id	sid	tid	token	lemma	pos
cana-5394	1	1	communications	communication	NOUN
cana-5394	1	2	on	on	ADP
cana-5394	1	3	applied	apply	VERB
cana-5394	1	4	nonlinear	nonlinear	ADJ
cana-5394	1	5	analysis	analysis	NOUN
cana-5394	1	6	issn	issn	NOUN
cana-5394	1	7	:	:	PUNCT
cana-5394	1	8	1074	1074	NUM
cana-5394	1	9	-	-	PUNCT
cana-5394	1	10	133x	133x	NUM
cana-5394	1	11	vol	vol	NOUN
cana-5394	1	12	32	32	NUM
cana-5394	1	13	no	no	NOUN
cana-5394	1	14	.	.	NOUN
cana-5394	1	15	3	3	NUM
cana-5394	1	16	(	(	PUNCT
cana-5394	1	17	2025	2025	NUM
cana-5394	1	18	)	)	PUNCT
cana-5394	1	19	930	930	NUM
cana-5394	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	1	21	on	on	ADP
cana-5394	1	22	functional	functional	ADJ
cana-5394	1	23	identities	identity	NOUN
cana-5394	1	24	involving	involve	VERB
cana-5394	1	25	n	n	PRON
cana-5394	1	26	-derivations	-derivation	NOUN
cana-5394	1	27	in	in	ADP
cana-5394	1	28	semiprime	semiprime	NOUN
cana-5394	1	29	rings	ring	NOUN
cana-5394	1	30	g.	g.	PROPN
cana-5394	1	31	naga	naga	PROPN
cana-5394	1	32	malleswari1	malleswari1	PROPN
cana-5394	1	33	,	,	PUNCT
cana-5394	1	34	g.	g.	PROPN
cana-5394	1	35	swapna2	swapna2	PROPN
cana-5394	1	36	,	,	PUNCT
cana-5394	1	37	p.	p.	PROPN
cana-5394	1	38	srilatha	srilatha	PROPN
cana-5394	2	1	devi3	devi3	PROPN
cana-5394	2	2	,	,	PUNCT
cana-5394	2	3	s.	s.	PROPN
cana-5394	2	4	sreenivasulu4	sreenivasulu4	PROPN
cana-5394	3	1	*	*	PROPN
cana-5394	3	2	*	*	PROPN
cana-5394	3	3	,	,	PUNCT
cana-5394	3	4	a.	a.	PROPN
cana-5394	3	5	mallikarjuna	mallikarjuna	PROPN
cana-5394	3	6	reddy5	reddy5	PROPN
cana-5394	3	7	1	1	NUM
cana-5394	3	8	,	,	PUNCT
cana-5394	3	9	2	2	NUM
cana-5394	3	10	department	department	NOUN
cana-5394	3	11	of	of	ADP
cana-5394	3	12	mathematics	mathematics	PROPN
cana-5394	3	13	,	,	PUNCT
cana-5394	3	14	vnr	vnr	PROPN
cana-5394	3	15	vignana	vignana	PROPN
cana-5394	3	16	jyothi	jyothi	PROPN
cana-5394	3	17	institute	institute	PROPN
cana-5394	3	18	of	of	ADP
cana-5394	3	19	engineering	engineering	NOUN
cana-5394	3	20	and	and	CCONJ
cana-5394	3	21	technology	technology	NOUN
cana-5394	3	22	,	,	PUNCT
cana-5394	3	23	hyderabad	hyderabad	PROPN
cana-5394	3	24	.	.	PUNCT
cana-5394	4	1	3research	3research	NUM
cana-5394	4	2	scholar	scholar	NOUN
cana-5394	4	3	,	,	PUNCT
cana-5394	4	4	department	department	NOUN
cana-5394	4	5	of	of	ADP
cana-5394	4	6	mathematics	mathematic	NOUN
cana-5394	4	7	,	,	PUNCT
cana-5394	4	8	sri	sri	PROPN
cana-5394	4	9	krishnadevaraya	krishnadevaraya	PROPN
cana-5394	4	10	university	university	PROPN
cana-5394	4	11	,	,	PUNCT
cana-5394	4	12	anantapuramu	anantapuramu	VERB
cana-5394	4	13	4department	4department	NUM
cana-5394	4	14	of	of	ADP
cana-5394	4	15	mathematics	mathematic	NOUN
cana-5394	4	16	,	,	PUNCT
cana-5394	4	17	kts	kts	NOUN
cana-5394	4	18	gdc	gdc	PROPN
cana-5394	4	19	,	,	PUNCT
cana-5394	4	20	rayadurg	rayadurg	NOUN
cana-5394	4	21	.	.	PUNCT
cana-5394	5	1	*	*	PUNCT
cana-5394	5	2	*	*	PUNCT
cana-5394	5	3	corresponding	correspond	VERB
cana-5394	5	4	author	author	NOUN
cana-5394	5	5	:	:	PUNCT
cana-5394	5	6	s.	s.	PROPN
cana-5394	5	7	sreenivasulu	sreenivasulu	PROPN
cana-5394	5	8	,	,	PUNCT
cana-5394	5	9	5department	5department	NUM
cana-5394	5	10	of	of	ADP
cana-5394	5	11	mathematics	mathematic	NOUN
cana-5394	5	12	,	,	PUNCT
cana-5394	5	13	sri	sri	PROPN
cana-5394	5	14	krishnadevaraya	krishnadevaraya	PROPN
cana-5394	5	15	university	university	PROPN
cana-5394	5	16	,	,	PUNCT
cana-5394	5	17	anantapuramu	anantapuramu	ADJ
cana-5394	5	18	article	article	NOUN
cana-5394	5	19	history	history	NOUN
cana-5394	5	20	:	:	PUNCT
cana-5394	5	21	received	receive	VERB
cana-5394	5	22	:	:	PUNCT
cana-5394	5	23	15	15	NUM
cana-5394	5	24	-	-	SYM
cana-5394	5	25	01	01	NUM
cana-5394	5	26	-	-	PUNCT
cana-5394	5	27	2025	2025	NUM
cana-5394	5	28	revised	revise	VERB
cana-5394	5	29	:	:	PUNCT
cana-5394	5	30	25	25	NUM
cana-5394	5	31	-	-	PUNCT
cana-5394	5	32	02	02	NUM
cana-5394	5	33	-	-	PUNCT
cana-5394	5	34	2025	2025	NUM
cana-5394	5	35	accepted	accept	VERB
cana-5394	5	36	:	:	PUNCT
cana-5394	5	37	04	04	NUM
cana-5394	5	38	-	-	PUNCT
cana-5394	5	39	03	03	NUM
cana-5394	5	40	-	-	PUNCT
cana-5394	5	41	2025	2025	NUM
cana-5394	5	42	abstract	abstract	NOUN
cana-5394	5	43	:	:	PUNCT
cana-5394	5	44	this	this	DET
cana-5394	5	45	paper	paper	NOUN
cana-5394	5	46	investigates	investigate	VERB
cana-5394	5	47	n	n	PRON
cana-5394	5	48	-additive	-additive	ADJ
cana-5394	5	49	maps	map	NOUN
cana-5394	5	50	,	,	PUNCT
cana-5394	5	51	known	know	VERB
cana-5394	5	52	as	as	ADP
cana-5394	5	53	symmetric	symmetric	ADJ
cana-5394	5	54	n	n	NUM
cana-5394	5	55	-derivations	-derivation	NOUN
cana-5394	5	56	whose	whose	DET
cana-5394	5	57	trace	trace	NOUN
cana-5394	5	58	satisfies	satisfy	VERB
cana-5394	5	59	certain	certain	ADJ
cana-5394	5	60	algebraic	algebraic	ADJ
cana-5394	5	61	identities	identity	NOUN
cana-5394	5	62	.	.	PUNCT
cana-5394	6	1	we	we	PRON
cana-5394	6	2	establish	establish	VERB
cana-5394	6	3	various	various	ADJ
cana-5394	6	4	results	result	NOUN
cana-5394	6	5	that	that	PRON
cana-5394	6	6	enhance	enhance	VERB
cana-5394	6	7	our	our	PRON
cana-5394	6	8	understanding	understanding	NOUN
cana-5394	6	9	of	of	ADP
cana-5394	6	10	these	these	DET
cana-5394	6	11	maps	map	NOUN
cana-5394	6	12	,	,	PUNCT
cana-5394	6	13	notably	notably	ADV
cana-5394	6	14	their	their	PRON
cana-5394	6	15	occurrence	occurrence	NOUN
cana-5394	6	16	in	in	ADP
cana-5394	6	17	semiprime	semiprime	NOUN
cana-5394	6	18	rings	ring	NOUN
cana-5394	6	19	and	and	CCONJ
cana-5394	6	20	the	the	DET
cana-5394	6	21	behavior	behavior	NOUN
cana-5394	6	22	of	of	ADP
cana-5394	6	23	rings	ring	NOUN
cana-5394	6	24	under	under	ADP
cana-5394	6	25	particular	particular	ADJ
cana-5394	6	26	algebraic	algebraic	ADJ
cana-5394	6	27	identities	identity	NOUN
cana-5394	6	28	involving	involve	VERB
cana-5394	6	29	nonzero	nonzero	PROPN
cana-5394	6	30	ideal	ideal	ADJ
cana-5394	6	31	.	.	PUNCT
cana-5394	7	1	furthermore	furthermore	ADV
cana-5394	7	2	,	,	PUNCT
cana-5394	7	3	we	we	PRON
cana-5394	7	4	prove	prove	VERB
cana-5394	7	5	that	that	SCONJ
cana-5394	7	6	the	the	DET
cana-5394	7	7	semiprime	semiprime	NOUN
cana-5394	7	8	ring	ring	NOUN
cana-5394	7	9	possesses	possess	VERB
cana-5394	7	10	a	a	DET
cana-5394	7	11	nonzero	nonzero	ADJ
cana-5394	7	12	central	central	ADJ
cana-5394	7	13	ideal	ideal	NOUN
cana-5394	7	14	under	under	ADP
cana-5394	7	15	certain	certain	ADJ
cana-5394	7	16	algebraic	algebraic	ADJ
cana-5394	7	17	conditions	condition	NOUN
cana-5394	7	18	.	.	PUNCT
cana-5394	8	1	keywords	keyword	NOUN
cana-5394	8	2	:	:	PUNCT
cana-5394	8	3	derivation	derivation	NOUN
cana-5394	8	4	,	,	PUNCT
cana-5394	8	5	symmetric	symmetric	ADJ
cana-5394	8	6	n	n	PRON
cana-5394	8	7	-derivation	-derivation	NOUN
cana-5394	8	8	,	,	PUNCT
cana-5394	8	9	semiprime	semiprime	NOUN
cana-5394	8	10	ring	ring	NOUN
cana-5394	8	11	,	,	PUNCT
cana-5394	8	12	ideal	ideal	ADJ
cana-5394	8	13	.	.	PUNCT
cana-5394	9	1	mathematics	mathematic	NOUN
cana-5394	9	2	subject	subject	PROPN
cana-5394	9	3	classification:16r50	classification:16r50	PROPN
cana-5394	9	4	,	,	PUNCT
cana-5394	9	5	16n60	16n60	NUM
cana-5394	9	6	,	,	PUNCT
cana-5394	9	7	16w25	16w25	NUM
cana-5394	9	8	.	.	PUNCT
cana-5394	10	1	1	1	X
cana-5394	10	2	.	.	X
cana-5394	10	3	introduction	introduction	NOUN
cana-5394	10	4	:	:	PUNCT
cana-5394	10	5	throughout	throughout	ADP
cana-5394	10	6	,	,	PUNCT
cana-5394	10	7	s	s	VERB
cana-5394	10	8	will	will	AUX
cana-5394	10	9	be	be	AUX
cana-5394	10	10	an	an	DET
cana-5394	10	11	associative	associative	ADJ
cana-5394	10	12	ring	ring	NOUN
cana-5394	10	13	with	with	ADP
cana-5394	10	14	(	(	PUNCT
cana-5394	10	15	)	)	PUNCT
cana-5394	10	16	z	z	NOUN
cana-5394	10	17	s	s	NOUN
cana-5394	10	18	as	as	ADP
cana-5394	10	19	its	its	PRON
cana-5394	10	20	center	center	NOUN
cana-5394	10	21	.	.	PUNCT
cana-5394	11	1	a	a	DET
cana-5394	11	2	ring	ring	NOUN
cana-5394	11	3	s	s	NOUN
cana-5394	11	4	is	be	AUX
cana-5394	11	5	said	say	VERB
cana-5394	11	6	to	to	PART
cana-5394	11	7	be	be	AUX
cana-5394	11	8	prime	prime	ADJ
cana-5394	11	9	if	if	SCONJ
cana-5394	11	10			NOUN
cana-5394	11	11	0s	0s	ADJ
cana-5394	11	12			PROPN
cana-5394	11	13	=	=	PRON
cana-5394	11	14	implies	imply	VERB
cana-5394	11	15	that	that	SCONJ
cana-5394	11	16	either	either	CCONJ
cana-5394	11	17	0	0	NUM
cana-5394	11	18	=	=	SYM
cana-5394	11	19	or	or	CCONJ
cana-5394	11	20	0	0	NUM
cana-5394	11	21	=	=	SYM
cana-5394	11	22	and	and	CCONJ
cana-5394	11	23	semiprime	semiprime	NOUN
cana-5394	11	24	if	if	SCONJ
cana-5394	11	25			PRON
cana-5394	11	26	0s	0s	ADJ
cana-5394	11	27			X
cana-5394	11	28	=	=	PRON
cana-5394	11	29	implies	imply	VERB
cana-5394	11	30	that	that	SCONJ
cana-5394	11	31	0	0	NUM
cana-5394	11	32	=	=	SYM
cana-5394	11	33	where	where	SCONJ
cana-5394	11	34	,	,	PUNCT
cana-5394	11	35	s	s	PROPN
cana-5394	11	36			PROPN
cana-5394	11	37			PROPN
cana-5394	11	38	.	.	PUNCT
cana-5394	12	1	the	the	DET
cana-5394	12	2	notion	notion	NOUN
cana-5394	12	3	[	[	PUNCT
cana-5394	12	4	,	,	PUNCT
cana-5394	12	5	]	]	X
cana-5394	12	6			PROPN
cana-5394	12	7			PROPN
cana-5394	12	8	and	and	CCONJ
cana-5394	12	9			PROPN
cana-5394	12	10			PROPN
cana-5394	12	11	denote	denote	VERB
cana-5394	12	12	the	the	DET
cana-5394	12	13	commutator	commutator	NOUN
cana-5394	12	14			PUNCT
cana-5394	12	15	−	−	NOUN
cana-5394	12	16	and	and	CCONJ
cana-5394	12	17	the	the	DET
cana-5394	12	18	anticommutator	anticommutator	NOUN
cana-5394	12	19			PUNCT
cana-5394	12	20	+	+	PROPN
cana-5394	12	21	,	,	PUNCT
cana-5394	12	22	respectively	respectively	ADV
cana-5394	12	23	,	,	PUNCT
cana-5394	12	24	for	for	ADP
cana-5394	12	25	any	any	PRON
cana-5394	12	26	,	,	PUNCT
cana-5394	12	27	s	s	PROPN
cana-5394	12	28			PROPN
cana-5394	12	29			PROPN
cana-5394	12	30	.	.	PUNCT
cana-5394	13	1	a	a	DET
cana-5394	13	2	ring	ring	NOUN
cana-5394	13	3	s	s	NOUN
cana-5394	13	4	is	be	AUX
cana-5394	13	5	said	say	VERB
cana-5394	13	6	to	to	PART
cana-5394	13	7	be	be	AUX
cana-5394	13	8	n	n	PRON
cana-5394	13	9	-torsion	-torsion	NOUN
cana-5394	13	10	free	free	ADJ
cana-5394	13	11	if	if	SCONJ
cana-5394	13	12	0nx	0nx	NOUN
cana-5394	13	13	=	=	PUNCT
cana-5394	13	14	implies	imply	VERB
cana-5394	13	15	that	that	SCONJ
cana-5394	13	16	0x	0x	NOUN
cana-5394	13	17	=	=	PUNCT
cana-5394	13	18	for	for	ADP
cana-5394	13	19	all	all	DET
cana-5394	13	20	.x	.x	NOUN
cana-5394	13	21	s	s	PROPN
cana-5394	13	22	if	if	SCONJ
cana-5394	13	23	s	s	VERB
cana-5394	13	24	is	be	AUX
cana-5394	13	25	!	!	PUNCT
cana-5394	14	1	n	n	CCONJ
cana-5394	14	2	-torsion	-torsion	NOUN
cana-5394	14	3	free	free	ADJ
cana-5394	14	4	,	,	PUNCT
cana-5394	14	5	then	then	ADV
cana-5394	14	6	it	it	PRON
cana-5394	14	7	is	be	AUX
cana-5394	14	8	m	m	NOUN
cana-5394	14	9	-torsion	-torsion	NOUN
cana-5394	14	10	free	free	ADJ
cana-5394	14	11	for	for	ADP
cana-5394	14	12	every	every	DET
cana-5394	14	13	divisor	divisor	NOUN
cana-5394	14	14	m	m	ADV
cana-5394	14	15	of	of	ADP
cana-5394	14	16	!	!	PUNCT
cana-5394	15	1	n	n	X
cana-5394	15	2	.	.	PUNCT
cana-5394	16	1	an	an	DET
cana-5394	16	2	additive	additive	ADJ
cana-5394	16	3	mapping	mapping	NOUN
cana-5394	16	4	:	:	PUNCT
cana-5394	16	5	d	d	X
cana-5394	16	6	s	s	VERB
cana-5394	16	7	s→	s→	VERB
cana-5394	16	8	is	be	AUX
cana-5394	16	9	called	call	VERB
cana-5394	16	10	a	a	DET
cana-5394	16	11	derivation	derivation	NOUN
cana-5394	16	12	if	if	SCONJ
cana-5394	16	13	(	(	PUNCT
cana-5394	16	14	)	)	PUNCT
cana-5394	16	15	(	(	PUNCT
cana-5394	16	16	)	)	PUNCT
cana-5394	16	17	(	(	PUNCT
cana-5394	16	18	)	)	PUNCT
cana-5394	16	19	d	d	X
cana-5394	16	20	d	d	X
cana-5394	16	21	d	d	PROPN
cana-5394	16	22			PROPN
cana-5394	16	23			PROPN
cana-5394	16	24			PROPN
cana-5394	16	25	=	=	PROPN
cana-5394	16	26	+	+	PROPN
cana-5394	16	27	holds	hold	VERB
cana-5394	16	28	for	for	ADP
cana-5394	16	29	all	all	PRON
cana-5394	16	30	,	,	PUNCT
cana-5394	16	31	s	s	PROPN
cana-5394	16	32			PROPN
cana-5394	16	33			NOUN
cana-5394	16	34	.to	.to	PUNCT
cana-5394	17	1	expand	expand	VERB
cana-5394	17	2	the	the	DET
cana-5394	17	3	theory	theory	NOUN
cana-5394	17	4	of	of	ADP
cana-5394	17	5	derivation	derivation	NOUN
cana-5394	17	6	,	,	PUNCT
cana-5394	17	7	maksa	maksa	PROPN
cana-5394	17	8	[	[	X
cana-5394	17	9	1	1	NUM
cana-5394	17	10	]	]	PUNCT
cana-5394	17	11	introduced	introduce	VERB
cana-5394	17	12	the	the	DET
cana-5394	17	13	concept	concept	NOUN
cana-5394	17	14	of	of	ADP
cana-5394	17	15	symmetric	symmetric	ADJ
cana-5394	17	16	bi	bi	NOUN
cana-5394	17	17	-	-	NOUN
cana-5394	17	18	derivations	derivation	NOUN
cana-5394	17	19	on	on	ADP
cana-5394	17	20	rings	ring	NOUN
cana-5394	17	21	,	,	PUNCT
cana-5394	17	22	which	which	DET
cana-5394	17	23	vukman	vukman	NOUN
cana-5394	17	24	carried	carry	VERB
cana-5394	17	25	out	out	ADP
cana-5394	17	26	a	a	DET
cana-5394	17	27	more	more	ADV
cana-5394	17	28	extensive	extensive	ADJ
cana-5394	17	29	investigation[see	investigation[see	NOUN
cana-5394	17	30	in	in	ADP
cana-5394	17	31	[	[	X
cana-5394	17	32	2],[3	2],[3	NUM
cana-5394	17	33	]	]	X
cana-5394	17	34	]	]	PUNCT
cana-5394	17	35	.	.	PUNCT
cana-5394	18	1	a	a	DET
cana-5394	18	2	bi	bi	ADJ
cana-5394	18	3	-	-	ADJ
cana-5394	18	4	additive	additive	ADJ
cana-5394	18	5	map	map	NOUN
cana-5394	18	6	:	:	PUNCT
cana-5394	18	7	d	d	X
cana-5394	18	8	s	s	X
cana-5394	18	9	s	s	PROPN
cana-5394	18	10	s	s	PROPN
cana-5394	18	11	→	→	PROPN
cana-5394	18	12	is	be	AUX
cana-5394	18	13	said	say	VERB
cana-5394	18	14	to	to	PART
cana-5394	18	15	be	be	AUX
cana-5394	18	16	a	a	DET
cana-5394	18	17	bi	bi	NOUN
cana-5394	18	18	-	-	NOUN
cana-5394	18	19	derivation	derivation	NOUN
cana-5394	18	20	if	if	SCONJ
cana-5394	18	21	(	(	PUNCT
cana-5394	18	22	,	,	PUNCT
cana-5394	18	23	)	)	PUNCT
cana-5394	18	24	(	(	PUNCT
cana-5394	18	25	,	,	PUNCT
cana-5394	18	26	)	)	PUNCT
cana-5394	18	27	(	(	PUNCT
cana-5394	18	28	,	,	PUNCT
cana-5394	18	29	)	)	PUNCT
cana-5394	18	30	,	,	PUNCT
cana-5394	18	31	(	(	PUNCT
cana-5394	18	32	,	,	PUNCT
cana-5394	18	33	)	)	PUNCT
cana-5394	18	34	(	(	PUNCT
cana-5394	18	35	,	,	PUNCT
cana-5394	18	36	)	)	PUNCT
cana-5394	18	37	(	(	PUNCT
cana-5394	18	38	,	,	PUNCT
cana-5394	18	39	)	)	PUNCT
cana-5394	19	1	d	d	PUNCT
cana-5394	20	1	d	d	PUNCT
cana-5394	20	2	d	d	X
cana-5394	20	3	d	d	PROPN
cana-5394	20	4	d	d	X
cana-5394	20	5	d	d	NOUN
cana-5394	20	6			PROPN
cana-5394	20	7			PROPN
cana-5394	20	8			PROPN
cana-5394	20	9			PROPN
cana-5394	20	10			X
cana-5394	20	11			PROPN
cana-5394	20	12			PROPN
cana-5394	20	13			PROPN
cana-5394	20	14			PROPN
cana-5394	20	15			PROPN
cana-5394	21	1			PROPN
cana-5394	21	2			PROPN
cana-5394	21	3			PROPN
cana-5394	21	4			PROPN
cana-5394	21	5			NOUN
cana-5394	21	6			CCONJ
cana-5394	21	7			CCONJ
cana-5394	22	1			ADV
cana-5394	22	2			ADV
cana-5394	22	3	=	=	ADJ
cana-5394	23	1	+	+	CCONJ
cana-5394	23	2	=	=	PUNCT
cana-5394	24	1	+	+	CCONJ
cana-5394	24	2	hold	hold	VERB
cana-5394	24	3	for	for	ADP
cana-5394	24	4	any	any	PRON
cana-5394	24	5	,	,	PUNCT
cana-5394	24	6	,	,	PUNCT
cana-5394	24	7	,	,	PUNCT
cana-5394	24	8	s	s	PROPN
cana-5394	24	9			PROPN
cana-5394	24	10			PROPN
cana-5394	24	11			NOUN
cana-5394	24	12			NOUN
cana-5394	24	13	.	.	PUNCT
cana-5394	25	1	the	the	DET
cana-5394	25	2	aforementioned	aforementioned	ADJ
cana-5394	25	3	conditions	condition	NOUN
cana-5394	25	4	are	be	AUX
cana-5394	25	5	identical	identical	ADJ
cana-5394	25	6	if	if	SCONJ
cana-5394	25	7	d	d	NOUN
cana-5394	25	8	is	be	AUX
cana-5394	25	9	also	also	ADV
cana-5394	25	10	a	a	DET
cana-5394	25	11	symmetric	symmetric	ADJ
cana-5394	25	12	map	map	NOUN
cana-5394	25	13	,	,	PUNCT
cana-5394	25	14	that	that	ADV
cana-5394	25	15	is	is	ADV
cana-5394	25	16	,	,	PUNCT
cana-5394	25	17	if	if	SCONJ
cana-5394	25	18	(	(	PUNCT
cana-5394	25	19	,	,	PUNCT
cana-5394	25	20	)	)	PUNCT
cana-5394	25	21	(	(	PUNCT
cana-5394	25	22	,	,	PUNCT
cana-5394	25	23	)	)	PUNCT
cana-5394	25	24	d	d	X
cana-5394	25	25	d	d	NOUN
cana-5394	25	26			PROPN
cana-5394	25	27			PROPN
cana-5394	25	28	=	=	PROPN
cana-5394	25	29	for	for	ADP
cana-5394	25	30	every	every	PRON
cana-5394	25	31	,	,	PUNCT
cana-5394	25	32	s	s	PROPN
cana-5394	25	33			PROPN
cana-5394	25	34			PROPN
cana-5394	25	35	.	.	PUNCT
cana-5394	26	1	in	in	ADP
cana-5394	26	2	this	this	DET
cana-5394	26	3	case	case	NOUN
cana-5394	26	4	,	,	PUNCT
cana-5394	26	5	d	d	PROPN
cana-5394	26	6	is	be	AUX
cana-5394	26	7	referred	refer	VERB
cana-5394	26	8	to	to	ADP
cana-5394	26	9	as	as	ADP
cana-5394	26	10	a	a	DET
cana-5394	26	11	symmetric	symmetric	ADJ
cana-5394	26	12	bi	bi	NOUN
cana-5394	26	13	-	-	NOUN
cana-5394	26	14	derivation	derivation	NOUN
cana-5394	26	15	on	on	ADP
cana-5394	26	16	s	s	PRON
cana-5394	26	17	.	.	PUNCT
cana-5394	27	1	several	several	ADJ
cana-5394	27	2	authors	author	NOUN
cana-5394	27	3	have	have	AUX
cana-5394	27	4	explored	explore	VERB
cana-5394	27	5	symmetric	symmetric	ADJ
cana-5394	27	6	bi	bi	NOUN
cana-5394	27	7	-	-	NOUN
cana-5394	27	8	derivations	derivation	NOUN
cana-5394	27	9	on	on	ADP
cana-5394	27	10	rings	ring	NOUN
cana-5394	27	11	(	(	PUNCT
cana-5394	27	12	see[4],[5],[6	see[4],[5],[6	NOUN
cana-5394	27	13	]	]	X
cana-5394	27	14	]	]	PUNCT
cana-5394	27	15	)	)	PUNCT
cana-5394	27	16	and	and	CCONJ
cana-5394	27	17	brought	bring	VERB
cana-5394	27	18	about	about	ADP
cana-5394	27	19	insightful	insightful	ADJ
cana-5394	27	20	conclusions	conclusion	NOUN
cana-5394	27	21	.	.	PUNCT
cana-5394	28	1	the	the	DET
cana-5394	28	2	study	study	NOUN
cana-5394	28	3	of	of	ADP
cana-5394	28	4	tri	tri	ADJ
cana-5394	28	5	-	-	NOUN
cana-5394	28	6	derivation	derivation	NOUN
cana-5394	28	7	was	be	AUX
cana-5394	28	8	initiated	initiate	VERB
cana-5394	28	9	by	by	ADP
cana-5394	28	10	ozturk	ozturk	NOUN
cana-5394	29	1	[	[	X
cana-5394	29	2	7	7	NUM
cana-5394	29	3	]	]	PUNCT
cana-5394	29	4	,	,	PUNCT
cana-5394	29	5	in	in	ADP
cana-5394	29	6	which	which	PRON
cana-5394	29	7	he	he	PRON
cana-5394	29	8	communications	communication	VERB
cana-5394	29	9	on	on	ADP
cana-5394	29	10	applied	apply	VERB
cana-5394	29	11	nonlinear	nonlinear	ADJ
cana-5394	29	12	analysis	analysis	NOUN
cana-5394	29	13	issn	issn	NOUN
cana-5394	29	14	:	:	PUNCT
cana-5394	29	15	1074	1074	NUM
cana-5394	29	16	-	-	PUNCT
cana-5394	29	17	133x	133x	NUM
cana-5394	29	18	vol	vol	NOUN
cana-5394	29	19	32	32	NUM
cana-5394	29	20	no	no	NOUN
cana-5394	29	21	.	.	NOUN
cana-5394	29	22	3	3	NUM
cana-5394	29	23	(	(	PUNCT
cana-5394	29	24	2025	2025	NUM
cana-5394	29	25	)	)	PUNCT
cana-5394	29	26	931	931	NUM
cana-5394	29	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	29	28	proved	prove	VERB
cana-5394	29	29	various	various	ADJ
cana-5394	29	30	results	result	NOUN
cana-5394	29	31	.	.	PUNCT
cana-5394	30	1	advancing	advance	VERB
cana-5394	30	2	this	this	DET
cana-5394	30	3	idea	idea	NOUN
cana-5394	30	4	,	,	PUNCT
cana-5394	30	5	park	park	NOUN
cana-5394	30	6	[	[	X
cana-5394	30	7	8	8	NUM
cana-5394	30	8	]	]	PUNCT
cana-5394	30	9	introduced	introduce	VERB
cana-5394	30	10	the	the	DET
cana-5394	30	11	concept	concept	NOUN
cana-5394	30	12	of	of	ADP
cana-5394	30	13	permuting	permute	VERB
cana-5394	30	14	n	n	PRON
cana-5394	30	15	derivation	derivation	NOUN
cana-5394	30	16	as	as	SCONJ
cana-5394	30	17	follows	follow	VERB
cana-5394	30	18	:	:	PUNCT
cana-5394	30	19	let	let	VERB
cana-5394	30	20	2n	2n	NUM
cana-5394	30	21			NUM
cana-5394	30	22	be	be	AUX
cana-5394	30	23	a	a	DET
cana-5394	30	24	fixed	fix	VERB
cana-5394	30	25	integer	integer	NOUN
cana-5394	30	26	,	,	PUNCT
cana-5394	30	27	and	and	CCONJ
cana-5394	30	28	·	·	PUNCT
cana-5394	30	29	·	·	PUNCT
cana-5394	30	30	·	·	SYM
cana-5394	30	31	n	n	CCONJ
cana-5394	30	32	n	n	NUM
cana-5394	30	33	times	time	NOUN
cana-5394	30	34	s	s	PART
cana-5394	30	35	s	s	X
cana-5394	30	36	s	s	X
cana-5394	30	37	s	s	X
cana-5394	30	38	−	−	NOUN
cana-5394	30	39	=	=	PUNCT
cana-5394	30	40			PROPN
cana-5394	30	41			PROPN
cana-5394	30	42			PROPN
cana-5394	30	43	.	.	PUNCT
cana-5394	31	1	a	a	DET
cana-5394	31	2	map	map	NOUN
cana-5394	31	3	:	:	PUNCT
cana-5394	31	4	snd	snd	PROPN
cana-5394	31	5	s→	s→	PROPN
cana-5394	31	6	is	be	AUX
cana-5394	31	7	said	say	VERB
cana-5394	31	8	to	to	PART
cana-5394	31	9	be	be	AUX
cana-5394	31	10	symmetric	symmetric	ADJ
cana-5394	31	11	(	(	PUNCT
cana-5394	31	12	permuting	permuting	NOUN
cana-5394	31	13	)	)	PUNCT
cana-5394	31	14	if	if	SCONJ
cana-5394	31	15	1	1	NUM
cana-5394	31	16	2	2	NUM
cana-5394	31	17	(	(	PUNCT
cana-5394	31	18	1	1	NUM
cana-5394	31	19	)	)	PUNCT
cana-5394	31	20	(	(	PUNCT
cana-5394	31	21	2	2	NUM
cana-5394	31	22	)	)	PUNCT
cana-5394	31	23	(	(	PUNCT
cana-5394	31	24	)	)	PUNCT
cana-5394	31	25	(	(	PUNCT
cana-5394	31	26	,	,	PUNCT
cana-5394	31	27	,	,	PUNCT
cana-5394	31	28	...	...	PUNCT
cana-5394	31	29	,	,	PUNCT
cana-5394	31	30	)	)	PUNCT
cana-5394	31	31	(	(	PUNCT
cana-5394	31	32	,	,	PUNCT
cana-5394	31	33	,	,	PUNCT
cana-5394	31	34	...	...	PUNCT
cana-5394	31	35	,	,	PUNCT
cana-5394	31	36	)	)	PUNCT
cana-5394	32	1	n	n	CCONJ
cana-5394	32	2	nd	nd	NOUN
cana-5394	32	3	d	d	X
cana-5394	32	4			PROPN
cana-5394	32	5			PROPN
cana-5394	32	6			X
cana-5394	32	7			X
cana-5394	32	8			X
cana-5394	32	9			X
cana-5394	32	10			PROPN
cana-5394	32	11	=	=	NOUN
cana-5394	32	12	for	for	ADP
cana-5394	32	13	all	all	DET
cana-5394	32	14	permutations	permutation	NOUN
cana-5394	32	15	(	(	PUNCT
cana-5394	32	16	)	)	PUNCT
cana-5394	32	17	nt	not	PART
cana-5394	32	18	s	s	NUM
cana-5394	32	19			PROPN
cana-5394	32	20	and	and	CCONJ
cana-5394	32	21	j	j	PROPN
cana-5394	32	22	s	s	PROPN
cana-5394	32	23			NOUN
cana-5394	32	24	where	where	SCONJ
cana-5394	32	25	1,2	1,2	NUM
cana-5394	32	26	,	,	PUNCT
cana-5394	32	27	...	...	PUNCT
cana-5394	32	28	,	,	PUNCT
cana-5394	32	29	.j	.j	PROPN
cana-5394	32	30	n=	n=	PROPN
cana-5394	32	31	let	let	VERB
cana-5394	32	32	2n	2n	NUM
cana-5394	32	33			NUM
cana-5394	32	34	be	be	AUX
cana-5394	32	35	a	a	DET
cana-5394	32	36	fixed	fix	VERB
cana-5394	32	37	integer	integer	NOUN
cana-5394	32	38	.	.	PUNCT
cana-5394	33	1	an	an	DET
cana-5394	33	2	n	n	CCONJ
cana-5394	33	3	-	-	PUNCT
cana-5394	33	4	additive	additive	ADJ
cana-5394	33	5	mapping	mapping	NOUN
cana-5394	33	6	(	(	PUNCT
cana-5394	33	7	i.e.	i.e.	X
cana-5394	33	8	,	,	PUNCT
cana-5394	33	9	additive	additive	VERB
cana-5394	33	10	in	in	ADP
cana-5394	33	11	each	each	DET
cana-5394	33	12	argument	argument	NOUN
cana-5394	33	13	)	)	PUNCT
cana-5394	33	14	:	:	PUNCT
cana-5394	33	15	s	s	VERB
cana-5394	33	16	nd	nd	PRON
cana-5394	33	17	s→	s→	NOUN
cana-5394	33	18	is	be	AUX
cana-5394	33	19	called	call	VERB
cana-5394	33	20	an	an	DET
cana-5394	33	21	n	n	PRON
cana-5394	33	22	-derivation	-derivation	NOUN
cana-5394	33	23	on	on	ADP
cana-5394	33	24	s	s	PRON
cana-5394	33	25	if	if	SCONJ
cana-5394	33	26	the	the	DET
cana-5394	33	27	relation	relation	NOUN
cana-5394	33	28	:	:	PUNCT
cana-5394	33	29	1	1	NUM
cana-5394	33	30	1	1	NUM
cana-5394	33	31	2	2	NUM
cana-5394	33	32	1	1	NUM
cana-5394	33	33	2	2	NUM
cana-5394	33	34	1	1	NUM
cana-5394	33	35	1	1	NUM
cana-5394	33	36	1	1	NUM
cana-5394	33	37	2	2	NUM
cana-5394	33	38	,	,	PUNCT
cana-5394	33	39	,	,	PUNCT
cana-5394	33	40	...	...	PUNCT
cana-5394	33	41	,	,	PUNCT
cana-5394	33	42	)	)	PUNCT
cana-5394	33	43	(	(	PUNCT
cana-5394	33	44	,	,	PUNCT
cana-5394	33	45	,	,	PUNCT
cana-5394	33	46	...	...	PUNCT
cana-5394	33	47	,	,	PUNCT
cana-5394	33	48	)	)	PUNCT
cana-5394	33	49	(	(	PUNCT
cana-5394	33	50	,	,	PUNCT
cana-5394	33	51	,	,	PUNCT
cana-5394	33	52	...	...	PUNCT
cana-5394	33	53	,	,	PUNCT
cana-5394	33	54	)	)	PUNCT
cana-5394	33	55	,	,	PUNCT
cana-5394	33	56	(	(	PUNCT
cana-5394	33	57	n	n	CCONJ
cana-5394	33	58	n	n	PRON
cana-5394	33	59	nd	nd	NOUN
cana-5394	33	60	d	d	X
cana-5394	33	61	d	d	PROPN
cana-5394	33	62			PROPN
cana-5394	33	63			X
cana-5394	33	64			X
cana-5394	33	65			X
cana-5394	33	66			X
cana-5394	33	67			X
cana-5394	33	68			X
cana-5394	33	69			X
cana-5394	33	70			X
cana-5394	33	71			PROPN
cana-5394	33	72			PROPN
cana-5394	33	73			PROPN
cana-5394	33	74	=	=	PROPN
cana-5394	34	1	+	+	NOUN
cana-5394	34	2	1	1	NUM
cana-5394	34	3	2	2	NUM
cana-5394	34	4	2	2	NUM
cana-5394	34	5	1	1	NUM
cana-5394	34	6	2	2	NUM
cana-5394	34	7	2	2	NUM
cana-5394	34	8	2	2	NUM
cana-5394	34	9	1	1	NUM
cana-5394	34	10	2	2	NUM
cana-5394	34	11	,	,	PUNCT
cana-5394	34	12	,	,	PUNCT
cana-5394	34	13	...	...	PUNCT
cana-5394	34	14	,	,	PUNCT
cana-5394	34	15	)	)	PUNCT
cana-5394	34	16	(	(	PUNCT
cana-5394	34	17	,	,	PUNCT
cana-5394	34	18	,	,	PUNCT
cana-5394	34	19	...	...	PUNCT
cana-5394	34	20	,	,	PUNCT
cana-5394	34	21	)	)	PUNCT
cana-5394	34	22	(	(	PUNCT
cana-5394	34	23	,	,	PUNCT
cana-5394	34	24	,	,	PUNCT
cana-5394	34	25	...	...	PUNCT
cana-5394	34	26	,	,	PUNCT
cana-5394	34	27	)	)	PUNCT
cana-5394	34	28	,	,	PUNCT
cana-5394	34	29	(	(	PUNCT
cana-5394	34	30	n	n	CCONJ
cana-5394	34	31	n	n	PRON
cana-5394	34	32	nd	nd	NOUN
cana-5394	34	33	d	d	X
cana-5394	34	34	d	d	PROPN
cana-5394	34	35			PROPN
cana-5394	34	36			X
cana-5394	34	37			X
cana-5394	34	38			X
cana-5394	34	39			X
cana-5394	34	40			X
cana-5394	34	41			X
cana-5394	34	42			X
cana-5394	34	43			X
cana-5394	34	44			PROPN
cana-5394	34	45			PROPN
cana-5394	34	46			PROPN
cana-5394	34	47	=	=	PROPN
cana-5394	35	1	+	+	NOUN
cana-5394	35	2	1	1	NUM
cana-5394	35	3	2	2	NUM
cana-5394	35	4	1	1	NUM
cana-5394	35	5	2	2	NUM
cana-5394	35	6	1	1	NUM
cana-5394	35	7	2	2	NUM
cana-5394	35	8	,	,	PUNCT
cana-5394	35	9	,	,	PUNCT
cana-5394	35	10	...	...	PUNCT
cana-5394	35	11	,	,	PUNCT
cana-5394	35	12	)	)	PUNCT
cana-5394	35	13	(	(	PUNCT
cana-5394	35	14	,	,	PUNCT
cana-5394	35	15	,	,	PUNCT
cana-5394	35	16	...	...	PUNCT
cana-5394	35	17	,	,	PUNCT
cana-5394	35	18	)	)	PUNCT
cana-5394	35	19	(	(	PUNCT
cana-5394	35	20	,	,	PUNCT
cana-5394	35	21	,	,	PUNCT
cana-5394	35	22	...	...	PUNCT
cana-5394	35	23	,	,	PUNCT
cana-5394	35	24	)	)	PUNCT
cana-5394	35	25	(	(	PUNCT
cana-5394	35	26	n	n	CCONJ
cana-5394	35	27	n	n	CCONJ
cana-5394	35	28	n	n	CCONJ
cana-5394	35	29	n	n	CCONJ
cana-5394	35	30	n	n	ADV
cana-5394	35	31	nd	nd	ADV
cana-5394	35	32	dd	dd	VERB
cana-5394	35	33			X
cana-5394	35	34			X
cana-5394	35	35			X
cana-5394	35	36			X
cana-5394	35	37			X
cana-5394	35	38			X
cana-5394	35	39			X
cana-5394	35	40			X
cana-5394	35	41			X
cana-5394	35	42			PROPN
cana-5394	35	43			PROPN
cana-5394	35	44	=	=	PUNCT
cana-5394	36	1			PROPN
cana-5394	36	2	+	+	CCONJ
cana-5394	36	3			ADV
cana-5394	36	4	hold	hold	VERB
cana-5394	36	5	for	for	ADP
cana-5394	36	6	all	all	PRON
cana-5394	36	7	,	,	PUNCT
cana-5394	36	8	,	,	PUNCT
cana-5394	36	9	j	j	PROPN
cana-5394	36	10	1,2,	1,2,	NUM
cana-5394	36	11	...	...	PUNCT
cana-5394	36	12	,j	,j	PUNCT
cana-5394	37	1	j	j	PROPN
cana-5394	37	2	s	s	PROPN
cana-5394	37	3	n	n	PROPN
cana-5394	37	4			PROPN
cana-5394	37	5			NOUN
cana-5394	37	6	=	=	SYM
cana-5394	38	1	.many	.many	NUM
cana-5394	38	2	authors	author	NOUN
cana-5394	38	3	have	have	AUX
cana-5394	38	4	examined	examine	VERB
cana-5394	38	5	numerous	numerous	ADJ
cana-5394	38	6	algebraic	algebraic	ADJ
cana-5394	38	7	identities	identity	NOUN
cana-5394	38	8	involving	involve	VERB
cana-5394	38	9	the	the	DET
cana-5394	38	10	traces	trace	NOUN
cana-5394	38	11	of	of	ADP
cana-5394	38	12	n	n	NOUN
cana-5394	38	13	-	-	PUNCT
cana-5394	38	14	derivations	derivation	NOUN
cana-5394	38	15	and	and	CCONJ
cana-5394	38	16	have	have	AUX
cana-5394	38	17	uncovered	uncover	VERB
cana-5394	38	18	several	several	ADJ
cana-5394	38	19	insightful	insightful	ADJ
cana-5394	38	20	results	result	NOUN
cana-5394	38	21	(	(	PUNCT
cana-5394	38	22	see[9],[10],[11],[12],[13	see[9],[10],[11],[12],[13	ADJ
cana-5394	38	23	]	]	X
cana-5394	38	24	]	]	X
cana-5394	38	25	)	)	PUNCT
cana-5394	38	26	.	.	PUNCT
cana-5394	39	1	a	a	DET
cana-5394	39	2	symmetric	symmetric	ADJ
cana-5394	39	3	2	2	NUM
cana-5394	39	4	-	-	PUNCT
cana-5394	39	5	derivation	derivation	NOUN
cana-5394	39	6	is	be	AUX
cana-5394	39	7	a	a	DET
cana-5394	39	8	bi	bi	NOUN
cana-5394	39	9	-	-	NOUN
cana-5394	39	10	derivation	derivation	NOUN
cana-5394	39	11	,	,	PUNCT
cana-5394	39	12	a	a	DET
cana-5394	39	13	symmetric	symmetric	ADJ
cana-5394	39	14	3	3	NUM
cana-5394	39	15	-	-	PUNCT
cana-5394	39	16	derivation	derivation	NOUN
cana-5394	39	17	(	(	PUNCT
cana-5394	39	18	or	or	CCONJ
cana-5394	39	19	tri	tri	ADJ
cana-5394	39	20	-	-	NOUN
cana-5394	39	21	derivation	derivation	NOUN
cana-5394	39	22	)	)	PUNCT
cana-5394	39	23	on	on	ADP
cana-5394	39	24	rings	ring	NOUN
cana-5394	39	25	,	,	PUNCT
cana-5394	39	26	and	and	CCONJ
cana-5394	39	27	a	a	DET
cana-5394	39	28	1	1	NUM
cana-5394	39	29	-	-	PUNCT
cana-5394	39	30	derivation	derivation	NOUN
cana-5394	39	31	is	be	AUX
cana-5394	39	32	clearly	clearly	ADV
cana-5394	39	33	a	a	DET
cana-5394	39	34	derivation	derivation	NOUN
cana-5394	39	35	.	.	PUNCT
cana-5394	40	1	posner	posner	NOUN
cana-5394	40	2	's	's	PART
cana-5394	40	3	finding	finding	NOUN
cana-5394	40	4	[	[	X
cana-5394	40	5	14	14	NUM
cana-5394	40	6	]	]	PUNCT
cana-5394	40	7	showed	show	VERB
cana-5394	40	8	that	that	SCONJ
cana-5394	40	9	if	if	SCONJ
cana-5394	40	10	a	a	DET
cana-5394	40	11	prime	prime	ADJ
cana-5394	40	12	ring	ring	NOUN
cana-5394	40	13	has	have	VERB
cana-5394	40	14	a	a	DET
cana-5394	40	15	nonzero	nonzero	NOUN
cana-5394	40	16	centralizing	centralize	VERB
cana-5394	40	17	derivation	derivation	NOUN
cana-5394	40	18	,	,	PUNCT
cana-5394	40	19	it	it	PRON
cana-5394	40	20	becomes	become	VERB
cana-5394	40	21	commutative	commutative	ADJ
cana-5394	40	22	.	.	PUNCT
cana-5394	41	1	this	this	PRON
cana-5394	41	2	led	lead	VERB
cana-5394	41	3	to	to	ADP
cana-5394	41	4	the	the	DET
cana-5394	41	5	study	study	NOUN
cana-5394	41	6	of	of	ADP
cana-5394	41	7	centralizing	centralize	VERB
cana-5394	41	8	maps	map	NOUN
cana-5394	41	9	on	on	ADP
cana-5394	41	10	prime	prime	ADJ
cana-5394	41	11	rings	ring	NOUN
cana-5394	41	12	.	.	PUNCT
cana-5394	42	1	since	since	SCONJ
cana-5394	42	2	then	then	ADV
cana-5394	42	3	,	,	PUNCT
cana-5394	42	4	researchers	researcher	NOUN
cana-5394	42	5	like	like	VERB
cana-5394	42	6	brešar	brešar	ADJ
cana-5394	42	7	[	[	X
cana-5394	42	8	15	15	NUM
cana-5394	42	9	]	]	X
cana-5394	42	10	,	,	PUNCT
cana-5394	42	11	deng	deng	PROPN
cana-5394	42	12	-	-	PUNCT
cana-5394	42	13	bell	bell	PROPN
cana-5394	42	14	[	[	X
cana-5394	42	15	16	16	NUM
cana-5394	42	16	]	]	PUNCT
cana-5394	42	17	,	,	PUNCT
cana-5394	42	18	lanski	lanski	NOUN
cana-5394	42	19	[	[	X
cana-5394	42	20	17	17	NUM
cana-5394	42	21	]	]	PUNCT
cana-5394	42	22	,	,	PUNCT
cana-5394	42	23	and	and	CCONJ
cana-5394	42	24	vukman	vukman	NOUN
cana-5394	43	1	[	[	X
cana-5394	43	2	18	18	NUM
cana-5394	43	3	]	]	PUNCT
cana-5394	43	4	have	have	AUX
cana-5394	43	5	made	make	VERB
cana-5394	43	6	important	important	ADJ
cana-5394	43	7	contributions	contribution	NOUN
cana-5394	43	8	.	.	PUNCT
cana-5394	44	1	vukman	vukman	ADJ
cana-5394	44	2	derived	derive	VERB
cana-5394	44	3	conclusions	conclusion	NOUN
cana-5394	44	4	about	about	ADP
cana-5394	44	5	the	the	DET
cana-5394	44	6	trace	trace	NOUN
cana-5394	44	7	of	of	ADP
cana-5394	44	8	symmetric	symmetric	ADJ
cana-5394	44	9	bi	bi	NOUN
cana-5394	44	10	-	-	NOUN
cana-5394	44	11	derivations	derivation	NOUN
cana-5394	44	12	in	in	ADP
cana-5394	44	13	prime	prime	ADJ
cana-5394	44	14	rings	ring	NOUN
cana-5394	44	15	(	(	PUNCT
cana-5394	44	16	see	see	VERB
cana-5394	44	17	[	[	X
cana-5394	44	18	19,20	19,20	X
cana-5394	44	19	]	]	PUNCT
cana-5394	44	20	for	for	ADP
cana-5394	44	21	more	more	ADJ
cana-5394	44	22	details	detail	NOUN
cana-5394	44	23	)	)	PUNCT
cana-5394	44	24	.	.	PUNCT
cana-5394	45	1	in	in	ADP
cana-5394	45	2	[	[	X
cana-5394	45	3	21	21	NUM
cana-5394	45	4	]	]	PUNCT
cana-5394	45	5	,	,	PUNCT
cana-5394	45	6	ashraf	ashraf	VERB
cana-5394	45	7	formulated	formulate	VERB
cana-5394	45	8	corresponding	corresponding	ADJ
cana-5394	45	9	results	result	NOUN
cana-5394	45	10	for	for	ADP
cana-5394	45	11	semiprime	semiprime	NOUN
cana-5394	45	12	rings	ring	NOUN
cana-5394	45	13	.	.	PUNCT
cana-5394	46	1	moreover	moreover	ADV
cana-5394	46	2	,	,	PUNCT
cana-5394	46	3	ashraf	ashraf	PROPN
cana-5394	46	4	et	et	PROPN
cana-5394	46	5	al	al	PROPN
cana-5394	46	6	.	.	PUNCT
cana-5394	47	1	[	[	X
cana-5394	47	2	22	22	NUM
cana-5394	47	3	,	,	PUNCT
cana-5394	47	4	23	23	NUM
cana-5394	47	5	]	]	PUNCT
cana-5394	47	6	derived	derive	VERB
cana-5394	47	7	the	the	DET
cana-5394	47	8	commutativity	commutativity	NOUN
cana-5394	47	9	of	of	ADP
cana-5394	47	10	rings	ring	NOUN
cana-5394	47	11	allowing	allow	VERB
cana-5394	47	12	n	n	ADP
cana-5394	47	13	-derivations	-derivation	NOUN
cana-5394	47	14	whose	whose	DET
cana-5394	47	15	traces	trace	NOUN
cana-5394	47	16	adhere	adhere	VERB
cana-5394	47	17	to	to	ADP
cana-5394	47	18	particular	particular	ADJ
cana-5394	47	19	polynomial	polynomial	ADJ
cana-5394	47	20	conditions	condition	NOUN
cana-5394	47	21	.	.	PUNCT
cana-5394	48	1	several	several	ADJ
cana-5394	48	2	authors	author	NOUN
cana-5394	48	3	have	have	AUX
cana-5394	48	4	explored	explore	VERB
cana-5394	48	5	different	different	ADJ
cana-5394	48	6	identities	identity	NOUN
cana-5394	48	7	related	relate	VERB
cana-5394	48	8	to	to	ADP
cana-5394	48	9	traces	trace	NOUN
cana-5394	48	10	of	of	ADP
cana-5394	48	11	bi	bi	NOUN
cana-5394	48	12	-	-	NOUN
cana-5394	48	13	derivations	derivation	NOUN
cana-5394	48	14	and	and	CCONJ
cana-5394	48	15	n	n	DET
cana-5394	48	16	-derivations	-derivation	NOUN
cana-5394	48	17	,	,	PUNCT
cana-5394	48	18	yielding	yield	VERB
cana-5394	48	19	numerous	numerous	ADJ
cana-5394	48	20	intriguing	intriguing	ADJ
cana-5394	48	21	findings	finding	NOUN
cana-5394	48	22	(	(	PUNCT
cana-5394	48	23	see	see	VERB
cana-5394	48	24	[	[	X
cana-5394	48	25	24	24	NUM
cana-5394	48	26	-	-	SYM
cana-5394	48	27	26	26	NUM
cana-5394	48	28	]	]	PUNCT
cana-5394	48	29	and	and	CCONJ
cana-5394	48	30	references	reference	NOUN
cana-5394	48	31	therein	therein	ADV
cana-5394	48	32	)	)	PUNCT
cana-5394	48	33	.	.	PUNCT
cana-5394	49	1	the	the	DET
cana-5394	49	2	main	main	ADJ
cana-5394	49	3	objective	objective	NOUN
cana-5394	49	4	of	of	ADP
cana-5394	49	5	this	this	DET
cana-5394	49	6	paper	paper	NOUN
cana-5394	49	7	is	be	AUX
cana-5394	49	8	to	to	PART
cana-5394	49	9	establish	establish	VERB
cana-5394	49	10	comparable	comparable	ADJ
cana-5394	49	11	results	result	NOUN
cana-5394	49	12	on	on	ADP
cana-5394	49	13	the	the	DET
cana-5394	49	14	permutation	permutation	NOUN
cana-5394	49	15	of	of	ADP
cana-5394	49	16	n	n	DET
cana-5394	49	17	derivations	derivation	NOUN
cana-5394	49	18	within	within	ADP
cana-5394	49	19	the	the	DET
cana-5394	49	20	semiprime	semiprime	NOUN
cana-5394	49	21	rings	ring	NOUN
cana-5394	49	22	.	.	PUNCT
cana-5394	50	1	in	in	ADP
cana-5394	50	2	particular	particular	ADJ
cana-5394	50	3	,	,	PUNCT
cana-5394	50	4	we	we	PRON
cana-5394	50	5	examine	examine	VERB
cana-5394	50	6	the	the	DET
cana-5394	50	7	structure	structure	NOUN
cana-5394	50	8	of	of	ADP
cana-5394	50	9	semiprime	semiprime	NOUN
cana-5394	50	10	rings	ring	NOUN
cana-5394	50	11	and	and	CCONJ
cana-5394	50	12	identify	identify	VERB
cana-5394	50	13	the	the	DET
cana-5394	50	14	forms	form	NOUN
cana-5394	50	15	of	of	ADP
cana-5394	50	16	mappings	mapping	NOUN
cana-5394	50	17	(	(	PUNCT
cana-5394	50	18	traces	trace	NOUN
cana-5394	50	19	of	of	ADP
cana-5394	50	20	n	n	PRON
cana-5394	50	21	-derivations	-derivation	NOUN
cana-5394	50	22	)	)	PUNCT
cana-5394	50	23	that	that	PRON
cana-5394	50	24	satisfy	satisfy	VERB
cana-5394	50	25	to	to	ADP
cana-5394	50	26	specific	specific	ADJ
cana-5394	50	27	functional	functional	ADJ
cana-5394	50	28	identities	identity	NOUN
cana-5394	50	29	.	.	PUNCT
cana-5394	51	1	more	more	ADV
cana-5394	51	2	precisely	precisely	ADV
cana-5394	51	3	,	,	PUNCT
cana-5394	51	4	we	we	PRON
cana-5394	51	5	establish	establish	VERB
cana-5394	51	6	that	that	SCONJ
cana-5394	51	7	:	:	PUNCT
cana-5394	51	8	for	for	ADP
cana-5394	51	9	a	a	DET
cana-5394	51	10	fixed	fix	VERB
cana-5394	51	11	integer	integer	NOUN
cana-5394	51	12	2,n	2,n	NUM
cana-5394	51	13			NUM
cana-5394	51	14	let	let	VERB
cana-5394	51	15	s	s	PRON
cana-5394	51	16	be	be	AUX
cana-5394	51	17	an	an	DET
cana-5394	51	18	!	!	PUNCT
cana-5394	51	19	n	n	CCONJ
cana-5394	51	20	-torsion	-torsion	PROPN
cana-5394	51	21	free	free	ADJ
cana-5394	51	22	semiprime	semiprime	NOUN
cana-5394	51	23	ring	ring	NOUN
cana-5394	51	24	and	and	CCONJ
cana-5394	51	25	m	m	AUX
cana-5394	51	26	be	be	AUX
cana-5394	51	27	a	a	DET
cana-5394	51	28	nonzero	nonzero	NOUN
cana-5394	51	29	ideal	ideal	NOUN
cana-5394	51	30	of	of	ADP
cana-5394	51	31	s	s	PRON
cana-5394	51	32	.	.	PUNCT
cana-5394	52	1	suppose	suppose	VERB
cana-5394	52	2	that	that	SCONJ
cana-5394	52	3	s	s	VERB
cana-5394	52	4	admits	admit	VERB
cana-5394	52	5	a	a	DET
cana-5394	52	6	nonzero	nonzero	ADJ
cana-5394	52	7	symmetric	symmetric	ADJ
cana-5394	52	8	n	n	PRON
cana-5394	52	9	derivation	derivation	NOUN
cana-5394	52	10	:	:	PUNCT
cana-5394	52	11	s	s	VERB
cana-5394	52	12	nd	nd	INTJ
cana-5394	52	13	s→	s→	NOUN
cana-5394	52	14	with	with	ADP
cana-5394	52	15	trace	trace	NOUN
cana-5394	52	16	:	:	PUNCT
cana-5394	52	17	s	s	VERB
cana-5394	52	18	s	s	PROPN
cana-5394	52	19	→	→	PUNCT
cana-5394	52	20	then	then	ADV
cana-5394	52	21	s	s	VERB
cana-5394	52	22	has	have	VERB
cana-5394	52	23	a	a	DET
cana-5394	52	24	nonzero	nonzero	ADJ
cana-5394	52	25	central	central	ADJ
cana-5394	52	26	ideal	ideal	NOUN
cana-5394	52	27	.	.	PUNCT
cana-5394	53	1	2	2	X
cana-5394	53	2	.	.	X
cana-5394	53	3	preliminary	preliminary	ADJ
cana-5394	53	4	results	result	NOUN
cana-5394	53	5	:	:	PUNCT
cana-5394	53	6	the	the	DET
cana-5394	53	7	following	follow	VERB
cana-5394	53	8	supporting	support	VERB
cana-5394	53	9	implications	implication	NOUN
cana-5394	53	10	are	be	AUX
cana-5394	53	11	crucial	crucial	ADJ
cana-5394	53	12	to	to	PART
cana-5394	53	13	derive	derive	VERB
cana-5394	53	14	the	the	DET
cana-5394	53	15	primary	primary	ADJ
cana-5394	53	16	results	result	NOUN
cana-5394	53	17	of	of	ADP
cana-5394	53	18	this	this	DET
cana-5394	53	19	articlewe	articlewe	NOUN
cana-5394	53	20	begin	begin	VERB
cana-5394	53	21	by	by	ADP
cana-5394	53	22	presenting	present	VERB
cana-5394	53	23	several	several	ADJ
cana-5394	53	24	well	well	ADV
cana-5394	53	25	-	-	PUNCT
cana-5394	53	26	established	establish	VERB
cana-5394	53	27	results	result	NOUN
cana-5394	53	28	.	.	PUNCT
cana-5394	54	1	lemma	lemma	PROPN
cana-5394	54	2	2.1	2.1	NUM
cana-5394	54	3	.	.	PUNCT
cana-5394	55	1	[	[	X
cana-5394	55	2	8	8	NUM
cana-5394	55	3	]	]	PUNCT
cana-5394	55	4	let	let	VERB
cana-5394	55	5	n	n	PRON
cana-5394	55	6	be	be	AUX
cana-5394	55	7	a	a	DET
cana-5394	55	8	fixed	fix	VERB
cana-5394	55	9	positive	positive	ADJ
cana-5394	55	10	integer	integer	NOUN
cana-5394	55	11	and	and	CCONJ
cana-5394	55	12	s	s	VERB
cana-5394	55	13	an	an	PRON
cana-5394	55	14	!	!	PUNCT
cana-5394	55	15	n	n	CCONJ
cana-5394	55	16	-torsion	-torsion	PROPN
cana-5394	55	17	free	free	ADJ
cana-5394	55	18	ring	ring	NOUN
cana-5394	55	19	.	.	PUNCT
cana-5394	56	1	assume	assume	VERB
cana-5394	56	2	that	that	SCONJ
cana-5394	56	3	1	1	NUM
cana-5394	56	4	2	2	NUM
cana-5394	56	5	,	,	PUNCT
cana-5394	56	6	,	,	PUNCT
cana-5394	56	7	.	.	PUNCT
cana-5394	56	8	.	.	PUNCT
cana-5394	57	1	.	.	PUNCT
cana-5394	58	1	,	,	PUNCT
cana-5394	58	2	nx	nx	X
cana-5394	58	3	x	x	X
cana-5394	58	4	x	x	X
cana-5394	58	5	s	s	PROPN
cana-5394	58	6	satisfy	satisfy	VERB
cana-5394	58	7	2	2	NUM
cana-5394	58	8	1	1	NUM
cana-5394	58	9	2	2	NUM
cana-5394	58	10	·	·	PUNCT
cana-5394	58	11	·	·	PUNCT
cana-5394	58	12	·	·	PUNCT
cana-5394	58	13	0n	0n	NOUN
cana-5394	58	14	nx	nx	NUM
cana-5394	58	15	x	x	X
cana-5394	58	16	x	x	PROPN
cana-5394	59	1			PROPN
cana-5394	60	1	+	+	NOUN
cana-5394	61	1	+	+	CCONJ
cana-5394	61	2	+	+	CCONJ
cana-5394	61	3	=	=	SYM
cana-5394	61	4	(	(	PUNCT
cana-5394	61	5	or	or	CCONJ
cana-5394	61	6	(	(	PUNCT
cana-5394	61	7	)	)	PUNCT
cana-5394	61	8	z	z	NOUN
cana-5394	61	9	s	s	PROPN
cana-5394	61	10	)	)	PUNCT
cana-5394	61	11	for	for	ADP
cana-5394	61	12	1,2,	1,2,	NUM
cana-5394	61	13	...	...	PUNCT
cana-5394	61	14	,n	,n	PUNCT
cana-5394	61	15	=	=	PUNCT
cana-5394	61	16	.	.	PUNCT
cana-5394	62	1	then	then	ADV
cana-5394	62	2	,	,	PUNCT
cana-5394	62	3	0jx	0jx	NOUN
cana-5394	62	4	=	=	SYM
cana-5394	62	5	(	(	PUNCT
cana-5394	62	6	or	or	CCONJ
cana-5394	62	7	(	(	PUNCT
cana-5394	62	8	)	)	PUNCT
cana-5394	62	9	z	z	NOUN
cana-5394	62	10	s	s	PROPN
cana-5394	62	11	)	)	PUNCT
cana-5394	62	12	for	for	ADP
cana-5394	62	13	1,2	1,2	NUM
cana-5394	62	14	,	,	PUNCT
cana-5394	62	15	...	...	PUNCT
cana-5394	62	16	,	,	PUNCT
cana-5394	62	17	.j	.j	PROPN
cana-5394	62	18	n=	n=	ADJ
cana-5394	62	19	communications	communication	NOUN
cana-5394	62	20	on	on	ADP
cana-5394	62	21	applied	apply	VERB
cana-5394	62	22	nonlinear	nonlinear	ADJ
cana-5394	62	23	analysis	analysis	NOUN
cana-5394	62	24	issn	issn	NOUN
cana-5394	62	25	:	:	PUNCT
cana-5394	62	26	1074	1074	NUM
cana-5394	62	27	-	-	PUNCT
cana-5394	62	28	133x	133x	NUM
cana-5394	62	29	vol	vol	NOUN
cana-5394	62	30	32	32	NUM
cana-5394	62	31	no	no	NOUN
cana-5394	62	32	.	.	NOUN
cana-5394	62	33	3	3	NUM
cana-5394	62	34	(	(	PUNCT
cana-5394	62	35	2025	2025	NUM
cana-5394	62	36	)	)	PUNCT
cana-5394	62	37	932	932	NUM
cana-5394	62	38	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	62	39	lemma	lemma	PROPN
cana-5394	62	40	2.2	2.2	NUM
cana-5394	62	41	.	.	PUNCT
cana-5394	63	1	[	[	X
cana-5394	63	2	[	[	X
cana-5394	63	3	6	6	NUM
cana-5394	63	4	]	]	PUNCT
cana-5394	63	5	,	,	PUNCT
cana-5394	63	6	lemma	lemma	PROPN
cana-5394	63	7	3	3	X
cana-5394	63	8	]	]	X
cana-5394	63	9	let	let	VERB
cana-5394	63	10	s	s	PRON
cana-5394	63	11	be	be	AUX
cana-5394	63	12	a	a	DET
cana-5394	63	13	2	2	NUM
cana-5394	63	14	-	-	PUNCT
cana-5394	63	15	torsion	torsion	NOUN
cana-5394	63	16	free	free	ADJ
cana-5394	63	17	semiprime	semiprime	NOUN
cana-5394	63	18	ring	ring	NOUN
cana-5394	63	19	and	and	CCONJ
cana-5394	63	20	m	m	AUX
cana-5394	63	21	be	be	AUX
cana-5394	63	22	a	a	DET
cana-5394	63	23	nonzero	nonzero	NOUN
cana-5394	63	24	ideal	ideal	NOUN
cana-5394	63	25	of	of	ADP
cana-5394	63	26	.s	.s	NOUN
cana-5394	63	27	if	if	SCONJ
cana-5394	63	28	[	[	PUNCT
cana-5394	63	29	,	,	PUNCT
cana-5394	63	30	]	]	X
cana-5394	63	31	(	(	PUNCT
cana-5394	63	32	)	)	PUNCT
cana-5394	63	33	,	,	PUNCT
cana-5394	63	34	m	m	VERB
cana-5394	63	35	m	m	VERB
cana-5394	63	36	z	z	NOUN
cana-5394	63	37	s	s	NOUN
cana-5394	63	38	then	then	ADV
cana-5394	63	39	s	s	VERB
cana-5394	63	40	contains	contain	VERB
cana-5394	63	41	a	a	DET
cana-5394	63	42	nonzero	nonzero	ADJ
cana-5394	63	43	central	central	ADJ
cana-5394	63	44	ideal	ideal	NOUN
cana-5394	63	45	.	.	PUNCT
cana-5394	64	1	lemma	lemma	PROPN
cana-5394	64	2	2.3	2.3	NUM
cana-5394	64	3	.	.	PUNCT
cana-5394	65	1	[	[	X
cana-5394	65	2	[	[	X
cana-5394	65	3	6	6	NUM
cana-5394	65	4	]	]	PUNCT
cana-5394	65	5	,	,	PUNCT
cana-5394	65	6	lemma	lemma	PROPN
cana-5394	65	7	4	4	X
cana-5394	65	8	]	]	PUNCT
cana-5394	65	9	let	let	VERB
cana-5394	65	10	s	s	PRON
cana-5394	65	11	be	be	AUX
cana-5394	65	12	a	a	DET
cana-5394	65	13	2	2	NUM
cana-5394	65	14	-	-	PUNCT
cana-5394	65	15	torsion	torsion	NOUN
cana-5394	65	16	free	free	ADJ
cana-5394	65	17	semiprime	semiprime	NOUN
cana-5394	65	18	ring	ring	NOUN
cana-5394	65	19	and	and	CCONJ
cana-5394	65	20	m	m	AUX
cana-5394	65	21	be	be	AUX
cana-5394	65	22	a	a	DET
cana-5394	65	23	nonzero	nonzero	NOUN
cana-5394	65	24	ideal	ideal	NOUN
cana-5394	65	25	of	of	ADP
cana-5394	65	26	.s	.s	NOUN
cana-5394	65	27	if	if	SCONJ
cana-5394	65	28	(	(	PUNCT
cana-5394	65	29	)	)	PUNCT
cana-5394	65	30	,	,	PUNCT
cana-5394	65	31	m	m	VERB
cana-5394	65	32	m	m	VERB
cana-5394	65	33	z	z	NOUN
cana-5394	65	34	s	s	NOUN
cana-5394	65	35	then	then	ADV
cana-5394	65	36	s	s	VERB
cana-5394	65	37	contains	contain	VERB
cana-5394	65	38	a	a	DET
cana-5394	65	39	nonzero	nonzero	ADJ
cana-5394	65	40	central	central	ADJ
cana-5394	65	41	ideal	ideal	NOUN
cana-5394	65	42	.	.	PUNCT
cana-5394	66	1	we	we	PRON
cana-5394	66	2	start	start	VERB
cana-5394	66	3	our	our	PRON
cana-5394	66	4	investigation	investigation	NOUN
cana-5394	66	5	of	of	ADP
cana-5394	66	6	symmetric	symmetric	ADJ
cana-5394	66	7	n	n	CCONJ
cana-5394	66	8	-	-	PUNCT
cana-5394	66	9	derivations	derivation	NOUN
cana-5394	66	10	with	with	ADP
cana-5394	66	11	the	the	DET
cana-5394	66	12	following	follow	VERB
cana-5394	66	13	result	result	NOUN
cana-5394	66	14	.	.	PUNCT
cana-5394	67	1	3	3	X
cana-5394	67	2	.	.	X
cana-5394	67	3	main	main	ADJ
cana-5394	67	4	results	result	NOUN
cana-5394	67	5	:	:	PUNCT
cana-5394	67	6	theorem	theorem	VERB
cana-5394	67	7	3.1	3.1	NUM
cana-5394	67	8	.	.	PUNCT
cana-5394	68	1	for	for	ADP
cana-5394	68	2	a	a	DET
cana-5394	68	3	fixed	fix	VERB
cana-5394	68	4	integer	integer	NOUN
cana-5394	68	5	2,n	2,n	NUM
cana-5394	68	6			NUM
cana-5394	68	7	let	let	VERB
cana-5394	68	8	s	s	PRON
cana-5394	68	9	be	be	AUX
cana-5394	68	10	an	an	DET
cana-5394	68	11	!	!	PUNCT
cana-5394	68	12	n	n	CCONJ
cana-5394	68	13	-torsion	-torsion	PROPN
cana-5394	68	14	free	free	ADJ
cana-5394	68	15	semiprime	semiprime	NOUN
cana-5394	68	16	ring	ring	NOUN
cana-5394	68	17	and	and	CCONJ
cana-5394	68	18	m	m	AUX
cana-5394	68	19	be	be	AUX
cana-5394	68	20	a	a	DET
cana-5394	68	21	nonzero	nonzero	NOUN
cana-5394	68	22	ideal	ideal	NOUN
cana-5394	68	23	of	of	ADP
cana-5394	68	24	.s	.s	NOUN
cana-5394	68	25	suppose	suppose	VERB
cana-5394	68	26	that	that	SCONJ
cana-5394	68	27	s	s	VERB
cana-5394	68	28	admits	admit	VERB
cana-5394	68	29	a	a	DET
cana-5394	68	30	nonzero	nonzero	ADJ
cana-5394	68	31	symmetric	symmetric	ADJ
cana-5394	68	32	n	n	CCONJ
cana-5394	68	33	-derivation	-derivation	NOUN
cana-5394	68	34	:	:	PUNCT
cana-5394	68	35	s	s	VERB
cana-5394	68	36	nd	nd	INTJ
cana-5394	68	37	s→	s→	NOUN
cana-5394	68	38	with	with	ADP
cana-5394	68	39	trace	trace	NOUN
cana-5394	68	40	:	:	PUNCT
cana-5394	68	41	s	s	AUX
cana-5394	68	42	s	s	NOUN
cana-5394	68	43	→	→	SYM
cana-5394	68	44	satisfying	satisfy	VERB
cana-5394	68	45	(	(	PUNCT
cana-5394	68	46	)	)	PUNCT
cana-5394	68	47	(	(	PUNCT
cana-5394	68	48	)	)	PUNCT
cana-5394	68	49	(	(	PUNCT
cana-5394	68	50	)	)	PUNCT
cana-5394	68	51	z	z	NOUN
cana-5394	69	1	s	s	PROPN
cana-5394	69	2			PROPN
cana-5394	69	3			NUM
cana-5394	69	4			PROPN
cana-5394	69	5			ADJ
cana-5394	69	6			NOUN
cana-5394	69	7	for	for	ADP
cana-5394	69	8	all	all	PRON
cana-5394	69	9	,	,	PUNCT
cana-5394	69	10	,	,	PUNCT
cana-5394	69	11	m	m	PROPN
cana-5394	69	12			PROPN
cana-5394	69	13			PROPN
cana-5394	69	14	then	then	ADV
cana-5394	69	15	s	s	AUX
cana-5394	69	16	has	have	VERB
cana-5394	69	17	a	a	DET
cana-5394	69	18	nonzero	nonzero	ADJ
cana-5394	69	19	central	central	ADJ
cana-5394	69	20	ideal	ideal	NOUN
cana-5394	69	21	.	.	PUNCT
cana-5394	70	1	proof	proof	NOUN
cana-5394	70	2	:	:	PUNCT
cana-5394	70	3	it	it	PRON
cana-5394	70	4	is	be	AUX
cana-5394	70	5	assumed	assume	VERB
cana-5394	70	6	that	that	SCONJ
cana-5394	70	7	(	(	PUNCT
cana-5394	70	8	)	)	PUNCT
cana-5394	70	9	(	(	PUNCT
cana-5394	70	10	)	)	PUNCT
cana-5394	70	11	(	(	PUNCT
cana-5394	70	12	)	)	PUNCT
cana-5394	71	1	z	z	NOUN
cana-5394	71	2	s	s	PROPN
cana-5394	71	3			PROPN
cana-5394	71	4			NUM
cana-5394	71	5			PROPN
cana-5394	71	6			ADJ
cana-5394	71	7			NOUN
cana-5394	71	8	for	for	ADP
cana-5394	71	9	all	all	PRON
cana-5394	71	10	,	,	PUNCT
cana-5394	71	11	.m	.m	PUNCT
cana-5394	72	1			PROPN
cana-5394	72	2			NOUN
cana-5394	72	3	(	(	PUNCT
cana-5394	72	4	3.1	3.1	NUM
cana-5394	72	5	)	)	PUNCT
cana-5394	72	6	replacing	replace	VERB
cana-5394	72	7			PROPN
cana-5394	72	8	by	by	ADP
cana-5394	72	9	mk	mk	NOUN
cana-5394	72	10	+	+	X
cana-5394	72	11	for	for	ADP
cana-5394	72	12	k	k	PROPN
cana-5394	72	13	m	m	PROPN
cana-5394	72	14	in	in	ADP
cana-5394	72	15	(	(	PUNCT
cana-5394	72	16	3.1	3.1	NUM
cana-5394	72	17	)	)	PUNCT
cana-5394	72	18	and	and	CCONJ
cana-5394	72	19	1	1	NUM
cana-5394	72	20	1,m	1,m	NOUN
cana-5394	72	21	n	n	X
cana-5394	72	22			NOUN
cana-5394	72	23	−	−	NOUN
cana-5394	73	1	we	we	PRON
cana-5394	73	2	have	have	VERB
cana-5394	73	3	(	(	PUNCT
cana-5394	73	4	)	)	PUNCT
cana-5394	73	5	1	1	NUM
cana-5394	73	6	1	1	NUM
cana-5394	73	7	(	(	PUNCT
cana-5394	73	8	)	)	PUNCT
cana-5394	73	9	(	(	PUNCT
cana-5394	73	10	(	(	PUNCT
cana-5394	73	11	)	)	PUNCT
cana-5394	73	12	(	(	PUNCT
cana-5394	73	13	)	)	PUNCT
cana-5394	74	1	d	d	X
cana-5394	74	2	(	(	PUNCT
cana-5394	74	3	,	,	PUNCT
cana-5394	74	4	...	...	PUNCT
cana-5394	74	5	,	,	PUNCT
cana-5394	74	6	,	,	PUNCT
cana-5394	74	7	,	,	PUNCT
cana-5394	74	8	...	...	PUNCT
cana-5394	74	9	,	,	PUNCT
cana-5394	74	10	)	)	PUNCT
cana-5394	74	11	)	)	PUNCT
cana-5394	75	1	(	(	PUNCT
cana-5394	75	2	)	)	PUNCT
cana-5394	75	3	n	n	CCONJ
cana-5394	75	4	n	n	PROPN
cana-5394	76	1	t	t	NOUN
cana-5394	76	2	t	t	PROPN
cana-5394	76	3	t	t	PROPN
cana-5394	76	4	timesn	timesn	PROPN
cana-5394	76	5	t	t	PROPN
cana-5394	76	6	times	times	PROPN
cana-5394	76	7	mk	mk	PROPN
cana-5394	76	8	c	c	PROPN
cana-5394	76	9	mk	mk	PROPN
cana-5394	76	10	mk	mk	PROPN
cana-5394	76	11	mk	mk	PROPN
cana-5394	76	12	z	z	PROPN
cana-5394	76	13	s	s	PROPN
cana-5394	76	14			PROPN
cana-5394	77	1			NUM
cana-5394	77	2			PROPN
cana-5394	77	3			PROPN
cana-5394	77	4			PROPN
cana-5394	77	5			PROPN
cana-5394	77	6			PUNCT
cana-5394	77	7			PROPN
cana-5394	77	8	−	−	NOUN
cana-5394	77	9	=	=	SYM
cana-5394	77	10	−−	−−	NOUN
cana-5394	77	11	−	−	NOUN
cana-5394	78	1	+	+	CCONJ
cana-5394	78	2	+	+	CCONJ
cana-5394	78	3			PROPN
cana-5394	78	4			PROPN
cana-5394	78	5			NOUN
cana-5394	78	6	for	for	ADP
cana-5394	78	7	all	all	PRON
cana-5394	78	8	,	,	PUNCT
cana-5394	78	9	,	,	PUNCT
cana-5394	78	10	.k	.k	PROPN
cana-5394	79	1	m	m	PROPN
cana-5394	80	1			PROPN
cana-5394	80	2			NOUN
cana-5394	80	3	(	(	PUNCT
cana-5394	80	4	3.2	3.2	NUM
cana-5394	80	5	)	)	PUNCT
cana-5394	80	6	with	with	ADP
cana-5394	80	7	the	the	DET
cana-5394	80	8	given	give	VERB
cana-5394	80	9	condition	condition	NOUN
cana-5394	80	10	,	,	PUNCT
cana-5394	80	11	we	we	PRON
cana-5394	80	12	arrive	arrive	VERB
cana-5394	80	13	at	at	ADP
cana-5394	80	14	(	(	PUNCT
cana-5394	80	15	)	)	PUNCT
cana-5394	80	16	1	1	NUM
cana-5394	80	17	1	1	NUM
cana-5394	80	18	(	(	PUNCT
cana-5394	80	19	)	)	PUNCT
cana-5394	80	20	d	d	NOUN
cana-5394	80	21	(	(	PUNCT
cana-5394	80	22	,	,	PUNCT
cana-5394	80	23	...	...	PUNCT
cana-5394	80	24	,	,	PUNCT
cana-5394	80	25	,	,	PUNCT
cana-5394	80	26	,	,	PUNCT
cana-5394	80	27	...	...	PUNCT
cana-5394	80	28	,	,	PUNCT
cana-5394	80	29	)	)	PUNCT
cana-5394	80	30	(	(	PUNCT
cana-5394	80	31	)	)	PUNCT
cana-5394	81	1	n	n	CCONJ
cana-5394	81	2	n	n	PROPN
cana-5394	81	3	t	t	NOUN
cana-5394	81	4	t	t	PROPN
cana-5394	81	5	t	t	PROPN
cana-5394	81	6	timesn	timesn	PROPN
cana-5394	81	7	t	t	PROPN
cana-5394	81	8	times	times	PROPN
cana-5394	81	9	c	c	PROPN
cana-5394	81	10	mk	mk	PROPN
cana-5394	81	11	mk	mk	PROPN
cana-5394	81	12	z	z	PROPN
cana-5394	81	13	s	s	PROPN
cana-5394	81	14			PROPN
cana-5394	82	1			PROPN
cana-5394	82	2			PROPN
cana-5394	82	3	−	−	PROPN
cana-5394	82	4	=	=	PUNCT
cana-5394	82	5	−−	−−	NOUN
cana-5394	82	6	−	−	NOUN
cana-5394	82	7			NOUN
cana-5394	82	8	for	for	ADP
cana-5394	82	9	all	all	PRON
cana-5394	82	10	,	,	PUNCT
cana-5394	82	11	,	,	PUNCT
cana-5394	82	12	.k	.k	PROPN
cana-5394	83	1	m	m	PROPN
cana-5394	84	1			PROPN
cana-5394	84	2			NOUN
cana-5394	84	3	using	use	VERB
cana-5394	84	4	lemma2.1	lemma2.1	PROPN
cana-5394	84	5	,	,	PUNCT
cana-5394	84	6	we	we	PRON
cana-5394	84	7	see	see	VERB
cana-5394	84	8	that	that	PRON
cana-5394	84	9	(	(	PUNCT
cana-5394	84	10	)	)	PUNCT
cana-5394	84	11	(	(	PUNCT
cana-5394	84	12	,	,	PUNCT
cana-5394	84	13	...	...	PUNCT
cana-5394	84	14	,	,	PUNCT
cana-5394	84	15	,	,	PUNCT
cana-5394	84	16	)	)	PUNCT
cana-5394	84	17	(	(	PUNCT
cana-5394	84	18	)	)	PUNCT
cana-5394	84	19	n	n	NOUN
cana-5394	85	1	d	d	X
cana-5394	85	2	k	k	PROPN
cana-5394	85	3	z	z	NOUN
cana-5394	85	4	s	s	PROPN
cana-5394	85	5			PROPN
cana-5394	86	1			PROPN
cana-5394	86	2			PROPN
cana-5394	86	3			NOUN
cana-5394	86	4	for	for	ADP
cana-5394	86	5	all	all	PRON
cana-5394	86	6	,	,	PUNCT
cana-5394	86	7	,	,	PUNCT
cana-5394	86	8	.k	.k	PROPN
cana-5394	87	1	m	m	PROPN
cana-5394	88	1			PROPN
cana-5394	88	2			NOUN
cana-5394	88	3	since	since	SCONJ
cana-5394	88	4	s	s	PRON
cana-5394	88	5	is	be	AUX
cana-5394	88	6	!	!	PUNCT
cana-5394	89	1	n	n	X
cana-5394	89	2	-torsion	-torsion	NOUN
cana-5394	89	3	free	free	ADJ
cana-5394	89	4	,	,	PUNCT
cana-5394	89	5	we	we	PRON
cana-5394	89	6	have	have	VERB
cana-5394	89	7	(	(	PUNCT
cana-5394	89	8	)	)	PUNCT
cana-5394	89	9	(	(	PUNCT
cana-5394	89	10	,	,	PUNCT
cana-5394	89	11	...	...	PUNCT
cana-5394	89	12	,	,	PUNCT
cana-5394	89	13	,	,	PUNCT
cana-5394	89	14	)	)	PUNCT
cana-5394	89	15	(	(	PUNCT
cana-5394	89	16	)	)	PUNCT
cana-5394	90	1	d	d	X
cana-5394	90	2	k	k	X
cana-5394	90	3	z	z	NOUN
cana-5394	90	4	s	s	PROPN
cana-5394	90	5			PROPN
cana-5394	90	6			PROPN
cana-5394	90	7			PROPN
cana-5394	90	8			NOUN
cana-5394	90	9	for	for	ADP
cana-5394	90	10	all	all	PRON
cana-5394	90	11	,	,	PUNCT
cana-5394	90	12	,	,	PUNCT
cana-5394	90	13	.k	.k	PROPN
cana-5394	90	14	m	m	PROPN
cana-5394	91	1			PROPN
cana-5394	91	2			NOUN
cana-5394	91	3	(	(	PUNCT
cana-5394	91	4	3.3	3.3	NUM
cana-5394	91	5	)	)	PUNCT
cana-5394	91	6	writing	write	VERB
cana-5394	91	7			PROPN
cana-5394	91	8	in	in	ADP
cana-5394	91	9	place	place	NOUN
cana-5394	91	10	of	of	ADP
cana-5394	91	11	k	k	PROPN
cana-5394	91	12	in	in	ADP
cana-5394	91	13	(	(	PUNCT
cana-5394	91	14	3.3	3.3	NUM
cana-5394	91	15	)	)	PUNCT
cana-5394	91	16	,	,	PUNCT
cana-5394	91	17	we	we	PRON
cana-5394	91	18	get	get	VERB
cana-5394	91	19	(	(	PUNCT
cana-5394	91	20	)	)	PUNCT
cana-5394	91	21	(	(	PUNCT
cana-5394	91	22	)	)	PUNCT
cana-5394	91	23	(	(	PUNCT
cana-5394	91	24	)	)	PUNCT
cana-5394	91	25	z	z	NOUN
cana-5394	91	26	s	s	PROPN
cana-5394	91	27			PROPN
cana-5394	91	28			NUM
cana-5394	91	29			PROPN
cana-5394	91	30			NOUN
cana-5394	91	31	for	for	ADP
cana-5394	91	32	all	all	PRON
cana-5394	91	33	,	,	PUNCT
cana-5394	91	34	.m	.m	PUNCT
cana-5394	92	1			PROPN
cana-5394	92	2			NOUN
cana-5394	92	3	(	(	PUNCT
cana-5394	92	4	3.4	3.4	NUM
cana-5394	92	5	)	)	PUNCT
cana-5394	92	6	by	by	ADP
cana-5394	92	7	utilizing	utilize	VERB
cana-5394	92	8	the	the	DET
cana-5394	92	9	hypothesis	hypothesis	NOUN
cana-5394	92	10	,	,	PUNCT
cana-5394	92	11	we	we	PRON
cana-5394	92	12	obtain	obtain	VERB
cana-5394	92	13	that	that	PRON
cana-5394	92	14	(	(	PUNCT
cana-5394	92	15	)	)	PUNCT
cana-5394	92	16	z	z	NOUN
cana-5394	92	17	s	s	VERB
cana-5394	92	18			NOUN
cana-5394	92	19	for	for	ADP
cana-5394	92	20	all	all	PRON
cana-5394	92	21	,	,	PUNCT
cana-5394	92	22	.m	.m	PUNCT
cana-5394	93	1			PROPN
cana-5394	93	2			NOUN
cana-5394	93	3	(	(	PUNCT
cana-5394	93	4	3.5	3.5	NUM
cana-5394	93	5	)	)	PUNCT
cana-5394	93	6	commuting	commute	VERB
cana-5394	93	7	with	with	ADP
cana-5394	93	8	,	,	PUNCT
cana-5394	93	9	r	r	NOUN
cana-5394	93	10	s	s	NOUN
cana-5394	93	11	we	we	PRON
cana-5394	93	12	arrive	arrive	VERB
cana-5394	93	13	[	[	PUNCT
cana-5394	93	14	,	,	PUNCT
cana-5394	93	15	]	]	PUNCT
cana-5394	93	16	0r	0r	PROPN
cana-5394	93	17	=	=	PUNCT
cana-5394	93	18	for	for	ADP
cana-5394	93	19	all	all	PRON
cana-5394	93	20	,	,	PUNCT
cana-5394	93	21	,	,	PUNCT
cana-5394	93	22	m	m	PROPN
cana-5394	93	23			PROPN
cana-5394	93	24			PROPN
cana-5394	93	25	.r	.r	NOUN
cana-5394	93	26	s	s	PROPN
cana-5394	93	27	(	(	PUNCT
cana-5394	93	28	3.6	3.6	NUM
cana-5394	93	29	)	)	PUNCT
cana-5394	93	30	and	and	CCONJ
cana-5394	93	31	so	so	ADV
cana-5394	93	32	,	,	PUNCT
cana-5394	93	33	[	[	PUNCT
cana-5394	93	34	,	,	PUNCT
cana-5394	93	35	]	]	PUNCT
cana-5394	93	36	0	0	PUNCT
cana-5394	93	37	[	[	PUNCT
cana-5394	93	38	,	,	PUNCT
cana-5394	93	39	]	]	X
cana-5394	93	40	r	r	NOUN
cana-5394	93	41	r	r	NOUN
cana-5394	93	42			PROPN
cana-5394	93	43			PROPN
cana-5394	93	44	+	+	NOUN
cana-5394	93	45	=	=	PUNCT
cana-5394	93	46	for	for	ADP
cana-5394	93	47	all	all	PRON
cana-5394	93	48	,	,	PUNCT
cana-5394	93	49	,	,	PUNCT
cana-5394	93	50	m	m	PROPN
cana-5394	93	51			PROPN
cana-5394	93	52			PROPN
cana-5394	93	53	.r	.r	NOUN
cana-5394	94	1	s	s	PROPN
cana-5394	94	2	(	(	PUNCT
cana-5394	94	3	3.7	3.7	NUM
cana-5394	94	4	)	)	PUNCT
cana-5394	94	5	replacing	replace	VERB
cana-5394	94	6			PROPN
cana-5394	94	7	by	by	ADP
cana-5394	94	8	k	k	NOUN
cana-5394	94	9	in	in	ADP
cana-5394	94	10	(	(	PUNCT
cana-5394	94	11	3.7	3.7	NUM
cana-5394	94	12	)	)	PUNCT
cana-5394	94	13	and	and	CCONJ
cana-5394	94	14	using	use	VERB
cana-5394	94	15	(	(	PUNCT
cana-5394	94	16	3.7	3.7	NUM
cana-5394	94	17	)	)	PUNCT
cana-5394	94	18	,	,	PUNCT
cana-5394	94	19	we	we	PRON
cana-5394	94	20	see	see	VERB
cana-5394	94	21	that	that	SCONJ
cana-5394	94	22	[	[	PUNCT
cana-5394	94	23	,	,	PUNCT
cana-5394	94	24	]	]	PUNCT
cana-5394	94	25	0k	0k	NOUN
cana-5394	94	26	r	r	NOUN
cana-5394	94	27	=	=	PUNCT
cana-5394	94	28	for	for	ADP
cana-5394	94	29	all	all	PRON
cana-5394	95	1	,	,	PUNCT
cana-5394	95	2	,	,	PUNCT
cana-5394	95	3	,	,	PUNCT
cana-5394	95	4	k	k	PROPN
cana-5394	95	5	m	m	PROPN
cana-5394	95	6			PROPN
cana-5394	95	7			PROPN
cana-5394	95	8	.r	.r	NOUN
cana-5394	96	1	s	s	ADJ
cana-5394	96	2	communications	communication	NOUN
cana-5394	96	3	on	on	ADP
cana-5394	96	4	applied	apply	VERB
cana-5394	96	5	nonlinear	nonlinear	ADJ
cana-5394	96	6	analysis	analysis	NOUN
cana-5394	96	7	issn	issn	NOUN
cana-5394	96	8	:	:	PUNCT
cana-5394	96	9	1074	1074	NUM
cana-5394	96	10	-	-	PUNCT
cana-5394	96	11	133x	133x	NUM
cana-5394	96	12	vol	vol	NOUN
cana-5394	96	13	32	32	NUM
cana-5394	96	14	no	no	NOUN
cana-5394	96	15	.	.	NOUN
cana-5394	96	16	3	3	NUM
cana-5394	96	17	(	(	PUNCT
cana-5394	96	18	2025	2025	NUM
cana-5394	96	19	)	)	PUNCT
cana-5394	96	20	933	933	NUM
cana-5394	96	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	96	22	on	on	ADP
cana-5394	96	23	replacing	replace	VERB
cana-5394	96	24			PROPN
cana-5394	96	25	by	by	ADP
cana-5394	96	26	[	[	PUNCT
cana-5394	96	27	,	,	PUNCT
cana-5394	96	28	]	]	X
cana-5394	96	29	k	k	X
cana-5394	96	30	r	r	NOUN
cana-5394	96	31	,	,	PUNCT
cana-5394	96	32	we	we	PRON
cana-5394	96	33	get	get	VERB
cana-5394	96	34	0	0	NUM
cana-5394	96	35	[	[	PUNCT
cana-5394	96	36	,	,	PUNCT
cana-5394	96	37	]	]	X
cana-5394	96	38	[	[	PUNCT
cana-5394	96	39	,	,	PUNCT
cana-5394	96	40	]	]	X
cana-5394	96	41	k	k	X
cana-5394	96	42	r	r	PROPN
cana-5394	96	43	k	k	PROPN
cana-5394	96	44	r	r	PROPN
cana-5394	96	45	=	=	NOUN
cana-5394	96	46	for	for	ADP
cana-5394	96	47	all	all	PRON
cana-5394	96	48	,	,	PUNCT
cana-5394	96	49	,	,	PUNCT
cana-5394	96	50	k	k	PROPN
cana-5394	96	51	m	m	PROPN
cana-5394	96	52			NOUN
cana-5394	96	53	.r	.r	NOUN
cana-5394	97	1	s	s	NOUN
cana-5394	97	2	that	that	PRON
cana-5394	97	3	is	be	AUX
cana-5394	97	4	,	,	PUNCT
cana-5394	97	5	)	)	PUNCT
cana-5394	97	6	[	[	PUNCT
cana-5394	97	7	,	,	PUNCT
cana-5394	97	8	]	]	X
cana-5394	97	9	[	[	PUNCT
cana-5394	97	10	,	,	PUNCT
cana-5394	97	11	]	]	X
cana-5394	97	12	(	(	PUNCT
cana-5394	97	13	0k	0k	NOUN
cana-5394	97	14	r	r	NOUN
cana-5394	97	15	s	s	PROPN
cana-5394	97	16	k	k	PROPN
cana-5394	97	17	r	r	PROPN
cana-5394	97	18			PROPN
cana-5394	97	19	=	=	PUNCT
cana-5394	97	20	for	for	ADP
cana-5394	97	21	all	all	PRON
cana-5394	97	22	,	,	PUNCT
cana-5394	97	23	,	,	PUNCT
cana-5394	97	24	k	k	PROPN
cana-5394	97	25	m	m	PROPN
cana-5394	97	26			NOUN
cana-5394	97	27	.r	.r	NOUN
cana-5394	98	1	s	s	PROPN
cana-5394	98	2	since	since	SCONJ
cana-5394	98	3	s	s	NOUN
cana-5394	98	4	is	be	AUX
cana-5394	98	5	a	a	DET
cana-5394	98	6	semiprime	semiprime	NOUN
cana-5394	98	7	ring	ring	NOUN
cana-5394	98	8	,	,	PUNCT
cana-5394	98	9	we	we	PRON
cana-5394	98	10	have	have	VERB
cana-5394	98	11	[	[	PUNCT
cana-5394	98	12	,	,	PUNCT
cana-5394	98	13	]	]	PUNCT
cana-5394	98	14	0k	0k	PRON
cana-5394	98	15	r	r	NOUN
cana-5394	98	16			PROPN
cana-5394	98	17	=	=	PUNCT
cana-5394	98	18	for	for	ADP
cana-5394	98	19	all	all	PRON
cana-5394	98	20	,	,	PUNCT
cana-5394	98	21	,	,	PUNCT
cana-5394	98	22	k	k	PROPN
cana-5394	98	23	m	m	PROPN
cana-5394	98	24			NOUN
cana-5394	98	25	.r	.r	NOUN
cana-5394	99	1	s	s	PROPN
cana-5394	99	2	taking	take	VERB
cana-5394	99	3			PROPN
cana-5394	99	4	to	to	PART
cana-5394	99	5	be	be	AUX
cana-5394	99	6	[	[	PUNCT
cana-5394	99	7	,	,	PUNCT
cana-5394	99	8	]	]	X
cana-5394	99	9	t	t	X
cana-5394	99	10	k	k	X
cana-5394	99	11	r	r	NOUN
cana-5394	99	12	,	,	PUNCT
cana-5394	99	13	,	,	PUNCT
cana-5394	99	14	t	t	PROPN
cana-5394	99	15	s	s	PROPN
cana-5394	99	16	we	we	PRON
cana-5394	99	17	see	see	VERB
cana-5394	99	18	that	that	PRON
cana-5394	99	19	0	0	NUM
cana-5394	99	20	[	[	PUNCT
cana-5394	99	21	,	,	PUNCT
cana-5394	99	22	]	]	X
cana-5394	99	23	[	[	PUNCT
cana-5394	99	24	,	,	PUNCT
cana-5394	99	25	]	]	X
cana-5394	99	26	k	k	X
cana-5394	99	27	r	r	NOUN
cana-5394	99	28	t	t	NOUN
cana-5394	99	29	k	k	NOUN
cana-5394	99	30	r	r	NOUN
cana-5394	99	31	=	=	PUNCT
cana-5394	99	32	.	.	PUNCT
cana-5394	100	1	by	by	ADP
cana-5394	100	2	the	the	DET
cana-5394	100	3	semiprimeness	semiprimeness	NOUN
cana-5394	100	4	of	of	ADP
cana-5394	100	5	,	,	PUNCT
cana-5394	100	6	s	s	VERB
cana-5394	100	7	we	we	PRON
cana-5394	100	8	get	get	VERB
cana-5394	100	9	(	(	PUNCT
cana-5394	100	10	)	)	PUNCT
cana-5394	100	11	.m	.m	PROPN
cana-5394	101	1	z	z	NOUN
cana-5394	101	2	s	s	PROPN
cana-5394	101	3	thus	thus	ADV
cana-5394	101	4	,	,	PUNCT
cana-5394	101	5	s	s	PROPN
cana-5394	101	6	contains	contain	VERB
cana-5394	101	7	a	a	DET
cana-5394	101	8	nonzero	nonzero	ADJ
cana-5394	101	9	central	central	ADJ
cana-5394	101	10	ideal	ideal	NOUN
cana-5394	101	11	.	.	PUNCT
cana-5394	102	1	the	the	DET
cana-5394	102	2	proof	proof	NOUN
cana-5394	102	3	is	be	AUX
cana-5394	102	4	complete	complete	ADJ
cana-5394	102	5	.	.	PUNCT
cana-5394	103	1	theorem	theorem	NOUN
cana-5394	103	2	3.2.for	3.2.for	ADP
cana-5394	103	3	a	a	DET
cana-5394	103	4	fixed	fix	VERB
cana-5394	103	5	integer	integer	NOUN
cana-5394	103	6	2,n	2,n	NUM
cana-5394	103	7			NUM
cana-5394	103	8	let	let	VERB
cana-5394	103	9	s	s	PRON
cana-5394	103	10	be	be	AUX
cana-5394	103	11	an	an	DET
cana-5394	103	12	!	!	PUNCT
cana-5394	103	13	n	n	CCONJ
cana-5394	103	14	-torsion	-torsion	PROPN
cana-5394	103	15	free	free	ADJ
cana-5394	103	16	semiprime	semiprime	NOUN
cana-5394	103	17	ring	ring	NOUN
cana-5394	103	18	and	and	CCONJ
cana-5394	103	19	m	m	AUX
cana-5394	103	20	be	be	AUX
cana-5394	103	21	a	a	DET
cana-5394	103	22	nonzero	nonzero	NOUN
cana-5394	103	23	ideal	ideal	NOUN
cana-5394	103	24	of	of	ADP
cana-5394	103	25	.s	.s	NOUN
cana-5394	103	26	suppose	suppose	VERB
cana-5394	103	27	that	that	SCONJ
cana-5394	103	28	s	s	VERB
cana-5394	103	29	admits	admit	VERB
cana-5394	103	30	a	a	DET
cana-5394	103	31	nonzero	nonzero	ADJ
cana-5394	103	32	symmetric	symmetric	ADJ
cana-5394	103	33	n	n	CCONJ
cana-5394	103	34	-derivation	-derivation	NOUN
cana-5394	103	35	:	:	PUNCT
cana-5394	103	36	s	s	VERB
cana-5394	103	37	nd	nd	INTJ
cana-5394	103	38	s→	s→	NOUN
cana-5394	103	39	with	with	ADP
cana-5394	103	40	trace	trace	NOUN
cana-5394	103	41	:	:	PUNCT
cana-5394	103	42	s	s	AUX
cana-5394	103	43	s	s	NOUN
cana-5394	103	44	→	→	SYM
cana-5394	103	45	satisfying	satisfy	VERB
cana-5394	103	46	(	(	PUNCT
cana-5394	103	47	)	)	PUNCT
cana-5394	103	48	(	(	PUNCT
cana-5394	103	49	)	)	PUNCT
cana-5394	103	50	z(s)	z(s)	PART
cana-5394	104	1			PROPN
cana-5394	105	1			NUM
cana-5394	105	2			PROPN
cana-5394	105	3			PROPN
cana-5394	105	4			PROPN
cana-5394	105	5			NOUN
cana-5394	105	6	for	for	ADP
cana-5394	105	7	all	all	PRON
cana-5394	105	8	,	,	PUNCT
cana-5394	105	9	,	,	PUNCT
cana-5394	105	10	m	m	PROPN
cana-5394	105	11			PROPN
cana-5394	105	12			PROPN
cana-5394	105	13	then	then	ADV
cana-5394	105	14	s	s	AUX
cana-5394	105	15	has	have	VERB
cana-5394	105	16	a	a	DET
cana-5394	105	17	nonzero	nonzero	ADJ
cana-5394	105	18	central	central	ADJ
cana-5394	105	19	ideal	ideal	NOUN
cana-5394	105	20	.	.	PUNCT
cana-5394	106	1	proof	proof	NOUN
cana-5394	106	2	:	:	PUNCT
cana-5394	106	3	it	it	PRON
cana-5394	106	4	is	be	AUX
cana-5394	106	5	given	give	VERB
cana-5394	106	6	that	that	PRON
cana-5394	106	7	(	(	PUNCT
cana-5394	106	8	)	)	PUNCT
cana-5394	106	9	(	(	PUNCT
cana-5394	106	10	)	)	PUNCT
cana-5394	106	11	z(s)	z(s)	PART
cana-5394	106	12			PROPN
cana-5394	107	1			NUM
cana-5394	107	2			PROPN
cana-5394	107	3			PROPN
cana-5394	107	4			PROPN
cana-5394	107	5			NOUN
cana-5394	107	6	for	for	ADP
cana-5394	107	7	all	all	PRON
cana-5394	107	8	,	,	PUNCT
cana-5394	107	9	.m	.m	PUNCT
cana-5394	108	1			PROPN
cana-5394	108	2			NOUN
cana-5394	108	3	(	(	PUNCT
cana-5394	108	4	3.8	3.8	NUM
cana-5394	108	5	)	)	PUNCT
cana-5394	108	6	replacing	replace	VERB
cana-5394	108	7			PROPN
cana-5394	108	8	by	by	ADP
cana-5394	108	9	mk	mk	NOUN
cana-5394	108	10	+	+	X
cana-5394	108	11	for	for	ADP
cana-5394	108	12	k	k	PROPN
cana-5394	108	13	m	m	PROPN
cana-5394	108	14	in	in	ADP
cana-5394	108	15	(	(	PUNCT
cana-5394	108	16	3.8	3.8	NUM
cana-5394	108	17	)	)	PUNCT
cana-5394	108	18	and	and	CCONJ
cana-5394	108	19	1	1	NUM
cana-5394	108	20	1,m	1,m	NOUN
cana-5394	108	21	n	n	X
cana-5394	108	22			NOUN
cana-5394	108	23	−	−	NOUN
cana-5394	108	24	we	we	PRON
cana-5394	108	25	arrive	arrive	VERB
cana-5394	108	26	at	at	ADP
cana-5394	108	27	(	(	PUNCT
cana-5394	108	28	)	)	PUNCT
cana-5394	108	29	1	1	NUM
cana-5394	108	30	1	1	NUM
cana-5394	108	31	(	(	PUNCT
cana-5394	108	32	)	)	PUNCT
cana-5394	108	33	(	(	PUNCT
cana-5394	108	34	(	(	PUNCT
cana-5394	108	35	)	)	PUNCT
cana-5394	108	36	(	(	PUNCT
cana-5394	108	37	)	)	PUNCT
cana-5394	108	38	d	d	X
cana-5394	108	39	(	(	PUNCT
cana-5394	108	40	,	,	PUNCT
cana-5394	108	41	...	...	PUNCT
cana-5394	108	42	,	,	PUNCT
cana-5394	108	43	,	,	PUNCT
cana-5394	108	44	,	,	PUNCT
cana-5394	108	45	...	...	PUNCT
cana-5394	108	46	,	,	PUNCT
cana-5394	108	47	)	)	PUNCT
cana-5394	108	48	)	)	PUNCT
cana-5394	108	49	(	(	PUNCT
cana-5394	108	50	)	)	PUNCT
cana-5394	108	51	n	n	CCONJ
cana-5394	108	52	n	n	PROPN
cana-5394	108	53	t	t	NOUN
cana-5394	108	54	t	t	PROPN
cana-5394	108	55	t	t	PROPN
cana-5394	108	56	timesn	timesn	PROPN
cana-5394	108	57	t	t	PROPN
cana-5394	108	58	times	times	PROPN
cana-5394	108	59	mk	mk	PROPN
cana-5394	108	60	c	c	PROPN
cana-5394	108	61	mk	mk	PROPN
cana-5394	108	62	mk	mk	PROPN
cana-5394	108	63	mk	mk	PROPN
cana-5394	108	64	z	z	PROPN
cana-5394	108	65	s	s	PROPN
cana-5394	108	66			PROPN
cana-5394	109	1			NUM
cana-5394	109	2			PROPN
cana-5394	109	3			PROPN
cana-5394	109	4			PROPN
cana-5394	109	5			PROPN
cana-5394	109	6			PROPN
cana-5394	109	7			PROPN
cana-5394	109	8			PROPN
cana-5394	109	9	−	−	NOUN
cana-5394	109	10	=	=	PUNCT
cana-5394	109	11	−−	−−	NOUN
cana-5394	109	12	−	−	NOUN
cana-5394	110	1	+	+	CCONJ
cana-5394	110	2	+	+	CCONJ
cana-5394	110	3			PROPN
cana-5394	110	4			PROPN
cana-5394	110	5			NOUN
cana-5394	110	6	for	for	ADP
cana-5394	110	7	all	all	PRON
cana-5394	110	8	,	,	PUNCT
cana-5394	110	9	,	,	PUNCT
cana-5394	110	10	.k	.k	PROPN
cana-5394	111	1	m	m	PROPN
cana-5394	112	1			PROPN
cana-5394	112	2			NOUN
cana-5394	112	3	(	(	PUNCT
cana-5394	112	4	3.9	3.9	NUM
cana-5394	112	5	)	)	PUNCT
cana-5394	112	6	considering	consider	VERB
cana-5394	112	7	the	the	DET
cana-5394	112	8	given	give	VERB
cana-5394	112	9	condition	condition	NOUN
cana-5394	112	10	,	,	PUNCT
cana-5394	112	11	we	we	PRON
cana-5394	112	12	establish	establish	VERB
cana-5394	112	13	(	(	PUNCT
cana-5394	112	14	)	)	PUNCT
cana-5394	112	15	1	1	NUM
cana-5394	112	16	1	1	NUM
cana-5394	112	17	(	(	PUNCT
cana-5394	112	18	)	)	PUNCT
cana-5394	112	19	d	d	NOUN
cana-5394	112	20	(	(	PUNCT
cana-5394	112	21	,	,	PUNCT
cana-5394	112	22	...	...	PUNCT
cana-5394	112	23	,	,	PUNCT
cana-5394	112	24	,	,	PUNCT
cana-5394	112	25	,	,	PUNCT
cana-5394	112	26	...	...	PUNCT
cana-5394	112	27	,	,	PUNCT
cana-5394	112	28	)	)	PUNCT
cana-5394	112	29	(	(	PUNCT
cana-5394	112	30	)	)	PUNCT
cana-5394	112	31	.	.	PUNCT
cana-5394	113	1	n	n	CCONJ
cana-5394	114	1	n	n	PROPN
cana-5394	114	2	t	t	PROPN
cana-5394	114	3	t	t	PROPN
cana-5394	114	4	t	t	PROPN
cana-5394	114	5	timesn	timesn	PROPN
cana-5394	114	6	t	t	PROPN
cana-5394	114	7	times	times	PROPN
cana-5394	114	8	c	c	PROPN
cana-5394	114	9	mk	mk	PROPN
cana-5394	114	10	mk	mk	PROPN
cana-5394	114	11	z	z	PROPN
cana-5394	114	12	s	s	PROPN
cana-5394	114	13			PROPN
cana-5394	114	14			PROPN
cana-5394	114	15			PROPN
cana-5394	114	16	−	−	PROPN
cana-5394	114	17	=	=	PUNCT
cana-5394	114	18	−−	−−	NOUN
cana-5394	114	19	−	−	NOUN
cana-5394	114	20			NOUN
cana-5394	114	21	(	(	PUNCT
cana-5394	114	22	3.10	3.10	NUM
cana-5394	114	23	)	)	PUNCT
cana-5394	114	24	with	with	ADP
cana-5394	114	25	the	the	DET
cana-5394	114	26	help	help	NOUN
cana-5394	114	27	of	of	ADP
cana-5394	114	28	lemma	lemma	PROPN
cana-5394	114	29	2.1	2.1	NUM
cana-5394	114	30	,	,	PUNCT
cana-5394	114	31	we	we	PRON
cana-5394	114	32	see	see	VERB
cana-5394	114	33	that	that	PRON
cana-5394	114	34	(	(	PUNCT
cana-5394	114	35	)	)	PUNCT
cana-5394	114	36	(	(	PUNCT
cana-5394	114	37	,	,	PUNCT
cana-5394	114	38	...	...	PUNCT
cana-5394	114	39	,	,	PUNCT
cana-5394	114	40	,	,	PUNCT
cana-5394	114	41	)	)	PUNCT
cana-5394	114	42	(	(	PUNCT
cana-5394	114	43	)	)	PUNCT
cana-5394	114	44	n	n	NOUN
cana-5394	115	1	d	d	X
cana-5394	115	2	k	k	PROPN
cana-5394	115	3	z	z	NOUN
cana-5394	115	4	s	s	PROPN
cana-5394	115	5			PROPN
cana-5394	115	6			PROPN
cana-5394	115	7			PROPN
cana-5394	115	8			NOUN
cana-5394	115	9	for	for	ADP
cana-5394	115	10	all	all	PRON
cana-5394	115	11	,	,	PUNCT
cana-5394	115	12	,	,	PUNCT
cana-5394	115	13	.k	.k	PROPN
cana-5394	115	14	m	m	PROPN
cana-5394	116	1			PROPN
cana-5394	116	2			NOUN
cana-5394	116	3	since	since	SCONJ
cana-5394	116	4	s	s	PRON
cana-5394	116	5	is	be	AUX
cana-5394	116	6	!	!	PUNCT
cana-5394	117	1	n	n	X
cana-5394	117	2	-torsion	-torsion	NOUN
cana-5394	117	3	free	free	ADJ
cana-5394	117	4	,	,	PUNCT
cana-5394	117	5	we	we	PRON
cana-5394	117	6	have	have	VERB
cana-5394	117	7	(	(	PUNCT
cana-5394	117	8	)	)	PUNCT
cana-5394	117	9	(	(	PUNCT
cana-5394	117	10	,	,	PUNCT
cana-5394	117	11	...	...	PUNCT
cana-5394	117	12	,	,	PUNCT
cana-5394	117	13	,	,	PUNCT
cana-5394	117	14	)	)	PUNCT
cana-5394	117	15	(	(	PUNCT
cana-5394	117	16	)	)	PUNCT
cana-5394	118	1	d	d	X
cana-5394	118	2	k	k	X
cana-5394	118	3	z	z	NOUN
cana-5394	118	4	s	s	PROPN
cana-5394	118	5			PROPN
cana-5394	118	6			PROPN
cana-5394	118	7			PROPN
cana-5394	118	8			NOUN
cana-5394	118	9	for	for	ADP
cana-5394	118	10	all	all	PRON
cana-5394	118	11	,	,	PUNCT
cana-5394	118	12	,	,	PUNCT
cana-5394	118	13	.k	.k	PROPN
cana-5394	118	14	m	m	PROPN
cana-5394	119	1			PROPN
cana-5394	119	2			NOUN
cana-5394	119	3	(	(	PUNCT
cana-5394	119	4	3.11	3.11	NUM
cana-5394	119	5	)	)	PUNCT
cana-5394	119	6	writing	write	VERB
cana-5394	119	7			PROPN
cana-5394	119	8	in	in	ADP
cana-5394	119	9	place	place	NOUN
cana-5394	119	10	of	of	ADP
cana-5394	119	11	k	k	PROPN
cana-5394	119	12	in	in	ADP
cana-5394	119	13	(	(	PUNCT
cana-5394	119	14	3.11	3.11	NUM
cana-5394	119	15	)	)	PUNCT
cana-5394	119	16	,	,	PUNCT
cana-5394	119	17	we	we	PRON
cana-5394	119	18	get	get	VERB
cana-5394	119	19	(	(	PUNCT
cana-5394	119	20	)	)	PUNCT
cana-5394	119	21	(	(	PUNCT
cana-5394	119	22	)	)	PUNCT
cana-5394	119	23	z(s)	z(s)	PART
cana-5394	120	1			PROPN
cana-5394	121	1			NUM
cana-5394	121	2			PROPN
cana-5394	121	3			NOUN
cana-5394	121	4	for	for	ADP
cana-5394	121	5	all	all	PRON
cana-5394	121	6	,	,	PUNCT
cana-5394	121	7	.m	.m	PUNCT
cana-5394	122	1			PROPN
cana-5394	122	2			NOUN
cana-5394	122	3	(	(	PUNCT
cana-5394	122	4	3.12	3.12	NUM
cana-5394	122	5	)	)	PUNCT
cana-5394	122	6	using	use	VERB
cana-5394	122	7	the	the	DET
cana-5394	122	8	hypothesis	hypothesis	NOUN
cana-5394	122	9	,	,	PUNCT
cana-5394	122	10	we	we	PRON
cana-5394	122	11	obtain	obtain	VERB
cana-5394	122	12	that	that	DET
cana-5394	122	13	(	(	PUNCT
cana-5394	122	14	)	)	PUNCT
cana-5394	122	15	z	z	NOUN
cana-5394	123	1	s	s	ADJ
cana-5394	123	2			PROPN
cana-5394	123	3			NOUN
cana-5394	123	4	for	for	ADP
cana-5394	123	5	all	all	PRON
cana-5394	123	6	,	,	PUNCT
cana-5394	123	7	.m	.m	PUNCT
cana-5394	124	1			PROPN
cana-5394	124	2			NOUN
cana-5394	124	3	(	(	PUNCT
cana-5394	124	4	3.13	3.13	NUM
cana-5394	124	5	)	)	PUNCT
cana-5394	124	6	that	that	PRON
cana-5394	124	7	is	be	AUX
cana-5394	124	8	,	,	PUNCT
cana-5394	124	9	(	(	PUNCT
cana-5394	124	10	)	)	PUNCT
cana-5394	124	11	.m	.m	PROPN
cana-5394	125	1	m	m	VERB
cana-5394	125	2	z	z	NOUN
cana-5394	125	3	s	s	PROPN
cana-5394	125	4	hence	hence	ADV
cana-5394	125	5	,	,	PUNCT
cana-5394	125	6	by	by	ADP
cana-5394	125	7	using	use	VERB
cana-5394	125	8	lemma	lemma	PROPN
cana-5394	125	9	2.3	2.3	NUM
cana-5394	125	10	,	,	PUNCT
cana-5394	125	11	s	s	NOUN
cana-5394	125	12	contains	contain	VERB
cana-5394	125	13	a	a	DET
cana-5394	125	14	nonzero	nonzero	ADJ
cana-5394	125	15	central	central	ADJ
cana-5394	125	16	ideal	ideal	NOUN
cana-5394	125	17	.	.	PUNCT
cana-5394	126	1	theorem	theorem	VERB
cana-5394	126	2	3.3	3.3	NUM
cana-5394	126	3	.	.	PUNCT
cana-5394	127	1	for	for	ADP
cana-5394	127	2	a	a	DET
cana-5394	127	3	fixed	fix	VERB
cana-5394	127	4	integer	integer	NOUN
cana-5394	127	5	2,n	2,n	NUM
cana-5394	127	6			NUM
cana-5394	127	7	let	let	VERB
cana-5394	127	8	s	s	PRON
cana-5394	127	9	be	be	AUX
cana-5394	127	10	an	an	DET
cana-5394	127	11	!	!	PUNCT
cana-5394	127	12	n	n	CCONJ
cana-5394	127	13	-torsion	-torsion	PROPN
cana-5394	127	14	free	free	ADJ
cana-5394	127	15	semiprime	semiprime	NOUN
cana-5394	127	16	ring	ring	NOUN
cana-5394	127	17	and	and	CCONJ
cana-5394	127	18	m	m	AUX
cana-5394	127	19	be	be	AUX
cana-5394	127	20	a	a	DET
cana-5394	127	21	nonzero	nonzero	NOUN
cana-5394	127	22	ideal	ideal	NOUN
cana-5394	127	23	of	of	ADP
cana-5394	127	24	.s	.s	NOUN
cana-5394	127	25	suppose	suppose	VERB
cana-5394	127	26	that	that	SCONJ
cana-5394	127	27	s	s	VERB
cana-5394	127	28	admits	admit	VERB
cana-5394	127	29	two	two	NUM
cana-5394	127	30	nonzero	nonzero	ADJ
cana-5394	127	31	symmetric	symmetric	ADJ
cana-5394	127	32	n	n	ADJ
cana-5394	127	33	-derivations	-derivation	NOUN
cana-5394	127	34	:	:	PUNCT
cana-5394	127	35	s	s	VERB
cana-5394	127	36	nd	nd	ADV
cana-5394	127	37	s→	s→	NOUN
cana-5394	127	38	with	with	SCONJ
cana-5394	127	39	communications	communication	NOUN
cana-5394	127	40	on	on	ADP
cana-5394	127	41	applied	apply	VERB
cana-5394	127	42	nonlinear	nonlinear	ADJ
cana-5394	127	43	analysis	analysis	NOUN
cana-5394	127	44	issn	issn	NOUN
cana-5394	127	45	:	:	PUNCT
cana-5394	127	46	1074	1074	NUM
cana-5394	127	47	-	-	PUNCT
cana-5394	127	48	133x	133x	NUM
cana-5394	127	49	vol	vol	NOUN
cana-5394	127	50	32	32	NUM
cana-5394	127	51	no	no	NOUN
cana-5394	127	52	.	.	NOUN
cana-5394	127	53	3	3	NUM
cana-5394	127	54	(	(	PUNCT
cana-5394	127	55	2025	2025	NUM
cana-5394	127	56	)	)	PUNCT
cana-5394	127	57	934	934	NUM
cana-5394	127	58	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	127	59	trace	trace	NOUN
cana-5394	127	60	:	:	PUNCT
cana-5394	128	1	s	s	AUX
cana-5394	128	2	s	s	NOUN
cana-5394	128	3	→	→	SYM
cana-5394	128	4	satisfying	satisfy	VERB
cana-5394	128	5	(	(	PUNCT
cana-5394	128	6	)	)	PUNCT
cana-5394	128	7	(	(	PUNCT
cana-5394	128	8	)	)	PUNCT
cana-5394	128	9	[	[	PUNCT
cana-5394	128	10	,	,	PUNCT
cana-5394	128	11	]	]	PUNCT
cana-5394	128	12	z(s)	z(s)	X
cana-5394	128	13			PROPN
cana-5394	129	1			NUM
cana-5394	129	2			PROPN
cana-5394	129	3			PROPN
cana-5394	129	4			PROPN
cana-5394	129	5			NOUN
cana-5394	129	6	for	for	ADP
cana-5394	129	7	all	all	PRON
cana-5394	129	8	,	,	PUNCT
cana-5394	129	9	,	,	PUNCT
cana-5394	129	10	m	m	PROPN
cana-5394	129	11			PROPN
cana-5394	129	12			PROPN
cana-5394	129	13	then	then	ADV
cana-5394	129	14	s	s	AUX
cana-5394	129	15	has	have	VERB
cana-5394	129	16	a	a	DET
cana-5394	129	17	nonzero	nonzero	ADJ
cana-5394	129	18	central	central	ADJ
cana-5394	129	19	ideal	ideal	NOUN
cana-5394	129	20	.	.	PUNCT
cana-5394	130	1	proof	proof	NOUN
cana-5394	130	2	:	:	PUNCT
cana-5394	130	3	it	it	PRON
cana-5394	130	4	is	be	AUX
cana-5394	130	5	assumed	assume	VERB
cana-5394	130	6	that	that	SCONJ
cana-5394	130	7	(	(	PUNCT
cana-5394	130	8	)	)	PUNCT
cana-5394	130	9	(	(	PUNCT
cana-5394	130	10	)	)	PUNCT
cana-5394	130	11	[	[	PUNCT
cana-5394	130	12	,	,	PUNCT
cana-5394	130	13	]	]	PUNCT
cana-5394	130	14	z(s)	z(s)	X
cana-5394	131	1			PROPN
cana-5394	132	1			NUM
cana-5394	132	2			PROPN
cana-5394	132	3			PROPN
cana-5394	132	4			PROPN
cana-5394	132	5			NOUN
cana-5394	132	6	for	for	ADP
cana-5394	132	7	all	all	PRON
cana-5394	132	8	,	,	PUNCT
cana-5394	132	9	.m	.m	PUNCT
cana-5394	133	1			PROPN
cana-5394	133	2			NOUN
cana-5394	133	3	(	(	PUNCT
cana-5394	133	4	3.14	3.14	NUM
cana-5394	133	5	)	)	PUNCT
cana-5394	133	6	replacing	replace	VERB
cana-5394	133	7			PROPN
cana-5394	133	8	by	by	ADP
cana-5394	133	9	mk	mk	NOUN
cana-5394	133	10	+	+	X
cana-5394	133	11	for	for	ADP
cana-5394	133	12	k	k	PROPN
cana-5394	133	13	m	m	PROPN
cana-5394	133	14	(	(	PUNCT
cana-5394	133	15	3.14	3.14	NUM
cana-5394	133	16	)	)	PUNCT
cana-5394	133	17	and	and	CCONJ
cana-5394	133	18	1	1	NUM
cana-5394	133	19	1,m	1,m	NOUN
cana-5394	133	20	n	n	X
cana-5394	133	21			NOUN
cana-5394	133	22	−	−	NOUN
cana-5394	133	23	we	we	PRON
cana-5394	133	24	arrive	arrive	VERB
cana-5394	133	25	at	at	ADP
cana-5394	133	26	(	(	PUNCT
cana-5394	133	27	)	)	PUNCT
cana-5394	133	28	1	1	NUM
cana-5394	133	29	1	1	NUM
cana-5394	133	30	(	(	PUNCT
cana-5394	133	31	)	)	PUNCT
cana-5394	133	32	(	(	PUNCT
cana-5394	133	33	(	(	PUNCT
cana-5394	133	34	)	)	PUNCT
cana-5394	133	35	(	(	PUNCT
cana-5394	133	36	)	)	PUNCT
cana-5394	133	37	d	d	X
cana-5394	133	38	(	(	PUNCT
cana-5394	133	39	,	,	PUNCT
cana-5394	133	40	...	...	PUNCT
cana-5394	133	41	,	,	PUNCT
cana-5394	133	42	,	,	PUNCT
cana-5394	133	43	,	,	PUNCT
cana-5394	133	44	...	...	PUNCT
cana-5394	133	45	,	,	PUNCT
cana-5394	133	46	)	)	PUNCT
cana-5394	133	47	)	)	PUNCT
cana-5394	134	1	[	[	PUNCT
cana-5394	134	2	,	,	PUNCT
cana-5394	134	3	]	]	X
cana-5394	134	4	[	[	PUNCT
cana-5394	134	5	,	,	PUNCT
cana-5394	134	6	]	]	X
cana-5394	134	7	(	(	PUNCT
cana-5394	134	8	)	)	PUNCT
cana-5394	134	9	n	n	CCONJ
cana-5394	134	10	n	n	PROPN
cana-5394	134	11	t	t	NOUN
cana-5394	134	12	t	t	PROPN
cana-5394	134	13	t	t	PROPN
cana-5394	134	14	timesn	timesn	PROPN
cana-5394	134	15	t	t	PROPN
cana-5394	134	16	times	times	PROPN
cana-5394	134	17	mk	mk	PROPN
cana-5394	134	18	c	c	PROPN
cana-5394	134	19	mk	mk	PROPN
cana-5394	134	20	mk	mk	PROPN
cana-5394	134	21	mk	mk	PROPN
cana-5394	134	22	z	z	PROPN
cana-5394	134	23	s	s	PROPN
cana-5394	134	24			PROPN
cana-5394	135	1			NUM
cana-5394	135	2			PROPN
cana-5394	135	3			PROPN
cana-5394	135	4			PROPN
cana-5394	135	5			PROPN
cana-5394	135	6			PROPN
cana-5394	135	7			PROPN
cana-5394	135	8			PROPN
cana-5394	135	9	−	−	NOUN
cana-5394	135	10	=	=	PUNCT
cana-5394	135	11	−−	−−	NOUN
cana-5394	135	12	−	−	NOUN
cana-5394	136	1	+	+	CCONJ
cana-5394	136	2	+	+	CCONJ
cana-5394	136	3			PROPN
cana-5394	136	4			PROPN
cana-5394	136	5			NOUN
cana-5394	136	6	for	for	ADP
cana-5394	136	7	all	all	PRON
cana-5394	136	8	,	,	PUNCT
cana-5394	136	9	,	,	PUNCT
cana-5394	136	10	k	k	PROPN
cana-5394	136	11	m	m	PROPN
cana-5394	136	12			PROPN
cana-5394	136	13			PROPN
cana-5394	136	14	.	.	PUNCT
cana-5394	137	1	(	(	PUNCT
cana-5394	137	2	3.15	3.15	NUM
cana-5394	137	3	)	)	PUNCT
cana-5394	137	4	using	use	VERB
cana-5394	137	5	the	the	DET
cana-5394	137	6	given	give	VERB
cana-5394	137	7	condition	condition	NOUN
cana-5394	137	8	,	,	PUNCT
cana-5394	137	9	we	we	PRON
cana-5394	137	10	get	get	VERB
cana-5394	137	11	(	(	PUNCT
cana-5394	137	12	)	)	PUNCT
cana-5394	137	13	1	1	NUM
cana-5394	137	14	1	1	NUM
cana-5394	137	15	(	(	PUNCT
cana-5394	137	16	)	)	PUNCT
cana-5394	138	1	d	d	NOUN
cana-5394	138	2	(	(	PUNCT
cana-5394	138	3	,	,	PUNCT
cana-5394	138	4	...	...	PUNCT
cana-5394	138	5	,	,	PUNCT
cana-5394	138	6	,	,	PUNCT
cana-5394	138	7	,	,	PUNCT
cana-5394	138	8	...	...	PUNCT
cana-5394	138	9	,	,	PUNCT
cana-5394	138	10	)	)	PUNCT
cana-5394	138	11	(	(	PUNCT
cana-5394	138	12	)	)	PUNCT
cana-5394	138	13	.	.	PUNCT
cana-5394	139	1	n	n	CCONJ
cana-5394	140	1	n	n	PROPN
cana-5394	140	2	t	t	PROPN
cana-5394	140	3	t	t	PROPN
cana-5394	140	4	t	t	PROPN
cana-5394	140	5	timesn	timesn	PROPN
cana-5394	140	6	t	t	PROPN
cana-5394	140	7	times	times	PROPN
cana-5394	140	8	c	c	PROPN
cana-5394	140	9	mk	mk	PROPN
cana-5394	140	10	mk	mk	PROPN
cana-5394	140	11	z	z	PROPN
cana-5394	140	12	s	s	PROPN
cana-5394	140	13			PROPN
cana-5394	140	14			PROPN
cana-5394	140	15			PROPN
cana-5394	140	16	−	−	PROPN
cana-5394	140	17	=	=	PUNCT
cana-5394	140	18	−−	−−	NOUN
cana-5394	140	19	−	−	NOUN
cana-5394	140	20			NOUN
cana-5394	140	21	(	(	PUNCT
cana-5394	140	22	3.16	3.16	NUM
cana-5394	140	23	)	)	PUNCT
cana-5394	140	24	with	with	ADP
cana-5394	140	25	the	the	DET
cana-5394	140	26	help	help	NOUN
cana-5394	140	27	of	of	ADP
cana-5394	140	28	lemma	lemma	PROPN
cana-5394	140	29	2.1	2.1	NUM
cana-5394	140	30	,	,	PUNCT
cana-5394	140	31	we	we	PRON
cana-5394	140	32	obtain	obtain	VERB
cana-5394	140	33	(	(	PUNCT
cana-5394	140	34	)	)	PUNCT
cana-5394	140	35	(	(	PUNCT
cana-5394	140	36	,	,	PUNCT
cana-5394	140	37	...	...	PUNCT
cana-5394	140	38	,	,	PUNCT
cana-5394	140	39	,	,	PUNCT
cana-5394	140	40	)	)	PUNCT
cana-5394	140	41	(	(	PUNCT
cana-5394	140	42	)	)	PUNCT
cana-5394	140	43	n	n	NOUN
cana-5394	141	1	d	d	X
cana-5394	141	2	k	k	PROPN
cana-5394	141	3	z	z	NOUN
cana-5394	141	4	s	s	PROPN
cana-5394	141	5			PROPN
cana-5394	141	6			PROPN
cana-5394	141	7			PROPN
cana-5394	141	8			NOUN
cana-5394	141	9	for	for	ADP
cana-5394	141	10	all	all	PRON
cana-5394	141	11	,	,	PUNCT
cana-5394	141	12	,	,	PUNCT
cana-5394	141	13	.k	.k	PROPN
cana-5394	142	1	m	m	PROPN
cana-5394	143	1			PROPN
cana-5394	143	2			NOUN
cana-5394	143	3	since	since	SCONJ
cana-5394	143	4	s	s	PRON
cana-5394	143	5	is	be	AUX
cana-5394	143	6	!	!	PUNCT
cana-5394	144	1	n	n	X
cana-5394	144	2	-torsionfree	-torsionfree	PROPN
cana-5394	144	3	,	,	PUNCT
cana-5394	144	4	we	we	PRON
cana-5394	144	5	have	have	VERB
cana-5394	144	6	(	(	PUNCT
cana-5394	144	7	)	)	PUNCT
cana-5394	144	8	(	(	PUNCT
cana-5394	144	9	,	,	PUNCT
cana-5394	144	10	...	...	PUNCT
cana-5394	144	11	,	,	PUNCT
cana-5394	144	12	,	,	PUNCT
cana-5394	144	13	)	)	PUNCT
cana-5394	144	14	(	(	PUNCT
cana-5394	144	15	)	)	PUNCT
cana-5394	145	1	d	d	X
cana-5394	145	2	k	k	X
cana-5394	145	3	z	z	NOUN
cana-5394	145	4	s	s	PROPN
cana-5394	145	5			PROPN
cana-5394	145	6			PROPN
cana-5394	145	7			PROPN
cana-5394	145	8			NOUN
cana-5394	145	9	for	for	ADP
cana-5394	145	10	all	all	PRON
cana-5394	145	11	,	,	PUNCT
cana-5394	145	12	,	,	PUNCT
cana-5394	145	13	.k	.k	PROPN
cana-5394	145	14	m	m	PROPN
cana-5394	146	1			PROPN
cana-5394	146	2			NOUN
cana-5394	146	3	(	(	PUNCT
cana-5394	146	4	3.17	3.17	NUM
cana-5394	146	5	)	)	PUNCT
cana-5394	146	6	writing	write	VERB
cana-5394	146	7			PROPN
cana-5394	146	8	in	in	ADP
cana-5394	146	9	place	place	NOUN
cana-5394	146	10	of	of	ADP
cana-5394	146	11	,	,	PUNCT
cana-5394	146	12	k	k	PROPN
cana-5394	146	13	we	we	PRON
cana-5394	146	14	get	get	VERB
cana-5394	146	15	(	(	PUNCT
cana-5394	146	16	)	)	PUNCT
cana-5394	146	17	(	(	PUNCT
cana-5394	146	18	)	)	PUNCT
cana-5394	146	19	z(s)	z(s)	PART
cana-5394	147	1			PROPN
cana-5394	148	1			NUM
cana-5394	148	2			PROPN
cana-5394	148	3			NOUN
cana-5394	148	4	for	for	ADP
cana-5394	148	5	all	all	PRON
cana-5394	148	6	,	,	PUNCT
cana-5394	148	7	.m	.m	PUNCT
cana-5394	149	1			PROPN
cana-5394	149	2			NOUN
cana-5394	149	3	(	(	PUNCT
cana-5394	149	4	3.18	3.18	NUM
cana-5394	149	5	)	)	PUNCT
cana-5394	149	6	from	from	ADP
cana-5394	149	7	the	the	DET
cana-5394	149	8	hypothesis	hypothesis	NOUN
cana-5394	149	9	,	,	PUNCT
cana-5394	149	10	we	we	PRON
cana-5394	149	11	attain	attain	VERB
cana-5394	149	12	[	[	PUNCT
cana-5394	149	13	,	,	PUNCT
cana-5394	149	14	]	]	PUNCT
cana-5394	149	15	z(s)	z(s)	PROPN
cana-5394	149	16			PROPN
cana-5394	149	17			PROPN
cana-5394	149	18	for	for	ADP
cana-5394	149	19	all	all	PRON
cana-5394	149	20	,	,	PUNCT
cana-5394	149	21	.m	.m	PUNCT
cana-5394	150	1			PROPN
cana-5394	150	2			NOUN
cana-5394	150	3	(	(	PUNCT
cana-5394	150	4	3.19	3.19	NUM
cana-5394	150	5	)	)	PUNCT
cana-5394	150	6	that	that	PRON
cana-5394	150	7	is	be	AUX
cana-5394	150	8	,	,	PUNCT
cana-5394	150	9	[	[	PUNCT
cana-5394	150	10	,	,	PUNCT
cana-5394	150	11	]	]	X
cana-5394	150	12	(	(	PUNCT
cana-5394	150	13	)	)	PUNCT
cana-5394	150	14	.m	.m	PROPN
cana-5394	151	1	m	m	VERB
cana-5394	151	2	z	z	NOUN
cana-5394	151	3	s	s	PROPN
cana-5394	151	4	hence	hence	ADV
cana-5394	151	5	,	,	PUNCT
cana-5394	151	6	by	by	ADP
cana-5394	151	7	using	use	VERB
cana-5394	151	8	lemma	lemma	PROPN
cana-5394	151	9	2.2	2.2	NUM
cana-5394	151	10	,	,	PUNCT
cana-5394	151	11	s	s	NOUN
cana-5394	151	12	contains	contain	VERB
cana-5394	151	13	a	a	DET
cana-5394	151	14	nonzero	nonzero	ADJ
cana-5394	151	15	central	central	ADJ
cana-5394	151	16	ideal	ideal	NOUN
cana-5394	151	17	.	.	PUNCT
cana-5394	152	1	from	from	ADP
cana-5394	152	2	the	the	DET
cana-5394	152	3	earlier	early	ADJ
cana-5394	152	4	results	result	NOUN
cana-5394	152	5	,	,	PUNCT
cana-5394	152	6	we	we	PRON
cana-5394	152	7	arrive	arrive	VERB
cana-5394	152	8	at	at	ADP
cana-5394	152	9	the	the	DET
cana-5394	152	10	following	follow	VERB
cana-5394	152	11	corollary	corollary	NOUN
cana-5394	152	12	:	:	PUNCT
cana-5394	152	13	corollary3.4	corollary3.4	NOUN
cana-5394	152	14	:	:	PUNCT
cana-5394	152	15	for	for	ADP
cana-5394	152	16	a	a	DET
cana-5394	152	17	fixed	fix	VERB
cana-5394	152	18	integer	integer	NOUN
cana-5394	152	19	2n	2n	NUM
cana-5394	152	20			NUM
cana-5394	152	21	,	,	PUNCT
cana-5394	152	22	let	let	VERB
cana-5394	152	23	s	s	PRON
cana-5394	152	24	be	be	AUX
cana-5394	152	25	an	an	DET
cana-5394	152	26	!	!	PUNCT
cana-5394	152	27	n	n	CCONJ
cana-5394	152	28	-torsion	-torsion	PROPN
cana-5394	152	29	free	free	ADJ
cana-5394	152	30	semiprime	semiprime	NOUN
cana-5394	152	31	ring	ring	NOUN
cana-5394	152	32	.	.	PUNCT
cana-5394	153	1	suppose	suppose	VERB
cana-5394	153	2	that	that	SCONJ
cana-5394	153	3	s	s	VERB
cana-5394	153	4	admits	admit	VERB
cana-5394	153	5	two	two	NUM
cana-5394	153	6	nonzero	nonzero	ADJ
cana-5394	153	7	symmetric	symmetric	ADJ
cana-5394	153	8	n	n	ADJ
cana-5394	153	9	-derivations	-derivation	NOUN
cana-5394	153	10	:	:	PUNCT
cana-5394	153	11	s	s	VERB
cana-5394	153	12	nd	nd	ADV
cana-5394	153	13	s→	s→	NOUN
cana-5394	153	14	with	with	ADP
cana-5394	153	15	trace	trace	NOUN
cana-5394	153	16	:	:	PUNCT
cana-5394	153	17	s	s	AUX
cana-5394	153	18	s	s	PROPN
cana-5394	153	19	→	→	SYM
cana-5394	153	20	satisfying	satisfy	VERB
cana-5394	153	21	any	any	DET
cana-5394	153	22	one	one	NUM
cana-5394	153	23	of	of	ADP
cana-5394	153	24	the	the	DET
cana-5394	153	25	conditions	condition	NOUN
cana-5394	153	26	:	:	PUNCT
cana-5394	153	27	(	(	PUNCT
cana-5394	153	28	)	)	PUNCT
cana-5394	153	29	(	(	PUNCT
cana-5394	153	30	)	)	PUNCT
cana-5394	153	31	(	(	PUNCT
cana-5394	153	32	)	)	PUNCT
cana-5394	153	33	z	z	NOUN
cana-5394	153	34	s	s	PROPN
cana-5394	153	35			PROPN
cana-5394	153	36			NUM
cana-5394	153	37			PROPN
cana-5394	153	38			ADJ
cana-5394	153	39			NOUN
cana-5394	153	40	for	for	ADP
cana-5394	153	41	all	all	PRON
cana-5394	153	42	,	,	PUNCT
cana-5394	153	43	,	,	PUNCT
cana-5394	153	44	s	s	PROPN
cana-5394	153	45			PROPN
cana-5394	153	46			PROPN
cana-5394	153	47	(	(	PUNCT
cana-5394	153	48	)	)	PUNCT
cana-5394	153	49	(	(	PUNCT
cana-5394	153	50	)	)	PUNCT
cana-5394	153	51	(	(	PUNCT
cana-5394	153	52	)	)	PUNCT
cana-5394	154	1	z	z	NOUN
cana-5394	154	2	s	s	PROPN
cana-5394	154	3			PROPN
cana-5394	154	4			ADJ
cana-5394	154	5			PROPN
cana-5394	154	6			PROPN
cana-5394	154	7			PROPN
cana-5394	154	8			NOUN
cana-5394	154	9	for	for	ADP
cana-5394	154	10	all	all	PRON
cana-5394	154	11	,	,	PUNCT
cana-5394	154	12	,	,	PUNCT
cana-5394	154	13	s	s	PROPN
cana-5394	154	14			PROPN
cana-5394	154	15			PROPN
cana-5394	154	16	(	(	PUNCT
cana-5394	154	17	)	)	PUNCT
cana-5394	154	18	(	(	PUNCT
cana-5394	154	19	)	)	PUNCT
cana-5394	154	20	[	[	PUNCT
cana-5394	154	21	,	,	PUNCT
cana-5394	154	22	]	]	X
cana-5394	154	23	(	(	PUNCT
cana-5394	154	24	)	)	PUNCT
cana-5394	154	25	z	z	NOUN
cana-5394	154	26	s	s	PROPN
cana-5394	154	27			PROPN
cana-5394	155	1			ADJ
cana-5394	155	2			PROPN
cana-5394	155	3			PROPN
cana-5394	155	4			PROPN
cana-5394	155	5			NOUN
cana-5394	155	6	for	for	ADP
cana-5394	155	7	all	all	PRON
cana-5394	155	8	,	,	PUNCT
cana-5394	155	9	s	s	PROPN
cana-5394	155	10			PROPN
cana-5394	155	11			NOUN
cana-5394	155	12	then	then	ADV
cana-5394	155	13	s	s	VERB
cana-5394	155	14	is	be	AUX
cana-5394	155	15	commutative	commutative	ADJ
cana-5394	155	16	.	.	PUNCT
cana-5394	156	1	theorem	theorem	VERB
cana-5394	156	2	3.5	3.5	NUM
cana-5394	156	3	.	.	PUNCT
cana-5394	157	1	for	for	ADP
cana-5394	157	2	a	a	DET
cana-5394	157	3	fixed	fix	VERB
cana-5394	157	4	integer	integer	NOUN
cana-5394	157	5	2,n	2,n	NUM
cana-5394	157	6			NUM
cana-5394	157	7	let	let	VERB
cana-5394	157	8	s	s	PRON
cana-5394	157	9	be	be	AUX
cana-5394	157	10	an	an	DET
cana-5394	157	11	!	!	PUNCT
cana-5394	157	12	n	n	CCONJ
cana-5394	157	13	-torsion	-torsion	PROPN
cana-5394	157	14	free	free	ADJ
cana-5394	157	15	semiprime	semiprime	NOUN
cana-5394	157	16	ring	ring	NOUN
cana-5394	157	17	and	and	CCONJ
cana-5394	157	18	m	m	AUX
cana-5394	157	19	be	be	AUX
cana-5394	157	20	a	a	DET
cana-5394	157	21	nonzero	nonzero	NOUN
cana-5394	157	22	ideal	ideal	NOUN
cana-5394	157	23	of	of	ADP
cana-5394	157	24	.s	.s	NOUN
cana-5394	157	25	suppose	suppose	VERB
cana-5394	157	26	that	that	SCONJ
cana-5394	157	27	s	s	VERB
cana-5394	157	28	admits	admit	VERB
cana-5394	157	29	two	two	NUM
cana-5394	157	30	nonzero	nonzero	ADJ
cana-5394	157	31	symmetric	symmetric	ADJ
cana-5394	157	32	n	n	ADJ
cana-5394	157	33	-derivations	-derivation	NOUN
cana-5394	157	34	:	:	PUNCT
cana-5394	157	35	s	s	VERB
cana-5394	157	36	nd	nd	ADV
cana-5394	157	37	s→	s→	NOUN
cana-5394	157	38	with	with	ADP
cana-5394	157	39	trace	trace	NOUN
cana-5394	157	40	:	:	PUNCT
cana-5394	157	41	s	s	AUX
cana-5394	157	42	s	s	NOUN
cana-5394	157	43	→	→	SYM
cana-5394	157	44	satisfying	satisfy	VERB
cana-5394	157	45	[	[	PUNCT
cana-5394	157	46	,	,	PUNCT
cana-5394	157	47	]	]	PUNCT
cana-5394	157	48	z(s)	z(s)	X
cana-5394	157	49			PROPN
cana-5394	158	1			PROPN
cana-5394	158	2			ADJ
cana-5394	158	3			NOUN
cana-5394	158	4	for	for	ADP
cana-5394	158	5	all	all	PRON
cana-5394	158	6	,	,	PUNCT
cana-5394	158	7	,	,	PUNCT
cana-5394	158	8	m	m	PROPN
cana-5394	158	9			PROPN
cana-5394	158	10			PROPN
cana-5394	158	11	then	then	ADV
cana-5394	158	12	s	s	AUX
cana-5394	158	13	has	have	VERB
cana-5394	158	14	a	a	DET
cana-5394	158	15	nonzero	nonzero	ADJ
cana-5394	158	16	central	central	ADJ
cana-5394	158	17	ideal	ideal	NOUN
cana-5394	158	18	.	.	PUNCT
cana-5394	159	1	proof	proof	NOUN
cana-5394	159	2	:	:	PUNCT
cana-5394	159	3	it	it	PRON
cana-5394	159	4	is	be	AUX
cana-5394	159	5	given	give	VERB
cana-5394	159	6	that	that	PRON
cana-5394	159	7	[	[	PUNCT
cana-5394	159	8	,	,	PUNCT
cana-5394	159	9	]	]	PUNCT
cana-5394	159	10	z(s)	z(s)	X
cana-5394	160	1			PROPN
cana-5394	161	1			PROPN
cana-5394	161	2			ADJ
cana-5394	161	3			NOUN
cana-5394	161	4	for	for	ADP
cana-5394	161	5	all	all	PRON
cana-5394	161	6	,	,	PUNCT
cana-5394	161	7	.m	.m	PUNCT
cana-5394	162	1			PROPN
cana-5394	162	2			NOUN
cana-5394	162	3	(	(	PUNCT
cana-5394	162	4	3.20	3.20	NUM
cana-5394	162	5	)	)	PUNCT
cana-5394	162	6	communications	communication	NOUN
cana-5394	162	7	on	on	ADP
cana-5394	162	8	applied	apply	VERB
cana-5394	162	9	nonlinear	nonlinear	ADJ
cana-5394	162	10	analysis	analysis	NOUN
cana-5394	162	11	issn	issn	NOUN
cana-5394	162	12	:	:	PUNCT
cana-5394	162	13	1074	1074	NUM
cana-5394	162	14	-	-	PUNCT
cana-5394	162	15	133x	133x	NUM
cana-5394	162	16	vol	vol	NOUN
cana-5394	162	17	32	32	NUM
cana-5394	162	18	no	no	NOUN
cana-5394	162	19	.	.	NOUN
cana-5394	162	20	3	3	NUM
cana-5394	162	21	(	(	PUNCT
cana-5394	162	22	2025	2025	NUM
cana-5394	162	23	)	)	PUNCT
cana-5394	162	24	935	935	NUM
cana-5394	162	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	162	26	replacing	replace	VERB
cana-5394	162	27			PROPN
cana-5394	162	28	by	by	ADP
cana-5394	162	29	mk	mk	NOUN
cana-5394	162	30	+	+	X
cana-5394	162	31	for	for	ADP
cana-5394	162	32	k	k	PROPN
cana-5394	162	33	m	m	PROPN
cana-5394	162	34	in	in	ADP
cana-5394	162	35	(	(	PUNCT
cana-5394	162	36	3.20	3.20	NUM
cana-5394	162	37	)	)	PUNCT
cana-5394	162	38	and	and	CCONJ
cana-5394	162	39	1	1	NUM
cana-5394	162	40	1,m	1,m	NOUN
cana-5394	162	41	n	n	X
cana-5394	162	42			NOUN
cana-5394	162	43	−	−	NOUN
cana-5394	163	1	we	we	PRON
cana-5394	163	2	get	get	VERB
cana-5394	163	3	(	(	PUNCT
cana-5394	163	4	[	[	PUNCT
cana-5394	163	5	,	,	PUNCT
cana-5394	163	6	]	]	X
cana-5394	163	7	[	[	PUNCT
cana-5394	163	8	,	,	PUNCT
cana-5394	163	9	]	]	X
cana-5394	163	10	)	)	PUNCT
cana-5394	163	11	z(s)mk	z(s)mk	NUM
cana-5394	164	1	mk	mk	PRON
cana-5394	164	2			PROPN
cana-5394	164	3			PROPN
cana-5394	164	4			PROPN
cana-5394	164	5			PUNCT
cana-5394	164	6	+	+	ADJ
cana-5394	165	1			PROPN
cana-5394	165	2	+	+	CCONJ
cana-5394	165	3			NOUN
cana-5394	165	4	for	for	ADP
cana-5394	165	5	all	all	PRON
cana-5394	165	6	,	,	PUNCT
cana-5394	165	7	,	,	PUNCT
cana-5394	165	8	.k	.k	PROPN
cana-5394	166	1	m	m	PROPN
cana-5394	167	1			PROPN
cana-5394	167	2			NOUN
cana-5394	167	3	(	(	PUNCT
cana-5394	167	4	3.21	3.21	NUM
cana-5394	167	5	)	)	PUNCT
cana-5394	167	6	by	by	ADP
cana-5394	167	7	reducing	reduce	VERB
cana-5394	167	8	,	,	PUNCT
cana-5394	167	9	we	we	PRON
cana-5394	167	10	arrive	arrive	VERB
cana-5394	167	11	at	at	ADP
cana-5394	167	12	(	(	PUNCT
cana-5394	167	13	)	)	PUNCT
cana-5394	167	14	1	1	NUM
cana-5394	167	15	1	1	NUM
cana-5394	167	16	(	(	PUNCT
cana-5394	167	17	[	[	PUNCT
cana-5394	167	18	,	,	PUNCT
cana-5394	167	19	]	]	X
cana-5394	167	20	[	[	PUNCT
cana-5394	167	21	,	,	PUNCT
cana-5394	167	22	]	]	X
cana-5394	167	23	)	)	PUNCT
cana-5394	168	1	d	d	X
cana-5394	168	2	(	(	PUNCT
cana-5394	168	3	[	[	PUNCT
cana-5394	168	4	,	,	PUNCT
cana-5394	168	5	]	]	X
cana-5394	168	6	,	,	PUNCT
cana-5394	168	7	...	...	PUNCT
cana-5394	168	8	,	,	PUNCT
cana-5394	168	9	[	[	PUNCT
cana-5394	168	10	,	,	PUNCT
cana-5394	168	11	]	]	X
cana-5394	168	12	,	,	PUNCT
cana-5394	168	13	[	[	PUNCT
cana-5394	168	14	,	,	PUNCT
cana-5394	168	15	]	]	X
cana-5394	168	16	,	,	PUNCT
cana-5394	168	17	...	...	PUNCT
cana-5394	168	18	,	,	PUNCT
cana-5394	168	19	[	[	PUNCT
cana-5394	168	20	,	,	PUNCT
cana-5394	168	21	]	]	PUNCT
cana-5394	168	22	)	)	PUNCT
cana-5394	168	23	[	[	PUNCT
cana-5394	168	24	,	,	PUNCT
cana-5394	168	25	]	]	X
cana-5394	168	26	[	[	PUNCT
cana-5394	168	27	,	,	PUNCT
cana-5394	168	28	]	]	X
cana-5394	168	29	(	(	PUNCT
cana-5394	168	30	)	)	PUNCT
cana-5394	168	31	n	n	CCONJ
cana-5394	168	32	n	n	PROPN
cana-5394	168	33	t	t	NOUN
cana-5394	168	34	t	t	PROPN
cana-5394	168	35	t	t	PROPN
cana-5394	168	36	timesn	timesn	PROPN
cana-5394	168	37	t	t	PROPN
cana-5394	168	38	times	times	PROPN
cana-5394	168	39	mk	mk	PROPN
cana-5394	168	40	c	c	PROPN
cana-5394	168	41	mk	mk	PROPN
cana-5394	168	42	mk	mk	PROPN
cana-5394	168	43	mk	mk	PROPN
cana-5394	168	44	z	z	PROPN
cana-5394	168	45	s	s	PROPN
cana-5394	168	46			PROPN
cana-5394	168	47			PROPN
cana-5394	168	48			PROPN
cana-5394	168	49			PROPN
cana-5394	168	50			PROPN
cana-5394	168	51			PROPN
cana-5394	168	52			PROPN
cana-5394	168	53			PROPN
cana-5394	168	54			X
cana-5394	168	55			PROPN
cana-5394	168	56			PROPN
cana-5394	168	57			PROPN
cana-5394	168	58	−	−	NOUN
cana-5394	168	59	=	=	PUNCT
cana-5394	168	60	−−	−−	NOUN
cana-5394	168	61	−	−	NOUN
cana-5394	169	1	+	+	CCONJ
cana-5394	169	2	+	+	CCONJ
cana-5394	169	3			PROPN
cana-5394	169	4			PROPN
cana-5394	169	5			NOUN
cana-5394	169	6	for	for	ADP
cana-5394	169	7	all	all	PRON
cana-5394	169	8	,	,	PUNCT
cana-5394	169	9	,	,	PUNCT
cana-5394	169	10	.k	.k	PROPN
cana-5394	170	1	m	m	PROPN
cana-5394	171	1			PROPN
cana-5394	171	2			NOUN
cana-5394	171	3	(	(	PUNCT
cana-5394	171	4	3.22	3.22	NUM
cana-5394	171	5	)	)	PUNCT
cana-5394	171	6	considering	consider	VERB
cana-5394	171	7	the	the	DET
cana-5394	171	8	hypothesis	hypothesis	NOUN
cana-5394	171	9	,	,	PUNCT
cana-5394	171	10	we	we	PRON
cana-5394	171	11	get	get	VERB
cana-5394	171	12	(	(	PUNCT
cana-5394	171	13	)	)	PUNCT
cana-5394	171	14	1	1	NUM
cana-5394	171	15	1	1	NUM
cana-5394	171	16	d	d	X
cana-5394	171	17	(	(	PUNCT
cana-5394	171	18	[	[	PUNCT
cana-5394	171	19	,	,	PUNCT
cana-5394	171	20	]	]	X
cana-5394	171	21	,	,	PUNCT
cana-5394	171	22	...	...	PUNCT
cana-5394	171	23	,	,	PUNCT
cana-5394	171	24	[	[	PUNCT
cana-5394	171	25	,	,	PUNCT
cana-5394	171	26	]	]	X
cana-5394	171	27	,	,	PUNCT
cana-5394	171	28	[	[	PUNCT
cana-5394	171	29	,	,	PUNCT
cana-5394	171	30	]	]	X
cana-5394	171	31	,	,	PUNCT
cana-5394	171	32	...	...	PUNCT
cana-5394	171	33	,	,	PUNCT
cana-5394	171	34	[	[	PUNCT
cana-5394	171	35	,	,	PUNCT
cana-5394	171	36	]	]	PUNCT
cana-5394	171	37	)	)	PUNCT
cana-5394	171	38	(	(	PUNCT
cana-5394	171	39	)	)	PUNCT
cana-5394	172	1	n	n	CCONJ
cana-5394	172	2	n	n	PROPN
cana-5394	172	3	t	t	NOUN
cana-5394	172	4	t	t	PROPN
cana-5394	172	5	t	t	PROPN
cana-5394	172	6	timesn	timesn	PROPN
cana-5394	172	7	t	t	PROPN
cana-5394	172	8	times	times	PROPN
cana-5394	172	9	c	c	PROPN
cana-5394	172	10	mk	mk	PROPN
cana-5394	172	11	mk	mk	PROPN
cana-5394	172	12	z	z	PROPN
cana-5394	172	13	s	s	PROPN
cana-5394	173	1			PROPN
cana-5394	173	2			PROPN
cana-5394	173	3			PROPN
cana-5394	173	4			PROPN
cana-5394	173	5			PROPN
cana-5394	173	6	−	−	NOUN
cana-5394	173	7	=	=	SYM
cana-5394	173	8	−−	−−	NOUN
cana-5394	173	9	−	−	NOUN
cana-5394	173	10			NOUN
cana-5394	173	11	for	for	ADP
cana-5394	173	12	all	all	PRON
cana-5394	173	13	,	,	PUNCT
cana-5394	173	14	,	,	PUNCT
cana-5394	173	15	.k	.k	PROPN
cana-5394	174	1	m	m	PROPN
cana-5394	175	1			PROPN
cana-5394	175	2			NOUN
cana-5394	175	3	(	(	PUNCT
cana-5394	175	4	3.23	3.23	NUM
cana-5394	175	5	)	)	PUNCT
cana-5394	175	6	consequently	consequently	ADV
cana-5394	175	7	,	,	PUNCT
cana-5394	175	8	we	we	PRON
cana-5394	175	9	obtain	obtain	VERB
cana-5394	175	10	the	the	DET
cana-5394	175	11	following	following	NOUN
cana-5394	175	12	:	:	PUNCT
cana-5394	175	13	2	2	NUM
cana-5394	175	14	1	1	NUM
cana-5394	175	15	1	1	NUM
cana-5394	175	16	2	2	NUM
cana-5394	175	17	1	1	NUM
cana-5394	175	18	(	(	PUNCT
cana-5394	175	19	,	,	PUNCT
cana-5394	175	20	,	,	PUNCT
cana-5394	175	21	)	)	PUNCT
cana-5394	175	22	(	(	PUNCT
cana-5394	175	23	,	,	PUNCT
cana-5394	175	24	,	,	PUNCT
cana-5394	175	25	)	)	PUNCT
cana-5394	175	26	·	·	PUNCT
cana-5394	175	27	·	·	PUNCT
cana-5394	175	28	·	·	PUNCT
cana-5394	175	29	(	(	PUNCT
cana-5394	175	30	,	,	PUNCT
cana-5394	175	31	,	,	PUNCT
cana-5394	175	32	)	)	PUNCT
cana-5394	175	33	0n	0n	PROPN
cana-5394	175	34	nmp	nmp	PROPN
cana-5394	176	1	k	k	PROPN
cana-5394	177	1	m	m	PROPN
cana-5394	177	2	p	p	X
cana-5394	178	1	k	k	X
cana-5394	178	2	m	m	VERB
cana-5394	178	3	p	p	NOUN
cana-5394	178	4	k	k	PROPN
cana-5394	178	5			PROPN
cana-5394	178	6			PROPN
cana-5394	178	7			PROPN
cana-5394	178	8			PROPN
cana-5394	178	9	−	−	NOUN
cana-5394	178	10	−+	−+	PROPN
cana-5394	178	11	+	+	PROPN
cana-5394	179	1	+	+	CCONJ
cana-5394	179	2	=	=	NOUN
cana-5394	179	3	for	for	ADP
cana-5394	179	4	all	all	PRON
cana-5394	179	5	,	,	PUNCT
cana-5394	179	6	,	,	PUNCT
cana-5394	179	7	.k	.k	PROPN
cana-5394	180	1	m	m	PROPN
cana-5394	181	1			PROPN
cana-5394	181	2			NOUN
cana-5394	181	3	where	where	SCONJ
cana-5394	181	4	(	(	PUNCT
cana-5394	181	5	)	)	PUNCT
cana-5394	181	6	(	(	PUNCT
cana-5394	181	7	)	)	PUNCT
cana-5394	181	8	(	(	PUNCT
cana-5394	181	9	,	,	PUNCT
cana-5394	181	10	,	,	PUNCT
cana-5394	181	11	)	)	PUNCT
cana-5394	182	1	d	d	X
cana-5394	182	2	(	(	PUNCT
cana-5394	182	3	[	[	PUNCT
cana-5394	182	4	,	,	PUNCT
cana-5394	182	5	]	]	X
cana-5394	182	6	,	,	PUNCT
cana-5394	182	7	...	...	PUNCT
cana-5394	182	8	,	,	PUNCT
cana-5394	182	9	[	[	PUNCT
cana-5394	182	10	,	,	PUNCT
cana-5394	182	11	]	]	PUNCT
cana-5394	182	12	,	,	PUNCT
cana-5394	182	13	[	[	PUNCT
cana-5394	182	14	,	,	PUNCT
cana-5394	182	15	]	]	X
cana-5394	182	16	,	,	PUNCT
cana-5394	182	17	...	...	PUNCT
cana-5394	182	18	,	,	PUNCT
cana-5394	182	19	[	[	PUNCT
cana-5394	182	20	,	,	PUNCT
cana-5394	182	21	]	]	PUNCT
cana-5394	182	22	)	)	PUNCT
cana-5394	183	1	[	[	PUNCT
cana-5394	183	2	d	d	X
cana-5394	183	3	(	(	PUNCT
cana-5394	183	4	,	,	PUNCT
cana-5394	183	5	...	...	PUNCT
cana-5394	183	6	,	,	PUNCT
cana-5394	183	7	,	,	PUNCT
cana-5394	183	8	,	,	PUNCT
cana-5394	183	9	...	...	PUNCT
cana-5394	183	10	,	,	PUNCT
cana-5394	183	11	)	)	PUNCT
cana-5394	183	12	,	,	PUNCT
cana-5394	183	13	]	]	PUNCT
cana-5394	183	14	n	n	CCONJ
cana-5394	183	15	n	n	PRON
cana-5394	183	16	t	t	NOUN
cana-5394	183	17	t	t	PROPN
cana-5394	183	18	t	t	PROPN
cana-5394	183	19	t	t	PROPN
cana-5394	183	20	timest	tim	ADJ
cana-5394	183	21	timesn	timesn	PROPN
cana-5394	183	22	t	t	PROPN
cana-5394	183	23	times	time	NOUN
cana-5394	183	24	n	n	ADP
cana-5394	183	25	t	t	NOUN
cana-5394	183	26	times	time	NOUN
cana-5394	184	1	p	p	PROPN
cana-5394	184	2	k	k	PROPN
cana-5394	184	3	c	c	PROPN
cana-5394	184	4	mk	mk	PROPN
cana-5394	184	5	mk	mk	PROPN
cana-5394	184	6	c	c	PROPN
cana-5394	184	7	k	k	PROPN
cana-5394	184	8	k	k	PROPN
cana-5394	185	1			PROPN
cana-5394	185	2			PROPN
cana-5394	185	3			PROPN
cana-5394	185	4			PROPN
cana-5394	185	5			PROPN
cana-5394	185	6			PROPN
cana-5394	185	7			PROPN
cana-5394	185	8			PROPN
cana-5394	185	9			PROPN
cana-5394	185	10			PROPN
cana-5394	185	11	−−−	−−−	NOUN
cana-5394	186	1	−	−	PROPN
cana-5394	186	2	−	−	NOUN
cana-5394	187	1	−	−	PROPN
cana-5394	187	2	=	=	SYM
cana-5394	187	3	−	−	PROPN
cana-5394	187	4	signifies	signify	VERB
cana-5394	187	5	the	the	DET
cana-5394	187	6	sum	sum	NOUN
cana-5394	187	7	of	of	ADP
cana-5394	187	8	terms	term	NOUN
cana-5394	187	9	contain	contain	VERB
cana-5394	187	10	k	k	PROPN
cana-5394	187	11	appears	appear	VERB
cana-5394	187	12	t	t	PROPN
cana-5394	187	13	-times	-time	NOUN
cana-5394	187	14	.	.	PUNCT
cana-5394	188	1	in	in	ADP
cana-5394	188	2	consideration	consideration	NOUN
cana-5394	188	3	of	of	ADP
cana-5394	188	4	lemma2.1	lemma2.1	PROPN
cana-5394	188	5	and	and	CCONJ
cana-5394	188	6	the	the	DET
cana-5394	188	7	restriction	restriction	NOUN
cana-5394	188	8	of	of	ADP
cana-5394	188	9	torsion	torsion	NOUN
cana-5394	188	10	-	-	PUNCT
cana-5394	188	11	freeness	freeness	NOUN
cana-5394	188	12	in	in	ADP
cana-5394	188	13	,	,	PUNCT
cana-5394	188	14	s	s	AUX
cana-5394	188	15	we	we	PRON
cana-5394	188	16	get	get	VERB
cana-5394	188	17	(	(	PUNCT
cana-5394	188	18	)	)	PUNCT
cana-5394	188	19	[	[	PUNCT
cana-5394	188	20	,	,	PUNCT
cana-5394	188	21	]	]	X
cana-5394	188	22	,	,	PUNCT
cana-5394	188	23	...	...	PUNCT
cana-5394	188	24	,	,	PUNCT
cana-5394	188	25	[	[	PUNCT
cana-5394	188	26	,	,	PUNCT
cana-5394	188	27	]	]	X
cana-5394	188	28	,	,	PUNCT
cana-5394	188	29	[	[	PUNCT
cana-5394	188	30	,	,	PUNCT
cana-5394	188	31	]	]	X
cana-5394	188	32	(	(	PUNCT
cana-5394	188	33	)	)	PUNCT
cana-5394	188	34	d	d	X
cana-5394	188	35	mk	mk	PROPN
cana-5394	188	36	z	z	PROPN
cana-5394	188	37	s	s	PROPN
cana-5394	189	1			PROPN
cana-5394	189	2			PROPN
cana-5394	189	3			PROPN
cana-5394	189	4			PROPN
cana-5394	189	5			NOUN
cana-5394	189	6	for	for	ADP
cana-5394	189	7	all	all	PRON
cana-5394	189	8	,	,	PUNCT
cana-5394	189	9	,	,	PUNCT
cana-5394	189	10	.k	.k	PROPN
cana-5394	190	1	m	m	PROPN
cana-5394	191	1			PROPN
cana-5394	191	2			NOUN
cana-5394	191	3	(	(	PUNCT
cana-5394	191	4	3.24	3.24	NUM
cana-5394	191	5	)	)	PUNCT
cana-5394	191	6	replacing	replace	VERB
cana-5394	191	7	k	k	X
cana-5394	191	8	by	by	ADV
cana-5394	191	9	,	,	PUNCT
cana-5394	191	10			PROPN
cana-5394	191	11	we	we	PRON
cana-5394	191	12	get	get	VERB
cana-5394	191	13	(	(	PUNCT
cana-5394	191	14	)	)	PUNCT
cana-5394	191	15	[	[	PUNCT
cana-5394	191	16	,	,	PUNCT
cana-5394	191	17	]	]	X
cana-5394	191	18	(	(	PUNCT
cana-5394	191	19	)	)	PUNCT
cana-5394	191	20	z	z	NOUN
cana-5394	191	21	s	s	PROPN
cana-5394	191	22			PROPN
cana-5394	191	23			PROPN
cana-5394	191	24			NOUN
cana-5394	191	25	for	for	ADP
cana-5394	191	26	all	all	PRON
cana-5394	191	27	,	,	PUNCT
cana-5394	191	28	.m	.m	PUNCT
cana-5394	192	1			PROPN
cana-5394	192	2			NOUN
cana-5394	192	3	(	(	PUNCT
cana-5394	192	4	3.25	3.25	NUM
cana-5394	192	5	)	)	PUNCT
cana-5394	192	6	applying	apply	VERB
cana-5394	192	7	the	the	DET
cana-5394	192	8	hypothesis	hypothesis	NOUN
cana-5394	192	9	once	once	ADV
cana-5394	192	10	again	again	ADV
cana-5394	192	11	,	,	PUNCT
cana-5394	192	12	we	we	PRON
cana-5394	192	13	obtain	obtain	VERB
cana-5394	192	14	z(s)	z(s)	PROPN
cana-5394	192	15			NOUN
cana-5394	192	16	for	for	ADP
cana-5394	192	17	all	all	PRON
cana-5394	192	18	,	,	PUNCT
cana-5394	192	19	.m	.m	PUNCT
cana-5394	193	1			PROPN
cana-5394	193	2			PROPN
cana-5394	193	3	consequently	consequently	ADV
cana-5394	193	4	,	,	PUNCT
cana-5394	193	5	the	the	DET
cana-5394	193	6	result	result	NOUN
cana-5394	193	7	can	can	AUX
cana-5394	193	8	be	be	AUX
cana-5394	193	9	derived	derive	VERB
cana-5394	193	10	using	use	VERB
cana-5394	193	11	the	the	DET
cana-5394	193	12	same	same	ADJ
cana-5394	193	13	approach	approach	NOUN
cana-5394	193	14	as	as	ADP
cana-5394	193	15	(	(	PUNCT
cana-5394	193	16	3.5	3.5	NUM
cana-5394	193	17	)	)	PUNCT
cana-5394	193	18	in	in	ADP
cana-5394	193	19	theorem	theorem	ADJ
cana-5394	193	20	3.1	3.1	NUM
cana-5394	193	21	.	.	PUNCT
cana-5394	194	1	following	follow	VERB
cana-5394	194	2	results	result	NOUN
cana-5394	194	3	can	can	AUX
cana-5394	194	4	analogously	analogously	ADV
cana-5394	194	5	be	be	AUX
cana-5394	194	6	obtained	obtain	VERB
cana-5394	194	7	.	.	PUNCT
cana-5394	195	1	theorem	theorem	VERB
cana-5394	195	2	3.6	3.6	NUM
cana-5394	195	3	.	.	PUNCT
cana-5394	196	1	for	for	ADP
cana-5394	196	2	a	a	DET
cana-5394	196	3	fixed	fix	VERB
cana-5394	196	4	integer	integer	NOUN
cana-5394	196	5	2,n	2,n	NUM
cana-5394	196	6			NUM
cana-5394	196	7	let	let	VERB
cana-5394	196	8	s	s	PRON
cana-5394	196	9	be	be	AUX
cana-5394	196	10	an	an	DET
cana-5394	196	11	!	!	PUNCT
cana-5394	196	12	n	n	CCONJ
cana-5394	196	13	-torsion	-torsion	PROPN
cana-5394	196	14	free	free	ADJ
cana-5394	196	15	semiprime	semiprime	NOUN
cana-5394	196	16	ring	ring	NOUN
cana-5394	196	17	and	and	CCONJ
cana-5394	196	18	m	m	AUX
cana-5394	196	19	be	be	AUX
cana-5394	196	20	a	a	DET
cana-5394	196	21	nonzero	nonzero	NOUN
cana-5394	196	22	ideal	ideal	NOUN
cana-5394	196	23	of	of	ADP
cana-5394	196	24	.s	.s	NOUN
cana-5394	196	25	suppose	suppose	VERB
cana-5394	196	26	that	that	SCONJ
cana-5394	196	27	s	s	VERB
cana-5394	196	28	admits	admit	VERB
cana-5394	196	29	two	two	NUM
cana-5394	196	30	nonzero	nonzero	ADJ
cana-5394	196	31	symmetric	symmetric	ADJ
cana-5394	196	32	n	n	ADJ
cana-5394	196	33	-derivations	-derivation	NOUN
cana-5394	196	34	:	:	PUNCT
cana-5394	196	35	s	s	VERB
cana-5394	196	36	nd	nd	ADV
cana-5394	196	37	s→	s→	NOUN
cana-5394	196	38	with	with	ADP
cana-5394	196	39	trace	trace	NOUN
cana-5394	196	40	:	:	PUNCT
cana-5394	196	41	s	s	AUX
cana-5394	196	42	s	s	NOUN
cana-5394	196	43	→	→	SYM
cana-5394	196	44	satisfying	satisfy	VERB
cana-5394	196	45	[	[	PUNCT
cana-5394	196	46	,	,	PUNCT
cana-5394	196	47	]	]	PUNCT
cana-5394	196	48	z(s)	z(s)	X
cana-5394	196	49			PROPN
cana-5394	197	1			PROPN
cana-5394	197	2			PROPN
cana-5394	197	3			PROPN
cana-5394	197	4			NOUN
cana-5394	197	5	for	for	ADP
cana-5394	197	6	all	all	PRON
cana-5394	197	7	,	,	PUNCT
cana-5394	197	8	,	,	PUNCT
cana-5394	197	9	m	m	PROPN
cana-5394	197	10			PROPN
cana-5394	197	11			PROPN
cana-5394	197	12	then	then	ADV
cana-5394	197	13	s	s	AUX
cana-5394	197	14	has	have	VERB
cana-5394	197	15	a	a	DET
cana-5394	197	16	nonzero	nonzero	ADJ
cana-5394	197	17	central	central	ADJ
cana-5394	197	18	ideal	ideal	NOUN
cana-5394	197	19	.	.	PUNCT
cana-5394	198	1	theorem	theorem	VERB
cana-5394	198	2	3.7	3.7	NUM
cana-5394	198	3	.	.	PUNCT
cana-5394	199	1	for	for	ADP
cana-5394	199	2	a	a	DET
cana-5394	199	3	fixed	fix	VERB
cana-5394	199	4	integer	integer	NOUN
cana-5394	199	5	2,n	2,n	NUM
cana-5394	199	6			NUM
cana-5394	199	7	let	let	VERB
cana-5394	199	8	s	s	PRON
cana-5394	199	9	be	be	AUX
cana-5394	199	10	an	an	DET
cana-5394	199	11	!	!	PUNCT
cana-5394	199	12	n	n	CCONJ
cana-5394	199	13	-torsion	-torsion	PROPN
cana-5394	199	14	free	free	ADJ
cana-5394	199	15	semiprime	semiprime	NOUN
cana-5394	199	16	ring	ring	NOUN
cana-5394	199	17	and	and	CCONJ
cana-5394	199	18	m	m	AUX
cana-5394	199	19	be	be	AUX
cana-5394	199	20	a	a	DET
cana-5394	199	21	nonzero	nonzero	NOUN
cana-5394	199	22	ideal	ideal	NOUN
cana-5394	199	23	of	of	ADP
cana-5394	199	24	.s	.s	NOUN
cana-5394	199	25	suppose	suppose	VERB
cana-5394	199	26	that	that	SCONJ
cana-5394	199	27	s	s	VERB
cana-5394	199	28	admits	admit	VERB
cana-5394	199	29	two	two	NUM
cana-5394	199	30	nonzero	nonzero	ADJ
cana-5394	199	31	symmetric	symmetric	ADJ
cana-5394	199	32	n	n	ADJ
cana-5394	199	33	-derivations	-derivation	NOUN
cana-5394	199	34	:	:	PUNCT
cana-5394	199	35	s	s	VERB
cana-5394	199	36	nd	nd	ADV
cana-5394	199	37	s→	s→	NOUN
cana-5394	199	38	with	with	ADP
cana-5394	199	39	trace	trace	NOUN
cana-5394	199	40	:	:	PUNCT
cana-5394	199	41	s	s	AUX
cana-5394	199	42	s	s	NOUN
cana-5394	199	43	→	→	SYM
cana-5394	199	44	satisfying	satisfy	VERB
cana-5394	199	45	[	[	PUNCT
cana-5394	199	46	,	,	PUNCT
cana-5394	199	47	]	]	X
cana-5394	199	48	[	[	PUNCT
cana-5394	199	49	,	,	PUNCT
cana-5394	199	50	]	]	PUNCT
cana-5394	199	51	z(s)	z(s)	X
cana-5394	199	52			PROPN
cana-5394	200	1			PROPN
cana-5394	200	2			PROPN
cana-5394	200	3			PROPN
cana-5394	200	4			NOUN
cana-5394	200	5	for	for	ADP
cana-5394	200	6	all	all	PRON
cana-5394	200	7	,	,	PUNCT
cana-5394	200	8	,	,	PUNCT
cana-5394	200	9	m	m	PROPN
cana-5394	200	10			PROPN
cana-5394	200	11			PROPN
cana-5394	200	12	then	then	ADV
cana-5394	200	13	s	s	AUX
cana-5394	200	14	has	have	VERB
cana-5394	200	15	a	a	DET
cana-5394	200	16	nonzero	nonzero	ADJ
cana-5394	200	17	central	central	ADJ
cana-5394	200	18	ideal	ideal	NOUN
cana-5394	200	19	.	.	PUNCT
cana-5394	201	1	communications	communication	NOUN
cana-5394	201	2	on	on	ADP
cana-5394	201	3	applied	apply	VERB
cana-5394	201	4	nonlinear	nonlinear	ADJ
cana-5394	201	5	analysis	analysis	NOUN
cana-5394	201	6	issn	issn	NOUN
cana-5394	201	7	:	:	PUNCT
cana-5394	201	8	1074	1074	NUM
cana-5394	201	9	-	-	PUNCT
cana-5394	201	10	133x	133x	NUM
cana-5394	201	11	vol	vol	NOUN
cana-5394	201	12	32	32	NUM
cana-5394	201	13	no	no	NOUN
cana-5394	201	14	.	.	NOUN
cana-5394	201	15	3	3	NUM
cana-5394	201	16	(	(	PUNCT
cana-5394	201	17	2025	2025	NUM
cana-5394	201	18	)	)	PUNCT
cana-5394	202	1	936	936	NUM
cana-5394	202	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	202	3	corollary3.8	corollary3.8	PROPN
cana-5394	202	4	:	:	PUNCT
cana-5394	202	5	for	for	ADP
cana-5394	202	6	a	a	DET
cana-5394	202	7	fixed	fix	VERB
cana-5394	202	8	integer	integer	NOUN
cana-5394	202	9	2,n	2,n	NUM
cana-5394	202	10			NUM
cana-5394	202	11	let	let	VERB
cana-5394	202	12	s	s	PRON
cana-5394	202	13	be	be	AUX
cana-5394	202	14	an	an	DET
cana-5394	202	15	!	!	PUNCT
cana-5394	202	16	n	n	CCONJ
cana-5394	202	17	-torsion	-torsion	PROPN
cana-5394	202	18	free	free	ADJ
cana-5394	202	19	semiprime	semiprime	NOUN
cana-5394	202	20	ring	ring	NOUN
cana-5394	202	21	.	.	PUNCT
cana-5394	203	1	assume	assume	VERB
cana-5394	203	2	that	that	SCONJ
cana-5394	203	3	s	s	VERB
cana-5394	203	4	admits	admit	VERB
cana-5394	203	5	two	two	NUM
cana-5394	203	6	nonzero	nonzero	ADJ
cana-5394	203	7	symmetric	symmetric	ADJ
cana-5394	203	8	n	n	ADJ
cana-5394	203	9	-derivations	-derivation	NOUN
cana-5394	203	10	:	:	PUNCT
cana-5394	203	11	s	s	VERB
cana-5394	203	12	nd	nd	ADV
cana-5394	203	13	s→	s→	NOUN
cana-5394	203	14	with	with	ADP
cana-5394	203	15	trace	trace	NOUN
cana-5394	203	16	:	:	PUNCT
cana-5394	203	17	s	s	VERB
cana-5394	203	18	s	s	PROPN
cana-5394	203	19	→	→	SYM
cana-5394	203	20	satisfying	satisfy	VERB
cana-5394	203	21	any	any	DET
cana-5394	203	22	one	one	NUM
cana-5394	203	23	of	of	ADP
cana-5394	203	24	the	the	DET
cana-5394	203	25	conditions	condition	NOUN
cana-5394	203	26	:	:	PUNCT
cana-5394	203	27	[	[	PUNCT
cana-5394	203	28	,	,	PUNCT
cana-5394	203	29	]	]	PUNCT
cana-5394	203	30	z(s)	z(s)	X
cana-5394	203	31			PROPN
cana-5394	204	1			PROPN
cana-5394	204	2			ADJ
cana-5394	204	3			NOUN
cana-5394	204	4	for	for	ADP
cana-5394	204	5	all	all	PRON
cana-5394	204	6	,	,	PUNCT
cana-5394	204	7	,	,	PUNCT
cana-5394	204	8	s	s	PROPN
cana-5394	204	9			PROPN
cana-5394	204	10			PROPN
cana-5394	204	11	[	[	PUNCT
cana-5394	204	12	,	,	PUNCT
cana-5394	204	13	]	]	PUNCT
cana-5394	204	14	z(s)	z(s)	X
cana-5394	205	1			PROPN
cana-5394	205	2			PROPN
cana-5394	205	3			PROPN
cana-5394	205	4			PROPN
cana-5394	205	5			NOUN
cana-5394	205	6	for	for	ADP
cana-5394	205	7	all	all	PRON
cana-5394	205	8	,	,	PUNCT
cana-5394	205	9	,	,	PUNCT
cana-5394	205	10	s	s	PROPN
cana-5394	205	11			PROPN
cana-5394	205	12			PROPN
cana-5394	205	13	[	[	PUNCT
cana-5394	205	14	,	,	PUNCT
cana-5394	205	15	]	]	X
cana-5394	205	16	[	[	PUNCT
cana-5394	205	17	,	,	PUNCT
cana-5394	205	18	]	]	PUNCT
cana-5394	205	19	z(s)	z(s)	X
cana-5394	205	20			PROPN
cana-5394	206	1			PROPN
cana-5394	206	2			PROPN
cana-5394	206	3			PROPN
cana-5394	206	4			NOUN
cana-5394	206	5	for	for	ADP
cana-5394	206	6	all	all	PRON
cana-5394	206	7	,	,	PUNCT
cana-5394	206	8	s	s	PROPN
cana-5394	206	9			PROPN
cana-5394	206	10			NOUN
cana-5394	206	11	thus	thus	ADV
cana-5394	206	12	,	,	PUNCT
cana-5394	206	13	s	s	VERB
cana-5394	206	14	is	be	AUX
cana-5394	206	15	commutative	commutative	ADJ
cana-5394	206	16	.	.	PUNCT
cana-5394	207	1	theorem	theorem	VERB
cana-5394	207	2	3.9	3.9	NUM
cana-5394	207	3	.	.	PUNCT
cana-5394	208	1	for	for	ADP
cana-5394	208	2	a	a	DET
cana-5394	208	3	fixed	fix	VERB
cana-5394	208	4	integer	integer	NOUN
cana-5394	208	5	2,n	2,n	NUM
cana-5394	208	6			NUM
cana-5394	208	7	let	let	VERB
cana-5394	208	8	s	s	PRON
cana-5394	208	9	be	be	AUX
cana-5394	208	10	an	an	DET
cana-5394	208	11	!	!	PUNCT
cana-5394	208	12	n	n	CCONJ
cana-5394	208	13	-torsion	-torsion	PROPN
cana-5394	208	14	free	free	ADJ
cana-5394	208	15	semiprime	semiprime	NOUN
cana-5394	208	16	ring	ring	NOUN
cana-5394	208	17	and	and	CCONJ
cana-5394	208	18	m	m	AUX
cana-5394	208	19	be	be	AUX
cana-5394	208	20	a	a	DET
cana-5394	208	21	nonzero	nonzero	NOUN
cana-5394	208	22	ideal	ideal	NOUN
cana-5394	208	23	of	of	ADP
cana-5394	208	24	.s	.s	NOUN
cana-5394	208	25	suppose	suppose	VERB
cana-5394	208	26	that	that	SCONJ
cana-5394	208	27	s	s	VERB
cana-5394	208	28	admits	admit	VERB
cana-5394	208	29	two	two	NUM
cana-5394	208	30	nonzero	nonzero	ADJ
cana-5394	208	31	symmetric	symmetric	ADJ
cana-5394	208	32	n	n	ADJ
cana-5394	208	33	-derivations	-derivation	NOUN
cana-5394	208	34	:	:	PUNCT
cana-5394	208	35	s	s	VERB
cana-5394	208	36	nd	nd	ADV
cana-5394	208	37	s→	s→	NOUN
cana-5394	208	38	with	with	ADP
cana-5394	208	39	trace	trace	NOUN
cana-5394	208	40	:	:	PUNCT
cana-5394	208	41	s	s	AUX
cana-5394	208	42	s	s	NOUN
cana-5394	208	43	→	→	SYM
cana-5394	208	44	satisfying	satisfy	VERB
cana-5394	208	45	(	(	PUNCT
cana-5394	208	46	)	)	PUNCT
cana-5394	208	47	(	(	PUNCT
cana-5394	208	48	)	)	PUNCT
cana-5394	208	49	(	(	PUNCT
cana-5394	208	50	)	)	PUNCT
cana-5394	208	51	z	z	NOUN
cana-5394	209	1	s	s	PROPN
cana-5394	209	2			PROPN
cana-5394	209	3			NUM
cana-5394	209	4			PROPN
cana-5394	209	5			ADJ
cana-5394	209	6			NOUN
cana-5394	209	7	for	for	ADP
cana-5394	209	8	all	all	PRON
cana-5394	209	9	,	,	PUNCT
cana-5394	209	10	,	,	PUNCT
cana-5394	209	11	m	m	PROPN
cana-5394	209	12			PROPN
cana-5394	209	13			PROPN
cana-5394	209	14	then	then	ADV
cana-5394	209	15	s	s	AUX
cana-5394	209	16	has	have	VERB
cana-5394	209	17	a	a	DET
cana-5394	209	18	nonzero	nonzero	ADJ
cana-5394	209	19	central	central	ADJ
cana-5394	209	20	ideal	ideal	NOUN
cana-5394	209	21	.	.	PUNCT
cana-5394	210	1	proof	proof	NOUN
cana-5394	210	2	:	:	PUNCT
cana-5394	210	3	it	it	PRON
cana-5394	210	4	is	be	AUX
cana-5394	210	5	given	give	VERB
cana-5394	210	6	that	that	PRON
cana-5394	210	7	(	(	PUNCT
cana-5394	210	8	)	)	PUNCT
cana-5394	210	9	(	(	PUNCT
cana-5394	210	10	)	)	PUNCT
cana-5394	210	11	(	(	PUNCT
cana-5394	210	12	)	)	PUNCT
cana-5394	210	13	z	z	NOUN
cana-5394	210	14	s	s	PROPN
cana-5394	210	15			PROPN
cana-5394	210	16			NUM
cana-5394	210	17			PROPN
cana-5394	210	18			ADJ
cana-5394	210	19			NOUN
cana-5394	210	20	for	for	ADP
cana-5394	210	21	all	all	PRON
cana-5394	210	22	,	,	PUNCT
cana-5394	210	23	.m	.m	PUNCT
cana-5394	211	1			PROPN
cana-5394	211	2			NOUN
cana-5394	211	3	(	(	PUNCT
cana-5394	211	4	3.26	3.26	NUM
cana-5394	211	5	)	)	PUNCT
cana-5394	211	6	replacing	replace	VERB
cana-5394	211	7			PROPN
cana-5394	211	8	by	by	ADP
cana-5394	211	9	mk	mk	NOUN
cana-5394	211	10	+	+	X
cana-5394	211	11	for	for	ADP
cana-5394	211	12	k	k	PROPN
cana-5394	211	13	m	m	PROPN
cana-5394	211	14	in	in	ADP
cana-5394	211	15	(	(	PUNCT
cana-5394	211	16	3.26	3.26	NUM
cana-5394	211	17	)	)	PUNCT
cana-5394	211	18	and	and	CCONJ
cana-5394	211	19	1	1	NUM
cana-5394	211	20	1,m	1,m	NOUN
cana-5394	211	21	n	n	X
cana-5394	211	22			NOUN
cana-5394	211	23	−	−	NOUN
cana-5394	211	24	we	we	PRON
cana-5394	211	25	arrive	arrive	VERB
cana-5394	211	26	at	at	ADP
cana-5394	211	27	(	(	PUNCT
cana-5394	211	28	)	)	PUNCT
cana-5394	211	29	1	1	NUM
cana-5394	211	30	1	1	NUM
cana-5394	211	31	(	(	PUNCT
cana-5394	211	32	)	)	PUNCT
cana-5394	211	33	(	(	PUNCT
cana-5394	211	34	)	)	PUNCT
cana-5394	211	35	(	(	PUNCT
cana-5394	211	36	)	)	PUNCT
cana-5394	211	37	(	(	PUNCT
cana-5394	211	38	)	)	PUNCT
cana-5394	211	39	(	(	PUNCT
cana-5394	211	40	)	)	PUNCT
cana-5394	211	41	d	d	X
cana-5394	211	42	(	(	PUNCT
cana-5394	211	43	,	,	PUNCT
cana-5394	211	44	...	...	PUNCT
cana-5394	211	45	,	,	PUNCT
cana-5394	211	46	,	,	PUNCT
cana-5394	211	47	,	,	PUNCT
cana-5394	211	48	...	...	PUNCT
cana-5394	211	49	,	,	PUNCT
cana-5394	211	50	)	)	PUNCT
cana-5394	211	51	(	(	PUNCT
cana-5394	211	52	)	)	PUNCT
cana-5394	212	1	n	n	CCONJ
cana-5394	212	2	n	n	PROPN
cana-5394	212	3	t	t	NOUN
cana-5394	212	4	t	t	PROPN
cana-5394	212	5	t	t	PROPN
cana-5394	212	6	timesn	timesn	PROPN
cana-5394	212	7	t	t	PROPN
cana-5394	212	8	times	times	PROPN
cana-5394	212	9	mk	mk	PROPN
cana-5394	212	10	c	c	PROPN
cana-5394	212	11	mk	mk	PROPN
cana-5394	212	12	mk	mk	PROPN
cana-5394	212	13	mk	mk	PROPN
cana-5394	212	14	z	z	PROPN
cana-5394	212	15	s	s	PROPN
cana-5394	212	16			PROPN
cana-5394	213	1			NUM
cana-5394	213	2			PROPN
cana-5394	213	3			NUM
cana-5394	213	4			PROPN
cana-5394	213	5			NUM
cana-5394	213	6			PROPN
cana-5394	213	7			PROPN
cana-5394	213	8			PROPN
cana-5394	213	9			PROPN
cana-5394	213	10			PUNCT
cana-5394	213	11			PROPN
cana-5394	213	12	−	−	NOUN
cana-5394	213	13	=	=	SYM
cana-5394	213	14	−−	−−	NOUN
cana-5394	213	15	−	−	NOUN
cana-5394	214	1	+	+	CCONJ
cana-5394	214	2	+	+	CCONJ
cana-5394	214	3			PROPN
cana-5394	214	4			PROPN
cana-5394	214	5			NOUN
cana-5394	214	6	for	for	ADP
cana-5394	214	7	all	all	PRON
cana-5394	214	8	,	,	PUNCT
cana-5394	214	9	,	,	PUNCT
cana-5394	214	10	.k	.k	PROPN
cana-5394	215	1	m	m	PROPN
cana-5394	216	1			PROPN
cana-5394	216	2			NOUN
cana-5394	216	3	(	(	PUNCT
cana-5394	216	4	3.27	3.27	NUM
cana-5394	216	5	)	)	PUNCT
cana-5394	216	6	on	on	ADP
cana-5394	216	7	using	use	VERB
cana-5394	216	8	the	the	DET
cana-5394	216	9	hypothesis	hypothesis	NOUN
cana-5394	216	10	,	,	PUNCT
cana-5394	216	11	we	we	PRON
cana-5394	216	12	arrive	arrive	VERB
cana-5394	216	13	at	at	ADP
cana-5394	216	14	(	(	PUNCT
cana-5394	216	15	)	)	PUNCT
cana-5394	216	16	1	1	NUM
cana-5394	216	17	1	1	NUM
cana-5394	216	18	(	(	PUNCT
cana-5394	216	19	)	)	PUNCT
cana-5394	216	20	d	d	NOUN
cana-5394	216	21	(	(	PUNCT
cana-5394	216	22	,	,	PUNCT
cana-5394	216	23	...	...	PUNCT
cana-5394	216	24	,	,	PUNCT
cana-5394	216	25	,	,	PUNCT
cana-5394	216	26	,	,	PUNCT
cana-5394	216	27	...	...	PUNCT
cana-5394	216	28	,	,	PUNCT
cana-5394	216	29	)	)	PUNCT
cana-5394	216	30	(	(	PUNCT
cana-5394	216	31	)	)	PUNCT
cana-5394	217	1	n	n	CCONJ
cana-5394	217	2	n	n	PROPN
cana-5394	217	3	t	t	NOUN
cana-5394	217	4	t	t	PROPN
cana-5394	217	5	t	t	PROPN
cana-5394	217	6	timesn	timesn	PROPN
cana-5394	217	7	t	t	PROPN
cana-5394	217	8	times	times	PROPN
cana-5394	217	9	c	c	PROPN
cana-5394	217	10	mk	mk	PROPN
cana-5394	217	11	mk	mk	PROPN
cana-5394	217	12	z	z	PROPN
cana-5394	217	13	s	s	PROPN
cana-5394	217	14			PROPN
cana-5394	218	1			PROPN
cana-5394	218	2			PROPN
cana-5394	218	3	−	−	PROPN
cana-5394	218	4	=	=	PUNCT
cana-5394	218	5	−−	−−	NOUN
cana-5394	218	6	−	−	NOUN
cana-5394	218	7			NOUN
cana-5394	218	8	for	for	ADP
cana-5394	218	9	all	all	PRON
cana-5394	218	10	,	,	PUNCT
cana-5394	218	11	,	,	PUNCT
cana-5394	218	12	.k	.k	PROPN
cana-5394	219	1	m	m	PROPN
cana-5394	220	1			PROPN
cana-5394	220	2			NOUN
cana-5394	220	3	(	(	PUNCT
cana-5394	220	4	3.28	3.28	NUM
cana-5394	220	5	)	)	PUNCT
cana-5394	220	6	using	use	VERB
cana-5394	220	7	lemma2.1	lemma2.1	PROPN
cana-5394	220	8	,	,	PUNCT
cana-5394	220	9	we	we	PRON
cana-5394	220	10	obtain	obtain	VERB
cana-5394	220	11	that	that	PRON
cana-5394	220	12	(	(	PUNCT
cana-5394	220	13	)	)	PUNCT
cana-5394	220	14	(	(	PUNCT
cana-5394	220	15	,	,	PUNCT
cana-5394	220	16	...	...	PUNCT
cana-5394	220	17	,	,	PUNCT
cana-5394	220	18	,	,	PUNCT
cana-5394	220	19	)	)	PUNCT
cana-5394	220	20	(	(	PUNCT
cana-5394	220	21	)	)	PUNCT
cana-5394	221	1	d	d	X
cana-5394	221	2	k	k	X
cana-5394	221	3	z	z	NOUN
cana-5394	221	4	s	s	PROPN
cana-5394	221	5			PROPN
cana-5394	221	6			PROPN
cana-5394	221	7			PROPN
cana-5394	221	8			NOUN
cana-5394	221	9	for	for	ADP
cana-5394	221	10	all	all	PRON
cana-5394	221	11	,	,	PUNCT
cana-5394	221	12	,	,	PUNCT
cana-5394	221	13	.k	.k	PROPN
cana-5394	221	14	m	m	PROPN
cana-5394	222	1			PROPN
cana-5394	222	2			NOUN
cana-5394	222	3	(	(	PUNCT
cana-5394	222	4	3.29	3.29	NUM
cana-5394	222	5	)	)	PUNCT
cana-5394	222	6	writing	write	VERB
cana-5394	222	7			PROPN
cana-5394	222	8	in	in	ADP
cana-5394	222	9	place	place	NOUN
cana-5394	222	10	of	of	ADP
cana-5394	222	11	k	k	PROPN
cana-5394	222	12	,	,	PUNCT
cana-5394	222	13	we	we	PRON
cana-5394	222	14	get	get	VERB
cana-5394	222	15	(	(	PUNCT
cana-5394	222	16	)	)	PUNCT
cana-5394	222	17	(	(	PUNCT
cana-5394	222	18	)	)	PUNCT
cana-5394	222	19	z(s)	z(s)	PART
cana-5394	223	1			PROPN
cana-5394	224	1			NUM
cana-5394	224	2			PROPN
cana-5394	224	3			NOUN
cana-5394	224	4	for	for	ADP
cana-5394	224	5	all	all	PRON
cana-5394	224	6	,	,	PUNCT
cana-5394	224	7	.m	.m	PUNCT
cana-5394	225	1			PROPN
cana-5394	225	2			NOUN
cana-5394	225	3	(	(	PUNCT
cana-5394	225	4	3.30	3.30	NUM
cana-5394	225	5	)	)	PUNCT
cana-5394	225	6	from	from	ADP
cana-5394	225	7	the	the	DET
cana-5394	225	8	hypothesis	hypothesis	NOUN
cana-5394	225	9	,	,	PUNCT
cana-5394	225	10	we	we	PRON
cana-5394	225	11	see	see	VERB
cana-5394	225	12	that	that	DET
cana-5394	225	13	(	(	PUNCT
cana-5394	225	14	)	)	PUNCT
cana-5394	225	15	z	z	NOUN
cana-5394	225	16	s	s	VERB
cana-5394	225	17			NOUN
cana-5394	225	18	for	for	ADP
cana-5394	225	19	all	all	PRON
cana-5394	225	20	,	,	PUNCT
cana-5394	225	21	.m	.m	PUNCT
cana-5394	226	1			PROPN
cana-5394	226	2			NOUN
cana-5394	226	3	(	(	PUNCT
cana-5394	226	4	3.31	3.31	NUM
cana-5394	226	5	)	)	PUNCT
cana-5394	226	6	consequently	consequently	ADV
cana-5394	226	7	,	,	PUNCT
cana-5394	226	8	the	the	DET
cana-5394	226	9	result	result	NOUN
cana-5394	226	10	can	can	AUX
cana-5394	226	11	be	be	AUX
cana-5394	226	12	derived	derive	VERB
cana-5394	226	13	using	use	VERB
cana-5394	226	14	the	the	DET
cana-5394	226	15	same	same	ADJ
cana-5394	226	16	approach	approach	NOUN
cana-5394	226	17	as	as	ADP
cana-5394	226	18	(	(	PUNCT
cana-5394	226	19	3.5	3.5	NUM
cana-5394	226	20	)	)	PUNCT
cana-5394	226	21	in	in	ADP
cana-5394	226	22	theorem	theorem	ADJ
cana-5394	226	23	3.1	3.1	NUM
cana-5394	226	24	.	.	PUNCT
cana-5394	227	1	following	follow	VERB
cana-5394	227	2	results	result	NOUN
cana-5394	227	3	can	can	AUX
cana-5394	227	4	analogously	analogously	ADV
cana-5394	227	5	be	be	AUX
cana-5394	227	6	obtained	obtain	VERB
cana-5394	227	7	.	.	PUNCT
cana-5394	228	1	theorem	theorem	VERB
cana-5394	228	2	3.10	3.10	NUM
cana-5394	228	3	.	.	PUNCT
cana-5394	229	1	for	for	ADP
cana-5394	229	2	a	a	DET
cana-5394	229	3	fixed	fix	VERB
cana-5394	229	4	integer	integer	NOUN
cana-5394	229	5	2,n	2,n	NUM
cana-5394	229	6			NUM
cana-5394	229	7	let	let	VERB
cana-5394	229	8	s	s	PRON
cana-5394	229	9	be	be	AUX
cana-5394	229	10	an	an	DET
cana-5394	229	11	!	!	PUNCT
cana-5394	229	12	n	n	CCONJ
cana-5394	229	13	-torsion	-torsion	PROPN
cana-5394	229	14	free	free	ADJ
cana-5394	229	15	semiprime	semiprime	NOUN
cana-5394	229	16	ring	ring	NOUN
cana-5394	229	17	and	and	CCONJ
cana-5394	229	18	m	m	AUX
cana-5394	229	19	be	be	AUX
cana-5394	229	20	a	a	DET
cana-5394	229	21	nonzero	nonzero	NOUN
cana-5394	229	22	ideal	ideal	NOUN
cana-5394	229	23	of	of	ADP
cana-5394	229	24	.s	.s	NOUN
cana-5394	229	25	suppose	suppose	VERB
cana-5394	229	26	that	that	SCONJ
cana-5394	229	27	s	s	VERB
cana-5394	229	28	admits	admit	VERB
cana-5394	229	29	two	two	NUM
cana-5394	229	30	nonzero	nonzero	ADJ
cana-5394	229	31	symmetric	symmetric	ADJ
cana-5394	229	32	n	n	ADJ
cana-5394	229	33	-derivations	-derivation	NOUN
cana-5394	229	34	:	:	PUNCT
cana-5394	229	35	s	s	VERB
cana-5394	229	36	nd	nd	ADV
cana-5394	229	37	s→	s→	NOUN
cana-5394	229	38	with	with	ADP
cana-5394	229	39	trace	trace	NOUN
cana-5394	229	40	:	:	PUNCT
cana-5394	229	41	s	s	AUX
cana-5394	229	42	s	s	NOUN
cana-5394	229	43	→	→	SYM
cana-5394	229	44	satisfying	satisfy	VERB
cana-5394	229	45	(	(	PUNCT
cana-5394	229	46	)	)	PUNCT
cana-5394	229	47	(	(	PUNCT
cana-5394	229	48	)	)	PUNCT
cana-5394	229	49	(	(	PUNCT
cana-5394	229	50	)	)	PUNCT
cana-5394	229	51	z	z	NOUN
cana-5394	230	1	s	s	PROPN
cana-5394	230	2			PROPN
cana-5394	230	3			ADJ
cana-5394	230	4			PROPN
cana-5394	230	5			PROPN
cana-5394	230	6			PROPN
cana-5394	230	7			NOUN
cana-5394	230	8	for	for	ADP
cana-5394	230	9	all	all	PRON
cana-5394	230	10	,	,	PUNCT
cana-5394	230	11	,	,	PUNCT
cana-5394	230	12	m	m	PROPN
cana-5394	230	13			PROPN
cana-5394	230	14			PROPN
cana-5394	230	15	then	then	ADV
cana-5394	230	16	s	s	AUX
cana-5394	230	17	has	have	VERB
cana-5394	230	18	a	a	DET
cana-5394	230	19	nonzero	nonzero	ADJ
cana-5394	230	20	central	central	ADJ
cana-5394	230	21	ideal	ideal	NOUN
cana-5394	230	22	.	.	PUNCT
cana-5394	231	1	communications	communication	NOUN
cana-5394	231	2	on	on	ADP
cana-5394	231	3	applied	apply	VERB
cana-5394	231	4	nonlinear	nonlinear	ADJ
cana-5394	231	5	analysis	analysis	NOUN
cana-5394	231	6	issn	issn	NOUN
cana-5394	231	7	:	:	PUNCT
cana-5394	231	8	1074	1074	NUM
cana-5394	231	9	-	-	PUNCT
cana-5394	231	10	133x	133x	NUM
cana-5394	231	11	vol	vol	NOUN
cana-5394	231	12	32	32	NUM
cana-5394	231	13	no	no	NOUN
cana-5394	231	14	.	.	NOUN
cana-5394	231	15	3	3	NUM
cana-5394	231	16	(	(	PUNCT
cana-5394	231	17	2025	2025	NUM
cana-5394	231	18	)	)	PUNCT
cana-5394	231	19	937	937	NUM
cana-5394	231	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	231	21	theorem	theorem	VERB
cana-5394	231	22	3.11	3.11	NUM
cana-5394	231	23	.	.	PUNCT
cana-5394	232	1	for	for	ADP
cana-5394	232	2	a	a	DET
cana-5394	232	3	fixed	fix	VERB
cana-5394	232	4	integer	integer	NOUN
cana-5394	232	5	2,n	2,n	NUM
cana-5394	232	6			NUM
cana-5394	232	7	let	let	VERB
cana-5394	232	8	s	s	PRON
cana-5394	232	9	be	be	AUX
cana-5394	232	10	an	an	DET
cana-5394	232	11	!	!	PUNCT
cana-5394	232	12	n	n	CCONJ
cana-5394	232	13	-torsion	-torsion	PROPN
cana-5394	232	14	free	free	ADJ
cana-5394	232	15	semiprime	semiprime	NOUN
cana-5394	232	16	ring	ring	NOUN
cana-5394	232	17	and	and	CCONJ
cana-5394	232	18	m	m	AUX
cana-5394	232	19	be	be	AUX
cana-5394	232	20	a	a	DET
cana-5394	232	21	nonzero	nonzero	NOUN
cana-5394	232	22	ideal	ideal	NOUN
cana-5394	232	23	of	of	ADP
cana-5394	232	24	.s	.s	NOUN
cana-5394	232	25	suppose	suppose	VERB
cana-5394	232	26	that	that	SCONJ
cana-5394	232	27	s	s	VERB
cana-5394	232	28	admits	admit	VERB
cana-5394	232	29	two	two	NUM
cana-5394	232	30	nonzero	nonzero	ADJ
cana-5394	232	31	symmetric	symmetric	ADJ
cana-5394	232	32	n	n	ADJ
cana-5394	232	33	-derivations	-derivation	NOUN
cana-5394	232	34	:	:	PUNCT
cana-5394	232	35	s	s	VERB
cana-5394	232	36	nd	nd	ADV
cana-5394	232	37	s→	s→	NOUN
cana-5394	232	38	with	with	ADP
cana-5394	232	39	trace	trace	NOUN
cana-5394	232	40	:	:	PUNCT
cana-5394	232	41	s	s	AUX
cana-5394	232	42	s	s	NOUN
cana-5394	232	43	→	→	SYM
cana-5394	232	44	satisfying	satisfy	VERB
cana-5394	232	45	(	(	PUNCT
cana-5394	232	46	)	)	PUNCT
cana-5394	232	47	(	(	PUNCT
cana-5394	232	48	)	)	PUNCT
cana-5394	232	49	[	[	PUNCT
cana-5394	232	50	,	,	PUNCT
cana-5394	232	51	]	]	X
cana-5394	232	52	(	(	PUNCT
cana-5394	232	53	)	)	PUNCT
cana-5394	232	54	z	z	NOUN
cana-5394	233	1	s	s	PROPN
cana-5394	233	2			PROPN
cana-5394	233	3			ADJ
cana-5394	233	4			PROPN
cana-5394	233	5			PROPN
cana-5394	233	6			PROPN
cana-5394	233	7			NOUN
cana-5394	233	8	for	for	ADP
cana-5394	233	9	all	all	PRON
cana-5394	233	10	,	,	PUNCT
cana-5394	233	11	,	,	PUNCT
cana-5394	233	12	m	m	PROPN
cana-5394	233	13			PROPN
cana-5394	233	14			PROPN
cana-5394	233	15	then	then	ADV
cana-5394	233	16	s	s	AUX
cana-5394	233	17	has	have	VERB
cana-5394	233	18	a	a	DET
cana-5394	233	19	nonzero	nonzero	ADJ
cana-5394	233	20	central	central	ADJ
cana-5394	233	21	ideal	ideal	NOUN
cana-5394	233	22	.	.	PUNCT
cana-5394	234	1	corollary3.12	corollary3.12	NOUN
cana-5394	234	2	:	:	PUNCT
cana-5394	234	3	for	for	ADP
cana-5394	234	4	a	a	DET
cana-5394	234	5	fixed	fix	VERB
cana-5394	234	6	integer	integer	NOUN
cana-5394	234	7	2,n	2,n	NUM
cana-5394	234	8			NUM
cana-5394	234	9	let	let	VERB
cana-5394	234	10	s	s	PRON
cana-5394	234	11	be	be	AUX
cana-5394	234	12	an	an	DET
cana-5394	234	13	!	!	PUNCT
cana-5394	234	14	n	n	CCONJ
cana-5394	234	15	-torsion	-torsion	PROPN
cana-5394	234	16	free	free	ADJ
cana-5394	234	17	semiprime	semiprime	NOUN
cana-5394	234	18	ring	ring	NOUN
cana-5394	234	19	.	.	PUNCT
cana-5394	235	1	assume	assume	VERB
cana-5394	235	2	that	that	SCONJ
cana-5394	235	3	s	s	VERB
cana-5394	235	4	admits	admit	VERB
cana-5394	235	5	two	two	NUM
cana-5394	235	6	nonzero	nonzero	ADJ
cana-5394	235	7	symmetric	symmetric	ADJ
cana-5394	235	8	n	n	ADJ
cana-5394	235	9	-derivations	-derivation	NOUN
cana-5394	235	10	:	:	PUNCT
cana-5394	235	11	s	s	VERB
cana-5394	235	12	nd	nd	ADV
cana-5394	235	13	s→	s→	NOUN
cana-5394	235	14	with	with	ADP
cana-5394	235	15	trace	trace	NOUN
cana-5394	235	16	:	:	PUNCT
cana-5394	235	17	s	s	VERB
cana-5394	235	18	s	s	PROPN
cana-5394	235	19	→	→	SYM
cana-5394	235	20	satisfying	satisfy	VERB
cana-5394	235	21	any	any	DET
cana-5394	235	22	one	one	NUM
cana-5394	235	23	of	of	ADP
cana-5394	235	24	the	the	DET
cana-5394	235	25	conditions	condition	NOUN
cana-5394	235	26	:	:	PUNCT
cana-5394	235	27	(	(	PUNCT
cana-5394	235	28	)	)	PUNCT
cana-5394	235	29	(	(	PUNCT
cana-5394	235	30	)	)	PUNCT
cana-5394	235	31	(	(	PUNCT
cana-5394	235	32	)	)	PUNCT
cana-5394	235	33	z	z	NOUN
cana-5394	235	34	s	s	PROPN
cana-5394	235	35			PROPN
cana-5394	235	36			NUM
cana-5394	235	37			PROPN
cana-5394	235	38			ADJ
cana-5394	235	39			NOUN
cana-5394	235	40	for	for	ADP
cana-5394	235	41	all	all	PRON
cana-5394	235	42	,	,	PUNCT
cana-5394	235	43	s	s	PROPN
cana-5394	235	44			PROPN
cana-5394	235	45			NOUN
cana-5394	235	46	,	,	PUNCT
cana-5394	235	47	(	(	PUNCT
cana-5394	235	48	)	)	PUNCT
cana-5394	235	49	(	(	PUNCT
cana-5394	235	50	)	)	PUNCT
cana-5394	235	51	(	(	PUNCT
cana-5394	235	52	)	)	PUNCT
cana-5394	236	1	z	z	NOUN
cana-5394	236	2	s	s	PROPN
cana-5394	236	3			PROPN
cana-5394	236	4			ADJ
cana-5394	236	5			PROPN
cana-5394	236	6			PROPN
cana-5394	236	7			PROPN
cana-5394	236	8			NOUN
cana-5394	236	9	for	for	ADP
cana-5394	236	10	all	all	PRON
cana-5394	236	11	,	,	PUNCT
cana-5394	236	12	s	s	PROPN
cana-5394	236	13			PROPN
cana-5394	236	14			NOUN
cana-5394	236	15	,	,	PUNCT
cana-5394	236	16	(	(	PUNCT
cana-5394	236	17	)	)	PUNCT
cana-5394	236	18	(	(	PUNCT
cana-5394	236	19	)	)	PUNCT
cana-5394	236	20	[	[	PUNCT
cana-5394	236	21	,	,	PUNCT
cana-5394	236	22	]	]	X
cana-5394	236	23	(	(	PUNCT
cana-5394	236	24	)	)	PUNCT
cana-5394	236	25	z	z	NOUN
cana-5394	236	26	s	s	PROPN
cana-5394	236	27			PROPN
cana-5394	237	1			ADJ
cana-5394	237	2			PROPN
cana-5394	237	3			PROPN
cana-5394	237	4			PROPN
cana-5394	237	5			NOUN
cana-5394	237	6	for	for	ADP
cana-5394	237	7	all	all	PRON
cana-5394	237	8	,	,	PUNCT
cana-5394	237	9	s	s	PROPN
cana-5394	237	10			PROPN
cana-5394	237	11			NOUN
cana-5394	237	12	thus	thus	ADV
cana-5394	237	13	,	,	PUNCT
cana-5394	237	14	s	s	VERB
cana-5394	237	15	is	be	AUX
cana-5394	237	16	commutative	commutative	ADJ
cana-5394	237	17	.	.	PUNCT
cana-5394	238	1	example	example	NOUN
cana-5394	238	2	:	:	PUNCT
cana-5394	238	3	suppose	suppose	VERB
cana-5394	238	4	the	the	DET
cana-5394	238	5	ring	ring	NOUN
cana-5394	238	6	/	/	SYM
cana-5394	238	7	,	,	PUNCT
cana-5394	238	8	0	0	NUM
cana-5394	238	9	0	0	NUM
cana-5394	238	10	u	u	NOUN
cana-5394	238	11	v	v	ADP
cana-5394	238	12	s	s	X
cana-5394	238	13	u	u	NOUN
cana-5394	238	14	v	v	ADP
cana-5394	238	15	z	z	NOUN
cana-5394	238	16			NOUN
cana-5394	238	17			PART
cana-5394	238	18			NOUN
cana-5394	238	19	=	=	PUNCT
cana-5394	238	20			NOUN
cana-5394	238	21			ADJ
cana-5394	238	22			NOUN
cana-5394	238	23			VERB
cana-5394	238	24			ADJ
cana-5394	238	25			PROPN
cana-5394	238	26	.	.	PUNCT
cana-5394	239	1	consider	consider	VERB
cana-5394	239	2	0	0	NUM
cana-5394	239	3	/	/	SYM
cana-5394	239	4	,	,	PUNCT
cana-5394	239	5	0	0	NUM
cana-5394	239	6	0	0	NUM
cana-5394	239	7	u	u	NOUN
cana-5394	239	8	m	m	NOUN
cana-5394	239	9	u	u	NOUN
cana-5394	239	10	v	v	ADP
cana-5394	239	11	z	z	NOUN
cana-5394	239	12			NOUN
cana-5394	239	13			PART
cana-5394	239	14			NOUN
cana-5394	239	15	=	=	PUNCT
cana-5394	239	16			NOUN
cana-5394	239	17			ADJ
cana-5394	239	18			NOUN
cana-5394	239	19			VERB
cana-5394	239	20			NOUN
cana-5394	239	21			PRON
cana-5394	239	22	be	be	VERB
cana-5394	239	23	a	a	DET
cana-5394	239	24	nonzero	nonzero	NOUN
cana-5394	239	25	ideal	ideal	NOUN
cana-5394	239	26	of	of	ADP
cana-5394	239	27	.s	.s	NOUN
cana-5394	239	28	denote	denote	NOUN
cana-5394	239	29	,	,	PUNCT
cana-5394	239	30	0	0	NUM
cana-5394	239	31	0	0	NUM
cana-5394	240	1	i	i	PRON
cana-5394	240	2	i	i	PRON
cana-5394	241	1	i	i	PRON
cana-5394	241	2	u	u	VERB
cana-5394	241	3	v	v	VERB
cana-5394	241	4	w	w	PROPN
cana-5394	241	5	s	s	PROPN
cana-5394	241	6			NOUN
cana-5394	241	7			NOUN
cana-5394	241	8	=	=	PUNCT
cana-5394	241	9			SYM
cana-5394	241	10			NOUN
cana-5394	241	11			PROPN
cana-5394	241	12			PROPN
cana-5394	241	13	,	,	PUNCT
cana-5394	241	14	,	,	PUNCT
cana-5394	241	15	i	i	PRON
cana-5394	241	16	iu	iu	ADP
cana-5394	241	17	v	v	NUM
cana-5394	241	18	z	z	PROPN
cana-5394	241	19	1	1	NUM
cana-5394	241	20	i	i	PRON
cana-5394	241	21	n	n	VERB
cana-5394	241	22			NOUN
cana-5394	241	23	and	and	CCONJ
cana-5394	241	24	define	define	VERB
cana-5394	241	25	as	as	ADP
cana-5394	241	26	:	:	PUNCT
cana-5394	241	27	s	s	VERB
cana-5394	241	28	nd	nd	ADV
cana-5394	241	29	s→	s→	NOUN
cana-5394	241	30	1	1	NUM
cana-5394	241	31	2	2	NUM
cana-5394	241	32	1	1	NUM
cana-5394	241	33	2	2	NUM
cana-5394	241	34	...	...	SYM
cana-5394	241	35	0	0	PUNCT
cana-5394	242	1	(	(	PUNCT
cana-5394	242	2	,	,	PUNCT
cana-5394	242	3	...	...	PUNCT
cana-5394	242	4	)	)	PUNCT
cana-5394	243	1	0	0	NUM
cana-5394	243	2	0	0	NUM
cana-5394	243	3	n	n	CCONJ
cana-5394	243	4	n	n	PROPN
cana-5394	243	5	w	w	PROPN
cana-5394	243	6	w	w	PROPN
cana-5394	243	7	w	w	PROPN
cana-5394	243	8	d	d	PROPN
cana-5394	243	9	w	w	PROPN
cana-5394	243	10	w	w	PROPN
cana-5394	243	11	w	w	PROPN
cana-5394	243	12			PROPN
cana-5394	243	13			NOUN
cana-5394	243	14	=	=	NOUN
cana-5394	243	15			NOUN
cana-5394	243	16			NOUN
cana-5394	243	17			VERB
cana-5394	243	18			PROPN
cana-5394	243	19	with	with	ADP
cana-5394	243	20	trace	trace	NOUN
cana-5394	243	21	:	:	PUNCT
cana-5394	243	22	s	s	AUX
cana-5394	243	23	s	s	PROPN
cana-5394	243	24	→	→	PUNCT
cana-5394	243	25	define	define	VERB
cana-5394	243	26	by	by	ADP
cana-5394	243	27	0	0	NUM
cana-5394	243	28	0	0	NUM
cana-5394	243	29	0	0	NUM
cana-5394	243	30	0	0	NUM
cana-5394	243	31	0	0	NUM
cana-5394	243	32	na	na	NOUN
cana-5394	243	33	b	b	PROPN
cana-5394	243	34	b	b	X
cana-5394	244	1			NUM
cana-5394	244	2			NOUN
cana-5394	244	3			PROPN
cana-5394	244	4			PROPN
cana-5394	244	5			ADJ
cana-5394	244	6			PUNCT
cana-5394	245	1	=	=	NOUN
cana-5394	245	2			PROPN
cana-5394	245	3			PROPN
cana-5394	245	4			NOUN
cana-5394	245	5			PROPN
cana-5394	245	6			NOUN
cana-5394	245	7			VERB
cana-5394	245	8			PROPN
cana-5394	245	9			VERB
cana-5394	245	10			NOUN
cana-5394	245	11			NOUN
cana-5394	245	12	.	.	PUNCT
cana-5394	246	1	it	it	PRON
cana-5394	246	2	is	be	AUX
cana-5394	246	3	evident	evident	ADJ
cana-5394	246	4	that	that	PRON
cana-5394	246	5	is	be	AUX
cana-5394	246	6	a	a	DET
cana-5394	246	7	symmetric	symmetric	ADJ
cana-5394	246	8	n	n	PRON
cana-5394	246	9	-derivation	-derivation	NOUN
cana-5394	246	10	,	,	PUNCT
cana-5394	246	11	and	and	CCONJ
cana-5394	246	12	all	all	DET
cana-5394	246	13	the	the	DET
cana-5394	246	14	conditions	condition	NOUN
cana-5394	246	15	in	in	ADP
cana-5394	246	16	the	the	DET
cana-5394	246	17	theorems	theorem	NOUN
cana-5394	246	18	are	be	AUX
cana-5394	246	19	satisfied	satisfied	ADJ
cana-5394	246	20	.	.	PUNCT
cana-5394	247	1	however	however	ADV
cana-5394	247	2	,	,	PUNCT
cana-5394	247	3	m	m	PROPN
cana-5394	247	4	is	be	AUX
cana-5394	247	5	nonzero	nonzero	X
cana-5394	247	6	ideal	ideal	NOUN
cana-5394	247	7	and	and	CCONJ
cana-5394	247	8	s	s	AUX
cana-5394	247	9	does	do	AUX
cana-5394	247	10	not	not	PART
cana-5394	247	11	contain	contain	VERB
cana-5394	247	12	any	any	DET
cana-5394	247	13	nonzero	nonzero	ADJ
cana-5394	247	14	central	central	ADJ
cana-5394	247	15	ideal	ideal	NOUN
cana-5394	247	16	.	.	PUNCT
cana-5394	248	1	acknowledgments	acknowledgment	NOUN
cana-5394	248	2	:	:	PUNCT
cana-5394	248	3	the	the	DET
cana-5394	248	4	authors	author	NOUN
cana-5394	248	5	are	be	AUX
cana-5394	248	6	very	very	ADV
cana-5394	248	7	thankful	thankful	ADJ
cana-5394	248	8	to	to	ADP
cana-5394	248	9	the	the	DET
cana-5394	248	10	referees	referee	NOUN
cana-5394	248	11	for	for	ADP
cana-5394	248	12	their	their	PRON
cana-5394	248	13	valuable	valuable	ADJ
cana-5394	248	14	comments	comment	NOUN
cana-5394	248	15	and	and	CCONJ
cana-5394	248	16	suggestions	suggestion	NOUN
cana-5394	248	17	which	which	PRON
cana-5394	248	18	have	have	AUX
cana-5394	248	19	improved	improve	VERB
cana-5394	248	20	the	the	DET
cana-5394	248	21	manuscript	manuscript	NOUN
cana-5394	248	22	immensely	immensely	ADV
cana-5394	248	23	.	.	PUNCT
cana-5394	249	1	conflict	conflict	NOUN
cana-5394	249	2	of	of	ADP
cana-5394	249	3	interest	interest	NOUN
cana-5394	249	4	:	:	PUNCT
cana-5394	249	5	the	the	DET
cana-5394	249	6	authors	author	NOUN
cana-5394	249	7	declare	declare	VERB
cana-5394	249	8	that	that	SCONJ
cana-5394	249	9	they	they	PRON
cana-5394	249	10	have	have	VERB
cana-5394	249	11	no	no	DET
cana-5394	249	12	conflicts	conflict	NOUN
cana-5394	249	13	of	of	ADP
cana-5394	249	14	interest	interest	NOUN
cana-5394	249	15	.	.	PUNCT
cana-5394	250	1	references	reference	NOUN
cana-5394	250	2	:	:	PUNCT
cana-5394	251	1	[	[	X
cana-5394	251	2	1	1	X
cana-5394	251	3	]	]	X
cana-5394	251	4	g.	g.	PROPN
cana-5394	251	5	maksa	maksa	PROPN
cana-5394	251	6	,	,	PUNCT
cana-5394	251	7	a	a	DET
cana-5394	251	8	remark	remark	NOUN
cana-5394	251	9	on	on	ADP
cana-5394	251	10	symmetric	symmetric	ADJ
cana-5394	251	11	bi	bi	ADJ
cana-5394	251	12	-	-	ADJ
cana-5394	251	13	additive	additive	ADJ
cana-5394	251	14	functions	function	NOUN
cana-5394	251	15	having	have	VERB
cana-5394	251	16	nonnegative	nonnegative	ADJ
cana-5394	251	17	diagonalization	diagonalization	NOUN
cana-5394	251	18	,	,	PUNCT
cana-5394	251	19	glas	glas	PROPN
cana-5394	251	20	.	.	PUNCT
cana-5394	251	21	math	math	PROPN
cana-5394	251	22	,	,	PUNCT
cana-5394	251	23	15	15	NUM
cana-5394	251	24	(	(	PUNCT
cana-5394	251	25	1980	1980	NUM
cana-5394	251	26	)	)	PUNCT
cana-5394	251	27	,	,	PUNCT
cana-5394	251	28	279–282	279–282	NUM
cana-5394	251	29	.	.	PUNCT
cana-5394	252	1	[	[	X
cana-5394	252	2	2	2	X
cana-5394	252	3	]	]	PUNCT
cana-5394	252	4	j.	j.	PROPN
cana-5394	252	5	vukman	vukman	PROPN
cana-5394	252	6	,	,	PUNCT
cana-5394	252	7	symmetric	symmetric	ADJ
cana-5394	252	8	bi	bi	NOUN
cana-5394	252	9	-	-	NOUN
cana-5394	252	10	derivations	derivation	NOUN
cana-5394	252	11	on	on	ADP
cana-5394	252	12	prime	prime	ADJ
cana-5394	252	13	and	and	CCONJ
cana-5394	252	14	semiprime	semiprime	NOUN
cana-5394	252	15	rings	ring	NOUN
cana-5394	252	16	,	,	PUNCT
cana-5394	252	17	aeq	aeq	PROPN
cana-5394	252	18	.	.	PUNCT
cana-5394	252	19	math	math	PROPN
cana-5394	252	20	.	.	PUNCT
cana-5394	252	21	,	,	PUNCT
cana-5394	252	22	38	38	NUM
cana-5394	252	23	(	(	PUNCT
cana-5394	252	24	1989	1989	NUM
cana-5394	252	25	)	)	PUNCT
cana-5394	252	26	,	,	PUNCT
cana-5394	252	27	245	245	NUM
cana-5394	252	28	254	254	NUM
cana-5394	252	29	.	.	PUNCT
cana-5394	253	1	[	[	X
cana-5394	253	2	3	3	X
cana-5394	253	3	]	]	X
cana-5394	253	4	j.	j.	PROPN
cana-5394	253	5	vukman	vukman	PROPN
cana-5394	253	6	,	,	PUNCT
cana-5394	253	7	two	two	NUM
cana-5394	253	8	results	result	NOUN
cana-5394	253	9	concerning	concern	VERB
cana-5394	253	10	symmetric	symmetric	ADJ
cana-5394	253	11	bi	bi	NOUN
cana-5394	253	12	-	-	NOUN
cana-5394	253	13	derivations	derivation	NOUN
cana-5394	253	14	on	on	ADP
cana-5394	253	15	prime	prime	ADJ
cana-5394	253	16	and	and	CCONJ
cana-5394	253	17	semiprime	semiprime	NOUN
cana-5394	253	18	rings	ring	NOUN
cana-5394	253	19	,	,	PUNCT
cana-5394	253	20	aeq	aeq	PROPN
cana-5394	253	21	.	.	PUNCT
cana-5394	253	22	math	math	NOUN
cana-5394	253	23	.	.	PUNCT
cana-5394	253	24	,	,	PUNCT
cana-5394	253	25	40	40	NUM
cana-5394	253	26	(	(	PUNCT
cana-5394	253	27	1990	1990	NUM
cana-5394	253	28	)	)	PUNCT
cana-5394	253	29	,	,	PUNCT
cana-5394	253	30	181–189	181–189	NUM
cana-5394	253	31	.	.	PUNCT
cana-5394	254	1	[	[	X
cana-5394	254	2	4	4	NUM
cana-5394	254	3	]	]	PUNCT
cana-5394	254	4	m.ashraf	m.ashraf	NOUN
cana-5394	254	5	,	,	PUNCT
cana-5394	254	6	on	on	ADP
cana-5394	254	7	symmetric	symmetric	ADJ
cana-5394	254	8	bi	bi	NOUN
cana-5394	254	9	-	-	NOUN
cana-5394	254	10	derivations	derivation	NOUN
cana-5394	254	11	in	in	ADP
cana-5394	254	12	rings	ring	NOUN
cana-5394	254	13	,	,	PUNCT
cana-5394	254	14	rend	rend	VERB
cana-5394	254	15	.	.	PUNCT
cana-5394	255	1	istit	istit	PROPN
cana-5394	255	2	.	.	PUNCT
cana-5394	256	1	mat	mat	NOUN
cana-5394	256	2	.	.	PROPN
cana-5394	256	3	univ	univ	PROPN
cana-5394	256	4	.	.	PUNCT
cana-5394	256	5	trieste	trieste	PROPN
cana-5394	256	6	,	,	PUNCT
cana-5394	256	7	31	31	NUM
cana-5394	256	8	(	(	PUNCT
cana-5394	256	9	1999	1999	NUM
cana-5394	256	10	)	)	PUNCT
cana-5394	256	11	,	,	PUNCT
cana-5394	256	12	25	25	NUM
cana-5394	256	13	–	–	PUNCT
cana-5394	256	14	36	36	NUM
cana-5394	256	15	.	.	PUNCT
cana-5394	257	1	[	[	X
cana-5394	257	2	5	5	NUM
cana-5394	257	3	]	]	X
cana-5394	257	4	g.	g.	PROPN
cana-5394	257	5	maksa	maksa	PROPN
cana-5394	257	6	,	,	PUNCT
cana-5394	257	7	on	on	ADP
cana-5394	257	8	the	the	DET
cana-5394	257	9	trace	trace	NOUN
cana-5394	257	10	of	of	ADP
cana-5394	257	11	symmetric	symmetric	ADJ
cana-5394	257	12	biderivations	biderivation	NOUN
cana-5394	257	13	,	,	PUNCT
cana-5394	257	14	c.	c.	PROPN
cana-5394	257	15	r.	r.	PROPN
cana-5394	257	16	math	math	PROPN
cana-5394	257	17	.	.	PUNCT
cana-5394	258	1	rep	rep	PROPN
cana-5394	258	2	.	.	PROPN
cana-5394	258	3	acad	acad	PROPN
cana-5394	258	4	.	.	PUNCT
cana-5394	259	1	sci	sci	PROPN
cana-5394	259	2	.	.	PUNCT
cana-5394	259	3	canada	canada	PROPN
cana-5394	259	4	ix	ix	PROPN
cana-5394	259	5	,	,	PUNCT
cana-5394	259	6	1987	1987	NUM
cana-5394	259	7	,	,	PUNCT
cana-5394	259	8	303–308	303–308	NUM
cana-5394	259	9	.	.	PUNCT
cana-5394	260	1	communications	communication	NOUN
cana-5394	260	2	on	on	ADP
cana-5394	260	3	applied	apply	VERB
cana-5394	260	4	nonlinear	nonlinear	ADJ
cana-5394	260	5	analysis	analysis	NOUN
cana-5394	260	6	issn	issn	NOUN
cana-5394	260	7	:	:	PUNCT
cana-5394	260	8	1074	1074	NUM
cana-5394	260	9	-	-	PUNCT
cana-5394	260	10	133x	133x	NUM
cana-5394	260	11	vol	vol	NOUN
cana-5394	260	12	32	32	NUM
cana-5394	260	13	no	no	NOUN
cana-5394	260	14	.	.	NOUN
cana-5394	260	15	3	3	NUM
cana-5394	260	16	(	(	PUNCT
cana-5394	260	17	2025	2025	NUM
cana-5394	260	18	)	)	PUNCT
cana-5394	260	19	938	938	NUM
cana-5394	260	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-5394	261	1	[	[	X
cana-5394	261	2	6	6	NUM
cana-5394	261	3	]	]	PUNCT
cana-5394	261	4	e.	e.	PROPN
cana-5394	261	5	koc	koc	PROPN
cana-5394	261	6	,	,	PUNCT
cana-5394	261	7	sogutc	sogutc	NOUN
cana-5394	261	8	u	u	PROPN
cana-5394	261	9	,	,	PUNCT
cana-5394	261	10	s.	s.	PROPN
cana-5394	261	11	huang	huang	PROPN
cana-5394	261	12	,	,	PUNCT
cana-5394	261	13	note	note	VERB
cana-5394	261	14	on	on	ADP
cana-5394	261	15	lie	lie	NOUN
cana-5394	261	16	ideals	ideal	NOUN
cana-5394	261	17	with	with	ADP
cana-5394	261	18	symmetric	symmetric	ADJ
cana-5394	261	19	bi	bi	NOUN
cana-5394	261	20	-	-	NOUN
cana-5394	261	21	derivations	derivation	NOUN
cana-5394	261	22	in	in	ADP
cana-5394	261	23	semiprime	semiprime	NOUN
cana-5394	261	24	rings	ring	NOUN
cana-5394	261	25	,	,	PUNCT
cana-5394	262	1	indian	indian	PROPN
cana-5394	262	2	j.	j.	PROPN
cana-5394	262	3	pure	pure	PROPN
cana-5394	262	4	appl	appl	PROPN
cana-5394	262	5	.	.	PUNCT
cana-5394	262	6	math	math	PROPN
cana-5394	262	7	.	.	PUNCT
cana-5394	262	8	,	,	PUNCT
cana-5394	262	9	54	54	NUM
cana-5394	262	10	(	(	PUNCT
cana-5394	262	11	2023	2023	NUM
cana-5394	262	12	)	)	PUNCT
cana-5394	262	13	,	,	PUNCT
cana-5394	262	14	608–618	608–618	NUM
cana-5394	262	15	.	.	PUNCT
cana-5394	263	1	[	[	X
cana-5394	263	2	7	7	X
cana-5394	263	3	]	]	X
cana-5394	263	4	m.	m.	NOUN
cana-5394	263	5	a.	a.	PROPN
cana-5394	263	6	ozturk	ozturk	PROPN
cana-5394	263	7	,	,	PUNCT
cana-5394	263	8	permuting	permute	VERB
cana-5394	263	9	tri	tri	NOUN
cana-5394	263	10	-	-	NOUN
cana-5394	263	11	derivations	derivation	NOUN
cana-5394	263	12	in	in	ADP
cana-5394	263	13	prime	prime	ADJ
cana-5394	263	14	and	and	CCONJ
cana-5394	263	15	semi	semi	ADJ
cana-5394	263	16	-	-	ADJ
cana-5394	263	17	prime	prime	ADJ
cana-5394	263	18	rings	ring	NOUN
cana-5394	263	19	,	,	PUNCT
cana-5394	263	20	east	east	PROPN
cana-5394	263	21	asian	asian	PROPN
cana-5394	263	22	math	math	PROPN
cana-5394	263	23	.	.	PUNCT
cana-5394	264	1	j.	j.	PROPN
cana-5394	264	2	,	,	PUNCT
cana-5394	264	3	15	15	NUM
cana-5394	264	4	(	(	PUNCT
cana-5394	264	5	1999	1999	NUM
cana-5394	264	6	)	)	PUNCT
cana-5394	264	7	,	,	PUNCT
cana-5394	264	8	177–190	177–190	NUM
cana-5394	264	9	.	.	PUNCT
cana-5394	265	1	[	[	X
cana-5394	265	2	8	8	NUM
cana-5394	265	3	]	]	PUNCT
cana-5394	265	4	k.	k.	PROPN
cana-5394	265	5	h.	h.	PROPN
cana-5394	265	6	park	park	PROPN
cana-5394	265	7	,	,	PUNCT
cana-5394	265	8	on	on	ADP
cana-5394	265	9	prime	prime	ADJ
cana-5394	265	10	and	and	CCONJ
cana-5394	265	11	semi	semi	ADJ
cana-5394	265	12	-	-	ADJ
cana-5394	265	13	prime	prime	ADJ
cana-5394	265	14	rings	ring	NOUN
cana-5394	265	15	with	with	ADP
cana-5394	265	16	symmetric	symmetric	ADJ
cana-5394	265	17	n	n	CCONJ
cana-5394	265	18	-	-	PUNCT
cana-5394	265	19	derivations	derivation	NOUN
cana-5394	265	20	,	,	PUNCT
cana-5394	265	21	j.	j.	PROPN
cana-5394	265	22	chungcheong	chungcheong	PROPN
cana-5394	265	23	math	math	PROPN
cana-5394	265	24	.	.	PUNCT
cana-5394	266	1	soc	soc	PROPN
cana-5394	266	2	.	.	PUNCT
cana-5394	266	3	,	,	PUNCT
cana-5394	266	4	22	22	NUM
cana-5394	266	5	(	(	PUNCT
cana-5394	266	6	2009	2009	NUM
cana-5394	266	7	)	)	PUNCT
cana-5394	266	8	,	,	PUNCT
cana-5394	266	9	451–458	451–458	NUM
cana-5394	266	10	.	.	PUNCT
cana-5394	267	1	[	[	X
cana-5394	267	2	9	9	NUM
cana-5394	267	3	]	]	PUNCT
cana-5394	267	4	s.	s.	PROPN
cana-5394	267	5	ali	ali	PROPN
cana-5394	267	6	,	,	PUNCT
cana-5394	267	7	t.	t.	PROPN
cana-5394	267	8	m.	m.	NOUN
cana-5394	267	9	alsuraiheed	alsuraiheed	PROPN
cana-5394	267	10	,	,	PUNCT
cana-5394	267	11	n.	n.	PROPN
cana-5394	267	12	parveen	parveen	PROPN
cana-5394	267	13	,	,	PUNCT
cana-5394	267	14	v.	v.	ADP
cana-5394	267	15	varshney	varshney	NOUN
cana-5394	267	16	,	,	PUNCT
cana-5394	267	17	action	action	NOUN
cana-5394	267	18	of	of	ADP
cana-5394	267	19	n	n	NOUN
cana-5394	267	20	-	-	PUNCT
cana-5394	267	21	derivations	derivation	NOUN
cana-5394	267	22	and	and	CCONJ
cana-5394	267	23	n	n	CCONJ
cana-5394	267	24	-	-	PUNCT
cana-5394	267	25	multipliers	multiplier	NOUN
cana-5394	267	26	on	on	ADP
cana-5394	267	27	ideals	ideal	NOUN
cana-5394	267	28	of	of	ADP
cana-5394	267	29	(	(	PUNCT
cana-5394	267	30	semi)-prime	semi)-prime	NOUN
cana-5394	267	31	rings	ring	NOUN
cana-5394	267	32	,	,	PUNCT
cana-5394	267	33	aims	aim	VERB
cana-5394	267	34	math	math	NOUN
cana-5394	267	35	.	.	PUNCT
cana-5394	267	36	,	,	PUNCT
cana-5394	267	37	8	8	NUM
cana-5394	267	38	(	(	PUNCT
cana-5394	267	39	2023	2023	NUM
cana-5394	267	40	)	)	PUNCT
cana-5394	267	41	,	,	PUNCT
cana-5394	267	42	17208–17228	17208–17228	NUM
cana-5394	267	43	.	.	PUNCT
cana-5394	268	1	[	[	X
cana-5394	268	2	10	10	NUM
cana-5394	268	3	]	]	SYM
cana-5394	268	4	m.ashraf	m.ashraf	NOUN
cana-5394	268	5	,	,	PUNCT
cana-5394	268	6	m.r.jamal	m.r.jamal	NOUN
cana-5394	268	7	,	,	PUNCT
cana-5394	268	8	traces	trace	NOUN
cana-5394	268	9	of	of	ADP
cana-5394	268	10	permuting	permute	VERB
cana-5394	268	11	n	n	CCONJ
cana-5394	268	12	-	-	PUNCT
cana-5394	268	13	additive	additive	ADJ
cana-5394	268	14	maps	map	NOUN
cana-5394	268	15	and	and	CCONJ
cana-5394	268	16	permuting	permute	VERB
cana-5394	268	17	n	n	CCONJ
cana-5394	268	18	-	-	PUNCT
cana-5394	268	19	derivations	derivation	NOUN
cana-5394	268	20	of	of	ADP
cana-5394	268	21	rings	ring	NOUN
cana-5394	268	22	,	,	PUNCT
cana-5394	268	23	mediterr	mediterr	PROPN
cana-5394	268	24	.	.	PUNCT
cana-5394	269	1	j.	j.	PROPN
cana-5394	269	2	math	math	PROPN
cana-5394	269	3	.	.	PUNCT
cana-5394	269	4	,	,	PUNCT
cana-5394	269	5	11	11	NUM
cana-5394	269	6	(	(	PUNCT
cana-5394	269	7	2014	2014	NUM
cana-5394	269	8	)	)	PUNCT
cana-5394	269	9	,	,	PUNCT
cana-5394	269	10	287–297	287–297	NUM
cana-5394	269	11	.	.	PUNCT
cana-5394	270	1	[	[	X
cana-5394	270	2	11	11	NUM
cana-5394	270	3	]	]	PUNCT
cana-5394	270	4	m.ashraf	m.ashraf	NOUN
cana-5394	270	5	,	,	PUNCT
cana-5394	270	6	m.	m.	PROPN
cana-5394	270	7	r.	r.	PROPN
cana-5394	270	8	jamal	jamal	PROPN
cana-5394	270	9	,	,	PUNCT
cana-5394	270	10	m.	m.	PROPN
cana-5394	270	11	r.	r.	PROPN
cana-5394	270	12	mozumder	mozumder	PROPN
cana-5394	270	13	,	,	PUNCT
cana-5394	270	14	on	on	ADP
cana-5394	270	15	the	the	DET
cana-5394	270	16	traces	trace	NOUN
cana-5394	270	17	of	of	ADP
cana-5394	270	18	certain	certain	ADJ
cana-5394	270	19	classes	class	NOUN
cana-5394	270	20	of	of	ADP
cana-5394	270	21	permuting	permute	VERB
cana-5394	270	22	mappings	mapping	NOUN
cana-5394	270	23	in	in	ADP
cana-5394	270	24	rings	ring	NOUN
cana-5394	270	25	,	,	PUNCT
cana-5394	270	26	georgian	georgian	PROPN
cana-5394	270	27	math	math	NOUN
cana-5394	270	28	.	.	PUNCT
cana-5394	271	1	j.	j.	PROPN
cana-5394	271	2	,	,	PUNCT
cana-5394	271	3	23	23	NUM
cana-5394	271	4	(	(	PUNCT
cana-5394	271	5	2016	2016	NUM
cana-5394	271	6	)	)	PUNCT
cana-5394	271	7	,	,	PUNCT
cana-5394	271	8	15–23	15–23	NUM
cana-5394	271	9	[	[	X
cana-5394	271	10	12	12	NUM
cana-5394	271	11	]	]	PUNCT
cana-5394	271	12	m.ashraf	m.ashraf	NOUN
cana-5394	271	13	,	,	PUNCT
cana-5394	271	14	a.	a.	PROPN
cana-5394	271	15	khan	khan	PROPN
cana-5394	271	16	,	,	PUNCT
cana-5394	271	17	m.	m.	PROPN
cana-5394	271	18	r.	r.	PROPN
cana-5394	271	19	jamal	jamal	PROPN
cana-5394	271	20	,	,	PUNCT
cana-5394	271	21	traces	trace	NOUN
cana-5394	271	22	of	of	ADP
cana-5394	271	23	permuting	permute	VERB
cana-5394	271	24	generalized	generalized	ADJ
cana-5394	271	25	n	n	CCONJ
cana-5394	271	26	-	-	PUNCT
cana-5394	271	27	derivations	derivation	NOUN
cana-5394	271	28	of	of	ADP
cana-5394	271	29	rings	ring	NOUN
cana-5394	271	30	,	,	PUNCT
cana-5394	271	31	miskolc	miskolc	ADJ
cana-5394	271	32	math	math	NOUN
cana-5394	271	33	.	.	PUNCT
cana-5394	272	1	notes	note	NOUN
cana-5394	272	2	,	,	PUNCT
cana-5394	272	3	19	19	NUM
cana-5394	272	4	(	(	PUNCT
cana-5394	272	5	2018	2018	NUM
cana-5394	272	6	)	)	PUNCT
cana-5394	272	7	,	,	PUNCT
cana-5394	272	8	731–740	731–740	NUM
cana-5394	272	9	.	.	PUNCT
cana-5394	273	1	[	[	X
cana-5394	273	2	13	13	NUM
cana-5394	273	3	]	]	X
cana-5394	273	4	n.	n.	PROPN
cana-5394	273	5	parveen	parveen	PROPN
cana-5394	273	6	,	,	PUNCT
cana-5394	273	7	product	product	NOUN
cana-5394	273	8	of	of	ADP
cana-5394	273	9	traces	trace	NOUN
cana-5394	273	10	of	of	ADP
cana-5394	273	11	symmetric	symmetric	ADJ
cana-5394	273	12	bi	bi	NOUN
cana-5394	273	13	-	-	NOUN
cana-5394	273	14	derivations	derivation	NOUN
cana-5394	273	15	in	in	ADP
cana-5394	273	16	rings	ring	NOUN
cana-5394	273	17	,	,	PUNCT
cana-5394	273	18	palest	pale	ADJ
cana-5394	273	19	.	.	PUNCT
cana-5394	274	1	j.	j.	PROPN
cana-5394	274	2	math	math	PROPN
cana-5394	274	3	.	.	PUNCT
cana-5394	274	4	,	,	PUNCT
cana-5394	274	5	11	11	NUM
cana-5394	274	6	(	(	PUNCT
cana-5394	274	7	2022	2022	NUM
cana-5394	274	8	)	)	PUNCT
cana-5394	274	9	,	,	PUNCT
cana-5394	274	10	210–216	210–216	NUM
cana-5394	274	11	.	.	PUNCT
cana-5394	275	1	[	[	X
cana-5394	275	2	14	14	NUM
cana-5394	275	3	]	]	X
cana-5394	275	4	e.	e.	PROPN
cana-5394	275	5	c.	c.	PROPN
cana-5394	275	6	posner	posner	PROPN
cana-5394	275	7	,	,	PUNCT
cana-5394	275	8	derivations	derivation	NOUN
cana-5394	275	9	in	in	ADP
cana-5394	275	10	prime	prime	ADJ
cana-5394	275	11	rings	ring	NOUN
cana-5394	275	12	,	,	PUNCT
cana-5394	275	13	proc	proc	NOUN
cana-5394	275	14	.	.	PUNCT
cana-5394	276	1	amer	amer	PROPN
cana-5394	276	2	.	.	PUNCT
cana-5394	276	3	math	math	PROPN
cana-5394	276	4	.	.	PUNCT
cana-5394	277	1	soc	soc	PROPN
cana-5394	277	2	.	.	PUNCT
cana-5394	277	3	,	,	PUNCT
cana-5394	277	4	8	8	NUM
cana-5394	277	5	(	(	PUNCT
cana-5394	277	6	1957	1957	NUM
cana-5394	277	7	)	)	PUNCT
cana-5394	277	8	,	,	PUNCT
cana-5394	277	9	1093–1100	1093–1100	NUM
cana-5394	277	10	.	.	PUNCT
cana-5394	278	1	[	[	X
cana-5394	278	2	15	15	NUM
cana-5394	278	3	]	]	X
cana-5394	278	4	m.	m.	NOUN
cana-5394	278	5	bresar	bresar	VERB
cana-5394	278	6	,	,	PUNCT
cana-5394	278	7	commuting	commuting	NOUN
cana-5394	278	8	maps	map	NOUN
cana-5394	278	9	:	:	PUNCT
cana-5394	278	10	a	a	DET
cana-5394	278	11	survey	survey	NOUN
cana-5394	278	12	,	,	PUNCT
cana-5394	278	13	taiwanese	taiwanese	PROPN
cana-5394	278	14	j.	j.	PROPN
cana-5394	278	15	math	math	PROPN
cana-5394	278	16	.	.	PUNCT
cana-5394	278	17	,	,	PUNCT
cana-5394	278	18	8	8	NUM
cana-5394	278	19	(	(	PUNCT
cana-5394	278	20	2004	2004	NUM
cana-5394	278	21	)	)	PUNCT
cana-5394	278	22	,	,	PUNCT
cana-5394	278	23	361–397	361–397	NUM
cana-5394	278	24	.	.	PUNCT
cana-5394	279	1	[	[	X
cana-5394	279	2	16	16	NUM
cana-5394	279	3	]	]	PUNCT
cana-5394	279	4	q.	q.	PROPN
cana-5394	279	5	deng	deng	PROPN
cana-5394	279	6	,	,	PUNCT
cana-5394	279	7	h.	h.	PROPN
cana-5394	279	8	e.	e.	PROPN
cana-5394	279	9	bell	bell	PROPN
cana-5394	279	10	,	,	PUNCT
cana-5394	279	11	on	on	ADP
cana-5394	279	12	derivations	derivation	NOUN
cana-5394	279	13	and	and	CCONJ
cana-5394	279	14	commutativity	commutativity	NOUN
cana-5394	279	15	in	in	ADP
cana-5394	279	16	semiprime	semiprime	NOUN
cana-5394	279	17	rings	ring	NOUN
cana-5394	279	18	,	,	PUNCT
cana-5394	279	19	commun	commun	PROPN
cana-5394	279	20	.	.	PUNCT
cana-5394	280	1	algebra	algebra	PROPN
cana-5394	280	2	,	,	PUNCT
cana-5394	280	3	23	23	NUM
cana-5394	280	4	(	(	PUNCT
cana-5394	280	5	1995	1995	NUM
cana-5394	280	6	)	)	PUNCT
cana-5394	280	7	,	,	PUNCT
cana-5394	280	8	3705–3713	3705–3713	NUM
cana-5394	280	9	.	.	PUNCT
cana-5394	281	1	[	[	X
cana-5394	281	2	17	17	NUM
cana-5394	281	3	]	]	X
cana-5394	281	4	c.	c.	NOUN
cana-5394	281	5	lanski	lanski	PROPN
cana-5394	281	6	,	,	PUNCT
cana-5394	281	7	differential	differential	ADJ
cana-5394	281	8	identities	identity	NOUN
cana-5394	281	9	,	,	PUNCT
cana-5394	281	10	lie	lie	VERB
cana-5394	281	11	ideals	ideal	NOUN
cana-5394	281	12	and	and	CCONJ
cana-5394	281	13	posner	posner	NOUN
cana-5394	281	14	’s	’s	PART
cana-5394	281	15	theorems	theorem	NOUN
cana-5394	281	16	,	,	PUNCT
cana-5394	281	17	pac	pac	PROPN
cana-5394	281	18	.	.	PUNCT
cana-5394	282	1	j.	j.	PROPN
cana-5394	282	2	math	math	PROPN
cana-5394	282	3	.	.	PROPN
cana-5394	282	4	,	,	PUNCT
cana-5394	282	5	134	134	NUM
cana-5394	282	6	(	(	PUNCT
cana-5394	282	7	1988	1988	NUM
cana-5394	282	8	)	)	PUNCT
cana-5394	282	9	,	,	PUNCT
cana-5394	282	10	275–297	275–297	NUM
cana-5394	282	11	.	.	PUNCT
cana-5394	283	1	[	[	X
cana-5394	283	2	18	18	NUM
cana-5394	283	3	]	]	X
cana-5394	283	4	j.	j.	PROPN
cana-5394	283	5	vukman	vukman	PROPN
cana-5394	283	6	,	,	PUNCT
cana-5394	283	7	commuting	commute	VERB
cana-5394	283	8	and	and	CCONJ
cana-5394	283	9	centralizing	centralize	VERB
cana-5394	283	10	mappings	mapping	NOUN
cana-5394	283	11	in	in	ADP
cana-5394	283	12	prime	prime	ADJ
cana-5394	283	13	rings	ring	NOUN
cana-5394	283	14	,	,	PUNCT
cana-5394	283	15	proc	proc	NOUN
cana-5394	283	16	.	.	PUNCT
cana-5394	284	1	amer	amer	PROPN
cana-5394	284	2	.	.	PUNCT
cana-5394	284	3	math	math	PROPN
cana-5394	284	4	.	.	PUNCT
cana-5394	285	1	soc	soc	PROPN
cana-5394	285	2	.	.	PUNCT
cana-5394	285	3	,	,	PUNCT
cana-5394	285	4	109	109	NUM
cana-5394	285	5	(	(	PUNCT
cana-5394	285	6	1990	1990	NUM
cana-5394	285	7	)	)	PUNCT
cana-5394	285	8	,	,	PUNCT
cana-5394	285	9	47–52	47–52	NUM
cana-5394	285	10	.	.	PUNCT
cana-5394	286	1	[	[	X
cana-5394	286	2	19	19	NUM
cana-5394	286	3	]	]	X
cana-5394	286	4	j.	j.	PROPN
cana-5394	286	5	vukman	vukman	PROPN
cana-5394	286	6	,	,	PUNCT
cana-5394	286	7	symmetric	symmetric	ADJ
cana-5394	286	8	bi	bi	NOUN
cana-5394	286	9	-	-	NOUN
cana-5394	286	10	derivations	derivation	NOUN
cana-5394	286	11	on	on	ADP
cana-5394	286	12	prime	prime	ADJ
cana-5394	286	13	and	and	CCONJ
cana-5394	286	14	semiprime	semiprime	NOUN
cana-5394	286	15	rings	ring	NOUN
cana-5394	286	16	,	,	PUNCT
cana-5394	286	17	aeq	aeq	PROPN
cana-5394	286	18	.	.	PUNCT
cana-5394	286	19	math	math	PROPN
cana-5394	286	20	.	.	PUNCT
cana-5394	286	21	,	,	PUNCT
cana-5394	286	22	38	38	NUM
cana-5394	286	23	(	(	PUNCT
cana-5394	286	24	1989	1989	NUM
cana-5394	286	25	)	)	PUNCT
cana-5394	286	26	,	,	PUNCT
cana-5394	286	27	245	245	NUM
cana-5394	286	28	254	254	NUM
cana-5394	286	29	.	.	PUNCT
cana-5394	287	1	[	[	X
cana-5394	287	2	20	20	NUM
cana-5394	287	3	]	]	PUNCT
cana-5394	287	4	j.	j.	PROPN
cana-5394	287	5	vukman	vukman	PROPN
cana-5394	287	6	,	,	PUNCT
cana-5394	287	7	two	two	NUM
cana-5394	287	8	results	result	NOUN
cana-5394	287	9	concerning	concern	VERB
cana-5394	287	10	symmetric	symmetric	ADJ
cana-5394	287	11	bi	bi	NOUN
cana-5394	287	12	-	-	NOUN
cana-5394	287	13	derivations	derivation	NOUN
cana-5394	287	14	on	on	ADP
cana-5394	287	15	prime	prime	ADJ
cana-5394	287	16	and	and	CCONJ
cana-5394	287	17	semiprime	semiprime	NOUN
cana-5394	287	18	rings	ring	NOUN
cana-5394	287	19	,	,	PUNCT
cana-5394	287	20	aeq	aeq	PROPN
cana-5394	287	21	.	.	PUNCT
cana-5394	287	22	math	math	NOUN
cana-5394	287	23	.	.	PUNCT
cana-5394	287	24	,	,	PUNCT
cana-5394	287	25	40	40	NUM
cana-5394	287	26	(	(	PUNCT
cana-5394	287	27	1990	1990	NUM
cana-5394	287	28	)	)	PUNCT
cana-5394	287	29	,	,	PUNCT
cana-5394	287	30	181–189	181–189	NUM
cana-5394	287	31	.	.	PUNCT
cana-5394	288	1	[	[	X
cana-5394	288	2	21	21	NUM
cana-5394	288	3	]	]	PUNCT
cana-5394	288	4	m.ashraf	m.ashraf	NOUN
cana-5394	288	5	,	,	PUNCT
cana-5394	288	6	on	on	ADP
cana-5394	288	7	symmetric	symmetric	ADJ
cana-5394	288	8	bi	bi	NOUN
cana-5394	288	9	-	-	NOUN
cana-5394	288	10	derivations	derivation	NOUN
cana-5394	288	11	in	in	ADP
cana-5394	288	12	rings	ring	NOUN
cana-5394	288	13	,	,	PUNCT
cana-5394	288	14	rend	rend	VERB
cana-5394	288	15	.	.	PUNCT
cana-5394	289	1	istit	istit	PROPN
cana-5394	289	2	.	.	PUNCT
cana-5394	290	1	mat	mat	NOUN
cana-5394	290	2	.	.	PROPN
cana-5394	290	3	univ	univ	PROPN
cana-5394	290	4	.	.	PUNCT
cana-5394	290	5	trieste	trieste	PROPN
cana-5394	290	6	,	,	PUNCT
cana-5394	290	7	31	31	NUM
cana-5394	290	8	(	(	PUNCT
cana-5394	290	9	1999	1999	NUM
cana-5394	290	10	)	)	PUNCT
cana-5394	290	11	,	,	PUNCT
cana-5394	290	12	25	25	NUM
cana-5394	290	13	–	–	PUNCT
cana-5394	290	14	36	36	NUM
cana-5394	290	15	.	.	PUNCT
cana-5394	291	1	[	[	X
cana-5394	291	2	22	22	NUM
cana-5394	291	3	]	]	PUNCT
cana-5394	291	4	m.ashraf	m.ashraf	NOUN
cana-5394	291	5	,	,	PUNCT
cana-5394	291	6	m.r.jamal	m.r.jamal	NOUN
cana-5394	291	7	,	,	PUNCT
cana-5394	291	8	traces	trace	NOUN
cana-5394	291	9	of	of	ADP
cana-5394	291	10	permuting	permute	VERB
cana-5394	291	11	n	n	CCONJ
cana-5394	291	12	-	-	PUNCT
cana-5394	291	13	additive	additive	ADJ
cana-5394	291	14	maps	map	NOUN
cana-5394	291	15	and	and	CCONJ
cana-5394	291	16	permuting	permute	VERB
cana-5394	291	17	n	n	CCONJ
cana-5394	291	18	-	-	PUNCT
cana-5394	291	19	derivations	derivation	NOUN
cana-5394	291	20	of	of	ADP
cana-5394	291	21	rings	ring	NOUN
cana-5394	291	22	,	,	PUNCT
cana-5394	291	23	mediterr	mediterr	PROPN
cana-5394	291	24	.	.	PUNCT
cana-5394	292	1	j.	j.	PROPN
cana-5394	292	2	math	math	PROPN
cana-5394	292	3	.	.	PUNCT
cana-5394	292	4	,	,	PUNCT
cana-5394	292	5	11	11	NUM
cana-5394	292	6	(	(	PUNCT
cana-5394	292	7	2014	2014	NUM
cana-5394	292	8	)	)	PUNCT
cana-5394	292	9	,	,	PUNCT
cana-5394	292	10	287–297	287–297	NUM
cana-5394	292	11	.	.	PUNCT
cana-5394	293	1	[	[	X
cana-5394	293	2	23	23	NUM
cana-5394	293	3	]	]	PUNCT
cana-5394	293	4	m.ashraf	m.ashraf	NOUN
cana-5394	293	5	,	,	PUNCT
cana-5394	293	6	n.parveen	n.parveen	NUM
cana-5394	293	7	,	,	PUNCT
cana-5394	293	8	m.	m.	NOUN
cana-5394	293	9	r.jamal	r.jamal	ADJ
cana-5394	293	10	,	,	PUNCT
cana-5394	293	11	traces	trace	NOUN
cana-5394	293	12	of	of	ADP
cana-5394	293	13	permuting	permute	VERB
cana-5394	293	14	n	n	CCONJ
cana-5394	293	15	-	-	PUNCT
cana-5394	293	16	derivations	derivation	NOUN
cana-5394	293	17	and	and	CCONJ
cana-5394	293	18	commutativity	commutativity	NOUN
cana-5394	293	19	of	of	ADP
cana-5394	293	20	rings	ring	NOUN
cana-5394	293	21	,	,	PUNCT
cana-5394	293	22	southeast	southeast	ADJ
cana-5394	293	23	asian	asian	ADJ
cana-5394	293	24	bull	bull	NOUN
cana-5394	293	25	.	.	PUNCT
cana-5394	294	1	math	math	NOUN
cana-5394	294	2	.	.	PUNCT
cana-5394	295	1	,	,	PUNCT
cana-5394	295	2	38	38	NUM
cana-5394	295	3	(	(	PUNCT
cana-5394	295	4	2014	2014	NUM
cana-5394	295	5	)	)	PUNCT
cana-5394	295	6	,	,	PUNCT
cana-5394	295	7	321–332	321–332	NUM
cana-5394	295	8	.	.	PUNCT
cana-5394	296	1	[	[	X
cana-5394	296	2	24	24	NUM
cana-5394	296	3	]	]	PUNCT
cana-5394	296	4	m.ashraf	m.ashraf	NOUN
cana-5394	296	5	,	,	PUNCT
cana-5394	296	6	on	on	ADP
cana-5394	296	7	symmetric	symmetric	ADJ
cana-5394	296	8	bi	bi	NOUN
cana-5394	296	9	-	-	NOUN
cana-5394	296	10	derivations	derivation	NOUN
cana-5394	296	11	in	in	ADP
cana-5394	296	12	rings	ring	NOUN
cana-5394	296	13	,	,	PUNCT
cana-5394	296	14	rend	rend	VERB
cana-5394	296	15	.	.	PUNCT
cana-5394	297	1	istit	istit	PROPN
cana-5394	297	2	.	.	PUNCT
cana-5394	298	1	mat	mat	NOUN
cana-5394	298	2	.	.	PROPN
cana-5394	298	3	univ	univ	PROPN
cana-5394	298	4	.	.	PUNCT
cana-5394	298	5	trieste	trieste	PROPN
cana-5394	298	6	,	,	PUNCT
cana-5394	298	7	31	31	NUM
cana-5394	298	8	(	(	PUNCT
cana-5394	298	9	1999	1999	NUM
cana-5394	298	10	)	)	PUNCT
cana-5394	298	11	,	,	PUNCT
cana-5394	298	12	25	25	NUM
cana-5394	298	13	–	–	PUNCT
cana-5394	298	14	36	36	NUM
cana-5394	298	15	.	.	PUNCT
cana-5394	299	1	[	[	X
cana-5394	299	2	25	25	NUM
cana-5394	299	3	]	]	X
cana-5394	299	4	g.	g.	PROPN
cana-5394	299	5	maksa	maksa	PROPN
cana-5394	299	6	,	,	PUNCT
cana-5394	299	7	on	on	ADP
cana-5394	299	8	the	the	DET
cana-5394	299	9	trace	trace	NOUN
cana-5394	299	10	of	of	ADP
cana-5394	299	11	symmetric	symmetric	ADJ
cana-5394	299	12	biderivations	biderivation	NOUN
cana-5394	299	13	,	,	PUNCT
cana-5394	299	14	c.	c.	PROPN
cana-5394	299	15	r.	r.	PROPN
cana-5394	299	16	math	math	PROPN
cana-5394	299	17	.	.	PUNCT
cana-5394	300	1	rep	rep	PROPN
cana-5394	300	2	.	.	PROPN
cana-5394	300	3	acad	acad	PROPN
cana-5394	300	4	.	.	PUNCT
cana-5394	301	1	sci	sci	PROPN
cana-5394	301	2	.	.	PUNCT
cana-5394	301	3	canada	canada	PROPN
cana-5394	301	4	ix	ix	PROPN
cana-5394	301	5	,	,	PUNCT
cana-5394	301	6	1987	1987	NUM
cana-5394	301	7	,	,	PUNCT
cana-5394	301	8	303–308	303–308	NUM
cana-5394	301	9	.	.	PUNCT
cana-5394	302	1	[	[	X
cana-5394	302	2	26	26	NUM
cana-5394	302	3	]	]	X
cana-5394	302	4	j.	j.	PROPN
cana-5394	302	5	vukman	vukman	PROPN
cana-5394	302	6	,	,	PUNCT
cana-5394	302	7	two	two	NUM
cana-5394	302	8	results	result	NOUN
cana-5394	302	9	concerning	concern	VERB
cana-5394	302	10	symmetric	symmetric	ADJ
cana-5394	302	11	bi	bi	NOUN
cana-5394	302	12	-	-	NOUN
cana-5394	302	13	derivations	derivation	NOUN
cana-5394	302	14	on	on	ADP
cana-5394	302	15	prime	prime	ADJ
cana-5394	302	16	and	and	CCONJ
cana-5394	302	17	semiprime	semiprime	NOUN
cana-5394	302	18	rings	ring	NOUN
cana-5394	302	19	,	,	PUNCT
cana-5394	302	20	aeq	aeq	PROPN
cana-5394	302	21	.	.	PUNCT
cana-5394	302	22	math	math	NOUN
cana-5394	302	23	.	.	PUNCT
cana-5394	302	24	,	,	PUNCT
cana-5394	302	25	40	40	NUM
cana-5394	302	26	(	(	PUNCT
cana-5394	302	27	1990	1990	NUM
cana-5394	302	28	)	)	PUNCT
cana-5394	302	29	,	,	PUNCT
cana-5394	302	30	181–189	181–189	NUM
cana-5394	302	31	.	.	PUNCT