id	sid	tid	token	lemma	pos
ejpam-3886	1	1	european	european	PROPN
ejpam-3886	1	2	journal	journal	PROPN
ejpam-3886	1	3	of	of	ADP
ejpam-3886	1	4	pure	pure	ADJ
ejpam-3886	1	5	and	and	CCONJ
ejpam-3886	1	6	applied	apply	VERB
ejpam-3886	1	7	mathematics	mathematic	NOUN
ejpam-3886	1	8	vol	vol	NOUN
ejpam-3886	1	9	.	.	PUNCT
ejpam-3886	2	1	14	14	NUM
ejpam-3886	2	2	,	,	PUNCT
ejpam-3886	2	3	no	no	INTJ
ejpam-3886	2	4	.	.	NOUN
ejpam-3886	2	5	1	1	NUM
ejpam-3886	2	6	,	,	PUNCT
ejpam-3886	2	7	2021	2021	NUM
ejpam-3886	2	8	,	,	PUNCT
ejpam-3886	2	9	164	164	NUM
ejpam-3886	2	10	-	-	SYM
ejpam-3886	2	11	172	172	NUM
ejpam-3886	2	12	issn	issn	PROPN
ejpam-3886	2	13	1307	1307	NUM
ejpam-3886	2	14	-	-	SYM
ejpam-3886	2	15	5543	5543	NUM
ejpam-3886	2	16	–	–	PUNCT
ejpam-3886	2	17	ejpam.com	ejpam.com	X
ejpam-3886	2	18	published	publish	VERB
ejpam-3886	2	19	by	by	ADP
ejpam-3886	2	20	new	new	PROPN
ejpam-3886	2	21	york	york	PROPN
ejpam-3886	2	22	business	business	PROPN
ejpam-3886	2	23	global	global	ADJ
ejpam-3886	2	24	ore	ore	NOUN
ejpam-3886	2	25	extension	extension	NOUN
ejpam-3886	2	26	rings	ring	NOUN
ejpam-3886	2	27	satisfy	satisfy	VERB
ejpam-3886	2	28	the	the	DET
ejpam-3886	2	29	weak	weak	ADJ
ejpam-3886	2	30	ps	ps	NOUN
ejpam-3886	2	31	-	-	PUNCT
ejpam-3886	2	32	rings	ring	NOUN
ejpam-3886	2	33	mohamed	mohamed	PROPN
ejpam-3886	2	34	a.	a.	PROPN
ejpam-3886	2	35	farahat1,2,∗	farahat1,2,∗	PROPN
ejpam-3886	2	36	,	,	PUNCT
ejpam-3886	2	37	salha	salha	NOUN
ejpam-3886	2	38	t.	t.	PROPN
ejpam-3886	2	39	al	al	PROPN
ejpam-3886	2	40	-	-	PUNCT
ejpam-3886	2	41	bogamy1	bogamy1	VERB
ejpam-3886	2	42	1	1	NUM
ejpam-3886	2	43	department	department	NOUN
ejpam-3886	2	44	of	of	ADP
ejpam-3886	2	45	mathematics	mathematic	NOUN
ejpam-3886	2	46	and	and	CCONJ
ejpam-3886	2	47	statistics	statistic	NOUN
ejpam-3886	2	48	,	,	PUNCT
ejpam-3886	2	49	faculty	faculty	NOUN
ejpam-3886	2	50	of	of	ADP
ejpam-3886	2	51	science	science	NOUN
ejpam-3886	2	52	,	,	PUNCT
ejpam-3886	2	53	taif	taif	PROPN
ejpam-3886	2	54	university	university	PROPN
ejpam-3886	2	55	,	,	PUNCT
ejpam-3886	2	56	taif	taif	PROPN
ejpam-3886	2	57	,	,	PUNCT
ejpam-3886	2	58	el	el	PROPN
ejpam-3886	2	59	-	-	PUNCT
ejpam-3886	2	60	haweiah	haweiah	NOUN
ejpam-3886	2	61	,	,	PUNCT
ejpam-3886	2	62	kingdom	kingdom	NOUN
ejpam-3886	2	63	of	of	ADP
ejpam-3886	2	64	saudi	saudi	PROPN
ejpam-3886	2	65	arabia	arabia	PROPN
ejpam-3886	2	66	(	(	PUNCT
ejpam-3886	2	67	ksa	ksa	PROPN
ejpam-3886	2	68	)	)	PUNCT
ejpam-3886	2	69	2	2	NUM
ejpam-3886	2	70	mathematics	mathematics	NOUN
ejpam-3886	2	71	department	department	NOUN
ejpam-3886	2	72	,	,	PUNCT
ejpam-3886	2	73	faculty	faculty	NOUN
ejpam-3886	2	74	of	of	ADP
ejpam-3886	2	75	science	science	NOUN
ejpam-3886	2	76	,	,	PUNCT
ejpam-3886	2	77	al	al	PROPN
ejpam-3886	2	78	-	-	PUNCT
ejpam-3886	2	79	azhar	azhar	PROPN
ejpam-3886	2	80	university	university	PROPN
ejpam-3886	2	81	,	,	PUNCT
ejpam-3886	2	82	cairo	cairo	PROPN
ejpam-3886	2	83	,	,	PUNCT
ejpam-3886	2	84	egypt	egypt	PROPN
ejpam-3886	2	85	abstract	abstract	PROPN
ejpam-3886	2	86	.	.	PUNCT
ejpam-3886	3	1	the	the	DET
ejpam-3886	3	2	main	main	ADJ
ejpam-3886	3	3	result	result	NOUN
ejpam-3886	3	4	of	of	ADP
ejpam-3886	3	5	this	this	DET
ejpam-3886	3	6	paper	paper	NOUN
ejpam-3886	3	7	is	be	AUX
ejpam-3886	3	8	that	that	SCONJ
ejpam-3886	3	9	:	:	PUNCT
ejpam-3886	3	10	if	if	SCONJ
ejpam-3886	3	11	r	r	NOUN
ejpam-3886	3	12	is	be	AUX
ejpam-3886	3	13	a	a	DET
ejpam-3886	3	14	weak	weak	ADJ
ejpam-3886	3	15	right	right	ADJ
ejpam-3886	3	16	ps	ps	NOUN
ejpam-3886	3	17	-	-	NOUN
ejpam-3886	3	18	ring	ring	NOUN
ejpam-3886	3	19	,	,	PUNCT
ejpam-3886	3	20	then	then	ADV
ejpam-3886	3	21	a	a	DET
ejpam-3886	3	22	=	=	PUNCT
ejpam-3886	3	23	r[x;α	r[x;α	PROPN
ejpam-3886	3	24	,	,	PUNCT
ejpam-3886	3	25	δ	δ	PROPN
ejpam-3886	3	26	]	]	X
ejpam-3886	3	27	,	,	PUNCT
ejpam-3886	3	28	the	the	DET
ejpam-3886	3	29	ore	ore	NOUN
ejpam-3886	3	30	extension	extension	NOUN
ejpam-3886	3	31	ring	ring	NOUN
ejpam-3886	3	32	,	,	PUNCT
ejpam-3886	3	33	is	be	AUX
ejpam-3886	3	34	a	a	DET
ejpam-3886	3	35	weak	weak	ADJ
ejpam-3886	3	36	right	right	ADJ
ejpam-3886	3	37	ps	ps	NOUN
ejpam-3886	3	38	-	-	NOUN
ejpam-3886	3	39	ring	ring	NOUN
ejpam-3886	3	40	whenever	whenever	SCONJ
ejpam-3886	3	41	the	the	DET
ejpam-3886	3	42	following	follow	VERB
ejpam-3886	3	43	conditions	condition	NOUN
ejpam-3886	3	44	hold	hold	VERB
ejpam-3886	3	45	on	on	ADP
ejpam-3886	3	46	r	r	NOUN
ejpam-3886	3	47	is	be	AUX
ejpam-3886	3	48	an	an	DET
ejpam-3886	3	49	(	(	PUNCT
ejpam-3886	3	50	α	α	NOUN
ejpam-3886	3	51	,	,	PUNCT
ejpam-3886	3	52	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	3	53	ni	ni	NOUN
ejpam-3886	3	54	-	-	NOUN
ejpam-3886	3	55	ring	ring	NOUN
ejpam-3886	3	56	with	with	ADP
ejpam-3886	3	57	nil(r	nil(r	NOUN
ejpam-3886	3	58	)	)	PUNCT
ejpam-3886	3	59	nilpotent	nilpotent	NOUN
ejpam-3886	3	60	,	,	PUNCT
ejpam-3886	3	61	α(e	α(e	PROPN
ejpam-3886	3	62	)	)	PUNCT
ejpam-3886	4	1	=	=	SYM
ejpam-3886	4	2	e	e	NOUN
ejpam-3886	4	3	and	and	CCONJ
ejpam-3886	4	4	δ(e	δ(e	NOUN
ejpam-3886	4	5	)	)	PUNCT
ejpam-3886	4	6	=	=	SYM
ejpam-3886	4	7	0	0	NUM
ejpam-3886	4	8	for	for	ADP
ejpam-3886	4	9	every	every	DET
ejpam-3886	4	10	idempotent	idempotent	ADJ
ejpam-3886	4	11	e	e	PROPN
ejpam-3886	4	12	∈	∈	PROPN
ejpam-3886	4	13	r.	r.	PROPN
ejpam-3886	4	14	2020	2020	NUM
ejpam-3886	5	1	mathematics	mathematics	PROPN
ejpam-3886	5	2	subject	subject	NOUN
ejpam-3886	5	3	classifications	classification	NOUN
ejpam-3886	5	4	:	:	PUNCT
ejpam-3886	5	5	16d25	16d25	NUM
ejpam-3886	5	6	,	,	PUNCT
ejpam-3886	5	7	16p60	16p60	NUM
ejpam-3886	5	8	,	,	PUNCT
ejpam-3886	5	9	16w60	16w60	NUM
ejpam-3886	5	10	key	key	ADJ
ejpam-3886	5	11	words	word	NOUN
ejpam-3886	5	12	and	and	CCONJ
ejpam-3886	5	13	phrases	phrase	NOUN
ejpam-3886	5	14	:	:	PUNCT
ejpam-3886	5	15	ps	ps	NOUN
ejpam-3886	5	16	-	-	PUNCT
ejpam-3886	5	17	ring	ring	NOUN
ejpam-3886	5	18	,	,	PUNCT
ejpam-3886	5	19	weak	weak	ADJ
ejpam-3886	5	20	ps	ps	NOUN
ejpam-3886	5	21	-	-	PUNCT
ejpam-3886	5	22	ring	ring	NOUN
ejpam-3886	5	23	,	,	PUNCT
ejpam-3886	5	24	ore	ore	NOUN
ejpam-3886	5	25	extensions	extension	NOUN
ejpam-3886	5	26	1	1	NUM
ejpam-3886	5	27	.	.	PUNCT
ejpam-3886	5	28	introduction	introduction	NOUN
ejpam-3886	5	29	throughout	throughout	ADP
ejpam-3886	5	30	this	this	DET
ejpam-3886	5	31	article	article	NOUN
ejpam-3886	5	32	,	,	PUNCT
ejpam-3886	5	33	all	all	DET
ejpam-3886	5	34	rings	ring	NOUN
ejpam-3886	5	35	are	be	AUX
ejpam-3886	5	36	associative	associative	ADJ
ejpam-3886	5	37	with	with	ADP
ejpam-3886	5	38	unity	unity	NOUN
ejpam-3886	5	39	(	(	PUNCT
ejpam-3886	5	40	r	r	NOUN
ejpam-3886	5	41	denotes	denote	VERB
ejpam-3886	5	42	such	such	DET
ejpam-3886	5	43	a	a	DET
ejpam-3886	5	44	ring	ring	NOUN
ejpam-3886	5	45	)	)	PUNCT
ejpam-3886	5	46	and	and	CCONJ
ejpam-3886	5	47	all	all	DET
ejpam-3886	5	48	modules	module	NOUN
ejpam-3886	5	49	are	be	AUX
ejpam-3886	5	50	unital	unital	ADJ
ejpam-3886	5	51	r	r	NOUN
ejpam-3886	5	52	-	-	PUNCT
ejpam-3886	5	53	modules	module	NOUN
ejpam-3886	5	54	unless	unless	SCONJ
ejpam-3886	5	55	explicitly	explicitly	ADV
ejpam-3886	5	56	indicated	indicate	VERB
ejpam-3886	5	57	otherwise	otherwise	ADV
ejpam-3886	5	58	.	.	PUNCT
ejpam-3886	6	1	according	accord	VERB
ejpam-3886	6	2	to	to	ADP
ejpam-3886	6	3	nicholson	nicholson	PROPN
ejpam-3886	6	4	and	and	CCONJ
ejpam-3886	6	5	watters	watter	NOUN
ejpam-3886	6	6	[	[	X
ejpam-3886	6	7	6	6	NUM
ejpam-3886	6	8	]	]	PUNCT
ejpam-3886	6	9	,	,	PUNCT
ejpam-3886	6	10	mr	mr	PROPN
ejpam-3886	6	11	is	be	AUX
ejpam-3886	6	12	called	call	VERB
ejpam-3886	6	13	a	a	DET
ejpam-3886	6	14	ps	ps	NOUN
ejpam-3886	6	15	-	-	PUNCT
ejpam-3886	6	16	module	module	NOUN
ejpam-3886	6	17	if	if	SCONJ
ejpam-3886	6	18	every	every	DET
ejpam-3886	6	19	simple	simple	ADJ
ejpam-3886	6	20	submodule	submodule	NOUN
ejpam-3886	6	21	is	be	AUX
ejpam-3886	6	22	projective	projective	ADJ
ejpam-3886	6	23	,	,	PUNCT
ejpam-3886	6	24	equivalently	equivalently	ADV
ejpam-3886	6	25	if	if	SCONJ
ejpam-3886	6	26	its	its	PRON
ejpam-3886	6	27	socle	socle	NOUN
ejpam-3886	6	28	,	,	PUNCT
ejpam-3886	6	29	soc	soc	NOUN
ejpam-3886	6	30	(	(	PUNCT
ejpam-3886	6	31	mr	mr	PROPN
ejpam-3886	6	32	)	)	PUNCT
ejpam-3886	6	33	,	,	PUNCT
ejpam-3886	6	34	is	be	AUX
ejpam-3886	6	35	projective	projective	ADJ
ejpam-3886	6	36	.	.	PUNCT
ejpam-3886	7	1	examples	example	NOUN
ejpam-3886	7	2	of	of	ADP
ejpam-3886	7	3	ps	ps	NOUN
ejpam-3886	7	4	-	-	PUNCT
ejpam-3886	7	5	modules	module	NOUN
ejpam-3886	7	6	include	include	VERB
ejpam-3886	7	7	nonsingular	nonsingular	ADJ
ejpam-3886	7	8	modules	module	NOUN
ejpam-3886	7	9	and	and	CCONJ
ejpam-3886	7	10	modules	module	NOUN
ejpam-3886	7	11	with	with	ADP
ejpam-3886	7	12	zero	zero	NUM
ejpam-3886	7	13	socle	socle	NOUN
ejpam-3886	7	14	.	.	PUNCT
ejpam-3886	8	1	a	a	DET
ejpam-3886	8	2	left	left	ADJ
ejpam-3886	8	3	ps	ps	NOUN
ejpam-3886	8	4	-	-	PUNCT
ejpam-3886	8	5	module	module	NOUN
ejpam-3886	8	6	rm	rm	NOUN
ejpam-3886	8	7	is	be	AUX
ejpam-3886	8	8	defined	define	VERB
ejpam-3886	8	9	analogously	analogously	ADV
ejpam-3886	8	10	.	.	PUNCT
ejpam-3886	9	1	a	a	DET
ejpam-3886	9	2	ring	ring	NOUN
ejpam-3886	9	3	r	r	NOUN
ejpam-3886	9	4	is	be	AUX
ejpam-3886	9	5	said	say	VERB
ejpam-3886	9	6	to	to	PART
ejpam-3886	9	7	be	be	AUX
ejpam-3886	9	8	a	a	DET
ejpam-3886	9	9	left	left	ADJ
ejpam-3886	9	10	ps	ps	NOUN
ejpam-3886	9	11	-	-	PUNCT
ejpam-3886	9	12	ring	ring	NOUN
ejpam-3886	9	13	if	if	SCONJ
ejpam-3886	9	14	rr	rr	PROPN
ejpam-3886	9	15	is	be	AUX
ejpam-3886	9	16	a	a	DET
ejpam-3886	9	17	ps	ps	NOUN
ejpam-3886	9	18	-	-	PUNCT
ejpam-3886	9	19	module	module	NOUN
ejpam-3886	9	20	.	.	PUNCT
ejpam-3886	10	1	equivalently	equivalently	ADV
ejpam-3886	10	2	,	,	PUNCT
ejpam-3886	10	3	if	if	SCONJ
ejpam-3886	10	4	the	the	DET
ejpam-3886	10	5	left	left	ADJ
ejpam-3886	10	6	annihilator	annihilator	NOUN
ejpam-3886	10	7	of	of	ADP
ejpam-3886	10	8	every	every	DET
ejpam-3886	10	9	maximal	maximal	ADJ
ejpam-3886	10	10	right	right	ADJ
ejpam-3886	10	11	ideal	ideal	NOUN
ejpam-3886	10	12	of	of	ADP
ejpam-3886	10	13	r	r	NOUN
ejpam-3886	10	14	is	be	AUX
ejpam-3886	10	15	a	a	DET
ejpam-3886	10	16	principal	principal	ADJ
ejpam-3886	10	17	left	leave	VERB
ejpam-3886	10	18	ideal	ideal	NOUN
ejpam-3886	10	19	generated	generate	VERB
ejpam-3886	10	20	by	by	ADP
ejpam-3886	10	21	an	an	DET
ejpam-3886	10	22	idempotent	idempotent	NOUN
ejpam-3886	10	23	.	.	PUNCT
ejpam-3886	11	1	some	some	DET
ejpam-3886	11	2	examples	example	NOUN
ejpam-3886	11	3	of	of	ADP
ejpam-3886	11	4	ps	ps	NOUN
ejpam-3886	11	5	-	-	PUNCT
ejpam-3886	11	6	rings	ring	NOUN
ejpam-3886	11	7	include	include	VERB
ejpam-3886	11	8	semiprime	semiprime	NOUN
ejpam-3886	11	9	and	and	CCONJ
ejpam-3886	11	10	p.p.-rings	p.p.-ring	NOUN
ejpam-3886	11	11	are	be	AUX
ejpam-3886	11	12	ps	ps	NOUN
ejpam-3886	11	13	-	-	PUNCT
ejpam-3886	11	14	rings	ring	NOUN
ejpam-3886	11	15	.	.	PUNCT
ejpam-3886	12	1	in	in	ADP
ejpam-3886	12	2	particular	particular	ADJ
ejpam-3886	12	3	every	every	DET
ejpam-3886	12	4	baer	baer	PROPN
ejpam-3886	12	5	ring	ring	NOUN
ejpam-3886	12	6	is	be	AUX
ejpam-3886	12	7	a	a	DET
ejpam-3886	12	8	ps	ps	NOUN
ejpam-3886	12	9	-	-	PUNCT
ejpam-3886	12	10	ring	ring	NOUN
ejpam-3886	12	11	.	.	PUNCT
ejpam-3886	13	1	the	the	DET
ejpam-3886	13	2	notion	notion	NOUN
ejpam-3886	13	3	of	of	ADP
ejpam-3886	13	4	ps	ps	NOUN
ejpam-3886	13	5	-	-	PUNCT
ejpam-3886	13	6	rings	ring	NOUN
ejpam-3886	13	7	is	be	AUX
ejpam-3886	13	8	not	not	PART
ejpam-3886	13	9	left	leave	VERB
ejpam-3886	13	10	-	-	PUNCT
ejpam-3886	13	11	right	right	NOUN
ejpam-3886	13	12	symmetric	symmetric	NOUN
ejpam-3886	13	13	(	(	PUNCT
ejpam-3886	13	14	cf	cf	NOUN
ejpam-3886	13	15	.	.	PUNCT
ejpam-3886	14	1	[	[	X
ejpam-3886	14	2	6	6	NUM
ejpam-3886	14	3	]	]	PUNCT
ejpam-3886	14	4	)	)	PUNCT
ejpam-3886	14	5	.	.	PUNCT
ejpam-3886	15	1	in	in	ADP
ejpam-3886	15	2	[	[	X
ejpam-3886	15	3	6	6	NUM
ejpam-3886	15	4	]	]	PUNCT
ejpam-3886	15	5	,	,	PUNCT
ejpam-3886	15	6	the	the	DET
ejpam-3886	15	7	authors	author	NOUN
ejpam-3886	15	8	proved	prove	VERB
ejpam-3886	15	9	that	that	SCONJ
ejpam-3886	15	10	,	,	PUNCT
ejpam-3886	15	11	if	if	SCONJ
ejpam-3886	15	12	r	r	NOUN
ejpam-3886	15	13	is	be	AUX
ejpam-3886	15	14	a	a	DET
ejpam-3886	15	15	ps	ps	NOUN
ejpam-3886	15	16	-	-	PUNCT
ejpam-3886	15	17	ring	ring	NOUN
ejpam-3886	15	18	so	so	ADV
ejpam-3886	15	19	also	also	ADV
ejpam-3886	15	20	are	be	AUX
ejpam-3886	15	21	r[x	r[x	NOUN
ejpam-3886	15	22	]	]	PUNCT
ejpam-3886	15	23	and	and	CCONJ
ejpam-3886	15	24	r[[x	r[[x	PROPN
ejpam-3886	15	25	]	]	X
ejpam-3886	15	26	]	]	PUNCT
ejpam-3886	15	27	.	.	PUNCT
ejpam-3886	16	1	the	the	DET
ejpam-3886	16	2	converse	converse	NOUN
ejpam-3886	16	3	of	of	ADP
ejpam-3886	16	4	this	this	DET
ejpam-3886	16	5	result	result	NOUN
ejpam-3886	16	6	is	be	AUX
ejpam-3886	16	7	false	false	ADJ
ejpam-3886	16	8	in	in	ADP
ejpam-3886	16	9	general	general	ADJ
ejpam-3886	16	10	by	by	ADP
ejpam-3886	16	11	the	the	DET
ejpam-3886	16	12	following	follow	VERB
ejpam-3886	16	13	example	example	NOUN
ejpam-3886	16	14	:	:	PUNCT
ejpam-3886	16	15	example	example	NOUN
ejpam-3886	16	16	1	1	NUM
ejpam-3886	16	17	(	(	PUNCT
ejpam-3886	16	18	[	[	X
ejpam-3886	16	19	6	6	NUM
ejpam-3886	16	20	]	]	PUNCT
ejpam-3886	16	21	,	,	PUNCT
ejpam-3886	16	22	example	example	NOUN
ejpam-3886	16	23	3.2	3.2	NUM
ejpam-3886	16	24	)	)	PUNCT
ejpam-3886	16	25	.	.	PUNCT
ejpam-3886	17	1	if	if	SCONJ
ejpam-3886	17	2	r	r	NOUN
ejpam-3886	17	3	=	=	SYM
ejpam-3886	17	4	z4	z4	X
ejpam-3886	17	5	,	,	PUNCT
ejpam-3886	17	6	then	then	ADV
ejpam-3886	17	7	r[x	r[x	NOUN
ejpam-3886	17	8	]	]	PUNCT
ejpam-3886	17	9	and	and	CCONJ
ejpam-3886	17	10	r[[x	r[[x	PROPN
ejpam-3886	17	11	]	]	X
ejpam-3886	17	12	]	]	X
ejpam-3886	17	13	are	be	AUX
ejpam-3886	17	14	ps	ps	NOUN
ejpam-3886	17	15	-	-	PUNCT
ejpam-3886	17	16	rings	ring	NOUN
ejpam-3886	17	17	but	but	CCONJ
ejpam-3886	17	18	r	r	NOUN
ejpam-3886	17	19	is	be	AUX
ejpam-3886	17	20	not	not	PART
ejpam-3886	17	21	ps	ps	NOUN
ejpam-3886	17	22	-	-	NOUN
ejpam-3886	17	23	ring	ring	NOUN
ejpam-3886	17	24	.	.	PUNCT
ejpam-3886	18	1	many	many	ADJ
ejpam-3886	18	2	authors	author	NOUN
ejpam-3886	18	3	investigated	investigate	VERB
ejpam-3886	18	4	the	the	DET
ejpam-3886	18	5	behavior	behavior	NOUN
ejpam-3886	18	6	of	of	ADP
ejpam-3886	18	7	ps	ps	NOUN
ejpam-3886	18	8	-	-	PUNCT
ejpam-3886	18	9	rings	ring	NOUN
ejpam-3886	18	10	with	with	ADP
ejpam-3886	18	11	respect	respect	NOUN
ejpam-3886	18	12	to	to	ADP
ejpam-3886	18	13	their	their	PRON
ejpam-3886	18	14	extensions	extension	NOUN
ejpam-3886	18	15	.	.	PUNCT
ejpam-3886	19	1	salem	salem	PROPN
ejpam-3886	19	2	et	et	PROPN
ejpam-3886	19	3	.	.	PUNCT
ejpam-3886	20	1	al	al	PROPN
ejpam-3886	20	2	.	.	PROPN
ejpam-3886	20	3	,	,	PUNCT
ejpam-3886	20	4	in	in	ADP
ejpam-3886	20	5	(	(	PUNCT
ejpam-3886	20	6	[	[	X
ejpam-3886	20	7	9	9	NUM
ejpam-3886	20	8	]	]	PUNCT
ejpam-3886	20	9	,	,	PUNCT
ejpam-3886	20	10	2015	2015	NUM
ejpam-3886	20	11	)	)	PUNCT
ejpam-3886	20	12	,	,	PUNCT
ejpam-3886	20	13	characterized	characterize	VERB
ejpam-3886	20	14	ps	ps	NOUN
ejpam-3886	20	15	-	-	PUNCT
ejpam-3886	20	16	modules	module	NOUN
ejpam-3886	20	17	over	over	ADP
ejpam-3886	20	18	ore	ore	NOUN
ejpam-3886	20	19	extensions	extension	NOUN
ejpam-3886	20	20	and	and	CCONJ
ejpam-3886	20	21	skew	skew	VERB
ejpam-3886	20	22	generalized	generalized	ADJ
ejpam-3886	20	23	power	power	NOUN
ejpam-3886	20	24	series	series	NOUN
ejpam-3886	20	25	extensions	extension	NOUN
ejpam-3886	20	26	.	.	PUNCT
ejpam-3886	21	1	also	also	ADV
ejpam-3886	21	2	,	,	PUNCT
ejpam-3886	21	3	farahat	farahat	NOUN
ejpam-3886	21	4	and	and	CCONJ
ejpam-3886	21	5	al	al	PROPN
ejpam-3886	21	6	-	-	PUNCT
ejpam-3886	21	7	harthy	harthy	ADJ
ejpam-3886	21	8	,	,	PUNCT
ejpam-3886	21	9	in	in	ADP
ejpam-3886	21	10	(	(	PUNCT
ejpam-3886	21	11	[	[	X
ejpam-3886	21	12	3	3	NUM
ejpam-3886	21	13	]	]	PUNCT
ejpam-3886	21	14	,	,	PUNCT
ejpam-3886	21	15	2017	2017	NUM
ejpam-3886	21	16	)	)	PUNCT
ejpam-3886	21	17	,	,	PUNCT
ejpam-3886	21	18	investigated	investigate	VERB
ejpam-3886	21	19	∗corresponding	∗corresponde	VERB
ejpam-3886	21	20	author	author	NOUN
ejpam-3886	21	21	.	.	PUNCT
ejpam-3886	22	1	doi	doi	NOUN
ejpam-3886	22	2	:	:	PUNCT
ejpam-3886	22	3	https://doi.org/10.29020/nybg.ejpam.v14i1.3886	https://doi.org/10.29020/nybg.ejpam.v14i1.3886	PROPN
ejpam-3886	22	4	email	email	NOUN
ejpam-3886	22	5	addresses	address	NOUN
ejpam-3886	22	6	:	:	PUNCT
ejpam-3886	22	7	m	m	VERB
ejpam-3886	22	8	farahat79@yahoo.com	farahat79@yahoo.com	PROPN
ejpam-3886	22	9	,	,	PUNCT
ejpam-3886	22	10	m.farahat	m.farahat	ADP
ejpam-3886	22	11	@tu.edu.sa	@tu.edu.sa	PROPN
ejpam-3886	22	12	(	(	PUNCT
ejpam-3886	22	13	m.	m.	NOUN
ejpam-3886	22	14	a.	a.	NOUN
ejpam-3886	22	15	farahat	farahat	PROPN
ejpam-3886	22	16	)	)	PUNCT
ejpam-3886	22	17	,	,	PUNCT
ejpam-3886	22	18	salhaalbogamy@hotmail.com	salhaalbogamy@hotmail.com	X
ejpam-3886	23	1	(	(	PUNCT
ejpam-3886	23	2	salha	salha	NOUN
ejpam-3886	23	3	t.	t.	PROPN
ejpam-3886	23	4	al	al	PROPN
ejpam-3886	23	5	-	-	PUNCT
ejpam-3886	23	6	bogamy	bogamy	NOUN
ejpam-3886	23	7	)	)	PUNCT
ejpam-3886	23	8	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3886	24	1	164	164	NUM
ejpam-3886	24	2	c	c	X
ejpam-3886	24	3	©	©	PROPN
ejpam-3886	24	4	2021	2021	NUM
ejpam-3886	24	5	ejpam	ejpam	VERB
ejpam-3886	24	6	all	all	DET
ejpam-3886	24	7	rights	right	NOUN
ejpam-3886	24	8	reserved	reserve	VERB
ejpam-3886	24	9	.	.	PUNCT
ejpam-3886	25	1	m.	m.	NOUN
ejpam-3886	25	2	a.	a.	PROPN
ejpam-3886	25	3	farahat	farahat	PROPN
ejpam-3886	25	4	,	,	PUNCT
ejpam-3886	25	5	salha	salha	NOUN
ejpam-3886	25	6	t.	t.	PROPN
ejpam-3886	25	7	al	al	PROPN
ejpam-3886	25	8	-	-	PUNCT
ejpam-3886	25	9	bogamy	bogamy	PROPN
ejpam-3886	25	10	/	/	SYM
ejpam-3886	25	11	eur	eur	PROPN
ejpam-3886	25	12	.	.	PUNCT
ejpam-3886	26	1	j.	j.	PROPN
ejpam-3886	26	2	pure	pure	PROPN
ejpam-3886	26	3	appl	appl	PROPN
ejpam-3886	26	4	.	.	PROPN
ejpam-3886	26	5	math	math	PROPN
ejpam-3886	26	6	,	,	PUNCT
ejpam-3886	26	7	14	14	NUM
ejpam-3886	26	8	(	(	PUNCT
ejpam-3886	26	9	1	1	NUM
ejpam-3886	26	10	)	)	PUNCT
ejpam-3886	26	11	(	(	PUNCT
ejpam-3886	26	12	2021	2021	NUM
ejpam-3886	26	13	)	)	PUNCT
ejpam-3886	26	14	,	,	PUNCT
ejpam-3886	26	15	164	164	NUM
ejpam-3886	26	16	-	-	SYM
ejpam-3886	26	17	172	172	NUM
ejpam-3886	26	18	165	165	NUM
ejpam-3886	26	19	ps	p	NOUN
ejpam-3886	26	20	-	-	PUNCT
ejpam-3886	26	21	modules	module	NOUN
ejpam-3886	26	22	over	over	ADP
ejpam-3886	26	23	generalized	generalize	VERB
ejpam-3886	26	24	mal’cev	mal’cev	PROPN
ejpam-3886	26	25	-	-	PUNCT
ejpam-3886	26	26	neumann	neumann	PROPN
ejpam-3886	26	27	series	series	PROPN
ejpam-3886	26	28	rings	ring	NOUN
ejpam-3886	26	29	.	.	PUNCT
ejpam-3886	27	1	in	in	ADP
ejpam-3886	27	2	(	(	PUNCT
ejpam-3886	27	3	[	[	X
ejpam-3886	27	4	8	8	NUM
ejpam-3886	27	5	]	]	SYM
ejpam-3886	27	6	,	,	PUNCT
ejpam-3886	27	7	2017	2017	NUM
ejpam-3886	27	8	)	)	PUNCT
ejpam-3886	27	9	,	,	PUNCT
ejpam-3886	27	10	paykan	paykan	PROPN
ejpam-3886	27	11	proved	prove	VERB
ejpam-3886	27	12	that	that	SCONJ
ejpam-3886	27	13	,	,	PUNCT
ejpam-3886	27	14	under	under	ADP
ejpam-3886	27	15	suitable	suitable	ADJ
ejpam-3886	27	16	conditions	condition	NOUN
ejpam-3886	27	17	,	,	PUNCT
ejpam-3886	27	18	if	if	SCONJ
ejpam-3886	27	19	r	r	NOUN
ejpam-3886	27	20	is	be	AUX
ejpam-3886	27	21	a	a	DET
ejpam-3886	27	22	right	right	ADJ
ejpam-3886	27	23	ps	ps	NOUN
ejpam-3886	27	24	-	-	NOUN
ejpam-3886	27	25	ring	ring	NOUN
ejpam-3886	27	26	,	,	PUNCT
ejpam-3886	27	27	then	then	ADV
ejpam-3886	27	28	so	so	ADV
ejpam-3886	27	29	the	the	DET
ejpam-3886	27	30	skew	skew	ADJ
ejpam-3886	27	31	inverse	inverse	NOUN
ejpam-3886	27	32	power	power	NOUN
ejpam-3886	27	33	series	series	PROPN
ejpam-3886	27	34	rings	ring	NOUN
ejpam-3886	27	35	.	.	PUNCT
ejpam-3886	28	1	recently	recently	ADV
ejpam-3886	28	2	,	,	PUNCT
ejpam-3886	28	3	farahat	farahat	NOUN
ejpam-3886	28	4	and	and	CCONJ
ejpam-3886	28	5	al	al	PROPN
ejpam-3886	28	6	-	-	PUNCT
ejpam-3886	28	7	bogamy	bogamy	PROPN
ejpam-3886	28	8	,	,	PUNCT
ejpam-3886	28	9	in	in	ADP
ejpam-3886	28	10	(	(	PUNCT
ejpam-3886	28	11	[	[	X
ejpam-3886	28	12	2	2	NUM
ejpam-3886	28	13	]	]	PUNCT
ejpam-3886	28	14	,	,	PUNCT
ejpam-3886	28	15	2018	2018	NUM
ejpam-3886	28	16	)	)	PUNCT
ejpam-3886	28	17	,	,	PUNCT
ejpam-3886	28	18	extend	extend	VERB
ejpam-3886	28	19	the	the	DET
ejpam-3886	28	20	notation	notation	NOUN
ejpam-3886	28	21	of	of	ADP
ejpam-3886	28	22	ps	ps	PROPN
ejpam-3886	28	23	-	-	PUNCT
ejpam-3886	28	24	rings	ring	NOUN
ejpam-3886	28	25	to	to	ADP
ejpam-3886	28	26	weak	weak	ADJ
ejpam-3886	28	27	ps	ps	NOUN
ejpam-3886	28	28	-	-	PUNCT
ejpam-3886	28	29	rings	ring	NOUN
ejpam-3886	28	30	.	.	PUNCT
ejpam-3886	29	1	recall	recall	VERB
ejpam-3886	29	2	the	the	DET
ejpam-3886	29	3	definition	definition	NOUN
ejpam-3886	29	4	of	of	ADP
ejpam-3886	29	5	weak	weak	ADJ
ejpam-3886	29	6	ps	ps	NOUN
ejpam-3886	29	7	-	-	PUNCT
ejpam-3886	29	8	rings	ring	NOUN
ejpam-3886	29	9	from	from	ADP
ejpam-3886	29	10	[	[	X
ejpam-3886	29	11	2	2	NUM
ejpam-3886	29	12	]	]	PUNCT
ejpam-3886	29	13	:	:	PUNCT
ejpam-3886	29	14	a	a	DET
ejpam-3886	29	15	ring	ring	NOUN
ejpam-3886	29	16	r	r	NOUN
ejpam-3886	29	17	satisfies	satisfie	NOUN
ejpam-3886	29	18	the	the	DET
ejpam-3886	29	19	right	right	ADJ
ejpam-3886	29	20	weak	weak	ADJ
ejpam-3886	29	21	ps	ps	NOUN
ejpam-3886	29	22	-	-	PUNCT
ejpam-3886	29	23	condition	condition	NOUN
ejpam-3886	29	24	if	if	SCONJ
ejpam-3886	29	25	,	,	PUNCT
ejpam-3886	29	26	the	the	DET
ejpam-3886	29	27	weak	weak	ADJ
ejpam-3886	29	28	annihilator	annihilator	NOUN
ejpam-3886	29	29	of	of	ADP
ejpam-3886	29	30	every	every	DET
ejpam-3886	29	31	maximal	maximal	ADJ
ejpam-3886	29	32	right	right	ADJ
ejpam-3886	29	33	ideal	ideal	NOUN
ejpam-3886	29	34	of	of	ADP
ejpam-3886	29	35	r	r	NOUN
ejpam-3886	29	36	is	be	AUX
ejpam-3886	29	37	a	a	DET
ejpam-3886	29	38	principal	principal	ADJ
ejpam-3886	29	39	left	leave	VERB
ejpam-3886	29	40	ideal	ideal	NOUN
ejpam-3886	29	41	generated	generate	VERB
ejpam-3886	29	42	by	by	ADP
ejpam-3886	29	43	an	an	DET
ejpam-3886	29	44	idempotent	idempotent	NOUN
ejpam-3886	29	45	.	.	PUNCT
ejpam-3886	30	1	similarly	similarly	ADV
ejpam-3886	30	2	,	,	PUNCT
ejpam-3886	30	3	the	the	DET
ejpam-3886	30	4	left	leave	VERB
ejpam-3886	30	5	weak	weak	ADJ
ejpam-3886	30	6	ps	ps	NOUN
ejpam-3886	30	7	-	-	PUNCT
ejpam-3886	30	8	condition	condition	NOUN
ejpam-3886	30	9	was	be	AUX
ejpam-3886	30	10	defined	define	VERB
ejpam-3886	30	11	.	.	PUNCT
ejpam-3886	31	1	a	a	DET
ejpam-3886	31	2	ring	ring	NOUN
ejpam-3886	31	3	r	r	NOUN
ejpam-3886	31	4	satisfies	satisfie	NOUN
ejpam-3886	31	5	the	the	DET
ejpam-3886	31	6	weak	weak	ADJ
ejpam-3886	31	7	ps	ps	NOUN
ejpam-3886	31	8	-	-	PUNCT
ejpam-3886	31	9	condition	condition	NOUN
ejpam-3886	31	10	if	if	SCONJ
ejpam-3886	31	11	it	it	PRON
ejpam-3886	31	12	satisfies	satisfy	VERB
ejpam-3886	31	13	both	both	CCONJ
ejpam-3886	31	14	the	the	DET
ejpam-3886	31	15	right	right	NOUN
ejpam-3886	31	16	and	and	CCONJ
ejpam-3886	31	17	the	the	DET
ejpam-3886	31	18	left	left	ADJ
ejpam-3886	31	19	weak	weak	ADJ
ejpam-3886	31	20	ps	ps	NOUN
ejpam-3886	31	21	-	-	PUNCT
ejpam-3886	31	22	conditions	condition	NOUN
ejpam-3886	31	23	.	.	PUNCT
ejpam-3886	32	1	the	the	DET
ejpam-3886	32	2	following	follow	VERB
ejpam-3886	32	3	are	be	AUX
ejpam-3886	32	4	some	some	DET
ejpam-3886	32	5	examples	example	NOUN
ejpam-3886	32	6	of	of	ADP
ejpam-3886	32	7	rings	ring	NOUN
ejpam-3886	32	8	satisfy	satisfy	VERB
ejpam-3886	32	9	the	the	DET
ejpam-3886	32	10	right	right	ADJ
ejpam-3886	32	11	weak	weak	ADJ
ejpam-3886	32	12	ps	ps	NOUN
ejpam-3886	32	13	-	-	NOUN
ejpam-3886	32	14	condition	condition	NOUN
ejpam-3886	32	15	.	.	PUNCT
ejpam-3886	33	1	example	example	NOUN
ejpam-3886	33	2	2	2	NUM
ejpam-3886	33	3	(	(	PUNCT
ejpam-3886	33	4	[	[	X
ejpam-3886	33	5	2	2	NUM
ejpam-3886	33	6	]	]	NUM
ejpam-3886	33	7	)	)	PUNCT
ejpam-3886	33	8	.	.	PUNCT
ejpam-3886	34	1	1	1	X
ejpam-3886	34	2	)	)	PUNCT
ejpam-3886	34	3	any	any	DET
ejpam-3886	34	4	local	local	ADJ
ejpam-3886	34	5	ring	ring	NOUN
ejpam-3886	34	6	is	be	AUX
ejpam-3886	34	7	a	a	DET
ejpam-3886	34	8	right	right	ADJ
ejpam-3886	34	9	weak	weak	ADJ
ejpam-3886	34	10	ps	ps	NOUN
ejpam-3886	34	11	-	-	NOUN
ejpam-3886	34	12	ring	ring	NOUN
ejpam-3886	34	13	.	.	PUNCT
ejpam-3886	35	1	2	2	X
ejpam-3886	35	2	)	)	PUNCT
ejpam-3886	35	3	the	the	DET
ejpam-3886	35	4	ring	ring	NOUN
ejpam-3886	35	5	zpq	zpq	NOUN
ejpam-3886	35	6	of	of	ADP
ejpam-3886	35	7	integers	integer	NOUN
ejpam-3886	35	8	modulo	modulo	PROPN
ejpam-3886	35	9	pq	pq	PROPN
ejpam-3886	35	10	,	,	PUNCT
ejpam-3886	35	11	where	where	SCONJ
ejpam-3886	35	12	p	p	NOUN
ejpam-3886	35	13	and	and	CCONJ
ejpam-3886	35	14	q	q	NOUN
ejpam-3886	35	15	are	be	AUX
ejpam-3886	35	16	distinct	distinct	ADJ
ejpam-3886	35	17	prime	prime	ADJ
ejpam-3886	35	18	numbers	number	NOUN
ejpam-3886	35	19	,	,	PUNCT
ejpam-3886	35	20	is	be	AUX
ejpam-3886	35	21	a	a	DET
ejpam-3886	35	22	reduced	reduce	VERB
ejpam-3886	35	23	(	(	PUNCT
ejpam-3886	35	24	weak	weak	ADJ
ejpam-3886	35	25	)	)	PUNCT
ejpam-3886	35	26	ps	ps	NOUN
ejpam-3886	35	27	-	-	PUNCT
ejpam-3886	35	28	ring	ring	NOUN
ejpam-3886	35	29	.	.	PUNCT
ejpam-3886	36	1	3	3	X
ejpam-3886	36	2	)	)	PUNCT
ejpam-3886	36	3	let	let	VERB
ejpam-3886	36	4	f	f	PRON
ejpam-3886	36	5	be	be	AUX
ejpam-3886	36	6	a	a	DET
ejpam-3886	36	7	field	field	NOUN
ejpam-3886	36	8	and	and	CCONJ
ejpam-3886	36	9	r	r	NOUN
ejpam-3886	36	10	=	=	SYM
ejpam-3886	36	11	(	(	PUNCT
ejpam-3886	36	12	f	f	NOUN
ejpam-3886	36	13	f	f	PROPN
ejpam-3886	36	14	f	f	PROPN
ejpam-3886	36	15	f	f	PROPN
ejpam-3886	36	16	)	)	PUNCT
ejpam-3886	36	17	is	be	AUX
ejpam-3886	36	18	a	a	DET
ejpam-3886	36	19	right	right	ADJ
ejpam-3886	36	20	weak	weak	ADJ
ejpam-3886	36	21	ps	ps	NOUN
ejpam-3886	36	22	-	-	NOUN
ejpam-3886	36	23	ring	ring	NOUN
ejpam-3886	36	24	.	.	PUNCT
ejpam-3886	37	1	4	4	NUM
ejpam-3886	37	2	)	)	PUNCT
ejpam-3886	37	3	a	a	DET
ejpam-3886	37	4	semisimple	semisimple	NOUN
ejpam-3886	37	5	ni	ni	PROPN
ejpam-3886	37	6	ring	ring	NOUN
ejpam-3886	37	7	is	be	AUX
ejpam-3886	37	8	a	a	DET
ejpam-3886	37	9	right	right	ADJ
ejpam-3886	37	10	weak	weak	ADJ
ejpam-3886	37	11	ps	ps	NOUN
ejpam-3886	37	12	-	-	PUNCT
ejpam-3886	37	13	ring	ring	NOUN
ejpam-3886	37	14	.	.	PUNCT
ejpam-3886	38	1	in	in	ADP
ejpam-3886	38	2	this	this	DET
ejpam-3886	38	3	paper	paper	NOUN
ejpam-3886	38	4	,	,	PUNCT
ejpam-3886	38	5	we	we	PRON
ejpam-3886	38	6	study	study	VERB
ejpam-3886	38	7	the	the	DET
ejpam-3886	38	8	transfer	transfer	NOUN
ejpam-3886	38	9	of	of	ADP
ejpam-3886	38	10	right	right	ADJ
ejpam-3886	38	11	weak	weak	ADJ
ejpam-3886	38	12	ps	ps	NOUN
ejpam-3886	38	13	-	-	PUNCT
ejpam-3886	38	14	condition	condition	NOUN
ejpam-3886	38	15	between	between	ADP
ejpam-3886	38	16	a	a	DET
ejpam-3886	38	17	base	base	NOUN
ejpam-3886	38	18	ring	ring	NOUN
ejpam-3886	38	19	r	r	NOUN
ejpam-3886	38	20	and	and	CCONJ
ejpam-3886	38	21	its	its	PRON
ejpam-3886	38	22	ore	ore	NOUN
ejpam-3886	38	23	extension	extension	NOUN
ejpam-3886	38	24	a	a	DET
ejpam-3886	38	25	=	=	SYM
ejpam-3886	38	26	r[x;α	r[x;α	PROPN
ejpam-3886	38	27	,	,	PUNCT
ejpam-3886	38	28	δ	δ	PROPN
ejpam-3886	38	29	]	]	PUNCT
ejpam-3886	38	30	.	.	PUNCT
ejpam-3886	39	1	2	2	X
ejpam-3886	39	2	.	.	X
ejpam-3886	39	3	notations	notation	NOUN
ejpam-3886	39	4	(	(	PUNCT
ejpam-3886	39	5	1	1	X
ejpam-3886	39	6	)	)	PUNCT
ejpam-3886	39	7	i	i	NOUN
ejpam-3886	39	8	d	d	NOUN
ejpam-3886	39	9	(	(	PUNCT
ejpam-3886	39	10	r	r	NOUN
ejpam-3886	39	11	)	)	PUNCT
ejpam-3886	39	12	denotes	denote	NOUN
ejpam-3886	39	13	idempotents	idempotent	NOUN
ejpam-3886	39	14	of	of	ADP
ejpam-3886	39	15	r.	r.	PROPN
ejpam-3886	39	16	(	(	PUNCT
ejpam-3886	39	17	2	2	NUM
ejpam-3886	39	18	)	)	PUNCT
ejpam-3886	39	19	nil	nil	NOUN
ejpam-3886	39	20	(	(	PUNCT
ejpam-3886	39	21	r	r	NOUN
ejpam-3886	39	22	)	)	PUNCT
ejpam-3886	39	23	denotes	denote	NOUN
ejpam-3886	39	24	nilpotents	nilpotent	NOUN
ejpam-3886	39	25	of	of	ADP
ejpam-3886	39	26	r.	r.	PROPN
ejpam-3886	39	27	(	(	PUNCT
ejpam-3886	39	28	3	3	NUM
ejpam-3886	39	29	)	)	PUNCT
ejpam-3886	39	30	for	for	ADP
ejpam-3886	39	31	a	a	DET
ejpam-3886	39	32	nonempty	nonempty	NOUN
ejpam-3886	39	33	subset	subset	NOUN
ejpam-3886	39	34	x	x	PUNCT
ejpam-3886	39	35	of	of	ADP
ejpam-3886	39	36	r	r	NOUN
ejpam-3886	39	37	,	,	PUNCT
ejpam-3886	39	38	nr(x	nr(x	NUM
ejpam-3886	39	39	)	)	PUNCT
ejpam-3886	39	40	denotes	denote	VERB
ejpam-3886	39	41	the	the	DET
ejpam-3886	39	42	weak	weak	ADJ
ejpam-3886	39	43	annihilator	annihilator	NOUN
ejpam-3886	39	44	of	of	ADP
ejpam-3886	39	45	x	x	PUNCT
ejpam-3886	39	46	over	over	ADP
ejpam-3886	39	47	r	r	NOUN
ejpam-3886	39	48	,	,	PUNCT
ejpam-3886	39	49	i.e.	i.e.	X
ejpam-3886	39	50	,	,	PUNCT
ejpam-3886	39	51	nr(x	nr(x	NUM
ejpam-3886	39	52	)	)	PUNCT
ejpam-3886	40	1	=	=	PRON
ejpam-3886	40	2	{	{	PUNCT
ejpam-3886	40	3	a	a	DET
ejpam-3886	40	4	∈	∈	NOUN
ejpam-3886	40	5	r	r	NOUN
ejpam-3886	40	6	|ax	|ax	X
ejpam-3886	40	7	∈	∈	NOUN
ejpam-3886	40	8	nil	nil	NOUN
ejpam-3886	40	9	(	(	PUNCT
ejpam-3886	40	10	r	r	NOUN
ejpam-3886	40	11	)	)	PUNCT
ejpam-3886	40	12	for	for	ADP
ejpam-3886	40	13	all	all	DET
ejpam-3886	40	14	x	x	SYM
ejpam-3886	40	15	∈	∈	NOUN
ejpam-3886	40	16	x	x	PUNCT
ejpam-3886	40	17	}	}	PUNCT
ejpam-3886	40	18	.	.	PUNCT
ejpam-3886	41	1	it	it	PRON
ejpam-3886	41	2	can	can	AUX
ejpam-3886	41	3	be	be	AUX
ejpam-3886	41	4	easily	easily	ADV
ejpam-3886	41	5	shown	show	VERB
ejpam-3886	41	6	that	that	SCONJ
ejpam-3886	41	7	ab	ab	PROPN
ejpam-3886	41	8	∈	∈	PROPN
ejpam-3886	41	9	nil	nil	NOUN
ejpam-3886	41	10	(	(	PUNCT
ejpam-3886	41	11	r)⇔	r)⇔	PROPN
ejpam-3886	41	12	ba	ba	PROPN
ejpam-3886	41	13	∈	∈	PROPN
ejpam-3886	41	14	nil	nil	NOUN
ejpam-3886	41	15	(	(	PUNCT
ejpam-3886	41	16	r	r	NOUN
ejpam-3886	41	17	)	)	PUNCT
ejpam-3886	41	18	for	for	ADP
ejpam-3886	41	19	all	all	DET
ejpam-3886	41	20	a	a	PRON
ejpam-3886	41	21	,	,	PUNCT
ejpam-3886	41	22	b	b	PROPN
ejpam-3886	41	23	∈	∈	PROPN
ejpam-3886	41	24	r.	r.	NOUN
ejpam-3886	41	25	(	(	PUNCT
ejpam-3886	41	26	4	4	X
ejpam-3886	41	27	)	)	PUNCT
ejpam-3886	41	28	r	r	NOUN
ejpam-3886	41	29	is	be	AUX
ejpam-3886	41	30	ni	ni	NOUN
ejpam-3886	41	31	if	if	SCONJ
ejpam-3886	41	32	nil	nil	NOUN
ejpam-3886	41	33	(	(	PUNCT
ejpam-3886	41	34	r	r	NOUN
ejpam-3886	41	35	)	)	PUNCT
ejpam-3886	41	36	is	be	AUX
ejpam-3886	41	37	a	a	DET
ejpam-3886	41	38	two	two	NUM
ejpam-3886	41	39	sided	sided	ADJ
ejpam-3886	41	40	ideal	ideal	NOUN
ejpam-3886	41	41	in	in	ADP
ejpam-3886	41	42	r.	r.	PROPN
ejpam-3886	41	43	3	3	NUM
ejpam-3886	41	44	.	.	PUNCT
ejpam-3886	41	45	ore	ore	NOUN
ejpam-3886	41	46	extension	extension	NOUN
ejpam-3886	41	47	rings	ring	NOUN
ejpam-3886	41	48	satisfy	satisfy	VERB
ejpam-3886	41	49	the	the	DET
ejpam-3886	41	50	weak	weak	ADJ
ejpam-3886	41	51	ps	ps	NOUN
ejpam-3886	41	52	-	-	PUNCT
ejpam-3886	41	53	condition	condition	NOUN
ejpam-3886	41	54	ore	ore	NOUN
ejpam-3886	41	55	extensions	extension	NOUN
ejpam-3886	41	56	,	,	PUNCT
ejpam-3886	41	57	named	name	VERB
ejpam-3886	41	58	after	after	ADP
ejpam-3886	41	59	øystein	øystein	ADJ
ejpam-3886	41	60	ore	ore	NOUN
ejpam-3886	41	61	(	(	PUNCT
ejpam-3886	41	62	1899–1968	1899–1968	NUM
ejpam-3886	41	63	)	)	PUNCT
ejpam-3886	41	64	,	,	PUNCT
ejpam-3886	41	65	are	be	AUX
ejpam-3886	41	66	special	special	ADJ
ejpam-3886	41	67	types	type	NOUN
ejpam-3886	41	68	of	of	ADP
ejpam-3886	41	69	ring	ring	NOUN
ejpam-3886	41	70	extensions	extension	NOUN
ejpam-3886	41	71	whose	whose	DET
ejpam-3886	41	72	properties	property	NOUN
ejpam-3886	41	73	are	be	AUX
ejpam-3886	41	74	relatively	relatively	ADV
ejpam-3886	41	75	well	well	ADV
ejpam-3886	41	76	understood	understand	VERB
ejpam-3886	41	77	.	.	PUNCT
ejpam-3886	42	1	these	these	DET
ejpam-3886	42	2	extensions	extension	NOUN
ejpam-3886	42	3	cover	cover	VERB
ejpam-3886	42	4	a	a	DET
ejpam-3886	42	5	large	large	ADJ
ejpam-3886	42	6	class	class	NOUN
ejpam-3886	42	7	of	of	ADP
ejpam-3886	42	8	noncommutative	noncommutative	ADJ
ejpam-3886	42	9	polynomial	polynomial	ADJ
ejpam-3886	42	10	extensions	extension	NOUN
ejpam-3886	42	11	.	.	PUNCT
ejpam-3886	43	1	are	be	AUX
ejpam-3886	43	2	special	special	ADJ
ejpam-3886	43	3	types	type	NOUN
ejpam-3886	43	4	of	of	ADP
ejpam-3886	43	5	ring	ring	NOUN
ejpam-3886	43	6	extensions	extension	NOUN
ejpam-3886	43	7	whose	whose	DET
ejpam-3886	43	8	properties	property	NOUN
ejpam-3886	43	9	are	be	AUX
ejpam-3886	43	10	relatively	relatively	ADV
ejpam-3886	43	11	well	well	ADV
ejpam-3886	43	12	understood	understand	VERB
ejpam-3886	43	13	.	.	PUNCT
ejpam-3886	44	1	the	the	DET
ejpam-3886	44	2	definition	definition	NOUN
ejpam-3886	44	3	of	of	ADP
ejpam-3886	44	4	noncommutative	noncommutative	ADJ
ejpam-3886	44	5	polynomial	polynomial	ADJ
ejpam-3886	44	6	rings	ring	NOUN
ejpam-3886	44	7	with	with	ADP
ejpam-3886	44	8	identity	identity	NOUN
ejpam-3886	44	9	was	be	AUX
ejpam-3886	44	10	first	first	ADV
ejpam-3886	44	11	introduced	introduce	VERB
ejpam-3886	44	12	by	by	ADP
ejpam-3886	44	13	øystein	øystein	ADJ
ejpam-3886	44	14	ore	ore	NOUN
ejpam-3886	44	15	[	[	X
ejpam-3886	44	16	7	7	NUM
ejpam-3886	44	17	]	]	PUNCT
ejpam-3886	44	18	.	.	PUNCT
ejpam-3886	45	1	ever	ever	ADV
ejpam-3886	45	2	since	since	SCONJ
ejpam-3886	45	3	the	the	DET
ejpam-3886	45	4	appearance	appearance	NOUN
ejpam-3886	45	5	of	of	ADP
ejpam-3886	45	6	ore	ore	NOUN
ejpam-3886	45	7	’s	’s	PART
ejpam-3886	45	8	fundamental	fundamental	ADJ
ejpam-3886	45	9	paper	paper	NOUN
ejpam-3886	45	10	[	[	X
ejpam-3886	45	11	7	7	NUM
ejpam-3886	45	12	]	]	PUNCT
ejpam-3886	45	13	,	,	PUNCT
ejpam-3886	45	14	ore	ore	NOUN
ejpam-3886	45	15	extensions	extension	NOUN
ejpam-3886	45	16	have	have	AUX
ejpam-3886	45	17	played	play	VERB
ejpam-3886	45	18	an	an	DET
ejpam-3886	45	19	important	important	ADJ
ejpam-3886	45	20	role	role	NOUN
ejpam-3886	45	21	in	in	ADP
ejpam-3886	45	22	noncommutative	noncommutative	ADJ
ejpam-3886	45	23	.	.	PUNCT
ejpam-3886	45	24	ore	ore	NOUN
ejpam-3886	45	25	extensions	extension	NOUN
ejpam-3886	45	26	have	have	VERB
ejpam-3886	45	27	wide	wide	ADJ
ejpam-3886	45	28	applications	application	NOUN
ejpam-3886	45	29	.	.	PUNCT
ejpam-3886	46	1	not	not	PART
ejpam-3886	46	2	only	only	ADV
ejpam-3886	46	3	do	do	AUX
ejpam-3886	46	4	they	they	PRON
ejpam-3886	46	5	provide	provide	VERB
ejpam-3886	46	6	interesting	interesting	ADJ
ejpam-3886	46	7	m.	m.	NOUN
ejpam-3886	46	8	a.	a.	NOUN
ejpam-3886	46	9	farahat	farahat	PROPN
ejpam-3886	46	10	,	,	PUNCT
ejpam-3886	46	11	salha	salha	NOUN
ejpam-3886	46	12	t.	t.	PROPN
ejpam-3886	46	13	al	al	PROPN
ejpam-3886	46	14	-	-	PUNCT
ejpam-3886	46	15	bogamy	bogamy	PROPN
ejpam-3886	46	16	/	/	SYM
ejpam-3886	46	17	eur	eur	PROPN
ejpam-3886	46	18	.	.	PUNCT
ejpam-3886	47	1	j.	j.	PROPN
ejpam-3886	47	2	pure	pure	PROPN
ejpam-3886	47	3	appl	appl	PROPN
ejpam-3886	47	4	.	.	PROPN
ejpam-3886	47	5	math	math	PROPN
ejpam-3886	47	6	,	,	PUNCT
ejpam-3886	47	7	14	14	NUM
ejpam-3886	47	8	(	(	PUNCT
ejpam-3886	47	9	1	1	NUM
ejpam-3886	47	10	)	)	PUNCT
ejpam-3886	47	11	(	(	PUNCT
ejpam-3886	47	12	2021	2021	NUM
ejpam-3886	47	13	)	)	PUNCT
ejpam-3886	47	14	,	,	PUNCT
ejpam-3886	47	15	164	164	NUM
ejpam-3886	47	16	-	-	SYM
ejpam-3886	47	17	172	172	NUM
ejpam-3886	47	18	166	166	NUM
ejpam-3886	47	19	examples	example	NOUN
ejpam-3886	47	20	in	in	ADP
ejpam-3886	47	21	noncommutative	noncommutative	ADJ
ejpam-3886	47	22	algebra	algebra	NOUN
ejpam-3886	47	23	,	,	PUNCT
ejpam-3886	47	24	they	they	PRON
ejpam-3886	47	25	have	have	AUX
ejpam-3886	47	26	also	also	ADV
ejpam-3886	47	27	been	be	AUX
ejpam-3886	47	28	a	a	DET
ejpam-3886	47	29	valuable	valuable	ADJ
ejpam-3886	47	30	tool	tool	NOUN
ejpam-3886	47	31	used	use	VERB
ejpam-3886	47	32	first	first	ADV
ejpam-3886	47	33	by	by	ADP
ejpam-3886	47	34	david	david	PROPN
ejpam-3886	47	35	hilbert	hilbert	PROPN
ejpam-3886	47	36	(	(	PUNCT
ejpam-3886	47	37	1862–1943	1862–1943	NUM
ejpam-3886	47	38	)	)	PUNCT
ejpam-3886	47	39	in	in	ADP
ejpam-3886	47	40	the	the	DET
ejpam-3886	47	41	study	study	NOUN
ejpam-3886	47	42	of	of	ADP
ejpam-3886	47	43	the	the	DET
ejpam-3886	47	44	independence	independence	NOUN
ejpam-3886	47	45	of	of	ADP
ejpam-3886	47	46	geometry	geometry	NOUN
ejpam-3886	47	47	axioms	axiom	NOUN
ejpam-3886	47	48	.	.	PUNCT
ejpam-3886	48	1	let	let	VERB
ejpam-3886	48	2	r	r	PRON
ejpam-3886	48	3	be	be	AUX
ejpam-3886	48	4	a	a	DET
ejpam-3886	48	5	ring	ring	NOUN
ejpam-3886	48	6	with	with	ADP
ejpam-3886	48	7	identity	identity	NOUN
ejpam-3886	48	8	1	1	NUM
ejpam-3886	48	9	and	and	CCONJ
ejpam-3886	48	10	α	α	PRON
ejpam-3886	48	11	an	an	DET
ejpam-3886	48	12	endomorphism	endomorphism	NOUN
ejpam-3886	48	13	of	of	ADP
ejpam-3886	48	14	r.	r.	PROPN
ejpam-3886	48	15	then	then	ADV
ejpam-3886	48	16	a	a	DET
ejpam-3886	48	17	map	map	NOUN
ejpam-3886	48	18	δ	δ	NOUN
ejpam-3886	48	19	:	:	PUNCT
ejpam-3886	48	20	r	r	NOUN
ejpam-3886	48	21	−→	−→	NOUN
ejpam-3886	48	22	r	r	NOUN
ejpam-3886	48	23	is	be	AUX
ejpam-3886	48	24	called	call	VERB
ejpam-3886	48	25	an	an	DET
ejpam-3886	48	26	α	α	NOUN
ejpam-3886	48	27	-	-	NOUN
ejpam-3886	48	28	derivation	derivation	NOUN
ejpam-3886	48	29	of	of	ADP
ejpam-3886	48	30	r	r	NOUN
ejpam-3886	49	1	if	if	SCONJ
ejpam-3886	49	2	δ	δ	PROPN
ejpam-3886	49	3	(	(	PUNCT
ejpam-3886	49	4	a+	a+	NOUN
ejpam-3886	49	5	b	b	NOUN
ejpam-3886	49	6	)	)	PUNCT
ejpam-3886	49	7	=	=	SYM
ejpam-3886	49	8	δ	δ	PROPN
ejpam-3886	49	9	(	(	PUNCT
ejpam-3886	49	10	a	a	NOUN
ejpam-3886	49	11	)	)	PUNCT
ejpam-3886	49	12	+	+	CCONJ
ejpam-3886	49	13	δ	δ	PROPN
ejpam-3886	49	14	(	(	PUNCT
ejpam-3886	49	15	b	b	NOUN
ejpam-3886	49	16	)	)	PUNCT
ejpam-3886	49	17	and	and	CCONJ
ejpam-3886	49	18	δ	δ	PROPN
ejpam-3886	49	19	(	(	PUNCT
ejpam-3886	49	20	ab	ab	PROPN
ejpam-3886	49	21	)	)	PUNCT
ejpam-3886	49	22	=	=	SYM
ejpam-3886	49	23	δ	δ	PROPN
ejpam-3886	49	24	(	(	PUNCT
ejpam-3886	49	25	a	a	NOUN
ejpam-3886	49	26	)	)	PUNCT
ejpam-3886	49	27	b+	b+	X
ejpam-3886	49	28	α	α	X
ejpam-3886	49	29	(	(	PUNCT
ejpam-3886	49	30	a	a	PROPN
ejpam-3886	49	31	)	)	PUNCT
ejpam-3886	49	32	δ	δ	NOUN
ejpam-3886	49	33	(	(	PUNCT
ejpam-3886	49	34	b	b	NOUN
ejpam-3886	49	35	)	)	PUNCT
ejpam-3886	49	36	,	,	PUNCT
ejpam-3886	49	37	for	for	ADP
ejpam-3886	49	38	all	all	DET
ejpam-3886	49	39	a	a	DET
ejpam-3886	49	40	,	,	PUNCT
ejpam-3886	49	41	b	b	X
ejpam-3886	49	42	∈	∈	PROPN
ejpam-3886	49	43	r.	r.	NOUN
ejpam-3886	49	44	we	we	PRON
ejpam-3886	49	45	denote	denote	VERB
ejpam-3886	49	46	by	by	ADP
ejpam-3886	49	47	a	a	DET
ejpam-3886	49	48	=	=	X
ejpam-3886	49	49	r	r	NOUN
ejpam-3886	49	50	[	[	X
ejpam-3886	49	51	x;α	x;α	PROPN
ejpam-3886	49	52	,	,	PUNCT
ejpam-3886	49	53	δ	δ	PROPN
ejpam-3886	49	54	]	]	PUNCT
ejpam-3886	49	55	,	,	PUNCT
ejpam-3886	49	56	the	the	DET
ejpam-3886	49	57	ore	ore	NOUN
ejpam-3886	49	58	extension	extension	NOUN
ejpam-3886	49	59	of	of	ADP
ejpam-3886	49	60	r	r	NOUN
ejpam-3886	49	61	whose	whose	DET
ejpam-3886	49	62	elements	element	NOUN
ejpam-3886	49	63	are	be	AUX
ejpam-3886	49	64	polynomials	polynomial	NOUN
ejpam-3886	49	65	over	over	ADP
ejpam-3886	49	66	r	r	NOUN
ejpam-3886	49	67	,	,	PUNCT
ejpam-3886	49	68	the	the	DET
ejpam-3886	49	69	addition	addition	NOUN
ejpam-3886	49	70	is	be	AUX
ejpam-3886	49	71	defined	define	VERB
ejpam-3886	49	72	as	as	ADP
ejpam-3886	49	73	usual	usual	ADJ
ejpam-3886	49	74	and	and	CCONJ
ejpam-3886	49	75	the	the	DET
ejpam-3886	49	76	multiplication	multiplication	NOUN
ejpam-3886	49	77	is	be	AUX
ejpam-3886	49	78	subject	subject	ADJ
ejpam-3886	49	79	to	to	ADP
ejpam-3886	49	80	the	the	DET
ejpam-3886	49	81	relation	relation	NOUN
ejpam-3886	49	82	(	(	PUNCT
ejpam-3886	49	83	ore	ore	NOUN
ejpam-3886	49	84	commutation	commutation	NOUN
ejpam-3886	49	85	rule	rule	NOUN
ejpam-3886	49	86	)	)	PUNCT
ejpam-3886	50	1	xa	xa	PROPN
ejpam-3886	51	1	=	=	SYM
ejpam-3886	51	2	α	α	PROPN
ejpam-3886	51	3	(	(	PUNCT
ejpam-3886	51	4	a)x+	a)x+	PROPN
ejpam-3886	51	5	δ	δ	PROPN
ejpam-3886	51	6	(	(	PUNCT
ejpam-3886	51	7	a	a	NOUN
ejpam-3886	51	8	)	)	PUNCT
ejpam-3886	51	9	,	,	PUNCT
ejpam-3886	51	10	for	for	ADP
ejpam-3886	51	11	each	each	DET
ejpam-3886	51	12	a	a	DET
ejpam-3886	51	13	∈	∈	PROPN
ejpam-3886	51	14	r.	r.	NOUN
ejpam-3886	51	15	we	we	PRON
ejpam-3886	51	16	assume	assume	VERB
ejpam-3886	51	17	that	that	SCONJ
ejpam-3886	51	18	1	1	NUM
ejpam-3886	51	19	is	be	AUX
ejpam-3886	51	20	the	the	DET
ejpam-3886	51	21	identity	identity	NOUN
ejpam-3886	51	22	element	element	NOUN
ejpam-3886	51	23	of	of	ADP
ejpam-3886	51	24	a	a	DET
ejpam-3886	51	25	=	=	SYM
ejpam-3886	51	26	r	r	NOUN
ejpam-3886	51	27	[	[	X
ejpam-3886	51	28	x;α	x;α	PROPN
ejpam-3886	51	29	,	,	PUNCT
ejpam-3886	51	30	δ	δ	PROPN
ejpam-3886	51	31	]	]	PUNCT
ejpam-3886	51	32	.	.	PUNCT
ejpam-3886	52	1	this	this	PRON
ejpam-3886	52	2	means	mean	VERB
ejpam-3886	52	3	that	that	SCONJ
ejpam-3886	52	4	α(1	α(1	PROPN
ejpam-3886	52	5	)	)	PUNCT
ejpam-3886	52	6	=	=	SYM
ejpam-3886	52	7	1	1	NUM
ejpam-3886	52	8	and	and	CCONJ
ejpam-3886	52	9	δ	δ	PROPN
ejpam-3886	52	10	(	(	PUNCT
ejpam-3886	52	11	1	1	NUM
ejpam-3886	52	12	)	)	PUNCT
ejpam-3886	52	13	=	=	SYM
ejpam-3886	52	14	0	0	NUM
ejpam-3886	52	15	,	,	PUNCT
ejpam-3886	52	16	since	since	SCONJ
ejpam-3886	52	17	x	x	X
ejpam-3886	52	18	=	=	SYM
ejpam-3886	53	1	x1	x1	PROPN
ejpam-3886	53	2	=	=	SYM
ejpam-3886	53	3	α	α	PROPN
ejpam-3886	53	4	(	(	PUNCT
ejpam-3886	53	5	1)x+	1)x+	NUM
ejpam-3886	53	6	δ	δ	NOUN
ejpam-3886	53	7	(	(	PUNCT
ejpam-3886	53	8	1)⇒	1)⇒	NUM
ejpam-3886	53	9	α(1	α(1	PROPN
ejpam-3886	53	10	)	)	PUNCT
ejpam-3886	53	11	=	=	SYM
ejpam-3886	53	12	1	1	NUM
ejpam-3886	53	13	and	and	CCONJ
ejpam-3886	53	14	δ	δ	PROPN
ejpam-3886	53	15	(	(	PUNCT
ejpam-3886	53	16	1	1	NUM
ejpam-3886	53	17	)	)	PUNCT
ejpam-3886	53	18	=	=	SYM
ejpam-3886	53	19	0	0	X
ejpam-3886	53	20	.	.	PUNCT
ejpam-3886	54	1	notation	notation	NOUN
ejpam-3886	54	2	(	(	PUNCT
ejpam-3886	54	3	[	[	X
ejpam-3886	54	4	5	5	NUM
ejpam-3886	54	5	]	]	PUNCT
ejpam-3886	54	6	)	)	PUNCT
ejpam-3886	54	7	.	.	PUNCT
ejpam-3886	55	1	for	for	ADP
ejpam-3886	55	2	integers	integer	NOUN
ejpam-3886	55	3	i	i	PRON
ejpam-3886	55	4	,	,	PUNCT
ejpam-3886	55	5	j	j	PROPN
ejpam-3886	55	6	with	with	ADP
ejpam-3886	55	7	j	j	PROPN
ejpam-3886	55	8	≥	≥	NUM
ejpam-3886	55	9	i	i	PRON
ejpam-3886	55	10	≥	≥	NOUN
ejpam-3886	55	11	0	0	NUM
ejpam-3886	55	12	,	,	PUNCT
ejpam-3886	55	13	λji	λji	PROPN
ejpam-3886	55	14	∈	∈	PROPN
ejpam-3886	55	15	end(r,+	end(r,+	PROPN
ejpam-3886	55	16	)	)	PUNCT
ejpam-3886	55	17	denotes	denote	VERB
ejpam-3886	55	18	the	the	DET
ejpam-3886	55	19	map	map	NOUN
ejpam-3886	55	20	which	which	PRON
ejpam-3886	55	21	is	be	AUX
ejpam-3886	55	22	the	the	DET
ejpam-3886	55	23	sum	sum	NOUN
ejpam-3886	55	24	of	of	ADP
ejpam-3886	55	25	all	all	DET
ejpam-3886	55	26	possible	possible	ADJ
ejpam-3886	55	27	”	"	PUNCT
ejpam-3886	55	28	words	word	NOUN
ejpam-3886	55	29	”	"	PUNCT
ejpam-3886	55	30	in	in	ADP
ejpam-3886	55	31	α	α	PROPN
ejpam-3886	55	32	and	and	CCONJ
ejpam-3886	55	33	δ	δ	PROPN
ejpam-3886	55	34	built	build	VERB
ejpam-3886	55	35	with	with	ADP
ejpam-3886	55	36	i	i	PROPN
ejpam-3886	55	37	letters	letter	NOUN
ejpam-3886	55	38	of	of	ADP
ejpam-3886	55	39	α	α	PROPN
ejpam-3886	55	40	and	and	CCONJ
ejpam-3886	55	41	j	j	PROPN
ejpam-3886	56	1	−	−	PROPN
ejpam-3886	57	1	i	i	PRON
ejpam-3886	57	2	letters	letter	NOUN
ejpam-3886	57	3	of	of	ADP
ejpam-3886	57	4	δ	δ	PROPN
ejpam-3886	57	5	.	.	PUNCT
ejpam-3886	58	1	for	for	ADP
ejpam-3886	58	2	instance	instance	NOUN
ejpam-3886	58	3	λ00	λ00	NOUN
ejpam-3886	58	4	=	=	SYM
ejpam-3886	58	5	idr	idr	PROPN
ejpam-3886	58	6	,	,	PUNCT
ejpam-3886	58	7	λ	λ	PROPN
ejpam-3886	58	8	j	j	PROPN
ejpam-3886	58	9	j	j	PROPN
ejpam-3886	58	10	=	=	SYM
ejpam-3886	58	11	αj	αj	PROPN
ejpam-3886	58	12	,	,	PUNCT
ejpam-3886	58	13	λj0	λj0	X
ejpam-3886	58	14	=	=	NOUN
ejpam-3886	58	15	δj	δj	NOUN
ejpam-3886	58	16	and	and	CCONJ
ejpam-3886	58	17	λjj−1	λjj−1	PROPN
ejpam-3886	58	18	=	=	PUNCT
ejpam-3886	59	1	αj−1δ	αj−1δ	PROPN
ejpam-3886	59	2	+	+	CCONJ
ejpam-3886	59	3	αj−2δα+	αj−2δα+	NOUN
ejpam-3886	59	4	...	...	PUNCT
ejpam-3886	59	5	+	+	PUNCT
ejpam-3886	59	6	δαj−1	δαj−1	NOUN
ejpam-3886	59	7	.	.	PUNCT
ejpam-3886	60	1	lemma	lemma	PROPN
ejpam-3886	60	2	1	1	NUM
ejpam-3886	60	3	(	(	PUNCT
ejpam-3886	60	4	[	[	X
ejpam-3886	60	5	5	5	NUM
ejpam-3886	60	6	]	]	PUNCT
ejpam-3886	60	7	)	)	PUNCT
ejpam-3886	60	8	.	.	PUNCT
ejpam-3886	61	1	for	for	ADP
ejpam-3886	61	2	any	any	DET
ejpam-3886	61	3	positive	positive	ADJ
ejpam-3886	61	4	integer	integer	NOUN
ejpam-3886	61	5	n	n	NOUN
ejpam-3886	61	6	and	and	CCONJ
ejpam-3886	61	7	r	r	NOUN
ejpam-3886	61	8	∈	∈	PROPN
ejpam-3886	61	9	r	r	NOUN
ejpam-3886	61	10	,	,	PUNCT
ejpam-3886	61	11	we	we	PRON
ejpam-3886	61	12	have	have	VERB
ejpam-3886	61	13	xnr	xnr	NUM
ejpam-3886	61	14	=	=	SYM
ejpam-3886	61	15	n∑	n∑	PROPN
ejpam-3886	61	16	i=0	i=0	PROPN
ejpam-3886	61	17	λni	λni	PROPN
ejpam-3886	61	18	(	(	PUNCT
ejpam-3886	61	19	r)xi	r)xi	PROPN
ejpam-3886	61	20	.	.	PUNCT
ejpam-3886	62	1	this	this	DET
ejpam-3886	62	2	formula	formula	NOUN
ejpam-3886	62	3	uniquely	uniquely	ADV
ejpam-3886	62	4	determines	determine	VERB
ejpam-3886	62	5	a	a	DET
ejpam-3886	62	6	general	general	ADJ
ejpam-3886	62	7	product	product	NOUN
ejpam-3886	62	8	of	of	ADP
ejpam-3886	62	9	(	(	PUNCT
ejpam-3886	62	10	left	left	ADJ
ejpam-3886	62	11	)	)	PUNCT
ejpam-3886	62	12	polynomials	polynomial	NOUN
ejpam-3886	62	13	in	in	ADP
ejpam-3886	62	14	r	r	NOUN
ejpam-3886	62	15	[	[	X
ejpam-3886	62	16	x;α	x;α	PROPN
ejpam-3886	62	17	,	,	PUNCT
ejpam-3886	62	18	δ	δ	PROPN
ejpam-3886	62	19	]	]	PUNCT
ejpam-3886	62	20	and	and	CCONJ
ejpam-3886	62	21	will	will	AUX
ejpam-3886	62	22	be	be	AUX
ejpam-3886	62	23	used	use	VERB
ejpam-3886	62	24	freely	freely	ADV
ejpam-3886	62	25	in	in	ADP
ejpam-3886	62	26	what	what	PRON
ejpam-3886	62	27	follows	follow	VERB
ejpam-3886	62	28	.	.	PUNCT
ejpam-3886	63	1	the	the	DET
ejpam-3886	63	2	ring	ring	NOUN
ejpam-3886	63	3	-	-	PUNCT
ejpam-3886	63	4	theoretical	theoretical	ADJ
ejpam-3886	63	5	properties	property	NOUN
ejpam-3886	63	6	of	of	ADP
ejpam-3886	63	7	ore	ore	NOUN
ejpam-3886	63	8	extension	extension	NOUN
ejpam-3886	63	9	have	have	AUX
ejpam-3886	63	10	been	be	AUX
ejpam-3886	63	11	investigated	investigate	VERB
ejpam-3886	63	12	by	by	ADP
ejpam-3886	63	13	many	many	ADJ
ejpam-3886	63	14	authors	author	NOUN
ejpam-3886	63	15	(	(	PUNCT
ejpam-3886	63	16	see	see	VERB
ejpam-3886	63	17	[	[	X
ejpam-3886	63	18	9	9	NUM
ejpam-3886	63	19	]	]	PUNCT
ejpam-3886	63	20	,	,	PUNCT
ejpam-3886	63	21	[	[	X
ejpam-3886	63	22	8	8	NUM
ejpam-3886	63	23	]	]	PUNCT
ejpam-3886	63	24	,	,	PUNCT
ejpam-3886	63	25	[	[	X
ejpam-3886	63	26	5	5	NUM
ejpam-3886	63	27	]	]	PUNCT
ejpam-3886	63	28	,	,	PUNCT
ejpam-3886	63	29	[	[	X
ejpam-3886	63	30	1	1	NUM
ejpam-3886	63	31	]	]	PUNCT
ejpam-3886	63	32	,	,	PUNCT
ejpam-3886	63	33	[	[	X
ejpam-3886	63	34	4	4	NUM
ejpam-3886	63	35	]	]	PUNCT
ejpam-3886	63	36	,	,	PUNCT
ejpam-3886	63	37	for	for	ADP
ejpam-3886	63	38	instance	instance	NOUN
ejpam-3886	63	39	)	)	PUNCT
ejpam-3886	63	40	.	.	PUNCT
ejpam-3886	64	1	there	there	PRON
ejpam-3886	64	2	are	be	VERB
ejpam-3886	64	3	many	many	ADJ
ejpam-3886	64	4	other	other	ADJ
ejpam-3886	64	5	papers	paper	NOUN
ejpam-3886	64	6	addressed	address	VERB
ejpam-3886	64	7	δ	δ	X
ejpam-3886	64	8	=	=	PUNCT
ejpam-3886	64	9	0	0	PROPN
ejpam-3886	64	10	and	and	CCONJ
ejpam-3886	64	11	α	α	PRON
ejpam-3886	64	12	an	an	DET
ejpam-3886	64	13	automorphism	automorphism	NOUN
ejpam-3886	64	14	or	or	CCONJ
ejpam-3886	64	15	the	the	DET
ejpam-3886	64	16	case	case	NOUN
ejpam-3886	64	17	where	where	SCONJ
ejpam-3886	64	18	α	α	NOUN
ejpam-3886	64	19	is	be	AUX
ejpam-3886	64	20	the	the	DET
ejpam-3886	64	21	identity	identity	NOUN
ejpam-3886	64	22	.	.	PUNCT
ejpam-3886	65	1	however	however	ADV
ejpam-3886	65	2	the	the	DET
ejpam-3886	65	3	recent	recent	ADJ
ejpam-3886	65	4	surge	surge	NOUN
ejpam-3886	65	5	of	of	ADP
ejpam-3886	65	6	interest	interest	NOUN
ejpam-3886	65	7	in	in	ADP
ejpam-3886	65	8	quantum	quantum	ADJ
ejpam-3886	65	9	groups	group	NOUN
ejpam-3886	65	10	and	and	CCONJ
ejpam-3886	65	11	quantized	quantize	VERB
ejpam-3886	65	12	algebras	algebras	PROPN
ejpam-3886	65	13	has	have	AUX
ejpam-3886	65	14	brought	bring	VERB
ejpam-3886	65	15	renewed	renew	VERB
ejpam-3886	65	16	interest	interest	NOUN
ejpam-3886	65	17	in	in	ADP
ejpam-3886	65	18	general	general	ADJ
ejpam-3886	65	19	ore	ore	NOUN
ejpam-3886	65	20	extensions	extension	NOUN
ejpam-3886	65	21	,	,	PUNCT
ejpam-3886	65	22	due	due	ADP
ejpam-3886	65	23	to	to	ADP
ejpam-3886	65	24	the	the	DET
ejpam-3886	65	25	fact	fact	NOUN
ejpam-3886	65	26	that	that	SCONJ
ejpam-3886	65	27	many	many	ADJ
ejpam-3886	65	28	of	of	ADP
ejpam-3886	65	29	these	these	DET
ejpam-3886	65	30	quantized	quantize	VERB
ejpam-3886	65	31	algebras	algebra	NOUN
ejpam-3886	65	32	and	and	CCONJ
ejpam-3886	65	33	their	their	PRON
ejpam-3886	65	34	representations	representation	NOUN
ejpam-3886	65	35	can	can	AUX
ejpam-3886	65	36	be	be	AUX
ejpam-3886	65	37	expressed	express	VERB
ejpam-3886	65	38	in	in	ADP
ejpam-3886	65	39	terms	term	NOUN
ejpam-3886	65	40	of	of	ADP
ejpam-3886	65	41	ore	ore	NOUN
ejpam-3886	65	42	extension	extension	NOUN
ejpam-3886	65	43	rings	ring	NOUN
ejpam-3886	65	44	.	.	PUNCT
ejpam-3886	66	1	when	when	SCONJ
ejpam-3886	66	2	we	we	PRON
ejpam-3886	66	3	move	move	VERB
ejpam-3886	66	4	from	from	ADP
ejpam-3886	66	5	these	these	DET
ejpam-3886	66	6	“	"	PUNCT
ejpam-3886	66	7	unmixed	unmixed	ADJ
ejpam-3886	66	8	”	"	PUNCT
ejpam-3886	66	9	polynomials	polynomial	NOUN
ejpam-3886	66	10	to	to	ADP
ejpam-3886	66	11	the	the	DET
ejpam-3886	66	12	general	general	ADJ
ejpam-3886	66	13	case	case	NOUN
ejpam-3886	66	14	with	with	ADP
ejpam-3886	66	15	an	an	DET
ejpam-3886	66	16	endomorphism	endomorphism	PROPN
ejpam-3886	66	17	α	α	NOUN
ejpam-3886	66	18	and	and	CCONJ
ejpam-3886	66	19	an	an	DET
ejpam-3886	66	20	α	α	NOUN
ejpam-3886	66	21	-	-	PUNCT
ejpam-3886	66	22	derivation	derivation	ADJ
ejpam-3886	66	23	δ	δ	NOUN
ejpam-3886	66	24	,	,	PUNCT
ejpam-3886	66	25	we	we	PRON
ejpam-3886	66	26	face	face	VERB
ejpam-3886	66	27	a	a	DET
ejpam-3886	66	28	much	much	ADV
ejpam-3886	66	29	greater	great	ADJ
ejpam-3886	66	30	challenge	challenge	NOUN
ejpam-3886	66	31	.	.	PUNCT
ejpam-3886	67	1	annin	annin	ADJ
ejpam-3886	68	1	[	[	X
ejpam-3886	68	2	1	1	NUM
ejpam-3886	68	3	]	]	PUNCT
ejpam-3886	68	4	introduced	introduce	VERB
ejpam-3886	68	5	the	the	DET
ejpam-3886	68	6	notion	notion	NOUN
ejpam-3886	68	7	of	of	ADP
ejpam-3886	68	8	(	(	PUNCT
ejpam-3886	68	9	α	α	PROPN
ejpam-3886	68	10	,	,	PUNCT
ejpam-3886	68	11	δ)-compatibility	δ)-compatibility	NOUN
ejpam-3886	68	12	as	as	SCONJ
ejpam-3886	68	13	follows	follow	VERB
ejpam-3886	68	14	:	:	PUNCT
ejpam-3886	68	15	definition	definition	NOUN
ejpam-3886	68	16	1	1	NUM
ejpam-3886	68	17	(	(	PUNCT
ejpam-3886	68	18	[	[	X
ejpam-3886	68	19	1	1	NUM
ejpam-3886	68	20	]	]	PUNCT
ejpam-3886	68	21	)	)	PUNCT
ejpam-3886	68	22	.	.	PUNCT
ejpam-3886	69	1	given	give	VERB
ejpam-3886	69	2	a	a	DET
ejpam-3886	69	3	module	module	NOUN
ejpam-3886	69	4	mr	mr	PROPN
ejpam-3886	69	5	,	,	PUNCT
ejpam-3886	69	6	an	an	DET
ejpam-3886	69	7	endomorphism	endomorphism	NOUN
ejpam-3886	69	8	α	α	NOUN
ejpam-3886	69	9	:	:	PUNCT
ejpam-3886	69	10	r	r	NOUN
ejpam-3886	69	11	−→	−→	NOUN
ejpam-3886	69	12	r	r	NOUN
ejpam-3886	69	13	and	and	CCONJ
ejpam-3886	69	14	an	an	DET
ejpam-3886	69	15	αderivation	αderivation	NOUN
ejpam-3886	69	16	δ	δ	NOUN
ejpam-3886	69	17	:	:	PUNCT
ejpam-3886	69	18	r	r	NOUN
ejpam-3886	69	19	−→	−→	PROPN
ejpam-3886	69	20	r.	r.	NOUN
ejpam-3886	69	21	we	we	PRON
ejpam-3886	69	22	say	say	VERB
ejpam-3886	69	23	that	that	SCONJ
ejpam-3886	69	24	mr	mr	PROPN
ejpam-3886	69	25	is	be	AUX
ejpam-3886	69	26	α	α	NOUN
ejpam-3886	69	27	-	-	ADJ
ejpam-3886	69	28	compatible	compatible	ADJ
ejpam-3886	69	29	if	if	SCONJ
ejpam-3886	69	30	for	for	ADP
ejpam-3886	69	31	each	each	DET
ejpam-3886	69	32	m	m	NOUN
ejpam-3886	69	33	∈m	∈m	NOUN
ejpam-3886	69	34	and	and	CCONJ
ejpam-3886	69	35	r	r	NOUN
ejpam-3886	69	36	∈	∈	PROPN
ejpam-3886	69	37	r	r	NOUN
ejpam-3886	69	38	,	,	PUNCT
ejpam-3886	69	39	we	we	PRON
ejpam-3886	69	40	have	have	VERB
ejpam-3886	69	41	mr	mr	PROPN
ejpam-3886	69	42	=	=	PUNCT
ejpam-3886	69	43	0⇔	0⇔	NOUN
ejpam-3886	69	44	mα(r	mα(r	NOUN
ejpam-3886	69	45	)	)	PUNCT
ejpam-3886	69	46	=	=	SYM
ejpam-3886	70	1	0	0	X
ejpam-3886	70	2	.	.	PUNCT
ejpam-3886	71	1	moreover	moreover	ADV
ejpam-3886	71	2	,	,	PUNCT
ejpam-3886	71	3	we	we	PRON
ejpam-3886	71	4	say	say	VERB
ejpam-3886	71	5	that	that	SCONJ
ejpam-3886	71	6	mr	mr	PROPN
ejpam-3886	71	7	is	be	AUX
ejpam-3886	71	8	δ	δ	NOUN
ejpam-3886	71	9	-	-	ADJ
ejpam-3886	71	10	compatible	compatible	ADJ
ejpam-3886	71	11	if	if	SCONJ
ejpam-3886	71	12	for	for	ADP
ejpam-3886	71	13	each	each	DET
ejpam-3886	71	14	m	m	NOUN
ejpam-3886	71	15	∈m	∈m	NOUN
ejpam-3886	71	16	and	and	CCONJ
ejpam-3886	71	17	r	r	NOUN
ejpam-3886	71	18	∈	∈	PROPN
ejpam-3886	71	19	r	r	NOUN
ejpam-3886	71	20	,	,	PUNCT
ejpam-3886	71	21	we	we	PRON
ejpam-3886	71	22	have	have	VERB
ejpam-3886	72	1	mr	mr	PROPN
ejpam-3886	72	2	=	=	SYM
ejpam-3886	72	3	0	0	PUNCT
ejpam-3886	72	4	=	=	NOUN
ejpam-3886	72	5	⇒	⇒	NOUN
ejpam-3886	72	6	mδ(r	mδ(r	PUNCT
ejpam-3886	72	7	)	)	PUNCT
ejpam-3886	72	8	=	=	SYM
ejpam-3886	73	1	0	0	X
ejpam-3886	73	2	.	.	PUNCT
ejpam-3886	74	1	if	if	SCONJ
ejpam-3886	74	2	mr	mr	PROPN
ejpam-3886	74	3	is	be	AUX
ejpam-3886	74	4	both	both	PRON
ejpam-3886	74	5	α	α	NOUN
ejpam-3886	74	6	-	-	ADJ
ejpam-3886	74	7	compatible	compatible	ADJ
ejpam-3886	74	8	and	and	CCONJ
ejpam-3886	74	9	δ	δ	NOUN
ejpam-3886	74	10	-	-	PUNCT
ejpam-3886	74	11	compatible	compatible	ADJ
ejpam-3886	74	12	,	,	PUNCT
ejpam-3886	74	13	we	we	PRON
ejpam-3886	74	14	say	say	VERB
ejpam-3886	74	15	that	that	SCONJ
ejpam-3886	74	16	mr	mr	PROPN
ejpam-3886	74	17	is	be	AUX
ejpam-3886	74	18	(	(	PUNCT
ejpam-3886	74	19	α	α	NOUN
ejpam-3886	74	20	,	,	PUNCT
ejpam-3886	74	21	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	74	22	.	.	PUNCT
ejpam-3886	74	23	m.	m.	NOUN
ejpam-3886	74	24	a.	a.	PROPN
ejpam-3886	74	25	farahat	farahat	PROPN
ejpam-3886	74	26	,	,	PUNCT
ejpam-3886	74	27	salha	salha	NOUN
ejpam-3886	74	28	t.	t.	PROPN
ejpam-3886	74	29	al	al	PROPN
ejpam-3886	74	30	-	-	PUNCT
ejpam-3886	74	31	bogamy	bogamy	PROPN
ejpam-3886	74	32	/	/	SYM
ejpam-3886	74	33	eur	eur	PROPN
ejpam-3886	74	34	.	.	PUNCT
ejpam-3886	75	1	j.	j.	PROPN
ejpam-3886	75	2	pure	pure	PROPN
ejpam-3886	75	3	appl	appl	PROPN
ejpam-3886	75	4	.	.	PROPN
ejpam-3886	75	5	math	math	PROPN
ejpam-3886	75	6	,	,	PUNCT
ejpam-3886	75	7	14	14	NUM
ejpam-3886	75	8	(	(	PUNCT
ejpam-3886	75	9	1	1	NUM
ejpam-3886	75	10	)	)	PUNCT
ejpam-3886	75	11	(	(	PUNCT
ejpam-3886	75	12	2021	2021	NUM
ejpam-3886	75	13	)	)	PUNCT
ejpam-3886	75	14	,	,	PUNCT
ejpam-3886	75	15	164	164	NUM
ejpam-3886	75	16	-	-	SYM
ejpam-3886	75	17	172	172	NUM
ejpam-3886	75	18	167	167	NUM
ejpam-3886	75	19	a	a	DET
ejpam-3886	75	20	ring	ring	NOUN
ejpam-3886	75	21	r	r	NOUN
ejpam-3886	75	22	is	be	AUX
ejpam-3886	75	23	called	call	VERB
ejpam-3886	75	24	(	(	PUNCT
ejpam-3886	75	25	α	α	X
ejpam-3886	75	26	,	,	PUNCT
ejpam-3886	75	27	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	75	28	if	if	SCONJ
ejpam-3886	75	29	rr	rr	PROPN
ejpam-3886	75	30	is	be	AUX
ejpam-3886	75	31	an	an	DET
ejpam-3886	75	32	(	(	PUNCT
ejpam-3886	75	33	α	α	NOUN
ejpam-3886	75	34	,	,	PUNCT
ejpam-3886	75	35	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	75	36	module	module	NOUN
ejpam-3886	75	37	.	.	PUNCT
ejpam-3886	76	1	the	the	DET
ejpam-3886	76	2	(	(	PUNCT
ejpam-3886	76	3	α	α	NOUN
ejpam-3886	76	4	,	,	PUNCT
ejpam-3886	76	5	δ)compatible	δ)compatible	ADJ
ejpam-3886	76	6	condition	condition	NOUN
ejpam-3886	76	7	on	on	ADP
ejpam-3886	76	8	the	the	DET
ejpam-3886	76	9	module	module	NOUN
ejpam-3886	76	10	mr	mr	PROPN
ejpam-3886	76	11	is	be	AUX
ejpam-3886	76	12	a	a	DET
ejpam-3886	76	13	natural	natural	ADJ
ejpam-3886	76	14	,	,	PUNCT
ejpam-3886	76	15	independently	independently	ADV
ejpam-3886	76	16	interesting	interesting	ADJ
ejpam-3886	76	17	condition	condition	NOUN
ejpam-3886	76	18	from	from	ADP
ejpam-3886	76	19	which	which	PRON
ejpam-3886	76	20	we	we	PRON
ejpam-3886	76	21	can	can	AUX
ejpam-3886	76	22	derive	derive	VERB
ejpam-3886	76	23	a	a	DET
ejpam-3886	76	24	number	number	NOUN
ejpam-3886	76	25	of	of	ADP
ejpam-3886	76	26	interesting	interesting	ADJ
ejpam-3886	76	27	properties	property	NOUN
ejpam-3886	76	28	,	,	PUNCT
ejpam-3886	76	29	and	and	CCONJ
ejpam-3886	76	30	it	it	PRON
ejpam-3886	76	31	will	will	AUX
ejpam-3886	76	32	be	be	AUX
ejpam-3886	76	33	of	of	ADP
ejpam-3886	76	34	invaluable	invaluable	ADJ
ejpam-3886	76	35	service	service	NOUN
ejpam-3886	76	36	in	in	ADP
ejpam-3886	76	37	the	the	DET
ejpam-3886	76	38	proof	proof	NOUN
ejpam-3886	76	39	of	of	ADP
ejpam-3886	76	40	our	our	PRON
ejpam-3886	76	41	main	main	ADJ
ejpam-3886	76	42	results	result	NOUN
ejpam-3886	76	43	.	.	PUNCT
ejpam-3886	77	1	remark	remark	NOUN
ejpam-3886	77	2	(	(	PUNCT
ejpam-3886	77	3	[	[	X
ejpam-3886	77	4	1	1	NUM
ejpam-3886	77	5	]	]	PUNCT
ejpam-3886	77	6	)	)	PUNCT
ejpam-3886	77	7	.	.	PUNCT
ejpam-3886	78	1	(	(	PUNCT
ejpam-3886	78	2	1	1	X
ejpam-3886	78	3	)	)	PUNCT
ejpam-3886	78	4	if	if	SCONJ
ejpam-3886	78	5	mr	mr	PROPN
ejpam-3886	78	6	is	be	AUX
ejpam-3886	78	7	α	α	NOUN
ejpam-3886	78	8	-	-	ADJ
ejpam-3886	78	9	compatible	compatible	ADJ
ejpam-3886	78	10	,	,	PUNCT
ejpam-3886	78	11	then	then	ADV
ejpam-3886	78	12	mr	mr	PROPN
ejpam-3886	78	13	is	be	AUX
ejpam-3886	78	14	αi	αi	NOUN
ejpam-3886	78	15	-	-	ADJ
ejpam-3886	78	16	compatible	compatible	ADJ
ejpam-3886	78	17	for	for	ADP
ejpam-3886	78	18	all	all	PRON
ejpam-3886	78	19	i	i	PRON
ejpam-3886	78	20	≥	≥	VERB
ejpam-3886	78	21	1	1	NUM
ejpam-3886	78	22	,	,	PUNCT
ejpam-3886	78	23	(	(	PUNCT
ejpam-3886	78	24	2	2	X
ejpam-3886	78	25	)	)	PUNCT
ejpam-3886	78	26	if	if	SCONJ
ejpam-3886	78	27	mr	mr	PROPN
ejpam-3886	78	28	is	be	AUX
ejpam-3886	78	29	δ	δ	PROPN
ejpam-3886	78	30	-	-	PUNCT
ejpam-3886	78	31	compatible	compatible	ADJ
ejpam-3886	78	32	,	,	PUNCT
ejpam-3886	78	33	then	then	ADV
ejpam-3886	78	34	mr	mr	PROPN
ejpam-3886	78	35	is	be	AUX
ejpam-3886	78	36	δi	δi	ADV
ejpam-3886	78	37	-	-	PUNCT
ejpam-3886	78	38	compatible	compatible	ADJ
ejpam-3886	78	39	for	for	ADP
ejpam-3886	78	40	all	all	PRON
ejpam-3886	78	41	i	i	PRON
ejpam-3886	78	42	≥	≥	VERB
ejpam-3886	78	43	1	1	NUM
ejpam-3886	78	44	,	,	PUNCT
ejpam-3886	78	45	(	(	PUNCT
ejpam-3886	78	46	3	3	X
ejpam-3886	78	47	)	)	PUNCT
ejpam-3886	78	48	if	if	SCONJ
ejpam-3886	78	49	mr	mr	PROPN
ejpam-3886	78	50	is	be	AUX
ejpam-3886	78	51	(	(	PUNCT
ejpam-3886	78	52	α	α	NOUN
ejpam-3886	78	53	,	,	PUNCT
ejpam-3886	78	54	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	78	55	,	,	PUNCT
ejpam-3886	78	56	then	then	ADV
ejpam-3886	78	57	for	for	ADP
ejpam-3886	78	58	each	each	DET
ejpam-3886	78	59	m	m	NOUN
ejpam-3886	78	60	∈	∈	NOUN
ejpam-3886	78	61	m	m	NOUN
ejpam-3886	78	62	and	and	CCONJ
ejpam-3886	78	63	r	r	NOUN
ejpam-3886	78	64	∈	∈	PROPN
ejpam-3886	78	65	r	r	NOUN
ejpam-3886	78	66	,	,	PUNCT
ejpam-3886	78	67	we	we	PRON
ejpam-3886	78	68	have	have	VERB
ejpam-3886	79	1	mr	mr	PROPN
ejpam-3886	79	2	=	=	SYM
ejpam-3886	79	3	0	0	PUNCT
ejpam-3886	79	4	=	=	NOUN
ejpam-3886	79	5	⇒	⇒	NOUN
ejpam-3886	79	6	mλji	mλji	ADJ
ejpam-3886	79	7	(	(	PUNCT
ejpam-3886	79	8	r	r	NOUN
ejpam-3886	79	9	)	)	PUNCT
ejpam-3886	79	10	=	=	SYM
ejpam-3886	79	11	0	0	NUM
ejpam-3886	79	12	for	for	ADP
ejpam-3886	79	13	all	all	DET
ejpam-3886	79	14	j	j	PROPN
ejpam-3886	79	15	≥	≥	NOUN
ejpam-3886	79	16	i	i	PRON
ejpam-3886	79	17	≥	≥	NOUN
ejpam-3886	79	18	0	0	NUM
ejpam-3886	79	19	.	.	PUNCT
ejpam-3886	80	1	in	in	ADP
ejpam-3886	80	2	what	what	PRON
ejpam-3886	80	3	follows	follow	VERB
ejpam-3886	80	4	,	,	PUNCT
ejpam-3886	80	5	we	we	PRON
ejpam-3886	80	6	characterize	characterize	VERB
ejpam-3886	80	7	ore	ore	NOUN
ejpam-3886	80	8	extension	extension	NOUN
ejpam-3886	80	9	rings	ring	NOUN
ejpam-3886	80	10	that	that	PRON
ejpam-3886	80	11	satisfy	satisfy	VERB
ejpam-3886	80	12	the	the	DET
ejpam-3886	80	13	weak	weak	ADJ
ejpam-3886	80	14	ps	ps	NOUN
ejpam-3886	80	15	-	-	NOUN
ejpam-3886	80	16	condition	condition	NOUN
ejpam-3886	80	17	.	.	PUNCT
ejpam-3886	81	1	we	we	PRON
ejpam-3886	81	2	need	need	VERB
ejpam-3886	81	3	first	first	ADV
ejpam-3886	81	4	the	the	DET
ejpam-3886	81	5	following	following	ADJ
ejpam-3886	81	6	lemmas	lemma	NOUN
ejpam-3886	81	7	which	which	PRON
ejpam-3886	81	8	will	will	AUX
ejpam-3886	81	9	help	help	VERB
ejpam-3886	81	10	us	we	PRON
ejpam-3886	81	11	in	in	ADP
ejpam-3886	81	12	our	our	PRON
ejpam-3886	81	13	target	target	NOUN
ejpam-3886	81	14	.	.	PUNCT
ejpam-3886	82	1	lemma	lemma	PROPN
ejpam-3886	82	2	2	2	X
ejpam-3886	82	3	.	.	PUNCT
ejpam-3886	83	1	let	let	VERB
ejpam-3886	83	2	r	r	PRON
ejpam-3886	83	3	be	be	AUX
ejpam-3886	83	4	an	an	DET
ejpam-3886	83	5	(	(	PUNCT
ejpam-3886	83	6	α	α	NOUN
ejpam-3886	83	7	,	,	PUNCT
ejpam-3886	83	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	83	9	ni	ni	NOUN
ejpam-3886	83	10	ring	ring	NOUN
ejpam-3886	83	11	.	.	PUNCT
ejpam-3886	84	1	if	if	SCONJ
ejpam-3886	84	2	a	a	DET
ejpam-3886	84	3	∈	∈	PROPN
ejpam-3886	84	4	nil	nil	NOUN
ejpam-3886	84	5	(	(	PUNCT
ejpam-3886	84	6	r	r	NOUN
ejpam-3886	84	7	)	)	PUNCT
ejpam-3886	84	8	,	,	PUNCT
ejpam-3886	84	9	then	then	ADV
ejpam-3886	84	10	λji	λji	PROPN
ejpam-3886	84	11	(	(	PUNCT
ejpam-3886	84	12	a	a	PRON
ejpam-3886	84	13	)	)	PUNCT
ejpam-3886	84	14	∈	∈	PROPN
ejpam-3886	84	15	nil	nil	NOUN
ejpam-3886	84	16	(	(	PUNCT
ejpam-3886	84	17	r	r	NOUN
ejpam-3886	84	18	)	)	PUNCT
ejpam-3886	84	19	for	for	ADP
ejpam-3886	84	20	all	all	DET
ejpam-3886	84	21	j	j	PROPN
ejpam-3886	84	22	≥	≥	NOUN
ejpam-3886	84	23	i	i	PRON
ejpam-3886	84	24	≥	≥	NOUN
ejpam-3886	84	25	0	0	NUM
ejpam-3886	84	26	.	.	PUNCT
ejpam-3886	84	27	proof	proof	NOUN
ejpam-3886	84	28	.	.	PUNCT
ejpam-3886	85	1	clearly	clearly	ADV
ejpam-3886	85	2	for	for	ADP
ejpam-3886	85	3	any	any	DET
ejpam-3886	85	4	r	r	NOUN
ejpam-3886	85	5	-	-	PUNCT
ejpam-3886	85	6	endomorphism	endomorphism	PROPN
ejpam-3886	85	7	α	α	NOUN
ejpam-3886	85	8	,	,	PUNCT
ejpam-3886	85	9	we	we	PRON
ejpam-3886	85	10	have	have	VERB
ejpam-3886	85	11	αk	αk	NOUN
ejpam-3886	85	12	(	(	PUNCT
ejpam-3886	85	13	nil	nil	NOUN
ejpam-3886	85	14	(	(	PUNCT
ejpam-3886	85	15	r	r	NOUN
ejpam-3886	85	16	)	)	PUNCT
ejpam-3886	85	17	)	)	PUNCT
ejpam-3886	86	1	⊆	⊆	NUM
ejpam-3886	86	2	nil	nil	NOUN
ejpam-3886	86	3	(	(	PUNCT
ejpam-3886	86	4	r	r	NOUN
ejpam-3886	86	5	)	)	PUNCT
ejpam-3886	86	6	,	,	PUNCT
ejpam-3886	86	7	for	for	ADP
ejpam-3886	86	8	any	any	DET
ejpam-3886	86	9	positive	positive	ADJ
ejpam-3886	86	10	integer	integer	NOUN
ejpam-3886	86	11	k.	k.	PROPN
ejpam-3886	86	12	since	since	SCONJ
ejpam-3886	86	13	r	r	NOUN
ejpam-3886	86	14	is	be	AUX
ejpam-3886	86	15	an	an	DET
ejpam-3886	86	16	(	(	PUNCT
ejpam-3886	86	17	α	α	NOUN
ejpam-3886	86	18	,	,	PUNCT
ejpam-3886	86	19	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	86	20	ring	ring	NOUN
ejpam-3886	86	21	,	,	PUNCT
ejpam-3886	86	22	we	we	PRON
ejpam-3886	86	23	have	have	VERB
ejpam-3886	86	24	also	also	ADV
ejpam-3886	86	25	that	that	SCONJ
ejpam-3886	86	26	δk	δk	PROPN
ejpam-3886	86	27	(	(	PUNCT
ejpam-3886	86	28	nil	nil	NOUN
ejpam-3886	86	29	(	(	PUNCT
ejpam-3886	86	30	r	r	NOUN
ejpam-3886	86	31	)	)	PUNCT
ejpam-3886	86	32	)	)	PUNCT
ejpam-3886	87	1	⊆	⊆	NUM
ejpam-3886	87	2	nil	nil	NOUN
ejpam-3886	87	3	(	(	PUNCT
ejpam-3886	87	4	r	r	NOUN
ejpam-3886	87	5	)	)	PUNCT
ejpam-3886	87	6	.	.	PUNCT
ejpam-3886	88	1	since	since	SCONJ
ejpam-3886	88	2	r	r	NOUN
ejpam-3886	88	3	is	be	AUX
ejpam-3886	88	4	an	an	DET
ejpam-3886	88	5	ni	ni	PROPN
ejpam-3886	88	6	ring	ring	NOUN
ejpam-3886	88	7	,	,	PUNCT
ejpam-3886	88	8	we	we	PRON
ejpam-3886	88	9	conclude	conclude	VERB
ejpam-3886	88	10	that	that	SCONJ
ejpam-3886	88	11	λji	λji	PROPN
ejpam-3886	88	12	(	(	PUNCT
ejpam-3886	88	13	a	a	PRON
ejpam-3886	88	14	)	)	PUNCT
ejpam-3886	88	15	∈	∈	PROPN
ejpam-3886	88	16	nil	nil	NOUN
ejpam-3886	88	17	(	(	PUNCT
ejpam-3886	88	18	r	r	NOUN
ejpam-3886	88	19	)	)	PUNCT
ejpam-3886	88	20	for	for	ADP
ejpam-3886	88	21	any	any	DET
ejpam-3886	88	22	a	a	DET
ejpam-3886	88	23	∈	∈	ADJ
ejpam-3886	88	24	nil	nil	NOUN
ejpam-3886	88	25	(	(	PUNCT
ejpam-3886	88	26	r	r	NOUN
ejpam-3886	88	27	)	)	PUNCT
ejpam-3886	88	28	and	and	CCONJ
ejpam-3886	88	29	for	for	ADP
ejpam-3886	88	30	all	all	DET
ejpam-3886	88	31	j	j	PROPN
ejpam-3886	88	32	≥	≥	NOUN
ejpam-3886	88	33	i	i	PRON
ejpam-3886	88	34	≥	≥	PROPN
ejpam-3886	88	35	0	0	NUM
ejpam-3886	88	36	.	.	PUNCT
ejpam-3886	89	1	lemma	lemma	PROPN
ejpam-3886	89	2	3	3	X
ejpam-3886	89	3	.	.	PUNCT
ejpam-3886	90	1	let	let	VERB
ejpam-3886	90	2	r	r	PRON
ejpam-3886	90	3	be	be	AUX
ejpam-3886	90	4	an	an	DET
ejpam-3886	90	5	(	(	PUNCT
ejpam-3886	90	6	α	α	NOUN
ejpam-3886	90	7	,	,	PUNCT
ejpam-3886	90	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	90	9	ni	ni	PROPN
ejpam-3886	90	10	ring	ring	NOUN
ejpam-3886	90	11	with	with	ADP
ejpam-3886	90	12	nil	nil	NOUN
ejpam-3886	90	13	(	(	PUNCT
ejpam-3886	90	14	r	r	NOUN
ejpam-3886	90	15	)	)	PUNCT
ejpam-3886	90	16	nilpotent	nilpotent	NOUN
ejpam-3886	90	17	and	and	CCONJ
ejpam-3886	90	18	f(x	f(x	NOUN
ejpam-3886	90	19	)	)	PUNCT
ejpam-3886	91	1	=	=	SYM
ejpam-3886	91	2	n∑	n∑	PROPN
ejpam-3886	91	3	i=0	i=0	PROPN
ejpam-3886	91	4	aix	aix	NOUN
ejpam-3886	91	5	i	i	PRON
ejpam-3886	91	6	∈	∈	VERB
ejpam-3886	91	7	a	a	DET
ejpam-3886	91	8	=	=	X
ejpam-3886	91	9	r	r	NOUN
ejpam-3886	92	1	[	[	X
ejpam-3886	92	2	x;α	x;α	PROPN
ejpam-3886	92	3	,	,	PUNCT
ejpam-3886	92	4	δ	δ	PROPN
ejpam-3886	92	5	]	]	PUNCT
ejpam-3886	92	6	.	.	PUNCT
ejpam-3886	93	1	then	then	ADV
ejpam-3886	93	2	f(x	f(x	PROPN
ejpam-3886	93	3	)	)	PUNCT
ejpam-3886	93	4	∈	∈	PROPN
ejpam-3886	93	5	nil	nil	NOUN
ejpam-3886	93	6	(	(	PUNCT
ejpam-3886	93	7	a	a	X
ejpam-3886	93	8	)	)	PUNCT
ejpam-3886	93	9	if	if	SCONJ
ejpam-3886	94	1	and	and	CCONJ
ejpam-3886	94	2	only	only	ADV
ejpam-3886	94	3	if	if	SCONJ
ejpam-3886	94	4	ai	ai	VERB
ejpam-3886	94	5	∈	∈	PROPN
ejpam-3886	94	6	nil	nil	NOUN
ejpam-3886	94	7	(	(	PUNCT
ejpam-3886	94	8	r	r	NOUN
ejpam-3886	94	9	)	)	PUNCT
ejpam-3886	94	10	for	for	ADP
ejpam-3886	94	11	all	all	DET
ejpam-3886	94	12	integers	integer	NOUN
ejpam-3886	94	13	0	0	NUM
ejpam-3886	94	14	≤	≤	NUM
ejpam-3886	95	1	i	i	PRON
ejpam-3886	95	2	≤	≤	ADJ
ejpam-3886	95	3	n.	n.	NOUN
ejpam-3886	95	4	proof	proof	NOUN
ejpam-3886	95	5	.	.	PUNCT
ejpam-3886	96	1	(=	(=	X
ejpam-3886	96	2	⇒	⇒	NOUN
ejpam-3886	96	3	)	)	PUNCT
ejpam-3886	96	4	suppose	suppose	VERB
ejpam-3886	96	5	that	that	SCONJ
ejpam-3886	96	6	f(x	f(x	PROPN
ejpam-3886	96	7	)	)	PUNCT
ejpam-3886	96	8	∈	∈	PROPN
ejpam-3886	96	9	nil	nil	NOUN
ejpam-3886	96	10	(	(	PUNCT
ejpam-3886	96	11	a	a	NOUN
ejpam-3886	96	12	)	)	PUNCT
ejpam-3886	96	13	.	.	PUNCT
ejpam-3886	97	1	then	then	ADV
ejpam-3886	97	2	there	there	PRON
ejpam-3886	97	3	exists	exist	VERB
ejpam-3886	97	4	some	some	DET
ejpam-3886	97	5	positive	positive	ADJ
ejpam-3886	97	6	integer	integer	NOUN
ejpam-3886	97	7	k	k	PROPN
ejpam-3886	97	8	such	such	ADJ
ejpam-3886	97	9	that	that	PRON
ejpam-3886	97	10	0	0	NUM
ejpam-3886	98	1	=	=	SYM
ejpam-3886	99	1	(	(	PUNCT
ejpam-3886	99	2	f(x))k	f(x))k	NOUN
ejpam-3886	99	3	=	=	PUNCT
ejpam-3886	99	4	(	(	PUNCT
ejpam-3886	99	5	a0	a0	NOUN
ejpam-3886	99	6	+	+	CCONJ
ejpam-3886	99	7	a1x+	a1x+	NOUN
ejpam-3886	99	8	a2x	a2x	PROPN
ejpam-3886	99	9	2	2	NUM
ejpam-3886	99	10	+	+	CCONJ
ejpam-3886	99	11	...	...	PUNCT
ejpam-3886	100	1	+	+	CCONJ
ejpam-3886	100	2	anx	anx	ADJ
ejpam-3886	100	3	n	n	NOUN
ejpam-3886	100	4	)	)	PUNCT
ejpam-3886	101	1	k	k	PROPN
ejpam-3886	101	2	.	.	PUNCT
ejpam-3886	102	1	then	then	ADV
ejpam-3886	102	2	0	0	NUM
ejpam-3886	102	3	=	=	SYM
ejpam-3886	102	4	(	(	PUNCT
ejpam-3886	102	5	f(x))k	f(x))k	NOUN
ejpam-3886	102	6	=	=	PUNCT
ejpam-3886	102	7	“	"	PUNCT
ejpam-3886	102	8	lower	low	ADJ
ejpam-3886	102	9	terms	term	NOUN
ejpam-3886	102	10	”	"	PUNCT
ejpam-3886	102	11	+	+	CCONJ
ejpam-3886	102	12	anα	anα	VERB
ejpam-3886	102	13	n(an)α2n(an)	n(an)α2n(an)	PROPN
ejpam-3886	102	14	...	...	PUNCT
ejpam-3886	102	15	α(k−1)n(an)xkn	α(k−1)n(an)xkn	NOUN
ejpam-3886	102	16	.	.	PUNCT
ejpam-3886	103	1	hence	hence	ADV
ejpam-3886	103	2	anα	anα	VERB
ejpam-3886	103	3	n(an)α2n(an)	n(an)α2n(an)	PART
ejpam-3886	103	4	...	...	PUNCT
ejpam-3886	103	5	α(k−1)n(an	α(k−1)n(an	NOUN
ejpam-3886	103	6	)	)	PUNCT
ejpam-3886	104	1	=	=	SYM
ejpam-3886	104	2	0	0	NUM
ejpam-3886	104	3	,	,	PUNCT
ejpam-3886	104	4	and	and	CCONJ
ejpam-3886	104	5	α	α	X
ejpam-3886	104	6	-	-	PUNCT
ejpam-3886	104	7	compatiblility	compatiblility	NOUN
ejpam-3886	104	8	of	of	ADP
ejpam-3886	104	9	r	r	NOUN
ejpam-3886	104	10	,	,	PUNCT
ejpam-3886	104	11	gives	give	VERB
ejpam-3886	104	12	an	an	DET
ejpam-3886	104	13	∈	∈	ADJ
ejpam-3886	104	14	nil	nil	NOUN
ejpam-3886	104	15	(	(	PUNCT
ejpam-3886	104	16	r	r	NOUN
ejpam-3886	104	17	)	)	PUNCT
ejpam-3886	104	18	.	.	PUNCT
ejpam-3886	105	1	so	so	ADV
ejpam-3886	105	2	λji	λji	PROPN
ejpam-3886	105	3	(	(	PUNCT
ejpam-3886	105	4	an	an	DET
ejpam-3886	105	5	)	)	PUNCT
ejpam-3886	105	6	∈	∈	PROPN
ejpam-3886	105	7	nil	nil	NOUN
ejpam-3886	105	8	(	(	PUNCT
ejpam-3886	105	9	r	r	NOUN
ejpam-3886	105	10	)	)	PUNCT
ejpam-3886	105	11	for	for	ADP
ejpam-3886	105	12	all	all	DET
ejpam-3886	105	13	j	j	PROPN
ejpam-3886	105	14	≥	≥	NOUN
ejpam-3886	105	15	i	i	PRON
ejpam-3886	105	16	≥	≥	NOUN
ejpam-3886	105	17	0	0	NUM
ejpam-3886	105	18	.	.	PUNCT
ejpam-3886	106	1	let	let	VERB
ejpam-3886	106	2	q	q	NOUN
ejpam-3886	106	3	=	=	SYM
ejpam-3886	106	4	a0	a0	PROPN
ejpam-3886	106	5	+	+	CCONJ
ejpam-3886	106	6	a1x+	a1x+	NOUN
ejpam-3886	106	7	a2x	a2x	PROPN
ejpam-3886	106	8	2	2	NUM
ejpam-3886	106	9	+	+	CCONJ
ejpam-3886	106	10	...	...	PUNCT
ejpam-3886	107	1	+	+	CCONJ
ejpam-3886	107	2	an−1x	an−1x	PROPN
ejpam-3886	107	3	n−1	n−1	PROPN
ejpam-3886	107	4	.	.	PUNCT
ejpam-3886	108	1	then	then	ADV
ejpam-3886	108	2	we	we	PRON
ejpam-3886	108	3	have	have	VERB
ejpam-3886	108	4	0	0	NUM
ejpam-3886	108	5	=	=	SYM
ejpam-3886	108	6	(	(	PUNCT
ejpam-3886	108	7	q+	q+	ADP
ejpam-3886	108	8	anx	anx	ADJ
ejpam-3886	108	9	n)k	n)k	ADV
ejpam-3886	108	10	=	=	X
ejpam-3886	108	11	(	(	PUNCT
ejpam-3886	108	12	q+	q+	NOUN
ejpam-3886	108	13	anx	anx	PROPN
ejpam-3886	108	14	n	n	CCONJ
ejpam-3886	108	15	)	)	PUNCT
ejpam-3886	108	16	(	(	PUNCT
ejpam-3886	108	17	q+	q+	X
ejpam-3886	108	18	anx	anx	PROPN
ejpam-3886	108	19	n	n	CCONJ
ejpam-3886	108	20	)	)	PUNCT
ejpam-3886	108	21	...	...	PUNCT
ejpam-3886	109	1	(	(	PUNCT
ejpam-3886	109	2	q+	q+	ADP
ejpam-3886	109	3	anx	anx	PROPN
ejpam-3886	109	4	n)︸	n)︸	PROPN
ejpam-3886	109	5	︷︷	︷︷	PROPN
ejpam-3886	109	6	︸	︸	ADP
ejpam-3886	109	7	k	k	ADJ
ejpam-3886	109	8	-	-	NOUN
ejpam-3886	109	9	factor	factor	NOUN
ejpam-3886	109	10	=	=	SYM
ejpam-3886	109	11	(	(	PUNCT
ejpam-3886	109	12	q2	q2	NOUN
ejpam-3886	109	13	+	+	ADP
ejpam-3886	109	14	qanx	qanx	ADJ
ejpam-3886	109	15	n	n	NOUN
ejpam-3886	109	16	+	+	CCONJ
ejpam-3886	109	17	anx	anx	ADJ
ejpam-3886	109	18	nq+	nq+	ADJ
ejpam-3886	109	19	anx	anx	PROPN
ejpam-3886	109	20	nanx	nanx	NOUN
ejpam-3886	109	21	n	n	PROPN
ejpam-3886	109	22	)	)	PUNCT
ejpam-3886	109	23	...	...	PUNCT
ejpam-3886	110	1	(	(	PUNCT
ejpam-3886	110	2	q+	q+	X
ejpam-3886	110	3	anx	anx	NOUN
ejpam-3886	110	4	n	n	CCONJ
ejpam-3886	110	5	)	)	PUNCT
ejpam-3886	110	6	=	=	SYM
ejpam-3886	110	7	qk	qk	NOUN
ejpam-3886	110	8	+	+	CCONJ
ejpam-3886	110	9	∆	∆	PROPN
ejpam-3886	110	10	,	,	PUNCT
ejpam-3886	110	11	where	where	SCONJ
ejpam-3886	110	12	∆	∆	PROPN
ejpam-3886	110	13	∈	∈	PROPN
ejpam-3886	110	14	a.	a.	NOUN
ejpam-3886	110	15	note	note	NOUN
ejpam-3886	110	16	that	that	SCONJ
ejpam-3886	110	17	the	the	DET
ejpam-3886	110	18	coefficients	coefficient	NOUN
ejpam-3886	110	19	of	of	ADP
ejpam-3886	110	20	∆	∆	PROPN
ejpam-3886	110	21	can	can	AUX
ejpam-3886	110	22	be	be	AUX
ejpam-3886	110	23	written	write	VERB
ejpam-3886	110	24	as	as	ADP
ejpam-3886	110	25	sums	sum	NOUN
ejpam-3886	110	26	of	of	ADP
ejpam-3886	110	27	monomials	monomial	NOUN
ejpam-3886	110	28	in	in	ADP
ejpam-3886	110	29	ai	ai	PROPN
ejpam-3886	110	30	and	and	CCONJ
ejpam-3886	110	31	λvu(aj	λvu(aj	PRON
ejpam-3886	110	32	)	)	PUNCT
ejpam-3886	110	33	,	,	PUNCT
ejpam-3886	110	34	where	where	SCONJ
ejpam-3886	110	35	ai	ai	VERB
ejpam-3886	110	36	,	,	PUNCT
ejpam-3886	110	37	aj	aj	PROPN
ejpam-3886	110	38	∈	∈	PROPN
ejpam-3886	110	39	{	{	PUNCT
ejpam-3886	110	40	a0	a0	PROPN
ejpam-3886	110	41	,	,	PUNCT
ejpam-3886	110	42	a1	a1	PROPN
ejpam-3886	110	43	,	,	PUNCT
ejpam-3886	110	44	a2	a2	PROPN
ejpam-3886	110	45	,	,	PUNCT
ejpam-3886	110	46	...	...	PUNCT
ejpam-3886	110	47	,	,	PUNCT
ejpam-3886	110	48	an	an	PRON
ejpam-3886	110	49	}	}	PUNCT
ejpam-3886	110	50	and	and	CCONJ
ejpam-3886	110	51	v	v	ADP
ejpam-3886	110	52	≥	≥	NOUN
ejpam-3886	110	53	u	u	NOUN
ejpam-3886	110	54	≥	≥	NOUN
ejpam-3886	110	55	0	0	NUM
ejpam-3886	110	56	,	,	PUNCT
ejpam-3886	110	57	and	and	CCONJ
ejpam-3886	110	58	each	each	DET
ejpam-3886	110	59	monomial	monomial	NOUN
ejpam-3886	110	60	has	have	VERB
ejpam-3886	110	61	an	an	PRON
ejpam-3886	110	62	and	and	CCONJ
ejpam-3886	110	63	λvu(an	λvu(an	NOUN
ejpam-3886	110	64	)	)	PUNCT
ejpam-3886	110	65	as	as	ADP
ejpam-3886	110	66	a	a	DET
ejpam-3886	110	67	factor	factor	NOUN
ejpam-3886	110	68	.	.	PUNCT
ejpam-3886	111	1	since	since	SCONJ
ejpam-3886	111	2	nil	nil	NOUN
ejpam-3886	111	3	(	(	PUNCT
ejpam-3886	111	4	r	r	NOUN
ejpam-3886	111	5	)	)	PUNCT
ejpam-3886	111	6	is	be	AUX
ejpam-3886	111	7	an	an	DET
ejpam-3886	111	8	ideal	ideal	NOUN
ejpam-3886	111	9	,	,	PUNCT
ejpam-3886	111	10	we	we	PRON
ejpam-3886	111	11	obtain	obtain	VERB
ejpam-3886	111	12	that	that	SCONJ
ejpam-3886	111	13	each	each	DET
ejpam-3886	111	14	monomial	monomial	NOUN
ejpam-3886	111	15	of	of	ADP
ejpam-3886	111	16	∆	∆	PROPN
ejpam-3886	111	17	is	be	AUX
ejpam-3886	111	18	in	in	ADP
ejpam-3886	111	19	nil	nil	ADJ
ejpam-3886	111	20	(	(	PUNCT
ejpam-3886	111	21	r	r	NOUN
ejpam-3886	111	22	)	)	PUNCT
ejpam-3886	111	23	and	and	CCONJ
ejpam-3886	112	1	so	so	ADV
ejpam-3886	112	2	∆	∆	PROPN
ejpam-3886	112	3	∈	∈	PROPN
ejpam-3886	112	4	nil	nil	NOUN
ejpam-3886	112	5	(	(	PUNCT
ejpam-3886	112	6	r	r	NOUN
ejpam-3886	112	7	)	)	PUNCT
ejpam-3886	113	1	[	[	X
ejpam-3886	113	2	x;α	x;α	PROPN
ejpam-3886	113	3	,	,	PUNCT
ejpam-3886	113	4	δ	δ	PROPN
ejpam-3886	113	5	]	]	PUNCT
ejpam-3886	113	6	.	.	PUNCT
ejpam-3886	114	1	thus	thus	ADV
ejpam-3886	114	2	we	we	PRON
ejpam-3886	114	3	obtain	obtain	VERB
ejpam-3886	114	4	qk	qk	NOUN
ejpam-3886	114	5	=	=	PUNCT
ejpam-3886	114	6	(	(	PUNCT
ejpam-3886	114	7	a0	a0	NOUN
ejpam-3886	114	8	+	+	CCONJ
ejpam-3886	114	9	a1x+	a1x+	NOUN
ejpam-3886	114	10	a2x	a2x	PROPN
ejpam-3886	114	11	2	2	NUM
ejpam-3886	114	12	+	+	CCONJ
ejpam-3886	114	13	...	...	PUNCT
ejpam-3886	115	1	+	+	CCONJ
ejpam-3886	115	2	an−1x	an−1x	PROPN
ejpam-3886	115	3	n−1	n−1	PROPN
ejpam-3886	115	4	)	)	PUNCT
ejpam-3886	115	5	k	k	PROPN
ejpam-3886	115	6	m.	m.	PROPN
ejpam-3886	115	7	a.	a.	PROPN
ejpam-3886	115	8	farahat	farahat	PROPN
ejpam-3886	115	9	,	,	PUNCT
ejpam-3886	115	10	salha	salha	NOUN
ejpam-3886	115	11	t.	t.	PROPN
ejpam-3886	115	12	al	al	PROPN
ejpam-3886	115	13	-	-	PUNCT
ejpam-3886	115	14	bogamy	bogamy	PROPN
ejpam-3886	115	15	/	/	SYM
ejpam-3886	115	16	eur	eur	PROPN
ejpam-3886	115	17	.	.	PUNCT
ejpam-3886	116	1	j.	j.	PROPN
ejpam-3886	116	2	pure	pure	PROPN
ejpam-3886	116	3	appl	appl	PROPN
ejpam-3886	116	4	.	.	PROPN
ejpam-3886	116	5	math	math	PROPN
ejpam-3886	116	6	,	,	PUNCT
ejpam-3886	116	7	14	14	NUM
ejpam-3886	116	8	(	(	PUNCT
ejpam-3886	116	9	1	1	NUM
ejpam-3886	116	10	)	)	PUNCT
ejpam-3886	116	11	(	(	PUNCT
ejpam-3886	116	12	2021	2021	NUM
ejpam-3886	116	13	)	)	PUNCT
ejpam-3886	116	14	,	,	PUNCT
ejpam-3886	116	15	164	164	NUM
ejpam-3886	116	16	-	-	SYM
ejpam-3886	116	17	172	172	NUM
ejpam-3886	116	18	168	168	NUM
ejpam-3886	116	19	=	=	PUNCT
ejpam-3886	116	20	“	"	PUNCT
ejpam-3886	116	21	lower	low	ADJ
ejpam-3886	116	22	terms	term	NOUN
ejpam-3886	116	23	”	"	PUNCT
ejpam-3886	116	24	+	+	CCONJ
ejpam-3886	116	25	an−1α	an−1α	NOUN
ejpam-3886	116	26	n−1(an−1)	n−1(an−1)	NUM
ejpam-3886	116	27	...	...	PUNCT
ejpam-3886	116	28	α	α	PROPN
ejpam-3886	116	29	(	(	PUNCT
ejpam-3886	116	30	k−1)(n−1)(an−1)x	k−1)(n−1)(an−1)x	X
ejpam-3886	116	31	k(n−1	k(n−1	PROPN
ejpam-3886	116	32	)	)	PUNCT
ejpam-3886	116	33	∈	∈	PROPN
ejpam-3886	116	34	nil	nil	NOUN
ejpam-3886	116	35	(	(	PUNCT
ejpam-3886	116	36	r	r	NOUN
ejpam-3886	116	37	)	)	PUNCT
ejpam-3886	117	1	[	[	X
ejpam-3886	117	2	x;α	x;α	PROPN
ejpam-3886	117	3	,	,	PUNCT
ejpam-3886	117	4	δ	δ	PROPN
ejpam-3886	117	5	]	]	PUNCT
ejpam-3886	117	6	.	.	PUNCT
ejpam-3886	118	1	therefore	therefore	ADV
ejpam-3886	118	2	an−1α	an−1α	NOUN
ejpam-3886	118	3	n−1(an−1)	n−1(an−1)	NUM
ejpam-3886	118	4	...	...	PUNCT
ejpam-3886	118	5	α	α	PROPN
ejpam-3886	118	6	(	(	PUNCT
ejpam-3886	118	7	k−1)(n−1)(an−1	k−1)(n−1)(an−1	PROPN
ejpam-3886	118	8	)	)	PUNCT
ejpam-3886	118	9	∈	∈	PROPN
ejpam-3886	118	10	nil	nil	NOUN
ejpam-3886	118	11	(	(	PUNCT
ejpam-3886	118	12	r	r	NOUN
ejpam-3886	118	13	)	)	PUNCT
ejpam-3886	118	14	and	and	CCONJ
ejpam-3886	118	15	so	so	ADV
ejpam-3886	118	16	an−1	an−1	PROPN
ejpam-3886	118	17	∈	∈	PROPN
ejpam-3886	118	18	nil	nil	NOUN
ejpam-3886	118	19	(	(	PUNCT
ejpam-3886	118	20	r	r	NOUN
ejpam-3886	118	21	)	)	PUNCT
ejpam-3886	118	22	.	.	PUNCT
ejpam-3886	119	1	by	by	ADP
ejpam-3886	119	2	using	use	VERB
ejpam-3886	119	3	induction	induction	NOUN
ejpam-3886	119	4	on	on	ADP
ejpam-3886	119	5	n	n	CCONJ
ejpam-3886	119	6	we	we	PRON
ejpam-3886	119	7	obtain	obtain	VERB
ejpam-3886	119	8	ai	ai	PROPN
ejpam-3886	119	9	∈	∈	PROPN
ejpam-3886	119	10	nil	nil	NOUN
ejpam-3886	119	11	(	(	PUNCT
ejpam-3886	119	12	r	r	NOUN
ejpam-3886	119	13	)	)	PUNCT
ejpam-3886	119	14	for	for	ADP
ejpam-3886	119	15	all	all	PRON
ejpam-3886	119	16	0	0	NUM
ejpam-3886	119	17	≤	≤	NUM
ejpam-3886	119	18	i	i	PRON
ejpam-3886	119	19	≤	≤	ADJ
ejpam-3886	119	20	n.	n.	NOUN
ejpam-3886	119	21	(	(	PUNCT
ejpam-3886	119	22	⇐	⇐	ADJ
ejpam-3886	119	23	=)	=)	PROPN
ejpam-3886	119	24	consider	consider	VERB
ejpam-3886	119	25	the	the	DET
ejpam-3886	119	26	finite	finite	NOUN
ejpam-3886	119	27	subset	subset	NOUN
ejpam-3886	119	28	s	s	PART
ejpam-3886	119	29	=	=	SYM
ejpam-3886	119	30	{	{	PUNCT
ejpam-3886	119	31	a0	a0	PROPN
ejpam-3886	119	32	,	,	PUNCT
ejpam-3886	119	33	a1	a1	PROPN
ejpam-3886	119	34	,	,	PUNCT
ejpam-3886	119	35	a2	a2	PROPN
ejpam-3886	119	36	,	,	PUNCT
ejpam-3886	119	37	...	...	PUNCT
ejpam-3886	119	38	,	,	PUNCT
ejpam-3886	119	39	an	an	DET
ejpam-3886	119	40	}	}	SYM
ejpam-3886	119	41	⊆	⊆	NUM
ejpam-3886	119	42	nil	nil	NOUN
ejpam-3886	119	43	(	(	PUNCT
ejpam-3886	119	44	r	r	NOUN
ejpam-3886	119	45	)	)	PUNCT
ejpam-3886	119	46	.	.	PUNCT
ejpam-3886	120	1	since	since	SCONJ
ejpam-3886	120	2	r	r	NOUN
ejpam-3886	120	3	is	be	AUX
ejpam-3886	120	4	an	an	DET
ejpam-3886	120	5	ni	ni	NOUN
ejpam-3886	120	6	ring	ring	NOUN
ejpam-3886	120	7	with	with	ADP
ejpam-3886	120	8	nil	nil	NOUN
ejpam-3886	120	9	(	(	PUNCT
ejpam-3886	120	10	r	r	NOUN
ejpam-3886	120	11	)	)	PUNCT
ejpam-3886	120	12	nilpotent	nilpotent	NOUN
ejpam-3886	120	13	,	,	PUNCT
ejpam-3886	120	14	there	there	PRON
ejpam-3886	120	15	exist	exist	VERB
ejpam-3886	120	16	integers	integer	NOUN
ejpam-3886	120	17	ki	ki	PROPN
ejpam-3886	120	18	such	such	ADJ
ejpam-3886	120	19	that	that	SCONJ
ejpam-3886	120	20	(	(	PUNCT
ejpam-3886	120	21	air)ki	air)ki	NOUN
ejpam-3886	120	22	=	=	SYM
ejpam-3886	120	23	0	0	NUM
ejpam-3886	120	24	,	,	PUNCT
ejpam-3886	120	25	0	0	NUM
ejpam-3886	120	26	≤	≤	NUM
ejpam-3886	120	27	i	i	PRON
ejpam-3886	120	28	≤	≤	ADJ
ejpam-3886	120	29	n.	n.	NOUN
ejpam-3886	120	30	let	let	VERB
ejpam-3886	120	31	k	k	PROPN
ejpam-3886	120	32	=	=	PROPN
ejpam-3886	120	33	k0	k0	PROPN
ejpam-3886	120	34	+	+	CCONJ
ejpam-3886	120	35	k1	k1	PROPN
ejpam-3886	120	36	+	+	CCONJ
ejpam-3886	120	37	...	...	PUNCT
ejpam-3886	121	1	+	+	CCONJ
ejpam-3886	121	2	kn	kn	NOUN
ejpam-3886	121	3	+	+	NOUN
ejpam-3886	121	4	1	1	X
ejpam-3886	121	5	.	.	PUNCT
ejpam-3886	121	6	then	then	ADV
ejpam-3886	121	7	we	we	PRON
ejpam-3886	121	8	have	have	VERB
ejpam-3886	121	9	(	(	PUNCT
ejpam-3886	121	10	air)k	air)k	PROPN
ejpam-3886	121	11	=	=	SYM
ejpam-3886	121	12	0	0	NUM
ejpam-3886	121	13	,	,	PUNCT
ejpam-3886	121	14	0	0	NUM
ejpam-3886	121	15	≤	≤	NUM
ejpam-3886	122	1	i	i	PRON
ejpam-3886	122	2	≤	≤	NUM
ejpam-3886	122	3	n.	n.	NOUN
ejpam-3886	122	4	we	we	PRON
ejpam-3886	122	5	have	have	VERB
ejpam-3886	122	6	(	(	PUNCT
ejpam-3886	122	7	f(x))k	f(x))k	NOUN
ejpam-3886	122	8	=	=	PUNCT
ejpam-3886	122	9	(	(	PUNCT
ejpam-3886	122	10	a0	a0	NOUN
ejpam-3886	122	11	+	+	CCONJ
ejpam-3886	122	12	a1x+	a1x+	NOUN
ejpam-3886	122	13	a2x	a2x	PROPN
ejpam-3886	122	14	2	2	NUM
ejpam-3886	122	15	+	+	CCONJ
ejpam-3886	122	16	...	...	PUNCT
ejpam-3886	122	17	+	+	CCONJ
ejpam-3886	122	18	anx	anx	ADJ
ejpam-3886	122	19	n	n	NOUN
ejpam-3886	122	20	)	)	PUNCT
ejpam-3886	122	21	k	k	X
ejpam-3886	123	1	=	=	PUNCT
ejpam-3886	123	2	n∑	n∑	PROPN
ejpam-3886	123	3	i=0	i=0	PROPN
ejpam-3886	123	4	aiλ	aiλ	NOUN
ejpam-3886	123	5	i	i	NOUN
ejpam-3886	123	6	0(a0	0(a0	NUM
ejpam-3886	123	7	)	)	PUNCT
ejpam-3886	123	8	+	+	CCONJ
ejpam-3886	124	1	(	(	PUNCT
ejpam-3886	124	2	n∑	n∑	NOUN
ejpam-3886	124	3	i=0	i=0	PROPN
ejpam-3886	124	4	a0λ	a0λ	NOUN
ejpam-3886	124	5	i	i	NOUN
ejpam-3886	124	6	0(a1	0(a1	PROPN
ejpam-3886	124	7	)	)	PUNCT
ejpam-3886	125	1	+	+	NUM
ejpam-3886	126	1	n∑	n∑	NOUN
ejpam-3886	126	2	i=1	i=1	PROPN
ejpam-3886	127	1	aiλ	aiλ	NOUN
ejpam-3886	127	2	i	i	PRON
ejpam-3886	127	3	1(a0	1(a0	NUM
ejpam-3886	127	4	)	)	PUNCT
ejpam-3886	127	5	)	)	PUNCT
ejpam-3886	128	1	x	x	PUNCT
ejpam-3886	129	1	+	+	PUNCT
ejpam-3886	129	2	(	(	PUNCT
ejpam-3886	129	3	n∑	n∑	PROPN
ejpam-3886	129	4	i=0	i=0	PROPN
ejpam-3886	129	5	aiλ	aiλ	NOUN
ejpam-3886	129	6	i	i	NOUN
ejpam-3886	129	7	0(a2	0(a2	PUNCT
ejpam-3886	129	8	)	)	PUNCT
ejpam-3886	130	1	+	+	CCONJ
ejpam-3886	131	1	n∑	n∑	X
ejpam-3886	131	2	i=1	i=1	PROPN
ejpam-3886	131	3	aiλ	aiλ	NOUN
ejpam-3886	131	4	i	i	PROPN
ejpam-3886	131	5	1(a1	1(a1	NUM
ejpam-3886	131	6	)	)	PUNCT
ejpam-3886	132	1	+	+	NUM
ejpam-3886	132	2	n∑	n∑	X
ejpam-3886	132	3	i=2	i=2	PROPN
ejpam-3886	132	4	aiλ	aiλ	NOUN
ejpam-3886	132	5	i	i	PROPN
ejpam-3886	132	6	2(a0	2(a0	NUM
ejpam-3886	132	7	)	)	PUNCT
ejpam-3886	132	8	)	)	PUNCT
ejpam-3886	133	1	x2	x2	PROPN
ejpam-3886	134	1	+	+	PROPN
ejpam-3886	134	2	...	...	PUNCT
ejpam-3886	134	3	+	+	CCONJ
ejpam-3886	134	4	(	(	PUNCT
ejpam-3886	134	5	k∑	k∑	ADJ
ejpam-3886	134	6	s=0	s=0	PROPN
ejpam-3886	134	7	(	(	PUNCT
ejpam-3886	134	8	n∑	n∑	INTJ
ejpam-3886	134	9	i	i	PROPN
ejpam-3886	134	10	=	=	NOUN
ejpam-3886	134	11	s	s	PART
ejpam-3886	134	12	aiλ	aiλ	NOUN
ejpam-3886	134	13	i	i	PRON
ejpam-3886	134	14	s(ak−s	s(ak−s	NOUN
ejpam-3886	134	15	)	)	PUNCT
ejpam-3886	134	16	)	)	PUNCT
ejpam-3886	134	17	)	)	PUNCT
ejpam-3886	134	18	xk	xk	PROPN
ejpam-3886	135	1	+	+	CCONJ
ejpam-3886	135	2	...	...	PUNCT
ejpam-3886	135	3	+	+	NUM
ejpam-3886	135	4	anα	anα	PROPN
ejpam-3886	135	5	n(an)xn	n(an)xn	NOUN
ejpam-3886	135	6	.	.	PUNCT
ejpam-3886	136	1	we	we	PRON
ejpam-3886	136	2	show	show	VERB
ejpam-3886	136	3	that	that	SCONJ
ejpam-3886	136	4	the	the	DET
ejpam-3886	136	5	coefficients	coefficient	NOUN
ejpam-3886	136	6	of	of	ADP
ejpam-3886	136	7	(	(	PUNCT
ejpam-3886	136	8	f(x))k	f(x))k	NOUN
ejpam-3886	136	9	can	can	AUX
ejpam-3886	136	10	be	be	AUX
ejpam-3886	136	11	written	write	VERB
ejpam-3886	136	12	as	as	ADP
ejpam-3886	136	13	sums	sum	NOUN
ejpam-3886	136	14	of	of	ADP
ejpam-3886	136	15	monomials	monomial	NOUN
ejpam-3886	136	16	of	of	ADP
ejpam-3886	136	17	length	length	NOUN
ejpam-3886	136	18	k	k	PROPN
ejpam-3886	136	19	in	in	ADP
ejpam-3886	136	20	ai	ai	PROPN
ejpam-3886	136	21	and	and	CCONJ
ejpam-3886	136	22	λvu(aj	λvu(aj	PRON
ejpam-3886	136	23	)	)	PUNCT
ejpam-3886	136	24	,	,	PUNCT
ejpam-3886	136	25	where	where	SCONJ
ejpam-3886	136	26	ai	ai	VERB
ejpam-3886	136	27	,	,	PUNCT
ejpam-3886	136	28	aj	aj	PROPN
ejpam-3886	136	29	∈	∈	PROPN
ejpam-3886	136	30	{	{	PUNCT
ejpam-3886	136	31	a0	a0	PROPN
ejpam-3886	136	32	,	,	PUNCT
ejpam-3886	136	33	a1	a1	PROPN
ejpam-3886	136	34	,	,	PUNCT
ejpam-3886	136	35	a2	a2	PROPN
ejpam-3886	136	36	,	,	PUNCT
ejpam-3886	136	37	...	...	PUNCT
ejpam-3886	136	38	,	,	PUNCT
ejpam-3886	136	39	an	an	PRON
ejpam-3886	136	40	}	}	PUNCT
ejpam-3886	136	41	and	and	CCONJ
ejpam-3886	136	42	v	v	ADP
ejpam-3886	136	43	≥	≥	NOUN
ejpam-3886	136	44	u	u	NOUN
ejpam-3886	136	45	≥	≥	NUM
ejpam-3886	136	46	0	0	NUM
ejpam-3886	136	47	are	be	AUX
ejpam-3886	136	48	integers	integer	NOUN
ejpam-3886	136	49	.	.	PUNCT
ejpam-3886	137	1	by	by	ADP
ejpam-3886	137	2	using	use	VERB
ejpam-3886	137	3	(	(	PUNCT
ejpam-3886	137	4	α	α	NOUN
ejpam-3886	137	5	,	,	PUNCT
ejpam-3886	137	6	δ)-compatiblility	δ)-compatiblility	NOUN
ejpam-3886	137	7	of	of	ADP
ejpam-3886	137	8	r	r	NOUN
ejpam-3886	137	9	and	and	CCONJ
ejpam-3886	137	10	(	(	PUNCT
ejpam-3886	137	11	air)k	air)k	PROPN
ejpam-3886	137	12	=	=	SYM
ejpam-3886	137	13	0	0	NUM
ejpam-3886	137	14	,	,	PUNCT
ejpam-3886	137	15	0	0	NUM
ejpam-3886	137	16	≤	≤	NUM
ejpam-3886	138	1	i	i	PRON
ejpam-3886	138	2	≤	≤	PROPN
ejpam-3886	138	3	n	n	CCONJ
ejpam-3886	138	4	,	,	PUNCT
ejpam-3886	138	5	we	we	PRON
ejpam-3886	138	6	have	have	VERB
ejpam-3886	138	7	ai1λ	ai1λ	PROPN
ejpam-3886	138	8	vi2	vi2	NOUN
ejpam-3886	138	9	ui2	ui2	INTJ
ejpam-3886	138	10	(	(	PUNCT
ejpam-3886	138	11	ai2)λ	ai2)λ	NOUN
ejpam-3886	138	12	vi3	vi3	VERB
ejpam-3886	138	13	ui3	ui3	PROPN
ejpam-3886	138	14	(	(	PUNCT
ejpam-3886	138	15	ai3)	ai3)	NOUN
ejpam-3886	138	16	...	...	PUNCT
ejpam-3886	138	17	λ	λ	PROPN
ejpam-3886	138	18	vik	vik	PROPN
ejpam-3886	138	19	uik	uik	NOUN
ejpam-3886	138	20	(	(	PUNCT
ejpam-3886	138	21	aik	aik	PROPN
ejpam-3886	138	22	)	)	PUNCT
ejpam-3886	138	23	=	=	SYM
ejpam-3886	139	1	0	0	NUM
ejpam-3886	139	2	,	,	PUNCT
ejpam-3886	139	3	where	where	SCONJ
ejpam-3886	139	4	{	{	PUNCT
ejpam-3886	139	5	ai1	ai1	X
ejpam-3886	139	6	,	,	PUNCT
ejpam-3886	139	7	ai2	ai2	INTJ
ejpam-3886	139	8	,	,	PUNCT
ejpam-3886	139	9	...	...	PUNCT
ejpam-3886	139	10	,	,	PUNCT
ejpam-3886	139	11	aik	aik	NOUN
ejpam-3886	139	12	}	}	PUNCT
ejpam-3886	139	13	⊆	⊆	NUM
ejpam-3886	139	14	s.	s.	PROPN
ejpam-3886	139	15	thus	thus	ADV
ejpam-3886	139	16	(	(	PUNCT
ejpam-3886	139	17	f(x))k	f(x))k	NOUN
ejpam-3886	139	18	=	=	NOUN
ejpam-3886	139	19	0	0	PROPN
ejpam-3886	139	20	.	.	PUNCT
ejpam-3886	140	1	hence	hence	ADV
ejpam-3886	140	2	f(x	f(x	PROPN
ejpam-3886	140	3	)	)	PUNCT
ejpam-3886	140	4	is	be	AUX
ejpam-3886	140	5	a	a	DET
ejpam-3886	140	6	nilpotent	nilpotent	NOUN
ejpam-3886	140	7	of	of	ADP
ejpam-3886	140	8	a	a	DET
ejpam-3886	140	9	=	=	SYM
ejpam-3886	140	10	r	r	NOUN
ejpam-3886	141	1	[	[	X
ejpam-3886	141	2	x;α	x;α	PROPN
ejpam-3886	141	3	,	,	PUNCT
ejpam-3886	141	4	δ	δ	PROPN
ejpam-3886	141	5	]	]	PUNCT
ejpam-3886	141	6	.	.	PUNCT
ejpam-3886	142	1	corollary	corollary	ADJ
ejpam-3886	142	2	1	1	NUM
ejpam-3886	142	3	.	.	PUNCT
ejpam-3886	143	1	if	if	SCONJ
ejpam-3886	143	2	f(x	f(x	PROPN
ejpam-3886	143	3	)	)	PUNCT
ejpam-3886	144	1	=	=	PROPN
ejpam-3886	144	2	a0	a0	PROPN
ejpam-3886	144	3	+	+	CCONJ
ejpam-3886	144	4	a1x	a1x	NOUN
ejpam-3886	144	5	+	+	CCONJ
ejpam-3886	144	6	a2x	a2x	ADP
ejpam-3886	144	7	2	2	NUM
ejpam-3886	144	8	+	+	NUM
ejpam-3886	144	9	...	...	PUNCT
ejpam-3886	145	1	+	+	CCONJ
ejpam-3886	145	2	anx	anx	ADJ
ejpam-3886	145	3	n	n	PRON
ejpam-3886	145	4	∈	∈	PROPN
ejpam-3886	145	5	a	a	DET
ejpam-3886	145	6	=	=	X
ejpam-3886	145	7	r	r	NOUN
ejpam-3886	146	1	[	[	X
ejpam-3886	146	2	x;α	x;α	PROPN
ejpam-3886	146	3	,	,	PUNCT
ejpam-3886	146	4	δ	δ	PROPN
ejpam-3886	146	5	]	]	PUNCT
ejpam-3886	146	6	,	,	PUNCT
ejpam-3886	146	7	r	r	NOUN
ejpam-3886	146	8	is	be	AUX
ejpam-3886	146	9	an	an	DET
ejpam-3886	146	10	(	(	PUNCT
ejpam-3886	146	11	α	α	NOUN
ejpam-3886	146	12	,	,	PUNCT
ejpam-3886	146	13	δ)compatible	δ)compatible	ADJ
ejpam-3886	146	14	ring	ring	NOUN
ejpam-3886	146	15	and	and	CCONJ
ejpam-3886	146	16	satisfies	satisfy	VERB
ejpam-3886	146	17	any	any	DET
ejpam-3886	146	18	one	one	NUM
ejpam-3886	146	19	of	of	ADP
ejpam-3886	146	20	the	the	DET
ejpam-3886	146	21	following	following	ADJ
ejpam-3886	146	22	conditions	condition	NOUN
ejpam-3886	146	23	:	:	PUNCT
ejpam-3886	146	24	1	1	X
ejpam-3886	146	25	)	)	PUNCT
ejpam-3886	146	26	r	r	NOUN
ejpam-3886	146	27	is	be	AUX
ejpam-3886	146	28	a	a	DET
ejpam-3886	146	29	noetherian	noetherian	ADJ
ejpam-3886	146	30	ring	ring	NOUN
ejpam-3886	146	31	,	,	PUNCT
ejpam-3886	146	32	2	2	NUM
ejpam-3886	146	33	)	)	PUNCT
ejpam-3886	146	34	r	r	NOUN
ejpam-3886	146	35	has	have	VERB
ejpam-3886	146	36	either	either	CCONJ
ejpam-3886	146	37	the	the	DET
ejpam-3886	146	38	acc	acc	PROPN
ejpam-3886	146	39	or	or	CCONJ
ejpam-3886	146	40	dcc	dcc	PROPN
ejpam-3886	146	41	on	on	ADP
ejpam-3886	146	42	left	left	ADJ
ejpam-3886	146	43	annihilators	annihilators	PROPN
ejpam-3886	146	44	,	,	PUNCT
ejpam-3886	146	45	then	then	ADV
ejpam-3886	146	46	f(x	f(x	PROPN
ejpam-3886	146	47	)	)	PUNCT
ejpam-3886	146	48	∈	∈	PROPN
ejpam-3886	146	49	nil	nil	NOUN
ejpam-3886	146	50	(	(	PUNCT
ejpam-3886	146	51	a	a	X
ejpam-3886	146	52	)	)	PUNCT
ejpam-3886	146	53	if	if	SCONJ
ejpam-3886	147	1	and	and	CCONJ
ejpam-3886	147	2	only	only	ADV
ejpam-3886	147	3	if	if	SCONJ
ejpam-3886	147	4	ai	ai	VERB
ejpam-3886	147	5	∈	∈	PROPN
ejpam-3886	147	6	nil	nil	NOUN
ejpam-3886	147	7	(	(	PUNCT
ejpam-3886	147	8	r	r	NOUN
ejpam-3886	147	9	)	)	PUNCT
ejpam-3886	147	10	for	for	ADP
ejpam-3886	147	11	all	all	PRON
ejpam-3886	147	12	0	0	NUM
ejpam-3886	147	13	≤	≤	NUM
ejpam-3886	147	14	i	i	PRON
ejpam-3886	147	15	≤	≤	ADJ
ejpam-3886	147	16	n.	n.	NOUN
ejpam-3886	147	17	proof	proof	NOUN
ejpam-3886	147	18	.	.	PUNCT
ejpam-3886	148	1	if	if	SCONJ
ejpam-3886	148	2	r	r	NOUN
ejpam-3886	148	3	satisfies	satisfy	VERB
ejpam-3886	148	4	any	any	DET
ejpam-3886	148	5	one	one	NUM
ejpam-3886	148	6	of	of	ADP
ejpam-3886	148	7	the	the	DET
ejpam-3886	148	8	conditions	condition	NOUN
ejpam-3886	148	9	(	(	PUNCT
ejpam-3886	148	10	1	1	NUM
ejpam-3886	148	11	)	)	PUNCT
ejpam-3886	148	12	and	and	CCONJ
ejpam-3886	148	13	(	(	PUNCT
ejpam-3886	148	14	2	2	NUM
ejpam-3886	148	15	)	)	PUNCT
ejpam-3886	148	16	,	,	PUNCT
ejpam-3886	148	17	then	then	ADV
ejpam-3886	148	18	r	r	NOUN
ejpam-3886	148	19	is	be	AUX
ejpam-3886	148	20	an	an	DET
ejpam-3886	148	21	ni	ni	NOUN
ejpam-3886	148	22	ring	ring	NOUN
ejpam-3886	148	23	with	with	ADP
ejpam-3886	148	24	nil	nil	NOUN
ejpam-3886	148	25	(	(	PUNCT
ejpam-3886	148	26	r	r	NOUN
ejpam-3886	148	27	)	)	PUNCT
ejpam-3886	148	28	nilpotent	nilpotent	NOUN
ejpam-3886	148	29	.	.	PUNCT
ejpam-3886	149	1	hence	hence	ADV
ejpam-3886	149	2	the	the	DET
ejpam-3886	149	3	result	result	NOUN
ejpam-3886	149	4	follows	follow	VERB
ejpam-3886	149	5	directly	directly	ADV
ejpam-3886	149	6	from	from	ADP
ejpam-3886	149	7	lemma	lemma	PROPN
ejpam-3886	149	8	3	3	NUM
ejpam-3886	149	9	.	.	PUNCT
ejpam-3886	150	1	lemma	lemma	PROPN
ejpam-3886	150	2	4	4	X
ejpam-3886	150	3	.	.	PUNCT
ejpam-3886	151	1	let	let	VERB
ejpam-3886	151	2	r	r	PRON
ejpam-3886	151	3	be	be	AUX
ejpam-3886	151	4	an	an	DET
ejpam-3886	151	5	(	(	PUNCT
ejpam-3886	151	6	α	α	NOUN
ejpam-3886	151	7	,	,	PUNCT
ejpam-3886	151	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	151	9	ni	ni	PROPN
ejpam-3886	151	10	ring	ring	NOUN
ejpam-3886	151	11	with	with	ADP
ejpam-3886	151	12	nil	nil	NOUN
ejpam-3886	151	13	(	(	PUNCT
ejpam-3886	151	14	r	r	NOUN
ejpam-3886	151	15	)	)	PUNCT
ejpam-3886	151	16	nilpotent	nilpotent	NOUN
ejpam-3886	151	17	and	and	CCONJ
ejpam-3886	151	18	a	a	DET
ejpam-3886	151	19	,	,	PUNCT
ejpam-3886	151	20	b	b	X
ejpam-3886	151	21	∈	∈	PROPN
ejpam-3886	151	22	r.	r.	PROPN
ejpam-3886	151	23	then	then	ADV
ejpam-3886	151	24	ab	ab	PROPN
ejpam-3886	151	25	∈	∈	PROPN
ejpam-3886	151	26	nil	nil	NOUN
ejpam-3886	151	27	(	(	PUNCT
ejpam-3886	151	28	r	r	NOUN
ejpam-3886	151	29	)	)	PUNCT
ejpam-3886	151	30	if	if	SCONJ
ejpam-3886	151	31	and	and	CCONJ
ejpam-3886	151	32	only	only	ADV
ejpam-3886	151	33	if	if	SCONJ
ejpam-3886	151	34	aλvu(b	aλvu(b	ADJ
ejpam-3886	151	35	)	)	PUNCT
ejpam-3886	151	36	∈	∈	PROPN
ejpam-3886	151	37	nil	nil	NOUN
ejpam-3886	151	38	(	(	PUNCT
ejpam-3886	151	39	r	r	NOUN
ejpam-3886	151	40	)	)	PUNCT
ejpam-3886	151	41	,	,	PUNCT
ejpam-3886	151	42	where	where	SCONJ
ejpam-3886	151	43	v	v	X
ejpam-3886	151	44	≥	≥	NOUN
ejpam-3886	151	45	u	u	NOUN
ejpam-3886	151	46	≥	≥	NUM
ejpam-3886	151	47	0	0	NUM
ejpam-3886	151	48	are	be	AUX
ejpam-3886	151	49	integers	integer	NOUN
ejpam-3886	151	50	.	.	PUNCT
ejpam-3886	152	1	proof	proof	NOUN
ejpam-3886	152	2	.	.	PUNCT
ejpam-3886	153	1	(=	(=	X
ejpam-3886	153	2	⇒	⇒	NOUN
ejpam-3886	153	3	)	)	PUNCT
ejpam-3886	153	4	suppose	suppose	VERB
ejpam-3886	153	5	that	that	SCONJ
ejpam-3886	153	6	ab	ab	PROPN
ejpam-3886	153	7	∈	∈	PROPN
ejpam-3886	153	8	nil	nil	NOUN
ejpam-3886	153	9	(	(	PUNCT
ejpam-3886	153	10	r	r	NOUN
ejpam-3886	153	11	)	)	PUNCT
ejpam-3886	153	12	,	,	PUNCT
ejpam-3886	153	13	so	so	CCONJ
ejpam-3886	153	14	ba	ba	PROPN
ejpam-3886	153	15	∈	∈	PROPN
ejpam-3886	153	16	nil	nil	NOUN
ejpam-3886	153	17	(	(	PUNCT
ejpam-3886	153	18	r	r	NOUN
ejpam-3886	153	19	)	)	PUNCT
ejpam-3886	153	20	.	.	PUNCT
ejpam-3886	154	1	assume	assume	VERB
ejpam-3886	154	2	that	that	SCONJ
ejpam-3886	154	3	f(x	f(x	PROPN
ejpam-3886	154	4	)	)	PUNCT
ejpam-3886	155	1	=	=	SYM
ejpam-3886	155	2	b	b	PROPN
ejpam-3886	155	3	and	and	CCONJ
ejpam-3886	155	4	g(x	g(x	NOUN
ejpam-3886	155	5	)	)	PUNCT
ejpam-3886	156	1	=	=	NOUN
ejpam-3886	156	2	ax	ax	NOUN
ejpam-3886	156	3	∈	∈	PROPN
ejpam-3886	156	4	a	a	DET
ejpam-3886	156	5	=	=	X
ejpam-3886	156	6	r	r	NOUN
ejpam-3886	157	1	[	[	X
ejpam-3886	157	2	x;α	x;α	PROPN
ejpam-3886	157	3	,	,	PUNCT
ejpam-3886	157	4	δ	δ	PROPN
ejpam-3886	157	5	]	]	PUNCT
ejpam-3886	157	6	.	.	PUNCT
ejpam-3886	158	1	then	then	ADV
ejpam-3886	158	2	f(x)g(x	f(x)g(x	X
ejpam-3886	158	3	)	)	PUNCT
ejpam-3886	158	4	∈	∈	PROPN
ejpam-3886	158	5	nil	nil	NOUN
ejpam-3886	158	6	(	(	PUNCT
ejpam-3886	158	7	a	a	NOUN
ejpam-3886	158	8	)	)	PUNCT
ejpam-3886	158	9	,	,	PUNCT
ejpam-3886	158	10	so	so	ADV
ejpam-3886	158	11	g(x)f(x	g(x)f(x	NOUN
ejpam-3886	158	12	)	)	PUNCT
ejpam-3886	158	13	=	=	SYM
ejpam-3886	158	14	aδ(b	aδ(b	X
ejpam-3886	158	15	)	)	PUNCT
ejpam-3886	159	1	+	+	CCONJ
ejpam-3886	159	2	aα(b)x	aα(b)x	PUNCT
ejpam-3886	159	3	∈	∈	PROPN
ejpam-3886	159	4	nil	nil	NOUN
ejpam-3886	159	5	(	(	PUNCT
ejpam-3886	159	6	r	r	NOUN
ejpam-3886	159	7	)	)	PUNCT
ejpam-3886	159	8	[	[	X
ejpam-3886	159	9	x;α	x;α	PROPN
ejpam-3886	159	10	,	,	PUNCT
ejpam-3886	159	11	δ	δ	PROPN
ejpam-3886	159	12	]	]	PUNCT
ejpam-3886	159	13	.	.	PUNCT
ejpam-3886	160	1	thus	thus	ADV
ejpam-3886	160	2	aδ(b	aδ(b	NOUN
ejpam-3886	160	3	)	)	PUNCT
ejpam-3886	160	4	,	,	PUNCT
ejpam-3886	160	5	aα(b	aα(b	PUNCT
ejpam-3886	160	6	)	)	PUNCT
ejpam-3886	160	7	∈	∈	PROPN
ejpam-3886	160	8	nil	nil	NOUN
ejpam-3886	160	9	(	(	PUNCT
ejpam-3886	160	10	r	r	NOUN
ejpam-3886	160	11	)	)	PUNCT
ejpam-3886	160	12	.	.	PUNCT
ejpam-3886	161	1	now	now	ADV
ejpam-3886	161	2	suppose	suppose	VERB
ejpam-3886	161	3	that	that	SCONJ
ejpam-3886	161	4	h(x	h(x	PROPN
ejpam-3886	161	5	)	)	PUNCT
ejpam-3886	161	6	=	=	PUNCT
ejpam-3886	161	7	α(b	α(b	NOUN
ejpam-3886	161	8	)	)	PUNCT
ejpam-3886	161	9	and	and	CCONJ
ejpam-3886	161	10	k(x	k(x	PROPN
ejpam-3886	161	11	)	)	PUNCT
ejpam-3886	162	1	=	=	SYM
ejpam-3886	162	2	m.	m.	NOUN
ejpam-3886	162	3	a.	a.	NOUN
ejpam-3886	162	4	farahat	farahat	PROPN
ejpam-3886	162	5	,	,	PUNCT
ejpam-3886	162	6	salha	salha	NOUN
ejpam-3886	162	7	t.	t.	PROPN
ejpam-3886	162	8	al	al	PROPN
ejpam-3886	162	9	-	-	PUNCT
ejpam-3886	162	10	bogamy	bogamy	PROPN
ejpam-3886	162	11	/	/	SYM
ejpam-3886	162	12	eur	eur	PROPN
ejpam-3886	162	13	.	.	PUNCT
ejpam-3886	163	1	j.	j.	PROPN
ejpam-3886	163	2	pure	pure	PROPN
ejpam-3886	163	3	appl	appl	PROPN
ejpam-3886	163	4	.	.	PROPN
ejpam-3886	163	5	math	math	PROPN
ejpam-3886	163	6	,	,	PUNCT
ejpam-3886	163	7	14	14	NUM
ejpam-3886	163	8	(	(	PUNCT
ejpam-3886	163	9	1	1	NUM
ejpam-3886	163	10	)	)	PUNCT
ejpam-3886	163	11	(	(	PUNCT
ejpam-3886	163	12	2021	2021	NUM
ejpam-3886	163	13	)	)	PUNCT
ejpam-3886	163	14	,	,	PUNCT
ejpam-3886	163	15	164	164	NUM
ejpam-3886	163	16	-	-	SYM
ejpam-3886	163	17	172	172	NUM
ejpam-3886	163	18	169	169	NUM
ejpam-3886	163	19	ax	ax	NOUN
ejpam-3886	163	20	∈	∈	PROPN
ejpam-3886	163	21	a	a	DET
ejpam-3886	163	22	=	=	X
ejpam-3886	163	23	r	r	NOUN
ejpam-3886	164	1	[	[	X
ejpam-3886	164	2	x;α	x;α	PROPN
ejpam-3886	164	3	,	,	PUNCT
ejpam-3886	164	4	δ	δ	PROPN
ejpam-3886	164	5	]	]	PUNCT
ejpam-3886	164	6	.	.	PUNCT
ejpam-3886	165	1	then	then	ADV
ejpam-3886	165	2	h(x)k(x	h(x)k(x	X
ejpam-3886	165	3	)	)	PUNCT
ejpam-3886	165	4	∈	∈	PROPN
ejpam-3886	165	5	nil	nil	NOUN
ejpam-3886	165	6	(	(	PUNCT
ejpam-3886	165	7	a	a	NOUN
ejpam-3886	165	8	)	)	PUNCT
ejpam-3886	165	9	,	,	PUNCT
ejpam-3886	165	10	so	so	ADV
ejpam-3886	165	11	k(x)h(x	k(x)h(x	X
ejpam-3886	165	12	)	)	PUNCT
ejpam-3886	165	13	=	=	PUNCT
ejpam-3886	165	14	aδ(α(b	aδ(α(b	X
ejpam-3886	165	15	)	)	PUNCT
ejpam-3886	165	16	)	)	PUNCT
ejpam-3886	166	1	+	+	PUNCT
ejpam-3886	166	2	aα2(b)x	aα2(b)x	PROPN
ejpam-3886	166	3	∈	∈	ADJ
ejpam-3886	166	4	nil	nil	NOUN
ejpam-3886	166	5	(	(	PUNCT
ejpam-3886	166	6	r	r	NOUN
ejpam-3886	166	7	)	)	PUNCT
ejpam-3886	167	1	[	[	X
ejpam-3886	167	2	x;α	x;α	PROPN
ejpam-3886	167	3	,	,	PUNCT
ejpam-3886	167	4	δ	δ	PROPN
ejpam-3886	167	5	]	]	PUNCT
ejpam-3886	167	6	.	.	PUNCT
ejpam-3886	168	1	thus	thus	ADV
ejpam-3886	168	2	aδ(α(b	aδ(α(b	NOUN
ejpam-3886	168	3	)	)	PUNCT
ejpam-3886	168	4	)	)	PUNCT
ejpam-3886	168	5	,	,	PUNCT
ejpam-3886	168	6	aα2(b	aα2(b	NOUN
ejpam-3886	168	7	)	)	PUNCT
ejpam-3886	168	8	∈	∈	PROPN
ejpam-3886	168	9	nil	nil	NOUN
ejpam-3886	168	10	(	(	PUNCT
ejpam-3886	168	11	r	r	NOUN
ejpam-3886	168	12	)	)	PUNCT
ejpam-3886	168	13	.	.	PUNCT
ejpam-3886	169	1	since	since	SCONJ
ejpam-3886	169	2	aδ(b	aδ(b	NOUN
ejpam-3886	169	3	)	)	PUNCT
ejpam-3886	169	4	∈	∈	PROPN
ejpam-3886	169	5	nil	nil	NOUN
ejpam-3886	169	6	(	(	PUNCT
ejpam-3886	169	7	r	r	NOUN
ejpam-3886	169	8	)	)	PUNCT
ejpam-3886	169	9	,	,	PUNCT
ejpam-3886	169	10	for	for	ADP
ejpam-3886	169	11	p(x	p(x	NOUN
ejpam-3886	169	12	)	)	PUNCT
ejpam-3886	169	13	=	=	SYM
ejpam-3886	169	14	δ(b	δ(b	PROPN
ejpam-3886	169	15	)	)	PUNCT
ejpam-3886	169	16	and	and	CCONJ
ejpam-3886	169	17	q(x	q(x	PROPN
ejpam-3886	169	18	)	)	PUNCT
ejpam-3886	170	1	=	=	NOUN
ejpam-3886	170	2	ax	ax	NOUN
ejpam-3886	170	3	∈	∈	PROPN
ejpam-3886	170	4	a	a	DET
ejpam-3886	170	5	=	=	X
ejpam-3886	170	6	r	r	NOUN
ejpam-3886	171	1	[	[	X
ejpam-3886	171	2	x;α	x;α	PROPN
ejpam-3886	171	3	,	,	PUNCT
ejpam-3886	171	4	δ	δ	PROPN
ejpam-3886	171	5	]	]	PUNCT
ejpam-3886	171	6	,	,	PUNCT
ejpam-3886	171	7	we	we	PRON
ejpam-3886	171	8	have	have	VERB
ejpam-3886	171	9	p(x)q(x	p(x)q(x	NUM
ejpam-3886	171	10	)	)	PUNCT
ejpam-3886	171	11	∈	∈	PROPN
ejpam-3886	171	12	nil	nil	NOUN
ejpam-3886	171	13	(	(	PUNCT
ejpam-3886	171	14	a	a	NOUN
ejpam-3886	171	15	)	)	PUNCT
ejpam-3886	171	16	,	,	PUNCT
ejpam-3886	171	17	so	so	ADV
ejpam-3886	171	18	q(x)p(x	q(x)p(x	VERB
ejpam-3886	171	19	)	)	PUNCT
ejpam-3886	171	20	=	=	PUNCT
ejpam-3886	171	21	aδ2(b)+aα(δ(b))x	aδ2(b)+aα(δ(b))x	PROPN
ejpam-3886	171	22	∈	∈	PROPN
ejpam-3886	171	23	nil	nil	NOUN
ejpam-3886	171	24	(	(	PUNCT
ejpam-3886	171	25	r	r	NOUN
ejpam-3886	171	26	)	)	PUNCT
ejpam-3886	172	1	[	[	X
ejpam-3886	172	2	x;α	x;α	PROPN
ejpam-3886	172	3	,	,	PUNCT
ejpam-3886	172	4	δ	δ	PROPN
ejpam-3886	172	5	]	]	PUNCT
ejpam-3886	172	6	.	.	PUNCT
ejpam-3886	173	1	thus	thus	ADV
ejpam-3886	173	2	aδ2(b	aδ2(b	PROPN
ejpam-3886	173	3	)	)	PUNCT
ejpam-3886	173	4	,	,	PUNCT
ejpam-3886	173	5	aα(δ(b	aα(δ(b	NOUN
ejpam-3886	173	6	)	)	PUNCT
ejpam-3886	173	7	)	)	PUNCT
ejpam-3886	174	1	∈	∈	PROPN
ejpam-3886	174	2	nil	nil	NOUN
ejpam-3886	174	3	(	(	PUNCT
ejpam-3886	174	4	r	r	NOUN
ejpam-3886	174	5	)	)	PUNCT
ejpam-3886	174	6	.	.	PUNCT
ejpam-3886	175	1	continuing	continue	VERB
ejpam-3886	175	2	in	in	ADP
ejpam-3886	175	3	this	this	DET
ejpam-3886	175	4	process	process	NOUN
ejpam-3886	175	5	we	we	PRON
ejpam-3886	175	6	get	get	VERB
ejpam-3886	175	7	aαn1(δm1(αn2(δm2	aαn1(δm1(αn2(δm2	X
ejpam-3886	175	8	...	...	PUNCT
ejpam-3886	176	1	αni(δmj	αni(δmj	NOUN
ejpam-3886	176	2	(	(	PUNCT
ejpam-3886	176	3	b	b	NOUN
ejpam-3886	176	4	)	)	PUNCT
ejpam-3886	176	5	)	)	PUNCT
ejpam-3886	176	6	)	)	PUNCT
ejpam-3886	176	7	)	)	PUNCT
ejpam-3886	176	8	)	)	PUNCT
ejpam-3886	177	1	∈	∈	PROPN
ejpam-3886	177	2	nil	nil	NOUN
ejpam-3886	177	3	(	(	PUNCT
ejpam-3886	177	4	r	r	NOUN
ejpam-3886	177	5	)	)	PUNCT
ejpam-3886	177	6	,	,	PUNCT
ejpam-3886	177	7	where	where	SCONJ
ejpam-3886	177	8	ni	ni	PROPN
ejpam-3886	177	9	,	,	PUNCT
ejpam-3886	177	10	mj	mj	PROPN
ejpam-3886	177	11	are	be	AUX
ejpam-3886	177	12	nonnegative	nonnegative	ADJ
ejpam-3886	177	13	integers	integer	NOUN
ejpam-3886	177	14	.	.	PUNCT
ejpam-3886	178	1	thus	thus	ADV
ejpam-3886	178	2	aλvu(b	aλvu(b	VERB
ejpam-3886	178	3	)	)	PUNCT
ejpam-3886	178	4	∈	∈	PROPN
ejpam-3886	178	5	nil	nil	NOUN
ejpam-3886	178	6	(	(	PUNCT
ejpam-3886	178	7	r	r	NOUN
ejpam-3886	178	8	)	)	PUNCT
ejpam-3886	178	9	,	,	PUNCT
ejpam-3886	178	10	where	where	SCONJ
ejpam-3886	178	11	v	v	X
ejpam-3886	178	12	≥	≥	NOUN
ejpam-3886	178	13	u	u	NOUN
ejpam-3886	178	14	≥	≥	NUM
ejpam-3886	178	15	0	0	NUM
ejpam-3886	178	16	are	be	AUX
ejpam-3886	178	17	integers	integer	NOUN
ejpam-3886	178	18	.	.	PUNCT
ejpam-3886	179	1	(	(	PUNCT
ejpam-3886	179	2	⇐	⇐	ADJ
ejpam-3886	179	3	=)	=)	PROPN
ejpam-3886	179	4	suppose	suppose	VERB
ejpam-3886	179	5	that	that	SCONJ
ejpam-3886	179	6	aλvu(b	aλvu(b	NOUN
ejpam-3886	179	7	)	)	PUNCT
ejpam-3886	179	8	∈	∈	PROPN
ejpam-3886	179	9	nil	nil	NOUN
ejpam-3886	179	10	(	(	PUNCT
ejpam-3886	179	11	r	r	NOUN
ejpam-3886	179	12	)	)	PUNCT
ejpam-3886	179	13	,	,	PUNCT
ejpam-3886	179	14	where	where	SCONJ
ejpam-3886	179	15	v	v	X
ejpam-3886	179	16	≥	≥	NOUN
ejpam-3886	179	17	u	u	NOUN
ejpam-3886	179	18	≥	≥	NUM
ejpam-3886	179	19	0	0	NUM
ejpam-3886	179	20	are	be	AUX
ejpam-3886	179	21	integers	integer	NOUN
ejpam-3886	179	22	.	.	PUNCT
ejpam-3886	180	1	by	by	ADP
ejpam-3886	180	2	using	use	VERB
ejpam-3886	180	3	(	(	PUNCT
ejpam-3886	180	4	α	α	NOUN
ejpam-3886	180	5	,	,	PUNCT
ejpam-3886	180	6	δ)compatiblility	δ)compatiblility	NOUN
ejpam-3886	180	7	of	of	ADP
ejpam-3886	180	8	r	r	NOUN
ejpam-3886	180	9	,	,	PUNCT
ejpam-3886	180	10	we	we	PRON
ejpam-3886	180	11	can	can	AUX
ejpam-3886	180	12	conclude	conclude	VERB
ejpam-3886	180	13	that	that	SCONJ
ejpam-3886	180	14	ab	ab	PROPN
ejpam-3886	180	15	∈	∈	PROPN
ejpam-3886	180	16	nil	nil	NOUN
ejpam-3886	180	17	(	(	PUNCT
ejpam-3886	180	18	r	r	NOUN
ejpam-3886	180	19	)	)	PUNCT
ejpam-3886	180	20	.	.	PUNCT
ejpam-3886	181	1	proposition	proposition	NOUN
ejpam-3886	181	2	1	1	NUM
ejpam-3886	181	3	.	.	PUNCT
ejpam-3886	182	1	let	let	VERB
ejpam-3886	182	2	r	r	PRON
ejpam-3886	182	3	be	be	AUX
ejpam-3886	182	4	an	an	DET
ejpam-3886	182	5	(	(	PUNCT
ejpam-3886	182	6	α	α	NOUN
ejpam-3886	182	7	,	,	PUNCT
ejpam-3886	182	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	182	9	ni	ni	PROPN
ejpam-3886	182	10	ring	ring	NOUN
ejpam-3886	182	11	with	with	ADP
ejpam-3886	182	12	nil	nil	NOUN
ejpam-3886	182	13	(	(	PUNCT
ejpam-3886	182	14	r	r	NOUN
ejpam-3886	182	15	)	)	PUNCT
ejpam-3886	182	16	nilpotent	nilpotent	NOUN
ejpam-3886	182	17	,	,	PUNCT
ejpam-3886	182	18	f(x	f(x	PROPN
ejpam-3886	182	19	)	)	PUNCT
ejpam-3886	183	1	=	=	SYM
ejpam-3886	183	2	n∑	n∑	PROPN
ejpam-3886	183	3	i=0	i=0	PROPN
ejpam-3886	183	4	aix	aix	NOUN
ejpam-3886	183	5	i	i	PROPN
ejpam-3886	183	6	and	and	CCONJ
ejpam-3886	183	7	g(x	g(x	NOUN
ejpam-3886	183	8	)	)	PUNCT
ejpam-3886	184	1	=	=	PUNCT
ejpam-3886	184	2	m∑	m∑	CCONJ
ejpam-3886	184	3	j=0	j=0	PROPN
ejpam-3886	184	4	bjx	bjx	VERB
ejpam-3886	184	5	j	j	PROPN
ejpam-3886	184	6	∈	∈	PROPN
ejpam-3886	185	1	a	a	DET
ejpam-3886	185	2	=	=	X
ejpam-3886	185	3	r	r	NOUN
ejpam-3886	185	4	[	[	X
ejpam-3886	185	5	x;α	x;α	PROPN
ejpam-3886	185	6	,	,	PUNCT
ejpam-3886	185	7	δ	δ	PROPN
ejpam-3886	185	8	]	]	PUNCT
ejpam-3886	185	9	.	.	PUNCT
ejpam-3886	186	1	then	then	ADV
ejpam-3886	186	2	f(x)g(x	f(x)g(x	X
ejpam-3886	186	3	)	)	PUNCT
ejpam-3886	186	4	∈	∈	PROPN
ejpam-3886	186	5	nil	nil	NOUN
ejpam-3886	186	6	(	(	PUNCT
ejpam-3886	186	7	a	a	X
ejpam-3886	186	8	)	)	PUNCT
ejpam-3886	186	9	if	if	SCONJ
ejpam-3886	187	1	and	and	CCONJ
ejpam-3886	187	2	only	only	ADV
ejpam-3886	187	3	if	if	SCONJ
ejpam-3886	187	4	aibj	aibj	PROPN
ejpam-3886	187	5	∈	∈	PROPN
ejpam-3886	187	6	nil	nil	NOUN
ejpam-3886	187	7	(	(	PUNCT
ejpam-3886	187	8	r	r	NOUN
ejpam-3886	187	9	)	)	PUNCT
ejpam-3886	187	10	for	for	ADP
ejpam-3886	187	11	all	all	DET
ejpam-3886	187	12	integers	integer	NOUN
ejpam-3886	187	13	0	0	NUM
ejpam-3886	187	14	≤	≤	NUM
ejpam-3886	187	15	i	i	PRON
ejpam-3886	187	16	≤	≤	ADJ
ejpam-3886	187	17	n	n	CCONJ
ejpam-3886	187	18	and	and	CCONJ
ejpam-3886	187	19	0	0	NUM
ejpam-3886	187	20	≤	≤	NUM
ejpam-3886	187	21	j	j	PROPN
ejpam-3886	187	22	≤	≤	PROPN
ejpam-3886	187	23	m.	m.	NOUN
ejpam-3886	187	24	proof	proof	NOUN
ejpam-3886	187	25	.	.	PUNCT
ejpam-3886	188	1	suppose	suppose	VERB
ejpam-3886	188	2	that	that	SCONJ
ejpam-3886	188	3	f(x	f(x	PROPN
ejpam-3886	188	4	)	)	PUNCT
ejpam-3886	189	1	=	=	SYM
ejpam-3886	189	2	n∑	n∑	PROPN
ejpam-3886	189	3	i=0	i=0	PROPN
ejpam-3886	189	4	aix	aix	NOUN
ejpam-3886	189	5	i	i	PROPN
ejpam-3886	189	6	and	and	CCONJ
ejpam-3886	189	7	g(x	g(x	NOUN
ejpam-3886	189	8	)	)	PUNCT
ejpam-3886	190	1	=	=	PUNCT
ejpam-3886	190	2	m∑	m∑	CCONJ
ejpam-3886	190	3	j=0	j=0	PROPN
ejpam-3886	190	4	bjx	bjx	VERB
ejpam-3886	190	5	j	j	PROPN
ejpam-3886	190	6	∈	∈	PROPN
ejpam-3886	190	7	a	a	DET
ejpam-3886	190	8	such	such	ADJ
ejpam-3886	190	9	that	that	SCONJ
ejpam-3886	190	10	f(x)g(x	f(x)g(x	ADJ
ejpam-3886	190	11	)	)	PUNCT
ejpam-3886	190	12	∈	∈	PROPN
ejpam-3886	190	13	nil	nil	NOUN
ejpam-3886	190	14	(	(	PUNCT
ejpam-3886	190	15	a	a	NOUN
ejpam-3886	190	16	)	)	PUNCT
ejpam-3886	190	17	.	.	PUNCT
ejpam-3886	191	1	since	since	SCONJ
ejpam-3886	191	2	r	r	NOUN
ejpam-3886	191	3	is	be	AUX
ejpam-3886	191	4	an	an	DET
ejpam-3886	191	5	(	(	PUNCT
ejpam-3886	191	6	α	α	NOUN
ejpam-3886	191	7	,	,	PUNCT
ejpam-3886	191	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	191	9	ni	ni	PROPN
ejpam-3886	191	10	ring	ring	NOUN
ejpam-3886	191	11	with	with	ADP
ejpam-3886	191	12	nil	nil	NOUN
ejpam-3886	191	13	(	(	PUNCT
ejpam-3886	191	14	r	r	NOUN
ejpam-3886	191	15	)	)	PUNCT
ejpam-3886	191	16	nilpotent	nilpotent	NOUN
ejpam-3886	191	17	,	,	PUNCT
ejpam-3886	191	18	we	we	PRON
ejpam-3886	191	19	get	get	VERB
ejpam-3886	191	20	,	,	PUNCT
ejpam-3886	191	21	from	from	ADP
ejpam-3886	191	22	lemma	lemma	PROPN
ejpam-3886	191	23	3	3	NUM
ejpam-3886	191	24	,	,	PUNCT
ejpam-3886	191	25	the	the	DET
ejpam-3886	191	26	following	follow	VERB
ejpam-3886	191	27	:	:	PUNCT
ejpam-3886	191	28	∆n+m	∆n+m	PROPN
ejpam-3886	191	29	=	=	SYM
ejpam-3886	191	30	anα	anα	PROPN
ejpam-3886	191	31	n(bm	n(bm	ADV
ejpam-3886	191	32	)	)	PUNCT
ejpam-3886	191	33	∈	∈	PROPN
ejpam-3886	191	34	nil	nil	NOUN
ejpam-3886	191	35	(	(	PUNCT
ejpam-3886	191	36	r	r	NOUN
ejpam-3886	191	37	)	)	PUNCT
ejpam-3886	191	38	,	,	PUNCT
ejpam-3886	191	39	(	(	PUNCT
ejpam-3886	191	40	1	1	X
ejpam-3886	191	41	)	)	PUNCT
ejpam-3886	191	42	∆n+m−1	∆n+m−1	PROPN
ejpam-3886	191	43	=	=	SYM
ejpam-3886	191	44	anα	anα	PROPN
ejpam-3886	191	45	n(bm−1	n(bm−1	PROPN
ejpam-3886	191	46	)	)	PUNCT
ejpam-3886	192	1	+	+	CCONJ
ejpam-3886	192	2	an−1α	an−1α	NOUN
ejpam-3886	192	3	n−1(bm	n−1(bm	NOUN
ejpam-3886	192	4	)	)	PUNCT
ejpam-3886	192	5	+	+	CCONJ
ejpam-3886	192	6	anλ	anλ	VERB
ejpam-3886	192	7	n	n	CCONJ
ejpam-3886	192	8	n−1(bm	n−1(bm	NOUN
ejpam-3886	192	9	)	)	PUNCT
ejpam-3886	192	10	∈	∈	PROPN
ejpam-3886	192	11	nil	nil	NOUN
ejpam-3886	192	12	(	(	PUNCT
ejpam-3886	192	13	r	r	NOUN
ejpam-3886	192	14	)	)	PUNCT
ejpam-3886	192	15	,	,	PUNCT
ejpam-3886	192	16	(	(	PUNCT
ejpam-3886	192	17	2	2	X
ejpam-3886	192	18	)	)	PUNCT
ejpam-3886	192	19	∆n+m−2	∆n+m−2	NOUN
ejpam-3886	193	1	=	=	PUNCT
ejpam-3886	193	2	anα	anα	PROPN
ejpam-3886	193	3	n(bm−2	n(bm−2	PUNCT
ejpam-3886	193	4	)	)	PUNCT
ejpam-3886	194	1	+	+	CCONJ
ejpam-3886	194	2	n∑	n∑	INTJ
ejpam-3886	194	3	i	i	PROPN
ejpam-3886	194	4	=	=	PROPN
ejpam-3886	194	5	n−1	n−1	PROPN
ejpam-3886	194	6	aiλ	aiλ	NOUN
ejpam-3886	194	7	i	i	PROPN
ejpam-3886	194	8	n−1(bm−1	n−1(bm−1	PROPN
ejpam-3886	194	9	)	)	PUNCT
ejpam-3886	195	1	+	+	CCONJ
ejpam-3886	195	2	n∑	n∑	PUNCT
ejpam-3886	195	3	i	i	PROPN
ejpam-3886	195	4	=	=	PROPN
ejpam-3886	195	5	n−2	n−2	PROPN
ejpam-3886	195	6	aiλ	aiλ	NOUN
ejpam-3886	195	7	i	i	PROPN
ejpam-3886	195	8	n−2(bm	n−2(bm	PROPN
ejpam-3886	195	9	)	)	PUNCT
ejpam-3886	195	10	∈	∈	PROPN
ejpam-3886	195	11	nil	nil	NOUN
ejpam-3886	195	12	(	(	PUNCT
ejpam-3886	195	13	r	r	NOUN
ejpam-3886	195	14	)	)	PUNCT
ejpam-3886	195	15	.	.	PUNCT
ejpam-3886	196	1	(	(	PUNCT
ejpam-3886	196	2	3	3	X
ejpam-3886	196	3	)	)	PUNCT
ejpam-3886	196	4	from	from	ADP
ejpam-3886	196	5	eq.(1	eq.(1	ADJ
ejpam-3886	196	6	)	)	PUNCT
ejpam-3886	196	7	and	and	CCONJ
ejpam-3886	196	8	lemma	lemma	PROPN
ejpam-3886	196	9	4	4	NUM
ejpam-3886	196	10	,	,	PUNCT
ejpam-3886	196	11	we	we	PRON
ejpam-3886	196	12	obtain	obtain	VERB
ejpam-3886	196	13	anbm	anbm	ADV
ejpam-3886	196	14	∈	∈	PROPN
ejpam-3886	196	15	nil	nil	NOUN
ejpam-3886	196	16	(	(	PUNCT
ejpam-3886	196	17	r	r	NOUN
ejpam-3886	196	18	)	)	PUNCT
ejpam-3886	196	19	.	.	PUNCT
ejpam-3886	197	1	so	so	ADV
ejpam-3886	197	2	,	,	PUNCT
ejpam-3886	197	3	bman	bman	PROPN
ejpam-3886	197	4	∈	∈	PROPN
ejpam-3886	197	5	nil	nil	PROPN
ejpam-3886	197	6	(	(	PUNCT
ejpam-3886	197	7	r	r	NOUN
ejpam-3886	197	8	)	)	PUNCT
ejpam-3886	197	9	.	.	PUNCT
ejpam-3886	198	1	if	if	SCONJ
ejpam-3886	198	2	we	we	PRON
ejpam-3886	198	3	multiply	multiply	VERB
ejpam-3886	198	4	eq.(2	eq.(2	ADJ
ejpam-3886	198	5	)	)	PUNCT
ejpam-3886	198	6	on	on	ADP
ejpam-3886	198	7	the	the	DET
ejpam-3886	198	8	left	left	ADJ
ejpam-3886	198	9	side	side	NOUN
ejpam-3886	198	10	by	by	ADP
ejpam-3886	198	11	bm	bm	PROPN
ejpam-3886	198	12	,	,	PUNCT
ejpam-3886	198	13	then	then	ADV
ejpam-3886	198	14	we	we	PRON
ejpam-3886	198	15	get	get	VERB
ejpam-3886	198	16	bman−1α	bman−1α	NUM
ejpam-3886	198	17	n−1(bm	n−1(bm	NOUN
ejpam-3886	198	18	)	)	PUNCT
ejpam-3886	198	19	∈	∈	PROPN
ejpam-3886	198	20	nil	nil	NOUN
ejpam-3886	198	21	(	(	PUNCT
ejpam-3886	198	22	r	r	NOUN
ejpam-3886	198	23	)	)	PUNCT
ejpam-3886	198	24	.	.	PUNCT
ejpam-3886	199	1	thus	thus	ADV
ejpam-3886	199	2	,	,	PUNCT
ejpam-3886	199	3	by	by	ADP
ejpam-3886	199	4	lemma	lemma	PROPN
ejpam-3886	199	5	4	4	NUM
ejpam-3886	199	6	,	,	PUNCT
ejpam-3886	199	7	we	we	PRON
ejpam-3886	199	8	obtain	obtain	VERB
ejpam-3886	199	9	an−1bm	an−1bm	NUM
ejpam-3886	199	10	∈	∈	PROPN
ejpam-3886	199	11	nil	nil	NOUN
ejpam-3886	199	12	(	(	PUNCT
ejpam-3886	199	13	r	r	NOUN
ejpam-3886	199	14	)	)	PUNCT
ejpam-3886	199	15	.	.	PUNCT
ejpam-3886	200	1	again	again	ADV
ejpam-3886	200	2	from	from	ADP
ejpam-3886	200	3	eq.(2	eq.(2	ADJ
ejpam-3886	200	4	)	)	PUNCT
ejpam-3886	200	5	and	and	CCONJ
ejpam-3886	200	6	lemma	lemma	PROPN
ejpam-3886	200	7	4	4	NUM
ejpam-3886	200	8	,	,	PUNCT
ejpam-3886	200	9	we	we	PRON
ejpam-3886	200	10	obtain	obtain	VERB
ejpam-3886	200	11	anbm−1	anbm−1	PROPN
ejpam-3886	200	12	∈	∈	PROPN
ejpam-3886	200	13	nil	nil	NOUN
ejpam-3886	200	14	(	(	PUNCT
ejpam-3886	200	15	r	r	NOUN
ejpam-3886	200	16	)	)	PUNCT
ejpam-3886	200	17	.	.	PUNCT
ejpam-3886	201	1	applying	apply	VERB
ejpam-3886	201	2	the	the	DET
ejpam-3886	201	3	preceding	precede	VERB
ejpam-3886	201	4	method	method	NOUN
ejpam-3886	201	5	repeatedly	repeatedly	ADV
ejpam-3886	201	6	,	,	PUNCT
ejpam-3886	201	7	we	we	PRON
ejpam-3886	201	8	deduce	deduce	VERB
ejpam-3886	201	9	that	that	SCONJ
ejpam-3886	201	10	aibj	aibj	PROPN
ejpam-3886	201	11	∈	∈	PROPN
ejpam-3886	201	12	nil	nil	NOUN
ejpam-3886	201	13	(	(	PUNCT
ejpam-3886	201	14	r	r	NOUN
ejpam-3886	201	15	)	)	PUNCT
ejpam-3886	201	16	for	for	ADP
ejpam-3886	201	17	all	all	DET
ejpam-3886	201	18	integers	integer	NOUN
ejpam-3886	201	19	0	0	NUM
ejpam-3886	201	20	≤	≤	NUM
ejpam-3886	201	21	i	i	PRON
ejpam-3886	201	22	≤	≤	ADJ
ejpam-3886	201	23	n	n	CCONJ
ejpam-3886	201	24	and	and	CCONJ
ejpam-3886	201	25	0	0	NUM
ejpam-3886	201	26	≤	≤	NUM
ejpam-3886	201	27	j	j	PROPN
ejpam-3886	201	28	≤	≤	PROPN
ejpam-3886	201	29	m.	m.	NOUN
ejpam-3886	201	30	conversely	conversely	ADV
ejpam-3886	201	31	,	,	PUNCT
ejpam-3886	201	32	suppose	suppose	VERB
ejpam-3886	201	33	that	that	SCONJ
ejpam-3886	201	34	f(x	f(x	PROPN
ejpam-3886	201	35	)	)	PUNCT
ejpam-3886	202	1	=	=	SYM
ejpam-3886	202	2	n∑	n∑	PROPN
ejpam-3886	202	3	i=0	i=0	PROPN
ejpam-3886	202	4	aix	aix	NOUN
ejpam-3886	202	5	i	i	PROPN
ejpam-3886	202	6	and	and	CCONJ
ejpam-3886	202	7	g(x	g(x	NOUN
ejpam-3886	202	8	)	)	PUNCT
ejpam-3886	203	1	=	=	PUNCT
ejpam-3886	203	2	m∑	m∑	CCONJ
ejpam-3886	203	3	j=0	j=0	PROPN
ejpam-3886	203	4	bjx	bjx	VERB
ejpam-3886	203	5	j	j	PROPN
ejpam-3886	203	6	∈	∈	PROPN
ejpam-3886	204	1	a	a	DET
ejpam-3886	204	2	such	such	ADJ
ejpam-3886	204	3	that	that	SCONJ
ejpam-3886	204	4	aibj	aibj	PROPN
ejpam-3886	204	5	∈	∈	PROPN
ejpam-3886	204	6	nil	nil	NOUN
ejpam-3886	204	7	(	(	PUNCT
ejpam-3886	204	8	r	r	NOUN
ejpam-3886	204	9	)	)	PUNCT
ejpam-3886	204	10	for	for	ADP
ejpam-3886	204	11	all	all	DET
ejpam-3886	204	12	integers	integer	NOUN
ejpam-3886	204	13	0	0	NUM
ejpam-3886	204	14	≤	≤	NUM
ejpam-3886	205	1	i	i	PRON
ejpam-3886	206	1	≤	≤	ADJ
ejpam-3886	206	2	n	n	CCONJ
ejpam-3886	206	3	and	and	CCONJ
ejpam-3886	206	4	0	0	NUM
ejpam-3886	206	5	≤	≤	NUM
ejpam-3886	206	6	j	j	PROPN
ejpam-3886	206	7	≤	≤	PROPN
ejpam-3886	206	8	m.	m.	NOUN
ejpam-3886	206	9	we	we	PRON
ejpam-3886	206	10	show	show	VERB
ejpam-3886	206	11	that	that	SCONJ
ejpam-3886	206	12	f(x)g(x	f(x)g(x	ADJ
ejpam-3886	206	13	)	)	PUNCT
ejpam-3886	206	14	∈	∈	PROPN
ejpam-3886	206	15	nil	nil	NOUN
ejpam-3886	206	16	(	(	PUNCT
ejpam-3886	206	17	a	a	NOUN
ejpam-3886	206	18	)	)	PUNCT
ejpam-3886	206	19	.	.	PUNCT
ejpam-3886	207	1	from	from	ADP
ejpam-3886	207	2	lemma	lemma	PROPN
ejpam-3886	207	3	3	3	PROPN
ejpam-3886	207	4	and	and	CCONJ
ejpam-3886	207	5	lemma	lemma	PROPN
ejpam-3886	207	6	4	4	NUM
ejpam-3886	207	7	,	,	PUNCT
ejpam-3886	207	8	we	we	PRON
ejpam-3886	207	9	get	get	VERB
ejpam-3886	207	10	the	the	DET
ejpam-3886	207	11	following	following	NOUN
ejpam-3886	207	12	:	:	PUNCT
ejpam-3886	208	1	aλvu(b	aλvu(b	X
ejpam-3886	208	2	)	)	PUNCT
ejpam-3886	208	3	∈	∈	PROPN
ejpam-3886	208	4	nil	nil	NOUN
ejpam-3886	208	5	(	(	PUNCT
ejpam-3886	208	6	r	r	NOUN
ejpam-3886	208	7	)	)	PUNCT
ejpam-3886	208	8	,	,	PUNCT
ejpam-3886	208	9	where	where	SCONJ
ejpam-3886	208	10	v	v	X
ejpam-3886	208	11	≥	≥	NOUN
ejpam-3886	208	12	u	u	NOUN
ejpam-3886	208	13	≥	≥	NUM
ejpam-3886	208	14	0	0	NUM
ejpam-3886	208	15	are	be	AUX
ejpam-3886	208	16	integers	integer	NOUN
ejpam-3886	208	17	.	.	PUNCT
ejpam-3886	209	1	hence	hence	ADV
ejpam-3886	209	2	f(x)g(x	f(x)g(x	ADJ
ejpam-3886	209	3	)	)	PUNCT
ejpam-3886	209	4	∈	∈	PROPN
ejpam-3886	209	5	nil	nil	NOUN
ejpam-3886	209	6	(	(	PUNCT
ejpam-3886	209	7	a	a	NOUN
ejpam-3886	209	8	)	)	PUNCT
ejpam-3886	209	9	.	.	PUNCT
ejpam-3886	210	1	now	now	ADV
ejpam-3886	210	2	we	we	PRON
ejpam-3886	210	3	can	can	AUX
ejpam-3886	210	4	turn	turn	VERB
ejpam-3886	210	5	to	to	ADP
ejpam-3886	210	6	our	our	PRON
ejpam-3886	210	7	main	main	ADJ
ejpam-3886	210	8	theorems	theorem	NOUN
ejpam-3886	210	9	in	in	ADP
ejpam-3886	210	10	the	the	DET
ejpam-3886	210	11	paper	paper	NOUN
ejpam-3886	210	12	.	.	PUNCT
ejpam-3886	211	1	theorem	theorem	NOUN
ejpam-3886	211	2	1	1	NUM
ejpam-3886	211	3	.	.	PUNCT
ejpam-3886	212	1	let	let	VERB
ejpam-3886	212	2	r	r	PRON
ejpam-3886	212	3	be	be	AUX
ejpam-3886	212	4	an	an	DET
ejpam-3886	212	5	(	(	PUNCT
ejpam-3886	212	6	α	α	NOUN
ejpam-3886	212	7	,	,	PUNCT
ejpam-3886	212	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	212	9	ni	ni	PROPN
ejpam-3886	212	10	ring	ring	NOUN
ejpam-3886	212	11	with	with	ADP
ejpam-3886	212	12	nil	nil	NOUN
ejpam-3886	212	13	(	(	PUNCT
ejpam-3886	212	14	r	r	NOUN
ejpam-3886	212	15	)	)	PUNCT
ejpam-3886	212	16	nilpotent	nilpotent	NOUN
ejpam-3886	212	17	,	,	PUNCT
ejpam-3886	212	18	such	such	ADJ
ejpam-3886	212	19	that	that	SCONJ
ejpam-3886	212	20	α(e	α(e	NOUN
ejpam-3886	212	21	)	)	PUNCT
ejpam-3886	213	1	=	=	SYM
ejpam-3886	213	2	e	e	NOUN
ejpam-3886	213	3	and	and	CCONJ
ejpam-3886	213	4	δ(e	δ(e	NOUN
ejpam-3886	213	5	)	)	PUNCT
ejpam-3886	213	6	=	=	SYM
ejpam-3886	213	7	0	0	NUM
ejpam-3886	213	8	for	for	ADP
ejpam-3886	213	9	every	every	DET
ejpam-3886	213	10	e	e	PROPN
ejpam-3886	213	11	∈	∈	PROPN
ejpam-3886	213	12	id(r	id(r	NOUN
ejpam-3886	213	13	)	)	PUNCT
ejpam-3886	213	14	.	.	PUNCT
ejpam-3886	214	1	if	if	SCONJ
ejpam-3886	214	2	r	r	NOUN
ejpam-3886	214	3	is	be	AUX
ejpam-3886	214	4	a	a	DET
ejpam-3886	214	5	weak	weak	ADJ
ejpam-3886	214	6	right	right	ADJ
ejpam-3886	214	7	ps	ps	NOUN
ejpam-3886	214	8	-	-	NOUN
ejpam-3886	214	9	ring	ring	NOUN
ejpam-3886	214	10	,	,	PUNCT
ejpam-3886	214	11	then	then	ADV
ejpam-3886	214	12	a	a	DET
ejpam-3886	214	13	=	=	X
ejpam-3886	214	14	r	r	NOUN
ejpam-3886	214	15	[	[	X
ejpam-3886	214	16	x;α	x;α	PROPN
ejpam-3886	214	17	,	,	PUNCT
ejpam-3886	214	18	δ	δ	PROPN
ejpam-3886	214	19	]	]	PUNCT
ejpam-3886	214	20	is	be	AUX
ejpam-3886	214	21	a	a	DET
ejpam-3886	214	22	weak	weak	ADJ
ejpam-3886	214	23	right	right	ADJ
ejpam-3886	214	24	ps	ps	NOUN
ejpam-3886	214	25	-	-	PUNCT
ejpam-3886	214	26	ring	ring	NOUN
ejpam-3886	214	27	.	.	PUNCT
ejpam-3886	215	1	m.	m.	NOUN
ejpam-3886	215	2	a.	a.	PROPN
ejpam-3886	215	3	farahat	farahat	PROPN
ejpam-3886	215	4	,	,	PUNCT
ejpam-3886	215	5	salha	salha	NOUN
ejpam-3886	215	6	t.	t.	PROPN
ejpam-3886	215	7	al	al	PROPN
ejpam-3886	215	8	-	-	PUNCT
ejpam-3886	215	9	bogamy	bogamy	PROPN
ejpam-3886	215	10	/	/	SYM
ejpam-3886	215	11	eur	eur	PROPN
ejpam-3886	215	12	.	.	PUNCT
ejpam-3886	216	1	j.	j.	PROPN
ejpam-3886	216	2	pure	pure	PROPN
ejpam-3886	216	3	appl	appl	PROPN
ejpam-3886	216	4	.	.	PROPN
ejpam-3886	216	5	math	math	PROPN
ejpam-3886	216	6	,	,	PUNCT
ejpam-3886	216	7	14	14	NUM
ejpam-3886	216	8	(	(	PUNCT
ejpam-3886	216	9	1	1	NUM
ejpam-3886	216	10	)	)	PUNCT
ejpam-3886	216	11	(	(	PUNCT
ejpam-3886	216	12	2021	2021	NUM
ejpam-3886	216	13	)	)	PUNCT
ejpam-3886	216	14	,	,	PUNCT
ejpam-3886	216	15	164	164	NUM
ejpam-3886	216	16	-	-	SYM
ejpam-3886	216	17	172	172	NUM
ejpam-3886	216	18	170	170	NUM
ejpam-3886	216	19	proof	proof	NOUN
ejpam-3886	216	20	.	.	PUNCT
ejpam-3886	217	1	let	let	VERB
ejpam-3886	217	2	l	l	NOUN
ejpam-3886	217	3	be	be	AUX
ejpam-3886	217	4	a	a	DET
ejpam-3886	217	5	maximal	maximal	ADJ
ejpam-3886	217	6	right	right	ADJ
ejpam-3886	217	7	ideal	ideal	NOUN
ejpam-3886	217	8	of	of	ADP
ejpam-3886	217	9	a	a	DET
ejpam-3886	217	10	=	=	SYM
ejpam-3886	217	11	r	r	NOUN
ejpam-3886	218	1	[	[	X
ejpam-3886	218	2	x;α	x;α	PROPN
ejpam-3886	218	3	,	,	PUNCT
ejpam-3886	218	4	δ	δ	PROPN
ejpam-3886	218	5	]	]	PUNCT
ejpam-3886	218	6	.	.	PUNCT
ejpam-3886	219	1	we	we	PRON
ejpam-3886	219	2	will	will	AUX
ejpam-3886	219	3	show	show	VERB
ejpam-3886	219	4	that	that	SCONJ
ejpam-3886	219	5	either	either	CCONJ
ejpam-3886	219	6	na	na	INTJ
ejpam-3886	219	7	(	(	PUNCT
ejpam-3886	219	8	l	l	NOUN
ejpam-3886	219	9	)	)	PUNCT
ejpam-3886	219	10	⊆	⊆	NUM
ejpam-3886	219	11	nil(a	nil(a	NUM
ejpam-3886	219	12	)	)	PUNCT
ejpam-3886	219	13	or	or	CCONJ
ejpam-3886	219	14	na	na	ADP
ejpam-3886	219	15	(	(	PUNCT
ejpam-3886	219	16	l	l	NOUN
ejpam-3886	219	17	)	)	PUNCT
ejpam-3886	219	18	=	=	SYM
ejpam-3886	219	19	aq	aq	PROPN
ejpam-3886	219	20	,	,	PUNCT
ejpam-3886	219	21	where	where	SCONJ
ejpam-3886	219	22	q	q	PROPN
ejpam-3886	219	23	∈	∈	PROPN
ejpam-3886	219	24	id(a	id(a	NOUN
ejpam-3886	219	25	)	)	PUNCT
ejpam-3886	219	26	.	.	PUNCT
ejpam-3886	220	1	let	let	VERB
ejpam-3886	220	2	i	i	PRON
ejpam-3886	220	3	be	be	AUX
ejpam-3886	220	4	the	the	DET
ejpam-3886	220	5	set	set	NOUN
ejpam-3886	220	6	of	of	ADP
ejpam-3886	220	7	all	all	DET
ejpam-3886	220	8	coefficients	coefficient	NOUN
ejpam-3886	220	9	of	of	ADP
ejpam-3886	220	10	all	all	DET
ejpam-3886	220	11	polynomials	polynomial	NOUN
ejpam-3886	220	12	in	in	ADP
ejpam-3886	220	13	l	l	NOUN
ejpam-3886	220	14	and	and	CCONJ
ejpam-3886	220	15	let	let	VERB
ejpam-3886	220	16	j	j	PROPN
ejpam-3886	220	17	be	be	AUX
ejpam-3886	220	18	the	the	DET
ejpam-3886	220	19	right	right	ADJ
ejpam-3886	220	20	ideal	ideal	NOUN
ejpam-3886	220	21	of	of	ADP
ejpam-3886	220	22	r	r	NOUN
ejpam-3886	220	23	generated	generate	VERB
ejpam-3886	220	24	by	by	ADP
ejpam-3886	220	25	i	i	PRON
ejpam-3886	220	26	,	,	PUNCT
ejpam-3886	221	1	i.e.	i.e.	X
ejpam-3886	221	2	,	,	PUNCT
ejpam-3886	221	3	j	j	PROPN
ejpam-3886	221	4	=	=	SYM
ejpam-3886	221	5	〈	〈	PROPN
ejpam-3886	221	6	i〉r	i〉r	PROPN
ejpam-3886	221	7	=	=	SYM
ejpam-3886	221	8	ir	ir	PROPN
ejpam-3886	221	9	.	.	PUNCT
ejpam-3886	222	1	if	if	SCONJ
ejpam-3886	222	2	j	j	PROPN
ejpam-3886	222	3	=	=	SYM
ejpam-3886	222	4	r	r	NOUN
ejpam-3886	222	5	,	,	PUNCT
ejpam-3886	222	6	then	then	ADV
ejpam-3886	222	7	there	there	PRON
ejpam-3886	222	8	exist	exist	VERB
ejpam-3886	222	9	a1	a1	NOUN
ejpam-3886	222	10	,	,	PUNCT
ejpam-3886	222	11	a2	a2	PROPN
ejpam-3886	222	12	,	,	PUNCT
ejpam-3886	222	13	...	...	PUNCT
ejpam-3886	222	14	,	,	PUNCT
ejpam-3886	222	15	an	an	DET
ejpam-3886	222	16	∈	∈	NOUN
ejpam-3886	222	17	i	i	PRON
ejpam-3886	222	18	and	and	CCONJ
ejpam-3886	222	19	r1	r1	PROPN
ejpam-3886	222	20	,	,	PUNCT
ejpam-3886	222	21	r2	r2	PROPN
ejpam-3886	222	22	,	,	PUNCT
ejpam-3886	222	23	...	...	PUNCT
ejpam-3886	222	24	,	,	PUNCT
ejpam-3886	222	25	rn	rn	PROPN
ejpam-3886	222	26	∈	∈	PROPN
ejpam-3886	222	27	r	r	NOUN
ejpam-3886	222	28	,	,	PUNCT
ejpam-3886	222	29	such	such	ADJ
ejpam-3886	222	30	that	that	SCONJ
ejpam-3886	222	31	1	1	NUM
ejpam-3886	222	32	=	=	SYM
ejpam-3886	222	33	a1r1	a1r1	PROPN
ejpam-3886	222	34	+	+	ADJ
ejpam-3886	222	35	a2r2	a2r2	X
ejpam-3886	222	36	+	+	NUM
ejpam-3886	222	37	...	...	PUNCT
ejpam-3886	222	38	+	+	NUM
ejpam-3886	222	39	anrn	anrn	NOUN
ejpam-3886	222	40	.	.	PUNCT
ejpam-3886	223	1	suppose	suppose	VERB
ejpam-3886	223	2	that	that	SCONJ
ejpam-3886	223	3	ϕ(x	ϕ(x	PROPN
ejpam-3886	223	4	)	)	PUNCT
ejpam-3886	223	5	=	=	SYM
ejpam-3886	224	1	k∑	k∑	PROPN
ejpam-3886	225	1	i=0	i=0	PROPN
ejpam-3886	225	2	bix	bix	PROPN
ejpam-3886	225	3	i	i	PRON
ejpam-3886	225	4	∈	∈	PROPN
ejpam-3886	225	5	na	na	X
ejpam-3886	225	6	(	(	PUNCT
ejpam-3886	225	7	l	l	NOUN
ejpam-3886	225	8	)	)	PUNCT
ejpam-3886	225	9	,	,	PUNCT
ejpam-3886	225	10	then	then	ADV
ejpam-3886	225	11	for	for	ADP
ejpam-3886	225	12	every	every	DET
ejpam-3886	225	13	f(x	f(x	PROPN
ejpam-3886	225	14	)	)	PUNCT
ejpam-3886	225	15	=	=	SYM
ejpam-3886	225	16	n∑	n∑	PROPN
ejpam-3886	225	17	j=0	j=0	PROPN
ejpam-3886	225	18	ajx	ajx	VERB
ejpam-3886	225	19	j	j	PROPN
ejpam-3886	225	20	∈	∈	PROPN
ejpam-3886	225	21	l	l	PROPN
ejpam-3886	225	22	,	,	PUNCT
ejpam-3886	225	23	we	we	PRON
ejpam-3886	225	24	have	have	VERB
ejpam-3886	225	25	ϕ(x)f(x	ϕ(x)f(x	NOUN
ejpam-3886	225	26	)	)	PUNCT
ejpam-3886	226	1	=	=	PRON
ejpam-3886	226	2	(	(	PUNCT
ejpam-3886	226	3	k∑	k∑	NOUN
ejpam-3886	226	4	i=0	i=0	PROPN
ejpam-3886	226	5	bix	bix	PROPN
ejpam-3886	226	6	i	i	PRON
ejpam-3886	226	7	)	)	PUNCT
ejpam-3886	227	1			PROPN
ejpam-3886	227	2	n∑	n∑	PROPN
ejpam-3886	227	3	j=0	j=0	PROPN
ejpam-3886	227	4	ajx	ajx	PROPN
ejpam-3886	227	5	j	j	PROPN
ejpam-3886	227	6			PROPN
ejpam-3886	227	7	∈	∈	PROPN
ejpam-3886	227	8	nil(a	nil(a	PROPN
ejpam-3886	227	9	)	)	PUNCT
ejpam-3886	227	10	.	.	PUNCT
ejpam-3886	228	1	since	since	SCONJ
ejpam-3886	228	2	r	r	NOUN
ejpam-3886	228	3	is	be	AUX
ejpam-3886	228	4	an	an	DET
ejpam-3886	228	5	(	(	PUNCT
ejpam-3886	228	6	α	α	NOUN
ejpam-3886	228	7	,	,	PUNCT
ejpam-3886	228	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	228	9	ni	ni	PROPN
ejpam-3886	228	10	ring	ring	NOUN
ejpam-3886	228	11	with	with	ADP
ejpam-3886	228	12	nil	nil	NOUN
ejpam-3886	228	13	(	(	PUNCT
ejpam-3886	228	14	r	r	NOUN
ejpam-3886	228	15	)	)	PUNCT
ejpam-3886	228	16	nilpotent	nilpotent	NOUN
ejpam-3886	228	17	,	,	PUNCT
ejpam-3886	228	18	from	from	ADP
ejpam-3886	228	19	lemma	lemma	PROPN
ejpam-3886	228	20	1	1	NUM
ejpam-3886	228	21	,	,	PUNCT
ejpam-3886	228	22	we	we	PRON
ejpam-3886	228	23	get	get	VERB
ejpam-3886	228	24	that	that	DET
ejpam-3886	228	25	biaj	biaj	PROPN
ejpam-3886	228	26	∈	∈	PROPN
ejpam-3886	228	27	nil(r	nil(r	PROPN
ejpam-3886	228	28	)	)	PUNCT
ejpam-3886	228	29	,	,	PUNCT
ejpam-3886	228	30	for	for	ADP
ejpam-3886	228	31	all	all	DET
ejpam-3886	228	32	integers	integer	NOUN
ejpam-3886	228	33	0	0	NUM
ejpam-3886	228	34	≤	≤	NUM
ejpam-3886	229	1	i	i	PRON
ejpam-3886	229	2	≤	≤	NOUN
ejpam-3886	230	1	k	k	PRON
ejpam-3886	230	2	and	and	CCONJ
ejpam-3886	230	3	0	0	NUM
ejpam-3886	230	4	≤	≤	NUM
ejpam-3886	230	5	j	j	PROPN
ejpam-3886	230	6	≤	≤	PROPN
ejpam-3886	230	7	n.	n.	NOUN
ejpam-3886	230	8	consequently	consequently	ADV
ejpam-3886	230	9	,	,	PUNCT
ejpam-3886	230	10	for	for	ADP
ejpam-3886	230	11	every	every	DET
ejpam-3886	230	12	a	a	DET
ejpam-3886	230	13	∈	∈	PROPN
ejpam-3886	230	14	i	i	NOUN
ejpam-3886	230	15	,	,	PUNCT
ejpam-3886	230	16	bia	bia	PROPN
ejpam-3886	230	17	∈	∈	PROPN
ejpam-3886	230	18	nil(r	nil(r	PROPN
ejpam-3886	230	19	)	)	PUNCT
ejpam-3886	230	20	,	,	PUNCT
ejpam-3886	230	21	for	for	ADP
ejpam-3886	230	22	all	all	DET
ejpam-3886	230	23	integers	integer	NOUN
ejpam-3886	230	24	0	0	NUM
ejpam-3886	230	25	≤	≤	NUM
ejpam-3886	230	26	i	i	PRON
ejpam-3886	230	27	≤	≤	PROPN
ejpam-3886	230	28	k.	k.	PROPN
ejpam-3886	231	1	hence	hence	ADV
ejpam-3886	231	2	bi	bi	PROPN
ejpam-3886	231	3	∈	∈	PROPN
ejpam-3886	231	4	nr	nr	PROPN
ejpam-3886	231	5	(	(	PUNCT
ejpam-3886	231	6	j	j	PROPN
ejpam-3886	231	7	)	)	PUNCT
ejpam-3886	231	8	=	=	SYM
ejpam-3886	231	9	nr	nr	PROPN
ejpam-3886	231	10	(	(	PUNCT
ejpam-3886	231	11	r	r	NOUN
ejpam-3886	231	12	)	)	PUNCT
ejpam-3886	231	13	=	=	SYM
ejpam-3886	231	14	nil(r	nil(r	NOUN
ejpam-3886	231	15	)	)	PUNCT
ejpam-3886	231	16	,	,	PUNCT
ejpam-3886	231	17	for	for	ADP
ejpam-3886	231	18	all	all	DET
ejpam-3886	231	19	integers	integer	NOUN
ejpam-3886	231	20	0	0	NUM
ejpam-3886	231	21	≤	≤	NUM
ejpam-3886	231	22	i	i	PRON
ejpam-3886	231	23	≤	≤	PROPN
ejpam-3886	231	24	k.	k.	PROPN
ejpam-3886	231	25	therefore	therefore	ADV
ejpam-3886	231	26	ϕ(x	ϕ(x	PROPN
ejpam-3886	231	27	)	)	PUNCT
ejpam-3886	231	28	∈	∈	PROPN
ejpam-3886	231	29	nil(a	nil(a	PROPN
ejpam-3886	231	30	)	)	PUNCT
ejpam-3886	231	31	.	.	PUNCT
ejpam-3886	232	1	hence	hence	ADV
ejpam-3886	232	2	na	na	PROPN
ejpam-3886	232	3	(	(	PUNCT
ejpam-3886	232	4	l	l	NOUN
ejpam-3886	232	5	)	)	PUNCT
ejpam-3886	232	6	⊆	⊆	NUM
ejpam-3886	232	7	nil(a	nil(a	NUM
ejpam-3886	232	8	)	)	PUNCT
ejpam-3886	232	9	.	.	PUNCT
ejpam-3886	233	1	if	if	SCONJ
ejpam-3886	233	2	j	j	PROPN
ejpam-3886	233	3	6=	6=	PROPN
ejpam-3886	233	4	r	r	NOUN
ejpam-3886	233	5	,	,	PUNCT
ejpam-3886	233	6	we	we	PRON
ejpam-3886	233	7	show	show	VERB
ejpam-3886	233	8	that	that	SCONJ
ejpam-3886	233	9	j	j	PROPN
ejpam-3886	233	10	is	be	AUX
ejpam-3886	233	11	a	a	DET
ejpam-3886	233	12	maximal	maximal	ADJ
ejpam-3886	233	13	right	right	ADJ
ejpam-3886	233	14	ideal	ideal	NOUN
ejpam-3886	233	15	of	of	ADP
ejpam-3886	233	16	r.	r.	PROPN
ejpam-3886	233	17	let	let	VERB
ejpam-3886	233	18	r	r	NOUN
ejpam-3886	233	19	∈	∈	PROPN
ejpam-3886	233	20	r−j	r−j	NOUN
ejpam-3886	233	21	.	.	PUNCT
ejpam-3886	234	1	if	if	SCONJ
ejpam-3886	234	2	r	r	NOUN
ejpam-3886	234	3	∈	∈	PROPN
ejpam-3886	234	4	l	l	NOUN
ejpam-3886	234	5	,	,	PUNCT
ejpam-3886	234	6	then	then	ADV
ejpam-3886	234	7	r	r	NOUN
ejpam-3886	234	8	∈	∈	PROPN
ejpam-3886	235	1	i	i	PRON
ejpam-3886	235	2	and	and	CCONJ
ejpam-3886	235	3	so	so	ADV
ejpam-3886	235	4	r	r	NOUN
ejpam-3886	235	5	∈	∈	PROPN
ejpam-3886	235	6	j	j	PROPN
ejpam-3886	235	7	,	,	PUNCT
ejpam-3886	235	8	which	which	PRON
ejpam-3886	235	9	is	be	AUX
ejpam-3886	235	10	a	a	DET
ejpam-3886	235	11	contradiction	contradiction	NOUN
ejpam-3886	235	12	.	.	PUNCT
ejpam-3886	236	1	thus	thus	ADV
ejpam-3886	236	2	r	r	NOUN
ejpam-3886	236	3	/∈	/∈	PUNCT
ejpam-3886	236	4	l.	l.	NOUN
ejpam-3886	236	5	since	since	SCONJ
ejpam-3886	236	6	l	l	PROPN
ejpam-3886	236	7	is	be	AUX
ejpam-3886	236	8	a	a	DET
ejpam-3886	236	9	maximal	maximal	ADJ
ejpam-3886	236	10	right	right	ADJ
ejpam-3886	236	11	ideal	ideal	NOUN
ejpam-3886	236	12	of	of	ADP
ejpam-3886	236	13	a	a	PRON
ejpam-3886	236	14	,	,	PUNCT
ejpam-3886	236	15	we	we	PRON
ejpam-3886	236	16	have	have	VERB
ejpam-3886	236	17	a	a	DET
ejpam-3886	236	18	=	=	PUNCT
ejpam-3886	236	19	l+	l+	PUNCT
ejpam-3886	236	20	ra	ra	NOUN
ejpam-3886	236	21	.	.	PUNCT
ejpam-3886	237	1	it	it	PRON
ejpam-3886	237	2	follows	follow	VERB
ejpam-3886	237	3	that	that	SCONJ
ejpam-3886	237	4	there	there	PRON
ejpam-3886	237	5	exist	exist	VERB
ejpam-3886	237	6	f(x	f(x	NOUN
ejpam-3886	237	7	)	)	PUNCT
ejpam-3886	238	1	=	=	SYM
ejpam-3886	238	2	n∑	n∑	PROPN
ejpam-3886	238	3	i=0	i=0	PROPN
ejpam-3886	238	4	aix	aix	NOUN
ejpam-3886	238	5	i	i	NOUN
ejpam-3886	238	6	∈	∈	PROPN
ejpam-3886	238	7	l	l	NOUN
ejpam-3886	238	8	and	and	CCONJ
ejpam-3886	238	9	h(x	h(x	PROPN
ejpam-3886	238	10	)	)	PUNCT
ejpam-3886	239	1	=	=	PUNCT
ejpam-3886	239	2	m∑	m∑	CCONJ
ejpam-3886	239	3	j=0	j=0	PROPN
ejpam-3886	239	4	bjx	bjx	VERB
ejpam-3886	239	5	j	j	PROPN
ejpam-3886	239	6	∈	∈	PROPN
ejpam-3886	239	7	a	a	PRON
ejpam-3886	239	8	,	,	PUNCT
ejpam-3886	239	9	such	such	ADJ
ejpam-3886	239	10	that	that	SCONJ
ejpam-3886	239	11	1	1	NUM
ejpam-3886	239	12	=	=	SYM
ejpam-3886	239	13	a0	a0	PROPN
ejpam-3886	239	14	+	+	CCONJ
ejpam-3886	239	15	rb0	rb0	PROPN
ejpam-3886	239	16	.	.	PUNCT
ejpam-3886	240	1	if	if	SCONJ
ejpam-3886	240	2	a0	a0	PROPN
ejpam-3886	240	3	=	=	SYM
ejpam-3886	240	4	0	0	PROPN
ejpam-3886	240	5	,	,	PUNCT
ejpam-3886	240	6	then	then	ADV
ejpam-3886	240	7	1	1	X
ejpam-3886	240	8	=	=	SYM
ejpam-3886	240	9	rb0	rb0	NOUN
ejpam-3886	240	10	∈	∈	PROPN
ejpam-3886	240	11	rr	rr	NOUN
ejpam-3886	241	1	and	and	CCONJ
ejpam-3886	241	2	so	so	ADV
ejpam-3886	241	3	r	r	NOUN
ejpam-3886	241	4	=	=	SYM
ejpam-3886	241	5	j	j	PROPN
ejpam-3886	241	6	+	+	NUM
ejpam-3886	241	7	rr	rr	PROPN
ejpam-3886	241	8	.	.	PUNCT
ejpam-3886	242	1	if	if	SCONJ
ejpam-3886	242	2	a0	a0	PROPN
ejpam-3886	242	3	6=	6=	PROPN
ejpam-3886	242	4	0	0	NUM
ejpam-3886	242	5	,	,	PUNCT
ejpam-3886	242	6	then	then	ADV
ejpam-3886	242	7	a0	a0	PROPN
ejpam-3886	242	8	∈	∈	PROPN
ejpam-3886	243	1	i	i	PRON
ejpam-3886	243	2	⊂	⊂	PROPN
ejpam-3886	243	3	j	j	PROPN
ejpam-3886	243	4	which	which	PRON
ejpam-3886	243	5	implies	imply	VERB
ejpam-3886	243	6	that	that	SCONJ
ejpam-3886	243	7	r	r	NOUN
ejpam-3886	243	8	=	=	SYM
ejpam-3886	243	9	j	j	PROPN
ejpam-3886	243	10	+	+	NUM
ejpam-3886	243	11	rr	rr	PROPN
ejpam-3886	243	12	.	.	PUNCT
ejpam-3886	244	1	hence	hence	ADV
ejpam-3886	244	2	j	j	PROPN
ejpam-3886	244	3	is	be	AUX
ejpam-3886	244	4	a	a	DET
ejpam-3886	244	5	maximal	maximal	ADJ
ejpam-3886	244	6	right	right	ADJ
ejpam-3886	244	7	ideal	ideal	NOUN
ejpam-3886	244	8	of	of	ADP
ejpam-3886	244	9	r.	r.	PROPN
ejpam-3886	244	10	since	since	SCONJ
ejpam-3886	244	11	r	r	NOUN
ejpam-3886	244	12	is	be	AUX
ejpam-3886	244	13	a	a	DET
ejpam-3886	244	14	weak	weak	ADJ
ejpam-3886	244	15	right	right	ADJ
ejpam-3886	244	16	ps	ps	NOUN
ejpam-3886	244	17	-	-	NOUN
ejpam-3886	244	18	ring	ring	NOUN
ejpam-3886	244	19	,	,	PUNCT
ejpam-3886	244	20	it	it	PRON
ejpam-3886	244	21	follows	follow	VERB
ejpam-3886	244	22	that	that	SCONJ
ejpam-3886	244	23	either	either	CCONJ
ejpam-3886	244	24	nr	nr	PROPN
ejpam-3886	244	25	(	(	PUNCT
ejpam-3886	244	26	j	j	PROPN
ejpam-3886	244	27	)	)	PUNCT
ejpam-3886	244	28	⊆	⊆	NUM
ejpam-3886	244	29	nil(r	nil(r	NUM
ejpam-3886	244	30	)	)	PUNCT
ejpam-3886	244	31	or	or	CCONJ
ejpam-3886	244	32	nr	nr	PRON
ejpam-3886	244	33	(	(	PUNCT
ejpam-3886	244	34	j	j	PROPN
ejpam-3886	244	35	)	)	PUNCT
ejpam-3886	244	36	=	=	PUNCT
ejpam-3886	244	37	re	re	PROPN
ejpam-3886	244	38	,	,	PUNCT
ejpam-3886	244	39	where	where	SCONJ
ejpam-3886	244	40	e	e	PROPN
ejpam-3886	244	41	∈	∈	PROPN
ejpam-3886	244	42	id(r	id(r	NOUN
ejpam-3886	244	43	)	)	PUNCT
ejpam-3886	244	44	.	.	PUNCT
ejpam-3886	245	1	case	case	NOUN
ejpam-3886	245	2	(	(	PUNCT
ejpam-3886	245	3	1	1	NUM
ejpam-3886	245	4	):	):	PUNCT
ejpam-3886	245	5	assume	assume	VERB
ejpam-3886	245	6	that	that	SCONJ
ejpam-3886	245	7	nr	nr	PRON
ejpam-3886	245	8	(	(	PUNCT
ejpam-3886	245	9	j	j	PROPN
ejpam-3886	245	10	)	)	PUNCT
ejpam-3886	245	11	⊆	⊆	NUM
ejpam-3886	245	12	nil(r	nil(r	NOUN
ejpam-3886	245	13	)	)	PUNCT
ejpam-3886	245	14	.	.	PUNCT
ejpam-3886	246	1	we	we	PRON
ejpam-3886	246	2	will	will	AUX
ejpam-3886	246	3	show	show	VERB
ejpam-3886	246	4	that	that	SCONJ
ejpam-3886	246	5	na	na	ADP
ejpam-3886	246	6	(	(	PUNCT
ejpam-3886	246	7	l	l	NOUN
ejpam-3886	246	8	)	)	PUNCT
ejpam-3886	246	9	⊆	⊆	NUM
ejpam-3886	246	10	nil(a	nil(a	NUM
ejpam-3886	246	11	)	)	PUNCT
ejpam-3886	246	12	.	.	PUNCT
ejpam-3886	247	1	let	let	VERB
ejpam-3886	247	2	ϕ	ϕ	NOUN
ejpam-3886	247	3	(	(	PUNCT
ejpam-3886	247	4	x	x	NOUN
ejpam-3886	247	5	)	)	PUNCT
ejpam-3886	247	6	=	=	SYM
ejpam-3886	247	7	k∑	k∑	NOUN
ejpam-3886	247	8	i=0	i=0	ADJ
ejpam-3886	247	9	mix	mix	NOUN
ejpam-3886	247	10	i	i	NOUN
ejpam-3886	247	11	∈	∈	NOUN
ejpam-3886	247	12	na	na	X
ejpam-3886	247	13	(	(	PUNCT
ejpam-3886	247	14	l	l	NOUN
ejpam-3886	247	15	)	)	PUNCT
ejpam-3886	247	16	.	.	PUNCT
ejpam-3886	248	1	then	then	ADV
ejpam-3886	248	2	for	for	ADP
ejpam-3886	248	3	every	every	DET
ejpam-3886	248	4	g	g	PROPN
ejpam-3886	248	5	(	(	PUNCT
ejpam-3886	248	6	x	x	NOUN
ejpam-3886	248	7	)	)	PUNCT
ejpam-3886	248	8	=	=	SYM
ejpam-3886	248	9	n∑	n∑	NOUN
ejpam-3886	248	10	j=0	j=0	PROPN
ejpam-3886	248	11	ajx	ajx	VERB
ejpam-3886	248	12	j	j	PROPN
ejpam-3886	248	13	∈	∈	PROPN
ejpam-3886	248	14	l	l	PROPN
ejpam-3886	248	15	,	,	PUNCT
ejpam-3886	248	16	we	we	PRON
ejpam-3886	248	17	have	have	VERB
ejpam-3886	248	18	ϕ	ϕ	NOUN
ejpam-3886	248	19	(	(	PUNCT
ejpam-3886	248	20	x	x	NOUN
ejpam-3886	248	21	)	)	PUNCT
ejpam-3886	248	22	g	g	NOUN
ejpam-3886	248	23	(	(	PUNCT
ejpam-3886	248	24	x	x	NOUN
ejpam-3886	248	25	)	)	PUNCT
ejpam-3886	248	26	=	=	SYM
ejpam-3886	248	27	(	(	PUNCT
ejpam-3886	248	28	k∑	k∑	NOUN
ejpam-3886	248	29	i=0	i=0	PUNCT
ejpam-3886	248	30	mix	mix	VERB
ejpam-3886	248	31	i	i	NOUN
ejpam-3886	248	32	)	)	PUNCT
ejpam-3886	248	33			PROPN
ejpam-3886	248	34	n∑	n∑	PROPN
ejpam-3886	248	35	j=0	j=0	PROPN
ejpam-3886	248	36	ajx	ajx	PROPN
ejpam-3886	248	37	j	j	PROPN
ejpam-3886	248	38			PROPN
ejpam-3886	248	39	∈	∈	PROPN
ejpam-3886	248	40	nil(a	nil(a	PROPN
ejpam-3886	248	41	)	)	PUNCT
ejpam-3886	248	42	.	.	PUNCT
ejpam-3886	249	1	since	since	SCONJ
ejpam-3886	249	2	r	r	NOUN
ejpam-3886	249	3	is	be	AUX
ejpam-3886	249	4	an	an	DET
ejpam-3886	249	5	(	(	PUNCT
ejpam-3886	249	6	α	α	NOUN
ejpam-3886	249	7	,	,	PUNCT
ejpam-3886	249	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	249	9	ni	ni	PROPN
ejpam-3886	249	10	ring	ring	NOUN
ejpam-3886	249	11	with	with	ADP
ejpam-3886	249	12	nil	nil	NOUN
ejpam-3886	249	13	(	(	PUNCT
ejpam-3886	249	14	r	r	NOUN
ejpam-3886	249	15	)	)	PUNCT
ejpam-3886	249	16	nilpotent	nilpotent	NOUN
ejpam-3886	249	17	,	,	PUNCT
ejpam-3886	249	18	from	from	ADP
ejpam-3886	249	19	lemma	lemma	PROPN
ejpam-3886	249	20	1	1	NUM
ejpam-3886	249	21	,	,	PUNCT
ejpam-3886	249	22	we	we	PRON
ejpam-3886	249	23	get	get	VERB
ejpam-3886	249	24	that	that	DET
ejpam-3886	249	25	biaj	biaj	PROPN
ejpam-3886	249	26	∈	∈	PROPN
ejpam-3886	249	27	nil(r	nil(r	PROPN
ejpam-3886	249	28	)	)	PUNCT
ejpam-3886	249	29	,	,	PUNCT
ejpam-3886	249	30	for	for	ADP
ejpam-3886	249	31	all	all	DET
ejpam-3886	249	32	integers	integer	NOUN
ejpam-3886	249	33	0	0	NUM
ejpam-3886	249	34	≤	≤	NUM
ejpam-3886	249	35	i	i	PRON
ejpam-3886	249	36	and	and	CCONJ
ejpam-3886	249	37	0	0	NUM
ejpam-3886	249	38	≤	≤	NOUN
ejpam-3886	249	39	j.	j.	PROPN
ejpam-3886	249	40	consequently	consequently	ADV
ejpam-3886	249	41	,	,	PUNCT
ejpam-3886	249	42	for	for	ADP
ejpam-3886	249	43	every	every	DET
ejpam-3886	249	44	a	a	DET
ejpam-3886	249	45	∈	∈	PROPN
ejpam-3886	249	46	i	i	NOUN
ejpam-3886	249	47	,	,	PUNCT
ejpam-3886	249	48	bia	bia	PROPN
ejpam-3886	249	49	∈	∈	PROPN
ejpam-3886	249	50	nil(r	nil(r	PROPN
ejpam-3886	249	51	)	)	PUNCT
ejpam-3886	249	52	,	,	PUNCT
ejpam-3886	249	53	m.	m.	NOUN
ejpam-3886	249	54	a.	a.	NOUN
ejpam-3886	249	55	farahat	farahat	PROPN
ejpam-3886	249	56	,	,	PUNCT
ejpam-3886	249	57	salha	salha	NOUN
ejpam-3886	249	58	t.	t.	PROPN
ejpam-3886	249	59	al	al	PROPN
ejpam-3886	249	60	-	-	PUNCT
ejpam-3886	249	61	bogamy	bogamy	PROPN
ejpam-3886	249	62	/	/	SYM
ejpam-3886	249	63	eur	eur	PROPN
ejpam-3886	249	64	.	.	PUNCT
ejpam-3886	250	1	j.	j.	PROPN
ejpam-3886	250	2	pure	pure	PROPN
ejpam-3886	250	3	appl	appl	PROPN
ejpam-3886	250	4	.	.	PROPN
ejpam-3886	250	5	math	math	PROPN
ejpam-3886	250	6	,	,	PUNCT
ejpam-3886	250	7	14	14	NUM
ejpam-3886	250	8	(	(	PUNCT
ejpam-3886	250	9	1	1	NUM
ejpam-3886	250	10	)	)	PUNCT
ejpam-3886	250	11	(	(	PUNCT
ejpam-3886	250	12	2021	2021	NUM
ejpam-3886	250	13	)	)	PUNCT
ejpam-3886	250	14	,	,	PUNCT
ejpam-3886	250	15	164	164	NUM
ejpam-3886	250	16	-	-	SYM
ejpam-3886	250	17	172	172	NUM
ejpam-3886	250	18	171	171	NUM
ejpam-3886	250	19	for	for	ADP
ejpam-3886	250	20	all	all	DET
ejpam-3886	250	21	integers	integer	NOUN
ejpam-3886	250	22	0	0	NUM
ejpam-3886	250	23	≤	≤	NUM
ejpam-3886	250	24	i.	i.	NOUN
ejpam-3886	250	25	hence	hence	ADV
ejpam-3886	251	1	bi	bi	PROPN
ejpam-3886	251	2	∈	∈	PROPN
ejpam-3886	251	3	nr	nr	PROPN
ejpam-3886	251	4	(	(	PUNCT
ejpam-3886	251	5	j	j	PROPN
ejpam-3886	251	6	)	)	PUNCT
ejpam-3886	251	7	=	=	SYM
ejpam-3886	251	8	nr	nr	PROPN
ejpam-3886	251	9	(	(	PUNCT
ejpam-3886	251	10	r	r	NOUN
ejpam-3886	251	11	)	)	PUNCT
ejpam-3886	251	12	=	=	SYM
ejpam-3886	251	13	nil(r	nil(r	NOUN
ejpam-3886	251	14	)	)	PUNCT
ejpam-3886	251	15	,	,	PUNCT
ejpam-3886	251	16	for	for	ADP
ejpam-3886	251	17	all	all	DET
ejpam-3886	251	18	integers	integer	NOUN
ejpam-3886	251	19	0	0	NUM
ejpam-3886	251	20	≤	≤	NUM
ejpam-3886	251	21	i.	i.	NOUN
ejpam-3886	251	22	therefore	therefore	ADV
ejpam-3886	251	23	ϕ(x	ϕ(x	PROPN
ejpam-3886	251	24	)	)	PUNCT
ejpam-3886	251	25	∈	∈	PROPN
ejpam-3886	251	26	nil(a	nil(a	PROPN
ejpam-3886	251	27	)	)	PUNCT
ejpam-3886	251	28	and	and	CCONJ
ejpam-3886	251	29	we	we	PRON
ejpam-3886	251	30	have	have	VERB
ejpam-3886	251	31	na	na	PART
ejpam-3886	251	32	(	(	PUNCT
ejpam-3886	251	33	l	l	NOUN
ejpam-3886	251	34	)	)	PUNCT
ejpam-3886	251	35	⊆	⊆	NUM
ejpam-3886	251	36	nil(a	nil(a	NUM
ejpam-3886	251	37	)	)	PUNCT
ejpam-3886	251	38	.	.	PUNCT
ejpam-3886	252	1	case	case	NOUN
ejpam-3886	252	2	(	(	PUNCT
ejpam-3886	252	3	2	2	NUM
ejpam-3886	252	4	):	):	PUNCT
ejpam-3886	252	5	assume	assume	VERB
ejpam-3886	252	6	that	that	SCONJ
ejpam-3886	252	7	nr	nr	PRON
ejpam-3886	252	8	(	(	PUNCT
ejpam-3886	252	9	j	j	PROPN
ejpam-3886	252	10	)	)	PUNCT
ejpam-3886	252	11	=	=	PUNCT
ejpam-3886	252	12	re	re	PROPN
ejpam-3886	252	13	,	,	PUNCT
ejpam-3886	252	14	where	where	SCONJ
ejpam-3886	252	15	e	e	PROPN
ejpam-3886	252	16	∈	∈	PROPN
ejpam-3886	252	17	id(r	id(r	NOUN
ejpam-3886	252	18	)	)	PUNCT
ejpam-3886	252	19	.	.	PUNCT
ejpam-3886	253	1	we	we	PRON
ejpam-3886	253	2	will	will	AUX
ejpam-3886	253	3	show	show	VERB
ejpam-3886	253	4	that	that	SCONJ
ejpam-3886	253	5	na	na	ADP
ejpam-3886	253	6	(	(	PUNCT
ejpam-3886	253	7	l	l	NOUN
ejpam-3886	253	8	)	)	PUNCT
ejpam-3886	253	9	=	=	SYM
ejpam-3886	253	10	ah	ah	INTJ
ejpam-3886	253	11	,	,	PUNCT
ejpam-3886	253	12	where	where	SCONJ
ejpam-3886	253	13	h	h	PROPN
ejpam-3886	253	14	∈	∈	PROPN
ejpam-3886	253	15	id(a	id(a	NOUN
ejpam-3886	253	16	)	)	PUNCT
ejpam-3886	253	17	.	.	PUNCT
ejpam-3886	254	1	let	let	VERB
ejpam-3886	254	2	ϕ(x	ϕ(x	PRON
ejpam-3886	254	3	)	)	PUNCT
ejpam-3886	254	4	=	=	SYM
ejpam-3886	255	1	k∑	k∑	PROPN
ejpam-3886	256	1	i=0	i=0	PROPN
ejpam-3886	256	2	bix	bix	PROPN
ejpam-3886	256	3	i	i	PRON
ejpam-3886	256	4	∈	∈	PROPN
ejpam-3886	256	5	na	na	X
ejpam-3886	256	6	(	(	PUNCT
ejpam-3886	256	7	l	l	NOUN
ejpam-3886	256	8	)	)	PUNCT
ejpam-3886	256	9	and	and	CCONJ
ejpam-3886	256	10	ϕ(x	ϕ(x	NOUN
ejpam-3886	256	11	)	)	PUNCT
ejpam-3886	256	12	/∈	/∈	PUNCT
ejpam-3886	257	1	nil(a	nil(a	NUM
ejpam-3886	257	2	)	)	PUNCT
ejpam-3886	257	3	,	,	PUNCT
ejpam-3886	257	4	then	then	ADV
ejpam-3886	257	5	for	for	ADP
ejpam-3886	257	6	every	every	DET
ejpam-3886	257	7	f(x	f(x	PROPN
ejpam-3886	257	8	)	)	PUNCT
ejpam-3886	258	1	=	=	SYM
ejpam-3886	258	2	n∑	n∑	PROPN
ejpam-3886	258	3	j=0	j=0	PROPN
ejpam-3886	258	4	ajx	ajx	VERB
ejpam-3886	258	5	j	j	PROPN
ejpam-3886	258	6	∈	∈	PROPN
ejpam-3886	258	7	l	l	PROPN
ejpam-3886	258	8	,	,	PUNCT
ejpam-3886	258	9	we	we	PRON
ejpam-3886	258	10	have	have	VERB
ejpam-3886	258	11	ϕ(x)f(x	ϕ(x)f(x	NOUN
ejpam-3886	258	12	)	)	PUNCT
ejpam-3886	259	1	=	=	PRON
ejpam-3886	259	2	(	(	PUNCT
ejpam-3886	259	3	k∑	k∑	NOUN
ejpam-3886	259	4	i=0	i=0	PROPN
ejpam-3886	259	5	bix	bix	PROPN
ejpam-3886	259	6	i	i	PRON
ejpam-3886	259	7	)	)	PUNCT
ejpam-3886	260	1			PROPN
ejpam-3886	260	2	n∑	n∑	PROPN
ejpam-3886	260	3	j=0	j=0	PROPN
ejpam-3886	260	4	ajx	ajx	PROPN
ejpam-3886	260	5	j	j	PROPN
ejpam-3886	260	6			PROPN
ejpam-3886	260	7	∈	∈	PROPN
ejpam-3886	260	8	nil(a	nil(a	PROPN
ejpam-3886	260	9	)	)	PUNCT
ejpam-3886	260	10	.	.	PUNCT
ejpam-3886	261	1	since	since	SCONJ
ejpam-3886	261	2	r	r	NOUN
ejpam-3886	261	3	is	be	AUX
ejpam-3886	261	4	an	an	DET
ejpam-3886	261	5	(	(	PUNCT
ejpam-3886	261	6	α	α	NOUN
ejpam-3886	261	7	,	,	PUNCT
ejpam-3886	261	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	261	9	ni	ni	PROPN
ejpam-3886	261	10	ring	ring	NOUN
ejpam-3886	261	11	with	with	ADP
ejpam-3886	261	12	nil	nil	NOUN
ejpam-3886	261	13	(	(	PUNCT
ejpam-3886	261	14	r	r	NOUN
ejpam-3886	261	15	)	)	PUNCT
ejpam-3886	261	16	nilpotent	nilpotent	NOUN
ejpam-3886	261	17	,	,	PUNCT
ejpam-3886	261	18	from	from	ADP
ejpam-3886	261	19	lemma	lemma	PROPN
ejpam-3886	261	20	1	1	NUM
ejpam-3886	261	21	,	,	PUNCT
ejpam-3886	261	22	we	we	PRON
ejpam-3886	261	23	get	get	VERB
ejpam-3886	261	24	that	that	DET
ejpam-3886	261	25	biaj	biaj	PROPN
ejpam-3886	261	26	∈	∈	PROPN
ejpam-3886	261	27	nil(r	nil(r	PROPN
ejpam-3886	261	28	)	)	PUNCT
ejpam-3886	261	29	,	,	PUNCT
ejpam-3886	261	30	for	for	ADP
ejpam-3886	261	31	all	all	DET
ejpam-3886	261	32	integers	integer	NOUN
ejpam-3886	261	33	0	0	NUM
ejpam-3886	261	34	≤	≤	NUM
ejpam-3886	262	1	i	i	PRON
ejpam-3886	262	2	≤	≤	NOUN
ejpam-3886	263	1	k	k	PRON
ejpam-3886	263	2	and	and	CCONJ
ejpam-3886	263	3	0	0	NUM
ejpam-3886	263	4	≤	≤	NUM
ejpam-3886	263	5	j	j	PROPN
ejpam-3886	263	6	≤	≤	PROPN
ejpam-3886	263	7	n.	n.	NOUN
ejpam-3886	263	8	consequently	consequently	ADV
ejpam-3886	263	9	,	,	PUNCT
ejpam-3886	263	10	for	for	ADP
ejpam-3886	263	11	every	every	DET
ejpam-3886	263	12	a	a	DET
ejpam-3886	263	13	∈	∈	PROPN
ejpam-3886	263	14	i	i	NOUN
ejpam-3886	263	15	,	,	PUNCT
ejpam-3886	263	16	bia	bia	PROPN
ejpam-3886	263	17	∈	∈	PROPN
ejpam-3886	263	18	nil(r	nil(r	PROPN
ejpam-3886	263	19	)	)	PUNCT
ejpam-3886	263	20	,	,	PUNCT
ejpam-3886	263	21	for	for	ADP
ejpam-3886	263	22	all	all	DET
ejpam-3886	263	23	integers	integer	NOUN
ejpam-3886	263	24	0	0	NUM
ejpam-3886	263	25	≤	≤	NUM
ejpam-3886	263	26	i	i	PRON
ejpam-3886	263	27	≤	≤	PROPN
ejpam-3886	263	28	k.	k.	NOUN
ejpam-3886	263	29	for	for	ADP
ejpam-3886	263	30	any	any	DET
ejpam-3886	263	31	m	m	PROPN
ejpam-3886	263	32	∈	∈	PROPN
ejpam-3886	263	33	j	j	NOUN
ejpam-3886	263	34	,	,	PUNCT
ejpam-3886	263	35	there	there	PRON
ejpam-3886	263	36	exist	exist	VERB
ejpam-3886	263	37	a1	a1	NOUN
ejpam-3886	263	38	,	,	PUNCT
ejpam-3886	263	39	a2	a2	PROPN
ejpam-3886	263	40	,	,	PUNCT
ejpam-3886	263	41	...	...	PUNCT
ejpam-3886	263	42	,	,	PUNCT
ejpam-3886	263	43	an	an	DET
ejpam-3886	263	44	∈	∈	NOUN
ejpam-3886	263	45	i	i	PRON
ejpam-3886	263	46	and	and	CCONJ
ejpam-3886	263	47	r1	r1	PROPN
ejpam-3886	263	48	,	,	PUNCT
ejpam-3886	263	49	r2	r2	PROPN
ejpam-3886	263	50	,	,	PUNCT
ejpam-3886	263	51	...	...	PUNCT
ejpam-3886	263	52	,	,	PUNCT
ejpam-3886	263	53	rn	rn	PROPN
ejpam-3886	263	54	∈	∈	PROPN
ejpam-3886	263	55	r	r	NOUN
ejpam-3886	263	56	,	,	PUNCT
ejpam-3886	263	57	such	such	ADJ
ejpam-3886	263	58	that	that	DET
ejpam-3886	263	59	q	q	NOUN
ejpam-3886	264	1	=	=	X
ejpam-3886	264	2	a1r1	a1r1	PROPN
ejpam-3886	264	3	+	+	ADJ
ejpam-3886	264	4	a2r2	a2r2	X
ejpam-3886	264	5	+	+	NUM
ejpam-3886	264	6	...	...	PUNCT
ejpam-3886	264	7	+	+	NUM
ejpam-3886	264	8	anrn	anrn	NOUN
ejpam-3886	264	9	,	,	PUNCT
ejpam-3886	264	10	biq	biq	NOUN
ejpam-3886	264	11	=	=	SYM
ejpam-3886	264	12	(	(	PUNCT
ejpam-3886	264	13	bia1	bia1	PROPN
ejpam-3886	264	14	)	)	PUNCT
ejpam-3886	264	15	r1	r1	PROPN
ejpam-3886	264	16	+	+	CCONJ
ejpam-3886	264	17	(	(	PUNCT
ejpam-3886	264	18	bia2	bia2	PROPN
ejpam-3886	264	19	)	)	PUNCT
ejpam-3886	264	20	r2	r2	PROPN
ejpam-3886	264	21	+	+	CCONJ
ejpam-3886	264	22	...	...	PUNCT
ejpam-3886	265	1	+	+	CCONJ
ejpam-3886	265	2	(	(	PUNCT
ejpam-3886	265	3	bian	bian	ADJ
ejpam-3886	265	4	)	)	PUNCT
ejpam-3886	265	5	rn	rn	PROPN
ejpam-3886	265	6	,	,	PUNCT
ejpam-3886	265	7	hence	hence	ADV
ejpam-3886	265	8	biq	biq	PROPN
ejpam-3886	265	9	∈	∈	PROPN
ejpam-3886	265	10	nil(r	nil(r	PROPN
ejpam-3886	265	11	)	)	PUNCT
ejpam-3886	265	12	,	,	PUNCT
ejpam-3886	265	13	for	for	ADP
ejpam-3886	265	14	all	all	DET
ejpam-3886	265	15	integers	integer	NOUN
ejpam-3886	265	16	0	0	NUM
ejpam-3886	265	17	≤	≤	NUM
ejpam-3886	265	18	i	i	NOUN
ejpam-3886	265	19	≤	≤	NUM
ejpam-3886	266	1	k	k	X
ejpam-3886	266	2	,	,	PUNCT
ejpam-3886	266	3	so	so	CCONJ
ejpam-3886	266	4	bi	bi	PROPN
ejpam-3886	266	5	∈	∈	PROPN
ejpam-3886	266	6	nr	nr	PROPN
ejpam-3886	266	7	(	(	PUNCT
ejpam-3886	266	8	j	j	PROPN
ejpam-3886	266	9	)	)	PUNCT
ejpam-3886	266	10	=	=	SYM
ejpam-3886	266	11	re	re	PROPN
ejpam-3886	266	12	,	,	PUNCT
ejpam-3886	266	13	for	for	ADP
ejpam-3886	266	14	all	all	DET
ejpam-3886	266	15	integers	integer	NOUN
ejpam-3886	266	16	0	0	NUM
ejpam-3886	266	17	≤	≤	NUM
ejpam-3886	266	18	i	i	PRON
ejpam-3886	266	19	≤	≤	PROPN
ejpam-3886	266	20	k.	k.	X
ejpam-3886	267	1	therefore	therefore	ADV
ejpam-3886	267	2	there	there	PRON
ejpam-3886	267	3	exist	exist	VERB
ejpam-3886	267	4	ti	ti	NOUN
ejpam-3886	267	5	∈	∈	NOUN
ejpam-3886	267	6	r	r	NOUN
ejpam-3886	267	7	such	such	ADJ
ejpam-3886	267	8	that	that	DET
ejpam-3886	267	9	bi	bi	NOUN
ejpam-3886	267	10	=	=	NOUN
ejpam-3886	267	11	tie	tie	NOUN
ejpam-3886	267	12	,	,	PUNCT
ejpam-3886	267	13	for	for	ADP
ejpam-3886	267	14	all	all	DET
ejpam-3886	267	15	integers	integer	NOUN
ejpam-3886	267	16	0	0	NUM
ejpam-3886	267	17	≤	≤	NUM
ejpam-3886	267	18	i	i	PRON
ejpam-3886	267	19	≤	≤	PROPN
ejpam-3886	267	20	k.	k.	INTJ
ejpam-3886	268	1	since	since	SCONJ
ejpam-3886	268	2	for	for	ADP
ejpam-3886	268	3	any	any	DET
ejpam-3886	268	4	idempotent	idempotent	NOUN
ejpam-3886	268	5	e	e	NOUN
ejpam-3886	268	6	∈	∈	NOUN
ejpam-3886	268	7	r	r	NOUN
ejpam-3886	268	8	we	we	PRON
ejpam-3886	268	9	have	have	VERB
ejpam-3886	268	10	α(e	α(e	NOUN
ejpam-3886	268	11	)	)	PUNCT
ejpam-3886	268	12	=	=	SYM
ejpam-3886	268	13	e	e	NOUN
ejpam-3886	268	14	and	and	CCONJ
ejpam-3886	268	15	δ(e	δ(e	NOUN
ejpam-3886	268	16	)	)	PUNCT
ejpam-3886	268	17	=	=	SYM
ejpam-3886	268	18	0	0	NUM
ejpam-3886	268	19	,	,	PUNCT
ejpam-3886	268	20	we	we	PRON
ejpam-3886	268	21	can	can	AUX
ejpam-3886	268	22	conclude	conclude	VERB
ejpam-3886	268	23	that	that	PRON
ejpam-3886	268	24	ϕ(x	ϕ(x	X
ejpam-3886	268	25	)	)	PUNCT
ejpam-3886	268	26	=	=	SYM
ejpam-3886	269	1	k∑	k∑	PROPN
ejpam-3886	270	1	i=0	i=0	PROPN
ejpam-3886	270	2	bix	bix	PROPN
ejpam-3886	270	3	i	i	PRON
ejpam-3886	270	4	=	=	SYM
ejpam-3886	270	5	k∑	k∑	PROPN
ejpam-3886	271	1	i=0	i=0	PROPN
ejpam-3886	272	1	tiex	tiex	ADV
ejpam-3886	273	1	i	i	PRON
ejpam-3886	273	2	=	=	PUNCT
ejpam-3886	273	3	(	(	PUNCT
ejpam-3886	273	4	k∑	k∑	PROPN
ejpam-3886	273	5	i=0	i=0	AUX
ejpam-3886	273	6	tix	tix	VERB
ejpam-3886	273	7	i	i	PRON
ejpam-3886	273	8	)	)	PUNCT
ejpam-3886	274	1	e	e	X
ejpam-3886	274	2	∈	∈	PROPN
ejpam-3886	274	3	ah	ah	INTJ
ejpam-3886	274	4	,	,	PUNCT
ejpam-3886	274	5	where	where	SCONJ
ejpam-3886	274	6	h	h	NOUN
ejpam-3886	274	7	=	=	SYM
ejpam-3886	274	8	e	e	PROPN
ejpam-3886	274	9	=	=	PROPN
ejpam-3886	274	10	e2	e2	PROPN
ejpam-3886	274	11	=	=	SYM
ejpam-3886	274	12	h2	h2	PROPN
ejpam-3886	274	13	∈	∈	PROPN
ejpam-3886	274	14	a.	a.	NOUN
ejpam-3886	274	15	therefore	therefore	ADV
ejpam-3886	274	16	na	na	X
ejpam-3886	274	17	(	(	PUNCT
ejpam-3886	274	18	l	l	NOUN
ejpam-3886	274	19	)	)	PUNCT
ejpam-3886	274	20	=	=	SYM
ejpam-3886	274	21	ah	ah	INTJ
ejpam-3886	274	22	,	,	PUNCT
ejpam-3886	274	23	where	where	SCONJ
ejpam-3886	274	24	h	h	PROPN
ejpam-3886	274	25	∈	∈	PROPN
ejpam-3886	274	26	id(a	id(a	PROPN
ejpam-3886	274	27	)	)	PUNCT
ejpam-3886	274	28	and	and	CCONJ
ejpam-3886	274	29	the	the	DET
ejpam-3886	274	30	result	result	NOUN
ejpam-3886	274	31	is	be	AUX
ejpam-3886	274	32	proved	prove	VERB
ejpam-3886	274	33	.	.	PUNCT
ejpam-3886	275	1	theorem	theorem	NOUN
ejpam-3886	275	2	2	2	NUM
ejpam-3886	275	3	.	.	PUNCT
ejpam-3886	276	1	let	let	VERB
ejpam-3886	276	2	r	r	PRON
ejpam-3886	276	3	be	be	AUX
ejpam-3886	276	4	an	an	DET
ejpam-3886	276	5	(	(	PUNCT
ejpam-3886	276	6	α	α	NOUN
ejpam-3886	276	7	,	,	PUNCT
ejpam-3886	276	8	δ)-compatible	δ)-compatible	ADJ
ejpam-3886	276	9	ni	ni	PROPN
ejpam-3886	276	10	ring	ring	NOUN
ejpam-3886	276	11	with	with	ADP
ejpam-3886	276	12	nil	nil	NOUN
ejpam-3886	276	13	(	(	PUNCT
ejpam-3886	276	14	r	r	NOUN
ejpam-3886	276	15	)	)	PUNCT
ejpam-3886	276	16	nilpotent	nilpotent	NOUN
ejpam-3886	276	17	.	.	PUNCT
ejpam-3886	277	1	if	if	SCONJ
ejpam-3886	277	2	r	r	NOUN
ejpam-3886	277	3	is	be	AUX
ejpam-3886	277	4	a	a	DET
ejpam-3886	277	5	weak	weak	ADJ
ejpam-3886	277	6	left	left	ADJ
ejpam-3886	277	7	ps	ps	NOUN
ejpam-3886	277	8	-	-	PUNCT
ejpam-3886	277	9	ring	ring	NOUN
ejpam-3886	277	10	,	,	PUNCT
ejpam-3886	277	11	then	then	ADV
ejpam-3886	277	12	a	a	DET
ejpam-3886	277	13	=	=	X
ejpam-3886	277	14	r	r	NOUN
ejpam-3886	277	15	[	[	X
ejpam-3886	277	16	x;α	x;α	PROPN
ejpam-3886	277	17	,	,	PUNCT
ejpam-3886	277	18	δ	δ	PROPN
ejpam-3886	277	19	]	]	PUNCT
ejpam-3886	277	20	is	be	AUX
ejpam-3886	277	21	a	a	DET
ejpam-3886	277	22	weak	weak	ADJ
ejpam-3886	277	23	left	left	ADJ
ejpam-3886	277	24	ps	ps	NOUN
ejpam-3886	277	25	-	-	PUNCT
ejpam-3886	277	26	ring	ring	NOUN
ejpam-3886	277	27	.	.	PUNCT
ejpam-3886	278	1	proof	proof	NOUN
ejpam-3886	278	2	.	.	PUNCT
ejpam-3886	279	1	the	the	DET
ejpam-3886	279	2	proof	proof	NOUN
ejpam-3886	279	3	is	be	AUX
ejpam-3886	279	4	similar	similar	ADJ
ejpam-3886	279	5	to	to	ADP
ejpam-3886	279	6	the	the	DET
ejpam-3886	279	7	previous	previous	ADJ
ejpam-3886	279	8	proof	proof	NOUN
ejpam-3886	279	9	of	of	ADP
ejpam-3886	279	10	theorem	theorem	NOUN
ejpam-3886	279	11	1	1	NUM
ejpam-3886	279	12	.	.	PUNCT
ejpam-3886	280	1	the	the	DET
ejpam-3886	280	2	only	only	ADJ
ejpam-3886	280	3	thing	thing	NOUN
ejpam-3886	280	4	we	we	PRON
ejpam-3886	280	5	need	need	VERB
ejpam-3886	280	6	to	to	PART
ejpam-3886	280	7	note	note	VERB
ejpam-3886	280	8	here	here	ADV
ejpam-3886	280	9	is	be	AUX
ejpam-3886	280	10	that	that	SCONJ
ejpam-3886	280	11	,	,	PUNCT
ejpam-3886	280	12	if	if	SCONJ
ejpam-3886	280	13	l	l	NOUN
ejpam-3886	280	14	is	be	AUX
ejpam-3886	280	15	a	a	DET
ejpam-3886	280	16	maximal	maximal	ADJ
ejpam-3886	280	17	left	leave	VERB
ejpam-3886	280	18	ideal	ideal	NOUN
ejpam-3886	280	19	of	of	ADP
ejpam-3886	280	20	a	a	DET
ejpam-3886	280	21	=	=	SYM
ejpam-3886	280	22	r	r	NOUN
ejpam-3886	281	1	[	[	X
ejpam-3886	281	2	x;α	x;α	PROPN
ejpam-3886	281	3	,	,	PUNCT
ejpam-3886	281	4	δ	δ	PROPN
ejpam-3886	281	5	]	]	PUNCT
ejpam-3886	281	6	,	,	PUNCT
ejpam-3886	281	7	then	then	ADV
ejpam-3886	281	8	,	,	PUNCT
ejpam-3886	281	9	by	by	ADP
ejpam-3886	281	10	analogue	analogue	NOUN
ejpam-3886	281	11	manner	manner	NOUN
ejpam-3886	281	12	as	as	ADP
ejpam-3886	281	13	above	above	ADV
ejpam-3886	281	14	,	,	PUNCT
ejpam-3886	281	15	we	we	PRON
ejpam-3886	281	16	get	get	VERB
ejpam-3886	281	17	in	in	ADP
ejpam-3886	281	18	case	case	NOUN
ejpam-3886	281	19	(	(	PUNCT
ejpam-3886	281	20	2	2	NUM
ejpam-3886	281	21	)	)	PUNCT
ejpam-3886	282	1	that	that	SCONJ
ejpam-3886	282	2	bi	bi	PROPN
ejpam-3886	282	3	∈	∈	PROPN
ejpam-3886	282	4	nr	nr	PROPN
ejpam-3886	282	5	(	(	PUNCT
ejpam-3886	282	6	j	j	PROPN
ejpam-3886	282	7	)	)	PUNCT
ejpam-3886	282	8	=	=	SYM
ejpam-3886	282	9	re	re	PROPN
ejpam-3886	282	10	,	,	PUNCT
ejpam-3886	282	11	for	for	ADP
ejpam-3886	282	12	all	all	DET
ejpam-3886	282	13	integers	integer	NOUN
ejpam-3886	282	14	0	0	NUM
ejpam-3886	282	15	≤	≤	NUM
ejpam-3886	282	16	i	i	PRON
ejpam-3886	282	17	≤	≤	PROPN
ejpam-3886	282	18	k.	k.	X
ejpam-3886	282	19	therefore	therefore	ADV
ejpam-3886	282	20	there	there	PRON
ejpam-3886	282	21	exist	exist	VERB
ejpam-3886	282	22	ti	ti	NOUN
ejpam-3886	282	23	∈	∈	NOUN
ejpam-3886	282	24	r	r	NOUN
ejpam-3886	282	25	such	such	ADJ
ejpam-3886	282	26	that	that	DET
ejpam-3886	282	27	bi	bi	NOUN
ejpam-3886	282	28	=	=	PROPN
ejpam-3886	282	29	eti	eti	PROPN
ejpam-3886	282	30	,	,	PUNCT
ejpam-3886	282	31	for	for	ADP
ejpam-3886	282	32	all	all	DET
ejpam-3886	282	33	integers	integer	NOUN
ejpam-3886	282	34	0	0	NUM
ejpam-3886	282	35	≤	≤	NUM
ejpam-3886	282	36	i	i	PRON
ejpam-3886	282	37	≤	≤	PROPN
ejpam-3886	282	38	k.	k.	PROPN
ejpam-3886	283	1	so	so	ADV
ejpam-3886	283	2	ϕ(x	ϕ(x	PROPN
ejpam-3886	283	3	)	)	PUNCT
ejpam-3886	284	1	=	=	SYM
ejpam-3886	284	2	k∑	k∑	PROPN
ejpam-3886	285	1	i=0	i=0	PROPN
ejpam-3886	285	2	bix	bix	PROPN
ejpam-3886	285	3	i	i	PRON
ejpam-3886	285	4	=	=	SYM
ejpam-3886	285	5	k∑	k∑	PROPN
ejpam-3886	286	1	i=0	i=0	ADJ
ejpam-3886	286	2	etix	etix	NOUN
ejpam-3886	286	3	i	i	PRON
ejpam-3886	286	4	=	=	SYM
ejpam-3886	286	5	e	e	X
ejpam-3886	286	6	(	(	PUNCT
ejpam-3886	286	7	k∑	k∑	PROPN
ejpam-3886	286	8	i=0	i=0	AUX
ejpam-3886	286	9	tix	tix	VERB
ejpam-3886	286	10	i	i	PRON
ejpam-3886	286	11	)	)	PUNCT
ejpam-3886	286	12	∈	∈	PROPN
ejpam-3886	287	1	ha	ha	INTJ
ejpam-3886	287	2	,	,	PUNCT
ejpam-3886	287	3	where	where	SCONJ
ejpam-3886	287	4	h	h	NOUN
ejpam-3886	287	5	=	=	SYM
ejpam-3886	287	6	e	e	PROPN
ejpam-3886	287	7	=	=	PROPN
ejpam-3886	287	8	e2	e2	PROPN
ejpam-3886	287	9	=	=	SYM
ejpam-3886	287	10	h2	h2	PROPN
ejpam-3886	287	11	∈	∈	PROPN
ejpam-3886	287	12	a.	a.	NOUN
ejpam-3886	287	13	therefore	therefore	ADV
ejpam-3886	287	14	na	na	X
ejpam-3886	287	15	(	(	PUNCT
ejpam-3886	287	16	l	l	NOUN
ejpam-3886	287	17	)	)	PUNCT
ejpam-3886	287	18	=	=	SYM
ejpam-3886	288	1	ha	ha	INTJ
ejpam-3886	288	2	,	,	PUNCT
ejpam-3886	288	3	where	where	SCONJ
ejpam-3886	288	4	h	h	PROPN
ejpam-3886	288	5	∈	∈	PROPN
ejpam-3886	288	6	id(a	id(a	PROPN
ejpam-3886	288	7	)	)	PUNCT
ejpam-3886	288	8	and	and	CCONJ
ejpam-3886	288	9	the	the	DET
ejpam-3886	288	10	result	result	NOUN
ejpam-3886	288	11	is	be	AUX
ejpam-3886	288	12	proved	prove	VERB
ejpam-3886	288	13	.	.	PUNCT
ejpam-3886	289	1	references	reference	NOUN
ejpam-3886	289	2	172	172	NUM
ejpam-3886	289	3	assume	assume	VERB
ejpam-3886	289	4	that	that	SCONJ
ejpam-3886	289	5	δ	δ	PROPN
ejpam-3886	289	6	is	be	AUX
ejpam-3886	289	7	the	the	DET
ejpam-3886	289	8	zero	zero	NUM
ejpam-3886	289	9	map	map	NOUN
ejpam-3886	289	10	,	,	PUNCT
ejpam-3886	289	11	then	then	ADV
ejpam-3886	289	12	a	a	DET
ejpam-3886	289	13	=	=	X
ejpam-3886	289	14	r	r	NOUN
ejpam-3886	290	1	[	[	X
ejpam-3886	290	2	x;α	x;α	X
ejpam-3886	290	3	]	]	PUNCT
ejpam-3886	290	4	,	,	PUNCT
ejpam-3886	290	5	the	the	DET
ejpam-3886	290	6	usual	usual	ADJ
ejpam-3886	290	7	skew	skew	ADJ
ejpam-3886	290	8	polynomial	polynomial	ADJ
ejpam-3886	290	9	ring	ring	NOUN
ejpam-3886	290	10	over	over	ADP
ejpam-3886	290	11	r	r	NOUN
ejpam-3886	290	12	,	,	PUNCT
ejpam-3886	290	13	and	and	CCONJ
ejpam-3886	290	14	we	we	PRON
ejpam-3886	290	15	get	get	VERB
ejpam-3886	290	16	the	the	DET
ejpam-3886	290	17	following	follow	VERB
ejpam-3886	290	18	corollaries	corollary	NOUN
ejpam-3886	290	19	:	:	PUNCT
ejpam-3886	291	1	corollary	corollary	ADJ
ejpam-3886	291	2	2	2	X
ejpam-3886	291	3	.	.	PUNCT
ejpam-3886	292	1	let	let	VERB
ejpam-3886	292	2	r	r	PRON
ejpam-3886	292	3	be	be	AUX
ejpam-3886	292	4	an	an	DET
ejpam-3886	292	5	α	α	NOUN
ejpam-3886	292	6	-	-	ADJ
ejpam-3886	292	7	compatible	compatible	ADJ
ejpam-3886	292	8	ni	ni	NOUN
ejpam-3886	292	9	ring	ring	NOUN
ejpam-3886	292	10	with	with	ADP
ejpam-3886	292	11	nil	nil	NOUN
ejpam-3886	292	12	(	(	PUNCT
ejpam-3886	292	13	r	r	NOUN
ejpam-3886	292	14	)	)	PUNCT
ejpam-3886	292	15	nilpotent	nilpotent	NOUN
ejpam-3886	292	16	,	,	PUNCT
ejpam-3886	292	17	such	such	ADJ
ejpam-3886	292	18	that	that	SCONJ
ejpam-3886	292	19	α(e	α(e	NOUN
ejpam-3886	292	20	)	)	PUNCT
ejpam-3886	293	1	=	=	PUNCT
ejpam-3886	293	2	e	e	NOUN
ejpam-3886	293	3	for	for	ADP
ejpam-3886	293	4	every	every	DET
ejpam-3886	293	5	e	e	PROPN
ejpam-3886	293	6	∈	∈	PROPN
ejpam-3886	293	7	id(r	id(r	NOUN
ejpam-3886	293	8	)	)	PUNCT
ejpam-3886	293	9	.	.	PUNCT
ejpam-3886	294	1	if	if	SCONJ
ejpam-3886	294	2	r	r	NOUN
ejpam-3886	294	3	is	be	AUX
ejpam-3886	294	4	a	a	DET
ejpam-3886	294	5	weak	weak	ADJ
ejpam-3886	294	6	right	right	ADJ
ejpam-3886	294	7	ps	ps	NOUN
ejpam-3886	294	8	-	-	NOUN
ejpam-3886	294	9	ring	ring	NOUN
ejpam-3886	294	10	,	,	PUNCT
ejpam-3886	294	11	then	then	ADV
ejpam-3886	294	12	a	a	DET
ejpam-3886	294	13	=	=	X
ejpam-3886	294	14	r	r	NOUN
ejpam-3886	294	15	[	[	X
ejpam-3886	294	16	x;α	x;α	X
ejpam-3886	294	17	]	]	X
ejpam-3886	294	18	is	be	AUX
ejpam-3886	294	19	a	a	DET
ejpam-3886	294	20	weak	weak	ADJ
ejpam-3886	294	21	right	right	ADJ
ejpam-3886	294	22	ps	ps	NOUN
ejpam-3886	294	23	-	-	PUNCT
ejpam-3886	294	24	ring	ring	NOUN
ejpam-3886	294	25	.	.	PUNCT
ejpam-3886	295	1	corollary	corollary	ADJ
ejpam-3886	295	2	3	3	NUM
ejpam-3886	295	3	.	.	PUNCT
ejpam-3886	296	1	let	let	VERB
ejpam-3886	296	2	r	r	PRON
ejpam-3886	296	3	be	be	AUX
ejpam-3886	296	4	an	an	DET
ejpam-3886	296	5	α	α	NOUN
ejpam-3886	296	6	-	-	ADJ
ejpam-3886	296	7	compatible	compatible	ADJ
ejpam-3886	296	8	ni	ni	NOUN
ejpam-3886	296	9	ring	ring	NOUN
ejpam-3886	296	10	with	with	ADP
ejpam-3886	296	11	nil	nil	NOUN
ejpam-3886	296	12	(	(	PUNCT
ejpam-3886	296	13	r	r	NOUN
ejpam-3886	296	14	)	)	PUNCT
ejpam-3886	296	15	nilpotent	nilpotent	NOUN
ejpam-3886	296	16	.	.	PUNCT
ejpam-3886	297	1	if	if	SCONJ
ejpam-3886	297	2	r	r	NOUN
ejpam-3886	297	3	is	be	AUX
ejpam-3886	297	4	a	a	DET
ejpam-3886	297	5	weak	weak	ADJ
ejpam-3886	297	6	left	left	ADJ
ejpam-3886	297	7	ps	ps	NOUN
ejpam-3886	297	8	-	-	PUNCT
ejpam-3886	297	9	ring	ring	NOUN
ejpam-3886	297	10	,	,	PUNCT
ejpam-3886	297	11	then	then	ADV
ejpam-3886	297	12	a	a	DET
ejpam-3886	297	13	=	=	X
ejpam-3886	297	14	r	r	NOUN
ejpam-3886	297	15	[	[	X
ejpam-3886	297	16	x;α	x;α	X
ejpam-3886	297	17	]	]	X
ejpam-3886	297	18	is	be	AUX
ejpam-3886	297	19	a	a	DET
ejpam-3886	297	20	weak	weak	ADJ
ejpam-3886	297	21	left	left	ADJ
ejpam-3886	297	22	ps	ps	NOUN
ejpam-3886	297	23	-	-	PUNCT
ejpam-3886	297	24	ring	ring	NOUN
ejpam-3886	297	25	.	.	PUNCT
ejpam-3886	298	1	assume	assume	VERB
ejpam-3886	298	2	that	that	SCONJ
ejpam-3886	298	3	α	α	PRON
ejpam-3886	298	4	is	be	AUX
ejpam-3886	298	5	the	the	DET
ejpam-3886	298	6	identity	identity	NOUN
ejpam-3886	298	7	map	map	NOUN
ejpam-3886	298	8	,	,	PUNCT
ejpam-3886	298	9	then	then	ADV
ejpam-3886	298	10	a	a	PRON
ejpam-3886	298	11	=	=	SYM
ejpam-3886	299	1	r	r	X
ejpam-3886	299	2	[	[	X
ejpam-3886	299	3	x	x	X
ejpam-3886	299	4	;	;	PUNCT
ejpam-3886	299	5	δ	δ	X
ejpam-3886	299	6	]	]	PUNCT
ejpam-3886	299	7	,	,	PUNCT
ejpam-3886	299	8	the	the	DET
ejpam-3886	299	9	differential	differential	ADJ
ejpam-3886	299	10	polynomial	polynomial	ADJ
ejpam-3886	299	11	ring	ring	NOUN
ejpam-3886	299	12	over	over	ADP
ejpam-3886	299	13	r	r	NOUN
ejpam-3886	299	14	,	,	PUNCT
ejpam-3886	299	15	and	and	CCONJ
ejpam-3886	299	16	we	we	PRON
ejpam-3886	299	17	get	get	VERB
ejpam-3886	299	18	the	the	DET
ejpam-3886	299	19	following	follow	VERB
ejpam-3886	299	20	corollaries	corollary	NOUN
ejpam-3886	299	21	:	:	PUNCT
ejpam-3886	299	22	corollary	corollary	ADJ
ejpam-3886	299	23	4	4	X
ejpam-3886	299	24	.	.	PUNCT
ejpam-3886	300	1	let	let	VERB
ejpam-3886	300	2	r	r	PRON
ejpam-3886	300	3	be	be	AUX
ejpam-3886	300	4	a	a	DET
ejpam-3886	300	5	δ	δ	NOUN
ejpam-3886	300	6	-	-	PUNCT
ejpam-3886	300	7	compatible	compatible	ADJ
ejpam-3886	300	8	ni	ni	NOUN
ejpam-3886	300	9	ring	ring	NOUN
ejpam-3886	300	10	with	with	ADP
ejpam-3886	300	11	nil	nil	NOUN
ejpam-3886	300	12	(	(	PUNCT
ejpam-3886	300	13	r	r	NOUN
ejpam-3886	300	14	)	)	PUNCT
ejpam-3886	300	15	nilpotent	nilpotent	NOUN
ejpam-3886	300	16	,	,	PUNCT
ejpam-3886	300	17	such	such	ADJ
ejpam-3886	300	18	that	that	SCONJ
ejpam-3886	300	19	δ(e	δ(e	ADJ
ejpam-3886	300	20	)	)	PUNCT
ejpam-3886	300	21	=	=	SYM
ejpam-3886	300	22	0	0	NUM
ejpam-3886	300	23	for	for	ADP
ejpam-3886	300	24	every	every	DET
ejpam-3886	300	25	e	e	PROPN
ejpam-3886	300	26	∈	∈	PROPN
ejpam-3886	300	27	id(r	id(r	NOUN
ejpam-3886	300	28	)	)	PUNCT
ejpam-3886	300	29	.	.	PUNCT
ejpam-3886	301	1	if	if	SCONJ
ejpam-3886	301	2	r	r	NOUN
ejpam-3886	301	3	is	be	AUX
ejpam-3886	301	4	a	a	DET
ejpam-3886	301	5	weak	weak	ADJ
ejpam-3886	301	6	right	right	ADJ
ejpam-3886	301	7	ps	ps	NOUN
ejpam-3886	301	8	-	-	NOUN
ejpam-3886	301	9	ring	ring	NOUN
ejpam-3886	301	10	,	,	PUNCT
ejpam-3886	301	11	then	then	ADV
ejpam-3886	301	12	a	a	DET
ejpam-3886	301	13	=	=	SYM
ejpam-3886	301	14	r	r	X
ejpam-3886	301	15	[	[	X
ejpam-3886	301	16	x	x	X
ejpam-3886	301	17	;	;	PUNCT
ejpam-3886	301	18	δ	δ	X
ejpam-3886	301	19	]	]	X
ejpam-3886	301	20	is	be	AUX
ejpam-3886	301	21	a	a	DET
ejpam-3886	301	22	weak	weak	ADJ
ejpam-3886	301	23	right	right	ADJ
ejpam-3886	301	24	ps	ps	NOUN
ejpam-3886	301	25	-	-	PUNCT
ejpam-3886	301	26	ring	ring	NOUN
ejpam-3886	301	27	.	.	PUNCT
ejpam-3886	302	1	corollary	corollary	ADJ
ejpam-3886	302	2	5	5	NUM
ejpam-3886	302	3	.	.	PUNCT
ejpam-3886	303	1	let	let	VERB
ejpam-3886	303	2	r	r	PRON
ejpam-3886	303	3	be	be	AUX
ejpam-3886	303	4	a	a	DET
ejpam-3886	303	5	δ	δ	NOUN
ejpam-3886	303	6	-	-	PUNCT
ejpam-3886	303	7	compatible	compatible	ADJ
ejpam-3886	303	8	ni	ni	NOUN
ejpam-3886	303	9	ring	ring	NOUN
ejpam-3886	303	10	with	with	ADP
ejpam-3886	303	11	nil	nil	NOUN
ejpam-3886	303	12	(	(	PUNCT
ejpam-3886	303	13	r	r	NOUN
ejpam-3886	303	14	)	)	PUNCT
ejpam-3886	303	15	nilpotent	nilpotent	NOUN
ejpam-3886	303	16	.	.	PUNCT
ejpam-3886	304	1	if	if	SCONJ
ejpam-3886	304	2	r	r	NOUN
ejpam-3886	304	3	is	be	AUX
ejpam-3886	304	4	a	a	DET
ejpam-3886	304	5	weak	weak	ADJ
ejpam-3886	304	6	left	left	ADJ
ejpam-3886	304	7	ps	ps	NOUN
ejpam-3886	304	8	-	-	PUNCT
ejpam-3886	304	9	ring	ring	NOUN
ejpam-3886	304	10	,	,	PUNCT
ejpam-3886	304	11	then	then	ADV
ejpam-3886	304	12	a	a	DET
ejpam-3886	304	13	=	=	SYM
ejpam-3886	304	14	r	r	X
ejpam-3886	304	15	[	[	X
ejpam-3886	304	16	x	x	X
ejpam-3886	304	17	;	;	PUNCT
ejpam-3886	304	18	δ	δ	X
ejpam-3886	304	19	]	]	X
ejpam-3886	304	20	is	be	AUX
ejpam-3886	304	21	a	a	DET
ejpam-3886	304	22	weak	weak	ADJ
ejpam-3886	304	23	left	left	ADJ
ejpam-3886	304	24	ps	ps	NOUN
ejpam-3886	304	25	-	-	PUNCT
ejpam-3886	304	26	ring	ring	NOUN
ejpam-3886	304	27	.	.	PUNCT
ejpam-3886	305	1	assume	assume	VERB
ejpam-3886	305	2	that	that	SCONJ
ejpam-3886	305	3	α	α	PRON
ejpam-3886	305	4	is	be	AUX
ejpam-3886	305	5	the	the	DET
ejpam-3886	305	6	identity	identity	NOUN
ejpam-3886	305	7	map	map	NOUN
ejpam-3886	305	8	and	and	CCONJ
ejpam-3886	305	9	δ	δ	PROPN
ejpam-3886	305	10	is	be	AUX
ejpam-3886	305	11	the	the	DET
ejpam-3886	305	12	zero	zero	NUM
ejpam-3886	305	13	map	map	NOUN
ejpam-3886	305	14	,	,	PUNCT
ejpam-3886	305	15	then	then	ADV
ejpam-3886	305	16	a	a	DET
ejpam-3886	305	17	=	=	SYM
ejpam-3886	305	18	r	r	X
ejpam-3886	306	1	[	[	X
ejpam-3886	306	2	x	x	X
ejpam-3886	306	3	]	]	X
ejpam-3886	306	4	,	,	PUNCT
ejpam-3886	306	5	the	the	DET
ejpam-3886	306	6	usual	usual	ADJ
ejpam-3886	306	7	polynomial	polynomial	ADJ
ejpam-3886	306	8	ring	ring	NOUN
ejpam-3886	306	9	over	over	ADP
ejpam-3886	306	10	r	r	NOUN
ejpam-3886	306	11	,	,	PUNCT
ejpam-3886	306	12	and	and	CCONJ
ejpam-3886	306	13	we	we	PRON
ejpam-3886	306	14	get	get	VERB
ejpam-3886	306	15	the	the	DET
ejpam-3886	306	16	following	follow	VERB
ejpam-3886	306	17	corollary	corollary	ADJ
ejpam-3886	306	18	:	:	PUNCT
ejpam-3886	306	19	corollary	corollary	ADJ
ejpam-3886	306	20	6	6	NUM
ejpam-3886	306	21	.	.	PUNCT
ejpam-3886	307	1	let	let	VERB
ejpam-3886	307	2	r	r	PRON
ejpam-3886	307	3	be	be	AUX
ejpam-3886	307	4	an	an	DET
ejpam-3886	307	5	ni	ni	NOUN
ejpam-3886	307	6	ring	ring	NOUN
ejpam-3886	307	7	with	with	ADP
ejpam-3886	307	8	nil	nil	NOUN
ejpam-3886	307	9	(	(	PUNCT
ejpam-3886	307	10	r	r	NOUN
ejpam-3886	307	11	)	)	PUNCT
ejpam-3886	307	12	nilpotent	nilpotent	NOUN
ejpam-3886	307	13	.	.	PUNCT
ejpam-3886	308	1	if	if	SCONJ
ejpam-3886	308	2	r	r	NOUN
ejpam-3886	308	3	is	be	AUX
ejpam-3886	308	4	a	a	DET
ejpam-3886	308	5	weak	weak	ADJ
ejpam-3886	308	6	right	right	NOUN
ejpam-3886	308	7	(	(	PUNCT
ejpam-3886	308	8	left	left	ADJ
ejpam-3886	308	9	)	)	PUNCT
ejpam-3886	308	10	ps	ps	NOUN
ejpam-3886	308	11	-	-	PUNCT
ejpam-3886	308	12	ring	ring	NOUN
ejpam-3886	308	13	,	,	PUNCT
ejpam-3886	308	14	then	then	ADV
ejpam-3886	308	15	a	a	DET
ejpam-3886	308	16	=	=	SYM
ejpam-3886	308	17	r	r	NOUN
ejpam-3886	308	18	[	[	X
ejpam-3886	308	19	x	x	X
ejpam-3886	308	20	]	]	X
ejpam-3886	308	21	is	be	AUX
ejpam-3886	308	22	a	a	DET
ejpam-3886	308	23	weak	weak	ADJ
ejpam-3886	308	24	right	right	NOUN
ejpam-3886	308	25	(	(	PUNCT
ejpam-3886	308	26	left	left	ADJ
ejpam-3886	308	27	)	)	PUNCT
ejpam-3886	308	28	ps	ps	NOUN
ejpam-3886	308	29	-	-	PUNCT
ejpam-3886	308	30	ring	ring	NOUN
ejpam-3886	308	31	.	.	PUNCT
ejpam-3886	309	1	references	reference	NOUN
ejpam-3886	309	2	[	[	X
ejpam-3886	309	3	1	1	X
ejpam-3886	309	4	]	]	PUNCT
ejpam-3886	309	5	s.	s.	PROPN
ejpam-3886	309	6	annin	annin	PROPN
ejpam-3886	309	7	.	.	PROPN
ejpam-3886	310	1	cassociated	cassociate	VERB
ejpam-3886	310	2	primes	prime	NOUN
ejpam-3886	310	3	over	over	ADP
ejpam-3886	310	4	ore	ore	NOUN
ejpam-3886	310	5	extension	extension	NOUN
ejpam-3886	310	6	rings	ring	NOUN
ejpam-3886	310	7	.	.	PUNCT
ejpam-3886	311	1	j.	j.	PROPN
ejpam-3886	311	2	alg	alg	PROPN
ejpam-3886	311	3	.	.	PUNCT
ejpam-3886	312	1	appl	appl	PROPN
ejpam-3886	312	2	.	.	PROPN
ejpam-3886	312	3	,	,	PUNCT
ejpam-3886	312	4	3:193–205	3:193–205	NUM
ejpam-3886	312	5	,	,	PUNCT
ejpam-3886	312	6	2004	2004	NUM
ejpam-3886	312	7	.	.	PUNCT
ejpam-3886	313	1	[	[	X
ejpam-3886	313	2	2	2	NUM
ejpam-3886	313	3	]	]	PUNCT
ejpam-3886	313	4	m.	m.	NOUN
ejpam-3886	313	5	farahat	farahat	PROPN
ejpam-3886	313	6	and	and	CCONJ
ejpam-3886	313	7	s.	s.	PROPN
ejpam-3886	313	8	al	al	PROPN
ejpam-3886	313	9	-	-	PUNCT
ejpam-3886	313	10	bogamy	bogamy	PROPN
ejpam-3886	313	11	.	.	PUNCT
ejpam-3886	313	12	ps	ps	NOUN
ejpam-3886	313	13	-	-	PUNCT
ejpam-3886	313	14	ringss	ringss	NOUN
ejpam-3886	313	15	over	over	ADP
ejpam-3886	313	16	skew	skew	ADJ
ejpam-3886	313	17	hurwitz	hurwitz	PROPN
ejpam-3886	313	18	series	series	PROPN
ejpam-3886	313	19	.	.	PUNCT
ejpam-3886	314	1	eur	eur	PROPN
ejpam-3886	314	2	.	.	PUNCT
ejpam-3886	315	1	j.	j.	PROPN
ejpam-3886	315	2	pure	pure	PROPN
ejpam-3886	315	3	appl	appl	PROPN
ejpam-3886	315	4	.	.	PUNCT
ejpam-3886	315	5	math	math	PROPN
ejpam-3886	315	6	.	.	PUNCT
ejpam-3886	315	7	,	,	PUNCT
ejpam-3886	315	8	11(1):244–259	11(1):244–259	NOUN
ejpam-3886	315	9	,	,	PUNCT
ejpam-3886	315	10	2018	2018	NUM
ejpam-3886	315	11	.	.	PUNCT
ejpam-3886	316	1	[	[	X
ejpam-3886	316	2	3	3	X
ejpam-3886	316	3	]	]	PUNCT
ejpam-3886	316	4	m.	m.	NOUN
ejpam-3886	316	5	farahat	farahat	PROPN
ejpam-3886	316	6	and	and	CCONJ
ejpam-3886	316	7	n.	n.	PROPN
ejpam-3886	316	8	al	al	PROPN
ejpam-3886	316	9	-	-	PUNCT
ejpam-3886	316	10	harthy	harthy	ADJ
ejpam-3886	316	11	.	.	PUNCT
ejpam-3886	317	1	ps	ps	NOUN
ejpam-3886	317	2	-	-	PUNCT
ejpam-3886	317	3	modules	module	NOUN
ejpam-3886	317	4	of	of	ADP
ejpam-3886	317	5	generalized	generalized	ADJ
ejpam-3886	317	6	mal’cev	mal’cev	PROPN
ejpam-3886	317	7	-	-	PUNCT
ejpam-3886	317	8	neumann	neumann	PROPN
ejpam-3886	317	9	series	series	PROPN
ejpam-3886	317	10	rings	rings	PROPN
ejpam-3886	317	11	.	.	PUNCT
ejpam-3886	318	1	hacet	hacet	PROPN
ejpam-3886	318	2	.	.	PUNCT
ejpam-3886	319	1	j.	j.	PROPN
ejpam-3886	319	2	math	math	PROPN
ejpam-3886	319	3	.	.	PUNCT
ejpam-3886	320	1	stat	stat	PROPN
ejpam-3886	320	2	.	.	PUNCT
ejpam-3886	320	3	,	,	PUNCT
ejpam-3886	320	4	46(5):1–6	46(5):1–6	PROPN
ejpam-3886	320	5	,	,	PUNCT
ejpam-3886	320	6	2017	2017	NUM
ejpam-3886	320	7	.	.	PUNCT
ejpam-3886	321	1	[	[	X
ejpam-3886	321	2	4	4	NUM
ejpam-3886	321	3	]	]	X
ejpam-3886	321	4	c.	c.	PROPN
ejpam-3886	321	5	hong	hong	PROPN
ejpam-3886	321	6	,	,	PUNCT
ejpam-3886	321	7	n.	n.	PROPN
ejpam-3886	321	8	kim	kim	PROPN
ejpam-3886	321	9	,	,	PUNCT
ejpam-3886	321	10	and	and	CCONJ
ejpam-3886	321	11	t.	t.	PROPN
ejpam-3886	321	12	kwak	kwak	PROPN
ejpam-3886	321	13	.	.	PUNCT
ejpam-3886	322	1	ore	ore	NOUN
ejpam-3886	322	2	extensions	extension	NOUN
ejpam-3886	322	3	of	of	ADP
ejpam-3886	322	4	baer	baer	PROPN
ejpam-3886	322	5	and	and	CCONJ
ejpam-3886	322	6	p.p.-rings	p.p.-ring	NOUN
ejpam-3886	322	7	.	.	PUNCT
ejpam-3886	323	1	j.	j.	PROPN
ejpam-3886	323	2	pure	pure	PROPN
ejpam-3886	323	3	and	and	CCONJ
ejpam-3886	323	4	appl	appl	PROPN
ejpam-3886	323	5	.	.	PUNCT
ejpam-3886	323	6	alg	alg	PROPN
ejpam-3886	323	7	.	.	PROPN
ejpam-3886	323	8	,	,	PUNCT
ejpam-3886	323	9	151:215–226	151:215–226	NUM
ejpam-3886	323	10	,	,	PUNCT
ejpam-3886	323	11	2000	2000	NUM
ejpam-3886	323	12	.	.	PUNCT
ejpam-3886	324	1	[	[	X
ejpam-3886	324	2	5	5	X
ejpam-3886	324	3	]	]	PUNCT
ejpam-3886	324	4	t.	t.	PROPN
ejpam-3886	324	5	lam	lam	PROPN
ejpam-3886	324	6	,	,	PUNCT
ejpam-3886	324	7	a.	a.	NOUN
ejpam-3886	324	8	leory	leory	PROPN
ejpam-3886	324	9	,	,	PUNCT
ejpam-3886	324	10	and	and	CCONJ
ejpam-3886	324	11	j.	j.	PROPN
ejpam-3886	324	12	matczuk	matczuk	PROPN
ejpam-3886	324	13	.	.	PUNCT
ejpam-3886	324	14	primeness	primeness	PROPN
ejpam-3886	324	15	,	,	PUNCT
ejpam-3886	324	16	semiprimeness	semiprimeness	NOUN
ejpam-3886	324	17	and	and	CCONJ
ejpam-3886	324	18	the	the	DET
ejpam-3886	324	19	prime	prime	ADJ
ejpam-3886	324	20	radical	radical	NOUN
ejpam-3886	324	21	of	of	ADP
ejpam-3886	324	22	ore	ore	NOUN
ejpam-3886	324	23	extensions	extension	NOUN
ejpam-3886	324	24	.	.	PUNCT
ejpam-3886	325	1	comm	comm	NOUN
ejpam-3886	325	2	.	.	PUNCT
ejpam-3886	326	1	alg	alg	PROPN
ejpam-3886	326	2	.	.	PROPN
ejpam-3886	326	3	,	,	PUNCT
ejpam-3886	326	4	25(8):2459–2516	25(8):2459–2516	NUM
ejpam-3886	326	5	,	,	PUNCT
ejpam-3886	326	6	1997	1997	NUM
ejpam-3886	326	7	.	.	PUNCT
ejpam-3886	327	1	[	[	X
ejpam-3886	327	2	6	6	NUM
ejpam-3886	327	3	]	]	PUNCT
ejpam-3886	327	4	w.	w.	PROPN
ejpam-3886	327	5	nicholson	nicholson	PROPN
ejpam-3886	327	6	and	and	CCONJ
ejpam-3886	327	7	j.	j.	PROPN
ejpam-3886	327	8	watters	watters	PROPN
ejpam-3886	327	9	.	.	PUNCT
ejpam-3886	328	1	rings	ring	NOUN
ejpam-3886	328	2	with	with	ADP
ejpam-3886	328	3	projective	projective	ADJ
ejpam-3886	328	4	socle	socle	NOUN
ejpam-3886	328	5	.	.	PUNCT
ejpam-3886	329	1	proc	proc	PROPN
ejpam-3886	329	2	.	.	PUNCT
ejpam-3886	330	1	amer	amer	PROPN
ejpam-3886	330	2	.	.	PUNCT
ejpam-3886	330	3	math	math	PROPN
ejpam-3886	330	4	.	.	PUNCT
ejpam-3886	331	1	soc	soc	PROPN
ejpam-3886	331	2	.	.	PUNCT
ejpam-3886	331	3	,	,	PUNCT
ejpam-3886	331	4	102:443–450	102:443–450	NUM
ejpam-3886	331	5	,	,	PUNCT
ejpam-3886	331	6	1988	1988	NUM
ejpam-3886	331	7	.	.	PUNCT
ejpam-3886	332	1	[	[	X
ejpam-3886	332	2	7	7	X
ejpam-3886	332	3	]	]	X
ejpam-3886	332	4	ø	ø	NOUN
ejpam-3886	332	5	.	.	PUNCT
ejpam-3886	333	1	ore	ore	PROPN
ejpam-3886	333	2	.	.	PUNCT
ejpam-3886	333	3	theory	theory	NOUN
ejpam-3886	333	4	of	of	ADP
ejpam-3886	333	5	noncommutative	noncommutative	ADJ
ejpam-3886	333	6	polynomials	polynomial	NOUN
ejpam-3886	333	7	.	.	PUNCT
ejpam-3886	334	1	ann	ann	PROPN
ejpam-3886	334	2	.	.	PUNCT
ejpam-3886	334	3	math	math	PROPN
ejpam-3886	334	4	.	.	PUNCT
ejpam-3886	334	5	,	,	PUNCT
ejpam-3886	334	6	34(3):480–508	34(3):480–508	NUM
ejpam-3886	334	7	,	,	PUNCT
ejpam-3886	334	8	1933	1933	NUM
ejpam-3886	334	9	.	.	PUNCT
ejpam-3886	335	1	[	[	X
ejpam-3886	335	2	8	8	NUM
ejpam-3886	335	3	]	]	PUNCT
ejpam-3886	335	4	k.	k.	PROPN
ejpam-3886	336	1	paykan	paykan	PROPN
ejpam-3886	336	2	.	.	PUNCT
ejpam-3886	337	1	skew	skew	ADJ
ejpam-3886	337	2	inverse	inverse	NOUN
ejpam-3886	337	3	power	power	NOUN
ejpam-3886	337	4	series	series	PROPN
ejpam-3886	337	5	rings	ring	NOUN
ejpam-3886	337	6	over	over	ADP
ejpam-3886	337	7	a	a	DET
ejpam-3886	337	8	ring	ring	NOUN
ejpam-3886	337	9	with	with	ADP
ejpam-3886	337	10	projective	projective	ADJ
ejpam-3886	337	11	socle	socle	NOUN
ejpam-3886	337	12	.	.	PUNCT
ejpam-3886	338	1	czec	czec	ADJ
ejpam-3886	338	2	.	.	PUNCT
ejpam-3886	339	1	math	math	NOUN
ejpam-3886	339	2	.	.	PUNCT
ejpam-3886	340	1	j.	j.	PROPN
ejpam-3886	340	2	,	,	PUNCT
ejpam-3886	340	3	67(142):389–398	67(142):389–398	PROPN
ejpam-3886	340	4	,	,	PUNCT
ejpam-3886	340	5	2017	2017	NUM
ejpam-3886	340	6	.	.	PUNCT
ejpam-3886	341	1	[	[	X
ejpam-3886	341	2	9	9	NUM
ejpam-3886	341	3	]	]	X
ejpam-3886	341	4	r.	r.	PROPN
ejpam-3886	341	5	salem	salem	PROPN
ejpam-3886	341	6	,	,	PUNCT
ejpam-3886	341	7	m.	m.	NOUN
ejpam-3886	341	8	farahat	farahat	NOUN
ejpam-3886	341	9	,	,	PUNCT
ejpam-3886	341	10	and	and	CCONJ
ejpam-3886	341	11	h.	h.	PROPN
ejpam-3886	341	12	abd	abd	PROPN
ejpam-3886	341	13	-	-	PUNCT
ejpam-3886	341	14	elmalk	elmalk	NOUN
ejpam-3886	341	15	.	.	PUNCT
ejpam-3886	342	1	ps	ps	NOUN
ejpam-3886	342	2	-	-	PUNCT
ejpam-3886	342	3	modules	module	NOUN
ejpam-3886	342	4	over	over	ADP
ejpam-3886	342	5	ore	ore	NOUN
ejpam-3886	342	6	extensions	extension	NOUN
ejpam-3886	342	7	and	and	CCONJ
ejpam-3886	342	8	skew	skew	VERB
ejpam-3886	342	9	generalized	generalized	ADJ
ejpam-3886	342	10	power	power	NOUN
ejpam-3886	342	11	series	series	PROPN
ejpam-3886	342	12	rings	rings	PROPN
ejpam-3886	342	13	.	.	PUNCT
ejpam-3886	343	1	int	int	NOUN
ejpam-3886	343	2	.	.	PUNCT
ejpam-3886	344	1	j.	j.	PROPN
ejpam-3886	344	2	math	math	PROPN
ejpam-3886	344	3	.	.	PUNCT
ejpam-3886	345	1	sci	sci	PROPN
ejpam-3886	345	2	.	.	PROPN
ejpam-3886	345	3	,	,	PUNCT
ejpam-3886	345	4	2015	2015	NUM
ejpam-3886	345	5	.	.	PUNCT