id	sid	tid	token	lemma	pos
ejpam-6134	1	1	european	european	PROPN
ejpam-6134	1	2	journal	journal	PROPN
ejpam-6134	1	3	of	of	ADP
ejpam-6134	1	4	pure	pure	ADJ
ejpam-6134	1	5	and	and	CCONJ
ejpam-6134	1	6	applied	applied	ADJ
ejpam-6134	1	7	mathematics	mathematic	NOUN
ejpam-6134	1	8	2025	2025	NUM
ejpam-6134	1	9	,	,	PUNCT
ejpam-6134	1	10	vol	vol	NOUN
ejpam-6134	1	11	.	.	PROPN
ejpam-6134	1	12	18	18	NUM
ejpam-6134	1	13	,	,	PUNCT
ejpam-6134	1	14	issue	issue	NOUN
ejpam-6134	1	15	4	4	NUM
ejpam-6134	1	16	,	,	PUNCT
ejpam-6134	1	17	article	article	NOUN
ejpam-6134	1	18	number	number	NOUN
ejpam-6134	1	19	6134	6134	NUM
ejpam-6134	1	20	issn	issn	PROPN
ejpam-6134	1	21	1307	1307	NUM
ejpam-6134	1	22	-	-	SYM
ejpam-6134	1	23	5543	5543	NUM
ejpam-6134	1	24	–	–	PUNCT
ejpam-6134	1	25	ejpam.com	ejpam.com	X
ejpam-6134	1	26	published	publish	VERB
ejpam-6134	1	27	by	by	ADP
ejpam-6134	1	28	new	new	PROPN
ejpam-6134	1	29	york	york	PROPN
ejpam-6134	1	30	business	business	PROPN
ejpam-6134	1	31	global	global	PROPN
ejpam-6134	1	32	on	on	ADP
ejpam-6134	1	33	(	(	PUNCT
ejpam-6134	1	34	φ	φ	NOUN
ejpam-6134	1	35	,	,	PUNCT
ejpam-6134	1	36	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	1	37	in	in	ADP
ejpam-6134	1	38	rings	ring	NOUN
ejpam-6134	1	39	mahmoud	mahmoud	PROPN
ejpam-6134	1	40	m.	m.	PROPN
ejpam-6134	1	41	el	el	PROPN
ejpam-6134	1	42	-	-	PROPN
ejpam-6134	1	43	soufi1,2	soufi1,2	PROPN
ejpam-6134	1	44	,	,	PUNCT
ejpam-6134	1	45	munerah	munerah	PROPN
ejpam-6134	1	46	almulhem3,∗	almulhem3,∗	PROPN
ejpam-6134	1	47	,	,	PUNCT
ejpam-6134	1	48	m.	m.	NOUN
ejpam-6134	1	49	s.	s.	PROPN
ejpam-6134	1	50	tammam	tammam	PROPN
ejpam-6134	1	51	el	el	PROPN
ejpam-6134	1	52	-	-	PUNCT
ejpam-6134	1	53	sayiad4	sayiad4	NOUN
ejpam-6134	1	54	1	1	NUM
ejpam-6134	1	55	department	department	NOUN
ejpam-6134	1	56	of	of	ADP
ejpam-6134	1	57	mathematics	mathematic	NOUN
ejpam-6134	1	58	,	,	PUNCT
ejpam-6134	1	59	faculty	faculty	NOUN
ejpam-6134	1	60	of	of	ADP
ejpam-6134	1	61	science	science	NOUN
ejpam-6134	1	62	,	,	PUNCT
ejpam-6134	1	63	fayoum	fayoum	PROPN
ejpam-6134	1	64	university	university	PROPN
ejpam-6134	1	65	,	,	PUNCT
ejpam-6134	1	66	fayoum	fayoum	PROPN
ejpam-6134	1	67	,	,	PUNCT
ejpam-6134	1	68	egypt	egypt	PROPN
ejpam-6134	1	69	2	2	NUM
ejpam-6134	1	70	department	department	NOUN
ejpam-6134	1	71	of	of	ADP
ejpam-6134	1	72	mathematics	mathematic	NOUN
ejpam-6134	1	73	,	,	PUNCT
ejpam-6134	1	74	faculty	faculty	NOUN
ejpam-6134	1	75	of	of	ADP
ejpam-6134	1	76	science	science	NOUN
ejpam-6134	1	77	,	,	PUNCT
ejpam-6134	1	78	al	al	PROPN
ejpam-6134	1	79	baha	baha	PROPN
ejpam-6134	1	80	university	university	PROPN
ejpam-6134	1	81	,	,	PUNCT
ejpam-6134	1	82	al	al	PROPN
ejpam-6134	1	83	baha	baha	PROPN
ejpam-6134	1	84	,	,	PUNCT
ejpam-6134	1	85	kingdom	kingdom	NOUN
ejpam-6134	1	86	of	of	ADP
ejpam-6134	1	87	saudi	saudi	PROPN
ejpam-6134	1	88	arabia	arabia	PROPN
ejpam-6134	1	89	3	3	NUM
ejpam-6134	1	90	department	department	NOUN
ejpam-6134	1	91	of	of	ADP
ejpam-6134	1	92	mathematics	mathematics	PROPN
ejpam-6134	1	93	,	,	PUNCT
ejpam-6134	1	94	college	college	NOUN
ejpam-6134	1	95	of	of	ADP
ejpam-6134	1	96	science	science	NOUN
ejpam-6134	1	97	and	and	CCONJ
ejpam-6134	1	98	humanities	humanity	NOUN
ejpam-6134	1	99	,	,	PUNCT
ejpam-6134	1	100	imam	imam	PROPN
ejpam-6134	1	101	abdulrahman	abdulrahman	PROPN
ejpam-6134	1	102	bin	bin	PROPN
ejpam-6134	1	103	faisal	faisal	PROPN
ejpam-6134	1	104	university	university	PROPN
ejpam-6134	1	105	,	,	PUNCT
ejpam-6134	1	106	jubail	jubail	PROPN
ejpam-6134	1	107	,	,	PUNCT
ejpam-6134	1	108	35811	35811	NUM
ejpam-6134	1	109	,	,	PUNCT
ejpam-6134	1	110	kingdom	kingdom	NOUN
ejpam-6134	1	111	of	of	ADP
ejpam-6134	1	112	saudi	saudi	PROPN
ejpam-6134	1	113	arabia	arabia	PROPN
ejpam-6134	1	114	4	4	NUM
ejpam-6134	1	115	department	department	NOUN
ejpam-6134	1	116	of	of	ADP
ejpam-6134	1	117	mathematics	mathematic	NOUN
ejpam-6134	1	118	and	and	CCONJ
ejpam-6134	1	119	computer	computer	NOUN
ejpam-6134	1	120	science	science	NOUN
ejpam-6134	1	121	,	,	PUNCT
ejpam-6134	1	122	faculty	faculty	NOUN
ejpam-6134	1	123	of	of	ADP
ejpam-6134	1	124	science	science	NOUN
ejpam-6134	1	125	,	,	PUNCT
ejpam-6134	1	126	beni	beni	ADJ
ejpam-6134	1	127	-	-	ADJ
ejpam-6134	1	128	suef	suef	ADJ
ejpam-6134	1	129	university	university	NOUN
ejpam-6134	1	130	,	,	PUNCT
ejpam-6134	1	131	beni	beni	NOUN
ejpam-6134	1	132	-	-	ADJ
ejpam-6134	1	133	suef	suef	ADJ
ejpam-6134	1	134	,	,	PUNCT
ejpam-6134	1	135	62111	62111	NUM
ejpam-6134	1	136	,	,	PUNCT
ejpam-6134	1	137	egypt	egypt	PROPN
ejpam-6134	1	138	abstract	abstract	NOUN
ejpam-6134	1	139	.	.	PUNCT
ejpam-6134	2	1	in	in	ADP
ejpam-6134	2	2	this	this	DET
ejpam-6134	2	3	article	article	NOUN
ejpam-6134	2	4	,	,	PUNCT
ejpam-6134	2	5	we	we	PRON
ejpam-6134	2	6	examine	examine	VERB
ejpam-6134	2	7	the	the	DET
ejpam-6134	2	8	commutativity	commutativity	NOUN
ejpam-6134	2	9	of	of	ADP
ejpam-6134	2	10	a	a	DET
ejpam-6134	2	11	ring	ring	NOUN
ejpam-6134	2	12	ω	ω	PROPN
ejpam-6134	2	13	endowed	endow	VERB
ejpam-6134	2	14	with	with	ADP
ejpam-6134	2	15	a	a	DET
ejpam-6134	2	16	specific	specific	ADJ
ejpam-6134	2	17	kind	kind	NOUN
ejpam-6134	2	18	of	of	ADP
ejpam-6134	2	19	mapping	mapping	NOUN
ejpam-6134	2	20	called	call	VERB
ejpam-6134	2	21	centrally	centrally	ADV
ejpam-6134	2	22	extended	extended	ADJ
ejpam-6134	2	23	(	(	PUNCT
ejpam-6134	2	24	φ	φ	NOUN
ejpam-6134	2	25	,	,	PUNCT
ejpam-6134	2	26	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	2	27	,	,	PUNCT
ejpam-6134	2	28	where	where	SCONJ
ejpam-6134	2	29	φ	φ	PROPN
ejpam-6134	2	30	is	be	AUX
ejpam-6134	2	31	a	a	DET
ejpam-6134	2	32	mapping	mapping	NOUN
ejpam-6134	2	33	on	on	ADP
ejpam-6134	2	34	ω	ω	NUM
ejpam-6134	2	35	,	,	PUNCT
ejpam-6134	2	36	and	and	CCONJ
ejpam-6134	2	37	m	m	PROPN
ejpam-6134	2	38	is	be	AUX
ejpam-6134	2	39	an	an	DET
ejpam-6134	2	40	integer	integer	NOUN
ejpam-6134	2	41	.	.	PUNCT
ejpam-6134	3	1	this	this	DET
ejpam-6134	3	2	mapping	mapping	NOUN
ejpam-6134	3	3	is	be	AUX
ejpam-6134	3	4	a	a	DET
ejpam-6134	3	5	comprehensive	comprehensive	ADJ
ejpam-6134	3	6	kind	kind	NOUN
ejpam-6134	3	7	of	of	ADP
ejpam-6134	3	8	the	the	DET
ejpam-6134	3	9	homoderivation	homoderivation	NOUN
ejpam-6134	3	10	,	,	PUNCT
ejpam-6134	3	11	φ−homoderivation	φ−homoderivation	NOUN
ejpam-6134	3	12	,	,	PUNCT
ejpam-6134	3	13	and	and	CCONJ
ejpam-6134	3	14	m−homoderivation	m−homoderivation	NOUN
ejpam-6134	3	15	.	.	PUNCT
ejpam-6134	4	1	besides	besides	SCONJ
ejpam-6134	4	2	,	,	PUNCT
ejpam-6134	4	3	we	we	PRON
ejpam-6134	4	4	provide	provide	VERB
ejpam-6134	4	5	some	some	DET
ejpam-6134	4	6	properties	property	NOUN
ejpam-6134	4	7	of	of	ADP
ejpam-6134	4	8	its	its	PRON
ejpam-6134	4	9	center	center	NOUN
ejpam-6134	4	10	.	.	PUNCT
ejpam-6134	5	1	2020	2020	NUM
ejpam-6134	5	2	mathematics	mathematic	NOUN
ejpam-6134	5	3	subject	subject	NOUN
ejpam-6134	5	4	classifications	classification	NOUN
ejpam-6134	5	5	:	:	PUNCT
ejpam-6134	5	6	16n60	16n60	NUM
ejpam-6134	5	7	,	,	PUNCT
ejpam-6134	5	8	16u80	16u80	NUM
ejpam-6134	5	9	,	,	PUNCT
ejpam-6134	5	10	16w20	16w20	NUM
ejpam-6134	5	11	,	,	PUNCT
ejpam-6134	5	12	16w25	16w25	NUM
ejpam-6134	5	13	key	key	ADJ
ejpam-6134	5	14	words	word	NOUN
ejpam-6134	5	15	and	and	CCONJ
ejpam-6134	5	16	phrases	phrase	NOUN
ejpam-6134	5	17	:	:	PUNCT
ejpam-6134	5	18	prime	prime	ADJ
ejpam-6134	5	19	and	and	CCONJ
ejpam-6134	5	20	semiprime	semiprime	NOUN
ejpam-6134	5	21	rings	ring	NOUN
ejpam-6134	5	22	,	,	PUNCT
ejpam-6134	5	23	homoderivation	homoderivation	PROPN
ejpam-6134	5	24	,	,	PUNCT
ejpam-6134	5	25	ce	ce	NOUN
ejpam-6134	5	26	-	-	NOUN
ejpam-6134	5	27	derivation	derivation	NOUN
ejpam-6134	5	28	,	,	PUNCT
ejpam-6134	5	29	ce(φ	ce(φ	ADJ
ejpam-6134	5	30	,	,	PUNCT
ejpam-6134	5	31	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	5	32	1	1	NUM
ejpam-6134	5	33	.	.	PUNCT
ejpam-6134	5	34	introduction	introduction	NOUN
ejpam-6134	5	35	the	the	DET
ejpam-6134	5	36	study	study	NOUN
ejpam-6134	5	37	of	of	ADP
ejpam-6134	5	38	derivations	derivation	NOUN
ejpam-6134	5	39	in	in	ADP
ejpam-6134	5	40	ring	ring	NOUN
ejpam-6134	5	41	theory	theory	NOUN
ejpam-6134	5	42	has	have	AUX
ejpam-6134	5	43	significantly	significantly	ADV
ejpam-6134	5	44	evolved	evolve	VERB
ejpam-6134	5	45	over	over	ADP
ejpam-6134	5	46	the	the	DET
ejpam-6134	5	47	years	year	NOUN
ejpam-6134	5	48	,	,	PUNCT
ejpam-6134	5	49	incorporating	incorporate	VERB
ejpam-6134	5	50	various	various	ADJ
ejpam-6134	5	51	extensions	extension	NOUN
ejpam-6134	5	52	and	and	CCONJ
ejpam-6134	5	53	generalizations	generalization	NOUN
ejpam-6134	5	54	to	to	PART
ejpam-6134	5	55	enhance	enhance	VERB
ejpam-6134	5	56	the	the	DET
ejpam-6134	5	57	understanding	understanding	NOUN
ejpam-6134	5	58	of	of	ADP
ejpam-6134	5	59	algebraic	algebraic	ADJ
ejpam-6134	5	60	structures	structure	NOUN
ejpam-6134	5	61	and	and	CCONJ
ejpam-6134	5	62	their	their	PRON
ejpam-6134	5	63	functional	functional	ADJ
ejpam-6134	5	64	properties	property	NOUN
ejpam-6134	5	65	.	.	PUNCT
ejpam-6134	6	1	derivations	derivation	NOUN
ejpam-6134	6	2	serve	serve	VERB
ejpam-6134	6	3	as	as	ADP
ejpam-6134	6	4	fundamental	fundamental	ADJ
ejpam-6134	6	5	tools	tool	NOUN
ejpam-6134	6	6	in	in	ADP
ejpam-6134	6	7	analyzing	analyze	VERB
ejpam-6134	6	8	ring	ring	NOUN
ejpam-6134	6	9	behavior	behavior	NOUN
ejpam-6134	6	10	,	,	PUNCT
ejpam-6134	6	11	particularly	particularly	ADV
ejpam-6134	6	12	in	in	ADP
ejpam-6134	6	13	prime	prime	ADJ
ejpam-6134	6	14	and	and	CCONJ
ejpam-6134	6	15	semiprime	semiprime	NOUN
ejpam-6134	6	16	rings	ring	NOUN
ejpam-6134	6	17	,	,	PUNCT
ejpam-6134	6	18	where	where	SCONJ
ejpam-6134	6	19	they	they	PRON
ejpam-6134	6	20	play	play	VERB
ejpam-6134	6	21	a	a	DET
ejpam-6134	6	22	crucial	crucial	ADJ
ejpam-6134	6	23	role	role	NOUN
ejpam-6134	6	24	in	in	ADP
ejpam-6134	6	25	investigating	investigate	VERB
ejpam-6134	6	26	commutativity	commutativity	NOUN
ejpam-6134	6	27	and	and	CCONJ
ejpam-6134	6	28	structural	structural	ADJ
ejpam-6134	6	29	integrity	integrity	NOUN
ejpam-6134	6	30	.	.	PUNCT
ejpam-6134	7	1	over	over	ADP
ejpam-6134	7	2	time	time	NOUN
ejpam-6134	7	3	,	,	PUNCT
ejpam-6134	7	4	researchers	researcher	NOUN
ejpam-6134	7	5	have	have	AUX
ejpam-6134	7	6	proposed	propose	VERB
ejpam-6134	7	7	numerous	numerous	ADJ
ejpam-6134	7	8	generalizations	generalization	NOUN
ejpam-6134	7	9	of	of	ADP
ejpam-6134	7	10	derivations	derivation	NOUN
ejpam-6134	7	11	,	,	PUNCT
ejpam-6134	7	12	including	include	VERB
ejpam-6134	7	13	centrally	centrally	ADV
ejpam-6134	7	14	extended	extended	ADJ
ejpam-6134	7	15	mappings	mapping	NOUN
ejpam-6134	7	16	,	,	PUNCT
ejpam-6134	7	17	homoderivations	homoderivation	NOUN
ejpam-6134	7	18	,	,	PUNCT
ejpam-6134	7	19	and	and	CCONJ
ejpam-6134	7	20	m	m	NOUN
ejpam-6134	7	21	-	-	NOUN
ejpam-6134	7	22	homoderivations	homoderivation	NOUN
ejpam-6134	7	23	,	,	PUNCT
ejpam-6134	7	24	each	each	PRON
ejpam-6134	7	25	contributing	contribute	VERB
ejpam-6134	7	26	to	to	ADP
ejpam-6134	7	27	a	a	DET
ejpam-6134	7	28	deeper	deep	ADJ
ejpam-6134	7	29	understanding	understanding	NOUN
ejpam-6134	7	30	of	of	ADP
ejpam-6134	7	31	ring	ring	NOUN
ejpam-6134	7	32	theory	theory	NOUN
ejpam-6134	7	33	.	.	PUNCT
ejpam-6134	8	1	a	a	DET
ejpam-6134	8	2	pivotal	pivotal	ADJ
ejpam-6134	8	3	breakthrough	breakthrough	NOUN
ejpam-6134	8	4	in	in	ADP
ejpam-6134	8	5	this	this	DET
ejpam-6134	8	6	field	field	NOUN
ejpam-6134	8	7	was	be	AUX
ejpam-6134	8	8	the	the	DET
ejpam-6134	8	9	introduction	introduction	NOUN
ejpam-6134	8	10	of	of	ADP
ejpam-6134	8	11	centrally	centrally	ADV
ejpam-6134	8	12	extended	extend	VERB
ejpam-6134	8	13	derivations	derivation	NOUN
ejpam-6134	8	14	(	(	PUNCT
ejpam-6134	8	15	ce	ce	NOUN
ejpam-6134	8	16	-	-	NOUN
ejpam-6134	8	17	derivations	derivation	NOUN
ejpam-6134	8	18	)	)	PUNCT
ejpam-6134	8	19	by	by	ADP
ejpam-6134	8	20	bell	bell	NOUN
ejpam-6134	8	21	and	and	CCONJ
ejpam-6134	8	22	daif	daif	NOUN
ejpam-6134	8	23	[	[	X
ejpam-6134	8	24	1	1	NUM
ejpam-6134	8	25	]	]	PUNCT
ejpam-6134	8	26	,	,	PUNCT
ejpam-6134	8	27	which	which	PRON
ejpam-6134	8	28	ensured	ensure	VERB
ejpam-6134	8	29	that	that	SCONJ
ejpam-6134	8	30	certain	certain	ADJ
ejpam-6134	8	31	derivation	derivation	NOUN
ejpam-6134	8	32	properties	property	NOUN
ejpam-6134	8	33	were	be	AUX
ejpam-6134	8	34	preserved	preserve	VERB
ejpam-6134	8	35	within	within	ADP
ejpam-6134	8	36	the	the	DET
ejpam-6134	8	37	center	center	NOUN
ejpam-6134	8	38	of	of	ADP
ejpam-6134	8	39	the	the	DET
ejpam-6134	8	40	ring	ring	NOUN
ejpam-6134	8	41	,	,	PUNCT
ejpam-6134	8	42	thereby	thereby	ADV
ejpam-6134	8	43	influencing	influence	VERB
ejpam-6134	8	44	its	its	PRON
ejpam-6134	8	45	commutativity	commutativity	NOUN
ejpam-6134	8	46	.	.	PUNCT
ejpam-6134	9	1	∗corresponding	∗corresponde	VERB
ejpam-6134	9	2	author	author	NOUN
ejpam-6134	9	3	.	.	PUNCT
ejpam-6134	10	1	doi	doi	NOUN
ejpam-6134	10	2	:	:	PUNCT
ejpam-6134	10	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6134	https://doi.org/10.29020/nybg.ejpam.v18i4.6134	PROPN
ejpam-6134	10	4	email	email	NOUN
ejpam-6134	10	5	addresses	address	NOUN
ejpam-6134	10	6	:	:	PUNCT
ejpam-6134	10	7	mms06@fayoum.edu.eg	mms06@fayoum.edu.eg	PROPN
ejpam-6134	10	8	(	(	PUNCT
ejpam-6134	10	9	m.	m.	NOUN
ejpam-6134	10	10	m.	m.	PROPN
ejpam-6134	10	11	el	el	PROPN
ejpam-6134	10	12	-	-	PUNCT
ejpam-6134	10	13	soufi	soufi	NOUN
ejpam-6134	10	14	)	)	PUNCT
ejpam-6134	10	15	,	,	PUNCT
ejpam-6134	10	16	malmulhim@iau.edu.sa	malmulhim@iau.edu.sa	PROPN
ejpam-6134	10	17	(	(	PUNCT
ejpam-6134	10	18	m.	m.	NOUN
ejpam-6134	10	19	almulhem	almulhem	PROPN
ejpam-6134	10	20	)	)	PUNCT
ejpam-6134	10	21	,	,	PUNCT
ejpam-6134	10	22	mtammam2012@yahoo.com	mtammam2012@yahoo.com	X
ejpam-6134	11	1	(	(	PUNCT
ejpam-6134	11	2	m.	m.	NOUN
ejpam-6134	11	3	s.	s.	PROPN
ejpam-6134	11	4	tammam	tammam	PROPN
ejpam-6134	11	5	el	el	PROPN
ejpam-6134	11	6	-	-	PUNCT
ejpam-6134	11	7	sayiad	sayiad	ADJ
ejpam-6134	11	8	)	)	PUNCT
ejpam-6134	11	9	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6134	12	1	1	1	NUM
ejpam-6134	12	2	copyright	copyright	NOUN
ejpam-6134	12	3	:	:	PUNCT
ejpam-6134	12	4	©	©	PROPN
ejpam-6134	12	5	2025	2025	NUM
ejpam-6134	12	6	the	the	DET
ejpam-6134	12	7	author(s	author(s	NOUN
ejpam-6134	12	8	)	)	PUNCT
ejpam-6134	12	9	.	.	PUNCT
ejpam-6134	13	1	(	(	PUNCT
ejpam-6134	13	2	cc	cc	NOUN
ejpam-6134	13	3	by	by	ADP
ejpam-6134	13	4	-	-	PUNCT
ejpam-6134	13	5	nc	nc	PROPN
ejpam-6134	13	6	4.0	4.0	NUM
ejpam-6134	13	7	)	)	PUNCT
ejpam-6134	13	8	m.m	m.m	PROPN
ejpam-6134	13	9	.	.	PROPN
ejpam-6134	13	10	el	el	PROPN
ejpam-6134	13	11	-	-	PUNCT
ejpam-6134	13	12	soufi	soufi	ADJ
ejpam-6134	13	13	,	,	PUNCT
ejpam-6134	13	14	m.	m.	NOUN
ejpam-6134	13	15	almulhem	almulhem	NOUN
ejpam-6134	13	16	,	,	PUNCT
ejpam-6134	13	17	m.	m.	NOUN
ejpam-6134	13	18	s.	s.	PROPN
ejpam-6134	13	19	tammam	tammam	PROPN
ejpam-6134	13	20	el	el	PROPN
ejpam-6134	13	21	-	-	PROPN
ejpam-6134	13	22	sayiad	sayiad	PROPN
ejpam-6134	13	23	/	/	SYM
ejpam-6134	13	24	eur	eur	PROPN
ejpam-6134	13	25	.	.	PUNCT
ejpam-6134	14	1	j.	j.	PROPN
ejpam-6134	14	2	pure	pure	PROPN
ejpam-6134	14	3	appl	appl	PROPN
ejpam-6134	14	4	.	.	PROPN
ejpam-6134	14	5	math	math	PROPN
ejpam-6134	14	6	,	,	PUNCT
ejpam-6134	14	7	18	18	NUM
ejpam-6134	14	8	(	(	PUNCT
ejpam-6134	14	9	4	4	NUM
ejpam-6134	14	10	)	)	PUNCT
ejpam-6134	14	11	(	(	PUNCT
ejpam-6134	14	12	2025	2025	NUM
ejpam-6134	14	13	)	)	PUNCT
ejpam-6134	14	14	,	,	PUNCT
ejpam-6134	14	15	6134	6134	NUM
ejpam-6134	14	16	2	2	NUM
ejpam-6134	14	17	of	of	ADP
ejpam-6134	14	18	12	12	NUM
ejpam-6134	14	19	throughout	throughout	NOUN
ejpam-6134	14	20	,	,	PUNCT
ejpam-6134	14	21	unless	unless	SCONJ
ejpam-6134	14	22	stated	state	VERB
ejpam-6134	14	23	otherwise	otherwise	ADV
ejpam-6134	14	24	,	,	PUNCT
ejpam-6134	14	25	ω	ω	PROPN
ejpam-6134	14	26	is	be	AUX
ejpam-6134	14	27	an	an	DET
ejpam-6134	14	28	associative	associative	ADJ
ejpam-6134	14	29	ring	ring	NOUN
ejpam-6134	14	30	with	with	ADP
ejpam-6134	14	31	center	center	NOUN
ejpam-6134	14	32	ζ(ω	ζ(ω	PROPN
ejpam-6134	14	33	)	)	PUNCT
ejpam-6134	14	34	.	.	PUNCT
ejpam-6134	15	1	for	for	ADP
ejpam-6134	15	2	a	a	DET
ejpam-6134	15	3	subset	subset	NOUN
ejpam-6134	15	4	s	s	PROPN
ejpam-6134	15	5	of	of	ADP
ejpam-6134	15	6	ω	ω	PROPN
ejpam-6134	15	7	,	,	PUNCT
ejpam-6134	15	8	the	the	DET
ejpam-6134	15	9	map	map	NOUN
ejpam-6134	15	10	d	d	AUX
ejpam-6134	15	11	preserves	preserve	VERB
ejpam-6134	15	12	s	s	X
ejpam-6134	15	13	if	if	SCONJ
ejpam-6134	15	14	d(s	d(s	PROPN
ejpam-6134	15	15	)	)	PUNCT
ejpam-6134	16	1	⊆	⊆	NUM
ejpam-6134	16	2	s.	s.	PROPN
ejpam-6134	16	3	suppose	suppose	VERB
ejpam-6134	16	4	that	that	SCONJ
ejpam-6134	16	5	d	d	PROPN
ejpam-6134	16	6	is	be	AUX
ejpam-6134	16	7	a	a	DET
ejpam-6134	16	8	mapping	mapping	NOUN
ejpam-6134	16	9	of	of	ADP
ejpam-6134	16	10	a	a	DET
ejpam-6134	16	11	ring	ring	NOUN
ejpam-6134	16	12	ω	ω	NOUN
ejpam-6134	16	13	.	.	PUNCT
ejpam-6134	17	1	if	if	SCONJ
ejpam-6134	17	2	d(s	d(s	PROPN
ejpam-6134	17	3	+	+	CCONJ
ejpam-6134	17	4	u	u	NOUN
ejpam-6134	17	5	)	)	PUNCT
ejpam-6134	17	6	−	−	ADP
ejpam-6134	17	7	d(s	d(s	PROPN
ejpam-6134	17	8	)	)	PUNCT
ejpam-6134	17	9	−	−	PROPN
ejpam-6134	17	10	d(u	d(u	PROPN
ejpam-6134	17	11	)	)	PUNCT
ejpam-6134	17	12	∈	∈	PROPN
ejpam-6134	17	13	ζ(ω	ζ(ω	PROPN
ejpam-6134	17	14	)	)	PUNCT
ejpam-6134	17	15	and	and	CCONJ
ejpam-6134	17	16	d(su)−d(s)u−	d(su)−d(s)u−	PROPN
ejpam-6134	17	17	sd(u	sd(u	NOUN
ejpam-6134	17	18	)	)	PUNCT
ejpam-6134	17	19	∈	∈	PROPN
ejpam-6134	17	20	ζ(ω	ζ(ω	PROPN
ejpam-6134	17	21	)	)	PUNCT
ejpam-6134	17	22	for	for	ADP
ejpam-6134	17	23	every	every	DET
ejpam-6134	17	24	s	s	PROPN
ejpam-6134	17	25	,	,	PUNCT
ejpam-6134	17	26	u	u	PROPN
ejpam-6134	17	27	∈	∈	PROPN
ejpam-6134	17	28	ω	ω	PROPN
ejpam-6134	17	29	,	,	PUNCT
ejpam-6134	17	30	then	then	ADV
ejpam-6134	17	31	d	d	PROPN
ejpam-6134	17	32	is	be	AUX
ejpam-6134	17	33	called	call	VERB
ejpam-6134	17	34	a	a	DET
ejpam-6134	17	35	ce	ce	NOUN
ejpam-6134	17	36	-	-	NOUN
ejpam-6134	17	37	derivation	derivation	NOUN
ejpam-6134	17	38	.	.	PUNCT
ejpam-6134	18	1	the	the	DET
ejpam-6134	18	2	ce-(φ	ce-(φ	NOUN
ejpam-6134	18	3	,	,	PUNCT
ejpam-6134	18	4	ψ)-derivation	ψ)-derivation	NOUN
ejpam-6134	18	5	on	on	ADP
ejpam-6134	18	6	ω	ω	PROPN
ejpam-6134	18	7	was	be	AUX
ejpam-6134	18	8	described	describe	VERB
ejpam-6134	18	9	by	by	ADP
ejpam-6134	18	10	tammam	tammam	NOUN
ejpam-6134	18	11	et	et	PROPN
ejpam-6134	18	12	al	al	PROPN
ejpam-6134	18	13	.	.	PUNCT
ejpam-6134	19	1	[	[	X
ejpam-6134	19	2	2	2	X
ejpam-6134	19	3	]	]	PUNCT
ejpam-6134	19	4	as	as	ADP
ejpam-6134	19	5	a	a	DET
ejpam-6134	19	6	map	map	NOUN
ejpam-6134	19	7	d	d	NOUN
ejpam-6134	19	8	on	on	ADP
ejpam-6134	19	9	ω	ω	NUM
ejpam-6134	19	10	such	such	ADJ
ejpam-6134	19	11	that	that	SCONJ
ejpam-6134	19	12	,	,	PUNCT
ejpam-6134	19	13	for	for	ADP
ejpam-6134	19	14	each	each	DET
ejpam-6134	19	15	s	s	NOUN
ejpam-6134	19	16	,	,	PUNCT
ejpam-6134	19	17	u	u	PROPN
ejpam-6134	19	18	∈	∈	PROPN
ejpam-6134	19	19	ω	ω	PROPN
ejpam-6134	19	20	,	,	PUNCT
ejpam-6134	19	21	both	both	PRON
ejpam-6134	19	22	d(s+u)−d(s)−d(u	d(s+u)−d(s)−d(u	NOUN
ejpam-6134	19	23	)	)	PUNCT
ejpam-6134	19	24	and	and	CCONJ
ejpam-6134	19	25	d(su)−d(s)φ(u)−ψ(s)d(u	d(su)−d(s)φ(u)−ψ(s)d(u	PROPN
ejpam-6134	19	26	)	)	PUNCT
ejpam-6134	19	27	are	be	AUX
ejpam-6134	19	28	in	in	ADP
ejpam-6134	19	29	ζ(ω	ζ(ω	PROPN
ejpam-6134	19	30	)	)	PUNCT
ejpam-6134	19	31	,	,	PUNCT
ejpam-6134	19	32	for	for	ADP
ejpam-6134	19	33	more	more	ADV
ejpam-6134	19	34	recent	recent	ADJ
ejpam-6134	19	35	work	work	NOUN
ejpam-6134	19	36	,	,	PUNCT
ejpam-6134	19	37	see	see	VERB
ejpam-6134	19	38	[	[	X
ejpam-6134	19	39	3	3	X
ejpam-6134	19	40	]	]	PUNCT
ejpam-6134	19	41	and	and	CCONJ
ejpam-6134	19	42	[	[	X
ejpam-6134	19	43	4	4	NUM
ejpam-6134	19	44	]	]	PUNCT
ejpam-6134	19	45	.	.	PUNCT
ejpam-6134	20	1	in	in	ADP
ejpam-6134	20	2	1998	1998	NUM
ejpam-6134	20	3	,	,	PUNCT
ejpam-6134	20	4	filippov	filippov	NOUN
ejpam-6134	20	5	’s	’s	PART
ejpam-6134	20	6	concept	concept	NOUN
ejpam-6134	20	7	of	of	ADP
ejpam-6134	20	8	δ	δ	NOUN
ejpam-6134	20	9	-	-	PUNCT
ejpam-6134	20	10	derivations	derivation	NOUN
ejpam-6134	20	11	broadened	broaden	VERB
ejpam-6134	20	12	the	the	DET
ejpam-6134	20	13	classical	classical	ADJ
ejpam-6134	20	14	definition	definition	NOUN
ejpam-6134	20	15	by	by	ADP
ejpam-6134	20	16	incorporating	incorporate	VERB
ejpam-6134	20	17	a	a	DET
ejpam-6134	20	18	mapping	mapping	NOUN
ejpam-6134	20	19	δ	δ	NOUN
ejpam-6134	20	20	into	into	ADP
ejpam-6134	20	21	a	a	DET
ejpam-6134	20	22	derivation	derivation	NOUN
ejpam-6134	20	23	as	as	SCONJ
ejpam-6134	20	24	follows	follow	VERB
ejpam-6134	20	25	:	:	PUNCT
ejpam-6134	20	26	suppose	suppose	VERB
ejpam-6134	20	27	b	b	NOUN
ejpam-6134	20	28	is	be	AUX
ejpam-6134	20	29	an	an	DET
ejpam-6134	20	30	algebra	algebra	NOUN
ejpam-6134	20	31	over	over	ADP
ejpam-6134	20	32	h	h	NOUN
ejpam-6134	20	33	,	,	PUNCT
ejpam-6134	20	34	where	where	SCONJ
ejpam-6134	20	35	h	h	NOUN
ejpam-6134	20	36	is	be	AUX
ejpam-6134	20	37	a	a	DET
ejpam-6134	20	38	unital	unital	ADJ
ejpam-6134	20	39	commutative	commutative	ADJ
ejpam-6134	20	40	associative	associative	ADJ
ejpam-6134	20	41	ring	ring	NOUN
ejpam-6134	20	42	.	.	PUNCT
ejpam-6134	21	1	given	give	VERB
ejpam-6134	21	2	an	an	DET
ejpam-6134	21	3	arbitrary	arbitrary	ADJ
ejpam-6134	21	4	δ	δ	PROPN
ejpam-6134	21	5	∈	∈	PROPN
ejpam-6134	21	6	h	h	NOUN
ejpam-6134	21	7	,	,	PUNCT
ejpam-6134	21	8	a	a	DET
ejpam-6134	21	9	δderivation	δderivation	NOUN
ejpam-6134	21	10	of	of	ADP
ejpam-6134	21	11	b	b	NOUN
ejpam-6134	21	12	is	be	AUX
ejpam-6134	21	13	defined	define	VERB
ejpam-6134	21	14	to	to	PART
ejpam-6134	21	15	be	be	AUX
ejpam-6134	21	16	an	an	DET
ejpam-6134	21	17	f	f	ADJ
ejpam-6134	21	18	-	-	PUNCT
ejpam-6134	21	19	linear	linear	PROPN
ejpam-6134	21	20	mapping	mapping	NOUN
ejpam-6134	22	1	f	f	NOUN
ejpam-6134	22	2	:	:	PUNCT
ejpam-6134	22	3	b	b	X
ejpam-6134	22	4	→	→	SYM
ejpam-6134	22	5	b	b	X
ejpam-6134	22	6	satisfying	satisfy	VERB
ejpam-6134	22	7	the	the	DET
ejpam-6134	22	8	identity	identity	NOUN
ejpam-6134	22	9	f(su	f(su	NOUN
ejpam-6134	22	10	)	)	PUNCT
ejpam-6134	22	11	=	=	PUNCT
ejpam-6134	22	12	δ(f(s)u+	δ(f(s)u+	NOUN
ejpam-6134	22	13	sf(u	sf(u	NUM
ejpam-6134	22	14	)	)	PUNCT
ejpam-6134	22	15	)	)	PUNCT
ejpam-6134	22	16	for	for	ADP
ejpam-6134	22	17	each	each	DET
ejpam-6134	22	18	s	s	PROPN
ejpam-6134	22	19	,	,	PUNCT
ejpam-6134	22	20	u	u	PROPN
ejpam-6134	22	21	∈	∈	PROPN
ejpam-6134	22	22	b.	b.	PROPN
ejpam-6134	22	23	filippov	filippov	PROPN
ejpam-6134	22	24	’s	’s	PART
ejpam-6134	22	25	work	work	NOUN
ejpam-6134	23	1	[	[	X
ejpam-6134	23	2	5	5	NUM
ejpam-6134	23	3	]	]	PUNCT
ejpam-6134	23	4	has	have	AUX
ejpam-6134	23	5	paved	pave	VERB
ejpam-6134	23	6	the	the	DET
ejpam-6134	23	7	way	way	NOUN
ejpam-6134	23	8	for	for	ADP
ejpam-6134	23	9	more	more	ADJ
ejpam-6134	23	10	complex	complex	ADJ
ejpam-6134	23	11	derivation	derivation	NOUN
ejpam-6134	23	12	structures	structure	NOUN
ejpam-6134	23	13	.	.	PUNCT
ejpam-6134	24	1	building	build	VERB
ejpam-6134	24	2	on	on	ADP
ejpam-6134	24	3	these	these	DET
ejpam-6134	24	4	developments	development	NOUN
ejpam-6134	24	5	,	,	PUNCT
ejpam-6134	24	6	el	el	NOUN
ejpam-6134	24	7	-	-	PUNCT
ejpam-6134	24	8	soufi	soufi	NOUN
ejpam-6134	25	1	[	[	X
ejpam-6134	25	2	6	6	NUM
ejpam-6134	25	3	]	]	PUNCT
ejpam-6134	25	4	pioneered	pioneer	VERB
ejpam-6134	25	5	the	the	DET
ejpam-6134	25	6	study	study	NOUN
ejpam-6134	25	7	of	of	ADP
ejpam-6134	25	8	homoderivations	homoderivation	NOUN
ejpam-6134	25	9	,	,	PUNCT
ejpam-6134	25	10	which	which	PRON
ejpam-6134	25	11	is	be	AUX
ejpam-6134	25	12	defined	define	VERB
ejpam-6134	25	13	as	as	SCONJ
ejpam-6134	25	14	follows	follow	VERB
ejpam-6134	25	15	:	:	PUNCT
ejpam-6134	25	16	if	if	SCONJ
ejpam-6134	25	17	f(su	f(su	ADJ
ejpam-6134	25	18	)	)	PUNCT
ejpam-6134	25	19	=	=	SYM
ejpam-6134	25	20	sf(u)+f(s)u+f(s)f(u	sf(u)+f(s)u+f(s)f(u	PROPN
ejpam-6134	25	21	)	)	PUNCT
ejpam-6134	25	22	holds	hold	VERB
ejpam-6134	25	23	for	for	ADP
ejpam-6134	25	24	each	each	DET
ejpam-6134	25	25	s	s	NOUN
ejpam-6134	25	26	,	,	PUNCT
ejpam-6134	25	27	u	u	PROPN
ejpam-6134	25	28	∈	∈	PROPN
ejpam-6134	25	29	ω	ω	PROPN
ejpam-6134	25	30	,	,	PUNCT
ejpam-6134	25	31	then	then	ADV
ejpam-6134	25	32	f	f	PROPN
ejpam-6134	25	33	is	be	AUX
ejpam-6134	25	34	said	say	VERB
ejpam-6134	25	35	to	to	PART
ejpam-6134	25	36	be	be	AUX
ejpam-6134	25	37	a	a	DET
ejpam-6134	25	38	homoderivation	homoderivation	NOUN
ejpam-6134	25	39	,	,	PUNCT
ejpam-6134	25	40	provided	provide	VERB
ejpam-6134	25	41	it	it	PRON
ejpam-6134	25	42	is	be	AUX
ejpam-6134	25	43	additive	additive	ADJ
ejpam-6134	25	44	.	.	PUNCT
ejpam-6134	26	1	this	this	DET
ejpam-6134	26	2	concept	concept	NOUN
ejpam-6134	26	3	has	have	AUX
ejpam-6134	26	4	been	be	AUX
ejpam-6134	26	5	instrumental	instrumental	ADJ
ejpam-6134	26	6	in	in	ADP
ejpam-6134	26	7	the	the	DET
ejpam-6134	26	8	study	study	NOUN
ejpam-6134	26	9	of	of	ADP
ejpam-6134	26	10	prime	prime	ADJ
ejpam-6134	26	11	and	and	CCONJ
ejpam-6134	26	12	semiprime	semiprime	NOUN
ejpam-6134	26	13	rings	ring	NOUN
ejpam-6134	26	14	.	.	PUNCT
ejpam-6134	27	1	in	in	ADP
ejpam-6134	27	2	2022	2022	NUM
ejpam-6134	27	3	,	,	PUNCT
ejpam-6134	27	4	further	further	ADJ
ejpam-6134	27	5	advancements	advancement	NOUN
ejpam-6134	27	6	by	by	ADP
ejpam-6134	27	7	tammam	tammam	NOUN
ejpam-6134	27	8	et	et	PROPN
ejpam-6134	27	9	al	al	PROPN
ejpam-6134	27	10	.	.	PUNCT
ejpam-6134	28	1	[	[	X
ejpam-6134	28	2	7	7	X
ejpam-6134	28	3	]	]	PUNCT
ejpam-6134	28	4	led	lead	VERB
ejpam-6134	28	5	to	to	ADP
ejpam-6134	28	6	the	the	DET
ejpam-6134	28	7	introduction	introduction	NOUN
ejpam-6134	28	8	of	of	ADP
ejpam-6134	28	9	mhomoderivations	mhomoderivation	NOUN
ejpam-6134	28	10	,	,	PUNCT
ejpam-6134	28	11	integrating	integrate	VERB
ejpam-6134	28	12	an	an	DET
ejpam-6134	28	13	integer	integer	NOUN
ejpam-6134	28	14	parameter	parameter	NOUN
ejpam-6134	28	15	into	into	ADP
ejpam-6134	28	16	the	the	DET
ejpam-6134	28	17	derivation	derivation	NOUN
ejpam-6134	28	18	process	process	NOUN
ejpam-6134	28	19	to	to	PART
ejpam-6134	28	20	provide	provide	VERB
ejpam-6134	28	21	a	a	DET
ejpam-6134	28	22	new	new	ADJ
ejpam-6134	28	23	perspective	perspective	NOUN
ejpam-6134	28	24	on	on	ADP
ejpam-6134	28	25	non	non	ADJ
ejpam-6134	28	26	-	-	ADJ
ejpam-6134	28	27	commutative	commutative	ADJ
ejpam-6134	28	28	ring	ring	NOUN
ejpam-6134	28	29	behavior	behavior	NOUN
ejpam-6134	28	30	.	.	PUNCT
ejpam-6134	29	1	if	if	SCONJ
ejpam-6134	29	2	,	,	PUNCT
ejpam-6134	29	3	for	for	ADP
ejpam-6134	29	4	each	each	DET
ejpam-6134	29	5	s	s	NOUN
ejpam-6134	29	6	,	,	PUNCT
ejpam-6134	29	7	u	u	PROPN
ejpam-6134	29	8	∈	∈	PROPN
ejpam-6134	29	9	ω	ω	PROPN
ejpam-6134	29	10	,	,	PUNCT
ejpam-6134	29	11	the	the	DET
ejpam-6134	29	12	map	map	NOUN
ejpam-6134	29	13	f	f	PROPN
ejpam-6134	29	14	satisfies	satisfie	NOUN
ejpam-6134	29	15	f(su	f(su	PROPN
ejpam-6134	29	16	)	)	PUNCT
ejpam-6134	29	17	=	=	PUNCT
ejpam-6134	29	18	sf(u	sf(u	NUM
ejpam-6134	29	19	)	)	PUNCT
ejpam-6134	29	20	+	+	CCONJ
ejpam-6134	29	21	f(s)u	f(s)u	PROPN
ejpam-6134	29	22	+	+	CCONJ
ejpam-6134	29	23	mf(s)f(u	mf(s)f(u	NOUN
ejpam-6134	29	24	)	)	PUNCT
ejpam-6134	29	25	,	,	PUNCT
ejpam-6134	29	26	then	then	ADV
ejpam-6134	29	27	the	the	DET
ejpam-6134	29	28	map	map	NOUN
ejpam-6134	29	29	f	f	NOUN
ejpam-6134	29	30	is	be	AUX
ejpam-6134	29	31	said	say	VERB
ejpam-6134	29	32	to	to	PART
ejpam-6134	29	33	be	be	AUX
ejpam-6134	29	34	an	an	DET
ejpam-6134	29	35	mhomoderivation	mhomoderivation	NOUN
ejpam-6134	29	36	,	,	PUNCT
ejpam-6134	29	37	provided	provide	VERB
ejpam-6134	29	38	it	it	PRON
ejpam-6134	29	39	is	be	AUX
ejpam-6134	29	40	additive	additive	ADJ
ejpam-6134	29	41	.	.	PUNCT
ejpam-6134	30	1	in	in	ADP
ejpam-6134	30	2	2022	2022	NUM
ejpam-6134	30	3	,	,	PUNCT
ejpam-6134	30	4	el	el	NOUN
ejpam-6134	30	5	-	-	PUNCT
ejpam-6134	30	6	soufi	soufi	PROPN
ejpam-6134	30	7	et	et	PROPN
ejpam-6134	30	8	al	al	PROPN
ejpam-6134	30	9	.	.	PUNCT
ejpam-6134	31	1	[	[	X
ejpam-6134	31	2	8	8	NUM
ejpam-6134	31	3	]	]	PUNCT
ejpam-6134	31	4	introduced	introduce	VERB
ejpam-6134	31	5	the	the	DET
ejpam-6134	31	6	concept	concept	NOUN
ejpam-6134	31	7	of	of	ADP
ejpam-6134	31	8	φ	φ	NOUN
ejpam-6134	31	9	-	-	PUNCT
ejpam-6134	31	10	homoderivations	homoderivation	NOUN
ejpam-6134	31	11	by	by	ADP
ejpam-6134	31	12	incorporating	incorporate	VERB
ejpam-6134	31	13	a	a	DET
ejpam-6134	31	14	map	map	NOUN
ejpam-6134	31	15	φ	φ	PROPN
ejpam-6134	31	16	defined	define	VERB
ejpam-6134	31	17	on	on	ADP
ejpam-6134	31	18	a	a	DET
ejpam-6134	31	19	ring	ring	NOUN
ejpam-6134	31	20	ω	ω	NOUN
ejpam-6134	31	21	into	into	ADP
ejpam-6134	31	22	the	the	DET
ejpam-6134	31	23	homoderivation	homoderivation	NOUN
ejpam-6134	31	24	framework	framework	NOUN
ejpam-6134	31	25	.	.	PUNCT
ejpam-6134	32	1	specifically	specifically	ADV
ejpam-6134	32	2	,	,	PUNCT
ejpam-6134	32	3	an	an	DET
ejpam-6134	32	4	additive	additive	ADJ
ejpam-6134	32	5	map	map	NOUN
ejpam-6134	32	6	f	f	PROPN
ejpam-6134	32	7	on	on	ADP
ejpam-6134	32	8	ω	ω	PROPN
ejpam-6134	32	9	is	be	AUX
ejpam-6134	32	10	called	call	VERB
ejpam-6134	32	11	a	a	DET
ejpam-6134	32	12	φ	φ	NOUN
ejpam-6134	32	13	-	-	PUNCT
ejpam-6134	32	14	homoderivation	homoderivation	NOUN
ejpam-6134	32	15	if	if	SCONJ
ejpam-6134	32	16	,	,	PUNCT
ejpam-6134	32	17	for	for	ADP
ejpam-6134	32	18	all	all	DET
ejpam-6134	32	19	s	s	NOUN
ejpam-6134	32	20	,	,	PUNCT
ejpam-6134	32	21	u	u	PROPN
ejpam-6134	32	22	∈	∈	PROPN
ejpam-6134	32	23	ω	ω	PROPN
ejpam-6134	32	24	,	,	PUNCT
ejpam-6134	32	25	it	it	PRON
ejpam-6134	32	26	satisfies	satisfy	VERB
ejpam-6134	32	27	f(su	f(su	NOUN
ejpam-6134	32	28	)	)	PUNCT
ejpam-6134	32	29	=	=	SYM
ejpam-6134	32	30	f(s)φ(u	f(s)φ(u	NOUN
ejpam-6134	32	31	)	)	PUNCT
ejpam-6134	32	32	+	+	X
ejpam-6134	32	33	φ(s)f(u	φ(s)f(u	ADJ
ejpam-6134	32	34	)	)	PUNCT
ejpam-6134	32	35	+	+	CCONJ
ejpam-6134	32	36	f(s)f(u	f(s)f(u	ADJ
ejpam-6134	32	37	)	)	PUNCT
ejpam-6134	32	38	.	.	PUNCT
ejpam-6134	33	1	a	a	DET
ejpam-6134	33	2	centrally	centrally	ADV
ejpam-6134	33	3	extended	extend	VERB
ejpam-6134	33	4	homoderivation	homoderivation	NOUN
ejpam-6134	33	5	f	f	PROPN
ejpam-6134	33	6	is	be	AUX
ejpam-6134	33	7	defined	define	VERB
ejpam-6134	33	8	as	as	ADP
ejpam-6134	33	9	a	a	DET
ejpam-6134	33	10	map	map	NOUN
ejpam-6134	33	11	for	for	ADP
ejpam-6134	33	12	which	which	PRON
ejpam-6134	33	13	f(s+	f(s+	NOUN
ejpam-6134	33	14	u)−f(s)−f(u	u)−f(s)−f(u	ADJ
ejpam-6134	33	15	)	)	PUNCT
ejpam-6134	33	16	and	and	CCONJ
ejpam-6134	33	17	f(su)−f(s)f(u)−f(s)u−	f(su)−f(s)f(u)−f(s)u−	PROPN
ejpam-6134	33	18	sf(u	sf(u	VERB
ejpam-6134	33	19	)	)	PUNCT
ejpam-6134	33	20	both	both	PRON
ejpam-6134	33	21	lie	lie	VERB
ejpam-6134	33	22	in	in	ADP
ejpam-6134	33	23	the	the	DET
ejpam-6134	33	24	center	center	NOUN
ejpam-6134	33	25	ζ(ω	ζ(ω	PROPN
ejpam-6134	33	26	)	)	PUNCT
ejpam-6134	33	27	for	for	ADP
ejpam-6134	33	28	all	all	DET
ejpam-6134	33	29	s	s	NOUN
ejpam-6134	33	30	,	,	PUNCT
ejpam-6134	33	31	u	u	PROPN
ejpam-6134	33	32	∈	∈	PROPN
ejpam-6134	33	33	ω	ω	NOUN
ejpam-6134	33	34	.	.	PUNCT
ejpam-6134	34	1	furthermore	furthermore	ADV
ejpam-6134	34	2	,	,	PUNCT
ejpam-6134	34	3	if	if	SCONJ
ejpam-6134	34	4	f(s+u)−f(s)−f(u	f(s+u)−f(s)−f(u	NUM
ejpam-6134	34	5	)	)	PUNCT
ejpam-6134	34	6	∈	∈	PROPN
ejpam-6134	34	7	ζ(ω	ζ(ω	PROPN
ejpam-6134	34	8	)	)	PUNCT
ejpam-6134	34	9	and	and	CCONJ
ejpam-6134	34	10	f(su)−f(s)f(u)−f(s)φ(u)−φ(s)f(u	f(su)−f(s)f(u)−f(s)φ(u)−φ(s)f(u	PROPN
ejpam-6134	34	11	)	)	PUNCT
ejpam-6134	34	12	∈	∈	PROPN
ejpam-6134	34	13	ζ(ω	ζ(ω	PROPN
ejpam-6134	34	14	)	)	PUNCT
ejpam-6134	34	15	,	,	PUNCT
ejpam-6134	34	16	then	then	ADV
ejpam-6134	34	17	f	f	PROPN
ejpam-6134	34	18	is	be	AUX
ejpam-6134	34	19	called	call	VERB
ejpam-6134	34	20	a	a	DET
ejpam-6134	34	21	centrally	centrally	ADV
ejpam-6134	34	22	extended	extended	ADJ
ejpam-6134	34	23	φ	φ	NOUN
ejpam-6134	34	24	-	-	NOUN
ejpam-6134	34	25	homoderivation	homoderivation	NOUN
ejpam-6134	34	26	(	(	PUNCT
ejpam-6134	34	27	ce	ce	PROPN
ejpam-6134	34	28	-	-	PUNCT
ejpam-6134	34	29	φ	φ	NOUN
ejpam-6134	34	30	-	-	PUNCT
ejpam-6134	34	31	homoderivation	homoderivation	NOUN
ejpam-6134	34	32	)	)	PUNCT
ejpam-6134	34	33	.	.	PUNCT
ejpam-6134	35	1	the	the	DET
ejpam-6134	35	2	authors	author	NOUN
ejpam-6134	35	3	proved	prove	VERB
ejpam-6134	35	4	several	several	ADJ
ejpam-6134	35	5	results	result	NOUN
ejpam-6134	35	6	in	in	ADP
ejpam-6134	35	7	[	[	X
ejpam-6134	35	8	8	8	NUM
ejpam-6134	35	9	,	,	PUNCT
ejpam-6134	35	10	corollaries	corollary	NOUN
ejpam-6134	35	11	1	1	NUM
ejpam-6134	35	12	6	6	NUM
ejpam-6134	35	13	]	]	PUNCT
ejpam-6134	35	14	,	,	PUNCT
ejpam-6134	35	15	which	which	PRON
ejpam-6134	35	16	appear	appear	VERB
ejpam-6134	35	17	as	as	ADP
ejpam-6134	35	18	special	special	ADJ
ejpam-6134	35	19	cases	case	NOUN
ejpam-6134	35	20	of	of	ADP
ejpam-6134	35	21	our	our	PRON
ejpam-6134	35	22	results	result	NOUN
ejpam-6134	35	23	and	and	CCONJ
ejpam-6134	35	24	are	be	AUX
ejpam-6134	35	25	summarized	summarize	VERB
ejpam-6134	35	26	as	as	SCONJ
ejpam-6134	35	27	follows	follow	VERB
ejpam-6134	35	28	.	.	PUNCT
ejpam-6134	36	1	let	let	VERB
ejpam-6134	36	2	ω	ω	PRON
ejpam-6134	36	3	be	be	AUX
ejpam-6134	36	4	a	a	DET
ejpam-6134	36	5	semiprime	semiprime	NOUN
ejpam-6134	36	6	ring	ring	NOUN
ejpam-6134	36	7	with	with	ADP
ejpam-6134	36	8	center	center	NOUN
ejpam-6134	36	9	ζ(ω	ζ(ω	PROPN
ejpam-6134	36	10	)	)	PUNCT
ejpam-6134	36	11	,	,	PUNCT
ejpam-6134	36	12	and	and	CCONJ
ejpam-6134	36	13	let	let	VERB
ejpam-6134	36	14	φ	φ	PROPN
ejpam-6134	36	15	be	be	AUX
ejpam-6134	36	16	an	an	DET
ejpam-6134	36	17	epimorphism	epimorphism	NOUN
ejpam-6134	36	18	of	of	ADP
ejpam-6134	36	19	ω	ω	PROPN
ejpam-6134	36	20	.	.	PUNCT
ejpam-6134	37	1	the	the	DET
ejpam-6134	37	2	following	follow	VERB
ejpam-6134	37	3	hold	hold	NOUN
ejpam-6134	37	4	:	:	PUNCT
ejpam-6134	37	5	(	(	PUNCT
ejpam-6134	37	6	i	i	NOUN
ejpam-6134	37	7	)	)	PUNCT
ejpam-6134	37	8	if	if	SCONJ
ejpam-6134	37	9	the	the	DET
ejpam-6134	37	10	unique	unique	ADJ
ejpam-6134	37	11	central	central	ADJ
ejpam-6134	37	12	ideal	ideal	NOUN
ejpam-6134	37	13	of	of	ADP
ejpam-6134	37	14	ω	ω	PROPN
ejpam-6134	37	15	is	be	AUX
ejpam-6134	37	16	the	the	DET
ejpam-6134	37	17	zero	zero	NUM
ejpam-6134	37	18	ideal	ideal	NOUN
ejpam-6134	37	19	,	,	PUNCT
ejpam-6134	37	20	then	then	ADV
ejpam-6134	37	21	each	each	DET
ejpam-6134	37	22	nilpotent	nilpotent	ADJ
ejpam-6134	37	23	ce	ce	PROPN
ejpam-6134	37	24	-	-	PUNCT
ejpam-6134	37	25	homoderivation	homoderivation	NOUN
ejpam-6134	37	26	is	be	AUX
ejpam-6134	37	27	an	an	DET
ejpam-6134	37	28	ordinary	ordinary	ADJ
ejpam-6134	37	29	homoderivation	homoderivation	NOUN
ejpam-6134	37	30	(	(	PUNCT
ejpam-6134	37	31	corollary	corollary	ADJ
ejpam-6134	37	32	1	1	NUM
ejpam-6134	37	33	)	)	PUNCT
ejpam-6134	37	34	.	.	PUNCT
ejpam-6134	38	1	m.m	m.m	PROPN
ejpam-6134	38	2	.	.	PROPN
ejpam-6134	38	3	el	el	PROPN
ejpam-6134	38	4	-	-	PUNCT
ejpam-6134	38	5	soufi	soufi	ADJ
ejpam-6134	38	6	,	,	PUNCT
ejpam-6134	38	7	m.	m.	NOUN
ejpam-6134	38	8	almulhem	almulhem	NOUN
ejpam-6134	38	9	,	,	PUNCT
ejpam-6134	38	10	m.	m.	NOUN
ejpam-6134	38	11	s.	s.	PROPN
ejpam-6134	38	12	tammam	tammam	PROPN
ejpam-6134	38	13	el	el	PROPN
ejpam-6134	38	14	-	-	PROPN
ejpam-6134	38	15	sayiad	sayiad	PROPN
ejpam-6134	38	16	/	/	SYM
ejpam-6134	38	17	eur	eur	PROPN
ejpam-6134	38	18	.	.	PUNCT
ejpam-6134	39	1	j.	j.	PROPN
ejpam-6134	39	2	pure	pure	PROPN
ejpam-6134	39	3	appl	appl	PROPN
ejpam-6134	39	4	.	.	PROPN
ejpam-6134	39	5	math	math	PROPN
ejpam-6134	39	6	,	,	PUNCT
ejpam-6134	39	7	18	18	NUM
ejpam-6134	39	8	(	(	PUNCT
ejpam-6134	39	9	4	4	NUM
ejpam-6134	39	10	)	)	PUNCT
ejpam-6134	39	11	(	(	PUNCT
ejpam-6134	39	12	2025	2025	NUM
ejpam-6134	39	13	)	)	PUNCT
ejpam-6134	39	14	,	,	PUNCT
ejpam-6134	39	15	6134	6134	NUM
ejpam-6134	39	16	3	3	NUM
ejpam-6134	39	17	of	of	ADP
ejpam-6134	39	18	12	12	NUM
ejpam-6134	39	19	(	(	PUNCT
ejpam-6134	39	20	ii	ii	NOUN
ejpam-6134	39	21	)	)	PUNCT
ejpam-6134	39	22	assuming	assume	VERB
ejpam-6134	39	23	that	that	SCONJ
ejpam-6134	39	24	ζ(ω	ζ(ω	PROPN
ejpam-6134	39	25	)	)	PUNCT
ejpam-6134	39	26	contains	contain	VERB
ejpam-6134	39	27	no	no	DET
ejpam-6134	39	28	nonzero	nonzero	ADJ
ejpam-6134	39	29	nilpotent	nilpotent	ADJ
ejpam-6134	39	30	elements	element	NOUN
ejpam-6134	39	31	,	,	PUNCT
ejpam-6134	39	32	then	then	ADV
ejpam-6134	39	33	every	every	DET
ejpam-6134	39	34	nilpotent	nilpotent	ADJ
ejpam-6134	39	35	ce	ce	PROPN
ejpam-6134	39	36	-	-	PUNCT
ejpam-6134	39	37	φ	φ	PROPN
ejpam-6134	39	38	-	-	PUNCT
ejpam-6134	39	39	homoderivation	homoderivation	NOUN
ejpam-6134	39	40	which	which	PRON
ejpam-6134	39	41	commute	commute	VERB
ejpam-6134	39	42	with	with	ADP
ejpam-6134	39	43	φ	φ	PROPN
ejpam-6134	39	44	preserves	preserves	PROPN
ejpam-6134	39	45	ζ(ω	ζ(ω	PROPN
ejpam-6134	39	46	)	)	PUNCT
ejpam-6134	39	47	(	(	PUNCT
ejpam-6134	39	48	corollary	corollary	ADJ
ejpam-6134	39	49	2	2	NUM
ejpam-6134	39	50	)	)	PUNCT
ejpam-6134	39	51	.	.	PUNCT
ejpam-6134	40	1	(	(	PUNCT
ejpam-6134	40	2	iii	iii	X
ejpam-6134	40	3	)	)	PUNCT
ejpam-6134	40	4	assuming	assume	VERB
ejpam-6134	40	5	that	that	SCONJ
ejpam-6134	40	6	ζ(ω	ζ(ω	PROPN
ejpam-6134	40	7	)	)	PUNCT
ejpam-6134	40	8	contains	contain	VERB
ejpam-6134	40	9	no	no	DET
ejpam-6134	40	10	nonzero	nonzero	ADJ
ejpam-6134	40	11	nilpotent	nilpotent	ADJ
ejpam-6134	40	12	elements	element	NOUN
ejpam-6134	40	13	,	,	PUNCT
ejpam-6134	40	14	then	then	ADV
ejpam-6134	40	15	every	every	DET
ejpam-6134	40	16	nilpotent	nilpotent	ADJ
ejpam-6134	40	17	ce	ce	PROPN
ejpam-6134	40	18	-	-	PUNCT
ejpam-6134	40	19	homoderivations	homoderivation	NOUN
ejpam-6134	40	20	stabilize	stabilize	VERB
ejpam-6134	40	21	the	the	DET
ejpam-6134	40	22	center	center	NOUN
ejpam-6134	40	23	ζ(ω	ζ(ω	PROPN
ejpam-6134	40	24	)	)	PUNCT
ejpam-6134	40	25	(	(	PUNCT
ejpam-6134	40	26	corollary	corollary	ADJ
ejpam-6134	40	27	3	3	NUM
ejpam-6134	40	28	)	)	PUNCT
ejpam-6134	40	29	.	.	PUNCT
ejpam-6134	41	1	moreover	moreover	ADV
ejpam-6134	41	2	,	,	PUNCT
ejpam-6134	41	3	when	when	SCONJ
ejpam-6134	41	4	a	a	DET
ejpam-6134	41	5	prime	prime	ADJ
ejpam-6134	41	6	ring	ring	NOUN
ejpam-6134	41	7	admits	admit	VERB
ejpam-6134	41	8	a	a	PRON
ejpam-6134	41	9	nonzero	nonzero	PROPN
ejpam-6134	41	10	nilpotent	nilpotent	ADJ
ejpam-6134	41	11	ce	ce	PROPN
ejpam-6134	41	12	-	-	PUNCT
ejpam-6134	41	13	homoderivation	homoderivation	NOUN
ejpam-6134	41	14	satisfying	satisfy	VERB
ejpam-6134	41	15	one	one	NUM
ejpam-6134	41	16	of	of	ADP
ejpam-6134	41	17	the	the	DET
ejpam-6134	41	18	following	following	ADJ
ejpam-6134	41	19	conditions	condition	NOUN
ejpam-6134	41	20	:	:	PUNCT
ejpam-6134	41	21	(	(	PUNCT
ejpam-6134	41	22	i	i	NOUN
ejpam-6134	41	23	)	)	PUNCT
ejpam-6134	41	24	the	the	DET
ejpam-6134	41	25	map	map	NOUN
ejpam-6134	41	26	is	be	AUX
ejpam-6134	41	27	not	not	PART
ejpam-6134	41	28	a	a	DET
ejpam-6134	41	29	homoderivation	homoderivation	NOUN
ejpam-6134	41	30	(	(	PUNCT
ejpam-6134	41	31	corollary	corollary	ADJ
ejpam-6134	41	32	4	4	NUM
ejpam-6134	41	33	)	)	PUNCT
ejpam-6134	41	34	,	,	PUNCT
ejpam-6134	41	35	(	(	PUNCT
ejpam-6134	41	36	ii	ii	NOUN
ejpam-6134	41	37	)	)	PUNCT
ejpam-6134	41	38	the	the	DET
ejpam-6134	41	39	map	map	NOUN
ejpam-6134	41	40	has	have	VERB
ejpam-6134	41	41	a	a	DET
ejpam-6134	41	42	nonzero	nonzero	ADJ
ejpam-6134	41	43	image	image	NOUN
ejpam-6134	41	44	of	of	ADP
ejpam-6134	41	45	zero	zero	NUM
ejpam-6134	41	46	(	(	PUNCT
ejpam-6134	41	47	corollary	corollary	ADJ
ejpam-6134	41	48	5	5	NUM
ejpam-6134	41	49	)	)	PUNCT
ejpam-6134	41	50	,	,	PUNCT
ejpam-6134	41	51	(	(	PUNCT
ejpam-6134	41	52	iii	iii	X
ejpam-6134	41	53	)	)	PUNCT
ejpam-6134	41	54	the	the	DET
ejpam-6134	41	55	map	map	NOUN
ejpam-6134	41	56	annihilates	annihilate	VERB
ejpam-6134	41	57	all	all	DET
ejpam-6134	41	58	lie	lie	NOUN
ejpam-6134	41	59	(	(	PUNCT
ejpam-6134	41	60	jordan	jordan	PROPN
ejpam-6134	41	61	)	)	PUNCT
ejpam-6134	41	62	products	product	NOUN
ejpam-6134	41	63	(	(	PUNCT
ejpam-6134	41	64	corollary	corollary	ADJ
ejpam-6134	41	65	6	6	NUM
ejpam-6134	41	66	)	)	PUNCT
ejpam-6134	41	67	,	,	PUNCT
ejpam-6134	41	68	then	then	ADV
ejpam-6134	41	69	the	the	DET
ejpam-6134	41	70	ring	ring	PROPN
ejpam-6134	41	71	ω	ω	PROPN
ejpam-6134	41	72	is	be	AUX
ejpam-6134	41	73	commutative	commutative	ADJ
ejpam-6134	41	74	.	.	PUNCT
ejpam-6134	42	1	these	these	DET
ejpam-6134	42	2	contributions	contribution	NOUN
ejpam-6134	42	3	culminated	culminate	VERB
ejpam-6134	42	4	in	in	ADP
ejpam-6134	42	5	the	the	DET
ejpam-6134	42	6	study	study	NOUN
ejpam-6134	42	7	of	of	ADP
ejpam-6134	42	8	centrally	centrally	ADV
ejpam-6134	42	9	extended	extended	ADJ
ejpam-6134	42	10	(	(	PUNCT
ejpam-6134	42	11	ϕ,m)homoderivations	ϕ,m)homoderivation	NOUN
ejpam-6134	42	12	,	,	PUNCT
ejpam-6134	42	13	which	which	PRON
ejpam-6134	42	14	elegantly	elegantly	ADV
ejpam-6134	42	15	unify	unify	VERB
ejpam-6134	42	16	prior	prior	ADJ
ejpam-6134	42	17	derivation	derivation	NOUN
ejpam-6134	42	18	concepts	concept	NOUN
ejpam-6134	42	19	while	while	SCONJ
ejpam-6134	42	20	preserving	preserve	VERB
ejpam-6134	42	21	essential	essential	ADJ
ejpam-6134	42	22	algebraic	algebraic	ADJ
ejpam-6134	42	23	properties	property	NOUN
ejpam-6134	42	24	.	.	PUNCT
ejpam-6134	43	1	this	this	DET
ejpam-6134	43	2	paper	paper	NOUN
ejpam-6134	43	3	focuses	focus	VERB
ejpam-6134	43	4	on	on	ADP
ejpam-6134	43	5	a	a	DET
ejpam-6134	43	6	thorough	thorough	ADJ
ejpam-6134	43	7	exploration	exploration	NOUN
ejpam-6134	43	8	of	of	ADP
ejpam-6134	43	9	centrally	centrally	ADV
ejpam-6134	43	10	extended	extended	ADJ
ejpam-6134	43	11	(	(	PUNCT
ejpam-6134	43	12	ϕ,m)−homoderivations	ϕ,m)−homoderivation	NOUN
ejpam-6134	43	13	,	,	PUNCT
ejpam-6134	43	14	which	which	PRON
ejpam-6134	43	15	incorporate	incorporate	VERB
ejpam-6134	43	16	both	both	PRON
ejpam-6134	43	17	a	a	DET
ejpam-6134	43	18	mapping	mapping	NOUN
ejpam-6134	43	19	ϕ	ϕ	NOUN
ejpam-6134	43	20	and	and	CCONJ
ejpam-6134	43	21	an	an	DET
ejpam-6134	43	22	integer	integer	NOUN
ejpam-6134	43	23	m	m	VERB
ejpam-6134	43	24	while	while	SCONJ
ejpam-6134	43	25	ensuring	ensure	VERB
ejpam-6134	43	26	that	that	SCONJ
ejpam-6134	43	27	the	the	DET
ejpam-6134	43	28	extended	extended	ADJ
ejpam-6134	43	29	map	map	NOUN
ejpam-6134	43	30	remains	remain	VERB
ejpam-6134	43	31	within	within	ADP
ejpam-6134	43	32	the	the	DET
ejpam-6134	43	33	center	center	NOUN
ejpam-6134	43	34	of	of	ADP
ejpam-6134	43	35	the	the	DET
ejpam-6134	43	36	ring	ring	NOUN
ejpam-6134	43	37	,	,	PUNCT
ejpam-6134	43	38	and	and	CCONJ
ejpam-6134	43	39	their	their	PRON
ejpam-6134	43	40	impact	impact	NOUN
ejpam-6134	43	41	on	on	ADP
ejpam-6134	43	42	ring	ring	NOUN
ejpam-6134	43	43	commutativity	commutativity	NOUN
ejpam-6134	43	44	.	.	PUNCT
ejpam-6134	44	1	definition	definition	NOUN
ejpam-6134	44	2	1	1	NUM
ejpam-6134	44	3	.	.	PUNCT
ejpam-6134	45	1	let	let	VERB
ejpam-6134	45	2	f	f	PRON
ejpam-6134	45	3	be	be	AUX
ejpam-6134	45	4	an	an	DET
ejpam-6134	45	5	additive	additive	ADJ
ejpam-6134	45	6	mapping	mapping	NOUN
ejpam-6134	45	7	on	on	ADP
ejpam-6134	45	8	a	a	DET
ejpam-6134	45	9	ring	ring	NOUN
ejpam-6134	45	10	ω	ω	PROPN
ejpam-6134	45	11	,	,	PUNCT
ejpam-6134	45	12	φ	φ	PROPN
ejpam-6134	45	13	a	a	DET
ejpam-6134	45	14	mapping	mapping	NOUN
ejpam-6134	45	15	on	on	ADP
ejpam-6134	45	16	ω	ω	PROPN
ejpam-6134	45	17	,	,	PUNCT
ejpam-6134	45	18	m	m	VERB
ejpam-6134	45	19	an	an	DET
ejpam-6134	45	20	integer	integer	NOUN
ejpam-6134	45	21	,	,	PUNCT
ejpam-6134	45	22	and	and	CCONJ
ejpam-6134	45	23	s	s	X
ejpam-6134	45	24	,	,	PUNCT
ejpam-6134	45	25	u	u	VERB
ejpam-6134	45	26	any	any	DET
ejpam-6134	45	27	two	two	NUM
ejpam-6134	45	28	elements	element	NOUN
ejpam-6134	45	29	in	in	ADP
ejpam-6134	45	30	ω	ω	PROPN
ejpam-6134	45	31	.	.	PUNCT
ejpam-6134	46	1	(	(	PUNCT
ejpam-6134	46	2	i	i	NOUN
ejpam-6134	46	3	)	)	PUNCT
ejpam-6134	46	4	if	if	SCONJ
ejpam-6134	46	5	f	f	PROPN
ejpam-6134	46	6	achieves	achieve	VERB
ejpam-6134	46	7	f(su	f(su	NOUN
ejpam-6134	46	8	)	)	PUNCT
ejpam-6134	46	9	=	=	SYM
ejpam-6134	46	10	f(s)φ(u	f(s)φ(u	NOUN
ejpam-6134	46	11	)	)	PUNCT
ejpam-6134	46	12	+	+	X
ejpam-6134	46	13	φ(s)f(u	φ(s)f(u	X
ejpam-6134	46	14	)	)	PUNCT
ejpam-6134	46	15	+	+	NOUN
ejpam-6134	46	16	mf(s)f(u	mf(s)f(u	NOUN
ejpam-6134	46	17	)	)	PUNCT
ejpam-6134	46	18	,	,	PUNCT
ejpam-6134	46	19	then	then	ADV
ejpam-6134	46	20	f	f	PROPN
ejpam-6134	46	21	is	be	AUX
ejpam-6134	46	22	called	call	VERB
ejpam-6134	46	23	a	a	DET
ejpam-6134	46	24	(	(	PUNCT
ejpam-6134	46	25	φ	φ	NOUN
ejpam-6134	46	26	,	,	PUNCT
ejpam-6134	46	27	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	46	28	.	.	PUNCT
ejpam-6134	47	1	(	(	PUNCT
ejpam-6134	47	2	ii	ii	NOUN
ejpam-6134	47	3	)	)	PUNCT
ejpam-6134	47	4	if	if	SCONJ
ejpam-6134	47	5	f	f	PROPN
ejpam-6134	47	6	achieves	achieve	VERB
ejpam-6134	47	7	f(s+	f(s+	PROPN
ejpam-6134	47	8	u)−f(s)−f(u	u)−f(s)−f(u	ADJ
ejpam-6134	47	9	)	)	PUNCT
ejpam-6134	47	10	∈	∈	PROPN
ejpam-6134	47	11	ζ(ω	ζ(ω	PROPN
ejpam-6134	47	12	)	)	PUNCT
ejpam-6134	47	13	and	and	CCONJ
ejpam-6134	47	14	f(su)−	f(su)−	NOUN
ejpam-6134	47	15	{	{	PUNCT
ejpam-6134	47	16	f(s)φ(u	f(s)φ(u	NOUN
ejpam-6134	47	17	)	)	PUNCT
ejpam-6134	47	18	+	+	X
ejpam-6134	47	19	φ(s)f(u	φ(s)f(u	X
ejpam-6134	47	20	)	)	PUNCT
ejpam-6134	48	1	+	+	NOUN
ejpam-6134	48	2	mf(s)f(u	mf(s)f(u	NOUN
ejpam-6134	48	3	)	)	PUNCT
ejpam-6134	48	4	}	}	PUNCT
ejpam-6134	48	5	∈	∈	PROPN
ejpam-6134	48	6	ζ(ω	ζ(ω	PROPN
ejpam-6134	48	7	)	)	PUNCT
ejpam-6134	48	8	,	,	PUNCT
ejpam-6134	48	9	then	then	ADV
ejpam-6134	48	10	f	f	PROPN
ejpam-6134	48	11	is	be	AUX
ejpam-6134	48	12	called	call	VERB
ejpam-6134	48	13	a	a	DET
ejpam-6134	48	14	centrally	centrally	ADV
ejpam-6134	48	15	extended	extended	ADJ
ejpam-6134	48	16	(	(	PUNCT
ejpam-6134	48	17	φ	φ	NOUN
ejpam-6134	48	18	,	,	PUNCT
ejpam-6134	48	19	m)-homoderivation	m)-homoderivation	PROPN
ejpam-6134	48	20	(	(	PUNCT
ejpam-6134	48	21	ce	ce	PROPN
ejpam-6134	48	22	−	−	PROPN
ejpam-6134	48	23	(	(	PUNCT
ejpam-6134	48	24	φ	φ	PROPN
ejpam-6134	48	25	,	,	PUNCT
ejpam-6134	48	26	m)homoderivation	m)homoderivation	NOUN
ejpam-6134	48	27	)	)	PUNCT
ejpam-6134	48	28	.	.	PUNCT
ejpam-6134	49	1	the	the	DET
ejpam-6134	49	2	concept	concept	NOUN
ejpam-6134	49	3	of	of	ADP
ejpam-6134	49	4	nil	nil	ADJ
ejpam-6134	49	5	and	and	CCONJ
ejpam-6134	49	6	nilpotent	nilpotent	ADJ
ejpam-6134	49	7	derivations	derivation	NOUN
ejpam-6134	49	8	was	be	AUX
ejpam-6134	49	9	first	first	ADV
ejpam-6134	49	10	developed	develop	VERB
ejpam-6134	49	11	by	by	ADP
ejpam-6134	49	12	chung	chung	PROPN
ejpam-6134	50	1	[	[	X
ejpam-6134	50	2	9	9	NUM
ejpam-6134	50	3	]	]	PUNCT
ejpam-6134	50	4	.	.	PUNCT
ejpam-6134	51	1	let	let	VERB
ejpam-6134	51	2	ω	ω	PRON
ejpam-6134	51	3	be	be	AUX
ejpam-6134	51	4	a	a	DET
ejpam-6134	51	5	ring	ring	NOUN
ejpam-6134	51	6	endowed	endow	VERB
ejpam-6134	51	7	with	with	ADP
ejpam-6134	51	8	a	a	DET
ejpam-6134	51	9	derivation	derivation	NOUN
ejpam-6134	51	10	δ	δ	NOUN
ejpam-6134	51	11	.	.	PUNCT
ejpam-6134	52	1	if	if	SCONJ
ejpam-6134	52	2	n	n	NOUN
ejpam-6134	52	3	=	=	SYM
ejpam-6134	52	4	n(r	n(r	NOUN
ejpam-6134	52	5	)	)	PUNCT
ejpam-6134	52	6	∈	∈	PROPN
ejpam-6134	52	7	z+	z+	NUM
ejpam-6134	52	8	exists	exist	VERB
ejpam-6134	52	9	for	for	ADP
ejpam-6134	52	10	each	each	DET
ejpam-6134	52	11	r	r	NOUN
ejpam-6134	52	12	∈	∈	PROPN
ejpam-6134	52	13	ω	ω	NOUN
ejpam-6134	52	14	with	with	ADP
ejpam-6134	52	15	δn(r	δn(r	NOUN
ejpam-6134	52	16	)	)	PUNCT
ejpam-6134	52	17	=	=	SYM
ejpam-6134	52	18	0	0	NUM
ejpam-6134	52	19	,	,	PUNCT
ejpam-6134	52	20	then	then	ADV
ejpam-6134	52	21	δ	δ	PROPN
ejpam-6134	52	22	is	be	AUX
ejpam-6134	52	23	said	say	VERB
ejpam-6134	52	24	to	to	PART
ejpam-6134	52	25	be	be	AUX
ejpam-6134	52	26	nil	nil	ADJ
ejpam-6134	52	27	.	.	PUNCT
ejpam-6134	53	1	here	here	ADV
ejpam-6134	53	2	,	,	PUNCT
ejpam-6134	53	3	the	the	DET
ejpam-6134	53	4	derivation	derivation	NOUN
ejpam-6134	53	5	δ	δ	PROPN
ejpam-6134	53	6	is	be	AUX
ejpam-6134	53	7	referred	refer	VERB
ejpam-6134	53	8	to	to	PART
ejpam-6134	53	9	be	be	AUX
ejpam-6134	53	10	nilpotent	nilpotent	ADJ
ejpam-6134	53	11	if	if	SCONJ
ejpam-6134	53	12	the	the	DET
ejpam-6134	53	13	integer	integer	NOUN
ejpam-6134	53	14	n	n	AUX
ejpam-6134	53	15	may	may	AUX
ejpam-6134	53	16	be	be	AUX
ejpam-6134	53	17	freely	freely	ADV
ejpam-6134	53	18	extracted	extract	VERB
ejpam-6134	53	19	from	from	ADP
ejpam-6134	53	20	r.	r.	PROPN
ejpam-6134	53	21	definition	definition	NOUN
ejpam-6134	53	22	2	2	NUM
ejpam-6134	53	23	.	.	PUNCT
ejpam-6134	54	1	let	let	VERB
ejpam-6134	54	2	f	f	PROPN
ejpam-6134	54	3	and	and	CCONJ
ejpam-6134	54	4	φ	φ	PROPN
ejpam-6134	54	5	be	be	VERB
ejpam-6134	54	6	two	two	NUM
ejpam-6134	54	7	maps	map	NOUN
ejpam-6134	54	8	on	on	ADP
ejpam-6134	54	9	a	a	DET
ejpam-6134	54	10	ring	ring	NOUN
ejpam-6134	54	11	ω	ω	NOUN
ejpam-6134	54	12	and	and	CCONJ
ejpam-6134	54	13	s	s	PROPN
ejpam-6134	54	14	⊆	⊆	NUM
ejpam-6134	54	15	ω	ω	NOUN
ejpam-6134	54	16	.	.	PUNCT
ejpam-6134	55	1	if	if	SCONJ
ejpam-6134	55	2	f	f	PROPN
ejpam-6134	55	3	n(s	n(s	PROPN
ejpam-6134	55	4	)	)	PUNCT
ejpam-6134	55	5	=	=	SYM
ejpam-6134	55	6	(	(	PUNCT
ejpam-6134	55	7	0	0	NUM
ejpam-6134	55	8	)	)	PUNCT
ejpam-6134	55	9	for	for	ADP
ejpam-6134	55	10	some	some	PRON
ejpam-6134	55	11	n	n	PRON
ejpam-6134	55	12	∈	∈	PROPN
ejpam-6134	55	13	z+	z+	NUM
ejpam-6134	55	14	−	−	PROPN
ejpam-6134	55	15	{	{	PUNCT
ejpam-6134	55	16	1	1	NUM
ejpam-6134	55	17	}	}	PUNCT
ejpam-6134	55	18	,	,	PUNCT
ejpam-6134	55	19	then	then	ADV
ejpam-6134	55	20	f	f	PROPN
ejpam-6134	55	21	is	be	AUX
ejpam-6134	55	22	said	say	VERB
ejpam-6134	55	23	to	to	PART
ejpam-6134	55	24	be	be	AUX
ejpam-6134	55	25	nilpotent	nilpotent	ADJ
ejpam-6134	55	26	on	on	ADP
ejpam-6134	55	27	s.	s.	PROPN
ejpam-6134	55	28	two	two	NUM
ejpam-6134	55	29	maps	map	NOUN
ejpam-6134	55	30	f	f	PROPN
ejpam-6134	55	31	and	and	CCONJ
ejpam-6134	55	32	φ	φ	PROPN
ejpam-6134	55	33	are	be	AUX
ejpam-6134	55	34	called	call	VERB
ejpam-6134	55	35	commuting	commute	VERB
ejpam-6134	55	36	on	on	ADP
ejpam-6134	55	37	s	s	PRON
ejpam-6134	55	38	if	if	SCONJ
ejpam-6134	55	39	φ(f(s	φ(f(	NOUN
ejpam-6134	55	40	)	)	PUNCT
ejpam-6134	55	41	)	)	PUNCT
ejpam-6134	56	1	=	=	SYM
ejpam-6134	56	2	f(φ(s	f(φ(s	PROPN
ejpam-6134	56	3	)	)	PUNCT
ejpam-6134	56	4	)	)	PUNCT
ejpam-6134	56	5	,	,	PUNCT
ejpam-6134	56	6	for	for	ADP
ejpam-6134	56	7	each	each	DET
ejpam-6134	56	8	s	s	PROPN
ejpam-6134	56	9	∈	∈	PROPN
ejpam-6134	56	10	s.	s.	PROPN
ejpam-6134	56	11	m.m	m.m	PROPN
ejpam-6134	56	12	.	.	PROPN
ejpam-6134	56	13	el	el	PROPN
ejpam-6134	56	14	-	-	PUNCT
ejpam-6134	56	15	soufi	soufi	ADJ
ejpam-6134	56	16	,	,	PUNCT
ejpam-6134	56	17	m.	m.	NOUN
ejpam-6134	56	18	almulhem	almulhem	NOUN
ejpam-6134	56	19	,	,	PUNCT
ejpam-6134	56	20	m.	m.	NOUN
ejpam-6134	56	21	s.	s.	PROPN
ejpam-6134	56	22	tammam	tammam	PROPN
ejpam-6134	56	23	el	el	PROPN
ejpam-6134	56	24	-	-	PROPN
ejpam-6134	56	25	sayiad	sayiad	PROPN
ejpam-6134	56	26	/	/	SYM
ejpam-6134	56	27	eur	eur	PROPN
ejpam-6134	56	28	.	.	PUNCT
ejpam-6134	57	1	j.	j.	PROPN
ejpam-6134	57	2	pure	pure	PROPN
ejpam-6134	57	3	appl	appl	PROPN
ejpam-6134	57	4	.	.	PROPN
ejpam-6134	57	5	math	math	PROPN
ejpam-6134	57	6	,	,	PUNCT
ejpam-6134	57	7	18	18	NUM
ejpam-6134	57	8	(	(	PUNCT
ejpam-6134	57	9	4	4	NUM
ejpam-6134	57	10	)	)	PUNCT
ejpam-6134	57	11	(	(	PUNCT
ejpam-6134	57	12	2025	2025	NUM
ejpam-6134	57	13	)	)	PUNCT
ejpam-6134	57	14	,	,	PUNCT
ejpam-6134	57	15	6134	6134	NUM
ejpam-6134	57	16	4	4	NUM
ejpam-6134	57	17	of	of	ADP
ejpam-6134	57	18	12	12	NUM
ejpam-6134	57	19	remark	remark	NOUN
ejpam-6134	57	20	1	1	NUM
ejpam-6134	57	21	.	.	PUNCT
ejpam-6134	58	1	(	(	PUNCT
ejpam-6134	58	2	1	1	X
ejpam-6134	58	3	)	)	PUNCT
ejpam-6134	58	4	any	any	DET
ejpam-6134	58	5	homoderivation	homoderivation	NOUN
ejpam-6134	58	6	is	be	AUX
ejpam-6134	58	7	an	an	DET
ejpam-6134	58	8	(	(	PUNCT
ejpam-6134	58	9	iid	iid	NOUN
ejpam-6134	58	10	,	,	PUNCT
ejpam-6134	58	11	1)-homoderivation	1)-homoderivation	NUM
ejpam-6134	58	12	.	.	PUNCT
ejpam-6134	59	1	(	(	PUNCT
ejpam-6134	59	2	2	2	X
ejpam-6134	59	3	)	)	PUNCT
ejpam-6134	59	4	any	any	DET
ejpam-6134	59	5	ce−homoderivation	ce−homoderivation	NOUN
ejpam-6134	59	6	on	on	ADP
ejpam-6134	59	7	ω	ω	PROPN
ejpam-6134	59	8	is	be	AUX
ejpam-6134	59	9	a	a	DET
ejpam-6134	59	10	ce	ce	NOUN
ejpam-6134	59	11	−	−	PROPN
ejpam-6134	59	12	(	(	PUNCT
ejpam-6134	59	13	iid	iid	PROPN
ejpam-6134	59	14	,	,	PUNCT
ejpam-6134	59	15	1)-homoderivation	1)-homoderivation	NUM
ejpam-6134	59	16	of	of	ADP
ejpam-6134	59	17	ω	ω	PROPN
ejpam-6134	59	18	,	,	PUNCT
ejpam-6134	59	19	where	where	SCONJ
ejpam-6134	59	20	iid	iid	NOUN
ejpam-6134	59	21	refers	refer	VERB
ejpam-6134	59	22	to	to	ADP
ejpam-6134	59	23	the	the	DET
ejpam-6134	59	24	identity	identity	NOUN
ejpam-6134	59	25	map	map	NOUN
ejpam-6134	59	26	.	.	PUNCT
ejpam-6134	60	1	(	(	PUNCT
ejpam-6134	60	2	3	3	X
ejpam-6134	60	3	)	)	PUNCT
ejpam-6134	60	4	according	accord	VERB
ejpam-6134	60	5	to	to	ADP
ejpam-6134	60	6	definition	definition	NOUN
ejpam-6134	60	7	1	1	NUM
ejpam-6134	60	8	,	,	PUNCT
ejpam-6134	60	9	any	any	DET
ejpam-6134	60	10	(	(	PUNCT
ejpam-6134	60	11	φ	φ	NOUN
ejpam-6134	60	12	,	,	PUNCT
ejpam-6134	60	13	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	60	14	is	be	AUX
ejpam-6134	60	15	a	a	DET
ejpam-6134	60	16	centrally	centrally	ADV
ejpam-6134	60	17	extended	extended	ADJ
ejpam-6134	60	18	(	(	PUNCT
ejpam-6134	60	19	φ	φ	NOUN
ejpam-6134	60	20	,	,	PUNCT
ejpam-6134	60	21	m)homoderivation	m)homoderivation	NOUN
ejpam-6134	60	22	,	,	PUNCT
ejpam-6134	60	23	but	but	CCONJ
ejpam-6134	60	24	the	the	DET
ejpam-6134	60	25	inverse	inverse	NOUN
ejpam-6134	60	26	(	(	PUNCT
ejpam-6134	60	27	in	in	ADP
ejpam-6134	60	28	general	general	ADJ
ejpam-6134	60	29	)	)	PUNCT
ejpam-6134	60	30	is	be	AUX
ejpam-6134	60	31	not	not	PART
ejpam-6134	60	32	true	true	ADJ
ejpam-6134	60	33	.	.	PUNCT
ejpam-6134	61	1	during	during	ADP
ejpam-6134	61	2	our	our	PRON
ejpam-6134	61	3	research	research	NOUN
ejpam-6134	61	4	,	,	PUNCT
ejpam-6134	61	5	we	we	PRON
ejpam-6134	61	6	will	will	AUX
ejpam-6134	61	7	use	use	VERB
ejpam-6134	61	8	the	the	DET
ejpam-6134	61	9	following	follow	VERB
ejpam-6134	61	10	facts	fact	NOUN
ejpam-6134	61	11	.	.	PUNCT
ejpam-6134	62	1	lemma	lemma	PROPN
ejpam-6134	62	2	1	1	NUM
ejpam-6134	62	3	.	.	PUNCT
ejpam-6134	63	1	[	[	X
ejpam-6134	63	2	10	10	NUM
ejpam-6134	63	3	,	,	PUNCT
ejpam-6134	63	4	lemma	lemma	PROPN
ejpam-6134	63	5	1(b	1(b	NUM
ejpam-6134	63	6	)	)	PUNCT
ejpam-6134	63	7	]	]	PUNCT
ejpam-6134	63	8	for	for	ADP
ejpam-6134	63	9	a	a	DET
ejpam-6134	63	10	prime	prime	ADJ
ejpam-6134	63	11	ring	ring	NOUN
ejpam-6134	63	12	ω	ω	PROPN
ejpam-6134	63	13	,	,	PUNCT
ejpam-6134	63	14	with	with	ADP
ejpam-6134	63	15	center	center	NOUN
ejpam-6134	63	16	ζ(ω	ζ(ω	PROPN
ejpam-6134	63	17	)	)	PUNCT
ejpam-6134	63	18	,	,	PUNCT
ejpam-6134	63	19	the	the	DET
ejpam-6134	63	20	centralizer	centralizer	NOUN
ejpam-6134	63	21	of	of	ADP
ejpam-6134	63	22	any	any	DET
ejpam-6134	63	23	nonzero	nonzero	ADJ
ejpam-6134	63	24	one	one	NUM
ejpam-6134	63	25	-	-	PUNCT
ejpam-6134	63	26	sided	sided	ADJ
ejpam-6134	63	27	ideal	ideal	NOUN
ejpam-6134	63	28	coincides	coincide	VERB
ejpam-6134	63	29	with	with	ADP
ejpam-6134	63	30	ζ(ω	ζ(ω	PROPN
ejpam-6134	63	31	)	)	PUNCT
ejpam-6134	63	32	.	.	PUNCT
ejpam-6134	64	1	consequently	consequently	ADV
ejpam-6134	64	2	,	,	PUNCT
ejpam-6134	64	3	if	if	SCONJ
ejpam-6134	64	4	there	there	PRON
ejpam-6134	64	5	exists	exist	VERB
ejpam-6134	64	6	a	a	DET
ejpam-6134	64	7	nonzero	nonzero	ADJ
ejpam-6134	64	8	right	right	ADJ
ejpam-6134	64	9	ideal	ideal	NOUN
ejpam-6134	64	10	lying	lie	VERB
ejpam-6134	64	11	in	in	ADP
ejpam-6134	64	12	ζ(ω	ζ(ω	PROPN
ejpam-6134	64	13	)	)	PUNCT
ejpam-6134	64	14	,	,	PUNCT
ejpam-6134	64	15	then	then	ADV
ejpam-6134	64	16	ω	ω	X
ejpam-6134	64	17	must	must	AUX
ejpam-6134	64	18	necessarily	necessarily	ADV
ejpam-6134	64	19	be	be	AUX
ejpam-6134	64	20	commutative	commutative	ADJ
ejpam-6134	64	21	.	.	PUNCT
ejpam-6134	65	1	lemma	lemma	PROPN
ejpam-6134	65	2	2	2	NUM
ejpam-6134	65	3	.	.	PUNCT
ejpam-6134	66	1	[	[	X
ejpam-6134	66	2	11	11	NUM
ejpam-6134	66	3	,	,	PUNCT
ejpam-6134	66	4	lemma	lemma	PROPN
ejpam-6134	66	5	4	4	NUM
ejpam-6134	66	6	]	]	PUNCT
ejpam-6134	66	7	in	in	ADP
ejpam-6134	66	8	a	a	DET
ejpam-6134	66	9	prime	prime	ADJ
ejpam-6134	66	10	ring	ring	NOUN
ejpam-6134	66	11	ω	ω	NOUN
ejpam-6134	66	12	,	,	PUNCT
ejpam-6134	66	13	if	if	SCONJ
ejpam-6134	66	14	both	both	DET
ejpam-6134	66	15	p	p	NOUN
ejpam-6134	66	16	and	and	CCONJ
ejpam-6134	66	17	qp	qp	ADV
ejpam-6134	66	18	lie	lie	VERB
ejpam-6134	66	19	in	in	ADP
ejpam-6134	66	20	the	the	DET
ejpam-6134	66	21	center	center	NOUN
ejpam-6134	66	22	and	and	CCONJ
ejpam-6134	66	23	p	p	NOUN
ejpam-6134	66	24	is	be	AUX
ejpam-6134	66	25	nonzero	nonzero	ADJ
ejpam-6134	66	26	,	,	PUNCT
ejpam-6134	66	27	then	then	ADV
ejpam-6134	66	28	the	the	DET
ejpam-6134	66	29	element	element	NOUN
ejpam-6134	66	30	q	q	PROPN
ejpam-6134	66	31	lies	lie	VERB
ejpam-6134	66	32	in	in	ADP
ejpam-6134	66	33	the	the	DET
ejpam-6134	66	34	center	center	NOUN
ejpam-6134	66	35	ζ(ω	ζ(ω	PROPN
ejpam-6134	66	36	)	)	PUNCT
ejpam-6134	66	37	.	.	PUNCT
ejpam-6134	67	1	2	2	X
ejpam-6134	67	2	.	.	X
ejpam-6134	67	3	examples	example	NOUN
ejpam-6134	67	4	of	of	ADP
ejpam-6134	67	5	ce	ce	PROPN
ejpam-6134	67	6	−	−	PROPN
ejpam-6134	67	7	(	(	PUNCT
ejpam-6134	67	8	φ	φ	PROPN
ejpam-6134	67	9	,	,	PUNCT
ejpam-6134	67	10	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	67	11	in	in	ADP
ejpam-6134	67	12	the	the	DET
ejpam-6134	67	13	following	following	ADJ
ejpam-6134	67	14	examples	example	NOUN
ejpam-6134	67	15	,	,	PUNCT
ejpam-6134	67	16	we	we	PRON
ejpam-6134	67	17	ensure	ensure	VERB
ejpam-6134	67	18	that	that	SCONJ
ejpam-6134	67	19	there	there	PRON
ejpam-6134	67	20	are	be	VERB
ejpam-6134	67	21	ce−(φ	ce−(φ	NOUN
ejpam-6134	67	22	,	,	PUNCT
ejpam-6134	67	23	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	67	24	maps	map	NOUN
ejpam-6134	67	25	.	.	PUNCT
ejpam-6134	68	1	example	example	NOUN
ejpam-6134	69	1	1	1	NUM
ejpam-6134	69	2	.	.	PUNCT
ejpam-6134	69	3	let	let	VERB
ejpam-6134	69	4	ω	ω	PROPN
ejpam-6134	69	5	=	=	PROPN
ejpam-6134	69	6	z6	z6	PROPN
ejpam-6134	69	7	.	.	PUNCT
ejpam-6134	70	1	suppose	suppose	VERB
ejpam-6134	70	2	that	that	SCONJ
ejpam-6134	70	3	f	f	PROPN
ejpam-6134	70	4	,	,	PUNCT
ejpam-6134	70	5	φ	φ	PROPN
ejpam-6134	70	6	:	:	PUNCT
ejpam-6134	70	7	z6	z6	PROPN
ejpam-6134	70	8	→	→	SYM
ejpam-6134	70	9	z6	z6	PROPN
ejpam-6134	70	10	are	be	AUX
ejpam-6134	70	11	mappings	mapping	NOUN
ejpam-6134	70	12	on	on	ADP
ejpam-6134	70	13	z6	z6	PROPN
ejpam-6134	70	14	so	so	SCONJ
ejpam-6134	70	15	that	that	DET
ejpam-6134	70	16	f(α	f(α	NOUN
ejpam-6134	70	17	)	)	PUNCT
ejpam-6134	70	18	=	=	SYM
ejpam-6134	70	19	4α	4α	NOUN
ejpam-6134	70	20	and	and	CCONJ
ejpam-6134	70	21	φ(α	φ(α	ADJ
ejpam-6134	70	22	)	)	PUNCT
ejpam-6134	71	1	=	=	SYM
ejpam-6134	71	2	3α	3α	NOUN
ejpam-6134	71	3	,	,	PUNCT
ejpam-6134	71	4	for	for	ADP
ejpam-6134	71	5	all	all	DET
ejpam-6134	71	6	α	α	PRON
ejpam-6134	71	7	∈	∈	PROPN
ejpam-6134	71	8	z6	z6	PROPN
ejpam-6134	71	9	.	.	PUNCT
ejpam-6134	72	1	so	so	ADV
ejpam-6134	72	2	,	,	PUNCT
ejpam-6134	72	3	f	f	PROPN
ejpam-6134	72	4	will	will	AUX
ejpam-6134	72	5	be	be	AUX
ejpam-6134	72	6	a	a	DET
ejpam-6134	72	7	(	(	PUNCT
ejpam-6134	72	8	φ	φ	PROPN
ejpam-6134	72	9	,	,	PUNCT
ejpam-6134	72	10	4)-homoderivation	4)-homoderivation	NOUN
ejpam-6134	72	11	,	,	PUNCT
ejpam-6134	72	12	where	where	SCONJ
ejpam-6134	72	13	φ	φ	PROPN
ejpam-6134	72	14	is	be	AUX
ejpam-6134	72	15	an	an	DET
ejpam-6134	72	16	endomorphism	endomorphism	NOUN
ejpam-6134	72	17	,	,	PUNCT
ejpam-6134	72	18	and	and	CCONJ
ejpam-6134	72	19	f	f	PROPN
ejpam-6134	72	20	,	,	PUNCT
ejpam-6134	72	21	φ	φ	PROPN
ejpam-6134	72	22	are	be	AUX
ejpam-6134	72	23	commuting	commute	VERB
ejpam-6134	72	24	on	on	ADP
ejpam-6134	72	25	z6	z6	PROPN
ejpam-6134	72	26	.	.	PUNCT
ejpam-6134	72	27	example	example	NOUN
ejpam-6134	73	1	2	2	NUM
ejpam-6134	73	2	.	.	PUNCT
ejpam-6134	73	3	let	let	VERB
ejpam-6134	73	4	ω	ω	NOUN
ejpam-6134	73	5	=	=	SYM
ejpam-6134	73	6	m2(z	m2(z	PROPN
ejpam-6134	73	7	)	)	PUNCT
ejpam-6134	73	8	⊕	⊕	PROPN
ejpam-6134	73	9	z6	z6	PROPN
ejpam-6134	73	10	be	be	AUX
ejpam-6134	73	11	the	the	DET
ejpam-6134	73	12	ring	ring	NOUN
ejpam-6134	73	13	with	with	ADP
ejpam-6134	73	14	center	center	NOUN
ejpam-6134	73	15	ζ(ω	ζ(ω	PROPN
ejpam-6134	73	16	)	)	PUNCT
ejpam-6134	73	17	=	=	PRON
ejpam-6134	73	18	{	{	PUNCT
ejpam-6134	73	19	(	(	PUNCT
ejpam-6134	73	20	ai	ai	NOUN
ejpam-6134	73	21	,	,	PUNCT
ejpam-6134	73	22	x	x	NOUN
ejpam-6134	73	23	)	)	PUNCT
ejpam-6134	73	24	:	:	PUNCT
ejpam-6134	73	25	a	a	DET
ejpam-6134	73	26	∈	∈	PROPN
ejpam-6134	73	27	z	z	NOUN
ejpam-6134	73	28	,	,	PUNCT
ejpam-6134	73	29	x	x	SYM
ejpam-6134	73	30	∈	∈	PROPN
ejpam-6134	73	31	z6and	z6and	NOUN
ejpam-6134	73	32	i	i	PRON
ejpam-6134	73	33	∈	∈	PROPN
ejpam-6134	73	34	m2(z	m2(z	X
ejpam-6134	73	35	)	)	PUNCT
ejpam-6134	73	36	be	be	VERB
ejpam-6134	73	37	the	the	DET
ejpam-6134	73	38	identity	identity	NOUN
ejpam-6134	73	39	matrix	matrix	NOUN
ejpam-6134	73	40	}	}	PUNCT
ejpam-6134	73	41	.	.	PUNCT
ejpam-6134	74	1	suppose	suppose	VERB
ejpam-6134	74	2	that	that	SCONJ
ejpam-6134	74	3	the	the	DET
ejpam-6134	74	4	maps	map	NOUN
ejpam-6134	74	5	φ	φ	PROPN
ejpam-6134	74	6	,	,	PUNCT
ejpam-6134	74	7	f	f	PROPN
ejpam-6134	74	8	:	:	PUNCT
ejpam-6134	74	9	ω	ω	PROPN
ejpam-6134	74	10	→	→	SYM
ejpam-6134	74	11	ω	ω	PROPN
ejpam-6134	74	12	so	so	SCONJ
ejpam-6134	74	13	that	that	SCONJ
ejpam-6134	74	14	φ(σ	φ(σ	PROPN
ejpam-6134	74	15	,	,	PUNCT
ejpam-6134	74	16	ρ	ρ	NOUN
ejpam-6134	74	17	)	)	PUNCT
ejpam-6134	74	18	=	=	SYM
ejpam-6134	74	19	(	(	PUNCT
ejpam-6134	74	20	σ	σ	PROPN
ejpam-6134	74	21	,	,	PUNCT
ejpam-6134	74	22	3ρ	3ρ	NUM
ejpam-6134	74	23	)	)	PUNCT
ejpam-6134	74	24	and	and	CCONJ
ejpam-6134	74	25	f(σ	f(σ	PROPN
ejpam-6134	74	26	,	,	PUNCT
ejpam-6134	74	27	ρ	ρ	NOUN
ejpam-6134	74	28	)	)	PUNCT
ejpam-6134	74	29	=	=	PRON
ejpam-6134	74	30	(	(	PUNCT
ejpam-6134	74	31	−σ	−σ	NOUN
ejpam-6134	74	32	,	,	PUNCT
ejpam-6134	74	33	3	3	NUM
ejpam-6134	74	34	−	−	PROPN
ejpam-6134	74	35	ρ	ρ	PROPN
ejpam-6134	74	36	)	)	PUNCT
ejpam-6134	74	37	for	for	ADP
ejpam-6134	74	38	any	any	DET
ejpam-6134	74	39	(	(	PUNCT
ejpam-6134	74	40	σ	σ	PROPN
ejpam-6134	74	41	,	,	PUNCT
ejpam-6134	74	42	ρ	ρ	PROPN
ejpam-6134	74	43	)	)	PUNCT
ejpam-6134	74	44	∈	∈	PROPN
ejpam-6134	74	45	ω	ω	PROPN
ejpam-6134	74	46	.	.	PUNCT
ejpam-6134	75	1	for	for	ADP
ejpam-6134	75	2	(	(	PUNCT
ejpam-6134	75	3	σ	σ	PROPN
ejpam-6134	75	4	,	,	PUNCT
ejpam-6134	75	5	ρ	ρ	PROPN
ejpam-6134	75	6	)	)	PUNCT
ejpam-6134	75	7	,	,	PUNCT
ejpam-6134	75	8	(	(	PUNCT
ejpam-6134	75	9	τ	τ	PROPN
ejpam-6134	75	10	,	,	PUNCT
ejpam-6134	75	11	µ	µ	NOUN
ejpam-6134	75	12	)	)	PUNCT
ejpam-6134	75	13	∈	∈	PROPN
ejpam-6134	75	14	ω	ω	PROPN
ejpam-6134	75	15	,	,	PUNCT
ejpam-6134	75	16	we	we	PRON
ejpam-6134	75	17	have	have	VERB
ejpam-6134	75	18	φ((σ	φ((σ	VERB
ejpam-6134	75	19	,	,	PUNCT
ejpam-6134	75	20	ρ	ρ	NOUN
ejpam-6134	75	21	)	)	PUNCT
ejpam-6134	75	22	+	+	CCONJ
ejpam-6134	75	23	(	(	PUNCT
ejpam-6134	75	24	τ	τ	PROPN
ejpam-6134	75	25	,	,	PUNCT
ejpam-6134	75	26	µ	µ	NOUN
ejpam-6134	75	27	)	)	PUNCT
ejpam-6134	75	28	)	)	PUNCT
ejpam-6134	76	1	=	=	VERB
ejpam-6134	76	2	φ((σ	φ((σ	NOUN
ejpam-6134	76	3	+	+	CCONJ
ejpam-6134	76	4	τ	τ	PROPN
ejpam-6134	76	5	,	,	PUNCT
ejpam-6134	76	6	ρ	ρ	PROPN
ejpam-6134	76	7	+	+	X
ejpam-6134	76	8	µ	µ	NOUN
ejpam-6134	76	9	)	)	PUNCT
ejpam-6134	76	10	)	)	PUNCT
ejpam-6134	77	1	=	=	SYM
ejpam-6134	77	2	(	(	PUNCT
ejpam-6134	77	3	σ	σ	PROPN
ejpam-6134	77	4	+	+	X
ejpam-6134	77	5	τ	τ	PROPN
ejpam-6134	77	6	,	,	PUNCT
ejpam-6134	77	7	3ρ	3ρ	NUM
ejpam-6134	77	8	+	+	CCONJ
ejpam-6134	77	9	3µ	3µ	NOUN
ejpam-6134	77	10	)	)	PUNCT
ejpam-6134	77	11	.	.	PUNCT
ejpam-6134	78	1	on	on	ADP
ejpam-6134	78	2	the	the	DET
ejpam-6134	78	3	other	other	ADJ
ejpam-6134	78	4	hand	hand	NOUN
ejpam-6134	78	5	,	,	PUNCT
ejpam-6134	78	6	φ((σ	φ((σ	ADJ
ejpam-6134	78	7	,	,	PUNCT
ejpam-6134	78	8	ρ	ρ	NOUN
ejpam-6134	78	9	)	)	PUNCT
ejpam-6134	78	10	)	)	PUNCT
ejpam-6134	79	1	+	+	CCONJ
ejpam-6134	79	2	φ((τ	φ((τ	NOUN
ejpam-6134	79	3	,	,	PUNCT
ejpam-6134	79	4	µ	µ	NOUN
ejpam-6134	79	5	)	)	PUNCT
ejpam-6134	79	6	)	)	PUNCT
ejpam-6134	80	1	=	=	SYM
ejpam-6134	80	2	(	(	PUNCT
ejpam-6134	80	3	σ	σ	NOUN
ejpam-6134	80	4	,	,	PUNCT
ejpam-6134	80	5	3ρ	3ρ	NUM
ejpam-6134	80	6	)	)	PUNCT
ejpam-6134	81	1	+	+	CCONJ
ejpam-6134	81	2	(	(	PUNCT
ejpam-6134	81	3	τ	τ	PROPN
ejpam-6134	81	4	,	,	PUNCT
ejpam-6134	81	5	3µ	3µ	NUM
ejpam-6134	81	6	)	)	PUNCT
ejpam-6134	81	7	=	=	SYM
ejpam-6134	81	8	(	(	PUNCT
ejpam-6134	81	9	σ	σ	PROPN
ejpam-6134	81	10	+	+	X
ejpam-6134	81	11	τ	τ	PROPN
ejpam-6134	81	12	,	,	PUNCT
ejpam-6134	81	13	3ρ	3ρ	NUM
ejpam-6134	81	14	+	+	CCONJ
ejpam-6134	81	15	3µ	3µ	NOUN
ejpam-6134	81	16	)	)	PUNCT
ejpam-6134	81	17	.	.	PUNCT
ejpam-6134	82	1	thus	thus	ADV
ejpam-6134	82	2	φ((σ	φ((σ	ADJ
ejpam-6134	82	3	,	,	PUNCT
ejpam-6134	82	4	ρ	ρ	NOUN
ejpam-6134	82	5	)	)	PUNCT
ejpam-6134	82	6	+	+	CCONJ
ejpam-6134	82	7	(	(	PUNCT
ejpam-6134	82	8	τ	τ	PROPN
ejpam-6134	82	9	,	,	PUNCT
ejpam-6134	82	10	µ	µ	NOUN
ejpam-6134	82	11	)	)	PUNCT
ejpam-6134	82	12	)	)	PUNCT
ejpam-6134	83	1	=	=	SYM
ejpam-6134	83	2	φ((σ	φ((σ	ADJ
ejpam-6134	83	3	,	,	PUNCT
ejpam-6134	83	4	ρ))+φ((τ	ρ))+φ((τ	PROPN
ejpam-6134	83	5	,	,	PUNCT
ejpam-6134	83	6	µ	µ	NOUN
ejpam-6134	83	7	)	)	PUNCT
ejpam-6134	83	8	)	)	PUNCT
ejpam-6134	83	9	.	.	PUNCT
ejpam-6134	84	1	also	also	ADV
ejpam-6134	84	2	,	,	PUNCT
ejpam-6134	84	3	we	we	PRON
ejpam-6134	84	4	have	have	VERB
ejpam-6134	84	5	φ((σ	φ((σ	VERB
ejpam-6134	84	6	,	,	PUNCT
ejpam-6134	84	7	ρ)(τ	ρ)(τ	PROPN
ejpam-6134	84	8	,	,	PUNCT
ejpam-6134	84	9	µ	µ	NOUN
ejpam-6134	84	10	)	)	PUNCT
ejpam-6134	84	11	)	)	PUNCT
ejpam-6134	85	1	=	=	SYM
ejpam-6134	85	2	φ((στ	φ((στ	PROPN
ejpam-6134	85	3	,	,	PUNCT
ejpam-6134	85	4	ρµ	ρµ	NOUN
ejpam-6134	85	5	)	)	PUNCT
ejpam-6134	85	6	)	)	PUNCT
ejpam-6134	86	1	=	=	PRON
ejpam-6134	86	2	(	(	PUNCT
ejpam-6134	86	3	στ	στ	INTJ
ejpam-6134	86	4	,	,	PUNCT
ejpam-6134	86	5	3ρµ	3ρµ	NOUN
ejpam-6134	86	6	)	)	PUNCT
ejpam-6134	86	7	.	.	PUNCT
ejpam-6134	87	1	in	in	ADP
ejpam-6134	87	2	contrast	contrast	NOUN
ejpam-6134	87	3	,	,	PUNCT
ejpam-6134	87	4	φ((σ	φ((σ	ADJ
ejpam-6134	87	5	,	,	PUNCT
ejpam-6134	87	6	ρ))φ((τ	ρ))φ((τ	PROPN
ejpam-6134	87	7	,	,	PUNCT
ejpam-6134	87	8	µ	µ	NOUN
ejpam-6134	87	9	)	)	PUNCT
ejpam-6134	87	10	)	)	PUNCT
ejpam-6134	88	1	=	=	SYM
ejpam-6134	88	2	(	(	PUNCT
ejpam-6134	88	3	σ	σ	PROPN
ejpam-6134	88	4	,	,	PUNCT
ejpam-6134	88	5	3ρ)(τ	3ρ)(τ	NUM
ejpam-6134	88	6	,	,	PUNCT
ejpam-6134	88	7	3µ	3µ	NUM
ejpam-6134	88	8	)	)	PUNCT
ejpam-6134	88	9	=	=	PRON
ejpam-6134	88	10	(	(	PUNCT
ejpam-6134	88	11	στ	στ	INTJ
ejpam-6134	88	12	,	,	PUNCT
ejpam-6134	88	13	3ρµ	3ρµ	NOUN
ejpam-6134	88	14	)	)	PUNCT
ejpam-6134	88	15	.	.	PUNCT
ejpam-6134	89	1	so	so	ADV
ejpam-6134	89	2	φ((σ	φ((σ	ADJ
ejpam-6134	89	3	,	,	PUNCT
ejpam-6134	89	4	ρ)(τ	ρ)(τ	PROPN
ejpam-6134	89	5	,	,	PUNCT
ejpam-6134	89	6	µ	µ	NOUN
ejpam-6134	89	7	)	)	PUNCT
ejpam-6134	89	8	)	)	PUNCT
ejpam-6134	90	1	=	=	PUNCT
ejpam-6134	90	2	φ((σ	φ((σ	ADJ
ejpam-6134	90	3	,	,	PUNCT
ejpam-6134	90	4	ρ))φ((τ	ρ))φ((τ	PROPN
ejpam-6134	90	5	,	,	PUNCT
ejpam-6134	90	6	µ	µ	NOUN
ejpam-6134	90	7	)	)	PUNCT
ejpam-6134	90	8	)	)	PUNCT
ejpam-6134	90	9	.	.	PUNCT
ejpam-6134	91	1	furthermore	furthermore	ADV
ejpam-6134	91	2	,	,	PUNCT
ejpam-6134	91	3	f((σ	f((σ	NOUN
ejpam-6134	91	4	,	,	PUNCT
ejpam-6134	91	5	ρ	ρ	PROPN
ejpam-6134	91	6	)	)	PUNCT
ejpam-6134	91	7	+	+	CCONJ
ejpam-6134	91	8	(	(	PUNCT
ejpam-6134	91	9	τ	τ	PROPN
ejpam-6134	91	10	,	,	PUNCT
ejpam-6134	91	11	µ))−f((σ	µ))−f((σ	ADJ
ejpam-6134	91	12	,	,	PUNCT
ejpam-6134	91	13	ρ))−f((τ	ρ))−f((τ	PROPN
ejpam-6134	91	14	,	,	PUNCT
ejpam-6134	91	15	µ	µ	NOUN
ejpam-6134	91	16	)	)	PUNCT
ejpam-6134	91	17	)	)	PUNCT
ejpam-6134	91	18	=	=	SYM
ejpam-6134	91	19	(	(	PUNCT
ejpam-6134	91	20	0	0	NUM
ejpam-6134	91	21	,	,	PUNCT
ejpam-6134	91	22	3	3	X
ejpam-6134	91	23	)	)	PUNCT
ejpam-6134	91	24	∈	∈	PROPN
ejpam-6134	91	25	ζ(ω	ζ(ω	PROPN
ejpam-6134	91	26	)	)	PUNCT
ejpam-6134	91	27	and	and	CCONJ
ejpam-6134	91	28	f((σ	f((σ	NUM
ejpam-6134	91	29	,	,	PUNCT
ejpam-6134	91	30	ρ)(τ	ρ)(τ	PROPN
ejpam-6134	91	31	,	,	PUNCT
ejpam-6134	91	32	µ))−f((σ	µ))−f((σ	ADJ
ejpam-6134	91	33	,	,	PUNCT
ejpam-6134	91	34	ρ))f((τ	ρ))f((τ	PROPN
ejpam-6134	91	35	,	,	PUNCT
ejpam-6134	91	36	µ	µ	NOUN
ejpam-6134	91	37	)	)	PUNCT
ejpam-6134	91	38	)	)	PUNCT
ejpam-6134	92	1	−f((σ	−f((σ	NUM
ejpam-6134	92	2	,	,	PUNCT
ejpam-6134	92	3	ρ))φ((τ	ρ))φ((τ	PROPN
ejpam-6134	92	4	,	,	PUNCT
ejpam-6134	92	5	µ))−	µ))−	PROPN
ejpam-6134	92	6	φ((σ	φ((σ	ADJ
ejpam-6134	92	7	,	,	PUNCT
ejpam-6134	92	8	ρ))f((τ	ρ))f((τ	PROPN
ejpam-6134	92	9	,	,	PUNCT
ejpam-6134	92	10	µ	µ	NOUN
ejpam-6134	92	11	)	)	PUNCT
ejpam-6134	92	12	)	)	PUNCT
ejpam-6134	92	13	=	=	SYM
ejpam-6134	92	14	(	(	PUNCT
ejpam-6134	92	15	0	0	NUM
ejpam-6134	92	16	,	,	PUNCT
ejpam-6134	92	17	2ρµ	2ρµ	ADJ
ejpam-6134	92	18	)	)	PUNCT
ejpam-6134	92	19	∈	∈	PROPN
ejpam-6134	92	20	ζ(ω)−	ζ(ω)−	NOUN
ejpam-6134	92	21	{	{	PUNCT
ejpam-6134	92	22	0	0	NUM
ejpam-6134	92	23	}	}	PUNCT
ejpam-6134	92	24	,	,	PUNCT
ejpam-6134	92	25	where	where	SCONJ
ejpam-6134	92	26	2ρµ	2ρµ	ADJ
ejpam-6134	92	27	̸=	̸=	PROPN
ejpam-6134	92	28	0	0	NUM
ejpam-6134	92	29	for	for	ADP
ejpam-6134	92	30	all	all	DET
ejpam-6134	92	31	ρ	ρ	NOUN
ejpam-6134	92	32	,	,	PUNCT
ejpam-6134	92	33	µ	µ	PROPN
ejpam-6134	92	34	∈	∈	PROPN
ejpam-6134	92	35	z6	z6	PROPN
ejpam-6134	92	36	.	.	PUNCT
ejpam-6134	93	1	therefore	therefore	ADV
ejpam-6134	93	2	,	,	PUNCT
ejpam-6134	93	3	φ	φ	PROPN
ejpam-6134	93	4	is	be	AUX
ejpam-6134	93	5	an	an	DET
ejpam-6134	93	6	endomorphism	endomorphism	NOUN
ejpam-6134	93	7	and	and	CCONJ
ejpam-6134	93	8	f	f	PROPN
ejpam-6134	93	9	is	be	AUX
ejpam-6134	93	10	a	a	DET
ejpam-6134	93	11	ce−(φ	ce−(φ	NOUN
ejpam-6134	93	12	,	,	PUNCT
ejpam-6134	93	13	3)homoderivation	3)homoderivation	NUM
ejpam-6134	93	14	but	but	CCONJ
ejpam-6134	93	15	not	not	PART
ejpam-6134	93	16	a	a	DET
ejpam-6134	93	17	(	(	PUNCT
ejpam-6134	93	18	φ	φ	PROPN
ejpam-6134	93	19	,	,	PUNCT
ejpam-6134	93	20	3)-homoderivation	3)-homoderivation	NUM
ejpam-6134	93	21	map	map	NOUN
ejpam-6134	93	22	.	.	PUNCT
ejpam-6134	94	1	m.m	m.m	PROPN
ejpam-6134	94	2	.	.	PROPN
ejpam-6134	94	3	el	el	PROPN
ejpam-6134	94	4	-	-	PUNCT
ejpam-6134	94	5	soufi	soufi	ADJ
ejpam-6134	94	6	,	,	PUNCT
ejpam-6134	94	7	m.	m.	NOUN
ejpam-6134	94	8	almulhem	almulhem	NOUN
ejpam-6134	94	9	,	,	PUNCT
ejpam-6134	94	10	m.	m.	NOUN
ejpam-6134	94	11	s.	s.	PROPN
ejpam-6134	94	12	tammam	tammam	PROPN
ejpam-6134	94	13	el	el	PROPN
ejpam-6134	94	14	-	-	PROPN
ejpam-6134	94	15	sayiad	sayiad	PROPN
ejpam-6134	94	16	/	/	SYM
ejpam-6134	94	17	eur	eur	PROPN
ejpam-6134	94	18	.	.	PUNCT
ejpam-6134	95	1	j.	j.	PROPN
ejpam-6134	95	2	pure	pure	PROPN
ejpam-6134	95	3	appl	appl	PROPN
ejpam-6134	95	4	.	.	PROPN
ejpam-6134	95	5	math	math	PROPN
ejpam-6134	95	6	,	,	PUNCT
ejpam-6134	95	7	18	18	NUM
ejpam-6134	95	8	(	(	PUNCT
ejpam-6134	95	9	4	4	NUM
ejpam-6134	95	10	)	)	PUNCT
ejpam-6134	95	11	(	(	PUNCT
ejpam-6134	95	12	2025	2025	NUM
ejpam-6134	95	13	)	)	PUNCT
ejpam-6134	95	14	,	,	PUNCT
ejpam-6134	95	15	6134	6134	NUM
ejpam-6134	95	16	5	5	NUM
ejpam-6134	95	17	of	of	ADP
ejpam-6134	95	18	12	12	NUM
ejpam-6134	95	19	3	3	NUM
ejpam-6134	95	20	.	.	PUNCT
ejpam-6134	95	21	rings	ring	NOUN
ejpam-6134	95	22	with	with	ADP
ejpam-6134	95	23	centrally	centrally	ADV
ejpam-6134	95	24	extended	extended	ADJ
ejpam-6134	95	25	(	(	PUNCT
ejpam-6134	95	26	φ	φ	NOUN
ejpam-6134	95	27	,	,	PUNCT
ejpam-6134	95	28	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	95	29	this	this	DET
ejpam-6134	95	30	section	section	NOUN
ejpam-6134	95	31	provides	provide	VERB
ejpam-6134	95	32	an	an	DET
ejpam-6134	95	33	answer	answer	NOUN
ejpam-6134	95	34	to	to	ADP
ejpam-6134	95	35	the	the	DET
ejpam-6134	95	36	question	question	NOUN
ejpam-6134	95	37	:	:	PUNCT
ejpam-6134	95	38	when	when	SCONJ
ejpam-6134	95	39	is	be	AUX
ejpam-6134	95	40	a	a	DET
ejpam-6134	95	41	ce−(φ	ce−(φ	NOUN
ejpam-6134	95	42	,	,	PUNCT
ejpam-6134	95	43	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	95	44	a	a	DET
ejpam-6134	95	45	(	(	PUNCT
ejpam-6134	95	46	φ	φ	NOUN
ejpam-6134	95	47	,	,	PUNCT
ejpam-6134	95	48	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	95	49	?	?	PUNCT
ejpam-6134	96	1	additionally	additionally	ADV
ejpam-6134	96	2	,	,	PUNCT
ejpam-6134	96	3	we	we	PRON
ejpam-6134	96	4	will	will	AUX
ejpam-6134	96	5	provide	provide	VERB
ejpam-6134	96	6	information	information	NOUN
ejpam-6134	96	7	about	about	ADP
ejpam-6134	96	8	a	a	DET
ejpam-6134	96	9	ce−(φ	ce−(φ	NOUN
ejpam-6134	96	10	,	,	PUNCT
ejpam-6134	96	11	m)homoderivation	m)homoderivation	NOUN
ejpam-6134	96	12	.	.	PUNCT
ejpam-6134	97	1	throughout	throughout	ADP
ejpam-6134	97	2	,	,	PUNCT
ejpam-6134	97	3	f	f	PROPN
ejpam-6134	97	4	is	be	AUX
ejpam-6134	97	5	a	a	DET
ejpam-6134	97	6	nilpotent	nilpotent	NOUN
ejpam-6134	97	7	centrally	centrally	ADV
ejpam-6134	97	8	extended	extended	ADJ
ejpam-6134	97	9	(	(	PUNCT
ejpam-6134	97	10	φ	φ	NOUN
ejpam-6134	97	11	,	,	PUNCT
ejpam-6134	97	12	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	97	13	of	of	ADP
ejpam-6134	97	14	a	a	DET
ejpam-6134	97	15	ring	ring	NOUN
ejpam-6134	97	16	ω	ω	PROPN
ejpam-6134	97	17	,	,	PUNCT
ejpam-6134	97	18	φ	φ	PROPN
ejpam-6134	97	19	is	be	AUX
ejpam-6134	97	20	an	an	DET
ejpam-6134	97	21	epimorphism	epimorphism	NOUN
ejpam-6134	97	22	on	on	ADP
ejpam-6134	97	23	ω	ω	PROPN
ejpam-6134	97	24	,	,	PUNCT
ejpam-6134	97	25	f	f	PROPN
ejpam-6134	97	26	and	and	CCONJ
ejpam-6134	97	27	φ	φ	PROPN
ejpam-6134	97	28	are	be	AUX
ejpam-6134	97	29	commuting	commute	VERB
ejpam-6134	97	30	on	on	ADP
ejpam-6134	97	31	ω	ω	NUM
ejpam-6134	97	32	,	,	PUNCT
ejpam-6134	97	33	and	and	CCONJ
ejpam-6134	97	34	m	m	PROPN
ejpam-6134	97	35	∈	∈	PROPN
ejpam-6134	97	36	z.	z.	PROPN
ejpam-6134	97	37	remark	remark	PROPN
ejpam-6134	97	38	2	2	NUM
ejpam-6134	97	39	.	.	NUM
ejpam-6134	97	40	φf	φf	ADP
ejpam-6134	97	41	(	(	PUNCT
ejpam-6134	97	42	r	r	NOUN
ejpam-6134	97	43	,	,	PUNCT
ejpam-6134	97	44	s,+	s,+	PRON
ejpam-6134	97	45	)	)	PUNCT
ejpam-6134	97	46	and	and	CCONJ
ejpam-6134	97	47	(	(	PUNCT
ejpam-6134	97	48	φf	φf	X
ejpam-6134	97	49	(	(	PUNCT
ejpam-6134	97	50	r	r	NOUN
ejpam-6134	97	51	,	,	PUNCT
ejpam-6134	97	52	s	s	PROPN
ejpam-6134	97	53	,	,	PUNCT
ejpam-6134	97	54	.	.	PUNCT
ejpam-6134	97	55	)	)	PUNCT
ejpam-6134	97	56	)	)	PUNCT
ejpam-6134	98	1	refer	refer	VERB
ejpam-6134	98	2	to	to	ADP
ejpam-6134	98	3	the	the	DET
ejpam-6134	98	4	central	central	ADJ
ejpam-6134	98	5	element	element	NOUN
ejpam-6134	98	6	generated	generate	VERB
ejpam-6134	98	7	through	through	ADP
ejpam-6134	98	8	the	the	DET
ejpam-6134	98	9	influence	influence	NOUN
ejpam-6134	98	10	of	of	ADP
ejpam-6134	98	11	f	f	PROPN
ejpam-6134	98	12	on	on	ADP
ejpam-6134	98	13	the	the	DET
ejpam-6134	98	14	sum	sum	NOUN
ejpam-6134	98	15	r	r	NOUN
ejpam-6134	98	16	+	+	SYM
ejpam-6134	98	17	s	s	X
ejpam-6134	98	18	(	(	PUNCT
ejpam-6134	98	19	the	the	DET
ejpam-6134	98	20	product	product	NOUN
ejpam-6134	98	21	r.s	r.s	PROPN
ejpam-6134	98	22	)	)	PUNCT
ejpam-6134	98	23	for	for	ADP
ejpam-6134	98	24	any	any	DET
ejpam-6134	98	25	two	two	NUM
ejpam-6134	98	26	elements	element	NOUN
ejpam-6134	98	27	r	r	NOUN
ejpam-6134	98	28	,	,	PUNCT
ejpam-6134	98	29	s	s	PART
ejpam-6134	98	30	∈	∈	PROPN
ejpam-6134	98	31	ω	ω	PROPN
ejpam-6134	98	32	.	.	PUNCT
ejpam-6134	99	1	theorem	theorem	NOUN
ejpam-6134	99	2	1	1	NUM
ejpam-6134	99	3	.	.	PUNCT
ejpam-6134	99	4	suppose	suppose	VERB
ejpam-6134	99	5	that	that	SCONJ
ejpam-6134	99	6	ω	ω	PROPN
ejpam-6134	99	7	is	be	AUX
ejpam-6134	99	8	a	a	DET
ejpam-6134	99	9	ring	ring	NOUN
ejpam-6134	99	10	.	.	PUNCT
ejpam-6134	100	1	if	if	SCONJ
ejpam-6134	100	2	the	the	DET
ejpam-6134	100	3	zero	zero	NUM
ejpam-6134	100	4	ideal	ideal	NOUN
ejpam-6134	100	5	is	be	AUX
ejpam-6134	100	6	the	the	DET
ejpam-6134	100	7	only	only	ADJ
ejpam-6134	100	8	ideal	ideal	NOUN
ejpam-6134	100	9	of	of	ADP
ejpam-6134	100	10	ω	ω	PROPN
ejpam-6134	100	11	contained	contain	VERB
ejpam-6134	100	12	in	in	ADP
ejpam-6134	100	13	ζ(ω	ζ(ω	PROPN
ejpam-6134	100	14	)	)	PUNCT
ejpam-6134	100	15	,	,	PUNCT
ejpam-6134	100	16	then	then	ADV
ejpam-6134	100	17	f	f	PROPN
ejpam-6134	100	18	is	be	AUX
ejpam-6134	100	19	additive	additive	ADJ
ejpam-6134	100	20	.	.	PUNCT
ejpam-6134	101	1	proof	proof	NOUN
ejpam-6134	101	2	.	.	PUNCT
ejpam-6134	102	1	let	let	VERB
ejpam-6134	102	2	s	s	X
ejpam-6134	102	3	,	,	PUNCT
ejpam-6134	102	4	u	u	PROPN
ejpam-6134	102	5	∈	∈	PROPN
ejpam-6134	102	6	ω	ω	NOUN
ejpam-6134	102	7	be	be	VERB
ejpam-6134	102	8	two	two	NUM
ejpam-6134	102	9	fixed	fix	VERB
ejpam-6134	102	10	elements	element	NOUN
ejpam-6134	102	11	.	.	PUNCT
ejpam-6134	103	1	by	by	ADP
ejpam-6134	103	2	assumption	assumption	NOUN
ejpam-6134	103	3	,	,	PUNCT
ejpam-6134	103	4	f(s+	f(s+	NOUN
ejpam-6134	103	5	u	u	NOUN
ejpam-6134	103	6	)	)	PUNCT
ejpam-6134	103	7	=	=	PUNCT
ejpam-6134	103	8	f(s	f(s	X
ejpam-6134	103	9	)	)	PUNCT
ejpam-6134	103	10	+	+	X
ejpam-6134	103	11	f(u	f(u	NOUN
ejpam-6134	103	12	)	)	PUNCT
ejpam-6134	103	13	+	+	CCONJ
ejpam-6134	103	14	φf	φf	X
ejpam-6134	103	15	(	(	PUNCT
ejpam-6134	103	16	s	s	NOUN
ejpam-6134	103	17	,	,	PUNCT
ejpam-6134	103	18	u,+	u,+	ADJ
ejpam-6134	103	19	)	)	PUNCT
ejpam-6134	103	20	.	.	PUNCT
ejpam-6134	104	1	(	(	PUNCT
ejpam-6134	104	2	1	1	X
ejpam-6134	104	3	)	)	PUNCT
ejpam-6134	104	4	so	so	ADV
ejpam-6134	104	5	,	,	PUNCT
ejpam-6134	104	6	for	for	ADP
ejpam-6134	104	7	each	each	DET
ejpam-6134	104	8	v	v	ADP
ejpam-6134	104	9	∈	∈	PROPN
ejpam-6134	104	10	ω	ω	NOUN
ejpam-6134	104	11	,	,	PUNCT
ejpam-6134	104	12	we	we	PRON
ejpam-6134	104	13	obtain	obtain	VERB
ejpam-6134	104	14	f((s+	f((s+	ADJ
ejpam-6134	104	15	u)v	u)v	PUNCT
ejpam-6134	104	16	)	)	PUNCT
ejpam-6134	105	1	=	=	PRON
ejpam-6134	105	2	φ(s+	φ(s+	VERB
ejpam-6134	105	3	u)f(v	u)f(v	ADV
ejpam-6134	105	4	)	)	PUNCT
ejpam-6134	106	1	+	+	CCONJ
ejpam-6134	106	2	f(s+	f(s+	VERB
ejpam-6134	106	3	u)φ(v	u)φ(v	NOUN
ejpam-6134	106	4	)	)	PUNCT
ejpam-6134	107	1	+	+	VERB
ejpam-6134	107	2	mf(s+	mf(s+	VERB
ejpam-6134	107	3	u)f(v	u)f(v	ADV
ejpam-6134	107	4	)	)	PUNCT
ejpam-6134	108	1	+	+	CCONJ
ejpam-6134	108	2	φf	φf	X
ejpam-6134	108	3	(	(	PUNCT
ejpam-6134	108	4	s+	s+	NUM
ejpam-6134	108	5	u	u	NOUN
ejpam-6134	108	6	,	,	PUNCT
ejpam-6134	108	7	v	v	NOUN
ejpam-6134	108	8	,	,	PUNCT
ejpam-6134	108	9	.	.	PUNCT
ejpam-6134	108	10	)	)	PUNCT
ejpam-6134	109	1	=	=	PRON
ejpam-6134	109	2	(	(	PUNCT
ejpam-6134	109	3	f(s	f(s	ADV
ejpam-6134	109	4	)	)	PUNCT
ejpam-6134	109	5	+	+	CCONJ
ejpam-6134	109	6	f(u	f(u	NOUN
ejpam-6134	109	7	)	)	PUNCT
ejpam-6134	109	8	+	+	CCONJ
ejpam-6134	109	9	φf	φf	X
ejpam-6134	109	10	(	(	PUNCT
ejpam-6134	109	11	s	s	PROPN
ejpam-6134	109	12	,	,	PUNCT
ejpam-6134	109	13	u,+))(mf(v	u,+))(mf(v	ADJ
ejpam-6134	109	14	)	)	PUNCT
ejpam-6134	109	15	+	+	NUM
ejpam-6134	109	16	φ(v	φ(v	NOUN
ejpam-6134	109	17	)	)	PUNCT
ejpam-6134	109	18	)	)	PUNCT
ejpam-6134	110	1	+	+	CCONJ
ejpam-6134	110	2	φ(u)f(v	φ(u)f(v	X
ejpam-6134	110	3	)	)	PUNCT
ejpam-6134	110	4	+	+	NUM
ejpam-6134	110	5	φ(s)f(v	φ(s)f(v	PRON
ejpam-6134	110	6	)	)	PUNCT
ejpam-6134	110	7	+	+	CCONJ
ejpam-6134	110	8	φf	φf	X
ejpam-6134	110	9	(	(	PUNCT
ejpam-6134	110	10	s+	s+	NUM
ejpam-6134	110	11	u	u	NOUN
ejpam-6134	110	12	,	,	PUNCT
ejpam-6134	110	13	v	v	NOUN
ejpam-6134	110	14	,	,	PUNCT
ejpam-6134	110	15	.	.	PUNCT
ejpam-6134	110	16	)	)	PUNCT
ejpam-6134	110	17	.	.	PUNCT
ejpam-6134	111	1	(	(	PUNCT
ejpam-6134	111	2	2	2	X
ejpam-6134	111	3	)	)	PUNCT
ejpam-6134	111	4	however	however	ADV
ejpam-6134	111	5	,	,	PUNCT
ejpam-6134	111	6	we	we	PRON
ejpam-6134	111	7	also	also	ADV
ejpam-6134	111	8	have	have	VERB
ejpam-6134	111	9	f((s+	f((s+	ADJ
ejpam-6134	111	10	u)v	u)v	PUNCT
ejpam-6134	111	11	)	)	PUNCT
ejpam-6134	112	1	=	=	SYM
ejpam-6134	112	2	f(sv	f(sv	AUX
ejpam-6134	112	3	+	+	CCONJ
ejpam-6134	112	4	uv	uv	NOUN
ejpam-6134	112	5	)	)	PUNCT
ejpam-6134	112	6	=	=	SYM
ejpam-6134	112	7	f(sv	f(sv	NOUN
ejpam-6134	112	8	)	)	PUNCT
ejpam-6134	112	9	+	+	NUM
ejpam-6134	112	10	f(uv	f(uv	NOUN
ejpam-6134	112	11	)	)	PUNCT
ejpam-6134	113	1	+	+	CCONJ
ejpam-6134	113	2	φf	φf	X
ejpam-6134	113	3	(	(	PUNCT
ejpam-6134	113	4	sv	sv	INTJ
ejpam-6134	113	5	,	,	PUNCT
ejpam-6134	113	6	uv,+	uv,+	PROPN
ejpam-6134	113	7	)	)	PUNCT
ejpam-6134	113	8	=	=	SYM
ejpam-6134	113	9	f(s)φ(v	f(s)φ(v	NUM
ejpam-6134	113	10	)	)	PUNCT
ejpam-6134	113	11	+	+	NUM
ejpam-6134	113	12	φ(s)f(v	φ(s)f(v	PRON
ejpam-6134	113	13	)	)	PUNCT
ejpam-6134	113	14	+	+	ADJ
ejpam-6134	113	15	mf(s)f(v	mf(s)f(v	NOUN
ejpam-6134	113	16	)	)	PUNCT
ejpam-6134	113	17	+	+	CCONJ
ejpam-6134	113	18	φ(u)f(v	φ(u)f(v	NUM
ejpam-6134	113	19	)	)	PUNCT
ejpam-6134	113	20	+	+	NUM
ejpam-6134	113	21	f(u)φ(v	f(u)φ(v	NOUN
ejpam-6134	113	22	)	)	PUNCT
ejpam-6134	114	1	+	+	ADJ
ejpam-6134	114	2	mf(u)f(v	mf(u)f(v	ADJ
ejpam-6134	114	3	)	)	PUNCT
ejpam-6134	114	4	+	+	NUM
ejpam-6134	114	5	φf	φf	X
ejpam-6134	114	6	(	(	PUNCT
ejpam-6134	114	7	sv	sv	INTJ
ejpam-6134	114	8	,	,	PUNCT
ejpam-6134	114	9	uv,+	uv,+	PROPN
ejpam-6134	114	10	)	)	PUNCT
ejpam-6134	114	11	+	+	CCONJ
ejpam-6134	114	12	φf	φf	X
ejpam-6134	114	13	(	(	PUNCT
ejpam-6134	114	14	s	s	PROPN
ejpam-6134	114	15	,	,	PUNCT
ejpam-6134	114	16	v	v	NOUN
ejpam-6134	114	17	,	,	PUNCT
ejpam-6134	114	18	.	.	PUNCT
ejpam-6134	114	19	)	)	PUNCT
ejpam-6134	115	1	+	+	CCONJ
ejpam-6134	115	2	φf	φf	X
ejpam-6134	115	3	(	(	PUNCT
ejpam-6134	115	4	u	u	NOUN
ejpam-6134	115	5	,	,	PUNCT
ejpam-6134	115	6	v	v	NOUN
ejpam-6134	115	7	,	,	PUNCT
ejpam-6134	115	8	.	.	PUNCT
ejpam-6134	115	9	)	)	PUNCT
ejpam-6134	115	10	.	.	PUNCT
ejpam-6134	116	1	(	(	PUNCT
ejpam-6134	116	2	3	3	X
ejpam-6134	116	3	)	)	PUNCT
ejpam-6134	116	4	comparing	compare	VERB
ejpam-6134	116	5	(	(	PUNCT
ejpam-6134	116	6	2	2	NUM
ejpam-6134	116	7	)	)	PUNCT
ejpam-6134	116	8	and	and	CCONJ
ejpam-6134	116	9	(	(	PUNCT
ejpam-6134	116	10	3	3	NUM
ejpam-6134	116	11	)	)	PUNCT
ejpam-6134	116	12	,	,	PUNCT
ejpam-6134	116	13	we	we	PRON
ejpam-6134	116	14	get	get	VERB
ejpam-6134	116	15	(	(	PUNCT
ejpam-6134	116	16	mf(v	mf(v	NUM
ejpam-6134	116	17	)	)	PUNCT
ejpam-6134	117	1	+	+	CCONJ
ejpam-6134	117	2	φ(v))φf	φ(v))φf	NOUN
ejpam-6134	117	3	(	(	PUNCT
ejpam-6134	117	4	s	s	NOUN
ejpam-6134	117	5	,	,	PUNCT
ejpam-6134	117	6	u,+	u,+	ADJ
ejpam-6134	117	7	)	)	PUNCT
ejpam-6134	117	8	∈	∈	PROPN
ejpam-6134	117	9	ζ(ω	ζ(ω	PROPN
ejpam-6134	117	10	)	)	PUNCT
ejpam-6134	117	11	,	,	PUNCT
ejpam-6134	117	12	∀	∀	X
ejpam-6134	117	13	v	v	ADP
ejpam-6134	117	14	∈	∈	PROPN
ejpam-6134	117	15	ω	ω	NOUN
ejpam-6134	117	16	.	.	PUNCT
ejpam-6134	117	17	(	(	PUNCT
ejpam-6134	117	18	4	4	NUM
ejpam-6134	117	19	)	)	PUNCT
ejpam-6134	117	20	due	due	ADP
ejpam-6134	117	21	to	to	ADP
ejpam-6134	117	22	the	the	DET
ejpam-6134	117	23	fact	fact	NOUN
ejpam-6134	117	24	that	that	SCONJ
ejpam-6134	117	25	f	f	PROPN
ejpam-6134	117	26	is	be	AUX
ejpam-6134	117	27	nilpotent	nilpotent	ADJ
ejpam-6134	117	28	,	,	PUNCT
ejpam-6134	117	29	∃n	∃n	PROPN
ejpam-6134	117	30	∈	∈	PROPN
ejpam-6134	117	31	z	z	PROPN
ejpam-6134	117	32	,	,	PUNCT
ejpam-6134	117	33	n	n	PROPN
ejpam-6134	117	34	>	>	X
ejpam-6134	117	35	1	1	NUM
ejpam-6134	117	36	so	so	SCONJ
ejpam-6134	117	37	that	that	PRON
ejpam-6134	117	38	fn(s	fn(s	PUNCT
ejpam-6134	117	39	)	)	PUNCT
ejpam-6134	117	40	=	=	SYM
ejpam-6134	117	41	0	0	NUM
ejpam-6134	117	42	for	for	ADP
ejpam-6134	117	43	all	all	DET
ejpam-6134	117	44	s	s	PROPN
ejpam-6134	117	45	∈	∈	PROPN
ejpam-6134	117	46	ω	ω	NOUN
ejpam-6134	117	47	.	.	PUNCT
ejpam-6134	118	1	by	by	ADP
ejpam-6134	118	2	putting	put	VERB
ejpam-6134	118	3	fn−1(v	fn−1(v	NOUN
ejpam-6134	118	4	)	)	PUNCT
ejpam-6134	118	5	instead	instead	ADV
ejpam-6134	118	6	of	of	ADP
ejpam-6134	118	7	v	v	NOUN
ejpam-6134	118	8	in	in	ADP
ejpam-6134	118	9	(	(	PUNCT
ejpam-6134	118	10	4	4	NUM
ejpam-6134	118	11	)	)	PUNCT
ejpam-6134	118	12	,	,	PUNCT
ejpam-6134	118	13	the	the	DET
ejpam-6134	118	14	result	result	NOUN
ejpam-6134	118	15	is	be	AUX
ejpam-6134	118	16	φ(fn−1(v))φf	φ(fn−1(v))φf	X
ejpam-6134	118	17	(	(	PUNCT
ejpam-6134	118	18	s	s	X
ejpam-6134	118	19	,	,	PUNCT
ejpam-6134	118	20	u,+	u,+	ADJ
ejpam-6134	118	21	)	)	PUNCT
ejpam-6134	118	22	∈	∈	PROPN
ejpam-6134	118	23	ζ(ω	ζ(ω	PROPN
ejpam-6134	118	24	)	)	PUNCT
ejpam-6134	118	25	,	,	PUNCT
ejpam-6134	118	26	for	for	ADP
ejpam-6134	118	27	each	each	DET
ejpam-6134	118	28	v	v	ADP
ejpam-6134	118	29	∈	∈	PROPN
ejpam-6134	118	30	ω	ω	NOUN
ejpam-6134	118	31	.	.	PUNCT
ejpam-6134	119	1	(	(	PUNCT
ejpam-6134	119	2	5	5	NUM
ejpam-6134	119	3	)	)	PUNCT
ejpam-6134	119	4	since	since	SCONJ
ejpam-6134	119	5	φ	φ	PROPN
ejpam-6134	119	6	is	be	AUX
ejpam-6134	119	7	an	an	DET
ejpam-6134	119	8	epimorphism	epimorphism	NOUN
ejpam-6134	119	9	,	,	PUNCT
ejpam-6134	119	10	then	then	ADV
ejpam-6134	119	11	fn−1(v)φf	fn−1(v)φf	NOUN
ejpam-6134	119	12	(	(	PUNCT
ejpam-6134	119	13	s	s	NOUN
ejpam-6134	119	14	,	,	PUNCT
ejpam-6134	119	15	u,+	u,+	ADJ
ejpam-6134	119	16	)	)	PUNCT
ejpam-6134	119	17	∈	∈	PROPN
ejpam-6134	119	18	ζ(ω	ζ(ω	PROPN
ejpam-6134	119	19	)	)	PUNCT
ejpam-6134	119	20	,	,	PUNCT
ejpam-6134	119	21	for	for	ADP
ejpam-6134	119	22	each	each	DET
ejpam-6134	119	23	v	v	ADP
ejpam-6134	119	24	∈	∈	PROPN
ejpam-6134	119	25	ω	ω	NOUN
ejpam-6134	119	26	.	.	PUNCT
ejpam-6134	120	1	(	(	PUNCT
ejpam-6134	120	2	6	6	X
ejpam-6134	120	3	)	)	PUNCT
ejpam-6134	120	4	m.m	m.m	PROPN
ejpam-6134	120	5	.	.	PROPN
ejpam-6134	120	6	el	el	PROPN
ejpam-6134	120	7	-	-	PUNCT
ejpam-6134	120	8	soufi	soufi	ADJ
ejpam-6134	120	9	,	,	PUNCT
ejpam-6134	120	10	m.	m.	NOUN
ejpam-6134	120	11	almulhem	almulhem	NOUN
ejpam-6134	120	12	,	,	PUNCT
ejpam-6134	120	13	m.	m.	NOUN
ejpam-6134	120	14	s.	s.	PROPN
ejpam-6134	120	15	tammam	tammam	PROPN
ejpam-6134	120	16	el	el	PROPN
ejpam-6134	120	17	-	-	PROPN
ejpam-6134	120	18	sayiad	sayiad	PROPN
ejpam-6134	120	19	/	/	SYM
ejpam-6134	120	20	eur	eur	PROPN
ejpam-6134	120	21	.	.	PUNCT
ejpam-6134	121	1	j.	j.	PROPN
ejpam-6134	121	2	pure	pure	PROPN
ejpam-6134	121	3	appl	appl	PROPN
ejpam-6134	121	4	.	.	PROPN
ejpam-6134	121	5	math	math	PROPN
ejpam-6134	121	6	,	,	PUNCT
ejpam-6134	121	7	18	18	NUM
ejpam-6134	121	8	(	(	PUNCT
ejpam-6134	121	9	4	4	NUM
ejpam-6134	121	10	)	)	PUNCT
ejpam-6134	121	11	(	(	PUNCT
ejpam-6134	121	12	2025	2025	NUM
ejpam-6134	121	13	)	)	PUNCT
ejpam-6134	121	14	,	,	PUNCT
ejpam-6134	121	15	6134	6134	NUM
ejpam-6134	121	16	6	6	NUM
ejpam-6134	121	17	of	of	ADP
ejpam-6134	121	18	12	12	NUM
ejpam-6134	121	19	putting	put	VERB
ejpam-6134	121	20	fn−2(v	fn−2(v	NOUN
ejpam-6134	121	21	)	)	PUNCT
ejpam-6134	121	22	instead	instead	ADV
ejpam-6134	121	23	of	of	ADP
ejpam-6134	121	24	v	v	NOUN
ejpam-6134	121	25	in	in	ADP
ejpam-6134	121	26	(	(	PUNCT
ejpam-6134	121	27	4	4	NUM
ejpam-6134	121	28	)	)	PUNCT
ejpam-6134	121	29	,	,	PUNCT
ejpam-6134	121	30	we	we	PRON
ejpam-6134	121	31	get	get	VERB
ejpam-6134	121	32	(	(	PUNCT
ejpam-6134	121	33	mfn−1(v	mfn−1(v	NUM
ejpam-6134	121	34	)	)	PUNCT
ejpam-6134	122	1	+	+	CCONJ
ejpam-6134	122	2	φ(fn−2(v)))φf	φ(fn−2(v)))φf	NUM
ejpam-6134	122	3	(	(	PUNCT
ejpam-6134	122	4	s	s	NOUN
ejpam-6134	122	5	,	,	PUNCT
ejpam-6134	122	6	u,+	u,+	ADJ
ejpam-6134	122	7	)	)	PUNCT
ejpam-6134	122	8	∈	∈	PROPN
ejpam-6134	122	9	ζ(ω	ζ(ω	PROPN
ejpam-6134	122	10	)	)	PUNCT
ejpam-6134	122	11	,	,	PUNCT
ejpam-6134	122	12	for	for	ADP
ejpam-6134	122	13	each	each	DET
ejpam-6134	122	14	v	v	ADP
ejpam-6134	122	15	∈	∈	PROPN
ejpam-6134	122	16	ω	ω	NOUN
ejpam-6134	122	17	.	.	PUNCT
ejpam-6134	122	18	(	(	PUNCT
ejpam-6134	122	19	7	7	X
ejpam-6134	122	20	)	)	PUNCT
ejpam-6134	122	21	once	once	ADV
ejpam-6134	122	22	more	more	ADV
ejpam-6134	122	23	,	,	PUNCT
ejpam-6134	122	24	using	use	VERB
ejpam-6134	122	25	(	(	PUNCT
ejpam-6134	122	26	6	6	NUM
ejpam-6134	122	27	)	)	PUNCT
ejpam-6134	122	28	and	and	CCONJ
ejpam-6134	122	29	the	the	DET
ejpam-6134	122	30	fact	fact	NOUN
ejpam-6134	122	31	that	that	SCONJ
ejpam-6134	122	32	φ	φ	PROPN
ejpam-6134	122	33	is	be	AUX
ejpam-6134	122	34	onto	onto	ADP
ejpam-6134	122	35	,	,	PUNCT
ejpam-6134	122	36	we	we	PRON
ejpam-6134	122	37	get	get	VERB
ejpam-6134	122	38	fn−2(v)φf	fn−2(v)φf	NOUN
ejpam-6134	122	39	(	(	PUNCT
ejpam-6134	122	40	s	s	NOUN
ejpam-6134	122	41	,	,	PUNCT
ejpam-6134	122	42	u,+	u,+	ADJ
ejpam-6134	122	43	)	)	PUNCT
ejpam-6134	122	44	∈	∈	PROPN
ejpam-6134	122	45	ζ(ω	ζ(ω	PROPN
ejpam-6134	122	46	)	)	PUNCT
ejpam-6134	122	47	,	,	PUNCT
ejpam-6134	122	48	for	for	ADP
ejpam-6134	122	49	each	each	DET
ejpam-6134	122	50	v	v	ADP
ejpam-6134	122	51	∈	∈	PROPN
ejpam-6134	122	52	ω	ω	NOUN
ejpam-6134	122	53	.	.	PUNCT
ejpam-6134	123	1	(	(	PUNCT
ejpam-6134	123	2	8)	8)	NUM
ejpam-6134	123	3	hence	hence	ADV
ejpam-6134	123	4	,	,	PUNCT
ejpam-6134	123	5	we	we	PRON
ejpam-6134	123	6	may	may	AUX
ejpam-6134	123	7	repeat	repeat	VERB
ejpam-6134	123	8	the	the	DET
ejpam-6134	123	9	preceding	precede	VERB
ejpam-6134	123	10	procedure	procedure	NOUN
ejpam-6134	123	11	to	to	PART
ejpam-6134	123	12	achieve	achieve	VERB
ejpam-6134	123	13	f(v)φf	f(v)φf	PROPN
ejpam-6134	123	14	(	(	PUNCT
ejpam-6134	123	15	s	s	NOUN
ejpam-6134	123	16	,	,	PUNCT
ejpam-6134	123	17	u,+	u,+	ADJ
ejpam-6134	123	18	)	)	PUNCT
ejpam-6134	123	19	∈	∈	PROPN
ejpam-6134	123	20	ζ(ω	ζ(ω	PROPN
ejpam-6134	123	21	)	)	PUNCT
ejpam-6134	123	22	,	,	PUNCT
ejpam-6134	123	23	for	for	ADP
ejpam-6134	123	24	each	each	DET
ejpam-6134	123	25	v	v	ADP
ejpam-6134	123	26	∈	∈	PROPN
ejpam-6134	123	27	ω	ω	NOUN
ejpam-6134	123	28	.	.	PUNCT
ejpam-6134	124	1	(	(	PUNCT
ejpam-6134	124	2	9	9	X
ejpam-6134	124	3	)	)	PUNCT
ejpam-6134	124	4	using	use	VERB
ejpam-6134	124	5	(	(	PUNCT
ejpam-6134	124	6	4	4	NUM
ejpam-6134	124	7	)	)	PUNCT
ejpam-6134	124	8	and	and	CCONJ
ejpam-6134	124	9	(	(	PUNCT
ejpam-6134	124	10	9	9	NUM
ejpam-6134	124	11	)	)	PUNCT
ejpam-6134	124	12	,	,	PUNCT
ejpam-6134	124	13	we	we	PRON
ejpam-6134	124	14	get	get	VERB
ejpam-6134	124	15	φ(v)φf	φ(v)φf	ADP
ejpam-6134	124	16	(	(	PUNCT
ejpam-6134	124	17	s	s	X
ejpam-6134	124	18	,	,	PUNCT
ejpam-6134	124	19	u,+	u,+	ADJ
ejpam-6134	124	20	)	)	PUNCT
ejpam-6134	124	21	∈	∈	PROPN
ejpam-6134	124	22	ζ(ω	ζ(ω	PROPN
ejpam-6134	124	23	)	)	PUNCT
ejpam-6134	124	24	,	,	PUNCT
ejpam-6134	124	25	∀	∀	X
ejpam-6134	124	26	v	v	ADP
ejpam-6134	124	27	∈	∈	PROPN
ejpam-6134	124	28	ω	ω	NOUN
ejpam-6134	124	29	.	.	PUNCT
ejpam-6134	125	1	but	but	CCONJ
ejpam-6134	125	2	φ	φ	PROPN
ejpam-6134	125	3	is	be	AUX
ejpam-6134	125	4	an	an	DET
ejpam-6134	125	5	epimorphism	epimorphism	NOUN
ejpam-6134	125	6	,	,	PUNCT
ejpam-6134	125	7	thus	thus	ADV
ejpam-6134	125	8	vφf	vφf	X
ejpam-6134	125	9	(	(	PUNCT
ejpam-6134	125	10	s	s	NOUN
ejpam-6134	125	11	,	,	PUNCT
ejpam-6134	125	12	u,+	u,+	ADJ
ejpam-6134	125	13	)	)	PUNCT
ejpam-6134	125	14	∈	∈	PROPN
ejpam-6134	125	15	ζ(ω	ζ(ω	PROPN
ejpam-6134	125	16	)	)	PUNCT
ejpam-6134	125	17	,	,	PUNCT
ejpam-6134	125	18	∀	∀	X
ejpam-6134	125	19	v	v	ADP
ejpam-6134	125	20	∈	∈	PROPN
ejpam-6134	125	21	ω	ω	NOUN
ejpam-6134	125	22	.	.	PUNCT
ejpam-6134	126	1	therefore	therefore	ADV
ejpam-6134	126	2	,	,	PUNCT
ejpam-6134	126	3	ωφf	ωφf	X
ejpam-6134	126	4	(	(	PUNCT
ejpam-6134	126	5	s	s	X
ejpam-6134	126	6	,	,	PUNCT
ejpam-6134	126	7	u,+	u,+	ADJ
ejpam-6134	126	8	)	)	PUNCT
ejpam-6134	126	9	⊆	⊆	NUM
ejpam-6134	126	10	ζ(ω	ζ(ω	PROPN
ejpam-6134	126	11	)	)	PUNCT
ejpam-6134	126	12	.	.	PUNCT
ejpam-6134	127	1	so	so	ADV
ejpam-6134	127	2	,	,	PUNCT
ejpam-6134	127	3	ωφf	ωφf	X
ejpam-6134	127	4	(	(	PUNCT
ejpam-6134	127	5	s	s	X
ejpam-6134	127	6	,	,	PUNCT
ejpam-6134	127	7	u,+	u,+	ADJ
ejpam-6134	127	8	)	)	PUNCT
ejpam-6134	127	9	=	=	SYM
ejpam-6134	127	10	(	(	PUNCT
ejpam-6134	127	11	0	0	NUM
ejpam-6134	127	12	)	)	PUNCT
ejpam-6134	127	13	.	.	PUNCT
ejpam-6134	128	1	if	if	SCONJ
ejpam-6134	128	2	ann(ω	ann(ω	PROPN
ejpam-6134	128	3	)	)	PUNCT
ejpam-6134	128	4	is	be	AUX
ejpam-6134	128	5	the	the	DET
ejpam-6134	128	6	2	2	NUM
ejpam-6134	128	7	-	-	PUNCT
ejpam-6134	128	8	sided	sided	ADJ
ejpam-6134	128	9	annihilator	annihilator	NOUN
ejpam-6134	128	10	of	of	ADP
ejpam-6134	128	11	ω	ω	PROPN
ejpam-6134	128	12	,	,	PUNCT
ejpam-6134	128	13	then	then	ADV
ejpam-6134	128	14	φf	φf	X
ejpam-6134	128	15	(	(	PUNCT
ejpam-6134	128	16	s	s	X
ejpam-6134	128	17	,	,	PUNCT
ejpam-6134	128	18	u,+	u,+	ADJ
ejpam-6134	128	19	)	)	PUNCT
ejpam-6134	128	20	∈	∈	PROPN
ejpam-6134	128	21	ann(ω	ann(ω	PROPN
ejpam-6134	128	22	)	)	PUNCT
ejpam-6134	128	23	.	.	PUNCT
ejpam-6134	129	1	but	but	CCONJ
ejpam-6134	129	2	ann(ω	ann(ω	PROPN
ejpam-6134	129	3	)	)	PUNCT
ejpam-6134	129	4	is	be	AUX
ejpam-6134	129	5	an	an	DET
ejpam-6134	129	6	ideal	ideal	NOUN
ejpam-6134	129	7	on	on	ADP
ejpam-6134	129	8	ω	ω	NUM
ejpam-6134	129	9	contained	contain	VERB
ejpam-6134	129	10	in	in	ADP
ejpam-6134	129	11	ζ(ω	ζ(ω	PROPN
ejpam-6134	129	12	)	)	PUNCT
ejpam-6134	129	13	,	,	PUNCT
ejpam-6134	129	14	thus	thus	ADV
ejpam-6134	129	15	φf	φf	X
ejpam-6134	129	16	(	(	PUNCT
ejpam-6134	129	17	s	s	X
ejpam-6134	129	18	,	,	PUNCT
ejpam-6134	129	19	u,+	u,+	ADJ
ejpam-6134	129	20	)	)	PUNCT
ejpam-6134	129	21	=	=	SYM
ejpam-6134	130	1	0	0	X
ejpam-6134	130	2	.	.	PUNCT
ejpam-6134	131	1	therefore	therefore	ADV
ejpam-6134	131	2	,	,	PUNCT
ejpam-6134	131	3	using	use	VERB
ejpam-6134	131	4	(	(	PUNCT
ejpam-6134	131	5	1	1	NUM
ejpam-6134	131	6	)	)	PUNCT
ejpam-6134	131	7	,	,	PUNCT
ejpam-6134	131	8	f(s+u	f(s+u	PUNCT
ejpam-6134	131	9	)	)	PUNCT
ejpam-6134	131	10	=	=	SYM
ejpam-6134	131	11	f(s)+f(u	f(s)+f(u	NOUN
ejpam-6134	131	12	)	)	PUNCT
ejpam-6134	131	13	.	.	PUNCT
ejpam-6134	132	1	theorem	theorem	NOUN
ejpam-6134	132	2	2	2	NUM
ejpam-6134	132	3	.	.	PUNCT
ejpam-6134	133	1	if	if	SCONJ
ejpam-6134	133	2	the	the	DET
ejpam-6134	133	3	only	only	ADJ
ejpam-6134	133	4	central	central	ADJ
ejpam-6134	133	5	ideal	ideal	NOUN
ejpam-6134	133	6	in	in	ADP
ejpam-6134	133	7	a	a	DET
ejpam-6134	133	8	semiprime	semiprime	NOUN
ejpam-6134	133	9	ring	ring	NOUN
ejpam-6134	133	10	ω	ω	PROPN
ejpam-6134	133	11	is	be	AUX
ejpam-6134	133	12	the	the	DET
ejpam-6134	133	13	zero	zero	NUM
ejpam-6134	133	14	ideal	ideal	NOUN
ejpam-6134	133	15	,	,	PUNCT
ejpam-6134	133	16	then	then	ADV
ejpam-6134	133	17	the	the	DET
ejpam-6134	133	18	map	map	NOUN
ejpam-6134	133	19	f	f	PROPN
ejpam-6134	133	20	is	be	AUX
ejpam-6134	133	21	a	a	DET
ejpam-6134	133	22	(	(	PUNCT
ejpam-6134	133	23	φ	φ	NOUN
ejpam-6134	133	24	,	,	PUNCT
ejpam-6134	133	25	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	133	26	on	on	ADP
ejpam-6134	133	27	ω	ω	NUM
ejpam-6134	133	28	.	.	PUNCT
ejpam-6134	134	1	proof	proof	NOUN
ejpam-6134	134	2	.	.	PUNCT
ejpam-6134	135	1	according	accord	VERB
ejpam-6134	135	2	to	to	ADP
ejpam-6134	135	3	theorem	theorem	NOUN
ejpam-6134	135	4	1	1	NUM
ejpam-6134	135	5	,	,	PUNCT
ejpam-6134	135	6	f	f	PROPN
ejpam-6134	135	7	is	be	AUX
ejpam-6134	135	8	additive	additive	ADJ
ejpam-6134	135	9	.	.	PUNCT
ejpam-6134	136	1	let	let	VERB
ejpam-6134	136	2	u	u	NOUN
ejpam-6134	136	3	,	,	PUNCT
ejpam-6134	136	4	s	s	PROPN
ejpam-6134	136	5	,	,	PUNCT
ejpam-6134	136	6	t	t	PROPN
ejpam-6134	136	7	∈	∈	PROPN
ejpam-6134	136	8	ω	ω	PROPN
ejpam-6134	136	9	be	be	AUX
ejpam-6134	136	10	any	any	DET
ejpam-6134	136	11	elements	element	NOUN
ejpam-6134	136	12	in	in	ADP
ejpam-6134	136	13	ω	ω	PROPN
ejpam-6134	136	14	.	.	PUNCT
ejpam-6134	137	1	we	we	PRON
ejpam-6134	137	2	have	have	VERB
ejpam-6134	137	3	f(us	f(us	NOUN
ejpam-6134	137	4	)	)	PUNCT
ejpam-6134	137	5	=	=	VERB
ejpam-6134	137	6	mf(u)f(s	mf(u)f(s	X
ejpam-6134	137	7	)	)	PUNCT
ejpam-6134	137	8	+	+	CCONJ
ejpam-6134	137	9	f(u)φ(s	f(u)φ(s	NOUN
ejpam-6134	137	10	)	)	PUNCT
ejpam-6134	137	11	+	+	NUM
ejpam-6134	137	12	φ(u)f(s	φ(u)f(s	NUM
ejpam-6134	137	13	)	)	PUNCT
ejpam-6134	137	14	+	+	CCONJ
ejpam-6134	137	15	φf	φf	X
ejpam-6134	137	16	(	(	PUNCT
ejpam-6134	137	17	u	u	NOUN
ejpam-6134	137	18	,	,	PUNCT
ejpam-6134	137	19	s	s	PROPN
ejpam-6134	137	20	,	,	PUNCT
ejpam-6134	137	21	.	.	PUNCT
ejpam-6134	137	22	)	)	PUNCT
ejpam-6134	137	23	,	,	PUNCT
ejpam-6134	137	24	(	(	PUNCT
ejpam-6134	137	25	10	10	NUM
ejpam-6134	137	26	)	)	PUNCT
ejpam-6134	137	27	and	and	CCONJ
ejpam-6134	137	28	f(st	f(st	PROPN
ejpam-6134	137	29	)	)	PUNCT
ejpam-6134	137	30	=	=	SYM
ejpam-6134	137	31	mf(s)f(t	mf(s)f(t	PROPN
ejpam-6134	137	32	)	)	PUNCT
ejpam-6134	138	1	+	+	NUM
ejpam-6134	138	2	φ(s)f(t	φ(s)f(t	PRON
ejpam-6134	138	3	)	)	PUNCT
ejpam-6134	138	4	+	+	X
ejpam-6134	138	5	f(s)φ(t	f(s)φ(t	NUM
ejpam-6134	138	6	)	)	PUNCT
ejpam-6134	139	1	+	+	CCONJ
ejpam-6134	139	2	φf	φf	X
ejpam-6134	139	3	(	(	PUNCT
ejpam-6134	139	4	s	s	PROPN
ejpam-6134	139	5	,	,	PUNCT
ejpam-6134	139	6	t	t	PROPN
ejpam-6134	139	7	,	,	PUNCT
ejpam-6134	139	8	.	.	PUNCT
ejpam-6134	139	9	)	)	PUNCT
ejpam-6134	139	10	.	.	PUNCT
ejpam-6134	140	1	(	(	PUNCT
ejpam-6134	140	2	11	11	X
ejpam-6134	140	3	)	)	PUNCT
ejpam-6134	140	4	using	use	VERB
ejpam-6134	140	5	the	the	DET
ejpam-6134	140	6	associativity	associativity	NOUN
ejpam-6134	140	7	of	of	ADP
ejpam-6134	140	8	ω	ω	PROPN
ejpam-6134	140	9	,	,	PUNCT
ejpam-6134	140	10	we	we	PRON
ejpam-6134	140	11	obtain	obtain	VERB
ejpam-6134	140	12	f((us)t	f((us)t	NUM
ejpam-6134	140	13	)	)	PUNCT
ejpam-6134	140	14	=	=	SYM
ejpam-6134	140	15	f(u(st	f(u(st	NOUN
ejpam-6134	140	16	)	)	PUNCT
ejpam-6134	140	17	)	)	PUNCT
ejpam-6134	140	18	,	,	PUNCT
ejpam-6134	140	19	so	so	SCONJ
ejpam-6134	140	20	φf	φf	PRON
ejpam-6134	140	21	(	(	PUNCT
ejpam-6134	140	22	u	u	NOUN
ejpam-6134	140	23	,	,	PUNCT
ejpam-6134	140	24	s	s	PART
ejpam-6134	140	25	,	,	PUNCT
ejpam-6134	140	26	.)(mf(t	.)(mf(t	PUNCT
ejpam-6134	140	27	)	)	PUNCT
ejpam-6134	140	28	+	+	CCONJ
ejpam-6134	140	29	φ(t))−	φ(t))−	NOUN
ejpam-6134	140	30	φf	φf	ADP
ejpam-6134	140	31	(	(	PUNCT
ejpam-6134	140	32	s	s	PROPN
ejpam-6134	140	33	,	,	PUNCT
ejpam-6134	140	34	t	t	PROPN
ejpam-6134	140	35	,	,	PUNCT
ejpam-6134	140	36	.)(mf(u	.)(mf(u	PROPN
ejpam-6134	140	37	)	)	PUNCT
ejpam-6134	141	1	+	+	CCONJ
ejpam-6134	141	2	φ(u	φ(u	NOUN
ejpam-6134	141	3	)	)	PUNCT
ejpam-6134	141	4	)	)	PUNCT
ejpam-6134	141	5	∈	∈	PROPN
ejpam-6134	141	6	ζ(ω	ζ(ω	PROPN
ejpam-6134	141	7	)	)	PUNCT
ejpam-6134	141	8	.	.	PUNCT
ejpam-6134	142	1	(	(	PUNCT
ejpam-6134	142	2	12	12	NUM
ejpam-6134	142	3	)	)	PUNCT
ejpam-6134	142	4	therefore	therefore	ADV
ejpam-6134	142	5	,	,	PUNCT
ejpam-6134	142	6	φf	φf	X
ejpam-6134	142	7	(	(	PUNCT
ejpam-6134	142	8	u	u	NOUN
ejpam-6134	142	9	,	,	PUNCT
ejpam-6134	142	10	s	s	PROPN
ejpam-6134	142	11	,	,	PUNCT
ejpam-6134	142	12	.)[mf(t	.)[mf(t	ADJ
ejpam-6134	142	13	)	)	PUNCT
ejpam-6134	143	1	+	+	CCONJ
ejpam-6134	143	2	φ(t),mf(u	φ(t),mf(u	ADJ
ejpam-6134	143	3	)	)	PUNCT
ejpam-6134	143	4	+	+	NUM
ejpam-6134	143	5	φ(u	φ(u	NOUN
ejpam-6134	143	6	)	)	PUNCT
ejpam-6134	143	7	]	]	PUNCT
ejpam-6134	144	1	=	=	PUNCT
ejpam-6134	144	2	0	0	X
ejpam-6134	144	3	.	.	PUNCT
ejpam-6134	145	1	(	(	PUNCT
ejpam-6134	145	2	13	13	NUM
ejpam-6134	145	3	)	)	PUNCT
ejpam-6134	145	4	replacing	replace	VERB
ejpam-6134	145	5	t	t	NOUN
ejpam-6134	145	6	by	by	ADP
ejpam-6134	145	7	fn−1(t	fn−1(t	PROPN
ejpam-6134	145	8	)	)	PUNCT
ejpam-6134	145	9	in	in	ADP
ejpam-6134	145	10	(	(	PUNCT
ejpam-6134	145	11	13	13	NUM
ejpam-6134	145	12	)	)	PUNCT
ejpam-6134	145	13	,	,	PUNCT
ejpam-6134	145	14	we	we	PRON
ejpam-6134	145	15	have	have	VERB
ejpam-6134	145	16	φf	φf	PRON
ejpam-6134	145	17	(	(	PUNCT
ejpam-6134	145	18	u	u	NOUN
ejpam-6134	145	19	,	,	PUNCT
ejpam-6134	145	20	s	s	PROPN
ejpam-6134	145	21	,	,	PUNCT
ejpam-6134	145	22	.)[φ(fn−1(t)),mf(u	.)[φ(fn−1(t)),mf(u	PUNCT
ejpam-6134	145	23	)	)	PUNCT
ejpam-6134	146	1	+	+	CCONJ
ejpam-6134	146	2	φ(u	φ(u	NOUN
ejpam-6134	146	3	)	)	PUNCT
ejpam-6134	146	4	]	]	PUNCT
ejpam-6134	147	1	=	=	PUNCT
ejpam-6134	147	2	0	0	X
ejpam-6134	147	3	.	.	PUNCT
ejpam-6134	148	1	(	(	PUNCT
ejpam-6134	148	2	14	14	NUM
ejpam-6134	148	3	)	)	PUNCT
ejpam-6134	148	4	since	since	SCONJ
ejpam-6134	148	5	φ	φ	PROPN
ejpam-6134	148	6	and	and	CCONJ
ejpam-6134	148	7	f	f	PROPN
ejpam-6134	148	8	are	be	AUX
ejpam-6134	148	9	commuting	commute	VERB
ejpam-6134	148	10	and	and	CCONJ
ejpam-6134	148	11	φ	φ	PROPN
ejpam-6134	148	12	is	be	AUX
ejpam-6134	148	13	surjective	surjective	ADJ
ejpam-6134	148	14	,	,	PUNCT
ejpam-6134	148	15	then	then	ADV
ejpam-6134	148	16	φf	φf	X
ejpam-6134	148	17	(	(	PUNCT
ejpam-6134	148	18	u	u	NOUN
ejpam-6134	148	19	,	,	PUNCT
ejpam-6134	148	20	s	s	X
ejpam-6134	148	21	,	,	PUNCT
ejpam-6134	148	22	.)[fn−1(t),mf(u	.)[fn−1(t),mf(u	ADJ
ejpam-6134	148	23	)	)	PUNCT
ejpam-6134	149	1	+	+	CCONJ
ejpam-6134	149	2	φ(u	φ(u	NOUN
ejpam-6134	149	3	)	)	PUNCT
ejpam-6134	149	4	]	]	PUNCT
ejpam-6134	150	1	=	=	PUNCT
ejpam-6134	150	2	0	0	X
ejpam-6134	150	3	.	.	PUNCT
ejpam-6134	151	1	(	(	PUNCT
ejpam-6134	151	2	15	15	X
ejpam-6134	151	3	)	)	PUNCT
ejpam-6134	151	4	replacing	replace	VERB
ejpam-6134	151	5	t	t	NOUN
ejpam-6134	151	6	by	by	ADP
ejpam-6134	151	7	fn−2(t	fn−2(t	NOUN
ejpam-6134	151	8	)	)	PUNCT
ejpam-6134	151	9	in	in	ADP
ejpam-6134	151	10	(	(	PUNCT
ejpam-6134	151	11	13	13	NUM
ejpam-6134	151	12	)	)	PUNCT
ejpam-6134	151	13	and	and	CCONJ
ejpam-6134	151	14	using	use	VERB
ejpam-6134	151	15	(	(	PUNCT
ejpam-6134	151	16	15	15	NUM
ejpam-6134	151	17	)	)	PUNCT
ejpam-6134	151	18	,	,	PUNCT
ejpam-6134	151	19	we	we	PRON
ejpam-6134	151	20	have	have	VERB
ejpam-6134	151	21	φf	φf	PRON
ejpam-6134	151	22	(	(	PUNCT
ejpam-6134	151	23	u	u	NOUN
ejpam-6134	151	24	,	,	PUNCT
ejpam-6134	151	25	s	s	PROPN
ejpam-6134	151	26	,	,	PUNCT
ejpam-6134	151	27	.)[fn−2(t),mf(u	.)[fn−2(t),mf(u	NUM
ejpam-6134	151	28	)	)	PUNCT
ejpam-6134	152	1	+	+	CCONJ
ejpam-6134	152	2	φ(u	φ(u	NOUN
ejpam-6134	152	3	)	)	PUNCT
ejpam-6134	152	4	]	]	PUNCT
ejpam-6134	153	1	=	=	PUNCT
ejpam-6134	153	2	0	0	X
ejpam-6134	153	3	.	.	PUNCT
ejpam-6134	154	1	(	(	PUNCT
ejpam-6134	154	2	16	16	NUM
ejpam-6134	154	3	)	)	PUNCT
ejpam-6134	154	4	repeating	repeat	VERB
ejpam-6134	154	5	the	the	DET
ejpam-6134	154	6	steps	step	NOUN
ejpam-6134	154	7	above	above	ADV
ejpam-6134	154	8	,	,	PUNCT
ejpam-6134	154	9	we	we	PRON
ejpam-6134	154	10	get	get	VERB
ejpam-6134	154	11	φf	φf	ADP
ejpam-6134	154	12	(	(	PUNCT
ejpam-6134	154	13	u	u	NOUN
ejpam-6134	154	14	,	,	PUNCT
ejpam-6134	154	15	s	s	PROPN
ejpam-6134	154	16	,	,	PUNCT
ejpam-6134	154	17	.)[f(t),mf(u	.)[f(t),mf(u	PROPN
ejpam-6134	154	18	)	)	PUNCT
ejpam-6134	155	1	+	+	CCONJ
ejpam-6134	155	2	φ(u	φ(u	NOUN
ejpam-6134	155	3	)	)	PUNCT
ejpam-6134	155	4	]	]	PUNCT
ejpam-6134	156	1	=	=	PUNCT
ejpam-6134	156	2	0	0	X
ejpam-6134	156	3	.	.	PUNCT
ejpam-6134	156	4	(	(	PUNCT
ejpam-6134	156	5	17	17	NUM
ejpam-6134	156	6	)	)	PUNCT
ejpam-6134	156	7	m.m	m.m	PROPN
ejpam-6134	156	8	.	.	PROPN
ejpam-6134	156	9	el	el	PROPN
ejpam-6134	156	10	-	-	PUNCT
ejpam-6134	156	11	soufi	soufi	ADJ
ejpam-6134	156	12	,	,	PUNCT
ejpam-6134	156	13	m.	m.	NOUN
ejpam-6134	156	14	almulhem	almulhem	NOUN
ejpam-6134	156	15	,	,	PUNCT
ejpam-6134	156	16	m.	m.	NOUN
ejpam-6134	156	17	s.	s.	PROPN
ejpam-6134	156	18	tammam	tammam	PROPN
ejpam-6134	156	19	el	el	PROPN
ejpam-6134	156	20	-	-	PROPN
ejpam-6134	156	21	sayiad	sayiad	PROPN
ejpam-6134	156	22	/	/	SYM
ejpam-6134	156	23	eur	eur	PROPN
ejpam-6134	156	24	.	.	PUNCT
ejpam-6134	157	1	j.	j.	PROPN
ejpam-6134	157	2	pure	pure	PROPN
ejpam-6134	157	3	appl	appl	PROPN
ejpam-6134	157	4	.	.	PROPN
ejpam-6134	157	5	math	math	PROPN
ejpam-6134	157	6	,	,	PUNCT
ejpam-6134	157	7	18	18	NUM
ejpam-6134	157	8	(	(	PUNCT
ejpam-6134	157	9	4	4	NUM
ejpam-6134	157	10	)	)	PUNCT
ejpam-6134	157	11	(	(	PUNCT
ejpam-6134	157	12	2025	2025	NUM
ejpam-6134	157	13	)	)	PUNCT
ejpam-6134	157	14	,	,	PUNCT
ejpam-6134	157	15	6134	6134	NUM
ejpam-6134	157	16	7	7	NUM
ejpam-6134	157	17	of	of	ADP
ejpam-6134	157	18	12	12	NUM
ejpam-6134	157	19	from	from	ADP
ejpam-6134	157	20	(	(	PUNCT
ejpam-6134	157	21	13	13	NUM
ejpam-6134	157	22	)	)	PUNCT
ejpam-6134	157	23	and	and	CCONJ
ejpam-6134	157	24	(	(	PUNCT
ejpam-6134	157	25	17	17	NUM
ejpam-6134	157	26	)	)	PUNCT
ejpam-6134	157	27	,	,	PUNCT
ejpam-6134	157	28	we	we	PRON
ejpam-6134	157	29	get	get	VERB
ejpam-6134	157	30	φf	φf	ADP
ejpam-6134	157	31	(	(	PUNCT
ejpam-6134	157	32	u	u	NOUN
ejpam-6134	157	33	,	,	PUNCT
ejpam-6134	157	34	s	s	PROPN
ejpam-6134	157	35	,	,	PUNCT
ejpam-6134	157	36	.)[φ(t),mf(u	.)[φ(t),mf(u	ADJ
ejpam-6134	157	37	)	)	PUNCT
ejpam-6134	157	38	+	+	CCONJ
ejpam-6134	157	39	φ(u	φ(u	NOUN
ejpam-6134	157	40	)	)	PUNCT
ejpam-6134	157	41	]	]	PUNCT
ejpam-6134	158	1	=	=	PUNCT
ejpam-6134	158	2	0	0	X
ejpam-6134	158	3	.	.	PUNCT
ejpam-6134	159	1	(	(	PUNCT
ejpam-6134	159	2	18	18	NUM
ejpam-6134	159	3	)	)	PUNCT
ejpam-6134	159	4	again	again	ADV
ejpam-6134	159	5	,	,	PUNCT
ejpam-6134	159	6	since	since	SCONJ
ejpam-6134	159	7	φ	φ	PROPN
ejpam-6134	159	8	is	be	AUX
ejpam-6134	159	9	surjective	surjective	ADJ
ejpam-6134	159	10	,	,	PUNCT
ejpam-6134	159	11	then	then	ADV
ejpam-6134	159	12	φf	φf	X
ejpam-6134	159	13	(	(	PUNCT
ejpam-6134	159	14	u	u	NOUN
ejpam-6134	159	15	,	,	PUNCT
ejpam-6134	159	16	s	s	PROPN
ejpam-6134	159	17	,	,	PUNCT
ejpam-6134	159	18	.)[t	.)[t	NOUN
ejpam-6134	159	19	,	,	PUNCT
ejpam-6134	159	20	mf(u	mf(u	NOUN
ejpam-6134	159	21	)	)	PUNCT
ejpam-6134	160	1	+	+	CCONJ
ejpam-6134	160	2	φ(u	φ(u	NOUN
ejpam-6134	160	3	)	)	PUNCT
ejpam-6134	160	4	]	]	PUNCT
ejpam-6134	161	1	=	=	PUNCT
ejpam-6134	161	2	0	0	X
ejpam-6134	161	3	.	.	PUNCT
ejpam-6134	162	1	(	(	PUNCT
ejpam-6134	162	2	19	19	NUM
ejpam-6134	162	3	)	)	PUNCT
ejpam-6134	162	4	replacing	replace	VERB
ejpam-6134	162	5	t	t	NOUN
ejpam-6134	162	6	by	by	ADP
ejpam-6134	162	7	tu	tu	PROPN
ejpam-6134	162	8	in	in	ADP
ejpam-6134	162	9	(	(	PUNCT
ejpam-6134	162	10	19	19	NUM
ejpam-6134	162	11	)	)	PUNCT
ejpam-6134	162	12	,	,	PUNCT
ejpam-6134	162	13	we	we	PRON
ejpam-6134	162	14	get	get	VERB
ejpam-6134	162	15	φf	φf	ADP
ejpam-6134	162	16	(	(	PUNCT
ejpam-6134	162	17	u	u	NOUN
ejpam-6134	162	18	,	,	PUNCT
ejpam-6134	162	19	s	s	PROPN
ejpam-6134	162	20	,	,	PUNCT
ejpam-6134	162	21	.)t[u	.)t[u	NOUN
ejpam-6134	162	22	,	,	PUNCT
ejpam-6134	162	23	mf(u	mf(u	NOUN
ejpam-6134	162	24	)	)	PUNCT
ejpam-6134	162	25	+	+	CCONJ
ejpam-6134	162	26	φ(u	φ(u	NOUN
ejpam-6134	162	27	)	)	PUNCT
ejpam-6134	162	28	]	]	PUNCT
ejpam-6134	163	1	=	=	PUNCT
ejpam-6134	163	2	0	0	X
ejpam-6134	163	3	.	.	PUNCT
ejpam-6134	164	1	therefore	therefore	ADV
ejpam-6134	164	2	,	,	PUNCT
ejpam-6134	164	3	φf	φf	X
ejpam-6134	164	4	(	(	PUNCT
ejpam-6134	164	5	u	u	NOUN
ejpam-6134	164	6	,	,	PUNCT
ejpam-6134	164	7	s	s	PROPN
ejpam-6134	164	8	,	,	PUNCT
ejpam-6134	164	9	.)ω[u	.)ω[u	ADJ
ejpam-6134	164	10	,	,	PUNCT
ejpam-6134	164	11	mf(u	mf(u	NOUN
ejpam-6134	164	12	)	)	PUNCT
ejpam-6134	164	13	+	+	CCONJ
ejpam-6134	164	14	φ(u	φ(u	NOUN
ejpam-6134	164	15	)	)	PUNCT
ejpam-6134	164	16	]	]	PUNCT
ejpam-6134	165	1	=	=	PUNCT
ejpam-6134	165	2	0	0	X
ejpam-6134	165	3	.	.	PUNCT
ejpam-6134	165	4	let	let	VERB
ejpam-6134	165	5	q	q	NOUN
ejpam-6134	166	1	=	=	PUNCT
ejpam-6134	166	2	{	{	PUNCT
ejpam-6134	166	3	aα|α	aα|α	ADV
ejpam-6134	166	4	∈	∈	PROPN
ejpam-6134	166	5	λ	λ	PROPN
ejpam-6134	166	6	,	,	PUNCT
ejpam-6134	166	7	aα	aα	PROPN
ejpam-6134	166	8	be	be	VERB
ejpam-6134	166	9	a	a	DET
ejpam-6134	166	10	prime	prime	ADJ
ejpam-6134	166	11	ideal	ideal	NOUN
ejpam-6134	166	12	in	in	ADP
ejpam-6134	166	13	ω	ω	NUM
ejpam-6134	166	14	}	}	PUNCT
ejpam-6134	166	15	and	and	CCONJ
ejpam-6134	166	16	∩aα	∩aα	NOUN
ejpam-6134	166	17	=	=	SYM
ejpam-6134	166	18	(	(	PUNCT
ejpam-6134	166	19	0	0	NUM
ejpam-6134	166	20	)	)	PUNCT
ejpam-6134	166	21	.	.	PUNCT
ejpam-6134	167	1	suppose	suppose	VERB
ejpam-6134	167	2	that	that	SCONJ
ejpam-6134	167	3	a	a	PRON
ejpam-6134	167	4	represents	represent	VERB
ejpam-6134	167	5	a	a	DET
ejpam-6134	167	6	standard	standard	ADJ
ejpam-6134	167	7	aα	aα	NOUN
ejpam-6134	167	8	in	in	ADP
ejpam-6134	167	9	q.	q.	NOUN
ejpam-6134	167	10	for	for	ADP
ejpam-6134	167	11	each	each	DET
ejpam-6134	167	12	u	u	PROPN
ejpam-6134	167	13	∈	∈	PROPN
ejpam-6134	167	14	ω	ω	PROPN
ejpam-6134	167	15	,	,	PUNCT
ejpam-6134	167	16	we	we	PRON
ejpam-6134	167	17	have	have	VERB
ejpam-6134	167	18	either	either	CCONJ
ejpam-6134	167	19	φf	φf	PRON
ejpam-6134	167	20	(	(	PUNCT
ejpam-6134	167	21	u	u	NOUN
ejpam-6134	167	22	,	,	PUNCT
ejpam-6134	167	23	s	s	PROPN
ejpam-6134	167	24	,	,	PUNCT
ejpam-6134	167	25	.	.	PUNCT
ejpam-6134	167	26	)	)	PUNCT
ejpam-6134	168	1	∈	∈	PROPN
ejpam-6134	168	2	a	a	PRON
ejpam-6134	168	3	,	,	PUNCT
ejpam-6134	168	4	∀	∀	NOUN
ejpam-6134	168	5	s	s	NOUN
ejpam-6134	168	6	∈	∈	PROPN
ejpam-6134	168	7	ω	ω	NOUN
ejpam-6134	168	8	or	or	CCONJ
ejpam-6134	168	9	[	[	X
ejpam-6134	168	10	x	x	X
ejpam-6134	168	11	,	,	PUNCT
ejpam-6134	168	12	mf(u)+φ(u	mf(u)+φ(u	X
ejpam-6134	168	13	)	)	PUNCT
ejpam-6134	168	14	]	]	PUNCT
ejpam-6134	169	1	∈	∈	PROPN
ejpam-6134	169	2	a	a	PRON
ejpam-6134	169	3	,	,	PUNCT
ejpam-6134	169	4	∀	∀	X
ejpam-6134	169	5	x	x	X
ejpam-6134	169	6	∈	∈	NOUN
ejpam-6134	169	7	ω	ω	NOUN
ejpam-6134	169	8	.	.	PUNCT
ejpam-6134	170	1	firstly	firstly	ADV
ejpam-6134	170	2	,	,	PUNCT
ejpam-6134	170	3	if	if	SCONJ
ejpam-6134	170	4	φf	φf	PRON
ejpam-6134	170	5	(	(	PUNCT
ejpam-6134	170	6	u	u	NOUN
ejpam-6134	170	7	,	,	PUNCT
ejpam-6134	170	8	s	s	PROPN
ejpam-6134	170	9	,	,	PUNCT
ejpam-6134	170	10	.	.	PUNCT
ejpam-6134	170	11	)	)	PUNCT
ejpam-6134	170	12	∈	∈	PROPN
ejpam-6134	170	13	a	a	PRON
ejpam-6134	170	14	,	,	PUNCT
ejpam-6134	170	15	∀	∀	NOUN
ejpam-6134	170	16	s	s	NOUN
ejpam-6134	170	17	∈	∈	PROPN
ejpam-6134	170	18	ω	ω	PROPN
ejpam-6134	170	19	,	,	PUNCT
ejpam-6134	170	20	then	then	ADV
ejpam-6134	170	21	a+φf	a+φf	NOUN
ejpam-6134	170	22	(	(	PUNCT
ejpam-6134	170	23	u	u	NOUN
ejpam-6134	170	24	,	,	PUNCT
ejpam-6134	170	25	s	s	PROPN
ejpam-6134	170	26	,	,	PUNCT
ejpam-6134	170	27	.	.	PUNCT
ejpam-6134	170	28	)	)	PUNCT
ejpam-6134	171	1	=	=	SYM
ejpam-6134	171	2	a	a	X
ejpam-6134	171	3	,	,	PUNCT
ejpam-6134	171	4	∀	∀	NOUN
ejpam-6134	171	5	s	s	NOUN
ejpam-6134	171	6	∈	∈	PROPN
ejpam-6134	171	7	ω	ω	NOUN
ejpam-6134	171	8	.	.	PUNCT
ejpam-6134	172	1	thus	thus	ADV
ejpam-6134	172	2	,	,	PUNCT
ejpam-6134	172	3	a	a	PRON
ejpam-6134	172	4	+	+	NUM
ejpam-6134	172	5	ωφf	ωφf	ADJ
ejpam-6134	172	6	(	(	PUNCT
ejpam-6134	172	7	u	u	NOUN
ejpam-6134	172	8	,	,	PUNCT
ejpam-6134	172	9	s	s	PROPN
ejpam-6134	172	10	,	,	PUNCT
ejpam-6134	172	11	.	.	PUNCT
ejpam-6134	172	12	)	)	PUNCT
ejpam-6134	173	1	=	=	SYM
ejpam-6134	173	2	a	a	X
ejpam-6134	173	3	,	,	PUNCT
ejpam-6134	173	4	∀	∀	NOUN
ejpam-6134	173	5	s	s	NOUN
ejpam-6134	173	6	∈	∈	PROPN
ejpam-6134	173	7	ω	ω	NOUN
ejpam-6134	173	8	.	.	PUNCT
ejpam-6134	174	1	so	so	ADV
ejpam-6134	174	2	,	,	PUNCT
ejpam-6134	174	3	(	(	PUNCT
ejpam-6134	174	4	a	a	PRON
ejpam-6134	174	5	+	+	X
ejpam-6134	174	6	ωφf	ωφf	ADJ
ejpam-6134	174	7	(	(	PUNCT
ejpam-6134	174	8	u	u	NOUN
ejpam-6134	174	9	,	,	PUNCT
ejpam-6134	174	10	s	s	PROPN
ejpam-6134	174	11	,	,	PUNCT
ejpam-6134	174	12	.))(a	.))(a	PUNCT
ejpam-6134	175	1	+	+	CCONJ
ejpam-6134	175	2	w	w	X
ejpam-6134	175	3	)	)	PUNCT
ejpam-6134	175	4	=	=	SYM
ejpam-6134	175	5	(	(	PUNCT
ejpam-6134	175	6	a	a	DET
ejpam-6134	175	7	+	+	X
ejpam-6134	175	8	w)(a	w)(a	ADP
ejpam-6134	175	9	+	+	CCONJ
ejpam-6134	175	10	ωφf	ωφf	PROPN
ejpam-6134	175	11	(	(	PUNCT
ejpam-6134	175	12	u	u	NOUN
ejpam-6134	175	13	,	,	PUNCT
ejpam-6134	175	14	s	s	PROPN
ejpam-6134	175	15	,	,	PUNCT
ejpam-6134	175	16	.	.	PUNCT
ejpam-6134	175	17	)	)	PUNCT
ejpam-6134	175	18	)	)	PUNCT
ejpam-6134	175	19	,	,	PUNCT
ejpam-6134	175	20	∀	∀	X
ejpam-6134	175	21	s	s	X
ejpam-6134	175	22	,	,	PUNCT
ejpam-6134	175	23	w	w	PROPN
ejpam-6134	175	24	∈	∈	PROPN
ejpam-6134	175	25	ω	ω	PROPN
ejpam-6134	175	26	.	.	PUNCT
ejpam-6134	176	1	therefore	therefore	ADV
ejpam-6134	176	2	,	,	PUNCT
ejpam-6134	176	3	a	a	PRON
ejpam-6134	176	4	+	+	X
ejpam-6134	176	5	[	[	X
ejpam-6134	176	6	ωφf	ωφf	INTJ
ejpam-6134	176	7	(	(	PUNCT
ejpam-6134	176	8	u	u	NOUN
ejpam-6134	176	9	,	,	PUNCT
ejpam-6134	176	10	s	s	PROPN
ejpam-6134	176	11	,	,	PUNCT
ejpam-6134	176	12	.	.	PUNCT
ejpam-6134	176	13	)	)	PUNCT
ejpam-6134	176	14	,	,	PUNCT
ejpam-6134	176	15	w	w	X
ejpam-6134	176	16	]	]	X
ejpam-6134	176	17	=	=	SYM
ejpam-6134	176	18	a	a	PRON
ejpam-6134	176	19	,	,	PUNCT
ejpam-6134	176	20	∀	∀	NOUN
ejpam-6134	176	21	s	s	NOUN
ejpam-6134	176	22	,	,	PUNCT
ejpam-6134	176	23	w	w	PROPN
ejpam-6134	176	24	∈	∈	PROPN
ejpam-6134	176	25	ω	ω	PROPN
ejpam-6134	176	26	.	.	PUNCT
ejpam-6134	177	1	thus	thus	ADV
ejpam-6134	177	2	,	,	PUNCT
ejpam-6134	177	3	[	[	X
ejpam-6134	177	4	ωφf	ωφf	X
ejpam-6134	177	5	(	(	PUNCT
ejpam-6134	177	6	u	u	NOUN
ejpam-6134	177	7	,	,	PUNCT
ejpam-6134	177	8	s	s	PROPN
ejpam-6134	177	9	,	,	PUNCT
ejpam-6134	177	10	.	.	PUNCT
ejpam-6134	177	11	)	)	PUNCT
ejpam-6134	177	12	,	,	PUNCT
ejpam-6134	177	13	w	w	X
ejpam-6134	177	14	]	]	X
ejpam-6134	177	15	∈	∈	PROPN
ejpam-6134	177	16	∩aα	∩aα	NOUN
ejpam-6134	177	17	=	=	SYM
ejpam-6134	177	18	(	(	PUNCT
ejpam-6134	177	19	0	0	NUM
ejpam-6134	177	20	)	)	PUNCT
ejpam-6134	177	21	,	,	PUNCT
ejpam-6134	177	22	∀	∀	X
ejpam-6134	177	23	s	s	X
ejpam-6134	177	24	,	,	PUNCT
ejpam-6134	177	25	w	w	PROPN
ejpam-6134	177	26	∈	∈	PROPN
ejpam-6134	177	27	ω	ω	NOUN
ejpam-6134	177	28	.	.	PUNCT
ejpam-6134	178	1	that	that	PRON
ejpam-6134	178	2	is	be	AUX
ejpam-6134	178	3	ωφf	ωφf	INTJ
ejpam-6134	178	4	(	(	PUNCT
ejpam-6134	178	5	u	u	NOUN
ejpam-6134	178	6	,	,	PUNCT
ejpam-6134	178	7	s	s	PROPN
ejpam-6134	178	8	,	,	PUNCT
ejpam-6134	178	9	.	.	PUNCT
ejpam-6134	178	10	)	)	PUNCT
ejpam-6134	179	1	⊆	⊆	NUM
ejpam-6134	179	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	179	3	)	)	PUNCT
ejpam-6134	179	4	,	,	PUNCT
ejpam-6134	179	5	∀	∀	X
ejpam-6134	179	6	s	s	PART
ejpam-6134	179	7	∈	∈	PROPN
ejpam-6134	179	8	ω	ω	NOUN
ejpam-6134	179	9	.	.	PUNCT
ejpam-6134	180	1	so	so	ADV
ejpam-6134	180	2	,	,	PUNCT
ejpam-6134	180	3	φf	φf	X
ejpam-6134	180	4	(	(	PUNCT
ejpam-6134	180	5	u	u	NOUN
ejpam-6134	180	6	,	,	PUNCT
ejpam-6134	180	7	s	s	PROPN
ejpam-6134	180	8	,	,	PUNCT
ejpam-6134	180	9	.	.	PUNCT
ejpam-6134	180	10	)	)	PUNCT
ejpam-6134	181	1	=	=	SYM
ejpam-6134	181	2	0	0	NUM
ejpam-6134	181	3	∀	∀	NOUN
ejpam-6134	181	4	s	s	NOUN
ejpam-6134	181	5	∈	∈	PROPN
ejpam-6134	181	6	ω	ω	NOUN
ejpam-6134	181	7	.	.	PUNCT
ejpam-6134	182	1	in	in	ADP
ejpam-6134	182	2	the	the	DET
ejpam-6134	182	3	other	other	ADJ
ejpam-6134	182	4	case	case	NOUN
ejpam-6134	182	5	,	,	PUNCT
ejpam-6134	182	6	if	if	SCONJ
ejpam-6134	182	7	[	[	X
ejpam-6134	182	8	x	x	X
ejpam-6134	182	9	,	,	PUNCT
ejpam-6134	182	10	mf(u	mf(u	NOUN
ejpam-6134	182	11	)	)	PUNCT
ejpam-6134	182	12	+	+	CCONJ
ejpam-6134	182	13	φ(u	φ(u	NOUN
ejpam-6134	182	14	)	)	PUNCT
ejpam-6134	182	15	]	]	PUNCT
ejpam-6134	182	16	∈	∈	PROPN
ejpam-6134	182	17	a	a	PRON
ejpam-6134	182	18	,	,	PUNCT
ejpam-6134	182	19	for	for	ADP
ejpam-6134	182	20	each	each	DET
ejpam-6134	182	21	x	x	SYM
ejpam-6134	182	22	∈	∈	PROPN
ejpam-6134	182	23	ω	ω	PROPN
ejpam-6134	182	24	,	,	PUNCT
ejpam-6134	182	25	then	then	ADV
ejpam-6134	182	26	[	[	X
ejpam-6134	182	27	x	x	X
ejpam-6134	182	28	,	,	PUNCT
ejpam-6134	182	29	φ(u	φ(u	NOUN
ejpam-6134	182	30	)	)	PUNCT
ejpam-6134	182	31	+	+	NOUN
ejpam-6134	182	32	mf(u	mf(u	NOUN
ejpam-6134	182	33	)	)	PUNCT
ejpam-6134	182	34	]	]	PUNCT
ejpam-6134	183	1	+	+	ADP
ejpam-6134	183	2	a	a	PRON
ejpam-6134	183	3	=	=	X
ejpam-6134	183	4	a	a	NOUN
ejpam-6134	183	5	,	,	PUNCT
ejpam-6134	183	6	for	for	ADP
ejpam-6134	183	7	each	each	DET
ejpam-6134	183	8	x	x	SYM
ejpam-6134	183	9	∈	∈	PROPN
ejpam-6134	183	10	ω	ω	PROPN
ejpam-6134	183	11	.	.	PUNCT
ejpam-6134	184	1	therefore	therefore	ADV
ejpam-6134	184	2	,	,	PUNCT
ejpam-6134	184	3	[	[	X
ejpam-6134	184	4	x+a	x+a	X
ejpam-6134	184	5	,	,	PUNCT
ejpam-6134	184	6	φ(u	φ(u	NOUN
ejpam-6134	184	7	)	)	PUNCT
ejpam-6134	184	8	+	+	NOUN
ejpam-6134	184	9	mf(u	mf(u	NOUN
ejpam-6134	184	10	)	)	PUNCT
ejpam-6134	184	11	+	+	ADP
ejpam-6134	184	12	a	a	PRON
ejpam-6134	184	13	]	]	X
ejpam-6134	184	14	=	=	SYM
ejpam-6134	184	15	a	a	NOUN
ejpam-6134	184	16	,	,	PUNCT
ejpam-6134	184	17	for	for	ADP
ejpam-6134	184	18	each	each	DET
ejpam-6134	184	19	x	x	SYM
ejpam-6134	184	20	∈	∈	PROPN
ejpam-6134	184	21	ω	ω	PROPN
ejpam-6134	184	22	.	.	PUNCT
ejpam-6134	185	1	(	(	PUNCT
ejpam-6134	185	2	20	20	NUM
ejpam-6134	185	3	)	)	PUNCT
ejpam-6134	185	4	from	from	ADP
ejpam-6134	185	5	(	(	PUNCT
ejpam-6134	185	6	12	12	NUM
ejpam-6134	185	7	)	)	PUNCT
ejpam-6134	185	8	and	and	CCONJ
ejpam-6134	185	9	(	(	PUNCT
ejpam-6134	185	10	20	20	NUM
ejpam-6134	185	11	)	)	PUNCT
ejpam-6134	185	12	,	,	PUNCT
ejpam-6134	185	13	we	we	PRON
ejpam-6134	185	14	have	have	VERB
ejpam-6134	185	15	a	a	PRON
ejpam-6134	185	16	=	=	PUNCT
ejpam-6134	186	1	[	[	X
ejpam-6134	186	2	φf	φf	X
ejpam-6134	186	3	(	(	PUNCT
ejpam-6134	186	4	u	u	NOUN
ejpam-6134	186	5	,	,	PUNCT
ejpam-6134	186	6	s	s	PART
ejpam-6134	186	7	,	,	PUNCT
ejpam-6134	186	8	.)(mf(t	.)(mf(t	NUM
ejpam-6134	186	9	)	)	PUNCT
ejpam-6134	186	10	+	+	NUM
ejpam-6134	186	11	φ(t	φ(t	NOUN
ejpam-6134	186	12	)	)	PUNCT
ejpam-6134	186	13	)	)	PUNCT
ejpam-6134	187	1	+	+	VERB
ejpam-6134	187	2	a−	a−	NOUN
ejpam-6134	187	3	φf	φf	X
ejpam-6134	187	4	(	(	PUNCT
ejpam-6134	187	5	s	s	PROPN
ejpam-6134	187	6	,	,	PUNCT
ejpam-6134	187	7	t	t	PROPN
ejpam-6134	187	8	,	,	PUNCT
ejpam-6134	187	9	.)(mf(u	.)(mf(u	PROPN
ejpam-6134	187	10	)	)	PUNCT
ejpam-6134	187	11	+	+	CCONJ
ejpam-6134	187	12	φ(u	φ(u	NOUN
ejpam-6134	187	13	)	)	PUNCT
ejpam-6134	187	14	)	)	PUNCT
ejpam-6134	188	1	+	+	ADP
ejpam-6134	188	2	a	a	X
ejpam-6134	188	3	,	,	PUNCT
ejpam-6134	188	4	x+a	x+a	PRON
ejpam-6134	188	5	]	]	X
ejpam-6134	188	6	=	=	X
ejpam-6134	189	1	[	[	X
ejpam-6134	189	2	φf	φf	X
ejpam-6134	189	3	(	(	PUNCT
ejpam-6134	189	4	u	u	NOUN
ejpam-6134	189	5	,	,	PUNCT
ejpam-6134	189	6	s	s	PART
ejpam-6134	189	7	,	,	PUNCT
ejpam-6134	189	8	.)(mf(t	.)(mf(t	NUM
ejpam-6134	189	9	)	)	PUNCT
ejpam-6134	189	10	+	+	NUM
ejpam-6134	189	11	φ(t	φ(t	NOUN
ejpam-6134	189	12	)	)	PUNCT
ejpam-6134	189	13	)	)	PUNCT
ejpam-6134	190	1	+	+	ADV
ejpam-6134	190	2	a	a	PRON
ejpam-6134	190	3	,	,	PUNCT
ejpam-6134	190	4	x+a	x+a	X
ejpam-6134	190	5	]	]	PUNCT
ejpam-6134	190	6	for	for	ADP
ejpam-6134	190	7	each	each	DET
ejpam-6134	190	8	s	s	PROPN
ejpam-6134	190	9	,	,	PUNCT
ejpam-6134	190	10	t	t	PROPN
ejpam-6134	190	11	,	,	PUNCT
ejpam-6134	190	12	x	x	X
ejpam-6134	190	13	∈	∈	PROPN
ejpam-6134	190	14	ω	ω	PROPN
ejpam-6134	190	15	.	.	PUNCT
ejpam-6134	191	1	(	(	PUNCT
ejpam-6134	191	2	21	21	NUM
ejpam-6134	191	3	)	)	PUNCT
ejpam-6134	191	4	as	as	ADP
ejpam-6134	191	5	above	above	ADV
ejpam-6134	191	6	in	in	ADP
ejpam-6134	191	7	equation	equation	NOUN
ejpam-6134	191	8	(	(	PUNCT
ejpam-6134	191	9	13	13	NUM
ejpam-6134	191	10	)	)	PUNCT
ejpam-6134	191	11	we	we	PRON
ejpam-6134	191	12	get	get	VERB
ejpam-6134	191	13	a	a	DET
ejpam-6134	191	14	=	=	NOUN
ejpam-6134	192	1	[	[	X
ejpam-6134	192	2	φf	φf	X
ejpam-6134	192	3	(	(	PUNCT
ejpam-6134	192	4	u	u	NOUN
ejpam-6134	192	5	,	,	PUNCT
ejpam-6134	192	6	s	s	PROPN
ejpam-6134	192	7	,	,	PUNCT
ejpam-6134	192	8	.)t	.)t	PUNCT
ejpam-6134	193	1	+	+	CCONJ
ejpam-6134	193	2	a	a	X
ejpam-6134	193	3	,	,	PUNCT
ejpam-6134	193	4	x	x	X
ejpam-6134	194	1	+	+	X
ejpam-6134	194	2	a	a	X
ejpam-6134	194	3	]	]	X
ejpam-6134	194	4	=	=	PUNCT
ejpam-6134	195	1	[	[	X
ejpam-6134	195	2	φf	φf	X
ejpam-6134	195	3	(	(	PUNCT
ejpam-6134	195	4	u	u	NOUN
ejpam-6134	195	5	,	,	PUNCT
ejpam-6134	195	6	s	s	PROPN
ejpam-6134	195	7	,	,	PUNCT
ejpam-6134	195	8	.)t	.)t	PRON
ejpam-6134	195	9	,	,	PUNCT
ejpam-6134	195	10	x	x	X
ejpam-6134	195	11	]	]	X
ejpam-6134	195	12	+	+	CCONJ
ejpam-6134	195	13	a	a	X
ejpam-6134	195	14	,	,	PUNCT
ejpam-6134	195	15	for	for	ADP
ejpam-6134	195	16	each	each	DET
ejpam-6134	195	17	s	s	PROPN
ejpam-6134	195	18	,	,	PUNCT
ejpam-6134	195	19	t	t	PROPN
ejpam-6134	195	20	,	,	PUNCT
ejpam-6134	195	21	x	x	X
ejpam-6134	195	22	∈	∈	PROPN
ejpam-6134	195	23	ω	ω	PROPN
ejpam-6134	195	24	.	.	PUNCT
ejpam-6134	196	1	thus	thus	ADV
ejpam-6134	196	2	,	,	PUNCT
ejpam-6134	196	3	[	[	X
ejpam-6134	196	4	φf	φf	X
ejpam-6134	196	5	(	(	PUNCT
ejpam-6134	196	6	u	u	NOUN
ejpam-6134	196	7	,	,	PUNCT
ejpam-6134	196	8	s	s	PROPN
ejpam-6134	196	9	,	,	PUNCT
ejpam-6134	196	10	.)t	.)t	PRON
ejpam-6134	196	11	,	,	PUNCT
ejpam-6134	196	12	x	x	X
ejpam-6134	196	13	]	]	X
ejpam-6134	196	14	∈	∈	X
ejpam-6134	196	15	a	a	PRON
ejpam-6134	196	16	,	,	PUNCT
ejpam-6134	196	17	for	for	ADP
ejpam-6134	196	18	each	each	DET
ejpam-6134	196	19	s	s	PROPN
ejpam-6134	196	20	,	,	PUNCT
ejpam-6134	196	21	t	t	PROPN
ejpam-6134	196	22	,	,	PUNCT
ejpam-6134	196	23	x	x	X
ejpam-6134	196	24	∈	∈	PROPN
ejpam-6134	196	25	ω	ω	NOUN
ejpam-6134	196	26	.	.	PUNCT
ejpam-6134	197	1	so	so	ADV
ejpam-6134	197	2	,	,	PUNCT
ejpam-6134	197	3	we	we	PRON
ejpam-6134	197	4	achieve	achieve	VERB
ejpam-6134	197	5	[	[	PUNCT
ejpam-6134	197	6	φf	φf	X
ejpam-6134	197	7	(	(	PUNCT
ejpam-6134	197	8	u	u	NOUN
ejpam-6134	197	9	,	,	PUNCT
ejpam-6134	197	10	s	s	PROPN
ejpam-6134	197	11	,	,	PUNCT
ejpam-6134	197	12	.)t	.)t	PRON
ejpam-6134	197	13	,	,	PUNCT
ejpam-6134	197	14	x	x	X
ejpam-6134	197	15	]	]	X
ejpam-6134	197	16	∈	∈	PROPN
ejpam-6134	197	17	∩aα	∩aα	NOUN
ejpam-6134	197	18	=	=	SYM
ejpam-6134	197	19	(	(	PUNCT
ejpam-6134	197	20	0	0	NUM
ejpam-6134	197	21	)	)	PUNCT
ejpam-6134	197	22	,	,	PUNCT
ejpam-6134	197	23	for	for	ADP
ejpam-6134	197	24	each	each	DET
ejpam-6134	197	25	u	u	NOUN
ejpam-6134	197	26	,	,	PUNCT
ejpam-6134	197	27	s	s	PROPN
ejpam-6134	197	28	,	,	PUNCT
ejpam-6134	197	29	t	t	PROPN
ejpam-6134	197	30	,	,	PUNCT
ejpam-6134	197	31	x	x	X
ejpam-6134	197	32	∈	∈	PROPN
ejpam-6134	197	33	ω	ω	X
ejpam-6134	197	34	.	.	PUNCT
ejpam-6134	198	1	again	again	ADV
ejpam-6134	198	2	,	,	PUNCT
ejpam-6134	198	3	φf	φf	X
ejpam-6134	198	4	(	(	PUNCT
ejpam-6134	198	5	u	u	NOUN
ejpam-6134	198	6	,	,	PUNCT
ejpam-6134	198	7	s	s	PROPN
ejpam-6134	198	8	,	,	PUNCT
ejpam-6134	198	9	.	.	PUNCT
ejpam-6134	198	10	)	)	PUNCT
ejpam-6134	199	1	=	=	SYM
ejpam-6134	199	2	0	0	NUM
ejpam-6134	199	3	,	,	PUNCT
ejpam-6134	199	4	∀	∀	NOUN
ejpam-6134	199	5	s	s	NOUN
ejpam-6134	199	6	∈	∈	PROPN
ejpam-6134	199	7	ω	ω	NOUN
ejpam-6134	199	8	.	.	PUNCT
ejpam-6134	200	1	so	so	ADV
ejpam-6134	200	2	,	,	PUNCT
ejpam-6134	200	3	we	we	PRON
ejpam-6134	200	4	have	have	VERB
ejpam-6134	200	5	φf	φf	PRON
ejpam-6134	200	6	(	(	PUNCT
ejpam-6134	200	7	u	u	NOUN
ejpam-6134	200	8	,	,	PUNCT
ejpam-6134	200	9	s	s	PROPN
ejpam-6134	200	10	,	,	PUNCT
ejpam-6134	200	11	.	.	PUNCT
ejpam-6134	200	12	)	)	PUNCT
ejpam-6134	201	1	=	=	SYM
ejpam-6134	201	2	0	0	NUM
ejpam-6134	201	3	,	,	PUNCT
ejpam-6134	201	4	∀	∀	X
ejpam-6134	201	5	u	u	NOUN
ejpam-6134	201	6	,	,	PUNCT
ejpam-6134	201	7	s	s	PROPN
ejpam-6134	201	8	∈	∈	PROPN
ejpam-6134	201	9	ω	ω	PROPN
ejpam-6134	201	10	.	.	PUNCT
ejpam-6134	202	1	from	from	ADP
ejpam-6134	202	2	(	(	PUNCT
ejpam-6134	202	3	10	10	NUM
ejpam-6134	202	4	)	)	PUNCT
ejpam-6134	202	5	,	,	PUNCT
ejpam-6134	202	6	we	we	PRON
ejpam-6134	202	7	have	have	VERB
ejpam-6134	202	8	f(us	f(us	NOUN
ejpam-6134	202	9	)	)	PUNCT
ejpam-6134	202	10	=	=	SYM
ejpam-6134	202	11	f(u)φ(s	f(u)φ(s	NOUN
ejpam-6134	202	12	)	)	PUNCT
ejpam-6134	202	13	+	+	NUM
ejpam-6134	202	14	φ(u)f(s	φ(u)f(s	NUM
ejpam-6134	202	15	)	)	PUNCT
ejpam-6134	202	16	+	+	X
ejpam-6134	202	17	mf(u)f(s	mf(u)f(s	X
ejpam-6134	202	18	)	)	PUNCT
ejpam-6134	202	19	.	.	PUNCT
ejpam-6134	203	1	therefore	therefore	ADV
ejpam-6134	203	2	,	,	PUNCT
ejpam-6134	203	3	f	f	PROPN
ejpam-6134	203	4	is	be	AUX
ejpam-6134	203	5	a	a	DET
ejpam-6134	203	6	(	(	PUNCT
ejpam-6134	203	7	φ	φ	NOUN
ejpam-6134	203	8	,	,	PUNCT
ejpam-6134	203	9	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	203	10	of	of	ADP
ejpam-6134	203	11	ω	ω	PROPN
ejpam-6134	203	12	.	.	PUNCT
ejpam-6134	204	1	theorem	theorem	VERB
ejpam-6134	204	2	2	2	NUM
ejpam-6134	204	3	yields	yield	NOUN
ejpam-6134	204	4	the	the	DET
ejpam-6134	204	5	following	following	ADJ
ejpam-6134	204	6	result	result	NOUN
ejpam-6134	204	7	from	from	ADP
ejpam-6134	204	8	[	[	X
ejpam-6134	204	9	8	8	NUM
ejpam-6134	204	10	,	,	PUNCT
ejpam-6134	204	11	corollary	corollary	ADJ
ejpam-6134	204	12	1	1	NUM
ejpam-6134	204	13	]	]	PUNCT
ejpam-6134	204	14	.	.	PUNCT
ejpam-6134	205	1	corollary	corollary	ADJ
ejpam-6134	205	2	1	1	NUM
ejpam-6134	205	3	.	.	PUNCT
ejpam-6134	206	1	any	any	DET
ejpam-6134	206	2	nilpotent	nilpotent	ADJ
ejpam-6134	206	3	ce−homoderivation	ce−homoderivation	NOUN
ejpam-6134	206	4	is	be	AUX
ejpam-6134	206	5	also	also	ADV
ejpam-6134	206	6	a	a	DET
ejpam-6134	206	7	homoderivation	homoderivation	NOUN
ejpam-6134	206	8	if	if	SCONJ
ejpam-6134	206	9	the	the	DET
ejpam-6134	206	10	only	only	ADJ
ejpam-6134	206	11	central	central	ADJ
ejpam-6134	206	12	ideal	ideal	NOUN
ejpam-6134	206	13	in	in	ADP
ejpam-6134	206	14	the	the	DET
ejpam-6134	206	15	semiprime	semiprime	NOUN
ejpam-6134	206	16	ring	ring	NOUN
ejpam-6134	206	17	is	be	AUX
ejpam-6134	206	18	the	the	DET
ejpam-6134	206	19	zero	zero	NUM
ejpam-6134	206	20	ideal	ideal	NOUN
ejpam-6134	206	21	.	.	PUNCT
ejpam-6134	207	1	a	a	DET
ejpam-6134	207	2	ce−(φ	ce−(φ	NOUN
ejpam-6134	207	3	,	,	PUNCT
ejpam-6134	207	4	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	207	5	preserves	preserve	VERB
ejpam-6134	207	6	the	the	DET
ejpam-6134	207	7	center	center	NOUN
ejpam-6134	207	8	under	under	ADP
ejpam-6134	207	9	certain	certain	ADJ
ejpam-6134	207	10	conditions	condition	NOUN
ejpam-6134	207	11	,	,	PUNCT
ejpam-6134	207	12	according	accord	VERB
ejpam-6134	207	13	to	to	ADP
ejpam-6134	207	14	the	the	DET
ejpam-6134	207	15	following	follow	VERB
ejpam-6134	207	16	theorem	theorem	PROPN
ejpam-6134	207	17	.	.	PROPN
ejpam-6134	208	1	m.m	m.m	PROPN
ejpam-6134	208	2	.	.	PROPN
ejpam-6134	208	3	el	el	PROPN
ejpam-6134	208	4	-	-	PUNCT
ejpam-6134	208	5	soufi	soufi	ADJ
ejpam-6134	208	6	,	,	PUNCT
ejpam-6134	208	7	m.	m.	NOUN
ejpam-6134	208	8	almulhem	almulhem	NOUN
ejpam-6134	208	9	,	,	PUNCT
ejpam-6134	208	10	m.	m.	NOUN
ejpam-6134	208	11	s.	s.	PROPN
ejpam-6134	208	12	tammam	tammam	PROPN
ejpam-6134	208	13	el	el	PROPN
ejpam-6134	208	14	-	-	PROPN
ejpam-6134	208	15	sayiad	sayiad	PROPN
ejpam-6134	208	16	/	/	SYM
ejpam-6134	208	17	eur	eur	PROPN
ejpam-6134	208	18	.	.	PUNCT
ejpam-6134	209	1	j.	j.	PROPN
ejpam-6134	209	2	pure	pure	PROPN
ejpam-6134	209	3	appl	appl	PROPN
ejpam-6134	209	4	.	.	PROPN
ejpam-6134	209	5	math	math	PROPN
ejpam-6134	209	6	,	,	PUNCT
ejpam-6134	209	7	18	18	NUM
ejpam-6134	209	8	(	(	PUNCT
ejpam-6134	209	9	4	4	NUM
ejpam-6134	209	10	)	)	PUNCT
ejpam-6134	209	11	(	(	PUNCT
ejpam-6134	209	12	2025	2025	NUM
ejpam-6134	209	13	)	)	PUNCT
ejpam-6134	209	14	,	,	PUNCT
ejpam-6134	209	15	6134	6134	NUM
ejpam-6134	209	16	8	8	NUM
ejpam-6134	209	17	of	of	ADP
ejpam-6134	209	18	12	12	NUM
ejpam-6134	209	19	theorem	theorem	NOUN
ejpam-6134	209	20	3	3	NUM
ejpam-6134	209	21	.	.	PUNCT
ejpam-6134	210	1	if	if	SCONJ
ejpam-6134	210	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	210	3	)	)	PUNCT
ejpam-6134	210	4	contains	contain	VERB
ejpam-6134	210	5	no	no	DET
ejpam-6134	210	6	non	non	ADJ
ejpam-6134	210	7	-	-	ADJ
ejpam-6134	210	8	zero	zero	ADJ
ejpam-6134	210	9	nilpotent	nilpotent	ADJ
ejpam-6134	210	10	elements	element	NOUN
ejpam-6134	210	11	,	,	PUNCT
ejpam-6134	210	12	then	then	ADV
ejpam-6134	210	13	f	f	PROPN
ejpam-6134	210	14	preserves	preserve	VERB
ejpam-6134	210	15	ζ(ω	ζ(ω	PROPN
ejpam-6134	210	16	)	)	PUNCT
ejpam-6134	210	17	.	.	PUNCT
ejpam-6134	211	1	proof	proof	NOUN
ejpam-6134	211	2	.	.	PUNCT
ejpam-6134	212	1	suppose	suppose	VERB
ejpam-6134	212	2	that	that	SCONJ
ejpam-6134	212	3	ξ	ξ	PROPN
ejpam-6134	212	4	∈	∈	PROPN
ejpam-6134	212	5	ζ(ω	ζ(ω	PROPN
ejpam-6134	212	6	)	)	PUNCT
ejpam-6134	212	7	and	and	CCONJ
ejpam-6134	212	8	r	r	NOUN
ejpam-6134	212	9	∈	∈	PROPN
ejpam-6134	212	10	ω	ω	NOUN
ejpam-6134	212	11	.	.	PUNCT
ejpam-6134	213	1	then	then	ADV
ejpam-6134	213	2	,	,	PUNCT
ejpam-6134	213	3	f(ξr)−mf(ξ)f(r)−f(ξ)φ(r)−	f(ξr)−mf(ξ)f(r)−f(ξ)φ(r)−	NOUN
ejpam-6134	213	4	φ(ξ)f(r	φ(ξ)f(r	NOUN
ejpam-6134	213	5	)	)	PUNCT
ejpam-6134	213	6	∈	∈	PROPN
ejpam-6134	213	7	ζ(ω	ζ(ω	PROPN
ejpam-6134	213	8	)	)	PUNCT
ejpam-6134	213	9	,	,	PUNCT
ejpam-6134	213	10	(	(	PUNCT
ejpam-6134	213	11	22	22	NUM
ejpam-6134	213	12	)	)	PUNCT
ejpam-6134	213	13	and	and	CCONJ
ejpam-6134	213	14	f(rξ)−mf(r)f(ξ)−f(r)φ(ξ)−	f(rξ)−mf(r)f(ξ)−f(r)φ(ξ)−	NOUN
ejpam-6134	213	15	φ(r)f(ξ	φ(r)f(ξ	NUM
ejpam-6134	213	16	)	)	PUNCT
ejpam-6134	213	17	∈	∈	PROPN
ejpam-6134	213	18	ζ(ω	ζ(ω	PROPN
ejpam-6134	213	19	)	)	PUNCT
ejpam-6134	213	20	.	.	PUNCT
ejpam-6134	214	1	(	(	PUNCT
ejpam-6134	214	2	23	23	NUM
ejpam-6134	214	3	)	)	PUNCT
ejpam-6134	214	4	from	from	ADP
ejpam-6134	214	5	(	(	PUNCT
ejpam-6134	214	6	22	22	NUM
ejpam-6134	214	7	)	)	PUNCT
ejpam-6134	214	8	and	and	CCONJ
ejpam-6134	214	9	(	(	PUNCT
ejpam-6134	214	10	23	23	NUM
ejpam-6134	214	11	)	)	PUNCT
ejpam-6134	214	12	we	we	PRON
ejpam-6134	214	13	get	get	VERB
ejpam-6134	214	14	[	[	NOUN
ejpam-6134	214	15	mf(r	mf(r	NOUN
ejpam-6134	214	16	)	)	PUNCT
ejpam-6134	215	1	+	+	CCONJ
ejpam-6134	215	2	φ(r),f(ξ	φ(r),f(ξ	NOUN
ejpam-6134	215	3	)	)	PUNCT
ejpam-6134	215	4	]	]	PUNCT
ejpam-6134	215	5	∈	∈	PROPN
ejpam-6134	215	6	ζ(ω	ζ(ω	PROPN
ejpam-6134	215	7	)	)	PUNCT
ejpam-6134	215	8	,	,	PUNCT
ejpam-6134	215	9	∀	∀	PUNCT
ejpam-6134	215	10	r	r	NOUN
ejpam-6134	215	11	∈	∈	PROPN
ejpam-6134	215	12	ω	ω	NOUN
ejpam-6134	215	13	.	.	PUNCT
ejpam-6134	216	1	(	(	PUNCT
ejpam-6134	216	2	24	24	NUM
ejpam-6134	216	3	)	)	PUNCT
ejpam-6134	216	4	putting	put	VERB
ejpam-6134	216	5	fn−1(r	fn−1(r	NOUN
ejpam-6134	216	6	)	)	PUNCT
ejpam-6134	216	7	instead	instead	ADV
ejpam-6134	216	8	of	of	ADP
ejpam-6134	216	9	r	r	NOUN
ejpam-6134	216	10	in	in	ADP
ejpam-6134	216	11	(	(	PUNCT
ejpam-6134	216	12	24	24	NUM
ejpam-6134	216	13	)	)	PUNCT
ejpam-6134	216	14	and	and	CCONJ
ejpam-6134	216	15	since	since	SCONJ
ejpam-6134	216	16	φ	φ	PROPN
ejpam-6134	216	17	is	be	AUX
ejpam-6134	216	18	onto	onto	ADP
ejpam-6134	216	19	,	,	PUNCT
ejpam-6134	216	20	and	and	CCONJ
ejpam-6134	216	21	f	f	PROPN
ejpam-6134	216	22	,	,	PUNCT
ejpam-6134	216	23	φ	φ	PROPN
ejpam-6134	216	24	are	be	AUX
ejpam-6134	216	25	commuting	commute	VERB
ejpam-6134	216	26	,	,	PUNCT
ejpam-6134	216	27	we	we	PRON
ejpam-6134	216	28	get	get	VERB
ejpam-6134	216	29	[	[	X
ejpam-6134	216	30	fn−1(r),f(ξ	fn−1(r),f(ξ	X
ejpam-6134	216	31	)	)	PUNCT
ejpam-6134	216	32	]	]	PUNCT
ejpam-6134	217	1	∈	∈	PROPN
ejpam-6134	217	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	217	3	)	)	PUNCT
ejpam-6134	217	4	,	,	PUNCT
ejpam-6134	217	5	∀	∀	PUNCT
ejpam-6134	217	6	r	r	NOUN
ejpam-6134	217	7	∈	∈	PROPN
ejpam-6134	217	8	ω	ω	NOUN
ejpam-6134	217	9	.	.	PUNCT
ejpam-6134	218	1	(	(	PUNCT
ejpam-6134	218	2	25	25	NUM
ejpam-6134	218	3	)	)	PUNCT
ejpam-6134	218	4	once	once	ADV
ejpam-6134	218	5	more	more	ADV
ejpam-6134	218	6	,	,	PUNCT
ejpam-6134	218	7	substituting	substitute	VERB
ejpam-6134	218	8	fn−2(r	fn−2(r	NOUN
ejpam-6134	218	9	)	)	PUNCT
ejpam-6134	218	10	for	for	ADP
ejpam-6134	218	11	r	r	NOUN
ejpam-6134	218	12	in	in	ADP
ejpam-6134	218	13	(	(	PUNCT
ejpam-6134	218	14	24	24	NUM
ejpam-6134	218	15	)	)	PUNCT
ejpam-6134	218	16	and	and	CCONJ
ejpam-6134	218	17	using	use	VERB
ejpam-6134	218	18	(	(	PUNCT
ejpam-6134	218	19	25	25	NUM
ejpam-6134	218	20	)	)	PUNCT
ejpam-6134	218	21	,	,	PUNCT
ejpam-6134	218	22	we	we	PRON
ejpam-6134	218	23	achieve	achieve	VERB
ejpam-6134	218	24	[	[	PRON
ejpam-6134	218	25	φ(fn−2(r	φ(fn−2(r	PROPN
ejpam-6134	218	26	)	)	PUNCT
ejpam-6134	218	27	)	)	PUNCT
ejpam-6134	218	28	,	,	PUNCT
ejpam-6134	218	29	f(ξ	f(ξ	PROPN
ejpam-6134	218	30	)	)	PUNCT
ejpam-6134	218	31	]	]	PUNCT
ejpam-6134	219	1	∈	∈	PROPN
ejpam-6134	219	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	219	3	)	)	PUNCT
ejpam-6134	219	4	,	,	PUNCT
ejpam-6134	219	5	for	for	ADP
ejpam-6134	219	6	each	each	DET
ejpam-6134	219	7	r	r	NOUN
ejpam-6134	219	8	∈	∈	PROPN
ejpam-6134	219	9	ω	ω	PROPN
ejpam-6134	219	10	.	.	PUNCT
ejpam-6134	220	1	based	base	VERB
ejpam-6134	220	2	on	on	ADP
ejpam-6134	220	3	the	the	DET
ejpam-6134	220	4	features	feature	NOUN
ejpam-6134	220	5	of	of	ADP
ejpam-6134	220	6	φ	φ	PROPN
ejpam-6134	220	7	and	and	CCONJ
ejpam-6134	220	8	f	f	PROPN
ejpam-6134	220	9	,	,	PUNCT
ejpam-6134	220	10	the	the	DET
ejpam-6134	220	11	result	result	NOUN
ejpam-6134	220	12	is	be	AUX
ejpam-6134	220	13	[	[	X
ejpam-6134	220	14	fn−2(r),f(ξ	fn−2(r),f(ξ	NOUN
ejpam-6134	220	15	)	)	PUNCT
ejpam-6134	220	16	]	]	PUNCT
ejpam-6134	220	17	∈	∈	PROPN
ejpam-6134	220	18	ζ(ω	ζ(ω	PROPN
ejpam-6134	220	19	)	)	PUNCT
ejpam-6134	220	20	,	,	PUNCT
ejpam-6134	220	21	for	for	ADP
ejpam-6134	220	22	each	each	DET
ejpam-6134	220	23	r	r	NOUN
ejpam-6134	220	24	∈	∈	PROPN
ejpam-6134	220	25	ω	ω	NOUN
ejpam-6134	220	26	.	.	PUNCT
ejpam-6134	221	1	(	(	PUNCT
ejpam-6134	221	2	26	26	NUM
ejpam-6134	221	3	)	)	PUNCT
ejpam-6134	221	4	using	use	VERB
ejpam-6134	221	5	the	the	DET
ejpam-6134	221	6	same	same	ADJ
ejpam-6134	221	7	procedure	procedure	NOUN
ejpam-6134	221	8	as	as	ADP
ejpam-6134	221	9	before	before	ADV
ejpam-6134	221	10	,	,	PUNCT
ejpam-6134	221	11	we	we	PRON
ejpam-6134	221	12	get	get	VERB
ejpam-6134	221	13	[	[	X
ejpam-6134	221	14	f(r),f(ξ	f(r),f(ξ	NOUN
ejpam-6134	221	15	)	)	PUNCT
ejpam-6134	221	16	]	]	PUNCT
ejpam-6134	222	1	∈	∈	PROPN
ejpam-6134	222	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	222	3	)	)	PUNCT
ejpam-6134	222	4	,	,	PUNCT
ejpam-6134	222	5	for	for	ADP
ejpam-6134	222	6	each	each	DET
ejpam-6134	222	7	r	r	NOUN
ejpam-6134	222	8	∈	∈	PROPN
ejpam-6134	222	9	ω	ω	NOUN
ejpam-6134	222	10	.	.	PUNCT
ejpam-6134	223	1	(	(	PUNCT
ejpam-6134	223	2	27	27	NUM
ejpam-6134	223	3	)	)	PUNCT
ejpam-6134	223	4	from	from	ADP
ejpam-6134	223	5	(	(	PUNCT
ejpam-6134	223	6	24	24	NUM
ejpam-6134	223	7	)	)	PUNCT
ejpam-6134	223	8	and	and	CCONJ
ejpam-6134	223	9	(	(	PUNCT
ejpam-6134	223	10	27	27	NUM
ejpam-6134	223	11	)	)	PUNCT
ejpam-6134	223	12	we	we	PRON
ejpam-6134	223	13	have	have	VERB
ejpam-6134	223	14	[	[	X
ejpam-6134	223	15	φ(r),f(ξ	φ(r),f(ξ	NOUN
ejpam-6134	223	16	)	)	PUNCT
ejpam-6134	223	17	]	]	PUNCT
ejpam-6134	223	18	∈	∈	PROPN
ejpam-6134	223	19	ζ(ω	ζ(ω	PROPN
ejpam-6134	223	20	)	)	PUNCT
ejpam-6134	223	21	,	,	PUNCT
ejpam-6134	223	22	∀	∀	PUNCT
ejpam-6134	223	23	r	r	NOUN
ejpam-6134	223	24	∈	∈	PROPN
ejpam-6134	223	25	ω	ω	NOUN
ejpam-6134	223	26	.	.	PUNCT
ejpam-6134	224	1	but	but	CCONJ
ejpam-6134	224	2	φ	φ	PROPN
ejpam-6134	224	3	is	be	AUX
ejpam-6134	224	4	surjective	surjective	ADJ
ejpam-6134	224	5	,	,	PUNCT
ejpam-6134	224	6	then	then	ADV
ejpam-6134	224	7	[	[	X
ejpam-6134	224	8	r	r	NOUN
ejpam-6134	224	9	,	,	PUNCT
ejpam-6134	224	10	f(ξ	f(ξ	PROPN
ejpam-6134	224	11	)	)	PUNCT
ejpam-6134	224	12	]	]	PUNCT
ejpam-6134	224	13	∈	∈	PROPN
ejpam-6134	224	14	ζ(ω	ζ(ω	PROPN
ejpam-6134	224	15	)	)	PUNCT
ejpam-6134	224	16	,	,	PUNCT
ejpam-6134	224	17	for	for	ADP
ejpam-6134	224	18	each	each	DET
ejpam-6134	224	19	r	r	NOUN
ejpam-6134	224	20	∈	∈	PROPN
ejpam-6134	224	21	ω	ω	NOUN
ejpam-6134	224	22	.	.	PUNCT
ejpam-6134	225	1	(	(	PUNCT
ejpam-6134	225	2	28	28	NUM
ejpam-6134	225	3	)	)	PUNCT
ejpam-6134	225	4	in	in	ADP
ejpam-6134	225	5	(	(	PUNCT
ejpam-6134	225	6	28	28	NUM
ejpam-6134	225	7	)	)	PUNCT
ejpam-6134	225	8	,	,	PUNCT
ejpam-6134	225	9	replacing	replace	VERB
ejpam-6134	225	10	r	r	NOUN
ejpam-6134	225	11	with	with	ADP
ejpam-6134	225	12	rf(ξ	rf(ξ	NOUN
ejpam-6134	225	13	)	)	PUNCT
ejpam-6134	225	14	gives	give	VERB
ejpam-6134	225	15	[	[	PRON
ejpam-6134	225	16	rf(ξ),f(ξ	rf(ξ),f(ξ	NOUN
ejpam-6134	225	17	)	)	PUNCT
ejpam-6134	225	18	]	]	PUNCT
ejpam-6134	226	1	=	=	PUNCT
ejpam-6134	227	1	[	[	X
ejpam-6134	227	2	r	r	X
ejpam-6134	227	3	,	,	PUNCT
ejpam-6134	227	4	f(ξ)]f(ξ	f(ξ)]f(ξ	PROPN
ejpam-6134	227	5	)	)	PUNCT
ejpam-6134	227	6	∈	∈	PROPN
ejpam-6134	227	7	ζ(ω	ζ(ω	PROPN
ejpam-6134	227	8	)	)	PUNCT
ejpam-6134	227	9	,	,	PUNCT
ejpam-6134	227	10	for	for	ADP
ejpam-6134	227	11	each	each	DET
ejpam-6134	227	12	r	r	NOUN
ejpam-6134	227	13	∈	∈	PROPN
ejpam-6134	227	14	ω	ω	NOUN
ejpam-6134	227	15	.	.	PUNCT
ejpam-6134	228	1	(	(	PUNCT
ejpam-6134	228	2	29	29	NUM
ejpam-6134	228	3	)	)	PUNCT
ejpam-6134	228	4	so	so	ADV
ejpam-6134	228	5	,	,	PUNCT
ejpam-6134	228	6	we	we	PRON
ejpam-6134	228	7	get	get	VERB
ejpam-6134	228	8	[	[	X
ejpam-6134	228	9	[	[	X
ejpam-6134	228	10	r	r	X
ejpam-6134	228	11	,	,	PUNCT
ejpam-6134	228	12	f(ξ)]f(ξ	f(ξ)]f(ξ	PROPN
ejpam-6134	228	13	)	)	PUNCT
ejpam-6134	228	14	,	,	PUNCT
ejpam-6134	228	15	r	r	X
ejpam-6134	228	16	]	]	X
ejpam-6134	228	17	=	=	SYM
ejpam-6134	228	18	0	0	NUM
ejpam-6134	228	19	,	,	PUNCT
ejpam-6134	228	20	∀	∀	X
ejpam-6134	229	1	r	r	NOUN
ejpam-6134	229	2	∈	∈	PROPN
ejpam-6134	229	3	ω	ω	NOUN
ejpam-6134	229	4	.	.	PUNCT
ejpam-6134	230	1	therefore	therefore	ADV
ejpam-6134	230	2	,	,	PUNCT
ejpam-6134	230	3	[	[	X
ejpam-6134	230	4	r	r	NOUN
ejpam-6134	230	5	,	,	PUNCT
ejpam-6134	230	6	f(ξ)]2	f(ξ)]2	NOUN
ejpam-6134	230	7	=	=	NOUN
ejpam-6134	230	8	0	0	NUM
ejpam-6134	230	9	,	,	PUNCT
ejpam-6134	230	10	for	for	ADP
ejpam-6134	230	11	each	each	DET
ejpam-6134	230	12	r	r	NOUN
ejpam-6134	230	13	∈	∈	PROPN
ejpam-6134	230	14	ω	ω	NOUN
ejpam-6134	230	15	.	.	PUNCT
ejpam-6134	231	1	(	(	PUNCT
ejpam-6134	231	2	30	30	NUM
ejpam-6134	231	3	)	)	PUNCT
ejpam-6134	231	4	however	however	ADV
ejpam-6134	231	5	,	,	PUNCT
ejpam-6134	231	6	the	the	DET
ejpam-6134	231	7	nilpotent	nilpotent	ADJ
ejpam-6134	231	8	elements	element	NOUN
ejpam-6134	231	9	in	in	ADP
ejpam-6134	231	10	the	the	DET
ejpam-6134	231	11	center	center	NOUN
ejpam-6134	231	12	ζ(ω	ζ(ω	PROPN
ejpam-6134	231	13	)	)	PUNCT
ejpam-6134	231	14	are	be	AUX
ejpam-6134	231	15	zero	zero	NUM
ejpam-6134	231	16	,	,	PUNCT
ejpam-6134	231	17	therefore	therefore	ADV
ejpam-6134	231	18	we	we	PRON
ejpam-6134	231	19	can	can	AUX
ejpam-6134	231	20	deduce	deduce	VERB
ejpam-6134	231	21	that	that	SCONJ
ejpam-6134	232	1	[	[	X
ejpam-6134	232	2	r	r	NOUN
ejpam-6134	232	3	,	,	PUNCT
ejpam-6134	232	4	f(ξ	f(ξ	PROPN
ejpam-6134	232	5	)	)	PUNCT
ejpam-6134	232	6	]	]	PUNCT
ejpam-6134	232	7	=	=	PUNCT
ejpam-6134	232	8	0	0	NUM
ejpam-6134	232	9	,	,	PUNCT
ejpam-6134	232	10	∀	∀	X
ejpam-6134	232	11	r	r	NOUN
ejpam-6134	232	12	∈	∈	PROPN
ejpam-6134	232	13	ω	ω	NOUN
ejpam-6134	232	14	from	from	ADP
ejpam-6134	232	15	(	(	PUNCT
ejpam-6134	232	16	28	28	NUM
ejpam-6134	232	17	)	)	PUNCT
ejpam-6134	232	18	and	and	CCONJ
ejpam-6134	232	19	(	(	PUNCT
ejpam-6134	232	20	30	30	NUM
ejpam-6134	232	21	)	)	PUNCT
ejpam-6134	232	22	.	.	PUNCT
ejpam-6134	233	1	hence	hence	ADV
ejpam-6134	233	2	,	,	PUNCT
ejpam-6134	233	3	f(ξ	f(ξ	PROPN
ejpam-6134	233	4	)	)	PUNCT
ejpam-6134	233	5	∈	∈	PROPN
ejpam-6134	233	6	ζ(ω	ζ(ω	PROPN
ejpam-6134	233	7	)	)	PUNCT
ejpam-6134	233	8	,	,	PUNCT
ejpam-6134	233	9	i.e.	i.e.	X
ejpam-6134	233	10	,	,	PUNCT
ejpam-6134	233	11	f	f	PROPN
ejpam-6134	233	12	preserves	preserve	VERB
ejpam-6134	233	13	the	the	DET
ejpam-6134	233	14	center	center	NOUN
ejpam-6134	233	15	.	.	PUNCT
ejpam-6134	234	1	as	as	ADP
ejpam-6134	234	2	a	a	DET
ejpam-6134	234	3	direct	direct	ADJ
ejpam-6134	234	4	consequence	consequence	NOUN
ejpam-6134	234	5	of	of	ADP
ejpam-6134	234	6	the	the	DET
ejpam-6134	234	7	above	above	ADJ
ejpam-6134	234	8	theorem	theorem	NOUN
ejpam-6134	234	9	,	,	PUNCT
ejpam-6134	234	10	we	we	PRON
ejpam-6134	234	11	recover	recover	VERB
ejpam-6134	234	12	the	the	DET
ejpam-6134	234	13	following	follow	VERB
ejpam-6134	234	14	result	result	NOUN
ejpam-6134	234	15	previously	previously	ADV
ejpam-6134	234	16	established	establish	VERB
ejpam-6134	234	17	in	in	ADP
ejpam-6134	234	18	[	[	X
ejpam-6134	234	19	8	8	NUM
ejpam-6134	234	20	,	,	PUNCT
ejpam-6134	234	21	corollaries	corollary	NOUN
ejpam-6134	234	22	2	2	NUM
ejpam-6134	234	23	and	and	CCONJ
ejpam-6134	234	24	3	3	NUM
ejpam-6134	234	25	]	]	PUNCT
ejpam-6134	234	26	.	.	PUNCT
ejpam-6134	235	1	corollary	corollary	ADJ
ejpam-6134	235	2	2	2	NUM
ejpam-6134	235	3	.	.	PUNCT
ejpam-6134	236	1	f	f	PROPN
ejpam-6134	236	2	preserves	preserve	VERB
ejpam-6134	236	3	ζ(ω	ζ(ω	PROPN
ejpam-6134	236	4	)	)	PUNCT
ejpam-6134	236	5	.	.	PUNCT
ejpam-6134	237	1	corollary	corollary	ADJ
ejpam-6134	237	2	3	3	X
ejpam-6134	237	3	.	.	PUNCT
ejpam-6134	238	1	if	if	SCONJ
ejpam-6134	238	2	there	there	PRON
ejpam-6134	238	3	are	be	VERB
ejpam-6134	238	4	no	no	DET
ejpam-6134	238	5	non	non	ADJ
ejpam-6134	238	6	-	-	ADJ
ejpam-6134	238	7	zero	zero	ADJ
ejpam-6134	238	8	nilpotent	nilpotent	ADJ
ejpam-6134	238	9	elements	element	NOUN
ejpam-6134	238	10	in	in	ADP
ejpam-6134	238	11	ζ(ω	ζ(ω	PROPN
ejpam-6134	238	12	)	)	PUNCT
ejpam-6134	238	13	,	,	PUNCT
ejpam-6134	238	14	then	then	ADV
ejpam-6134	238	15	every	every	DET
ejpam-6134	238	16	nilpotent	nilpotent	NOUN
ejpam-6134	238	17	centrally	centrally	ADV
ejpam-6134	238	18	extended	extend	VERB
ejpam-6134	238	19	homoderivation	homoderivation	NOUN
ejpam-6134	238	20	of	of	ADP
ejpam-6134	238	21	ω	ω	PROPN
ejpam-6134	238	22	preserves	preserves	PROPN
ejpam-6134	238	23	ζ(ω	ζ(ω	PROPN
ejpam-6134	238	24	)	)	PUNCT
ejpam-6134	238	25	.	.	PUNCT
ejpam-6134	239	1	m.m	m.m	PROPN
ejpam-6134	239	2	.	.	PROPN
ejpam-6134	239	3	el	el	PROPN
ejpam-6134	239	4	-	-	PUNCT
ejpam-6134	239	5	soufi	soufi	ADJ
ejpam-6134	239	6	,	,	PUNCT
ejpam-6134	239	7	m.	m.	NOUN
ejpam-6134	239	8	almulhem	almulhem	NOUN
ejpam-6134	239	9	,	,	PUNCT
ejpam-6134	239	10	m.	m.	NOUN
ejpam-6134	239	11	s.	s.	PROPN
ejpam-6134	239	12	tammam	tammam	PROPN
ejpam-6134	239	13	el	el	PROPN
ejpam-6134	239	14	-	-	PROPN
ejpam-6134	239	15	sayiad	sayiad	PROPN
ejpam-6134	239	16	/	/	SYM
ejpam-6134	239	17	eur	eur	PROPN
ejpam-6134	239	18	.	.	PUNCT
ejpam-6134	240	1	j.	j.	PROPN
ejpam-6134	240	2	pure	pure	PROPN
ejpam-6134	240	3	appl	appl	PROPN
ejpam-6134	240	4	.	.	PROPN
ejpam-6134	240	5	math	math	PROPN
ejpam-6134	240	6	,	,	PUNCT
ejpam-6134	240	7	18	18	NUM
ejpam-6134	240	8	(	(	PUNCT
ejpam-6134	240	9	4	4	NUM
ejpam-6134	240	10	)	)	PUNCT
ejpam-6134	240	11	(	(	PUNCT
ejpam-6134	240	12	2025	2025	NUM
ejpam-6134	240	13	)	)	PUNCT
ejpam-6134	240	14	,	,	PUNCT
ejpam-6134	240	15	6134	6134	NUM
ejpam-6134	240	16	9	9	NUM
ejpam-6134	240	17	of	of	ADP
ejpam-6134	240	18	12	12	NUM
ejpam-6134	240	19	4	4	NUM
ejpam-6134	240	20	.	.	PUNCT
ejpam-6134	240	21	ce	ce	PROPN
ejpam-6134	241	1	−	−	PROPN
ejpam-6134	241	2	(	(	PUNCT
ejpam-6134	241	3	φ	φ	PROPN
ejpam-6134	241	4	,	,	PUNCT
ejpam-6134	241	5	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	241	6	and	and	CCONJ
ejpam-6134	241	7	commutativity	commutativity	NOUN
ejpam-6134	241	8	of	of	ADP
ejpam-6134	241	9	prime	prime	ADJ
ejpam-6134	241	10	rings	ring	NOUN
ejpam-6134	241	11	the	the	DET
ejpam-6134	241	12	primary	primary	ADJ
ejpam-6134	241	13	purpose	purpose	NOUN
ejpam-6134	241	14	of	of	ADP
ejpam-6134	241	15	this	this	DET
ejpam-6134	241	16	section	section	NOUN
ejpam-6134	241	17	is	be	AUX
ejpam-6134	241	18	to	to	PART
ejpam-6134	241	19	demonstrate	demonstrate	VERB
ejpam-6134	241	20	conditions	condition	NOUN
ejpam-6134	241	21	that	that	PRON
ejpam-6134	241	22	assure	assure	VERB
ejpam-6134	241	23	a	a	DET
ejpam-6134	241	24	prime	prime	ADJ
ejpam-6134	241	25	ring	ring	NOUN
ejpam-6134	241	26	’s	’s	PART
ejpam-6134	241	27	commutativity	commutativity	NOUN
ejpam-6134	241	28	when	when	SCONJ
ejpam-6134	241	29	it	it	PRON
ejpam-6134	241	30	admits	admit	VERB
ejpam-6134	241	31	a	a	DET
ejpam-6134	241	32	ce	ce	PROPN
ejpam-6134	241	33	−	−	PROPN
ejpam-6134	241	34	(	(	PUNCT
ejpam-6134	241	35	φ	φ	PROPN
ejpam-6134	241	36	,	,	PUNCT
ejpam-6134	241	37	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	241	38	.	.	PUNCT
ejpam-6134	242	1	throughout	throughout	ADP
ejpam-6134	242	2	,	,	PUNCT
ejpam-6134	242	3	ω	ω	PROPN
ejpam-6134	242	4	will	will	AUX
ejpam-6134	242	5	be	be	AUX
ejpam-6134	242	6	a	a	DET
ejpam-6134	242	7	prime	prime	ADJ
ejpam-6134	242	8	ring	ring	NOUN
ejpam-6134	242	9	,	,	PUNCT
ejpam-6134	242	10	φ	φ	X
ejpam-6134	242	11	an	an	DET
ejpam-6134	242	12	epimorphism	epimorphism	NOUN
ejpam-6134	242	13	on	on	ADP
ejpam-6134	242	14	ω	ω	PROPN
ejpam-6134	242	15	,	,	PUNCT
ejpam-6134	242	16	m	m	PROPN
ejpam-6134	242	17	∈	∈	PROPN
ejpam-6134	242	18	z	z	PROPN
ejpam-6134	242	19	,	,	PUNCT
ejpam-6134	242	20	f	f	PROPN
ejpam-6134	242	21	:	:	PUNCT
ejpam-6134	242	22	ω	ω	PROPN
ejpam-6134	242	23	→	→	SYM
ejpam-6134	242	24	ω	ω	PROPN
ejpam-6134	242	25	a	a	DET
ejpam-6134	242	26	nilpotent	nilpotent	NOUN
ejpam-6134	242	27	centrally	centrally	ADV
ejpam-6134	242	28	extended	extended	ADJ
ejpam-6134	242	29	(	(	PUNCT
ejpam-6134	242	30	φ	φ	NOUN
ejpam-6134	242	31	,	,	PUNCT
ejpam-6134	242	32	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	242	33	,	,	PUNCT
ejpam-6134	242	34	and	and	CCONJ
ejpam-6134	242	35	f	f	PROPN
ejpam-6134	242	36	,	,	PUNCT
ejpam-6134	242	37	φ	φ	PROPN
ejpam-6134	242	38	are	be	AUX
ejpam-6134	242	39	commuting	commute	VERB
ejpam-6134	242	40	.	.	PUNCT
ejpam-6134	243	1	theorem	theorem	ADJ
ejpam-6134	243	2	4	4	NUM
ejpam-6134	243	3	.	.	PUNCT
ejpam-6134	244	1	if	if	SCONJ
ejpam-6134	244	2	f	f	PROPN
ejpam-6134	244	3	is	be	AUX
ejpam-6134	244	4	not	not	PART
ejpam-6134	244	5	a	a	DET
ejpam-6134	244	6	(	(	PUNCT
ejpam-6134	244	7	φ	φ	NOUN
ejpam-6134	244	8	,	,	PUNCT
ejpam-6134	244	9	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	244	10	of	of	ADP
ejpam-6134	244	11	ω	ω	NUM
ejpam-6134	244	12	,	,	PUNCT
ejpam-6134	244	13	then	then	ADV
ejpam-6134	244	14	ω	ω	PROPN
ejpam-6134	244	15	is	be	AUX
ejpam-6134	244	16	commutative	commutative	ADJ
ejpam-6134	244	17	.	.	PUNCT
ejpam-6134	245	1	proof	proof	NOUN
ejpam-6134	245	2	.	.	PUNCT
ejpam-6134	246	1	if	if	SCONJ
ejpam-6134	246	2	ω	ω	PROPN
ejpam-6134	246	3	includes	include	VERB
ejpam-6134	246	4	no	no	DET
ejpam-6134	246	5	non	non	ADJ
ejpam-6134	246	6	-	-	ADJ
ejpam-6134	246	7	zero	zero	ADJ
ejpam-6134	246	8	central	central	ADJ
ejpam-6134	246	9	ideals	ideal	NOUN
ejpam-6134	246	10	,	,	PUNCT
ejpam-6134	246	11	according	accord	VERB
ejpam-6134	246	12	to	to	ADP
ejpam-6134	246	13	theorem	theorem	NOUN
ejpam-6134	246	14	2	2	NUM
ejpam-6134	246	15	,	,	PUNCT
ejpam-6134	246	16	f	f	PROPN
ejpam-6134	246	17	is	be	AUX
ejpam-6134	246	18	a	a	DET
ejpam-6134	246	19	(	(	PUNCT
ejpam-6134	246	20	φ	φ	NOUN
ejpam-6134	246	21	,	,	PUNCT
ejpam-6134	246	22	m)homoderivation	m)homoderivation	NOUN
ejpam-6134	246	23	on	on	ADP
ejpam-6134	246	24	ω	ω	PROPN
ejpam-6134	246	25	,	,	PUNCT
ejpam-6134	246	26	which	which	PRON
ejpam-6134	246	27	is	be	AUX
ejpam-6134	246	28	a	a	DET
ejpam-6134	246	29	contradiction	contradiction	NOUN
ejpam-6134	246	30	.	.	PUNCT
ejpam-6134	247	1	as	as	ADP
ejpam-6134	247	2	a	a	DET
ejpam-6134	247	3	consequence	consequence	NOUN
ejpam-6134	247	4	,	,	PUNCT
ejpam-6134	247	5	ω	ω	PROPN
ejpam-6134	247	6	has	have	VERB
ejpam-6134	247	7	a	a	DET
ejpam-6134	247	8	non	non	ADJ
ejpam-6134	247	9	-	-	ADJ
ejpam-6134	247	10	zero	zero	NUM
ejpam-6134	247	11	ideal	ideal	NOUN
ejpam-6134	247	12	that	that	PRON
ejpam-6134	247	13	is	be	AUX
ejpam-6134	247	14	contained	contain	VERB
ejpam-6134	247	15	in	in	ADP
ejpam-6134	247	16	the	the	DET
ejpam-6134	247	17	center	center	NOUN
ejpam-6134	247	18	ζ(ω	ζ(ω	PROPN
ejpam-6134	247	19	)	)	PUNCT
ejpam-6134	247	20	.	.	PUNCT
ejpam-6134	248	1	so	so	ADV
ejpam-6134	248	2	,	,	PUNCT
ejpam-6134	248	3	ω	ω	PROPN
ejpam-6134	248	4	is	be	AUX
ejpam-6134	248	5	commutative	commutative	ADJ
ejpam-6134	248	6	,	,	PUNCT
ejpam-6134	248	7	by	by	ADP
ejpam-6134	248	8	using	use	VERB
ejpam-6134	248	9	lemma	lemma	PROPN
ejpam-6134	248	10	1	1	NUM
ejpam-6134	248	11	.	.	PUNCT
ejpam-6134	249	1	theorem	theorem	NOUN
ejpam-6134	249	2	5	5	NUM
ejpam-6134	249	3	.	.	PUNCT
ejpam-6134	250	1	if	if	SCONJ
ejpam-6134	250	2	f(0	f(0	NOUN
ejpam-6134	250	3	)	)	PUNCT
ejpam-6134	250	4	is	be	AUX
ejpam-6134	250	5	non	non	ADJ
ejpam-6134	250	6	-	-	ADJ
ejpam-6134	250	7	zero	zero	NUM
ejpam-6134	250	8	,	,	PUNCT
ejpam-6134	250	9	then	then	ADV
ejpam-6134	250	10	ω	ω	PROPN
ejpam-6134	250	11	is	be	AUX
ejpam-6134	250	12	commutative	commutative	ADJ
ejpam-6134	250	13	.	.	PUNCT
ejpam-6134	251	1	proof	proof	NOUN
ejpam-6134	251	2	.	.	PUNCT
ejpam-6134	252	1	due	due	ADP
ejpam-6134	252	2	to	to	ADP
ejpam-6134	252	3	f	f	PROPN
ejpam-6134	252	4	is	be	AUX
ejpam-6134	252	5	a	a	DET
ejpam-6134	252	6	ce−(φ	ce−(φ	NOUN
ejpam-6134	252	7	,	,	PUNCT
ejpam-6134	252	8	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	252	9	,	,	PUNCT
ejpam-6134	252	10	then	then	ADV
ejpam-6134	252	11	f(0	f(0	NOUN
ejpam-6134	252	12	+	+	NOUN
ejpam-6134	252	13	0)−f(0)−f(0	0)−f(0)−f(0	NOUN
ejpam-6134	252	14	)	)	PUNCT
ejpam-6134	252	15	∈	∈	PROPN
ejpam-6134	252	16	ζ(ω	ζ(ω	PROPN
ejpam-6134	252	17	)	)	PUNCT
ejpam-6134	252	18	.	.	PUNCT
ejpam-6134	253	1	it	it	PRON
ejpam-6134	253	2	indicates	indicate	VERB
ejpam-6134	253	3	that	that	SCONJ
ejpam-6134	253	4	f(0	f(0	NOUN
ejpam-6134	253	5	)	)	PUNCT
ejpam-6134	253	6	∈	∈	PROPN
ejpam-6134	253	7	ζ(ω	ζ(ω	PROPN
ejpam-6134	253	8	)	)	PUNCT
ejpam-6134	253	9	.	.	PUNCT
ejpam-6134	254	1	using	use	VERB
ejpam-6134	254	2	the	the	DET
ejpam-6134	254	3	property	property	NOUN
ejpam-6134	254	4	of	of	ADP
ejpam-6134	254	5	f	f	PROPN
ejpam-6134	254	6	,	,	PUNCT
ejpam-6134	254	7	f(0t)−mf(0)f(t)−f(0)φ(t)−	f(0t)−mf(0)f(t)−f(0)φ(t)−	PROPN
ejpam-6134	254	8	φ(0)f(t	φ(0)f(t	PROPN
ejpam-6134	254	9	)	)	PUNCT
ejpam-6134	254	10	∈	∈	PROPN
ejpam-6134	254	11	ζ(ω	ζ(ω	PROPN
ejpam-6134	254	12	)	)	PUNCT
ejpam-6134	254	13	,	,	PUNCT
ejpam-6134	254	14	for	for	ADP
ejpam-6134	254	15	each	each	DET
ejpam-6134	254	16	t	t	PROPN
ejpam-6134	254	17	∈	∈	PROPN
ejpam-6134	254	18	ω	ω	PROPN
ejpam-6134	254	19	,	,	PUNCT
ejpam-6134	254	20	it	it	PRON
ejpam-6134	254	21	indicates	indicate	VERB
ejpam-6134	254	22	f(0)(mf(t	f(0)(mf(t	NOUN
ejpam-6134	254	23	)	)	PUNCT
ejpam-6134	254	24	+	+	NUM
ejpam-6134	255	1	φ(t	φ(t	NOUN
ejpam-6134	255	2	)	)	PUNCT
ejpam-6134	255	3	)	)	PUNCT
ejpam-6134	256	1	∈	∈	PROPN
ejpam-6134	256	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	256	3	)	)	PUNCT
ejpam-6134	256	4	,	,	PUNCT
ejpam-6134	256	5	for	for	ADP
ejpam-6134	256	6	each	each	DET
ejpam-6134	256	7	t	t	PROPN
ejpam-6134	256	8	∈	∈	PROPN
ejpam-6134	256	9	ω	ω	PROPN
ejpam-6134	256	10	.	.	PUNCT
ejpam-6134	257	1	knowing	know	VERB
ejpam-6134	257	2	that	that	SCONJ
ejpam-6134	257	3	f	f	PROPN
ejpam-6134	257	4	,	,	PUNCT
ejpam-6134	257	5	φ	φ	PROPN
ejpam-6134	257	6	are	be	AUX
ejpam-6134	257	7	commuting	commute	VERB
ejpam-6134	257	8	,	,	PUNCT
ejpam-6134	257	9	f	f	PROPN
ejpam-6134	257	10	is	be	AUX
ejpam-6134	257	11	nilpotent	nilpotent	ADJ
ejpam-6134	257	12	on	on	ADP
ejpam-6134	257	13	ω	ω	NUM
ejpam-6134	257	14	,	,	PUNCT
ejpam-6134	257	15	and	and	CCONJ
ejpam-6134	257	16	φ	φ	PROPN
ejpam-6134	257	17	is	be	AUX
ejpam-6134	257	18	surjective	surjective	ADJ
ejpam-6134	257	19	,	,	PUNCT
ejpam-6134	257	20	then	then	ADV
ejpam-6134	257	21	f(0)t	f(0)t	X
ejpam-6134	257	22	∈	∈	PROPN
ejpam-6134	257	23	ζ(ω	ζ(ω	PROPN
ejpam-6134	257	24	)	)	PUNCT
ejpam-6134	257	25	,	,	PUNCT
ejpam-6134	258	1	∀	∀	X
ejpam-6134	258	2	t	t	NOUN
ejpam-6134	258	3	∈	∈	PROPN
ejpam-6134	258	4	ω	ω	PROPN
ejpam-6134	258	5	.	.	PUNCT
ejpam-6134	259	1	therefore	therefore	ADV
ejpam-6134	259	2	,	,	PUNCT
ejpam-6134	259	3	[	[	X
ejpam-6134	259	4	f(0)t	f(0)t	X
ejpam-6134	259	5	,	,	PUNCT
ejpam-6134	259	6	v	v	NOUN
ejpam-6134	259	7	]	]	X
ejpam-6134	259	8	=	=	SYM
ejpam-6134	259	9	0	0	NUM
ejpam-6134	259	10	,	,	PUNCT
ejpam-6134	259	11	∀	∀	X
ejpam-6134	259	12	t	t	PROPN
ejpam-6134	259	13	,	,	PUNCT
ejpam-6134	259	14	v	v	NOUN
ejpam-6134	259	15	∈	∈	PROPN
ejpam-6134	259	16	ω	ω	NOUN
ejpam-6134	259	17	.	.	PUNCT
ejpam-6134	260	1	since	since	SCONJ
ejpam-6134	260	2	f(0	f(0	NOUN
ejpam-6134	260	3	)	)	PUNCT
ejpam-6134	260	4	∈	∈	PROPN
ejpam-6134	260	5	ζ(ω	ζ(ω	PROPN
ejpam-6134	260	6	)	)	PUNCT
ejpam-6134	260	7	,	,	PUNCT
ejpam-6134	260	8	we	we	PRON
ejpam-6134	260	9	get	get	VERB
ejpam-6134	260	10	f(0)[t	f(0)[t	ADJ
ejpam-6134	260	11	,	,	PUNCT
ejpam-6134	260	12	v	v	NOUN
ejpam-6134	260	13	]	]	X
ejpam-6134	260	14	=	=	SYM
ejpam-6134	260	15	0	0	NUM
ejpam-6134	260	16	,	,	PUNCT
ejpam-6134	260	17	for	for	ADP
ejpam-6134	260	18	all	all	DET
ejpam-6134	260	19	v	v	NOUN
ejpam-6134	260	20	,	,	PUNCT
ejpam-6134	260	21	t	t	PROPN
ejpam-6134	260	22	∈	∈	PROPN
ejpam-6134	260	23	ω	ω	PROPN
ejpam-6134	260	24	.	.	PUNCT
ejpam-6134	261	1	replacing	replace	VERB
ejpam-6134	261	2	t	t	PROPN
ejpam-6134	261	3	by	by	ADP
ejpam-6134	261	4	wt	wt	ADP
ejpam-6134	261	5	,	,	PUNCT
ejpam-6134	261	6	we	we	PRON
ejpam-6134	261	7	arrive	arrive	VERB
ejpam-6134	261	8	at	at	ADP
ejpam-6134	261	9	f(0)w[t	f(0)w[t	PROPN
ejpam-6134	261	10	,	,	PUNCT
ejpam-6134	261	11	v	v	NOUN
ejpam-6134	261	12	]	]	X
ejpam-6134	261	13	=	=	SYM
ejpam-6134	261	14	0	0	NUM
ejpam-6134	261	15	,	,	PUNCT
ejpam-6134	261	16	for	for	ADP
ejpam-6134	261	17	each	each	DET
ejpam-6134	261	18	v	v	NOUN
ejpam-6134	261	19	,	,	PUNCT
ejpam-6134	261	20	t	t	PROPN
ejpam-6134	261	21	,	,	PUNCT
ejpam-6134	261	22	w	w	PROPN
ejpam-6134	261	23	∈	∈	PROPN
ejpam-6134	261	24	ω	ω	NOUN
ejpam-6134	261	25	.	.	PUNCT
ejpam-6134	262	1	so	so	ADV
ejpam-6134	262	2	,	,	PUNCT
ejpam-6134	262	3	f(0)ω[t	f(0)ω[t	ADJ
ejpam-6134	262	4	,	,	PUNCT
ejpam-6134	262	5	v	v	NOUN
ejpam-6134	262	6	]	]	X
ejpam-6134	262	7	=	=	SYM
ejpam-6134	262	8	0	0	NUM
ejpam-6134	262	9	,	,	PUNCT
ejpam-6134	262	10	for	for	ADP
ejpam-6134	262	11	all	all	DET
ejpam-6134	262	12	v	v	NOUN
ejpam-6134	262	13	,	,	PUNCT
ejpam-6134	262	14	t	t	PROPN
ejpam-6134	262	15	∈	∈	PROPN
ejpam-6134	262	16	ω	ω	PROPN
ejpam-6134	262	17	.	.	PUNCT
ejpam-6134	263	1	using	use	VERB
ejpam-6134	263	2	the	the	DET
ejpam-6134	263	3	primeness	primeness	NOUN
ejpam-6134	263	4	of	of	ADP
ejpam-6134	263	5	ω	ω	NUM
ejpam-6134	263	6	and	and	CCONJ
ejpam-6134	263	7	f(0	f(0	NOUN
ejpam-6134	263	8	)	)	PUNCT
ejpam-6134	263	9	̸=	̸=	PROPN
ejpam-6134	263	10	0	0	NUM
ejpam-6134	263	11	,	,	PUNCT
ejpam-6134	263	12	[	[	X
ejpam-6134	263	13	t	t	X
ejpam-6134	263	14	,	,	PUNCT
ejpam-6134	263	15	v	v	NOUN
ejpam-6134	263	16	]	]	X
ejpam-6134	263	17	=	=	SYM
ejpam-6134	263	18	0	0	NUM
ejpam-6134	263	19	,	,	PUNCT
ejpam-6134	263	20	for	for	ADP
ejpam-6134	263	21	all	all	DET
ejpam-6134	263	22	v	v	NOUN
ejpam-6134	263	23	,	,	PUNCT
ejpam-6134	263	24	t	t	PROPN
ejpam-6134	263	25	∈	∈	PROPN
ejpam-6134	263	26	ω	ω	PROPN
ejpam-6134	263	27	,	,	PUNCT
ejpam-6134	263	28	i.e.	i.e.	X
ejpam-6134	263	29	,	,	PUNCT
ejpam-6134	263	30	ω	ω	PROPN
ejpam-6134	263	31	is	be	AUX
ejpam-6134	263	32	commutative	commutative	ADJ
ejpam-6134	263	33	.	.	PUNCT
ejpam-6134	264	1	theorem	theorem	NOUN
ejpam-6134	264	2	6	6	NUM
ejpam-6134	264	3	.	.	PUNCT
ejpam-6134	265	1	let	let	VERB
ejpam-6134	265	2	f	f	PROPN
ejpam-6134	265	3	̸=	̸=	PROPN
ejpam-6134	265	4	0	0	NUM
ejpam-6134	265	5	.	.	PUNCT
ejpam-6134	266	1	if	if	SCONJ
ejpam-6134	266	2	f([u	f([u	PROPN
ejpam-6134	266	3	,	,	PUNCT
ejpam-6134	266	4	s	s	PART
ejpam-6134	266	5	]	]	X
ejpam-6134	266	6	)	)	PUNCT
ejpam-6134	267	1	=	=	SYM
ejpam-6134	267	2	0	0	PUNCT
ejpam-6134	267	3	(	(	PUNCT
ejpam-6134	267	4	or	or	CCONJ
ejpam-6134	267	5	f(u	f(u	PROPN
ejpam-6134	267	6	◦	◦	PROPN
ejpam-6134	267	7	s	s	PART
ejpam-6134	267	8	)	)	PUNCT
ejpam-6134	267	9	=	=	SYM
ejpam-6134	267	10	0	0	NUM
ejpam-6134	267	11	)	)	PUNCT
ejpam-6134	267	12	,	,	PUNCT
ejpam-6134	267	13	for	for	ADP
ejpam-6134	267	14	each	each	DET
ejpam-6134	267	15	u	u	NOUN
ejpam-6134	267	16	,	,	PUNCT
ejpam-6134	267	17	s	s	PROPN
ejpam-6134	267	18	∈	∈	PROPN
ejpam-6134	267	19	ω	ω	PROPN
ejpam-6134	267	20	,	,	PUNCT
ejpam-6134	267	21	then	then	ADV
ejpam-6134	267	22	ω	ω	PROPN
ejpam-6134	267	23	is	be	AUX
ejpam-6134	267	24	commutative	commutative	ADJ
ejpam-6134	267	25	.	.	PUNCT
ejpam-6134	268	1	proof	proof	NOUN
ejpam-6134	268	2	.	.	PUNCT
ejpam-6134	269	1	if	if	SCONJ
ejpam-6134	269	2	ω	ω	PROPN
ejpam-6134	269	3	has	have	VERB
ejpam-6134	269	4	a	a	DET
ejpam-6134	269	5	non	non	ADJ
ejpam-6134	269	6	-	-	ADJ
ejpam-6134	269	7	zero	zero	ADJ
ejpam-6134	269	8	central	central	ADJ
ejpam-6134	269	9	ideal	ideal	NOUN
ejpam-6134	269	10	,	,	PUNCT
ejpam-6134	269	11	by	by	ADP
ejpam-6134	269	12	lemma	lemma	PROPN
ejpam-6134	269	13	1	1	NUM
ejpam-6134	269	14	,	,	PUNCT
ejpam-6134	269	15	ω	ω	PROPN
ejpam-6134	269	16	is	be	AUX
ejpam-6134	269	17	commutative	commutative	ADJ
ejpam-6134	269	18	.	.	PUNCT
ejpam-6134	270	1	now	now	ADV
ejpam-6134	270	2	,	,	PUNCT
ejpam-6134	270	3	assume	assume	VERB
ejpam-6134	270	4	that	that	SCONJ
ejpam-6134	270	5	the	the	DET
ejpam-6134	270	6	only	only	ADJ
ejpam-6134	270	7	central	central	ADJ
ejpam-6134	270	8	ideal	ideal	NOUN
ejpam-6134	270	9	in	in	ADP
ejpam-6134	270	10	ω	ω	PROPN
ejpam-6134	270	11	is	be	AUX
ejpam-6134	270	12	the	the	DET
ejpam-6134	270	13	zero	zero	NUM
ejpam-6134	270	14	ideal	ideal	NOUN
ejpam-6134	270	15	.	.	PUNCT
ejpam-6134	271	1	due	due	ADP
ejpam-6134	271	2	to	to	ADP
ejpam-6134	271	3	theorem	theorem	NOUN
ejpam-6134	271	4	1	1	NUM
ejpam-6134	271	5	,	,	PUNCT
ejpam-6134	271	6	f	f	PROPN
ejpam-6134	271	7	is	be	AUX
ejpam-6134	271	8	additive	additive	ADJ
ejpam-6134	271	9	.	.	PUNCT
ejpam-6134	272	1	firstly	firstly	ADV
ejpam-6134	272	2	,	,	PUNCT
ejpam-6134	272	3	assume	assume	VERB
ejpam-6134	272	4	that	that	SCONJ
ejpam-6134	272	5	f([u	f([u	PROPN
ejpam-6134	272	6	,	,	PUNCT
ejpam-6134	272	7	s	s	X
ejpam-6134	272	8	]	]	X
ejpam-6134	272	9	)	)	PUNCT
ejpam-6134	272	10	=	=	SYM
ejpam-6134	272	11	0	0	NUM
ejpam-6134	272	12	,	,	PUNCT
ejpam-6134	272	13	∀	∀	X
ejpam-6134	272	14	u	u	NOUN
ejpam-6134	272	15	,	,	PUNCT
ejpam-6134	272	16	s	s	PROPN
ejpam-6134	272	17	∈	∈	PROPN
ejpam-6134	272	18	ω	ω	PROPN
ejpam-6134	272	19	.	.	PUNCT
ejpam-6134	273	1	substituting	substitute	VERB
ejpam-6134	273	2	su	su	PROPN
ejpam-6134	273	3	for	for	ADP
ejpam-6134	273	4	u	u	PROPN
ejpam-6134	273	5	,	,	PUNCT
ejpam-6134	273	6	we	we	PRON
ejpam-6134	273	7	get	get	VERB
ejpam-6134	273	8	f([su	f([su	ADJ
ejpam-6134	273	9	,	,	PUNCT
ejpam-6134	273	10	s	s	NOUN
ejpam-6134	273	11	]	]	X
ejpam-6134	273	12	)	)	PUNCT
ejpam-6134	273	13	=	=	SYM
ejpam-6134	273	14	0	0	X
ejpam-6134	274	1	=	=	SYM
ejpam-6134	274	2	f(s[u	f(s[u	NOUN
ejpam-6134	274	3	,	,	PUNCT
ejpam-6134	274	4	s	s	PROPN
ejpam-6134	274	5	]	]	X
ejpam-6134	274	6	)	)	PUNCT
ejpam-6134	274	7	,	,	PUNCT
ejpam-6134	274	8	for	for	ADP
ejpam-6134	274	9	each	each	DET
ejpam-6134	274	10	u	u	NOUN
ejpam-6134	274	11	,	,	PUNCT
ejpam-6134	274	12	s	s	PROPN
ejpam-6134	274	13	in	in	ADP
ejpam-6134	274	14	ω	ω	PROPN
ejpam-6134	274	15	.	.	PUNCT
ejpam-6134	275	1	so	so	ADV
ejpam-6134	275	2	f(s)φ([u	f(s)φ([u	NUM
ejpam-6134	275	3	,	,	PUNCT
ejpam-6134	275	4	s	s	X
ejpam-6134	275	5	]	]	PUNCT
ejpam-6134	275	6	)	)	PUNCT
ejpam-6134	275	7	∈	∈	PROPN
ejpam-6134	275	8	ζ(ω	ζ(ω	PROPN
ejpam-6134	275	9	)	)	PUNCT
ejpam-6134	275	10	.	.	PUNCT
ejpam-6134	276	1	since	since	SCONJ
ejpam-6134	276	2	φ	φ	PROPN
ejpam-6134	276	3	is	be	AUX
ejpam-6134	276	4	an	an	DET
ejpam-6134	276	5	epimorphism	epimorphism	NOUN
ejpam-6134	276	6	,	,	PUNCT
ejpam-6134	276	7	we	we	PRON
ejpam-6134	276	8	get	get	VERB
ejpam-6134	276	9	f(u)[s	f(u)[s	NOUN
ejpam-6134	276	10	,	,	PUNCT
ejpam-6134	276	11	φ(u	φ(u	NOUN
ejpam-6134	276	12	)	)	PUNCT
ejpam-6134	276	13	]	]	PUNCT
ejpam-6134	277	1	∈	∈	PROPN
ejpam-6134	277	2	ζ(ω	ζ(ω	PROPN
ejpam-6134	277	3	)	)	PUNCT
ejpam-6134	277	4	,	,	PUNCT
ejpam-6134	277	5	∀	∀	X
ejpam-6134	277	6	u	u	NOUN
ejpam-6134	277	7	,	,	PUNCT
ejpam-6134	277	8	s	s	PROPN
ejpam-6134	277	9	∈	∈	PROPN
ejpam-6134	277	10	ω	ω	PROPN
ejpam-6134	277	11	.	.	PUNCT
ejpam-6134	278	1	(	(	PUNCT
ejpam-6134	278	2	31	31	NUM
ejpam-6134	278	3	)	)	PUNCT
ejpam-6134	278	4	in	in	ADP
ejpam-6134	278	5	(	(	PUNCT
ejpam-6134	278	6	31	31	NUM
ejpam-6134	278	7	)	)	PUNCT
ejpam-6134	278	8	,	,	PUNCT
ejpam-6134	278	9	putting	put	VERB
ejpam-6134	278	10	sφ(u	sφ(u	NOUN
ejpam-6134	278	11	)	)	PUNCT
ejpam-6134	278	12	instead	instead	ADV
ejpam-6134	278	13	of	of	ADP
ejpam-6134	278	14	s	s	PRON
ejpam-6134	278	15	,	,	PUNCT
ejpam-6134	278	16	the	the	DET
ejpam-6134	278	17	result	result	NOUN
ejpam-6134	278	18	is	be	AUX
ejpam-6134	278	19	f(u)[s	f(u)[s	NOUN
ejpam-6134	278	20	,	,	PUNCT
ejpam-6134	278	21	φ(u)]φ(u	φ(u)]φ(u	PROPN
ejpam-6134	278	22	)	)	PUNCT
ejpam-6134	278	23	∈	∈	PROPN
ejpam-6134	278	24	ζ(ω	ζ(ω	PROPN
ejpam-6134	278	25	)	)	PUNCT
ejpam-6134	278	26	,	,	PUNCT
ejpam-6134	278	27	∀	∀	X
ejpam-6134	278	28	u	u	NOUN
ejpam-6134	278	29	,	,	PUNCT
ejpam-6134	278	30	s	s	PROPN
ejpam-6134	278	31	∈	∈	PROPN
ejpam-6134	278	32	ω	ω	NOUN
ejpam-6134	278	33	.	.	PUNCT
ejpam-6134	279	1	thus	thus	ADV
ejpam-6134	279	2	,	,	PUNCT
ejpam-6134	279	3	[	[	X
ejpam-6134	279	4	t	t	PROPN
ejpam-6134	279	5	,	,	PUNCT
ejpam-6134	279	6	f(u)[s	f(u)[s	PROPN
ejpam-6134	279	7	,	,	PUNCT
ejpam-6134	279	8	φ(u)]φ(u	φ(u)]φ(u	PROPN
ejpam-6134	279	9	)	)	PUNCT
ejpam-6134	279	10	]	]	PUNCT
ejpam-6134	280	1	=	=	PUNCT
ejpam-6134	280	2	0	0	NUM
ejpam-6134	280	3	,	,	PUNCT
ejpam-6134	280	4	∀	∀	X
ejpam-6134	280	5	u	u	NOUN
ejpam-6134	280	6	,	,	PUNCT
ejpam-6134	280	7	s	s	PART
ejpam-6134	280	8	,	,	PUNCT
ejpam-6134	280	9	t	t	PROPN
ejpam-6134	280	10	∈	∈	PROPN
ejpam-6134	280	11	ω	ω	PROPN
ejpam-6134	280	12	,	,	PUNCT
ejpam-6134	280	13	which	which	PRON
ejpam-6134	280	14	leads	lead	VERB
ejpam-6134	280	15	to	to	ADP
ejpam-6134	280	16	f(u)[s	f(u)[s	PROPN
ejpam-6134	280	17	,	,	PUNCT
ejpam-6134	280	18	φ(u)][t	φ(u)][t	PROPN
ejpam-6134	280	19	,	,	PUNCT
ejpam-6134	280	20	φ(u	φ(u	NOUN
ejpam-6134	280	21	)	)	PUNCT
ejpam-6134	280	22	]	]	PUNCT
ejpam-6134	281	1	=	=	PUNCT
ejpam-6134	281	2	0	0	NUM
ejpam-6134	281	3	,	,	PUNCT
ejpam-6134	281	4	∀	∀	X
ejpam-6134	281	5	u	u	NOUN
ejpam-6134	281	6	,	,	PUNCT
ejpam-6134	281	7	s	s	PART
ejpam-6134	281	8	,	,	PUNCT
ejpam-6134	281	9	t	t	PROPN
ejpam-6134	281	10	∈	∈	PROPN
ejpam-6134	281	11	ω	ω	PROPN
ejpam-6134	281	12	.	.	PUNCT
ejpam-6134	282	1	(	(	PUNCT
ejpam-6134	282	2	32	32	NUM
ejpam-6134	282	3	)	)	PUNCT
ejpam-6134	282	4	putting	put	VERB
ejpam-6134	282	5	tw	tw	NOUN
ejpam-6134	282	6	in	in	ADP
ejpam-6134	282	7	place	place	NOUN
ejpam-6134	282	8	t	t	X
ejpam-6134	282	9	in	in	ADP
ejpam-6134	282	10	(	(	PUNCT
ejpam-6134	282	11	32	32	NUM
ejpam-6134	282	12	)	)	PUNCT
ejpam-6134	282	13	and	and	CCONJ
ejpam-6134	282	14	using	use	VERB
ejpam-6134	282	15	(	(	PUNCT
ejpam-6134	282	16	32	32	NUM
ejpam-6134	282	17	)	)	PUNCT
ejpam-6134	282	18	,	,	PUNCT
ejpam-6134	282	19	we	we	PRON
ejpam-6134	282	20	get	get	VERB
ejpam-6134	282	21	f(u)[s	f(u)[s	PROPN
ejpam-6134	282	22	,	,	PUNCT
ejpam-6134	282	23	φ(u)]t[w	φ(u)]t[w	PROPN
ejpam-6134	282	24	,	,	PUNCT
ejpam-6134	282	25	φ(u	φ(u	NOUN
ejpam-6134	282	26	)	)	PUNCT
ejpam-6134	282	27	]	]	PUNCT
ejpam-6134	283	1	=	=	PUNCT
ejpam-6134	283	2	0	0	NUM
ejpam-6134	283	3	,	,	PUNCT
ejpam-6134	283	4	∀	∀	X
ejpam-6134	283	5	u	u	NOUN
ejpam-6134	283	6	,	,	PUNCT
ejpam-6134	283	7	s	s	PROPN
ejpam-6134	283	8	,	,	PUNCT
ejpam-6134	283	9	w	w	PROPN
ejpam-6134	283	10	,	,	PUNCT
ejpam-6134	283	11	t	t	PROPN
ejpam-6134	283	12	∈	∈	PROPN
ejpam-6134	283	13	ω	ω	PROPN
ejpam-6134	283	14	.	.	PUNCT
ejpam-6134	284	1	(	(	PUNCT
ejpam-6134	284	2	33	33	NUM
ejpam-6134	284	3	)	)	PUNCT
ejpam-6134	284	4	using	use	VERB
ejpam-6134	284	5	the	the	DET
ejpam-6134	284	6	primeness	primeness	NOUN
ejpam-6134	284	7	of	of	ADP
ejpam-6134	284	8	ω	ω	NUM
ejpam-6134	284	9	,	,	PUNCT
ejpam-6134	284	10	for	for	ADP
ejpam-6134	284	11	each	each	DET
ejpam-6134	284	12	u	u	PROPN
ejpam-6134	284	13	∈	∈	PROPN
ejpam-6134	284	14	ω	ω	NUM
ejpam-6134	284	15	either	either	CCONJ
ejpam-6134	284	16	φ(u	φ(u	NOUN
ejpam-6134	284	17	)	)	PUNCT
ejpam-6134	284	18	∈	∈	PROPN
ejpam-6134	284	19	ζ(ω	ζ(ω	PROPN
ejpam-6134	284	20	)	)	PUNCT
ejpam-6134	284	21	or	or	CCONJ
ejpam-6134	284	22	f(u)[s	f(u)[s	NOUN
ejpam-6134	284	23	,	,	PUNCT
ejpam-6134	284	24	φ(u	φ(u	NOUN
ejpam-6134	284	25	)	)	PUNCT
ejpam-6134	284	26	]	]	PUNCT
ejpam-6134	285	1	=	=	PUNCT
ejpam-6134	285	2	0	0	NUM
ejpam-6134	285	3	,	,	PUNCT
ejpam-6134	285	4	∀	∀	NOUN
ejpam-6134	285	5	s	s	PART
ejpam-6134	285	6	∈	∈	PROPN
ejpam-6134	285	7	ω	ω	PROPN
ejpam-6134	285	8	.	.	PUNCT
ejpam-6134	285	9	assume	assume	VERB
ejpam-6134	285	10	that	that	SCONJ
ejpam-6134	285	11	u	u	PROPN
ejpam-6134	285	12	∈	∈	PROPN
ejpam-6134	285	13	ω	ω	PROPN
ejpam-6134	285	14	with	with	ADP
ejpam-6134	285	15	f(u)[s	f(u)[s	PROPN
ejpam-6134	285	16	,	,	PUNCT
ejpam-6134	285	17	φ(u	φ(u	NOUN
ejpam-6134	285	18	)	)	PUNCT
ejpam-6134	285	19	]	]	PUNCT
ejpam-6134	286	1	=	=	SYM
ejpam-6134	286	2	0	0	NUM
ejpam-6134	286	3	∀	∀	NOUN
ejpam-6134	286	4	s	s	NOUN
ejpam-6134	286	5	∈	∈	PROPN
ejpam-6134	286	6	ω	ω	NOUN
ejpam-6134	286	7	.	.	PUNCT
ejpam-6134	287	1	replacing	replace	VERB
ejpam-6134	287	2	s	s	PRON
ejpam-6134	287	3	by	by	ADP
ejpam-6134	287	4	st	st	PROPN
ejpam-6134	287	5	,	,	PUNCT
ejpam-6134	287	6	we	we	PRON
ejpam-6134	287	7	get	get	VERB
ejpam-6134	287	8	m.m	m.m	PROPN
ejpam-6134	287	9	.	.	PROPN
ejpam-6134	287	10	el	el	PROPN
ejpam-6134	287	11	-	-	PUNCT
ejpam-6134	287	12	soufi	soufi	ADJ
ejpam-6134	287	13	,	,	PUNCT
ejpam-6134	287	14	m.	m.	NOUN
ejpam-6134	287	15	almulhem	almulhem	NOUN
ejpam-6134	287	16	,	,	PUNCT
ejpam-6134	287	17	m.	m.	NOUN
ejpam-6134	287	18	s.	s.	PROPN
ejpam-6134	287	19	tammam	tammam	PROPN
ejpam-6134	287	20	el	el	PROPN
ejpam-6134	287	21	-	-	PROPN
ejpam-6134	287	22	sayiad	sayiad	PROPN
ejpam-6134	287	23	/	/	SYM
ejpam-6134	287	24	eur	eur	PROPN
ejpam-6134	287	25	.	.	PUNCT
ejpam-6134	288	1	j.	j.	PROPN
ejpam-6134	288	2	pure	pure	PROPN
ejpam-6134	288	3	appl	appl	PROPN
ejpam-6134	288	4	.	.	PROPN
ejpam-6134	288	5	math	math	PROPN
ejpam-6134	288	6	,	,	PUNCT
ejpam-6134	288	7	18	18	NUM
ejpam-6134	288	8	(	(	PUNCT
ejpam-6134	288	9	4	4	NUM
ejpam-6134	288	10	)	)	PUNCT
ejpam-6134	288	11	(	(	PUNCT
ejpam-6134	288	12	2025	2025	NUM
ejpam-6134	288	13	)	)	PUNCT
ejpam-6134	288	14	,	,	PUNCT
ejpam-6134	288	15	6134	6134	NUM
ejpam-6134	288	16	10	10	NUM
ejpam-6134	288	17	of	of	ADP
ejpam-6134	288	18	12	12	NUM
ejpam-6134	288	19	f(u)s[t	f(u)s[t	NOUN
ejpam-6134	288	20	,	,	PUNCT
ejpam-6134	288	21	φ(u	φ(u	NOUN
ejpam-6134	288	22	)	)	PUNCT
ejpam-6134	288	23	]	]	PUNCT
ejpam-6134	289	1	=	=	PUNCT
ejpam-6134	289	2	0	0	NUM
ejpam-6134	289	3	,	,	PUNCT
ejpam-6134	289	4	∀	∀	X
ejpam-6134	289	5	t	t	PROPN
ejpam-6134	289	6	,	,	PUNCT
ejpam-6134	289	7	s	s	PART
ejpam-6134	289	8	∈	∈	PROPN
ejpam-6134	289	9	ω	ω	PROPN
ejpam-6134	289	10	.	.	PUNCT
ejpam-6134	290	1	thus	thus	ADV
ejpam-6134	290	2	,	,	PUNCT
ejpam-6134	290	3	for	for	ADP
ejpam-6134	290	4	each	each	DET
ejpam-6134	290	5	u	u	PROPN
ejpam-6134	290	6	∈	∈	PROPN
ejpam-6134	290	7	ω	ω	NUM
ejpam-6134	290	8	either	either	CCONJ
ejpam-6134	290	9	φ(u	φ(u	NOUN
ejpam-6134	290	10	)	)	PUNCT
ejpam-6134	290	11	∈	∈	PROPN
ejpam-6134	290	12	ζ(ω	ζ(ω	PROPN
ejpam-6134	290	13	)	)	PUNCT
ejpam-6134	290	14	or	or	CCONJ
ejpam-6134	290	15	f(u	f(u	PROPN
ejpam-6134	290	16	)	)	PUNCT
ejpam-6134	291	1	=	=	SYM
ejpam-6134	291	2	0	0	X
ejpam-6134	291	3	.	.	X
ejpam-6134	291	4	consider	consider	VERB
ejpam-6134	291	5	that	that	PRON
ejpam-6134	291	6	ℵ	ℵ	NOUN
ejpam-6134	291	7	=	=	SYM
ejpam-6134	291	8	{	{	PUNCT
ejpam-6134	291	9	u	u	NOUN
ejpam-6134	291	10	∈	∈	PROPN
ejpam-6134	291	11	ω	ω	NOUN
ejpam-6134	291	12	:	:	PUNCT
ejpam-6134	291	13	φ(u	φ(u	PROPN
ejpam-6134	291	14	)	)	PUNCT
ejpam-6134	291	15	∈	∈	PROPN
ejpam-6134	291	16	ζ(ω	ζ(ω	PROPN
ejpam-6134	291	17	)	)	PUNCT
ejpam-6134	291	18	}	}	PUNCT
ejpam-6134	291	19	and	and	CCONJ
ejpam-6134	291	20	æ	æ	X
ejpam-6134	291	21	=	=	PRON
ejpam-6134	291	22	{	{	PUNCT
ejpam-6134	291	23	u	u	NOUN
ejpam-6134	291	24	∈	∈	PROPN
ejpam-6134	291	25	ω	ω	NOUN
ejpam-6134	291	26	:	:	PUNCT
ejpam-6134	291	27	f(u	f(u	PROPN
ejpam-6134	291	28	)	)	PUNCT
ejpam-6134	291	29	=	=	PUNCT
ejpam-6134	292	1	0	0	NUM
ejpam-6134	292	2	}	}	PUNCT
ejpam-6134	292	3	.	.	PUNCT
ejpam-6134	293	1	then	then	ADV
ejpam-6134	293	2	,	,	PUNCT
ejpam-6134	293	3	(	(	PUNCT
ejpam-6134	293	4	ℵ,+	ℵ,+	NOUN
ejpam-6134	293	5	)	)	PUNCT
ejpam-6134	293	6	and	and	CCONJ
ejpam-6134	293	7	(	(	PUNCT
ejpam-6134	293	8	æ,+	æ,+	NUM
ejpam-6134	293	9	)	)	PUNCT
ejpam-6134	293	10	are	be	AUX
ejpam-6134	293	11	additive	additive	ADJ
ejpam-6134	293	12	subgroups	subgroup	NOUN
ejpam-6134	293	13	of	of	ADP
ejpam-6134	293	14	the	the	DET
ejpam-6134	293	15	group	group	NOUN
ejpam-6134	293	16	(	(	PUNCT
ejpam-6134	293	17	ω,+	ω,+	NOUN
ejpam-6134	293	18	)	)	PUNCT
ejpam-6134	293	19	,	,	PUNCT
ejpam-6134	293	20	and	and	CCONJ
ejpam-6134	293	21	the	the	DET
ejpam-6134	293	22	union	union	NOUN
ejpam-6134	293	23	of	of	ADP
ejpam-6134	293	24	ℵ	ℵ	PROPN
ejpam-6134	293	25	and	and	CCONJ
ejpam-6134	293	26	æ	æ	PROPN
ejpam-6134	293	27	gives	give	VERB
ejpam-6134	293	28	the	the	DET
ejpam-6134	293	29	whole	whole	ADJ
ejpam-6134	293	30	ring	ring	NOUN
ejpam-6134	293	31	ω	ω	PROPN
ejpam-6134	293	32	.	.	PUNCT
ejpam-6134	294	1	so	so	ADV
ejpam-6134	294	2	either	either	CCONJ
ejpam-6134	294	3	ℵ	ℵ	X
ejpam-6134	294	4	=	=	SYM
ejpam-6134	294	5	ω	ω	PROPN
ejpam-6134	294	6	implies	imply	VERB
ejpam-6134	294	7	ω	ω	PROPN
ejpam-6134	294	8	is	be	AUX
ejpam-6134	294	9	commutative	commutative	ADJ
ejpam-6134	294	10	or	or	CCONJ
ejpam-6134	294	11	æ	æ	X
ejpam-6134	294	12	=	=	SYM
ejpam-6134	294	13	ω	ω	PROPN
ejpam-6134	294	14	implies	imply	VERB
ejpam-6134	294	15	f	f	PROPN
ejpam-6134	294	16	=	=	SYM
ejpam-6134	294	17	0	0	PROPN
ejpam-6134	294	18	.	.	PUNCT
ejpam-6134	295	1	secondly	secondly	ADV
ejpam-6134	295	2	,	,	PUNCT
ejpam-6134	295	3	let	let	VERB
ejpam-6134	295	4	f(u	f(u	PROPN
ejpam-6134	295	5	◦	◦	PROPN
ejpam-6134	295	6	s	s	PART
ejpam-6134	295	7	)	)	PUNCT
ejpam-6134	295	8	=	=	SYM
ejpam-6134	295	9	0	0	NUM
ejpam-6134	295	10	,	,	PUNCT
ejpam-6134	295	11	for	for	SCONJ
ejpam-6134	295	12	all	all	DET
ejpam-6134	295	13	u	u	NOUN
ejpam-6134	295	14	,	,	PUNCT
ejpam-6134	295	15	s	s	PROPN
ejpam-6134	295	16	in	in	ADP
ejpam-6134	295	17	ω	ω	NUM
ejpam-6134	295	18	.	.	PUNCT
ejpam-6134	296	1	putting	put	VERB
ejpam-6134	296	2	su	su	NOUN
ejpam-6134	296	3	instead	instead	ADV
ejpam-6134	296	4	of	of	ADP
ejpam-6134	296	5	u	u	NOUN
ejpam-6134	296	6	in	in	ADP
ejpam-6134	296	7	f(u	f(u	PROPN
ejpam-6134	296	8	◦	◦	PROPN
ejpam-6134	296	9	s	s	PART
ejpam-6134	296	10	)	)	PUNCT
ejpam-6134	296	11	=	=	SYM
ejpam-6134	296	12	0	0	NUM
ejpam-6134	296	13	,	,	PUNCT
ejpam-6134	296	14	then	then	ADV
ejpam-6134	296	15	f(su	f(su	PROPN
ejpam-6134	296	16	◦	◦	NOUN
ejpam-6134	296	17	s	s	PART
ejpam-6134	296	18	)	)	PUNCT
ejpam-6134	297	1	=	=	SYM
ejpam-6134	297	2	f(s(u	f(s(u	PROPN
ejpam-6134	297	3	◦	◦	PROPN
ejpam-6134	297	4	s	s	PART
ejpam-6134	297	5	)	)	PUNCT
ejpam-6134	297	6	)	)	PUNCT
ejpam-6134	298	1	=	=	SYM
ejpam-6134	298	2	0	0	NUM
ejpam-6134	298	3	∀	∀	NOUN
ejpam-6134	298	4	u	u	NOUN
ejpam-6134	298	5	,	,	PUNCT
ejpam-6134	298	6	s	s	PROPN
ejpam-6134	298	7	∈	∈	PROPN
ejpam-6134	298	8	ω	ω	PROPN
ejpam-6134	298	9	.	.	PUNCT
ejpam-6134	299	1	so	so	ADV
ejpam-6134	299	2	,	,	PUNCT
ejpam-6134	299	3	f(s)φ(u	f(s)φ(u	VERB
ejpam-6134	299	4	◦	◦	NOUN
ejpam-6134	299	5	s	s	PART
ejpam-6134	299	6	)	)	PUNCT
ejpam-6134	299	7	∈	∈	PROPN
ejpam-6134	299	8	ζ(ω	ζ(ω	PROPN
ejpam-6134	299	9	)	)	PUNCT
ejpam-6134	299	10	,	,	PUNCT
ejpam-6134	299	11	∀	∀	X
ejpam-6134	299	12	u	u	NOUN
ejpam-6134	299	13	,	,	PUNCT
ejpam-6134	299	14	s	s	PROPN
ejpam-6134	299	15	∈	∈	PROPN
ejpam-6134	299	16	ω	ω	PROPN
ejpam-6134	299	17	.	.	PUNCT
ejpam-6134	300	1	(	(	PUNCT
ejpam-6134	300	2	34	34	NUM
ejpam-6134	300	3	)	)	PUNCT
ejpam-6134	300	4	substituting	substitute	VERB
ejpam-6134	300	5	us	we	PRON
ejpam-6134	300	6	for	for	ADP
ejpam-6134	300	7	u	u	PROPN
ejpam-6134	300	8	in	in	ADP
ejpam-6134	300	9	(	(	PUNCT
ejpam-6134	300	10	34	34	NUM
ejpam-6134	300	11	)	)	PUNCT
ejpam-6134	300	12	,	,	PUNCT
ejpam-6134	300	13	we	we	PRON
ejpam-6134	300	14	get	get	VERB
ejpam-6134	300	15	f(s)φ(u	f(s)φ(u	NUM
ejpam-6134	300	16	◦	◦	PROPN
ejpam-6134	300	17	s)φ(s	s)φ(s	NOUN
ejpam-6134	300	18	)	)	PUNCT
ejpam-6134	300	19	∈	∈	PROPN
ejpam-6134	300	20	ζ(ω	ζ(ω	PROPN
ejpam-6134	300	21	)	)	PUNCT
ejpam-6134	300	22	,	,	PUNCT
ejpam-6134	300	23	∀	∀	X
ejpam-6134	300	24	u	u	NOUN
ejpam-6134	300	25	,	,	PUNCT
ejpam-6134	300	26	s	s	PROPN
ejpam-6134	300	27	∈	∈	PROPN
ejpam-6134	300	28	ω	ω	NOUN
ejpam-6134	300	29	.	.	PUNCT
ejpam-6134	301	1	by	by	ADP
ejpam-6134	301	2	lemma	lemma	PROPN
ejpam-6134	301	3	2	2	NUM
ejpam-6134	301	4	for	for	ADP
ejpam-6134	301	5	each	each	DET
ejpam-6134	301	6	s	s	PROPN
ejpam-6134	301	7	∈	∈	PROPN
ejpam-6134	301	8	ω	ω	PROPN
ejpam-6134	301	9	,	,	PUNCT
ejpam-6134	301	10	either	either	CCONJ
ejpam-6134	301	11	f(s)φ(u	f(s)φ(u	NOUN
ejpam-6134	301	12	◦	◦	PROPN
ejpam-6134	301	13	s	s	PART
ejpam-6134	301	14	)	)	PUNCT
ejpam-6134	301	15	=	=	SYM
ejpam-6134	301	16	0	0	NUM
ejpam-6134	301	17	∀	∀	NOUN
ejpam-6134	301	18	u	u	NOUN
ejpam-6134	301	19	∈	∈	PROPN
ejpam-6134	301	20	ω	ω	NOUN
ejpam-6134	301	21	or	or	CCONJ
ejpam-6134	301	22	φ(s	φ(s	NOUN
ejpam-6134	301	23	)	)	PUNCT
ejpam-6134	301	24	∈	∈	PROPN
ejpam-6134	301	25	ζ(ω	ζ(ω	PROPN
ejpam-6134	301	26	)	)	PUNCT
ejpam-6134	301	27	.	.	PUNCT
ejpam-6134	302	1	assume	assume	VERB
ejpam-6134	302	2	that	that	SCONJ
ejpam-6134	302	3	s	s	VERB
ejpam-6134	302	4	∈	∈	PROPN
ejpam-6134	302	5	ω	ω	NUM
ejpam-6134	302	6	where	where	SCONJ
ejpam-6134	302	7	f(s)φ(u	f(s)φ(u	VERB
ejpam-6134	302	8	◦	◦	NOUN
ejpam-6134	302	9	s	s	PART
ejpam-6134	302	10	)	)	PUNCT
ejpam-6134	302	11	=	=	SYM
ejpam-6134	303	1	0	0	NUM
ejpam-6134	303	2	∀	∀	NOUN
ejpam-6134	303	3	u	u	NOUN
ejpam-6134	303	4	∈	∈	PROPN
ejpam-6134	303	5	ω	ω	PROPN
ejpam-6134	303	6	.	.	PUNCT
ejpam-6134	303	7	(	(	PUNCT
ejpam-6134	303	8	35	35	NUM
ejpam-6134	303	9	)	)	PUNCT
ejpam-6134	303	10	putting	put	VERB
ejpam-6134	303	11	su	su	NOUN
ejpam-6134	303	12	instead	instead	ADV
ejpam-6134	303	13	of	of	ADP
ejpam-6134	303	14	u	u	NOUN
ejpam-6134	303	15	in	in	ADP
ejpam-6134	303	16	(	(	PUNCT
ejpam-6134	303	17	35	35	NUM
ejpam-6134	303	18	)	)	PUNCT
ejpam-6134	303	19	,	,	PUNCT
ejpam-6134	303	20	we	we	PRON
ejpam-6134	303	21	get	get	VERB
ejpam-6134	303	22	f(s)φ(s)φ([u	f(s)φ(s)φ([u	ADJ
ejpam-6134	303	23	,	,	PUNCT
ejpam-6134	303	24	s	s	X
ejpam-6134	303	25	]	]	X
ejpam-6134	303	26	)	)	PUNCT
ejpam-6134	303	27	=	=	SYM
ejpam-6134	303	28	0	0	NUM
ejpam-6134	303	29	,	,	PUNCT
ejpam-6134	303	30	∀	∀	X
ejpam-6134	304	1	u	u	NOUN
ejpam-6134	304	2	∈	∈	PROPN
ejpam-6134	304	3	ω	ω	NOUN
ejpam-6134	304	4	.	.	PUNCT
ejpam-6134	305	1	since	since	SCONJ
ejpam-6134	305	2	φ	φ	PROPN
ejpam-6134	305	3	is	be	AUX
ejpam-6134	305	4	an	an	DET
ejpam-6134	305	5	epimorphism	epimorphism	NOUN
ejpam-6134	305	6	,	,	PUNCT
ejpam-6134	305	7	f(s)φ(s)[u	f(s)φ(s)[u	NOUN
ejpam-6134	305	8	,	,	PUNCT
ejpam-6134	305	9	φ(s	φ(s	NOUN
ejpam-6134	305	10	)	)	PUNCT
ejpam-6134	305	11	]	]	PUNCT
ejpam-6134	306	1	=	=	PUNCT
ejpam-6134	306	2	0	0	NUM
ejpam-6134	306	3	,	,	PUNCT
ejpam-6134	306	4	∀	∀	X
ejpam-6134	306	5	u	u	NOUN
ejpam-6134	306	6	∈	∈	PROPN
ejpam-6134	306	7	ω	ω	NOUN
ejpam-6134	306	8	.	.	PUNCT
ejpam-6134	307	1	putting	put	VERB
ejpam-6134	307	2	ut	ut	PROPN
ejpam-6134	307	3	instead	instead	ADV
ejpam-6134	307	4	of	of	ADP
ejpam-6134	307	5	u	u	NOUN
ejpam-6134	307	6	,	,	PUNCT
ejpam-6134	307	7	we	we	PRON
ejpam-6134	307	8	get	get	VERB
ejpam-6134	307	9	f(s)φ(s)u[t	f(s)φ(s)u[t	NOUN
ejpam-6134	307	10	,	,	PUNCT
ejpam-6134	307	11	φ(s	φ(s	NOUN
ejpam-6134	307	12	)	)	PUNCT
ejpam-6134	307	13	]	]	PUNCT
ejpam-6134	308	1	=	=	PUNCT
ejpam-6134	308	2	0	0	NUM
ejpam-6134	308	3	,	,	PUNCT
ejpam-6134	308	4	∀	∀	X
ejpam-6134	308	5	u	u	NOUN
ejpam-6134	308	6	,	,	PUNCT
ejpam-6134	308	7	t	t	PROPN
ejpam-6134	308	8	∈	∈	PROPN
ejpam-6134	308	9	ω	ω	PROPN
ejpam-6134	308	10	.	.	PUNCT
ejpam-6134	309	1	by	by	ADP
ejpam-6134	309	2	the	the	DET
ejpam-6134	309	3	primeness	primeness	NOUN
ejpam-6134	309	4	of	of	ADP
ejpam-6134	309	5	ω	ω	PROPN
ejpam-6134	309	6	,	,	PUNCT
ejpam-6134	309	7	either	either	CCONJ
ejpam-6134	309	8	f(s)φ(s	f(s)φ(s	NOUN
ejpam-6134	309	9	)	)	PUNCT
ejpam-6134	309	10	=	=	SYM
ejpam-6134	309	11	0	0	NUM
ejpam-6134	309	12	or	or	CCONJ
ejpam-6134	309	13	φ(s	φ(s	NOUN
ejpam-6134	309	14	)	)	PUNCT
ejpam-6134	309	15	∈	∈	PROPN
ejpam-6134	309	16	ζ(ω	ζ(ω	PROPN
ejpam-6134	309	17	)	)	PUNCT
ejpam-6134	309	18	.	.	PUNCT
ejpam-6134	310	1	suppose	suppose	VERB
ejpam-6134	310	2	that	that	SCONJ
ejpam-6134	310	3	f(s)φ(s	f(s)φ(s	NOUN
ejpam-6134	310	4	)	)	PUNCT
ejpam-6134	310	5	=	=	SYM
ejpam-6134	311	1	0	0	X
ejpam-6134	311	2	.	.	PUNCT
ejpam-6134	311	3	by	by	ADP
ejpam-6134	311	4	(	(	PUNCT
ejpam-6134	311	5	35	35	NUM
ejpam-6134	311	6	)	)	PUNCT
ejpam-6134	311	7	,	,	PUNCT
ejpam-6134	311	8	we	we	PRON
ejpam-6134	311	9	get	get	VERB
ejpam-6134	311	10	f(s)φ(u)φ(s	f(s)φ(u)φ(s	NOUN
ejpam-6134	311	11	)	)	PUNCT
ejpam-6134	311	12	=	=	SYM
ejpam-6134	312	1	0	0	NUM
ejpam-6134	312	2	∀	∀	NOUN
ejpam-6134	312	3	u	u	NOUN
ejpam-6134	312	4	∈	∈	PROPN
ejpam-6134	312	5	ω	ω	PROPN
ejpam-6134	312	6	.	.	PUNCT
ejpam-6134	312	7	again	again	ADV
ejpam-6134	312	8	,	,	PUNCT
ejpam-6134	312	9	by	by	ADP
ejpam-6134	312	10	the	the	DET
ejpam-6134	312	11	primeness	primeness	NOUN
ejpam-6134	312	12	of	of	ADP
ejpam-6134	312	13	ω	ω	PROPN
ejpam-6134	312	14	and	and	CCONJ
ejpam-6134	312	15	φ	φ	PROPN
ejpam-6134	312	16	is	be	AUX
ejpam-6134	312	17	an	an	DET
ejpam-6134	312	18	epimorphism	epimorphism	NOUN
ejpam-6134	312	19	,	,	PUNCT
ejpam-6134	312	20	either	either	CCONJ
ejpam-6134	312	21	f(s	f(	NOUN
ejpam-6134	312	22	)	)	PUNCT
ejpam-6134	312	23	=	=	SYM
ejpam-6134	312	24	0	0	NUM
ejpam-6134	312	25	or	or	CCONJ
ejpam-6134	312	26	φ(s	φ(s	NOUN
ejpam-6134	312	27	)	)	PUNCT
ejpam-6134	313	1	=	=	SYM
ejpam-6134	313	2	0	0	X
ejpam-6134	313	3	.	.	PUNCT
ejpam-6134	314	1	therefore	therefore	ADV
ejpam-6134	314	2	,	,	PUNCT
ejpam-6134	314	3	for	for	ADP
ejpam-6134	314	4	each	each	DET
ejpam-6134	314	5	s	s	PROPN
ejpam-6134	314	6	∈	∈	PROPN
ejpam-6134	314	7	ω	ω	NOUN
ejpam-6134	314	8	,	,	PUNCT
ejpam-6134	314	9	there	there	PRON
ejpam-6134	314	10	are	be	VERB
ejpam-6134	314	11	two	two	NUM
ejpam-6134	314	12	cases	case	NOUN
ejpam-6134	314	13	:	:	PUNCT
ejpam-6134	314	14	either	either	PRON
ejpam-6134	314	15	f(s	f(	NOUN
ejpam-6134	314	16	)	)	PUNCT
ejpam-6134	314	17	=	=	SYM
ejpam-6134	314	18	0	0	NUM
ejpam-6134	314	19	or	or	CCONJ
ejpam-6134	314	20	φ(s	φ(s	NOUN
ejpam-6134	314	21	)	)	PUNCT
ejpam-6134	314	22	∈	∈	PROPN
ejpam-6134	314	23	ζ(ω	ζ(ω	PROPN
ejpam-6134	314	24	)	)	PUNCT
ejpam-6134	314	25	.	.	PUNCT
ejpam-6134	315	1	thus	thus	ADV
ejpam-6134	315	2	,	,	PUNCT
ejpam-6134	315	3	as	as	ADP
ejpam-6134	315	4	above	above	ADV
ejpam-6134	315	5	,	,	PUNCT
ejpam-6134	315	6	f	f	PROPN
ejpam-6134	315	7	=	=	SYM
ejpam-6134	315	8	0	0	PROPN
ejpam-6134	315	9	or	or	CCONJ
ejpam-6134	315	10	ω	ω	PROPN
ejpam-6134	315	11	is	be	AUX
ejpam-6134	315	12	commutative	commutative	ADJ
ejpam-6134	315	13	.	.	PUNCT
ejpam-6134	316	1	the	the	DET
ejpam-6134	316	2	primeness	primeness	NOUN
ejpam-6134	316	3	postulate	postulate	NOUN
ejpam-6134	316	4	in	in	ADP
ejpam-6134	316	5	theorem	theorem	NOUN
ejpam-6134	316	6	6	6	NUM
ejpam-6134	316	7	can	can	AUX
ejpam-6134	316	8	not	not	PART
ejpam-6134	316	9	be	be	AUX
ejpam-6134	316	10	disregarded	disregard	VERB
ejpam-6134	316	11	which	which	PRON
ejpam-6134	316	12	is	be	AUX
ejpam-6134	316	13	demonstrated	demonstrate	VERB
ejpam-6134	316	14	by	by	ADP
ejpam-6134	316	15	the	the	DET
ejpam-6134	316	16	counterexample	counterexample	NOUN
ejpam-6134	316	17	that	that	PRON
ejpam-6134	316	18	follows	follow	VERB
ejpam-6134	316	19	.	.	PUNCT
ejpam-6134	317	1	example	example	NOUN
ejpam-6134	318	1	3	3	X
ejpam-6134	318	2	.	.	PUNCT
ejpam-6134	318	3	let	let	VERB
ejpam-6134	318	4	ω	ω	PROPN
ejpam-6134	318	5	=	=	PRON
ejpam-6134	318	6	{	{	PUNCT
ejpam-6134	318	7	(	(	PUNCT
ejpam-6134	318	8	a	a	DET
ejpam-6134	318	9	b	b	NOUN
ejpam-6134	318	10	0	0	NUM
ejpam-6134	318	11	c	c	NOUN
ejpam-6134	318	12	)	)	PUNCT
ejpam-6134	318	13	:	:	PUNCT
ejpam-6134	319	1	0	0	NUM
ejpam-6134	319	2	,	,	PUNCT
ejpam-6134	319	3	a	a	DET
ejpam-6134	319	4	,	,	PUNCT
ejpam-6134	319	5	b	b	NOUN
ejpam-6134	319	6	,	,	PUNCT
ejpam-6134	319	7	c	c	PROPN
ejpam-6134	319	8	∈	∈	PROPN
ejpam-6134	319	9	r	r	NOUN
ejpam-6134	319	10	}	}	PUNCT
ejpam-6134	319	11	be	be	AUX
ejpam-6134	319	12	the	the	DET
ejpam-6134	319	13	ring	ring	NOUN
ejpam-6134	319	14	of	of	ADP
ejpam-6134	319	15	2	2	NUM
ejpam-6134	319	16	×	×	NOUN
ejpam-6134	319	17	2	2	NUM
ejpam-6134	319	18	upper	upper	ADJ
ejpam-6134	319	19	triangular	triangular	NOUN
ejpam-6134	319	20	matrices	matrix	NOUN
ejpam-6134	319	21	over	over	ADP
ejpam-6134	319	22	the	the	DET
ejpam-6134	319	23	field	field	NOUN
ejpam-6134	319	24	of	of	ADP
ejpam-6134	319	25	real	real	ADJ
ejpam-6134	319	26	numbers	number	NOUN
ejpam-6134	319	27	r.	r.	X
ejpam-6134	319	28	consider	consider	VERB
ejpam-6134	319	29	the	the	DET
ejpam-6134	319	30	matrix	matrix	NOUN
ejpam-6134	319	31	e12	e12	NOUN
ejpam-6134	319	32	=	=	SYM
ejpam-6134	319	33	(	(	PUNCT
ejpam-6134	319	34	0	0	NUM
ejpam-6134	319	35	1	1	NUM
ejpam-6134	319	36	0	0	NUM
ejpam-6134	319	37	0	0	NUM
ejpam-6134	319	38	)	)	PUNCT
ejpam-6134	319	39	∈	∈	PROPN
ejpam-6134	319	40	ω	ω	PROPN
ejpam-6134	319	41	,	,	PUNCT
ejpam-6134	319	42	and	and	CCONJ
ejpam-6134	319	43	define	define	VERB
ejpam-6134	319	44	the	the	DET
ejpam-6134	319	45	map	map	NOUN
ejpam-6134	320	1	f	f	X
ejpam-6134	320	2	:	:	PUNCT
ejpam-6134	320	3	ω	ω	PROPN
ejpam-6134	320	4	→	→	SYM
ejpam-6134	320	5	ω	ω	PROPN
ejpam-6134	320	6	by	by	ADP
ejpam-6134	320	7	f	f	PROPN
ejpam-6134	320	8	(	(	PUNCT
ejpam-6134	320	9	(	(	PUNCT
ejpam-6134	320	10	a	a	DET
ejpam-6134	320	11	b	b	NOUN
ejpam-6134	320	12	0	0	NUM
ejpam-6134	320	13	c	c	NOUN
ejpam-6134	320	14	)	)	PUNCT
ejpam-6134	320	15	)	)	PUNCT
ejpam-6134	321	1	=	=	PUNCT
ejpam-6134	321	2	(	(	PUNCT
ejpam-6134	321	3	c	c	X
ejpam-6134	321	4	−	−	PROPN
ejpam-6134	321	5	a)e12	a)e12	PROPN
ejpam-6134	321	6	.	.	PUNCT
ejpam-6134	322	1	take	take	VERB
ejpam-6134	322	2	φ	φ	PROPN
ejpam-6134	322	3	=	=	SYM
ejpam-6134	322	4	idω	idω	PROPN
ejpam-6134	322	5	,	,	PUNCT
ejpam-6134	322	6	the	the	DET
ejpam-6134	322	7	identity	identity	NOUN
ejpam-6134	322	8	map	map	NOUN
ejpam-6134	322	9	on	on	ADP
ejpam-6134	322	10	ω	ω	NUM
ejpam-6134	322	11	,	,	PUNCT
ejpam-6134	322	12	and	and	CCONJ
ejpam-6134	322	13	let	let	VERB
ejpam-6134	322	14	m	m	PRON
ejpam-6134	322	15	∈	∈	PROPN
ejpam-6134	322	16	z	z	PROPN
ejpam-6134	322	17	,	,	PUNCT
ejpam-6134	322	18	the	the	DET
ejpam-6134	322	19	set	set	NOUN
ejpam-6134	322	20	of	of	ADP
ejpam-6134	322	21	integers	integer	NOUN
ejpam-6134	322	22	.	.	PUNCT
ejpam-6134	323	1	it	it	PRON
ejpam-6134	323	2	is	be	AUX
ejpam-6134	323	3	clear	clear	ADJ
ejpam-6134	323	4	that	that	SCONJ
ejpam-6134	323	5	,	,	PUNCT
ejpam-6134	323	6	ω	ω	PROPN
ejpam-6134	323	7	is	be	AUX
ejpam-6134	323	8	not	not	PART
ejpam-6134	323	9	a	a	DET
ejpam-6134	323	10	prime	prime	ADJ
ejpam-6134	323	11	ring	ring	NOUN
ejpam-6134	323	12	.	.	PUNCT
ejpam-6134	324	1	moreover	moreover	ADV
ejpam-6134	324	2	,	,	PUNCT
ejpam-6134	324	3	f	f	PROPN
ejpam-6134	324	4	̸=	̸=	PROPN
ejpam-6134	324	5	0	0	NUM
ejpam-6134	324	6	,	,	PUNCT
ejpam-6134	324	7	f	f	PROPN
ejpam-6134	324	8	◦	◦	NOUN
ejpam-6134	324	9	φ	φ	PROPN
ejpam-6134	324	10	=	=	SYM
ejpam-6134	324	11	φ	φ	PROPN
ejpam-6134	324	12	◦	◦	PROPN
ejpam-6134	324	13	f	f	X
ejpam-6134	324	14	,	,	PUNCT
ejpam-6134	324	15	f	f	PROPN
ejpam-6134	324	16	2	2	NUM
ejpam-6134	324	17	=	=	SYM
ejpam-6134	324	18	0	0	NUM
ejpam-6134	324	19	,	,	PUNCT
ejpam-6134	324	20	and	and	CCONJ
ejpam-6134	324	21	f	f	X
ejpam-6134	324	22	(	(	PUNCT
ejpam-6134	324	23	[	[	X
ejpam-6134	324	24	a	a	X
ejpam-6134	324	25	,	,	PUNCT
ejpam-6134	324	26	b	b	NOUN
ejpam-6134	324	27	]	]	X
ejpam-6134	324	28	)	)	PUNCT
ejpam-6134	324	29	=	=	SYM
ejpam-6134	324	30	0	0	NUM
ejpam-6134	324	31	for	for	ADP
ejpam-6134	324	32	all	all	DET
ejpam-6134	324	33	a	a	DET
ejpam-6134	324	34	,	,	PUNCT
ejpam-6134	324	35	b	b	PROPN
ejpam-6134	324	36	∈	∈	PROPN
ejpam-6134	324	37	ω	ω	PROPN
ejpam-6134	324	38	.	.	PUNCT
ejpam-6134	325	1	the	the	DET
ejpam-6134	325	2	map	map	NOUN
ejpam-6134	325	3	f	f	PROPN
ejpam-6134	325	4	is	be	AUX
ejpam-6134	325	5	a	a	DET
ejpam-6134	325	6	ce−	ce−	PROPN
ejpam-6134	325	7	(	(	PUNCT
ejpam-6134	325	8	φ	φ	NOUN
ejpam-6134	325	9	,	,	PUNCT
ejpam-6134	325	10	m)-homoderivation	m)-homoderivation	NOUN
ejpam-6134	325	11	,	,	PUNCT
ejpam-6134	325	12	while	while	SCONJ
ejpam-6134	325	13	ω	ω	NOUN
ejpam-6134	325	14	is	be	AUX
ejpam-6134	325	15	not	not	PART
ejpam-6134	325	16	a	a	DET
ejpam-6134	325	17	commutative	commutative	ADJ
ejpam-6134	325	18	ring	ring	NOUN
ejpam-6134	325	19	.	.	PUNCT
ejpam-6134	326	1	as	as	ADP
ejpam-6134	326	2	an	an	DET
ejpam-6134	326	3	immediate	immediate	ADJ
ejpam-6134	326	4	consequence	consequence	NOUN
ejpam-6134	326	5	of	of	ADP
ejpam-6134	326	6	theorems	theorem	NOUN
ejpam-6134	326	7	4	4	NUM
ejpam-6134	326	8	,	,	PUNCT
ejpam-6134	326	9	5	5	NUM
ejpam-6134	326	10	,	,	PUNCT
ejpam-6134	326	11	and	and	CCONJ
ejpam-6134	326	12	6	6	NUM
ejpam-6134	326	13	,	,	PUNCT
ejpam-6134	326	14	we	we	PRON
ejpam-6134	326	15	obtain	obtain	VERB
ejpam-6134	326	16	the	the	DET
ejpam-6134	326	17	following	following	ADJ
ejpam-6134	326	18	result	result	NOUN
ejpam-6134	326	19	previously	previously	ADV
ejpam-6134	326	20	established	establish	VERB
ejpam-6134	326	21	in	in	ADP
ejpam-6134	326	22	[	[	X
ejpam-6134	326	23	8	8	NUM
ejpam-6134	326	24	,	,	PUNCT
ejpam-6134	326	25	corollaries	corollary	NOUN
ejpam-6134	326	26	4	4	NUM
ejpam-6134	326	27	,	,	PUNCT
ejpam-6134	326	28	5	5	NUM
ejpam-6134	326	29	,	,	PUNCT
ejpam-6134	326	30	and	and	CCONJ
ejpam-6134	326	31	6	6	NUM
ejpam-6134	326	32	]	]	PUNCT
ejpam-6134	326	33	.	.	PUNCT
ejpam-6134	327	1	m.m	m.m	PROPN
ejpam-6134	327	2	.	.	PROPN
ejpam-6134	327	3	el	el	PROPN
ejpam-6134	327	4	-	-	PUNCT
ejpam-6134	327	5	soufi	soufi	ADJ
ejpam-6134	327	6	,	,	PUNCT
ejpam-6134	327	7	m.	m.	NOUN
ejpam-6134	327	8	almulhem	almulhem	NOUN
ejpam-6134	327	9	,	,	PUNCT
ejpam-6134	327	10	m.	m.	NOUN
ejpam-6134	327	11	s.	s.	PROPN
ejpam-6134	327	12	tammam	tammam	PROPN
ejpam-6134	327	13	el	el	PROPN
ejpam-6134	327	14	-	-	PROPN
ejpam-6134	327	15	sayiad	sayiad	PROPN
ejpam-6134	327	16	/	/	SYM
ejpam-6134	327	17	eur	eur	PROPN
ejpam-6134	327	18	.	.	PUNCT
ejpam-6134	328	1	j.	j.	PROPN
ejpam-6134	328	2	pure	pure	PROPN
ejpam-6134	328	3	appl	appl	PROPN
ejpam-6134	328	4	.	.	PROPN
ejpam-6134	328	5	math	math	PROPN
ejpam-6134	328	6	,	,	PUNCT
ejpam-6134	328	7	18	18	NUM
ejpam-6134	328	8	(	(	PUNCT
ejpam-6134	328	9	4	4	NUM
ejpam-6134	328	10	)	)	PUNCT
ejpam-6134	328	11	(	(	PUNCT
ejpam-6134	328	12	2025	2025	NUM
ejpam-6134	328	13	)	)	PUNCT
ejpam-6134	328	14	,	,	PUNCT
ejpam-6134	328	15	6134	6134	NUM
ejpam-6134	328	16	11	11	NUM
ejpam-6134	328	17	of	of	ADP
ejpam-6134	328	18	12	12	NUM
ejpam-6134	328	19	corollary	corollary	ADJ
ejpam-6134	328	20	4	4	NUM
ejpam-6134	328	21	.	.	PUNCT
ejpam-6134	329	1	a	a	DET
ejpam-6134	329	2	prime	prime	ADJ
ejpam-6134	329	3	ring	ring	NOUN
ejpam-6134	329	4	ω	ω	PROPN
ejpam-6134	329	5	is	be	AUX
ejpam-6134	329	6	commutative	commutative	ADJ
ejpam-6134	329	7	if	if	SCONJ
ejpam-6134	329	8	ω	ω	PROPN
ejpam-6134	329	9	has	have	VERB
ejpam-6134	329	10	a	a	DET
ejpam-6134	329	11	nilpotent	nilpotent	ADJ
ejpam-6134	329	12	ce−homoderivation	ce−homoderivation	NOUN
ejpam-6134	329	13	f	f	PROPN
ejpam-6134	329	14	and	and	CCONJ
ejpam-6134	329	15	any	any	PRON
ejpam-6134	329	16	of	of	ADP
ejpam-6134	329	17	the	the	DET
ejpam-6134	329	18	following	follow	VERB
ejpam-6134	329	19	is	be	AUX
ejpam-6134	329	20	true	true	ADJ
ejpam-6134	329	21	:	:	PUNCT
ejpam-6134	329	22	(	(	PUNCT
ejpam-6134	329	23	i	i	NOUN
ejpam-6134	329	24	)	)	PUNCT
ejpam-6134	329	25	f	f	PROPN
ejpam-6134	329	26	is	be	AUX
ejpam-6134	329	27	not	not	PART
ejpam-6134	329	28	homoderivation	homoderivation	ADJ
ejpam-6134	329	29	.	.	PUNCT
ejpam-6134	330	1	(	(	PUNCT
ejpam-6134	330	2	ii	ii	NOUN
ejpam-6134	330	3	)	)	PUNCT
ejpam-6134	330	4	f(0	f(0	NOUN
ejpam-6134	330	5	)	)	PUNCT
ejpam-6134	330	6	is	be	AUX
ejpam-6134	330	7	non	non	ADJ
ejpam-6134	330	8	-	-	ADJ
ejpam-6134	330	9	zero	zero	NUM
ejpam-6134	330	10	.	.	PUNCT
ejpam-6134	331	1	(	(	PUNCT
ejpam-6134	331	2	iii	iii	X
ejpam-6134	331	3	)	)	PUNCT
ejpam-6134	331	4	f([p	f([p	PROPN
ejpam-6134	331	5	,	,	PUNCT
ejpam-6134	331	6	q	q	NOUN
ejpam-6134	331	7	]	]	X
ejpam-6134	331	8	)	)	PUNCT
ejpam-6134	331	9	=	=	SYM
ejpam-6134	331	10	0	0	PUNCT
ejpam-6134	331	11	(	(	PUNCT
ejpam-6134	331	12	or	or	CCONJ
ejpam-6134	331	13	f(p	f(p	PROPN
ejpam-6134	331	14	◦	◦	NOUN
ejpam-6134	331	15	q	q	PUNCT
ejpam-6134	331	16	)	)	PUNCT
ejpam-6134	331	17	=	=	SYM
ejpam-6134	331	18	0	0	NUM
ejpam-6134	331	19	)	)	PUNCT
ejpam-6134	331	20	for	for	ADP
ejpam-6134	331	21	each	each	DET
ejpam-6134	331	22	p	p	NOUN
ejpam-6134	331	23	,	,	PUNCT
ejpam-6134	331	24	q	q	PROPN
ejpam-6134	331	25	∈	∈	PROPN
ejpam-6134	331	26	ω	ω	PROPN
ejpam-6134	331	27	.	.	PROPN
ejpam-6134	332	1	5	5	NUM
ejpam-6134	332	2	.	.	X
ejpam-6134	332	3	conclusion	conclusion	NOUN
ejpam-6134	332	4	this	this	DET
ejpam-6134	332	5	study	study	NOUN
ejpam-6134	332	6	establishes	establish	VERB
ejpam-6134	332	7	key	key	ADJ
ejpam-6134	332	8	structural	structural	ADJ
ejpam-6134	332	9	properties	property	NOUN
ejpam-6134	332	10	of	of	ADP
ejpam-6134	332	11	ce-(φ	ce-(φ	NOUN
ejpam-6134	332	12	,	,	PUNCT
ejpam-6134	332	13	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	332	14	and	and	CCONJ
ejpam-6134	332	15	clarifies	clarify	VERB
ejpam-6134	332	16	their	their	PRON
ejpam-6134	332	17	reduction	reduction	NOUN
ejpam-6134	332	18	to	to	ADP
ejpam-6134	332	19	classical	classical	ADJ
ejpam-6134	332	20	(	(	PUNCT
ejpam-6134	332	21	φ	φ	NUM
ejpam-6134	332	22	,	,	PUNCT
ejpam-6134	332	23	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	332	24	under	under	ADP
ejpam-6134	332	25	central	central	ADJ
ejpam-6134	332	26	ideal	ideal	ADJ
ejpam-6134	332	27	constraints	constraint	NOUN
ejpam-6134	332	28	.	.	PUNCT
ejpam-6134	333	1	for	for	ADP
ejpam-6134	333	2	prime	prime	ADJ
ejpam-6134	333	3	rings	ring	NOUN
ejpam-6134	333	4	,	,	PUNCT
ejpam-6134	333	5	we	we	PRON
ejpam-6134	333	6	proved	prove	VERB
ejpam-6134	333	7	that	that	SCONJ
ejpam-6134	333	8	the	the	DET
ejpam-6134	333	9	presence	presence	NOUN
ejpam-6134	333	10	of	of	ADP
ejpam-6134	333	11	such	such	ADJ
ejpam-6134	333	12	mappings	mapping	NOUN
ejpam-6134	333	13	enforces	enforce	VERB
ejpam-6134	333	14	commutativity	commutativity	NOUN
ejpam-6134	333	15	,	,	PUNCT
ejpam-6134	333	16	while	while	SCONJ
ejpam-6134	333	17	also	also	ADV
ejpam-6134	333	18	demonstrating	demonstrate	VERB
ejpam-6134	333	19	through	through	ADP
ejpam-6134	333	20	a	a	DET
ejpam-6134	333	21	counterexample	counterexample	NOUN
ejpam-6134	333	22	that	that	SCONJ
ejpam-6134	333	23	the	the	DET
ejpam-6134	333	24	primeness	primeness	NOUN
ejpam-6134	333	25	condition	condition	NOUN
ejpam-6134	333	26	is	be	AUX
ejpam-6134	333	27	indispensable	indispensable	ADJ
ejpam-6134	333	28	.	.	PUNCT
ejpam-6134	334	1	these	these	DET
ejpam-6134	334	2	results	result	VERB
ejpam-6134	334	3	both	both	PRON
ejpam-6134	334	4	extend	extend	VERB
ejpam-6134	334	5	existing	exist	VERB
ejpam-6134	334	6	derivation	derivation	NOUN
ejpam-6134	334	7	theory	theory	NOUN
ejpam-6134	334	8	and	and	CCONJ
ejpam-6134	334	9	provide	provide	VERB
ejpam-6134	334	10	a	a	DET
ejpam-6134	334	11	solid	solid	ADJ
ejpam-6134	334	12	foundation	foundation	NOUN
ejpam-6134	334	13	for	for	ADP
ejpam-6134	334	14	further	further	ADJ
ejpam-6134	334	15	investigations	investigation	NOUN
ejpam-6134	334	16	into	into	ADP
ejpam-6134	334	17	the	the	DET
ejpam-6134	334	18	interplay	interplay	NOUN
ejpam-6134	334	19	between	between	ADP
ejpam-6134	334	20	generalized	generalized	ADJ
ejpam-6134	334	21	homoderivations	homoderivation	NOUN
ejpam-6134	334	22	and	and	CCONJ
ejpam-6134	334	23	ring	ring	NOUN
ejpam-6134	334	24	commutativity	commutativity	NOUN
ejpam-6134	334	25	.	.	PUNCT
ejpam-6134	335	1	acknowledegments	acknowledegment	VERB
ejpam-6134	335	2	the	the	DET
ejpam-6134	335	3	authors	author	NOUN
ejpam-6134	335	4	sincerely	sincerely	ADV
ejpam-6134	335	5	appreciate	appreciate	VERB
ejpam-6134	335	6	the	the	DET
ejpam-6134	335	7	constructive	constructive	ADJ
ejpam-6134	335	8	comments	comment	NOUN
ejpam-6134	335	9	and	and	CCONJ
ejpam-6134	335	10	suggestions	suggestion	NOUN
ejpam-6134	335	11	provided	provide	VERB
ejpam-6134	335	12	by	by	ADP
ejpam-6134	335	13	the	the	DET
ejpam-6134	335	14	reviewers	reviewer	NOUN
ejpam-6134	335	15	and	and	CCONJ
ejpam-6134	335	16	editors	editor	NOUN
ejpam-6134	335	17	,	,	PUNCT
ejpam-6134	335	18	which	which	PRON
ejpam-6134	335	19	have	have	AUX
ejpam-6134	335	20	significantly	significantly	ADV
ejpam-6134	335	21	contributed	contribute	VERB
ejpam-6134	335	22	to	to	ADP
ejpam-6134	335	23	improving	improve	VERB
ejpam-6134	335	24	the	the	DET
ejpam-6134	335	25	quality	quality	NOUN
ejpam-6134	335	26	of	of	ADP
ejpam-6134	335	27	this	this	DET
ejpam-6134	335	28	research	research	NOUN
ejpam-6134	335	29	.	.	PUNCT
ejpam-6134	336	1	references	reference	NOUN
ejpam-6134	336	2	[	[	X
ejpam-6134	336	3	1	1	NUM
ejpam-6134	336	4	]	]	X
ejpam-6134	336	5	h	h	NOUN
ejpam-6134	336	6	e	e	NOUN
ejpam-6134	336	7	bell	bell	NOUN
ejpam-6134	336	8	and	and	CCONJ
ejpam-6134	336	9	m	m	PROPN
ejpam-6134	336	10	n	n	PRON
ejpam-6134	336	11	daif	daif	NOUN
ejpam-6134	336	12	.	.	PUNCT
ejpam-6134	337	1	on	on	ADP
ejpam-6134	337	2	centrally	centrally	ADV
ejpam-6134	337	3	-	-	PUNCT
ejpam-6134	337	4	extended	extend	VERB
ejpam-6134	337	5	maps	map	NOUN
ejpam-6134	337	6	on	on	ADP
ejpam-6134	337	7	rings	ring	NOUN
ejpam-6134	337	8	.	.	PUNCT
ejpam-6134	338	1	beitr	beitr	NOUN
ejpam-6134	338	2	.	.	PUNCT
ejpam-6134	339	1	algebra	algebra	PROPN
ejpam-6134	339	2	geom	geom	PROPN
ejpam-6134	339	3	.	.	PROPN
ejpam-6134	339	4	,	,	PUNCT
ejpam-6134	339	5	57(1):129–136	57(1):129–136	PROPN
ejpam-6134	339	6	,	,	PUNCT
ejpam-6134	339	7	2016	2016	NUM
ejpam-6134	339	8	.	.	PUNCT
ejpam-6134	340	1	https://doi.org/10.1007/s13366-015-0244-8	https://doi.org/10.1007/s13366-015-0244-8	NUM
ejpam-6134	340	2	.	.	PUNCT
ejpam-6134	341	1	[	[	X
ejpam-6134	341	2	2	2	NUM
ejpam-6134	341	3	]	]	PUNCT
ejpam-6134	341	4	m	m	PROPN
ejpam-6134	341	5	s	s	NOUN
ejpam-6134	341	6	tammam	tammam	NOUN
ejpam-6134	341	7	el	el	PROPN
ejpam-6134	341	8	-	-	PUNCT
ejpam-6134	341	9	sayiad	sayiad	PROPN
ejpam-6134	341	10	,	,	PUNCT
ejpam-6134	341	11	n	n	PRON
ejpam-6134	341	12	m	m	VERB
ejpam-6134	341	13	muthana	muthana	NOUN
ejpam-6134	341	14	,	,	PUNCT
ejpam-6134	341	15	and	and	CCONJ
ejpam-6134	341	16	z	z	NOUN
ejpam-6134	341	17	s	s	AUX
ejpam-6134	341	18	alkhamisi	alkhamisi	VERB
ejpam-6134	341	19	.	.	PUNCT
ejpam-6134	342	1	on	on	ADP
ejpam-6134	342	2	right	right	ADJ
ejpam-6134	342	3	generalized	generalized	ADJ
ejpam-6134	342	4	(	(	PUNCT
ejpam-6134	342	5	α	α	X
ejpam-6134	342	6	,	,	PUNCT
ejpam-6134	342	7	β)-derivations	β)-derivations	PROPN
ejpam-6134	342	8	in	in	ADP
ejpam-6134	342	9	prime	prime	ADJ
ejpam-6134	342	10	rings	ring	NOUN
ejpam-6134	342	11	.	.	PUNCT
ejpam-6134	343	1	east	east	PROPN
ejpam-6134	343	2	-	-	PUNCT
ejpam-6134	343	3	west	west	PROPN
ejpam-6134	343	4	j.	j.	PROPN
ejpam-6134	343	5	math	math	PROPN
ejpam-6134	343	6	.	.	PUNCT
ejpam-6134	343	7	,	,	PUNCT
ejpam-6134	343	8	18(1):47–51	18(1):47–51	NUM
ejpam-6134	343	9	,	,	PUNCT
ejpam-6134	343	10	2016	2016	NUM
ejpam-6134	343	11	.	.	PUNCT
ejpam-6134	344	1	[	[	X
ejpam-6134	344	2	3	3	X
ejpam-6134	344	3	]	]	X
ejpam-6134	344	4	m	m	PROPN
ejpam-6134	344	5	s	s	NOUN
ejpam-6134	344	6	tammam	tammam	NOUN
ejpam-6134	344	7	el	el	PROPN
ejpam-6134	344	8	-	-	PROPN
ejpam-6134	344	9	sayiad	sayiad	PROPN
ejpam-6134	344	10	and	and	CCONJ
ejpam-6134	344	11	munerah	munerah	PROPN
ejpam-6134	344	12	almulhem	almulhem	NOUN
ejpam-6134	344	13	.	.	PUNCT
ejpam-6134	345	1	on	on	ADP
ejpam-6134	345	2	centrally	centrally	ADV
ejpam-6134	345	3	extended	extend	VERB
ejpam-6134	345	4	mappings	mapping	NOUN
ejpam-6134	345	5	that	that	PRON
ejpam-6134	345	6	are	be	AUX
ejpam-6134	345	7	centrally	centrally	ADV
ejpam-6134	345	8	extended	extend	VERB
ejpam-6134	345	9	additive	additive	NOUN
ejpam-6134	345	10	.	.	PUNCT
ejpam-6134	346	1	aims	aim	VERB
ejpam-6134	346	2	mathematics	mathematic	NOUN
ejpam-6134	346	3	,	,	PUNCT
ejpam-6134	346	4	9(11):33254–33262	9(11):33254–33262	NUM
ejpam-6134	346	5	,	,	PUNCT
ejpam-6134	346	6	2024	2024	NUM
ejpam-6134	346	7	.	.	PUNCT
ejpam-6134	347	1	https://doi.org/10.3934/math.20241586	https://doi.org/10.3934/math.20241586	NOUN
ejpam-6134	347	2	.	.	PUNCT
ejpam-6134	348	1	[	[	X
ejpam-6134	348	2	4	4	X
ejpam-6134	348	3	]	]	X
ejpam-6134	348	4	m	m	VERB
ejpam-6134	348	5	s	s	NOUN
ejpam-6134	348	6	tammam	tammam	NOUN
ejpam-6134	348	7	el	el	PROPN
ejpam-6134	348	8	-	-	PROPN
ejpam-6134	348	9	sayiad	sayiad	PROPN
ejpam-6134	348	10	and	and	CCONJ
ejpam-6134	348	11	munerah	munerah	PROPN
ejpam-6134	348	12	almulhem	almulhem	NOUN
ejpam-6134	348	13	.	.	PUNCT
ejpam-6134	349	1	on	on	ADP
ejpam-6134	349	2	centrally	centrally	ADV
ejpam-6134	349	3	extended	extended	ADJ
ejpam-6134	349	4	n	n	CCONJ
ejpam-6134	349	5	-	-	PUNCT
ejpam-6134	349	6	homoderivations	homoderivation	NOUN
ejpam-6134	349	7	on	on	ADP
ejpam-6134	349	8	rings	ring	NOUN
ejpam-6134	349	9	.	.	PUNCT
ejpam-6134	350	1	aims	aim	VERB
ejpam-6134	350	2	mathematics	mathematic	NOUN
ejpam-6134	350	3	,	,	PUNCT
ejpam-6134	350	4	10(3):7191–7205	10(3):7191–7205	NUM
ejpam-6134	350	5	,	,	PUNCT
ejpam-6134	350	6	2025	2025	NUM
ejpam-6134	350	7	.	.	PUNCT
ejpam-6134	351	1	https://doi.org/10.3934/math.2025328	https://doi.org/10.3934/math.2025328	NOUN
ejpam-6134	351	2	.	.	PUNCT
ejpam-6134	352	1	[	[	X
ejpam-6134	352	2	5	5	NUM
ejpam-6134	352	3	]	]	SYM
ejpam-6134	352	4	v	v	ADP
ejpam-6134	352	5	t	t	PROPN
ejpam-6134	352	6	filippov	filippov	NOUN
ejpam-6134	352	7	.	.	PUNCT
ejpam-6134	353	1	on	on	ADP
ejpam-6134	353	2	δ	δ	PROPN
ejpam-6134	353	3	-	-	PUNCT
ejpam-6134	353	4	derivations	derivation	NOUN
ejpam-6134	353	5	of	of	ADP
ejpam-6134	353	6	lie	lie	NOUN
ejpam-6134	353	7	algebras	algebra	NOUN
ejpam-6134	353	8	.	.	PUNCT
ejpam-6134	353	9	sib	sib	PROPN
ejpam-6134	353	10	.	.	PUNCT
ejpam-6134	353	11	math	math	PROPN
ejpam-6134	353	12	.	.	PUNCT
ejpam-6134	354	1	j.	j.	PROPN
ejpam-6134	354	2	,	,	PUNCT
ejpam-6134	354	3	39(3):1218–1230	39(3):1218–1230	NUM
ejpam-6134	354	4	,	,	PUNCT
ejpam-6134	354	5	1998	1998	NUM
ejpam-6134	354	6	.	.	PUNCT
ejpam-6134	355	1	[	[	X
ejpam-6134	355	2	6	6	NUM
ejpam-6134	355	3	]	]	X
ejpam-6134	355	4	mahmoud	mahmoud	PROPN
ejpam-6134	355	5	m	m	PROPN
ejpam-6134	355	6	el	el	PROPN
ejpam-6134	355	7	-	-	PUNCT
ejpam-6134	355	8	soufi	soufi	NOUN
ejpam-6134	355	9	.	.	PUNCT
ejpam-6134	356	1	rings	ring	NOUN
ejpam-6134	356	2	with	with	ADP
ejpam-6134	356	3	some	some	DET
ejpam-6134	356	4	kinds	kind	NOUN
ejpam-6134	356	5	of	of	ADP
ejpam-6134	356	6	mappings	mapping	NOUN
ejpam-6134	356	7	.	.	PUNCT
ejpam-6134	357	1	master	master	NOUN
ejpam-6134	357	2	’s	’s	PART
ejpam-6134	357	3	thesis	thesis	NOUN
ejpam-6134	357	4	,	,	PUNCT
ejpam-6134	357	5	cairo	cairo	PROPN
ejpam-6134	357	6	university	university	PROPN
ejpam-6134	357	7	,	,	PUNCT
ejpam-6134	357	8	branch	branch	NOUN
ejpam-6134	357	9	of	of	ADP
ejpam-6134	357	10	fayoum	fayoum	PROPN
ejpam-6134	357	11	,	,	PUNCT
ejpam-6134	357	12	cairo	cairo	PROPN
ejpam-6134	357	13	,	,	PUNCT
ejpam-6134	357	14	egypt	egypt	PROPN
ejpam-6134	357	15	,	,	PUNCT
ejpam-6134	357	16	2000	2000	NUM
ejpam-6134	357	17	.	.	PUNCT
ejpam-6134	358	1	[	[	X
ejpam-6134	358	2	7	7	X
ejpam-6134	358	3	]	]	X
ejpam-6134	358	4	m	m	VERB
ejpam-6134	358	5	s	s	NOUN
ejpam-6134	358	6	tammam	tammam	NOUN
ejpam-6134	358	7	el	el	PROPN
ejpam-6134	358	8	-	-	PUNCT
ejpam-6134	358	9	sayiad	sayiad	PROPN
ejpam-6134	358	10	,	,	PUNCT
ejpam-6134	358	11	a	a	DET
ejpam-6134	358	12	ageeb	ageeb	NOUN
ejpam-6134	358	13	,	,	PUNCT
ejpam-6134	358	14	and	and	CCONJ
ejpam-6134	358	15	a	a	DET
ejpam-6134	358	16	ghareeb	ghareeb	NOUN
ejpam-6134	358	17	.	.	PUNCT
ejpam-6134	359	1	centralizing	centralize	VERB
ejpam-6134	359	2	n	n	CCONJ
ejpam-6134	359	3	-	-	PUNCT
ejpam-6134	359	4	homoderivations	homoderivation	NOUN
ejpam-6134	359	5	of	of	ADP
ejpam-6134	359	6	semiprime	semiprime	NOUN
ejpam-6134	359	7	rings	ring	NOUN
ejpam-6134	359	8	.	.	PUNCT
ejpam-6134	360	1	journal	journal	PROPN
ejpam-6134	360	2	of	of	ADP
ejpam-6134	360	3	mathematics	mathematic	NOUN
ejpam-6134	360	4	,	,	PUNCT
ejpam-6134	360	5	2022(3):article	2022(3):article	NUM
ejpam-6134	360	6	i	i	PROPN
ejpam-6134	360	7	d	d	PROPN
ejpam-6134	360	8	1112183	1112183	NUM
ejpam-6134	360	9	,	,	PUNCT
ejpam-6134	360	10	8	8	NUM
ejpam-6134	360	11	pages	page	NOUN
ejpam-6134	360	12	.	.	PUNCT
ejpam-6134	360	13	,	,	PUNCT
ejpam-6134	360	14	2022	2022	NUM
ejpam-6134	360	15	.	.	PUNCT
ejpam-6134	361	1	https://doi.org/10.1155/2022/1112183	https://doi.org/10.1155/2022/1112183	PROPN
ejpam-6134	361	2	.	.	PUNCT
ejpam-6134	362	1	m.m	m.m	PROPN
ejpam-6134	362	2	.	.	PROPN
ejpam-6134	362	3	el	el	PROPN
ejpam-6134	362	4	-	-	PUNCT
ejpam-6134	362	5	soufi	soufi	ADJ
ejpam-6134	362	6	,	,	PUNCT
ejpam-6134	362	7	m.	m.	NOUN
ejpam-6134	362	8	almulhem	almulhem	NOUN
ejpam-6134	362	9	,	,	PUNCT
ejpam-6134	362	10	m.	m.	NOUN
ejpam-6134	362	11	s.	s.	PROPN
ejpam-6134	362	12	tammam	tammam	PROPN
ejpam-6134	362	13	el	el	PROPN
ejpam-6134	362	14	-	-	PROPN
ejpam-6134	362	15	sayiad	sayiad	PROPN
ejpam-6134	362	16	/	/	SYM
ejpam-6134	362	17	eur	eur	PROPN
ejpam-6134	362	18	.	.	PUNCT
ejpam-6134	363	1	j.	j.	PROPN
ejpam-6134	363	2	pure	pure	PROPN
ejpam-6134	363	3	appl	appl	PROPN
ejpam-6134	363	4	.	.	PROPN
ejpam-6134	363	5	math	math	PROPN
ejpam-6134	363	6	,	,	PUNCT
ejpam-6134	363	7	18	18	NUM
ejpam-6134	363	8	(	(	PUNCT
ejpam-6134	363	9	4	4	NUM
ejpam-6134	363	10	)	)	PUNCT
ejpam-6134	363	11	(	(	PUNCT
ejpam-6134	363	12	2025	2025	NUM
ejpam-6134	363	13	)	)	PUNCT
ejpam-6134	363	14	,	,	PUNCT
ejpam-6134	363	15	6134	6134	NUM
ejpam-6134	363	16	12	12	NUM
ejpam-6134	363	17	of	of	ADP
ejpam-6134	363	18	12	12	NUM
ejpam-6134	364	1	[	[	SYM
ejpam-6134	364	2	8	8	NUM
ejpam-6134	364	3	]	]	X
ejpam-6134	364	4	mahmoud	mahmoud	PROPN
ejpam-6134	364	5	m	m	PROPN
ejpam-6134	364	6	el	el	PROPN
ejpam-6134	364	7	-	-	PUNCT
ejpam-6134	364	8	soufi	soufi	PROPN
ejpam-6134	364	9	and	and	CCONJ
ejpam-6134	364	10	a	a	DET
ejpam-6134	364	11	ghareeb	ghareeb	NOUN
ejpam-6134	364	12	.	.	PUNCT
ejpam-6134	365	1	centrally	centrally	ADV
ejpam-6134	365	2	-	-	PUNCT
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ejpam-6134	365	4	α	α	NOUN
ejpam-6134	365	5	-	-	PUNCT
ejpam-6134	365	6	homo	homo	NOUN
ejpam-6134	365	7	-	-	PUNCT
ejpam-6134	365	8	derivations	derivation	NOUN
ejpam-6134	365	9	on	on	ADP
ejpam-6134	365	10	prime	prime	ADJ
ejpam-6134	365	11	and	and	CCONJ
ejpam-6134	365	12	semiprime	semiprime	NOUN
ejpam-6134	365	13	rings	ring	NOUN
ejpam-6134	365	14	.	.	PUNCT
ejpam-6134	366	1	journal	journal	PROPN
ejpam-6134	366	2	of	of	ADP
ejpam-6134	366	3	mathematics	mathematic	NOUN
ejpam-6134	366	4	,	,	PUNCT
ejpam-6134	366	5	2022	2022	NUM
ejpam-6134	366	6	:	:	PUNCT
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ejpam-6134	366	8	i	i	PROPN
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ejpam-6134	366	12	5	5	NUM
ejpam-6134	366	13	page	page	NOUN
ejpam-6134	366	14	.	.	PUNCT
ejpam-6134	366	15	,	,	PUNCT
ejpam-6134	366	16	2022	2022	NUM
ejpam-6134	366	17	.	.	PUNCT
ejpam-6134	367	1	https://doi.org/10.1155/2022/2584177	https://doi.org/10.1155/2022/2584177	PROPN
ejpam-6134	367	2	.	.	PUNCT
ejpam-6134	368	1	[	[	X
ejpam-6134	368	2	9	9	NUM
ejpam-6134	368	3	]	]	PUNCT
ejpam-6134	368	4	l	l	NOUN
ejpam-6134	368	5	o	o	X
ejpam-6134	368	6	chung	chung	PROPN
ejpam-6134	368	7	.	.	PUNCT
ejpam-6134	369	1	nil	nil	NOUN
ejpam-6134	369	2	derivations	derivation	NOUN
ejpam-6134	369	3	.	.	PUNCT
ejpam-6134	370	1	j.	j.	PROPN
ejpam-6134	370	2	algebra	algebra	PROPN
ejpam-6134	370	3	,	,	PUNCT
ejpam-6134	370	4	95(1):20–30	95(1):20–30	NUM
ejpam-6134	370	5	.	.	NUM
ejpam-6134	370	6	,	,	PUNCT
ejpam-6134	370	7	1985	1985	NUM
ejpam-6134	370	8	.	.	PUNCT
ejpam-6134	371	1	https://doi.org/10.1016/0021-8693(85)90089-4	https://doi.org/10.1016/0021-8693(85)90089-4	NOUN
ejpam-6134	371	2	.	.	PUNCT
ejpam-6134	372	1	[	[	X
ejpam-6134	372	2	10	10	NUM
ejpam-6134	372	3	]	]	X
ejpam-6134	372	4	h	h	NOUN
ejpam-6134	372	5	e	e	NOUN
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ejpam-6134	372	7	and	and	CCONJ
ejpam-6134	372	8	m	m	PROPN
ejpam-6134	372	9	n	n	PRON
ejpam-6134	372	10	daif	daif	NOUN
ejpam-6134	372	11	.	.	PUNCT
ejpam-6134	373	1	on	on	ADP
ejpam-6134	373	2	commutativity	commutativity	NOUN
ejpam-6134	373	3	and	and	CCONJ
ejpam-6134	373	4	strong	strong	ADJ
ejpam-6134	373	5	commutativity	commutativity	NOUN
ejpam-6134	373	6	preserving	preserve	VERB
ejpam-6134	373	7	maps	map	NOUN
ejpam-6134	373	8	.	.	PUNCT
ejpam-6134	374	1	canad	canad	PROPN
ejpam-6134	374	2	.	.	PUNCT
ejpam-6134	375	1	math	math	NOUN
ejpam-6134	375	2	.	.	PUNCT
ejpam-6134	376	1	bull	bull	PROPN
ejpam-6134	376	2	.	.	PUNCT
ejpam-6134	376	3	,	,	PUNCT
ejpam-6134	377	1	37(4):443–447	37(4):443–447	PROPN
ejpam-6134	377	2	.	.	PROPN
ejpam-6134	377	3	,	,	PUNCT
ejpam-6134	377	4	1994	1994	NUM
ejpam-6134	377	5	.	.	PUNCT
ejpam-6134	378	1	https://doi.org/10.4153/cmb-1994064-x	https://doi.org/10.4153/cmb-1994064-x	PROPN
ejpam-6134	378	2	.	.	PUNCT
ejpam-6134	379	1	[	[	X
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ejpam-6134	379	4	j	j	PROPN
ejpam-6134	379	5	h	h	PROPN
ejpam-6134	379	6	mayne	mayne	PROPN
ejpam-6134	379	7	.	.	PUNCT
ejpam-6134	380	1	centralizing	centralize	VERB
ejpam-6134	380	2	mappings	mapping	NOUN
ejpam-6134	380	3	of	of	ADP
ejpam-6134	380	4	prime	prime	ADJ
ejpam-6134	380	5	rings	ring	NOUN
ejpam-6134	380	6	.	.	PUNCT
ejpam-6134	381	1	canad	canad	PROPN
ejpam-6134	381	2	.	.	PUNCT
ejpam-6134	382	1	math	math	NOUN
ejpam-6134	382	2	.	.	PUNCT
ejpam-6134	383	1	bull	bull	PROPN
ejpam-6134	383	2	.	.	PUNCT
ejpam-6134	383	3	,	,	PUNCT
ejpam-6134	383	4	26(1):122–126	26(1):122–126	NUM
ejpam-6134	383	5	,	,	PUNCT
ejpam-6134	383	6	1984	1984	NUM
ejpam-6134	383	7	.	.	PUNCT
ejpam-6134	384	1	https://doi.org/10.4153/cmb-1984-018-2	https://doi.org/10.4153/cmb-1984-018-2	X
ejpam-6134	384	2	.	.	PUNCT
ejpam-6134	384	3	introduction	introduction	NOUN
ejpam-6134	384	4	examples	example	NOUN
ejpam-6134	384	5	of	of	ADP
ejpam-6134	384	6	ce-	ce-	PROPN
ejpam-6134	384	7	(	(	PUNCT
ejpam-6134	384	8	,	,	PUNCT
ejpam-6134	384	9	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	384	10	rings	ring	NOUN
ejpam-6134	384	11	with	with	ADP
ejpam-6134	384	12	centrally	centrally	ADV
ejpam-6134	384	13	extended	extended	ADJ
ejpam-6134	384	14	(	(	PUNCT
ejpam-6134	384	15	,	,	PUNCT
ejpam-6134	384	16	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	384	17	ce-	ce-	X
ejpam-6134	384	18	(	(	PUNCT
ejpam-6134	384	19	,	,	PUNCT
ejpam-6134	384	20	m)-homoderivations	m)-homoderivation	NOUN
ejpam-6134	384	21	and	and	CCONJ
ejpam-6134	384	22	commutativity	commutativity	NOUN
ejpam-6134	384	23	of	of	ADP
ejpam-6134	384	24	prime	prime	ADJ
ejpam-6134	384	25	rings	ring	NOUN
ejpam-6134	384	26	conclusion	conclusion	NOUN