id	sid	tid	token	lemma	pos
iajs-3594	1	1	330	330	NUM
iajs-3594	1	2	©	©	PROPN
iajs-3594	1	3	2025	2025	NUM
iajs-3594	1	4	the	the	DET
iajs-3594	1	5	author(s	author(s	NOUN
iajs-3594	1	6	)	)	PUNCT
iajs-3594	1	7	.	.	PUNCT
iajs-3594	2	1	published	publish	VERB
iajs-3594	2	2	by	by	ADP
iajs-3594	2	3	college	college	NOUN
iajs-3594	2	4	of	of	ADP
iajs-3594	2	5	education	education	NOUN
iajs-3594	2	6	for	for	ADP
iajs-3594	2	7	pure	pure	ADJ
iajs-3594	2	8	science	science	NOUN
iajs-3594	2	9	(	(	PUNCT
iajs-3594	2	10	ibn	ibn	PROPN
iajs-3594	2	11	al	al	PROPN
iajs-3594	2	12	-	-	PUNCT
iajs-3594	2	13	haitham	haitham	PROPN
iajs-3594	2	14	)	)	PUNCT
iajs-3594	2	15	,	,	PUNCT
iajs-3594	2	16	university	university	NOUN
iajs-3594	2	17	of	of	ADP
iajs-3594	2	18	baghdad	baghdad	PROPN
iajs-3594	2	19	.	.	PUNCT
iajs-3594	3	1	this	this	PRON
iajs-3594	3	2	is	be	AUX
iajs-3594	3	3	an	an	DET
iajs-3594	3	4	open	open	ADJ
iajs-3594	3	5	-	-	PUNCT
iajs-3594	3	6	access	access	NOUN
iajs-3594	3	7	article	article	NOUN
iajs-3594	3	8	distributed	distribute	VERB
iajs-3594	3	9	under	under	ADP
iajs-3594	3	10	the	the	DET
iajs-3594	3	11	terms	term	NOUN
iajs-3594	3	12	of	of	ADP
iajs-3594	3	13	the	the	DET
iajs-3594	3	14	creative	creative	ADJ
iajs-3594	3	15	commons	common	NOUN
iajs-3594	3	16	attribution	attribution	NOUN
iajs-3594	3	17	4.0	4.0	NUM
iajs-3594	3	18	international	international	ADJ
iajs-3594	3	19	license	license	NOUN
iajs-3594	3	20	some	some	DET
iajs-3594	3	21	results	result	NOUN
iajs-3594	3	22	on	on	ADP
iajs-3594	3	23	double	double	ADJ
iajs-3594	3	24	centralizer	centralizer	NOUN
iajs-3594	3	25	for	for	ADP
iajs-3594	3	26	prime	prime	ADJ
iajs-3594	3	27	and	and	CCONJ
iajs-3594	3	28	semiprime	semiprime	NOUN
iajs-3594	3	29	гrings	гring	NOUN
iajs-3594	3	30	aya	aya	PROPN
iajs-3594	3	31	hussein	hussein	PROPN
iajs-3594	3	32	khudair	khudair	NOUN
iajs-3594	3	33	1	1	NUM
iajs-3594	3	34	*	*	PUNCT
iajs-3594	3	35	,	,	PUNCT
iajs-3594	3	36	abdulrahman	abdulrahman	PROPN
iajs-3594	3	37	h.	h.	PROPN
iajs-3594	3	38	majeed	majeed	PROPN
iajs-3594	3	39	2	2	NUM
iajs-3594	3	40	,	,	PUNCT
iajs-3594	3	41	shrooq	shrooq	NOUN
iajs-3594	3	42	bahjat	bahjat	PROPN
iajs-3594	3	43	smeein	smeein	PROPN
iajs-3594	3	44	3	3	NUM
iajs-3594	3	45	and	and	CCONJ
iajs-3594	3	46	azza	azza	NOUN
iajs-3594	3	47	i.m.s	i.m.s	PROPN
iajs-3594	3	48	.	.	PUNCT
iajs-3594	4	1	abu	abu	PROPN
iajs-3594	4	2	-	-	PUNCT
iajs-3594	4	3	shams	sham	VERB
iajs-3594	4	4	4	4	NUM
iajs-3594	4	5	1	1	NUM
iajs-3594	4	6	department	department	NOUN
iajs-3594	4	7	of	of	ADP
iajs-3594	4	8	mathematics	mathematic	NOUN
iajs-3594	4	9	,	,	PUNCT
iajs-3594	4	10	college	college	NOUN
iajs-3594	4	11	of	of	ADP
iajs-3594	4	12	science	science	NOUN
iajs-3594	4	13	,	,	PUNCT
iajs-3594	4	14	university	university	NOUN
iajs-3594	4	15	of	of	ADP
iajs-3594	4	16	baghdad	baghdad	PROPN
iajs-3594	4	17	,	,	PUNCT
iajs-3594	4	18	baghdad	baghdad	PROPN
iajs-3594	4	19	,	,	PUNCT
iajs-3594	4	20	iraq	iraq	PROPN
iajs-3594	4	21	.	.	PUNCT
iajs-3594	5	1	2	2	NUM
iajs-3594	5	2	department	department	NOUN
iajs-3594	5	3	of	of	ADP
iajs-3594	5	4	mathematic	mathematic	PROPN
iajs-3594	5	5	,	,	PUNCT
iajs-3594	5	6	al	al	PROPN
iajs-3594	5	7	-	-	PUNCT
iajs-3594	5	8	mamoun	mamoun	PROPN
iajs-3594	5	9	university	university	PROPN
iajs-3594	5	10	college	college	NOUN
iajs-3594	5	11	,	,	PUNCT
iajs-3594	5	12	baghdad	baghdad	PROPN
iajs-3594	5	13	,	,	PUNCT
iajs-3594	5	14	iraq	iraq	PROPN
iajs-3594	5	15	.	.	PUNCT
iajs-3594	6	1	3	3	NUM
iajs-3594	6	2	department	department	NOUN
iajs-3594	6	3	of	of	ADP
iajs-3594	6	4	information	information	NOUN
iajs-3594	6	5	technology	technology	PROPN
iajs-3594	6	6	-section	-section	PROPN
iajs-3594	6	7	mathematics	mathematic	NOUN
iajs-3594	6	8	,	,	PUNCT
iajs-3594	6	9	university	university	NOUN
iajs-3594	6	10	of	of	ADP
iajs-3594	6	11	technology	technology	NOUN
iajs-3594	6	12	and	and	CCONJ
iajs-3594	6	13	applied	apply	VERB
iajs-3594	6	14	science	science	NOUN
iajs-3594	6	15	muscat	muscat	PROPN
iajs-3594	6	16	,	,	PUNCT
iajs-3594	6	17	sultanate	sultanate	NOUN
iajs-3594	6	18	of	of	ADP
iajs-3594	6	19	oman	oman	NOUN
iajs-3594	6	20	.	.	PUNCT
iajs-3594	7	1	doi.org/10.30526/38.2.3594	doi.org/10.30526/38.2.3594	NOUN
iajs-3594	7	2	abstract	abstract	VERB
iajs-3594	7	3	the	the	DET
iajs-3594	7	4	goal	goal	NOUN
iajs-3594	7	5	of	of	ADP
iajs-3594	7	6	this	this	DET
iajs-3594	7	7	work	work	NOUN
iajs-3594	7	8	,	,	PUNCT
iajs-3594	7	9	is	be	AUX
iajs-3594	7	10	to	to	PART
iajs-3594	7	11	examine	examine	VERB
iajs-3594	7	12	the	the	DET
iajs-3594	7	13	concept	concept	NOUN
iajs-3594	7	14	of	of	ADP
iajs-3594	7	15	a	a	DET
iajs-3594	7	16	double	double	ADJ
iajs-3594	7	17	centralizer	centralizer	NOUN
iajs-3594	7	18	(	(	PUNCT
iajs-3594	7	19	t	t	PROPN
iajs-3594	7	20	,	,	PUNCT
iajs-3594	7	21	s	s	PART
iajs-3594	7	22	)	)	PUNCT
iajs-3594	7	23	,	,	PUNCT
iajs-3594	7	24	and	and	CCONJ
iajs-3594	7	25	double	double	ADJ
iajs-3594	7	26	jordan	jordan	PROPN
iajs-3594	7	27	centralizer	centralizer	NOUN
iajs-3594	7	28	on	on	ADP
iajs-3594	7	29	prime	prime	NOUN
iajs-3594	7	30	and	and	CCONJ
iajs-3594	7	31	semiprime	semiprime	NOUN
iajs-3594	7	32	г	г	PROPN
iajs-3594	7	33	-	-	PUNCT
iajs-3594	7	34	rings	ring	NOUN
iajs-3594	7	35	,	,	PUNCT
iajs-3594	7	36	this	this	PRON
iajs-3594	7	37	is	be	AUX
iajs-3594	7	38	done	do	VERB
iajs-3594	7	39	by	by	ADP
iajs-3594	7	40	studying	study	VERB
iajs-3594	7	41	examples	example	NOUN
iajs-3594	7	42	,	,	PUNCT
iajs-3594	7	43	remarks	remark	NOUN
iajs-3594	7	44	and	and	CCONJ
iajs-3594	7	45	results	result	NOUN
iajs-3594	7	46	related	relate	VERB
iajs-3594	7	47	to	to	ADP
iajs-3594	7	48	that	that	DET
iajs-3594	7	49	concepts	concept	NOUN
iajs-3594	7	50	and	and	CCONJ
iajs-3594	7	51	looking	look	VERB
iajs-3594	7	52	for	for	ADP
iajs-3594	7	53	the	the	DET
iajs-3594	7	54	conditions	condition	NOUN
iajs-3594	7	55	under	under	ADP
iajs-3594	7	56	which	which	PRON
iajs-3594	7	57	t	t	NOUN
iajs-3594	7	58	equal	equal	ADJ
iajs-3594	7	59	s	s	NOUN
iajs-3594	7	60	,	,	PUNCT
iajs-3594	7	61	we	we	PRON
iajs-3594	7	62	prove	prove	VERB
iajs-3594	7	63	the	the	DET
iajs-3594	7	64	results	result	NOUN
iajs-3594	7	65	,	,	PUNCT
iajs-3594	7	66	the	the	DET
iajs-3594	7	67	first	first	ADJ
iajs-3594	7	68	result	result	NOUN
iajs-3594	7	69	,	,	PUNCT
iajs-3594	7	70	let	let	VERB
iajs-3594	7	71	a	a	PRON
iajs-3594	7	72	be	be	AUX
iajs-3594	7	73	a	a	DET
iajs-3594	7	74	semiprime	semiprime	NOUN
iajs-3594	7	75	γ	γ	X
iajs-3594	7	76	-	-	PUNCT
iajs-3594	7	77	ring	ring	NOUN
iajs-3594	7	78	and	and	CCONJ
iajs-3594	7	79	t	t	NOUN
iajs-3594	7	80	is	be	AUX
iajs-3594	7	81	a	a	DET
iajs-3594	7	82	left	left	ADJ
iajs-3594	7	83	centralizer	centralizer	NOUN
iajs-3594	7	84	,	,	PUNCT
iajs-3594	7	85	s	s	PART
iajs-3594	7	86	is	be	AUX
iajs-3594	7	87	a	a	DET
iajs-3594	7	88	right	right	ADJ
iajs-3594	7	89	centralizer	centralizer	NOUN
iajs-3594	7	90	,	,	PUNCT
iajs-3594	7	91	and	and	CCONJ
iajs-3594	7	92	they	they	PRON
iajs-3594	7	93	fulfilling	fulfil	VERB
iajs-3594	7	94	x	x	X
iajs-3594	7	95	𝛼	𝛼	X
iajs-3594	7	96	t(y	t(y	PROPN
iajs-3594	7	97	)	)	PUNCT
iajs-3594	7	98	=	=	SYM
iajs-3594	7	99	s	s	X
iajs-3594	7	100	(	(	PUNCT
iajs-3594	7	101	x	x	X
iajs-3594	7	102	)	)	PUNCT
iajs-3594	7	103	𝛼	𝛼	PROPN
iajs-3594	7	104	y	y	PROPN
iajs-3594	7	105	,	,	PUNCT
iajs-3594	7	106	for	for	ADP
iajs-3594	7	107	each	each	DET
iajs-3594	7	108	x	x	SYM
iajs-3594	7	109	∈	∈	PROPN
iajs-3594	7	110	a	a	X
iajs-3594	7	111	,	,	PUNCT
iajs-3594	7	112	𝛼	𝛼	PROPN
iajs-3594	7	113	∈	∈	PROPN
iajs-3594	7	114	γ	γ	X
iajs-3594	7	115	,	,	PUNCT
iajs-3594	7	116	thence	thence	NOUN
iajs-3594	7	117	(	(	PUNCT
iajs-3594	7	118	t	t	PROPN
iajs-3594	7	119	,	,	PUNCT
iajs-3594	7	120	s	s	PART
iajs-3594	7	121	)	)	PUNCT
iajs-3594	7	122	is	be	AUX
iajs-3594	7	123	a	a	DET
iajs-3594	7	124	double	double	ADJ
iajs-3594	7	125	centralizer	centralizer	NOUN
iajs-3594	7	126	.	.	PUNCT
iajs-3594	8	1	the	the	DET
iajs-3594	8	2	second	second	ADJ
iajs-3594	8	3	,	,	PUNCT
iajs-3594	8	4	let	let	VERB
iajs-3594	8	5	a	a	PRON
iajs-3594	8	6	be	be	AUX
iajs-3594	8	7	a	a	DET
iajs-3594	8	8	prime	prime	ADJ
iajs-3594	8	9	γ	γ	X
iajs-3594	8	10	-	-	NOUN
iajs-3594	8	11	ring	ring	NOUN
iajs-3594	8	12	,	,	PUNCT
iajs-3594	8	13	u	u	PRON
iajs-3594	8	14	be	be	VERB
iajs-3594	8	15	a	a	DET
iajs-3594	8	16	not	not	PART
iajs-3594	8	17	equal	equal	ADJ
iajs-3594	8	18	zero	zero	NUM
iajs-3594	8	19	ideal	ideal	NOUN
iajs-3594	8	20	of	of	ADP
iajs-3594	8	21	a	a	PRON
iajs-3594	8	22	,	,	PUNCT
iajs-3594	8	23	such	such	ADJ
iajs-3594	8	24	that	that	SCONJ
iajs-3594	8	25	,	,	PUNCT
iajs-3594	8	26	t	t	PROPN
iajs-3594	8	27	is	be	AUX
iajs-3594	8	28	a	a	DET
iajs-3594	8	29	left	left	ADJ
iajs-3594	8	30	centralizer	centralizer	NOUN
iajs-3594	8	31	,	,	PUNCT
iajs-3594	8	32	s	s	PART
iajs-3594	8	33	is	be	AUX
iajs-3594	8	34	a	a	DET
iajs-3594	8	35	right	right	ADJ
iajs-3594	8	36	centralizer	centralizer	NOUN
iajs-3594	8	37	,	,	PUNCT
iajs-3594	8	38	and	and	CCONJ
iajs-3594	8	39	fulfilling	fulfil	VERB
iajs-3594	8	40	x	x	X
iajs-3594	8	41	𝛼t(y	𝛼t(y	X
iajs-3594	8	42	)	)	PUNCT
iajs-3594	8	43	=	=	SYM
iajs-3594	8	44	s	s	X
iajs-3594	8	45	(	(	PUNCT
iajs-3594	8	46	x	x	X
iajs-3594	8	47	)	)	PUNCT
iajs-3594	8	48	𝛼	𝛼	PROPN
iajs-3594	8	49	y	y	PROPN
iajs-3594	8	50	,	,	PUNCT
iajs-3594	8	51	for	for	ADP
iajs-3594	8	52	each	each	DET
iajs-3594	8	53	x	x	NOUN
iajs-3594	8	54	,	,	PUNCT
iajs-3594	8	55	y	y	PROPN
iajs-3594	8	56	∈	∈	PROPN
iajs-3594	8	57	u	u	PROPN
iajs-3594	8	58	,	,	PUNCT
iajs-3594	8	59	𝛼	𝛼	PROPN
iajs-3594	8	60	∈	∈	PROPN
iajs-3594	8	61	γ	γ	X
iajs-3594	8	62	,	,	PUNCT
iajs-3594	8	63	thence	thence	NOUN
iajs-3594	8	64	(	(	PUNCT
iajs-3594	8	65	t	t	PROPN
iajs-3594	8	66	,	,	PUNCT
iajs-3594	8	67	s	s	PART
iajs-3594	8	68	)	)	PUNCT
iajs-3594	8	69	is	be	AUX
iajs-3594	8	70	a	a	DET
iajs-3594	8	71	double	double	ADJ
iajs-3594	8	72	centralizer	centralizer	NOUN
iajs-3594	8	73	.	.	PUNCT
iajs-3594	9	1	the	the	DET
iajs-3594	9	2	third	third	ADJ
iajs-3594	9	3	,	,	PUNCT
iajs-3594	9	4	let	let	VERB
iajs-3594	9	5	a	a	PRON
iajs-3594	9	6	be	be	AUX
iajs-3594	9	7	a	a	DET
iajs-3594	9	8	prime	prime	ADJ
iajs-3594	9	9	γ	γ	X
iajs-3594	9	10	-	-	NOUN
iajs-3594	9	11	ring	ring	NOUN
iajs-3594	9	12	,	,	PUNCT
iajs-3594	9	13	u	u	PRON
iajs-3594	9	14	be	be	VERB
iajs-3594	9	15	a	a	DET
iajs-3594	9	16	not	not	PART
iajs-3594	9	17	equal	equal	ADJ
iajs-3594	9	18	zero	zero	NUM
iajs-3594	9	19	ideal	ideal	NOUN
iajs-3594	9	20	of	of	ADP
iajs-3594	9	21	a	a	PRON
iajs-3594	9	22	and	and	CCONJ
iajs-3594	9	23	we	we	PRON
iajs-3594	9	24	get	get	VERB
iajs-3594	9	25	,	,	PUNCT
iajs-3594	9	26	if	if	SCONJ
iajs-3594	9	27	t	t	X
iajs-3594	9	28	=	=	SYM
iajs-3594	9	29	s	s	X
iajs-3594	9	30	on	on	ADP
iajs-3594	9	31	u	u	NOUN
iajs-3594	9	32	,	,	PUNCT
iajs-3594	9	33	thence	thence	NOUN
iajs-3594	9	34	t	t	PROPN
iajs-3594	9	35	=	=	SYM
iajs-3594	9	36	s	s	X
iajs-3594	9	37	on	on	ADP
iajs-3594	9	38	a.	a.	NOUN
iajs-3594	9	39	keywords	keyword	NOUN
iajs-3594	9	40	:	:	PUNCT
iajs-3594	9	41	prime	prime	ADJ
iajs-3594	9	42	г	г	PROPN
iajs-3594	9	43	-	-	PUNCT
iajs-3594	9	44	rings	ring	NOUN
iajs-3594	9	45	,	,	PUNCT
iajs-3594	9	46	semiprime	semiprime	NOUN
iajs-3594	9	47	г	г	PROPN
iajs-3594	9	48	-	-	PUNCT
iajs-3594	9	49	rings	ring	NOUN
iajs-3594	9	50	,	,	PUNCT
iajs-3594	9	51	centralizer	centralizer	NOUN
iajs-3594	9	52	,	,	PUNCT
iajs-3594	9	53	jordan	jordan	PROPN
iajs-3594	9	54	centralizer	centralizer	NOUN
iajs-3594	9	55	,	,	PUNCT
iajs-3594	9	56	double	double	ADJ
iajs-3594	9	57	centralizer	centralizer	NOUN
iajs-3594	9	58	,	,	PUNCT
iajs-3594	9	59	double	double	ADJ
iajs-3594	9	60	jordan	jordan	PROPN
iajs-3594	9	61	centralizer	centralizer	NOUN
iajs-3594	9	62	.	.	PUNCT
iajs-3594	10	1	1	1	X
iajs-3594	10	2	.	.	X
iajs-3594	10	3	introduction	introduction	NOUN
iajs-3594	10	4	barnes	barne	NOUN
iajs-3594	10	5	(	(	PUNCT
iajs-3594	10	6	1	1	X
iajs-3594	10	7	)	)	PUNCT
iajs-3594	10	8	defined	define	VERB
iajs-3594	10	9	𝛤-ring	𝛤-ring	PROPN
iajs-3594	10	10	.	.	PUNCT
iajs-3594	11	1	let	let	VERB
iajs-3594	11	2	a	a	PRON
iajs-3594	11	3	and	and	CCONJ
iajs-3594	11	4	γ	γ	NOUN
iajs-3594	11	5	be	be	AUX
iajs-3594	11	6	two	two	NUM
iajs-3594	11	7	additive	additive	ADJ
iajs-3594	11	8	abelian	abelian	ADJ
iajs-3594	11	9	groups	group	NOUN
iajs-3594	11	10	.	.	PUNCT
iajs-3594	12	1	it	it	PRON
iajs-3594	12	2	there	there	PRON
iajs-3594	12	3	is	be	VERB
iajs-3594	12	4	a	a	DET
iajs-3594	12	5	mapping	mapping	NOUN
iajs-3594	12	6	(	(	PUNCT
iajs-3594	12	7	𝑥	𝑥	NOUN
iajs-3594	12	8	,	,	PUNCT
iajs-3594	12	9	𝛼	𝛼	PROPN
iajs-3594	12	10	,	,	PUNCT
iajs-3594	12	11	𝑦	𝑦	NOUN
iajs-3594	12	12	)	)	PUNCT
iajs-3594	12	13	→	→	SYM
iajs-3594	12	14	(	(	PUNCT
iajs-3594	12	15	𝑥	𝑥	PRON
iajs-3594	12	16	𝛼	𝛼	NOUN
iajs-3594	12	17	𝑦	𝑦	NOUN
iajs-3594	12	18	)	)	PUNCT
iajs-3594	12	19	of	of	ADP
iajs-3594	12	20	𝐴	𝐴	PROPN
iajs-3594	12	21	×	×	PROPN
iajs-3594	12	22	г	г	PROPN
iajs-3594	12	23	×	×	PROPN
iajs-3594	12	24	𝐴	𝐴	PROPN
iajs-3594	12	25	→	→	SYM
iajs-3594	12	26	𝐴	𝐴	PROPN
iajs-3594	12	27	,	,	PUNCT
iajs-3594	12	28	satisfying	satisfy	VERB
iajs-3594	12	29	the	the	DET
iajs-3594	12	30	following	following	NOUN
iajs-3594	12	31	,	,	PUNCT
iajs-3594	12	32	for	for	ADP
iajs-3594	12	33	any	any	DET
iajs-3594	12	34	𝑥	𝑥	PROPN
iajs-3594	12	35	,	,	PUNCT
iajs-3594	12	36	𝑦	𝑦	NOUN
iajs-3594	12	37	,	,	PUNCT
iajs-3594	12	38	𝑧	𝑧	PROPN
iajs-3594	12	39	∈	∈	PROPN
iajs-3594	12	40	a	a	NOUN
iajs-3594	12	41	and	and	CCONJ
iajs-3594	12	42	α	α	NOUN
iajs-3594	12	43	,	,	PUNCT
iajs-3594	12	44	β	β	X
iajs-3594	12	45	∈𝛤.	∈𝛤.	PROPN
iajs-3594	12	46	i.	i.	NOUN
iajs-3594	12	47	(	(	PUNCT
iajs-3594	12	48	𝑥	𝑥	PROPN
iajs-3594	13	1	+	+	PUNCT
iajs-3594	14	1	𝑦)𝛼𝑧	𝑦)𝛼𝑧	PROPN
iajs-3594	14	2	=	=	SYM
iajs-3594	14	3	𝑥𝛼𝑧	𝑥𝛼𝑧	PROPN
iajs-3594	14	4	+	+	CCONJ
iajs-3594	14	5	𝑦𝛼𝑧	𝑦𝛼𝑧	PROPN
iajs-3594	14	6	,	,	PUNCT
iajs-3594	14	7	𝑥(α	𝑥(α	PROPN
iajs-3594	14	8	+	+	CCONJ
iajs-3594	14	9	β)y	β)y	SYM
iajs-3594	14	10	=	=	PUNCT
iajs-3594	14	11	xαy	xαy	X
iajs-3594	14	12	+	+	CCONJ
iajs-3594	14	13	xβy	xβy	PROPN
iajs-3594	14	14	,	,	PUNCT
iajs-3594	14	15	𝑥𝛼(y	𝑥𝛼(y	PUNCT
iajs-3594	14	16	+	+	X
iajs-3594	14	17	z	z	X
iajs-3594	14	18	)	)	PUNCT
iajs-3594	14	19	=	=	PUNCT
iajs-3594	14	20	xαy	xαy	PROPN
iajs-3594	15	1	+	+	CCONJ
iajs-3594	15	2	xαz	xαz	PROPN
iajs-3594	15	3	,	,	PUNCT
iajs-3594	15	4	ii	ii	PROPN
iajs-3594	15	5	.	.	PUNCT
iajs-3594	16	1	(	(	PUNCT
iajs-3594	16	2	𝑥𝛼𝑦)𝛽𝑧	𝑥𝛼𝑦)𝛽𝑧	NOUN
iajs-3594	16	3	=	=	SYM
iajs-3594	16	4	𝑥𝛼(𝑦𝛽𝑧	𝑥𝛼(𝑦𝛽𝑧	PROPN
iajs-3594	16	5	)	)	PUNCT
iajs-3594	16	6	,	,	PUNCT
iajs-3594	16	7	thence	thence	NOUN
iajs-3594	16	8	a	a	PRON
iajs-3594	16	9	is	be	AUX
iajs-3594	16	10	named	name	VERB
iajs-3594	16	11	a	a	DET
iajs-3594	16	12	𝛤-ring	𝛤-ring	PROPN
iajs-3594	16	13	.	.	PUNCT
iajs-3594	17	1	*	*	PUNCT
iajs-3594	17	2	corresponding	correspond	VERB
iajs-3594	17	3	author	author	NOUN
iajs-3594	17	4	.	.	PUNCT
iajs-3594	18	1	received	receive	VERB
iajs-3594	18	2	:	:	PUNCT
iajs-3594	18	3	11	11	NUM
iajs-3594	18	4	june	june	PROPN
iajs-3594	18	5	2023	2023	NUM
iajs-3594	18	6	accepted	accept	VERB
iajs-3594	18	7	:	:	PUNCT
iajs-3594	18	8	10	10	NUM
iajs-3594	18	9	september	september	PROPN
iajs-3594	18	10	2023	2023	NUM
iajs-3594	18	11	published	publish	VERB
iajs-3594	18	12	:	:	PUNCT
iajs-3594	18	13	20	20	NUM
iajs-3594	18	14	april	april	PROPN
iajs-3594	18	15	2025	2025	NUM
iajs-3594	18	16	4	4	NUM
iajs-3594	18	17	department	department	NOUN
iajs-3594	18	18	of	of	ADP
iajs-3594	18	19	mathematics	mathematic	NOUN
iajs-3594	18	20	,	,	PUNCT
iajs-3594	18	21	college	college	NOUN
iajs-3594	18	22	of	of	ADP
iajs-3594	18	23	science	science	NOUN
iajs-3594	18	24	,	,	PUNCT
iajs-3594	18	25	philadelphia	philadelphia	PROPN
iajs-3594	18	26	university	university	PROPN
iajs-3594	18	27	,	,	PUNCT
iajs-3594	18	28	ammaan	ammaan	PROPN
iajs-3594	18	29	,	,	PUNCT
iajs-3594	18	30	jordan	jordan	PROPN
iajs-3594	18	31	.	.	PUNCT
iajs-3594	19	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3594	19	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3594	19	3	https://orcid.org/0009-0002-8259-4959	https://orcid.org/0009-0002-8259-4959	VERB
iajs-3594	19	4	mailto:aya.hussein.g@gmail.com	mailto:aya.hussein.g@gmail.com	X
iajs-3594	19	5	https://orcid.org/0000-0001-8534-0749	https://orcid.org/0000-0001-8534-0749	VERB
iajs-3594	19	6	mailto:dulrahman.h.majeed@almamonuc.edu.iq	mailto:dulrahman.h.majeed@almamonuc.edu.iq	NOUN
iajs-3594	19	7	https://orcid.org/0009-0002-9351-4176	https://orcid.org/0009-0002-9351-4176	PROPN
iajs-3594	19	8	mailto:shrooq.smeein@hct.edu.om	mailto:shrooq.smeein@hct.edu.om	PROPN
iajs-3594	19	9	https://orcid.org/0009-0003-1741-1447	https://orcid.org/0009-0003-1741-1447	NOUN
iajs-3594	19	10	mailto:aabushams@philadelphia.edu.jo	mailto:aabushams@philadelphia.edu.jo	ADJ
iajs-3594	19	11	ihjpas	ihjpa	NOUN
iajs-3594	19	12	.	.	PUNCT
iajs-3594	20	1	2025	2025	NUM
iajs-3594	20	2	,	,	PUNCT
iajs-3594	20	3	38(2	38(2	NUM
iajs-3594	20	4	)	)	PUNCT
iajs-3594	20	5	331	331	NUM
iajs-3594	20	6	ozden	ozden	ADJ
iajs-3594	20	7	et	et	PROPN
iajs-3594	20	8	al	al	PROPN
iajs-3594	20	9	.	.	PROPN
iajs-3594	21	1	(	(	PUNCT
iajs-3594	21	2	2	2	X
iajs-3594	21	3	)	)	PUNCT
iajs-3594	21	4	defined	define	VERB
iajs-3594	21	5	the	the	DET
iajs-3594	21	6	subring	subring	NOUN
iajs-3594	21	7	.	.	PUNCT
iajs-3594	22	1	a	a	DET
iajs-3594	22	2	subring	subring	NOUN
iajs-3594	22	3	of	of	ADP
iajs-3594	22	4	г	г	NOUN
iajs-3594	22	5	-	-	PUNCT
iajs-3594	22	6	ring	ring	NOUN
iajs-3594	22	7	a	a	PRON
iajs-3594	22	8	is	be	AUX
iajs-3594	22	9	additive	additive	ADJ
iajs-3594	22	10	subgroup	subgroup	NOUN
iajs-3594	22	11	s	s	PROPN
iajs-3594	22	12	of	of	ADP
iajs-3594	22	13	a	a	DET
iajs-3594	22	14	such	such	ADJ
iajs-3594	22	15	that	that	SCONJ
iajs-3594	22	16	𝑆г𝑆	𝑆г𝑆	NOUN
iajs-3594	22	17	⸦	⸦	NOUN
iajs-3594	22	18	𝑆	𝑆	PROPN
iajs-3594	22	19	.	.	PUNCT
iajs-3594	23	1	let	let	VERB
iajs-3594	23	2	a	a	PRON
iajs-3594	23	3	be	be	AUX
iajs-3594	23	4	a	a	DET
iajs-3594	23	5	𝛤-ring	𝛤-ring	NOUN
iajs-3594	23	6	,	,	PUNCT
iajs-3594	23	7	thence	thence	NOUN
iajs-3594	23	8	a	a	PRON
iajs-3594	23	9	is	be	AUX
iajs-3594	23	10	named	name	VERB
iajs-3594	23	11	a	a	DET
iajs-3594	23	12	commutative	commutative	ADJ
iajs-3594	23	13	gamma	gamma	NOUN
iajs-3594	23	14	-	-	PUNCT
iajs-3594	23	15	ring	ring	NOUN
iajs-3594	23	16	if	if	SCONJ
iajs-3594	23	17	,	,	PUNCT
iajs-3594	23	18	𝑥α𝑦	𝑥α𝑦	NOUN
iajs-3594	23	19	=	=	SYM
iajs-3594	23	20	𝑦𝛼𝑥	𝑦𝛼𝑥	PROPN
iajs-3594	23	21	,	,	PUNCT
iajs-3594	23	22	holds	hold	VERB
iajs-3594	23	23	for	for	ADP
iajs-3594	23	24	any	any	DET
iajs-3594	23	25	𝑥	𝑥	PROPN
iajs-3594	23	26	,	,	PUNCT
iajs-3594	23	27	𝑦	𝑦	PROPN
iajs-3594	23	28	∈	∈	PROPN
iajs-3594	23	29	𝐴	𝐴	PROPN
iajs-3594	23	30	and	and	CCONJ
iajs-3594	23	31	α	α	PRON
iajs-3594	23	32	∈𝛤	∈𝛤	PROPN
iajs-3594	23	33	,	,	PUNCT
iajs-3594	23	34	kandamar	kandamar	PROPN
iajs-3594	23	35	et	et	PROPN
iajs-3594	23	36	al	al	PROPN
iajs-3594	23	37	.	.	PROPN
iajs-3594	24	1	(	(	PUNCT
iajs-3594	24	2	3	3	NUM
iajs-3594	24	3	)	)	PUNCT
iajs-3594	24	4	.	.	PUNCT
iajs-3594	25	1	a	a	DET
iajs-3594	25	2	subset	subset	ADJ
iajs-3594	25	3	𝑈	𝑈	PROPN
iajs-3594	25	4	of	of	ADP
iajs-3594	25	5	the	the	DET
iajs-3594	25	6	𝛤-ring	𝛤-ring	PROPN
iajs-3594	25	7	a	a	PRON
iajs-3594	25	8	is	be	AUX
iajs-3594	25	9	a	a	DET
iajs-3594	25	10	right	right	NOUN
iajs-3594	25	11	(	(	PUNCT
iajs-3594	25	12	left	left	ADJ
iajs-3594	25	13	)	)	PUNCT
iajs-3594	25	14	ideal	ideal	NOUN
iajs-3594	25	15	of	of	ADP
iajs-3594	25	16	a	a	PRON
iajs-3594	25	17	if	if	SCONJ
iajs-3594	25	18	u	u	NOUN
iajs-3594	25	19	is	be	AUX
iajs-3594	25	20	an	an	DET
iajs-3594	25	21	additive	additive	ADJ
iajs-3594	25	22	subgroup	subgroup	NOUN
iajs-3594	25	23	of	of	ADP
iajs-3594	25	24	a	a	PRON
iajs-3594	25	25	and	and	CCONJ
iajs-3594	25	26	𝑈г𝐴	𝑈г𝐴	NOUN
iajs-3594	25	27	=	=	SYM
iajs-3594	25	28	{	{	PUNCT
iajs-3594	25	29	𝑎αx	𝑎αx	NOUN
iajs-3594	25	30	:	:	PUNCT
iajs-3594	25	31	a	a	DET
iajs-3594	25	32	∈	∈	PROPN
iajs-3594	25	33	𝑈	𝑈	PROPN
iajs-3594	25	34	,	,	PUNCT
iajs-3594	25	35	𝛼	𝛼	PROPN
iajs-3594	25	36	∈	∈	PROPN
iajs-3594	25	37	γ	γ	X
iajs-3594	25	38	,	,	PUNCT
iajs-3594	25	39	𝑥	𝑥	PROPN
iajs-3594	25	40	∈	∈	PROPN
iajs-3594	25	41	a	a	DET
iajs-3594	25	42	}	}	PUNCT
iajs-3594	25	43	(	(	PUNCT
iajs-3594	25	44	aгu	aгu	NOUN
iajs-3594	25	45	)	)	PUNCT
iajs-3594	25	46	is	be	AUX
iajs-3594	25	47	contained	contain	VERB
iajs-3594	25	48	in	in	ADP
iajs-3594	25	49	u.	u.	NOUN
iajs-3594	25	50	if	if	SCONJ
iajs-3594	25	51	u	u	NOUN
iajs-3594	25	52	is	be	AUX
iajs-3594	25	53	both	both	CCONJ
iajs-3594	25	54	a	a	DET
iajs-3594	25	55	left	left	NOUN
iajs-3594	25	56	and	and	CCONJ
iajs-3594	25	57	a	a	DET
iajs-3594	25	58	right	right	ADJ
iajs-3594	25	59	ideal	ideal	NOUN
iajs-3594	25	60	,	,	PUNCT
iajs-3594	25	61	thence	thence	NOUN
iajs-3594	25	62	u	u	NOUN
iajs-3594	25	63	is	be	AUX
iajs-3594	25	64	a	a	DET
iajs-3594	25	65	two	two	NUM
iajs-3594	25	66	-	-	PUNCT
iajs-3594	25	67	sided	sided	ADJ
iajs-3594	25	68	ideal	ideal	NOUN
iajs-3594	25	69	,	,	PUNCT
iajs-3594	25	70	or	or	CCONJ
iajs-3594	25	71	simply	simply	ADV
iajs-3594	25	72	is	be	AUX
iajs-3594	25	73	an	an	DET
iajs-3594	25	74	ideal	ideal	NOUN
iajs-3594	25	75	of	of	ADP
iajs-3594	25	76	a.	a.	NOUN
iajs-3594	25	77	barnes	barnes	PROPN
iajs-3594	25	78	(	(	PUNCT
iajs-3594	25	79	1	1	NUM
iajs-3594	25	80	)	)	PUNCT
iajs-3594	25	81	.	.	PUNCT
iajs-3594	26	1	a	a	DET
iajs-3594	26	2	𝛤-ring	𝛤-ring	PROPN
iajs-3594	26	3	a	a	PRON
iajs-3594	26	4	is	be	AUX
iajs-3594	26	5	named	name	VERB
iajs-3594	26	6	prime	prime	ADJ
iajs-3594	26	7	if	if	SCONJ
iajs-3594	26	8	𝑚г𝐴г𝑛	𝑚г𝐴г𝑛	NOUN
iajs-3594	26	9	=	=	SYM
iajs-3594	26	10	(	(	PUNCT
iajs-3594	26	11	0	0	NUM
iajs-3594	26	12	)	)	PUNCT
iajs-3594	26	13	with	with	ADP
iajs-3594	26	14	𝑚	𝑚	PROPN
iajs-3594	26	15	,	,	PUNCT
iajs-3594	26	16	𝑛	𝑛	PRON
iajs-3594	26	17	∈	∈	PROPN
iajs-3594	26	18	a	a	DET
iajs-3594	26	19	implies	imply	VERB
iajs-3594	26	20	𝑚	𝑚	X
iajs-3594	26	21	=	=	SYM
iajs-3594	26	22	0	0	NUM
iajs-3594	26	23	𝑜𝑟	𝑜𝑟	X
iajs-3594	26	24	𝑛	𝑛	PROPN
iajs-3594	26	25	=	=	SYM
iajs-3594	26	26	0	0	NUM
iajs-3594	26	27	and	and	CCONJ
iajs-3594	26	28	semiprime	semiprime	NOUN
iajs-3594	26	29	if	if	SCONJ
iajs-3594	26	30	𝑚г𝐴г𝑚	𝑚г𝐴г𝑚	X
iajs-3594	26	31	=	=	SYM
iajs-3594	26	32	0	0	NUM
iajs-3594	26	33	with	with	ADP
iajs-3594	26	34	𝑚	𝑚	PROPN
iajs-3594	26	35	∈	∈	PROPN
iajs-3594	26	36	𝐴	𝐴	PROPN
iajs-3594	26	37	implies	imply	VERB
iajs-3594	26	38	𝑚	𝑚	X
iajs-3594	26	39	=	=	SYM
iajs-3594	26	40	0	0	NUM
iajs-3594	26	41	(	(	PUNCT
iajs-3594	26	42	4	4	NUM
iajs-3594	26	43	,	,	PUNCT
iajs-3594	26	44	5	5	NUM
iajs-3594	26	45	)	)	PUNCT
iajs-3594	26	46	.	.	PUNCT
iajs-3594	27	1	an	an	DET
iajs-3594	27	2	ideal	ideal	ADJ
iajs-3594	27	3	p	p	NOUN
iajs-3594	27	4	of	of	ADP
iajs-3594	27	5	a	a	DET
iajs-3594	27	6	gamma	gamma	NOUN
iajs-3594	27	7	-	-	PUNCT
iajs-3594	27	8	ring	ring	NOUN
iajs-3594	27	9	a	a	PRON
iajs-3594	27	10	is	be	AUX
iajs-3594	27	11	prime	prime	ADJ
iajs-3594	27	12	ideal	ideal	NOUN
iajs-3594	27	13	if	if	SCONJ
iajs-3594	27	14	for	for	ADP
iajs-3594	27	15	any	any	DET
iajs-3594	27	16	ideals	ideal	NOUN
iajs-3594	27	17	n	n	CCONJ
iajs-3594	27	18	,	,	PUNCT
iajs-3594	27	19	m⊆a	m⊆a	NOUN
iajs-3594	27	20	,	,	PUNCT
iajs-3594	27	21	n𝛤m⊆	n𝛤m⊆	NOUN
iajs-3594	28	1	𝑃	𝑃	NOUN
iajs-3594	28	2	implies	imply	VERB
iajs-3594	28	3	n⊆	n⊆	ADV
iajs-3594	28	4	𝑃	𝑃	NOUN
iajs-3594	28	5	or	or	CCONJ
iajs-3594	28	6	m⊆	m⊆	PROPN
iajs-3594	28	7	𝑃	𝑃	NOUN
iajs-3594	28	8	,	,	PUNCT
iajs-3594	28	9	kyuno	kyuno	NOUN
iajs-3594	28	10	(	(	PUNCT
iajs-3594	28	11	5	5	NUM
iajs-3594	28	12	)	)	PUNCT
iajs-3594	28	13	.	.	PUNCT
iajs-3594	29	1	a	a	DET
iajs-3594	29	2	gamma	gamma	NOUN
iajs-3594	29	3	-	-	PUNCT
iajs-3594	29	4	ring	ring	NOUN
iajs-3594	29	5	a	a	PRON
iajs-3594	29	6	is	be	AUX
iajs-3594	29	7	said	say	VERB
iajs-3594	29	8	to	to	PART
iajs-3594	29	9	be	be	AUX
iajs-3594	29	10	prime	prime	ADJ
iajs-3594	29	11	𝛤-ring	𝛤-ring	PROPN
iajs-3594	29	12	if	if	SCONJ
iajs-3594	29	13	the	the	DET
iajs-3594	29	14	zero	zero	NUM
iajs-3594	29	15	ideal	ideal	NOUN
iajs-3594	29	16	is	be	AUX
iajs-3594	29	17	prime	prime	ADJ
iajs-3594	29	18	ideal	ideal	NOUN
iajs-3594	29	19	,	,	PUNCT
iajs-3594	29	20	kyuno	kyuno	X
iajs-3594	29	21	(	(	PUNCT
iajs-3594	29	22	5	5	NUM
iajs-3594	29	23	)	)	PUNCT
iajs-3594	29	24	.	.	PUNCT
iajs-3594	30	1	let	let	VERB
iajs-3594	30	2	a	a	PRON
iajs-3594	30	3	be	be	AUX
iajs-3594	30	4	a	a	DET
iajs-3594	30	5	𝛤-ring	𝛤-ring	NOUN
iajs-3594	30	6	,	,	PUNCT
iajs-3594	30	7	thence	thence	NOUN
iajs-3594	30	8	a	a	PRON
iajs-3594	30	9	is	be	AUX
iajs-3594	30	10	named	name	VERB
iajs-3594	30	11	n	n	CCONJ
iajs-3594	30	12	-	-	PUNCT
iajs-3594	30	13	torsion	torsion	NOUN
iajs-3594	30	14	free	free	ADJ
iajs-3594	30	15	if	if	SCONJ
iajs-3594	30	16	𝑛	𝑛	PRON
iajs-3594	30	17	𝑥	𝑥	X
iajs-3594	30	18	=	=	SYM
iajs-3594	30	19	0	0	NUM
iajs-3594	30	20	,	,	PUNCT
iajs-3594	30	21	yields	yield	VERB
iajs-3594	30	22	𝑥	𝑥	NOUN
iajs-3594	30	23	=	=	SYM
iajs-3594	30	24	0	0	NUM
iajs-3594	30	25	,	,	PUNCT
iajs-3594	30	26	for	for	ADP
iajs-3594	30	27	every	every	DET
iajs-3594	30	28	𝑥	𝑥	PROPN
iajs-3594	30	29	∈	∈	PROPN
iajs-3594	30	30	a	a	PRON
iajs-3594	30	31	,	,	PUNCT
iajs-3594	30	32	where	where	SCONJ
iajs-3594	30	33	𝑛	𝑛	PROPN
iajs-3594	30	34	is	be	AUX
iajs-3594	30	35	positive	positive	ADJ
iajs-3594	30	36	integer	integer	NOUN
iajs-3594	30	37	,	,	PUNCT
iajs-3594	30	38	chakraborty	chakraborty	PROPN
iajs-3594	30	39	et	et	PROPN
iajs-3594	30	40	al	al	PROPN
iajs-3594	30	41	.	.	PROPN
iajs-3594	31	1	(	(	PUNCT
iajs-3594	31	2	6	6	NUM
iajs-3594	31	3	)	)	PUNCT
iajs-3594	31	4	.	.	PUNCT
iajs-3594	32	1	let	let	VERB
iajs-3594	32	2	a	a	PRON
iajs-3594	32	3	be	be	AUX
iajs-3594	32	4	a	a	DET
iajs-3594	32	5	gamma	gamma	NOUN
iajs-3594	32	6	-	-	PUNCT
iajs-3594	32	7	semiring	semiring	NOUN
iajs-3594	32	8	,	,	PUNCT
iajs-3594	32	9	an	an	DET
iajs-3594	32	10	element	element	NOUN
iajs-3594	32	11	1	1	NUM
iajs-3594	32	12	∈	∈	NOUN
iajs-3594	32	13	a	a	PRON
iajs-3594	32	14	,	,	PUNCT
iajs-3594	32	15	is	be	AUX
iajs-3594	32	16	named	name	VERB
iajs-3594	32	17	unity	unity	NOUN
iajs-3594	32	18	for	for	ADP
iajs-3594	32	19	any	any	DET
iajs-3594	32	20	𝑥	𝑥	PRON
iajs-3594	32	21	∈	∈	PROPN
iajs-3594	32	22	𝐴	𝐴	PROPN
iajs-3594	32	23	there	there	PRON
iajs-3594	32	24	are	be	VERB
iajs-3594	32	25	𝛼	𝛼	DET
iajs-3594	32	26	∈	∈	NOUN
iajs-3594	32	27	𝛤	𝛤	NOUN
iajs-3594	32	28	such	such	ADJ
iajs-3594	32	29	that	that	SCONJ
iajs-3594	32	30	𝑥	𝑥	PROPN
iajs-3594	32	31	𝛼	𝛼	SYM
iajs-3594	32	32	1	1	NUM
iajs-3594	32	33	=	=	SYM
iajs-3594	32	34	1	1	NUM
iajs-3594	32	35	𝛼	𝛼	NOUN
iajs-3594	32	36	𝑥	𝑥	NOUN
iajs-3594	32	37	=	=	SYM
iajs-3594	32	38	𝑥	𝑥	PROPN
iajs-3594	32	39	,	,	PUNCT
iajs-3594	32	40	rao	rao	NOUN
iajs-3594	32	41	(	(	PUNCT
iajs-3594	32	42	7	7	NUM
iajs-3594	32	43	)	)	PUNCT
iajs-3594	32	44	.	.	PUNCT
iajs-3594	33	1	özkum	özkum	NOUN
iajs-3594	33	2	et	et	PROPN
iajs-3594	33	3	al	al	PROPN
iajs-3594	33	4	.	.	PROPN
iajs-3594	34	1	(	(	PUNCT
iajs-3594	34	2	8)	8)	NUM
iajs-3594	34	3	defined	define	VERB
iajs-3594	34	4	the	the	DET
iajs-3594	34	5	derivation	derivation	NOUN
iajs-3594	34	6	and	and	CCONJ
iajs-3594	34	7	(	(	PUNCT
iajs-3594	34	8	jordan	jordan	PROPN
iajs-3594	34	9	derivation	derivation	PROPN
iajs-3594	34	10	)	)	PUNCT
iajs-3594	34	11	,	,	PUNCT
iajs-3594	34	12	let	let	VERB
iajs-3594	34	13	a	a	PRON
iajs-3594	34	14	be	be	AUX
iajs-3594	34	15	a	a	DET
iajs-3594	34	16	gamma	gamma	NOUN
iajs-3594	34	17	-	-	PUNCT
iajs-3594	34	18	ring	ring	NOUN
iajs-3594	34	19	and	and	CCONJ
iajs-3594	34	20	𝐷	𝐷	NOUN
iajs-3594	34	21	∶	∶	NOUN
iajs-3594	34	22	𝐴	𝐴	PROPN
iajs-3594	34	23	→	→	PUNCT
iajs-3594	34	24	a	a	DET
iajs-3594	34	25	and	and	CCONJ
iajs-3594	34	26	additive	additive	ADJ
iajs-3594	34	27	map	map	NOUN
iajs-3594	34	28	.	.	PUNCT
iajs-3594	35	1	thence	thence	NOUN
iajs-3594	35	2	d	d	NOUN
iajs-3594	35	3	is	be	AUX
iajs-3594	35	4	derivation	derivation	NOUN
iajs-3594	35	5	(	(	PUNCT
iajs-3594	35	6	resp	resp	NOUN
iajs-3594	35	7	.	.	PUNCT
iajs-3594	36	1	jordan	jordan	PROPN
iajs-3594	36	2	derivation	derivation	PROPN
iajs-3594	36	3	)	)	PUNCT
iajs-3594	36	4	,	,	PUNCT
iajs-3594	36	5	if	if	SCONJ
iajs-3594	36	6	𝐷	𝐷	PROPN
iajs-3594	36	7	(	(	PUNCT
iajs-3594	36	8	𝑚	𝑚	PROPN
iajs-3594	36	9	𝛼𝑛	𝛼𝑛	PROPN
iajs-3594	36	10	)	)	PUNCT
iajs-3594	36	11	=	=	SYM
iajs-3594	36	12	𝐷	𝐷	PROPN
iajs-3594	36	13	(	(	PUNCT
iajs-3594	36	14	𝑚	𝑚	NOUN
iajs-3594	36	15	)	)	PUNCT
iajs-3594	36	16	𝛼	𝛼	NOUN
iajs-3594	36	17	𝑛	𝑛	NOUN
iajs-3594	36	18	+	+	NUM
iajs-3594	36	19	𝑚𝛼	𝑚𝛼	ADP
iajs-3594	36	20	𝐷	𝐷	NOUN
iajs-3594	36	21	(	(	PUNCT
iajs-3594	36	22	𝑛	𝑛	NOUN
iajs-3594	36	23	)	)	PUNCT
iajs-3594	36	24	(	(	PUNCT
iajs-3594	36	25	resp	resp	NOUN
iajs-3594	36	26	.	.	PUNCT
iajs-3594	37	1	𝐷	𝐷	PROPN
iajs-3594	37	2	(	(	PUNCT
iajs-3594	37	3	𝑚	𝑚	PROPN
iajs-3594	37	4	𝛼	𝛼	PART
iajs-3594	37	5	𝑚	𝑚	NOUN
iajs-3594	37	6	)	)	PUNCT
iajs-3594	37	7	=	=	SYM
iajs-3594	37	8	𝐷	𝐷	PROPN
iajs-3594	37	9	(	(	PUNCT
iajs-3594	37	10	𝑚)𝛼	𝑚)𝛼	X
iajs-3594	37	11	𝑚	𝑚	X
iajs-3594	37	12	+	+	X
iajs-3594	37	13	𝑚𝛼	𝑚𝛼	NOUN
iajs-3594	37	14	𝐷(𝑚	𝐷(𝑚	NOUN
iajs-3594	37	15	)	)	PUNCT
iajs-3594	37	16	)	)	PUNCT
iajs-3594	37	17	,	,	PUNCT
iajs-3594	37	18	for	for	ADP
iajs-3594	37	19	any	any	DET
iajs-3594	37	20	𝑚	𝑚	NOUN
iajs-3594	37	21	,	,	PUNCT
iajs-3594	37	22	𝑛	𝑛	PRON
iajs-3594	37	23	∈	∈	PROPN
iajs-3594	37	24	a	a	PRON
iajs-3594	37	25	and	and	CCONJ
iajs-3594	37	26	α	α	NOUN
iajs-3594	37	27	∈	∈	PROPN
iajs-3594	37	28	г	г	NOUN
iajs-3594	37	29	,	,	PUNCT
iajs-3594	37	30	özkum	özkum	NOUN
iajs-3594	37	31	et	et	NOUN
iajs-3594	37	32	al	al	PROPN
iajs-3594	37	33	.	.	PROPN
iajs-3594	38	1	(	(	PUNCT
iajs-3594	38	2	8)	8)	NUM
iajs-3594	38	3	.	.	PUNCT
iajs-3594	39	1	every	every	DET
iajs-3594	39	2	derivation	derivation	NOUN
iajs-3594	39	3	of	of	ADP
iajs-3594	39	4	a	a	PRON
iajs-3594	39	5	,	,	PUNCT
iajs-3594	39	6	is	be	AUX
iajs-3594	39	7	jordan	jordan	PROPN
iajs-3594	39	8	derivation	derivation	PROPN
iajs-3594	39	9	but	but	CCONJ
iajs-3594	39	10	the	the	DET
iajs-3594	39	11	converse	converse	NOUN
iajs-3594	39	12	in	in	ADP
iajs-3594	39	13	general	general	ADJ
iajs-3594	39	14	is	be	AUX
iajs-3594	39	15	not	not	PART
iajs-3594	39	16	true	true	ADJ
iajs-3594	39	17	,	,	PUNCT
iajs-3594	39	18	see	see	VERB
iajs-3594	39	19	saleh	saleh	NOUN
iajs-3594	39	20	(	(	PUNCT
iajs-3594	39	21	9	9	NUM
iajs-3594	39	22	)	)	PUNCT
iajs-3594	39	23	.	.	PUNCT
iajs-3594	40	1	barnes	barne	NOUN
iajs-3594	40	2	(	(	PUNCT
iajs-3594	40	3	1	1	X
iajs-3594	40	4	)	)	PUNCT
iajs-3594	40	5	defined	define	VERB
iajs-3594	40	6	the	the	DET
iajs-3594	40	7	𝛤-homomorphism	𝛤-homomorphism	PROPN
iajs-3594	40	8	.	.	PUNCT
iajs-3594	41	1	let	let	VERB
iajs-3594	41	2	a	a	PRON
iajs-3594	41	3	and	and	CCONJ
iajs-3594	41	4	y	y	PRON
iajs-3594	41	5	both	both	PRON
iajs-3594	41	6	be	be	AUX
iajs-3594	41	7	𝛤-rings	𝛤-rings	PROPN
iajs-3594	41	8	,	,	PUNCT
iajs-3594	41	9	and	and	CCONJ
iajs-3594	41	10	∅	∅	VERB
iajs-3594	41	11	a	a	DET
iajs-3594	41	12	map	map	NOUN
iajs-3594	41	13	of	of	ADP
iajs-3594	41	14	a	a	PRON
iajs-3594	41	15	in	in	ADP
iajs-3594	41	16	to	to	ADP
iajs-3594	41	17	a.	a.	NOUN
iajs-3594	41	18	thence	thence	NOUN
iajs-3594	41	19	∅	∅	NOUN
iajs-3594	41	20	is	be	AUX
iajs-3594	41	21	a	a	DET
iajs-3594	41	22	𝛤-homomorphism	𝛤-homomorphism	NOUN
iajs-3594	41	23	,	,	PUNCT
iajs-3594	41	24	if	if	SCONJ
iajs-3594	41	25	and	and	CCONJ
iajs-3594	41	26	only	only	ADV
iajs-3594	41	27	𝑖𝑓	𝑖𝑓	ADP
iajs-3594	41	28	∅(𝑥𝛼𝑦	∅(𝑥𝛼𝑦	NOUN
iajs-3594	41	29	)	)	PUNCT
iajs-3594	42	1	=	=	SYM
iajs-3594	42	2	∅(𝑥)𝛼	∅(𝑥)𝛼	ADJ
iajs-3594	42	3	∅	∅	NOUN
iajs-3594	42	4	(	(	PUNCT
iajs-3594	42	5	𝑦	𝑦	NOUN
iajs-3594	42	6	)	)	PUNCT
iajs-3594	42	7	,	,	PUNCT
iajs-3594	42	8	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3594	42	9	𝑎𝑙𝑙𝑥	𝑎𝑙𝑙𝑥	NOUN
iajs-3594	42	10	,	,	PUNCT
iajs-3594	42	11	𝑦	𝑦	NOUN
iajs-3594	42	12	∈	∈	PROPN
iajs-3594	42	13	𝐴	𝐴	PROPN
iajs-3594	42	14	and	and	CCONJ
iajs-3594	42	15	𝛼	𝛼	ADP
iajs-3594	42	16	∈	∈	PROPN
iajs-3594	42	17	г	г	NOUN
iajs-3594	42	18	.	.	PUNCT
iajs-3594	43	1	if	if	SCONJ
iajs-3594	43	2	∅	∅	NOUN
iajs-3594	43	3	is	be	AUX
iajs-3594	43	4	also	also	ADV
iajs-3594	43	5	one	one	NUM
iajs-3594	43	6	-	-	PUNCT
iajs-3594	43	7	to	to	ADP
iajs-3594	43	8	one	one	NUM
iajs-3594	43	9	and	and	CCONJ
iajs-3594	43	10	onto	onto	ADP
iajs-3594	43	11	thence	thence	NOUN
iajs-3594	43	12	∅	∅	NOUN
iajs-3594	43	13	is	be	AUX
iajs-3594	43	14	a	a	DET
iajs-3594	43	15	𝛤isomorphism	𝛤isomorphism	NOUN
iajs-3594	43	16	.	.	PUNCT
iajs-3594	44	1	an	an	DET
iajs-3594	44	2	additive	additive	ADJ
iajs-3594	44	3	mapping	mapping	NOUN
iajs-3594	44	4	∅	∅	NOUN
iajs-3594	44	5	of	of	ADP
iajs-3594	44	6	𝛤-ring	𝛤-re	VERB
iajs-3594	44	7	a	a	PRON
iajs-3594	44	8	into	into	ADP
iajs-3594	44	9	a	a	DET
iajs-3594	44	10	𝛤-ring	𝛤-ring	NOUN
iajs-3594	44	11	a	a	PRON
iajs-3594	44	12	'	'	PUNCT
iajs-3594	44	13	is	be	AUX
iajs-3594	44	14	named	name	VERB
iajs-3594	44	15	jordan	jordan	PROPN
iajs-3594	44	16	homomorphism	homomorphism	PROPN
iajs-3594	44	17	if	if	SCONJ
iajs-3594	44	18	∅(𝑥𝛼𝑦	∅(𝑥𝛼𝑦	PROPN
iajs-3594	44	19	+	+	CCONJ
iajs-3594	44	20	𝑦𝛼𝑥	𝑦𝛼𝑥	VERB
iajs-3594	44	21	)	)	PUNCT
iajs-3594	44	22	=	=	SYM
iajs-3594	44	23	∅(x)α∅(𝑦	∅(x)α∅(𝑦	NOUN
iajs-3594	44	24	)	)	PUNCT
iajs-3594	45	1	+	+	NUM
iajs-3594	45	2	∅(y)α∅(𝑥	∅(y)α∅(𝑥	NOUN
iajs-3594	45	3	)	)	PUNCT
iajs-3594	45	4	,	,	PUNCT
iajs-3594	45	5	for	for	ADP
iajs-3594	45	6	each	each	DET
iajs-3594	45	7	𝑥	𝑥	PROPN
iajs-3594	45	8	,	,	PUNCT
iajs-3594	45	9	𝑦	𝑦	NOUN
iajs-3594	45	10	∈	∈	PROPN
iajs-3594	45	11	a	a	PRON
iajs-3594	45	12	and	and	CCONJ
iajs-3594	45	13	𝛼	𝛼	NOUN
iajs-3594	45	14	∈	∈	PROPN
iajs-3594	45	15	г	г	PROPN
iajs-3594	45	16	,	,	PUNCT
iajs-3594	45	17	shaheen	shaheen	PROPN
iajs-3594	45	18	(	(	PUNCT
iajs-3594	45	19	10	10	NUM
iajs-3594	45	20	)	)	PUNCT
iajs-3594	45	21	.	.	PUNCT
iajs-3594	46	1	let	let	VERB
iajs-3594	46	2	a	a	PRON
iajs-3594	46	3	be	be	AUX
iajs-3594	46	4	a	a	DET
iajs-3594	46	5	𝛤-ring	𝛤-ring	PROPN
iajs-3594	46	6	,	,	PUNCT
iajs-3594	46	7	a	a	DET
iajs-3594	46	8	mapping	mapping	NOUN
iajs-3594	46	9	d	d	NOUN
iajs-3594	46	10	of	of	ADP
iajs-3594	46	11	a	a	PRON
iajs-3594	46	12	,	,	PUNCT
iajs-3594	46	13	to	to	ADP
iajs-3594	46	14	itself	itself	PRON
iajs-3594	46	15	is	be	AUX
iajs-3594	46	16	named	name	VERB
iajs-3594	46	17	𝛤-centralizing	𝛤-centralize	VERB
iajs-3594	46	18	on	on	ADP
iajs-3594	46	19	a	a	DET
iajs-3594	46	20	subset	subset	ADJ
iajs-3594	46	21	𝑆	𝑆	PROPN
iajs-3594	46	22	of	of	ADP
iajs-3594	46	23	a	a	DET
iajs-3594	46	24	if	if	SCONJ
iajs-3594	46	25	[	[	PUNCT
iajs-3594	46	26	𝑥	𝑥	NOUN
iajs-3594	46	27	,	,	PUNCT
iajs-3594	46	28	𝑑	𝑑	PROPN
iajs-3594	46	29	(	(	PUNCT
iajs-3594	46	30	𝑥)]𝛼	𝑥)]𝛼	PROPN
iajs-3594	46	31	∈	∈	PROPN
iajs-3594	46	32	z(a	z(a	NOUN
iajs-3594	46	33	)	)	PUNCT
iajs-3594	46	34	,	,	PUNCT
iajs-3594	46	35	for	for	ADP
iajs-3594	46	36	every	every	DET
iajs-3594	46	37	𝑥	𝑥	PROPN
iajs-3594	46	38	∈	∈	PROPN
iajs-3594	46	39	𝑆	𝑆	PROPN
iajs-3594	46	40	and	and	CCONJ
iajs-3594	46	41	α	α	PRON
iajs-3594	46	42	∈	∈	PROPN
iajs-3594	46	43	г	г	NOUN
iajs-3594	46	44	,	,	PUNCT
iajs-3594	46	45	in	in	ADP
iajs-3594	46	46	the	the	DET
iajs-3594	46	47	special	special	ADJ
iajs-3594	46	48	case	case	NOUN
iajs-3594	46	49	when	when	SCONJ
iajs-3594	46	50	[	[	PUNCT
iajs-3594	46	51	𝑥	𝑥	X
iajs-3594	46	52	,	,	PUNCT
iajs-3594	46	53	𝑑	𝑑	PROPN
iajs-3594	46	54	(	(	PUNCT
iajs-3594	46	55	𝑥)]𝛼	𝑥)]𝛼	X
iajs-3594	46	56	=	=	SYM
iajs-3594	46	57	0	0	NUM
iajs-3594	46	58	,	,	PUNCT
iajs-3594	46	59	hold	hold	VERB
iajs-3594	46	60	for	for	ADP
iajs-3594	46	61	any	any	DET
iajs-3594	46	62	𝑥	𝑥	PRON
iajs-3594	46	63	∈	∈	PROPN
iajs-3594	46	64	s	s	X
iajs-3594	46	65	and	and	CCONJ
iajs-3594	46	66	𝛼	𝛼	NOUN
iajs-3594	46	67	∈	∈	PROPN
iajs-3594	46	68	г	г	PROPN
iajs-3594	46	69	,	,	PUNCT
iajs-3594	46	70	the	the	DET
iajs-3594	46	71	mapping	mapping	NOUN
iajs-3594	46	72	d	d	NOUN
iajs-3594	46	73	is	be	AUX
iajs-3594	46	74	named	name	VERB
iajs-3594	46	75	𝛤-commuting	𝛤-commute	VERB
iajs-3594	46	76	on	on	ADP
iajs-3594	46	77	𝑆	𝑆	PROPN
iajs-3594	46	78	,	,	PUNCT
iajs-3594	46	79	sameer	sameer	NOUN
iajs-3594	46	80	et	et	PROPN
iajs-3594	46	81	al	al	PROPN
iajs-3594	46	82	.	.	PROPN
iajs-3594	47	1	(	(	PUNCT
iajs-3594	47	2	11	11	NUM
iajs-3594	47	3	)	)	PUNCT
iajs-3594	47	4	.	.	PUNCT
iajs-3594	48	1	many	many	ADJ
iajs-3594	48	2	researchers	researcher	NOUN
iajs-3594	48	3	have	have	AUX
iajs-3594	48	4	studied	study	VERB
iajs-3594	48	5	centralizers	centralizer	NOUN
iajs-3594	48	6	and	and	CCONJ
iajs-3594	48	7	derivations	derivation	NOUN
iajs-3594	48	8	in	in	ADP
iajs-3594	48	9	prime	prime	ADJ
iajs-3594	48	10	and	and	CCONJ
iajs-3594	48	11	semiprime	semiprime	NOUN
iajs-3594	48	12	𝛤rings	𝛤rings	PROPN
iajs-3594	48	13	(	(	PUNCT
iajs-3594	48	14	1221	1221	NUM
iajs-3594	48	15	)	)	PUNCT
iajs-3594	48	16	and	and	CCONJ
iajs-3594	48	17	(	(	PUNCT
iajs-3594	48	18	22	22	NUM
iajs-3594	48	19	-	-	SYM
iajs-3594	48	20	30	30	NUM
iajs-3594	48	21	)	)	PUNCT
iajs-3594	48	22	.	.	PUNCT
iajs-3594	49	1	the	the	DET
iajs-3594	49	2	objective	objective	NOUN
iajs-3594	49	3	of	of	ADP
iajs-3594	49	4	this	this	DET
iajs-3594	49	5	paper	paper	NOUN
iajs-3594	49	6	is	be	AUX
iajs-3594	49	7	to	to	PART
iajs-3594	49	8	debate	debate	VERB
iajs-3594	49	9	,	,	PUNCT
iajs-3594	49	10	double	double	ADJ
iajs-3594	49	11	centralizer	centralizer	NOUN
iajs-3594	49	12	(	(	PUNCT
iajs-3594	49	13	t	t	PROPN
iajs-3594	49	14	,	,	PUNCT
iajs-3594	49	15	s	s	PART
iajs-3594	49	16	)	)	PUNCT
iajs-3594	49	17	,	,	PUNCT
iajs-3594	49	18	and	and	CCONJ
iajs-3594	49	19	double	double	ADJ
iajs-3594	49	20	jordan	jordan	PROPN
iajs-3594	49	21	centralizer	centralizer	NOUN
iajs-3594	49	22	on	on	ADP
iajs-3594	49	23	prime	prime	ADJ
iajs-3594	49	24	and	and	CCONJ
iajs-3594	49	25	semiprime	semiprime	NOUN
iajs-3594	49	26	𝛤rings	𝛤rings	PROPN
iajs-3594	49	27	,	,	PUNCT
iajs-3594	49	28	with	with	ADP
iajs-3594	49	29	fulfilling	fulfil	VERB
iajs-3594	49	30	certain	certain	ADJ
iajs-3594	49	31	identities	identity	NOUN
iajs-3594	49	32	.	.	PUNCT
iajs-3594	50	1	2	2	X
iajs-3594	50	2	.	.	NUM
iajs-3594	50	3	preliminaries	preliminary	NOUN
iajs-3594	50	4	and	and	CCONJ
iajs-3594	50	5	fundamentals	fundamental	NOUN
iajs-3594	50	6	2.1	2.1	NUM
iajs-3594	50	7	definition	definition	NOUN
iajs-3594	50	8	ali	ali	PROPN
iajs-3594	50	9	et	et	PROPN
iajs-3594	50	10	al	al	PROPN
iajs-3594	50	11	.	.	PROPN
iajs-3594	51	1	(	(	PUNCT
iajs-3594	51	2	18	18	NUM
iajs-3594	51	3	)	)	PUNCT
iajs-3594	51	4	let	let	VERB
iajs-3594	51	5	a	a	PRON
iajs-3594	51	6	be	be	AUX
iajs-3594	51	7	a	a	DET
iajs-3594	51	8	gamma	gamma	NOUN
iajs-3594	51	9	-	-	PUNCT
iajs-3594	51	10	ring	ring	NOUN
iajs-3594	51	11	,	,	PUNCT
iajs-3594	51	12	for	for	ADP
iajs-3594	51	13	any	any	DET
iajs-3594	51	14	𝑥	𝑥	PROPN
iajs-3594	51	15	,	,	PUNCT
iajs-3594	51	16	𝑦	𝑦	NOUN
iajs-3594	51	17	∈	∈	PROPN
iajs-3594	51	18	a	a	PRON
iajs-3594	51	19	and	and	CCONJ
iajs-3594	51	20	𝛼	𝛼	NOUN
iajs-3594	51	21	∈	∈	PROPN
iajs-3594	51	22	г	г	PROPN
iajs-3594	51	23	,	,	PUNCT
iajs-3594	51	24	the	the	DET
iajs-3594	51	25	symbol	symbol	NOUN
iajs-3594	51	26	[	[	X
iajs-3594	51	27	𝑟	𝑟	NOUN
iajs-3594	51	28	,	,	PUNCT
iajs-3594	51	29	𝑡]𝑎	𝑡]𝑎	NOUN
iajs-3594	51	30	=	=	PUNCT
iajs-3594	51	31	𝑟	𝑟	NOUN
iajs-3594	51	32	α	α	X
iajs-3594	51	33	𝑡	𝑡	PROPN
iajs-3594	51	34	‒	‒	X
iajs-3594	51	35	𝑡	𝑡	X
iajs-3594	51	36	𝛼	𝛼	PROPN
iajs-3594	51	37	𝑟	𝑟	NOUN
iajs-3594	51	38	,	,	PUNCT
iajs-3594	51	39	to	to	PART
iajs-3594	51	40	symbolize	symbolize	VERB
iajs-3594	51	41	the	the	DET
iajs-3594	51	42	commutator	commutator	NOUN
iajs-3594	51	43	.	.	PUNCT
iajs-3594	52	1	𝑇(𝑟	𝑇(𝑟	PUNCT
iajs-3594	52	2	⃘	⃘	PROPN
iajs-3594	52	3	𝑡	𝑡	NUM
iajs-3594	52	4	)	)	PUNCT
iajs-3594	52	5	=	=	PUNCT
iajs-3594	53	1	𝑟	𝑟	NOUN
iajs-3594	53	2	α	α	X
iajs-3594	53	3	𝑡	𝑡	X
iajs-3594	53	4	+	+	X
iajs-3594	53	5	𝑡	𝑡	X
iajs-3594	53	6	𝛼	𝛼	PROPN
iajs-3594	53	7	𝑟.	𝑟.	NOUN
iajs-3594	53	8	2.2	2.2	NUM
iajs-3594	53	9	lemma	lemma	PROPN
iajs-3594	53	10	ali	ali	PROPN
iajs-3594	53	11	et	et	PROPN
iajs-3594	53	12	al	al	PROPN
iajs-3594	53	13	.	.	PROPN
iajs-3594	54	1	(	(	PUNCT
iajs-3594	54	2	18	18	NUM
iajs-3594	54	3	)	)	PUNCT
iajs-3594	54	4	if	if	SCONJ
iajs-3594	54	5	a	a	PRON
iajs-3594	54	6	is	be	AUX
iajs-3594	54	7	a	a	DET
iajs-3594	54	8	gamma	gamma	NOUN
iajs-3594	54	9	-	-	PUNCT
iajs-3594	54	10	ring	ring	NOUN
iajs-3594	54	11	,	,	PUNCT
iajs-3594	54	12	for	for	ADP
iajs-3594	54	13	any	any	DET
iajs-3594	54	14	𝑟	𝑟	NOUN
iajs-3594	54	15	,	,	PUNCT
iajs-3594	54	16	𝑡	𝑡	PROPN
iajs-3594	54	17	,	,	PUNCT
iajs-3594	54	18	𝑠	𝑠	PROPN
iajs-3594	54	19	∈	∈	PROPN
iajs-3594	54	20	a	a	PRON
iajs-3594	54	21	and	and	CCONJ
iajs-3594	54	22	α	α	NOUN
iajs-3594	54	23	,	,	PUNCT
iajs-3594	54	24	𝛽	𝛽	PROPN
iajs-3594	54	25	∈	∈	PROPN
iajs-3594	54	26	г	г	PROPN
iajs-3594	54	27	thence	thence	NOUN
iajs-3594	54	28	:	:	PUNCT
iajs-3594	55	1	i.	i.	NOUN
iajs-3594	56	1	[	[	X
iajs-3594	56	2	𝑟	𝑟	X
iajs-3594	56	3	,	,	PUNCT
iajs-3594	56	4	𝑡]α	𝑡]α	ADJ
iajs-3594	56	5	+	+	PROPN
iajs-3594	56	6	[	[	X
iajs-3594	56	7	𝑡	𝑡	NOUN
iajs-3594	56	8	,	,	PUNCT
iajs-3594	56	9	𝑟]α	𝑟]α	NOUN
iajs-3594	56	10	=	=	SYM
iajs-3594	56	11	0	0	NUM
iajs-3594	56	12	ii	ii	NOUN
iajs-3594	56	13	.	.	PUNCT
iajs-3594	57	1	[	[	X
iajs-3594	57	2	𝑟	𝑟	X
iajs-3594	57	3	+	+	SYM
iajs-3594	57	4	𝑡	𝑡	PROPN
iajs-3594	57	5	,	,	PUNCT
iajs-3594	57	6	𝑠	𝑠	X
iajs-3594	57	7	]	]	X
iajs-3594	57	8	α	α	NOUN
iajs-3594	57	9	=	=	PUNCT
iajs-3594	58	1	[	[	X
iajs-3594	58	2	𝑟	𝑟	NOUN
iajs-3594	58	3	,	,	PUNCT
iajs-3594	58	4	𝑠]α+[𝑡	𝑠]α+[𝑡	PROPN
iajs-3594	58	5	,	,	PUNCT
iajs-3594	58	6	𝑠]α	𝑠]α	PROPN
iajs-3594	58	7	iii	iii	X
iajs-3594	58	8	.	.	PUNCT
iajs-3594	59	1	[	[	X
iajs-3594	59	2	𝑟	𝑟	X
iajs-3594	59	3	,	,	PUNCT
iajs-3594	59	4	𝑡	𝑡	PROPN
iajs-3594	59	5	+	+	PROPN
iajs-3594	59	6	𝑠]α=	𝑠]α=	PROPN
iajs-3594	59	7	[	[	X
iajs-3594	59	8	𝑟	𝑟	X
iajs-3594	59	9	,	,	PUNCT
iajs-3594	59	10	𝑡]α+[𝑟	𝑡]α+[𝑟	PROPN
iajs-3594	59	11	,	,	PUNCT
iajs-3594	59	12	𝑠]α	𝑠]α	PROPN
iajs-3594	59	13	iv	iv	NUM
iajs-3594	59	14	.	.	PUNCT
iajs-3594	60	1	[	[	X
iajs-3594	60	2	𝑟	𝑟	X
iajs-3594	60	3	,	,	PUNCT
iajs-3594	60	4	𝑡]α+β=	𝑡]α+β=	NOUN
iajs-3594	60	5	[	[	X
iajs-3594	60	6	𝑟	𝑟	X
iajs-3594	60	7	,	,	PUNCT
iajs-3594	60	8	𝑡]α+[𝑟	𝑡]α+[𝑟	PROPN
iajs-3594	60	9	,	,	PUNCT
iajs-3594	60	10	𝑡]ᵦ	𝑡]ᵦ	NOUN
iajs-3594	60	11	v.	v.	ADP
iajs-3594	61	1	[	[	X
iajs-3594	61	2	𝑟𝛽𝑡	𝑟𝛽𝑡	X
iajs-3594	61	3	,	,	PUNCT
iajs-3594	61	4	𝑠]α	𝑠]α	PROPN
iajs-3594	61	5	=	=	SYM
iajs-3594	61	6	𝑟𝛽[𝑡	𝑟𝛽[𝑡	PROPN
iajs-3594	61	7	,	,	PUNCT
iajs-3594	61	8	𝑠]α	𝑠]α	VERB
iajs-3594	61	9	+	+	CCONJ
iajs-3594	62	1	[	[	X
iajs-3594	62	2	𝑟	𝑟	NOUN
iajs-3594	62	3	,	,	PUNCT
iajs-3594	62	4	𝑠]α	𝑠]α	ADJ
iajs-3594	62	5	𝛽𝑡	𝛽𝑡	ADP
iajs-3594	62	6	+	+	NOUN
iajs-3594	62	7	𝑟	𝑟	X
iajs-3594	62	8	𝛽	𝛽	NOUN
iajs-3594	62	9	𝑠	𝑠	INTJ
iajs-3594	62	10	α	α	PROPN
iajs-3594	62	11	t	t	PROPN
iajs-3594	62	12	‒	‒	ADP
iajs-3594	62	13	𝑟	𝑟	X
iajs-3594	62	14	α	α	PROPN
iajs-3594	62	15	s	s	VERB
iajs-3594	62	16	𝛽	𝛽	X
iajs-3594	62	17	𝑡	𝑡	NOUN
iajs-3594	62	18	.	.	PUNCT
iajs-3594	63	1	2.3	2.3	NUM
iajs-3594	63	2	definition	definition	NOUN
iajs-3594	63	3	hoque	hoque	NOUN
iajs-3594	63	4	20(19	20(19	NUM
iajs-3594	63	5	)	)	PUNCT
iajs-3594	63	6	an	an	DET
iajs-3594	63	7	additive	additive	ADJ
iajs-3594	63	8	mapping	map	VERB
iajs-3594	63	9	𝑇	𝑇	PROPN
iajs-3594	63	10	:	:	PUNCT
iajs-3594	63	11	a	a	PRON
iajs-3594	63	12	→	→	X
iajs-3594	63	13	a	a	PRON
iajs-3594	63	14	is	be	AUX
iajs-3594	63	15	a	a	DET
iajs-3594	63	16	left	left	ADJ
iajs-3594	63	17	(	(	PUNCT
iajs-3594	63	18	right)centralizer	right)centralizer	NOUN
iajs-3594	63	19	,	,	PUNCT
iajs-3594	63	20	if	if	SCONJ
iajs-3594	63	21	𝑇(𝑟𝛼	𝑇(𝑟𝛼	ADV
iajs-3594	63	22	𝑡	𝑡	X
iajs-3594	63	23	)	)	PUNCT
iajs-3594	63	24	=	=	SYM
iajs-3594	63	25	𝑇	𝑇	PROPN
iajs-3594	63	26	(	(	PUNCT
iajs-3594	63	27	𝑟	𝑟	NOUN
iajs-3594	63	28	)	)	PUNCT
iajs-3594	64	1	𝛼	𝛼	PRON
iajs-3594	64	2	𝑡	𝑡	PROPN
iajs-3594	64	3	(	(	PUNCT
iajs-3594	64	4	𝑇	𝑇	PROPN
iajs-3594	64	5	(	(	PUNCT
iajs-3594	64	6	𝑟	𝑟	NOUN
iajs-3594	64	7	𝛼	𝛼	PART
iajs-3594	64	8	𝑡	𝑡	NOUN
iajs-3594	64	9	)	)	PUNCT
iajs-3594	64	10	=	=	PUNCT
iajs-3594	64	11	𝑟	𝑟	X
iajs-3594	64	12	𝛼	𝛼	X
iajs-3594	64	13	𝑇	𝑇	PROPN
iajs-3594	64	14	(	(	PUNCT
iajs-3594	64	15	𝑡	𝑡	NOUN
iajs-3594	64	16	)	)	PUNCT
iajs-3594	64	17	)	)	PUNCT
iajs-3594	64	18	holds	hold	VERB
iajs-3594	64	19	for	for	ADP
iajs-3594	64	20	any	any	DET
iajs-3594	64	21	𝑟	𝑟	NOUN
iajs-3594	64	22	,	,	PUNCT
iajs-3594	64	23	𝑡	𝑡	PROPN
iajs-3594	64	24	∈	∈	PROPN
iajs-3594	64	25	a	a	PRON
iajs-3594	64	26	and	and	CCONJ
iajs-3594	64	27	𝛼	𝛼	NOUN
iajs-3594	64	28	∈	∈	PROPN
iajs-3594	64	29	г	г	PROPN
iajs-3594	64	30	.	.	PUNCT
iajs-3594	65	1	a	a	DET
iajs-3594	65	2	centralizer	centralizer	NOUN
iajs-3594	65	3	is	be	AUX
iajs-3594	65	4	both	both	CCONJ
iajs-3594	65	5	a	a	DET
iajs-3594	65	6	left	left	ADJ
iajs-3594	65	7	and	and	CCONJ
iajs-3594	65	8	right	right	ADJ
iajs-3594	65	9	centralizer	centralizer	NOUN
iajs-3594	65	10	.	.	PUNCT
iajs-3594	66	1	ihjpas	ihjpas	PROPN
iajs-3594	66	2	.	.	PUNCT
iajs-3594	67	1	2025	2025	NUM
iajs-3594	67	2	,	,	PUNCT
iajs-3594	67	3	38(2	38(2	NUM
iajs-3594	67	4	)	)	PUNCT
iajs-3594	67	5	332	332	NUM
iajs-3594	67	6	2.4	2.4	NUM
iajs-3594	67	7	example	example	NOUN
iajs-3594	67	8	let	let	VERB
iajs-3594	67	9	f	f	PRON
iajs-3594	67	10	be	be	AUX
iajs-3594	67	11	a	a	DET
iajs-3594	67	12	field	field	NOUN
iajs-3594	67	13	,	,	PUNCT
iajs-3594	67	14	and	and	CCONJ
iajs-3594	67	15	𝐷2	𝐷2	NOUN
iajs-3594	67	16	(	(	PUNCT
iajs-3594	67	17	𝐹)be	𝐹)be	NOUN
iajs-3594	67	18	a	a	DET
iajs-3594	67	19	diagonal	diagonal	ADJ
iajs-3594	67	20	matrices	matrix	NOUN
iajs-3594	67	21	2	2	NUM
iajs-3594	67	22	by	by	ADP
iajs-3594	67	23	2	2	NUM
iajs-3594	67	24	over	over	ADP
iajs-3594	67	25	f	f	PROPN
iajs-3594	67	26	and	and	CCONJ
iajs-3594	67	27	γ	γ	X
iajs-3594	67	28	=	=	X
iajs-3594	67	29	{	{	PUNCT
iajs-3594	67	30	[	[	PUNCT
iajs-3594	67	31	0	0	NUM
iajs-3594	67	32	0	0	NUM
iajs-3594	67	33	0	0	NUM
iajs-3594	67	34	𝑛	𝑛	NOUN
iajs-3594	67	35	]	]	PUNCT
iajs-3594	67	36	,	,	PUNCT
iajs-3594	67	37	𝑛	𝑛	DET
iajs-3594	67	38	∈	∈	PROPN
iajs-3594	67	39	𝑍	𝑍	PROPN
iajs-3594	67	40	}	}	PUNCT
iajs-3594	67	41	,	,	PUNCT
iajs-3594	67	42	define	define	VERB
iajs-3594	67	43	𝑇	𝑇	PROPN
iajs-3594	67	44	:	:	PUNCT
iajs-3594	67	45	𝐷2	𝐷2	PROPN
iajs-3594	67	46	(	(	PUNCT
iajs-3594	67	47	𝐹	𝐹	PROPN
iajs-3594	67	48	)	)	PUNCT
iajs-3594	67	49	→	→	SYM
iajs-3594	67	50	𝐷2	𝐷2	PROPN
iajs-3594	67	51	(	(	PUNCT
iajs-3594	67	52	𝐹	𝐹	PROPN
iajs-3594	67	53	)	)	PUNCT
iajs-3594	67	54	as	as	ADP
iajs-3594	67	55	𝑇	𝑇	PROPN
iajs-3594	67	56	(	(	PUNCT
iajs-3594	67	57	[	[	PUNCT
iajs-3594	67	58	𝑎	𝑎	X
iajs-3594	67	59	0	0	NUM
iajs-3594	67	60	0	0	NUM
iajs-3594	67	61	𝑏	𝑏	NOUN
iajs-3594	67	62	]	]	PUNCT
iajs-3594	67	63	)	)	PUNCT
iajs-3594	68	1	=	=	PUNCT
iajs-3594	69	1	[	[	PUNCT
iajs-3594	69	2	0	0	NUM
iajs-3594	69	3	0	0	NUM
iajs-3594	69	4	0	0	NUM
iajs-3594	69	5	𝑏	𝑏	NOUN
iajs-3594	69	6	]	]	PUNCT
iajs-3594	69	7	,	,	PUNCT
iajs-3594	69	8	for	for	ADP
iajs-3594	69	9	any	any	DET
iajs-3594	69	10	𝑎	𝑎	NOUN
iajs-3594	69	11	,	,	PUNCT
iajs-3594	69	12	𝑏	𝑏	PROPN
iajs-3594	69	13	∈	∈	PROPN
iajs-3594	69	14	𝐹.	𝐹.	PROPN
iajs-3594	69	15	thence	thence	NOUN
iajs-3594	69	16	t	t	NOUN
iajs-3594	69	17	is	be	AUX
iajs-3594	69	18	a	a	DET
iajs-3594	69	19	centralizer	centralizer	NOUN
iajs-3594	69	20	.	.	PUNCT
iajs-3594	70	1	2.5	2.5	NUM
iajs-3594	70	2	definition	definition	NOUN
iajs-3594	70	3	hoque	hoque	NOUN
iajs-3594	70	4	(	(	PUNCT
iajs-3594	70	5	19)an	19)an	NUM
iajs-3594	70	6	additive	additive	ADJ
iajs-3594	70	7	mapping	mapping	NOUN
iajs-3594	70	8	𝑇	𝑇	PROPN
iajs-3594	70	9	:	:	PUNCT
iajs-3594	70	10	𝐴	𝐴	PROPN
iajs-3594	70	11	→	→	SYM
iajs-3594	70	12	𝐴	𝐴	PROPN
iajs-3594	70	13	,	,	PUNCT
iajs-3594	70	14	is	be	AUX
iajs-3594	70	15	jordan	jordan	PROPN
iajs-3594	70	16	left	left	PROPN
iajs-3594	70	17	(	(	PUNCT
iajs-3594	70	18	right	right	ADJ
iajs-3594	70	19	)	)	PUNCT
iajs-3594	70	20	centralizer	centralizer	NOUN
iajs-3594	70	21	,	,	PUNCT
iajs-3594	70	22	if	if	SCONJ
iajs-3594	70	23	𝑇(𝑥α	𝑇(𝑥α	ADJ
iajs-3594	70	24	x	x	X
iajs-3594	70	25	)	)	PUNCT
iajs-3594	70	26	=	=	SYM
iajs-3594	70	27	t(𝑥)𝛼𝑥	t(𝑥)𝛼𝑥	PROPN
iajs-3594	70	28	(	(	PUNCT
iajs-3594	70	29	𝑇(𝑥𝛼𝑥	𝑇(𝑥𝛼𝑥	ADJ
iajs-3594	70	30	)	)	PUNCT
iajs-3594	70	31	=	=	PUNCT
iajs-3594	70	32	𝑥𝛼𝑇(𝑥),for	𝑥𝛼𝑇(𝑥),for	ADP
iajs-3594	70	33	any	any	DET
iajs-3594	70	34	𝑥	𝑥	PRON
iajs-3594	70	35	∈	∈	PROPN
iajs-3594	70	36	a	a	PRON
iajs-3594	70	37	and	and	CCONJ
iajs-3594	70	38	α	α	NOUN
iajs-3594	70	39	∈	∈	PROPN
iajs-3594	70	40	г	г	PROPN
iajs-3594	70	41	.	.	PROPN
iajs-3594	70	42	2.6	2.6	NUM
iajs-3594	70	43	definition	definition	NOUN
iajs-3594	70	44	let	let	VERB
iajs-3594	70	45	a	a	PRON
iajs-3594	70	46	be	be	AUX
iajs-3594	70	47	gamma	gamma	NOUN
iajs-3594	70	48	-	-	PUNCT
iajs-3594	70	49	ring	ring	NOUN
iajs-3594	70	50	,	,	PUNCT
iajs-3594	70	51	let	let	VERB
iajs-3594	70	52	𝑇	𝑇	PROPN
iajs-3594	70	53	,	,	PUNCT
iajs-3594	70	54	𝑆	𝑆	PROPN
iajs-3594	70	55	:	:	PUNCT
iajs-3594	70	56	𝐴	𝐴	PROPN
iajs-3594	70	57	→	→	SYM
iajs-3594	70	58	𝐴	𝐴	PROPN
iajs-3594	70	59	,	,	PUNCT
iajs-3594	70	60	be	be	AUX
iajs-3594	70	61	an	an	DET
iajs-3594	70	62	additive	additive	ADJ
iajs-3594	70	63	mappings	mapping	NOUN
iajs-3594	70	64	,	,	PUNCT
iajs-3594	70	65	thence	thence	NOUN
iajs-3594	70	66	a	a	DET
iajs-3594	70	67	mate	mate	NOUN
iajs-3594	70	68	(	(	PUNCT
iajs-3594	70	69	t	t	PROPN
iajs-3594	70	70	,	,	PUNCT
iajs-3594	70	71	s	s	PART
iajs-3594	70	72	)	)	PUNCT
iajs-3594	70	73	is	be	AUX
iajs-3594	70	74	named	name	VERB
iajs-3594	70	75	a	a	DET
iajs-3594	70	76	double	double	ADJ
iajs-3594	70	77	centralizer	centralizer	NOUN
iajs-3594	70	78	,	,	PUNCT
iajs-3594	70	79	if	if	SCONJ
iajs-3594	70	80	𝑇	𝑇	PROPN
iajs-3594	70	81	is	be	AUX
iajs-3594	70	82	a	a	DET
iajs-3594	70	83	left	left	ADJ
iajs-3594	70	84	centralizer	centralizer	NOUN
iajs-3594	70	85	,	,	PUNCT
iajs-3594	70	86	s	s	PART
iajs-3594	70	87	is	be	AUX
iajs-3594	70	88	a	a	DET
iajs-3594	70	89	right	right	ADJ
iajs-3594	70	90	centralizer	centralizer	NOUN
iajs-3594	70	91	,	,	PUNCT
iajs-3594	70	92	and	and	CCONJ
iajs-3594	70	93	they	they	PRON
iajs-3594	70	94	satisfy	satisfy	VERB
iajs-3594	70	95	a	a	DET
iajs-3594	70	96	balanced	balanced	ADJ
iajs-3594	70	97	requirement	requirement	NOUN
iajs-3594	70	98	,	,	PUNCT
iajs-3594	70	99	𝑥α	𝑥α	ADP
iajs-3594	70	100	t(𝑦	t(𝑦	NOUN
iajs-3594	70	101	)	)	PUNCT
iajs-3594	70	102	=	=	SYM
iajs-3594	70	103	𝑆(𝑥	𝑆(𝑥	X
iajs-3594	70	104	)	)	PUNCT
iajs-3594	70	105	𝛼𝑦	𝛼𝑦	NOUN
iajs-3594	70	106	,	,	PUNCT
iajs-3594	70	107	for	for	ADP
iajs-3594	70	108	any	any	DET
iajs-3594	70	109	𝑥	𝑥	PROPN
iajs-3594	70	110	,	,	PUNCT
iajs-3594	70	111	𝑦	𝑦	PROPN
iajs-3594	70	112	∈	∈	PROPN
iajs-3594	70	113	𝐴	𝐴	PROPN
iajs-3594	70	114	,	,	PUNCT
iajs-3594	70	115	𝛼	𝛼	PROPN
iajs-3594	70	116	∈	∈	PROPN
iajs-3594	70	117	г	г	PROPN
iajs-3594	70	118	.	.	PROPN
iajs-3594	70	119	2.7	2.7	NUM
iajs-3594	70	120	definition	definition	NOUN
iajs-3594	70	121	let	let	VERB
iajs-3594	70	122	a	a	PRON
iajs-3594	70	123	be	be	AUX
iajs-3594	70	124	gamma	gamma	NOUN
iajs-3594	70	125	-	-	PUNCT
iajs-3594	70	126	ring	ring	NOUN
iajs-3594	70	127	,	,	PUNCT
iajs-3594	70	128	and	and	CCONJ
iajs-3594	70	129	let	let	VERB
iajs-3594	70	130	𝑇	𝑇	PROPN
iajs-3594	70	131	,	,	PUNCT
iajs-3594	70	132	𝑆	𝑆	PROPN
iajs-3594	70	133	:	:	PUNCT
iajs-3594	70	134	𝐴	𝐴	PROPN
iajs-3594	70	135	→	→	SYM
iajs-3594	70	136	𝐴	𝐴	PROPN
iajs-3594	70	137	,	,	PUNCT
iajs-3594	70	138	be	be	AUX
iajs-3594	70	139	an	an	DET
iajs-3594	70	140	additive	additive	ADJ
iajs-3594	70	141	mapping	mapping	NOUN
iajs-3594	70	142	,	,	PUNCT
iajs-3594	70	143	thence	thence	NOUN
iajs-3594	70	144	a	a	DET
iajs-3594	70	145	mate	mate	NOUN
iajs-3594	70	146	(	(	PUNCT
iajs-3594	70	147	t	t	PROPN
iajs-3594	70	148	,	,	PUNCT
iajs-3594	70	149	s	s	PART
iajs-3594	70	150	)	)	PUNCT
iajs-3594	70	151	is	be	AUX
iajs-3594	70	152	named	name	VERB
iajs-3594	70	153	a	a	DET
iajs-3594	70	154	double	double	ADJ
iajs-3594	70	155	jordan	jordan	PROPN
iajs-3594	70	156	centralizer	centralizer	NOUN
iajs-3594	70	157	,	,	PUNCT
iajs-3594	70	158	if	if	SCONJ
iajs-3594	70	159	t	t	PROPN
iajs-3594	70	160	is	be	AUX
iajs-3594	70	161	a	a	DET
iajs-3594	70	162	left	left	ADJ
iajs-3594	70	163	jordan	jordan	PROPN
iajs-3594	70	164	centralizer	centralizer	NOUN
iajs-3594	70	165	,	,	PUNCT
iajs-3594	70	166	s	s	PART
iajs-3594	70	167	is	be	AUX
iajs-3594	70	168	a	a	DET
iajs-3594	70	169	right	right	ADJ
iajs-3594	70	170	jordan	jordan	PROPN
iajs-3594	70	171	centralizer	centralizer	NOUN
iajs-3594	70	172	,	,	PUNCT
iajs-3594	70	173	and	and	CCONJ
iajs-3594	70	174	they	they	PRON
iajs-3594	70	175	satisfy	satisfy	VERB
iajs-3594	70	176	a	a	DET
iajs-3594	70	177	balanced	balanced	ADJ
iajs-3594	70	178	requirement	requirement	NOUN
iajs-3594	70	179	,	,	PUNCT
iajs-3594	70	180	(	(	PUNCT
iajs-3594	70	181	𝑥𝛼	𝑥𝛼	ADP
iajs-3594	70	182	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	70	183	)	)	PUNCT
iajs-3594	70	184	=	=	SYM
iajs-3594	70	185	𝑆(𝑥	𝑆(𝑥	X
iajs-3594	70	186	)	)	PUNCT
iajs-3594	70	187	𝛼	𝛼	PRON
iajs-3594	70	188	𝑥	𝑥	NOUN
iajs-3594	70	189	)	)	PUNCT
iajs-3594	70	190	,	,	PUNCT
iajs-3594	70	191	for	for	ADP
iajs-3594	70	192	any	any	DET
iajs-3594	70	193	𝑥	𝑥	PRON
iajs-3594	70	194	∈	∈	PROPN
iajs-3594	70	195	𝐴	𝐴	PROPN
iajs-3594	70	196	,	,	PUNCT
iajs-3594	70	197	𝛼	𝛼	PROPN
iajs-3594	70	198	∈	∈	PROPN
iajs-3594	70	199	г	г	PROPN
iajs-3594	70	200	.	.	PROPN
iajs-3594	70	201	3	3	NUM
iajs-3594	70	202	.	.	X
iajs-3594	70	203	main	main	ADJ
iajs-3594	70	204	results	result	NOUN
iajs-3594	70	205	in	in	ADP
iajs-3594	70	206	the	the	DET
iajs-3594	70	207	following	following	NOUN
iajs-3594	70	208	,	,	PUNCT
iajs-3594	70	209	we	we	PRON
iajs-3594	70	210	give	give	VERB
iajs-3594	70	211	the	the	DET
iajs-3594	70	212	definition	definition	NOUN
iajs-3594	70	213	of	of	ADP
iajs-3594	70	214	commuting	commute	VERB
iajs-3594	70	215	double	double	ADJ
iajs-3594	70	216	centralizer	centralizer	NOUN
iajs-3594	70	217	:	:	PUNCT
iajs-3594	70	218	3.1	3.1	NUM
iajs-3594	70	219	definition	definition	NOUN
iajs-3594	70	220	let	let	VERB
iajs-3594	70	221	a	a	PRON
iajs-3594	70	222	be	be	AUX
iajs-3594	70	223	a	a	DET
iajs-3594	70	224	𝛤-ring	𝛤-ring	PROPN
iajs-3594	70	225	,	,	PUNCT
iajs-3594	70	226	and	and	CCONJ
iajs-3594	70	227	(	(	PUNCT
iajs-3594	70	228	t	t	PROPN
iajs-3594	70	229	,	,	PUNCT
iajs-3594	70	230	s	s	PART
iajs-3594	70	231	)	)	PUNCT
iajs-3594	70	232	,	,	PUNCT
iajs-3594	70	233	be	be	AUX
iajs-3594	70	234	a	a	DET
iajs-3594	70	235	double	double	ADJ
iajs-3594	70	236	centralizer	centralizer	NOUN
iajs-3594	70	237	.	.	PUNCT
iajs-3594	71	1	thence	thence	NOUN
iajs-3594	71	2	(	(	PUNCT
iajs-3594	71	3	t	t	PROPN
iajs-3594	71	4	,	,	PUNCT
iajs-3594	71	5	s	s	PART
iajs-3594	71	6	)	)	PUNCT
iajs-3594	71	7	,	,	PUNCT
iajs-3594	71	8	is	be	AUX
iajs-3594	71	9	named	name	VERB
iajs-3594	71	10	commuting	commute	VERB
iajs-3594	71	11	double	double	ADJ
iajs-3594	71	12	centralizer	centralizer	NOUN
iajs-3594	71	13	,	,	PUNCT
iajs-3594	71	14	if	if	SCONJ
iajs-3594	71	15	t	t	PROPN
iajs-3594	71	16	and	and	CCONJ
iajs-3594	71	17	s	s	VERB
iajs-3594	71	18	are	be	AUX
iajs-3594	71	19	commuting	commute	VERB
iajs-3594	71	20	.	.	PUNCT
iajs-3594	72	1	now	now	ADV
iajs-3594	72	2	,	,	PUNCT
iajs-3594	72	3	we	we	PRON
iajs-3594	72	4	shall	shall	AUX
iajs-3594	72	5	give	give	VERB
iajs-3594	72	6	an	an	DET
iajs-3594	72	7	example	example	NOUN
iajs-3594	72	8	for	for	ADP
iajs-3594	72	9	a	a	DET
iajs-3594	72	10	commuting	commute	VERB
iajs-3594	72	11	double	double	ADJ
iajs-3594	72	12	centralizer	centralizer	NOUN
iajs-3594	72	13	.	.	PUNCT
iajs-3594	73	1	3.2	3.2	NUM
iajs-3594	73	2	example	example	NOUN
iajs-3594	73	3	let	let	VERB
iajs-3594	73	4	f	f	PRON
iajs-3594	73	5	be	be	AUX
iajs-3594	73	6	a	a	DET
iajs-3594	73	7	field	field	NOUN
iajs-3594	73	8	,	,	PUNCT
iajs-3594	73	9	and	and	CCONJ
iajs-3594	73	10	a	a	DET
iajs-3594	73	11	be	be	AUX
iajs-3594	73	12	a	a	DET
iajs-3594	73	13	𝛤-ring	𝛤-ring	NOUN
iajs-3594	73	14	of	of	ADP
iajs-3594	73	15	all	all	DET
iajs-3594	73	16	triangular	triangular	NOUN
iajs-3594	73	17	matrices	matrix	NOUN
iajs-3594	73	18	of	of	ADP
iajs-3594	73	19	the	the	PRON
iajs-3594	73	20	from	from	ADP
iajs-3594	73	21	𝑥	𝑥	PROPN
iajs-3594	73	22	=	=	PUNCT
iajs-3594	73	23	{	{	PUNCT
iajs-3594	73	24	[	[	PUNCT
iajs-3594	73	25	𝑑	𝑑	NOUN
iajs-3594	73	26	0	0	NUM
iajs-3594	73	27	0	0	NUM
iajs-3594	73	28	0	0	NUM
iajs-3594	74	1	𝑎	𝑎	PRON
iajs-3594	74	2	𝑑	𝑑	NOUN
iajs-3594	74	3	0	0	NUM
iajs-3594	74	4	0	0	NUM
iajs-3594	74	5	𝑐	𝑐	NOUN
iajs-3594	74	6	0	0	NUM
iajs-3594	75	1	𝑑	𝑑	NOUN
iajs-3594	75	2	0	0	NUM
iajs-3594	75	3	𝑏	𝑏	PROPN
iajs-3594	75	4	𝑐	𝑐	PROPN
iajs-3594	75	5	−𝑎	−𝑎	NOUN
iajs-3594	75	6	𝑑	𝑑	X
iajs-3594	75	7	]	]	PUNCT
iajs-3594	75	8	,	,	PUNCT
iajs-3594	75	9	𝑓𝑜𝑟𝑎𝑙𝑙	𝑓𝑜𝑟𝑎𝑙𝑙	PROPN
iajs-3594	75	10	𝑎	𝑎	PROPN
iajs-3594	75	11	,	,	PUNCT
iajs-3594	75	12	𝑏	𝑏	NOUN
iajs-3594	75	13	,	,	PUNCT
iajs-3594	75	14	𝑐	𝑐	PROPN
iajs-3594	75	15	,	,	PUNCT
iajs-3594	75	16	𝑑	𝑑	PROPN
iajs-3594	75	17	∈	∈	PROPN
iajs-3594	75	18	𝐹	𝐹	PROPN
iajs-3594	75	19	}	}	PUNCT
iajs-3594	75	20	,	,	PUNCT
iajs-3594	75	21	and	and	CCONJ
iajs-3594	75	22	г	г	PROPN
iajs-3594	75	23	=	=	X
iajs-3594	75	24	{	{	PUNCT
iajs-3594	75	25	[	[	PUNCT
iajs-3594	75	26	0	0	NUM
iajs-3594	75	27	0	0	NUM
iajs-3594	75	28	0	0	NUM
iajs-3594	75	29	0	0	NUM
iajs-3594	75	30	0	0	NUM
iajs-3594	75	31	0	0	NUM
iajs-3594	75	32	0	0	NUM
iajs-3594	75	33	0	0	NUM
iajs-3594	75	34	0	0	NUM
iajs-3594	75	35	0	0	NUM
iajs-3594	75	36	0	0	NUM
iajs-3594	75	37	0	0	NUM
iajs-3594	76	1	𝑛	𝑛	PRON
iajs-3594	76	2	0	0	NUM
iajs-3594	76	3	0	0	NUM
iajs-3594	76	4	0	0	NUM
iajs-3594	76	5	]	]	PUNCT
iajs-3594	76	6	,	,	PUNCT
iajs-3594	76	7	𝑓𝑜𝑟𝑎𝑙𝑙	𝑓𝑜𝑟𝑎𝑙𝑙	NOUN
iajs-3594	76	8	𝑛	𝑛	PRON
iajs-3594	76	9	∈	∈	PROPN
iajs-3594	76	10	z	z	PROPN
iajs-3594	76	11	}	}	PUNCT
iajs-3594	76	12	.	.	PUNCT
iajs-3594	77	1	in	in	ADP
iajs-3594	77	2	connection	connection	NOUN
iajs-3594	77	3	to	to	ADP
iajs-3594	77	4	the	the	DET
iajs-3594	77	5	frequent	frequent	ADJ
iajs-3594	77	6	process	process	NOUN
iajs-3594	77	7	of	of	ADP
iajs-3594	77	8	addition	addition	NOUN
iajs-3594	77	9	and	and	CCONJ
iajs-3594	77	10	multiplication	multiplication	NOUN
iajs-3594	77	11	and	and	CCONJ
iajs-3594	77	12	let	let	VERB
iajs-3594	77	13	t	t	PROPN
iajs-3594	77	14	,	,	PUNCT
iajs-3594	77	15	s	s	PART
iajs-3594	77	16	:	:	PUNCT
iajs-3594	77	17	a→a	a→a	ADV
iajs-3594	77	18	be	be	AUX
iajs-3594	77	19	additive	additive	ADJ
iajs-3594	77	20	mappings	mapping	NOUN
iajs-3594	77	21	defined	define	VERB
iajs-3594	77	22	by	by	ADP
iajs-3594	77	23	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	77	24	)	)	PUNCT
iajs-3594	77	25	=	=	PRON
iajs-3594	77	26	𝑦𝛼𝑥	𝑦𝛼𝑥	NOUN
iajs-3594	77	27	and	and	CCONJ
iajs-3594	77	28	𝑆(𝑥	𝑆(𝑥	NUM
iajs-3594	77	29	)	)	PUNCT
iajs-3594	77	30	=	=	SYM
iajs-3594	77	31	𝑥𝛼𝑦	𝑥𝛼𝑦	PROPN
iajs-3594	77	32	,	,	PUNCT
iajs-3594	77	33	for	for	ADP
iajs-3594	77	34	each	each	DET
iajs-3594	77	35	𝑥	𝑥	PROPN
iajs-3594	77	36	,	,	PUNCT
iajs-3594	77	37	𝑦	𝑦	NOUN
iajs-3594	77	38	∈	∈	PROPN
iajs-3594	77	39	a	a	PRON
iajs-3594	77	40	and	and	CCONJ
iajs-3594	77	41	𝛼	𝛼	NOUN
iajs-3594	77	42	∈	∈	PROPN
iajs-3594	77	43	г	г	PROPN
iajs-3594	77	44	.	.	PROPN
iajs-3594	78	1	where	where	SCONJ
iajs-3594	78	2	;	;	PUNCT
iajs-3594	79	1	𝑦	𝑦	X
iajs-3594	79	2	=	=	PUNCT
iajs-3594	79	3	[	[	PUNCT
iajs-3594	79	4	0	0	NUM
iajs-3594	79	5	0	0	NUM
iajs-3594	79	6	0	0	NUM
iajs-3594	79	7	0	0	NUM
iajs-3594	79	8	0	0	NUM
iajs-3594	79	9	0	0	NUM
iajs-3594	79	10	0	0	NUM
iajs-3594	79	11	0	0	NUM
iajs-3594	79	12	0	0	NUM
iajs-3594	79	13	0	0	NUM
iajs-3594	79	14	0	0	NUM
iajs-3594	79	15	0	0	NUM
iajs-3594	80	1	𝑏	𝑏	NOUN
iajs-3594	80	2	0	0	NUM
iajs-3594	80	3	0	0	NUM
iajs-3594	80	4	0	0	NUM
iajs-3594	80	5	]	]	PUNCT
iajs-3594	80	6	,	,	PUNCT
iajs-3594	80	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3594	80	8	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3594	80	9	𝑏	𝑏	PROPN
iajs-3594	80	10	∈	∈	PROPN
iajs-3594	80	11	f.	f.	NOUN
iajs-3594	80	12	it	it	PRON
iajs-3594	80	13	is	be	AUX
iajs-3594	80	14	clear	clear	ADJ
iajs-3594	80	15	that	that	SCONJ
iajs-3594	80	16	t	t	PROPN
iajs-3594	80	17	and	and	CCONJ
iajs-3594	80	18	s	s	VERB
iajs-3594	80	19	are	be	AUX
iajs-3594	80	20	commuting	commute	VERB
iajs-3594	80	21	double	double	ADJ
iajs-3594	80	22	centralizer	centralizer	NOUN
iajs-3594	80	23	.	.	PUNCT
iajs-3594	81	1	in	in	ADP
iajs-3594	81	2	the	the	DET
iajs-3594	81	3	following	follow	VERB
iajs-3594	81	4	results	result	NOUN
iajs-3594	81	5	,	,	PUNCT
iajs-3594	81	6	we	we	PRON
iajs-3594	81	7	give	give	VERB
iajs-3594	81	8	some	some	DET
iajs-3594	81	9	certain	certain	ADJ
iajs-3594	81	10	conditions	condition	NOUN
iajs-3594	81	11	to	to	PART
iajs-3594	81	12	obtain	obtain	VERB
iajs-3594	81	13	(	(	PUNCT
iajs-3594	81	14	t	t	PROPN
iajs-3594	81	15	,	,	PUNCT
iajs-3594	81	16	s	s	PART
iajs-3594	81	17	)	)	PUNCT
iajs-3594	81	18	is	be	AUX
iajs-3594	81	19	a	a	DET
iajs-3594	81	20	double	double	ADJ
iajs-3594	81	21	centralizer	centralizer	NOUN
iajs-3594	81	22	,	,	PUNCT
iajs-3594	81	23	where	where	SCONJ
iajs-3594	81	24	t	t	NOUN
iajs-3594	81	25	and	and	CCONJ
iajs-3594	81	26	s	s	PRON
iajs-3594	81	27	are	be	AUX
iajs-3594	81	28	from	from	ADP
iajs-3594	81	29	a	a	PRON
iajs-3594	81	30	to	to	ADP
iajs-3594	81	31	a.	a.	NOUN
iajs-3594	81	32	3.3	3.3	NUM
iajs-3594	81	33	theorem	theorem	NOUN
iajs-3594	81	34	let	let	VERB
iajs-3594	81	35	a	a	PRON
iajs-3594	81	36	be	be	AUX
iajs-3594	81	37	a	a	DET
iajs-3594	81	38	semiprime	semiprime	NOUN
iajs-3594	81	39	𝛤-ring	𝛤-ring	PROPN
iajs-3594	81	40	and	and	CCONJ
iajs-3594	81	41	t	t	PROPN
iajs-3594	81	42	,	,	PUNCT
iajs-3594	81	43	s	s	PART
iajs-3594	81	44	:	:	PUNCT
iajs-3594	81	45	a→	a→	PUNCT
iajs-3594	81	46	a	a	PRON
iajs-3594	81	47	be	be	AUX
iajs-3594	81	48	a	a	DET
iajs-3594	81	49	mapping	mapping	NOUN
iajs-3594	81	50	fulfilling	fulfil	VERB
iajs-3594	81	51	.	.	PUNCT
iajs-3594	82	1	𝑥𝛼𝑇(𝑦	𝑥𝛼𝑇(𝑦	NOUN
iajs-3594	82	2	)	)	PUNCT
iajs-3594	82	3	=	=	SYM
iajs-3594	82	4	𝑆(𝑥)𝛼𝑦	𝑆(𝑥)𝛼𝑦	PROPN
iajs-3594	82	5	,	,	PUNCT
iajs-3594	82	6	for	for	ADP
iajs-3594	82	7	each	each	DET
iajs-3594	82	8	𝑥	𝑥	PROPN
iajs-3594	82	9	,	,	PUNCT
iajs-3594	82	10	𝑦	𝑦	NOUN
iajs-3594	82	11	∈	∈	PROPN
iajs-3594	82	12	a	a	PRON
iajs-3594	82	13	and	and	CCONJ
iajs-3594	82	14	α	α	NOUN
iajs-3594	82	15	∈	∈	PROPN
iajs-3594	82	16	г	г	PROPN
iajs-3594	82	17	.	.	PUNCT
iajs-3594	83	1	(	(	PUNCT
iajs-3594	83	2	1	1	X
iajs-3594	83	3	)	)	PUNCT
iajs-3594	83	4	thence	thence	NOUN
iajs-3594	83	5	(	(	PUNCT
iajs-3594	83	6	t	t	PROPN
iajs-3594	83	7	,	,	PUNCT
iajs-3594	83	8	s	s	PART
iajs-3594	83	9	)	)	PUNCT
iajs-3594	83	10	is	be	AUX
iajs-3594	83	11	a	a	DET
iajs-3594	83	12	double	double	ADJ
iajs-3594	83	13	centralizer	centralizer	NOUN
iajs-3594	83	14	.	.	PUNCT
iajs-3594	84	1	ihjpas	ihjpas	PROPN
iajs-3594	84	2	.	.	PUNCT
iajs-3594	85	1	2025	2025	NUM
iajs-3594	85	2	,	,	PUNCT
iajs-3594	85	3	38(2	38(2	NUM
iajs-3594	85	4	)	)	PUNCT
iajs-3594	85	5	333	333	NUM
iajs-3594	85	6	proof	proof	NOUN
iajs-3594	85	7	:	:	PUNCT
iajs-3594	85	8	we	we	PRON
iajs-3594	85	9	need	need	VERB
iajs-3594	85	10	to	to	PART
iajs-3594	85	11	show	show	VERB
iajs-3594	85	12	that	that	SCONJ
iajs-3594	85	13	t	t	NOUN
iajs-3594	85	14	,	,	PUNCT
iajs-3594	85	15	s	s	VERB
iajs-3594	85	16	are	be	AUX
iajs-3594	85	17	additive	additive	ADJ
iajs-3594	85	18	mapping	mapping	NOUN
iajs-3594	85	19	,	,	PUNCT
iajs-3594	85	20	and	and	CCONJ
iajs-3594	85	21	𝑇	𝑇	PROPN
iajs-3594	85	22	(	(	PUNCT
iajs-3594	85	23	𝑥	𝑥	PROPN
iajs-3594	85	24	𝛼	𝛼	NOUN
iajs-3594	85	25	𝑦	𝑦	NOUN
iajs-3594	85	26	)	)	PUNCT
iajs-3594	85	27	=	=	SYM
iajs-3594	85	28	𝑇	𝑇	PROPN
iajs-3594	85	29	(	(	PUNCT
iajs-3594	85	30	𝑥	𝑥	NOUN
iajs-3594	85	31	)	)	PUNCT
iajs-3594	85	32	𝛼	𝛼	NOUN
iajs-3594	85	33	𝑦	𝑦	NOUN
iajs-3594	85	34	,	,	PUNCT
iajs-3594	85	35	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3594	85	36	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3594	85	37	𝑥	𝑥	PROPN
iajs-3594	85	38	,	,	PUNCT
iajs-3594	85	39	𝑦	𝑦	PROPN
iajs-3594	85	40	∈	∈	PROPN
iajs-3594	85	41	𝐴	𝐴	PROPN
iajs-3594	85	42	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	85	43	𝛼	𝛼	SYM
iajs-3594	85	44	∈	∈	PROPN
iajs-3594	85	45	𝛤.	𝛤.	PROPN
iajs-3594	85	46	𝑆	𝑆	PROPN
iajs-3594	85	47	(	(	PUNCT
iajs-3594	85	48	𝑥	𝑥	PROPN
iajs-3594	85	49	𝛼	𝛼	NOUN
iajs-3594	85	50	𝑦	𝑦	NOUN
iajs-3594	85	51	)	)	PUNCT
iajs-3594	85	52	=	=	PUNCT
iajs-3594	85	53	𝑥	𝑥	DET
iajs-3594	85	54	𝛼	𝛼	X
iajs-3594	85	55	𝑆	𝑆	PROPN
iajs-3594	85	56	(	(	PUNCT
iajs-3594	85	57	𝑦	𝑦	NOUN
iajs-3594	85	58	)	)	PUNCT
iajs-3594	85	59	,	,	PUNCT
iajs-3594	85	60	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3594	85	61	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3594	85	62	𝑥	𝑥	PROPN
iajs-3594	85	63	,	,	PUNCT
iajs-3594	85	64	𝑦	𝑦	PROPN
iajs-3594	85	65	∈	∈	PROPN
iajs-3594	85	66	𝐴	𝐴	PROPN
iajs-3594	85	67	,	,	PUNCT
iajs-3594	85	68	𝛼	𝛼	AUX
iajs-3594	85	69	∈	∈	PROPN
iajs-3594	85	70	𝛤.	𝛤.	PROPN
iajs-3594	85	71	now	now	ADV
iajs-3594	85	72	replace	replace	VERB
iajs-3594	85	73	𝑦	𝑦	NOUN
iajs-3594	85	74	by	by	ADP
iajs-3594	85	75	𝑦	𝑦	NOUN
iajs-3594	85	76	+	+	X
iajs-3594	85	77	z	z	NOUN
iajs-3594	85	78	in	in	ADP
iajs-3594	85	79	(	(	PUNCT
iajs-3594	85	80	1	1	NUM
iajs-3594	85	81	)	)	PUNCT
iajs-3594	85	82	,	,	PUNCT
iajs-3594	85	83	we	we	PRON
iajs-3594	85	84	imply	imply	VERB
iajs-3594	85	85	𝑥𝛼	𝑥𝛼	ADP
iajs-3594	85	86	𝑇(𝑦	𝑇(𝑦	PROPN
iajs-3594	85	87	+	+	CCONJ
iajs-3594	85	88	𝑧	𝑧	X
iajs-3594	85	89	)	)	PUNCT
iajs-3594	85	90	=	=	SYM
iajs-3594	85	91	𝑆(𝑥)𝛼𝑦	𝑆(𝑥)𝛼𝑦	PROPN
iajs-3594	85	92	+	+	CCONJ
iajs-3594	85	93	𝑆	𝑆	PROPN
iajs-3594	85	94	(	(	PUNCT
iajs-3594	85	95	𝑥)𝛼𝑧	𝑥)𝛼𝑧	PROPN
iajs-3594	85	96	,	,	PUNCT
iajs-3594	85	97	for	for	ADP
iajs-3594	85	98	each	each	DET
iajs-3594	85	99	𝑥	𝑥	PROPN
iajs-3594	85	100	,	,	PUNCT
iajs-3594	85	101	𝑦	𝑦	NOUN
iajs-3594	85	102	,	,	PUNCT
iajs-3594	85	103	𝑧	𝑧	DET
iajs-3594	85	104	∈	∈	PROPN
iajs-3594	85	105	𝐴	𝐴	NOUN
iajs-3594	85	106	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	85	107	𝛼	𝛼	SYM
iajs-3594	85	108	∈	∈	NOUN
iajs-3594	85	109	𝛤.	𝛤.	PROPN
iajs-3594	85	110	hence	hence	ADV
iajs-3594	85	111	𝑥𝛼	𝑥𝛼	AUX
iajs-3594	85	112	(	(	PUNCT
iajs-3594	85	113	𝑇(𝑦	𝑇(𝑦	X
iajs-3594	85	114	+	+	CCONJ
iajs-3594	85	115	𝑧	𝑧	X
iajs-3594	85	116	)	)	PUNCT
iajs-3594	85	117	−	−	PROPN
iajs-3594	85	118	𝑇	𝑇	PROPN
iajs-3594	85	119	(	(	PUNCT
iajs-3594	85	120	𝑦	𝑦	NOUN
iajs-3594	85	121	)	)	PUNCT
iajs-3594	85	122	−	−	PROPN
iajs-3594	85	123	𝑇(𝑧	𝑇(𝑧	NOUN
iajs-3594	85	124	)	)	PUNCT
iajs-3594	85	125	)	)	PUNCT
iajs-3594	85	126	=	=	SYM
iajs-3594	86	1	0	0	NUM
iajs-3594	86	2	,	,	PUNCT
iajs-3594	86	3	for	for	ADP
iajs-3594	86	4	𝑎𝑙𝑙	𝑎𝑙𝑙	PROPN
iajs-3594	86	5	𝑥	𝑥	PROPN
iajs-3594	86	6	,	,	PUNCT
iajs-3594	86	7	𝑦	𝑦	NOUN
iajs-3594	86	8	,	,	PUNCT
iajs-3594	86	9	𝑧	𝑧	DET
iajs-3594	86	10	∈	∈	PROPN
iajs-3594	86	11	𝐴	𝐴	NOUN
iajs-3594	86	12	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	86	13	𝛼	𝛼	SYM
iajs-3594	86	14	∈	∈	PROPN
iajs-3594	86	15	𝛤.	𝛤.	PROPN
iajs-3594	86	16	by	by	ADP
iajs-3594	86	17	the	the	DET
iajs-3594	86	18	semiprimeness	semiprimeness	NOUN
iajs-3594	86	19	of	of	ADP
iajs-3594	86	20	a	a	PRON
iajs-3594	86	21	,	,	PUNCT
iajs-3594	86	22	we	we	PRON
iajs-3594	86	23	imply	imply	VERB
iajs-3594	86	24	𝑇(𝑦	𝑇(𝑦	X
iajs-3594	86	25	+	+	CCONJ
iajs-3594	86	26	𝑧	𝑧	X
iajs-3594	86	27	)	)	PUNCT
iajs-3594	86	28	=	=	SYM
iajs-3594	86	29	𝑇(𝑦	𝑇(𝑦	VERB
iajs-3594	86	30	)	)	PUNCT
iajs-3594	86	31	+	+	CCONJ
iajs-3594	86	32	𝑇	𝑇	PROPN
iajs-3594	86	33	(	(	PUNCT
iajs-3594	86	34	𝑧	𝑧	NOUN
iajs-3594	86	35	)	)	PUNCT
iajs-3594	86	36	,	,	PUNCT
iajs-3594	86	37	for	for	ADP
iajs-3594	86	38	each	each	DET
iajs-3594	86	39	𝑦	𝑦	NOUN
iajs-3594	86	40	,	,	PUNCT
iajs-3594	86	41	𝑧	𝑧	DET
iajs-3594	86	42	∈	∈	NOUN
iajs-3594	86	43	𝐴.	𝐴.	NOUN
iajs-3594	86	44	similarly	similarly	ADV
iajs-3594	86	45	,	,	PUNCT
iajs-3594	86	46	we	we	PRON
iajs-3594	86	47	can	can	AUX
iajs-3594	86	48	show	show	VERB
iajs-3594	86	49	that	that	SCONJ
iajs-3594	86	50	𝑆(𝑥	𝑆(𝑥	NUM
iajs-3594	86	51	+	+	NUM
iajs-3594	86	52	𝑦	𝑦	X
iajs-3594	86	53	)	)	PUNCT
iajs-3594	86	54	=	=	SYM
iajs-3594	86	55	𝑆	𝑆	PROPN
iajs-3594	86	56	(	(	PUNCT
iajs-3594	86	57	𝑥	𝑥	NOUN
iajs-3594	86	58	)	)	PUNCT
iajs-3594	86	59	+	+	CCONJ
iajs-3594	86	60	𝑆	𝑆	PROPN
iajs-3594	86	61	(	(	PUNCT
iajs-3594	86	62	𝑦	𝑦	NOUN
iajs-3594	86	63	)	)	PUNCT
iajs-3594	86	64	,	,	PUNCT
iajs-3594	86	65	for	for	ADP
iajs-3594	86	66	each	each	DET
iajs-3594	86	67	𝑥	𝑥	PROPN
iajs-3594	86	68	,	,	PUNCT
iajs-3594	86	69	𝑦	𝑦	NOUN
iajs-3594	86	70	,	,	PUNCT
iajs-3594	86	71	∈	∈	PROPN
iajs-3594	86	72	𝐴.	𝐴.	PROPN
iajs-3594	86	73	now	now	ADV
iajs-3594	86	74	,	,	PUNCT
iajs-3594	86	75	replacing	replace	VERB
iajs-3594	86	76	𝑦	𝑦	NOUN
iajs-3594	86	77	with	with	ADP
iajs-3594	86	78	𝑦𝛽𝑧	𝑦𝛽𝑧	NOUN
iajs-3594	86	79	in	in	ADP
iajs-3594	86	80	(	(	PUNCT
iajs-3594	86	81	1	1	X
iajs-3594	86	82	)	)	PUNCT
iajs-3594	86	83	we	we	PRON
iajs-3594	86	84	obtain	obtain	VERB
iajs-3594	86	85	𝑥	𝑥	PRON
iajs-3594	86	86	𝛼	𝛼	NOUN
iajs-3594	86	87	(	(	PUNCT
iajs-3594	86	88	𝑇	𝑇	PROPN
iajs-3594	86	89	(	(	PUNCT
iajs-3594	86	90	𝑦	𝑦	NOUN
iajs-3594	86	91	𝛽𝑧	𝛽𝑧	NOUN
iajs-3594	86	92	)	)	PUNCT
iajs-3594	86	93	−	−	PROPN
iajs-3594	86	94	𝑇	𝑇	PROPN
iajs-3594	86	95	(	(	PUNCT
iajs-3594	86	96	𝑦	𝑦	NOUN
iajs-3594	86	97	)	)	PUNCT
iajs-3594	86	98	𝛽𝑧	𝛽𝑧	NOUN
iajs-3594	86	99	)	)	PUNCT
iajs-3594	86	100	=	=	SYM
iajs-3594	86	101	0	0	NUM
iajs-3594	86	102	,	,	PUNCT
iajs-3594	86	103	for	for	ADP
iajs-3594	86	104	each	each	DET
iajs-3594	86	105	𝑥	𝑥	PROPN
iajs-3594	86	106	,	,	PUNCT
iajs-3594	86	107	𝑦	𝑦	NOUN
iajs-3594	86	108	,	,	PUNCT
iajs-3594	86	109	𝑧	𝑧	DET
iajs-3594	86	110	∈	∈	PROPN
iajs-3594	86	111	𝐴	𝐴	NOUN
iajs-3594	86	112	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	86	113	𝛼	𝛼	NOUN
iajs-3594	86	114	,	,	PUNCT
iajs-3594	86	115	𝛽	𝛽	PROPN
iajs-3594	86	116	∈	∈	NOUN
iajs-3594	86	117	𝛤.	𝛤.	PROPN
iajs-3594	86	118	by	by	ADP
iajs-3594	86	119	the	the	DET
iajs-3594	86	120	semiprimeness	semiprimeness	NOUN
iajs-3594	86	121	of	of	ADP
iajs-3594	86	122	a	a	PRON
iajs-3594	86	123	,	,	PUNCT
iajs-3594	86	124	we	we	PRON
iajs-3594	86	125	imply	imply	VERB
iajs-3594	86	126	𝑇	𝑇	PROPN
iajs-3594	86	127	(	(	PUNCT
iajs-3594	86	128	𝑦	𝑦	NOUN
iajs-3594	86	129	𝛽	𝛽	NOUN
iajs-3594	86	130	𝑧	𝑧	X
iajs-3594	86	131	)	)	PUNCT
iajs-3594	86	132	=	=	SYM
iajs-3594	86	133	𝑇	𝑇	PROPN
iajs-3594	86	134	(	(	PUNCT
iajs-3594	86	135	𝑦	𝑦	NOUN
iajs-3594	86	136	)	)	PUNCT
iajs-3594	86	137	𝛽	𝛽	NOUN
iajs-3594	86	138	𝑧	𝑧	PROPN
iajs-3594	86	139	,	,	PUNCT
iajs-3594	86	140	for	for	ADP
iajs-3594	86	141	each	each	DET
iajs-3594	86	142	𝑦	𝑦	NOUN
iajs-3594	86	143	,	,	PUNCT
iajs-3594	86	144	𝑧	𝑧	DET
iajs-3594	86	145	∈	∈	PROPN
iajs-3594	86	146	𝐴	𝐴	NOUN
iajs-3594	86	147	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	86	148	𝛽	𝛽	PROPN
iajs-3594	86	149	∈	∈	NOUN
iajs-3594	86	150	𝛤.	𝛤.	PROPN
iajs-3594	86	151	similarly	similarly	ADV
iajs-3594	86	152	,	,	PUNCT
iajs-3594	86	153	we	we	PRON
iajs-3594	86	154	can	can	AUX
iajs-3594	86	155	show	show	VERB
iajs-3594	86	156	𝑆	𝑆	PROPN
iajs-3594	86	157	(	(	PUNCT
iajs-3594	86	158	𝑥	𝑥	PROPN
iajs-3594	86	159	𝛼	𝛼	NOUN
iajs-3594	86	160	𝑦	𝑦	NOUN
iajs-3594	86	161	)	)	PUNCT
iajs-3594	86	162	=	=	PUNCT
iajs-3594	87	1	𝑥	𝑥	DET
iajs-3594	87	2	𝛼	𝛼	X
iajs-3594	87	3	𝑆	𝑆	PROPN
iajs-3594	87	4	(	(	PUNCT
iajs-3594	87	5	𝑦	𝑦	NOUN
iajs-3594	87	6	)	)	PUNCT
iajs-3594	87	7	,	,	PUNCT
iajs-3594	87	8	for	for	ADP
iajs-3594	87	9	each	each	DET
iajs-3594	87	10	𝑥	𝑥	PROPN
iajs-3594	87	11	,	,	PUNCT
iajs-3594	87	12	𝑦	𝑦	PROPN
iajs-3594	87	13	∈	∈	PROPN
iajs-3594	87	14	𝐴	𝐴	PROPN
iajs-3594	87	15	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	87	16	𝛼	𝛼	SYM
iajs-3594	87	17	∈	∈	PROPN
iajs-3594	87	18	𝛤.	𝛤.	PROPN
iajs-3594	87	19	thence	thence	NOUN
iajs-3594	87	20	(	(	PUNCT
iajs-3594	87	21	t	t	PROPN
iajs-3594	87	22	,	,	PUNCT
iajs-3594	87	23	s	s	PART
iajs-3594	87	24	)	)	PUNCT
iajs-3594	87	25	is	be	AUX
iajs-3594	87	26	a	a	DET
iajs-3594	87	27	double	double	ADJ
iajs-3594	87	28	centralizer	centralizer	NOUN
iajs-3594	87	29	.	.	PUNCT
iajs-3594	88	1	now	now	ADV
iajs-3594	88	2	,	,	PUNCT
iajs-3594	88	3	we	we	PRON
iajs-3594	88	4	give	give	VERB
iajs-3594	88	5	some	some	DET
iajs-3594	88	6	results	result	NOUN
iajs-3594	88	7	which	which	PRON
iajs-3594	88	8	make	make	VERB
iajs-3594	88	9	t	t	X
iajs-3594	88	10	=	=	SYM
iajs-3594	88	11	s	s	NOUN
iajs-3594	88	12	under	under	ADP
iajs-3594	88	13	different	different	ADJ
iajs-3594	88	14	conditions	condition	NOUN
iajs-3594	88	15	,	,	PUNCT
iajs-3594	88	16	where	where	SCONJ
iajs-3594	88	17	(	(	PUNCT
iajs-3594	88	18	t	t	PROPN
iajs-3594	88	19	,	,	PUNCT
iajs-3594	88	20	s	s	PART
iajs-3594	88	21	)	)	PUNCT
iajs-3594	88	22	is	be	AUX
iajs-3594	88	23	double	double	ADJ
iajs-3594	88	24	centralizer	centralizer	NOUN
iajs-3594	88	25	.	.	PUNCT
iajs-3594	89	1	3.4	3.4	NUM
iajs-3594	89	2	theorem	theorem	NOUN
iajs-3594	89	3	let	let	VERB
iajs-3594	89	4	a	a	PRON
iajs-3594	89	5	be	be	AUX
iajs-3594	89	6	a	a	DET
iajs-3594	89	7	prime	prime	ADJ
iajs-3594	89	8	𝛤-ring	𝛤-ring	PROPN
iajs-3594	89	9	,	,	PUNCT
iajs-3594	89	10	u	u	PRON
iajs-3594	89	11	be	be	VERB
iajs-3594	89	12	a	a	DET
iajs-3594	89	13	not	not	PART
iajs-3594	89	14	equal	equal	ADJ
iajs-3594	89	15	zero	zero	NUM
iajs-3594	89	16	ideal	ideal	NOUN
iajs-3594	89	17	of	of	ADP
iajs-3594	89	18	a.	a.	NOUN
iajs-3594	89	19	let	let	VERB
iajs-3594	89	20	𝑇	𝑇	PROPN
iajs-3594	89	21	,	,	PUNCT
iajs-3594	89	22	𝑆	𝑆	PROPN
iajs-3594	89	23	:	:	PUNCT
iajs-3594	89	24	𝐴	𝐴	PROPN
iajs-3594	89	25	→	→	SYM
iajs-3594	89	26	𝐴	𝐴	PROPN
iajs-3594	89	27	be	be	AUX
iajs-3594	89	28	additive	additive	ADJ
iajs-3594	89	29	mappings	mapping	NOUN
iajs-3594	89	30	such	such	ADJ
iajs-3594	89	31	that	that	SCONJ
iajs-3594	89	32	t	t	PROPN
iajs-3594	89	33	is	be	AUX
iajs-3594	89	34	a	a	DET
iajs-3594	89	35	left	left	ADJ
iajs-3594	89	36	centralizer	centralizer	NOUN
iajs-3594	89	37	,	,	PUNCT
iajs-3594	89	38	s	s	PART
iajs-3594	89	39	is	be	AUX
iajs-3594	89	40	a	a	DET
iajs-3594	89	41	right	right	ADJ
iajs-3594	89	42	centralizer	centralizer	NOUN
iajs-3594	89	43	and	and	CCONJ
iajs-3594	89	44	they	they	PRON
iajs-3594	89	45	gratify	gratify	VERB
iajs-3594	89	46	𝑥𝛼𝑇(𝑦	𝑥𝛼𝑇(𝑦	PROPN
iajs-3594	89	47	)	)	PUNCT
iajs-3594	89	48	=	=	SYM
iajs-3594	89	49	𝑆(𝑥)𝛼𝑦	𝑆(𝑥)𝛼𝑦	PROPN
iajs-3594	89	50	,	,	PUNCT
iajs-3594	89	51	for	for	ADP
iajs-3594	89	52	each	each	DET
iajs-3594	89	53	𝑥	𝑥	PROPN
iajs-3594	89	54	,	,	PUNCT
iajs-3594	89	55	𝑦	𝑦	PRON
iajs-3594	89	56	∈	∈	NOUN
iajs-3594	89	57	𝑈	𝑈	PROPN
iajs-3594	89	58	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	89	59	𝛼	𝛼	PROPN
iajs-3594	89	60	∈	∈	PROPN
iajs-3594	90	1	г	г	PROPN
iajs-3594	90	2	.	.	PUNCT
iajs-3594	90	3	thence	thence	NOUN
iajs-3594	90	4	(	(	PUNCT
iajs-3594	90	5	t	t	PROPN
iajs-3594	90	6	,	,	PUNCT
iajs-3594	90	7	s	s	PART
iajs-3594	90	8	)	)	PUNCT
iajs-3594	90	9	is	be	AUX
iajs-3594	90	10	a	a	DET
iajs-3594	90	11	double	double	ADJ
iajs-3594	90	12	centralizer	centralizer	NOUN
iajs-3594	90	13	.	.	PUNCT
iajs-3594	91	1	proof	proof	NOUN
iajs-3594	91	2	:	:	PUNCT
iajs-3594	91	3	we	we	PRON
iajs-3594	91	4	have	have	VERB
iajs-3594	91	5	𝑥	𝑥	DET
iajs-3594	91	6	𝛼	𝛼	NOUN
iajs-3594	91	7	𝑇(𝑦	𝑇(𝑦	NOUN
iajs-3594	91	8	)	)	PUNCT
iajs-3594	91	9	=	=	SYM
iajs-3594	91	10	𝑆	𝑆	PROPN
iajs-3594	91	11	(	(	PUNCT
iajs-3594	91	12	𝑥	𝑥	NOUN
iajs-3594	91	13	)	)	PUNCT
iajs-3594	91	14	𝛼	𝛼	PRON
iajs-3594	91	15	𝑦	𝑦	NOUN
iajs-3594	91	16	,	,	PUNCT
iajs-3594	91	17	for	for	ADP
iajs-3594	91	18	each	each	DET
iajs-3594	91	19	𝑥	𝑥	PROPN
iajs-3594	91	20	,	,	PUNCT
iajs-3594	91	21	𝑦	𝑦	PROPN
iajs-3594	91	22	∈	∈	PROPN
iajs-3594	91	23	𝑈	𝑈	PROPN
iajs-3594	91	24	,	,	PUNCT
iajs-3594	91	25	𝛼	𝛼	PROPN
iajs-3594	91	26	∈	∈	PROPN
iajs-3594	91	27	𝛤.	𝛤.	PROPN
iajs-3594	91	28	(	(	PUNCT
iajs-3594	91	29	2	2	NUM
iajs-3594	91	30	)	)	PUNCT
iajs-3594	91	31	replace	replace	NOUN
iajs-3594	91	32	x	x	PUNCT
iajs-3594	91	33	with	with	ADP
iajs-3594	91	34	𝑥𝛽𝑟	𝑥𝛽𝑟	VERB
iajs-3594	91	35	in	in	ADP
iajs-3594	91	36	(	(	PUNCT
iajs-3594	91	37	2	2	NUM
iajs-3594	91	38	)	)	PUNCT
iajs-3594	91	39	when	when	SCONJ
iajs-3594	91	40	𝑥	𝑥	PRON
iajs-3594	91	41	∈	∈	PROPN
iajs-3594	91	42	u	u	PROPN
iajs-3594	91	43	,	,	PUNCT
iajs-3594	91	44	β	β	X
iajs-3594	91	45	∈	∈	PROPN
iajs-3594	91	46	г	г	PROPN
iajs-3594	91	47	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	91	48	𝑟	𝑟	NOUN
iajs-3594	91	49	∈	∈	PROPN
iajs-3594	91	50	𝐴	𝐴	PROPN
iajs-3594	91	51	,	,	PUNCT
iajs-3594	91	52	we	we	PRON
iajs-3594	91	53	imply	imply	VERB
iajs-3594	91	54	𝑥	𝑥	PRON
iajs-3594	91	55	𝛽	𝛽	NOUN
iajs-3594	91	56	(	(	PUNCT
iajs-3594	91	57	𝑟	𝑟	X
iajs-3594	91	58	𝛼	𝛼	PRON
iajs-3594	91	59	𝑇(𝑦	𝑇(𝑦	NOUN
iajs-3594	91	60	)	)	PUNCT
iajs-3594	91	61	−	−	PROPN
iajs-3594	91	62	𝑆	𝑆	PROPN
iajs-3594	91	63	(	(	PUNCT
iajs-3594	91	64	𝑟	𝑟	NOUN
iajs-3594	91	65	)	)	PUNCT
iajs-3594	91	66	𝛼	𝛼	PRON
iajs-3594	91	67	𝑦	𝑦	NOUN
iajs-3594	91	68	)	)	PUNCT
iajs-3594	91	69	=	=	SYM
iajs-3594	91	70	0	0	NUM
iajs-3594	91	71	,	,	PUNCT
iajs-3594	91	72	for	for	ADP
iajs-3594	91	73	each	each	DET
iajs-3594	91	74	𝑟	𝑟	PRON
iajs-3594	91	75	∈	∈	PROPN
iajs-3594	91	76	𝐴	𝐴	PROPN
iajs-3594	91	77	,	,	PUNCT
iajs-3594	91	78	𝑥	𝑥	PROPN
iajs-3594	91	79	,	,	PUNCT
iajs-3594	91	80	𝑦	𝑦	NOUN
iajs-3594	91	81	∈	∈	PROPN
iajs-3594	91	82	𝑈𝑎𝑛𝑑	𝑈𝑎𝑛𝑑	PROPN
iajs-3594	91	83	𝛼	𝛼	NOUN
iajs-3594	91	84	,	,	PUNCT
iajs-3594	91	85	𝛽	𝛽	PROPN
iajs-3594	91	86	∈	∈	NOUN
iajs-3594	91	87	𝛤.	𝛤.	PROPN
iajs-3594	91	88	i.e.	i.e.	X
iajs-3594	91	89	𝑥	𝑥	X
iajs-3594	91	90	𝛾	𝛾	ADP
iajs-3594	91	91	𝐴	𝐴	PROPN
iajs-3594	91	92	𝛽	𝛽	PROPN
iajs-3594	91	93	(	(	PUNCT
iajs-3594	91	94	𝑟	𝑟	X
iajs-3594	91	95	𝛼	𝛼	PRON
iajs-3594	91	96	𝑇(𝑦	𝑇(𝑦	NOUN
iajs-3594	91	97	)	)	PUNCT
iajs-3594	91	98	−	−	PROPN
iajs-3594	91	99	𝑆	𝑆	PROPN
iajs-3594	91	100	(	(	PUNCT
iajs-3594	91	101	𝑟)𝛼	𝑟)𝛼	NOUN
iajs-3594	91	102	𝑦	𝑦	X
iajs-3594	91	103	)	)	PUNCT
iajs-3594	91	104	=	=	SYM
iajs-3594	91	105	0	0	NUM
iajs-3594	91	106	,	,	PUNCT
iajs-3594	91	107	for	for	ADP
iajs-3594	91	108	each	each	DET
iajs-3594	91	109	𝑟	𝑟	PRON
iajs-3594	91	110	∈	∈	PROPN
iajs-3594	91	111	𝐴	𝐴	PROPN
iajs-3594	91	112	,	,	PUNCT
iajs-3594	91	113	𝑥	𝑥	PROPN
iajs-3594	91	114	,	,	PUNCT
iajs-3594	91	115	𝑦	𝑦	NOUN
iajs-3594	91	116	∈	∈	PROPN
iajs-3594	91	117	𝑈𝑎𝑛𝑑	𝑈𝑎𝑛𝑑	PROPN
iajs-3594	91	118	𝛼	𝛼	PROPN
iajs-3594	91	119	,	,	PUNCT
iajs-3594	91	120	𝛽	𝛽	PROPN
iajs-3594	91	121	,	,	PUNCT
iajs-3594	91	122	𝛾	𝛾	ADP
iajs-3594	91	123	∈	∈	X
iajs-3594	91	124	𝛤.	𝛤.	PROPN
iajs-3594	91	125	by	by	ADP
iajs-3594	91	126	primeness	primeness	NOUN
iajs-3594	91	127	of	of	ADP
iajs-3594	91	128	a	a	PRON
iajs-3594	91	129	and	and	CCONJ
iajs-3594	91	130	since	since	SCONJ
iajs-3594	91	131	u	u	PRON
iajs-3594	91	132	be	be	VERB
iajs-3594	91	133	a	a	DET
iajs-3594	91	134	not	not	PART
iajs-3594	91	135	equal	equal	ADJ
iajs-3594	91	136	zero	zero	NUM
iajs-3594	91	137	ideal	ideal	NOUN
iajs-3594	91	138	of	of	ADP
iajs-3594	91	139	a	a	PRON
iajs-3594	91	140	,	,	PUNCT
iajs-3594	91	141	we	we	PRON
iajs-3594	91	142	imply	imply	VERB
iajs-3594	91	143	𝑟𝛼𝑇(𝑦	𝑟𝛼𝑇(𝑦	NOUN
iajs-3594	91	144	)	)	PUNCT
iajs-3594	92	1	=	=	SYM
iajs-3594	92	2	𝑆(𝑟)𝛼𝑦	𝑆(𝑟)𝛼𝑦	PROPN
iajs-3594	92	3	,	,	PUNCT
iajs-3594	92	4	for	for	ADP
iajs-3594	92	5	each	each	DET
iajs-3594	92	6	𝑟	𝑟	PRON
iajs-3594	92	7	∈	∈	PROPN
iajs-3594	92	8	𝐴	𝐴	PROPN
iajs-3594	92	9	,	,	PUNCT
iajs-3594	92	10	𝑦	𝑦	NOUN
iajs-3594	92	11	∈	∈	NOUN
iajs-3594	92	12	𝑈	𝑈	PROPN
iajs-3594	92	13	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	92	14	𝛼	𝛼	PROPN
iajs-3594	92	15	∈	∈	PROPN
iajs-3594	92	16	𝛤.	𝛤.	PROPN
iajs-3594	92	17	(	(	PUNCT
iajs-3594	92	18	3	3	X
iajs-3594	92	19	)	)	PUNCT
iajs-3594	92	20	replacing	replace	VERB
iajs-3594	92	21	𝑦	𝑦	NOUN
iajs-3594	92	22	with	with	ADP
iajs-3594	92	23	𝑡𝜎𝑦	𝑡𝜎𝑦	NOUN
iajs-3594	92	24	in	in	ADP
iajs-3594	92	25	(	(	PUNCT
iajs-3594	92	26	3	3	NUM
iajs-3594	92	27	)	)	PUNCT
iajs-3594	92	28	,	,	PUNCT
iajs-3594	92	29	where	where	SCONJ
iajs-3594	92	30	𝑡	𝑡	PROPN
iajs-3594	92	31	∈	∈	PROPN
iajs-3594	92	32	𝐴	𝐴	PROPN
iajs-3594	92	33	,	,	PUNCT
iajs-3594	92	34	𝑦	𝑦	NOUN
iajs-3594	92	35	∈	∈	PROPN
iajs-3594	92	36	𝑈	𝑈	PROPN
iajs-3594	92	37	,	,	PUNCT
iajs-3594	92	38	and	and	CCONJ
iajs-3594	92	39	σ	σ	NUM
iajs-3594	92	40	∈	∈	PROPN
iajs-3594	92	41	г	г	PROPN
iajs-3594	92	42	.	.	PUNCT
iajs-3594	93	1	(	(	PUNCT
iajs-3594	93	2	𝑟	𝑟	X
iajs-3594	93	3	𝛼	𝛼	NOUN
iajs-3594	93	4	𝑇(𝑡	𝑇(𝑡	NOUN
iajs-3594	93	5	)	)	PUNCT
iajs-3594	93	6	−	−	PROPN
iajs-3594	93	7	𝑆	𝑆	PROPN
iajs-3594	93	8	(	(	PUNCT
iajs-3594	93	9	𝑟)𝛼	𝑟)𝛼	NOUN
iajs-3594	93	10	𝑡	𝑡	PROPN
iajs-3594	93	11	)	)	PUNCT
iajs-3594	93	12	𝜎	𝜎	PROPN
iajs-3594	93	13	𝑦	𝑦	NOUN
iajs-3594	93	14	=	=	SYM
iajs-3594	93	15	0	0	NUM
iajs-3594	93	16	,	,	PUNCT
iajs-3594	93	17	for	for	ADP
iajs-3594	93	18	each	each	DET
iajs-3594	93	19	t	t	PROPN
iajs-3594	93	20	,	,	PUNCT
iajs-3594	93	21	𝑟	𝑟	NOUN
iajs-3594	93	22	∈	∈	PROPN
iajs-3594	93	23	𝐴	𝐴	PROPN
iajs-3594	93	24	,	,	PUNCT
iajs-3594	93	25	𝑦	𝑦	NOUN
iajs-3594	93	26	∈	∈	PROPN
iajs-3594	93	27	𝑈	𝑈	PROPN
iajs-3594	93	28	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	93	29	𝛼	𝛼	NOUN
iajs-3594	93	30	,	,	PUNCT
iajs-3594	93	31	𝜎	𝜎	PROPN
iajs-3594	93	32	∈	∈	PROPN
iajs-3594	93	33	𝛤.	𝛤.	PROPN
iajs-3594	93	34	implies	imply	VERB
iajs-3594	93	35	that	that	SCONJ
iajs-3594	93	36	(	(	PUNCT
iajs-3594	93	37	𝑟𝛼𝑇(𝑡	𝑟𝛼𝑇(𝑡	INTJ
iajs-3594	93	38	)	)	PUNCT
iajs-3594	93	39	−	−	NOUN
iajs-3594	93	40	s(r)αt)𝜎u𝛿𝐴	s(r)αt)𝜎u𝛿𝐴	PROPN
iajs-3594	93	41	=	=	SYM
iajs-3594	93	42	0	0	NUM
iajs-3594	93	43	,	,	PUNCT
iajs-3594	93	44	for	for	ADP
iajs-3594	93	45	each	each	DET
iajs-3594	93	46	𝑡	𝑡	NOUN
iajs-3594	93	47	,	,	PUNCT
iajs-3594	93	48	𝑟	𝑟	X
iajs-3594	93	49	∈	∈	PROPN
iajs-3594	93	50	a	a	PRON
iajs-3594	93	51	and	and	CCONJ
iajs-3594	93	52	𝛼	𝛼	ADJ
iajs-3594	93	53	,	,	PUNCT
iajs-3594	93	54	𝜎	𝜎	PROPN
iajs-3594	93	55	,	,	PUNCT
iajs-3594	93	56	δ	δ	PROPN
iajs-3594	93	57	∈	∈	PROPN
iajs-3594	93	58	г	г	PROPN
iajs-3594	93	59	.	.	PUNCT
iajs-3594	93	60	by	by	ADP
iajs-3594	93	61	the	the	DET
iajs-3594	93	62	primeness	primeness	NOUN
iajs-3594	93	63	of	of	ADP
iajs-3594	93	64	a	a	PRON
iajs-3594	93	65	,	,	PUNCT
iajs-3594	93	66	we	we	PRON
iajs-3594	93	67	imply	imply	VERB
iajs-3594	93	68	𝑟	𝑟	PRON
iajs-3594	93	69	𝛼	𝛼	NOUN
iajs-3594	93	70	𝑇(𝑡	𝑇(𝑡	NOUN
iajs-3594	93	71	)	)	PUNCT
iajs-3594	93	72	=	=	SYM
iajs-3594	93	73	𝑆	𝑆	PROPN
iajs-3594	93	74	(	(	PUNCT
iajs-3594	93	75	𝑟)𝛼	𝑟)𝛼	NOUN
iajs-3594	93	76	𝑡	𝑡	PROPN
iajs-3594	93	77	,	,	PUNCT
iajs-3594	93	78	for	for	ADP
iajs-3594	93	79	each	each	DET
iajs-3594	93	80	𝑡	𝑡	NOUN
iajs-3594	93	81	,	,	PUNCT
iajs-3594	93	82	𝑟	𝑟	NOUN
iajs-3594	93	83	∈	∈	PROPN
iajs-3594	93	84	𝐴	𝐴	PROPN
iajs-3594	93	85	,	,	PUNCT
iajs-3594	93	86	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	93	87	,	,	PUNCT
iajs-3594	93	88	𝛼	𝛼	PROPN
iajs-3594	93	89	∈	∈	PROPN
iajs-3594	93	90	𝛤.	𝛤.	PROPN
iajs-3594	93	91	3.5	3.5	NUM
iajs-3594	93	92	theorem	theorem	NOUN
iajs-3594	93	93	let	let	VERB
iajs-3594	93	94	a	a	PRON
iajs-3594	93	95	be	be	AUX
iajs-3594	93	96	a	a	DET
iajs-3594	93	97	prime	prime	ADJ
iajs-3594	93	98	gamma	gamma	NOUN
iajs-3594	93	99	-	-	PUNCT
iajs-3594	93	100	ring	ring	NOUN
iajs-3594	93	101	,	,	PUNCT
iajs-3594	93	102	u	u	PRON
iajs-3594	93	103	be	be	VERB
iajs-3594	93	104	a	a	DET
iajs-3594	93	105	not	not	PART
iajs-3594	93	106	equal	equal	ADJ
iajs-3594	93	107	zero	zero	NUM
iajs-3594	93	108	ideal	ideal	NOUN
iajs-3594	93	109	of	of	ADP
iajs-3594	93	110	a	a	PRON
iajs-3594	93	111	,	,	PUNCT
iajs-3594	93	112	and	and	CCONJ
iajs-3594	93	113	(	(	PUNCT
iajs-3594	93	114	t	t	PROPN
iajs-3594	93	115	,	,	PUNCT
iajs-3594	93	116	s	s	PART
iajs-3594	93	117	)	)	PUNCT
iajs-3594	93	118	be	be	AUX
iajs-3594	93	119	a	a	DET
iajs-3594	93	120	double	double	ADJ
iajs-3594	93	121	centralizer	centralizer	NOUN
iajs-3594	93	122	.	.	PUNCT
iajs-3594	94	1	if	if	SCONJ
iajs-3594	94	2	t	t	PROPN
iajs-3594	94	3	=	=	SYM
iajs-3594	94	4	s	s	X
iajs-3594	94	5	on	on	ADP
iajs-3594	94	6	u	u	NOUN
iajs-3594	94	7	,	,	PUNCT
iajs-3594	94	8	thence	thence	NOUN
iajs-3594	94	9	t	t	PROPN
iajs-3594	94	10	=	=	SYM
iajs-3594	94	11	s	s	X
iajs-3594	94	12	on	on	ADP
iajs-3594	94	13	a.	a.	NOUN
iajs-3594	94	14	proof	proof	NOUN
iajs-3594	94	15	:	:	PUNCT
iajs-3594	94	16	we	we	PRON
iajs-3594	94	17	have	have	VERB
iajs-3594	94	18	𝑇	𝑇	PROPN
iajs-3594	94	19	(	(	PUNCT
iajs-3594	94	20	𝑥	𝑥	NOUN
iajs-3594	94	21	)	)	PUNCT
iajs-3594	94	22	=	=	SYM
iajs-3594	94	23	𝑆	𝑆	PROPN
iajs-3594	94	24	(	(	PUNCT
iajs-3594	94	25	𝑥	𝑥	NOUN
iajs-3594	94	26	)	)	PUNCT
iajs-3594	94	27	,	,	PUNCT
iajs-3594	94	28	for	for	ADP
iajs-3594	94	29	each	each	PRON
iajs-3594	95	1	𝑥	𝑥	DET
iajs-3594	95	2	∈	∈	PROPN
iajs-3594	95	3	𝑈.	𝑈.	NOUN
iajs-3594	95	4	(	(	PUNCT
iajs-3594	95	5	4	4	NUM
iajs-3594	95	6	)	)	PUNCT
iajs-3594	95	7	by	by	ADP
iajs-3594	95	8	replacing	replace	VERB
iajs-3594	95	9	𝑥	𝑥	PRON
iajs-3594	95	10	with	with	ADP
iajs-3594	95	11	𝑟𝛼𝑥	𝑟𝛼𝑥	PRON
iajs-3594	95	12	in	in	ADP
iajs-3594	95	13	(	(	PUNCT
iajs-3594	95	14	4	4	NUM
iajs-3594	95	15	)	)	PUNCT
iajs-3594	95	16	,	,	PUNCT
iajs-3594	95	17	when	when	SCONJ
iajs-3594	95	18	𝑟	𝑟	X
iajs-3594	95	19	∈	∈	VERB
iajs-3594	95	20	a	a	DET
iajs-3594	95	21	,	,	PUNCT
iajs-3594	95	22	𝑥	𝑥	PROPN
iajs-3594	95	23	∈	∈	PROPN
iajs-3594	95	24	u	u	NOUN
iajs-3594	95	25	and	and	CCONJ
iajs-3594	95	26	𝛼	𝛼	ADP
iajs-3594	95	27	∈	∈	PROPN
iajs-3594	95	28	г	г	PROPN
iajs-3594	95	29	,	,	PUNCT
iajs-3594	95	30	we	we	PRON
iajs-3594	95	31	imply	imply	VERB
iajs-3594	95	32	ihjpas	ihjpa	VERB
iajs-3594	95	33	.	.	PUNCT
iajs-3594	96	1	2025	2025	NUM
iajs-3594	96	2	,	,	PUNCT
iajs-3594	96	3	38(2	38(2	NUM
iajs-3594	96	4	)	)	PUNCT
iajs-3594	96	5	334	334	NUM
iajs-3594	96	6	𝑇(𝑟)𝛼	𝑇(𝑟)𝛼	NOUN
iajs-3594	96	7	𝑥	𝑥	NOUN
iajs-3594	96	8	=	=	PUNCT
iajs-3594	96	9	𝑟	𝑟	X
iajs-3594	96	10	𝛼	𝛼	NOUN
iajs-3594	96	11	𝑆(𝑥	𝑆(𝑥	NOUN
iajs-3594	96	12	)	)	PUNCT
iajs-3594	96	13	=	=	SYM
iajs-3594	96	14	𝑟𝛼𝑇(𝑥	𝑟𝛼𝑇(𝑥	PROPN
iajs-3594	96	15	)	)	PUNCT
iajs-3594	96	16	,	,	PUNCT
iajs-3594	96	17	for	for	ADP
iajs-3594	96	18	each	each	DET
iajs-3594	96	19	∈	∈	PROPN
iajs-3594	96	20	𝐴	𝐴	PROPN
iajs-3594	96	21	,	,	PUNCT
iajs-3594	96	22	𝑥	𝑥	PRON
iajs-3594	96	23	∈	∈	PROPN
iajs-3594	96	24	𝑈	𝑈	PROPN
iajs-3594	96	25	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	96	26	𝛼	𝛼	PROPN
iajs-3594	96	27	∈	∈	PROPN
iajs-3594	96	28	𝛤	𝛤	PROPN
iajs-3594	96	29	.	.	PUNCT
iajs-3594	97	1	(	(	PUNCT
iajs-3594	97	2	5	5	NUM
iajs-3594	97	3	)	)	PUNCT
iajs-3594	97	4	since	since	SCONJ
iajs-3594	97	5	(	(	PUNCT
iajs-3594	97	6	t	t	PROPN
iajs-3594	97	7	,	,	PUNCT
iajs-3594	97	8	s	s	PART
iajs-3594	97	9	)	)	PUNCT
iajs-3594	97	10	are	be	AUX
iajs-3594	97	11	a	a	DET
iajs-3594	97	12	double	double	ADJ
iajs-3594	97	13	centralizer	centralizer	NOUN
iajs-3594	97	14	,	,	PUNCT
iajs-3594	97	15	(	(	PUNCT
iajs-3594	97	16	5	5	X
iajs-3594	97	17	)	)	PUNCT
iajs-3594	97	18	leads	lead	VERB
iajs-3594	97	19	to	to	ADP
iajs-3594	97	20	𝑇(𝑟)𝛼	𝑇(𝑟)𝛼	NOUN
iajs-3594	97	21	𝑥	𝑥	PROPN
iajs-3594	97	22	=	=	SYM
iajs-3594	97	23	𝑆	𝑆	PROPN
iajs-3594	97	24	(	(	PUNCT
iajs-3594	97	25	𝑟)𝛼	𝑟)𝛼	NOUN
iajs-3594	97	26	𝑥	𝑥	NOUN
iajs-3594	97	27	,	,	PUNCT
iajs-3594	97	28	for	for	ADP
iajs-3594	97	29	each	each	DET
iajs-3594	97	30	𝑥	𝑥	PRON
iajs-3594	97	31	∈	∈	PROPN
iajs-3594	97	32	𝑈	𝑈	PROPN
iajs-3594	97	33	,	,	PUNCT
iajs-3594	97	34	𝑟	𝑟	NOUN
iajs-3594	97	35	∈	∈	PROPN
iajs-3594	97	36	𝐴	𝐴	PROPN
iajs-3594	97	37	,	,	PUNCT
iajs-3594	97	38	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	97	39	𝛼	𝛼	SYM
iajs-3594	97	40	∈	∈	PROPN
iajs-3594	97	41	𝛤.i.e	𝛤.i.e	PRON
iajs-3594	97	42	.	.	PUNCT
iajs-3594	98	1	(	(	PUNCT
iajs-3594	98	2	𝑇	𝑇	PROPN
iajs-3594	98	3	(	(	PUNCT
iajs-3594	98	4	𝑟	𝑟	NOUN
iajs-3594	98	5	)	)	PUNCT
iajs-3594	98	6	−	−	PROPN
iajs-3594	98	7	𝑆	𝑆	PROPN
iajs-3594	98	8	(	(	PUNCT
iajs-3594	98	9	𝑟))𝛼𝑈𝛽𝐴	𝑟))𝛼𝑈𝛽𝐴	X
iajs-3594	98	10	=	=	SYM
iajs-3594	98	11	0	0	NUM
iajs-3594	98	12	,	,	PUNCT
iajs-3594	98	13	for	for	ADP
iajs-3594	98	14	each	each	DET
iajs-3594	98	15	𝑟	𝑟	PRON
iajs-3594	98	16	∈	∈	PROPN
iajs-3594	98	17	a	a	PRON
iajs-3594	98	18	,	,	PUNCT
iajs-3594	98	19	and	and	CCONJ
iajs-3594	98	20	𝛼	𝛼	X
iajs-3594	98	21	,	,	PUNCT
iajs-3594	98	22	𝛽	𝛽	PROPN
iajs-3594	98	23	∈	∈	PROPN
iajs-3594	98	24	г	г	PROPN
iajs-3594	98	25	.	.	PUNCT
iajs-3594	99	1	since	since	SCONJ
iajs-3594	99	2	a	a	PRON
iajs-3594	99	3	is	be	AUX
iajs-3594	99	4	a	a	DET
iajs-3594	99	5	prime	prime	ADJ
iajs-3594	99	6	𝛤-ring	𝛤-ring	PROPN
iajs-3594	99	7	and	and	CCONJ
iajs-3594	99	8	u	u	NOUN
iajs-3594	99	9	be	be	VERB
iajs-3594	99	10	a	a	DET
iajs-3594	99	11	not	not	PART
iajs-3594	99	12	equal	equal	ADJ
iajs-3594	99	13	zero	zero	NUM
iajs-3594	99	14	ideal	ideal	NOUN
iajs-3594	99	15	of	of	ADP
iajs-3594	99	16	a	a	PRON
iajs-3594	99	17	,	,	PUNCT
iajs-3594	99	18	we	we	PRON
iajs-3594	99	19	imply	imply	VERB
iajs-3594	99	20	t	t	PROPN
iajs-3594	99	21	=	=	PUNCT
iajs-3594	99	22	s.	s.	PROPN
iajs-3594	99	23	from	from	ADP
iajs-3594	99	24	theorem	theorem	PROPN
iajs-3594	99	25	above	above	ADV
iajs-3594	99	26	,	,	PUNCT
iajs-3594	99	27	we	we	PRON
iajs-3594	99	28	imply	imply	VERB
iajs-3594	99	29	the	the	DET
iajs-3594	99	30	following	following	NOUN
iajs-3594	99	31	:	:	PUNCT
iajs-3594	99	32	3.6	3.6	NUM
iajs-3594	99	33	corollary	corollary	NOUN
iajs-3594	99	34	let	let	VERB
iajs-3594	99	35	a	a	PRON
iajs-3594	99	36	be	be	AUX
iajs-3594	99	37	a	a	DET
iajs-3594	99	38	prime	prime	ADJ
iajs-3594	99	39	gamma	gamma	NOUN
iajs-3594	99	40	-	-	PUNCT
iajs-3594	99	41	ring	ring	NOUN
iajs-3594	99	42	,	,	PUNCT
iajs-3594	99	43	u	u	PRON
iajs-3594	99	44	be	be	VERB
iajs-3594	99	45	an	an	DET
iajs-3594	99	46	ideal	ideal	NOUN
iajs-3594	99	47	of	of	ADP
iajs-3594	99	48	a	a	PRON
iajs-3594	99	49	and	and	CCONJ
iajs-3594	99	50	(	(	PUNCT
iajs-3594	99	51	t	t	PROPN
iajs-3594	99	52	,	,	PUNCT
iajs-3594	99	53	s	s	AUX
iajs-3594	99	54	)	)	PUNCT
iajs-3594	99	55	be	be	AUX
iajs-3594	99	56	a	a	DET
iajs-3594	99	57	double	double	ADJ
iajs-3594	99	58	centralizer	centralizer	NOUN
iajs-3594	99	59	.	.	PUNCT
iajs-3594	100	1	if	if	SCONJ
iajs-3594	100	2	,	,	PUNCT
iajs-3594	100	3	t	t	PROPN
iajs-3594	100	4	=	=	SYM
iajs-3594	100	5	s	s	PART
iajs-3594	100	6	=	=	NOUN
iajs-3594	100	7	0	0	NUM
iajs-3594	100	8	on	on	ADP
iajs-3594	100	9	u	u	NOUN
iajs-3594	100	10	,	,	PUNCT
iajs-3594	100	11	thence	thence	NOUN
iajs-3594	100	12	t	t	NOUN
iajs-3594	100	13	=	=	SYM
iajs-3594	100	14	s	s	PART
iajs-3594	100	15	=	=	NOUN
iajs-3594	100	16	0	0	NUM
iajs-3594	100	17	on	on	ADP
iajs-3594	100	18	a.	a.	NOUN
iajs-3594	100	19	in	in	ADP
iajs-3594	100	20	the	the	DET
iajs-3594	100	21	following	following	NOUN
iajs-3594	100	22	theorem	theorem	NOUN
iajs-3594	100	23	,	,	PUNCT
iajs-3594	100	24	we	we	PRON
iajs-3594	100	25	shall	shall	AUX
iajs-3594	100	26	prove	prove	VERB
iajs-3594	100	27	that	that	SCONJ
iajs-3594	100	28	t	t	PROPN
iajs-3594	100	29	=	=	SYM
iajs-3594	100	30	s	s	NOUN
iajs-3594	100	31	in	in	ADP
iajs-3594	100	32	case	case	NOUN
iajs-3594	100	33	t	t	NOUN
iajs-3594	100	34	acts	act	VERB
iajs-3594	100	35	as	as	ADP
iajs-3594	100	36	a	a	DET
iajs-3594	100	37	homomorphism	homomorphism	NOUN
iajs-3594	100	38	on	on	ADP
iajs-3594	100	39	a.	a.	NOUN
iajs-3594	100	40	3.7	3.7	NUM
iajs-3594	100	41	theorem	theorem	NOUN
iajs-3594	100	42	let	let	VERB
iajs-3594	100	43	a	a	PRON
iajs-3594	100	44	be	be	AUX
iajs-3594	100	45	a	a	DET
iajs-3594	100	46	semiprime	semiprime	NOUN
iajs-3594	100	47	gamma	gamma	NOUN
iajs-3594	100	48	-	-	PUNCT
iajs-3594	100	49	ring	ring	NOUN
iajs-3594	100	50	and	and	CCONJ
iajs-3594	100	51	let	let	VERB
iajs-3594	100	52	(	(	PUNCT
iajs-3594	100	53	t	t	PROPN
iajs-3594	100	54	,	,	PUNCT
iajs-3594	100	55	s	s	PART
iajs-3594	100	56	)	)	PUNCT
iajs-3594	100	57	be	be	AUX
iajs-3594	100	58	a	a	DET
iajs-3594	100	59	double	double	ADJ
iajs-3594	100	60	centralizer	centralizer	NOUN
iajs-3594	100	61	,	,	PUNCT
iajs-3594	100	62	if	if	SCONJ
iajs-3594	100	63	t	t	PROPN
iajs-3594	100	64	acts	act	VERB
iajs-3594	100	65	as	as	ADP
iajs-3594	100	66	a	a	DET
iajs-3594	100	67	homomorphism	homomorphism	NOUN
iajs-3594	100	68	on	on	ADP
iajs-3594	100	69	a	a	DET
iajs-3594	100	70	,	,	PUNCT
iajs-3594	100	71	thence	thence	NOUN
iajs-3594	100	72	t	t	NOUN
iajs-3594	100	73	=	=	PUNCT
iajs-3594	100	74	s.	s.	PROPN
iajs-3594	100	75	proof	proof	NOUN
iajs-3594	100	76	:	:	PUNCT
iajs-3594	100	77	we	we	PRON
iajs-3594	100	78	have	have	VERB
iajs-3594	100	79	𝑇(𝑥𝛼𝑦	𝑇(𝑥𝛼𝑦	ADJ
iajs-3594	100	80	)	)	PUNCT
iajs-3594	100	81	=	=	PUNCT
iajs-3594	101	1	𝑇(𝑥)𝛼	𝑇(𝑥)𝛼	PROPN
iajs-3594	101	2	𝑦	𝑦	NOUN
iajs-3594	101	3	,	,	PUNCT
iajs-3594	101	4	for	for	ADP
iajs-3594	101	5	each	each	DET
iajs-3594	101	6	𝑥	𝑥	PROPN
iajs-3594	101	7	,	,	PUNCT
iajs-3594	101	8	𝑦	𝑦	PROPN
iajs-3594	101	9	∈	∈	PROPN
iajs-3594	101	10	𝐴	𝐴	PROPN
iajs-3594	101	11	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	101	12	𝛼	𝛼	PROPN
iajs-3594	101	13	∈	∈	PROPN
iajs-3594	101	14	г	г	PROPN
iajs-3594	101	15	.	.	PUNCT
iajs-3594	102	1	since	since	SCONJ
iajs-3594	102	2	t	t	PROPN
iajs-3594	102	3	is	be	AUX
iajs-3594	102	4	acts	act	VERB
iajs-3594	102	5	homomorphism	homomorphism	PROPN
iajs-3594	102	6	on	on	ADP
iajs-3594	102	7	a	a	DET
iajs-3594	102	8	,	,	PUNCT
iajs-3594	102	9	thence	thence	NOUN
iajs-3594	102	10	t	t	NOUN
iajs-3594	102	11	(	(	PUNCT
iajs-3594	102	12	x	x	X
iajs-3594	102	13	)	)	PUNCT
iajs-3594	102	14	𝛼	𝛼	NOUN
iajs-3594	102	15	t(y	t(y	PROPN
iajs-3594	102	16	)	)	PUNCT
iajs-3594	102	17	=	=	SYM
iajs-3594	102	18	t	t	PROPN
iajs-3594	102	19	(	(	PUNCT
iajs-3594	102	20	x	x	X
iajs-3594	102	21	)	)	PUNCT
iajs-3594	102	22	𝛼	𝛼	PROPN
iajs-3594	102	23	y	y	PROPN
iajs-3594	102	24	,	,	PUNCT
iajs-3594	102	25	for	for	ADP
iajs-3594	102	26	any	any	DET
iajs-3594	102	27	𝑥	𝑥	PROPN
iajs-3594	102	28	,	,	PUNCT
iajs-3594	102	29	𝑦	𝑦	PROPN
iajs-3594	102	30	∈	∈	PROPN
iajs-3594	102	31	𝐴	𝐴	PROPN
iajs-3594	102	32	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	102	33	𝛼	𝛼	SYM
iajs-3594	102	34	∈	∈	PROPN
iajs-3594	102	35	𝛤.	𝛤.	PROPN
iajs-3594	102	36	(	(	PUNCT
iajs-3594	102	37	6	6	NUM
iajs-3594	102	38	)	)	PUNCT
iajs-3594	102	39	on	on	ADP
iajs-3594	102	40	the	the	DET
iajs-3594	102	41	other	other	ADJ
iajs-3594	102	42	hand	hand	NOUN
iajs-3594	102	43	;	;	PUNCT
iajs-3594	102	44	𝑥	𝑥	DET
iajs-3594	102	45	𝛼	𝛼	NOUN
iajs-3594	102	46	𝑇(𝑦	𝑇(𝑦	NOUN
iajs-3594	102	47	)	)	PUNCT
iajs-3594	102	48	=	=	SYM
iajs-3594	102	49	𝑆	𝑆	PROPN
iajs-3594	102	50	(	(	PUNCT
iajs-3594	102	51	𝑥	𝑥	NOUN
iajs-3594	102	52	)	)	PUNCT
iajs-3594	102	53	𝛼	𝛼	NOUN
iajs-3594	102	54	𝑦	𝑦	NOUN
iajs-3594	102	55	,	,	PUNCT
iajs-3594	102	56	for	for	ADP
iajs-3594	102	57	𝑎𝑙𝑙	𝑎𝑙𝑙	PROPN
iajs-3594	102	58	𝑥	𝑥	PROPN
iajs-3594	102	59	,	,	PUNCT
iajs-3594	102	60	𝑦	𝑦	PROPN
iajs-3594	102	61	∈	∈	PROPN
iajs-3594	102	62	𝐴	𝐴	PROPN
iajs-3594	102	63	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	102	64	𝛼	𝛼	SYM
iajs-3594	102	65	∈	∈	PROPN
iajs-3594	102	66	𝛤.	𝛤.	PROPN
iajs-3594	102	67	(	(	PUNCT
iajs-3594	102	68	7	7	NUM
iajs-3594	102	69	)	)	PUNCT
iajs-3594	102	70	the	the	DET
iajs-3594	102	71	substation	substation	NOUN
iajs-3594	102	72	𝑇(𝑥	𝑇(𝑥	PROPN
iajs-3594	102	73	)	)	PUNCT
iajs-3594	102	74	for	for	ADP
iajs-3594	102	75	𝑥	𝑥	PROPN
iajs-3594	102	76	in	in	ADP
iajs-3594	102	77	(	(	PUNCT
iajs-3594	102	78	7	7	NUM
iajs-3594	102	79	)	)	PUNCT
iajs-3594	102	80	,	,	PUNCT
iajs-3594	102	81	gives	give	VERB
iajs-3594	102	82	𝑇(𝑥	𝑇(𝑥	PRON
iajs-3594	102	83	)	)	PUNCT
iajs-3594	102	84	𝛼	𝛼	NOUN
iajs-3594	102	85	𝑇(𝑦	𝑇(𝑦	NOUN
iajs-3594	102	86	)	)	PUNCT
iajs-3594	102	87	=	=	SYM
iajs-3594	102	88	𝑆(𝑇(𝑥	𝑆(𝑇(𝑥	NOUN
iajs-3594	102	89	)	)	PUNCT
iajs-3594	102	90	)	)	PUNCT
iajs-3594	103	1	𝛼	𝛼	PRON
iajs-3594	103	2	𝑦	𝑦	NOUN
iajs-3594	103	3	,	,	PUNCT
iajs-3594	103	4	for	for	SCONJ
iajs-3594	103	5	each	each	DET
iajs-3594	103	6	𝑥	𝑥	PROPN
iajs-3594	103	7	,	,	PUNCT
iajs-3594	103	8	𝑦	𝑦	PROPN
iajs-3594	103	9	∈	∈	PROPN
iajs-3594	103	10	𝐴	𝐴	PROPN
iajs-3594	103	11	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	103	12	𝛼	𝛼	SYM
iajs-3594	103	13	∈	∈	PROPN
iajs-3594	103	14	𝛤.	𝛤.	PROPN
iajs-3594	103	15	(	(	PUNCT
iajs-3594	103	16	8)	8)	NUM
iajs-3594	103	17	by	by	ADP
iajs-3594	103	18	comparing	compare	VERB
iajs-3594	103	19	(	(	PUNCT
iajs-3594	103	20	6	6	NUM
iajs-3594	103	21	)	)	PUNCT
iajs-3594	103	22	with	with	ADP
iajs-3594	103	23	(	(	PUNCT
iajs-3594	103	24	8)	8)	NUM
iajs-3594	103	25	,	,	PUNCT
iajs-3594	103	26	we	we	PRON
iajs-3594	103	27	arrive	arrive	VERB
iajs-3594	103	28	at	at	ADP
iajs-3594	103	29	(	(	PUNCT
iajs-3594	103	30	𝑆(𝑇(𝑥	𝑆(𝑇(𝑥	NOUN
iajs-3594	103	31	)	)	PUNCT
iajs-3594	103	32	)	)	PUNCT
iajs-3594	104	1	−	−	PROPN
iajs-3594	104	2	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	104	3	)	)	PUNCT
iajs-3594	104	4	)	)	PUNCT
iajs-3594	105	1	𝛼	𝛼	PRON
iajs-3594	105	2	𝑦	𝑦	NOUN
iajs-3594	105	3	=	=	SYM
iajs-3594	105	4	0	0	NUM
iajs-3594	105	5	multiply	multiply	NOUN
iajs-3594	105	6	from	from	ADP
iajs-3594	105	7	the	the	DET
iajs-3594	105	8	right	right	NOUN
iajs-3594	105	9	by	by	ADP
iajs-3594	105	10	(	(	PUNCT
iajs-3594	105	11	𝑆(𝑇(𝑥	𝑆(𝑇(𝑥	NOUN
iajs-3594	105	12	)	)	PUNCT
iajs-3594	105	13	)	)	PUNCT
iajs-3594	106	1	−	−	PROPN
iajs-3594	106	2	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	106	3	)	)	PUNCT
iajs-3594	106	4	)	)	PUNCT
iajs-3594	106	5	,	,	PUNCT
iajs-3594	106	6	we	we	PRON
iajs-3594	106	7	get	get	VERB
iajs-3594	106	8	(	(	PUNCT
iajs-3594	106	9	𝑆(𝑇(𝑥	𝑆(𝑇(𝑥	NOUN
iajs-3594	106	10	)	)	PUNCT
iajs-3594	106	11	)	)	PUNCT
iajs-3594	107	1	−	−	ADP
iajs-3594	108	1	𝑇(𝑥))𝛽	𝑇(𝑥))𝛽	ADV
iajs-3594	108	2	𝑦	𝑦	NUM
iajs-3594	108	3	𝛼	𝛼	NOUN
iajs-3594	108	4	(	(	PUNCT
iajs-3594	108	5	𝑆(𝑇(𝑥	𝑆(𝑇(𝑥	NOUN
iajs-3594	108	6	)	)	PUNCT
iajs-3594	108	7	)	)	PUNCT
iajs-3594	109	1	−	−	PROPN
iajs-3594	109	2	(	(	PUNCT
iajs-3594	109	3	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	109	4	)	)	PUNCT
iajs-3594	109	5	)	)	PUNCT
iajs-3594	110	1	=	=	PUNCT
iajs-3594	110	2	0	0	NUM
iajs-3594	110	3	,	,	PUNCT
iajs-3594	110	4	for	for	ADP
iajs-3594	110	5	each	each	DET
iajs-3594	110	6	𝑥	𝑥	PROPN
iajs-3594	110	7	,	,	PUNCT
iajs-3594	110	8	𝑦	𝑦	PROPN
iajs-3594	110	9	∈	∈	PROPN
iajs-3594	110	10	𝐴	𝐴	PROPN
iajs-3594	110	11	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	110	12	𝛼	𝛼	NOUN
iajs-3594	110	13	,	,	PUNCT
iajs-3594	110	14	𝛽	𝛽	PROPN
iajs-3594	110	15	∈	∈	PROPN
iajs-3594	110	16	г	г	PROPN
iajs-3594	110	17	.	.	PUNCT
iajs-3594	110	18	by	by	ADP
iajs-3594	110	19	semiprimeness	semiprimeness	NOUN
iajs-3594	110	20	of	of	ADP
iajs-3594	110	21	a	a	PRON
iajs-3594	110	22	,	,	PUNCT
iajs-3594	110	23	we	we	PRON
iajs-3594	110	24	have	have	VERB
iajs-3594	110	25	𝑆	𝑆	PROPN
iajs-3594	110	26	(	(	PUNCT
iajs-3594	110	27	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	110	28	)	)	PUNCT
iajs-3594	110	29	)	)	PUNCT
iajs-3594	111	1	=	=	SYM
iajs-3594	111	2	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	111	3	)	)	PUNCT
iajs-3594	111	4	,	,	PUNCT
iajs-3594	111	5	for	for	ADP
iajs-3594	111	6	each	each	DET
iajs-3594	111	7	𝑥	𝑥	PRON
iajs-3594	111	8	∈	∈	PROPN
iajs-3594	111	9	𝐴.	𝐴.	NOUN
iajs-3594	111	10	(	(	PUNCT
iajs-3594	111	11	9	9	NUM
iajs-3594	111	12	)	)	PUNCT
iajs-3594	111	13	from	from	ADP
iajs-3594	111	14	(	(	PUNCT
iajs-3594	111	15	9	9	NUM
iajs-3594	111	16	)	)	PUNCT
iajs-3594	111	17	and	and	CCONJ
iajs-3594	111	18	using	use	VERB
iajs-3594	111	19	(	(	PUNCT
iajs-3594	111	20	7	7	NUM
iajs-3594	111	21	)	)	PUNCT
iajs-3594	111	22	,	,	PUNCT
iajs-3594	111	23	we	we	PRON
iajs-3594	111	24	imply	imply	VERB
iajs-3594	111	25	𝑇(𝑥)𝛼𝑇(𝑦	𝑇(𝑥)𝛼𝑇(𝑦	ADJ
iajs-3594	111	26	)	)	PUNCT
iajs-3594	112	1	=	=	PUNCT
iajs-3594	112	2	𝑇(𝑥)𝛼𝑆(𝑦	𝑇(𝑥)𝛼𝑆(𝑦	VERB
iajs-3594	112	3	)	)	PUNCT
iajs-3594	112	4	,	,	PUNCT
iajs-3594	112	5	for	for	ADP
iajs-3594	112	6	each	each	DET
iajs-3594	112	7	𝑥	𝑥	PROPN
iajs-3594	112	8	,	,	PUNCT
iajs-3594	112	9	𝑦	𝑦	PROPN
iajs-3594	112	10	∈	∈	PROPN
iajs-3594	112	11	𝐴	𝐴	PROPN
iajs-3594	112	12	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	112	13	𝛼	𝛼	SYM
iajs-3594	112	14	∈	∈	PROPN
iajs-3594	112	15	𝛤.	𝛤.	PROPN
iajs-3594	112	16	(	(	PUNCT
iajs-3594	112	17	10	10	NUM
iajs-3594	112	18	)	)	PUNCT
iajs-3594	112	19	replace	replace	NOUN
iajs-3594	112	20	𝑥	𝑥	NOUN
iajs-3594	112	21	by	by	ADP
iajs-3594	112	22	𝑥𝛽𝑧	𝑥𝛽𝑧	NOUN
iajs-3594	112	23	and	and	CCONJ
iajs-3594	112	24	𝑦	𝑦	NOUN
iajs-3594	112	25	by	by	ADP
iajs-3594	112	26	𝑦𝜎𝑤	𝑦𝜎𝑤	NOUN
iajs-3594	112	27	in	in	ADP
iajs-3594	112	28	(	(	PUNCT
iajs-3594	112	29	7	7	NUM
iajs-3594	112	30	)	)	PUNCT
iajs-3594	112	31	and	and	CCONJ
iajs-3594	112	32	using	use	VERB
iajs-3594	112	33	(	(	PUNCT
iajs-3594	112	34	10	10	NUM
iajs-3594	112	35	)	)	PUNCT
iajs-3594	112	36	,	,	PUNCT
iajs-3594	112	37	we	we	PRON
iajs-3594	112	38	arrive	arrive	VERB
iajs-3594	112	39	at	at	ADP
iajs-3594	112	40	𝑥𝛽𝑧	𝑥𝛽𝑧	NOUN
iajs-3594	112	41	𝛼𝑇(𝑦)𝜎𝑆	𝛼𝑇(𝑦)𝜎𝑆	PROPN
iajs-3594	112	42	(	(	PUNCT
iajs-3594	112	43	𝑤	𝑤	X
iajs-3594	112	44	)	)	PUNCT
iajs-3594	112	45	=	=	PUNCT
iajs-3594	113	1	𝑆(𝑥	𝑆(𝑥	PUNCT
iajs-3594	113	2	𝛽	𝛽	NOUN
iajs-3594	113	3	𝑧	𝑧	X
iajs-3594	113	4	)	)	PUNCT
iajs-3594	113	5	𝛼	𝛼	PROPN
iajs-3594	113	6	𝑦	𝑦	NUM
iajs-3594	113	7	𝜎	𝜎	SYM
iajs-3594	113	8	𝑤	𝑤	ADP
iajs-3594	113	9	,	,	PUNCT
iajs-3594	113	10	for	for	ADP
iajs-3594	113	11	each	each	DET
iajs-3594	113	12	𝑥	𝑥	PROPN
iajs-3594	113	13	,	,	PUNCT
iajs-3594	113	14	𝑦	𝑦	NOUN
iajs-3594	113	15	,	,	PUNCT
iajs-3594	113	16	𝑧	𝑧	VERB
iajs-3594	113	17	,	,	PUNCT
iajs-3594	113	18	𝑤	𝑤	ADP
iajs-3594	113	19	∈	∈	PROPN
iajs-3594	113	20	𝐴	𝐴	PROPN
iajs-3594	113	21	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	113	22	𝛼	𝛼	PROPN
iajs-3594	113	23	,	,	PUNCT
iajs-3594	113	24	𝛽	𝛽	PROPN
iajs-3594	113	25	,	,	PUNCT
iajs-3594	113	26	𝜎	𝜎	PROPN
iajs-3594	113	27	,	,	PUNCT
iajs-3594	113	28	∈	∈	PROPN
iajs-3594	113	29	𝛤.	𝛤.	PROPN
iajs-3594	113	30	(	(	PUNCT
iajs-3594	113	31	11	11	NUM
iajs-3594	113	32	)	)	PUNCT
iajs-3594	113	33	thence	thence	NOUN
iajs-3594	113	34	by	by	ADP
iajs-3594	113	35	(	(	PUNCT
iajs-3594	113	36	11	11	NUM
iajs-3594	113	37	)	)	PUNCT
iajs-3594	113	38	,	,	PUNCT
iajs-3594	113	39	we	we	PRON
iajs-3594	113	40	imply	imply	VERB
iajs-3594	113	41	𝑥𝛽𝑆(𝑧)𝛼	𝑥𝛽𝑆(𝑧)𝛼	VERB
iajs-3594	114	1	𝑦	𝑦	PRON
iajs-3594	114	2	𝜎	𝜎	PROPN
iajs-3594	114	3	𝑆	𝑆	PROPN
iajs-3594	114	4	(	(	PUNCT
iajs-3594	114	5	𝑤	𝑤	NOUN
iajs-3594	114	6	)	)	PUNCT
iajs-3594	114	7	=	=	SYM
iajs-3594	114	8	𝑆	𝑆	PROPN
iajs-3594	114	9	(	(	PUNCT
iajs-3594	114	10	𝑥	𝑥	NOUN
iajs-3594	114	11	𝛽	𝛽	PROPN
iajs-3594	114	12	𝑧)𝛼𝑦𝜎𝑤	𝑧)𝛼𝑦𝜎𝑤	PROPN
iajs-3594	114	13	,	,	PUNCT
iajs-3594	114	14	for	for	ADP
iajs-3594	114	15	𝑎𝑙𝑙	𝑎𝑙𝑙	PROPN
iajs-3594	114	16	𝑥	𝑥	PROPN
iajs-3594	114	17	,	,	PUNCT
iajs-3594	114	18	𝑦	𝑦	PROPN
iajs-3594	114	19	,	,	PUNCT
iajs-3594	114	20	𝑧	𝑧	VERB
iajs-3594	114	21	,	,	PUNCT
iajs-3594	114	22	𝑤	𝑤	ADP
iajs-3594	114	23	∈	∈	PROPN
iajs-3594	114	24	𝐴	𝐴	PROPN
iajs-3594	114	25	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	114	26	𝛼	𝛼	PROPN
iajs-3594	114	27	,	,	PUNCT
iajs-3594	114	28	𝛽	𝛽	PROPN
iajs-3594	114	29	,	,	PUNCT
iajs-3594	114	30	𝜎	𝜎	PROPN
iajs-3594	114	31	∈	∈	PROPN
iajs-3594	114	32	𝛤.	𝛤.	PROPN
iajs-3594	114	33	whence	whence	NOUN
iajs-3594	114	34	it	it	PRON
iajs-3594	114	35	follows	follow	VERB
iajs-3594	114	36	that	that	SCONJ
iajs-3594	114	37	𝑥𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤	𝑥𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤	PROPN
iajs-3594	114	38	)	)	PUNCT
iajs-3594	114	39	−	−	PROPN
iajs-3594	114	40	w	w	NOUN
iajs-3594	114	41	)	)	PUNCT
iajs-3594	114	42	=	=	SYM
iajs-3594	114	43	0	0	NUM
iajs-3594	114	44	,	,	PUNCT
iajs-3594	114	45	for	for	ADP
iajs-3594	114	46	each	each	DET
iajs-3594	114	47	𝑥	𝑥	PROPN
iajs-3594	114	48	,	,	PUNCT
iajs-3594	114	49	𝑦	𝑦	NOUN
iajs-3594	114	50	,	,	PUNCT
iajs-3594	114	51	𝑧	𝑧	VERB
iajs-3594	114	52	,	,	PUNCT
iajs-3594	114	53	𝑤	𝑤	ADP
iajs-3594	114	54	∈	∈	PROPN
iajs-3594	114	55	𝐴𝑎𝑛𝑑	𝐴𝑎𝑛𝑑	PROPN
iajs-3594	114	56	𝛼	𝛼	PROPN
iajs-3594	114	57	,	,	PUNCT
iajs-3594	114	58	𝛽	𝛽	PROPN
iajs-3594	114	59	,	,	PUNCT
iajs-3594	114	60	𝜎	𝜎	PROPN
iajs-3594	114	61	∈	∈	PROPN
iajs-3594	114	62	𝛤.	𝛤.	PROPN
iajs-3594	114	63	(	(	PUNCT
iajs-3594	114	64	12	12	NUM
iajs-3594	114	65	)	)	PUNCT
iajs-3594	114	66	the	the	DET
iajs-3594	114	67	substitution	substitution	NOUN
iajs-3594	114	68	on	on	ADP
iajs-3594	114	69	s	s	PROPN
iajs-3594	114	70	(	(	PUNCT
iajs-3594	114	71	w)−	w)−	PROPN
iajs-3594	114	72	w	w	PROPN
iajs-3594	114	73	for	for	ADP
iajs-3594	114	74	x	x	SYM
iajs-3594	114	75	in	in	ADP
iajs-3594	114	76	(	(	PUNCT
iajs-3594	114	77	12	12	NUM
iajs-3594	114	78	)	)	PUNCT
iajs-3594	114	79	,	,	PUNCT
iajs-3594	114	80	gives	give	VERB
iajs-3594	114	81	us	we	PRON
iajs-3594	114	82	(	(	PUNCT
iajs-3594	114	83	𝑆(𝑤	𝑆(𝑤	PROPN
iajs-3594	114	84	)	)	PUNCT
iajs-3594	114	85	−	−	PROPN
iajs-3594	114	86	𝑤)𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤	𝑤)𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤	NUM
iajs-3594	114	87	)	)	PUNCT
iajs-3594	114	88	−	−	PROPN
iajs-3594	114	89	𝑤	𝑤	X
iajs-3594	114	90	)	)	PUNCT
iajs-3594	114	91	=	=	SYM
iajs-3594	114	92	0	0	NUM
iajs-3594	114	93	,	,	PUNCT
iajs-3594	114	94	for	for	ADP
iajs-3594	114	95	each	each	DET
iajs-3594	114	96	𝑦	𝑦	NOUN
iajs-3594	114	97	,	,	PUNCT
iajs-3594	114	98	𝑧	𝑧	PROPN
iajs-3594	114	99	,	,	PUNCT
iajs-3594	114	100	𝑤	𝑤	ADP
iajs-3594	114	101	∈	∈	PROPN
iajs-3594	114	102	𝐴	𝐴	PROPN
iajs-3594	114	103	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	114	104	𝛼	𝛼	PROPN
iajs-3594	114	105	,	,	PUNCT
iajs-3594	114	106	𝛽	𝛽	PROPN
iajs-3594	114	107	,	,	PUNCT
iajs-3594	114	108	𝜎	𝜎	PROPN
iajs-3594	114	109	∈	∈	PROPN
iajs-3594	114	110	γ	γ	X
iajs-3594	114	111	.	.	PUNCT
iajs-3594	115	1	right	right	ADJ
iajs-3594	115	2	multiplication	multiplication	NOUN
iajs-3594	115	3	the	the	DET
iajs-3594	115	4	above	above	ADJ
iajs-3594	115	5	relation	relation	NOUN
iajs-3594	115	6	by	by	ADP
iajs-3594	115	7	s(z	s(z	PROPN
iajs-3594	115	8	)	)	PUNCT
iajs-3594	115	9	,	,	PUNCT
iajs-3594	115	10	yields	yield	NOUN
iajs-3594	115	11	(	(	PUNCT
iajs-3594	115	12	𝑆(𝑤	𝑆(𝑤	NUM
iajs-3594	115	13	)	)	PUNCT
iajs-3594	115	14	−	−	PROPN
iajs-3594	115	15	𝑤)𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤	𝑤)𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤	NUM
iajs-3594	115	16	)	)	PUNCT
iajs-3594	115	17	−	−	PROPN
iajs-3594	116	1	𝑤)𝛾𝑆(𝑧	𝑤)𝛾𝑆(𝑧	NOUN
iajs-3594	116	2	)	)	PUNCT
iajs-3594	116	3	=	=	SYM
iajs-3594	116	4	0	0	NUM
iajs-3594	116	5	,	,	PUNCT
iajs-3594	116	6	for	for	ADP
iajs-3594	116	7	𝑎𝑙𝑙	𝑎𝑙𝑙	PROPN
iajs-3594	116	8	𝑦	𝑦	PROPN
iajs-3594	116	9	,	,	PUNCT
iajs-3594	116	10	𝑧	𝑧	PROPN
iajs-3594	116	11	,	,	PUNCT
iajs-3594	116	12	𝑤	𝑤	ADP
iajs-3594	116	13	∈	∈	PROPN
iajs-3594	116	14	𝐴	𝐴	PROPN
iajs-3594	116	15	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	116	16	𝛼	𝛼	PROPN
iajs-3594	116	17	,	,	PUNCT
iajs-3594	116	18	𝛽	𝛽	PROPN
iajs-3594	116	19	,	,	PUNCT
iajs-3594	116	20	𝜎	𝜎	PROPN
iajs-3594	116	21	,	,	PUNCT
iajs-3594	116	22	𝛾	𝛾	PROPN
iajs-3594	116	23	∈	∈	PROPN
iajs-3594	116	24	γ	γ	X
iajs-3594	116	25	.	.	PROPN
iajs-3594	116	26	by	by	ADP
iajs-3594	116	27	the	the	DET
iajs-3594	116	28	semiprimeness	semiprimeness	NOUN
iajs-3594	116	29	of	of	ADP
iajs-3594	116	30	a	a	PRON
iajs-3594	116	31	,	,	PUNCT
iajs-3594	116	32	we	we	PRON
iajs-3594	116	33	obtain	obtain	VERB
iajs-3594	116	34	(	(	PUNCT
iajs-3594	116	35	𝑆(𝑤	𝑆(𝑤	NUM
iajs-3594	116	36	)	)	PUNCT
iajs-3594	116	37	−	−	NOUN
iajs-3594	116	38	w)βs(𝑧	w)βs(𝑧	ADV
iajs-3594	116	39	)	)	PUNCT
iajs-3594	116	40	=	=	SYM
iajs-3594	116	41	0	0	NUM
iajs-3594	116	42	,	,	PUNCT
iajs-3594	116	43	for	for	ADP
iajs-3594	116	44	each	each	DET
iajs-3594	116	45	𝑤	𝑤	ADP
iajs-3594	116	46	,	,	PUNCT
iajs-3594	116	47	𝑧	𝑧	PROPN
iajs-3594	116	48	∈	∈	PROPN
iajs-3594	116	49	a	a	NOUN
iajs-3594	116	50	and	and	CCONJ
iajs-3594	116	51	β	β	X
iajs-3594	116	52	∈	∈	PROPN
iajs-3594	116	53	г	г	PROPN
iajs-3594	116	54	.	.	PUNCT
iajs-3594	117	1	this	this	PRON
iajs-3594	117	2	gives	give	VERB
iajs-3594	117	3	𝑆(𝑤	𝑆(𝑤	PROPN
iajs-3594	117	4	)	)	PUNCT
iajs-3594	117	5	𝛽	𝛽	NOUN
iajs-3594	117	6	𝑆(𝑧	𝑆(𝑧	NOUN
iajs-3594	117	7	)	)	PUNCT
iajs-3594	117	8	=	=	PUNCT
iajs-3594	117	9	𝑤	𝑤	ADP
iajs-3594	117	10	𝛽	𝛽	NOUN
iajs-3594	117	11	𝑆(𝑧	𝑆(𝑧	NUM
iajs-3594	117	12	)	)	PUNCT
iajs-3594	117	13	,	,	PUNCT
iajs-3594	117	14	for	for	ADP
iajs-3594	117	15	each	each	DET
iajs-3594	117	16	𝑤	𝑤	ADP
iajs-3594	117	17	,	,	PUNCT
iajs-3594	117	18	𝑧	𝑧	DET
iajs-3594	117	19	∈	∈	PROPN
iajs-3594	117	20	𝐴	𝐴	PROPN
iajs-3594	117	21	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	117	22	𝛽	𝛽	PROPN
iajs-3594	117	23	∈	∈	PROPN
iajs-3594	117	24	𝛤.	𝛤.	PROPN
iajs-3594	117	25	(	(	PUNCT
iajs-3594	117	26	13	13	NUM
iajs-3594	117	27	)	)	PUNCT
iajs-3594	117	28	from	from	ADP
iajs-3594	117	29	(	(	PUNCT
iajs-3594	117	30	7	7	NUM
iajs-3594	117	31	)	)	PUNCT
iajs-3594	117	32	and	and	CCONJ
iajs-3594	117	33	(	(	PUNCT
iajs-3594	117	34	13	13	NUM
iajs-3594	117	35	)	)	PUNCT
iajs-3594	117	36	,	,	PUNCT
iajs-3594	117	37	we	we	PRON
iajs-3594	117	38	obtain	obtain	VERB
iajs-3594	117	39	ihjpas	ihjpa	NOUN
iajs-3594	117	40	.	.	PUNCT
iajs-3594	118	1	2025	2025	NUM
iajs-3594	118	2	,	,	PUNCT
iajs-3594	118	3	38(2	38(2	NUM
iajs-3594	118	4	)	)	PUNCT
iajs-3594	118	5	335	335	NUM
iajs-3594	118	6	𝑥	𝑥	DET
iajs-3594	118	7	𝛼	𝛼	PRON
iajs-3594	118	8	𝑇(𝑦	𝑇(𝑦	NOUN
iajs-3594	118	9	)	)	PUNCT
iajs-3594	118	10	=	=	SYM
iajs-3594	118	11	𝑆(𝑥	𝑆(𝑥	X
iajs-3594	118	12	)	)	PUNCT
iajs-3594	118	13	𝛼	𝛼	PRON
iajs-3594	118	14	𝑆(𝑦	𝑆(𝑦	NOUN
iajs-3594	118	15	)	)	PUNCT
iajs-3594	118	16	=	=	PUNCT
iajs-3594	119	1	𝑥	𝑥	DET
iajs-3594	119	2	𝛼	𝛼	X
iajs-3594	119	3	𝑆(𝑦	𝑆(𝑦	NOUN
iajs-3594	119	4	)	)	PUNCT
iajs-3594	119	5	,	,	PUNCT
iajs-3594	119	6	for	for	ADP
iajs-3594	119	7	each	each	DET
iajs-3594	119	8	𝑥	𝑥	PROPN
iajs-3594	119	9	,	,	PUNCT
iajs-3594	119	10	𝑦	𝑦	PROPN
iajs-3594	119	11	∈	∈	PROPN
iajs-3594	119	12	𝐴	𝐴	PROPN
iajs-3594	119	13	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	119	14	𝛼	𝛼	PROPN
iajs-3594	119	15	∈	∈	PROPN
iajs-3594	119	16	г	г	PROPN
iajs-3594	119	17	.	.	PUNCT
iajs-3594	120	1	of	of	ADP
iajs-3594	120	2	course	course	ADV
iajs-3594	120	3	,	,	PUNCT
iajs-3594	120	4	we	we	PRON
iajs-3594	120	5	have	have	VERB
iajs-3594	120	6	also	also	ADV
iajs-3594	120	7	,	,	PUNCT
iajs-3594	120	8	𝑥𝛼(𝑇(𝑦	𝑥𝛼(𝑇(𝑦	ADJ
iajs-3594	120	9	)	)	PUNCT
iajs-3594	120	10	−s(y	−s(y	ADJ
iajs-3594	120	11	)	)	PUNCT
iajs-3594	120	12	)	)	PUNCT
iajs-3594	121	1	=	=	SYM
iajs-3594	121	2	0	0	NUM
iajs-3594	121	3	,	,	PUNCT
iajs-3594	121	4	for	for	ADP
iajs-3594	121	5	each	each	DET
iajs-3594	121	6	x	x	NOUN
iajs-3594	121	7	,	,	PUNCT
iajs-3594	121	8	y	y	PROPN
iajs-3594	121	9	∈	∈	PROPN
iajs-3594	121	10	a	a	PRON
iajs-3594	121	11	,	,	PUNCT
iajs-3594	121	12	and	and	CCONJ
iajs-3594	121	13	𝛼	𝛼	X
iajs-3594	121	14	∈	∈	PROPN
iajs-3594	121	15	г	г	PROPN
iajs-3594	121	16	.	.	PUNCT
iajs-3594	121	17	by	by	ADP
iajs-3594	121	18	the	the	DET
iajs-3594	121	19	semiprimeness	semiprimeness	NOUN
iajs-3594	121	20	of	of	ADP
iajs-3594	121	21	a	a	PRON
iajs-3594	121	22	,	,	PUNCT
iajs-3594	121	23	we	we	PRON
iajs-3594	121	24	imply	imply	VERB
iajs-3594	121	25	t	t	PROPN
iajs-3594	121	26	=	=	PROPN
iajs-3594	121	27	s.	s.	PROPN
iajs-3594	121	28	in	in	ADP
iajs-3594	121	29	the	the	DET
iajs-3594	121	30	following	following	NOUN
iajs-3594	121	31	theorem	theorem	NOUN
iajs-3594	121	32	,	,	PUNCT
iajs-3594	121	33	we	we	PRON
iajs-3594	121	34	gives	give	VERB
iajs-3594	121	35	a	a	DET
iajs-3594	121	36	relation	relation	NOUN
iajs-3594	121	37	between	between	ADP
iajs-3594	121	38	t	t	PROPN
iajs-3594	121	39	and	and	CCONJ
iajs-3594	121	40	s	s	PROPN
iajs-3594	121	41	,	,	PUNCT
iajs-3594	121	42	where	where	SCONJ
iajs-3594	121	43	(	(	PUNCT
iajs-3594	121	44	t	t	PROPN
iajs-3594	121	45	,	,	PUNCT
iajs-3594	121	46	s	s	PART
iajs-3594	121	47	)	)	PUNCT
iajs-3594	121	48	is	be	AUX
iajs-3594	121	49	a	a	DET
iajs-3594	121	50	double	double	ADJ
iajs-3594	121	51	centralizer	centralizer	NOUN
iajs-3594	121	52	on	on	ADP
iajs-3594	121	53	prime	prime	ADJ
iajs-3594	121	54	γ	γ	X
iajs-3594	121	55	-	-	NOUN
iajs-3594	121	56	ring	ring	NOUN
iajs-3594	121	57	.	.	PUNCT
iajs-3594	122	1	3.8	3.8	NUM
iajs-3594	122	2	theorem	theorem	NOUN
iajs-3594	122	3	let	let	VERB
iajs-3594	122	4	a	a	PRON
iajs-3594	122	5	be	be	AUX
iajs-3594	122	6	a	a	DET
iajs-3594	122	7	prime	prime	ADJ
iajs-3594	122	8	𝛤-ring	𝛤-ring	PROPN
iajs-3594	122	9	,	,	PUNCT
iajs-3594	122	10	and	and	CCONJ
iajs-3594	122	11	u	u	PRON
iajs-3594	122	12	be	be	VERB
iajs-3594	122	13	a	a	DET
iajs-3594	122	14	not	not	PART
iajs-3594	122	15	equal	equal	ADJ
iajs-3594	122	16	zero	zero	NUM
iajs-3594	122	17	ideal	ideal	NOUN
iajs-3594	122	18	of	of	ADP
iajs-3594	122	19	a	a	PRON
iajs-3594	122	20	,	,	PUNCT
iajs-3594	122	21	we	we	PRON
iajs-3594	122	22	imply	imply	VERB
iajs-3594	122	23	(	(	PUNCT
iajs-3594	122	24	t	t	PROPN
iajs-3594	122	25	,	,	PUNCT
iajs-3594	122	26	s	s	PART
iajs-3594	122	27	)	)	PUNCT
iajs-3594	122	28	be	be	AUX
iajs-3594	122	29	a	a	DET
iajs-3594	122	30	double	double	ADJ
iajs-3594	122	31	centralizer	centralizer	NOUN
iajs-3594	122	32	.	.	PUNCT
iajs-3594	123	1	if	if	SCONJ
iajs-3594	123	2	𝑇(𝑟	𝑇(𝑟	PUNCT
iajs-3594	123	3	𝛼	𝛼	PRON
iajs-3594	123	4	𝑥	𝑥	NOUN
iajs-3594	123	5	)	)	PUNCT
iajs-3594	123	6	=	=	SYM
iajs-3594	123	7	𝑆(𝑟)𝛼𝑥	𝑆(𝑟)𝛼𝑥	NOUN
iajs-3594	123	8	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3594	123	9	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3594	123	10	𝑟	𝑟	X
iajs-3594	123	11	∈	∈	PROPN
iajs-3594	123	12	𝐴	𝐴	PROPN
iajs-3594	123	13	,	,	PUNCT
iajs-3594	123	14	𝑥	𝑥	PRON
iajs-3594	123	15	∈	∈	PROPN
iajs-3594	123	16	𝑈	𝑈	PROPN
iajs-3594	123	17	,	,	PUNCT
iajs-3594	123	18	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
iajs-3594	123	19	𝑇	𝑇	NOUN
iajs-3594	123	20	=	=	SYM
iajs-3594	123	21	𝑆.	𝑆.	PROPN
iajs-3594	123	22	proof	proof	NOUN
iajs-3594	123	23	:	:	PUNCT
iajs-3594	123	24	we	we	PRON
iajs-3594	123	25	have	have	VERB
iajs-3594	123	26	𝑇(𝑟𝛼𝑥	𝑇(𝑟𝛼𝑥	NOUN
iajs-3594	123	27	)	)	PUNCT
iajs-3594	124	1	=	=	SYM
iajs-3594	124	2	𝑇(𝑟)𝛼𝑥	𝑇(𝑟)𝛼𝑥	NOUN
iajs-3594	124	3	=	=	PUNCT
iajs-3594	124	4	𝑆(𝑟)𝛼𝑥	𝑆(𝑟)𝛼𝑥	PROPN
iajs-3594	124	5	for	for	ADP
iajs-3594	124	6	each	each	DET
iajs-3594	124	7	𝑟	𝑟	NOUN
iajs-3594	124	8	,	,	PUNCT
iajs-3594	124	9	𝑡	𝑡	PROPN
iajs-3594	124	10	∈	∈	PROPN
iajs-3594	124	11	a	a	DET
iajs-3594	124	12	,	,	PUNCT
iajs-3594	124	13	𝑥	𝑥	PROPN
iajs-3594	124	14	∈	∈	PROPN
iajs-3594	124	15	u	u	NOUN
iajs-3594	124	16	and	and	CCONJ
iajs-3594	124	17	𝛼	𝛼	NOUN
iajs-3594	124	18	∈	∈	PROPN
iajs-3594	124	19	г	г	PROPN
iajs-3594	124	20	.	.	PUNCT
iajs-3594	125	1	this	this	PRON
iajs-3594	125	2	reduces	reduce	VERB
iajs-3594	125	3	to	to	ADP
iajs-3594	125	4	(	(	PUNCT
iajs-3594	125	5	𝑇(𝑟	𝑇(𝑟	NUM
iajs-3594	125	6	)	)	PUNCT
iajs-3594	125	7	−	−	PROPN
iajs-3594	125	8	𝑆(𝑟))𝛼𝑥	𝑆(𝑟))𝛼𝑥	NOUN
iajs-3594	126	1	=	=	SYM
iajs-3594	126	2	0	0	PROPN
iajs-3594	126	3	,	,	PUNCT
iajs-3594	126	4	for	for	ADP
iajs-3594	126	5	each	each	DET
iajs-3594	126	6	𝑟	𝑟	PRON
iajs-3594	126	7	∈	∈	PROPN
iajs-3594	126	8	a	a	DET
iajs-3594	126	9	,	,	PUNCT
iajs-3594	126	10	𝑥	𝑥	PROPN
iajs-3594	126	11	∈	∈	PROPN
iajs-3594	126	12	u	u	NOUN
iajs-3594	126	13	and	and	CCONJ
iajs-3594	126	14	𝛼	𝛼	PRON
iajs-3594	126	15	∈	∈	PROPN
iajs-3594	126	16	г	г	PROPN
iajs-3594	126	17	.	.	PUNCT
iajs-3594	127	1	(	(	PUNCT
iajs-3594	127	2	14	14	NUM
iajs-3594	127	3	)	)	PUNCT
iajs-3594	127	4	replacing	replace	VERB
iajs-3594	127	5	x	x	PUNCT
iajs-3594	127	6	with	with	ADP
iajs-3594	127	7	𝑡𝛽𝑥	𝑡𝛽𝑥	PROPN
iajs-3594	127	8	in	in	ADP
iajs-3594	127	9	(	(	PUNCT
iajs-3594	127	10	14	14	NUM
iajs-3594	127	11	)	)	PUNCT
iajs-3594	127	12	,	,	PUNCT
iajs-3594	127	13	when	when	SCONJ
iajs-3594	127	14	𝑡	𝑡	PROPN
iajs-3594	127	15	∈	∈	PROPN
iajs-3594	127	16	a	a	PRON
iajs-3594	127	17	,	,	PUNCT
iajs-3594	127	18	𝑥	𝑥	PROPN
iajs-3594	127	19	∈	∈	PROPN
iajs-3594	127	20	u	u	NOUN
iajs-3594	127	21	,	,	PUNCT
iajs-3594	127	22	𝛼	𝛼	PROPN
iajs-3594	127	23	∈	∈	PROPN
iajs-3594	127	24	г	г	PROPN
iajs-3594	127	25	,	,	PUNCT
iajs-3594	127	26	leads	lead	VERB
iajs-3594	127	27	to	to	ADP
iajs-3594	127	28	(	(	PUNCT
iajs-3594	127	29	𝑇(𝑟	𝑇(𝑟	NUM
iajs-3594	127	30	)	)	PUNCT
iajs-3594	127	31	−	−	PROPN
iajs-3594	127	32	𝑆(𝑟))𝛼𝑡𝛽𝑥	𝑆(𝑟))𝛼𝑡𝛽𝑥	NOUN
iajs-3594	127	33	=	=	NOUN
iajs-3594	127	34	0	0	PROPN
iajs-3594	127	35	,	,	PUNCT
iajs-3594	127	36	for	for	ADP
iajs-3594	127	37	each	each	DET
iajs-3594	127	38	𝑟	𝑟	PRON
iajs-3594	127	39	∈	∈	PROPN
iajs-3594	127	40	𝐴	𝐴	PROPN
iajs-3594	127	41	,	,	PUNCT
iajs-3594	127	42	𝑥	𝑥	DET
iajs-3594	127	43	∈	∈	PROPN
iajs-3594	127	44	𝑈	𝑈	PROPN
iajs-3594	127	45	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	127	46	𝛼	𝛼	NOUN
iajs-3594	127	47	,	,	PUNCT
iajs-3594	127	48	𝛽	𝛽	PROPN
iajs-3594	127	49	∈	∈	PROPN
iajs-3594	127	50	г	г	PROPN
iajs-3594	127	51	.	.	PUNCT
iajs-3594	128	1	i.e.	i.e.	X
iajs-3594	128	2	(	(	PUNCT
iajs-3594	128	3	𝑇(𝑟	𝑇(𝑟	NUM
iajs-3594	128	4	)	)	PUNCT
iajs-3594	128	5	−	−	PROPN
iajs-3594	128	6	𝑆(𝑟))𝛼𝐴	𝑆(𝑟))𝛼𝐴	PROPN
iajs-3594	128	7	𝛽𝑈	𝛽𝑈	PROPN
iajs-3594	128	8	=	=	NOUN
iajs-3594	128	9	,	,	PUNCT
iajs-3594	128	10	for	for	ADP
iajs-3594	128	11	each	each	DET
iajs-3594	128	12	𝑟	𝑟	PRON
iajs-3594	128	13	∈	∈	PROPN
iajs-3594	128	14	a	a	PRON
iajs-3594	128	15	and	and	CCONJ
iajs-3594	128	16	𝛼	𝛼	NOUN
iajs-3594	128	17	,	,	PUNCT
iajs-3594	128	18	𝛽	𝛽	PROPN
iajs-3594	128	19	∈	∈	PROPN
iajs-3594	128	20	г	г	PROPN
iajs-3594	128	21	.	.	PUNCT
iajs-3594	128	22	since	since	SCONJ
iajs-3594	128	23	a	a	PRON
iajs-3594	128	24	is	be	AUX
iajs-3594	128	25	a	a	DET
iajs-3594	128	26	prime	prime	ADJ
iajs-3594	128	27	gamma	gamma	NOUN
iajs-3594	128	28	-	-	PUNCT
iajs-3594	128	29	ring	ring	NOUN
iajs-3594	128	30	,	,	PUNCT
iajs-3594	128	31	and	and	CCONJ
iajs-3594	128	32	u	u	PRON
iajs-3594	128	33	be	be	VERB
iajs-3594	128	34	a	a	DET
iajs-3594	128	35	not	not	PART
iajs-3594	128	36	equal	equal	ADJ
iajs-3594	128	37	zero	zero	NUM
iajs-3594	128	38	ideal	ideal	NOUN
iajs-3594	128	39	,	,	PUNCT
iajs-3594	128	40	we	we	PRON
iajs-3594	128	41	have	have	VERB
iajs-3594	128	42	t	t	PROPN
iajs-3594	128	43	=	=	PUNCT
iajs-3594	128	44	s.	s.	PROPN
iajs-3594	128	45	3.9	3.9	NUM
iajs-3594	128	46	theorem	theorem	NOUN
iajs-3594	128	47	let	let	VERB
iajs-3594	128	48	a	a	PRON
iajs-3594	128	49	be	be	AUX
iajs-3594	128	50	a	a	DET
iajs-3594	128	51	prime	prime	ADJ
iajs-3594	128	52	𝛤-ring	𝛤-ring	PROPN
iajs-3594	128	53	,	,	PUNCT
iajs-3594	128	54	and	and	CCONJ
iajs-3594	128	55	(	(	PUNCT
iajs-3594	128	56	t	t	PROPN
iajs-3594	128	57	,	,	PUNCT
iajs-3594	128	58	s	s	AUX
iajs-3594	128	59	)	)	PUNCT
iajs-3594	128	60	be	be	AUX
iajs-3594	128	61	a	a	DET
iajs-3594	128	62	double	double	ADJ
iajs-3594	128	63	centralizer	centralizer	NOUN
iajs-3594	128	64	,	,	PUNCT
iajs-3594	128	65	if	if	SCONJ
iajs-3594	128	66	t	t	PROPN
iajs-3594	128	67	acts	act	VERB
iajs-3594	128	68	as	as	ADP
iajs-3594	128	69	a	a	DET
iajs-3594	128	70	not	not	PART
iajs-3594	128	71	equal	equal	ADJ
iajs-3594	128	72	zero	zero	NUM
iajs-3594	128	73	jordan	jordan	PROPN
iajs-3594	128	74	homomorphism	homomorphism	PROPN
iajs-3594	128	75	(	(	PUNCT
iajs-3594	128	76	𝑇(𝑥𝛼𝑥	𝑇(𝑥𝛼𝑥	ADJ
iajs-3594	128	77	)	)	PUNCT
iajs-3594	128	78	=	=	PUNCT
iajs-3594	129	1	𝑇(𝑥)𝛼	𝑇(𝑥)𝛼	PROPN
iajs-3594	129	2	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	129	3	)	)	PUNCT
iajs-3594	129	4	)	)	PUNCT
iajs-3594	129	5	for	for	ADP
iajs-3594	129	6	each	each	DET
iajs-3594	129	7	𝑥	𝑥	PROPN
iajs-3594	129	8	∈	∈	PROPN
iajs-3594	129	9	𝐴	𝐴	PROPN
iajs-3594	129	10	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	129	11	𝛼	𝛼	PROPN
iajs-3594	129	12	∈	∈	PROPN
iajs-3594	129	13	𝛤.	𝛤.	PROPN
iajs-3594	129	14	thence	thence	NOUN
iajs-3594	129	15	t	t	PROPN
iajs-3594	129	16	=	=	SYM
iajs-3594	129	17	s	s	NOUN
iajs-3594	129	18	=	=	NOUN
iajs-3594	129	19	id	id	NOUN
iajs-3594	129	20	.	.	PUNCT
iajs-3594	129	21	proof	proof	NOUN
iajs-3594	129	22	:	:	PUNCT
iajs-3594	129	23	we	we	PRON
iajs-3594	129	24	have	have	VERB
iajs-3594	129	25	𝑇(𝑥𝛼𝑥	𝑇(𝑥𝛼𝑥	NOUN
iajs-3594	129	26	)	)	PUNCT
iajs-3594	130	1	=	=	SYM
iajs-3594	130	2	𝑇(𝑥)𝛼𝑥	𝑇(𝑥)𝛼𝑥	NOUN
iajs-3594	130	3	,	,	PUNCT
iajs-3594	130	4	for	for	ADP
iajs-3594	130	5	each	each	DET
iajs-3594	130	6	𝑥	𝑥	PRON
iajs-3594	130	7	∈	∈	PROPN
iajs-3594	130	8	a	a	PRON
iajs-3594	130	9	and	and	CCONJ
iajs-3594	130	10	𝛼	𝛼	NOUN
iajs-3594	130	11	∈	∈	PROPN
iajs-3594	130	12	г	г	PROPN
iajs-3594	130	13	.	.	PUNCT
iajs-3594	130	14	thence	thence	NOUN
iajs-3594	130	15	from	from	ADP
iajs-3594	130	16	above	above	ADP
iajs-3594	130	17	relation	relation	NOUN
iajs-3594	130	18	and	and	CCONJ
iajs-3594	130	19	since	since	SCONJ
iajs-3594	130	20	t	t	PROPN
iajs-3594	130	21	is	be	AUX
iajs-3594	130	22	acts	act	NOUN
iajs-3594	130	23	as	as	ADP
iajs-3594	130	24	jordan	jordan	PROPN
iajs-3594	130	25	homomorphism	homomorphism	PROPN
iajs-3594	130	26	.	.	PUNCT
iajs-3594	131	1	yields	yield	NOUN
iajs-3594	131	2	,	,	PUNCT
iajs-3594	131	3	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	131	4	)	)	PUNCT
iajs-3594	131	5	𝛼	𝛼	PROPN
iajs-3594	131	6	(	(	PUNCT
iajs-3594	131	7	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	131	8	)	)	PUNCT
iajs-3594	131	9	–	–	PUNCT
iajs-3594	131	10	𝑥	𝑥	X
iajs-3594	131	11	)	)	PUNCT
iajs-3594	131	12	=	=	SYM
iajs-3594	131	13	0	0	NUM
iajs-3594	131	14	,	,	PUNCT
iajs-3594	131	15	for	for	ADP
iajs-3594	131	16	each	each	DET
iajs-3594	131	17	𝑥	𝑥	PROPN
iajs-3594	131	18	∈	∈	PROPN
iajs-3594	131	19	𝐴	𝐴	PROPN
iajs-3594	131	20	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	131	21	𝛼	𝛼	PROPN
iajs-3594	131	22	∈	∈	PROPN
iajs-3594	131	23	г	г	PROPN
iajs-3594	131	24	.	.	PUNCT
iajs-3594	132	1	(	(	PUNCT
iajs-3594	132	2	15	15	NUM
iajs-3594	132	3	)	)	PUNCT
iajs-3594	132	4	replace	replace	NOUN
iajs-3594	132	5	𝑥	𝑥	NOUN
iajs-3594	132	6	by	by	ADP
iajs-3594	132	7	𝑥𝛽	𝑥𝛽	NOUN
iajs-3594	132	8	𝑦	𝑦	NOUN
iajs-3594	132	9	in	in	ADP
iajs-3594	132	10	(	(	PUNCT
iajs-3594	132	11	15	15	NUM
iajs-3594	132	12	)	)	PUNCT
iajs-3594	132	13	,	,	PUNCT
iajs-3594	132	14	we	we	PRON
iajs-3594	132	15	imply	imply	VERB
iajs-3594	132	16	𝑇(𝑥)𝛽	𝑇(𝑥)𝛽	PROPN
iajs-3594	132	17	𝑦	𝑦	X
iajs-3594	132	18	𝛼	𝛼	X
iajs-3594	132	19	(	(	PUNCT
iajs-3594	132	20	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	21	)	)	PUNCT
iajs-3594	132	22	–	–	PUNCT
iajs-3594	132	23	𝑥)𝛽	𝑥)𝛽	X
iajs-3594	132	24	𝑦	𝑦	NOUN
iajs-3594	132	25	=	=	SYM
iajs-3594	132	26	0	0	NUM
iajs-3594	132	27	,	,	PUNCT
iajs-3594	132	28	for	for	ADP
iajs-3594	132	29	each	each	DET
iajs-3594	132	30	𝑥	𝑥	PROPN
iajs-3594	132	31	,	,	PUNCT
iajs-3594	132	32	𝑦	𝑦	PROPN
iajs-3594	132	33	∈	∈	PROPN
iajs-3594	132	34	𝐴	𝐴	PROPN
iajs-3594	132	35	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3594	132	36	𝛼	𝛼	NOUN
iajs-3594	132	37	,	,	PUNCT
iajs-3594	132	38	𝛽	𝛽	PROPN
iajs-3594	132	39	∈	∈	PROPN
iajs-3594	132	40	𝛤.	𝛤.	PROPN
iajs-3594	132	41	(	(	PUNCT
iajs-3594	132	42	16	16	NUM
iajs-3594	132	43	)	)	PUNCT
iajs-3594	132	44	linearization	linearization	NOUN
iajs-3594	132	45	(	(	PUNCT
iajs-3594	132	46	16	16	NUM
iajs-3594	132	47	)	)	PUNCT
iajs-3594	132	48	,	,	PUNCT
iajs-3594	132	49	we	we	PRON
iajs-3594	132	50	imply	imply	VERB
iajs-3594	132	51	(	(	PUNCT
iajs-3594	132	52	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	53	)	)	PUNCT
iajs-3594	132	54	𝛽	𝛽	NOUN
iajs-3594	132	55	𝑦	𝑦	X
iajs-3594	132	56	𝛼	𝛼	X
iajs-3594	132	57	(	(	PUNCT
iajs-3594	132	58	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	59	)	)	PUNCT
iajs-3594	132	60	–	–	PUNCT
iajs-3594	132	61	𝑥)𝛽	𝑥)𝛽	PUNCT
iajs-3594	132	62	𝑧	𝑧	X
iajs-3594	132	63	+	+	NUM
iajs-3594	132	64	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	65	)	)	PUNCT
iajs-3594	132	66	𝛽	𝛽	NOUN
iajs-3594	132	67	𝑧	𝑧	ADJ
iajs-3594	132	68	𝛼	𝛼	X
iajs-3594	132	69	(	(	PUNCT
iajs-3594	132	70	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	71	)	)	PUNCT
iajs-3594	132	72	–	–	PUNCT
iajs-3594	132	73	𝑥)𝛽	𝑥)𝛽	X
iajs-3594	132	74	𝑦	𝑦	NOUN
iajs-3594	132	75	=	=	SYM
iajs-3594	132	76	0	0	NUM
iajs-3594	132	77	(	(	PUNCT
iajs-3594	132	78	17	17	NUM
iajs-3594	132	79	)	)	PUNCT
iajs-3594	132	80	now	now	ADV
iajs-3594	132	81	,	,	PUNCT
iajs-3594	132	82	replacing	replace	VERB
iajs-3594	132	83	𝑧	𝑧	PRON
iajs-3594	132	84	by	by	ADP
iajs-3594	132	85	𝑦𝜎𝑧	𝑦𝜎𝑧	NOUN
iajs-3594	132	86	in	in	ADP
iajs-3594	132	87	(	(	PUNCT
iajs-3594	132	88	17	17	NUM
iajs-3594	132	89	)	)	PUNCT
iajs-3594	132	90	and	and	CCONJ
iajs-3594	132	91	using	use	VERB
iajs-3594	132	92	(	(	PUNCT
iajs-3594	132	93	16	16	NUM
iajs-3594	132	94	)	)	PUNCT
iajs-3594	132	95	,	,	PUNCT
iajs-3594	132	96	we	we	PRON
iajs-3594	132	97	obtain	obtain	VERB
iajs-3594	132	98	(	(	PUNCT
iajs-3594	132	99	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	100	)	)	PUNCT
iajs-3594	132	101	𝛽	𝛽	NOUN
iajs-3594	132	102	𝑦	𝑦	X
iajs-3594	132	103	𝛼	𝛼	NOUN
iajs-3594	132	104	𝐴	𝐴	NOUN
iajs-3594	132	105	𝛾	𝛾	PROPN
iajs-3594	132	106	(	(	PUNCT
iajs-3594	132	107	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	132	108	)	)	PUNCT
iajs-3594	132	109	–	–	PUNCT
iajs-3594	133	1	𝑥)𝛽𝑦	𝑥)𝛽𝑦	PROPN
iajs-3594	133	2	,	,	PUNCT
iajs-3594	133	3	for	for	ADP
iajs-3594	133	4	each	each	DET
iajs-3594	133	5	𝑥	𝑥	PROPN
iajs-3594	133	6	,	,	PUNCT
iajs-3594	133	7	𝑦	𝑦	PROPN
iajs-3594	133	8	∈	∈	PROPN
iajs-3594	133	9	𝐴	𝐴	PROPN
iajs-3594	133	10	,	,	PUNCT
iajs-3594	133	11	𝛼	𝛼	PROPN
iajs-3594	133	12	,	,	PUNCT
iajs-3594	133	13	𝛽	𝛽	NOUN
iajs-3594	133	14	,	,	PUNCT
iajs-3594	133	15	𝛾	𝛾	PROPN
iajs-3594	133	16	∈	∈	PROPN
iajs-3594	133	17	г	г	PROPN
iajs-3594	133	18	.	.	PUNCT
iajs-3594	133	19	by	by	ADP
iajs-3594	133	20	the	the	DET
iajs-3594	133	21	primeness	primeness	NOUN
iajs-3594	133	22	of	of	ADP
iajs-3594	133	23	a	a	PRON
iajs-3594	133	24	,	,	PUNCT
iajs-3594	133	25	we	we	PRON
iajs-3594	133	26	imply	imply	VERB
iajs-3594	133	27	𝑇(𝑥	𝑇(𝑥	PRON
iajs-3594	133	28	)	)	PUNCT
iajs-3594	133	29	=	=	SYM
iajs-3594	134	1	𝑥	𝑥	NOUN
iajs-3594	134	2	,	,	PUNCT
iajs-3594	134	3	for	for	ADP
iajs-3594	134	4	each	each	DET
iajs-3594	134	5	𝑥	𝑥	PRON
iajs-3594	134	6	∈	∈	PROPN
iajs-3594	134	7	a.	a.	NOUN
iajs-3594	134	8	(	(	PUNCT
iajs-3594	134	9	18	18	NUM
iajs-3594	134	10	)	)	PUNCT
iajs-3594	134	11	otherwise	otherwise	ADV
iajs-3594	134	12	,	,	PUNCT
iajs-3594	134	13	t=0	t=0	PROPN
iajs-3594	134	14	.	.	PROPN
iajs-3594	134	15	from	from	ADP
iajs-3594	134	16	𝑥𝛼𝑇(𝑦	𝑥𝛼𝑇(𝑦	PROPN
iajs-3594	134	17	)	)	PUNCT
iajs-3594	134	18	=	=	PUNCT
iajs-3594	134	19	s(x)𝛼𝑦	s(x)𝛼𝑦	PROPN
iajs-3594	134	20	,	,	PUNCT
iajs-3594	134	21	and	and	CCONJ
iajs-3594	134	22	by	by	ADP
iajs-3594	134	23	(	(	PUNCT
iajs-3594	134	24	18	18	NUM
iajs-3594	134	25	)	)	PUNCT
iajs-3594	134	26	,	,	PUNCT
iajs-3594	134	27	we	we	PRON
iajs-3594	134	28	imply	imply	VERB
iajs-3594	134	29	t	t	NOUN
iajs-3594	134	30	=	=	SYM
iajs-3594	134	31	s	s	PART
iajs-3594	134	32	=	=	PUNCT
iajs-3594	135	1	i	i	PROPN
iajs-3594	135	2	d.	d.	PROPN
iajs-3594	135	3	4	4	NUM
iajs-3594	135	4	.	.	PUNCT
iajs-3594	136	1	conclusion	conclusion	NOUN
iajs-3594	136	2	this	this	DET
iajs-3594	136	3	work	work	NOUN
iajs-3594	136	4	is	be	AUX
iajs-3594	136	5	to	to	PART
iajs-3594	136	6	discuss	discuss	VERB
iajs-3594	136	7	double	double	ADJ
iajs-3594	136	8	centralizer	centralizer	NOUN
iajs-3594	136	9	(	(	PUNCT
iajs-3594	136	10	t	t	PROPN
iajs-3594	136	11	,	,	PUNCT
iajs-3594	136	12	s	s	PART
iajs-3594	136	13	)	)	PUNCT
iajs-3594	136	14	,	,	PUNCT
iajs-3594	136	15	and	and	CCONJ
iajs-3594	136	16	double	double	ADJ
iajs-3594	136	17	jordan	jordan	PROPN
iajs-3594	136	18	centralizer	centralizer	NOUN
iajs-3594	136	19	on	on	ADP
iajs-3594	136	20	prime	prime	ADJ
iajs-3594	136	21	and	and	CCONJ
iajs-3594	136	22	semiprime	semiprime	NOUN
iajs-3594	136	23	гrings	гring	NOUN
iajs-3594	136	24	,	,	PUNCT
iajs-3594	136	25	with	with	ADP
iajs-3594	136	26	fulfilling	fulfil	VERB
iajs-3594	136	27	certain	certain	ADJ
iajs-3594	136	28	identities	identity	NOUN
iajs-3594	136	29	.	.	PUNCT
iajs-3594	137	1	we	we	PRON
iajs-3594	137	2	prove	prove	VERB
iajs-3594	137	3	that	that	SCONJ
iajs-3594	137	4	;	;	PUNCT
iajs-3594	137	5	when	when	SCONJ
iajs-3594	137	6	t	t	PROPN
iajs-3594	137	7	is	be	AUX
iajs-3594	137	8	a	a	DET
iajs-3594	137	9	left	left	ADJ
iajs-3594	137	10	centralizer	centralizer	NOUN
iajs-3594	137	11	,	,	PUNCT
iajs-3594	137	12	s	s	PART
iajs-3594	137	13	is	be	AUX
iajs-3594	137	14	a	a	DET
iajs-3594	137	15	right	right	ADJ
iajs-3594	137	16	centralizer	centralizer	NOUN
iajs-3594	137	17	,	,	PUNCT
iajs-3594	137	18	and	and	CCONJ
iajs-3594	137	19	they	they	PRON
iajs-3594	137	20	fulfilling	fulfil	VERB
iajs-3594	137	21	𝑥𝛼𝑇(𝑦	𝑥𝛼𝑇(𝑦	PROPN
iajs-3594	137	22	)	)	PUNCT
iajs-3594	137	23	=	=	SYM
iajs-3594	137	24	𝑆(𝑥)𝛼𝑦	𝑆(𝑥)𝛼𝑦	PROPN
iajs-3594	137	25	,	,	PUNCT
iajs-3594	137	26	for	for	ADP
iajs-3594	137	27	each	each	DET
iajs-3594	137	28	𝑥	𝑥	PROPN
iajs-3594	137	29	,	,	PUNCT
iajs-3594	137	30	𝑦	𝑦	PRON
iajs-3594	137	31	∈	∈	NOUN
iajs-3594	137	32	𝑈	𝑈	PROPN
iajs-3594	137	33	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	137	34	𝛼	𝛼	PROPN
iajs-3594	137	35	∈	∈	PROPN
iajs-3594	137	36	г	г	PROPN
iajs-3594	137	37	.	.	PUNCT
iajs-3594	138	1	then	then	ADV
iajs-3594	138	2	(	(	PUNCT
iajs-3594	138	3	t	t	PROPN
iajs-3594	138	4	,	,	PUNCT
iajs-3594	138	5	s	s	PART
iajs-3594	138	6	)	)	PUNCT
iajs-3594	138	7	is	be	AUX
iajs-3594	138	8	a	a	DET
iajs-3594	138	9	double	double	ADJ
iajs-3594	138	10	centralizer	centralizer	NOUN
iajs-3594	138	11	.	.	PUNCT
iajs-3594	139	1	also	also	ADV
iajs-3594	139	2	if	if	SCONJ
iajs-3594	139	3	(	(	PUNCT
iajs-3594	139	4	t	t	PROPN
iajs-3594	139	5	,	,	PUNCT
iajs-3594	139	6	s	s	AUX
iajs-3594	139	7	)	)	PUNCT
iajs-3594	139	8	be	be	AUX
iajs-3594	139	9	a	a	DET
iajs-3594	139	10	double	double	ADJ
iajs-3594	139	11	centralizer	centralizer	NOUN
iajs-3594	139	12	,	,	PUNCT
iajs-3594	139	13	t	t	PROPN
iajs-3594	139	14	acts	act	VERB
iajs-3594	139	15	as	as	ADP
iajs-3594	139	16	a	a	DET
iajs-3594	139	17	homomorphism	homomorphism	NOUN
iajs-3594	139	18	on	on	ADP
iajs-3594	139	19	a	a	DET
iajs-3594	139	20	,	,	PUNCT
iajs-3594	139	21	then	then	ADV
iajs-3594	139	22	t	t	PROPN
iajs-3594	139	23	=	=	SYM
iajs-3594	139	24	s	s	PROPN
iajs-3594	139	25	,	,	PUNCT
iajs-3594	139	26	and	and	CCONJ
iajs-3594	139	27	if	if	SCONJ
iajs-3594	139	28	t	t	PROPN
iajs-3594	139	29	acts	act	VERB
iajs-3594	139	30	as	as	ADP
iajs-3594	139	31	a	a	DET
iajs-3594	139	32	not	not	PART
iajs-3594	139	33	equal	equal	ADJ
iajs-3594	139	34	zero	zero	NUM
iajs-3594	139	35	jordan	jordan	PROPN
iajs-3594	139	36	homomorphism	homomorphism	PROPN
iajs-3594	139	37	(	(	PUNCT
iajs-3594	139	38	𝑇(𝑥𝛼𝑥	𝑇(𝑥𝛼𝑥	ADJ
iajs-3594	139	39	)	)	PUNCT
iajs-3594	139	40	=	=	PUNCT
iajs-3594	139	41	𝑇(𝑥)𝛼	𝑇(𝑥)𝛼	PROPN
iajs-3594	139	42	𝑇(𝑥	𝑇(𝑥	NOUN
iajs-3594	139	43	)	)	PUNCT
iajs-3594	139	44	)	)	PUNCT
iajs-3594	139	45	for	for	SCONJ
iajs-3594	139	46	each	each	DET
iajs-3594	139	47	𝑥	𝑥	PROPN
iajs-3594	139	48	∈	∈	PROPN
iajs-3594	139	49	𝐴	𝐴	PROPN
iajs-3594	139	50	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-3594	139	51	𝛼	𝛼	PROPN
iajs-3594	139	52	∈	∈	PROPN
iajs-3594	139	53	𝛤.	𝛤.	PROPN
iajs-3594	139	54	then	then	ADV
iajs-3594	139	55	t	t	PROPN
iajs-3594	139	56	=	=	SYM
iajs-3594	139	57	s	s	PART
iajs-3594	139	58	=	=	X
iajs-3594	139	59	i	i	PROPN
iajs-3594	139	60	d.	d.	PROPN
iajs-3594	139	61	ihjpas	ihjpas	PROPN
iajs-3594	139	62	.	.	PUNCT
iajs-3594	140	1	2025	2025	NUM
iajs-3594	140	2	,	,	PUNCT
iajs-3594	140	3	38(2	38(2	NUM
iajs-3594	140	4	)	)	PUNCT
iajs-3594	140	5	336	336	NUM
iajs-3594	140	6	acknowledgment	acknowledgment	NOUN
iajs-3594	140	7	our	our	PRON
iajs-3594	140	8	researcher	researcher	NOUN
iajs-3594	140	9	extends	extend	VERB
iajs-3594	140	10	his	his	PRON
iajs-3594	140	11	sincere	sincere	ADJ
iajs-3594	140	12	thanks	thank	NOUN
iajs-3594	140	13	to	to	ADP
iajs-3594	140	14	the	the	DET
iajs-3594	140	15	editor	editor	NOUN
iajs-3594	140	16	and	and	CCONJ
iajs-3594	140	17	members	member	NOUN
iajs-3594	140	18	of	of	ADP
iajs-3594	140	19	the	the	DET
iajs-3594	140	20	preparatory	preparatory	PROPN
iajs-3594	140	21	committee	committee	NOUN
iajs-3594	140	22	of	of	ADP
iajs-3594	140	23	the	the	DET
iajs-3594	140	24	ibn	ibn	PROPN
iajs-3594	140	25	al	al	PROPN
iajs-3594	140	26	-	-	PUNCT
iajs-3594	140	27	haitham	haitham	PROPN
iajs-3594	140	28	journal	journal	PROPN
iajs-3594	140	29	of	of	ADP
iajs-3594	140	30	pure	pure	ADJ
iajs-3594	140	31	and	and	CCONJ
iajs-3594	140	32	applied	applied	ADJ
iajs-3594	140	33	sciences	science	NOUN
iajs-3594	140	34	.	.	PUNCT
iajs-3594	141	1	conflict	conflict	NOUN
iajs-3594	141	2	of	of	ADP
iajs-3594	141	3	interest	interest	NOUN
iajs-3594	141	4	there	there	PRON
iajs-3594	141	5	are	be	VERB
iajs-3594	141	6	no	no	DET
iajs-3594	141	7	conflicts	conflict	NOUN
iajs-3594	141	8	of	of	ADP
iajs-3594	141	9	interest	interest	NOUN
iajs-3594	141	10	.	.	PUNCT
iajs-3594	142	1	funding	funding	NOUN
iajs-3594	142	2	there	there	PRON
iajs-3594	142	3	is	be	VERB
iajs-3594	142	4	no	no	DET
iajs-3594	142	5	funding	funding	NOUN
iajs-3594	142	6	for	for	ADP
iajs-3594	142	7	the	the	DET
iajs-3594	142	8	article	article	NOUN
iajs-3594	142	9	.	.	PUNCT
iajs-3594	143	1	references	reference	NOUN
iajs-3594	143	2	1	1	NUM
iajs-3594	143	3	.	.	PUNCT
iajs-3594	144	1	barnes	barne	NOUN
iajs-3594	144	2	we	we	PRON
iajs-3594	144	3	.	.	PUNCT
iajs-3594	145	1	on	on	ADP
iajs-3594	145	2	the	the	DET
iajs-3594	145	3	г	г	NOUN
iajs-3594	145	4	-	-	PUNCT
iajs-3594	145	5	rings	ring	NOUN
iajs-3594	145	6	of	of	ADP
iajs-3594	145	7	nobusawa	nobusawa	PROPN
iajs-3594	145	8	.	.	PUNCT
iajs-3594	146	1	pacific	pacific	PROPN
iajs-3594	146	2	j	j	PROPN
iajs-3594	146	3	math	math	PROPN
iajs-3594	146	4	.	.	PUNCT
iajs-3594	147	1	1966;18:411	1966;18:411	PROPN
iajs-3594	147	2	-	-	PUNCT
iajs-3594	147	3	422	422	NUM
iajs-3594	147	4	.	.	NOUN
iajs-3594	148	1	2	2	NUM
iajs-3594	148	2	.	.	X
iajs-3594	148	3	özden	özden	ADJ
iajs-3594	149	1	d	d	PROPN
iajs-3594	149	2	,	,	PUNCT
iajs-3594	149	3	öztürk	öztürk	PROPN
iajs-3594	149	4	ma	ma	PROPN
iajs-3594	149	5	,	,	PUNCT
iajs-3594	149	6	jun	jun	PROPN
iajs-3594	149	7	yb	yb	PROPN
iajs-3594	149	8	.	.	PUNCT
iajs-3594	150	1	permuting	permute	VERB
iajs-3594	150	2	tri	tri	NOUN
iajs-3594	150	3	-	-	NOUN
iajs-3594	150	4	derivations	derivation	NOUN
iajs-3594	150	5	in	in	ADP
iajs-3594	150	6	prime	prime	ADJ
iajs-3594	150	7	and	and	CCONJ
iajs-3594	150	8	semi	semi	ADJ
iajs-3594	150	9	-	-	ADJ
iajs-3594	150	10	prime	prime	ADJ
iajs-3594	150	11	gamma	gamma	NOUN
iajs-3594	150	12	rings	ring	NOUN
iajs-3594	150	13	.	.	PUNCT
iajs-3594	151	1	kyungpook	kyungpook	PROPN
iajs-3594	151	2	math	math	PROPN
iajs-3594	151	3	j.	j.	PROPN
iajs-3594	151	4	2006	2006	NUM
iajs-3594	151	5	;	;	PUNCT
iajs-3594	152	1	46(2):153	46(2):153	NUM
iajs-3594	152	2	-	-	PUNCT
iajs-3594	152	3	167	167	NUM
iajs-3594	152	4	.	.	PUNCT
iajs-3594	153	1	3	3	X
iajs-3594	153	2	.	.	X
iajs-3594	153	3	kandamar	kandamar	PROPN
iajs-3594	153	4	h	h	PROPN
iajs-3594	153	5	,	,	PUNCT
iajs-3594	153	6	arslan	arslan	PROPN
iajs-3594	153	7	o.	o.	PROPN
iajs-3594	153	8	on	on	ADP
iajs-3594	153	9	the	the	DET
iajs-3594	153	10	commutativity	commutativity	NOUN
iajs-3594	153	11	conditions	condition	NOUN
iajs-3594	153	12	for	for	ADP
iajs-3594	153	13	rings	ring	NOUN
iajs-3594	153	14	and	and	CCONJ
iajs-3594	153	15	γ	γ	NOUN
iajs-3594	153	16	-	-	PUNCT
iajs-3594	153	17	rings	ring	NOUN
iajs-3594	153	18	.	.	PUNCT
iajs-3594	154	1	hacettepe	hacettepe	PROPN
iajs-3594	154	2	j	j	PROPN
iajs-3594	154	3	math	math	PROPN
iajs-3594	154	4	stat	stat	PROPN
iajs-3594	154	5	.	.	PUNCT
iajs-3594	155	1	2020;49(5):1660	2020;49(5):1660	NUM
iajs-3594	155	2	-	-	SYM
iajs-3594	155	3	1666	1666	NUM
iajs-3594	155	4	.	.	PUNCT
iajs-3594	156	1	4	4	X
iajs-3594	156	2	.	.	X
iajs-3594	156	3	kamali	kamali	PROPN
iajs-3594	156	4	ardakani	ardakani	PROPN
iajs-3594	156	5	l	l	PROPN
iajs-3594	156	6	,	,	PUNCT
iajs-3594	156	7	davvaz	davvaz	PROPN
iajs-3594	156	8	b	b	PROPN
iajs-3594	156	9	,	,	PUNCT
iajs-3594	156	10	huang	huang	PROPN
iajs-3594	156	11	s.	s.	PROPN
iajs-3594	156	12	on	on	ADP
iajs-3594	156	13	derivations	derivation	NOUN
iajs-3594	156	14	of	of	ADP
iajs-3594	156	15	prime	prime	ADJ
iajs-3594	156	16	and	and	CCONJ
iajs-3594	156	17	semiprime	semiprime	NOUN
iajs-3594	156	18	gamma	gamma	PROPN
iajs-3594	156	19	rings	rings	PROPN
iajs-3594	156	20	.	.	PUNCT
iajs-3594	157	1	bol	bol	PROPN
iajs-3594	157	2	soc	soc	PROPN
iajs-3594	157	3	paran	paran	PROPN
iajs-3594	157	4	mat	mat	PROPN
iajs-3594	157	5	.	.	PUNCT
iajs-3594	158	1	2019;37(2):157	2019;37(2):157	NUM
iajs-3594	158	2	-	-	SYM
iajs-3594	158	3	166	166	NUM
iajs-3594	158	4	.	.	PUNCT
iajs-3594	158	5	5	5	NUM
iajs-3594	158	6	.	.	PUNCT
iajs-3594	158	7	kyuno	kyuno	PROPN
iajs-3594	158	8	s.	s.	PROPN
iajs-3594	158	9	prime	prime	PROPN
iajs-3594	158	10	ideals	ideal	NOUN
iajs-3594	158	11	in	in	ADP
iajs-3594	158	12	gamma	gamma	NOUN
iajs-3594	158	13	rings	ring	NOUN
iajs-3594	158	14	.	.	PUNCT
iajs-3594	159	1	pacific	pacific	PROPN
iajs-3594	159	2	j	j	PROPN
iajs-3594	159	3	math	math	NOUN
iajs-3594	159	4	.	.	PUNCT
iajs-3594	160	1	1982;98(2):375	1982;98(2):375	NUM
iajs-3594	160	2	-	-	SYM
iajs-3594	160	3	379	379	NUM
iajs-3594	160	4	.	.	PUNCT
iajs-3594	161	1	6	6	NUM
iajs-3594	161	2	.	.	X
iajs-3594	161	3	chakraborty	chakraborty	PROPN
iajs-3594	161	4	s	s	PROPN
iajs-3594	161	5	,	,	PUNCT
iajs-3594	161	6	paul	paul	PROPN
iajs-3594	161	7	ac	ac	PROPN
iajs-3594	161	8	.	.	PROPN
iajs-3594	162	1	on	on	ADP
iajs-3594	162	2	jordan	jordan	PROPN
iajs-3594	162	3	k	k	PROPN
iajs-3594	162	4	-	-	PUNCT
iajs-3594	162	5	derivations	derivation	NOUN
iajs-3594	162	6	of	of	ADP
iajs-3594	162	7	2	2	NUM
iajs-3594	162	8	-	-	PUNCT
iajs-3594	162	9	torsion	torsion	NOUN
iajs-3594	162	10	free	free	ADJ
iajs-3594	162	11	prime	prime	ADJ
iajs-3594	162	12	гn	гn	NOUN
iajs-3594	162	13	-	-	PUNCT
iajs-3594	162	14	rings	ring	NOUN
iajs-3594	162	15	.	.	PUNCT
iajs-3594	163	1	punjab	punjab	PROPN
iajs-3594	163	2	univ	univ	PROPN
iajs-3594	163	3	j	j	PROPN
iajs-3594	163	4	math	math	PROPN
iajs-3594	163	5	.	.	PUNCT
iajs-3594	164	1	2008;40:97	2008;40:97	NUM
iajs-3594	164	2	-	-	SYM
iajs-3594	164	3	101	101	NUM
iajs-3594	164	4	.	.	PUNCT
iajs-3594	165	1	7	7	X
iajs-3594	165	2	.	.	X
iajs-3594	165	3	marapureddy	marapureddy	PROPN
iajs-3594	165	4	,	,	PUNCT
iajs-3594	165	5	m.	m.	PROPN
iajs-3594	165	6	k.	k.	PROPN
iajs-3594	165	7	r.	r.	PROPN
iajs-3594	165	8	on	on	ADP
iajs-3594	165	9	γ	γ	PROPN
iajs-3594	165	10	-	-	PUNCT
iajs-3594	165	11	semiring	semiring	NOUN
iajs-3594	165	12	with	with	ADP
iajs-3594	165	13	identity	identity	NOUN
iajs-3594	165	14	.	.	PUNCT
iajs-3594	166	1	discussiones	discussione	NOUN
iajs-3594	166	2	mathematicae	mathematicae	ADJ
iajs-3594	166	3	-	-	ADJ
iajs-3594	166	4	general	general	ADJ
iajs-3594	166	5	algebra	algebra	NOUN
iajs-3594	166	6	and	and	CCONJ
iajs-3594	166	7	applications	application	NOUN
iajs-3594	166	8	,	,	PUNCT
iajs-3594	166	9	2017	2017	NUM
iajs-3594	166	10	;	;	PUNCT
iajs-3594	166	11	37(2	37(2	NUM
iajs-3594	166	12	)	)	PUNCT
iajs-3594	166	13	,	,	PUNCT
iajs-3594	166	14	189	189	NUM
iajs-3594	166	15	-	-	SYM
iajs-3594	166	16	207	207	NUM
iajs-3594	166	17	.	.	PUNCT
iajs-3594	167	1	http://dx.doi.org/10.7151/dmgaa.1276	http://dx.doi.org/10.7151/dmgaa.1276	X
iajs-3594	167	2	.	.	NOUN
iajs-3594	167	3	8	8	X
iajs-3594	167	4	.	.	X
iajs-3594	167	5	özkum	özkum	PROPN
iajs-3594	167	6	g	g	PROPN
iajs-3594	167	7	,	,	PUNCT
iajs-3594	167	8	soytürk	soytürk	PROPN
iajs-3594	167	9	m.	m.	PROPN
iajs-3594	167	10	gamma	gamma	PROPN
iajs-3594	167	11	rings	ring	NOUN
iajs-3594	167	12	with	with	ADP
iajs-3594	167	13	derivation	derivation	NOUN
iajs-3594	167	14	.	.	PUNCT
iajs-3594	168	1	eur	eur	PROPN
iajs-3594	168	2	int	int	PROPN
iajs-3594	168	3	j	j	PROPN
iajs-3594	168	4	sci	sci	PROPN
iajs-3594	168	5	technol	technol	NOUN
iajs-3594	168	6	.	.	PUNCT
iajs-3594	169	1	2021;10(7):125	2021;10(7):125	PROPN
iajs-3594	169	2	-	-	PUNCT
iajs-3594	169	3	138	138	NUM
iajs-3594	169	4	.	.	NOUN
iajs-3594	170	1	9	9	X
iajs-3594	170	2	.	.	X
iajs-3594	171	1	saleh	saleh	PROPN
iajs-3594	171	2	sm	sm	PROPN
iajs-3594	171	3	.	.	PUNCT
iajs-3594	172	1	on	on	ADP
iajs-3594	172	2	prime	prime	ADJ
iajs-3594	172	3	г	г	PROPN
iajs-3594	172	4	-	-	PUNCT
iajs-3594	172	5	rings	ring	NOUN
iajs-3594	172	6	with	with	ADP
iajs-3594	172	7	derivation	derivation	NOUN
iajs-3594	172	8	.	.	PUNCT
iajs-3594	173	1	phd	phd	NOUN
iajs-3594	173	2	thesis	thesis	PROPN
iajs-3594	173	3	.	.	PUNCT
iajs-3594	174	1	al	al	PROPN
iajs-3594	174	2	-	-	PUNCT
iajs-3594	174	3	mustansiry	mustansiry	PROPN
iajs-3594	174	4	university	university	NOUN
iajs-3594	174	5	;	;	PUNCT
iajs-3594	174	6	2010	2010	NUM
iajs-3594	174	7	.	.	PUNCT
iajs-3594	175	1	10	10	NUM
iajs-3594	175	2	.	.	X
iajs-3594	176	1	shaheen	shaheen	PROPN
iajs-3594	176	2	rc	rc	PROPN
iajs-3594	176	3	.	.	PUNCT
iajs-3594	177	1	on	on	ADP
iajs-3594	177	2	higher	high	ADJ
iajs-3594	177	3	homomorphism	homomorphism	NOUN
iajs-3594	177	4	of	of	ADP
iajs-3594	177	5	completely	completely	ADV
iajs-3594	177	6	prime	prime	ADJ
iajs-3594	177	7	gamma	gamma	NOUN
iajs-3594	177	8	rings	ring	NOUN
iajs-3594	177	9	.	.	PUNCT
iajs-3594	178	1	j	j	PROPN
iajs-3594	178	2	al	al	PROPN
iajs-3594	178	3	-	-	PUNCT
iajs-3594	178	4	qadisiyah	qadisiyah	NOUN
iajs-3594	178	5	pure	pure	ADJ
iajs-3594	178	6	sci	sci	PROPN
iajs-3594	178	7	.	.	PROPN
iajs-3594	179	1	2008;13(2):1	2008;13(2):1	NUM
iajs-3594	179	2	-	-	SYM
iajs-3594	179	3	9	9	NUM
iajs-3594	179	4	.	.	NOUN
iajs-3594	179	5	11	11	NUM
iajs-3594	179	6	.	.	PUNCT
iajs-3594	180	1	motashar	motashar	PROPN
iajs-3594	180	2	sk	sk	PROPN
iajs-3594	180	3	,	,	PUNCT
iajs-3594	180	4	majeed	majeed	PROPN
iajs-3594	180	5	ah	ah	INTJ
iajs-3594	180	6	.	.	PUNCT
iajs-3594	181	1	г	г	ADV
iajs-3594	181	2	-	-	PUNCT
iajs-3594	181	3	centralizing	centralize	VERB
iajs-3594	181	4	mappings	mapping	NOUN
iajs-3594	181	5	of	of	ADP
iajs-3594	181	6	semiprime	semiprime	NOUN
iajs-3594	181	7	г	г	PROPN
iajs-3594	181	8	-	-	PUNCT
iajs-3594	181	9	rings	ring	NOUN
iajs-3594	181	10	.	.	PUNCT
iajs-3594	182	1	iraqi	iraqi	PROPN
iajs-3594	182	2	j	j	PROPN
iajs-3594	182	3	sci	sci	PROPN
iajs-3594	182	4	.	.	PUNCT
iajs-3594	183	1	2012;53(3):657	2012;53(3):657	NUM
iajs-3594	183	2	-	-	SYM
iajs-3594	183	3	662	662	NUM
iajs-3594	183	4	.	.	PUNCT
iajs-3594	184	1	12	12	NUM
iajs-3594	184	2	.	.	PUNCT
iajs-3594	185	1	mutlak	mutlak	ADV
iajs-3594	185	2	at	at	ADP
iajs-3594	185	3	,	,	PUNCT
iajs-3594	185	4	majeed	majeed	PROPN
iajs-3594	185	5	ah	ah	INTJ
iajs-3594	185	6	.	.	PUNCT
iajs-3594	186	1	on	on	ADP
iajs-3594	186	2	centralizers	centralizer	NOUN
iajs-3594	186	3	of	of	ADP
iajs-3594	186	4	2	2	NUM
iajs-3594	186	5	-	-	PUNCT
iajs-3594	186	6	torsion	torsion	NOUN
iajs-3594	186	7	free	free	ADJ
iajs-3594	186	8	semiprime	semiprime	NOUN
iajs-3594	186	9	gamma	gamma	PROPN
iajs-3594	186	10	rings	ring	NOUN
iajs-3594	186	11	.	.	PUNCT
iajs-3594	187	1	iraqi	iraqi	PROPN
iajs-3594	187	2	j	j	PROPN
iajs-3594	187	3	sci	sci	PROPN
iajs-3594	187	4	.	.	PUNCT
iajs-3594	188	1	2021;(7):2351	2021;(7):2351	NUM
iajs-3594	188	2	-	-	PUNCT
iajs-3594	188	3	2356	2356	NUM
iajs-3594	188	4	.	.	PUNCT
iajs-3594	189	1	13	13	NUM
iajs-3594	189	2	.	.	PUNCT
iajs-3594	190	1	majeed	majeed	PROPN
iajs-3594	190	2	ah	ah	INTJ
iajs-3594	190	3	,	,	PUNCT
iajs-3594	190	4	hamil	hamil	PROPN
iajs-3594	190	5	sa	sa	PROPN
iajs-3594	190	6	.	.	PUNCT
iajs-3594	190	7	-orthogonal	-orthogonal	PUNCT
iajs-3594	191	1	for	for	ADP
iajs-3594	191	2	k	k	NOUN
iajs-3594	191	3	-	-	PUNCT
iajs-3594	191	4	derivations	derivation	NOUN
iajs-3594	191	5	and	and	CCONJ
iajs-3594	191	6	k	k	ADJ
iajs-3594	191	7	-	-	PUNCT
iajs-3594	191	8	reverse	reverse	ADJ
iajs-3594	191	9	derivations	derivation	NOUN
iajs-3594	191	10	.	.	PUNCT
iajs-3594	192	1	j	j	PROPN
iajs-3594	192	2	phys	phys	PROPN
iajs-3594	192	3	.	.	PUNCT
iajs-3594	193	1	2020;1530:1	2020;1530:1	NUM
iajs-3594	193	2	-	-	SYM
iajs-3594	193	3	6	6	NUM
iajs-3594	193	4	.	.	NOUN
iajs-3594	193	5	14	14	NUM
iajs-3594	193	6	.	.	PUNCT
iajs-3594	194	1	majeed	majeed	PROPN
iajs-3594	194	2	ah	ah	INTJ
iajs-3594	194	3	,	,	PUNCT
iajs-3594	194	4	hamil	hamil	PROPN
iajs-3594	194	5	sa	sa	PROPN
iajs-3594	194	6	.	.	PUNCT
iajs-3594	195	1	derivations	derivation	NOUN
iajs-3594	195	2	in	in	ADP
iajs-3594	195	3	gamma	gamma	PROPN
iajs-3594	195	4	rings	ring	NOUN
iajs-3594	195	5	with	with	ADP
iajs-3594	195	6	γ	γ	NOUN
iajs-3594	195	7	-	-	ADJ
iajs-3594	195	8	lie	lie	NOUN
iajs-3594	195	9	and	and	CCONJ
iajs-3594	195	10	γ	γ	PROPN
iajs-3594	195	11	-	-	PUNCT
iajs-3594	195	12	jordan	jordan	PROPN
iajs-3594	195	13	structures	structure	NOUN
iajs-3594	195	14	.	.	PUNCT
iajs-3594	196	1	j	j	PROPN
iajs-3594	196	2	phys	phys	PROPN
iajs-3594	196	3	conf	conf	NOUN
iajs-3594	196	4	ser	ser	NOUN
iajs-3594	196	5	.	.	PUNCT
iajs-3594	196	6	2020;1530:012049	2020;1530:012049	NUM
iajs-3594	196	7	.	.	PUNCT
iajs-3594	197	1	http://doi.org/10.1088/1742-6596/1530/1/012049	http://doi.org/10.1088/1742-6596/1530/1/012049	DET
iajs-3594	197	2	15	15	NUM
iajs-3594	197	3	.	.	PUNCT
iajs-3594	198	1	majeed	majeed	PROPN
iajs-3594	198	2	ah	ah	INTJ
iajs-3594	198	3	,	,	PUNCT
iajs-3594	198	4	hamil	hamil	PROPN
iajs-3594	198	5	sa	sa	PROPN
iajs-3594	198	6	.	.	PUNCT
iajs-3594	199	1	derivations	derivation	NOUN
iajs-3594	199	2	and	and	CCONJ
iajs-3594	199	3	reverse	reverse	ADJ
iajs-3594	199	4	derivations	derivation	NOUN
iajs-3594	199	5	on	on	ADP
iajs-3594	199	6	γ	γ	NOUN
iajs-3594	199	7	-	-	ADJ
iajs-3594	199	8	prime	prime	ADJ
iajs-3594	199	9	and	and	CCONJ
iajs-3594	199	10	γ	γ	NOUN
iajs-3594	199	11	-	-	PUNCT
iajs-3594	199	12	semiprime	semiprime	ADJ
iajs-3594	199	13	gamma	gamma	NOUN
iajs-3594	199	14	semirings	semirings	PROPN
iajs-3594	199	15	.	.	PUNCT
iajs-3594	200	1	j	j	PROPN
iajs-3594	200	2	phys	phys	PROPN
iajs-3594	200	3	conf	conf	NOUN
iajs-3594	200	4	ser	ser	PROPN
iajs-3594	200	5	.	.	PUNCT
iajs-3594	201	1	2020;1530:012050	2020;1530:012050	NUM
iajs-3594	201	2	.	.	PUNCT
iajs-3594	202	1	http://doi.org/10.1088/1742-6596/1530/1/012050	http://doi.org/10.1088/1742-6596/1530/1/012050	PROPN
iajs-3594	202	2	16	16	NUM
iajs-3594	202	3	.	.	PUNCT
iajs-3594	203	1	majeed	majeed	PROPN
iajs-3594	203	2	ah	ah	INTJ
iajs-3594	203	3	,	,	PUNCT
iajs-3594	203	4	hamil	hamil	PROPN
iajs-3594	203	5	sa	sa	PROPN
iajs-3594	203	6	.	.	PUNCT
iajs-3594	204	1	on	on	ADP
iajs-3594	204	2	commutativity	commutativity	NOUN
iajs-3594	204	3	of	of	ADP
iajs-3594	204	4	prime	prime	ADJ
iajs-3594	204	5	and	and	CCONJ
iajs-3594	204	6	semiprime	semiprime	NOUN
iajs-3594	204	7	gamma	gamma	PROPN
iajs-3594	204	8	rings	ring	NOUN
iajs-3594	204	9	with	with	ADP
iajs-3594	204	10	reverse	reverse	ADJ
iajs-3594	204	11	derivations	derivation	NOUN
iajs-3594	204	12	.	.	PUNCT
iajs-3594	205	1	iraqi	iraqi	PROPN
iajs-3594	205	2	j	j	PROPN
iajs-3594	205	3	sci	sci	PROPN
iajs-3594	205	4	.	.	PUNCT
iajs-3594	206	1	2019;60(7):1546	2019;60(7):1546	NUM
iajs-3594	206	2	-	-	SYM
iajs-3594	206	3	1550	1550	NUM
iajs-3594	206	4	.	.	PUNCT
iajs-3594	207	1	17	17	NUM
iajs-3594	207	2	.	.	PUNCT
iajs-3594	208	1	chakraborty	chakraborty	PROPN
iajs-3594	208	2	s	s	PROPN
iajs-3594	208	3	,	,	PUNCT
iajs-3594	208	4	rashid	rashid	PROPN
iajs-3594	208	5	mm	mm	PROPN
iajs-3594	208	6	,	,	PUNCT
iajs-3594	208	7	paul	paul	PROPN
iajs-3594	208	8	ac	ac	PROPN
iajs-3594	208	9	.	.	PUNCT
iajs-3594	209	1	inner	inner	ADJ
iajs-3594	209	2	derivations	derivation	NOUN
iajs-3594	209	3	on	on	ADP
iajs-3594	209	4	semiprime	semiprime	NOUN
iajs-3594	209	5	gamma	gamma	PROPN
iajs-3594	209	6	rings	rings	PROPN
iajs-3594	209	7	.	.	PUNCT
iajs-3594	210	1	ganit	ganit	PROPN
iajs-3594	210	2	j	j	PROPN
iajs-3594	210	3	bangladesh	bangladesh	PROPN
iajs-3594	210	4	math	math	PROPN
iajs-3594	210	5	soc	soc	PROPN
iajs-3594	210	6	.	.	PUNCT
iajs-3594	211	1	2019;39:101	2019;39:101	NUM
iajs-3594	211	2	-	-	SYM
iajs-3594	211	3	110	110	NUM
iajs-3594	211	4	.	.	PUNCT
iajs-3594	212	1	18	18	NUM
iajs-3594	212	2	.	.	PUNCT
iajs-3594	212	3	kadhim	kadhim	PROPN
iajs-3594	212	4	ak	ak	PROPN
iajs-3594	212	5	,	,	PUNCT
iajs-3594	212	6	sulaiman	sulaiman	PROPN
iajs-3594	212	7	h	h	PROPN
iajs-3594	212	8	,	,	PUNCT
iajs-3594	212	9	majeed	majeed	PROPN
iajs-3594	212	10	arh	arh	PROPN
iajs-3594	212	11	.	.	PUNCT
iajs-3594	213	1	г	г	NOUN
iajs-3594	213	2	-	-	PUNCT
iajs-3594	213	3	centralizer	centralizer	NOUN
iajs-3594	213	4	and	and	CCONJ
iajs-3594	213	5	reverse	reverse	ADJ
iajs-3594	213	6	г*-centralizers	г*-centralizer	NOUN
iajs-3594	213	7	on	on	ADP
iajs-3594	213	8	semiprime	semiprime	NOUN
iajs-3594	213	9	гring	гre	VERB
iajs-3594	213	10	with	with	ADP
iajs-3594	213	11	involution	involution	NOUN
iajs-3594	213	12	.	.	PUNCT
iajs-3594	214	1	int	int	PROPN
iajs-3594	214	2	math	math	PROPN
iajs-3594	214	3	forum	forum	PROPN
iajs-3594	214	4	.	.	PUNCT
iajs-3594	215	1	2015;10(8):385	2015;10(8):385	NUM
iajs-3594	215	2	-	-	SYM
iajs-3594	215	3	393	393	NUM
iajs-3594	215	4	.	.	PUNCT
iajs-3594	216	1	19	19	NUM
iajs-3594	216	2	.	.	NOUN
iajs-3594	216	3	hoque	hoque	NOUN
iajs-3594	216	4	mf	mf	PROPN
iajs-3594	216	5	,	,	PUNCT
iajs-3594	216	6	paul	paul	PROPN
iajs-3594	216	7	ac	ac	PROPN
iajs-3594	216	8	.	.	PUNCT
iajs-3594	217	1	on	on	ADP
iajs-3594	217	2	centralizers	centralizer	NOUN
iajs-3594	217	3	of	of	ADP
iajs-3594	217	4	semiprime	semiprime	NOUN
iajs-3594	217	5	gamma	gamma	PROPN
iajs-3594	217	6	rings	rings	PROPN
iajs-3594	217	7	.	.	PUNCT
iajs-3594	218	1	int	int	PROPN
iajs-3594	218	2	math	math	PROPN
iajs-3594	218	3	forum	forum	PROPN
iajs-3594	218	4	.	.	PUNCT
iajs-3594	219	1	2011;6(13):627638	2011;6(13):627638	NUM
iajs-3594	219	2	.	.	PUNCT
iajs-3594	220	1	http://dx.doi.org/10.7151/dmgaa.1276	http://dx.doi.org/10.7151/dmgaa.1276	X
iajs-3594	221	1	http://doi.org/10.1088/1742-6596/1530/1/012049	http://doi.org/10.1088/1742-6596/1530/1/012049	PRON
iajs-3594	221	2	http://doi.org/10.1088/1742-6596/1530/1/012050	http://doi.org/10.1088/1742-6596/1530/1/012050	PROPN
iajs-3594	221	3	ihjpas	ihjpas	PROPN
iajs-3594	221	4	.	.	PUNCT
iajs-3594	222	1	2025	2025	NUM
iajs-3594	222	2	,	,	PUNCT
iajs-3594	222	3	38(2	38(2	NUM
iajs-3594	222	4	)	)	PUNCT
iajs-3594	222	5	337	337	NUM
iajs-3594	222	6	20	20	NUM
iajs-3594	222	7	.	.	PUNCT
iajs-3594	223	1	ibraheem	ibraheem	PROPN
iajs-3594	223	2	rk	rk	PROPN
iajs-3594	223	3	,	,	PUNCT
iajs-3594	223	4	majeed	majeed	PROPN
iajs-3594	223	5	ah	ah	INTJ
iajs-3594	223	6	.	.	PUNCT
iajs-3594	224	1	on	on	ADP
iajs-3594	224	2	lie	lie	NOUN
iajs-3594	224	3	structure	structure	NOUN
iajs-3594	224	4	in	in	ADP
iajs-3594	224	5	semi	semi	ADJ
iajs-3594	224	6	-	-	ADJ
iajs-3594	224	7	prime	prime	ADJ
iajs-3594	224	8	inverse	inverse	NOUN
iajs-3594	224	9	semi	semi	NOUN
iajs-3594	224	10	-	-	NOUN
iajs-3594	224	11	rings	ring	NOUN
iajs-3594	224	12	.	.	PUNCT
iajs-3594	225	1	iraqi	iraqi	PROPN
iajs-3594	225	2	j	j	PROPN
iajs-3594	225	3	sci	sci	PROPN
iajs-3594	225	4	.	.	PUNCT
iajs-3594	226	1	2019;60(12):2711	2019;60(12):2711	PROPN
iajs-3594	226	2	-	-	SYM
iajs-3594	226	3	2718	2718	NUM
iajs-3594	226	4	.	.	PUNCT
iajs-3594	227	1	http://doi.org/10.24996/ijs.2019.60.12.21	http://doi.org/10.24996/ijs.2019.60.12.21	PROPN
iajs-3594	227	2	21	21	NUM
iajs-3594	227	3	.	.	PUNCT
iajs-3594	228	1	dimitrov	dimitrov	PROPN
iajs-3594	228	2	s.	s.	PROPN
iajs-3594	228	3	derivations	derivation	NOUN
iajs-3594	228	4	on	on	ADP
iajs-3594	228	5	semirings	semirings	PROPN
iajs-3594	228	6	.	.	PUNCT
iajs-3594	229	1	aip	aip	PROPN
iajs-3594	229	2	conf	conf	PROPN
iajs-3594	229	3	.	.	PUNCT
iajs-3594	230	1	proc.2017;1910:060011	proc.2017;1910:060011	PROPN
iajs-3594	230	2	http://doi.org/10.1063/1.5014005	http://doi.org/10.1063/1.5014005	PROPN
iajs-3594	230	3	22	22	NUM
iajs-3594	230	4	.	.	PUNCT
iajs-3594	231	1	golan	golan	PROPN
iajs-3594	231	2	js	js	PROPN
iajs-3594	231	3	.	.	PUNCT
iajs-3594	232	1	semirings	semiring	NOUN
iajs-3594	232	2	and	and	CCONJ
iajs-3594	232	3	their	their	PRON
iajs-3594	232	4	applications	application	NOUN
iajs-3594	232	5	.	.	PUNCT
iajs-3594	233	1	university	university	NOUN
iajs-3594	233	2	of	of	ADP
iajs-3594	233	3	haifa	haifa	PROPN
iajs-3594	233	4	,	,	PUNCT
iajs-3594	233	5	haifa	haifa	PROPN
iajs-3594	233	6	,	,	PUNCT
iajs-3594	233	7	palestine	palestine	PROPN
iajs-3594	233	8	;	;	PUNCT
iajs-3594	233	9	1992	1992	NUM
iajs-3594	233	10	.	.	PUNCT
iajs-3594	234	1	23	23	NUM
iajs-3594	234	2	.	.	PUNCT
iajs-3594	235	1	golan	golan	PROPN
iajs-3594	235	2	js	js	PROPN
iajs-3594	235	3	.	.	PUNCT
iajs-3594	236	1	the	the	DET
iajs-3594	236	2	theory	theory	NOUN
iajs-3594	236	3	of	of	ADP
iajs-3594	236	4	semirings	semiring	NOUN
iajs-3594	236	5	with	with	ADP
iajs-3594	236	6	applications	application	NOUN
iajs-3594	236	7	mathematics	mathematic	NOUN
iajs-3594	236	8	and	and	CCONJ
iajs-3594	236	9	theoretical	theoretical	ADJ
iajs-3594	236	10	computer	computer	NOUN
iajs-3594	236	11	science	science	NOUN
iajs-3594	236	12	.	.	PUNCT
iajs-3594	237	1	john	john	PROPN
iajs-3594	237	2	wiley	wiley	PROPN
iajs-3594	237	3	and	and	CCONJ
iajs-3594	237	4	sons	son	NOUN
iajs-3594	237	5	,	,	PUNCT
iajs-3594	237	6	new	new	PROPN
iajs-3594	237	7	york	york	PROPN
iajs-3594	237	8	;	;	PUNCT
iajs-3594	237	9	1992	1992	NUM
iajs-3594	237	10	.	.	PUNCT
iajs-3594	238	1	24	24	NUM
iajs-3594	238	2	.	.	PUNCT
iajs-3594	239	1	golan	golan	PROPN
iajs-3594	239	2	js	js	PROPN
iajs-3594	239	3	.	.	PUNCT
iajs-3594	240	1	semirings	semiring	NOUN
iajs-3594	240	2	and	and	CCONJ
iajs-3594	240	3	their	their	PRON
iajs-3594	240	4	applications	application	NOUN
iajs-3594	240	5	.	.	PUNCT
iajs-3594	241	1	university	university	NOUN
iajs-3594	241	2	of	of	ADP
iajs-3594	241	3	haifa	haifa	PROPN
iajs-3594	241	4	,	,	PUNCT
iajs-3594	241	5	haifa	haifa	PROPN
iajs-3594	241	6	,	,	PUNCT
iajs-3594	241	7	palestine	palestine	PROPN
iajs-3594	241	8	;	;	PUNCT
iajs-3594	241	9	1992	1992	NUM
iajs-3594	241	10	.	.	PUNCT
iajs-3594	242	1	25	25	NUM
iajs-3594	242	2	.	.	PUNCT
iajs-3594	243	1	joseph	joseph	PROPN
iajs-3594	243	2	hm	hm	INTJ
iajs-3594	243	3	.	.	PUNCT
iajs-3594	244	1	centralizing	centralize	VERB
iajs-3594	244	2	mapping	mapping	NOUN
iajs-3594	244	3	of	of	ADP
iajs-3594	244	4	prime	prime	ADJ
iajs-3594	244	5	rings	ring	NOUN
iajs-3594	244	6	.	.	PUNCT
iajs-3594	245	1	can	can	AUX
iajs-3594	245	2	math	math	VERB
iajs-3594	245	3	bull	bull	NOUN
iajs-3594	245	4	.	.	PUNCT
iajs-3594	246	1	1984;27(1):122126	1984;27(1):122126	X
iajs-3594	246	2	.	.	PUNCT
iajs-3594	247	1	http://doi.org/10.4153/cmb-1984-018-2	http://doi.org/10.4153/cmb-1984-018-2	ADP
iajs-3594	247	2	26	26	NUM
iajs-3594	247	3	.	.	PUNCT
iajs-3594	248	1	mary	mary	PROPN
iajs-3594	248	2	d	d	PROPN
iajs-3594	248	3	,	,	PUNCT
iajs-3594	248	4	murugensan	murugensan	ADJ
iajs-3594	248	5	r	r	NOUN
iajs-3594	248	6	,	,	PUNCT
iajs-3594	248	7	namasivayam	namasivayam	ADJ
iajs-3594	248	8	p.	p.	NOUN
iajs-3594	248	9	centralizers	centralizer	NOUN
iajs-3594	248	10	on	on	ADP
iajs-3594	248	11	semiprime	semiprime	NOUN
iajs-3594	248	12	semirings	semirings	PROPN
iajs-3594	248	13	.	.	PUNCT
iajs-3594	249	1	iosr	iosr	PROPN
iajs-3594	249	2	j	j	PROPN
iajs-3594	249	3	math	math	NOUN
iajs-3594	249	4	.	.	PUNCT
iajs-3594	250	1	2016;12(3):86	2016;12(3):86	NUM
iajs-3594	250	2	-	-	SYM
iajs-3594	250	3	93	93	NUM
iajs-3594	250	4	.	.	PUNCT
iajs-3594	251	1	27	27	NUM
iajs-3594	251	2	.	.	PUNCT
iajs-3594	252	1	sara	sara	PROPN
iajs-3594	252	2	a	a	PROPN
iajs-3594	252	3	,	,	PUNCT
iajs-3594	252	4	aslam	aslam	PROPN
iajs-3594	252	5	m.	m.	PROPN
iajs-3594	252	6	on	on	ADP
iajs-3594	252	7	li	li	PROPN
iajs-3594	252	8	ideal	ideal	PROPN
iajs-3594	252	9	of	of	ADP
iajs-3594	252	10	inverse	inverse	NOUN
iajs-3594	252	11	semirings	semiring	NOUN
iajs-3594	252	12	.	.	PUNCT
iajs-3594	253	1	ital	ital	PROPN
iajs-3594	253	2	j	j	PROPN
iajs-3594	253	3	pure	pure	ADJ
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