id	sid	tid	token	lemma	pos
cana-5404	1	1	communications	communication	NOUN
cana-5404	1	2	on	on	ADP
cana-5404	1	3	applied	apply	VERB
cana-5404	1	4	nonlinear	nonlinear	ADJ
cana-5404	1	5	analysis	analysis	NOUN
cana-5404	1	6	issn	issn	NOUN
cana-5404	1	7	:	:	PUNCT
cana-5404	1	8	1074	1074	NUM
cana-5404	1	9	-	-	PUNCT
cana-5404	1	10	133x	133x	NUM
cana-5404	1	11	vol	vol	VERB
cana-5404	1	12	32	32	NUM
cana-5404	1	13	no	no	NOUN
cana-5404	1	14	.	.	PUNCT
cana-5404	2	1	10s	10	NOUN
cana-5404	2	2	(	(	PUNCT
cana-5404	2	3	2025	2025	NUM
cana-5404	2	4	)	)	PUNCT
cana-5404	2	5	2138	2138	NUM
cana-5404	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	3	2	scp	scp	PROPN
cana-5404	3	3	identities	identity	NOUN
cana-5404	3	4	of	of	ADP
cana-5404	3	5	multiplicative	multiplicative	ADJ
cana-5404	3	6	(	(	PUNCT
cana-5404	3	7	generalized)-derivations	generalized)-derivation	NOUN
cana-5404	3	8	on	on	ADP
cana-5404	3	9	one	one	NUM
cana-5404	3	10	-	-	PUNCT
cana-5404	3	11	sided	sided	ADJ
cana-5404	3	12	ideals	ideal	NOUN
cana-5404	3	13	in	in	ADP
cana-5404	3	14	semiprime	semiprime	NOUN
cana-5404	3	15	rings	ring	NOUN
cana-5404	3	16	ahmed	ahmed	PROPN
cana-5404	3	17	aboubakr1,2	aboubakr1,2	PROPN
cana-5404	3	18	*	*	PROPN
cana-5404	3	19	and	and	CCONJ
cana-5404	3	20	nawal	nawal	PROPN
cana-5404	3	21	m.	m.	NOUN
cana-5404	3	22	noureldeen1,3	noureldeen1,3	PROPN
cana-5404	3	23	1department	1department	NUM
cana-5404	3	24	of	of	ADP
cana-5404	3	25	mathematics	mathematic	NOUN
cana-5404	3	26	,	,	PUNCT
cana-5404	3	27	collage	collage	NOUN
cana-5404	3	28	of	of	ADP
cana-5404	3	29	science	science	NOUN
cana-5404	3	30	,	,	PUNCT
cana-5404	3	31	taibah	taibah	PROPN
cana-5404	3	32	university	university	PROPN
cana-5404	3	33	,	,	PUNCT
cana-5404	3	34	medina	medina	PROPN
cana-5404	3	35	,	,	PUNCT
cana-5404	3	36	saudi	saudi	PROPN
cana-5404	3	37	arabia	arabia	PROPN
cana-5404	3	38	.	.	PUNCT
cana-5404	4	1	2department	2department	NUM
cana-5404	4	2	of	of	ADP
cana-5404	4	3	mathematics	mathematic	NOUN
cana-5404	4	4	,	,	PUNCT
cana-5404	4	5	faculty	faculty	NOUN
cana-5404	4	6	of	of	ADP
cana-5404	4	7	science	science	NOUN
cana-5404	4	8	,	,	PUNCT
cana-5404	4	9	fayoum	fayoum	PROPN
cana-5404	4	10	university	university	PROPN
cana-5404	4	11	,	,	PUNCT
cana-5404	4	12	63514	63514	NUM
cana-5404	4	13	fayoum	fayoum	PROPN
cana-5404	4	14	,	,	PUNCT
cana-5404	4	15	egypt	egypt	PROPN
cana-5404	4	16	.	.	PUNCT
cana-5404	5	1	aaboubakr@taibahu.edu.sa	aaboubakr@taibahu.edu.sa	PROPN
cana-5404	5	2	&	&	CCONJ
cana-5404	5	3	afs00@fayoum.edu.eg	afs00@fayoum.edu.eg	PROPN
cana-5404	5	4	3department	3department	NUM
cana-5404	5	5	of	of	ADP
cana-5404	5	6	mathematics	mathematic	NOUN
cana-5404	5	7	,	,	PUNCT
cana-5404	5	8	women	woman	NOUN
cana-5404	5	9	’s	’s	PART
cana-5404	5	10	college	college	PROPN
cana-5404	5	11	of	of	ADP
cana-5404	5	12	arts	art	NOUN
cana-5404	5	13	,	,	PUNCT
cana-5404	5	14	sciences	science	NOUN
cana-5404	5	15	and	and	CCONJ
cana-5404	5	16	education	education	NOUN
cana-5404	5	17	,	,	PUNCT
cana-5404	5	18	ain	ain	PROPN
cana-5404	5	19	shams	shams	PROPN
cana-5404	5	20	university	university	PROPN
cana-5404	5	21	,	,	PUNCT
cana-5404	5	22	egypt	egypt	PROPN
cana-5404	5	23	.	.	PUNCT
cana-5404	6	1	neldeen@taibahu.edu.sa	neldeen@taibahu.edu.sa	PROPN
cana-5404	6	2	article	article	PROPN
cana-5404	6	3	history	history	NOUN
cana-5404	6	4	:	:	PUNCT
cana-5404	6	5	received	receive	VERB
cana-5404	6	6	:	:	PUNCT
cana-5404	6	7	12	12	NUM
cana-5404	6	8	-	-	SYM
cana-5404	6	9	01	01	NUM
cana-5404	6	10	-	-	PUNCT
cana-5404	6	11	2025	2025	NUM
cana-5404	6	12	revised	revise	VERB
cana-5404	6	13	:	:	PUNCT
cana-5404	6	14	15	15	NUM
cana-5404	6	15	-	-	NUM
cana-5404	6	16	02	02	NUM
cana-5404	6	17	-	-	PUNCT
cana-5404	6	18	2025	2025	NUM
cana-5404	6	19	accepted	accept	VERB
cana-5404	6	20	:	:	PUNCT
cana-5404	6	21	01	01	NUM
cana-5404	6	22	-	-	SYM
cana-5404	6	23	03	03	NUM
cana-5404	6	24	-	-	PUNCT
cana-5404	6	25	2025	2025	NUM
cana-5404	6	26	abstract	abstract	NOUN
cana-5404	6	27	:	:	PUNCT
cana-5404	6	28	let	let	VERB
cana-5404	6	29	ℬ	ℬ	NOUN
cana-5404	6	30	be	be	AUX
cana-5404	6	31	a	a	DET
cana-5404	6	32	ring	ring	NOUN
cana-5404	6	33	.	.	PUNCT
cana-5404	7	1	a	a	DET
cana-5404	7	2	map	map	NOUN
cana-5404	7	3	γ	γ	X
cana-5404	7	4	:	:	PUNCT
cana-5404	7	5	ℬ	ℬ	NOUN
cana-5404	7	6	→	→	SYM
cana-5404	7	7	ℬ	ℬ	NOUN
cana-5404	7	8	is	be	AUX
cana-5404	7	9	termed	term	VERB
cana-5404	7	10	a	a	DET
cana-5404	7	11	multiplicative	multiplicative	ADJ
cana-5404	7	12	(	(	PUNCT
cana-5404	7	13	generalized)-derivation	generalized)-derivation	NOUN
cana-5404	7	14	(	(	PUNCT
cana-5404	7	15	abbreviated	abbreviate	VERB
cana-5404	7	16	as	as	ADP
cana-5404	7	17	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	7	18	(	(	PUNCT
cana-5404	7	19	𝐺	𝐺	NOUN
cana-5404	7	20	)	)	PUNCT
cana-5404	7	21	−	−	PROPN
cana-5404	7	22	𝐷	𝐷	NOUN
cana-5404	7	23	throughout	throughout	ADP
cana-5404	7	24	this	this	DET
cana-5404	7	25	paper	paper	NOUN
cana-5404	7	26	)	)	PUNCT
cana-5404	7	27	if	if	SCONJ
cana-5404	7	28	γ(𝜈𝜔	γ(𝜈𝜔	NOUN
cana-5404	7	29	)	)	PUNCT
cana-5404	7	30	=	=	SYM
cana-5404	7	31	γ(𝜈)𝜔	γ(𝜈)𝜔	PROPN
cana-5404	7	32	+	+	CCONJ
cana-5404	7	33	𝜈𝛿(𝜔	𝜈𝛿(𝜔	PRON
cana-5404	7	34	)	)	PUNCT
cana-5404	7	35	holds	hold	VERB
cana-5404	7	36	∀	∀	NOUN
cana-5404	7	37	𝜈,𝜔	𝜈,𝜔	NOUN
cana-5404	7	38	∈	∈	PROPN
cana-5404	7	39	ℬ	ℬ	NOUN
cana-5404	7	40	where	where	SCONJ
cana-5404	7	41	𝛿	𝛿	ADJ
cana-5404	7	42	:	:	PUNCT
cana-5404	7	43	ℬ	ℬ	NOUN
cana-5404	7	44	→	→	SYM
cana-5404	7	45	ℬ	ℬ	NOUN
cana-5404	7	46	is	be	AUX
cana-5404	7	47	any	any	DET
cana-5404	7	48	map	map	NOUN
cana-5404	7	49	(	(	PUNCT
cana-5404	7	50	not	not	PART
cana-5404	7	51	necessarily	necessarily	ADV
cana-5404	7	52	a	a	DET
cana-5404	7	53	derivation	derivation	NOUN
cana-5404	7	54	)	)	PUNCT
cana-5404	7	55	.	.	PUNCT
cana-5404	8	1	if	if	SCONJ
cana-5404	8	2	γ	γ	PROPN
cana-5404	8	3	,	,	PUNCT
cana-5404	8	4	δ	δ	PROPN
cana-5404	8	5	are	be	AUX
cana-5404	8	6	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	8	7	(	(	PUNCT
cana-5404	8	8	𝐺	𝐺	NOUN
cana-5404	8	9	)	)	PUNCT
cana-5404	8	10	−	−	PROPN
cana-5404	8	11	𝐷	𝐷	NOUN
cana-5404	8	12	associated	associate	VERB
cana-5404	8	13	with	with	ADP
cana-5404	8	14	maps	map	NOUN
cana-5404	8	15	𝛿	𝛿	PROPN
cana-5404	8	16	,	,	PUNCT
cana-5404	8	17	𝜉	𝜉	X
cana-5404	8	18	respectively	respectively	ADV
cana-5404	8	19	.	.	PUNCT
cana-5404	9	1	this	this	DET
cana-5404	9	2	paper	paper	NOUN
cana-5404	9	3	aims	aim	VERB
cana-5404	9	4	to	to	PART
cana-5404	9	5	investigate	investigate	VERB
cana-5404	9	6	the	the	DET
cana-5404	9	7	following	follow	VERB
cana-5404	9	8	algebric	algebric	ADJ
cana-5404	9	9	identities	identity	NOUN
cana-5404	9	10	:	:	PUNCT
cana-5404	9	11	(	(	PUNCT
cana-5404	9	12	i	i	NOUN
cana-5404	9	13	)	)	PUNCT
cana-5404	10	1	[	[	X
cana-5404	10	2	γ(𝜈	γ(𝜈	NOUN
cana-5404	10	3	)	)	PUNCT
cana-5404	10	4	,	,	PUNCT
cana-5404	10	5	γ(𝜔	γ(𝜔	NOUN
cana-5404	10	6	)	)	PUNCT
cana-5404	10	7	]	]	PUNCT
cana-5404	10	8	=	=	SYM
cana-5404	10	9	±[𝜈,𝜔	±[𝜈,𝜔	NOUN
cana-5404	10	10	]	]	PUNCT
cana-5404	10	11	(	(	PUNCT
cana-5404	10	12	scp	scp	PROPN
cana-5404	10	13	map	map	VERB
cana-5404	10	14	γ	γ	NOUN
cana-5404	10	15	)	)	PUNCT
cana-5404	10	16	,	,	PUNCT
cana-5404	10	17	(	(	PUNCT
cana-5404	10	18	ii)[γ(𝜈),𝜔	ii)[γ(𝜈),𝜔	X
cana-5404	10	19	]	]	X
cana-5404	10	20	=	=	SYM
cana-5404	10	21	±[𝜈	±[𝜈	ADJ
cana-5404	10	22	,	,	PUNCT
cana-5404	10	23	δ(𝜔	δ(𝜔	PROPN
cana-5404	10	24	)	)	PUNCT
cana-5404	10	25	]	]	PUNCT
cana-5404	10	26	,	,	PUNCT
cana-5404	10	27	(	(	PUNCT
cana-5404	10	28	iii	iii	NOUN
cana-5404	10	29	)	)	PUNCT
cana-5404	11	1	[	[	X
cana-5404	11	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	11	3	)	)	PUNCT
cana-5404	11	4	,	,	PUNCT
cana-5404	11	5	γ(𝜔	γ(𝜔	NOUN
cana-5404	11	6	)	)	PUNCT
cana-5404	11	7	]	]	PUNCT
cana-5404	12	1	=	=	SYM
cana-5404	12	2	±[𝜈,𝜔	±[𝜈,𝜔	NOUN
cana-5404	12	3	]	]	PUNCT
cana-5404	12	4	and	and	CCONJ
cana-5404	12	5	(	(	PUNCT
cana-5404	12	6	iv	iv	X
cana-5404	12	7	)	)	PUNCT
cana-5404	13	1	[	[	X
cana-5404	13	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	13	3	)	)	PUNCT
cana-5404	13	4	,	,	PUNCT
cana-5404	13	5	𝜔	𝜔	PROPN
cana-5404	13	6	]	]	X
cana-5404	13	7	=	=	SYM
cana-5404	13	8	±[𝜈	±[𝜈	NOUN
cana-5404	13	9	,	,	PUNCT
cana-5404	13	10	γ(𝜔	γ(𝜔	NOUN
cana-5404	13	11	)	)	PUNCT
cana-5404	13	12	]	]	PUNCT
cana-5404	13	13	for	for	ADP
cana-5404	13	14	all	all	DET
cana-5404	13	15	𝜈,𝜔	𝜈,𝜔	NOUN
cana-5404	13	16	in	in	ADP
cana-5404	13	17	a	a	DET
cana-5404	13	18	left	left	ADJ
cana-5404	13	19	ideal	ideal	NOUN
cana-5404	13	20	of	of	ADP
cana-5404	13	21	a	a	DET
cana-5404	13	22	semiprime	semiprime	NOUN
cana-5404	13	23	ring	ring	NOUN
cana-5404	13	24	ℬ.	ℬ.	PROPN
cana-5404	13	25	additionally	additionally	ADV
cana-5404	13	26	,	,	PUNCT
cana-5404	13	27	we	we	PRON
cana-5404	13	28	present	present	VERB
cana-5404	13	29	an	an	DET
cana-5404	13	30	example	example	NOUN
cana-5404	13	31	illustrating	illustrate	VERB
cana-5404	13	32	that	that	SCONJ
cana-5404	13	33	the	the	DET
cana-5404	13	34	semiprimeness	semiprimeness	NOUN
cana-5404	13	35	condition	condition	NOUN
cana-5404	13	36	in	in	ADP
cana-5404	13	37	our	our	PRON
cana-5404	13	38	theorems	theorem	NOUN
cana-5404	13	39	can	can	AUX
cana-5404	13	40	not	not	PART
cana-5404	13	41	be	be	AUX
cana-5404	13	42	omitted	omit	VERB
cana-5404	13	43	.	.	PUNCT
cana-5404	14	1	mathematics	mathematic	NOUN
cana-5404	14	2	subject	subject	ADJ
cana-5404	14	3	classification	classification	NOUN
cana-5404	14	4	(	(	PUNCT
cana-5404	14	5	msc2020	msc2020	PROPN
cana-5404	14	6	):	):	PUNCT
cana-5404	14	7	16w25	16w25	NUM
cana-5404	14	8	,	,	PUNCT
cana-5404	14	9	16n60	16n60	NUM
cana-5404	14	10	,	,	PUNCT
cana-5404	14	11	16u80	16u80	NUM
cana-5404	14	12	.	.	PUNCT
cana-5404	15	1	keywords	keyword	NOUN
cana-5404	15	2	:	:	PUNCT
cana-5404	15	3	semiprime	semiprime	NOUN
cana-5404	15	4	rings	ring	NOUN
cana-5404	15	5	,	,	PUNCT
cana-5404	15	6	l	l	NOUN
cana-5404	15	7	-	-	NOUN
cana-5404	15	8	ideals	ideal	NOUN
cana-5404	15	9	,	,	PUNCT
cana-5404	15	10	generalized	generalized	ADJ
cana-5404	15	11	derivation	derivation	NOUN
cana-5404	15	12	,	,	PUNCT
cana-5404	15	13	multiplicative	multiplicative	X
cana-5404	15	14	(	(	PUNCT
cana-5404	15	15	generalized)derivation	generalized)derivation	NOUN
cana-5404	15	16	.	.	NOUN
cana-5404	16	1	1introduction	1introduction	NUM
cana-5404	16	2	the	the	DET
cana-5404	16	3	concept	concept	NOUN
cana-5404	16	4	of	of	ADP
cana-5404	16	5	multiplicative	multiplicative	ADJ
cana-5404	16	6	derivation	derivation	NOUN
cana-5404	16	7	was	be	AUX
cana-5404	16	8	first	first	ADV
cana-5404	16	9	established	establish	VERB
cana-5404	16	10	in	in	ADP
cana-5404	16	11	1991	1991	NUM
cana-5404	16	12	by	by	ADP
cana-5404	16	13	daif	daif	NOUN
cana-5404	16	14	[	[	X
cana-5404	16	15	1	1	X
cana-5404	16	16	]	]	PUNCT
cana-5404	16	17	who	who	PRON
cana-5404	16	18	defined	define	VERB
cana-5404	16	19	it	it	PRON
cana-5404	16	20	as	as	ADP
cana-5404	16	21	:	:	PUNCT
cana-5404	16	22	a	a	DET
cana-5404	16	23	function	function	NOUN
cana-5404	16	24	𝛿	𝛿	ADJ
cana-5404	16	25	:	:	PUNCT
cana-5404	16	26	ℬ	ℬ	NOUN
cana-5404	16	27	→	→	SYM
cana-5404	16	28	ℬ	ℬ	NOUN
cana-5404	16	29	(	(	PUNCT
cana-5404	16	30	not	not	PART
cana-5404	16	31	necessarily	necessarily	ADV
cana-5404	16	32	additive	additive	VERB
cana-5404	16	33	)	)	PUNCT
cana-5404	16	34	is	be	AUX
cana-5404	16	35	termed	term	VERB
cana-5404	16	36	a	a	DET
cana-5404	16	37	multiplicative	multiplicative	ADJ
cana-5404	16	38	derivation	derivation	NOUN
cana-5404	16	39	of	of	ADP
cana-5404	16	40	ℬ	ℬ	NOUN
cana-5404	16	41	if	if	SCONJ
cana-5404	16	42	𝛿(𝜈𝜔	𝛿(𝜈𝜔	NOUN
cana-5404	16	43	)	)	PUNCT
cana-5404	16	44	=	=	VERB
cana-5404	17	1	𝛿(𝜈)𝜔	𝛿(𝜈)𝜔	NOUN
cana-5404	17	2	+	+	CCONJ
cana-5404	17	3	𝜈𝛿(𝜔	𝜈𝛿(𝜔	X
cana-5404	17	4	)	)	PUNCT
cana-5404	17	5	for	for	ADP
cana-5404	17	6	all	all	DET
cana-5404	17	7	𝜈,𝜔	𝜈,𝜔	PROPN
cana-5404	17	8	∈	∈	PROPN
cana-5404	17	9	ℬ.	ℬ.	NOUN
cana-5404	17	10	the	the	DET
cana-5404	17	11	research	research	NOUN
cana-5404	17	12	by	by	ADP
cana-5404	17	13	daif	daif	NOUN
cana-5404	17	14	[	[	X
cana-5404	17	15	1	1	X
cana-5404	17	16	]	]	PUNCT
cana-5404	17	17	drew	draw	VERB
cana-5404	17	18	inspiration	inspiration	NOUN
cana-5404	17	19	from	from	ADP
cana-5404	17	20	martindale	martindale	PROPN
cana-5404	17	21	’s	’s	PART
cana-5404	17	22	work	work	NOUN
cana-5404	17	23	[	[	X
cana-5404	17	24	2	2	NUM
cana-5404	17	25	]	]	PUNCT
cana-5404	17	26	.	.	PUNCT
cana-5404	18	1	subsequently	subsequently	ADV
cana-5404	18	2	,	,	PUNCT
cana-5404	18	3	goldmann	goldmann	NOUN
cana-5404	18	4	and	and	CCONJ
cana-5404	18	5	ŝemrl	ŝemrl	NOUN
cana-5404	18	6	provided	provide	VERB
cana-5404	18	7	a	a	DET
cana-5404	18	8	thorough	thorough	ADJ
cana-5404	18	9	characterization	characterization	NOUN
cana-5404	18	10	of	of	ADP
cana-5404	18	11	these	these	DET
cana-5404	18	12	functions	function	NOUN
cana-5404	18	13	in	in	ADP
cana-5404	18	14	[	[	X
cana-5404	18	15	3	3	NUM
cana-5404	18	16	]	]	PUNCT
cana-5404	18	17	.	.	PUNCT
cana-5404	19	1	this	this	DET
cana-5404	19	2	initial	initial	ADJ
cana-5404	19	3	definition	definition	NOUN
cana-5404	19	4	was	be	AUX
cana-5404	19	5	later	later	ADV
cana-5404	19	6	broadened	broaden	VERB
cana-5404	19	7	to	to	PART
cana-5404	19	8	multiplicative	multiplicative	VERB
cana-5404	19	9	generalized	generalized	ADJ
cana-5404	19	10	derivation	derivation	NOUN
cana-5404	19	11	by	by	ADP
cana-5404	19	12	daif	daif	NOUN
cana-5404	19	13	and	and	CCONJ
cana-5404	19	14	tammam	tammam	NOUN
cana-5404	19	15	in	in	ADP
cana-5404	19	16	[	[	X
cana-5404	19	17	4	4	NUM
cana-5404	19	18	]	]	PUNCT
cana-5404	19	19	,	,	PUNCT
cana-5404	19	20	who	who	PRON
cana-5404	19	21	characterized	characterize	VERB
cana-5404	19	22	it	it	PRON
cana-5404	19	23	as	as	ADP
cana-5404	19	24	:	:	PUNCT
cana-5404	19	25	a	a	DET
cana-5404	19	26	function	function	NOUN
cana-5404	19	27	γ	γ	NOUN
cana-5404	19	28	:	:	PUNCT
cana-5404	19	29	𝑅	𝑅	PROPN
cana-5404	19	30	→	→	SYM
cana-5404	19	31	ℬ	ℬ	NOUN
cana-5404	19	32	(	(	PUNCT
cana-5404	19	33	not	not	PART
cana-5404	19	34	necessarily	necessarily	ADV
cana-5404	19	35	additive	additive	VERB
cana-5404	19	36	)	)	PUNCT
cana-5404	19	37	qualifies	qualifie	NOUN
cana-5404	19	38	as	as	ADP
cana-5404	19	39	a	a	DET
cana-5404	19	40	multiplicative	multiplicative	ADJ
cana-5404	19	41	generalized	generalized	ADJ
cana-5404	19	42	derivation	derivation	NOUN
cana-5404	19	43	if	if	SCONJ
cana-5404	19	44	there	there	PRON
cana-5404	19	45	exists	exist	VERB
cana-5404	19	46	a	a	DET
cana-5404	19	47	multiplicative	multiplicative	ADJ
cana-5404	19	48	derivation	derivation	NOUN
cana-5404	19	49	𝛿	𝛿	ADJ
cana-5404	19	50	:	:	PUNCT
cana-5404	19	51	ℬ	ℬ	NOUN
cana-5404	19	52	→	→	SYM
cana-5404	19	53	ℬ	ℬ	NOUN
cana-5404	19	54	such	such	ADJ
cana-5404	19	55	that	that	PRON
cana-5404	19	56	γ(𝜈𝜔	γ(𝜈𝜔	NOUN
cana-5404	19	57	)	)	PUNCT
cana-5404	19	58	=	=	SYM
cana-5404	19	59	γ(𝜈)𝜔	γ(𝜈)𝜔	PROPN
cana-5404	19	60	+	+	NUM
cana-5404	19	61	𝜈𝛿(𝜔	𝜈𝛿(𝜔	NUM
cana-5404	19	62	)	)	PUNCT
cana-5404	19	63	∀	∀	PUNCT
cana-5404	19	64	𝜈,𝜔	𝜈,𝜔	PROPN
cana-5404	19	65	∈	∈	PROPN
cana-5404	19	66	ℬ.	ℬ.	PROPN
cana-5404	19	67	later	later	ADV
cana-5404	19	68	,	,	PUNCT
cana-5404	19	69	dhara	dhara	PROPN
cana-5404	19	70	and	and	CCONJ
cana-5404	19	71	ali	ali	PROPN
cana-5404	20	1	[	[	X
cana-5404	20	2	5	5	NUM
cana-5404	20	3	]	]	PUNCT
cana-5404	20	4	expanded	expand	VERB
cana-5404	20	5	this	this	DET
cana-5404	20	6	definition	definition	NOUN
cana-5404	20	7	of	of	ADP
cana-5404	20	8	multiplicative	multiplicative	ADJ
cana-5404	20	9	generalized	generalize	VERB
cana-5404	20	10	derivation	derivation	NOUN
cana-5404	20	11	by	by	ADP
cana-5404	20	12	allowing	allow	VERB
cana-5404	20	13	𝛿	𝛿	NOUN
cana-5404	20	14	to	to	PART
cana-5404	20	15	be	be	AUX
cana-5404	20	16	any	any	DET
cana-5404	20	17	function	function	NOUN
cana-5404	20	18	on	on	ADP
cana-5404	20	19	ℬ.	ℬ.	PROPN
cana-5404	20	20	therefore	therefore	ADV
cana-5404	20	21	,	,	PUNCT
cana-5404	20	22	a	a	DET
cana-5404	20	23	function	function	NOUN
cana-5404	20	24	γ	γ	NOUN
cana-5404	20	25	:	:	PUNCT
cana-5404	20	26	𝑅	𝑅	PROPN
cana-5404	20	27	→	→	SYM
cana-5404	20	28	ℬ	ℬ	NOUN
cana-5404	20	29	(	(	PUNCT
cana-5404	20	30	not	not	PART
cana-5404	20	31	necessarily	necessarily	ADV
cana-5404	20	32	additive	additive	VERB
cana-5404	20	33	)	)	PUNCT
cana-5404	20	34	is	be	AUX
cana-5404	20	35	designated	designate	VERB
cana-5404	20	36	a	a	DET
cana-5404	20	37	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	20	38	(	(	PUNCT
cana-5404	20	39	𝐺	𝐺	NOUN
cana-5404	20	40	)	)	PUNCT
cana-5404	20	41	−	−	PROPN
cana-5404	20	42	𝐷	𝐷	PROPN
cana-5404	20	43	(	(	PUNCT
cana-5404	20	44	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	20	45	(	(	PUNCT
cana-5404	20	46	𝐺	𝐺	NOUN
cana-5404	20	47	)	)	PUNCT
cana-5404	20	48	−	−	PROPN
cana-5404	20	49	𝐷	𝐷	NOUN
cana-5404	20	50	)	)	PUNCT
cana-5404	20	51	if	if	SCONJ
cana-5404	20	52	γ(𝜈𝜔	γ(𝜈𝜔	NOUN
cana-5404	20	53	)	)	PUNCT
cana-5404	20	54	=	=	SYM
cana-5404	20	55	γ(𝜈)𝜔	γ(𝜈)𝜔	PROPN
cana-5404	20	56	+	+	CCONJ
cana-5404	20	57	𝜈𝛿(𝜔	𝜈𝛿(𝜔	X
cana-5404	20	58	)	)	PUNCT
cana-5404	20	59	is	be	AUX
cana-5404	20	60	satisfied	satisfied	ADJ
cana-5404	20	61	∀	∀	X
cana-5404	20	62	𝜈	𝜈	X
cana-5404	20	63	,	,	PUNCT
cana-5404	20	64	𝜔	𝜔	PROPN
cana-5404	20	65	∈	∈	PROPN
cana-5404	20	66	ℬ	ℬ	NOUN
cana-5404	20	67	,	,	PUNCT
cana-5404	20	68	where	where	SCONJ
cana-5404	20	69	𝛿	𝛿	ADJ
cana-5404	20	70	:	:	PUNCT
cana-5404	20	71	𝑅	𝑅	NOUN
cana-5404	20	72	→	→	SYM
cana-5404	20	73	ℬ	ℬ	NOUN
cana-5404	20	74	represents	represent	VERB
cana-5404	20	75	any	any	DET
cana-5404	20	76	function	function	NOUN
cana-5404	20	77	(	(	PUNCT
cana-5404	20	78	not	not	PART
cana-5404	20	79	necessarily	necessarily	ADV
cana-5404	20	80	a	a	DET
cana-5404	20	81	derivation	derivation	NOUN
cana-5404	20	82	nor	nor	CCONJ
cana-5404	20	83	additive	additive	NOUN
cana-5404	20	84	)	)	PUNCT
cana-5404	20	85	.	.	PUNCT
cana-5404	21	1	consequently	consequently	ADV
cana-5404	21	2	,	,	PUNCT
cana-5404	21	3	the	the	DET
cana-5404	21	4	framework	framework	NOUN
cana-5404	21	5	of	of	ADP
cana-5404	21	6	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	21	7	(	(	PUNCT
cana-5404	21	8	𝐺	𝐺	NOUN
cana-5404	21	9	)	)	PUNCT
cana-5404	21	10	−	−	PROPN
cana-5404	21	11	𝐷	𝐷	PROPN
cana-5404	21	12	encompasses	encompass	VERB
cana-5404	21	13	the	the	DET
cana-5404	21	14	framework	framework	NOUN
cana-5404	21	15	of	of	ADP
cana-5404	21	16	multiplicative	multiplicative	ADJ
cana-5404	21	17	derivation	derivation	NOUN
cana-5404	21	18	.	.	PUNCT
cana-5404	22	1	additionally	additionally	ADV
cana-5404	22	2	,	,	PUNCT
cana-5404	22	3	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	22	4	(	(	PUNCT
cana-5404	22	5	𝐺	𝐺	NOUN
cana-5404	22	6	)	)	PUNCT
cana-5404	22	7	−	−	PROPN
cana-5404	22	8	𝐷	𝐷	NOUN
cana-5404	22	9	with	with	ADP
cana-5404	22	10	𝛿	𝛿	PROPN
cana-5404	22	11	=	=	SYM
cana-5404	22	12	0	0	NUM
cana-5404	22	13	encompasses	encompass	VERB
cana-5404	22	14	the	the	DET
cana-5404	22	15	notion	notion	NOUN
cana-5404	22	16	of	of	ADP
cana-5404	22	17	multiplicative	multiplicative	ADJ
cana-5404	22	18	centralizer	centralizer	NOUN
cana-5404	22	19	(	(	PUNCT
cana-5404	22	20	not	not	PART
cana-5404	22	21	necessarily	necessarily	ADV
cana-5404	22	22	additive	additive	VERB
cana-5404	22	23	)	)	PUNCT
cana-5404	22	24	.	.	PUNCT
cana-5404	23	1	mailto:afs00@fayoum.edu.eg	mailto:afs00@fayoum.edu.eg	PROPN
cana-5404	23	2	communications	communication	NOUN
cana-5404	23	3	on	on	ADP
cana-5404	23	4	applied	apply	VERB
cana-5404	23	5	nonlinear	nonlinear	ADJ
cana-5404	23	6	analysis	analysis	NOUN
cana-5404	23	7	issn	issn	NOUN
cana-5404	23	8	:	:	PUNCT
cana-5404	23	9	1074	1074	NUM
cana-5404	23	10	-	-	PUNCT
cana-5404	23	11	133x	133x	NUM
cana-5404	23	12	vol	vol	VERB
cana-5404	23	13	32	32	NUM
cana-5404	23	14	no	no	NOUN
cana-5404	23	15	.	.	PUNCT
cana-5404	24	1	10s	10	NOUN
cana-5404	24	2	(	(	PUNCT
cana-5404	24	3	2025	2025	NUM
cana-5404	24	4	)	)	PUNCT
cana-5404	24	5	2139	2139	NUM
cana-5404	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	25	1	it	it	PRON
cana-5404	25	2	is	be	AUX
cana-5404	25	3	evident	evident	ADJ
cana-5404	25	4	that	that	SCONJ
cana-5404	25	5	every	every	DET
cana-5404	25	6	generalized	generalized	ADJ
cana-5404	25	7	derivation	derivation	NOUN
cana-5404	25	8	constitutes	constitute	VERB
cana-5404	25	9	a	a	DET
cana-5404	25	10	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	25	11	(	(	PUNCT
cana-5404	25	12	𝐺	𝐺	NOUN
cana-5404	25	13	)	)	PUNCT
cana-5404	25	14	−	−	PROPN
cana-5404	25	15	𝐷	𝐷	NOUN
cana-5404	25	16	on	on	ADP
cana-5404	25	17	ℬ.	ℬ.	PROPN
cana-5404	25	18	nonetheless	nonetheless	ADV
cana-5404	25	19	,	,	PUNCT
cana-5404	25	20	the	the	DET
cana-5404	25	21	converse	converse	NOUN
cana-5404	25	22	does	do	AUX
cana-5404	25	23	not	not	PART
cana-5404	25	24	necessarily	necessarily	ADV
cana-5404	25	25	hold	hold	VERB
cana-5404	25	26	in	in	ADP
cana-5404	25	27	all	all	DET
cana-5404	25	28	instances	instance	NOUN
cana-5404	25	29	,	,	PUNCT
cana-5404	25	30	as	as	SCONJ
cana-5404	25	31	illustrated	illustrate	VERB
cana-5404	25	32	by	by	ADP
cana-5404	25	33	the	the	DET
cana-5404	25	34	subsequent	subsequent	ADJ
cana-5404	25	35	examples	example	NOUN
cana-5404	25	36	:	:	PUNCT
cana-5404	25	37	example	example	NOUN
cana-5404	25	38	1.1	1.1	NUM
cana-5404	25	39	consider	consider	VERB
cana-5404	25	40	ℬ	ℬ	NOUN
cana-5404	25	41	=	=	SYM
cana-5404	25	42	𝐶[0,1	𝐶[0,1	ADP
cana-5404	25	43	]	]	PUNCT
cana-5404	25	44	,	,	PUNCT
cana-5404	25	45	the	the	DET
cana-5404	25	46	ring	ring	NOUN
cana-5404	25	47	of	of	ADP
cana-5404	25	48	all	all	DET
cana-5404	25	49	continuous	continuous	ADJ
cana-5404	25	50	real	real	ADJ
cana-5404	25	51	functions	function	NOUN
cana-5404	25	52	and	and	CCONJ
cana-5404	25	53	define	define	VERB
cana-5404	25	54	maps	map	NOUN
cana-5404	25	55	𝛿	𝛿	ADJ
cana-5404	25	56	:	:	PUNCT
cana-5404	25	57	ℬ	ℬ	PROPN
cana-5404	25	58	→	→	SYM
cana-5404	25	59	ℬ	ℬ	NOUN
cana-5404	25	60	,	,	PUNCT
cana-5404	25	61	as	as	SCONJ
cana-5404	25	62	follows	follow	VERB
cana-5404	25	63	:	:	PUNCT
cana-5404	25	64	𝛿(𝑓)(𝜈	𝛿(𝑓)(𝜈	X
cana-5404	25	65	)	)	PUNCT
cana-5404	25	66	=	=	PRON
cana-5404	25	67	{	{	PUNCT
cana-5404	25	68	0	0	NUM
cana-5404	25	69	,	,	PUNCT
cana-5404	25	70	otherwise	otherwise	ADV
cana-5404	25	71	.	.	PUNCT
cana-5404	26	1	𝑓(𝜈)𝑙𝑜𝑔|𝑓(𝜈)|	𝑓(𝜈)𝑙𝑜𝑔|𝑓(𝜈)|	X
cana-5404	26	2	∀	∀	NOUN
cana-5404	26	3	𝑓(𝜈)≠0	𝑓(𝜈)≠0	NOUN
cana-5404	26	4	and	and	CCONJ
cana-5404	26	5	γ	γ	X
cana-5404	26	6	:	:	PUNCT
cana-5404	26	7	ℬ	ℬ	PROPN
cana-5404	26	8	→	→	SYM
cana-5404	26	9	ℬ	ℬ	NOUN
cana-5404	26	10	,	,	PUNCT
cana-5404	26	11	as	as	SCONJ
cana-5404	26	12	follows	follow	VERB
cana-5404	26	13	,	,	PUNCT
cana-5404	26	14	γ(𝑓)(𝜈	γ(𝑓)(𝜈	X
cana-5404	26	15	)	)	PUNCT
cana-5404	26	16	=	=	SYM
cana-5404	26	17	{	{	PUNCT
cana-5404	26	18	0	0	NUM
cana-5404	26	19	,	,	PUNCT
cana-5404	26	20	otherwise	otherwise	ADV
cana-5404	26	21	.	.	PUNCT
cana-5404	27	1	𝑓(𝜈)(1+𝑙𝑜𝑔|𝑓(𝜈)|	𝑓(𝜈)(1+𝑙𝑜𝑔|𝑓(𝜈)|	PROPN
cana-5404	27	2	)	)	PUNCT
cana-5404	27	3	∀	∀	PUNCT
cana-5404	28	1	𝑓(𝜈)≠0	𝑓(𝜈)≠0	NOUN
cana-5404	28	2	it	it	PRON
cana-5404	28	3	is	be	AUX
cana-5404	28	4	evident	evident	ADJ
cana-5404	28	5	that	that	SCONJ
cana-5404	28	6	𝛿	𝛿	NOUN
cana-5404	28	7	and	and	CCONJ
cana-5404	28	8	γ	γ	NOUN
cana-5404	28	9	do	do	AUX
cana-5404	28	10	not	not	PART
cana-5404	28	11	exhibit	exhibit	VERB
cana-5404	28	12	additive	additive	ADJ
cana-5404	28	13	properties	property	NOUN
cana-5404	28	14	,	,	PUNCT
cana-5404	28	15	while	while	SCONJ
cana-5404	28	16	𝐷	𝐷	NOUN
cana-5404	28	17	functions	function	NOUN
cana-5404	28	18	as	as	ADP
cana-5404	28	19	a	a	DET
cana-5404	28	20	multiplicative	multiplicative	ADJ
cana-5404	28	21	derivation	derivation	NOUN
cana-5404	28	22	,	,	PUNCT
cana-5404	28	23	and	and	CCONJ
cana-5404	28	24	γ	γ	NOUN
cana-5404	28	25	is	be	AUX
cana-5404	28	26	classified	classify	VERB
cana-5404	28	27	as	as	ADP
cana-5404	28	28	a	a	DET
cana-5404	28	29	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	28	30	(	(	PUNCT
cana-5404	28	31	𝐺	𝐺	NOUN
cana-5404	28	32	)	)	PUNCT
cana-5404	28	33	−	−	PROPN
cana-5404	28	34	𝐷	𝐷	NOUN
cana-5404	28	35	associated	associate	VERB
cana-5404	28	36	with	with	ADP
cana-5404	28	37	𝐷.	𝐷.	PROPN
cana-5404	28	38	example	example	NOUN
cana-5404	28	39	1.2	1.2	NUM
cana-5404	28	40	consider	consider	VERB
cana-5404	28	41	the	the	DET
cana-5404	28	42	ring	ring	NOUN
cana-5404	28	43	ℬ	ℬ	NOUN
cana-5404	28	44	=	=	PRON
cana-5404	28	45	{	{	PUNCT
cana-5404	28	46	,	,	PUNCT
cana-5404	28	47	(	(	PUNCT
cana-5404	28	48	0	0	NUM
cana-5404	28	49	0	0	NUM
cana-5404	29	1	𝑏	𝑏	PROPN
cana-5404	29	2	𝑐	𝑐	PROPN
cana-5404	29	3	0	0	NUM
cana-5404	29	4	0	0	NUM
cana-5404	29	5	0	0	NUM
cana-5404	30	1	𝑑	𝑑	NOUN
cana-5404	30	2	0	0	NUM
cana-5404	30	3	0	0	NUM
cana-5404	30	4	0	0	NUM
cana-5404	30	5	0	0	NUM
cana-5404	30	6	0	0	NUM
cana-5404	30	7	0	0	NUM
cana-5404	30	8	0	0	NUM
cana-5404	30	9	0	0	NUM
cana-5404	30	10	)	)	PUNCT
cana-5404	30	11	|𝑏	|𝑏	PROPN
cana-5404	30	12	,	,	PUNCT
cana-5404	30	13	𝑑	𝑑	PROPN
cana-5404	30	14	,	,	PUNCT
cana-5404	30	15	𝑐	𝑐	PROPN
cana-5404	30	16	∈	∈	PROPN
cana-5404	30	17	ℝ	ℝ	PROPN
cana-5404	30	18	}	}	PUNCT
cana-5404	30	19	.	.	PUNCT
cana-5404	31	1	define	define	VERB
cana-5404	31	2	maps	maps	PROPN
cana-5404	31	3	𝛤	𝛤	PROPN
cana-5404	31	4	:	:	PUNCT
cana-5404	31	5	ℬ	ℬ	NOUN
cana-5404	31	6	→	→	SYM
cana-5404	31	7	ℬ	ℬ	NOUN
cana-5404	31	8	and	and	CCONJ
cana-5404	31	9	𝛿:ℬ	𝛿:ℬ	NOUN
cana-5404	31	10	→	→	SYM
cana-5404	31	11	ℬ	ℬ	NOUN
cana-5404	31	12	as	as	SCONJ
cana-5404	31	13	follows	follow	VERB
cana-5404	31	14	:	:	PUNCT
cana-5404	31	15	𝛤	𝛤	PROPN
cana-5404	31	16	(	(	PUNCT
cana-5404	31	17	(	(	PUNCT
cana-5404	31	18	0	0	NUM
cana-5404	31	19	0	0	NUM
cana-5404	31	20	𝑏	𝑏	PROPN
cana-5404	31	21	𝑐	𝑐	PROPN
cana-5404	31	22	0	0	NUM
cana-5404	31	23	0	0	NUM
cana-5404	31	24	0	0	NUM
cana-5404	32	1	𝑑	𝑑	NOUN
cana-5404	32	2	0	0	NUM
cana-5404	32	3	0	0	NUM
cana-5404	32	4	0	0	NUM
cana-5404	32	5	0	0	NUM
cana-5404	32	6	0	0	NUM
cana-5404	32	7	0	0	NUM
cana-5404	32	8	0	0	NUM
cana-5404	32	9	0	0	NUM
cana-5404	32	10	)	)	PUNCT
cana-5404	32	11	)	)	PUNCT
cana-5404	33	1	=	=	PUNCT
cana-5404	33	2	(	(	PUNCT
cana-5404	33	3	0	0	NUM
cana-5404	33	4	0	0	NUM
cana-5404	33	5	0	0	NUM
cana-5404	34	1	𝑏𝑑	𝑏𝑑	PROPN
cana-5404	34	2	0	0	NUM
cana-5404	34	3	0	0	NUM
cana-5404	34	4	0	0	NUM
cana-5404	34	5	0	0	NUM
cana-5404	34	6	0	0	NUM
cana-5404	34	7	0	0	NUM
cana-5404	34	8	0	0	NUM
cana-5404	34	9	0	0	NUM
cana-5404	34	10	0	0	NUM
cana-5404	34	11	0	0	NUM
cana-5404	34	12	0	0	NUM
cana-5404	34	13	0	0	NUM
cana-5404	34	14	)	)	PUNCT
cana-5404	34	15	,	,	PUNCT
cana-5404	34	16	and	and	CCONJ
cana-5404	34	17	𝛿	𝛿	ADJ
cana-5404	34	18	(	(	PUNCT
cana-5404	34	19	(	(	PUNCT
cana-5404	34	20	0	0	NUM
cana-5404	34	21	0	0	NUM
cana-5404	34	22	𝑏	𝑏	PROPN
cana-5404	34	23	𝑐	𝑐	PROPN
cana-5404	34	24	0	0	NUM
cana-5404	34	25	0	0	NUM
cana-5404	34	26	0	0	NUM
cana-5404	34	27	𝑑	𝑑	NOUN
cana-5404	34	28	0	0	NUM
cana-5404	34	29	0	0	NUM
cana-5404	34	30	0	0	NUM
cana-5404	34	31	0	0	NUM
cana-5404	34	32	0	0	NUM
cana-5404	34	33	0	0	NUM
cana-5404	34	34	0	0	NUM
cana-5404	34	35	0	0	NUM
cana-5404	34	36	)	)	PUNCT
cana-5404	34	37	)	)	PUNCT
cana-5404	35	1	=	=	PUNCT
cana-5404	35	2	(	(	PUNCT
cana-5404	35	3	0	0	NUM
cana-5404	35	4	0	0	NUM
cana-5404	35	5	0	0	NUM
cana-5404	35	6	𝑐2	𝑐2	NOUN
cana-5404	35	7	0	0	NUM
cana-5404	35	8	0	0	NUM
cana-5404	35	9	0	0	NUM
cana-5404	35	10	0	0	NUM
cana-5404	35	11	0	0	NUM
cana-5404	35	12	0	0	NUM
cana-5404	35	13	0	0	NUM
cana-5404	35	14	0	0	NUM
cana-5404	35	15	0	0	NUM
cana-5404	35	16	0	0	NUM
cana-5404	35	17	0	0	NUM
cana-5404	35	18	0	0	NUM
cana-5404	35	19	)	)	PUNCT
cana-5404	35	20	.	.	PUNCT
cana-5404	36	1	it	it	PRON
cana-5404	36	2	is	be	AUX
cana-5404	36	3	evident	evident	ADJ
cana-5404	36	4	that	that	SCONJ
cana-5404	36	5	𝛿	𝛿	NOUN
cana-5404	36	6	and	and	CCONJ
cana-5404	36	7	γ	γ	NOUN
cana-5404	36	8	do	do	AUX
cana-5404	36	9	not	not	PART
cana-5404	36	10	exhibit	exhibit	VERB
cana-5404	36	11	additive	additive	ADJ
cana-5404	36	12	properties	property	NOUN
cana-5404	36	13	,	,	PUNCT
cana-5404	36	14	while	while	SCONJ
cana-5404	36	15	𝐷	𝐷	NOUN
cana-5404	36	16	functions	function	NOUN
cana-5404	36	17	as	as	ADP
cana-5404	36	18	a	a	DET
cana-5404	36	19	multiplicative	multiplicative	ADJ
cana-5404	36	20	derivation	derivation	NOUN
cana-5404	36	21	,	,	PUNCT
cana-5404	36	22	and	and	CCONJ
cana-5404	36	23	γ	γ	NOUN
cana-5404	36	24	is	be	AUX
cana-5404	36	25	classified	classify	VERB
cana-5404	36	26	as	as	ADP
cana-5404	36	27	a	a	DET
cana-5404	36	28	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	36	29	(	(	PUNCT
cana-5404	36	30	𝐺	𝐺	NOUN
cana-5404	36	31	)	)	PUNCT
cana-5404	36	32	−	−	PROPN
cana-5404	36	33	𝐷	𝐷	NOUN
cana-5404	36	34	associated	associate	VERB
cana-5404	36	35	with	with	ADP
cana-5404	36	36	𝐷.	𝐷.	PROPN
cana-5404	36	37	in	in	ADP
cana-5404	36	38	the	the	DET
cana-5404	36	39	last	last	ADJ
cana-5404	36	40	thirty	thirty	NUM
cana-5404	36	41	years	year	NOUN
cana-5404	36	42	,	,	PUNCT
cana-5404	36	43	some	some	DET
cana-5404	36	44	researchers	researcher	NOUN
cana-5404	36	45	have	have	AUX
cana-5404	36	46	demonstrated	demonstrate	VERB
cana-5404	36	47	commutativity	commutativity	NOUN
cana-5404	36	48	theorems	theorem	NOUN
cana-5404	36	49	for	for	ADP
cana-5404	36	50	certain	certain	ADJ
cana-5404	36	51	prime	prime	ADJ
cana-5404	36	52	or	or	CCONJ
cana-5404	36	53	semiprime	semiprime	NOUN
cana-5404	36	54	rings	ring	NOUN
cana-5404	36	55	having	have	VERB
cana-5404	36	56	automorphisms	automorphism	NOUN
cana-5404	36	57	or	or	CCONJ
cana-5404	36	58	derivations	derivation	NOUN
cana-5404	36	59	which	which	PRON
cana-5404	36	60	are	be	AUX
cana-5404	36	61	centralizing	centralize	VERB
cana-5404	36	62	or	or	CCONJ
cana-5404	36	63	commuting	commute	VERB
cana-5404	36	64	on	on	ADP
cana-5404	36	65	certain	certain	ADJ
cana-5404	36	66	ordered	order	VERB
cana-5404	36	67	subsets	subset	NOUN
cana-5404	36	68	of	of	ADP
cana-5404	36	69	ℬ.	ℬ.	PROPN
cana-5404	36	70	(	(	PUNCT
cana-5404	36	71	see	see	VERB
cana-5404	36	72	[	[	X
cana-5404	36	73	6	6	NUM
cana-5404	36	74	]	]	PUNCT
cana-5404	36	75	,	,	PUNCT
cana-5404	37	1	[	[	X
cana-5404	37	2	7	7	NUM
cana-5404	37	3	]	]	PUNCT
cana-5404	37	4	,	,	PUNCT
cana-5404	37	5	[	[	X
cana-5404	37	6	8	8	NUM
cana-5404	37	7	]	]	PUNCT
cana-5404	37	8	,	,	PUNCT
cana-5404	37	9	[	[	X
cana-5404	37	10	9	9	NUM
cana-5404	37	11	]	]	PUNCT
cana-5404	37	12	,	,	PUNCT
cana-5404	37	13	[	[	X
cana-5404	37	14	10	10	NUM
cana-5404	37	15	]	]	PUNCT
cana-5404	37	16	and	and	CCONJ
cana-5404	38	1	[	[	X
cana-5404	38	2	11	11	NUM
cana-5404	38	3	]	]	PUNCT
cana-5404	38	4	,	,	PUNCT
cana-5404	38	5	where	where	SCONJ
cana-5404	38	6	further	further	ADJ
cana-5404	38	7	references	reference	NOUN
cana-5404	38	8	can	can	AUX
cana-5404	38	9	be	be	AUX
cana-5404	38	10	found	find	VERB
cana-5404	38	11	)	)	PUNCT
cana-5404	38	12	.	.	PUNCT
cana-5404	39	1	let	let	VERB
cana-5404	39	2	𝑆	𝑆	PROPN
cana-5404	39	3	be	be	AUX
cana-5404	39	4	a	a	DET
cana-5404	39	5	non	non	ADJ
cana-5404	39	6	-	-	ADJ
cana-5404	39	7	empty	empty	ADJ
cana-5404	39	8	subset	subset	NOUN
cana-5404	39	9	of	of	ADP
cana-5404	39	10	ℬ.	ℬ.	PROPN
cana-5404	39	11	we	we	PRON
cana-5404	39	12	define	define	VERB
cana-5404	39	13	a	a	DET
cana-5404	39	14	function	function	NOUN
cana-5404	39	15	γ:ℬ	γ:ℬ	PROPN
cana-5404	39	16	→	→	SYM
cana-5404	39	17	ℬ	ℬ	NOUN
cana-5404	39	18	as	as	ADP
cana-5404	39	19	commutativity	commutativity	NOUN
cana-5404	39	20	preserving	preserve	VERB
cana-5404	39	21	on	on	ADP
cana-5404	39	22	a	a	DET
cana-5404	39	23	subset	subset	ADJ
cana-5404	39	24	𝑆	𝑆	PROPN
cana-5404	39	25	of	of	ADP
cana-5404	39	26	ℬ	ℬ	PROPN
cana-5404	39	27	when	when	SCONJ
cana-5404	39	28	the	the	DET
cana-5404	39	29	following	follow	VERB
cana-5404	39	30	condition	condition	NOUN
cana-5404	39	31	holds	hold	VERB
cana-5404	39	32	:	:	PUNCT
cana-5404	39	33	for	for	ADP
cana-5404	39	34	any	any	DET
cana-5404	39	35	𝜈	𝜈	NOUN
cana-5404	39	36	,	,	PUNCT
cana-5404	39	37	𝜔	𝜔	PROPN
cana-5404	39	38	∈	∈	PROPN
cana-5404	39	39	𝑆	𝑆	PROPN
cana-5404	39	40	,	,	PUNCT
cana-5404	39	41	if	if	SCONJ
cana-5404	39	42	[	[	X
cana-5404	39	43	𝜈	𝜈	X
cana-5404	39	44	,	,	PUNCT
cana-5404	39	45	𝜔	𝜔	X
cana-5404	39	46	]	]	X
cana-5404	39	47	=	=	SYM
cana-5404	39	48	0	0	NUM
cana-5404	39	49	,	,	PUNCT
cana-5404	39	50	then	then	ADV
cana-5404	39	51	[	[	X
cana-5404	39	52	γ(𝜈	γ(𝜈	NOUN
cana-5404	39	53	)	)	PUNCT
cana-5404	39	54	,	,	PUNCT
cana-5404	39	55	γ(𝜔	γ(𝜔	NOUN
cana-5404	39	56	)	)	PUNCT
cana-5404	39	57	]	]	PUNCT
cana-5404	40	1	=	=	PUNCT
cana-5404	40	2	0	0	X
cana-5404	40	3	.	.	PUNCT
cana-5404	41	1	furthermore	furthermore	ADV
cana-5404	41	2	,	,	PUNCT
cana-5404	41	3	the	the	DET
cana-5404	41	4	function	function	NOUN
cana-5404	41	5	γ	γ	PROPN
cana-5404	41	6	is	be	AUX
cana-5404	41	7	termed	term	VERB
cana-5404	41	8	strong	strong	ADJ
cana-5404	41	9	commutativity	commutativity	NOUN
cana-5404	41	10	preserving	preserve	VERB
cana-5404	41	11	(	(	PUNCT
cana-5404	41	12	abbreviated	abbreviate	VERB
cana-5404	41	13	as	as	ADP
cana-5404	41	14	scp	scp	NOUN
cana-5404	41	15	)	)	PUNCT
cana-5404	41	16	on	on	ADP
cana-5404	41	17	𝑆	𝑆	PROPN
cana-5404	41	18	if	if	SCONJ
cana-5404	41	19	∀	∀	NOUN
cana-5404	41	20	elements	element	VERB
cana-5404	41	21	𝜈,𝜔	𝜈,𝜔	PROPN
cana-5404	41	22	∈	∈	PROPN
cana-5404	41	23	𝑆	𝑆	PROPN
cana-5404	41	24	,	,	PUNCT
cana-5404	41	25	the	the	DET
cana-5404	41	26	equality	equality	NOUN
cana-5404	41	27	[	[	X
cana-5404	41	28	𝜈	𝜈	X
cana-5404	41	29	,	,	PUNCT
cana-5404	41	30	𝜔	𝜔	X
cana-5404	41	31	]	]	PUNCT
cana-5404	41	32	=	=	PUNCT
cana-5404	42	1	[	[	X
cana-5404	42	2	γ(𝜈	γ(𝜈	NUM
cana-5404	42	3	)	)	PUNCT
cana-5404	42	4	,	,	PUNCT
cana-5404	42	5	γ(𝜔	γ(𝜔	PROPN
cana-5404	42	6	)	)	PUNCT
cana-5404	42	7	]	]	PUNCT
cana-5404	42	8	is	be	AUX
cana-5404	42	9	satisfied	satisfied	ADJ
cana-5404	42	10	.	.	PUNCT
cana-5404	43	1	an	an	DET
cana-5404	43	2	expanding	expand	VERB
cana-5404	43	3	body	body	NOUN
cana-5404	43	4	of	of	ADP
cana-5404	43	5	research	research	NOUN
cana-5404	43	6	exists	exist	VERB
cana-5404	43	7	concerning	concern	VERB
cana-5404	43	8	strong	strong	ADJ
cana-5404	43	9	commutativity	commutativity	NOUN
cana-5404	43	10	preserving	preserve	VERB
cana-5404	43	11	(	(	PUNCT
cana-5404	43	12	scp	scp	NOUN
cana-5404	43	13	)	)	PUNCT
cana-5404	43	14	functions	function	NOUN
cana-5404	43	15	and	and	CCONJ
cana-5404	43	16	derivations	derivation	NOUN
cana-5404	43	17	(	(	PUNCT
cana-5404	43	18	see	see	VERB
cana-5404	43	19	works	work	VERB
cana-5404	43	20	by	by	ADP
cana-5404	43	21	[	[	X
cana-5404	43	22	12	12	NUM
cana-5404	43	23	]	]	PUNCT
cana-5404	43	24	,	,	PUNCT
cana-5404	43	25	[	[	X
cana-5404	43	26	13	13	NUM
cana-5404	43	27	]	]	PUNCT
cana-5404	43	28	,	,	PUNCT
cana-5404	43	29	[	[	X
cana-5404	43	30	14	14	NUM
cana-5404	43	31	]	]	X
cana-5404	43	32	,	,	PUNCT
cana-5404	43	33	among	among	ADP
cana-5404	43	34	others	other	NOUN
cana-5404	43	35	)	)	PUNCT
cana-5404	43	36	in	in	ADP
cana-5404	43	37	[	[	X
cana-5404	43	38	15	15	NUM
cana-5404	43	39	]	]	PUNCT
cana-5404	43	40	,	,	PUNCT
cana-5404	43	41	ali	ali	PROPN
cana-5404	43	42	demonstrated	demonstrate	VERB
cana-5404	43	43	that	that	SCONJ
cana-5404	43	44	when	when	SCONJ
cana-5404	43	45	ℬ	ℬ	PRON
cana-5404	43	46	represents	represent	VERB
cana-5404	43	47	a	a	DET
cana-5404	43	48	semiprime	semiprime	NOUN
cana-5404	43	49	ring	ring	NOUN
cana-5404	43	50	and	and	CCONJ
cana-5404	43	51	𝑓	𝑓	PRON
cana-5404	43	52	constitutes	constitute	VERB
cana-5404	43	53	an	an	DET
cana-5404	43	54	endomorphism	endomorphism	NOUN
cana-5404	43	55	that	that	SCONJ
cana-5404	43	56	functions	function	NOUN
cana-5404	43	57	as	as	ADP
cana-5404	43	58	a	a	DET
cana-5404	43	59	scp	scp	NOUN
cana-5404	43	60	map	map	NOUN
cana-5404	43	61	on	on	ADP
cana-5404	43	62	a	a	DET
cana-5404	43	63	nonzero	nonzero	NOUN
cana-5404	43	64	ideal	ideal	ADJ
cana-5404	43	65	𝑈	𝑈	PROPN
cana-5404	43	66	of	of	ADP
cana-5404	43	67	ℬ	ℬ	PROPN
cana-5404	43	68	,	,	PUNCT
cana-5404	43	69	then	then	ADV
cana-5404	43	70	𝑓	𝑓	PRON
cana-5404	43	71	necessarily	necessarily	ADV
cana-5404	43	72	operates	operate	VERB
cana-5404	43	73	as	as	ADP
cana-5404	43	74	a	a	DET
cana-5404	43	75	commuting	commuting	NOUN
cana-5404	43	76	map	map	NOUN
cana-5404	43	77	on	on	ADP
cana-5404	43	78	𝑈.	𝑈.	PROPN
cana-5404	43	79	additionally	additionally	ADV
cana-5404	43	80	,	,	PUNCT
cana-5404	43	81	in	in	ADP
cana-5404	43	82	[	[	X
cana-5404	43	83	16	16	NUM
cana-5404	43	84	]	]	PUNCT
cana-5404	43	85	,	,	PUNCT
cana-5404	43	86	samman	samman	PROPN
cana-5404	43	87	established	establish	VERB
cana-5404	43	88	that	that	SCONJ
cana-5404	43	89	an	an	DET
cana-5404	43	90	epimorphism	epimorphism	NOUN
cana-5404	43	91	of	of	ADP
cana-5404	43	92	a	a	DET
cana-5404	43	93	semiprime	semiprime	NOUN
cana-5404	43	94	ring	ring	NOUN
cana-5404	43	95	exhibits	exhibit	VERB
cana-5404	43	96	communications	communication	NOUN
cana-5404	43	97	on	on	ADP
cana-5404	43	98	applied	apply	VERB
cana-5404	43	99	nonlinear	nonlinear	ADJ
cana-5404	43	100	analysis	analysis	NOUN
cana-5404	43	101	issn	issn	NOUN
cana-5404	43	102	:	:	PUNCT
cana-5404	43	103	1074	1074	NUM
cana-5404	43	104	-	-	PUNCT
cana-5404	43	105	133x	133x	NUM
cana-5404	43	106	vol	vol	VERB
cana-5404	43	107	32	32	NUM
cana-5404	43	108	no	no	NOUN
cana-5404	43	109	.	.	PUNCT
cana-5404	44	1	10s	10	NOUN
cana-5404	44	2	(	(	PUNCT
cana-5404	44	3	2025	2025	NUM
cana-5404	44	4	)	)	PUNCT
cana-5404	44	5	2140	2140	NUM
cana-5404	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	44	7	strong	strong	ADJ
cana-5404	44	8	commutativity	commutativity	NOUN
cana-5404	44	9	preserving	preserve	VERB
cana-5404	44	10	properties	property	NOUN
cana-5404	44	11	if	if	SCONJ
cana-5404	44	12	and	and	CCONJ
cana-5404	44	13	only	only	ADV
cana-5404	44	14	if	if	SCONJ
cana-5404	44	15	it	it	PRON
cana-5404	44	16	is	be	AUX
cana-5404	44	17	centralizing	centralize	VERB
cana-5404	44	18	.	.	PUNCT
cana-5404	45	1	both	both	DET
cana-5404	45	2	derivations	derivation	NOUN
cana-5404	45	3	and	and	CCONJ
cana-5404	45	4	scp	scp	PROPN
cana-5404	45	5	maps	map	NOUN
cana-5404	45	6	have	have	AUX
cana-5404	45	7	been	be	AUX
cana-5404	45	8	thoroughly	thoroughly	ADV
cana-5404	45	9	investigated	investigate	VERB
cana-5404	45	10	by	by	ADP
cana-5404	45	11	numerous	numerous	ADJ
cana-5404	45	12	scholars	scholar	NOUN
cana-5404	45	13	within	within	ADP
cana-5404	45	14	the	the	DET
cana-5404	45	15	domains	domain	NOUN
cana-5404	45	16	of	of	ADP
cana-5404	45	17	operator	operator	NOUN
cana-5404	45	18	algebras	algebra	NOUN
cana-5404	45	19	,	,	PUNCT
cana-5404	45	20	prime	prime	ADJ
cana-5404	45	21	rings	ring	NOUN
cana-5404	45	22	,	,	PUNCT
cana-5404	45	23	and	and	CCONJ
cana-5404	45	24	semiprime	semiprime	NOUN
cana-5404	45	25	rings	ring	NOUN
cana-5404	45	26	as	as	ADV
cana-5404	45	27	well	well	ADV
cana-5404	45	28	.	.	PUNCT
cana-5404	46	1	this	this	DET
cana-5404	46	2	article	article	NOUN
cana-5404	46	3	aims	aim	VERB
cana-5404	46	4	to	to	PART
cana-5404	46	5	establish	establish	VERB
cana-5404	46	6	several	several	ADJ
cana-5404	46	7	theorems	theorem	NOUN
cana-5404	46	8	concerning	concern	VERB
cana-5404	46	9	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	46	10	(	(	PUNCT
cana-5404	46	11	𝐺	𝐺	NOUN
cana-5404	46	12	)	)	PUNCT
cana-5404	46	13	−	−	PROPN
cana-5404	46	14	𝐷	𝐷	PROPN
cana-5404	46	15	on	on	ADP
cana-5404	46	16	semiprime	semiprime	NOUN
cana-5404	46	17	rings	ring	NOUN
cana-5404	46	18	that	that	PRON
cana-5404	46	19	hold	hold	VERB
cana-5404	46	20	significant	significant	ADJ
cana-5404	46	21	independent	independent	ADJ
cana-5404	46	22	value	value	NOUN
cana-5404	46	23	.	.	PUNCT
cana-5404	47	1	specifically	specifically	ADV
cana-5404	47	2	,	,	PUNCT
cana-5404	47	3	our	our	PRON
cana-5404	47	4	investigations	investigation	NOUN
cana-5404	47	5	extend	extend	VERB
cana-5404	47	6	previously	previously	ADV
cana-5404	47	7	established	establish	VERB
cana-5404	47	8	results	result	NOUN
cana-5404	47	9	by	by	ADP
cana-5404	47	10	generalizing	generalize	VERB
cana-5404	47	11	in	in	ADP
cana-5404	47	12	two	two	NUM
cana-5404	47	13	directions	direction	NOUN
cana-5404	47	14	:	:	PUNCT
cana-5404	47	15	first	first	ADV
cana-5404	47	16	,	,	PUNCT
cana-5404	47	17	by	by	ADP
cana-5404	47	18	replacing	replace	VERB
cana-5404	47	19	a	a	DET
cana-5404	47	20	two	two	NUM
cana-5404	47	21	-	-	PUNCT
cana-5404	47	22	sided	sided	ADJ
cana-5404	47	23	ideal	ideal	NOUN
cana-5404	47	24	with	with	ADP
cana-5404	47	25	a	a	DET
cana-5404	47	26	left	left	ADJ
cana-5404	47	27	-	-	PUNCT
cana-5404	47	28	sided	side	VERB
cana-5404	47	29	ideal	ideal	ADJ
cana-5404	47	30	𝐿	𝐿	PROPN
cana-5404	47	31	(	(	PUNCT
cana-5404	47	32	abbreviated	abbreviate	VERB
cana-5404	47	33	as	as	ADP
cana-5404	47	34	l	l	NOUN
cana-5404	47	35	-	-	NOUN
cana-5404	47	36	ideal	ideal	NOUN
cana-5404	47	37	)	)	PUNCT
cana-5404	47	38	,	,	PUNCT
cana-5404	47	39	and	and	CCONJ
cana-5404	47	40	second	second	ADJ
cana-5404	47	41	,	,	PUNCT
cana-5404	47	42	by	by	ADP
cana-5404	47	43	substituting	substitute	VERB
cana-5404	47	44	a	a	DET
cana-5404	47	45	generalized	generalized	ADJ
cana-5404	47	46	derivation	derivation	NOUN
cana-5404	47	47	with	with	ADP
cana-5404	47	48	a	a	DET
cana-5404	47	49	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	47	50	(	(	PUNCT
cana-5404	47	51	𝐺	𝐺	NOUN
cana-5404	47	52	)	)	PUNCT
cana-5404	47	53	−	−	PROPN
cana-5404	47	54	𝐷	𝐷	NOUN
cana-5404	47	55	within	within	ADP
cana-5404	47	56	the	the	DET
cana-5404	47	57	framework	framework	NOUN
cana-5404	47	58	of	of	ADP
cana-5404	47	59	semiprime	semiprime	NOUN
cana-5404	47	60	rings	ring	NOUN
cana-5404	47	61	.	.	PUNCT
cana-5404	48	1	for	for	ADP
cana-5404	48	2	our	our	PRON
cana-5404	48	3	analysis	analysis	NOUN
cana-5404	48	4	,	,	PUNCT
cana-5404	48	5	we	we	PRON
cana-5404	48	6	will	will	AUX
cana-5404	48	7	examine	examine	VERB
cana-5404	48	8	γ	γ	NOUN
cana-5404	48	9	and	and	CCONJ
cana-5404	48	10	δ	δ	PROPN
cana-5404	48	11	as	as	ADP
cana-5404	48	12	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	48	13	(	(	PUNCT
cana-5404	48	14	𝐺	𝐺	NOUN
cana-5404	48	15	)	)	PUNCT
cana-5404	48	16	−	−	PROPN
cana-5404	48	17	𝐷	𝐷	NOUN
cana-5404	48	18	associated	associate	VERB
cana-5404	48	19	with	with	ADP
cana-5404	48	20	the	the	DET
cana-5404	48	21	maps	map	NOUN
cana-5404	48	22	𝛿	𝛿	NOUN
cana-5404	48	23	and	and	CCONJ
cana-5404	48	24	𝜉	𝜉	X
cana-5404	48	25	respectively	respectively	ADV
cana-5404	48	26	.	.	PUNCT
cana-5404	49	1	our	our	PRON
cana-5404	49	2	research	research	NOUN
cana-5404	49	3	will	will	AUX
cana-5404	49	4	focus	focus	VERB
cana-5404	49	5	on	on	ADP
cana-5404	49	6	investigating	investigate	VERB
cana-5404	49	7	the	the	DET
cana-5404	49	8	following	follow	VERB
cana-5404	49	9	algebraic	algebraic	ADJ
cana-5404	49	10	conditions	condition	NOUN
cana-5404	49	11	:	:	PUNCT
cana-5404	50	1	[	[	X
cana-5404	50	2	γ(𝜈	γ(𝜈	NOUN
cana-5404	50	3	)	)	PUNCT
cana-5404	50	4	,	,	PUNCT
cana-5404	50	5	γ(𝜔	γ(𝜔	NOUN
cana-5404	50	6	)	)	PUNCT
cana-5404	50	7	]	]	PUNCT
cana-5404	50	8	=	=	SYM
cana-5404	50	9	±[𝜈,𝜔	±[𝜈,𝜔	NOUN
cana-5404	50	10	]	]	PUNCT
cana-5404	50	11	(	(	PUNCT
cana-5404	50	12	scp	scp	PROPN
cana-5404	50	13	map	map	VERB
cana-5404	50	14	γ	γ	NOUN
cana-5404	50	15	)	)	PUNCT
cana-5404	50	16	,	,	PUNCT
cana-5404	51	1	[	[	X
cana-5404	51	2	γ(𝜈	γ(𝜈	NOUN
cana-5404	51	3	)	)	PUNCT
cana-5404	51	4	,	,	PUNCT
cana-5404	51	5	𝜔	𝜔	PROPN
cana-5404	51	6	]	]	X
cana-5404	51	7	=	=	SYM
cana-5404	51	8	±[𝜈	±[𝜈	NOUN
cana-5404	51	9	,	,	PUNCT
cana-5404	51	10	δ(𝜔	δ(𝜔	PROPN
cana-5404	51	11	)	)	PUNCT
cana-5404	51	12	]	]	PUNCT
cana-5404	51	13	,	,	PUNCT
cana-5404	51	14	[	[	X
cana-5404	51	15	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	51	16	)	)	PUNCT
cana-5404	51	17	,	,	PUNCT
cana-5404	51	18	γ(𝜔	γ(𝜔	NOUN
cana-5404	51	19	)	)	PUNCT
cana-5404	51	20	]	]	PUNCT
cana-5404	51	21	=	=	SYM
cana-5404	51	22	±[𝜈,𝜔	±[𝜈,𝜔	NOUN
cana-5404	51	23	]	]	PUNCT
cana-5404	51	24	and	and	CCONJ
cana-5404	51	25	[	[	X
cana-5404	51	26	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	51	27	)	)	PUNCT
cana-5404	51	28	,	,	PUNCT
cana-5404	51	29	𝜔	𝜔	PROPN
cana-5404	51	30	]	]	X
cana-5404	51	31	=	=	SYM
cana-5404	51	32	±[𝜈	±[𝜈	NOUN
cana-5404	51	33	,	,	PUNCT
cana-5404	51	34	γ(𝜔	γ(𝜔	NOUN
cana-5404	51	35	)	)	PUNCT
cana-5404	51	36	]	]	PUNCT
cana-5404	51	37	∀𝜈,𝜔	∀𝜈,𝜔	PROPN
cana-5404	51	38	∈	∈	PROPN
cana-5404	51	39	𝐿.	𝐿.	VERB
cana-5404	51	40	the	the	DET
cana-5404	51	41	following	follow	VERB
cana-5404	51	42	lemmas	lemmas	PROPN
cana-5404	51	43	will	will	AUX
cana-5404	51	44	be	be	AUX
cana-5404	51	45	essential	essential	ADJ
cana-5404	51	46	for	for	ADP
cana-5404	51	47	establishing	establish	VERB
cana-5404	51	48	our	our	PRON
cana-5404	51	49	results	result	NOUN
cana-5404	51	50	.	.	PUNCT
cana-5404	52	1	lemma	lemma	PROPN
cana-5404	52	2	1.3	1.3	NUM
cana-5404	53	1	[	[	X
cana-5404	53	2	17	17	NUM
cana-5404	53	3	]	]	PUNCT
cana-5404	53	4	.	.	PUNCT
cana-5404	54	1	let	let	VERB
cana-5404	54	2	ℬ	ℬ	PRON
cana-5404	54	3	be	be	AUX
cana-5404	54	4	a	a	DET
cana-5404	54	5	2	2	NUM
cana-5404	54	6	-	-	PUNCT
cana-5404	54	7	torsion	torsion	NOUN
cana-5404	54	8	free	free	ADJ
cana-5404	54	9	semiprime	semiprime	NOUN
cana-5404	54	10	ring	ring	NOUN
cana-5404	54	11	and	and	CCONJ
cana-5404	54	12	𝐿	𝐿	PROPN
cana-5404	54	13	a	a	DET
cana-5404	54	14	l	l	NOUN
cana-5404	54	15	-	-	NOUN
cana-5404	54	16	ideal	ideal	NOUN
cana-5404	54	17	of	of	ADP
cana-5404	54	18	ℬ.	ℬ.	PROPN
cana-5404	54	19	if	if	SCONJ
cana-5404	54	20	elements	element	NOUN
cana-5404	54	21	𝑎	𝑎	VERB
cana-5404	54	22	,	,	PUNCT
cana-5404	54	23	𝑏	𝑏	PRON
cana-5404	54	24	∈	∈	PROPN
cana-5404	54	25	ℬ	ℬ	NOUN
cana-5404	54	26	satisfy	satisfy	VERB
cana-5404	54	27	the	the	DET
cana-5404	54	28	condition	condition	NOUN
cana-5404	54	29	𝑎𝜈𝑏	𝑎𝜈𝑏	NOUN
cana-5404	54	30	+	+	CCONJ
cana-5404	54	31	𝑏𝜈𝑎	𝑏𝜈𝑎	PROPN
cana-5404	54	32	=	=	SYM
cana-5404	54	33	0	0	NUM
cana-5404	54	34	∀𝜈	∀𝜈	PROPN
cana-5404	54	35	∈	∈	PROPN
cana-5404	54	36	𝐿	𝐿	PROPN
cana-5404	54	37	,	,	PUNCT
cana-5404	54	38	then	then	ADV
cana-5404	54	39	𝑎𝜈𝑏	𝑎𝜈𝑏	PROPN
cana-5404	54	40	=	=	SYM
cana-5404	54	41	0	0	PROPN
cana-5404	54	42	and	and	CCONJ
cana-5404	54	43	𝑏𝜈𝑎	𝑏𝜈𝑎	PRON
cana-5404	55	1	=	=	NOUN
cana-5404	55	2	0	0	PUNCT
cana-5404	55	3	∀𝜈	∀𝜈	PUNCT
cana-5404	55	4	∈	∈	PROPN
cana-5404	55	5	𝐿.	𝐿.	PROPN
cana-5404	55	6	lemma	lemma	PROPN
cana-5404	55	7	1.4	1.4	NUM
cana-5404	56	1	[	[	SYM
cana-5404	56	2	11	11	NUM
cana-5404	56	3	,	,	PUNCT
cana-5404	56	4	lemma	lemma	PROPN
cana-5404	56	5	2.1	2.1	NUM
cana-5404	56	6	]	]	PUNCT
cana-5404	56	7	.	.	PUNCT
cana-5404	57	1	let	let	VERB
cana-5404	57	2	ℬ	ℬ	PRON
cana-5404	57	3	be	be	AUX
cana-5404	57	4	a	a	DET
cana-5404	57	5	semiprime	semiprime	NOUN
cana-5404	57	6	ring	ring	NOUN
cana-5404	57	7	,	,	PUNCT
cana-5404	57	8	ℒ	ℒ	DET
cana-5404	57	9	a	a	DET
cana-5404	57	10	nonzero	nonzero	ADJ
cana-5404	57	11	two	two	NUM
cana-5404	57	12	-	-	PUNCT
cana-5404	57	13	sided	sided	ADJ
cana-5404	57	14	ideal	ideal	NOUN
cana-5404	57	15	of	of	ADP
cana-5404	57	16	ℬ.	ℬ.	PROPN
cana-5404	57	17	if	if	SCONJ
cana-5404	57	18	an	an	DET
cana-5404	57	19	element	element	NOUN
cana-5404	57	20	𝑎	𝑎	PROPN
cana-5404	57	21	∈	∈	NOUN
cana-5404	57	22	ℒ	ℒ	NOUN
cana-5404	57	23	satisfies	satisfy	VERB
cana-5404	57	24	the	the	DET
cana-5404	57	25	condition	condition	NOUN
cana-5404	57	26	𝑎𝜈𝑎	𝑎𝜈𝑎	NOUN
cana-5404	57	27	=	=	SYM
cana-5404	57	28	0	0	NUM
cana-5404	57	29	∀𝜈	∀𝜈	X
cana-5404	57	30	∈	∈	PROPN
cana-5404	57	31	ℒ	ℒ	PROPN
cana-5404	57	32	,	,	PUNCT
cana-5404	57	33	then	then	ADV
cana-5404	57	34	𝑎	𝑎	X
cana-5404	57	35	=	=	SYM
cana-5404	57	36	0	0	NUM
cana-5404	57	37	.	.	PUNCT
cana-5404	58	1	2scp	2scp	ADJ
cana-5404	58	2	condition	condition	NOUN
cana-5404	58	3	on	on	ADP
cana-5404	58	4	𝑴𝒖𝒍𝒕.	𝑴𝒖𝒍𝒕.	PROPN
cana-5404	58	5	(	(	PUNCT
cana-5404	58	6	𝑮	𝑮	PROPN
cana-5404	58	7	)	)	PUNCT
cana-5404	58	8	−	−	PROPN
cana-5404	58	9	𝑫	𝑫	NOUN
cana-5404	58	10	on	on	ADP
cana-5404	58	11	l	l	ADJ
cana-5404	58	12	-	-	ADJ
cana-5404	58	13	ideal	ideal	ADJ
cana-5404	58	14	theorem	theorem	ADJ
cana-5404	58	15	2.1	2.1	NUM
cana-5404	58	16	consider	consider	VERB
cana-5404	58	17	a	a	DET
cana-5404	58	18	2	2	NUM
cana-5404	58	19	-	-	PUNCT
cana-5404	58	20	torsion	torsion	NOUN
cana-5404	58	21	free	free	ADJ
cana-5404	58	22	semiprime	semiprime	NOUN
cana-5404	58	23	ring	ring	NOUN
cana-5404	58	24	ℬ	ℬ	PROPN
cana-5404	58	25	,	,	PUNCT
cana-5404	58	26	𝐿	𝐿	PROPN
cana-5404	58	27	a	a	DET
cana-5404	58	28	nonzero	nonzero	ADJ
cana-5404	58	29	l	l	NOUN
cana-5404	58	30	-	-	NOUN
cana-5404	58	31	ideal	ideal	NOUN
cana-5404	58	32	of	of	ADP
cana-5404	58	33	ℬ	ℬ	NOUN
cana-5404	58	34	and	and	CCONJ
cana-5404	58	35	𝛤:ℬ	𝛤:ℬ	NOUN
cana-5404	58	36	→	→	SYM
cana-5404	58	37	ℬ	ℬ	NOUN
cana-5404	58	38	a	a	DET
cana-5404	58	39	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	58	40	(	(	PUNCT
cana-5404	58	41	𝐺	𝐺	NOUN
cana-5404	58	42	)	)	PUNCT
cana-5404	58	43	−	−	PROPN
cana-5404	58	44	𝐷	𝐷	NOUN
cana-5404	58	45	linked	link	VERB
cana-5404	58	46	to	to	ADP
cana-5404	58	47	the	the	DET
cana-5404	58	48	map	map	NOUN
cana-5404	58	49	𝜉.	𝜉.	ADV
cana-5404	58	50	if	if	SCONJ
cana-5404	58	51	𝛤(𝜈𝜔	𝛤(𝜈𝜔	NOUN
cana-5404	58	52	)	)	PUNCT
cana-5404	58	53	=	=	SYM
cana-5404	58	54	𝜈𝛤(𝜔	𝜈𝛤(𝜔	NOUN
cana-5404	58	55	)	)	PUNCT
cana-5404	59	1	+	+	CCONJ
cana-5404	59	2	𝜉(𝜈)𝜔	𝜉(𝜈)𝜔	X
cana-5404	59	3	𝜈,𝜔	𝜈,𝜔	NOUN
cana-5404	59	4	∈	∈	PROPN
cana-5404	59	5	𝐿	𝐿	PROPN
cana-5404	59	6	and	and	CCONJ
cana-5404	59	7	𝛤	𝛤	PROPN
cana-5404	59	8	is	be	AUX
cana-5404	59	9	scp	scp	NOUN
cana-5404	59	10	on	on	ADP
cana-5404	59	11	𝐿	𝐿	PROPN
cana-5404	59	12	,	,	PUNCT
cana-5404	59	13	then	then	ADV
cana-5404	59	14	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	59	15	)	)	PUNCT
cana-5404	59	16	,	,	PUNCT
cana-5404	59	17	𝜈	𝜈	X
cana-5404	59	18	]	]	X
cana-5404	59	19	=	=	SYM
cana-5404	59	20	0	0	NUM
cana-5404	59	21	and	and	CCONJ
cana-5404	59	22	𝐿[𝛤(𝜈	𝐿[𝛤(𝜈	NOUN
cana-5404	59	23	)	)	PUNCT
cana-5404	59	24	,	,	PUNCT
cana-5404	59	25	𝜈	𝜈	X
cana-5404	59	26	]	]	X
cana-5404	59	27	=	=	SYM
cana-5404	59	28	0	0	NUM
cana-5404	59	29	,	,	PUNCT
cana-5404	59	30	∀𝜈	∀𝜈	X
cana-5404	59	31	∈	∈	NOUN
cana-5404	59	32	𝐿.	𝐿.	ADJ
cana-5404	59	33	proof	proof	NOUN
cana-5404	59	34	.	.	PUNCT
cana-5404	60	1	given	give	VERB
cana-5404	60	2	that	that	SCONJ
cana-5404	60	3	γ	γ	PROPN
cana-5404	60	4	exhibits	exhibit	VERB
cana-5404	60	5	the	the	DET
cana-5404	60	6	scp	scp	PROPN
cana-5404	60	7	property	property	NOUN
cana-5404	60	8	on	on	ADP
cana-5404	60	9	𝐿	𝐿	PROPN
cana-5404	60	10	,	,	PUNCT
cana-5404	60	11	it	it	PRON
cana-5404	60	12	follows	follow	VERB
cana-5404	60	13	that	that	SCONJ
cana-5404	60	14	[	[	X
cana-5404	60	15	γ(𝜈	γ(𝜈	NOUN
cana-5404	60	16	)	)	PUNCT
cana-5404	60	17	,	,	PUNCT
cana-5404	60	18	γ(𝜔	γ(𝜔	NOUN
cana-5404	60	19	)	)	PUNCT
cana-5404	60	20	]	]	PUNCT
cana-5404	61	1	=	=	PUNCT
cana-5404	62	1	[	[	X
cana-5404	62	2	𝜈	𝜈	X
cana-5404	62	3	,	,	PUNCT
cana-5404	62	4	𝜔	𝜔	X
cana-5404	62	5	]	]	X
cana-5404	62	6	∀	∀	X
cana-5404	62	7	𝜈	𝜈	X
cana-5404	62	8	,	,	PUNCT
cana-5404	62	9	𝜔	𝜔	PROPN
cana-5404	62	10	∈	∈	NOUN
cana-5404	62	11	𝐿.	𝐿.	VERB
cana-5404	62	12	(	(	PUNCT
cana-5404	62	13	1	1	NUM
cana-5404	62	14	)	)	PUNCT
cana-5404	62	15	by	by	ADP
cana-5404	62	16	substituting	substitute	VERB
cana-5404	62	17	𝜔	𝜔	NOUN
cana-5404	62	18	with	with	ADP
cana-5404	62	19	𝜔𝜈	𝜔𝜈	INTJ
cana-5404	62	20	in	in	ADP
cana-5404	62	21	(	(	PUNCT
cana-5404	62	22	1	1	NUM
cana-5404	62	23	)	)	PUNCT
cana-5404	62	24	,	,	PUNCT
cana-5404	62	25	we	we	PRON
cana-5404	62	26	obtain	obtain	VERB
cana-5404	62	27	[	[	X
cana-5404	62	28	γ(𝜈	γ(𝜈	NOUN
cana-5404	62	29	)	)	PUNCT
cana-5404	62	30	,	,	PUNCT
cana-5404	62	31	γ(𝜔)]𝜈	γ(𝜔)]𝜈	PROPN
cana-5404	62	32	+	+	NUM
cana-5404	62	33	γ(𝜔)[γ(𝜈	γ(𝜔)[γ(𝜈	NUM
cana-5404	62	34	)	)	PUNCT
cana-5404	62	35	,	,	PUNCT
cana-5404	62	36	𝜈	𝜈	X
cana-5404	62	37	]	]	X
cana-5404	63	1	+	+	CCONJ
cana-5404	63	2	[	[	X
cana-5404	63	3	γ(𝜈	γ(𝜈	NOUN
cana-5404	63	4	)	)	PUNCT
cana-5404	63	5	,	,	PUNCT
cana-5404	63	6	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	PROPN
cana-5404	63	7	)	)	PUNCT
cana-5404	63	8	+	+	CCONJ
cana-5404	63	9	𝜔[γ(𝜈	𝜔[γ(𝜈	NUM
cana-5404	63	10	)	)	PUNCT
cana-5404	63	11	,	,	PUNCT
cana-5404	63	12	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	63	13	)	)	PUNCT
cana-5404	63	14	]	]	PUNCT
cana-5404	64	1	=	=	PUNCT
cana-5404	65	1	[	[	X
cana-5404	65	2	𝜈	𝜈	X
cana-5404	65	3	,	,	PUNCT
cana-5404	65	4	𝜔]𝜈	𝜔]𝜈	X
cana-5404	65	5	∀	∀	X
cana-5404	65	6	𝜈	𝜈	X
cana-5404	65	7	,	,	PUNCT
cana-5404	65	8	𝜔	𝜔	PROPN
cana-5404	65	9	∈	∈	NOUN
cana-5404	65	10	𝐿.	𝐿.	VERB
cana-5404	65	11	(	(	PUNCT
cana-5404	65	12	2	2	NUM
cana-5404	65	13	)	)	PUNCT
cana-5404	65	14	multiplying	multiplying	NOUN
cana-5404	65	15	(	(	PUNCT
cana-5404	65	16	1	1	NUM
cana-5404	65	17	)	)	PUNCT
cana-5404	65	18	to	to	ADP
cana-5404	65	19	the	the	DET
cana-5404	65	20	right	right	NOUN
cana-5404	65	21	by	by	ADP
cana-5404	65	22	𝜈	𝜈	X
cana-5404	65	23	,	,	PUNCT
cana-5404	65	24	we	we	PRON
cana-5404	65	25	have	have	VERB
cana-5404	65	26	[	[	X
cana-5404	65	27	γ(𝜈	γ(𝜈	NOUN
cana-5404	65	28	)	)	PUNCT
cana-5404	65	29	,	,	PUNCT
cana-5404	65	30	γ(𝜔)]𝜈	γ(𝜔)]𝜈	PROPN
cana-5404	66	1	=	=	PUNCT
cana-5404	67	1	[	[	X
cana-5404	67	2	𝜈	𝜈	X
cana-5404	67	3	,	,	PUNCT
cana-5404	67	4	𝜔]𝜈	𝜔]𝜈	X
cana-5404	67	5	∀	∀	X
cana-5404	67	6	𝜈	𝜈	X
cana-5404	67	7	,	,	PUNCT
cana-5404	67	8	𝜔	𝜔	PROPN
cana-5404	67	9	∈	∈	NOUN
cana-5404	67	10	𝐿.	𝐿.	NOUN
cana-5404	67	11	(	(	PUNCT
cana-5404	67	12	3	3	X
cana-5404	67	13	)	)	PUNCT
cana-5404	67	14	combining	combine	VERB
cana-5404	67	15	(	(	PUNCT
cana-5404	67	16	2	2	NUM
cana-5404	67	17	)	)	PUNCT
cana-5404	67	18	and	and	CCONJ
cana-5404	67	19	(	(	PUNCT
cana-5404	67	20	3	3	NUM
cana-5404	67	21	)	)	PUNCT
cana-5404	67	22	,	,	PUNCT
cana-5404	67	23	we	we	PRON
cana-5404	67	24	obtain	obtain	VERB
cana-5404	67	25	γ(𝜔)[γ(𝜈	γ(𝜔)[γ(𝜈	NUM
cana-5404	67	26	)	)	PUNCT
cana-5404	67	27	,	,	PUNCT
cana-5404	67	28	𝜈	𝜈	X
cana-5404	67	29	]	]	X
cana-5404	68	1	+	+	CCONJ
cana-5404	68	2	[	[	X
cana-5404	68	3	γ(𝜈	γ(𝜈	NOUN
cana-5404	68	4	)	)	PUNCT
cana-5404	68	5	,	,	PUNCT
cana-5404	68	6	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	PROPN
cana-5404	68	7	)	)	PUNCT
cana-5404	68	8	+	+	CCONJ
cana-5404	68	9	𝜔[γ(𝜈	𝜔[γ(𝜈	NUM
cana-5404	68	10	)	)	PUNCT
cana-5404	68	11	,	,	PUNCT
cana-5404	68	12	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	68	13	)	)	PUNCT
cana-5404	68	14	]	]	PUNCT
cana-5404	69	1	=	=	SYM
cana-5404	69	2	0	0	NUM
cana-5404	69	3	∀	∀	X
cana-5404	69	4	𝜈	𝜈	X
cana-5404	69	5	,	,	PUNCT
cana-5404	69	6	𝜔	𝜔	PROPN
cana-5404	69	7	∈	∈	NOUN
cana-5404	69	8	𝐿.	𝐿.	NOUN
cana-5404	69	9	(	(	PUNCT
cana-5404	69	10	4	4	NUM
cana-5404	69	11	)	)	PUNCT
cana-5404	69	12	now	now	ADV
cana-5404	69	13	,	,	PUNCT
cana-5404	69	14	if	if	SCONJ
cana-5404	69	15	we	we	PRON
cana-5404	69	16	replace	replace	VERB
cana-5404	69	17	𝜔	𝜔	SYM
cana-5404	69	18	with	with	ADP
cana-5404	69	19	𝑧𝜔	𝑧𝜔	NOUN
cana-5404	69	20	in	in	ADP
cana-5404	69	21	(	(	PUNCT
cana-5404	69	22	4	4	NUM
cana-5404	69	23	)	)	PUNCT
cana-5404	69	24	and	and	CCONJ
cana-5404	69	25	apply	apply	VERB
cana-5404	69	26	(	(	PUNCT
cana-5404	69	27	4	4	NUM
cana-5404	69	28	)	)	PUNCT
cana-5404	69	29	,	,	PUNCT
cana-5404	69	30	we	we	PRON
cana-5404	69	31	obtain	obtain	VERB
cana-5404	69	32	𝜉(𝑧)𝜔[γ(𝜈	𝜉(𝑧)𝜔[γ(𝜈	PROPN
cana-5404	69	33	)	)	PUNCT
cana-5404	69	34	,	,	PUNCT
cana-5404	69	35	𝜈	𝜈	X
cana-5404	69	36	]	]	X
cana-5404	70	1	+	+	CCONJ
cana-5404	70	2	[	[	X
cana-5404	70	3	γ(𝜈	γ(𝜈	NOUN
cana-5404	70	4	)	)	PUNCT
cana-5404	70	5	,	,	PUNCT
cana-5404	70	6	𝑧]𝜔𝜉(𝜈	𝑧]𝜔𝜉(𝜈	PROPN
cana-5404	70	7	)	)	PUNCT
cana-5404	70	8	=	=	SYM
cana-5404	71	1	0	0	NUM
cana-5404	71	2	∀	∀	NUM
cana-5404	71	3	𝜈	𝜈	X
cana-5404	71	4	,	,	PUNCT
cana-5404	71	5	𝜔	𝜔	VERB
cana-5404	71	6	,	,	PUNCT
cana-5404	71	7	𝑧	𝑧	DET
cana-5404	71	8	∈	∈	PROPN
cana-5404	71	9	𝐿.	𝐿.	NOUN
cana-5404	71	10	(	(	PUNCT
cana-5404	71	11	5	5	NUM
cana-5404	71	12	)	)	PUNCT
cana-5404	71	13	take	take	VERB
cana-5404	71	14	𝑧	𝑧	NOUN
cana-5404	71	15	=	=	SYM
cana-5404	71	16	𝜈	𝜈	NOUN
cana-5404	71	17	,	,	PUNCT
cana-5404	71	18	we	we	PRON
cana-5404	71	19	have	have	VERB
cana-5404	71	20	𝜉(𝜈)𝜔[γ(𝜈	𝜉(𝜈)𝜔[γ(𝜈	NOUN
cana-5404	71	21	)	)	PUNCT
cana-5404	71	22	,	,	PUNCT
cana-5404	71	23	𝜈	𝜈	X
cana-5404	71	24	]	]	X
cana-5404	72	1	+	+	CCONJ
cana-5404	72	2	[	[	X
cana-5404	72	3	γ(𝜈	γ(𝜈	NOUN
cana-5404	72	4	)	)	PUNCT
cana-5404	72	5	,	,	PUNCT
cana-5404	72	6	𝜈]𝜔𝜉(𝜈	𝜈]𝜔𝜉(𝜈	X
cana-5404	72	7	)	)	PUNCT
cana-5404	72	8	=	=	SYM
cana-5404	72	9	0	0	NUM
cana-5404	72	10	∀𝜈	∀𝜈	NOUN
cana-5404	72	11	,	,	PUNCT
cana-5404	72	12	𝜔	𝜔	AUX
cana-5404	72	13	∈	∈	NOUN
cana-5404	72	14	𝐿.	𝐿.	VERB
cana-5404	72	15	lemma	lemma	PROPN
cana-5404	72	16	1.3	1.3	NUM
cana-5404	72	17	,	,	PUNCT
cana-5404	72	18	gives	give	VERB
cana-5404	72	19	[	[	PRON
cana-5404	72	20	γ(𝜈	γ(𝜈	NOUN
cana-5404	72	21	)	)	PUNCT
cana-5404	72	22	,	,	PUNCT
cana-5404	72	23	𝜈]𝜔𝜉(𝜈	𝜈]𝜔𝜉(𝜈	X
cana-5404	72	24	)	)	PUNCT
cana-5404	73	1	=	=	SYM
cana-5404	73	2	0	0	NUM
cana-5404	73	3	for	for	ADP
cana-5404	73	4	all	all	DET
cana-5404	73	5	𝜈	𝜈	NOUN
cana-5404	73	6	,	,	PUNCT
cana-5404	73	7	𝜔	𝜔	PROPN
cana-5404	73	8	∈	∈	NOUN
cana-5404	73	9	𝐿.	𝐿.	NOUN
cana-5404	73	10	(	(	PUNCT
cana-5404	73	11	6	6	NUM
cana-5404	73	12	)	)	PUNCT
cana-5404	73	13	since	since	SCONJ
cana-5404	73	14	γ(𝜈2	γ(𝜈2	VERB
cana-5404	73	15	)	)	PUNCT
cana-5404	73	16	=	=	SYM
cana-5404	74	1	γ(𝜈)𝜈	γ(𝜈)𝜈	PROPN
cana-5404	74	2	+	+	CCONJ
cana-5404	74	3	𝜈𝜉(𝜈	𝜈𝜉(𝜈	NOUN
cana-5404	74	4	)	)	PUNCT
cana-5404	74	5	=	=	SYM
cana-5404	74	6	𝜈γ(𝜈	𝜈γ(𝜈	NOUN
cana-5404	74	7	)	)	PUNCT
cana-5404	74	8	+	+	CCONJ
cana-5404	74	9	𝜉(𝜈)𝜈	𝜉(𝜈)𝜈	PROPN
cana-5404	74	10	∀𝜈	∀𝜈	PROPN
cana-5404	74	11	∈	∈	PROPN
cana-5404	74	12	𝐿	𝐿	PROPN
cana-5404	74	13	,	,	PUNCT
cana-5404	74	14	this	this	PRON
cana-5404	74	15	gives	give	VERB
cana-5404	74	16	[	[	PRON
cana-5404	74	17	γ(𝜈	γ(𝜈	NOUN
cana-5404	74	18	)	)	PUNCT
cana-5404	74	19	,	,	PUNCT
cana-5404	74	20	𝜈	𝜈	X
cana-5404	74	21	]	]	PUNCT
cana-5404	74	22	=	=	PUNCT
cana-5404	75	1	[	[	X
cana-5404	75	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	75	3	)	)	PUNCT
cana-5404	75	4	,	,	PUNCT
cana-5404	75	5	𝜈	𝜈	X
cana-5404	75	6	]	]	X
cana-5404	75	7	∀	∀	X
cana-5404	75	8	𝜈	𝜈	X
cana-5404	75	9	∈	∈	PROPN
cana-5404	75	10	𝐿.	𝐿.	PROPN
cana-5404	75	11	(	(	PUNCT
cana-5404	75	12	7	7	NUM
cana-5404	75	13	)	)	PUNCT
cana-5404	75	14	communications	communication	NOUN
cana-5404	75	15	on	on	ADP
cana-5404	75	16	applied	apply	VERB
cana-5404	75	17	nonlinear	nonlinear	ADJ
cana-5404	75	18	analysis	analysis	NOUN
cana-5404	75	19	issn	issn	NOUN
cana-5404	75	20	:	:	PUNCT
cana-5404	75	21	1074	1074	NUM
cana-5404	75	22	-	-	PUNCT
cana-5404	75	23	133x	133x	NUM
cana-5404	75	24	vol	vol	VERB
cana-5404	75	25	32	32	NUM
cana-5404	75	26	no	no	NOUN
cana-5404	75	27	.	.	PUNCT
cana-5404	76	1	10s	10	NOUN
cana-5404	76	2	(	(	PUNCT
cana-5404	76	3	2025	2025	NUM
cana-5404	76	4	)	)	PUNCT
cana-5404	76	5	2141	2141	NUM
cana-5404	76	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	76	7	using	use	VERB
cana-5404	76	8	(	(	PUNCT
cana-5404	76	9	7	7	NUM
cana-5404	76	10	)	)	PUNCT
cana-5404	76	11	in	in	ADP
cana-5404	76	12	(	(	PUNCT
cana-5404	76	13	6	6	X
cana-5404	76	14	)	)	PUNCT
cana-5404	76	15	we	we	PRON
cana-5404	76	16	have	have	VERB
cana-5404	76	17	[	[	X
cana-5404	76	18	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	76	19	)	)	PUNCT
cana-5404	76	20	,	,	PUNCT
cana-5404	76	21	𝜈]𝜔𝜉(𝜈	𝜈]𝜔𝜉(𝜈	X
cana-5404	76	22	)	)	PUNCT
cana-5404	76	23	=	=	PUNCT
cana-5404	77	1	0	0	NUM
cana-5404	77	2	∀𝜈	∀𝜈	PUNCT
cana-5404	77	3	∈	∈	PROPN
cana-5404	77	4	𝐿	𝐿	PROPN
cana-5404	77	5	,	,	PUNCT
cana-5404	77	6	that	that	PRON
cana-5404	77	7	is	is	ADV
cana-5404	77	8	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	77	9	)	)	PUNCT
cana-5404	77	10	,	,	PUNCT
cana-5404	77	11	𝜈]ℬ𝐿[𝜉(𝜈	𝜈]ℬ𝐿[𝜉(𝜈	NOUN
cana-5404	77	12	)	)	PUNCT
cana-5404	77	13	,	,	PUNCT
cana-5404	77	14	𝜈	𝜈	X
cana-5404	77	15	]	]	X
cana-5404	77	16	=	=	SYM
cana-5404	77	17	0	0	NUM
cana-5404	77	18	∀𝜈	∀𝜈	PUNCT
cana-5404	77	19	∈	∈	NOUN
cana-5404	77	20	𝐿.	𝐿.	VERB
cana-5404	77	21	using	use	VERB
cana-5404	77	22	the	the	DET
cana-5404	77	23	semiprimeness	semiprimeness	NOUN
cana-5404	77	24	of	of	ADP
cana-5404	77	25	ℬ	ℬ	NOUN
cana-5404	77	26	,	,	PUNCT
cana-5404	77	27	we	we	PRON
cana-5404	77	28	deduce	deduce	VERB
cana-5404	77	29	that	that	SCONJ
cana-5404	77	30	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	77	31	)	)	PUNCT
cana-5404	77	32	,	,	PUNCT
cana-5404	77	33	𝜈	𝜈	X
cana-5404	77	34	]	]	X
cana-5404	77	35	=	=	SYM
cana-5404	77	36	0	0	NUM
cana-5404	77	37	∀	∀	NOUN
cana-5404	77	38	𝜈	𝜈	X
cana-5404	77	39	∈	∈	NOUN
cana-5404	77	40	𝐿.	𝐿.	PROPN
cana-5404	77	41	substituting	substituting	NOUN
cana-5404	77	42	into	into	ADP
cana-5404	77	43	(	(	PUNCT
cana-5404	77	44	7	7	NUM
cana-5404	77	45	)	)	PUNCT
cana-5404	77	46	,	,	PUNCT
cana-5404	77	47	we	we	PRON
cana-5404	77	48	obtain	obtain	VERB
cana-5404	77	49	𝐿[γ(𝜈	𝐿[γ(𝜈	NOUN
cana-5404	77	50	)	)	PUNCT
cana-5404	77	51	,	,	PUNCT
cana-5404	77	52	𝜈	𝜈	X
cana-5404	77	53	]	]	X
cana-5404	77	54	=	=	SYM
cana-5404	77	55	0	0	NUM
cana-5404	77	56	∀𝜈	∀𝜈	X
cana-5404	77	57	∈	∈	NOUN
cana-5404	77	58	𝐿.	𝐿.	VERB
cana-5404	77	59	the	the	DET
cana-5404	77	60	case	case	NOUN
cana-5404	77	61	[	[	X
cana-5404	77	62	γ(𝜈	γ(𝜈	NOUN
cana-5404	77	63	)	)	PUNCT
cana-5404	77	64	,	,	PUNCT
cana-5404	77	65	γ(𝜔	γ(𝜔	NOUN
cana-5404	77	66	)	)	PUNCT
cana-5404	77	67	]	]	PUNCT
cana-5404	78	1	=	=	PUNCT
cana-5404	78	2	−[𝜈	−[𝜈	NOUN
cana-5404	78	3	,	,	PUNCT
cana-5404	78	4	𝜔	𝜔	PROPN
cana-5404	78	5	]	]	X
cana-5404	78	6	∀𝜈,𝜔	∀𝜈,𝜔	PROPN
cana-5404	78	7	∈	∈	PROPN
cana-5404	78	8	𝐿	𝐿	PROPN
cana-5404	78	9	is	be	AUX
cana-5404	78	10	similar	similar	ADJ
cana-5404	78	11	.	.	PUNCT
cana-5404	79	1	remark	remark	VERB
cana-5404	79	2	2.2	2.2	NUM
cana-5404	79	3	the	the	DET
cana-5404	79	4	result	result	NOUN
cana-5404	79	5	can	can	AUX
cana-5404	79	6	be	be	AUX
cana-5404	79	7	established	establish	VERB
cana-5404	79	8	in	in	ADP
cana-5404	79	9	a	a	DET
cana-5404	79	10	similar	similar	ADJ
cana-5404	79	11	manner	manner	NOUN
cana-5404	79	12	for	for	ADP
cana-5404	79	13	the	the	DET
cana-5404	79	14	case	case	NOUN
cana-5404	79	15	where	where	SCONJ
cana-5404	79	16	[	[	X
cana-5404	79	17	𝛤(𝜈	𝛤(𝜈	NOUN
cana-5404	79	18	)	)	PUNCT
cana-5404	79	19	,	,	PUNCT
cana-5404	79	20	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	79	21	)	)	PUNCT
cana-5404	79	22	]	]	PUNCT
cana-5404	80	1	=	=	PUNCT
cana-5404	80	2	−[𝜈,𝜔	−[𝜈,𝜔	X
cana-5404	80	3	]	]	PUNCT
cana-5404	80	4	holds	hold	VERB
cana-5404	80	5	∀𝜈	∀𝜈	PROPN
cana-5404	80	6	,	,	PUNCT
cana-5404	80	7	𝜔	𝜔	PROPN
cana-5404	80	8	∈	∈	NOUN
cana-5404	80	9	𝐿.	𝐿.	VERB
cana-5404	80	10	corollary	corollary	NOUN
cana-5404	80	11	2.3	2.3	NUM
cana-5404	80	12	let	let	VERB
cana-5404	80	13	ℬ	ℬ	PRON
cana-5404	80	14	be	be	AUX
cana-5404	80	15	a	a	DET
cana-5404	80	16	2	2	NUM
cana-5404	80	17	-	-	PUNCT
cana-5404	80	18	torsion	torsion	NOUN
cana-5404	80	19	free	free	ADJ
cana-5404	80	20	semiprime	semiprime	NOUN
cana-5404	80	21	ring	ring	NOUN
cana-5404	80	22	,	,	PUNCT
cana-5404	80	23	and	and	CCONJ
cana-5404	80	24	let	let	VERB
cana-5404	80	25	𝛤	𝛤	PRON
cana-5404	80	26	:	:	PUNCT
cana-5404	80	27	ℬ	ℬ	NOUN
cana-5404	80	28	→	→	SYM
cana-5404	80	29	ℬ	ℬ	NOUN
cana-5404	80	30	be	be	AUX
cana-5404	80	31	a	a	DET
cana-5404	80	32	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	80	33	(	(	PUNCT
cana-5404	80	34	𝐺	𝐺	NOUN
cana-5404	80	35	)	)	PUNCT
cana-5404	80	36	−	−	PROPN
cana-5404	80	37	𝐷	𝐷	NOUN
cana-5404	80	38	associated	associate	VERB
cana-5404	80	39	with	with	ADP
cana-5404	80	40	the	the	DET
cana-5404	80	41	map	map	NOUN
cana-5404	80	42	𝜉.	𝜉.	ADV
cana-5404	80	43	if	if	SCONJ
cana-5404	80	44	𝛤(𝜈𝜔	𝛤(𝜈𝜔	NOUN
cana-5404	80	45	)	)	PUNCT
cana-5404	80	46	=	=	SYM
cana-5404	81	1	𝜈𝛤(𝜔	𝜈𝛤(𝜔	NOUN
cana-5404	81	2	)	)	PUNCT
cana-5404	82	1	+	+	CCONJ
cana-5404	82	2	𝜉(𝜈)𝜔	𝜉(𝜈)𝜔	PROPN
cana-5404	82	3	∀𝜈,𝜔	∀𝜈,𝜔	PROPN
cana-5404	82	4	∈	∈	PROPN
cana-5404	82	5	ℬ	ℬ	PROPN
cana-5404	82	6	,	,	PUNCT
cana-5404	82	7	and	and	CCONJ
cana-5404	82	8	if	if	SCONJ
cana-5404	82	9	𝛤	𝛤	PROPN
cana-5404	82	10	is	be	AUX
cana-5404	82	11	scp	scp	NOUN
cana-5404	82	12	on	on	ADP
cana-5404	82	13	𝐿	𝐿	PROPN
cana-5404	82	14	or	or	CCONJ
cana-5404	82	15	satisfies	satisfie	NOUN
cana-5404	82	16	[	[	X
cana-5404	82	17	𝛤(𝜈	𝛤(𝜈	NOUN
cana-5404	82	18	)	)	PUNCT
cana-5404	82	19	,	,	PUNCT
cana-5404	82	20	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	82	21	)	)	PUNCT
cana-5404	82	22	]	]	PUNCT
cana-5404	83	1	=	=	PUNCT
cana-5404	83	2	−[𝜈,𝜔	−[𝜈,𝜔	X
cana-5404	83	3	]	]	X
cana-5404	83	4	∀𝜈,𝜔	∀𝜈,𝜔	PROPN
cana-5404	83	5	∈	∈	PROPN
cana-5404	83	6	ℬ	ℬ	PROPN
cana-5404	83	7	,	,	PUNCT
cana-5404	83	8	then	then	ADV
cana-5404	83	9	𝜉	𝜉	PROPN
cana-5404	83	10	and	and	CCONJ
cana-5404	83	11	𝛤	𝛤	PROPN
cana-5404	83	12	commute	commute	NOUN
cana-5404	83	13	on	on	ADP
cana-5404	83	14	ℬ.	ℬ.	PROPN
cana-5404	83	15	theorem	theorem	ADJ
cana-5404	83	16	2.4	2.4	NUM
cana-5404	83	17	let	let	VERB
cana-5404	83	18	ℬ	ℬ	PRON
cana-5404	83	19	be	be	AUX
cana-5404	83	20	a	a	DET
cana-5404	83	21	semiprime	semiprime	NOUN
cana-5404	83	22	ring	ring	NOUN
cana-5404	83	23	,	,	PUNCT
cana-5404	83	24	𝐿	𝐿	PROPN
cana-5404	83	25	a	a	DET
cana-5404	83	26	nonzero	nonzero	NOUN
cana-5404	83	27	left	leave	VERB
cana-5404	83	28	ideal	ideal	NOUN
cana-5404	83	29	of	of	ADP
cana-5404	83	30	ℬ	ℬ	NOUN
cana-5404	83	31	,	,	PUNCT
cana-5404	83	32	and	and	CCONJ
cana-5404	83	33	𝛤	𝛤	PROPN
cana-5404	83	34	,	,	PUNCT
cana-5404	83	35	𝛥	𝛥	PROPN
cana-5404	83	36	two	two	NUM
cana-5404	83	37	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	83	38	(	(	PUNCT
cana-5404	83	39	𝐺	𝐺	NOUN
cana-5404	83	40	)	)	PUNCT
cana-5404	83	41	−	−	PROPN
cana-5404	83	42	𝐷	𝐷	NOUN
cana-5404	83	43	associated	associate	VERB
cana-5404	83	44	with	with	ADP
cana-5404	83	45	the	the	DET
cana-5404	83	46	maps	map	NOUN
cana-5404	83	47	𝛿	𝛿	ADJ
cana-5404	83	48	and	and	CCONJ
cana-5404	83	49	𝜉	𝜉	NOUN
cana-5404	83	50	,	,	PUNCT
cana-5404	83	51	respectively	respectively	ADV
cana-5404	83	52	.	.	PUNCT
cana-5404	84	1	if	if	SCONJ
cana-5404	84	2	[	[	X
cana-5404	84	3	𝛤(𝜈	𝛤(𝜈	NOUN
cana-5404	84	4	)	)	PUNCT
cana-5404	84	5	,	,	PUNCT
cana-5404	84	6	𝜔	𝜔	ADP
cana-5404	84	7	]	]	PUNCT
cana-5404	84	8	=	=	PUNCT
cana-5404	85	1	[	[	X
cana-5404	85	2	𝜈	𝜈	X
cana-5404	85	3	,	,	PUNCT
cana-5404	85	4	𝛥(𝜔	𝛥(𝜔	NUM
cana-5404	85	5	)	)	PUNCT
cana-5404	85	6	]	]	PUNCT
cana-5404	85	7	or	or	CCONJ
cana-5404	85	8	[	[	X
cana-5404	85	9	𝛤(𝜈	𝛤(𝜈	X
cana-5404	85	10	)	)	PUNCT
cana-5404	85	11	,	,	PUNCT
cana-5404	85	12	𝜔	𝜔	ADP
cana-5404	85	13	]	]	X
cana-5404	85	14	=	=	SYM
cana-5404	85	15	−[𝜈	−[𝜈	PROPN
cana-5404	85	16	,	,	PUNCT
cana-5404	85	17	𝛥(𝜔	𝛥(𝜔	NOUN
cana-5404	85	18	)	)	PUNCT
cana-5404	85	19	]	]	X
cana-5404	85	20	,	,	PUNCT
cana-5404	85	21	∀𝜈	∀𝜈	PROPN
cana-5404	85	22	,	,	PUNCT
cana-5404	85	23	𝜔	𝜔	PROPN
cana-5404	85	24	∈	∈	PROPN
cana-5404	85	25	𝐿	𝐿	PROPN
cana-5404	85	26	,	,	PUNCT
cana-5404	85	27	then	then	ADV
cana-5404	85	28	𝐿[𝛿(𝜈	𝐿[𝛿(𝜈	PROPN
cana-5404	85	29	)	)	PUNCT
cana-5404	85	30	,	,	PUNCT
cana-5404	85	31	𝜈	𝜈	X
cana-5404	85	32	]	]	X
cana-5404	85	33	=	=	SYM
cana-5404	85	34	0	0	NUM
cana-5404	85	35	and	and	CCONJ
cana-5404	85	36	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	85	37	)	)	PUNCT
cana-5404	85	38	,	,	PUNCT
cana-5404	85	39	𝜈	𝜈	X
cana-5404	85	40	]	]	X
cana-5404	85	41	=	=	SYM
cana-5404	85	42	0	0	NUM
cana-5404	85	43	,	,	PUNCT
cana-5404	85	44	∀𝜈	∀𝜈	X
cana-5404	85	45	∈	∈	NOUN
cana-5404	85	46	𝐿.	𝐿.	ADJ
cana-5404	85	47	proof	proof	NOUN
cana-5404	85	48	.	.	PUNCT
cana-5404	86	1	assume	assume	VERB
cana-5404	86	2	[	[	X
cana-5404	86	3	γ(𝜈	γ(𝜈	NOUN
cana-5404	86	4	)	)	PUNCT
cana-5404	86	5	,	,	PUNCT
cana-5404	86	6	𝜔	𝜔	ADP
cana-5404	86	7	]	]	PUNCT
cana-5404	86	8	=	=	PUNCT
cana-5404	87	1	[	[	X
cana-5404	87	2	𝜈	𝜈	X
cana-5404	87	3	,	,	PUNCT
cana-5404	87	4	δ(𝜔	δ(𝜔	PROPN
cana-5404	87	5	)	)	PUNCT
cana-5404	87	6	]	]	PUNCT
cana-5404	87	7	∀	∀	PUNCT
cana-5404	87	8	𝜈	𝜈	X
cana-5404	87	9	,	,	PUNCT
cana-5404	87	10	𝜔	𝜔	PROPN
cana-5404	87	11	∈	∈	NOUN
cana-5404	87	12	𝐿.	𝐿.	NOUN
cana-5404	87	13	(	(	PUNCT
cana-5404	87	14	8)	8)	NUM
cana-5404	87	15	if	if	SCONJ
cana-5404	87	16	we	we	PRON
cana-5404	87	17	replace	replace	VERB
cana-5404	87	18	in	in	ADP
cana-5404	87	19	(	(	PUNCT
cana-5404	87	20	8)	8)	NUM
cana-5404	87	21	𝜈	𝜈	NOUN
cana-5404	87	22	by	by	ADP
cana-5404	87	23	𝜈𝜔	𝜈𝜔	NOUN
cana-5404	87	24	,	,	PUNCT
cana-5404	87	25	we	we	PRON
cana-5404	87	26	get	get	VERB
cana-5404	87	27	[	[	NOUN
cana-5404	87	28	γ(𝜈	γ(𝜈	NOUN
cana-5404	87	29	)	)	PUNCT
cana-5404	87	30	,	,	PUNCT
cana-5404	87	31	𝜔]𝜔	𝜔]𝜔	PROPN
cana-5404	88	1	+	+	X
cana-5404	89	1	[	[	X
cana-5404	89	2	𝜈,𝜔]𝛿(𝜔	𝜈,𝜔]𝛿(𝜔	ADJ
cana-5404	89	3	)	)	PUNCT
cana-5404	89	4	+	+	NUM
cana-5404	89	5	𝜈[𝛿(𝜔),𝜔	𝜈[𝛿(𝜔),𝜔	X
cana-5404	89	6	]	]	X
cana-5404	89	7	=	=	PUNCT
cana-5404	90	1	[	[	X
cana-5404	90	2	𝜈	𝜈	X
cana-5404	90	3	,	,	PUNCT
cana-5404	90	4	δ(𝜔)]𝜔	δ(𝜔)]𝜔	ADP
cana-5404	90	5	+	+	NUM
cana-5404	90	6	𝜈[𝜔	𝜈[𝜔	NUM
cana-5404	90	7	,	,	PUNCT
cana-5404	90	8	δ(𝜔	δ(𝜔	PROPN
cana-5404	90	9	)	)	PUNCT
cana-5404	90	10	]	]	PUNCT
cana-5404	90	11	∀	∀	PUNCT
cana-5404	90	12	𝜈	𝜈	X
cana-5404	90	13	,	,	PUNCT
cana-5404	90	14	𝜔	𝜔	PROPN
cana-5404	90	15	∈	∈	NOUN
cana-5404	90	16	𝐿.	𝐿.	NOUN
cana-5404	90	17	(	(	PUNCT
cana-5404	90	18	9	9	NUM
cana-5404	90	19	)	)	PUNCT
cana-5404	90	20	on	on	ADP
cana-5404	90	21	the	the	DET
cana-5404	90	22	other	other	ADJ
cana-5404	90	23	hand	hand	NOUN
cana-5404	90	24	,	,	PUNCT
cana-5404	90	25	if	if	SCONJ
cana-5404	90	26	we	we	PRON
cana-5404	90	27	multiply	multiply	VERB
cana-5404	90	28	(	(	PUNCT
cana-5404	90	29	8)	8)	NUM
cana-5404	90	30	on	on	ADP
cana-5404	90	31	the	the	DET
cana-5404	90	32	right	right	NOUN
cana-5404	90	33	by	by	ADP
cana-5404	90	34	𝜔	𝜔	ADP
cana-5404	90	35	,	,	PUNCT
cana-5404	90	36	we	we	PRON
cana-5404	90	37	obtain	obtain	VERB
cana-5404	90	38	[	[	X
cana-5404	90	39	γ(𝜈	γ(𝜈	NOUN
cana-5404	90	40	)	)	PUNCT
cana-5404	90	41	,	,	PUNCT
cana-5404	90	42	𝜔]𝜔	𝜔]𝜔	NOUN
cana-5404	90	43	=	=	PUNCT
cana-5404	91	1	[	[	X
cana-5404	91	2	𝜈	𝜈	X
cana-5404	91	3	,	,	PUNCT
cana-5404	91	4	δ(𝜔)]𝜔	δ(𝜔)]𝜔	ADJ
cana-5404	91	5	∀	∀	X
cana-5404	91	6	𝜈	𝜈	NOUN
cana-5404	91	7	,	,	PUNCT
cana-5404	91	8	𝜔	𝜔	PROPN
cana-5404	91	9	∈	∈	NOUN
cana-5404	91	10	𝐿.	𝐿.	NOUN
cana-5404	91	11	(	(	PUNCT
cana-5404	91	12	10	10	NUM
cana-5404	91	13	)	)	PUNCT
cana-5404	91	14	subtracting	subtract	VERB
cana-5404	91	15	(	(	PUNCT
cana-5404	91	16	10	10	NUM
cana-5404	91	17	)	)	PUNCT
cana-5404	91	18	from	from	ADP
cana-5404	91	19	(	(	PUNCT
cana-5404	91	20	9	9	NUM
cana-5404	91	21	)	)	PUNCT
cana-5404	91	22	,	,	PUNCT
cana-5404	91	23	we	we	PRON
cana-5404	91	24	get	get	VERB
cana-5404	91	25	[	[	X
cana-5404	91	26	𝜈	𝜈	X
cana-5404	91	27	,	,	PUNCT
cana-5404	91	28	𝜔]𝛿(𝜔	𝜔]𝛿(𝜔	ADJ
cana-5404	91	29	)	)	PUNCT
cana-5404	92	1	+	+	CCONJ
cana-5404	92	2	𝜈[𝛿(𝜔	𝜈[𝛿(𝜔	NOUN
cana-5404	92	3	)	)	PUNCT
cana-5404	92	4	,	,	PUNCT
cana-5404	92	5	𝜔	𝜔	ADP
cana-5404	92	6	]	]	X
cana-5404	92	7	=	=	SYM
cana-5404	92	8	𝜈[𝜔	𝜈[𝜔	PROPN
cana-5404	92	9	,	,	PUNCT
cana-5404	92	10	δ(𝜔	δ(𝜔	PROPN
cana-5404	92	11	)	)	PUNCT
cana-5404	92	12	]	]	PUNCT
cana-5404	92	13	,	,	PUNCT
cana-5404	92	14	∀𝜈	∀𝜈	PROPN
cana-5404	92	15	,	,	PUNCT
cana-5404	92	16	𝜔	𝜔	PROPN
cana-5404	92	17	∈	∈	NOUN
cana-5404	92	18	𝐿.	𝐿.	NOUN
cana-5404	92	19	(	(	PUNCT
cana-5404	92	20	11	11	NUM
cana-5404	92	21	)	)	PUNCT
cana-5404	92	22	now	now	ADV
cana-5404	92	23	if	if	SCONJ
cana-5404	92	24	we	we	PRON
cana-5404	92	25	substitute	substitute	VERB
cana-5404	92	26	𝜈	𝜈	X
cana-5404	92	27	by	by	ADP
cana-5404	92	28	𝑟𝜈	𝑟𝜈	PROPN
cana-5404	92	29	in	in	ADP
cana-5404	92	30	(	(	PUNCT
cana-5404	92	31	11	11	NUM
cana-5404	92	32	)	)	PUNCT
cana-5404	92	33	,	,	PUNCT
cana-5404	92	34	we	we	PRON
cana-5404	92	35	have	have	VERB
cana-5404	92	36	𝑟[𝜈	𝑟[𝜈	NUM
cana-5404	92	37	,	,	PUNCT
cana-5404	92	38	𝜔]𝛿(𝜔	𝜔]𝛿(𝜔	PROPN
cana-5404	92	39	)	)	PUNCT
cana-5404	93	1	+	+	CCONJ
cana-5404	94	1	[	[	X
cana-5404	94	2	𝑟	𝑟	NOUN
cana-5404	94	3	,	,	PUNCT
cana-5404	94	4	𝜔]𝜈𝛿(𝜔	𝜔]𝜈𝛿(𝜔	ADJ
cana-5404	94	5	)	)	PUNCT
cana-5404	94	6	+	+	NUM
cana-5404	94	7	𝑟𝜈[𝛿(𝜔),𝜔	𝑟𝜈[𝛿(𝜔),𝜔	NOUN
cana-5404	94	8	]	]	X
cana-5404	94	9	=	=	SYM
cana-5404	94	10	𝑟𝜈[𝜔	𝑟𝜈[𝜔	NOUN
cana-5404	94	11	,	,	PUNCT
cana-5404	94	12	δ(𝜔	δ(𝜔	PROPN
cana-5404	94	13	)	)	PUNCT
cana-5404	94	14	]	]	PUNCT
cana-5404	94	15	∀	∀	PUNCT
cana-5404	94	16	𝜈	𝜈	X
cana-5404	94	17	,	,	PUNCT
cana-5404	94	18	𝜔	𝜔	PROPN
cana-5404	94	19	∈	∈	PROPN
cana-5404	94	20	𝐿	𝐿	PROPN
cana-5404	94	21	,	,	PUNCT
cana-5404	94	22	𝑟	𝑟	X
cana-5404	94	23	∈	∈	PROPN
cana-5404	94	24	ℬ.	ℬ.	PROPN
cana-5404	94	25	(	(	PUNCT
cana-5404	94	26	12	12	NUM
cana-5404	94	27	)	)	PUNCT
cana-5404	94	28	if	if	SCONJ
cana-5404	94	29	we	we	PRON
cana-5404	94	30	multiply	multiply	VERB
cana-5404	94	31	(	(	PUNCT
cana-5404	94	32	11	11	NUM
cana-5404	94	33	)	)	PUNCT
cana-5404	94	34	on	on	ADP
cana-5404	94	35	the	the	DET
cana-5404	94	36	left	left	NOUN
cana-5404	94	37	by	by	ADP
cana-5404	94	38	𝑟	𝑟	NOUN
cana-5404	94	39	,	,	PUNCT
cana-5404	94	40	we	we	PRON
cana-5404	94	41	get	get	VERB
cana-5404	94	42	𝑟[𝜈	𝑟[𝜈	NOUN
cana-5404	94	43	,	,	PUNCT
cana-5404	94	44	𝜔]𝛿(𝜔	𝜔]𝛿(𝜔	PROPN
cana-5404	94	45	)	)	PUNCT
cana-5404	95	1	+	+	CCONJ
cana-5404	95	2	𝑟𝜈[𝛿(𝜔),𝜔	𝑟𝜈[𝛿(𝜔),𝜔	NOUN
cana-5404	95	3	]	]	X
cana-5404	95	4	=	=	SYM
cana-5404	95	5	𝑟𝜈[𝜔	𝑟𝜈[𝜔	NOUN
cana-5404	95	6	,	,	PUNCT
cana-5404	95	7	δ(𝜔	δ(𝜔	PROPN
cana-5404	95	8	)	)	PUNCT
cana-5404	95	9	]	]	PUNCT
cana-5404	95	10	∀	∀	PUNCT
cana-5404	95	11	𝜈	𝜈	X
cana-5404	95	12	,	,	PUNCT
cana-5404	95	13	𝜔	𝜔	PROPN
cana-5404	95	14	∈	∈	PROPN
cana-5404	95	15	𝐿	𝐿	PROPN
cana-5404	95	16	,	,	PUNCT
cana-5404	95	17	𝑟	𝑟	X
cana-5404	95	18	∈	∈	PROPN
cana-5404	95	19	ℬ.	ℬ.	PROPN
cana-5404	95	20	(	(	PUNCT
cana-5404	95	21	13	13	NUM
cana-5404	95	22	)	)	PUNCT
cana-5404	95	23	from	from	ADP
cana-5404	95	24	(	(	PUNCT
cana-5404	95	25	12	12	NUM
cana-5404	95	26	)	)	PUNCT
cana-5404	95	27	and	and	CCONJ
cana-5404	95	28	(	(	PUNCT
cana-5404	95	29	13	13	NUM
cana-5404	95	30	)	)	PUNCT
cana-5404	95	31	,	,	PUNCT
cana-5404	95	32	we	we	PRON
cana-5404	95	33	obtain	obtain	VERB
cana-5404	95	34	[	[	X
cana-5404	95	35	𝑟	𝑟	NOUN
cana-5404	95	36	,	,	PUNCT
cana-5404	95	37	𝜔]𝜈𝛿(𝜔	𝜔]𝜈𝛿(𝜔	ADJ
cana-5404	95	38	)	)	PUNCT
cana-5404	95	39	=	=	SYM
cana-5404	96	1	0	0	X
cana-5404	96	2	.	.	PUNCT
cana-5404	97	1	if	if	SCONJ
cana-5404	97	2	we	we	PRON
cana-5404	97	3	substitute	substitute	VERB
cana-5404	97	4	𝑟	𝑟	NOUN
cana-5404	97	5	with	with	ADP
cana-5404	97	6	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	97	7	)	)	PUNCT
cana-5404	97	8	,	,	PUNCT
cana-5404	97	9	we	we	PRON
cana-5404	97	10	have	have	VERB
cana-5404	97	11	[	[	X
cana-5404	97	12	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	97	13	)	)	PUNCT
cana-5404	97	14	,	,	PUNCT
cana-5404	97	15	𝜔]𝜈𝛿(𝜔	𝜔]𝜈𝛿(𝜔	ADJ
cana-5404	97	16	)	)	PUNCT
cana-5404	98	1	=	=	SYM
cana-5404	98	2	0	0	NUM
cana-5404	98	3	∀	∀	X
cana-5404	98	4	𝜈	𝜈	X
cana-5404	98	5	,	,	PUNCT
cana-5404	98	6	𝜔	𝜔	PROPN
cana-5404	98	7	∈	∈	NOUN
cana-5404	98	8	𝐿.	𝐿.	NOUN
cana-5404	98	9	(	(	PUNCT
cana-5404	98	10	14	14	NUM
cana-5404	98	11	)	)	PUNCT
cana-5404	98	12	if	if	SCONJ
cana-5404	98	13	we	we	PRON
cana-5404	98	14	multiply	multiply	VERB
cana-5404	98	15	,	,	PUNCT
cana-5404	98	16	(	(	PUNCT
cana-5404	98	17	14	14	NUM
cana-5404	98	18	)	)	PUNCT
cana-5404	98	19	to	to	ADP
cana-5404	98	20	the	the	DET
cana-5404	98	21	right	right	NOUN
cana-5404	98	22	by	by	ADP
cana-5404	98	23	𝜔	𝜔	ADP
cana-5404	98	24	,	,	PUNCT
cana-5404	98	25	we	we	PRON
cana-5404	98	26	get	get	VERB
cana-5404	98	27	[	[	X
cana-5404	98	28	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	98	29	)	)	PUNCT
cana-5404	98	30	,	,	PUNCT
cana-5404	98	31	𝜔]𝜈𝛿(𝜔)𝜔	𝜔]𝜈𝛿(𝜔)𝜔	PROPN
cana-5404	99	1	=	=	SYM
cana-5404	99	2	0	0	X
cana-5404	99	3	.	.	PUNCT
cana-5404	100	1	replacing	replace	VERB
cana-5404	100	2	𝜈	𝜈	PRON
cana-5404	100	3	by	by	ADP
cana-5404	100	4	𝜈𝜔	𝜈𝜔	NOUN
cana-5404	100	5	in	in	ADP
cana-5404	100	6	(	(	PUNCT
cana-5404	100	7	14	14	NUM
cana-5404	100	8	)	)	PUNCT
cana-5404	100	9	,	,	PUNCT
cana-5404	100	10	to	to	PART
cana-5404	100	11	get	get	VERB
cana-5404	100	12	[	[	X
cana-5404	100	13	𝛿(𝜔),𝜔]𝜈𝜔𝛿(𝜔	𝛿(𝜔),𝜔]𝜈𝜔𝛿(𝜔	ADV
cana-5404	100	14	)	)	PUNCT
cana-5404	100	15	=	=	SYM
cana-5404	101	1	0	0	X
cana-5404	101	2	.	.	PUNCT
cana-5404	101	3	by	by	ADP
cana-5404	101	4	subtracting	subtract	VERB
cana-5404	101	5	the	the	DET
cana-5404	101	6	last	last	ADJ
cana-5404	101	7	two	two	NUM
cana-5404	101	8	identities	identity	NOUN
cana-5404	101	9	,	,	PUNCT
cana-5404	101	10	we	we	PRON
cana-5404	101	11	arrive	arrive	VERB
cana-5404	101	12	at	at	ADP
cana-5404	101	13	[	[	X
cana-5404	101	14	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	101	15	)	)	PUNCT
cana-5404	101	16	,	,	PUNCT
cana-5404	101	17	𝜔]𝜈[𝛿(𝜔),𝜔	𝜔]𝜈[𝛿(𝜔),𝜔	PROPN
cana-5404	101	18	]	]	X
cana-5404	101	19	=	=	SYM
cana-5404	102	1	0	0	X
cana-5404	102	2	.	.	PUNCT
cana-5404	103	1	since	since	SCONJ
cana-5404	103	2	𝐿	𝐿	PROPN
cana-5404	103	3	is	be	AUX
cana-5404	103	4	a	a	DET
cana-5404	103	5	left	left	ADJ
cana-5404	103	6	ideal	ideal	NOUN
cana-5404	103	7	,	,	PUNCT
cana-5404	103	8	it	it	PRON
cana-5404	103	9	follows	follow	VERB
cana-5404	103	10	that	that	SCONJ
cana-5404	103	11	𝐿[𝛿(𝜔	𝐿[𝛿(𝜔	ADJ
cana-5404	103	12	)	)	PUNCT
cana-5404	103	13	,	,	PUNCT
cana-5404	103	14	𝜔]ℬ𝐿[𝛿(𝜔),𝜔	𝜔]ℬ𝐿[𝛿(𝜔),𝜔	X
cana-5404	103	15	]	]	X
cana-5404	103	16	=	=	SYM
cana-5404	104	1	0	0	NUM
cana-5404	104	2	∀	∀	NOUN
cana-5404	104	3	𝜔	𝜔	PRON
cana-5404	104	4	∈	∈	NOUN
cana-5404	104	5	𝐿.	𝐿.	NOUN
cana-5404	104	6	using	use	VERB
cana-5404	104	7	the	the	DET
cana-5404	104	8	semiprimeness	semiprimeness	NOUN
cana-5404	104	9	of	of	ADP
cana-5404	104	10	ℬ	ℬ	PROPN
cana-5404	104	11	,	,	PUNCT
cana-5404	104	12	we	we	PRON
cana-5404	104	13	conclude	conclude	VERB
cana-5404	104	14	that	that	PRON
cana-5404	104	15	𝐿[𝛿(𝜔),𝜔	𝐿[𝛿(𝜔),𝜔	PROPN
cana-5404	104	16	]	]	X
cana-5404	104	17	=	=	SYM
cana-5404	104	18	0	0	NUM
cana-5404	104	19	,	,	PUNCT
cana-5404	104	20	∀𝜔	∀𝜔	PROPN
cana-5404	104	21	∈	∈	PROPN
cana-5404	104	22	𝐿.	𝐿.	VERB
cana-5404	104	23	on	on	ADP
cana-5404	104	24	the	the	DET
cana-5404	104	25	other	other	ADJ
cana-5404	104	26	hand	hand	NOUN
cana-5404	104	27	,	,	PUNCT
cana-5404	104	28	if	if	SCONJ
cana-5404	104	29	we	we	PRON
cana-5404	104	30	replace	replace	VERB
cana-5404	104	31	𝜔	𝜔	VERB
cana-5404	104	32	with	with	ADP
cana-5404	104	33	𝜔𝜈	𝜔𝜈	INTJ
cana-5404	104	34	in	in	ADP
cana-5404	104	35	(	(	PUNCT
cana-5404	104	36	8)	8)	NUM
cana-5404	104	37	,	,	PUNCT
cana-5404	104	38	we	we	PRON
cana-5404	104	39	obtain	obtain	VERB
cana-5404	104	40	[	[	X
cana-5404	104	41	γ(𝜈	γ(𝜈	NOUN
cana-5404	104	42	)	)	PUNCT
cana-5404	104	43	,	,	PUNCT
cana-5404	104	44	𝜔]𝜈	𝜔]𝜈	X
cana-5404	104	45	+	+	CCONJ
cana-5404	104	46	𝜔[γ(𝜈	𝜔[γ(𝜈	NUM
cana-5404	104	47	)	)	PUNCT
cana-5404	104	48	,	,	PUNCT
cana-5404	104	49	𝜈	𝜈	X
cana-5404	104	50	]	]	X
cana-5404	104	51	=	=	PUNCT
cana-5404	105	1	[	[	X
cana-5404	105	2	𝜈	𝜈	X
cana-5404	105	3	,	,	PUNCT
cana-5404	105	4	δ(𝜔)]𝜈	δ(𝜔)]𝜈	X
cana-5404	105	5	+	+	SYM
cana-5404	105	6	𝜔[𝜈	𝜔[𝜈	NOUN
cana-5404	105	7	,	,	PUNCT
cana-5404	105	8	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	105	9	)	)	PUNCT
cana-5404	105	10	]	]	PUNCT
cana-5404	106	1	+	+	CCONJ
cana-5404	106	2	[	[	X
cana-5404	106	3	𝜈	𝜈	X
cana-5404	106	4	,	,	PUNCT
cana-5404	106	5	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	NOUN
cana-5404	106	6	)	)	PUNCT
cana-5404	106	7	∀	∀	PUNCT
cana-5404	106	8	𝜈	𝜈	X
cana-5404	106	9	,	,	PUNCT
cana-5404	106	10	𝜔	𝜔	PROPN
cana-5404	106	11	∈	∈	NOUN
cana-5404	106	12	𝐿.	𝐿.	NOUN
cana-5404	106	13	(	(	PUNCT
cana-5404	106	14	15	15	NUM
cana-5404	106	15	)	)	PUNCT
cana-5404	106	16	using	use	VERB
cana-5404	106	17	(	(	PUNCT
cana-5404	106	18	8)	8)	NUM
cana-5404	106	19	in	in	ADP
cana-5404	106	20	(	(	PUNCT
cana-5404	106	21	15	15	NUM
cana-5404	106	22	)	)	PUNCT
cana-5404	106	23	we	we	PRON
cana-5404	106	24	get	get	VERB
cana-5404	106	25	𝜔[γ(𝜈	𝜔[γ(𝜈	NUM
cana-5404	106	26	)	)	PUNCT
cana-5404	106	27	,	,	PUNCT
cana-5404	106	28	𝜈	𝜈	X
cana-5404	106	29	]	]	X
cana-5404	106	30	=	=	SYM
cana-5404	106	31	𝜔[𝜈	𝜔[𝜈	NOUN
cana-5404	106	32	,	,	PUNCT
cana-5404	106	33	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	106	34	)	)	PUNCT
cana-5404	106	35	]	]	PUNCT
cana-5404	107	1	+	+	CCONJ
cana-5404	107	2	[	[	X
cana-5404	107	3	𝜈	𝜈	X
cana-5404	107	4	,	,	PUNCT
cana-5404	107	5	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	NOUN
cana-5404	107	6	)	)	PUNCT
cana-5404	107	7	∀	∀	PUNCT
cana-5404	107	8	𝜈	𝜈	X
cana-5404	107	9	,	,	PUNCT
cana-5404	107	10	𝜔	𝜔	PROPN
cana-5404	107	11	∈	∈	NOUN
cana-5404	107	12	𝐿.	𝐿.	NOUN
cana-5404	107	13	(	(	PUNCT
cana-5404	107	14	16	16	NUM
cana-5404	107	15	)	)	PUNCT
cana-5404	107	16	communications	communication	NOUN
cana-5404	107	17	on	on	ADP
cana-5404	107	18	applied	apply	VERB
cana-5404	107	19	nonlinear	nonlinear	ADJ
cana-5404	107	20	analysis	analysis	NOUN
cana-5404	107	21	issn	issn	NOUN
cana-5404	107	22	:	:	PUNCT
cana-5404	107	23	1074	1074	NUM
cana-5404	107	24	-	-	PUNCT
cana-5404	107	25	133x	133x	NUM
cana-5404	107	26	vol	vol	VERB
cana-5404	107	27	32	32	NUM
cana-5404	107	28	no	no	NOUN
cana-5404	107	29	.	.	PUNCT
cana-5404	108	1	10s	10	NOUN
cana-5404	108	2	(	(	PUNCT
cana-5404	108	3	2025	2025	NUM
cana-5404	108	4	)	)	PUNCT
cana-5404	108	5	2142	2142	NUM
cana-5404	108	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	109	1	we	we	PRON
cana-5404	109	2	can	can	AUX
cana-5404	109	3	proceed	proceed	VERB
cana-5404	109	4	in	in	ADP
cana-5404	109	5	a	a	DET
cana-5404	109	6	similar	similar	ADJ
cana-5404	109	7	manner	manner	NOUN
cana-5404	109	8	as	as	ADP
cana-5404	109	9	above	above	ADV
cana-5404	109	10	and	and	CCONJ
cana-5404	109	11	deduce	deduce	VERB
cana-5404	109	12	that	that	SCONJ
cana-5404	109	13	𝐿[𝜉(𝜔	𝐿[𝜉(𝜔	NOUN
cana-5404	109	14	)	)	PUNCT
cana-5404	109	15	,	,	PUNCT
cana-5404	109	16	𝜔	𝜔	ADP
cana-5404	109	17	]	]	X
cana-5404	109	18	=	=	SYM
cana-5404	109	19	0	0	NUM
cana-5404	109	20	,	,	PUNCT
cana-5404	109	21	∀𝜔	∀𝜔	PROPN
cana-5404	109	22	∈	∈	PROPN
cana-5404	109	23	𝐿.	𝐿.	VERB
cana-5404	109	24	the	the	DET
cana-5404	109	25	case	case	NOUN
cana-5404	109	26	where	where	SCONJ
cana-5404	109	27	[	[	X
cana-5404	109	28	γ(𝜈	γ(𝜈	NOUN
cana-5404	109	29	)	)	PUNCT
cana-5404	109	30	,	,	PUNCT
cana-5404	109	31	𝜔	𝜔	ADP
cana-5404	109	32	]	]	X
cana-5404	109	33	=	=	SYM
cana-5404	109	34	−[𝜈	−[𝜈	PROPN
cana-5404	109	35	,	,	PUNCT
cana-5404	109	36	δ(𝜔	δ(𝜔	PROPN
cana-5404	109	37	)	)	PUNCT
cana-5404	109	38	]	]	PUNCT
cana-5404	109	39	,	,	PUNCT
cana-5404	109	40	∀𝜈	∀𝜈	PROPN
cana-5404	109	41	,	,	PUNCT
cana-5404	109	42	𝜔	𝜔	AUX
cana-5404	109	43	∈	∈	NOUN
cana-5404	109	44	𝐿	𝐿	PROPN
cana-5404	109	45	follows	follow	VERB
cana-5404	109	46	analogously	analogously	ADV
cana-5404	109	47	.	.	PUNCT
cana-5404	110	1	corollary	corollary	ADJ
cana-5404	110	2	2.5	2.5	NUM
cana-5404	110	3	let	let	VERB
cana-5404	110	4	ℬ	ℬ	PRON
cana-5404	110	5	be	be	AUX
cana-5404	110	6	a	a	DET
cana-5404	110	7	semiprime	semiprime	NOUN
cana-5404	110	8	ring	ring	NOUN
cana-5404	110	9	,	,	PUNCT
cana-5404	110	10	ℒ	ℒ	DET
cana-5404	110	11	a	a	DET
cana-5404	110	12	nonzero	nonzero	ADJ
cana-5404	110	13	two	two	NUM
cana-5404	110	14	-	-	PUNCT
cana-5404	110	15	sided	sided	ADJ
cana-5404	110	16	ideal	ideal	NOUN
cana-5404	110	17	,	,	PUNCT
cana-5404	110	18	and	and	CCONJ
cana-5404	110	19	𝛤	𝛤	PROPN
cana-5404	110	20	,	,	PUNCT
cana-5404	110	21	𝛥	𝛥	PROPN
cana-5404	110	22	two	two	NUM
cana-5404	110	23	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	110	24	(	(	PUNCT
cana-5404	110	25	𝐺	𝐺	NOUN
cana-5404	110	26	)	)	PUNCT
cana-5404	110	27	−	−	PROPN
cana-5404	110	28	𝐷	𝐷	NOUN
cana-5404	110	29	associated	associate	VERB
cana-5404	110	30	with	with	ADP
cana-5404	110	31	the	the	DET
cana-5404	110	32	maps	map	NOUN
cana-5404	110	33	𝛿	𝛿	ADJ
cana-5404	110	34	and	and	CCONJ
cana-5404	110	35	𝜉	𝜉	NOUN
cana-5404	110	36	,	,	PUNCT
cana-5404	110	37	respectively	respectively	ADV
cana-5404	110	38	.	.	PUNCT
cana-5404	111	1	if	if	SCONJ
cana-5404	111	2	[	[	X
cana-5404	111	3	𝛤(𝜈	𝛤(𝜈	NOUN
cana-5404	111	4	)	)	PUNCT
cana-5404	111	5	,	,	PUNCT
cana-5404	111	6	𝜔	𝜔	ADP
cana-5404	111	7	]	]	PUNCT
cana-5404	111	8	=	=	PUNCT
cana-5404	112	1	[	[	X
cana-5404	112	2	𝜈	𝜈	X
cana-5404	112	3	,	,	PUNCT
cana-5404	112	4	𝛥(𝜔	𝛥(𝜔	NUM
cana-5404	112	5	)	)	PUNCT
cana-5404	112	6	]	]	PUNCT
cana-5404	112	7	or	or	CCONJ
cana-5404	112	8	[	[	X
cana-5404	112	9	𝛤(𝜈	𝛤(𝜈	X
cana-5404	112	10	)	)	PUNCT
cana-5404	112	11	,	,	PUNCT
cana-5404	112	12	𝜔	𝜔	ADP
cana-5404	112	13	]	]	X
cana-5404	112	14	=	=	SYM
cana-5404	112	15	−[𝜈	−[𝜈	PROPN
cana-5404	112	16	,	,	PUNCT
cana-5404	112	17	𝛥(𝜔)]∀𝜈	𝛥(𝜔)]∀𝜈	NUM
cana-5404	112	18	,	,	PUNCT
cana-5404	112	19	𝜔	𝜔	PROPN
cana-5404	112	20	∈	∈	NOUN
cana-5404	112	21	ℒ	ℒ	NOUN
cana-5404	112	22	,	,	PUNCT
cana-5404	112	23	then	then	ADV
cana-5404	112	24	𝛿	𝛿	ADJ
cana-5404	112	25	and	and	CCONJ
cana-5404	112	26	𝜉	𝜉	ADP
cana-5404	112	27	commute	commute	NOUN
cana-5404	112	28	on	on	ADP
cana-5404	112	29	ℒ.	ℒ.	NOUN
cana-5404	112	30	proof	proof	NOUN
cana-5404	112	31	.	.	PUNCT
cana-5404	113	1	by	by	ADP
cana-5404	113	2	theorem	theorem	NOUN
cana-5404	113	3	2.4	2.4	NUM
cana-5404	113	4	,	,	PUNCT
cana-5404	113	5	we	we	PRON
cana-5404	113	6	have	have	VERB
cana-5404	113	7	ℒ[𝛿(𝜔),𝜔	ℒ[𝛿(𝜔),𝜔	ADV
cana-5404	113	8	]	]	X
cana-5404	113	9	=	=	SYM
cana-5404	113	10	0∀𝜔	0∀𝜔	NOUN
cana-5404	113	11	∈	∈	PROPN
cana-5404	113	12	ℒ.	ℒ.	PROPN
cana-5404	113	13	multiplying	multiplying	NOUN
cana-5404	113	14	on	on	ADP
cana-5404	113	15	the	the	DET
cana-5404	113	16	left	left	NOUN
cana-5404	113	17	by	by	ADP
cana-5404	113	18	[	[	X
cana-5404	113	19	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	113	20	)	)	PUNCT
cana-5404	113	21	,	,	PUNCT
cana-5404	113	22	𝜔	𝜔	ADP
cana-5404	113	23	]	]	PUNCT
cana-5404	113	24	,	,	PUNCT
cana-5404	113	25	we	we	PRON
cana-5404	113	26	obtain	obtain	VERB
cana-5404	113	27	[	[	X
cana-5404	113	28	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	113	29	)	)	PUNCT
cana-5404	113	30	,	,	PUNCT
cana-5404	113	31	𝜔]ℒ[𝛿(𝜔),𝜔	𝜔]ℒ[𝛿(𝜔),𝜔	X
cana-5404	113	32	]	]	PUNCT
cana-5404	114	1	=	=	SYM
cana-5404	114	2	0	0	X
cana-5404	114	3	.	.	PUNCT
cana-5404	115	1	by	by	ADP
cana-5404	115	2	lemma	lemma	PROPN
cana-5404	115	3	1.4	1.4	NUM
cana-5404	115	4	,	,	PUNCT
cana-5404	115	5	it	it	PRON
cana-5404	115	6	follows	follow	VERB
cana-5404	115	7	that	that	SCONJ
cana-5404	116	1	[	[	X
cana-5404	116	2	𝛿(𝜔	𝛿(𝜔	NOUN
cana-5404	116	3	)	)	PUNCT
cana-5404	116	4	,	,	PUNCT
cana-5404	116	5	𝜔	𝜔	PROPN
cana-5404	116	6	]	]	PUNCT
cana-5404	116	7	=	=	SYM
cana-5404	116	8	0∀𝜔	0∀𝜔	NOUN
cana-5404	116	9	∈	∈	PROPN
cana-5404	116	10	ℒ	ℒ	PROPN
cana-5404	116	11	,	,	PUNCT
cana-5404	116	12	which	which	PRON
cana-5404	116	13	implies	imply	VERB
cana-5404	116	14	that	that	SCONJ
cana-5404	116	15	𝛿	𝛿	ADJ
cana-5404	116	16	commutes	commute	NOUN
cana-5404	116	17	on	on	ADP
cana-5404	116	18	ℒ.	ℒ.	PROPN
cana-5404	116	19	similarly	similarly	ADV
cana-5404	116	20	,	,	PUNCT
cana-5404	116	21	we	we	PRON
cana-5404	116	22	can	can	AUX
cana-5404	116	23	show	show	VERB
cana-5404	116	24	that	that	SCONJ
cana-5404	116	25	𝜉	𝜉	NOUN
cana-5404	116	26	also	also	ADV
cana-5404	116	27	commutes	commute	VERB
cana-5404	116	28	on	on	ADP
cana-5404	116	29	ℒ.	ℒ.	PROPN
cana-5404	116	30	theorem	theorem	NOUN
cana-5404	116	31	2.6	2.6	NUM
cana-5404	116	32	let	let	VERB
cana-5404	116	33	ℬ	ℬ	PRON
cana-5404	116	34	represent	represent	VERB
cana-5404	116	35	a	a	DET
cana-5404	116	36	semiprime	semiprime	NOUN
cana-5404	116	37	ring	ring	NOUN
cana-5404	116	38	that	that	PRON
cana-5404	116	39	is	be	AUX
cana-5404	116	40	free	free	ADJ
cana-5404	116	41	of	of	ADP
cana-5404	116	42	2	2	NUM
cana-5404	116	43	-	-	PUNCT
cana-5404	116	44	torsion	torsion	NOUN
cana-5404	116	45	,	,	PUNCT
cana-5404	116	46	𝐿	𝐿	PROPN
cana-5404	116	47	a	a	DET
cana-5404	116	48	nonzero	nonzero	ADJ
cana-5404	116	49	l	l	NOUN
cana-5404	116	50	-	-	NOUN
cana-5404	116	51	ideal	ideal	NOUN
cana-5404	116	52	within	within	ADP
cana-5404	116	53	ℬ	ℬ	PROPN
cana-5404	116	54	,	,	PUNCT
cana-5404	116	55	and	and	CCONJ
cana-5404	116	56	𝛤	𝛤	PROPN
cana-5404	116	57	:	:	PUNCT
cana-5404	116	58	ℬ	ℬ	NOUN
cana-5404	116	59	→	→	SYM
cana-5404	116	60	ℬ	ℬ	DET
cana-5404	116	61	a	a	DET
cana-5404	116	62	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	116	63	(	(	PUNCT
cana-5404	116	64	𝐺	𝐺	NOUN
cana-5404	116	65	)	)	PUNCT
cana-5404	116	66	−	−	PROPN
cana-5404	116	67	𝐷	𝐷	NOUN
cana-5404	116	68	mapping	mapping	NOUN
cana-5404	116	69	linked	link	VERB
cana-5404	116	70	to	to	ADP
cana-5404	116	71	𝜉.	𝜉.	PROPN
cana-5404	116	72	suppose	suppose	VERB
cana-5404	116	73	that	that	SCONJ
cana-5404	116	74	∀𝜈	∀𝜈	PROPN
cana-5404	116	75	,	,	PUNCT
cana-5404	116	76	𝜔	𝜔	PROPN
cana-5404	116	77	∈	∈	PROPN
cana-5404	116	78	𝐿	𝐿	PROPN
cana-5404	116	79	,	,	PUNCT
cana-5404	116	80	the	the	DET
cana-5404	116	81	relation	relation	NOUN
cana-5404	116	82	𝛤(𝜈𝜔	𝛤(𝜈𝜔	PROPN
cana-5404	116	83	)	)	PUNCT
cana-5404	116	84	=	=	PUNCT
cana-5404	116	85	𝜈𝛤(𝜔	𝜈𝛤(𝜔	NOUN
cana-5404	116	86	)	)	PUNCT
cana-5404	117	1	+	+	CCONJ
cana-5404	117	2	𝜉(𝜈)𝜔	𝜉(𝜈)𝜔	PROPN
cana-5404	117	3	holds	hold	VERB
cana-5404	117	4	,	,	PUNCT
cana-5404	117	5	along	along	ADP
cana-5404	117	6	with	with	ADP
cana-5404	117	7	either	either	DET
cana-5404	117	8	[	[	X
cana-5404	117	9	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	117	10	)	)	PUNCT
cana-5404	117	11	,	,	PUNCT
cana-5404	117	12	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	117	13	)	)	PUNCT
cana-5404	117	14	]	]	PUNCT
cana-5404	118	1	=	=	PUNCT
cana-5404	119	1	[	[	X
cana-5404	119	2	𝜈	𝜈	X
cana-5404	119	3	,	,	PUNCT
cana-5404	119	4	𝜔	𝜔	X
cana-5404	119	5	]	]	PUNCT
cana-5404	119	6	or	or	CCONJ
cana-5404	119	7	[	[	X
cana-5404	119	8	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	119	9	)	)	PUNCT
cana-5404	119	10	,	,	PUNCT
cana-5404	119	11	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	119	12	)	)	PUNCT
cana-5404	119	13	]	]	PUNCT
cana-5404	120	1	=	=	PUNCT
cana-5404	120	2	−[𝜈	−[𝜈	PROPN
cana-5404	120	3	,	,	PUNCT
cana-5404	120	4	𝜔	𝜔	PROPN
cana-5404	120	5	]	]	PUNCT
cana-5404	120	6	.	.	PUNCT
cana-5404	121	1	then	then	ADV
cana-5404	121	2	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	121	3	)	)	PUNCT
cana-5404	121	4	,	,	PUNCT
cana-5404	121	5	𝜈	𝜈	X
cana-5404	121	6	]	]	X
cana-5404	121	7	=	=	SYM
cana-5404	121	8	0	0	NUM
cana-5404	121	9	and	and	CCONJ
cana-5404	121	10	𝐿[𝛤(𝜈	𝐿[𝛤(𝜈	NOUN
cana-5404	121	11	)	)	PUNCT
cana-5404	121	12	,	,	PUNCT
cana-5404	121	13	𝜈	𝜈	X
cana-5404	121	14	]	]	X
cana-5404	121	15	=	=	SYM
cana-5404	121	16	0	0	NUM
cana-5404	121	17	,	,	PUNCT
cana-5404	121	18	∀𝜈	∀𝜈	X
cana-5404	121	19	∈	∈	NOUN
cana-5404	121	20	𝐿.	𝐿.	ADJ
cana-5404	121	21	proof	proof	NOUN
cana-5404	121	22	.	.	PUNCT
cana-5404	122	1	assume	assume	VERB
cana-5404	122	2	that	that	SCONJ
cana-5404	123	1	[	[	X
cana-5404	123	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	123	3	)	)	PUNCT
cana-5404	123	4	,	,	PUNCT
cana-5404	123	5	γ(𝜔	γ(𝜔	NOUN
cana-5404	123	6	)	)	PUNCT
cana-5404	123	7	]	]	PUNCT
cana-5404	124	1	=	=	PUNCT
cana-5404	125	1	[	[	X
cana-5404	125	2	𝜈	𝜈	X
cana-5404	125	3	,	,	PUNCT
cana-5404	125	4	𝜔	𝜔	X
cana-5404	125	5	]	]	X
cana-5404	125	6	∀	∀	X
cana-5404	125	7	𝜈	𝜈	X
cana-5404	125	8	,	,	PUNCT
cana-5404	125	9	𝜔	𝜔	AUX
cana-5404	125	10	∈	∈	NOUN
cana-5404	125	11	𝐿.	𝐿.	NOUN
cana-5404	125	12	(	(	PUNCT
cana-5404	125	13	17	17	NUM
cana-5404	125	14	)	)	PUNCT
cana-5404	125	15	substituting	substitute	VERB
cana-5404	125	16	𝜔	𝜔	NOUN
cana-5404	125	17	with	with	ADP
cana-5404	125	18	𝜔𝜈	𝜔𝜈	INTJ
cana-5404	125	19	in	in	ADP
cana-5404	125	20	(	(	PUNCT
cana-5404	125	21	17	17	NUM
cana-5404	125	22	)	)	PUNCT
cana-5404	125	23	,	,	PUNCT
cana-5404	125	24	we	we	PRON
cana-5404	125	25	obtain	obtain	VERB
cana-5404	125	26	[	[	X
cana-5404	125	27	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	125	28	)	)	PUNCT
cana-5404	125	29	,	,	PUNCT
cana-5404	125	30	γ(𝜔)]𝜈	γ(𝜔)]𝜈	PROPN
cana-5404	125	31	+	+	CCONJ
cana-5404	125	32	γ(𝜔)[𝜉(𝜈	γ(𝜔)[𝜉(𝜈	PROPN
cana-5404	125	33	)	)	PUNCT
cana-5404	125	34	,	,	PUNCT
cana-5404	125	35	𝜈	𝜈	X
cana-5404	125	36	]	]	X
cana-5404	126	1	+	+	CCONJ
cana-5404	126	2	[	[	X
cana-5404	126	3	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	126	4	)	)	PUNCT
cana-5404	126	5	,	,	PUNCT
cana-5404	126	6	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	PROPN
cana-5404	126	7	)	)	PUNCT
cana-5404	126	8	=	=	PUNCT
cana-5404	127	1	[	[	X
cana-5404	127	2	𝜈	𝜈	X
cana-5404	127	3	,	,	PUNCT
cana-5404	127	4	𝜔]𝜈	𝜔]𝜈	X
cana-5404	127	5	∀	∀	X
cana-5404	127	6	𝜈	𝜈	X
cana-5404	127	7	,	,	PUNCT
cana-5404	127	8	𝜔	𝜔	PROPN
cana-5404	127	9	∈	∈	NOUN
cana-5404	127	10	𝐿.	𝐿.	NOUN
cana-5404	127	11	(	(	PUNCT
cana-5404	127	12	18	18	NUM
cana-5404	127	13	)	)	PUNCT
cana-5404	127	14	multiplying	multiplying	NOUN
cana-5404	127	15	(	(	PUNCT
cana-5404	127	16	17	17	NUM
cana-5404	127	17	)	)	PUNCT
cana-5404	127	18	to	to	ADP
cana-5404	127	19	the	the	DET
cana-5404	127	20	right	right	NOUN
cana-5404	127	21	by	by	ADP
cana-5404	127	22	𝜈	𝜈	X
cana-5404	127	23	,	,	PUNCT
cana-5404	127	24	we	we	PRON
cana-5404	127	25	obtain	obtain	VERB
cana-5404	127	26	[	[	X
cana-5404	127	27	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	127	28	)	)	PUNCT
cana-5404	127	29	,	,	PUNCT
cana-5404	127	30	γ(𝜔)]𝜈	γ(𝜔)]𝜈	PROPN
cana-5404	128	1	=	=	PUNCT
cana-5404	129	1	[	[	X
cana-5404	129	2	𝜈	𝜈	X
cana-5404	129	3	,	,	PUNCT
cana-5404	129	4	𝜔]𝜈	𝜔]𝜈	X
cana-5404	129	5	∀	∀	X
cana-5404	129	6	𝜈	𝜈	X
cana-5404	129	7	,	,	PUNCT
cana-5404	129	8	𝜔	𝜔	PROPN
cana-5404	129	9	∈	∈	NOUN
cana-5404	129	10	𝐿.	𝐿.	NOUN
cana-5404	129	11	(	(	PUNCT
cana-5404	129	12	19	19	NUM
cana-5404	129	13	)	)	PUNCT
cana-5404	129	14	combining	combine	VERB
cana-5404	129	15	(	(	PUNCT
cana-5404	129	16	18	18	NUM
cana-5404	129	17	)	)	PUNCT
cana-5404	129	18	and	and	CCONJ
cana-5404	129	19	(	(	PUNCT
cana-5404	129	20	19	19	NUM
cana-5404	129	21	)	)	PUNCT
cana-5404	129	22	we	we	PRON
cana-5404	129	23	obtain	obtain	VERB
cana-5404	129	24	γ(𝜔)[𝜉(𝜈	γ(𝜔)[𝜉(𝜈	NOUN
cana-5404	129	25	)	)	PUNCT
cana-5404	129	26	,	,	PUNCT
cana-5404	129	27	𝜈	𝜈	X
cana-5404	129	28	]	]	X
cana-5404	129	29	+	+	CCONJ
cana-5404	129	30	[	[	X
cana-5404	129	31	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	129	32	)	)	PUNCT
cana-5404	129	33	,	,	PUNCT
cana-5404	129	34	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	PROPN
cana-5404	129	35	)	)	PUNCT
cana-5404	129	36	=	=	SYM
cana-5404	129	37	0	0	NUM
cana-5404	129	38	∀	∀	X
cana-5404	129	39	𝜈	𝜈	X
cana-5404	129	40	,	,	PUNCT
cana-5404	129	41	𝜔	𝜔	AUX
cana-5404	129	42	∈	∈	NOUN
cana-5404	129	43	𝐿.	𝐿.	NOUN
cana-5404	129	44	(	(	PUNCT
cana-5404	129	45	20	20	NUM
cana-5404	129	46	)	)	PUNCT
cana-5404	129	47	if	if	SCONJ
cana-5404	129	48	we	we	PRON
cana-5404	129	49	substitute	substitute	VERB
cana-5404	129	50	𝑧𝜔	𝑧𝜔	NOUN
cana-5404	129	51	for	for	ADP
cana-5404	129	52	𝜔	𝜔	PRON
cana-5404	129	53	in	in	X
cana-5404	129	54	(	(	PUNCT
cana-5404	129	55	20	20	NUM
cana-5404	129	56	)	)	PUNCT
cana-5404	129	57	,	,	PUNCT
cana-5404	129	58	the	the	DET
cana-5404	129	59	result	result	NOUN
cana-5404	129	60	is	be	AUX
cana-5404	129	61	𝑧γ(𝜔)[𝜉(𝜈	𝑧γ(𝜔)[𝜉(𝜈	PROPN
cana-5404	129	62	)	)	PUNCT
cana-5404	129	63	,	,	PUNCT
cana-5404	129	64	𝜈	𝜈	X
cana-5404	129	65	]	]	X
cana-5404	130	1	+	+	NUM
cana-5404	130	2	𝜉(𝑧)𝜔[𝜉(𝜈	𝜉(𝑧)𝜔[𝜉(𝜈	PROPN
cana-5404	130	3	)	)	PUNCT
cana-5404	130	4	,	,	PUNCT
cana-5404	130	5	𝜈	𝜈	X
cana-5404	130	6	]	]	X
cana-5404	130	7	+	+	NUM
cana-5404	130	8	𝑧[𝜉(𝜈	𝑧[𝜉(𝜈	NOUN
cana-5404	130	9	)	)	PUNCT
cana-5404	130	10	,	,	PUNCT
cana-5404	130	11	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	PROPN
cana-5404	130	12	)	)	PUNCT
cana-5404	130	13	+	+	CCONJ
cana-5404	131	1	[	[	X
cana-5404	131	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	131	3	)	)	PUNCT
cana-5404	131	4	,	,	PUNCT
cana-5404	131	5	𝑧]𝜔𝜉(𝜈	𝑧]𝜔𝜉(𝜈	PROPN
cana-5404	131	6	)	)	PUNCT
cana-5404	131	7	=	=	SYM
cana-5404	131	8	0	0	NUM
cana-5404	131	9	∀	∀	NUM
cana-5404	131	10	𝜈	𝜈	X
cana-5404	131	11	,	,	PUNCT
cana-5404	131	12	𝜔	𝜔	VERB
cana-5404	131	13	,	,	PUNCT
cana-5404	131	14	𝑧	𝑧	PRON
cana-5404	131	15	∈	∈	PROPN
cana-5404	131	16	𝐿.	𝐿.	PROPN
cana-5404	131	17	(	(	PUNCT
cana-5404	131	18	21	21	NUM
cana-5404	131	19	)	)	PUNCT
cana-5404	131	20	using	use	VERB
cana-5404	131	21	(	(	PUNCT
cana-5404	131	22	20	20	NUM
cana-5404	131	23	)	)	PUNCT
cana-5404	131	24	in	in	ADP
cana-5404	131	25	(	(	PUNCT
cana-5404	131	26	21	21	NUM
cana-5404	131	27	)	)	PUNCT
cana-5404	131	28	and	and	CCONJ
cana-5404	131	29	take	take	VERB
cana-5404	131	30	𝑧	𝑧	PRON
cana-5404	131	31	=	=	SYM
cana-5404	131	32	𝜈	𝜈	NOUN
cana-5404	131	33	,	,	PUNCT
cana-5404	131	34	we	we	PRON
cana-5404	131	35	get	get	VERB
cana-5404	131	36	𝜉(𝜈)𝜔[𝜉(𝜈	𝜉(𝜈)𝜔[𝜉(𝜈	PROPN
cana-5404	131	37	)	)	PUNCT
cana-5404	131	38	,	,	PUNCT
cana-5404	131	39	𝜈	𝜈	X
cana-5404	131	40	]	]	X
cana-5404	132	1	+	+	CCONJ
cana-5404	132	2	[	[	X
cana-5404	132	3	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	132	4	)	)	PUNCT
cana-5404	132	5	,	,	PUNCT
cana-5404	132	6	𝜈]𝜔𝜉(𝜈	𝜈]𝜔𝜉(𝜈	X
cana-5404	132	7	)	)	PUNCT
cana-5404	132	8	=	=	SYM
cana-5404	133	1	0	0	NUM
cana-5404	133	2	∀	∀	X
cana-5404	133	3	𝜈	𝜈	X
cana-5404	133	4	,	,	PUNCT
cana-5404	133	5	𝜔	𝜔	PROPN
cana-5404	133	6	∈	∈	NOUN
cana-5404	133	7	𝐿.	𝐿.	NOUN
cana-5404	133	8	(	(	PUNCT
cana-5404	133	9	22	22	NUM
cana-5404	133	10	)	)	PUNCT
cana-5404	133	11	lemma	lemma	PROPN
cana-5404	133	12	1.3	1.3	NUM
cana-5404	133	13	,	,	PUNCT
cana-5404	133	14	gives	give	VERB
cana-5404	133	15	[	[	PRON
cana-5404	133	16	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	133	17	)	)	PUNCT
cana-5404	133	18	,	,	PUNCT
cana-5404	133	19	𝜈]𝜔𝜉(𝜈	𝜈]𝜔𝜉(𝜈	X
cana-5404	133	20	)	)	PUNCT
cana-5404	133	21	=	=	SYM
cana-5404	133	22	0	0	NUM
cana-5404	133	23	,	,	PUNCT
cana-5404	133	24	∀𝜈	∀𝜈	PROPN
cana-5404	133	25	,	,	PUNCT
cana-5404	133	26	𝜔	𝜔	PROPN
cana-5404	133	27	∈	∈	PROPN
cana-5404	133	28	𝐿	𝐿	PROPN
cana-5404	133	29	,	,	PUNCT
cana-5404	133	30	this	this	PRON
cana-5404	133	31	gives	give	VERB
cana-5404	133	32	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	133	33	)	)	PUNCT
cana-5404	133	34	,	,	PUNCT
cana-5404	133	35	𝜈]ℬ𝐿[𝜉(𝜈	𝜈]ℬ𝐿[𝜉(𝜈	NOUN
cana-5404	133	36	)	)	PUNCT
cana-5404	133	37	,	,	PUNCT
cana-5404	133	38	𝜈	𝜈	X
cana-5404	133	39	]	]	X
cana-5404	133	40	=	=	SYM
cana-5404	133	41	0	0	NUM
cana-5404	133	42	,	,	PUNCT
cana-5404	133	43	∀𝜈	∀𝜈	X
cana-5404	133	44	∈	∈	PROPN
cana-5404	133	45	𝐿.	𝐿.	VERB
cana-5404	133	46	given	give	VERB
cana-5404	133	47	that	that	SCONJ
cana-5404	133	48	ℬ	ℬ	NOUN
cana-5404	133	49	is	be	AUX
cana-5404	133	50	semiprime	semiprime	NOUN
cana-5404	133	51	,	,	PUNCT
cana-5404	133	52	it	it	PRON
cana-5404	133	53	follows	follow	VERB
cana-5404	133	54	that	that	SCONJ
cana-5404	133	55	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	133	56	)	)	PUNCT
cana-5404	133	57	,	,	PUNCT
cana-5404	133	58	𝜈	𝜈	X
cana-5404	133	59	]	]	X
cana-5404	133	60	=	=	SYM
cana-5404	133	61	0	0	NUM
cana-5404	133	62	,	,	PUNCT
cana-5404	133	63	∀𝜈	∀𝜈	X
cana-5404	133	64	∈	∈	PROPN
cana-5404	133	65	𝐿.	𝐿.	VERB
cana-5404	133	66	since	since	SCONJ
cana-5404	133	67	γ(𝜈2	γ(𝜈2	NOUN
cana-5404	133	68	)	)	PUNCT
cana-5404	133	69	=	=	SYM
cana-5404	134	1	γ(𝜈)𝜈	γ(𝜈)𝜈	PROPN
cana-5404	134	2	+	+	CCONJ
cana-5404	134	3	𝜈𝜉(𝜈	𝜈𝜉(𝜈	NOUN
cana-5404	134	4	)	)	PUNCT
cana-5404	134	5	=	=	SYM
cana-5404	134	6	𝜈γ(𝜈	𝜈γ(𝜈	NOUN
cana-5404	134	7	)	)	PUNCT
cana-5404	134	8	+	+	NUM
cana-5404	134	9	𝜉(𝜈)𝜈	𝜉(𝜈)𝜈	NOUN
cana-5404	134	10	∀	∀	X
cana-5404	134	11	𝜈	𝜈	X
cana-5404	134	12	∈	∈	PROPN
cana-5404	134	13	𝐿	𝐿	PROPN
cana-5404	134	14	,	,	PUNCT
cana-5404	134	15	this	this	PRON
cana-5404	134	16	gives	give	VERB
cana-5404	134	17	[	[	PRON
cana-5404	134	18	γ(𝜈	γ(𝜈	NOUN
cana-5404	134	19	)	)	PUNCT
cana-5404	134	20	,	,	PUNCT
cana-5404	134	21	𝜈	𝜈	X
cana-5404	134	22	]	]	PUNCT
cana-5404	134	23	=	=	PUNCT
cana-5404	135	1	[	[	X
cana-5404	135	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	135	3	)	)	PUNCT
cana-5404	135	4	,	,	PUNCT
cana-5404	135	5	𝜈	𝜈	X
cana-5404	135	6	]	]	X
cana-5404	135	7	,	,	PUNCT
cana-5404	135	8	thus	thus	ADV
cana-5404	135	9	𝐿[γ(𝜈	𝐿[γ(𝜈	NOUN
cana-5404	135	10	)	)	PUNCT
cana-5404	135	11	,	,	PUNCT
cana-5404	135	12	𝜈	𝜈	X
cana-5404	135	13	]	]	X
cana-5404	135	14	=	=	SYM
cana-5404	135	15	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	PROPN
cana-5404	135	16	)	)	PUNCT
cana-5404	135	17	,	,	PUNCT
cana-5404	135	18	𝜈	𝜈	X
cana-5404	135	19	]	]	X
cana-5404	135	20	=	=	SYM
cana-5404	135	21	0	0	NUM
cana-5404	135	22	,	,	PUNCT
cana-5404	135	23	∀𝜈	∀𝜈	X
cana-5404	135	24	∈	∈	PROPN
cana-5404	135	25	𝐿.	𝐿.	VERB
cana-5404	135	26	the	the	DET
cana-5404	135	27	case	case	NOUN
cana-5404	135	28	[	[	X
cana-5404	135	29	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	135	30	)	)	PUNCT
cana-5404	135	31	,	,	PUNCT
cana-5404	135	32	γ(𝜔	γ(𝜔	NOUN
cana-5404	135	33	)	)	PUNCT
cana-5404	135	34	]	]	PUNCT
cana-5404	136	1	=	=	PUNCT
cana-5404	136	2	−[𝜈	−[𝜈	NOUN
cana-5404	136	3	,	,	PUNCT
cana-5404	136	4	𝜔	𝜔	PROPN
cana-5404	136	5	]	]	PUNCT
cana-5404	136	6	,	,	PUNCT
cana-5404	136	7	∀𝜈	∀𝜈	PROPN
cana-5404	136	8	,	,	PUNCT
cana-5404	136	9	𝜔	𝜔	PROPN
cana-5404	136	10	∈	∈	NOUN
cana-5404	136	11	𝐿	𝐿	PROPN
cana-5404	136	12	is	be	AUX
cana-5404	136	13	similar	similar	ADJ
cana-5404	136	14	.	.	PUNCT
cana-5404	137	1	corollary	corollary	ADJ
cana-5404	137	2	2.7	2.7	NUM
cana-5404	137	3	let	let	VERB
cana-5404	137	4	ℬ	ℬ	PRON
cana-5404	137	5	be	be	AUX
cana-5404	137	6	a	a	DET
cana-5404	137	7	semiprime	semiprime	NOUN
cana-5404	137	8	ring	ring	NOUN
cana-5404	137	9	that	that	PRON
cana-5404	137	10	is	be	AUX
cana-5404	137	11	free	free	ADJ
cana-5404	137	12	of	of	ADP
cana-5404	137	13	2	2	NUM
cana-5404	137	14	-	-	PUNCT
cana-5404	137	15	torsion	torsion	NOUN
cana-5404	137	16	,	,	PUNCT
cana-5404	137	17	and	and	CCONJ
cana-5404	137	18	let	let	VERB
cana-5404	137	19	𝛤	𝛤	PRON
cana-5404	137	20	:	:	PUNCT
cana-5404	137	21	ℬ	ℬ	NOUN
cana-5404	137	22	→	→	SYM
cana-5404	137	23	ℬ	ℬ	NOUN
cana-5404	137	24	be	be	AUX
cana-5404	137	25	a	a	DET
cana-5404	137	26	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	137	27	(	(	PUNCT
cana-5404	137	28	𝐺	𝐺	NOUN
cana-5404	137	29	)	)	PUNCT
cana-5404	137	30	−	−	PROPN
cana-5404	137	31	𝐷	𝐷	NOUN
cana-5404	137	32	mapping	mapping	NOUN
cana-5404	137	33	associated	associate	VERB
cana-5404	137	34	with	with	ADP
cana-5404	137	35	𝜉.	𝜉.	PROPN
cana-5404	137	36	if	if	SCONJ
cana-5404	137	37	𝛤(𝜈𝜔	𝛤(𝜈𝜔	NOUN
cana-5404	137	38	)	)	PUNCT
cana-5404	137	39	=	=	SYM
cana-5404	138	1	𝜈𝛤(𝜔	𝜈𝛤(𝜔	PROPN
cana-5404	138	2	)	)	PUNCT
cana-5404	139	1	+	+	CCONJ
cana-5404	139	2	𝜉(𝜈)𝜔	𝜉(𝜈)𝜔	ADJ
cana-5404	139	3	and	and	CCONJ
cana-5404	139	4	[	[	X
cana-5404	139	5	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	139	6	)	)	PUNCT
cana-5404	139	7	,	,	PUNCT
cana-5404	139	8	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	139	9	)	)	PUNCT
cana-5404	139	10	]	]	PUNCT
cana-5404	140	1	=	=	PUNCT
cana-5404	141	1	[	[	X
cana-5404	141	2	𝜈	𝜈	X
cana-5404	141	3	,	,	PUNCT
cana-5404	141	4	𝜔	𝜔	X
cana-5404	141	5	]	]	PUNCT
cana-5404	141	6	or	or	CCONJ
cana-5404	141	7	[	[	X
cana-5404	141	8	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	141	9	)	)	PUNCT
cana-5404	141	10	,	,	PUNCT
cana-5404	141	11	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	141	12	)	)	PUNCT
cana-5404	141	13	]	]	PUNCT
cana-5404	141	14	=	=	PUNCT
cana-5404	141	15	−[𝜈,𝜔	−[𝜈,𝜔	X
cana-5404	141	16	]	]	PUNCT
cana-5404	141	17	holds	hold	VERB
cana-5404	141	18	∀𝜈,𝜔	∀𝜈,𝜔	PROPN
cana-5404	141	19	∈	∈	PROPN
cana-5404	141	20	ℬ	ℬ	PROPN
cana-5404	141	21	,	,	PUNCT
cana-5404	141	22	then	then	ADV
cana-5404	141	23	𝜉	𝜉	PROPN
cana-5404	141	24	and	and	CCONJ
cana-5404	141	25	𝛤	𝛤	PROPN
cana-5404	141	26	commute	commute	VERB
cana-5404	141	27	on	on	ADP
cana-5404	141	28	ℬ.	ℬ.	PROPN
cana-5404	141	29	communications	communication	NOUN
cana-5404	141	30	on	on	ADP
cana-5404	141	31	applied	apply	VERB
cana-5404	141	32	nonlinear	nonlinear	ADJ
cana-5404	141	33	analysis	analysis	NOUN
cana-5404	141	34	issn	issn	NOUN
cana-5404	141	35	:	:	PUNCT
cana-5404	141	36	1074	1074	NUM
cana-5404	141	37	-	-	PUNCT
cana-5404	141	38	133x	133x	NUM
cana-5404	141	39	vol	vol	VERB
cana-5404	141	40	32	32	NUM
cana-5404	141	41	no	no	NOUN
cana-5404	141	42	.	.	PUNCT
cana-5404	142	1	10s	10	NOUN
cana-5404	142	2	(	(	PUNCT
cana-5404	142	3	2025	2025	NUM
cana-5404	142	4	)	)	PUNCT
cana-5404	142	5	2143	2143	NUM
cana-5404	142	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	142	7	theorem	theorem	VERB
cana-5404	142	8	2.8	2.8	NUM
cana-5404	142	9	let	let	VERB
cana-5404	142	10	ℬ	ℬ	PRON
cana-5404	142	11	represent	represent	VERB
cana-5404	142	12	a	a	DET
cana-5404	142	13	semiprime	semiprime	NOUN
cana-5404	142	14	ring	ring	NOUN
cana-5404	142	15	,	,	PUNCT
cana-5404	142	16	𝐿	𝐿	PROPN
cana-5404	142	17	a	a	DET
cana-5404	142	18	nonzero	nonzero	ADJ
cana-5404	142	19	l	l	NOUN
cana-5404	142	20	-	-	NOUN
cana-5404	142	21	ideal	ideal	NOUN
cana-5404	142	22	of	of	ADP
cana-5404	142	23	ℬ	ℬ	NOUN
cana-5404	142	24	,	,	PUNCT
cana-5404	142	25	and	and	CCONJ
cana-5404	142	26	𝛤	𝛤	PROPN
cana-5404	142	27	a	a	DET
cana-5404	142	28	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	142	29	(	(	PUNCT
cana-5404	142	30	𝐺	𝐺	NOUN
cana-5404	142	31	)	)	PUNCT
cana-5404	142	32	−	−	PROPN
cana-5404	142	33	𝐷	𝐷	NOUN
cana-5404	142	34	linked	link	VERB
cana-5404	142	35	to	to	ADP
cana-5404	142	36	a	a	DET
cana-5404	142	37	mapping	mapping	NOUN
cana-5404	142	38	𝜉.	𝜉.	ADV
cana-5404	142	39	if	if	SCONJ
cana-5404	142	40	it	it	PRON
cana-5404	142	41	holds	hold	VERB
cana-5404	142	42	that	that	SCONJ
cana-5404	142	43	[	[	X
cana-5404	142	44	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	142	45	)	)	PUNCT
cana-5404	142	46	,	,	PUNCT
cana-5404	142	47	𝜔	𝜔	ADP
cana-5404	142	48	]	]	PUNCT
cana-5404	142	49	=	=	PUNCT
cana-5404	143	1	[	[	X
cana-5404	143	2	𝜈	𝜈	X
cana-5404	143	3	,	,	PUNCT
cana-5404	143	4	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	143	5	)	)	PUNCT
cana-5404	143	6	]	]	PUNCT
cana-5404	143	7	or	or	CCONJ
cana-5404	143	8	[	[	X
cana-5404	143	9	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	143	10	)	)	PUNCT
cana-5404	143	11	,	,	PUNCT
cana-5404	143	12	𝜔	𝜔	PROPN
cana-5404	143	13	]	]	X
cana-5404	143	14	=	=	SYM
cana-5404	143	15	−[𝜈	−[𝜈	PROPN
cana-5404	143	16	,	,	PUNCT
cana-5404	143	17	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	143	18	)	)	PUNCT
cana-5404	143	19	]	]	PUNCT
cana-5404	143	20	∀	∀	NOUN
cana-5404	143	21	elements	element	NOUN
cana-5404	143	22	𝜈	𝜈	X
cana-5404	143	23	,	,	PUNCT
cana-5404	143	24	𝜔	𝜔	PROPN
cana-5404	143	25	∈	∈	PROPN
cana-5404	143	26	𝐿	𝐿	PROPN
cana-5404	143	27	,	,	PUNCT
cana-5404	143	28	then	then	ADV
cana-5404	143	29	it	it	PRON
cana-5404	143	30	follows	follow	VERB
cana-5404	143	31	that	that	SCONJ
cana-5404	143	32	𝐿[𝛿(𝜈	𝐿[𝛿(𝜈	PROPN
cana-5404	143	33	)	)	PUNCT
cana-5404	143	34	,	,	PUNCT
cana-5404	143	35	𝜈	𝜈	X
cana-5404	143	36	]	]	X
cana-5404	143	37	=	=	SYM
cana-5404	143	38	0	0	NUM
cana-5404	143	39	and	and	CCONJ
cana-5404	143	40	𝐿[𝛤(𝜈	𝐿[𝛤(𝜈	NOUN
cana-5404	143	41	)	)	PUNCT
cana-5404	143	42	,	,	PUNCT
cana-5404	143	43	𝜈	𝜈	X
cana-5404	143	44	]	]	X
cana-5404	143	45	=	=	SYM
cana-5404	143	46	0	0	NUM
cana-5404	143	47	,	,	PUNCT
cana-5404	143	48	∀𝜈	∀𝜈	X
cana-5404	143	49	∈	∈	NOUN
cana-5404	143	50	𝐿.	𝐿.	ADJ
cana-5404	143	51	proof	proof	NOUN
cana-5404	143	52	.	.	PUNCT
cana-5404	144	1	assume	assume	VERB
cana-5404	144	2	that	that	SCONJ
cana-5404	145	1	[	[	X
cana-5404	145	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	145	3	)	)	PUNCT
cana-5404	145	4	,	,	PUNCT
cana-5404	145	5	𝜔	𝜔	ADP
cana-5404	145	6	]	]	PUNCT
cana-5404	145	7	=	=	PUNCT
cana-5404	145	8	[	[	X
cana-5404	145	9	𝜈	𝜈	X
cana-5404	145	10	,	,	PUNCT
cana-5404	145	11	γ(𝜔	γ(𝜔	NOUN
cana-5404	145	12	)	)	PUNCT
cana-5404	145	13	]	]	PUNCT
cana-5404	145	14	∀	∀	PUNCT
cana-5404	145	15	𝜈	𝜈	X
cana-5404	145	16	,	,	PUNCT
cana-5404	145	17	𝜔	𝜔	PROPN
cana-5404	145	18	∈	∈	NOUN
cana-5404	145	19	𝐿.	𝐿.	NOUN
cana-5404	145	20	(	(	PUNCT
cana-5404	145	21	23	23	NUM
cana-5404	145	22	)	)	PUNCT
cana-5404	145	23	by	by	ADP
cana-5404	145	24	substituting	substitute	VERB
cana-5404	145	25	𝜔	𝜔	NOUN
cana-5404	145	26	with	with	ADP
cana-5404	145	27	𝜔𝜈	𝜔𝜈	INTJ
cana-5404	145	28	in	in	ADP
cana-5404	145	29	(	(	PUNCT
cana-5404	145	30	23	23	NUM
cana-5404	145	31	)	)	PUNCT
cana-5404	145	32	,	,	PUNCT
cana-5404	145	33	we	we	PRON
cana-5404	145	34	derive	derive	VERB
cana-5404	145	35	[	[	X
cana-5404	145	36	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	145	37	)	)	PUNCT
cana-5404	145	38	,	,	PUNCT
cana-5404	145	39	𝜔]𝜈	𝜔]𝜈	X
cana-5404	145	40	+	+	CCONJ
cana-5404	145	41	𝜔[𝜉(𝜈	𝜔[𝜉(𝜈	NOUN
cana-5404	145	42	)	)	PUNCT
cana-5404	145	43	,	,	PUNCT
cana-5404	145	44	𝜈	𝜈	X
cana-5404	145	45	]	]	X
cana-5404	145	46	=	=	PUNCT
cana-5404	146	1	[	[	X
cana-5404	146	2	𝜈	𝜈	X
cana-5404	146	3	,	,	PUNCT
cana-5404	146	4	γ(𝜔)]𝜈	γ(𝜔)]𝜈	PROPN
cana-5404	146	5	+	+	CCONJ
cana-5404	146	6	𝜔[𝜈	𝜔[𝜈	NOUN
cana-5404	146	7	,	,	PUNCT
cana-5404	146	8	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	146	9	)	)	PUNCT
cana-5404	146	10	]	]	PUNCT
cana-5404	147	1	+	+	CCONJ
cana-5404	147	2	[	[	X
cana-5404	147	3	𝜈	𝜈	X
cana-5404	147	4	,	,	PUNCT
cana-5404	147	5	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	NOUN
cana-5404	147	6	)	)	PUNCT
cana-5404	147	7	∀	∀	PUNCT
cana-5404	147	8	𝜈	𝜈	X
cana-5404	147	9	,	,	PUNCT
cana-5404	147	10	𝜔	𝜔	PROPN
cana-5404	147	11	∈	∈	NOUN
cana-5404	147	12	𝐿.	𝐿.	NOUN
cana-5404	147	13	(	(	PUNCT
cana-5404	147	14	24	24	NUM
cana-5404	147	15	)	)	PUNCT
cana-5404	147	16	multiplying	multiplying	NOUN
cana-5404	147	17	(	(	PUNCT
cana-5404	147	18	23	23	NUM
cana-5404	147	19	)	)	PUNCT
cana-5404	147	20	to	to	ADP
cana-5404	147	21	the	the	DET
cana-5404	147	22	right	right	NOUN
cana-5404	147	23	by	by	ADP
cana-5404	147	24	𝜈	𝜈	X
cana-5404	147	25	,	,	PUNCT
cana-5404	147	26	we	we	PRON
cana-5404	147	27	obtain	obtain	VERB
cana-5404	147	28	[	[	X
cana-5404	147	29	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	147	30	)	)	PUNCT
cana-5404	147	31	,	,	PUNCT
cana-5404	147	32	𝜔]𝜈	𝜔]𝜈	X
cana-5404	147	33	=	=	PUNCT
cana-5404	148	1	[	[	X
cana-5404	148	2	𝜈	𝜈	X
cana-5404	148	3	,	,	PUNCT
cana-5404	148	4	γ(𝜔)]𝜈	γ(𝜔)]𝜈	NUM
cana-5404	148	5	∀	∀	NUM
cana-5404	148	6	𝜈	𝜈	NOUN
cana-5404	148	7	,	,	PUNCT
cana-5404	148	8	𝜔	𝜔	AUX
cana-5404	148	9	∈	∈	NOUN
cana-5404	148	10	𝐿.	𝐿.	NOUN
cana-5404	148	11	(	(	PUNCT
cana-5404	148	12	25	25	NUM
cana-5404	148	13	)	)	PUNCT
cana-5404	148	14	combining	combine	VERB
cana-5404	148	15	(	(	PUNCT
cana-5404	148	16	24	24	NUM
cana-5404	148	17	)	)	PUNCT
cana-5404	148	18	and	and	CCONJ
cana-5404	148	19	(	(	PUNCT
cana-5404	148	20	25	25	NUM
cana-5404	148	21	)	)	PUNCT
cana-5404	148	22	,	,	PUNCT
cana-5404	148	23	we	we	PRON
cana-5404	148	24	have	have	VERB
cana-5404	148	25	2𝜔[𝜉(𝜈	2𝜔[𝜉(𝜈	NUM
cana-5404	148	26	)	)	PUNCT
cana-5404	148	27	,	,	PUNCT
cana-5404	148	28	𝜈	𝜈	X
cana-5404	148	29	]	]	X
cana-5404	148	30	=	=	PUNCT
cana-5404	149	1	[	[	X
cana-5404	149	2	𝜈	𝜈	X
cana-5404	149	3	,	,	PUNCT
cana-5404	149	4	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	NOUN
cana-5404	149	5	)	)	PUNCT
cana-5404	149	6	∀	∀	PUNCT
cana-5404	149	7	𝜈	𝜈	X
cana-5404	149	8	,	,	PUNCT
cana-5404	149	9	𝜔	𝜔	PROPN
cana-5404	149	10	∈	∈	NOUN
cana-5404	149	11	𝐿.	𝐿.	NOUN
cana-5404	149	12	(	(	PUNCT
cana-5404	149	13	26	26	NUM
cana-5404	149	14	)	)	PUNCT
cana-5404	149	15	now	now	ADV
cana-5404	149	16	,	,	PUNCT
cana-5404	149	17	by	by	ADP
cana-5404	149	18	substituting	substitute	VERB
cana-5404	149	19	𝜔	𝜔	X
cana-5404	149	20	with	with	ADP
cana-5404	149	21	𝑧𝜔	𝑧𝜔	NOUN
cana-5404	149	22	in	in	ADP
cana-5404	149	23	(	(	PUNCT
cana-5404	149	24	26	26	NUM
cana-5404	149	25	)	)	PUNCT
cana-5404	149	26	and	and	CCONJ
cana-5404	149	27	applying	apply	VERB
cana-5404	149	28	(	(	PUNCT
cana-5404	149	29	26	26	NUM
cana-5404	149	30	)	)	PUNCT
cana-5404	149	31	,	,	PUNCT
cana-5404	149	32	we	we	PRON
cana-5404	149	33	obtain	obtain	VERB
cana-5404	149	34	[	[	X
cana-5404	149	35	𝜈	𝜈	X
cana-5404	149	36	,	,	PUNCT
cana-5404	149	37	𝑧]𝜔𝜉(𝜈	𝑧]𝜔𝜉(𝜈	PROPN
cana-5404	149	38	)	)	PUNCT
cana-5404	149	39	=	=	SYM
cana-5404	150	1	0	0	NUM
cana-5404	150	2	∀	∀	NUM
cana-5404	150	3	𝜈	𝜈	X
cana-5404	150	4	,	,	PUNCT
cana-5404	150	5	𝜔	𝜔	VERB
cana-5404	150	6	,	,	PUNCT
cana-5404	150	7	𝑧	𝑧	PRON
cana-5404	150	8	∈	∈	PROPN
cana-5404	150	9	𝐿.	𝐿.	NOUN
cana-5404	150	10	(	(	PUNCT
cana-5404	150	11	27	27	NUM
cana-5404	150	12	)	)	PUNCT
cana-5404	150	13	by	by	ADP
cana-5404	150	14	replacing	replace	VERB
cana-5404	150	15	𝑧	𝑧	PRON
cana-5404	150	16	with	with	ADP
cana-5404	150	17	𝑟𝑧	𝑟𝑧	INTJ
cana-5404	150	18	in	in	ADP
cana-5404	150	19	(	(	PUNCT
cana-5404	150	20	27	27	NUM
cana-5404	150	21	)	)	PUNCT
cana-5404	150	22	,	,	PUNCT
cana-5404	150	23	we	we	PRON
cana-5404	150	24	obtain	obtain	VERB
cana-5404	150	25	[	[	X
cana-5404	150	26	𝜈	𝜈	X
cana-5404	150	27	,	,	PUNCT
cana-5404	150	28	𝑟]𝑧𝜔𝜉(𝜈	𝑟]𝑧𝜔𝜉(𝜈	NOUN
cana-5404	150	29	)	)	PUNCT
cana-5404	150	30	=	=	SYM
cana-5404	150	31	0	0	NUM
cana-5404	150	32	∀	∀	X
cana-5404	150	33	𝜈	𝜈	X
cana-5404	150	34	,	,	PUNCT
cana-5404	150	35	𝜔	𝜔	VERB
cana-5404	150	36	,	,	PUNCT
cana-5404	150	37	𝑧	𝑧	PRON
cana-5404	150	38	∈	∈	NOUN
cana-5404	151	1	𝐿𝑟	𝐿𝑟	PROPN
cana-5404	151	2	∈	∈	PROPN
cana-5404	151	3	ℬ.	ℬ.	PROPN
cana-5404	151	4	(	(	PUNCT
cana-5404	151	5	28	28	NUM
cana-5404	151	6	)	)	PUNCT
cana-5404	151	7	replacing	replace	VERB
cana-5404	151	8	𝑧	𝑧	PRON
cana-5404	151	9	by	by	ADP
cana-5404	151	10	𝜉(𝜈)𝑧	𝜉(𝜈)𝑧	PROPN
cana-5404	151	11	in	in	ADP
cana-5404	151	12	(	(	PUNCT
cana-5404	151	13	28	28	NUM
cana-5404	151	14	)	)	PUNCT
cana-5404	151	15	,	,	PUNCT
cana-5404	151	16	to	to	PART
cana-5404	151	17	get	get	VERB
cana-5404	151	18	0	0	NUM
cana-5404	151	19	=	=	PUNCT
cana-5404	152	1	[	[	X
cana-5404	152	2	𝜈	𝜈	X
cana-5404	152	3	,	,	PUNCT
cana-5404	152	4	𝑟]𝜉(𝜈)𝑧𝜔𝜉(𝜈	𝑟]𝜉(𝜈)𝑧𝜔𝜉(𝜈	PROPN
cana-5404	152	5	)	)	PUNCT
cana-5404	152	6	,	,	PUNCT
cana-5404	152	7	that	that	PRON
cana-5404	152	8	is	be	AUX
cana-5404	152	9	[	[	X
cana-5404	152	10	𝜈	𝜈	X
cana-5404	152	11	,	,	PUNCT
cana-5404	152	12	𝑟]𝜉(𝜈)ℬ𝑧𝜔𝜉(𝜈	𝑟]𝜉(𝜈)ℬ𝑧𝜔𝜉(𝜈	NUM
cana-5404	152	13	)	)	PUNCT
cana-5404	152	14	=	=	SYM
cana-5404	152	15	(	(	PUNCT
cana-5404	152	16	0	0	NUM
cana-5404	152	17	)	)	PUNCT
cana-5404	152	18	∀	∀	PUNCT
cana-5404	153	1	𝜈	𝜈	X
cana-5404	153	2	,	,	PUNCT
cana-5404	153	3	𝜔	𝜔	VERB
cana-5404	153	4	,	,	PUNCT
cana-5404	153	5	𝑧	𝑧	DET
cana-5404	153	6	∈	∈	PROPN
cana-5404	153	7	𝐿	𝐿	PROPN
cana-5404	153	8	,	,	PUNCT
cana-5404	153	9	𝑟	𝑟	X
cana-5404	153	10	∈	∈	NOUN
cana-5404	153	11	𝑅.	𝑅.	NOUN
cana-5404	153	12	interchanging	interchange	VERB
cana-5404	153	13	𝑧	𝑧	PROPN
cana-5404	153	14	and	and	CCONJ
cana-5404	153	15	𝜔	𝜔	VERB
cana-5404	153	16	and	and	CCONJ
cana-5404	153	17	then	then	ADV
cana-5404	153	18	subtracting	subtract	VERB
cana-5404	153	19	one	one	NUM
cana-5404	153	20	from	from	ADP
cana-5404	153	21	the	the	DET
cana-5404	153	22	other	other	ADJ
cana-5404	153	23	,	,	PUNCT
cana-5404	153	24	we	we	PRON
cana-5404	153	25	have	have	VERB
cana-5404	153	26	[	[	X
cana-5404	153	27	𝜈	𝜈	X
cana-5404	153	28	,	,	PUNCT
cana-5404	153	29	𝑟]𝜉(𝜈)ℬ[𝑧	𝑟]𝜉(𝜈)ℬ[𝑧	NOUN
cana-5404	153	30	,	,	PUNCT
cana-5404	153	31	𝜔]𝜉(𝜈	𝜔]𝜉(𝜈	NOUN
cana-5404	153	32	)	)	PUNCT
cana-5404	153	33	=	=	SYM
cana-5404	153	34	(	(	PUNCT
cana-5404	153	35	0	0	NUM
cana-5404	153	36	)	)	PUNCT
cana-5404	153	37	∀	∀	PUNCT
cana-5404	154	1	𝜈	𝜈	X
cana-5404	154	2	,	,	PUNCT
cana-5404	154	3	𝜔	𝜔	VERB
cana-5404	154	4	,	,	PUNCT
cana-5404	154	5	𝑧	𝑧	DET
cana-5404	154	6	∈	∈	PROPN
cana-5404	154	7	𝐿.	𝐿.	VERB
cana-5404	154	8	in	in	ADP
cana-5404	154	9	particular	particular	ADJ
cana-5404	154	10	,	,	PUNCT
cana-5404	154	11	by	by	ADP
cana-5404	154	12	setting	set	VERB
cana-5404	154	13	𝑟	𝑟	NOUN
cana-5404	154	14	=	=	SYM
cana-5404	154	15	𝑧	𝑧	PROPN
cana-5404	154	16	and	and	CCONJ
cana-5404	154	17	𝜔	𝜔	AUX
cana-5404	154	18	=	=	SYM
cana-5404	154	19	𝜈	𝜈	NOUN
cana-5404	154	20	,	,	PUNCT
cana-5404	154	21	we	we	PRON
cana-5404	154	22	have	have	VERB
cana-5404	154	23	[	[	X
cana-5404	154	24	𝜈	𝜈	X
cana-5404	154	25	,	,	PUNCT
cana-5404	154	26	𝑧]𝜉(𝜈)ℬ[𝜈	𝑧]𝜉(𝜈)ℬ[𝜈	NOUN
cana-5404	154	27	,	,	PUNCT
cana-5404	154	28	𝑧]𝜉(𝜈	𝑧]𝜉(𝜈	PROPN
cana-5404	154	29	)	)	PUNCT
cana-5404	154	30	=	=	SYM
cana-5404	154	31	(	(	PUNCT
cana-5404	154	32	0	0	NUM
cana-5404	154	33	)	)	PUNCT
cana-5404	154	34	,	,	PUNCT
cana-5404	154	35	∀𝜈	∀𝜈	PROPN
cana-5404	154	36	,	,	PUNCT
cana-5404	154	37	𝑧	𝑧	PROPN
cana-5404	154	38	∈	∈	PROPN
cana-5404	154	39	𝐿.	𝐿.	VERB
cana-5404	154	40	since	since	SCONJ
cana-5404	154	41	ℬ	ℬ	NOUN
cana-5404	154	42	is	be	AUX
cana-5404	154	43	semiprime	semiprime	NOUN
cana-5404	154	44	,	,	PUNCT
cana-5404	154	45	it	it	PRON
cana-5404	154	46	follows	follow	VERB
cana-5404	154	47	that	that	SCONJ
cana-5404	155	1	[	[	X
cana-5404	155	2	𝜈	𝜈	X
cana-5404	155	3	,	,	PUNCT
cana-5404	155	4	𝑧]𝜉(𝜈	𝑧]𝜉(𝜈	PROPN
cana-5404	155	5	)	)	PUNCT
cana-5404	155	6	=	=	SYM
cana-5404	155	7	0	0	NUM
cana-5404	155	8	∀	∀	X
cana-5404	155	9	𝜈	𝜈	X
cana-5404	155	10	,	,	PUNCT
cana-5404	155	11	𝑧	𝑧	PROPN
cana-5404	155	12	∈	∈	PROPN
cana-5404	155	13	𝐿.	𝐿.	NOUN
cana-5404	155	14	(	(	PUNCT
cana-5404	155	15	29	29	NUM
cana-5404	155	16	)	)	PUNCT
cana-5404	155	17	by	by	ADP
cana-5404	155	18	multiplying	multiply	VERB
cana-5404	155	19	(	(	PUNCT
cana-5404	155	20	29	29	NUM
cana-5404	155	21	)	)	PUNCT
cana-5404	155	22	on	on	ADP
cana-5404	155	23	the	the	DET
cana-5404	155	24	right	right	NOUN
cana-5404	155	25	by	by	ADP
cana-5404	155	26	𝜈	𝜈	X
cana-5404	155	27	,	,	PUNCT
cana-5404	155	28	we	we	PRON
cana-5404	155	29	deduce	deduce	VERB
cana-5404	155	30	that	that	SCONJ
cana-5404	156	1	[	[	X
cana-5404	156	2	𝜈	𝜈	X
cana-5404	156	3	,	,	PUNCT
cana-5404	156	4	𝑧]𝜉(𝜈)𝜈	𝑧]𝜉(𝜈)𝜈	X
cana-5404	156	5	=	=	SYM
cana-5404	156	6	0	0	NUM
cana-5404	156	7	,	,	PUNCT
cana-5404	156	8	∀𝜈	∀𝜈	PROPN
cana-5404	156	9	,	,	PUNCT
cana-5404	156	10	𝑧	𝑧	PROPN
cana-5404	156	11	∈	∈	PROPN
cana-5404	156	12	𝐿.	𝐿.	PROPN
cana-5404	156	13	substituting	substituting	NOUN
cana-5404	156	14	𝑧	𝑧	NOUN
cana-5404	156	15	with	with	ADP
cana-5404	156	16	𝑧𝜈	𝑧𝜈	PROPN
cana-5404	156	17	,	,	PUNCT
cana-5404	156	18	we	we	PRON
cana-5404	156	19	find	find	VERB
cana-5404	156	20	that	that	SCONJ
cana-5404	156	21	[	[	X
cana-5404	156	22	𝜈	𝜈	X
cana-5404	156	23	,	,	PUNCT
cana-5404	156	24	𝑧]𝜈𝜉(𝜈	𝑧]𝜈𝜉(𝜈	PROPN
cana-5404	156	25	)	)	PUNCT
cana-5404	156	26	=	=	SYM
cana-5404	156	27	0	0	NUM
cana-5404	156	28	,	,	PUNCT
cana-5404	156	29	∀𝜈	∀𝜈	PROPN
cana-5404	156	30	,	,	PUNCT
cana-5404	156	31	𝑧	𝑧	PROPN
cana-5404	156	32	∈	∈	PROPN
cana-5404	156	33	𝐿.	𝐿.	NOUN
cana-5404	156	34	subtracting	subtract	VERB
cana-5404	156	35	these	these	DET
cana-5404	156	36	two	two	NUM
cana-5404	156	37	equations	equation	NOUN
cana-5404	156	38	yields	yield	VERB
cana-5404	156	39	[	[	X
cana-5404	156	40	𝜈	𝜈	X
cana-5404	156	41	,	,	PUNCT
cana-5404	156	42	𝑧][𝜉(𝜈	𝑧][𝜉(𝜈	PROPN
cana-5404	156	43	)	)	PUNCT
cana-5404	156	44	,	,	PUNCT
cana-5404	156	45	𝜈	𝜈	X
cana-5404	156	46	]	]	X
cana-5404	156	47	=	=	SYM
cana-5404	156	48	0	0	NUM
cana-5404	156	49	,	,	PUNCT
cana-5404	156	50	∀𝜈	∀𝜈	PROPN
cana-5404	156	51	,	,	PUNCT
cana-5404	156	52	𝑧	𝑧	PROPN
cana-5404	156	53	∈	∈	PROPN
cana-5404	156	54	𝐿.	𝐿.	NOUN
cana-5404	156	55	replacing	replace	VERB
cana-5404	156	56	𝑧	𝑧	NOUN
cana-5404	156	57	with	with	ADP
cana-5404	156	58	𝜉(𝜈)𝑧	𝜉(𝜈)𝑧	PROPN
cana-5404	156	59	in	in	ADP
cana-5404	156	60	this	this	DET
cana-5404	156	61	result	result	NOUN
cana-5404	156	62	gives	give	VERB
cana-5404	156	63	[	[	PRON
cana-5404	156	64	𝜈	𝜈	X
cana-5404	156	65	,	,	PUNCT
cana-5404	156	66	𝜉(𝜈)]𝑧[𝜈	𝜉(𝜈)]𝑧[𝜈	PROPN
cana-5404	156	67	,	,	PUNCT
cana-5404	156	68	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	156	69	)	)	PUNCT
cana-5404	156	70	]	]	PUNCT
cana-5404	157	1	=	=	SYM
cana-5404	157	2	0	0	NUM
cana-5404	157	3	,	,	PUNCT
cana-5404	157	4	∀𝜈	∀𝜈	PROPN
cana-5404	157	5	,	,	PUNCT
cana-5404	157	6	𝑧	𝑧	PROPN
cana-5404	157	7	∈	∈	PROPN
cana-5404	157	8	𝐿	𝐿	PROPN
cana-5404	157	9	,	,	PUNCT
cana-5404	157	10	which	which	PRON
cana-5404	157	11	can	can	AUX
cana-5404	157	12	be	be	AUX
cana-5404	157	13	rewritten	rewrite	VERB
cana-5404	157	14	as	as	ADP
cana-5404	157	15	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	157	16	)	)	PUNCT
cana-5404	157	17	,	,	PUNCT
cana-5404	157	18	𝜈]ℬ𝐿[𝜉(𝜈	𝜈]ℬ𝐿[𝜉(𝜈	NOUN
cana-5404	157	19	)	)	PUNCT
cana-5404	157	20	,	,	PUNCT
cana-5404	157	21	𝜈	𝜈	X
cana-5404	157	22	]	]	X
cana-5404	157	23	=	=	SYM
cana-5404	157	24	(	(	PUNCT
cana-5404	157	25	0	0	NUM
cana-5404	157	26	)	)	PUNCT
cana-5404	157	27	,	,	PUNCT
cana-5404	157	28	∀𝜈	∀𝜈	X
cana-5404	157	29	∈	∈	PROPN
cana-5404	157	30	𝐿.	𝐿.	VERB
cana-5404	157	31	the	the	DET
cana-5404	157	32	semiprimeness	semiprimeness	NOUN
cana-5404	157	33	of	of	ADP
cana-5404	157	34	ℬ	ℬ	PROPN
cana-5404	157	35	implies	imply	VERB
cana-5404	157	36	that	that	SCONJ
cana-5404	157	37	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	NOUN
cana-5404	157	38	)	)	PUNCT
cana-5404	157	39	,	,	PUNCT
cana-5404	157	40	𝜈	𝜈	X
cana-5404	157	41	]	]	X
cana-5404	157	42	=	=	SYM
cana-5404	157	43	(	(	PUNCT
cana-5404	157	44	0	0	NUM
cana-5404	157	45	)	)	PUNCT
cana-5404	157	46	,	,	PUNCT
cana-5404	157	47	∀𝜈	∀𝜈	X
cana-5404	157	48	∈	∈	PROPN
cana-5404	157	49	𝐿.	𝐿.	VERB
cana-5404	157	50	from	from	ADP
cana-5404	157	51	our	our	PRON
cana-5404	157	52	assumption	assumption	NOUN
cana-5404	157	53	,	,	PUNCT
cana-5404	157	54	we	we	PRON
cana-5404	157	55	have	have	VERB
cana-5404	157	56	[	[	X
cana-5404	157	57	𝜈	𝜈	X
cana-5404	157	58	,	,	PUNCT
cana-5404	157	59	γ(𝜈	γ(𝜈	NUM
cana-5404	157	60	)	)	PUNCT
cana-5404	157	61	]	]	PUNCT
cana-5404	158	1	=	=	PUNCT
cana-5404	159	1	[	[	X
cana-5404	159	2	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	159	3	)	)	PUNCT
cana-5404	159	4	,	,	PUNCT
cana-5404	159	5	𝜈	𝜈	X
cana-5404	159	6	]	]	X
cana-5404	159	7	,	,	PUNCT
cana-5404	159	8	and	and	CCONJ
cana-5404	159	9	thus	thus	ADV
cana-5404	159	10	𝐿[𝜈	𝐿[𝜈	NOUN
cana-5404	159	11	,	,	PUNCT
cana-5404	159	12	γ(𝜈	γ(𝜈	NUM
cana-5404	159	13	)	)	PUNCT
cana-5404	159	14	]	]	PUNCT
cana-5404	159	15	=	=	PUNCT
cana-5404	159	16	𝐿[𝜉(𝜈	𝐿[𝜉(𝜈	PROPN
cana-5404	159	17	)	)	PUNCT
cana-5404	159	18	,	,	PUNCT
cana-5404	159	19	𝜈	𝜈	X
cana-5404	159	20	]	]	X
cana-5404	159	21	=	=	SYM
cana-5404	159	22	0	0	NUM
cana-5404	159	23	,	,	PUNCT
cana-5404	159	24	∀𝜈	∀𝜈	X
cana-5404	159	25	∈	∈	PROPN
cana-5404	159	26	𝐿.	𝐿.	VERB
cana-5404	159	27	the	the	DET
cana-5404	159	28	case	case	NOUN
cana-5404	159	29	where	where	SCONJ
cana-5404	159	30	[	[	X
cana-5404	159	31	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	159	32	)	)	PUNCT
cana-5404	159	33	,	,	PUNCT
cana-5404	159	34	𝜔	𝜔	PROPN
cana-5404	159	35	]	]	X
cana-5404	159	36	=	=	SYM
cana-5404	159	37	−[𝜈	−[𝜈	PROPN
cana-5404	159	38	,	,	PUNCT
cana-5404	159	39	γ(𝜔	γ(𝜔	PROPN
cana-5404	159	40	)	)	PUNCT
cana-5404	159	41	]	]	PUNCT
cana-5404	159	42	∀	∀	PUNCT
cana-5404	159	43	𝜈,𝜔	𝜈,𝜔	PROPN
cana-5404	159	44	∈	∈	PROPN
cana-5404	159	45	𝐿	𝐿	PROPN
cana-5404	159	46	follows	follow	VERB
cana-5404	159	47	similarly	similarly	ADV
cana-5404	159	48	.	.	PUNCT
cana-5404	160	1	corollary	corollary	ADJ
cana-5404	160	2	2.9	2.9	NUM
cana-5404	160	3	let	let	VERB
cana-5404	160	4	ℬ	ℬ	PRON
cana-5404	160	5	be	be	AUX
cana-5404	160	6	a	a	DET
cana-5404	160	7	semiprime	semiprime	NOUN
cana-5404	160	8	ring	ring	NOUN
cana-5404	160	9	,	,	PUNCT
cana-5404	160	10	and	and	CCONJ
cana-5404	160	11	let	let	VERB
cana-5404	160	12	𝛤	𝛤	PRON
cana-5404	160	13	be	be	AUX
cana-5404	160	14	a	a	DET
cana-5404	160	15	𝑚𝑢𝑙𝑡.	𝑚𝑢𝑙𝑡.	NOUN
cana-5404	160	16	(	(	PUNCT
cana-5404	160	17	𝐺	𝐺	NOUN
cana-5404	160	18	)	)	PUNCT
cana-5404	160	19	−	−	PROPN
cana-5404	160	20	𝐷	𝐷	NOUN
cana-5404	160	21	associated	associate	VERB
cana-5404	160	22	with	with	ADP
cana-5404	160	23	a	a	DET
cana-5404	160	24	map	map	NOUN
cana-5404	161	1	𝜉.	𝜉.	ADV
cana-5404	161	2	if	if	SCONJ
cana-5404	161	3	[	[	X
cana-5404	161	4	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	161	5	)	)	PUNCT
cana-5404	161	6	,	,	PUNCT
cana-5404	161	7	𝜔	𝜔	ADP
cana-5404	161	8	]	]	PUNCT
cana-5404	161	9	=	=	PUNCT
cana-5404	162	1	[	[	X
cana-5404	162	2	𝜈	𝜈	X
cana-5404	162	3	,	,	PUNCT
cana-5404	162	4	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	162	5	)	)	PUNCT
cana-5404	162	6	]	]	PUNCT
cana-5404	162	7	or	or	CCONJ
cana-5404	162	8	[	[	X
cana-5404	162	9	𝜉(𝜈	𝜉(𝜈	NOUN
cana-5404	162	10	)	)	PUNCT
cana-5404	162	11	,	,	PUNCT
cana-5404	162	12	𝜔	𝜔	PROPN
cana-5404	162	13	]	]	X
cana-5404	162	14	=	=	SYM
cana-5404	162	15	−[𝜈	−[𝜈	PROPN
cana-5404	162	16	,	,	PUNCT
cana-5404	162	17	𝛤(𝜔	𝛤(𝜔	NOUN
cana-5404	162	18	)	)	PUNCT
cana-5404	162	19	]	]	PUNCT
cana-5404	162	20	,	,	PUNCT
cana-5404	162	21	∀𝜈	∀𝜈	PROPN
cana-5404	162	22	,	,	PUNCT
cana-5404	162	23	𝜔	𝜔	PROPN
cana-5404	162	24	∈	∈	PROPN
cana-5404	162	25	ℬ	ℬ	NOUN
cana-5404	162	26	,	,	PUNCT
cana-5404	162	27	then	then	ADV
cana-5404	162	28	𝛤	𝛤	PROPN
cana-5404	162	29	and	and	CCONJ
cana-5404	162	30	𝜉	𝜉	PROPN
cana-5404	162	31	are	be	AUX
cana-5404	162	32	commuting	commute	VERB
cana-5404	162	33	maps	map	NOUN
cana-5404	162	34	on	on	ADP
cana-5404	162	35	ℬ.	ℬ.	NOUN
cana-5404	162	36	communications	communication	NOUN
cana-5404	162	37	on	on	ADP
cana-5404	162	38	applied	apply	VERB
cana-5404	162	39	nonlinear	nonlinear	ADJ
cana-5404	162	40	analysis	analysis	NOUN
cana-5404	162	41	issn	issn	NOUN
cana-5404	162	42	:	:	PUNCT
cana-5404	162	43	1074	1074	NUM
cana-5404	162	44	-	-	PUNCT
cana-5404	162	45	133x	133x	NUM
cana-5404	162	46	vol	vol	VERB
cana-5404	162	47	32	32	NUM
cana-5404	162	48	no	no	NOUN
cana-5404	162	49	.	.	PUNCT
cana-5404	163	1	10s	10	NOUN
cana-5404	163	2	(	(	PUNCT
cana-5404	163	3	2025	2025	NUM
cana-5404	163	4	)	)	PUNCT
cana-5404	163	5	2144	2144	NUM
cana-5404	163	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5404	163	7	3example	3example	NUM
cana-5404	163	8	the	the	DET
cana-5404	163	9	following	follow	VERB
cana-5404	163	10	example	example	NOUN
cana-5404	163	11	demonstrates	demonstrate	VERB
cana-5404	163	12	that	that	SCONJ
cana-5404	163	13	the	the	DET
cana-5404	163	14	semiprimeness	semiprimeness	NOUN
cana-5404	163	15	condition	condition	NOUN
cana-5404	163	16	in	in	ADP
cana-5404	163	17	the	the	DET
cana-5404	163	18	preceding	precede	VERB
cana-5404	163	19	theorems	theorem	NOUN
cana-5404	163	20	is	be	AUX
cana-5404	163	21	essential	essential	ADJ
cana-5404	163	22	and	and	CCONJ
cana-5404	163	23	can	can	AUX
cana-5404	163	24	not	not	PART
cana-5404	163	25	be	be	AUX
cana-5404	163	26	omitted	omit	VERB
cana-5404	163	27	.	.	PUNCT
cana-5404	163	28	example	example	NOUN
cana-5404	163	29	3.1	3.1	NUM
cana-5404	163	30	let	let	VERB
cana-5404	163	31	ℬ	ℬ	NOUN
cana-5404	163	32	=	=	PUNCT
cana-5404	163	33	{	{	PUNCT
cana-5404	163	34	(	(	PUNCT
cana-5404	163	35	0	0	NUM
cana-5404	163	36	𝑎	𝑎	X
cana-5404	163	37	𝑏	𝑏	NOUN
cana-5404	163	38	0	0	NUM
cana-5404	163	39	0	0	NUM
cana-5404	163	40	𝑐	𝑐	NOUN
cana-5404	163	41	0	0	NUM
cana-5404	163	42	0	0	NUM
cana-5404	163	43	0	0	NUM
cana-5404	163	44	)	)	PUNCT
cana-5404	163	45	|𝑎	|𝑎	PROPN
cana-5404	163	46	,	,	PUNCT
cana-5404	163	47	𝑏	𝑏	NOUN
cana-5404	163	48	,	,	PUNCT
cana-5404	163	49	𝑐	𝑐	PROPN
cana-5404	163	50	∈	∈	PROPN
cana-5404	163	51	ℤ	ℤ	PROPN
cana-5404	163	52	(	(	PUNCT
cana-5404	163	53	𝑡ℎ𝑒	𝑡ℎ𝑒	INTJ
cana-5404	163	54	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-5404	163	55	𝑜𝑓	𝑜𝑓	X
cana-5404	163	56	𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠	𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠	PROPN
cana-5404	163	57	)	)	PUNCT
cana-5404	163	58	}	}	PUNCT
cana-5404	163	59	.	.	PUNCT
cana-5404	164	1	for	for	ADP
cana-5404	164	2	any	any	DET
cana-5404	164	3	0	0	NUM
cana-5404	164	4	≠	≠	PROPN
cana-5404	164	5	𝑏	𝑏	PROPN
cana-5404	164	6	∈	∈	PROPN
cana-5404	164	7	ℤ	ℤ	PROPN
cana-5404	164	8	,	,	PUNCT
cana-5404	164	9	(	(	PUNCT
cana-5404	164	10	0	0	NUM
cana-5404	164	11	0	0	NUM
cana-5404	165	1	𝑏	𝑏	NOUN
cana-5404	165	2	0	0	NUM
cana-5404	165	3	0	0	NUM
cana-5404	165	4	0	0	NUM
cana-5404	165	5	0	0	NUM
cana-5404	165	6	0	0	NUM
cana-5404	165	7	0	0	NUM
cana-5404	165	8	)	)	PUNCT
cana-5404	165	9	ℬ	ℬ	NOUN
cana-5404	165	10	(	(	PUNCT
cana-5404	165	11	0	0	NUM
cana-5404	165	12	0	0	NUM
cana-5404	165	13	𝑏	𝑏	NOUN
cana-5404	165	14	0	0	NUM
cana-5404	165	15	0	0	NUM
cana-5404	165	16	0	0	NUM
cana-5404	165	17	0	0	NUM
cana-5404	165	18	0	0	NUM
cana-5404	165	19	0	0	NUM
cana-5404	165	20	)	)	PUNCT
cana-5404	165	21	=	=	SYM
cana-5404	165	22	(	(	PUNCT
cana-5404	165	23	0	0	NUM
cana-5404	165	24	)	)	PUNCT
cana-5404	165	25	,	,	PUNCT
cana-5404	165	26	then	then	ADV
cana-5404	165	27	ℬ	ℬ	NOUN
cana-5404	165	28	is	be	AUX
cana-5404	165	29	not	not	PART
cana-5404	165	30	a	a	DET
cana-5404	165	31	semiprime	semiprime	NOUN
cana-5404	165	32	ring	ring	NOUN
cana-5404	165	33	.	.	PUNCT
cana-5404	166	1	define	define	VERB
cana-5404	166	2	𝛤	𝛤	PROPN
cana-5404	166	3	:	:	PUNCT
cana-5404	166	4	ℬ	ℬ	NOUN
cana-5404	166	5	→	→	SYM
cana-5404	166	6	ℬ	ℬ	NOUN
cana-5404	166	7	and	and	CCONJ
cana-5404	166	8	𝜉	𝜉	ADP
cana-5404	166	9	:	:	PUNCT
cana-5404	166	10	ℬ	ℬ	NOUN
cana-5404	166	11	→	→	SYM
cana-5404	166	12	ℬ	ℬ	NOUN
cana-5404	166	13	is	be	AUX
cana-5404	166	14	given	give	VERB
cana-5404	166	15	by	by	ADP
cana-5404	166	16	:	:	PUNCT
cana-5404	166	17	γ	γ	X
cana-5404	166	18	(	(	PUNCT
cana-5404	166	19	(	(	PUNCT
cana-5404	166	20	0	0	NUM
cana-5404	166	21	𝑎	𝑎	X
cana-5404	166	22	𝑏	𝑏	NOUN
cana-5404	166	23	0	0	NUM
cana-5404	166	24	0	0	NUM
cana-5404	166	25	𝑐	𝑐	NOUN
cana-5404	166	26	0	0	NUM
cana-5404	166	27	0	0	NUM
cana-5404	166	28	0	0	NUM
cana-5404	166	29	)	)	PUNCT
cana-5404	166	30	)	)	PUNCT
cana-5404	167	1	=	=	PUNCT
cana-5404	167	2	(	(	PUNCT
cana-5404	167	3	0	0	NUM
cana-5404	167	4	0	0	NUM
cana-5404	168	1	𝑏	𝑏	NOUN
cana-5404	168	2	0	0	NUM
cana-5404	168	3	0	0	NUM
cana-5404	168	4	0	0	NUM
cana-5404	168	5	0	0	NUM
cana-5404	168	6	0	0	NUM
cana-5404	168	7	0	0	NUM
cana-5404	168	8	)	)	PUNCT
cana-5404	168	9	,	,	PUNCT
cana-5404	168	10	and	and	CCONJ
cana-5404	168	11	𝜉	𝜉	X
cana-5404	168	12	(	(	PUNCT
cana-5404	168	13	(	(	PUNCT
cana-5404	168	14	0	0	NUM
cana-5404	168	15	𝑎	𝑎	X
cana-5404	168	16	𝑏	𝑏	NOUN
cana-5404	168	17	0	0	NUM
cana-5404	168	18	0	0	NUM
cana-5404	168	19	𝑐	𝑐	NOUN
cana-5404	168	20	0	0	NUM
cana-5404	168	21	0	0	NUM
cana-5404	168	22	0	0	NUM
cana-5404	168	23	)	)	PUNCT
cana-5404	168	24	)	)	PUNCT
cana-5404	169	1	=	=	PUNCT
cana-5404	169	2	(	(	PUNCT
cana-5404	169	3	0	0	NUM
cana-5404	169	4	𝑎2	𝑎2	NOUN
cana-5404	169	5	0	0	NUM
cana-5404	169	6	0	0	SYM
cana-5404	169	7	0	0	NUM
cana-5404	169	8	𝑐	𝑐	NOUN
cana-5404	169	9	0	0	NUM
cana-5404	169	10	0	0	NUM
cana-5404	169	11	0	0	NUM
cana-5404	169	12	)	)	PUNCT
cana-5404	169	13	.	.	PUNCT
cana-5404	170	1	it	it	PRON
cana-5404	170	2	is	be	AUX
cana-5404	170	3	easy	easy	ADJ
cana-5404	170	4	to	to	PART
cana-5404	170	5	verify	verify	VERB
cana-5404	170	6	that	that	SCONJ
cana-5404	170	7	γ	γ	PROPN
cana-5404	170	8	is	be	AUX
cana-5404	170	9	a	a	DET
cana-5404	170	10	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	170	11	(	(	PUNCT
cana-5404	170	12	𝐺	𝐺	NOUN
cana-5404	170	13	)	)	PUNCT
cana-5404	170	14	−	−	PROPN
cana-5404	170	15	𝐷	𝐷	NOUN
cana-5404	170	16	associated	associate	VERB
cana-5404	170	17	with	with	ADP
cana-5404	170	18	the	the	DET
cana-5404	170	19	map	map	NOUN
cana-5404	171	1	𝜉.	𝜉.	PROPN
cana-5404	171	2	furthermore	furthermore	ADV
cana-5404	171	3	,	,	PUNCT
cana-5404	171	4	it	it	PRON
cana-5404	171	5	can	can	AUX
cana-5404	171	6	be	be	AUX
cana-5404	171	7	directly	directly	ADV
cana-5404	171	8	confirmed	confirm	VERB
cana-5404	171	9	that	that	SCONJ
cana-5404	171	10	:	:	PUNCT
cana-5404	171	11	γ(𝜈)𝜔	γ(𝜈)𝜔	PROPN
cana-5404	171	12	=	=	SYM
cana-5404	171	13	𝜔γ(𝜈	𝜔γ(𝜈	NUM
cana-5404	171	14	)	)	PUNCT
cana-5404	171	15	=	=	SYM
cana-5404	171	16	0	0	NUM
cana-5404	171	17	,	,	PUNCT
cana-5404	171	18	∀𝜈	∀𝜈	PROPN
cana-5404	171	19	,	,	PUNCT
cana-5404	171	20	𝜔	𝜔	PROPN
cana-5404	171	21	∈	∈	NOUN
cana-5404	171	22	ℬ.	ℬ.	NOUN
cana-5404	171	23	consequently	consequently	ADV
cana-5404	171	24	,	,	PUNCT
cana-5404	171	25	[	[	X
cana-5404	171	26	γ(𝜈	γ(𝜈	NOUN
cana-5404	171	27	)	)	PUNCT
cana-5404	171	28	,	,	PUNCT
cana-5404	171	29	𝜔	𝜔	PROPN
cana-5404	171	30	]	]	X
cana-5404	171	31	=	=	SYM
cana-5404	171	32	±[𝜈	±[𝜈	NOUN
cana-5404	171	33	,	,	PUNCT
cana-5404	171	34	γ(𝜔	γ(𝜔	NOUN
cana-5404	171	35	)	)	PUNCT
cana-5404	171	36	]	]	PUNCT
cana-5404	171	37	,	,	PUNCT
cana-5404	171	38	is	be	AUX
cana-5404	171	39	hold	hold	ADJ
cana-5404	171	40	for	for	ADP
cana-5404	171	41	all	all	DET
cana-5404	171	42	𝜈,𝜔	𝜈,𝜔	PROPN
cana-5404	171	43	∈	∈	PROPN
cana-5404	171	44	ℬ.	ℬ.	NOUN
cana-5404	171	45	however	however	ADV
cana-5404	171	46	,	,	PUNCT
cana-5404	171	47	despite	despite	SCONJ
cana-5404	171	48	these	these	DET
cana-5404	171	49	properties	property	NOUN
cana-5404	171	50	,	,	PUNCT
cana-5404	171	51	𝜉	𝜉	ADP
cana-5404	171	52	≠	≠	PROPN
cana-5404	171	53	0	0	NUM
cana-5404	171	54	,	,	PUNCT
cana-5404	171	55	𝜉	𝜉	X
cana-5404	171	56	is	be	AUX
cana-5404	171	57	not	not	PART
cana-5404	171	58	commuting	commute	VERB
cana-5404	171	59	on	on	ADP
cana-5404	171	60	ℬ	ℬ	NOUN
cana-5404	171	61	,	,	PUNCT
cana-5404	171	62	and	and	CCONJ
cana-5404	171	63	γ(𝜈𝜔	γ(𝜈𝜔	PROPN
cana-5404	171	64	)	)	PUNCT
cana-5404	171	65	≠	≠	PROPN
cana-5404	171	66	γ(𝜈)𝜔.	γ(𝜈)𝜔.	PROPN
cana-5404	171	67	references	reference	VERB
cana-5404	171	68	[	[	X
cana-5404	171	69	1	1	NUM
cana-5404	171	70	]	]	X
cana-5404	171	71	daif	daif	NOUN
cana-5404	171	72	,	,	PUNCT
cana-5404	171	73	m.	m.	NOUN
cana-5404	171	74	n.	n.	NOUN
cana-5404	171	75	:	:	PUNCT
cana-5404	171	76	when	when	SCONJ
cana-5404	171	77	is	be	AUX
cana-5404	171	78	a	a	DET
cana-5404	171	79	multiplicative	multiplicative	ADJ
cana-5404	171	80	derivation	derivation	NOUN
cana-5404	171	81	additive	additive	NOUN
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cana-5404	171	83	,	,	PUNCT
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cana-5404	172	1	j.	j.	PROPN
cana-5404	172	2	math	math	PROPN
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cana-5404	173	2	.	.	PROPN
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cana-5404	173	4	14(3)(1991	14(3)(1991	NUM
cana-5404	173	5	)	)	PUNCT
cana-5404	173	6	615618	615618	NUM
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cana-5404	174	5	iii	iii	PROPN
cana-5404	174	6	,	,	PUNCT
cana-5404	174	7	w.	w.	PROPN
cana-5404	174	8	s.	s.	PROPN
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cana-5404	174	10	when	when	SCONJ
cana-5404	174	11	are	be	AUX
cana-5404	174	12	multiplicative	multiplicative	ADJ
cana-5404	174	13	maps	map	NOUN
cana-5404	174	14	additive	additive	VERB
cana-5404	174	15	?	?	PUNCT
cana-5404	174	16	,	,	PUNCT
cana-5404	174	17	proc	proc	PROPN
cana-5404	174	18	.	.	PUNCT
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cana-5404	175	2	.	.	PUNCT
cana-5404	175	3	math	math	PROPN
cana-5404	175	4	.	.	PUNCT
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cana-5404	176	2	.	.	PUNCT
cana-5404	176	3	,	,	PUNCT
cana-5404	176	4	21(1969	21(1969	NUM
cana-5404	176	5	)	)	PUNCT
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cana-5404	176	7	-	-	NUM
cana-5404	176	8	698	698	NUM
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cana-5404	177	2	3	3	NUM
cana-5404	177	3	]	]	X
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cana-5404	177	6	h.	h.	PROPN
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cana-5404	177	9	,	,	PUNCT
cana-5404	177	10	p.	p.	NOUN
cana-5404	177	11	:	:	PUNCT
cana-5404	177	12	multiplicative	multiplicative	ADJ
cana-5404	177	13	derivations	derivation	NOUN
cana-5404	177	14	on	on	ADP
cana-5404	177	15	𝐶(𝑋	𝐶(𝑋	ADJ
cana-5404	177	16	)	)	PUNCT
cana-5404	177	17	,	,	PUNCT
cana-5404	177	18	monatsh	monatsh	ADJ
cana-5404	177	19	.	.	PUNCT
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cana-5404	177	21	.	.	PUNCT
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cana-5404	177	23	121(3)(1996	121(3)(1996	X
cana-5404	177	24	)	)	PUNCT
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cana-5404	177	27	197	197	NUM
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cana-5404	178	2	4	4	NUM
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cana-5404	178	10	,	,	PUNCT
cana-5404	178	11	m.	m.	NOUN
cana-5404	178	12	s.	s.	PROPN
cana-5404	178	13	:	:	PUNCT
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cana-5404	178	15	generalized	generalize	VERB
cana-5404	178	16	derivations	derivation	NOUN
cana-5404	178	17	which	which	PRON
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cana-5404	178	19	additive	additive	ADJ
cana-5404	178	20	,	,	PUNCT
cana-5404	178	21	eastwest	eastwest	PROPN
cana-5404	178	22	j.	j.	PROPN
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cana-5404	178	27	)	)	PUNCT
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cana-5404	178	29	-	-	SYM
cana-5404	178	30	37	37	NUM
cana-5404	178	31	.	.	PUNCT
cana-5404	179	1	[	[	X
cana-5404	179	2	5	5	NUM
cana-5404	179	3	]	]	X
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cana-5404	179	10	s.	s.	PROPN
cana-5404	179	11	:	:	PUNCT
cana-5404	179	12	on	on	ADP
cana-5404	179	13	𝑀𝑢𝑙𝑡.	𝑀𝑢𝑙𝑡.	PROPN
cana-5404	179	14	(	(	PUNCT
cana-5404	179	15	𝐺	𝐺	NOUN
cana-5404	179	16	)	)	PUNCT
cana-5404	179	17	−	−	PROPN
cana-5404	180	1	𝐷s	𝐷s	PROPN
cana-5404	180	2	in	in	ADP
cana-5404	180	3	prime	prime	ADJ
cana-5404	180	4	and	and	CCONJ
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cana-5404	180	6	rings	ring	NOUN
cana-5404	180	7	,	,	PUNCT
cana-5404	180	8	aequat	aequat	PROPN
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cana-5404	181	1	math	math	PROPN
cana-5404	181	2	.	.	PUNCT
cana-5404	181	3	,	,	PUNCT
cana-5404	181	4	86(2013	86(2013	VERB
cana-5404	181	5	)	)	PUNCT
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cana-5404	181	7	-	-	SYM
cana-5404	181	8	79	79	NUM
cana-5404	181	9	.	.	PUNCT
cana-5404	182	1	[	[	X
cana-5404	182	2	6	6	NUM
cana-5404	182	3	]	]	X
cana-5404	182	4	argac	argac	PROPN
cana-5404	182	5	,	,	PUNCT
cana-5404	182	6	n.	n.	NOUN
cana-5404	182	7	:	:	PUNCT
cana-5404	182	8	on	on	ADP
cana-5404	182	9	prime	prime	ADJ
cana-5404	182	10	and	and	CCONJ
cana-5404	182	11	semiprime	semiprime	NOUN
cana-5404	182	12	rings	ring	NOUN
cana-5404	182	13	with	with	ADP
cana-5404	182	14	derivations	derivation	NOUN
cana-5404	182	15	,	,	PUNCT
cana-5404	182	16	algebra	algebra	NOUN
cana-5404	182	17	colloq	colloq	PROPN
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cana-5404	182	19	,	,	PUNCT
cana-5404	182	20	13(3)(2006	13(3)(2006	NUM
cana-5404	182	21	)	)	PUNCT
cana-5404	182	22	371380	371380	NUM
cana-5404	182	23	.	.	PUNCT
cana-5404	183	1	[	[	X
cana-5404	183	2	7	7	NUM
cana-5404	183	3	]	]	PUNCT
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cana-5404	183	5	,	,	PUNCT
cana-5404	183	6	m.	m.	NOUN
cana-5404	183	7	,	,	PUNCT
cana-5404	183	8	ali	ali	PROPN
cana-5404	183	9	,	,	PUNCT
cana-5404	183	10	a.	a.	NOUN
cana-5404	183	11	and	and	CCONJ
cana-5404	183	12	rani	rani	PROPN
cana-5404	183	13	,	,	PUNCT
cana-5404	183	14	r.	r.	PROPN
cana-5404	183	15	:	:	PUNCT
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cana-5404	183	18	derivations	derivation	NOUN
cana-5404	183	19	of	of	ADP
cana-5404	183	20	prime	prime	ADJ
cana-5404	183	21	rings	ring	NOUN
cana-5404	183	22	,	,	PUNCT
cana-5404	183	23	southeast	southeast	ADJ
cana-5404	183	24	asian	asian	ADJ
cana-5404	183	25	bull	bull	NOUN
cana-5404	183	26	.	.	PUNCT
cana-5404	183	27	math	math	NOUN
cana-5404	183	28	.	.	PUNCT
cana-5404	183	29	,	,	PUNCT
cana-5404	183	30	29(2005	29(2005	NUM
cana-5404	183	31	)	)	PUNCT
cana-5404	183	32	669	669	NUM
cana-5404	183	33	-	-	SYM
cana-5404	183	34	675	675	NUM
cana-5404	183	35	.	.	PUNCT
cana-5404	184	1	[	[	X
cana-5404	184	2	8	8	NUM
cana-5404	184	3	]	]	PUNCT
cana-5404	184	4	ashraf	ashraf	NOUN
cana-5404	184	5	,	,	PUNCT
cana-5404	184	6	m.	m.	NOUN
cana-5404	184	7	,	,	PUNCT
cana-5404	184	8	ur	ur	NOUN
cana-5404	184	9	-	-	NOUN
cana-5404	184	10	rehman	rehman	ADJ
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cana-5404	184	12	n.	n.	PROPN
cana-5404	184	13	,	,	PUNCT
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cana-5404	184	21	:	:	PUNCT
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cana-5404	184	23	semiprime	semiprime	NOUN
cana-5404	184	24	rings	ring	NOUN
cana-5404	184	25	with	with	ADP
cana-5404	184	26	generalized	generalized	ADJ
cana-5404	184	27	derivations	derivation	NOUN
cana-5404	184	28	,	,	PUNCT
cana-5404	184	29	bol	bol	NOUN
cana-5404	184	30	.	.	PUNCT
cana-5404	185	1	soc	soc	PROPN
cana-5404	185	2	.	.	PUNCT
cana-5404	186	1	paran	paran	PROPN
cana-5404	186	2	.	.	PUNCT
cana-5404	187	1	mat	mat	PROPN
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cana-5404	187	5	)	)	PUNCT
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