797	312	30	no	no	INTJ
ejpam-4797	312	31	.	.	NOUN
ejpam-4797	312	32	4	4	NUM
ejpam-4797	312	33	,	,	PUNCT
ejpam-4797	312	34	pp	pp	ADJ
ejpam-4797	312	35	.	.	PUNCT
ejpam-4797	313	1	949–955	949–955	NUM
ejpam-4797	313	2	,	,	PUNCT
ejpam-4797	313	3	2016	2016	NUM
ejpam-4797	313	4	.	.	PUNCT
ejpam-4797	314	1	[	[	X
ejpam-4797	314	2	5	5	X
ejpam-4797	314	3	]	]	X
ejpam-4797	314	4	y.	y.	PROPN
ejpam-4797	314	5	hirano	hirano	PROPN
ejpam-4797	314	6	,	,	PUNCT
ejpam-4797	314	7	h.	h.	PROPN
ejpam-4797	314	8	tominaga	tominaga	PROPN
ejpam-4797	314	9	,	,	PUNCT
ejpam-4797	314	10	and	and	CCONJ
ejpam-4797	314	11	a.	a.	NOUN
ejpam-4797	314	12	yaqub	yaqub	NOUN
ejpam-4797	314	13	,	,	PUNCT
ejpam-4797	314	14	“	"	PUNCT
ejpam-4797	314	15	on	on	ADP
ejpam-4797	314	16	rings	ring	NOUN
ejpam-4797	314	17	in	in	ADP
ejpam-4797	314	18	which	which	PRON
ejpam-4797	314	19	every	every	DET
ejpam-4797	314	20	element	element	NOUN
ejpam-4797	314	21	is	be	AUX
ejpam-4797	314	22	uniquely	uniquely	ADV
ejpam-4797	314	23	expressible	expressible	ADJ
ejpam-4797	314	24	as	as	ADP
ejpam-4797	314	25	a	a	DET
ejpam-4797	314	26	sum	sum	NOUN
ejpam-4797	314	27	of	of	ADP
ejpam-4797	314	28	a	a	DET
ejpam-4797	314	29	nilpotent	nilpotent	ADJ
ejpam-4797	314	30	element	element	NOUN
ejpam-4797	314	31	and	and	CCONJ
ejpam-4797	314	32	a	a	DET
ejpam-4797	314	33	certain	certain	ADJ
ejpam-4797	314	34	potent	potent	ADJ
ejpam-4797	314	35	element	element	NOUN
ejpam-4797	314	36	,	,	PUNCT
ejpam-4797	314	37	”	"	PUNCT
ejpam-4797	314	38	mathematical	mathematical	ADJ
ejpam-4797	314	39	journal	journal	NOUN
ejpam-4797	314	40	of	of	ADP
ejpam-4797	314	41	okayama	okayama	PROPN
ejpam-4797	314	42	university	university	PROPN
ejpam-4797	314	43	,	,	PUNCT
ejpam-4797	314	44	vol	vol	NOUN
ejpam-4797	314	45	.	.	PROPN
ejpam-4797	314	46	30	30	NUM
ejpam-4797	314	47	,	,	PUNCT
ejpam-4797	314	48	no	no	INTJ
ejpam-4797	314	49	.	.	NOUN
ejpam-4797	314	50	1	1	NUM
ejpam-4797	314	51	,	,	PUNCT
ejpam-4797	314	52	pp	pp	ADJ
ejpam-4797	314	53	.	.	PUNCT
ejpam-4797	315	1	33–40	33–40	NUM
ejpam-4797	315	2	,	,	PUNCT
ejpam-4797	315	3	1988	1988	NUM
ejpam-4797	315	4	.	.	PUNCT
ejpam-4797	316	1	[	[	X
ejpam-4797	316	2	6	6	NUM
ejpam-4797	316	3	]	]	PUNCT
ejpam-4797	316	4	t.	t.	PROPN
ejpam-4797	316	5	koşan	koşan	PROPN
ejpam-4797	316	6	,	,	PUNCT
ejpam-4797	316	7	z.	z.	PROPN
ejpam-4797	316	8	wang	wang	PROPN
ejpam-4797	316	9	,	,	PUNCT
ejpam-4797	316	10	and	and	CCONJ
ejpam-4797	316	11	y.	y.	PROPN
ejpam-4797	316	12	zhou	zhou	PROPN
ejpam-4797	316	13	,	,	PUNCT
ejpam-4797	316	14	“	"	PUNCT
ejpam-4797	316	15	nil	nil	ADJ
ejpam-4797	316	16	-	-	ADJ
ejpam-4797	316	17	clean	clean	ADJ
ejpam-4797	316	18	and	and	CCONJ
ejpam-4797	316	19	strongly	strongly	ADV
ejpam-4797	316	20	nil	nil	ADJ
ejpam-4797	316	21	-	-	PUNCT
ejpam-4797	316	22	clean	clean	ADJ
ejpam-4797	316	23	rings	ring	NOUN
ejpam-4797	316	24	,	,	PUNCT
ejpam-4797	316	25	”	"	PUNCT
ejpam-4797	316	26	journal	journal	NOUN
ejpam-4797	316	27	of	of	ADP
ejpam-4797	316	28	pure	pure	ADJ
ejpam-4797	316	29	and	and	CCONJ
ejpam-4797	316	30	applied	applied	ADJ
ejpam-4797	316	31	algebra	algebra	NOUN
ejpam-4797	316	32	,	,	PUNCT
ejpam-4797	316	33	vol	vol	NOUN
ejpam-4797	316	34	.	.	NOUN
ejpam-4797	316	35	220	220	NUM
ejpam-4797	316	36	,	,	PUNCT
ejpam-4797	316	37	no	no	INTJ
ejpam-4797	316	38	.	.	NOUN
ejpam-4797	316	39	2	2	NUM
ejpam-4797	316	40	,	,	PUNCT
ejpam-4797	316	41	pp	pp	ADJ
ejpam-4797	316	42	.	.	PUNCT
ejpam-4797	317	1	633–646	633–646	NUM
ejpam-4797	317	2	,	,	PUNCT
ejpam-4797	317	3	2016	2016	NUM
ejpam-4797	317	4	.	.	PUNCT
ejpam-4797	318	1	[	[	X
ejpam-4797	318	2	7	7	X
ejpam-4797	318	3	]	]	X
ejpam-4797	318	4	h.	h.	PROPN
ejpam-4797	318	5	chen	chen	PROPN
ejpam-4797	318	6	and	and	CCONJ
ejpam-4797	318	7	m.	m.	NOUN
ejpam-4797	318	8	sheibani	sheibani	NOUN
ejpam-4797	318	9	,	,	PUNCT
ejpam-4797	318	10	“	"	PUNCT
ejpam-4797	318	11	strongly	strongly	ADV
ejpam-4797	318	12	2	2	NUM
ejpam-4797	318	13	-	-	PUNCT
ejpam-4797	318	14	nil	nil	ADJ
ejpam-4797	318	15	-	-	PUNCT
ejpam-4797	318	16	clean	clean	ADJ
ejpam-4797	318	17	rings	ring	NOUN
ejpam-4797	318	18	,	,	PUNCT
ejpam-4797	318	19	”	"	PUNCT
ejpam-4797	318	20	journal	journal	NOUN
ejpam-4797	318	21	of	of	ADP
ejpam-4797	318	22	algebra	algebra	PROPN
ejpam-4797	318	23	and	and	CCONJ
ejpam-4797	318	24	its	its	PRON
ejpam-4797	318	25	applications	application	NOUN
ejpam-4797	318	26	,	,	PUNCT
ejpam-4797	318	27	vol	vol	NOUN
ejpam-4797	318	28	.	.	PROPN
ejpam-4797	318	29	16	16	NUM
ejpam-4797	318	30	,	,	PUNCT
ejpam-4797	318	31	no	no	INTJ
ejpam-4797	318	32	.	.	NOUN
ejpam-4797	318	33	09	09	NUM
ejpam-4797	318	34	,	,	PUNCT
ejpam-4797	318	35	p.	p.	NOUN
ejpam-4797	318	36	1750178	1750178	NUM
ejpam-4797	318	37	,	,	PUNCT
ejpam-4797	318	38	2017	2017	NUM
ejpam-4797	318	39	.	.	PUNCT
ejpam-4797	319	1	[	[	X
ejpam-4797	319	2	8	8	X
ejpam-4797	319	3	]	]	X
ejpam-4797	319	4	h.	h.	PROPN
ejpam-4797	319	5	chen	chen	PROPN
ejpam-4797	319	6	and	and	CCONJ
ejpam-4797	319	7	m.	m.	PROPN
ejpam-4797	319	8	s.	s.	PROPN
ejpam-4797	319	9	abdolyousefi	abdolyousefi	PROPN
ejpam-4797	319	10	,	,	PUNCT
ejpam-4797	319	11	“	"	PUNCT
ejpam-4797	319	12	strongly	strongly	ADV
ejpam-4797	319	13	2	2	NUM
ejpam-4797	319	14	-	-	PUNCT
ejpam-4797	319	15	nil	nil	ADV
ejpam-4797	319	16	-	-	PUNCT
ejpam-4797	319	17	clean	clean	ADJ
ejpam-4797	319	18	rings	ring	NOUN
ejpam-4797	319	19	with	with	ADP
ejpam-4797	319	20	involutions	involution	NOUN
ejpam-4797	319	21	,	,	PUNCT
ejpam-4797	319	22	”	"	PUNCT
ejpam-4797	319	23	czechoslovak	czechoslovak	ADJ
ejpam-4797	319	24	mathematical	mathematical	ADJ
ejpam-4797	319	25	journal	journal	PROPN
ejpam-4797	319	26	,	,	PUNCT
ejpam-4797	319	27	vol	vol	NOUN
ejpam-4797	319	28	.	.	PROPN
ejpam-4797	319	29	69	69	NUM
ejpam-4797	319	30	,	,	PUNCT
ejpam-4797	319	31	no	no	INTJ
ejpam-4797	319	32	.	.	NOUN
ejpam-4797	319	33	2	2	NUM
ejpam-4797	319	34	,	,	PUNCT
ejpam-4797	319	35	pp	pp	ADJ
ejpam-4797	319	36	.	.	PUNCT
ejpam-4797	320	1	317–330	317–330	NUM
ejpam-4797	320	2	,	,	PUNCT
ejpam-4797	320	3	2019	2019	NUM
ejpam-4797	320	4	.	.	PUNCT
ejpam-4797	321	1	[	[	X
ejpam-4797	321	2	9	9	X
ejpam-4797	321	3	]	]	PUNCT
ejpam-4797	321	4	j.	j.	PROPN
ejpam-4797	321	5	chen	chen	PROPN
ejpam-4797	321	6	,	,	PUNCT
ejpam-4797	321	7	y.	y.	PROPN
ejpam-4797	321	8	wang	wang	PROPN
ejpam-4797	321	9	,	,	PUNCT
ejpam-4797	321	10	and	and	CCONJ
ejpam-4797	321	11	y.	y.	PROPN
ejpam-4797	321	12	ren	ren	PROPN
ejpam-4797	321	13	,	,	PUNCT
ejpam-4797	321	14	“	"	PUNCT
ejpam-4797	321	15	2	2	NUM
ejpam-4797	321	16	-	-	PUNCT
ejpam-4797	321	17	nil	nil	ADJ
ejpam-4797	321	18	-	-	PUNCT
ejpam-4797	321	19	clean	clean	ADJ
ejpam-4797	321	20	rings	ring	NOUN
ejpam-4797	321	21	,	,	PUNCT
ejpam-4797	321	22	”	"	PUNCT
ejpam-4797	321	23	journal	journal	NOUN
ejpam-4797	321	24	of	of	ADP
ejpam-4797	321	25	shandong	shandong	PROPN
ejpam-4797	321	26	university(natural	university(natural	ADJ
ejpam-4797	321	27	science	science	NOUN
ejpam-4797	321	28	)	)	PUNCT
ejpam-4797	321	29	,	,	PUNCT
ejpam-4797	321	30	vol	vol	NOUN
ejpam-4797	321	31	.	.	PROPN
ejpam-4797	321	32	57	57	NUM
ejpam-4797	321	33	,	,	PUNCT
ejpam-4797	321	34	no	no	INTJ
ejpam-4797	321	35	.	.	NOUN
ejpam-4797	321	36	2	2	NUM
ejpam-4797	321	37	,	,	PUNCT
ejpam-4797	321	38	p.	p.	NOUN
ejpam-4797	321	39	14	14	NUM
ejpam-4797	321	40	,	,	PUNCT
ejpam-4797	321	41	2022	2022	NUM
ejpam-4797	321	42	.	.	PUNCT
ejpam-4797	322	1	[	[	X
ejpam-4797	322	2	10	10	NUM
ejpam-4797	322	3	]	]	PUNCT
ejpam-4797	322	4	m.	m.	NOUN
ejpam-4797	322	5	s.	s.	PROPN
ejpam-4797	322	6	abdolyousefi	abdolyousefi	PROPN
ejpam-4797	322	7	,	,	PUNCT
ejpam-4797	322	8	n.	n.	PROPN
ejpam-4797	322	9	ashrafi	ashrafi	PROPN
ejpam-4797	322	10	,	,	PUNCT
ejpam-4797	322	11	and	and	CCONJ
ejpam-4797	322	12	h.	h.	PROPN
ejpam-4797	322	13	chen	chen	PROPN
ejpam-4797	322	14	,	,	PUNCT
ejpam-4797	322	15	“	"	PUNCT
ejpam-4797	322	16	on	on	ADP
ejpam-4797	322	17	2	2	NUM
ejpam-4797	322	18	-	-	PUNCT
ejpam-4797	322	19	nil	nil	ADJ
ejpam-4797	322	20	-	-	PUNCT
ejpam-4797	322	21	good	good	ADJ
ejpam-4797	322	22	rings	ring	NOUN
ejpam-4797	322	23	,	,	PUNCT
ejpam-4797	322	24	”	"	PUNCT
ejpam-4797	322	25	journal	journal	NOUN
ejpam-4797	322	26	of	of	ADP
ejpam-4797	322	27	algebra	algebra	PROPN
ejpam-4797	322	28	and	and	CCONJ
ejpam-4797	322	29	its	its	PRON
ejpam-4797	322	30	applications	application	NOUN
ejpam-4797	322	31	,	,	PUNCT
ejpam-4797	322	32	vol	vol	NOUN
ejpam-4797	322	33	.	.	PROPN
ejpam-4797	322	34	17	17	NUM
ejpam-4797	322	35	,	,	PUNCT
ejpam-4797	322	36	no	no	INTJ
ejpam-4797	322	37	.	.	NOUN
ejpam-4797	322	38	06	06	NUM
ejpam-4797	322	39	,	,	PUNCT
ejpam-4797	322	40	p.	p.	NOUN
ejpam-4797	322	41	1850110	1850110	NUM
ejpam-4797	322	42	,	,	PUNCT
ejpam-4797	322	43	2018	2018	NUM
ejpam-4797	322	44	.	.	PUNCT
ejpam-4797	323	1	[	[	X
ejpam-4797	323	2	11	11	NUM
ejpam-4797	323	3	]	]	X
ejpam-4797	323	4	y.	y.	PROPN
ejpam-4797	323	5	rao	rao	PROPN
ejpam-4797	323	6	,	,	PUNCT
ejpam-4797	323	7	s.	s.	PROPN
ejpam-4797	323	8	kosari	kosari	PROPN
ejpam-4797	323	9	,	,	PUNCT
ejpam-4797	323	10	h.	h.	PROPN
ejpam-4797	323	11	guan	guan	PROPN
ejpam-4797	323	12	,	,	PUNCT
ejpam-4797	323	13	m.	m.	NOUN
ejpam-4797	323	14	akhoundi	akhoundi	PROPN
ejpam-4797	323	15	,	,	PUNCT
ejpam-4797	323	16	and	and	CCONJ
ejpam-4797	323	17	s.	s.	PROPN
ejpam-4797	323	18	omidi	omidi	PROPN
ejpam-4797	323	19	,	,	PUNCT
ejpam-4797	323	20	“	"	PUNCT
ejpam-4797	323	21	a	a	DET
ejpam-4797	323	22	short	short	ADJ
ejpam-4797	323	23	note	note	NOUN
ejpam-4797	323	24	on	on	ADP
ejpam-4797	323	25	the	the	DET
ejpam-4797	323	26	left	left	ADJ
ejpam-4797	323	27	a	a	DET
ejpam-4797	323	28	-	-	PUNCT
ejpam-4797	323	29	γ	γ	NOUN
ejpam-4797	323	30	-	-	PUNCT
ejpam-4797	323	31	hyperideals	hyperideal	NOUN
ejpam-4797	323	32	in	in	ADP
ejpam-4797	323	33	ordered	order	VERB
ejpam-4797	323	34	γ	γ	NOUN
ejpam-4797	323	35	-	-	PUNCT
ejpam-4797	323	36	semihypergroups	semihypergroup	NOUN
ejpam-4797	323	37	,	,	PUNCT
ejpam-4797	323	38	”	"	PUNCT
ejpam-4797	323	39	akce	akce	PROPN
ejpam-4797	323	40	international	international	ADJ
ejpam-4797	323	41	journal	journal	NOUN
ejpam-4797	323	42	of	of	ADP
ejpam-4797	323	43	graphs	graph	NOUN
ejpam-4797	323	44	and	and	CCONJ
ejpam-4797	323	45	combinatorics	combinatoric	NOUN
ejpam-4797	323	46	,	,	PUNCT
ejpam-4797	323	47	vol	vol	NOUN
ejpam-4797	323	48	.	.	PROPN
ejpam-4797	323	49	19	19	NUM
ejpam-4797	323	50	,	,	PUNCT
ejpam-4797	323	51	no	no	INTJ
ejpam-4797	323	52	.	.	NOUN
ejpam-4797	323	53	1	1	NUM
ejpam-4797	323	54	,	,	PUNCT
ejpam-4797	323	55	pp	pp	ADJ
ejpam-4797	323	56	.	.	PUNCT
ejpam-4797	324	1	49–53	49–53	NUM
ejpam-4797	324	2	,	,	PUNCT
ejpam-4797	324	3	2022	2022	NUM
ejpam-4797	324	4	.	.	PUNCT
ejpam-4797	325	1	[	[	X
ejpam-4797	325	2	12	12	NUM
ejpam-4797	325	3	]	]	PUNCT
ejpam-4797	325	4	z.	z.	PROPN
ejpam-4797	325	5	shao	shao	PROPN
ejpam-4797	325	6	,	,	PUNCT
ejpam-4797	325	7	x.	x.	PROPN
ejpam-4797	325	8	chen	chen	PROPN
ejpam-4797	325	9	,	,	PUNCT
ejpam-4797	325	10	s.	s.	PROPN
ejpam-4797	325	11	kosari	kosari	PROPN
ejpam-4797	325	12	,	,	PUNCT
ejpam-4797	325	13	and	and	CCONJ
ejpam-4797	325	14	s.	s.	PROPN
ejpam-4797	325	15	omidi	omidi	PROPN
ejpam-4797	325	16	,	,	PUNCT
ejpam-4797	325	17	“	"	PUNCT
ejpam-4797	325	18	on	on	ADP
ejpam-4797	325	19	some	some	DET
ejpam-4797	325	20	properties	property	NOUN
ejpam-4797	325	21	of	of	ADP
ejpam-4797	325	22	right	right	ADJ
ejpam-4797	325	23	pure	pure	ADJ
ejpam-4797	325	24	(	(	PUNCT
ejpam-4797	325	25	biquasi-	biquasi-	NUM
ejpam-4797	325	26	)	)	PUNCT
ejpam-4797	325	27	hyperideals	hyperideal	NOUN
ejpam-4797	325	28	in	in	ADP
ejpam-4797	325	29	ordered	order	VERB
ejpam-4797	325	30	semihyperrings	semihyperring	NOUN
ejpam-4797	325	31	,	,	PUNCT
ejpam-4797	325	32	”	"	PUNCT
ejpam-4797	325	33	politehn	politehn	NOUN
ejpam-4797	325	34	.	.	PUNCT
ejpam-4797	326	1	univ	univ	PROPN
ejpam-4797	326	2	.	.	PUNCT
ejpam-4797	327	1	bucharest	bucharest	PROPN
ejpam-4797	327	2	sci	sci	PROPN
ejpam-4797	327	3	.	.	PUNCT
ejpam-4797	327	4	bull	bull	PROPN
ejpam-4797	327	5	.	.	PUNCT
ejpam-4797	328	1	ser	ser	PROPN
ejpam-4797	328	2	.	.	PUNCT
ejpam-4797	329	1	a	a	DET
ejpam-4797	329	2	appl	appl	PROPN
ejpam-4797	329	3	.	.	PUNCT
ejpam-4797	329	4	math	math	NOUN
ejpam-4797	329	5	.	.	PUNCT
ejpam-4797	330	1	phys	phy	NOUN
ejpam-4797	330	2	,	,	PUNCT
ejpam-4797	330	3	vol	vol	NOUN
ejpam-4797	330	4	.	.	PROPN
ejpam-4797	330	5	83	83	NUM
ejpam-4797	330	6	,	,	PUNCT
ejpam-4797	330	7	no	no	INTJ
ejpam-4797	330	8	.	.	NOUN
ejpam-4797	330	9	4	4	NUM
ejpam-4797	330	10	,	,	PUNCT
ejpam-4797	330	11	pp	pp	ADJ
ejpam-4797	330	12	.	.	PUNCT
ejpam-4797	331	1	95–104	95–104	NOUN
ejpam-4797	331	2	,	,	PUNCT
ejpam-4797	331	3	2021	2021	NUM
ejpam-4797	331	4	.	.	PUNCT
ejpam-4797	332	1	[	[	X
ejpam-4797	332	2	13	13	NUM
ejpam-4797	332	3	]	]	PUNCT
ejpam-4797	332	4	z.	z.	PROPN
ejpam-4797	332	5	kou	kou	PROPN
ejpam-4797	332	6	,	,	PUNCT
ejpam-4797	332	7	s.	s.	PROPN
ejpam-4797	332	8	kosari	kosari	PROPN
ejpam-4797	332	9	,	,	PUNCT
ejpam-4797	332	10	m.	m.	NOUN
ejpam-4797	332	11	monemrad	monemrad	PROPN
ejpam-4797	332	12	,	,	PUNCT
ejpam-4797	332	13	m.	m.	NOUN
ejpam-4797	332	14	akhoundi	akhoundi	PROPN
ejpam-4797	332	15	,	,	PUNCT
ejpam-4797	332	16	and	and	CCONJ
ejpam-4797	332	17	s.	s.	PROPN
ejpam-4797	332	18	omidi	omidi	PROPN
ejpam-4797	332	19	,	,	PUNCT
ejpam-4797	332	20	“	"	PUNCT
ejpam-4797	332	21	a	a	DET
ejpam-4797	332	22	note	note	NOUN
ejpam-4797	332	23	on	on	ADP
ejpam-4797	332	24	the	the	DET
ejpam-4797	332	25	connection	connection	NOUN
ejpam-4797	332	26	between	between	ADP
ejpam-4797	332	27	ordered	order	VERB
ejpam-4797	332	28	semihyperrings	semihyperring	NOUN
ejpam-4797	332	29	,	,	PUNCT
ejpam-4797	332	30	”	"	PUNCT
ejpam-4797	332	31	symmetry	symmetry	NOUN
ejpam-4797	332	32	,	,	PUNCT
ejpam-4797	332	33	vol	vol	NOUN
ejpam-4797	332	34	.	.	PROPN
ejpam-4797	332	35	13	13	NUM
ejpam-4797	332	36	,	,	PUNCT
ejpam-4797	332	37	no	no	INTJ
ejpam-4797	332	38	.	.	NOUN
ejpam-4797	332	39	11	11	NUM
ejpam-4797	332	40	,	,	PUNCT
ejpam-4797	332	41	p.	p.	NOUN
ejpam-4797	332	42	2035	2035	NUM
ejpam-4797	332	43	,	,	PUNCT
ejpam-4797	332	44	2021	2021	NUM
ejpam-4797	332	45	.	.	PUNCT
ejpam-4797	333	1	references	reference	NOUN
ejpam-4797	333	2	1684	1684	NUM
ejpam-4797	334	1	[	[	X
ejpam-4797	334	2	14	14	NUM
ejpam-4797	334	3	]	]	X
ejpam-4797	334	4	p.	p.	NOUN
ejpam-4797	334	5	vámos	vámos	PROPN
ejpam-4797	334	6	,	,	PUNCT
ejpam-4797	334	7	“	"	PUNCT
ejpam-4797	334	8	2	2	NUM
ejpam-4797	334	9	-	-	PUNCT
ejpam-4797	334	10	good	good	ADJ
ejpam-4797	334	11	rings	ring	NOUN
ejpam-4797	334	12	,	,	PUNCT
ejpam-4797	334	13	”	"	PUNCT
ejpam-4797	334	14	quarterly	quarterly	ADJ
ejpam-4797	334	15	journal	journal	NOUN
ejpam-4797	334	16	of	of	ADP
ejpam-4797	334	17	mathematics	mathematic	NOUN
ejpam-4797	334	18	,	,	PUNCT
ejpam-4797	334	19	vol	vol	NOUN
ejpam-4797	334	20	.	.	PROPN
ejpam-4797	335	1	56	56	NUM
ejpam-4797	335	2	,	,	PUNCT
ejpam-4797	335	3	no	no	INTJ
ejpam-4797	335	4	.	.	NOUN
ejpam-4797	335	5	3	3	NUM
ejpam-4797	335	6	,	,	PUNCT
ejpam-4797	335	7	pp	pp	ADJ
ejpam-4797	335	8	.	.	PUNCT
ejpam-4797	336	1	417	417	NUM
ejpam-4797	336	2	–	–	PUNCT
ejpam-4797	336	3	430	430	NUM
ejpam-4797	336	4	,	,	PUNCT
ejpam-4797	336	5	2005	2005	NUM
ejpam-4797	336	6	.	.	PUNCT
ejpam-4797	337	1	[	[	X
ejpam-4797	337	2	15	15	NUM
ejpam-4797	337	3	]	]	X
ejpam-4797	337	4	z.	z.	PROPN
ejpam-4797	337	5	ying	ying	PROPN
ejpam-4797	337	6	,	,	PUNCT
ejpam-4797	337	7	t.	t.	PROPN
ejpam-4797	337	8	koşan	koşan	PROPN
ejpam-4797	337	9	,	,	PUNCT
ejpam-4797	337	10	and	and	CCONJ
ejpam-4797	337	11	y.	y.	PROPN
ejpam-4797	337	12	zhou	zhou	PROPN
ejpam-4797	337	13	,	,	PUNCT
ejpam-4797	337	14	“	"	PUNCT
ejpam-4797	337	15	rings	ring	NOUN
ejpam-4797	337	16	in	in	ADP
ejpam-4797	337	17	which	which	PRON
ejpam-4797	337	18	every	every	DET
ejpam-4797	337	19	element	element	NOUN
ejpam-4797	337	20	is	be	AUX
ejpam-4797	337	21	a	a	DET
ejpam-4797	337	22	sum	sum	NOUN
ejpam-4797	337	23	of	of	ADP
ejpam-4797	337	24	two	two	NUM
ejpam-4797	337	25	tripotents	tripotent	NOUN
ejpam-4797	337	26	,	,	PUNCT
ejpam-4797	337	27	”	"	PUNCT
ejpam-4797	337	28	canadian	canadian	ADJ
ejpam-4797	337	29	mathematical	mathematical	ADJ
ejpam-4797	337	30	bulletin	bulletin	NOUN
ejpam-4797	337	31	,	,	PUNCT
ejpam-4797	337	32	vol	vol	NOUN
ejpam-4797	337	33	.	.	PROPN
ejpam-4797	337	34	59	59	NUM
ejpam-4797	337	35	,	,	PUNCT
ejpam-4797	337	36	no	no	INTJ
ejpam-4797	337	37	.	.	NOUN
ejpam-4797	337	38	3	3	NUM
ejpam-4797	337	39	,	,	PUNCT
ejpam-4797	337	40	pp	pp	ADJ
ejpam-4797	337	41	.	.	PUNCT
ejpam-4797	338	1	661–672	661–672	NUM
ejpam-4797	338	2	,	,	PUNCT
ejpam-4797	338	3	2016	2016	NUM
ejpam-4797	338	4	.	.	PUNCT
ejpam-4797	339	1	[	[	X
ejpam-4797	339	2	16	16	X
ejpam-4797	339	3	]	]	X
ejpam-4797	339	4	y.	y.	PROPN
ejpam-4797	339	5	zhou	zhou	PROPN
ejpam-4797	339	6	,	,	PUNCT
ejpam-4797	339	7	“	"	PUNCT
ejpam-4797	339	8	rings	ring	NOUN
ejpam-4797	339	9	in	in	ADP
ejpam-4797	339	10	which	which	PRON
ejpam-4797	339	11	elements	element	NOUN
ejpam-4797	339	12	are	be	AUX
ejpam-4797	339	13	sums	sum	NOUN
ejpam-4797	339	14	of	of	ADP
ejpam-4797	339	15	nilpotents	nilpotent	NOUN
ejpam-4797	339	16	,	,	PUNCT
ejpam-4797	339	17	idempotents	idempotent	NOUN
ejpam-4797	339	18	and	and	CCONJ
ejpam-4797	339	19	tripotents	tripotent	NOUN
ejpam-4797	339	20	,	,	PUNCT
ejpam-4797	339	21	”	"	PUNCT
ejpam-4797	339	22	journal	journal	NOUN
ejpam-4797	339	23	of	of	ADP
ejpam-4797	339	24	algebra	algebra	PROPN
ejpam-4797	339	25	and	and	CCONJ
ejpam-4797	339	26	its	its	PRON
ejpam-4797	339	27	applications	application	NOUN
ejpam-4797	339	28	,	,	PUNCT
ejpam-4797	339	29	vol	vol	NOUN
ejpam-4797	339	30	.	.	PROPN
ejpam-4797	339	31	17	17	NUM
ejpam-4797	339	32	,	,	PUNCT
ejpam-4797	339	33	no	no	INTJ
ejpam-4797	339	34	.	.	NOUN
ejpam-4797	339	35	01	01	NUM
ejpam-4797	339	36	,	,	PUNCT
ejpam-4797	339	37	p.	p.	NOUN
ejpam-4797	339	38	1850009	1850009	NUM
ejpam-4797	339	39	,	,	PUNCT
ejpam-4797	339	40	2018	2018	NUM
ejpam-4797	339	41	.	.	PUNCT
ejpam-4797	340	1	[	[	X
ejpam-4797	340	2	17	17	NUM
ejpam-4797	340	3	]	]	PUNCT
ejpam-4797	340	4	t.	t.	PROPN
ejpam-4797	340	5	y.	y.	PROPN
ejpam-4797	340	6	lam	lam	PROPN
ejpam-4797	340	7	,	,	PUNCT
ejpam-4797	340	8	a	a	DET
ejpam-4797	340	9	first	first	ADJ
ejpam-4797	340	10	course	course	NOUN
ejpam-4797	340	11	in	in	ADP
ejpam-4797	340	12	noncommutative	noncommutative	ADJ
ejpam-4797	340	13	rings	ring	NOUN
ejpam-4797	340	14	,	,	PUNCT
ejpam-4797	340	15	vol	vol	NOUN
ejpam-4797	340	16	.	.	PROPN
ejpam-4797	340	17	131	131	NUM
ejpam-4797	340	18	.	.	PUNCT
ejpam-4797	340	19	springer	springer	NOUN
ejpam-4797	340	20	,	,	PUNCT
ejpam-4797	340	21	1991	1991	NUM
ejpam-4797	340	22	.	.	PUNCT
ejpam-4797	341	1	[	[	X
ejpam-4797	341	2	18	18	NUM
ejpam-4797	341	3	]	]	PUNCT
ejpam-4797	341	4	p.	p.	NOUN
ejpam-4797	341	5	v.	v.	PROPN
ejpam-4797	341	6	danchev	danchev	PROPN
ejpam-4797	341	7	,	,	PUNCT
ejpam-4797	341	8	“	"	PUNCT
ejpam-4797	341	9	a	a	DET
ejpam-4797	341	10	note	note	NOUN
ejpam-4797	341	11	on	on	ADP
ejpam-4797	341	12	nil	nil	ADJ
ejpam-4797	341	13	-	-	PUNCT
ejpam-4797	341	14	clean	clean	ADJ
ejpam-4797	341	15	rings	ring	NOUN
ejpam-4797	341	16	,	,	PUNCT
ejpam-4797	341	17	”	"	PUNCT
ejpam-4797	341	18	acta	acta	PROPN
ejpam-4797	341	19	universitatis	universitatis	PROPN
ejpam-4797	341	20	sapientiae	sapientiae	PROPN
ejpam-4797	341	21	,	,	PUNCT
ejpam-4797	341	22	mathematica	mathematica	PROPN
ejpam-4797	341	23	,	,	PUNCT
ejpam-4797	341	24	vol	vol	NOUN
ejpam-4797	341	25	.	.	PROPN
ejpam-4797	341	26	12	12	NUM
ejpam-4797	341	27	,	,	PUNCT
ejpam-4797	341	28	no	no	INTJ
ejpam-4797	341	29	.	.	NOUN
ejpam-4797	341	30	2	2	NUM
ejpam-4797	341	31	,	,	PUNCT
ejpam-4797	341	32	pp	pp	ADJ
ejpam-4797	341	33	.	.	PUNCT
ejpam-4797	342	1	287–293	287–293	NUM
ejpam-4797	342	2	,	,	PUNCT
ejpam-4797	342	3	2020	2020	NUM
ejpam-4797	342	4	.	.	PUNCT