id	sid	tid	token	lemma	pos
ejpam-6044	1	1	european	european	PROPN
ejpam-6044	1	2	journal	journal	PROPN
ejpam-6044	1	3	of	of	ADP
ejpam-6044	1	4	pure	pure	ADJ
ejpam-6044	1	5	and	and	CCONJ
ejpam-6044	1	6	applied	applied	ADJ
ejpam-6044	1	7	mathematics	mathematic	NOUN
ejpam-6044	1	8	2025	2025	NUM
ejpam-6044	1	9	,	,	PUNCT
ejpam-6044	1	10	vol	vol	NOUN
ejpam-6044	1	11	.	.	PROPN
ejpam-6044	1	12	18	18	NUM
ejpam-6044	1	13	,	,	PUNCT
ejpam-6044	1	14	issue	issue	NOUN
ejpam-6044	1	15	2	2	NUM
ejpam-6044	1	16	,	,	PUNCT
ejpam-6044	1	17	article	article	NOUN
ejpam-6044	1	18	number	number	NOUN
ejpam-6044	1	19	6044	6044	NUM
ejpam-6044	1	20	issn	issn	VERB
ejpam-6044	1	21	1307	1307	NUM
ejpam-6044	1	22	-	-	SYM
ejpam-6044	1	23	5543	5543	NUM
ejpam-6044	1	24	–	–	PUNCT
ejpam-6044	1	25	ejpam.com	ejpam.com	X
ejpam-6044	1	26	published	publish	VERB
ejpam-6044	1	27	by	by	ADP
ejpam-6044	1	28	new	new	PROPN
ejpam-6044	1	29	york	york	PROPN
ejpam-6044	1	30	business	business	PROPN
ejpam-6044	1	31	global	global	PROPN
ejpam-6044	1	32	on	on	ADP
ejpam-6044	1	33	generalized	generalized	ADJ
ejpam-6044	1	34	(	(	PUNCT
ejpam-6044	1	35	α	α	NOUN
ejpam-6044	1	36	,	,	PUNCT
ejpam-6044	1	37	∗)-derivations	∗)-derivation	NOUN
ejpam-6044	1	38	and	and	CCONJ
ejpam-6044	1	39	α	α	NOUN
ejpam-6044	1	40	-	-	PUNCT
ejpam-6044	1	41	centralizers	centralizer	NOUN
ejpam-6044	1	42	on	on	ADP
ejpam-6044	1	43	rings	ring	NOUN
ejpam-6044	1	44	faiza	faiza	PROPN
ejpam-6044	1	45	shujat1,∗	shujat1,∗	PROPN
ejpam-6044	1	46	,	,	PUNCT
ejpam-6044	1	47	salwa	salwa	PROPN
ejpam-6044	1	48	alharbi1	alharbi1	PROPN
ejpam-6044	1	49	1	1	NUM
ejpam-6044	1	50	department	department	NOUN
ejpam-6044	1	51	of	of	ADP
ejpam-6044	1	52	mathematics	mathematic	NOUN
ejpam-6044	1	53	,	,	PUNCT
ejpam-6044	1	54	faculty	faculty	NOUN
ejpam-6044	1	55	of	of	ADP
ejpam-6044	1	56	science	science	NOUN
ejpam-6044	1	57	,	,	PUNCT
ejpam-6044	1	58	taibah	taibah	PROPN
ejpam-6044	1	59	university	university	PROPN
ejpam-6044	1	60	,	,	PUNCT
ejpam-6044	1	61	madinah	madinah	PROPN
ejpam-6044	1	62	,	,	PUNCT
ejpam-6044	1	63	saudi	saudi	PROPN
ejpam-6044	1	64	arabia	arabia	PROPN
ejpam-6044	1	65	abstract	abstract	NOUN
ejpam-6044	1	66	.	.	PUNCT
ejpam-6044	2	1	the	the	DET
ejpam-6044	2	2	intention	intention	NOUN
ejpam-6044	2	3	of	of	ADP
ejpam-6044	2	4	the	the	DET
ejpam-6044	2	5	current	current	ADJ
ejpam-6044	2	6	research	research	NOUN
ejpam-6044	2	7	is	be	AUX
ejpam-6044	2	8	to	to	PART
ejpam-6044	2	9	define	define	VERB
ejpam-6044	2	10	the	the	DET
ejpam-6044	2	11	concept	concept	NOUN
ejpam-6044	2	12	of	of	ADP
ejpam-6044	2	13	(	(	PUNCT
ejpam-6044	2	14	α	α	X
ejpam-6044	2	15	,	,	PUNCT
ejpam-6044	2	16	∗)-derivations	∗)-derivation	NOUN
ejpam-6044	2	17	on	on	ADP
ejpam-6044	2	18	ring	ring	NOUN
ejpam-6044	2	19	r	r	NOUN
ejpam-6044	2	20	,	,	PUNCT
ejpam-6044	2	21	where	where	SCONJ
ejpam-6044	2	22	α	α	NOUN
ejpam-6044	2	23	is	be	AUX
ejpam-6044	2	24	an	an	DET
ejpam-6044	2	25	automorphism	automorphism	NOUN
ejpam-6044	2	26	of	of	ADP
ejpam-6044	2	27	r	r	NOUN
ejpam-6044	2	28	and	and	CCONJ
ejpam-6044	2	29	∗	∗	NOUN
ejpam-6044	2	30	represents	represent	VERB
ejpam-6044	2	31	involution	involution	NOUN
ejpam-6044	2	32	on	on	ADP
ejpam-6044	2	33	r.	r.	PROPN
ejpam-6044	2	34	we	we	PRON
ejpam-6044	2	35	obtain	obtain	VERB
ejpam-6044	2	36	some	some	DET
ejpam-6044	2	37	commutativity	commutativity	NOUN
ejpam-6044	2	38	theorems	theorem	NOUN
ejpam-6044	2	39	in	in	ADP
ejpam-6044	2	40	case	case	NOUN
ejpam-6044	2	41	of	of	ADP
ejpam-6044	2	42	prime	prime	ADJ
ejpam-6044	2	43	ring	ring	NOUN
ejpam-6044	2	44	by	by	ADP
ejpam-6044	2	45	utilizing	utilize	VERB
ejpam-6044	2	46	the	the	DET
ejpam-6044	2	47	role	role	NOUN
ejpam-6044	2	48	of	of	ADP
ejpam-6044	2	49	α	α	PROPN
ejpam-6044	2	50	and	and	CCONJ
ejpam-6044	2	51	∗.	∗.	NUM
ejpam-6044	2	52	we	we	PRON
ejpam-6044	2	53	will	will	AUX
ejpam-6044	2	54	also	also	ADV
ejpam-6044	2	55	discuss	discuss	VERB
ejpam-6044	2	56	the	the	DET
ejpam-6044	2	57	proofs	proof	NOUN
ejpam-6044	2	58	of	of	ADP
ejpam-6044	2	59	theorems	theorem	NOUN
ejpam-6044	2	60	in	in	ADP
ejpam-6044	2	61	case	case	NOUN
ejpam-6044	2	62	of	of	ADP
ejpam-6044	2	63	non	non	ADJ
ejpam-6044	2	64	-	-	ADJ
ejpam-6044	2	65	commutative	commutative	ADJ
ejpam-6044	2	66	prime	prime	ADJ
ejpam-6044	2	67	ring	ring	NOUN
ejpam-6044	2	68	and	and	CCONJ
ejpam-6044	2	69	under	under	ADP
ejpam-6044	2	70	which	which	PRON
ejpam-6044	2	71	condition	condition	NOUN
ejpam-6044	2	72	generalized	generalize	VERB
ejpam-6044	2	73	(	(	PUNCT
ejpam-6044	2	74	α	α	NOUN
ejpam-6044	2	75	,	,	PUNCT
ejpam-6044	2	76	∗)-derivation	∗)-derivation	NOUN
ejpam-6044	2	77	behaves	behave	VERB
ejpam-6044	2	78	like	like	ADP
ejpam-6044	2	79	an	an	DET
ejpam-6044	2	80	α	α	NOUN
ejpam-6044	2	81	-	-	PUNCT
ejpam-6044	2	82	centralizers	centralizer	NOUN
ejpam-6044	2	83	.	.	PUNCT
ejpam-6044	3	1	suitable	suitable	ADJ
ejpam-6044	3	2	examples	example	NOUN
ejpam-6044	3	3	are	be	AUX
ejpam-6044	3	4	given	give	VERB
ejpam-6044	3	5	in	in	ADP
ejpam-6044	3	6	favor	favor	NOUN
ejpam-6044	3	7	of	of	ADP
ejpam-6044	3	8	introduced	introduce	VERB
ejpam-6044	3	9	concept	concept	NOUN
ejpam-6044	3	10	.	.	PUNCT
ejpam-6044	4	1	2020	2020	NUM
ejpam-6044	4	2	mathematics	mathematic	NOUN
ejpam-6044	4	3	subject	subject	NOUN
ejpam-6044	4	4	classifications	classification	NOUN
ejpam-6044	4	5	:	:	PUNCT
ejpam-6044	4	6	ams	am	NOUN
ejpam-6044	4	7	16n60	16n60	NUM
ejpam-6044	4	8	,	,	PUNCT
ejpam-6044	4	9	16w10	16w10	NUM
ejpam-6044	4	10	,	,	PUNCT
ejpam-6044	4	11	16r50	16r50	NUM
ejpam-6044	4	12	,	,	PUNCT
ejpam-6044	4	13	47b47	47b47	VERB
ejpam-6044	4	14	key	key	ADJ
ejpam-6044	4	15	words	word	NOUN
ejpam-6044	4	16	and	and	CCONJ
ejpam-6044	4	17	phrases	phrase	NOUN
ejpam-6044	4	18	:	:	PUNCT
ejpam-6044	4	19	generalized	generalize	VERB
ejpam-6044	4	20	(	(	PUNCT
ejpam-6044	4	21	α	α	X
ejpam-6044	4	22	,	,	PUNCT
ejpam-6044	4	23	∗)-derivations	∗)-derivations	PROPN
ejpam-6044	4	24	,	,	PUNCT
ejpam-6044	4	25	prime	prime	ADJ
ejpam-6044	4	26	∗-ring	∗-ring	NOUN
ejpam-6044	4	27	,	,	PUNCT
ejpam-6044	4	28	α	α	NOUN
ejpam-6044	4	29	-	-	PUNCT
ejpam-6044	4	30	centralizer	centralizer	NOUN
ejpam-6044	4	31	1	1	NUM
ejpam-6044	4	32	.	.	PUNCT
ejpam-6044	4	33	introduction	introduction	NOUN
ejpam-6044	4	34	through	through	ADP
ejpam-6044	4	35	out	out	ADP
ejpam-6044	4	36	the	the	DET
ejpam-6044	4	37	manuscript	manuscript	NOUN
ejpam-6044	4	38	,	,	PUNCT
ejpam-6044	4	39	the	the	DET
ejpam-6044	4	40	notation	notation	NOUN
ejpam-6044	4	41	z(r	z(r	PROPN
ejpam-6044	4	42	)	)	PUNCT
ejpam-6044	4	43	stands	stand	VERB
ejpam-6044	4	44	for	for	ADP
ejpam-6044	4	45	the	the	DET
ejpam-6044	4	46	center	center	NOUN
ejpam-6044	4	47	of	of	ADP
ejpam-6044	4	48	an	an	DET
ejpam-6044	4	49	associative	associative	ADJ
ejpam-6044	4	50	ring	ring	NOUN
ejpam-6044	4	51	r.	r.	NOUN
ejpam-6044	5	1	the	the	DET
ejpam-6044	5	2	symbol	symbol	NOUN
ejpam-6044	5	3	[	[	X
ejpam-6044	5	4	b	b	X
ejpam-6044	5	5	,	,	PUNCT
ejpam-6044	5	6	d	d	X
ejpam-6044	5	7	]	]	X
ejpam-6044	5	8	specifies	specify	VERB
ejpam-6044	5	9	the	the	DET
ejpam-6044	5	10	commutator	commutator	NOUN
ejpam-6044	5	11	of	of	ADP
ejpam-6044	5	12	b	b	PROPN
ejpam-6044	5	13	,	,	PUNCT
ejpam-6044	5	14	d	d	PROPN
ejpam-6044	5	15	∈	∈	PROPN
ejpam-6044	5	16	r	r	NOUN
ejpam-6044	5	17	,	,	PUNCT
ejpam-6044	5	18	which	which	PRON
ejpam-6044	5	19	is	be	AUX
ejpam-6044	5	20	represented	represent	VERB
ejpam-6044	5	21	by	by	ADP
ejpam-6044	5	22	the	the	DET
ejpam-6044	5	23	mathematical	mathematical	ADJ
ejpam-6044	5	24	formula	formula	NOUN
ejpam-6044	5	25	bd	bd	PROPN
ejpam-6044	5	26	−	−	NOUN
ejpam-6044	5	27	db	db	PROPN
ejpam-6044	5	28	.	.	PUNCT
ejpam-6044	6	1	if	if	SCONJ
ejpam-6044	6	2	pr	pr	NOUN
ejpam-6044	6	3	=	=	SYM
ejpam-6044	6	4	0	0	NUM
ejpam-6044	6	5	implies	imply	VERB
ejpam-6044	6	6	r	r	NOUN
ejpam-6044	6	7	=	=	SYM
ejpam-6044	6	8	0	0	NUM
ejpam-6044	6	9	for	for	ADP
ejpam-6044	6	10	every	every	DET
ejpam-6044	6	11	r	r	NOUN
ejpam-6044	6	12	∈	∈	NOUN
ejpam-6044	6	13	r	r	NOUN
ejpam-6044	6	14	and	and	CCONJ
ejpam-6044	6	15	p	p	NOUN
ejpam-6044	6	16	>	>	X
ejpam-6044	6	17	1	1	NUM
ejpam-6044	6	18	is	be	AUX
ejpam-6044	6	19	a	a	DET
ejpam-6044	6	20	fixed	fix	VERB
ejpam-6044	6	21	integer	integer	NOUN
ejpam-6044	6	22	,	,	PUNCT
ejpam-6044	6	23	then	then	ADV
ejpam-6044	6	24	a	a	DET
ejpam-6044	6	25	ring	ring	NOUN
ejpam-6044	6	26	r	r	NOUN
ejpam-6044	6	27	is	be	AUX
ejpam-6044	6	28	a	a	DET
ejpam-6044	6	29	p	p	ADJ
ejpam-6044	6	30	-	-	PUNCT
ejpam-6044	6	31	torsion	torsion	NOUN
ejpam-6044	6	32	free	free	ADJ
ejpam-6044	6	33	ring	ring	NOUN
ejpam-6044	6	34	.	.	PUNCT
ejpam-6044	7	1	a	a	DET
ejpam-6044	7	2	ring	ring	NOUN
ejpam-6044	7	3	r	r	NOUN
ejpam-6044	7	4	is	be	AUX
ejpam-6044	7	5	a	a	DET
ejpam-6044	7	6	prime	prime	NOUN
ejpam-6044	7	7	if	if	SCONJ
ejpam-6044	7	8	rrt	rrt	VERB
ejpam-6044	7	9	=	=	SYM
ejpam-6044	7	10	{	{	PUNCT
ejpam-6044	7	11	0	0	NUM
ejpam-6044	7	12	}	}	PUNCT
ejpam-6044	7	13	gives	give	VERB
ejpam-6044	7	14	that	that	SCONJ
ejpam-6044	7	15	either	either	CCONJ
ejpam-6044	7	16	t	t	PROPN
ejpam-6044	7	17	=	=	SYM
ejpam-6044	7	18	0	0	NUM
ejpam-6044	7	19	or	or	CCONJ
ejpam-6044	7	20	r	r	NOUN
ejpam-6044	7	21	=	=	SYM
ejpam-6044	7	22	0	0	NUM
ejpam-6044	7	23	.	.	PUNCT
ejpam-6044	8	1	it	it	PRON
ejpam-6044	8	2	is	be	AUX
ejpam-6044	8	3	called	call	VERB
ejpam-6044	8	4	semiprime	semiprime	NOUN
ejpam-6044	8	5	if	if	SCONJ
ejpam-6044	8	6	it	it	PRON
ejpam-6044	8	7	fulfills	fulfill	VERB
ejpam-6044	8	8	the	the	DET
ejpam-6044	8	9	requirement	requirement	NOUN
ejpam-6044	8	10	that	that	SCONJ
ejpam-6044	8	11	crc	crc	NOUN
ejpam-6044	8	12	=	=	PUNCT
ejpam-6044	8	13	{	{	PUNCT
ejpam-6044	8	14	0	0	NUM
ejpam-6044	8	15	}	}	PUNCT
ejpam-6044	8	16	yields	yield	NOUN
ejpam-6044	8	17	that	that	PRON
ejpam-6044	8	18	c	c	AUX
ejpam-6044	8	19	=	=	PUNCT
ejpam-6044	8	20	0	0	PROPN
ejpam-6044	8	21	.	.	PUNCT
ejpam-6044	9	1	in	in	ADP
ejpam-6044	9	2	simple	simple	ADJ
ejpam-6044	9	3	terms	term	NOUN
ejpam-6044	9	4	,	,	PUNCT
ejpam-6044	9	5	the	the	DET
ejpam-6044	9	6	mapping	mapping	NOUN
ejpam-6044	9	7	ζ	ζ	NOUN
ejpam-6044	9	8	is	be	AUX
ejpam-6044	9	9	(	(	PUNCT
ejpam-6044	9	10	skew)-commuting	skew)-commute	VERB
ejpam-6044	9	11	on	on	ADP
ejpam-6044	9	12	r	r	NOUN
ejpam-6044	9	13	if	if	SCONJ
ejpam-6044	9	14	ζ(c)c	ζ(c)c	PROPN
ejpam-6044	9	15	+	+	CCONJ
ejpam-6044	9	16	cζ(c	cζ(c	X
ejpam-6044	9	17	)	)	PUNCT
ejpam-6044	9	18	=	=	SYM
ejpam-6044	9	19	0	0	NUM
ejpam-6044	9	20	for	for	ADP
ejpam-6044	9	21	each	each	PRON
ejpam-6044	9	22	of	of	ADP
ejpam-6044	9	23	c	c	PROPN
ejpam-6044	9	24	∈	∈	PROPN
ejpam-6044	9	25	r.	r.	PROPN
ejpam-6044	9	26	if	if	SCONJ
ejpam-6044	9	27	ζ(c)c	ζ(c)c	PROPN
ejpam-6044	9	28	+	+	CCONJ
ejpam-6044	9	29	cζ(c	cζ(c	X
ejpam-6044	9	30	)	)	PUNCT
ejpam-6044	9	31	∈	∈	PROPN
ejpam-6044	9	32	z(r	z(r	PROPN
ejpam-6044	9	33	)	)	PUNCT
ejpam-6044	9	34	for	for	ADP
ejpam-6044	9	35	each	each	DET
ejpam-6044	9	36	c	c	NOUN
ejpam-6044	9	37	∈	∈	PROPN
ejpam-6044	9	38	r	r	NOUN
ejpam-6044	9	39	,	,	PUNCT
ejpam-6044	9	40	then	then	ADV
ejpam-6044	9	41	a	a	DET
ejpam-6044	9	42	map	map	NOUN
ejpam-6044	9	43	ζ	ζ	NOUN
ejpam-6044	9	44	from	from	ADP
ejpam-6044	9	45	r	r	NOUN
ejpam-6044	9	46	to	to	ADP
ejpam-6044	9	47	r	r	NOUN
ejpam-6044	9	48	is	be	AUX
ejpam-6044	9	49	thought	think	VERB
ejpam-6044	9	50	to	to	PART
ejpam-6044	9	51	be	be	AUX
ejpam-6044	9	52	(	(	PUNCT
ejpam-6044	9	53	skew)-centralizing	skew)-centralize	VERB
ejpam-6044	9	54	on	on	ADP
ejpam-6044	9	55	r.	r.	PROPN
ejpam-6044	9	56	if	if	SCONJ
ejpam-6044	9	57	the	the	DET
ejpam-6044	9	58	mapping	mapping	NOUN
ejpam-6044	9	59	η	η	PROPN
ejpam-6044	9	60	from	from	ADP
ejpam-6044	9	61	r	r	NOUN
ejpam-6044	9	62	to	to	ADP
ejpam-6044	9	63	r	r	NOUN
ejpam-6044	9	64	fulfills	fulfill	VERB
ejpam-6044	9	65	the	the	DET
ejpam-6044	9	66	equation	equation	NOUN
ejpam-6044	9	67	η(ce	η(ce	PROPN
ejpam-6044	9	68	)	)	PUNCT
ejpam-6044	9	69	=	=	SYM
ejpam-6044	9	70	η(c)e+	η(c)e+	X
ejpam-6044	9	71	cη(e	cη(e	NUM
ejpam-6044	9	72	)	)	PUNCT
ejpam-6044	9	73	,	,	PUNCT
ejpam-6044	9	74	for	for	ADP
ejpam-6044	9	75	each	each	PRON
ejpam-6044	9	76	of	of	ADP
ejpam-6044	9	77	c	c	NOUN
ejpam-6044	9	78	,	,	PUNCT
ejpam-6044	9	79	e	e	PROPN
ejpam-6044	9	80	∈	∈	PROPN
ejpam-6044	9	81	r	r	NOUN
ejpam-6044	9	82	,	,	PUNCT
ejpam-6044	9	83	then	then	ADV
ejpam-6044	9	84	it	it	PRON
ejpam-6044	9	85	is	be	AUX
ejpam-6044	9	86	regarded	regard	VERB
ejpam-6044	9	87	as	as	ADP
ejpam-6044	9	88	a	a	DET
ejpam-6044	9	89	derivation	derivation	NOUN
ejpam-6044	9	90	on	on	ADP
ejpam-6044	9	91	r.	r.	PROPN
ejpam-6044	9	92	let	let	VERB
ejpam-6044	9	93	r	r	PRON
ejpam-6044	9	94	be	be	AUX
ejpam-6044	9	95	a	a	DET
ejpam-6044	9	96	ring	ring	NOUN
ejpam-6044	9	97	whose	whose	DET
ejpam-6044	9	98	automorphism	automorphism	NOUN
ejpam-6044	9	99	is	be	AUX
ejpam-6044	9	100	β	β	NOUN
ejpam-6044	9	101	.	.	PUNCT
ejpam-6044	10	1	a	a	DET
ejpam-6044	10	2	map	map	NOUN
ejpam-6044	10	3	h	h	NOUN
ejpam-6044	10	4	on	on	ADP
ejpam-6044	10	5	r	r	NOUN
ejpam-6044	10	6	satisfying	satisfy	VERB
ejpam-6044	10	7	h(dk	h(dk	NOUN
ejpam-6044	10	8	)	)	PUNCT
ejpam-6044	10	9	=	=	SYM
ejpam-6044	10	10	h(d)β(k	h(d)β(k	NOUN
ejpam-6044	10	11	)	)	PUNCT
ejpam-6044	10	12	+	+	CCONJ
ejpam-6044	10	13	dh(k	dh(k	X
ejpam-6044	10	14	)	)	PUNCT
ejpam-6044	10	15	is	be	AUX
ejpam-6044	10	16	recognized	recognize	VERB
ejpam-6044	10	17	as	as	ADP
ejpam-6044	10	18	the	the	DET
ejpam-6044	10	19	β	β	NOUN
ejpam-6044	10	20	-	-	NOUN
ejpam-6044	10	21	derivation	derivation	NOUN
ejpam-6044	10	22	(	(	PUNCT
ejpam-6044	10	23	skew	skew	NOUN
ejpam-6044	10	24	-	-	PUNCT
ejpam-6044	10	25	derivation	derivation	NOUN
ejpam-6044	10	26	)	)	PUNCT
ejpam-6044	10	27	if	if	SCONJ
ejpam-6044	10	28	it	it	PRON
ejpam-6044	10	29	holds	hold	VERB
ejpam-6044	10	30	for	for	ADP
ejpam-6044	10	31	any	any	DET
ejpam-6044	10	32	∗corresponding	∗corresponde	VERB
ejpam-6044	10	33	author	author	NOUN
ejpam-6044	10	34	.	.	PUNCT
ejpam-6044	11	1	doi	doi	NOUN
ejpam-6044	11	2	:	:	PUNCT
ejpam-6044	11	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6044	https://doi.org/10.29020/nybg.ejpam.v18i2.6044	NUM
ejpam-6044	11	4	email	email	NOUN
ejpam-6044	11	5	addresses	address	NOUN
ejpam-6044	11	6	:	:	PUNCT
ejpam-6044	11	7	faiza.shujat@gmail.com	faiza.shujat@gmail.com	X
ejpam-6044	11	8	,	,	PUNCT
ejpam-6044	11	9	fullahkhan@taibahu.edu.sa	fullahkhan@taibahu.edu.sa	PROPN
ejpam-6044	11	10	(	(	PUNCT
ejpam-6044	11	11	f.	f.	PROPN
ejpam-6044	11	12	shujat	shujat	PROPN
ejpam-6044	11	13	)	)	PUNCT
ejpam-6044	11	14	,	,	PUNCT
ejpam-6044	11	15	salwa.alharbi1990@gmail.com	salwa.alharbi1990@gmail.com	PROPN
ejpam-6044	11	16	(	(	PUNCT
ejpam-6044	11	17	s.	s.	PROPN
ejpam-6044	11	18	alharbi	alharbi	PROPN
ejpam-6044	11	19	)	)	PUNCT
ejpam-6044	11	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6044	12	1	1	1	NUM
ejpam-6044	12	2	copyright	copyright	NOUN
ejpam-6044	12	3	:	:	PUNCT
ejpam-6044	12	4	©	©	PROPN
ejpam-6044	12	5	2025	2025	NUM
ejpam-6044	12	6	the	the	DET
ejpam-6044	12	7	author(s	author(s	NOUN
ejpam-6044	12	8	)	)	PUNCT
ejpam-6044	12	9	.	.	PUNCT
ejpam-6044	13	1	(	(	PUNCT
ejpam-6044	13	2	cc	cc	NOUN
ejpam-6044	13	3	by	by	ADP
ejpam-6044	13	4	-	-	PUNCT
ejpam-6044	13	5	nc	nc	PROPN
ejpam-6044	13	6	4.0	4.0	NUM
ejpam-6044	13	7	)	)	PUNCT
ejpam-6044	13	8	f.	f.	PROPN
ejpam-6044	13	9	shujat	shujat	PROPN
ejpam-6044	13	10	,	,	PUNCT
ejpam-6044	13	11	s.	s.	PROPN
ejpam-6044	13	12	alharbi	alharbi	PROPN
ejpam-6044	13	13	/	/	SYM
ejpam-6044	13	14	eur	eur	PROPN
ejpam-6044	13	15	.	.	PUNCT
ejpam-6044	14	1	j.	j.	PROPN
ejpam-6044	14	2	pure	pure	PROPN
ejpam-6044	14	3	appl	appl	PROPN
ejpam-6044	14	4	.	.	PROPN
ejpam-6044	14	5	math	math	PROPN
ejpam-6044	14	6	,	,	PUNCT
ejpam-6044	14	7	18	18	NUM
ejpam-6044	14	8	(	(	PUNCT
ejpam-6044	14	9	2	2	NUM
ejpam-6044	14	10	)	)	PUNCT
ejpam-6044	14	11	(	(	PUNCT
ejpam-6044	14	12	2025	2025	NUM
ejpam-6044	14	13	)	)	PUNCT
ejpam-6044	14	14	,	,	PUNCT
ejpam-6044	14	15	6044	6044	NUM
ejpam-6044	14	16	2	2	NUM
ejpam-6044	14	17	of	of	ADP
ejpam-6044	14	18	11	11	NUM
ejpam-6044	14	19	pair	pair	NOUN
ejpam-6044	14	20	d	d	NOUN
ejpam-6044	14	21	,	,	PUNCT
ejpam-6044	14	22	k	k	PROPN
ejpam-6044	14	23	in	in	ADP
ejpam-6044	14	24	r	r	NOUN
ejpam-6044	14	25	and	and	CCONJ
ejpam-6044	14	26	h	h	NOUN
ejpam-6044	14	27	has	have	VERB
ejpam-6044	14	28	additivity	additivity	NOUN
ejpam-6044	14	29	.	.	PUNCT
ejpam-6044	15	1	the	the	DET
ejpam-6044	15	2	combination	combination	NOUN
ejpam-6044	15	3	form	form	NOUN
ejpam-6044	15	4	h	h	NOUN
ejpam-6044	15	5	=	=	NOUN
ejpam-6044	16	1	β	β	X
ejpam-6044	16	2	−	−	NOUN
ejpam-6044	17	1	i	i	PRON
ejpam-6044	17	2	served	serve	VERB
ejpam-6044	17	3	as	as	ADP
ejpam-6044	17	4	the	the	DET
ejpam-6044	17	5	βderivation	βderivation	NOUN
ejpam-6044	17	6	if	if	SCONJ
ejpam-6044	17	7	we	we	PRON
ejpam-6044	17	8	symbolize	symbolize	VERB
ejpam-6044	17	9	the	the	DET
ejpam-6044	17	10	identity	identity	NOUN
ejpam-6044	17	11	map	map	NOUN
ejpam-6044	17	12	on	on	ADP
ejpam-6044	17	13	r	r	NOUN
ejpam-6044	17	14	by	by	ADP
ejpam-6044	17	15	i.	i.	NOUN
ejpam-6044	17	16	before	before	ADP
ejpam-6044	17	17	moving	move	VERB
ejpam-6044	17	18	onto	onto	ADP
ejpam-6044	17	19	this	this	DET
ejpam-6044	17	20	section	section	NOUN
ejpam-6044	17	21	’s	’s	PART
ejpam-6044	17	22	primary	primary	ADJ
ejpam-6044	17	23	findings	finding	NOUN
ejpam-6044	17	24	,	,	PUNCT
ejpam-6044	17	25	we	we	PRON
ejpam-6044	17	26	clarify	clarify	VERB
ejpam-6044	17	27	a	a	DET
ejpam-6044	17	28	few	few	ADJ
ejpam-6044	17	29	fundamental	fundamental	ADJ
ejpam-6044	17	30	concepts	concept	NOUN
ejpam-6044	17	31	and	and	CCONJ
ejpam-6044	17	32	terminologies	terminology	NOUN
ejpam-6044	17	33	.	.	PUNCT
ejpam-6044	18	1	involution	involution	NOUN
ejpam-6044	18	2	is	be	AUX
ejpam-6044	18	3	an	an	DET
ejpam-6044	18	4	additive	additive	ADJ
ejpam-6044	18	5	mapping	mapping	NOUN
ejpam-6044	18	6	defined	define	VERB
ejpam-6044	18	7	as	as	ADP
ejpam-6044	18	8	∗	∗	NOUN
ejpam-6044	18	9	from	from	ADP
ejpam-6044	18	10	r	r	NOUN
ejpam-6044	18	11	to	to	ADP
ejpam-6044	18	12	r	r	NOUN
ejpam-6044	18	13	that	that	PRON
ejpam-6044	18	14	fulfills	fulfill	VERB
ejpam-6044	18	15	the	the	DET
ejpam-6044	18	16	following	follow	VERB
ejpam-6044	18	17	two	two	NUM
ejpam-6044	18	18	requirements	requirement	NOUN
ejpam-6044	18	19	:	:	PUNCT
ejpam-6044	18	20	(	(	PUNCT
ejpam-6044	18	21	dj)∗	dj)∗	PROPN
ejpam-6044	18	22	=	=	SYM
ejpam-6044	18	23	j∗d∗	j∗d∗	NOUN
ejpam-6044	18	24	and	and	CCONJ
ejpam-6044	18	25	(	(	PUNCT
ejpam-6044	18	26	d∗)∗	d∗)∗	PROPN
ejpam-6044	18	27	=	=	SYM
ejpam-6044	18	28	d	d	PROPN
ejpam-6044	18	29	for	for	ADP
ejpam-6044	18	30	each	each	DET
ejpam-6044	18	31	d	d	PROPN
ejpam-6044	18	32	,	,	PUNCT
ejpam-6044	18	33	j	j	PROPN
ejpam-6044	18	34	∈	∈	PROPN
ejpam-6044	18	35	r.	r.	PROPN
ejpam-6044	18	36	invertible	invertible	ADJ
ejpam-6044	18	37	matrices	matrix	NOUN
ejpam-6044	18	38	and	and	CCONJ
ejpam-6044	18	39	identity	identity	NOUN
ejpam-6044	18	40	matrices	matrix	NOUN
ejpam-6044	18	41	are	be	AUX
ejpam-6044	18	42	the	the	DET
ejpam-6044	18	43	most	most	ADV
ejpam-6044	18	44	common	common	ADJ
ejpam-6044	18	45	examples	example	NOUN
ejpam-6044	18	46	of	of	ADP
ejpam-6044	18	47	involution	involution	NOUN
ejpam-6044	18	48	over	over	ADP
ejpam-6044	18	49	the	the	DET
ejpam-6044	18	50	matrix	matrix	NOUN
ejpam-6044	18	51	ring	ring	NOUN
ejpam-6044	18	52	.	.	PUNCT
ejpam-6044	19	1	a	a	DET
ejpam-6044	19	2	∗-ring	∗-ring	NOUN
ejpam-6044	19	3	,	,	PUNCT
ejpam-6044	19	4	sometimes	sometimes	ADV
ejpam-6044	19	5	referred	refer	VERB
ejpam-6044	19	6	to	to	ADP
ejpam-6044	19	7	as	as	ADP
ejpam-6044	19	8	an	an	DET
ejpam-6044	19	9	involution	involution	NOUN
ejpam-6044	19	10	ring	ring	NOUN
ejpam-6044	19	11	(	(	PUNCT
ejpam-6044	19	12	or	or	CCONJ
ejpam-6044	19	13	ring	ring	NOUN
ejpam-6044	19	14	combined	combine	VERB
ejpam-6044	19	15	with	with	ADP
ejpam-6044	19	16	an	an	DET
ejpam-6044	19	17	involution	involution	NOUN
ejpam-6044	19	18	∗	∗	NOUN
ejpam-6044	19	19	)	)	PUNCT
ejpam-6044	19	20	.	.	PUNCT
ejpam-6044	20	1	the	the	DET
ejpam-6044	20	2	references	reference	NOUN
ejpam-6044	20	3	[	[	X
ejpam-6044	20	4	1	1	NUM
ejpam-6044	20	5	]	]	PUNCT
ejpam-6044	20	6	,	,	PUNCT
ejpam-6044	20	7	[	[	X
ejpam-6044	20	8	2	2	NUM
ejpam-6044	20	9	]	]	PUNCT
ejpam-6044	20	10	,	,	PUNCT
ejpam-6044	20	11	[	[	X
ejpam-6044	20	12	3	3	NUM
ejpam-6044	20	13	]	]	PUNCT
ejpam-6044	20	14	,	,	PUNCT
ejpam-6044	20	15	[	[	X
ejpam-6044	20	16	4	4	NUM
ejpam-6044	20	17	]	]	PUNCT
ejpam-6044	20	18	,	,	PUNCT
ejpam-6044	20	19	[	[	X
ejpam-6044	20	20	5	5	NUM
ejpam-6044	20	21	]	]	PUNCT
ejpam-6044	20	22	,	,	PUNCT
ejpam-6044	20	23	[	[	X
ejpam-6044	20	24	6	6	NUM
ejpam-6044	20	25	]	]	PUNCT
ejpam-6044	20	26	are	be	AUX
ejpam-6044	20	27	ideal	ideal	ADJ
ejpam-6044	20	28	places	place	NOUN
ejpam-6044	20	29	for	for	ADP
ejpam-6044	20	30	start	start	NOUN
ejpam-6044	20	31	reading	read	VERB
ejpam-6044	20	32	about	about	ADP
ejpam-6044	20	33	generalized	generalized	ADJ
ejpam-6044	20	34	derivations	derivation	NOUN
ejpam-6044	20	35	,	,	PUNCT
ejpam-6044	20	36	involution	involution	NOUN
ejpam-6044	20	37	,	,	PUNCT
ejpam-6044	20	38	centralizers	centralizer	NOUN
ejpam-6044	20	39	and	and	CCONJ
ejpam-6044	20	40	their	their	PRON
ejpam-6044	20	41	related	related	ADJ
ejpam-6044	20	42	topics	topic	NOUN
ejpam-6044	20	43	.	.	PUNCT
ejpam-6044	21	1	a	a	DET
ejpam-6044	21	2	ring	ring	NOUN
ejpam-6044	21	3	possessing	possess	VERB
ejpam-6044	21	4	involution	involution	NOUN
ejpam-6044	21	5	∗	∗	NOUN
ejpam-6044	21	6	is	be	AUX
ejpam-6044	21	7	called	call	VERB
ejpam-6044	21	8	a	a	DET
ejpam-6044	21	9	∗-prime	∗-prime	ADJ
ejpam-6044	21	10	ring	ring	NOUN
ejpam-6044	21	11	if	if	SCONJ
ejpam-6044	21	12	arb	arb	PROPN
ejpam-6044	21	13	=	=	PUNCT
ejpam-6044	21	14	arb∗	arb∗	PROPN
ejpam-6044	21	15	=	=	SYM
ejpam-6044	21	16	{	{	PUNCT
ejpam-6044	21	17	0	0	NUM
ejpam-6044	21	18	}	}	PUNCT
ejpam-6044	21	19	,	,	PUNCT
ejpam-6044	21	20	or	or	CCONJ
ejpam-6044	21	21	arb	arb	X
ejpam-6044	21	22	=	=	PUNCT
ejpam-6044	21	23	a∗rb	a∗rb	PROPN
ejpam-6044	21	24	=	=	PUNCT
ejpam-6044	21	25	{	{	PUNCT
ejpam-6044	21	26	0	0	NUM
ejpam-6044	21	27	}	}	PUNCT
ejpam-6044	21	28	,	,	PUNCT
ejpam-6044	21	29	where	where	SCONJ
ejpam-6044	21	30	a	a	DET
ejpam-6044	21	31	,	,	PUNCT
ejpam-6044	21	32	b	b	X
ejpam-6044	21	33	∈	∈	PROPN
ejpam-6044	21	34	r	r	NOUN
ejpam-6044	21	35	,	,	PUNCT
ejpam-6044	21	36	implies	imply	VERB
ejpam-6044	21	37	that	that	SCONJ
ejpam-6044	21	38	either	either	CCONJ
ejpam-6044	21	39	a	a	DET
ejpam-6044	21	40	=	=	SYM
ejpam-6044	21	41	0	0	NUM
ejpam-6044	21	42	or	or	CCONJ
ejpam-6044	21	43	b	b	NOUN
ejpam-6044	21	44	=	=	SYM
ejpam-6044	21	45	0	0	PROPN
ejpam-6044	21	46	.	.	PUNCT
ejpam-6044	22	1	it	it	PRON
ejpam-6044	22	2	is	be	AUX
ejpam-6044	22	3	a	a	DET
ejpam-6044	22	4	noticeable	noticeable	ADJ
ejpam-6044	22	5	fact	fact	NOUN
ejpam-6044	22	6	that	that	SCONJ
ejpam-6044	22	7	all	all	DET
ejpam-6044	22	8	prime	prime	ADJ
ejpam-6044	22	9	rings	ring	NOUN
ejpam-6044	22	10	possessing	possess	VERB
ejpam-6044	22	11	involution	involution	NOUN
ejpam-6044	22	12	∗	∗	NOUN
ejpam-6044	22	13	are	be	AUX
ejpam-6044	22	14	∗-prime	∗-prime	ADJ
ejpam-6044	22	15	but	but	CCONJ
ejpam-6044	22	16	not	not	PART
ejpam-6044	22	17	necessarily	necessarily	ADV
ejpam-6044	22	18	prime	prime	ADJ
ejpam-6044	22	19	.	.	PUNCT
ejpam-6044	23	1	for	for	ADP
ejpam-6044	23	2	example	example	NOUN
ejpam-6044	23	3	,	,	PUNCT
ejpam-6044	23	4	r0	r0	NOUN
ejpam-6044	23	5	denotes	denote	VERB
ejpam-6044	23	6	the	the	DET
ejpam-6044	23	7	opposite	opposite	ADJ
ejpam-6044	23	8	ring	ring	NOUN
ejpam-6044	23	9	of	of	ADP
ejpam-6044	23	10	a	a	DET
ejpam-6044	23	11	prime	prime	ADJ
ejpam-6044	23	12	ring	ring	NOUN
ejpam-6044	23	13	r	r	NOUN
ejpam-6044	23	14	,	,	PUNCT
ejpam-6044	23	15	then	then	ADV
ejpam-6044	23	16	r	r	NOUN
ejpam-6044	23	17	×	×	PROPN
ejpam-6044	23	18	r0	r0	NOUN
ejpam-6044	23	19	having	have	VERB
ejpam-6044	23	20	exchange	exchange	NOUN
ejpam-6044	23	21	involution	involution	NOUN
ejpam-6044	23	22	∗xe	∗xe	PROPN
ejpam-6044	23	23	defined	define	VERB
ejpam-6044	23	24	as	as	ADP
ejpam-6044	23	25	∗xe(x	∗xe(x	PROPN
ejpam-6044	23	26	,	,	PUNCT
ejpam-6044	23	27	y	y	NOUN
ejpam-6044	23	28	)	)	PUNCT
ejpam-6044	23	29	=	=	SYM
ejpam-6044	23	30	(	(	PUNCT
ejpam-6044	23	31	y	y	PROPN
ejpam-6044	23	32	,	,	PUNCT
ejpam-6044	23	33	x	x	X
ejpam-6044	23	34	)	)	PUNCT
ejpam-6044	23	35	is	be	AUX
ejpam-6044	23	36	a	a	DET
ejpam-6044	23	37	∗xe	∗xe	NOUN
ejpam-6044	23	38	-	-	PUNCT
ejpam-6044	23	39	prime	prime	ADJ
ejpam-6044	23	40	but	but	CCONJ
ejpam-6044	23	41	not	not	PART
ejpam-6044	23	42	prime	prime	ADJ
ejpam-6044	23	43	.	.	PUNCT
ejpam-6044	24	1	let	let	VERB
ejpam-6044	24	2	r	r	PRON
ejpam-6044	24	3	be	be	AUX
ejpam-6044	24	4	a	a	DET
ejpam-6044	24	5	∗-ring	∗-ring	NOUN
ejpam-6044	24	6	.	.	PUNCT
ejpam-6044	25	1	a	a	DET
ejpam-6044	25	2	mapping	mapping	NOUN
ejpam-6044	25	3	d	d	NOUN
ejpam-6044	25	4	:	:	PUNCT
ejpam-6044	25	5	r	r	NOUN
ejpam-6044	25	6	→	→	SYM
ejpam-6044	25	7	r	r	NOUN
ejpam-6044	25	8	is	be	AUX
ejpam-6044	25	9	said	say	VERB
ejpam-6044	25	10	to	to	PART
ejpam-6044	25	11	be	be	AUX
ejpam-6044	25	12	a	a	DET
ejpam-6044	25	13	∗-derivation	∗-derivation	NOUN
ejpam-6044	25	14	if	if	SCONJ
ejpam-6044	25	15	it	it	PRON
ejpam-6044	25	16	satisfies	satisfy	VERB
ejpam-6044	25	17	:	:	PUNCT
ejpam-6044	25	18	(	(	PUNCT
ejpam-6044	25	19	i	i	NOUN
ejpam-6044	25	20	)	)	PUNCT
ejpam-6044	25	21	additivity	additivity	NOUN
ejpam-6044	25	22	and	and	CCONJ
ejpam-6044	25	23	(	(	PUNCT
ejpam-6044	25	24	ii	ii	NOUN
ejpam-6044	25	25	)	)	PUNCT
ejpam-6044	25	26	d(xy	d(xy	PROPN
ejpam-6044	25	27	)	)	PUNCT
ejpam-6044	25	28	=	=	SYM
ejpam-6044	25	29	d(x)y∗	d(x)y∗	PROPN
ejpam-6044	25	30	+	+	NUM
ejpam-6044	25	31	xd(y	xd(y	NUM
ejpam-6044	25	32	)	)	PUNCT
ejpam-6044	25	33	for	for	ADP
ejpam-6044	25	34	all	all	DET
ejpam-6044	25	35	x	x	NOUN
ejpam-6044	25	36	,	,	PUNCT
ejpam-6044	25	37	y	y	PROPN
ejpam-6044	25	38	∈	∈	PROPN
ejpam-6044	25	39	r.	r.	PROPN
ejpam-6044	25	40	in	in	ADP
ejpam-6044	25	41	the	the	DET
ejpam-6044	25	42	case	case	NOUN
ejpam-6044	25	43	where	where	SCONJ
ejpam-6044	25	44	r	r	NOUN
ejpam-6044	25	45	is	be	AUX
ejpam-6044	25	46	a	a	DET
ejpam-6044	25	47	commutative	commutative	ADJ
ejpam-6044	25	48	∗-ring	∗-ring	NOUN
ejpam-6044	25	49	,	,	PUNCT
ejpam-6044	25	50	d	d	PROPN
ejpam-6044	25	51	has	have	VERB
ejpam-6044	25	52	the	the	DET
ejpam-6044	25	53	form	form	NOUN
ejpam-6044	25	54	d(x	d(x	NOUN
ejpam-6044	25	55	)	)	PUNCT
ejpam-6044	26	1	=	=	PUNCT
ejpam-6044	26	2	a(x	a(x	NOUN
ejpam-6044	26	3	−	−	NOUN
ejpam-6044	26	4	x∗	x∗	NOUN
ejpam-6044	26	5	)	)	PUNCT
ejpam-6044	26	6	for	for	ADP
ejpam-6044	26	7	some	some	PRON
ejpam-6044	26	8	a	a	DET
ejpam-6044	26	9	∈	∈	PROPN
ejpam-6044	26	10	r	r	NOUN
ejpam-6044	26	11	,	,	PUNCT
ejpam-6044	26	12	which	which	PRON
ejpam-6044	26	13	is	be	AUX
ejpam-6044	26	14	a	a	DET
ejpam-6044	26	15	∗-derivation	∗-derivation	NOUN
ejpam-6044	26	16	on	on	ADP
ejpam-6044	26	17	r.	r.	NOUN
ejpam-6044	26	18	following	follow	VERB
ejpam-6044	26	19	[	[	X
ejpam-6044	26	20	7	7	NUM
ejpam-6044	26	21	]	]	PUNCT
ejpam-6044	26	22	,	,	PUNCT
ejpam-6044	26	23	a	a	DET
ejpam-6044	26	24	mapping	mapping	NOUN
ejpam-6044	26	25	t	t	NOUN
ejpam-6044	26	26	:	:	PUNCT
ejpam-6044	26	27	r	r	NOUN
ejpam-6044	26	28	→	→	SYM
ejpam-6044	26	29	r	r	NOUN
ejpam-6044	26	30	is	be	AUX
ejpam-6044	26	31	called	call	VERB
ejpam-6044	26	32	a	a	DET
ejpam-6044	26	33	left	left	ADJ
ejpam-6044	26	34	(	(	PUNCT
ejpam-6044	26	35	right	right	ADJ
ejpam-6044	26	36	)	)	PUNCT
ejpam-6044	26	37	centralizer	centralizer	NOUN
ejpam-6044	26	38	if	if	SCONJ
ejpam-6044	26	39	t	t	PROPN
ejpam-6044	26	40	(	(	PUNCT
ejpam-6044	26	41	xy	xy	NOUN
ejpam-6044	26	42	)	)	PUNCT
ejpam-6044	26	43	=	=	SYM
ejpam-6044	26	44	t	t	PROPN
ejpam-6044	26	45	(	(	PUNCT
ejpam-6044	26	46	x)y	x)y	PUNCT
ejpam-6044	26	47	(	(	PUNCT
ejpam-6044	26	48	t	t	X
ejpam-6044	26	49	(	(	PUNCT
ejpam-6044	26	50	xy	xy	NOUN
ejpam-6044	26	51	)	)	PUNCT
ejpam-6044	26	52	=	=	SYM
ejpam-6044	26	53	xt	xt	X
ejpam-6044	26	54	(	(	PUNCT
ejpam-6044	26	55	y	y	NOUN
ejpam-6044	26	56	)	)	PUNCT
ejpam-6044	26	57	)	)	PUNCT
ejpam-6044	26	58	holds	hold	VERB
ejpam-6044	26	59	for	for	ADP
ejpam-6044	26	60	all	all	DET
ejpam-6044	26	61	x	x	NOUN
ejpam-6044	26	62	,	,	PUNCT
ejpam-6044	26	63	y	y	PROPN
ejpam-6044	26	64	∈	∈	PROPN
ejpam-6044	26	65	r	r	NOUN
ejpam-6044	26	66	and	and	CCONJ
ejpam-6044	26	67	t	t	PROPN
ejpam-6044	26	68	is	be	AUX
ejpam-6044	26	69	also	also	ADV
ejpam-6044	26	70	additive	additive	ADJ
ejpam-6044	26	71	.	.	PUNCT
ejpam-6044	27	1	in	in	ADP
ejpam-6044	27	2	the	the	DET
ejpam-6044	27	3	same	same	ADJ
ejpam-6044	27	4	line	line	NOUN
ejpam-6044	27	5	of	of	ADP
ejpam-6044	27	6	investigation	investigation	NOUN
ejpam-6044	27	7	,	,	PUNCT
ejpam-6044	27	8	the	the	DET
ejpam-6044	27	9	expression	expression	NOUN
ejpam-6044	27	10	for	for	ADP
ejpam-6044	27	11	∗-centralizer	∗-centralizer	VERB
ejpam-6044	27	12	comes	come	VERB
ejpam-6044	27	13	out	out	ADP
ejpam-6044	27	14	as	as	SCONJ
ejpam-6044	27	15	follows	follow	VERB
ejpam-6044	27	16	:	:	PUNCT
ejpam-6044	27	17	a	a	DET
ejpam-6044	27	18	mapping	mapping	NOUN
ejpam-6044	27	19	t	t	NOUN
ejpam-6044	27	20	on	on	ADP
ejpam-6044	27	21	r	r	NOUN
ejpam-6044	27	22	,	,	PUNCT
ejpam-6044	27	23	additive	additive	ADJ
ejpam-6044	27	24	and	and	CCONJ
ejpam-6044	27	25	satisfying	satisfying	ADJ
ejpam-6044	27	26	t	t	PROPN
ejpam-6044	27	27	(	(	PUNCT
ejpam-6044	27	28	xy	xy	NOUN
ejpam-6044	27	29	)	)	PUNCT
ejpam-6044	28	1	=	=	SYM
ejpam-6044	28	2	t	t	PROPN
ejpam-6044	28	3	(	(	PUNCT
ejpam-6044	28	4	x)y∗	x)y∗	PROPN
ejpam-6044	28	5	and	and	CCONJ
ejpam-6044	28	6	t	t	PROPN
ejpam-6044	28	7	(	(	PUNCT
ejpam-6044	28	8	xy	xy	PROPN
ejpam-6044	28	9	)	)	PUNCT
ejpam-6044	28	10	=	=	SYM
ejpam-6044	28	11	x∗t	x∗t	NUM
ejpam-6044	28	12	(	(	PUNCT
ejpam-6044	28	13	y	y	NOUN
ejpam-6044	28	14	)	)	PUNCT
ejpam-6044	28	15	for	for	ADP
ejpam-6044	28	16	all	all	DET
ejpam-6044	28	17	x	x	NOUN
ejpam-6044	28	18	,	,	PUNCT
ejpam-6044	28	19	y	y	PROPN
ejpam-6044	28	20	∈	∈	PROPN
ejpam-6044	28	21	r	r	NOUN
ejpam-6044	28	22	will	will	AUX
ejpam-6044	28	23	be	be	AUX
ejpam-6044	28	24	called	call	VERB
ejpam-6044	28	25	left	left	ADJ
ejpam-6044	28	26	∗-centralizer	∗-centralizer	NOUN
ejpam-6044	28	27	and	and	CCONJ
ejpam-6044	28	28	right	right	ADJ
ejpam-6044	28	29	∗-centralizer	∗-centralizer	NOUN
ejpam-6044	28	30	respectively	respectively	ADV
ejpam-6044	28	31	on	on	ADP
ejpam-6044	28	32	r.	r.	PROPN
ejpam-6044	28	33	a	a	DET
ejpam-6044	28	34	remarkable	remarkable	ADJ
ejpam-6044	28	35	investigation	investigation	NOUN
ejpam-6044	28	36	on	on	ADP
ejpam-6044	28	37	the	the	DET
ejpam-6044	28	38	theory	theory	NOUN
ejpam-6044	28	39	of	of	ADP
ejpam-6044	28	40	centralizers	centralizer	NOUN
ejpam-6044	28	41	and	and	CCONJ
ejpam-6044	28	42	∗-centralizers	∗-centralizer	NOUN
ejpam-6044	28	43	presented	present	VERB
ejpam-6044	28	44	in	in	ADP
ejpam-6044	28	45	[	[	X
ejpam-6044	28	46	8–11	8–11	NOUN
ejpam-6044	28	47	]	]	PUNCT
ejpam-6044	28	48	.	.	PUNCT
ejpam-6044	29	1	in	in	ADP
ejpam-6044	29	2	[	[	X
ejpam-6044	29	3	4	4	NUM
ejpam-6044	29	4	]	]	PUNCT
ejpam-6044	29	5	,	,	PUNCT
ejpam-6044	29	6	authors	author	NOUN
ejpam-6044	29	7	proved	prove	VERB
ejpam-6044	29	8	an	an	DET
ejpam-6044	29	9	advancement	advancement	NOUN
ejpam-6044	29	10	of	of	ADP
ejpam-6044	29	11	the	the	DET
ejpam-6044	29	12	generalized	generalize	VERB
ejpam-6044	29	13	concept	concept	NOUN
ejpam-6044	29	14	of	of	ADP
ejpam-6044	29	15	∗-derivation	∗-derivation	NOUN
ejpam-6044	29	16	on	on	ADP
ejpam-6044	29	17	standard	standard	ADJ
ejpam-6044	29	18	operator	operator	NOUN
ejpam-6044	29	19	algebra	algebra	NOUN
ejpam-6044	29	20	.	.	PUNCT
ejpam-6044	30	1	a	a	DET
ejpam-6044	30	2	ring	ring	NOUN
ejpam-6044	30	3	with	with	ADP
ejpam-6044	30	4	endomorphism	endomorphism	PROPN
ejpam-6044	30	5	α	α	PROPN
ejpam-6044	30	6	,	,	PUNCT
ejpam-6044	30	7	if	if	SCONJ
ejpam-6044	30	8	we	we	PRON
ejpam-6044	30	9	take	take	VERB
ejpam-6044	30	10	γ	γ	NOUN
ejpam-6044	30	11	=	=	SYM
ejpam-6044	30	12	ς	ς	PROPN
ejpam-6044	30	13	−	−	PROPN
ejpam-6044	30	14	α	α	NOUN
ejpam-6044	30	15	,	,	PUNCT
ejpam-6044	30	16	then	then	ADV
ejpam-6044	30	17	γ	γ	PROPN
ejpam-6044	30	18	is	be	AUX
ejpam-6044	30	19	an	an	DET
ejpam-6044	30	20	(	(	PUNCT
ejpam-6044	30	21	α	α	NOUN
ejpam-6044	30	22	,	,	PUNCT
ejpam-6044	30	23	i)-derivation	i)-derivation	NOUN
ejpam-6044	30	24	,	,	PUNCT
ejpam-6044	30	25	but	but	CCONJ
ejpam-6044	30	26	not	not	PART
ejpam-6044	30	27	a	a	DET
ejpam-6044	30	28	derivation	derivation	NOUN
ejpam-6044	30	29	when	when	SCONJ
ejpam-6044	30	30	r	r	NOUN
ejpam-6044	30	31	is	be	AUX
ejpam-6044	30	32	semiprime	semiprime	NOUN
ejpam-6044	30	33	and	and	CCONJ
ejpam-6044	30	34	i	i	PRON
ejpam-6044	30	35	=	=	NOUN
ejpam-6044	31	1	α	α	X
ejpam-6044	31	2	.	.	PUNCT
ejpam-6044	32	1	the	the	DET
ejpam-6044	32	2	inclusive	inclusive	ADJ
ejpam-6044	32	3	information	information	NOUN
ejpam-6044	32	4	can	can	AUX
ejpam-6044	32	5	be	be	AUX
ejpam-6044	32	6	found	find	VERB
ejpam-6044	32	7	in	in	ADP
ejpam-6044	32	8	[	[	X
ejpam-6044	32	9	12	12	NUM
ejpam-6044	32	10	]	]	PUNCT
ejpam-6044	32	11	.	.	PUNCT
ejpam-6044	33	1	some	some	DET
ejpam-6044	33	2	commutativity	commutativity	NOUN
ejpam-6044	33	3	results	result	VERB
ejpam-6044	33	4	about	about	ADP
ejpam-6044	33	5	∗-bimultipliers	∗-bimultiplier	NOUN
ejpam-6044	33	6	and	and	CCONJ
ejpam-6044	33	7	generalized	generalized	ADJ
ejpam-6044	33	8	∗-biderivations	∗-biderivation	NOUN
ejpam-6044	33	9	can	can	AUX
ejpam-6044	33	10	be	be	AUX
ejpam-6044	33	11	viewed	view	VERB
ejpam-6044	33	12	in	in	ADP
ejpam-6044	33	13	[	[	X
ejpam-6044	33	14	3	3	NUM
ejpam-6044	33	15	]	]	PUNCT
ejpam-6044	33	16	.	.	PUNCT
ejpam-6044	34	1	we	we	PRON
ejpam-6044	34	2	review	review	VERB
ejpam-6044	34	3	the	the	DET
ejpam-6044	34	4	concept	concept	NOUN
ejpam-6044	34	5	of	of	ADP
ejpam-6044	34	6	such	such	ADJ
ejpam-6044	34	7	γ	γ	NOUN
ejpam-6044	34	8	and	and	CCONJ
ejpam-6044	34	9	introduce	introduce	VERB
ejpam-6044	34	10	the	the	DET
ejpam-6044	34	11	concept	concept	NOUN
ejpam-6044	34	12	of	of	ADP
ejpam-6044	34	13	(	(	PUNCT
ejpam-6044	34	14	α	α	NOUN
ejpam-6044	34	15	,	,	PUNCT
ejpam-6044	34	16	∗)derivation	∗)derivation	NOUN
ejpam-6044	34	17	and	and	CCONJ
ejpam-6044	34	18	generalized	generalize	VERB
ejpam-6044	34	19	(	(	PUNCT
ejpam-6044	34	20	α	α	NOUN
ejpam-6044	34	21	,	,	PUNCT
ejpam-6044	34	22	∗)-derivation	∗)-derivation	NOUN
ejpam-6044	34	23	on	on	ADP
ejpam-6044	34	24	r	r	NOUN
ejpam-6044	34	25	as	as	SCONJ
ejpam-6044	34	26	follows	follow	VERB
ejpam-6044	34	27	:	:	PUNCT
ejpam-6044	34	28	definition	definition	NOUN
ejpam-6044	34	29	1	1	NUM
ejpam-6044	34	30	.	.	PUNCT
ejpam-6044	35	1	let	let	VERB
ejpam-6044	35	2	d	d	NOUN
ejpam-6044	35	3	:	:	PUNCT
ejpam-6044	35	4	r	r	AUX
ejpam-6044	35	5	−→	−→	NOUN
ejpam-6044	35	6	r	r	NOUN
ejpam-6044	35	7	be	be	VERB
ejpam-6044	35	8	an	an	DET
ejpam-6044	35	9	additive	additive	ADJ
ejpam-6044	35	10	map	map	NOUN
ejpam-6044	35	11	.	.	PUNCT
ejpam-6044	36	1	d	d	NOUN
ejpam-6044	36	2	is	be	AUX
ejpam-6044	36	3	said	say	VERB
ejpam-6044	36	4	to	to	PART
ejpam-6044	36	5	be	be	AUX
ejpam-6044	36	6	(	(	PUNCT
ejpam-6044	36	7	α	α	NOUN
ejpam-6044	36	8	,	,	PUNCT
ejpam-6044	36	9	∗)-derivation	∗)-derivation	NOUN
ejpam-6044	36	10	on	on	ADP
ejpam-6044	36	11	r	r	NOUN
ejpam-6044	36	12	if	if	SCONJ
ejpam-6044	36	13	it	it	PRON
ejpam-6044	36	14	satisfy	satisfy	VERB
ejpam-6044	36	15	the	the	DET
ejpam-6044	36	16	conditions	condition	NOUN
ejpam-6044	36	17	:	:	PUNCT
ejpam-6044	36	18	d(νk	d(νk	NOUN
ejpam-6044	36	19	)	)	PUNCT
ejpam-6044	36	20	=	=	SYM
ejpam-6044	36	21	d(ν)α(k	d(ν)α(k	NOUN
ejpam-6044	36	22	)	)	PUNCT
ejpam-6044	36	23	+	+	CCONJ
ejpam-6044	36	24	ν∗d(k	ν∗d(k	PROPN
ejpam-6044	36	25	)	)	PUNCT
ejpam-6044	36	26	,	,	PUNCT
ejpam-6044	36	27	for	for	ADP
ejpam-6044	36	28	every	every	DET
ejpam-6044	36	29	ν	ν	NOUN
ejpam-6044	36	30	,	,	PUNCT
ejpam-6044	36	31	k	k	PROPN
ejpam-6044	36	32	∈	∈	PROPN
ejpam-6044	36	33	r	r	PROPN
ejpam-6044	36	34	,	,	PUNCT
ejpam-6044	36	35	f.	f.	PROPN
ejpam-6044	36	36	shujat	shujat	PROPN
ejpam-6044	36	37	,	,	PUNCT
ejpam-6044	36	38	s.	s.	PROPN
ejpam-6044	36	39	alharbi	alharbi	PROPN
ejpam-6044	36	40	/	/	SYM
ejpam-6044	36	41	eur	eur	PROPN
ejpam-6044	36	42	.	.	PUNCT
ejpam-6044	37	1	j.	j.	PROPN
ejpam-6044	37	2	pure	pure	PROPN
ejpam-6044	37	3	appl	appl	PROPN
ejpam-6044	37	4	.	.	PROPN
ejpam-6044	37	5	math	math	PROPN
ejpam-6044	37	6	,	,	PUNCT
ejpam-6044	37	7	18	18	NUM
ejpam-6044	37	8	(	(	PUNCT
ejpam-6044	37	9	2	2	NUM
ejpam-6044	37	10	)	)	PUNCT
ejpam-6044	37	11	(	(	PUNCT
ejpam-6044	37	12	2025	2025	NUM
ejpam-6044	37	13	)	)	PUNCT
ejpam-6044	37	14	,	,	PUNCT
ejpam-6044	37	15	6044	6044	NUM
ejpam-6044	37	16	3	3	NUM
ejpam-6044	37	17	of	of	ADP
ejpam-6044	37	18	11	11	NUM
ejpam-6044	37	19	where	where	SCONJ
ejpam-6044	37	20	∗	∗	NOUN
ejpam-6044	37	21	is	be	AUX
ejpam-6044	37	22	an	an	DET
ejpam-6044	37	23	involution	involution	NOUN
ejpam-6044	37	24	on	on	ADP
ejpam-6044	37	25	r	r	NOUN
ejpam-6044	37	26	and	and	CCONJ
ejpam-6044	37	27	α	α	NOUN
ejpam-6044	37	28	is	be	AUX
ejpam-6044	37	29	the	the	DET
ejpam-6044	37	30	automorphism	automorphism	NOUN
ejpam-6044	37	31	on	on	ADP
ejpam-6044	37	32	r.	r.	PROPN
ejpam-6044	37	33	example	example	NOUN
ejpam-6044	37	34	1	1	NUM
ejpam-6044	37	35	.	.	X
ejpam-6044	37	36	consider	consider	VERB
ejpam-6044	37	37	a	a	DET
ejpam-6044	37	38	∗-ring	∗-ring	NOUN
ejpam-6044	37	39	r	r	NOUN
ejpam-6044	37	40	=	=	PUNCT
ejpam-6044	37	41	{	{	PUNCT
ejpam-6044	37	42	(	(	PUNCT
ejpam-6044	37	43	p	p	NOUN
ejpam-6044	37	44	0	0	NUM
ejpam-6044	37	45	q	q	NOUN
ejpam-6044	37	46	r	r	NOUN
ejpam-6044	37	47	)	)	PUNCT
ejpam-6044	38	1	|	|	ADV
ejpam-6044	38	2	p	p	X
ejpam-6044	38	3	,	,	PUNCT
ejpam-6044	38	4	q	q	INTJ
ejpam-6044	38	5	,	,	PUNCT
ejpam-6044	38	6	r	r	NOUN
ejpam-6044	38	7	∈	∈	PROPN
ejpam-6044	38	8	2z4	2z4	NUM
ejpam-6044	38	9	}	}	PUNCT
ejpam-6044	38	10	.	.	PUNCT
ejpam-6044	39	1	define	define	VERB
ejpam-6044	39	2	involution	involution	NOUN
ejpam-6044	39	3	mapping	mapping	NOUN
ejpam-6044	39	4	∗	∗	NOUN
ejpam-6044	39	5	from	from	ADP
ejpam-6044	39	6	r	r	NOUN
ejpam-6044	39	7	to	to	ADP
ejpam-6044	39	8	itself	itself	PRON
ejpam-6044	39	9	by	by	ADP
ejpam-6044	39	10	(	(	PUNCT
ejpam-6044	39	11	p	p	NOUN
ejpam-6044	39	12	0	0	NUM
ejpam-6044	39	13	q	q	NOUN
ejpam-6044	39	14	r	r	NOUN
ejpam-6044	39	15	)	)	PUNCT
ejpam-6044	39	16	∗	∗	NOUN
ejpam-6044	39	17	=	=	SYM
ejpam-6044	39	18	(	(	PUNCT
ejpam-6044	39	19	−p	−p	NOUN
ejpam-6044	39	20	0	0	NUM
ejpam-6044	39	21	0	0	NUM
ejpam-6044	39	22	0	0	NUM
ejpam-6044	39	23	)	)	PUNCT
ejpam-6044	39	24	for	for	ADP
ejpam-6044	39	25	all	all	DET
ejpam-6044	39	26	p	p	NOUN
ejpam-6044	39	27	∈	∈	PROPN
ejpam-6044	39	28	2z4	2z4	NUM
ejpam-6044	39	29	under	under	ADP
ejpam-6044	39	30	matrix	matrix	NOUN
ejpam-6044	39	31	addition	addition	NOUN
ejpam-6044	39	32	and	and	CCONJ
ejpam-6044	39	33	matrix	matrix	NOUN
ejpam-6044	39	34	multiplication	multiplication	NOUN
ejpam-6044	39	35	,	,	PUNCT
ejpam-6044	39	36	where	where	SCONJ
ejpam-6044	39	37	z4	z4	PROPN
ejpam-6044	39	38	has	have	VERB
ejpam-6044	39	39	its	its	PRON
ejpam-6044	39	40	usual	usual	ADJ
ejpam-6044	39	41	notation	notation	NOUN
ejpam-6044	39	42	.	.	PUNCT
ejpam-6044	40	1	take	take	VERB
ejpam-6044	40	2	a	a	DET
ejpam-6044	40	3	mapping	mapping	NOUN
ejpam-6044	40	4	α	α	NOUN
ejpam-6044	40	5	,	,	PUNCT
ejpam-6044	40	6	d	d	NOUN
ejpam-6044	40	7	:	:	PUNCT
ejpam-6044	40	8	r	r	NOUN
ejpam-6044	40	9	→	→	SYM
ejpam-6044	40	10	r	r	NOUN
ejpam-6044	40	11	defined	define	VERB
ejpam-6044	40	12	by	by	ADP
ejpam-6044	40	13	α	α	PROPN
ejpam-6044	40	14	[	[	X
ejpam-6044	40	15	(	(	PUNCT
ejpam-6044	40	16	p	p	NOUN
ejpam-6044	40	17	0	0	NUM
ejpam-6044	40	18	q	q	NOUN
ejpam-6044	40	19	r	r	NOUN
ejpam-6044	40	20	)	)	PUNCT
ejpam-6044	40	21	]	]	PUNCT
ejpam-6044	41	1	=	=	PUNCT
ejpam-6044	41	2	(	(	PUNCT
ejpam-6044	41	3	r	r	NOUN
ejpam-6044	41	4	0	0	NUM
ejpam-6044	41	5	q	q	NOUN
ejpam-6044	41	6	p	p	NOUN
ejpam-6044	41	7	)	)	PUNCT
ejpam-6044	41	8	and	and	CCONJ
ejpam-6044	41	9	d	d	X
ejpam-6044	42	1	[	[	X
ejpam-6044	42	2	(	(	PUNCT
ejpam-6044	42	3	p	p	NOUN
ejpam-6044	42	4	0	0	NUM
ejpam-6044	42	5	q	q	NOUN
ejpam-6044	42	6	r	r	NOUN
ejpam-6044	42	7	)	)	PUNCT
ejpam-6044	42	8	]	]	PUNCT
ejpam-6044	43	1	=	=	PUNCT
ejpam-6044	43	2	(	(	PUNCT
ejpam-6044	43	3	0	0	NUM
ejpam-6044	43	4	0	0	NUM
ejpam-6044	43	5	q	q	NOUN
ejpam-6044	43	6	0	0	NUM
ejpam-6044	43	7	)	)	PUNCT
ejpam-6044	43	8	for	for	ADP
ejpam-6044	43	9	all	all	DET
ejpam-6044	43	10	p	p	NOUN
ejpam-6044	43	11	,	,	PUNCT
ejpam-6044	43	12	q	q	ADJ
ejpam-6044	43	13	,	,	PUNCT
ejpam-6044	43	14	r	r	NOUN
ejpam-6044	43	15	∈	∈	PROPN
ejpam-6044	43	16	2z4	2z4	NUM
ejpam-6044	43	17	.	.	PUNCT
ejpam-6044	44	1	it	it	PRON
ejpam-6044	44	2	is	be	AUX
ejpam-6044	44	3	clear	clear	ADJ
ejpam-6044	44	4	that	that	SCONJ
ejpam-6044	44	5	d	d	AUX
ejpam-6044	44	6	satisfy	satisfy	VERB
ejpam-6044	44	7	the	the	DET
ejpam-6044	44	8	above	above	ADJ
ejpam-6044	44	9	definition	definition	NOUN
ejpam-6044	44	10	,	,	PUNCT
ejpam-6044	44	11	hence	hence	ADV
ejpam-6044	44	12	it	it	PRON
ejpam-6044	44	13	is	be	AUX
ejpam-6044	44	14	(	(	PUNCT
ejpam-6044	44	15	α	α	NOUN
ejpam-6044	44	16	,	,	PUNCT
ejpam-6044	44	17	∗)-derivation	∗)-derivation	NOUN
ejpam-6044	44	18	on	on	ADP
ejpam-6044	44	19	r.	r.	NOUN
ejpam-6044	44	20	we	we	PRON
ejpam-6044	44	21	observe	observe	VERB
ejpam-6044	44	22	that	that	SCONJ
ejpam-6044	44	23	if	if	SCONJ
ejpam-6044	44	24	∗	∗	NOUN
ejpam-6044	44	25	=	=	PUNCT
ejpam-6044	44	26	i	i	PRON
ejpam-6044	44	27	,	,	PUNCT
ejpam-6044	44	28	the	the	DET
ejpam-6044	44	29	definition	definition	NOUN
ejpam-6044	44	30	1	1	NUM
ejpam-6044	44	31	will	will	AUX
ejpam-6044	44	32	be	be	AUX
ejpam-6044	44	33	set	set	VERB
ejpam-6044	44	34	as	as	ADP
ejpam-6044	44	35	skew	skew	ADJ
ejpam-6044	44	36	derivation	derivation	NOUN
ejpam-6044	44	37	with	with	ADP
ejpam-6044	44	38	automorphism	automorphism	NOUN
ejpam-6044	44	39	α	α	NOUN
ejpam-6044	44	40	.	.	PUNCT
ejpam-6044	45	1	it	it	PRON
ejpam-6044	45	2	is	be	AUX
ejpam-6044	45	3	somewhat	somewhat	ADV
ejpam-6044	45	4	a	a	DET
ejpam-6044	45	5	unified	unified	ADJ
ejpam-6044	45	6	notion	notion	NOUN
ejpam-6044	45	7	of	of	ADP
ejpam-6044	45	8	skew	skew	ADJ
ejpam-6044	45	9	derivation	derivation	NOUN
ejpam-6044	45	10	and	and	CCONJ
ejpam-6044	45	11	∗-derivation	∗-derivation	NOUN
ejpam-6044	45	12	.	.	PUNCT
ejpam-6044	46	1	next	next	ADV
ejpam-6044	46	2	we	we	PRON
ejpam-6044	46	3	extend	extend	VERB
ejpam-6044	46	4	our	our	PRON
ejpam-6044	46	5	definition	definition	NOUN
ejpam-6044	46	6	to	to	ADP
ejpam-6044	46	7	the	the	DET
ejpam-6044	46	8	case	case	NOUN
ejpam-6044	46	9	of	of	ADP
ejpam-6044	46	10	generalized	generalized	ADJ
ejpam-6044	46	11	derivation	derivation	NOUN
ejpam-6044	46	12	.	.	PUNCT
ejpam-6044	47	1	definition	definition	NOUN
ejpam-6044	47	2	2	2	NUM
ejpam-6044	47	3	.	.	PUNCT
ejpam-6044	48	1	let	let	VERB
ejpam-6044	48	2	f	f	X
ejpam-6044	48	3	,	,	PUNCT
ejpam-6044	48	4	d	d	NOUN
ejpam-6044	48	5	:	:	PUNCT
ejpam-6044	48	6	r	r	NOUN
ejpam-6044	48	7	−→	−→	NOUN
ejpam-6044	48	8	r	r	NOUN
ejpam-6044	48	9	be	be	VERB
ejpam-6044	48	10	additive	additive	ADJ
ejpam-6044	48	11	maps	map	NOUN
ejpam-6044	48	12	.	.	PUNCT
ejpam-6044	49	1	f	f	PROPN
ejpam-6044	49	2	is	be	AUX
ejpam-6044	49	3	said	say	VERB
ejpam-6044	49	4	to	to	PART
ejpam-6044	49	5	be	be	AUX
ejpam-6044	49	6	generalized	generalize	VERB
ejpam-6044	49	7	(	(	PUNCT
ejpam-6044	49	8	α	α	NOUN
ejpam-6044	49	9	,	,	PUNCT
ejpam-6044	49	10	∗)derivation	∗)derivation	NOUN
ejpam-6044	49	11	associated	associate	VERB
ejpam-6044	49	12	with	with	ADP
ejpam-6044	49	13	d	d	PROPN
ejpam-6044	49	14	on	on	ADP
ejpam-6044	49	15	r	r	NOUN
ejpam-6044	49	16	if	if	SCONJ
ejpam-6044	49	17	it	it	PRON
ejpam-6044	49	18	satisfy	satisfy	VERB
ejpam-6044	49	19	the	the	DET
ejpam-6044	49	20	below	below	ADP
ejpam-6044	49	21	condition	condition	NOUN
ejpam-6044	49	22	f(νk	f(νk	NOUN
ejpam-6044	49	23	)	)	PUNCT
ejpam-6044	49	24	=	=	SYM
ejpam-6044	49	25	f(ν)α(k	f(ν)α(k	ADJ
ejpam-6044	49	26	)	)	PUNCT
ejpam-6044	49	27	+	+	CCONJ
ejpam-6044	49	28	ν∗d(k	ν∗d(k	PROPN
ejpam-6044	49	29	)	)	PUNCT
ejpam-6044	49	30	,	,	PUNCT
ejpam-6044	49	31	for	for	ADP
ejpam-6044	49	32	every	every	DET
ejpam-6044	49	33	ν	ν	NOUN
ejpam-6044	49	34	,	,	PUNCT
ejpam-6044	49	35	k	k	PROPN
ejpam-6044	49	36	∈	∈	PROPN
ejpam-6044	49	37	r	r	NOUN
ejpam-6044	49	38	,	,	PUNCT
ejpam-6044	49	39	where	where	SCONJ
ejpam-6044	49	40	∗	∗	NOUN
ejpam-6044	49	41	is	be	AUX
ejpam-6044	49	42	an	an	DET
ejpam-6044	49	43	involution	involution	NOUN
ejpam-6044	49	44	on	on	ADP
ejpam-6044	49	45	r	r	NOUN
ejpam-6044	49	46	and	and	CCONJ
ejpam-6044	49	47	α	α	NOUN
ejpam-6044	49	48	is	be	AUX
ejpam-6044	49	49	the	the	DET
ejpam-6044	49	50	automorphism	automorphism	NOUN
ejpam-6044	49	51	on	on	ADP
ejpam-6044	49	52	r.	r.	PROPN
ejpam-6044	49	53	example	example	NOUN
ejpam-6044	50	1	2	2	NUM
ejpam-6044	50	2	.	.	PUNCT
ejpam-6044	50	3	let	let	VERB
ejpam-6044	50	4	r	r	NOUN
ejpam-6044	50	5	=	=	PUNCT
ejpam-6044	50	6			PUNCT
ejpam-6044	50	7	0	0	ADV
ejpam-6044	50	8	a	a	DET
ejpam-6044	50	9	b	b	NOUN
ejpam-6044	50	10	0	0	NUM
ejpam-6044	50	11	0	0	NUM
ejpam-6044	50	12	c	c	NOUN
ejpam-6044	50	13	0	0	NUM
ejpam-6044	50	14	0	0	NUM
ejpam-6044	50	15	0	0	NUM
ejpam-6044	51	1			PROPN
ejpam-6044	51	2	∣∣∣∣∣	∣∣∣∣∣	PROPN
ejpam-6044	51	3	a	a	PRON
ejpam-6044	51	4	,	,	PUNCT
ejpam-6044	51	5	b	b	NOUN
ejpam-6044	51	6	,	,	PUNCT
ejpam-6044	51	7	c	c	PROPN
ejpam-6044	51	8	∈	∈	PROPN
ejpam-6044	51	9	2z4	2z4	NUM
ejpam-6044	52	1			NOUN
ejpam-6044	52	2	be	be	VERB
ejpam-6044	52	3	a	a	DET
ejpam-6044	52	4	ring	ring	NOUN
ejpam-6044	52	5	with	with	ADP
ejpam-6044	52	6	usual	usual	ADJ
ejpam-6044	52	7	operation	operation	NOUN
ejpam-6044	52	8	of	of	ADP
ejpam-6044	52	9	matrix	matrix	NOUN
ejpam-6044	52	10	addition	addition	NOUN
ejpam-6044	52	11	and	and	CCONJ
ejpam-6044	52	12	multiplication	multiplication	NOUN
ejpam-6044	52	13	.	.	PUNCT
ejpam-6044	53	1	define	define	VERB
ejpam-6044	53	2	f	f	PROPN
ejpam-6044	53	3	,	,	PUNCT
ejpam-6044	54	1	d	d	NOUN
ejpam-6044	54	2	:	:	PUNCT
ejpam-6044	54	3	r	r	NOUN
ejpam-6044	54	4	→	→	SYM
ejpam-6044	54	5	r	r	NOUN
ejpam-6044	54	6	as	as	ADP
ejpam-6044	54	7	f(r	f(r	NOUN
ejpam-6044	54	8	)	)	PUNCT
ejpam-6044	55	1	=	=	PUNCT
ejpam-6044	56	1	0	0	ADP
ejpam-6044	56	2	0	0	NUM
ejpam-6044	56	3	b	b	X
ejpam-6044	56	4	0	0	NUM
ejpam-6044	56	5	0	0	NUM
ejpam-6044	56	6	0	0	NUM
ejpam-6044	56	7	0	0	NUM
ejpam-6044	56	8	0	0	NUM
ejpam-6044	56	9	0	0	NUM
ejpam-6044	57	1			PROPN
ejpam-6044	57	2	,	,	PUNCT
ejpam-6044	57	3	and	and	CCONJ
ejpam-6044	57	4	a	a	DET
ejpam-6044	57	5	map	map	NOUN
ejpam-6044	57	6	d	d	NOUN
ejpam-6044	57	7	is	be	AUX
ejpam-6044	57	8	given	give	VERB
ejpam-6044	57	9	by	by	ADP
ejpam-6044	57	10	d(r	d(r	NOUN
ejpam-6044	57	11	)	)	PUNCT
ejpam-6044	57	12	=	=	PUNCT
ejpam-6044	57	13	0	0	ADP
ejpam-6044	57	14	0	0	NUM
ejpam-6044	57	15	0	0	NUM
ejpam-6044	57	16	0	0	NUM
ejpam-6044	57	17	0	0	NUM
ejpam-6044	57	18	c	c	NOUN
ejpam-6044	57	19	0	0	NUM
ejpam-6044	57	20	0	0	NUM
ejpam-6044	57	21	0	0	NUM
ejpam-6044	57	22			PROPN
ejpam-6044	57	23	,	,	PUNCT
ejpam-6044	57	24	r	r	PROPN
ejpam-6044	57	25	∈	∈	PROPN
ejpam-6044	57	26	r.	r.	NOUN
ejpam-6044	57	27	define	define	VERB
ejpam-6044	57	28	α	α	NOUN
ejpam-6044	57	29	:	:	PUNCT
ejpam-6044	57	30	r	r	NOUN
ejpam-6044	57	31	→	→	SYM
ejpam-6044	57	32	r	r	NOUN
ejpam-6044	57	33	by	by	ADP
ejpam-6044	57	34	α(r	α(r	NOUN
ejpam-6044	57	35	)	)	PUNCT
ejpam-6044	57	36	=	=	SYM
ejpam-6044	57	37	0	0	ADV
ejpam-6044	57	38	−a	−a	VERB
ejpam-6044	57	39	−b	−b	ADP
ejpam-6044	57	40	0	0	NUM
ejpam-6044	57	41	0	0	NUM
ejpam-6044	57	42	−c	−c	NOUN
ejpam-6044	57	43	0	0	NUM
ejpam-6044	57	44	0	0	NUM
ejpam-6044	57	45	0	0	NUM
ejpam-6044	58	1			PROPN
ejpam-6044	58	2	.	.	PUNCT
ejpam-6044	59	1	the	the	DET
ejpam-6044	59	2	involution	involution	NOUN
ejpam-6044	59	3	is	be	AUX
ejpam-6044	59	4	given	give	VERB
ejpam-6044	59	5	by	by	ADP
ejpam-6044	59	6	r∗	r∗	PROPN
ejpam-6044	59	7	=	=	PUNCT
ejpam-6044	59	8	0	0	ADP
ejpam-6044	59	9	c	c	PROPN
ejpam-6044	59	10	b	b	PROPN
ejpam-6044	59	11	0	0	NUM
ejpam-6044	59	12	0	0	NUM
ejpam-6044	59	13	a	a	DET
ejpam-6044	59	14	0	0	NUM
ejpam-6044	59	15	0	0	NUM
ejpam-6044	59	16	0	0	NUM
ejpam-6044	59	17			PROPN
ejpam-6044	59	18	.	.	PUNCT
ejpam-6044	60	1	f	f	PROPN
ejpam-6044	60	2	will	will	AUX
ejpam-6044	60	3	be	be	AUX
ejpam-6044	60	4	a	a	DET
ejpam-6044	60	5	generalized	generalized	ADJ
ejpam-6044	60	6	(	(	PUNCT
ejpam-6044	60	7	α	α	NOUN
ejpam-6044	60	8	,	,	PUNCT
ejpam-6044	60	9	∗)-derivation	∗)-derivation	NOUN
ejpam-6044	60	10	associated	associate	VERB
ejpam-6044	60	11	with	with	ADP
ejpam-6044	60	12	d.	d.	PROPN
ejpam-6044	60	13	f.	f.	PROPN
ejpam-6044	60	14	shujat	shujat	PROPN
ejpam-6044	60	15	,	,	PUNCT
ejpam-6044	60	16	s.	s.	PROPN
ejpam-6044	60	17	alharbi	alharbi	PROPN
ejpam-6044	60	18	/	/	SYM
ejpam-6044	60	19	eur	eur	PROPN
ejpam-6044	60	20	.	.	PUNCT
ejpam-6044	61	1	j.	j.	PROPN
ejpam-6044	61	2	pure	pure	PROPN
ejpam-6044	61	3	appl	appl	PROPN
ejpam-6044	61	4	.	.	PROPN
ejpam-6044	61	5	math	math	PROPN
ejpam-6044	61	6	,	,	PUNCT
ejpam-6044	61	7	18	18	NUM
ejpam-6044	61	8	(	(	PUNCT
ejpam-6044	61	9	2	2	NUM
ejpam-6044	61	10	)	)	PUNCT
ejpam-6044	61	11	(	(	PUNCT
ejpam-6044	61	12	2025	2025	NUM
ejpam-6044	61	13	)	)	PUNCT
ejpam-6044	61	14	,	,	PUNCT
ejpam-6044	61	15	6044	6044	NUM
ejpam-6044	61	16	4	4	NUM
ejpam-6044	61	17	of	of	ADP
ejpam-6044	61	18	11	11	NUM
ejpam-6044	61	19	a	a	DET
ejpam-6044	61	20	lot	lot	NOUN
ejpam-6044	61	21	of	of	ADP
ejpam-6044	61	22	research	research	NOUN
ejpam-6044	61	23	has	have	AUX
ejpam-6044	61	24	been	be	AUX
ejpam-6044	61	25	done	do	VERB
ejpam-6044	61	26	in	in	ADP
ejpam-6044	61	27	the	the	DET
ejpam-6044	61	28	context	context	NOUN
ejpam-6044	61	29	of	of	ADP
ejpam-6044	61	30	involution	involution	NOUN
ejpam-6044	61	31	involved	involve	VERB
ejpam-6044	61	32	with	with	ADP
ejpam-6044	61	33	derivation	derivation	NOUN
ejpam-6044	61	34	,	,	PUNCT
ejpam-6044	61	35	generalized	generalized	ADJ
ejpam-6044	61	36	derivation	derivation	NOUN
ejpam-6044	61	37	,	,	PUNCT
ejpam-6044	61	38	jordan	jordan	PROPN
ejpam-6044	61	39	derivation	derivation	PROPN
ejpam-6044	61	40	,	,	PUNCT
ejpam-6044	61	41	left	leave	VERB
ejpam-6044	61	42	derivation	derivation	NOUN
ejpam-6044	61	43	,	,	PUNCT
ejpam-6044	61	44	etc	etc	X
ejpam-6044	61	45	.	.	X
ejpam-6044	62	1	our	our	PRON
ejpam-6044	62	2	present	present	ADJ
ejpam-6044	62	3	research	research	NOUN
ejpam-6044	62	4	is	be	AUX
ejpam-6044	62	5	motivated	motivate	VERB
ejpam-6044	62	6	by	by	ADP
ejpam-6044	62	7	all	all	DET
ejpam-6044	62	8	the	the	DET
ejpam-6044	62	9	above	above	ADJ
ejpam-6044	62	10	theories	theory	NOUN
ejpam-6044	62	11	and	and	CCONJ
ejpam-6044	62	12	the	the	DET
ejpam-6044	62	13	role	role	NOUN
ejpam-6044	62	14	of	of	ADP
ejpam-6044	62	15	automorphism	automorphism	NOUN
ejpam-6044	62	16	on	on	ADP
ejpam-6044	62	17	r	r	NOUN
ejpam-6044	62	18	and	and	CCONJ
ejpam-6044	62	19	involution	involution	NOUN
ejpam-6044	62	20	.	.	PUNCT
ejpam-6044	63	1	we	we	PRON
ejpam-6044	63	2	will	will	AUX
ejpam-6044	63	3	prove	prove	VERB
ejpam-6044	63	4	some	some	DET
ejpam-6044	63	5	commutativity	commutativity	NOUN
ejpam-6044	63	6	theorems	theorem	NOUN
ejpam-6044	63	7	in	in	ADP
ejpam-6044	63	8	the	the	DET
ejpam-6044	63	9	setting	setting	NOUN
ejpam-6044	63	10	of	of	ADP
ejpam-6044	63	11	prime	prime	ADJ
ejpam-6044	63	12	and	and	CCONJ
ejpam-6044	63	13	semiprime	semiprime	NOUN
ejpam-6044	63	14	rings	ring	NOUN
ejpam-6044	63	15	.	.	PUNCT
ejpam-6044	64	1	we	we	PRON
ejpam-6044	64	2	will	will	AUX
ejpam-6044	64	3	observe	observe	VERB
ejpam-6044	64	4	that	that	SCONJ
ejpam-6044	64	5	α	α	PROPN
ejpam-6044	64	6	and	and	CCONJ
ejpam-6044	64	7	∗	∗	NOUN
ejpam-6044	64	8	play	play	VERB
ejpam-6044	64	9	a	a	DET
ejpam-6044	64	10	crucial	crucial	ADJ
ejpam-6044	64	11	role	role	NOUN
ejpam-6044	64	12	in	in	ADP
ejpam-6044	64	13	our	our	PRON
ejpam-6044	64	14	proofs	proof	NOUN
ejpam-6044	64	15	.	.	PUNCT
ejpam-6044	65	1	the	the	DET
ejpam-6044	65	2	commutativity	commutativity	NOUN
ejpam-6044	65	3	theorem	theorem	VERB
ejpam-6044	65	4	on	on	ADP
ejpam-6044	65	5	prime	prime	ADJ
ejpam-6044	65	6	rings	ring	NOUN
ejpam-6044	65	7	possessing	possess	VERB
ejpam-6044	65	8	automorphisms	automorphisms	PROPN
ejpam-6044	65	9	(	(	PUNCT
ejpam-6044	65	10	or	or	CCONJ
ejpam-6044	65	11	endomorphisms	endomorphism	NOUN
ejpam-6044	65	12	)	)	PUNCT
ejpam-6044	65	13	proved	prove	VERB
ejpam-6044	65	14	in	in	ADP
ejpam-6044	65	15	[	[	X
ejpam-6044	65	16	5	5	NUM
ejpam-6044	65	17	,	,	PUNCT
ejpam-6044	65	18	6	6	NUM
ejpam-6044	65	19	,	,	PUNCT
ejpam-6044	65	20	9	9	NUM
ejpam-6044	65	21	,	,	PUNCT
ejpam-6044	65	22	13	13	NUM
ejpam-6044	65	23	]	]	PUNCT
ejpam-6044	65	24	.	.	PUNCT
ejpam-6044	66	1	further	far	ADV
ejpam-6044	66	2	we	we	PRON
ejpam-6044	66	3	refer	refer	VERB
ejpam-6044	66	4	the	the	DET
ejpam-6044	66	5	reader	reader	NOUN
ejpam-6044	66	6	to	to	ADP
ejpam-6044	66	7	the	the	DET
ejpam-6044	66	8	extensive	extensive	ADJ
ejpam-6044	66	9	bibliography	bibliography	NOUN
ejpam-6044	66	10	contained	contain	VERB
ejpam-6044	66	11	in	in	ADP
ejpam-6044	66	12	it	it	PRON
ejpam-6044	66	13	.	.	PUNCT
ejpam-6044	67	1	motivated	motivate	VERB
ejpam-6044	67	2	by	by	ADP
ejpam-6044	67	3	the	the	DET
ejpam-6044	67	4	above	above	PROPN
ejpam-6044	67	5	literature	literature	NOUN
ejpam-6044	67	6	review	review	NOUN
ejpam-6044	67	7	and	and	CCONJ
ejpam-6044	67	8	concepts	concept	NOUN
ejpam-6044	67	9	,	,	PUNCT
ejpam-6044	67	10	we	we	PRON
ejpam-6044	67	11	putout	putout	VERB
ejpam-6044	67	12	the	the	DET
ejpam-6044	67	13	extension	extension	NOUN
ejpam-6044	67	14	of	of	ADP
ejpam-6044	67	15	the	the	DET
ejpam-6044	67	16	notion	notion	NOUN
ejpam-6044	67	17	of	of	ADP
ejpam-6044	67	18	generalized	generalized	ADJ
ejpam-6044	67	19	(	(	PUNCT
ejpam-6044	67	20	α	α	NOUN
ejpam-6044	67	21	,	,	PUNCT
ejpam-6044	67	22	∗)-derivation	∗)-derivation	NOUN
ejpam-6044	67	23	to	to	PART
ejpam-6044	67	24	generalized	generalize	VERB
ejpam-6044	67	25	(	(	PUNCT
ejpam-6044	67	26	α	α	NOUN
ejpam-6044	67	27	,	,	PUNCT
ejpam-6044	67	28	∗)-n	∗)-n	NOUN
ejpam-6044	67	29	-	-	PUNCT
ejpam-6044	67	30	derivation	derivation	NOUN
ejpam-6044	67	31	on	on	ADP
ejpam-6044	67	32	rings	ring	NOUN
ejpam-6044	67	33	as	as	SCONJ
ejpam-6044	67	34	follows	follow	VERB
ejpam-6044	67	35	:	:	PUNCT
ejpam-6044	67	36	definition	definition	NOUN
ejpam-6044	67	37	3	3	NUM
ejpam-6044	67	38	.	.	PUNCT
ejpam-6044	68	1	a	a	DET
ejpam-6044	68	2	mapping	mapping	NOUN
ejpam-6044	68	3	f	f	X
ejpam-6044	68	4	:	:	PUNCT
ejpam-6044	68	5	rn	rn	PROPN
ejpam-6044	68	6	→	→	SYM
ejpam-6044	68	7	r	r	NOUN
ejpam-6044	68	8	is	be	AUX
ejpam-6044	68	9	called	call	VERB
ejpam-6044	68	10	a	a	DET
ejpam-6044	68	11	generalized	generalized	ADJ
ejpam-6044	68	12	(	(	PUNCT
ejpam-6044	68	13	α	α	NOUN
ejpam-6044	68	14	,	,	PUNCT
ejpam-6044	68	15	∗)-n	∗)-n	ADJ
ejpam-6044	68	16	-derivation	-derivation	NOUN
ejpam-6044	68	17	if	if	SCONJ
ejpam-6044	68	18	there	there	PRON
ejpam-6044	68	19	exists	exist	VERB
ejpam-6044	68	20	an	an	DET
ejpam-6044	68	21	(	(	PUNCT
ejpam-6044	68	22	α	α	NOUN
ejpam-6044	68	23	,	,	PUNCT
ejpam-6044	68	24	∗)-n	∗)-n	NOUN
ejpam-6044	68	25	-	-	PUNCT
ejpam-6044	68	26	derivation	derivation	NOUN
ejpam-6044	68	27	d	d	NOUN
ejpam-6044	68	28	:	:	PUNCT
ejpam-6044	68	29	rn	rn	PROPN
ejpam-6044	68	30	→	→	SYM
ejpam-6044	68	31	r	r	VERB
ejpam-6044	68	32	such	such	ADJ
ejpam-6044	68	33	that	that	SCONJ
ejpam-6044	68	34	f	f	PROPN
ejpam-6044	68	35	(	(	PUNCT
ejpam-6044	68	36	ς1	ς1	NOUN
ejpam-6044	68	37	,	,	PUNCT
ejpam-6044	68	38	.	.	PUNCT
ejpam-6044	68	39	.	.	PUNCT
ejpam-6044	69	1	.	.	PUNCT
ejpam-6044	70	1	,	,	PUNCT
ejpam-6044	70	2	ςkς	ςkς	X
ejpam-6044	70	3	′	′	NUM
ejpam-6044	71	1	k	k	NOUN
ejpam-6044	71	2	,	,	PUNCT
ejpam-6044	71	3	.	.	PUNCT
ejpam-6044	71	4	.	.	PUNCT
ejpam-6044	71	5	.	.	PUNCT
ejpam-6044	72	1	,	,	PUNCT
ejpam-6044	72	2	ςn	ςn	X
ejpam-6044	72	3	)	)	PUNCT
ejpam-6044	73	1	=	=	SYM
ejpam-6044	73	2	f	f	PROPN
ejpam-6044	73	3	(	(	PUNCT
ejpam-6044	73	4	ς1	ς1	NOUN
ejpam-6044	73	5	,	,	PUNCT
ejpam-6044	73	6	.	.	PUNCT
ejpam-6044	73	7	.	.	PUNCT
ejpam-6044	73	8	.	.	PUNCT
ejpam-6044	73	9	,	,	PUNCT
ejpam-6044	73	10	ςk	ςk	NUM
ejpam-6044	73	11	,	,	PUNCT
ejpam-6044	73	12	.	.	PUNCT
ejpam-6044	73	13	.	.	PUNCT
ejpam-6044	73	14	.	.	PUNCT
ejpam-6044	74	1	,	,	PUNCT
ejpam-6044	74	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	74	3	′	′	NUM
ejpam-6044	75	1	k	k	NOUN
ejpam-6044	75	2	)	)	PUNCT
ejpam-6044	76	1	+	+	CCONJ
ejpam-6044	76	2	ς∗kd(ς1	ς∗kd(ς1	NUM
ejpam-6044	76	3	,	,	PUNCT
ejpam-6044	76	4	.	.	PUNCT
ejpam-6044	76	5	.	.	PUNCT
ejpam-6044	76	6	.	.	PUNCT
ejpam-6044	77	1	,	,	PUNCT
ejpam-6044	77	2	ς	ς	PROPN
ejpam-6044	77	3	′	′	NUM
ejpam-6044	77	4	k	k	NOUN
ejpam-6044	77	5	,	,	PUNCT
ejpam-6044	77	6	.	.	PUNCT
ejpam-6044	77	7	.	.	PUNCT
ejpam-6044	78	1	.	.	PUNCT
ejpam-6044	79	1	,	,	PUNCT
ejpam-6044	79	2	ςn	ςn	X
ejpam-6044	79	3	)	)	PUNCT
ejpam-6044	79	4	for	for	ADP
ejpam-6044	79	5	all	all	DET
ejpam-6044	79	6	ς1	ς1	NOUN
ejpam-6044	79	7	,	,	PUNCT
ejpam-6044	79	8	.	.	PUNCT
ejpam-6044	79	9	.	.	PUNCT
ejpam-6044	80	1	.	.	PUNCT
ejpam-6044	81	1	,	,	PUNCT
ejpam-6044	81	2	ςk	ςk	NUM
ejpam-6044	81	3	,	,	PUNCT
ejpam-6044	81	4	ς	ς	PROPN
ejpam-6044	81	5	′	′	NUM
ejpam-6044	81	6	k	k	NOUN
ejpam-6044	81	7	,	,	PUNCT
ejpam-6044	81	8	.	.	PUNCT
ejpam-6044	81	9	.	.	PUNCT
ejpam-6044	81	10	.	.	PUNCT
ejpam-6044	82	1	,	,	PUNCT
ejpam-6044	82	2	ςn	ςn	PROPN
ejpam-6044	82	3	∈	∈	PROPN
ejpam-6044	82	4	r.	r.	PROPN
ejpam-6044	82	5	2	2	NUM
ejpam-6044	82	6	.	.	PUNCT
ejpam-6044	82	7	results	result	NOUN
ejpam-6044	82	8	on	on	ADP
ejpam-6044	82	9	prime	prime	ADJ
ejpam-6044	82	10	∗-ring	∗-ring	NOUN
ejpam-6044	82	11	we	we	PRON
ejpam-6044	82	12	start	start	VERB
ejpam-6044	82	13	with	with	ADP
ejpam-6044	82	14	the	the	DET
ejpam-6044	82	15	following	follow	VERB
ejpam-6044	82	16	results	result	NOUN
ejpam-6044	82	17	:	:	PUNCT
ejpam-6044	82	18	lemma	lemma	PROPN
ejpam-6044	82	19	1	1	NUM
ejpam-6044	82	20	.	.	PUNCT
ejpam-6044	83	1	[	[	X
ejpam-6044	83	2	14	14	NUM
ejpam-6044	83	3	]	]	PUNCT
ejpam-6044	83	4	the	the	DET
ejpam-6044	83	5	center	center	NOUN
ejpam-6044	83	6	of	of	ADP
ejpam-6044	83	7	r	r	NOUN
ejpam-6044	83	8	includes	include	VERB
ejpam-6044	83	9	the	the	DET
ejpam-6044	83	10	center	center	NOUN
ejpam-6044	83	11	of	of	ADP
ejpam-6044	83	12	a	a	DET
ejpam-6044	83	13	nonzero	nonzero	NOUN
ejpam-6044	83	14	ideal	ideal	NOUN
ejpam-6044	83	15	(	(	PUNCT
ejpam-6044	83	16	one	one	NUM
ejpam-6044	83	17	-	-	PUNCT
ejpam-6044	83	18	sided	sided	ADJ
ejpam-6044	83	19	)	)	PUNCT
ejpam-6044	83	20	for	for	ADP
ejpam-6044	83	21	a	a	DET
ejpam-6044	83	22	semiprime	semiprime	NOUN
ejpam-6044	83	23	ring	ring	NOUN
ejpam-6044	83	24	r.	r.	PROPN
ejpam-6044	83	25	any	any	DET
ejpam-6044	83	26	commutative	commutative	ADJ
ejpam-6044	83	27	ideal	ideal	NOUN
ejpam-6044	83	28	(	(	PUNCT
ejpam-6044	83	29	one	one	NUM
ejpam-6044	83	30	-	-	PUNCT
ejpam-6044	83	31	sided	sided	ADJ
ejpam-6044	83	32	)	)	PUNCT
ejpam-6044	83	33	is	be	AUX
ejpam-6044	83	34	immediately	immediately	ADV
ejpam-6044	83	35	enclosed	enclose	VERB
ejpam-6044	83	36	z(r	z(r	NOUN
ejpam-6044	83	37	)	)	PUNCT
ejpam-6044	83	38	.	.	PUNCT
ejpam-6044	84	1	theorem	theorem	NOUN
ejpam-6044	84	2	1	1	NUM
ejpam-6044	84	3	.	.	PUNCT
ejpam-6044	85	1	let	let	VERB
ejpam-6044	85	2	a	a	DET
ejpam-6044	85	3	semiprime	semiprime	NOUN
ejpam-6044	85	4	∗-ring	∗-re	VERB
ejpam-6044	85	5	be	be	AUX
ejpam-6044	85	6	r	r	NOUN
ejpam-6044	85	7	,	,	PUNCT
ejpam-6044	85	8	∗	∗	NOUN
ejpam-6044	85	9	be	be	VERB
ejpam-6044	85	10	an	an	DET
ejpam-6044	85	11	involution	involution	NOUN
ejpam-6044	85	12	,	,	PUNCT
ejpam-6044	85	13	and	and	CCONJ
ejpam-6044	85	14	α	α	PRON
ejpam-6044	85	15	be	be	VERB
ejpam-6044	85	16	an	an	DET
ejpam-6044	85	17	automorphism	automorphism	NOUN
ejpam-6044	85	18	on	on	ADP
ejpam-6044	85	19	r.	r.	PROPN
ejpam-6044	85	20	if	if	SCONJ
ejpam-6044	85	21	f1,f2	f1,f2	PROPN
ejpam-6044	85	22	are	be	AUX
ejpam-6044	85	23	two	two	NUM
ejpam-6044	85	24	generalized	generalized	ADJ
ejpam-6044	85	25	(	(	PUNCT
ejpam-6044	85	26	α	α	NOUN
ejpam-6044	85	27	,	,	PUNCT
ejpam-6044	85	28	∗)-n	∗)-n	NOUN
ejpam-6044	85	29	-	-	PUNCT
ejpam-6044	85	30	derivations	derivation	NOUN
ejpam-6044	85	31	on	on	ADP
ejpam-6044	85	32	r	r	NOUN
ejpam-6044	85	33	associated	associate	VERB
ejpam-6044	85	34	with	with	ADP
ejpam-6044	85	35	(	(	PUNCT
ejpam-6044	85	36	α	α	NOUN
ejpam-6044	85	37	,	,	PUNCT
ejpam-6044	85	38	∗)-nderivation	∗)-nderivation	NOUN
ejpam-6044	85	39	d1,d2	d1,d2	PROPN
ejpam-6044	85	40	respectively	respectively	ADV
ejpam-6044	85	41	,	,	PUNCT
ejpam-6044	85	42	such	such	ADJ
ejpam-6044	85	43	that	that	DET
ejpam-6044	85	44	f1(ς1	f1(ς1	NOUN
ejpam-6044	85	45	,	,	PUNCT
ejpam-6044	85	46	ς2	ς2	PROPN
ejpam-6044	85	47	,	,	PUNCT
ejpam-6044	85	48	...	...	PUNCT
ejpam-6044	85	49	,	,	PUNCT
ejpam-6044	86	1	ςn	ςn	X
ejpam-6044	86	2	)	)	PUNCT
ejpam-6044	86	3	=	=	NOUN
ejpam-6044	86	4	f2(ς1	f2(ς1	NUM
ejpam-6044	86	5	,	,	PUNCT
ejpam-6044	86	6	ς2	ς2	PROPN
ejpam-6044	86	7	,	,	PUNCT
ejpam-6044	86	8	...	...	PUNCT
ejpam-6044	86	9	,	,	PUNCT
ejpam-6044	86	10	ςn	ςn	NOUN
ejpam-6044	86	11	)	)	PUNCT
ejpam-6044	86	12	for	for	ADP
ejpam-6044	86	13	all	all	DET
ejpam-6044	86	14	ς1	ς1	NOUN
ejpam-6044	86	15	,	,	PUNCT
ejpam-6044	86	16	ς2	ς2	PROPN
ejpam-6044	86	17	,	,	PUNCT
ejpam-6044	86	18	...	...	PUNCT
ejpam-6044	86	19	,	,	PUNCT
ejpam-6044	86	20	ςn	ςn	PROPN
ejpam-6044	86	21	∈	∈	PROPN
ejpam-6044	86	22	r	r	NOUN
ejpam-6044	86	23	,	,	PUNCT
ejpam-6044	86	24	then	then	ADV
ejpam-6044	86	25	d1	d1	PROPN
ejpam-6044	86	26	=	=	SYM
ejpam-6044	86	27	d2	d2	PROPN
ejpam-6044	86	28	.	.	PUNCT
ejpam-6044	87	1	proof	proof	NOUN
ejpam-6044	87	2	.	.	PUNCT
ejpam-6044	88	1	by	by	ADP
ejpam-6044	88	2	the	the	DET
ejpam-6044	88	3	hypothesis	hypothesis	NOUN
ejpam-6044	88	4	,	,	PUNCT
ejpam-6044	88	5	we	we	PRON
ejpam-6044	88	6	are	be	AUX
ejpam-6044	88	7	given	give	VERB
ejpam-6044	88	8	that	that	DET
ejpam-6044	88	9	f1(ς1	f1(ς1	NOUN
ejpam-6044	88	10	,	,	PUNCT
ejpam-6044	88	11	ς2	ς2	PROPN
ejpam-6044	88	12	,	,	PUNCT
ejpam-6044	88	13	...	...	PUNCT
ejpam-6044	88	14	,	,	PUNCT
ejpam-6044	88	15	ςn	ςn	X
ejpam-6044	88	16	)	)	PUNCT
ejpam-6044	88	17	=	=	NOUN
ejpam-6044	88	18	f2(ς1	f2(ς1	NUM
ejpam-6044	88	19	,	,	PUNCT
ejpam-6044	88	20	ς2	ς2	PROPN
ejpam-6044	88	21	,	,	PUNCT
ejpam-6044	88	22	...	...	PUNCT
ejpam-6044	88	23	,	,	PUNCT
ejpam-6044	88	24	ςn	ςn	PROPN
ejpam-6044	88	25	)	)	PUNCT
ejpam-6044	88	26	,	,	PUNCT
ejpam-6044	88	27	for	for	ADP
ejpam-6044	88	28	every	every	DET
ejpam-6044	88	29	ς1	ς1	NOUN
ejpam-6044	88	30	,	,	PUNCT
ejpam-6044	88	31	ς2	ς2	PROPN
ejpam-6044	88	32	,	,	PUNCT
ejpam-6044	88	33	...	...	PUNCT
ejpam-6044	88	34	,	,	PUNCT
ejpam-6044	88	35	ςn	ςn	PROPN
ejpam-6044	88	36	∈	∈	PROPN
ejpam-6044	88	37	r.	r.	PROPN
ejpam-6044	88	38	(	(	PUNCT
ejpam-6044	88	39	1	1	X
ejpam-6044	88	40	)	)	PUNCT
ejpam-6044	88	41	rewrite	rewrite	NOUN
ejpam-6044	88	42	(	(	PUNCT
ejpam-6044	88	43	1	1	NUM
ejpam-6044	88	44	)	)	PUNCT
ejpam-6044	88	45	to	to	PART
ejpam-6044	88	46	get	get	VERB
ejpam-6044	88	47	the	the	DET
ejpam-6044	88	48	form	form	NOUN
ejpam-6044	88	49	f1(ς1	f1(ς1	NOUN
ejpam-6044	88	50	,	,	PUNCT
ejpam-6044	88	51	...	...	PUNCT
ejpam-6044	88	52	,	,	PUNCT
ejpam-6044	88	53	ςkς	ςkς	X
ejpam-6044	88	54	′	′	NUM
ejpam-6044	89	1	k	k	PROPN
ejpam-6044	89	2	,	,	PUNCT
ejpam-6044	89	3	...	...	PUNCT
ejpam-6044	89	4	,	,	PUNCT
ejpam-6044	89	5	ςn	ςn	X
ejpam-6044	89	6	)	)	PUNCT
ejpam-6044	89	7	=	=	NOUN
ejpam-6044	89	8	f2(ς1	f2(ς1	NOUN
ejpam-6044	89	9	,	,	PUNCT
ejpam-6044	89	10	...	...	PUNCT
ejpam-6044	89	11	,	,	PUNCT
ejpam-6044	89	12	ςkς	ςkς	X
ejpam-6044	89	13	′	′	NUM
ejpam-6044	90	1	k	k	PROPN
ejpam-6044	90	2	,	,	PUNCT
ejpam-6044	90	3	...	...	PUNCT
ejpam-6044	90	4	,	,	PUNCT
ejpam-6044	90	5	n	n	PROPN
ejpam-6044	90	6	)	)	PUNCT
ejpam-6044	90	7	,	,	PUNCT
ejpam-6044	90	8	for	for	ADP
ejpam-6044	90	9	every	every	DET
ejpam-6044	90	10	ς1	ς1	NOUN
ejpam-6044	90	11	,	,	PUNCT
ejpam-6044	90	12	ς2	ς2	PROPN
ejpam-6044	90	13	,	,	PUNCT
ejpam-6044	90	14	...	...	PUNCT
ejpam-6044	90	15	,	,	PUNCT
ejpam-6044	90	16	ςk	ςk	PROPN
ejpam-6044	90	17	,	,	PUNCT
ejpam-6044	90	18	ς	ς	PROPN
ejpam-6044	90	19	′	′	NUM
ejpam-6044	90	20	k	k	NOUN
ejpam-6044	90	21	,	,	PUNCT
ejpam-6044	90	22	...	...	PUNCT
ejpam-6044	90	23	,	,	PUNCT
ejpam-6044	90	24	ςn	ςn	PROPN
ejpam-6044	90	25	∈	∈	PROPN
ejpam-6044	90	26	r.	r.	PROPN
ejpam-6044	90	27	(	(	PUNCT
ejpam-6044	90	28	2	2	NUM
ejpam-6044	90	29	)	)	PUNCT
ejpam-6044	90	30	by	by	ADP
ejpam-6044	90	31	definition	definition	NOUN
ejpam-6044	90	32	the	the	DET
ejpam-6044	90	33	above	above	ADJ
ejpam-6044	90	34	equation	equation	NOUN
ejpam-6044	90	35	reword	reword	NOUN
ejpam-6044	90	36	as	as	ADP
ejpam-6044	90	37	f1(ς1	f1(ς1	NOUN
ejpam-6044	90	38	,	,	PUNCT
ejpam-6044	90	39	.	.	PUNCT
ejpam-6044	90	40	.	.	PUNCT
ejpam-6044	91	1	.	.	PUNCT
ejpam-6044	92	1	,	,	PUNCT
ejpam-6044	92	2	ςk	ςk	NUM
ejpam-6044	92	3	,	,	PUNCT
ejpam-6044	92	4	.	.	PUNCT
ejpam-6044	92	5	.	.	PUNCT
ejpam-6044	93	1	.	.	PUNCT
ejpam-6044	94	1	,	,	PUNCT
ejpam-6044	94	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	94	3	′	′	NUM
ejpam-6044	95	1	k	k	NOUN
ejpam-6044	95	2	)	)	PUNCT
ejpam-6044	96	1	+	+	CCONJ
ejpam-6044	96	2	ς∗kd1(ς1	ς∗kd1(ς1	NOUN
ejpam-6044	96	3	,	,	PUNCT
ejpam-6044	96	4	.	.	PUNCT
ejpam-6044	96	5	.	.	PUNCT
ejpam-6044	96	6	.	.	PUNCT
ejpam-6044	97	1	,	,	PUNCT
ejpam-6044	97	2	ς	ς	PROPN
ejpam-6044	97	3	′	′	NUM
ejpam-6044	97	4	k	k	NOUN
ejpam-6044	97	5	,	,	PUNCT
ejpam-6044	97	6	.	.	PUNCT
ejpam-6044	97	7	.	.	PUNCT
ejpam-6044	98	1	.	.	PUNCT
ejpam-6044	99	1	,	,	PUNCT
ejpam-6044	99	2	ςn	ςn	X
ejpam-6044	99	3	)	)	PUNCT
ejpam-6044	99	4	=	=	NOUN
ejpam-6044	99	5	f2(ς1	f2(ς1	NOUN
ejpam-6044	99	6	,	,	PUNCT
ejpam-6044	99	7	.	.	PUNCT
ejpam-6044	99	8	.	.	PUNCT
ejpam-6044	100	1	.	.	PUNCT
ejpam-6044	101	1	,	,	PUNCT
ejpam-6044	101	2	ςk	ςk	NUM
ejpam-6044	101	3	,	,	PUNCT
ejpam-6044	101	4	.	.	PUNCT
ejpam-6044	101	5	.	.	PUNCT
ejpam-6044	102	1	.	.	PUNCT
ejpam-6044	103	1	,	,	PUNCT
ejpam-6044	103	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	103	3	′	′	NUM
ejpam-6044	104	1	k	k	NOUN
ejpam-6044	104	2	)	)	PUNCT
ejpam-6044	105	1	+	+	NOUN
ejpam-6044	105	2	ς∗kd2(ς1	ς∗kd2(ς1	NOUN
ejpam-6044	105	3	,	,	PUNCT
ejpam-6044	105	4	.	.	PUNCT
ejpam-6044	105	5	.	.	PUNCT
ejpam-6044	105	6	.	.	PUNCT
ejpam-6044	106	1	,	,	PUNCT
ejpam-6044	106	2	ς	ς	PROPN
ejpam-6044	106	3	′	′	NUM
ejpam-6044	106	4	k	k	NOUN
ejpam-6044	106	5	,	,	PUNCT
ejpam-6044	106	6	.	.	PUNCT
ejpam-6044	106	7	.	.	PUNCT
ejpam-6044	107	1	.	.	PUNCT
ejpam-6044	108	1	,	,	PUNCT
ejpam-6044	108	2	ςn	ςn	PROPN
ejpam-6044	108	3	)	)	PUNCT
ejpam-6044	108	4	.	.	PUNCT
ejpam-6044	109	1	f.	f.	PROPN
ejpam-6044	109	2	shujat	shujat	PROPN
ejpam-6044	109	3	,	,	PUNCT
ejpam-6044	109	4	s.	s.	PROPN
ejpam-6044	109	5	alharbi	alharbi	PROPN
ejpam-6044	109	6	/	/	SYM
ejpam-6044	109	7	eur	eur	PROPN
ejpam-6044	109	8	.	.	PUNCT
ejpam-6044	110	1	j.	j.	PROPN
ejpam-6044	110	2	pure	pure	PROPN
ejpam-6044	110	3	appl	appl	PROPN
ejpam-6044	110	4	.	.	PROPN
ejpam-6044	110	5	math	math	PROPN
ejpam-6044	110	6	,	,	PUNCT
ejpam-6044	110	7	18	18	NUM
ejpam-6044	110	8	(	(	PUNCT
ejpam-6044	110	9	2	2	NUM
ejpam-6044	110	10	)	)	PUNCT
ejpam-6044	110	11	(	(	PUNCT
ejpam-6044	110	12	2025	2025	NUM
ejpam-6044	110	13	)	)	PUNCT
ejpam-6044	110	14	,	,	PUNCT
ejpam-6044	110	15	6044	6044	NUM
ejpam-6044	110	16	5	5	NUM
ejpam-6044	110	17	of	of	ADP
ejpam-6044	110	18	11	11	NUM
ejpam-6044	110	19	application	application	NOUN
ejpam-6044	110	20	of	of	ADP
ejpam-6044	110	21	(	(	PUNCT
ejpam-6044	110	22	1	1	NUM
ejpam-6044	110	23	)	)	PUNCT
ejpam-6044	110	24	with	with	ADP
ejpam-6044	110	25	the	the	DET
ejpam-6044	110	26	last	last	ADJ
ejpam-6044	110	27	expression	expression	NOUN
ejpam-6044	110	28	to	to	PART
ejpam-6044	110	29	find	find	VERB
ejpam-6044	110	30	ς∗kd1(ς1	ς∗kd1(ς1	NOUN
ejpam-6044	110	31	,	,	PUNCT
ejpam-6044	110	32	.	.	PUNCT
ejpam-6044	110	33	.	.	PUNCT
ejpam-6044	111	1	.	.	PUNCT
ejpam-6044	112	1	,	,	PUNCT
ejpam-6044	112	2	ς	ς	PROPN
ejpam-6044	112	3	′	′	NUM
ejpam-6044	112	4	k	k	NOUN
ejpam-6044	112	5	,	,	PUNCT
ejpam-6044	112	6	.	.	PUNCT
ejpam-6044	112	7	.	.	PUNCT
ejpam-6044	113	1	.	.	PUNCT
ejpam-6044	114	1	,	,	PUNCT
ejpam-6044	114	2	ςn	ςn	X
ejpam-6044	114	3	)	)	PUNCT
ejpam-6044	114	4	=	=	SYM
ejpam-6044	115	1	ς∗kd2(ς1	ς∗kd2(ς1	PROPN
ejpam-6044	115	2	,	,	PUNCT
ejpam-6044	115	3	.	.	PUNCT
ejpam-6044	115	4	.	.	PUNCT
ejpam-6044	115	5	.	.	PUNCT
ejpam-6044	116	1	,	,	PUNCT
ejpam-6044	116	2	ς	ς	PROPN
ejpam-6044	116	3	′	′	NUM
ejpam-6044	116	4	k	k	NOUN
ejpam-6044	116	5	,	,	PUNCT
ejpam-6044	116	6	.	.	PUNCT
ejpam-6044	116	7	.	.	PUNCT
ejpam-6044	117	1	.	.	PUNCT
ejpam-6044	118	1	,	,	PUNCT
ejpam-6044	118	2	ςn	ςn	X
ejpam-6044	118	3	)	)	PUNCT
ejpam-6044	118	4	=	=	SYM
ejpam-6044	119	1	ς∗k	ς∗k	NUM
ejpam-6044	119	2	(	(	PUNCT
ejpam-6044	119	3	d1(ς1	d1(ς1	NOUN
ejpam-6044	119	4	,	,	PUNCT
ejpam-6044	119	5	.	.	PUNCT
ejpam-6044	119	6	.	.	PUNCT
ejpam-6044	119	7	.	.	PUNCT
ejpam-6044	120	1	,	,	PUNCT
ejpam-6044	120	2	ς	ς	PROPN
ejpam-6044	120	3	′	′	NUM
ejpam-6044	120	4	k	k	NOUN
ejpam-6044	120	5	,	,	PUNCT
ejpam-6044	120	6	.	.	PUNCT
ejpam-6044	120	7	.	.	PUNCT
ejpam-6044	121	1	.	.	PUNCT
ejpam-6044	122	1	,	,	PUNCT
ejpam-6044	122	2	ςn)−d2(ς1	ςn)−d2(ς1	NOUN
ejpam-6044	122	3	,	,	PUNCT
ejpam-6044	122	4	.	.	PUNCT
ejpam-6044	122	5	.	.	PUNCT
ejpam-6044	122	6	.	.	PUNCT
ejpam-6044	123	1	,	,	PUNCT
ejpam-6044	123	2	ς	ς	PROPN
ejpam-6044	123	3	′	′	NUM
ejpam-6044	123	4	k	k	NOUN
ejpam-6044	123	5	,	,	PUNCT
ejpam-6044	123	6	.	.	PUNCT
ejpam-6044	123	7	.	.	PUNCT
ejpam-6044	124	1	.	.	PUNCT
ejpam-6044	125	1	,	,	PUNCT
ejpam-6044	125	2	ςn	ςn	NOUN
ejpam-6044	125	3	)	)	PUNCT
ejpam-6044	125	4	)	)	PUNCT
ejpam-6044	126	1	=	=	PUNCT
ejpam-6044	126	2	0	0	NUM
ejpam-6044	126	3	,	,	PUNCT
ejpam-6044	126	4	for	for	ADP
ejpam-6044	126	5	each	each	DET
ejpam-6044	126	6	ς1	ς1	NOUN
ejpam-6044	126	7	,	,	PUNCT
ejpam-6044	126	8	.	.	PUNCT
ejpam-6044	126	9	.	.	PUNCT
ejpam-6044	126	10	.	.	PUNCT
ejpam-6044	127	1	,	,	PUNCT
ejpam-6044	127	2	ςk	ςk	NUM
ejpam-6044	127	3	,	,	PUNCT
ejpam-6044	127	4	.	.	PUNCT
ejpam-6044	127	5	.	.	PUNCT
ejpam-6044	128	1	.	.	PUNCT
ejpam-6044	129	1	,	,	PUNCT
ejpam-6044	129	2	ςn	ςn	PROPN
ejpam-6044	129	3	∈	∈	PROPN
ejpam-6044	129	4	r.	r.	PROPN
ejpam-6044	129	5	(	(	PUNCT
ejpam-6044	129	6	3	3	X
ejpam-6044	129	7	)	)	PUNCT
ejpam-6044	129	8	particularly	particularly	ADV
ejpam-6044	129	9	consider	consider	VERB
ejpam-6044	129	10	ςk	ςk	NOUN
ejpam-6044	129	11	=	=	NOUN
ejpam-6044	129	12	ς∗k	ς∗k	NUM
ejpam-6044	129	13	for	for	ADP
ejpam-6044	129	14	the	the	DET
ejpam-6044	129	15	case	case	NOUN
ejpam-6044	129	16	of	of	ADP
ejpam-6044	129	17	∗	∗	NOUN
ejpam-6044	129	18	−	−	PROPN
ejpam-6044	129	19	ring	ring	NOUN
ejpam-6044	129	20	to	to	PART
ejpam-6044	129	21	obtain	obtain	VERB
ejpam-6044	129	22	ςkr	ςkr	NOUN
ejpam-6044	129	23	(	(	PUNCT
ejpam-6044	129	24	d1(ς1	d1(ς1	NOUN
ejpam-6044	129	25	,	,	PUNCT
ejpam-6044	129	26	.	.	PUNCT
ejpam-6044	129	27	.	.	PUNCT
ejpam-6044	130	1	.	.	PUNCT
ejpam-6044	131	1	,	,	PUNCT
ejpam-6044	131	2	ς	ς	PROPN
ejpam-6044	131	3	′	′	NUM
ejpam-6044	131	4	k	k	NOUN
ejpam-6044	131	5	,	,	PUNCT
ejpam-6044	131	6	.	.	PUNCT
ejpam-6044	131	7	.	.	PUNCT
ejpam-6044	132	1	.	.	PUNCT
ejpam-6044	133	1	,	,	PUNCT
ejpam-6044	133	2	ςn)−d2(ς1	ςn)−d2(ς1	NOUN
ejpam-6044	133	3	,	,	PUNCT
ejpam-6044	133	4	.	.	PUNCT
ejpam-6044	133	5	.	.	PUNCT
ejpam-6044	133	6	.	.	PUNCT
ejpam-6044	134	1	,	,	PUNCT
ejpam-6044	134	2	ς	ς	PROPN
ejpam-6044	134	3	′	′	NUM
ejpam-6044	134	4	k	k	NOUN
ejpam-6044	134	5	,	,	PUNCT
ejpam-6044	134	6	.	.	PUNCT
ejpam-6044	134	7	.	.	PUNCT
ejpam-6044	135	1	.	.	PUNCT
ejpam-6044	136	1	,	,	PUNCT
ejpam-6044	136	2	ςn	ςn	NOUN
ejpam-6044	136	3	)	)	PUNCT
ejpam-6044	136	4	)	)	PUNCT
ejpam-6044	137	1	=	=	SYM
ejpam-6044	137	2	0	0	NUM
ejpam-6044	138	1	for	for	ADP
ejpam-6044	138	2	each	each	DET
ejpam-6044	138	3	ς1	ς1	NOUN
ejpam-6044	138	4	,	,	PUNCT
ejpam-6044	138	5	.	.	PUNCT
ejpam-6044	138	6	.	.	PUNCT
ejpam-6044	138	7	.	.	PUNCT
ejpam-6044	139	1	,	,	PUNCT
ejpam-6044	139	2	ςk	ςk	NUM
ejpam-6044	139	3	,	,	PUNCT
ejpam-6044	139	4	.	.	PUNCT
ejpam-6044	139	5	.	.	PUNCT
ejpam-6044	140	1	.	.	PUNCT
ejpam-6044	141	1	,	,	PUNCT
ejpam-6044	141	2	ςn	ςn	PROPN
ejpam-6044	141	3	∈	∈	PROPN
ejpam-6044	141	4	r.	r.	NOUN
ejpam-6044	141	5	condition	condition	NOUN
ejpam-6044	141	6	of	of	ADP
ejpam-6044	141	7	semiprimeness	semiprimeness	NOUN
ejpam-6044	141	8	of	of	ADP
ejpam-6044	141	9	r	r	NOUN
ejpam-6044	141	10	implies	imply	VERB
ejpam-6044	141	11	that	that	SCONJ
ejpam-6044	141	12	d1(ς1	d1(ς1	NOUN
ejpam-6044	141	13	,	,	PUNCT
ejpam-6044	141	14	.	.	PUNCT
ejpam-6044	141	15	.	.	PUNCT
ejpam-6044	142	1	.	.	PUNCT
ejpam-6044	143	1	,	,	PUNCT
ejpam-6044	143	2	ς	ς	PROPN
ejpam-6044	143	3	′	′	NUM
ejpam-6044	143	4	k	k	NOUN
ejpam-6044	143	5	,	,	PUNCT
ejpam-6044	143	6	.	.	PUNCT
ejpam-6044	143	7	.	.	PUNCT
ejpam-6044	144	1	.	.	PUNCT
ejpam-6044	145	1	,	,	PUNCT
ejpam-6044	145	2	ςn	ςn	X
ejpam-6044	145	3	)	)	PUNCT
ejpam-6044	145	4	=	=	SYM
ejpam-6044	145	5	d2(ς1	d2(ς1	NOUN
ejpam-6044	145	6	,	,	PUNCT
ejpam-6044	145	7	.	.	PUNCT
ejpam-6044	145	8	.	.	PUNCT
ejpam-6044	145	9	.	.	PUNCT
ejpam-6044	146	1	,	,	PUNCT
ejpam-6044	146	2	ς	ς	PROPN
ejpam-6044	146	3	′	′	NUM
ejpam-6044	146	4	k	k	NOUN
ejpam-6044	146	5	,	,	PUNCT
ejpam-6044	146	6	.	.	PUNCT
ejpam-6044	146	7	.	.	PUNCT
ejpam-6044	147	1	.	.	PUNCT
ejpam-6044	148	1	,	,	PUNCT
ejpam-6044	148	2	ςn	ςn	NOUN
ejpam-6044	148	3	)	)	PUNCT
ejpam-6044	148	4	,	,	PUNCT
ejpam-6044	148	5	for	for	ADP
ejpam-6044	148	6	every	every	DET
ejpam-6044	148	7	ς1	ς1	NOUN
ejpam-6044	148	8	,	,	PUNCT
ejpam-6044	148	9	.	.	PUNCT
ejpam-6044	148	10	.	.	PUNCT
ejpam-6044	149	1	.	.	PUNCT
ejpam-6044	150	1	,	,	PUNCT
ejpam-6044	150	2	ς	ς	PROPN
ejpam-6044	150	3	′	′	NUM
ejpam-6044	150	4	k	k	NOUN
ejpam-6044	150	5	,	,	PUNCT
ejpam-6044	150	6	.	.	PUNCT
ejpam-6044	150	7	.	.	PUNCT
ejpam-6044	151	1	.	.	PUNCT
ejpam-6044	152	1	,	,	PUNCT
ejpam-6044	152	2	ςn	ςn	PROPN
ejpam-6044	152	3	∈	∈	PROPN
ejpam-6044	152	4	r	r	NOUN
ejpam-6044	152	5	therefore	therefore	ADV
ejpam-6044	152	6	,	,	PUNCT
ejpam-6044	152	7	d1	d1	PROPN
ejpam-6044	152	8	=	=	SYM
ejpam-6044	152	9	d2	d2	PROPN
ejpam-6044	152	10	.	.	PUNCT
ejpam-6044	153	1	this	this	PRON
ejpam-6044	153	2	completes	complete	VERB
ejpam-6044	153	3	the	the	DET
ejpam-6044	153	4	proof	proof	NOUN
ejpam-6044	153	5	.	.	PUNCT
ejpam-6044	154	1	theorem	theorem	NOUN
ejpam-6044	154	2	2	2	NUM
ejpam-6044	154	3	.	.	PUNCT
ejpam-6044	155	1	if	if	SCONJ
ejpam-6044	155	2	a	a	DET
ejpam-6044	155	3	prime	prime	NOUN
ejpam-6044	155	4	∗-ring	∗-ring	NOUN
ejpam-6044	155	5	r	r	NOUN
ejpam-6044	155	6	admits	admit	VERB
ejpam-6044	155	7	a	a	DET
ejpam-6044	155	8	nonzero	nonzero	X
ejpam-6044	155	9	(	(	PUNCT
ejpam-6044	155	10	α	α	NOUN
ejpam-6044	155	11	,	,	PUNCT
ejpam-6044	155	12	∗)-n	∗)-n	NOUN
ejpam-6044	155	13	-	-	PUNCT
ejpam-6044	155	14	derivation	derivation	NOUN
ejpam-6044	155	15	d	d	NOUN
ejpam-6044	155	16	,	,	PUNCT
ejpam-6044	155	17	then	then	ADV
ejpam-6044	155	18	r	r	NOUN
ejpam-6044	155	19	is	be	AUX
ejpam-6044	155	20	commutative	commutative	ADJ
ejpam-6044	155	21	.	.	PUNCT
ejpam-6044	156	1	proof	proof	NOUN
ejpam-6044	156	2	.	.	PUNCT
ejpam-6044	157	1	given	give	VERB
ejpam-6044	157	2	that	that	SCONJ
ejpam-6044	157	3	d	d	NOUN
ejpam-6044	157	4	is	be	AUX
ejpam-6044	157	5	a	a	DET
ejpam-6044	157	6	(	(	PUNCT
ejpam-6044	157	7	α	α	NOUN
ejpam-6044	157	8	,	,	PUNCT
ejpam-6044	157	9	∗)-n	∗)-n	NOUN
ejpam-6044	157	10	-	-	NOUN
ejpam-6044	157	11	derivation	derivation	NOUN
ejpam-6044	157	12	,	,	PUNCT
ejpam-6044	157	13	by	by	ADP
ejpam-6044	157	14	definition	definition	NOUN
ejpam-6044	157	15	we	we	PRON
ejpam-6044	157	16	have	have	VERB
ejpam-6044	157	17	d(ς1	d(ς1	NOUN
ejpam-6044	157	18	,	,	PUNCT
ejpam-6044	157	19	.	.	PUNCT
ejpam-6044	157	20	.	.	PUNCT
ejpam-6044	157	21	.	.	PUNCT
ejpam-6044	158	1	,	,	PUNCT
ejpam-6044	158	2	ςkς	ςkς	X
ejpam-6044	158	3	′	′	NUM
ejpam-6044	159	1	k	k	NOUN
ejpam-6044	159	2	,	,	PUNCT
ejpam-6044	159	3	.	.	PUNCT
ejpam-6044	159	4	.	.	PUNCT
ejpam-6044	159	5	.	.	PUNCT
ejpam-6044	160	1	,	,	PUNCT
ejpam-6044	160	2	ςn	ςn	X
ejpam-6044	160	3	)	)	PUNCT
ejpam-6044	160	4	=	=	SYM
ejpam-6044	160	5	d(ς1	d(ς1	NOUN
ejpam-6044	160	6	,	,	PUNCT
ejpam-6044	160	7	.	.	PUNCT
ejpam-6044	160	8	.	.	PUNCT
ejpam-6044	161	1	.	.	PUNCT
ejpam-6044	162	1	,	,	PUNCT
ejpam-6044	162	2	ςk	ςk	NUM
ejpam-6044	162	3	,	,	PUNCT
ejpam-6044	162	4	.	.	PUNCT
ejpam-6044	162	5	.	.	PUNCT
ejpam-6044	163	1	.	.	PUNCT
ejpam-6044	164	1	,	,	PUNCT
ejpam-6044	164	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	164	3	′	′	NUM
ejpam-6044	165	1	k	k	NOUN
ejpam-6044	165	2	)	)	PUNCT
ejpam-6044	166	1	+	+	CCONJ
ejpam-6044	166	2	ς∗kd(ς1	ς∗kd(ς1	NUM
ejpam-6044	166	3	,	,	PUNCT
ejpam-6044	166	4	.	.	PUNCT
ejpam-6044	166	5	.	.	PUNCT
ejpam-6044	166	6	.	.	PUNCT
ejpam-6044	167	1	,	,	PUNCT
ejpam-6044	167	2	ς	ς	PROPN
ejpam-6044	167	3	′	′	NUM
ejpam-6044	167	4	k	k	NOUN
ejpam-6044	167	5	,	,	PUNCT
ejpam-6044	167	6	.	.	PUNCT
ejpam-6044	167	7	.	.	PUNCT
ejpam-6044	168	1	.	.	PUNCT
ejpam-6044	169	1	,	,	PUNCT
ejpam-6044	169	2	ςn	ςn	X
ejpam-6044	169	3	)	)	PUNCT
ejpam-6044	169	4	for	for	ADP
ejpam-6044	169	5	every	every	DET
ejpam-6044	169	6	ςk	ςk	PROPN
ejpam-6044	169	7	,	,	PUNCT
ejpam-6044	169	8	ς	ς	PROPN
ejpam-6044	169	9	′	′	NUM
ejpam-6044	169	10	k	k	PROPN
ejpam-6044	169	11	∈	∈	PROPN
ejpam-6044	169	12	r.	r.	PROPN
ejpam-6044	169	13	now	now	ADV
ejpam-6044	169	14	substitute	substitute	VERB
ejpam-6044	169	15	ςk	ςk	PROPN
ejpam-6044	169	16	=	=	SYM
ejpam-6044	169	17	ςky	ςky	PROPN
ejpam-6044	169	18	,	,	PUNCT
ejpam-6044	169	19	where	where	SCONJ
ejpam-6044	169	20	y	y	PROPN
ejpam-6044	169	21	∈	∈	PROPN
ejpam-6044	169	22	r	r	NOUN
ejpam-6044	169	23	,	,	PUNCT
ejpam-6044	169	24	we	we	PRON
ejpam-6044	169	25	get	get	VERB
ejpam-6044	169	26	d(ς1	d(ς1	NOUN
ejpam-6044	169	27	,	,	PUNCT
ejpam-6044	169	28	.	.	PUNCT
ejpam-6044	169	29	.	.	PUNCT
ejpam-6044	170	1	.	.	PUNCT
ejpam-6044	171	1	,	,	PUNCT
ejpam-6044	171	2	ςkyς	ςkyς	NOUN
ejpam-6044	171	3	′	′	NUM
ejpam-6044	172	1	k	k	PROPN
ejpam-6044	172	2	,	,	PUNCT
ejpam-6044	172	3	.	.	PUNCT
ejpam-6044	172	4	.	.	PUNCT
ejpam-6044	172	5	.	.	PUNCT
ejpam-6044	173	1	,	,	PUNCT
ejpam-6044	173	2	ςn	ςn	X
ejpam-6044	173	3	)	)	PUNCT
ejpam-6044	173	4	=	=	SYM
ejpam-6044	173	5	d(ς1	d(ς1	NOUN
ejpam-6044	173	6	,	,	PUNCT
ejpam-6044	173	7	.	.	PUNCT
ejpam-6044	173	8	.	.	PUNCT
ejpam-6044	174	1	.	.	PUNCT
ejpam-6044	175	1	,	,	PUNCT
ejpam-6044	175	2	ςky	ςky	ADJ
ejpam-6044	175	3	,	,	PUNCT
ejpam-6044	175	4	.	.	PUNCT
ejpam-6044	175	5	.	.	PUNCT
ejpam-6044	176	1	.	.	PUNCT
ejpam-6044	177	1	,	,	PUNCT
ejpam-6044	177	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	177	3	′	′	NUM
ejpam-6044	178	1	k	k	NOUN
ejpam-6044	178	2	)	)	PUNCT
ejpam-6044	179	1	+	+	CCONJ
ejpam-6044	179	2	(	(	PUNCT
ejpam-6044	179	3	ςky	ςky	ADJ
ejpam-6044	179	4	)	)	PUNCT
ejpam-6044	179	5	∗d(ς1	∗d(ς1	NOUN
ejpam-6044	179	6	,	,	PUNCT
ejpam-6044	179	7	.	.	PUNCT
ejpam-6044	179	8	.	.	PUNCT
ejpam-6044	179	9	.	.	PUNCT
ejpam-6044	180	1	,	,	PUNCT
ejpam-6044	180	2	ς	ς	PROPN
ejpam-6044	180	3	′	′	NUM
ejpam-6044	180	4	k	k	NOUN
ejpam-6044	180	5	,	,	PUNCT
ejpam-6044	180	6	.	.	PUNCT
ejpam-6044	180	7	.	.	PUNCT
ejpam-6044	181	1	.	.	PUNCT
ejpam-6044	182	1	,	,	PUNCT
ejpam-6044	182	2	ςn	ςn	X
ejpam-6044	182	3	)	)	PUNCT
ejpam-6044	182	4	for	for	ADP
ejpam-6044	182	5	every	every	DET
ejpam-6044	182	6	ςk	ςk	PROPN
ejpam-6044	182	7	,	,	PUNCT
ejpam-6044	182	8	ς	ς	PROPN
ejpam-6044	183	1	′	′	NUM
ejpam-6044	183	2	k	k	NOUN
ejpam-6044	183	3	,	,	PUNCT
ejpam-6044	183	4	y	y	PROPN
ejpam-6044	183	5	∈	∈	PROPN
ejpam-6044	183	6	r.	r.	NOUN
ejpam-6044	183	7	expanding	expand	VERB
ejpam-6044	183	8	the	the	DET
ejpam-6044	183	9	terms	term	NOUN
ejpam-6044	183	10	,	,	PUNCT
ejpam-6044	183	11	d(ς1	d(ς1	NOUN
ejpam-6044	183	12	,	,	PUNCT
ejpam-6044	183	13	.	.	PUNCT
ejpam-6044	183	14	.	.	PUNCT
ejpam-6044	183	15	.	.	PUNCT
ejpam-6044	184	1	,	,	PUNCT
ejpam-6044	184	2	ςkyς	ςkyς	NOUN
ejpam-6044	184	3	′	′	NUM
ejpam-6044	185	1	k	k	PROPN
ejpam-6044	185	2	,	,	PUNCT
ejpam-6044	185	3	.	.	PUNCT
ejpam-6044	185	4	.	.	PUNCT
ejpam-6044	185	5	.	.	PUNCT
ejpam-6044	186	1	,	,	PUNCT
ejpam-6044	186	2	ςn	ςn	X
ejpam-6044	186	3	)	)	PUNCT
ejpam-6044	186	4	=	=	SYM
ejpam-6044	186	5	d(ς1	d(ς1	NOUN
ejpam-6044	186	6	,	,	PUNCT
ejpam-6044	186	7	.	.	PUNCT
ejpam-6044	186	8	.	.	PUNCT
ejpam-6044	187	1	.	.	PUNCT
ejpam-6044	188	1	,	,	PUNCT
ejpam-6044	188	2	ςk	ςk	NUM
ejpam-6044	188	3	,	,	PUNCT
ejpam-6044	188	4	.	.	PUNCT
ejpam-6044	188	5	.	.	PUNCT
ejpam-6044	189	1	.	.	PUNCT
ejpam-6044	190	1	,	,	PUNCT
ejpam-6044	190	2	ςn)α(y)α(ς	ςn)α(y)α(ς	NUM
ejpam-6044	190	3	′	′	NUM
ejpam-6044	190	4	k	k	NOUN
ejpam-6044	190	5	)	)	PUNCT
ejpam-6044	191	1	+	+	CCONJ
ejpam-6044	191	2	ς∗kd(ς1	ς∗kd(ς1	NUM
ejpam-6044	191	3	,	,	PUNCT
ejpam-6044	191	4	.	.	PUNCT
ejpam-6044	191	5	.	.	PUNCT
ejpam-6044	191	6	.	.	PUNCT
ejpam-6044	192	1	,	,	PUNCT
ejpam-6044	192	2	y	y	PROPN
ejpam-6044	192	3	,	,	PUNCT
ejpam-6044	192	4	.	.	PUNCT
ejpam-6044	192	5	.	.	PUNCT
ejpam-6044	193	1	.	.	PUNCT
ejpam-6044	194	1	,	,	PUNCT
ejpam-6044	194	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	194	3	′	′	NUM
ejpam-6044	195	1	k	k	NOUN
ejpam-6044	195	2	)	)	PUNCT
ejpam-6044	196	1	+	+	CCONJ
ejpam-6044	196	2	y∗ς∗kd(ς1	y∗ς∗kd(ς1	PROPN
ejpam-6044	196	3	,	,	PUNCT
ejpam-6044	196	4	.	.	PUNCT
ejpam-6044	196	5	.	.	PUNCT
ejpam-6044	197	1	.	.	PUNCT
ejpam-6044	198	1	,	,	PUNCT
ejpam-6044	198	2	ς	ς	PROPN
ejpam-6044	198	3	′	′	NUM
ejpam-6044	198	4	k	k	NOUN
ejpam-6044	198	5	,	,	PUNCT
ejpam-6044	198	6	.	.	PUNCT
ejpam-6044	198	7	.	.	PUNCT
ejpam-6044	199	1	.	.	PUNCT
ejpam-6044	200	1	,	,	PUNCT
ejpam-6044	200	2	ςn	ςn	X
ejpam-6044	200	3	)	)	PUNCT
ejpam-6044	200	4	(	(	PUNCT
ejpam-6044	200	5	4	4	X
ejpam-6044	200	6	)	)	PUNCT
ejpam-6044	200	7	for	for	ADP
ejpam-6044	200	8	every	every	DET
ejpam-6044	200	9	ς1	ς1	NOUN
ejpam-6044	200	10	,	,	PUNCT
ejpam-6044	200	11	..	..	PUNCT
ejpam-6044	200	12	,	,	PUNCT
ejpam-6044	200	13	ςk	ςk	NUM
ejpam-6044	200	14	,	,	PUNCT
ejpam-6044	200	15	ς	ς	PROPN
ejpam-6044	200	16	′	′	NUM
ejpam-6044	201	1	k	k	NOUN
ejpam-6044	201	2	,	,	PUNCT
ejpam-6044	201	3	...	...	PUNCT
ejpam-6044	201	4	,	,	PUNCT
ejpam-6044	201	5	ςn	ςn	PROPN
ejpam-6044	201	6	,	,	PUNCT
ejpam-6044	201	7	y	y	PROPN
ejpam-6044	201	8	∈	∈	PROPN
ejpam-6044	201	9	r.	r.	PROPN
ejpam-6044	201	10	alternative	alternative	ADJ
ejpam-6044	201	11	expression	expression	NOUN
ejpam-6044	201	12	for	for	ADP
ejpam-6044	201	13	the	the	DET
ejpam-6044	201	14	left	left	ADJ
ejpam-6044	201	15	hand	hand	NOUN
ejpam-6044	201	16	side	side	NOUN
ejpam-6044	201	17	of	of	ADP
ejpam-6044	201	18	above	above	ADJ
ejpam-6044	201	19	equation	equation	NOUN
ejpam-6044	201	20	is	be	AUX
ejpam-6044	201	21	given	give	VERB
ejpam-6044	201	22	by	by	ADP
ejpam-6044	201	23	d(ς1	d(ς1	NOUN
ejpam-6044	201	24	,	,	PUNCT
ejpam-6044	201	25	.	.	PUNCT
ejpam-6044	201	26	.	.	PUNCT
ejpam-6044	201	27	.	.	PUNCT
ejpam-6044	202	1	,	,	PUNCT
ejpam-6044	202	2	ςk(yς	ςk(yς	NOUN
ejpam-6044	202	3	′	′	NUM
ejpam-6044	203	1	k	k	X
ejpam-6044	203	2	)	)	PUNCT
ejpam-6044	203	3	,	,	PUNCT
ejpam-6044	203	4	.	.	PUNCT
ejpam-6044	203	5	.	.	PUNCT
ejpam-6044	204	1	.	.	PUNCT
ejpam-6044	205	1	,	,	PUNCT
ejpam-6044	205	2	ςn	ςn	X
ejpam-6044	205	3	)	)	PUNCT
ejpam-6044	205	4	=	=	SYM
ejpam-6044	205	5	d(ς1	d(ς1	NOUN
ejpam-6044	205	6	,	,	PUNCT
ejpam-6044	205	7	.	.	PUNCT
ejpam-6044	205	8	.	.	PUNCT
ejpam-6044	206	1	.	.	PUNCT
ejpam-6044	207	1	,	,	PUNCT
ejpam-6044	207	2	ςk	ςk	NUM
ejpam-6044	207	3	,	,	PUNCT
ejpam-6044	207	4	.	.	PUNCT
ejpam-6044	207	5	.	.	PUNCT
ejpam-6044	208	1	.	.	PUNCT
ejpam-6044	209	1	,	,	PUNCT
ejpam-6044	209	2	ςn)α(yς	ςn)α(yς	NUM
ejpam-6044	209	3	′	′	NUM
ejpam-6044	210	1	k	k	X
ejpam-6044	210	2	)	)	PUNCT
ejpam-6044	211	1	+	+	CCONJ
ejpam-6044	211	2	ς∗kd(ς1	ς∗kd(ς1	NUM
ejpam-6044	211	3	,	,	PUNCT
ejpam-6044	211	4	.	.	PUNCT
ejpam-6044	211	5	.	.	PUNCT
ejpam-6044	211	6	.	.	PUNCT
ejpam-6044	212	1	,	,	PUNCT
ejpam-6044	212	2	yς	yς	VERB
ejpam-6044	212	3	′	′	NUM
ejpam-6044	213	1	k	k	PROPN
ejpam-6044	213	2	,	,	PUNCT
ejpam-6044	213	3	.	.	PUNCT
ejpam-6044	213	4	.	.	PUNCT
ejpam-6044	213	5	.	.	PUNCT
ejpam-6044	214	1	,	,	PUNCT
ejpam-6044	214	2	ςn	ςn	NOUN
ejpam-6044	214	3	)	)	PUNCT
ejpam-6044	214	4	,	,	PUNCT
ejpam-6044	214	5	for	for	ADP
ejpam-6044	214	6	every	every	DET
ejpam-6044	214	7	ς1	ς1	NOUN
ejpam-6044	214	8	,	,	PUNCT
ejpam-6044	214	9	.	.	PUNCT
ejpam-6044	214	10	.	.	PUNCT
ejpam-6044	215	1	.	.	PUNCT
ejpam-6044	216	1	,	,	PUNCT
ejpam-6044	216	2	ςk	ςk	NUM
ejpam-6044	216	3	,	,	PUNCT
ejpam-6044	216	4	ς	ς	PROPN
ejpam-6044	216	5	′	′	NUM
ejpam-6044	216	6	k	k	NOUN
ejpam-6044	216	7	,	,	PUNCT
ejpam-6044	216	8	.	.	PUNCT
ejpam-6044	216	9	.	.	PUNCT
ejpam-6044	216	10	.	.	PUNCT
ejpam-6044	217	1	,	,	PUNCT
ejpam-6044	217	2	ςn	ςn	PROPN
ejpam-6044	217	3	,	,	PUNCT
ejpam-6044	217	4	y	y	PROPN
ejpam-6044	217	5	,	,	PUNCT
ejpam-6044	217	6	ς	ς	PROPN
ejpam-6044	217	7	∗	∗	NOUN
ejpam-6044	217	8	k	k	PROPN
ejpam-6044	217	9	∈	∈	PROPN
ejpam-6044	217	10	r.	r.	PROPN
ejpam-6044	217	11	expanding	expand	VERB
ejpam-6044	217	12	the	the	DET
ejpam-6044	217	13	terms	term	NOUN
ejpam-6044	217	14	d(ς1	d(ς1	NOUN
ejpam-6044	217	15	,	,	PUNCT
ejpam-6044	217	16	.	.	PUNCT
ejpam-6044	217	17	.	.	PUNCT
ejpam-6044	217	18	.	.	PUNCT
ejpam-6044	218	1	,	,	PUNCT
ejpam-6044	218	2	ςk(yς	ςk(yς	NOUN
ejpam-6044	218	3	′	′	NUM
ejpam-6044	219	1	k	k	X
ejpam-6044	219	2	)	)	PUNCT
ejpam-6044	219	3	,	,	PUNCT
ejpam-6044	219	4	.	.	PUNCT
ejpam-6044	219	5	.	.	PUNCT
ejpam-6044	220	1	.	.	PUNCT
ejpam-6044	221	1	,	,	PUNCT
ejpam-6044	221	2	ςn	ςn	X
ejpam-6044	221	3	)	)	PUNCT
ejpam-6044	221	4	=	=	SYM
ejpam-6044	221	5	d(ς1	d(ς1	NOUN
ejpam-6044	221	6	,	,	PUNCT
ejpam-6044	221	7	.	.	PUNCT
ejpam-6044	221	8	.	.	PUNCT
ejpam-6044	222	1	.	.	PUNCT
ejpam-6044	223	1	,	,	PUNCT
ejpam-6044	223	2	ςk	ςk	NUM
ejpam-6044	223	3	,	,	PUNCT
ejpam-6044	223	4	.	.	PUNCT
ejpam-6044	223	5	.	.	PUNCT
ejpam-6044	224	1	.	.	PUNCT
ejpam-6044	225	1	,	,	PUNCT
ejpam-6044	225	2	ςn)α(y)α(ς	ςn)α(y)α(ς	NUM
ejpam-6044	225	3	′	′	NUM
ejpam-6044	225	4	k	k	NOUN
ejpam-6044	225	5	)	)	PUNCT
ejpam-6044	226	1	+	+	CCONJ
ejpam-6044	226	2	ς∗kd(ς1	ς∗kd(ς1	NUM
ejpam-6044	226	3	,	,	PUNCT
ejpam-6044	226	4	.	.	PUNCT
ejpam-6044	226	5	.	.	PUNCT
ejpam-6044	226	6	.	.	PUNCT
ejpam-6044	227	1	,	,	PUNCT
ejpam-6044	227	2	y	y	PROPN
ejpam-6044	227	3	,	,	PUNCT
ejpam-6044	227	4	.	.	PUNCT
ejpam-6044	227	5	.	.	PUNCT
ejpam-6044	228	1	.	.	PUNCT
ejpam-6044	229	1	,	,	PUNCT
ejpam-6044	229	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	229	3	′	′	NUM
ejpam-6044	230	1	k	k	NOUN
ejpam-6044	230	2	)	)	PUNCT
ejpam-6044	231	1	+	+	CCONJ
ejpam-6044	231	2	ς∗ky	ς∗ky	NUM
ejpam-6044	231	3	∗d(ς1	∗d(ς1	NOUN
ejpam-6044	231	4	,	,	PUNCT
ejpam-6044	231	5	.	.	PUNCT
ejpam-6044	231	6	.	.	PUNCT
ejpam-6044	231	7	.	.	PUNCT
ejpam-6044	232	1	,	,	PUNCT
ejpam-6044	232	2	ς	ς	PROPN
ejpam-6044	232	3	′	′	NUM
ejpam-6044	232	4	k	k	NOUN
ejpam-6044	232	5	,	,	PUNCT
ejpam-6044	232	6	.	.	PUNCT
ejpam-6044	232	7	.	.	PUNCT
ejpam-6044	233	1	.	.	PUNCT
ejpam-6044	234	1	,	,	PUNCT
ejpam-6044	234	2	ςn	ςn	PROPN
ejpam-6044	234	3	)	)	PUNCT
ejpam-6044	234	4	.	.	PUNCT
ejpam-6044	235	1	(	(	PUNCT
ejpam-6044	235	2	5	5	X
ejpam-6044	235	3	)	)	PUNCT
ejpam-6044	235	4	substituting	substitute	VERB
ejpam-6044	235	5	equation	equation	NOUN
ejpam-6044	235	6	(	(	PUNCT
ejpam-6044	235	7	4	4	NUM
ejpam-6044	235	8	)	)	PUNCT
ejpam-6044	235	9	into	into	ADP
ejpam-6044	235	10	equation	equation	NOUN
ejpam-6044	235	11	(	(	PUNCT
ejpam-6044	235	12	5	5	NUM
ejpam-6044	235	13	)	)	PUNCT
ejpam-6044	235	14	,	,	PUNCT
ejpam-6044	235	15	we	we	PRON
ejpam-6044	235	16	get	get	VERB
ejpam-6044	235	17	0	0	NUM
ejpam-6044	236	1	=	=	SYM
ejpam-6044	236	2	(	(	PUNCT
ejpam-6044	236	3	ς∗ky	ς∗ky	NOUN
ejpam-6044	236	4	∗	∗	NOUN
ejpam-6044	236	5	−	−	PROPN
ejpam-6044	236	6	y∗ς∗k)d(ς1	y∗ς∗k)d(ς1	NOUN
ejpam-6044	236	7	,	,	PUNCT
ejpam-6044	236	8	.	.	PUNCT
ejpam-6044	236	9	.	.	PUNCT
ejpam-6044	237	1	.	.	PUNCT
ejpam-6044	238	1	,	,	PUNCT
ejpam-6044	238	2	ς	ς	PROPN
ejpam-6044	238	3	′	′	NUM
ejpam-6044	238	4	k	k	NOUN
ejpam-6044	238	5	,	,	PUNCT
ejpam-6044	238	6	.	.	PUNCT
ejpam-6044	238	7	.	.	PUNCT
ejpam-6044	239	1	.	.	PUNCT
ejpam-6044	240	1	,	,	PUNCT
ejpam-6044	240	2	ςn	ςn	PROPN
ejpam-6044	240	3	)	)	PUNCT
ejpam-6044	240	4	.	.	PUNCT
ejpam-6044	241	1	for	for	ADP
ejpam-6044	241	2	every	every	DET
ejpam-6044	241	3	ς1	ς1	NOUN
ejpam-6044	241	4	,	,	PUNCT
ejpam-6044	241	5	...	...	PUNCT
ejpam-6044	241	6	,	,	PUNCT
ejpam-6044	241	7	ςn	ςn	PROPN
ejpam-6044	241	8	,	,	PUNCT
ejpam-6044	241	9	y	y	PROPN
ejpam-6044	241	10	∈	∈	PROPN
ejpam-6044	241	11	r.	r.	PROPN
ejpam-6044	241	12	f.	f.	PROPN
ejpam-6044	241	13	shujat	shujat	PROPN
ejpam-6044	241	14	,	,	PUNCT
ejpam-6044	241	15	s.	s.	PROPN
ejpam-6044	241	16	alharbi	alharbi	PROPN
ejpam-6044	241	17	/	/	SYM
ejpam-6044	241	18	eur	eur	PROPN
ejpam-6044	241	19	.	.	PUNCT
ejpam-6044	242	1	j.	j.	PROPN
ejpam-6044	242	2	pure	pure	PROPN
ejpam-6044	242	3	appl	appl	PROPN
ejpam-6044	242	4	.	.	PROPN
ejpam-6044	242	5	math	math	PROPN
ejpam-6044	242	6	,	,	PUNCT
ejpam-6044	242	7	18	18	NUM
ejpam-6044	242	8	(	(	PUNCT
ejpam-6044	242	9	2	2	NUM
ejpam-6044	242	10	)	)	PUNCT
ejpam-6044	242	11	(	(	PUNCT
ejpam-6044	242	12	2025	2025	NUM
ejpam-6044	242	13	)	)	PUNCT
ejpam-6044	242	14	,	,	PUNCT
ejpam-6044	242	15	6044	6044	NUM
ejpam-6044	242	16	6	6	NUM
ejpam-6044	242	17	of	of	ADP
ejpam-6044	242	18	11	11	NUM
ejpam-6044	242	19	this	this	DET
ejpam-6044	242	20	simplifies	simplifie	NOUN
ejpam-6044	242	21	to	to	ADP
ejpam-6044	242	22	0	0	NUM
ejpam-6044	242	23	=	=	SYM
ejpam-6044	243	1	[	[	X
ejpam-6044	243	2	ς∗k	ς∗k	NUM
ejpam-6044	243	3	,	,	PUNCT
ejpam-6044	243	4	y	y	PROPN
ejpam-6044	243	5	∗]d(ς1	∗]d(ς1	NOUN
ejpam-6044	243	6	,	,	PUNCT
ejpam-6044	243	7	.	.	PUNCT
ejpam-6044	243	8	.	.	PUNCT
ejpam-6044	243	9	.	.	PUNCT
ejpam-6044	244	1	,	,	PUNCT
ejpam-6044	244	2	ς	ς	PROPN
ejpam-6044	244	3	′	′	NUM
ejpam-6044	244	4	k	k	NOUN
ejpam-6044	244	5	,	,	PUNCT
ejpam-6044	244	6	.	.	PUNCT
ejpam-6044	244	7	.	.	PUNCT
ejpam-6044	245	1	.	.	PUNCT
ejpam-6044	246	1	,	,	PUNCT
ejpam-6044	246	2	ςn	ςn	PROPN
ejpam-6044	246	3	)	)	PUNCT
ejpam-6044	246	4	.	.	PUNCT
ejpam-6044	247	1	now	now	ADV
ejpam-6044	247	2	,	,	PUNCT
ejpam-6044	247	3	substitute	substitute	NOUN
ejpam-6044	247	4	ς∗k	ς∗k	NUM
ejpam-6044	247	5	=	=	SYM
ejpam-6044	247	6	ςk	ςk	PROPN
ejpam-6044	247	7	and	and	CCONJ
ejpam-6044	247	8	y∗	y∗	PROPN
ejpam-6044	248	1	=	=	SYM
ejpam-6044	248	2	y	y	PROPN
ejpam-6044	248	3	into	into	ADP
ejpam-6044	248	4	the	the	DET
ejpam-6044	248	5	above	above	ADJ
ejpam-6044	248	6	equation	equation	NOUN
ejpam-6044	248	7	,	,	PUNCT
ejpam-6044	248	8	[	[	X
ejpam-6044	248	9	ςk	ςk	NUM
ejpam-6044	248	10	,	,	PUNCT
ejpam-6044	248	11	y]d(ς1	y]d(ς1	NOUN
ejpam-6044	248	12	,	,	PUNCT
ejpam-6044	248	13	.	.	PUNCT
ejpam-6044	248	14	.	.	PUNCT
ejpam-6044	248	15	.	.	PUNCT
ejpam-6044	249	1	,	,	PUNCT
ejpam-6044	249	2	ς	ς	PROPN
ejpam-6044	249	3	′	′	NUM
ejpam-6044	249	4	k	k	NOUN
ejpam-6044	249	5	,	,	PUNCT
ejpam-6044	249	6	.	.	PUNCT
ejpam-6044	249	7	.	.	PUNCT
ejpam-6044	250	1	.	.	PUNCT
ejpam-6044	251	1	,	,	PUNCT
ejpam-6044	251	2	ςn	ςn	X
ejpam-6044	251	3	)	)	PUNCT
ejpam-6044	251	4	=	=	SYM
ejpam-6044	251	5	0	0	NUM
ejpam-6044	251	6	for	for	ADP
ejpam-6044	251	7	everyς1	everyς1	PROPN
ejpam-6044	251	8	,	,	PUNCT
ejpam-6044	251	9	.	.	PUNCT
ejpam-6044	251	10	.	.	PUNCT
ejpam-6044	252	1	.	.	PUNCT
ejpam-6044	253	1	,	,	PUNCT
ejpam-6044	253	2	ς	ς	PROPN
ejpam-6044	253	3	′	′	NUM
ejpam-6044	253	4	k	k	NOUN
ejpam-6044	253	5	,	,	PUNCT
ejpam-6044	253	6	.	.	PUNCT
ejpam-6044	253	7	.	.	PUNCT
ejpam-6044	254	1	.	.	PUNCT
ejpam-6044	255	1	,	,	PUNCT
ejpam-6044	255	2	ςn	ςn	PROPN
ejpam-6044	255	3	∈	∈	PROPN
ejpam-6044	255	4	r.	r.	PROPN
ejpam-6044	255	5	replace	replace	VERB
ejpam-6044	255	6	y	y	PROPN
ejpam-6044	255	7	by	by	ADP
ejpam-6044	255	8	yr	yr	NOUN
ejpam-6044	255	9	to	to	PART
ejpam-6044	255	10	obtain	obtain	VERB
ejpam-6044	255	11	[	[	X
ejpam-6044	255	12	ςk	ςk	NOUN
ejpam-6044	255	13	,	,	PUNCT
ejpam-6044	255	14	y]rd(ς1	y]rd(ς1	NOUN
ejpam-6044	255	15	,	,	PUNCT
ejpam-6044	255	16	.	.	PUNCT
ejpam-6044	255	17	.	.	PUNCT
ejpam-6044	255	18	.	.	PUNCT
ejpam-6044	256	1	,	,	PUNCT
ejpam-6044	256	2	ς	ς	PROPN
ejpam-6044	256	3	′	′	NUM
ejpam-6044	256	4	k	k	NOUN
ejpam-6044	256	5	,	,	PUNCT
ejpam-6044	256	6	.	.	PUNCT
ejpam-6044	256	7	.	.	PUNCT
ejpam-6044	257	1	.	.	PUNCT
ejpam-6044	258	1	,	,	PUNCT
ejpam-6044	258	2	ςn	ςn	X
ejpam-6044	258	3	)	)	PUNCT
ejpam-6044	258	4	=	=	SYM
ejpam-6044	258	5	0	0	NUM
ejpam-6044	258	6	for	for	ADP
ejpam-6044	258	7	every	every	DET
ejpam-6044	258	8	ς1	ς1	NOUN
ejpam-6044	258	9	,	,	PUNCT
ejpam-6044	258	10	.	.	PUNCT
ejpam-6044	258	11	.	.	PUNCT
ejpam-6044	259	1	.	.	PUNCT
ejpam-6044	260	1	,	,	PUNCT
ejpam-6044	260	2	ς	ς	PROPN
ejpam-6044	260	3	′	′	NUM
ejpam-6044	260	4	k	k	NOUN
ejpam-6044	260	5	,	,	PUNCT
ejpam-6044	260	6	.	.	PUNCT
ejpam-6044	260	7	.	.	PUNCT
ejpam-6044	261	1	.	.	PUNCT
ejpam-6044	262	1	,	,	PUNCT
ejpam-6044	262	2	ςn	ςn	PROPN
ejpam-6044	262	3	,	,	PUNCT
ejpam-6044	262	4	y	y	PROPN
ejpam-6044	262	5	,	,	PUNCT
ejpam-6044	262	6	r	r	PROPN
ejpam-6044	262	7	∈	∈	PROPN
ejpam-6044	262	8	r.	r.	NOUN
ejpam-6044	262	9	(	(	PUNCT
ejpam-6044	262	10	6	6	NUM
ejpam-6044	262	11	)	)	PUNCT
ejpam-6044	262	12	by	by	ADP
ejpam-6044	262	13	primeness	primeness	NOUN
ejpam-6044	262	14	,	,	PUNCT
ejpam-6044	262	15	we	we	PRON
ejpam-6044	262	16	conclude	conclude	VERB
ejpam-6044	262	17	that	that	SCONJ
ejpam-6044	262	18	either	either	DET
ejpam-6044	262	19	d(ς1	d(ς1	NOUN
ejpam-6044	262	20	,	,	PUNCT
ejpam-6044	262	21	.	.	PUNCT
ejpam-6044	262	22	.	.	PUNCT
ejpam-6044	263	1	.	.	PUNCT
ejpam-6044	264	1	,	,	PUNCT
ejpam-6044	264	2	ς	ς	PROPN
ejpam-6044	264	3	′	′	NUM
ejpam-6044	264	4	k	k	NOUN
ejpam-6044	264	5	,	,	PUNCT
ejpam-6044	264	6	.	.	PUNCT
ejpam-6044	264	7	.	.	PUNCT
ejpam-6044	265	1	.	.	PUNCT
ejpam-6044	266	1	,	,	PUNCT
ejpam-6044	266	2	ςn	ςn	X
ejpam-6044	266	3	)	)	PUNCT
ejpam-6044	266	4	=	=	SYM
ejpam-6044	266	5	0	0	NUM
ejpam-6044	266	6	for	for	ADP
ejpam-6044	266	7	every	every	DET
ejpam-6044	266	8	ς1	ς1	NOUN
ejpam-6044	266	9	,	,	PUNCT
ejpam-6044	266	10	.	.	PUNCT
ejpam-6044	266	11	.	.	PUNCT
ejpam-6044	267	1	.	.	PUNCT
ejpam-6044	268	1	,	,	PUNCT
ejpam-6044	268	2	ς	ς	PROPN
ejpam-6044	268	3	′	′	NUM
ejpam-6044	268	4	k	k	NOUN
ejpam-6044	268	5	,	,	PUNCT
ejpam-6044	268	6	.	.	PUNCT
ejpam-6044	268	7	.	.	PUNCT
ejpam-6044	268	8	.	.	PUNCT
ejpam-6044	269	1	,	,	PUNCT
ejpam-6044	269	2	ςn	ςn	PROPN
ejpam-6044	269	3	∈	∈	PROPN
ejpam-6044	269	4	r	r	NOUN
ejpam-6044	269	5	or	or	CCONJ
ejpam-6044	269	6	[	[	X
ejpam-6044	269	7	ςk	ςk	PROPN
ejpam-6044	269	8	,	,	PUNCT
ejpam-6044	269	9	y	y	NOUN
ejpam-6044	269	10	]	]	X
ejpam-6044	269	11	=	=	SYM
ejpam-6044	269	12	0	0	NUM
ejpam-6044	269	13	for	for	ADP
ejpam-6044	269	14	every	every	DET
ejpam-6044	269	15	ςk	ςk	PROPN
ejpam-6044	269	16	,	,	PUNCT
ejpam-6044	269	17	y	y	PROPN
ejpam-6044	269	18	∈	∈	PROPN
ejpam-6044	269	19	r.	r.	PROPN
ejpam-6044	269	20	since	since	SCONJ
ejpam-6044	269	21	d(ς1	d(ς1	NOUN
ejpam-6044	269	22	,	,	PUNCT
ejpam-6044	269	23	.	.	PUNCT
ejpam-6044	269	24	.	.	PUNCT
ejpam-6044	269	25	.	.	PUNCT
ejpam-6044	270	1	,	,	PUNCT
ejpam-6044	270	2	ς	ς	PROPN
ejpam-6044	270	3	′	′	NUM
ejpam-6044	270	4	k	k	NOUN
ejpam-6044	270	5	,	,	PUNCT
ejpam-6044	270	6	.	.	PUNCT
ejpam-6044	270	7	.	.	PUNCT
ejpam-6044	271	1	.	.	PUNCT
ejpam-6044	272	1	,	,	PUNCT
ejpam-6044	272	2	ςn	ςn	X
ejpam-6044	272	3	)	)	PUNCT
ejpam-6044	272	4	̸=	̸=	PROPN
ejpam-6044	272	5	0	0	NUM
ejpam-6044	272	6	,	,	PUNCT
ejpam-6044	272	7	it	it	PRON
ejpam-6044	272	8	follows	follow	VERB
ejpam-6044	272	9	that	that	SCONJ
ejpam-6044	273	1	[	[	X
ejpam-6044	273	2	ςk	ςk	NUM
ejpam-6044	273	3	,	,	PUNCT
ejpam-6044	273	4	y	y	NOUN
ejpam-6044	273	5	]	]	X
ejpam-6044	273	6	=	=	SYM
ejpam-6044	273	7	0	0	NUM
ejpam-6044	273	8	for	for	ADP
ejpam-6044	273	9	every	every	DET
ejpam-6044	273	10	ςk	ςk	PROPN
ejpam-6044	273	11	,	,	PUNCT
ejpam-6044	273	12	y	y	PROPN
ejpam-6044	273	13	∈	∈	PROPN
ejpam-6044	273	14	r.	r.	PROPN
ejpam-6044	273	15	thus	thus	ADV
ejpam-6044	273	16	,	,	PUNCT
ejpam-6044	273	17	r	r	NOUN
ejpam-6044	273	18	is	be	AUX
ejpam-6044	273	19	commutative	commutative	ADJ
ejpam-6044	273	20	.	.	PUNCT
ejpam-6044	274	1	corollary	corollary	ADJ
ejpam-6044	274	2	1	1	NUM
ejpam-6044	274	3	.	.	PUNCT
ejpam-6044	275	1	if	if	SCONJ
ejpam-6044	275	2	a	a	DET
ejpam-6044	275	3	non	non	ADJ
ejpam-6044	275	4	-	-	ADJ
ejpam-6044	275	5	commutative	commutative	ADJ
ejpam-6044	275	6	prime	prime	NOUN
ejpam-6044	275	7	∗-ring	∗-ring	NOUN
ejpam-6044	275	8	r	r	NOUN
ejpam-6044	275	9	admits	admit	VERB
ejpam-6044	275	10	a	a	DET
ejpam-6044	275	11	(	(	PUNCT
ejpam-6044	275	12	α	α	NOUN
ejpam-6044	275	13	,	,	PUNCT
ejpam-6044	275	14	∗)-n	∗)-n	NOUN
ejpam-6044	275	15	-	-	PUNCT
ejpam-6044	275	16	derivation	derivation	NOUN
ejpam-6044	275	17	d	d	NOUN
ejpam-6044	275	18	,	,	PUNCT
ejpam-6044	275	19	then	then	ADV
ejpam-6044	275	20	d	d	X
ejpam-6044	275	21	=	=	NOUN
ejpam-6044	275	22	0	0	NUM
ejpam-6044	275	23	.	.	PUNCT
ejpam-6044	276	1	theorem	theorem	NOUN
ejpam-6044	276	2	3	3	X
ejpam-6044	276	3	.	.	PUNCT
ejpam-6044	277	1	let	let	VERB
ejpam-6044	277	2	r	r	PRON
ejpam-6044	277	3	be	be	AUX
ejpam-6044	277	4	a	a	DET
ejpam-6044	277	5	prime	prime	ADJ
ejpam-6044	277	6	∗-ring	∗-ring	NOUN
ejpam-6044	277	7	.	.	PUNCT
ejpam-6044	278	1	if	if	SCONJ
ejpam-6044	278	2	r	r	NOUN
ejpam-6044	278	3	admits	admit	VERB
ejpam-6044	278	4	a	a	DET
ejpam-6044	278	5	nonzero	nonzero	NOUN
ejpam-6044	278	6	generalized	generalized	ADJ
ejpam-6044	278	7	(	(	PUNCT
ejpam-6044	278	8	α	α	NOUN
ejpam-6044	278	9	,	,	PUNCT
ejpam-6044	278	10	∗)-nderivation	∗)-nderivation	NOUN
ejpam-6044	278	11	f	f	NOUN
ejpam-6044	278	12	associated	associate	VERB
ejpam-6044	278	13	with	with	ADP
ejpam-6044	278	14	an	an	DET
ejpam-6044	278	15	(	(	PUNCT
ejpam-6044	278	16	α	α	NOUN
ejpam-6044	278	17	,	,	PUNCT
ejpam-6044	278	18	∗)-n	∗)-n	NOUN
ejpam-6044	278	19	-	-	PUNCT
ejpam-6044	278	20	derivation	derivation	NOUN
ejpam-6044	278	21	d	d	NOUN
ejpam-6044	278	22	,	,	PUNCT
ejpam-6044	278	23	then	then	ADV
ejpam-6044	278	24	one	one	NUM
ejpam-6044	278	25	of	of	ADP
ejpam-6044	278	26	the	the	DET
ejpam-6044	278	27	conditions	condition	NOUN
ejpam-6044	278	28	hold	hold	VERB
ejpam-6044	278	29	:	:	PUNCT
ejpam-6044	278	30	1	1	X
ejpam-6044	278	31	.	.	X
ejpam-6044	278	32	r	r	NOUN
ejpam-6044	278	33	is	be	AUX
ejpam-6044	278	34	commutative	commutative	ADJ
ejpam-6044	278	35	.	.	PUNCT
ejpam-6044	279	1	2	2	NUM
ejpam-6044	279	2	.	.	X
ejpam-6044	279	3	f	f	PROPN
ejpam-6044	279	4	acts	act	VERB
ejpam-6044	279	5	as	as	SCONJ
ejpam-6044	279	6	left	leave	VERB
ejpam-6044	279	7	α	α	NOUN
ejpam-6044	279	8	-	-	PUNCT
ejpam-6044	279	9	centralizer	centralizer	NOUN
ejpam-6044	279	10	.	.	PUNCT
ejpam-6044	280	1	proof	proof	NOUN
ejpam-6044	280	2	.	.	PUNCT
ejpam-6044	281	1	since	since	SCONJ
ejpam-6044	281	2	f	f	PROPN
ejpam-6044	281	3	is	be	AUX
ejpam-6044	281	4	a	a	DET
ejpam-6044	281	5	generalized	generalized	ADJ
ejpam-6044	281	6	(	(	PUNCT
ejpam-6044	281	7	α	α	NOUN
ejpam-6044	281	8	,	,	PUNCT
ejpam-6044	281	9	∗)-n	∗)-n	NOUN
ejpam-6044	281	10	-	-	NOUN
ejpam-6044	281	11	derivation	derivation	NOUN
ejpam-6044	281	12	,	,	PUNCT
ejpam-6044	281	13	then	then	ADV
ejpam-6044	281	14	we	we	PRON
ejpam-6044	281	15	obtain	obtain	VERB
ejpam-6044	281	16	for	for	ADP
ejpam-6044	281	17	all	all	DET
ejpam-6044	281	18	ς1	ς1	NOUN
ejpam-6044	281	19	,	,	PUNCT
ejpam-6044	281	20	...	...	PUNCT
ejpam-6044	281	21	,	,	PUNCT
ejpam-6044	281	22	ςk	ςk	PROPN
ejpam-6044	281	23	,	,	PUNCT
ejpam-6044	281	24	ς	ς	PROPN
ejpam-6044	281	25	′	′	NUM
ejpam-6044	281	26	k	k	NOUN
ejpam-6044	281	27	,	,	PUNCT
ejpam-6044	281	28	...	...	PUNCT
ejpam-6044	281	29	,	,	PUNCT
ejpam-6044	281	30	ςn	ςn	PROPN
ejpam-6044	281	31	∈	∈	PROPN
ejpam-6044	281	32	r	r	NOUN
ejpam-6044	281	33	f(ς1	f(ς1	NOUN
ejpam-6044	281	34	,	,	PUNCT
ejpam-6044	281	35	...	...	PUNCT
ejpam-6044	281	36	,	,	PUNCT
ejpam-6044	281	37	ςkyς	ςkyς	PROPN
ejpam-6044	281	38	′	′	NUM
ejpam-6044	282	1	k	k	PROPN
ejpam-6044	282	2	,	,	PUNCT
ejpam-6044	282	3	...	...	PUNCT
ejpam-6044	282	4	,	,	PUNCT
ejpam-6044	282	5	ςn	ςn	NOUN
ejpam-6044	282	6	)	)	PUNCT
ejpam-6044	282	7	=	=	NOUN
ejpam-6044	282	8	f(ς1	f(ς1	NOUN
ejpam-6044	282	9	,	,	PUNCT
ejpam-6044	282	10	.	.	PUNCT
ejpam-6044	282	11	.	.	PUNCT
ejpam-6044	282	12	.	.	PUNCT
ejpam-6044	283	1	,	,	PUNCT
ejpam-6044	283	2	ςky	ςky	ADJ
ejpam-6044	283	3	,	,	PUNCT
ejpam-6044	283	4	.	.	PUNCT
ejpam-6044	283	5	.	.	PUNCT
ejpam-6044	284	1	.	.	PUNCT
ejpam-6044	285	1	,	,	PUNCT
ejpam-6044	285	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	285	3	′	′	NUM
ejpam-6044	286	1	k	k	NOUN
ejpam-6044	286	2	)	)	PUNCT
ejpam-6044	287	1	+	+	CCONJ
ejpam-6044	287	2	(	(	PUNCT
ejpam-6044	287	3	ςky	ςky	ADJ
ejpam-6044	287	4	)	)	PUNCT
ejpam-6044	287	5	∗d(ς1	∗d(ς1	NOUN
ejpam-6044	287	6	,	,	PUNCT
ejpam-6044	287	7	.	.	PUNCT
ejpam-6044	287	8	.	.	PUNCT
ejpam-6044	287	9	.	.	PUNCT
ejpam-6044	288	1	,	,	PUNCT
ejpam-6044	288	2	ς	ς	PROPN
ejpam-6044	288	3	′	′	NUM
ejpam-6044	288	4	k	k	NOUN
ejpam-6044	288	5	,	,	PUNCT
ejpam-6044	288	6	.	.	PUNCT
ejpam-6044	288	7	.	.	PUNCT
ejpam-6044	289	1	.	.	PUNCT
ejpam-6044	290	1	,	,	PUNCT
ejpam-6044	290	2	ςn	ςn	PROPN
ejpam-6044	290	3	)	)	PUNCT
ejpam-6044	290	4	.	.	PUNCT
ejpam-6044	291	1	(	(	PUNCT
ejpam-6044	291	2	7	7	X
ejpam-6044	291	3	)	)	PUNCT
ejpam-6044	291	4	simplify	simplify	ADJ
ejpam-6044	291	5	above	above	ADP
ejpam-6044	291	6	expression	expression	NOUN
ejpam-6044	291	7	to	to	PART
ejpam-6044	291	8	get	get	VERB
ejpam-6044	291	9	f(ς1	f(ς1	NOUN
ejpam-6044	291	10	,	,	PUNCT
ejpam-6044	291	11	...	...	PUNCT
ejpam-6044	291	12	,	,	PUNCT
ejpam-6044	291	13	ςkyς	ςkyς	PROPN
ejpam-6044	291	14	′	′	NUM
ejpam-6044	292	1	k	k	PROPN
ejpam-6044	292	2	,	,	PUNCT
ejpam-6044	292	3	...	...	PUNCT
ejpam-6044	292	4	,	,	PUNCT
ejpam-6044	292	5	ςn	ςn	NOUN
ejpam-6044	292	6	)	)	PUNCT
ejpam-6044	292	7	=	=	NOUN
ejpam-6044	292	8	f(ς1	f(ς1	NOUN
ejpam-6044	292	9	,	,	PUNCT
ejpam-6044	292	10	.	.	PUNCT
ejpam-6044	292	11	.	.	PUNCT
ejpam-6044	293	1	.	.	PUNCT
ejpam-6044	294	1	,	,	PUNCT
ejpam-6044	294	2	ςk	ςk	NUM
ejpam-6044	294	3	,	,	PUNCT
ejpam-6044	294	4	.	.	PUNCT
ejpam-6044	294	5	.	.	PUNCT
ejpam-6044	295	1	.	.	PUNCT
ejpam-6044	296	1	,	,	PUNCT
ejpam-6044	296	2	ςn)α(y)α(ς	ςn)α(y)α(ς	NUM
ejpam-6044	296	3	′	′	NUM
ejpam-6044	296	4	k	k	NOUN
ejpam-6044	296	5	)	)	PUNCT
ejpam-6044	297	1	+	+	PROPN
ejpam-6044	297	2	(	(	PUNCT
ejpam-6044	297	3	ςk	ςk	NOUN
ejpam-6044	297	4	)	)	PUNCT
ejpam-6044	297	5	∗d(ς1	∗d(ς1	NOUN
ejpam-6044	297	6	,	,	PUNCT
ejpam-6044	297	7	.	.	PUNCT
ejpam-6044	297	8	.	.	PUNCT
ejpam-6044	298	1	.	.	PUNCT
ejpam-6044	299	1	,	,	PUNCT
ejpam-6044	299	2	y	y	PROPN
ejpam-6044	299	3	,	,	PUNCT
ejpam-6044	299	4	.	.	PUNCT
ejpam-6044	299	5	.	.	PUNCT
ejpam-6044	300	1	.	.	PUNCT
ejpam-6044	301	1	,	,	PUNCT
ejpam-6044	301	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	301	3	′	′	NUM
ejpam-6044	302	1	k	k	NOUN
ejpam-6044	302	2	)	)	PUNCT
ejpam-6044	303	1	+	+	PROPN
ejpam-6044	303	2	y∗ς∗kd(ς1	y∗ς∗kd(ς1	PROPN
ejpam-6044	303	3	,	,	PUNCT
ejpam-6044	303	4	.	.	PUNCT
ejpam-6044	303	5	.	.	PUNCT
ejpam-6044	304	1	.	.	PUNCT
ejpam-6044	305	1	,	,	PUNCT
ejpam-6044	305	2	ς	ς	PROPN
ejpam-6044	305	3	′	′	NUM
ejpam-6044	305	4	k	k	NOUN
ejpam-6044	305	5	,	,	PUNCT
ejpam-6044	305	6	.	.	PUNCT
ejpam-6044	305	7	.	.	PUNCT
ejpam-6044	306	1	.	.	PUNCT
ejpam-6044	307	1	,	,	PUNCT
ejpam-6044	307	2	ςn	ςn	X
ejpam-6044	307	3	)	)	PUNCT
ejpam-6044	307	4	(	(	PUNCT
ejpam-6044	307	5	8)	8)	NUM
ejpam-6044	307	6	for	for	ADP
ejpam-6044	307	7	every	every	DET
ejpam-6044	307	8	ς1	ς1	NOUN
ejpam-6044	307	9	,	,	PUNCT
ejpam-6044	307	10	...	...	PUNCT
ejpam-6044	307	11	,	,	PUNCT
ejpam-6044	307	12	ςk	ςk	PROPN
ejpam-6044	307	13	,	,	PUNCT
ejpam-6044	307	14	y	y	PROPN
ejpam-6044	307	15	,	,	PUNCT
ejpam-6044	307	16	ς	ς	PROPN
ejpam-6044	307	17	′	′	NUM
ejpam-6044	307	18	k	k	NOUN
ejpam-6044	307	19	,	,	PUNCT
ejpam-6044	307	20	...	...	PUNCT
ejpam-6044	307	21	,	,	PUNCT
ejpam-6044	307	22	ςn	ςn	PROPN
ejpam-6044	307	23	in	in	ADP
ejpam-6044	307	24	r.	r.	PROPN
ejpam-6044	307	25	alternatively	alternatively	ADV
ejpam-6044	307	26	in	in	ADP
ejpam-6044	307	27	view	view	NOUN
ejpam-6044	307	28	of	of	ADP
ejpam-6044	307	29	(	(	PUNCT
ejpam-6044	307	30	7	7	X
ejpam-6044	307	31	)	)	PUNCT
ejpam-6044	307	32	we	we	PRON
ejpam-6044	307	33	find	find	VERB
ejpam-6044	307	34	f	f	PROPN
ejpam-6044	307	35	(	(	PUNCT
ejpam-6044	307	36	ς1	ς1	NOUN
ejpam-6044	307	37	,	,	PUNCT
ejpam-6044	307	38	.	.	PUNCT
ejpam-6044	307	39	.	.	PUNCT
ejpam-6044	308	1	.	.	PUNCT
ejpam-6044	309	1	,	,	PUNCT
ejpam-6044	309	2	ςkyς	ςkyς	NOUN
ejpam-6044	309	3	′	′	NUM
ejpam-6044	310	1	k	k	PROPN
ejpam-6044	310	2	,	,	PUNCT
ejpam-6044	310	3	.	.	PUNCT
ejpam-6044	310	4	.	.	PUNCT
ejpam-6044	310	5	.	.	PUNCT
ejpam-6044	311	1	,	,	PUNCT
ejpam-6044	311	2	ςn	ςn	X
ejpam-6044	311	3	)	)	PUNCT
ejpam-6044	312	1	=	=	SYM
ejpam-6044	312	2	f	f	PROPN
ejpam-6044	312	3	(	(	PUNCT
ejpam-6044	312	4	ς1	ς1	NOUN
ejpam-6044	312	5	,	,	PUNCT
ejpam-6044	312	6	.	.	PUNCT
ejpam-6044	312	7	.	.	PUNCT
ejpam-6044	312	8	.	.	PUNCT
ejpam-6044	312	9	,	,	PUNCT
ejpam-6044	312	10	ςk	ςk	NUM
ejpam-6044	312	11	,	,	PUNCT
ejpam-6044	312	12	.	.	PUNCT
ejpam-6044	312	13	.	.	PUNCT
ejpam-6044	312	14	.	.	PUNCT
ejpam-6044	313	1	,	,	PUNCT
ejpam-6044	313	2	ςn)α(yς	ςn)α(yς	NUM
ejpam-6044	313	3	′	′	NUM
ejpam-6044	314	1	k	k	X
ejpam-6044	314	2	)	)	PUNCT
ejpam-6044	315	1	+	+	CCONJ
ejpam-6044	315	2	ς∗kd(ς1	ς∗kd(ς1	NUM
ejpam-6044	315	3	,	,	PUNCT
ejpam-6044	315	4	.	.	PUNCT
ejpam-6044	315	5	.	.	PUNCT
ejpam-6044	315	6	.	.	PUNCT
ejpam-6044	316	1	,	,	PUNCT
ejpam-6044	316	2	yς	yς	VERB
ejpam-6044	316	3	′	′	NUM
ejpam-6044	317	1	k	k	PROPN
ejpam-6044	317	2	,	,	PUNCT
ejpam-6044	317	3	.	.	PUNCT
ejpam-6044	317	4	.	.	PUNCT
ejpam-6044	317	5	.	.	PUNCT
ejpam-6044	318	1	,	,	PUNCT
ejpam-6044	318	2	ςn	ςn	NOUN
ejpam-6044	318	3	)	)	PUNCT
ejpam-6044	318	4	.	.	PUNCT
ejpam-6044	319	1	expand	expand	VERB
ejpam-6044	319	2	the	the	DET
ejpam-6044	319	3	right	right	ADJ
ejpam-6044	319	4	hand	hand	NOUN
ejpam-6044	319	5	side	side	NOUN
ejpam-6044	319	6	of	of	ADP
ejpam-6044	319	7	last	last	ADJ
ejpam-6044	319	8	expression	expression	NOUN
ejpam-6044	319	9	to	to	PART
ejpam-6044	319	10	get	get	VERB
ejpam-6044	319	11	f	f	PROPN
ejpam-6044	319	12	(	(	PUNCT
ejpam-6044	319	13	ς1	ς1	NOUN
ejpam-6044	319	14	,	,	PUNCT
ejpam-6044	319	15	.	.	PUNCT
ejpam-6044	319	16	.	.	PUNCT
ejpam-6044	320	1	.	.	PUNCT
ejpam-6044	321	1	,	,	PUNCT
ejpam-6044	321	2	ςk	ςk	NUM
ejpam-6044	321	3	,	,	PUNCT
ejpam-6044	321	4	.	.	PUNCT
ejpam-6044	321	5	.	.	PUNCT
ejpam-6044	322	1	.	.	PUNCT
ejpam-6044	323	1	,	,	PUNCT
ejpam-6044	323	2	ςn)α(y)α(ς	ςn)α(y)α(ς	NUM
ejpam-6044	323	3	′	′	NUM
ejpam-6044	323	4	k)+	k)+	NOUN
ejpam-6044	323	5	ς∗kd(ς1	ς∗kd(ς1	PROPN
ejpam-6044	323	6	,	,	PUNCT
ejpam-6044	323	7	.	.	PUNCT
ejpam-6044	323	8	.	.	PUNCT
ejpam-6044	324	1	.	.	PUNCT
ejpam-6044	325	1	,	,	PUNCT
ejpam-6044	325	2	y	y	PROPN
ejpam-6044	325	3	,	,	PUNCT
ejpam-6044	325	4	.	.	PUNCT
ejpam-6044	325	5	.	.	PUNCT
ejpam-6044	326	1	.	.	PUNCT
ejpam-6044	327	1	,	,	PUNCT
ejpam-6044	327	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	327	3	′	′	NUM
ejpam-6044	327	4	k)+	k)+	NOUN
ejpam-6044	327	5	ς∗ky	ς∗ky	PROPN
ejpam-6044	327	6	∗d(ς1	∗d(ς1	PROPN
ejpam-6044	327	7	,	,	PUNCT
ejpam-6044	327	8	.	.	PUNCT
ejpam-6044	327	9	.	.	PUNCT
ejpam-6044	328	1	.	.	PUNCT
ejpam-6044	329	1	,	,	PUNCT
ejpam-6044	329	2	ς	ς	PROPN
ejpam-6044	329	3	′	′	NUM
ejpam-6044	329	4	k	k	NOUN
ejpam-6044	329	5	,	,	PUNCT
ejpam-6044	329	6	.	.	PUNCT
ejpam-6044	329	7	.	.	PUNCT
ejpam-6044	330	1	.	.	PUNCT
ejpam-6044	331	1	,	,	PUNCT
ejpam-6044	331	2	ςn	ςn	X
ejpam-6044	331	3	)	)	PUNCT
ejpam-6044	331	4	(	(	PUNCT
ejpam-6044	331	5	9	9	X
ejpam-6044	331	6	)	)	PUNCT
ejpam-6044	331	7	f.	f.	PROPN
ejpam-6044	331	8	shujat	shujat	PROPN
ejpam-6044	331	9	,	,	PUNCT
ejpam-6044	331	10	s.	s.	PROPN
ejpam-6044	331	11	alharbi	alharbi	PROPN
ejpam-6044	331	12	/	/	SYM
ejpam-6044	331	13	eur	eur	PROPN
ejpam-6044	331	14	.	.	PUNCT
ejpam-6044	332	1	j.	j.	PROPN
ejpam-6044	332	2	pure	pure	PROPN
ejpam-6044	332	3	appl	appl	PROPN
ejpam-6044	332	4	.	.	PROPN
ejpam-6044	332	5	math	math	PROPN
ejpam-6044	332	6	,	,	PUNCT
ejpam-6044	332	7	18	18	NUM
ejpam-6044	332	8	(	(	PUNCT
ejpam-6044	332	9	2	2	NUM
ejpam-6044	332	10	)	)	PUNCT
ejpam-6044	332	11	(	(	PUNCT
ejpam-6044	332	12	2025	2025	NUM
ejpam-6044	332	13	)	)	PUNCT
ejpam-6044	332	14	,	,	PUNCT
ejpam-6044	332	15	6044	6044	NUM
ejpam-6044	332	16	7	7	NUM
ejpam-6044	332	17	of	of	ADP
ejpam-6044	332	18	11	11	NUM
ejpam-6044	332	19	combining	combine	VERB
ejpam-6044	332	20	(	(	PUNCT
ejpam-6044	332	21	15	15	NUM
ejpam-6044	332	22	)	)	PUNCT
ejpam-6044	332	23	and	and	CCONJ
ejpam-6044	332	24	(	(	PUNCT
ejpam-6044	332	25	9	9	X
ejpam-6044	332	26	)	)	PUNCT
ejpam-6044	332	27	together	together	ADV
ejpam-6044	332	28	,	,	PUNCT
ejpam-6044	332	29	we	we	PRON
ejpam-6044	332	30	get	get	VERB
ejpam-6044	332	31	for	for	ADP
ejpam-6044	332	32	every	every	DET
ejpam-6044	332	33	ς1	ς1	NOUN
ejpam-6044	332	34	,	,	PUNCT
ejpam-6044	332	35	...	...	PUNCT
ejpam-6044	332	36	,	,	PUNCT
ejpam-6044	332	37	ςk	ςk	PROPN
ejpam-6044	332	38	,	,	PUNCT
ejpam-6044	332	39	y	y	PROPN
ejpam-6044	332	40	,	,	PUNCT
ejpam-6044	332	41	ς	ς	PROPN
ejpam-6044	332	42	′	′	NUM
ejpam-6044	332	43	k	k	NOUN
ejpam-6044	332	44	,	,	PUNCT
ejpam-6044	332	45	...	...	PUNCT
ejpam-6044	332	46	,	,	PUNCT
ejpam-6044	332	47	ςn	ςn	VERB
ejpam-6044	332	48	in	in	ADP
ejpam-6044	332	49	r	r	PROPN
ejpam-6044	332	50	y∗ς∗kd(ς1	y∗ς∗kd(ς1	PROPN
ejpam-6044	332	51	,	,	PUNCT
ejpam-6044	332	52	.	.	PUNCT
ejpam-6044	332	53	.	.	PUNCT
ejpam-6044	332	54	.	.	PUNCT
ejpam-6044	333	1	,	,	PUNCT
ejpam-6044	333	2	ς	ς	PROPN
ejpam-6044	333	3	′	′	NUM
ejpam-6044	333	4	k	k	NOUN
ejpam-6044	333	5	,	,	PUNCT
ejpam-6044	333	6	.	.	PUNCT
ejpam-6044	333	7	.	.	PUNCT
ejpam-6044	334	1	.	.	PUNCT
ejpam-6044	335	1	,	,	PUNCT
ejpam-6044	335	2	ςn	ςn	X
ejpam-6044	335	3	)	)	PUNCT
ejpam-6044	335	4	=	=	PUNCT
ejpam-6044	335	5	ς∗ky	ς∗ky	NUM
ejpam-6044	335	6	∗d(ς1	∗d(ς1	NOUN
ejpam-6044	335	7	,	,	PUNCT
ejpam-6044	335	8	.	.	PUNCT
ejpam-6044	335	9	.	.	PUNCT
ejpam-6044	335	10	.	.	PUNCT
ejpam-6044	336	1	,	,	PUNCT
ejpam-6044	336	2	ς	ς	PROPN
ejpam-6044	336	3	′	′	NUM
ejpam-6044	336	4	k	k	NOUN
ejpam-6044	336	5	,	,	PUNCT
ejpam-6044	336	6	.	.	PUNCT
ejpam-6044	336	7	.	.	PUNCT
ejpam-6044	337	1	.	.	PUNCT
ejpam-6044	338	1	,	,	PUNCT
ejpam-6044	338	2	ςn	ςn	X
ejpam-6044	338	3	)	)	PUNCT
ejpam-6044	338	4	(	(	PUNCT
ejpam-6044	338	5	10	10	NUM
ejpam-6044	338	6	)	)	PUNCT
ejpam-6044	338	7	put	put	VERB
ejpam-6044	338	8	ςk	ςk	NUM
ejpam-6044	338	9	and	and	CCONJ
ejpam-6044	338	10	y	y	PROPN
ejpam-6044	338	11	instead	instead	ADV
ejpam-6044	338	12	of	of	ADP
ejpam-6044	338	13	ς∗k	ς∗k	NUM
ejpam-6044	338	14	and	and	CCONJ
ejpam-6044	338	15	y∗	y∗	PROPN
ejpam-6044	338	16	to	to	PART
ejpam-6044	338	17	observe	observe	VERB
ejpam-6044	338	18	that	that	SCONJ
ejpam-6044	339	1	[	[	X
ejpam-6044	339	2	ςk	ςk	NUM
ejpam-6044	339	3	,	,	PUNCT
ejpam-6044	339	4	y](d(ς1	y](d(ς1	NOUN
ejpam-6044	339	5	,	,	PUNCT
ejpam-6044	339	6	.	.	PUNCT
ejpam-6044	339	7	.	.	PUNCT
ejpam-6044	340	1	.	.	PUNCT
ejpam-6044	341	1	,	,	PUNCT
ejpam-6044	341	2	ς	ς	PROPN
ejpam-6044	341	3	′	′	NUM
ejpam-6044	341	4	k	k	NOUN
ejpam-6044	341	5	,	,	PUNCT
ejpam-6044	341	6	.	.	PUNCT
ejpam-6044	341	7	.	.	PUNCT
ejpam-6044	342	1	.	.	PUNCT
ejpam-6044	343	1	,	,	PUNCT
ejpam-6044	343	2	ςn	ςn	NOUN
ejpam-6044	343	3	)	)	PUNCT
ejpam-6044	343	4	)	)	PUNCT
ejpam-6044	344	1	=	=	SYM
ejpam-6044	344	2	0	0	NUM
ejpam-6044	345	1	for	for	ADP
ejpam-6044	345	2	every	every	DET
ejpam-6044	345	3	ς1	ς1	NOUN
ejpam-6044	345	4	,	,	PUNCT
ejpam-6044	345	5	...	...	PUNCT
ejpam-6044	345	6	,	,	PUNCT
ejpam-6044	345	7	ςk	ςk	PROPN
ejpam-6044	345	8	,	,	PUNCT
ejpam-6044	345	9	y	y	PROPN
ejpam-6044	345	10	,	,	PUNCT
ejpam-6044	345	11	ς	ς	PROPN
ejpam-6044	345	12	′	′	NUM
ejpam-6044	345	13	k	k	NOUN
ejpam-6044	345	14	,	,	PUNCT
ejpam-6044	345	15	...	...	PUNCT
ejpam-6044	345	16	,	,	PUNCT
ejpam-6044	345	17	ςn	ςn	PROPN
ejpam-6044	345	18	∈	∈	PROPN
ejpam-6044	345	19	r.	r.	PROPN
ejpam-6044	345	20	(	(	PUNCT
ejpam-6044	345	21	11	11	NUM
ejpam-6044	345	22	)	)	PUNCT
ejpam-6044	345	23	again	again	ADV
ejpam-6044	345	24	replace	replace	VERB
ejpam-6044	345	25	y	y	NOUN
ejpam-6044	345	26	by	by	ADP
ejpam-6044	345	27	yr	yr	NOUN
ejpam-6044	345	28	where	where	SCONJ
ejpam-6044	345	29	r	r	NOUN
ejpam-6044	345	30	∈	∈	NOUN
ejpam-6044	345	31	r	r	NOUN
ejpam-6044	345	32	in	in	ADP
ejpam-6044	345	33	(	(	PUNCT
ejpam-6044	345	34	11	11	NUM
ejpam-6044	345	35	)	)	PUNCT
ejpam-6044	345	36	and	and	CCONJ
ejpam-6044	345	37	using	use	VERB
ejpam-6044	345	38	it	it	PRON
ejpam-6044	345	39	[	[	X
ejpam-6044	345	40	ςk	ςk	NUM
ejpam-6044	345	41	,	,	PUNCT
ejpam-6044	345	42	y]r(d(ς1	y]r(d(ς1	PROPN
ejpam-6044	345	43	,	,	PUNCT
ejpam-6044	345	44	.	.	PUNCT
ejpam-6044	345	45	.	.	PUNCT
ejpam-6044	346	1	.	.	PUNCT
ejpam-6044	347	1	,	,	PUNCT
ejpam-6044	347	2	ς	ς	PROPN
ejpam-6044	347	3	′	′	NUM
ejpam-6044	347	4	k	k	NOUN
ejpam-6044	347	5	,	,	PUNCT
ejpam-6044	347	6	.	.	PUNCT
ejpam-6044	347	7	.	.	PUNCT
ejpam-6044	348	1	.	.	PUNCT
ejpam-6044	349	1	,	,	PUNCT
ejpam-6044	349	2	ςn	ςn	NOUN
ejpam-6044	349	3	)	)	PUNCT
ejpam-6044	349	4	)	)	PUNCT
ejpam-6044	350	1	=	=	SYM
ejpam-6044	350	2	0	0	NUM
ejpam-6044	351	1	for	for	ADP
ejpam-6044	351	2	every	every	DET
ejpam-6044	351	3	ς1	ς1	NOUN
ejpam-6044	351	4	,	,	PUNCT
ejpam-6044	351	5	...	...	PUNCT
ejpam-6044	351	6	,	,	PUNCT
ejpam-6044	351	7	ςk	ςk	PROPN
ejpam-6044	351	8	,	,	PUNCT
ejpam-6044	351	9	y	y	PROPN
ejpam-6044	351	10	,	,	PUNCT
ejpam-6044	351	11	ς	ς	PROPN
ejpam-6044	351	12	′	′	NUM
ejpam-6044	351	13	k	k	NOUN
ejpam-6044	351	14	,	,	PUNCT
ejpam-6044	351	15	...	...	PUNCT
ejpam-6044	351	16	,	,	PUNCT
ejpam-6044	351	17	ςn	ςn	PROPN
ejpam-6044	351	18	∈	∈	PROPN
ejpam-6044	351	19	r.	r.	PROPN
ejpam-6044	351	20	(	(	PUNCT
ejpam-6044	351	21	12	12	NUM
ejpam-6044	351	22	)	)	PUNCT
ejpam-6044	351	23	from	from	ADP
ejpam-6044	351	24	(	(	PUNCT
ejpam-6044	351	25	12	12	NUM
ejpam-6044	351	26	)	)	PUNCT
ejpam-6044	351	27	,	,	PUNCT
ejpam-6044	351	28	we	we	PRON
ejpam-6044	351	29	say	say	VERB
ejpam-6044	351	30	that	that	SCONJ
ejpam-6044	351	31	r	r	NOUN
ejpam-6044	351	32	can	can	AUX
ejpam-6044	351	33	be	be	AUX
ejpam-6044	351	34	written	write	VERB
ejpam-6044	351	35	as	as	ADP
ejpam-6044	351	36	k+	k+	PROPN
ejpam-6044	351	37	1	1	NUM
ejpam-6044	351	38	∪k+	∪k+	NUM
ejpam-6044	351	39	2	2	NUM
ejpam-6044	351	40	,	,	PUNCT
ejpam-6044	351	41	where	where	SCONJ
ejpam-6044	351	42	k+	k+	X
ejpam-6044	351	43	1	1	X
ejpam-6044	351	44	=	=	SYM
ejpam-6044	351	45	{	{	PUNCT
ejpam-6044	352	1	[	[	X
ejpam-6044	352	2	ςk	ςk	NUM
ejpam-6044	352	3	,	,	PUNCT
ejpam-6044	352	4	y	y	NOUN
ejpam-6044	352	5	]	]	X
ejpam-6044	352	6	=	=	SYM
ejpam-6044	352	7	0	0	NUM
ejpam-6044	353	1	|	|	ADV
ejpam-6044	353	2	ςk	ςk	NUM
ejpam-6044	353	3	,	,	PUNCT
ejpam-6044	353	4	y	y	PROPN
ejpam-6044	353	5	∈	∈	PROPN
ejpam-6044	353	6	r	r	NOUN
ejpam-6044	353	7	}	}	PUNCT
ejpam-6044	353	8	and	and	CCONJ
ejpam-6044	353	9	k+	k+	X
ejpam-6044	353	10	2	2	X
ejpam-6044	353	11	=	=	SYM
ejpam-6044	353	12	{	{	PUNCT
ejpam-6044	353	13	ς1	ς1	NOUN
ejpam-6044	353	14	...	...	PUNCT
ejpam-6044	353	15	,	,	PUNCT
ejpam-6044	353	16	ςn	ςn	PROPN
ejpam-6044	353	17	∈	∈	PROPN
ejpam-6044	353	18	r	r	NOUN
ejpam-6044	353	19	|	|	ADV
ejpam-6044	353	20	d(ς1	d(ς1	NOUN
ejpam-6044	353	21	,	,	PUNCT
ejpam-6044	353	22	.	.	PUNCT
ejpam-6044	353	23	.	.	PUNCT
ejpam-6044	353	24	.	.	PUNCT
ejpam-6044	354	1	,	,	PUNCT
ejpam-6044	354	2	ς	ς	PROPN
ejpam-6044	354	3	′	′	NUM
ejpam-6044	354	4	k	k	NOUN
ejpam-6044	354	5	,	,	PUNCT
ejpam-6044	354	6	.	.	PUNCT
ejpam-6044	354	7	.	.	PUNCT
ejpam-6044	355	1	.	.	PUNCT
ejpam-6044	356	1	,	,	PUNCT
ejpam-6044	356	2	ςn	ςn	X
ejpam-6044	356	3	)	)	PUNCT
ejpam-6044	356	4	=	=	PUNCT
ejpam-6044	357	1	0	0	NUM
ejpam-6044	357	2	}	}	PUNCT
ejpam-6044	357	3	.	.	PUNCT
ejpam-6044	358	1	which	which	PRON
ejpam-6044	358	2	is	be	AUX
ejpam-6044	358	3	a	a	DET
ejpam-6044	358	4	contradiction	contradiction	NOUN
ejpam-6044	358	5	to	to	ADP
ejpam-6044	358	6	the	the	DET
ejpam-6044	358	7	fact	fact	NOUN
ejpam-6044	358	8	that	that	SCONJ
ejpam-6044	358	9	r	r	NOUN
ejpam-6044	358	10	can	can	AUX
ejpam-6044	358	11	not	not	PART
ejpam-6044	358	12	be	be	AUX
ejpam-6044	358	13	determined	determine	VERB
ejpam-6044	358	14	by	by	ADP
ejpam-6044	358	15	the	the	DET
ejpam-6044	358	16	union	union	NOUN
ejpam-6044	358	17	of	of	ADP
ejpam-6044	358	18	two	two	NUM
ejpam-6044	358	19	additive	additive	ADJ
ejpam-6044	358	20	subgroups	subgroup	NOUN
ejpam-6044	358	21	,	,	PUNCT
ejpam-6044	358	22	namely	namely	ADV
ejpam-6044	358	23	k+	k+	X
ejpam-6044	358	24	1	1	NUM
ejpam-6044	358	25	and	and	CCONJ
ejpam-6044	358	26	k+	k+	NOUN
ejpam-6044	358	27	2	2	X
ejpam-6044	358	28	.	.	PUNCT
ejpam-6044	359	1	hence	hence	ADV
ejpam-6044	359	2	,	,	PUNCT
ejpam-6044	359	3	primeness	primeness	NOUN
ejpam-6044	359	4	implies	imply	VERB
ejpam-6044	359	5	that	that	SCONJ
ejpam-6044	359	6	either	either	CCONJ
ejpam-6044	359	7	k+	k+	X
ejpam-6044	359	8	1	1	NUM
ejpam-6044	359	9	=	=	SYM
ejpam-6044	359	10	r	r	NOUN
ejpam-6044	359	11	or	or	CCONJ
ejpam-6044	359	12	k+	k+	X
ejpam-6044	359	13	2	2	NUM
ejpam-6044	359	14	=	=	SYM
ejpam-6044	359	15	r.	r.	NOUN
ejpam-6044	359	16	if	if	SCONJ
ejpam-6044	359	17	k+	k+	PROPN
ejpam-6044	359	18	1	1	X
ejpam-6044	359	19	=	=	SYM
ejpam-6044	359	20	r	r	NOUN
ejpam-6044	359	21	,	,	PUNCT
ejpam-6044	359	22	then	then	ADV
ejpam-6044	359	23	r	r	NOUN
ejpam-6044	359	24	is	be	AUX
ejpam-6044	359	25	commutative	commutative	ADJ
ejpam-6044	359	26	by	by	ADP
ejpam-6044	359	27	lemma	lemma	PROPN
ejpam-6044	359	28	1	1	NUM
ejpam-6044	359	29	.	.	PUNCT
ejpam-6044	360	1	in	in	ADP
ejpam-6044	360	2	case	case	NOUN
ejpam-6044	360	3	k+	k+	NOUN
ejpam-6044	360	4	2	2	NUM
ejpam-6044	360	5	=	=	SYM
ejpam-6044	360	6	r	r	NOUN
ejpam-6044	360	7	,	,	PUNCT
ejpam-6044	360	8	we	we	PRON
ejpam-6044	360	9	say	say	VERB
ejpam-6044	360	10	that	that	SCONJ
ejpam-6044	360	11	after	after	ADP
ejpam-6044	360	12	simple	simple	ADJ
ejpam-6044	360	13	manipulation	manipulation	NOUN
ejpam-6044	360	14	[	[	X
ejpam-6044	360	15	d(ς1	d(ς1	NOUN
ejpam-6044	360	16	,	,	PUNCT
ejpam-6044	360	17	.	.	PUNCT
ejpam-6044	360	18	.	.	PUNCT
ejpam-6044	361	1	.	.	PUNCT
ejpam-6044	362	1	,	,	PUNCT
ejpam-6044	362	2	ςn	ςn	NOUN
ejpam-6044	362	3	)	)	PUNCT
ejpam-6044	362	4	,	,	PUNCT
ejpam-6044	362	5	r	r	X
ejpam-6044	362	6	]	]	X
ejpam-6044	362	7	=	=	SYM
ejpam-6044	362	8	0	0	NUM
ejpam-6044	362	9	for	for	ADP
ejpam-6044	362	10	every	every	DET
ejpam-6044	362	11	r	r	NOUN
ejpam-6044	362	12	∈	∈	PROPN
ejpam-6044	362	13	r.	r.	NOUN
ejpam-6044	362	14	hence	hence	ADV
ejpam-6044	362	15	,	,	PUNCT
ejpam-6044	362	16	d	d	ADP
ejpam-6044	362	17	commutes	commute	NOUN
ejpam-6044	362	18	with	with	ADP
ejpam-6044	362	19	r.	r.	PROPN
ejpam-6044	362	20	an	an	DET
ejpam-6044	362	21	application	application	NOUN
ejpam-6044	362	22	of	of	ADP
ejpam-6044	362	23	theorem	theorem	ADJ
ejpam-6044	362	24	2	2	NUM
ejpam-6044	362	25	guarantees	guarantee	VERB
ejpam-6044	362	26	that	that	SCONJ
ejpam-6044	362	27	either	either	CCONJ
ejpam-6044	362	28	d	d	PROPN
ejpam-6044	362	29	=	=	SYM
ejpam-6044	362	30	0	0	NUM
ejpam-6044	362	31	or	or	CCONJ
ejpam-6044	362	32	r	r	NOUN
ejpam-6044	362	33	is	be	AUX
ejpam-6044	362	34	commutative	commutative	ADJ
ejpam-6044	362	35	.	.	PUNCT
ejpam-6044	363	1	again	again	ADV
ejpam-6044	363	2	,	,	PUNCT
ejpam-6044	363	3	we	we	PRON
ejpam-6044	363	4	are	be	AUX
ejpam-6044	363	5	done	do	VERB
ejpam-6044	363	6	in	in	ADP
ejpam-6044	363	7	the	the	DET
ejpam-6044	363	8	second	second	ADJ
ejpam-6044	363	9	case	case	NOUN
ejpam-6044	363	10	.	.	PUNCT
ejpam-6044	364	1	on	on	ADP
ejpam-6044	364	2	the	the	DET
ejpam-6044	364	3	other	other	ADJ
ejpam-6044	364	4	hand	hand	NOUN
ejpam-6044	364	5	take	take	VERB
ejpam-6044	364	6	d	d	NOUN
ejpam-6044	364	7	=	=	SYM
ejpam-6044	364	8	0	0	NUM
ejpam-6044	364	9	,	,	PUNCT
ejpam-6044	364	10	and	and	CCONJ
ejpam-6044	364	11	use	use	VERB
ejpam-6044	364	12	the	the	DET
ejpam-6044	364	13	definition	definition	NOUN
ejpam-6044	364	14	to	to	PART
ejpam-6044	364	15	get	get	VERB
ejpam-6044	364	16	the	the	DET
ejpam-6044	364	17	expression	expression	NOUN
ejpam-6044	364	18	f	f	X
ejpam-6044	364	19	(	(	PUNCT
ejpam-6044	364	20	ς1	ς1	NOUN
ejpam-6044	364	21	,	,	PUNCT
ejpam-6044	364	22	.	.	PUNCT
ejpam-6044	364	23	.	.	PUNCT
ejpam-6044	364	24	.	.	PUNCT
ejpam-6044	365	1	,	,	PUNCT
ejpam-6044	365	2	ςkr	ςkr	VERB
ejpam-6044	365	3	,	,	PUNCT
ejpam-6044	365	4	.	.	PUNCT
ejpam-6044	365	5	.	.	PUNCT
ejpam-6044	366	1	.	.	PUNCT
ejpam-6044	367	1	,	,	PUNCT
ejpam-6044	367	2	ςn	ςn	X
ejpam-6044	367	3	)	)	PUNCT
ejpam-6044	368	1	=	=	SYM
ejpam-6044	368	2	f	f	PROPN
ejpam-6044	368	3	(	(	PUNCT
ejpam-6044	368	4	ς1	ς1	NOUN
ejpam-6044	368	5	,	,	PUNCT
ejpam-6044	368	6	.	.	PUNCT
ejpam-6044	368	7	.	.	PUNCT
ejpam-6044	368	8	.	.	PUNCT
ejpam-6044	368	9	,	,	PUNCT
ejpam-6044	368	10	ςk	ςk	NUM
ejpam-6044	368	11	,	,	PUNCT
ejpam-6044	368	12	.	.	PUNCT
ejpam-6044	368	13	.	.	PUNCT
ejpam-6044	368	14	.	.	PUNCT
ejpam-6044	369	1	,	,	PUNCT
ejpam-6044	369	2	ςn)α(r	ςn)α(r	NOUN
ejpam-6044	369	3	)	)	PUNCT
ejpam-6044	369	4	for	for	ADP
ejpam-6044	369	5	every	every	DET
ejpam-6044	369	6	ς1	ς1	NOUN
ejpam-6044	369	7	,	,	PUNCT
ejpam-6044	369	8	.	.	PUNCT
ejpam-6044	369	9	.	.	PUNCT
ejpam-6044	370	1	.	.	PUNCT
ejpam-6044	371	1	,	,	PUNCT
ejpam-6044	371	2	ςk	ςk	NUM
ejpam-6044	371	3	,	,	PUNCT
ejpam-6044	371	4	.	.	PUNCT
ejpam-6044	371	5	.	.	PUNCT
ejpam-6044	372	1	.	.	PUNCT
ejpam-6044	373	1	,	,	PUNCT
ejpam-6044	373	2	ςn	ςn	PROPN
ejpam-6044	373	3	,	,	PUNCT
ejpam-6044	373	4	r	r	NOUN
ejpam-6044	373	5	∈	∈	PROPN
ejpam-6044	373	6	r	r	NOUN
ejpam-6044	373	7	,	,	PUNCT
ejpam-6044	373	8	where	where	SCONJ
ejpam-6044	373	9	f	f	PROPN
ejpam-6044	373	10	acts	act	VERB
ejpam-6044	373	11	as	as	ADP
ejpam-6044	373	12	a	a	DET
ejpam-6044	373	13	left	left	ADJ
ejpam-6044	373	14	α	α	NOUN
ejpam-6044	373	15	-	-	PUNCT
ejpam-6044	373	16	centralizer	centralizer	NOUN
ejpam-6044	373	17	.	.	PUNCT
ejpam-6044	374	1	theorem	theorem	ADJ
ejpam-6044	374	2	4	4	NUM
ejpam-6044	374	3	.	.	PUNCT
ejpam-6044	375	1	let	let	VERB
ejpam-6044	375	2	r	r	PRON
ejpam-6044	375	3	be	be	AUX
ejpam-6044	375	4	a	a	DET
ejpam-6044	375	5	non	non	ADJ
ejpam-6044	375	6	-	-	ADJ
ejpam-6044	375	7	commutative	commutative	ADJ
ejpam-6044	375	8	prime	prime	ADJ
ejpam-6044	375	9	∗-ring	∗-ring	NOUN
ejpam-6044	375	10	.	.	PUNCT
ejpam-6044	376	1	if	if	SCONJ
ejpam-6044	376	2	r	r	NOUN
ejpam-6044	376	3	admits	admit	VERB
ejpam-6044	376	4	a	a	DET
ejpam-6044	376	5	nonzero	nonzero	NOUN
ejpam-6044	376	6	generalized	generalized	ADJ
ejpam-6044	376	7	(	(	PUNCT
ejpam-6044	376	8	α	α	NOUN
ejpam-6044	376	9	,	,	PUNCT
ejpam-6044	376	10	∗)-n	∗)-n	NOUN
ejpam-6044	376	11	-	-	PUNCT
ejpam-6044	376	12	derivation	derivation	NOUN
ejpam-6044	376	13	f	f	PROPN
ejpam-6044	376	14	associated	associate	VERB
ejpam-6044	376	15	with	with	ADP
ejpam-6044	376	16	an	an	DET
ejpam-6044	376	17	(	(	PUNCT
ejpam-6044	376	18	α	α	NOUN
ejpam-6044	376	19	,	,	PUNCT
ejpam-6044	376	20	∗)-n	∗)-n	NOUN
ejpam-6044	376	21	-	-	PUNCT
ejpam-6044	376	22	derivation	derivation	NOUN
ejpam-6044	376	23	d	d	NOUN
ejpam-6044	376	24	,	,	PUNCT
ejpam-6044	376	25	then	then	ADV
ejpam-6044	376	26	f	f	PROPN
ejpam-6044	376	27	acts	act	VERB
ejpam-6044	376	28	as	as	ADP
ejpam-6044	376	29	left	leave	VERB
ejpam-6044	376	30	αcentralizer	αcentralizer	NOUN
ejpam-6044	376	31	.	.	PUNCT
ejpam-6044	377	1	proof	proof	NOUN
ejpam-6044	377	2	.	.	PUNCT
ejpam-6044	378	1	the	the	DET
ejpam-6044	378	2	proof	proof	NOUN
ejpam-6044	378	3	is	be	AUX
ejpam-6044	378	4	straight	straight	ADV
ejpam-6044	378	5	forward	forward	ADV
ejpam-6044	378	6	by	by	ADP
ejpam-6044	378	7	the	the	DET
ejpam-6044	378	8	application	application	NOUN
ejpam-6044	378	9	of	of	ADP
ejpam-6044	378	10	theorem	theorem	ADJ
ejpam-6044	378	11	3	3	NUM
ejpam-6044	378	12	.	.	PUNCT
ejpam-6044	378	13	theorem	theorem	NOUN
ejpam-6044	378	14	5	5	NUM
ejpam-6044	378	15	.	.	PUNCT
ejpam-6044	379	1	let	let	VERB
ejpam-6044	379	2	r	r	PRON
ejpam-6044	379	3	be	be	AUX
ejpam-6044	379	4	a	a	DET
ejpam-6044	379	5	2	2	NUM
ejpam-6044	379	6	-	-	PUNCT
ejpam-6044	379	7	torsion	torsion	NOUN
ejpam-6044	379	8	-	-	PUNCT
ejpam-6044	379	9	free	free	ADJ
ejpam-6044	379	10	prime	prime	NOUN
ejpam-6044	379	11	∗-ring	∗-ring	NOUN
ejpam-6044	379	12	having	having	AUX
ejpam-6044	379	13	generalized	generalize	VERB
ejpam-6044	379	14	(	(	PUNCT
ejpam-6044	379	15	α	α	NOUN
ejpam-6044	379	16	,	,	PUNCT
ejpam-6044	379	17	∗)-n	∗)-n	NOUN
ejpam-6044	379	18	-	-	PUNCT
ejpam-6044	379	19	derivations	derivation	NOUN
ejpam-6044	379	20	f1	f1	NOUN
ejpam-6044	379	21	and	and	CCONJ
ejpam-6044	379	22	f2	f2	PROPN
ejpam-6044	379	23	associated	associate	VERB
ejpam-6044	379	24	with	with	ADP
ejpam-6044	379	25	(	(	PUNCT
ejpam-6044	379	26	α	α	NOUN
ejpam-6044	379	27	,	,	PUNCT
ejpam-6044	379	28	∗)-n	∗)-n	NOUN
ejpam-6044	379	29	-	-	PUNCT
ejpam-6044	379	30	derivations	derivation	NOUN
ejpam-6044	379	31	d1	d1	NOUN
ejpam-6044	379	32	and	and	CCONJ
ejpam-6044	379	33	d2	d2	PROPN
ejpam-6044	379	34	respectively	respectively	ADV
ejpam-6044	379	35	.	.	PUNCT
ejpam-6044	380	1	if	if	SCONJ
ejpam-6044	380	2	f1(ς1	f1(ς1	NOUN
ejpam-6044	380	3	,	,	PUNCT
ejpam-6044	380	4	.	.	PUNCT
ejpam-6044	380	5	.	.	PUNCT
ejpam-6044	381	1	.	.	PUNCT
ejpam-6044	382	1	,	,	PUNCT
ejpam-6044	382	2	ςk	ςk	NUM
ejpam-6044	382	3	,	,	PUNCT
ejpam-6044	382	4	.	.	PUNCT
ejpam-6044	382	5	.	.	PUNCT
ejpam-6044	383	1	.	.	PUNCT
ejpam-6044	384	1	,	,	PUNCT
ejpam-6044	385	1	ςn)d2(y1	ςn)d2(y1	NUM
ejpam-6044	385	2	,	,	PUNCT
ejpam-6044	385	3	.	.	PUNCT
ejpam-6044	385	4	.	.	PUNCT
ejpam-6044	386	1	.	.	PUNCT
ejpam-6044	387	1	,	,	PUNCT
ejpam-6044	387	2	yk	yk	PROPN
ejpam-6044	387	3	,	,	PUNCT
ejpam-6044	387	4	.	.	PUNCT
ejpam-6044	387	5	.	.	PUNCT
ejpam-6044	388	1	.	.	PUNCT
ejpam-6044	389	1	,	,	PUNCT
ejpam-6044	389	2	yn	yn	PROPN
ejpam-6044	389	3	)	)	PUNCT
ejpam-6044	389	4	−f2(ς1	−f2(ς1	PROPN
ejpam-6044	389	5	,	,	PUNCT
ejpam-6044	389	6	.	.	PUNCT
ejpam-6044	389	7	.	.	PUNCT
ejpam-6044	390	1	.	.	PUNCT
ejpam-6044	391	1	,	,	PUNCT
ejpam-6044	391	2	ςk	ςk	NUM
ejpam-6044	391	3	,	,	PUNCT
ejpam-6044	391	4	.	.	PUNCT
ejpam-6044	391	5	.	.	PUNCT
ejpam-6044	392	1	.	.	PUNCT
ejpam-6044	393	1	,	,	PUNCT
ejpam-6044	394	1	ςn)d1(y1	ςn)d1(y1	NOUN
ejpam-6044	394	2	,	,	PUNCT
ejpam-6044	394	3	.	.	PUNCT
ejpam-6044	394	4	.	.	PUNCT
ejpam-6044	394	5	.	.	PUNCT
ejpam-6044	395	1	,	,	PUNCT
ejpam-6044	395	2	yk	yk	PROPN
ejpam-6044	395	3	,	,	PUNCT
ejpam-6044	395	4	.	.	PUNCT
ejpam-6044	395	5	.	.	PUNCT
ejpam-6044	396	1	.	.	PUNCT
ejpam-6044	397	1	,	,	PUNCT
ejpam-6044	397	2	yn	yn	PROPN
ejpam-6044	397	3	)	)	PUNCT
ejpam-6044	397	4	=	=	SYM
ejpam-6044	397	5	0	0	NUM
ejpam-6044	397	6	,	,	PUNCT
ejpam-6044	397	7	for	for	ADP
ejpam-6044	397	8	each	each	DET
ejpam-6044	397	9	ς1	ς1	NOUN
ejpam-6044	397	10	,	,	PUNCT
ejpam-6044	397	11	...	...	PUNCT
ejpam-6044	397	12	,	,	PUNCT
ejpam-6044	397	13	ςn	ςn	PROPN
ejpam-6044	397	14	,	,	PUNCT
ejpam-6044	397	15	y1	y1	PROPN
ejpam-6044	397	16	,	,	PUNCT
ejpam-6044	397	17	...	...	PUNCT
ejpam-6044	397	18	,	,	PUNCT
ejpam-6044	397	19	yn	yn	PROPN
ejpam-6044	397	20	∈	∈	PROPN
ejpam-6044	397	21	r	r	NOUN
ejpam-6044	397	22	,	,	PUNCT
ejpam-6044	397	23	then	then	ADV
ejpam-6044	397	24	one	one	NUM
ejpam-6044	397	25	of	of	ADP
ejpam-6044	397	26	the	the	DET
ejpam-6044	397	27	following	follow	VERB
ejpam-6044	397	28	holds	hold	NOUN
ejpam-6044	397	29	:	:	PUNCT
ejpam-6044	397	30	f.	f.	PROPN
ejpam-6044	397	31	shujat	shujat	PROPN
ejpam-6044	397	32	,	,	PUNCT
ejpam-6044	397	33	s.	s.	PROPN
ejpam-6044	397	34	alharbi	alharbi	PROPN
ejpam-6044	397	35	/	/	SYM
ejpam-6044	397	36	eur	eur	PROPN
ejpam-6044	397	37	.	.	PUNCT
ejpam-6044	398	1	j.	j.	PROPN
ejpam-6044	398	2	pure	pure	PROPN
ejpam-6044	398	3	appl	appl	PROPN
ejpam-6044	398	4	.	.	PROPN
ejpam-6044	398	5	math	math	PROPN
ejpam-6044	398	6	,	,	PUNCT
ejpam-6044	398	7	18	18	NUM
ejpam-6044	398	8	(	(	PUNCT
ejpam-6044	398	9	2	2	NUM
ejpam-6044	398	10	)	)	PUNCT
ejpam-6044	398	11	(	(	PUNCT
ejpam-6044	398	12	2025	2025	NUM
ejpam-6044	398	13	)	)	PUNCT
ejpam-6044	398	14	,	,	PUNCT
ejpam-6044	398	15	6044	6044	NUM
ejpam-6044	398	16	8	8	NUM
ejpam-6044	398	17	of	of	ADP
ejpam-6044	398	18	11	11	NUM
ejpam-6044	398	19	(	(	PUNCT
ejpam-6044	398	20	i	i	NOUN
ejpam-6044	398	21	)	)	PUNCT
ejpam-6044	398	22	f1	f1	NOUN
ejpam-6044	398	23	=	=	SYM
ejpam-6044	398	24	0	0	NUM
ejpam-6044	398	25	or	or	CCONJ
ejpam-6044	398	26	f2	f2	PROPN
ejpam-6044	398	27	acts	act	VERB
ejpam-6044	398	28	as	as	ADP
ejpam-6044	398	29	a	a	DET
ejpam-6044	398	30	left	left	ADJ
ejpam-6044	398	31	α	α	NOUN
ejpam-6044	398	32	-	-	PUNCT
ejpam-6044	398	33	centralizer	centralizer	NOUN
ejpam-6044	398	34	.	.	PUNCT
ejpam-6044	399	1	(	(	PUNCT
ejpam-6044	399	2	ii	ii	NOUN
ejpam-6044	399	3	)	)	PUNCT
ejpam-6044	399	4	f2	f2	PROPN
ejpam-6044	399	5	=	=	SYM
ejpam-6044	399	6	0	0	NUM
ejpam-6044	399	7	or	or	CCONJ
ejpam-6044	399	8	f1	f1	NOUN
ejpam-6044	399	9	acts	act	NOUN
ejpam-6044	399	10	as	as	ADP
ejpam-6044	399	11	a	a	DET
ejpam-6044	399	12	left	left	ADJ
ejpam-6044	399	13	α	α	NOUN
ejpam-6044	399	14	-	-	PUNCT
ejpam-6044	399	15	centralizer	centralizer	NOUN
ejpam-6044	399	16	.	.	PUNCT
ejpam-6044	400	1	proof	proof	NOUN
ejpam-6044	400	2	.	.	PUNCT
ejpam-6044	401	1	suppose	suppose	VERB
ejpam-6044	401	2	that	that	SCONJ
ejpam-6044	401	3	f1(ς1	f1(ς1	NOUN
ejpam-6044	401	4	,	,	PUNCT
ejpam-6044	401	5	.	.	PUNCT
ejpam-6044	401	6	.	.	PUNCT
ejpam-6044	402	1	.	.	PUNCT
ejpam-6044	403	1	,	,	PUNCT
ejpam-6044	403	2	ςk	ςk	NUM
ejpam-6044	403	3	,	,	PUNCT
ejpam-6044	403	4	.	.	PUNCT
ejpam-6044	403	5	.	.	PUNCT
ejpam-6044	404	1	.	.	PUNCT
ejpam-6044	405	1	,	,	PUNCT
ejpam-6044	406	1	ςn)d2(y1	ςn)d2(y1	NUM
ejpam-6044	406	2	,	,	PUNCT
ejpam-6044	406	3	.	.	PUNCT
ejpam-6044	406	4	.	.	PUNCT
ejpam-6044	407	1	.	.	PUNCT
ejpam-6044	408	1	,	,	PUNCT
ejpam-6044	408	2	yk	yk	PROPN
ejpam-6044	408	3	,	,	PUNCT
ejpam-6044	408	4	.	.	PUNCT
ejpam-6044	408	5	.	.	PUNCT
ejpam-6044	409	1	.	.	PUNCT
ejpam-6044	410	1	,	,	PUNCT
ejpam-6044	410	2	yn	yn	PROPN
ejpam-6044	410	3	)	)	PUNCT
ejpam-6044	410	4	−f2(ς1	−f2(ς1	PROPN
ejpam-6044	410	5	,	,	PUNCT
ejpam-6044	410	6	.	.	PUNCT
ejpam-6044	410	7	.	.	PUNCT
ejpam-6044	411	1	.	.	PUNCT
ejpam-6044	412	1	,	,	PUNCT
ejpam-6044	412	2	ςk	ςk	NUM
ejpam-6044	412	3	,	,	PUNCT
ejpam-6044	412	4	.	.	PUNCT
ejpam-6044	412	5	.	.	PUNCT
ejpam-6044	413	1	.	.	PUNCT
ejpam-6044	414	1	,	,	PUNCT
ejpam-6044	415	1	ςn)d1(y1	ςn)d1(y1	NOUN
ejpam-6044	415	2	,	,	PUNCT
ejpam-6044	415	3	.	.	PUNCT
ejpam-6044	415	4	.	.	PUNCT
ejpam-6044	415	5	.	.	PUNCT
ejpam-6044	416	1	,	,	PUNCT
ejpam-6044	416	2	yk	yk	PROPN
ejpam-6044	416	3	,	,	PUNCT
ejpam-6044	416	4	.	.	PUNCT
ejpam-6044	416	5	.	.	PUNCT
ejpam-6044	417	1	.	.	PUNCT
ejpam-6044	418	1	,	,	PUNCT
ejpam-6044	418	2	yn	yn	PROPN
ejpam-6044	418	3	)	)	PUNCT
ejpam-6044	418	4	=	=	SYM
ejpam-6044	419	1	0	0	X
ejpam-6044	419	2	.	.	PUNCT
ejpam-6044	420	1	(	(	PUNCT
ejpam-6044	420	2	13	13	NUM
ejpam-6044	420	3	)	)	PUNCT
ejpam-6044	420	4	put	put	VERB
ejpam-6044	420	5	ykz	ykz	NOUN
ejpam-6044	420	6	in	in	ADP
ejpam-6044	420	7	place	place	NOUN
ejpam-6044	420	8	of	of	ADP
ejpam-6044	420	9	yk	yk	PROPN
ejpam-6044	420	10	,	,	PUNCT
ejpam-6044	420	11	we	we	PRON
ejpam-6044	420	12	have	have	VERB
ejpam-6044	420	13	for	for	ADP
ejpam-6044	420	14	each	each	DET
ejpam-6044	420	15	ς1	ς1	NOUN
ejpam-6044	420	16	,	,	PUNCT
ejpam-6044	420	17	...	...	PUNCT
ejpam-6044	420	18	,	,	PUNCT
ejpam-6044	420	19	ςn	ςn	PROPN
ejpam-6044	420	20	,	,	PUNCT
ejpam-6044	420	21	y1	y1	PROPN
ejpam-6044	420	22	,	,	PUNCT
ejpam-6044	420	23	...	...	PUNCT
ejpam-6044	420	24	,	,	PUNCT
ejpam-6044	420	25	yn	yn	PROPN
ejpam-6044	420	26	∈	∈	PROPN
ejpam-6044	420	27	r	r	NOUN
ejpam-6044	420	28	f1(ς1	f1(ς1	NOUN
ejpam-6044	420	29	,	,	PUNCT
ejpam-6044	420	30	.	.	PUNCT
ejpam-6044	420	31	.	.	PUNCT
ejpam-6044	421	1	.	.	PUNCT
ejpam-6044	422	1	,	,	PUNCT
ejpam-6044	422	2	ςk	ςk	NUM
ejpam-6044	422	3	,	,	PUNCT
ejpam-6044	422	4	.	.	PUNCT
ejpam-6044	422	5	.	.	PUNCT
ejpam-6044	423	1	.	.	PUNCT
ejpam-6044	424	1	,	,	PUNCT
ejpam-6044	425	1	ςn)d2(y1	ςn)d2(y1	NUM
ejpam-6044	425	2	,	,	PUNCT
ejpam-6044	425	3	.	.	PUNCT
ejpam-6044	425	4	.	.	PUNCT
ejpam-6044	426	1	.	.	PUNCT
ejpam-6044	427	1	,	,	PUNCT
ejpam-6044	427	2	ykz	ykz	INTJ
ejpam-6044	427	3	,	,	PUNCT
ejpam-6044	427	4	.	.	PUNCT
ejpam-6044	427	5	.	.	PUNCT
ejpam-6044	428	1	.	.	PUNCT
ejpam-6044	429	1	,	,	PUNCT
ejpam-6044	429	2	yn	yn	PROPN
ejpam-6044	429	3	)	)	PUNCT
ejpam-6044	429	4	−f2(ς1	−f2(ς1	PROPN
ejpam-6044	429	5	,	,	PUNCT
ejpam-6044	429	6	.	.	PUNCT
ejpam-6044	429	7	.	.	PUNCT
ejpam-6044	430	1	.	.	PUNCT
ejpam-6044	431	1	,	,	PUNCT
ejpam-6044	431	2	ςk	ςk	NUM
ejpam-6044	431	3	,	,	PUNCT
ejpam-6044	431	4	.	.	PUNCT
ejpam-6044	431	5	.	.	PUNCT
ejpam-6044	432	1	.	.	PUNCT
ejpam-6044	433	1	,	,	PUNCT
ejpam-6044	434	1	ςn)d1(y1	ςn)d1(y1	NOUN
ejpam-6044	434	2	,	,	PUNCT
ejpam-6044	434	3	.	.	PUNCT
ejpam-6044	434	4	.	.	PUNCT
ejpam-6044	434	5	.	.	PUNCT
ejpam-6044	435	1	,	,	PUNCT
ejpam-6044	435	2	ykz	ykz	INTJ
ejpam-6044	435	3	,	,	PUNCT
ejpam-6044	435	4	.	.	PUNCT
ejpam-6044	435	5	.	.	PUNCT
ejpam-6044	436	1	.	.	PUNCT
ejpam-6044	437	1	,	,	PUNCT
ejpam-6044	437	2	yn	yn	PROPN
ejpam-6044	437	3	)	)	PUNCT
ejpam-6044	437	4	=	=	SYM
ejpam-6044	437	5	0	0	X
ejpam-6044	437	6	.	.	PUNCT
ejpam-6044	437	7	explore	explore	VERB
ejpam-6044	437	8	the	the	DET
ejpam-6044	437	9	above	above	ADJ
ejpam-6044	437	10	equation	equation	NOUN
ejpam-6044	437	11	f1ς1	f1ς1	NOUN
ejpam-6044	437	12	,	,	PUNCT
ejpam-6044	437	13	.	.	PUNCT
ejpam-6044	437	14	.	.	PUNCT
ejpam-6044	438	1	.	.	PUNCT
ejpam-6044	439	1	,	,	PUNCT
ejpam-6044	439	2	ςk	ςk	NUM
ejpam-6044	439	3	,	,	PUNCT
ejpam-6044	439	4	.	.	PUNCT
ejpam-6044	439	5	.	.	PUNCT
ejpam-6044	440	1	.	.	PUNCT
ejpam-6044	441	1	,	,	PUNCT
ejpam-6044	441	2	ςn){d2(y1	ςn){d2(y1	NUM
ejpam-6044	441	3	,	,	PUNCT
ejpam-6044	441	4	.	.	PUNCT
ejpam-6044	441	5	.	.	PUNCT
ejpam-6044	442	1	.	.	PUNCT
ejpam-6044	443	1	,	,	PUNCT
ejpam-6044	443	2	yk	yk	PROPN
ejpam-6044	443	3	,	,	PUNCT
ejpam-6044	443	4	.	.	PUNCT
ejpam-6044	443	5	.	.	PUNCT
ejpam-6044	444	1	.	.	PUNCT
ejpam-6044	445	1	,	,	PUNCT
ejpam-6044	445	2	yn)α(z	yn)α(z	PROPN
ejpam-6044	445	3	)	)	PUNCT
ejpam-6044	446	1	+	+	CCONJ
ejpam-6044	446	2	y∗kd2(y1	y∗kd2(y1	ADJ
ejpam-6044	446	3	,	,	PUNCT
ejpam-6044	446	4	.	.	PUNCT
ejpam-6044	446	5	.	.	PUNCT
ejpam-6044	447	1	.	.	PUNCT
ejpam-6044	448	1	,	,	PUNCT
ejpam-6044	449	1	z	z	X
ejpam-6044	449	2	,	,	PUNCT
ejpam-6044	449	3	.	.	PUNCT
ejpam-6044	449	4	.	.	PUNCT
ejpam-6044	449	5	.	.	PUNCT
ejpam-6044	450	1	,	,	PUNCT
ejpam-6044	450	2	yn	yn	PROPN
ejpam-6044	450	3	)	)	PUNCT
ejpam-6044	450	4	}	}	PUNCT
ejpam-6044	450	5	−	−	ADP
ejpam-6044	450	6	f2(ς1	f2(ς1	NOUN
ejpam-6044	450	7	,	,	PUNCT
ejpam-6044	450	8	.	.	PUNCT
ejpam-6044	450	9	.	.	PUNCT
ejpam-6044	451	1	.	.	PUNCT
ejpam-6044	452	1	,	,	PUNCT
ejpam-6044	452	2	ςk	ςk	NUM
ejpam-6044	452	3	,	,	PUNCT
ejpam-6044	452	4	.	.	PUNCT
ejpam-6044	452	5	.	.	PUNCT
ejpam-6044	453	1	.	.	PUNCT
ejpam-6044	454	1	,	,	PUNCT
ejpam-6044	454	2	ςn){d1(y1	ςn){d1(y1	PROPN
ejpam-6044	454	3	,	,	PUNCT
ejpam-6044	454	4	.	.	PUNCT
ejpam-6044	454	5	.	.	PUNCT
ejpam-6044	455	1	.	.	PUNCT
ejpam-6044	456	1	,	,	PUNCT
ejpam-6044	456	2	yk	yk	PROPN
ejpam-6044	456	3	,	,	PUNCT
ejpam-6044	456	4	.	.	PUNCT
ejpam-6044	456	5	.	.	PUNCT
ejpam-6044	457	1	.	.	PUNCT
ejpam-6044	458	1	,	,	PUNCT
ejpam-6044	458	2	yn)α(z	yn)α(z	PROPN
ejpam-6044	458	3	)	)	PUNCT
ejpam-6044	459	1	+	+	X
ejpam-6044	459	2	y∗kd1(y1	y∗kd1(y1	ADJ
ejpam-6044	459	3	,	,	PUNCT
ejpam-6044	459	4	.	.	PUNCT
ejpam-6044	459	5	.	.	PUNCT
ejpam-6044	460	1	.	.	PUNCT
ejpam-6044	461	1	,	,	PUNCT
ejpam-6044	462	1	z	z	X
ejpam-6044	462	2	,	,	PUNCT
ejpam-6044	462	3	.	.	PUNCT
ejpam-6044	462	4	.	.	PUNCT
ejpam-6044	462	5	.	.	PUNCT
ejpam-6044	463	1	,	,	PUNCT
ejpam-6044	463	2	yn	yn	PROPN
ejpam-6044	463	3	)	)	PUNCT
ejpam-6044	463	4	}	}	PUNCT
ejpam-6044	464	1	=	=	PUNCT
ejpam-6044	464	2	0	0	X
ejpam-6044	464	3	.	.	PUNCT
ejpam-6044	465	1	from	from	ADP
ejpam-6044	465	2	(	(	PUNCT
ejpam-6044	465	3	13	13	NUM
ejpam-6044	465	4	)	)	PUNCT
ejpam-6044	465	5	,	,	PUNCT
ejpam-6044	465	6	we	we	PRON
ejpam-6044	465	7	arrive	arrive	VERB
ejpam-6044	465	8	at	at	ADP
ejpam-6044	465	9	f1(ς1	f1(ς1	NOUN
ejpam-6044	465	10	,	,	PUNCT
ejpam-6044	465	11	.	.	PUNCT
ejpam-6044	465	12	.	.	PUNCT
ejpam-6044	466	1	.	.	PUNCT
ejpam-6044	467	1	,	,	PUNCT
ejpam-6044	467	2	ςk	ςk	NUM
ejpam-6044	467	3	,	,	PUNCT
ejpam-6044	467	4	.	.	PUNCT
ejpam-6044	467	5	.	.	PUNCT
ejpam-6044	468	1	.	.	PUNCT
ejpam-6044	469	1	,	,	PUNCT
ejpam-6044	469	2	ςn)y	ςn)y	NUM
ejpam-6044	469	3	∗	∗	NUM
ejpam-6044	469	4	kd2(y1	kd2(y1	NOUN
ejpam-6044	469	5	,	,	PUNCT
ejpam-6044	469	6	.	.	PUNCT
ejpam-6044	469	7	.	.	PUNCT
ejpam-6044	470	1	.	.	PUNCT
ejpam-6044	471	1	,	,	PUNCT
ejpam-6044	472	1	z	z	X
ejpam-6044	472	2	,	,	PUNCT
ejpam-6044	472	3	.	.	PUNCT
ejpam-6044	472	4	.	.	PUNCT
ejpam-6044	472	5	.	.	PUNCT
ejpam-6044	473	1	,	,	PUNCT
ejpam-6044	473	2	yn	yn	PROPN
ejpam-6044	473	3	)	)	PUNCT
ejpam-6044	473	4	−f2(ς1	−f2(ς1	PROPN
ejpam-6044	473	5	,	,	PUNCT
ejpam-6044	473	6	.	.	PUNCT
ejpam-6044	473	7	.	.	PUNCT
ejpam-6044	474	1	.	.	PUNCT
ejpam-6044	475	1	,	,	PUNCT
ejpam-6044	475	2	ςk	ςk	NUM
ejpam-6044	475	3	,	,	PUNCT
ejpam-6044	475	4	.	.	PUNCT
ejpam-6044	475	5	.	.	PUNCT
ejpam-6044	476	1	.	.	PUNCT
ejpam-6044	477	1	,	,	PUNCT
ejpam-6044	477	2	ςn)y	ςn)y	NUM
ejpam-6044	477	3	∗	∗	NOUN
ejpam-6044	477	4	kd1(y1	kd1(y1	NOUN
ejpam-6044	477	5	,	,	PUNCT
ejpam-6044	477	6	.	.	PUNCT
ejpam-6044	477	7	.	.	PUNCT
ejpam-6044	478	1	.	.	PUNCT
ejpam-6044	479	1	,	,	PUNCT
ejpam-6044	480	1	z	z	X
ejpam-6044	480	2	,	,	PUNCT
ejpam-6044	480	3	.	.	PUNCT
ejpam-6044	480	4	.	.	PUNCT
ejpam-6044	480	5	.	.	PUNCT
ejpam-6044	481	1	,	,	PUNCT
ejpam-6044	481	2	yn	yn	PROPN
ejpam-6044	481	3	)	)	PUNCT
ejpam-6044	481	4	=	=	SYM
ejpam-6044	481	5	0	0	NUM
ejpam-6044	481	6	,	,	PUNCT
ejpam-6044	481	7	(	(	PUNCT
ejpam-6044	481	8	14	14	NUM
ejpam-6044	481	9	)	)	PUNCT
ejpam-6044	481	10	for	for	ADP
ejpam-6044	481	11	each	each	DET
ejpam-6044	481	12	ς1	ς1	NOUN
ejpam-6044	481	13	,	,	PUNCT
ejpam-6044	481	14	...	...	PUNCT
ejpam-6044	481	15	,	,	PUNCT
ejpam-6044	481	16	ςn	ςn	PROPN
ejpam-6044	481	17	,	,	PUNCT
ejpam-6044	481	18	y1	y1	PROPN
ejpam-6044	481	19	,	,	PUNCT
ejpam-6044	481	20	...	...	PUNCT
ejpam-6044	481	21	,	,	PUNCT
ejpam-6044	481	22	yn	yn	PROPN
ejpam-6044	481	23	∈	∈	PROPN
ejpam-6044	481	24	r.	r.	PROPN
ejpam-6044	481	25	multiplying	multiplying	NOUN
ejpam-6044	481	26	(	(	PUNCT
ejpam-6044	481	27	14	14	NUM
ejpam-6044	481	28	)	)	PUNCT
ejpam-6044	481	29	from	from	ADP
ejpam-6044	481	30	the	the	DET
ejpam-6044	481	31	right	right	NOUN
ejpam-6044	481	32	by	by	ADP
ejpam-6044	481	33	pd1(y	pd1(y	PROPN
ejpam-6044	481	34	′	′	NUM
ejpam-6044	481	35	1	1	NUM
ejpam-6044	481	36	,	,	PUNCT
ejpam-6044	481	37	.	.	PUNCT
ejpam-6044	481	38	.	.	PUNCT
ejpam-6044	481	39	.	.	PUNCT
ejpam-6044	482	1	,	,	PUNCT
ejpam-6044	482	2	y	y	PROPN
ejpam-6044	482	3	′	′	NUM
ejpam-6044	483	1	k	k	ADP
ejpam-6044	483	2	,	,	PUNCT
ejpam-6044	483	3	.	.	PUNCT
ejpam-6044	483	4	.	.	PUNCT
ejpam-6044	483	5	.	.	PUNCT
ejpam-6044	484	1	,	,	PUNCT
ejpam-6044	484	2	y	y	PROPN
ejpam-6044	484	3	′	′	NUM
ejpam-6044	484	4	n	n	CCONJ
ejpam-6044	484	5	)	)	PUNCT
ejpam-6044	484	6	where	where	SCONJ
ejpam-6044	484	7	p	p	X
ejpam-6044	484	8	,	,	PUNCT
ejpam-6044	484	9	y′k	y′k	NOUN
ejpam-6044	484	10	∈	∈	PROPN
ejpam-6044	484	11	r	r	NOUN
ejpam-6044	484	12	,	,	PUNCT
ejpam-6044	484	13	we	we	PRON
ejpam-6044	484	14	obtain	obtain	VERB
ejpam-6044	484	15	(	(	PUNCT
ejpam-6044	484	16	f1(ς1	f1(ς1	NOUN
ejpam-6044	484	17	,	,	PUNCT
ejpam-6044	484	18	.	.	PUNCT
ejpam-6044	484	19	.	.	PUNCT
ejpam-6044	484	20	.	.	PUNCT
ejpam-6044	485	1	,	,	PUNCT
ejpam-6044	485	2	ςk	ςk	NUM
ejpam-6044	485	3	,	,	PUNCT
ejpam-6044	485	4	.	.	PUNCT
ejpam-6044	485	5	.	.	PUNCT
ejpam-6044	486	1	.	.	PUNCT
ejpam-6044	487	1	,	,	PUNCT
ejpam-6044	487	2	ςn)y	ςn)y	NUM
ejpam-6044	487	3	∗	∗	NUM
ejpam-6044	487	4	kd2(y1	kd2(y1	NOUN
ejpam-6044	487	5	,	,	PUNCT
ejpam-6044	487	6	.	.	PUNCT
ejpam-6044	487	7	.	.	PUNCT
ejpam-6044	488	1	.	.	PUNCT
ejpam-6044	489	1	,	,	PUNCT
ejpam-6044	490	1	z	z	X
ejpam-6044	490	2	,	,	PUNCT
ejpam-6044	490	3	.	.	PUNCT
ejpam-6044	490	4	.	.	PUNCT
ejpam-6044	490	5	.	.	PUNCT
ejpam-6044	491	1	,	,	PUNCT
ejpam-6044	491	2	yn	yn	PROPN
ejpam-6044	491	3	)	)	PUNCT
ejpam-6044	491	4	−f2(ς1	−f2(ς1	PROPN
ejpam-6044	491	5	,	,	PUNCT
ejpam-6044	491	6	.	.	PUNCT
ejpam-6044	491	7	.	.	PUNCT
ejpam-6044	492	1	.	.	PUNCT
ejpam-6044	493	1	,	,	PUNCT
ejpam-6044	493	2	ςk	ςk	NUM
ejpam-6044	493	3	,	,	PUNCT
ejpam-6044	493	4	.	.	PUNCT
ejpam-6044	493	5	.	.	PUNCT
ejpam-6044	494	1	.	.	PUNCT
ejpam-6044	495	1	,	,	PUNCT
ejpam-6044	495	2	ςn)y	ςn)y	NUM
ejpam-6044	495	3	∗	∗	NOUN
ejpam-6044	495	4	kd1(y1	kd1(y1	NOUN
ejpam-6044	495	5	,	,	PUNCT
ejpam-6044	495	6	.	.	PUNCT
ejpam-6044	495	7	.	.	PUNCT
ejpam-6044	496	1	.	.	PUNCT
ejpam-6044	497	1	,	,	PUNCT
ejpam-6044	498	1	z	z	X
ejpam-6044	498	2	,	,	PUNCT
ejpam-6044	498	3	.	.	PUNCT
ejpam-6044	498	4	.	.	PUNCT
ejpam-6044	498	5	.	.	PUNCT
ejpam-6044	499	1	,	,	PUNCT
ejpam-6044	499	2	yn))pd1(y	yn))pd1(y	PRON
ejpam-6044	499	3	′	′	NOUN
ejpam-6044	499	4	1	1	NUM
ejpam-6044	499	5	,	,	PUNCT
ejpam-6044	499	6	.	.	PUNCT
ejpam-6044	499	7	.	.	PUNCT
ejpam-6044	500	1	.	.	PUNCT
ejpam-6044	501	1	,	,	PUNCT
ejpam-6044	501	2	y	y	PROPN
ejpam-6044	501	3	′	′	NUM
ejpam-6044	502	1	k	k	ADP
ejpam-6044	502	2	,	,	PUNCT
ejpam-6044	502	3	.	.	PUNCT
ejpam-6044	502	4	.	.	PUNCT
ejpam-6044	502	5	.	.	PUNCT
ejpam-6044	503	1	,	,	PUNCT
ejpam-6044	503	2	y	y	PROPN
ejpam-6044	503	3	′	′	NUM
ejpam-6044	503	4	n	n	CCONJ
ejpam-6044	503	5	)	)	PUNCT
ejpam-6044	503	6	=	=	SYM
ejpam-6044	503	7	0	0	NUM
ejpam-6044	503	8	,	,	PUNCT
ejpam-6044	503	9	(	(	PUNCT
ejpam-6044	503	10	15	15	NUM
ejpam-6044	503	11	)	)	PUNCT
ejpam-6044	503	12	for	for	ADP
ejpam-6044	503	13	each	each	DET
ejpam-6044	503	14	ς1	ς1	NOUN
ejpam-6044	503	15	,	,	PUNCT
ejpam-6044	503	16	...	...	PUNCT
ejpam-6044	503	17	,	,	PUNCT
ejpam-6044	503	18	ςn	ςn	PROPN
ejpam-6044	503	19	,	,	PUNCT
ejpam-6044	503	20	y1	y1	PROPN
ejpam-6044	503	21	,	,	PUNCT
ejpam-6044	503	22	...	...	PUNCT
ejpam-6044	503	23	,	,	PUNCT
ejpam-6044	503	24	yn	yn	PROPN
ejpam-6044	503	25	in	in	ADP
ejpam-6044	503	26	r.	r.	PROPN
ejpam-6044	503	27	case	case	NOUN
ejpam-6044	503	28	1	1	NUM
ejpam-6044	503	29	in	in	ADP
ejpam-6044	503	30	view	view	NOUN
ejpam-6044	503	31	of	of	ADP
ejpam-6044	503	32	(	(	PUNCT
ejpam-6044	503	33	14	14	NUM
ejpam-6044	503	34	)	)	PUNCT
ejpam-6044	503	35	,	,	PUNCT
ejpam-6044	503	36	(	(	PUNCT
ejpam-6044	503	37	15	15	X
ejpam-6044	503	38	)	)	PUNCT
ejpam-6044	503	39	takes	take	VERB
ejpam-6044	503	40	the	the	DET
ejpam-6044	503	41	form	form	NOUN
ejpam-6044	503	42	2f2(ς1	2f2(ς1	NUM
ejpam-6044	503	43	,	,	PUNCT
ejpam-6044	503	44	.	.	PUNCT
ejpam-6044	503	45	.	.	PUNCT
ejpam-6044	504	1	.	.	PUNCT
ejpam-6044	505	1	,	,	PUNCT
ejpam-6044	505	2	ςk	ςk	NUM
ejpam-6044	505	3	,	,	PUNCT
ejpam-6044	505	4	.	.	PUNCT
ejpam-6044	505	5	.	.	PUNCT
ejpam-6044	506	1	.	.	PUNCT
ejpam-6044	507	1	,	,	PUNCT
ejpam-6044	507	2	ςn)ykd1(y1	ςn)ykd1(y1	NOUN
ejpam-6044	507	3	,	,	PUNCT
ejpam-6044	507	4	.	.	PUNCT
ejpam-6044	507	5	.	.	PUNCT
ejpam-6044	508	1	.	.	PUNCT
ejpam-6044	509	1	,	,	PUNCT
ejpam-6044	510	1	z	z	X
ejpam-6044	510	2	,	,	PUNCT
ejpam-6044	510	3	.	.	PUNCT
ejpam-6044	510	4	.	.	PUNCT
ejpam-6044	510	5	.	.	PUNCT
ejpam-6044	511	1	,	,	PUNCT
ejpam-6044	511	2	yn)pd1(y	yn)pd1(y	VERB
ejpam-6044	511	3	′	′	NUM
ejpam-6044	511	4	1	1	NUM
ejpam-6044	511	5	,	,	PUNCT
ejpam-6044	511	6	.	.	PUNCT
ejpam-6044	511	7	.	.	PUNCT
ejpam-6044	512	1	.	.	PUNCT
ejpam-6044	513	1	,	,	PUNCT
ejpam-6044	513	2	y	y	PROPN
ejpam-6044	513	3	′	′	NUM
ejpam-6044	514	1	k	k	ADP
ejpam-6044	514	2	,	,	PUNCT
ejpam-6044	514	3	.	.	PUNCT
ejpam-6044	514	4	.	.	PUNCT
ejpam-6044	514	5	.	.	PUNCT
ejpam-6044	515	1	,	,	PUNCT
ejpam-6044	515	2	y	y	PROPN
ejpam-6044	515	3	′	′	NUM
ejpam-6044	515	4	n	n	CCONJ
ejpam-6044	515	5	)	)	PUNCT
ejpam-6044	515	6	=	=	SYM
ejpam-6044	515	7	0	0	NUM
ejpam-6044	515	8	using	use	VERB
ejpam-6044	515	9	∗-primeness	∗-primeness	NOUN
ejpam-6044	515	10	of	of	ADP
ejpam-6044	515	11	r	r	NOUN
ejpam-6044	515	12	and	and	CCONJ
ejpam-6044	515	13	2	2	NUM
ejpam-6044	515	14	-	-	PUNCT
ejpam-6044	515	15	torsion	torsion	NOUN
ejpam-6044	515	16	-	-	PUNCT
ejpam-6044	515	17	freeness	freeness	NOUN
ejpam-6044	515	18	of	of	ADP
ejpam-6044	515	19	r	r	PROPN
ejpam-6044	515	20	,	,	PUNCT
ejpam-6044	515	21	we	we	PRON
ejpam-6044	515	22	find	find	VERB
ejpam-6044	515	23	f2(ς1	f2(ς1	PRON
ejpam-6044	515	24	,	,	PUNCT
ejpam-6044	515	25	.	.	PUNCT
ejpam-6044	515	26	.	.	PUNCT
ejpam-6044	516	1	.	.	PUNCT
ejpam-6044	517	1	,	,	PUNCT
ejpam-6044	517	2	ςk	ςk	NUM
ejpam-6044	517	3	,	,	PUNCT
ejpam-6044	517	4	.	.	PUNCT
ejpam-6044	517	5	.	.	PUNCT
ejpam-6044	518	1	.	.	PUNCT
ejpam-6044	519	1	,	,	PUNCT
ejpam-6044	519	2	ςn)ykd1(y1	ςn)ykd1(y1	NOUN
ejpam-6044	519	3	,	,	PUNCT
ejpam-6044	519	4	.	.	PUNCT
ejpam-6044	519	5	.	.	PUNCT
ejpam-6044	520	1	.	.	PUNCT
ejpam-6044	521	1	,	,	PUNCT
ejpam-6044	522	1	z	z	X
ejpam-6044	522	2	,	,	PUNCT
ejpam-6044	522	3	.	.	PUNCT
ejpam-6044	522	4	.	.	PUNCT
ejpam-6044	522	5	.	.	PUNCT
ejpam-6044	523	1	,	,	PUNCT
ejpam-6044	523	2	yn	yn	PROPN
ejpam-6044	523	3	)	)	PUNCT
ejpam-6044	523	4	=	=	SYM
ejpam-6044	523	5	0	0	NUM
ejpam-6044	523	6	,	,	PUNCT
ejpam-6044	523	7	for	for	ADP
ejpam-6044	523	8	every	every	DET
ejpam-6044	523	9	ς1	ς1	NOUN
ejpam-6044	523	10	,	,	PUNCT
ejpam-6044	523	11	...	...	PUNCT
ejpam-6044	523	12	,	,	PUNCT
ejpam-6044	523	13	ςn	ςn	PROPN
ejpam-6044	523	14	,	,	PUNCT
ejpam-6044	523	15	y1	y1	PROPN
ejpam-6044	523	16	,	,	PUNCT
ejpam-6044	523	17	...	...	PUNCT
ejpam-6044	523	18	,	,	PUNCT
ejpam-6044	523	19	yn	yn	PROPN
ejpam-6044	523	20	,	,	PUNCT
ejpam-6044	523	21	z	z	PROPN
ejpam-6044	523	22	∈	∈	PROPN
ejpam-6044	523	23	r.	r.	PROPN
ejpam-6044	523	24	again	again	ADV
ejpam-6044	523	25	,	,	PUNCT
ejpam-6044	523	26	making	make	VERB
ejpam-6044	523	27	use	use	NOUN
ejpam-6044	523	28	of	of	ADP
ejpam-6044	523	29	primeness	primeness	NOUN
ejpam-6044	523	30	,	,	PUNCT
ejpam-6044	523	31	we	we	PRON
ejpam-6044	523	32	can	can	AUX
ejpam-6044	523	33	conclude	conclude	VERB
ejpam-6044	523	34	either	either	CCONJ
ejpam-6044	523	35	f2	f2	PROPN
ejpam-6044	523	36	=	=	SYM
ejpam-6044	523	37	0	0	NUM
ejpam-6044	523	38	or	or	CCONJ
ejpam-6044	523	39	d1	d1	PROPN
ejpam-6044	523	40	=	=	SYM
ejpam-6044	523	41	0	0	X
ejpam-6044	523	42	.	.	PUNCT
ejpam-6044	524	1	in	in	ADP
ejpam-6044	524	2	case	case	NOUN
ejpam-6044	524	3	d1	d1	NOUN
ejpam-6044	524	4	=	=	SYM
ejpam-6044	524	5	0	0	NUM
ejpam-6044	524	6	,	,	PUNCT
ejpam-6044	524	7	we	we	PRON
ejpam-6044	524	8	obtain	obtain	VERB
ejpam-6044	524	9	f1(ς1	f1(ς1	NOUN
ejpam-6044	524	10	,	,	PUNCT
ejpam-6044	524	11	.	.	PUNCT
ejpam-6044	524	12	.	.	PUNCT
ejpam-6044	525	1	.	.	PUNCT
ejpam-6044	526	1	,	,	PUNCT
ejpam-6044	526	2	ςkς	ςkς	X
ejpam-6044	526	3	′	′	NUM
ejpam-6044	527	1	k	k	NOUN
ejpam-6044	527	2	,	,	PUNCT
ejpam-6044	527	3	.	.	PUNCT
ejpam-6044	527	4	.	.	PUNCT
ejpam-6044	527	5	.	.	PUNCT
ejpam-6044	528	1	,	,	PUNCT
ejpam-6044	528	2	ςn	ςn	X
ejpam-6044	528	3	)	)	PUNCT
ejpam-6044	528	4	=	=	SYM
ejpam-6044	528	5	f1(ς1	f1(ς1	NOUN
ejpam-6044	528	6	,	,	PUNCT
ejpam-6044	528	7	.	.	PUNCT
ejpam-6044	528	8	.	.	PUNCT
ejpam-6044	529	1	.	.	PUNCT
ejpam-6044	530	1	,	,	PUNCT
ejpam-6044	530	2	ςk	ςk	NUM
ejpam-6044	530	3	,	,	PUNCT
ejpam-6044	530	4	.	.	PUNCT
ejpam-6044	530	5	.	.	PUNCT
ejpam-6044	531	1	.	.	PUNCT
ejpam-6044	532	1	,	,	PUNCT
ejpam-6044	532	2	ςn)α(ς	ςn)α(ς	VERB
ejpam-6044	532	3	′	′	NUM
ejpam-6044	533	1	k	k	NOUN
ejpam-6044	533	2	)	)	PUNCT
ejpam-6044	533	3	,	,	PUNCT
ejpam-6044	533	4	which	which	PRON
ejpam-6044	533	5	implies	imply	VERB
ejpam-6044	533	6	that	that	SCONJ
ejpam-6044	533	7	f1	f1	NOUN
ejpam-6044	533	8	acts	act	VERB
ejpam-6044	533	9	as	as	ADP
ejpam-6044	533	10	an	an	DET
ejpam-6044	533	11	α	α	NOUN
ejpam-6044	533	12	-	-	PUNCT
ejpam-6044	533	13	centralizer	centralizer	NOUN
ejpam-6044	533	14	on	on	ADP
ejpam-6044	533	15	r.	r.	PROPN
ejpam-6044	533	16	case	case	NOUN
ejpam-6044	533	17	2	2	NUM
ejpam-6044	533	18	multiply	multiply	NOUN
ejpam-6044	533	19	(	(	PUNCT
ejpam-6044	533	20	15	15	NUM
ejpam-6044	533	21	)	)	PUNCT
ejpam-6044	533	22	by	by	ADP
ejpam-6044	533	23	pd2(y	pd2(y	PROPN
ejpam-6044	533	24	′	′	NUM
ejpam-6044	533	25	1	1	NUM
ejpam-6044	533	26	,	,	PUNCT
ejpam-6044	533	27	.	.	PUNCT
ejpam-6044	533	28	.	.	PUNCT
ejpam-6044	534	1	.	.	PUNCT
ejpam-6044	535	1	,	,	PUNCT
ejpam-6044	535	2	y	y	PROPN
ejpam-6044	535	3	′	′	NUM
ejpam-6044	536	1	k	k	ADP
ejpam-6044	536	2	,	,	PUNCT
ejpam-6044	536	3	.	.	PUNCT
ejpam-6044	536	4	.	.	PUNCT
ejpam-6044	536	5	.	.	PUNCT
ejpam-6044	537	1	,	,	PUNCT
ejpam-6044	537	2	y	y	PROPN
ejpam-6044	537	3	′	′	NUM
ejpam-6044	537	4	n	n	CCONJ
ejpam-6044	537	5	)	)	PUNCT
ejpam-6044	537	6	from	from	ADP
ejpam-6044	537	7	the	the	DET
ejpam-6044	537	8	right	right	NOUN
ejpam-6044	537	9	and	and	CCONJ
ejpam-6044	537	10	use	use	VERB
ejpam-6044	537	11	it	it	PRON
ejpam-6044	537	12	again	again	ADV
ejpam-6044	537	13	to	to	PART
ejpam-6044	537	14	obtain	obtain	VERB
ejpam-6044	537	15	2f1(ς1	2f1(ς1	NUM
ejpam-6044	537	16	,	,	PUNCT
ejpam-6044	537	17	.	.	PUNCT
ejpam-6044	537	18	.	.	PUNCT
ejpam-6044	538	1	.	.	PUNCT
ejpam-6044	539	1	,	,	PUNCT
ejpam-6044	539	2	ςk	ςk	NUM
ejpam-6044	539	3	,	,	PUNCT
ejpam-6044	539	4	.	.	PUNCT
ejpam-6044	539	5	.	.	PUNCT
ejpam-6044	540	1	.	.	PUNCT
ejpam-6044	541	1	,	,	PUNCT
ejpam-6044	541	2	ςn)ykd2(y1	ςn)ykd2(y1	PROPN
ejpam-6044	541	3	,	,	PUNCT
ejpam-6044	541	4	.	.	PUNCT
ejpam-6044	541	5	.	.	PUNCT
ejpam-6044	542	1	.	.	PUNCT
ejpam-6044	543	1	,	,	PUNCT
ejpam-6044	544	1	z	z	X
ejpam-6044	544	2	,	,	PUNCT
ejpam-6044	544	3	.	.	PUNCT
ejpam-6044	544	4	.	.	PUNCT
ejpam-6044	544	5	.	.	PUNCT
ejpam-6044	545	1	,	,	PUNCT
ejpam-6044	545	2	yn)pd2(y	yn)pd2(y	PROPN
ejpam-6044	545	3	′	′	NUM
ejpam-6044	545	4	1	1	NUM
ejpam-6044	545	5	,	,	PUNCT
ejpam-6044	545	6	.	.	PUNCT
ejpam-6044	545	7	.	.	PUNCT
ejpam-6044	546	1	.	.	PUNCT
ejpam-6044	547	1	,	,	PUNCT
ejpam-6044	547	2	y	y	PROPN
ejpam-6044	547	3	′	′	NUM
ejpam-6044	548	1	k	k	ADP
ejpam-6044	548	2	,	,	PUNCT
ejpam-6044	548	3	.	.	PUNCT
ejpam-6044	548	4	.	.	PUNCT
ejpam-6044	548	5	.	.	PUNCT
ejpam-6044	549	1	,	,	PUNCT
ejpam-6044	549	2	y	y	PROPN
ejpam-6044	549	3	′	′	NUM
ejpam-6044	549	4	n	n	CCONJ
ejpam-6044	549	5	)	)	PUNCT
ejpam-6044	549	6	=	=	SYM
ejpam-6044	549	7	0	0	NUM
ejpam-6044	549	8	,	,	PUNCT
ejpam-6044	549	9	for	for	ADP
ejpam-6044	549	10	each	each	DET
ejpam-6044	549	11	ς1	ς1	NOUN
ejpam-6044	549	12	,	,	PUNCT
ejpam-6044	549	13	...	...	PUNCT
ejpam-6044	549	14	,	,	PUNCT
ejpam-6044	549	15	ςn	ςn	PROPN
ejpam-6044	549	16	,	,	PUNCT
ejpam-6044	549	17	y1	y1	PROPN
ejpam-6044	549	18	,	,	PUNCT
ejpam-6044	549	19	...	...	PUNCT
ejpam-6044	549	20	,	,	PUNCT
ejpam-6044	549	21	yn	yn	PROPN
ejpam-6044	549	22	inr	inr	PROPN
ejpam-6044	549	23	.	.	PUNCT
ejpam-6044	550	1	in	in	ADP
ejpam-6044	550	2	view	view	NOUN
ejpam-6044	550	3	of	of	ADP
ejpam-6044	550	4	(	(	PUNCT
ejpam-6044	550	5	15	15	NUM
ejpam-6044	550	6	)	)	PUNCT
ejpam-6044	550	7	,	,	PUNCT
ejpam-6044	550	8	the	the	DET
ejpam-6044	550	9	equation	equation	NOUN
ejpam-6044	550	10	takes	take	VERB
ejpam-6044	550	11	the	the	DET
ejpam-6044	550	12	form	form	NOUN
ejpam-6044	550	13	after	after	ADP
ejpam-6044	550	14	repeating	repeat	VERB
ejpam-6044	550	15	the	the	DET
ejpam-6044	550	16	similar	similar	ADJ
ejpam-6044	550	17	footsteps	footstep	NOUN
ejpam-6044	550	18	as	as	ADP
ejpam-6044	550	19	in	in	ADP
ejpam-6044	550	20	case	case	NOUN
ejpam-6044	550	21	1	1	NUM
ejpam-6044	550	22	,	,	PUNCT
ejpam-6044	550	23	we	we	PRON
ejpam-6044	550	24	conclude	conclude	VERB
ejpam-6044	550	25	either	either	CCONJ
ejpam-6044	550	26	f1	f1	PROPN
ejpam-6044	550	27	=	=	SYM
ejpam-6044	550	28	0	0	NUM
ejpam-6044	550	29	or	or	CCONJ
ejpam-6044	550	30	d2	d2	PROPN
ejpam-6044	550	31	=	=	SYM
ejpam-6044	550	32	0	0	PROPN
ejpam-6044	550	33	.	.	PUNCT
ejpam-6044	551	1	in	in	ADP
ejpam-6044	551	2	case	case	NOUN
ejpam-6044	551	3	d2	d2	PROPN
ejpam-6044	551	4	=	=	SYM
ejpam-6044	551	5	0	0	PROPN
ejpam-6044	551	6	,	,	PUNCT
ejpam-6044	551	7	f2	f2	PROPN
ejpam-6044	551	8	acts	act	VERB
ejpam-6044	551	9	as	as	ADP
ejpam-6044	551	10	a	a	DET
ejpam-6044	551	11	left	left	ADJ
ejpam-6044	551	12	α	α	NOUN
ejpam-6044	551	13	-	-	PUNCT
ejpam-6044	551	14	centralizer	centralizer	NOUN
ejpam-6044	551	15	.	.	PUNCT
ejpam-6044	552	1	f.	f.	PROPN
ejpam-6044	552	2	shujat	shujat	PROPN
ejpam-6044	552	3	,	,	PUNCT
ejpam-6044	552	4	s.	s.	PROPN
ejpam-6044	552	5	alharbi	alharbi	PROPN
ejpam-6044	552	6	/	/	SYM
ejpam-6044	552	7	eur	eur	PROPN
ejpam-6044	552	8	.	.	PUNCT
ejpam-6044	553	1	j.	j.	PROPN
ejpam-6044	553	2	pure	pure	PROPN
ejpam-6044	553	3	appl	appl	PROPN
ejpam-6044	553	4	.	.	PROPN
ejpam-6044	553	5	math	math	PROPN
ejpam-6044	553	6	,	,	PUNCT
ejpam-6044	553	7	18	18	NUM
ejpam-6044	553	8	(	(	PUNCT
ejpam-6044	553	9	2	2	NUM
ejpam-6044	553	10	)	)	PUNCT
ejpam-6044	553	11	(	(	PUNCT
ejpam-6044	553	12	2025	2025	NUM
ejpam-6044	553	13	)	)	PUNCT
ejpam-6044	553	14	,	,	PUNCT
ejpam-6044	553	15	6044	6044	NUM
ejpam-6044	553	16	9	9	NUM
ejpam-6044	553	17	of	of	ADP
ejpam-6044	553	18	11	11	NUM
ejpam-6044	553	19	theorem	theorem	NOUN
ejpam-6044	553	20	6	6	NUM
ejpam-6044	553	21	.	.	PUNCT
ejpam-6044	554	1	let	let	VERB
ejpam-6044	554	2	r	r	PRON
ejpam-6044	554	3	be	be	AUX
ejpam-6044	554	4	a	a	DET
ejpam-6044	554	5	semi	semi	ADJ
ejpam-6044	554	6	-	-	ADJ
ejpam-6044	554	7	prime	prime	ADJ
ejpam-6044	554	8	∗-ring	∗-ring	NOUN
ejpam-6044	554	9	admitting	admit	VERB
ejpam-6044	554	10	a	a	DET
ejpam-6044	554	11	generalized	generalized	ADJ
ejpam-6044	554	12	(	(	PUNCT
ejpam-6044	554	13	α	α	NOUN
ejpam-6044	554	14	,	,	PUNCT
ejpam-6044	554	15	∗)-n	∗)-n	NOUN
ejpam-6044	554	16	-	-	PUNCT
ejpam-6044	554	17	derivation	derivation	NOUN
ejpam-6044	554	18	f	f	NOUN
ejpam-6044	554	19	with	with	ADP
ejpam-6044	554	20	associated	associated	ADJ
ejpam-6044	554	21	(	(	PUNCT
ejpam-6044	554	22	α	α	NOUN
ejpam-6044	554	23	,	,	PUNCT
ejpam-6044	554	24	∗)-n	∗)-n	NOUN
ejpam-6044	554	25	-	-	PUNCT
ejpam-6044	554	26	derivation	derivation	ADJ
ejpam-6044	554	27	d.	d.	NOUN
ejpam-6044	554	28	then	then	ADV
ejpam-6044	554	29	d(r	d(r	PROPN
ejpam-6044	554	30	,	,	PUNCT
ejpam-6044	554	31	r	r	NOUN
ejpam-6044	554	32	,	,	PUNCT
ejpam-6044	554	33	.	.	PUNCT
ejpam-6044	554	34	.	.	PUNCT
ejpam-6044	555	1	.	.	PUNCT
ejpam-6044	556	1	,	,	PUNCT
ejpam-6044	556	2	r	r	X
ejpam-6044	556	3	)	)	PUNCT
ejpam-6044	556	4	⊆	⊆	NUM
ejpam-6044	556	5	z(r	z(r	NOUN
ejpam-6044	556	6	)	)	PUNCT
ejpam-6044	556	7	.	.	PUNCT
ejpam-6044	557	1	proof	proof	NOUN
ejpam-6044	557	2	.	.	PUNCT
ejpam-6044	558	1	since	since	SCONJ
ejpam-6044	558	2	r	r	NOUN
ejpam-6044	558	3	is	be	AUX
ejpam-6044	558	4	a	a	DET
ejpam-6044	558	5	∗-ring	∗-ring	NOUN
ejpam-6044	558	6	and	and	CCONJ
ejpam-6044	558	7	admits	admit	VERB
ejpam-6044	558	8	a	a	DET
ejpam-6044	558	9	generalized	generalized	ADJ
ejpam-6044	558	10	(	(	PUNCT
ejpam-6044	558	11	α	α	NOUN
ejpam-6044	558	12	,	,	PUNCT
ejpam-6044	558	13	∗)-n	∗)-n	NOUN
ejpam-6044	558	14	-	-	PUNCT
ejpam-6044	558	15	derivation	derivation	NOUN
ejpam-6044	558	16	f	f	NOUN
ejpam-6044	558	17	,	,	PUNCT
ejpam-6044	558	18	we	we	PRON
ejpam-6044	558	19	take	take	VERB
ejpam-6044	558	20	a	a	DET
ejpam-6044	558	21	quick	quick	ADJ
ejpam-6044	558	22	start	start	NOUN
ejpam-6044	558	23	from	from	ADP
ejpam-6044	558	24	equation	equation	NOUN
ejpam-6044	558	25	(	(	PUNCT
ejpam-6044	558	26	6	6	NUM
ejpam-6044	558	27	)	)	PUNCT
ejpam-6044	558	28	in	in	ADP
ejpam-6044	558	29	theorem	theorem	ADJ
ejpam-6044	558	30	2	2	NUM
ejpam-6044	558	31	[	[	X
ejpam-6044	558	32	x	x	X
ejpam-6044	558	33	,	,	PUNCT
ejpam-6044	558	34	y]rd(ς1	y]rd(ς1	NOUN
ejpam-6044	558	35	,	,	PUNCT
ejpam-6044	558	36	.	.	PUNCT
ejpam-6044	558	37	.	.	PUNCT
ejpam-6044	558	38	.	.	PUNCT
ejpam-6044	559	1	,	,	PUNCT
ejpam-6044	559	2	ς	ς	PROPN
ejpam-6044	559	3	′	′	NUM
ejpam-6044	559	4	k	k	NOUN
ejpam-6044	559	5	,	,	PUNCT
ejpam-6044	559	6	.	.	PUNCT
ejpam-6044	559	7	.	.	PUNCT
ejpam-6044	560	1	.	.	PUNCT
ejpam-6044	561	1	,	,	PUNCT
ejpam-6044	561	2	ςn	ςn	X
ejpam-6044	561	3	)	)	PUNCT
ejpam-6044	561	4	=	=	SYM
ejpam-6044	561	5	0	0	NUM
ejpam-6044	561	6	for	for	ADP
ejpam-6044	561	7	every	every	DET
ejpam-6044	561	8	ς1	ς1	NOUN
ejpam-6044	561	9	,	,	PUNCT
ejpam-6044	561	10	...	...	PUNCT
ejpam-6044	561	11	,	,	PUNCT
ejpam-6044	561	12	ςn	ςn	PROPN
ejpam-6044	561	13	∈	∈	PROPN
ejpam-6044	561	14	r.	r.	PROPN
ejpam-6044	561	15	(	(	PUNCT
ejpam-6044	561	16	16	16	NUM
ejpam-6044	561	17	)	)	PUNCT
ejpam-6044	561	18	replacing	replace	VERB
ejpam-6044	561	19	x	x	PUNCT
ejpam-6044	561	20	by	by	ADP
ejpam-6044	561	21	d(ς1	d(ς1	NOUN
ejpam-6044	561	22	,	,	PUNCT
ejpam-6044	561	23	.	.	PUNCT
ejpam-6044	561	24	.	.	PUNCT
ejpam-6044	562	1	.	.	PUNCT
ejpam-6044	563	1	,	,	PUNCT
ejpam-6044	563	2	ς	ς	PROPN
ejpam-6044	563	3	′	′	NUM
ejpam-6044	563	4	k	k	NOUN
ejpam-6044	563	5	,	,	PUNCT
ejpam-6044	563	6	.	.	PUNCT
ejpam-6044	563	7	.	.	PUNCT
ejpam-6044	564	1	.	.	PUNCT
ejpam-6044	565	1	,	,	PUNCT
ejpam-6044	565	2	ςn)x	ςn)x	PROPN
ejpam-6044	565	3	,	,	PUNCT
ejpam-6044	565	4	we	we	PRON
ejpam-6044	565	5	obtain	obtain	VERB
ejpam-6044	565	6	for	for	ADP
ejpam-6044	565	7	every	every	DET
ejpam-6044	565	8	ς1	ς1	NOUN
ejpam-6044	565	9	,	,	PUNCT
ejpam-6044	565	10	...	...	PUNCT
ejpam-6044	565	11	,	,	PUNCT
ejpam-6044	565	12	ςn	ςn	PROPN
ejpam-6044	565	13	∈	∈	PROPN
ejpam-6044	565	14	r	r	NOUN
ejpam-6044	565	15	d(ς1	d(ς1	NOUN
ejpam-6044	565	16	,	,	PUNCT
ejpam-6044	565	17	.	.	PUNCT
ejpam-6044	565	18	.	.	PUNCT
ejpam-6044	566	1	.	.	PUNCT
ejpam-6044	567	1	,	,	PUNCT
ejpam-6044	567	2	ς	ς	PROPN
ejpam-6044	567	3	′	′	NUM
ejpam-6044	567	4	k	k	NOUN
ejpam-6044	567	5	,	,	PUNCT
ejpam-6044	567	6	.	.	PUNCT
ejpam-6044	567	7	.	.	PUNCT
ejpam-6044	568	1	.	.	PUNCT
ejpam-6044	569	1	,	,	PUNCT
ejpam-6044	569	2	ςn)[x	ςn)[x	PROPN
ejpam-6044	569	3	,	,	PUNCT
ejpam-6044	569	4	y]rd(ς1	y]rd(ς1	NOUN
ejpam-6044	569	5	,	,	PUNCT
ejpam-6044	569	6	.	.	PUNCT
ejpam-6044	569	7	.	.	PUNCT
ejpam-6044	570	1	.	.	PUNCT
ejpam-6044	571	1	,	,	PUNCT
ejpam-6044	571	2	ς	ς	PROPN
ejpam-6044	571	3	′	′	NUM
ejpam-6044	571	4	k	k	NOUN
ejpam-6044	571	5	,	,	PUNCT
ejpam-6044	571	6	.	.	PUNCT
ejpam-6044	571	7	.	.	PUNCT
ejpam-6044	572	1	.	.	PUNCT
ejpam-6044	573	1	,	,	PUNCT
ejpam-6044	573	2	ςn	ςn	X
ejpam-6044	573	3	)	)	PUNCT
ejpam-6044	574	1	+	+	PROPN
ejpam-6044	574	2	[	[	X
ejpam-6044	574	3	d(ς1	d(ς1	NOUN
ejpam-6044	574	4	,	,	PUNCT
ejpam-6044	574	5	.	.	PUNCT
ejpam-6044	574	6	.	.	PUNCT
ejpam-6044	575	1	.	.	PUNCT
ejpam-6044	576	1	,	,	PUNCT
ejpam-6044	576	2	ς	ς	PROPN
ejpam-6044	576	3	′	′	NUM
ejpam-6044	576	4	k	k	NOUN
ejpam-6044	576	5	,	,	PUNCT
ejpam-6044	576	6	.	.	PUNCT
ejpam-6044	576	7	.	.	PUNCT
ejpam-6044	577	1	.	.	PUNCT
ejpam-6044	578	1	,	,	PUNCT
ejpam-6044	578	2	ςn	ςn	NOUN
ejpam-6044	578	3	)	)	PUNCT
ejpam-6044	578	4	,	,	PUNCT
ejpam-6044	578	5	y]xrd(ς1	y]xrd(ς1	PROPN
ejpam-6044	578	6	,	,	PUNCT
ejpam-6044	578	7	.	.	PUNCT
ejpam-6044	578	8	.	.	PUNCT
ejpam-6044	579	1	.	.	PUNCT
ejpam-6044	580	1	,	,	PUNCT
ejpam-6044	580	2	ς	ς	PROPN
ejpam-6044	580	3	′	′	NUM
ejpam-6044	580	4	k	k	NOUN
ejpam-6044	580	5	,	,	PUNCT
ejpam-6044	580	6	.	.	PUNCT
ejpam-6044	580	7	.	.	PUNCT
ejpam-6044	581	1	.	.	PUNCT
ejpam-6044	582	1	,	,	PUNCT
ejpam-6044	582	2	ςn	ςn	X
ejpam-6044	582	3	)	)	PUNCT
ejpam-6044	582	4	=	=	SYM
ejpam-6044	583	1	0	0	X
ejpam-6044	583	2	.	.	PUNCT
ejpam-6044	584	1	(	(	PUNCT
ejpam-6044	584	2	17	17	NUM
ejpam-6044	584	3	)	)	PUNCT
ejpam-6044	584	4	from	from	ADP
ejpam-6044	584	5	(	(	PUNCT
ejpam-6044	584	6	16	16	NUM
ejpam-6044	584	7	)	)	PUNCT
ejpam-6044	584	8	and	and	CCONJ
ejpam-6044	584	9	(	(	PUNCT
ejpam-6044	584	10	17	17	NUM
ejpam-6044	584	11	)	)	PUNCT
ejpam-6044	584	12	,	,	PUNCT
ejpam-6044	584	13	we	we	PRON
ejpam-6044	584	14	conclude	conclude	VERB
ejpam-6044	584	15	[	[	X
ejpam-6044	584	16	d(ς1	d(ς1	NOUN
ejpam-6044	584	17	,	,	PUNCT
ejpam-6044	584	18	.	.	PUNCT
ejpam-6044	584	19	.	.	PUNCT
ejpam-6044	585	1	.	.	PUNCT
ejpam-6044	586	1	,	,	PUNCT
ejpam-6044	586	2	ς	ς	PROPN
ejpam-6044	586	3	′	′	NUM
ejpam-6044	586	4	k	k	NOUN
ejpam-6044	586	5	,	,	PUNCT
ejpam-6044	586	6	.	.	PUNCT
ejpam-6044	586	7	.	.	PUNCT
ejpam-6044	587	1	.	.	PUNCT
ejpam-6044	588	1	,	,	PUNCT
ejpam-6044	588	2	ςn	ςn	NOUN
ejpam-6044	588	3	)	)	PUNCT
ejpam-6044	588	4	,	,	PUNCT
ejpam-6044	588	5	y]xrd(ς1	y]xrd(ς1	PROPN
ejpam-6044	588	6	,	,	PUNCT
ejpam-6044	588	7	.	.	PUNCT
ejpam-6044	588	8	.	.	PUNCT
ejpam-6044	589	1	.	.	PUNCT
ejpam-6044	590	1	,	,	PUNCT
ejpam-6044	590	2	ς	ς	PROPN
ejpam-6044	590	3	′	′	NUM
ejpam-6044	590	4	k	k	NOUN
ejpam-6044	590	5	,	,	PUNCT
ejpam-6044	590	6	.	.	PUNCT
ejpam-6044	590	7	.	.	PUNCT
ejpam-6044	591	1	.	.	PUNCT
ejpam-6044	592	1	,	,	PUNCT
ejpam-6044	592	2	ςn	ςn	X
ejpam-6044	592	3	)	)	PUNCT
ejpam-6044	592	4	=	=	SYM
ejpam-6044	592	5	0	0	NUM
ejpam-6044	592	6	for	for	ADP
ejpam-6044	592	7	every	every	DET
ejpam-6044	592	8	ς1	ς1	NOUN
ejpam-6044	592	9	,	,	PUNCT
ejpam-6044	592	10	...	...	PUNCT
ejpam-6044	592	11	,	,	PUNCT
ejpam-6044	592	12	ςn	ςn	PROPN
ejpam-6044	592	13	∈	∈	PROPN
ejpam-6044	592	14	r.	r.	PROPN
ejpam-6044	592	15	(	(	PUNCT
ejpam-6044	592	16	18	18	NUM
ejpam-6044	592	17	)	)	PUNCT
ejpam-6044	592	18	we	we	PRON
ejpam-6044	592	19	can	can	AUX
ejpam-6044	592	20	write	write	VERB
ejpam-6044	592	21	also	also	ADV
ejpam-6044	592	22	[	[	X
ejpam-6044	592	23	d(ς1	d(ς1	NOUN
ejpam-6044	592	24	,	,	PUNCT
ejpam-6044	592	25	.	.	PUNCT
ejpam-6044	592	26	.	.	PUNCT
ejpam-6044	593	1	.	.	PUNCT
ejpam-6044	594	1	,	,	PUNCT
ejpam-6044	594	2	ς	ς	PROPN
ejpam-6044	594	3	′	′	NUM
ejpam-6044	594	4	k	k	NOUN
ejpam-6044	594	5	,	,	PUNCT
ejpam-6044	594	6	.	.	PUNCT
ejpam-6044	594	7	.	.	PUNCT
ejpam-6044	595	1	.	.	PUNCT
ejpam-6044	596	1	,	,	PUNCT
ejpam-6044	596	2	ςn	ςn	NOUN
ejpam-6044	596	3	)	)	PUNCT
ejpam-6044	596	4	,	,	PUNCT
ejpam-6044	596	5	y]xryd(ς1	y]xryd(ς1	NOUN
ejpam-6044	596	6	,	,	PUNCT
ejpam-6044	596	7	.	.	PUNCT
ejpam-6044	596	8	.	.	PUNCT
ejpam-6044	597	1	.	.	PUNCT
ejpam-6044	598	1	,	,	PUNCT
ejpam-6044	598	2	ς	ς	PROPN
ejpam-6044	598	3	′	′	NUM
ejpam-6044	598	4	k	k	NOUN
ejpam-6044	598	5	,	,	PUNCT
ejpam-6044	598	6	.	.	PUNCT
ejpam-6044	598	7	.	.	PUNCT
ejpam-6044	599	1	.	.	PUNCT
ejpam-6044	600	1	,	,	PUNCT
ejpam-6044	600	2	ςn	ςn	X
ejpam-6044	600	3	)	)	PUNCT
ejpam-6044	600	4	=	=	SYM
ejpam-6044	600	5	0	0	NUM
ejpam-6044	600	6	for	for	ADP
ejpam-6044	600	7	every	every	DET
ejpam-6044	600	8	ς1	ς1	NOUN
ejpam-6044	600	9	,	,	PUNCT
ejpam-6044	600	10	...	...	PUNCT
ejpam-6044	600	11	,	,	PUNCT
ejpam-6044	600	12	ςn	ςn	PROPN
ejpam-6044	600	13	,	,	PUNCT
ejpam-6044	600	14	y	y	PROPN
ejpam-6044	600	15	,	,	PUNCT
ejpam-6044	600	16	x	x	PROPN
ejpam-6044	600	17	∈	∈	PROPN
ejpam-6044	600	18	r.	r.	PROPN
ejpam-6044	600	19	(	(	PUNCT
ejpam-6044	600	20	19	19	NUM
ejpam-6044	600	21	)	)	PUNCT
ejpam-6044	600	22	now	now	ADV
ejpam-6044	600	23	multiply	multiply	ADV
ejpam-6044	600	24	(	(	PUNCT
ejpam-6044	600	25	18)by	18)by	NUM
ejpam-6044	600	26	y	y	PROPN
ejpam-6044	600	27	from	from	ADP
ejpam-6044	600	28	right	right	ADV
ejpam-6044	600	29	and	and	CCONJ
ejpam-6044	600	30	subtract	subtract	VERB
ejpam-6044	600	31	the	the	DET
ejpam-6044	600	32	resulting	result	VERB
ejpam-6044	600	33	equation	equation	NOUN
ejpam-6044	600	34	with	with	ADP
ejpam-6044	600	35	(	(	PUNCT
ejpam-6044	600	36	19)to	19)to	NUM
ejpam-6044	600	37	find	find	VERB
ejpam-6044	600	38	[	[	X
ejpam-6044	600	39	d(ς1	d(ς1	NOUN
ejpam-6044	600	40	,	,	PUNCT
ejpam-6044	600	41	.	.	PUNCT
ejpam-6044	600	42	.	.	PUNCT
ejpam-6044	600	43	.	.	PUNCT
ejpam-6044	601	1	,	,	PUNCT
ejpam-6044	601	2	ς	ς	PROPN
ejpam-6044	601	3	′	′	NUM
ejpam-6044	601	4	k	k	NOUN
ejpam-6044	601	5	,	,	PUNCT
ejpam-6044	601	6	.	.	PUNCT
ejpam-6044	601	7	.	.	PUNCT
ejpam-6044	602	1	.	.	PUNCT
ejpam-6044	603	1	,	,	PUNCT
ejpam-6044	603	2	ςn	ςn	PROPN
ejpam-6044	603	3	)	)	PUNCT
ejpam-6044	603	4	,	,	PUNCT
ejpam-6044	603	5	y]xr[d(ς1	y]xr[d(ς1	NOUN
ejpam-6044	603	6	,	,	PUNCT
ejpam-6044	603	7	.	.	PUNCT
ejpam-6044	603	8	.	.	PUNCT
ejpam-6044	604	1	.	.	PUNCT
ejpam-6044	605	1	,	,	PUNCT
ejpam-6044	605	2	ς	ς	PROPN
ejpam-6044	605	3	′	′	NUM
ejpam-6044	605	4	k	k	NOUN
ejpam-6044	605	5	,	,	PUNCT
ejpam-6044	605	6	.	.	PUNCT
ejpam-6044	605	7	.	.	PUNCT
ejpam-6044	606	1	.	.	PUNCT
ejpam-6044	607	1	,	,	PUNCT
ejpam-6044	607	2	ςn	ςn	PROPN
ejpam-6044	607	3	)	)	PUNCT
ejpam-6044	607	4	,	,	PUNCT
ejpam-6044	607	5	y	y	PROPN
ejpam-6044	607	6	]	]	X
ejpam-6044	607	7	=	=	SYM
ejpam-6044	607	8	0	0	NUM
ejpam-6044	608	1	for	for	ADP
ejpam-6044	608	2	every	every	DET
ejpam-6044	608	3	ς1	ς1	NOUN
ejpam-6044	608	4	,	,	PUNCT
ejpam-6044	608	5	...	...	PUNCT
ejpam-6044	608	6	,	,	PUNCT
ejpam-6044	608	7	ςn	ςn	PROPN
ejpam-6044	608	8	,	,	PUNCT
ejpam-6044	608	9	x	x	PRON
ejpam-6044	608	10	,	,	PUNCT
ejpam-6044	608	11	y	y	PROPN
ejpam-6044	608	12	∈	∈	PROPN
ejpam-6044	608	13	r.	r.	PROPN
ejpam-6044	608	14	since	since	SCONJ
ejpam-6044	608	15	this	this	PRON
ejpam-6044	608	16	holds	hold	VERB
ejpam-6044	608	17	for	for	ADP
ejpam-6044	608	18	all	all	DET
ejpam-6044	608	19	y	y	PROPN
ejpam-6044	608	20	∈	∈	PROPN
ejpam-6044	608	21	r	r	NOUN
ejpam-6044	608	22	,	,	PUNCT
ejpam-6044	608	23	it	it	PRON
ejpam-6044	608	24	follows	follow	VERB
ejpam-6044	608	25	that	that	SCONJ
ejpam-6044	609	1	[	[	X
ejpam-6044	609	2	d(ς1	d(ς1	NOUN
ejpam-6044	609	3	,	,	PUNCT
ejpam-6044	609	4	.	.	PUNCT
ejpam-6044	609	5	.	.	PUNCT
ejpam-6044	610	1	.	.	PUNCT
ejpam-6044	611	1	,	,	PUNCT
ejpam-6044	611	2	ς	ς	PROPN
ejpam-6044	611	3	′	′	NUM
ejpam-6044	611	4	k	k	NOUN
ejpam-6044	611	5	,	,	PUNCT
ejpam-6044	611	6	.	.	PUNCT
ejpam-6044	611	7	.	.	PUNCT
ejpam-6044	612	1	.	.	PUNCT
ejpam-6044	613	1	,	,	PUNCT
ejpam-6044	613	2	ςn	ςn	PROPN
ejpam-6044	613	3	)	)	PUNCT
ejpam-6044	613	4	,	,	PUNCT
ejpam-6044	613	5	y	y	PROPN
ejpam-6044	613	6	]	]	X
ejpam-6044	613	7	=	=	SYM
ejpam-6044	613	8	0	0	NUM
ejpam-6044	614	1	for	for	ADP
ejpam-6044	614	2	every	every	DET
ejpam-6044	614	3	y	y	PROPN
ejpam-6044	614	4	∈	∈	PROPN
ejpam-6044	614	5	r.	r.	PROPN
ejpam-6044	614	6	thus	thus	ADV
ejpam-6044	614	7	,	,	PUNCT
ejpam-6044	614	8	we	we	PRON
ejpam-6044	614	9	conclude	conclude	VERB
ejpam-6044	614	10	that	that	DET
ejpam-6044	614	11	d(ς1	d(ς1	NOUN
ejpam-6044	614	12	,	,	PUNCT
ejpam-6044	614	13	.	.	PUNCT
ejpam-6044	614	14	.	.	PUNCT
ejpam-6044	615	1	.	.	PUNCT
ejpam-6044	616	1	,	,	PUNCT
ejpam-6044	616	2	ς	ς	PROPN
ejpam-6044	616	3	′	′	NUM
ejpam-6044	616	4	k	k	NOUN
ejpam-6044	616	5	,	,	PUNCT
ejpam-6044	616	6	.	.	PUNCT
ejpam-6044	616	7	.	.	PUNCT
ejpam-6044	617	1	.	.	PUNCT
ejpam-6044	618	1	,	,	PUNCT
ejpam-6044	618	2	ςn	ςn	X
ejpam-6044	618	3	)	)	PUNCT
ejpam-6044	618	4	⊆	⊆	NUM
ejpam-6044	618	5	z(r	z(r	NOUN
ejpam-6044	618	6	)	)	PUNCT
ejpam-6044	618	7	.	.	PUNCT
ejpam-6044	619	1	theorem	theorem	VERB
ejpam-6044	619	2	7	7	NUM
ejpam-6044	619	3	.	.	PUNCT
ejpam-6044	620	1	let	let	VERB
ejpam-6044	620	2	r	r	PRON
ejpam-6044	620	3	be	be	AUX
ejpam-6044	620	4	a	a	DET
ejpam-6044	620	5	semiprime	semiprime	NOUN
ejpam-6044	620	6	ring	ring	NOUN
ejpam-6044	620	7	with	with	ADP
ejpam-6044	620	8	involution	involution	NOUN
ejpam-6044	620	9	∗.	∗.	PUNCT
ejpam-6044	620	10	if	if	SCONJ
ejpam-6044	620	11	f	f	PROPN
ejpam-6044	620	12	is	be	AUX
ejpam-6044	620	13	a	a	DET
ejpam-6044	620	14	generalized	generalized	ADJ
ejpam-6044	620	15	(	(	PUNCT
ejpam-6044	620	16	α	α	NOUN
ejpam-6044	620	17	,	,	PUNCT
ejpam-6044	620	18	∗)-nderivation	∗)-nderivation	NOUN
ejpam-6044	620	19	of	of	ADP
ejpam-6044	620	20	r	r	NOUN
ejpam-6044	620	21	associated	associate	VERB
ejpam-6044	620	22	with	with	ADP
ejpam-6044	620	23	an	an	DET
ejpam-6044	620	24	(	(	PUNCT
ejpam-6044	620	25	α	α	NOUN
ejpam-6044	620	26	,	,	PUNCT
ejpam-6044	620	27	∗)-n	∗)-n	NOUN
ejpam-6044	620	28	-	-	PUNCT
ejpam-6044	620	29	derivation	derivation	NOUN
ejpam-6044	621	1	d	d	NOUN
ejpam-6044	621	2	such	such	ADJ
ejpam-6044	621	3	that	that	SCONJ
ejpam-6044	621	4	f	f	PROPN
ejpam-6044	621	5	(	(	PUNCT
ejpam-6044	621	6	ς1	ς1	NOUN
ejpam-6044	621	7	,	,	PUNCT
ejpam-6044	621	8	.	.	PUNCT
ejpam-6044	621	9	.	.	PUNCT
ejpam-6044	622	1	.	.	PUNCT
ejpam-6044	623	1	,	,	PUNCT
ejpam-6044	623	2	ςk	ςk	NUM
ejpam-6044	623	3	,	,	PUNCT
ejpam-6044	623	4	.	.	PUNCT
ejpam-6044	623	5	.	.	PUNCT
ejpam-6044	624	1	.	.	PUNCT
ejpam-6044	625	1	,	,	PUNCT
ejpam-6044	625	2	ςn)yi	ςn)yi	PUNCT
ejpam-6044	626	1	=	=	SYM
ejpam-6044	626	2	ςif	ςif	X
ejpam-6044	626	3	(	(	PUNCT
ejpam-6044	626	4	y1	y1	INTJ
ejpam-6044	626	5	,	,	PUNCT
ejpam-6044	626	6	.	.	PUNCT
ejpam-6044	626	7	.	.	PUNCT
ejpam-6044	626	8	.	.	PUNCT
ejpam-6044	627	1	,	,	PUNCT
ejpam-6044	627	2	yk	yk	PROPN
ejpam-6044	627	3	,	,	PUNCT
ejpam-6044	627	4	.	.	PUNCT
ejpam-6044	627	5	.	.	PUNCT
ejpam-6044	628	1	.	.	PUNCT
ejpam-6044	629	1	,	,	PUNCT
ejpam-6044	629	2	yn	yn	PROPN
ejpam-6044	629	3	)	)	PUNCT
ejpam-6044	629	4	for	for	ADP
ejpam-6044	629	5	every	every	DET
ejpam-6044	629	6	ς1	ς1	NOUN
ejpam-6044	629	7	,	,	PUNCT
ejpam-6044	629	8	.	.	PUNCT
ejpam-6044	629	9	.	.	PUNCT
ejpam-6044	630	1	.	.	PUNCT
ejpam-6044	631	1	,	,	PUNCT
ejpam-6044	631	2	ςk	ςk	NUM
ejpam-6044	631	3	,	,	PUNCT
ejpam-6044	631	4	.	.	PUNCT
ejpam-6044	631	5	.	.	PUNCT
ejpam-6044	632	1	.	.	PUNCT
ejpam-6044	633	1	,	,	PUNCT
ejpam-6044	633	2	ςn	ςn	PROPN
ejpam-6044	633	3	,	,	PUNCT
ejpam-6044	633	4	y1	y1	NOUN
ejpam-6044	633	5	,	,	PUNCT
ejpam-6044	633	6	.	.	PUNCT
ejpam-6044	633	7	.	.	PUNCT
ejpam-6044	633	8	.	.	PUNCT
ejpam-6044	634	1	,	,	PUNCT
ejpam-6044	634	2	yk	yk	PROPN
ejpam-6044	634	3	,	,	PUNCT
ejpam-6044	634	4	.	.	PUNCT
ejpam-6044	634	5	.	.	PUNCT
ejpam-6044	635	1	.	.	PUNCT
ejpam-6044	636	1	,	,	PUNCT
ejpam-6044	636	2	yn	yn	PROPN
ejpam-6044	636	3	∈	∈	PROPN
ejpam-6044	636	4	r	r	NOUN
ejpam-6044	636	5	,	,	PUNCT
ejpam-6044	636	6	then	then	ADV
ejpam-6044	636	7	f	f	PROPN
ejpam-6044	636	8	is	be	AUX
ejpam-6044	636	9	a	a	DET
ejpam-6044	636	10	left	left	ADJ
ejpam-6044	636	11	α	α	NOUN
ejpam-6044	636	12	-	-	PUNCT
ejpam-6044	636	13	centralizer	centralizer	NOUN
ejpam-6044	636	14	.	.	PUNCT
ejpam-6044	637	1	f.	f.	PROPN
ejpam-6044	637	2	shujat	shujat	PROPN
ejpam-6044	637	3	,	,	PUNCT
ejpam-6044	637	4	s.	s.	PROPN
ejpam-6044	637	5	alharbi	alharbi	PROPN
ejpam-6044	637	6	/	/	SYM
ejpam-6044	637	7	eur	eur	PROPN
ejpam-6044	637	8	.	.	PUNCT
ejpam-6044	638	1	j.	j.	PROPN
ejpam-6044	638	2	pure	pure	PROPN
ejpam-6044	638	3	appl	appl	PROPN
ejpam-6044	638	4	.	.	PROPN
ejpam-6044	638	5	math	math	PROPN
ejpam-6044	638	6	,	,	PUNCT
ejpam-6044	638	7	18	18	NUM
ejpam-6044	638	8	(	(	PUNCT
ejpam-6044	638	9	2	2	NUM
ejpam-6044	638	10	)	)	PUNCT
ejpam-6044	638	11	(	(	PUNCT
ejpam-6044	638	12	2025	2025	NUM
ejpam-6044	638	13	)	)	PUNCT
ejpam-6044	638	14	,	,	PUNCT
ejpam-6044	638	15	6044	6044	NUM
ejpam-6044	638	16	10	10	NUM
ejpam-6044	638	17	of	of	ADP
ejpam-6044	638	18	11	11	NUM
ejpam-6044	638	19	proof	proof	NOUN
ejpam-6044	638	20	.	.	PUNCT
ejpam-6044	639	1	by	by	ADP
ejpam-6044	639	2	the	the	DET
ejpam-6044	639	3	given	give	VERB
ejpam-6044	639	4	hypotheses	hypothesis	NOUN
ejpam-6044	639	5	,	,	PUNCT
ejpam-6044	639	6	we	we	PRON
ejpam-6044	639	7	have	have	VERB
ejpam-6044	639	8	f	f	PROPN
ejpam-6044	639	9	(	(	PUNCT
ejpam-6044	639	10	ς1	ς1	NOUN
ejpam-6044	639	11	,	,	PUNCT
ejpam-6044	639	12	.	.	PUNCT
ejpam-6044	639	13	.	.	PUNCT
ejpam-6044	640	1	.	.	PUNCT
ejpam-6044	641	1	,	,	PUNCT
ejpam-6044	641	2	ςk	ςk	NUM
ejpam-6044	641	3	,	,	PUNCT
ejpam-6044	641	4	.	.	PUNCT
ejpam-6044	641	5	.	.	PUNCT
ejpam-6044	642	1	.	.	PUNCT
ejpam-6044	643	1	,	,	PUNCT
ejpam-6044	643	2	ςn)yi	ςn)yi	PUNCT
ejpam-6044	644	1	=	=	SYM
ejpam-6044	644	2	ςif	ςif	X
ejpam-6044	644	3	(	(	PUNCT
ejpam-6044	644	4	y1	y1	INTJ
ejpam-6044	644	5	,	,	PUNCT
ejpam-6044	644	6	.	.	PUNCT
ejpam-6044	644	7	.	.	PUNCT
ejpam-6044	644	8	.	.	PUNCT
ejpam-6044	645	1	,	,	PUNCT
ejpam-6044	645	2	yk	yk	PROPN
ejpam-6044	645	3	,	,	PUNCT
ejpam-6044	645	4	.	.	PUNCT
ejpam-6044	645	5	.	.	PUNCT
ejpam-6044	646	1	.	.	PUNCT
ejpam-6044	647	1	,	,	PUNCT
ejpam-6044	647	2	yn	yn	PROPN
ejpam-6044	647	3	)	)	PUNCT
ejpam-6044	647	4	(	(	PUNCT
ejpam-6044	647	5	20	20	NUM
ejpam-6044	647	6	)	)	PUNCT
ejpam-6044	647	7	for	for	ADP
ejpam-6044	647	8	every	every	DET
ejpam-6044	647	9	ς1	ς1	NOUN
ejpam-6044	647	10	,	,	PUNCT
ejpam-6044	647	11	.	.	PUNCT
ejpam-6044	647	12	.	.	PUNCT
ejpam-6044	648	1	.	.	PUNCT
ejpam-6044	649	1	,	,	PUNCT
ejpam-6044	649	2	ςk	ςk	NUM
ejpam-6044	649	3	,	,	PUNCT
ejpam-6044	649	4	.	.	PUNCT
ejpam-6044	649	5	.	.	PUNCT
ejpam-6044	650	1	.	.	PUNCT
ejpam-6044	651	1	,	,	PUNCT
ejpam-6044	651	2	ςn	ςn	PROPN
ejpam-6044	651	3	,	,	PUNCT
ejpam-6044	651	4	y1	y1	NOUN
ejpam-6044	651	5	,	,	PUNCT
ejpam-6044	651	6	.	.	PUNCT
ejpam-6044	651	7	.	.	PUNCT
ejpam-6044	651	8	.	.	PUNCT
ejpam-6044	652	1	,	,	PUNCT
ejpam-6044	652	2	yk	yk	PROPN
ejpam-6044	652	3	,	,	PUNCT
ejpam-6044	652	4	.	.	PUNCT
ejpam-6044	652	5	.	.	PUNCT
ejpam-6044	653	1	.	.	PUNCT
ejpam-6044	654	1	,	,	PUNCT
ejpam-6044	654	2	yn	yn	PROPN
ejpam-6044	654	3	∈	∈	PROPN
ejpam-6044	654	4	r.	r.	PROPN
ejpam-6044	654	5	substituting	substitute	VERB
ejpam-6044	654	6	yk	yk	PROPN
ejpam-6044	654	7	=	=	PUNCT
ejpam-6044	654	8	ykz	ykz	PROPN
ejpam-6044	654	9	,	,	PUNCT
ejpam-6044	654	10	where	where	SCONJ
ejpam-6044	654	11	z	z	PROPN
ejpam-6044	654	12	∈	∈	PROPN
ejpam-6044	654	13	r	r	NOUN
ejpam-6044	654	14	,	,	PUNCT
ejpam-6044	654	15	we	we	PRON
ejpam-6044	654	16	obtain	obtain	VERB
ejpam-6044	654	17	f	f	PROPN
ejpam-6044	654	18	(	(	PUNCT
ejpam-6044	654	19	ς1	ς1	NOUN
ejpam-6044	654	20	,	,	PUNCT
ejpam-6044	654	21	.	.	PUNCT
ejpam-6044	654	22	.	.	PUNCT
ejpam-6044	655	1	.	.	PUNCT
ejpam-6044	656	1	,	,	PUNCT
ejpam-6044	656	2	ςk	ςk	NUM
ejpam-6044	656	3	,	,	PUNCT
ejpam-6044	656	4	.	.	PUNCT
ejpam-6044	656	5	.	.	PUNCT
ejpam-6044	657	1	.	.	PUNCT
ejpam-6044	658	1	,	,	PUNCT
ejpam-6044	658	2	ςn)yi	ςn)yi	PUNCT
ejpam-6044	659	1	=	=	SYM
ejpam-6044	659	2	ςif	ςif	X
ejpam-6044	659	3	(	(	PUNCT
ejpam-6044	659	4	y1	y1	INTJ
ejpam-6044	659	5	,	,	PUNCT
ejpam-6044	659	6	.	.	PUNCT
ejpam-6044	659	7	.	.	PUNCT
ejpam-6044	659	8	.	.	PUNCT
ejpam-6044	660	1	,	,	PUNCT
ejpam-6044	660	2	ykz	ykz	INTJ
ejpam-6044	660	3	,	,	PUNCT
ejpam-6044	660	4	.	.	PUNCT
ejpam-6044	660	5	.	.	PUNCT
ejpam-6044	661	1	.	.	PUNCT
ejpam-6044	662	1	,	,	PUNCT
ejpam-6044	662	2	yn	yn	PROPN
ejpam-6044	662	3	)	)	PUNCT
ejpam-6044	662	4	.	.	PUNCT
ejpam-6044	663	1	using	use	VERB
ejpam-6044	663	2	definition	definition	NOUN
ejpam-6044	663	3	on	on	ADP
ejpam-6044	663	4	the	the	DET
ejpam-6044	663	5	right	right	ADJ
ejpam-6044	663	6	hand	hand	NOUN
ejpam-6044	663	7	side	side	NOUN
ejpam-6044	663	8	,	,	PUNCT
ejpam-6044	663	9	we	we	PRON
ejpam-6044	663	10	have	have	VERB
ejpam-6044	663	11	f	f	PROPN
ejpam-6044	663	12	(	(	PUNCT
ejpam-6044	663	13	ς1	ς1	NOUN
ejpam-6044	663	14	,	,	PUNCT
ejpam-6044	663	15	.	.	PUNCT
ejpam-6044	663	16	.	.	PUNCT
ejpam-6044	664	1	.	.	PUNCT
ejpam-6044	665	1	,	,	PUNCT
ejpam-6044	665	2	ςk	ςk	NUM
ejpam-6044	665	3	,	,	PUNCT
ejpam-6044	665	4	.	.	PUNCT
ejpam-6044	665	5	.	.	PUNCT
ejpam-6044	666	1	.	.	PUNCT
ejpam-6044	667	1	,	,	PUNCT
ejpam-6044	667	2	ςn)yi	ςn)yi	PUNCT
ejpam-6044	668	1	=	=	SYM
ejpam-6044	668	2	ςif	ςif	X
ejpam-6044	668	3	(	(	PUNCT
ejpam-6044	668	4	y1	y1	INTJ
ejpam-6044	668	5	,	,	PUNCT
ejpam-6044	668	6	.	.	PUNCT
ejpam-6044	668	7	.	.	PUNCT
ejpam-6044	668	8	.	.	PUNCT
ejpam-6044	669	1	,	,	PUNCT
ejpam-6044	669	2	yk	yk	PROPN
ejpam-6044	669	3	,	,	PUNCT
ejpam-6044	669	4	.	.	PUNCT
ejpam-6044	669	5	.	.	PUNCT
ejpam-6044	670	1	.	.	PUNCT
ejpam-6044	671	1	,	,	PUNCT
ejpam-6044	671	2	yn)α(z	yn)α(z	PROPN
ejpam-6044	671	3	)	)	PUNCT
ejpam-6044	672	1	+	+	CCONJ
ejpam-6044	672	2	ςiy	ςiy	PROPN
ejpam-6044	672	3	∗	∗	PROPN
ejpam-6044	672	4	kd(y1	kd(y1	PROPN
ejpam-6044	672	5	,	,	PUNCT
ejpam-6044	672	6	.	.	PUNCT
ejpam-6044	672	7	.	.	PUNCT
ejpam-6044	673	1	.	.	PUNCT
ejpam-6044	674	1	,	,	PUNCT
ejpam-6044	675	1	z	z	X
ejpam-6044	675	2	,	,	PUNCT
ejpam-6044	675	3	.	.	PUNCT
ejpam-6044	675	4	.	.	PUNCT
ejpam-6044	675	5	.	.	PUNCT
ejpam-6044	676	1	,	,	PUNCT
ejpam-6044	676	2	yn	yn	PROPN
ejpam-6044	676	3	)	)	PUNCT
ejpam-6044	676	4	.	.	PUNCT
ejpam-6044	677	1	since	since	SCONJ
ejpam-6044	677	2	α	α	PROPN
ejpam-6044	677	3	is	be	AUX
ejpam-6044	677	4	an	an	DET
ejpam-6044	677	5	automorphism	automorphism	NOUN
ejpam-6044	677	6	,	,	PUNCT
ejpam-6044	677	7	we	we	PRON
ejpam-6044	677	8	may	may	AUX
ejpam-6044	677	9	put	put	VERB
ejpam-6044	677	10	z	z	NOUN
ejpam-6044	677	11	=	=	SYM
ejpam-6044	677	12	α−1(z	α−1(z	PROPN
ejpam-6044	677	13	)	)	PUNCT
ejpam-6044	677	14	to	to	PART
ejpam-6044	677	15	have	have	VERB
ejpam-6044	677	16	f	f	PROPN
ejpam-6044	677	17	(	(	PUNCT
ejpam-6044	677	18	ς1	ς1	NOUN
ejpam-6044	677	19	,	,	PUNCT
ejpam-6044	677	20	.	.	PUNCT
ejpam-6044	677	21	.	.	PUNCT
ejpam-6044	678	1	.	.	PUNCT
ejpam-6044	679	1	,	,	PUNCT
ejpam-6044	679	2	ςk	ςk	NUM
ejpam-6044	679	3	,	,	PUNCT
ejpam-6044	679	4	.	.	PUNCT
ejpam-6044	679	5	.	.	PUNCT
ejpam-6044	680	1	.	.	PUNCT
ejpam-6044	681	1	,	,	PUNCT
ejpam-6044	681	2	ςn)yi	ςn)yi	PUNCT
ejpam-6044	682	1	=	=	SYM
ejpam-6044	682	2	ςif	ςif	X
ejpam-6044	682	3	(	(	PUNCT
ejpam-6044	682	4	y1	y1	INTJ
ejpam-6044	682	5	,	,	PUNCT
ejpam-6044	682	6	.	.	PUNCT
ejpam-6044	682	7	.	.	PUNCT
ejpam-6044	682	8	.	.	PUNCT
ejpam-6044	683	1	,	,	PUNCT
ejpam-6044	683	2	yk	yk	PROPN
ejpam-6044	683	3	,	,	PUNCT
ejpam-6044	683	4	.	.	PUNCT
ejpam-6044	683	5	.	.	PUNCT
ejpam-6044	684	1	.	.	PUNCT
ejpam-6044	685	1	,	,	PUNCT
ejpam-6044	685	2	yn)z	yn)z	PROPN
ejpam-6044	685	3	+	+	CCONJ
ejpam-6044	685	4	ςiy	ςiy	PROPN
ejpam-6044	685	5	∗	∗	PROPN
ejpam-6044	685	6	kd(y1	kd(y1	PROPN
ejpam-6044	685	7	,	,	PUNCT
ejpam-6044	685	8	.	.	PUNCT
ejpam-6044	685	9	.	.	PUNCT
ejpam-6044	686	1	.	.	PUNCT
ejpam-6044	687	1	,	,	PUNCT
ejpam-6044	687	2	α	α	PROPN
ejpam-6044	687	3	−1(z	−1(z	PROPN
ejpam-6044	687	4	)	)	PUNCT
ejpam-6044	687	5	,	,	PUNCT
ejpam-6044	687	6	.	.	PUNCT
ejpam-6044	687	7	.	.	PUNCT
ejpam-6044	688	1	.	.	PUNCT
ejpam-6044	689	1	,	,	PUNCT
ejpam-6044	689	2	yn	yn	PROPN
ejpam-6044	689	3	)	)	PUNCT
ejpam-6044	689	4	.	.	PUNCT
ejpam-6044	690	1	rearranging	rearrange	VERB
ejpam-6044	690	2	by	by	ADP
ejpam-6044	690	3	putting	put	VERB
ejpam-6044	690	4	yi	yi	NOUN
ejpam-6044	690	5	=	=	PUNCT
ejpam-6044	690	6	yiz	yiz	PROPN
ejpam-6044	690	7	in	in	ADP
ejpam-6044	690	8	the	the	DET
ejpam-6044	690	9	above	above	ADJ
ejpam-6044	690	10	equation	equation	NOUN
ejpam-6044	690	11	,	,	PUNCT
ejpam-6044	690	12	we	we	PRON
ejpam-6044	690	13	get	get	VERB
ejpam-6044	690	14	f	f	PROPN
ejpam-6044	690	15	(	(	PUNCT
ejpam-6044	690	16	ς1	ς1	NOUN
ejpam-6044	690	17	,	,	PUNCT
ejpam-6044	690	18	.	.	PUNCT
ejpam-6044	690	19	.	.	PUNCT
ejpam-6044	691	1	.	.	PUNCT
ejpam-6044	692	1	,	,	PUNCT
ejpam-6044	692	2	ςk	ςk	NUM
ejpam-6044	692	3	,	,	PUNCT
ejpam-6044	692	4	.	.	PUNCT
ejpam-6044	692	5	.	.	PUNCT
ejpam-6044	693	1	.	.	PUNCT
ejpam-6044	694	1	,	,	PUNCT
ejpam-6044	695	1	ςn)yi	ςn)yi	NUM
ejpam-6044	695	2	−	−	PROPN
ejpam-6044	696	1	ςif	ςif	NOUN
ejpam-6044	696	2	(	(	PUNCT
ejpam-6044	696	3	y1	y1	NOUN
ejpam-6044	696	4	,	,	PUNCT
ejpam-6044	696	5	.	.	PUNCT
ejpam-6044	696	6	.	.	PUNCT
ejpam-6044	696	7	.	.	PUNCT
ejpam-6044	697	1	,	,	PUNCT
ejpam-6044	697	2	yk	yk	PROPN
ejpam-6044	697	3	,	,	PUNCT
ejpam-6044	697	4	.	.	PUNCT
ejpam-6044	697	5	.	.	PUNCT
ejpam-6044	698	1	.	.	PUNCT
ejpam-6044	699	1	,	,	PUNCT
ejpam-6044	699	2	yn)z	yn)z	PROPN
ejpam-6044	699	3	=	=	SYM
ejpam-6044	699	4	ςiy	ςiy	PROPN
ejpam-6044	699	5	∗	∗	PROPN
ejpam-6044	699	6	kd(y1	kd(y1	PROPN
ejpam-6044	699	7	,	,	PUNCT
ejpam-6044	699	8	.	.	PUNCT
ejpam-6044	699	9	.	.	PUNCT
ejpam-6044	700	1	.	.	PUNCT
ejpam-6044	701	1	,	,	PUNCT
ejpam-6044	701	2	α	α	PROPN
ejpam-6044	701	3	−1(z	−1(z	PROPN
ejpam-6044	701	4	)	)	PUNCT
ejpam-6044	701	5	,	,	PUNCT
ejpam-6044	701	6	.	.	PUNCT
ejpam-6044	701	7	.	.	PUNCT
ejpam-6044	702	1	.	.	PUNCT
ejpam-6044	703	1	,	,	PUNCT
ejpam-6044	703	2	yn	yn	PROPN
ejpam-6044	703	3	)	)	PUNCT
ejpam-6044	703	4	,	,	PUNCT
ejpam-6044	703	5	for	for	ADP
ejpam-6044	703	6	every	every	DET
ejpam-6044	703	7	ς1	ς1	NOUN
ejpam-6044	703	8	,	,	PUNCT
ejpam-6044	703	9	.	.	PUNCT
ejpam-6044	703	10	.	.	PUNCT
ejpam-6044	704	1	.	.	PUNCT
ejpam-6044	705	1	,	,	PUNCT
ejpam-6044	705	2	ςk	ςk	NUM
ejpam-6044	705	3	,	,	PUNCT
ejpam-6044	705	4	.	.	PUNCT
ejpam-6044	705	5	.	.	PUNCT
ejpam-6044	706	1	.	.	PUNCT
ejpam-6044	707	1	,	,	PUNCT
ejpam-6044	707	2	ςn	ςn	PROPN
ejpam-6044	707	3	,	,	PUNCT
ejpam-6044	707	4	y1	y1	NOUN
ejpam-6044	707	5	,	,	PUNCT
ejpam-6044	707	6	.	.	PUNCT
ejpam-6044	707	7	.	.	PUNCT
ejpam-6044	707	8	.	.	PUNCT
ejpam-6044	708	1	,	,	PUNCT
ejpam-6044	708	2	yk	yk	PROPN
ejpam-6044	708	3	,	,	PUNCT
ejpam-6044	708	4	.	.	PUNCT
ejpam-6044	708	5	.	.	PUNCT
ejpam-6044	709	1	.	.	PUNCT
ejpam-6044	710	1	,	,	PUNCT
ejpam-6044	710	2	yn	yn	PROPN
ejpam-6044	710	3	∈	∈	PROPN
ejpam-6044	710	4	r.	r.	NOUN
ejpam-6044	710	5	taking	take	VERB
ejpam-6044	710	6	z	z	PROPN
ejpam-6044	710	7	as	as	ADP
ejpam-6044	710	8	a	a	DET
ejpam-6044	710	9	common	common	ADJ
ejpam-6044	710	10	factor	factor	NOUN
ejpam-6044	710	11	and	and	CCONJ
ejpam-6044	710	12	using	use	VERB
ejpam-6044	710	13	(	(	PUNCT
ejpam-6044	710	14	20	20	NUM
ejpam-6044	710	15	)	)	PUNCT
ejpam-6044	710	16	,	,	PUNCT
ejpam-6044	710	17	we	we	PRON
ejpam-6044	710	18	have	have	VERB
ejpam-6044	710	19	ςiy	ςiy	PROPN
ejpam-6044	710	20	∗	∗	PROPN
ejpam-6044	710	21	kd(y1	kd(y1	NOUN
ejpam-6044	710	22	,	,	PUNCT
ejpam-6044	710	23	.	.	PUNCT
ejpam-6044	710	24	.	.	PUNCT
ejpam-6044	711	1	.	.	PUNCT
ejpam-6044	712	1	,	,	PUNCT
ejpam-6044	712	2	α	α	PROPN
ejpam-6044	712	3	−1(z	−1(z	PROPN
ejpam-6044	712	4	)	)	PUNCT
ejpam-6044	712	5	,	,	PUNCT
ejpam-6044	712	6	.	.	PUNCT
ejpam-6044	712	7	.	.	PUNCT
ejpam-6044	713	1	.	.	PUNCT
ejpam-6044	714	1	,	,	PUNCT
ejpam-6044	714	2	yn	yn	PROPN
ejpam-6044	714	3	)	)	PUNCT
ejpam-6044	714	4	=	=	SYM
ejpam-6044	714	5	0	0	NUM
ejpam-6044	714	6	,	,	PUNCT
ejpam-6044	714	7	for	for	ADP
ejpam-6044	714	8	every	every	DET
ejpam-6044	714	9	ςi	ςi	PROPN
ejpam-6044	714	10	,	,	PUNCT
ejpam-6044	714	11	y1	y1	PROPN
ejpam-6044	714	12	,	,	PUNCT
ejpam-6044	714	13	.	.	PUNCT
ejpam-6044	714	14	.	.	PUNCT
ejpam-6044	714	15	.	.	PUNCT
ejpam-6044	715	1	,	,	PUNCT
ejpam-6044	715	2	yk	yk	PROPN
ejpam-6044	715	3	,	,	PUNCT
ejpam-6044	715	4	.	.	PUNCT
ejpam-6044	715	5	.	.	PUNCT
ejpam-6044	716	1	.	.	PUNCT
ejpam-6044	717	1	,	,	PUNCT
ejpam-6044	717	2	yn	yn	PROPN
ejpam-6044	717	3	∈	∈	PROPN
ejpam-6044	717	4	r.	r.	PROPN
ejpam-6044	717	5	it	it	PRON
ejpam-6044	717	6	follows	follow	VERB
ejpam-6044	717	7	that	that	DET
ejpam-6044	717	8	ςiykd(y1	ςiykd(y1	NOUN
ejpam-6044	717	9	,	,	PUNCT
ejpam-6044	717	10	.	.	PUNCT
ejpam-6044	717	11	.	.	PUNCT
ejpam-6044	718	1	.	.	PUNCT
ejpam-6044	719	1	,	,	PUNCT
ejpam-6044	720	1	z	z	X
ejpam-6044	720	2	,	,	PUNCT
ejpam-6044	720	3	.	.	PUNCT
ejpam-6044	720	4	.	.	PUNCT
ejpam-6044	720	5	.	.	PUNCT
ejpam-6044	721	1	,	,	PUNCT
ejpam-6044	721	2	yn	yn	PROPN
ejpam-6044	721	3	)	)	PUNCT
ejpam-6044	721	4	=	=	SYM
ejpam-6044	722	1	0	0	X
ejpam-6044	722	2	.	.	PUNCT
ejpam-6044	723	1	for	for	ADP
ejpam-6044	723	2	every	every	DET
ejpam-6044	723	3	ςi	ςi	PROPN
ejpam-6044	723	4	,	,	PUNCT
ejpam-6044	723	5	y1	y1	PROPN
ejpam-6044	723	6	,	,	PUNCT
ejpam-6044	723	7	.	.	PUNCT
ejpam-6044	723	8	.	.	PUNCT
ejpam-6044	723	9	.	.	PUNCT
ejpam-6044	724	1	,	,	PUNCT
ejpam-6044	724	2	yk	yk	PROPN
ejpam-6044	724	3	,	,	PUNCT
ejpam-6044	724	4	.	.	PUNCT
ejpam-6044	724	5	.	.	PUNCT
ejpam-6044	725	1	.	.	PUNCT
ejpam-6044	726	1	,	,	PUNCT
ejpam-6044	726	2	yn	yn	PROPN
ejpam-6044	726	3	∈	∈	PROPN
ejpam-6044	726	4	r.	r.	PROPN
ejpam-6044	726	5	a	a	DET
ejpam-6044	726	6	simple	simple	ADJ
ejpam-6044	726	7	manipulation	manipulation	NOUN
ejpam-6044	726	8	yields	yield	VERB
ejpam-6044	726	9	that	that	DET
ejpam-6044	726	10	d(y1	d(y1	NOUN
ejpam-6044	726	11	,	,	PUNCT
ejpam-6044	726	12	.	.	PUNCT
ejpam-6044	726	13	.	.	PUNCT
ejpam-6044	726	14	.	.	PUNCT
ejpam-6044	727	1	,	,	PUNCT
ejpam-6044	728	1	z	z	X
ejpam-6044	728	2	,	,	PUNCT
ejpam-6044	728	3	.	.	PUNCT
ejpam-6044	728	4	.	.	PUNCT
ejpam-6044	728	5	.	.	PUNCT
ejpam-6044	729	1	,	,	PUNCT
ejpam-6044	729	2	yn)ςid(y1	yn)ςid(y1	PROPN
ejpam-6044	729	3	,	,	PUNCT
ejpam-6044	729	4	.	.	PUNCT
ejpam-6044	729	5	.	.	PUNCT
ejpam-6044	730	1	.	.	PUNCT
ejpam-6044	731	1	,	,	PUNCT
ejpam-6044	732	1	z	z	X
ejpam-6044	732	2	,	,	PUNCT
ejpam-6044	732	3	.	.	PUNCT
ejpam-6044	732	4	.	.	PUNCT
ejpam-6044	732	5	.	.	PUNCT
ejpam-6044	733	1	,	,	PUNCT
ejpam-6044	733	2	yn)ykd(y1	yn)ykd(y1	NOUN
ejpam-6044	733	3	,	,	PUNCT
ejpam-6044	733	4	.	.	PUNCT
ejpam-6044	733	5	.	.	PUNCT
ejpam-6044	734	1	.	.	PUNCT
ejpam-6044	735	1	,	,	PUNCT
ejpam-6044	736	1	z	z	X
ejpam-6044	736	2	,	,	PUNCT
ejpam-6044	736	3	.	.	PUNCT
ejpam-6044	736	4	.	.	PUNCT
ejpam-6044	736	5	.	.	PUNCT
ejpam-6044	737	1	,	,	PUNCT
ejpam-6044	737	2	yn)ςi	yn)ςi	PUNCT
ejpam-6044	738	1	=	=	SYM
ejpam-6044	738	2	0	0	NUM
ejpam-6044	738	3	,	,	PUNCT
ejpam-6044	738	4	for	for	ADP
ejpam-6044	738	5	every	every	DET
ejpam-6044	738	6	ςi	ςi	PROPN
ejpam-6044	738	7	,	,	PUNCT
ejpam-6044	738	8	y1	y1	PROPN
ejpam-6044	738	9	,	,	PUNCT
ejpam-6044	738	10	..	..	PUNCT
ejpam-6044	738	11	,	,	PUNCT
ejpam-6044	738	12	yk	yk	PROPN
ejpam-6044	738	13	,	,	PUNCT
ejpam-6044	738	14	z	z	PROPN
ejpam-6044	738	15	∈	∈	PROPN
ejpam-6044	738	16	r.	r.	NOUN
ejpam-6044	738	17	making	make	VERB
ejpam-6044	738	18	use	use	NOUN
ejpam-6044	738	19	of	of	ADP
ejpam-6044	738	20	semi	semi	ADJ
ejpam-6044	738	21	-	-	NOUN
ejpam-6044	738	22	primeness	primeness	NOUN
ejpam-6044	738	23	of	of	ADP
ejpam-6044	738	24	r	r	NOUN
ejpam-6044	738	25	,	,	PUNCT
ejpam-6044	738	26	it	it	PRON
ejpam-6044	738	27	follows	follow	VERB
ejpam-6044	738	28	that	that	DET
ejpam-6044	738	29	d(y1	d(y1	NOUN
ejpam-6044	738	30	,	,	PUNCT
ejpam-6044	738	31	.	.	PUNCT
ejpam-6044	738	32	.	.	PUNCT
ejpam-6044	739	1	.	.	PUNCT
ejpam-6044	740	1	,	,	PUNCT
ejpam-6044	741	1	z	z	X
ejpam-6044	741	2	,	,	PUNCT
ejpam-6044	741	3	.	.	PUNCT
ejpam-6044	741	4	.	.	PUNCT
ejpam-6044	741	5	.	.	PUNCT
ejpam-6044	742	1	,	,	PUNCT
ejpam-6044	742	2	yn	yn	PROPN
ejpam-6044	742	3	)	)	PUNCT
ejpam-6044	742	4	=	=	SYM
ejpam-6044	742	5	0	0	NUM
ejpam-6044	742	6	for	for	ADP
ejpam-6044	742	7	every	every	DET
ejpam-6044	742	8	y1	y1	NOUN
ejpam-6044	742	9	,	,	PUNCT
ejpam-6044	742	10	..	..	PUNCT
ejpam-6044	742	11	,	,	PUNCT
ejpam-6044	742	12	yi	yi	PROPN
ejpam-6044	742	13	,	,	PUNCT
ejpam-6044	742	14	z	z	PROPN
ejpam-6044	742	15	∈	∈	PROPN
ejpam-6044	742	16	r.	r.	NOUN
ejpam-6044	742	17	hence	hence	ADV
ejpam-6044	743	1	f	f	PROPN
ejpam-6044	743	2	acting	act	VERB
ejpam-6044	743	3	as	as	ADP
ejpam-6044	743	4	a	a	DET
ejpam-6044	743	5	left	left	ADJ
ejpam-6044	743	6	α	α	NOUN
ejpam-6044	743	7	-	-	PUNCT
ejpam-6044	743	8	centralizer	centralizer	NOUN
ejpam-6044	743	9	.	.	PUNCT
ejpam-6044	744	1	acknowledgements	acknowledgement	NOUN
ejpam-6044	744	2	the	the	DET
ejpam-6044	744	3	authors	author	NOUN
ejpam-6044	744	4	are	be	AUX
ejpam-6044	744	5	extremely	extremely	ADV
ejpam-6044	744	6	grateful	grateful	ADJ
ejpam-6044	744	7	to	to	ADP
ejpam-6044	744	8	the	the	DET
ejpam-6044	744	9	reviewers	reviewer	NOUN
ejpam-6044	744	10	and	and	CCONJ
ejpam-6044	744	11	editor	editor	NOUN
ejpam-6044	744	12	for	for	ADP
ejpam-6044	744	13	their	their	PRON
ejpam-6044	744	14	generous	generous	ADJ
ejpam-6044	744	15	suggestions	suggestion	NOUN
ejpam-6044	744	16	,	,	PUNCT
ejpam-6044	744	17	insightful	insightful	ADJ
ejpam-6044	744	18	remarks	remark	NOUN
ejpam-6044	744	19	and	and	CCONJ
ejpam-6044	744	20	recommendations	recommendation	NOUN
ejpam-6044	744	21	to	to	PART
ejpam-6044	744	22	make	make	VERB
ejpam-6044	744	23	this	this	DET
ejpam-6044	744	24	manuscript	manuscript	NOUN
ejpam-6044	744	25	well	well	ADV
ejpam-6044	744	26	organized	organize	VERB
ejpam-6044	744	27	.	.	PUNCT
ejpam-6044	745	1	competing	compete	VERB
ejpam-6044	745	2	interests	interest	NOUN
ejpam-6044	745	3	:	:	PUNCT
ejpam-6044	745	4	regarding	regard	VERB
ejpam-6044	745	5	the	the	DET
ejpam-6044	745	6	publication	publication	NOUN
ejpam-6044	745	7	of	of	ADP
ejpam-6044	745	8	this	this	DET
ejpam-6044	745	9	work	work	NOUN
ejpam-6044	745	10	,	,	PUNCT
ejpam-6044	745	11	the	the	DET
ejpam-6044	745	12	authors	author	NOUN
ejpam-6044	745	13	affirm	affirm	VERB
ejpam-6044	745	14	that	that	SCONJ
ejpam-6044	745	15	they	they	PRON
ejpam-6044	745	16	have	have	VERB
ejpam-6044	745	17	no	no	DET
ejpam-6044	745	18	conflicts	conflict	NOUN
ejpam-6044	745	19	of	of	ADP
ejpam-6044	745	20	interest	interest	NOUN
ejpam-6044	745	21	.	.	PUNCT
ejpam-6044	746	1	f.	f.	PROPN
ejpam-6044	746	2	shujat	shujat	PROPN
ejpam-6044	746	3	,	,	PUNCT
ejpam-6044	746	4	s.	s.	PROPN
ejpam-6044	746	5	alharbi	alharbi	PROPN
ejpam-6044	746	6	/	/	SYM
ejpam-6044	746	7	eur	eur	PROPN
ejpam-6044	746	8	.	.	PUNCT
ejpam-6044	747	1	j.	j.	PROPN
ejpam-6044	747	2	pure	pure	PROPN
ejpam-6044	747	3	appl	appl	PROPN
ejpam-6044	747	4	.	.	PROPN
ejpam-6044	747	5	math	math	PROPN
ejpam-6044	747	6	,	,	PUNCT
ejpam-6044	747	7	18	18	NUM
ejpam-6044	747	8	(	(	PUNCT
ejpam-6044	747	9	2	2	NUM
ejpam-6044	747	10	)	)	PUNCT
ejpam-6044	747	11	(	(	PUNCT
ejpam-6044	747	12	2025	2025	NUM
ejpam-6044	747	13	)	)	PUNCT
ejpam-6044	747	14	,	,	PUNCT
ejpam-6044	747	15	6044	6044	NUM
ejpam-6044	747	16	11	11	NUM
ejpam-6044	747	17	of	of	ADP
ejpam-6044	747	18	11	11	NUM
ejpam-6044	747	19	references	reference	NOUN
ejpam-6044	747	20	[	[	X
ejpam-6044	747	21	1	1	NUM
ejpam-6044	747	22	]	]	PUNCT
ejpam-6044	747	23	a.	a.	PROPN
ejpam-6044	747	24	ali	ali	PROPN
ejpam-6044	747	25	,	,	PUNCT
ejpam-6044	747	26	v.	v.	PROPN
ejpam-6044	747	27	de	de	X
ejpam-6044	747	28	filippis	filippis	PROPN
ejpam-6044	747	29	,	,	PUNCT
ejpam-6044	747	30	and	and	CCONJ
ejpam-6044	747	31	f.	f.	PROPN
ejpam-6044	747	32	shujat	shujat	PROPN
ejpam-6044	747	33	.	.	PUNCT
ejpam-6044	748	1	on	on	ADP
ejpam-6044	748	2	one	one	NUM
ejpam-6044	748	3	sided	sided	ADJ
ejpam-6044	748	4	ideals	ideal	NOUN
ejpam-6044	748	5	of	of	ADP
ejpam-6044	748	6	a	a	DET
ejpam-6044	748	7	semiprime	semiprime	NOUN
ejpam-6044	748	8	ring	ring	NOUN
ejpam-6044	748	9	with	with	ADP
ejpam-6044	748	10	generalized	generalized	ADJ
ejpam-6044	748	11	derivations	derivation	NOUN
ejpam-6044	748	12	.	.	PUNCT
ejpam-6044	749	1	aequationes	aequatione	NOUN
ejpam-6044	749	2	mathematicae	mathematicae	PROPN
ejpam-6044	749	3	,	,	PUNCT
ejpam-6044	749	4	86(1	86(1	PROPN
ejpam-6044	749	5	-	-	PUNCT
ejpam-6044	749	6	2):1–9	2):1–9	NOUN
ejpam-6044	749	7	,	,	PUNCT
ejpam-6044	749	8	2012	2012	NUM
ejpam-6044	749	9	.	.	PUNCT
ejpam-6044	750	1	[	[	X
ejpam-6044	750	2	2	2	NUM
ejpam-6044	750	3	]	]	PUNCT
ejpam-6044	750	4	a.	a.	NOUN
ejpam-6044	750	5	z.	z.	PROPN
ejpam-6044	750	6	ansari	ansari	PROPN
ejpam-6044	750	7	,	,	PUNCT
ejpam-6044	750	8	f.	f.	PROPN
ejpam-6044	750	9	shujat	shujat	PROPN
ejpam-6044	750	10	,	,	PUNCT
ejpam-6044	750	11	a.	a.	PROPN
ejpam-6044	750	12	kamil	kamil	PROPN
ejpam-6044	750	13	,	,	PUNCT
ejpam-6044	750	14	and	and	CCONJ
ejpam-6044	750	15	a.	a.	NOUN
ejpam-6044	750	16	fallatah	fallatah	PROPN
ejpam-6044	750	17	.	.	PUNCT
ejpam-6044	751	1	jordan	jordan	PROPN
ejpam-6044	751	2	φ	φ	PROPN
ejpam-6044	751	3	-	-	PUNCT
ejpam-6044	751	4	centralizers	centralizer	NOUN
ejpam-6044	751	5	on	on	ADP
ejpam-6044	751	6	semiprime	semiprime	NOUN
ejpam-6044	751	7	and	and	CCONJ
ejpam-6044	751	8	involution	involution	NOUN
ejpam-6044	751	9	rings	ring	NOUN
ejpam-6044	751	10	.	.	PUNCT
ejpam-6044	752	1	european	european	PROPN
ejpam-6044	752	2	journal	journal	PROPN
ejpam-6044	752	3	of	of	ADP
ejpam-6044	752	4	pure	pure	ADJ
ejpam-6044	752	5	and	and	CCONJ
ejpam-6044	752	6	applied	applied	ADJ
ejpam-6044	752	7	mathematics	mathematic	NOUN
ejpam-6044	752	8	,	,	PUNCT
ejpam-6044	752	9	18(1):5493–5508	18(1):5493–5508	NUM
ejpam-6044	752	10	,	,	PUNCT
ejpam-6044	752	11	2025	2025	NUM
ejpam-6044	752	12	.	.	PUNCT
ejpam-6044	753	1	[	[	X
ejpam-6044	753	2	3	3	X
ejpam-6044	753	3	]	]	X
ejpam-6044	753	4	s.	s.	PROPN
ejpam-6044	753	5	ali	ali	PROPN
ejpam-6044	753	6	and	and	CCONJ
ejpam-6044	753	7	m.	m.	PROPN
ejpam-6044	753	8	s.	s.	PROPN
ejpam-6044	753	9	khan	khan	PROPN
ejpam-6044	753	10	.	.	PUNCT
ejpam-6044	754	1	on	on	ADP
ejpam-6044	754	2	∗-bimultipliers	∗-bimultiplier	NOUN
ejpam-6044	754	3	,	,	PUNCT
ejpam-6044	754	4	generalized	generalized	ADJ
ejpam-6044	754	5	∗-biderivations	∗-biderivation	NOUN
ejpam-6044	754	6	and	and	CCONJ
ejpam-6044	754	7	related	related	ADJ
ejpam-6044	754	8	mappings	mapping	NOUN
ejpam-6044	754	9	.	.	PUNCT
ejpam-6044	755	1	kyungpook	kyungpook	PROPN
ejpam-6044	755	2	mathematical	mathematical	PROPN
ejpam-6044	755	3	journal	journal	PROPN
ejpam-6044	755	4	,	,	PUNCT
ejpam-6044	755	5	51(3):301–309	51(3):301–309	PROPN
ejpam-6044	755	6	,	,	PUNCT
ejpam-6044	755	7	2011	2011	NUM
ejpam-6044	755	8	.	.	PUNCT
ejpam-6044	756	1	[	[	X
ejpam-6044	756	2	4	4	X
ejpam-6044	756	3	]	]	PUNCT
ejpam-6044	756	4	a.	a.	NOUN
ejpam-6044	756	5	z.	z.	PROPN
ejpam-6044	756	6	ansari	ansari	PROPN
ejpam-6044	756	7	and	and	CCONJ
ejpam-6044	756	8	f.	f.	PROPN
ejpam-6044	756	9	shujat	shujat	PROPN
ejpam-6044	756	10	.	.	PUNCT
ejpam-6044	757	1	jordan	jordan	PROPN
ejpam-6044	757	2	∗-derivations	∗-derivations	PROPN
ejpam-6044	757	3	on	on	ADP
ejpam-6044	757	4	standard	standard	ADJ
ejpam-6044	757	5	operator	operator	NOUN
ejpam-6044	757	6	algebras	algebra	NOUN
ejpam-6044	757	7	.	.	PUNCT
ejpam-6044	758	1	filomat	filomat	PROPN
ejpam-6044	758	2	,	,	PUNCT
ejpam-6044	758	3	37(1):37–41	37(1):37–41	NUM
ejpam-6044	758	4	,	,	PUNCT
ejpam-6044	758	5	2023	2023	NUM
ejpam-6044	758	6	.	.	PUNCT
ejpam-6044	759	1	[	[	X
ejpam-6044	759	2	5	5	NUM
ejpam-6044	759	3	]	]	X
ejpam-6044	759	4	n.	n.	NOUN
ejpam-6044	759	5	divinsky	divinsky	NOUN
ejpam-6044	759	6	.	.	PUNCT
ejpam-6044	760	1	on	on	ADP
ejpam-6044	760	2	commuting	commute	VERB
ejpam-6044	760	3	automorphisms	automorphism	NOUN
ejpam-6044	760	4	of	of	ADP
ejpam-6044	760	5	rings	ring	NOUN
ejpam-6044	760	6	.	.	PUNCT
ejpam-6044	761	1	transactions	transaction	NOUN
ejpam-6044	761	2	of	of	ADP
ejpam-6044	761	3	the	the	DET
ejpam-6044	761	4	royal	royal	ADJ
ejpam-6044	761	5	society	society	NOUN
ejpam-6044	761	6	of	of	ADP
ejpam-6044	761	7	canada	canada	PROPN
ejpam-6044	761	8	,	,	PUNCT
ejpam-6044	761	9	section	section	NOUN
ejpam-6044	761	10	iii	iii	PROPN
ejpam-6044	761	11	,	,	PUNCT
ejpam-6044	761	12	49:19–22	49:19–22	NUM
ejpam-6044	761	13	,	,	PUNCT
ejpam-6044	761	14	1955	1955	NUM
ejpam-6044	761	15	.	.	PUNCT
ejpam-6044	762	1	[	[	X
ejpam-6044	762	2	6	6	NUM
ejpam-6044	762	3	]	]	PUNCT
ejpam-6044	762	4	j.	j.	PROPN
ejpam-6044	762	5	luh	luh	PROPN
ejpam-6044	762	6	.	.	PUNCT
ejpam-6044	763	1	a	a	DET
ejpam-6044	763	2	note	note	NOUN
ejpam-6044	763	3	on	on	ADP
ejpam-6044	763	4	commuting	commute	VERB
ejpam-6044	763	5	automorphisms	automorphism	NOUN
ejpam-6044	763	6	of	of	ADP
ejpam-6044	763	7	rings	ring	NOUN
ejpam-6044	763	8	.	.	PUNCT
ejpam-6044	764	1	the	the	DET
ejpam-6044	764	2	american	american	PROPN
ejpam-6044	764	3	mathematical	mathematical	PROPN
ejpam-6044	764	4	monthly	monthly	ADV
ejpam-6044	764	5	,	,	PUNCT
ejpam-6044	764	6	77(1):61–62	77(1):61–62	NUM
ejpam-6044	764	7	,	,	PUNCT
ejpam-6044	764	8	1970	1970	NUM
ejpam-6044	764	9	.	.	PUNCT
ejpam-6044	765	1	[	[	X
ejpam-6044	765	2	7	7	X
ejpam-6044	765	3	]	]	X
ejpam-6044	765	4	b.	b.	NOUN
ejpam-6044	765	5	zalar	zalar	PROPN
ejpam-6044	765	6	.	.	PUNCT
ejpam-6044	766	1	on	on	ADP
ejpam-6044	766	2	multipliers	multiplier	NOUN
ejpam-6044	766	3	of	of	ADP
ejpam-6044	766	4	semiprime	semiprime	NOUN
ejpam-6044	766	5	rings	ring	NOUN
ejpam-6044	766	6	.	.	PUNCT
ejpam-6044	767	1	commentationes	commentatione	NOUN
ejpam-6044	767	2	mathematicae	mathematicae	VERB
ejpam-6044	767	3	universitatis	universitatis	PROPN
ejpam-6044	767	4	carolinae	carolinae	PROPN
ejpam-6044	767	5	,	,	PUNCT
ejpam-6044	767	6	32(4):609–614	32(4):609–614	NUM
ejpam-6044	767	7	,	,	PUNCT
ejpam-6044	767	8	1991	1991	NUM
ejpam-6044	767	9	.	.	PUNCT
ejpam-6044	768	1	[	[	X
ejpam-6044	768	2	8	8	NUM
ejpam-6044	768	3	]	]	X
ejpam-6044	768	4	b.	b.	PROPN
ejpam-6044	768	5	e.	e.	PROPN
ejpam-6044	768	6	johnson	johnson	PROPN
ejpam-6044	768	7	.	.	PUNCT
ejpam-6044	769	1	an	an	DET
ejpam-6044	769	2	introduction	introduction	NOUN
ejpam-6044	769	3	to	to	ADP
ejpam-6044	769	4	the	the	DET
ejpam-6044	769	5	theory	theory	NOUN
ejpam-6044	769	6	of	of	ADP
ejpam-6044	769	7	centralizers	centralizer	NOUN
ejpam-6044	769	8	.	.	PUNCT
ejpam-6044	770	1	proceedings	proceeding	NOUN
ejpam-6044	770	2	of	of	ADP
ejpam-6044	770	3	the	the	DET
ejpam-6044	770	4	london	london	PROPN
ejpam-6044	770	5	mathematical	mathematical	ADJ
ejpam-6044	770	6	society	society	NOUN
ejpam-6044	770	7	,	,	PUNCT
ejpam-6044	770	8	14(3):299–320	14(3):299–320	PROPN
ejpam-6044	770	9	,	,	PUNCT
ejpam-6044	770	10	1964	1964	NUM
ejpam-6044	770	11	.	.	PUNCT
ejpam-6044	771	1	[	[	X
ejpam-6044	771	2	9	9	NUM
ejpam-6044	771	3	]	]	SYM
ejpam-6044	771	4	a.	a.	NOUN
ejpam-6044	771	5	n.	n.	PROPN
ejpam-6044	771	6	khan	khan	PROPN
ejpam-6044	771	7	and	and	CCONJ
ejpam-6044	771	8	s.	s.	PROPN
ejpam-6044	771	9	ali	ali	PROPN
ejpam-6044	771	10	.	.	PROPN
ejpam-6044	771	11	involution	involution	PROPN
ejpam-6044	771	12	on	on	ADP
ejpam-6044	771	13	prime	prime	ADJ
ejpam-6044	771	14	rings	ring	NOUN
ejpam-6044	771	15	with	with	ADP
ejpam-6044	771	16	endomorphisms	endomorphism	NOUN
ejpam-6044	771	17	.	.	PUNCT
ejpam-6044	772	1	aims	aim	VERB
ejpam-6044	772	2	mathematics	mathematic	NOUN
ejpam-6044	772	3	,	,	PUNCT
ejpam-6044	772	4	5(4):3274–3283	5(4):3274–3283	NUM
ejpam-6044	772	5	,	,	PUNCT
ejpam-6044	772	6	2020	2020	NUM
ejpam-6044	772	7	.	.	PUNCT
ejpam-6044	773	1	[	[	X
ejpam-6044	773	2	10	10	NUM
ejpam-6044	773	3	]	]	X
ejpam-6044	773	4	r.	r.	PROPN
ejpam-6044	773	5	larsen	larsen	PROPN
ejpam-6044	773	6	.	.	PUNCT
ejpam-6044	774	1	an	an	DET
ejpam-6044	774	2	introduction	introduction	NOUN
ejpam-6044	774	3	to	to	ADP
ejpam-6044	774	4	the	the	DET
ejpam-6044	774	5	theory	theory	NOUN
ejpam-6044	774	6	of	of	ADP
ejpam-6044	774	7	multipliers	multiplier	NOUN
ejpam-6044	774	8	.	.	PUNCT
ejpam-6044	775	1	springer	springer	NOUN
ejpam-6044	775	2	-	-	PUNCT
ejpam-6044	775	3	verlag	verlag	PROPN
ejpam-6044	775	4	,	,	PUNCT
ejpam-6044	775	5	berlin	berlin	PROPN
ejpam-6044	775	6	,	,	PUNCT
ejpam-6044	775	7	1971	1971	NUM
ejpam-6044	775	8	.	.	PUNCT
ejpam-6044	776	1	[	[	X
ejpam-6044	776	2	11	11	NUM
ejpam-6044	776	3	]	]	PUNCT
ejpam-6044	776	4	j.	j.	PROPN
ejpam-6044	776	5	k.	k.	PROPN
ejpam-6044	776	6	wang	wang	PROPN
ejpam-6044	776	7	.	.	PUNCT
ejpam-6044	777	1	multipliers	multiplier	NOUN
ejpam-6044	777	2	of	of	ADP
ejpam-6044	777	3	commutative	commutative	ADJ
ejpam-6044	777	4	banach	banach	NOUN
ejpam-6044	777	5	algebras	algebra	NOUN
ejpam-6044	777	6	.	.	PUNCT
ejpam-6044	778	1	pacific	pacific	PROPN
ejpam-6044	778	2	journal	journal	PROPN
ejpam-6044	778	3	of	of	ADP
ejpam-6044	778	4	mathematics	mathematic	NOUN
ejpam-6044	778	5	,	,	PUNCT
ejpam-6044	778	6	11(3):1131–1149	11(3):1131–1149	NUM
ejpam-6044	778	7	,	,	PUNCT
ejpam-6044	778	8	1961	1961	NUM
ejpam-6044	778	9	.	.	PUNCT
ejpam-6044	779	1	[	[	X
ejpam-6044	779	2	12	12	NUM
ejpam-6044	779	3	]	]	PUNCT
ejpam-6044	779	4	m.	m.	NOUN
ejpam-6044	779	5	brešar	brešar	PROPN
ejpam-6044	779	6	and	and	CCONJ
ejpam-6044	779	7	j.	j.	PROPN
ejpam-6044	779	8	vukman	vukman	PROPN
ejpam-6044	779	9	.	.	PUNCT
ejpam-6044	780	1	on	on	ADP
ejpam-6044	780	2	some	some	DET
ejpam-6044	780	3	additive	additive	ADJ
ejpam-6044	780	4	mappings	mapping	NOUN
ejpam-6044	780	5	in	in	ADP
ejpam-6044	780	6	rings	ring	NOUN
ejpam-6044	780	7	with	with	ADP
ejpam-6044	780	8	involution	involution	NOUN
ejpam-6044	780	9	.	.	PUNCT
ejpam-6044	781	1	aequationes	aequatione	NOUN
ejpam-6044	781	2	mathematicae	mathematicae	PROPN
ejpam-6044	781	3	,	,	PUNCT
ejpam-6044	781	4	38(2	38(2	NOUN
ejpam-6044	781	5	-	-	SYM
ejpam-6044	781	6	3):178–185	3):178–185	NUM
ejpam-6044	781	7	,	,	PUNCT
ejpam-6044	781	8	1989	1989	NUM
ejpam-6044	781	9	.	.	PUNCT
ejpam-6044	782	1	[	[	X
ejpam-6044	782	2	13	13	NUM
ejpam-6044	782	3	]	]	PUNCT
ejpam-6044	782	4	j.	j.	PROPN
ejpam-6044	782	5	h.	h.	PROPN
ejpam-6044	782	6	mayne	mayne	PROPN
ejpam-6044	782	7	.	.	PUNCT
ejpam-6044	783	1	centralizing	centralize	VERB
ejpam-6044	783	2	automorphisms	automorphism	NOUN
ejpam-6044	783	3	of	of	ADP
ejpam-6044	783	4	prime	prime	ADJ
ejpam-6044	783	5	rings	ring	NOUN
ejpam-6044	783	6	.	.	PUNCT
ejpam-6044	784	1	canadian	canadian	ADJ
ejpam-6044	784	2	mathematical	mathematical	ADJ
ejpam-6044	784	3	bulletin	bulletin	NOUN
ejpam-6044	784	4	,	,	PUNCT
ejpam-6044	784	5	19(1):113–115	19(1):113–115	PROPN
ejpam-6044	784	6	,	,	PUNCT
ejpam-6044	784	7	1976	1976	NUM
ejpam-6044	784	8	.	.	PUNCT
ejpam-6044	785	1	[	[	X
ejpam-6044	785	2	14	14	NUM
ejpam-6044	785	3	]	]	PUNCT
ejpam-6044	785	4	i.	i.	PROPN
ejpam-6044	785	5	n.	n.	PROPN
ejpam-6044	785	6	herstein	herstein	PROPN
ejpam-6044	785	7	.	.	PUNCT
ejpam-6044	786	1	rings	ring	NOUN
ejpam-6044	786	2	with	with	ADP
ejpam-6044	786	3	involution	involution	NOUN
ejpam-6044	786	4	.	.	PUNCT
ejpam-6044	787	1	university	university	NOUN
ejpam-6044	787	2	of	of	ADP
ejpam-6044	787	3	chicago	chicago	PROPN
ejpam-6044	787	4	press	press	PROPN
ejpam-6044	787	5	,	,	PUNCT
ejpam-6044	787	6	chicago	chicago	PROPN
ejpam-6044	787	7	,	,	PUNCT
ejpam-6044	787	8	1976	1976	NUM
ejpam-6044	787	9	.	.	PUNCT