id	sid	tid	token	lemma	pos
ejpam-4602	1	1	european	european	PROPN
ejpam-4602	1	2	journal	journal	PROPN
ejpam-4602	1	3	of	of	ADP
ejpam-4602	1	4	pure	pure	ADJ
ejpam-4602	1	5	and	and	CCONJ
ejpam-4602	1	6	applied	apply	VERB
ejpam-4602	1	7	mathematics	mathematic	NOUN
ejpam-4602	1	8	vol	vol	NOUN
ejpam-4602	1	9	.	.	PROPN
ejpam-4602	2	1	15	15	NUM
ejpam-4602	2	2	,	,	PUNCT
ejpam-4602	2	3	no	no	INTJ
ejpam-4602	2	4	.	.	NOUN
ejpam-4602	2	5	4	4	NUM
ejpam-4602	2	6	,	,	PUNCT
ejpam-4602	2	7	2022	2022	NUM
ejpam-4602	2	8	,	,	PUNCT
ejpam-4602	2	9	2032	2032	NUM
ejpam-4602	2	10	-	-	SYM
ejpam-4602	2	11	2042	2042	NUM
ejpam-4602	2	12	issn	issn	PROPN
ejpam-4602	2	13	1307	1307	NUM
ejpam-4602	2	14	-	-	SYM
ejpam-4602	2	15	5543	5543	NUM
ejpam-4602	2	16	–	–	PUNCT
ejpam-4602	2	17	ejpam.com	ejpam.com	X
ejpam-4602	2	18	published	publish	VERB
ejpam-4602	2	19	by	by	ADP
ejpam-4602	2	20	new	new	PROPN
ejpam-4602	2	21	york	york	PROPN
ejpam-4602	2	22	business	business	PROPN
ejpam-4602	2	23	global	global	ADJ
ejpam-4602	2	24	reverse	reverse	ADJ
ejpam-4602	2	25	derivations	derivation	NOUN
ejpam-4602	2	26	on	on	ADP
ejpam-4602	2	27	δ	δ	NOUN
ejpam-4602	2	28	-	-	PUNCT
ejpam-4602	2	29	prime	prime	PROPN
ejpam-4602	2	30	rings	ring	NOUN
ejpam-4602	2	31	iman	iman	NOUN
ejpam-4602	2	32	taha1,∗	taha1,∗	NOUN
ejpam-4602	2	33	,	,	PUNCT
ejpam-4602	2	34	rohaidah	rohaidah	NOUN
ejpam-4602	2	35	masri1	masri1	NOUN
ejpam-4602	2	36	,	,	PUNCT
ejpam-4602	2	37	ahmad	ahmad	PROPN
ejpam-4602	2	38	al	al	PROPN
ejpam-4602	2	39	khalaf2	khalaf2	PROPN
ejpam-4602	2	40	,	,	PUNCT
ejpam-4602	2	41	rawdah	rawdah	NOUN
ejpam-4602	2	42	tarmizi1	tarmizi1	NOUN
ejpam-4602	2	43	1	1	NUM
ejpam-4602	2	44	department	department	NOUN
ejpam-4602	2	45	of	of	ADP
ejpam-4602	2	46	mathematics	mathematic	NOUN
ejpam-4602	2	47	,	,	PUNCT
ejpam-4602	2	48	faculty	faculty	NOUN
ejpam-4602	2	49	of	of	ADP
ejpam-4602	2	50	sciences	science	NOUN
ejpam-4602	2	51	and	and	CCONJ
ejpam-4602	2	52	mathematics	mathematic	NOUN
ejpam-4602	2	53	,	,	PUNCT
ejpam-4602	2	54	sultan	sultan	PROPN
ejpam-4602	2	55	idris	idris	PROPN
ejpam-4602	2	56	universiti	universiti	PROPN
ejpam-4602	2	57	,	,	PUNCT
ejpam-4602	2	58	tanjong	tanjong	PROPN
ejpam-4602	2	59	malim	malim	PROPN
ejpam-4602	2	60	,	,	PUNCT
ejpam-4602	2	61	perak	perak	PROPN
ejpam-4602	2	62	,	,	PUNCT
ejpam-4602	2	63	malaysia	malaysia	PROPN
ejpam-4602	2	64	2	2	NUM
ejpam-4602	2	65	department	department	NOUN
ejpam-4602	2	66	of	of	ADP
ejpam-4602	2	67	mathematics	mathematic	NOUN
ejpam-4602	2	68	and	and	CCONJ
ejpam-4602	2	69	statistics	statistic	NOUN
ejpam-4602	2	70	,	,	PUNCT
ejpam-4602	2	71	faculty	faculty	NOUN
ejpam-4602	2	72	of	of	ADP
ejpam-4602	2	73	sciences	science	NOUN
ejpam-4602	2	74	,	,	PUNCT
ejpam-4602	2	75	imam	imam	PROPN
ejpam-4602	2	76	mohammad	mohammad	PROPN
ejpam-4602	2	77	ibn	ibn	PROPN
ejpam-4602	2	78	saud	saud	PROPN
ejpam-4602	2	79	islamic	islamic	PROPN
ejpam-4602	2	80	university	university	PROPN
ejpam-4602	2	81	,	,	PUNCT
ejpam-4602	2	82	riyadh	riyadh	PROPN
ejpam-4602	2	83	,	,	PUNCT
ejpam-4602	2	84	riyadh	riyadh	PROPN
ejpam-4602	2	85	,	,	PUNCT
ejpam-4602	2	86	saudi	saudi	PROPN
ejpam-4602	2	87	arabia	arabia	PROPN
ejpam-4602	2	88	abstract	abstract	NOUN
ejpam-4602	2	89	.	.	PUNCT
ejpam-4602	3	1	in	in	ADP
ejpam-4602	3	2	this	this	DET
ejpam-4602	3	3	paper	paper	NOUN
ejpam-4602	3	4	,	,	PUNCT
ejpam-4602	3	5	we	we	PRON
ejpam-4602	3	6	generalized	generalize	VERB
ejpam-4602	3	7	posner	posner	NOUN
ejpam-4602	3	8	’s	’s	PART
ejpam-4602	3	9	theorem	theorem	ADJ
ejpam-4602	3	10	,	,	PUNCT
ejpam-4602	3	11	then	then	ADV
ejpam-4602	3	12	,	,	PUNCT
ejpam-4602	3	13	mayne	mayne	PROPN
ejpam-4602	3	14	’s	’s	PART
ejpam-4602	3	15	theorem	theorem	NOUN
ejpam-4602	3	16	has	have	AUX
ejpam-4602	3	17	been	be	AUX
ejpam-4602	3	18	extended	extend	VERB
ejpam-4602	3	19	to	to	PART
ejpam-4602	3	20	get	get	VERB
ejpam-4602	3	21	a	a	DET
ejpam-4602	3	22	main	main	ADJ
ejpam-4602	3	23	result	result	NOUN
ejpam-4602	3	24	,	,	PUNCT
ejpam-4602	3	25	and	and	CCONJ
ejpam-4602	3	26	presented	present	VERB
ejpam-4602	3	27	by	by	ADP
ejpam-4602	3	28	the	the	DET
ejpam-4602	3	29	following	following	NOUN
ejpam-4602	3	30	theorem	theorem	NOUN
ejpam-4602	3	31	,	,	PUNCT
ejpam-4602	3	32	if	if	SCONJ
ejpam-4602	3	33	δ	δ	PROPN
ejpam-4602	3	34	is	be	AUX
ejpam-4602	3	35	a	a	DET
ejpam-4602	3	36	nonzero	nonzero	NOUN
ejpam-4602	3	37	centralizing	centralize	VERB
ejpam-4602	3	38	reverse	reverse	ADJ
ejpam-4602	3	39	derivations	derivation	NOUN
ejpam-4602	3	40	on	on	ADP
ejpam-4602	3	41	a	a	DET
ejpam-4602	3	42	nonzero	nonzero	ADJ
ejpam-4602	3	43	δ	δ	NOUN
ejpam-4602	3	44	-	-	PUNCT
ejpam-4602	3	45	ideal	ideal	ADJ
ejpam-4602	3	46	u	u	NOUN
ejpam-4602	3	47	of	of	ADP
ejpam-4602	3	48	δ	δ	PROPN
ejpam-4602	3	49	–	–	PUNCT
ejpam-4602	3	50	prime	prime	ADJ
ejpam-4602	3	51	ring	ring	NOUN
ejpam-4602	3	52	r	r	NOUN
ejpam-4602	3	53	,	,	PUNCT
ejpam-4602	3	54	then	then	ADV
ejpam-4602	3	55	r	r	NOUN
ejpam-4602	3	56	is	be	AUX
ejpam-4602	3	57	commutative	commutative	ADJ
ejpam-4602	3	58	.	.	PUNCT
ejpam-4602	4	1	2020	2020	NUM
ejpam-4602	4	2	mathematics	mathematic	NOUN
ejpam-4602	4	3	subject	subject	NOUN
ejpam-4602	4	4	classifications	classification	NOUN
ejpam-4602	4	5	:	:	PUNCT
ejpam-4602	4	6	16w25	16w25	NUM
ejpam-4602	4	7	,	,	PUNCT
ejpam-4602	4	8	16n60	16n60	NUM
ejpam-4602	4	9	key	key	ADJ
ejpam-4602	4	10	words	word	NOUN
ejpam-4602	4	11	and	and	CCONJ
ejpam-4602	4	12	phrases	phrase	NOUN
ejpam-4602	4	13	:	:	PUNCT
ejpam-4602	4	14	reverse	reverse	ADJ
ejpam-4602	4	15	derivation	derivation	NOUN
ejpam-4602	4	16	,	,	PUNCT
ejpam-4602	4	17	δ	δ	NOUN
ejpam-4602	4	18	-	-	PUNCT
ejpam-4602	4	19	prime	prime	ADJ
ejpam-4602	4	20	ring	ring	NOUN
ejpam-4602	4	21	,	,	PUNCT
ejpam-4602	4	22	δ	δ	PROPN
ejpam-4602	4	23	-	-	PUNCT
ejpam-4602	4	24	ideal	ideal	ADJ
ejpam-4602	4	25	1	1	NUM
ejpam-4602	4	26	.	.	PUNCT
ejpam-4602	4	27	introduction	introduction	NOUN
ejpam-4602	4	28	throughout	throughout	ADP
ejpam-4602	4	29	this	this	DET
ejpam-4602	4	30	paper	paper	NOUN
ejpam-4602	4	31	,	,	PUNCT
ejpam-4602	4	32	r	r	NOUN
ejpam-4602	4	33	assumed	assume	VERB
ejpam-4602	4	34	to	to	PART
ejpam-4602	4	35	be	be	AUX
ejpam-4602	4	36	an	an	DET
ejpam-4602	4	37	associative	associative	ADJ
ejpam-4602	4	38	ring	ring	NOUN
ejpam-4602	4	39	with	with	ADP
ejpam-4602	4	40	unity	unity	NOUN
ejpam-4602	4	41	and	and	CCONJ
ejpam-4602	4	42	the	the	DET
ejpam-4602	4	43	center	center	NOUN
ejpam-4602	4	44	z(r	z(r	PROPN
ejpam-4602	4	45	)	)	PUNCT
ejpam-4602	4	46	,	,	PUNCT
ejpam-4602	4	47	the	the	DET
ejpam-4602	4	48	commutator	commutator	NOUN
ejpam-4602	4	49	is	be	AUX
ejpam-4602	4	50	defined	define	VERB
ejpam-4602	4	51	as	as	ADP
ejpam-4602	4	52	[	[	X
ejpam-4602	4	53	u	u	NOUN
ejpam-4602	4	54	,	,	PUNCT
ejpam-4602	4	55	v	v	NOUN
ejpam-4602	4	56	]	]	X
ejpam-4602	4	57	=	=	PUNCT
ejpam-4602	4	58	uv	uv	NOUN
ejpam-4602	4	59	−	−	PROPN
ejpam-4602	4	60	vu	vu	NOUN
ejpam-4602	4	61	,	,	PUNCT
ejpam-4602	4	62	is	be	AUX
ejpam-4602	4	63	also	also	ADV
ejpam-4602	4	64	called	call	VERB
ejpam-4602	4	65	a	a	DET
ejpam-4602	4	66	lie	lie	NOUN
ejpam-4602	4	67	commutator	commutator	NOUN
ejpam-4602	4	68	for	for	ADP
ejpam-4602	4	69	elements	element	NOUN
ejpam-4602	4	70	u	u	NOUN
ejpam-4602	4	71	,	,	PUNCT
ejpam-4602	4	72	v	v	NOUN
ejpam-4602	4	73	∈	∈	NOUN
ejpam-4602	4	74	r	r	NOUN
ejpam-4602	4	75	,	,	PUNCT
ejpam-4602	4	76	the	the	DET
ejpam-4602	4	77	symbol	symbol	NOUN
ejpam-4602	4	78	c(r	c(r	NOUN
ejpam-4602	4	79	)	)	PUNCT
ejpam-4602	4	80	stand	stand	VERB
ejpam-4602	4	81	for	for	ADP
ejpam-4602	4	82	the	the	DET
ejpam-4602	4	83	set	set	NOUN
ejpam-4602	4	84	of	of	ADP
ejpam-4602	4	85	all	all	DET
ejpam-4602	4	86	commutator	commutator	NOUN
ejpam-4602	4	87	ideals	ideal	NOUN
ejpam-4602	4	88	generated	generate	VERB
ejpam-4602	4	89	by	by	ADP
ejpam-4602	4	90	[	[	X
ejpam-4602	4	91	u	u	NOUN
ejpam-4602	4	92	,	,	PUNCT
ejpam-4602	4	93	v	v	ADP
ejpam-4602	4	94	]	]	PUNCT
ejpam-4602	4	95	,	,	PUNCT
ejpam-4602	4	96	the	the	DET
ejpam-4602	4	97	set	set	NOUN
ejpam-4602	4	98	annu	annu	X
ejpam-4602	4	99	=	=	SYM
ejpam-4602	4	100	{	{	PUNCT
ejpam-4602	4	101	r	r	NOUN
ejpam-4602	4	102	∈	∈	PROPN
ejpam-4602	4	103	r	r	NOUN
ejpam-4602	4	104	:	:	PUNCT
ejpam-4602	4	105	ru	ru	PROPN
ejpam-4602	4	106	=	=	SYM
ejpam-4602	4	107	0	0	NUM
ejpam-4602	4	108	}	}	PUNCT
ejpam-4602	4	109	is	be	AUX
ejpam-4602	4	110	the	the	DET
ejpam-4602	4	111	annihilator	annihilator	NOUN
ejpam-4602	4	112	of	of	ADP
ejpam-4602	4	113	u	u	PROPN
ejpam-4602	4	114	of	of	ADP
ejpam-4602	4	115	r.	r.	PROPN
ejpam-4602	4	116	the	the	DET
ejpam-4602	4	117	smallest	small	ADJ
ejpam-4602	4	118	positive	positive	ADJ
ejpam-4602	4	119	integer	integer	NOUN
ejpam-4602	4	120	n	n	CCONJ
ejpam-4602	4	121	such	such	ADJ
ejpam-4602	4	122	that	that	SCONJ
ejpam-4602	4	123	n	n	NOUN
ejpam-4602	4	124	·	·	PUNCT
ejpam-4602	4	125	u	u	NOUN
ejpam-4602	4	126	=	=	NOUN
ejpam-4602	4	127	0	0	NUM
ejpam-4602	4	128	for	for	SCONJ
ejpam-4602	4	129	all	all	PRON
ejpam-4602	4	130	u	u	NOUN
ejpam-4602	4	131	∈	∈	NOUN
ejpam-4602	4	132	r	r	NOUN
ejpam-4602	4	133	is	be	AUX
ejpam-4602	4	134	the	the	DET
ejpam-4602	4	135	characteristic	characteristic	NOUN
ejpam-4602	4	136	of	of	ADP
ejpam-4602	4	137	the	the	DET
ejpam-4602	4	138	ring	ring	NOUN
ejpam-4602	4	139	r.	r.	PROPN
ejpam-4602	4	140	we	we	PRON
ejpam-4602	4	141	will	will	AUX
ejpam-4602	4	142	employ	employ	VERB
ejpam-4602	4	143	some	some	DET
ejpam-4602	4	144	commutator	commutator	NOUN
ejpam-4602	4	145	properties	property	NOUN
ejpam-4602	4	146	like	like	ADP
ejpam-4602	4	147	[	[	X
ejpam-4602	4	148	uv	uv	NOUN
ejpam-4602	4	149	,	,	PUNCT
ejpam-4602	4	150	w	w	NOUN
ejpam-4602	4	151	]	]	X
ejpam-4602	4	152	=	=	SYM
ejpam-4602	4	153	u[v	u[v	PROPN
ejpam-4602	4	154	,	,	PUNCT
ejpam-4602	4	155	w]+[u	w]+[u	PROPN
ejpam-4602	4	156	,	,	PUNCT
ejpam-4602	4	157	w]v	w]v	VERB
ejpam-4602	4	158	and	and	CCONJ
ejpam-4602	4	159	[	[	X
ejpam-4602	4	160	u	u	NOUN
ejpam-4602	4	161	,	,	PUNCT
ejpam-4602	4	162	vw	vw	PROPN
ejpam-4602	4	163	]	]	X
ejpam-4602	4	164	=	=	SYM
ejpam-4602	4	165	v[u	v[u	PROPN
ejpam-4602	4	166	,	,	PUNCT
ejpam-4602	4	167	w	w	NOUN
ejpam-4602	4	168	]	]	X
ejpam-4602	5	1	+	+	CCONJ
ejpam-4602	5	2	[	[	X
ejpam-4602	5	3	u	u	NOUN
ejpam-4602	5	4	,	,	PUNCT
ejpam-4602	5	5	v]w	v]w	ADP
ejpam-4602	5	6	∀u	∀u	NOUN
ejpam-4602	5	7	,	,	PUNCT
ejpam-4602	5	8	v	v	NOUN
ejpam-4602	5	9	,	,	PUNCT
ejpam-4602	5	10	w	w	PROPN
ejpam-4602	5	11	∈	∈	PROPN
ejpam-4602	5	12	r.	r.	PROPN
ejpam-4602	5	13	furthermore	furthermore	ADV
ejpam-4602	5	14	,	,	PUNCT
ejpam-4602	5	15	we	we	PRON
ejpam-4602	5	16	recall	recall	VERB
ejpam-4602	5	17	that	that	SCONJ
ejpam-4602	5	18	r	r	NOUN
ejpam-4602	5	19	is	be	AUX
ejpam-4602	5	20	called	call	VERB
ejpam-4602	5	21	a	a	DET
ejpam-4602	5	22	prime	prime	ADJ
ejpam-4602	5	23	ring	ring	NOUN
ejpam-4602	5	24	if	if	SCONJ
ejpam-4602	5	25	urv	urv	ADP
ejpam-4602	5	26	=	=	SYM
ejpam-4602	5	27	{	{	PUNCT
ejpam-4602	5	28	0	0	NUM
ejpam-4602	5	29	}	}	PUNCT
ejpam-4602	5	30	,	,	PUNCT
ejpam-4602	5	31	then	then	ADV
ejpam-4602	5	32	either	either	CCONJ
ejpam-4602	5	33	u	u	PROPN
ejpam-4602	5	34	=	=	NOUN
ejpam-4602	5	35	0	0	NUM
ejpam-4602	5	36	or	or	CCONJ
ejpam-4602	5	37	v	v	NOUN
ejpam-4602	5	38	=	=	SYM
ejpam-4602	5	39	0	0	NUM
ejpam-4602	5	40	,	,	PUNCT
ejpam-4602	5	41	and	and	CCONJ
ejpam-4602	5	42	by	by	ADP
ejpam-4602	5	43	analogy	analogy	NOUN
ejpam-4602	5	44	,	,	PUNCT
ejpam-4602	5	45	r	r	NOUN
ejpam-4602	5	46	is	be	AUX
ejpam-4602	5	47	called	call	VERB
ejpam-4602	5	48	a	a	DET
ejpam-4602	5	49	semiprime	semiprime	NOUN
ejpam-4602	5	50	ring	ring	NOUN
ejpam-4602	5	51	,	,	PUNCT
ejpam-4602	5	52	for	for	ADP
ejpam-4602	5	53	u	u	PROPN
ejpam-4602	5	54	∈	∈	PROPN
ejpam-4602	5	55	r	r	NOUN
ejpam-4602	5	56	,	,	PUNCT
ejpam-4602	5	57	if	if	SCONJ
ejpam-4602	5	58	uru	uru	PROPN
ejpam-4602	5	59	=	=	SYM
ejpam-4602	5	60	{	{	PUNCT
ejpam-4602	5	61	0	0	NUM
ejpam-4602	5	62	}	}	PUNCT
ejpam-4602	5	63	,	,	PUNCT
ejpam-4602	5	64	then	then	ADV
ejpam-4602	5	65	u	u	X
ejpam-4602	5	66	=	=	NOUN
ejpam-4602	5	67	0	0	PROPN
ejpam-4602	5	68	.	.	PUNCT
ejpam-4602	6	1	a	a	DET
ejpam-4602	6	2	map	map	NOUN
ejpam-4602	6	3	f	f	NOUN
ejpam-4602	6	4	from	from	ADP
ejpam-4602	6	5	r	r	NOUN
ejpam-4602	6	6	to	to	ADP
ejpam-4602	6	7	r	r	NOUN
ejpam-4602	6	8	is	be	AUX
ejpam-4602	6	9	said	say	VERB
ejpam-4602	6	10	to	to	PART
ejpam-4602	6	11	be	be	AUX
ejpam-4602	6	12	a	a	DET
ejpam-4602	6	13	centralizing	centralizing	NOUN
ejpam-4602	6	14	on	on	ADP
ejpam-4602	6	15	u	u	PRON
ejpam-4602	6	16	if	if	SCONJ
ejpam-4602	6	17	[	[	X
ejpam-4602	6	18	u	u	NOUN
ejpam-4602	6	19	,	,	PUNCT
ejpam-4602	6	20	f	f	PROPN
ejpam-4602	6	21	(	(	PUNCT
ejpam-4602	6	22	u	u	NOUN
ejpam-4602	6	23	)	)	PUNCT
ejpam-4602	6	24	]	]	PUNCT
ejpam-4602	7	1	∈	∈	PROPN
ejpam-4602	7	2	z(r	z(r	PROPN
ejpam-4602	7	3	)	)	PUNCT
ejpam-4602	7	4	for	for	ADP
ejpam-4602	7	5	all	all	DET
ejpam-4602	7	6	u	u	PRON
ejpam-4602	7	7	∈	∈	PROPN
ejpam-4602	7	8	u	u	NOUN
ejpam-4602	7	9	.	.	PUNCT
ejpam-4602	8	1	an	an	DET
ejpam-4602	8	2	additive	additive	ADJ
ejpam-4602	8	3	map	map	NOUN
ejpam-4602	8	4	δ	δ	NOUN
ejpam-4602	8	5	:	:	PUNCT
ejpam-4602	8	6	r	r	NOUN
ejpam-4602	8	7	→	→	SYM
ejpam-4602	8	8	r	r	NOUN
ejpam-4602	8	9	is	be	AUX
ejpam-4602	8	10	called	call	VERB
ejpam-4602	8	11	a	a	DET
ejpam-4602	8	12	derivation	derivation	NOUN
ejpam-4602	8	13	on	on	ADP
ejpam-4602	8	14	r	r	NOUN
ejpam-4602	8	15	,	,	PUNCT
ejpam-4602	8	16	if	if	SCONJ
ejpam-4602	8	17	the	the	DET
ejpam-4602	8	18	condition	condition	NOUN
ejpam-4602	8	19	δ(uv	δ(uv	NOUN
ejpam-4602	8	20	)	)	PUNCT
ejpam-4602	8	21	=	=	SYM
ejpam-4602	8	22	δ(u)v	δ(u)v	PROPN
ejpam-4602	9	1	+	+	CCONJ
ejpam-4602	10	1	uδ(v	uδ(v	PROPN
ejpam-4602	10	2	)	)	PUNCT
ejpam-4602	10	3	,	,	PUNCT
ejpam-4602	10	4	∀u	∀u	NOUN
ejpam-4602	10	5	,	,	PUNCT
ejpam-4602	10	6	v	v	NOUN
ejpam-4602	10	7	∈	∈	NOUN
ejpam-4602	10	8	r	r	NOUN
ejpam-4602	10	9	holds	hold	NOUN
ejpam-4602	10	10	,	,	PUNCT
ejpam-4602	10	11	while	while	SCONJ
ejpam-4602	10	12	an	an	DET
ejpam-4602	10	13	additive	additive	ADJ
ejpam-4602	10	14	map	map	NOUN
ejpam-4602	10	15	δ	δ	NOUN
ejpam-4602	10	16	:	:	PUNCT
ejpam-4602	10	17	r	r	NOUN
ejpam-4602	10	18	→	→	SYM
ejpam-4602	10	19	r	r	NOUN
ejpam-4602	10	20	is	be	AUX
ejpam-4602	10	21	said	say	VERB
ejpam-4602	10	22	to	to	PART
ejpam-4602	10	23	be	be	AUX
ejpam-4602	10	24	a	a	DET
ejpam-4602	10	25	reverse	reverse	ADJ
ejpam-4602	10	26	derivation	derivation	NOUN
ejpam-4602	10	27	on	on	ADP
ejpam-4602	10	28	r	r	NOUN
ejpam-4602	10	29	if	if	SCONJ
ejpam-4602	10	30	satisfies	satisfy	VERB
ejpam-4602	10	31	the	the	DET
ejpam-4602	10	32	rule	rule	NOUN
ejpam-4602	10	33	δ(uv	δ(uv	NOUN
ejpam-4602	10	34	)	)	PUNCT
ejpam-4602	10	35	=	=	SYM
ejpam-4602	10	36	δ(v)u+	δ(v)u+	X
ejpam-4602	10	37	vδ(u	vδ(u	NOUN
ejpam-4602	10	38	)	)	PUNCT
ejpam-4602	10	39	,	,	PUNCT
ejpam-4602	10	40	∀u	∀u	NOUN
ejpam-4602	10	41	,	,	PUNCT
ejpam-4602	10	42	v	v	ADP
ejpam-4602	10	43	∈	∈	PROPN
ejpam-4602	10	44	r.	r.	PROPN
ejpam-4602	10	45	furthermore	furthermore	ADV
ejpam-4602	10	46	,	,	PUNCT
ejpam-4602	10	47	for	for	ADP
ejpam-4602	10	48	a	a	DET
ejpam-4602	10	49	fixed	fix	VERB
ejpam-4602	10	50	element	element	NOUN
ejpam-4602	10	51	u	u	NOUN
ejpam-4602	10	52	∈	∈	PROPN
ejpam-4602	10	53	r	r	NOUN
ejpam-4602	10	54	the	the	DET
ejpam-4602	10	55	additive	additive	ADJ
ejpam-4602	10	56	map	map	NOUN
ejpam-4602	10	57	∂u	∂u	NOUN
ejpam-4602	10	58	:	:	PUNCT
ejpam-4602	10	59	r	r	NOUN
ejpam-4602	10	60	→	→	SYM
ejpam-4602	10	61	r	r	NOUN
ejpam-4602	10	62	defined	define	VERB
ejpam-4602	10	63	by	by	ADP
ejpam-4602	10	64	,	,	PUNCT
ejpam-4602	10	65	∂u(v	∂u(v	PROPN
ejpam-4602	10	66	)	)	PUNCT
ejpam-4602	11	1	=	=	NOUN
ejpam-4602	11	2	uv	uv	NOUN
ejpam-4602	11	3	−	−	PROPN
ejpam-4602	11	4	vu	vu	NOUN
ejpam-4602	11	5	,	,	PUNCT
ejpam-4602	11	6	where	where	SCONJ
ejpam-4602	11	7	v	v	X
ejpam-4602	11	8	∈	∈	PROPN
ejpam-4602	11	9	r	r	NOUN
ejpam-4602	11	10	is	be	AUX
ejpam-4602	11	11	called	call	VERB
ejpam-4602	11	12	a	a	DET
ejpam-4602	11	13	partial	partial	ADJ
ejpam-4602	11	14	derivation	derivation	NOUN
ejpam-4602	11	15	generated	generate	VERB
ejpam-4602	11	16	by	by	ADP
ejpam-4602	11	17	u	u	PROPN
ejpam-4602	11	18	∈	∈	PROPN
ejpam-4602	11	19	r	r	NOUN
ejpam-4602	11	20	(	(	PUNCT
ejpam-4602	11	21	∂u(u	∂u(u	NOUN
ejpam-4602	11	22	)	)	PUNCT
ejpam-4602	11	23	=	=	PUNCT
ejpam-4602	12	1	[	[	X
ejpam-4602	12	2	u	u	NOUN
ejpam-4602	12	3	,	,	PUNCT
ejpam-4602	12	4	u	u	NOUN
ejpam-4602	12	5	]	]	X
ejpam-4602	12	6	=	=	PUNCT
ejpam-4602	12	7	{	{	PUNCT
ejpam-4602	12	8	[	[	X
ejpam-4602	12	9	u	u	NOUN
ejpam-4602	12	10	,	,	PUNCT
ejpam-4602	12	11	t	t	PROPN
ejpam-4602	12	12	]	]	PUNCT
ejpam-4602	12	13	:	:	PUNCT
ejpam-4602	12	14	t	t	PROPN
ejpam-4602	12	15	∈	∈	PROPN
ejpam-4602	12	16	u	u	NOUN
ejpam-4602	12	17	}	}	PUNCT
ejpam-4602	12	18	)	)	PUNCT
ejpam-4602	12	19	.	.	PUNCT
ejpam-4602	13	1	∗corresponding	∗corresponde	VERB
ejpam-4602	13	2	author	author	NOUN
ejpam-4602	13	3	.	.	PUNCT
ejpam-4602	14	1	doi	doi	NOUN
ejpam-4602	14	2	:	:	PUNCT
ejpam-4602	14	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4602	https://doi.org/10.29020/nybg.ejpam.v15i4.4602	PROPN
ejpam-4602	14	4	email	email	NOUN
ejpam-4602	14	5	addresses	address	NOUN
ejpam-4602	14	6	:	:	PUNCT
ejpam-4602	14	7	tfaith80gmail.com	tfaith80gmail.com	X
ejpam-4602	14	8	(	(	PUNCT
ejpam-4602	14	9	i.	i.	PROPN
ejpam-4602	14	10	taha	taha	PROPN
ejpam-4602	14	11	)	)	PUNCT
ejpam-4602	14	12	,	,	PUNCT
ejpam-4602	14	13	ajalkalaf@imamu.edu.sa	ajalkalaf@imamu.edu.sa	NOUN
ejpam-4602	14	14	(	(	PUNCT
ejpam-4602	14	15	a.	a.	PROPN
ejpam-4602	14	16	al	al	PROPN
ejpam-4602	14	17	khalaf	khalaf	PROPN
ejpam-4602	14	18	)	)	PUNCT
ejpam-4602	14	19	,	,	PUNCT
ejpam-4602	14	20	rohaidah@fsmt.upsi.edu.my	rohaidah@fsmt.upsi.edu.my	PROPN
ejpam-4602	14	21	(	(	PUNCT
ejpam-4602	14	22	r.	r.	PROPN
ejpam-4602	14	23	masri	masri	PROPN
ejpam-4602	14	24	)	)	PUNCT
ejpam-4602	14	25	,	,	PUNCT
ejpam-4602	14	26	rawdah@fsmt.upsi.edu.my	rawdah@fsmt.upsi.edu.my	NOUN
ejpam-4602	14	27	(	(	PUNCT
ejpam-4602	14	28	r.	r.	PROPN
ejpam-4602	14	29	tarmizi	tarmizi	NOUN
ejpam-4602	14	30	)	)	PUNCT
ejpam-4602	14	31	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4602	14	32	2032	2032	NUM
ejpam-4602	15	1	©	©	PROPN
ejpam-4602	15	2	2022	2022	NUM
ejpam-4602	15	3	ejpam	ejpam	VERB
ejpam-4602	15	4	all	all	DET
ejpam-4602	15	5	rights	right	NOUN
ejpam-4602	15	6	reserved	reserve	VERB
ejpam-4602	15	7	.	.	PUNCT
ejpam-4602	16	1	i.	i.	PROPN
ejpam-4602	16	2	taha	taha	PROPN
ejpam-4602	16	3	et	et	PROPN
ejpam-4602	16	4	al	al	PROPN
ejpam-4602	16	5	.	.	PUNCT
ejpam-4602	16	6	/	/	SYM
ejpam-4602	16	7	eur	eur	PROPN
ejpam-4602	16	8	.	.	PUNCT
ejpam-4602	17	1	j.	j.	PROPN
ejpam-4602	17	2	pure	pure	PROPN
ejpam-4602	17	3	appl	appl	PROPN
ejpam-4602	17	4	.	.	PROPN
ejpam-4602	17	5	math	math	PROPN
ejpam-4602	17	6	,	,	PUNCT
ejpam-4602	17	7	15	15	NUM
ejpam-4602	17	8	(	(	PUNCT
ejpam-4602	17	9	4	4	NUM
ejpam-4602	17	10	)	)	PUNCT
ejpam-4602	17	11	(	(	PUNCT
ejpam-4602	17	12	2022	2022	NUM
ejpam-4602	17	13	)	)	PUNCT
ejpam-4602	17	14	,	,	PUNCT
ejpam-4602	17	15	2032	2032	NUM
ejpam-4602	17	16	-	-	SYM
ejpam-4602	17	17	2042	2042	NUM
ejpam-4602	17	18	2033	2033	NUM
ejpam-4602	17	19	let	let	VERB
ejpam-4602	17	20	∆	∆	PROPN
ejpam-4602	17	21	be	be	AUX
ejpam-4602	17	22	a	a	DET
ejpam-4602	17	23	subset	subset	NOUN
ejpam-4602	17	24	of	of	ADP
ejpam-4602	17	25	the	the	DET
ejpam-4602	17	26	set	set	NOUN
ejpam-4602	17	27	of	of	ADP
ejpam-4602	17	28	all	all	DET
ejpam-4602	17	29	derivations	derivation	NOUN
ejpam-4602	17	30	d	d	X
ejpam-4602	17	31	on	on	ADP
ejpam-4602	17	32	r.	r.	PROPN
ejpam-4602	17	33	an	an	DET
ejpam-4602	17	34	ideal	ideal	ADJ
ejpam-4602	17	35	u	u	NOUN
ejpam-4602	17	36	of	of	ADP
ejpam-4602	17	37	r	r	NOUN
ejpam-4602	17	38	such	such	ADJ
ejpam-4602	17	39	that	that	PRON
ejpam-4602	17	40	δ(u	δ(u	PROPN
ejpam-4602	17	41	)	)	PUNCT
ejpam-4602	17	42	⊆	⊆	NUM
ejpam-4602	17	43	u	u	NOUN
ejpam-4602	17	44	is	be	AUX
ejpam-4602	17	45	used	use	VERB
ejpam-4602	17	46	to	to	PART
ejpam-4602	17	47	be	be	AUX
ejpam-4602	17	48	called	call	VERB
ejpam-4602	17	49	a	a	DET
ejpam-4602	17	50	δideal	δideal	NOUN
ejpam-4602	17	51	of	of	ADP
ejpam-4602	17	52	r.	r.	PROPN
ejpam-4602	17	53	a	a	DET
ejpam-4602	17	54	ring	ring	NOUN
ejpam-4602	17	55	r	r	NOUN
ejpam-4602	17	56	is	be	AUX
ejpam-4602	17	57	called	call	VERB
ejpam-4602	17	58	δ	δ	NOUN
ejpam-4602	17	59	-	-	NOUN
ejpam-4602	17	60	prime	prime	NOUN
ejpam-4602	17	61	if	if	SCONJ
ejpam-4602	17	62	,	,	PUNCT
ejpam-4602	17	63	for	for	ADP
ejpam-4602	17	64	any	any	DET
ejpam-4602	17	65	two	two	NUM
ejpam-4602	17	66	δideals	δideal	NOUN
ejpam-4602	17	67	u	u	NOUN
ejpam-4602	17	68	,	,	PUNCT
ejpam-4602	17	69	v	v	NOUN
ejpam-4602	17	70	of	of	ADP
ejpam-4602	17	71	r	r	NOUN
ejpam-4602	17	72	,	,	PUNCT
ejpam-4602	17	73	the	the	DET
ejpam-4602	17	74	condition	condition	NOUN
ejpam-4602	17	75	uv	uv	NOUN
ejpam-4602	17	76	=	=	NOUN
ejpam-4602	17	77	0	0	NUM
ejpam-4602	17	78	infers	infer	NOUN
ejpam-4602	17	79	that	that	SCONJ
ejpam-4602	17	80	either	either	CCONJ
ejpam-4602	17	81	u	u	X
ejpam-4602	17	82	=	=	NOUN
ejpam-4602	17	83	0	0	NUM
ejpam-4602	17	84	or	or	CCONJ
ejpam-4602	17	85	v	v	NOUN
ejpam-4602	17	86	=	=	SYM
ejpam-4602	17	87	0	0	NUM
ejpam-4602	17	88	,	,	PUNCT
ejpam-4602	17	89	equivalently	equivalently	ADV
ejpam-4602	17	90	,	,	PUNCT
ejpam-4602	17	91	we	we	PRON
ejpam-4602	17	92	call	call	VERB
ejpam-4602	17	93	a	a	DET
ejpam-4602	17	94	ring	ring	NOUN
ejpam-4602	17	95	r	r	NOUN
ejpam-4602	17	96	that	that	DET
ejpam-4602	17	97	δsemiprime	δsemiprime	NOUN
ejpam-4602	17	98	,	,	PUNCT
ejpam-4602	17	99	for	for	ADP
ejpam-4602	17	100	u	u	NOUN
ejpam-4602	17	101	⊆	⊆	NUM
ejpam-4602	17	102	d	d	PROPN
ejpam-4602	17	103	,	,	PUNCT
ejpam-4602	17	104	if	if	SCONJ
ejpam-4602	17	105	[	[	X
ejpam-4602	17	106	u	u	NOUN
ejpam-4602	17	107	,	,	PUNCT
ejpam-4602	17	108	u	u	NOUN
ejpam-4602	17	109	]	]	PUNCT
ejpam-4602	17	110	̸=	̸=	PROPN
ejpam-4602	17	111	0	0	NUM
ejpam-4602	17	112	,	,	PUNCT
ejpam-4602	17	113	implies	imply	VERB
ejpam-4602	17	114	u	u	PRON
ejpam-4602	17	115	̸=	̸=	PROPN
ejpam-4602	17	116	0	0	NUM
ejpam-4602	17	117	.	.	PUNCT
ejpam-4602	18	1	moreover	moreover	ADV
ejpam-4602	18	2	,	,	PUNCT
ejpam-4602	18	3	other	other	ADJ
ejpam-4602	18	4	terminologies	terminology	NOUN
ejpam-4602	18	5	are	be	AUX
ejpam-4602	18	6	standard	standard	ADJ
ejpam-4602	18	7	and	and	CCONJ
ejpam-4602	18	8	they	they	PRON
ejpam-4602	18	9	were	be	AUX
ejpam-4602	18	10	considered	consider	VERB
ejpam-4602	18	11	as	as	ADP
ejpam-4602	18	12	in	in	ADP
ejpam-4602	18	13	[	[	X
ejpam-4602	18	14	10],[11	10],[11	NUM
ejpam-4602	18	15	]	]	PUNCT
ejpam-4602	18	16	and	and	CCONJ
ejpam-4602	18	17	[	[	X
ejpam-4602	18	18	12	12	NUM
ejpam-4602	18	19	]	]	PUNCT
ejpam-4602	18	20	.	.	PUNCT
ejpam-4602	19	1	posner	posner	PROPN
ejpam-4602	19	2	’s	’s	PART
ejpam-4602	19	3	first	first	ADJ
ejpam-4602	19	4	theorem	theorem	NOUN
ejpam-4602	19	5	demonstrated	demonstrate	VERB
ejpam-4602	19	6	that	that	SCONJ
ejpam-4602	19	7	the	the	DET
ejpam-4602	19	8	composition	composition	NOUN
ejpam-4602	19	9	of	of	ADP
ejpam-4602	19	10	two	two	NUM
ejpam-4602	19	11	nonzero	nonzero	ADJ
ejpam-4602	19	12	derivations	derivation	NOUN
ejpam-4602	19	13	on	on	ADP
ejpam-4602	19	14	a	a	DET
ejpam-4602	19	15	prime	prime	ADJ
ejpam-4602	19	16	ring	ring	NOUN
ejpam-4602	19	17	r	r	NOUN
ejpam-4602	19	18	,	,	PUNCT
ejpam-4602	19	19	with	with	ADP
ejpam-4602	19	20	charr	charr	NOUN
ejpam-4602	19	21	̸=	̸=	PROPN
ejpam-4602	19	22	2	2	NUM
ejpam-4602	19	23	,	,	PUNCT
ejpam-4602	19	24	is	be	AUX
ejpam-4602	19	25	not	not	PART
ejpam-4602	19	26	a	a	DET
ejpam-4602	19	27	derivation	derivation	NOUN
ejpam-4602	19	28	.	.	PUNCT
ejpam-4602	20	1	the	the	DET
ejpam-4602	20	2	second	second	ADJ
ejpam-4602	20	3	theorem	theorem	NOUN
ejpam-4602	20	4	of	of	ADP
ejpam-4602	20	5	posner	posner	NOUN
ejpam-4602	20	6	proved	prove	VERB
ejpam-4602	20	7	that	that	SCONJ
ejpam-4602	20	8	if	if	SCONJ
ejpam-4602	20	9	the	the	DET
ejpam-4602	20	10	nonzero	nonzero	NOUN
ejpam-4602	20	11	derivation	derivation	NOUN
ejpam-4602	20	12	d	d	NOUN
ejpam-4602	20	13	on	on	ADP
ejpam-4602	20	14	a	a	DET
ejpam-4602	20	15	prime	prime	ADJ
ejpam-4602	20	16	ring	ring	NOUN
ejpam-4602	20	17	r	r	NOUN
ejpam-4602	20	18	is	be	AUX
ejpam-4602	20	19	a	a	DET
ejpam-4602	20	20	centralizing	centralizing	NOUN
ejpam-4602	20	21	on	on	ADP
ejpam-4602	20	22	r	r	NOUN
ejpam-4602	20	23	,	,	PUNCT
ejpam-4602	20	24	then	then	ADV
ejpam-4602	20	25	r	r	NOUN
ejpam-4602	20	26	is	be	AUX
ejpam-4602	20	27	commutative	commutative	ADJ
ejpam-4602	21	1	[	[	X
ejpam-4602	21	2	26	26	NUM
ejpam-4602	21	3	]	]	PUNCT
ejpam-4602	21	4	.	.	PUNCT
ejpam-4602	22	1	mayne	mayne	PROPN
ejpam-4602	22	2	generalized	generalize	VERB
ejpam-4602	22	3	posner	posner	NOUN
ejpam-4602	22	4	’s	’s	PART
ejpam-4602	22	5	theorem	theorem	NOUN
ejpam-4602	22	6	when	when	SCONJ
ejpam-4602	22	7	a	a	DET
ejpam-4602	22	8	ring	ring	NOUN
ejpam-4602	22	9	r	r	NOUN
ejpam-4602	22	10	has	have	VERB
ejpam-4602	22	11	an	an	DET
ejpam-4602	22	12	automorphism	automorphism	NOUN
ejpam-4602	22	13	or	or	CCONJ
ejpam-4602	22	14	a	a	DET
ejpam-4602	22	15	nonzero	nonzero	NOUN
ejpam-4602	22	16	centralizing	centralize	VERB
ejpam-4602	22	17	derivation	derivation	NOUN
ejpam-4602	22	18	on	on	ADP
ejpam-4602	22	19	some	some	DET
ejpam-4602	22	20	ideal	ideal	ADJ
ejpam-4602	22	21	u	u	NOUN
ejpam-4602	22	22	̸=	̸=	PROPN
ejpam-4602	22	23	0	0	NUM
ejpam-4602	22	24	,	,	PUNCT
ejpam-4602	22	25	concluding	conclude	VERB
ejpam-4602	22	26	that	that	SCONJ
ejpam-4602	22	27	r	r	NOUN
ejpam-4602	22	28	is	be	AUX
ejpam-4602	22	29	commutative	commutative	ADJ
ejpam-4602	23	1	[	[	X
ejpam-4602	23	2	22	22	NUM
ejpam-4602	23	3	]	]	PUNCT
ejpam-4602	23	4	.	.	PUNCT
ejpam-4602	24	1	likewise	likewise	ADV
ejpam-4602	24	2	,	,	PUNCT
ejpam-4602	24	3	some	some	DET
ejpam-4602	24	4	generalizations	generalization	NOUN
ejpam-4602	24	5	of	of	ADP
ejpam-4602	24	6	these	these	DET
ejpam-4602	24	7	results	result	NOUN
ejpam-4602	24	8	with	with	ADP
ejpam-4602	24	9	different	different	ADJ
ejpam-4602	24	10	ways	way	NOUN
ejpam-4602	24	11	for	for	ADP
ejpam-4602	24	12	a	a	DET
ejpam-4602	24	13	prime	prime	ADJ
ejpam-4602	24	14	and	and	CCONJ
ejpam-4602	24	15	semiprime	semiprime	NOUN
ejpam-4602	24	16	rings	ring	NOUN
ejpam-4602	24	17	are	be	AUX
ejpam-4602	24	18	condidered	condidere	VERB
ejpam-4602	24	19	in	in	ADP
ejpam-4602	24	20	[	[	X
ejpam-4602	24	21	6	6	NUM
ejpam-4602	24	22	,	,	PUNCT
ejpam-4602	24	23	7	7	NUM
ejpam-4602	24	24	,	,	PUNCT
ejpam-4602	24	25	13	13	NUM
ejpam-4602	24	26	,	,	PUNCT
ejpam-4602	24	27	14	14	NUM
ejpam-4602	24	28	,	,	PUNCT
ejpam-4602	24	29	21–24	21–24	NUM
ejpam-4602	24	30	,	,	PUNCT
ejpam-4602	24	31	26	26	NUM
ejpam-4602	24	32	,	,	PUNCT
ejpam-4602	24	33	28	28	NUM
ejpam-4602	24	34	,	,	PUNCT
ejpam-4602	24	35	29	29	NUM
ejpam-4602	24	36	]	]	PUNCT
ejpam-4602	24	37	.	.	PUNCT
ejpam-4602	25	1	in	in	ADP
ejpam-4602	25	2	general	general	ADJ
ejpam-4602	25	3	,	,	PUNCT
ejpam-4602	25	4	they	they	PRON
ejpam-4602	25	5	have	have	AUX
ejpam-4602	25	6	showed	show	VERB
ejpam-4602	25	7	commutativity	commutativity	NOUN
ejpam-4602	25	8	of	of	ADP
ejpam-4602	25	9	prime	prime	ADJ
ejpam-4602	25	10	and	and	CCONJ
ejpam-4602	25	11	semiprime	semiprime	NOUN
ejpam-4602	25	12	rings	ring	NOUN
ejpam-4602	25	13	admit	admit	VERB
ejpam-4602	25	14	centralizing	centralize	VERB
ejpam-4602	25	15	derivations	derivation	NOUN
ejpam-4602	25	16	on	on	ADP
ejpam-4602	25	17	specific	specific	ADJ
ejpam-4602	25	18	subsets	subset	NOUN
ejpam-4602	25	19	of	of	ADP
ejpam-4602	25	20	r.	r.	PROPN
ejpam-4602	25	21	overall	overall	PROPN
ejpam-4602	25	22	,	,	PUNCT
ejpam-4602	25	23	brešar	brešar	ADJ
ejpam-4602	25	24	and	and	CCONJ
ejpam-4602	25	25	vukman	vukman	NOUN
ejpam-4602	25	26	who	who	PRON
ejpam-4602	25	27	started	start	VERB
ejpam-4602	25	28	the	the	DET
ejpam-4602	25	29	researching	researching	NOUN
ejpam-4602	25	30	on	on	ADP
ejpam-4602	25	31	reverse	reverse	ADJ
ejpam-4602	25	32	derivation	derivation	NOUN
ejpam-4602	25	33	concept	concept	NOUN
ejpam-4602	25	34	(	(	PUNCT
ejpam-4602	25	35	see	see	VERB
ejpam-4602	25	36	[	[	X
ejpam-4602	25	37	8	8	NUM
ejpam-4602	25	38	]	]	NUM
ejpam-4602	25	39	)	)	PUNCT
ejpam-4602	25	40	,	,	PUNCT
ejpam-4602	25	41	recently	recently	ADV
ejpam-4602	25	42	,	,	PUNCT
ejpam-4602	25	43	samman	samman	NOUN
ejpam-4602	25	44	and	and	CCONJ
ejpam-4602	25	45	alyamani	alyamani	PROPN
ejpam-4602	26	1	[	[	X
ejpam-4602	26	2	27	27	NUM
ejpam-4602	26	3	]	]	PUNCT
ejpam-4602	26	4	came	come	VERB
ejpam-4602	26	5	,	,	PUNCT
ejpam-4602	26	6	with	with	ADP
ejpam-4602	26	7	many	many	ADJ
ejpam-4602	26	8	properties	property	NOUN
ejpam-4602	26	9	of	of	ADP
ejpam-4602	26	10	reverse	reverse	ADJ
ejpam-4602	26	11	derivations	derivation	NOUN
ejpam-4602	26	12	in	in	ADP
ejpam-4602	26	13	prime	prime	ADJ
ejpam-4602	26	14	(	(	PUNCT
ejpam-4602	26	15	resp	resp	NOUN
ejpam-4602	26	16	.	.	PUNCT
ejpam-4602	26	17	semiprime	semiprime	NOUN
ejpam-4602	26	18	)	)	PUNCT
ejpam-4602	26	19	rings	ring	NOUN
ejpam-4602	26	20	.	.	PUNCT
ejpam-4602	27	1	2	2	X
ejpam-4602	27	2	.	.	X
ejpam-4602	27	3	preliminaries	preliminary	NOUN
ejpam-4602	27	4	many	many	ADJ
ejpam-4602	27	5	researchers	researcher	NOUN
ejpam-4602	27	6	studied	study	VERB
ejpam-4602	27	7	the	the	DET
ejpam-4602	27	8	properties	property	NOUN
ejpam-4602	27	9	of	of	ADP
ejpam-4602	27	10	lie	lie	NOUN
ejpam-4602	27	11	rings	ring	NOUN
ejpam-4602	27	12	with	with	ADP
ejpam-4602	27	13	derivations	derivation	NOUN
ejpam-4602	27	14	d	d	X
ejpam-4602	27	15	of	of	ADP
ejpam-4602	27	16	differentially	differentially	ADV
ejpam-4602	27	17	simple	simple	ADJ
ejpam-4602	27	18	,	,	PUNCT
ejpam-4602	27	19	prime	prime	ADJ
ejpam-4602	27	20	and	and	CCONJ
ejpam-4602	27	21	semiprime	semiprime	NOUN
ejpam-4602	27	22	rings	ring	NOUN
ejpam-4602	27	23	(	(	PUNCT
ejpam-4602	27	24	see	see	VERB
ejpam-4602	27	25	for	for	ADP
ejpam-4602	27	26	example	example	NOUN
ejpam-4602	27	27	[	[	X
ejpam-4602	27	28	1–4	1–4	X
ejpam-4602	27	29	]	]	X
ejpam-4602	27	30	,	,	PUNCT
ejpam-4602	27	31	[	[	X
ejpam-4602	27	32	14	14	NUM
ejpam-4602	27	33	,	,	PUNCT
ejpam-4602	27	34	15	15	NUM
ejpam-4602	27	35	]	]	PUNCT
ejpam-4602	27	36	,	,	PUNCT
ejpam-4602	27	37	[	[	X
ejpam-4602	27	38	16	16	NUM
ejpam-4602	27	39	,	,	PUNCT
ejpam-4602	27	40	17	17	NUM
ejpam-4602	27	41	]	]	PUNCT
ejpam-4602	27	42	and	and	CCONJ
ejpam-4602	27	43	[	[	X
ejpam-4602	27	44	18	18	NUM
ejpam-4602	27	45	,	,	PUNCT
ejpam-4602	27	46	28	28	NUM
ejpam-4602	27	47	]	]	PUNCT
ejpam-4602	27	48	,	,	PUNCT
ejpam-4602	27	49	where	where	SCONJ
ejpam-4602	27	50	further	further	ADJ
ejpam-4602	27	51	references	reference	NOUN
ejpam-4602	27	52	can	can	AUX
ejpam-4602	27	53	be	be	AUX
ejpam-4602	27	54	found	find	VERB
ejpam-4602	27	55	for	for	ADP
ejpam-4602	27	56	the	the	DET
ejpam-4602	27	57	widening	widening	NOUN
ejpam-4602	27	58	in	in	ADP
ejpam-4602	27	59	this	this	DET
ejpam-4602	27	60	field	field	NOUN
ejpam-4602	27	61	.	.	PUNCT
ejpam-4602	28	1	passman	passman	NOUN
ejpam-4602	28	2	in	in	ADP
ejpam-4602	28	3	[	[	X
ejpam-4602	28	4	25	25	NUM
ejpam-4602	28	5	]	]	PUNCT
ejpam-4602	28	6	,	,	PUNCT
ejpam-4602	28	7	has	have	AUX
ejpam-4602	28	8	investigated	investigate	VERB
ejpam-4602	28	9	the	the	DET
ejpam-4602	28	10	commutative	commutative	ADJ
ejpam-4602	28	11	rings	ring	NOUN
ejpam-4602	28	12	with	with	ADP
ejpam-4602	28	13	semiprime	semiprime	NOUN
ejpam-4602	28	14	lie	lie	NOUN
ejpam-4602	28	15	ring	ring	PROPN
ejpam-4602	28	16	d.	d.	PROPN
ejpam-4602	28	17	there	there	PRON
ejpam-4602	28	18	are	be	VERB
ejpam-4602	28	19	many	many	ADJ
ejpam-4602	28	20	papers	paper	NOUN
ejpam-4602	28	21	in	in	ADP
ejpam-4602	28	22	this	this	DET
ejpam-4602	28	23	line	line	NOUN
ejpam-4602	28	24	,	,	PUNCT
ejpam-4602	28	25	as	as	ADP
ejpam-4602	28	26	[	[	X
ejpam-4602	28	27	9	9	NUM
ejpam-4602	28	28	,	,	PUNCT
ejpam-4602	28	29	19	19	NUM
ejpam-4602	28	30	,	,	PUNCT
ejpam-4602	28	31	20	20	NUM
ejpam-4602	28	32	]	]	PUNCT
ejpam-4602	28	33	.	.	PUNCT
ejpam-4602	29	1	the	the	DET
ejpam-4602	29	2	following	follow	VERB
ejpam-4602	29	3	main	main	ADJ
ejpam-4602	29	4	lemmas	lemma	NOUN
ejpam-4602	29	5	that	that	PRON
ejpam-4602	29	6	will	will	AUX
ejpam-4602	29	7	be	be	AUX
ejpam-4602	29	8	used	use	VERB
ejpam-4602	29	9	to	to	PART
ejpam-4602	29	10	prove	prove	VERB
ejpam-4602	29	11	our	our	PRON
ejpam-4602	29	12	new	new	ADJ
ejpam-4602	29	13	results	result	NOUN
ejpam-4602	29	14	,	,	PUNCT
ejpam-4602	29	15	to	to	PART
ejpam-4602	29	16	which	which	PRON
ejpam-4602	29	17	we	we	PRON
ejpam-4602	29	18	shall	shall	AUX
ejpam-4602	29	19	refer	refer	VERB
ejpam-4602	29	20	,	,	PUNCT
ejpam-4602	29	21	stated	state	VERB
ejpam-4602	29	22	as	as	ADP
ejpam-4602	29	23	in	in	ADP
ejpam-4602	29	24	the	the	DET
ejpam-4602	29	25	following	following	NOUN
ejpam-4602	29	26	:	:	PUNCT
ejpam-4602	29	27	lemma	lemma	PROPN
ejpam-4602	29	28	1	1	NUM
ejpam-4602	29	29	.	.	PUNCT
ejpam-4602	30	1	[	[	X
ejpam-4602	30	2	26	26	NUM
ejpam-4602	30	3	,	,	PUNCT
ejpam-4602	30	4	lemma	lemma	PROPN
ejpam-4602	30	5	3	3	X
ejpam-4602	30	6	]	]	PUNCT
ejpam-4602	30	7	let	let	VERB
ejpam-4602	30	8	r	r	PRON
ejpam-4602	30	9	be	be	AUX
ejpam-4602	30	10	a	a	DET
ejpam-4602	30	11	prime	prime	ADJ
ejpam-4602	30	12	ring	ring	NOUN
ejpam-4602	30	13	,	,	PUNCT
ejpam-4602	30	14	and	and	CCONJ
ejpam-4602	30	15	d	d	ADP
ejpam-4602	30	16	a	a	DET
ejpam-4602	30	17	derivation	derivation	NOUN
ejpam-4602	30	18	of	of	ADP
ejpam-4602	30	19	r	r	NOUN
ejpam-4602	30	20	such	such	ADJ
ejpam-4602	30	21	that	that	SCONJ
ejpam-4602	30	22	ad(a)−	ad(a)−	ADP
ejpam-4602	30	23	d(a)a	d(a)a	PROPN
ejpam-4602	30	24	=	=	SYM
ejpam-4602	30	25	0	0	NUM
ejpam-4602	30	26	for	for	ADP
ejpam-4602	30	27	all	all	DET
ejpam-4602	30	28	a	a	DET
ejpam-4602	30	29	∈	∈	PROPN
ejpam-4602	30	30	r.	r.	NOUN
ejpam-4602	30	31	then	then	ADV
ejpam-4602	30	32	r	r	NOUN
ejpam-4602	30	33	is	be	AUX
ejpam-4602	30	34	commutative	commutative	ADJ
ejpam-4602	30	35	.	.	PUNCT
ejpam-4602	31	1	lemma	lemma	PROPN
ejpam-4602	31	2	2	2	NUM
ejpam-4602	31	3	.	.	PUNCT
ejpam-4602	32	1	[	[	X
ejpam-4602	32	2	21	21	NUM
ejpam-4602	32	3	,	,	PUNCT
ejpam-4602	32	4	theorem	theorem	VERB
ejpam-4602	32	5	]	]	PUNCT
ejpam-4602	32	6	let	let	VERB
ejpam-4602	32	7	r	r	PRON
ejpam-4602	32	8	be	be	AUX
ejpam-4602	32	9	a	a	DET
ejpam-4602	32	10	prime	prime	ADJ
ejpam-4602	32	11	ring	ring	NOUN
ejpam-4602	32	12	with	with	ADP
ejpam-4602	32	13	a	a	DET
ejpam-4602	32	14	nontrivial	nontrivial	ADJ
ejpam-4602	32	15	centralizing	centralize	VERB
ejpam-4602	32	16	automorphism	automorphism	NOUN
ejpam-4602	32	17	.	.	PUNCT
ejpam-4602	33	1	then	then	ADV
ejpam-4602	33	2	,	,	PUNCT
ejpam-4602	33	3	r	r	NOUN
ejpam-4602	33	4	is	be	AUX
ejpam-4602	33	5	a	a	DET
ejpam-4602	33	6	commutative	commutative	ADJ
ejpam-4602	33	7	integral	integral	ADJ
ejpam-4602	33	8	domain	domain	NOUN
ejpam-4602	33	9	.	.	PUNCT
ejpam-4602	34	1	lemma	lemma	PROPN
ejpam-4602	34	2	3	3	NUM
ejpam-4602	34	3	.	.	PUNCT
ejpam-4602	35	1	[	[	X
ejpam-4602	35	2	22	22	NUM
ejpam-4602	35	3	,	,	PUNCT
ejpam-4602	35	4	theorem	theorem	VERB
ejpam-4602	35	5	]	]	PUNCT
ejpam-4602	35	6	let	let	VERB
ejpam-4602	35	7	r	r	PRON
ejpam-4602	35	8	be	be	AUX
ejpam-4602	35	9	a	a	DET
ejpam-4602	35	10	prime	prime	ADJ
ejpam-4602	35	11	ring	ring	NOUN
ejpam-4602	35	12	and	and	CCONJ
ejpam-4602	35	13	a	a	DET
ejpam-4602	35	14	̸=	̸=	PROPN
ejpam-4602	35	15	{	{	PUNCT
ejpam-4602	35	16	0	0	NUM
ejpam-4602	35	17	}	}	PUNCT
ejpam-4602	35	18	be	be	AUX
ejpam-4602	35	19	an	an	DET
ejpam-4602	35	20	ideal	ideal	NOUN
ejpam-4602	35	21	of	of	ADP
ejpam-4602	35	22	r.	r.	PROPN
ejpam-4602	35	23	if	if	SCONJ
ejpam-4602	35	24	r	r	NOUN
ejpam-4602	35	25	has	have	VERB
ejpam-4602	35	26	a	a	DET
ejpam-4602	35	27	nontrivial	nontrivial	ADJ
ejpam-4602	35	28	automorphisim	automorphisim	NOUN
ejpam-4602	35	29	or	or	CCONJ
ejpam-4602	35	30	derivation	derivation	NOUN
ejpam-4602	35	31	t	t	NOUN
ejpam-4602	35	32	such	such	ADJ
ejpam-4602	35	33	that	that	DET
ejpam-4602	35	34	uut	uut	PROPN
ejpam-4602	35	35	−	−	PROPN
ejpam-4602	35	36	utu	utu	PROPN
ejpam-4602	35	37	is	be	AUX
ejpam-4602	35	38	in	in	ADP
ejpam-4602	35	39	the	the	DET
ejpam-4602	35	40	center	center	NOUN
ejpam-4602	35	41	of	of	ADP
ejpam-4602	35	42	r	r	NOUN
ejpam-4602	35	43	and	and	CCONJ
ejpam-4602	35	44	ut	ut	PROPN
ejpam-4602	35	45	is	be	AUX
ejpam-4602	35	46	in	in	ADP
ejpam-4602	35	47	u	u	NOUN
ejpam-4602	35	48	for	for	ADP
ejpam-4602	35	49	every	every	DET
ejpam-4602	35	50	u	u	NOUN
ejpam-4602	35	51	in	in	ADP
ejpam-4602	35	52	u	u	PROPN
ejpam-4602	35	53	,	,	PUNCT
ejpam-4602	35	54	then	then	ADV
ejpam-4602	35	55	r	r	NOUN
ejpam-4602	35	56	is	be	AUX
ejpam-4602	35	57	commutative	commutative	ADJ
ejpam-4602	35	58	.	.	PUNCT
ejpam-4602	36	1	lemma	lemma	PROPN
ejpam-4602	36	2	4	4	NUM
ejpam-4602	36	3	.	.	PUNCT
ejpam-4602	37	1	[	[	X
ejpam-4602	37	2	4	4	NUM
ejpam-4602	37	3	,	,	PUNCT
ejpam-4602	37	4	lemma	lemma	PROPN
ejpam-4602	37	5	13	13	NUM
ejpam-4602	37	6	]	]	PUNCT
ejpam-4602	37	7	let	let	VERB
ejpam-4602	37	8	a	a	DET
ejpam-4602	37	9	̸=	̸=	PROPN
ejpam-4602	37	10	0	0	NUM
ejpam-4602	37	11	be	be	AUX
ejpam-4602	37	12	a	a	DET
ejpam-4602	37	13	lie	lie	NOUN
ejpam-4602	37	14	δ	δ	NOUN
ejpam-4602	37	15	-	-	NOUN
ejpam-4602	37	16	ideal	ideal	NOUN
ejpam-4602	37	17	of	of	ADP
ejpam-4602	37	18	a	a	DET
ejpam-4602	37	19	δ	δ	NOUN
ejpam-4602	37	20	-	-	PUNCT
ejpam-4602	37	21	semiprime	semiprime	NOUN
ejpam-4602	37	22	and	and	CCONJ
ejpam-4602	37	23	subring	subring	NOUN
ejpam-4602	37	24	of	of	ADP
ejpam-4602	37	25	a	a	DET
ejpam-4602	37	26	ring	ring	NOUN
ejpam-4602	37	27	r	r	NOUN
ejpam-4602	37	28	of	of	ADP
ejpam-4602	37	29	charr	charr	NOUN
ejpam-4602	37	30	̸=	̸=	PROPN
ejpam-4602	37	31	2	2	NUM
ejpam-4602	37	32	.	.	PUNCT
ejpam-4602	38	1	then	then	ADV
ejpam-4602	38	2	a	a	DET
ejpam-4602	38	3	⊆	⊆	NUM
ejpam-4602	38	4	z(r	z(r	NOUN
ejpam-4602	38	5	)	)	PUNCT
ejpam-4602	38	6	or	or	CCONJ
ejpam-4602	38	7	a	a	PRON
ejpam-4602	38	8	contains	contain	VERB
ejpam-4602	38	9	a	a	DET
ejpam-4602	38	10	non	non	ADJ
ejpam-4602	38	11	-	-	ADJ
ejpam-4602	38	12	central	central	ADJ
ejpam-4602	38	13	δ	δ	PROPN
ejpam-4602	38	14	-	-	PUNCT
ejpam-4602	38	15	ideal	ideal	NOUN
ejpam-4602	38	16	of	of	ADP
ejpam-4602	38	17	r.	r.	PROPN
ejpam-4602	38	18	i.	i.	PROPN
ejpam-4602	38	19	taha	taha	PROPN
ejpam-4602	38	20	et	et	PROPN
ejpam-4602	38	21	al	al	PROPN
ejpam-4602	38	22	.	.	PUNCT
ejpam-4602	38	23	/	/	SYM
ejpam-4602	38	24	eur	eur	PROPN
ejpam-4602	38	25	.	.	PUNCT
ejpam-4602	39	1	j.	j.	PROPN
ejpam-4602	39	2	pure	pure	PROPN
ejpam-4602	39	3	appl	appl	PROPN
ejpam-4602	39	4	.	.	PROPN
ejpam-4602	39	5	math	math	PROPN
ejpam-4602	39	6	,	,	PUNCT
ejpam-4602	39	7	15	15	NUM
ejpam-4602	39	8	(	(	PUNCT
ejpam-4602	39	9	4	4	NUM
ejpam-4602	39	10	)	)	PUNCT
ejpam-4602	39	11	(	(	PUNCT
ejpam-4602	39	12	2022	2022	NUM
ejpam-4602	39	13	)	)	PUNCT
ejpam-4602	39	14	,	,	PUNCT
ejpam-4602	39	15	2032	2032	NUM
ejpam-4602	39	16	-	-	SYM
ejpam-4602	39	17	2042	2042	NUM
ejpam-4602	39	18	2034	2034	NUM
ejpam-4602	39	19	therefore	therefore	ADV
ejpam-4602	39	20	the	the	DET
ejpam-4602	39	21	purpose	purpose	NOUN
ejpam-4602	39	22	of	of	ADP
ejpam-4602	39	23	our	our	PRON
ejpam-4602	39	24	research	research	NOUN
ejpam-4602	39	25	is	be	AUX
ejpam-4602	39	26	to	to	PART
ejpam-4602	39	27	study	study	VERB
ejpam-4602	39	28	the	the	DET
ejpam-4602	39	29	structure	structure	NOUN
ejpam-4602	39	30	of	of	ADP
ejpam-4602	39	31	reverse	reverse	ADJ
ejpam-4602	39	32	derivation	derivation	NOUN
ejpam-4602	39	33	on	on	ADP
ejpam-4602	39	34	δ	δ	PROPN
ejpam-4602	39	35	–	–	PUNCT
ejpam-4602	39	36	prime	prime	ADJ
ejpam-4602	39	37	ring	ring	NOUN
ejpam-4602	39	38	and	and	CCONJ
ejpam-4602	39	39	some	some	DET
ejpam-4602	39	40	properties	property	NOUN
ejpam-4602	39	41	of	of	ADP
ejpam-4602	39	42	centralizing	centralize	VERB
ejpam-4602	39	43	reverse	reverse	ADJ
ejpam-4602	39	44	derivations	derivation	NOUN
ejpam-4602	39	45	on	on	ADP
ejpam-4602	39	46	nonzero	nonzero	PROPN
ejpam-4602	39	47	δ	δ	PROPN
ejpam-4602	39	48	–	–	PUNCT
ejpam-4602	39	49	ideal	ideal	NOUN
ejpam-4602	39	50	of	of	ADP
ejpam-4602	39	51	δ	δ	PROPN
ejpam-4602	39	52	–	–	PUNCT
ejpam-4602	39	53	prime	prime	ADJ
ejpam-4602	39	54	ring	ring	NOUN
ejpam-4602	39	55	.	.	PUNCT
ejpam-4602	40	1	3	3	X
ejpam-4602	40	2	.	.	X
ejpam-4602	41	1	some	some	DET
ejpam-4602	41	2	properties	property	NOUN
ejpam-4602	41	3	and	and	CCONJ
ejpam-4602	41	4	examples	example	NOUN
ejpam-4602	41	5	of	of	ADP
ejpam-4602	41	6	reverse	reverse	ADJ
ejpam-4602	41	7	derivation	derivation	NOUN
ejpam-4602	41	8	in	in	ADP
ejpam-4602	41	9	fact	fact	NOUN
ejpam-4602	41	10	,	,	PUNCT
ejpam-4602	41	11	it	it	PRON
ejpam-4602	41	12	is	be	AUX
ejpam-4602	41	13	not	not	PART
ejpam-4602	41	14	necessary	necessary	ADJ
ejpam-4602	41	15	that	that	SCONJ
ejpam-4602	41	16	every	every	DET
ejpam-4602	41	17	derivation	derivation	NOUN
ejpam-4602	41	18	is	be	AUX
ejpam-4602	41	19	a	a	DET
ejpam-4602	41	20	reverse	reverse	ADJ
ejpam-4602	41	21	derivation	derivation	NOUN
ejpam-4602	41	22	on	on	ADP
ejpam-4602	41	23	a	a	DET
ejpam-4602	41	24	ring	ring	NOUN
ejpam-4602	41	25	r	r	NOUN
ejpam-4602	41	26	or	or	CCONJ
ejpam-4602	41	27	vice	vice	NOUN
ejpam-4602	41	28	versa	versa	ADV
ejpam-4602	41	29	.	.	PUNCT
ejpam-4602	42	1	taking	take	VERB
ejpam-4602	42	2	into	into	ADP
ejpam-4602	42	3	consideration	consideration	NOUN
ejpam-4602	42	4	when	when	SCONJ
ejpam-4602	42	5	the	the	DET
ejpam-4602	42	6	ring	ring	NOUN
ejpam-4602	42	7	r	r	NOUN
ejpam-4602	42	8	is	be	AUX
ejpam-4602	42	9	commutative	commutative	ADJ
ejpam-4602	42	10	,	,	PUNCT
ejpam-4602	42	11	then	then	ADV
ejpam-4602	42	12	the	the	DET
ejpam-4602	42	13	derivation	derivation	NOUN
ejpam-4602	42	14	and	and	CCONJ
ejpam-4602	42	15	the	the	DET
ejpam-4602	42	16	reverse	reverse	ADJ
ejpam-4602	42	17	derivation	derivation	NOUN
ejpam-4602	42	18	are	be	AUX
ejpam-4602	42	19	coincides	coincide	NOUN
ejpam-4602	42	20	.	.	PUNCT
ejpam-4602	43	1	therefore	therefore	ADV
ejpam-4602	43	2	it	it	PRON
ejpam-4602	43	3	is	be	AUX
ejpam-4602	43	4	possible	possible	ADJ
ejpam-4602	43	5	to	to	PART
ejpam-4602	43	6	define	define	VERB
ejpam-4602	43	7	an	an	DET
ejpam-4602	43	8	example	example	NOUN
ejpam-4602	43	9	about	about	ADP
ejpam-4602	43	10	this	this	DET
ejpam-4602	43	11	case	case	NOUN
ejpam-4602	43	12	.	.	PUNCT
ejpam-4602	43	13	example	example	NOUN
ejpam-4602	44	1	1	1	NUM
ejpam-4602	44	2	.	.	PUNCT
ejpam-4602	45	1	let	let	VERB
ejpam-4602	45	2	r	r	NOUN
ejpam-4602	45	3	=	=	PRON
ejpam-4602	45	4	{	{	PUNCT
ejpam-4602	45	5	[	[	PUNCT
ejpam-4602	45	6	u	u	NOUN
ejpam-4602	45	7	v	v	ADP
ejpam-4602	45	8	0	0	NUM
ejpam-4602	45	9	0	0	NUM
ejpam-4602	45	10	]	]	PUNCT
ejpam-4602	45	11	:	:	PUNCT
ejpam-4602	45	12	u	u	NOUN
ejpam-4602	45	13	,	,	PUNCT
ejpam-4602	45	14	v	v	PROPN
ejpam-4602	45	15	∈	∈	PROPN
ejpam-4602	45	16	t	t	PROPN
ejpam-4602	45	17	}	}	PUNCT
ejpam-4602	45	18	,	,	PUNCT
ejpam-4602	45	19	such	such	ADJ
ejpam-4602	45	20	that	that	SCONJ
ejpam-4602	45	21	t	t	PROPN
ejpam-4602	45	22	is	be	AUX
ejpam-4602	45	23	a	a	DET
ejpam-4602	45	24	ring	ring	NOUN
ejpam-4602	45	25	with	with	ADP
ejpam-4602	45	26	t	t	PROPN
ejpam-4602	45	27	2	2	NUM
ejpam-4602	45	28	̸=	̸=	PROPN
ejpam-4602	45	29	{	{	PUNCT
ejpam-4602	45	30	0	0	NUM
ejpam-4602	45	31	}	}	PUNCT
ejpam-4602	45	32	.	.	PUNCT
ejpam-4602	46	1	let	let	VERB
ejpam-4602	46	2	δ	δ	NOUN
ejpam-4602	46	3	:	:	PUNCT
ejpam-4602	46	4	r	r	X
ejpam-4602	46	5	→	→	SYM
ejpam-4602	46	6	r	r	NOUN
ejpam-4602	46	7	be	be	AUX
ejpam-4602	46	8	an	an	DET
ejpam-4602	46	9	additive	additive	ADJ
ejpam-4602	46	10	mapping	mapping	NOUN
ejpam-4602	46	11	defined	define	VERB
ejpam-4602	46	12	as	as	ADP
ejpam-4602	46	13	δ	δ	PROPN
ejpam-4602	46	14	(	(	PUNCT
ejpam-4602	46	15	[	[	PUNCT
ejpam-4602	46	16	u	u	NOUN
ejpam-4602	46	17	v	v	ADP
ejpam-4602	46	18	0	0	NUM
ejpam-4602	46	19	0	0	NUM
ejpam-4602	46	20	]	]	PUNCT
ejpam-4602	46	21	)	)	PUNCT
ejpam-4602	47	1	=	=	PUNCT
ejpam-4602	47	2	[	[	PUNCT
ejpam-4602	47	3	0	0	NUM
ejpam-4602	47	4	v	v	NUM
ejpam-4602	47	5	0	0	NUM
ejpam-4602	47	6	0	0	NUM
ejpam-4602	47	7	]	]	PUNCT
ejpam-4602	47	8	:	:	PUNCT
ejpam-4602	47	9	∀u	∀u	NOUN
ejpam-4602	47	10	,	,	PUNCT
ejpam-4602	47	11	v	v	NOUN
ejpam-4602	47	12	∈	∈	NOUN
ejpam-4602	47	13	t.	t.	NOUN
ejpam-4602	47	14	and	and	CCONJ
ejpam-4602	47	15	δ̀	δ̀	NOUN
ejpam-4602	47	16	:	:	PUNCT
ejpam-4602	47	17	r	r	X
ejpam-4602	47	18	→	→	SYM
ejpam-4602	47	19	r	r	NOUN
ejpam-4602	47	20	be	be	AUX
ejpam-4602	47	21	an	an	DET
ejpam-4602	47	22	additive	additive	ADJ
ejpam-4602	47	23	mapping	mapping	NOUN
ejpam-4602	47	24	defined	define	VERB
ejpam-4602	47	25	as	as	ADP
ejpam-4602	47	26	δ̀	δ̀	PROPN
ejpam-4602	47	27	(	(	PUNCT
ejpam-4602	47	28	[	[	PUNCT
ejpam-4602	47	29	u	u	NOUN
ejpam-4602	47	30	v	v	ADP
ejpam-4602	47	31	0	0	NUM
ejpam-4602	47	32	0	0	NUM
ejpam-4602	47	33	]	]	PUNCT
ejpam-4602	47	34	)	)	PUNCT
ejpam-4602	48	1	=	=	PUNCT
ejpam-4602	48	2	[	[	PUNCT
ejpam-4602	48	3	0	0	NUM
ejpam-4602	48	4	u	u	NOUN
ejpam-4602	48	5	0	0	NUM
ejpam-4602	48	6	0	0	NUM
ejpam-4602	48	7	]	]	PUNCT
ejpam-4602	48	8	:	:	PUNCT
ejpam-4602	48	9	∀u	∀u	NOUN
ejpam-4602	48	10	,	,	PUNCT
ejpam-4602	48	11	v	v	NOUN
ejpam-4602	48	12	∈	∈	NOUN
ejpam-4602	48	13	t.	t.	NOUN
ejpam-4602	48	14	it	it	PRON
ejpam-4602	48	15	is	be	AUX
ejpam-4602	48	16	easy	easy	ADJ
ejpam-4602	48	17	to	to	PART
ejpam-4602	48	18	see	see	VERB
ejpam-4602	48	19	that	that	SCONJ
ejpam-4602	48	20	δ	δ	PROPN
ejpam-4602	48	21	is	be	AUX
ejpam-4602	48	22	a	a	DET
ejpam-4602	48	23	derivation	derivation	NOUN
ejpam-4602	48	24	,	,	PUNCT
ejpam-4602	48	25	but	but	CCONJ
ejpam-4602	48	26	not	not	PART
ejpam-4602	48	27	a	a	DET
ejpam-4602	48	28	reverse	reverse	ADJ
ejpam-4602	48	29	derivation	derivation	NOUN
ejpam-4602	48	30	,	,	PUNCT
ejpam-4602	48	31	while	while	SCONJ
ejpam-4602	48	32	δ̀	δ̀	PRON
ejpam-4602	48	33	is	be	AUX
ejpam-4602	48	34	both	both	PRON
ejpam-4602	48	35	a	a	DET
ejpam-4602	48	36	derivation	derivation	NOUN
ejpam-4602	48	37	and	and	CCONJ
ejpam-4602	48	38	a	a	DET
ejpam-4602	48	39	reverse	reverse	ADJ
ejpam-4602	48	40	derivation	derivation	NOUN
ejpam-4602	48	41	.	.	PUNCT
ejpam-4602	49	1	now	now	ADV
ejpam-4602	49	2	,	,	PUNCT
ejpam-4602	49	3	if	if	SCONJ
ejpam-4602	49	4	the	the	DET
ejpam-4602	49	5	ring	ring	NOUN
ejpam-4602	49	6	r	r	NOUN
ejpam-4602	49	7	is	be	AUX
ejpam-4602	49	8	commutative	commutative	ADJ
ejpam-4602	49	9	,	,	PUNCT
ejpam-4602	49	10	then	then	ADV
ejpam-4602	49	11	δ(uv	δ(uv	NOUN
ejpam-4602	49	12	)	)	PUNCT
ejpam-4602	49	13	=	=	SYM
ejpam-4602	49	14	δ(vu	δ(vu	NOUN
ejpam-4602	49	15	)	)	PUNCT
ejpam-4602	49	16	.	.	PUNCT
ejpam-4602	50	1	if	if	SCONJ
ejpam-4602	50	2	δ	δ	PROPN
ejpam-4602	50	3	is	be	AUX
ejpam-4602	50	4	a	a	DET
ejpam-4602	50	5	derivation	derivation	NOUN
ejpam-4602	50	6	on	on	ADP
ejpam-4602	50	7	a	a	DET
ejpam-4602	50	8	ring	ring	NOUN
ejpam-4602	50	9	r	r	NOUN
ejpam-4602	50	10	,	,	PUNCT
ejpam-4602	50	11	so	so	ADV
ejpam-4602	50	12	δ(uv	δ(uv	NOUN
ejpam-4602	50	13	)	)	PUNCT
ejpam-4602	50	14	=	=	SYM
ejpam-4602	50	15	δ(u)v	δ(u)v	PROPN
ejpam-4602	51	1	+	+	CCONJ
ejpam-4602	51	2	uδ(v	uδ(v	PUNCT
ejpam-4602	51	3	)	)	PUNCT
ejpam-4602	51	4	and	and	CCONJ
ejpam-4602	51	5	δ(vu	δ(vu	NOUN
ejpam-4602	51	6	)	)	PUNCT
ejpam-4602	51	7	=	=	SYM
ejpam-4602	51	8	δ(v)u+	δ(v)u+	NOUN
ejpam-4602	51	9	vδ(u	vδ(u	NUM
ejpam-4602	51	10	)	)	PUNCT
ejpam-4602	51	11	.	.	PUNCT
ejpam-4602	52	1	this	this	PRON
ejpam-4602	52	2	means	mean	VERB
ejpam-4602	52	3	δ(u)v	δ(u)v	PROPN
ejpam-4602	52	4	+	+	CCONJ
ejpam-4602	52	5	uδ(v	uδ(v	NUM
ejpam-4602	52	6	)	)	PUNCT
ejpam-4602	52	7	=	=	SYM
ejpam-4602	52	8	δ(v)u+	δ(v)u+	NOUN
ejpam-4602	52	9	vδ(u	vδ(u	NUM
ejpam-4602	52	10	)	)	PUNCT
ejpam-4602	52	11	.	.	PUNCT
ejpam-4602	53	1	thus	thus	ADV
ejpam-4602	53	2	,	,	PUNCT
ejpam-4602	53	3	δ	δ	PROPN
ejpam-4602	53	4	is	be	AUX
ejpam-4602	53	5	a	a	DET
ejpam-4602	53	6	reverse	reverse	ADJ
ejpam-4602	53	7	derivation	derivation	NOUN
ejpam-4602	53	8	too	too	ADV
ejpam-4602	53	9	.	.	PUNCT
ejpam-4602	54	1	lemma	lemma	PROPN
ejpam-4602	54	2	5	5	X
ejpam-4602	54	3	.	.	PUNCT
ejpam-4602	55	1	let	let	VERB
ejpam-4602	55	2	r	r	PRON
ejpam-4602	55	3	be	be	AUX
ejpam-4602	55	4	a	a	DET
ejpam-4602	55	5	δ	δ	NOUN
ejpam-4602	55	6	-	-	ADJ
ejpam-4602	55	7	prime	prime	ADJ
ejpam-4602	55	8	ring	ring	NOUN
ejpam-4602	55	9	,	,	PUNCT
ejpam-4602	55	10	u	u	PROPN
ejpam-4602	55	11	̸=	̸=	PROPN
ejpam-4602	55	12	{	{	PUNCT
ejpam-4602	55	13	0	0	NUM
ejpam-4602	55	14	}	}	PUNCT
ejpam-4602	55	15	a	a	DET
ejpam-4602	55	16	δ	δ	NOUN
ejpam-4602	55	17	-	-	PUNCT
ejpam-4602	55	18	ideal	ideal	NOUN
ejpam-4602	55	19	of	of	ADP
ejpam-4602	55	20	r	r	NOUN
ejpam-4602	55	21	,	,	PUNCT
ejpam-4602	55	22	which	which	PRON
ejpam-4602	55	23	δ	δ	PROPN
ejpam-4602	55	24	̸=	̸=	PROPN
ejpam-4602	55	25	0	0	NUM
ejpam-4602	56	1	a	a	DET
ejpam-4602	56	2	reverse	reverse	ADJ
ejpam-4602	56	3	derivation	derivation	NOUN
ejpam-4602	56	4	on	on	ADP
ejpam-4602	56	5	a	a	DET
ejpam-4602	56	6	r.	r.	NOUN
ejpam-4602	56	7	if	if	SCONJ
ejpam-4602	56	8	δ(u	δ(u	PROPN
ejpam-4602	56	9	)	)	PUNCT
ejpam-4602	56	10	=	=	SYM
ejpam-4602	56	11	0	0	NUM
ejpam-4602	56	12	,	,	PUNCT
ejpam-4602	56	13	then	then	ADV
ejpam-4602	56	14	δ(r	δ(r	NOUN
ejpam-4602	56	15	)	)	PUNCT
ejpam-4602	57	1	=	=	SYM
ejpam-4602	57	2	0	0	X
ejpam-4602	57	3	.	.	PUNCT
ejpam-4602	58	1	proof	proof	NOUN
ejpam-4602	58	2	.	.	PUNCT
ejpam-4602	59	1	since	since	SCONJ
ejpam-4602	59	2	u	u	NOUN
ejpam-4602	59	3	is	be	AUX
ejpam-4602	59	4	δ	δ	NOUN
ejpam-4602	59	5	-	-	PUNCT
ejpam-4602	59	6	ideal	ideal	NOUN
ejpam-4602	59	7	of	of	ADP
ejpam-4602	59	8	r	r	NOUN
ejpam-4602	59	9	,	,	PUNCT
ejpam-4602	59	10	then	then	ADV
ejpam-4602	59	11	ru	ru	VERB
ejpam-4602	59	12	⊆	⊆	NUM
ejpam-4602	59	13	u	u	NOUN
ejpam-4602	59	14	and	and	CCONJ
ejpam-4602	59	15	ur	ur	PRON
ejpam-4602	59	16	⊆	⊆	NUM
ejpam-4602	59	17	u.	u.	PROPN
ejpam-4602	59	18	i.	i.	PROPN
ejpam-4602	59	19	taha	taha	PROPN
ejpam-4602	59	20	et	et	PROPN
ejpam-4602	59	21	al	al	PROPN
ejpam-4602	59	22	.	.	PUNCT
ejpam-4602	59	23	/	/	SYM
ejpam-4602	59	24	eur	eur	PROPN
ejpam-4602	59	25	.	.	PUNCT
ejpam-4602	60	1	j.	j.	PROPN
ejpam-4602	60	2	pure	pure	PROPN
ejpam-4602	60	3	appl	appl	PROPN
ejpam-4602	60	4	.	.	PROPN
ejpam-4602	60	5	math	math	PROPN
ejpam-4602	60	6	,	,	PUNCT
ejpam-4602	60	7	15	15	NUM
ejpam-4602	60	8	(	(	PUNCT
ejpam-4602	60	9	4	4	NUM
ejpam-4602	60	10	)	)	PUNCT
ejpam-4602	60	11	(	(	PUNCT
ejpam-4602	60	12	2022	2022	NUM
ejpam-4602	60	13	)	)	PUNCT
ejpam-4602	60	14	,	,	PUNCT
ejpam-4602	60	15	2032	2032	NUM
ejpam-4602	60	16	-	-	SYM
ejpam-4602	60	17	2042	2042	NUM
ejpam-4602	60	18	2035	2035	NUM
ejpam-4602	60	19	so	so	ADV
ejpam-4602	60	20	δ(ru	δ(ru	NUM
ejpam-4602	60	21	)	)	PUNCT
ejpam-4602	60	22	⊆	⊆	NUM
ejpam-4602	60	23	δ(u	δ(u	NUM
ejpam-4602	60	24	)	)	PUNCT
ejpam-4602	60	25	=	=	SYM
ejpam-4602	60	26	0	0	NUM
ejpam-4602	60	27	and	and	CCONJ
ejpam-4602	60	28	δ(ur	δ(ur	NUM
ejpam-4602	60	29	)	)	PUNCT
ejpam-4602	60	30	⊆	⊆	NUM
ejpam-4602	60	31	δ(u	δ(u	NUM
ejpam-4602	60	32	)	)	PUNCT
ejpam-4602	60	33	=	=	SYM
ejpam-4602	61	1	0	0	NUM
ejpam-4602	61	2	,	,	PUNCT
ejpam-4602	61	3	then	then	ADV
ejpam-4602	61	4	δ(ru	δ(ru	NUM
ejpam-4602	61	5	)	)	PUNCT
ejpam-4602	61	6	=	=	PUNCT
ejpam-4602	61	7	δ(ur	δ(ur	X
ejpam-4602	61	8	)	)	PUNCT
ejpam-4602	61	9	=	=	SYM
ejpam-4602	62	1	0	0	X
ejpam-4602	62	2	.	.	PUNCT
ejpam-4602	63	1	since	since	SCONJ
ejpam-4602	63	2	δ	δ	PROPN
ejpam-4602	63	3	is	be	AUX
ejpam-4602	63	4	a	a	DET
ejpam-4602	63	5	reverse	reverse	ADJ
ejpam-4602	63	6	derivation	derivation	NOUN
ejpam-4602	63	7	then	then	ADV
ejpam-4602	63	8	,	,	PUNCT
ejpam-4602	63	9	δ(ru	δ(ru	NUM
ejpam-4602	63	10	)	)	PUNCT
ejpam-4602	63	11	=	=	SYM
ejpam-4602	63	12	δ(u)r+	δ(u)r+	NOUN
ejpam-4602	63	13	uδ(r	uδ(r	NUM
ejpam-4602	63	14	)	)	PUNCT
ejpam-4602	63	15	=	=	SYM
ejpam-4602	63	16	0	0	NUM
ejpam-4602	63	17	and	and	CCONJ
ejpam-4602	63	18	δ(ur	δ(ur	NUM
ejpam-4602	63	19	)	)	PUNCT
ejpam-4602	63	20	=	=	PUNCT
ejpam-4602	63	21	δ(r)u	δ(r)u	X
ejpam-4602	63	22	+	+	NOUN
ejpam-4602	63	23	rδ(u	rδ(u	NOUN
ejpam-4602	63	24	)	)	PUNCT
ejpam-4602	63	25	=	=	SYM
ejpam-4602	63	26	0	0	NUM
ejpam-4602	63	27	,	,	PUNCT
ejpam-4602	63	28	this	this	PRON
ejpam-4602	63	29	means	mean	VERB
ejpam-4602	63	30	uδ(r	uδ(r	PUNCT
ejpam-4602	63	31	)	)	PUNCT
ejpam-4602	64	1	=	=	SYM
ejpam-4602	64	2	δ(r)u	δ(r)u	X
ejpam-4602	64	3	=	=	NOUN
ejpam-4602	64	4	0	0	NUM
ejpam-4602	64	5	.	.	PUNCT
ejpam-4602	65	1	thus	thus	ADV
ejpam-4602	65	2	,	,	PUNCT
ejpam-4602	65	3	we	we	PRON
ejpam-4602	65	4	deduce	deduce	VERB
ejpam-4602	65	5	that	that	SCONJ
ejpam-4602	65	6	δ(r	δ(r	NOUN
ejpam-4602	65	7	)	)	PUNCT
ejpam-4602	65	8	⊆	⊆	NUM
ejpam-4602	65	9	annu	annu	NOUN
ejpam-4602	65	10	,	,	PUNCT
ejpam-4602	65	11	but	but	CCONJ
ejpam-4602	65	12	u	u	NOUN
ejpam-4602	65	13	is	be	AUX
ejpam-4602	65	14	a	a	DET
ejpam-4602	65	15	δ	δ	NOUN
ejpam-4602	65	16	-	-	PUNCT
ejpam-4602	65	17	ideal	ideal	ADJ
ejpam-4602	65	18	,	,	PUNCT
ejpam-4602	65	19	hence	hence	ADV
ejpam-4602	65	20	δ(r	δ(r	NUM
ejpam-4602	65	21	)	)	PUNCT
ejpam-4602	65	22	=	=	SYM
ejpam-4602	66	1	0	0	X
ejpam-4602	66	2	.	.	PUNCT
ejpam-4602	67	1	lemma	lemma	PROPN
ejpam-4602	67	2	6	6	NUM
ejpam-4602	67	3	.	.	PUNCT
ejpam-4602	68	1	let	let	VERB
ejpam-4602	68	2	δ	δ	PROPN
ejpam-4602	68	3	̸=	̸=	PROPN
ejpam-4602	68	4	0	0	NUM
ejpam-4602	68	5	be	be	AUX
ejpam-4602	68	6	a	a	DET
ejpam-4602	68	7	reverse	reverse	ADJ
ejpam-4602	68	8	derivation	derivation	NOUN
ejpam-4602	68	9	on	on	ADP
ejpam-4602	68	10	a	a	DET
ejpam-4602	68	11	δ	δ	NOUN
ejpam-4602	68	12	-	-	PUNCT
ejpam-4602	68	13	ring	ring	NOUN
ejpam-4602	68	14	r	r	NOUN
ejpam-4602	68	15	and	and	CCONJ
ejpam-4602	68	16	let	let	VERB
ejpam-4602	68	17	u	u	PRON
ejpam-4602	68	18	̸=	̸=	PROPN
ejpam-4602	68	19	{	{	PUNCT
ejpam-4602	68	20	0	0	NUM
ejpam-4602	68	21	}	}	PUNCT
ejpam-4602	68	22	be	be	AUX
ejpam-4602	68	23	δ	δ	NOUN
ejpam-4602	68	24	-	-	PUNCT
ejpam-4602	68	25	ideal	ideal	NOUN
ejpam-4602	68	26	of	of	ADP
ejpam-4602	68	27	r.	r.	PROPN
ejpam-4602	69	1	if	if	SCONJ
ejpam-4602	69	2	[	[	X
ejpam-4602	69	3	δ(u	δ(u	NUM
ejpam-4602	69	4	)	)	PUNCT
ejpam-4602	69	5	,	,	PUNCT
ejpam-4602	69	6	u	u	NOUN
ejpam-4602	69	7	]	]	X
ejpam-4602	69	8	=	=	SYM
ejpam-4602	69	9	0	0	NUM
ejpam-4602	69	10	∀u	∀u	PROPN
ejpam-4602	69	11	∈	∈	NUM
ejpam-4602	69	12	u	u	NOUN
ejpam-4602	69	13	,	,	PUNCT
ejpam-4602	69	14	(	(	PUNCT
ejpam-4602	69	15	1	1	X
ejpam-4602	69	16	)	)	PUNCT
ejpam-4602	69	17	then	then	ADV
ejpam-4602	69	18	r	r	NOUN
ejpam-4602	69	19	is	be	AUX
ejpam-4602	69	20	commutative	commutative	ADJ
ejpam-4602	69	21	.	.	PUNCT
ejpam-4602	70	1	proof	proof	NOUN
ejpam-4602	70	2	.	.	PUNCT
ejpam-4602	71	1	now	now	ADV
ejpam-4602	71	2	,	,	PUNCT
ejpam-4602	71	3	let	let	VERB
ejpam-4602	71	4	linearize	linearize	VERB
ejpam-4602	71	5	the	the	DET
ejpam-4602	71	6	identity	identity	NOUN
ejpam-4602	71	7	(	(	PUNCT
ejpam-4602	71	8	1	1	NUM
ejpam-4602	71	9	)	)	PUNCT
ejpam-4602	71	10	on	on	ADP
ejpam-4602	71	11	u	u	NOUN
ejpam-4602	71	12	,	,	PUNCT
ejpam-4602	71	13	then	then	ADV
ejpam-4602	71	14	we	we	PRON
ejpam-4602	71	15	have	have	VERB
ejpam-4602	71	16	for	for	ADP
ejpam-4602	71	17	all	all	DET
ejpam-4602	71	18	u	u	NOUN
ejpam-4602	71	19	,	,	PUNCT
ejpam-4602	71	20	v	v	NOUN
ejpam-4602	71	21	∈	∈	PROPN
ejpam-4602	71	22	u	u	NOUN
ejpam-4602	71	23	0	0	PUNCT
ejpam-4602	72	1	=	=	PUNCT
ejpam-4602	73	1	[	[	X
ejpam-4602	73	2	δ(u+	δ(u+	NUM
ejpam-4602	73	3	v	v	NOUN
ejpam-4602	73	4	)	)	PUNCT
ejpam-4602	73	5	,	,	PUNCT
ejpam-4602	73	6	u+	u+	NOUN
ejpam-4602	73	7	v	v	ADP
ejpam-4602	73	8	]	]	PUNCT
ejpam-4602	73	9	=	=	PUNCT
ejpam-4602	74	1	[	[	X
ejpam-4602	74	2	δ(u	δ(u	NUM
ejpam-4602	74	3	)	)	PUNCT
ejpam-4602	74	4	,	,	PUNCT
ejpam-4602	74	5	u	u	X
ejpam-4602	74	6	]	]	X
ejpam-4602	74	7	+	+	CCONJ
ejpam-4602	75	1	[	[	X
ejpam-4602	75	2	δ(u	δ(u	NUM
ejpam-4602	75	3	)	)	PUNCT
ejpam-4602	75	4	,	,	PUNCT
ejpam-4602	76	1	v]+	v]+	PROPN
ejpam-4602	77	1	[	[	X
ejpam-4602	77	2	δ(v	δ(v	PROPN
ejpam-4602	77	3	)	)	PUNCT
ejpam-4602	77	4	,	,	PUNCT
ejpam-4602	77	5	u	u	X
ejpam-4602	77	6	]	]	X
ejpam-4602	77	7	+	+	CCONJ
ejpam-4602	77	8	[	[	X
ejpam-4602	77	9	δ(v	δ(v	PROPN
ejpam-4602	77	10	)	)	PUNCT
ejpam-4602	77	11	,	,	PUNCT
ejpam-4602	77	12	v	v	ADP
ejpam-4602	77	13	]	]	X
ejpam-4602	77	14	=	=	PUNCT
ejpam-4602	78	1	[	[	X
ejpam-4602	78	2	δ(u	δ(u	NUM
ejpam-4602	78	3	)	)	PUNCT
ejpam-4602	78	4	,	,	PUNCT
ejpam-4602	78	5	v	v	ADP
ejpam-4602	78	6	]	]	PUNCT
ejpam-4602	78	7	+	+	CCONJ
ejpam-4602	79	1	[	[	X
ejpam-4602	79	2	δ(v	δ(v	PROPN
ejpam-4602	79	3	)	)	PUNCT
ejpam-4602	79	4	,	,	PUNCT
ejpam-4602	79	5	u	u	NOUN
ejpam-4602	79	6	]	]	X
ejpam-4602	79	7	.	.	PUNCT
ejpam-4602	80	1	[	[	X
ejpam-4602	80	2	δ(u	δ(u	NUM
ejpam-4602	80	3	)	)	PUNCT
ejpam-4602	80	4	,	,	PUNCT
ejpam-4602	80	5	v	v	ADP
ejpam-4602	80	6	]	]	X
ejpam-4602	80	7	=	=	PUNCT
ejpam-4602	81	1	[	[	X
ejpam-4602	81	2	u	u	NOUN
ejpam-4602	81	3	,	,	PUNCT
ejpam-4602	81	4	δ(v	δ(v	PROPN
ejpam-4602	81	5	)	)	PUNCT
ejpam-4602	81	6	]	]	PUNCT
ejpam-4602	81	7	.	.	PUNCT
ejpam-4602	82	1	(	(	PUNCT
ejpam-4602	82	2	2	2	X
ejpam-4602	82	3	)	)	PUNCT
ejpam-4602	82	4	write	write	VERB
ejpam-4602	82	5	vδ(u	vδ(u	NOUN
ejpam-4602	82	6	)	)	PUNCT
ejpam-4602	82	7	instead	instead	ADV
ejpam-4602	82	8	of	of	ADP
ejpam-4602	82	9	δ(u	δ(u	NOUN
ejpam-4602	82	10	)	)	PUNCT
ejpam-4602	82	11	in	in	ADP
ejpam-4602	82	12	(	(	PUNCT
ejpam-4602	82	13	2	2	NUM
ejpam-4602	82	14	)	)	PUNCT
ejpam-4602	82	15	,	,	PUNCT
ejpam-4602	82	16	then	then	ADV
ejpam-4602	82	17	we	we	PRON
ejpam-4602	82	18	get	get	VERB
ejpam-4602	82	19	[	[	X
ejpam-4602	82	20	u	u	NOUN
ejpam-4602	82	21	,	,	PUNCT
ejpam-4602	82	22	δ(v	δ(v	PROPN
ejpam-4602	82	23	)	)	PUNCT
ejpam-4602	82	24	]	]	PUNCT
ejpam-4602	83	1	=	=	PUNCT
ejpam-4602	84	1	[	[	X
ejpam-4602	84	2	vδ(u	vδ(u	NOUN
ejpam-4602	84	3	)	)	PUNCT
ejpam-4602	84	4	,	,	PUNCT
ejpam-4602	84	5	v	v	ADP
ejpam-4602	84	6	]	]	X
ejpam-4602	84	7	=	=	PUNCT
ejpam-4602	84	8	v[δ(u	v[δ(u	NUM
ejpam-4602	84	9	)	)	PUNCT
ejpam-4602	84	10	,	,	PUNCT
ejpam-4602	84	11	v	v	ADP
ejpam-4602	84	12	]	]	PUNCT
ejpam-4602	84	13	+	+	CCONJ
ejpam-4602	84	14	[	[	X
ejpam-4602	84	15	v	v	NOUN
ejpam-4602	84	16	,	,	PUNCT
ejpam-4602	84	17	v]δ(u	v]δ(u	NOUN
ejpam-4602	84	18	)	)	PUNCT
ejpam-4602	84	19	=	=	PUNCT
ejpam-4602	84	20	v[δ(u	v[δ(u	NOUN
ejpam-4602	84	21	)	)	PUNCT
ejpam-4602	84	22	,	,	PUNCT
ejpam-4602	84	23	v	v	ADP
ejpam-4602	84	24	]	]	X
ejpam-4602	84	25	=	=	SYM
ejpam-4602	84	26	v[u	v[u	PROPN
ejpam-4602	84	27	,	,	PUNCT
ejpam-4602	84	28	δ(v	δ(v	PROPN
ejpam-4602	84	29	)	)	PUNCT
ejpam-4602	84	30	]	]	PUNCT
ejpam-4602	85	1	=	=	PUNCT
ejpam-4602	85	2	vuδ(v)−	vuδ(v)−	PROPN
ejpam-4602	85	3	vδ(v)u	vδ(v)u	PROPN
ejpam-4602	85	4	=	=	SYM
ejpam-4602	85	5	vuδ(v)−	vuδ(v)−	PROPN
ejpam-4602	85	6	δ(v)vu	δ(v)vu	NOUN
ejpam-4602	86	1	=	=	PUNCT
ejpam-4602	87	1	[	[	X
ejpam-4602	87	2	vu	vu	X
ejpam-4602	87	3	,	,	PUNCT
ejpam-4602	87	4	δ(v	δ(v	PROPN
ejpam-4602	87	5	)	)	PUNCT
ejpam-4602	87	6	]	]	PUNCT
ejpam-4602	88	1	=	=	PUNCT
ejpam-4602	88	2	i.	i.	PROPN
ejpam-4602	88	3	taha	taha	PROPN
ejpam-4602	88	4	et	et	PROPN
ejpam-4602	88	5	al	al	PROPN
ejpam-4602	88	6	.	.	PUNCT
ejpam-4602	88	7	/	/	SYM
ejpam-4602	88	8	eur	eur	PROPN
ejpam-4602	88	9	.	.	PUNCT
ejpam-4602	89	1	j.	j.	PROPN
ejpam-4602	89	2	pure	pure	PROPN
ejpam-4602	89	3	appl	appl	PROPN
ejpam-4602	89	4	.	.	PROPN
ejpam-4602	89	5	math	math	PROPN
ejpam-4602	89	6	,	,	PUNCT
ejpam-4602	89	7	15	15	NUM
ejpam-4602	89	8	(	(	PUNCT
ejpam-4602	89	9	4	4	NUM
ejpam-4602	89	10	)	)	PUNCT
ejpam-4602	89	11	(	(	PUNCT
ejpam-4602	89	12	2022	2022	NUM
ejpam-4602	89	13	)	)	PUNCT
ejpam-4602	89	14	,	,	PUNCT
ejpam-4602	89	15	2032	2032	NUM
ejpam-4602	89	16	-	-	SYM
ejpam-4602	89	17	2042	2042	NUM
ejpam-4602	89	18	2036	2036	NUM
ejpam-4602	90	1	[	[	X
ejpam-4602	90	2	δ(vu	δ(vu	NOUN
ejpam-4602	90	3	)	)	PUNCT
ejpam-4602	90	4	,	,	PUNCT
ejpam-4602	90	5	v	v	ADP
ejpam-4602	90	6	]	]	X
ejpam-4602	90	7	=	=	PUNCT
ejpam-4602	91	1	[	[	X
ejpam-4602	91	2	δ(u)v	δ(u)v	X
ejpam-4602	91	3	+	+	CCONJ
ejpam-4602	91	4	uδ(v	uδ(v	PROPN
ejpam-4602	91	5	)	)	PUNCT
ejpam-4602	91	6	,	,	PUNCT
ejpam-4602	91	7	v	v	ADP
ejpam-4602	91	8	]	]	X
ejpam-4602	91	9	=	=	PUNCT
ejpam-4602	92	1	[	[	X
ejpam-4602	92	2	δ(u)v	δ(u)v	PROPN
ejpam-4602	92	3	,	,	PUNCT
ejpam-4602	92	4	v	v	ADP
ejpam-4602	92	5	]	]	PUNCT
ejpam-4602	92	6	+	+	CCONJ
ejpam-4602	93	1	[	[	X
ejpam-4602	93	2	uδ(v	uδ(v	NUM
ejpam-4602	93	3	)	)	PUNCT
ejpam-4602	93	4	,	,	PUNCT
ejpam-4602	93	5	v	v	ADP
ejpam-4602	93	6	]	]	X
ejpam-4602	93	7	=	=	PUNCT
ejpam-4602	94	1	[	[	X
ejpam-4602	94	2	δ(u	δ(u	NUM
ejpam-4602	94	3	)	)	PUNCT
ejpam-4602	94	4	,	,	PUNCT
ejpam-4602	94	5	v]v	v]v	ADJ
ejpam-4602	94	6	+	+	CCONJ
ejpam-4602	95	1	[	[	X
ejpam-4602	95	2	u	u	NOUN
ejpam-4602	95	3	,	,	PUNCT
ejpam-4602	95	4	v]δ(v	v]δ(v	NOUN
ejpam-4602	95	5	)	)	PUNCT
ejpam-4602	96	1	[	[	X
ejpam-4602	96	2	v	v	NOUN
ejpam-4602	96	3	,	,	PUNCT
ejpam-4602	96	4	u]δ(v	u]δ(v	NOUN
ejpam-4602	96	5	)	)	PUNCT
ejpam-4602	96	6	=	=	SYM
ejpam-4602	97	1	0	0	X
ejpam-4602	97	2	.	.	PUNCT
ejpam-4602	98	1	(	(	PUNCT
ejpam-4602	98	2	3	3	X
ejpam-4602	98	3	)	)	PUNCT
ejpam-4602	98	4	now	now	ADV
ejpam-4602	98	5	,	,	PUNCT
ejpam-4602	98	6	replace	replace	VERB
ejpam-4602	98	7	u	u	NOUN
ejpam-4602	98	8	by	by	ADP
ejpam-4602	98	9	uv	uv	NOUN
ejpam-4602	98	10	in	in	ADP
ejpam-4602	98	11	(	(	PUNCT
ejpam-4602	98	12	3	3	NUM
ejpam-4602	98	13	)	)	PUNCT
ejpam-4602	98	14	,	,	PUNCT
ejpam-4602	98	15	then	then	ADV
ejpam-4602	98	16	,	,	PUNCT
ejpam-4602	98	17	[	[	X
ejpam-4602	98	18	v	v	NOUN
ejpam-4602	98	19	,	,	PUNCT
ejpam-4602	98	20	uv]δ(v	uv]δ(v	PROPN
ejpam-4602	98	21	)	)	PUNCT
ejpam-4602	98	22	=	=	SYM
ejpam-4602	99	1	0	0	NUM
ejpam-4602	99	2	,	,	PUNCT
ejpam-4602	99	3	[	[	X
ejpam-4602	99	4	v	v	NOUN
ejpam-4602	99	5	,	,	PUNCT
ejpam-4602	99	6	u]vδ(v	u]vδ(v	PROPN
ejpam-4602	99	7	)	)	PUNCT
ejpam-4602	99	8	=	=	SYM
ejpam-4602	99	9	0	0	NUM
ejpam-4602	99	10	,	,	PUNCT
ejpam-4602	99	11	thus	thus	ADV
ejpam-4602	99	12	[	[	X
ejpam-4602	99	13	v	v	NOUN
ejpam-4602	99	14	,	,	PUNCT
ejpam-4602	99	15	u]uδ(v	u]uδ(v	NOUN
ejpam-4602	99	16	)	)	PUNCT
ejpam-4602	99	17	=	=	SYM
ejpam-4602	100	1	0	0	X
ejpam-4602	100	2	.	.	PUNCT
ejpam-4602	101	1	hence	hence	ADV
ejpam-4602	101	2	,	,	PUNCT
ejpam-4602	101	3	since	since	SCONJ
ejpam-4602	101	4	r	r	NOUN
ejpam-4602	101	5	is	be	AUX
ejpam-4602	101	6	a	a	DET
ejpam-4602	101	7	δ	δ	NOUN
ejpam-4602	101	8	-	-	ADJ
ejpam-4602	101	9	prime	prime	ADJ
ejpam-4602	101	10	ring	ring	NOUN
ejpam-4602	101	11	,	,	PUNCT
ejpam-4602	101	12	either	either	CCONJ
ejpam-4602	101	13	δ(u	δ(u	NOUN
ejpam-4602	101	14	)	)	PUNCT
ejpam-4602	101	15	=	=	SYM
ejpam-4602	101	16	0	0	NUM
ejpam-4602	101	17	,	,	PUNCT
ejpam-4602	101	18	then	then	ADV
ejpam-4602	101	19	u	u	NOUN
ejpam-4602	101	20	=	=	PROPN
ejpam-4602	101	21	0	0	PUNCT
ejpam-4602	101	22	and	and	CCONJ
ejpam-4602	101	23	this	this	PRON
ejpam-4602	101	24	contradicts	contradict	VERB
ejpam-4602	101	25	the	the	DET
ejpam-4602	101	26	assumption	assumption	NOUN
ejpam-4602	101	27	,	,	PUNCT
ejpam-4602	101	28	or	or	CCONJ
ejpam-4602	101	29	[	[	X
ejpam-4602	101	30	v	v	NOUN
ejpam-4602	101	31	,	,	PUNCT
ejpam-4602	101	32	u	u	NOUN
ejpam-4602	101	33	]	]	X
ejpam-4602	101	34	=	=	SYM
ejpam-4602	101	35	0	0	NUM
ejpam-4602	101	36	for	for	ADP
ejpam-4602	101	37	all	all	DET
ejpam-4602	101	38	u	u	NOUN
ejpam-4602	101	39	,	,	PUNCT
ejpam-4602	101	40	v	v	PROPN
ejpam-4602	101	41	∈	∈	PROPN
ejpam-4602	101	42	u	u	NOUN
ejpam-4602	101	43	,	,	PUNCT
ejpam-4602	101	44	therefore	therefore	ADV
ejpam-4602	101	45	u	u	NOUN
ejpam-4602	101	46	is	be	AUX
ejpam-4602	101	47	commutative	commutative	ADJ
ejpam-4602	101	48	.	.	PUNCT
ejpam-4602	102	1	so	so	ADV
ejpam-4602	102	2	we	we	PRON
ejpam-4602	102	3	get	get	VERB
ejpam-4602	102	4	uc(r	uc(r	NOUN
ejpam-4602	102	5	)	)	PUNCT
ejpam-4602	102	6	=	=	SYM
ejpam-4602	102	7	0	0	NUM
ejpam-4602	102	8	,	,	PUNCT
ejpam-4602	102	9	then	then	ADV
ejpam-4602	102	10	c(r	c(r	NOUN
ejpam-4602	102	11	)	)	PUNCT
ejpam-4602	103	1	=	=	SYM
ejpam-4602	103	2	0	0	NUM
ejpam-4602	103	3	,	,	PUNCT
ejpam-4602	103	4	hence	hence	ADV
ejpam-4602	103	5	r	r	NOUN
ejpam-4602	103	6	is	be	AUX
ejpam-4602	103	7	commutative	commutative	ADJ
ejpam-4602	103	8	.	.	PUNCT
ejpam-4602	104	1	4	4	X
ejpam-4602	104	2	.	.	X
ejpam-4602	104	3	reverse	reverse	ADJ
ejpam-4602	104	4	derivation	derivation	NOUN
ejpam-4602	104	5	on	on	ADP
ejpam-4602	104	6	δideal	δideal	ADJ
ejpam-4602	104	7	through	through	ADP
ejpam-4602	104	8	this	this	DET
ejpam-4602	104	9	section	section	NOUN
ejpam-4602	104	10	,	,	PUNCT
ejpam-4602	104	11	we	we	PRON
ejpam-4602	104	12	will	will	AUX
ejpam-4602	104	13	prove	prove	VERB
ejpam-4602	104	14	several	several	ADJ
ejpam-4602	104	15	lemmas	lemma	NOUN
ejpam-4602	104	16	arriving	arrive	VERB
ejpam-4602	104	17	to	to	ADP
ejpam-4602	104	18	the	the	DET
ejpam-4602	104	19	extension	extension	NOUN
ejpam-4602	104	20	of	of	ADP
ejpam-4602	104	21	lemma	lemma	PROPN
ejpam-4602	104	22	(	(	PUNCT
ejpam-4602	104	23	1	1	NUM
ejpam-4602	104	24	)	)	PUNCT
ejpam-4602	104	25	,	,	PUNCT
ejpam-4602	104	26	presented	present	VERB
ejpam-4602	104	27	by	by	ADP
ejpam-4602	104	28	the	the	DET
ejpam-4602	104	29	main	main	ADJ
ejpam-4602	104	30	theorem	theorem	NOUN
ejpam-4602	104	31	.	.	PUNCT
ejpam-4602	105	1	lemma	lemma	PROPN
ejpam-4602	105	2	7	7	X
ejpam-4602	105	3	.	.	PUNCT
ejpam-4602	106	1	let	let	VERB
ejpam-4602	106	2	r	r	PRON
ejpam-4602	106	3	be	be	AUX
ejpam-4602	106	4	a	a	DET
ejpam-4602	106	5	δ	δ	NOUN
ejpam-4602	106	6	-	-	ADJ
ejpam-4602	106	7	prime	prime	ADJ
ejpam-4602	106	8	ring	ring	NOUN
ejpam-4602	106	9	and	and	CCONJ
ejpam-4602	106	10	let	let	VERB
ejpam-4602	106	11	δ	δ	PROPN
ejpam-4602	106	12	̸=	̸=	PROPN
ejpam-4602	106	13	0	0	NUM
ejpam-4602	106	14	be	be	AUX
ejpam-4602	106	15	a	a	DET
ejpam-4602	106	16	reverse	reverse	ADJ
ejpam-4602	106	17	derivation	derivation	NOUN
ejpam-4602	106	18	on	on	ADP
ejpam-4602	106	19	r.	r.	PROPN
ejpam-4602	106	20	if	if	SCONJ
ejpam-4602	106	21	u[δn(u	u[δn(u	NOUN
ejpam-4602	106	22	)	)	PUNCT
ejpam-4602	106	23	,	,	PUNCT
ejpam-4602	107	1	r	r	X
ejpam-4602	107	2	]	]	X
ejpam-4602	107	3	=	=	SYM
ejpam-4602	107	4	0	0	NUM
ejpam-4602	107	5	,	,	PUNCT
ejpam-4602	107	6	∀u	∀u	NOUN
ejpam-4602	107	7	∈	∈	NOUN
ejpam-4602	107	8	r	r	NOUN
ejpam-4602	107	9	or	or	CCONJ
ejpam-4602	107	10	[	[	X
ejpam-4602	107	11	δn(v	δn(v	X
ejpam-4602	107	12	)	)	PUNCT
ejpam-4602	107	13	,	,	PUNCT
ejpam-4602	107	14	r]u	r]u	ADJ
ejpam-4602	107	15	=	=	SYM
ejpam-4602	107	16	0	0	NUM
ejpam-4602	107	17	,	,	PUNCT
ejpam-4602	107	18	∀u	∀u	NOUN
ejpam-4602	107	19	,	,	PUNCT
ejpam-4602	107	20	v	v	NOUN
ejpam-4602	107	21	∈	∈	NOUN
ejpam-4602	107	22	r	r	NOUN
ejpam-4602	107	23	,	,	PUNCT
ejpam-4602	107	24	n	n	PRON
ejpam-4602	107	25	∈	∈	PROPN
ejpam-4602	107	26	z+	z+	NUM
ejpam-4602	107	27	.	.	PUNCT
ejpam-4602	108	1	then	then	ADV
ejpam-4602	108	2	either	either	CCONJ
ejpam-4602	108	3	u	u	X
ejpam-4602	108	4	=	=	NOUN
ejpam-4602	108	5	0	0	NUM
ejpam-4602	108	6	or	or	CCONJ
ejpam-4602	108	7	v	v	ADP
ejpam-4602	108	8	∈	∈	PROPN
ejpam-4602	108	9	z(r	z(r	NOUN
ejpam-4602	108	10	)	)	PUNCT
ejpam-4602	108	11	.	.	PUNCT
ejpam-4602	109	1	proof	proof	NOUN
ejpam-4602	109	2	.	.	PUNCT
ejpam-4602	110	1	assume	assume	VERB
ejpam-4602	110	2	u	u	NOUN
ejpam-4602	110	3	,	,	PUNCT
ejpam-4602	110	4	v	v	NOUN
ejpam-4602	110	5	∈	∈	NOUN
ejpam-4602	110	6	r	r	NOUN
ejpam-4602	110	7	and	and	CCONJ
ejpam-4602	110	8	n	n	CCONJ
ejpam-4602	110	9	∈	∈	PROPN
ejpam-4602	110	10	z+	z+	NUM
ejpam-4602	110	11	,	,	PUNCT
ejpam-4602	110	12	then	then	ADV
ejpam-4602	110	13	from	from	ADP
ejpam-4602	110	14	the	the	DET
ejpam-4602	110	15	assumption	assumption	NOUN
ejpam-4602	110	16	u[δn(u	u[δn(u	NOUN
ejpam-4602	110	17	)	)	PUNCT
ejpam-4602	110	18	,	,	PUNCT
ejpam-4602	110	19	r	r	X
ejpam-4602	110	20	]	]	X
ejpam-4602	110	21	=	=	SYM
ejpam-4602	110	22	0	0	NUM
ejpam-4602	110	23	,	,	PUNCT
ejpam-4602	110	24	this	this	DET
ejpam-4602	110	25	equivalents	equivalent	NOUN
ejpam-4602	110	26	to	to	ADP
ejpam-4602	110	27	u∂δn(v)(r	u∂δn(v)(r	PROPN
ejpam-4602	110	28	)	)	PUNCT
ejpam-4602	110	29	=	=	SYM
ejpam-4602	110	30	0	0	NUM
ejpam-4602	110	31	,	,	PUNCT
ejpam-4602	110	32	(	(	PUNCT
ejpam-4602	110	33	4	4	NUM
ejpam-4602	110	34	)	)	PUNCT
ejpam-4602	110	35	then	then	ADV
ejpam-4602	110	36	0	0	X
ejpam-4602	110	37	=	=	SYM
ejpam-4602	110	38	u∂δn(v)(ab	u∂δn(v)(ab	PROPN
ejpam-4602	110	39	)	)	PUNCT
ejpam-4602	111	1	=	=	VERB
ejpam-4602	111	2	u∂δn(v)(b)a+	u∂δn(v)(b)a+	NOUN
ejpam-4602	111	3	ub∂δn(v)(a	ub∂δn(v)(a	ADJ
ejpam-4602	111	4	)	)	PUNCT
ejpam-4602	111	5	,	,	PUNCT
ejpam-4602	111	6	∀a	∀a	X
ejpam-4602	111	7	,	,	PUNCT
ejpam-4602	111	8	b	b	X
ejpam-4602	111	9	∈	∈	NOUN
ejpam-4602	111	10	r	r	NOUN
ejpam-4602	111	11	now	now	ADV
ejpam-4602	111	12	from	from	ADP
ejpam-4602	111	13	(	(	PUNCT
ejpam-4602	111	14	4	4	X
ejpam-4602	111	15	)	)	PUNCT
ejpam-4602	111	16	ub∂δn(v)(a	ub∂δn(v)(a	ADJ
ejpam-4602	111	17	)	)	PUNCT
ejpam-4602	111	18	=	=	SYM
ejpam-4602	111	19	0	0	NUM
ejpam-4602	111	20	,	,	PUNCT
ejpam-4602	111	21	i.	i.	PROPN
ejpam-4602	111	22	taha	taha	PROPN
ejpam-4602	111	23	et	et	PROPN
ejpam-4602	111	24	al	al	PROPN
ejpam-4602	111	25	.	.	PUNCT
ejpam-4602	111	26	/	/	SYM
ejpam-4602	111	27	eur	eur	PROPN
ejpam-4602	111	28	.	.	PUNCT
ejpam-4602	112	1	j.	j.	PROPN
ejpam-4602	112	2	pure	pure	PROPN
ejpam-4602	112	3	appl	appl	PROPN
ejpam-4602	112	4	.	.	PROPN
ejpam-4602	112	5	math	math	PROPN
ejpam-4602	112	6	,	,	PUNCT
ejpam-4602	112	7	15	15	NUM
ejpam-4602	112	8	(	(	PUNCT
ejpam-4602	112	9	4	4	NUM
ejpam-4602	112	10	)	)	PUNCT
ejpam-4602	112	11	(	(	PUNCT
ejpam-4602	112	12	2022	2022	NUM
ejpam-4602	112	13	)	)	PUNCT
ejpam-4602	112	14	,	,	PUNCT
ejpam-4602	112	15	2032	2032	NUM
ejpam-4602	112	16	-	-	SYM
ejpam-4602	112	17	2042	2042	NUM
ejpam-4602	112	18	2037	2037	NUM
ejpam-4602	112	19	this	this	PRON
ejpam-4602	112	20	means	mean	VERB
ejpam-4602	112	21	ur[δn(v	ur[δn(v	PROPN
ejpam-4602	112	22	)	)	PUNCT
ejpam-4602	112	23	,	,	PUNCT
ejpam-4602	112	24	a	a	DET
ejpam-4602	112	25	]	]	X
ejpam-4602	112	26	=	=	SYM
ejpam-4602	112	27	0	0	X
ejpam-4602	112	28	.	.	PUNCT
ejpam-4602	113	1	consequently	consequently	ADV
ejpam-4602	113	2	,	,	PUNCT
ejpam-4602	113	3	urδk([δn(v	urδk([δn(v	PROPN
ejpam-4602	113	4	)	)	PUNCT
ejpam-4602	113	5	,	,	PUNCT
ejpam-4602	113	6	a	a	DET
ejpam-4602	113	7	]	]	X
ejpam-4602	113	8	)	)	PUNCT
ejpam-4602	113	9	=	=	SYM
ejpam-4602	114	1	0	0	X
ejpam-4602	114	2	.	.	PUNCT
ejpam-4602	115	1	what	what	PRON
ejpam-4602	115	2	forces	force	VERB
ejpam-4602	115	3	that	that	PRON
ejpam-4602	115	4	u	u	NOUN
ejpam-4602	115	5	=	=	NOUN
ejpam-4602	115	6	0	0	NUM
ejpam-4602	115	7	or	or	CCONJ
ejpam-4602	115	8	[	[	X
ejpam-4602	115	9	δn(v	δn(v	X
ejpam-4602	115	10	)	)	PUNCT
ejpam-4602	115	11	,	,	PUNCT
ejpam-4602	115	12	a	a	PRON
ejpam-4602	115	13	]	]	X
ejpam-4602	115	14	=	=	SYM
ejpam-4602	115	15	0	0	X
ejpam-4602	115	16	.	.	PUNCT
ejpam-4602	116	1	hence	hence	ADV
ejpam-4602	116	2	v	v	ADP
ejpam-4602	116	3	∈	∈	PROPN
ejpam-4602	116	4	z(r	z(r	PROPN
ejpam-4602	116	5	)	)	PUNCT
ejpam-4602	116	6	.	.	PUNCT
ejpam-4602	117	1	lemma	lemma	PROPN
ejpam-4602	117	2	8	8	NUM
ejpam-4602	117	3	.	.	PUNCT
ejpam-4602	118	1	let	let	VERB
ejpam-4602	118	2	r	r	PRON
ejpam-4602	118	3	be	be	AUX
ejpam-4602	118	4	a	a	DET
ejpam-4602	118	5	δ	δ	NOUN
ejpam-4602	118	6	-	-	ADJ
ejpam-4602	118	7	prime	prime	ADJ
ejpam-4602	118	8	ring	ring	NOUN
ejpam-4602	118	9	and	and	CCONJ
ejpam-4602	118	10	let	let	VERB
ejpam-4602	118	11	u	u	PRON
ejpam-4602	118	12	̸=	̸=	PROPN
ejpam-4602	118	13	{	{	PUNCT
ejpam-4602	118	14	0	0	NUM
ejpam-4602	118	15	}	}	PUNCT
ejpam-4602	118	16	be	be	AUX
ejpam-4602	118	17	a	a	DET
ejpam-4602	118	18	right	right	ADJ
ejpam-4602	118	19	δ	δ	NOUN
ejpam-4602	118	20	-	-	PUNCT
ejpam-4602	118	21	ideal	ideal	ADJ
ejpam-4602	118	22	,	,	PUNCT
ejpam-4602	118	23	which	which	PRON
ejpam-4602	118	24	δ	δ	PROPN
ejpam-4602	118	25	is	be	AUX
ejpam-4602	118	26	a	a	DET
ejpam-4602	118	27	reverse	reverse	ADJ
ejpam-4602	118	28	derivation	derivation	NOUN
ejpam-4602	118	29	on	on	ADP
ejpam-4602	118	30	r.	r.	PROPN
ejpam-4602	118	31	if	if	SCONJ
ejpam-4602	118	32	u	u	NOUN
ejpam-4602	118	33	is	be	AUX
ejpam-4602	118	34	commutative	commutative	ADJ
ejpam-4602	118	35	,	,	PUNCT
ejpam-4602	118	36	then	then	ADV
ejpam-4602	118	37	r	r	NOUN
ejpam-4602	118	38	is	be	AUX
ejpam-4602	118	39	commutative	commutative	ADJ
ejpam-4602	118	40	.	.	PUNCT
ejpam-4602	119	1	proof	proof	NOUN
ejpam-4602	119	2	.	.	PUNCT
ejpam-4602	120	1	assume	assume	VERB
ejpam-4602	120	2	that	that	SCONJ
ejpam-4602	120	3	u	u	PROPN
ejpam-4602	120	4	∈	∈	PROPN
ejpam-4602	120	5	u	u	NOUN
ejpam-4602	120	6	.	.	PUNCT
ejpam-4602	121	1	since	since	SCONJ
ejpam-4602	121	2	u	u	NOUN
ejpam-4602	121	3	is	be	AUX
ejpam-4602	121	4	commutative	commutative	ADJ
ejpam-4602	121	5	,	,	PUNCT
ejpam-4602	121	6	then	then	ADV
ejpam-4602	121	7	∂u(u	∂u(u	NOUN
ejpam-4602	121	8	)	)	PUNCT
ejpam-4602	121	9	=	=	PUNCT
ejpam-4602	122	1	[	[	X
ejpam-4602	122	2	u	u	NOUN
ejpam-4602	122	3	,	,	PUNCT
ejpam-4602	122	4	u	u	NOUN
ejpam-4602	122	5	]	]	X
ejpam-4602	122	6	=	=	SYM
ejpam-4602	122	7	0	0	X
ejpam-4602	122	8	.	.	PUNCT
ejpam-4602	122	9	now	now	ADV
ejpam-4602	122	10	by	by	ADP
ejpam-4602	122	11	lemma	lemma	PROPN
ejpam-4602	122	12	(	(	PUNCT
ejpam-4602	122	13	5	5	NUM
ejpam-4602	122	14	)	)	PUNCT
ejpam-4602	122	15	,	,	PUNCT
ejpam-4602	122	16	∂u(r	∂u(r	PROPN
ejpam-4602	122	17	)	)	PUNCT
ejpam-4602	122	18	=	=	SYM
ejpam-4602	122	19	0	0	NUM
ejpam-4602	122	20	,	,	PUNCT
ejpam-4602	122	21	then	then	ADV
ejpam-4602	122	22	u	u	PROPN
ejpam-4602	122	23	∈	∈	PROPN
ejpam-4602	122	24	z(r),∀u	z(r),∀u	PROPN
ejpam-4602	122	25	∈	∈	PROPN
ejpam-4602	122	26	u	u	NOUN
ejpam-4602	122	27	,	,	PUNCT
ejpam-4602	122	28	hence	hence	ADV
ejpam-4602	122	29	u	u	NOUN
ejpam-4602	122	30	⊆	⊆	NUM
ejpam-4602	122	31	z(r	z(r	NOUN
ejpam-4602	122	32	)	)	PUNCT
ejpam-4602	122	33	.	.	PUNCT
ejpam-4602	123	1	thus	thus	ADV
ejpam-4602	123	2	c(r	c(r	NOUN
ejpam-4602	123	3	)	)	PUNCT
ejpam-4602	123	4	=	=	SYM
ejpam-4602	123	5	0	0	NUM
ejpam-4602	123	6	,	,	PUNCT
ejpam-4602	123	7	this	this	PRON
ejpam-4602	123	8	means	mean	VERB
ejpam-4602	123	9	r	r	NOUN
ejpam-4602	123	10	is	be	AUX
ejpam-4602	123	11	commutative	commutative	ADJ
ejpam-4602	123	12	.	.	PUNCT
ejpam-4602	124	1	lemma	lemma	PROPN
ejpam-4602	124	2	9	9	NUM
ejpam-4602	124	3	.	.	PUNCT
ejpam-4602	125	1	let	let	VERB
ejpam-4602	125	2	r	r	PRON
ejpam-4602	125	3	be	be	AUX
ejpam-4602	125	4	a	a	DET
ejpam-4602	125	5	δ	δ	NOUN
ejpam-4602	125	6	-	-	ADJ
ejpam-4602	125	7	prime	prime	ADJ
ejpam-4602	125	8	ring	ring	NOUN
ejpam-4602	125	9	and	and	CCONJ
ejpam-4602	125	10	δ	δ	PROPN
ejpam-4602	125	11	̸=	̸=	PROPN
ejpam-4602	125	12	0	0	NUM
ejpam-4602	125	13	a	a	DET
ejpam-4602	125	14	reverse	reverse	ADJ
ejpam-4602	125	15	derivation	derivation	NOUN
ejpam-4602	125	16	on	on	ADP
ejpam-4602	125	17	r.	r.	PROPN
ejpam-4602	126	1	if	if	SCONJ
ejpam-4602	126	2	[	[	X
ejpam-4602	126	3	v	v	NOUN
ejpam-4602	126	4	,	,	PUNCT
ejpam-4602	126	5	δn(u)v	δn(u)v	PRON
ejpam-4602	126	6	]	]	X
ejpam-4602	126	7	∈	∈	PROPN
ejpam-4602	126	8	z(r),∀u	z(r),∀u	PROPN
ejpam-4602	126	9	,	,	PUNCT
ejpam-4602	126	10	v	v	ADP
ejpam-4602	126	11	̸=	̸=	PROPN
ejpam-4602	126	12	0	0	NUM
ejpam-4602	126	13	∈	∈	PROPN
ejpam-4602	126	14	r	r	NOUN
ejpam-4602	126	15	,	,	PUNCT
ejpam-4602	126	16	n	n	PRON
ejpam-4602	126	17	∈	∈	NOUN
ejpam-4602	126	18	z+	z+	NUM
ejpam-4602	126	19	,	,	PUNCT
ejpam-4602	126	20	then	then	ADV
ejpam-4602	126	21	u	u	PROPN
ejpam-4602	126	22	∈	∈	PROPN
ejpam-4602	126	23	z(r	z(r	PROPN
ejpam-4602	126	24	)	)	PUNCT
ejpam-4602	126	25	.	.	PUNCT
ejpam-4602	127	1	proof	proof	NOUN
ejpam-4602	127	2	.	.	PUNCT
ejpam-4602	128	1	assume	assume	VERB
ejpam-4602	128	2	a	a	DET
ejpam-4602	128	3	∈	∈	PROPN
ejpam-4602	128	4	r	r	NOUN
ejpam-4602	128	5	,	,	PUNCT
ejpam-4602	128	6	then	then	ADV
ejpam-4602	128	7	we	we	PRON
ejpam-4602	128	8	get	get	VERB
ejpam-4602	128	9	0	0	NUM
ejpam-4602	129	1	=	=	PUNCT
ejpam-4602	130	1	[	[	X
ejpam-4602	130	2	δn(u)v	δn(u)v	X
ejpam-4602	130	3	,	,	PUNCT
ejpam-4602	130	4	a	a	PRON
ejpam-4602	130	5	]	]	X
ejpam-4602	130	6	=	=	SYM
ejpam-4602	130	7	δn(u)[v	δn(u)[v	ADJ
ejpam-4602	130	8	,	,	PUNCT
ejpam-4602	130	9	a	a	PRON
ejpam-4602	130	10	]	]	X
ejpam-4602	130	11	+	+	CCONJ
ejpam-4602	130	12	[	[	X
ejpam-4602	130	13	δn(u	δn(u	NOUN
ejpam-4602	130	14	)	)	PUNCT
ejpam-4602	130	15	,	,	PUNCT
ejpam-4602	130	16	a]v	a]v	NOUN
ejpam-4602	130	17	=	=	PUNCT
ejpam-4602	131	1	[	[	X
ejpam-4602	131	2	δn(u	δn(u	NOUN
ejpam-4602	131	3	)	)	PUNCT
ejpam-4602	131	4	,	,	PUNCT
ejpam-4602	131	5	a]v	a]v	PROPN
ejpam-4602	131	6	.	.	PUNCT
ejpam-4602	132	1	then	then	ADV
ejpam-4602	132	2	,	,	PUNCT
ejpam-4602	132	3	by	by	ADP
ejpam-4602	132	4	lemma	lemma	PROPN
ejpam-4602	132	5	(	(	PUNCT
ejpam-4602	132	6	7	7	NUM
ejpam-4602	132	7	)	)	PUNCT
ejpam-4602	132	8	,	,	PUNCT
ejpam-4602	132	9	u	u	PROPN
ejpam-4602	132	10	∈	∈	PROPN
ejpam-4602	132	11	z(r	z(r	PROPN
ejpam-4602	132	12	)	)	PUNCT
ejpam-4602	132	13	.	.	PUNCT
ejpam-4602	133	1	lemma	lemma	PROPN
ejpam-4602	133	2	10	10	NUM
ejpam-4602	133	3	.	.	PUNCT
ejpam-4602	134	1	let	let	VERB
ejpam-4602	134	2	r	r	PRON
ejpam-4602	134	3	be	be	AUX
ejpam-4602	134	4	a	a	DET
ejpam-4602	134	5	δ	δ	NOUN
ejpam-4602	134	6	-	-	ADJ
ejpam-4602	134	7	prime	prime	ADJ
ejpam-4602	134	8	ring	ring	NOUN
ejpam-4602	134	9	of	of	ADP
ejpam-4602	134	10	charr	charr	NOUN
ejpam-4602	134	11	̸=	̸=	PROPN
ejpam-4602	134	12	2	2	NUM
ejpam-4602	134	13	and	and	CCONJ
ejpam-4602	134	14	let	let	VERB
ejpam-4602	134	15	u	u	PRON
ejpam-4602	134	16	be	be	AUX
ejpam-4602	134	17	a	a	DET
ejpam-4602	134	18	δ	δ	NOUN
ejpam-4602	134	19	-	-	PUNCT
ejpam-4602	134	20	ideal	ideal	NOUN
ejpam-4602	134	21	of	of	ADP
ejpam-4602	134	22	r	r	NOUN
ejpam-4602	134	23	,	,	PUNCT
ejpam-4602	134	24	which	which	PRON
ejpam-4602	134	25	δ	δ	PROPN
ejpam-4602	134	26	̸=	̸=	PROPN
ejpam-4602	134	27	0	0	NUM
ejpam-4602	134	28	is	be	AUX
ejpam-4602	134	29	.	.	PUNCT
ejpam-4602	135	1	if	if	SCONJ
ejpam-4602	135	2	[	[	X
ejpam-4602	135	3	a	a	X
ejpam-4602	135	4	,	,	PUNCT
ejpam-4602	135	5	δ(a	δ(a	PROPN
ejpam-4602	135	6	)	)	PUNCT
ejpam-4602	135	7	]	]	PUNCT
ejpam-4602	135	8	∈	∈	PROPN
ejpam-4602	135	9	z(r	z(r	PROPN
ejpam-4602	135	10	)	)	PUNCT
ejpam-4602	135	11	∀a	∀a	NOUN
ejpam-4602	136	1	∈	∈	PROPN
ejpam-4602	136	2	u	u	NOUN
ejpam-4602	136	3	,	,	PUNCT
ejpam-4602	136	4	(	(	PUNCT
ejpam-4602	136	5	5	5	NUM
ejpam-4602	136	6	)	)	PUNCT
ejpam-4602	136	7	then	then	ADV
ejpam-4602	136	8	[	[	X
ejpam-4602	136	9	a	a	X
ejpam-4602	136	10	,	,	PUNCT
ejpam-4602	136	11	δ(a	δ(a	PROPN
ejpam-4602	136	12	)	)	PUNCT
ejpam-4602	136	13	]	]	PUNCT
ejpam-4602	137	1	=	=	PUNCT
ejpam-4602	137	2	0	0	X
ejpam-4602	137	3	.	.	PUNCT
ejpam-4602	138	1	proof	proof	NOUN
ejpam-4602	138	2	.	.	PUNCT
ejpam-4602	139	1	assume	assume	VERB
ejpam-4602	139	2	that	that	SCONJ
ejpam-4602	139	3	a	a	DET
ejpam-4602	139	4	,	,	PUNCT
ejpam-4602	139	5	b	b	PROPN
ejpam-4602	139	6	∈	∈	PROPN
ejpam-4602	139	7	u	u	NOUN
ejpam-4602	139	8	.	.	PUNCT
ejpam-4602	140	1	now	now	ADV
ejpam-4602	140	2	by	by	ADP
ejpam-4602	140	3	linearlizing	linearlize	VERB
ejpam-4602	140	4	the	the	DET
ejpam-4602	140	5	identity	identity	NOUN
ejpam-4602	140	6	(	(	PUNCT
ejpam-4602	140	7	5	5	NUM
ejpam-4602	140	8	)	)	PUNCT
ejpam-4602	140	9	,	,	PUNCT
ejpam-4602	140	10	we	we	PRON
ejpam-4602	140	11	see	see	VERB
ejpam-4602	140	12	[	[	X
ejpam-4602	140	13	a+	a+	X
ejpam-4602	140	14	b	b	NOUN
ejpam-4602	140	15	,	,	PUNCT
ejpam-4602	140	16	δ(a+	δ(a+	NOUN
ejpam-4602	140	17	b	b	NOUN
ejpam-4602	140	18	)	)	PUNCT
ejpam-4602	140	19	]	]	PUNCT
ejpam-4602	141	1	=	=	PUNCT
ejpam-4602	142	1	[	[	X
ejpam-4602	142	2	a	a	X
ejpam-4602	142	3	,	,	PUNCT
ejpam-4602	142	4	δ(a	δ(a	PROPN
ejpam-4602	142	5	)	)	PUNCT
ejpam-4602	142	6	]	]	PUNCT
ejpam-4602	143	1	+	+	CCONJ
ejpam-4602	143	2	[	[	X
ejpam-4602	143	3	a	a	DET
ejpam-4602	143	4	,	,	PUNCT
ejpam-4602	143	5	δ(b	δ(b	PROPN
ejpam-4602	143	6	)	)	PUNCT
ejpam-4602	143	7	]	]	PUNCT
ejpam-4602	144	1	+	+	CCONJ
ejpam-4602	144	2	[	[	X
ejpam-4602	144	3	b	b	X
ejpam-4602	144	4	,	,	PUNCT
ejpam-4602	144	5	δ(a	δ(a	PROPN
ejpam-4602	144	6	)	)	PUNCT
ejpam-4602	144	7	]	]	PUNCT
ejpam-4602	145	1	+	+	CCONJ
ejpam-4602	145	2	[	[	X
ejpam-4602	145	3	b	b	NOUN
ejpam-4602	145	4	,	,	PUNCT
ejpam-4602	145	5	δ(b	δ(b	PROPN
ejpam-4602	145	6	)	)	PUNCT
ejpam-4602	145	7	]	]	PUNCT
ejpam-4602	145	8	,	,	PUNCT
ejpam-4602	145	9	=	=	PUNCT
ejpam-4602	146	1	[	[	X
ejpam-4602	146	2	a	a	PRON
ejpam-4602	146	3	,	,	PUNCT
ejpam-4602	146	4	δ(b	δ(b	PROPN
ejpam-4602	146	5	)	)	PUNCT
ejpam-4602	146	6	]	]	PUNCT
ejpam-4602	147	1	+	+	CCONJ
ejpam-4602	147	2	[	[	X
ejpam-4602	147	3	b	b	X
ejpam-4602	147	4	,	,	PUNCT
ejpam-4602	147	5	δ(a	δ(a	PROPN
ejpam-4602	147	6	)	)	PUNCT
ejpam-4602	147	7	]	]	PUNCT
ejpam-4602	147	8	,	,	PUNCT
ejpam-4602	147	9	then	then	ADV
ejpam-4602	147	10	,	,	PUNCT
ejpam-4602	147	11	from	from	ADP
ejpam-4602	147	12	(	(	PUNCT
ejpam-4602	147	13	5	5	NUM
ejpam-4602	147	14	)	)	PUNCT
ejpam-4602	147	15	[	[	X
ejpam-4602	147	16	a	a	DET
ejpam-4602	147	17	,	,	PUNCT
ejpam-4602	147	18	δ(b	δ(b	PROPN
ejpam-4602	147	19	)	)	PUNCT
ejpam-4602	147	20	]	]	PUNCT
ejpam-4602	148	1	+	+	CCONJ
ejpam-4602	148	2	[	[	X
ejpam-4602	148	3	b	b	X
ejpam-4602	148	4	,	,	PUNCT
ejpam-4602	148	5	δ(a	δ(a	PROPN
ejpam-4602	148	6	)	)	PUNCT
ejpam-4602	148	7	]	]	PUNCT
ejpam-4602	148	8	∈	∈	PROPN
ejpam-4602	148	9	z(r	z(r	PROPN
ejpam-4602	148	10	)	)	PUNCT
ejpam-4602	148	11	.	.	PUNCT
ejpam-4602	149	1	(	(	PUNCT
ejpam-4602	149	2	6	6	X
ejpam-4602	149	3	)	)	PUNCT
ejpam-4602	149	4	now	now	ADV
ejpam-4602	149	5	replace	replace	VERB
ejpam-4602	149	6	b	b	NOUN
ejpam-4602	149	7	by	by	ADP
ejpam-4602	149	8	a2	a2	PROPN
ejpam-4602	149	9	in	in	ADP
ejpam-4602	149	10	(	(	PUNCT
ejpam-4602	149	11	6	6	NUM
ejpam-4602	149	12	)	)	PUNCT
ejpam-4602	149	13	,	,	PUNCT
ejpam-4602	149	14	we	we	PRON
ejpam-4602	149	15	have	have	VERB
ejpam-4602	149	16	[	[	X
ejpam-4602	149	17	a	a	DET
ejpam-4602	149	18	,	,	PUNCT
ejpam-4602	149	19	δ(a2	δ(a2	NOUN
ejpam-4602	149	20	)	)	PUNCT
ejpam-4602	149	21	]	]	PUNCT
ejpam-4602	150	1	+	+	CCONJ
ejpam-4602	150	2	[	[	X
ejpam-4602	150	3	a2	a2	PROPN
ejpam-4602	150	4	,	,	PUNCT
ejpam-4602	150	5	δ(a	δ(a	PROPN
ejpam-4602	150	6	)	)	PUNCT
ejpam-4602	150	7	]	]	PUNCT
ejpam-4602	151	1	=	=	PUNCT
ejpam-4602	152	1	[	[	X
ejpam-4602	152	2	a	a	PRON
ejpam-4602	152	3	,	,	PUNCT
ejpam-4602	152	4	δ(a)a+	δ(a)a+	PUNCT
ejpam-4602	152	5	aδ(a	aδ(a	PUNCT
ejpam-4602	152	6	)	)	PUNCT
ejpam-4602	152	7	]	]	PUNCT
ejpam-4602	153	1	+	+	CCONJ
ejpam-4602	153	2	[	[	X
ejpam-4602	153	3	a2	a2	PROPN
ejpam-4602	153	4	,	,	PUNCT
ejpam-4602	153	5	δ(a	δ(a	PROPN
ejpam-4602	153	6	)	)	PUNCT
ejpam-4602	153	7	]	]	PUNCT
ejpam-4602	153	8	.	.	PUNCT
ejpam-4602	154	1	i.	i.	PROPN
ejpam-4602	154	2	taha	taha	PROPN
ejpam-4602	154	3	et	et	PROPN
ejpam-4602	154	4	al	al	PROPN
ejpam-4602	154	5	.	.	PUNCT
ejpam-4602	154	6	/	/	SYM
ejpam-4602	154	7	eur	eur	PROPN
ejpam-4602	154	8	.	.	PUNCT
ejpam-4602	155	1	j.	j.	PROPN
ejpam-4602	155	2	pure	pure	PROPN
ejpam-4602	155	3	appl	appl	PROPN
ejpam-4602	155	4	.	.	PROPN
ejpam-4602	155	5	math	math	PROPN
ejpam-4602	155	6	,	,	PUNCT
ejpam-4602	155	7	15	15	NUM
ejpam-4602	155	8	(	(	PUNCT
ejpam-4602	155	9	4	4	NUM
ejpam-4602	155	10	)	)	PUNCT
ejpam-4602	155	11	(	(	PUNCT
ejpam-4602	155	12	2022	2022	NUM
ejpam-4602	155	13	)	)	PUNCT
ejpam-4602	155	14	,	,	PUNCT
ejpam-4602	155	15	2032	2032	NUM
ejpam-4602	155	16	-	-	SYM
ejpam-4602	155	17	2042	2042	NUM
ejpam-4602	155	18	2038	2038	NUM
ejpam-4602	155	19	then	then	ADV
ejpam-4602	155	20	,	,	PUNCT
ejpam-4602	155	21	[	[	X
ejpam-4602	155	22	a	a	DET
ejpam-4602	155	23	,	,	PUNCT
ejpam-4602	155	24	δ(a2	δ(a2	NOUN
ejpam-4602	155	25	)	)	PUNCT
ejpam-4602	155	26	]	]	PUNCT
ejpam-4602	156	1	+	+	CCONJ
ejpam-4602	156	2	[	[	X
ejpam-4602	156	3	a2	a2	PROPN
ejpam-4602	156	4	,	,	PUNCT
ejpam-4602	156	5	δ(a	δ(a	PROPN
ejpam-4602	156	6	)	)	PUNCT
ejpam-4602	156	7	]	]	PUNCT
ejpam-4602	157	1	=	=	SYM
ejpam-4602	157	2	4a[a	4a[a	NUM
ejpam-4602	157	3	,	,	PUNCT
ejpam-4602	157	4	δ(a	δ(a	PROPN
ejpam-4602	157	5	)	)	PUNCT
ejpam-4602	157	6	]	]	PUNCT
ejpam-4602	158	1	∈	∈	PROPN
ejpam-4602	158	2	z(r	z(r	PROPN
ejpam-4602	158	3	)	)	PUNCT
ejpam-4602	158	4	.	.	PUNCT
ejpam-4602	159	1	hence	hence	ADV
ejpam-4602	159	2	,	,	PUNCT
ejpam-4602	159	3	a[a	a[a	PROPN
ejpam-4602	159	4	,	,	PUNCT
ejpam-4602	159	5	δ(a	δ(a	PROPN
ejpam-4602	159	6	)	)	PUNCT
ejpam-4602	159	7	]	]	PUNCT
ejpam-4602	159	8	∈	∈	PROPN
ejpam-4602	159	9	z(r	z(r	PROPN
ejpam-4602	159	10	)	)	PUNCT
ejpam-4602	159	11	.	.	PUNCT
ejpam-4602	160	1	(	(	PUNCT
ejpam-4602	160	2	7	7	NUM
ejpam-4602	160	3	)	)	PUNCT
ejpam-4602	160	4	therefore	therefore	ADV
ejpam-4602	160	5	,	,	PUNCT
ejpam-4602	160	6	[	[	X
ejpam-4602	160	7	a[a	a[a	NOUN
ejpam-4602	160	8	,	,	PUNCT
ejpam-4602	160	9	δ(a	δ(a	PROPN
ejpam-4602	160	10	)	)	PUNCT
ejpam-4602	160	11	]	]	PUNCT
ejpam-4602	160	12	,	,	PUNCT
ejpam-4602	160	13	δ(a	δ(a	PROPN
ejpam-4602	160	14	)	)	PUNCT
ejpam-4602	160	15	]	]	PUNCT
ejpam-4602	161	1	=	=	SYM
ejpam-4602	161	2	0	0	NUM
ejpam-4602	161	3	a[a	a[a	NOUN
ejpam-4602	161	4	,	,	PUNCT
ejpam-4602	161	5	δ(a)]δ(a)−	δ(a)]δ(a)−	NOUN
ejpam-4602	161	6	δ(a)a[a	δ(a)a[a	NOUN
ejpam-4602	161	7	,	,	PUNCT
ejpam-4602	161	8	δ(a	δ(a	PROPN
ejpam-4602	161	9	)	)	PUNCT
ejpam-4602	161	10	]	]	PUNCT
ejpam-4602	162	1	=	=	PUNCT
ejpam-4602	162	2	0	0	PUNCT
ejpam-4602	163	1	[	[	X
ejpam-4602	163	2	a	a	X
ejpam-4602	163	3	,	,	PUNCT
ejpam-4602	163	4	δ(a)](aδ(a)−	δ(a)](aδ(a)−	PUNCT
ejpam-4602	163	5	δ(a)a	δ(a)a	NUM
ejpam-4602	163	6	)	)	PUNCT
ejpam-4602	163	7	=	=	SYM
ejpam-4602	163	8	0	0	PUNCT
ejpam-4602	164	1	[	[	X
ejpam-4602	164	2	a	a	X
ejpam-4602	164	3	,	,	PUNCT
ejpam-4602	164	4	δ(a)]2	δ(a)]2	PROPN
ejpam-4602	164	5	=	=	SYM
ejpam-4602	164	6	0	0	X
ejpam-4602	164	7	.	.	PUNCT
ejpam-4602	165	1	from	from	ADP
ejpam-4602	165	2	(	(	PUNCT
ejpam-4602	165	3	5	5	NUM
ejpam-4602	165	4	)	)	PUNCT
ejpam-4602	165	5	,	,	PUNCT
ejpam-4602	165	6	δ([a	δ([a	PROPN
ejpam-4602	165	7	,	,	PUNCT
ejpam-4602	165	8	δ(a	δ(a	PROPN
ejpam-4602	165	9	)	)	PUNCT
ejpam-4602	165	10	]	]	PUNCT
ejpam-4602	165	11	)	)	PUNCT
ejpam-4602	165	12	∈	∈	PROPN
ejpam-4602	165	13	z(r	z(r	PROPN
ejpam-4602	165	14	)	)	PUNCT
ejpam-4602	165	15	and	and	CCONJ
ejpam-4602	165	16	δ([a	δ([a	PROPN
ejpam-4602	165	17	,	,	PUNCT
ejpam-4602	165	18	δ(a	δ(a	PROPN
ejpam-4602	165	19	)	)	PUNCT
ejpam-4602	165	20	]	]	PUNCT
ejpam-4602	165	21	)	)	PUNCT
ejpam-4602	165	22	=	=	PUNCT
ejpam-4602	166	1	[	[	X
ejpam-4602	166	2	δ(a	δ(a	PROPN
ejpam-4602	166	3	)	)	PUNCT
ejpam-4602	166	4	,	,	PUNCT
ejpam-4602	166	5	δ(a	δ(a	PROPN
ejpam-4602	166	6	)	)	PUNCT
ejpam-4602	166	7	]	]	PUNCT
ejpam-4602	167	1	+	+	CCONJ
ejpam-4602	167	2	[	[	X
ejpam-4602	167	3	a	a	X
ejpam-4602	167	4	,	,	PUNCT
ejpam-4602	167	5	δ2(a	δ2(a	PROPN
ejpam-4602	167	6	)	)	PUNCT
ejpam-4602	167	7	]	]	PUNCT
ejpam-4602	168	1	=	=	PUNCT
ejpam-4602	169	1	[	[	X
ejpam-4602	169	2	a	a	X
ejpam-4602	169	3	,	,	PUNCT
ejpam-4602	169	4	δ2(a	δ2(a	PROPN
ejpam-4602	169	5	)	)	PUNCT
ejpam-4602	169	6	]	]	PUNCT
ejpam-4602	169	7	.	.	PUNCT
ejpam-4602	170	1	we	we	PRON
ejpam-4602	170	2	obtain	obtain	VERB
ejpam-4602	170	3	,	,	PUNCT
ejpam-4602	170	4	δ([a	δ([a	ADV
ejpam-4602	170	5	,	,	PUNCT
ejpam-4602	170	6	δ2(a	δ2(a	PROPN
ejpam-4602	170	7	)	)	PUNCT
ejpam-4602	170	8	]	]	PUNCT
ejpam-4602	170	9	)	)	PUNCT
ejpam-4602	170	10	∈	∈	PROPN
ejpam-4602	170	11	z(r	z(r	PROPN
ejpam-4602	170	12	)	)	PUNCT
ejpam-4602	170	13	.	.	PUNCT
ejpam-4602	171	1	now	now	ADV
ejpam-4602	171	2	δ([a	δ([a	PROPN
ejpam-4602	171	3	,	,	PUNCT
ejpam-4602	171	4	δ2(a	δ2(a	PROPN
ejpam-4602	171	5	)	)	PUNCT
ejpam-4602	171	6	]	]	PUNCT
ejpam-4602	171	7	)	)	PUNCT
ejpam-4602	172	1	=	=	PUNCT
ejpam-4602	173	1	[	[	X
ejpam-4602	173	2	δ(a	δ(a	PROPN
ejpam-4602	173	3	)	)	PUNCT
ejpam-4602	173	4	,	,	PUNCT
ejpam-4602	173	5	δ2(a	δ2(a	PROPN
ejpam-4602	173	6	)	)	PUNCT
ejpam-4602	173	7	]	]	PUNCT
ejpam-4602	174	1	+	+	CCONJ
ejpam-4602	174	2	[	[	X
ejpam-4602	174	3	a	a	X
ejpam-4602	174	4	,	,	PUNCT
ejpam-4602	174	5	δ3(a	δ3(a	PROPN
ejpam-4602	174	6	)	)	PUNCT
ejpam-4602	174	7	]	]	PUNCT
ejpam-4602	175	1	=	=	PUNCT
ejpam-4602	176	1	[	[	X
ejpam-4602	176	2	a	a	X
ejpam-4602	176	3	,	,	PUNCT
ejpam-4602	176	4	δ3(a	δ3(a	PROPN
ejpam-4602	176	5	)	)	PUNCT
ejpam-4602	176	6	]	]	PUNCT
ejpam-4602	176	7	.	.	PUNCT
ejpam-4602	177	1	then	then	ADV
ejpam-4602	177	2	,	,	PUNCT
ejpam-4602	177	3	[	[	X
ejpam-4602	177	4	a	a	X
ejpam-4602	177	5	,	,	PUNCT
ejpam-4602	177	6	δ3(a	δ3(a	PROPN
ejpam-4602	177	7	)	)	PUNCT
ejpam-4602	177	8	]	]	PUNCT
ejpam-4602	177	9	∈	∈	PROPN
ejpam-4602	177	10	z(r	z(r	PROPN
ejpam-4602	177	11	)	)	PUNCT
ejpam-4602	177	12	.	.	PUNCT
ejpam-4602	178	1	(	(	PUNCT
ejpam-4602	178	2	8)	8)	NUM
ejpam-4602	178	3	now	now	ADV
ejpam-4602	178	4	,	,	PUNCT
ejpam-4602	178	5	using	use	VERB
ejpam-4602	178	6	the	the	DET
ejpam-4602	178	7	induction	induction	NOUN
ejpam-4602	178	8	on	on	ADP
ejpam-4602	178	9	a	a	DET
ejpam-4602	178	10	number	number	NOUN
ejpam-4602	178	11	n	n	CCONJ
ejpam-4602	178	12	,	,	PUNCT
ejpam-4602	178	13	so	so	ADV
ejpam-4602	178	14	we	we	PRON
ejpam-4602	178	15	have	have	VERB
ejpam-4602	178	16	[	[	X
ejpam-4602	178	17	a	a	PRON
ejpam-4602	178	18	,	,	PUNCT
ejpam-4602	178	19	δn(a	δn(a	NUM
ejpam-4602	178	20	)	)	PUNCT
ejpam-4602	178	21	]	]	PUNCT
ejpam-4602	179	1	∈	∈	PROPN
ejpam-4602	179	2	z(r	z(r	PROPN
ejpam-4602	179	3	)	)	PUNCT
ejpam-4602	179	4	.	.	PUNCT
ejpam-4602	180	1	(	(	PUNCT
ejpam-4602	180	2	9	9	X
ejpam-4602	180	3	)	)	PUNCT
ejpam-4602	180	4	replacing	replace	VERB
ejpam-4602	180	5	b	b	NOUN
ejpam-4602	180	6	in	in	ADP
ejpam-4602	180	7	(	(	PUNCT
ejpam-4602	180	8	6	6	NUM
ejpam-4602	180	9	)	)	PUNCT
ejpam-4602	180	10	by	by	ADP
ejpam-4602	180	11	aδn(a	aδn(a	NOUN
ejpam-4602	180	12	)	)	PUNCT
ejpam-4602	180	13	,	,	PUNCT
ejpam-4602	180	14	we	we	PRON
ejpam-4602	180	15	have	have	VERB
ejpam-4602	180	16	[	[	X
ejpam-4602	180	17	a	a	DET
ejpam-4602	180	18	,	,	PUNCT
ejpam-4602	180	19	δ(aδn(a	δ(aδn(a	NOUN
ejpam-4602	180	20	)	)	PUNCT
ejpam-4602	180	21	)	)	PUNCT
ejpam-4602	180	22	]	]	PUNCT
ejpam-4602	181	1	+	+	CCONJ
ejpam-4602	181	2	[	[	X
ejpam-4602	181	3	aδn(a	aδn(a	NOUN
ejpam-4602	181	4	)	)	PUNCT
ejpam-4602	181	5	,	,	PUNCT
ejpam-4602	181	6	δ(a	δ(a	PROPN
ejpam-4602	181	7	)	)	PUNCT
ejpam-4602	181	8	]	]	PUNCT
ejpam-4602	182	1	∈	∈	PROPN
ejpam-4602	182	2	z(r	z(r	PROPN
ejpam-4602	182	3	)	)	PUNCT
ejpam-4602	182	4	.	.	PUNCT
ejpam-4602	183	1	i.	i.	PROPN
ejpam-4602	183	2	taha	taha	PROPN
ejpam-4602	183	3	et	et	PROPN
ejpam-4602	183	4	al	al	PROPN
ejpam-4602	183	5	.	.	PUNCT
ejpam-4602	183	6	/	/	SYM
ejpam-4602	183	7	eur	eur	PROPN
ejpam-4602	183	8	.	.	PUNCT
ejpam-4602	184	1	j.	j.	PROPN
ejpam-4602	184	2	pure	pure	PROPN
ejpam-4602	184	3	appl	appl	PROPN
ejpam-4602	184	4	.	.	PROPN
ejpam-4602	184	5	math	math	PROPN
ejpam-4602	184	6	,	,	PUNCT
ejpam-4602	184	7	15	15	NUM
ejpam-4602	184	8	(	(	PUNCT
ejpam-4602	184	9	4	4	NUM
ejpam-4602	184	10	)	)	PUNCT
ejpam-4602	184	11	(	(	PUNCT
ejpam-4602	184	12	2022	2022	NUM
ejpam-4602	184	13	)	)	PUNCT
ejpam-4602	184	14	,	,	PUNCT
ejpam-4602	184	15	2032	2032	NUM
ejpam-4602	184	16	-	-	SYM
ejpam-4602	184	17	2042	2042	NUM
ejpam-4602	184	18	2039	2039	NUM
ejpam-4602	184	19	then	then	ADV
ejpam-4602	184	20	,	,	PUNCT
ejpam-4602	184	21	[	[	X
ejpam-4602	184	22	a	a	DET
ejpam-4602	184	23	,	,	PUNCT
ejpam-4602	184	24	δ(aδn(a	δ(aδn(a	NOUN
ejpam-4602	184	25	)	)	PUNCT
ejpam-4602	184	26	)	)	PUNCT
ejpam-4602	184	27	]	]	PUNCT
ejpam-4602	185	1	+	+	CCONJ
ejpam-4602	185	2	[	[	X
ejpam-4602	185	3	aδn(a	aδn(a	NOUN
ejpam-4602	185	4	)	)	PUNCT
ejpam-4602	185	5	,	,	PUNCT
ejpam-4602	185	6	δ(a	δ(a	PROPN
ejpam-4602	185	7	)	)	PUNCT
ejpam-4602	185	8	]	]	PUNCT
ejpam-4602	186	1	=	=	PUNCT
ejpam-4602	187	1	[	[	X
ejpam-4602	187	2	a	a	DET
ejpam-4602	187	3	,	,	PUNCT
ejpam-4602	187	4	δn+1(a)a+	δn+1(a)a+	NOUN
ejpam-4602	187	5	δn(a)δ(a	δn(a)δ(a	NOUN
ejpam-4602	187	6	)	)	PUNCT
ejpam-4602	187	7	]	]	PUNCT
ejpam-4602	188	1	+	+	CCONJ
ejpam-4602	188	2	[	[	X
ejpam-4602	188	3	aδn(a	aδn(a	NOUN
ejpam-4602	188	4	)	)	PUNCT
ejpam-4602	188	5	,	,	PUNCT
ejpam-4602	188	6	δ(a	δ(a	PROPN
ejpam-4602	188	7	)	)	PUNCT
ejpam-4602	188	8	]	]	PUNCT
ejpam-4602	189	1	=	=	PUNCT
ejpam-4602	190	1	[	[	X
ejpam-4602	190	2	a	a	X
ejpam-4602	190	3	,	,	PUNCT
ejpam-4602	190	4	δn+1(a)a	δn+1(a)a	NOUN
ejpam-4602	190	5	]	]	PUNCT
ejpam-4602	191	1	+	+	CCONJ
ejpam-4602	192	1	[	[	X
ejpam-4602	192	2	a	a	X
ejpam-4602	192	3	,	,	PUNCT
ejpam-4602	192	4	δn(a)δ(a)]−	δn(a)δ(a)]−	PROPN
ejpam-4602	192	5	[	[	X
ejpam-4602	192	6	δ(a	δ(a	PROPN
ejpam-4602	192	7	)	)	PUNCT
ejpam-4602	192	8	,	,	PUNCT
ejpam-4602	192	9	aδn(a	aδn(a	PROPN
ejpam-4602	192	10	)	)	PUNCT
ejpam-4602	192	11	]	]	PUNCT
ejpam-4602	193	1	=	=	PUNCT
ejpam-4602	194	1	[	[	X
ejpam-4602	194	2	a	a	DET
ejpam-4602	194	3	,	,	PUNCT
ejpam-4602	194	4	δn+1(a)]a+	δn+1(a)]a+	ADJ
ejpam-4602	194	5	δn(a)[a	δn(a)[a	ADV
ejpam-4602	194	6	,	,	PUNCT
ejpam-4602	194	7	δ(a	δ(a	PROPN
ejpam-4602	194	8	)	)	PUNCT
ejpam-4602	194	9	]	]	PUNCT
ejpam-4602	195	1	+	+	CCONJ
ejpam-4602	195	2	[	[	X
ejpam-4602	195	3	a	a	DET
ejpam-4602	195	4	,	,	PUNCT
ejpam-4602	195	5	δn(a)]δ(a)−	δn(a)]δ(a)−	NOUN
ejpam-4602	195	6	a[δ(a	a[δ(a	NOUN
ejpam-4602	195	7	)	)	PUNCT
ejpam-4602	195	8	,	,	PUNCT
ejpam-4602	195	9	δn(a)]−	δn(a)]−	PROPN
ejpam-4602	195	10	[	[	X
ejpam-4602	195	11	δ(a	δ(a	PROPN
ejpam-4602	195	12	)	)	PUNCT
ejpam-4602	195	13	,	,	PUNCT
ejpam-4602	195	14	a]δn(a	a]δn(a	PROPN
ejpam-4602	195	15	)	)	PUNCT
ejpam-4602	195	16	=	=	PUNCT
ejpam-4602	196	1	s.	s.	PROPN
ejpam-4602	196	2	then	then	ADV
ejpam-4602	196	3	,	,	PUNCT
ejpam-4602	196	4	0	0	PUNCT
ejpam-4602	197	1	=	=	PUNCT
ejpam-4602	198	1	[	[	X
ejpam-4602	198	2	s	s	X
ejpam-4602	198	3	,	,	PUNCT
ejpam-4602	198	4	δn(a	δn(a	NUM
ejpam-4602	198	5	)	)	PUNCT
ejpam-4602	198	6	]	]	PUNCT
ejpam-4602	199	1	=	=	PUNCT
ejpam-4602	199	2	[	[	X
ejpam-4602	199	3	[	[	X
ejpam-4602	199	4	a	a	PRON
ejpam-4602	199	5	,	,	PUNCT
ejpam-4602	199	6	δn+1(a)]a	δn+1(a)]a	NUM
ejpam-4602	199	7	,	,	PUNCT
ejpam-4602	199	8	δn(a)]+	δn(a)]+	X
ejpam-4602	199	9	(	(	PUNCT
ejpam-4602	199	10	10	10	NUM
ejpam-4602	199	11	)	)	PUNCT
ejpam-4602	200	1	[	[	X
ejpam-4602	200	2	δn(a)[a	δn(a)[a	ADV
ejpam-4602	200	3	,	,	PUNCT
ejpam-4602	200	4	δ(a	δ(a	PROPN
ejpam-4602	200	5	)	)	PUNCT
ejpam-4602	200	6	]	]	PUNCT
ejpam-4602	200	7	,	,	PUNCT
ejpam-4602	200	8	δn(a)]+	δn(a)]+	VERB
ejpam-4602	201	1	[	[	X
ejpam-4602	201	2	[	[	X
ejpam-4602	201	3	a	a	PRON
ejpam-4602	201	4	,	,	PUNCT
ejpam-4602	201	5	δn(a)]δ(a	δn(a)]δ(a	NOUN
ejpam-4602	201	6	)	)	PUNCT
ejpam-4602	201	7	,	,	PUNCT
ejpam-4602	201	8	δn(a)]−	δn(a)]−	PROPN
ejpam-4602	201	9	[	[	X
ejpam-4602	201	10	a[δ(a	a[δ(a	ADP
ejpam-4602	201	11	)	)	PUNCT
ejpam-4602	201	12	,	,	PUNCT
ejpam-4602	201	13	δn(a	δn(a	NUM
ejpam-4602	201	14	)	)	PUNCT
ejpam-4602	201	15	]	]	PUNCT
ejpam-4602	201	16	,	,	PUNCT
ejpam-4602	201	17	δn(a)]−	δn(a)]−	PROPN
ejpam-4602	201	18	[	[	X
ejpam-4602	201	19	[	[	X
ejpam-4602	201	20	δ(a	δ(a	PROPN
ejpam-4602	201	21	)	)	PUNCT
ejpam-4602	201	22	,	,	PUNCT
ejpam-4602	201	23	a]δn(a	a]δn(a	PROPN
ejpam-4602	201	24	)	)	PUNCT
ejpam-4602	201	25	,	,	PUNCT
ejpam-4602	201	26	δn(a	δn(a	NUM
ejpam-4602	201	27	)	)	PUNCT
ejpam-4602	201	28	]	]	PUNCT
ejpam-4602	201	29	.	.	PUNCT
ejpam-4602	202	1	now	now	ADV
ejpam-4602	202	2	substituting	substitute	VERB
ejpam-4602	202	3	instead	instead	ADV
ejpam-4602	202	4	b	b	X
ejpam-4602	202	5	in	in	ADP
ejpam-4602	202	6	(	(	PUNCT
ejpam-4602	202	7	6	6	NUM
ejpam-4602	202	8	)	)	PUNCT
ejpam-4602	202	9	by	by	ADP
ejpam-4602	202	10	a2δn(a	a2δn(a	NOUN
ejpam-4602	202	11	)	)	PUNCT
ejpam-4602	202	12	,	,	PUNCT
ejpam-4602	202	13	then	then	ADV
ejpam-4602	202	14	we	we	PRON
ejpam-4602	202	15	get	get	VERB
ejpam-4602	202	16	[	[	X
ejpam-4602	202	17	a	a	PRON
ejpam-4602	202	18	,	,	PUNCT
ejpam-4602	202	19	δ(a2δn(a	δ(a2δn(a	NOUN
ejpam-4602	202	20	)	)	PUNCT
ejpam-4602	202	21	)	)	PUNCT
ejpam-4602	202	22	]	]	PUNCT
ejpam-4602	203	1	+	+	CCONJ
ejpam-4602	203	2	[	[	X
ejpam-4602	203	3	a2δn(a	a2δn(a	NOUN
ejpam-4602	203	4	)	)	PUNCT
ejpam-4602	203	5	,	,	PUNCT
ejpam-4602	203	6	δ(a	δ(a	PROPN
ejpam-4602	203	7	)	)	PUNCT
ejpam-4602	203	8	]	]	PUNCT
ejpam-4602	204	1	∈	∈	PROPN
ejpam-4602	204	2	z(r	z(r	PROPN
ejpam-4602	204	3	)	)	PUNCT
ejpam-4602	204	4	.	.	PUNCT
ejpam-4602	205	1	then	then	ADV
ejpam-4602	205	2	,	,	PUNCT
ejpam-4602	205	3	[	[	X
ejpam-4602	205	4	a	a	X
ejpam-4602	205	5	,	,	PUNCT
ejpam-4602	205	6	δ(a2δn(a	δ(a2δn(a	NOUN
ejpam-4602	205	7	)	)	PUNCT
ejpam-4602	205	8	)	)	PUNCT
ejpam-4602	205	9	]	]	PUNCT
ejpam-4602	206	1	+	+	CCONJ
ejpam-4602	206	2	[	[	X
ejpam-4602	206	3	a2δn(a	a2δn(a	NOUN
ejpam-4602	206	4	)	)	PUNCT
ejpam-4602	206	5	,	,	PUNCT
ejpam-4602	206	6	δ(a	δ(a	PROPN
ejpam-4602	206	7	)	)	PUNCT
ejpam-4602	206	8	]	]	PUNCT
ejpam-4602	207	1	=	=	PUNCT
ejpam-4602	208	1	[	[	X
ejpam-4602	208	2	a	a	X
ejpam-4602	208	3	,	,	PUNCT
ejpam-4602	208	4	δ(a)aδn(a	δ(a)aδn(a	NOUN
ejpam-4602	208	5	)	)	PUNCT
ejpam-4602	208	6	)	)	PUNCT
ejpam-4602	208	7	]	]	PUNCT
ejpam-4602	209	1	+	+	CCONJ
ejpam-4602	209	2	[	[	X
ejpam-4602	209	3	a	a	X
ejpam-4602	209	4	,	,	PUNCT
ejpam-4602	209	5	aδ(a)δn(a	aδ(a)δn(a	PROPN
ejpam-4602	209	6	)	)	PUNCT
ejpam-4602	209	7	]	]	PUNCT
ejpam-4602	210	1	+	+	CCONJ
ejpam-4602	210	2	[	[	X
ejpam-4602	210	3	a	a	X
ejpam-4602	210	4	,	,	PUNCT
ejpam-4602	210	5	a2δn+1(a)]−	a2δn+1(a)]−	PROPN
ejpam-4602	211	1	[	[	X
ejpam-4602	211	2	δ(a	δ(a	PROPN
ejpam-4602	211	3	)	)	PUNCT
ejpam-4602	211	4	,	,	PUNCT
ejpam-4602	211	5	a2δn(a	a2δn(a	PROPN
ejpam-4602	211	6	)	)	PUNCT
ejpam-4602	211	7	]	]	PUNCT
ejpam-4602	212	1	=	=	SYM
ejpam-4602	212	2	4[a	4[a	NUM
ejpam-4602	212	3	,	,	PUNCT
ejpam-4602	212	4	δ(a)]aδn(a	δ(a)]aδn(a	NOUN
ejpam-4602	212	5	)	)	PUNCT
ejpam-4602	212	6	+	+	CCONJ
ejpam-4602	213	1	[	[	X
ejpam-4602	213	2	a	a	X
ejpam-4602	213	3	,	,	PUNCT
ejpam-4602	213	4	δn(a)]aδ(a	δn(a)]aδ(a	NOUN
ejpam-4602	213	5	)	)	PUNCT
ejpam-4602	214	1	+	+	CCONJ
ejpam-4602	215	1	[	[	X
ejpam-4602	215	2	a	a	DET
ejpam-4602	215	3	,	,	PUNCT
ejpam-4602	215	4	δn(a)]δ(a)a+	δn(a)]δ(a)a+	NOUN
ejpam-4602	215	5	[	[	X
ejpam-4602	215	6	a	a	X
ejpam-4602	215	7	,	,	PUNCT
ejpam-4602	215	8	δn+1(a)]a2	δn+1(a)]a2	NOUN
ejpam-4602	215	9	−	−	PROPN
ejpam-4602	216	1	[	[	X
ejpam-4602	216	2	δ(a	δ(a	PROPN
ejpam-4602	216	3	)	)	PUNCT
ejpam-4602	216	4	,	,	PUNCT
ejpam-4602	216	5	δn(a)]a2	δn(a)]a2	NOUN
ejpam-4602	216	6	=	=	SYM
ejpam-4602	216	7	m.	m.	NOUN
ejpam-4602	216	8	now	now	ADV
ejpam-4602	216	9	,	,	PUNCT
ejpam-4602	216	10	multiply	multiply	VERB
ejpam-4602	216	11	m	m	VERB
ejpam-4602	216	12	by	by	ADP
ejpam-4602	216	13	[	[	X
ejpam-4602	216	14	a	a	PRON
ejpam-4602	216	15	,	,	PUNCT
ejpam-4602	216	16	δn(a	δn(a	NUM
ejpam-4602	216	17	)	)	PUNCT
ejpam-4602	216	18	]	]	PUNCT
ejpam-4602	216	19	and	and	CCONJ
ejpam-4602	216	20	in	in	ADP
ejpam-4602	216	21	view	view	NOUN
ejpam-4602	216	22	of	of	ADP
ejpam-4602	216	23	(	(	PUNCT
ejpam-4602	216	24	10	10	NUM
ejpam-4602	216	25	)	)	PUNCT
ejpam-4602	216	26	,	,	PUNCT
ejpam-4602	216	27	w	w	PROPN
ejpam-4602	216	28	have	have	VERB
ejpam-4602	216	29	[	[	X
ejpam-4602	216	30	a	a	DET
ejpam-4602	216	31	,	,	PUNCT
ejpam-4602	216	32	δn(a)]2aδ(a	δn(a)]2aδ(a	NOUN
ejpam-4602	216	33	)	)	PUNCT
ejpam-4602	217	1	+	+	CCONJ
ejpam-4602	218	1	[	[	X
ejpam-4602	218	2	a	a	X
ejpam-4602	218	3	,	,	PUNCT
ejpam-4602	218	4	δn(a)]2δ(a)a−	δn(a)]2δ(a)a−	NOUN
ejpam-4602	218	5	[	[	X
ejpam-4602	218	6	δ(a	δ(a	PROPN
ejpam-4602	218	7	)	)	PUNCT
ejpam-4602	218	8	,	,	PUNCT
ejpam-4602	218	9	δn(a)][a	δn(a)][a	PROPN
ejpam-4602	218	10	,	,	PUNCT
ejpam-4602	218	11	δn(a)]a2	δn(a)]a2	PROPN
ejpam-4602	218	12	.	.	PUNCT
ejpam-4602	219	1	then	then	ADV
ejpam-4602	219	2	,	,	PUNCT
ejpam-4602	219	3	by	by	AUX
ejpam-4602	219	4	continue	continue	VERB
ejpam-4602	219	5	the	the	DET
ejpam-4602	219	6	processes	process	NOUN
ejpam-4602	219	7	,	,	PUNCT
ejpam-4602	219	8	we	we	PRON
ejpam-4602	219	9	obtain	obtain	VERB
ejpam-4602	219	10	[	[	PUNCT
ejpam-4602	219	11	a	a	DET
ejpam-4602	219	12	,	,	PUNCT
ejpam-4602	219	13	δn(a)]3δ(a	δn(a)]3δ(a	NOUN
ejpam-4602	219	14	)	)	PUNCT
ejpam-4602	219	15	=	=	SYM
ejpam-4602	219	16	0	0	NUM
ejpam-4602	219	17	,	,	PUNCT
ejpam-4602	219	18	and	and	CCONJ
ejpam-4602	219	19	so	so	ADV
ejpam-4602	219	20	[	[	X
ejpam-4602	219	21	a	a	X
ejpam-4602	219	22	,	,	PUNCT
ejpam-4602	219	23	δn+1(a)]4δ(a	δn+1(a)]4δ(a	PROPN
ejpam-4602	219	24	)	)	PUNCT
ejpam-4602	219	25	=	=	SYM
ejpam-4602	219	26	0	0	NUM
ejpam-4602	219	27	,	,	PUNCT
ejpam-4602	219	28	references	reference	NOUN
ejpam-4602	219	29	2040	2040	NUM
ejpam-4602	219	30	then	then	ADV
ejpam-4602	219	31	,	,	PUNCT
ejpam-4602	219	32	[	[	X
ejpam-4602	219	33	a	a	X
ejpam-4602	219	34	,	,	PUNCT
ejpam-4602	219	35	δn+1(a)]4r	δn+1(a)]4r	PROPN
ejpam-4602	219	36	=	=	SYM
ejpam-4602	219	37	0	0	NUM
ejpam-4602	219	38	,	,	PUNCT
ejpam-4602	219	39	this	this	PRON
ejpam-4602	219	40	means	mean	VERB
ejpam-4602	219	41	b	b	NOUN
ejpam-4602	219	42	=	=	SYM
ejpam-4602	219	43	∞∑	∞∑	NUM
ejpam-4602	219	44	n=1	n=1	PROPN
ejpam-4602	219	45	∑	∑	AUX
ejpam-4602	219	46	a∈u	a∈u	VERB
ejpam-4602	220	1	[	[	X
ejpam-4602	220	2	a	a	X
ejpam-4602	220	3	,	,	PUNCT
ejpam-4602	220	4	δn(a)]r	δn(a)]r	SCONJ
ejpam-4602	220	5	is	be	AUX
ejpam-4602	220	6	a	a	DET
ejpam-4602	220	7	sum	sum	NOUN
ejpam-4602	220	8	of	of	ADP
ejpam-4602	220	9	nilpotent	nilpotent	ADJ
ejpam-4602	220	10	ideals	ideal	NOUN
ejpam-4602	220	11	and	and	CCONJ
ejpam-4602	220	12	so	so	ADV
ejpam-4602	220	13	u	u	NOUN
ejpam-4602	220	14	is	be	AUX
ejpam-4602	220	15	a	a	DET
ejpam-4602	220	16	nil	nil	ADJ
ejpam-4602	220	17	ideal	ideal	NOUN
ejpam-4602	220	18	,	,	PUNCT
ejpam-4602	220	19	then	then	ADV
ejpam-4602	220	20	b	b	X
ejpam-4602	220	21	=	=	SYM
ejpam-4602	220	22	0	0	PROPN
ejpam-4602	220	23	.	.	PUNCT
ejpam-4602	221	1	this	this	PRON
ejpam-4602	221	2	means	mean	VERB
ejpam-4602	221	3	[	[	X
ejpam-4602	221	4	a	a	X
ejpam-4602	221	5	,	,	PUNCT
ejpam-4602	221	6	δ(a	δ(a	PROPN
ejpam-4602	221	7	)	)	PUNCT
ejpam-4602	221	8	]	]	PUNCT
ejpam-4602	222	1	=	=	PUNCT
ejpam-4602	222	2	0	0	X
ejpam-4602	222	3	.	.	PUNCT
ejpam-4602	223	1	lemma	lemma	PROPN
ejpam-4602	223	2	11	11	NUM
ejpam-4602	223	3	.	.	PUNCT
ejpam-4602	224	1	let	let	VERB
ejpam-4602	224	2	r	r	NOUN
ejpam-4602	224	3	be	be	AUX
ejpam-4602	224	4	δprime	δprime	NOUN
ejpam-4602	224	5	ring	ring	NOUN
ejpam-4602	224	6	of	of	ADP
ejpam-4602	224	7	charr	charr	NOUN
ejpam-4602	224	8	̸=	̸=	PROPN
ejpam-4602	224	9	2	2	NUM
ejpam-4602	224	10	and	and	CCONJ
ejpam-4602	224	11	let	let	VERB
ejpam-4602	224	12	δ	δ	PRON
ejpam-4602	224	13	be	be	AUX
ejpam-4602	224	14	a	a	DET
ejpam-4602	224	15	reverse	reverse	ADJ
ejpam-4602	224	16	derivation	derivation	NOUN
ejpam-4602	224	17	,	,	PUNCT
ejpam-4602	224	18	such	such	ADJ
ejpam-4602	224	19	that	that	SCONJ
ejpam-4602	224	20	[	[	X
ejpam-4602	224	21	a	a	X
ejpam-4602	224	22	,	,	PUNCT
ejpam-4602	224	23	δ(a	δ(a	PROPN
ejpam-4602	224	24	)	)	PUNCT
ejpam-4602	224	25	]	]	PUNCT
ejpam-4602	225	1	∈	∈	PROPN
ejpam-4602	225	2	z(r),∀a	z(r),∀a	PROPN
ejpam-4602	225	3	∈	∈	PROPN
ejpam-4602	225	4	r.	r.	PROPN
ejpam-4602	225	5	then	then	ADV
ejpam-4602	225	6	r	r	NOUN
ejpam-4602	225	7	is	be	AUX
ejpam-4602	225	8	commutative	commutative	ADJ
ejpam-4602	225	9	.	.	PUNCT
ejpam-4602	226	1	proof	proof	NOUN
ejpam-4602	226	2	.	.	PUNCT
ejpam-4602	227	1	recall	recall	VERB
ejpam-4602	227	2	that	that	PRON
ejpam-4602	227	3	u	u	NOUN
ejpam-4602	228	1	=	=	PUNCT
ejpam-4602	228	2	[	[	X
ejpam-4602	228	3	r	r	X
ejpam-4602	228	4	,	,	PUNCT
ejpam-4602	228	5	r	r	NOUN
ejpam-4602	228	6	]	]	X
ejpam-4602	228	7	is	be	AUX
ejpam-4602	228	8	a	a	DET
ejpam-4602	228	9	lie	lie	NOUN
ejpam-4602	228	10	ideal	ideal	NOUN
ejpam-4602	228	11	of	of	ADP
ejpam-4602	228	12	a	a	DET
ejpam-4602	228	13	prime	prime	ADJ
ejpam-4602	228	14	ring	ring	NOUN
ejpam-4602	228	15	r.	r.	PROPN
ejpam-4602	228	16	moreover	moreover	ADV
ejpam-4602	228	17	,	,	PUNCT
ejpam-4602	228	18	δ([r	δ([r	PROPN
ejpam-4602	228	19	,	,	PUNCT
ejpam-4602	228	20	r	r	NOUN
ejpam-4602	228	21	]	]	PUNCT
ejpam-4602	228	22	)	)	PUNCT
ejpam-4602	229	1	⊆	⊆	NUM
ejpam-4602	229	2	[	[	X
ejpam-4602	229	3	r	r	NOUN
ejpam-4602	229	4	,	,	PUNCT
ejpam-4602	229	5	r	r	NOUN
ejpam-4602	229	6	]	]	PUNCT
ejpam-4602	229	7	.	.	PUNCT
ejpam-4602	230	1	since	since	SCONJ
ejpam-4602	230	2	every	every	DET
ejpam-4602	230	3	ideal	ideal	NOUN
ejpam-4602	230	4	in	in	ADP
ejpam-4602	230	5	δprime	δprime	NOUN
ejpam-4602	230	6	ring	ring	NOUN
ejpam-4602	230	7	is	be	AUX
ejpam-4602	230	8	a	a	DET
ejpam-4602	230	9	δideal	δideal	NOUN
ejpam-4602	230	10	,	,	PUNCT
ejpam-4602	230	11	now	now	ADV
ejpam-4602	230	12	,	,	PUNCT
ejpam-4602	230	13	if	if	SCONJ
ejpam-4602	230	14	u	u	PRON
ejpam-4602	230	15	=	=	PUNCT
ejpam-4602	231	1	[	[	X
ejpam-4602	231	2	r	r	X
ejpam-4602	231	3	,	,	PUNCT
ejpam-4602	231	4	r	r	NOUN
ejpam-4602	231	5	]	]	X
ejpam-4602	231	6	is	be	AUX
ejpam-4602	231	7	commutative	commutative	ADJ
ejpam-4602	231	8	,	,	PUNCT
ejpam-4602	231	9	then	then	ADV
ejpam-4602	231	10	,	,	PUNCT
ejpam-4602	231	11	c(r	c(r	NOUN
ejpam-4602	231	12	)	)	PUNCT
ejpam-4602	231	13	is	be	AUX
ejpam-4602	231	14	a	a	DET
ejpam-4602	231	15	nil	nil	ADJ
ejpam-4602	231	16	ideal	ideal	NOUN
ejpam-4602	231	17	(	(	PUNCT
ejpam-4602	231	18	see	see	VERB
ejpam-4602	231	19	[	[	X
ejpam-4602	231	20	5	5	NUM
ejpam-4602	231	21	]	]	PUNCT
ejpam-4602	231	22	,	,	PUNCT
ejpam-4602	231	23	lemma	lemma	PROPN
ejpam-4602	231	24	1.7	1.7	NUM
ejpam-4602	231	25	)	)	PUNCT
ejpam-4602	231	26	.	.	PUNCT
ejpam-4602	232	1	hence	hence	ADV
ejpam-4602	232	2	c(r	c(r	NOUN
ejpam-4602	232	3	)	)	PUNCT
ejpam-4602	233	1	=	=	SYM
ejpam-4602	233	2	0	0	NUM
ejpam-4602	234	1	and	and	CCONJ
ejpam-4602	234	2	r	r	NOUN
ejpam-4602	234	3	is	be	AUX
ejpam-4602	234	4	commutative	commutative	ADJ
ejpam-4602	234	5	.	.	PUNCT
ejpam-4602	235	1	therefore	therefore	ADV
ejpam-4602	235	2	,	,	PUNCT
ejpam-4602	235	3	by	by	ADP
ejpam-4602	235	4	using	use	VERB
ejpam-4602	235	5	lemma	lemma	PROPN
ejpam-4602	235	6	(	(	PUNCT
ejpam-4602	235	7	4	4	NUM
ejpam-4602	235	8	)	)	PUNCT
ejpam-4602	235	9	,	,	PUNCT
ejpam-4602	235	10	u	u	NOUN
ejpam-4602	235	11	=	=	PUNCT
ejpam-4602	236	1	[	[	X
ejpam-4602	236	2	r	r	X
ejpam-4602	236	3	,	,	PUNCT
ejpam-4602	236	4	r	r	NOUN
ejpam-4602	236	5	]	]	PUNCT
ejpam-4602	236	6	contains	contain	VERB
ejpam-4602	236	7	a	a	DET
ejpam-4602	236	8	δideal	δideal	NOUN
ejpam-4602	236	9	of	of	ADP
ejpam-4602	236	10	r.	r.	PROPN
ejpam-4602	236	11	thus	thus	ADV
ejpam-4602	236	12	,	,	PUNCT
ejpam-4602	236	13	by	by	ADP
ejpam-4602	236	14	(	(	PUNCT
ejpam-4602	236	15	6	6	NUM
ejpam-4602	236	16	)	)	PUNCT
ejpam-4602	236	17	,	,	PUNCT
ejpam-4602	236	18	this	this	PRON
ejpam-4602	236	19	implies	imply	VERB
ejpam-4602	236	20	[	[	X
ejpam-4602	236	21	a	a	X
ejpam-4602	236	22	,	,	PUNCT
ejpam-4602	236	23	δ(a	δ(a	PROPN
ejpam-4602	236	24	)	)	PUNCT
ejpam-4602	236	25	]	]	PUNCT
ejpam-4602	237	1	∈	∈	PROPN
ejpam-4602	237	2	z(r	z(r	PROPN
ejpam-4602	237	3	)	)	PUNCT
ejpam-4602	237	4	,	,	PUNCT
ejpam-4602	237	5	∀a	∀a	NOUN
ejpam-4602	237	6	∈	∈	PROPN
ejpam-4602	237	7	r	r	NOUN
ejpam-4602	237	8	,	,	PUNCT
ejpam-4602	237	9	this	this	PRON
ejpam-4602	237	10	gives	give	VERB
ejpam-4602	237	11	δ(u	δ(u	PROPN
ejpam-4602	237	12	)	)	PUNCT
ejpam-4602	237	13	∈	∈	PROPN
ejpam-4602	237	14	z(r	z(r	PROPN
ejpam-4602	237	15	)	)	PUNCT
ejpam-4602	237	16	,	,	PUNCT
ejpam-4602	237	17	then	then	ADV
ejpam-4602	237	18	for	for	ADP
ejpam-4602	237	19	all	all	DET
ejpam-4602	237	20	a	a	DET
ejpam-4602	237	21	∈	∈	PROPN
ejpam-4602	237	22	u	u	NOUN
ejpam-4602	237	23	,	,	PUNCT
ejpam-4602	237	24	(	(	PUNCT
ejpam-4602	237	25	[	[	X
ejpam-4602	237	26	a	a	X
ejpam-4602	237	27	,	,	PUNCT
ejpam-4602	237	28	δ(a	δ(a	PROPN
ejpam-4602	237	29	)	)	PUNCT
ejpam-4602	237	30	]	]	PUNCT
ejpam-4602	238	1	=	=	PUNCT
ejpam-4602	238	2	0	0	NUM
ejpam-4602	238	3	,	,	PUNCT
ejpam-4602	238	4	and	and	CCONJ
ejpam-4602	238	5	based	base	VERB
ejpam-4602	238	6	on	on	ADP
ejpam-4602	238	7	lemma	lemma	PROPN
ejpam-4602	238	8	(	(	PUNCT
ejpam-4602	238	9	8)	8)	NUM
ejpam-4602	238	10	r	r	NOUN
ejpam-4602	238	11	will	will	AUX
ejpam-4602	238	12	be	be	AUX
ejpam-4602	238	13	commutative	commutative	ADJ
ejpam-4602	238	14	.	.	PUNCT
ejpam-4602	239	1	now	now	ADV
ejpam-4602	239	2	,	,	PUNCT
ejpam-4602	239	3	the	the	DET
ejpam-4602	239	4	proof	proof	NOUN
ejpam-4602	239	5	of	of	ADP
ejpam-4602	239	6	the	the	DET
ejpam-4602	239	7	next	next	ADJ
ejpam-4602	239	8	and	and	CCONJ
ejpam-4602	239	9	main	main	ADJ
ejpam-4602	239	10	theorem	theorem	NOUN
ejpam-4602	239	11	is	be	AUX
ejpam-4602	239	12	just	just	ADV
ejpam-4602	239	13	a	a	DET
ejpam-4602	239	14	generalization	generalization	NOUN
ejpam-4602	239	15	of	of	ADP
ejpam-4602	239	16	the	the	DET
ejpam-4602	239	17	lemmas	lemmas	ADJ
ejpam-4602	239	18	(	(	PUNCT
ejpam-4602	239	19	2	2	NUM
ejpam-4602	239	20	)	)	PUNCT
ejpam-4602	239	21	and	and	CCONJ
ejpam-4602	239	22	(	(	PUNCT
ejpam-4602	239	23	3	3	X
ejpam-4602	239	24	)	)	PUNCT
ejpam-4602	239	25	which	which	PRON
ejpam-4602	239	26	is	be	AUX
ejpam-4602	239	27	represented	represent	VERB
ejpam-4602	239	28	as	as	ADP
ejpam-4602	239	29	the	the	DET
ejpam-4602	239	30	following	following	NOUN
ejpam-4602	239	31	:	:	PUNCT
ejpam-4602	239	32	theorem	theorem	NOUN
ejpam-4602	239	33	1	1	X
ejpam-4602	239	34	.	.	PUNCT
ejpam-4602	240	1	let	let	VERB
ejpam-4602	240	2	r	r	NOUN
ejpam-4602	240	3	be	be	AUX
ejpam-4602	240	4	δprime	δprime	NOUN
ejpam-4602	240	5	ring	ring	NOUN
ejpam-4602	240	6	of	of	ADP
ejpam-4602	240	7	charr	charr	NOUN
ejpam-4602	240	8	̸=	̸=	PROPN
ejpam-4602	240	9	2	2	NUM
ejpam-4602	240	10	and	and	CCONJ
ejpam-4602	240	11	u	u	PROPN
ejpam-4602	240	12	̸=	̸=	PROPN
ejpam-4602	240	13	{	{	PUNCT
ejpam-4602	240	14	0	0	NUM
ejpam-4602	240	15	}	}	PUNCT
ejpam-4602	240	16	be	be	AUX
ejpam-4602	240	17	a	a	DET
ejpam-4602	240	18	δideal	δideal	NOUN
ejpam-4602	240	19	,	,	PUNCT
ejpam-4602	240	20	which	which	PRON
ejpam-4602	240	21	δ	δ	PROPN
ejpam-4602	240	22	̸=	̸=	PROPN
ejpam-4602	240	23	0	0	NUM
ejpam-4602	240	24	is	be	AUX
ejpam-4602	240	25	a	a	DET
ejpam-4602	240	26	reverse	reverse	ADJ
ejpam-4602	240	27	derivation	derivation	NOUN
ejpam-4602	240	28	.	.	PUNCT
ejpam-4602	241	1	if	if	SCONJ
ejpam-4602	241	2	[	[	X
ejpam-4602	241	3	u	u	NOUN
ejpam-4602	241	4	,	,	PUNCT
ejpam-4602	241	5	δ(u	δ(u	PROPN
ejpam-4602	241	6	)	)	PUNCT
ejpam-4602	241	7	]	]	PUNCT
ejpam-4602	242	1	∈	∈	PROPN
ejpam-4602	242	2	z(r),∀u	z(r),∀u	PROPN
ejpam-4602	242	3	∈	∈	PROPN
ejpam-4602	242	4	u.	u.	NOUN
ejpam-4602	242	5	then	then	ADV
ejpam-4602	242	6	r	r	NOUN
ejpam-4602	242	7	is	be	AUX
ejpam-4602	242	8	commutative	commutative	ADJ
ejpam-4602	242	9	.	.	PUNCT
ejpam-4602	243	1	proof	proof	NOUN
ejpam-4602	243	2	.	.	PUNCT
ejpam-4602	244	1	from	from	ADP
ejpam-4602	244	2	[	[	X
ejpam-4602	244	3	u	u	NOUN
ejpam-4602	244	4	,	,	PUNCT
ejpam-4602	244	5	δ(u	δ(u	PROPN
ejpam-4602	244	6	)	)	PUNCT
ejpam-4602	244	7	]	]	PUNCT
ejpam-4602	244	8	∈	∈	PROPN
ejpam-4602	244	9	z(r	z(r	PROPN
ejpam-4602	244	10	)	)	PUNCT
ejpam-4602	244	11	,	,	PUNCT
ejpam-4602	244	12	then	then	ADV
ejpam-4602	244	13	,	,	PUNCT
ejpam-4602	244	14	by	by	ADP
ejpam-4602	244	15	lemma	lemma	PROPN
ejpam-4602	244	16	(	(	PUNCT
ejpam-4602	244	17	6	6	NUM
ejpam-4602	244	18	)	)	PUNCT
ejpam-4602	244	19	,	,	PUNCT
ejpam-4602	244	20	we	we	PRON
ejpam-4602	244	21	have	have	VERB
ejpam-4602	244	22	[	[	X
ejpam-4602	244	23	u	u	NOUN
ejpam-4602	244	24	,	,	PUNCT
ejpam-4602	244	25	δ(u	δ(u	PROPN
ejpam-4602	244	26	)	)	PUNCT
ejpam-4602	244	27	]	]	PUNCT
ejpam-4602	245	1	=	=	PUNCT
ejpam-4602	245	2	0	0	X
ejpam-4602	245	3	.	.	PUNCT
ejpam-4602	245	4	then	then	ADV
ejpam-4602	245	5	using	use	VERB
ejpam-4602	245	6	lemma	lemma	PROPN
ejpam-4602	245	7	(	(	PUNCT
ejpam-4602	245	8	8)	8)	NUM
ejpam-4602	245	9	and	and	CCONJ
ejpam-4602	245	10	since	since	SCONJ
ejpam-4602	245	11	r	r	NOUN
ejpam-4602	245	12	is	be	AUX
ejpam-4602	245	13	δprime	δprime	NOUN
ejpam-4602	245	14	ring	ring	NOUN
ejpam-4602	245	15	,	,	PUNCT
ejpam-4602	245	16	which	which	PRON
ejpam-4602	245	17	[	[	X
ejpam-4602	245	18	u	u	NOUN
ejpam-4602	245	19	,	,	PUNCT
ejpam-4602	245	20	δ(u	δ(u	PROPN
ejpam-4602	245	21	)	)	PUNCT
ejpam-4602	245	22	]	]	PUNCT
ejpam-4602	246	1	=	=	PUNCT
ejpam-4602	246	2	0	0	X
ejpam-4602	246	3	.	.	PUNCT
ejpam-4602	247	1	then	then	ADV
ejpam-4602	247	2	r	r	NOUN
ejpam-4602	247	3	is	be	AUX
ejpam-4602	247	4	commutative	commutative	ADJ
ejpam-4602	247	5	.	.	PUNCT
ejpam-4602	248	1	references	reference	NOUN
ejpam-4602	248	2	[	[	X
ejpam-4602	248	3	1	1	NUM
ejpam-4602	248	4	]	]	X
ejpam-4602	248	5	ahmad	ahmad	PROPN
ejpam-4602	248	6	al	al	PROPN
ejpam-4602	248	7	khalaf	khalaf	PROPN
ejpam-4602	248	8	,	,	PUNCT
ejpam-4602	248	9	iman	iman	PROPN
ejpam-4602	248	10	taha	taha	PROPN
ejpam-4602	248	11	,	,	PUNCT
ejpam-4602	248	12	and	and	CCONJ
ejpam-4602	248	13	orest	or	ADJ
ejpam-4602	248	14	d	d	NOUN
ejpam-4602	248	15	artemovych	artemovych	NOUN
ejpam-4602	248	16	.	.	PUNCT
ejpam-4602	249	1	commutators	commutator	NOUN
ejpam-4602	249	2	in	in	ADP
ejpam-4602	249	3	semiprime	semiprime	NOUN
ejpam-4602	249	4	gamma	gamma	PROPN
ejpam-4602	249	5	rings	ring	NOUN
ejpam-4602	249	6	.	.	PUNCT
ejpam-4602	250	1	asian	asian	ADJ
ejpam-4602	250	2	-	-	PUNCT
ejpam-4602	250	3	european	european	ADJ
ejpam-4602	250	4	journal	journal	NOUN
ejpam-4602	250	5	of	of	ADP
ejpam-4602	250	6	mathematics	mathematic	NOUN
ejpam-4602	250	7	,	,	PUNCT
ejpam-4602	250	8	13(04):2050078	13(04):2050078	NUM
ejpam-4602	250	9	,	,	PUNCT
ejpam-4602	250	10	2020	2020	NUM
ejpam-4602	250	11	.	.	PUNCT
ejpam-4602	251	1	[	[	X
ejpam-4602	251	2	2	2	NUM
ejpam-4602	251	3	]	]	PUNCT
ejpam-4602	251	4	a	a	DET
ejpam-4602	251	5	alkhalaf	alkhalaf	PROPN
ejpam-4602	251	6	,	,	PUNCT
ejpam-4602	251	7	o	o	NOUN
ejpam-4602	251	8	artemovych	artemovych	NOUN
ejpam-4602	251	9	,	,	PUNCT
ejpam-4602	251	10	and	and	CCONJ
ejpam-4602	251	11	i	i	PRON
ejpam-4602	251	12	taha	taha	PROPN
ejpam-4602	251	13	.	.	PUNCT
ejpam-4602	252	1	derivations	derivation	NOUN
ejpam-4602	252	2	in	in	ADP
ejpam-4602	252	3	differentially	differentially	ADV
ejpam-4602	252	4	prime	prime	ADJ
ejpam-4602	252	5	rings	ring	NOUN
ejpam-4602	252	6	.	.	PUNCT
ejpam-4602	253	1	journal	journal	PROPN
ejpam-4602	253	2	of	of	ADP
ejpam-4602	253	3	algebra	algebra	PROPN
ejpam-4602	253	4	and	and	CCONJ
ejpam-4602	253	5	its	its	PRON
ejpam-4602	253	6	applications	application	NOUN
ejpam-4602	253	7	,	,	PUNCT
ejpam-4602	253	8	17(07):1850129	17(07):1850129	NUM
ejpam-4602	253	9	,	,	PUNCT
ejpam-4602	253	10	2018	2018	NUM
ejpam-4602	253	11	.	.	PUNCT
ejpam-4602	254	1	[	[	X
ejpam-4602	254	2	3	3	X
ejpam-4602	254	3	]	]	PUNCT
ejpam-4602	254	4	a	a	DET
ejpam-4602	254	5	alkhalaf	alkhalaf	PROPN
ejpam-4602	254	6	,	,	PUNCT
ejpam-4602	254	7	o	o	PROPN
ejpam-4602	254	8	artemovych	artemovych	NOUN
ejpam-4602	254	9	,	,	PUNCT
ejpam-4602	254	10	i	i	PRON
ejpam-4602	254	11	taha	taha	PROPN
ejpam-4602	254	12	,	,	PUNCT
ejpam-4602	254	13	and	and	CCONJ
ejpam-4602	254	14	a	a	DET
ejpam-4602	254	15	aljouiiee	aljouiiee	NOUN
ejpam-4602	254	16	.	.	PUNCT
ejpam-4602	255	1	derivations	derivation	NOUN
ejpam-4602	255	2	of	of	ADP
ejpam-4602	255	3	differentially	differentially	ADV
ejpam-4602	255	4	semiprime	semiprime	NOUN
ejpam-4602	255	5	rings	ring	NOUN
ejpam-4602	255	6	.	.	PUNCT
ejpam-4602	256	1	asian	asian	ADJ
ejpam-4602	256	2	-	-	PUNCT
ejpam-4602	256	3	european	european	ADJ
ejpam-4602	256	4	journal	journal	NOUN
ejpam-4602	256	5	of	of	ADP
ejpam-4602	256	6	mathematics	mathematic	NOUN
ejpam-4602	256	7	,	,	PUNCT
ejpam-4602	256	8	12(05):1950079	12(05):1950079	NUM
ejpam-4602	256	9	,	,	PUNCT
ejpam-4602	256	10	2019	2019	NUM
ejpam-4602	256	11	.	.	PUNCT
ejpam-4602	257	1	[	[	X
ejpam-4602	257	2	4	4	NUM
ejpam-4602	257	3	]	]	X
ejpam-4602	257	4	o	o	X
ejpam-4602	257	5	artemovych	artemovych	NOUN
ejpam-4602	257	6	and	and	CCONJ
ejpam-4602	257	7	m	m	PROPN
ejpam-4602	257	8	lukashenko	lukashenko	PROPN
ejpam-4602	257	9	.	.	PUNCT
ejpam-4602	258	1	lie	lie	NOUN
ejpam-4602	258	2	and	and	CCONJ
ejpam-4602	258	3	jordan	jordan	PROPN
ejpam-4602	258	4	structures	structure	NOUN
ejpam-4602	258	5	of	of	ADP
ejpam-4602	258	6	differentially	differentially	ADV
ejpam-4602	258	7	semiprime	semiprime	NOUN
ejpam-4602	258	8	rings	ring	NOUN
ejpam-4602	258	9	.	.	PUNCT
ejpam-4602	259	1	algebra	algebra	NOUN
ejpam-4602	259	2	and	and	CCONJ
ejpam-4602	259	3	discrete	discrete	ADJ
ejpam-4602	259	4	mathematics	mathematic	NOUN
ejpam-4602	259	5	,	,	PUNCT
ejpam-4602	259	6	20(1	20(1	NUM
ejpam-4602	259	7	)	)	PUNCT
ejpam-4602	259	8	,	,	PUNCT
ejpam-4602	259	9	2015	2015	NUM
ejpam-4602	259	10	.	.	PUNCT
ejpam-4602	260	1	references	reference	NOUN
ejpam-4602	260	2	2041	2041	NUM
ejpam-4602	261	1	[	[	X
ejpam-4602	261	2	5	5	NUM
ejpam-4602	261	3	]	]	PUNCT
ejpam-4602	261	4	h	h	NOUN
ejpam-4602	261	5	bell	bell	NOUN
ejpam-4602	261	6	and	and	CCONJ
ejpam-4602	261	7	a	a	DET
ejpam-4602	261	8	klein	klein	PROPN
ejpam-4602	261	9	.	.	PUNCT
ejpam-4602	262	1	combinatorial	combinatorial	PROPN
ejpam-4602	262	2	commutativity	commutativity	NOUN
ejpam-4602	262	3	and	and	CCONJ
ejpam-4602	262	4	finiteness	finiteness	NOUN
ejpam-4602	262	5	conditions	condition	NOUN
ejpam-4602	262	6	for	for	ADP
ejpam-4602	262	7	rings	ring	NOUN
ejpam-4602	262	8	.	.	PUNCT
ejpam-4602	263	1	communications	communication	NOUN
ejpam-4602	263	2	in	in	ADP
ejpam-4602	263	3	algebra	algebra	NOUN
ejpam-4602	263	4	,	,	PUNCT
ejpam-4602	263	5	29(7):2935–2943	29(7):2935–2943	PROPN
ejpam-4602	263	6	,	,	PUNCT
ejpam-4602	263	7	2001	2001	NUM
ejpam-4602	263	8	.	.	PUNCT
ejpam-4602	264	1	[	[	X
ejpam-4602	264	2	6	6	NUM
ejpam-4602	264	3	]	]	PUNCT
ejpam-4602	264	4	m	m	VERB
ejpam-4602	264	5	brešar	brešar	ADJ
ejpam-4602	264	6	.	.	PUNCT
ejpam-4602	265	1	on	on	ADP
ejpam-4602	265	2	a	a	DET
ejpam-4602	265	3	generalization	generalization	NOUN
ejpam-4602	265	4	of	of	ADP
ejpam-4602	265	5	the	the	DET
ejpam-4602	265	6	notion	notion	NOUN
ejpam-4602	265	7	of	of	ADP
ejpam-4602	265	8	centralizing	centralize	VERB
ejpam-4602	265	9	mappings	mapping	NOUN
ejpam-4602	265	10	.	.	PUNCT
ejpam-4602	266	1	proceedings	proceeding	NOUN
ejpam-4602	266	2	of	of	ADP
ejpam-4602	266	3	the	the	DET
ejpam-4602	266	4	american	american	PROPN
ejpam-4602	266	5	mathematical	mathematical	PROPN
ejpam-4602	266	6	society	society	NOUN
ejpam-4602	266	7	,	,	PUNCT
ejpam-4602	266	8	114(3):641–649	114(3):641–649	NUM
ejpam-4602	266	9	,	,	PUNCT
ejpam-4602	266	10	1992	1992	NUM
ejpam-4602	266	11	.	.	PUNCT
ejpam-4602	267	1	[	[	X
ejpam-4602	267	2	7	7	X
ejpam-4602	267	3	]	]	SYM
ejpam-4602	267	4	m	m	VERB
ejpam-4602	267	5	brešar	brešar	ADJ
ejpam-4602	267	6	.	.	PUNCT
ejpam-4602	268	1	centralizing	centralize	VERB
ejpam-4602	268	2	mappings	mapping	NOUN
ejpam-4602	268	3	and	and	CCONJ
ejpam-4602	268	4	derivations	derivation	NOUN
ejpam-4602	268	5	in	in	ADP
ejpam-4602	268	6	prime	prime	ADJ
ejpam-4602	268	7	rings	ring	NOUN
ejpam-4602	268	8	.	.	PUNCT
ejpam-4602	269	1	j.	j.	PROPN
ejpam-4602	269	2	algebra	algebra	PROPN
ejpam-4602	269	3	,	,	PUNCT
ejpam-4602	269	4	156(2):385–394	156(2):385–394	NUM
ejpam-4602	269	5	,	,	PUNCT
ejpam-4602	269	6	1993	1993	NUM
ejpam-4602	269	7	.	.	PUNCT
ejpam-4602	270	1	[	[	X
ejpam-4602	270	2	8	8	NUM
ejpam-4602	270	3	]	]	X
ejpam-4602	270	4	matej	matej	PROPN
ejpam-4602	270	5	brešar	brešar	PROPN
ejpam-4602	270	6	and	and	CCONJ
ejpam-4602	270	7	joso	joso	NOUN
ejpam-4602	270	8	vukman	vukman	ADJ
ejpam-4602	270	9	.	.	PUNCT
ejpam-4602	271	1	on	on	ADP
ejpam-4602	271	2	some	some	DET
ejpam-4602	271	3	additive	additive	ADJ
ejpam-4602	271	4	mappings	mapping	NOUN
ejpam-4602	271	5	in	in	ADP
ejpam-4602	271	6	rings	ring	NOUN
ejpam-4602	271	7	with	with	ADP
ejpam-4602	271	8	involution	involution	NOUN
ejpam-4602	271	9	.	.	PUNCT
ejpam-4602	272	1	aequationes	aequatione	NOUN
ejpam-4602	272	2	mathematicae	mathematicae	PROPN
ejpam-4602	272	3	,	,	PUNCT
ejpam-4602	272	4	38(2):178–185	38(2):178–185	NUM
ejpam-4602	272	5	,	,	PUNCT
ejpam-4602	272	6	1989	1989	NUM
ejpam-4602	272	7	.	.	PUNCT
ejpam-4602	273	1	[	[	X
ejpam-4602	273	2	9	9	NUM
ejpam-4602	273	3	]	]	X
ejpam-4602	273	4	mikhail	mikhail	NOUN
ejpam-4602	273	5	a	a	DET
ejpam-4602	273	6	chebotar	chebotar	ADJ
ejpam-4602	273	7	and	and	CCONJ
ejpam-4602	273	8	pjek	pjek	NOUN
ejpam-4602	273	9	-	-	PUNCT
ejpam-4602	273	10	hwee	hwee	PROPN
ejpam-4602	273	11	lee	lee	PROPN
ejpam-4602	273	12	.	.	PROPN
ejpam-4602	274	1	prime	prime	PROPN
ejpam-4602	274	2	lie	lie	NOUN
ejpam-4602	274	3	rings	ring	NOUN
ejpam-4602	274	4	of	of	ADP
ejpam-4602	274	5	derivations	derivation	NOUN
ejpam-4602	274	6	of	of	ADP
ejpam-4602	274	7	commutative	commutative	ADJ
ejpam-4602	274	8	rings	ring	NOUN
ejpam-4602	274	9	.	.	PUNCT
ejpam-4602	275	1	communications	communication	NOUN
ejpam-4602	275	2	in	in	ADP
ejpam-4602	275	3	algebra	algebra	PROPN
ejpam-4602	275	4	®	®	NOUN
ejpam-4602	275	5	,	,	PUNCT
ejpam-4602	275	6	34(12):4339–4344	34(12):4339–4344	NUM
ejpam-4602	275	7	,	,	PUNCT
ejpam-4602	275	8	2006	2006	NUM
ejpam-4602	275	9	.	.	PUNCT
ejpam-4602	276	1	[	[	X
ejpam-4602	276	2	10	10	NUM
ejpam-4602	276	3	]	]	X
ejpam-4602	276	4	i	i	PRON
ejpam-4602	276	5	herstein	herstein	NOUN
ejpam-4602	276	6	.	.	PUNCT
ejpam-4602	277	1	on	on	ADP
ejpam-4602	277	2	the	the	DET
ejpam-4602	277	3	lie	lie	NOUN
ejpam-4602	277	4	and	and	CCONJ
ejpam-4602	277	5	jordan	jordan	PROPN
ejpam-4602	277	6	rings	ring	NOUN
ejpam-4602	277	7	of	of	ADP
ejpam-4602	277	8	a	a	DET
ejpam-4602	277	9	simple	simple	ADJ
ejpam-4602	277	10	associative	associative	ADJ
ejpam-4602	277	11	ring	ring	NOUN
ejpam-4602	277	12	.	.	PUNCT
ejpam-4602	278	1	american	american	PROPN
ejpam-4602	278	2	journal	journal	PROPN
ejpam-4602	278	3	of	of	ADP
ejpam-4602	278	4	mathematics	mathematic	NOUN
ejpam-4602	278	5	,	,	PUNCT
ejpam-4602	278	6	77(2):279–285	77(2):279–285	PROPN
ejpam-4602	278	7	,	,	PUNCT
ejpam-4602	278	8	1955	1955	NUM
ejpam-4602	278	9	.	.	PUNCT
ejpam-4602	279	1	[	[	X
ejpam-4602	279	2	11	11	NUM
ejpam-4602	279	3	]	]	X
ejpam-4602	279	4	i	i	PRON
ejpam-4602	279	5	herstein	herstein	NOUN
ejpam-4602	279	6	.	.	PUNCT
ejpam-4602	280	1	topics	topic	NOUN
ejpam-4602	280	2	in	in	ADP
ejpam-4602	280	3	ring	ring	NOUN
ejpam-4602	280	4	theory	theory	NOUN
ejpam-4602	280	5	.	.	PUNCT
ejpam-4602	281	1	e	e	PROPN
ejpam-4602	281	2	university	university	PROPN
ejpam-4602	281	3	of	of	ADP
ejpam-4602	281	4	chicago	chicago	PROPN
ejpam-4602	281	5	press	press	NOUN
ejpam-4602	281	6	.	.	PUNCT
ejpam-4602	282	1	chicago	chicago	PROPN
ejpam-4602	282	2	,	,	PUNCT
ejpam-4602	282	3	il	il	PROPN
ejpam-4602	282	4	,	,	PUNCT
ejpam-4602	282	5	1965	1965	NUM
ejpam-4602	282	6	.	.	PUNCT
ejpam-4602	283	1	[	[	X
ejpam-4602	283	2	12	12	NUM
ejpam-4602	283	3	]	]	X
ejpam-4602	283	4	i	i	PROPN
ejpam-4602	283	5	herstein	herstein	NOUN
ejpam-4602	283	6	.	.	PUNCT
ejpam-4602	284	1	noncommutative	noncommutative	ADJ
ejpam-4602	284	2	rings	ring	NOUN
ejpam-4602	284	3	,	,	PUNCT
ejpam-4602	284	4	volume	volume	NOUN
ejpam-4602	284	5	15	15	NUM
ejpam-4602	284	6	.	.	PUNCT
ejpam-4602	285	1	american	american	PROPN
ejpam-4602	285	2	mathematical	mathematical	PROPN
ejpam-4602	285	3	soc	soc	PROPN
ejpam-4602	285	4	.	.	PUNCT
ejpam-4602	285	5	,	,	PUNCT
ejpam-4602	285	6	1994	1994	NUM
ejpam-4602	285	7	.	.	PUNCT
ejpam-4602	286	1	[	[	X
ejpam-4602	286	2	13	13	NUM
ejpam-4602	286	3	]	]	X
ejpam-4602	286	4	y	y	PROPN
ejpam-4602	286	5	hirano	hirano	PROPN
ejpam-4602	286	6	,	,	PUNCT
ejpam-4602	286	7	a	a	DET
ejpam-4602	286	8	kaya	kaya	PROPN
ejpam-4602	286	9	,	,	PUNCT
ejpam-4602	286	10	and	and	CCONJ
ejpam-4602	286	11	h	h	PROPN
ejpam-4602	286	12	tominaga	tominaga	NOUN
ejpam-4602	286	13	.	.	PUNCT
ejpam-4602	287	1	on	on	ADP
ejpam-4602	287	2	a	a	DET
ejpam-4602	287	3	theorem	theorem	NOUN
ejpam-4602	287	4	of	of	ADP
ejpam-4602	287	5	mayne	mayne	NOUN
ejpam-4602	287	6	.	.	PUNCT
ejpam-4602	288	1	mathematical	mathematical	ADJ
ejpam-4602	288	2	journal	journal	PROPN
ejpam-4602	288	3	of	of	ADP
ejpam-4602	288	4	okayama	okayama	PROPN
ejpam-4602	288	5	university	university	PROPN
ejpam-4602	288	6	,	,	PUNCT
ejpam-4602	288	7	25(2):125–132	25(2):125–132	PROPN
ejpam-4602	288	8	,	,	PUNCT
ejpam-4602	288	9	1983	1983	NUM
ejpam-4602	288	10	.	.	PUNCT
ejpam-4602	289	1	[	[	X
ejpam-4602	289	2	14	14	NUM
ejpam-4602	289	3	]	]	X
ejpam-4602	289	4	y	y	PROPN
ejpam-4602	289	5	hirano	hirano	PROPN
ejpam-4602	289	6	,	,	PUNCT
ejpam-4602	289	7	h	h	NOUN
ejpam-4602	289	8	tominaga	tominaga	NOUN
ejpam-4602	289	9	,	,	PUNCT
ejpam-4602	289	10	and	and	CCONJ
ejpam-4602	289	11	a	a	DET
ejpam-4602	289	12	trzepizur	trzepizur	NOUN
ejpam-4602	289	13	.	.	PUNCT
ejpam-4602	290	1	on	on	ADP
ejpam-4602	290	2	a	a	DET
ejpam-4602	290	3	theorem	theorem	NOUN
ejpam-4602	290	4	of	of	ADP
ejpam-4602	290	5	posner	posner	NOUN
ejpam-4602	290	6	.	.	PUNCT
ejpam-4602	291	1	mathematical	mathematical	ADJ
ejpam-4602	291	2	journal	journal	PROPN
ejpam-4602	291	3	of	of	ADP
ejpam-4602	291	4	okayama	okayama	PROPN
ejpam-4602	291	5	university	university	PROPN
ejpam-4602	291	6	,	,	PUNCT
ejpam-4602	291	7	27(1):19–23	27(1):19–23	NUM
ejpam-4602	291	8	,	,	PUNCT
ejpam-4602	291	9	1985	1985	NUM
ejpam-4602	291	10	.	.	PUNCT
ejpam-4602	292	1	[	[	X
ejpam-4602	292	2	15	15	NUM
ejpam-4602	292	3	]	]	X
ejpam-4602	292	4	m	m	VERB
ejpam-4602	292	5	hongan	hongan	ADJ
ejpam-4602	292	6	and	and	CCONJ
ejpam-4602	292	7	a	a	DET
ejpam-4602	292	8	trzepizur	trzepizur	NOUN
ejpam-4602	292	9	.	.	PUNCT
ejpam-4602	293	1	on	on	ADP
ejpam-4602	293	2	generalization	generalization	NOUN
ejpam-4602	293	3	of	of	ADP
ejpam-4602	293	4	a	a	DET
ejpam-4602	293	5	theorem	theorem	NOUN
ejpam-4602	293	6	of	of	ADP
ejpam-4602	293	7	posner	posner	NOUN
ejpam-4602	293	8	.	.	PUNCT
ejpam-4602	294	1	mathematical	mathematical	ADJ
ejpam-4602	294	2	journal	journal	PROPN
ejpam-4602	294	3	of	of	ADP
ejpam-4602	294	4	okayama	okayama	PROPN
ejpam-4602	294	5	university	university	PROPN
ejpam-4602	294	6	,	,	PUNCT
ejpam-4602	294	7	27(1):19–23	27(1):19–23	NUM
ejpam-4602	294	8	,	,	PUNCT
ejpam-4602	294	9	1985	1985	NUM
ejpam-4602	294	10	.	.	PUNCT
ejpam-4602	295	1	[	[	X
ejpam-4602	295	2	16	16	NUM
ejpam-4602	295	3	]	]	PUNCT
ejpam-4602	295	4	cr	cr	PROPN
ejpam-4602	295	5	jordan	jordan	PROPN
ejpam-4602	295	6	and	and	CCONJ
ejpam-4602	295	7	da	da	PROPN
ejpam-4602	295	8	jordan	jordan	PROPN
ejpam-4602	295	9	.	.	PUNCT
ejpam-4602	296	1	lie	lie	PROPN
ejpam-4602	296	2	rings	ring	NOUN
ejpam-4602	296	3	of	of	ADP
ejpam-4602	296	4	derivations	derivation	NOUN
ejpam-4602	296	5	of	of	ADP
ejpam-4602	296	6	associative	associative	ADJ
ejpam-4602	296	7	rings	ring	NOUN
ejpam-4602	296	8	.	.	PUNCT
ejpam-4602	297	1	journal	journal	PROPN
ejpam-4602	297	2	of	of	ADP
ejpam-4602	297	3	the	the	DET
ejpam-4602	297	4	london	london	PROPN
ejpam-4602	297	5	mathematical	mathematical	ADJ
ejpam-4602	297	6	society	society	NOUN
ejpam-4602	297	7	,	,	PUNCT
ejpam-4602	297	8	2(17):33–41	2(17):33–41	NUM
ejpam-4602	297	9	,	,	PUNCT
ejpam-4602	297	10	1978	1978	NUM
ejpam-4602	297	11	.	.	PUNCT
ejpam-4602	298	1	[	[	X
ejpam-4602	298	2	17	17	NUM
ejpam-4602	298	3	]	]	X
ejpam-4602	298	4	david	david	PROPN
ejpam-4602	298	5	alan	alan	PROPN
ejpam-4602	298	6	jordan	jordan	PROPN
ejpam-4602	298	7	.	.	PUNCT
ejpam-4602	298	8	noetherian	noetherian	ADJ
ejpam-4602	298	9	ore	ore	NOUN
ejpam-4602	298	10	extensions	extension	NOUN
ejpam-4602	298	11	and	and	CCONJ
ejpam-4602	298	12	jacobson	jacobson	PROPN
ejpam-4602	298	13	rings	ring	NOUN
ejpam-4602	298	14	.	.	PUNCT
ejpam-4602	299	1	journal	journal	PROPN
ejpam-4602	299	2	of	of	ADP
ejpam-4602	299	3	the	the	DET
ejpam-4602	299	4	london	london	PROPN
ejpam-4602	299	5	mathematical	mathematical	ADJ
ejpam-4602	299	6	society	society	NOUN
ejpam-4602	299	7	,	,	PUNCT
ejpam-4602	299	8	2(3):281–291	2(3):281–291	NUM
ejpam-4602	299	9	,	,	PUNCT
ejpam-4602	299	10	1975	1975	NUM
ejpam-4602	299	11	.	.	PUNCT
ejpam-4602	300	1	[	[	X
ejpam-4602	300	2	18	18	NUM
ejpam-4602	300	3	]	]	X
ejpam-4602	300	4	ahmad	ahmad	PROPN
ejpam-4602	300	5	al	al	PROPN
ejpam-4602	300	6	khalaf	khalaf	PROPN
ejpam-4602	300	7	,	,	PUNCT
ejpam-4602	300	8	orest	or	ADJ
ejpam-4602	300	9	d	d	X
ejpam-4602	300	10	artemovych	artemovych	NOUN
ejpam-4602	300	11	,	,	PUNCT
ejpam-4602	300	12	and	and	CCONJ
ejpam-4602	300	13	iman	iman	PROPN
ejpam-4602	300	14	taha	taha	PROPN
ejpam-4602	300	15	.	.	PUNCT
ejpam-4602	300	16	rings	ring	NOUN
ejpam-4602	300	17	with	with	ADP
ejpam-4602	300	18	simple	simple	ADJ
ejpam-4602	300	19	lie	lie	NOUN
ejpam-4602	300	20	rings	ring	NOUN
ejpam-4602	300	21	of	of	ADP
ejpam-4602	300	22	lie	lie	NOUN
ejpam-4602	300	23	and	and	CCONJ
ejpam-4602	300	24	jordan	jordan	PROPN
ejpam-4602	300	25	derivations	derivations	PROPN
ejpam-4602	300	26	.	.	PUNCT
ejpam-4602	301	1	journal	journal	PROPN
ejpam-4602	301	2	of	of	ADP
ejpam-4602	301	3	algebra	algebra	PROPN
ejpam-4602	301	4	and	and	CCONJ
ejpam-4602	301	5	its	its	PRON
ejpam-4602	301	6	applications	application	NOUN
ejpam-4602	301	7	,	,	PUNCT
ejpam-4602	301	8	17(04):1850078	17(04):1850078	NUM
ejpam-4602	301	9	,	,	PUNCT
ejpam-4602	301	10	2018	2018	NUM
ejpam-4602	301	11	.	.	PUNCT
ejpam-4602	302	1	[	[	X
ejpam-4602	302	2	19	19	NUM
ejpam-4602	302	3	]	]	X
ejpam-4602	302	4	tsiu	tsiu	ADJ
ejpam-4602	302	5	-	-	PUNCT
ejpam-4602	302	6	kwen	kwen	NOUN
ejpam-4602	302	7	lee	lee	PROPN
ejpam-4602	302	8	and	and	CCONJ
ejpam-4602	302	9	kun	kun	PROPN
ejpam-4602	302	10	-	-	PUNCT
ejpam-4602	302	11	shan	shan	PROPN
ejpam-4602	302	12	liu	liu	PROPN
ejpam-4602	302	13	.	.	PUNCT
ejpam-4602	303	1	the	the	DET
ejpam-4602	303	2	skolem	skolem	NOUN
ejpam-4602	303	3	–	–	PUNCT
ejpam-4602	303	4	noether	noether	PROPN
ejpam-4602	303	5	theorem	theorem	NOUN
ejpam-4602	303	6	for	for	ADP
ejpam-4602	303	7	semiprime	semiprime	NOUN
ejpam-4602	303	8	rings	ring	NOUN
ejpam-4602	303	9	satisfying	satisfy	VERB
ejpam-4602	303	10	a	a	DET
ejpam-4602	303	11	strict	strict	ADJ
ejpam-4602	303	12	identity	identity	NOUN
ejpam-4602	303	13	.	.	PUNCT
ejpam-4602	304	1	communications	communication	NOUN
ejpam-4602	304	2	in	in	ADP
ejpam-4602	304	3	algebra	algebra	PROPN
ejpam-4602	304	4	®	®	NOUN
ejpam-4602	304	5	,	,	PUNCT
ejpam-4602	304	6	35(6):1949–1955	35(6):1949–1955	NUM
ejpam-4602	304	7	,	,	PUNCT
ejpam-4602	304	8	2007	2007	NUM
ejpam-4602	304	9	.	.	PUNCT
ejpam-4602	305	1	[	[	X
ejpam-4602	305	2	20	20	NUM
ejpam-4602	305	3	]	]	X
ejpam-4602	305	4	chia	chia	PROPN
ejpam-4602	305	5	-	-	PUNCT
ejpam-4602	305	6	hsin	hsin	PROPN
ejpam-4602	305	7	liu	liu	PROPN
ejpam-4602	305	8	.	.	PROPN
ejpam-4602	305	9	semiprime	semiprime	PROPN
ejpam-4602	305	10	lie	lie	PROPN
ejpam-4602	305	11	rings	ring	NOUN
ejpam-4602	305	12	of	of	ADP
ejpam-4602	305	13	derivations	derivation	NOUN
ejpam-4602	305	14	of	of	ADP
ejpam-4602	305	15	commutative	commutative	ADJ
ejpam-4602	305	16	rings	ring	NOUN
ejpam-4602	305	17	.	.	PUNCT
ejpam-4602	306	1	contemporary	contemporary	ADJ
ejpam-4602	306	2	mathematics	mathematics	PROPN
ejpam-4602	306	3	,	,	PUNCT
ejpam-4602	306	4	420:259	420:259	NOUN
ejpam-4602	306	5	,	,	PUNCT
ejpam-4602	306	6	2006	2006	NUM
ejpam-4602	306	7	.	.	PUNCT
ejpam-4602	307	1	[	[	X
ejpam-4602	307	2	21	21	NUM
ejpam-4602	307	3	]	]	X
ejpam-4602	307	4	j	j	PROPN
ejpam-4602	307	5	mayne	mayne	NOUN
ejpam-4602	307	6	.	.	PUNCT
ejpam-4602	308	1	centralizing	centralize	VERB
ejpam-4602	308	2	automorphisms	automorphism	NOUN
ejpam-4602	308	3	of	of	ADP
ejpam-4602	308	4	prime	prime	ADJ
ejpam-4602	308	5	rings	ring	NOUN
ejpam-4602	308	6	.	.	PUNCT
ejpam-4602	309	1	canadian	canadian	ADJ
ejpam-4602	309	2	mathematical	mathematical	ADJ
ejpam-4602	309	3	bulletin	bulletin	NOUN
ejpam-4602	309	4	,	,	PUNCT
ejpam-4602	309	5	19(1):113–115	19(1):113–115	PROPN
ejpam-4602	309	6	,	,	PUNCT
ejpam-4602	309	7	1976	1976	NUM
ejpam-4602	309	8	.	.	PUNCT
ejpam-4602	310	1	references	reference	NOUN
ejpam-4602	310	2	2042	2042	NUM
ejpam-4602	311	1	[	[	X
ejpam-4602	311	2	22	22	NUM
ejpam-4602	311	3	]	]	X
ejpam-4602	311	4	j	j	PROPN
ejpam-4602	311	5	mayne	mayne	PROPN
ejpam-4602	311	6	.	.	PUNCT
ejpam-4602	312	1	ideals	ideal	NOUN
ejpam-4602	312	2	and	and	CCONJ
ejpam-4602	312	3	centralizing	centralize	VERB
ejpam-4602	312	4	mappings	mapping	NOUN
ejpam-4602	312	5	in	in	ADP
ejpam-4602	312	6	prime	prime	ADJ
ejpam-4602	312	7	rings	ring	NOUN
ejpam-4602	312	8	.	.	PUNCT
ejpam-4602	313	1	proceedings	proceeding	NOUN
ejpam-4602	313	2	of	of	ADP
ejpam-4602	313	3	the	the	DET
ejpam-4602	313	4	american	american	PROPN
ejpam-4602	313	5	mathematical	mathematical	PROPN
ejpam-4602	313	6	society	society	NOUN
ejpam-4602	313	7	,	,	PUNCT
ejpam-4602	313	8	86(2):211–212	86(2):211–212	NUM
ejpam-4602	313	9	,	,	PUNCT
ejpam-4602	313	10	1982	1982	NUM
ejpam-4602	313	11	.	.	PUNCT
ejpam-4602	314	1	[	[	X
ejpam-4602	314	2	23	23	NUM
ejpam-4602	314	3	]	]	X
ejpam-4602	314	4	j	j	PROPN
ejpam-4602	314	5	mayne	mayne	NOUN
ejpam-4602	314	6	.	.	PUNCT
ejpam-4602	315	1	centralizing	centralize	VERB
ejpam-4602	315	2	mappings	mapping	NOUN
ejpam-4602	315	3	of	of	ADP
ejpam-4602	315	4	prime	prime	ADJ
ejpam-4602	315	5	rings	ring	NOUN
ejpam-4602	315	6	.	.	PUNCT
ejpam-4602	316	1	canadian	canadian	ADJ
ejpam-4602	316	2	mathematical	mathematical	ADJ
ejpam-4602	316	3	bulletin	bulletin	NOUN
ejpam-4602	316	4	,	,	PUNCT
ejpam-4602	316	5	27(1):122–126	27(1):122–126	NOUN
ejpam-4602	316	6	,	,	PUNCT
ejpam-4602	316	7	1984	1984	NUM
ejpam-4602	316	8	.	.	PUNCT
ejpam-4602	317	1	[	[	X
ejpam-4602	317	2	24	24	NUM
ejpam-4602	317	3	]	]	X
ejpam-4602	317	4	r	r	NOUN
ejpam-4602	317	5	miers	mier	NOUN
ejpam-4602	317	6	.	.	PUNCT
ejpam-4602	318	1	centralizing	centralize	VERB
ejpam-4602	318	2	mappings	mapping	NOUN
ejpam-4602	318	3	of	of	ADP
ejpam-4602	318	4	operator	operator	NOUN
ejpam-4602	318	5	algebras	algebra	NOUN
ejpam-4602	318	6	.	.	PUNCT
ejpam-4602	318	7	journal	journal	PROPN
ejpam-4602	318	8	of	of	ADP
ejpam-4602	318	9	algebra	algebra	PROPN
ejpam-4602	318	10	,	,	PUNCT
ejpam-4602	318	11	59(1):56–64	59(1):56–64	NUM
ejpam-4602	318	12	,	,	PUNCT
ejpam-4602	318	13	1979	1979	NUM
ejpam-4602	318	14	.	.	PUNCT
ejpam-4602	319	1	[	[	X
ejpam-4602	319	2	25	25	NUM
ejpam-4602	319	3	]	]	X
ejpam-4602	319	4	ds	ds	ADJ
ejpam-4602	319	5	passman	passman	NOUN
ejpam-4602	319	6	.	.	PUNCT
ejpam-4602	320	1	simple	simple	ADJ
ejpam-4602	320	2	lie	lie	NOUN
ejpam-4602	320	3	algebras	algebra	NOUN
ejpam-4602	320	4	of	of	ADP
ejpam-4602	320	5	witt	witt	PROPN
ejpam-4602	320	6	type	type	NOUN
ejpam-4602	320	7	.	.	PUNCT
ejpam-4602	321	1	journal	journal	PROPN
ejpam-4602	321	2	of	of	ADP
ejpam-4602	321	3	algebra	algebra	PROPN
ejpam-4602	321	4	,	,	PUNCT
ejpam-4602	321	5	206(2):682–692	206(2):682–692	NUM
ejpam-4602	321	6	,	,	PUNCT
ejpam-4602	321	7	1998	1998	NUM
ejpam-4602	321	8	.	.	PUNCT
ejpam-4602	322	1	[	[	X
ejpam-4602	322	2	26	26	NUM
ejpam-4602	322	3	]	]	X
ejpam-4602	322	4	e	e	NOUN
ejpam-4602	322	5	posner	posner	NOUN
ejpam-4602	322	6	.	.	PUNCT
ejpam-4602	323	1	derivations	derivation	NOUN
ejpam-4602	323	2	in	in	ADP
ejpam-4602	323	3	prime	prime	ADJ
ejpam-4602	323	4	rings	ring	NOUN
ejpam-4602	323	5	.	.	PUNCT
ejpam-4602	324	1	proceedings	proceeding	NOUN
ejpam-4602	324	2	of	of	ADP
ejpam-4602	324	3	the	the	DET
ejpam-4602	324	4	american	american	PROPN
ejpam-4602	324	5	mathematical	mathematical	PROPN
ejpam-4602	324	6	society	society	NOUN
ejpam-4602	324	7	,	,	PUNCT
ejpam-4602	324	8	8(6):1093–1100	8(6):1093–1100	PROPN
ejpam-4602	324	9	,	,	PUNCT
ejpam-4602	324	10	1957	1957	NUM
ejpam-4602	324	11	.	.	PUNCT
ejpam-4602	325	1	[	[	X
ejpam-4602	325	2	27	27	NUM
ejpam-4602	325	3	]	]	X
ejpam-4602	325	4	mohammad	mohammad	PROPN
ejpam-4602	325	5	samman	samman	PROPN
ejpam-4602	325	6	and	and	CCONJ
ejpam-4602	325	7	nouf	nouf	PROPN
ejpam-4602	325	8	alyamani	alyamani	PROPN
ejpam-4602	325	9	.	.	PUNCT
ejpam-4602	326	1	derivations	derivation	NOUN
ejpam-4602	326	2	and	and	CCONJ
ejpam-4602	326	3	reverse	reverse	ADJ
ejpam-4602	326	4	derivations	derivation	NOUN
ejpam-4602	326	5	in	in	ADP
ejpam-4602	326	6	semiprime	semiprime	NOUN
ejpam-4602	326	7	rings	ring	NOUN
ejpam-4602	326	8	.	.	PUNCT
ejpam-4602	327	1	in	in	ADP
ejpam-4602	327	2	international	international	PROPN
ejpam-4602	327	3	mathematical	mathematical	ADJ
ejpam-4602	327	4	forum	forum	PROPN
ejpam-4602	327	5	,	,	PUNCT
ejpam-4602	327	6	volume	volume	NOUN
ejpam-4602	327	7	2	2	NUM
ejpam-4602	327	8	,	,	PUNCT
ejpam-4602	327	9	pages	page	NOUN
ejpam-4602	327	10	1895–1902	1895–1902	NUM
ejpam-4602	327	11	,	,	PUNCT
ejpam-4602	327	12	2007	2007	NUM
ejpam-4602	327	13	.	.	PUNCT
ejpam-4602	328	1	[	[	X
ejpam-4602	328	2	28	28	NUM
ejpam-4602	328	3	]	]	X
ejpam-4602	328	4	iman	iman	PROPN
ejpam-4602	328	5	taha	taha	PROPN
ejpam-4602	328	6	,	,	PUNCT
ejpam-4602	328	7	rohaidah	rohaidah	PROPN
ejpam-4602	328	8	masri	masri	PROPN
ejpam-4602	328	9	,	,	PUNCT
ejpam-4602	328	10	ahmad	ahmad	PROPN
ejpam-4602	328	11	alkhalaf	alkhalaf	PROPN
ejpam-4602	328	12	,	,	PUNCT
ejpam-4602	328	13	and	and	CCONJ
ejpam-4602	328	14	rawdah	rawdah	PROPN
ejpam-4602	328	15	tarmizi	tarmizi	PROPN
ejpam-4602	328	16	.	.	PUNCT
ejpam-4602	329	1	derivations	derivation	NOUN
ejpam-4602	329	2	in	in	ADP
ejpam-4602	329	3	differentially	differentially	ADV
ejpam-4602	329	4	δ	δ	NOUN
ejpam-4602	329	5	-	-	ADJ
ejpam-4602	329	6	prime	prime	NOUN
ejpam-4602	329	7	rings	ring	NOUN
ejpam-4602	329	8	.	.	PUNCT
ejpam-4602	330	1	european	european	PROPN
ejpam-4602	330	2	journal	journal	PROPN
ejpam-4602	330	3	of	of	ADP
ejpam-4602	330	4	pure	pure	ADJ
ejpam-4602	330	5	and	and	CCONJ
ejpam-4602	330	6	applied	applied	ADJ
ejpam-4602	330	7	mathematics	mathematic	NOUN
ejpam-4602	330	8	,	,	PUNCT
ejpam-4602	330	9	15(2):454–466	15(2):454–466	PROPN
ejpam-4602	330	10	,	,	PUNCT
ejpam-4602	330	11	2022	2022	NUM
ejpam-4602	330	12	.	.	PUNCT
ejpam-4602	331	1	[	[	X
ejpam-4602	331	2	29	29	NUM
ejpam-4602	331	3	]	]	X
ejpam-4602	331	4	j	j	PROPN
ejpam-4602	331	5	vukman	vukman	NOUN
ejpam-4602	331	6	.	.	PUNCT
ejpam-4602	332	1	commuting	commute	VERB
ejpam-4602	332	2	and	and	CCONJ
ejpam-4602	332	3	centralizing	centralize	VERB
ejpam-4602	332	4	mappings	mapping	NOUN
ejpam-4602	332	5	in	in	ADP
ejpam-4602	332	6	prime	prime	ADJ
ejpam-4602	332	7	rings	ring	NOUN
ejpam-4602	332	8	.	.	PUNCT
ejpam-4602	333	1	proceedings	proceeding	NOUN
ejpam-4602	333	2	of	of	ADP
ejpam-4602	333	3	the	the	DET
ejpam-4602	333	4	american	american	PROPN
ejpam-4602	333	5	mathematical	mathematical	PROPN
ejpam-4602	333	6	society	society	NOUN
ejpam-4602	333	7	,	,	PUNCT
ejpam-4602	333	8	109(1):47–52	109(1):47–52	NUM
ejpam-4602	333	9	,	,	PUNCT
ejpam-4602	333	10	1990	1990	NUM
ejpam-4602	333	11	.	.	PUNCT