id	sid	tid	token	lemma	pos
ma-208	1	1	2024	2024	NUM
ma-208	1	2	ada	ada	PROPN
ma-208	1	3	academica	academica	PROPN
ma-208	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-208	1	5	.	.	PUNCT
ma-208	2	1	j.	j.	PROPN
ma-208	2	2	math	math	PROPN
ma-208	2	3	.	.	PUNCT
ma-208	3	1	anal	anal	ADJ
ma-208	3	2	.	.	PUNCT
ma-208	4	1	4	4	NUM
ma-208	4	2	(	(	PUNCT
ma-208	4	3	2024	2024	NUM
ma-208	4	4	)	)	PUNCT
ma-208	5	1	5doi	5doi	NOUN
ma-208	5	2	:	:	PUNCT
ma-208	5	3	10.28924	10.28924	NUM
ma-208	5	4	/	/	SYM
ma-208	5	5	ada	ada	PROPN
ma-208	5	6	/	/	SYM
ma-208	5	7	ma.4.5	ma.4.5	PROPN
ma-208	5	8	a	a	DET
ma-208	5	9	new	new	ADJ
ma-208	5	10	study	study	NOUN
ma-208	5	11	on	on	ADP
ma-208	5	12	generalized	generalized	ADJ
ma-208	5	13	reverse	reverse	ADJ
ma-208	5	14	derivations	derivation	NOUN
ma-208	5	15	of	of	ADP
ma-208	5	16	semi	semi	ADJ
ma-208	5	17	-	-	ADJ
ma-208	5	18	prime	prime	ADJ
ma-208	5	19	ring	ring	NOUN
ma-208	5	20	muhammad	muhammad	PROPN
ma-208	5	21	naeem	naeem	PROPN
ma-208	5	22	abbas1,∗	abbas1,∗	PROPN
ma-208	5	23	,	,	PUNCT
ma-208	5	24	mukhtar	mukhtar	PROPN
ma-208	5	25	ahmad1,∗	ahmad1,∗	PROPN
ma-208	5	26	,	,	PUNCT
ma-208	5	27	abdul	abdul	PROPN
ma-208	5	28	rauf	rauf	PROPN
ma-208	5	29	khan2	khan2	PROPN
ma-208	5	30	,	,	PUNCT
ma-208	5	31	ather	ather	PROPN
ma-208	5	32	qayyum3	qayyum3	PROPN
ma-208	5	33	,	,	PUNCT
ma-208	5	34	siti	siti	NOUN
ma-208	5	35	suzlinsupadi3	suzlinsupadi3	NOUN
ma-208	6	1	1department	1department	NUM
ma-208	6	2	of	of	ADP
ma-208	6	3	mathematics	mathematic	NOUN
ma-208	6	4	,	,	PUNCT
ma-208	6	5	khawaja	khawaja	PROPN
ma-208	6	6	fareed	fareed	PROPN
ma-208	6	7	university	university	PROPN
ma-208	6	8	of	of	ADP
ma-208	6	9	engineering	engineering	NOUN
ma-208	6	10	and	and	CCONJ
ma-208	6	11	information	information	NOUN
ma-208	6	12	technology	technology	PROPN
ma-208	6	13	rahim	rahim	PROPN
ma-208	6	14	yar	yar	PROPN
ma-208	6	15	khan	khan	PROPN
ma-208	6	16	,	,	PUNCT
ma-208	6	17	pakistan	pakistan	PROPN
ma-208	6	18	itxmemuktar@gmail.com	itxmemuktar@gmail.com	PROPN
ma-208	6	19	,	,	PUNCT
ma-208	6	20	naeemabbas995@gmail.com	naeemabbas995@gmail.com	X
ma-208	7	1	2department	2department	NUM
ma-208	7	2	of	of	ADP
ma-208	7	3	mathematics	mathematic	NOUN
ma-208	7	4	,	,	PUNCT
ma-208	7	5	ghazi	ghazi	PROPN
ma-208	7	6	university	university	PROPN
ma-208	7	7	,	,	PUNCT
ma-208	7	8	d.g.khan	d.g.khan	NOUN
ma-208	7	9	,	,	PUNCT
ma-208	7	10	pakistan	pakistan	PROPN
ma-208	7	11	arkhan@gudgk.edu.pk	arkhan@gudgk.edu.pk	NOUN
ma-208	7	12	3institute	3institute	NUM
ma-208	7	13	of	of	ADP
ma-208	7	14	mathematical	mathematical	ADJ
ma-208	7	15	sciences	sciences	PROPN
ma-208	7	16	,	,	PUNCT
ma-208	7	17	universiti	universiti	PROPN
ma-208	7	18	malaya	malaya	PROPN
ma-208	7	19	,	,	PUNCT
ma-208	7	20	malaysia	malaysia	PROPN
ma-208	7	21	dratherqayyum@um.edu.my	dratherqayyum@um.edu.my	PROPN
ma-208	7	22	,	,	PUNCT
ma-208	7	23	suzlin@um.edu.my	suzlin@um.edu.my	ADJ
ma-208	7	24	∗correspondence	∗correspondence	NOUN
ma-208	7	25	:	:	PUNCT
ma-208	7	26	naeemabbas995@gmail.com	naeemabbas995@gmail.com	NUM
ma-208	7	27	,	,	PUNCT
ma-208	7	28	itxmemuktar@gmail.com	itxmemuktar@gmail.com	PROPN
ma-208	8	1	abstract	abstract	PROPN
ma-208	8	2	.	.	PUNCT
ma-208	9	1	the	the	DET
ma-208	9	2	aim	aim	NOUN
ma-208	9	3	of	of	ADP
ma-208	9	4	this	this	DET
ma-208	9	5	paper	paper	NOUN
ma-208	9	6	is	be	AUX
ma-208	9	7	to	to	PART
ma-208	9	8	extend	extend	VERB
ma-208	9	9	the	the	DET
ma-208	9	10	ideas	idea	NOUN
ma-208	9	11	from	from	ADP
ma-208	9	12	generalized	generalized	ADJ
ma-208	9	13	reverse	reverse	ADJ
ma-208	9	14	derivation	derivation	NOUN
ma-208	9	15	to	to	ADP
ma-208	9	16	gener	gener	NOUN
ma-208	9	17	-	-	PUNCT
ma-208	9	18	alized	alize	VERB
ma-208	9	19	(	(	PUNCT
ma-208	9	20	α	α	NOUN
ma-208	9	21	,	,	PUNCT
ma-208	9	22	β)-reverse	β)-reverse	PUNCT
ma-208	9	23	derivations	derivation	NOUN
ma-208	9	24	on	on	ADP
ma-208	9	25	semi	semi	ADJ
ma-208	9	26	-	-	ADJ
ma-208	9	27	prime	prime	ADJ
ma-208	9	28	ring	ring	NOUN
ma-208	9	29	.	.	PUNCT
ma-208	10	1	we	we	PRON
ma-208	10	2	prove	prove	VERB
ma-208	10	3	that	that	SCONJ
ma-208	10	4	,	,	PUNCT
ma-208	10	5	if	if	SCONJ
ma-208	10	6	0	0	NUM
ma-208	10	7	6=	6=	NUM
ma-208	11	1	d	d	NOUN
ma-208	11	2	be	be	AUX
ma-208	11	3	reverse	reverse	ADJ
ma-208	11	4	derivationin	derivationin	NOUN
ma-208	11	5	r	r	NOUN
ma-208	11	6	and	and	CCONJ
ma-208	11	7	a	a	DET
ma-208	11	8	generalized	generalize	VERB
ma-208	11	9	(	(	PUNCT
ma-208	11	10	α	α	NOUN
ma-208	11	11	,	,	PUNCT
ma-208	11	12	β)-reverse	β)-reverse	PUNCT
ma-208	11	13	derivation	derivation	NOUN
ma-208	11	14	g	g	NOUN
ma-208	11	15	,	,	PUNCT
ma-208	11	16	then	then	ADV
ma-208	11	17	g	g	PROPN
ma-208	11	18	is	be	AUX
ma-208	11	19	β	β	NOUN
ma-208	11	20	-	-	ADJ
ma-208	11	21	strong	strong	ADJ
ma-208	11	22	commutative	commutative	ADJ
ma-208	11	23	preserved	preserve	VERB
ma-208	11	24	.	.	PUNCT
ma-208	12	1	nextwe	nextwe	PROPN
ma-208	12	2	can	can	AUX
ma-208	12	3	prove	prove	VERB
ma-208	12	4	that	that	SCONJ
ma-208	12	5	r	r	NOUN
ma-208	12	6	is	be	AUX
ma-208	12	7	commutative	commutative	ADJ
ma-208	12	8	.	.	PUNCT
ma-208	13	1	1	1	X
ma-208	13	2	.	.	X
ma-208	13	3	introduction	introduction	NOUN
ma-208	13	4	the	the	DET
ma-208	13	5	study	study	NOUN
ma-208	13	6	of	of	ADP
ma-208	13	7	centralizing	centralize	VERB
ma-208	13	8	mapping	mapping	NOUN
ma-208	13	9	of	of	ADP
ma-208	13	10	semi	semi	ADJ
ma-208	13	11	-	-	ADJ
ma-208	13	12	prime	prime	ADJ
ma-208	13	13	rings	ring	NOUN
ma-208	13	14	given	give	VERB
ma-208	13	15	by	by	ADP
ma-208	13	16	bell	bell	NOUN
ma-208	13	17	and	and	CCONJ
ma-208	13	18	martindale	martindale	PROPN
ma-208	14	1	[	[	X
ma-208	14	2	3	3	NUM
ma-208	14	3	]	]	PUNCT
ma-208	14	4	.	.	PUNCT
ma-208	15	1	belland	belland	PROPN
ma-208	15	2	martindale	martindale	PROPN
ma-208	16	1	[	[	X
ma-208	16	2	3	3	NUM
ma-208	16	3	]	]	PUNCT
ma-208	16	4	proved	prove	VERB
ma-208	16	5	that	that	SCONJ
ma-208	16	6	[	[	X
ma-208	16	7	d(u1	d(u1	NOUN
ma-208	16	8	)	)	PUNCT
ma-208	16	9	,	,	PUNCT
ma-208	16	10	u1]α	u1]α	PROPN
ma-208	16	11	,	,	PUNCT
ma-208	16	12	β	β	X
ma-208	16	13	=	=	SYM
ma-208	16	14	0	0	NUM
ma-208	16	15	∀	∀	NOUN
ma-208	16	16	u1	u1	NOUN
ma-208	16	17	∈	∈	PROPN
ma-208	16	18	b	b	PROPN
ma-208	16	19	,	,	PUNCT
ma-208	16	20	where	where	SCONJ
ma-208	16	21	0	0	X
ma-208	17	1	6=	6=	NUM
ma-208	17	2	d	d	ADP
ma-208	17	3	a	a	DET
ma-208	17	4	derivation	derivation	NOUN
ma-208	17	5	of	of	ADP
ma-208	17	6	r	r	NOUN
ma-208	17	7	and	and	CCONJ
ma-208	17	8	r	r	NOUN
ma-208	17	9	is	be	AUX
ma-208	17	10	semi	semi	ADJ
ma-208	17	11	-	-	ADJ
ma-208	17	12	prime	prime	ADJ
ma-208	17	13	ring	ring	NOUN
ma-208	17	14	,	,	PUNCT
ma-208	17	15	then	then	ADV
ma-208	17	16	commutativity	commutativity	NOUN
ma-208	17	17	holds	hold	VERB
ma-208	17	18	in	in	ADP
ma-208	17	19	r.	r.	PROPN
ma-208	17	20	bell	bell	PROPN
ma-208	17	21	and	and	CCONJ
ma-208	17	22	daif	daif	NOUN
ma-208	17	23	were	be	AUX
ma-208	17	24	studied	study	VERB
ma-208	17	25	the	the	DET
ma-208	17	26	commu	commu	NOUN
ma-208	17	27	-	-	PUNCT
ma-208	17	28	tativity	tativity	NOUN
ma-208	17	29	in	in	ADP
ma-208	17	30	prime	prime	ADJ
ma-208	17	31	and	and	CCONJ
ma-208	17	32	semi	semi	ADJ
ma-208	17	33	-	-	ADJ
ma-208	17	34	prime	prime	ADJ
ma-208	17	35	rings	ring	NOUN
ma-208	17	36	that	that	PRON
ma-208	17	37	bind	bind	VERB
ma-208	17	38	endomorphism	endomorphism	NOUN
ma-208	17	39	or	or	CCONJ
ma-208	17	40	a	a	DET
ma-208	17	41	derivation	derivation	NOUN
ma-208	17	42	that	that	PRON
ma-208	17	43	preserves	preserve	VERB
ma-208	17	44	a	a	DET
ma-208	17	45	β	β	NOUN
ma-208	17	46	-	-	ADJ
ma-208	17	47	strong	strong	ADJ
ma-208	17	48	commutativity	commutativity	NOUN
ma-208	17	49	on	on	ADP
ma-208	17	50	a	a	DET
ma-208	17	51	non	non	ADJ
ma-208	17	52	-	-	ADJ
ma-208	17	53	zero	zero	NUM
ma-208	17	54	ideal	ideal	NOUN
ma-208	17	55	right	right	ADV
ma-208	17	56	in	in	ADP
ma-208	17	57	[	[	X
ma-208	17	58	2	2	NUM
ma-208	17	59	]	]	PUNCT
ma-208	17	60	.	.	PUNCT
ma-208	18	1	further	far	ADV
ma-208	18	2	,	,	PUNCT
ma-208	18	3	ali	ali	PROPN
ma-208	18	4	and	and	CCONJ
ma-208	18	5	shah	shah	PROPN
ma-208	18	6	[	[	X
ma-208	18	7	1	1	X
ma-208	18	8	]	]	PUNCT
ma-208	18	9	extend	extend	VERB
ma-208	18	10	some	some	DET
ma-208	18	11	con	con	NOUN
ma-208	18	12	-	-	PUNCT
ma-208	18	13	sequences	sequence	NOUN
ma-208	18	14	for	for	ADP
ma-208	18	15	generalized	generalized	ADJ
ma-208	18	16	derivation	derivation	NOUN
ma-208	18	17	of	of	ADP
ma-208	18	18	bell	bell	NOUN
ma-208	18	19	and	and	CCONJ
ma-208	18	20	martindale	martindale	PROPN
ma-208	19	1	[	[	X
ma-208	19	2	3	3	NUM
ma-208	19	3	]	]	PUNCT
ma-208	19	4	.	.	PUNCT
ma-208	20	1	bresar	bresar	VERB
ma-208	20	2	established	establish	VERB
ma-208	20	3	that	that	SCONJ
ma-208	20	4	,	,	PUNCT
ma-208	20	5	if	if	SCONJ
ma-208	20	6	b	b	PROPN
ma-208	20	7	6=	6=	ADP
ma-208	20	8	0is	0is	ADJ
ma-208	20	9	left	leave	VERB
ma-208	20	10	ideal	ideal	NOUN
ma-208	20	11	in	in	ADP
ma-208	20	12	r	r	NOUN
ma-208	20	13	a	a	DET
ma-208	20	14	prime	prime	ADJ
ma-208	20	15	ring	ring	NOUN
ma-208	20	16	,	,	PUNCT
ma-208	20	17	and	and	CCONJ
ma-208	20	18	two	two	NUM
ma-208	20	19	mappings	mapping	NOUN
ma-208	20	20	d1	d1	PROPN
ma-208	20	21	and	and	CCONJ
ma-208	20	22	d2	d2	PROPN
ma-208	20	23	are	be	AUX
ma-208	20	24	(	(	PUNCT
ma-208	20	25	α	α	X
ma-208	20	26	,	,	PUNCT
ma-208	20	27	β)-derivations	β)-derivations	PROPN
ma-208	20	28	in	in	ADP
ma-208	20	29	r	r	NOUN
ma-208	20	30	satisfies	satisfie	NOUN
ma-208	20	31	(	(	PUNCT
ma-208	20	32	d1α(a)−	d1α(a)−	NOUN
ma-208	20	33	β(a)d2	β(a)d2	PROPN
ma-208	20	34	)	)	PUNCT
ma-208	20	35	∈	∈	PROPN
ma-208	20	36	z(r	z(r	PROPN
ma-208	20	37	)	)	PUNCT
ma-208	20	38	,	,	PUNCT
ma-208	20	39	for	for	ADP
ma-208	20	40	each	each	DET
ma-208	20	41	a	a	DET
ma-208	20	42	∈	∈	PROPN
ma-208	20	43	b	b	NOUN
ma-208	20	44	,	,	PUNCT
ma-208	20	45	so	so	ADV
ma-208	20	46	commutativity	commutativity	NOUN
ma-208	20	47	holds	hold	VERB
ma-208	20	48	in	in	ADP
ma-208	20	49	r	r	NOUN
ma-208	20	50	[	[	X
ma-208	20	51	5	5	NUM
ma-208	20	52	]	]	PUNCT
ma-208	20	53	.	.	PUNCT
ma-208	21	1	some	some	DET
ma-208	21	2	properties	property	NOUN
ma-208	21	3	arestudied	arestudie	VERB
ma-208	21	4	by	by	ADP
ma-208	21	5	vukman	vukman	NOUN
ma-208	21	6	in	in	ADP
ma-208	21	7	[	[	X
ma-208	21	8	12	12	NUM
ma-208	21	9	]	]	PUNCT
ma-208	21	10	and	and	CCONJ
ma-208	21	11	[	[	X
ma-208	21	12	4	4	NUM
ma-208	21	13	]	]	PUNCT
ma-208	21	14	.	.	PUNCT
ma-208	22	1	m.	m.	NOUN
ma-208	22	2	samman	samman	NOUN
ma-208	22	3	and	and	CCONJ
ma-208	22	4	n.	n.	PROPN
ma-208	22	5	al	al	PROPN
ma-208	22	6	yamani	yamani	PROPN
ma-208	23	1	[	[	X
ma-208	23	2	8	8	NUM
ma-208	23	3	]	]	PUNCT
ma-208	23	4	studied	study	VERB
ma-208	23	5	reverse	reverse	ADJ
ma-208	23	6	derivation	derivation	NOUN
ma-208	23	7	onsemi	onsemi	ADP
ma-208	23	8	prime	prime	ADJ
ma-208	23	9	rings	ring	NOUN
ma-208	23	10	.	.	PUNCT
ma-208	24	1	they	they	PRON
ma-208	24	2	proved	prove	VERB
ma-208	24	3	that	that	SCONJ
ma-208	24	4	the	the	DET
ma-208	24	5	mapping	mapping	NOUN
ma-208	24	6	d	d	NOUN
ma-208	24	7	:	:	PUNCT
ma-208	24	8	r	r	NOUN
ma-208	24	9	→	→	SYM
ma-208	24	10	r	r	NOUN
ma-208	24	11	is	be	AUX
ma-208	24	12	central	central	ADJ
ma-208	24	13	derivation	derivation	NOUN
ma-208	24	14	iff	iff	NOUN
ma-208	24	15	it	it	PRON
ma-208	24	16	is	be	AUX
ma-208	24	17	reversederivation	reversederivation	NOUN
ma-208	24	18	and	and	CCONJ
ma-208	24	19	also	also	ADV
ma-208	24	20	that	that	SCONJ
ma-208	24	21	d	d	PROPN
ma-208	25	1	6=	6=	ADP
ma-208	25	2	0	0	NUM
ma-208	25	3	a	a	DET
ma-208	25	4	reverse	reverse	ADJ
ma-208	25	5	derivation	derivation	NOUN
ma-208	25	6	in	in	ADP
ma-208	25	7	semi	semi	ADJ
ma-208	25	8	-	-	ADJ
ma-208	25	9	prime	prime	ADJ
ma-208	25	10	ring	ring	NOUN
ma-208	25	11	r	r	NOUN
ma-208	25	12	,	,	PUNCT
ma-208	25	13	then	then	ADV
ma-208	25	14	the	the	DET
ma-208	25	15	commutativityexists	commutativityexist	NOUN
ma-208	25	16	in	in	ADP
ma-208	25	17	r	r	NOUN
ma-208	25	18	resently	resently	ADV
ma-208	25	19	mukhtar	mukhtar	PROPN
ma-208	25	20	ahmad	ahmad	PROPN
ma-208	25	21	et.al[9	et.al[9	PROPN
ma-208	25	22	]	]	PUNCT
ma-208	25	23	.	.	PUNCT
ma-208	26	1	later	later	ADV
ma-208	26	2	,	,	PUNCT
ma-208	26	3	the	the	DET
ma-208	26	4	idea	idea	NOUN
ma-208	26	5	of	of	ADP
ma-208	26	6	revers	rever	NOUN
ma-208	26	7	derivation	derivation	NOUN
ma-208	26	8	and	and	CCONJ
ma-208	26	9	some	some	DET
ma-208	26	10	properties	property	NOUN
ma-208	26	11	received	receive	VERB
ma-208	26	12	:	:	PUNCT
ma-208	26	13	19	19	NUM
ma-208	26	14	nov	nov	PROPN
ma-208	26	15	2023	2023	NUM
ma-208	26	16	.	.	PUNCT
ma-208	27	1	key	key	ADJ
ma-208	27	2	words	word	NOUN
ma-208	27	3	and	and	CCONJ
ma-208	27	4	phrases	phrase	NOUN
ma-208	27	5	.	.	PUNCT
ma-208	28	1	semi	semi	ADJ
ma-208	28	2	-	-	ADJ
ma-208	28	3	prime	prime	ADJ
ma-208	28	4	ring	ring	NOUN
ma-208	28	5	;	;	PUNCT
ma-208	28	6	ideal	ideal	ADJ
ma-208	28	7	;	;	PUNCT
ma-208	28	8	r	r	X
ma-208	28	9	-	-	PUNCT
ma-208	28	10	generalized	generalize	VERB
ma-208	28	11	reverse	reverse	ADJ
ma-208	28	12	derivation	derivation	NOUN
ma-208	28	13	;	;	PUNCT
ma-208	28	14	derivation	derivation	NOUN
ma-208	28	15	;	;	PUNCT
ma-208	28	16	reverse	reverse	VERB
ma-208	28	17	derivation;r	derivation;r	NOUN
ma-208	28	18	-	-	PUNCT
ma-208	28	19	generalized	generalize	VERB
ma-208	28	20	derivation	derivation	NOUN
ma-208	28	21	and	and	CCONJ
ma-208	28	22	reverse	reverse	ADJ
ma-208	28	23	derivation	derivation	NOUN
ma-208	28	24	.	.	PUNCT
ma-208	29	1	1	1	NUM
ma-208	29	2	https://adac.ee	https://adac.ee	PROPN
ma-208	29	3	https://doi.org/10.28924/ada/ma.4.5	https://doi.org/10.28924/ada/ma.4.5	PROPN
ma-208	29	4	eur	eur	PROPN
ma-208	29	5	.	.	PUNCT
ma-208	30	1	j.	j.	PROPN
ma-208	30	2	math	math	PROPN
ma-208	30	3	.	.	PUNCT
ma-208	31	1	anal	anal	PROPN
ma-208	31	2	.	.	PUNCT
ma-208	32	1	10.28924	10.28924	NUM
ma-208	32	2	/	/	SYM
ma-208	32	3	ada	ada	PROPN
ma-208	32	4	/	/	SYM
ma-208	32	5	ma.4.5	ma.4.5	PROPN
ma-208	32	6	2of	2of	ADJ
ma-208	32	7	reverse	reverse	ADJ
ma-208	32	8	derivation	derivation	NOUN
ma-208	32	9	were	be	AUX
ma-208	32	10	studied	study	VERB
ma-208	32	11	by	by	ADP
ma-208	32	12	bresar	bresar	VERB
ma-208	32	13	and	and	CCONJ
ma-208	32	14	vukman	vukman	VERB
ma-208	33	1	[	[	X
ma-208	33	2	4	4	NUM
ma-208	33	3	]	]	PUNCT
ma-208	33	4	.	.	PUNCT
ma-208	34	1	the	the	DET
ma-208	34	2	aim	aim	NOUN
ma-208	34	3	of	of	ADP
ma-208	34	4	this	this	DET
ma-208	34	5	paper	paper	NOUN
ma-208	34	6	is	be	AUX
ma-208	34	7	extention	extention	NOUN
ma-208	34	8	thenotion	thenotion	NOUN
ma-208	34	9	of	of	ADP
ma-208	34	10	generalized	generalized	ADJ
ma-208	34	11	reverse	reverse	ADJ
ma-208	34	12	derivation	derivation	NOUN
ma-208	34	13	to	to	AUX
ma-208	34	14	generalized	generalize	VERB
ma-208	34	15	(	(	PUNCT
ma-208	34	16	α	α	NOUN
ma-208	34	17	,	,	PUNCT
ma-208	34	18	β)-reverse	β)-reverse	PUNCT
ma-208	34	19	derivation	derivation	NOUN
ma-208	34	20	resently	resently	ADV
ma-208	34	21	mukhtarahmad	mukhtarahmad	VERB
ma-208	34	22	et.al[10	et.al[10	ADJ
ma-208	34	23	]	]	X
ma-208	34	24	.	.	PUNCT
ma-208	35	1	a	a	DET
ma-208	35	2	mapping	mapping	NOUN
ma-208	35	3	g	g	NOUN
ma-208	35	4	:	:	PUNCT
ma-208	35	5	r	r	NOUN
ma-208	35	6	→	→	SYM
ma-208	35	7	r	r	NOUN
ma-208	35	8	which	which	PRON
ma-208	35	9	associate	associate	VERB
ma-208	35	10	with	with	ADP
ma-208	35	11	(	(	PUNCT
ma-208	35	12	α	α	NOUN
ma-208	35	13	,	,	PUNCT
ma-208	35	14	β)-reverse	β)-reverse	PUNCT
ma-208	35	15	derivation	derivation	NOUN
ma-208	35	16	d	d	NOUN
ma-208	35	17	is	be	AUX
ma-208	35	18	saidto	saidto	NOUN
ma-208	35	19	be	be	AUX
ma-208	35	20	a	a	DET
ma-208	35	21	generalized	generalized	ADJ
ma-208	35	22	(	(	PUNCT
ma-208	35	23	α	α	NOUN
ma-208	35	24	,	,	PUNCT
ma-208	35	25	β)-reverse	β)-reverse	PUNCT
ma-208	35	26	derivation	derivation	NOUN
ma-208	35	27	if	if	SCONJ
ma-208	35	28	,	,	PUNCT
ma-208	35	29	g(u1v1	g(u1v1	NOUN
ma-208	35	30	)	)	PUNCT
ma-208	35	31	=	=	SYM
ma-208	35	32	g(v1)α(u1	g(v1)α(u1	PROPN
ma-208	35	33	)	)	PUNCT
ma-208	36	1	+	+	CCONJ
ma-208	36	2	β(v1)d(u1)resently	β(v1)d(u1)resently	ADV
ma-208	36	3	r.m	r.m	PROPN
ma-208	36	4	.	.	PROPN
ma-208	36	5	kashif	kashif	PROPN
ma-208	36	6	et.al[11	et.al[11	PROPN
ma-208	36	7	]	]	PUNCT
ma-208	36	8	.	.	PUNCT
ma-208	37	1	2	2	X
ma-208	37	2	.	.	X
ma-208	37	3	preliminaries	preliminary	NOUN
ma-208	37	4	throughout	throughout	ADP
ma-208	37	5	this	this	DET
ma-208	37	6	paper	paper	NOUN
ma-208	37	7	,	,	PUNCT
ma-208	37	8	definition	definition	NOUN
ma-208	37	9	2.1	2.1	NUM
ma-208	37	10	.	.	PUNCT
ma-208	38	1	let	let	VERB
ma-208	38	2	r	r	NOUN
ma-208	38	3	is	be	AUX
ma-208	38	4	ring	ring	NOUN
ma-208	38	5	and	and	CCONJ
ma-208	38	6	it	it	PRON
ma-208	38	7	is	be	AUX
ma-208	38	8	considered	consider	VERB
ma-208	38	9	as	as	ADP
ma-208	38	10	a	a	DET
ma-208	38	11	semi	semi	ADJ
ma-208	38	12	-	-	ADJ
ma-208	38	13	prime	prime	ADJ
ma-208	38	14	ring	ring	NOUN
ma-208	38	15	iff	iff	NOUN
ma-208	38	16	for	for	ADP
ma-208	38	17	any	any	DET
ma-208	38	18	u1	u1	NOUN
ma-208	38	19	;	;	PUNCT
ma-208	38	20	u1	u1	PROPN
ma-208	38	21	6=	6=	NUM
ma-208	38	22	0such	0such	PROPN
ma-208	38	23	that	that	SCONJ
ma-208	38	24	u1ru1	u1ru1	PROPN
ma-208	38	25	=	=	SYM
ma-208	38	26	0	0	NUM
ma-208	38	27	implies	imply	VERB
ma-208	38	28	u1	u1	NOUN
ma-208	38	29	=	=	SYM
ma-208	38	30	0	0	PROPN
ma-208	38	31	.	.	PUNCT
ma-208	39	1	definition	definition	NOUN
ma-208	39	2	2.2	2.2	NUM
ma-208	39	3	.	.	PUNCT
ma-208	40	1	the	the	DET
ma-208	40	2	additive	additive	ADJ
ma-208	40	3	mapping	mapping	NOUN
ma-208	40	4	d1	d1	PROPN
ma-208	40	5	:	:	PUNCT
ma-208	40	6	r	r	NOUN
ma-208	40	7	→	→	SYM
ma-208	40	8	r	r	NOUN
ma-208	40	9	is	be	AUX
ma-208	40	10	known	know	VERB
ma-208	40	11	as	as	ADP
ma-208	40	12	(	(	PUNCT
ma-208	40	13	α	α	NOUN
ma-208	40	14	,	,	PUNCT
ma-208	40	15	β)-derivation	β)-derivation	NOUN
ma-208	40	16	,	,	PUNCT
ma-208	40	17	if	if	SCONJ
ma-208	40	18	d1(u1v1	d1(u1v1	NOUN
ma-208	40	19	)	)	PUNCT
ma-208	40	20	=	=	SYM
ma-208	40	21	d1(u1)α(v1	d1(u1)α(v1	X
ma-208	40	22	)	)	PUNCT
ma-208	41	1	+	+	SYM
ma-208	41	2	β(u1)d1(v1	β(u1)d1(v1	NOUN
ma-208	41	3	)	)	PUNCT
ma-208	41	4	hold	hold	VERB
ma-208	41	5	∀	∀	NUM
ma-208	41	6	u1	u1	NOUN
ma-208	41	7	,	,	PUNCT
ma-208	41	8	v1	v1	NOUN
ma-208	41	9	∈	∈	PROPN
ma-208	41	10	r	r	NOUN
ma-208	41	11	,	,	PUNCT
ma-208	41	12	where	where	SCONJ
ma-208	41	13	α	α	NOUN
ma-208	41	14	and	and	CCONJ
ma-208	41	15	β	β	X
ma-208	41	16	are	be	AUX
ma-208	41	17	automorphism	automorphism	NOUN
ma-208	41	18	.	.	PUNCT
ma-208	42	1	definition	definition	NOUN
ma-208	42	2	2.3	2.3	NUM
ma-208	42	3	.	.	PUNCT
ma-208	43	1	the	the	DET
ma-208	43	2	mapping	mapping	NOUN
ma-208	43	3	d1	d1	PROPN
ma-208	43	4	:	:	PUNCT
ma-208	43	5	r	r	NOUN
ma-208	43	6	→	→	SYM
ma-208	43	7	r	r	NOUN
ma-208	43	8	is	be	AUX
ma-208	43	9	called	call	VERB
ma-208	43	10	a	a	DET
ma-208	43	11	(	(	PUNCT
ma-208	43	12	α	α	NOUN
ma-208	43	13	,	,	PUNCT
ma-208	43	14	β)-reverse	β)-reverse	PUNCT
ma-208	43	15	derivation	derivation	NOUN
ma-208	43	16	if	if	SCONJ
ma-208	43	17	d1(u1v1	d1(u1v1	NOUN
ma-208	43	18	)	)	PUNCT
ma-208	43	19	=	=	PUNCT
ma-208	43	20	d1(v1)α(u1	d1(v1)α(u1	NOUN
ma-208	43	21	)	)	PUNCT
ma-208	44	1	+	+	SYM
ma-208	44	2	β(v1)d1(u1	β(v1)d1(u1	NOUN
ma-208	44	3	)	)	PUNCT
ma-208	44	4	holds	hold	VERB
ma-208	44	5	∀	∀	NOUN
ma-208	44	6	u1	u1	NOUN
ma-208	44	7	,	,	PUNCT
ma-208	44	8	v1	v1	NOUN
ma-208	44	9	∈	∈	PROPN
ma-208	44	10	r	r	NOUN
ma-208	44	11	,	,	PUNCT
ma-208	44	12	where	where	SCONJ
ma-208	44	13	α	α	NOUN
ma-208	44	14	and	and	CCONJ
ma-208	44	15	β	β	X
ma-208	44	16	are	be	AUX
ma-208	44	17	automorphism	automorphism	NOUN
ma-208	44	18	.	.	PUNCT
ma-208	45	1	definition	definition	NOUN
ma-208	45	2	2.4	2.4	NUM
ma-208	45	3	.	.	PUNCT
ma-208	46	1	an	an	DET
ma-208	46	2	additive	additive	ADJ
ma-208	46	3	mapping	mapping	NOUN
ma-208	46	4	h	h	NOUN
ma-208	46	5	:	:	PUNCT
ma-208	46	6	r	r	X
ma-208	46	7	→	→	SYM
ma-208	46	8	r	r	NOUN
ma-208	46	9	be	be	AUX
ma-208	46	10	a	a	DET
ma-208	46	11	right	right	NOUN
ma-208	46	12	(	(	PUNCT
ma-208	46	13	left	left	ADJ
ma-208	46	14	)	)	PUNCT
ma-208	46	15	generalized	generalize	VERB
ma-208	46	16	(	(	PUNCT
ma-208	46	17	α	α	NOUN
ma-208	46	18	,	,	PUNCT
ma-208	46	19	β)-reversederivation	β)-reversederivation	PUNCT
ma-208	46	20	if	if	SCONJ
ma-208	46	21	there	there	PRON
ma-208	46	22	is	be	VERB
ma-208	46	23	a	a	DET
ma-208	46	24	derivation	derivation	NOUN
ma-208	46	25	d	d	NOUN
ma-208	46	26	from	from	ADP
ma-208	46	27	r	r	NOUN
ma-208	46	28	to	to	ADP
ma-208	46	29	r	r	NOUN
ma-208	46	30	such	such	ADJ
ma-208	46	31	thath(u1v1	thath(u1v1	NOUN
ma-208	46	32	)	)	PUNCT
ma-208	46	33	=	=	SYM
ma-208	46	34	h(v1)α(u1)+β(v1)d(u1	h(v1)α(u1)+β(v1)d(u1	NOUN
ma-208	46	35	)	)	PUNCT
ma-208	46	36	(	(	PUNCT
ma-208	46	37	h(u1v1	h(u1v1	X
ma-208	46	38	)	)	PUNCT
ma-208	46	39	=	=	SYM
ma-208	46	40	d(v1)α(u1	d(v1)α(u1	NOUN
ma-208	46	41	)	)	PUNCT
ma-208	46	42	+	+	NUM
ma-208	46	43	β(v1)h(u1	β(v1)h(u1	NOUN
ma-208	46	44	)	)	PUNCT
ma-208	46	45	for	for	ADP
ma-208	46	46	all	all	DET
ma-208	46	47	u1	u1	NOUN
ma-208	46	48	,	,	PUNCT
ma-208	46	49	v1	v1	PROPN
ma-208	46	50	∈	∈	PROPN
ma-208	46	51	r.	r.	PROPN
ma-208	46	52	h	h	PROPN
ma-208	46	53	be	be	AUX
ma-208	46	54	a	a	DET
ma-208	46	55	generalized	generalized	ADJ
ma-208	46	56	reverse	reverse	NOUN
ma-208	46	57	(	(	PUNCT
ma-208	46	58	α	α	NOUN
ma-208	46	59	,	,	PUNCT
ma-208	46	60	β	β	NOUN
ma-208	46	61	)	)	PUNCT
ma-208	46	62	of	of	ADP
ma-208	46	63	r	r	NOUN
ma-208	46	64	associatedwith	associatedwith	NOUN
ma-208	46	65	(	(	PUNCT
ma-208	46	66	α	α	NOUN
ma-208	46	67	,	,	PUNCT
ma-208	46	68	β	β	NOUN
ma-208	46	69	)	)	PUNCT
ma-208	46	70	derivation	derivation	NOUN
ma-208	46	71	.	.	PUNCT
ma-208	47	1	definition	definition	NOUN
ma-208	47	2	2.5	2.5	NUM
ma-208	47	3	.	.	PUNCT
ma-208	48	1	some	some	DET
ma-208	48	2	identities	identity	NOUN
ma-208	48	3	holds	hold	VERB
ma-208	48	4	for	for	ADP
ma-208	48	5	every	every	DET
ma-208	48	6	u1	u1	NOUN
ma-208	48	7	,	,	PUNCT
ma-208	48	8	v1	v1	NOUN
ma-208	48	9	,	,	PUNCT
ma-208	48	10	w1	w1	NOUN
ma-208	48	11	∈	∈	PROPN
ma-208	48	12	r	r	NOUN
ma-208	49	1	[	[	X
ma-208	49	2	u1	u1	NOUN
ma-208	49	3	,	,	PUNCT
ma-208	49	4	v1w1	v1w1	X
ma-208	49	5	]	]	PUNCT
ma-208	49	6	=	=	SYM
ma-208	49	7	v1[u1	v1[u1	PROPN
ma-208	49	8	,	,	PUNCT
ma-208	49	9	w1]+[u1	w1]+[u1	PROPN
ma-208	49	10	,	,	PUNCT
ma-208	49	11	v1]w1	v1]w1	NOUN
ma-208	49	12	[	[	X
ma-208	49	13	u1v1	u1v1	NOUN
ma-208	49	14	,	,	PUNCT
ma-208	49	15	w1	w1	NOUN
ma-208	49	16	]	]	PUNCT
ma-208	49	17	=	=	PUNCT
ma-208	50	1	[	[	X
ma-208	50	2	u1	u1	NOUN
ma-208	50	3	,	,	PUNCT
ma-208	50	4	w1]v1	w1]v1	PROPN
ma-208	50	5	+	+	PROPN
ma-208	50	6	u1[v1	u1[v1	PROPN
ma-208	50	7	,	,	PUNCT
ma-208	50	8	w1	w1	NOUN
ma-208	50	9	]	]	PUNCT
ma-208	51	1	[	[	X
ma-208	51	2	u1v1	u1v1	X
ma-208	51	3	,	,	PUNCT
ma-208	51	4	w1]α	w1]α	PROPN
ma-208	51	5	,	,	PUNCT
ma-208	51	6	β	β	X
ma-208	51	7	=	=	SYM
ma-208	51	8	u1[v1	u1[v1	PROPN
ma-208	51	9	,	,	PUNCT
ma-208	51	10	w1]α	w1]α	PROPN
ma-208	51	11	,	,	PUNCT
ma-208	51	12	β	β	X
ma-208	51	13	+	+	PROPN
ma-208	52	1	[	[	X
ma-208	52	2	u1	u1	NOUN
ma-208	52	3	,	,	PUNCT
ma-208	52	4	β(w1)]v1	β(w1)]v1	X
ma-208	52	5	=	=	SYM
ma-208	52	6	u1[v1	u1[v1	PROPN
ma-208	52	7	,	,	PUNCT
ma-208	52	8	α(w1	α(w1	NUM
ma-208	52	9	)	)	PUNCT
ma-208	52	10	]	]	PUNCT
ma-208	53	1	+	+	CCONJ
ma-208	53	2	[	[	X
ma-208	53	3	u1	u1	NOUN
ma-208	53	4	,	,	PUNCT
ma-208	53	5	w1]α	w1]α	PROPN
ma-208	53	6	,	,	PUNCT
ma-208	53	7	β	β	X
ma-208	53	8	v1	v1	PROPN
ma-208	53	9	[	[	X
ma-208	53	10	u1	u1	NOUN
ma-208	53	11	,	,	PUNCT
ma-208	53	12	v1w1]α	v1w1]α	NOUN
ma-208	53	13	,	,	PUNCT
ma-208	53	14	β	β	X
ma-208	53	15	=	=	SYM
ma-208	53	16	β(v1)[u1	β(v1)[u1	PROPN
ma-208	53	17	,	,	PUNCT
ma-208	53	18	w1]α	w1]α	PROPN
ma-208	53	19	,	,	PUNCT
ma-208	53	20	β	β	X
ma-208	53	21	+	+	PROPN
ma-208	54	1	[	[	X
ma-208	54	2	u1	u1	NOUN
ma-208	54	3	,	,	PUNCT
ma-208	54	4	v1]α	v1]α	NOUN
ma-208	54	5	,	,	PUNCT
ma-208	54	6	β	β	X
ma-208	54	7	α(w1	α(w1	NOUN
ma-208	54	8	)	)	PUNCT
ma-208	54	9	definition	definition	NOUN
ma-208	54	10	2.6	2.6	NUM
ma-208	54	11	.	.	PUNCT
ma-208	55	1	the	the	DET
ma-208	55	2	derivation	derivation	NOUN
ma-208	55	3	h	h	NOUN
ma-208	55	4	would	would	AUX
ma-208	55	5	be	be	AUX
ma-208	55	6	commuting	commute	VERB
ma-208	55	7	,	,	PUNCT
ma-208	55	8	if	if	SCONJ
ma-208	55	9	0	0	NUM
ma-208	55	10	=	=	SYM
ma-208	56	1	[	[	X
ma-208	56	2	v1	v1	NOUN
ma-208	56	3	,	,	PUNCT
ma-208	56	4	h(u1	h(u1	NOUN
ma-208	56	5	)	)	PUNCT
ma-208	56	6	]	]	PUNCT
ma-208	56	7	,	,	PUNCT
ma-208	56	8	∀	∀	NOUN
ma-208	56	9	u1	u1	NOUN
ma-208	56	10	,	,	PUNCT
ma-208	56	11	v1	v1	PROPN
ma-208	56	12	∈	∈	PROPN
ma-208	56	13	r.	r.	PROPN
ma-208	56	14	definition	definition	NOUN
ma-208	56	15	2.7	2.7	NUM
ma-208	56	16	.	.	PUNCT
ma-208	57	1	the	the	DET
ma-208	57	2	strong	strong	ADJ
ma-208	57	3	commutativity	commutativity	NOUN
ma-208	57	4	preserving	preserve	VERB
ma-208	57	5	is	be	AUX
ma-208	57	6	defined	define	VERB
ma-208	57	7	as	as	ADP
ma-208	57	8	[	[	X
ma-208	57	9	g(u1	g(u1	NOUN
ma-208	57	10	)	)	PUNCT
ma-208	57	11	,	,	PUNCT
ma-208	57	12	g(v1	g(v1	NOUN
ma-208	57	13	)	)	PUNCT
ma-208	57	14	]	]	PUNCT
ma-208	58	1	=	=	PUNCT
ma-208	59	1	[	[	X
ma-208	59	2	u1	u1	NOUN
ma-208	59	3	,	,	PUNCT
ma-208	59	4	v1	v1	NOUN
ma-208	59	5	]	]	PUNCT
ma-208	59	6	forall	forall	NOUN
ma-208	59	7	u1	u1	NOUN
ma-208	59	8	,	,	PUNCT
ma-208	59	9	v1	v1	NOUN
ma-208	59	10	∈	∈	PROPN
ma-208	59	11	r	r	NOUN
ma-208	59	12	,	,	PUNCT
ma-208	59	13	where	where	SCONJ
ma-208	59	14	g	g	NOUN
ma-208	59	15	:	:	PUNCT
ma-208	59	16	r→	r→	PROPN
ma-208	59	17	r	r	NOUN
ma-208	59	18	is	be	AUX
ma-208	59	19	a	a	DET
ma-208	59	20	mapping	mapping	NOUN
ma-208	59	21	on	on	ADP
ma-208	59	22	r.	r.	PROPN
ma-208	59	23	lemma	lemma	PROPN
ma-208	59	24	2.8	2.8	NUM
ma-208	59	25	.	.	PUNCT
ma-208	60	1	let	let	VERB
ma-208	60	2	u1	u1	NOUN
ma-208	60	3	6=	6=	PRON
ma-208	60	4	0	0	NUM
ma-208	60	5	in	in	ADP
ma-208	60	6	z(center	z(center	NOUN
ma-208	60	7	of	of	ADP
ma-208	60	8	ring	ring	NOUN
ma-208	60	9	)	)	PUNCT
ma-208	60	10	,	,	PUNCT
ma-208	60	11	if	if	SCONJ
ma-208	60	12	u1	u1	NOUN
ma-208	60	13	,	,	PUNCT
ma-208	60	14	v1	v1	PROPN
ma-208	60	15	∈	∈	PROPN
ma-208	60	16	z	z	NOUN
ma-208	60	17	,	,	PUNCT
ma-208	60	18	then	then	ADV
ma-208	60	19	v1	v1	VERB
ma-208	60	20	∈	∈	PROPN
ma-208	60	21	z.	z.	PROPN
ma-208	60	22	lemma	lemma	PROPN
ma-208	61	1	2.9	2.9	NUM
ma-208	61	2	.	.	PUNCT
ma-208	62	1	let	let	VERB
ma-208	62	2	g	g	NOUN
ma-208	62	3	:	:	PUNCT
ma-208	62	4	r	r	NOUN
ma-208	62	5	→	→	SYM
ma-208	62	6	r	r	NOUN
ma-208	62	7	be	be	AUX
ma-208	62	8	an	an	DET
ma-208	62	9	additive	additive	ADJ
ma-208	62	10	map	map	NOUN
ma-208	62	11	and	and	CCONJ
ma-208	62	12	on	on	ADP
ma-208	62	13	a	a	DET
ma-208	62	14	left	left	ADJ
ma-208	62	15	ideal	ideal	NOUN
ma-208	62	16	b	b	PROPN
ma-208	62	17	of	of	ADP
ma-208	62	18	r	r	NOUN
ma-208	62	19	,	,	PUNCT
ma-208	62	20	g	g	PROPN
ma-208	62	21	is	be	AUX
ma-208	62	22	centralizing	centralize	VERB
ma-208	62	23	,	,	PUNCT
ma-208	62	24	then	then	ADV
ma-208	62	25	g(u1	g(u1	NOUN
ma-208	62	26	)	)	PUNCT
ma-208	62	27	∈	∈	PROPN
ma-208	62	28	r	r	NOUN
ma-208	62	29	∀	∀	X
ma-208	62	30	u1	u1	NOUN
ma-208	62	31	∈	∈	PROPN
ma-208	62	32	b	b	PROPN
ma-208	62	33	∪	∪	PROPN
ma-208	62	34	z.	z.	PROPN
ma-208	62	35	lemma	lemma	PROPN
ma-208	62	36	2.10	2.10	NUM
ma-208	62	37	.	.	PUNCT
ma-208	63	1	let	let	VERB
ma-208	63	2	0	0	NUM
ma-208	64	1	6=	6=	NUM
ma-208	64	2	b	b	X
ma-208	64	3	be	be	AUX
ma-208	64	4	an	an	DET
ma-208	64	5	ideal	ideal	NOUN
ma-208	64	6	of	of	ADP
ma-208	64	7	a	a	DET
ma-208	64	8	semi	semi	ADJ
ma-208	64	9	-	-	ADJ
ma-208	64	10	prime	prime	ADJ
ma-208	64	11	ring	ring	NOUN
ma-208	64	12	r.	r.	PROPN
ma-208	64	13	if	if	SCONJ
ma-208	64	14	the	the	DET
ma-208	64	15	set	set	NOUN
ma-208	64	16	[	[	X
ma-208	64	17	b	b	X
ma-208	64	18	,	,	PUNCT
ma-208	64	19	b	b	NOUN
ma-208	64	20	]	]	X
ma-208	64	21	centralizes	centralize	VERB
ma-208	64	22	z	z	NOUN
ma-208	64	23	in	in	ADP
ma-208	64	24	r	r	NOUN
ma-208	64	25	,	,	PUNCT
ma-208	64	26	then	then	ADV
ma-208	64	27	b	b	NOUN
ma-208	64	28	centralizes	centralize	VERB
ma-208	64	29	z.	z.	PROPN
ma-208	64	30	2.1	2.1	NUM
ma-208	64	31	.	.	PUNCT
ma-208	65	1	point	point	NOUN
ma-208	65	2	-	-	PUNCT
ma-208	65	3	wise	wise	ADJ
ma-208	65	4	operation	operation	NOUN
ma-208	65	5	.	.	PUNCT
ma-208	66	1	theorem	theorem	VERB
ma-208	66	2	2.11	2.11	NUM
ma-208	66	3	.	.	PUNCT
ma-208	67	1	suppose	suppose	VERB
ma-208	67	2	0	0	PUNCT
ma-208	68	1	6=	6=	NUM
ma-208	68	2	d	d	NOUN
ma-208	68	3	from	from	ADP
ma-208	68	4	r	r	NOUN
ma-208	68	5	to	to	ADP
ma-208	68	6	r	r	NOUN
ma-208	68	7	a	a	DET
ma-208	68	8	derivation	derivation	NOUN
ma-208	68	9	in	in	ADP
ma-208	68	10	a	a	DET
ma-208	68	11	semi	semi	ADJ
ma-208	68	12	-	-	ADJ
ma-208	68	13	prime	prime	ADJ
ma-208	68	14	ring	ring	NOUN
ma-208	68	15	r.	r.	PROPN
ma-208	68	16	let	let	VERB
ma-208	68	17	generalized	generalize	VERB
ma-208	68	18	(	(	PUNCT
ma-208	68	19	α	α	NOUN
ma-208	68	20	,	,	PUNCT
ma-208	68	21	β)-reverse	β)-reverse	PUNCT
ma-208	68	22	derivation	derivation	NOUN
ma-208	68	23	g	g	NOUN
ma-208	68	24	on	on	ADP
ma-208	68	25	a	a	DET
ma-208	68	26	left	left	ADJ
ma-208	68	27	ideal	ideal	NOUN
ma-208	68	28	b	b	PROPN
ma-208	68	29	6=	6=	ADP
ma-208	68	30	0	0	NUM
ma-208	68	31	of	of	ADP
ma-208	68	32	r.	r.	PROPN
ma-208	68	33	then	then	ADV
ma-208	68	34	g	g	PROPN
ma-208	68	35	satisfies	satisfie	NOUN
ma-208	68	36	[	[	X
ma-208	68	37	g(w1	g(w1	X
ma-208	68	38	)	)	PUNCT
ma-208	68	39	,	,	PUNCT
ma-208	68	40	g(v1	g(v1	NOUN
ma-208	68	41	)	)	PUNCT
ma-208	68	42	]	]	PUNCT
ma-208	69	1	=	=	PUNCT
ma-208	69	2	β([w1	β([w1	NOUN
ma-208	69	3	,	,	PUNCT
ma-208	69	4	v1	v1	NOUN
ma-208	69	5	]	]	PUNCT
ma-208	69	6	)	)	PUNCT
ma-208	69	7	for	for	ADP
ma-208	69	8	all	all	DET
ma-208	69	9	v1	v1	NOUN
ma-208	69	10	,	,	PUNCT
ma-208	69	11	w1	w1	NOUN
ma-208	69	12	∈	∈	PROPN
ma-208	69	13	b	b	PROPN
ma-208	69	14	(	(	PUNCT
ma-208	69	15	that	that	PRON
ma-208	69	16	is	is	ADV
ma-208	69	17	,	,	PUNCT
ma-208	69	18	g	g	PROPN
ma-208	69	19	is	be	AUX
ma-208	69	20	β	β	NOUN
ma-208	69	21	-	-	ADJ
ma-208	69	22	strong	strong	ADJ
ma-208	69	23	commutativitypreserved	commutativitypreserve	VERB
ma-208	69	24	)	)	PUNCT
ma-208	69	25	,	,	PUNCT
ma-208	69	26	when	when	SCONJ
ma-208	69	27	g	g	PROPN
ma-208	69	28	is	be	AUX
ma-208	69	29	a	a	DET
ma-208	69	30	homomorphism	homomorphism	NOUN
ma-208	69	31	on	on	ADP
ma-208	69	32	b.	b.	PROPN
ma-208	69	33	proof	proof	NOUN
ma-208	69	34	.	.	PUNCT
ma-208	70	1	since	since	SCONJ
ma-208	70	2	g	g	PROPN
ma-208	70	3	is	be	AUX
ma-208	70	4	generalized	generalize	VERB
ma-208	70	5	(	(	PUNCT
ma-208	70	6	α	α	NOUN
ma-208	70	7	,	,	PUNCT
ma-208	70	8	β)-reverse	β)-reverse	PUNCT
ma-208	70	9	derivation	derivation	NOUN
ma-208	70	10	and	and	CCONJ
ma-208	70	11	homomorphism	homomorphism	NOUN
ma-208	70	12	on	on	ADP
ma-208	70	13	b	b	NOUN
ma-208	70	14	,	,	PUNCT
ma-208	70	15	such	such	ADJ
ma-208	70	16	that	that	DET
ma-208	70	17	g(u1v1	g(u1v1	NOUN
ma-208	70	18	)	)	PUNCT
ma-208	70	19	=	=	SYM
ma-208	70	20	g(u1)g(v1)∀	g(u1)g(v1)∀	NOUN
ma-208	70	21	u1	u1	NOUN
ma-208	70	22	,	,	PUNCT
ma-208	70	23	v1	v1	PROPN
ma-208	70	24	∈	∈	PROPN
ma-208	70	25	b.	b.	PROPN
ma-208	70	26	https://doi.org/10.28924/ada/ma.4.5	https://doi.org/10.28924/ada/ma.4.5	PROPN
ma-208	70	27	eur	eur	PROPN
ma-208	70	28	.	.	PUNCT
ma-208	71	1	j.	j.	PROPN
ma-208	71	2	math	math	PROPN
ma-208	71	3	.	.	PUNCT
ma-208	72	1	anal	anal	PROPN
ma-208	72	2	.	.	PUNCT
ma-208	73	1	10.28924	10.28924	NUM
ma-208	73	2	/	/	SYM
ma-208	73	3	ada	ada	PROPN
ma-208	73	4	/	/	SYM
ma-208	73	5	ma.4.5	ma.4.5	PROPN
ma-208	73	6	3this	3this	NUM
ma-208	73	7	implies	imply	VERB
ma-208	73	8	g(u1v1	g(u1v1	NOUN
ma-208	73	9	)	)	PUNCT
ma-208	73	10	=	=	SYM
ma-208	73	11	g(u1)g(v1	g(u1)g(v1	NOUN
ma-208	73	12	)	)	PUNCT
ma-208	73	13	=	=	SYM
ma-208	73	14	g(v1)α(u1	g(v1)α(u1	PROPN
ma-208	73	15	)	)	PUNCT
ma-208	74	1	+	+	SYM
ma-208	74	2	β(v1)d(u1	β(v1)d(u1	NOUN
ma-208	74	3	)	)	PUNCT
ma-208	74	4	,	,	PUNCT
ma-208	74	5	f	f	PROPN
ma-208	74	6	or	or	CCONJ
ma-208	74	7	al	al	PROPN
ma-208	74	8	l	l	PROPN
ma-208	74	9	u1	u1	NOUN
ma-208	74	10	,	,	PUNCT
ma-208	74	11	v1	v1	PROPN
ma-208	74	12	∈	∈	PROPN
ma-208	74	13	b.	b.	PROPN
ma-208	74	14	(	(	PUNCT
ma-208	74	15	1	1	X
ma-208	74	16	)	)	PUNCT
ma-208	74	17	we	we	PRON
ma-208	74	18	replace	replace	VERB
ma-208	74	19	v1	v1	NOUN
ma-208	74	20	by	by	ADP
ma-208	74	21	v1w1	v1w1	DET
ma-208	74	22	where	where	SCONJ
ma-208	74	23	w1	w1	NOUN
ma-208	74	24	∈	∈	PROPN
ma-208	74	25	b	b	PROPN
ma-208	74	26	,	,	PUNCT
ma-208	74	27	in	in	ADP
ma-208	74	28	equation	equation	NOUN
ma-208	74	29	(	(	PUNCT
ma-208	74	30	1	1	NUM
ma-208	74	31	)	)	PUNCT
ma-208	74	32	,	,	PUNCT
ma-208	74	33	we	we	PRON
ma-208	74	34	obtain	obtain	VERB
ma-208	74	35	g(u1)g(v1w1	g(u1)g(v1w1	PUNCT
ma-208	74	36	)	)	PUNCT
ma-208	74	37	=	=	SYM
ma-208	74	38	g(v1w1)α(u1	g(v1w1)α(u1	X
ma-208	74	39	)	)	PUNCT
ma-208	75	1	+	+	CCONJ
ma-208	75	2	β(v1w1)d(u1)this	β(v1w1)d(u1)this	PRON
ma-208	75	3	gives	give	VERB
ma-208	75	4	g(u1)g(v1w1	g(u1)g(v1w1	ADV
ma-208	75	5	)	)	PUNCT
ma-208	75	6	=	=	SYM
ma-208	75	7	g(u1v1w1	g(u1v1w1	PROPN
ma-208	75	8	)	)	PUNCT
ma-208	75	9	=	=	SYM
ma-208	75	10	g(v1)g(w1)α(u1	g(v1)g(w1)α(u1	NOUN
ma-208	75	11	)	)	PUNCT
ma-208	76	1	+	+	CCONJ
ma-208	76	2	β(v1w1)d(u1	β(v1w1)d(u1	X
ma-208	76	3	)	)	PUNCT
ma-208	76	4	,	,	PUNCT
ma-208	76	5	f	f	PROPN
ma-208	76	6	or	or	CCONJ
ma-208	76	7	al	al	PROPN
ma-208	76	8	l	l	PROPN
ma-208	76	9	u1	u1	NOUN
ma-208	76	10	,	,	PUNCT
ma-208	76	11	v1	v1	PROPN
ma-208	76	12	∈	∈	PROPN
ma-208	76	13	b.	b.	PROPN
ma-208	76	14	(	(	PUNCT
ma-208	76	15	2	2	NUM
ma-208	76	16	)	)	PUNCT
ma-208	76	17	as	as	SCONJ
ma-208	76	18	g	g	PROPN
ma-208	76	19	is	be	AUX
ma-208	76	20	homomorphism	homomorphism	NOUN
ma-208	76	21	,	,	PUNCT
ma-208	76	22	so	so	ADV
ma-208	76	23	we	we	PRON
ma-208	76	24	get	get	VERB
ma-208	76	25	g(u1)g(v1w1	g(u1)g(v1w1	PUNCT
ma-208	76	26	)	)	PUNCT
ma-208	76	27	=	=	SYM
ma-208	77	1	g(u1)g(v1)g(w1	g(u1)g(v1)g(w1	NOUN
ma-208	77	2	)	)	PUNCT
ma-208	77	3	=	=	PUNCT
ma-208	77	4	g(u1v1)g(w1)this	g(u1v1)g(w1)thi	NOUN
ma-208	77	5	equalized	equalize	VERB
ma-208	77	6	to	to	ADP
ma-208	77	7	g(u1v1)g(w1	g(u1v1)g(w1	NOUN
ma-208	77	8	)	)	PUNCT
ma-208	77	9	=	=	SYM
ma-208	77	10	(	(	PUNCT
ma-208	77	11	g(v1)α(u1	g(v1)α(u1	PROPN
ma-208	77	12	)	)	PUNCT
ma-208	78	1	+	+	CCONJ
ma-208	78	2	β(v1)d(u1))g(w1)this	β(v1)d(u1))g(w1)this	PRON
ma-208	78	3	relates	relate	VERB
ma-208	78	4	to	to	ADP
ma-208	78	5	g(u1v1)g(w1	g(u1v1)g(w1	NOUN
ma-208	78	6	)	)	PUNCT
ma-208	78	7	=	=	SYM
ma-208	78	8	g(v1)α(u1)g(w1	g(v1)α(u1)g(w1	NOUN
ma-208	78	9	)	)	PUNCT
ma-208	78	10	+	+	CCONJ
ma-208	78	11	β(v1)d(u1)g(w1)by	β(v1)d(u1)g(w1)by	X
ma-208	78	12	the	the	DET
ma-208	78	13	equation	equation	NOUN
ma-208	78	14	(	(	PUNCT
ma-208	78	15	2	2	NUM
ma-208	78	16	)	)	PUNCT
ma-208	78	17	,	,	PUNCT
ma-208	78	18	we	we	PRON
ma-208	78	19	get	get	VERB
ma-208	78	20	g(u1v1)g(w1	g(u1v1)g(w1	NOUN
ma-208	78	21	)	)	PUNCT
ma-208	78	22	=	=	SYM
ma-208	78	23	g(v1)g(w1)α(u1	g(v1)g(w1)α(u1	NOUN
ma-208	78	24	)	)	PUNCT
ma-208	79	1	+	+	NUM
ma-208	79	2	β(v1)d(u1)g(w1	β(v1)d(u1)g(w1	NOUN
ma-208	79	3	)	)	PUNCT
ma-208	79	4	,	,	PUNCT
ma-208	79	5	f	f	PROPN
ma-208	79	6	or	or	CCONJ
ma-208	79	7	al	al	PROPN
ma-208	79	8	l	l	PROPN
ma-208	79	9	u1	u1	NOUN
ma-208	79	10	,	,	PUNCT
ma-208	79	11	v1	v1	PROPN
ma-208	79	12	∈	∈	PROPN
ma-208	79	13	b.	b.	PROPN
ma-208	79	14	(	(	PUNCT
ma-208	79	15	3	3	NUM
ma-208	79	16	)	)	PUNCT
ma-208	79	17	from	from	ADP
ma-208	79	18	equation	equation	NOUN
ma-208	79	19	(	(	PUNCT
ma-208	79	20	2	2	NUM
ma-208	79	21	)	)	PUNCT
ma-208	79	22	and	and	CCONJ
ma-208	79	23	equation	equation	NOUN
ma-208	79	24	(	(	PUNCT
ma-208	79	25	3	3	NUM
ma-208	79	26	)	)	PUNCT
ma-208	79	27	,	,	PUNCT
ma-208	79	28	we	we	PRON
ma-208	79	29	obtain	obtain	VERB
ma-208	79	30	β(v1)d(u1)g(w1	β(v1)d(u1)g(w1	PUNCT
ma-208	79	31	)	)	PUNCT
ma-208	80	1	=	=	SYM
ma-208	80	2	β(v1)d(u1)β(w1)this	β(v1)d(u1)β(w1)this	PROPN
ma-208	80	3	implies	imply	VERB
ma-208	80	4	β(v1)d(u1)(g(w1)−	β(v1)d(u1)(g(w1)−	PROPN
ma-208	80	5	β(w1	β(w1	NOUN
ma-208	80	6	)	)	PUNCT
ma-208	80	7	)	)	PUNCT
ma-208	81	1	=	=	SYM
ma-208	81	2	0	0	NUM
ma-208	81	3	,	,	PUNCT
ma-208	81	4	f	f	PROPN
ma-208	81	5	or	or	CCONJ
ma-208	81	6	al	al	PROPN
ma-208	81	7	l	l	PROPN
ma-208	81	8	u1	u1	NOUN
ma-208	81	9	,	,	PUNCT
ma-208	81	10	v1	v1	PROPN
ma-208	81	11	∈	∈	PROPN
ma-208	81	12	b.	b.	PROPN
ma-208	81	13	(	(	PUNCT
ma-208	81	14	4	4	X
ma-208	81	15	)	)	PUNCT
ma-208	81	16	put	put	VERB
ma-208	81	17	w1	w1	NOUN
ma-208	81	18	=	=	PUNCT
ma-208	82	1	[	[	X
ma-208	82	2	w1	w1	NOUN
ma-208	82	3	,	,	PUNCT
ma-208	82	4	v1	v1	NOUN
ma-208	82	5	]	]	PUNCT
ma-208	82	6	in	in	ADP
ma-208	82	7	equation	equation	NOUN
ma-208	82	8	(	(	PUNCT
ma-208	82	9	4	4	NUM
ma-208	82	10	)	)	PUNCT
ma-208	82	11	,	,	PUNCT
ma-208	82	12	we	we	PRON
ma-208	82	13	have	have	VERB
ma-208	82	14	β(v1)d(u1)(g([w1	β(v1)d(u1)(g([w1	NOUN
ma-208	82	15	,	,	PUNCT
ma-208	82	16	v1])−	v1])−	VERB
ma-208	82	17	β([w1	β([w1	NOUN
ma-208	82	18	,	,	PUNCT
ma-208	82	19	v1	v1	NOUN
ma-208	82	20	]	]	PUNCT
ma-208	82	21	)	)	PUNCT
ma-208	82	22	)	)	PUNCT
ma-208	83	1	=	=	PUNCT
ma-208	83	2	0,we	0,we	PROPN
ma-208	83	3	arrives	arrive	VERB
ma-208	83	4	to	to	ADP
ma-208	83	5	d(u1)β(v1)(g([w1	d(u1)β(v1)(g([w1	NOUN
ma-208	83	6	,	,	PUNCT
ma-208	83	7	v1])−	v1])−	NOUN
ma-208	83	8	β([w1	β([w1	NOUN
ma-208	83	9	,	,	PUNCT
ma-208	83	10	v1	v1	NOUN
ma-208	83	11	]	]	PUNCT
ma-208	83	12	)	)	PUNCT
ma-208	83	13	)	)	PUNCT
ma-208	84	1	=	=	PUNCT
ma-208	84	2	0.by	0.by	PUNCT
ma-208	84	3	replacing	replace	VERB
ma-208	84	4	β(v1	β(v1	NOUN
ma-208	84	5	)	)	PUNCT
ma-208	84	6	by	by	ADP
ma-208	84	7	(	(	PUNCT
ma-208	84	8	g([w1	g([w1	NOUN
ma-208	84	9	,	,	PUNCT
ma-208	84	10	v1])−	v1])−	NOUN
ma-208	84	11	β([w1	β([w1	NOUN
ma-208	84	12	,	,	PUNCT
ma-208	84	13	v1]))α(r)d(u1	v1]))α(r)d(u1	PROPN
ma-208	84	14	)	)	PUNCT
ma-208	84	15	,	,	PUNCT
ma-208	84	16	we	we	PRON
ma-208	84	17	obtain	obtain	VERB
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ma-208	84	20	v1])−	v1])−	VERB
ma-208	84	21	β([w1	β([w1	NOUN
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ma-208	84	24	,	,	PUNCT
ma-208	84	25	v1])−	v1])−	VERB
ma-208	84	26	β([w1	β([w1	NOUN
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ma-208	84	28	v1	v1	NOUN
ma-208	84	29	]	]	PUNCT
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ma-208	84	31	)	)	PUNCT
ma-208	85	1	=	=	PUNCT
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ma-208	85	6	v1])−	v1])−	VERB
ma-208	85	7	β([w1	β([w1	NOUN
ma-208	85	8	,	,	PUNCT
ma-208	85	9	v1]))rd(u1)(g([w1	v1]))rd(u1)(g([w1	PROPN
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ma-208	85	11	v1])−	v1])−	VERB
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ma-208	85	17	)	)	PUNCT
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ma-208	86	5	-	-	ADJ
ma-208	86	6	prime	prime	ADJ
ma-208	86	7	,	,	PUNCT
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ma-208	86	9	we	we	PRON
ma-208	86	10	obtain	obtain	VERB
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ma-208	86	12	,	,	PUNCT
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ma-208	86	19	)	)	PUNCT
ma-208	87	1	=	=	PUNCT
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ma-208	88	3	6=	6=	NOUN
ma-208	88	4	0	0	NUM
ma-208	88	5	,	,	PUNCT
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ma-208	88	8	g([w1	g([w1	NOUN
ma-208	88	9	,	,	PUNCT
ma-208	88	10	v1])−	v1])−	VERB
ma-208	88	11	β([w1	β([w1	NOUN
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ma-208	88	13	v1	v1	NOUN
ma-208	88	14	]	]	PUNCT
ma-208	88	15	)	)	PUNCT
ma-208	89	1	=	=	SYM
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ma-208	89	4	g([w1	g([w1	NOUN
ma-208	89	5	,	,	PUNCT
ma-208	89	6	v1	v1	NOUN
ma-208	89	7	]	]	PUNCT
ma-208	89	8	)	)	PUNCT
ma-208	90	1	=	=	SYM
ma-208	90	2	β([w1	β([w1	NOUN
ma-208	90	3	,	,	PUNCT
ma-208	90	4	v1	v1	NOUN
ma-208	90	5	]	]	PUNCT
ma-208	90	6	)	)	PUNCT
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ma-208	90	9	.	.	PUNCT
ma-208	91	1	j.	j.	PROPN
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ma-208	91	3	.	.	PUNCT
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ma-208	92	2	.	.	PUNCT
ma-208	93	1	10.28924	10.28924	NUM
ma-208	93	2	/	/	SYM
ma-208	93	3	ada	ada	PROPN
ma-208	93	4	/	/	SYM
ma-208	93	5	ma.4.5	ma.4.5	PROPN
ma-208	93	6	4as	4as	PROPN
ma-208	93	7	g	g	PROPN
ma-208	93	8	is	be	AUX
ma-208	93	9	homomorphism	homomorphism	NOUN
ma-208	93	10	,	,	PUNCT
ma-208	93	11	so	so	ADV
ma-208	93	12	we	we	PRON
ma-208	93	13	have	have	VERB
ma-208	93	14	[	[	X
ma-208	93	15	g(w1	g(w1	X
ma-208	93	16	)	)	PUNCT
ma-208	93	17	,	,	PUNCT
ma-208	93	18	g(v1	g(v1	NOUN
ma-208	93	19	)	)	PUNCT
ma-208	93	20	]	]	PUNCT
ma-208	94	1	=	=	PUNCT
ma-208	94	2	β([w1	β([w1	NOUN
ma-208	94	3	,	,	PUNCT
ma-208	94	4	v1])so	v1])so	PROPN
ma-208	94	5	g	g	PROPN
ma-208	94	6	is	be	AUX
ma-208	94	7	β	β	NOUN
ma-208	94	8	-	-	ADJ
ma-208	94	9	strong	strong	ADJ
ma-208	94	10	commutative	commutative	ADJ
ma-208	94	11	preserved	preserve	VERB
ma-208	94	12	on	on	ADP
ma-208	94	13	b.	b.	PROPN
ma-208	94	14	theorem	theorem	PROPN
ma-208	94	15	2.12	2.12	NUM
ma-208	94	16	.	.	PUNCT
ma-208	95	1	let	let	VERB
ma-208	95	2	g	g	NOUN
ma-208	95	3	on	on	ADP
ma-208	95	4	a	a	DET
ma-208	95	5	left	left	ADJ
ma-208	95	6	ideal	ideal	NOUN
ma-208	95	7	b	b	PROPN
ma-208	95	8	6=	6=	ADP
ma-208	95	9	0	0	NUM
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ma-208	95	11	r	r	NOUN
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ma-208	95	17	,	,	PUNCT
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ma-208	95	20	.	.	PUNCT
ma-208	96	1	if	if	SCONJ
ma-208	96	2	gis	gis	PROPN
ma-208	96	3	homomorphism	homomorphism	NOUN
ma-208	96	4	on	on	ADP
ma-208	96	5	b	b	NOUN
ma-208	96	6	,	,	PUNCT
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ma-208	96	9	b	b	NOUN
ma-208	96	10	,	,	PUNCT
ma-208	96	11	g	g	PROPN
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ma-208	96	14	.	.	PUNCT
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ma-208	97	2	.	.	PUNCT
ma-208	98	1	by	by	ADP
ma-208	98	2	theorem	theorem	NOUN
ma-208	98	3	2.11	2.11	NUM
ma-208	98	4	,	,	PUNCT
ma-208	98	5	g	g	PROPN
ma-208	98	6	is	be	AUX
ma-208	98	7	β	β	NOUN
ma-208	98	8	-	-	ADJ
ma-208	98	9	strong	strong	ADJ
ma-208	98	10	commutative	commutative	ADJ
ma-208	98	11	preserved	preserve	VERB
ma-208	98	12	,	,	PUNCT
ma-208	98	13	then	then	ADV
ma-208	98	14	∀	∀	NOUN
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ma-208	98	16	,	,	PUNCT
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ma-208	98	18	∈	∈	PROPN
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ma-208	98	20	,	,	PUNCT
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ma-208	98	23	β([u1	β([u1	ADJ
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ma-208	98	25	v1	v1	NOUN
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ma-208	99	4	,	,	PUNCT
ma-208	99	5	g(v1	g(v1	NOUN
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ma-208	99	7	]	]	PUNCT
ma-208	99	8	(	(	PUNCT
ma-208	99	9	5	5	X
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ma-208	99	12	v1	v1	NOUN
ma-208	99	13	=	=	SYM
ma-208	99	14	v1u1	v1u1	NOUN
ma-208	99	15	in	in	ADP
ma-208	99	16	equation	equation	NOUN
ma-208	99	17	(	(	PUNCT
ma-208	99	18	5	5	X
ma-208	99	19	)	)	PUNCT
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ma-208	99	22	β([u1	β([u1	NOUN
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ma-208	101	4	,	,	PUNCT
ma-208	101	5	g(v1u1	g(v1u1	NOUN
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ma-208	101	7	]	]	PUNCT
ma-208	102	1	β([u1	β([u1	X
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ma-208	102	3	v1])β(u1	v1])β(u1	NOUN
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ma-208	102	5	=	=	PUNCT
ma-208	103	1	[	[	X
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ma-208	103	4	,	,	PUNCT
ma-208	103	5	g(v1)]g(u1)by	g(v1)]g(u1)by	PROPN
ma-208	103	6	equation	equation	NOUN
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ma-208	103	8	5	5	NUM
ma-208	103	9	)	)	PUNCT
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ma-208	104	7	v1])(g(u1)−	v1])(g(u1)−	NOUN
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ma-208	106	1	(	(	PUNCT
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ma-208	106	3	)	)	PUNCT
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ma-208	107	2	in	in	ADP
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ma-208	107	4	(	(	PUNCT
ma-208	107	5	5	5	X
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ma-208	107	7	we	we	PRON
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ma-208	107	9	β([u1	β([u1	NOUN
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ma-208	108	4	,	,	PUNCT
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ma-208	108	10	v1	v1	NOUN
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ma-208	108	22	,	,	PUNCT
ma-208	108	23	we	we	PRON
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ma-208	108	25	β(u1)β([u1	β(u1)β([u1	NUM
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ma-208	108	27	v1	v1	NOUN
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ma-208	108	32	,	,	PUNCT
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ma-208	108	34	implies	implie	NOUN
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ma-208	108	37	β(u1))β([u1	β(u1))β([u1	NOUN
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ma-208	108	39	v1	v1	NOUN
ma-208	108	40	]	]	PUNCT
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ma-208	108	42	=	=	SYM
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ma-208	110	1	(	(	PUNCT
ma-208	110	2	7	7	X
ma-208	110	3	)	)	PUNCT
ma-208	110	4	put	put	VERB
ma-208	110	5	v1	v1	NOUN
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ma-208	110	8	in	in	ADP
ma-208	110	9	equation	equation	NOUN
ma-208	110	10	(	(	PUNCT
ma-208	110	11	6	6	NUM
ma-208	110	12	)	)	PUNCT
ma-208	110	13	,	,	PUNCT
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ma-208	110	15	have	have	VERB
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ma-208	110	21	)	)	PUNCT
ma-208	111	1	=	=	PUNCT
ma-208	112	1	0,this	0,thi	NOUN
ma-208	112	2	implies	imply	VERB
ma-208	112	3	β([u1	β([u1	PROPN
ma-208	112	4	,	,	PUNCT
ma-208	112	5	r1])β(v1)(g(u1)−	r1])β(v1)(g(u1)−	ADJ
ma-208	112	6	β(u1	β(u1	NOUN
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ma-208	112	8	)	)	PUNCT
ma-208	113	1	=	=	PUNCT
ma-208	113	2	0,we	0,we	NUM
ma-208	113	3	get	get	VERB
ma-208	113	4	β([u1	β([u1	ADJ
ma-208	113	5	,	,	PUNCT
ma-208	113	6	r1])b(g(u1)−	r1])b(g(u1)−	NOUN
ma-208	113	7	β(u1	β(u1	NOUN
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ma-208	113	9	)	)	PUNCT
ma-208	114	1	=	=	PUNCT
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ma-208	114	3	that	that	SCONJ
ma-208	114	4	β([u1	β([u1	ADV
ma-208	114	5	,	,	PUNCT
ma-208	114	6	r1])rb(g(u1)−	r1])rb(g(u1)−	NOUN
ma-208	114	7	β(u1	β(u1	NOUN
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ma-208	114	9	)	)	PUNCT
ma-208	115	1	=	=	PUNCT
ma-208	116	1	0,by	0,by	NUM
ma-208	116	2	semi	semi	ADJ
ma-208	116	3	-	-	NOUN
ma-208	116	4	primeness	primeness	NOUN
ma-208	116	5	of	of	ADP
ma-208	116	6	r	r	NOUN
ma-208	116	7	,	,	PUNCT
ma-208	116	8	there	there	PRON
ma-208	116	9	exist	exist	VERB
ma-208	116	10	a	a	DET
ma-208	116	11	family	family	NOUN
ma-208	116	12	w	w	NOUN
ma-208	116	13	=	=	PUNCT
ma-208	116	14	{	{	PUNCT
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ma-208	116	16	/	/	SYM
ma-208	116	17	θ	θ	PROPN
ma-208	116	18	∈	∈	PROPN
ma-208	116	19	∧	∧	PROPN
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ma-208	116	21	of	of	ADP
ma-208	116	22	prime	prime	ADJ
ma-208	116	23	ideals	ideal	NOUN
ma-208	116	24	such	such	ADJ
ma-208	116	25	that	that	SCONJ
ma-208	116	26	⋂	⋂	PROPN
ma-208	116	27	pθ	pθ	NOUN
ma-208	116	28	=	=	NOUN
ma-208	116	29	0.if	0.if	NUM
ma-208	116	30	w	w	VERB
ma-208	116	31	has	have	VERB
ma-208	116	32	a	a	DET
ma-208	116	33	member	member	NOUN
ma-208	116	34	p	p	NOUN
ma-208	116	35	and	and	CCONJ
ma-208	116	36	u1	u1	PROPN
ma-208	116	37	∈	∈	PROPN
ma-208	116	38	b	b	PROPN
ma-208	116	39	,	,	PUNCT
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ma-208	116	41	last	last	ADJ
ma-208	116	42	relation	relation	NOUN
ma-208	116	43	,	,	PUNCT
ma-208	116	44	we	we	PRON
ma-208	116	45	get	get	VERB
ma-208	116	46	,	,	PUNCT
ma-208	116	47	b(g(u1	b(g(u1	NOUN
ma-208	116	48	)	)	PUNCT
ma-208	116	49	−	−	NOUN
ma-208	116	50	β(u1	β(u1	NOUN
ma-208	116	51	)	)	PUNCT
ma-208	116	52	)	)	PUNCT
ma-208	117	1	not	not	PART
ma-208	117	2	in	in	ADP
ma-208	117	3	p	p	NOUN
ma-208	117	4	or	or	CCONJ
ma-208	117	5	[	[	X
ma-208	117	6	β(u1	β(u1	NOUN
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ma-208	117	8	,	,	PUNCT
ma-208	117	9	r	r	X
ma-208	117	10	]	]	X
ma-208	117	11	⊆	⊆	NUM
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ma-208	118	2	∃	∃	PROPN
ma-208	118	3	v1	v1	PROPN
ma-208	118	4	∈	∈	PROPN
ma-208	118	5	b	b	NOUN
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ma-208	118	7	that	that	SCONJ
ma-208	118	8	[	[	X
ma-208	118	9	β(u1	β(u1	NOUN
ma-208	118	10	)	)	PUNCT
ma-208	118	11	,	,	PUNCT
ma-208	118	12	r	r	NOUN
ma-208	118	13	]	]	X
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ma-208	118	15	in	in	ADP
ma-208	118	16	p	p	NOUN
ma-208	118	17	.	.	PUNCT
ma-208	119	1	it	it	PRON
ma-208	119	2	implies	imply	VERB
ma-208	119	3	b(g(v1	b(g(v1	NOUN
ma-208	119	4	)	)	PUNCT
ma-208	119	5	−	−	PROPN
ma-208	119	6	β(v1	β(v1	NOUN
ma-208	119	7	)	)	PUNCT
ma-208	119	8	)	)	PUNCT
ma-208	120	1	⊆	⊆	NUM
ma-208	120	2	p	p	NOUN
ma-208	120	3	.	.	PUNCT
ma-208	121	1	let	let	VERB
ma-208	121	2	w1	w1	PROPN
ma-208	121	3	∈	∈	PROPN
ma-208	121	4	b	b	NOUN
ma-208	121	5	is	be	AUX
ma-208	121	6	arbitrary	arbitrary	ADJ
ma-208	121	7	such	such	ADJ
ma-208	121	8	that	that	SCONJ
ma-208	122	1	[	[	X
ma-208	122	2	β(v1+w1	β(v1+w1	NOUN
ma-208	122	3	)	)	PUNCT
ma-208	122	4	,	,	PUNCT
ma-208	122	5	r	r	X
ma-208	122	6	]	]	X
ma-208	122	7	⊆	⊆	NUM
ma-208	122	8	p	p	NOUN
ma-208	122	9	.	.	PUNCT
ma-208	123	1	this	this	PRON
ma-208	123	2	means	mean	VERB
ma-208	123	3	that	that	SCONJ
ma-208	123	4	[	[	X
ma-208	123	5	β(w1	β(w1	NOUN
ma-208	123	6	)	)	PUNCT
ma-208	123	7	,	,	PUNCT
ma-208	123	8	r	r	X
ma-208	123	9	]	]	X
ma-208	123	10	not	not	PART
ma-208	123	11	in	in	ADP
ma-208	123	12	p	p	NOUN
ma-208	123	13	and	and	CCONJ
ma-208	123	14	hence	hence	ADV
ma-208	123	15	(	(	PUNCT
ma-208	123	16	g(w1)−	g(w1)−	ADJ
ma-208	123	17	β(w1	β(w1	NOUN
ma-208	123	18	)	)	PUNCT
ma-208	123	19	)	)	PUNCT
ma-208	124	1	⊆	⊆	NUM
ma-208	124	2	p	p	NOUN
ma-208	124	3	.	.	PUNCT
ma-208	125	1	in	in	ADP
ma-208	125	2	other	other	ADJ
ma-208	125	3	ways	way	NOUN
ma-208	125	4	[	[	X
ma-208	125	5	β(v1	β(v1	X
ma-208	125	6	+	+	ADJ
ma-208	125	7	w1	w1	NOUN
ma-208	125	8	)	)	PUNCT
ma-208	125	9	,	,	PUNCT
ma-208	126	1	r	r	X
ma-208	126	2	]	]	X
ma-208	126	3	⊆	⊆	NUM
ma-208	126	4	p	p	NOUN
ma-208	126	5	,	,	PUNCT
ma-208	126	6	then	then	ADV
ma-208	126	7	b(g(v1	b(g(v1	VERB
ma-208	126	8	+	+	NOUN
ma-208	126	9	w1)−	w1)−	ADJ
ma-208	126	10	β(v1	β(v1	ADJ
ma-208	126	11	+	+	NOUN
ma-208	126	12	w1	w1	NOUN
ma-208	126	13	)	)	PUNCT
ma-208	126	14	)	)	PUNCT
ma-208	127	1	⊆	⊆	NUM
ma-208	127	2	p	p	NOUN
ma-208	127	3	.	.	PUNCT
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ma-208	128	2	eur	eur	PROPN
ma-208	128	3	.	.	PUNCT
ma-208	129	1	j.	j.	PROPN
ma-208	129	2	math	math	PROPN
ma-208	129	3	.	.	PUNCT
ma-208	130	1	anal	anal	PROPN
ma-208	130	2	.	.	PUNCT
ma-208	131	1	10.28924	10.28924	NUM
ma-208	131	2	/	/	SYM
ma-208	131	3	ada	ada	PROPN
ma-208	131	4	/	/	SYM
ma-208	131	5	ma.4.5	ma.4.5	PROPN
ma-208	131	6	5it	5it	PROPN
ma-208	131	7	gives	give	VERB
ma-208	131	8	b(g(w1)−	b(g(w1)−	NOUN
ma-208	131	9	β(w1	β(w1	NOUN
ma-208	131	10	)	)	PUNCT
ma-208	131	11	)	)	PUNCT
ma-208	132	1	⊆	⊆	NUM
ma-208	132	2	p	p	NOUN
ma-208	132	3	.we	.we	PUNCT
ma-208	132	4	obtain	obtain	VERB
ma-208	132	5	b(g(w1)−β(w1	b(g(w1)−β(w1	NOUN
ma-208	132	6	)	)	PUNCT
ma-208	132	7	)	)	PUNCT
ma-208	133	1	⊆	⊆	NUM
ma-208	133	2	p	p	NOUN
ma-208	133	3	for	for	ADP
ma-208	133	4	every	every	DET
ma-208	133	5	w1	w1	NOUN
ma-208	133	6	∈	∈	PROPN
ma-208	133	7	b	b	PROPN
ma-208	133	8	and	and	CCONJ
ma-208	133	9	hence	hence	ADV
ma-208	133	10	[	[	X
ma-208	133	11	b	b	X
ma-208	133	12	,	,	PUNCT
ma-208	133	13	b](g(w1)−β(w1	b](g(w1)−β(w1	NOUN
ma-208	133	14	)	)	PUNCT
ma-208	133	15	)	)	PUNCT
ma-208	134	1	⊆	⊆	NUM
ma-208	134	2	p	p	NOUN
ma-208	134	3	∀	∀	NOUN
ma-208	134	4	w1	w1	NOUN
ma-208	134	5	∈	∈	PROPN
ma-208	134	6	b.as	b.as	NOUN
ma-208	134	7	p	p	NOUN
ma-208	134	8	is	be	AUX
ma-208	134	9	arbitrary	arbitrary	ADJ
ma-208	134	10	and	and	CCONJ
ma-208	134	11	⋂	⋂	PROPN
ma-208	134	12	pθ	pθ	PROPN
ma-208	134	13	=	=	SYM
ma-208	134	14	0	0	PROPN
ma-208	134	15	,	,	PUNCT
ma-208	134	16	this	this	PRON
ma-208	134	17	implies	imply	VERB
ma-208	134	18	[	[	X
ma-208	134	19	b	b	X
ma-208	134	20	,	,	PUNCT
ma-208	134	21	b](g(w1)−	b](g(w1)−	ADJ
ma-208	134	22	β(w1	β(w1	NOUN
ma-208	134	23	)	)	PUNCT
ma-208	134	24	)	)	PUNCT
ma-208	135	1	=	=	SYM
ma-208	135	2	0	0	NUM
ma-208	135	3	for	for	ADP
ma-208	135	4	all	all	DET
ma-208	135	5	w1	w1	PROPN
ma-208	135	6	∈	∈	PROPN
ma-208	135	7	b.	b.	PROPN
ma-208	135	8	similarly	similarly	ADV
ma-208	135	9	,	,	PUNCT
ma-208	135	10	we	we	PRON
ma-208	135	11	can	can	AUX
ma-208	135	12	show	show	VERB
ma-208	135	13	that	that	SCONJ
ma-208	135	14	(	(	PUNCT
ma-208	135	15	g(w1)−	g(w1)−	ADJ
ma-208	135	16	β(w1))[b	β(w1))[b	PROPN
ma-208	135	17	,	,	PUNCT
ma-208	135	18	b	b	X
ma-208	135	19	]	]	X
ma-208	135	20	=	=	SYM
ma-208	135	21	0	0	NUM
ma-208	135	22	for	for	ADP
ma-208	135	23	all	all	DET
ma-208	135	24	w1	w1	PROPN
ma-208	135	25	∈	∈	PROPN
ma-208	135	26	b.	b.	NOUN
ma-208	135	27	this	this	PRON
ma-208	135	28	implies	imply	VERB
ma-208	135	29	that	that	SCONJ
ma-208	135	30	(	(	PUNCT
ma-208	135	31	g(w1)−	g(w1)−	ADJ
ma-208	135	32	β(w1	β(w1	NOUN
ma-208	135	33	)	)	PUNCT
ma-208	135	34	)	)	PUNCT
ma-208	136	1	∈	∈	PROPN
ma-208	136	2	cr[b	cr[b	PROPN
ma-208	136	3	,	,	PUNCT
ma-208	136	4	b	b	NOUN
ma-208	136	5	]	]	X
ma-208	136	6	,	,	PUNCT
ma-208	136	7	for	for	ADP
ma-208	136	8	all	all	DET
ma-208	136	9	w1	w1	NOUN
ma-208	136	10	∈	∈	PROPN
ma-208	136	11	b.	b.	PROPN
ma-208	136	12	by	by	ADP
ma-208	136	13	lemma	lemma	PROPN
ma-208	136	14	2.10	2.10	NUM
ma-208	136	15	and	and	CCONJ
ma-208	136	16	[	[	X
ma-208	136	17	6	6	NUM
ma-208	136	18	]	]	PUNCT
ma-208	136	19	,	,	PUNCT
ma-208	136	20	we	we	PRON
ma-208	136	21	have	have	VERB
ma-208	136	22	(	(	PUNCT
ma-208	136	23	g(u1	g(u1	NOUN
ma-208	136	24	)	)	PUNCT
ma-208	136	25	,	,	PUNCT
ma-208	136	26	β(u1	β(u1	NOUN
ma-208	136	27	)	)	PUNCT
ma-208	136	28	)	)	PUNCT
ma-208	137	1	∈	∈	PROPN
ma-208	137	2	cr(b	cr(b	NOUN
ma-208	137	3	)	)	PUNCT
ma-208	137	4	,	,	PUNCT
ma-208	137	5	∀	∀	X
ma-208	137	6	w1	w1	NOUN
ma-208	137	7	∈	∈	PROPN
ma-208	137	8	b.thus	b.thus	ADP
ma-208	138	1	we	we	PRON
ma-208	138	2	have	have	VERB
ma-208	138	3	[	[	X
ma-208	138	4	g(u1	g(u1	NOUN
ma-208	138	5	)	)	PUNCT
ma-208	138	6	−	−	PROPN
ma-208	138	7	u1	u1	NOUN
ma-208	138	8	,	,	PUNCT
ma-208	138	9	β(u1	β(u1	PUNCT
ma-208	138	10	)	)	PUNCT
ma-208	138	11	]	]	PUNCT
ma-208	139	1	=	=	SYM
ma-208	139	2	0	0	NUM
ma-208	139	3	∀	∀	NOUN
ma-208	139	4	u1	u1	PROPN
ma-208	139	5	∈	∈	PROPN
ma-208	139	6	b.	b.	PROPN
ma-208	139	7	this	this	PRON
ma-208	139	8	implies	imply	VERB
ma-208	139	9	that	that	SCONJ
ma-208	139	10	[	[	X
ma-208	139	11	g(u1	g(u1	NOUN
ma-208	139	12	)	)	PUNCT
ma-208	139	13	,	,	PUNCT
ma-208	139	14	β(u1	β(u1	NOUN
ma-208	139	15	)	)	PUNCT
ma-208	139	16	]	]	PUNCT
ma-208	140	1	=	=	SYM
ma-208	140	2	0	0	NUM
ma-208	140	3	∀	∀	NOUN
ma-208	140	4	u1	u1	PROPN
ma-208	140	5	∈	∈	PROPN
ma-208	140	6	b.this	b.this	PROPN
ma-208	140	7	shows	show	VERB
ma-208	140	8	that	that	SCONJ
ma-208	140	9	g	g	PROPN
ma-208	140	10	is	be	AUX
ma-208	140	11	commuting	commute	VERB
ma-208	140	12	on	on	ADP
ma-208	140	13	b.	b.	PROPN
ma-208	140	14	theorem	theorem	PROPN
ma-208	140	15	2.13	2.13	NUM
ma-208	140	16	.	.	PUNCT
ma-208	141	1	suppose	suppose	VERB
ma-208	141	2	a	a	DET
ma-208	141	3	derivation	derivation	NOUN
ma-208	141	4	,	,	PUNCT
ma-208	142	1	d	d	X
ma-208	142	2	:	:	PUNCT
ma-208	142	3	r	r	NOUN
ma-208	142	4	→	→	SYM
ma-208	142	5	r	r	NOUN
ma-208	142	6	where	where	SCONJ
ma-208	142	7	0	0	X
ma-208	142	8	6=	6=	NUM
ma-208	142	9	d	d	PROPN
ma-208	142	10	,	,	PUNCT
ma-208	142	11	in	in	ADP
ma-208	142	12	r	r	NOUN
ma-208	142	13	and	and	CCONJ
ma-208	142	14	a	a	DET
ma-208	142	15	generalized	generalize	VERB
ma-208	142	16	(	(	PUNCT
ma-208	142	17	α	α	NOUN
ma-208	142	18	,	,	PUNCT
ma-208	142	19	β)-reverse	β)-reverse	PUNCT
ma-208	142	20	derivation	derivation	NOUN
ma-208	142	21	g	g	NOUN
ma-208	142	22	on	on	ADP
ma-208	142	23	left	leave	VERB
ma-208	142	24	ideal	ideal	PROPN
ma-208	142	25	b	b	PROPN
ma-208	142	26	6=	6=	PROPN
ma-208	142	27	0	0	NUM
ma-208	142	28	.	.	PUNCT
ma-208	143	1	if	if	SCONJ
ma-208	143	2	g	g	PROPN
ma-208	143	3	is	be	AUX
ma-208	143	4	a	a	DET
ma-208	143	5	homomorphism	homomorphism	NOUN
ma-208	143	6	on	on	ADP
ma-208	143	7	b	b	NOUN
ma-208	143	8	,	,	PUNCT
ma-208	143	9	then	then	ADV
ma-208	143	10	commutativity	commutativity	NOUN
ma-208	143	11	existsin	existsin	VERB
ma-208	143	12	r.	r.	PROPN
ma-208	143	13	proof	proof	PROPN
ma-208	143	14	.	.	PUNCT
ma-208	144	1	by	by	ADP
ma-208	144	2	our	our	PRON
ma-208	144	3	hypothesis	hypothesis	NOUN
ma-208	144	4	[	[	X
ma-208	144	5	g(u1	g(u1	NOUN
ma-208	144	6	)	)	PUNCT
ma-208	144	7	,	,	PUNCT
ma-208	144	8	u1]α	u1]α	PROPN
ma-208	144	9	,	,	PUNCT
ma-208	144	10	β	β	X
ma-208	144	11	=	=	SYM
ma-208	144	12	0	0	NUM
ma-208	144	13	,	,	PUNCT
ma-208	144	14	f	f	PROPN
ma-208	144	15	or	or	CCONJ
ma-208	144	16	al	al	PROPN
ma-208	144	17	l	l	PROPN
ma-208	144	18	u1	u1	PROPN
ma-208	144	19	∈	∈	PROPN
ma-208	144	20	b.	b.	PROPN
ma-208	145	1	(	(	PUNCT
ma-208	145	2	8)	8)	NUM
ma-208	145	3	we	we	PRON
ma-208	145	4	replace	replace	VERB
ma-208	145	5	u1	u1	NOUN
ma-208	145	6	by	by	ADP
ma-208	145	7	u1	u1	NOUN
ma-208	145	8	+	+	CCONJ
ma-208	145	9	v1	v1	NOUN
ma-208	145	10	,	,	PUNCT
ma-208	145	11	in	in	ADP
ma-208	145	12	equation	equation	NOUN
ma-208	145	13	(	(	PUNCT
ma-208	145	14	2.1),we	2.1),we	NUM
ma-208	145	15	get	get	VERB
ma-208	145	16	[	[	X
ma-208	145	17	g(u1	g(u1	NOUN
ma-208	145	18	+	+	CCONJ
ma-208	145	19	v1	v1	NOUN
ma-208	145	20	)	)	PUNCT
ma-208	145	21	,	,	PUNCT
ma-208	145	22	u1	u1	NOUN
ma-208	145	23	+	+	CCONJ
ma-208	145	24	v1]α	v1]α	NOUN
ma-208	145	25	,	,	PUNCT
ma-208	145	26	β	β	X
ma-208	145	27	=	=	SYM
ma-208	145	28	0,we	0,we	NUM
ma-208	145	29	have	have	VERB
ma-208	145	30	[	[	X
ma-208	145	31	g(u1	g(u1	NOUN
ma-208	145	32	)	)	PUNCT
ma-208	145	33	+	+	NUM
ma-208	145	34	g(v1	g(v1	NOUN
ma-208	145	35	)	)	PUNCT
ma-208	145	36	,	,	PUNCT
ma-208	145	37	u1	u1	NOUN
ma-208	145	38	+	+	CCONJ
ma-208	145	39	v1]α	v1]α	NOUN
ma-208	145	40	,	,	PUNCT
ma-208	145	41	β	β	X
ma-208	145	42	=	=	PUNCT
ma-208	145	43	0,we	0,we	NUM
ma-208	145	44	arrives	arrive	VERB
ma-208	145	45	to	to	ADP
ma-208	145	46	[	[	X
ma-208	145	47	g(u1	g(u1	NOUN
ma-208	145	48	)	)	PUNCT
ma-208	145	49	+	+	NUM
ma-208	145	50	g(v1	g(v1	NOUN
ma-208	145	51	)	)	PUNCT
ma-208	145	52	,	,	PUNCT
ma-208	145	53	u1]α	u1]α	PROPN
ma-208	145	54	,	,	PUNCT
ma-208	145	55	β	β	X
ma-208	146	1	+	+	ADJ
ma-208	146	2	[	[	X
ma-208	146	3	g(u1	g(u1	NOUN
ma-208	146	4	)	)	PUNCT
ma-208	146	5	+	+	NUM
ma-208	146	6	g(v1	g(v1	NOUN
ma-208	146	7	)	)	PUNCT
ma-208	146	8	,	,	PUNCT
ma-208	146	9	v1]α	v1]α	NOUN
ma-208	146	10	,	,	PUNCT
ma-208	146	11	β	β	X
ma-208	146	12	=	=	SYM
ma-208	146	13	0,this	0,this	PRON
ma-208	146	14	gives	give	VERB
ma-208	146	15	[	[	NOUN
ma-208	146	16	g(u1	g(u1	NOUN
ma-208	146	17	)	)	PUNCT
ma-208	146	18	,	,	PUNCT
ma-208	146	19	u1]α	u1]α	PROPN
ma-208	146	20	,	,	PUNCT
ma-208	146	21	β	β	X
ma-208	147	1	+	+	PROPN
ma-208	147	2	[	[	X
ma-208	147	3	g(v1	g(v1	NOUN
ma-208	147	4	)	)	PUNCT
ma-208	147	5	,	,	PUNCT
ma-208	147	6	u1]α	u1]α	PROPN
ma-208	147	7	,	,	PUNCT
ma-208	147	8	β	β	X
ma-208	148	1	+	+	ADJ
ma-208	148	2	[	[	X
ma-208	148	3	g(u1	g(u1	NOUN
ma-208	148	4	)	)	PUNCT
ma-208	148	5	,	,	PUNCT
ma-208	148	6	v1]α	v1]α	NOUN
ma-208	148	7	,	,	PUNCT
ma-208	148	8	β	β	X
ma-208	148	9	+	+	PROPN
ma-208	148	10	[	[	X
ma-208	148	11	g(v1	g(v1	NOUN
ma-208	148	12	)	)	PUNCT
ma-208	148	13	,	,	PUNCT
ma-208	148	14	v1]α	v1]α	NOUN
ma-208	148	15	,	,	PUNCT
ma-208	148	16	β	β	X
ma-208	148	17	=	=	SYM
ma-208	148	18	0.by	0.by	NUM
ma-208	148	19	equation	equation	NOUN
ma-208	148	20	(	(	PUNCT
ma-208	148	21	)	)	PUNCT
ma-208	148	22	,	,	PUNCT
ma-208	148	23	we	we	PRON
ma-208	148	24	obtain	obtain	VERB
ma-208	148	25	[	[	X
ma-208	148	26	g(u1	g(u1	NOUN
ma-208	148	27	)	)	PUNCT
ma-208	148	28	,	,	PUNCT
ma-208	148	29	v1]α	v1]α	NOUN
ma-208	148	30	,	,	PUNCT
ma-208	148	31	β	β	X
ma-208	148	32	+	+	PROPN
ma-208	148	33	[	[	X
ma-208	148	34	g(v1	g(v1	NOUN
ma-208	148	35	)	)	PUNCT
ma-208	148	36	,	,	PUNCT
ma-208	148	37	u1]α	u1]α	PROPN
ma-208	148	38	,	,	PUNCT
ma-208	148	39	β	β	X
ma-208	148	40	=	=	SYM
ma-208	148	41	0	0	NUM
ma-208	148	42	,	,	PUNCT
ma-208	148	43	f	f	PROPN
ma-208	148	44	or	or	CCONJ
ma-208	148	45	al	al	PROPN
ma-208	148	46	l	l	PROPN
ma-208	148	47	u1	u1	PROPN
ma-208	148	48	∈	∈	PROPN
ma-208	148	49	b.	b.	PROPN
ma-208	148	50	(	(	PUNCT
ma-208	148	51	9	9	NUM
ma-208	148	52	)	)	PUNCT
ma-208	148	53	by	by	ADP
ma-208	148	54	substituting	substitute	VERB
ma-208	148	55	v1	v1	NOUN
ma-208	148	56	=	=	SYM
ma-208	149	1	u1v1	u1v1	NOUN
ma-208	149	2	in	in	ADP
ma-208	149	3	equation	equation	NOUN
ma-208	149	4	(	(	PUNCT
ma-208	149	5	9	9	NUM
ma-208	149	6	)	)	PUNCT
ma-208	149	7	,	,	PUNCT
ma-208	149	8	we	we	PRON
ma-208	149	9	have	have	VERB
ma-208	149	10	[	[	X
ma-208	149	11	g(u1	g(u1	NOUN
ma-208	149	12	)	)	PUNCT
ma-208	149	13	,	,	PUNCT
ma-208	149	14	u1v1]α	u1v1]α	PRON
ma-208	149	15	,	,	PUNCT
ma-208	149	16	β	β	X
ma-208	149	17	+	+	PROPN
ma-208	149	18	[	[	X
ma-208	149	19	g(u1v1	g(u1v1	NOUN
ma-208	149	20	)	)	PUNCT
ma-208	149	21	,	,	PUNCT
ma-208	149	22	u1]α	u1]α	PROPN
ma-208	149	23	,	,	PUNCT
ma-208	149	24	β	β	X
ma-208	149	25	=	=	SYM
ma-208	149	26	0,we	0,we	NUM
ma-208	149	27	have	have	VERB
ma-208	149	28	β(u1)[g(u1	β(u1)[g(u1	NOUN
ma-208	149	29	)	)	PUNCT
ma-208	149	30	,	,	PUNCT
ma-208	149	31	v1]α	v1]α	NOUN
ma-208	149	32	,	,	PUNCT
ma-208	149	33	β	β	X
ma-208	150	1	+	+	ADJ
ma-208	150	2	[	[	X
ma-208	150	3	g(u1	g(u1	NOUN
ma-208	150	4	)	)	PUNCT
ma-208	150	5	,	,	PUNCT
ma-208	150	6	u1]α	u1]α	PROPN
ma-208	150	7	,	,	PUNCT
ma-208	150	8	β	β	NOUN
ma-208	150	9	α(v1	α(v1	NOUN
ma-208	150	10	)	)	PUNCT
ma-208	150	11	+	+	CCONJ
ma-208	151	1	[	[	X
ma-208	151	2	g(v1)α(u1	g(v1)α(u1	X
ma-208	151	3	)	)	PUNCT
ma-208	151	4	+	+	SYM
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ma-208	151	6	)	)	PUNCT
ma-208	151	7	,	,	PUNCT
ma-208	151	8	u1]α	u1]α	PROPN
ma-208	151	9	,	,	PUNCT
ma-208	151	10	β	β	X
ma-208	151	11	=	=	SYM
ma-208	151	12	0.this	0.this	PROPN
ma-208	151	13	implies	imply	VERB
ma-208	151	14	us	we	PRON
ma-208	151	15	by	by	ADP
ma-208	151	16	the	the	DET
ma-208	151	17	equation	equation	NOUN
ma-208	151	18	(	(	PUNCT
ma-208	151	19	2.1	2.1	NUM
ma-208	151	20	)	)	PUNCT
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ma-208	151	22	β(u1)[g(u1	β(u1)[g(u1	NOUN
ma-208	151	23	)	)	PUNCT
ma-208	151	24	,	,	PUNCT
ma-208	151	25	v1]α	v1]α	NOUN
ma-208	151	26	,	,	PUNCT
ma-208	151	27	β	β	X
ma-208	152	1	+	+	PROPN
ma-208	152	2	[	[	X
ma-208	152	3	g(v1)α(u1	g(v1)α(u1	NOUN
ma-208	152	4	)	)	PUNCT
ma-208	152	5	,	,	PUNCT
ma-208	152	6	u1]α	u1]α	PROPN
ma-208	152	7	,	,	PUNCT
ma-208	152	8	β	β	X
ma-208	153	1	+	+	PROPN
ma-208	153	2	[	[	X
ma-208	153	3	β(v1)d(u1	β(v1)d(u1	X
ma-208	153	4	)	)	PUNCT
ma-208	153	5	,	,	PUNCT
ma-208	153	6	u1]α	u1]α	PROPN
ma-208	153	7	,	,	PUNCT
ma-208	153	8	β	β	X
ma-208	153	9	=	=	SYM
ma-208	153	10	0.this	0.this	PRON
ma-208	153	11	gives	give	VERB
ma-208	153	12	us	we	PRON
ma-208	153	13	by	by	ADP
ma-208	153	14	[	[	X
ma-208	153	15	α(u1	α(u1	NOUN
ma-208	153	16	)	)	PUNCT
ma-208	153	17	,	,	PUNCT
ma-208	153	18	α(u1	α(u1	NOUN
ma-208	153	19	)	)	PUNCT
ma-208	153	20	]	]	PUNCT
ma-208	154	1	=	=	PUNCT
ma-208	154	2	0	0	NUM
ma-208	154	3	,	,	PUNCT
ma-208	154	4	β(u1)[g(u1	β(u1)[g(u1	NUM
ma-208	154	5	)	)	PUNCT
ma-208	154	6	,	,	PUNCT
ma-208	154	7	v1]α	v1]α	NOUN
ma-208	154	8	,	,	PUNCT
ma-208	154	9	β	β	X
ma-208	154	10	+	+	PROPN
ma-208	154	11	[	[	X
ma-208	154	12	g(v1	g(v1	NOUN
ma-208	154	13	)	)	PUNCT
ma-208	154	14	,	,	PUNCT
ma-208	154	15	u1]α	u1]α	PROPN
ma-208	154	16	,	,	PUNCT
ma-208	154	17	β	β	X
ma-208	154	18	α(u1	α(u1	NOUN
ma-208	154	19	)	)	PUNCT
ma-208	154	20	+	+	CCONJ
ma-208	155	1	[	[	X
ma-208	155	2	β(v1)d(u1	β(v1)d(u1	NOUN
ma-208	155	3	)	)	PUNCT
ma-208	155	4	,	,	PUNCT
ma-208	155	5	u1]α	u1]α	PROPN
ma-208	155	6	,	,	PUNCT
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ma-208	155	8	=	=	SYM
ma-208	155	9	0,since	0,since	NUM
ma-208	155	10	g	g	PROPN
ma-208	155	11	is	be	AUX
ma-208	155	12	commuting	commute	VERB
ma-208	155	13	on	on	ADP
ma-208	155	14	b	b	NUM
ma-208	155	15	,	,	PUNCT
ma-208	155	16	we	we	PRON
ma-208	155	17	have	have	VERB
ma-208	155	18	[	[	X
ma-208	155	19	β(v1)d(u1	β(v1)d(u1	NOUN
ma-208	155	20	)	)	PUNCT
ma-208	155	21	,	,	PUNCT
ma-208	155	22	u1]α	u1]α	PROPN
ma-208	155	23	,	,	PUNCT
ma-208	155	24	β	β	X
ma-208	155	25	=	=	SYM
ma-208	155	26	0	0	NUM
ma-208	155	27	,	,	PUNCT
ma-208	155	28	f	f	PROPN
ma-208	155	29	or	or	CCONJ
ma-208	155	30	al	al	PROPN
ma-208	155	31	l	l	PROPN
ma-208	155	32	u1	u1	PROPN
ma-208	155	33	∈	∈	PROPN
ma-208	155	34	b.	b.	PROPN
ma-208	156	1	(	(	PUNCT
ma-208	156	2	10	10	NUM
ma-208	156	3	)	)	PUNCT
ma-208	156	4	we	we	PRON
ma-208	156	5	replace	replace	VERB
ma-208	156	6	v1	v1	NOUN
ma-208	156	7	by	by	ADP
ma-208	156	8	r1v1	r1v1	PUNCT
ma-208	156	9	in	in	ADP
ma-208	156	10	equation	equation	NOUN
ma-208	156	11	(	(	PUNCT
ma-208	156	12	10	10	NUM
ma-208	156	13	)	)	PUNCT
ma-208	156	14	,	,	PUNCT
ma-208	156	15	we	we	PRON
ma-208	156	16	have	have	VERB
ma-208	156	17	[	[	X
ma-208	156	18	β(r1v1)d(u1	β(r1v1)d(u1	X
ma-208	156	19	)	)	PUNCT
ma-208	156	20	,	,	PUNCT
ma-208	156	21	u1]α	u1]α	PROPN
ma-208	156	22	,	,	PUNCT
ma-208	156	23	β	β	X
ma-208	156	24	=	=	SYM
ma-208	156	25	0	0	NUM
ma-208	156	26	,	,	PUNCT
ma-208	156	27	https://doi.org/10.28924/ada/ma.4.5	https://doi.org/10.28924/ada/ma.4.5	PROPN
ma-208	156	28	eur	eur	PROPN
ma-208	156	29	.	.	PUNCT
ma-208	157	1	j.	j.	PROPN
ma-208	157	2	math	math	PROPN
ma-208	157	3	.	.	PUNCT
ma-208	158	1	anal	anal	PROPN
ma-208	158	2	.	.	PUNCT
ma-208	159	1	10.28924	10.28924	NUM
ma-208	159	2	/	/	SYM
ma-208	159	3	ada	ada	PROPN
ma-208	159	4	/	/	SYM
ma-208	159	5	ma.4.5	ma.4.5	PROPN
ma-208	159	6	6we	6we	PROPN
ma-208	159	7	get	get	VERB
ma-208	159	8	β(r1)[β(v1)d(u1	β(r1)[β(v1)d(u1	PUNCT
ma-208	159	9	)	)	PUNCT
ma-208	159	10	,	,	PUNCT
ma-208	159	11	u1]α	u1]α	PROPN
ma-208	159	12	,	,	PUNCT
ma-208	159	13	β	β	X
ma-208	160	1	+	+	PROPN
ma-208	160	2	[	[	NOUN
ma-208	160	3	β(r1	β(r1	NOUN
ma-208	160	4	)	)	PUNCT
ma-208	160	5	,	,	PUNCT
ma-208	160	6	β(u1)]β(v1)d(u1	β(u1)]β(v1)d(u1	NOUN
ma-208	160	7	)	)	PUNCT
ma-208	160	8	=	=	PUNCT
ma-208	160	9	0.by	0.by	NUM
ma-208	160	10	equation	equation	NOUN
ma-208	160	11	(	(	PUNCT
ma-208	160	12	10	10	NUM
ma-208	160	13	)	)	PUNCT
ma-208	160	14	,	,	PUNCT
ma-208	160	15	we	we	PRON
ma-208	160	16	have	have	VERB
ma-208	160	17	[	[	NOUN
ma-208	160	18	β(r1	β(r1	NOUN
ma-208	160	19	)	)	PUNCT
ma-208	160	20	,	,	PUNCT
ma-208	160	21	β(u1)]β(v1)d(u1	β(u1)]β(v1)d(u1	NOUN
ma-208	160	22	)	)	PUNCT
ma-208	160	23	=	=	SYM
ma-208	161	1	0,this	0,this	PRON
ma-208	161	2	gives	give	VERB
ma-208	161	3	[	[	PRON
ma-208	161	4	β(r1	β(r1	NOUN
ma-208	161	5	)	)	PUNCT
ma-208	161	6	,	,	PUNCT
ma-208	161	7	β(u1)]bd(u1	β(u1)]bd(u1	PUNCT
ma-208	161	8	)	)	PUNCT
ma-208	162	1	=	=	SYM
ma-208	162	2	0	0	NUM
ma-208	162	3	,	,	PUNCT
ma-208	162	4	for	for	ADP
ma-208	162	5	all	all	DET
ma-208	162	6	u1	u1	NOUN
ma-208	162	7	∈	∈	PROPN
ma-208	162	8	b	b	PROPN
ma-208	162	9	and	and	CCONJ
ma-208	162	10	r1	r1	PROPN
ma-208	162	11	∈	∈	PROPN
ma-208	162	12	r	r	NOUN
ma-208	162	13	,	,	PUNCT
ma-208	162	14	by	by	ADP
ma-208	162	15	the	the	DET
ma-208	162	16	semi	semi	NOUN
ma-208	162	17	-	-	NOUN
ma-208	162	18	primeness	primeness	NOUN
ma-208	162	19	of	of	ADP
ma-208	162	20	r	r	NOUN
ma-208	162	21	,	,	PUNCT
ma-208	162	22	∃	∃	PROPN
ma-208	162	23	a	a	DET
ma-208	162	24	set	set	NOUN
ma-208	162	25	ω	ω	NOUN
ma-208	162	26	=	=	SYM
ma-208	162	27	{	{	PUNCT
ma-208	162	28	pα	pα	PROPN
ma-208	162	29	/	/	SYM
ma-208	162	30	α	α	NOUN
ma-208	162	31	∈	∈	PROPN
ma-208	162	32	∧	∧	PROPN
ma-208	162	33	}	}	PUNCT
ma-208	162	34	of	of	ADP
ma-208	162	35	prime	prime	ADJ
ma-208	162	36	ideals	ideal	NOUN
ma-208	162	37	and	and	CCONJ
ma-208	162	38	⋂	⋂	PROPN
ma-208	162	39	pα	pα	NOUN
ma-208	162	40	=	=	PUNCT
ma-208	162	41	(	(	PUNCT
ma-208	162	42	0).if	0).if	INTJ
ma-208	162	43	p	p	NOUN
ma-208	162	44	∈	∈	PROPN
ma-208	162	45	ω	ω	NOUN
ma-208	162	46	and	and	CCONJ
ma-208	162	47	u1	u1	PROPN
ma-208	162	48	∈	∈	PROPN
ma-208	162	49	b	b	PROPN
ma-208	162	50	,	,	PUNCT
ma-208	162	51	then	then	ADV
ma-208	162	52	by	by	ADP
ma-208	162	53	equation	equation	NOUN
ma-208	162	54	(	(	PUNCT
ma-208	162	55	10	10	NUM
ma-208	162	56	)	)	PUNCT
ma-208	162	57	,	,	PUNCT
ma-208	163	1	[	[	X
ma-208	163	2	r	r	NOUN
ma-208	163	3	,	,	PUNCT
ma-208	163	4	β(u1	β(u1	ADJ
ma-208	163	5	)	)	PUNCT
ma-208	163	6	]	]	PUNCT
ma-208	164	1	⊆	⊆	NUM
ma-208	164	2	p	p	NOUN
ma-208	164	3	or	or	CCONJ
ma-208	164	4	p	p	NOUN
ma-208	164	5	⊇	⊇	PROPN
ma-208	164	6	d(u1	d(u1	NOUN
ma-208	164	7	)	)	PUNCT
ma-208	164	8	.	.	PUNCT
ma-208	165	1	since	since	SCONJ
ma-208	165	2	0	0	NUM
ma-208	165	3	6=	6=	NUM
ma-208	165	4	d	d	NOUN
ma-208	165	5	on	on	ADP
ma-208	165	6	r	r	NOUN
ma-208	165	7	,	,	PUNCT
ma-208	165	8	soby	soby	NOUN
ma-208	165	9	[	[	X
ma-208	165	10	7	7	NUM
ma-208	165	11	]	]	PUNCT
ma-208	165	12	,	,	PUNCT
ma-208	165	13	0	0	PUNCT
ma-208	165	14	6=	6=	NUM
ma-208	165	15	d	d	PROPN
ma-208	165	16	on	on	ADP
ma-208	165	17	b.	b.	PROPN
ma-208	165	18	consider	consider	VERB
ma-208	165	19	d(u1)p	d(u1)p	PROPN
ma-208	165	20	,	,	PUNCT
ma-208	165	21	where	where	SCONJ
ma-208	165	22	u1	u1	PROPN
ma-208	165	23	∈	∈	PROPN
ma-208	165	24	b	b	PROPN
ma-208	165	25	,	,	PUNCT
ma-208	165	26	then	then	ADV
ma-208	165	27	p	p	PROPN
ma-208	165	28	⊇	⊇	NOUN
ma-208	165	29	[	[	X
ma-208	165	30	r	r	NOUN
ma-208	165	31	,	,	PUNCT
ma-208	165	32	β(u1	β(u1	NUM
ma-208	165	33	)	)	PUNCT
ma-208	165	34	]	]	PUNCT
ma-208	165	35	.	.	PUNCT
ma-208	166	1	suppose	suppose	VERB
ma-208	166	2	w1	w1	PROPN
ma-208	166	3	∈	∈	PROPN
ma-208	166	4	b	b	PROPN
ma-208	166	5	,	,	PUNCT
ma-208	166	6	wesee	wesee	NOUN
ma-208	166	7	that	that	DET
ma-208	166	8	w1	w1	NOUN
ma-208	166	9	not	not	PART
ma-208	166	10	in	in	ADP
ma-208	166	11	z	z	PROPN
ma-208	166	12	,	,	PUNCT
ma-208	166	13	then	then	ADV
ma-208	166	14	d(w1	d(w1	NOUN
ma-208	166	15	)	)	PUNCT
ma-208	166	16	⊆	⊆	NUM
ma-208	166	17	p	p	NOUN
ma-208	166	18	and	and	CCONJ
ma-208	166	19	u1	u1	NOUN
ma-208	166	20	+	+	CCONJ
ma-208	166	21	w1	w1	NOUN
ma-208	166	22	not	not	PART
ma-208	166	23	in	in	ADP
ma-208	166	24	z.	z.	PROPN
ma-208	166	25	this	this	PRON
ma-208	166	26	gives	give	VERB
ma-208	166	27	that	that	DET
ma-208	166	28	d(u1	d(u1	NOUN
ma-208	166	29	+	+	CCONJ
ma-208	166	30	w1	w1	NOUN
ma-208	166	31	)	)	PUNCT
ma-208	166	32	⊆	⊆	NUM
ma-208	166	33	p	p	DET
ma-208	166	34	andthen	andthen	ADJ
ma-208	166	35	d(u1	d(u1	NOUN
ma-208	166	36	)	)	PUNCT
ma-208	166	37	⊆	⊆	NUM
ma-208	166	38	p	p	NOUN
ma-208	166	39	,	,	PUNCT
ma-208	166	40	which	which	PRON
ma-208	166	41	contradicts	contradict	VERB
ma-208	166	42	to	to	ADP
ma-208	166	43	our	our	PRON
ma-208	166	44	consideration	consideration	NOUN
ma-208	166	45	that	that	SCONJ
ma-208	166	46	d(u1)p	d(u1)p	PROPN
ma-208	166	47	.	.	PUNCT
ma-208	167	1	so	so	ADV
ma-208	167	2	,	,	PUNCT
ma-208	167	3	this	this	PRON
ma-208	167	4	gives	give	VERB
ma-208	167	5	us	we	PRON
ma-208	167	6	w1	w1	NOUN
ma-208	167	7	∈	∈	PROPN
ma-208	167	8	z	z	PROPN
ma-208	167	9	,	,	PUNCT
ma-208	167	10	∀	∀	X
ma-208	167	11	w1	w1	NOUN
ma-208	167	12	∈	∈	PROPN
ma-208	167	13	b.this	b.this	PROPN
ma-208	167	14	implies	imply	VERB
ma-208	167	15	that	that	SCONJ
ma-208	167	16	b	b	NOUN
ma-208	167	17	is	be	AUX
ma-208	167	18	commutative	commutative	ADJ
ma-208	167	19	also	also	ADV
ma-208	167	20	that	that	SCONJ
ma-208	167	21	by	by	ADP
ma-208	167	22	the	the	DET
ma-208	167	23	[	[	X
ma-208	167	24	7	7	NUM
ma-208	167	25	]	]	PUNCT
ma-208	167	26	,	,	PUNCT
ma-208	167	27	then	then	ADV
ma-208	167	28	commutativity	commutativity	NOUN
ma-208	167	29	holds	hold	VERB
ma-208	167	30	in	in	ADP
ma-208	167	31	r.	r.	PROPN
ma-208	167	32	theorem	theorem	NOUN
ma-208	167	33	2.14	2.14	NUM
ma-208	167	34	.	.	PUNCT
ma-208	168	1	suppose	suppose	VERB
ma-208	168	2	a	a	DET
ma-208	168	3	semi	semi	ADJ
ma-208	168	4	-	-	ADJ
ma-208	168	5	prime	prime	ADJ
ma-208	168	6	ring	ring	NOUN
ma-208	168	7	r	r	NOUN
ma-208	168	8	and	and	CCONJ
ma-208	168	9	a	a	DET
ma-208	168	10	left	left	ADJ
ma-208	168	11	ideal	ideal	NOUN
ma-208	168	12	b	b	PROPN
ma-208	168	13	of	of	ADP
ma-208	168	14	r	r	PROPN
ma-208	168	15	,	,	PUNCT
ma-208	168	16	s.t	s.t	PROPN
ma-208	168	17	.	.	PROPN
ma-208	168	18	b⋂	b⋂	PROPN
ma-208	168	19	z	z	PROPN
ma-208	168	20	6=	6=	ADP
ma-208	168	21	0	0	NUM
ma-208	168	22	for	for	ADP
ma-208	168	23	center	center	PROPN
ma-208	168	24	z	z	PROPN
ma-208	168	25	of	of	ADP
ma-208	168	26	r.	r.	PROPN
ma-208	168	27	let	let	VERB
ma-208	168	28	a	a	DET
ma-208	168	29	generalized	generalized	ADJ
ma-208	168	30	(	(	PUNCT
ma-208	168	31	α	α	NOUN
ma-208	168	32	,	,	PUNCT
ma-208	168	33	β)-reverse	β)-reverse	PUNCT
ma-208	168	34	derivation	derivation	NOUN
ma-208	168	35	g	g	NOUN
ma-208	168	36	on	on	ADP
ma-208	168	37	r	r	NOUN
ma-208	168	38	and	and	CCONJ
ma-208	168	39	d	d	NOUN
ma-208	168	40	6=	6=	ADP
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ma-208	168	42	a	a	DET
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ma-208	168	44	and	and	CCONJ
ma-208	168	45	g	g	PROPN
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ma-208	168	47	on	on	ADP
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ma-208	168	52	in	in	ADP
ma-208	168	53	r.	r.	NOUN
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ma-208	169	6	g	g	PROPN
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ma-208	169	10	b	b	NOUN
ma-208	169	11	,	,	PUNCT
ma-208	169	12	so	so	ADV
ma-208	169	13	our	our	PRON
ma-208	169	14	proof	proof	NOUN
ma-208	169	15	is	be	AUX
ma-208	169	16	complete.as	complete.as	PRON
ma-208	169	17	g	g	NOUN
ma-208	169	18	is	be	AUX
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ma-208	169	20	b	b	NOUN
ma-208	169	21	and	and	CCONJ
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ma-208	169	23	theorem	theorem	NOUN
ma-208	169	24	2.12	2.12	NUM
ma-208	169	25	,	,	PUNCT
ma-208	169	26	we	we	PRON
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ma-208	169	28	[	[	X
ma-208	169	29	g(u1	g(u1	NOUN
ma-208	169	30	)	)	PUNCT
ma-208	169	31	,	,	PUNCT
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ma-208	170	11	(	(	PUNCT
ma-208	170	12	11	11	NUM
ma-208	170	13	)	)	PUNCT
ma-208	170	14	put	put	VERB
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ma-208	170	17	(	(	PUNCT
ma-208	170	18	u1	u1	NOUN
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ma-208	170	20	v1	v1	NOUN
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ma-208	170	23	equation	equation	NOUN
ma-208	170	24	(	(	PUNCT
ma-208	170	25	11	11	NUM
ma-208	170	26	)	)	PUNCT
ma-208	170	27	,	,	PUNCT
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ma-208	170	32	v1	v1	NOUN
ma-208	170	33	)	)	PUNCT
ma-208	170	34	,	,	PUNCT
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ma-208	170	37	v1]α	v1]α	NUM
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ma-208	171	2	,	,	PUNCT
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ma-208	171	4	all	all	DET
ma-208	171	5	u1	u1	NOUN
ma-208	171	6	∈	∈	PROPN
ma-208	171	7	b	b	PROPN
ma-208	171	8	,	,	PUNCT
ma-208	171	9	this	this	PRON
ma-208	171	10	relates	relate	VERB
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ma-208	171	12	[	[	X
ma-208	171	13	g(u1	g(u1	NOUN
ma-208	171	14	)	)	PUNCT
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ma-208	171	20	β	β	X
ma-208	172	1	+	+	PROPN
ma-208	172	2	[	[	X
ma-208	172	3	g(v1	g(v1	NOUN
ma-208	172	4	)	)	PUNCT
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ma-208	172	7	+	+	CCONJ
ma-208	172	8	v1]α	v1]α	NUM
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ma-208	172	10	β	β	X
ma-208	172	11	∈	∈	PROPN
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ma-208	172	16	∈	∈	PROPN
ma-208	172	17	b.it	b.it	NOUN
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ma-208	172	19	β(u1)[g(u1	β(u1)[g(u1	NOUN
ma-208	172	20	)	)	PUNCT
ma-208	172	21	,	,	PUNCT
ma-208	172	22	u1]α	u1]α	PROPN
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ma-208	172	24	β	β	X
ma-208	172	25	+	+	NOUN
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ma-208	172	27	)	)	PUNCT
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ma-208	173	1	+	+	PROPN
ma-208	173	2	[	[	X
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ma-208	173	4	)	)	PUNCT
ma-208	173	5	,	,	PUNCT
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ma-208	174	3	)	)	PUNCT
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ma-208	174	8	α(u1	α(u1	NOUN
ma-208	174	9	)	)	PUNCT
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ma-208	174	21	)	)	PUNCT
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ma-208	184	34	[	[	X
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ma-208	184	40	β	β	X
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ma-208	185	4	every	every	DET
ma-208	185	5	v1	v1	NOUN
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ma-208	185	8	we	we	PRON
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ma-208	185	10	d	d	NOUN
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ma-208	185	17	by	by	ADP
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ma-208	185	20	[	[	X
ma-208	185	21	3	3	NUM
ma-208	185	22	]	]	PUNCT
ma-208	185	23	,	,	PUNCT
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ma-208	187	3	1	1	NUM
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ma-208	187	12	and	and	CCONJ
ma-208	187	13	commuting	commute	VERB
ma-208	187	14	generalized	generalized	ADJ
ma-208	187	15	derivations	derivation	NOUN
ma-208	187	16	on	on	ADP
ma-208	187	17	prime	prime	ADJ
ma-208	187	18	rings	ring	NOUN
ma-208	187	19	,	,	PUNCT
ma-208	187	20	mat	mat	PROPN
ma-208	187	21	.	.	PROPN
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ma-208	187	23	,	,	PUNCT
ma-208	187	24	60	60	NUM
ma-208	187	25	(	(	PUNCT
ma-208	187	26	2008	2008	NUM
ma-208	187	27	)	)	PUNCT
ma-208	187	28	,	,	PUNCT
ma-208	187	29	1	1	NUM
ma-208	187	30	-	-	SYM
ma-208	187	31	2.[2	2.[2	NUM
ma-208	187	32	]	]	X
ma-208	187	33	h.e	h.e	PROPN
ma-208	187	34	.	.	PROPN
ma-208	187	35	bell	bell	PROPN
ma-208	187	36	,	,	PUNCT
ma-208	187	37	m.n	m.n	PROPN
ma-208	187	38	.	.	PROPN
ma-208	187	39	daig	daig	PROPN
ma-208	187	40	,	,	PUNCT
ma-208	187	41	on	on	ADP
ma-208	187	42	commutativity	commutativity	NOUN
ma-208	187	43	and	and	CCONJ
ma-208	187	44	strong	strong	ADJ
ma-208	187	45	commutativity	commutativity	NOUN
ma-208	187	46	preserving	preserve	VERB
ma-208	187	47	maps	map	NOUN
ma-208	187	48	,	,	PUNCT
ma-208	187	49	can	can	AUX
ma-208	187	50	.	.	PUNCT
ma-208	188	1	math	math	NOUN
ma-208	188	2	.	.	PUNCT
ma-208	189	1	bull	bull	NOUN
ma-208	189	2	.	.	PUNCT
ma-208	190	1	37	37	NUM
ma-208	190	2	(	(	PUNCT
ma-208	190	3	1994),443	1994),443	NUM
ma-208	190	4	-	-	SYM
ma-208	190	5	447.[3	447.[3	NUM
ma-208	190	6	]	]	X
ma-208	190	7	h.e	h.e	PROPN
ma-208	190	8	.	.	PROPN
ma-208	190	9	bell	bell	PROPN
ma-208	190	10	,	,	PUNCT
ma-208	190	11	w.s	w.s	PROPN
ma-208	190	12	.	.	PROPN
ma-208	190	13	martindale	martindale	PROPN
ma-208	190	14	,	,	PUNCT
ma-208	190	15	iii	iii	PROPN
ma-208	190	16	,	,	PUNCT
ma-208	190	17	centralizing	centralize	VERB
ma-208	190	18	mapping	mapping	NOUN
ma-208	190	19	of	of	ADP
ma-208	190	20	semi	semi	ADJ
ma-208	190	21	-	-	ADJ
ma-208	190	22	prime	prime	ADJ
ma-208	190	23	rings	ring	NOUN
ma-208	190	24	,	,	PUNCT
ma-208	190	25	can	can	AUX
ma-208	190	26	.	.	PUNCT
ma-208	191	1	math	math	NOUN
ma-208	191	2	.	.	PUNCT
ma-208	192	1	soc	soc	PROPN
ma-208	192	2	.	.	PUNCT
ma-208	193	1	30	30	NUM
ma-208	193	2	(	(	PUNCT
ma-208	193	3	1987	1987	NUM
ma-208	193	4	)	)	PUNCT
ma-208	193	5	,	,	PUNCT
ma-208	193	6	92	92	NUM
ma-208	193	7	-	-	SYM
ma-208	193	8	101.[4	101.[4	NUM
ma-208	193	9	]	]	PUNCT
ma-208	193	10	m.	m.	NOUN
ma-208	193	11	bresar	bresar	PROPN
ma-208	193	12	,	,	PUNCT
ma-208	193	13	j.	j.	PROPN
ma-208	193	14	vukman	vukman	PROPN
ma-208	193	15	,	,	PUNCT
ma-208	193	16	on	on	ADP
ma-208	193	17	some	some	DET
ma-208	193	18	additive	additive	ADJ
ma-208	193	19	mappings	mapping	NOUN
ma-208	193	20	in	in	ADP
ma-208	193	21	rings	ring	NOUN
ma-208	193	22	with	with	ADP
ma-208	193	23	involution	involution	NOUN
ma-208	193	24	,	,	PUNCT
ma-208	193	25	aequat	aequat	PROPN
ma-208	193	26	.	.	PUNCT
ma-208	194	1	math	math	NOUN
ma-208	194	2	.	.	PUNCT
ma-208	195	1	38	38	NUM
ma-208	195	2	(	(	PUNCT
ma-208	195	3	1989	1989	NUM
ma-208	195	4	)	)	PUNCT
ma-208	195	5	,	,	PUNCT
ma-208	195	6	178	178	NUM
ma-208	195	7	-	-	SYM
ma-208	195	8	185.[5	185.[5	NUM
ma-208	195	9	]	]	PUNCT
ma-208	195	10	m.	m.	NOUN
ma-208	195	11	bresar	bresar	VERB
ma-208	195	12	,	,	PUNCT
ma-208	195	13	centralizing	centralize	VERB
ma-208	195	14	mappings	mapping	NOUN
ma-208	195	15	and	and	CCONJ
ma-208	195	16	derivations	derivation	NOUN
ma-208	195	17	in	in	ADP
ma-208	195	18	prime	prime	ADJ
ma-208	195	19	rings	ring	NOUN
ma-208	195	20	,	,	PUNCT
ma-208	195	21	j.	j.	PROPN
ma-208	195	22	algebra	algebra	PROPN
ma-208	195	23	,	,	PUNCT
ma-208	195	24	156	156	NUM
ma-208	195	25	(	(	PUNCT
ma-208	195	26	1993	1993	NUM
ma-208	195	27	)	)	PUNCT
ma-208	195	28	,	,	PUNCT
ma-208	195	29	385	385	NUM
ma-208	195	30	-	-	SYM
ma-208	195	31	394.[6	394.[6	NUM
ma-208	195	32	]	]	X
ma-208	195	33	m.n	m.n	PROPN
ma-208	195	34	.	.	PROPN
ma-208	195	35	daig	daig	PROPN
ma-208	195	36	,	,	PUNCT
ma-208	195	37	h.e	h.e	PROPN
ma-208	195	38	.	.	PROPN
ma-208	195	39	bell	bell	PROPN
ma-208	195	40	,	,	PUNCT
ma-208	195	41	remarks	remark	NOUN
ma-208	195	42	on	on	ADP
ma-208	195	43	derivations	derivation	NOUN
ma-208	195	44	on	on	ADP
ma-208	195	45	semi	semi	ADJ
ma-208	195	46	-	-	ADJ
ma-208	195	47	prime	prime	ADJ
ma-208	195	48	rings	ring	NOUN
ma-208	195	49	,	,	PUNCT
ma-208	195	50	int	int	NOUN
ma-208	195	51	.	.	PUNCT
ma-208	196	1	j.	j.	PROPN
ma-208	196	2	math	math	PROPN
ma-208	196	3	.	.	PUNCT
ma-208	197	1	math	math	NOUN
ma-208	197	2	.	.	PUNCT
ma-208	198	1	sci	sci	PROPN
ma-208	198	2	.	.	PROPN
ma-208	198	3	15	15	NUM
ma-208	198	4	(	(	PUNCT
ma-208	198	5	1992	1992	NUM
ma-208	198	6	)	)	PUNCT
ma-208	198	7	,	,	PUNCT
ma-208	198	8	205	205	NUM
ma-208	198	9	-	-	SYM
ma-208	198	10	206.[7	206.[7	NUM
ma-208	198	11	]	]	X
ma-208	198	12	j.h	j.h	PROPN
ma-208	198	13	.	.	PROPN
ma-208	198	14	mayne	mayne	PROPN
ma-208	198	15	,	,	PUNCT
ma-208	198	16	centralizing	centralize	VERB
ma-208	198	17	mappings	mapping	NOUN
ma-208	198	18	of	of	ADP
ma-208	198	19	prime	prime	ADJ
ma-208	198	20	rings	ring	NOUN
ma-208	198	21	,	,	PUNCT
ma-208	198	22	can	can	AUX
ma-208	198	23	.	.	PUNCT
ma-208	199	1	math	math	NOUN
ma-208	199	2	.	.	PUNCT
ma-208	200	1	bull	bull	NOUN
ma-208	200	2	.	.	PUNCT
ma-208	201	1	27	27	NUM
ma-208	201	2	(	(	PUNCT
ma-208	201	3	1984	1984	NUM
ma-208	201	4	)	)	PUNCT
ma-208	201	5	,	,	PUNCT
ma-208	201	6	122	122	NUM
ma-208	201	7	-	-	SYM
ma-208	201	8	126.[8	126.[8	NUM
ma-208	201	9	]	]	X
ma-208	201	10	z.h	z.h	PROPN
ma-208	201	11	.	.	PROPN
ma-208	201	12	niazi	niazi	PROPN
ma-208	201	13	,	,	PUNCT
ma-208	201	14	m.a.t	m.a.t	NOUN
ma-208	201	15	.	.	PUNCT
ma-208	201	16	bhatti	bhatti	PROPN
ma-208	201	17	,	,	PUNCT
ma-208	201	18	m.	m.	NOUN
ma-208	201	19	aslam	aslam	PROPN
ma-208	201	20	,	,	PUNCT
ma-208	201	21	y.	y.	PROPN
ma-208	201	22	qayyum	qayyum	PROPN
ma-208	201	23	,	,	PUNCT
ma-208	201	24	m.	m.	PROPN
ma-208	201	25	ibrahim	ibrahim	PROPN
ma-208	201	26	,	,	PUNCT
ma-208	201	27	a.	a.	PROPN
ma-208	201	28	qayyum	qayyum	PROPN
ma-208	201	29	,	,	PUNCT
ma-208	201	30	d	d	ADJ
ma-208	201	31	-	-	PUNCT
ma-208	201	32	lucky	lucky	ADJ
ma-208	201	33	labelling	labelling	NOUN
ma-208	201	34	of	of	ADP
ma-208	201	35	some	some	DET
ma-208	201	36	special	special	ADJ
ma-208	201	37	graphs	graph	NOUN
ma-208	201	38	,	,	PUNCT
ma-208	201	39	amer	amer	PROPN
ma-208	201	40	.	.	PUNCT
ma-208	202	1	j.	j.	PROPN
ma-208	202	2	math	math	PROPN
ma-208	202	3	.	.	PUNCT
ma-208	203	1	anal	anal	ADJ
ma-208	203	2	.	.	PUNCT
ma-208	204	1	10	10	NUM
ma-208	204	2	(	(	PUNCT
ma-208	204	3	2022	2022	NUM
ma-208	204	4	)	)	PUNCT
ma-208	204	5	,	,	PUNCT
ma-208	204	6	3	3	NUM
ma-208	204	7	-	-	SYM
ma-208	204	8	11.[9	11.[9	NUM
ma-208	204	9	]	]	PUNCT
ma-208	204	10	m.	m.	NOUN
ma-208	204	11	ahmad	ahmad	PROPN
ma-208	204	12	,	,	PUNCT
ma-208	204	13	s.	s.	PROPN
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ma-208	204	15	,	,	PUNCT
ma-208	204	16	i.	i.	PROPN
ma-208	204	17	zahid	zahid	PROPN
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ma-208	204	19	u.	u.	PROPN
ma-208	204	20	parveen	parveen	PROPN
ma-208	204	21	,	,	PUNCT
ma-208	204	22	m.	m.	NOUN
ma-208	204	23	sultan	sultan	PROPN
ma-208	204	24	,	,	PUNCT
ma-208	204	25	a.	a.	PROPN
ma-208	204	26	qayyum	qayyum	PROPN
ma-208	204	27	,	,	PUNCT
ma-208	204	28	on	on	ADP
ma-208	204	29	degree	degree	NOUN
ma-208	204	30	based	base	VERB
ma-208	204	31	topological	topological	ADJ
ma-208	204	32	indices	index	NOUN
ma-208	204	33	of	of	ADP
ma-208	204	34	petersensubdivision	petersensubdivision	NOUN
ma-208	204	35	graph	graph	NOUN
ma-208	204	36	,	,	PUNCT
ma-208	204	37	eur	eur	PROPN
ma-208	204	38	.	.	PUNCT
ma-208	205	1	j.	j.	PROPN
ma-208	205	2	math	math	PROPN
ma-208	205	3	.	.	PUNCT
ma-208	206	1	anal	anal	ADJ
ma-208	206	2	.	.	PUNCT
ma-208	207	1	3	3	NUM
ma-208	207	2	(	(	PUNCT
ma-208	207	3	2023	2023	NUM
ma-208	207	4	)	)	PUNCT
ma-208	207	5	,	,	PUNCT
ma-208	207	6	20[10	20[10	NUM
ma-208	207	7	]	]	PUNCT
ma-208	207	8	m.	m.	PROPN
ma-208	207	9	ahmad	ahmad	PROPN
ma-208	207	10	,	,	PUNCT
ma-208	207	11	m.j	m.j	PROPN
ma-208	207	12	.	.	PROPN
ma-208	207	13	hussain	hussain	PROPN
ma-208	207	14	,	,	PUNCT
ma-208	207	15	g.	g.	PROPN
ma-208	207	16	atta	atta	PROPN
ma-208	207	17	,	,	PUNCT
ma-208	207	18	s.	s.	PROPN
ma-208	207	19	raza	raza	PROPN
ma-208	207	20	,	,	PUNCT
ma-208	207	21	i.	i.	PROPN
ma-208	207	22	waheed	waheed	PROPN
ma-208	207	23	,	,	PUNCT
ma-208	207	24	a.	a.	PROPN
ma-208	207	25	qayyum	qayyum	PROPN
ma-208	207	26	,	,	PUNCT
ma-208	207	27	topological	topological	ADJ
ma-208	207	28	evaluation	evaluation	NOUN
ma-208	207	29	of	of	ADP
ma-208	207	30	four	four	NUM
ma-208	207	31	para	para	ADJ
ma-208	207	32	-	-	PUNCT
ma-208	207	33	line	line	NOUN
ma-208	207	34	graphsabsolute	graphsabsolute	ADJ
ma-208	207	35	pentacene	pentacene	NOUN
ma-208	207	36	graphs	graph	NOUN
ma-208	207	37	using	use	VERB
ma-208	207	38	topological	topological	ADJ
ma-208	207	39	indices	index	NOUN
ma-208	207	40	,	,	PUNCT
ma-208	207	41	int	int	NOUN
ma-208	207	42	.	.	PUNCT
ma-208	208	1	j.	j.	PROPN
ma-208	208	2	anal	anal	PROPN
ma-208	208	3	.	.	PUNCT
ma-208	209	1	appl	appl	PROPN
ma-208	209	2	.	.	PROPN
ma-208	210	1	21	21	NUM
ma-208	210	2	(	(	PUNCT
ma-208	210	3	2023	2023	NUM
ma-208	210	4	)	)	PUNCT
ma-208	210	5	,	,	PUNCT
ma-208	210	6	66.[11	66.[11	PROPN
ma-208	210	7	]	]	X
ma-208	211	1	r.m.k	r.m.k	PROPN
ma-208	211	2	.	.	PROPN
ma-208	211	3	iqbal	iqbal	PROPN
ma-208	211	4	,	,	PUNCT
ma-208	211	5	m.	m.	NOUN
ma-208	211	6	ahmad	ahmad	PROPN
ma-208	211	7	,	,	PUNCT
ma-208	211	8	a.	a.	PROPN
ma-208	211	9	qayyum	qayyum	PROPN
ma-208	211	10	,	,	PUNCT
ma-208	211	11	s.s	s.s	PROPN
ma-208	211	12	.	.	PROPN
ma-208	211	13	supadi	supadi	PROPN
ma-208	211	14	,	,	PUNCT
ma-208	211	15	m.j	m.j	PROPN
ma-208	211	16	.	.	PROPN
ma-208	211	17	hussain	hussain	PROPN
ma-208	211	18	,	,	PUNCT
ma-208	211	19	s.	s.	PROPN
ma-208	211	20	raza	raza	PROPN
ma-208	211	21	,	,	PUNCT
ma-208	211	22	on	on	ADP
ma-208	211	23	degree	degree	NOUN
ma-208	211	24	-	-	PUNCT
ma-208	211	25	based	base	VERB
ma-208	211	26	topological	topological	ADJ
ma-208	211	27	indices	index	NOUN
ma-208	211	28	oftoeplitz	oftoeplitz	PROPN
ma-208	211	29	graphs	graph	NOUN
ma-208	211	30	,	,	PUNCT
ma-208	211	31	int	int	NOUN
ma-208	211	32	.	.	PUNCT
ma-208	212	1	j.	j.	PROPN
ma-208	212	2	anal	anal	PROPN
ma-208	212	3	.	.	PUNCT
ma-208	213	1	appl	appl	PROPN
ma-208	213	2	.	.	PROPN
ma-208	214	1	21	21	NUM
ma-208	214	2	(	(	PUNCT
ma-208	214	3	2023	2023	NUM
ma-208	214	4	)	)	PUNCT
ma-208	214	5	,	,	PUNCT
ma-208	214	6	111.[12	111.[12	NUM
ma-208	214	7	]	]	X
ma-208	214	8	a.	a.	NOUN
ma-208	214	9	asghar	asghar	PROPN
ma-208	214	10	,	,	PUNCT
ma-208	214	11	a.	a.	PROPN
ma-208	214	12	qayyum	qayyum	PROPN
ma-208	214	13	,	,	PUNCT
ma-208	214	14	n.	n.	PROPN
ma-208	214	15	muhammad	muhammad	PROPN
ma-208	214	16	,	,	PUNCT
ma-208	214	17	different	different	ADJ
ma-208	214	18	types	type	NOUN
ma-208	214	19	of	of	ADP
ma-208	214	20	topological	topological	ADJ
ma-208	214	21	structures	structure	NOUN
ma-208	214	22	by	by	ADP
ma-208	214	23	graphs	graph	NOUN
ma-208	214	24	,	,	PUNCT
ma-208	214	25	eur	eur	PROPN
ma-208	214	26	.	.	PUNCT
ma-208	215	1	j.	j.	PROPN
ma-208	215	2	math	math	PROPN
ma-208	215	3	.	.	PUNCT
ma-208	216	1	anal	anal	PROPN
ma-208	216	2	.	.	PUNCT
ma-208	217	1	3(2022	3(2022	NUM
ma-208	217	2	)	)	PUNCT
ma-208	217	3	,	,	PUNCT
ma-208	218	1	3	3	X
ma-208	218	2	.	.	X
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ma-208	218	4	1	1	NUM
ma-208	218	5	.	.	PUNCT
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ma-208	219	2	2	2	NUM
ma-208	219	3	.	.	PUNCT
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ma-208	219	5	2.1	2.1	NUM
ma-208	219	6	.	.	PUNCT
ma-208	220	1	point	point	NOUN
ma-208	220	2	-	-	PUNCT
ma-208	220	3	wise	wise	ADJ
ma-208	220	4	operation	operation	NOUN
ma-208	220	5	references	reference	NOUN