id	sid	tid	token	lemma	pos
cana-6071	1	1	communications	communication	NOUN
cana-6071	1	2	on	on	ADP
cana-6071	1	3	applied	apply	VERB
cana-6071	1	4	nonlinear	nonlinear	ADJ
cana-6071	1	5	analysis	analysis	NOUN
cana-6071	1	6	issn	issn	NOUN
cana-6071	1	7	:	:	PUNCT
cana-6071	1	8	1074	1074	NUM
cana-6071	1	9	-	-	PUNCT
cana-6071	1	10	133x	133x	NUM
cana-6071	1	11	vol	vol	NOUN
cana-6071	1	12	31	31	NUM
cana-6071	1	13	no	no	NOUN
cana-6071	1	14	.	.	PUNCT
cana-6071	2	1	8s	8s	PROPN
cana-6071	2	2	(	(	PUNCT
cana-6071	2	3	2024	2024	NUM
cana-6071	2	4	)	)	PUNCT
cana-6071	2	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-6071	2	6	1180	1180	NUM
cana-6071	2	7	a	a	DET
cana-6071	2	8	note	note	NOUN
cana-6071	2	9	on	on	ADP
cana-6071	2	10	radicals	radical	NOUN
cana-6071	2	11	of	of	ADP
cana-6071	2	12	semiring	semiring	NOUN
cana-6071	2	13	of	of	ADP
cana-6071	2	14	matrices	matrix	NOUN
cana-6071	2	15	dr	dr	PROPN
cana-6071	2	16	.	.	PROPN
cana-6071	2	17	manohar	manohar	PROPN
cana-6071	2	18	b.	b.	PROPN
cana-6071	2	19	bhagirath1	bhagirath1	PROPN
cana-6071	2	20	,	,	PUNCT
cana-6071	2	21	dr	dr	PROPN
cana-6071	2	22	.	.	PROPN
cana-6071	2	23	narendrakumar	narendrakumar	PROPN
cana-6071	2	24	r.	r.	PROPN
cana-6071	2	25	dasre2	dasre2	PROPN
cana-6071	2	26	,	,	PUNCT
cana-6071	2	27	dr	dr	PROPN
cana-6071	2	28	.	.	PROPN
cana-6071	2	29	pritam	pritam	PROPN
cana-6071	2	30	gujarathi	gujarathi	PROPN
cana-6071	2	31	-	-	PUNCT
cana-6071	2	32	wani3	wani3	PROPN
cana-6071	2	33	*	*	PROPN
cana-6071	2	34	1associate	1associate	NUM
cana-6071	2	35	professor	professor	NOUN
cana-6071	2	36	,	,	PUNCT
cana-6071	2	37	head	head	NOUN
cana-6071	2	38	,	,	PUNCT
cana-6071	2	39	department	department	NOUN
cana-6071	2	40	of	of	ADP
cana-6071	2	41	mathematics	mathematic	NOUN
cana-6071	2	42	,	,	PUNCT
cana-6071	2	43	annasaheb	annasaheb	NOUN
cana-6071	2	44	vartak	vartak	NOUN
cana-6071	2	45	college	college	PROPN
cana-6071	2	46	of	of	ADP
cana-6071	2	47	arts	art	NOUN
cana-6071	2	48	,	,	PUNCT
cana-6071	2	49	science	science	NOUN
cana-6071	2	50	and	and	CCONJ
cana-6071	2	51	commerce	commerce	PROPN
cana-6071	2	52	,	,	PUNCT
cana-6071	2	53	vasai	vasai	PROPN
cana-6071	2	54	,	,	PUNCT
cana-6071	2	55	dist	dist	NOUN
cana-6071	2	56	.	.	PUNCT
cana-6071	3	1	palghar	palghar	PROPN
cana-6071	3	2	401202	401202	NUM
cana-6071	3	3	.	.	PUNCT
cana-6071	4	1	email	email	NOUN
cana-6071	4	2	-	-	PUNCT
cana-6071	4	3	id	id	NOUN
cana-6071	4	4	:	:	PUNCT
cana-6071	4	5	manoharbhagirath@gmail.com	manoharbhagirath@gmail.com	X
cana-6071	5	1	2associate	2associate	NUM
cana-6071	5	2	professor	professor	NOUN
cana-6071	5	3	,	,	PUNCT
cana-6071	5	4	head	head	NOUN
cana-6071	5	5	,	,	PUNCT
cana-6071	5	6	department	department	NOUN
cana-6071	5	7	of	of	ADP
cana-6071	5	8	engineering	engineering	NOUN
cana-6071	5	9	sciences	science	NOUN
cana-6071	5	10	,	,	PUNCT
cana-6071	5	11	ramrao	ramrao	VERB
cana-6071	5	12	adik	adik	PROPN
cana-6071	5	13	institute	institute	PROPN
cana-6071	5	14	of	of	ADP
cana-6071	5	15	technology	technology	PROPN
cana-6071	5	16	,	,	PUNCT
cana-6071	5	17	nerul	nerul	PROPN
cana-6071	5	18	,	,	PUNCT
cana-6071	5	19	navi	navi	PROPN
cana-6071	5	20	mumbai-400706	mumbai-400706	NOUN
cana-6071	5	21	,	,	PUNCT
cana-6071	5	22	email	email	NOUN
cana-6071	5	23	-	-	PUNCT
cana-6071	5	24	id	id	NOUN
cana-6071	5	25	:	:	PUNCT
cana-6071	5	26	narendasre@rait.ac.in	narendasre@rait.ac.in	PROPN
cana-6071	5	27	3*assistant	3*assistant	ADJ
cana-6071	5	28	professor	professor	NOUN
cana-6071	5	29	,	,	PUNCT
cana-6071	5	30	department	department	NOUN
cana-6071	5	31	of	of	ADP
cana-6071	5	32	engineering	engineering	NOUN
cana-6071	5	33	sciences	science	NOUN
cana-6071	5	34	,	,	PUNCT
cana-6071	5	35	ramrao	ramrao	VERB
cana-6071	5	36	adik	adik	PROPN
cana-6071	5	37	institute	institute	PROPN
cana-6071	5	38	of	of	ADP
cana-6071	5	39	technology	technology	PROPN
cana-6071	5	40	,	,	PUNCT
cana-6071	5	41	nerul	nerul	PROPN
cana-6071	5	42	,	,	PUNCT
cana-6071	5	43	navi	navi	PROPN
cana-6071	5	44	mumbai-400706	mumbai-400706	NOUN
cana-6071	5	45	,	,	PUNCT
cana-6071	5	46	email	email	NOUN
cana-6071	5	47	-	-	PUNCT
cana-6071	5	48	id	id	NOUN
cana-6071	5	49	:	:	PUNCT
cana-6071	5	50	pritam.wani@rait.ac.in	pritam.wani@rait.ac.in	PRON
cana-6071	5	51	*	*	PUNCT
cana-6071	5	52	corresponding	correspond	VERB
cana-6071	5	53	author	author	NOUN
cana-6071	5	54	:	:	PUNCT
cana-6071	5	55	dr	dr	PROPN
cana-6071	5	56	.	.	PROPN
cana-6071	5	57	pritam	pritam	PROPN
cana-6071	5	58	gujarathi	gujarathi	PROPN
cana-6071	5	59	-	-	PUNCT
cana-6071	5	60	wani	wani	PROPN
cana-6071	5	61	email	email	NOUN
cana-6071	5	62	-	-	PUNCT
cana-6071	5	63	id	id	X
cana-6071	5	64	:	:	PUNCT
cana-6071	5	65	pritam.wani@rait.ac.in	pritam.wani@rait.ac.in	NUM
cana-6071	5	66	article	article	NOUN
cana-6071	5	67	history	history	NOUN
cana-6071	5	68	:	:	PUNCT
cana-6071	5	69	received:01/11/2024	received:01/11/2024	NOUN
cana-6071	5	70	revised	revise	VERB
cana-6071	5	71	:	:	PUNCT
cana-6071	5	72	06/12/2024	06/12/2024	NUM
cana-6071	5	73	accepted	accept	VERB
cana-6071	5	74	:	:	PUNCT
cana-6071	5	75	30/12/2024	30/12/2024	NUM
cana-6071	5	76	abstract	abstract	NOUN
cana-6071	5	77	:	:	PUNCT
cana-6071	5	78	in	in	ADP
cana-6071	5	79	this	this	DET
cana-6071	5	80	article	article	NOUN
cana-6071	5	81	we	we	PRON
cana-6071	5	82	introduce	introduce	VERB
cana-6071	5	83	and	and	CCONJ
cana-6071	5	84	investigate	investigate	VERB
cana-6071	5	85	radicals	radical	NOUN
cana-6071	5	86	of	of	ADP
cana-6071	5	87	semiring	semiring	NOUN
cana-6071	5	88	of	of	ADP
cana-6071	5	89	matrices	matrix	NOUN
cana-6071	5	90	.	.	PUNCT
cana-6071	6	1	we	we	PRON
cana-6071	6	2	establish	establish	VERB
cana-6071	6	3	that	that	SCONJ
cana-6071	6	4	if	if	SCONJ
cana-6071	6	5	r	r	NOUN
cana-6071	6	6	is	be	AUX
cana-6071	6	7	a	a	DET
cana-6071	6	8	radical	radical	ADJ
cana-6071	6	9	class	class	NOUN
cana-6071	6	10	which	which	PRON
cana-6071	6	11	is	be	AUX
cana-6071	6	12	(	(	PUNCT
cana-6071	6	13	right	right	ADJ
cana-6071	6	14	or	or	CCONJ
cana-6071	6	15	left)-hereditary	left)-hereditary	PROPN
cana-6071	6	16	and	and	CCONJ
cana-6071	6	17	(	(	PUNCT
cana-6071	6	18	right	right	NOUN
cana-6071	6	19	or	or	CCONJ
cana-6071	6	20	left)-strong	left)-strong	PROPN
cana-6071	6	21	,	,	PUNCT
cana-6071	6	22	then	then	ADV
cana-6071	6	23	r	r	NOUN
cana-6071	6	24	has	have	VERB
cana-6071	6	25	the	the	DET
cana-6071	6	26	property	property	NOUN
cana-6071	6	27	that	that	PRON
cana-6071	6	28	the	the	DET
cana-6071	6	29	r	r	NOUN
cana-6071	6	30	-	-	PUNCT
cana-6071	6	31	radical	radical	ADJ
cana-6071	6	32	of	of	ADP
cana-6071	6	33	the	the	DET
cana-6071	6	34	semiring	semiring	NOUN
cana-6071	6	35	of	of	ADP
cana-6071	6	36	matrices	matrix	NOUN
cana-6071	6	37	of	of	ADP
cana-6071	6	38	order	order	NOUN
cana-6071	6	39	n	n	NOUN
cana-6071	6	40	over	over	ADP
cana-6071	6	41	a	a	DET
cana-6071	6	42	semiring	semire	VERB
cana-6071	6	43	r	r	NOUN
cana-6071	6	44	is	be	AUX
cana-6071	6	45	equal	equal	ADJ
cana-6071	6	46	to	to	ADP
cana-6071	6	47	the	the	DET
cana-6071	6	48	semiring	semiring	NOUN
cana-6071	6	49	of	of	ADP
cana-6071	6	50	n×n	n×n	NOUN
cana-6071	6	51	-	-	PUNCT
cana-6071	6	52	matrices	matrix	NOUN
cana-6071	6	53	over	over	ADP
cana-6071	6	54	the	the	DET
cana-6071	6	55	semiring	semire	VERB
cana-6071	6	56	r(r	r(r	PROPN
cana-6071	6	57	)	)	PUNCT
cana-6071	6	58	.	.	PUNCT
cana-6071	7	1	keywords	keyword	NOUN
cana-6071	7	2	:	:	PUNCT
cana-6071	7	3	semirings	semiring	NOUN
cana-6071	7	4	,	,	PUNCT
cana-6071	7	5	radical	radical	ADJ
cana-6071	7	6	classes	class	NOUN
cana-6071	7	7	,	,	PUNCT
cana-6071	7	8	semiring	semire	VERB
cana-6071	7	9	of	of	ADP
cana-6071	7	10	matrices	matrix	NOUN
cana-6071	7	11	.	.	PUNCT
cana-6071	8	1	1	1	X
cana-6071	8	2	.	.	X
cana-6071	8	3	introduction	introduction	NOUN
cana-6071	8	4	throughout	throughout	ADP
cana-6071	8	5	this	this	DET
cana-6071	8	6	article	article	NOUN
cana-6071	8	7	semirings	semiring	NOUN
cana-6071	8	8	will	will	AUX
cana-6071	8	9	be	be	AUX
cana-6071	8	10	associative	associative	ADJ
cana-6071	8	11	,	,	PUNCT
cana-6071	8	12	not	not	PART
cana-6071	8	13	necessariliy	necessariliy	ADJ
cana-6071	8	14	with	with	ADP
cana-6071	8	15	unity	unity	NOUN
cana-6071	8	16	element	element	NOUN
cana-6071	8	17	and	and	CCONJ
cana-6071	8	18	radicals	radical	NOUN
cana-6071	8	19	in	in	ADP
cana-6071	8	20	the	the	DET
cana-6071	8	21	sense	sense	NOUN
cana-6071	8	22	of	of	ADP
cana-6071	8	23	kurosh	kurosh	NOUN
cana-6071	8	24	amitusar	amitusar	VERB
cana-6071	8	25	as	as	SCONJ
cana-6071	8	26	defined	define	VERB
cana-6071	8	27	in	in	ADP
cana-6071	8	28	[	[	X
cana-6071	8	29	9	9	NUM
cana-6071	8	30	]	]	PUNCT
cana-6071	8	31	.	.	PUNCT
cana-6071	9	1	in	in	ADP
cana-6071	9	2	this	this	DET
cana-6071	9	3	article	article	NOUN
cana-6071	9	4	we	we	PRON
cana-6071	9	5	have	have	AUX
cana-6071	9	6	introduced	introduce	VERB
cana-6071	9	7	and	and	CCONJ
cana-6071	9	8	investigated	investigate	VERB
cana-6071	9	9	radicals	radical	NOUN
cana-6071	9	10	of	of	ADP
cana-6071	9	11	semiring	semiring	NOUN
cana-6071	9	12	of	of	ADP
cana-6071	9	13	matrices	matrix	NOUN
cana-6071	9	14	and	and	CCONJ
cana-6071	9	15	of	of	ADP
cana-6071	9	16	polynomial	polynomial	ADJ
cana-6071	9	17	semirings	semiring	NOUN
cana-6071	9	18	.	.	PUNCT
cana-6071	10	1	in	in	ADP
cana-6071	10	2	this	this	DET
cana-6071	10	3	article	article	NOUN
cana-6071	10	4	,	,	PUNCT
cana-6071	10	5	we	we	PRON
cana-6071	10	6	have	have	AUX
cana-6071	10	7	shown	show	VERB
cana-6071	10	8	that	that	SCONJ
cana-6071	10	9	if	if	SCONJ
cana-6071	10	10	r	r	NOUN
cana-6071	10	11	is	be	AUX
cana-6071	10	12	a	a	DET
cana-6071	10	13	radical	radical	ADJ
cana-6071	10	14	class	class	NOUN
cana-6071	10	15	which	which	PRON
cana-6071	10	16	is	be	AUX
cana-6071	10	17	(	(	PUNCT
cana-6071	10	18	right	right	ADJ
cana-6071	10	19	or	or	CCONJ
cana-6071	10	20	left)hereditary	left)hereditary	ADJ
cana-6071	10	21	and	and	CCONJ
cana-6071	10	22	(	(	PUNCT
cana-6071	10	23	right	right	NOUN
cana-6071	10	24	or	or	CCONJ
cana-6071	10	25	left)-strong	left)-strong	PROPN
cana-6071	10	26	,	,	PUNCT
cana-6071	10	27	then	then	ADV
cana-6071	10	28	r	r	NOUN
cana-6071	10	29	has	have	VERB
cana-6071	10	30	the	the	DET
cana-6071	10	31	property	property	NOUN
cana-6071	10	32	that	that	PRON
cana-6071	10	33	the	the	DET
cana-6071	10	34	r	r	NOUN
cana-6071	10	35	-	-	PUNCT
cana-6071	10	36	radical	radical	ADJ
cana-6071	10	37	of	of	ADP
cana-6071	10	38	the	the	DET
cana-6071	10	39	semiring	semiring	NOUN
cana-6071	10	40	of	of	ADP
cana-6071	10	41	matrices	matrix	NOUN
cana-6071	10	42	of	of	ADP
cana-6071	10	43	order	order	NOUN
cana-6071	10	44	n	n	NOUN
cana-6071	10	45	over	over	ADP
cana-6071	10	46	a	a	DET
cana-6071	10	47	semiring	semire	VERB
cana-6071	10	48	r	r	NOUN
cana-6071	10	49	is	be	AUX
cana-6071	10	50	equal	equal	ADJ
cana-6071	10	51	to	to	ADP
cana-6071	10	52	the	the	DET
cana-6071	10	53	semiring	semiring	NOUN
cana-6071	10	54	of	of	ADP
cana-6071	10	55	matrices	matrix	NOUN
cana-6071	10	56	of	of	ADP
cana-6071	10	57	order	order	NOUN
cana-6071	10	58	n	n	NOUN
cana-6071	10	59	over	over	ADP
cana-6071	10	60	the	the	DET
cana-6071	10	61	semiring	semire	VERB
cana-6071	10	62	r(r	r(r	PROPN
cana-6071	10	63	)	)	PUNCT
cana-6071	10	64	.	.	PUNCT
cana-6071	11	1	the	the	DET
cana-6071	11	2	interrelation	interrelation	NOUN
cana-6071	11	3	and	and	CCONJ
cana-6071	11	4	independence	independence	NOUN
cana-6071	11	5	of	of	ADP
cana-6071	11	6	polynomial	polynomial	ADJ
cana-6071	11	7	extensibility	extensibility	NOUN
cana-6071	11	8	of	of	ADP
cana-6071	11	9	radical	radical	ADJ
cana-6071	11	10	and	and	CCONJ
cana-6071	11	11	semisimple	semisimple	NOUN
cana-6071	11	12	classes	class	NOUN
cana-6071	11	13	and	and	CCONJ
cana-6071	11	14	of	of	ADP
cana-6071	11	15	the	the	DET
cana-6071	11	16	amitsur	amitsur	ADJ
cana-6071	11	17	property	property	NOUN
cana-6071	11	18	are	be	AUX
cana-6071	11	19	investigated	investigate	VERB
cana-6071	11	20	for	for	ADP
cana-6071	11	21	associative	associative	ADJ
cana-6071	11	22	rings	ring	NOUN
cana-6071	11	23	in	in	ADP
cana-6071	11	24	[	[	X
cana-6071	11	25	12	12	NUM
cana-6071	11	26	]	]	PUNCT
cana-6071	11	27	.	.	PUNCT
cana-6071	12	1	in	in	ADP
cana-6071	12	2	this	this	DET
cana-6071	12	3	article	article	NOUN
cana-6071	12	4	,	,	PUNCT
cana-6071	12	5	we	we	PRON
cana-6071	12	6	have	have	AUX
cana-6071	12	7	tried	try	VERB
cana-6071	12	8	to	to	PART
cana-6071	12	9	generalize	generalize	VERB
cana-6071	12	10	some	some	DET
cana-6071	12	11	results	result	NOUN
cana-6071	12	12	for	for	ADP
cana-6071	12	13	semirings	semiring	NOUN
cana-6071	12	14	.	.	PUNCT
cana-6071	13	1	throughout	throughout	ADP
cana-6071	13	2	this	this	DET
cana-6071	13	3	article	article	NOUN
cana-6071	13	4	↦	↦	PROPN
cana-6071	13	5	stands	stand	VERB
cana-6071	13	6	for	for	ADP
cana-6071	13	7	onto	onto	ADP
cana-6071	13	8	homomorphism	homomorphism	NOUN
cana-6071	13	9	and	and	CCONJ
cana-6071	13	10	⊲	⊲	NOUN
cana-6071	13	11	stands	stand	VERB
cana-6071	13	12	for	for	ADP
cana-6071	13	13	an	an	DET
cana-6071	13	14	ideal	ideal	NOUN
cana-6071	13	15	of	of	ADP
cana-6071	13	16	a	a	DET
cana-6071	13	17	semiring	semire	VERB
cana-6071	13	18	r.	r.	NOUN
cana-6071	13	19	for	for	ADP
cana-6071	13	20	details	detail	NOUN
cana-6071	13	21	of	of	ADP
cana-6071	13	22	semiring	semire	VERB
cana-6071	13	23	theory	theory	NOUN
cana-6071	13	24	and	and	CCONJ
cana-6071	13	25	more	more	ADJ
cana-6071	13	26	on	on	ADP
cana-6071	13	27	radical	radical	ADJ
cana-6071	13	28	theory	theory	NOUN
cana-6071	13	29	for	for	ADP
cana-6071	13	30	associative	associative	ADJ
cana-6071	13	31	semirings	semiring	NOUN
cana-6071	13	32	the	the	DET
cana-6071	13	33	readers	reader	NOUN
cana-6071	13	34	are	be	AUX
cana-6071	13	35	referred	refer	VERB
cana-6071	13	36	to	to	ADP
cana-6071	13	37	[	[	X
cana-6071	13	38	1	1	NUM
cana-6071	13	39	]	]	PUNCT
cana-6071	13	40	,	,	PUNCT
cana-6071	14	1	[	[	X
cana-6071	14	2	2	2	NUM
cana-6071	14	3	]	]	PUNCT
cana-6071	14	4	,	,	PUNCT
cana-6071	14	5	[	[	X
cana-6071	14	6	6	6	NUM
cana-6071	14	7	]	]	PUNCT
cana-6071	14	8	,	,	PUNCT
cana-6071	14	9	[	[	X
cana-6071	14	10	8	8	NUM
cana-6071	14	11	]	]	PUNCT
cana-6071	14	12	and	and	CCONJ
cana-6071	14	13	[	[	X
cana-6071	14	14	10	10	NUM
cana-6071	14	15	]	]	PUNCT
cana-6071	14	16	.	.	PUNCT
cana-6071	15	1	2	2	X
cana-6071	15	2	.	.	X
cana-6071	15	3	radical	radical	ADJ
cana-6071	15	4	of	of	ADP
cana-6071	15	5	semiring	semiring	NOUN
cana-6071	15	6	of	of	ADP
cana-6071	15	7	matrices	matrix	NOUN
cana-6071	15	8	.	.	PUNCT
cana-6071	16	1	definition	definition	NOUN
cana-6071	16	2	2.1	2.1	NUM
cana-6071	16	3	.	.	PUNCT
cana-6071	17	1	the	the	DET
cana-6071	17	2	additive	additive	ADJ
cana-6071	17	3	semigroup	semigroup	NOUN
cana-6071	17	4	(	(	PUNCT
cana-6071	17	5	r	r	NOUN
cana-6071	17	6	,	,	PUNCT
cana-6071	17	7	+	+	NOUN
cana-6071	17	8	)	)	PUNCT
cana-6071	17	9	of	of	ADP
cana-6071	17	10	a	a	DET
cana-6071	17	11	semiring	semiring	NOUN
cana-6071	17	12	r	r	NOUN
cana-6071	17	13	will	will	AUX
cana-6071	17	14	be	be	AUX
cana-6071	17	15	denoted	denote	VERB
cana-6071	17	16	by	by	ADP
cana-6071	17	17	r	r	NOUN
cana-6071	17	18	+	+	NOUN
cana-6071	17	19	and	and	CCONJ
cana-6071	17	20	for	for	ADP
cana-6071	17	21	an	an	DET
cana-6071	17	22	abelian	abelian	ADJ
cana-6071	17	23	semigroup	semigroup	NOUN
cana-6071	17	24	r	r	NOUN
cana-6071	17	25	+	+	PROPN
cana-6071	17	26	,	,	PUNCT
cana-6071	17	27	we	we	PRON
cana-6071	17	28	may	may	AUX
cana-6071	17	29	define	define	VERB
cana-6071	17	30	always	always	ADV
cana-6071	17	31	a	a	DET
cana-6071	17	32	semiring	semire	VERB
cana-6071	17	33	r	r	NOUN
cana-6071	17	34	0	0	NUM
cana-6071	17	35	with	with	ADP
cana-6071	17	36	zero	zero	NUM
cana-6071	17	37	multiplication	multiplication	NOUN
cana-6071	17	38	,	,	PUNCT
cana-6071	17	39	called	call	VERB
cana-6071	17	40	a	a	DET
cana-6071	17	41	zerosemiring	zerosemiring	NOUN
cana-6071	17	42	by	by	ADP
cana-6071	17	43	the	the	DET
cana-6071	17	44	rule	rule	NOUN
cana-6071	17	45	xy	xy	PROPN
cana-6071	17	46	=	=	NOUN
cana-6071	17	47	0	0	PROPN
cana-6071	17	48	for	for	ADP
cana-6071	17	49	all	all	DET
cana-6071	17	50	x	x	NOUN
cana-6071	17	51	,	,	PUNCT
cana-6071	17	52	y	y	PROPN
cana-6071	17	53	∈	∈	PROPN
cana-6071	17	54	r.	r.	PROPN
cana-6071	17	55	definition	definition	NOUN
cana-6071	17	56	2.2	2.2	NUM
cana-6071	17	57	.	.	PUNCT
cana-6071	18	1	a	a	DET
cana-6071	18	2	radical	radical	ADJ
cana-6071	18	3	r	r	NOUN
cana-6071	18	4	is	be	AUX
cana-6071	18	5	called	call	VERB
cana-6071	18	6	an	an	DET
cana-6071	18	7	a	a	PRON
cana-6071	18	8	-	-	PUNCT
cana-6071	18	9	radical	radical	ADJ
cana-6071	18	10	if	if	SCONJ
cana-6071	18	11	for	for	ADP
cana-6071	18	12	any	any	DET
cana-6071	18	13	semiring	semire	VERB
cana-6071	18	14	r	r	NOUN
cana-6071	18	15	∈	∈	NOUN
cana-6071	18	16	r	r	NOUN
cana-6071	18	17	and	and	CCONJ
cana-6071	18	18	any	any	DET
cana-6071	18	19	additive	additive	ADJ
cana-6071	18	20	homomorphism	homomorphism	NOUN
cana-6071	18	21	f	f	X
cana-6071	18	22	:	:	PUNCT
cana-6071	18	23	r	r	X
cana-6071	18	24	→	→	SYM
cana-6071	18	25	s	s	VERB
cana-6071	18	26	such	such	ADJ
cana-6071	18	27	that	that	SCONJ
cana-6071	18	28	f	f	PROPN
cana-6071	18	29	(	(	PUNCT
cana-6071	18	30	r	r	NOUN
cana-6071	18	31	)	)	PUNCT
cana-6071	18	32	is	be	AUX
cana-6071	18	33	a	a	DET
cana-6071	18	34	subsemiring	subsemiring	NOUN
cana-6071	18	35	of	of	ADP
cana-6071	18	36	s	s	PRON
cana-6071	18	37	also	also	ADV
cana-6071	18	38	f	f	PROPN
cana-6071	18	39	(	(	PUNCT
cana-6071	18	40	s	s	X
cana-6071	18	41	)	)	PUNCT
cana-6071	18	42	∈	∈	PROPN
cana-6071	18	43	r.	r.	PROPN
cana-6071	18	44	definition	definition	NOUN
cana-6071	18	45	2.3	2.3	NUM
cana-6071	18	46	.	.	PUNCT
cana-6071	19	1	a	a	DET
cana-6071	19	2	semiring	semire	VERB
cana-6071	19	3	r	r	NOUN
cana-6071	19	4	is	be	AUX
cana-6071	19	5	said	say	VERB
cana-6071	19	6	to	to	PART
cana-6071	19	7	be	be	AUX
cana-6071	19	8	simple	simple	ADJ
cana-6071	19	9	if	if	SCONJ
cana-6071	19	10	it	it	PRON
cana-6071	19	11	has	have	VERB
cana-6071	19	12	no	no	DET
cana-6071	19	13	proper	proper	ADJ
cana-6071	19	14	ideals	ideal	NOUN
cana-6071	19	15	.	.	PUNCT
cana-6071	20	1	definition	definition	NOUN
cana-6071	20	2	2.4	2.4	NUM
cana-6071	20	3	.	.	PUNCT
cana-6071	21	1	if	if	SCONJ
cana-6071	21	2	m	m	NOUN
cana-6071	21	3	is	be	AUX
cana-6071	21	4	the	the	DET
cana-6071	21	5	class	class	NOUN
cana-6071	21	6	of	of	ADP
cana-6071	21	7	all	all	DET
cana-6071	21	8	simple	simple	ADJ
cana-6071	21	9	semirings	semiring	NOUN
cana-6071	21	10	with	with	ADP
cana-6071	21	11	unity	unity	NOUN
cana-6071	21	12	,	,	PUNCT
cana-6071	21	13	then	then	ADV
cana-6071	21	14	um	um	INTJ
cana-6071	21	15	is	be	AUX
cana-6071	21	16	called	call	VERB
cana-6071	21	17	the	the	DET
cana-6071	21	18	brown	brown	PROPN
cana-6071	21	19	mccoy	mccoy	PROPN
cana-6071	21	20	radical	radical	ADJ
cana-6071	21	21	class	class	NOUN
cana-6071	21	22	.	.	PUNCT
cana-6071	22	1	proposition	proposition	NOUN
cana-6071	22	2	2.5	2.5	NUM
cana-6071	22	3	.	.	PUNCT
cana-6071	23	1	let	let	VERB
cana-6071	23	2	ρ	ρ	NOUN
cana-6071	23	3	be	be	AUX
cana-6071	23	4	a	a	DET
cana-6071	23	5	regular	regular	ADJ
cana-6071	23	6	class	class	NOUN
cana-6071	23	7	of	of	ADP
cana-6071	23	8	semirings	semiring	NOUN
cana-6071	23	9	.	.	PUNCT
cana-6071	24	1	the	the	DET
cana-6071	24	2	upper	upper	ADJ
cana-6071	24	3	radical	radical	ADJ
cana-6071	24	4	uρ	uρ	NOUN
cana-6071	24	5	is	be	AUX
cana-6071	24	6	hereditary	hereditary	ADJ
cana-6071	24	7	if	if	SCONJ
cana-6071	25	1	and	and	CCONJ
cana-6071	25	2	only	only	ADV
cana-6071	25	3	if	if	SCONJ
cana-6071	25	4	ρ	ρ	PROPN
cana-6071	25	5	satisfies	satisfy	VERB
cana-6071	25	6	the	the	DET
cana-6071	25	7	condition	condition	NOUN
cana-6071	25	8	(	(	PUNCT
cana-6071	25	9	i	i	NOUN
cana-6071	25	10	)	)	PUNCT
cana-6071	25	11	:	:	PUNCT
cana-6071	25	12	if	if	SCONJ
cana-6071	25	13	0	0	NUM
cana-6071	25	14	≠	≠	PROPN
cana-6071	25	15	𝐼	𝐼	ADP
cana-6071	25	16	⊲	⊲	NOUN
cana-6071	25	17	r	r	NOUN
cana-6071	26	1	and	and	CCONJ
cana-6071	26	2	there	there	PRON
cana-6071	26	3	is	be	VERB
cana-6071	26	4	a	a	DET
cana-6071	26	5	i	i	PRON
cana-6071	26	6	↦	↦	PROPN
cana-6071	26	7	t	t	NOUN
cana-6071	26	8	such	such	ADJ
cana-6071	26	9	that	that	SCONJ
cana-6071	26	10	0	0	NUM
cana-6071	26	11	≠	≠	PROPN
cana-6071	26	12	t	t	PROPN
cana-6071	26	13	∈	∈	PROPN
cana-6071	26	14	ρ	ρ	PROPN
cana-6071	26	15	,	,	PUNCT
cana-6071	26	16	then	then	ADV
cana-6071	26	17	there	there	PRON
cana-6071	26	18	exists	exist	VERB
cana-6071	26	19	an	an	DET
cana-6071	26	20	r	r	NOUN
cana-6071	26	21	↦	↦	PRON
cana-6071	26	22	s	s	VERB
cana-6071	26	23	such	such	ADJ
cana-6071	26	24	that	that	DET
cana-6071	26	25	0	0	NUM
cana-6071	26	26	≠	≠	PROPN
cana-6071	26	27	s	s	PART
cana-6071	26	28	∈	∈	PROPN
cana-6071	26	29	ρ	ρ	NOUN
cana-6071	26	30	.	.	PUNCT
cana-6071	27	1	mailto:manoharbhagirath@gmail.com	mailto:manoharbhagirath@gmail.com	PROPN
cana-6071	27	2	mailto:%20narendasre@rait.ac.in	mailto:%20narendasre@rait.ac.in	VERB
cana-6071	27	3	mailto:pritam.wani@rait.ac.in	mailto:pritam.wani@rait.ac.in	PROPN
cana-6071	27	4	mailto:pritam.wani@rait.ac.in	mailto:pritam.wani@rait.ac.in	PROPN
cana-6071	27	5	communications	communication	NOUN
cana-6071	27	6	on	on	ADP
cana-6071	27	7	applied	apply	VERB
cana-6071	27	8	nonlinear	nonlinear	ADJ
cana-6071	27	9	analysis	analysis	NOUN
cana-6071	27	10	issn	issn	NOUN
cana-6071	27	11	:	:	PUNCT
cana-6071	27	12	1074	1074	NUM
cana-6071	27	13	-	-	PUNCT
cana-6071	27	14	133x	133x	NUM
cana-6071	27	15	vol	vol	NOUN
cana-6071	27	16	31	31	NUM
cana-6071	27	17	no	no	NOUN
cana-6071	27	18	.	.	PUNCT
cana-6071	28	1	8s	8s	PROPN
cana-6071	28	2	(	(	PUNCT
cana-6071	28	3	2024	2024	NUM
cana-6071	28	4	)	)	PUNCT
cana-6071	28	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-6071	28	6	1181	1181	NUM
cana-6071	28	7	proof	proof	NOUN
cana-6071	28	8	.	.	PUNCT
cana-6071	28	9	suppose	suppose	VERB
cana-6071	28	10	that	that	SCONJ
cana-6071	28	11	uρ	uρ	INTJ
cana-6071	28	12	is	be	AUX
cana-6071	28	13	hereditary	hereditary	ADJ
cana-6071	28	14	.	.	PUNCT
cana-6071	29	1	if	if	SCONJ
cana-6071	29	2	0	0	NUM
cana-6071	29	3	≠	≠	PROPN
cana-6071	29	4	𝐼	𝐼	ADP
cana-6071	29	5	⊲	⊲	NOUN
cana-6071	29	6	r	r	NOUN
cana-6071	30	1	and	and	CCONJ
cana-6071	30	2	there	there	PRON
cana-6071	30	3	is	be	VERB
cana-6071	30	4	a	a	DET
cana-6071	30	5	i	i	PRON
cana-6071	30	6	↦	↦	NOUN
cana-6071	30	7	𝑇	𝑇	PROPN
cana-6071	30	8	such	such	ADJ
cana-6071	30	9	that	that	DET
cana-6071	30	10	0	0	NUM
cana-6071	30	11	≠	≠	PROPN
cana-6071	30	12	t	t	PROPN
cana-6071	30	13	∈	∈	PROPN
cana-6071	30	14	ρ	ρ	PROPN
cana-6071	30	15	,	,	PUNCT
cana-6071	30	16	then	then	ADV
cana-6071	30	17	there	there	PRON
cana-6071	30	18	exists	exist	VERB
cana-6071	30	19	an	an	DET
cana-6071	30	20	r	r	NOUN
cana-6071	30	21	↦	↦	PRON
cana-6071	30	22	s	s	VERB
cana-6071	30	23	such	such	ADJ
cana-6071	30	24	that	that	DET
cana-6071	30	25	0	0	NUM
cana-6071	30	26	≠	≠	PROPN
cana-6071	30	27	s	s	PART
cana-6071	30	28	∈	∈	PROPN
cana-6071	30	29	ρ	ρ	NOUN
cana-6071	30	30	.	.	PUNCT
cana-6071	31	1	conversely	conversely	ADV
cana-6071	31	2	,	,	PUNCT
cana-6071	31	3	by	by	ADP
cana-6071	31	4	(	(	PUNCT
cana-6071	31	5	i	i	NOUN
cana-6071	31	6	)	)	PUNCT
cana-6071	31	7	,	,	PUNCT
cana-6071	31	8	i	i	PRON
cana-6071	31	9	∉	∉	VERB
cana-6071	31	10	uρ	uρ	INTJ
cana-6071	31	11	⇒	⇒	PROPN
cana-6071	31	12	r	r	NOUN
cana-6071	31	13	∉	∉	PROPN
cana-6071	31	14	uρ	uρ	PROPN
cana-6071	31	15	.	.	PUNCT
cana-6071	32	1	but	but	CCONJ
cana-6071	32	2	then	then	ADV
cana-6071	32	3	r	r	NOUN
cana-6071	32	4	∈	∈	PROPN
cana-6071	32	5	uρ	uρ	ADP
cana-6071	32	6	⇒	⇒	NOUN
cana-6071	33	1	i	i	PRON
cana-6071	33	2	∈	∈	VERB
cana-6071	33	3	uρ	uρ	VERB
cana-6071	33	4	,	,	PUNCT
cana-6071	33	5	hence	hence	ADV
cana-6071	33	6	uρ	uρ	INTJ
cana-6071	33	7	is	be	AUX
cana-6071	33	8	hereditary	hereditary	ADJ
cana-6071	33	9	.	.	PUNCT
cana-6071	34	1	proposition	proposition	NOUN
cana-6071	34	2	2.6	2.6	NUM
cana-6071	34	3	.	.	PUNCT
cana-6071	35	1	if	if	SCONJ
cana-6071	35	2	k	k	PROPN
cana-6071	35	3	is	be	AUX
cana-6071	35	4	a	a	DET
cana-6071	35	5	subtractive	subtractive	NOUN
cana-6071	35	6	ideal	ideal	NOUN
cana-6071	35	7	in	in	ADP
cana-6071	35	8	an	an	DET
cana-6071	35	9	ideal	ideal	ADJ
cana-6071	35	10	i	i	PRON
cana-6071	35	11	of	of	ADP
cana-6071	35	12	a	a	DET
cana-6071	35	13	semiring	semire	VERB
cana-6071	35	14	r	r	NOUN
cana-6071	35	15	and	and	CCONJ
cana-6071	35	16	i	i	PRON
cana-6071	35	17	/	/	SYM
cana-6071	35	18	k	k	PROPN
cana-6071	35	19	is	be	AUX
cana-6071	35	20	a	a	DET
cana-6071	35	21	semiring	semiring	NOUN
cana-6071	35	22	with	with	ADP
cana-6071	35	23	unity	unity	NOUN
cana-6071	35	24	element	element	NOUN
cana-6071	35	25	,	,	PUNCT
cana-6071	35	26	then	then	ADV
cana-6071	35	27	k	k	PROPN
cana-6071	35	28	is	be	AUX
cana-6071	35	29	an	an	DET
cana-6071	35	30	ideal	ideal	NOUN
cana-6071	35	31	in	in	ADP
cana-6071	35	32	r.	r.	PROPN
cana-6071	35	33	proof	proof	NOUN
cana-6071	35	34	.	.	PUNCT
cana-6071	36	1	let	let	VERB
cana-6071	36	2	e	e	PROPN
cana-6071	36	3	+	+	CCONJ
cana-6071	36	4	k	k	X
cana-6071	36	5	be	be	VERB
cana-6071	36	6	the	the	DET
cana-6071	36	7	identity	identity	NOUN
cana-6071	36	8	in	in	ADP
cana-6071	36	9	i	i	PROPN
cana-6071	36	10	/	/	SYM
cana-6071	36	11	k.	k.	PROPN
cana-6071	36	12	then	then	ADV
cana-6071	36	13	any	any	DET
cana-6071	36	14	a	a	DET
cana-6071	36	15	∈	∈	NOUN
cana-6071	36	16	r	r	NOUN
cana-6071	36	17	and	and	CCONJ
cana-6071	36	18	any	any	DET
cana-6071	36	19	k	k	PROPN
cana-6071	36	20	∈	∈	PROPN
cana-6071	37	1	k.	k.	NOUN
cana-6071	37	2	then	then	ADV
cana-6071	37	3	we	we	PRON
cana-6071	37	4	have	have	VERB
cana-6071	37	5	ak	ak	PROPN
cana-6071	37	6	∈	∈	PROPN
cana-6071	37	7	i	i	PRON
cana-6071	37	8	as	as	ADP
cana-6071	37	9	k	k	PROPN
cana-6071	37	10	⊲	⊲	PROPN
cana-6071	37	11	i.	i.	PROPN
cana-6071	37	12	therefore	therefore	ADV
cana-6071	37	13	ak	ak	PROPN
cana-6071	38	1	+	+	CCONJ
cana-6071	38	2	k	k	PROPN
cana-6071	38	3	=	=	PRON
cana-6071	38	4	(	(	PUNCT
cana-6071	38	5	e	e	NOUN
cana-6071	38	6	+	+	NOUN
cana-6071	38	7	k)(ak	k)(ak	VERB
cana-6071	38	8	+	+	X
cana-6071	38	9	k	k	X
cana-6071	38	10	)	)	PUNCT
cana-6071	38	11	=	=	SYM
cana-6071	38	12	e(ak	e(ak	PROPN
cana-6071	38	13	)	)	PUNCT
cana-6071	39	1	+	+	NUM
cana-6071	40	1	k	k	NOUN
cana-6071	40	2	=	=	SYM
cana-6071	40	3	(	(	PUNCT
cana-6071	40	4	ea)k	ea)k	PROPN
cana-6071	40	5	+	+	PROPN
cana-6071	40	6	k	k	PROPN
cana-6071	40	7	=	=	SYM
cana-6071	40	8	k.	k.	PROPN
cana-6071	40	9	ak	ak	PROPN
cana-6071	41	1	+	+	PROPN
cana-6071	41	2	k	k	PROPN
cana-6071	41	3	=	=	SYM
cana-6071	41	4	k	k	PROPN
cana-6071	41	5	⇒	⇒	PROPN
cana-6071	41	6	ak	ak	PROPN
cana-6071	41	7	+	+	CCONJ
cana-6071	41	8	k1	k1	NOUN
cana-6071	41	9	=	=	SYM
cana-6071	41	10	k2	k2	PROPN
cana-6071	41	11	∈	∈	PROPN
cana-6071	41	12	k	k	PROPN
cana-6071	41	13	for	for	ADP
cana-6071	41	14	some	some	DET
cana-6071	41	15	k1	k1	NOUN
cana-6071	41	16	,	,	PUNCT
cana-6071	41	17	k2	k2	PROPN
cana-6071	41	18	∈	∈	PROPN
cana-6071	41	19	k	k	PROPN
cana-6071	41	20	⇒	⇒	PROPN
cana-6071	41	21	ak	ak	PROPN
cana-6071	41	22	∈	∈	PROPN
cana-6071	41	23	k	k	PROPN
cana-6071	42	1	(	(	PUNCT
cana-6071	42	2	k	k	PROPN
cana-6071	42	3	is	be	AUX
cana-6071	42	4	subtractive	subtractive	NOUN
cana-6071	42	5	)	)	PUNCT
cana-6071	42	6	.	.	PUNCT
cana-6071	43	1	similarly	similarly	ADV
cana-6071	43	2	we	we	PRON
cana-6071	43	3	get	get	VERB
cana-6071	43	4	ka	ka	PROPN
cana-6071	43	5	∈	∈	PROPN
cana-6071	43	6	k	k	PROPN
cana-6071	43	7	and	and	CCONJ
cana-6071	43	8	k	k	PROPN
cana-6071	43	9	⊲	⊲	PROPN
cana-6071	43	10	r.	r.	PROPN
cana-6071	43	11	proposition	proposition	PROPN
cana-6071	43	12	2.7	2.7	NUM
cana-6071	43	13	.	.	PUNCT
cana-6071	44	1	if	if	SCONJ
cana-6071	44	2	ρ	ρ	PROPN
cana-6071	44	3	is	be	AUX
cana-6071	44	4	any	any	DET
cana-6071	44	5	regular	regular	ADJ
cana-6071	44	6	class	class	NOUN
cana-6071	44	7	of	of	ADP
cana-6071	44	8	semirings	semiring	NOUN
cana-6071	44	9	with	with	ADP
cana-6071	44	10	unity	unity	NOUN
cana-6071	44	11	element	element	NOUN
cana-6071	44	12	,	,	PUNCT
cana-6071	44	13	then	then	ADV
cana-6071	44	14	the	the	DET
cana-6071	44	15	upper	upper	ADJ
cana-6071	44	16	radical	radical	NOUN
cana-6071	44	17	uρ	uρ	NOUN
cana-6071	44	18	is	be	AUX
cana-6071	44	19	hereditary	hereditary	ADJ
cana-6071	44	20	.	.	PUNCT
cana-6071	45	1	proof	proof	NOUN
cana-6071	45	2	.	.	PUNCT
cana-6071	46	1	we	we	PRON
cana-6071	46	2	must	must	AUX
cana-6071	46	3	show	show	VERB
cana-6071	46	4	that	that	SCONJ
cana-6071	46	5	ρ	ρ	PROPN
cana-6071	46	6	satisfies	satisfie	NOUN
cana-6071	46	7	condition	condition	NOUN
cana-6071	46	8	(	(	PUNCT
cana-6071	46	9	i	i	NOUN
cana-6071	46	10	)	)	PUNCT
cana-6071	46	11	of	of	ADP
cana-6071	46	12	proposition	proposition	NOUN
cana-6071	46	13	2.5	2.5	NUM
cana-6071	46	14	that	that	PRON
cana-6071	46	15	i	i	PRON
cana-6071	46	16	is	be	AUX
cana-6071	46	17	an	an	DET
cana-6071	46	18	ideal	ideal	NOUN
cana-6071	46	19	in	in	ADP
cana-6071	46	20	a	a	DET
cana-6071	46	21	semiring	semiring	NOUN
cana-6071	46	22	r	r	NOUN
cana-6071	46	23	such	such	ADJ
cana-6071	46	24	that	that	SCONJ
cana-6071	46	25	i	i	PRON
cana-6071	46	26	has	have	VERB
cana-6071	46	27	a	a	DET
cana-6071	46	28	non	non	ADJ
cana-6071	46	29	-	-	ADJ
cana-6071	46	30	zero	zero	ADJ
cana-6071	46	31	homomorphic	homomorphic	ADJ
cana-6071	46	32	image	image	NOUN
cana-6071	46	33	c	c	NOUN
cana-6071	46	34	=	=	SYM
cana-6071	46	35	i	i	PROPN
cana-6071	46	36	/	/	SYM
cana-6071	46	37	k,(k	k,(k	PROPN
cana-6071	46	38	is	be	AUX
cana-6071	46	39	subtractive	subtractive	NOUN
cana-6071	46	40	)	)	PUNCT
cana-6071	46	41	in	in	ADP
cana-6071	46	42	ρ	ρ	PROPN
cana-6071	46	43	.	.	PUNCT
cana-6071	47	1	then	then	ADV
cana-6071	47	2	i	i	PRON
cana-6071	47	3	/	/	SYM
cana-6071	47	4	k	k	PROPN
cana-6071	47	5	has	have	VERB
cana-6071	47	6	a	a	DET
cana-6071	47	7	unity	unity	NOUN
cana-6071	47	8	and	and	CCONJ
cana-6071	47	9	by	by	ADP
cana-6071	47	10	proposition	proposition	NOUN
cana-6071	47	11	2.6	2.6	NUM
cana-6071	47	12	,	,	PUNCT
cana-6071	47	13	k	k	PROPN
cana-6071	47	14	is	be	AUX
cana-6071	47	15	an	an	DET
cana-6071	47	16	ideal	ideal	NOUN
cana-6071	47	17	of	of	ADP
cana-6071	47	18	r.	r.	PROPN
cana-6071	47	19	since	since	SCONJ
cana-6071	47	20	i	i	PROPN
cana-6071	47	21	/	/	SYM
cana-6071	47	22	k	k	PROPN
cana-6071	47	23	has	have	VERB
cana-6071	47	24	a	a	DET
cana-6071	47	25	unity	unity	NOUN
cana-6071	47	26	,	,	PUNCT
cana-6071	47	27	it	it	PRON
cana-6071	47	28	must	must	AUX
cana-6071	47	29	be	be	AUX
cana-6071	47	30	a	a	DET
cana-6071	47	31	direct	direct	ADJ
cana-6071	47	32	summed	summed	NOUN
cana-6071	47	33	of	of	ADP
cana-6071	47	34	r	r	PROPN
cana-6071	47	35	/	/	SYM
cana-6071	47	36	k.	k.	NOUN
cana-6071	47	37	then	then	ADV
cana-6071	47	38	r	r	PROPN
cana-6071	47	39	/	/	SYM
cana-6071	47	40	k	k	NOUN
cana-6071	47	41	can	can	AUX
cana-6071	47	42	be	be	AUX
cana-6071	47	43	mapped	map	VERB
cana-6071	47	44	homomorphically	homomorphically	ADV
cana-6071	47	45	to	to	ADP
cana-6071	47	46	i	i	PRON
cana-6071	47	47	/	/	SYM
cana-6071	47	48	k	k	PROPN
cana-6071	47	49	and	and	CCONJ
cana-6071	47	50	this	this	PRON
cana-6071	47	51	is	be	AUX
cana-6071	47	52	in	in	ADP
cana-6071	47	53	ρ	ρ	PROPN
cana-6071	47	54	and	and	CCONJ
cana-6071	47	55	non	non	ADJ
cana-6071	47	56	-	-	ADJ
cana-6071	47	57	zero	zero	NUM
cana-6071	47	58	.	.	PUNCT
cana-6071	48	1	hence	hence	ADV
cana-6071	48	2	(	(	PUNCT
cana-6071	48	3	i	i	NOUN
cana-6071	48	4	)	)	PUNCT
cana-6071	48	5	holds	hold	VERB
cana-6071	48	6	and	and	CCONJ
cana-6071	48	7	uρ	uρ	INTJ
cana-6071	48	8	is	be	AUX
cana-6071	48	9	hereditary	hereditary	ADJ
cana-6071	48	10	.	.	PUNCT
cana-6071	49	1	theorem	theorem	VERB
cana-6071	49	2	2.8	2.8	NUM
cana-6071	49	3	.	.	PUNCT
cana-6071	49	4	brown	brown	PROPN
cana-6071	49	5	mccoy	mccoy	PROPN
cana-6071	49	6	radical	radical	PROPN
cana-6071	49	7	is	be	AUX
cana-6071	49	8	hereditary	hereditary	ADJ
cana-6071	49	9	.	.	PUNCT
cana-6071	50	1	theorem	theorem	VERB
cana-6071	50	2	2.9	2.9	NUM
cana-6071	50	3	.	.	PUNCT
cana-6071	51	1	dorroh	dorroh	PROPN
cana-6071	51	2	’s	’s	PART
cana-6071	51	3	extension	extension	NOUN
cana-6071	51	4	theorem	theorem	NOUN
cana-6071	51	5	:	:	PUNCT
cana-6071	51	6	every	every	DET
cana-6071	51	7	semiring	semire	VERB
cana-6071	51	8	r	r	NOUN
cana-6071	51	9	can	can	AUX
cana-6071	51	10	be	be	AUX
cana-6071	51	11	embedded	embed	VERB
cana-6071	51	12	as	as	ADP
cana-6071	51	13	an	an	DET
cana-6071	51	14	ideal	ideal	NOUN
cana-6071	51	15	into	into	ADP
cana-6071	51	16	a	a	DET
cana-6071	51	17	semiring	semiring	NOUN
cana-6071	51	18	r	r	NOUN
cana-6071	51	19	with	with	ADP
cana-6071	51	20	unity	unity	NOUN
cana-6071	51	21	element	element	NOUN
cana-6071	51	22	.	.	PUNCT
cana-6071	52	1	proof	proof	NOUN
cana-6071	52	2	.	.	PUNCT
cana-6071	53	1	on	on	ADP
cana-6071	53	2	the	the	DET
cana-6071	53	3	set	set	NOUN
cana-6071	53	4	r	r	NOUN
cana-6071	53	5	’	'	PUNCT
cana-6071	53	6	=	=	SYM
cana-6071	53	7	{	{	PUNCT
cana-6071	53	8	(	(	PUNCT
cana-6071	53	9	a	a	X
cana-6071	53	10	,	,	PUNCT
cana-6071	53	11	n	n	CCONJ
cana-6071	53	12	)	)	PUNCT
cana-6071	53	13	/	/	PUNCT
cana-6071	53	14	a	a	DET
cana-6071	53	15	∈	∈	NOUN
cana-6071	53	16	r	r	NOUN
cana-6071	53	17	and	and	CCONJ
cana-6071	53	18	n	n	CCONJ
cana-6071	53	19	∈	∈	NOUN
cana-6071	53	20	z+	z+	NUM
cana-6071	53	21	∪	∪	X
cana-6071	53	22	{	{	PUNCT
cana-6071	53	23	0	0	NUM
cana-6071	53	24	}	}	PUNCT
cana-6071	53	25	}	}	PUNCT
cana-6071	53	26	.	.	PUNCT
cana-6071	54	1	define	define	VERB
cana-6071	54	2	(	(	PUNCT
cana-6071	54	3	a	a	PRON
cana-6071	54	4	,	,	PUNCT
cana-6071	54	5	n	n	CCONJ
cana-6071	54	6	)	)	PUNCT
cana-6071	54	7	+	+	CCONJ
cana-6071	54	8	(	(	PUNCT
cana-6071	54	9	a	a	DET
cana-6071	54	10	’	'	PUNCT
cana-6071	54	11	,	,	PUNCT
cana-6071	54	12	n	n	CCONJ
cana-6071	54	13	’	'	PUNCT
cana-6071	54	14	)	)	PUNCT
cana-6071	55	1	=	=	PRON
cana-6071	55	2	(	(	PUNCT
cana-6071	55	3	a	a	DET
cana-6071	55	4	+	+	NOUN
cana-6071	55	5	a	a	X
cana-6071	55	6	’	'	PUNCT
cana-6071	55	7	,	,	PUNCT
cana-6071	55	8	n	n	PROPN
cana-6071	55	9	+	+	CCONJ
cana-6071	55	10	n	n	CCONJ
cana-6071	55	11	’	'	PUNCT
cana-6071	55	12	)	)	PUNCT
cana-6071	56	1	and	and	CCONJ
cana-6071	56	2	(	(	PUNCT
cana-6071	56	3	a	a	PRON
cana-6071	56	4	,	,	PUNCT
cana-6071	56	5	n	n	CCONJ
cana-6071	56	6	)	)	PUNCT
cana-6071	56	7	(	(	PUNCT
cana-6071	56	8	a	a	PRON
cana-6071	56	9	’	'	PUNCT
cana-6071	56	10	,	,	PUNCT
cana-6071	56	11	n	n	CCONJ
cana-6071	56	12	’	'	PUNCT
cana-6071	56	13	)	)	PUNCT
cana-6071	57	1	=	=	PRON
cana-6071	58	1	(	(	PUNCT
cana-6071	58	2	aa	aa	NOUN
cana-6071	58	3	’	'	PUNCT
cana-6071	58	4	+	+	CCONJ
cana-6071	58	5	n’a	n’a	PROPN
cana-6071	58	6	+	+	PROPN
cana-6071	58	7	nb	nb	NOUN
cana-6071	58	8	’	'	PUNCT
cana-6071	58	9	,	,	PUNCT
cana-6071	58	10	nn	nn	NOUN
cana-6071	58	11	’	'	PUNCT
cana-6071	58	12	)	)	PUNCT
cana-6071	58	13	.	.	PUNCT
cana-6071	59	1	r	r	X
cana-6071	59	2	’	'	PUNCT
cana-6071	59	3	is	be	AUX
cana-6071	59	4	a	a	DET
cana-6071	59	5	semiring	semiring	NOUN
cana-6071	59	6	with	with	ADP
cana-6071	59	7	unity	unity	NOUN
cana-6071	59	8	(	(	PUNCT
cana-6071	59	9	0	0	NUM
cana-6071	59	10	,	,	PUNCT
cana-6071	59	11	1	1	NUM
cana-6071	59	12	)	)	PUNCT
cana-6071	59	13	and	and	CCONJ
cana-6071	59	14	r	r	NOUN
cana-6071	59	15	≅	≅	PROPN
cana-6071	59	16	(	(	PUNCT
cana-6071	59	17	r	r	NOUN
cana-6071	59	18	,	,	PUNCT
cana-6071	59	19	0	0	NUM
cana-6071	59	20	)	)	PUNCT
cana-6071	59	21	⊲	⊲	NOUN
cana-6071	60	1	r	r	NOUN
cana-6071	60	2	’	'	PUNCT
cana-6071	60	3	note	note	NOUN
cana-6071	60	4	:	:	PUNCT
cana-6071	60	5	the	the	DET
cana-6071	60	6	ring	ring	NOUN
cana-6071	60	7	r	r	NOUN
cana-6071	60	8	’	'	PUNCT
cana-6071	60	9	is	be	AUX
cana-6071	60	10	refereed	refereed	ADJ
cana-6071	60	11	to	to	PART
cana-6071	60	12	be	be	AUX
cana-6071	60	13	the	the	DET
cana-6071	60	14	dorroh	dorroh	NOUN
cana-6071	60	15	’s	’s	PART
cana-6071	60	16	extension	extension	NOUN
cana-6071	60	17	.	.	PUNCT
cana-6071	61	1	we	we	PRON
cana-6071	61	2	shall	shall	AUX
cana-6071	61	3	use	use	VERB
cana-6071	61	4	the	the	DET
cana-6071	61	5	following	following	ADJ
cana-6071	61	6	notations	notation	NOUN
cana-6071	61	7	.	.	PUNCT
cana-6071	62	1	if	if	SCONJ
cana-6071	62	2	r	r	NOUN
cana-6071	62	3	is	be	AUX
cana-6071	62	4	a	a	DET
cana-6071	62	5	semiring	semiring	NOUN
cana-6071	62	6	and	and	CCONJ
cana-6071	62	7	n	n	PRON
cana-6071	62	8	is	be	AUX
cana-6071	62	9	a	a	DET
cana-6071	62	10	positive	positive	ADJ
cana-6071	62	11	integer	integer	NOUN
cana-6071	62	12	.	.	PUNCT
cana-6071	63	1	r	r	NOUN
cana-6071	63	2	(	(	PUNCT
cana-6071	63	3	n	n	CCONJ
cana-6071	63	4	)	)	PUNCT
cana-6071	63	5	denotes	denote	VERB
cana-6071	63	6	the	the	DET
cana-6071	63	7	semiring	semiring	NOUN
cana-6071	63	8	of	of	ADP
cana-6071	63	9	matrices	matrix	NOUN
cana-6071	63	10	of	of	ADP
cana-6071	63	11	order	order	NOUN
cana-6071	63	12	n	n	NOUN
cana-6071	63	13	over	over	ADP
cana-6071	63	14	r.	r.	PROPN
cana-6071	63	15	for	for	ADP
cana-6071	63	16	i	i	PRON
cana-6071	63	17	,	,	PUNCT
cana-6071	63	18	j	j	PROPN
cana-6071	63	19	∈	∈	PROPN
cana-6071	63	20	{	{	PUNCT
cana-6071	63	21	1	1	NUM
cana-6071	63	22	,	,	PUNCT
cana-6071	63	23	2	2	NUM
cana-6071	63	24	,	,	PUNCT
cana-6071	63	25	3	3	NUM
cana-6071	63	26	,	,	PUNCT
cana-6071	63	27	...	...	PUNCT
cana-6071	63	28	,	,	PUNCT
cana-6071	63	29	n	n	CCONJ
cana-6071	63	30	}	}	PUNCT
cana-6071	63	31	,	,	PUNCT
cana-6071	63	32	r	r	NOUN
cana-6071	63	33	(	(	PUNCT
cana-6071	63	34	j	j	PROPN
cana-6071	63	35	)	)	PUNCT
cana-6071	63	36	denotes	denote	VERB
cana-6071	63	37	the	the	DET
cana-6071	63	38	subsemiring	subsemiring	NOUN
cana-6071	63	39	of	of	ADP
cana-6071	63	40	r	r	NOUN
cana-6071	63	41	(	(	PUNCT
cana-6071	63	42	n	n	CCONJ
cana-6071	63	43	)	)	PUNCT
cana-6071	63	44	consisting	consist	VERB
cana-6071	63	45	of	of	ADP
cana-6071	63	46	all	all	DET
cana-6071	63	47	matrices	matrix	NOUN
cana-6071	63	48	with	with	ADP
cana-6071	63	49	elements	element	NOUN
cana-6071	63	50	from	from	ADP
cana-6071	63	51	r	r	NOUN
cana-6071	63	52	in	in	ADP
cana-6071	63	53	the	the	DET
cana-6071	63	54	(	(	PUNCT
cana-6071	63	55	i	i	NOUN
cana-6071	63	56	,	,	PUNCT
cana-6071	63	57	j)th	j)th	ADJ
cana-6071	63	58	position	position	NOUN
cana-6071	63	59	and	and	CCONJ
cana-6071	63	60	with	with	ADP
cana-6071	63	61	0	0	NUM
cana-6071	63	62	’s	’s	NOUN
cana-6071	63	63	elsewhere	elsewhere	ADV
cana-6071	63	64	.	.	PUNCT
cana-6071	64	1	for	for	ADP
cana-6071	64	2	i	i	PROPN
cana-6071	64	3	∈	∈	PROPN
cana-6071	64	4	{	{	PUNCT
cana-6071	64	5	1	1	NUM
cana-6071	64	6	,	,	PUNCT
cana-6071	64	7	2	2	NUM
cana-6071	64	8	,	,	PUNCT
cana-6071	64	9	3	3	NUM
cana-6071	64	10	,	,	PUNCT
cana-6071	64	11	...	...	PUNCT
cana-6071	64	12	,	,	PUNCT
cana-6071	64	13	n	n	CCONJ
cana-6071	64	14	}	}	PUNCT
cana-6071	64	15	,	,	PUNCT
cana-6071	64	16	we	we	PRON
cana-6071	64	17	define	define	VERB
cana-6071	64	18	r	r	NOUN
cana-6071	64	19	(	(	PUNCT
cana-6071	64	20	i	i	NOUN
cana-6071	64	21	)	)	PUNCT
cana-6071	64	22	as	as	ADP
cana-6071	64	23	the	the	DET
cana-6071	64	24	right	right	ADJ
cana-6071	64	25	ideal	ideal	NOUN
cana-6071	64	26	∑	∑	PUNCT
cana-6071	64	27	r(𝑖𝑗)𝑛	r(𝑖𝑗)𝑛	PROPN
cana-6071	64	28	𝑗=1	𝑗=1	PROPN
cana-6071	64	29	of	of	ADP
cana-6071	64	30	r	r	PROPN
cana-6071	64	31	(	(	PUNCT
cana-6071	64	32	n	n	CCONJ
cana-6071	64	33	)	)	PUNCT
cana-6071	64	34	and	and	CCONJ
cana-6071	64	35	we	we	PRON
cana-6071	64	36	define	define	VERB
cana-6071	64	37	li	li	PROPN
cana-6071	64	38	as	as	ADP
cana-6071	64	39	the	the	DET
cana-6071	64	40	left	left	ADJ
cana-6071	64	41	ideal	ideal	NOUN
cana-6071	64	42	∑	∑	PUNCT
cana-6071	64	43	r(𝑘𝑖)𝑛	r(𝑘𝑖)𝑛	ADP
cana-6071	64	44	𝑘=1	𝑘=1	PRON
cana-6071	64	45	of	of	ADP
cana-6071	64	46	r	r	NOUN
cana-6071	64	47	(	(	PUNCT
cana-6071	64	48	n	n	CCONJ
cana-6071	64	49	)	)	PUNCT
cana-6071	64	50	.	.	PUNCT
cana-6071	65	1	if	if	SCONJ
cana-6071	65	2	x	x	SYM
cana-6071	65	3	∈	∈	PROPN
cana-6071	65	4	r	r	NOUN
cana-6071	65	5	and	and	CCONJ
cana-6071	65	6	j	j	PROPN
cana-6071	65	7	is	be	AUX
cana-6071	65	8	a	a	DET
cana-6071	65	9	non	non	ADJ
cana-6071	65	10	-	-	ADJ
cana-6071	65	11	empty	empty	ADJ
cana-6071	65	12	subset	subset	NOUN
cana-6071	65	13	of	of	ADP
cana-6071	65	14	{	{	PUNCT
cana-6071	65	15	1	1	NUM
cana-6071	65	16	,	,	PUNCT
cana-6071	65	17	2	2	NUM
cana-6071	65	18	,	,	PUNCT
cana-6071	65	19	3	3	NUM
cana-6071	65	20	,	,	PUNCT
cana-6071	65	21	...	...	PUNCT
cana-6071	65	22	,	,	PUNCT
cana-6071	65	23	n	n	CCONJ
cana-6071	65	24	}	}	PUNCT
cana-6071	65	25	with	with	ADP
cana-6071	65	26	i	i	PROPN
cana-6071	65	27	∈	∈	PROPN
cana-6071	65	28	j	j	PROPN
cana-6071	65	29	,	,	PUNCT
cana-6071	65	30	then	then	ADV
cana-6071	65	31	xj	xj	PROPN
cana-6071	65	32	(	(	PUNCT
cana-6071	65	33	i)(x	i)(x	NOUN
cana-6071	65	34	)	)	PUNCT
cana-6071	65	35	denotes	denote	VERB
cana-6071	65	36	the	the	DET
cana-6071	65	37	n	n	NUM
cana-6071	65	38	×	×	NOUN
cana-6071	65	39	n	n	NOUN
cana-6071	65	40	matrix	matrix	NOUN
cana-6071	65	41	with	with	ADP
cana-6071	65	42	x	x	PUNCT
cana-6071	65	43	in	in	ADP
cana-6071	65	44	the	the	DET
cana-6071	65	45	(	(	PUNCT
cana-6071	65	46	i	i	PROPN
cana-6071	65	47	,	,	PUNCT
cana-6071	65	48	j)-th	j)-th	NOUN
cana-6071	65	49	position	position	NOUN
cana-6071	65	50	for	for	ADP
cana-6071	65	51	all	all	DET
cana-6071	65	52	j	j	PROPN
cana-6071	65	53	∈	∈	PROPN
cana-6071	65	54	j	j	PROPN
cana-6071	65	55	and	and	CCONJ
cana-6071	65	56	with	with	ADP
cana-6071	65	57	0	0	NUM
cana-6071	65	58	’s	’s	NOUN
cana-6071	65	59	elsewhere	elsewhere	ADV
cana-6071	65	60	.	.	PUNCT
cana-6071	66	1	then	then	ADV
cana-6071	66	2	x(i	x(i	PROPN
cana-6071	66	3	)	)	PUNCT
cana-6071	66	4	=	=	SYM
cana-6071	66	5	⋃	⋃	PROPN
cana-6071	66	6	𝑥∈𝑅	𝑥∈𝑅	PROPN
cana-6071	66	7	xj	xj	PROPN
cana-6071	66	8	(	(	PUNCT
cana-6071	66	9	i)(x	i)(x	NOUN
cana-6071	66	10	)	)	PUNCT
cana-6071	66	11	is	be	AUX
cana-6071	66	12	a	a	DET
cana-6071	66	13	left	left	ADJ
cana-6071	66	14	ideal	ideal	NOUN
cana-6071	66	15	of	of	ADP
cana-6071	66	16	the	the	DET
cana-6071	66	17	semiring	semire	VERB
cana-6071	66	18	r	r	NOUN
cana-6071	66	19	(	(	PUNCT
cana-6071	66	20	n	n	CCONJ
cana-6071	66	21	)	)	PUNCT
cana-6071	66	22	.	.	PUNCT
cana-6071	67	1	moreover	moreover	ADV
cana-6071	67	2	r	r	NOUN
cana-6071	67	3	≅xj	≅xj	X
cana-6071	67	4	(	(	PUNCT
cana-6071	67	5	i	i	NOUN
cana-6071	67	6	,	,	PUNCT
cana-6071	67	7	x	x	NOUN
cana-6071	67	8	)	)	PUNCT
cana-6071	67	9	under	under	ADP
cana-6071	67	10	the	the	DET
cana-6071	67	11	natural	natural	ADJ
cana-6071	67	12	map	map	NOUN
cana-6071	67	13	.	.	PUNCT
cana-6071	68	1	if	if	SCONJ
cana-6071	68	2	r	r	NOUN
cana-6071	68	3	is	be	AUX
cana-6071	68	4	the	the	DET
cana-6071	68	5	brown	brown	PROPN
cana-6071	68	6	mccoy	mccoy	PROPN
cana-6071	68	7	radical	radical	NOUN
cana-6071	68	8	,	,	PUNCT
cana-6071	68	9	then	then	ADV
cana-6071	68	10	r	r	NOUN
cana-6071	68	11	is	be	AUX
cana-6071	68	12	hereditary	hereditary	ADJ
cana-6071	68	13	and	and	CCONJ
cana-6071	68	14	satisfies	satisfie	NOUN
cana-6071	68	15	r(r	r(r	PROPN
cana-6071	68	16	(	(	PUNCT
cana-6071	68	17	n	n	CCONJ
cana-6071	68	18	)	)	PUNCT
cana-6071	68	19	)	)	PUNCT
cana-6071	69	1	=	=	SYM
cana-6071	69	2	(	(	PUNCT
cana-6071	69	3	r(r))(n	r(r))(n	PROPN
cana-6071	69	4	)	)	PUNCT
cana-6071	69	5	.	.	PUNCT
cana-6071	70	1	proposition	proposition	NOUN
cana-6071	70	2	2.10	2.10	NUM
cana-6071	70	3	.	.	PUNCT
cana-6071	71	1	if	if	SCONJ
cana-6071	71	2	i	i	PRON
cana-6071	71	3	⊲	⊲	NOUN
cana-6071	71	4	r	r	X
cana-6071	71	5	,	,	PUNCT
cana-6071	71	6	then	then	ADV
cana-6071	71	7	i	i	PRON
cana-6071	71	8	(	(	PUNCT
cana-6071	71	9	n	n	CCONJ
cana-6071	71	10	)	)	PUNCT
cana-6071	71	11	.	.	PUNCT
cana-6071	72	1	moreover	moreover	ADV
cana-6071	72	2	if	if	SCONJ
cana-6071	72	3	i	i	PRON
cana-6071	72	4	is	be	AUX
cana-6071	72	5	a	a	DET
cana-6071	72	6	subtractive	subtractive	NOUN
cana-6071	72	7	ideal	ideal	NOUN
cana-6071	72	8	,	,	PUNCT
cana-6071	72	9	then	then	ADV
cana-6071	72	10	so	so	ADV
cana-6071	72	11	is	be	AUX
cana-6071	72	12	i	i	PRON
cana-6071	72	13	(	(	PUNCT
cana-6071	72	14	n	n	CCONJ
cana-6071	72	15	)	)	PUNCT
cana-6071	72	16	.	.	PUNCT
cana-6071	73	1	proposition	proposition	NOUN
cana-6071	73	2	2.11	2.11	NUM
cana-6071	73	3	.	.	PUNCT
cana-6071	74	1	if	if	SCONJ
cana-6071	74	2	r	r	NOUN
cana-6071	74	3	is	be	AUX
cana-6071	74	4	a	a	DET
cana-6071	74	5	semiring	semiring	NOUN
cana-6071	74	6	with	with	ADP
cana-6071	74	7	unity	unity	NOUN
cana-6071	74	8	element	element	NOUN
cana-6071	74	9	and	and	CCONJ
cana-6071	74	10	k	k	NOUN
cana-6071	74	11	⊲	⊲	NOUN
cana-6071	74	12	r	r	NOUN
cana-6071	74	13	(	(	PUNCT
cana-6071	74	14	n	n	CCONJ
cana-6071	74	15	)	)	PUNCT
cana-6071	74	16	,	,	PUNCT
cana-6071	74	17	then	then	ADV
cana-6071	74	18	k	k	PROPN
cana-6071	74	19	=	=	PUNCT
cana-6071	74	20	i	i	PROPN
cana-6071	74	21	(	(	PUNCT
cana-6071	74	22	n	n	CCONJ
cana-6071	74	23	)	)	PUNCT
cana-6071	74	24	with	with	ADP
cana-6071	74	25	some	some	DET
cana-6071	74	26	i	i	PRON
cana-6071	74	27	⊲	⊲	PROPN
cana-6071	74	28	r.	r.	PROPN
cana-6071	74	29	proposition	proposition	PROPN
cana-6071	74	30	2.12	2.12	NUM
cana-6071	74	31	.	.	PUNCT
cana-6071	75	1	if	if	SCONJ
cana-6071	75	2	r	r	NOUN
cana-6071	75	3	is	be	AUX
cana-6071	75	4	a	a	DET
cana-6071	75	5	radical	radical	ADJ
cana-6071	75	6	,	,	PUNCT
cana-6071	75	7	then	then	ADV
cana-6071	75	8	r(r(n	r(r(n	PROPN
cana-6071	75	9	)	)	PUNCT
cana-6071	75	10	)	)	PUNCT
cana-6071	76	1	=	=	PUNCT
cana-6071	76	2	i	i	PROPN
cana-6071	76	3	(	(	PUNCT
cana-6071	76	4	n	n	CCONJ
cana-6071	76	5	)	)	PUNCT
cana-6071	76	6	for	for	ADP
cana-6071	76	7	some	some	DET
cana-6071	76	8	ideal	ideal	ADJ
cana-6071	76	9	i	i	PRON
cana-6071	76	10	of	of	ADP
cana-6071	76	11	r	r	NOUN
cana-6071	76	12	and	and	CCONJ
cana-6071	76	13	for	for	ADP
cana-6071	76	14	every	every	DET
cana-6071	76	15	semiring	semire	VERB
cana-6071	76	16	r.	r.	NOUN
cana-6071	76	17	proof	proof	NOUN
cana-6071	76	18	.	.	PUNCT
cana-6071	77	1	for	for	ADP
cana-6071	77	2	any	any	DET
cana-6071	77	3	radical	radical	ADJ
cana-6071	77	4	r	r	NOUN
cana-6071	77	5	and	and	CCONJ
cana-6071	77	6	for	for	ADP
cana-6071	77	7	any	any	DET
cana-6071	77	8	semiring	semiring	NOUN
cana-6071	77	9	r	r	NOUN
cana-6071	77	10	,	,	PUNCT
cana-6071	77	11	if	if	SCONJ
cana-6071	77	12	i	i	PRON
cana-6071	77	13	⊲	⊲	NOUN
cana-6071	77	14	r	r	X
cana-6071	77	15	,	,	PUNCT
cana-6071	77	16	then	then	ADV
cana-6071	77	17	r(i	r(i	NOUN
cana-6071	77	18	)	)	PUNCT
cana-6071	77	19	⊲	⊲	NOUN
cana-6071	77	20	r.	r.	PROPN
cana-6071	77	21	if	if	SCONJ
cana-6071	77	22	i	i	PRON
cana-6071	77	23	⊲	⊲	NOUN
cana-6071	78	1	r	r	X
cana-6071	78	2	,	,	PUNCT
cana-6071	78	3	then	then	ADV
cana-6071	78	4	i	i	PRON
cana-6071	78	5	(	(	PUNCT
cana-6071	78	6	n	n	CCONJ
cana-6071	78	7	)	)	PUNCT
cana-6071	78	8	⊲	⊲	NOUN
cana-6071	79	1	r	r	NOUN
cana-6071	79	2	(	(	PUNCT
cana-6071	79	3	n	n	CCONJ
cana-6071	79	4	)	)	PUNCT
cana-6071	79	5	⇒	⇒	NOUN
cana-6071	79	6	r(i	r(i	PROPN
cana-6071	79	7	(	(	PUNCT
cana-6071	79	8	n	n	CCONJ
cana-6071	79	9	)	)	PUNCT
cana-6071	79	10	)	)	PUNCT
cana-6071	80	1	⊲	⊲	NOUN
cana-6071	80	2	r(n	r(n	NOUN
cana-6071	80	3	)	)	PUNCT
cana-6071	80	4	⇒	⇒	NOUN
cana-6071	80	5	r(i	r(i	PROPN
cana-6071	80	6	(	(	PUNCT
cana-6071	80	7	n	n	CCONJ
cana-6071	80	8	)	)	PUNCT
cana-6071	80	9	)	)	PUNCT
cana-6071	81	1	⊆	⊆	NUM
cana-6071	81	2	r(r	r(r	PROPN
cana-6071	81	3	(	(	PUNCT
cana-6071	81	4	n	n	CCONJ
cana-6071	81	5	)	)	PUNCT
cana-6071	81	6	)	)	PUNCT
cana-6071	81	7	.	.	PUNCT
cana-6071	82	1	in	in	ADP
cana-6071	82	2	particular	particular	ADJ
cana-6071	82	3	,	,	PUNCT
cana-6071	82	4	r(r(n	r(r(n	PROPN
cana-6071	82	5	)	)	PUNCT
cana-6071	82	6	)	)	PUNCT
cana-6071	82	7	is	be	AUX
cana-6071	82	8	an	an	DET
cana-6071	82	9	ideal	ideal	NOUN
cana-6071	82	10	in	in	ADP
cana-6071	82	11	r(n	r(n	PROPN
cana-6071	82	12	)	)	PUNCT
cana-6071	82	13	.	.	PUNCT
cana-6071	83	1	therefore	therefore	ADV
cana-6071	83	2	r(r(n	r(r(n	ADJ
cana-6071	83	3	)	)	PUNCT
cana-6071	83	4	)	)	PUNCT
cana-6071	84	1	=	=	PUNCT
cana-6071	84	2	i	i	PROPN
cana-6071	84	3	(	(	PUNCT
cana-6071	84	4	n	n	CCONJ
cana-6071	84	5	)	)	PUNCT
cana-6071	84	6	for	for	ADP
cana-6071	84	7	some	some	DET
cana-6071	84	8	ideal	ideal	NOUN
cana-6071	84	9	i	i	PRON
cana-6071	84	10	in	in	ADP
cana-6071	84	11	r	r	NOUN
cana-6071	84	12	’	'	PUNCT
cana-6071	84	13	,	,	PUNCT
cana-6071	84	14	where	where	SCONJ
cana-6071	84	15	r	r	X
cana-6071	84	16	’	'	PUNCT
cana-6071	84	17	is	be	AUX
cana-6071	84	18	dorroh	dorroh	NOUN
cana-6071	84	19	’s	’s	PART
cana-6071	84	20	extension	extension	NOUN
cana-6071	84	21	.	.	PUNCT
cana-6071	85	1	but	but	CCONJ
cana-6071	85	2	i	i	PRON
cana-6071	85	3	(	(	PUNCT
cana-6071	85	4	n	n	CCONJ
cana-6071	85	5	)	)	PUNCT
cana-6071	85	6	⊆	⊆	NUM
cana-6071	85	7	r	r	NOUN
cana-6071	85	8	(	(	PUNCT
cana-6071	85	9	n	n	CCONJ
cana-6071	85	10	)	)	PUNCT
cana-6071	85	11	⇒	⇒	NOUN
cana-6071	85	12	i	i	PRON
cana-6071	85	13	⊆	⊆	NUM
cana-6071	85	14	r	r	NOUN
cana-6071	85	15	⇒	⇒	NOUN
cana-6071	85	16	i	i	PRON
cana-6071	85	17	⊲	⊲	PROPN
cana-6071	85	18	r.	r.	PROPN
cana-6071	85	19	theorem	theorem	VERB
cana-6071	85	20	2.13	2.13	NUM
cana-6071	85	21	.	.	PUNCT
cana-6071	86	1	let	let	VERB
cana-6071	86	2	r	r	PRON
cana-6071	86	3	be	be	AUX
cana-6071	86	4	a	a	DET
cana-6071	86	5	radical	radical	ADJ
cana-6071	86	6	class	class	NOUN
cana-6071	86	7	,	,	PUNCT
cana-6071	86	8	let	let	VERB
cana-6071	86	9	r	r	PRON
cana-6071	86	10	be	be	AUX
cana-6071	86	11	a	a	DET
cana-6071	86	12	semiring	semiring	NOUN
cana-6071	86	13	,	,	PUNCT
cana-6071	86	14	and	and	CCONJ
cana-6071	86	15	let	let	VERB
cana-6071	86	16	n	n	PRON
cana-6071	86	17	be	be	AUX
cana-6071	86	18	a	a	DET
cana-6071	86	19	positive	positive	ADJ
cana-6071	86	20	integer	integer	NOUN
cana-6071	86	21	.	.	PUNCT
cana-6071	87	1	the	the	DET
cana-6071	87	2	following	follow	VERB
cana-6071	87	3	statements	statement	NOUN
cana-6071	87	4	are	be	AUX
cana-6071	87	5	equivalent	equivalent	ADJ
cana-6071	87	6	.	.	PUNCT
cana-6071	88	1	(	(	PUNCT
cana-6071	88	2	1	1	X
cana-6071	88	3	)	)	PUNCT
cana-6071	88	4	if	if	SCONJ
cana-6071	88	5	r	r	NOUN
cana-6071	88	6	∈	∈	PROPN
cana-6071	88	7	r	r	NOUN
cana-6071	88	8	,	,	PUNCT
cana-6071	88	9	then	then	ADV
cana-6071	88	10	r	r	NOUN
cana-6071	88	11	(	(	PUNCT
cana-6071	88	12	n	n	CCONJ
cana-6071	88	13	)	)	PUNCT
cana-6071	88	14	∈	∈	PROPN
cana-6071	88	15	r.	r.	NOUN
cana-6071	88	16	(	(	PUNCT
cana-6071	88	17	2	2	NUM
cana-6071	88	18	)	)	PUNCT
cana-6071	88	19	(	(	PUNCT
cana-6071	88	20	r(r))(n	r(r))(n	PROPN
cana-6071	88	21	)	)	PUNCT
cana-6071	88	22	⊆	⊆	NUM
cana-6071	88	23	r(r	r(r	PROPN
cana-6071	88	24	(	(	PUNCT
cana-6071	88	25	n	n	CCONJ
cana-6071	88	26	)	)	PUNCT
cana-6071	88	27	)	)	PUNCT
cana-6071	88	28	.	.	PUNCT
cana-6071	89	1	communications	communication	NOUN
cana-6071	89	2	on	on	ADP
cana-6071	89	3	applied	apply	VERB
cana-6071	89	4	nonlinear	nonlinear	ADJ
cana-6071	89	5	analysis	analysis	NOUN
cana-6071	89	6	issn	issn	NOUN
cana-6071	89	7	:	:	PUNCT
cana-6071	89	8	1074	1074	NUM
cana-6071	89	9	-	-	PUNCT
cana-6071	89	10	133x	133x	NUM
cana-6071	89	11	vol	vol	NOUN
cana-6071	89	12	31	31	NUM
cana-6071	89	13	no	no	NOUN
cana-6071	89	14	.	.	PUNCT
cana-6071	90	1	8s	8s	PROPN
cana-6071	90	2	(	(	PUNCT
cana-6071	90	3	2024	2024	NUM
cana-6071	90	4	)	)	PUNCT
cana-6071	90	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-6071	90	6	1182	1182	NUM
cana-6071	90	7	proof	proof	NOUN
cana-6071	90	8	.	.	PUNCT
cana-6071	91	1	now	now	ADV
cana-6071	91	2	r(r	r(r	VERB
cana-6071	91	3	)	)	PUNCT
cana-6071	91	4	∈	∈	PROPN
cana-6071	91	5	r	r	NOUN
cana-6071	91	6	so	so	SCONJ
cana-6071	91	7	that	that	SCONJ
cana-6071	91	8	by	by	ADP
cana-6071	91	9	(	(	PUNCT
cana-6071	91	10	1	1	NUM
cana-6071	91	11	)	)	PUNCT
cana-6071	91	12	(	(	PUNCT
cana-6071	91	13	r(r))(n	r(r))(n	NOUN
cana-6071	91	14	)	)	PUNCT
cana-6071	91	15	∈	∈	PROPN
cana-6071	91	16	r.	r.	PROPN
cana-6071	91	17	since	since	SCONJ
cana-6071	91	18	(	(	PUNCT
cana-6071	91	19	r(r))(n	r(r))(n	PROPN
cana-6071	91	20	)	)	PUNCT
cana-6071	91	21	is	be	AUX
cana-6071	91	22	an	an	DET
cana-6071	91	23	ideal	ideal	NOUN
cana-6071	91	24	in	in	ADP
cana-6071	91	25	r	r	NOUN
cana-6071	91	26	(	(	PUNCT
cana-6071	91	27	n	n	CCONJ
cana-6071	91	28	)	)	PUNCT
cana-6071	91	29	,	,	PUNCT
cana-6071	91	30	hence	hence	ADV
cana-6071	91	31	(	(	PUNCT
cana-6071	91	32	r(r))(n	r(r))(n	PROPN
cana-6071	91	33	)	)	PUNCT
cana-6071	91	34	⊆	⊆	NUM
cana-6071	91	35	r(r	r(r	PROPN
cana-6071	91	36	(	(	PUNCT
cana-6071	91	37	n	n	CCONJ
cana-6071	91	38	)	)	PUNCT
cana-6071	91	39	)	)	PUNCT
cana-6071	91	40	.	.	PUNCT
cana-6071	92	1	now	now	ADV
cana-6071	92	2	r∈	r∈	PROPN
cana-6071	92	3	r	r	NOUN
cana-6071	92	4	implies	imply	VERB
cana-6071	92	5	that	that	SCONJ
cana-6071	92	6	r(r	r(r	NOUN
cana-6071	92	7	)	)	PUNCT
cana-6071	92	8	=	=	VERB
cana-6071	93	1	r.	r.	PROPN
cana-6071	94	1	so	so	ADV
cana-6071	94	2	that	that	SCONJ
cana-6071	94	3	r	r	NOUN
cana-6071	94	4	(	(	PUNCT
cana-6071	94	5	n	n	CCONJ
cana-6071	94	6	)	)	PUNCT
cana-6071	94	7	=	=	SYM
cana-6071	94	8	(	(	PUNCT
cana-6071	94	9	r(r))(n	r(r))(n	PROPN
cana-6071	94	10	)	)	PUNCT
cana-6071	94	11	.	.	PUNCT
cana-6071	95	1	by	by	ADP
cana-6071	95	2	(	(	PUNCT
cana-6071	95	3	2	2	NUM
cana-6071	95	4	)	)	PUNCT
cana-6071	95	5	(	(	PUNCT
cana-6071	95	6	r(r))(n	r(r))(n	PROPN
cana-6071	95	7	)	)	PUNCT
cana-6071	95	8	⊆	⊆	NUM
cana-6071	95	9	r(r	r(r	PROPN
cana-6071	95	10	(	(	PUNCT
cana-6071	95	11	n	n	CCONJ
cana-6071	95	12	)	)	PUNCT
cana-6071	95	13	)	)	PUNCT
cana-6071	95	14	.	.	PUNCT
cana-6071	96	1	hence	hence	ADV
cana-6071	96	2	r(n)=	r(n)=	ADP
cana-6071	96	3	r(rn	r(rn	NUM
cana-6071	96	4	)	)	PUNCT
cana-6071	96	5	and	and	CCONJ
cana-6071	96	6	rn	rn	PROPN
cana-6071	96	7	∈	∈	PROPN
cana-6071	96	8	r.	r.	PROPN
cana-6071	96	9	theorem	theorem	VERB
cana-6071	96	10	2.14	2.14	NUM
cana-6071	96	11	.	.	PUNCT
cana-6071	97	1	let	let	VERB
cana-6071	97	2	r	r	PRON
cana-6071	97	3	be	be	AUX
cana-6071	97	4	a	a	DET
cana-6071	97	5	radical	radical	ADJ
cana-6071	97	6	class	class	NOUN
cana-6071	97	7	,	,	PUNCT
cana-6071	97	8	let	let	VERB
cana-6071	97	9	r	r	PRON
cana-6071	97	10	be	be	AUX
cana-6071	97	11	a	a	DET
cana-6071	97	12	semiring	semiring	NOUN
cana-6071	97	13	,	,	PUNCT
cana-6071	97	14	and	and	CCONJ
cana-6071	97	15	let	let	VERB
cana-6071	97	16	n	n	PRON
cana-6071	97	17	be	be	AUX
cana-6071	97	18	a	a	DET
cana-6071	97	19	positive	positive	ADJ
cana-6071	97	20	integer	integer	NOUN
cana-6071	97	21	.	.	PUNCT
cana-6071	98	1	the	the	DET
cana-6071	98	2	following	follow	VERB
cana-6071	98	3	statements	statement	NOUN
cana-6071	98	4	are	be	AUX
cana-6071	98	5	equivalent	equivalent	ADJ
cana-6071	98	6	.	.	PUNCT
cana-6071	99	1	1	1	X
cana-6071	99	2	.	.	X
cana-6071	100	1	if	if	SCONJ
cana-6071	100	2	r	r	NOUN
cana-6071	100	3	(	(	PUNCT
cana-6071	100	4	n	n	CCONJ
cana-6071	100	5	)	)	PUNCT
cana-6071	100	6	∈	∈	PROPN
cana-6071	100	7	r	r	NOUN
cana-6071	100	8	,	,	PUNCT
cana-6071	100	9	then	then	ADV
cana-6071	100	10	r	r	PROPN
cana-6071	100	11	∈	∈	PROPN
cana-6071	100	12	r.	r.	PROPN
cana-6071	100	13	2	2	NUM
cana-6071	100	14	.	.	PUNCT
cana-6071	101	1	r(r	r(r	NOUN
cana-6071	101	2	(	(	PUNCT
cana-6071	101	3	n	n	CCONJ
cana-6071	101	4	)	)	PUNCT
cana-6071	101	5	)	)	PUNCT
cana-6071	102	1	⊆	⊆	X
cana-6071	102	2	(	(	PUNCT
cana-6071	102	3	r(r))(n	r(r))(n	NOUN
cana-6071	102	4	)	)	PUNCT
cana-6071	102	5	proof	proof	NOUN
cana-6071	102	6	.	.	PUNCT
cana-6071	103	1	by	by	ADP
cana-6071	103	2	proposition	proposition	NOUN
cana-6071	103	3	2.12	2.12	NUM
cana-6071	103	4	r(r	r(r	PROPN
cana-6071	103	5	(	(	PUNCT
cana-6071	103	6	n	n	CCONJ
cana-6071	103	7	)	)	PUNCT
cana-6071	103	8	)	)	PUNCT
cana-6071	103	9	=	=	SYM
cana-6071	103	10	i(n	i(n	NOUN
cana-6071	103	11	)	)	PUNCT
cana-6071	103	12	,	,	PUNCT
cana-6071	103	13	for	for	ADP
cana-6071	103	14	some	some	DET
cana-6071	103	15	ideal	ideal	ADJ
cana-6071	103	16	i	i	PRON
cana-6071	103	17	of	of	ADP
cana-6071	103	18	r.	r.	PROPN
cana-6071	103	19	from	from	ADP
cana-6071	103	20	(	(	PUNCT
cana-6071	103	21	1	1	X
cana-6071	103	22	)	)	PUNCT
cana-6071	103	23	we	we	PRON
cana-6071	103	24	have	have	VERB
cana-6071	103	25	i	i	PROPN
cana-6071	103	26	∈	∈	PROPN
cana-6071	103	27	r.	r.	NOUN
cana-6071	103	28	hence	hence	ADV
cana-6071	103	29	i	i	PROPN
cana-6071	103	30	⊆	⊆	NUM
cana-6071	103	31	r(r	r(r	PROPN
cana-6071	103	32	)	)	PUNCT
cana-6071	103	33	and	and	CCONJ
cana-6071	103	34	so	so	ADV
cana-6071	103	35	r(r	r(r	PROPN
cana-6071	103	36	(	(	PUNCT
cana-6071	103	37	n	n	CCONJ
cana-6071	103	38	)	)	PUNCT
cana-6071	103	39	)	)	PUNCT
cana-6071	104	1	=	=	PUNCT
cana-6071	104	2	i	i	PROPN
cana-6071	104	3	(	(	PUNCT
cana-6071	104	4	n	n	CCONJ
cana-6071	104	5	)	)	PUNCT
cana-6071	104	6	⊆	⊆	NUM
cana-6071	104	7	(	(	PUNCT
cana-6071	104	8	r(r))(n	r(r))(n	NOUN
cana-6071	104	9	)	)	PUNCT
cana-6071	104	10	.	.	PUNCT
cana-6071	105	1	now	now	ADV
cana-6071	105	2	r	r	NOUN
cana-6071	105	3	(	(	PUNCT
cana-6071	105	4	n	n	CCONJ
cana-6071	105	5	)	)	PUNCT
cana-6071	105	6	∈	∈	NOUN
cana-6071	105	7	r	r	NOUN
cana-6071	105	8	implies	imply	VERB
cana-6071	105	9	that	that	SCONJ
cana-6071	105	10	r(r	r(r	PROPN
cana-6071	105	11	(	(	PUNCT
cana-6071	105	12	n	n	CCONJ
cana-6071	105	13	)	)	PUNCT
cana-6071	105	14	)	)	PUNCT
cana-6071	106	1	=	=	SYM
cana-6071	106	2	r	r	NOUN
cana-6071	106	3	(	(	PUNCT
cana-6071	106	4	n	n	CCONJ
cana-6071	106	5	)	)	PUNCT
cana-6071	106	6	.	.	PUNCT
cana-6071	107	1	thus	thus	ADV
cana-6071	107	2	by	by	ADP
cana-6071	107	3	(	(	PUNCT
cana-6071	107	4	2	2	NUM
cana-6071	107	5	)	)	PUNCT
cana-6071	107	6	,	,	PUNCT
cana-6071	107	7	r	r	NOUN
cana-6071	107	8	(	(	PUNCT
cana-6071	107	9	n	n	CCONJ
cana-6071	107	10	)	)	PUNCT
cana-6071	107	11	=	=	SYM
cana-6071	107	12	r(r	r(r	NOUN
cana-6071	107	13	(	(	PUNCT
cana-6071	107	14	n	n	CCONJ
cana-6071	107	15	)	)	PUNCT
cana-6071	107	16	)	)	PUNCT
cana-6071	108	1	⊆	⊆	X
cana-6071	108	2	(	(	PUNCT
cana-6071	108	3	r(r))(n	r(r))(n	NOUN
cana-6071	108	4	)	)	PUNCT
cana-6071	108	5	and	and	CCONJ
cana-6071	108	6	so	so	ADV
cana-6071	108	7	r	r	NOUN
cana-6071	108	8	(	(	PUNCT
cana-6071	108	9	n	n	CCONJ
cana-6071	108	10	)	)	PUNCT
cana-6071	108	11	=	=	SYM
cana-6071	108	12	(	(	PUNCT
cana-6071	108	13	r(r))(n	r(r))(n	PROPN
cana-6071	108	14	)	)	PUNCT
cana-6071	108	15	.	.	PUNCT
cana-6071	109	1	hence	hence	ADV
cana-6071	109	2	r(r	r(r	NOUN
cana-6071	109	3	)	)	PUNCT
cana-6071	110	1	=	=	SYM
cana-6071	110	2	r	r	NOUN
cana-6071	110	3	and	and	CCONJ
cana-6071	110	4	r	r	PROPN
cana-6071	110	5	∈	∈	PROPN
cana-6071	110	6	r.	r.	PROPN
cana-6071	110	7	theorem	theorem	VERB
cana-6071	110	8	2.15	2.15	NUM
cana-6071	110	9	.	.	PUNCT
cana-6071	111	1	let	let	VERB
cana-6071	111	2	r	r	PRON
cana-6071	111	3	be	be	AUX
cana-6071	111	4	a	a	DET
cana-6071	111	5	strong	strong	ADJ
cana-6071	111	6	radical	radical	ADJ
cana-6071	111	7	class	class	NOUN
cana-6071	111	8	.	.	PUNCT
cana-6071	112	1	then	then	ADV
cana-6071	112	2	r	r	NOUN
cana-6071	112	3	∈	∈	NOUN
cana-6071	112	4	r	r	NOUN
cana-6071	112	5	implies	imply	VERB
cana-6071	112	6	r	r	NOUN
cana-6071	112	7	(	(	PUNCT
cana-6071	112	8	n	n	CCONJ
cana-6071	112	9	)	)	PUNCT
cana-6071	112	10	∈	∈	PROPN
cana-6071	112	11	r.	r.	NOUN
cana-6071	112	12	proof	proof	NOUN
cana-6071	112	13	.	.	PUNCT
cana-6071	113	1	let	let	VERB
cana-6071	113	2	n	n	PRON
cana-6071	113	3	>	>	X
cana-6071	113	4	1	1	NUM
cana-6071	113	5	,	,	PUNCT
cana-6071	113	6	let	let	VERB
cana-6071	113	7	i	i	PRON
cana-6071	113	8	∈	∈	PROPN
cana-6071	113	9	{	{	PUNCT
cana-6071	113	10	1	1	NUM
cana-6071	113	11	,	,	PUNCT
cana-6071	113	12	2	2	NUM
cana-6071	113	13	,	,	PUNCT
cana-6071	113	14	3	3	NUM
cana-6071	113	15	,	,	PUNCT
cana-6071	113	16	...	...	PUNCT
cana-6071	113	17	,	,	PUNCT
cana-6071	113	18	n	n	CCONJ
cana-6071	113	19	}	}	PUNCT
cana-6071	113	20	be	be	AUX
cana-6071	113	21	fixed	fix	VERB
cana-6071	113	22	and	and	CCONJ
cana-6071	113	23	let	let	VERB
cana-6071	113	24	j	j	PROPN
cana-6071	113	25	∈	∈	PROPN
cana-6071	113	26	{	{	PUNCT
cana-6071	113	27	1	1	NUM
cana-6071	113	28	,	,	PUNCT
cana-6071	113	29	2	2	NUM
cana-6071	113	30	,	,	PUNCT
cana-6071	113	31	3	3	NUM
cana-6071	113	32	,	,	PUNCT
cana-6071	113	33	...	...	PUNCT
cana-6071	113	34	,	,	PUNCT
cana-6071	113	35	n	n	CCONJ
cana-6071	113	36	}	}	PUNCT
cana-6071	113	37	with	with	ADP
cana-6071	113	38	i≠	i≠	PROPN
cana-6071	113	39	j.	j.	PROPN
cana-6071	113	40	set	set	VERB
cana-6071	113	41	j	j	PROPN
cana-6071	114	1	=	=	PRON
cana-6071	114	2	{	{	PUNCT
cana-6071	114	3	i	i	PROPN
cana-6071	114	4	,	,	PUNCT
cana-6071	114	5	j	j	PROPN
cana-6071	114	6	}	}	PUNCT
cana-6071	114	7	.	.	PUNCT
cana-6071	115	1	then	then	ADV
cana-6071	115	2	xj	xj	PROPN
cana-6071	115	3	(	(	PUNCT
cana-6071	115	4	i)(∈	i)(∈	PROPN
cana-6071	115	5	r.	r.	PROPN
cana-6071	115	6	since	since	SCONJ
cana-6071	115	7	r	r	NOUN
cana-6071	115	8	is	be	AUX
cana-6071	115	9	strong	strong	ADJ
cana-6071	115	10	,	,	PUNCT
cana-6071	115	11	therefore	therefore	ADV
cana-6071	115	12	xj	xj	PROPN
cana-6071	115	13	(	(	PUNCT
cana-6071	115	14	i	i	NOUN
cana-6071	115	15	)	)	PUNCT
cana-6071	115	16	⊆	⊆	NUM
cana-6071	115	17	r(r	r(r	PROPN
cana-6071	115	18	(	(	PUNCT
cana-6071	115	19	i	i	NOUN
cana-6071	115	20	)	)	PUNCT
cana-6071	115	21	)	)	PUNCT
cana-6071	115	22	.	.	PUNCT
cana-6071	116	1	setting	set	VERB
cana-6071	116	2	k	k	X
cana-6071	116	3	=	=	PRON
cana-6071	116	4	{	{	PUNCT
cana-6071	116	5	i	i	NOUN
cana-6071	116	6	}	}	PUNCT
cana-6071	116	7	.	.	PUNCT
cana-6071	117	1	we	we	PRON
cana-6071	117	2	like	like	VERB
cana-6071	117	3	wise	wise	ADV
cana-6071	117	4	obtain	obtain	VERB
cana-6071	117	5	xk	xk	PROPN
cana-6071	117	6	(	(	PUNCT
cana-6071	117	7	i	i	NOUN
cana-6071	117	8	)	)	PUNCT
cana-6071	117	9	⊆	⊆	NUM
cana-6071	117	10	r(r(i	r(r(i	NOUN
cana-6071	117	11	)	)	PUNCT
cana-6071	117	12	)	)	PUNCT
cana-6071	117	13	.	.	PUNCT
cana-6071	118	1	hence	hence	ADV
cana-6071	118	2	xj	xj	PROPN
cana-6071	118	3	(	(	PUNCT
cana-6071	118	4	i	i	NOUN
cana-6071	118	5	)	)	PUNCT
cana-6071	119	1	+	+	CCONJ
cana-6071	119	2	xk	xk	PROPN
cana-6071	119	3	(	(	PUNCT
cana-6071	119	4	i	i	NOUN
cana-6071	119	5	)	)	PUNCT
cana-6071	119	6	⊆	⊆	NUM
cana-6071	119	7	r(r(i	r(r(i	NOUN
cana-6071	119	8	)	)	PUNCT
cana-6071	119	9	)	)	PUNCT
cana-6071	119	10	.	.	PUNCT
cana-6071	120	1	since	since	SCONJ
cana-6071	120	2	i	i	PRON
cana-6071	120	3	≠	≠	PROPN
cana-6071	120	4	j.	j.	PROPN
cana-6071	120	5	and	and	CCONJ
cana-6071	120	6	j	j	PROPN
cana-6071	120	7	was	be	AUX
cana-6071	120	8	otherwise	otherwise	ADV
cana-6071	120	9	arbitary	arbitary	ADJ
cana-6071	120	10	,	,	PUNCT
cana-6071	120	11	then	then	ADV
cana-6071	120	12	ri	ri	PROPN
cana-6071	120	13	⊆	⊆	NUM
cana-6071	120	14	r(r	r(r	PROPN
cana-6071	120	15	(	(	PUNCT
cana-6071	120	16	i	i	NOUN
cana-6071	120	17	)	)	PUNCT
cana-6071	120	18	)	)	PUNCT
cana-6071	120	19	.	.	PUNCT
cana-6071	121	1	now	now	ADV
cana-6071	121	2	r	r	NOUN
cana-6071	121	3	(	(	PUNCT
cana-6071	121	4	i)is	i)is	PROPN
cana-6071	121	5	r	r	NOUN
cana-6071	121	6	-	-	PUNCT
cana-6071	121	7	right	right	ADJ
cana-6071	121	8	ideal	ideal	NOUN
cana-6071	121	9	of	of	ADP
cana-6071	121	10	r(n	r(n	PROPN
cana-6071	121	11	)	)	PUNCT
cana-6071	121	12	so	so	SCONJ
cana-6071	121	13	that	that	SCONJ
cana-6071	121	14	,	,	PUNCT
cana-6071	121	15	since	since	SCONJ
cana-6071	121	16	r	r	NOUN
cana-6071	121	17	is	be	AUX
cana-6071	121	18	strong	strong	ADJ
cana-6071	121	19	,	,	PUNCT
cana-6071	121	20	we	we	PRON
cana-6071	121	21	have	have	VERB
cana-6071	121	22	r(i	r(i	NOUN
cana-6071	121	23	)	)	PUNCT
cana-6071	122	1	⊆	⊆	NUM
cana-6071	122	2	r(r	r(r	PROPN
cana-6071	122	3	(	(	PUNCT
cana-6071	122	4	n	n	CCONJ
cana-6071	122	5	)	)	PUNCT
cana-6071	122	6	)	)	PUNCT
cana-6071	122	7	.	.	PUNCT
cana-6071	123	1	this	this	PRON
cana-6071	123	2	is	be	AUX
cana-6071	123	3	true	true	ADJ
cana-6071	123	4	for	for	ADP
cana-6071	123	5	j	j	PROPN
cana-6071	123	6	∈	∈	PROPN
cana-6071	123	7	{	{	PUNCT
cana-6071	123	8	1	1	NUM
cana-6071	123	9	,	,	PUNCT
cana-6071	123	10	2	2	NUM
cana-6071	123	11	,	,	PUNCT
cana-6071	123	12	3	3	NUM
cana-6071	123	13	,	,	PUNCT
cana-6071	123	14	...	...	PUNCT
cana-6071	123	15	,	,	PUNCT
cana-6071	123	16	n	n	CCONJ
cana-6071	123	17	}	}	PUNCT
cana-6071	123	18	.	.	PUNCT
cana-6071	124	1	we	we	PRON
cana-6071	124	2	obtain	obtain	VERB
cana-6071	124	3	∑	∑	PUNCT
cana-6071	124	4	r(𝑖)𝑛	r(𝑖)𝑛	NOUN
cana-6071	124	5	𝑖=1	𝑖=1	PROPN
cana-6071	125	1	⊆	⊆	NUM
cana-6071	125	2	r(r(n	r(r(n	NUM
cana-6071	125	3	)	)	PUNCT
cana-6071	125	4	)	)	PUNCT
cana-6071	125	5	.	.	PUNCT
cana-6071	126	1	hence	hence	ADV
cana-6071	126	2	r	r	NOUN
cana-6071	126	3	(	(	PUNCT
cana-6071	126	4	n	n	CCONJ
cana-6071	126	5	)	)	PUNCT
cana-6071	126	6	=	=	SYM
cana-6071	126	7	r(r	r(r	NOUN
cana-6071	126	8	(	(	PUNCT
cana-6071	126	9	n	n	CCONJ
cana-6071	126	10	)	)	PUNCT
cana-6071	126	11	)	)	PUNCT
cana-6071	126	12	and	and	CCONJ
cana-6071	126	13	r	r	NOUN
cana-6071	126	14	(	(	PUNCT
cana-6071	126	15	n	n	CCONJ
cana-6071	126	16	)	)	PUNCT
cana-6071	126	17	∈	∈	PROPN
cana-6071	126	18	r.	r.	PROPN
cana-6071	126	19	theorem2.16	theorem2.16	PROPN
cana-6071	126	20	.	.	PUNCT
cana-6071	127	1	let	let	VERB
cana-6071	127	2	r	r	PRON
cana-6071	127	3	be	be	AUX
cana-6071	127	4	a	a	DET
cana-6071	127	5	hereditary	hereditary	ADJ
cana-6071	127	6	(	(	PUNCT
cana-6071	127	7	or	or	CCONJ
cana-6071	127	8	right	right	ADJ
cana-6071	127	9	hereditary	hereditary	ADJ
cana-6071	127	10	)	)	PUNCT
cana-6071	127	11	radical	radical	ADJ
cana-6071	127	12	class	class	NOUN
cana-6071	127	13	.	.	PUNCT
cana-6071	128	1	then	then	ADV
cana-6071	128	2	r	r	NOUN
cana-6071	128	3	(	(	PUNCT
cana-6071	128	4	n	n	CCONJ
cana-6071	128	5	)	)	PUNCT
cana-6071	128	6	∈	∈	NOUN
cana-6071	128	7	r	r	NOUN
cana-6071	128	8	implies	imply	VERB
cana-6071	128	9	r	r	PROPN
cana-6071	128	10	∈	∈	PROPN
cana-6071	128	11	r.	r.	PROPN
cana-6071	128	12	theorem2.17	theorem2.17	PROPN
cana-6071	128	13	.	.	PUNCT
cana-6071	129	1	let	let	VERB
cana-6071	129	2	r	r	PRON
cana-6071	129	3	be	be	AUX
cana-6071	129	4	a	a	DET
cana-6071	129	5	hereditary	hereditary	ADJ
cana-6071	129	6	and	and	CCONJ
cana-6071	129	7	left	left	ADJ
cana-6071	129	8	-	-	PUNCT
cana-6071	129	9	strong	strong	ADJ
cana-6071	129	10	(	(	PUNCT
cana-6071	129	11	right	right	ADV
cana-6071	129	12	strong	strong	ADJ
cana-6071	129	13	)	)	PUNCT
cana-6071	129	14	radical	radical	ADJ
cana-6071	129	15	class	class	NOUN
cana-6071	129	16	.	.	PUNCT
cana-6071	130	1	then	then	ADV
cana-6071	130	2	r	r	NOUN
cana-6071	130	3	∈	∈	PROPN
cana-6071	130	4	r	r	NOUN
cana-6071	130	5	implies	imply	VERB
cana-6071	130	6	r+	r+	PUNCT
cana-6071	130	7	∈	∈	PROPN
cana-6071	130	8	r.	r.	PROPN
cana-6071	130	9	theorem	theorem	VERB
cana-6071	130	10	2.18	2.18	NUM
cana-6071	130	11	.	.	PUNCT
cana-6071	131	1	let	let	VERB
cana-6071	131	2	r	r	PRON
cana-6071	131	3	be	be	AUX
cana-6071	131	4	a	a	DET
cana-6071	131	5	hereditary	hereditary	ADJ
cana-6071	131	6	and	and	CCONJ
cana-6071	131	7	left	left	ADJ
cana-6071	131	8	-	-	PUNCT
cana-6071	131	9	strong	strong	ADJ
cana-6071	131	10	(	(	PUNCT
cana-6071	131	11	right	right	ADV
cana-6071	131	12	strong	strong	ADJ
cana-6071	131	13	)	)	PUNCT
cana-6071	131	14	radical	radical	ADJ
cana-6071	131	15	class	class	NOUN
cana-6071	131	16	.	.	PUNCT
cana-6071	132	1	then	then	ADV
cana-6071	132	2	r	r	NOUN
cana-6071	132	3	∈	∈	NOUN
cana-6071	132	4	r	r	NOUN
cana-6071	132	5	implies	imply	VERB
cana-6071	132	6	r	r	NOUN
cana-6071	132	7	(	(	PUNCT
cana-6071	132	8	n)∈	n)∈	PROPN
cana-6071	132	9	r.	r.	PROPN
cana-6071	132	10	theorem	theorem	VERB
cana-6071	132	11	2.19	2.19	NUM
cana-6071	132	12	.	.	PUNCT
cana-6071	133	1	let	let	VERB
cana-6071	133	2	r	r	PRON
cana-6071	133	3	be	be	AUX
cana-6071	133	4	a	a	DET
cana-6071	133	5	radical	radical	ADJ
cana-6071	133	6	class	class	NOUN
cana-6071	133	7	which	which	PRON
cana-6071	133	8	is	be	AUX
cana-6071	133	9	(	(	PUNCT
cana-6071	133	10	left	left	ADJ
cana-6071	133	11	or	or	CCONJ
cana-6071	133	12	right)-hereditary	right)-hereditary	ADJ
cana-6071	133	13	and	and	CCONJ
cana-6071	133	14	(	(	PUNCT
cana-6071	133	15	left	left	ADJ
cana-6071	133	16	or	or	CCONJ
cana-6071	133	17	right	right	ADJ
cana-6071	133	18	)	)	PUNCT
cana-6071	133	19	-strong	-strong	NOUN
cana-6071	133	20	.	.	PUNCT
cana-6071	134	1	then	then	ADV
cana-6071	134	2	r(r	r(r	PROPN
cana-6071	134	3	(	(	PUNCT
cana-6071	134	4	n	n	CCONJ
cana-6071	134	5	)	)	PUNCT
cana-6071	134	6	)	)	PUNCT
cana-6071	135	1	=	=	PUNCT
cana-6071	135	2	(	(	PUNCT
cana-6071	135	3	r(r	r(r	PROPN
cana-6071	135	4	(	(	PUNCT
cana-6071	135	5	n)))n	n)))n	PROPN
cana-6071	135	6	.	.	PUNCT
cana-6071	136	1	acknowledgement	acknowledgement	NOUN
cana-6071	136	2	:	:	PUNCT
cana-6071	136	3	i	i	PRON
cana-6071	136	4	would	would	AUX
cana-6071	136	5	like	like	VERB
cana-6071	136	6	to	to	PART
cana-6071	136	7	extend	extend	VERB
cana-6071	136	8	my	my	PRON
cana-6071	136	9	heartfelt	heartfelt	ADJ
cana-6071	136	10	gratitude	gratitude	NOUN
cana-6071	136	11	to	to	ADP
cana-6071	136	12	dr	dr	PROPN
cana-6071	136	13	.	.	PROPN
cana-6071	136	14	rajendra	rajendra	PROPN
cana-6071	136	15	p.	p.	PROPN
cana-6071	136	16	deore	deore	PROPN
cana-6071	136	17	,	,	PUNCT
cana-6071	136	18	our	our	PRON
cana-6071	136	19	respected	respected	ADJ
cana-6071	136	20	mentor	mentor	NOUN
cana-6071	136	21	,	,	PUNCT
cana-6071	136	22	for	for	ADP
cana-6071	136	23	his	his	PRON
cana-6071	136	24	invaluable	invaluable	ADJ
cana-6071	136	25	guidance	guidance	NOUN
cana-6071	136	26	and	and	CCONJ
cana-6071	136	27	unwavering	unwavere	VERB
cana-6071	136	28	support	support	NOUN
cana-6071	136	29	throughout	throughout	ADP
cana-6071	136	30	this	this	DET
cana-6071	136	31	research	research	NOUN
cana-6071	136	32	work	work	NOUN
cana-6071	136	33	.	.	PUNCT
cana-6071	137	1	references	reference	NOUN
cana-6071	137	2	[	[	X
cana-6071	137	3	1	1	NUM
cana-6071	137	4	]	]	X
cana-6071	137	5	al	al	PROPN
cana-6071	137	6	-	-	PUNCT
cana-6071	137	7	thani	thani	PROPN
cana-6071	137	8	h.	h.	PROPN
cana-6071	137	9	m.	m.	PROPN
cana-6071	137	10	j.	j.	PROPN
cana-6071	137	11	weak	weak	ADJ
cana-6071	137	12	radical	radical	ADJ
cana-6071	137	13	classes	class	NOUN
cana-6071	137	14	,	,	PUNCT
cana-6071	137	15	tamkang	tamkang	PROPN
cana-6071	137	16	journal	journal	PROPN
cana-6071	137	17	of	of	ADP
cana-6071	137	18	mathematics	mathematic	NOUN
cana-6071	137	19	,	,	PUNCT
cana-6071	137	20	35(4	35(4	NUM
cana-6071	137	21	)	)	PUNCT
cana-6071	137	22	,	,	PUNCT
cana-6071	137	23	359	359	NUM
cana-6071	137	24	-	-	SYM
cana-6071	137	25	369	369	NUM
cana-6071	137	26	(	(	PUNCT
cana-6071	137	27	2004	2004	NUM
cana-6071	137	28	)	)	PUNCT
cana-6071	137	29	.	.	PUNCT
cana-6071	138	1	[	[	X
cana-6071	138	2	2	2	X
cana-6071	138	3	]	]	X
cana-6071	138	4	dutta	dutta	PROPN
cana-6071	138	5	t.	t.	PROPN
cana-6071	138	6	k.	k.	PROPN
cana-6071	138	7	and	and	CCONJ
cana-6071	138	8	das	das	PROPN
cana-6071	138	9	m.	m.	PROPN
cana-6071	138	10	l.	l.	PROPN
cana-6071	138	11	normal	normal	ADJ
cana-6071	138	12	radical	radical	ADJ
cana-6071	138	13	class	class	NOUN
cana-6071	138	14	of	of	ADP
cana-6071	138	15	semirings	semiring	NOUN
cana-6071	138	16	,	,	PUNCT
cana-6071	138	17	siutheast	siutheast	ADJ
cana-6071	138	18	asian	asian	ADJ
cana-6071	138	19	bulletin	bulletin	NOUN
cana-6071	138	20	of	of	ADP
cana-6071	138	21	mathematics	mathematic	NOUN
cana-6071	138	22	,	,	PUNCT
cana-6071	138	23	35	35	NUM
cana-6071	138	24	,	,	PUNCT
cana-6071	138	25	389	389	NUM
cana-6071	138	26	-	-	SYM
cana-6071	138	27	400	400	NUM
cana-6071	138	28	(	(	PUNCT
cana-6071	138	29	2011	2011	NUM
cana-6071	138	30	)	)	PUNCT
cana-6071	138	31	.	.	PUNCT
cana-6071	139	1	[	[	X
cana-6071	139	2	3	3	X
cana-6071	139	3	]	]	X
cana-6071	139	4	b.	b.	PROPN
cana-6071	139	5	j.	j.	PROPN
cana-6071	139	6	gardner	gardner	PROPN
cana-6071	139	7	,	,	PUNCT
cana-6071	139	8	a	a	DET
cana-6071	139	9	note	note	NOUN
cana-6071	139	10	on	on	ADP
cana-6071	139	11	radicals	radical	NOUN
cana-6071	139	12	and	and	CCONJ
cana-6071	139	13	polynomial	polynomial	ADJ
cana-6071	139	14	rings	ring	NOUN
cana-6071	139	15	,	,	PUNCT
cana-6071	139	16	math	math	NOUN
cana-6071	139	17	.	.	PUNCT
cana-6071	140	1	scand	scand	PROPN
cana-6071	140	2	.	.	PROPN
cana-6071	140	3	,	,	PUNCT
cana-6071	140	4	31	31	NUM
cana-6071	140	5	(	(	PUNCT
cana-6071	140	6	1972	1972	NUM
cana-6071	140	7	)	)	PUNCT
cana-6071	140	8	,	,	PUNCT
cana-6071	140	9	83	83	NUM
cana-6071	140	10	-	-	SYM
cana-6071	140	11	88	88	NUM
cana-6071	140	12	.	.	PUNCT
cana-6071	141	1	[	[	X
cana-6071	141	2	4	4	NUM
cana-6071	141	3	]	]	PUNCT
cana-6071	141	4	gardner	gardner	PROPN
cana-6071	141	5	b.	b.	PROPN
cana-6071	141	6	j.	j.	PROPN
cana-6071	141	7	and	and	CCONJ
cana-6071	141	8	wiegandt	wiegandt	PROPN
cana-6071	141	9	r.	r.	PROPN
cana-6071	141	10	radical	radical	PROPN
cana-6071	141	11	theory	theory	NOUN
cana-6071	141	12	of	of	ADP
cana-6071	141	13	rings	ring	NOUN
cana-6071	141	14	,	,	PUNCT
cana-6071	141	15	marcel	marcel	PROPN
cana-6071	141	16	dekker	dekker	PROPN
cana-6071	141	17	,	,	PUNCT
cana-6071	141	18	(	(	PUNCT
cana-6071	141	19	2004	2004	NUM
cana-6071	141	20	)	)	PUNCT
cana-6071	141	21	.	.	PUNCT
cana-6071	142	1	[	[	X
cana-6071	142	2	5	5	X
cana-6071	142	3	]	]	PUNCT
cana-6071	142	4	b.	b.	PROPN
cana-6071	142	5	j.	j.	PROPN
cana-6071	142	6	gardner	gardner	PROPN
cana-6071	142	7	,	,	PUNCT
cana-6071	142	8	radicals	radical	NOUN
cana-6071	142	9	of	of	ADP
cana-6071	142	10	abelian	abelian	ADJ
cana-6071	142	11	groups	group	NOUN
cana-6071	142	12	and	and	CCONJ
cana-6071	142	13	associative	associative	ADJ
cana-6071	142	14	rings	ring	NOUN
cana-6071	142	15	,	,	PUNCT
cana-6071	142	16	acta	acta	PROPN
cana-6071	142	17	math	math	PROPN
cana-6071	142	18	.	.	PUNCT
cana-6071	143	1	acad	acad	PROPN
cana-6071	143	2	.	.	PUNCT
cana-6071	144	1	sci	sci	PROPN
cana-6071	144	2	.	.	PROPN
cana-6071	144	3	hung	hung	PROPN
cana-6071	144	4	.	.	PROPN
cana-6071	144	5	,	,	PUNCT
cana-6071	144	6	24(3	24(3	NUM
cana-6071	144	7	-	-	SYM
cana-6071	144	8	4	4	NUM
cana-6071	144	9	)	)	PUNCT
cana-6071	144	10	,	,	PUNCT
cana-6071	144	11	259	259	NUM
cana-6071	144	12	-	-	SYM
cana-6071	144	13	268	268	NUM
cana-6071	144	14	(	(	PUNCT
cana-6071	144	15	1973	1973	NUM
cana-6071	144	16	)	)	PUNCT
cana-6071	144	17	.	.	PUNCT
cana-6071	145	1	[	[	X
cana-6071	145	2	6	6	NUM
cana-6071	145	3	]	]	X
cana-6071	145	4	golan	golan	PROPN
cana-6071	145	5	j.	j.	PROPN
cana-6071	145	6	s.	s.	PROPN
cana-6071	145	7	semirings	semirings	PROPN
cana-6071	145	8	and	and	CCONJ
cana-6071	145	9	their	their	PRON
cana-6071	145	10	applications	application	NOUN
cana-6071	145	11	,	,	PUNCT
cana-6071	145	12	kluwer	kluwer	NOUN
cana-6071	145	13	academic	academic	ADJ
cana-6071	145	14	publisher	publisher	NOUN
cana-6071	145	15	(	(	PUNCT
cana-6071	145	16	1999	1999	NUM
cana-6071	145	17	)	)	PUNCT
cana-6071	145	18	.	.	PUNCT
cana-6071	146	1	[	[	X
cana-6071	146	2	7	7	X
cana-6071	146	3	]	]	X
cana-6071	146	4	n.	n.	PROPN
cana-6071	146	5	v.	v.	PROPN
cana-6071	146	6	loi	loi	PROPN
cana-6071	146	7	,	,	PUNCT
cana-6071	146	8	r.	r.	PROPN
cana-6071	146	9	wiegandt	wiegandt	PROPN
cana-6071	146	10	on	on	ADP
cana-6071	146	11	the	the	DET
cana-6071	146	12	amitsur	amitsur	ADJ
cana-6071	146	13	properties	property	NOUN
cana-6071	146	14	of	of	ADP
cana-6071	146	15	radicals	radical	NOUN
cana-6071	146	16	,	,	PUNCT
cana-6071	146	17	algebra	algebra	NOUN
cana-6071	146	18	and	and	CCONJ
cana-6071	146	19	discrete	discrete	ADJ
cana-6071	146	20	mathe	mathe	NOUN
cana-6071	146	21	matics	matic	NOUN
cana-6071	146	22	,	,	PUNCT
cana-6071	146	23	3	3	NUM
cana-6071	146	24	,	,	PUNCT
cana-6071	146	25	92	92	NUM
cana-6071	146	26	-	-	SYM
cana-6071	146	27	100	100	NUM
cana-6071	146	28	(	(	PUNCT
cana-6071	146	29	2006	2006	NUM
cana-6071	146	30	)	)	PUNCT
cana-6071	146	31	.	.	PUNCT
cana-6071	147	1	[	[	X
cana-6071	147	2	8	8	NUM
cana-6071	147	3	]	]	PUNCT
cana-6071	147	4	morak	morak	PROPN
cana-6071	147	5	b.	b.	PROPN
cana-6071	147	6	on	on	ADP
cana-6071	147	7	the	the	DET
cana-6071	147	8	radical	radical	ADJ
cana-6071	147	9	theory	theory	NOUN
cana-6071	147	10	of	of	ADP
cana-6071	147	11	semirings	semiring	NOUN
cana-6071	147	12	,	,	PUNCT
cana-6071	147	13	,	,	PUNCT
cana-6071	147	14	beitrage	beitrage	NOUN
cana-6071	147	15	alg	alg	PROPN
cana-6071	147	16	.	.	PUNCT
cana-6071	148	1	und	und	PROPN
cana-6071	148	2	.	.	PUNCT
cana-6071	149	1	geom	geom	PROPN
cana-6071	149	2	.	.	PROPN
cana-6071	149	3	,	,	PUNCT
cana-6071	149	4	40,533	40,533	NUM
cana-6071	149	5	-	-	SYM
cana-6071	149	6	549	549	NUM
cana-6071	149	7	(	(	PUNCT
cana-6071	149	8	1999	1999	NUM
cana-6071	149	9	)	)	PUNCT
cana-6071	149	10	.	.	PUNCT
cana-6071	150	1	[	[	X
cana-6071	150	2	9	9	NUM
cana-6071	150	3	]	]	X
cana-6071	150	4	olson	olson	NOUN
cana-6071	150	5	d.	d.	PROPN
cana-6071	150	6	m.	m.	PROPN
cana-6071	150	7	and	and	CCONJ
cana-6071	150	8	jenkins	jenkins	PROPN
cana-6071	150	9	t.	t.	PROPN
cana-6071	150	10	l.	l.	PROPN
cana-6071	150	11	radical	radical	PROPN
cana-6071	150	12	theory	theory	NOUN
cana-6071	150	13	for	for	ADP
cana-6071	150	14	hemirings	hemiring	NOUN
cana-6071	150	15	,	,	PUNCT
cana-6071	150	16	communications	communication	NOUN
cana-6071	150	17	on	on	ADP
cana-6071	150	18	applied	apply	VERB
cana-6071	150	19	nonlinear	nonlinear	ADJ
cana-6071	150	20	analysis	analysis	NOUN
cana-6071	150	21	issn	issn	NOUN
cana-6071	150	22	:	:	PUNCT
cana-6071	150	23	1074	1074	NUM
cana-6071	150	24	-	-	PUNCT
cana-6071	150	25	133x	133x	NUM
cana-6071	150	26	vol	vol	NOUN
cana-6071	150	27	31	31	NUM
cana-6071	150	28	no	no	NOUN
cana-6071	150	29	.	.	PUNCT
cana-6071	151	1	8s	8s	PROPN
cana-6071	151	2	(	(	PUNCT
cana-6071	151	3	2024	2024	NUM
cana-6071	151	4	)	)	PUNCT
cana-6071	151	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-6071	151	6	1183	1183	NUM
cana-6071	152	1	[	[	X
cana-6071	152	2	10	10	NUM
cana-6071	152	3	]	]	X
cana-6071	152	4	pawar	pawar	PROPN
cana-6071	152	5	k.	k.	PROPN
cana-6071	152	6	f.	f.	PROPN
cana-6071	152	7	deore	deore	PROPN
cana-6071	152	8	r.	r.	PROPN
cana-6071	152	9	p.	p.	PROPN
cana-6071	152	10	on	on	ADP
cana-6071	152	11	lower	low	ADJ
cana-6071	152	12	radical	radical	ADJ
cana-6071	152	13	constructions	construction	NOUN
cana-6071	152	14	in	in	ADP
cana-6071	152	15	non	non	ADJ
cana-6071	152	16	-	-	ADJ
cana-6071	152	17	associative	associative	ADJ
cana-6071	152	18	semirings	semiring	NOUN
cana-6071	152	19	,	,	PUNCT
cana-6071	152	20	int	int	NOUN
cana-6071	152	21	.	.	PUNCT
cana-6071	153	1	mathematical	mathematical	PROPN
cana-6071	153	2	forum	forum	PROPN
cana-6071	153	3	,	,	PUNCT
cana-6071	153	4	14(4	14(4	NUM
cana-6071	153	5	)	)	PUNCT
cana-6071	153	6	,	,	PUNCT
cana-6071	153	7	697	697	NUM
cana-6071	153	8	-	-	SYM
cana-6071	153	9	704	704	NUM
cana-6071	153	10	(	(	PUNCT
cana-6071	153	11	2009	2009	NUM
cana-6071	153	12	)	)	PUNCT
cana-6071	153	13	.	.	PUNCT
cana-6071	154	1	[	[	X
cana-6071	154	2	11	11	NUM
cana-6071	154	3	]	]	PUNCT
cana-6071	154	4	pawar	pawar	PROPN
cana-6071	154	5	k.	k.	PROPN
cana-6071	154	6	f.	f.	PROPN
cana-6071	154	7	radical	radical	PROPN
cana-6071	154	8	theory	theory	NOUN
cana-6071	154	9	for	for	ADP
cana-6071	154	10	associative	associative	ADJ
cana-6071	154	11	semirings	semiring	NOUN
cana-6071	154	12	and	and	CCONJ
cana-6071	154	13	structure	structure	NOUN
cana-6071	154	14	theorems	theorem	NOUN
cana-6071	154	15	,	,	PUNCT
cana-6071	154	16	unpublished	unpublished	ADJ
cana-6071	154	17	ph	ph	NOUN
cana-6071	154	18	.	.	PUNCT
cana-6071	155	1	d	d	X
cana-6071	155	2	thesis	thesis	NOUN
cana-6071	155	3	,	,	PUNCT
cana-6071	155	4	(	(	PUNCT
cana-6071	155	5	2011	2011	NUM
cana-6071	155	6	)	)	PUNCT
cana-6071	155	7	.	.	PUNCT
cana-6071	156	1	[	[	X
cana-6071	156	2	12	12	NUM
cana-6071	156	3	]	]	X
cana-6071	156	4	s.	s.	PROPN
cana-6071	156	5	tumurbat	tumurbat	PROPN
cana-6071	156	6	,	,	PUNCT
cana-6071	156	7	r.	r.	PROPN
cana-6071	156	8	wiegandt	wiegandt	PROPN
cana-6071	156	9	radicals	radical	NOUN
cana-6071	156	10	of	of	ADP
cana-6071	156	11	polynomial	polynomial	ADJ
cana-6071	156	12	rings	ring	NOUN
cana-6071	156	13	,	,	PUNCT
cana-6071	156	14	soochow	soochow	PROPN
cana-6071	156	15	j.	j.	PROPN
cana-6071	156	16	maths	maths	PROPN
cana-6071	156	17	.	.	PROPN
cana-6071	156	18	,	,	PUNCT
cana-6071	156	19	29	29	NUM
cana-6071	156	20	(	(	PUNCT
cana-6071	156	21	4	4	NUM
cana-6071	156	22	)	)	PUNCT
cana-6071	156	23	,	,	PUNCT
cana-6071	156	24	425	425	NUM
cana-6071	156	25	-	-	SYM
cana-6071	156	26	434	434	NUM
cana-6071	156	27	(	(	PUNCT
cana-6071	156	28	2003	2003	NUM
cana-6071	156	29	)	)	PUNCT
cana-6071	156	30	.	.	PUNCT