id	sid	tid	token	lemma	pos
iajs-3482	1	1	397	397	NUM
iajs-3482	1	2	©	©	ADP
iajs-3482	1	3	2025	2025	NUM
iajs-3482	1	4	the	the	DET
iajs-3482	1	5	author(s	author(s	NOUN
iajs-3482	1	6	)	)	PUNCT
iajs-3482	1	7	.	.	PUNCT
iajs-3482	2	1	published	publish	VERB
iajs-3482	2	2	by	by	ADP
iajs-3482	2	3	college	college	NOUN
iajs-3482	2	4	of	of	ADP
iajs-3482	2	5	education	education	NOUN
iajs-3482	2	6	for	for	ADP
iajs-3482	2	7	pure	pure	ADJ
iajs-3482	2	8	science	science	NOUN
iajs-3482	2	9	(	(	PUNCT
iajs-3482	2	10	ibn	ibn	PROPN
iajs-3482	2	11	al	al	PROPN
iajs-3482	2	12	-	-	PUNCT
iajs-3482	2	13	haitham	haitham	PROPN
iajs-3482	2	14	)	)	PUNCT
iajs-3482	2	15	,	,	PUNCT
iajs-3482	2	16	university	university	NOUN
iajs-3482	2	17	of	of	ADP
iajs-3482	2	18	baghdad	baghdad	PROPN
iajs-3482	2	19	.	.	PUNCT
iajs-3482	3	1	this	this	PRON
iajs-3482	3	2	is	be	AUX
iajs-3482	3	3	an	an	DET
iajs-3482	3	4	open	open	ADJ
iajs-3482	3	5	-	-	PUNCT
iajs-3482	3	6	access	access	NOUN
iajs-3482	3	7	article	article	NOUN
iajs-3482	3	8	distributed	distribute	VERB
iajs-3482	3	9	under	under	ADP
iajs-3482	3	10	the	the	DET
iajs-3482	3	11	terms	term	NOUN
iajs-3482	3	12	of	of	ADP
iajs-3482	3	13	the	the	DET
iajs-3482	3	14	creative	creative	ADJ
iajs-3482	3	15	commons	common	NOUN
iajs-3482	3	16	attribution	attribution	NOUN
iajs-3482	3	17	4.0	4.0	NUM
iajs-3482	3	18	international	international	ADJ
iajs-3482	3	19	license	license	NOUN
iajs-3482	3	20	centralizer	centralizer	NOUN
iajs-3482	3	21	on	on	ADP
iajs-3482	3	22	lie	lie	NOUN
iajs-3482	3	23	-	-	PUNCT
iajs-3482	3	24	ideal	ideal	NOUN
iajs-3482	3	25	of	of	ADP
iajs-3482	3	26	semi	semi	ADJ
iajs-3482	3	27	-	-	ADJ
iajs-3482	3	28	prime	prime	ADJ
iajs-3482	3	29	inverse	inverse	NOUN
iajs-3482	3	30	semi	semi	ADJ
iajs-3482	3	31	-	-	ADJ
iajs-3482	3	32	ring	ring	ADJ
iajs-3482	3	33	ali	ali	PROPN
iajs-3482	3	34	ja	ja	PROPN
iajs-3482	3	35	.	.	PUNCT
iajs-3482	4	1	abass1	abass1	PROPN
iajs-3482	4	2	*	*	VERB
iajs-3482	4	3	,	,	PUNCT
iajs-3482	4	4	abdulahman	abdulahman	PROPN
iajs-3482	4	5	h.	h.	PROPN
iajs-3482	4	6	majeed	majeed	PROPN
iajs-3482	4	7	2	2	PROPN
iajs-3482	4	8	,	,	PUNCT
iajs-3482	4	9	mohammed	mohammed	PROPN
iajs-3482	4	10	yasin	yasin	PROPN
iajs-3482	4	11	3	3	NUM
iajs-3482	4	12	and	and	CCONJ
iajs-3482	4	13	shrooq	shrooq	NOUN
iajs-3482	4	14	bahjat	bahjat	PROPN
iajs-3482	4	15	smeein4	smeein4	PROPN
iajs-3482	4	16	1	1	NUM
iajs-3482	4	17	department	department	NOUN
iajs-3482	4	18	of	of	ADP
iajs-3482	4	19	mathematics	mathematics	PROPN
iajs-3482	4	20	,	,	PUNCT
iajs-3482	4	21	college	college	NOUN
iajs-3482	4	22	,	,	PUNCT
iajs-3482	4	23	of	of	ADP
iajs-3482	4	24	science	science	NOUN
iajs-3482	4	25	,	,	PUNCT
iajs-3482	4	26	university	university	NOUN
iajs-3482	4	27	of	of	ADP
iajs-3482	4	28	baghdad	baghdad	PROPN
iajs-3482	4	29	,	,	PUNCT
iajs-3482	4	30	baghdad	baghdad	PROPN
iajs-3482	4	31	,	,	PUNCT
iajs-3482	4	32	iraq	iraq	PROPN
iajs-3482	4	33	.	.	PUNCT
iajs-3482	5	1	2	2	NUM
iajs-3482	5	2	department	department	NOUN
iajs-3482	5	3	of	of	ADP
iajs-3482	5	4	mathematic	mathematic	PROPN
iajs-3482	5	5	,	,	PUNCT
iajs-3482	5	6	al	al	PROPN
iajs-3482	5	7	-	-	PUNCT
iajs-3482	5	8	mamoun	mamoun	PROPN
iajs-3482	5	9	university	university	PROPN
iajs-3482	5	10	college	college	NOUN
iajs-3482	5	11	,	,	PUNCT
iajs-3482	5	12	baghdad	baghdad	PROPN
iajs-3482	5	13	,	,	PUNCT
iajs-3482	5	14	iraq	iraq	PROPN
iajs-3482	5	15	.	.	PUNCT
iajs-3482	6	1	3	3	NUM
iajs-3482	6	2	department	department	NOUN
iajs-3482	6	3	of	of	ADP
iajs-3482	6	4	mathematics	mathematic	NOUN
iajs-3482	6	5	,	,	PUNCT
iajs-3482	6	6	an	an	DET
iajs-3482	6	7	-	-	PUNCT
iajs-3482	6	8	najah	najah	ADJ
iajs-3482	6	9	national	national	ADJ
iajs-3482	6	10	university	university	NOUN
iajs-3482	6	11	,	,	PUNCT
iajs-3482	6	12	nablus	nablus	X
iajs-3482	6	13	p400	p400	PROPN
iajs-3482	6	14	,	,	PUNCT
iajs-3482	6	15	palestine	palestine	PROPN
iajs-3482	6	16	.	.	PROPN
iajs-3482	7	1	4	4	NUM
iajs-3482	7	2	information	information	NOUN
iajs-3482	7	3	department	department	NOUN
iajs-3482	7	4	,	,	PUNCT
iajs-3482	7	5	section	section	NOUN
iajs-3482	7	6	mathematics	mathematic	NOUN
iajs-3482	7	7	,	,	PUNCT
iajs-3482	7	8	university	university	NOUN
iajs-3482	7	9	of	of	ADP
iajs-3482	7	10	technology	technology	NOUN
iajs-3482	7	11	and	and	CCONJ
iajs-3482	7	12	applied	apply	VERB
iajs-3482	7	13	science	science	NOUN
iajs-3482	7	14	muscat	muscat	PROPN
iajs-3482	7	15	,	,	PUNCT
iajs-3482	7	16	sultanate	sultanate	NOUN
iajs-3482	7	17	of	of	ADP
iajs-3482	7	18	oman	oman	NOUN
iajs-3482	7	19	.	.	PUNCT
iajs-3482	8	1	*	*	PUNCT
iajs-3482	8	2	corresponding	correspond	VERB
iajs-3482	8	3	author	author	NOUN
iajs-3482	8	4	.	.	PUNCT
iajs-3482	9	1	received	receive	VERB
iajs-3482	9	2	:	:	PUNCT
iajs-3482	9	3	10	10	NUM
iajs-3482	9	4	may	may	PROPN
iajs-3482	9	5	2023	2023	NUM
iajs-3482	9	6	accepted	accept	VERB
iajs-3482	9	7	:	:	PUNCT
iajs-3482	9	8	27	27	NUM
iajs-3482	9	9	august	august	PROPN
iajs-3482	9	10	2023	2023	NUM
iajs-3482	9	11	published	publish	VERB
iajs-3482	9	12	:	:	PUNCT
iajs-3482	9	13	20	20	NUM
iajs-3482	9	14	january	january	PROPN
iajs-3482	9	15	2025	2025	NUM
iajs-3482	9	16	doi.org/10.30526/38.1.3482	doi.org/10.30526/38.1.3482	NOUN
iajs-3482	9	17	abstract	abstract	ADJ
iajs-3482	9	18	the	the	DET
iajs-3482	9	19	summary	summary	NOUN
iajs-3482	9	20	purpose	purpose	NOUN
iajs-3482	9	21	of	of	ADP
iajs-3482	9	22	this	this	DET
iajs-3482	9	23	work	work	NOUN
iajs-3482	9	24	:	:	PUNCT
iajs-3482	9	25	we	we	PRON
iajs-3482	9	26	extending	extend	VERB
iajs-3482	9	27	certain	certain	ADJ
iajs-3482	9	28	results	result	NOUN
iajs-3482	9	29	on	on	ADP
iajs-3482	9	30	α	α	NOUN
iajs-3482	9	31	-	-	PUNCT
iajs-3482	9	32	centralizer	centralizer	NOUN
iajs-3482	9	33	of	of	ADP
iajs-3482	9	34	inverse	inverse	NOUN
iajs-3482	9	35	semiring	semire	VERB
iajs-3482	9	36	under	under	ADP
iajs-3482	9	37	specific	specific	ADJ
iajs-3482	9	38	conditions	condition	NOUN
iajs-3482	9	39	,	,	PUNCT
iajs-3482	9	40	achieve	achieve	VERB
iajs-3482	9	41	new	new	ADJ
iajs-3482	9	42	results	result	NOUN
iajs-3482	9	43	on	on	ADP
iajs-3482	9	44	lie	lie	NOUN
iajs-3482	9	45	ideal	ideal	NOUN
iajs-3482	9	46	of	of	ADP
iajs-3482	9	47	inverse	inverse	NOUN
iajs-3482	9	48	semiring	semire	VERB
iajs-3482	9	49	with	with	ADP
iajs-3482	9	50	some	some	DET
iajs-3482	9	51	consequent	consequent	ADJ
iajs-3482	9	52	collieries	colliery	NOUN
iajs-3482	9	53	,	,	PUNCT
iajs-3482	9	54	generalize	generalize	VERB
iajs-3482	9	55	assorted	assorted	ADJ
iajs-3482	9	56	α	α	NOUN
iajs-3482	9	57	-	-	PUNCT
iajs-3482	9	58	centralizer	centralizer	NOUN
iajs-3482	9	59	for	for	ADP
iajs-3482	9	60	lie	lie	NOUN
iajs-3482	9	61	ideal	ideal	NOUN
iajs-3482	9	62	of	of	ADP
iajs-3482	9	63	inverse	inverse	NOUN
iajs-3482	9	64	semiring	semire	VERB
iajs-3482	9	65	with	with	ADP
iajs-3482	9	66	some	some	DET
iajs-3482	9	67	collieries	colliery	NOUN
iajs-3482	9	68	,	,	PUNCT
iajs-3482	9	69	investigate	investigate	VERB
iajs-3482	9	70	significant	significant	ADJ
iajs-3482	9	71	theorems	theorem	NOUN
iajs-3482	9	72	on	on	ADP
iajs-3482	9	73	jordan	jordan	PROPN
iajs-3482	9	74	α	α	PROPN
iajs-3482	9	75	-	-	PUNCT
iajs-3482	9	76	centralizer	centralizer	NOUN
iajs-3482	9	77	of	of	ADP
iajs-3482	9	78	prime	prime	ADJ
iajs-3482	9	79	inverse	inverse	NOUN
iajs-3482	9	80	semiring	semiring	NOUN
iajs-3482	9	81	and	and	CCONJ
iajs-3482	9	82	we	we	PRON
iajs-3482	9	83	extend	extend	VERB
iajs-3482	9	84	certain	certain	ADJ
iajs-3482	9	85	results	result	NOUN
iajs-3482	9	86	of	of	ADP
iajs-3482	9	87	𝛼	𝛼	PROPN
iajs-3482	9	88	−centralizers	−centralizers	PUNCT
iajs-3482	9	89	and	and	CCONJ
iajs-3482	9	90	jordan	jordan	PROPN
iajs-3482	9	91	𝛼	𝛼	PROPN
iajs-3482	9	92	−centralizers	−centralizer	NOUN
iajs-3482	9	93	on	on	ADP
iajs-3482	9	94	lie	lie	NOUN
iajs-3482	9	95	-	-	PUNCT
iajs-3482	9	96	ideals	ideal	NOUN
iajs-3482	9	97	of	of	ADP
iajs-3482	9	98	prime	prime	ADJ
iajs-3482	9	99	semi	semi	NOUN
iajs-3482	9	100	-	-	NOUN
iajs-3482	9	101	rings	ring	NOUN
iajs-3482	9	102	to	to	ADP
iajs-3482	9	103	prime	prime	ADJ
iajs-3482	9	104	inverse	inverse	NOUN
iajs-3482	9	105	semi	semi	NOUN
iajs-3482	9	106	-	-	NOUN
iajs-3482	9	107	ring	ring	ADJ
iajs-3482	9	108	,	,	PUNCT
iajs-3482	9	109	we	we	PRON
iajs-3482	9	110	generalizing	generalize	VERB
iajs-3482	9	111	the	the	DET
iajs-3482	9	112	results	result	NOUN
iajs-3482	9	113	of	of	ADP
iajs-3482	9	114	mary	mary	PROPN
iajs-3482	9	115	in	in	ADP
iajs-3482	9	116	to	to	ADP
iajs-3482	9	117	α	α	NOUN
iajs-3482	9	118	-	-	PUNCT
iajs-3482	9	119	centralizer	centralizer	NOUN
iajs-3482	9	120	on	on	ADP
iajs-3482	9	121	semiring	semiring	NOUN
iajs-3482	9	122	,	,	PUNCT
iajs-3482	9	123	also	also	ADV
iajs-3482	9	124	we	we	PRON
iajs-3482	9	125	generalize	generalize	VERB
iajs-3482	9	126	our	our	PRON
iajs-3482	9	127	results	result	NOUN
iajs-3482	9	128	on	on	ADP
iajs-3482	9	129	lie	lie	NOUN
iajs-3482	9	130	ideals	ideal	NOUN
iajs-3482	9	131	of	of	ADP
iajs-3482	9	132	inverse	inverse	NOUN
iajs-3482	9	133	semiring	semiring	NOUN
iajs-3482	9	134	.	.	PUNCT
iajs-3482	10	1	we	we	PRON
iajs-3482	10	2	extending	extend	VERB
iajs-3482	10	3	the	the	DET
iajs-3482	10	4	results	result	NOUN
iajs-3482	10	5	of	of	ADP
iajs-3482	10	6	shafiq	shafiq	PROPN
iajs-3482	10	7	,	,	PUNCT
iajs-3482	10	8	aslam	aslam	PROPN
iajs-3482	10	9	,	,	PUNCT
iajs-3482	10	10	javed	jave	VERB
iajs-3482	10	11	to	to	ADP
iajs-3482	10	12	𝛼	𝛼	PRON
iajs-3482	10	13	−	−	PROPN
iajs-3482	10	14	centralizer	centralizer	NOUN
iajs-3482	10	15	of	of	ADP
iajs-3482	10	16	inverse	inverse	NOUN
iajs-3482	10	17	semiring	semiring	NOUN
iajs-3482	10	18	.	.	PUNCT
iajs-3482	11	1	𝑠𝑖𝑛𝑐𝑒	𝑠𝑖𝑛𝑐𝑒	PROPN
iajs-3482	11	2	𝑅	𝑅	PROPN
iajs-3482	11	3	is	be	AUX
iajs-3482	11	4	left	leave	VERB
iajs-3482	11	5	(	(	PUNCT
iajs-3482	11	6	right	right	ADJ
iajs-3482	11	7	)	)	PUNCT
iajs-3482	11	8	jordan	jordan	PROPN
iajs-3482	11	9	𝛼	𝛼	PROPN
iajs-3482	11	10	−	−	PROPN
iajs-3482	11	11	centralizer	centralizer	NOUN
iajs-3482	11	12	on”v	on”v	PROPN
iajs-3482	11	13	,	,	PUNCT
iajs-3482	11	14	we	we	PRON
iajs-3482	11	15	get	get	VERB
iajs-3482	11	16	the	the	DET
iajs-3482	11	17	output	output	NOUN
iajs-3482	11	18	r	r	NOUN
iajs-3482	11	19	is	be	AUX
iajs-3482	11	20	a	a	DET
iajs-3482	11	21	left	left	ADJ
iajs-3482	11	22	(	(	PUNCT
iajs-3482	11	23	right	right	ADJ
iajs-3482	11	24	)	)	PUNCT
iajs-3482	11	25	𝛼	𝛼	NOUN
iajs-3482	11	26	−	−	NOUN
iajs-3482	11	27	centralizer	centralizer	NOUN
iajs-3482	11	28	on	on	ADP
iajs-3482	11	29	𝑉.”if	𝑉.”if	PROPN
iajs-3482	11	30	it	it	PRON
iajs-3482	11	31	where	where	SCONJ
iajs-3482	11	32	𝛼	𝛼	PRON
iajs-3482	11	33	is	be	AUX
iajs-3482	11	34	an	an	DET
iajs-3482	11	35	automorphism	automorphism	NOUN
iajs-3482	11	36	of	of	ADP
iajs-3482	11	37	v,𝑅(𝑢	v,𝑅(𝑢	PROPN
iajs-3482	11	38	)	)	PUNCT
iajs-3482	11	39	∈	∈	PROPN
iajs-3482	11	40	𝑉	𝑉	PROPN
iajs-3482	11	41	,	,	PUNCT
iajs-3482	11	42	for	for	ADP
iajs-3482	11	43	any	any	DET
iajs-3482	11	44	𝑢	𝑢	PROPN
iajs-3482	11	45	∈	∈	PROPN
iajs-3482	11	46	𝑉	𝑉	PROPN
iajs-3482	11	47	,	,	PUNCT
iajs-3482	11	48	and	and	CCONJ
iajs-3482	11	49	𝛼(𝑍(𝑉	𝛼(𝑍(𝑉	NOUN
iajs-3482	11	50	)	)	PUNCT
iajs-3482	11	51	)	)	PUNCT
iajs-3482	12	1	=	=	SYM
iajs-3482	12	2	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	12	3	)	)	PUNCT
iajs-3482	12	4	.	.	PUNCT
iajs-3482	13	1	we	we	PRON
iajs-3482	13	2	also	also	ADV
iajs-3482	13	3	get	get	VERB
iajs-3482	13	4	the	the	DET
iajs-3482	13	5	following	follow	VERB
iajs-3482	13	6	output	output	NOUN
iajs-3482	13	7	r	r	NOUN
iajs-3482	13	8	is	be	AUX
iajs-3482	13	9	𝑎	𝑎	DET
iajs-3482	13	10	𝛼	𝛼	NOUN
iajs-3482	13	11	−	−	NOUN
iajs-3482	13	12	centralizer	centralizer	NOUN
iajs-3482	13	13	on	on	ADP
iajs-3482	13	14	𝑉.	𝑉.	NOUN
iajs-3482	13	15	keywords	keyword	NOUN
iajs-3482	13	16	:	:	PUNCT
iajs-3482	13	17	lie	lie	NOUN
iajs-3482	13	18	-	-	PUNCT
iajs-3482	13	19	ideal	ideal	ADJ
iajs-3482	13	20	,	,	PUNCT
iajs-3482	13	21	prime	prime	ADJ
iajs-3482	13	22	inverse	inverse	NOUN
iajs-3482	13	23	semi	semi	NOUN
iajs-3482	13	24	-	-	ADJ
iajs-3482	13	25	ring	ring	ADJ
iajs-3482	13	26	,	,	PUNCT
iajs-3482	13	27	semi	semi	ADJ
iajs-3482	13	28	-	-	ADJ
iajs-3482	13	29	prime	prime	ADJ
iajs-3482	13	30	inverse	inverse	NOUN
iajs-3482	13	31	semi	semi	NOUN
iajs-3482	13	32	-	-	NOUN
iajs-3482	13	33	ring	ring	ADJ
iajs-3482	13	34	,	,	PUNCT
iajs-3482	13	35	𝛼	𝛼	NOUN
iajs-3482	13	36	−centralizer	−centralizer	PROPN
iajs-3482	13	37	,	,	PUNCT
iajs-3482	13	38	jordan	jordan	PROPN
iajs-3482	13	39	α	α	PROPN
iajs-3482	13	40	-	-	PUNCT
iajs-3482	13	41	centralizer	centralizer	NOUN
iajs-3482	13	42	.	.	PUNCT
iajs-3482	14	1	1	1	X
iajs-3482	14	2	.	.	X
iajs-3482	14	3	introduction	introduction	NOUN
iajs-3482	14	4	let	let	VERB
iajs-3482	14	5	𝑀	𝑀	PROPN
iajs-3482	14	6	be	be	AUX
iajs-3482	14	7	a	a	DET
iajs-3482	14	8	non	non	ADJ
iajs-3482	14	9	-	-	ADJ
iajs-3482	14	10	empty	empty	ADJ
iajs-3482	14	11	set	set	NOUN
iajs-3482	14	12	with	with	ADP
iajs-3482	14	13	binary	binary	ADJ
iajs-3482	14	14	operation	operation	NOUN
iajs-3482	14	15	(	(	PUNCT
iajs-3482	14	16	•	•	NOUN
iajs-3482	14	17	)	)	PUNCT
iajs-3482	14	18	defined	define	VERB
iajs-3482	14	19	on	on	ADP
iajs-3482	14	20	𝑀	𝑀	PROPN
iajs-3482	14	21	,	,	PUNCT
iajs-3482	14	22	then	then	ADV
iajs-3482	14	23	(	(	PUNCT
iajs-3482	14	24	𝑀,•	𝑀,•	NOUN
iajs-3482	14	25	)	)	PUNCT
iajs-3482	14	26	is	be	AUX
iajs-3482	14	27	named	name	VERB
iajs-3482	14	28	semi	semi	ADV
iajs-3482	14	29	−	−	PROPN
iajs-3482	14	30	group	group	NOUN
iajs-3482	14	31	iff	iff	NOUN
iajs-3482	14	32	𝑘	𝑘	PRON
iajs-3482	14	33	•	•	NOUN
iajs-3482	14	34	(	(	PUNCT
iajs-3482	14	35	𝑠	𝑠	INTJ
iajs-3482	14	36	•	•	NUM
iajs-3482	14	37	𝑡	𝑡	PROPN
iajs-3482	14	38	)	)	PUNCT
iajs-3482	14	39	=	=	SYM
iajs-3482	15	1	(	(	PUNCT
iajs-3482	15	2	𝑘	𝑘	PRON
iajs-3482	15	3	•	•	NUM
iajs-3482	15	4	𝑠	𝑠	NOUN
iajs-3482	15	5	)	)	PUNCT
iajs-3482	15	6	•	•	NUM
iajs-3482	15	7	𝑡	𝑡	PROPN
iajs-3482	15	8	for	for	ADP
iajs-3482	15	9	any	any	DET
iajs-3482	15	10	𝑘	𝑘	PROPN
iajs-3482	15	11	,	,	PUNCT
iajs-3482	15	12	𝑠	𝑠	PROPN
iajs-3482	15	13	,	,	PUNCT
iajs-3482	15	14	𝑡	𝑡	PROPN
iajs-3482	15	15	∈	∈	PROPN
iajs-3482	15	16	𝑀(1	𝑀(1	NOUN
iajs-3482	15	17	)	)	PUNCT
iajs-3482	15	18	,	,	PUNCT
iajs-3482	15	19	a	a	DET
iajs-3482	15	20	semi	semi	ADJ
iajs-3482	15	21	−	−	PROPN
iajs-3482	15	22	group	group	NOUN
iajs-3482	15	23	𝑀	𝑀	PROPN
iajs-3482	15	24	is	be	AUX
iajs-3482	15	25	named	name	VERB
iajs-3482	15	26	commutative	commutative	ADJ
iajs-3482	15	27	semi	semi	ADJ
iajs-3482	15	28	−	−	PROPN
iajs-3482	15	29	group	group	NOUN
iajs-3482	15	30	if	if	SCONJ
iajs-3482	15	31	𝑘	𝑘	PROPN
iajs-3482	15	32	•	•	NOUN
iajs-3482	15	33	𝑠	𝑠	INTJ
iajs-3482	15	34	=	=	SYM
iajs-3482	15	35	𝑠	𝑠	PROPN
iajs-3482	15	36	•	•	NUM
iajs-3482	15	37	𝑘	𝑘	PROPN
iajs-3482	15	38	,	,	PUNCT
iajs-3482	15	39	holds	hold	VERB
iajs-3482	15	40	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	15	41	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	15	42	𝑘	𝑘	PROPN
iajs-3482	15	43	,	,	PUNCT
iajs-3482	15	44	𝑠	𝑠	PROPN
iajs-3482	15	45	∈	∈	PROPN
iajs-3482	15	46	𝑀	𝑀	PROPN
iajs-3482	15	47	(	(	PUNCT
iajs-3482	15	48	1	1	NUM
iajs-3482	15	49	)	)	PUNCT
iajs-3482	15	50	,	,	PUNCT
iajs-3482	15	51	a	a	DET
iajs-3482	15	52	non	non	NOUN
iajs-3482	15	53	−	−	NOUN
iajs-3482	15	54	empty	empty	ADJ
iajs-3482	15	55	set	set	NOUN
iajs-3482	15	56	with	with	ADP
iajs-3482	15	57	two	two	NUM
iajs-3482	15	58	−	−	PROPN
iajs-3482	15	59	binary	binary	ADJ
iajs-3482	15	60	operations(+	operations(+	NOUN
iajs-3482	15	61	)	)	PUNCT
iajs-3482	15	62	and	and	CCONJ
iajs-3482	15	63	(	(	PUNCT
iajs-3482	15	64	•	•	X
iajs-3482	15	65	)	)	PUNCT
iajs-3482	15	66	is	be	AUX
iajs-3482	15	67	named	name	VERB
iajs-3482	15	68	semi	semi	ADJ
iajs-3482	15	69	-	-	ADJ
iajs-3482	15	70	ring	ring	ADJ
iajs-3482	15	71	iff	iff	NOUN
iajs-3482	15	72	the	the	DET
iajs-3482	15	73	following	follow	VERB
iajs-3482	15	74	requirements	requirement	NOUN
iajs-3482	15	75	hold	hold	VERB
iajs-3482	15	76	:	:	PUNCT
iajs-3482	15	77	i	i	X
iajs-3482	15	78	)	)	PUNCT
iajs-3482	15	79	(	(	PUNCT
iajs-3482	15	80	𝑀	𝑀	PROPN
iajs-3482	15	81	,	,	PUNCT
iajs-3482	15	82	+	+	PUNCT
iajs-3482	15	83	)	)	PUNCT
iajs-3482	15	84	is	be	AUX
iajs-3482	15	85	commutative	commutative	ADJ
iajs-3482	15	86	semi	semi	ADJ
iajs-3482	15	87	−	−	PROPN
iajs-3482	15	88	group	group	NOUN
iajs-3482	15	89	.	.	PUNCT
iajs-3482	16	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3482	16	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3482	16	3	https://doi.org/10.30526/38.1.3501	https://doi.org/10.30526/38.1.3501	PROPN
iajs-3482	16	4	https://orcid.org/0009-0005-8166-2106	https://orcid.org/0009-0005-8166-2106	NOUN
iajs-3482	16	5	mailto:ali.jaafar1603b@sc.uobaghdad.edu.iq	mailto:ali.jaafar1603b@sc.uobaghdad.edu.iq	PROPN
iajs-3482	16	6	https://orcid.org/0000-0001-8534-0749	https://orcid.org/0000-0001-8534-0749	PROPN
iajs-3482	16	7	mailto:dulrahman.h.majeed@almamonuc.edu.iq	mailto:dulrahman.h.majeed@almamonuc.edu.iq	NOUN
iajs-3482	16	8	https://orcid.org/0009-0009-9394-5698	https://orcid.org/0009-0009-9394-5698	NOUN
iajs-3482	16	9	mailto:m.yasin@najah.edu	mailto:m.yasin@najah.edu	PROPN
iajs-3482	16	10	https://orcid.org/0009-0002-9351-4176	https://orcid.org/0009-0002-9351-4176	PROPN
iajs-3482	16	11	mailto:shrooq.smeein@hct.edu.om	mailto:shrooq.smeein@hct.edu.om	PROPN
iajs-3482	16	12	ihjpas	ihjpas	PROPN
iajs-3482	16	13	.	.	PUNCT
iajs-3482	17	1	2025	2025	NUM
iajs-3482	17	2	,	,	PUNCT
iajs-3482	17	3	38	38	NUM
iajs-3482	17	4	(	(	PUNCT
iajs-3482	17	5	1	1	NUM
iajs-3482	17	6	)	)	PUNCT
iajs-3482	17	7	398	398	NUM
iajs-3482	17	8	ii	ii	NOUN
iajs-3482	17	9	)	)	PUNCT
iajs-3482	17	10	(	(	PUNCT
iajs-3482	17	11	𝑀,•	𝑀,•	NOUN
iajs-3482	17	12	)	)	PUNCT
iajs-3482	17	13	semi	semi	ADV
iajs-3482	17	14	−	−	PROPN
iajs-3482	17	15	group	group	NOUN
iajs-3482	17	16	.	.	PUNCT
iajs-3482	18	1	iii	iii	X
iajs-3482	18	2	)	)	PUNCT
iajs-3482	18	3	𝑎	𝑎	NOUN
iajs-3482	18	4	•	•	NOUN
iajs-3482	18	5	(	(	PUNCT
iajs-3482	18	6	𝑘	𝑘	X
iajs-3482	18	7	+	+	NOUN
iajs-3482	18	8	𝑠	𝑠	NOUN
iajs-3482	18	9	)	)	PUNCT
iajs-3482	18	10	=	=	PUNCT
iajs-3482	19	1	𝑎	𝑎	PRON
iajs-3482	19	2	•	•	NOUN
iajs-3482	19	3	𝑘	𝑘	VERB
iajs-3482	20	1	+	+	NOUN
iajs-3482	20	2	𝑎	𝑎	NUM
iajs-3482	20	3	•	•	NOUN
iajs-3482	20	4	𝑠	𝑠	X
iajs-3482	20	5	and(𝑘	and(𝑘	PROPN
iajs-3482	20	6	+	+	CCONJ
iajs-3482	20	7	𝑠	𝑠	NOUN
iajs-3482	20	8	)	)	PUNCT
iajs-3482	20	9	•	•	NUM
iajs-3482	21	1	𝑎	𝑎	X
iajs-3482	21	2	=	=	SYM
iajs-3482	21	3	𝑘	𝑘	DET
iajs-3482	21	4	•	•	NOUN
iajs-3482	21	5	𝑎	𝑎	VERB
iajs-3482	21	6	+	+	NUM
iajs-3482	21	7	𝑠	𝑠	NUM
iajs-3482	21	8	•	•	NOUN
iajs-3482	21	9	𝑎𝑓𝑜𝑟	𝑎𝑓𝑜𝑟	NOUN
iajs-3482	21	10	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	21	11	𝑎	𝑎	PROPN
iajs-3482	21	12	,	,	PUNCT
iajs-3482	21	13	𝑘	𝑘	INTJ
iajs-3482	21	14	,	,	PUNCT
iajs-3482	21	15	𝑠	𝑠	PROPN
iajs-3482	21	16	∈	∈	PROPN
iajs-3482	21	17	𝑀	𝑀	PROPN
iajs-3482	21	18	(	(	PUNCT
iajs-3482	21	19	2	2	NUM
iajs-3482	21	20	)	)	PUNCT
iajs-3482	21	21	,	,	PUNCT
iajs-3482	21	22	(	(	PUNCT
iajs-3482	21	23	𝑀	𝑀	PROPN
iajs-3482	21	24	,	,	PUNCT
iajs-3482	21	25	+	+	PUNCT
iajs-3482	21	26	)	)	PUNCT
iajs-3482	21	27	is	be	AUX
iajs-3482	21	28	named	name	VERB
iajs-3482	21	29	additive	additive	ADJ
iajs-3482	21	30	commutative	commutative	ADJ
iajs-3482	21	31	with	with	ADP
iajs-3482	21	32	neutral	neutral	ADJ
iajs-3482	21	33	element	element	NOUN
iajs-3482	21	34	0	0	NUM
iajs-3482	21	35	.	.	PUNCT
iajs-3482	22	1	(	(	PUNCT
iajs-3482	22	2	i.	i.	PROPN
iajs-3482	22	3	e.	e.	PROPN
iajs-3482	22	4	𝑓𝑜𝑟	𝑓𝑜𝑟	PROPN
iajs-3482	22	5	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	22	6	𝑘	𝑘	PRON
iajs-3482	22	7	∈	∈	PROPN
iajs-3482	22	8	𝑀	𝑀	PROPN
iajs-3482	22	9	,	,	PUNCT
iajs-3482	22	10	𝑘	𝑘	X
iajs-3482	22	11	+	+	NOUN
iajs-3482	22	12	0	0	NUM
iajs-3482	22	13	=	=	SYM
iajs-3482	22	14	0	0	PUNCT
iajs-3482	23	1	+	+	CCONJ
iajs-3482	23	2	𝑘	𝑘	X
iajs-3482	23	3	=	=	SYM
iajs-3482	23	4	𝑘	𝑘	X
iajs-3482	23	5	)	)	PUNCT
iajs-3482	23	6	iff	iff	VERB
iajs-3482	23	7	𝑘	𝑘	PROPN
iajs-3482	24	1	+	+	PROPN
iajs-3482	24	2	𝑠	𝑠	PROPN
iajs-3482	24	3	=	=	SYM
iajs-3482	24	4	𝑘	𝑘	PROPN
iajs-3482	24	5	+	+	NOUN
iajs-3482	24	6	𝑛	𝑛	PROPN
iajs-3482	24	7	holds	hold	VERB
iajs-3482	24	8	for	for	ADP
iajs-3482	24	9	any	any	DET
iajs-3482	24	10	𝑘	𝑘	NOUN
iajs-3482	24	11	,	,	PUNCT
iajs-3482	24	12	𝑠	𝑠	PROPN
iajs-3482	24	13	∈	∈	PROPN
iajs-3482	24	14	𝑀	𝑀	PROPN
iajs-3482	24	15	,	,	PUNCT
iajs-3482	24	16	and	and	CCONJ
iajs-3482	24	17	(	(	PUNCT
iajs-3482	24	18	𝑀,•	𝑀,•	PROPN
iajs-3482	24	19	)	)	PUNCT
iajs-3482	24	20	is	be	AUX
iajs-3482	24	21	a	a	DET
iajs-3482	24	22	semi	semi	ADJ
iajs-3482	24	23	−	−	PROPN
iajs-3482	24	24	group	group	NOUN
iajs-3482	24	25	with	with	ADP
iajs-3482	24	26	zero	zero	NUM
iajs-3482	24	27	0	0	NUM
iajs-3482	24	28	,	,	PUNCT
iajs-3482	24	29	𝑖.	𝑖.	ADV
iajs-3482	24	30	𝑒.	𝑒.	PROPN
iajs-3482	24	31	,	,	PUNCT
iajs-3482	25	1	0	0	X
iajs-3482	25	2	.	.	PUNCT
iajs-3482	26	1	𝑎	𝑎	X
iajs-3482	26	2	=	=	PUNCT
iajs-3482	26	3	𝑎.	𝑎.	NOUN
iajs-3482	26	4	0	0	PUNCT
iajs-3482	27	1	=	=	SYM
iajs-3482	27	2	0	0	NUM
iajs-3482	27	3	for	for	ADP
iajs-3482	27	4	any	any	DET
iajs-3482	27	5	𝑎	𝑎	PROPN
iajs-3482	27	6	∈	∈	NOUN
iajs-3482	27	7	𝑀.	𝑀.	NOUN
iajs-3482	27	8	a	a	DET
iajs-3482	27	9	semi	semi	ADJ
iajs-3482	27	10	−	−	NOUN
iajs-3482	27	11	ring	ring	NOUN
iajs-3482	27	12	(	(	PUNCT
iajs-3482	27	13	𝑀	𝑀	PROPN
iajs-3482	27	14	,	,	PUNCT
iajs-3482	27	15	+	+	NOUN
iajs-3482	27	16	,	,	PUNCT
iajs-3482	27	17	•	•	NUM
iajs-3482	27	18	)	)	PUNCT
iajs-3482	27	19	is	be	AUX
iajs-3482	27	20	named	name	VERB
iajs-3482	27	21	commutative	commutative	ADJ
iajs-3482	27	22	iff	iff	PROPN
iajs-3482	27	23	𝑘	𝑘	PROPN
iajs-3482	27	24	•	•	NOUN
iajs-3482	27	25	𝑠	𝑠	PROPN
iajs-3482	27	26	=	=	SYM
iajs-3482	27	27	𝑠	𝑠	PROPN
iajs-3482	27	28	•	•	NOUN
iajs-3482	27	29	𝑘	𝑘	PRON
iajs-3482	27	30	holds	hold	VERB
iajs-3482	27	31	for	for	ADP
iajs-3482	27	32	any	any	DET
iajs-3482	27	33	𝑘	𝑘	NOUN
iajs-3482	27	34	,	,	PUNCT
iajs-3482	27	35	𝑠	𝑠	PROPN
iajs-3482	27	36	∈	∈	PROPN
iajs-3482	27	37	𝑀	𝑀	PROPN
iajs-3482	27	38	(	(	PUNCT
iajs-3482	27	39	2	2	NUM
iajs-3482	27	40	)	)	PUNCT
iajs-3482	27	41	,	,	PUNCT
iajs-3482	27	42	let	let	VERB
iajs-3482	27	43	(	(	PUNCT
iajs-3482	27	44	m	m	NOUN
iajs-3482	27	45	,	,	PUNCT
iajs-3482	27	46	+	+	ADJ
iajs-3482	27	47	,	,	PUNCT
iajs-3482	27	48	•	•	PRON
iajs-3482	27	49	)	)	PUNCT
iajs-3482	27	50	be	be	AUX
iajs-3482	27	51	an	an	DET
iajs-3482	27	52	additively	additively	ADV
iajs-3482	27	53	commutative	commutative	ADJ
iajs-3482	27	54	semiring	semiring	NOUN
iajs-3482	27	55	.	.	PUNCT
iajs-3482	28	1	then	then	ADV
iajs-3482	28	2	m	m	VERB
iajs-3482	28	3	is	be	AUX
iajs-3482	28	4	named	name	VERB
iajs-3482	28	5	inverse	inverse	ADJ
iajs-3482	28	6	semi	semi	ADJ
iajs-3482	28	7	-	-	NOUN
iajs-3482	28	8	ring	ring	ADJ
iajs-3482	28	9	,	,	PUNCT
iajs-3482	28	10	if	if	SCONJ
iajs-3482	28	11	(	(	PUNCT
iajs-3482	28	12	m	m	NOUN
iajs-3482	28	13	,	,	PUNCT
iajs-3482	28	14	+	+	PUNCT
iajs-3482	28	15	)	)	PUNCT
iajs-3482	28	16	is	be	AUX
iajs-3482	28	17	an	an	DET
iajs-3482	28	18	inverse	inverse	NOUN
iajs-3482	28	19	semi	semi	NOUN
iajs-3482	28	20	-	-	NOUN
iajs-3482	28	21	group	group	NOUN
iajs-3482	28	22	(	(	PUNCT
iajs-3482	28	23	i.e	i.e	PROPN
iajs-3482	28	24	)	)	PUNCT
iajs-3482	28	25	for	for	ADP
iajs-3482	28	26	each	each	DET
iajs-3482	28	27	𝑘	𝑘	PROPN
iajs-3482	28	28	∈	∈	PROPN
iajs-3482	28	29	𝑀	𝑀	PROPN
iajs-3482	28	30	there	there	PRON
iajs-3482	28	31	are	be	VERB
iajs-3482	28	32	a	a	DET
iajs-3482	28	33	unique	unique	ADJ
iajs-3482	28	34	𝑘′	𝑘′	NUM
iajs-3482	28	35	∈	∈	NOUN
iajs-3482	28	36	𝑀	𝑀	PROPN
iajs-3482	28	37	such	such	ADJ
iajs-3482	28	38	that	that	PRON
iajs-3482	28	39	,	,	PUNCT
iajs-3482	28	40	𝑘	𝑘	X
iajs-3482	28	41	=	=	SYM
iajs-3482	28	42	𝑘	𝑘	PROPN
iajs-3482	28	43	+	+	NOUN
iajs-3482	28	44	𝑘′	𝑘′	NUM
iajs-3482	28	45	+	+	CCONJ
iajs-3482	28	46	𝑘	𝑘	PROPN
iajs-3482	28	47	and	and	CCONJ
iajs-3482	28	48	𝑘′	𝑘′	NUM
iajs-3482	29	1	+	+	CCONJ
iajs-3482	29	2	𝑘	𝑘	PROPN
iajs-3482	29	3	+	+	CCONJ
iajs-3482	29	4	𝑘′	𝑘′	NUM
iajs-3482	29	5	=	=	SYM
iajs-3482	29	6	𝑘′	𝑘′	NUM
iajs-3482	29	7	(	(	PUNCT
iajs-3482	29	8	2	2	NUM
iajs-3482	29	9	)	)	PUNCT
iajs-3482	29	10	,	,	PUNCT
iajs-3482	29	11	and	and	CCONJ
iajs-3482	29	12	is	be	AUX
iajs-3482	29	13	called	call	VERB
iajs-3482	29	14	cancellative	cancellative	ADJ
iajs-3482	29	15	semi	semi	ADV
iajs-3482	29	16	−	−	PROPN
iajs-3482	29	17	ring	ring	NOUN
iajs-3482	29	18	iff	iff	PROPN
iajs-3482	29	19	for	for	ADP
iajs-3482	29	20	any	any	DET
iajs-3482	29	21	𝑘	𝑘	PROPN
iajs-3482	29	22	,	,	PUNCT
iajs-3482	29	23	𝑠	𝑠	PROPN
iajs-3482	29	24	,	,	PUNCT
iajs-3482	29	25	𝑚	𝑚	PROPN
iajs-3482	29	26	∈	∈	PROPN
iajs-3482	29	27	𝑀	𝑀	PROPN
iajs-3482	29	28	,	,	PUNCT
iajs-3482	29	29	such	such	ADJ
iajs-3482	29	30	that	that	SCONJ
iajs-3482	29	31	𝑘	𝑘	PROPN
iajs-3482	30	1	+	+	X
iajs-3482	30	2	𝑠	𝑠	PROPN
iajs-3482	30	3	=	=	SYM
iajs-3482	30	4	𝑘	𝑘	PROPN
iajs-3482	30	5	+	+	CCONJ
iajs-3482	30	6	𝑚	𝑚	X
iajs-3482	30	7	,	,	PUNCT
iajs-3482	30	8	then	then	ADV
iajs-3482	30	9	𝑠	𝑠	PROPN
iajs-3482	30	10	=	=	SYM
iajs-3482	30	11	𝑚.a	𝑚.a	PROPN
iajs-3482	30	12	semi	semi	ADJ
iajs-3482	30	13	-	-	ADJ
iajs-3482	30	14	ring𝑀	ring𝑀	PROPN
iajs-3482	30	15	is	be	AUX
iajs-3482	30	16	named	name	VERB
iajs-3482	30	17	prime	prime	ADJ
iajs-3482	30	18	semi	semi	ADJ
iajs-3482	30	19	-	-	NOUN
iajs-3482	30	20	ring	ring	ADJ
iajs-3482	30	21	if	if	SCONJ
iajs-3482	30	22	for	for	ADP
iajs-3482	30	23	any	any	DET
iajs-3482	30	24	𝑘	𝑘	NOUN
iajs-3482	30	25	,	,	PUNCT
iajs-3482	30	26	𝑠	𝑠	PROPN
iajs-3482	30	27	∈	∈	PROPN
iajs-3482	30	28	𝑀	𝑀	PROPN
iajs-3482	30	29	,	,	PUNCT
iajs-3482	30	30	𝑘	𝑘	PROPN
iajs-3482	30	31	𝑀	𝑀	PROPN
iajs-3482	30	32	𝑠	𝑠	PROPN
iajs-3482	30	33	=	=	SYM
iajs-3482	30	34	0	0	PROPN
iajs-3482	30	35	implies	imply	VERB
iajs-3482	30	36	that	that	SCONJ
iajs-3482	30	37	either	either	CCONJ
iajs-3482	30	38	𝑘	𝑘	PROPN
iajs-3482	30	39	=	=	SYM
iajs-3482	30	40	0	0	NUM
iajs-3482	30	41	𝑜𝑟	𝑜𝑟	PRON
iajs-3482	30	42	𝑠	𝑠	PROPN
iajs-3482	30	43	=	=	SYM
iajs-3482	30	44	0	0	PROPN
iajs-3482	30	45	.	.	PUNCT
iajs-3482	31	1	a	a	DET
iajs-3482	31	2	semi	semi	ADJ
iajs-3482	31	3	−	−	NOUN
iajs-3482	31	4	ring	ring	NOUN
iajs-3482	31	5	𝑀	𝑀	PROPN
iajs-3482	31	6	is	be	AUX
iajs-3482	31	7	named	name	VERB
iajs-3482	31	8	a	a	DET
iajs-3482	31	9	semi	semi	ADJ
iajs-3482	31	10	-	-	ADJ
iajs-3482	31	11	prime	prime	ADJ
iajs-3482	31	12	if	if	SCONJ
iajs-3482	31	13	for	for	ADP
iajs-3482	31	14	any	any	DET
iajs-3482	31	15	𝑘	𝑘	PROPN
iajs-3482	31	16	∈	∈	PROPN
iajs-3482	31	17	𝑀	𝑀	PROPN
iajs-3482	31	18	,	,	PUNCT
iajs-3482	31	19	𝑘	𝑘	PROPN
iajs-3482	31	20	𝑀	𝑀	PROPN
iajs-3482	31	21	𝑘	𝑘	NOUN
iajs-3482	31	22	=	=	SYM
iajs-3482	31	23	0	0	NUM
iajs-3482	31	24	mplies	mplie	NOUN
iajs-3482	31	25	that	that	PRON
iajs-3482	31	26	𝑘	𝑘	ADP
iajs-3482	31	27	=	=	NOUN
iajs-3482	31	28	0	0	PROPN
iajs-3482	31	29	.	.	PUNCT
iajs-3482	32	1	(	(	PUNCT
iajs-3482	32	2	3	3	NUM
iajs-3482	32	3	)	)	PUNCT
iajs-3482	32	4	,	,	PUNCT
iajs-3482	32	5	a	a	DET
iajs-3482	32	6	semi	semi	ADJ
iajs-3482	32	7	-	-	ADJ
iajs-3482	32	8	ring	ring	ADJ
iajs-3482	32	9	m	m	VERB
iajs-3482	32	10	is	be	AUX
iajs-3482	32	11	named	name	VERB
iajs-3482	32	12	𝑞	𝑞	X
iajs-3482	32	13	−	−	PROPN
iajs-3482	32	14	torsion	torsion	NOUN
iajs-3482	32	15	free	free	ADJ
iajs-3482	32	16	where	where	SCONJ
iajs-3482	32	17	𝑞	𝑞	X
iajs-3482	32	18	≠	≠	PROPN
iajs-3482	32	19	0	0	NUM
iajs-3482	32	20	is	be	AUX
iajs-3482	32	21	an	an	DET
iajs-3482	32	22	integer	integer	NOUN
iajs-3482	32	23	if	if	SCONJ
iajs-3482	32	24	whenever	whenever	SCONJ
iajs-3482	32	25	q𝑘	q𝑘	NOUN
iajs-3482	32	26	=	=	NOUN
iajs-3482	32	27	0	0	NUM
iajs-3482	32	28	with	with	ADP
iajs-3482	32	29	𝑘	𝑘	PROPN
iajs-3482	32	30	∈	∈	PROPN
iajs-3482	32	31	𝑀	𝑀	PROPN
iajs-3482	32	32	,	,	PUNCT
iajs-3482	32	33	then	then	ADV
iajs-3482	32	34	𝑘	𝑘	X
iajs-3482	32	35	=	=	NOUN
iajs-3482	32	36	0	0	PROPN
iajs-3482	32	37	.	.	PUNCT
iajs-3482	33	1	a	a	DET
iajs-3482	33	2	commutator	commutator	NOUN
iajs-3482	33	3	[	[	X
iajs-3482	33	4	.	.	PUNCT
iajs-3482	33	5	,	,	PUNCT
iajs-3482	33	6	.	.	PUNCT
iajs-3482	33	7	]	]	PUNCT
iajs-3482	34	1	in	in	ADP
iajs-3482	34	2	inverse	inverse	NOUN
iajs-3482	34	3	semi	semi	ADV
iajs-3482	34	4	−	−	PROPN
iajs-3482	34	5	rings	ring	NOUN
iajs-3482	34	6	defines	define	NOUN
iajs-3482	34	7	as	as	ADP
iajs-3482	34	8	[	[	X
iajs-3482	34	9	𝑘	𝑘	X
iajs-3482	34	10	,	,	PUNCT
iajs-3482	34	11	𝑠	𝑠	X
iajs-3482	34	12	]	]	X
iajs-3482	34	13	=	=	PUNCT
iajs-3482	34	14	𝑘𝑠	𝑘𝑠	INTJ
iajs-3482	34	15	+	+	NUM
iajs-3482	34	16	𝑘𝑠	𝑘𝑠	NUM
iajs-3482	34	17	´	´	NOUN
iajs-3482	34	18	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3482	34	19	,	,	PUNCT
iajs-3482	34	20	𝑘	𝑘	PROPN
iajs-3482	34	21	𝑜	𝑜	NOUN
iajs-3482	34	22	𝑠	𝑠	NOUN
iajs-3482	34	23	=	=	PUNCT
iajs-3482	34	24	𝑘𝑠	𝑘𝑠	INTJ
iajs-3482	35	1	+	+	NUM
iajs-3482	35	2	𝑘𝑠	𝑘𝑠	INTJ
iajs-3482	35	3	(	(	PUNCT
iajs-3482	35	4	3	3	NUM
iajs-3482	35	5	)	)	PUNCT
iajs-3482	35	6	.	.	PUNCT
iajs-3482	36	1	in	in	ADP
iajs-3482	36	2	(	(	PUNCT
iajs-3482	36	3	4	4	X
iajs-3482	36	4	)	)	PUNCT
iajs-3482	36	5	albas	alba	NOUN
iajs-3482	36	6	presented	present	VERB
iajs-3482	36	7	the	the	DET
iajs-3482	36	8	𝛼	𝛼	PROPN
iajs-3482	36	9	−	−	NOUN
iajs-3482	36	10	centralizer	centralizer	NOUN
iajs-3482	36	11	concept	concept	NOUN
iajs-3482	36	12	and	and	CCONJ
iajs-3482	36	13	the	the	DET
iajs-3482	36	14	jordan	jordan	PROPN
iajs-3482	36	15	α	α	PROPN
iajs-3482	36	16	−centralizer	−centralizer	PROPN
iajs-3482	36	17	concept	concept	NOUN
iajs-3482	36	18	,	,	PUNCT
iajs-3482	36	19	which	which	PRON
iajs-3482	36	20	could	could	AUX
iajs-3482	36	21	be	be	AUX
iajs-3482	36	22	a	a	DET
iajs-3482	36	23	generalization	generalization	NOUN
iajs-3482	36	24	of	of	ADP
iajs-3482	36	25	jordan	jordan	PROPN
iajs-3482	36	26	centralizer	centralizer	NOUN
iajs-3482	36	27	and	and	CCONJ
iajs-3482	36	28	centralizer	centralizer	NOUN
iajs-3482	36	29	and	and	CCONJ
iajs-3482	36	30	tried	try	VERB
iajs-3482	36	31	beneath	beneath	ADP
iajs-3482	36	32	particular	particular	ADJ
iajs-3482	36	33	requirements	requirement	NOUN
iajs-3482	36	34	on	on	ADP
iajs-3482	36	35	a	a	DET
iajs-3482	36	36	2	2	NUM
iajs-3482	36	37	−torsion	−torsion	NOUN
iajs-3482	36	38	free	free	ADJ
iajs-3482	36	39	semi	semi	ADJ
iajs-3482	36	40	−	−	PROPN
iajs-3482	36	41	prime	prime	ADJ
iajs-3482	36	42	ring	ring	NOUN
iajs-3482	36	43	,	,	PUNCT
iajs-3482	36	44	each	each	DET
iajs-3482	36	45	jordan	jordan	PROPN
iajs-3482	36	46	α	α	PROPN
iajs-3482	36	47	-	-	PUNCT
iajs-3482	36	48	centralizer	centralizer	NOUN
iajs-3482	36	49	is	be	AUX
iajs-3482	36	50	α	α	PRON
iajs-3482	36	51	centralizer	centralizer	NOUN
iajs-3482	36	52	,	,	PUNCT
iajs-3482	36	53	where	where	SCONJ
iajs-3482	36	54	α	α	PRON
iajs-3482	36	55	could	could	AUX
iajs-3482	36	56	be	be	AUX
iajs-3482	36	57	a	a	DET
iajs-3482	36	58	surjective	surjective	ADJ
iajs-3482	36	59	homomorphism	homomorphism	NOUN
iajs-3482	36	60	.	.	PUNCT
iajs-3482	37	1	inverse	inverse	ADJ
iajs-3482	37	2	semi	semi	NOUN
iajs-3482	37	3	-	-	NOUN
iajs-3482	37	4	rings	ring	NOUN
iajs-3482	37	5	considered	consider	VERB
iajs-3482	37	6	in	in	ADP
iajs-3482	37	7	different	different	ADJ
iajs-3482	37	8	directions	direction	NOUN
iajs-3482	37	9	by	by	ADP
iajs-3482	37	10	numerous	numerous	ADJ
iajs-3482	37	11	authors	author	NOUN
iajs-3482	37	12	,	,	PUNCT
iajs-3482	37	13	see	see	VERB
iajs-3482	37	14	(	(	PUNCT
iajs-3482	37	15	5	5	NUM
iajs-3482	37	16	-	-	SYM
iajs-3482	37	17	12	12	NUM
iajs-3482	37	18	)	)	PUNCT
iajs-3482	37	19	.	.	PUNCT
iajs-3482	38	1	in	in	ADP
iajs-3482	38	2	this	this	DET
iajs-3482	38	3	work	work	NOUN
iajs-3482	38	4	our	our	PRON
iajs-3482	38	5	aim	aim	NOUN
iajs-3482	38	6	is	be	AUX
iajs-3482	38	7	to	to	PART
iajs-3482	38	8	consider	consider	VERB
iajs-3482	38	9	the	the	DET
iajs-3482	38	10	results	result	NOUN
iajs-3482	38	11	of	of	ADP
iajs-3482	38	12	majeed	majeed	NOUN
iajs-3482	38	13	and	and	CCONJ
iajs-3482	38	14	meften	meften	ADJ
iajs-3482	38	15	(	(	PUNCT
iajs-3482	38	16	13	13	NUM
iajs-3482	38	17	)	)	PUNCT
iajs-3482	38	18	in	in	ADP
iajs-3482	38	19	the	the	DET
iajs-3482	38	20	inverse	inverse	NOUN
iajs-3482	38	21	semi	semi	NOUN
iajs-3482	38	22	-	-	NOUN
iajs-3482	38	23	ring	ring	NOUN
iajs-3482	38	24	.	.	PUNCT
iajs-3482	39	1	in	in	ADP
iajs-3482	39	2	this	this	DET
iajs-3482	39	3	article	article	NOUN
iajs-3482	39	4	,	,	PUNCT
iajs-3482	39	5	m	m	PROPN
iajs-3482	39	6	will	will	AUX
iajs-3482	39	7	represent	represent	VERB
iajs-3482	39	8	additive	additive	ADJ
iajs-3482	39	9	inverse	inverse	NOUN
iajs-3482	39	10	semi	semi	NOUN
iajs-3482	39	11	-	-	NOUN
iajs-3482	39	12	ring	ring	NOUN
iajs-3482	39	13	that	that	PRON
iajs-3482	39	14	satisfies	satisfy	VERB
iajs-3482	39	15	the	the	DET
iajs-3482	39	16	requirement	requirement	NOUN
iajs-3482	39	17	that	that	SCONJ
iajs-3482	39	18	for	for	ADP
iajs-3482	39	19	any	any	DET
iajs-3482	39	20	r	r	NOUN
iajs-3482	39	21	∈	∈	PROPN
iajs-3482	39	22	m	m	NOUN
iajs-3482	39	23	,	,	PUNCT
iajs-3482	39	24	𝑘	𝑘	PROPN
iajs-3482	39	25	+	+	PROPN
iajs-3482	39	26	�	�	PROPN
iajs-3482	39	27	́	́	PROPN
iajs-3482	39	28	�	�	PROPN
iajs-3482	39	29	is	be	AUX
iajs-3482	39	30	located	locate	VERB
iajs-3482	39	31	in	in	ADP
iajs-3482	39	32	the	the	DET
iajs-3482	39	33	center	center	NOUN
iajs-3482	39	34	z(m	z(m	PROPN
iajs-3482	39	35	)	)	PUNCT
iajs-3482	39	36	of	of	ADP
iajs-3482	39	37	m	m	PROPN
iajs-3482	39	38	.	.	PUNCT
iajs-3482	40	1	2	2	X
iajs-3482	40	2	.	.	X
iajs-3482	40	3	preliminaries	preliminary	NOUN
iajs-3482	40	4	we	we	PRON
iajs-3482	40	5	recalled	recall	VERB
iajs-3482	40	6	the	the	DET
iajs-3482	40	7	definitions	definition	NOUN
iajs-3482	40	8	of	of	ADP
iajs-3482	40	9	lie	lie	NOUN
iajs-3482	41	1	−	−	PROPN
iajs-3482	41	2	ideal	ideal	ADJ
iajs-3482	41	3	,	,	PUNCT
iajs-3482	41	4	square	square	ADJ
iajs-3482	41	5	closed	closed	ADJ
iajs-3482	41	6	lie	lie	NOUN
iajs-3482	41	7	−	−	PROPN
iajs-3482	41	8	ideal	ideal	NOUN
iajs-3482	41	9	of	of	ADP
iajs-3482	41	10	a	a	DET
iajs-3482	41	11	semiring	semire	VERB
iajs-3482	41	12	𝑀	𝑀	PROPN
iajs-3482	41	13	,	,	PUNCT
iajs-3482	41	14	and	and	CCONJ
iajs-3482	41	15	some	some	DET
iajs-3482	41	16	definitions	definition	NOUN
iajs-3482	41	17	,	,	PUNCT
iajs-3482	41	18	lemmas	lemma	VERB
iajs-3482	41	19	that	that	PRON
iajs-3482	41	20	will	will	AUX
iajs-3482	41	21	be	be	AUX
iajs-3482	41	22	used	use	VERB
iajs-3482	41	23	later	later	ADV
iajs-3482	41	24	.	.	PUNCT
iajs-3482	42	1	definition	definition	NOUN
iajs-3482	42	2	(	(	PUNCT
iajs-3482	42	3	2.1):(14	2.1):(14	NOUN
iajs-3482	42	4	)	)	PUNCT
iajs-3482	42	5	an	an	DET
iajs-3482	42	6	additive	additive	ADJ
iajs-3482	42	7	sub	sub	NOUN
iajs-3482	42	8	semi	semi	ADJ
iajs-3482	42	9	−	−	PROPN
iajs-3482	42	10	group	group	NOUN
iajs-3482	42	11	of	of	ADP
iajs-3482	42	12	inverse	inverse	NOUN
iajs-3482	42	13	semi	semi	ADV
iajs-3482	42	14	−	−	PROPN
iajs-3482	42	15	ring	ring	NOUN
iajs-3482	42	16	𝑀	𝑀	PROPN
iajs-3482	42	17	satisfies[𝑛	satisfies[𝑛	NOUN
iajs-3482	42	18	,	,	PUNCT
iajs-3482	42	19	q	q	X
iajs-3482	42	20	]	]	X
iajs-3482	42	21	=	=	PUNCT
iajs-3482	42	22	𝑛q	𝑛q	NOUN
iajs-3482	42	23	+	+	CCONJ
iajs-3482	42	24	q′𝑘	q′𝑘	PROPN
iajs-3482	42	25	∈	∈	PROPN
iajs-3482	42	26	𝑉	𝑉	PROPN
iajs-3482	42	27	for	for	ADP
iajs-3482	42	28	any	any	DET
iajs-3482	42	29	𝑘	𝑘	PROPN
iajs-3482	42	30	∈	∈	PROPN
iajs-3482	42	31	𝑉	𝑉	PROPN
iajs-3482	42	32	,	,	PUNCT
iajs-3482	42	33	q	q	PROPN
iajs-3482	42	34	∈	∈	PROPN
iajs-3482	42	35	𝑀	𝑀	PROPN
iajs-3482	42	36	,	,	PUNCT
iajs-3482	42	37	is	be	AUX
iajs-3482	42	38	named	name	VERB
iajs-3482	42	39	a	a	DET
iajs-3482	42	40	lie	lie	NOUN
iajs-3482	42	41	-	-	PUNCT
iajs-3482	42	42	ideal	ideal	NOUN
iajs-3482	42	43	of	of	ADP
iajs-3482	42	44	m	m	PROPN
iajs-3482	42	45	.	.	PUNCT
iajs-3482	43	1	definition	definition	NOUN
iajs-3482	43	2	(	(	PUNCT
iajs-3482	43	3	2.2):(14	2.2):(14	NOUN
iajs-3482	43	4	)	)	PUNCT
iajs-3482	43	5	let	let	VERB
iajs-3482	43	6	v	v	PART
iajs-3482	43	7	be	be	AUX
iajs-3482	43	8	a	a	DET
iajs-3482	43	9	lie	lie	NOUN
iajs-3482	43	10	−	−	NOUN
iajs-3482	43	11	ideal	ideal	NOUN
iajs-3482	43	12	of	of	ADP
iajs-3482	43	13	a	a	DET
iajs-3482	43	14	ring	ring	NOUN
iajs-3482	43	15	,	,	PUNCT
iajs-3482	43	16	"	"	PUNCT
iajs-3482	43	17	then	then	ADV
iajs-3482	43	18	𝑉	𝑉	PROPN
iajs-3482	43	19	is	be	AUX
iajs-3482	43	20	named	name	VERB
iajs-3482	43	21	a	a	DET
iajs-3482	43	22	squane	squane	NOUN
iajs-3482	43	23	closed	close	VERB
iajs-3482	43	24	lie	lie	NOUN
iajs-3482	43	25	−	−	PROPN
iajs-3482	43	26	ideal	ideal	NOUN
iajs-3482	43	27	"	"	PUNCT
iajs-3482	43	28	of	of	ADP
iajs-3482	43	29	𝑀	𝑀	PROPN
iajs-3482	43	30	if	if	SCONJ
iajs-3482	43	31	𝑘2	𝑘2	PROPN
iajs-3482	43	32	∈	∈	PROPN
iajs-3482	43	33	𝑉	𝑉	PROPN
iajs-3482	43	34	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	43	35	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	43	36	𝑘	𝑘	DET
iajs-3482	43	37	∈	∈	PROPN
iajs-3482	43	38	𝑉.	𝑉.	NOUN
iajs-3482	43	39	note	note	NOUN
iajs-3482	43	40	that	that	SCONJ
iajs-3482	43	41	if	if	SCONJ
iajs-3482	43	42	v	v	NOUN
iajs-3482	43	43	is	be	AUX
iajs-3482	43	44	a	a	DET
iajs-3482	43	45	square	square	ADJ
iajs-3482	43	46	closed	close	VERB
iajs-3482	43	47	lie	lie	NOUN
iajs-3482	43	48	-	-	PUNCT
iajs-3482	43	49	idealof	idealof	PROPN
iajs-3482	43	50	𝑀	𝑀	PROPN
iajs-3482	43	51	,	,	PUNCT
iajs-3482	43	52	then	then	ADV
iajs-3482	43	53	2𝑘q	2𝑘q	PROPN
iajs-3482	43	54	∈	∈	PROPN
iajs-3482	43	55	𝑉	𝑉	PROPN
iajs-3482	43	56	for	for	ADP
iajs-3482	43	57	any	any	DET
iajs-3482	43	58	𝑘	𝑘	NOUN
iajs-3482	43	59	,	,	PUNCT
iajs-3482	43	60	q	q	PROPN
iajs-3482	43	61	∈	∈	PROPN
iajs-3482	43	62	𝑉.	𝑉.	NOUN
iajs-3482	43	63	definition	definition	NOUN
iajs-3482	43	64	(	(	PUNCT
iajs-3482	43	65	2.3):(2	2.3):(2	NUM
iajs-3482	43	66	)	)	PUNCT
iajs-3482	43	67	,	,	PUNCT
iajs-3482	43	68	(	(	PUNCT
iajs-3482	43	69	15	15	X
iajs-3482	43	70	)	)	PUNCT
iajs-3482	43	71	let	let	VERB
iajs-3482	43	72	i	i	PRON
iajs-3482	43	73	be	be	AUX
iajs-3482	43	74	a	a	DET
iajs-3482	43	75	nonzero	nonzero	ADJ
iajs-3482	43	76	ideal	ideal	NOUN
iajs-3482	43	77	of	of	ADP
iajs-3482	43	78	𝑀	𝑀	PROPN
iajs-3482	43	79	,	,	PUNCT
iajs-3482	43	80	the	the	DET
iajs-3482	43	81	set	set	NOUN
iajs-3482	43	82	𝑍(𝐼	𝑍(𝐼	ADJ
iajs-3482	43	83	)	)	PUNCT
iajs-3482	43	84	=	=	PRON
iajs-3482	43	85	{	{	PUNCT
iajs-3482	43	86	𝑘	𝑘	NOUN
iajs-3482	43	87	∈	∈	PROPN
iajs-3482	43	88	𝐼	𝐼	PROPN
iajs-3482	43	89	,	,	PUNCT
iajs-3482	43	90	𝑘q	𝑘q	NOUN
iajs-3482	43	91	=	=	NOUN
iajs-3482	43	92	q𝑘	q𝑘	NOUN
iajs-3482	43	93	,	,	PUNCT
iajs-3482	43	94	for	for	SCONJ
iajs-3482	43	95	any	any	DET
iajs-3482	43	96	q	q	PROPN
iajs-3482	43	97	∈	∈	PROPN
iajs-3482	43	98	𝐼	𝐼	PROPN
iajs-3482	43	99	}	}	PUNCT
iajs-3482	43	100	is	be	AUX
iajs-3482	43	101	named	name	VERB
iajs-3482	43	102	the	the	DET
iajs-3482	43	103	center	center	NOUN
iajs-3482	43	104	of	of	ADP
iajs-3482	43	105	𝐼.	𝐼.	PROPN
iajs-3482	43	106	definition	definition	NOUN
iajs-3482	43	107	(	(	PUNCT
iajs-3482	43	108	2.4):(2	2.4):(2	NUM
iajs-3482	43	109	)	)	PUNCT
iajs-3482	43	110	,	,	PUNCT
iajs-3482	43	111	(	(	PUNCT
iajs-3482	43	112	16	16	NUM
iajs-3482	43	113	)	)	PUNCT
iajs-3482	43	114	let	let	VERB
iajs-3482	43	115	𝑞	𝑞	PROPN
iajs-3482	43	116	∈	∈	PROPN
iajs-3482	43	117	𝑀	𝑀	PROPN
iajs-3482	43	118	,	,	PUNCT
iajs-3482	43	119	the	the	DET
iajs-3482	43	120	set	set	NOUN
iajs-3482	43	121	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	43	122	)	)	PUNCT
iajs-3482	44	1	=	=	PRON
iajs-3482	44	2	{	{	PUNCT
iajs-3482	44	3	𝑘	𝑘	PROPN
iajs-3482	44	4	∈	∈	PROPN
iajs-3482	44	5	𝑀	𝑀	PROPN
iajs-3482	44	6	,	,	PUNCT
iajs-3482	44	7	𝑘𝑞	𝑘𝑞	X
iajs-3482	44	8	=	=	SYM
iajs-3482	44	9	𝑞𝑘	𝑞𝑘	ADP
iajs-3482	44	10	,	,	PUNCT
iajs-3482	44	11	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	44	12	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	44	13	𝑞	𝑞	PROPN
iajs-3482	44	14	∈	∈	PROPN
iajs-3482	44	15	𝑀	𝑀	PROPN
iajs-3482	44	16	}	}	PUNCT
iajs-3482	44	17	is	be	AUX
iajs-3482	44	18	named	name	VERB
iajs-3482	44	19	the	the	DET
iajs-3482	44	20	center	center	NOUN
iajs-3482	44	21	of	of	ADP
iajs-3482	44	22	the	the	DET
iajs-3482	44	23	semi	semi	ADJ
iajs-3482	44	24	−	−	PROPN
iajs-3482	44	25	ring	ring	NOUN
iajs-3482	44	26	m.	m.	NOUN
iajs-3482	44	27	clearly	clearly	ADV
iajs-3482	44	28	that	that	DET
iajs-3482	44	29	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	44	30	)	)	PUNCT
iajs-3482	44	31	is	be	AUX
iajs-3482	44	32	a	a	DET
iajs-3482	44	33	subsemi	subsemi	NOUN
iajs-3482	44	34	−	−	NOUN
iajs-3482	44	35	ring	ring	NOUN
iajs-3482	44	36	of	of	ADP
iajs-3482	44	37	𝑀.	𝑀.	PROPN
iajs-3482	44	38	note	note	NOUN
iajs-3482	45	1	that	that	SCONJ
iajs-3482	45	2	if	if	SCONJ
iajs-3482	45	3	m	m	NOUN
iajs-3482	45	4	is	be	AUX
iajs-3482	45	5	multiplicatively	multiplicatively	ADV
iajs-3482	45	6	commutative	commutative	ADJ
iajs-3482	45	7	then	then	ADV
iajs-3482	45	8	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	45	9	)	)	PUNCT
iajs-3482	45	10	=	=	SYM
iajs-3482	45	11	𝑀.	𝑀.	NOUN
iajs-3482	45	12	ihjpas	ihjpa	NOUN
iajs-3482	45	13	.	.	PUNCT
iajs-3482	46	1	2025	2025	NUM
iajs-3482	46	2	,	,	PUNCT
iajs-3482	46	3	38	38	NUM
iajs-3482	46	4	(	(	PUNCT
iajs-3482	46	5	1	1	NUM
iajs-3482	46	6	)	)	PUNCT
iajs-3482	46	7	399	399	NUM
iajs-3482	46	8	lemma	lemma	PROPN
iajs-3482	46	9	(	(	PUNCT
iajs-3482	46	10	2.5):(10	2.5):(10	NOUN
iajs-3482	46	11	)	)	PUNCT
iajs-3482	46	12	,	,	PUNCT
iajs-3482	46	13	(	(	PUNCT
iajs-3482	46	14	17	17	NUM
iajs-3482	46	15	)	)	PUNCT
iajs-3482	46	16	let	let	VERB
iajs-3482	46	17	m	m	PRON
iajs-3482	46	18	be	be	AUX
iajs-3482	46	19	an	an	DET
iajs-3482	46	20	additive	additive	ADJ
iajs-3482	46	21	inverse	inverse	NOUN
iajs-3482	46	22	semi	semi	NOUN
iajs-3482	46	23	-	-	NOUN
iajs-3482	46	24	ring	ring	ADJ
iajs-3482	46	25	,	,	PUNCT
iajs-3482	46	26	for	for	ADP
iajs-3482	46	27	any	any	DET
iajs-3482	46	28	k	k	NOUN
iajs-3482	46	29	,	,	PUNCT
iajs-3482	46	30	𝑞	𝑞	PROPN
iajs-3482	46	31	∈	∈	PROPN
iajs-3482	46	32	𝑀	𝑀	PROPN
iajs-3482	46	33	,	,	PUNCT
iajs-3482	46	34	𝑖𝑓	𝑖𝑓	ADP
iajs-3482	46	35	𝑘	𝑘	X
iajs-3482	47	1	+	+	CCONJ
iajs-3482	47	2	𝑞	𝑞	X
iajs-3482	47	3	=	=	SYM
iajs-3482	47	4	0	0	PROPN
iajs-3482	47	5	then	then	ADV
iajs-3482	47	6	𝑘	𝑘	X
iajs-3482	47	7	=	=	SYM
iajs-3482	47	8	𝑞′.	𝑞′.	PROPN
iajs-3482	47	9	note	note	VERB
iajs-3482	47	10	that	that	SCONJ
iajs-3482	47	11	in	in	ADP
iajs-3482	47	12	general	general	ADJ
iajs-3482	47	13	𝑘	𝑘	PROPN
iajs-3482	48	1	+	+	CCONJ
iajs-3482	48	2	𝑘	𝑘	DET
iajs-3482	48	3	′	′	ADJ
iajs-3482	48	4	≠	≠	PROPN
iajs-3482	48	5	0	0	NUM
iajs-3482	48	6	,	,	PUNCT
iajs-3482	48	7	𝑘	𝑘	PROPN
iajs-3482	49	1	+	+	NOUN
iajs-3482	49	2	𝑘	𝑘	DET
iajs-3482	49	3	′	′	NOUN
iajs-3482	49	4	=	=	SYM
iajs-3482	49	5	0	0	NUM
iajs-3482	49	6	,	,	PUNCT
iajs-3482	49	7	iff	iff	VERB
iajs-3482	49	8	there	there	PRON
iajs-3482	49	9	are	be	VERB
iajs-3482	49	10	some	some	PRON
iajs-3482	49	11	𝑞	𝑞	X
iajs-3482	49	12	𝜖	𝜖	PROPN
iajs-3482	49	13	𝑀	𝑀	PROPN
iajs-3482	49	14	with	with	ADP
iajs-3482	49	15	𝑘	𝑘	PRON
iajs-3482	50	1	+	+	NOUN
iajs-3482	50	2	𝑞	𝑞	X
iajs-3482	50	3	=	=	SYM
iajs-3482	50	4	0	0	PUNCT
iajs-3482	51	1	[	[	X
iajs-3482	51	2	2	2	NUM
iajs-3482	51	3	]	]	X
iajs-3482	51	4	proposition	proposition	NOUN
iajs-3482	51	5	(	(	PUNCT
iajs-3482	51	6	2.6):(12),(18	2.6):(12),(18	NUM
iajs-3482	51	7	)	)	PUNCT
iajs-3482	51	8	for	for	ADP
iajs-3482	51	9	any	any	DET
iajs-3482	51	10	r	r	NOUN
iajs-3482	51	11	,	,	PUNCT
iajs-3482	51	12	s	s	PART
iajs-3482	51	13	∈	∈	PROPN
iajs-3482	51	14	m	m	PROPN
iajs-3482	51	15	,	,	PUNCT
iajs-3482	51	16	the	the	DET
iajs-3482	51	17	following	following	NOUN
iajs-3482	51	18	are	be	AUX
iajs-3482	51	19	holds	hold	NOUN
iajs-3482	51	20	:	:	PUNCT
iajs-3482	51	21	i.	i.	NOUN
iajs-3482	51	22	(	(	PUNCT
iajs-3482	51	23	𝑘	𝑘	PROPN
iajs-3482	51	24	+	+	CCONJ
iajs-3482	51	25	𝑞)′	𝑞)′	ADJ
iajs-3482	51	26	=	=	PUNCT
iajs-3482	51	27	𝑘′	𝑘′	NUM
iajs-3482	51	28	+	+	NUM
iajs-3482	51	29	𝑞′	𝑞′	X
iajs-3482	51	30	ii	ii	NOUN
iajs-3482	51	31	.	.	PUNCT
iajs-3482	52	1	(	(	PUNCT
iajs-3482	52	2	𝑘	𝑘	X
iajs-3482	52	3	𝑞)′′	𝑞)′′	NOUN
iajs-3482	52	4	=	=	PUNCT
iajs-3482	52	5	𝑘′𝑞	𝑘′𝑞	PROPN
iajs-3482	52	6	=	=	SYM
iajs-3482	52	7	𝑘𝑞′	𝑘𝑞′	X
iajs-3482	52	8	iii	iii	PROPN
iajs-3482	52	9	.	.	PUNCT
iajs-3482	53	1	𝑘′′	𝑘′′	PROPN
iajs-3482	53	2	=	=	PUNCT
iajs-3482	53	3	𝑘	𝑘	DET
iajs-3482	53	4	iv	iv	X
iajs-3482	53	5	.	.	PUNCT
iajs-3482	54	1	𝑘′𝑞′	𝑘′𝑞′	PROPN
iajs-3482	54	2	=	=	SYM
iajs-3482	54	3	(	(	PUNCT
iajs-3482	54	4	𝑘′𝑞)′	𝑘′𝑞)′	PROPN
iajs-3482	54	5	=	=	SYM
iajs-3482	54	6	(	(	PUNCT
iajs-3482	54	7	𝑘𝑞)′′	𝑘𝑞)′′	PROPN
iajs-3482	54	8	=	=	SYM
iajs-3482	54	9	𝑘𝑞.	𝑘𝑞.	PROPN
iajs-3482	54	10	lemma	lemma	PROPN
iajs-3482	54	11	(	(	PUNCT
iajs-3482	54	12	2.7):(12),(19	2.7):(12),(19	NUM
iajs-3482	54	13	)	)	PUNCT
iajs-3482	54	14	let	let	VERB
iajs-3482	54	15	m	m	PRON
iajs-3482	54	16	be	be	AUX
iajs-3482	54	17	ring	ring	NOUN
iajs-3482	54	18	and	and	CCONJ
iajs-3482	54	19	k	k	NOUN
iajs-3482	54	20	,	,	PUNCT
iajs-3482	54	21	𝑞	𝑞	PROPN
iajs-3482	54	22	,	,	PUNCT
iajs-3482	54	23	𝑤	𝑤	ADP
iajs-3482	54	24	∈	∈	PROPN
iajs-3482	54	25	𝑀	𝑀	PROPN
iajs-3482	55	1	then	then	ADV
iajs-3482	55	2	i.	i.	PROPN
iajs-3482	56	1	[	[	X
iajs-3482	56	2	𝑘	𝑘	X
iajs-3482	56	3	,	,	PUNCT
iajs-3482	56	4	𝑘	𝑘	X
iajs-3482	56	5	]	]	X
iajs-3482	56	6	=	=	SYM
iajs-3482	56	7	0	0	NUM
iajs-3482	56	8	ii	ii	NOUN
iajs-3482	56	9	.	.	PUNCT
iajs-3482	57	1	[	[	X
iajs-3482	57	2	𝑘	𝑘	X
iajs-3482	57	3	+	+	SYM
iajs-3482	57	4	𝑞	𝑞	PROPN
iajs-3482	57	5	,	,	PUNCT
iajs-3482	57	6	𝑤	𝑤	ADP
iajs-3482	57	7	]	]	PUNCT
iajs-3482	57	8	=	=	PUNCT
iajs-3482	58	1	[	[	X
iajs-3482	58	2	𝑘	𝑘	X
iajs-3482	58	3	,	,	PUNCT
iajs-3482	58	4	𝑤	𝑤	ADP
iajs-3482	58	5	]	]	PUNCT
iajs-3482	59	1	+	+	CCONJ
iajs-3482	59	2	[	[	X
iajs-3482	59	3	𝑞	𝑞	X
iajs-3482	59	4	,	,	PUNCT
iajs-3482	59	5	𝑤	𝑤	ADP
iajs-3482	59	6	]	]	X
iajs-3482	59	7	iii	iii	X
iajs-3482	59	8	.	.	PUNCT
iajs-3482	60	1	[	[	X
iajs-3482	60	2	𝑘𝑞	𝑘𝑞	X
iajs-3482	60	3	,	,	PUNCT
iajs-3482	60	4	𝑤	𝑤	ADP
iajs-3482	60	5	]	]	PUNCT
iajs-3482	60	6	=	=	SYM
iajs-3482	60	7	𝑘[𝑞	𝑘[𝑞	NOUN
iajs-3482	60	8	,	,	PUNCT
iajs-3482	60	9	𝑤	𝑤	ADP
iajs-3482	60	10	]	]	PUNCT
iajs-3482	61	1	+	+	CCONJ
iajs-3482	61	2	[	[	X
iajs-3482	61	3	𝑘	𝑘	X
iajs-3482	61	4	,	,	PUNCT
iajs-3482	61	5	𝑤]𝑞	𝑤]𝑞	NOUN
iajs-3482	61	6	iv	iv	NOUN
iajs-3482	61	7	.	.	PUNCT
iajs-3482	62	1	[	[	X
iajs-3482	62	2	𝑘	𝑘	X
iajs-3482	62	3	,	,	PUNCT
iajs-3482	62	4	𝑞𝑤	𝑞𝑤	VERB
iajs-3482	62	5	]	]	PUNCT
iajs-3482	62	6	=	=	SYM
iajs-3482	62	7	𝑞[𝑘	𝑞[𝑘	X
iajs-3482	62	8	,	,	PUNCT
iajs-3482	62	9	𝑤	𝑤	ADP
iajs-3482	62	10	]	]	PUNCT
iajs-3482	63	1	+	+	CCONJ
iajs-3482	63	2	[	[	X
iajs-3482	63	3	𝑘	𝑘	X
iajs-3482	63	4	,	,	PUNCT
iajs-3482	63	5	𝑞]𝑤.	𝑞]𝑤.	VERB
iajs-3482	63	6	definition	definition	NOUN
iajs-3482	63	7	(	(	PUNCT
iajs-3482	63	8	2.8):(15),(20	2.8):(15),(20	X
iajs-3482	63	9	)	)	PUNCT
iajs-3482	63	10	let	let	VERB
iajs-3482	63	11	m	m	PRON
iajs-3482	63	12	be	be	AUX
iajs-3482	63	13	a	a	DET
iajs-3482	63	14	semi	semi	ADJ
iajs-3482	63	15	-	-	NOUN
iajs-3482	63	16	ring	ring	NOUN
iajs-3482	63	17	,	,	PUNCT
iajs-3482	63	18	an	an	DET
iajs-3482	63	19	additive	additive	ADJ
iajs-3482	63	20	mapping	mapping	NOUN
iajs-3482	63	21	𝑅	𝑅	PROPN
iajs-3482	63	22	:	:	PUNCT
iajs-3482	63	23	𝑀	𝑀	PROPN
iajs-3482	63	24	→	→	SYM
iajs-3482	63	25	𝑀	𝑀	PROPN
iajs-3482	63	26	is	be	AUX
iajs-3482	63	27	nameda	nameda	ADJ
iajs-3482	63	28	(	(	PUNCT
iajs-3482	63	29	𝛼	𝛼	NOUN
iajs-3482	63	30	,	,	PUNCT
iajs-3482	63	31	𝛼	𝛼	NOUN
iajs-3482	63	32	)	)	PUNCT
iajs-3482	63	33	−	−	PROPN
iajs-3482	63	34	derivation	derivation	NOUN
iajs-3482	63	35	"	"	PUNCT
iajs-3482	63	36	𝑖𝑓	𝑖𝑓	ADP
iajs-3482	63	37	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	63	38	)	)	PUNCT
iajs-3482	63	39	=	=	SYM
iajs-3482	63	40	𝑅(𝑘)𝛼(𝑞	𝑅(𝑘)𝛼(𝑞	NOUN
iajs-3482	63	41	)	)	PUNCT
iajs-3482	64	1	+	+	CCONJ
iajs-3482	64	2	𝛼(𝑘)𝑅(𝑞	𝛼(𝑘)𝑅(𝑞	NOUN
iajs-3482	64	3	)	)	PUNCT
iajs-3482	64	4	for	for	ADP
iajs-3482	64	5	any	any	DET
iajs-3482	64	6	𝑘	𝑘	NOUN
iajs-3482	64	7	,	,	PUNCT
iajs-3482	64	8	𝑞	𝑞	PROPN
iajs-3482	64	9	∈	∈	PROPN
iajs-3482	64	10	𝑀	𝑀	PROPN
iajs-3482	64	11	,	,	PUNCT
iajs-3482	64	12	and	and	CCONJ
iajs-3482	64	13	we	we	PRON
iajs-3482	64	14	say	say	VERB
iajs-3482	64	15	that	that	SCONJ
iajs-3482	64	16	r	r	NOUN
iajs-3482	64	17	is	be	AUX
iajs-3482	64	18	jordan	jordan	PROPN
iajs-3482	64	19	(	(	PUNCT
iajs-3482	64	20	𝛼	𝛼	PROPN
iajs-3482	64	21	,	,	PUNCT
iajs-3482	64	22	𝛼	𝛼	NOUN
iajs-3482	64	23	)	)	PUNCT
iajs-3482	64	24	−	−	NOUN
iajs-3482	64	25	derivation	derivation	NOUN
iajs-3482	64	26	"	"	PUNCT
iajs-3482	64	27	if	if	SCONJ
iajs-3482	64	28	𝑅(𝑘2	𝑅(𝑘2	NOUN
iajs-3482	64	29	)	)	PUNCT
iajs-3482	64	30	=	=	SYM
iajs-3482	64	31	𝑅(𝑘)𝛼(𝑘	𝑅(𝑘)𝛼(𝑘	NOUN
iajs-3482	64	32	)	)	PUNCT
iajs-3482	65	1	+	+	CCONJ
iajs-3482	65	2	𝛼(𝑘)𝑅(𝑘	𝛼(𝑘)𝑅(𝑘	NOUN
iajs-3482	65	3	)	)	PUNCT
iajs-3482	65	4	for	for	ADP
iajs-3482	65	5	any	any	DET
iajs-3482	65	6	𝑘	𝑘	PROPN
iajs-3482	65	7	∈	∈	PROPN
iajs-3482	65	8	𝑀,where	𝑀,where	PUNCT
iajs-3482	65	9	𝛼	𝛼	PRON
iajs-3482	65	10	be	be	AUX
iajs-3482	65	11	additive	additive	ADJ
iajs-3482	65	12	mapping	mapping	NOUN
iajs-3482	65	13	on	on	ADP
iajs-3482	65	14	𝑀.	𝑀.	PROPN
iajs-3482	65	15	every	every	DET
iajs-3482	65	16	derivation	derivation	NOUN
iajs-3482	65	17	is	be	AUX
iajs-3482	65	18	(	(	PUNCT
iajs-3482	65	19	𝛼	𝛼	NOUN
iajs-3482	65	20	,	,	PUNCT
iajs-3482	65	21	𝛼	𝛼	NOUN
iajs-3482	65	22	)	)	PUNCT
iajs-3482	65	23	−	−	NOUN
iajs-3482	65	24	derivation	derivation	NOUN
iajs-3482	65	25	is	be	AUX
iajs-3482	65	26	jordan	jordan	PROPN
iajs-3482	65	27	(	(	PUNCT
iajs-3482	65	28	𝛼	𝛼	PROPN
iajs-3482	65	29	,	,	PUNCT
iajs-3482	65	30	𝛼	𝛼	NOUN
iajs-3482	65	31	)	)	PUNCT
iajs-3482	66	1	−	−	NOUN
iajs-3482	66	2	derivation	derivation	NOUN
iajs-3482	66	3	,	,	PUNCT
iajs-3482	66	4	but	but	CCONJ
iajs-3482	66	5	the	the	DET
iajs-3482	66	6	converse	converse	NOUN
iajs-3482	66	7	in	in	ADP
iajs-3482	66	8	general	general	ADJ
iajs-3482	66	9	is	be	AUX
iajs-3482	66	10	not	not	PART
iajs-3482	66	11	true	true	ADJ
iajs-3482	66	12	.	.	PUNCT
iajs-3482	67	1	definition	definition	NOUN
iajs-3482	67	2	(	(	PUNCT
iajs-3482	67	3	2.9):(3),(21	2.9):(3),(21	NOUN
iajs-3482	67	4	)	)	PUNCT
iajs-3482	67	5	a	a	DET
iajs-3482	67	6	left	left	ADJ
iajs-3482	67	7	(	(	PUNCT
iajs-3482	67	8	right)𝛼	right)𝛼	NOUN
iajs-3482	67	9	−	−	NOUN
iajs-3482	67	10	centralizer	centralizer	NOUN
iajs-3482	67	11	"	"	PUNCT
iajs-3482	67	12	"	"	PUNCT
iajs-3482	67	13	of	of	ADP
iajs-3482	67	14	a	a	DET
iajs-3482	67	15	semi	semi	ADJ
iajs-3482	67	16	-	-	ADJ
iajs-3482	67	17	ring	ring	ADJ
iajs-3482	67	18	𝑀	𝑀	PROPN
iajs-3482	67	19	is	be	AUX
iajs-3482	67	20	an	an	DET
iajs-3482	67	21	“	"	PUNCT
iajs-3482	67	22	additive	additive	ADJ
iajs-3482	67	23	mapping	mapping	NOUN
iajs-3482	67	24	”	"	PUNCT
iajs-3482	67	25	𝑅	𝑅	PROPN
iajs-3482	67	26	:	:	PUNCT
iajs-3482	67	27	𝑀	𝑀	PROPN
iajs-3482	67	28	→	→	SYM
iajs-3482	67	29	𝑀	𝑀	PROPN
iajs-3482	67	30	which	which	PRON
iajs-3482	67	31	satisfies	satisfy	VERB
iajs-3482	67	32	𝑅(𝑘𝑞	𝑅(𝑘𝑞	VERB
iajs-3482	67	33	)	)	PUNCT
iajs-3482	68	1	+	+	CCONJ
iajs-3482	68	2	𝑅(𝑘)𝛼(𝑞)′	𝑅(𝑘)𝛼(𝑞)′	PROPN
iajs-3482	68	3	=	=	SYM
iajs-3482	68	4	0	0	NUM
iajs-3482	68	5	,	,	PUNCT
iajs-3482	68	6	(	(	PUNCT
iajs-3482	68	7	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	68	8	)	)	PUNCT
iajs-3482	68	9	+	+	CCONJ
iajs-3482	68	10	𝛼(𝑘)′𝑅(𝑞	𝛼(𝑘)′𝑅(𝑞	NOUN
iajs-3482	68	11	)	)	PUNCT
iajs-3482	68	12	=	=	SYM
iajs-3482	68	13	0	0	NUM
iajs-3482	68	14	)	)	PUNCT
iajs-3482	68	15	for	for	ADP
iajs-3482	68	16	any	any	DET
iajs-3482	68	17	𝑘	𝑘	NOUN
iajs-3482	68	18	,	,	PUNCT
iajs-3482	68	19	𝑞	𝑞	PROPN
iajs-3482	68	20	∈	∈	PROPN
iajs-3482	68	21	𝑀.	𝑀.	PROPN
iajs-3482	68	22	"	"	PUNCT
iajs-3482	68	23	𝛼	𝛼	NOUN
iajs-3482	68	24	−centralizer	−centralizer	NOUN
iajs-3482	68	25	of	of	ADP
iajs-3482	68	26	a	a	DET
iajs-3482	68	27	ring	ring	NOUN
iajs-3482	68	28	𝑀	𝑀	PROPN
iajs-3482	68	29	is	be	AUX
iajs-3482	68	30	“	"	PUNCT
iajs-3482	68	31	both	both	PRON
iajs-3482	68	32	left	left	ADJ
iajs-3482	68	33	and	and	CCONJ
iajs-3482	68	34	right	right	ADJ
iajs-3482	68	35	”	"	PUNCT
iajs-3482	68	36	𝛼	𝛼	PROPN
iajs-3482	68	37	−	−	NOUN
iajs-3482	68	38	centralizer	centralizer	NOUN
iajs-3482	68	39	"	"	PUNCT
iajs-3482	68	40	,	,	PUNCT
iajs-3482	68	41	"	"	PUNCT
iajs-3482	68	42	where	where	SCONJ
iajs-3482	68	43	𝛼	𝛼	PRON
iajs-3482	68	44	is	be	AUX
iajs-3482	68	45	an	an	DET
iajs-3482	68	46	additive	additive	ADJ
iajs-3482	68	47	mapping	mapping	NOUN
iajs-3482	68	48	”	"	PUNCT
iajs-3482	68	49	on	on	ADP
iajs-3482	68	50	m	m	PROPN
iajs-3482	68	51	.	.	PUNCT
iajs-3482	69	1	definition	definition	NOUN
iajs-3482	69	2	(	(	PUNCT
iajs-3482	69	3	2.10):(3),(22	2.10):(3),(22	NOUN
iajs-3482	69	4	)	)	PUNCT
iajs-3482	69	5	a	a	DET
iajs-3482	69	6	left	left	ADJ
iajs-3482	69	7	(	(	PUNCT
iajs-3482	69	8	right	right	ADJ
iajs-3482	69	9	)	)	PUNCT
iajs-3482	69	10	jordan	jordan	PROPN
iajs-3482	69	11	”	"	PUNCT
iajs-3482	69	12	𝛼	𝛼	PROPN
iajs-3482	69	13	−	−	PROPN
iajs-3482	69	14	"	"	PUNCT
iajs-3482	69	15	centralizer	centralizer	NOUN
iajs-3482	69	16	"	"	PUNCT
iajs-3482	69	17	"	"	PUNCT
iajs-3482	69	18	of	of	ADP
iajs-3482	69	19	a	a	DET
iajs-3482	69	20	semi	semi	ADJ
iajs-3482	69	21	-	-	ADJ
iajs-3482	69	22	ring	ring	ADJ
iajs-3482	69	23	m	m	NOUN
iajs-3482	69	24	is	be	AUX
iajs-3482	69	25	an	an	DET
iajs-3482	69	26	“	"	PUNCT
iajs-3482	69	27	addittive	addittive	ADJ
iajs-3482	69	28	mapping	mapping	NOUN
iajs-3482	69	29	”	"	PUNCT
iajs-3482	69	30	𝑅	𝑅	PROPN
iajs-3482	69	31	:	:	PUNCT
iajs-3482	69	32	𝑀	𝑀	PROPN
iajs-3482	69	33	→	→	SYM
iajs-3482	69	34	𝑀	𝑀	PROPN
iajs-3482	69	35	which	which	PRON
iajs-3482	69	36	satisfy	satisfy	VERB
iajs-3482	69	37	𝑅(𝑘2	𝑅(𝑘2	VERB
iajs-3482	69	38	)	)	PUNCT
iajs-3482	69	39	+	+	CCONJ
iajs-3482	69	40	𝑅(𝑘	𝑅(𝑘	NOUN
iajs-3482	69	41	)	)	PUNCT
iajs-3482	69	42	𝛼(𝑘)′	𝛼(𝑘)′	PROPN
iajs-3482	69	43	=	=	SYM
iajs-3482	69	44	0	0	NUM
iajs-3482	69	45	,	,	PUNCT
iajs-3482	69	46	(	(	PUNCT
iajs-3482	69	47	𝑅(𝑘2	𝑅(𝑘2	NOUN
iajs-3482	69	48	)	)	PUNCT
iajs-3482	69	49	+	+	NUM
iajs-3482	69	50	𝛼(𝑘)′𝑅(𝑘	𝛼(𝑘)′𝑅(𝑘	NOUN
iajs-3482	69	51	)	)	PUNCT
iajs-3482	69	52	=	=	SYM
iajs-3482	69	53	0	0	NUM
iajs-3482	69	54	)	)	PUNCT
iajs-3482	69	55	for	for	ADP
iajs-3482	69	56	any	any	DET
iajs-3482	69	57	𝑘	𝑘	PROPN
iajs-3482	69	58	∈	∈	PROPN
iajs-3482	69	59	𝑀	𝑀	PROPN
iajs-3482	69	60	,	,	PUNCT
iajs-3482	69	61	𝛼	𝛼	PROPN
iajs-3482	69	62	−	−	PROPN
iajs-3482	69	63	"	"	PUNCT
iajs-3482	69	64	"	"	PUNCT
iajs-3482	69	65	jordan	jordan	PROPN
iajs-3482	69	66	"	"	PUNCT
iajs-3482	69	67	centralizer	centralizer	NOUN
iajs-3482	69	68	of	of	ADP
iajs-3482	69	69	a	a	DET
iajs-3482	69	70	ring	ring	NOUN
iajs-3482	69	71	m	m	NOUN
iajs-3482	69	72	is	be	AUX
iajs-3482	69	73	both	both	PRON
iajs-3482	69	74	“	"	PUNCT
iajs-3482	69	75	left	left	ADJ
iajs-3482	69	76	and	and	CCONJ
iajs-3482	69	77	right	right	ADJ
iajs-3482	69	78	jordan	jordan	PROPN
iajs-3482	69	79	”	"	PUNCT
iajs-3482	70	1	𝛼	𝛼	PROPN
iajs-3482	70	2	−	−	NOUN
iajs-3482	70	3	centralizer	centralizer	NOUN
iajs-3482	70	4	,	,	PUNCT
iajs-3482	70	5	where	where	SCONJ
iajs-3482	70	6	𝛼	𝛼	PRON
iajs-3482	70	7	be	be	AUX
iajs-3482	70	8	“	"	PUNCT
iajs-3482	70	9	additive	additive	ADJ
iajs-3482	70	10	mapping	mapping	NOUN
iajs-3482	70	11	”	"	PUNCT
iajs-3482	70	12	on	on	ADP
iajs-3482	70	13	m	m	PROPN
iajs-3482	70	14	.	.	PUNCT
iajs-3482	71	1	3	3	X
iajs-3482	71	2	.	.	X
iajs-3482	71	3	main	main	ADJ
iajs-3482	71	4	results	result	NOUN
iajs-3482	71	5	to	to	PART
iajs-3482	71	6	verify	verify	VERB
iajs-3482	71	7	our	our	PRON
iajs-3482	71	8	main	main	ADJ
iajs-3482	71	9	results	result	NOUN
iajs-3482	71	10	,	,	PUNCT
iajs-3482	71	11	we	we	PRON
iajs-3482	71	12	must	must	AUX
iajs-3482	71	13	utilize	utilize	VERB
iajs-3482	71	14	the	the	DET
iajs-3482	71	15	following	following	NOUN
iajs-3482	71	16	.	.	PUNCT
iajs-3482	72	1	lemma	lemma	PROPN
iajs-3482	72	2	(	(	PUNCT
iajs-3482	72	3	3.1):(4),(23	3.1):(4),(23	NOUN
iajs-3482	72	4	)	)	PUNCT
iajs-3482	72	5	if	if	SCONJ
iajs-3482	72	6	𝑉	𝑉	PROPN
iajs-3482	72	7	⊄	⊄	NOUN
iajs-3482	72	8	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	72	9	)	)	PUNCT
iajs-3482	72	10	is	be	AUX
iajs-3482	72	11	a	a	DET
iajs-3482	72	12	lie	lie	NOUN
iajs-3482	72	13	-	-	PUNCT
iajs-3482	72	14	ideal	ideal	NOUN
iajs-3482	72	15	of	of	ADP
iajs-3482	72	16	a	a	DET
iajs-3482	72	17	2	2	NUM
iajs-3482	72	18	−	−	NOUN
iajs-3482	72	19	tortion	tortion	NOUN
iajs-3482	72	20	free	free	ADJ
iajs-3482	72	21	prime	prime	NOUN
iajs-3482	72	22	"	"	PUNCT
iajs-3482	72	23	semirig	semirig	NOUN
iajs-3482	72	24	𝑀	𝑀	PROPN
iajs-3482	72	25	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-3482	72	26	𝑘	𝑘	PROPN
iajs-3482	72	27	,	,	PUNCT
iajs-3482	72	28	𝑞	𝑞	PROPN
iajs-3482	72	29	∈	∈	PROPN
iajs-3482	72	30	𝑀	𝑀	PROPN
iajs-3482	72	31	such	such	ADJ
iajs-3482	72	32	that	that	SCONJ
iajs-3482	72	33	𝑘	𝑘	PROPN
iajs-3482	72	34	𝑉	𝑉	PROPN
iajs-3482	72	35	𝑞	𝑞	NOUN
iajs-3482	72	36	=	=	SYM
iajs-3482	72	37	0	0	PROPN
iajs-3482	72	38	,	,	PUNCT
iajs-3482	72	39	then	then	ADV
iajs-3482	72	40	𝑘	𝑘	X
iajs-3482	72	41	=	=	SYM
iajs-3482	72	42	0	0	NUM
iajs-3482	72	43	or	or	CCONJ
iajs-3482	72	44	𝑚	𝑚	X
iajs-3482	72	45	=	=	SYM
iajs-3482	72	46	0	0	NUM
iajs-3482	72	47	.	.	PUNCT
iajs-3482	73	1	from	from	ADP
iajs-3482	73	2	this	this	PRON
iajs-3482	73	3	we	we	PRON
iajs-3482	73	4	mean	mean	VERB
iajs-3482	73	5	by	by	ADP
iajs-3482	73	6	v	v	NUM
iajs-3482	73	7	is	be	AUX
iajs-3482	73	8	a	a	DET
iajs-3482	73	9	square	square	ADJ
iajs-3482	73	10	closed	closed	ADJ
iajs-3482	73	11	lie	lie	NOUN
iajs-3482	74	1	−	−	PROPN
iajs-3482	74	2	ideal	ideal	NOUN
iajs-3482	74	3	of	of	ADP
iajs-3482	74	4	𝑀.	𝑀.	PROPN
iajs-3482	74	5	"	"	PUNCT
iajs-3482	74	6	lemma	lemma	PROPN
iajs-3482	74	7	(	(	PUNCT
iajs-3482	74	8	3.2	3.2	NUM
iajs-3482	74	9	)	)	PUNCT
iajs-3482	74	10	ihjpas	ihjpa	NOUN
iajs-3482	74	11	.	.	PUNCT
iajs-3482	75	1	2025	2025	NUM
iajs-3482	75	2	,	,	PUNCT
iajs-3482	75	3	38	38	NUM
iajs-3482	75	4	(	(	PUNCT
iajs-3482	75	5	1	1	NUM
iajs-3482	75	6	)	)	PUNCT
iajs-3482	75	7	400	400	NUM
iajs-3482	75	8	let	let	VERB
iajs-3482	75	9	m	m	PRON
iajs-3482	75	10	be	be	AUX
iajs-3482	75	11	a	a	DET
iajs-3482	75	12	2	2	NUM
iajs-3482	75	13	−	−	NOUN
iajs-3482	75	14	tortion	tortion	NOUN
iajs-3482	75	15	free	free	ADJ
iajs-3482	75	16	prime	prime	ADJ
iajs-3482	75	17	semi	semi	ADJ
iajs-3482	75	18	-	-	NOUN
iajs-3482	75	19	ring	ring	NOUN
iajs-3482	75	20	.	.	PUNCT
iajs-3482	76	1	suppose	suppose	VERB
iajs-3482	76	2	that	that	SCONJ
iajs-3482	76	3	𝐹	𝐹	PROPN
iajs-3482	76	4	,	,	PUNCT
iajs-3482	76	5	𝐺	𝐺	PROPN
iajs-3482	76	6	∶	∶	NOUN
iajs-3482	76	7	𝑉𝑥𝑉	𝑉𝑥𝑉	PROPN
iajs-3482	76	8	→	→	PUNCT
iajs-3482	76	9	𝑉	𝑉	ADJ
iajs-3482	76	10	biadditive	biadditive	ADJ
iajs-3482	76	11	mappings	mapping	NOUN
iajs-3482	76	12	.	.	PUNCT
iajs-3482	77	1	if	if	SCONJ
iajs-3482	77	2	𝐹(𝑘	𝐹(𝑘	NUM
iajs-3482	77	3	,	,	PUNCT
iajs-3482	77	4	𝑞	𝑞	NOUN
iajs-3482	77	5	)	)	PUNCT
iajs-3482	77	6	𝑤	𝑤	ADP
iajs-3482	77	7	𝐺(𝑘	𝐺(𝑘	PROPN
iajs-3482	77	8	,	,	PUNCT
iajs-3482	77	9	𝑞	𝑞	NOUN
iajs-3482	77	10	)	)	PUNCT
iajs-3482	77	11	=	=	SYM
iajs-3482	77	12	0	0	NUM
iajs-3482	78	1	for	for	ADP
iajs-3482	78	2	any	any	DET
iajs-3482	78	3	𝑘	𝑘	NOUN
iajs-3482	78	4	,	,	PUNCT
iajs-3482	78	5	𝑞	𝑞	X
iajs-3482	78	6	,	,	PUNCT
iajs-3482	78	7	𝑤	𝑤	ADP
iajs-3482	78	8	∈	∈	PROPN
iajs-3482	78	9	𝑉	𝑉	PROPN
iajs-3482	78	10	,	,	PUNCT
iajs-3482	78	11	then	then	ADV
iajs-3482	78	12	𝐹(𝑘	𝐹(𝑘	NUM
iajs-3482	78	13	,	,	PUNCT
iajs-3482	78	14	𝑞	𝑞	NOUN
iajs-3482	78	15	)	)	PUNCT
iajs-3482	78	16	𝑤	𝑤	ADP
iajs-3482	78	17	𝐺(𝑢	𝐺(𝑢	PROPN
iajs-3482	78	18	,	,	PUNCT
iajs-3482	78	19	𝑣	𝑣	NOUN
iajs-3482	78	20	)	)	PUNCT
iajs-3482	78	21	=	=	SYM
iajs-3482	78	22	0	0	NUM
iajs-3482	78	23	for	for	ADP
iajs-3482	78	24	any	any	DET
iajs-3482	78	25	𝑘	𝑘	NOUN
iajs-3482	78	26	,	,	PUNCT
iajs-3482	78	27	𝑞	𝑞	X
iajs-3482	78	28	,	,	PUNCT
iajs-3482	78	29	𝑢	𝑢	PROPN
iajs-3482	78	30	,	,	PUNCT
iajs-3482	78	31	𝑣	𝑣	NOUN
iajs-3482	78	32	,	,	PUNCT
iajs-3482	78	33	𝑤	𝑤	ADP
iajs-3482	78	34	∈	∈	NOUN
iajs-3482	78	35	𝑉.	𝑉.	NOUN
iajs-3482	78	36	proof	proof	NOUN
iajs-3482	78	37	:	:	PUNCT
iajs-3482	78	38	𝐹(𝑘	𝐹(𝑘	NUM
iajs-3482	78	39	,	,	PUNCT
iajs-3482	78	40	𝑞	𝑞	NOUN
iajs-3482	78	41	)	)	PUNCT
iajs-3482	78	42	𝑤	𝑤	ADP
iajs-3482	78	43	𝐺(𝑘	𝐺(𝑘	PROPN
iajs-3482	78	44	,	,	PUNCT
iajs-3482	78	45	𝑞	𝑞	NOUN
iajs-3482	78	46	)	)	PUNCT
iajs-3482	78	47	=	=	SYM
iajs-3482	78	48	0	0	NUM
iajs-3482	78	49	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	78	50	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	78	51	𝑘	𝑘	PROPN
iajs-3482	78	52	,	,	PUNCT
iajs-3482	78	53	𝑞	𝑞	X
iajs-3482	78	54	,	,	PUNCT
iajs-3482	78	55	𝑤	𝑤	ADP
iajs-3482	78	56	∈	∈	PROPN
iajs-3482	78	57	𝑉	𝑉	PROPN
iajs-3482	78	58	(	(	PUNCT
iajs-3482	78	59	*	*	NOUN
iajs-3482	78	60	)	)	PUNCT
iajs-3482	78	61	replace	replace	VERB
iajs-3482	78	62	𝑘	𝑘	NOUN
iajs-3482	78	63	with	with	ADP
iajs-3482	78	64	𝑘	𝑘	PRON
iajs-3482	79	1	+	+	CCONJ
iajs-3482	79	2	𝑢	𝑢	X
iajs-3482	79	3	,	,	PUNCT
iajs-3482	79	4	we	we	PRON
iajs-3482	79	5	have	have	VERB
iajs-3482	79	6	𝐹(𝑘	𝐹(𝑘	PRON
iajs-3482	79	7	+	+	NUM
iajs-3482	79	8	𝑢	𝑢	PROPN
iajs-3482	79	9	,	,	PUNCT
iajs-3482	79	10	𝑞	𝑞	NOUN
iajs-3482	79	11	)	)	PUNCT
iajs-3482	79	12	𝑤	𝑤	ADP
iajs-3482	79	13	𝐺(𝑘	𝐺(𝑘	X
iajs-3482	79	14	+	+	CCONJ
iajs-3482	79	15	𝑢	𝑢	PROPN
iajs-3482	79	16	,	,	PUNCT
iajs-3482	79	17	𝑞	𝑞	X
iajs-3482	79	18	)	)	PUNCT
iajs-3482	79	19	=	=	SYM
iajs-3482	79	20	0	0	NUM
iajs-3482	80	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	80	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	80	3	𝑘	𝑘	PROPN
iajs-3482	80	4	,	,	PUNCT
iajs-3482	80	5	𝑞	𝑞	PROPN
iajs-3482	80	6	,	,	PUNCT
iajs-3482	80	7	𝑤	𝑤	PROPN
iajs-3482	80	8	,	,	PUNCT
iajs-3482	80	9	𝑢	𝑢	PROPN
iajs-3482	80	10	∈	∈	PROPN
iajs-3482	80	11	𝑉	𝑉	PROPN
iajs-3482	80	12	by	by	ADP
iajs-3482	80	13	using	use	VERB
iajs-3482	80	14	the	the	DET
iajs-3482	80	15	additive	additive	NOUN
iajs-3482	80	16	of	of	ADP
iajs-3482	80	17	f	f	PROPN
iajs-3482	80	18	and	and	CCONJ
iajs-3482	80	19	g	g	PROPN
iajs-3482	80	20	𝐹(𝑘	𝐹(𝑘	PROPN
iajs-3482	80	21	,	,	PUNCT
iajs-3482	80	22	𝑞	𝑞	NOUN
iajs-3482	80	23	)	)	PUNCT
iajs-3482	80	24	𝑤	𝑤	ADP
iajs-3482	80	25	𝐺(𝑢	𝐺(𝑢	PROPN
iajs-3482	80	26	,	,	PUNCT
iajs-3482	80	27	𝑞	𝑞	NOUN
iajs-3482	80	28	)	)	PUNCT
iajs-3482	80	29	=	=	SYM
iajs-3482	80	30	𝐹(𝑢	𝐹(𝑢	PROPN
iajs-3482	80	31	,	,	PUNCT
iajs-3482	80	32	𝑞)′	𝑞)′	VERB
iajs-3482	80	33	𝑤	𝑤	ADP
iajs-3482	80	34	𝐺(𝑘	𝐺(𝑘	PROPN
iajs-3482	80	35	,	,	PUNCT
iajs-3482	80	36	𝑞	𝑞	NOUN
iajs-3482	80	37	)	)	PUNCT
iajs-3482	80	38	replace	replace	NOUN
iajs-3482	80	39	w	w	NOUN
iajs-3482	80	40	by	by	ADP
iajs-3482	80	41	24	24	NUM
iajs-3482	80	42	𝐹(𝑘	𝐹(𝑘	PROPN
iajs-3482	80	43	,	,	PUNCT
iajs-3482	80	44	𝑞	𝑞	NOUN
iajs-3482	80	45	)	)	PUNCT
iajs-3482	80	46	𝑧	𝑧	PRON
iajs-3482	80	47	𝐺(𝑢	𝐺(𝑢	NUM
iajs-3482	80	48	,	,	PUNCT
iajs-3482	80	49	𝑞	𝑞	NOUN
iajs-3482	80	50	)	)	PUNCT
iajs-3482	80	51	(	(	PUNCT
iajs-3482	80	52	𝐹(𝑘	𝐹(𝑘	NOUN
iajs-3482	80	53	,	,	PUNCT
iajs-3482	80	54	𝑞)𝑤	𝑞)𝑤	ADJ
iajs-3482	80	55	24𝐺(𝑢	24𝐺(𝑢	PROPN
iajs-3482	80	56	,	,	PUNCT
iajs-3482	80	57	𝑞	𝑞	NOUN
iajs-3482	80	58	)	)	PUNCT
iajs-3482	80	59	)	)	PUNCT
iajs-3482	81	1	𝑧	𝑧	PRON
iajs-3482	81	2	𝐹(𝑘	𝐹(𝑘	PROPN
iajs-3482	81	3	,	,	PUNCT
iajs-3482	81	4	𝑞	𝑞	NOUN
iajs-3482	81	5	)	)	PUNCT
iajs-3482	81	6	𝑤	𝑤	ADP
iajs-3482	81	7	𝐺(𝑢	𝐺(𝑢	PROPN
iajs-3482	81	8	,	,	PUNCT
iajs-3482	81	9	𝑞	𝑞	NOUN
iajs-3482	81	10	)	)	PUNCT
iajs-3482	81	11	=	=	SYM
iajs-3482	82	1	𝐹(𝑢	𝐹(𝑢	PROPN
iajs-3482	82	2	,	,	PUNCT
iajs-3482	82	3	𝑞)′	𝑞)′	VERB
iajs-3482	82	4	𝑤	𝑤	ADP
iajs-3482	82	5	24𝐺(𝑢	24𝐺(𝑢	PROPN
iajs-3482	82	6	,	,	PUNCT
iajs-3482	82	7	𝑞	𝑞	NOUN
iajs-3482	82	8	)	)	PUNCT
iajs-3482	82	9	𝑧	𝑧	PRON
iajs-3482	82	10	𝐹(𝑘	𝐹(𝑘	PROPN
iajs-3482	82	11	,	,	PUNCT
iajs-3482	82	12	𝑞	𝑞	NOUN
iajs-3482	82	13	)	)	PUNCT
iajs-3482	82	14	𝑤	𝑤	ADP
iajs-3482	82	15	𝐺(𝑘	𝐺(𝑘	PROPN
iajs-3482	82	16	,	,	PUNCT
iajs-3482	82	17	𝑞	𝑞	NOUN
iajs-3482	82	18	)	)	PUNCT
iajs-3482	82	19	=	=	SYM
iajs-3482	82	20	0	0	NUM
iajs-3482	82	21	by	by	ADP
iajs-3482	82	22	(	(	PUNCT
iajs-3482	82	23	*	*	NOUN
iajs-3482	82	24	)	)	PUNCT
iajs-3482	82	25	,	,	PUNCT
iajs-3482	82	26	we	we	PRON
iajs-3482	82	27	get	get	VERB
iajs-3482	82	28	24𝐹(𝑘	24𝐹(𝑘	NUM
iajs-3482	82	29	,	,	PUNCT
iajs-3482	82	30	𝑞)𝑤𝐺(𝑢	𝑞)𝑤𝐺(𝑢	PROPN
iajs-3482	82	31	,	,	PUNCT
iajs-3482	82	32	𝑞)𝑧𝐹(𝑘	𝑞)𝑧𝐹(𝑘	PROPN
iajs-3482	82	33	,	,	PUNCT
iajs-3482	82	34	𝑞)𝑤𝐺(𝑢	𝑞)𝑤𝐺(𝑢	PROPN
iajs-3482	82	35	,	,	PUNCT
iajs-3482	82	36	𝑞	𝑞	PROPN
iajs-3482	82	37	)	)	PUNCT
iajs-3482	82	38	=	=	SYM
iajs-3482	82	39	0	0	NUM
iajs-3482	83	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	83	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	83	3	𝑘	𝑘	PROPN
iajs-3482	83	4	,	,	PUNCT
iajs-3482	83	5	𝑞	𝑞	PROPN
iajs-3482	83	6	,	,	PUNCT
iajs-3482	83	7	𝑢	𝑢	PROPN
iajs-3482	83	8	,	,	PUNCT
iajs-3482	83	9	𝑧	𝑧	PRON
iajs-3482	83	10	∈	∈	PROPN
iajs-3482	83	11	𝑉	𝑉	PROPN
iajs-3482	83	12	(	(	PUNCT
iajs-3482	83	13	*	*	PUNCT
iajs-3482	83	14	*	*	PUNCT
iajs-3482	83	15	)	)	PUNCT
iajs-3482	83	16	if	if	SCONJ
iajs-3482	83	17	𝑉	𝑉	PROPN
iajs-3482	83	18	⊄	⊄	NOUN
iajs-3482	83	19	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	83	20	)	)	PUNCT
iajs-3482	83	21	,	,	PUNCT
iajs-3482	83	22	by	by	ADP
iajs-3482	83	23	lemma(3.1	lemma(3.1	NOUN
iajs-3482	83	24	)	)	PUNCT
iajs-3482	83	25	,	,	PUNCT
iajs-3482	83	26	we	we	PRON
iajs-3482	83	27	get	get	VERB
iajs-3482	83	28	”	"	PUNCT
iajs-3482	83	29	𝐹(𝑘	𝐹(𝑘	PROPN
iajs-3482	83	30	,	,	PUNCT
iajs-3482	83	31	𝑞	𝑞	NOUN
iajs-3482	83	32	)	)	PUNCT
iajs-3482	83	33	𝑤	𝑤	ADP
iajs-3482	83	34	𝐺(𝑢	𝐺(𝑢	PROPN
iajs-3482	83	35	,	,	PUNCT
iajs-3482	83	36	𝑞	𝑞	X
iajs-3482	83	37	)	)	PUNCT
iajs-3482	83	38	=	=	SYM
iajs-3482	83	39	0	0	NUM
iajs-3482	84	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	84	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	84	3	𝑘	𝑘	PROPN
iajs-3482	84	4	,	,	PUNCT
iajs-3482	84	5	𝑞	𝑞	X
iajs-3482	84	6	,	,	PUNCT
iajs-3482	84	7	𝑢	𝑢	PROPN
iajs-3482	84	8	,	,	PUNCT
iajs-3482	84	9	𝑤	𝑤	ADP
iajs-3482	84	10	∈	∈	PROPN
iajs-3482	84	11	𝑉	𝑉	PROPN
iajs-3482	84	12	”	"	PUNCT
iajs-3482	84	13	if	if	SCONJ
iajs-3482	84	14	𝑉	𝑉	PROPN
iajs-3482	84	15	⊂	⊂	PROPN
iajs-3482	84	16	𝑍(𝑀	𝑍(𝑀	PROPN
iajs-3482	84	17	)	)	PUNCT
iajs-3482	84	18	,	,	PUNCT
iajs-3482	84	19	multiply	multiply	VERB
iajs-3482	84	20	the	the	DET
iajs-3482	84	21	relation	relation	NOUN
iajs-3482	84	22	(	(	PUNCT
iajs-3482	84	23	*	*	PUNCT
iajs-3482	84	24	*	*	PUNCT
iajs-3482	84	25	)	)	PUNCT
iajs-3482	84	26	from	from	ADP
iajs-3482	84	27	the	the	DET
iajs-3482	84	28	right	right	NOUN
iajs-3482	84	29	by	by	ADP
iajs-3482	84	30	𝑧𝑡	𝑧𝑡	NOUN
iajs-3482	84	31	,	,	PUNCT
iajs-3482	84	32	where	where	SCONJ
iajs-3482	84	33	𝑡	𝑡	PROPN
iajs-3482	84	34	∈	∈	PROPN
iajs-3482	84	35	𝑀	𝑀	PROPN
iajs-3482	84	36	,	,	PUNCT
iajs-3482	84	37	we	we	PRON
iajs-3482	84	38	get	get	VERB
iajs-3482	84	39	24𝐹(𝑘	24𝐹(𝑘	NUM
iajs-3482	84	40	,	,	PUNCT
iajs-3482	84	41	𝑞)𝑤	𝑞)𝑤	ADJ
iajs-3482	84	42	𝐺(𝑢	𝐺(𝑢	ADJ
iajs-3482	84	43	,	,	PUNCT
iajs-3482	84	44	𝑞	𝑞	NOUN
iajs-3482	84	45	)	)	PUNCT
iajs-3482	84	46	𝑧	𝑧	PROPN
iajs-3482	84	47	𝑡	𝑡	PROPN
iajs-3482	84	48	𝐹(𝑘	𝐹(𝑘	NOUN
iajs-3482	84	49	,	,	PUNCT
iajs-3482	84	50	𝑞)𝑤𝐺(𝑢	𝑞)𝑤𝐺(𝑢	PROPN
iajs-3482	84	51	,	,	PUNCT
iajs-3482	84	52	𝑞)𝑧	𝑞)𝑧	X
iajs-3482	84	53	=	=	SYM
iajs-3482	84	54	0	0	NUM
iajs-3482	84	55	,	,	PUNCT
iajs-3482	84	56	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	84	57	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	84	58	𝑘	𝑘	PROPN
iajs-3482	84	59	,	,	PUNCT
iajs-3482	84	60	𝑞	𝑞	PROPN
iajs-3482	84	61	,	,	PUNCT
iajs-3482	84	62	𝑢	𝑢	PROPN
iajs-3482	84	63	,	,	PUNCT
iajs-3482	84	64	𝑧	𝑧	VERB
iajs-3482	84	65	,	,	PUNCT
iajs-3482	84	66	𝑤	𝑤	ADP
iajs-3482	84	67	∈	∈	PROPN
iajs-3482	84	68	𝑉	𝑉	PROPN
iajs-3482	84	69	,	,	PUNCT
iajs-3482	84	70	𝑡	𝑡	PROPN
iajs-3482	84	71	∈	∈	PROPN
iajs-3482	84	72	𝑀	𝑀	PROPN
iajs-3482	84	73	since	since	SCONJ
iajs-3482	84	74	m	m	PROPN
iajs-3482	84	75	is	be	AUX
iajs-3482	84	76	2	2	NUM
iajs-3482	84	77	−	−	NOUN
iajs-3482	84	78	tortion	tortion	NOUN
iajs-3482	84	79	free	free	ADJ
iajs-3482	84	80	prime	prime	ADJ
iajs-3482	84	81	semi	semi	ADJ
iajs-3482	84	82	-	-	NOUN
iajs-3482	84	83	ring	ring	NOUN
iajs-3482	84	84	,	,	PUNCT
iajs-3482	84	85	we	we	PRON
iajs-3482	84	86	have	have	VERB
iajs-3482	84	87	𝐹(𝑘	𝐹(𝑘	NUM
iajs-3482	84	88	,	,	PUNCT
iajs-3482	84	89	𝑞	𝑞	NOUN
iajs-3482	84	90	)	)	PUNCT
iajs-3482	84	91	𝑤	𝑤	ADP
iajs-3482	84	92	𝐺(𝑢	𝐺(𝑢	PROPN
iajs-3482	84	93	,	,	PUNCT
iajs-3482	84	94	𝑞	𝑞	NOUN
iajs-3482	84	95	)	)	PUNCT
iajs-3482	84	96	𝑧	𝑧	PROPN
iajs-3482	84	97	=	=	SYM
iajs-3482	84	98	0	0	NUM
iajs-3482	84	99	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	84	100	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	84	101	𝑘	𝑘	PROPN
iajs-3482	84	102	,	,	PUNCT
iajs-3482	84	103	𝑞	𝑞	PROPN
iajs-3482	84	104	,	,	PUNCT
iajs-3482	84	105	𝑢	𝑢	PROPN
iajs-3482	84	106	,	,	PUNCT
iajs-3482	84	107	𝑧	𝑧	VERB
iajs-3482	84	108	,	,	PUNCT
iajs-3482	84	109	𝑤	𝑤	ADP
iajs-3482	84	110	∈	∈	PROPN
iajs-3482	84	111	𝑉	𝑉	PROPN
iajs-3482	84	112	if	if	SCONJ
iajs-3482	84	113	we	we	PRON
iajs-3482	84	114	multiply	multiply	VERB
iajs-3482	84	115	the	the	DET
iajs-3482	84	116	relation	relation	NOUN
iajs-3482	84	117	by	by	ADP
iajs-3482	84	118	t	t	PROPN
iajs-3482	84	119	an	an	DET
iajs-3482	84	120	element	element	NOUN
iajs-3482	84	121	of	of	ADP
iajs-3482	84	122	m	m	PROPN
iajs-3482	84	123	,	,	PUNCT
iajs-3482	84	124	which	which	PRON
iajs-3482	84	125	is	be	AUX
iajs-3482	84	126	prime	prime	ADJ
iajs-3482	84	127	,	,	PUNCT
iajs-3482	84	128	and	and	CCONJ
iajs-3482	84	129	do	do	VERB
iajs-3482	84	130	a	a	DET
iajs-3482	84	131	right	right	ADJ
iajs-3482	84	132	multiplication	multiplication	NOUN
iajs-3482	84	133	,	,	PUNCT
iajs-3482	84	134	the	the	DET
iajs-3482	84	135	result	result	NOUN
iajs-3482	84	136	is	be	AUX
iajs-3482	84	137	𝐹(𝑘	𝐹(𝑘	NUM
iajs-3482	84	138	,	,	PUNCT
iajs-3482	84	139	𝑞	𝑞	NOUN
iajs-3482	84	140	)	)	PUNCT
iajs-3482	84	141	𝑤	𝑤	ADP
iajs-3482	84	142	𝐺(𝑢	𝐺(𝑢	PROPN
iajs-3482	84	143	,	,	PUNCT
iajs-3482	84	144	𝑞	𝑞	X
iajs-3482	84	145	)	)	PUNCT
iajs-3482	84	146	=	=	SYM
iajs-3482	84	147	0	0	NUM
iajs-3482	84	148	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	84	149	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	84	150	𝑘	𝑘	PROPN
iajs-3482	84	151	,	,	PUNCT
iajs-3482	84	152	𝑞	𝑞	X
iajs-3482	84	153	,	,	PUNCT
iajs-3482	84	154	𝑢	𝑢	PROPN
iajs-3482	84	155	,	,	PUNCT
iajs-3482	84	156	𝑤	𝑤	ADP
iajs-3482	84	157	∈	∈	PROPN
iajs-3482	84	158	𝑉	𝑉	PROPN
iajs-3482	84	159	we	we	PRON
iajs-3482	84	160	can	can	AUX
iajs-3482	84	161	acquire	acquire	VERB
iajs-3482	84	162	the	the	DET
iajs-3482	84	163	lemma	lemma	PROPN
iajs-3482	84	164	's	's	PART
iajs-3482	84	165	claim	claim	NOUN
iajs-3482	84	166	by	by	ADP
iajs-3482	84	167	exchanging	exchange	VERB
iajs-3482	84	168	𝑞	𝑞	PRON
iajs-3482	84	169	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	84	170	𝑞	𝑞	PROPN
iajs-3482	84	171	+	+	X
iajs-3482	84	172	𝑣	𝑣	X
iajs-3482	84	173	,	,	PUNCT
iajs-3482	84	174	in	in	ADP
iajs-3482	84	175	a	a	DET
iajs-3482	84	176	way	way	NOUN
iajs-3482	84	177	analogous	analogous	ADJ
iajs-3482	84	178	to	to	ADP
iajs-3482	84	179	the	the	DET
iajs-3482	84	180	one	one	NOUN
iajs-3482	84	181	used	use	VERB
iajs-3482	84	182	above	above	ADV
iajs-3482	84	183	.	.	PUNCT
iajs-3482	85	1	theorem	theorem	NOUN
iajs-3482	85	2	(	(	PUNCT
iajs-3482	85	3	3.3	3.3	NUM
iajs-3482	85	4	)	)	PUNCT
iajs-3482	85	5	let	let	VERB
iajs-3482	85	6	𝑀	𝑀	PRON
iajs-3482	85	7	be	be	AUX
iajs-3482	85	8	2	2	NUM
iajs-3482	85	9	−	−	NOUN
iajs-3482	85	10	tortion	tortion	NOUN
iajs-3482	85	11	free	free	ADJ
iajs-3482	85	12	prime	prime	ADJ
iajs-3482	85	13	semi	semi	ADJ
iajs-3482	85	14	-	-	NOUN
iajs-3482	85	15	ring	ring	NOUN
iajs-3482	85	16	.	.	PUNCT
iajs-3482	86	1	if	if	SCONJ
iajs-3482	86	2	𝑅	𝑅	PROPN
iajs-3482	86	3	is	be	AUX
iajs-3482	86	4	left	leave	VERB
iajs-3482	86	5	(	(	PUNCT
iajs-3482	86	6	right	right	ADJ
iajs-3482	86	7	)	)	PUNCT
iajs-3482	86	8	jordan	jordan	PROPN
iajs-3482	86	9	𝛼	𝛼	PROPN
iajs-3482	86	10	−	−	PROPN
iajs-3482	86	11	centralizer	centralizer	NOUN
iajs-3482	86	12	on”v	on”v	PROPN
iajs-3482	86	13	,	,	PUNCT
iajs-3482	86	14	then	then	ADV
iajs-3482	86	15	r	r	NOUN
iajs-3482	86	16	is	be	AUX
iajs-3482	86	17	a	a	DET
iajs-3482	86	18	left	left	ADJ
iajs-3482	86	19	(	(	PUNCT
iajs-3482	86	20	right	right	ADJ
iajs-3482	86	21	)	)	PUNCT
iajs-3482	86	22	𝛼	𝛼	NOUN
iajs-3482	86	23	−	−	NOUN
iajs-3482	86	24	centralizer	centralizer	NOUN
iajs-3482	86	25	on	on	ADP
iajs-3482	86	26	𝑉.	𝑉.	NOUN
iajs-3482	86	27	”	"	PUNCT
iajs-3482	86	28	proof	proof	NOUN
iajs-3482	86	29	:	:	PUNCT
iajs-3482	86	30	𝑅(𝑘2	𝑅(𝑘2	VERB
iajs-3482	86	31	)	)	PUNCT
iajs-3482	86	32	+	+	NUM
iajs-3482	86	33	𝑅(𝑘)′𝛼(𝑘	𝑅(𝑘)′𝛼(𝑘	NOUN
iajs-3482	86	34	)	)	PUNCT
iajs-3482	87	1	=	=	PUNCT
iajs-3482	87	2	0	0	NUM
iajs-3482	88	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	88	2	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	88	3	𝑘	𝑘	PRON
iajs-3482	88	4	∈	∈	PROPN
iajs-3482	88	5	𝑉	𝑉	PROPN
iajs-3482	88	6	(	(	PUNCT
iajs-3482	88	7	1	1	NUM
iajs-3482	88	8	)	)	PUNCT
iajs-3482	88	9	we	we	PRON
iajs-3482	88	10	replace	replace	VERB
iajs-3482	88	11	𝑘	𝑘	PRON
iajs-3482	88	12	by	by	ADP
iajs-3482	88	13	𝑘	𝑘	PRON
iajs-3482	89	1	+	+	NOUN
iajs-3482	89	2	𝑞	𝑞	X
iajs-3482	89	3	when	when	SCONJ
iajs-3482	89	4	𝑘	𝑘	X
iajs-3482	89	5	,	,	PUNCT
iajs-3482	89	6	𝑞	𝑞	X
iajs-3482	89	7	in	in	ADP
iajs-3482	89	8	𝑈	𝑈	PROPN
iajs-3482	89	9	,	,	PUNCT
iajs-3482	89	10	we	we	PRON
iajs-3482	89	11	get	get	VERB
iajs-3482	89	12	𝑅((𝑘	𝑅((𝑘	PROPN
iajs-3482	89	13	+	+	CCONJ
iajs-3482	89	14	𝑞)2	𝑞)2	NOUN
iajs-3482	89	15	)	)	PUNCT
iajs-3482	89	16	=	=	PUNCT
iajs-3482	90	1	𝑅(𝑘	𝑅(𝑘	DET
iajs-3482	90	2	+	+	NUM
iajs-3482	90	3	𝑞	𝑞	X
iajs-3482	90	4	)	)	PUNCT
iajs-3482	90	5	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	90	6	+	+	NUM
iajs-3482	90	7	𝑞	𝑞	NOUN
iajs-3482	90	8	)	)	PUNCT
iajs-3482	90	9	𝑅(𝑘2	𝑅(𝑘2	PROPN
iajs-3482	90	10	+	+	CCONJ
iajs-3482	90	11	𝑘𝑞	𝑘𝑞	NOUN
iajs-3482	90	12	+	+	CCONJ
iajs-3482	90	13	𝑞𝑘	𝑞𝑘	ADP
iajs-3482	90	14	+	+	ADJ
iajs-3482	90	15	𝑞2	𝑞2	NOUN
iajs-3482	90	16	)	)	PUNCT
iajs-3482	90	17	=	=	SYM
iajs-3482	90	18	𝑅(𝑘2	𝑅(𝑘2	PROPN
iajs-3482	90	19	)	)	PUNCT
iajs-3482	90	20	+	+	CCONJ
iajs-3482	90	21	𝑅(𝑘𝑞	𝑅(𝑘𝑞	X
iajs-3482	90	22	+	+	CCONJ
iajs-3482	90	23	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	90	24	)	)	PUNCT
iajs-3482	90	25	+	+	CCONJ
iajs-3482	90	26	𝑅(𝑞2	𝑅(𝑞2	NOUN
iajs-3482	90	27	)	)	PUNCT
iajs-3482	90	28	=	=	SYM
iajs-3482	90	29	𝑅(𝑘)𝛼(𝑘	𝑅(𝑘)𝛼(𝑘	NOUN
iajs-3482	90	30	)	)	PUNCT
iajs-3482	91	1	+	+	CCONJ
iajs-3482	91	2	𝑅(𝑘𝑞	𝑅(𝑘𝑞	X
iajs-3482	91	3	+	+	CCONJ
iajs-3482	91	4	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	91	5	)	)	PUNCT
iajs-3482	91	6	+	+	NOUN
iajs-3482	91	7	𝑅(𝑞)𝛼(𝑞	𝑅(𝑞)𝛼(𝑞	NOUN
iajs-3482	91	8	)	)	PUNCT
iajs-3482	91	9	“	"	PUNCT
iajs-3482	91	10	𝑅(𝑘	𝑅(𝑘	X
iajs-3482	91	11	+	+	X
iajs-3482	91	12	𝑞)𝛼(𝑘	𝑞)𝛼(𝑘	ADJ
iajs-3482	91	13	+	+	CCONJ
iajs-3482	91	14	𝑞	𝑞	X
iajs-3482	91	15	)	)	PUNCT
iajs-3482	91	16	=	=	SYM
iajs-3482	91	17	𝑅(𝑘)𝛼(𝑘	𝑅(𝑘)𝛼(𝑘	NOUN
iajs-3482	91	18	)	)	PUNCT
iajs-3482	92	1	+	+	CCONJ
iajs-3482	92	2	𝑅(𝑘)𝛼(𝑞	𝑅(𝑘)𝛼(𝑞	NOUN
iajs-3482	92	3	)	)	PUNCT
iajs-3482	92	4	+	+	SYM
iajs-3482	92	5	𝑅(𝑞)𝛼(𝑘	𝑅(𝑞)𝛼(𝑘	X
iajs-3482	92	6	)	)	PUNCT
iajs-3482	92	7	+	+	NUM
iajs-3482	92	8	𝑅(𝑞)𝛼(𝑞	𝑅(𝑞)𝛼(𝑞	NOUN
iajs-3482	92	9	)	)	PUNCT
iajs-3482	92	10	”	"	PUNCT
iajs-3482	92	11	we	we	PRON
iajs-3482	92	12	get	get	VERB
iajs-3482	92	13	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	92	14	+	+	CCONJ
iajs-3482	92	15	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	92	16	)	)	PUNCT
iajs-3482	92	17	+	+	CCONJ
iajs-3482	92	18	𝑅(𝑘)𝛼(𝑞)′	𝑅(𝑘)𝛼(𝑞)′	PROPN
iajs-3482	92	19	+	+	CCONJ
iajs-3482	92	20	𝑅(𝑞)𝛼(𝑘	𝑅(𝑞)𝛼(𝑘	ADJ
iajs-3482	92	21	)	)	PUNCT
iajs-3482	92	22	′	′	NUM
iajs-3482	93	1	=	=	SYM
iajs-3482	93	2	0	0	NUM
iajs-3482	93	3	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	93	4	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	93	5	𝑘	𝑘	PROPN
iajs-3482	93	6	,	,	PUNCT
iajs-3482	93	7	𝑞	𝑞	PROPN
iajs-3482	93	8	∈	∈	PROPN
iajs-3482	93	9	𝑉	𝑉	PROPN
iajs-3482	93	10	(	(	PUNCT
iajs-3482	93	11	2	2	NUM
iajs-3482	93	12	)	)	PUNCT
iajs-3482	93	13	by	by	ADP
iajs-3482	93	14	replacing	replace	VERB
iajs-3482	93	15	𝑞	𝑞	PRON
iajs-3482	93	16	with	with	ADP
iajs-3482	93	17	2(𝑘𝑞	2(𝑘𝑞	NUM
iajs-3482	93	18	+	+	SYM
iajs-3482	93	19	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	93	20	)	)	PUNCT
iajs-3482	93	21	and	and	CCONJ
iajs-3482	93	22	using	use	VERB
iajs-3482	93	23	(	(	PUNCT
iajs-3482	93	24	2	2	NUM
iajs-3482	93	25	)	)	PUNCT
iajs-3482	93	26	,	,	PUNCT
iajs-3482	93	27	we	we	PRON
iajs-3482	93	28	get	get	VERB
iajs-3482	93	29	2𝑅(𝑘(𝑘𝑞	2𝑅(𝑘(𝑘𝑞	NOUN
iajs-3482	93	30	+	+	CCONJ
iajs-3482	93	31	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	93	32	)	)	PUNCT
iajs-3482	93	33	+	+	CCONJ
iajs-3482	93	34	(	(	PUNCT
iajs-3482	93	35	𝑘𝑞	𝑘𝑞	X
iajs-3482	93	36	+	+	ADJ
iajs-3482	93	37	𝑞𝑘)𝑘	𝑞𝑘)𝑘	PROPN
iajs-3482	93	38	)	)	PUNCT
iajs-3482	94	1	+	+	NOUN
iajs-3482	95	1	2𝑅(𝑘)𝛼(𝑘𝑞	2𝑅(𝑘)𝛼(𝑘𝑞	NUM
iajs-3482	95	2	)	)	PUNCT
iajs-3482	95	3	′	′	NUM
iajs-3482	96	1	+	+	CCONJ
iajs-3482	97	1	2𝑅(𝑘)𝛼(𝑞𝑘)′	2𝑅(𝑘)𝛼(𝑞𝑘)′	NUM
iajs-3482	97	2	+	+	CCONJ
iajs-3482	97	3	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	97	4	+	+	CCONJ
iajs-3482	97	5	𝑞𝑘)𝛼(𝑘)′	𝑞𝑘)𝛼(𝑘)′	NOUN
iajs-3482	97	6	=	=	SYM
iajs-3482	97	7	0	0	NUM
iajs-3482	97	8	2𝑅(𝑘(𝑘𝑞	2𝑅(𝑘(𝑘𝑞	NUM
iajs-3482	97	9	+	+	CCONJ
iajs-3482	97	10	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	97	11	)	)	PUNCT
iajs-3482	97	12	+	+	CCONJ
iajs-3482	97	13	(	(	PUNCT
iajs-3482	97	14	𝑘𝑞	𝑘𝑞	X
iajs-3482	97	15	+	+	ADJ
iajs-3482	97	16	𝑞𝑘)𝑘	𝑞𝑘)𝑘	NUM
iajs-3482	97	17	)	)	PUNCT
iajs-3482	97	18	=	=	SYM
iajs-3482	98	1	2𝑅(𝑘)𝛼(𝑘𝑞	2𝑅(𝑘)𝛼(𝑘𝑞	NUM
iajs-3482	98	2	)	)	PUNCT
iajs-3482	98	3	+	+	CCONJ
iajs-3482	98	4	2𝑅(𝑘)𝛼(𝑞𝑘	2𝑅(𝑘)𝛼(𝑞𝑘	NUM
iajs-3482	98	5	)	)	PUNCT
iajs-3482	99	1	+	+	CCONJ
iajs-3482	99	2	2	2	NUM
iajs-3482	99	3	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	99	4	+	+	CCONJ
iajs-3482	99	5	𝑞𝑘)𝛼(𝑘	𝑞𝑘)𝛼(𝑘	NOUN
iajs-3482	99	6	)	)	PUNCT
iajs-3482	99	7	ihjpas	ihjpa	NOUN
iajs-3482	99	8	.	.	PUNCT
iajs-3482	100	1	2025	2025	NUM
iajs-3482	100	2	,	,	PUNCT
iajs-3482	100	3	38	38	NUM
iajs-3482	100	4	(	(	PUNCT
iajs-3482	100	5	1	1	NUM
iajs-3482	100	6	)	)	PUNCT
iajs-3482	100	7	401	401	NUM
iajs-3482	100	8	(	(	PUNCT
iajs-3482	100	9	3	3	NUM
iajs-3482	100	10	)	)	PUNCT
iajs-3482	100	11	this	this	PRON
iajs-3482	100	12	can	can	AUX
iajs-3482	100	13	also	also	ADV
iajs-3482	100	14	be	be	AUX
iajs-3482	100	15	computed	compute	VERB
iajs-3482	100	16	using	use	VERB
iajs-3482	100	17	an	an	DET
iajs-3482	100	18	alternate	alternate	ADJ
iajs-3482	100	19	way	way	NOUN
iajs-3482	100	20	2𝑅(𝑘2𝑞	2𝑅(𝑘2𝑞	NUM
iajs-3482	100	21	+	+	NUM
iajs-3482	100	22	𝑞𝑘2	𝑞𝑘2	NOUN
iajs-3482	100	23	)	)	PUNCT
iajs-3482	101	1	+	+	CCONJ
iajs-3482	101	2	4𝑅(𝑘𝑞𝑘	4𝑅(𝑘𝑞𝑘	NUM
iajs-3482	101	3	)	)	PUNCT
iajs-3482	102	1	+	+	CCONJ
iajs-3482	102	2	2	2	NUM
iajs-3482	102	3	𝑅(𝑘)𝛼(𝑘𝑞)′	𝑅(𝑘)𝛼(𝑘𝑞)′	PUNCT
iajs-3482	102	4	+	+	NUM
iajs-3482	102	5	2𝑅(𝑞)𝛼(𝑘2)′	2𝑅(𝑞)𝛼(𝑘2)′	NUM
iajs-3482	102	6	=	=	SYM
iajs-3482	102	7	0	0	NUM
iajs-3482	102	8	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	102	9	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	102	10	𝑘	𝑘	PROPN
iajs-3482	102	11	,	,	PUNCT
iajs-3482	102	12	𝑞	𝑞	PROPN
iajs-3482	102	13	∈	∈	PROPN
iajs-3482	102	14	𝑉	𝑉	PROPN
iajs-3482	102	15	(	(	PUNCT
iajs-3482	102	16	4	4	NUM
iajs-3482	102	17	)	)	PUNCT
iajs-3482	102	18	from	from	ADP
iajs-3482	102	19	(	(	PUNCT
iajs-3482	102	20	3	3	NUM
iajs-3482	102	21	)	)	PUNCT
iajs-3482	102	22	and	and	CCONJ
iajs-3482	102	23	(	(	PUNCT
iajs-3482	102	24	4	4	NUM
iajs-3482	102	25	)	)	PUNCT
iajs-3482	102	26	,	,	PUNCT
iajs-3482	102	27	we	we	PRON
iajs-3482	102	28	obtain	obtain	VERB
iajs-3482	102	29	𝑅(𝑘𝑞𝑘	𝑅(𝑘𝑞𝑘	NUM
iajs-3482	102	30	)	)	PUNCT
iajs-3482	103	1	+	+	CCONJ
iajs-3482	103	2	𝑅(𝑘)𝛼(𝑞𝑘)′	𝑅(𝑘)𝛼(𝑞𝑘)′	X
iajs-3482	103	3	=	=	SYM
iajs-3482	103	4	0	0	NUM
iajs-3482	103	5	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	103	6	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	103	7	𝑘	𝑘	PROPN
iajs-3482	103	8	,	,	PUNCT
iajs-3482	103	9	𝑞	𝑞	PROPN
iajs-3482	103	10	∈	∈	PROPN
iajs-3482	103	11	𝑉	𝑉	PROPN
iajs-3482	103	12	(	(	PUNCT
iajs-3482	103	13	5	5	NUM
iajs-3482	103	14	)	)	PUNCT
iajs-3482	103	15	if	if	SCONJ
iajs-3482	103	16	we	we	PRON
iajs-3482	103	17	linearize	linearize	VERB
iajs-3482	103	18	(	(	PUNCT
iajs-3482	103	19	5	5	NUM
iajs-3482	103	20	)	)	PUNCT
iajs-3482	103	21	,	,	PUNCT
iajs-3482	103	22	we	we	PRON
iajs-3482	103	23	get	get	VERB
iajs-3482	103	24	𝑅(𝑘𝑞𝑡	𝑅(𝑘𝑞𝑡	ADJ
iajs-3482	103	25	+	+	CCONJ
iajs-3482	103	26	𝑡𝑞𝑘	𝑡𝑞𝑘	ADJ
iajs-3482	103	27	)	)	PUNCT
iajs-3482	103	28	+	+	CCONJ
iajs-3482	103	29	𝑅(𝑘)𝛼(𝑞𝑡)′	𝑅(𝑘)𝛼(𝑞𝑡)′	ADJ
iajs-3482	103	30	+	+	CCONJ
iajs-3482	103	31	𝑅(𝑡)𝛼(𝑞𝑘)′	𝑅(𝑡)𝛼(𝑞𝑘)′	X
iajs-3482	103	32	=	=	SYM
iajs-3482	103	33	0	0	NUM
iajs-3482	103	34	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	103	35	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	103	36	𝑘	𝑘	PROPN
iajs-3482	103	37	,	,	PUNCT
iajs-3482	103	38	𝑞	𝑞	PROPN
iajs-3482	103	39	,	,	PUNCT
iajs-3482	103	40	𝑡	𝑡	PROPN
iajs-3482	103	41	∈	∈	PROPN
iajs-3482	103	42	𝑉	𝑉	PROPN
iajs-3482	103	43	(	(	PUNCT
iajs-3482	103	44	6	6	NUM
iajs-3482	103	45	)	)	PUNCT
iajs-3482	103	46	since	since	SCONJ
iajs-3482	103	47	v	v	NOUN
iajs-3482	103	48	is	be	AUX
iajs-3482	103	49	a	a	DET
iajs-3482	103	50	square	square	ADJ
iajs-3482	103	51	closed	close	VERB
iajs-3482	103	52	lie	lie	NOUN
iajs-3482	103	53	-	-	PUNCT
iajs-3482	103	54	ideal	ideal	ADJ
iajs-3482	103	55	,	,	PUNCT
iajs-3482	103	56	we	we	PRON
iajs-3482	103	57	have	have	VERB
iajs-3482	103	58	24(𝑘𝑞𝑡𝑞𝑘	24(𝑘𝑞𝑡𝑞𝑘	NUM
iajs-3482	103	59	+	+	NUM
iajs-3482	103	60	𝑞𝑘𝑡𝑘𝑞	𝑞𝑘𝑡𝑘𝑞	NOUN
iajs-3482	103	61	)	)	PUNCT
iajs-3482	103	62	∈	∈	PROPN
iajs-3482	103	63	𝑉.	𝑉.	NOUN
iajs-3482	103	64	now	now	ADV
iajs-3482	103	65	we	we	PRON
iajs-3482	103	66	shall	shall	AUX
iajs-3482	103	67	compute	compute	VERB
iajs-3482	103	68	𝑓	𝑓	PRON
iajs-3482	103	69	=	=	SYM
iajs-3482	103	70	24𝑅(𝑘𝑞𝑡𝑞𝑘	24𝑅(𝑘𝑞𝑡𝑞𝑘	NUM
iajs-3482	103	71	+	+	NUM
iajs-3482	103	72	𝑞𝑘𝑡𝑘𝑞	𝑞𝑘𝑡𝑘𝑞	NOUN
iajs-3482	103	73	)	)	PUNCT
iajs-3482	103	74	in	in	ADP
iajs-3482	103	75	two	two	NUM
iajs-3482	103	76	different	different	ADJ
iajs-3482	103	77	ways	way	NOUN
iajs-3482	103	78	,	,	PUNCT
iajs-3482	103	79	using	use	VERB
iajs-3482	103	80	(	(	PUNCT
iajs-3482	103	81	5	5	NUM
iajs-3482	103	82	)	)	PUNCT
iajs-3482	103	83	we	we	PRON
iajs-3482	103	84	have	have	VERB
iajs-3482	104	1	𝑓	𝑓	DET
iajs-3482	104	2	+	+	NUM
iajs-3482	104	3	24𝑅(𝑘)𝛼(𝑞𝑡𝑞𝑘)′	24𝑅(𝑘)𝛼(𝑞𝑡𝑞𝑘)′	NUM
iajs-3482	104	4	+	+	CCONJ
iajs-3482	104	5	𝑅(𝑞)𝛼(𝑘𝑡𝑘𝑞)′	𝑅(𝑞)𝛼(𝑘𝑡𝑘𝑞)′	X
iajs-3482	104	6	=	=	SYM
iajs-3482	104	7	0	0	NUM
iajs-3482	104	8	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	104	9	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	104	10	𝑘	𝑘	PROPN
iajs-3482	104	11	,	,	PUNCT
iajs-3482	104	12	𝑞	𝑞	PROPN
iajs-3482	104	13	,	,	PUNCT
iajs-3482	104	14	𝑡	𝑡	PROPN
iajs-3482	104	15	∈	∈	PROPN
iajs-3482	104	16	𝑉	𝑉	PROPN
iajs-3482	104	17	(	(	PUNCT
iajs-3482	104	18	7	7	X
iajs-3482	104	19	)	)	PUNCT
iajs-3482	104	20	using	use	VERB
iajs-3482	104	21	(	(	PUNCT
iajs-3482	104	22	6	6	NUM
iajs-3482	104	23	)	)	PUNCT
iajs-3482	104	24	we	we	PRON
iajs-3482	104	25	have	have	VERB
iajs-3482	104	26	𝑓	𝑓	DET
iajs-3482	104	27	+	+	NUM
iajs-3482	104	28	24𝑅(𝑘𝑞)𝛼(𝑡𝑞𝑘)′	24𝑅(𝑘𝑞)𝛼(𝑡𝑞𝑘)′	NUM
iajs-3482	104	29	+	+	CCONJ
iajs-3482	104	30	𝑅(𝑞𝑘)𝛼(𝑡𝑘𝑞)′	𝑅(𝑞𝑘)𝛼(𝑡𝑘𝑞)′	X
iajs-3482	104	31	=	=	SYM
iajs-3482	104	32	0	0	NUM
iajs-3482	104	33	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	104	34	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	104	35	𝑘	𝑘	PROPN
iajs-3482	104	36	,	,	PUNCT
iajs-3482	104	37	𝑞	𝑞	PROPN
iajs-3482	104	38	,	,	PUNCT
iajs-3482	104	39	𝑡	𝑡	PROPN
iajs-3482	104	40	∈	∈	PROPN
iajs-3482	104	41	𝑉	𝑉	PROPN
iajs-3482	104	42	(	(	PUNCT
iajs-3482	104	43	8)	8)	NUM
iajs-3482	104	44	comparing	compare	VERB
iajs-3482	104	45	(	(	PUNCT
iajs-3482	104	46	7	7	NUM
iajs-3482	104	47	)	)	PUNCT
iajs-3482	104	48	and	and	CCONJ
iajs-3482	104	49	(	(	PUNCT
iajs-3482	104	50	8)	8)	NUM
iajs-3482	104	51	𝑅(𝑘)𝛼(𝑞𝑡𝑞𝑘)′	𝑅(𝑘)𝛼(𝑞𝑡𝑞𝑘)′	NOUN
iajs-3482	104	52	+	+	CCONJ
iajs-3482	104	53	𝑅(𝑞)𝛼(𝑘𝑡𝑘𝑞)′	𝑅(𝑞)𝛼(𝑘𝑡𝑘𝑞)′	PROPN
iajs-3482	104	54	+	+	CCONJ
iajs-3482	104	55	𝑅(𝑘𝑞)𝛼(𝑡𝑞𝑘	𝑅(𝑘𝑞)𝛼(𝑡𝑞𝑘	PROPN
iajs-3482	104	56	)	)	PUNCT
iajs-3482	104	57	+	+	NUM
iajs-3482	104	58	𝑅(𝑞𝑘)𝛼(𝑡𝑞𝑘	𝑅(𝑞𝑘)𝛼(𝑡𝑞𝑘	NOUN
iajs-3482	104	59	)	)	PUNCT
iajs-3482	104	60	=	=	SYM
iajs-3482	104	61	0	0	NUM
iajs-3482	104	62	(	(	PUNCT
iajs-3482	104	63	𝑅(𝑘𝑞	𝑅(𝑘𝑞	PROPN
iajs-3482	104	64	)	)	PUNCT
iajs-3482	104	65	+	+	PUNCT
iajs-3482	104	66	𝑅(𝑘)𝛼(𝑞)′)𝛼(𝑡𝑞𝑘	𝑅(𝑘)𝛼(𝑞)′)𝛼(𝑡𝑞𝑘	X
iajs-3482	104	67	)	)	PUNCT
iajs-3482	104	68	+	+	CCONJ
iajs-3482	104	69	(	(	PUNCT
iajs-3482	104	70	𝑅(𝑞𝑘	𝑅(𝑞𝑘	ADV
iajs-3482	104	71	)	)	PUNCT
iajs-3482	104	72	+	+	CCONJ
iajs-3482	104	73	𝑅(𝑞)𝛼(𝑘)′	𝑅(𝑞)𝛼(𝑘)′	ADJ
iajs-3482	104	74	)	)	PUNCT
iajs-3482	104	75	𝛼(𝑡𝑘𝑞	𝛼(𝑡𝑘𝑞	PROPN
iajs-3482	104	76	)	)	PUNCT
iajs-3482	104	77	=	=	SYM
iajs-3482	104	78	0	0	NUM
iajs-3482	104	79	introducing	introduce	VERB
iajs-3482	104	80	a	a	DET
iajs-3482	104	81	additive	additive	ADJ
iajs-3482	104	82	mapping	mapping	NOUN
iajs-3482	104	83	,	,	PUNCT
iajs-3482	104	84	𝐺(𝑘	𝐺(𝑘	PROPN
iajs-3482	104	85	,	,	PUNCT
iajs-3482	104	86	𝑞	𝑞	NOUN
iajs-3482	104	87	)	)	PUNCT
iajs-3482	104	88	=	=	SYM
iajs-3482	104	89	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	104	90	)	)	PUNCT
iajs-3482	105	1	+	+	CCONJ
iajs-3482	105	2	𝑅(𝑘)𝛼(𝑞)′	𝑅(𝑘)𝛼(𝑞)′	PROPN
iajs-3482	105	3	,	,	PUNCT
iajs-3482	105	4	we	we	PRON
iajs-3482	105	5	arrive	arrive	VERB
iajs-3482	105	6	at	at	ADP
iajs-3482	105	7	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	105	8	,	,	PUNCT
iajs-3482	105	9	𝑞)𝛼(𝑡𝑞𝑘	𝑞)𝛼(𝑡𝑞𝑘	NOUN
iajs-3482	105	10	)	)	PUNCT
iajs-3482	106	1	+	+	CCONJ
iajs-3482	106	2	𝐺(𝑞	𝐺(𝑞	ADV
iajs-3482	106	3	,	,	PUNCT
iajs-3482	106	4	𝑘)(𝑡𝑘𝑞	𝑘)(𝑡𝑘𝑞	NOUN
iajs-3482	106	5	)	)	PUNCT
iajs-3482	107	1	=	=	SYM
iajs-3482	107	2	0	0	NUM
iajs-3482	107	3	by	by	ADP
iajs-3482	107	4	lemma	lemma	PROPN
iajs-3482	107	5	(	(	PUNCT
iajs-3482	107	6	2.5	2.5	NUM
iajs-3482	107	7	)	)	PUNCT
iajs-3482	107	8	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	107	9	,	,	PUNCT
iajs-3482	107	10	𝑞)𝛼(𝑡𝑞𝑘	𝑞)𝛼(𝑡𝑞𝑘	NUM
iajs-3482	107	11	)	)	PUNCT
iajs-3482	107	12	=	=	SYM
iajs-3482	108	1	𝐺(𝑞	𝐺(𝑞	ADV
iajs-3482	108	2	,	,	PUNCT
iajs-3482	108	3	𝑘)′𝛼(𝑡𝑘𝑞	𝑘)′𝛼(𝑡𝑘𝑞	NOUN
iajs-3482	108	4	)	)	PUNCT
iajs-3482	108	5	(	(	PUNCT
iajs-3482	108	6	9	9	X
iajs-3482	108	7	)	)	PUNCT
iajs-3482	108	8	we	we	PRON
iajs-3482	108	9	can	can	AUX
iajs-3482	108	10	be	be	AUX
iajs-3482	108	11	rewritten	rewrite	VERB
iajs-3482	108	12	equality	equality	NOUN
iajs-3482	108	13	(	(	PUNCT
iajs-3482	108	14	2)in	2)in	NUM
iajs-3482	108	15	this	this	DET
iajs-3482	108	16	notation	notation	NOUN
iajs-3482	108	17	as	as	ADP
iajs-3482	108	18	𝐺(𝑘	𝐺(𝑘	PROPN
iajs-3482	108	19	,	,	PUNCT
iajs-3482	108	20	𝑞	𝑞	NOUN
iajs-3482	108	21	)	)	PUNCT
iajs-3482	108	22	+	+	CCONJ
iajs-3482	108	23	𝐺(𝑞	𝐺(𝑞	ADV
iajs-3482	108	24	,	,	PUNCT
iajs-3482	108	25	𝑘)′	𝑘)′	NOUN
iajs-3482	108	26	=	=	NOUN
iajs-3482	108	27	0	0	X
iajs-3482	108	28	.	.	PUNCT
iajs-3482	109	1	using	use	VERB
iajs-3482	109	2	equality	equality	NOUN
iajs-3482	109	3	(	(	PUNCT
iajs-3482	109	4	9	9	NUM
iajs-3482	109	5	)	)	PUNCT
iajs-3482	109	6	and	and	CCONJ
iajs-3482	109	7	this	this	DET
iajs-3482	109	8	fact	fact	NOUN
iajs-3482	109	9	,	,	PUNCT
iajs-3482	109	10	we	we	PRON
iajs-3482	109	11	obtain	obtain	VERB
iajs-3482	109	12	𝐺(𝑘	𝐺(𝑘	ADP
iajs-3482	109	13	,	,	PUNCT
iajs-3482	109	14	𝑞)𝛼	𝑞)𝛼	PRON
iajs-3482	109	15	(	(	PUNCT
iajs-3482	109	16	𝑡	𝑡	X
iajs-3482	109	17	[	[	X
iajs-3482	109	18	𝑘	𝑘	X
iajs-3482	109	19	,	,	PUNCT
iajs-3482	109	20	𝑞	𝑞	X
iajs-3482	109	21	]	]	X
iajs-3482	109	22	)	)	PUNCT
iajs-3482	109	23	=	=	SYM
iajs-3482	110	1	0	0	NUM
iajs-3482	111	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	111	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	111	3	𝑘	𝑘	PROPN
iajs-3482	111	4	,	,	PUNCT
iajs-3482	111	5	𝑞	𝑞	PROPN
iajs-3482	111	6	,	,	PUNCT
iajs-3482	111	7	𝑡	𝑡	PROPN
iajs-3482	111	8	,	,	PUNCT
iajs-3482	111	9	𝑧	𝑧	DET
iajs-3482	111	10	∈	∈	PROPN
iajs-3482	111	11	𝑉	𝑉	PROPN
iajs-3482	111	12	(	(	PUNCT
iajs-3482	111	13	10	10	NUM
iajs-3482	111	14	)	)	PUNCT
iajs-3482	111	15	now	now	ADV
iajs-3482	111	16	using	use	VERB
iajs-3482	111	17	lemma	lemma	PROPN
iajs-3482	111	18	(	(	PUNCT
iajs-3482	111	19	3.2	3.2	NUM
iajs-3482	111	20	)	)	PUNCT
iajs-3482	111	21	,	,	PUNCT
iajs-3482	111	22	we	we	PRON
iajs-3482	111	23	have	have	VERB
iajs-3482	111	24	𝐺(𝑘	𝐺(𝑘	X
iajs-3482	111	25	,	,	PUNCT
iajs-3482	111	26	𝑞)𝛼	𝑞)𝛼	PRON
iajs-3482	111	27	(	(	PUNCT
iajs-3482	111	28	𝑧	𝑧	X
iajs-3482	111	29	[	[	X
iajs-3482	111	30	𝑢	𝑢	X
iajs-3482	111	31	,	,	PUNCT
iajs-3482	111	32	𝑣	𝑣	NOUN
iajs-3482	111	33	]	]	X
iajs-3482	111	34	)	)	PUNCT
iajs-3482	111	35	=	=	SYM
iajs-3482	112	1	0	0	NUM
iajs-3482	113	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	113	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	113	3	𝑘	𝑘	PROPN
iajs-3482	113	4	,	,	PUNCT
iajs-3482	113	5	𝑞	𝑞	X
iajs-3482	113	6	,	,	PUNCT
iajs-3482	113	7	𝑧	𝑧	PROPN
iajs-3482	113	8	,	,	PUNCT
iajs-3482	113	9	𝑢	𝑢	PROPN
iajs-3482	113	10	,	,	PUNCT
iajs-3482	113	11	𝑣	𝑣	PRON
iajs-3482	113	12	∈	∈	PROPN
iajs-3482	113	13	𝑉	𝑉	PROPN
iajs-3482	113	14	(	(	PUNCT
iajs-3482	113	15	11	11	NUM
iajs-3482	113	16	)	)	PUNCT
iajs-3482	113	17	(	(	PUNCT
iajs-3482	113	18	i	i	NOUN
iajs-3482	113	19	)	)	PUNCT
iajs-3482	113	20	if	if	SCONJ
iajs-3482	113	21	𝑉	𝑉	PROPN
iajs-3482	113	22	is	be	AUX
iajs-3482	113	23	non	non	ADJ
iajs-3482	113	24	commutative	commutative	ADJ
iajs-3482	113	25	”	"	PUNCT
iajs-3482	113	26	since	since	SCONJ
iajs-3482	113	27	𝛼	𝛼	PROPN
iajs-3482	113	28	is	be	AUX
iajs-3482	113	29	surjective	surjective	ADJ
iajs-3482	113	30	and	and	CCONJ
iajs-3482	113	31	using	use	VERB
iajs-3482	113	32	lemma	lemma	PROPN
iajs-3482	113	33	(	(	PUNCT
iajs-3482	113	34	3.1	3.1	NUM
iajs-3482	113	35	)	)	PUNCT
iajs-3482	113	36	,	,	PUNCT
iajs-3482	113	37	we	we	PRON
iajs-3482	113	38	have	have	VERB
iajs-3482	113	39	𝐺(𝑘	𝐺(𝑘	X
iajs-3482	113	40	,	,	PUNCT
iajs-3482	113	41	𝑞	𝑞	NOUN
iajs-3482	113	42	)	)	PUNCT
iajs-3482	113	43	=	=	SYM
iajs-3482	113	44	0	0	NUM
iajs-3482	113	45	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	113	46	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	113	47	𝑘	𝑘	PROPN
iajs-3482	113	48	,	,	PUNCT
iajs-3482	113	49	𝑞	𝑞	PROPN
iajs-3482	113	50	∈	∈	PROPN
iajs-3482	113	51	𝑉	𝑉	PROPN
iajs-3482	113	52	(	(	PUNCT
iajs-3482	113	53	ii	ii	NOUN
iajs-3482	113	54	)	)	PUNCT
iajs-3482	113	55	if	if	SCONJ
iajs-3482	113	56	𝑉	𝑉	PROPN
iajs-3482	113	57	is	be	AUX
iajs-3482	113	58	commutative	commutative	ADJ
iajs-3482	113	59	and	and	CCONJ
iajs-3482	113	60	𝑉	𝑉	PROPN
iajs-3482	113	61	⊄	⊄	NOUN
iajs-3482	113	62	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	113	63	)	)	PUNCT
iajs-3482	113	64	”	"	PUNCT
iajs-3482	113	65	compute	compute	NOUN
iajs-3482	113	66	𝑁	𝑁	PROPN
iajs-3482	113	67	=	=	PROPN
iajs-3482	113	68	24	24	NUM
iajs-3482	113	69	𝑅(𝑘𝑞𝑧𝑞𝑘	𝑅(𝑘𝑞𝑧𝑞𝑘	PROPN
iajs-3482	113	70	)	)	PUNCT
iajs-3482	113	71	in	in	ADP
iajs-3482	113	72	two	two	NUM
iajs-3482	113	73	different	different	ADJ
iajs-3482	113	74	ways	way	NOUN
iajs-3482	113	75	.	.	PUNCT
iajs-3482	114	1	using	use	VERB
iajs-3482	114	2	(	(	PUNCT
iajs-3482	114	3	5	5	NUM
iajs-3482	114	4	)	)	PUNCT
iajs-3482	114	5	,	,	PUNCT
iajs-3482	114	6	we	we	PRON
iajs-3482	114	7	have	have	VERB
iajs-3482	114	8	𝑁	𝑁	PROPN
iajs-3482	114	9	+	+	CCONJ
iajs-3482	114	10	24	24	NUM
iajs-3482	114	11	𝑅(𝑘)′𝛼(𝑞𝑧𝑞𝑘	𝑅(𝑘)′𝛼(𝑞𝑧𝑞𝑘	NOUN
iajs-3482	114	12	)	)	PUNCT
iajs-3482	114	13	=	=	SYM
iajs-3482	114	14	0	0	NUM
iajs-3482	115	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	115	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	115	3	𝑘	𝑘	PROPN
iajs-3482	115	4	,	,	PUNCT
iajs-3482	115	5	𝑞	𝑞	X
iajs-3482	115	6	,	,	PUNCT
iajs-3482	115	7	𝑧	𝑧	PROPN
iajs-3482	115	8	∈	∈	PROPN
iajs-3482	115	9	𝑉	𝑉	PROPN
iajs-3482	115	10	(	(	PUNCT
iajs-3482	115	11	12	12	NUM
iajs-3482	115	12	)	)	PUNCT
iajs-3482	115	13	𝑁	𝑁	PROPN
iajs-3482	115	14	+	+	CCONJ
iajs-3482	115	15	24	24	NUM
iajs-3482	115	16	𝑅(𝑘𝑚)′𝛼(𝑧𝑚𝑘	𝑅(𝑘𝑚)′𝛼(𝑧𝑚𝑘	NOUN
iajs-3482	115	17	)	)	PUNCT
iajs-3482	115	18	=	=	SYM
iajs-3482	115	19	0	0	NUM
iajs-3482	115	20	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	115	21	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	115	22	𝑘	𝑘	PROPN
iajs-3482	115	23	,	,	PUNCT
iajs-3482	115	24	𝑞	𝑞	X
iajs-3482	115	25	,	,	PUNCT
iajs-3482	115	26	𝑧	𝑧	PROPN
iajs-3482	115	27	∈	∈	PROPN
iajs-3482	115	28	𝑉	𝑉	PROPN
iajs-3482	115	29	(	(	PUNCT
iajs-3482	115	30	13	13	NUM
iajs-3482	115	31	)	)	PUNCT
iajs-3482	115	32	from	from	ADP
iajs-3482	115	33	(	(	PUNCT
iajs-3482	115	34	12	12	NUM
iajs-3482	115	35	)	)	PUNCT
iajs-3482	115	36	and	and	CCONJ
iajs-3482	115	37	(	(	PUNCT
iajs-3482	115	38	13	13	NUM
iajs-3482	115	39	)	)	PUNCT
iajs-3482	115	40	,	,	PUNCT
iajs-3482	115	41	we	we	PRON
iajs-3482	115	42	arrive	arrive	VERB
iajs-3482	115	43	at	at	ADP
iajs-3482	115	44	𝑅(𝑘𝑞)𝛼(𝑧𝑞𝑘	𝑅(𝑘𝑞)𝛼(𝑧𝑞𝑘	PROPN
iajs-3482	115	45	)	)	PUNCT
iajs-3482	115	46	+	+	SYM
iajs-3482	115	47	𝑅(𝑘)′𝛼(𝑞𝑧𝑞𝑘	𝑅(𝑘)′𝛼(𝑞𝑧𝑞𝑘	NOUN
iajs-3482	115	48	)	)	PUNCT
iajs-3482	116	1	=	=	SYM
iajs-3482	116	2	0	0	NUM
iajs-3482	116	3	(	(	PUNCT
iajs-3482	116	4	𝑅(𝑘𝑞	𝑅(𝑘𝑞	PROPN
iajs-3482	116	5	)	)	PUNCT
iajs-3482	116	6	+	+	NUM
iajs-3482	116	7	𝑅(𝑘)′𝛼(𝑞))𝛼(𝑧𝑞𝑘	𝑅(𝑘)′𝛼(𝑞))𝛼(𝑧𝑞𝑘	NOUN
iajs-3482	116	8	)	)	PUNCT
iajs-3482	116	9	=	=	SYM
iajs-3482	116	10	0	0	NUM
iajs-3482	117	1	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	117	2	,	,	PUNCT
iajs-3482	117	3	𝑞)𝛼(𝑧𝑞𝑘	𝑞)𝛼(𝑧𝑞𝑘	NOUN
iajs-3482	117	4	)	)	PUNCT
iajs-3482	117	5	=	=	SYM
iajs-3482	117	6	0	0	NUM
iajs-3482	118	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	118	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	118	3	𝑘	𝑘	PROPN
iajs-3482	118	4	,	,	PUNCT
iajs-3482	118	5	𝑞	𝑞	X
iajs-3482	118	6	,	,	PUNCT
iajs-3482	118	7	𝑧	𝑧	PROPN
iajs-3482	118	8	∈	∈	PROPN
iajs-3482	118	9	𝑉	𝑉	PROPN
iajs-3482	118	10	(	(	PUNCT
iajs-3482	118	11	14	14	NUM
iajs-3482	118	12	)	)	PUNCT
iajs-3482	118	13	let	let	VERB
iajs-3482	118	14	𝜓	𝜓	X
iajs-3482	118	15	(	(	PUNCT
iajs-3482	118	16	𝑘	𝑘	X
iajs-3482	118	17	,	,	PUNCT
iajs-3482	118	18	𝑞	𝑞	NOUN
iajs-3482	118	19	)	)	PUNCT
iajs-3482	118	20	=	=	PUNCT
iajs-3482	119	1	𝛼(𝑞𝑘),“it	𝛼(𝑞𝑘),“it	NOUN
iajs-3482	119	2	's	's	PART
iajs-3482	119	3	clear	clear	ADJ
iajs-3482	119	4	that	that	SCONJ
iajs-3482	119	5	𝜓	𝜓	PROPN
iajs-3482	119	6	is	be	AUX
iajs-3482	119	7	additive	additive	ADJ
iajs-3482	119	8	mapping	mapping	NOUN
iajs-3482	119	9	,	,	PUNCT
iajs-3482	119	10	therefore	therefore	ADV
iajs-3482	119	11	ihjpas	ihjpa	VERB
iajs-3482	119	12	.	.	PUNCT
iajs-3482	120	1	2025	2025	NUM
iajs-3482	120	2	,	,	PUNCT
iajs-3482	120	3	38	38	NUM
iajs-3482	120	4	(	(	PUNCT
iajs-3482	120	5	1	1	NUM
iajs-3482	120	6	)	)	PUNCT
iajs-3482	120	7	402	402	NUM
iajs-3482	120	8	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	120	9	,	,	PUNCT
iajs-3482	120	10	𝑞)𝛼(𝑧)𝜓(𝑘	𝑞)𝛼(𝑧)𝜓(𝑘	PRON
iajs-3482	120	11	,	,	PUNCT
iajs-3482	120	12	𝑞	𝑞	X
iajs-3482	120	13	)	)	PUNCT
iajs-3482	120	14	=	=	SYM
iajs-3482	120	15	0	0	NUM
iajs-3482	120	16	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	120	17	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	120	18	𝑘	𝑘	PROPN
iajs-3482	120	19	,	,	PUNCT
iajs-3482	120	20	𝑞	𝑞	X
iajs-3482	120	21	,	,	PUNCT
iajs-3482	120	22	𝑧	𝑧	DET
iajs-3482	120	23	∈	∈	PROPN
iajs-3482	120	24	𝑉	𝑉	PROPN
iajs-3482	120	25	using	use	VERB
iajs-3482	120	26	lemma	lemma	PROPN
iajs-3482	120	27	(	(	PUNCT
iajs-3482	120	28	3.2	3.2	NUM
iajs-3482	120	29	)	)	PUNCT
iajs-3482	120	30	,	,	PUNCT
iajs-3482	120	31	we	we	PRON
iajs-3482	120	32	have	have	VERB
iajs-3482	120	33	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	120	34	,	,	PUNCT
iajs-3482	120	35	𝑞)𝛼(𝑧)𝜓(𝑢	𝑞)𝛼(𝑧)𝜓(𝑢	PRON
iajs-3482	120	36	,	,	PUNCT
iajs-3482	120	37	𝑣	𝑣	NOUN
iajs-3482	120	38	)	)	PUNCT
iajs-3482	120	39	=	=	SYM
iajs-3482	120	40	0	0	NUM
iajs-3482	121	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	121	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	121	3	𝑘	𝑘	PROPN
iajs-3482	121	4	,	,	PUNCT
iajs-3482	121	5	𝑞	𝑞	X
iajs-3482	121	6	,	,	PUNCT
iajs-3482	121	7	𝑧	𝑧	PROPN
iajs-3482	121	8	,	,	PUNCT
iajs-3482	121	9	𝑢	𝑢	PROPN
iajs-3482	121	10	,	,	PUNCT
iajs-3482	121	11	𝑣	𝑣	PRON
iajs-3482	121	12	∈	∈	PROPN
iajs-3482	121	13	𝑉	𝑉	PROPN
iajs-3482	121	14	implies	imply	VERB
iajs-3482	121	15	that	that	SCONJ
iajs-3482	121	16	𝐺(𝑘	𝐺(𝑘	ADP
iajs-3482	121	17	,	,	PUNCT
iajs-3482	121	18	𝑞)𝛼(𝑧𝑢𝑣	𝑞)𝛼(𝑧𝑢𝑣	NUM
iajs-3482	121	19	)	)	PUNCT
iajs-3482	121	20	=	=	PUNCT
iajs-3482	121	21	0	0	NUM
iajs-3482	121	22	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	121	23	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	121	24	𝑘	𝑘	PROPN
iajs-3482	121	25	,	,	PUNCT
iajs-3482	121	26	𝑞	𝑞	X
iajs-3482	121	27	,	,	PUNCT
iajs-3482	121	28	𝑧	𝑧	PROPN
iajs-3482	121	29	,	,	PUNCT
iajs-3482	121	30	𝑢	𝑢	PROPN
iajs-3482	121	31	,	,	PUNCT
iajs-3482	121	32	𝑣	𝑣	PRON
iajs-3482	121	33	∈	∈	PROPN
iajs-3482	121	34	𝑉	𝑉	PROPN
iajs-3482	121	35	(	(	PUNCT
iajs-3482	121	36	15	15	NUM
iajs-3482	121	37	)	)	PUNCT
iajs-3482	121	38	replacing	replace	VERB
iajs-3482	121	39	𝛼(𝑣	𝛼(𝑣	NUM
iajs-3482	121	40	)	)	PUNCT
iajs-3482	121	41	with	with	ADP
iajs-3482	121	42	2𝐺(𝑘	2𝐺(𝑘	NUM
iajs-3482	121	43	,	,	PUNCT
iajs-3482	121	44	𝑞)𝛼(𝑧	𝑞)𝛼(𝑧	ADJ
iajs-3482	121	45	)	)	PUNCT
iajs-3482	121	46	,	,	PUNCT
iajs-3482	121	47	𝑢sing	𝑢se	VERB
iajs-3482	121	48	lemma	lemma	PROPN
iajs-3482	121	49	(	(	PUNCT
iajs-3482	121	50	3.1	3.1	NUM
iajs-3482	121	51	)	)	PUNCT
iajs-3482	121	52	and	and	CCONJ
iajs-3482	121	53	m	m	PROPN
iajs-3482	121	54	is	be	AUX
iajs-3482	121	55	prime	prime	ADJ
iajs-3482	121	56	semi	semi	ADJ
iajs-3482	121	57	-	-	ADJ
iajs-3482	121	58	ring	ring	NOUN
iajs-3482	121	59	,	,	PUNCT
iajs-3482	121	60	we	we	PRON
iajs-3482	121	61	have	have	VERB
iajs-3482	121	62	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	121	63	,	,	PUNCT
iajs-3482	121	64	𝑞)𝛼(𝑧	𝑞)𝛼(𝑧	ADJ
iajs-3482	121	65	)	)	PUNCT
iajs-3482	121	66	=	=	PUNCT
iajs-3482	121	67	0	0	NUM
iajs-3482	122	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	122	2	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	122	3	𝑘	𝑘	PROPN
iajs-3482	122	4	,	,	PUNCT
iajs-3482	122	5	𝑞	𝑞	X
iajs-3482	122	6	,	,	PUNCT
iajs-3482	122	7	𝑧	𝑧	DET
iajs-3482	122	8	∈	∈	PROPN
iajs-3482	122	9	𝑉	𝑉	PROPN
iajs-3482	122	10	using	use	VERB
iajs-3482	122	11	lemma	lemma	PROPN
iajs-3482	122	12	(	(	PUNCT
iajs-3482	122	13	3.1	3.1	NUM
iajs-3482	122	14	)	)	PUNCT
iajs-3482	122	15	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	122	16	,	,	PUNCT
iajs-3482	122	17	𝑞	𝑞	NOUN
iajs-3482	122	18	)	)	PUNCT
iajs-3482	122	19	=	=	SYM
iajs-3482	122	20	0	0	NUM
iajs-3482	122	21	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	122	22	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	122	23	𝑘	𝑘	PROPN
iajs-3482	122	24	,	,	PUNCT
iajs-3482	122	25	𝑞	𝑞	PROPN
iajs-3482	122	26	∈	∈	PROPN
iajs-3482	122	27	𝑉	𝑉	PROPN
iajs-3482	122	28	(	(	PUNCT
iajs-3482	122	29	i	i	NOUN
iajs-3482	122	30	)	)	PUNCT
iajs-3482	123	1	if	if	SCONJ
iajs-3482	123	2	𝑉	𝑉	PROPN
iajs-3482	123	3	⊂	⊂	PROPN
iajs-3482	123	4	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	123	5	)	)	PUNCT
iajs-3482	123	6	multiplying	multiply	VERB
iajs-3482	123	7	relation	relation	NOUN
iajs-3482	123	8	(	(	PUNCT
iajs-3482	123	9	15	15	NUM
iajs-3482	123	10	)	)	PUNCT
iajs-3482	123	11	on	on	ADP
iajs-3482	123	12	the	the	DET
iajs-3482	123	13	right	right	NOUN
iajs-3482	123	14	by	by	ADP
iajs-3482	123	15	t	t	PROPN
iajs-3482	123	16	,	,	PUNCT
iajs-3482	123	17	where	where	SCONJ
iajs-3482	123	18	t	t	PROPN
iajs-3482	123	19	∈	∈	PROPN
iajs-3482	123	20	𝑀	𝑀	PROPN
iajs-3482	123	21	and	and	CCONJ
iajs-3482	123	22	since	since	SCONJ
iajs-3482	123	23	m	m	PROPN
iajs-3482	123	24	is	be	AUX
iajs-3482	123	25	a	a	DET
iajs-3482	123	26	prime	prime	NOUN
iajs-3482	123	27	,	,	PUNCT
iajs-3482	123	28	we	we	PRON
iajs-3482	123	29	can	can	AUX
iajs-3482	123	30	obtain	obtain	VERB
iajs-3482	123	31	the	the	DET
iajs-3482	123	32	result	result	NOUN
iajs-3482	123	33	.	.	PUNCT
iajs-3482	124	1	𝐺(𝑘	𝐺(𝑘	NOUN
iajs-3482	124	2	,	,	PUNCT
iajs-3482	124	3	𝑞	𝑞	NOUN
iajs-3482	124	4	)	)	PUNCT
iajs-3482	124	5	=	=	SYM
iajs-3482	124	6	0	0	NUM
iajs-3482	124	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	124	8	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	124	9	𝑘	𝑘	PROPN
iajs-3482	124	10	,	,	PUNCT
iajs-3482	124	11	𝑞	𝑞	PROPN
iajs-3482	124	12	∈	∈	PROPN
iajs-3482	124	13	𝑉	𝑉	PROPN
iajs-3482	124	14	if	if	SCONJ
iajs-3482	124	15	𝑅(𝑘2	𝑅(𝑘2	NOUN
iajs-3482	124	16	)	)	PUNCT
iajs-3482	124	17	+	+	NUM
iajs-3482	124	18	𝛼(𝑘)′𝑅(𝑘	𝛼(𝑘)′𝑅(𝑘	NOUN
iajs-3482	124	19	)	)	PUNCT
iajs-3482	124	20	=	=	SYM
iajs-3482	124	21	0	0	NUM
iajs-3482	124	22	,	,	PUNCT
iajs-3482	124	23	reaching	reach	VERB
iajs-3482	124	24	the	the	DET
iajs-3482	124	25	conclusion	conclusion	NOUN
iajs-3482	124	26	of	of	ADP
iajs-3482	124	27	the	the	DET
iajs-3482	124	28	theorem	theorem	NOUN
iajs-3482	124	29	with	with	ADP
iajs-3482	124	30	the	the	DET
iajs-3482	124	31	same	same	ADJ
iajs-3482	124	32	procedure	procedure	NOUN
iajs-3482	124	33	as	as	SCONJ
iajs-3482	124	34	before	before	ADV
iajs-3482	124	35	completes	complete	VERB
iajs-3482	124	36	the	the	DET
iajs-3482	124	37	proof	proof	NOUN
iajs-3482	124	38	.	.	PUNCT
iajs-3482	125	1	lemma	lemma	PROPN
iajs-3482	125	2	(	(	PUNCT
iajs-3482	125	3	3.4	3.4	NUM
iajs-3482	125	4	)	)	PUNCT
iajs-3482	125	5	let	let	VERB
iajs-3482	125	6	m	m	PRON
iajs-3482	125	7	be	be	AUX
iajs-3482	125	8	a	a	DET
iajs-3482	125	9	2	2	NUM
iajs-3482	125	10	−	−	NOUN
iajs-3482	125	11	tortion	tortion	NOUN
iajs-3482	125	12	free	free	ADJ
iajs-3482	125	13	prime	prime	NOUN
iajs-3482	125	14	semi	semi	ADJ
iajs-3482	125	15	−	−	PROPN
iajs-3482	125	16	ring	ring	NOUN
iajs-3482	125	17	,	,	PUNCT
iajs-3482	125	18	𝐻	𝐻	PROPN
iajs-3482	125	19	,	,	PUNCT
iajs-3482	125	20	𝛼	𝛼	PROPN
iajs-3482	125	21	:	:	PUNCT
iajs-3482	125	22	𝑀	𝑀	PROPN
iajs-3482	125	23	→	→	SYM
iajs-3482	125	24	𝑀	𝑀	PROPN
iajs-3482	125	25	,	,	PUNCT
iajs-3482	125	26	h	h	NOUN
iajs-3482	125	27	is	be	AUX
iajs-3482	125	28	(	(	PUNCT
iajs-3482	125	29	𝛼	𝛼	NOUN
iajs-3482	125	30	,	,	PUNCT
iajs-3482	125	31	𝛼	𝛼	NOUN
iajs-3482	125	32	)	)	PUNCT
iajs-3482	125	33	−	−	NOUN
iajs-3482	125	34	derivation	derivation	NOUN
iajs-3482	125	35	on	on	ADP
iajs-3482	125	36	𝑉	𝑉	PROPN
iajs-3482	125	37	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3482	125	38	𝑎	𝑎	PROPN
iajs-3482	125	39	∈	∈	PROPN
iajs-3482	125	40	𝑉	𝑉	PROPN
iajs-3482	125	41	some	some	DET
iajs-3482	125	42	fixed	fix	VERB
iajs-3482	125	43	element	element	NOUN
iajs-3482	125	44	,	,	PUNCT
iajs-3482	125	45	where	where	SCONJ
iajs-3482	125	46	𝛼	𝛼	NOUN
iajs-3482	125	47	is	be	AUX
iajs-3482	125	48	automorphism	automorphism	NOUN
iajs-3482	125	49	of	of	ADP
iajs-3482	125	50	𝑉	𝑉	PROPN
iajs-3482	125	51	,	,	PUNCT
iajs-3482	125	52	such	such	ADJ
iajs-3482	125	53	that	that	PRON
iajs-3482	125	54	𝛼(𝑉	𝛼(𝑉	NUM
iajs-3482	125	55	)	)	PUNCT
iajs-3482	126	1	=	=	PUNCT
iajs-3482	126	2	𝑉	𝑉	PROPN
iajs-3482	126	3	then	then	ADV
iajs-3482	126	4	(	(	PUNCT
iajs-3482	126	5	ii	ii	NOUN
iajs-3482	126	6	)	)	PUNCT
iajs-3482	126	7	𝐻(𝑘)𝐻(𝑞	𝐻(𝑘)𝐻(𝑞	NOUN
iajs-3482	126	8	)	)	PUNCT
iajs-3482	126	9	=	=	SYM
iajs-3482	126	10	0	0	NUM
iajs-3482	126	11	for	for	ADP
iajs-3482	126	12	any	any	DET
iajs-3482	126	13	𝑘	𝑘	NOUN
iajs-3482	126	14	,	,	PUNCT
iajs-3482	126	15	𝑞	𝑞	PROPN
iajs-3482	126	16	∈	∈	PROPN
iajs-3482	126	17	𝑈	𝑈	PROPN
iajs-3482	126	18	implies	imply	VERB
iajs-3482	126	19	𝐻	𝐻	PROPN
iajs-3482	126	20	=	=	SYM
iajs-3482	126	21	0	0	NUM
iajs-3482	126	22	on	on	ADP
iajs-3482	126	23	𝑉.	𝑉.	PROPN
iajs-3482	126	24	(	(	PUNCT
iajs-3482	126	25	iii)𝑎𝛼(𝑘	iii)𝑎𝛼(𝑘	PROPN
iajs-3482	126	26	)	)	PUNCT
iajs-3482	126	27	+	+	NUM
iajs-3482	126	28	𝛼(𝑘)′𝑎	𝛼(𝑘)′𝑎	PROPN
iajs-3482	126	29	∈	∈	NOUN
iajs-3482	126	30	𝑍(𝑉	𝑍(𝑉	NOUN
iajs-3482	126	31	)	)	PUNCT
iajs-3482	126	32	for	for	ADP
iajs-3482	126	33	any	any	PRON
iajs-3482	126	34	𝑘	𝑘	PRON
iajs-3482	126	35	∈	∈	PROPN
iajs-3482	126	36	𝑉	𝑉	PROPN
iajs-3482	126	37	implies	imply	VERB
iajs-3482	126	38	𝑎	𝑎	NOUN
iajs-3482	126	39	∈	∈	NOUN
iajs-3482	126	40	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	126	41	)	)	PUNCT
iajs-3482	126	42	.	.	PUNCT
iajs-3482	127	1	proof	proof	NOUN
iajs-3482	127	2	:	:	PUNCT
iajs-3482	127	3	(	(	PUNCT
iajs-3482	127	4	i	i	NOUN
iajs-3482	127	5	)	)	PUNCT
iajs-3482	127	6	𝐻(𝑘)𝛼(𝑞)𝐻(𝑘	𝐻(𝑘)𝛼(𝑞)𝐻(𝑘	NOUN
iajs-3482	127	7	)	)	PUNCT
iajs-3482	127	8	=	=	SYM
iajs-3482	127	9	𝐻(𝑘)𝐻(𝑞𝑘	𝐻(𝑘)𝐻(𝑞𝑘	PROPN
iajs-3482	127	10	)	)	PUNCT
iajs-3482	127	11	+	+	CCONJ
iajs-3482	127	12	𝐻(𝑘)′𝐻(𝑞)𝛼(𝑘	𝐻(𝑘)′𝐻(𝑞)𝛼(𝑘	NOUN
iajs-3482	127	13	)	)	PUNCT
iajs-3482	127	14	𝐻(𝑘)(𝐻(𝑞)𝛼(𝑘	𝐻(𝑘)(𝐻(𝑞)𝛼(𝑘	X
iajs-3482	127	15	)	)	PUNCT
iajs-3482	128	1	+	+	CCONJ
iajs-3482	128	2	𝛼(𝑞)𝐻(𝑘	𝛼(𝑞)𝐻(𝑘	NOUN
iajs-3482	128	3	)	)	PUNCT
iajs-3482	128	4	)	)	PUNCT
iajs-3482	129	1	+	+	CCONJ
iajs-3482	129	2	𝐻(𝑘)′𝐻(𝑞)𝛼(𝑘	𝐻(𝑘)′𝐻(𝑞)𝛼(𝑘	NOUN
iajs-3482	129	3	)	)	PUNCT
iajs-3482	129	4	=	=	SYM
iajs-3482	129	5	0	0	NUM
iajs-3482	129	6	𝐻(𝑘)𝐻(𝑞)𝛼(𝑘	𝐻(𝑘)𝐻(𝑞)𝛼(𝑘	PROPN
iajs-3482	129	7	)	)	PUNCT
iajs-3482	129	8	+	+	CCONJ
iajs-3482	129	9	𝐻(𝑘)𝛼(𝑞)𝐻(𝑘	𝐻(𝑘)𝛼(𝑞)𝐻(𝑘	NOUN
iajs-3482	129	10	)	)	PUNCT
iajs-3482	129	11	+	+	SYM
iajs-3482	129	12	𝐻(𝑘)′𝐻(𝑞)𝛼(𝑘	𝐻(𝑘)′𝐻(𝑞)𝛼(𝑘	NOUN
iajs-3482	129	13	)	)	PUNCT
iajs-3482	130	1	=	=	SYM
iajs-3482	130	2	0	0	NUM
iajs-3482	130	3	by	by	ADP
iajs-3482	130	4	hypothesis	hypothesis	NOUN
iajs-3482	130	5	,	,	PUNCT
iajs-3482	130	6	and	and	CCONJ
iajs-3482	130	7	m	m	PROPN
iajs-3482	130	8	is	be	AUX
iajs-3482	130	9	inverse	inverse	ADJ
iajs-3482	130	10	semi	semi	ADJ
iajs-3482	130	11	-	-	NOUN
iajs-3482	130	12	ring	ring	ADJ
iajs-3482	130	13	,	,	PUNCT
iajs-3482	130	14	we	we	PRON
iajs-3482	130	15	get	get	VERB
iajs-3482	130	16	𝐻(𝑘)𝛼(𝑞)𝐻(𝑘	𝐻(𝑘)𝛼(𝑞)𝐻(𝑘	NOUN
iajs-3482	130	17	)	)	PUNCT
iajs-3482	130	18	=	=	SYM
iajs-3482	131	1	0	0	PUNCT
iajs-3482	131	2	since	since	SCONJ
iajs-3482	131	3	𝛼	𝛼	PROPN
iajs-3482	131	4	is	be	AUX
iajs-3482	131	5	automorphism	automorphism	NOUN
iajs-3482	131	6	of	of	ADP
iajs-3482	131	7	𝑉	𝑉	PROPN
iajs-3482	131	8	,	,	PUNCT
iajs-3482	131	9	such	such	ADJ
iajs-3482	131	10	that	that	PRON
iajs-3482	131	11	𝛼(𝑉	𝛼(𝑉	NUM
iajs-3482	131	12	)	)	PUNCT
iajs-3482	131	13	=	=	SYM
iajs-3482	131	14	𝑉	𝑉	PROPN
iajs-3482	131	15	,	,	PUNCT
iajs-3482	131	16	we	we	PRON
iajs-3482	131	17	get	get	VERB
iajs-3482	131	18	𝐻(𝑘	𝐻(𝑘	PRON
iajs-3482	131	19	)	)	PUNCT
iajs-3482	131	20	𝑉	𝑉	PROPN
iajs-3482	131	21	𝐻(𝑘	𝐻(𝑘	NOUN
iajs-3482	131	22	)	)	PUNCT
iajs-3482	131	23	=	=	SYM
iajs-3482	131	24	0	0	NUM
iajs-3482	131	25	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	131	26	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	131	27	𝑘	𝑘	PRON
iajs-3482	131	28	∈	∈	PROPN
iajs-3482	131	29	𝑉	𝑉	PROPN
iajs-3482	131	30	if	if	SCONJ
iajs-3482	131	31	𝑉	𝑉	PROPN
iajs-3482	131	32	⊄	⊄	NOUN
iajs-3482	131	33	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	131	34	)	)	PUNCT
iajs-3482	131	35	,	,	PUNCT
iajs-3482	131	36	and	and	CCONJ
iajs-3482	131	37	𝛼	𝛼	PROPN
iajs-3482	131	38	is	be	AUX
iajs-3482	131	39	automorphism	automorphism	NOUN
iajs-3482	131	40	of	of	ADP
iajs-3482	131	41	𝑉	𝑉	PROPN
iajs-3482	131	42	,	,	PUNCT
iajs-3482	131	43	lemma	lemma	PROPN
iajs-3482	131	44	(	(	PUNCT
iajs-3482	131	45	3.2	3.2	NUM
iajs-3482	131	46	)	)	PUNCT
iajs-3482	131	47	we	we	PRON
iajs-3482	131	48	have	have	VERB
iajs-3482	131	49	𝐻	𝐻	NOUN
iajs-3482	131	50	=	=	SYM
iajs-3482	131	51	0	0	NUM
iajs-3482	131	52	𝑜𝑛	𝑜𝑛	PROPN
iajs-3482	131	53	𝑉.	𝑉.	NOUN
iajs-3482	131	54	if	if	SCONJ
iajs-3482	131	55	v	v	ADP
iajs-3482	131	56	⊂	⊂	PROPN
iajs-3482	131	57	z(m	z(m	PROPN
iajs-3482	131	58	)	)	PUNCT
iajs-3482	131	59	𝐻(𝑘)𝑡𝐻(𝑘	𝐻(𝑘)𝑡𝐻(𝑘	PROPN
iajs-3482	131	60	)	)	PUNCT
iajs-3482	131	61	=	=	SYM
iajs-3482	131	62	0	0	NUM
iajs-3482	132	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	132	2	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	132	3	𝑘	𝑘	PRON
iajs-3482	132	4	∈	∈	PROPN
iajs-3482	132	5	𝑉	𝑉	PROPN
iajs-3482	132	6	,	,	PUNCT
iajs-3482	132	7	𝑡	𝑡	PROPN
iajs-3482	132	8	∈	∈	PROPN
iajs-3482	132	9	𝑀	𝑀	PROPN
iajs-3482	132	10	so	so	ADV
iajs-3482	132	11	,	,	PUNCT
iajs-3482	132	12	by	by	ADP
iajs-3482	132	13	primness	primness	NOUN
iajs-3482	132	14	of	of	ADP
iajs-3482	132	15	m	m	PROPN
iajs-3482	132	16	,	,	PUNCT
iajs-3482	132	17	we	we	PRON
iajs-3482	132	18	have	have	VERB
iajs-3482	132	19	𝐻	𝐻	NOUN
iajs-3482	132	20	=	=	SYM
iajs-3482	132	21	0	0	PUNCT
iajs-3482	132	22	𝑜𝑛	𝑜𝑛	PROPN
iajs-3482	132	23	𝑉	𝑉	PROPN
iajs-3482	132	24	(	(	PUNCT
iajs-3482	132	25	ii	ii	NOUN
iajs-3482	132	26	)	)	PUNCT
iajs-3482	132	27	define	define	VERB
iajs-3482	132	28	𝐻(𝑘	𝐻(𝑘	PRON
iajs-3482	132	29	)	)	PUNCT
iajs-3482	132	30	=	=	SYM
iajs-3482	132	31	𝑎𝛼(𝑘	𝑎𝛼(𝑘	X
iajs-3482	132	32	)	)	PUNCT
iajs-3482	133	1	+	+	CCONJ
iajs-3482	133	2	𝛼(𝑘)𝑎′	𝛼(𝑘)𝑎′	NOUN
iajs-3482	133	3	“	"	PUNCT
iajs-3482	133	4	it	it	PRON
iajs-3482	133	5	is	be	AUX
iajs-3482	133	6	easy	easy	ADJ
iajs-3482	133	7	to	to	PART
iajs-3482	133	8	see	see	VERB
iajs-3482	133	9	that	that	SCONJ
iajs-3482	133	10	”	"	PUNCT
iajs-3482	133	11	h	h	NOUN
iajs-3482	133	12	is	be	AUX
iajs-3482	133	13	a	a	DET
iajs-3482	133	14	(	(	PUNCT
iajs-3482	133	15	𝛼	𝛼	NOUN
iajs-3482	133	16	,	,	PUNCT
iajs-3482	133	17	𝛼	𝛼	NOUN
iajs-3482	133	18	)	)	PUNCT
iajs-3482	133	19	−	−	NOUN
iajs-3482	133	20	derivations	derivation	NOUN
iajs-3482	133	21	,	,	PUNCT
iajs-3482	133	22	since	since	SCONJ
iajs-3482	133	23	𝐻(𝑘	𝐻(𝑘	PRON
iajs-3482	133	24	)	)	PUNCT
iajs-3482	133	25	∈	∈	NOUN
iajs-3482	133	26	𝑍(𝑉	𝑍(𝑉	NOUN
iajs-3482	133	27	)	)	PUNCT
iajs-3482	133	28	for	for	ADP
iajs-3482	133	29	any	any	DET
iajs-3482	133	30	𝑘	𝑘	PROPN
iajs-3482	133	31	∈	∈	PROPN
iajs-3482	133	32	𝑉	𝑉	PROPN
iajs-3482	133	33	,	,	PUNCT
iajs-3482	133	34	we	we	PRON
iajs-3482	133	35	have	have	AUX
iajs-3482	133	36	𝐻(𝑞)𝛼(𝑘	𝐻(𝑞)𝛼(𝑘	VERB
iajs-3482	133	37	)	)	PUNCT
iajs-3482	134	1	=	=	SYM
iajs-3482	134	2	𝛼(𝑘)𝐻(𝑞	𝛼(𝑘)𝐻(𝑞	NOUN
iajs-3482	134	3	)	)	PUNCT
iajs-3482	134	4	and	and	CCONJ
iajs-3482	134	5	also	also	ADV
iajs-3482	134	6	2𝐻(𝑞𝑧)𝛼(𝑘	2𝐻(𝑞𝑧)𝛼(𝑘	X
iajs-3482	134	7	)	)	PUNCT
iajs-3482	134	8	=	=	SYM
iajs-3482	134	9	2	2	NUM
iajs-3482	134	10	𝛼(𝑘)𝐻(𝑞𝑧	𝛼(𝑘)𝐻(𝑞𝑧	NUM
iajs-3482	134	11	)	)	PUNCT
iajs-3482	134	12	since	since	SCONJ
iajs-3482	134	13	m	m	PROPN
iajs-3482	134	14	is	be	AUX
iajs-3482	134	15	prime	prime	ADJ
iajs-3482	134	16	,	,	PUNCT
iajs-3482	134	17	we	we	PRON
iajs-3482	134	18	get	get	VERB
iajs-3482	134	19	𝐻(𝑞)𝛼(𝑧𝑘	𝐻(𝑞)𝛼(𝑧𝑘	PRON
iajs-3482	134	20	)	)	PUNCT
iajs-3482	135	1	+	+	CCONJ
iajs-3482	135	2	𝛼(𝑞)𝐻(𝑧)𝛼(𝑘	𝛼(𝑞)𝐻(𝑧)𝛼(𝑘	NOUN
iajs-3482	135	3	)	)	PUNCT
iajs-3482	135	4	=	=	SYM
iajs-3482	136	1	𝛼(𝑘)𝐻(𝑞)𝛼(𝑧	𝛼(𝑘)𝐻(𝑞)𝛼(𝑧	NUM
iajs-3482	136	2	)	)	PUNCT
iajs-3482	137	1	+	+	X
iajs-3482	137	2	𝛼(𝑘𝑞)𝐻(𝑧	𝛼(𝑘𝑞)𝐻(𝑧	NUM
iajs-3482	137	3	)	)	PUNCT
iajs-3482	137	4	𝐻(𝑞)(𝛼(𝑧)𝛼(𝑘	𝐻(𝑞)(𝛼(𝑧)𝛼(𝑘	NOUN
iajs-3482	137	5	)	)	PUNCT
iajs-3482	138	1	+	+	CCONJ
iajs-3482	138	2	𝛼(𝑘)𝛼(𝑧)′	𝛼(𝑘)𝛼(𝑧)′	X
iajs-3482	138	3	)	)	PUNCT
iajs-3482	138	4	=	=	PUNCT
iajs-3482	138	5	𝐻(𝑧)(𝛼(𝑞)𝛼(𝑘)′	𝐻(𝑧)(𝛼(𝑞)𝛼(𝑘)′	X
iajs-3482	138	6	+	+	NOUN
iajs-3482	138	7	𝛼(𝑘)(𝑞	𝛼(𝑘)(𝑞	NUM
iajs-3482	138	8	)	)	PUNCT
iajs-3482	138	9	)	)	PUNCT
iajs-3482	139	1	𝐻(𝑞)[𝛼(𝑧	𝐻(𝑞)[𝛼(𝑧	PROPN
iajs-3482	139	2	)	)	PUNCT
iajs-3482	139	3	,	,	PUNCT
iajs-3482	139	4	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	139	5	)	)	PUNCT
iajs-3482	139	6	]	]	PUNCT
iajs-3482	140	1	=	=	PUNCT
iajs-3482	140	2	𝐻(𝑧)[𝛼(𝑞	𝐻(𝑧)[𝛼(𝑞	NOUN
iajs-3482	140	3	)	)	PUNCT
iajs-3482	140	4	,	,	PUNCT
iajs-3482	140	5	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	140	6	)	)	PUNCT
iajs-3482	140	7	]	]	PUNCT
iajs-3482	140	8	ihjpas	ihjpa	VERB
iajs-3482	140	9	.	.	PUNCT
iajs-3482	141	1	2025	2025	NUM
iajs-3482	141	2	,	,	PUNCT
iajs-3482	141	3	38	38	NUM
iajs-3482	141	4	(	(	PUNCT
iajs-3482	141	5	1	1	NUM
iajs-3482	141	6	)	)	PUNCT
iajs-3482	141	7	403	403	NUM
iajs-3482	141	8	since	since	SCONJ
iajs-3482	141	9	𝛼	𝛼	PRON
iajs-3482	141	10	is	be	AUX
iajs-3482	141	11	automorphism	automorphism	NOUN
iajs-3482	141	12	,	,	PUNCT
iajs-3482	141	13	take	take	VERB
iajs-3482	141	14	𝛼(𝑧	𝛼(𝑧	NOUN
iajs-3482	141	15	)	)	PUNCT
iajs-3482	142	1	=	=	SYM
iajs-3482	142	2	𝑎.	𝑎.	VERB
iajs-3482	142	3	obviously	obviously	ADV
iajs-3482	142	4	𝐻(𝑎	𝐻(𝑎	NOUN
iajs-3482	142	5	)	)	PUNCT
iajs-3482	142	6	=	=	SYM
iajs-3482	142	7	0	0	NUM
iajs-3482	142	8	,	,	PUNCT
iajs-3482	142	9	so	so	ADV
iajs-3482	142	10	,	,	PUNCT
iajs-3482	142	11	we	we	PRON
iajs-3482	142	12	obtain	obtain	VERB
iajs-3482	142	13	by	by	ADP
iajs-3482	142	14	(	(	PUNCT
iajs-3482	142	15	i	i	NOUN
iajs-3482	142	16	)	)	PUNCT
iajs-3482	142	17	𝐻(𝑞)𝐻(𝑘	𝐻(𝑞)𝐻(𝑘	NOUN
iajs-3482	142	18	)	)	PUNCT
iajs-3482	142	19	=	=	SYM
iajs-3482	142	20	0	0	PUNCT
iajs-3482	143	1	“	"	PUNCT
iajs-3482	143	2	by	by	ADP
iajs-3482	143	3	virtue	virtue	NOUN
iajs-3482	143	4	of	of	ADP
iajs-3482	143	5	(	(	PUNCT
iajs-3482	143	6	i	i	NOUN
iajs-3482	143	7	)	)	PUNCT
iajs-3482	143	8	we	we	PRON
iajs-3482	143	9	get	get	VERB
iajs-3482	143	10	”	"	PUNCT
iajs-3482	143	11	h	h	NOUN
iajs-3482	143	12	=	=	NOUN
iajs-3482	143	13	0	0	NUM
iajs-3482	143	14	and	and	CCONJ
iajs-3482	143	15	hence	hence	ADV
iajs-3482	143	16	a	a	DET
iajs-3482	143	17	∈	∈	NOUN
iajs-3482	143	18	z(m	z(m	NUM
iajs-3482	143	19	)	)	PUNCT
iajs-3482	143	20	.	.	PUNCT
iajs-3482	144	1	lemma	lemma	PROPN
iajs-3482	144	2	(	(	PUNCT
iajs-3482	144	3	3.5	3.5	NUM
iajs-3482	144	4	)	)	PUNCT
iajs-3482	144	5	let	let	VERB
iajs-3482	144	6	m	m	PRON
iajs-3482	144	7	be	be	AUX
iajs-3482	144	8	a	a	DET
iajs-3482	144	9	2	2	NUM
iajs-3482	144	10	−	−	NOUN
iajs-3482	144	11	tortion	tortion	NOUN
iajs-3482	144	12	free	free	ADJ
iajs-3482	144	13	prime	prime	NOUN
iajs-3482	144	14	semi	semi	ADJ
iajs-3482	144	15	−	−	PROPN
iajs-3482	144	16	ring	ring	NOUN
iajs-3482	144	17	,	,	PUNCT
iajs-3482	144	18	r	r	NOUN
iajs-3482	144	19	and	and	CCONJ
iajs-3482	144	20	α	α	NOUN
iajs-3482	144	21	are	be	AUX
iajs-3482	144	22	additive	additive	ADJ
iajs-3482	144	23	mappings	mapping	NOUN
iajs-3482	144	24	on	on	ADP
iajs-3482	144	25	m	m	PROPN
iajs-3482	144	26	,	,	PUNCT
iajs-3482	144	27	and	and	CCONJ
iajs-3482	144	28	𝑎	𝑎	PRON
iajs-3482	144	29	∈	∈	NOUN
iajs-3482	144	30	𝑉	𝑉	PROPN
iajs-3482	144	31	some	some	DET
iajs-3482	144	32	fixed	fix	VERB
iajs-3482	144	33	element	element	NOUN
iajs-3482	144	34	.	.	PUNCT
iajs-3482	145	1	if	if	SCONJ
iajs-3482	145	2	𝑅(𝑘	𝑅(𝑘	NOUN
iajs-3482	145	3	)	)	PUNCT
iajs-3482	145	4	=	=	PUNCT
iajs-3482	145	5	𝑎	𝑎	PROPN
iajs-3482	145	6	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	145	7	)	)	PUNCT
iajs-3482	146	1	+	+	CCONJ
iajs-3482	146	2	𝛼(𝑘)𝑎and𝑅(𝑘	𝛼(𝑘)𝑎and𝑅(𝑘	NUM
iajs-3482	146	3	𝑜	𝑜	X
iajs-3482	146	4	𝑞	𝑞	X
iajs-3482	146	5	)	)	PUNCT
iajs-3482	146	6	+	+	CCONJ
iajs-3482	146	7	𝑅(𝑘)𝑜	𝑅(𝑘)𝑜	ADJ
iajs-3482	146	8	𝛼(𝑞)′	𝛼(𝑞)′	PROPN
iajs-3482	146	9	=	=	SYM
iajs-3482	146	10	0	0	PUNCT
iajs-3482	146	11	and	and	CCONJ
iajs-3482	146	12	𝑅(𝑘	𝑅(𝑘	ADP
iajs-3482	146	13	𝑜	𝑜	NOUN
iajs-3482	146	14	𝑞	𝑞	X
iajs-3482	146	15	)	)	PUNCT
iajs-3482	146	16	+	+	NUM
iajs-3482	146	17	𝛼(𝑘)′𝑜𝑅(𝑞	𝛼(𝑘)′𝑜𝑅(𝑞	NUM
iajs-3482	146	18	)	)	PUNCT
iajs-3482	146	19	=	=	SYM
iajs-3482	146	20	0	0	NUM
iajs-3482	146	21	for	for	ADP
iajs-3482	146	22	any	any	DET
iajs-3482	146	23	𝑘	𝑘	NOUN
iajs-3482	146	24	,	,	PUNCT
iajs-3482	146	25	𝑞	𝑞	PROPN
iajs-3482	146	26	∈	∈	PROPN
iajs-3482	146	27	𝑉	𝑉	PROPN
iajs-3482	146	28	then	then	ADV
iajs-3482	146	29	“	"	PUNCT
iajs-3482	146	30	𝑎	𝑎	PROPN
iajs-3482	146	31	∈	∈	NOUN
iajs-3482	146	32	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	146	33	)	)	PUNCT
iajs-3482	146	34	,	,	PUNCT
iajs-3482	146	35	“	"	PUNCT
iajs-3482	146	36	where	where	SCONJ
iajs-3482	146	37	𝛼	𝛼	PRON
iajs-3482	146	38	is	be	AUX
iajs-3482	146	39	a	a	DET
iajs-3482	146	40	surjective	surjective	ADJ
iajs-3482	146	41	”	"	PUNCT
iajs-3482	146	42	endomorphism	endomorphism	NOUN
iajs-3482	146	43	of	of	ADP
iajs-3482	146	44	𝑉	𝑉	PROPN
iajs-3482	146	45	.	.	PUNCT
iajs-3482	147	1	proof	proof	NOUN
iajs-3482	147	2	:	:	PUNCT
iajs-3482	147	3	by	by	ADP
iajs-3482	147	4	hypothesis	hypothesis	NOUN
iajs-3482	147	5	“	"	PUNCT
iajs-3482	147	6	𝑅(𝑘𝑞	𝑅(𝑘𝑞	X
iajs-3482	147	7	+	+	CCONJ
iajs-3482	147	8	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	147	9	)	)	PUNCT
iajs-3482	147	10	=	=	SYM
iajs-3482	147	11	𝑅(𝑘)𝛼(𝑞	𝑅(𝑘)𝛼(𝑞	NOUN
iajs-3482	147	12	)	)	PUNCT
iajs-3482	148	1	+	+	CCONJ
iajs-3482	148	2	𝛼(𝑞)𝑅(𝑘	𝛼(𝑞)𝑅(𝑘	NOUN
iajs-3482	148	3	)	)	PUNCT
iajs-3482	148	4	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	148	5	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	148	6	𝑘	𝑘	PROPN
iajs-3482	148	7	,	,	PUNCT
iajs-3482	148	8	𝑞	𝑞	PROPN
iajs-3482	148	9	∈	∈	PROPN
iajs-3482	148	10	𝑉	𝑉	PROPN
iajs-3482	148	11	”	"	PUNCT
iajs-3482	148	12	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	148	13	)	)	PUNCT
iajs-3482	148	14	+	+	CCONJ
iajs-3482	148	15	𝑅(𝑞𝑘	𝑅(𝑞𝑘	X
iajs-3482	148	16	)	)	PUNCT
iajs-3482	148	17	=	=	SYM
iajs-3482	148	18	𝑅(𝑘)𝛼(𝑞	𝑅(𝑘)𝛼(𝑞	NOUN
iajs-3482	148	19	)	)	PUNCT
iajs-3482	148	20	+	+	CCONJ
iajs-3482	148	21	𝛼(𝑞)𝑅(𝑘	𝛼(𝑞)𝑅(𝑘	NOUN
iajs-3482	148	22	)	)	PUNCT
iajs-3482	148	23	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-3482	148	24	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	148	25	𝑘	𝑘	PROPN
iajs-3482	148	26	,	,	PUNCT
iajs-3482	148	27	𝑞	𝑞	PROPN
iajs-3482	148	28	∈	∈	PROPN
iajs-3482	148	29	𝑉	𝑉	PROPN
iajs-3482	148	30	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	148	31	)	)	PUNCT
iajs-3482	148	32	+	+	CCONJ
iajs-3482	148	33	𝑅(𝑞𝑘	𝑅(𝑞𝑘	X
iajs-3482	148	34	)	)	PUNCT
iajs-3482	148	35	=	=	SYM
iajs-3482	148	36	𝑎𝛼(𝑘𝑞	𝑎𝛼(𝑘𝑞	X
iajs-3482	148	37	)	)	PUNCT
iajs-3482	148	38	+	+	CCONJ
iajs-3482	148	39	𝛼(𝑘𝑞)𝑎	𝛼(𝑘𝑞)𝑎	NOUN
iajs-3482	148	40	+	+	SYM
iajs-3482	148	41	𝑎𝛼(𝑞𝑘	𝑎𝛼(𝑞𝑘	PROPN
iajs-3482	148	42	)	)	PUNCT
iajs-3482	149	1	+	+	CCONJ
iajs-3482	149	2	𝛼(𝑞𝑘)𝑎	𝛼(𝑞𝑘)𝑎	NOUN
iajs-3482	149	3	=	=	SYM
iajs-3482	149	4	𝑎𝛼(𝑘)𝛼(𝑞	𝑎𝛼(𝑘)𝛼(𝑞	NOUN
iajs-3482	149	5	)	)	PUNCT
iajs-3482	150	1	+	+	CCONJ
iajs-3482	151	1	𝛼(𝑘)𝛼(𝑞)𝑎	𝛼(𝑘)𝛼(𝑞)𝑎	NUM
iajs-3482	151	2	+	+	NUM
iajs-3482	151	3	𝑎𝛼(𝑞)𝛼(𝑘	𝑎𝛼(𝑞)𝛼(𝑘	NOUN
iajs-3482	151	4	)	)	PUNCT
iajs-3482	152	1	+	+	PROPN
iajs-3482	152	2	𝛼(𝑞)𝛼(𝑘)𝑎	𝛼(𝑞)𝛼(𝑘)𝑎	PROPN
iajs-3482	152	3	𝑅(𝑘)𝛼(𝑞	𝑅(𝑘)𝛼(𝑞	NOUN
iajs-3482	152	4	)	)	PUNCT
iajs-3482	153	1	+	+	CCONJ
iajs-3482	153	2	𝛼(𝑞)𝑅(𝑘	𝛼(𝑞)𝑅(𝑘	NOUN
iajs-3482	153	3	)	)	PUNCT
iajs-3482	153	4	=	=	SYM
iajs-3482	153	5	𝑎𝛼(𝑘)𝛼(𝑞	𝑎𝛼(𝑘)𝛼(𝑞	NOUN
iajs-3482	153	6	)	)	PUNCT
iajs-3482	154	1	+	+	NUM
iajs-3482	154	2	𝛼(𝑘)𝑎𝛼(𝑞	𝛼(𝑘)𝑎𝛼(𝑞	NOUN
iajs-3482	154	3	)	)	PUNCT
iajs-3482	154	4	+	+	NUM
iajs-3482	154	5	𝛼(𝑞)𝑎𝛼(𝑘	𝛼(𝑞)𝑎𝛼(𝑘	X
iajs-3482	154	6	)	)	PUNCT
iajs-3482	154	7	+	+	PROPN
iajs-3482	154	8	𝛼(𝑞)𝛼(𝑘)𝑎	𝛼(𝑞)𝛼(𝑘)𝑎	PROPN
iajs-3482	154	9	+	+	NUM
iajs-3482	154	10	𝑎𝛼(𝑘)𝛼(𝑞	𝑎𝛼(𝑘)𝛼(𝑞	NOUN
iajs-3482	154	11	)	)	PUNCT
iajs-3482	155	1	+	+	CCONJ
iajs-3482	156	1	𝛼(𝑘)𝛼(𝑞)𝑎	𝛼(𝑘)𝛼(𝑞)𝑎	NUM
iajs-3482	156	2	+	+	NUM
iajs-3482	156	3	𝑎𝛼(𝑞)𝛼(𝑘	𝑎𝛼(𝑞)𝛼(𝑘	NOUN
iajs-3482	156	4	)	)	PUNCT
iajs-3482	157	1	+	+	PROPN
iajs-3482	157	2	𝛼(𝑞)𝛼(𝑘)𝑎	𝛼(𝑞)𝛼(𝑘)𝑎	PROPN
iajs-3482	157	3	=	=	SYM
iajs-3482	157	4	𝑎𝛼(𝑘)𝛼(𝑞	𝑎𝛼(𝑘)𝛼(𝑞	NOUN
iajs-3482	157	5	)	)	PUNCT
iajs-3482	158	1	+	+	NUM
iajs-3482	158	2	𝛼(𝑘)𝑎𝛼(𝑞	𝛼(𝑘)𝑎𝛼(𝑞	NOUN
iajs-3482	158	3	)	)	PUNCT
iajs-3482	159	1	+	+	NUM
iajs-3482	159	2	𝛼(𝑞)𝑎𝛼(𝑘	𝛼(𝑞)𝑎𝛼(𝑘	X
iajs-3482	159	3	)	)	PUNCT
iajs-3482	159	4	+	+	SYM
iajs-3482	160	1	𝛼(𝑞)𝛼(𝑘)𝑎	𝛼(𝑞)𝛼(𝑘)𝑎	PROPN
iajs-3482	160	2	(	(	PUNCT
iajs-3482	160	3	𝑎	𝑎	PROPN
iajs-3482	160	4	+	+	NUM
iajs-3482	160	5	𝑎′)𝛼(𝑘)𝛼(𝑞	𝑎′)𝛼(𝑘)𝛼(𝑞	NOUN
iajs-3482	160	6	)	)	PUNCT
iajs-3482	160	7	+	+	NUM
iajs-3482	160	8	𝛼(𝑞)𝛼(𝑘)(𝑎	𝛼(𝑞)𝛼(𝑘)(𝑎	PROPN
iajs-3482	160	9	+	+	CCONJ
iajs-3482	160	10	𝑎′	𝑎′	NUM
iajs-3482	160	11	)	)	PUNCT
iajs-3482	160	12	+	+	CCONJ
iajs-3482	160	13	𝛼(𝑘)𝛼(𝑞)𝑎	𝛼(𝑘)𝛼(𝑞)𝑎	NUM
iajs-3482	160	14	+	+	NUM
iajs-3482	160	15	𝑎𝛼(𝑞)𝛼(𝑘	𝑎𝛼(𝑞)𝛼(𝑘	NOUN
iajs-3482	160	16	)	)	PUNCT
iajs-3482	160	17	+	+	NUM
iajs-3482	160	18	𝛼(𝑘)𝑎′𝛼(𝑞	𝛼(𝑘)𝑎′𝛼(𝑞	NOUN
iajs-3482	160	19	)	)	PUNCT
iajs-3482	160	20	+	+	CCONJ
iajs-3482	160	21	𝛼(𝑞)𝑎′𝛼(𝑘	𝛼(𝑞)𝑎′𝛼(𝑘	X
iajs-3482	160	22	)	)	PUNCT
iajs-3482	160	23	=	=	SYM
iajs-3482	160	24	0	0	NUM
iajs-3482	160	25	since	since	SCONJ
iajs-3482	160	26	𝑎	𝑎	PROPN
iajs-3482	160	27	+	+	NUM
iajs-3482	160	28	𝑎′	𝑎′	PRON
iajs-3482	160	29	∈	∈	NOUN
iajs-3482	160	30	𝑍(𝑉	𝑍(𝑉	NOUN
iajs-3482	160	31	)	)	PUNCT
iajs-3482	160	32	𝛼(𝑘)𝛼(𝑞)(𝑎	𝛼(𝑘)𝛼(𝑞)(𝑎	PROPN
iajs-3482	160	33	+	+	CCONJ
iajs-3482	160	34	𝑎′	𝑎′	NUM
iajs-3482	160	35	+	+	NUM
iajs-3482	160	36	𝑎	𝑎	X
iajs-3482	160	37	)	)	PUNCT
iajs-3482	160	38	+	+	CCONJ
iajs-3482	160	39	(	(	PUNCT
iajs-3482	160	40	𝑎	𝑎	X
iajs-3482	160	41	+	+	NUM
iajs-3482	160	42	𝑎′	𝑎′	PRON
iajs-3482	160	43	+	+	NUM
iajs-3482	160	44	𝑎)𝛼(𝑞)𝛼(𝑘	𝑎)𝛼(𝑞)𝛼(𝑘	NUM
iajs-3482	160	45	)	)	PUNCT
iajs-3482	160	46	+	+	NUM
iajs-3482	160	47	𝑎𝛼(𝑞)𝛼(𝑘	𝑎𝛼(𝑞)𝛼(𝑘	X
iajs-3482	160	48	)	)	PUNCT
iajs-3482	160	49	+	+	NUM
iajs-3482	160	50	𝛼(𝑘)𝑎′𝛼(𝑞	𝛼(𝑘)𝑎′𝛼(𝑞	NOUN
iajs-3482	160	51	)	)	PUNCT
iajs-3482	160	52	=	=	PUNCT
iajs-3482	160	53	0	0	X
iajs-3482	161	1	𝛼(𝑘)𝛼(𝑞)𝑎	𝛼(𝑘)𝛼(𝑞)𝑎	ADJ
iajs-3482	161	2	+	+	NUM
iajs-3482	161	3	𝛼(𝑘)𝑎′𝛼(𝑞	𝛼(𝑘)𝑎′𝛼(𝑞	NOUN
iajs-3482	161	4	)	)	PUNCT
iajs-3482	162	1	+	+	NOUN
iajs-3482	162	2	𝑎𝛼(𝑞)𝛼(𝑘	𝑎𝛼(𝑞)𝛼(𝑘	X
iajs-3482	162	3	)	)	PUNCT
iajs-3482	163	1	+	+	NUM
iajs-3482	163	2	𝛼(𝑞)𝑎′𝛼(𝑘	𝛼(𝑞)𝑎′𝛼(𝑘	X
iajs-3482	163	3	)	)	PUNCT
iajs-3482	163	4	=	=	SYM
iajs-3482	163	5	0	0	NUM
iajs-3482	164	1	𝛼(𝑘)(𝛼(𝑞)𝑎	𝛼(𝑘)(𝛼(𝑞)𝑎	ADJ
iajs-3482	164	2	+	+	CCONJ
iajs-3482	164	3	𝑎′𝛼(𝑞	𝑎′𝛼(𝑞	NOUN
iajs-3482	164	4	)	)	PUNCT
iajs-3482	164	5	)	)	PUNCT
iajs-3482	165	1	+	+	CCONJ
iajs-3482	165	2	(	(	PUNCT
iajs-3482	165	3	𝛼(𝑞)𝑎	𝛼(𝑞)𝑎	X
iajs-3482	165	4	+	+	CCONJ
iajs-3482	165	5	𝑎′𝛼(𝑞))𝛼(𝑘)′	𝑎′𝛼(𝑞))𝛼(𝑘)′	X
iajs-3482	165	6	=	=	SYM
iajs-3482	165	7	0	0	NUM
iajs-3482	166	1	but	but	CCONJ
iajs-3482	166	2	𝛼	𝛼	PROPN
iajs-3482	166	3	is	be	AUX
iajs-3482	166	4	a	a	DET
iajs-3482	166	5	surjective	surjective	ADJ
iajs-3482	166	6	𝑎𝛼(𝑘	𝑎𝛼(𝑘	NOUN
iajs-3482	166	7	)	)	PUNCT
iajs-3482	167	1	+	+	CCONJ
iajs-3482	167	2	𝛼(𝑘)𝑎′	𝛼(𝑘)𝑎′	PROPN
iajs-3482	167	3	∈	∈	NOUN
iajs-3482	167	4	𝑍(𝑉	𝑍(𝑉	NOUN
iajs-3482	167	5	)	)	PUNCT
iajs-3482	167	6	by	by	ADP
iajs-3482	167	7	lemma	lemma	PROPN
iajs-3482	167	8	(	(	PUNCT
iajs-3482	167	9	3.4	3.4	NUM
iajs-3482	167	10	)	)	PUNCT
iajs-3482	167	11	(	(	PUNCT
iajs-3482	167	12	ii	ii	NOUN
iajs-3482	167	13	)	)	PUNCT
iajs-3482	167	14	,	,	PUNCT
iajs-3482	167	15	we	we	PRON
iajs-3482	167	16	get	get	VERB
iajs-3482	167	17	𝑎	𝑎	NOUN
iajs-3482	167	18	∈	∈	NOUN
iajs-3482	167	19	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	167	20	)	)	PUNCT
iajs-3482	167	21	𝑅(𝑘	𝑅(𝑘	DET
iajs-3482	167	22	𝑜	𝑜	NOUN
iajs-3482	167	23	𝑞	𝑞	X
iajs-3482	167	24	)	)	PUNCT
iajs-3482	167	25	+	+	CCONJ
iajs-3482	167	26	𝑅(𝑘)𝑜	𝑅(𝑘)𝑜	ADJ
iajs-3482	167	27	𝛼(𝑞)′	𝛼(𝑞)′	PROPN
iajs-3482	167	28	=	=	SYM
iajs-3482	167	29	0	0	PUNCT
iajs-3482	168	1	and	and	CCONJ
iajs-3482	168	2	𝑅(𝑘	𝑅(𝑘	ADP
iajs-3482	168	3	𝑜	𝑜	NOUN
iajs-3482	168	4	𝑞	𝑞	X
iajs-3482	168	5	)	)	PUNCT
iajs-3482	169	1	+	+	CCONJ
iajs-3482	169	2	𝛼(𝑘)′𝑜	𝛼(𝑘)′𝑜	PROPN
iajs-3482	169	3	𝑅(𝑞	𝑅(𝑞	NUM
iajs-3482	169	4	)	)	PUNCT
iajs-3482	169	5	=	=	SYM
iajs-3482	170	1	0	0	X
iajs-3482	170	2	.	.	PUNCT
iajs-3482	171	1	lemma	lemma	PROPN
iajs-3482	171	2	(	(	PUNCT
iajs-3482	171	3	3.6	3.6	NUM
iajs-3482	171	4	)	)	PUNCT
iajs-3482	171	5	let	let	VERB
iajs-3482	171	6	𝑀	𝑀	PRON
iajs-3482	171	7	be	be	AUX
iajs-3482	171	8	𝑎	𝑎	PROPN
iajs-3482	171	9	2	2	NUM
iajs-3482	171	10	−	−	NOUN
iajs-3482	171	11	tortion	tortion	NOUN
iajs-3482	171	12	free	free	ADJ
iajs-3482	171	13	prime	prime	ADJ
iajs-3482	171	14	semi	semi	ADJ
iajs-3482	171	15	-	-	NOUN
iajs-3482	171	16	ring	ring	ADJ
iajs-3482	171	17	,	,	PUNCT
iajs-3482	171	18	and	and	CCONJ
iajs-3482	171	19	𝑅	𝑅	NOUN
iajs-3482	171	20	,	,	PUNCT
iajs-3482	171	21	𝛼	𝛼	NOUN
iajs-3482	171	22	are	be	AUX
iajs-3482	171	23	additive	additive	ADJ
iajs-3482	171	24	mappings	mapping	NOUN
iajs-3482	171	25	on	on	ADP
iajs-3482	171	26	𝑀	𝑀	PROPN
iajs-3482	171	27	,	,	PUNCT
iajs-3482	171	28	𝑅	𝑅	PROPN
iajs-3482	171	29	satisfies	satisfie	NOUN
iajs-3482	171	30	𝑅(𝑘	𝑅(𝑘	PRON
iajs-3482	171	31	o	o	X
iajs-3482	171	32	𝑞	𝑞	X
iajs-3482	171	33	)	)	PUNCT
iajs-3482	171	34	+	+	CCONJ
iajs-3482	171	35	𝑅(𝑘)o	𝑅(𝑘)o	ADJ
iajs-3482	171	36	𝛼(𝑞)′	𝛼(𝑞)′	PROPN
iajs-3482	171	37	=	=	SYM
iajs-3482	171	38	0	0	PUNCT
iajs-3482	171	39	and	and	CCONJ
iajs-3482	171	40	𝑅(𝑘	𝑅(𝑘	ADP
iajs-3482	171	41	𝑜	𝑜	NOUN
iajs-3482	171	42	𝑞	𝑞	X
iajs-3482	171	43	)	)	PUNCT
iajs-3482	171	44	+	+	CCONJ
iajs-3482	172	1	𝛼(𝑘)′	𝛼(𝑘)′	NUM
iajs-3482	172	2	𝑜	𝑜	PRON
iajs-3482	172	3	𝑅(𝑞	𝑅(𝑞	NUM
iajs-3482	172	4	)	)	PUNCT
iajs-3482	172	5	=	=	SYM
iajs-3482	172	6	0	0	NUM
iajs-3482	172	7	for	for	ADP
iajs-3482	172	8	any𝑘	any𝑘	NOUN
iajs-3482	172	9	,	,	PUNCT
iajs-3482	172	10	𝑞	𝑞	PROPN
iajs-3482	172	11	∈	∈	PROPN
iajs-3482	172	12	𝑉	𝑉	PROPN
iajs-3482	172	13	,	,	PUNCT
iajs-3482	172	14	then	then	ADV
iajs-3482	172	15	𝑅(𝑧	𝑅(𝑧	NUM
iajs-3482	172	16	)	)	PUNCT
iajs-3482	172	17	∈	∈	NOUN
iajs-3482	172	18	𝑍(𝑉	𝑍(𝑉	NOUN
iajs-3482	172	19	)	)	PUNCT
iajs-3482	172	20	for	for	ADP
iajs-3482	172	21	any	any	DET
iajs-3482	172	22	𝑧	𝑧	DET
iajs-3482	172	23	∈	∈	PROPN
iajs-3482	172	24	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	172	25	)	)	PUNCT
iajs-3482	172	26	,	,	PUNCT
iajs-3482	172	27	where	where	SCONJ
iajs-3482	172	28	𝛼	𝛼	PRON
iajs-3482	172	29	is	be	AUX
iajs-3482	172	30	a	a	DET
iajs-3482	172	31	surjective	surjective	ADJ
iajs-3482	172	32	endomorphism	endomorphism	NOUN
iajs-3482	172	33	of	of	ADP
iajs-3482	172	34	𝑉	𝑉	PROPN
iajs-3482	172	35	..	..	PUNCT
iajs-3482	172	36	proof	proof	NOUN
iajs-3482	172	37	:	:	PUNCT
iajs-3482	172	38	𝑅(𝑘𝑞	𝑅(𝑘𝑞	X
iajs-3482	172	39	+	+	CCONJ
iajs-3482	172	40	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	172	41	)	)	PUNCT
iajs-3482	173	1	+	+	CCONJ
iajs-3482	173	2	𝑅(𝑘)𝛼(𝑞)′	𝑅(𝑘)𝛼(𝑞)′	PROPN
iajs-3482	173	3	+	+	NOUN
iajs-3482	173	4	𝛼(𝑞)′𝑅(𝑘	𝛼(𝑞)′𝑅(𝑘	NOUN
iajs-3482	173	5	)	)	PUNCT
iajs-3482	173	6	=	=	SYM
iajs-3482	173	7	0	0	PUNCT
iajs-3482	173	8	𝑅(𝑘𝑞	𝑅(𝑘𝑞	NOUN
iajs-3482	173	9	+	+	CCONJ
iajs-3482	173	10	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	173	11	)	)	PUNCT
iajs-3482	173	12	+	+	CCONJ
iajs-3482	173	13	𝛼(𝑘)′𝑅(𝑞	𝛼(𝑘)′𝑅(𝑞	NOUN
iajs-3482	173	14	)	)	PUNCT
iajs-3482	173	15	+	+	NUM
iajs-3482	173	16	𝑅(𝑞)𝛼(𝑘)′	𝑅(𝑞)𝛼(𝑘)′	NOUN
iajs-3482	173	17	=	=	SYM
iajs-3482	173	18	0	0	NUM
iajs-3482	173	19	.	.	PUNCT
iajs-3482	174	1	because	because	SCONJ
iajs-3482	174	2	𝑅(𝑧	𝑅(𝑧	NUM
iajs-3482	174	3	)	)	PUNCT
iajs-3482	174	4	∈	∈	NOUN
iajs-3482	174	5	𝑍(𝑉	𝑍(𝑉	NOUN
iajs-3482	174	6	)	)	PUNCT
iajs-3482	175	1	take	take	VERB
iajs-3482	175	2	any	any	DET
iajs-3482	175	3	t	t	NOUN
iajs-3482	175	4	∈	∈	PROPN
iajs-3482	175	5	z(u	z(u	PROPN
iajs-3482	175	6	)	)	PUNCT
iajs-3482	175	7	and	and	CCONJ
iajs-3482	175	8	denote	denote	VERB
iajs-3482	175	9	a	a	DET
iajs-3482	175	10	=	=	NOUN
iajs-3482	175	11	r(t	r(t	NOUN
iajs-3482	175	12	)	)	PUNCT
iajs-3482	175	13	”	"	PUNCT
iajs-3482	176	1	2𝑅(𝑡𝑘	2𝑅(𝑡𝑘	X
iajs-3482	176	2	)	)	PUNCT
iajs-3482	176	3	=	=	SYM
iajs-3482	176	4	𝑅(𝑡𝑘	𝑅(𝑡𝑘	X
iajs-3482	176	5	+	+	NUM
iajs-3482	176	6	𝑘𝑡	𝑘𝑡	PROPN
iajs-3482	176	7	)	)	PUNCT
iajs-3482	176	8	=	=	PUNCT
iajs-3482	176	9	𝑅(𝑡)𝛼(𝑘	𝑅(𝑡)𝛼(𝑘	ADJ
iajs-3482	176	10	)	)	PUNCT
iajs-3482	176	11	+	+	NUM
iajs-3482	176	12	𝛼(𝑘)𝑅(𝑡	𝛼(𝑘)𝑅(𝑡	NOUN
iajs-3482	176	13	)	)	PUNCT
iajs-3482	176	14	=	=	SYM
iajs-3482	176	15	𝑎𝛼(𝑘	𝑎𝛼(𝑘	X
iajs-3482	176	16	)	)	PUNCT
iajs-3482	177	1	+	+	X
iajs-3482	177	2	𝛼(𝑘)𝑎	𝛼(𝑘)𝑎	ADV
iajs-3482	177	3	a	a	DET
iajs-3482	177	4	simple	simple	ADJ
iajs-3482	177	5	check	check	NOUN
iajs-3482	177	6	reveals	reveal	VERB
iajs-3482	177	7	that	that	SCONJ
iajs-3482	177	8	𝑀(𝑘	𝑀(𝑘	NOUN
iajs-3482	177	9	)	)	PUNCT
iajs-3482	177	10	=	=	SYM
iajs-3482	178	1	2𝑅(𝑡𝑘	2𝑅(𝑡𝑘	X
iajs-3482	178	2	)	)	PUNCT
iajs-3482	178	3	is	be	AUX
iajs-3482	178	4	satisfies	satisfie	NOUN
iajs-3482	178	5	𝑀(𝑘	𝑀(𝑘	PROPN
iajs-3482	178	6	𝑜	𝑜	NOUN
iajs-3482	178	7	𝑞	𝑞	PROPN
iajs-3482	178	8	)	)	PUNCT
iajs-3482	178	9	=	=	SYM
iajs-3482	178	10	2𝑅(𝑡(𝑘𝑞	2𝑅(𝑡(𝑘𝑞	NUM
iajs-3482	178	11	+	+	CCONJ
iajs-3482	178	12	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	178	13	)	)	PUNCT
iajs-3482	178	14	=	=	SYM
iajs-3482	179	1	2𝑅(𝑡𝑘𝑞	2𝑅(𝑡𝑘𝑞	NUM
iajs-3482	179	2	+	+	NUM
iajs-3482	179	3	𝑞𝑡𝑘	𝑞𝑡𝑘	NOUN
iajs-3482	179	4	)	)	PUNCT
iajs-3482	179	5	=	=	SYM
iajs-3482	179	6	2𝑅(𝑡𝑘)𝛼(𝑞	2𝑅(𝑡𝑘)𝛼(𝑞	NUM
iajs-3482	179	7	)	)	PUNCT
iajs-3482	179	8	+	+	CCONJ
iajs-3482	179	9	2	2	NUM
iajs-3482	179	10	𝛼(𝑞)𝑅(𝑡𝑘	𝛼(𝑞)𝑅(𝑡𝑘	NUM
iajs-3482	179	11	)	)	PUNCT
iajs-3482	179	12	ihjpas	ihjpa	NOUN
iajs-3482	179	13	.	.	PUNCT
iajs-3482	180	1	2025	2025	NUM
iajs-3482	180	2	,	,	PUNCT
iajs-3482	180	3	38	38	NUM
iajs-3482	180	4	(	(	PUNCT
iajs-3482	180	5	1	1	NUM
iajs-3482	180	6	)	)	PUNCT
iajs-3482	180	7	404	404	NUM
iajs-3482	180	8	=	=	SYM
iajs-3482	180	9	𝑀(𝑘)𝛼(𝑞	𝑀(𝑘)𝛼(𝑞	NOUN
iajs-3482	180	10	)	)	PUNCT
iajs-3482	181	1	+	+	CCONJ
iajs-3482	181	2	𝛼(𝑞)𝑀(𝑘	𝛼(𝑞)𝑀(𝑘	NOUN
iajs-3482	181	3	)	)	PUNCT
iajs-3482	181	4	=	=	SYM
iajs-3482	181	5	𝑀(𝑘)𝑜𝛼(𝑞	𝑀(𝑘)𝑜𝛼(𝑞	NOUN
iajs-3482	181	6	)	)	PUNCT
iajs-3482	181	7	𝑀(𝑘	𝑀(𝑘	PROPN
iajs-3482	182	1	𝑜𝑞	𝑜𝑞	PROPN
iajs-3482	182	2	)	)	PUNCT
iajs-3482	182	3	=	=	SYM
iajs-3482	182	4	2𝑅(𝑡(𝑘𝑞	2𝑅(𝑡(𝑘𝑞	NUM
iajs-3482	182	5	+	+	CCONJ
iajs-3482	182	6	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	182	7	)	)	PUNCT
iajs-3482	182	8	=	=	SYM
iajs-3482	182	9	2𝑅(𝑘(𝑡𝑞	2𝑅(𝑘(𝑡𝑞	NOUN
iajs-3482	182	10	)	)	PUNCT
iajs-3482	182	11	+	+	CCONJ
iajs-3482	182	12	(	(	PUNCT
iajs-3482	182	13	𝑡𝑞)𝑘	𝑡𝑞)𝑘	NOUN
iajs-3482	182	14	)	)	PUNCT
iajs-3482	182	15	=	=	SYM
iajs-3482	182	16	2𝛼(𝑘)𝑅((𝑡𝑞	2𝛼(𝑘)𝑅((𝑡𝑞	X
iajs-3482	182	17	)	)	PUNCT
iajs-3482	182	18	+	+	NUM
iajs-3482	182	19	2𝑅(𝑡𝑞)𝛼(𝑘	2𝑅(𝑡𝑞)𝛼(𝑘	X
iajs-3482	182	20	)	)	PUNCT
iajs-3482	182	21	=	=	SYM
iajs-3482	182	22	𝛼(𝑘)𝑀(𝑞	𝛼(𝑘)𝑀(𝑞	PROPN
iajs-3482	182	23	)	)	PUNCT
iajs-3482	182	24	+	+	CCONJ
iajs-3482	182	25	𝑀(𝑞)𝛼(𝑘	𝑀(𝑞)𝛼(𝑘	X
iajs-3482	182	26	)	)	PUNCT
iajs-3482	182	27	=	=	SYM
iajs-3482	182	28	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	182	29	)	)	PUNCT
iajs-3482	182	30	𝑜	𝑜	PROPN
iajs-3482	182	31	𝑀(𝑞	𝑀(𝑞	X
iajs-3482	182	32	)	)	PUNCT
iajs-3482	182	33	𝑀(𝑘	𝑀(𝑘	PROPN
iajs-3482	182	34	𝑜	𝑜	NOUN
iajs-3482	182	35	𝑞	𝑞	PROPN
iajs-3482	182	36	)	)	PUNCT
iajs-3482	182	37	=	=	SYM
iajs-3482	182	38	𝑀(𝑘	𝑀(𝑘	NUM
iajs-3482	182	39	)	)	PUNCT
iajs-3482	182	40	𝑜	𝑜	NOUN
iajs-3482	182	41	𝛼(𝑞	𝛼(𝑞	NOUN
iajs-3482	182	42	)	)	PUNCT
iajs-3482	182	43	=	=	SYM
iajs-3482	182	44	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	182	45	)	)	PUNCT
iajs-3482	182	46	𝑜	𝑜	PROPN
iajs-3482	182	47	𝑀(𝑞	𝑀(𝑞	VERB
iajs-3482	182	48	)	)	PUNCT
iajs-3482	182	49	𝑓𝑜𝑛	𝑓𝑜𝑛	AUX
iajs-3482	182	50	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	182	51	𝑘	𝑘	X
iajs-3482	182	52	,	,	PUNCT
iajs-3482	182	53	𝑞	𝑞	PROPN
iajs-3482	182	54	∈	∈	PROPN
iajs-3482	182	55	𝑀	𝑀	PROPN
iajs-3482	182	56	by	by	ADP
iajs-3482	182	57	lemma	lemma	PROPN
iajs-3482	182	58	(	(	PUNCT
iajs-3482	182	59	3.5	3.5	NUM
iajs-3482	182	60	)	)	PUNCT
iajs-3482	182	61	,	,	PUNCT
iajs-3482	182	62	we	we	PRON
iajs-3482	182	63	have	have	VERB
iajs-3482	182	64	𝑅(𝑡	𝑅(𝑡	NOUN
iajs-3482	182	65	)	)	PUNCT
iajs-3482	182	66	∈	∈	PROPN
iajs-3482	182	67	𝑍(𝑀	𝑍(𝑀	NOUN
iajs-3482	182	68	)	)	PUNCT
iajs-3482	182	69	.	.	PUNCT
iajs-3482	183	1	theorem	theorem	NOUN
iajs-3482	183	2	(	(	PUNCT
iajs-3482	183	3	3.7	3.7	NUM
iajs-3482	183	4	)	)	PUNCT
iajs-3482	183	5	let	let	VERB
iajs-3482	183	6	𝑀	𝑀	PRON
iajs-3482	183	7	be	be	AUX
iajs-3482	183	8	2	2	NUM
iajs-3482	183	9	−	−	NOUN
iajs-3482	183	10	tortion	tortion	NOUN
iajs-3482	183	11	free	free	ADJ
iajs-3482	183	12	prime	prime	NOUN
iajs-3482	183	13	semi	semi	ADJ
iajs-3482	183	14	−	−	PROPN
iajs-3482	183	15	ring	ring	NOUN
iajs-3482	183	16	and	and	CCONJ
iajs-3482	183	17	𝑅	𝑅	NOUN
iajs-3482	183	18	,	,	PUNCT
iajs-3482	183	19	𝛼	𝛼	PROPN
iajs-3482	183	20	:	:	PUNCT
iajs-3482	183	21	𝑀	𝑀	PROPN
iajs-3482	183	22	→	→	SYM
iajs-3482	183	23	𝑀	𝑀	PROPN
iajs-3482	183	24	additive	additive	ADJ
iajs-3482	183	25	mappings	mapping	NOUN
iajs-3482	183	26	,	,	PUNCT
iajs-3482	183	27	r	r	NOUN
iajs-3482	183	28	satisfies	satisfie	NOUN
iajs-3482	183	29	𝑅(𝑘	𝑅(𝑘	PRON
iajs-3482	183	30	𝑜	𝑜	NOUN
iajs-3482	183	31	𝑞	𝑞	X
iajs-3482	183	32	)	)	PUNCT
iajs-3482	184	1	+	+	CCONJ
iajs-3482	184	2	𝑅(𝑘)𝑜𝛼(𝑞)′	𝑅(𝑘)𝑜𝛼(𝑞)′	PROPN
iajs-3482	184	3	=	=	SYM
iajs-3482	184	4	0	0	NUM
iajs-3482	184	5	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3482	184	6	𝑅(𝑘	𝑅(𝑘	PUNCT
iajs-3482	184	7	𝑜	𝑜	PROPN
iajs-3482	184	8	𝑞	𝑞	X
iajs-3482	184	9	)	)	PUNCT
iajs-3482	184	10	+	+	NUM
iajs-3482	184	11	𝛼(𝑘)′𝑜𝑅(𝑞	𝛼(𝑘)′𝑜𝑅(𝑞	NUM
iajs-3482	184	12	)	)	PUNCT
iajs-3482	184	13	=	=	SYM
iajs-3482	184	14	0	0	NUM
iajs-3482	184	15	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	184	16	𝑎𝑙𝑙	𝑎𝑙𝑙	X
iajs-3482	184	17	𝑘	𝑘	PROPN
iajs-3482	184	18	,	,	PUNCT
iajs-3482	184	19	𝑞	𝑞	PROPN
iajs-3482	184	20	∈	∈	PROPN
iajs-3482	184	21	𝑉	𝑉	PROPN
iajs-3482	184	22	then	then	ADV
iajs-3482	184	23	𝑅	𝑅	PROPN
iajs-3482	184	24	is	be	AUX
iajs-3482	184	25	𝑎	𝑎	PROPN
iajs-3482	184	26	𝛼	𝛼	NOUN
iajs-3482	184	27	−	−	NOUN
iajs-3482	184	28	centralizer	centralizer	NOUN
iajs-3482	184	29	on	on	ADP
iajs-3482	184	30	𝑉	𝑉	PROPN
iajs-3482	184	31	,	,	PUNCT
iajs-3482	184	32	where	where	SCONJ
iajs-3482	184	33	𝛼	𝛼	PRON
iajs-3482	184	34	is	be	AUX
iajs-3482	184	35	an	an	DET
iajs-3482	184	36	automorphism	automorphism	NOUN
iajs-3482	184	37	of	of	ADP
iajs-3482	184	38	𝑉	𝑉	PROPN
iajs-3482	184	39	,	,	PUNCT
iajs-3482	184	40	𝑅(𝑢	𝑅(𝑢	NUM
iajs-3482	184	41	)	)	PUNCT
iajs-3482	184	42	∈	∈	PROPN
iajs-3482	184	43	𝑉	𝑉	PROPN
iajs-3482	184	44	,	,	PUNCT
iajs-3482	184	45	for	for	ADP
iajs-3482	184	46	any	any	DET
iajs-3482	184	47	𝑢	𝑢	PROPN
iajs-3482	184	48	∈	∈	PROPN
iajs-3482	184	49	𝑉	𝑉	PROPN
iajs-3482	184	50	,	,	PUNCT
iajs-3482	184	51	and	and	CCONJ
iajs-3482	184	52	𝛼(𝑍(𝑉	𝛼(𝑍(𝑉	NOUN
iajs-3482	184	53	)	)	PUNCT
iajs-3482	184	54	)	)	PUNCT
iajs-3482	185	1	=	=	SYM
iajs-3482	185	2	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	185	3	)	)	PUNCT
iajs-3482	185	4	.	.	PUNCT
iajs-3482	186	1	proof	proof	NOUN
iajs-3482	186	2	:	:	PUNCT
iajs-3482	186	3	since	since	SCONJ
iajs-3482	186	4	u	u	NOUN
iajs-3482	186	5	is	be	AUX
iajs-3482	186	6	a	a	DET
iajs-3482	186	7	square	square	ADJ
iajs-3482	186	8	closed	closed	ADJ
iajs-3482	186	9	lie	lie	NOUN
iajs-3482	186	10	−	−	PROPN
iajs-3482	186	11	ideal	ideal	NOUN
iajs-3482	186	12	of	of	ADP
iajs-3482	186	13	𝑀	𝑀	PROPN
iajs-3482	186	14	,	,	PUNCT
iajs-3482	186	15	and	and	CCONJ
iajs-3482	186	16	by	by	ADP
iajs-3482	186	17	lemma	lemma	PROPN
iajs-3482	186	18	(	(	PUNCT
iajs-3482	186	19	2.5	2.5	NUM
iajs-3482	186	20	)	)	PUNCT
iajs-3482	186	21	,	,	PUNCT
iajs-3482	186	22	we	we	PRON
iajs-3482	186	23	get	get	VERB
iajs-3482	186	24	2𝑅(𝑘𝑞	2𝑅(𝑘𝑞	NUM
iajs-3482	186	25	+	+	NOUN
iajs-3482	186	26	𝑞𝑘	𝑞𝑘	NOUN
iajs-3482	186	27	)	)	PUNCT
iajs-3482	186	28	=	=	SYM
iajs-3482	186	29	2𝑅(𝑘)𝛼(𝑞	2𝑅(𝑘)𝛼(𝑞	PROPN
iajs-3482	186	30	)	)	PUNCT
iajs-3482	187	1	+	+	NUM
iajs-3482	187	2	2𝛼(𝑞)𝑅(𝑘	2𝛼(𝑞)𝑅(𝑘	X
iajs-3482	188	1	)	)	PUNCT
iajs-3482	188	2	=	=	SYM
iajs-3482	188	3	2𝛼(𝑘)𝑅(𝑞	2𝛼(𝑘)𝑅(𝑞	NUM
iajs-3482	188	4	)	)	PUNCT
iajs-3482	189	1	+	+	CCONJ
iajs-3482	189	2	2𝑅(𝑞)𝛼(𝑘	2𝑅(𝑞)𝛼(𝑘	NOUN
iajs-3482	189	3	)	)	PUNCT
iajs-3482	189	4	if	if	SCONJ
iajs-3482	189	5	v	v	NOUN
iajs-3482	189	6	is	be	AUX
iajs-3482	189	7	a	a	DET
iajs-3482	189	8	commutative	commutative	ADJ
iajs-3482	189	9	,	,	PUNCT
iajs-3482	189	10	we	we	PRON
iajs-3482	189	11	have	have	VERB
iajs-3482	189	12	r(r2	r(r2	NOUN
iajs-3482	189	13	)	)	PUNCT
iajs-3482	189	14	=	=	SYM
iajs-3482	189	15	r(r)α(r	r(r)α(r	NOUN
iajs-3482	189	16	)	)	PUNCT
iajs-3482	189	17	=	=	SYM
iajs-3482	189	18	α(r)r(r	α(r)r(r	NOUN
iajs-3482	189	19	)	)	PUNCT
iajs-3482	189	20	if	if	SCONJ
iajs-3482	189	21	v	v	NOUN
iajs-3482	189	22	is	be	AUX
iajs-3482	189	23	a	a	DET
iajs-3482	189	24	non	non	ADJ
iajs-3482	189	25	-	-	ADJ
iajs-3482	189	26	commutative	commutative	ADJ
iajs-3482	189	27	replace	replace	NOUN
iajs-3482	189	28	𝑞	𝑞	X
iajs-3482	189	29	by	by	ADP
iajs-3482	189	30	2𝑘𝑞	2𝑘𝑞	NOUN
iajs-3482	189	31	+	+	CCONJ
iajs-3482	189	32	2𝑞𝑘	2𝑞𝑘	NOUN
iajs-3482	189	33	in	in	ADP
iajs-3482	189	34	(	(	PUNCT
iajs-3482	189	35	2	2	NUM
iajs-3482	189	36	)	)	PUNCT
iajs-3482	189	37	,	,	PUNCT
iajs-3482	189	38	we	we	PRON
iajs-3482	189	39	get	get	VERB
iajs-3482	189	40	,	,	PUNCT
iajs-3482	189	41	4𝑅(𝑘)𝛼(𝑘𝑞	4𝑅(𝑘)𝛼(𝑘𝑞	NUM
iajs-3482	189	42	+	+	CCONJ
iajs-3482	189	43	𝑞𝑘	𝑞𝑘	ADP
iajs-3482	189	44	)	)	PUNCT
iajs-3482	189	45	+	+	CCONJ
iajs-3482	189	46	4𝛼(𝑘𝑞	4𝛼(𝑘𝑞	NUM
iajs-3482	189	47	+	+	CCONJ
iajs-3482	189	48	𝑞𝑘)𝑅(𝑘	𝑞𝑘)𝑅(𝑘	NOUN
iajs-3482	189	49	)	)	PUNCT
iajs-3482	189	50	=	=	PUNCT
iajs-3482	190	1	4𝛼(𝑘)𝑅(𝑘𝑞	4𝛼(𝑘)𝑅(𝑘𝑞	NUM
iajs-3482	190	2	+	+	CCONJ
iajs-3482	190	3	𝑞𝑘	𝑞𝑘	ADP
iajs-3482	190	4	)	)	PUNCT
iajs-3482	190	5	+	+	CCONJ
iajs-3482	190	6	4𝑅(𝑘𝑞	4𝑅(𝑘𝑞	NUM
iajs-3482	190	7	+	+	NUM
iajs-3482	190	8	𝑞𝑘)𝛼(𝑘	𝑞𝑘)𝛼(𝑘	NOUN
iajs-3482	190	9	)	)	PUNCT
iajs-3482	190	10	4𝑅(𝑘)𝛼(𝑘)𝛼(𝑞	4𝑅(𝑘)𝛼(𝑘)𝛼(𝑞	NUM
iajs-3482	190	11	)	)	PUNCT
iajs-3482	190	12	+	+	NOUN
iajs-3482	190	13	4𝑅(𝑘)𝛼(𝑞)𝛼(𝑘	4𝑅(𝑘)𝛼(𝑞)𝛼(𝑘	NUM
iajs-3482	190	14	)	)	PUNCT
iajs-3482	190	15	+	+	NUM
iajs-3482	190	16	4𝛼(𝑘)𝛼(𝑞)𝑅(𝑘	4𝛼(𝑘)𝛼(𝑞)𝑅(𝑘	NUM
iajs-3482	190	17	)	)	PUNCT
iajs-3482	191	1	+	+	NUM
iajs-3482	191	2	4𝛼(𝑞)𝛼(𝑘)𝑅(𝑘	4𝛼(𝑞)𝛼(𝑘)𝑅(𝑘	NOUN
iajs-3482	191	3	)	)	PUNCT
iajs-3482	191	4	=	=	SYM
iajs-3482	191	5	4𝛼(𝑘)𝑅(𝑘)𝛼(𝑞	4𝛼(𝑘)𝑅(𝑘)𝛼(𝑞	NUM
iajs-3482	191	6	)	)	PUNCT
iajs-3482	192	1	+	+	CCONJ
iajs-3482	192	2	4𝛼(𝑘)𝛼(𝑞)𝑅(𝑘	4𝛼(𝑘)𝛼(𝑞)𝑅(𝑘	X
iajs-3482	192	3	)	)	PUNCT
iajs-3482	193	1	+	+	NOUN
iajs-3482	194	1	4𝑅(𝑘)𝛼(𝑞)𝛼(𝑘	4𝑅(𝑘)𝛼(𝑞)𝛼(𝑘	NUM
iajs-3482	194	2	)	)	PUNCT
iajs-3482	194	3	+	+	NUM
iajs-3482	194	4	4𝛼(𝑞)𝑅(𝑘)𝛼(𝑘	4𝛼(𝑞)𝑅(𝑘)𝛼(𝑘	NUM
iajs-3482	194	5	)	)	PUNCT
iajs-3482	194	6	by	by	ADP
iajs-3482	194	7	using	use	VERB
iajs-3482	194	8	the	the	DET
iajs-3482	194	9	property	property	NOUN
iajs-3482	194	10	of	of	ADP
iajs-3482	194	11	2	2	NUM
iajs-3482	194	12	−	−	NOUN
iajs-3482	194	13	tortion	tortion	NOUN
iajs-3482	194	14	free	free	ADJ
iajs-3482	194	15	semi	semi	NOUN
iajs-3482	194	16	−	−	PROPN
iajs-3482	194	17	ring	ring	NOUN
iajs-3482	194	18	,	,	PUNCT
iajs-3482	194	19	we	we	PRON
iajs-3482	194	20	obtain	obtain	VERB
iajs-3482	194	21	𝑅(𝑘)𝛼(𝑘)𝛼(𝑞	𝑅(𝑘)𝛼(𝑘)𝛼(𝑞	NUM
iajs-3482	194	22	)	)	PUNCT
iajs-3482	195	1	+	+	CCONJ
iajs-3482	195	2	𝛼(𝑞)𝛼(𝑘)𝑅(𝑘	𝛼(𝑞)𝛼(𝑘)𝑅(𝑘	X
iajs-3482	195	3	)	)	PUNCT
iajs-3482	195	4	+	+	NUM
iajs-3482	195	5	𝛼(𝑘)′𝑅(𝑘)𝛼(𝑞	𝛼(𝑘)′𝑅(𝑘)𝛼(𝑞	NOUN
iajs-3482	195	6	)	)	PUNCT
iajs-3482	196	1	+	+	CCONJ
iajs-3482	197	1	𝛼(𝑞)𝑅(𝑘)𝛼(𝑘)′	𝛼(𝑞)𝑅(𝑘)𝛼(𝑘)′	PROPN
iajs-3482	197	2	=	=	SYM
iajs-3482	197	3	0	0	NUM
iajs-3482	198	1	now	now	ADV
iajs-3482	198	2	it	it	PRON
iajs-3482	198	3	follows	follow	VERB
iajs-3482	198	4	that	that	SCONJ
iajs-3482	198	5	[	[	X
iajs-3482	198	6	𝑅(𝑘	𝑅(𝑘	NOUN
iajs-3482	198	7	)	)	PUNCT
iajs-3482	198	8	,	,	PUNCT
iajs-3482	198	9	𝛼(𝑘)]𝛼(𝑞	𝛼(𝑘)]𝛼(𝑞	PROPN
iajs-3482	198	10	)	)	PUNCT
iajs-3482	198	11	=	=	PUNCT
iajs-3482	199	1	𝛼(𝑞)[𝑅(𝑘	𝛼(𝑞)[𝑅(𝑘	NOUN
iajs-3482	199	2	)	)	PUNCT
iajs-3482	199	3	,	,	PUNCT
iajs-3482	199	4	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	199	5	)	)	PUNCT
iajs-3482	199	6	]	]	PUNCT
iajs-3482	200	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	200	2	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	200	3	𝑘	𝑘	PROPN
iajs-3482	200	4	,	,	PUNCT
iajs-3482	200	5	𝑞	𝑞	PROPN
iajs-3482	200	6	∈	∈	PROPN
iajs-3482	200	7	𝑉	𝑉	PROPN
iajs-3482	200	8	but	but	CCONJ
iajs-3482	200	9	α	α	PROPN
iajs-3482	200	10	is	be	AUX
iajs-3482	200	11	surjective	surjective	ADJ
iajs-3482	200	12	,	,	PUNCT
iajs-3482	200	13	then	then	ADV
iajs-3482	200	14	we	we	PRON
iajs-3482	200	15	get	get	VERB
iajs-3482	200	16	[	[	PUNCT
iajs-3482	200	17	𝑅(𝑘	𝑅(𝑘	NOUN
iajs-3482	200	18	)	)	PUNCT
iajs-3482	200	19	,	,	PUNCT
iajs-3482	200	20	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	200	21	)	)	PUNCT
iajs-3482	200	22	]	]	PUNCT
iajs-3482	201	1	∈	∈	PROPN
iajs-3482	201	2	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	201	3	)	)	PUNCT
iajs-3482	201	4	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	201	5	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	201	6	𝑘	𝑘	PROPN
iajs-3482	201	7	,	,	PUNCT
iajs-3482	201	8	𝑞	𝑞	PROPN
iajs-3482	201	9	∈	∈	PROPN
iajs-3482	201	10	𝑉	𝑉	PROPN
iajs-3482	201	11	the	the	DET
iajs-3482	201	12	next	next	ADJ
iajs-3482	201	13	goal	goal	NOUN
iajs-3482	201	14	is	be	AUX
iajs-3482	201	15	to	to	PART
iajs-3482	201	16	show	show	VERB
iajs-3482	201	17	that	that	SCONJ
iajs-3482	201	18	[	[	X
iajs-3482	201	19	𝑅(𝑘	𝑅(𝑘	NOUN
iajs-3482	201	20	)	)	PUNCT
iajs-3482	201	21	,	,	PUNCT
iajs-3482	201	22	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	201	23	)	)	PUNCT
iajs-3482	201	24	]	]	PUNCT
iajs-3482	202	1	=	=	SYM
iajs-3482	202	2	0	0	NUM
iajs-3482	202	3	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	202	4	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	202	5	𝑘	𝑘	PRON
iajs-3482	202	6	∈	∈	PROPN
iajs-3482	202	7	𝑉.	𝑉.	NOUN
iajs-3482	202	8	take	take	VERB
iajs-3482	202	9	any	any	DET
iajs-3482	202	10	t	t	NOUN
iajs-3482	202	11	∈	∈	PROPN
iajs-3482	202	12	z(u	z(u	PROPN
iajs-3482	202	13	)	)	PUNCT
iajs-3482	202	14	4𝑅(𝑡𝑘	4𝑅(𝑡𝑘	NUM
iajs-3482	202	15	)	)	PUNCT
iajs-3482	202	16	=	=	SYM
iajs-3482	203	1	2𝑅(𝑡𝑘	2𝑅(𝑡𝑘	NUM
iajs-3482	203	2	+	+	CCONJ
iajs-3482	203	3	𝑘𝑡	𝑘𝑡	NOUN
iajs-3482	203	4	)	)	PUNCT
iajs-3482	203	5	=	=	SYM
iajs-3482	203	6	2𝑅(𝑡)𝛼(𝑘	2𝑅(𝑡)𝛼(𝑘	X
iajs-3482	203	7	)	)	PUNCT
iajs-3482	204	1	+	+	CCONJ
iajs-3482	204	2	2𝛼(𝑘)𝑅(𝑡	2𝛼(𝑘)𝑅(𝑡	X
iajs-3482	204	3	)	)	PUNCT
iajs-3482	204	4	=	=	SYM
iajs-3482	204	5	2𝑅(𝑘)𝛼(𝑡	2𝑅(𝑘)𝛼(𝑡	NUM
iajs-3482	204	6	)	)	PUNCT
iajs-3482	204	7	+	+	CCONJ
iajs-3482	204	8	2𝛼(𝑡)𝑅(𝑘	2𝛼(𝑡)𝑅(𝑘	X
iajs-3482	204	9	)	)	PUNCT
iajs-3482	204	10	using	use	VERB
iajs-3482	204	11	lemma	lemma	PROPN
iajs-3482	204	12	(	(	PUNCT
iajs-3482	204	13	3.6	3.6	NUM
iajs-3482	204	14	)	)	PUNCT
iajs-3482	204	15	,	,	PUNCT
iajs-3482	204	16	we	we	PRON
iajs-3482	204	17	get	get	VERB
iajs-3482	204	18	𝑅(𝑡𝑘	𝑅(𝑡𝑘	NOUN
iajs-3482	204	19	)	)	PUNCT
iajs-3482	204	20	=	=	SYM
iajs-3482	204	21	𝑅(𝑘)𝛼(𝑡	𝑅(𝑘)𝛼(𝑡	NOUN
iajs-3482	204	22	)	)	PUNCT
iajs-3482	204	23	=	=	SYM
iajs-3482	204	24	𝑅(𝑡)𝛼(𝑘	𝑅(𝑡)𝛼(𝑘	ADJ
iajs-3482	204	25	)	)	PUNCT
iajs-3482	204	26	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3482	204	27	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3482	204	28	𝑘	𝑘	PROPN
iajs-3482	204	29	,	,	PUNCT
iajs-3482	204	30	𝑡	𝑡	PROPN
iajs-3482	204	31	∈	∈	PROPN
iajs-3482	204	32	𝑉	𝑉	PROPN
iajs-3482	204	33	4[𝑅(𝑘	4[𝑅(𝑘	NOUN
iajs-3482	204	34	)	)	PUNCT
iajs-3482	204	35	,	,	PUNCT
iajs-3482	204	36	𝛼(𝑘)]𝛼(𝑡	𝛼(𝑘)]𝛼(𝑡	NOUN
iajs-3482	204	37	)	)	PUNCT
iajs-3482	204	38	=	=	SYM
iajs-3482	204	39	4𝑅(𝑘)𝛼(𝑘𝑡	4𝑅(𝑘)𝛼(𝑘𝑡	X
iajs-3482	204	40	)	)	PUNCT
iajs-3482	204	41	+	+	NOUN
iajs-3482	204	42	4𝛼(𝑘)′𝑅(𝑘)𝛼(𝑡	4𝛼(𝑘)′𝑅(𝑘)𝛼(𝑡	X
iajs-3482	204	43	)	)	PUNCT
iajs-3482	204	44	=	=	SYM
iajs-3482	205	1	4𝑅(𝑘)𝛼(𝑡𝑘	4𝑅(𝑘)𝛼(𝑡𝑘	X
iajs-3482	205	2	)	)	PUNCT
iajs-3482	206	1	+	+	PROPN
iajs-3482	206	2	4𝑅(𝑘)𝛼(𝑡)𝛼(𝑘)′	4𝑅(𝑘)𝛼(𝑡)𝛼(𝑘)′	NUM
iajs-3482	206	3	=	=	SYM
iajs-3482	206	4	0	0	NUM
iajs-3482	206	5	since	since	SCONJ
iajs-3482	206	6	𝛼(𝑍(𝑉	𝛼(𝑍(𝑉	NUM
iajs-3482	206	7	)	)	PUNCT
iajs-3482	206	8	)	)	PUNCT
iajs-3482	207	1	=	=	SYM
iajs-3482	207	2	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	207	3	)	)	PUNCT
iajs-3482	207	4	,	,	PUNCT
iajs-3482	207	5	and	and	CCONJ
iajs-3482	207	6	[	[	X
iajs-3482	207	7	𝑅(𝑘	𝑅(𝑘	NOUN
iajs-3482	207	8	)	)	PUNCT
iajs-3482	207	9	,	,	PUNCT
iajs-3482	207	10	𝛼(𝑘	𝛼(𝑘	PROPN
iajs-3482	207	11	)	)	PUNCT
iajs-3482	207	12	]	]	PUNCT
iajs-3482	208	1	itself	itself	PRON
iajs-3482	208	2	is	be	AUX
iajs-3482	208	3	central	central	ADJ
iajs-3482	208	4	element	element	NOUN
iajs-3482	208	5	,	,	PUNCT
iajs-3482	208	6	by	by	ADP
iajs-3482	208	7	lemma	lemma	PROPN
iajs-3482	208	8	(	(	PUNCT
iajs-3482	208	9	3.1	3.1	NUM
iajs-3482	208	10	)	)	PUNCT
iajs-3482	208	11	,	,	PUNCT
iajs-3482	208	12	we	we	PRON
iajs-3482	208	13	get	get	VERB
iajs-3482	208	14	our	our	PRON
iajs-3482	208	15	goal	goal	NOUN
iajs-3482	208	16	.	.	PUNCT
iajs-3482	209	1	ihjpas	ihjpas	PROPN
iajs-3482	209	2	.	.	PUNCT
iajs-3482	210	1	2025	2025	NUM
iajs-3482	210	2	,	,	PUNCT
iajs-3482	210	3	38	38	NUM
iajs-3482	210	4	(	(	PUNCT
iajs-3482	210	5	1	1	NUM
iajs-3482	210	6	)	)	PUNCT
iajs-3482	210	7	405	405	NUM
iajs-3482	210	8	2𝑅(𝑘2	2𝑅(𝑘2	NOUN
iajs-3482	210	9	)	)	PUNCT
iajs-3482	210	10	=	=	PUNCT
iajs-3482	210	11	𝑅(𝑘𝑘	𝑅(𝑘𝑘	X
iajs-3482	210	12	+	+	SYM
iajs-3482	210	13	𝑘𝑘	𝑘𝑘	NOUN
iajs-3482	210	14	)	)	PUNCT
iajs-3482	210	15	=	=	PUNCT
iajs-3482	210	16	𝑅(𝑘)𝛼(𝑘	𝑅(𝑘)𝛼(𝑘	NOUN
iajs-3482	210	17	)	)	PUNCT
iajs-3482	211	1	+	+	CCONJ
iajs-3482	211	2	𝛼(𝑘)𝑅(𝑘	𝛼(𝑘)𝑅(𝑘	NOUN
iajs-3482	211	3	)	)	PUNCT
iajs-3482	211	4	=	=	SYM
iajs-3482	211	5	2𝑅(𝑘)𝛼(𝑘	2𝑅(𝑘)𝛼(𝑘	PROPN
iajs-3482	211	6	)	)	PUNCT
iajs-3482	211	7	=	=	SYM
iajs-3482	211	8	2𝛼(𝑘)𝑅(𝑘	2𝛼(𝑘)𝑅(𝑘	NUM
iajs-3482	211	9	)	)	PUNCT
iajs-3482	211	10	.	.	PUNCT
iajs-3482	212	1	by	by	ADP
iajs-3482	212	2	theorem	theorem	NOUN
iajs-3482	212	3	3.3	3.3	NUM
iajs-3482	212	4	,	,	PUNCT
iajs-3482	212	5	we	we	PRON
iajs-3482	212	6	get	get	VERB
iajs-3482	212	7	our	our	PRON
iajs-3482	212	8	result	result	NOUN
iajs-3482	212	9	.	.	PUNCT
iajs-3482	213	1	4	4	X
iajs-3482	213	2	.	.	X
iajs-3482	213	3	conclusion	conclusion	NOUN
iajs-3482	213	4	in	in	ADP
iajs-3482	213	5	this	this	DET
iajs-3482	213	6	work	work	NOUN
iajs-3482	213	7	,	,	PUNCT
iajs-3482	213	8	we	we	PRON
iajs-3482	213	9	extend	extend	VERB
iajs-3482	213	10	certain	certain	ADJ
iajs-3482	213	11	results	result	NOUN
iajs-3482	213	12	of	of	ADP
iajs-3482	213	13	𝛼-centralizers	𝛼-centralizer	NOUN
iajs-3482	213	14	and	and	CCONJ
iajs-3482	213	15	jordan	jordan	PROPN
iajs-3482	213	16	𝛼-centralizers	𝛼-centralizer	NOUN
iajs-3482	213	17	on	on	ADP
iajs-3482	213	18	lie	lie	NOUN
iajs-3482	213	19	ideals	ideal	NOUN
iajs-3482	213	20	of	of	ADP
iajs-3482	213	21	prime	prime	ADJ
iajs-3482	213	22	rings	ring	NOUN
iajs-3482	213	23	to	to	ADP
iajs-3482	213	24	prime	prime	ADJ
iajs-3482	213	25	inverse	inverse	NOUN
iajs-3482	213	26	semirings	semiring	NOUN
iajs-3482	213	27	.	.	PUNCT
iajs-3482	214	1	we	we	PRON
iajs-3482	214	2	got	get	VERB
iajs-3482	214	3	the	the	DET
iajs-3482	214	4	output	output	NOUN
iajs-3482	214	5	r	r	NOUN
iajs-3482	214	6	is	be	AUX
iajs-3482	214	7	a	a	DET
iajs-3482	214	8	left	left	ADJ
iajs-3482	214	9	(	(	PUNCT
iajs-3482	214	10	right	right	ADJ
iajs-3482	214	11	)	)	PUNCT
iajs-3482	214	12	𝛼	𝛼	NOUN
iajs-3482	214	13	−	−	NOUN
iajs-3482	214	14	centralizer	centralizer	NOUN
iajs-3482	214	15	on	on	ADP
iajs-3482	214	16	𝑉.”if	𝑉.”if	PROPN
iajs-3482	214	17	it	it	PRON
iajs-3482	214	18	where	where	SCONJ
iajs-3482	214	19	𝛼	𝛼	PRON
iajs-3482	214	20	is	be	AUX
iajs-3482	214	21	an	an	DET
iajs-3482	214	22	automorphism	automorphism	NOUN
iajs-3482	214	23	of	of	ADP
iajs-3482	214	24	v,𝑅(𝑢	v,𝑅(𝑢	PROPN
iajs-3482	214	25	)	)	PUNCT
iajs-3482	214	26	∈	∈	PROPN
iajs-3482	214	27	𝑉	𝑉	PROPN
iajs-3482	214	28	,	,	PUNCT
iajs-3482	214	29	for	for	ADP
iajs-3482	214	30	any	any	DET
iajs-3482	214	31	𝑢	𝑢	PROPN
iajs-3482	214	32	∈	∈	PROPN
iajs-3482	214	33	𝑉	𝑉	PROPN
iajs-3482	214	34	,	,	PUNCT
iajs-3482	214	35	and	and	CCONJ
iajs-3482	214	36	𝛼(𝑍(𝑉	𝛼(𝑍(𝑉	NOUN
iajs-3482	214	37	)	)	PUNCT
iajs-3482	214	38	)	)	PUNCT
iajs-3482	215	1	=	=	SYM
iajs-3482	215	2	𝑍(𝑉	𝑍(𝑉	NUM
iajs-3482	215	3	)	)	PUNCT
iajs-3482	215	4	.	.	PUNCT
iajs-3482	216	1	we	we	PRON
iajs-3482	216	2	also	also	ADV
iajs-3482	216	3	get	get	VERB
iajs-3482	216	4	the	the	DET
iajs-3482	216	5	following	follow	VERB
iajs-3482	216	6	output	output	NOUN
iajs-3482	216	7	r	r	NOUN
iajs-3482	216	8	is	be	AUX
iajs-3482	216	9	𝑎	𝑎	DET
iajs-3482	216	10	𝛼	𝛼	NOUN
iajs-3482	216	11	−	−	NOUN
iajs-3482	216	12	centralizer	centralizer	NOUN
iajs-3482	216	13	on	on	ADP
iajs-3482	216	14	𝑉	𝑉	PROPN
iajs-3482	216	15	acknowledgment	acknowledgment	NOUN
iajs-3482	216	16	our	our	PRON
iajs-3482	216	17	researcher	researcher	NOUN
iajs-3482	216	18	extends	extend	VERB
iajs-3482	216	19	his	his	PRON
iajs-3482	216	20	sincere	sincere	ADJ
iajs-3482	216	21	thanks	thank	NOUN
iajs-3482	216	22	to	to	ADP
iajs-3482	216	23	the	the	DET
iajs-3482	216	24	editor	editor	NOUN
iajs-3482	216	25	and	and	CCONJ
iajs-3482	216	26	members	member	NOUN
iajs-3482	216	27	of	of	ADP
iajs-3482	216	28	the	the	DET
iajs-3482	216	29	preparatory	preparatory	PROPN
iajs-3482	216	30	committee	committee	NOUN
iajs-3482	216	31	of	of	ADP
iajs-3482	216	32	the	the	DET
iajs-3482	216	33	ibn	ibn	PROPN
iajs-3482	216	34	al	al	PROPN
iajs-3482	216	35	-	-	PUNCT
iajs-3482	216	36	haitham	haitham	PROPN
iajs-3482	216	37	journal	journal	PROPN
iajs-3482	216	38	of	of	ADP
iajs-3482	216	39	pure	pure	ADJ
iajs-3482	216	40	and	and	CCONJ
iajs-3482	216	41	applied	applied	ADJ
iajs-3482	216	42	sciences	science	NOUN
iajs-3482	216	43	.	.	PUNCT
iajs-3482	217	1	conflict	conflict	NOUN
iajs-3482	217	2	of	of	ADP
iajs-3482	217	3	interest	interest	NOUN
iajs-3482	217	4	there	there	PRON
iajs-3482	217	5	are	be	VERB
iajs-3482	217	6	no	no	DET
iajs-3482	217	7	conflicts	conflict	NOUN
iajs-3482	217	8	of	of	ADP
iajs-3482	217	9	interest	interest	NOUN
iajs-3482	217	10	.	.	PUNCT
iajs-3482	218	1	funding	funding	NOUN
iajs-3482	218	2	there	there	PRON
iajs-3482	218	3	is	be	VERB
iajs-3482	218	4	no	no	DET
iajs-3482	218	5	funding	funding	NOUN
iajs-3482	218	6	for	for	ADP
iajs-3482	218	7	the	the	DET
iajs-3482	218	8	article	article	NOUN
iajs-3482	218	9	.	.	PUNCT
iajs-3482	219	1	references	reference	NOUN
iajs-3482	219	2	1	1	NUM
iajs-3482	219	3	.	.	PUNCT
iajs-3482	219	4	vandiver	vandiver	ADJ
iajs-3482	219	5	s.h	s.h	PROPN
iajs-3482	219	6	.	.	PROPN
iajs-3482	219	7	note	note	NOUN
iajs-3482	219	8	on	on	ADP
iajs-3482	219	9	a	a	DET
iajs-3482	219	10	simple	simple	ADJ
iajs-3482	219	11	type	type	NOUN
iajs-3482	219	12	of	of	ADP
iajs-3482	219	13	algebra	algebra	NOUN
iajs-3482	219	14	in	in	ADP
iajs-3482	219	15	which	which	PRON
iajs-3482	219	16	the	the	DET
iajs-3482	219	17	cancellation	cancellation	NOUN
iajs-3482	219	18	law	law	NOUN
iajs-3482	219	19	of	of	ADP
iajs-3482	219	20	addition	addition	NOUN
iajs-3482	219	21	does	do	AUX
iajs-3482	219	22	not	not	PART
iajs-3482	219	23	hold	hold	VERB
iajs-3482	219	24	.	.	PUNCT
iajs-3482	220	1	bull	bull	NOUN
iajs-3482	220	2	.	.	PUNCT
iajs-3482	221	1	amer	amer	PROPN
iajs-3482	221	2	.	.	PUNCT
iajs-3482	221	3	math	math	PROPN
iajs-3482	221	4	.	.	PUNCT
iajs-3482	222	1	soc	soc	PROPN
iajs-3482	222	2	.	.	PUNCT
iajs-3482	223	1	nature	nature	NOUN
iajs-3482	223	2	.	.	PUNCT
iajs-3482	224	1	1934	1934	NUM
iajs-3482	224	2	;	;	PUNCT
iajs-3482	224	3	40	40	NUM
iajs-3482	224	4	:	:	PUNCT
iajs-3482	224	5	914	914	NUM
iajs-3482	224	6	-	-	SYM
iajs-3482	224	7	920	920	NUM
iajs-3482	224	8	.	.	PUNCT
iajs-3482	225	1	2	2	X
iajs-3482	225	2	.	.	X
iajs-3482	225	3	karvellas	karvellas	PROPN
iajs-3482	225	4	p.h	p.h	PROPN
iajs-3482	225	5	.	.	PROPN
iajs-3482	225	6	:	:	PUNCT
iajs-3482	226	1	inversive	inversive	ADJ
iajs-3482	226	2	semi	semi	NOUN
iajs-3482	226	3	-	-	NOUN
iajs-3482	226	4	rings	ring	NOUN
iajs-3482	226	5	.	.	PUNCT
iajs-3482	227	1	j.	j.	PROPN
iajs-3482	227	2	aust	aust	PROPN
iajs-3482	227	3	.	.	PUNCT
iajs-3482	228	1	math	math	PROPN
iajs-3482	228	2	.	.	PUNCT
iajs-3482	229	1	science	science	NOUN
iajs-3482	229	2	.	.	PUNCT
iajs-3482	230	1	1974	1974	NUM
iajs-3482	230	2	;	;	PUNCT
iajs-3482	230	3	18	18	NUM
iajs-3482	230	4	:	:	SYM
iajs-3482	230	5	277	277	NUM
iajs-3482	230	6	-	-	SYM
iajs-3482	230	7	288	288	NUM
iajs-3482	230	8	.	.	PUNCT
iajs-3482	231	1	3	3	X
iajs-3482	231	2	.	.	X
iajs-3482	231	3	javed	javed	PROPN
iajs-3482	231	4	m.a	m.a	PROPN
iajs-3482	231	5	.	.	PROPN
iajs-3482	231	6	,	,	PUNCT
iajs-3482	231	7	aslam	aslam	PROPN
iajs-3482	231	8	m.	m.	PROPN
iajs-3482	231	9	,	,	PUNCT
iajs-3482	231	10	hussain	hussain	PROPN
iajs-3482	231	11	,	,	PUNCT
iajs-3482	231	12	m.	m.	NOUN
iajs-3482	231	13	on	on	ADP
iajs-3482	231	14	requirement	requirement	NOUN
iajs-3482	231	15	(	(	PUNCT
iajs-3482	231	16	a2	a2	PROPN
iajs-3482	231	17	)	)	PUNCT
iajs-3482	231	18	of	of	ADP
iajs-3482	231	19	bandlet	bandlet	NOUN
iajs-3482	231	20	,	,	PUNCT
iajs-3482	231	21	petrich	petrich	NOUN
iajs-3482	231	22	for	for	ADP
iajs-3482	231	23	inverse	inverse	NOUN
iajs-3482	231	24	semi	semi	NOUN
iajs-3482	231	25	-	-	NOUN
iajs-3482	231	26	rings	ring	NOUN
iajs-3482	231	27	.	.	PUNCT
iajs-3482	232	1	int	int	NOUN
iajs-3482	232	2	.	.	PUNCT
iajs-3482	233	1	mathematical	mathematical	PROPN
iajs-3482	233	2	forum	forum	PROPN
iajs-3482	233	3	.	.	PUNCT
iajs-3482	234	1	nature	nature	NOUN
iajs-3482	234	2	.	.	PUNCT
iajs-3482	235	1	2012	2012	NUM
iajs-3482	235	2	;	;	PUNCT
iajs-3482	235	3	59(7	59(7	NUM
iajs-3482	235	4	):	):	PUNCT
iajs-3482	235	5	2903–2914	2903–2914	NOUN
iajs-3482	235	6	.	.	PUNCT
iajs-3482	236	1	4	4	X
iajs-3482	236	2	.	.	X
iajs-3482	236	3	albas	albas	PROPN
iajs-3482	236	4	e.:on	e.:on	PROPN
iajs-3482	236	5	τ	τ	PROPN
iajs-3482	236	6	-centralizers	-centralizer	NOUN
iajs-3482	236	7	of	of	ADP
iajs-3482	236	8	semi	semi	ADJ
iajs-3482	236	9	-	-	ADJ
iajs-3482	236	10	prime	prime	ADJ
iajs-3482	236	11	rings	ring	NOUN
iajs-3482	236	12	.	.	PUNCT
iajs-3482	237	1	siberian	siberian	ADJ
iajs-3482	237	2	math	math	PROPN
iajs-3482	237	3	.	.	PUNCT
iajs-3482	238	1	j.	j.	PROPN
iajs-3482	238	2	science	science	PROPN
iajs-3482	238	3	.	.	PUNCT
iajs-3482	239	1	2007	2007	NUM
iajs-3482	239	2	;	;	PUNCT
iajs-3482	239	3	48	48	NUM
iajs-3482	239	4	:	:	SYM
iajs-3482	239	5	191	191	NUM
iajs-3482	239	6	-	-	SYM
iajs-3482	239	7	196	196	NUM
iajs-3482	239	8	.	.	PUNCT
iajs-3482	240	1	5	5	X
iajs-3482	240	2	.	.	X
iajs-3482	240	3	ibraheem	ibraheem	PROPN
iajs-3482	240	4	r.kh	r.kh	PROPN
iajs-3482	240	5	.	.	PUNCT
iajs-3482	240	6	;	;	PUNCT
iajs-3482	240	7	majeed	majeed	PROPN
iajs-3482	240	8	a.h	a.h	PROPN
iajs-3482	240	9	.	.	PROPN
iajs-3482	240	10	:	:	PUNCT
iajs-3482	241	1	u	u	X
iajs-3482	241	2	-	-	PROPN
iajs-3482	241	3	s	s	PART
iajs-3482	241	4	jordan	jordan	PROPN
iajs-3482	241	5	homomorphisim	homomorphisim	PROPN
iajs-3482	241	6	of	of	ADP
iajs-3482	241	7	inverse	inverse	NOUN
iajs-3482	241	8	semi	semi	NOUN
iajs-3482	241	9	-	-	NOUN
iajs-3482	241	10	rings	ring	NOUN
iajs-3482	241	11	.	.	PUNCT
iajs-3482	242	1	iraqi	iraqi	ADJ
iajs-3482	242	2	journal	journal	PROPN
iajs-3482	242	3	of	of	ADP
iajs-3482	242	4	science	science	NOUN
iajs-3482	242	5	.	.	PUNCT
iajs-3482	243	1	nature	nature	NOUN
iajs-3482	243	2	.	.	PUNCT
iajs-3482	244	1	2019	2019	NUM
iajs-3482	244	2	,	,	PUNCT
iajs-3482	244	3	60	60	NUM
iajs-3482	244	4	(	(	PUNCT
iajs-3482	244	5	8):1783	8):1783	NUM
iajs-3482	244	6	-	-	SYM
iajs-3482	244	7	1790	1790	NUM
iajs-3482	244	8	.	.	PUNCT
iajs-3482	245	1	https://doi.org/10.24996/ijs.2019.60.8.15	https://doi.org/10.24996/ijs.2019.60.8.15	PROPN
iajs-3482	245	2	6	6	NUM
iajs-3482	245	3	.	.	PUNCT
iajs-3482	246	1	rasheed	rasheed	PROPN
iajs-3482	246	2	m.k	m.k	PROPN
iajs-3482	246	3	.	.	PROPN
iajs-3482	246	4	,	,	PUNCT
iajs-3482	246	5	hameed	hameed	PROPN
iajs-3482	246	6	f.a	f.a	PROPN
iajs-3482	246	7	.	.	PROPN
iajs-3482	246	8	,	,	PUNCT
iajs-3482	246	9	majeed	majeed	PROPN
iajs-3482	246	10	a.h	a.h	PROPN
iajs-3482	246	11	.	.	PROPN
iajs-3482	246	12	:	:	PUNCT
iajs-3482	246	13	on	on	ADP
iajs-3482	246	14	generalized	generalized	ADJ
iajs-3482	246	15	(	(	PUNCT
iajs-3482	246	16	α	α	NOUN
iajs-3482	246	17	,	,	PUNCT
iajs-3482	246	18	β	β	NOUN
iajs-3482	246	19	)	)	PUNCT
iajs-3482	246	20	derivation	derivation	NOUN
iajs-3482	246	21	on	on	ADP
iajs-3482	246	22	prime	prime	ADJ
iajs-3482	246	23	semi	semi	NOUN
iajs-3482	246	24	-	-	NOUN
iajs-3482	246	25	rings	ring	NOUN
iajs-3482	246	26	.	.	PUNCT
iajs-3482	247	1	j.	j.	PROPN
iajs-3482	247	2	phys	phys	PROPN
iajs-3482	247	3	.	.	PUNCT
iajs-3482	247	4	:	:	PUNCT
iajs-3482	248	1	conf	conf	PROPN
iajs-3482	248	2	.	.	PUNCT
iajs-3482	248	3	ser	ser	PROPN
iajs-3482	248	4	.	.	PROPN
iajs-3482	249	1	2020	2020	NUM
iajs-3482	249	2	,	,	PUNCT
iajs-3482	249	3	1291	1291	NUM
iajs-3482	249	4	-	-	SYM
iajs-3482	249	5	1508	1508	NUM
iajs-3482	249	6	.	.	PUNCT
iajs-3482	250	1	https://doi.org/	https://doi.org/	VERB
iajs-3482	250	2	10.1088/1742	10.1088/1742	NUM
iajs-3482	250	3	-	-	SYM
iajs-3482	250	4	6596/1591/1/012080	6596/1591/1/012080	NUM
iajs-3482	250	5	7	7	NUM
iajs-3482	250	6	.	.	PUNCT
iajs-3482	251	1	rasheed	rasheed	PROPN
iajs-3482	251	2	m.k	m.k	PROPN
iajs-3482	251	3	.	.	PROPN
iajs-3482	251	4	,	,	PUNCT
iajs-3482	251	5	majeed	majeed	PROPN
iajs-3482	251	6	a.h	a.h	PROPN
iajs-3482	251	7	.	.	PROPN
iajs-3482	251	8	:	:	PUNCT
iajs-3482	252	1	some	some	DET
iajs-3482	252	2	results	result	NOUN
iajs-3482	252	3	of	of	ADP
iajs-3482	252	4	(	(	PUNCT
iajs-3482	252	5	α	α	NOUN
iajs-3482	252	6	,	,	PUNCT
iajs-3482	252	7	β	β	NOUN
iajs-3482	252	8	)	)	PUNCT
iajs-3482	252	9	derivations	derivation	NOUN
iajs-3482	252	10	on	on	ADP
iajs-3482	252	11	prime	prime	ADJ
iajs-3482	252	12	semi	semi	NOUN
iajs-3482	252	13	-	-	NOUN
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iajs-3482	254	8	.	.	PUNCT
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iajs-3482	256	5	.	.	PUNCT
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iajs-3482	257	8	.	.	PUNCT
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iajs-3482	260	6	-	-	SYM
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iajs-3482	260	8	.	.	PUNCT
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iajs-3482	260	11	.	.	PUNCT
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iajs-3482	260	28	.	.	PUNCT
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iajs-3482	261	9	;	;	PUNCT
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iajs-3482	261	11	:	:	SYM
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iajs-3482	261	13	-	-	SYM
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iajs-3482	262	2	.	.	PUNCT
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iajs-3482	268	6	.	.	PUNCT
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iajs-3482	274	9	.	.	PUNCT
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iajs-3482	284	5	.	.	PUNCT
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