id	sid	tid	token	lemma	pos
ejpam-1797	1	1	1_dutta.dvi	1_dutta.dvi	NUM
ejpam-1797	1	2	european	european	ADJ
ejpam-1797	1	3	journal	journal	NOUN
ejpam-1797	1	4	of	of	ADP
ejpam-1797	1	5	pure	pure	ADJ
ejpam-1797	1	6	and	and	CCONJ
ejpam-1797	1	7	applied	apply	VERB
ejpam-1797	1	8	mathematics	mathematic	NOUN
ejpam-1797	1	9	vol	vol	NOUN
ejpam-1797	1	10	.	.	PROPN
ejpam-1797	1	11	5	5	NUM
ejpam-1797	1	12	,	,	PUNCT
ejpam-1797	1	13	no	no	INTJ
ejpam-1797	1	14	.	.	NOUN
ejpam-1797	1	15	4	4	NUM
ejpam-1797	1	16	,	,	PUNCT
ejpam-1797	1	17	2012	2012	NUM
ejpam-1797	1	18	,	,	PUNCT
ejpam-1797	1	19	401	401	NUM
ejpam-1797	1	20	-	-	SYM
ejpam-1797	1	21	413	413	NUM
ejpam-1797	1	22	issn	issn	PROPN
ejpam-1797	1	23	1307	1307	NUM
ejpam-1797	1	24	-	-	SYM
ejpam-1797	1	25	5543	5543	NUM
ejpam-1797	1	26	–	–	PUNCT
ejpam-1797	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-1797	1	28	weakly	weakly	ADJ
ejpam-1797	1	29	special	special	ADJ
ejpam-1797	1	30	radical	radical	ADJ
ejpam-1797	1	31	class	class	NOUN
ejpam-1797	1	32	and	and	CCONJ
ejpam-1797	1	33	special	special	ADJ
ejpam-1797	1	34	radical	radical	ADJ
ejpam-1797	1	35	class	class	NOUN
ejpam-1797	1	36	of	of	ADP
ejpam-1797	1	37	ternary	ternary	ADJ
ejpam-1797	1	38	semirings	semiring	NOUN
ejpam-1797	1	39	tapan	tapan	VERB
ejpam-1797	1	40	k.	k.	PROPN
ejpam-1797	1	41	dutta1	dutta1	PROPN
ejpam-1797	1	42	,	,	PUNCT
ejpam-1797	1	43	kar	kar	PROPN
ejpam-1797	1	44	ping	ping	PROPN
ejpam-1797	1	45	shum	shum	PROPN
ejpam-1797	1	46	2,∗	2,∗	PROPN
ejpam-1797	1	47	,	,	PUNCT
ejpam-1797	1	48	shobhan	shobhan	ADV
ejpam-1797	1	49	mandal1	mandal1	ADJ
ejpam-1797	1	50	1	1	NUM
ejpam-1797	1	51	department	department	NOUN
ejpam-1797	1	52	of	of	ADP
ejpam-1797	1	53	pure	pure	ADJ
ejpam-1797	1	54	mathematics	mathematic	NOUN
ejpam-1797	1	55	,	,	PUNCT
ejpam-1797	1	56	university	university	NOUN
ejpam-1797	1	57	of	of	ADP
ejpam-1797	1	58	calcutta	calcutta	PROPN
ejpam-1797	1	59	,	,	PUNCT
ejpam-1797	1	60	35	35	NUM
ejpam-1797	1	61	,	,	PUNCT
ejpam-1797	1	62	ballygunge	ballygunge	VERB
ejpam-1797	1	63	circular	circular	ADJ
ejpam-1797	1	64	road	road	NOUN
ejpam-1797	1	65	,	,	PUNCT
ejpam-1797	1	66	kolkata710019,india	kolkata710019,india	PROPN
ejpam-1797	1	67	2	2	NUM
ejpam-1797	1	68	institute	institute	PROPN
ejpam-1797	1	69	of	of	ADP
ejpam-1797	1	70	mathematics	mathematics	PROPN
ejpam-1797	1	71	,	,	PUNCT
ejpam-1797	1	72	yunnan	yunnan	PROPN
ejpam-1797	1	73	university	university	PROPN
ejpam-1797	1	74	,	,	PUNCT
ejpam-1797	1	75	kunming	kunming	NOUN
ejpam-1797	1	76	,	,	PUNCT
ejpam-1797	1	77	650091	650091	NUM
ejpam-1797	1	78	,	,	PUNCT
ejpam-1797	1	79	people	people	NOUN
ejpam-1797	1	80	republic	republic	NOUN
ejpam-1797	1	81	of	of	ADP
ejpam-1797	1	82	china	china	PROPN
ejpam-1797	1	83	abstract	abstract	PROPN
ejpam-1797	1	84	.	.	PUNCT
ejpam-1797	2	1	in	in	ADP
ejpam-1797	2	2	this	this	DET
ejpam-1797	2	3	paper	paper	NOUN
ejpam-1797	2	4	,	,	PUNCT
ejpam-1797	2	5	we	we	PRON
ejpam-1797	2	6	consider	consider	VERB
ejpam-1797	2	7	the	the	DET
ejpam-1797	2	8	weakly	weakly	ADJ
ejpam-1797	2	9	special	special	ADJ
ejpam-1797	2	10	radical	radical	ADJ
ejpam-1797	2	11	classes	class	NOUN
ejpam-1797	2	12	and	and	CCONJ
ejpam-1797	2	13	special	special	ADJ
ejpam-1797	2	14	radical	radical	ADJ
ejpam-1797	2	15	classes	class	NOUN
ejpam-1797	2	16	of	of	ADP
ejpam-1797	2	17	ternary	ternary	ADJ
ejpam-1797	2	18	semirings	semiring	NOUN
ejpam-1797	2	19	.	.	PUNCT
ejpam-1797	3	1	some	some	PRON
ejpam-1797	3	2	of	of	ADP
ejpam-1797	3	3	our	our	PRON
ejpam-1797	3	4	results	result	NOUN
ejpam-1797	3	5	are	be	AUX
ejpam-1797	3	6	similar	similar	ADJ
ejpam-1797	3	7	to	to	ADP
ejpam-1797	3	8	those	those	PRON
ejpam-1797	3	9	in	in	ADP
ejpam-1797	3	10	rings	ring	NOUN
ejpam-1797	3	11	theory	theory	NOUN
ejpam-1797	3	12	as	as	ADV
ejpam-1797	3	13	well	well	ADV
ejpam-1797	3	14	as	as	ADP
ejpam-1797	3	15	in	in	ADP
ejpam-1797	3	16	semiring	semire	VERB
ejpam-1797	3	17	theory	theory	NOUN
ejpam-1797	3	18	.	.	PUNCT
ejpam-1797	4	1	in	in	ADP
ejpam-1797	4	2	particular	particular	ADJ
ejpam-1797	4	3	,	,	PUNCT
ejpam-1797	4	4	the	the	DET
ejpam-1797	4	5	upper	upper	ADJ
ejpam-1797	4	6	radicals	radical	NOUN
ejpam-1797	4	7	of	of	ADP
ejpam-1797	4	8	the	the	DET
ejpam-1797	4	9	above	above	ADJ
ejpam-1797	4	10	two	two	NUM
ejpam-1797	4	11	classes	class	NOUN
ejpam-1797	4	12	are	be	AUX
ejpam-1797	4	13	determined	determine	VERB
ejpam-1797	4	14	.	.	PUNCT
ejpam-1797	5	1	2010	2010	NUM
ejpam-1797	5	2	mathematics	mathematic	NOUN
ejpam-1797	5	3	subject	subject	NOUN
ejpam-1797	5	4	classifications	classification	NOUN
ejpam-1797	5	5	:	:	PUNCT
ejpam-1797	5	6	16y60	16y60	NUM
ejpam-1797	5	7	,	,	PUNCT
ejpam-1797	5	8	16y99	16y99	NUM
ejpam-1797	5	9	key	key	ADJ
ejpam-1797	5	10	words	word	NOUN
ejpam-1797	5	11	and	and	CCONJ
ejpam-1797	5	12	phrases	phrase	NOUN
ejpam-1797	5	13	:	:	PUNCT
ejpam-1797	5	14	weakly	weakly	ADJ
ejpam-1797	5	15	special	special	ADJ
ejpam-1797	5	16	radical	radical	ADJ
ejpam-1797	5	17	classes	class	NOUN
ejpam-1797	5	18	,	,	PUNCT
ejpam-1797	5	19	special	special	ADJ
ejpam-1797	5	20	radical	radical	ADJ
ejpam-1797	5	21	classes	class	NOUN
ejpam-1797	5	22	,	,	PUNCT
ejpam-1797	5	23	singular	singular	ADJ
ejpam-1797	5	24	radicals	radical	NOUN
ejpam-1797	5	25	,	,	PUNCT
ejpam-1797	5	26	special	special	ADJ
ejpam-1797	5	27	singular	singular	ADJ
ejpam-1797	5	28	radical	radical	NOUN
ejpam-1797	5	29	of	of	ADP
ejpam-1797	5	30	ternary	ternary	ADJ
ejpam-1797	5	31	semirings	semiring	NOUN
ejpam-1797	5	32	.	.	PUNCT
ejpam-1797	6	1	1	1	X
ejpam-1797	6	2	.	.	X
ejpam-1797	6	3	introduction	introduction	NOUN
ejpam-1797	6	4	in	in	ADP
ejpam-1797	6	5	this	this	DET
ejpam-1797	6	6	paper	paper	NOUN
ejpam-1797	6	7	,	,	PUNCT
ejpam-1797	6	8	we	we	PRON
ejpam-1797	6	9	consider	consider	VERB
ejpam-1797	6	10	“	"	PUNCT
ejpam-1797	6	11	the	the	DET
ejpam-1797	6	12	singular	singular	ADJ
ejpam-1797	6	13	ideal	ideal	NOUN
ejpam-1797	6	14	of	of	ADP
ejpam-1797	6	15	a	a	DET
ejpam-1797	6	16	ternary	ternary	ADJ
ejpam-1797	6	17	semiring	semiring	NOUN
ejpam-1797	6	18	”	"	PUNCT
ejpam-1797	6	19	mentioned	mention	VERB
ejpam-1797	6	20	in	in	ADP
ejpam-1797	6	21	[	[	X
ejpam-1797	6	22	3	3	NUM
ejpam-1797	6	23	]	]	PUNCT
ejpam-1797	6	24	.	.	PUNCT
ejpam-1797	7	1	the	the	DET
ejpam-1797	7	2	notion	notion	NOUN
ejpam-1797	7	3	of	of	ADP
ejpam-1797	7	4	a	a	DET
ejpam-1797	7	5	ternary	ternary	ADJ
ejpam-1797	7	6	semiring	semiring	NOUN
ejpam-1797	7	7	was	be	AUX
ejpam-1797	7	8	first	first	ADV
ejpam-1797	7	9	introduced	introduce	VERB
ejpam-1797	7	10	by	by	ADP
ejpam-1797	7	11	t.	t.	PROPN
ejpam-1797	7	12	k.	k.	PROPN
ejpam-1797	7	13	dutta	dutta	PROPN
ejpam-1797	7	14	and	and	CCONJ
ejpam-1797	7	15	s.	s.	PROPN
ejpam-1797	7	16	kar	kar	PROPN
ejpam-1797	7	17	in	in	ADP
ejpam-1797	7	18	[	[	X
ejpam-1797	7	19	3	3	NUM
ejpam-1797	7	20	]	]	PUNCT
ejpam-1797	7	21	.	.	PUNCT
ejpam-1797	8	1	subsequently	subsequently	ADV
ejpam-1797	8	2	,	,	PUNCT
ejpam-1797	8	3	many	many	ADJ
ejpam-1797	8	4	related	related	ADJ
ejpam-1797	8	5	notions	notion	NOUN
ejpam-1797	8	6	of	of	ADP
ejpam-1797	8	7	semiring	semiring	NOUN
ejpam-1797	8	8	and	and	CCONJ
ejpam-1797	8	9	ring	ring	NOUN
ejpam-1797	8	10	have	have	AUX
ejpam-1797	8	11	been	be	AUX
ejpam-1797	8	12	generalized	generalize	VERB
ejpam-1797	8	13	to	to	ADP
ejpam-1797	8	14	ternary	ternary	ADJ
ejpam-1797	8	15	semirings	semiring	NOUN
ejpam-1797	8	16	.	.	PUNCT
ejpam-1797	9	1	some	some	DET
ejpam-1797	9	2	earlier	early	ADJ
ejpam-1797	9	3	works	work	NOUN
ejpam-1797	9	4	of	of	ADP
ejpam-1797	9	5	ternary	ternary	ADJ
ejpam-1797	9	6	semiring	semiring	NOUN
ejpam-1797	9	7	may	may	AUX
ejpam-1797	9	8	be	be	AUX
ejpam-1797	9	9	found	find	VERB
ejpam-1797	9	10	in	in	ADP
ejpam-1797	9	11	[	[	X
ejpam-1797	9	12	4]-[12	4]-[12	NOUN
ejpam-1797	9	13	]	]	PUNCT
ejpam-1797	9	14	and	and	CCONJ
ejpam-1797	9	15	[	[	X
ejpam-1797	9	16	13	13	NUM
ejpam-1797	9	17	,	,	PUNCT
ejpam-1797	9	18	14	14	NUM
ejpam-1797	9	19	,	,	PUNCT
ejpam-1797	9	20	15	15	NUM
ejpam-1797	9	21	,	,	PUNCT
ejpam-1797	9	22	16	16	NUM
ejpam-1797	9	23	]	]	PUNCT
ejpam-1797	9	24	.	.	PUNCT
ejpam-1797	10	1	the	the	DET
ejpam-1797	10	2	partitioning	partitioning	NOUN
ejpam-1797	10	3	and	and	CCONJ
ejpam-1797	10	4	subtractive	subtractive	ADJ
ejpam-1797	10	5	ideals	ideal	NOUN
ejpam-1797	10	6	of	of	ADP
ejpam-1797	10	7	ternary	ternary	ADJ
ejpam-1797	10	8	semirings	semiring	NOUN
ejpam-1797	10	9	were	be	AUX
ejpam-1797	10	10	considered	consider	VERB
ejpam-1797	10	11	by	by	ADP
ejpam-1797	10	12	j.	j.	PROPN
ejpam-1797	10	13	n.	n.	PROPN
ejpam-1797	10	14	chaudhari	chaudhari	PROPN
ejpam-1797	10	15	and	and	CCONJ
ejpam-1797	10	16	k.	k.	PROPN
ejpam-1797	10	17	j.	j.	PROPN
ejpam-1797	10	18	ingale	ingale	PROPN
ejpam-1797	10	19	in	in	ADP
ejpam-1797	10	20	[	[	X
ejpam-1797	10	21	2	2	NUM
ejpam-1797	10	22	]	]	PUNCT
ejpam-1797	10	23	.	.	PUNCT
ejpam-1797	11	1	for	for	ADP
ejpam-1797	11	2	the	the	DET
ejpam-1797	11	3	general	general	ADJ
ejpam-1797	11	4	radical	radical	ADJ
ejpam-1797	11	5	theory	theory	NOUN
ejpam-1797	11	6	of	of	ADP
ejpam-1797	11	7	rings	ring	NOUN
ejpam-1797	11	8	,	,	PUNCT
ejpam-1797	11	9	the	the	DET
ejpam-1797	11	10	reader	reader	NOUN
ejpam-1797	11	11	is	be	AUX
ejpam-1797	11	12	referred	refer	VERB
ejpam-1797	11	13	to	to	ADP
ejpam-1797	11	14	the	the	DET
ejpam-1797	11	15	classical	classical	ADJ
ejpam-1797	11	16	monograph	monograph	NOUN
ejpam-1797	11	17	of	of	ADP
ejpam-1797	11	18	n.	n.	PROPN
ejpam-1797	11	19	j.	j.	PROPN
ejpam-1797	11	20	divinsky	divinsky	PROPN
ejpam-1797	12	1	[	[	X
ejpam-1797	12	2	17	17	NUM
ejpam-1797	12	3	]	]	PUNCT
ejpam-1797	12	4	.	.	PUNCT
ejpam-1797	13	1	for	for	ADP
ejpam-1797	13	2	definitions	definition	NOUN
ejpam-1797	13	3	and	and	CCONJ
ejpam-1797	13	4	properties	property	NOUN
ejpam-1797	13	5	of	of	ADP
ejpam-1797	13	6	ideals	ideal	NOUN
ejpam-1797	13	7	,	,	PUNCT
ejpam-1797	13	8	homomorphism	homomorphism	NOUN
ejpam-1797	13	9	,	,	PUNCT
ejpam-1797	13	10	quotient	quotient	NOUN
ejpam-1797	13	11	for	for	ADP
ejpam-1797	13	12	ternary	ternary	ADJ
ejpam-1797	13	13	semirings	semiring	NOUN
ejpam-1797	13	14	,	,	PUNCT
ejpam-1797	13	15	singular	singular	ADJ
ejpam-1797	13	16	ideals	ideal	NOUN
ejpam-1797	13	17	,	,	PUNCT
ejpam-1797	13	18	singular	singular	ADJ
ejpam-1797	13	19	ternary	ternary	ADJ
ejpam-1797	13	20	semirings	semiring	NOUN
ejpam-1797	13	21	,	,	PUNCT
ejpam-1797	13	22	non	non	ADJ
ejpam-1797	13	23	-	-	ADJ
ejpam-1797	13	24	singular	singular	ADJ
ejpam-1797	13	25	ternary	ternary	ADJ
ejpam-1797	13	26	semiring	semiring	NOUN
ejpam-1797	13	27	,	,	PUNCT
ejpam-1797	13	28	the	the	DET
ejpam-1797	13	29	reader	reader	NOUN
ejpam-1797	13	30	is	be	AUX
ejpam-1797	13	31	referred	refer	VERB
ejpam-1797	13	32	to	to	ADP
ejpam-1797	13	33	[	[	X
ejpam-1797	13	34	2	2	NUM
ejpam-1797	13	35	]	]	PUNCT
ejpam-1797	13	36	.	.	PUNCT
ejpam-1797	14	1	the	the	DET
ejpam-1797	14	2	concepts	concept	NOUN
ejpam-1797	14	3	of	of	ADP
ejpam-1797	14	4	radical	radical	ADJ
ejpam-1797	14	5	class	class	NOUN
ejpam-1797	14	6	for	for	ADP
ejpam-1797	14	7	hemirings	hemiring	NOUN
ejpam-1797	14	8	were	be	AUX
ejpam-1797	14	9	given	give	VERB
ejpam-1797	14	10	by	by	ADP
ejpam-1797	14	11	d.	d.	PROPN
ejpam-1797	14	12	m.	m.	PROPN
ejpam-1797	14	13	olson	olson	PROPN
ejpam-1797	14	14	and	and	CCONJ
ejpam-1797	14	15	t.	t.	PROPN
ejpam-1797	14	16	l.	l.	PROPN
ejpam-1797	14	17	jenkins	jenkins	PROPN
ejpam-1797	14	18	in	in	ADP
ejpam-1797	14	19	1983	1983	NUM
ejpam-1797	14	20	,	,	PUNCT
ejpam-1797	14	21	see	see	VERB
ejpam-1797	14	22	[	[	X
ejpam-1797	14	23	27	27	NUM
ejpam-1797	14	24	]	]	PUNCT
ejpam-1797	14	25	.	.	PUNCT
ejpam-1797	15	1	moreover	moreover	ADV
ejpam-1797	15	2	,	,	PUNCT
ejpam-1797	15	3	the	the	DET
ejpam-1797	15	4	general	general	ADJ
ejpam-1797	15	5	theory	theory	NOUN
ejpam-1797	15	6	to	to	ADP
ejpam-1797	15	7	upper	upper	ADJ
ejpam-1797	15	8	radicals	radical	NOUN
ejpam-1797	15	9	was	be	AUX
ejpam-1797	15	10	extended	extend	VERB
ejpam-1797	15	11	by	by	ADP
ejpam-1797	15	12	a.	a.	NOUN
ejpam-1797	15	13	c.	c.	PROPN
ejpam-1797	15	14	nance	nance	NOUN
ejpam-1797	15	15	in	in	ADP
ejpam-1797	15	16	[	[	X
ejpam-1797	15	17	27	27	NUM
ejpam-1797	15	18	]	]	PUNCT
ejpam-1797	15	19	and	and	CCONJ
ejpam-1797	15	20	the	the	DET
ejpam-1797	15	21	special	special	ADJ
ejpam-1797	15	22	radical	radical	ADJ
ejpam-1797	15	23	classes	class	NOUN
ejpam-1797	15	24	and	and	CCONJ
ejpam-1797	15	25	properties	property	NOUN
ejpam-1797	15	26	of	of	ADP
ejpam-1797	15	27	special	special	ADJ
ejpam-1797	15	28	radicals	radical	NOUN
ejpam-1797	15	29	were	be	AUX
ejpam-1797	15	30	investigated	investigate	VERB
ejpam-1797	15	31	by	by	ADP
ejpam-1797	15	32	m.	m.	PROPN
ejpam-1797	15	33	d.	d.	PROPN
ejpam-1797	15	34	olson	olson	PROPN
ejpam-1797	15	35	,	,	PUNCT
ejpam-1797	15	36	g.a.p	g.a.p	PROPN
ejpam-1797	15	37	.	.	PUNCT
ejpam-1797	15	38	heyman	heyman	PROPN
ejpam-1797	15	39	and	and	CCONJ
ejpam-1797	15	40	h.	h.	PROPN
ejpam-1797	15	41	j.	j.	PROPN
ejpam-1797	15	42	l.	l.	PROPN
ejpam-1797	15	43	roux	roux	PROPN
ejpam-1797	15	44	in	in	ADP
ejpam-1797	15	45	[	[	X
ejpam-1797	15	46	26	26	NUM
ejpam-1797	15	47	]	]	PUNCT
ejpam-1797	15	48	.	.	PUNCT
ejpam-1797	16	1	the	the	DET
ejpam-1797	16	2	properties	property	NOUN
ejpam-1797	16	3	of	of	ADP
ejpam-1797	16	4	the	the	DET
ejpam-1797	16	5	weakly	weakly	ADJ
ejpam-1797	16	6	special	special	ADJ
ejpam-1797	16	7	radical	radical	ADJ
ejpam-1797	16	8	class	class	NOUN
ejpam-1797	16	9	of	of	ADP
ejpam-1797	16	10	hemirings	hemiring	NOUN
ejpam-1797	16	11	were	be	AUX
ejpam-1797	16	12	also	also	ADV
ejpam-1797	16	13	studied	study	VERB
ejpam-1797	16	14	by	by	ADP
ejpam-1797	16	15	the	the	DET
ejpam-1797	16	16	above	above	ADJ
ejpam-1797	16	17	authors	author	NOUN
ejpam-1797	16	18	.	.	PUNCT
ejpam-1797	17	1	∗corresponding	∗corresponde	VERB
ejpam-1797	17	2	author	author	NOUN
ejpam-1797	17	3	.	.	PUNCT
ejpam-1797	18	1	email	email	NOUN
ejpam-1797	18	2	addresses	address	NOUN
ejpam-1797	18	3	:	:	PUNCT
ejpam-1797	19	1	duttatapankumar	duttatapankumar	PROPN
ejpam-1797	19	2	�	�	PROPN
ejpam-1797	19	3	yahoo	yahoo	PROPN
ejpam-1797	19	4	.	.	PUNCT
ejpam-1797	20	1	o.in	o.in	PROPN
ejpam-1797	20	2	(	(	PUNCT
ejpam-1797	20	3	t.	t.	PROPN
ejpam-1797	20	4	dutta	dutta	PROPN
ejpam-1797	20	5	)	)	PUNCT
ejpam-1797	20	6	,	,	PUNCT
ejpam-1797	20	7	kpshum�ynu.edu	kpshum�ynu.edu	PROPN
ejpam-1797	20	8	.	.	PROPN
ejpam-1797	21	1	n	n	PROPN
ejpam-1797	21	2	(	(	PUNCT
ejpam-1797	21	3	k.	k.	PROPN
ejpam-1797	21	4	shum	shum	PROPN
ejpam-1797	21	5	)	)	PUNCT
ejpam-1797	21	6	,	,	PUNCT
ejpam-1797	21	7	shobhan139	shobhan139	PROPN
ejpam-1797	21	8	�	�	NOUN
ejpam-1797	21	9	gmail	gmail	NOUN
ejpam-1797	21	10	.	.	PUNCT
ejpam-1797	22	1	om	om	PROPN
ejpam-1797	22	2	(	(	PUNCT
ejpam-1797	22	3	s.	s.	PROPN
ejpam-1797	22	4	mandal	mandal	PROPN
ejpam-1797	22	5	)	)	PUNCT
ejpam-1797	22	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1797	23	1	401	401	NUM
ejpam-1797	23	2	c	c	X
ejpam-1797	23	3	©	©	PROPN
ejpam-1797	23	4	2012	2012	NUM
ejpam-1797	23	5	ejpam	ejpam	VERB
ejpam-1797	23	6	all	all	DET
ejpam-1797	23	7	rights	right	NOUN
ejpam-1797	23	8	reserved	reserve	VERB
ejpam-1797	23	9	.	.	PUNCT
ejpam-1797	24	1	t.	t.	PROPN
ejpam-1797	24	2	dutta	dutta	PROPN
ejpam-1797	24	3	,	,	PUNCT
ejpam-1797	24	4	k.	k.	PROPN
ejpam-1797	24	5	shum	shum	PROPN
ejpam-1797	24	6	,	,	PUNCT
ejpam-1797	24	7	s.	s.	PROPN
ejpam-1797	24	8	mandal	mandal	PROPN
ejpam-1797	24	9	/	/	SYM
ejpam-1797	24	10	eur	eur	PROPN
ejpam-1797	24	11	.	.	PUNCT
ejpam-1797	25	1	j.	j.	PROPN
ejpam-1797	25	2	pure	pure	PROPN
ejpam-1797	25	3	appl	appl	PROPN
ejpam-1797	25	4	.	.	PROPN
ejpam-1797	25	5	math	math	PROPN
ejpam-1797	25	6	,	,	PUNCT
ejpam-1797	25	7	5	5	NUM
ejpam-1797	25	8	(	(	PUNCT
ejpam-1797	25	9	2012	2012	NUM
ejpam-1797	25	10	)	)	PUNCT
ejpam-1797	25	11	,	,	PUNCT
ejpam-1797	25	12	401	401	NUM
ejpam-1797	25	13	-	-	SYM
ejpam-1797	25	14	413	413	NUM
ejpam-1797	25	15	402	402	NUM
ejpam-1797	25	16	in	in	ADP
ejpam-1797	25	17	this	this	DET
ejpam-1797	25	18	paper	paper	NOUN
ejpam-1797	25	19	,	,	PUNCT
ejpam-1797	25	20	we	we	PRON
ejpam-1797	25	21	first	first	ADV
ejpam-1797	25	22	extend	extend	VERB
ejpam-1797	25	23	the	the	DET
ejpam-1797	25	24	above	above	ADJ
ejpam-1797	25	25	notions	notion	NOUN
ejpam-1797	25	26	to	to	ADP
ejpam-1797	25	27	ternary	ternary	ADJ
ejpam-1797	25	28	semirings	semiring	NOUN
ejpam-1797	25	29	.	.	PUNCT
ejpam-1797	26	1	then	then	ADV
ejpam-1797	26	2	,	,	PUNCT
ejpam-1797	26	3	we	we	PRON
ejpam-1797	26	4	define	define	VERB
ejpam-1797	26	5	again	again	ADV
ejpam-1797	26	6	the	the	DET
ejpam-1797	26	7	weakly	weakly	ADJ
ejpam-1797	26	8	special	special	ADJ
ejpam-1797	26	9	radical	radical	ADJ
ejpam-1797	26	10	class	class	NOUN
ejpam-1797	26	11	and	and	CCONJ
ejpam-1797	26	12	special	special	ADJ
ejpam-1797	26	13	radical	radical	ADJ
ejpam-1797	26	14	class	class	NOUN
ejpam-1797	26	15	for	for	ADP
ejpam-1797	26	16	ternary	ternary	ADJ
ejpam-1797	26	17	semirings	semiring	NOUN
ejpam-1797	26	18	.	.	PUNCT
ejpam-1797	27	1	we	we	PRON
ejpam-1797	27	2	will	will	AUX
ejpam-1797	27	3	show	show	VERB
ejpam-1797	27	4	that	that	SCONJ
ejpam-1797	27	5	the	the	DET
ejpam-1797	27	6	class	class	NOUN
ejpam-1797	27	7	of	of	ADP
ejpam-1797	27	8	all	all	DET
ejpam-1797	27	9	semiprime	semiprime	NOUN
ejpam-1797	27	10	and	and	CCONJ
ejpam-1797	27	11	non	non	ADJ
ejpam-1797	27	12	singular	singular	ADJ
ejpam-1797	27	13	ternary	ternary	ADJ
ejpam-1797	27	14	semirings	semiring	NOUN
ejpam-1797	27	15	forms	form	VERB
ejpam-1797	27	16	a	a	DET
ejpam-1797	27	17	weakly	weakly	ADJ
ejpam-1797	27	18	special	special	ADJ
ejpam-1797	27	19	radical	radical	ADJ
ejpam-1797	27	20	class	class	NOUN
ejpam-1797	27	21	whereas	whereas	SCONJ
ejpam-1797	27	22	the	the	DET
ejpam-1797	27	23	class	class	NOUN
ejpam-1797	27	24	of	of	ADP
ejpam-1797	27	25	all	all	DET
ejpam-1797	27	26	prime	prime	ADJ
ejpam-1797	27	27	non	non	ADJ
ejpam-1797	27	28	-	-	ADJ
ejpam-1797	27	29	singular	singular	ADJ
ejpam-1797	27	30	ternary	ternary	ADJ
ejpam-1797	27	31	semirings	semiring	NOUN
ejpam-1797	27	32	forms	form	VERB
ejpam-1797	27	33	a	a	DET
ejpam-1797	27	34	special	special	ADJ
ejpam-1797	27	35	radical	radical	ADJ
ejpam-1797	27	36	class	class	NOUN
ejpam-1797	27	37	.	.	PUNCT
ejpam-1797	28	1	as	as	ADP
ejpam-1797	28	2	a	a	DET
ejpam-1797	28	3	consequence	consequence	NOUN
ejpam-1797	28	4	of	of	ADP
ejpam-1797	28	5	this	this	DET
ejpam-1797	28	6	result	result	NOUN
ejpam-1797	28	7	,	,	PUNCT
ejpam-1797	28	8	the	the	DET
ejpam-1797	28	9	upper	upper	ADJ
ejpam-1797	28	10	radicals	radical	NOUN
ejpam-1797	28	11	will	will	AUX
ejpam-1797	28	12	be	be	AUX
ejpam-1797	28	13	determined	determine	VERB
ejpam-1797	28	14	by	by	ADP
ejpam-1797	28	15	the	the	DET
ejpam-1797	28	16	above	above	ADJ
ejpam-1797	28	17	two	two	NUM
ejpam-1797	28	18	classes	class	NOUN
ejpam-1797	28	19	which	which	PRON
ejpam-1797	28	20	are	be	AUX
ejpam-1797	28	21	called	call	VERB
ejpam-1797	28	22	the	the	DET
ejpam-1797	28	23	singular	singular	ADJ
ejpam-1797	28	24	radical	radical	ADJ
ejpam-1797	28	25	and	and	CCONJ
ejpam-1797	28	26	special	special	ADJ
ejpam-1797	28	27	singular	singular	ADJ
ejpam-1797	28	28	radical	radical	NOUN
ejpam-1797	28	29	of	of	ADP
ejpam-1797	28	30	ternary	ternary	ADJ
ejpam-1797	28	31	semirings	semiring	NOUN
ejpam-1797	28	32	,	,	PUNCT
ejpam-1797	28	33	respectively	respectively	ADV
ejpam-1797	28	34	.	.	PUNCT
ejpam-1797	29	1	for	for	ADP
ejpam-1797	29	2	the	the	DET
ejpam-1797	29	3	radical	radical	ADJ
ejpam-1797	29	4	properties	property	NOUN
ejpam-1797	29	5	,	,	PUNCT
ejpam-1797	29	6	the	the	DET
ejpam-1797	29	7	reader	reader	NOUN
ejpam-1797	29	8	is	be	AUX
ejpam-1797	29	9	referred	refer	VERB
ejpam-1797	29	10	to	to	ADP
ejpam-1797	29	11	the	the	DET
ejpam-1797	29	12	well	well	ADV
ejpam-1797	29	13	known	know	VERB
ejpam-1797	29	14	monograph	monograph	NOUN
ejpam-1797	29	15	of	of	ADP
ejpam-1797	29	16	n.	n.	PROPN
ejpam-1797	29	17	j.	j.	PROPN
ejpam-1797	29	18	divinsky	divinsky	PROPN
ejpam-1797	29	19	.	.	PUNCT
ejpam-1797	30	1	throughout	throughout	ADP
ejpam-1797	30	2	this	this	DET
ejpam-1797	30	3	paper	paper	NOUN
ejpam-1797	30	4	,	,	PUNCT
ejpam-1797	30	5	s	s	VERB
ejpam-1797	30	6	will	will	AUX
ejpam-1797	30	7	be	be	AUX
ejpam-1797	30	8	used	use	VERB
ejpam-1797	30	9	to	to	PART
ejpam-1797	30	10	denote	denote	VERB
ejpam-1797	30	11	a	a	DET
ejpam-1797	30	12	ternary	ternary	ADJ
ejpam-1797	30	13	semiring	semiring	NOUN
ejpam-1797	30	14	with	with	ADP
ejpam-1797	30	15	zero	zero	NUM
ejpam-1797	30	16	and	and	CCONJ
ejpam-1797	30	17	s∗	s∗	PROPN
ejpam-1797	30	18	=	=	SYM
ejpam-1797	30	19	s	s	PART
ejpam-1797	30	20	\	\	X
ejpam-1797	30	21	{	{	PUNCT
ejpam-1797	30	22	0	0	NUM
ejpam-1797	30	23	}	}	PUNCT
ejpam-1797	30	24	.	.	PUNCT
ejpam-1797	31	1	also	also	ADV
ejpam-1797	31	2	we	we	PRON
ejpam-1797	31	3	use	use	VERB
ejpam-1797	31	4	m	m	PRON
ejpam-1797	31	5	to	to	PART
ejpam-1797	31	6	denote	denote	VERB
ejpam-1797	31	7	a	a	DET
ejpam-1797	31	8	right	right	ADJ
ejpam-1797	31	9	ternary	ternary	ADJ
ejpam-1797	31	10	s	s	NOUN
ejpam-1797	31	11	-	-	PUNCT
ejpam-1797	31	12	semimodule	semimodule	NOUN
ejpam-1797	31	13	with	with	ADP
ejpam-1797	31	14	zero	zero	NUM
ejpam-1797	31	15	.	.	PUNCT
ejpam-1797	32	1	2	2	X
ejpam-1797	32	2	.	.	X
ejpam-1797	32	3	the	the	DET
ejpam-1797	32	4	radical	radical	ADJ
ejpam-1797	32	5	class	class	NOUN
ejpam-1797	32	6	,	,	PUNCT
ejpam-1797	32	7	weakly	weakly	ADJ
ejpam-1797	32	8	special	special	ADJ
ejpam-1797	32	9	radical	radical	ADJ
ejpam-1797	32	10	class	class	NOUN
ejpam-1797	32	11	and	and	CCONJ
ejpam-1797	32	12	special	special	ADJ
ejpam-1797	32	13	radical	radical	ADJ
ejpam-1797	32	14	class	class	NOUN
ejpam-1797	32	15	let	let	VERB
ejpam-1797	32	16	s	s	PRON
ejpam-1797	32	17	be	be	AUX
ejpam-1797	32	18	a	a	DET
ejpam-1797	32	19	ternary	ternary	ADJ
ejpam-1797	32	20	semiring	semiring	NOUN
ejpam-1797	32	21	and	and	CCONJ
ejpam-1797	32	22	m	m	VERB
ejpam-1797	32	23	a	a	DET
ejpam-1797	32	24	right	right	ADJ
ejpam-1797	32	25	ternary	ternary	ADJ
ejpam-1797	32	26	s	s	NOUN
ejpam-1797	32	27	-	-	NOUN
ejpam-1797	32	28	semimodule	semimodule	NOUN
ejpam-1797	32	29	.	.	PUNCT
ejpam-1797	33	1	we	we	PRON
ejpam-1797	33	2	define	define	VERB
ejpam-1797	33	3	zs(m	zs(m	NUM
ejpam-1797	33	4	)	)	PUNCT
ejpam-1797	34	1	[	[	X
ejpam-1797	34	2	2	2	X
ejpam-1797	34	3	]	]	PUNCT
ejpam-1797	34	4	by	by	ADP
ejpam-1797	34	5	{	{	PUNCT
ejpam-1797	34	6	m	m	PROPN
ejpam-1797	34	7	∈	∈	NOUN
ejpam-1797	34	8	m	m	NOUN
ejpam-1797	34	9	:	:	PUNCT
ejpam-1797	34	10	rs(m	rs(m	X
ejpam-1797	34	11	)	)	PUNCT
ejpam-1797	34	12	be	be	AUX
ejpam-1797	34	13	an	an	DET
ejpam-1797	34	14	essential	essential	ADJ
ejpam-1797	34	15	right	right	ADJ
ejpam-1797	34	16	ideal	ideal	NOUN
ejpam-1797	34	17	of	of	ADP
ejpam-1797	34	18	s	s	NOUN
ejpam-1797	34	19	}	}	PUNCT
ejpam-1797	34	20	.	.	PUNCT
ejpam-1797	35	1	we	we	PRON
ejpam-1797	35	2	first	first	ADV
ejpam-1797	35	3	give	give	VERB
ejpam-1797	35	4	the	the	DET
ejpam-1797	35	5	following	follow	VERB
ejpam-1797	35	6	crucial	crucial	ADJ
ejpam-1797	35	7	definition	definition	NOUN
ejpam-1797	35	8	.	.	PUNCT
ejpam-1797	36	1	definition	definition	NOUN
ejpam-1797	36	2	1	1	NUM
ejpam-1797	36	3	(	(	PUNCT
ejpam-1797	36	4	[	[	X
ejpam-1797	36	5	2	2	NUM
ejpam-1797	36	6	]	]	NUM
ejpam-1797	36	7	)	)	PUNCT
ejpam-1797	36	8	.	.	PUNCT
ejpam-1797	37	1	a	a	DET
ejpam-1797	37	2	ternary	ternary	ADJ
ejpam-1797	37	3	subsemimodule	subsemimodule	NOUN
ejpam-1797	37	4	zs(m	zs(m	NUM
ejpam-1797	37	5	)	)	PUNCT
ejpam-1797	37	6	of	of	ADP
ejpam-1797	37	7	m	m	PROPN
ejpam-1797	37	8	is	be	AUX
ejpam-1797	37	9	called	call	VERB
ejpam-1797	37	10	a	a	DET
ejpam-1797	37	11	singular	singular	ADJ
ejpam-1797	37	12	ternary	ternary	ADJ
ejpam-1797	37	13	subsemimodule	subsemimodule	NOUN
ejpam-1797	37	14	of	of	ADP
ejpam-1797	37	15	the	the	DET
ejpam-1797	37	16	right	right	ADJ
ejpam-1797	37	17	ternary	ternary	ADJ
ejpam-1797	37	18	s	s	NOUN
ejpam-1797	37	19	-	-	PUNCT
ejpam-1797	37	20	semimodule	semimodule	NOUN
ejpam-1797	37	21	m.	m.	NOUN
ejpam-1797	37	22	we	we	PRON
ejpam-1797	37	23	call	call	VERB
ejpam-1797	37	24	a	a	DET
ejpam-1797	37	25	singular	singular	ADJ
ejpam-1797	37	26	ternary	ternary	ADJ
ejpam-1797	37	27	subsemimodule	subsemimodule	NOUN
ejpam-1797	37	28	zs(s	zs(s	NUM
ejpam-1797	37	29	)	)	PUNCT
ejpam-1797	37	30	a	a	DET
ejpam-1797	37	31	right	right	ADJ
ejpam-1797	37	32	ideal	ideal	NOUN
ejpam-1797	37	33	of	of	ADP
ejpam-1797	37	34	a	a	DET
ejpam-1797	37	35	ternary	ternary	ADJ
ejpam-1797	37	36	semiring	semiring	NOUN
ejpam-1797	37	37	s	s	NOUN
ejpam-1797	37	38	and	and	CCONJ
ejpam-1797	37	39	call	call	VERB
ejpam-1797	37	40	this	this	DET
ejpam-1797	37	41	kind	kind	NOUN
ejpam-1797	37	42	of	of	ADP
ejpam-1797	37	43	right	right	ADJ
ejpam-1797	37	44	ideal	ideal	NOUN
ejpam-1797	37	45	the	the	DET
ejpam-1797	37	46	(	(	PUNCT
ejpam-1797	37	47	right	right	ADJ
ejpam-1797	37	48	)	)	PUNCT
ejpam-1797	37	49	singular	singular	PROPN
ejpam-1797	37	50	ideal	ideal	NOUN
ejpam-1797	38	1	[	[	X
ejpam-1797	38	2	2	2	NUM
ejpam-1797	38	3	]	]	PUNCT
ejpam-1797	38	4	of	of	ADP
ejpam-1797	38	5	a	a	DET
ejpam-1797	38	6	ternary	ternary	ADJ
ejpam-1797	38	7	semiring	semiring	NOUN
ejpam-1797	38	8	s	s	X
ejpam-1797	38	9	which	which	PRON
ejpam-1797	38	10	is	be	AUX
ejpam-1797	38	11	denoted	denote	VERB
ejpam-1797	38	12	by	by	ADP
ejpam-1797	38	13	z(s	z(s	PROPN
ejpam-1797	38	14	)	)	PUNCT
ejpam-1797	38	15	,	,	PUNCT
ejpam-1797	38	16	that	that	ADV
ejpam-1797	38	17	is	is	ADV
ejpam-1797	38	18	,	,	PUNCT
ejpam-1797	38	19	z(s	z(s	PROPN
ejpam-1797	38	20	)	)	PUNCT
ejpam-1797	38	21	=	=	PRON
ejpam-1797	39	1	{	{	PUNCT
ejpam-1797	39	2	t	t	PROPN
ejpam-1797	39	3	∈	∈	PROPN
ejpam-1797	39	4	s	s	PART
ejpam-1797	39	5	:	:	PUNCT
ejpam-1797	39	6	rs(t	rs(t	X
ejpam-1797	39	7	)	)	PUNCT
ejpam-1797	39	8	is	be	AUX
ejpam-1797	39	9	an	an	DET
ejpam-1797	39	10	essential	essential	ADJ
ejpam-1797	39	11	right	right	ADJ
ejpam-1797	39	12	ideal	ideal	NOUN
ejpam-1797	39	13	of	of	ADP
ejpam-1797	39	14	s	s	NOUN
ejpam-1797	39	15	}	}	PUNCT
ejpam-1797	39	16	.	.	PUNCT
ejpam-1797	40	1	a	a	DET
ejpam-1797	40	2	ternary	ternary	ADJ
ejpam-1797	40	3	semiring	semiring	NOUN
ejpam-1797	40	4	s	s	NOUN
ejpam-1797	40	5	is	be	AUX
ejpam-1797	40	6	said	say	VERB
ejpam-1797	40	7	to	to	PART
ejpam-1797	40	8	satisfy	satisfy	VERB
ejpam-1797	40	9	the	the	DET
ejpam-1797	40	10	condition	condition	NOUN
ejpam-1797	40	11	α[2	α[2	X
ejpam-1797	40	12	]	]	X
ejpam-1797	40	13	if	if	SCONJ
ejpam-1797	40	14	for	for	ADP
ejpam-1797	40	15	any	any	DET
ejpam-1797	40	16	nonzero	nonzero	NOUN
ejpam-1797	40	17	element	element	NOUN
ejpam-1797	40	18	a	a	PRON
ejpam-1797	40	19	in	in	ADP
ejpam-1797	40	20	s	s	NOUN
ejpam-1797	40	21	,	,	PUNCT
ejpam-1797	40	22	rs(a	rs(a	NOUN
ejpam-1797	40	23	)	)	PUNCT
ejpam-1797	40	24	6=	6=	SYM
ejpam-1797	40	25	s	s	VERB
ejpam-1797	40	26	or	or	CCONJ
ejpam-1797	40	27	equivalently	equivalently	ADV
ejpam-1797	40	28	ass	ass	NOUN
ejpam-1797	40	29	=	=	SYM
ejpam-1797	40	30	0	0	NUM
ejpam-1797	40	31	implies	imply	VERB
ejpam-1797	40	32	that	that	SCONJ
ejpam-1797	40	33	a	a	DET
ejpam-1797	40	34	=	=	NOUN
ejpam-1797	40	35	0	0	NUM
ejpam-1797	40	36	.	.	PUNCT
ejpam-1797	41	1	the	the	DET
ejpam-1797	41	2	properties	property	NOUN
ejpam-1797	41	3	of	of	ADP
ejpam-1797	41	4	singular	singular	ADJ
ejpam-1797	41	5	modules	module	NOUN
ejpam-1797	41	6	of	of	ADP
ejpam-1797	41	7	a	a	DET
ejpam-1797	41	8	ternary	ternary	ADJ
ejpam-1797	41	9	modules	module	NOUN
ejpam-1797	41	10	are	be	AUX
ejpam-1797	41	11	given	give	VERB
ejpam-1797	41	12	in	in	ADP
ejpam-1797	41	13	the	the	DET
ejpam-1797	41	14	following	follow	VERB
ejpam-1797	41	15	propositions	proposition	NOUN
ejpam-1797	41	16	,	,	PUNCT
ejpam-1797	41	17	see	see	VERB
ejpam-1797	41	18	[	[	X
ejpam-1797	41	19	2	2	NUM
ejpam-1797	41	20	]	]	PUNCT
ejpam-1797	41	21	.	.	PUNCT
ejpam-1797	42	1	proposition	proposition	NOUN
ejpam-1797	42	2	1	1	NUM
ejpam-1797	42	3	.	.	NUM
ejpam-1797	42	4	zs(m	zs(m	NUM
ejpam-1797	42	5	)	)	PUNCT
ejpam-1797	42	6	is	be	AUX
ejpam-1797	42	7	a	a	DET
ejpam-1797	42	8	ternary	ternary	ADJ
ejpam-1797	42	9	subsemimodule	subsemimodule	NOUN
ejpam-1797	42	10	of	of	ADP
ejpam-1797	42	11	m.	m.	NOUN
ejpam-1797	42	12	proposition	proposition	NOUN
ejpam-1797	42	13	2	2	NUM
ejpam-1797	42	14	.	.	PUNCT
ejpam-1797	43	1	let	let	VERB
ejpam-1797	43	2	s	s	PRON
ejpam-1797	43	3	be	be	AUX
ejpam-1797	43	4	a	a	DET
ejpam-1797	43	5	ternary	ternary	ADJ
ejpam-1797	43	6	semiring	semiring	NOUN
ejpam-1797	43	7	with	with	ADP
ejpam-1797	43	8	condition	condition	NOUN
ejpam-1797	43	9	α	α	NOUN
ejpam-1797	43	10	.	.	PUNCT
ejpam-1797	44	1	then	then	ADV
ejpam-1797	44	2	the	the	DET
ejpam-1797	44	3	singular	singular	PROPN
ejpam-1797	44	4	ideal	ideal	PROPN
ejpam-1797	44	5	z(s	z(s	PROPN
ejpam-1797	44	6	)	)	PUNCT
ejpam-1797	44	7	is	be	AUX
ejpam-1797	44	8	a	a	DET
ejpam-1797	44	9	k	k	NOUN
ejpam-1797	44	10	-	-	NOUN
ejpam-1797	44	11	ideal	ideal	NOUN
ejpam-1797	44	12	of	of	ADP
ejpam-1797	44	13	s.	s.	PROPN
ejpam-1797	44	14	proposition	proposition	PROPN
ejpam-1797	44	15	3	3	X
ejpam-1797	44	16	.	.	PUNCT
ejpam-1797	45	1	if	if	SCONJ
ejpam-1797	45	2	i	i	PRON
ejpam-1797	45	3	is	be	AUX
ejpam-1797	45	4	an	an	DET
ejpam-1797	45	5	ideal	ideal	NOUN
ejpam-1797	45	6	of	of	ADP
ejpam-1797	45	7	a	a	DET
ejpam-1797	45	8	ternary	ternary	ADJ
ejpam-1797	45	9	semiring	semiring	NOUN
ejpam-1797	45	10	s	s	X
ejpam-1797	45	11	and	and	CCONJ
ejpam-1797	45	12	as	as	ADP
ejpam-1797	45	13	a	a	DET
ejpam-1797	45	14	ternary	ternary	ADJ
ejpam-1797	45	15	semiring	semiring	NOUN
ejpam-1797	45	16	,	,	PUNCT
ejpam-1797	45	17	i	i	PRON
ejpam-1797	45	18	is	be	AUX
ejpam-1797	45	19	semiprime	semiprime	NOUN
ejpam-1797	45	20	,	,	PUNCT
ejpam-1797	45	21	then	then	ADV
ejpam-1797	45	22	z(i	z(i	NUM
ejpam-1797	45	23	)	)	PUNCT
ejpam-1797	46	1	=	=	PUNCT
ejpam-1797	46	2	i	i	PRON
ejpam-1797	46	3	∩	∩	ADJ
ejpam-1797	46	4	z(s	z(s	PROPN
ejpam-1797	46	5	)	)	PUNCT
ejpam-1797	46	6	.	.	PUNCT
ejpam-1797	47	1	we	we	PRON
ejpam-1797	47	2	give	give	VERB
ejpam-1797	47	3	the	the	DET
ejpam-1797	47	4	following	follow	VERB
ejpam-1797	47	5	example	example	NOUN
ejpam-1797	47	6	to	to	PART
ejpam-1797	47	7	show	show	VERB
ejpam-1797	47	8	that	that	SCONJ
ejpam-1797	47	9	there	there	PRON
ejpam-1797	47	10	exists	exist	VERB
ejpam-1797	47	11	a	a	DET
ejpam-1797	47	12	prime(semiprime	prime(semiprime	NOUN
ejpam-1797	47	13	)	)	PUNCT
ejpam-1797	47	14	ternary	ternary	ADJ
ejpam-1797	47	15	semiring	semiring	NOUN
ejpam-1797	47	16	satisfying	satisfy	VERB
ejpam-1797	47	17	the	the	DET
ejpam-1797	47	18	condition	condition	NOUN
ejpam-1797	47	19	α	α	NOUN
ejpam-1797	47	20	.	.	PUNCT
ejpam-1797	47	21	example	example	NOUN
ejpam-1797	48	1	1	1	NUM
ejpam-1797	48	2	(	(	PUNCT
ejpam-1797	48	3	[	[	X
ejpam-1797	48	4	2	2	NUM
ejpam-1797	48	5	]	]	PUNCT
ejpam-1797	48	6	)	)	PUNCT
ejpam-1797	48	7	.	.	PUNCT
ejpam-1797	49	1	let	let	VERB
ejpam-1797	49	2	s	s	PRON
ejpam-1797	49	3	be	be	AUX
ejpam-1797	49	4	a	a	DET
ejpam-1797	49	5	prime(semiprime	prime(semiprime	ADJ
ejpam-1797	49	6	)	)	PUNCT
ejpam-1797	49	7	ternary	ternary	ADJ
ejpam-1797	49	8	semiring	semiring	NOUN
ejpam-1797	49	9	.	.	PUNCT
ejpam-1797	50	1	then	then	ADV
ejpam-1797	50	2	s	s	VERB
ejpam-1797	50	3	satisfies	satisfie	NOUN
ejpam-1797	50	4	the	the	DET
ejpam-1797	50	5	condition	condition	NOUN
ejpam-1797	50	6	α	α	NOUN
ejpam-1797	50	7	.	.	PUNCT
ejpam-1797	51	1	the	the	DET
ejpam-1797	51	2	following	follow	VERB
ejpam-1797	51	3	theorem	theorem	NOUN
ejpam-1797	51	4	of	of	ADP
ejpam-1797	51	5	ternary	ternary	ADJ
ejpam-1797	51	6	semirings	semiring	NOUN
ejpam-1797	51	7	can	can	AUX
ejpam-1797	51	8	be	be	AUX
ejpam-1797	51	9	found	find	VERB
ejpam-1797	51	10	in	in	ADP
ejpam-1797	51	11	[	[	X
ejpam-1797	51	12	2	2	NUM
ejpam-1797	51	13	]	]	PUNCT
ejpam-1797	51	14	.	.	PUNCT
ejpam-1797	52	1	t.	t.	PROPN
ejpam-1797	52	2	dutta	dutta	PROPN
ejpam-1797	52	3	,	,	PUNCT
ejpam-1797	52	4	k.	k.	PROPN
ejpam-1797	52	5	shum	shum	PROPN
ejpam-1797	52	6	,	,	PUNCT
ejpam-1797	52	7	s.	s.	PROPN
ejpam-1797	52	8	mandal	mandal	PROPN
ejpam-1797	52	9	/	/	SYM
ejpam-1797	52	10	eur	eur	PROPN
ejpam-1797	52	11	.	.	PUNCT
ejpam-1797	53	1	j.	j.	PROPN
ejpam-1797	53	2	pure	pure	PROPN
ejpam-1797	53	3	appl	appl	PROPN
ejpam-1797	53	4	.	.	PROPN
ejpam-1797	53	5	math	math	PROPN
ejpam-1797	53	6	,	,	PUNCT
ejpam-1797	53	7	5	5	NUM
ejpam-1797	53	8	(	(	PUNCT
ejpam-1797	53	9	2012	2012	NUM
ejpam-1797	53	10	)	)	PUNCT
ejpam-1797	53	11	,	,	PUNCT
ejpam-1797	53	12	401	401	NUM
ejpam-1797	53	13	-	-	SYM
ejpam-1797	53	14	413	413	NUM
ejpam-1797	53	15	403	403	NUM
ejpam-1797	53	16	theorem	theorem	NOUN
ejpam-1797	53	17	1	1	NUM
ejpam-1797	53	18	.	.	PUNCT
ejpam-1797	54	1	let	let	VERB
ejpam-1797	54	2	s	s	PRON
ejpam-1797	54	3	and	and	CCONJ
ejpam-1797	54	4	s′	s′	ADJ
ejpam-1797	54	5	be	be	AUX
ejpam-1797	54	6	two	two	NUM
ejpam-1797	54	7	semi	semi	ADJ
ejpam-1797	54	8	-	-	ADJ
ejpam-1797	54	9	isomorphic	isomorphic	ADJ
ejpam-1797	54	10	ternary	ternary	ADJ
ejpam-1797	54	11	semirings	semiring	NOUN
ejpam-1797	54	12	.	.	PUNCT
ejpam-1797	55	1	then	then	ADV
ejpam-1797	55	2	s	s	VERB
ejpam-1797	55	3	is	be	AUX
ejpam-1797	55	4	singular(nonsingular	singular(nonsingular	PROPN
ejpam-1797	55	5	)	)	PUNCT
ejpam-1797	56	1	if	if	SCONJ
ejpam-1797	56	2	and	and	CCONJ
ejpam-1797	56	3	only	only	ADV
ejpam-1797	56	4	if	if	SCONJ
ejpam-1797	56	5	s′	s′	ADJ
ejpam-1797	56	6	is	be	AUX
ejpam-1797	56	7	singular(resp	singular(resp	PROPN
ejpam-1797	56	8	.	.	PUNCT
ejpam-1797	56	9	nonsingular	nonsingular	ADJ
ejpam-1797	56	10	)	)	PUNCT
ejpam-1797	56	11	.	.	PUNCT
ejpam-1797	57	1	we	we	PRON
ejpam-1797	57	2	state	state	VERB
ejpam-1797	57	3	the	the	DET
ejpam-1797	57	4	following	follow	VERB
ejpam-1797	57	5	known	know	VERB
ejpam-1797	57	6	definition	definition	NOUN
ejpam-1797	57	7	.	.	PUNCT
ejpam-1797	58	1	definition	definition	NOUN
ejpam-1797	58	2	2	2	NUM
ejpam-1797	58	3	(	(	PUNCT
ejpam-1797	58	4	[	[	X
ejpam-1797	58	5	5	5	NUM
ejpam-1797	58	6	]	]	NUM
ejpam-1797	58	7	)	)	PUNCT
ejpam-1797	58	8	.	.	PUNCT
ejpam-1797	59	1	a	a	DET
ejpam-1797	59	2	non	non	ADJ
ejpam-1797	59	3	-	-	ADJ
ejpam-1797	59	4	empty	empty	ADJ
ejpam-1797	59	5	subset	subset	NOUN
ejpam-1797	59	6	a	a	PRON
ejpam-1797	59	7	of	of	ADP
ejpam-1797	59	8	a	a	DET
ejpam-1797	59	9	ternary	ternary	ADJ
ejpam-1797	59	10	semiring	semiring	NOUN
ejpam-1797	59	11	s	s	VERB
ejpam-1797	59	12	is	be	AUX
ejpam-1797	59	13	called	call	VERB
ejpam-1797	59	14	a	a	DET
ejpam-1797	59	15	p	p	NOUN
ejpam-1797	59	16	-	-	PUNCT
ejpam-1797	59	17	system	system	NOUN
ejpam-1797	59	18	if	if	SCONJ
ejpam-1797	59	19	for	for	ADP
ejpam-1797	59	20	each	each	DET
ejpam-1797	59	21	a	a	DET
ejpam-1797	59	22	∈	∈	PROPN
ejpam-1797	59	23	a	a	DET
ejpam-1797	59	24	there	there	PRON
ejpam-1797	59	25	exist	exist	VERB
ejpam-1797	59	26	elements	element	NOUN
ejpam-1797	59	27	x1	x1	NUM
ejpam-1797	59	28	,	,	PUNCT
ejpam-1797	59	29	x2	x2	PROPN
ejpam-1797	59	30	,	,	PUNCT
ejpam-1797	59	31	x3	x3	ADJ
ejpam-1797	59	32	,	,	PUNCT
ejpam-1797	59	33	x4	x4	PROPN
ejpam-1797	59	34	of	of	ADP
ejpam-1797	59	35	s	s	PRON
ejpam-1797	59	36	such	such	ADJ
ejpam-1797	59	37	that	that	SCONJ
ejpam-1797	59	38	ax1ax2a	ax1ax2a	NOUN
ejpam-1797	59	39	∈	∈	PROPN
ejpam-1797	59	40	a	a	DET
ejpam-1797	59	41	or	or	CCONJ
ejpam-1797	59	42	ax1	ax1	NOUN
ejpam-1797	59	43	x2ax3	x2ax3	NOUN
ejpam-1797	59	44	x4a	x4a	PUNCT
ejpam-1797	60	1	∈	∈	PROPN
ejpam-1797	60	2	a	a	PRON
ejpam-1797	60	3	or	or	CCONJ
ejpam-1797	60	4	ax1	ax1	NUM
ejpam-1797	60	5	x2ax3ax4	x2ax3ax4	X
ejpam-1797	60	6	∈	∈	PROPN
ejpam-1797	60	7	a	a	PRON
ejpam-1797	60	8	or	or	CCONJ
ejpam-1797	60	9	x1ax2ax3	x1ax2ax3	NUM
ejpam-1797	60	10	x4a	x4a	PUNCT
ejpam-1797	61	1	∈	∈	PROPN
ejpam-1797	61	2	a.	a.	NOUN
ejpam-1797	61	3	in	in	ADP
ejpam-1797	61	4	the	the	DET
ejpam-1797	61	5	following	following	NOUN
ejpam-1797	61	6	theorem	theorem	NOUN
ejpam-1797	61	7	,	,	PUNCT
ejpam-1797	61	8	we	we	PRON
ejpam-1797	61	9	characterize	characterize	VERB
ejpam-1797	61	10	the	the	DET
ejpam-1797	61	11	semiprime	semiprime	NOUN
ejpam-1797	61	12	ideal	ideal	NOUN
ejpam-1797	61	13	of	of	ADP
ejpam-1797	61	14	a	a	DET
ejpam-1797	61	15	ternary	ternary	ADJ
ejpam-1797	61	16	semiring	semiring	NOUN
ejpam-1797	61	17	.	.	PUNCT
ejpam-1797	62	1	theorem	theorem	ADJ
ejpam-1797	62	2	2	2	NUM
ejpam-1797	62	3	(	(	PUNCT
ejpam-1797	62	4	[	[	X
ejpam-1797	62	5	5	5	NUM
ejpam-1797	62	6	]	]	NUM
ejpam-1797	62	7	)	)	PUNCT
ejpam-1797	62	8	.	.	PUNCT
ejpam-1797	63	1	a	a	DET
ejpam-1797	63	2	proper	proper	ADJ
ejpam-1797	63	3	ideal	ideal	NOUN
ejpam-1797	63	4	q	q	NOUN
ejpam-1797	63	5	of	of	ADP
ejpam-1797	63	6	a	a	DET
ejpam-1797	63	7	ternary	ternary	ADJ
ejpam-1797	63	8	semiring	semiring	NOUN
ejpam-1797	63	9	s	s	VERB
ejpam-1797	63	10	is	be	AUX
ejpam-1797	63	11	semiprime	semiprime	NOUN
ejpam-1797	63	12	if	if	SCONJ
ejpam-1797	64	1	and	and	CCONJ
ejpam-1797	64	2	only	only	ADV
ejpam-1797	64	3	if	if	SCONJ
ejpam-1797	64	4	its	its	PRON
ejpam-1797	64	5	complement	complement	NOUN
ejpam-1797	64	6	p	p	NOUN
ejpam-1797	64	7	c	c	NOUN
ejpam-1797	64	8	is	be	AUX
ejpam-1797	64	9	a	a	DET
ejpam-1797	64	10	p	p	NOUN
ejpam-1797	64	11	-	-	PUNCT
ejpam-1797	64	12	system	system	NOUN
ejpam-1797	64	13	.	.	PUNCT
ejpam-1797	65	1	proposition	proposition	NOUN
ejpam-1797	65	2	4	4	NUM
ejpam-1797	65	3	.	.	PUNCT
ejpam-1797	66	1	let	let	VERB
ejpam-1797	66	2	s	s	PRON
ejpam-1797	66	3	be	be	AUX
ejpam-1797	66	4	a	a	DET
ejpam-1797	66	5	ternary	ternary	ADJ
ejpam-1797	66	6	semiring	semiring	NOUN
ejpam-1797	66	7	.	.	PUNCT
ejpam-1797	67	1	if	if	SCONJ
ejpam-1797	67	2	q	q	NOUN
ejpam-1797	67	3	is	be	AUX
ejpam-1797	67	4	a	a	DET
ejpam-1797	67	5	semiprime	semiprime	NOUN
ejpam-1797	67	6	ideal	ideal	NOUN
ejpam-1797	67	7	of	of	ADP
ejpam-1797	67	8	s	s	PRON
ejpam-1797	68	1	and	and	CCONJ
ejpam-1797	68	2	i	i	PRON
ejpam-1797	68	3	is	be	AUX
ejpam-1797	68	4	an	an	DET
ejpam-1797	68	5	ideal	ideal	NOUN
ejpam-1797	68	6	of	of	ADP
ejpam-1797	68	7	s	s	PRON
ejpam-1797	68	8	then	then	ADV
ejpam-1797	68	9	q	q	PROPN
ejpam-1797	68	10	∩	∩	NOUN
ejpam-1797	68	11	i	i	PRON
ejpam-1797	68	12	is	be	AUX
ejpam-1797	68	13	a	a	DET
ejpam-1797	68	14	semiprime	semiprime	NOUN
ejpam-1797	68	15	ideal	ideal	NOUN
ejpam-1797	68	16	of	of	ADP
ejpam-1797	68	17	i.	i.	PROPN
ejpam-1797	68	18	proof	proof	PROPN
ejpam-1797	68	19	.	.	PUNCT
ejpam-1797	69	1	let	let	VERB
ejpam-1797	69	2	j	j	PROPN
ejpam-1797	69	3	be	be	AUX
ejpam-1797	69	4	an	an	DET
ejpam-1797	69	5	ideal	ideal	NOUN
ejpam-1797	69	6	of	of	ADP
ejpam-1797	69	7	i	i	PRON
ejpam-1797	69	8	such	such	ADJ
ejpam-1797	69	9	that	that	SCONJ
ejpam-1797	69	10	j3	j3	PROPN
ejpam-1797	69	11	⊆	⊆	NUM
ejpam-1797	69	12	i	i	PROPN
ejpam-1797	69	13	∩q	∩q	PROPN
ejpam-1797	69	14	.	.	PUNCT
ejpam-1797	70	1	then	then	ADV
ejpam-1797	70	2	j3	j3	PROPN
ejpam-1797	70	3	⊆	⊆	NUM
ejpam-1797	70	4	q.	q.	PROPN
ejpam-1797	70	5	if	if	SCONJ
ejpam-1797	70	6	possible	possible	ADJ
ejpam-1797	70	7	,	,	PUNCT
ejpam-1797	70	8	let	let	VERB
ejpam-1797	70	9	j	j	PROPN
ejpam-1797	70	10	6⊆	6⊆	VERB
ejpam-1797	70	11	i	i	PRON
ejpam-1797	70	12	∩q	∩q	PROPN
ejpam-1797	70	13	.	.	PUNCT
ejpam-1797	71	1	then	then	ADV
ejpam-1797	71	2	j	j	PROPN
ejpam-1797	71	3	6⊆	6⊆	PROPN
ejpam-1797	71	4	q.	q.	PROPN
ejpam-1797	71	5	hence	hence	ADV
ejpam-1797	71	6	,	,	PUNCT
ejpam-1797	71	7	there	there	PRON
ejpam-1797	71	8	exists	exist	VERB
ejpam-1797	71	9	an	an	DET
ejpam-1797	71	10	element	element	NOUN
ejpam-1797	71	11	a	a	DET
ejpam-1797	71	12	∈	∈	PROPN
ejpam-1797	71	13	j	j	NOUN
ejpam-1797	71	14	but	but	CCONJ
ejpam-1797	71	15	a	a	DET
ejpam-1797	71	16	6∈	6∈	PROPN
ejpam-1797	71	17	q.	q.	NOUN
ejpam-1797	71	18	now	now	ADV
ejpam-1797	71	19	by	by	ADP
ejpam-1797	71	20	theorem	theorem	NOUN
ejpam-1797	71	21	2	2	NUM
ejpam-1797	71	22	,	,	PUNCT
ejpam-1797	71	23	q	q	PRON
ejpam-1797	71	24	is	be	AUX
ejpam-1797	71	25	a	a	DET
ejpam-1797	71	26	p	p	NOUN
ejpam-1797	71	27	-	-	PUNCT
ejpam-1797	71	28	system	system	NOUN
ejpam-1797	71	29	.	.	PUNCT
ejpam-1797	72	1	then	then	ADV
ejpam-1797	72	2	a	a	DET
ejpam-1797	72	3	∈	∈	PROPN
ejpam-1797	72	4	qc	qc	PROPN
ejpam-1797	72	5	implies	imply	VERB
ejpam-1797	72	6	that	that	SCONJ
ejpam-1797	72	7	there	there	PRON
ejpam-1797	72	8	exist	exist	VERB
ejpam-1797	72	9	elements	element	NOUN
ejpam-1797	72	10	x1	x1	NUM
ejpam-1797	72	11	,	,	PUNCT
ejpam-1797	72	12	x2	x2	PROPN
ejpam-1797	72	13	,	,	PUNCT
ejpam-1797	72	14	x3	x3	ADJ
ejpam-1797	72	15	,	,	PUNCT
ejpam-1797	72	16	x4	x4	PROPN
ejpam-1797	72	17	of	of	ADP
ejpam-1797	72	18	s	s	PRON
ejpam-1797	72	19	such	such	ADJ
ejpam-1797	72	20	that	that	SCONJ
ejpam-1797	72	21	ax1ax2a	ax1ax2a	PROPN
ejpam-1797	72	22	∈	∈	PROPN
ejpam-1797	72	23	qc	qc	PROPN
ejpam-1797	72	24	or	or	CCONJ
ejpam-1797	72	25	ax1	ax1	NUM
ejpam-1797	72	26	x2ax3	x2ax3	NOUN
ejpam-1797	72	27	x4a	x4a	PUNCT
ejpam-1797	73	1	∈	∈	PROPN
ejpam-1797	73	2	qc	qc	PROPN
ejpam-1797	73	3	or	or	CCONJ
ejpam-1797	73	4	ax1	ax1	PRON
ejpam-1797	73	5	x2ax3ax4	x2ax3ax4	X
ejpam-1797	74	1	∈	∈	PROPN
ejpam-1797	74	2	qc	qc	PROPN
ejpam-1797	74	3	or	or	CCONJ
ejpam-1797	74	4	x1ax2ax3	x1ax2ax3	NUM
ejpam-1797	75	1	x4a	x4a	PROPN
ejpam-1797	75	2	∈	∈	PROPN
ejpam-1797	75	3	qc	qc	PROPN
ejpam-1797	75	4	.	.	PUNCT
ejpam-1797	76	1	we	we	PRON
ejpam-1797	76	2	consider	consider	VERB
ejpam-1797	76	3	the	the	DET
ejpam-1797	76	4	following	follow	VERB
ejpam-1797	76	5	situation	situation	NOUN
ejpam-1797	76	6	:	:	PUNCT
ejpam-1797	76	7	if	if	SCONJ
ejpam-1797	76	8	ax1ax2a	ax1ax2a	PROPN
ejpam-1797	76	9	∈qc	∈qc	NUM
ejpam-1797	76	10	,	,	PUNCT
ejpam-1797	76	11	then	then	ADV
ejpam-1797	76	12	there	there	PRON
ejpam-1797	76	13	exist	exist	VERB
ejpam-1797	76	14	elements	element	NOUN
ejpam-1797	76	15	s1	s1	NOUN
ejpam-1797	76	16	,	,	PUNCT
ejpam-1797	76	17	s2	s2	PROPN
ejpam-1797	76	18	,	,	PUNCT
ejpam-1797	76	19	s3	s3	PROPN
ejpam-1797	76	20	,	,	PUNCT
ejpam-1797	76	21	s4	s4	PROPN
ejpam-1797	76	22	of	of	ADP
ejpam-1797	76	23	s	s	PRON
ejpam-1797	76	24	such	such	ADJ
ejpam-1797	76	25	that	that	SCONJ
ejpam-1797	76	26	ax1ax2as1ax1ax2as2ax1ax2a	ax1ax2as1ax1ax2as2ax1ax2a	X
ejpam-1797	76	27	∈qc	∈qc	PROPN
ejpam-1797	76	28	or	or	CCONJ
ejpam-1797	76	29	ax1ax2as1s2ax1ax2as3s4ax1ax2a	ax1ax2as1s2ax1ax2as3s4ax1ax2a	PROPN
ejpam-1797	76	30	∈qc	∈qc	PUNCT
ejpam-1797	76	31	or	or	CCONJ
ejpam-1797	76	32	ax1ax2as1s2ax1ax2as3ax1ax2as4	ax1ax2as1s2ax1ax2as3ax1ax2as4	PROPN
ejpam-1797	76	33	∈	∈	PROPN
ejpam-1797	76	34	qc	qc	PROPN
ejpam-1797	76	35	or	or	CCONJ
ejpam-1797	76	36	s1ax1ax2as2ax1ax2as3s4ax1ax2a	s1ax1ax2as2ax1ax2as3s4ax1ax2a	PROPN
ejpam-1797	76	37	∈qc	∈qc	PROPN
ejpam-1797	76	38	.	.	PUNCT
ejpam-1797	77	1	now	now	ADV
ejpam-1797	77	2	consider	consider	VERB
ejpam-1797	77	3	the	the	DET
ejpam-1797	77	4	following	follow	VERB
ejpam-1797	77	5	cases	case	NOUN
ejpam-1797	77	6	:	:	PUNCT
ejpam-1797	77	7	(	(	PUNCT
ejpam-1797	77	8	i	i	NOUN
ejpam-1797	77	9	)	)	PUNCT
ejpam-1797	77	10	ax1ax2as1ax1ax2as2ax1ax2a	ax1ax2as1ax1ax2as2ax1ax2a	PROPN
ejpam-1797	78	1	=	=	VERB
ejpam-1797	78	2	a(x1ax2as1ax1ax2as2)a(x1ax2)a	a(x1ax2as1ax1ax2as2)a(x1ax2)a	PROPN
ejpam-1797	78	3	=	=	SYM
ejpam-1797	78	4	ai1ai2a	ai1ai2a	PROPN
ejpam-1797	78	5	∈	∈	PROPN
ejpam-1797	78	6	j3	j3	PROPN
ejpam-1797	78	7	⊆	⊆	NUM
ejpam-1797	78	8	q	q	PROPN
ejpam-1797	78	9	as	as	ADP
ejpam-1797	78	10	i1	i1	PROPN
ejpam-1797	78	11	,	,	PUNCT
ejpam-1797	78	12	i2	i2	PROPN
ejpam-1797	78	13	∈	∈	PROPN
ejpam-1797	78	14	i	i	PRON
ejpam-1797	78	15	where	where	SCONJ
ejpam-1797	78	16	i1	i1	PROPN
ejpam-1797	78	17	=	=	PUNCT
ejpam-1797	78	18	x1ax2as1ax1ax2as2	x1ax2as1ax1ax2as2	PROPN
ejpam-1797	78	19	and	and	CCONJ
ejpam-1797	78	20	i2	i2	PROPN
ejpam-1797	78	21	=	=	PUNCT
ejpam-1797	78	22	x1ax2	x1ax2	PROPN
ejpam-1797	78	23	.	.	PUNCT
ejpam-1797	79	1	(	(	PUNCT
ejpam-1797	79	2	ii	ii	NOUN
ejpam-1797	79	3	)	)	PUNCT
ejpam-1797	79	4	ax1ax2as1s2ax1ax2as3s4ax1ax2a	ax1ax2as1s2ax1ax2as3s4ax1ax2a	PROPN
ejpam-1797	80	1	=	=	PUNCT
ejpam-1797	80	2	a(x1ax2as1s2ax1ax2as3s4)a(x1ax2)a	a(x1ax2as1s2ax1ax2as3s4)a(x1ax2)a	NOUN
ejpam-1797	80	3	=	=	NUM
ejpam-1797	80	4	ai1ai2a	ai1ai2a	PROPN
ejpam-1797	80	5	∈	∈	PROPN
ejpam-1797	80	6	j3	j3	PROPN
ejpam-1797	80	7	⊆	⊆	NUM
ejpam-1797	80	8	q	q	PROPN
ejpam-1797	80	9	as	as	ADP
ejpam-1797	80	10	i1	i1	PROPN
ejpam-1797	80	11	,	,	PUNCT
ejpam-1797	80	12	i2	i2	PROPN
ejpam-1797	80	13	∈	∈	PROPN
ejpam-1797	80	14	i	i	PRON
ejpam-1797	80	15	where	where	SCONJ
ejpam-1797	80	16	i1	i1	PROPN
ejpam-1797	80	17	=	=	PROPN
ejpam-1797	80	18	x1ax2as1s2ax1ax2as3s4	x1ax2as1s2ax1ax2as3s4	PROPN
ejpam-1797	80	19	and	and	CCONJ
ejpam-1797	80	20	i2	i2	PROPN
ejpam-1797	80	21	=	=	PUNCT
ejpam-1797	80	22	x1ax2	x1ax2	PROPN
ejpam-1797	80	23	.	.	PUNCT
ejpam-1797	81	1	(	(	PUNCT
ejpam-1797	81	2	iii	iii	NOUN
ejpam-1797	81	3	)	)	PUNCT
ejpam-1797	81	4	ax1ax2as1s2ax1ax2as3ax1ax2as4	ax1ax2as1s2ax1ax2as3ax1ax2as4	NOUN
ejpam-1797	81	5	=	=	PRON
ejpam-1797	81	6	a(x1ax2)a(s1s2a)(x1ax2as3ax1)a(x2as4	a(x1ax2)a(s1s2a)(x1ax2as3ax1)a(x2as4	X
ejpam-1797	81	7	)	)	PUNCT
ejpam-1797	81	8	=	=	PROPN
ejpam-1797	81	9	ai1ai2i3ai4	ai1ai2i3ai4	NOUN
ejpam-1797	81	10	∈	∈	PROPN
ejpam-1797	81	11	j3	j3	PROPN
ejpam-1797	81	12	⊆	⊆	NUM
ejpam-1797	81	13	q	q	PROPN
ejpam-1797	81	14	as	as	ADP
ejpam-1797	81	15	i1	i1	PROPN
ejpam-1797	81	16	,	,	PUNCT
ejpam-1797	81	17	i2	i2	PROPN
ejpam-1797	81	18	,	,	PUNCT
ejpam-1797	81	19	i3	i3	NOUN
ejpam-1797	81	20	,	,	PUNCT
ejpam-1797	81	21	i4	i4	PROPN
ejpam-1797	81	22	∈	∈	PROPN
ejpam-1797	82	1	i	i	PRON
ejpam-1797	82	2	where	where	SCONJ
ejpam-1797	82	3	i1	i1	PROPN
ejpam-1797	82	4	=	=	PUNCT
ejpam-1797	82	5	x1ax2	x1ax2	PROPN
ejpam-1797	82	6	,	,	PUNCT
ejpam-1797	82	7	i2	i2	PROPN
ejpam-1797	82	8	=	=	SYM
ejpam-1797	82	9	s1s2a	s1s2a	X
ejpam-1797	82	10	,	,	PUNCT
ejpam-1797	82	11	i3	i3	NOUN
ejpam-1797	82	12	=	=	SYM
ejpam-1797	82	13	x1ax2as3ax1	x1ax2as3ax1	PROPN
ejpam-1797	82	14	and	and	CCONJ
ejpam-1797	82	15	i4	i4	PROPN
ejpam-1797	82	16	=	=	SYM
ejpam-1797	82	17	x2as4	x2as4	PROPN
ejpam-1797	82	18	.	.	PUNCT
ejpam-1797	83	1	t.	t.	PROPN
ejpam-1797	83	2	dutta	dutta	PROPN
ejpam-1797	83	3	,	,	PUNCT
ejpam-1797	83	4	k.	k.	PROPN
ejpam-1797	83	5	shum	shum	PROPN
ejpam-1797	83	6	,	,	PUNCT
ejpam-1797	83	7	s.	s.	PROPN
ejpam-1797	83	8	mandal	mandal	PROPN
ejpam-1797	83	9	/	/	SYM
ejpam-1797	83	10	eur	eur	PROPN
ejpam-1797	83	11	.	.	PUNCT
ejpam-1797	84	1	j.	j.	PROPN
ejpam-1797	84	2	pure	pure	PROPN
ejpam-1797	84	3	appl	appl	PROPN
ejpam-1797	84	4	.	.	PROPN
ejpam-1797	84	5	math	math	PROPN
ejpam-1797	84	6	,	,	PUNCT
ejpam-1797	84	7	5	5	NUM
ejpam-1797	84	8	(	(	PUNCT
ejpam-1797	84	9	2012	2012	NUM
ejpam-1797	84	10	)	)	PUNCT
ejpam-1797	84	11	,	,	PUNCT
ejpam-1797	84	12	401	401	NUM
ejpam-1797	84	13	-	-	SYM
ejpam-1797	84	14	413	413	NUM
ejpam-1797	84	15	404	404	NUM
ejpam-1797	84	16	(	(	PUNCT
ejpam-1797	84	17	iv	iv	X
ejpam-1797	84	18	)	)	PUNCT
ejpam-1797	84	19	s1ax1ax2as2ax1ax2as3s4ax1ax2a	s1ax1ax2as2ax1ax2as3s4ax1ax2a	NOUN
ejpam-1797	84	20	=(	=(	NOUN
ejpam-1797	84	21	s1ax1)a(x2as2)a(x1ax2)(as3s4ax1ax2)a	s1ax1)a(x2as2)a(x1ax2)(as3s4ax1ax2)a	ADJ
ejpam-1797	84	22	=	=	SYM
ejpam-1797	84	23	i1ai2ai3i4a	i1ai2ai3i4a	NOUN
ejpam-1797	84	24	∈	∈	PROPN
ejpam-1797	84	25	j3	j3	PROPN
ejpam-1797	84	26	⊆	⊆	NUM
ejpam-1797	84	27	q	q	PROPN
ejpam-1797	84	28	as	as	ADP
ejpam-1797	84	29	i1	i1	PROPN
ejpam-1797	84	30	,	,	PUNCT
ejpam-1797	84	31	i2	i2	PROPN
ejpam-1797	84	32	,	,	PUNCT
ejpam-1797	84	33	i3	i3	NOUN
ejpam-1797	84	34	,	,	PUNCT
ejpam-1797	84	35	i4	i4	PROPN
ejpam-1797	84	36	∈	∈	PROPN
ejpam-1797	85	1	i	i	PRON
ejpam-1797	85	2	where	where	SCONJ
ejpam-1797	85	3	i1	i1	PROPN
ejpam-1797	85	4	=	=	PROPN
ejpam-1797	85	5	s1ax1	s1ax1	PROPN
ejpam-1797	85	6	,	,	PUNCT
ejpam-1797	85	7	i2	i2	PROPN
ejpam-1797	85	8	=	=	SYM
ejpam-1797	85	9	x2as2	x2as2	PROPN
ejpam-1797	85	10	,	,	PUNCT
ejpam-1797	85	11	i3	i3	NOUN
ejpam-1797	85	12	=	=	SYM
ejpam-1797	85	13	x1ax2	x1ax2	NOUN
ejpam-1797	85	14	and	and	CCONJ
ejpam-1797	85	15	i4	i4	PROPN
ejpam-1797	85	16	=	=	PUNCT
ejpam-1797	85	17	as3s4ax1ax2	as3s4ax1ax2	NOUN
ejpam-1797	85	18	.	.	PUNCT
ejpam-1797	86	1	now	now	ADV
ejpam-1797	86	2	,	,	PUNCT
ejpam-1797	86	3	from	from	ADP
ejpam-1797	86	4	(	(	PUNCT
ejpam-1797	86	5	i	i	NOUN
ejpam-1797	86	6	)	)	PUNCT
ejpam-1797	86	7	,	,	PUNCT
ejpam-1797	86	8	(	(	PUNCT
ejpam-1797	86	9	ii	ii	NOUN
ejpam-1797	86	10	)	)	PUNCT
ejpam-1797	86	11	,	,	PUNCT
ejpam-1797	86	12	(	(	PUNCT
ejpam-1797	86	13	iii	iii	NOUN
ejpam-1797	86	14	)	)	PUNCT
ejpam-1797	86	15	and	and	CCONJ
ejpam-1797	86	16	(	(	PUNCT
ejpam-1797	86	17	iv	iv	X
ejpam-1797	86	18	)	)	PUNCT
ejpam-1797	86	19	,	,	PUNCT
ejpam-1797	86	20	we	we	PRON
ejpam-1797	86	21	can	can	AUX
ejpam-1797	86	22	easily	easily	ADV
ejpam-1797	86	23	see	see	VERB
ejpam-1797	86	24	that	that	SCONJ
ejpam-1797	86	25	ax1ax2a	ax1ax2a	PROPN
ejpam-1797	86	26	6∈qc	6∈qc	NUM
ejpam-1797	86	27	.	.	PUNCT
ejpam-1797	87	1	hence	hence	ADV
ejpam-1797	87	2	,	,	PUNCT
ejpam-1797	87	3	j	j	PROPN
ejpam-1797	87	4	⊆	⊆	NUM
ejpam-1797	87	5	q	q	NOUN
ejpam-1797	87	6	,	,	PUNCT
ejpam-1797	87	7	and	and	CCONJ
ejpam-1797	87	8	whence	whence	NOUN
ejpam-1797	87	9	,	,	PUNCT
ejpam-1797	87	10	j	j	PROPN
ejpam-1797	87	11	⊆	⊆	NUM
ejpam-1797	87	12	i	i	PROPN
ejpam-1797	87	13	∩q	∩q	PROPN
ejpam-1797	87	14	.	.	PUNCT
ejpam-1797	88	1	this	this	PRON
ejpam-1797	88	2	proves	prove	VERB
ejpam-1797	88	3	that	that	SCONJ
ejpam-1797	88	4	i	i	PRON
ejpam-1797	88	5	∩q	∩q	PROPN
ejpam-1797	88	6	is	be	AUX
ejpam-1797	88	7	a	a	DET
ejpam-1797	88	8	semiprime	semiprime	NOUN
ejpam-1797	88	9	ideal	ideal	NOUN
ejpam-1797	88	10	of	of	ADP
ejpam-1797	88	11	s.	s.	PROPN
ejpam-1797	88	12	we	we	PRON
ejpam-1797	88	13	state	state	VERB
ejpam-1797	88	14	below	below	ADP
ejpam-1797	88	15	the	the	DET
ejpam-1797	88	16	following	follow	VERB
ejpam-1797	88	17	definition	definition	NOUN
ejpam-1797	88	18	of	of	ADP
ejpam-1797	88	19	m	m	NOUN
ejpam-1797	88	20	-	-	NOUN
ejpam-1797	88	21	system	system	NOUN
ejpam-1797	88	22	of	of	ADP
ejpam-1797	88	23	a	a	DET
ejpam-1797	88	24	ternary	ternary	ADJ
ejpam-1797	88	25	semiring	semiring	NOUN
ejpam-1797	88	26	.	.	PUNCT
ejpam-1797	89	1	definition	definition	NOUN
ejpam-1797	89	2	3	3	NUM
ejpam-1797	89	3	(	(	PUNCT
ejpam-1797	89	4	[	[	X
ejpam-1797	89	5	4	4	NUM
ejpam-1797	89	6	]	]	NUM
ejpam-1797	89	7	)	)	PUNCT
ejpam-1797	89	8	.	.	PUNCT
ejpam-1797	90	1	a	a	DET
ejpam-1797	90	2	non	non	ADJ
ejpam-1797	90	3	-	-	ADJ
ejpam-1797	90	4	empty	empty	ADJ
ejpam-1797	90	5	subset	subset	NOUN
ejpam-1797	90	6	a	a	PRON
ejpam-1797	90	7	of	of	ADP
ejpam-1797	90	8	a	a	DET
ejpam-1797	90	9	ternary	ternary	ADJ
ejpam-1797	90	10	semiring	semiring	NOUN
ejpam-1797	90	11	s	s	VERB
ejpam-1797	90	12	is	be	AUX
ejpam-1797	90	13	called	call	VERB
ejpam-1797	90	14	an	an	DET
ejpam-1797	90	15	m	m	NOUN
ejpam-1797	90	16	-	-	NOUN
ejpam-1797	90	17	system	system	NOUN
ejpam-1797	90	18	if	if	SCONJ
ejpam-1797	90	19	for	for	ADP
ejpam-1797	90	20	each	each	DET
ejpam-1797	90	21	a	a	DET
ejpam-1797	90	22	,	,	PUNCT
ejpam-1797	90	23	b	b	NOUN
ejpam-1797	90	24	,	,	PUNCT
ejpam-1797	90	25	c	c	PROPN
ejpam-1797	90	26	∈	∈	PROPN
ejpam-1797	90	27	a	a	PRON
ejpam-1797	90	28	there	there	PRON
ejpam-1797	90	29	exist	exist	VERB
ejpam-1797	90	30	elements	element	NOUN
ejpam-1797	90	31	x1	x1	NUM
ejpam-1797	90	32	,	,	PUNCT
ejpam-1797	90	33	x2	x2	PROPN
ejpam-1797	90	34	,	,	PUNCT
ejpam-1797	90	35	x3	x3	ADJ
ejpam-1797	90	36	,	,	PUNCT
ejpam-1797	90	37	x4	x4	PROPN
ejpam-1797	90	38	of	of	ADP
ejpam-1797	90	39	s	s	PRON
ejpam-1797	90	40	such	such	ADJ
ejpam-1797	90	41	that	that	SCONJ
ejpam-1797	90	42	ax1	ax1	PROPN
ejpam-1797	90	43	bx2c	bx2c	PROPN
ejpam-1797	90	44	∈	∈	PROPN
ejpam-1797	90	45	a	a	PRON
ejpam-1797	90	46	or	or	CCONJ
ejpam-1797	90	47	ax1	ax1	NOUN
ejpam-1797	90	48	x2	x2	NOUN
ejpam-1797	90	49	bx3	bx3	VERB
ejpam-1797	90	50	x4c	x4c	PROPN
ejpam-1797	90	51	∈	∈	PROPN
ejpam-1797	90	52	a	a	PRON
ejpam-1797	90	53	or	or	CCONJ
ejpam-1797	90	54	ax1	ax1	NOUN
ejpam-1797	90	55	x2	x2	ADJ
ejpam-1797	90	56	bx3cx4	bx3cx4	NOUN
ejpam-1797	90	57	∈	∈	PROPN
ejpam-1797	90	58	a	a	PRON
ejpam-1797	90	59	or	or	CCONJ
ejpam-1797	90	60	x1ax2	x1ax2	NOUN
ejpam-1797	90	61	bx3	bx3	VERB
ejpam-1797	90	62	x4c	x4c	PUNCT
ejpam-1797	90	63	∈	∈	PROPN
ejpam-1797	90	64	a.	a.	NOUN
ejpam-1797	90	65	we	we	PRON
ejpam-1797	90	66	now	now	ADV
ejpam-1797	90	67	characterize	characterize	VERB
ejpam-1797	90	68	the	the	DET
ejpam-1797	90	69	prime	prime	ADJ
ejpam-1797	90	70	ideals	ideal	NOUN
ejpam-1797	90	71	of	of	ADP
ejpam-1797	90	72	a	a	DET
ejpam-1797	90	73	ternary	ternary	ADJ
ejpam-1797	90	74	semiring	semiring	NOUN
ejpam-1797	90	75	.	.	PUNCT
ejpam-1797	91	1	theorem	theorem	VERB
ejpam-1797	91	2	3	3	NUM
ejpam-1797	91	3	(	(	PUNCT
ejpam-1797	91	4	[	[	X
ejpam-1797	91	5	4	4	NUM
ejpam-1797	91	6	]	]	NUM
ejpam-1797	91	7	)	)	PUNCT
ejpam-1797	91	8	.	.	PUNCT
ejpam-1797	92	1	a	a	DET
ejpam-1797	92	2	proper	proper	ADJ
ejpam-1797	92	3	ideal	ideal	NOUN
ejpam-1797	92	4	p	p	NOUN
ejpam-1797	92	5	of	of	ADP
ejpam-1797	92	6	a	a	DET
ejpam-1797	92	7	ternary	ternary	ADJ
ejpam-1797	92	8	semiring	semiring	NOUN
ejpam-1797	92	9	s	s	X
ejpam-1797	92	10	is	be	AUX
ejpam-1797	92	11	prime	prime	ADJ
ejpam-1797	92	12	if	if	SCONJ
ejpam-1797	92	13	and	and	CCONJ
ejpam-1797	92	14	only	only	ADV
ejpam-1797	92	15	if	if	SCONJ
ejpam-1797	92	16	its	its	PRON
ejpam-1797	92	17	complement	complement	NOUN
ejpam-1797	92	18	p	p	NOUN
ejpam-1797	92	19	c	c	NOUN
ejpam-1797	92	20	is	be	AUX
ejpam-1797	92	21	an	an	DET
ejpam-1797	92	22	m	m	NOUN
ejpam-1797	92	23	-	-	NOUN
ejpam-1797	92	24	system	system	NOUN
ejpam-1797	92	25	.	.	PUNCT
ejpam-1797	93	1	proposition	proposition	NOUN
ejpam-1797	93	2	5	5	NUM
ejpam-1797	93	3	.	.	PUNCT
ejpam-1797	94	1	let	let	VERB
ejpam-1797	94	2	s	s	PRON
ejpam-1797	94	3	be	be	AUX
ejpam-1797	94	4	a	a	DET
ejpam-1797	94	5	ternary	ternary	ADJ
ejpam-1797	94	6	semiring	semiring	NOUN
ejpam-1797	94	7	and	and	CCONJ
ejpam-1797	94	8	p	p	X
ejpam-1797	94	9	a	a	DET
ejpam-1797	94	10	prime	prime	ADJ
ejpam-1797	94	11	ideal	ideal	NOUN
ejpam-1797	94	12	of	of	ADP
ejpam-1797	94	13	s.	s.	PROPN
ejpam-1797	94	14	if	if	SCONJ
ejpam-1797	94	15	i	i	PRON
ejpam-1797	94	16	is	be	AUX
ejpam-1797	94	17	an	an	DET
ejpam-1797	94	18	ideal	ideal	NOUN
ejpam-1797	94	19	of	of	ADP
ejpam-1797	94	20	s.	s.	PROPN
ejpam-1797	94	21	then	then	ADV
ejpam-1797	94	22	p	p	X
ejpam-1797	94	23	∩	∩	NOUN
ejpam-1797	94	24	i	i	PRON
ejpam-1797	94	25	is	be	AUX
ejpam-1797	94	26	a	a	DET
ejpam-1797	94	27	prime	prime	ADJ
ejpam-1797	94	28	ideal	ideal	NOUN
ejpam-1797	94	29	of	of	ADP
ejpam-1797	94	30	i.	i.	PROPN
ejpam-1797	94	31	proof	proof	PROPN
ejpam-1797	94	32	.	.	PUNCT
ejpam-1797	95	1	proceeding	proceed	VERB
ejpam-1797	95	2	as	as	ADP
ejpam-1797	95	3	in	in	ADP
ejpam-1797	95	4	proposition	proposition	NOUN
ejpam-1797	95	5	4	4	NUM
ejpam-1797	95	6	,	,	PUNCT
ejpam-1797	95	7	the	the	DET
ejpam-1797	95	8	proposition	proposition	NOUN
ejpam-1797	95	9	follows	follow	VERB
ejpam-1797	95	10	immediately	immediately	ADV
ejpam-1797	95	11	.	.	PUNCT
ejpam-1797	96	1	in	in	ADP
ejpam-1797	96	2	below	below	ADV
ejpam-1797	96	3	,	,	PUNCT
ejpam-1797	96	4	we	we	PRON
ejpam-1797	96	5	let	let	VERB
ejpam-1797	96	6	s	s	PRON
ejpam-1797	96	7	=	=	X
ejpam-1797	96	8	{	{	PUNCT
ejpam-1797	96	9	s	s	X
ejpam-1797	96	10	:	:	PUNCT
ejpam-1797	96	11	s	s	AUX
ejpam-1797	96	12	be	be	AUX
ejpam-1797	96	13	a	a	DET
ejpam-1797	96	14	ternary	ternary	ADJ
ejpam-1797	96	15	semiring	semiring	NOUN
ejpam-1797	96	16	such	such	ADJ
ejpam-1797	96	17	that	that	SCONJ
ejpam-1797	96	18	every	every	DET
ejpam-1797	96	19	nonzero	nonzero	ADJ
ejpam-1797	96	20	homomorphic	homomorphic	ADJ
ejpam-1797	96	21	image	image	NOUN
ejpam-1797	96	22	s′	s′	NUM
ejpam-1797	96	23	of	of	ADP
ejpam-1797	96	24	s	s	PROPN
ejpam-1797	96	25	contains	contain	VERB
ejpam-1797	96	26	a	a	DET
ejpam-1797	96	27	nonzero	nonzero	PROPN
ejpam-1797	96	28	ideal	ideal	NOUN
ejpam-1797	96	29	which	which	PRON
ejpam-1797	96	30	is	be	AUX
ejpam-1797	96	31	singular	singular	ADJ
ejpam-1797	96	32	as	as	ADP
ejpam-1797	96	33	a	a	DET
ejpam-1797	96	34	ternary	ternary	ADJ
ejpam-1797	96	35	semiring	semiring	NOUN
ejpam-1797	96	36	}	}	PUNCT
ejpam-1797	96	37	.	.	PUNCT
ejpam-1797	97	1	proposition	proposition	NOUN
ejpam-1797	97	2	6	6	NUM
ejpam-1797	97	3	.	.	PUNCT
ejpam-1797	97	4	s	s	PART
ejpam-1797	97	5	=	=	X
ejpam-1797	97	6	{	{	PUNCT
ejpam-1797	97	7	s	s	X
ejpam-1797	97	8	:	:	PUNCT
ejpam-1797	97	9	s	s	VERB
ejpam-1797	97	10	is	be	AUX
ejpam-1797	97	11	a	a	DET
ejpam-1797	97	12	ternary	ternary	ADJ
ejpam-1797	97	13	semiring	semiring	NOUN
ejpam-1797	97	14	such	such	ADJ
ejpam-1797	97	15	that	that	PRON
ejpam-1797	97	16	for	for	ADP
ejpam-1797	97	17	every	every	DET
ejpam-1797	97	18	nonzero	nonzero	ADJ
ejpam-1797	97	19	homomorphic	homomorphic	ADJ
ejpam-1797	97	20	image	image	NOUN
ejpam-1797	97	21	s′	s′	NUM
ejpam-1797	97	22	of	of	ADP
ejpam-1797	97	23	s	s	PROPN
ejpam-1797	97	24	,	,	PUNCT
ejpam-1797	97	25	β(s′	β(s′	PROPN
ejpam-1797	97	26	)	)	PUNCT
ejpam-1797	97	27	6=	6=	ADP
ejpam-1797	97	28	0	0	NUM
ejpam-1797	97	29	or	or	CCONJ
ejpam-1797	97	30	z(s′	z(s′	PROPN
ejpam-1797	97	31	)	)	PUNCT
ejpam-1797	97	32	6=	6=	ADP
ejpam-1797	97	33	0	0	NUM
ejpam-1797	97	34	}	}	PUNCT
ejpam-1797	97	35	.	.	PUNCT
ejpam-1797	98	1	proof	proof	NOUN
ejpam-1797	98	2	.	.	PUNCT
ejpam-1797	99	1	let	let	VERB
ejpam-1797	99	2	s	s	PRON
ejpam-1797	99	3	′	′	VERB
ejpam-1797	99	4	=	=	PUNCT
ejpam-1797	99	5	{	{	PUNCT
ejpam-1797	99	6	s	s	X
ejpam-1797	99	7	:	:	PUNCT
ejpam-1797	99	8	s	s	VERB
ejpam-1797	99	9	is	be	AUX
ejpam-1797	99	10	a	a	DET
ejpam-1797	99	11	ternary	ternary	ADJ
ejpam-1797	99	12	semiring	semiring	NOUN
ejpam-1797	99	13	such	such	ADJ
ejpam-1797	99	14	that	that	PRON
ejpam-1797	99	15	for	for	ADP
ejpam-1797	99	16	every	every	DET
ejpam-1797	99	17	nonzero	nonzero	ADJ
ejpam-1797	99	18	homomorphic	homomorphic	ADJ
ejpam-1797	99	19	image	image	NOUN
ejpam-1797	99	20	s′	s′	NUM
ejpam-1797	99	21	of	of	ADP
ejpam-1797	99	22	s	s	PROPN
ejpam-1797	99	23	,	,	PUNCT
ejpam-1797	99	24	β(s′	β(s′	PROPN
ejpam-1797	99	25	)	)	PUNCT
ejpam-1797	99	26	6=	6=	ADP
ejpam-1797	99	27	0	0	NUM
ejpam-1797	99	28	or	or	CCONJ
ejpam-1797	99	29	z(s′	z(s′	PROPN
ejpam-1797	99	30	)	)	PUNCT
ejpam-1797	99	31	6=	6=	ADP
ejpam-1797	99	32	0	0	NUM
ejpam-1797	99	33	}	}	PUNCT
ejpam-1797	99	34	.	.	PUNCT
ejpam-1797	100	1	let	let	VERB
ejpam-1797	100	2	s	s	PRON
ejpam-1797	100	3	∈	∈	NOUN
ejpam-1797	100	4	s	s	NOUN
ejpam-1797	100	5	and	and	CCONJ
ejpam-1797	100	6	s′	s′	ADJ
ejpam-1797	100	7	be	be	VERB
ejpam-1797	100	8	a	a	DET
ejpam-1797	100	9	nonzero	nonzero	ADJ
ejpam-1797	100	10	homomorphic	homomorphic	ADJ
ejpam-1797	100	11	image	image	NOUN
ejpam-1797	100	12	of	of	ADP
ejpam-1797	100	13	s.	s.	PROPN
ejpam-1797	100	14	so	so	SCONJ
ejpam-1797	100	15	s′	s′	PROPN
ejpam-1797	100	16	contains	contain	VERB
ejpam-1797	100	17	a	a	DET
ejpam-1797	100	18	nonzero	nonzero	NOUN
ejpam-1797	100	19	ideal	ideal	NOUN
ejpam-1797	101	1	i	i	PRON
ejpam-1797	101	2	(	(	PUNCT
ejpam-1797	101	3	say	say	PROPN
ejpam-1797	101	4	)	)	PUNCT
ejpam-1797	101	5	which	which	PRON
ejpam-1797	101	6	is	be	AUX
ejpam-1797	101	7	singular	singular	ADJ
ejpam-1797	101	8	as	as	ADP
ejpam-1797	101	9	a	a	DET
ejpam-1797	101	10	ternary	ternary	ADJ
ejpam-1797	101	11	semiring	semiring	NOUN
ejpam-1797	101	12	i.e.	i.e.	X
ejpam-1797	101	13	z(i	z(i	NOUN
ejpam-1797	101	14	)	)	PUNCT
ejpam-1797	101	15	=	=	NOUN
ejpam-1797	102	1	i	i	PRON
ejpam-1797	102	2	.	.	PUNCT
ejpam-1797	102	3	suppose	suppose	VERB
ejpam-1797	102	4	β(s′	β(s′	NOUN
ejpam-1797	102	5	)	)	PUNCT
ejpam-1797	102	6	=	=	SYM
ejpam-1797	103	1	0	0	X
ejpam-1797	103	2	.	.	PUNCT
ejpam-1797	103	3	then	then	ADV
ejpam-1797	103	4	s′	s′	PROPN
ejpam-1797	103	5	is	be	AUX
ejpam-1797	103	6	a	a	DET
ejpam-1797	103	7	semiprime	semiprime	NOUN
ejpam-1797	103	8	ternary	ternary	ADJ
ejpam-1797	103	9	semiring	semiring	NOUN
ejpam-1797	103	10	;	;	PUNCT
ejpam-1797	103	11	so	so	CCONJ
ejpam-1797	103	12	i	i	PRON
ejpam-1797	103	13	is	be	AUX
ejpam-1797	103	14	a	a	DET
ejpam-1797	103	15	semiprime	semiprime	NOUN
ejpam-1797	103	16	ternary	ternary	NOUN
ejpam-1797	103	17	semiring	semiring	NOUN
ejpam-1797	103	18	by	by	ADP
ejpam-1797	103	19	proposition	proposition	NOUN
ejpam-1797	103	20	4	4	NUM
ejpam-1797	103	21	.	.	PUNCT
ejpam-1797	103	22	hence	hence	ADV
ejpam-1797	103	23	,	,	PUNCT
ejpam-1797	103	24	by	by	ADP
ejpam-1797	103	25	proposition	proposition	NOUN
ejpam-1797	103	26	3	3	NUM
ejpam-1797	103	27	,	,	PUNCT
ejpam-1797	103	28	z(i	z(i	NUM
ejpam-1797	103	29	)	)	PUNCT
ejpam-1797	103	30	=	=	PUNCT
ejpam-1797	104	1	i	i	PRON
ejpam-1797	104	2	∩	∩	ADJ
ejpam-1797	104	3	z(s′	z(s′	NUM
ejpam-1797	104	4	)	)	PUNCT
ejpam-1797	104	5	i.e.	i.e.	X
ejpam-1797	104	6	i	i	PRON
ejpam-1797	104	7	∩	∩	ADJ
ejpam-1797	104	8	z(s′	z(s′	PROPN
ejpam-1797	104	9	)	)	PUNCT
ejpam-1797	105	1	=	=	VERB
ejpam-1797	106	1	i	i	PRON
ejpam-1797	106	2	which	which	PRON
ejpam-1797	106	3	implies	imply	VERB
ejpam-1797	106	4	that	that	SCONJ
ejpam-1797	106	5	i	i	PRON
ejpam-1797	106	6	⊆	⊆	NUM
ejpam-1797	106	7	z(s′	z(s′	NUM
ejpam-1797	106	8	)	)	PUNCT
ejpam-1797	106	9	.	.	PUNCT
ejpam-1797	107	1	consequently	consequently	ADV
ejpam-1797	107	2	z(s′	z(s′	NUM
ejpam-1797	107	3	)	)	PUNCT
ejpam-1797	107	4	6=	6=	ADP
ejpam-1797	107	5	0	0	NUM
ejpam-1797	107	6	.	.	PUNCT
ejpam-1797	108	1	hence	hence	ADV
ejpam-1797	108	2	β(s′	β(s′	NUM
ejpam-1797	108	3	)	)	PUNCT
ejpam-1797	109	1	6=	6=	ADP
ejpam-1797	109	2	0	0	NUM
ejpam-1797	109	3	or	or	CCONJ
ejpam-1797	109	4	z(s′	z(s′	PROPN
ejpam-1797	109	5	)	)	PUNCT
ejpam-1797	109	6	6=	6=	ADP
ejpam-1797	109	7	0	0	X
ejpam-1797	109	8	.	.	PUNCT
ejpam-1797	110	1	therefore	therefore	ADV
ejpam-1797	110	2	,	,	PUNCT
ejpam-1797	110	3	s	s	VERB
ejpam-1797	110	4	∈	∈	PROPN
ejpam-1797	110	5	s	s	PART
ejpam-1797	110	6	′.	′.	NOUN
ejpam-1797	110	7	conversely	conversely	ADV
ejpam-1797	110	8	,	,	PUNCT
ejpam-1797	110	9	let	let	VERB
ejpam-1797	110	10	s	s	PRON
ejpam-1797	110	11	∈	∈	NOUN
ejpam-1797	110	12	s	s	PART
ejpam-1797	110	13	′	′	NUM
ejpam-1797	110	14	and	and	CCONJ
ejpam-1797	110	15	s′	s′	ADJ
ejpam-1797	110	16	be	be	VERB
ejpam-1797	110	17	a	a	DET
ejpam-1797	110	18	nonzero	nonzero	ADJ
ejpam-1797	110	19	homomorphic	homomorphic	ADJ
ejpam-1797	110	20	image	image	NOUN
ejpam-1797	110	21	of	of	ADP
ejpam-1797	110	22	s	s	PRON
ejpam-1797	110	23	such	such	ADJ
ejpam-1797	110	24	that	that	PRON
ejpam-1797	110	25	β(s′	β(s′	PROPN
ejpam-1797	110	26	)	)	PUNCT
ejpam-1797	110	27	6=	6=	ADP
ejpam-1797	110	28	0	0	NUM
ejpam-1797	110	29	or	or	CCONJ
ejpam-1797	110	30	z(s′	z(s′	PROPN
ejpam-1797	110	31	)	)	PUNCT
ejpam-1797	110	32	6=	6=	ADP
ejpam-1797	110	33	0	0	X
ejpam-1797	110	34	.	.	PUNCT
ejpam-1797	110	35	suppose	suppose	VERB
ejpam-1797	110	36	that	that	SCONJ
ejpam-1797	110	37	β(s′	β(s′	PROPN
ejpam-1797	110	38	)	)	PUNCT
ejpam-1797	110	39	6=	6=	ADP
ejpam-1797	110	40	0	0	X
ejpam-1797	110	41	.	.	PUNCT
ejpam-1797	111	1	then	then	ADV
ejpam-1797	111	2	s′	s′	PROPN
ejpam-1797	111	3	contains	contain	VERB
ejpam-1797	111	4	a	a	DET
ejpam-1797	111	5	nonzero	nonzero	NOUN
ejpam-1797	111	6	ideal	ideal	NOUN
ejpam-1797	111	7	i	i	PRON
ejpam-1797	111	8	such	such	ADJ
ejpam-1797	111	9	that	that	DET
ejpam-1797	111	10	i3	i3	NOUN
ejpam-1797	111	11	=	=	NOUN
ejpam-1797	111	12	0	0	X
ejpam-1797	111	13	.	.	PUNCT
ejpam-1797	112	1	we	we	PRON
ejpam-1797	112	2	now	now	ADV
ejpam-1797	112	3	prove	prove	VERB
ejpam-1797	112	4	that	that	SCONJ
ejpam-1797	112	5	z(i	z(i	NOUN
ejpam-1797	112	6	)	)	PUNCT
ejpam-1797	113	1	=	=	NOUN
ejpam-1797	113	2	i	i	INTJ
ejpam-1797	113	3	.	.	PUNCT
ejpam-1797	114	1	for	for	ADP
ejpam-1797	114	2	this	this	DET
ejpam-1797	114	3	purpose	purpose	NOUN
ejpam-1797	114	4	,	,	PUNCT
ejpam-1797	114	5	let	let	VERB
ejpam-1797	114	6	x	x	SYM
ejpam-1797	114	7	∈	∈	PROPN
ejpam-1797	115	1	i	i	PRON
ejpam-1797	115	2	and	and	CCONJ
ejpam-1797	115	3	h	h	DET
ejpam-1797	115	4	a	a	DET
ejpam-1797	115	5	nonzero	nonzero	ADJ
ejpam-1797	115	6	right	right	ADJ
ejpam-1797	115	7	ideal	ideal	NOUN
ejpam-1797	115	8	of	of	ADP
ejpam-1797	115	9	i	i	PRON
ejpam-1797	115	10	.	.	PUNCT
ejpam-1797	116	1	then	then	ADV
ejpam-1797	116	2	xhi	xhi	PROPN
ejpam-1797	117	1	⊆	⊆	NUM
ejpam-1797	117	2	x	x	SYM
ejpam-1797	117	3	i	i	PRON
ejpam-1797	117	4	i	i	VERB
ejpam-1797	117	5	⊆	⊆	NUM
ejpam-1797	118	1	i	i	PRON
ejpam-1797	119	1	i	i	PRON
ejpam-1797	120	1	i	i	VERB
ejpam-1797	120	2	=	=	NOUN
ejpam-1797	121	1	0	0	X
ejpam-1797	121	2	.	.	PUNCT
ejpam-1797	122	1	this	this	PRON
ejpam-1797	122	2	leads	lead	VERB
ejpam-1797	122	3	h	h	NOUN
ejpam-1797	122	4	⊆	⊆	NUM
ejpam-1797	122	5	ri(x),that	ri(x),that	ADV
ejpam-1797	122	6	is	be	AUX
ejpam-1797	122	7	,	,	PUNCT
ejpam-1797	122	8	ri	ri	PROPN
ejpam-1797	122	9	(	(	PUNCT
ejpam-1797	122	10	x)∩	x)∩	PROPN
ejpam-1797	122	11	h	h	NOUN
ejpam-1797	122	12	=	=	PROPN
ejpam-1797	122	13	h	h	PROPN
ejpam-1797	123	1	6=	6=	PROPN
ejpam-1797	123	2	0	0	X
ejpam-1797	123	3	.	.	PUNCT
ejpam-1797	124	1	therefore	therefore	ADV
ejpam-1797	124	2	,	,	PUNCT
ejpam-1797	124	3	x	x	PROPN
ejpam-1797	124	4	∈	∈	PROPN
ejpam-1797	124	5	z(i	z(i	NUM
ejpam-1797	124	6	)	)	PUNCT
ejpam-1797	124	7	.	.	PUNCT
ejpam-1797	125	1	hence	hence	ADV
ejpam-1797	125	2	,	,	PUNCT
ejpam-1797	125	3	z(i	z(i	NUM
ejpam-1797	125	4	)	)	PUNCT
ejpam-1797	126	1	=	=	SYM
ejpam-1797	126	2	i	i	INTJ
ejpam-1797	126	3	.	.	PUNCT
ejpam-1797	127	1	if	if	SCONJ
ejpam-1797	127	2	β(s′	β(s′	PROPN
ejpam-1797	127	3	)	)	PUNCT
ejpam-1797	127	4	=	=	SYM
ejpam-1797	127	5	0	0	NUM
ejpam-1797	127	6	,	,	PUNCT
ejpam-1797	127	7	then	then	ADV
ejpam-1797	127	8	z(s′	z(s′	PROPN
ejpam-1797	127	9	)	)	PUNCT
ejpam-1797	127	10	6=	6=	ADP
ejpam-1797	127	11	0	0	X
ejpam-1797	127	12	.	.	PUNCT
ejpam-1797	128	1	since	since	SCONJ
ejpam-1797	128	2	β(s′	β(s′	PROPN
ejpam-1797	128	3	)	)	PUNCT
ejpam-1797	128	4	=	=	SYM
ejpam-1797	128	5	0	0	NUM
ejpam-1797	128	6	,	,	PUNCT
ejpam-1797	128	7	s′	s′	PROPN
ejpam-1797	128	8	is	be	AUX
ejpam-1797	128	9	a	a	DET
ejpam-1797	128	10	semiprime	semiprime	NOUN
ejpam-1797	128	11	ternary	ternary	ADJ
ejpam-1797	128	12	semiring	semiring	NOUN
ejpam-1797	128	13	.	.	PUNCT
ejpam-1797	129	1	let	let	VERB
ejpam-1797	129	2	z(s′	z(s′	NUM
ejpam-1797	129	3	)	)	PUNCT
ejpam-1797	130	1	=	=	SYM
ejpam-1797	130	2	i	i	PRON
ejpam-1797	130	3	.	.	PUNCT
ejpam-1797	131	1	then	then	ADV
ejpam-1797	131	2	,	,	PUNCT
ejpam-1797	131	3	by	by	ADP
ejpam-1797	131	4	example	example	NOUN
ejpam-1797	131	5	1	1	NUM
ejpam-1797	131	6	and	and	CCONJ
ejpam-1797	131	7	proposition	proposition	NOUN
ejpam-1797	131	8	2	2	NUM
ejpam-1797	131	9	,	,	PUNCT
ejpam-1797	131	10	i	i	PRON
ejpam-1797	131	11	is	be	AUX
ejpam-1797	131	12	a	a	DET
ejpam-1797	131	13	nonzero	nonzero	ADJ
ejpam-1797	131	14	ideal	ideal	NOUN
ejpam-1797	131	15	of	of	ADP
ejpam-1797	131	16	the	the	DET
ejpam-1797	131	17	t.	t.	PROPN
ejpam-1797	131	18	dutta	dutta	PROPN
ejpam-1797	131	19	,	,	PUNCT
ejpam-1797	131	20	k.	k.	PROPN
ejpam-1797	131	21	shum	shum	PROPN
ejpam-1797	131	22	,	,	PUNCT
ejpam-1797	131	23	s.	s.	PROPN
ejpam-1797	131	24	mandal	mandal	PROPN
ejpam-1797	131	25	/	/	SYM
ejpam-1797	131	26	eur	eur	PROPN
ejpam-1797	131	27	.	.	PUNCT
ejpam-1797	132	1	j.	j.	PROPN
ejpam-1797	132	2	pure	pure	PROPN
ejpam-1797	132	3	appl	appl	PROPN
ejpam-1797	132	4	.	.	PROPN
ejpam-1797	132	5	math	math	PROPN
ejpam-1797	132	6	,	,	PUNCT
ejpam-1797	132	7	5	5	NUM
ejpam-1797	132	8	(	(	PUNCT
ejpam-1797	132	9	2012	2012	NUM
ejpam-1797	132	10	)	)	PUNCT
ejpam-1797	132	11	,	,	PUNCT
ejpam-1797	132	12	401	401	NUM
ejpam-1797	132	13	-	-	SYM
ejpam-1797	132	14	413	413	NUM
ejpam-1797	132	15	405	405	NUM
ejpam-1797	132	16	semiprime	semiprime	NOUN
ejpam-1797	132	17	ternary	ternary	ADJ
ejpam-1797	132	18	semiring	semire	VERB
ejpam-1797	132	19	s′.	s′.	PROPN
ejpam-1797	132	20	hence	hence	ADV
ejpam-1797	132	21	.	.	PUNCT
ejpam-1797	133	1	by	by	ADP
ejpam-1797	133	2	proposition	proposition	NOUN
ejpam-1797	133	3	4	4	NUM
ejpam-1797	133	4	,	,	PUNCT
ejpam-1797	133	5	i	i	PRON
ejpam-1797	133	6	is	be	AUX
ejpam-1797	133	7	a	a	DET
ejpam-1797	133	8	semiprime	semiprime	NOUN
ejpam-1797	133	9	ternary	ternary	ADJ
ejpam-1797	133	10	semiring	semiring	NOUN
ejpam-1797	133	11	.	.	PUNCT
ejpam-1797	134	1	thus	thus	ADV
ejpam-1797	134	2	,	,	PUNCT
ejpam-1797	134	3	by	by	ADP
ejpam-1797	134	4	prop	prop	NOUN
ejpam-1797	134	5	3	3	NUM
ejpam-1797	134	6	,	,	PUNCT
ejpam-1797	134	7	z(i	z(i	NUM
ejpam-1797	134	8	)	)	PUNCT
ejpam-1797	134	9	=	=	PUNCT
ejpam-1797	134	10	i	i	PRON
ejpam-1797	134	11	∩	∩	ADJ
ejpam-1797	134	12	z(s′	z(s′	VERB
ejpam-1797	134	13	)	)	PUNCT
ejpam-1797	135	1	=	=	VERB
ejpam-1797	135	2	i	i	PRON
ejpam-1797	135	3	that	that	ADV
ejpam-1797	135	4	is	be	AUX
ejpam-1797	135	5	,	,	PUNCT
ejpam-1797	135	6	i	i	PRON
ejpam-1797	135	7	is	be	AUX
ejpam-1797	135	8	singular	singular	ADJ
ejpam-1797	135	9	as	as	ADP
ejpam-1797	135	10	a	a	DET
ejpam-1797	135	11	ternary	ternary	ADJ
ejpam-1797	135	12	semiring	semiring	NOUN
ejpam-1797	135	13	.	.	PUNCT
ejpam-1797	136	1	in	in	ADP
ejpam-1797	136	2	both	both	DET
ejpam-1797	136	3	cases	case	NOUN
ejpam-1797	136	4	s	s	VERB
ejpam-1797	136	5	∈	∈	NOUN
ejpam-1797	136	6	s	s	NOUN
ejpam-1797	136	7	.	.	PUNCT
ejpam-1797	137	1	thus	thus	ADV
ejpam-1797	137	2	,	,	PUNCT
ejpam-1797	137	3	s	s	VERB
ejpam-1797	137	4	=	=	NOUN
ejpam-1797	137	5	s	s	PROPN
ejpam-1797	137	6	′.	′.	NOUN
ejpam-1797	137	7	recall	recall	VERB
ejpam-1797	137	8	the	the	DET
ejpam-1797	137	9	following	follow	VERB
ejpam-1797	137	10	definition	definition	NOUN
ejpam-1797	137	11	of	of	ADP
ejpam-1797	137	12	hereditary	hereditary	ADJ
ejpam-1797	137	13	class	class	NOUN
ejpam-1797	137	14	given	give	VERB
ejpam-1797	137	15	in	in	ADP
ejpam-1797	137	16	n.	n.	PROPN
ejpam-1797	137	17	j.	j.	PROPN
ejpam-1797	137	18	divinsky	divinsky	PROPN
ejpam-1797	138	1	[	[	X
ejpam-1797	138	2	3	3	NUM
ejpam-1797	138	3	]	]	PUNCT
ejpam-1797	138	4	.	.	PUNCT
ejpam-1797	139	1	definition	definition	NOUN
ejpam-1797	139	2	4	4	NUM
ejpam-1797	139	3	.	.	PUNCT
ejpam-1797	140	1	a	a	DET
ejpam-1797	140	2	class	class	NOUN
ejpam-1797	140	3	ρ	ρ	NOUN
ejpam-1797	140	4	of	of	ADP
ejpam-1797	140	5	ternary	ternary	ADJ
ejpam-1797	140	6	semirings	semiring	NOUN
ejpam-1797	140	7	is	be	AUX
ejpam-1797	140	8	called	call	VERB
ejpam-1797	140	9	hereditary	hereditary	ADJ
ejpam-1797	140	10	if	if	SCONJ
ejpam-1797	140	11	i	i	PRON
ejpam-1797	140	12	is	be	AUX
ejpam-1797	140	13	an	an	DET
ejpam-1797	140	14	ideal	ideal	NOUN
ejpam-1797	140	15	of	of	ADP
ejpam-1797	140	16	a	a	DET
ejpam-1797	140	17	ternary	ternary	ADJ
ejpam-1797	140	18	semiring	semiring	NOUN
ejpam-1797	140	19	s	s	X
ejpam-1797	140	20	and	and	CCONJ
ejpam-1797	141	1	s	s	PROPN
ejpam-1797	141	2	∈	∈	NOUN
ejpam-1797	141	3	ρ	ρ	NOUN
ejpam-1797	141	4	then	then	ADV
ejpam-1797	141	5	i	i	PROPN
ejpam-1797	141	6	∈	∈	PROPN
ejpam-1797	141	7	ρ	ρ	NOUN
ejpam-1797	141	8	.	.	PUNCT
ejpam-1797	142	1	now	now	ADV
ejpam-1797	142	2	,	,	PUNCT
ejpam-1797	142	3	by	by	ADP
ejpam-1797	142	4	d.	d.	PROPN
ejpam-1797	142	5	m.	m.	PROPN
ejpam-1797	142	6	olson	olson	PROPN
ejpam-1797	142	7	and	and	CCONJ
ejpam-1797	142	8	a.	a.	PROPN
ejpam-1797	142	9	c.	c.	PROPN
ejpam-1797	142	10	nance	nance	PROPN
ejpam-1797	143	1	[	[	X
ejpam-1797	143	2	27	27	NUM
ejpam-1797	143	3	]	]	PUNCT
ejpam-1797	143	4	,	,	PUNCT
ejpam-1797	143	5	we	we	PRON
ejpam-1797	143	6	define	define	VERB
ejpam-1797	143	7	the	the	DET
ejpam-1797	143	8	regular	regular	ADJ
ejpam-1797	143	9	class	class	NOUN
ejpam-1797	143	10	in	in	ADP
ejpam-1797	143	11	a	a	DET
ejpam-1797	143	12	ternary	ternary	ADJ
ejpam-1797	143	13	semiring	semiring	NOUN
ejpam-1797	143	14	as	as	SCONJ
ejpam-1797	143	15	follows	follow	VERB
ejpam-1797	143	16	:	:	PUNCT
ejpam-1797	143	17	definition	definition	NOUN
ejpam-1797	143	18	5	5	NUM
ejpam-1797	143	19	.	.	PUNCT
ejpam-1797	144	1	a	a	DET
ejpam-1797	144	2	classm	classm	NOUN
ejpam-1797	144	3	of	of	ADP
ejpam-1797	144	4	ternary	ternary	ADJ
ejpam-1797	144	5	semirings	semiring	NOUN
ejpam-1797	144	6	is	be	AUX
ejpam-1797	144	7	called	call	VERB
ejpam-1797	144	8	regular	regular	ADV
ejpam-1797	144	9	if	if	SCONJ
ejpam-1797	144	10	s	s	NOUN
ejpam-1797	144	11	∈m	∈m	NOUN
ejpam-1797	144	12	and	and	CCONJ
ejpam-1797	144	13	i	i	PRON
ejpam-1797	144	14	is	be	AUX
ejpam-1797	144	15	a	a	DET
ejpam-1797	144	16	nonzero	nonzero	ADJ
ejpam-1797	144	17	ideal	ideal	NOUN
ejpam-1797	144	18	of	of	ADP
ejpam-1797	144	19	the	the	DET
ejpam-1797	144	20	ternary	ternary	ADJ
ejpam-1797	144	21	semiring	semire	VERB
ejpam-1797	144	22	s	s	PART
ejpam-1797	144	23	,	,	PUNCT
ejpam-1797	144	24	then	then	ADV
ejpam-1797	144	25	there	there	PRON
ejpam-1797	144	26	is	be	VERB
ejpam-1797	144	27	a	a	DET
ejpam-1797	144	28	nonzero	nonzero	ADJ
ejpam-1797	144	29	homomorphic	homomorphic	ADJ
ejpam-1797	144	30	image	image	NOUN
ejpam-1797	144	31	of	of	ADP
ejpam-1797	144	32	i	i	PRON
ejpam-1797	144	33	inm	inm	PROPN
ejpam-1797	144	34	.	.	PUNCT
ejpam-1797	145	1	following	follow	VERB
ejpam-1797	145	2	d.	d.	PROPN
ejpam-1797	145	3	m.	m.	PROPN
ejpam-1797	145	4	olson	olson	PROPN
ejpam-1797	145	5	and	and	CCONJ
ejpam-1797	145	6	a.	a.	PROPN
ejpam-1797	145	7	c.	c.	PROPN
ejpam-1797	145	8	nance	nance	PROPN
ejpam-1797	145	9	[	[	X
ejpam-1797	145	10	27	27	NUM
ejpam-1797	145	11	]	]	PUNCT
ejpam-1797	145	12	,	,	PUNCT
ejpam-1797	145	13	we	we	PRON
ejpam-1797	145	14	define	define	VERB
ejpam-1797	145	15	the	the	DET
ejpam-1797	145	16	radical	radical	ADJ
ejpam-1797	145	17	class	class	NOUN
ejpam-1797	145	18	in	in	ADP
ejpam-1797	145	19	ternary	ternary	ADJ
ejpam-1797	145	20	semiring	semiring	NOUN
ejpam-1797	145	21	.	.	PUNCT
ejpam-1797	146	1	definition	definition	NOUN
ejpam-1797	146	2	6	6	NUM
ejpam-1797	146	3	.	.	PUNCT
ejpam-1797	147	1	a	a	DET
ejpam-1797	147	2	nonempty	nonempty	ADJ
ejpam-1797	147	3	class	class	NOUN
ejpam-1797	147	4	γ	γ	NOUN
ejpam-1797	147	5	of	of	ADP
ejpam-1797	147	6	ternary	ternary	ADJ
ejpam-1797	147	7	semirings	semiring	NOUN
ejpam-1797	147	8	is	be	AUX
ejpam-1797	147	9	called	call	VERB
ejpam-1797	147	10	a	a	DET
ejpam-1797	147	11	radical	radical	ADJ
ejpam-1797	147	12	class	class	NOUN
ejpam-1797	147	13	if	if	SCONJ
ejpam-1797	147	14	the	the	DET
ejpam-1797	147	15	following	follow	VERB
ejpam-1797	147	16	conditions	condition	NOUN
ejpam-1797	147	17	hold	hold	VERB
ejpam-1797	147	18	:	:	PUNCT
ejpam-1797	147	19	(	(	PUNCT
ejpam-1797	147	20	r1	r1	PROPN
ejpam-1797	147	21	)	)	PUNCT
ejpam-1797	147	22	γ	γ	PROPN
ejpam-1797	147	23	is	be	AUX
ejpam-1797	147	24	homomorphically	homomorphically	ADV
ejpam-1797	147	25	closed	close	VERB
ejpam-1797	147	26	.	.	PUNCT
ejpam-1797	148	1	(	(	PUNCT
ejpam-1797	148	2	r2	r2	PROPN
ejpam-1797	148	3	)	)	PUNCT
ejpam-1797	149	1	if	if	SCONJ
ejpam-1797	149	2	s	s	VERB
ejpam-1797	149	3	6∈	6∈	PROPN
ejpam-1797	149	4	γ	γ	NOUN
ejpam-1797	149	5	,	,	PUNCT
ejpam-1797	149	6	then	then	ADV
ejpam-1797	149	7	s	s	VERB
ejpam-1797	149	8	contains	contain	VERB
ejpam-1797	149	9	a	a	DET
ejpam-1797	149	10	proper	proper	ADJ
ejpam-1797	149	11	k	k	NOUN
ejpam-1797	149	12	-	-	NOUN
ejpam-1797	149	13	ideal	ideal	NOUN
ejpam-1797	149	14	k	k	ADP
ejpam-1797	149	15	such	such	ADJ
ejpam-1797	149	16	that	that	PRON
ejpam-1797	149	17	s	s	PROPN
ejpam-1797	149	18	/	/	SYM
ejpam-1797	149	19	k	k	PROPN
ejpam-1797	149	20	has	have	VERB
ejpam-1797	149	21	no	no	DET
ejpam-1797	149	22	nonzero	nonzero	NOUN
ejpam-1797	149	23	γ	γ	NOUN
ejpam-1797	149	24	-	-	PUNCT
ejpam-1797	149	25	ideals	ideal	NOUN
ejpam-1797	149	26	(	(	PUNCT
ejpam-1797	149	27	ideals	ideal	NOUN
ejpam-1797	149	28	which	which	PRON
ejpam-1797	149	29	are	be	AUX
ejpam-1797	149	30	as	as	SCONJ
ejpam-1797	149	31	ternary	ternary	ADJ
ejpam-1797	149	32	semirings	semiring	NOUN
ejpam-1797	149	33	are	be	AUX
ejpam-1797	149	34	in	in	ADP
ejpam-1797	149	35	the	the	DET
ejpam-1797	149	36	class	class	NOUN
ejpam-1797	149	37	γ	γ	NOUN
ejpam-1797	149	38	)	)	PUNCT
ejpam-1797	149	39	.	.	PUNCT
ejpam-1797	150	1	as	as	ADP
ejpam-1797	150	2	an	an	DET
ejpam-1797	150	3	example	example	NOUN
ejpam-1797	150	4	of	of	ADP
ejpam-1797	150	5	the	the	DET
ejpam-1797	150	6	radical	radical	ADJ
ejpam-1797	150	7	class	class	NOUN
ejpam-1797	150	8	in	in	ADP
ejpam-1797	150	9	ternary	ternary	ADJ
ejpam-1797	150	10	semiring	semiring	NOUN
ejpam-1797	150	11	,	,	PUNCT
ejpam-1797	150	12	we	we	PRON
ejpam-1797	150	13	have	have	VERB
ejpam-1797	150	14	the	the	DET
ejpam-1797	150	15	following	follow	VERB
ejpam-1797	150	16	lemma	lemma	PROPN
ejpam-1797	150	17	.	.	PUNCT
ejpam-1797	151	1	lemma	lemma	PROPN
ejpam-1797	151	2	1	1	NUM
ejpam-1797	151	3	.	.	PUNCT
ejpam-1797	152	1	if	if	SCONJ
ejpam-1797	152	2	i	i	PRON
ejpam-1797	152	3	is	be	AUX
ejpam-1797	152	4	a	a	DET
ejpam-1797	152	5	nil	nil	ADJ
ejpam-1797	152	6	ideal	ideal	NOUN
ejpam-1797	152	7	of	of	ADP
ejpam-1797	152	8	a	a	DET
ejpam-1797	152	9	ternary	ternary	ADJ
ejpam-1797	152	10	semiring	semire	VERB
ejpam-1797	152	11	s	s	PART
ejpam-1797	152	12	,	,	PUNCT
ejpam-1797	152	13	then	then	ADV
ejpam-1797	152	14	i	i	PRON
ejpam-1797	152	15	is	be	AUX
ejpam-1797	152	16	also	also	ADV
ejpam-1797	152	17	a	a	DET
ejpam-1797	152	18	nil	nil	ADJ
ejpam-1797	152	19	ideal	ideal	NOUN
ejpam-1797	152	20	of	of	ADP
ejpam-1797	152	21	s.	s.	PROPN
ejpam-1797	152	22	proof	proof	PROPN
ejpam-1797	152	23	.	.	PUNCT
ejpam-1797	153	1	let	let	VERB
ejpam-1797	153	2	i	i	PRON
ejpam-1797	153	3	be	be	AUX
ejpam-1797	153	4	a	a	DET
ejpam-1797	153	5	nil	nil	ADJ
ejpam-1797	153	6	ideal	ideal	NOUN
ejpam-1797	153	7	of	of	ADP
ejpam-1797	153	8	s	s	PRON
ejpam-1797	153	9	and	and	CCONJ
ejpam-1797	153	10	x	x	SYM
ejpam-1797	153	11	∈	∈	PROPN
ejpam-1797	154	1	i	i	PRON
ejpam-1797	154	2	.	.	PUNCT
ejpam-1797	155	1	then	then	ADV
ejpam-1797	155	2	x	x	X
ejpam-1797	156	1	+	+	CCONJ
ejpam-1797	156	2	i	i	PRON
ejpam-1797	156	3	∈	∈	VERB
ejpam-1797	156	4	i	i	PRON
ejpam-1797	156	5	for	for	ADP
ejpam-1797	156	6	some	some	PRON
ejpam-1797	156	7	i	i	PRON
ejpam-1797	156	8	∈	∈	PROPN
ejpam-1797	157	1	i	i	PRON
ejpam-1797	157	2	since	since	SCONJ
ejpam-1797	157	3	i	i	PRON
ejpam-1797	157	4	is	be	AUX
ejpam-1797	157	5	nil	nil	ADJ
ejpam-1797	157	6	,	,	PUNCT
ejpam-1797	157	7	for	for	ADP
ejpam-1797	157	8	each	each	DET
ejpam-1797	157	9	t	t	NOUN
ejpam-1797	157	10	in	in	ADP
ejpam-1797	157	11	s	s	PRON
ejpam-1797	157	12	there	there	PRON
ejpam-1797	157	13	exists	exist	VERB
ejpam-1797	157	14	a	a	DET
ejpam-1797	157	15	positive	positive	ADJ
ejpam-1797	157	16	integer	integer	NOUN
ejpam-1797	157	17	n(depending	n(depende	VERB
ejpam-1797	157	18	on	on	ADP
ejpam-1797	157	19	t	t	PROPN
ejpam-1797	157	20	)	)	PUNCT
ejpam-1797	157	21	such	such	ADJ
ejpam-1797	157	22	that	that	SCONJ
ejpam-1797	157	23	[	[	X
ejpam-1797	157	24	(	(	PUNCT
ejpam-1797	157	25	x+	x+	ADJ
ejpam-1797	157	26	i)t]n(x+	i)t]n(x+	NUM
ejpam-1797	157	27	i	i	PROPN
ejpam-1797	157	28	)	)	PUNCT
ejpam-1797	157	29	=	=	PUNCT
ejpam-1797	158	1	0	0	X
ejpam-1797	158	2	.	.	PUNCT
ejpam-1797	159	1	since	since	SCONJ
ejpam-1797	159	2	i	i	PRON
ejpam-1797	159	3	is	be	AUX
ejpam-1797	159	4	an	an	DET
ejpam-1797	159	5	ideal	ideal	NOUN
ejpam-1797	159	6	of	of	ADP
ejpam-1797	159	7	s	s	PROPN
ejpam-1797	159	8	,	,	PUNCT
ejpam-1797	159	9	so	so	SCONJ
ejpam-1797	159	10	[	[	X
ejpam-1797	159	11	(	(	PUNCT
ejpam-1797	159	12	x+	x+	ADJ
ejpam-1797	159	13	i)t]n(x+	i)t]n(x+	NUM
ejpam-1797	159	14	i	i	PROPN
ejpam-1797	159	15	)	)	PUNCT
ejpam-1797	159	16	can	can	AUX
ejpam-1797	159	17	be	be	AUX
ejpam-1797	159	18	written	write	VERB
ejpam-1797	159	19	as	as	ADP
ejpam-1797	159	20	(	(	PUNCT
ejpam-1797	159	21	x	x	NOUN
ejpam-1797	159	22	t)n	t)n	NOUN
ejpam-1797	159	23	x+h	x+h	NUM
ejpam-1797	159	24	for	for	ADP
ejpam-1797	159	25	a	a	DET
ejpam-1797	159	26	particular	particular	ADJ
ejpam-1797	159	27	h	h	NOUN
ejpam-1797	159	28	∈	∈	PROPN
ejpam-1797	160	1	i	i	PRON
ejpam-1797	160	2	.	.	PUNCT
ejpam-1797	161	1	thus	thus	ADV
ejpam-1797	161	2	,	,	PUNCT
ejpam-1797	161	3	(	(	PUNCT
ejpam-1797	161	4	x	x	NOUN
ejpam-1797	161	5	t)n	t)n	PUNCT
ejpam-1797	161	6	x+h=	x+h=	PROPN
ejpam-1797	161	7	0	0	NUM
ejpam-1797	161	8	,	,	PUNCT
ejpam-1797	161	9	and	and	CCONJ
ejpam-1797	161	10	as	as	ADP
ejpam-1797	161	11	a	a	DET
ejpam-1797	161	12	result	result	NOUN
ejpam-1797	161	13	,	,	PUNCT
ejpam-1797	161	14	we	we	PRON
ejpam-1797	161	15	have	have	VERB
ejpam-1797	161	16	(	(	PUNCT
ejpam-1797	161	17	x	x	NOUN
ejpam-1797	161	18	t)n+1h+hth=	t)n+1h+hth=	NOUN
ejpam-1797	161	19	0	0	NUM
ejpam-1797	161	20	.	.	PUNCT
ejpam-1797	162	1	now	now	ADV
ejpam-1797	162	2	,	,	PUNCT
ejpam-1797	162	3	(	(	PUNCT
ejpam-1797	162	4	x	x	SYM
ejpam-1797	162	5	t)2n+1	t)2n+1	NOUN
ejpam-1797	162	6	x	x	SYM
ejpam-1797	162	7	=	=	PUNCT
ejpam-1797	162	8	(	(	PUNCT
ejpam-1797	162	9	x	x	X
ejpam-1797	162	10	t)n+1[(x	t)n+1[(x	NOUN
ejpam-1797	162	11	t)n	t)n	NOUN
ejpam-1797	162	12	x	x	PUNCT
ejpam-1797	163	1	+	+	NUM
ejpam-1797	163	2	h	h	NOUN
ejpam-1797	163	3	]	]	X
ejpam-1797	164	1	+	+	CCONJ
ejpam-1797	164	2	hth	hth	PROPN
ejpam-1797	164	3	=	=	SYM
ejpam-1797	164	4	hth	hth	PROPN
ejpam-1797	164	5	.	.	PUNCT
ejpam-1797	165	1	but	but	CCONJ
ejpam-1797	165	2	,	,	PUNCT
ejpam-1797	165	3	hth	hth	PROPN
ejpam-1797	165	4	∈	∈	PROPN
ejpam-1797	166	1	i	i	PRON
ejpam-1797	166	2	,	,	PUNCT
ejpam-1797	166	3	so	so	CCONJ
ejpam-1797	166	4	(	(	PUNCT
ejpam-1797	166	5	x	x	SYM
ejpam-1797	166	6	t)2n+1	t)2n+1	NOUN
ejpam-1797	166	7	x	x	SYM
ejpam-1797	166	8	∈	∈	PROPN
ejpam-1797	166	9	i	i	PRON
ejpam-1797	166	10	.	.	PUNCT
ejpam-1797	167	1	(	(	PUNCT
ejpam-1797	167	2	x	x	X
ejpam-1797	167	3	t)2n+1	t)2n+1	NOUN
ejpam-1797	167	4	x	x	PUNCT
ejpam-1797	167	5	is	be	AUX
ejpam-1797	167	6	nilpotent	nilpotent	ADJ
ejpam-1797	167	7	.	.	PUNCT
ejpam-1797	168	1	hence	hence	ADV
ejpam-1797	168	2	,	,	PUNCT
ejpam-1797	168	3	for	for	ADP
ejpam-1797	168	4	each	each	DET
ejpam-1797	168	5	t	t	NOUN
ejpam-1797	168	6	∈	∈	PROPN
ejpam-1797	168	7	s	s	PART
ejpam-1797	168	8	,	,	PUNCT
ejpam-1797	168	9	[	[	X
ejpam-1797	168	10	(	(	PUNCT
ejpam-1797	168	11	x	x	PART
ejpam-1797	168	12	t)2n+1	t)2n+1	NOUN
ejpam-1797	168	13	x	x	SYM
ejpam-1797	168	14	t]k[(x	t]k[(x	NOUN
ejpam-1797	168	15	t)2n+1	t)2n+1	NOUN
ejpam-1797	168	16	x	x	X
ejpam-1797	168	17	]	]	X
ejpam-1797	168	18	=	=	SYM
ejpam-1797	168	19	0	0	NUM
ejpam-1797	168	20	for	for	ADP
ejpam-1797	168	21	some	some	DET
ejpam-1797	168	22	positive	positive	ADJ
ejpam-1797	168	23	integer	integer	NOUN
ejpam-1797	168	24	k	k	PROPN
ejpam-1797	168	25	⇒	⇒	PROPN
ejpam-1797	168	26	(	(	PUNCT
ejpam-1797	168	27	x	x	X
ejpam-1797	168	28	t)(2n+2)k+2n+1	t)(2n+2)k+2n+1	NOUN
ejpam-1797	168	29	x	x	X
ejpam-1797	168	30	=	=	NOUN
ejpam-1797	168	31	0	0	NUM
ejpam-1797	168	32	.	.	PUNCT
ejpam-1797	169	1	thus	thus	ADV
ejpam-1797	169	2	,	,	PUNCT
ejpam-1797	169	3	x	x	PRON
ejpam-1797	169	4	is	be	AUX
ejpam-1797	169	5	nilpotent	nilpotent	ADJ
ejpam-1797	169	6	and	and	CCONJ
ejpam-1797	169	7	thus	thus	ADV
ejpam-1797	169	8	i	i	PRON
ejpam-1797	169	9	is	be	AUX
ejpam-1797	169	10	a	a	DET
ejpam-1797	169	11	nil	nil	ADJ
ejpam-1797	169	12	ideal	ideal	NOUN
ejpam-1797	169	13	of	of	ADP
ejpam-1797	169	14	s.	s.	PROPN
ejpam-1797	169	15	theorem	theorem	VERB
ejpam-1797	169	16	4	4	NUM
ejpam-1797	169	17	.	.	PUNCT
ejpam-1797	170	1	the	the	DET
ejpam-1797	170	2	class	class	NOUN
ejpam-1797	170	3	n	n	PROPN
ejpam-1797	170	4	of	of	ADP
ejpam-1797	170	5	all	all	DET
ejpam-1797	170	6	nil	nil	ADJ
ejpam-1797	170	7	ternary	ternary	ADJ
ejpam-1797	170	8	semirings	semiring	NOUN
ejpam-1797	170	9	is	be	AUX
ejpam-1797	170	10	a	a	DET
ejpam-1797	170	11	radical	radical	ADJ
ejpam-1797	170	12	class	class	NOUN
ejpam-1797	170	13	.	.	PUNCT
ejpam-1797	171	1	proof	proof	NOUN
ejpam-1797	171	2	.	.	PUNCT
ejpam-1797	172	1	clearly	clearly	ADV
ejpam-1797	172	2	,	,	PUNCT
ejpam-1797	172	3	the	the	DET
ejpam-1797	172	4	class	class	NOUN
ejpam-1797	172	5	n	n	PART
ejpam-1797	172	6	is	be	AUX
ejpam-1797	172	7	homomorphically	homomorphically	ADV
ejpam-1797	172	8	closed	close	VERB
ejpam-1797	172	9	.	.	PUNCT
ejpam-1797	173	1	let	let	VERB
ejpam-1797	173	2	s	s	PRON
ejpam-1797	173	3	be	be	AUX
ejpam-1797	173	4	a	a	DET
ejpam-1797	173	5	ternary	ternary	ADJ
ejpam-1797	173	6	semiring	semiring	NOUN
ejpam-1797	173	7	such	such	ADJ
ejpam-1797	173	8	that	that	PRON
ejpam-1797	173	9	s	s	VERB
ejpam-1797	173	10	6∈	6∈	NOUN
ejpam-1797	173	11	n	n	NOUN
ejpam-1797	173	12	.	.	PUNCT
ejpam-1797	174	1	now	now	ADV
ejpam-1797	174	2	,	,	PUNCT
ejpam-1797	174	3	using	use	VERB
ejpam-1797	174	4	zorn	zorn	PROPN
ejpam-1797	174	5	’s	’s	PART
ejpam-1797	174	6	lemma	lemma	PROPN
ejpam-1797	174	7	,	,	PUNCT
ejpam-1797	174	8	we	we	PRON
ejpam-1797	174	9	choose	choose	VERB
ejpam-1797	174	10	an	an	DET
ejpam-1797	174	11	ideal	ideal	ADJ
ejpam-1797	174	12	m	m	NOUN
ejpam-1797	174	13	of	of	ADP
ejpam-1797	174	14	s	s	PRON
ejpam-1797	174	15	which	which	PRON
ejpam-1797	174	16	is	be	AUX
ejpam-1797	174	17	maximal	maximal	ADJ
ejpam-1797	174	18	with	with	ADP
ejpam-1797	174	19	respect	respect	NOUN
ejpam-1797	174	20	to	to	ADP
ejpam-1797	174	21	being	be	AUX
ejpam-1797	174	22	a	a	DET
ejpam-1797	174	23	nil	nil	ADJ
ejpam-1797	174	24	ideal	ideal	NOUN
ejpam-1797	174	25	.	.	PUNCT
ejpam-1797	175	1	since	since	SCONJ
ejpam-1797	175	2	s	s	PROPN
ejpam-1797	175	3	6∈	6∈	NUM
ejpam-1797	175	4	n	n	NOUN
ejpam-1797	175	5	,	,	PUNCT
ejpam-1797	175	6	m	m	VERB
ejpam-1797	175	7	is	be	AUX
ejpam-1797	175	8	a	a	DET
ejpam-1797	175	9	proper	proper	ADJ
ejpam-1797	175	10	ideal	ideal	NOUN
ejpam-1797	175	11	of	of	ADP
ejpam-1797	175	12	s.	s.	PROPN
ejpam-1797	175	13	by	by	ADP
ejpam-1797	175	14	lemma	lemma	PROPN
ejpam-1797	175	15	1	1	NUM
ejpam-1797	175	16	and	and	CCONJ
ejpam-1797	175	17	the	the	DET
ejpam-1797	175	18	maximality	maximality	NOUN
ejpam-1797	175	19	of	of	ADP
ejpam-1797	175	20	m	m	PROPN
ejpam-1797	175	21	,	,	PUNCT
ejpam-1797	175	22	we	we	PRON
ejpam-1797	175	23	see	see	VERB
ejpam-1797	175	24	that	that	SCONJ
ejpam-1797	175	25	m	m	VERB
ejpam-1797	175	26	=	=	ADJ
ejpam-1797	175	27	m	m	VERB
ejpam-1797	175	28	.	.	PUNCT
ejpam-1797	176	1	this	this	PRON
ejpam-1797	176	2	means	mean	VERB
ejpam-1797	176	3	that	that	SCONJ
ejpam-1797	176	4	m	m	PROPN
ejpam-1797	176	5	is	be	AUX
ejpam-1797	176	6	a	a	DET
ejpam-1797	176	7	k	k	NOUN
ejpam-1797	176	8	-	-	NOUN
ejpam-1797	176	9	ideal	ideal	NOUN
ejpam-1797	176	10	of	of	ADP
ejpam-1797	176	11	s.	s.	PROPN
ejpam-1797	176	12	if	if	SCONJ
ejpam-1797	176	13	i	i	PRON
ejpam-1797	176	14	/	/	SYM
ejpam-1797	176	15	m	m	VERB
ejpam-1797	176	16	is	be	AUX
ejpam-1797	176	17	any	any	DET
ejpam-1797	176	18	n	n	DET
ejpam-1797	176	19	-ideal	-ideal	NOUN
ejpam-1797	176	20	of	of	ADP
ejpam-1797	176	21	s	s	NOUN
ejpam-1797	176	22	/	/	SYM
ejpam-1797	176	23	m	m	VERB
ejpam-1797	176	24	then	then	ADV
ejpam-1797	176	25	for	for	ADP
ejpam-1797	176	26	any	any	DET
ejpam-1797	176	27	x	x	SYM
ejpam-1797	176	28	∈	∈	PROPN
ejpam-1797	176	29	i	i	PRON
ejpam-1797	176	30	,	,	PUNCT
ejpam-1797	176	31	x	x	X
ejpam-1797	176	32	/	/	SYM
ejpam-1797	176	33	m	m	VERB
ejpam-1797	176	34	is	be	AUX
ejpam-1797	176	35	nilpotent	nilpotent	ADJ
ejpam-1797	176	36	.	.	PUNCT
ejpam-1797	177	1	now	now	ADV
ejpam-1797	177	2	,	,	PUNCT
ejpam-1797	177	3	for	for	ADP
ejpam-1797	177	4	each	each	DET
ejpam-1797	177	5	t	t	PROPN
ejpam-1797	177	6	/	/	SYM
ejpam-1797	177	7	m	m	PROPN
ejpam-1797	177	8	∈	∈	NOUN
ejpam-1797	177	9	s	s	NOUN
ejpam-1797	177	10	/	/	SYM
ejpam-1797	177	11	m	m	PROPN
ejpam-1797	177	12	,	,	PUNCT
ejpam-1797	177	13	there	there	PRON
ejpam-1797	177	14	exists	exist	VERB
ejpam-1797	177	15	a	a	DET
ejpam-1797	177	16	positive	positive	ADJ
ejpam-1797	177	17	integer	integer	NOUN
ejpam-1797	177	18	n	n	CCONJ
ejpam-1797	177	19	such	such	ADJ
ejpam-1797	177	20	that	that	SCONJ
ejpam-1797	177	21	[	[	X
ejpam-1797	177	22	(	(	PUNCT
ejpam-1797	177	23	x	x	X
ejpam-1797	177	24	/	/	SYM
ejpam-1797	177	25	m)(t	m)(t	ADJ
ejpam-1797	177	26	/	/	SYM
ejpam-1797	177	27	m)]n	m)]n	NOUN
ejpam-1797	177	28	x	x	SYM
ejpam-1797	177	29	/	/	SYM
ejpam-1797	177	30	m	m	VERB
ejpam-1797	177	31	=	=	PUNCT
ejpam-1797	177	32	(	(	PUNCT
ejpam-1797	177	33	(	(	PUNCT
ejpam-1797	177	34	x	x	NOUN
ejpam-1797	177	35	t)n	t)n	NOUN
ejpam-1797	177	36	x)/m	x)/m	PROPN
ejpam-1797	177	37	=	=	SYM
ejpam-1797	177	38	0	0	NUM
ejpam-1797	177	39	/	/	SYM
ejpam-1797	177	40	m	m	PROPN
ejpam-1797	177	41	.	.	PUNCT
ejpam-1797	178	1	but	but	CCONJ
ejpam-1797	178	2	then	then	ADV
ejpam-1797	178	3	(	(	PUNCT
ejpam-1797	178	4	x	x	NOUN
ejpam-1797	178	5	t)n	t)n	NOUN
ejpam-1797	178	6	x	x	PUNCT
ejpam-1797	178	7	∈	∈	PROPN
ejpam-1797	178	8	m	m	VERB
ejpam-1797	178	9	which	which	PRON
ejpam-1797	178	10	makes	make	VERB
ejpam-1797	178	11	(	(	PUNCT
ejpam-1797	178	12	x	x	NOUN
ejpam-1797	178	13	t)n	t)n	X
ejpam-1797	178	14	x	x	PUNCT
ejpam-1797	178	15	and	and	CCONJ
ejpam-1797	178	16	therefore	therefore	ADV
ejpam-1797	178	17	,	,	PUNCT
ejpam-1797	178	18	x	x	PUNCT
ejpam-1797	178	19	is	be	AUX
ejpam-1797	178	20	nilpotent	nilpotent	ADJ
ejpam-1797	178	21	.	.	PUNCT
ejpam-1797	179	1	thus	thus	ADV
ejpam-1797	179	2	,	,	PUNCT
ejpam-1797	179	3	i	i	PRON
ejpam-1797	179	4	is	be	AUX
ejpam-1797	179	5	a	a	DET
ejpam-1797	179	6	nil	nil	ADJ
ejpam-1797	179	7	ideal	ideal	NOUN
ejpam-1797	179	8	of	of	ADP
ejpam-1797	179	9	s	s	PROPN
ejpam-1797	179	10	,	,	PUNCT
ejpam-1797	179	11	and	and	CCONJ
ejpam-1797	179	12	so	so	ADV
ejpam-1797	179	13	i	i	PRON
ejpam-1797	179	14	⊆	⊆	NUM
ejpam-1797	179	15	m	m	VERB
ejpam-1797	179	16	since	since	SCONJ
ejpam-1797	179	17	m	m	PROPN
ejpam-1797	179	18	is	be	AUX
ejpam-1797	179	19	maximal	maximal	ADJ
ejpam-1797	179	20	.	.	PUNCT
ejpam-1797	180	1	hence	hence	ADV
ejpam-1797	180	2	,	,	PUNCT
ejpam-1797	180	3	i	i	PRON
ejpam-1797	180	4	/	/	SYM
ejpam-1797	180	5	m	m	VERB
ejpam-1797	180	6	=	=	PUNCT
ejpam-1797	180	7	(	(	PUNCT
ejpam-1797	180	8	0	0	NUM
ejpam-1797	180	9	)	)	PUNCT
ejpam-1797	180	10	,	,	PUNCT
ejpam-1797	180	11	and	and	CCONJ
ejpam-1797	180	12	so	so	ADV
ejpam-1797	180	13	s	s	NOUN
ejpam-1797	180	14	/	/	SYM
ejpam-1797	180	15	m	m	VERB
ejpam-1797	180	16	has	have	VERB
ejpam-1797	180	17	a	a	DET
ejpam-1797	180	18	no	no	DET
ejpam-1797	180	19	non	non	ADJ
ejpam-1797	180	20	-	-	ADJ
ejpam-1797	180	21	zero	zero	NUM
ejpam-1797	180	22	n	n	DET
ejpam-1797	180	23	-ideal	-ideal	NOUN
ejpam-1797	180	24	.	.	PUNCT
ejpam-1797	181	1	this	this	PRON
ejpam-1797	181	2	shows	show	VERB
ejpam-1797	181	3	that	that	SCONJ
ejpam-1797	181	4	n	n	PRON
ejpam-1797	181	5	is	be	AUX
ejpam-1797	181	6	a	a	DET
ejpam-1797	181	7	radical	radical	ADJ
ejpam-1797	181	8	class	class	NOUN
ejpam-1797	181	9	.	.	PUNCT
ejpam-1797	182	1	t.	t.	PROPN
ejpam-1797	182	2	dutta	dutta	PROPN
ejpam-1797	182	3	,	,	PUNCT
ejpam-1797	182	4	k.	k.	PROPN
ejpam-1797	182	5	shum	shum	PROPN
ejpam-1797	182	6	,	,	PUNCT
ejpam-1797	182	7	s.	s.	PROPN
ejpam-1797	182	8	mandal	mandal	PROPN
ejpam-1797	182	9	/	/	SYM
ejpam-1797	182	10	eur	eur	PROPN
ejpam-1797	182	11	.	.	PUNCT
ejpam-1797	183	1	j.	j.	PROPN
ejpam-1797	183	2	pure	pure	PROPN
ejpam-1797	183	3	appl	appl	PROPN
ejpam-1797	183	4	.	.	PROPN
ejpam-1797	183	5	math	math	PROPN
ejpam-1797	183	6	,	,	PUNCT
ejpam-1797	183	7	5	5	NUM
ejpam-1797	183	8	(	(	PUNCT
ejpam-1797	183	9	2012	2012	NUM
ejpam-1797	183	10	)	)	PUNCT
ejpam-1797	183	11	,	,	PUNCT
ejpam-1797	183	12	401	401	NUM
ejpam-1797	183	13	-	-	SYM
ejpam-1797	183	14	413	413	NUM
ejpam-1797	183	15	406	406	NUM
ejpam-1797	183	16	lemma	lemma	PROPN
ejpam-1797	183	17	2	2	NUM
ejpam-1797	183	18	.	.	PUNCT
ejpam-1797	184	1	if	if	SCONJ
ejpam-1797	184	2	φ	φ	PROPN
ejpam-1797	184	3	is	be	AUX
ejpam-1797	184	4	a	a	DET
ejpam-1797	184	5	semi	semi	NOUN
ejpam-1797	184	6	-	-	NOUN
ejpam-1797	184	7	isomorphism	isomorphism	NOUN
ejpam-1797	184	8	from	from	ADP
ejpam-1797	184	9	a	a	DET
ejpam-1797	184	10	ternary	ternary	ADJ
ejpam-1797	184	11	semiring	semiring	NOUN
ejpam-1797	184	12	s	s	VERB
ejpam-1797	184	13	onto	onto	ADP
ejpam-1797	184	14	a	a	DET
ejpam-1797	184	15	ternary	ternary	ADJ
ejpam-1797	184	16	semiring	semiring	NOUN
ejpam-1797	184	17	t	t	PROPN
ejpam-1797	185	1	and	and	CCONJ
ejpam-1797	185	2	i	i	PRON
ejpam-1797	185	3	is	be	AUX
ejpam-1797	185	4	a	a	DET
ejpam-1797	185	5	nonzero	nonzero	ADJ
ejpam-1797	185	6	ideal	ideal	NOUN
ejpam-1797	185	7	of	of	ADP
ejpam-1797	185	8	s	s	PROPN
ejpam-1797	185	9	,	,	PUNCT
ejpam-1797	185	10	then	then	ADV
ejpam-1797	185	11	φ(i	φ(i	NUM
ejpam-1797	185	12	)	)	PUNCT
ejpam-1797	186	1	is	be	AUX
ejpam-1797	186	2	a	a	DET
ejpam-1797	186	3	nonzero	nonzero	NOUN
ejpam-1797	186	4	ideal	ideal	NOUN
ejpam-1797	186	5	of	of	ADP
ejpam-1797	186	6	t	t	PROPN
ejpam-1797	186	7	.	.	PUNCT
ejpam-1797	187	1	proof	proof	NOUN
ejpam-1797	187	2	.	.	PUNCT
ejpam-1797	188	1	clearly	clearly	ADV
ejpam-1797	188	2	,	,	PUNCT
ejpam-1797	188	3	φ(i	φ(i	PROPN
ejpam-1797	188	4	)	)	PUNCT
ejpam-1797	188	5	is	be	AUX
ejpam-1797	188	6	an	an	DET
ejpam-1797	188	7	ideal	ideal	NOUN
ejpam-1797	188	8	of	of	ADP
ejpam-1797	188	9	t	t	PROPN
ejpam-1797	188	10	.	.	PUNCT
ejpam-1797	189	1	if	if	SCONJ
ejpam-1797	189	2	φ(i	φ(i	NUM
ejpam-1797	189	3	)	)	PUNCT
ejpam-1797	189	4	=	=	SYM
ejpam-1797	189	5	(	(	PUNCT
ejpam-1797	189	6	0	0	NUM
ejpam-1797	189	7	)	)	PUNCT
ejpam-1797	189	8	,	,	PUNCT
ejpam-1797	189	9	then	then	ADV
ejpam-1797	189	10	,	,	PUNCT
ejpam-1797	189	11	i	i	PROPN
ejpam-1797	189	12	⊆	⊆	NUM
ejpam-1797	189	13	kerφ	kerφ	NOUN
ejpam-1797	189	14	=	=	SYM
ejpam-1797	189	15	(	(	PUNCT
ejpam-1797	189	16	0	0	NUM
ejpam-1797	189	17	)	)	PUNCT
ejpam-1797	189	18	,	,	PUNCT
ejpam-1797	189	19	as	as	SCONJ
ejpam-1797	189	20	φ	φ	PROPN
ejpam-1797	189	21	is	be	AUX
ejpam-1797	189	22	semiisomorphism	semiisomorphism	ADJ
ejpam-1797	189	23	.	.	PUNCT
ejpam-1797	190	1	thus	thus	ADV
ejpam-1797	190	2	i	i	PRON
ejpam-1797	190	3	=	=	SYM
ejpam-1797	190	4	(	(	PUNCT
ejpam-1797	190	5	0	0	NUM
ejpam-1797	190	6	)	)	PUNCT
ejpam-1797	190	7	,	,	PUNCT
ejpam-1797	190	8	a	a	DET
ejpam-1797	190	9	contradiction	contradiction	NOUN
ejpam-1797	190	10	.	.	PUNCT
ejpam-1797	191	1	this	this	PRON
ejpam-1797	191	2	shows	show	VERB
ejpam-1797	191	3	that	that	SCONJ
ejpam-1797	191	4	φ(i	φ(i	PROPN
ejpam-1797	191	5	)	)	PUNCT
ejpam-1797	191	6	is	be	AUX
ejpam-1797	191	7	a	a	DET
ejpam-1797	191	8	nonzero	nonzero	NOUN
ejpam-1797	191	9	ideal	ideal	NOUN
ejpam-1797	191	10	of	of	ADP
ejpam-1797	191	11	t	t	PROPN
ejpam-1797	191	12	.	.	PUNCT
ejpam-1797	192	1	the	the	DET
ejpam-1797	192	2	following	follow	VERB
ejpam-1797	192	3	theorem	theorem	NOUN
ejpam-1797	192	4	is	be	AUX
ejpam-1797	192	5	a	a	DET
ejpam-1797	192	6	theorem	theorem	NOUN
ejpam-1797	192	7	for	for	ADP
ejpam-1797	192	8	the	the	DET
ejpam-1797	192	9	regular	regular	ADJ
ejpam-1797	192	10	radical	radical	ADJ
ejpam-1797	192	11	class	class	NOUN
ejpam-1797	192	12	of	of	ADP
ejpam-1797	192	13	the	the	DET
ejpam-1797	192	14	ternary	ternary	ADJ
ejpam-1797	192	15	semirings	semiring	NOUN
ejpam-1797	192	16	.	.	PUNCT
ejpam-1797	193	1	theorem	theorem	VERB
ejpam-1797	193	2	5	5	NUM
ejpam-1797	193	3	.	.	PUNCT
ejpam-1797	194	1	ifm	ifm	PROPN
ejpam-1797	194	2	is	be	AUX
ejpam-1797	194	3	a	a	DET
ejpam-1797	194	4	regular	regular	ADJ
ejpam-1797	194	5	class	class	NOUN
ejpam-1797	194	6	of	of	ADP
ejpam-1797	194	7	ternary	ternary	ADJ
ejpam-1797	194	8	semirings	semiring	NOUN
ejpam-1797	194	9	,	,	PUNCT
ejpam-1797	194	10	thenum	thenum	NOUN
ejpam-1797	194	11	=	=	NOUN
ejpam-1797	194	12	{	{	PUNCT
ejpam-1797	194	13	ternary	ternary	ADJ
ejpam-1797	194	14	semirings	semiring	NOUN
ejpam-1797	194	15	s	s	PART
ejpam-1797	194	16	:	:	PUNCT
ejpam-1797	194	17	no	no	DET
ejpam-1797	194	18	nonzero	nonzero	ADJ
ejpam-1797	194	19	homomorphic	homomorphic	ADJ
ejpam-1797	194	20	image	image	NOUN
ejpam-1797	194	21	of	of	ADP
ejpam-1797	194	22	s	s	PROPN
ejpam-1797	194	23	is	be	AUX
ejpam-1797	194	24	inm	inm	PROPN
ejpam-1797	194	25	}	}	PUNCT
ejpam-1797	194	26	is	be	AUX
ejpam-1797	194	27	a	a	DET
ejpam-1797	194	28	radical	radical	ADJ
ejpam-1797	194	29	class	class	NOUN
ejpam-1797	194	30	.	.	PUNCT
ejpam-1797	195	1	proof	proof	NOUN
ejpam-1797	195	2	.	.	PUNCT
ejpam-1797	196	1	suppose	suppose	VERB
ejpam-1797	196	2	that	that	SCONJ
ejpam-1797	196	3	s	s	VERB
ejpam-1797	196	4	∈	∈	X
ejpam-1797	196	5	um	um	INTJ
ejpam-1797	196	6	and	and	CCONJ
ejpam-1797	196	7	φ(s	φ(s	NOUN
ejpam-1797	196	8	)	)	PUNCT
ejpam-1797	196	9	is	be	AUX
ejpam-1797	196	10	a	a	DET
ejpam-1797	196	11	nonzero	nonzero	ADJ
ejpam-1797	196	12	homomorphic	homomorphic	ADJ
ejpam-1797	196	13	image	image	NOUN
ejpam-1797	196	14	of	of	ADP
ejpam-1797	196	15	s.	s.	PROPN
ejpam-1797	196	16	let	let	VERB
ejpam-1797	196	17	ψ(φ(s	ψ(φ(s	PROPN
ejpam-1797	196	18	)	)	PUNCT
ejpam-1797	196	19	)	)	PUNCT
ejpam-1797	196	20	be	be	AUX
ejpam-1797	196	21	a	a	DET
ejpam-1797	196	22	nonzero	nonzero	ADJ
ejpam-1797	196	23	homomorphic	homomorphic	ADJ
ejpam-1797	196	24	image	image	NOUN
ejpam-1797	196	25	of	of	ADP
ejpam-1797	196	26	φ(s	φ(s	NOUN
ejpam-1797	196	27	)	)	PUNCT
ejpam-1797	196	28	.	.	PUNCT
ejpam-1797	197	1	then	then	ADV
ejpam-1797	197	2	ψ(φ(s	ψ(φ(s	PROPN
ejpam-1797	197	3	)	)	PUNCT
ejpam-1797	197	4	)	)	PUNCT
ejpam-1797	198	1	=	=	PRON
ejpam-1797	198	2	(	(	PUNCT
ejpam-1797	198	3	ψφ)(s	ψφ)(s	PROPN
ejpam-1797	198	4	)	)	PUNCT
ejpam-1797	198	5	is	be	AUX
ejpam-1797	198	6	a	a	DET
ejpam-1797	198	7	nonzero	nonzero	ADJ
ejpam-1797	198	8	homomorphic	homomorphic	ADJ
ejpam-1797	198	9	image	image	NOUN
ejpam-1797	198	10	of	of	ADP
ejpam-1797	198	11	s.	s.	PROPN
ejpam-1797	198	12	since	since	SCONJ
ejpam-1797	198	13	s	s	PROPN
ejpam-1797	198	14	∈	∈	PROPN
ejpam-1797	198	15	um	um	INTJ
ejpam-1797	198	16	,	,	PUNCT
ejpam-1797	198	17	(	(	PUNCT
ejpam-1797	198	18	ψφ)(s	ψφ)(s	PROPN
ejpam-1797	198	19	)	)	PUNCT
ejpam-1797	198	20	6∈	6∈	PROPN
ejpam-1797	198	21	m	m	VERB
ejpam-1797	198	22	.	.	PUNCT
ejpam-1797	199	1	hence	hence	ADV
ejpam-1797	199	2	φ(s	φ(s	NOUN
ejpam-1797	199	3	)	)	PUNCT
ejpam-1797	199	4	∈	∈	PROPN
ejpam-1797	199	5	um	um	INTJ
ejpam-1797	199	6	.	.	PUNCT
ejpam-1797	200	1	thus	thus	ADV
ejpam-1797	200	2	um	um	INTJ
ejpam-1797	200	3	is	be	AUX
ejpam-1797	200	4	homomorphically	homomorphically	ADV
ejpam-1797	200	5	closed	close	VERB
ejpam-1797	200	6	.	.	PUNCT
ejpam-1797	201	1	next	next	ADV
ejpam-1797	201	2	,	,	PUNCT
ejpam-1797	201	3	we	we	PRON
ejpam-1797	201	4	suppose	suppose	VERB
ejpam-1797	201	5	that	that	SCONJ
ejpam-1797	201	6	s	s	VERB
ejpam-1797	201	7	6∈	6∈	PROPN
ejpam-1797	201	8	um	um	INTJ
ejpam-1797	201	9	.	.	PUNCT
ejpam-1797	202	1	then	then	ADV
ejpam-1797	202	2	there	there	PRON
ejpam-1797	202	3	exists	exist	VERB
ejpam-1797	202	4	a	a	DET
ejpam-1797	202	5	nonzero	nonzero	ADJ
ejpam-1797	202	6	homomorphic	homomorphic	ADJ
ejpam-1797	202	7	image	image	NOUN
ejpam-1797	202	8	φ(s	φ(s	NOUN
ejpam-1797	202	9	)	)	PUNCT
ejpam-1797	202	10	∈m	∈m	NOUN
ejpam-1797	202	11	.	.	PUNCT
ejpam-1797	203	1	now	now	ADV
ejpam-1797	203	2	as	as	SCONJ
ejpam-1797	203	3	φ	φ	PROPN
ejpam-1797	203	4	is	be	AUX
ejpam-1797	203	5	nonzero	nonzero	NOUN
ejpam-1797	203	6	,	,	PUNCT
ejpam-1797	203	7	kerφ	kerφ	PROPN
ejpam-1797	203	8	is	be	AUX
ejpam-1797	203	9	a	a	DET
ejpam-1797	203	10	proper	proper	ADJ
ejpam-1797	203	11	k	k	NOUN
ejpam-1797	203	12	-	-	NOUN
ejpam-1797	203	13	ideal	ideal	NOUN
ejpam-1797	203	14	of	of	ADP
ejpam-1797	203	15	s	s	PRON
ejpam-1797	203	16	and	and	CCONJ
ejpam-1797	203	17	s	s	PROPN
ejpam-1797	203	18	/	/	SYM
ejpam-1797	203	19	kerφ	kerφ	PROPN
ejpam-1797	203	20	≃	≃	PROPN
ejpam-1797	203	21	φ(s	φ(s	NOUN
ejpam-1797	203	22	)	)	PUNCT
ejpam-1797	203	23	,	,	PUNCT
ejpam-1797	203	24	and	and	CCONJ
ejpam-1797	203	25	let	let	VERB
ejpam-1797	203	26	the	the	DET
ejpam-1797	203	27	semi	semi	ADJ
ejpam-1797	203	28	-	-	ADJ
ejpam-1797	203	29	isomorphism	isomorphism	ADJ
ejpam-1797	203	30	beψ	beψ	NOUN
ejpam-1797	203	31	.	.	PUNCT
ejpam-1797	204	1	if	if	SCONJ
ejpam-1797	204	2	i	i	PRON
ejpam-1797	204	3	is	be	AUX
ejpam-1797	204	4	a	a	DET
ejpam-1797	204	5	nonzeroum	nonzeroum	NOUN
ejpam-1797	204	6	-ideal	-ideal	NOUN
ejpam-1797	204	7	of	of	ADP
ejpam-1797	204	8	s	s	PROPN
ejpam-1797	204	9	/	/	SYM
ejpam-1797	204	10	kerφ	kerφ	PROPN
ejpam-1797	204	11	,	,	PUNCT
ejpam-1797	204	12	then	then	ADV
ejpam-1797	204	13	by	by	ADP
ejpam-1797	204	14	lemma	lemma	PROPN
ejpam-1797	204	15	2,ψ(i	2,ψ(i	NUM
ejpam-1797	204	16	)	)	PUNCT
ejpam-1797	204	17	is	be	AUX
ejpam-1797	204	18	a	a	DET
ejpam-1797	204	19	nonzero	nonzero	ADJ
ejpam-1797	204	20	ideal	ideal	NOUN
ejpam-1797	204	21	of	of	ADP
ejpam-1797	204	22	φ(s	φ(s	NOUN
ejpam-1797	204	23	)	)	PUNCT
ejpam-1797	204	24	.	.	PUNCT
ejpam-1797	205	1	since	since	SCONJ
ejpam-1797	205	2	i	i	PRON
ejpam-1797	205	3	∈	∈	PROPN
ejpam-1797	205	4	um	um	INTJ
ejpam-1797	205	5	and	and	CCONJ
ejpam-1797	205	6	um	um	INTJ
ejpam-1797	205	7	is	be	AUX
ejpam-1797	205	8	homomorphically	homomorphically	ADV
ejpam-1797	205	9	closed	close	VERB
ejpam-1797	205	10	,	,	PUNCT
ejpam-1797	205	11	ψ(i	ψ(i	NOUN
ejpam-1797	205	12	)	)	PUNCT
ejpam-1797	205	13	∈	∈	PROPN
ejpam-1797	205	14	um	um	INTJ
ejpam-1797	205	15	.	.	PUNCT
ejpam-1797	206	1	since	since	SCONJ
ejpam-1797	206	2	φ(s	φ(	VERB
ejpam-1797	206	3	)	)	PUNCT
ejpam-1797	206	4	∈	∈	PROPN
ejpam-1797	206	5	m	m	PROPN
ejpam-1797	206	6	and	and	CCONJ
ejpam-1797	206	7	m	m	PROPN
ejpam-1797	206	8	is	be	AUX
ejpam-1797	206	9	regular	regular	ADJ
ejpam-1797	206	10	,	,	PUNCT
ejpam-1797	206	11	ψ(i	ψ(i	PROPN
ejpam-1797	206	12	)	)	PUNCT
ejpam-1797	206	13	has	have	VERB
ejpam-1797	206	14	a	a	DET
ejpam-1797	206	15	nonzero	nonzero	ADJ
ejpam-1797	206	16	homomorphic	homomorphic	ADJ
ejpam-1797	206	17	image	image	NOUN
ejpam-1797	206	18	in	in	ADP
ejpam-1797	206	19	m	m	PROPN
ejpam-1797	206	20	.	.	PUNCT
ejpam-1797	207	1	this	this	PRON
ejpam-1797	207	2	,	,	PUNCT
ejpam-1797	207	3	however	however	ADV
ejpam-1797	207	4	,	,	PUNCT
ejpam-1797	207	5	contradicts	contradict	VERB
ejpam-1797	207	6	toψ(i	toψ(i	PROPN
ejpam-1797	207	7	)	)	PUNCT
ejpam-1797	207	8	∈	∈	PROPN
ejpam-1797	207	9	um	um	INTJ
ejpam-1797	207	10	,	,	PUNCT
ejpam-1797	207	11	as	as	ADP
ejpam-1797	207	12	a	a	DET
ejpam-1797	207	13	result	result	NOUN
ejpam-1797	207	14	,	,	PUNCT
ejpam-1797	207	15	we	we	PRON
ejpam-1797	207	16	have	have	AUX
ejpam-1797	207	17	shown	show	VERB
ejpam-1797	207	18	that	that	SCONJ
ejpam-1797	207	19	s	s	PROPN
ejpam-1797	207	20	/	/	SYM
ejpam-1797	207	21	kerφ	kerφ	PROPN
ejpam-1797	207	22	has	have	VERB
ejpam-1797	207	23	no	no	DET
ejpam-1797	207	24	nonzero	nonzero	NOUN
ejpam-1797	207	25	um	um	INTJ
ejpam-1797	207	26	-ideals.thus	-ideals.thus	NOUN
ejpam-1797	207	27	,	,	PUNCT
ejpam-1797	207	28	um	um	INTJ
ejpam-1797	207	29	is	be	AUX
ejpam-1797	207	30	indeed	indeed	ADV
ejpam-1797	207	31	a	a	DET
ejpam-1797	207	32	radical	radical	ADJ
ejpam-1797	207	33	class	class	NOUN
ejpam-1797	207	34	.	.	PUNCT
ejpam-1797	208	1	definition	definition	NOUN
ejpam-1797	208	2	7	7	NUM
ejpam-1797	208	3	.	.	PUNCT
ejpam-1797	209	1	let	let	VERB
ejpam-1797	209	2	s	s	PRON
ejpam-1797	209	3	be	be	AUX
ejpam-1797	209	4	a	a	DET
ejpam-1797	209	5	ternary	ternary	ADJ
ejpam-1797	209	6	semiring	semiring	NOUN
ejpam-1797	209	7	and	and	CCONJ
ejpam-1797	209	8	a	a	DET
ejpam-1797	209	9	be	be	AUX
ejpam-1797	209	10	a	a	DET
ejpam-1797	209	11	nonempty	nonempty	ADJ
ejpam-1797	209	12	subset	subset	NOUN
ejpam-1797	209	13	of	of	ADP
ejpam-1797	209	14	s.	s.	PROPN
ejpam-1797	209	15	then	then	ADV
ejpam-1797	209	16	,	,	PUNCT
ejpam-1797	209	17	the	the	DET
ejpam-1797	209	18	annihilator	annihilator	NOUN
ejpam-1797	209	19	of	of	ADP
ejpam-1797	209	20	a	a	PRON
ejpam-1797	209	21	in	in	ADP
ejpam-1797	209	22	s	s	NOUN
ejpam-1797	209	23	,	,	PUNCT
ejpam-1797	209	24	denoted	denote	VERB
ejpam-1797	209	25	by	by	ADP
ejpam-1797	209	26	anns(a	anns(a	NOUN
ejpam-1797	209	27	)	)	PUNCT
ejpam-1797	209	28	,	,	PUNCT
ejpam-1797	209	29	is	be	AUX
ejpam-1797	209	30	defined	define	VERB
ejpam-1797	209	31	by	by	ADP
ejpam-1797	209	32	{	{	PUNCT
ejpam-1797	209	33	x	x	SYM
ejpam-1797	209	34	∈	∈	PROPN
ejpam-1797	209	35	s	s	PART
ejpam-1797	209	36	:	:	PUNCT
ejpam-1797	209	37	axs	axs	X
ejpam-1797	209	38	=	=	SYM
ejpam-1797	209	39	0	0	NUM
ejpam-1797	209	40	and	and	CCONJ
ejpam-1797	209	41	asx	asx	NOUN
ejpam-1797	209	42	=	=	NOUN
ejpam-1797	209	43	0	0	NUM
ejpam-1797	209	44	for	for	ADP
ejpam-1797	209	45	all	all	PRON
ejpam-1797	209	46	s	s	PART
ejpam-1797	209	47	∈	∈	NOUN
ejpam-1797	209	48	s	s	PART
ejpam-1797	209	49	}	}	PUNCT
ejpam-1797	209	50	.	.	PUNCT
ejpam-1797	210	1	proposition	proposition	NOUN
ejpam-1797	210	2	7	7	NUM
ejpam-1797	210	3	.	.	PUNCT
ejpam-1797	211	1	let	let	VERB
ejpam-1797	211	2	s	s	PRON
ejpam-1797	211	3	be	be	AUX
ejpam-1797	211	4	a	a	DET
ejpam-1797	211	5	ternary	ternary	ADJ
ejpam-1797	211	6	semiring	semiring	NOUN
ejpam-1797	211	7	and	and	CCONJ
ejpam-1797	211	8	a	a	DET
ejpam-1797	211	9	be	be	AUX
ejpam-1797	211	10	a	a	DET
ejpam-1797	211	11	right	right	ADJ
ejpam-1797	211	12	ideal	ideal	NOUN
ejpam-1797	211	13	of	of	ADP
ejpam-1797	211	14	s.	s.	PROPN
ejpam-1797	211	15	then	then	ADV
ejpam-1797	211	16	anns(a	anns(a	NOUN
ejpam-1797	211	17	)	)	PUNCT
ejpam-1797	211	18	is	be	AUX
ejpam-1797	211	19	a	a	DET
ejpam-1797	211	20	k	k	NOUN
ejpam-1797	211	21	-	-	NOUN
ejpam-1797	211	22	ideal	ideal	NOUN
ejpam-1797	211	23	of	of	ADP
ejpam-1797	211	24	s.	s.	PROPN
ejpam-1797	211	25	proof	proof	PROPN
ejpam-1797	211	26	.	.	PUNCT
ejpam-1797	212	1	obviously	obviously	ADV
ejpam-1797	212	2	,	,	PUNCT
ejpam-1797	212	3	anns(a	anns(a	NOUN
ejpam-1797	212	4	)	)	PUNCT
ejpam-1797	212	5	is	be	AUX
ejpam-1797	212	6	nonempty	nonempty	ADJ
ejpam-1797	212	7	since	since	SCONJ
ejpam-1797	212	8	0	0	NUM
ejpam-1797	212	9	∈	∈	PROPN
ejpam-1797	212	10	anns(a	anns(a	NOUN
ejpam-1797	212	11	)	)	PUNCT
ejpam-1797	212	12	.	.	PUNCT
ejpam-1797	213	1	also	also	ADV
ejpam-1797	213	2	,	,	PUNCT
ejpam-1797	213	3	if	if	SCONJ
ejpam-1797	213	4	a	a	DET
ejpam-1797	213	5	,	,	PUNCT
ejpam-1797	213	6	b	b	PROPN
ejpam-1797	213	7	∈	∈	PROPN
ejpam-1797	213	8	anns(a	anns(a	NOUN
ejpam-1797	213	9	)	)	PUNCT
ejpam-1797	213	10	,	,	PUNCT
ejpam-1797	213	11	then	then	ADV
ejpam-1797	213	12	a+	a+	PUNCT
ejpam-1797	213	13	b	b	X
ejpam-1797	213	14	∈	∈	PROPN
ejpam-1797	213	15	anns(a	anns(a	NOUN
ejpam-1797	213	16	)	)	PUNCT
ejpam-1797	213	17	.	.	PUNCT
ejpam-1797	214	1	let	let	VERB
ejpam-1797	214	2	x	x	PUNCT
ejpam-1797	214	3	∈	∈	PROPN
ejpam-1797	214	4	anns(a	anns(a	NOUN
ejpam-1797	214	5	)	)	PUNCT
ejpam-1797	214	6	and	and	CCONJ
ejpam-1797	214	7	s1	s1	NOUN
ejpam-1797	214	8	,	,	PUNCT
ejpam-1797	214	9	s2	s2	PROPN
ejpam-1797	214	10	∈	∈	PROPN
ejpam-1797	214	11	s.	s.	PROPN
ejpam-1797	214	12	then	then	ADV
ejpam-1797	214	13	,	,	PUNCT
ejpam-1797	214	14	axs	axs	PROPN
ejpam-1797	214	15	=	=	SYM
ejpam-1797	214	16	0	0	NUM
ejpam-1797	214	17	and	and	CCONJ
ejpam-1797	214	18	asx	asx	NOUN
ejpam-1797	214	19	=	=	NOUN
ejpam-1797	214	20	0	0	NUM
ejpam-1797	214	21	for	for	ADP
ejpam-1797	214	22	all	all	DET
ejpam-1797	214	23	s	s	PART
ejpam-1797	214	24	∈	∈	PROPN
ejpam-1797	214	25	s	s	X
ejpam-1797	214	26	and	and	CCONJ
ejpam-1797	214	27	so	so	ADV
ejpam-1797	214	28	axs1s2s	axs1s2s	NOUN
ejpam-1797	214	29	=	=	PUNCT
ejpam-1797	214	30	0	0	PUNCT
ejpam-1797	214	31	and	and	CCONJ
ejpam-1797	214	32	asxs1s2	asxs1s2	X
ejpam-1797	214	33	=	=	NOUN
ejpam-1797	214	34	0	0	NUM
ejpam-1797	214	35	for	for	ADP
ejpam-1797	214	36	all	all	DET
ejpam-1797	214	37	s	s	PROPN
ejpam-1797	214	38	∈	∈	PROPN
ejpam-1797	214	39	s.	s.	PROPN
ejpam-1797	214	40	this	this	PRON
ejpam-1797	214	41	leads	lead	VERB
ejpam-1797	214	42	to	to	ADP
ejpam-1797	214	43	xs1s2	xs1s2	PROPN
ejpam-1797	214	44	∈	∈	PROPN
ejpam-1797	214	45	anns(a	anns(a	NOUN
ejpam-1797	214	46	)	)	PUNCT
ejpam-1797	214	47	.	.	PUNCT
ejpam-1797	215	1	hence	hence	ADV
ejpam-1797	215	2	,	,	PUNCT
ejpam-1797	215	3	anns(a	anns(a	NOUN
ejpam-1797	215	4	)	)	PUNCT
ejpam-1797	215	5	is	be	AUX
ejpam-1797	215	6	a	a	DET
ejpam-1797	215	7	right	right	ADJ
ejpam-1797	215	8	ideal	ideal	NOUN
ejpam-1797	215	9	of	of	ADP
ejpam-1797	215	10	s.	s.	PROPN
ejpam-1797	215	11	also	also	ADV
ejpam-1797	215	12	as1s2	as1s2	PROPN
ejpam-1797	215	13	xs	xs	PROPN
ejpam-1797	215	14	⊆	⊆	NUM
ejpam-1797	215	15	axs	axs	PROPN
ejpam-1797	215	16	=	=	SYM
ejpam-1797	215	17	0	0	NUM
ejpam-1797	215	18	and	and	CCONJ
ejpam-1797	215	19	ass1s2	ass1s2	PROPN
ejpam-1797	215	20	x	x	SYM
ejpam-1797	215	21	⊆	⊆	NUM
ejpam-1797	215	22	asx	asx	NOUN
ejpam-1797	215	23	=	=	NOUN
ejpam-1797	215	24	0	0	NUM
ejpam-1797	215	25	for	for	ADP
ejpam-1797	215	26	all	all	DET
ejpam-1797	215	27	s	s	PART
ejpam-1797	215	28	∈	∈	NOUN
ejpam-1797	215	29	s	s	NOUN
ejpam-1797	215	30	as	as	SCONJ
ejpam-1797	215	31	a	a	PRON
ejpam-1797	215	32	is	be	AUX
ejpam-1797	215	33	a	a	DET
ejpam-1797	215	34	right	right	ADJ
ejpam-1797	215	35	ideal	ideal	NOUN
ejpam-1797	215	36	of	of	ADP
ejpam-1797	215	37	s.	s.	PROPN
ejpam-1797	215	38	so	so	ADV
ejpam-1797	215	39	s1s2	s1s2	PROPN
ejpam-1797	215	40	x	x	SYM
ejpam-1797	215	41	∈	∈	PROPN
ejpam-1797	215	42	anns(a	anns(a	NOUN
ejpam-1797	215	43	)	)	PUNCT
ejpam-1797	215	44	.	.	PUNCT
ejpam-1797	216	1	hence	hence	ADV
ejpam-1797	216	2	,	,	PUNCT
ejpam-1797	216	3	anns(a	anns(a	NOUN
ejpam-1797	216	4	)	)	PUNCT
ejpam-1797	216	5	is	be	AUX
ejpam-1797	216	6	a	a	DET
ejpam-1797	216	7	left	left	ADJ
ejpam-1797	216	8	ideal	ideal	NOUN
ejpam-1797	216	9	of	of	ADP
ejpam-1797	216	10	s.	s.	PROPN
ejpam-1797	216	11	again	again	ADV
ejpam-1797	216	12	,	,	PUNCT
ejpam-1797	216	13	we	we	PRON
ejpam-1797	216	14	have	have	VERB
ejpam-1797	216	15	as1	as1	NOUN
ejpam-1797	216	16	xs2s	xs2s	PROPN
ejpam-1797	216	17	=	=	PROPN
ejpam-1797	216	18	(	(	PUNCT
ejpam-1797	216	19	as1	as1	NOUN
ejpam-1797	216	20	x)s2s	x)s2s	PROPN
ejpam-1797	216	21	=	=	PROPN
ejpam-1797	216	22	0	0	PUNCT
ejpam-1797	216	23	and	and	CCONJ
ejpam-1797	216	24	ass1	ass1	PROPN
ejpam-1797	216	25	xs2	xs2	PROPN
ejpam-1797	217	1	⊆	⊆	NUM
ejpam-1797	217	2	axs	axs	X
ejpam-1797	217	3	=	=	PUNCT
ejpam-1797	217	4	0	0	NUM
ejpam-1797	217	5	for	for	ADP
ejpam-1797	217	6	all	all	PRON
ejpam-1797	217	7	s	s	PART
ejpam-1797	217	8	∈	∈	PROPN
ejpam-1797	217	9	s.	s.	PROPN
ejpam-1797	218	1	so	so	ADV
ejpam-1797	218	2	s1	s1	PROPN
ejpam-1797	218	3	xs2	xs2	PROPN
ejpam-1797	218	4	∈	∈	PROPN
ejpam-1797	218	5	anns(a	anns(a	NOUN
ejpam-1797	218	6	)	)	PUNCT
ejpam-1797	218	7	.	.	PUNCT
ejpam-1797	219	1	hence	hence	ADV
ejpam-1797	219	2	,	,	PUNCT
ejpam-1797	219	3	anns(a	anns(a	NOUN
ejpam-1797	219	4	)	)	PUNCT
ejpam-1797	219	5	is	be	AUX
ejpam-1797	219	6	a	a	DET
ejpam-1797	219	7	lateral	lateral	ADJ
ejpam-1797	219	8	ideal	ideal	NOUN
ejpam-1797	219	9	of	of	ADP
ejpam-1797	219	10	s.	s.	PROPN
ejpam-1797	219	11	thus	thus	ADV
ejpam-1797	219	12	,	,	PUNCT
ejpam-1797	219	13	anns(a	anns(a	NOUN
ejpam-1797	219	14	)	)	PUNCT
ejpam-1797	219	15	is	be	AUX
ejpam-1797	219	16	an	an	DET
ejpam-1797	219	17	ideal	ideal	NOUN
ejpam-1797	219	18	of	of	ADP
ejpam-1797	219	19	s.	s.	PROPN
ejpam-1797	219	20	now	now	ADV
ejpam-1797	219	21	let	let	VERB
ejpam-1797	219	22	a	a	DET
ejpam-1797	219	23	,	,	PUNCT
ejpam-1797	219	24	a	a	DET
ejpam-1797	219	25	+	+	NOUN
ejpam-1797	219	26	b	b	NOUN
ejpam-1797	219	27	∈	∈	PROPN
ejpam-1797	219	28	anns(a	anns(a	NOUN
ejpam-1797	219	29	)	)	PUNCT
ejpam-1797	219	30	.	.	PUNCT
ejpam-1797	220	1	then	then	ADV
ejpam-1797	220	2	aas	aas	PROPN
ejpam-1797	220	3	=	=	NOUN
ejpam-1797	220	4	0	0	NUM
ejpam-1797	220	5	=	=	SYM
ejpam-1797	220	6	a(a+	a(a+	NOUN
ejpam-1797	220	7	b)s	b)s	NOUN
ejpam-1797	220	8	and	and	CCONJ
ejpam-1797	220	9	asa	asa	PROPN
ejpam-1797	220	10	=	=	SYM
ejpam-1797	220	11	0	0	NUM
ejpam-1797	220	12	=	=	PUNCT
ejpam-1797	220	13	as(a	as(a	NOUN
ejpam-1797	220	14	+	+	SYM
ejpam-1797	220	15	b	b	X
ejpam-1797	220	16	)	)	PUNCT
ejpam-1797	220	17	for	for	ADP
ejpam-1797	220	18	all	all	DET
ejpam-1797	220	19	s	s	PROPN
ejpam-1797	220	20	∈	∈	PROPN
ejpam-1797	220	21	s.	s.	PROPN
ejpam-1797	220	22	this	this	PRON
ejpam-1797	220	23	implies	imply	VERB
ejpam-1797	220	24	abs	ab	NOUN
ejpam-1797	220	25	=	=	SYM
ejpam-1797	220	26	0	0	NUM
ejpam-1797	220	27	and	and	CCONJ
ejpam-1797	220	28	asb	asb	PROPN
ejpam-1797	220	29	=	=	PROPN
ejpam-1797	220	30	0	0	NUM
ejpam-1797	220	31	for	for	ADP
ejpam-1797	220	32	all	all	PRON
ejpam-1797	220	33	s	s	PART
ejpam-1797	220	34	∈	∈	PROPN
ejpam-1797	220	35	s.	s.	PROPN
ejpam-1797	221	1	so	so	ADV
ejpam-1797	221	2	,	,	PUNCT
ejpam-1797	221	3	b	b	PROPN
ejpam-1797	221	4	∈	∈	PROPN
ejpam-1797	221	5	anns(a	anns(a	NOUN
ejpam-1797	221	6	)	)	PUNCT
ejpam-1797	221	7	.	.	PUNCT
ejpam-1797	222	1	this	this	PRON
ejpam-1797	222	2	proves	prove	VERB
ejpam-1797	222	3	that	that	SCONJ
ejpam-1797	222	4	anns(a	anns(a	NOUN
ejpam-1797	222	5	)	)	PUNCT
ejpam-1797	222	6	is	be	AUX
ejpam-1797	222	7	a	a	DET
ejpam-1797	222	8	k	k	NOUN
ejpam-1797	222	9	-	-	NOUN
ejpam-1797	222	10	ideal	ideal	NOUN
ejpam-1797	222	11	of	of	ADP
ejpam-1797	222	12	s.	s.	PROPN
ejpam-1797	222	13	following	follow	VERB
ejpam-1797	222	14	d.	d.	PROPN
ejpam-1797	222	15	m.	m.	PROPN
ejpam-1797	222	16	olson	olson	PROPN
ejpam-1797	222	17	,	,	PUNCT
ejpam-1797	222	18	g.a.p	g.a.p	PROPN
ejpam-1797	222	19	.	.	PUNCT
ejpam-1797	222	20	heyman	heyman	PROPN
ejpam-1797	222	21	and	and	CCONJ
ejpam-1797	222	22	h.	h.	PROPN
ejpam-1797	222	23	j.	j.	PROPN
ejpam-1797	222	24	l.	l.	PROPN
ejpam-1797	222	25	roux	roux	PROPN
ejpam-1797	223	1	[	[	X
ejpam-1797	223	2	26	26	NUM
ejpam-1797	223	3	]	]	PUNCT
ejpam-1797	223	4	,	,	PUNCT
ejpam-1797	223	5	we	we	PRON
ejpam-1797	223	6	define	define	VERB
ejpam-1797	223	7	he	he	PRON
ejpam-1797	223	8	weakly	weakly	ADJ
ejpam-1797	223	9	special	special	ADJ
ejpam-1797	223	10	radical	radical	ADJ
ejpam-1797	223	11	class	class	NOUN
ejpam-1797	223	12	in	in	ADP
ejpam-1797	223	13	ternary	ternary	ADJ
ejpam-1797	223	14	semirings	semiring	NOUN
ejpam-1797	223	15	as	as	SCONJ
ejpam-1797	223	16	follows	follow	VERB
ejpam-1797	223	17	:	:	PUNCT
ejpam-1797	223	18	definition	definition	NOUN
ejpam-1797	223	19	8	8	NUM
ejpam-1797	223	20	.	.	PUNCT
ejpam-1797	224	1	a	a	DET
ejpam-1797	224	2	classm	classm	NOUN
ejpam-1797	224	3	of	of	ADP
ejpam-1797	224	4	ternary	ternary	ADJ
ejpam-1797	224	5	semirings	semiring	NOUN
ejpam-1797	224	6	is	be	AUX
ejpam-1797	224	7	called	call	VERB
ejpam-1797	224	8	a	a	DET
ejpam-1797	224	9	weakly	weakly	ADJ
ejpam-1797	224	10	special	special	ADJ
ejpam-1797	224	11	radical	radical	ADJ
ejpam-1797	224	12	class	class	NOUN
ejpam-1797	224	13	ifm	ifm	NOUN
ejpam-1797	224	14	is	be	AUX
ejpam-1797	224	15	a	a	DET
ejpam-1797	224	16	hereditary	hereditary	ADJ
ejpam-1797	224	17	class	class	NOUN
ejpam-1797	224	18	of	of	ADP
ejpam-1797	224	19	semiprime	semiprime	NOUN
ejpam-1797	224	20	ternary	ternary	ADJ
ejpam-1797	224	21	semirings	semiring	NOUN
ejpam-1797	224	22	satisfying	satisfy	VERB
ejpam-1797	224	23	the	the	DET
ejpam-1797	224	24	following	follow	VERB
ejpam-1797	224	25	conditions	condition	NOUN
ejpam-1797	224	26	:	:	PUNCT
ejpam-1797	224	27	(	(	PUNCT
ejpam-1797	224	28	x	x	X
ejpam-1797	224	29	)	)	PUNCT
ejpam-1797	224	30	if	if	SCONJ
ejpam-1797	224	31	s	s	VERB
ejpam-1797	224	32	is	be	AUX
ejpam-1797	224	33	a	a	DET
ejpam-1797	224	34	ternary	ternary	ADJ
ejpam-1797	224	35	semiring	semiring	NOUN
ejpam-1797	224	36	and	and	CCONJ
ejpam-1797	224	37	s	s	NOUN
ejpam-1797	224	38	is	be	AUX
ejpam-1797	224	39	semi	semi	ADJ
ejpam-1797	224	40	-	-	ADJ
ejpam-1797	224	41	isomorphic	isomorphic	ADJ
ejpam-1797	224	42	to	to	ADP
ejpam-1797	224	43	t	t	PROPN
ejpam-1797	224	44	with	with	ADP
ejpam-1797	224	45	t	t	PROPN
ejpam-1797	224	46	∈m	∈m	NOUN
ejpam-1797	224	47	,	,	PUNCT
ejpam-1797	224	48	then	then	ADV
ejpam-1797	224	49	s	s	VERB
ejpam-1797	224	50	∈m	∈m	NOUN
ejpam-1797	224	51	.	.	PUNCT
ejpam-1797	225	1	t.	t.	PROPN
ejpam-1797	225	2	dutta	dutta	PROPN
ejpam-1797	225	3	,	,	PUNCT
ejpam-1797	225	4	k.	k.	PROPN
ejpam-1797	225	5	shum	shum	PROPN
ejpam-1797	225	6	,	,	PUNCT
ejpam-1797	225	7	s.	s.	PROPN
ejpam-1797	225	8	mandal	mandal	PROPN
ejpam-1797	225	9	/	/	SYM
ejpam-1797	225	10	eur	eur	PROPN
ejpam-1797	225	11	.	.	PUNCT
ejpam-1797	226	1	j.	j.	PROPN
ejpam-1797	226	2	pure	pure	PROPN
ejpam-1797	226	3	appl	appl	PROPN
ejpam-1797	226	4	.	.	PROPN
ejpam-1797	226	5	math	math	PROPN
ejpam-1797	226	6	,	,	PUNCT
ejpam-1797	226	7	5	5	NUM
ejpam-1797	226	8	(	(	PUNCT
ejpam-1797	226	9	2012	2012	NUM
ejpam-1797	226	10	)	)	PUNCT
ejpam-1797	226	11	,	,	PUNCT
ejpam-1797	226	12	401	401	NUM
ejpam-1797	226	13	-	-	SYM
ejpam-1797	226	14	413	413	NUM
ejpam-1797	226	15	407	407	NUM
ejpam-1797	226	16	(	(	PUNCT
ejpam-1797	226	17	z	z	NOUN
ejpam-1797	226	18	)	)	PUNCT
ejpam-1797	226	19	if	if	SCONJ
ejpam-1797	226	20	i	i	PRON
ejpam-1797	226	21	∈m	∈m	VERB
ejpam-1797	227	1	and	and	CCONJ
ejpam-1797	227	2	i	i	PRON
ejpam-1797	227	3	is	be	AUX
ejpam-1797	227	4	an	an	DET
ejpam-1797	227	5	ideal	ideal	NOUN
ejpam-1797	227	6	of	of	ADP
ejpam-1797	227	7	a	a	DET
ejpam-1797	227	8	ternary	ternary	ADJ
ejpam-1797	227	9	semiring	semire	VERB
ejpam-1797	227	10	s	s	NOUN
ejpam-1797	227	11	,	,	PUNCT
ejpam-1797	227	12	then	then	ADV
ejpam-1797	227	13	s	s	PROPN
ejpam-1797	227	14	/	/	SYM
ejpam-1797	227	15	anns(i	anns(i	NOUN
ejpam-1797	227	16	)	)	PUNCT
ejpam-1797	227	17	∈m	∈m	NOUN
ejpam-1797	227	18	.	.	PUNCT
ejpam-1797	228	1	we	we	PRON
ejpam-1797	228	2	give	give	VERB
ejpam-1797	228	3	the	the	DET
ejpam-1797	228	4	following	follow	VERB
ejpam-1797	228	5	crucial	crucial	ADJ
ejpam-1797	228	6	lemma	lemma	PROPN
ejpam-1797	228	7	.	.	PUNCT
ejpam-1797	229	1	it	it	PRON
ejpam-1797	229	2	is	be	AUX
ejpam-1797	229	3	noted	note	VERB
ejpam-1797	229	4	that	that	SCONJ
ejpam-1797	229	5	the	the	DET
ejpam-1797	229	6	h	h	NOUN
ejpam-1797	229	7	-	-	PUNCT
ejpam-1797	229	8	prime	prime	ADJ
ejpam-1797	229	9	and	and	CCONJ
ejpam-1797	229	10	h	h	NOUN
ejpam-1797	229	11	-	-	PUNCT
ejpam-1797	229	12	semiprime	semiprime	NOUN
ejpam-1797	229	13	ideals	ideal	NOUN
ejpam-1797	229	14	in	in	ADP
ejpam-1797	229	15	semirings	semiring	NOUN
ejpam-1797	229	16	and	and	CCONJ
ejpam-1797	229	17	γ	γ	NOUN
ejpam-1797	229	18	-	-	PUNCT
ejpam-1797	229	19	semirings	semiring	NOUN
ejpam-1797	229	20	have	have	AUX
ejpam-1797	229	21	been	be	AUX
ejpam-1797	229	22	recently	recently	ADV
ejpam-1797	229	23	investigated	investigate	VERB
ejpam-1797	229	24	by	by	ADP
ejpam-1797	229	25	s.	s.	PROPN
ejpam-1797	229	26	sardar	sardar	PROPN
ejpam-1797	229	27	and	and	CCONJ
ejpam-1797	229	28	others	other	NOUN
ejpam-1797	229	29	in	in	ADP
ejpam-1797	229	30	[	[	X
ejpam-1797	229	31	29	29	NUM
ejpam-1797	229	32	]	]	PUNCT
ejpam-1797	229	33	.	.	PUNCT
ejpam-1797	230	1	for	for	ADP
ejpam-1797	230	2	semiprime	semiprime	NOUN
ejpam-1797	230	3	ternary	ternary	PROPN
ejpam-1797	230	4	semirings	semiring	NOUN
ejpam-1797	230	5	,	,	PUNCT
ejpam-1797	230	6	we	we	PRON
ejpam-1797	230	7	have	have	VERB
ejpam-1797	230	8	the	the	DET
ejpam-1797	230	9	following	follow	VERB
ejpam-1797	230	10	main	main	ADJ
ejpam-1797	230	11	theorem	theorem	NOUN
ejpam-1797	230	12	.	.	PUNCT
ejpam-1797	230	13	theorem	theorem	NOUN
ejpam-1797	230	14	6	6	NUM
ejpam-1797	230	15	.	.	PUNCT
ejpam-1797	231	1	if	if	SCONJ
ejpam-1797	231	2	m	m	NOUN
ejpam-1797	231	3	is	be	AUX
ejpam-1797	231	4	a	a	DET
ejpam-1797	231	5	hereditary	hereditary	ADJ
ejpam-1797	231	6	class	class	NOUN
ejpam-1797	231	7	of	of	ADP
ejpam-1797	231	8	semiprime	semiprime	NOUN
ejpam-1797	231	9	ternary	ternary	ADJ
ejpam-1797	231	10	semirings	semiring	NOUN
ejpam-1797	231	11	which	which	PRON
ejpam-1797	231	12	satisfies	satisfy	VERB
ejpam-1797	231	13	properties	property	NOUN
ejpam-1797	231	14	:	:	PUNCT
ejpam-1797	231	15	“	"	PUNCT
ejpam-1797	231	16	if	if	SCONJ
ejpam-1797	231	17	s	s	VERB
ejpam-1797	231	18	∈	∈	PROPN
ejpam-1797	231	19	m	m	NOUN
ejpam-1797	231	20	and	and	CCONJ
ejpam-1797	231	21	s	s	VERB
ejpam-1797	231	22	is	be	AUX
ejpam-1797	231	23	semi	semi	ADJ
ejpam-1797	231	24	-	-	ADJ
ejpam-1797	231	25	isomorphic	isomorphic	ADJ
ejpam-1797	231	26	to	to	ADP
ejpam-1797	231	27	t	t	PROPN
ejpam-1797	231	28	then	then	ADV
ejpam-1797	231	29	t	t	PROPN
ejpam-1797	231	30	∈	∈	PROPN
ejpam-1797	231	31	m	m	INTJ
ejpam-1797	231	32	”	"	PUNCT
ejpam-1797	231	33	then	then	ADV
ejpam-1797	231	34	the	the	DET
ejpam-1797	231	35	following	follow	VERB
ejpam-1797	231	36	conditions	condition	NOUN
ejpam-1797	231	37	are	be	AUX
ejpam-1797	231	38	equivalent	equivalent	ADJ
ejpam-1797	231	39	:	:	PUNCT
ejpam-1797	231	40	(	(	PUNCT
ejpam-1797	231	41	1	1	X
ejpam-1797	231	42	)	)	PUNCT
ejpam-1797	231	43	if	if	SCONJ
ejpam-1797	231	44	a∈m	a∈m	NOUN
ejpam-1797	231	45	and	and	CCONJ
ejpam-1797	231	46	a	a	PRON
ejpam-1797	231	47	is	be	AUX
ejpam-1797	231	48	an	an	DET
ejpam-1797	231	49	ideal	ideal	NOUN
ejpam-1797	231	50	of	of	ADP
ejpam-1797	231	51	s	s	PROPN
ejpam-1797	231	52	,	,	PUNCT
ejpam-1797	231	53	then	then	ADV
ejpam-1797	231	54	s	s	PROPN
ejpam-1797	231	55	/	/	SYM
ejpam-1797	231	56	anns(a	anns(a	NOUN
ejpam-1797	231	57	)	)	PUNCT
ejpam-1797	231	58	∈m	∈m	NOUN
ejpam-1797	231	59	;	;	PUNCT
ejpam-1797	231	60	(	(	PUNCT
ejpam-1797	231	61	2	2	X
ejpam-1797	231	62	)	)	PUNCT
ejpam-1797	231	63	if	if	SCONJ
ejpam-1797	231	64	a∈m	a∈m	NOUN
ejpam-1797	231	65	with	with	ADP
ejpam-1797	231	66	a	a	DET
ejpam-1797	231	67	an	an	DET
ejpam-1797	231	68	ideal	ideal	NOUN
ejpam-1797	231	69	of	of	ADP
ejpam-1797	231	70	s	s	NOUN
ejpam-1797	231	71	and	and	CCONJ
ejpam-1797	231	72	anns(a	anns(a	NOUN
ejpam-1797	231	73	)	)	PUNCT
ejpam-1797	231	74	=	=	SYM
ejpam-1797	231	75	0	0	NUM
ejpam-1797	231	76	,	,	PUNCT
ejpam-1797	231	77	then	then	ADV
ejpam-1797	231	78	s	s	VERB
ejpam-1797	231	79	∈m	∈m	NOUN
ejpam-1797	231	80	.	.	PUNCT
ejpam-1797	232	1	(	(	PUNCT
ejpam-1797	232	2	3	3	X
ejpam-1797	232	3	)	)	PUNCT
ejpam-1797	232	4	if	if	SCONJ
ejpam-1797	232	5	a∈m	a∈m	NOUN
ejpam-1797	232	6	and	and	CCONJ
ejpam-1797	232	7	a	a	PRON
ejpam-1797	232	8	is	be	AUX
ejpam-1797	232	9	an	an	DET
ejpam-1797	232	10	essential	essential	ADJ
ejpam-1797	232	11	ideal	ideal	NOUN
ejpam-1797	232	12	of	of	ADP
ejpam-1797	232	13	s	s	PROPN
ejpam-1797	232	14	,	,	PUNCT
ejpam-1797	232	15	then	then	ADV
ejpam-1797	232	16	s	s	VERB
ejpam-1797	232	17	∈m	∈m	NOUN
ejpam-1797	232	18	.	.	PUNCT
ejpam-1797	233	1	proof	proof	NOUN
ejpam-1797	233	2	.	.	PUNCT
ejpam-1797	234	1	for	for	ADP
ejpam-1797	234	2	an	an	DET
ejpam-1797	234	3	hereditary	hereditary	ADJ
ejpam-1797	234	4	radical	radical	ADJ
ejpam-1797	234	5	class	class	NOUN
ejpam-1797	234	6	of	of	ADP
ejpam-1797	234	7	semiprime	semiprime	NOUN
ejpam-1797	234	8	ternary	ternary	PROPN
ejpam-1797	234	9	semirings	semiring	NOUN
ejpam-1797	234	10	,	,	PUNCT
ejpam-1797	234	11	it	it	PRON
ejpam-1797	234	12	is	be	AUX
ejpam-1797	234	13	clear	clear	ADJ
ejpam-1797	234	14	that	that	SCONJ
ejpam-1797	234	15	(	(	PUNCT
ejpam-1797	234	16	1	1	X
ejpam-1797	234	17	)	)	PUNCT
ejpam-1797	234	18	⇒	⇒	NOUN
ejpam-1797	234	19	(	(	PUNCT
ejpam-1797	234	20	2	2	NUM
ejpam-1797	234	21	)	)	PUNCT
ejpam-1797	234	22	.	.	PUNCT
ejpam-1797	235	1	for	for	ADP
ejpam-1797	235	2	(	(	PUNCT
ejpam-1797	235	3	2	2	X
ejpam-1797	235	4	)	)	PUNCT
ejpam-1797	235	5	⇒	⇒	NOUN
ejpam-1797	235	6	(	(	PUNCT
ejpam-1797	235	7	3	3	NUM
ejpam-1797	235	8	)	)	PUNCT
ejpam-1797	235	9	,	,	PUNCT
ejpam-1797	235	10	suppose	suppose	VERB
ejpam-1797	235	11	that	that	SCONJ
ejpam-1797	235	12	a	a	PRON
ejpam-1797	235	13	is	be	AUX
ejpam-1797	235	14	an	an	DET
ejpam-1797	235	15	essential	essential	ADJ
ejpam-1797	235	16	ideal	ideal	NOUN
ejpam-1797	235	17	of	of	ADP
ejpam-1797	235	18	s	s	PRON
ejpam-1797	235	19	and	and	CCONJ
ejpam-1797	235	20	a	a	DET
ejpam-1797	235	21	∈	∈	NOUN
ejpam-1797	235	22	m	m	VERB
ejpam-1797	235	23	.	.	PUNCT
ejpam-1797	236	1	now	now	ADV
ejpam-1797	236	2	let	let	VERB
ejpam-1797	236	3	a	a	DET
ejpam-1797	236	4	∈	∈	PROPN
ejpam-1797	236	5	a∩	a∩	PROPN
ejpam-1797	236	6	anns(a	anns(a	NOUN
ejpam-1797	236	7	)	)	PUNCT
ejpam-1797	236	8	,	,	PUNCT
ejpam-1797	236	9	then	then	ADV
ejpam-1797	236	10	aas	aas	PROPN
ejpam-1797	236	11	=	=	PROPN
ejpam-1797	236	12	0	0	PROPN
ejpam-1797	236	13	and	and	CCONJ
ejpam-1797	236	14	asa	asa	PROPN
ejpam-1797	236	15	=	=	SYM
ejpam-1797	236	16	0	0	PROPN
ejpam-1797	236	17	.	.	PUNCT
ejpam-1797	237	1	this	this	PRON
ejpam-1797	237	2	implies	imply	VERB
ejpam-1797	237	3	that	that	DET
ejpam-1797	237	4	aaa	aaa	NOUN
ejpam-1797	237	5	=	=	SYM
ejpam-1797	237	6	0	0	NUM
ejpam-1797	237	7	and	and	CCONJ
ejpam-1797	237	8	aaa	aaa	NOUN
ejpam-1797	237	9	=	=	SYM
ejpam-1797	237	10	0	0	NUM
ejpam-1797	237	11	since	since	SCONJ
ejpam-1797	237	12	a	a	DET
ejpam-1797	237	13	∈	∈	PROPN
ejpam-1797	237	14	a	a	PRON
ejpam-1797	237	15	and	and	CCONJ
ejpam-1797	237	16	a⊆	a⊆	PROPN
ejpam-1797	237	17	s.	s.	PROPN
ejpam-1797	237	18	thus	thus	ADV
ejpam-1797	237	19	aaaaa	aaaaa	VERB
ejpam-1797	237	20	=	=	SYM
ejpam-1797	237	21	0	0	NUM
ejpam-1797	237	22	,	,	PUNCT
ejpam-1797	237	23	aaaaaaa	aaaaaaa	NOUN
ejpam-1797	237	24	=	=	SYM
ejpam-1797	237	25	0	0	NUM
ejpam-1797	237	26	,	,	PUNCT
ejpam-1797	237	27	aaaaaaa=	aaaaaaa=	X
ejpam-1797	237	28	0	0	NUM
ejpam-1797	237	29	and	and	CCONJ
ejpam-1797	237	30	aaaaaaa	aaaaaaa	NOUN
ejpam-1797	237	31	=	=	SYM
ejpam-1797	237	32	0.thus	0.thus	NUM
ejpam-1797	237	33	,	,	PUNCT
ejpam-1797	237	34	a	a	DET
ejpam-1797	237	35	=	=	NOUN
ejpam-1797	237	36	0	0	NUM
ejpam-1797	237	37	since	since	SCONJ
ejpam-1797	237	38	a	a	PRON
ejpam-1797	237	39	is	be	AUX
ejpam-1797	237	40	semiprime	semiprime	NOUN
ejpam-1797	237	41	.	.	PUNCT
ejpam-1797	238	1	because	because	SCONJ
ejpam-1797	238	2	a	a	PRON
ejpam-1797	238	3	is	be	AUX
ejpam-1797	238	4	essential	essential	ADJ
ejpam-1797	238	5	,	,	PUNCT
ejpam-1797	238	6	anns(a	anns(a	NOUN
ejpam-1797	238	7	)	)	PUNCT
ejpam-1797	238	8	=	=	SYM
ejpam-1797	239	1	0	0	X
ejpam-1797	239	2	.	.	PUNCT
ejpam-1797	239	3	now	now	ADV
ejpam-1797	239	4	by	by	ADP
ejpam-1797	239	5	(	(	PUNCT
ejpam-1797	239	6	2	2	NUM
ejpam-1797	239	7	)	)	PUNCT
ejpam-1797	239	8	,	,	PUNCT
ejpam-1797	239	9	s	s	VERB
ejpam-1797	239	10	∈	∈	PROPN
ejpam-1797	239	11	m	m	VERB
ejpam-1797	239	12	.	.	PUNCT
ejpam-1797	240	1	assume	assume	VERB
ejpam-1797	240	2	that	that	SCONJ
ejpam-1797	240	3	(	(	PUNCT
ejpam-1797	240	4	3	3	X
ejpam-1797	240	5	)	)	PUNCT
ejpam-1797	240	6	holds	hold	VERB
ejpam-1797	240	7	and	and	CCONJ
ejpam-1797	240	8	consider	consider	VERB
ejpam-1797	240	9	a	a	DET
ejpam-1797	240	10	ternary	ternary	ADJ
ejpam-1797	240	11	semiring	semire	VERB
ejpam-1797	240	12	a	a	DET
ejpam-1797	240	13	∈m	∈m	NOUN
ejpam-1797	240	14	with	with	ADP
ejpam-1797	240	15	a	a	DET
ejpam-1797	240	16	an	an	DET
ejpam-1797	240	17	ideal	ideal	NOUN
ejpam-1797	240	18	of	of	ADP
ejpam-1797	240	19	s.	s.	PROPN
ejpam-1797	240	20	then	then	ADV
ejpam-1797	240	21	,	,	PUNCT
ejpam-1797	240	22	by	by	ADP
ejpam-1797	240	23	using	use	VERB
ejpam-1797	240	24	the	the	DET
ejpam-1797	240	25	arguments	argument	NOUN
ejpam-1797	240	26	as	as	ADP
ejpam-1797	240	27	in	in	ADP
ejpam-1797	240	28	above	above	ADV
ejpam-1797	240	29	,	,	PUNCT
ejpam-1797	240	30	we	we	PRON
ejpam-1797	240	31	see	see	VERB
ejpam-1797	240	32	that	that	SCONJ
ejpam-1797	240	33	a∩	a∩	PROPN
ejpam-1797	240	34	anns(a	anns(a	NOUN
ejpam-1797	240	35	)	)	PUNCT
ejpam-1797	240	36	=	=	PUNCT
ejpam-1797	241	1	(	(	PUNCT
ejpam-1797	241	2	0	0	NUM
ejpam-1797	241	3	)	)	PUNCT
ejpam-1797	241	4	since	since	SCONJ
ejpam-1797	241	5	a	a	PRON
ejpam-1797	241	6	is	be	AUX
ejpam-1797	241	7	semiprime	semiprime	NOUN
ejpam-1797	241	8	.	.	PUNCT
ejpam-1797	242	1	now	now	ADV
ejpam-1797	242	2	(	(	PUNCT
ejpam-1797	242	3	a+	a+	PUNCT
ejpam-1797	242	4	anns(a))/anns(a	anns(a))/anns(a	NOUN
ejpam-1797	242	5	)	)	PUNCT
ejpam-1797	242	6	is	be	AUX
ejpam-1797	242	7	a	a	DET
ejpam-1797	242	8	nonzero	nonzero	ADJ
ejpam-1797	242	9	ideal	ideal	NOUN
ejpam-1797	242	10	of	of	ADP
ejpam-1797	242	11	s	s	NOUN
ejpam-1797	242	12	/	/	SYM
ejpam-1797	242	13	anns(a	anns(a	NOUN
ejpam-1797	242	14	)	)	PUNCT
ejpam-1797	242	15	.	.	PUNCT
ejpam-1797	243	1	also	also	ADV
ejpam-1797	243	2	,	,	PUNCT
ejpam-1797	243	3	we	we	PRON
ejpam-1797	243	4	have	have	AUX
ejpam-1797	243	5	a=	a=	VERB
ejpam-1797	243	6	a/(a∩	a/(a∩	X
ejpam-1797	243	7	anns(a))≃	anns(a))≃	PROPN
ejpam-1797	243	8	(	(	PUNCT
ejpam-1797	243	9	a+	a+	PUNCT
ejpam-1797	243	10	anns(a))/anns(a	anns(a))/anns(a	NOUN
ejpam-1797	243	11	)	)	PUNCT
ejpam-1797	243	12	,	,	PUNCT
ejpam-1797	243	13	and	and	CCONJ
ejpam-1797	243	14	so	so	ADV
ejpam-1797	243	15	by	by	ADP
ejpam-1797	243	16	the	the	DET
ejpam-1797	243	17	given	give	VERB
ejpam-1797	243	18	condition	condition	NOUN
ejpam-1797	243	19	,	,	PUNCT
ejpam-1797	243	20	we	we	PRON
ejpam-1797	243	21	deduce	deduce	VERB
ejpam-1797	243	22	that	that	PRON
ejpam-1797	243	23	(	(	PUNCT
ejpam-1797	243	24	a+	a+	PUNCT
ejpam-1797	243	25	anns(a))/anns(a	anns(a))/anns(a	PROPN
ejpam-1797	243	26	)	)	PUNCT
ejpam-1797	243	27	∈	∈	PROPN
ejpam-1797	243	28	m	m	NOUN
ejpam-1797	243	29	.	.	PUNCT
ejpam-1797	244	1	however,(a+	however,(a+	ADJ
ejpam-1797	244	2	anns(a))/anns(a	anns(a))/anns(a	NOUN
ejpam-1797	244	3	)	)	PUNCT
ejpam-1797	244	4	is	be	AUX
ejpam-1797	244	5	an	an	DET
ejpam-1797	244	6	essential	essential	ADJ
ejpam-1797	244	7	ideal	ideal	NOUN
ejpam-1797	244	8	in	in	ADP
ejpam-1797	244	9	s	s	NOUN
ejpam-1797	244	10	/	/	SYM
ejpam-1797	244	11	anns(a),for	anns(a),for	ADJ
ejpam-1797	244	12	if	if	SCONJ
ejpam-1797	244	13	h	h	NOUN
ejpam-1797	244	14	/	/	SYM
ejpam-1797	244	15	anns(a	anns(a	NOUN
ejpam-1797	244	16	)	)	PUNCT
ejpam-1797	244	17	is	be	AUX
ejpam-1797	244	18	a	a	DET
ejpam-1797	244	19	nonzero	nonzero	ADJ
ejpam-1797	244	20	ideal	ideal	NOUN
ejpam-1797	244	21	of	of	ADP
ejpam-1797	244	22	s	s	NOUN
ejpam-1797	244	23	/	/	SYM
ejpam-1797	244	24	anns(a	anns(a	NOUN
ejpam-1797	244	25	)	)	PUNCT
ejpam-1797	244	26	,	,	PUNCT
ejpam-1797	244	27	then	then	ADV
ejpam-1797	244	28	h	h	PROPN
ejpam-1797	244	29	∩	∩	PROPN
ejpam-1797	244	30	a	a	PRON
ejpam-1797	244	31	6=	6=	NUM
ejpam-1797	244	32	0	0	NUM
ejpam-1797	244	33	,	,	PUNCT
ejpam-1797	244	34	otherwise	otherwise	ADV
ejpam-1797	244	35	,	,	PUNCT
ejpam-1797	244	36	ahs	ahs	INTJ
ejpam-1797	244	37	,	,	PUNCT
ejpam-1797	244	38	ash	ash	NOUN
ejpam-1797	244	39	⊆	⊆	NUM
ejpam-1797	244	40	h	h	NOUN
ejpam-1797	244	41	∩	∩	NOUN
ejpam-1797	244	42	a	a	PRON
ejpam-1797	244	43	would	would	AUX
ejpam-1797	244	44	be	be	AUX
ejpam-1797	244	45	zero	zero	NUM
ejpam-1797	244	46	which	which	PRON
ejpam-1797	244	47	implies	imply	VERB
ejpam-1797	244	48	that	that	SCONJ
ejpam-1797	244	49	h	h	NOUN
ejpam-1797	244	50	⊆	⊆	NUM
ejpam-1797	244	51	anns(a	anns(a	NOUN
ejpam-1797	244	52	)	)	PUNCT
ejpam-1797	244	53	which	which	PRON
ejpam-1797	244	54	is	be	AUX
ejpam-1797	244	55	impossible	impossible	ADJ
ejpam-1797	244	56	.	.	PUNCT
ejpam-1797	245	1	also	also	ADV
ejpam-1797	245	2	,	,	PUNCT
ejpam-1797	245	3	(	(	PUNCT
ejpam-1797	245	4	h∩a)∩anns(a	h∩a)∩anns(a	PROPN
ejpam-1797	245	5	)	)	PUNCT
ejpam-1797	245	6	=	=	PUNCT
ejpam-1797	245	7	(	(	PUNCT
ejpam-1797	245	8	0	0	NUM
ejpam-1797	245	9	)	)	PUNCT
ejpam-1797	245	10	.	.	PUNCT
ejpam-1797	246	1	thus	thus	ADV
ejpam-1797	246	2	,	,	PUNCT
ejpam-1797	246	3	there	there	PRON
ejpam-1797	246	4	exits	exit	VERB
ejpam-1797	246	5	an	an	DET
ejpam-1797	246	6	a	a	PRON
ejpam-1797	246	7	(	(	PUNCT
ejpam-1797	246	8	6=	6=	NOUN
ejpam-1797	246	9	0	0	NUM
ejpam-1797	246	10	)	)	PUNCT
ejpam-1797	246	11	∈	∈	PROPN
ejpam-1797	246	12	h∩a	h∩a	NOUN
ejpam-1797	246	13	such	such	ADJ
ejpam-1797	246	14	that	that	SCONJ
ejpam-1797	246	15	a	a	DET
ejpam-1797	246	16	/	/	SYM
ejpam-1797	246	17	anns(a	anns(a	NOUN
ejpam-1797	246	18	)	)	PUNCT
ejpam-1797	246	19	6=	6=	ADP
ejpam-1797	246	20	0	0	NUM
ejpam-1797	246	21	/	/	SYM
ejpam-1797	246	22	anns(a	anns(a	NOUN
ejpam-1797	246	23	)	)	PUNCT
ejpam-1797	246	24	.	.	PUNCT
ejpam-1797	247	1	also	also	ADV
ejpam-1797	247	2	,	,	PUNCT
ejpam-1797	247	3	a	a	DET
ejpam-1797	247	4	/	/	SYM
ejpam-1797	247	5	anns(a	anns(a	NOUN
ejpam-1797	247	6	)	)	PUNCT
ejpam-1797	247	7	∈	∈	PROPN
ejpam-1797	248	1	[	[	X
ejpam-1797	248	2	a+	a+	X
ejpam-1797	248	3	anns(a)/anns(a	anns(a)/anns(a	NOUN
ejpam-1797	248	4	)	)	PUNCT
ejpam-1797	248	5	]	]	PUNCT
ejpam-1797	248	6	.	.	PUNCT
ejpam-1797	249	1	hence	hence	ADV
ejpam-1797	249	2	,	,	PUNCT
ejpam-1797	249	3	[	[	X
ejpam-1797	249	4	a+	a+	X
ejpam-1797	249	5	anns(a)/anns(a	anns(a)/anns(a	NOUN
ejpam-1797	249	6	)	)	PUNCT
ejpam-1797	249	7	]	]	PUNCT
ejpam-1797	250	1	⋂	⋂	PROPN
ejpam-1797	250	2	(	(	PUNCT
ejpam-1797	250	3	h	h	NOUN
ejpam-1797	250	4	/	/	SYM
ejpam-1797	250	5	anns(a	anns(a	NOUN
ejpam-1797	250	6	)	)	PUNCT
ejpam-1797	250	7	)	)	PUNCT
ejpam-1797	251	1	6=	6=	ADP
ejpam-1797	251	2	0	0	NUM
ejpam-1797	251	3	/	/	SYM
ejpam-1797	251	4	anns(a	anns(a	NOUN
ejpam-1797	251	5	)	)	PUNCT
ejpam-1797	251	6	.	.	PUNCT
ejpam-1797	252	1	but	but	CCONJ
ejpam-1797	252	2	then	then	ADV
ejpam-1797	252	3	,	,	PUNCT
ejpam-1797	252	4	by	by	ADP
ejpam-1797	252	5	the	the	DET
ejpam-1797	252	6	condition	condition	NOUN
ejpam-1797	252	7	(	(	PUNCT
ejpam-1797	252	8	3	3	NUM
ejpam-1797	252	9	)	)	PUNCT
ejpam-1797	252	10	,	,	PUNCT
ejpam-1797	252	11	we	we	PRON
ejpam-1797	252	12	have	have	VERB
ejpam-1797	252	13	s	s	NOUN
ejpam-1797	252	14	/	/	SYM
ejpam-1797	252	15	anns(a	anns(a	NOUN
ejpam-1797	252	16	)	)	PUNCT
ejpam-1797	252	17	∈m	∈m	NOUN
ejpam-1797	252	18	and	and	CCONJ
ejpam-1797	252	19	so	so	ADV
ejpam-1797	252	20	,	,	PUNCT
ejpam-1797	252	21	condition	condition	NOUN
ejpam-1797	252	22	(	(	PUNCT
ejpam-1797	252	23	1	1	X
ejpam-1797	252	24	)	)	PUNCT
ejpam-1797	252	25	follows	follow	VERB
ejpam-1797	252	26	,	,	PUNCT
ejpam-1797	252	27	as	as	SCONJ
ejpam-1797	252	28	desired	desire	VERB
ejpam-1797	252	29	.	.	PUNCT
ejpam-1797	253	1	lemma	lemma	PROPN
ejpam-1797	253	2	3	3	X
ejpam-1797	253	3	.	.	PUNCT
ejpam-1797	254	1	if	if	SCONJ
ejpam-1797	254	2	the	the	DET
ejpam-1797	254	3	ternary	ternary	ADJ
ejpam-1797	254	4	semirings	semiring	NOUN
ejpam-1797	254	5	s	s	PART
ejpam-1797	254	6	≃	≃	PROPN
ejpam-1797	254	7	t	t	PROPN
ejpam-1797	254	8	and	and	CCONJ
ejpam-1797	254	9	t	t	PROPN
ejpam-1797	254	10	are	be	AUX
ejpam-1797	254	11	semiprime	semiprime	NOUN
ejpam-1797	254	12	,	,	PUNCT
ejpam-1797	254	13	then	then	ADV
ejpam-1797	254	14	s	s	VERB
ejpam-1797	254	15	is	be	AUX
ejpam-1797	254	16	semiprime	semiprime	NOUN
ejpam-1797	254	17	.	.	PUNCT
ejpam-1797	255	1	proof	proof	NOUN
ejpam-1797	255	2	.	.	PUNCT
ejpam-1797	256	1	suppose	suppose	VERB
ejpam-1797	256	2	that	that	SCONJ
ejpam-1797	256	3	φ	φ	PROPN
ejpam-1797	256	4	is	be	AUX
ejpam-1797	256	5	a	a	DET
ejpam-1797	256	6	semi	semi	NOUN
ejpam-1797	256	7	-	-	NOUN
ejpam-1797	256	8	isomorphism	isomorphism	NOUN
ejpam-1797	256	9	and	and	CCONJ
ejpam-1797	256	10	a	a	PRON
ejpam-1797	256	11	is	be	AUX
ejpam-1797	256	12	an	an	DET
ejpam-1797	256	13	ideal	ideal	NOUN
ejpam-1797	256	14	of	of	ADP
ejpam-1797	256	15	s	s	PRON
ejpam-1797	256	16	such	such	ADJ
ejpam-1797	256	17	that	that	DET
ejpam-1797	256	18	a3	a3	NOUN
ejpam-1797	256	19	=	=	PUNCT
ejpam-1797	256	20	(	(	PUNCT
ejpam-1797	256	21	0	0	NUM
ejpam-1797	256	22	)	)	PUNCT
ejpam-1797	256	23	.	.	PUNCT
ejpam-1797	257	1	then	then	ADV
ejpam-1797	257	2	,	,	PUNCT
ejpam-1797	257	3	φ(a	φ(a	ADJ
ejpam-1797	257	4	)	)	PUNCT
ejpam-1797	257	5	is	be	AUX
ejpam-1797	257	6	an	an	DET
ejpam-1797	257	7	ideal	ideal	NOUN
ejpam-1797	257	8	of	of	ADP
ejpam-1797	257	9	t	t	PROPN
ejpam-1797	257	10	and	and	CCONJ
ejpam-1797	257	11	φ(a3	φ(a3	NOUN
ejpam-1797	257	12	)	)	PUNCT
ejpam-1797	257	13	=	=	PUNCT
ejpam-1797	258	1	(	(	PUNCT
ejpam-1797	258	2	0)⇒	0)⇒	X
ejpam-1797	258	3	[	[	X
ejpam-1797	258	4	φ(a)]3	φ(a)]3	NOUN
ejpam-1797	258	5	=	=	SYM
ejpam-1797	258	6	(	(	PUNCT
ejpam-1797	258	7	0	0	NUM
ejpam-1797	258	8	)	)	PUNCT
ejpam-1797	258	9	.	.	PUNCT
ejpam-1797	259	1	thus	thus	ADV
ejpam-1797	259	2	,	,	PUNCT
ejpam-1797	259	3	φ(a	φ(a	ADJ
ejpam-1797	259	4	)	)	PUNCT
ejpam-1797	259	5	=	=	SYM
ejpam-1797	260	1	(	(	PUNCT
ejpam-1797	260	2	0	0	NUM
ejpam-1797	260	3	)	)	PUNCT
ejpam-1797	260	4	since	since	SCONJ
ejpam-1797	260	5	t	t	PROPN
ejpam-1797	260	6	is	be	AUX
ejpam-1797	260	7	semiprime⇒	semiprime⇒	PROPN
ejpam-1797	260	8	a⊆	a⊆	PROPN
ejpam-1797	260	9	kerφ	kerφ	NOUN
ejpam-1797	260	10	=	=	PUNCT
ejpam-1797	260	11	(	(	PUNCT
ejpam-1797	260	12	0)⇒	0)⇒	X
ejpam-1797	260	13	a=	a=	X
ejpam-1797	260	14	(	(	PUNCT
ejpam-1797	260	15	0	0	NUM
ejpam-1797	260	16	)	)	PUNCT
ejpam-1797	260	17	.	.	PUNCT
ejpam-1797	261	1	this	this	PRON
ejpam-1797	261	2	shows	show	VERB
ejpam-1797	261	3	that	that	SCONJ
ejpam-1797	261	4	s	s	VERB
ejpam-1797	261	5	is	be	AUX
ejpam-1797	261	6	a	a	DET
ejpam-1797	261	7	semiprime	semiprime	NOUN
ejpam-1797	261	8	ternary	ternary	ADJ
ejpam-1797	261	9	semiring	semiring	NOUN
ejpam-1797	261	10	.	.	PUNCT
ejpam-1797	262	1	the	the	DET
ejpam-1797	262	2	semiprime	semiprime	NOUN
ejpam-1797	262	3	ternary	ternary	ADJ
ejpam-1797	262	4	semirings	semiring	NOUN
ejpam-1797	262	5	is	be	AUX
ejpam-1797	262	6	described	describe	VERB
ejpam-1797	262	7	in	in	ADP
ejpam-1797	262	8	the	the	DET
ejpam-1797	262	9	following	follow	VERB
ejpam-1797	262	10	lemma	lemma	PROPN
ejpam-1797	262	11	.	.	PUNCT
ejpam-1797	263	1	lemma	lemma	PROPN
ejpam-1797	263	2	4	4	NUM
ejpam-1797	263	3	.	.	PUNCT
ejpam-1797	264	1	if	if	SCONJ
ejpam-1797	264	2	the	the	DET
ejpam-1797	264	3	ternary	ternary	ADJ
ejpam-1797	264	4	semirings	semiring	NOUN
ejpam-1797	264	5	s	s	PART
ejpam-1797	264	6	≃	≃	PROPN
ejpam-1797	264	7	t	t	PROPN
ejpam-1797	264	8	and	and	CCONJ
ejpam-1797	264	9	s	s	PRON
ejpam-1797	264	10	are	be	AUX
ejpam-1797	264	11	semiprime	semiprime	NOUN
ejpam-1797	264	12	,	,	PUNCT
ejpam-1797	264	13	then	then	ADV
ejpam-1797	264	14	t	t	PROPN
ejpam-1797	264	15	is	be	AUX
ejpam-1797	264	16	semiprime	semiprime	NOUN
ejpam-1797	264	17	.	.	PUNCT
ejpam-1797	265	1	t.	t.	PROPN
ejpam-1797	265	2	dutta	dutta	PROPN
ejpam-1797	265	3	,	,	PUNCT
ejpam-1797	265	4	k.	k.	PROPN
ejpam-1797	265	5	shum	shum	PROPN
ejpam-1797	265	6	,	,	PUNCT
ejpam-1797	265	7	s.	s.	PROPN
ejpam-1797	265	8	mandal	mandal	PROPN
ejpam-1797	265	9	/	/	SYM
ejpam-1797	265	10	eur	eur	PROPN
ejpam-1797	265	11	.	.	PUNCT
ejpam-1797	266	1	j.	j.	PROPN
ejpam-1797	266	2	pure	pure	PROPN
ejpam-1797	266	3	appl	appl	PROPN
ejpam-1797	266	4	.	.	PROPN
ejpam-1797	266	5	math	math	PROPN
ejpam-1797	266	6	,	,	PUNCT
ejpam-1797	266	7	5	5	NUM
ejpam-1797	266	8	(	(	PUNCT
ejpam-1797	266	9	2012	2012	NUM
ejpam-1797	266	10	)	)	PUNCT
ejpam-1797	266	11	,	,	PUNCT
ejpam-1797	266	12	401	401	NUM
ejpam-1797	266	13	-	-	SYM
ejpam-1797	266	14	413	413	NUM
ejpam-1797	266	15	408	408	NUM
ejpam-1797	266	16	proof	proof	NOUN
ejpam-1797	266	17	.	.	PUNCT
ejpam-1797	266	18	suppose	suppose	VERB
ejpam-1797	266	19	that	that	SCONJ
ejpam-1797	266	20	φ	φ	PROPN
ejpam-1797	266	21	:	:	PUNCT
ejpam-1797	266	22	s	s	X
ejpam-1797	266	23	→	→	SYM
ejpam-1797	266	24	t	t	PROPN
ejpam-1797	266	25	is	be	AUX
ejpam-1797	266	26	a	a	DET
ejpam-1797	266	27	semi	semi	NOUN
ejpam-1797	266	28	-	-	NOUN
ejpam-1797	266	29	isomorphism	isomorphism	ADJ
ejpam-1797	266	30	and	and	CCONJ
ejpam-1797	266	31	a′	a′	NOUN
ejpam-1797	266	32	is	be	AUX
ejpam-1797	266	33	an	an	DET
ejpam-1797	266	34	ideal	ideal	NOUN
ejpam-1797	266	35	of	of	ADP
ejpam-1797	266	36	t	t	NOUN
ejpam-1797	266	37	such	such	DET
ejpam-1797	266	38	that	that	DET
ejpam-1797	266	39	a′3	a′3	NOUN
ejpam-1797	266	40	=	=	SYM
ejpam-1797	266	41	(	(	PUNCT
ejpam-1797	266	42	0	0	NUM
ejpam-1797	266	43	)	)	PUNCT
ejpam-1797	266	44	.	.	PUNCT
ejpam-1797	267	1	then	then	ADV
ejpam-1797	267	2	,	,	PUNCT
ejpam-1797	267	3	there	there	PRON
ejpam-1797	267	4	exists	exist	VERB
ejpam-1797	267	5	an	an	DET
ejpam-1797	267	6	ideal	ideal	NOUN
ejpam-1797	267	7	a	a	PRON
ejpam-1797	267	8	of	of	ADP
ejpam-1797	267	9	s	s	PRON
ejpam-1797	267	10	such	such	ADJ
ejpam-1797	267	11	that	that	PRON
ejpam-1797	267	12	φ(a	φ(a	ADJ
ejpam-1797	267	13	)	)	PUNCT
ejpam-1797	267	14	=	=	PUNCT
ejpam-1797	267	15	a′	a′	PROPN
ejpam-1797	267	16	,	,	PUNCT
ejpam-1797	267	17	as	as	SCONJ
ejpam-1797	267	18	φ	φ	PROPN
ejpam-1797	267	19	is	be	AUX
ejpam-1797	267	20	surjective	surjective	ADJ
ejpam-1797	267	21	.	.	PUNCT
ejpam-1797	268	1	now	now	ADV
ejpam-1797	268	2	φ(a3	φ(a3	NOUN
ejpam-1797	268	3	)	)	PUNCT
ejpam-1797	269	1	=	=	PUNCT
ejpam-1797	270	1	[	[	X
ejpam-1797	270	2	φ(a)]3	φ(a)]3	X
ejpam-1797	270	3	=	=	PUNCT
ejpam-1797	270	4	a′3	a′3	NOUN
ejpam-1797	270	5	=	=	SYM
ejpam-1797	270	6	(	(	PUNCT
ejpam-1797	270	7	0	0	NUM
ejpam-1797	270	8	)	)	PUNCT
ejpam-1797	270	9	.	.	PUNCT
ejpam-1797	271	1	thus	thus	ADV
ejpam-1797	271	2	a3	a3	VERB
ejpam-1797	271	3	=	=	SYM
ejpam-1797	271	4	(	(	PUNCT
ejpam-1797	271	5	0	0	NUM
ejpam-1797	271	6	)	)	PUNCT
ejpam-1797	271	7	,	,	PUNCT
ejpam-1797	271	8	as	as	ADP
ejpam-1797	271	9	kerφ	kerφ	PROPN
ejpam-1797	271	10	=	=	PROPN
ejpam-1797	271	11	0	0	X
ejpam-1797	271	12	.	.	PUNCT
ejpam-1797	272	1	since	since	SCONJ
ejpam-1797	272	2	s	s	PROPN
ejpam-1797	272	3	is	be	AUX
ejpam-1797	272	4	semiprime	semiprime	NOUN
ejpam-1797	272	5	,	,	PUNCT
ejpam-1797	272	6	a=	a=	X
ejpam-1797	272	7	(	(	PUNCT
ejpam-1797	272	8	0	0	NUM
ejpam-1797	272	9	)	)	PUNCT
ejpam-1797	272	10	,	,	PUNCT
ejpam-1797	272	11	we	we	PRON
ejpam-1797	272	12	have	have	VERB
ejpam-1797	272	13	a′	a′	NOUN
ejpam-1797	272	14	=	=	SYM
ejpam-1797	272	15	(	(	PUNCT
ejpam-1797	272	16	0	0	NUM
ejpam-1797	272	17	)	)	PUNCT
ejpam-1797	272	18	.	.	PUNCT
ejpam-1797	273	1	thus	thus	ADV
ejpam-1797	273	2	,	,	PUNCT
ejpam-1797	273	3	t	t	PROPN
ejpam-1797	273	4	is	be	AUX
ejpam-1797	273	5	a	a	DET
ejpam-1797	273	6	semiprime	semiprime	NOUN
ejpam-1797	273	7	ternary	ternary	ADJ
ejpam-1797	273	8	semiring	semiring	NOUN
ejpam-1797	273	9	.	.	PUNCT
ejpam-1797	274	1	in	in	ADP
ejpam-1797	274	2	the	the	DET
ejpam-1797	274	3	following	following	NOUN
ejpam-1797	274	4	theorem	theorem	NOUN
ejpam-1797	274	5	,	,	PUNCT
ejpam-1797	274	6	we	we	PRON
ejpam-1797	274	7	consider	consider	VERB
ejpam-1797	274	8	the	the	DET
ejpam-1797	274	9	semiprime	semiprime	NOUN
ejpam-1797	274	10	non	non	ADJ
ejpam-1797	274	11	-	-	ADJ
ejpam-1797	274	12	singular	singular	ADJ
ejpam-1797	274	13	ternary	ternary	ADJ
ejpam-1797	274	14	semirings	semiring	NOUN
ejpam-1797	274	15	.	.	PUNCT
ejpam-1797	275	1	theorem	theorem	VERB
ejpam-1797	275	2	7	7	NUM
ejpam-1797	275	3	.	.	PUNCT
ejpam-1797	276	1	the	the	DET
ejpam-1797	276	2	class	class	NOUN
ejpam-1797	276	3	℘	℘	PROPN
ejpam-1797	276	4	of	of	ADP
ejpam-1797	276	5	semiprime	semiprime	NOUN
ejpam-1797	276	6	non	non	ADJ
ejpam-1797	276	7	-	-	ADJ
ejpam-1797	276	8	singular	singular	ADJ
ejpam-1797	276	9	ternary	ternary	ADJ
ejpam-1797	276	10	semirings	semiring	NOUN
ejpam-1797	276	11	forms	form	VERB
ejpam-1797	276	12	a	a	DET
ejpam-1797	276	13	weakly	weakly	ADJ
ejpam-1797	276	14	special	special	ADJ
ejpam-1797	276	15	radical	radical	ADJ
ejpam-1797	276	16	class	class	NOUN
ejpam-1797	276	17	.	.	PUNCT
ejpam-1797	277	1	proof	proof	NOUN
ejpam-1797	277	2	.	.	PUNCT
ejpam-1797	278	1	let	let	VERB
ejpam-1797	278	2	s	s	PRON
ejpam-1797	278	3	∈	∈	NOUN
ejpam-1797	278	4	℘	℘	PROPN
ejpam-1797	278	5	and	and	CCONJ
ejpam-1797	278	6	i	i	PRON
ejpam-1797	278	7	be	be	VERB
ejpam-1797	278	8	a	a	DET
ejpam-1797	278	9	nonzero	nonzero	NOUN
ejpam-1797	278	10	ideal	ideal	NOUN
ejpam-1797	278	11	of	of	ADP
ejpam-1797	278	12	s.	s.	PROPN
ejpam-1797	278	13	then	then	ADV
ejpam-1797	278	14	by	by	ADP
ejpam-1797	278	15	lemma	lemma	PROPN
ejpam-1797	278	16	4	4	NUM
ejpam-1797	278	17	,	,	PUNCT
ejpam-1797	278	18	i	i	PRON
ejpam-1797	278	19	is	be	AUX
ejpam-1797	278	20	a	a	DET
ejpam-1797	278	21	semiprime	semiprime	NOUN
ejpam-1797	278	22	ternary	ternary	ADJ
ejpam-1797	278	23	semiring	semiring	NOUN
ejpam-1797	278	24	.	.	PUNCT
ejpam-1797	279	1	also	also	ADV
ejpam-1797	279	2	,	,	PUNCT
ejpam-1797	279	3	from	from	ADP
ejpam-1797	279	4	proposition	proposition	NOUN
ejpam-1797	279	5	3	3	NUM
ejpam-1797	279	6	,	,	PUNCT
ejpam-1797	279	7	z(i	z(i	NUM
ejpam-1797	279	8	)	)	PUNCT
ejpam-1797	279	9	=	=	SYM
ejpam-1797	279	10	i	i	PRON
ejpam-1797	279	11	∩	∩	ADJ
ejpam-1797	279	12	z(s	z(s	PROPN
ejpam-1797	279	13	)	)	PUNCT
ejpam-1797	279	14	=	=	PUNCT
ejpam-1797	279	15	(	(	PUNCT
ejpam-1797	279	16	0	0	NUM
ejpam-1797	279	17	)	)	PUNCT
ejpam-1797	279	18	,	,	PUNCT
ejpam-1797	279	19	since	since	SCONJ
ejpam-1797	279	20	z(s)=0	z(s)=0	PRON
ejpam-1797	279	21	and	and	CCONJ
ejpam-1797	279	22	s	s	NOUN
ejpam-1797	279	23	is	be	AUX
ejpam-1797	279	24	nonsingular	nonsingular	ADJ
ejpam-1797	279	25	,	,	PUNCT
ejpam-1797	279	26	i	i	PROPN
ejpam-1797	279	27	∈	∈	PROPN
ejpam-1797	279	28	℘.	℘.	PROPN
ejpam-1797	279	29	thus	thus	ADV
ejpam-1797	279	30	,	,	PUNCT
ejpam-1797	279	31	the	the	DET
ejpam-1797	279	32	class	class	NOUN
ejpam-1797	279	33	℘	℘	PROPN
ejpam-1797	279	34	is	be	AUX
ejpam-1797	279	35	a	a	DET
ejpam-1797	279	36	hereditary	hereditary	ADJ
ejpam-1797	279	37	class	class	NOUN
ejpam-1797	279	38	of	of	ADP
ejpam-1797	279	39	semiprime	semiprime	NOUN
ejpam-1797	279	40	nonsingular	nonsingular	ADJ
ejpam-1797	279	41	ternary	ternary	ADJ
ejpam-1797	279	42	semirings	semiring	NOUN
ejpam-1797	279	43	.	.	PUNCT
ejpam-1797	280	1	by	by	ADP
ejpam-1797	280	2	theorem	theorem	NOUN
ejpam-1797	280	3	1	1	NUM
ejpam-1797	280	4	and	and	CCONJ
ejpam-1797	280	5	lemma	lemma	PROPN
ejpam-1797	280	6	3	3	NUM
ejpam-1797	280	7	,	,	PUNCT
ejpam-1797	280	8	the	the	DET
ejpam-1797	280	9	class	class	NOUN
ejpam-1797	280	10	℘	℘	PROPN
ejpam-1797	280	11	satisfies	satisfy	VERB
ejpam-1797	280	12	the	the	DET
ejpam-1797	280	13	property	property	NOUN
ejpam-1797	280	14	(	(	PUNCT
ejpam-1797	280	15	x	x	NOUN
ejpam-1797	280	16	)	)	PUNCT
ejpam-1797	280	17	of	of	ADP
ejpam-1797	280	18	the	the	DET
ejpam-1797	280	19	definition	definition	NOUN
ejpam-1797	280	20	of	of	ADP
ejpam-1797	280	21	weakly	weakly	ADJ
ejpam-1797	280	22	special	special	ADJ
ejpam-1797	280	23	radical	radical	ADJ
ejpam-1797	280	24	class	class	NOUN
ejpam-1797	280	25	.	.	PUNCT
ejpam-1797	281	1	in	in	ADP
ejpam-1797	281	2	order	order	NOUN
ejpam-1797	281	3	to	to	PART
ejpam-1797	281	4	prove	prove	VERB
ejpam-1797	281	5	the	the	DET
ejpam-1797	281	6	condition	condition	NOUN
ejpam-1797	281	7	(	(	PUNCT
ejpam-1797	281	8	z	z	NOUN
ejpam-1797	281	9	)	)	PUNCT
ejpam-1797	281	10	of	of	ADP
ejpam-1797	281	11	the	the	DET
ejpam-1797	281	12	weakly	weakly	ADJ
ejpam-1797	281	13	special	special	ADJ
ejpam-1797	281	14	radical	radical	ADJ
ejpam-1797	281	15	class	class	NOUN
ejpam-1797	281	16	,	,	PUNCT
ejpam-1797	281	17	in	in	ADP
ejpam-1797	281	18	view	view	NOUN
ejpam-1797	281	19	of	of	ADP
ejpam-1797	281	20	theorem	theorem	NOUN
ejpam-1797	281	21	1	1	NUM
ejpam-1797	281	22	,	,	PUNCT
ejpam-1797	281	23	lemma	lemma	PROPN
ejpam-1797	281	24	4	4	NUM
ejpam-1797	281	25	and	and	CCONJ
ejpam-1797	281	26	lemma	lemma	PROPN
ejpam-1797	281	27	6	6	NUM
ejpam-1797	281	28	,	,	PUNCT
ejpam-1797	281	29	it	it	PRON
ejpam-1797	281	30	suffices	suffice	VERB
ejpam-1797	281	31	to	to	PART
ejpam-1797	281	32	prove	prove	VERB
ejpam-1797	281	33	that	that	SCONJ
ejpam-1797	281	34	if	if	SCONJ
ejpam-1797	281	35	i	i	PRON
ejpam-1797	281	36	∈	∈	VERB
ejpam-1797	281	37	℘	℘	PROPN
ejpam-1797	281	38	and	and	CCONJ
ejpam-1797	281	39	i	i	PRON
ejpam-1797	281	40	is	be	AUX
ejpam-1797	281	41	an	an	DET
ejpam-1797	281	42	essential	essential	ADJ
ejpam-1797	281	43	ideal	ideal	NOUN
ejpam-1797	281	44	of	of	ADP
ejpam-1797	281	45	a	a	DET
ejpam-1797	281	46	ternary	ternary	ADJ
ejpam-1797	281	47	semiring	semiring	NOUN
ejpam-1797	281	48	s	s	VERB
ejpam-1797	281	49	then	then	ADV
ejpam-1797	281	50	s	s	PART
ejpam-1797	281	51	∈	∈	PROPN
ejpam-1797	281	52	℘.	℘.	PROPN
ejpam-1797	281	53	to	to	PART
ejpam-1797	281	54	prove	prove	VERB
ejpam-1797	281	55	that	that	SCONJ
ejpam-1797	281	56	s	s	VERB
ejpam-1797	281	57	is	be	AUX
ejpam-1797	281	58	a	a	DET
ejpam-1797	281	59	semiprime	semiprime	NOUN
ejpam-1797	281	60	ternary	ternary	ADJ
ejpam-1797	281	61	semiring	semiring	NOUN
ejpam-1797	281	62	,	,	PUNCT
ejpam-1797	281	63	we	we	PRON
ejpam-1797	281	64	let	let	VERB
ejpam-1797	281	65	k3	k3	VERB
ejpam-1797	281	66	=	=	NOUN
ejpam-1797	281	67	0	0	NUM
ejpam-1797	281	68	,	,	PUNCT
ejpam-1797	281	69	where	where	SCONJ
ejpam-1797	281	70	k	k	PROPN
ejpam-1797	281	71	is	be	AUX
ejpam-1797	281	72	an	an	DET
ejpam-1797	281	73	ideal	ideal	NOUN
ejpam-1797	281	74	of	of	ADP
ejpam-1797	281	75	s.	s.	PROPN
ejpam-1797	281	76	let	let	VERB
ejpam-1797	282	1	k	k	NOUN
ejpam-1797	282	2	′	′	NUM
ejpam-1797	283	1	=	=	SYM
ejpam-1797	283	2	k	k	PROPN
ejpam-1797	283	3	∩	∩	PROPN
ejpam-1797	283	4	i	i	PRON
ejpam-1797	283	5	.	.	PUNCT
ejpam-1797	284	1	then	then	ADV
ejpam-1797	284	2	k	k	PROPN
ejpam-1797	284	3	′	′	PROPN
ejpam-1797	284	4	is	be	AUX
ejpam-1797	284	5	an	an	DET
ejpam-1797	284	6	ideal	ideal	NOUN
ejpam-1797	284	7	of	of	ADP
ejpam-1797	284	8	i	i	PRON
ejpam-1797	284	9	.	.	PUNCT
ejpam-1797	285	1	now	now	ADV
ejpam-1797	285	2	k	k	X
ejpam-1797	285	3	′3	′3	NOUN
ejpam-1797	285	4	⊆	⊆	NUM
ejpam-1797	285	5	k3	k3	ADJ
ejpam-1797	285	6	=	=	SYM
ejpam-1797	285	7	0	0	NUM
ejpam-1797	285	8	implies	imply	VERB
ejpam-1797	285	9	k	k	X
ejpam-1797	285	10	′	′	NUM
ejpam-1797	286	1	=	=	NOUN
ejpam-1797	286	2	0	0	PUNCT
ejpam-1797	286	3	since	since	SCONJ
ejpam-1797	286	4	i	i	PRON
ejpam-1797	286	5	is	be	AUX
ejpam-1797	286	6	a	a	DET
ejpam-1797	286	7	semiprime	semiprime	NOUN
ejpam-1797	286	8	ternary	ternary	ADJ
ejpam-1797	286	9	semiring	semiring	NOUN
ejpam-1797	286	10	.	.	PUNCT
ejpam-1797	287	1	since	since	SCONJ
ejpam-1797	287	2	i	i	PRON
ejpam-1797	287	3	is	be	AUX
ejpam-1797	287	4	an	an	DET
ejpam-1797	287	5	essential	essential	ADJ
ejpam-1797	287	6	ideal	ideal	NOUN
ejpam-1797	287	7	and	and	CCONJ
ejpam-1797	287	8	so	so	ADV
ejpam-1797	287	9	k	k	PROPN
ejpam-1797	287	10	=	=	PUNCT
ejpam-1797	287	11	0	0	X
ejpam-1797	287	12	.	.	PUNCT
ejpam-1797	288	1	this	this	PRON
ejpam-1797	288	2	shows	show	VERB
ejpam-1797	288	3	that	that	SCONJ
ejpam-1797	288	4	s	s	VERB
ejpam-1797	288	5	is	be	AUX
ejpam-1797	288	6	a	a	DET
ejpam-1797	288	7	semiprime	semiprime	NOUN
ejpam-1797	288	8	ternary	ternary	ADJ
ejpam-1797	288	9	semiring	semiring	NOUN
ejpam-1797	288	10	.	.	PUNCT
ejpam-1797	289	1	again	again	ADV
ejpam-1797	289	2	if	if	SCONJ
ejpam-1797	289	3	s	s	VERB
ejpam-1797	289	4	is	be	AUX
ejpam-1797	289	5	not	not	PART
ejpam-1797	289	6	a	a	DET
ejpam-1797	289	7	nonsingular	nonsingular	ADJ
ejpam-1797	289	8	ternary	ternary	ADJ
ejpam-1797	289	9	semiring	semiring	NOUN
ejpam-1797	289	10	,	,	PUNCT
ejpam-1797	289	11	then	then	ADV
ejpam-1797	289	12	by	by	ADP
ejpam-1797	289	13	example	example	NOUN
ejpam-1797	289	14	1	1	NUM
ejpam-1797	289	15	and	and	CCONJ
ejpam-1797	289	16	proposition	proposition	NOUN
ejpam-1797	289	17	2	2	NUM
ejpam-1797	289	18	,	,	PUNCT
ejpam-1797	289	19	we	we	PRON
ejpam-1797	289	20	can	can	AUX
ejpam-1797	289	21	easily	easily	ADV
ejpam-1797	289	22	see	see	VERB
ejpam-1797	289	23	that	that	SCONJ
ejpam-1797	289	24	z(s	z(s	PROPN
ejpam-1797	289	25	)	)	PUNCT
ejpam-1797	289	26	is	be	AUX
ejpam-1797	289	27	a	a	DET
ejpam-1797	289	28	nonzero	nonzero	NOUN
ejpam-1797	289	29	ideal	ideal	NOUN
ejpam-1797	289	30	of	of	ADP
ejpam-1797	289	31	s	s	PRON
ejpam-1797	289	32	and	and	CCONJ
ejpam-1797	289	33	thus	thus	ADV
ejpam-1797	289	34	i	i	PRON
ejpam-1797	289	35	∩	∩	ADJ
ejpam-1797	289	36	z(s	z(s	PROPN
ejpam-1797	289	37	)	)	PUNCT
ejpam-1797	289	38	6=	6=	ADP
ejpam-1797	289	39	0	0	NUM
ejpam-1797	289	40	,	,	PUNCT
ejpam-1797	289	41	since	since	SCONJ
ejpam-1797	289	42	i	i	PRON
ejpam-1797	289	43	is	be	AUX
ejpam-1797	289	44	an	an	DET
ejpam-1797	289	45	essential	essential	ADJ
ejpam-1797	289	46	ideal	ideal	NOUN
ejpam-1797	289	47	of	of	ADP
ejpam-1797	289	48	s.	s.	PROPN
ejpam-1797	289	49	hence	hence	ADV
ejpam-1797	289	50	,	,	PUNCT
ejpam-1797	289	51	by	by	ADP
ejpam-1797	289	52	proposition	proposition	NOUN
ejpam-1797	289	53	3	3	NUM
ejpam-1797	289	54	,	,	PUNCT
ejpam-1797	289	55	z(i	z(i	NUM
ejpam-1797	289	56	)	)	PUNCT
ejpam-1797	289	57	6=	6=	ADP
ejpam-1797	289	58	0	0	NUM
ejpam-1797	289	59	which	which	PRON
ejpam-1797	289	60	is	be	AUX
ejpam-1797	289	61	a	a	DET
ejpam-1797	289	62	contradiction	contradiction	NOUN
ejpam-1797	289	63	,	,	PUNCT
ejpam-1797	289	64	since	since	SCONJ
ejpam-1797	289	65	i	i	PRON
ejpam-1797	289	66	∈	∈	PROPN
ejpam-1797	289	67	℘.	℘.	PROPN
ejpam-1797	289	68	therefore	therefore	ADV
ejpam-1797	289	69	,	,	PUNCT
ejpam-1797	289	70	s	s	X
ejpam-1797	289	71	is	be	AUX
ejpam-1797	289	72	nonsingular	nonsingular	ADJ
ejpam-1797	289	73	and	and	CCONJ
ejpam-1797	289	74	hence	hence	ADV
ejpam-1797	289	75	,	,	PUNCT
ejpam-1797	289	76	s	s	PROPN
ejpam-1797	289	77	∈	∈	PROPN
ejpam-1797	289	78	℘.	℘.	PROPN
ejpam-1797	289	79	this	this	PRON
ejpam-1797	289	80	shows	show	VERB
ejpam-1797	289	81	that	that	SCONJ
ejpam-1797	289	82	℘	℘	PROPN
ejpam-1797	289	83	is	be	AUX
ejpam-1797	289	84	a	a	DET
ejpam-1797	289	85	weakly	weakly	ADJ
ejpam-1797	289	86	special	special	ADJ
ejpam-1797	289	87	radical	radical	ADJ
ejpam-1797	289	88	class	class	NOUN
ejpam-1797	289	89	.	.	PUNCT
ejpam-1797	290	1	3	3	X
ejpam-1797	290	2	.	.	X
ejpam-1797	290	3	supernilpotent	supernilpotent	PROPN
ejpam-1797	290	4	radical	radical	ADJ
ejpam-1797	290	5	class	class	NOUN
ejpam-1797	290	6	and	and	CCONJ
ejpam-1797	290	7	weakly	weakly	ADJ
ejpam-1797	290	8	special	special	ADJ
ejpam-1797	290	9	radical	radical	ADJ
ejpam-1797	290	10	class	class	NOUN
ejpam-1797	290	11	of	of	ADP
ejpam-1797	290	12	ternary	ternary	ADJ
ejpam-1797	290	13	semirings	semiring	NOUN
ejpam-1797	290	14	we	we	PRON
ejpam-1797	290	15	first	first	ADV
ejpam-1797	290	16	give	give	VERB
ejpam-1797	290	17	the	the	DET
ejpam-1797	290	18	following	follow	VERB
ejpam-1797	290	19	useful	useful	ADJ
ejpam-1797	290	20	definition	definition	NOUN
ejpam-1797	290	21	.	.	PUNCT
ejpam-1797	291	1	definition	definition	NOUN
ejpam-1797	291	2	9	9	NUM
ejpam-1797	291	3	.	.	PUNCT
ejpam-1797	292	1	let	let	VERB
ejpam-1797	292	2	s	s	PRON
ejpam-1797	292	3	be	be	AUX
ejpam-1797	292	4	a	a	DET
ejpam-1797	292	5	ternary	ternary	ADJ
ejpam-1797	292	6	semiring	semiring	NOUN
ejpam-1797	292	7	.	.	PUNCT
ejpam-1797	293	1	then	then	ADV
ejpam-1797	293	2	,	,	PUNCT
ejpam-1797	293	3	we	we	PRON
ejpam-1797	293	4	call	call	VERB
ejpam-1797	293	5	an	an	DET
ejpam-1797	293	6	ideal	ideal	NOUN
ejpam-1797	293	7	i	i	PRON
ejpam-1797	293	8	of	of	ADP
ejpam-1797	293	9	s	s	PART
ejpam-1797	293	10	nilpotent	nilpotent	ADJ
ejpam-1797	293	11	if	if	SCONJ
ejpam-1797	293	12	there	there	PRON
ejpam-1797	293	13	exists	exist	VERB
ejpam-1797	293	14	a	a	DET
ejpam-1797	293	15	positive	positive	ADJ
ejpam-1797	293	16	integer	integer	NOUN
ejpam-1797	293	17	n	n	CCONJ
ejpam-1797	293	18	such	such	ADJ
ejpam-1797	293	19	that	that	SCONJ
ejpam-1797	293	20	i2n+1	i2n+1	PROPN
ejpam-1797	294	1	=	=	SYM
ejpam-1797	294	2	0	0	X
ejpam-1797	294	3	.	.	PUNCT
ejpam-1797	295	1	the	the	DET
ejpam-1797	295	2	semiring	semire	VERB
ejpam-1797	295	3	s	s	NOUN
ejpam-1797	295	4	is	be	AUX
ejpam-1797	295	5	said	say	VERB
ejpam-1797	295	6	to	to	PART
ejpam-1797	295	7	be	be	AUX
ejpam-1797	295	8	a	a	DET
ejpam-1797	295	9	nilpotent	nilpotent	ADJ
ejpam-1797	295	10	ternary	ternary	ADJ
ejpam-1797	295	11	semiring	semiring	NOUN
ejpam-1797	295	12	if	if	SCONJ
ejpam-1797	295	13	s	s	VERB
ejpam-1797	295	14	is	be	AUX
ejpam-1797	295	15	nilpotent	nilpotent	ADJ
ejpam-1797	295	16	as	as	ADP
ejpam-1797	295	17	an	an	DET
ejpam-1797	295	18	ideal	ideal	NOUN
ejpam-1797	295	19	of	of	ADP
ejpam-1797	295	20	itself	itself	PRON
ejpam-1797	295	21	.	.	PUNCT
ejpam-1797	296	1	following	follow	VERB
ejpam-1797	296	2	d.	d.	PROPN
ejpam-1797	296	3	m.	m.	PROPN
ejpam-1797	296	4	olson	olson	PROPN
ejpam-1797	296	5	and	and	CCONJ
ejpam-1797	296	6	a.	a.	PROPN
ejpam-1797	296	7	c.	c.	PROPN
ejpam-1797	296	8	nance	nance	PROPN
ejpam-1797	296	9	[	[	X
ejpam-1797	296	10	27	27	NUM
ejpam-1797	296	11	]	]	PUNCT
ejpam-1797	296	12	,	,	PUNCT
ejpam-1797	296	13	we	we	PRON
ejpam-1797	296	14	define	define	VERB
ejpam-1797	296	15	the	the	DET
ejpam-1797	296	16	supernilpotent	supernilpotent	ADJ
ejpam-1797	296	17	radical	radical	ADJ
ejpam-1797	296	18	class	class	NOUN
ejpam-1797	296	19	in	in	ADP
ejpam-1797	296	20	ternary	ternary	ADJ
ejpam-1797	296	21	semiring	semiring	NOUN
ejpam-1797	296	22	as	as	SCONJ
ejpam-1797	296	23	follows	follow	VERB
ejpam-1797	296	24	:	:	PUNCT
ejpam-1797	296	25	definition	definition	NOUN
ejpam-1797	296	26	10	10	NUM
ejpam-1797	296	27	.	.	PUNCT
ejpam-1797	297	1	a	a	DET
ejpam-1797	297	2	radical	radical	ADJ
ejpam-1797	297	3	class	class	NOUN
ejpam-1797	297	4	of	of	ADP
ejpam-1797	297	5	ternary	ternary	ADJ
ejpam-1797	297	6	semirings	semiring	NOUN
ejpam-1797	297	7	is	be	AUX
ejpam-1797	297	8	called	call	VERB
ejpam-1797	297	9	a	a	DET
ejpam-1797	297	10	supernilpotent	supernilpotent	ADJ
ejpam-1797	297	11	radical	radical	ADJ
ejpam-1797	297	12	class	class	NOUN
ejpam-1797	297	13	if	if	SCONJ
ejpam-1797	297	14	it	it	PRON
ejpam-1797	297	15	is	be	AUX
ejpam-1797	297	16	hereditary	hereditary	ADJ
ejpam-1797	297	17	and	and	CCONJ
ejpam-1797	297	18	contains	contain	VERB
ejpam-1797	297	19	all	all	DET
ejpam-1797	297	20	the	the	DET
ejpam-1797	297	21	nilpotent	nilpotent	ADJ
ejpam-1797	297	22	ternary	ternary	ADJ
ejpam-1797	297	23	semirings	semiring	NOUN
ejpam-1797	297	24	.	.	PUNCT
ejpam-1797	298	1	the	the	DET
ejpam-1797	298	2	following	follow	VERB
ejpam-1797	298	3	lemma	lemma	PROPN
ejpam-1797	298	4	is	be	AUX
ejpam-1797	298	5	a	a	DET
ejpam-1797	298	6	crucial	crucial	ADJ
ejpam-1797	298	7	lemma	lemma	PROPN
ejpam-1797	298	8	.	.	PUNCT
ejpam-1797	299	1	lemma	lemma	PROPN
ejpam-1797	299	2	5	5	X
ejpam-1797	299	3	.	.	PUNCT
ejpam-1797	300	1	let	let	VERB
ejpam-1797	300	2	s	s	PRON
ejpam-1797	300	3	be	be	AUX
ejpam-1797	300	4	a	a	DET
ejpam-1797	300	5	ternary	ternary	ADJ
ejpam-1797	300	6	semiring	semiring	NOUN
ejpam-1797	300	7	.	.	PUNCT
ejpam-1797	301	1	if	if	SCONJ
ejpam-1797	301	2	i	i	PRON
ejpam-1797	301	3	is	be	AUX
ejpam-1797	301	4	a	a	DET
ejpam-1797	301	5	semiprime	semiprime	NOUN
ejpam-1797	301	6	k	k	X
ejpam-1797	301	7	-	-	PUNCT
ejpam-1797	301	8	ideal	ideal	ADJ
ejpam-1797	301	9	j	j	PROPN
ejpam-1797	301	10	and	and	CCONJ
ejpam-1797	301	11	j	j	PROPN
ejpam-1797	301	12	is	be	AUX
ejpam-1797	301	13	an	an	DET
ejpam-1797	301	14	ideal	ideal	NOUN
ejpam-1797	301	15	of	of	ADP
ejpam-1797	301	16	s	s	PROPN
ejpam-1797	301	17	,	,	PUNCT
ejpam-1797	301	18	then	then	ADV
ejpam-1797	301	19	i	i	PRON
ejpam-1797	301	20	is	be	AUX
ejpam-1797	301	21	an	an	DET
ejpam-1797	301	22	ideal	ideal	NOUN
ejpam-1797	301	23	of	of	ADP
ejpam-1797	301	24	s.	s.	PROPN
ejpam-1797	301	25	t.	t.	PROPN
ejpam-1797	301	26	dutta	dutta	PROPN
ejpam-1797	301	27	,	,	PUNCT
ejpam-1797	301	28	k.	k.	PROPN
ejpam-1797	301	29	shum	shum	PROPN
ejpam-1797	301	30	,	,	PUNCT
ejpam-1797	301	31	s.	s.	PROPN
ejpam-1797	301	32	mandal	mandal	PROPN
ejpam-1797	301	33	/	/	SYM
ejpam-1797	301	34	eur	eur	PROPN
ejpam-1797	301	35	.	.	PUNCT
ejpam-1797	302	1	j.	j.	PROPN
ejpam-1797	302	2	pure	pure	PROPN
ejpam-1797	302	3	appl	appl	PROPN
ejpam-1797	302	4	.	.	PROPN
ejpam-1797	302	5	math	math	PROPN
ejpam-1797	302	6	,	,	PUNCT
ejpam-1797	302	7	5	5	NUM
ejpam-1797	302	8	(	(	PUNCT
ejpam-1797	302	9	2012	2012	NUM
ejpam-1797	302	10	)	)	PUNCT
ejpam-1797	302	11	,	,	PUNCT
ejpam-1797	302	12	401	401	NUM
ejpam-1797	302	13	-	-	SYM
ejpam-1797	302	14	413	413	NUM
ejpam-1797	302	15	409	409	NUM
ejpam-1797	302	16	proof	proof	NOUN
ejpam-1797	302	17	.	.	PUNCT
ejpam-1797	303	1	since	since	SCONJ
ejpam-1797	303	2	i	i	PRON
ejpam-1797	303	3	is	be	AUX
ejpam-1797	303	4	a	a	DET
ejpam-1797	303	5	semiprime	semiprime	NOUN
ejpam-1797	303	6	k	k	X
ejpam-1797	303	7	-	-	NOUN
ejpam-1797	303	8	ideal	ideal	NOUN
ejpam-1797	303	9	of	of	ADP
ejpam-1797	303	10	j	j	PROPN
ejpam-1797	303	11	,	,	PUNCT
ejpam-1797	303	12	it	it	PRON
ejpam-1797	303	13	is	be	AUX
ejpam-1797	303	14	easy	easy	ADJ
ejpam-1797	303	15	to	to	PART
ejpam-1797	303	16	see	see	VERB
ejpam-1797	303	17	that	that	PRON
ejpam-1797	303	18	j	j	PROPN
ejpam-1797	303	19	/	/	SYM
ejpam-1797	303	20	i	i	PROPN
ejpam-1797	303	21	is	be	AUX
ejpam-1797	303	22	a	a	DET
ejpam-1797	303	23	semiprime	semiprime	NOUN
ejpam-1797	303	24	ternary	ternary	ADJ
ejpam-1797	303	25	semiring	semiring	NOUN
ejpam-1797	303	26	.	.	PUNCT
ejpam-1797	304	1	now	now	ADV
ejpam-1797	304	2	,	,	PUNCT
ejpam-1797	304	3	(	(	PUNCT
ejpam-1797	304	4	i+sis+jsisj	i+sis+jsisj	NOUN
ejpam-1797	304	5	)	)	PUNCT
ejpam-1797	304	6	is	be	AUX
ejpam-1797	304	7	an	an	DET
ejpam-1797	304	8	ideal	ideal	NOUN
ejpam-1797	304	9	of	of	ADP
ejpam-1797	304	10	j	j	PROPN
ejpam-1797	304	11	and	and	CCONJ
ejpam-1797	304	12	we	we	PRON
ejpam-1797	304	13	have	have	VERB
ejpam-1797	304	14	(	(	PUNCT
ejpam-1797	304	15	i+sis+jsisj	i+sis+jsisj	NOUN
ejpam-1797	304	16	/	/	SYM
ejpam-1797	304	17	i)5	i)5	NOUN
ejpam-1797	305	1	⊆	⊆	NUM
ejpam-1797	305	2	i	i	NOUN
ejpam-1797	305	3	/	/	SYM
ejpam-1797	305	4	i	i	NOUN
ejpam-1797	305	5	=	=	NOUN
ejpam-1797	305	6	0	0	PUNCT
ejpam-1797	305	7	/	/	SYM
ejpam-1797	305	8	i	i	PRON
ejpam-1797	305	9	.	.	PUNCT
ejpam-1797	306	1	thus	thus	ADV
ejpam-1797	306	2	,	,	PUNCT
ejpam-1797	306	3	i	i	PRON
ejpam-1797	306	4	+	+	NUM
ejpam-1797	306	5	sis	sis	NOUN
ejpam-1797	306	6	+	+	X
ejpam-1797	306	7	jsisj	jsisj	NOUN
ejpam-1797	306	8	/	/	SYM
ejpam-1797	306	9	i	i	NOUN
ejpam-1797	306	10	=	=	PUNCT
ejpam-1797	306	11	0	0	PUNCT
ejpam-1797	306	12	/	/	SYM
ejpam-1797	306	13	i	i	PRON
ejpam-1797	306	14	as	as	ADP
ejpam-1797	306	15	j	j	PROPN
ejpam-1797	306	16	/	/	SYM
ejpam-1797	306	17	i	i	PROPN
ejpam-1797	306	18	is	be	AUX
ejpam-1797	306	19	semiprime	semiprime	NOUN
ejpam-1797	306	20	ternary	ternary	ADJ
ejpam-1797	306	21	semiring	semiring	NOUN
ejpam-1797	306	22	.	.	PUNCT
ejpam-1797	307	1	since	since	SCONJ
ejpam-1797	307	2	i	i	PRON
ejpam-1797	307	3	is	be	AUX
ejpam-1797	307	4	k	k	NOUN
ejpam-1797	307	5	-	-	NOUN
ejpam-1797	307	6	ideal	ideal	NOUN
ejpam-1797	307	7	of	of	ADP
ejpam-1797	307	8	j	j	PROPN
ejpam-1797	307	9	,	,	PUNCT
ejpam-1797	307	10	sis	sis	NOUN
ejpam-1797	307	11	⊆	⊆	NUM
ejpam-1797	307	12	i	i	PRON
ejpam-1797	307	13	.	.	PUNCT
ejpam-1797	308	1	again	again	ADV
ejpam-1797	308	2	,	,	PUNCT
ejpam-1797	308	3	(	(	PUNCT
ejpam-1797	308	4	i	i	PRON
ejpam-1797	308	5	+	+	CCONJ
ejpam-1797	308	6	iss	iss	PROPN
ejpam-1797	308	7	)	)	PUNCT
ejpam-1797	308	8	is	be	AUX
ejpam-1797	308	9	an	an	DET
ejpam-1797	308	10	ideal	ideal	NOUN
ejpam-1797	308	11	of	of	ADP
ejpam-1797	308	12	j	j	PROPN
ejpam-1797	308	13	and	and	CCONJ
ejpam-1797	308	14	(	(	PUNCT
ejpam-1797	308	15	i	i	PROPN
ejpam-1797	308	16	+	+	CCONJ
ejpam-1797	308	17	iss	iss	PROPN
ejpam-1797	308	18	/	/	SYM
ejpam-1797	308	19	i)3	i)3	PROPN
ejpam-1797	308	20	/	/	SYM
ejpam-1797	308	21	i	i	PROPN
ejpam-1797	308	22	⊆	⊆	NUM
ejpam-1797	309	1	i	i	NOUN
ejpam-1797	309	2	/	/	SYM
ejpam-1797	309	3	i	i	NOUN
ejpam-1797	309	4	=	=	SYM
ejpam-1797	309	5	(	(	PUNCT
ejpam-1797	309	6	0	0	NUM
ejpam-1797	309	7	)	)	PUNCT
ejpam-1797	309	8	.	.	PUNCT
ejpam-1797	310	1	thus	thus	ADV
ejpam-1797	310	2	,	,	PUNCT
ejpam-1797	310	3	(	(	PUNCT
ejpam-1797	310	4	i	i	NOUN
ejpam-1797	310	5	+	+	NOUN
ejpam-1797	310	6	iss)/i	iss)/i	VERB
ejpam-1797	310	7	=	=	SYM
ejpam-1797	310	8	0	0	NUM
ejpam-1797	310	9	/	/	SYM
ejpam-1797	310	10	i	i	PRON
ejpam-1797	310	11	as	as	ADP
ejpam-1797	310	12	j	j	PROPN
ejpam-1797	310	13	/	/	SYM
ejpam-1797	310	14	i	i	PROPN
ejpam-1797	310	15	is	be	AUX
ejpam-1797	310	16	a	a	DET
ejpam-1797	310	17	semiprime	semiprime	NOUN
ejpam-1797	310	18	ternary	ternary	ADJ
ejpam-1797	310	19	semiring	semiring	NOUN
ejpam-1797	310	20	.	.	PUNCT
ejpam-1797	311	1	now	now	ADV
ejpam-1797	311	2	as	as	SCONJ
ejpam-1797	311	3	i	i	PRON
ejpam-1797	311	4	is	be	AUX
ejpam-1797	311	5	a	a	DET
ejpam-1797	311	6	k	k	NOUN
ejpam-1797	311	7	-	-	NOUN
ejpam-1797	311	8	ideal	ideal	NOUN
ejpam-1797	311	9	of	of	ADP
ejpam-1797	311	10	j	j	PROPN
ejpam-1797	311	11	,	,	PUNCT
ejpam-1797	311	12	iss	iss	PROPN
ejpam-1797	311	13	⊆	⊆	NUM
ejpam-1797	311	14	i	i	PRON
ejpam-1797	311	15	.	.	PUNCT
ejpam-1797	312	1	also	also	ADV
ejpam-1797	312	2	,	,	PUNCT
ejpam-1797	312	3	(	(	PUNCT
ejpam-1797	312	4	i+ssi	i+ssi	X
ejpam-1797	312	5	)	)	PUNCT
ejpam-1797	312	6	is	be	AUX
ejpam-1797	312	7	an	an	DET
ejpam-1797	312	8	ideal	ideal	NOUN
ejpam-1797	312	9	of	of	ADP
ejpam-1797	312	10	j	j	PROPN
ejpam-1797	312	11	and	and	CCONJ
ejpam-1797	312	12	(	(	PUNCT
ejpam-1797	312	13	i+ssi	i+ssi	NUM
ejpam-1797	312	14	/	/	SYM
ejpam-1797	312	15	i)3	i)3	PROPN
ejpam-1797	312	16	⊆	⊆	NUM
ejpam-1797	312	17	i	i	PROPN
ejpam-1797	312	18	/	/	SYM
ejpam-1797	312	19	i	i	NOUN
ejpam-1797	312	20	=	=	SYM
ejpam-1797	312	21	(	(	PUNCT
ejpam-1797	312	22	0	0	NUM
ejpam-1797	312	23	)	)	PUNCT
ejpam-1797	312	24	.	.	PUNCT
ejpam-1797	313	1	hence	hence	ADV
ejpam-1797	313	2	,	,	PUNCT
ejpam-1797	313	3	we	we	PRON
ejpam-1797	313	4	have	have	VERB
ejpam-1797	313	5	(	(	PUNCT
ejpam-1797	313	6	i+ssi)/i	i+ssi)/i	NUM
ejpam-1797	313	7	=	=	SYM
ejpam-1797	313	8	(	(	PUNCT
ejpam-1797	313	9	0	0	NUM
ejpam-1797	313	10	)	)	PUNCT
ejpam-1797	313	11	as	as	SCONJ
ejpam-1797	313	12	j	j	PROPN
ejpam-1797	313	13	/	/	SYM
ejpam-1797	313	14	i	i	PROPN
ejpam-1797	313	15	is	be	AUX
ejpam-1797	313	16	a	a	DET
ejpam-1797	313	17	semiprime	semiprime	NOUN
ejpam-1797	313	18	ternary	ternary	ADJ
ejpam-1797	313	19	semiring	semiring	NOUN
ejpam-1797	313	20	.	.	PUNCT
ejpam-1797	314	1	now	now	ADV
ejpam-1797	314	2	as	as	SCONJ
ejpam-1797	314	3	i	i	PRON
ejpam-1797	314	4	is	be	AUX
ejpam-1797	314	5	a	a	DET
ejpam-1797	314	6	k	k	NOUN
ejpam-1797	314	7	-	-	NOUN
ejpam-1797	314	8	ideal	ideal	NOUN
ejpam-1797	314	9	of	of	ADP
ejpam-1797	314	10	j	j	PROPN
ejpam-1797	314	11	,	,	PUNCT
ejpam-1797	314	12	we	we	PRON
ejpam-1797	314	13	have	have	VERB
ejpam-1797	314	14	ssi	ssi	PROPN
ejpam-1797	314	15	⊆	⊆	NUM
ejpam-1797	314	16	i	i	PRON
ejpam-1797	314	17	.	.	PUNCT
ejpam-1797	315	1	this	this	PRON
ejpam-1797	315	2	proves	prove	VERB
ejpam-1797	315	3	that	that	SCONJ
ejpam-1797	315	4	i	i	PRON
ejpam-1797	315	5	is	be	AUX
ejpam-1797	315	6	an	an	DET
ejpam-1797	315	7	ideal	ideal	NOUN
ejpam-1797	315	8	of	of	ADP
ejpam-1797	315	9	s.	s.	PROPN
ejpam-1797	315	10	corollary	corollary	PROPN
ejpam-1797	315	11	1	1	NUM
ejpam-1797	315	12	.	.	PUNCT
ejpam-1797	316	1	let	let	VERB
ejpam-1797	316	2	s	s	PRON
ejpam-1797	316	3	be	be	AUX
ejpam-1797	316	4	a	a	DET
ejpam-1797	316	5	ternary	ternary	ADJ
ejpam-1797	316	6	semiring	semiring	NOUN
ejpam-1797	316	7	.	.	PUNCT
ejpam-1797	317	1	if	if	SCONJ
ejpam-1797	317	2	i	i	PRON
ejpam-1797	317	3	is	be	AUX
ejpam-1797	317	4	a	a	DET
ejpam-1797	317	5	prime	prime	ADJ
ejpam-1797	317	6	k	k	NOUN
ejpam-1797	317	7	-	-	NOUN
ejpam-1797	317	8	ideal	ideal	NOUN
ejpam-1797	317	9	of	of	ADP
ejpam-1797	317	10	j	j	PROPN
ejpam-1797	317	11	and	and	CCONJ
ejpam-1797	317	12	j	j	PROPN
ejpam-1797	317	13	is	be	AUX
ejpam-1797	317	14	an	an	DET
ejpam-1797	317	15	ideal	ideal	NOUN
ejpam-1797	317	16	of	of	ADP
ejpam-1797	317	17	s	s	PROPN
ejpam-1797	317	18	,	,	PUNCT
ejpam-1797	317	19	then	then	ADV
ejpam-1797	317	20	i	i	PRON
ejpam-1797	317	21	is	be	AUX
ejpam-1797	317	22	an	an	DET
ejpam-1797	317	23	ideal	ideal	NOUN
ejpam-1797	317	24	of	of	ADP
ejpam-1797	317	25	s.	s.	PROPN
ejpam-1797	317	26	we	we	PRON
ejpam-1797	317	27	now	now	ADV
ejpam-1797	317	28	formulate	formulate	VERB
ejpam-1797	317	29	a	a	DET
ejpam-1797	317	30	theorem	theorem	NOUN
ejpam-1797	317	31	of	of	ADP
ejpam-1797	317	32	weakly	weakly	ADJ
ejpam-1797	317	33	special	special	ADJ
ejpam-1797	317	34	radical	radical	ADJ
ejpam-1797	317	35	class	class	NOUN
ejpam-1797	317	36	of	of	ADP
ejpam-1797	317	37	ternary	ternary	ADJ
ejpam-1797	317	38	semirings	semiring	NOUN
ejpam-1797	317	39	.	.	PUNCT
ejpam-1797	318	1	theorem	theorem	VERB
ejpam-1797	318	2	8	8	NUM
ejpam-1797	318	3	.	.	PUNCT
ejpam-1797	319	1	ifm	ifm	PROPN
ejpam-1797	319	2	is	be	AUX
ejpam-1797	319	3	a	a	DET
ejpam-1797	319	4	weakly	weakly	ADJ
ejpam-1797	319	5	special	special	ADJ
ejpam-1797	319	6	radical	radical	ADJ
ejpam-1797	319	7	class	class	NOUN
ejpam-1797	319	8	of	of	ADP
ejpam-1797	319	9	ternary	ternary	ADJ
ejpam-1797	319	10	semirings	semiring	NOUN
ejpam-1797	319	11	,	,	PUNCT
ejpam-1797	319	12	then	then	ADV
ejpam-1797	319	13	um	um	INTJ
ejpam-1797	319	14	=	=	SYM
ejpam-1797	319	15	{	{	PUNCT
ejpam-1797	319	16	ternary	ternary	ADJ
ejpam-1797	319	17	semirings	semiring	NOUN
ejpam-1797	319	18	s	s	PART
ejpam-1797	319	19	:	:	PUNCT
ejpam-1797	319	20	no	no	DET
ejpam-1797	319	21	nonzero	nonzero	ADJ
ejpam-1797	319	22	homomorphic	homomorphic	ADJ
ejpam-1797	319	23	image	image	NOUN
ejpam-1797	319	24	of	of	ADP
ejpam-1797	319	25	s	s	PROPN
ejpam-1797	319	26	is	be	AUX
ejpam-1797	319	27	inm	inm	PROPN
ejpam-1797	319	28	}	}	PUNCT
ejpam-1797	319	29	is	be	AUX
ejpam-1797	319	30	a	a	DET
ejpam-1797	319	31	supernilpotent	supernilpotent	ADJ
ejpam-1797	319	32	radical	radical	ADJ
ejpam-1797	319	33	class	class	NOUN
ejpam-1797	319	34	.	.	PUNCT
ejpam-1797	320	1	proof	proof	NOUN
ejpam-1797	320	2	.	.	PUNCT
ejpam-1797	321	1	sincem	sincem	PROPN
ejpam-1797	321	2	is	be	AUX
ejpam-1797	321	3	a	a	DET
ejpam-1797	321	4	weakly	weakly	ADJ
ejpam-1797	321	5	special	special	ADJ
ejpam-1797	321	6	radical	radical	ADJ
ejpam-1797	321	7	class	class	NOUN
ejpam-1797	321	8	of	of	ADP
ejpam-1797	321	9	ternary	ternary	ADJ
ejpam-1797	321	10	semirings	semiring	NOUN
ejpam-1797	321	11	,	,	PUNCT
ejpam-1797	321	12	m	m	VERB
ejpam-1797	321	13	is	be	AUX
ejpam-1797	321	14	a	a	DET
ejpam-1797	321	15	hereditary	hereditary	ADJ
ejpam-1797	321	16	class	class	NOUN
ejpam-1797	321	17	.	.	PUNCT
ejpam-1797	322	1	let	let	VERB
ejpam-1797	322	2	s	s	PRON
ejpam-1797	322	3	∈m	∈m	VERB
ejpam-1797	322	4	and	and	CCONJ
ejpam-1797	322	5	i	i	PRON
ejpam-1797	322	6	be	be	VERB
ejpam-1797	322	7	a	a	DET
ejpam-1797	322	8	nonzero	nonzero	NOUN
ejpam-1797	322	9	ideal	ideal	NOUN
ejpam-1797	322	10	of	of	ADP
ejpam-1797	322	11	s.	s.	PROPN
ejpam-1797	323	1	then	then	ADV
ejpam-1797	323	2	i	i	PRON
ejpam-1797	323	3	∈m	∈m	NOUN
ejpam-1797	323	4	.	.	PUNCT
ejpam-1797	324	1	now	now	ADV
ejpam-1797	324	2	i	i	PRON
ejpam-1797	324	3	is	be	AUX
ejpam-1797	324	4	a	a	DET
ejpam-1797	324	5	homomorphic	homomorphic	ADJ
ejpam-1797	324	6	image	image	NOUN
ejpam-1797	324	7	of	of	ADP
ejpam-1797	324	8	itself	itself	PRON
ejpam-1797	324	9	.	.	PUNCT
ejpam-1797	325	1	this	this	PRON
ejpam-1797	325	2	means	mean	VERB
ejpam-1797	325	3	that	that	SCONJ
ejpam-1797	325	4	m	m	NOUN
ejpam-1797	325	5	is	be	AUX
ejpam-1797	325	6	regular	regular	ADJ
ejpam-1797	325	7	.	.	PUNCT
ejpam-1797	326	1	hence	hence	ADV
ejpam-1797	326	2	,	,	PUNCT
ejpam-1797	326	3	by	by	ADP
ejpam-1797	326	4	theorem	theorem	NOUN
ejpam-1797	326	5	5	5	NUM
ejpam-1797	326	6	,	,	PUNCT
ejpam-1797	326	7	um	um	INTJ
ejpam-1797	326	8	is	be	AUX
ejpam-1797	326	9	a	a	DET
ejpam-1797	326	10	radical	radical	ADJ
ejpam-1797	326	11	class	class	NOUN
ejpam-1797	326	12	.	.	PUNCT
ejpam-1797	327	1	in	in	ADP
ejpam-1797	327	2	order	order	NOUN
ejpam-1797	327	3	to	to	PART
ejpam-1797	327	4	show	show	VERB
ejpam-1797	327	5	that	that	SCONJ
ejpam-1797	327	6	um	um	INTJ
ejpam-1797	327	7	is	be	AUX
ejpam-1797	327	8	a	a	DET
ejpam-1797	327	9	hereditary	hereditary	ADJ
ejpam-1797	327	10	class	class	NOUN
ejpam-1797	327	11	,	,	PUNCT
ejpam-1797	327	12	let	let	VERB
ejpam-1797	327	13	s	s	PRON
ejpam-1797	327	14	∈	∈	VERB
ejpam-1797	327	15	um	um	INTJ
ejpam-1797	327	16	and	and	CCONJ
ejpam-1797	327	17	j	j	PROPN
ejpam-1797	327	18	be	be	AUX
ejpam-1797	327	19	a	a	DET
ejpam-1797	327	20	nonzero	nonzero	NOUN
ejpam-1797	327	21	ideal	ideal	NOUN
ejpam-1797	327	22	of	of	ADP
ejpam-1797	327	23	s.	s.	PROPN
ejpam-1797	327	24	if	if	SCONJ
ejpam-1797	327	25	j	j	PROPN
ejpam-1797	327	26	6∈	6∈	PROPN
ejpam-1797	327	27	um	um	INTJ
ejpam-1797	327	28	,	,	PUNCT
ejpam-1797	327	29	then	then	ADV
ejpam-1797	327	30	there	there	PRON
ejpam-1797	327	31	is	be	VERB
ejpam-1797	327	32	a	a	DET
ejpam-1797	327	33	nonzero	nonzero	ADJ
ejpam-1797	327	34	homomorphic	homomorphic	ADJ
ejpam-1797	327	35	image	image	NOUN
ejpam-1797	327	36	φ(j	φ(j	PROPN
ejpam-1797	327	37	)	)	PUNCT
ejpam-1797	327	38	of	of	ADP
ejpam-1797	327	39	j	j	PROPN
ejpam-1797	327	40	inm	inm	PROPN
ejpam-1797	327	41	.	.	PUNCT
ejpam-1797	328	1	let	let	VERB
ejpam-1797	328	2	k	k	PROPN
ejpam-1797	328	3	=	=	SYM
ejpam-1797	328	4	kerφ	kerφ	PROPN
ejpam-1797	328	5	.	.	PUNCT
ejpam-1797	329	1	then	then	ADV
ejpam-1797	329	2	k	k	PROPN
ejpam-1797	329	3	is	be	AUX
ejpam-1797	329	4	a	a	DET
ejpam-1797	329	5	k	k	NOUN
ejpam-1797	329	6	-	-	NOUN
ejpam-1797	329	7	ideal	ideal	NOUN
ejpam-1797	329	8	of	of	ADP
ejpam-1797	329	9	j	j	PROPN
ejpam-1797	329	10	and	and	CCONJ
ejpam-1797	329	11	j	j	PROPN
ejpam-1797	329	12	/	/	SYM
ejpam-1797	329	13	k	k	PROPN
ejpam-1797	329	14	≃	≃	ADJ
ejpam-1797	329	15	φ(j	φ(j	PROPN
ejpam-1797	329	16	)	)	PUNCT
ejpam-1797	329	17	.	.	PUNCT
ejpam-1797	330	1	since	since	SCONJ
ejpam-1797	330	2	φ(j	φ(j	NOUN
ejpam-1797	330	3	)	)	PUNCT
ejpam-1797	330	4	∈m	∈m	NOUN
ejpam-1797	330	5	,	,	PUNCT
ejpam-1797	330	6	φ(j	φ(j	PROPN
ejpam-1797	330	7	)	)	PUNCT
ejpam-1797	330	8	is	be	AUX
ejpam-1797	330	9	semiprime	semiprime	NOUN
ejpam-1797	330	10	,	,	PUNCT
ejpam-1797	330	11	and	and	CCONJ
ejpam-1797	330	12	so	so	ADV
ejpam-1797	330	13	by	by	ADP
ejpam-1797	330	14	lemma	lemma	PROPN
ejpam-1797	330	15	3	3	NUM
ejpam-1797	330	16	,	,	PUNCT
ejpam-1797	330	17	j	j	PROPN
ejpam-1797	330	18	/	/	SYM
ejpam-1797	330	19	k	k	PROPN
ejpam-1797	330	20	is	be	AUX
ejpam-1797	330	21	semiprime	semiprime	NOUN
ejpam-1797	330	22	.	.	PUNCT
ejpam-1797	331	1	now	now	ADV
ejpam-1797	331	2	φ(j	φ(j	PROPN
ejpam-1797	331	3	)	)	PUNCT
ejpam-1797	331	4	is	be	AUX
ejpam-1797	331	5	nonzero	nonzero	ADJ
ejpam-1797	331	6	,	,	PUNCT
ejpam-1797	331	7	and	and	CCONJ
ejpam-1797	331	8	hence	hence	ADV
ejpam-1797	331	9	,	,	PUNCT
ejpam-1797	331	10	j	j	PROPN
ejpam-1797	331	11	/	/	SYM
ejpam-1797	331	12	k	k	PROPN
ejpam-1797	331	13	6=	6=	PROPN
ejpam-1797	331	14	(	(	PUNCT
ejpam-1797	331	15	0	0	NUM
ejpam-1797	331	16	)	)	PUNCT
ejpam-1797	331	17	.	.	PUNCT
ejpam-1797	332	1	since	since	SCONJ
ejpam-1797	332	2	k	k	PROPN
ejpam-1797	332	3	is	be	AUX
ejpam-1797	332	4	a	a	DET
ejpam-1797	332	5	k	k	NOUN
ejpam-1797	332	6	-	-	NOUN
ejpam-1797	332	7	ideal	ideal	ADJ
ejpam-1797	332	8	,	,	PUNCT
ejpam-1797	332	9	k	k	PROPN
ejpam-1797	332	10	is	be	AUX
ejpam-1797	332	11	a	a	DET
ejpam-1797	332	12	semiprime	semiprime	NOUN
ejpam-1797	332	13	ideal	ideal	NOUN
ejpam-1797	332	14	of	of	ADP
ejpam-1797	332	15	j	j	PROPN
ejpam-1797	332	16	and	and	CCONJ
ejpam-1797	332	17	,	,	PUNCT
ejpam-1797	332	18	whence	whence	PROPN
ejpam-1797	332	19	k	k	PROPN
ejpam-1797	332	20	is	be	AUX
ejpam-1797	332	21	an	an	DET
ejpam-1797	332	22	ideal	ideal	NOUN
ejpam-1797	332	23	of	of	ADP
ejpam-1797	332	24	s	s	PROPN
ejpam-1797	332	25	,	,	PUNCT
ejpam-1797	332	26	by	by	ADP
ejpam-1797	332	27	proposition	proposition	NOUN
ejpam-1797	332	28	5	5	NUM
ejpam-1797	332	29	.	.	PUNCT
ejpam-1797	333	1	now	now	ADV
ejpam-1797	333	2	j	j	PROPN
ejpam-1797	333	3	/	/	SYM
ejpam-1797	333	4	k	k	PROPN
ejpam-1797	333	5	is	be	AUX
ejpam-1797	333	6	a	a	DET
ejpam-1797	333	7	nonzero	nonzero	ADJ
ejpam-1797	333	8	ideal	ideal	NOUN
ejpam-1797	333	9	of	of	ADP
ejpam-1797	333	10	s	s	PROPN
ejpam-1797	333	11	/	/	SYM
ejpam-1797	333	12	k	k	NOUN
ejpam-1797	333	13	,	,	PUNCT
ejpam-1797	333	14	and	and	CCONJ
ejpam-1797	333	15	so	so	ADV
ejpam-1797	333	16	by	by	ADP
ejpam-1797	333	17	the	the	DET
ejpam-1797	333	18	property	property	NOUN
ejpam-1797	333	19	(	(	PUNCT
ejpam-1797	333	20	z	z	NOUN
ejpam-1797	333	21	)	)	PUNCT
ejpam-1797	333	22	,	,	PUNCT
ejpam-1797	333	23	(	(	PUNCT
ejpam-1797	333	24	s	s	X
ejpam-1797	333	25	/	/	SYM
ejpam-1797	333	26	k)/anns(j	k)/anns(j	NOUN
ejpam-1797	333	27	/	/	SYM
ejpam-1797	333	28	k	k	NOUN
ejpam-1797	333	29	)	)	PUNCT
ejpam-1797	333	30	∈	∈	PROPN
ejpam-1797	333	31	m	m	VERB
ejpam-1797	333	32	,	,	PUNCT
ejpam-1797	333	33	since	since	SCONJ
ejpam-1797	333	34	by	by	ADP
ejpam-1797	333	35	the	the	DET
ejpam-1797	333	36	property	property	NOUN
ejpam-1797	333	37	(	(	PUNCT
ejpam-1797	333	38	x	x	NOUN
ejpam-1797	333	39	)	)	PUNCT
ejpam-1797	333	40	,	,	PUNCT
ejpam-1797	333	41	j	j	PROPN
ejpam-1797	333	42	/	/	SYM
ejpam-1797	333	43	k	k	PROPN
ejpam-1797	333	44	∈	∈	PROPN
ejpam-1797	333	45	m	m	VERB
ejpam-1797	333	46	.	.	PUNCT
ejpam-1797	334	1	but	but	CCONJ
ejpam-1797	334	2	,	,	PUNCT
ejpam-1797	334	3	if	if	SCONJ
ejpam-1797	334	4	s	s	PROPN
ejpam-1797	334	5	/	/	SYM
ejpam-1797	334	6	k	k	PROPN
ejpam-1797	334	7	⊆	⊆	NUM
ejpam-1797	334	8	anns(j	anns(j	PROPN
ejpam-1797	334	9	/	/	SYM
ejpam-1797	334	10	k	k	NOUN
ejpam-1797	334	11	)	)	PUNCT
ejpam-1797	334	12	,	,	PUNCT
ejpam-1797	334	13	then	then	ADV
ejpam-1797	334	14	we	we	PRON
ejpam-1797	334	15	have	have	VERB
ejpam-1797	334	16	(	(	PUNCT
ejpam-1797	334	17	j	j	NOUN
ejpam-1797	334	18	/	/	SYM
ejpam-1797	334	19	k)3	k)3	PROPN
ejpam-1797	334	20	=	=	SYM
ejpam-1797	334	21	(	(	PUNCT
ejpam-1797	334	22	0	0	NUM
ejpam-1797	334	23	)	)	PUNCT
ejpam-1797	334	24	which	which	PRON
ejpam-1797	334	25	can	can	AUX
ejpam-1797	334	26	not	not	PART
ejpam-1797	334	27	happen	happen	VERB
ejpam-1797	334	28	since	since	SCONJ
ejpam-1797	334	29	j	j	PROPN
ejpam-1797	334	30	/	/	SYM
ejpam-1797	334	31	k	k	PROPN
ejpam-1797	334	32	is	be	AUX
ejpam-1797	334	33	a	a	DET
ejpam-1797	334	34	semiprime	semiprime	NOUN
ejpam-1797	334	35	ternary	ternary	ADJ
ejpam-1797	334	36	semiring	semiring	NOUN
ejpam-1797	334	37	.	.	PUNCT
ejpam-1797	335	1	this	this	PRON
ejpam-1797	335	2	leads	lead	VERB
ejpam-1797	335	3	to	to	ADP
ejpam-1797	335	4	anns(j	anns(j	PROPN
ejpam-1797	335	5	/	/	SYM
ejpam-1797	335	6	k	k	NOUN
ejpam-1797	335	7	)	)	PUNCT
ejpam-1797	335	8	6=	6=	ADP
ejpam-1797	335	9	s	s	X
ejpam-1797	335	10	/	/	SYM
ejpam-1797	335	11	k	k	PROPN
ejpam-1797	335	12	.	.	PUNCT
ejpam-1797	336	1	thus	thus	ADV
ejpam-1797	336	2	,	,	PUNCT
ejpam-1797	336	3	(	(	PUNCT
ejpam-1797	336	4	s	s	X
ejpam-1797	336	5	/	/	SYM
ejpam-1797	336	6	k)/anns(j	k)/anns(j	NOUN
ejpam-1797	336	7	/	/	SYM
ejpam-1797	336	8	k	k	NOUN
ejpam-1797	336	9	)	)	PUNCT
ejpam-1797	336	10	is	be	AUX
ejpam-1797	336	11	a	a	DET
ejpam-1797	336	12	homomorphic	homomorphic	ADJ
ejpam-1797	336	13	image	image	NOUN
ejpam-1797	336	14	of	of	ADP
ejpam-1797	336	15	s	s	PRON
ejpam-1797	336	16	which	which	PRON
ejpam-1797	336	17	is	be	AUX
ejpam-1797	336	18	inm	inm	PROPN
ejpam-1797	336	19	and	and	CCONJ
ejpam-1797	336	20	(	(	PUNCT
ejpam-1797	336	21	s	s	X
ejpam-1797	336	22	/	/	SYM
ejpam-1797	336	23	k)/anns(j	k)/anns(j	NOUN
ejpam-1797	336	24	/	/	SYM
ejpam-1797	336	25	k	k	NOUN
ejpam-1797	336	26	)	)	PUNCT
ejpam-1797	336	27	6=	6=	ADP
ejpam-1797	336	28	(	(	PUNCT
ejpam-1797	336	29	0	0	NUM
ejpam-1797	336	30	)	)	PUNCT
ejpam-1797	336	31	,	,	PUNCT
ejpam-1797	336	32	since	since	SCONJ
ejpam-1797	336	33	any	any	DET
ejpam-1797	336	34	annihilator	annihilator	PROPN
ejpam-1797	336	35	ideal	ideal	NOUN
ejpam-1797	336	36	is	be	AUX
ejpam-1797	336	37	necessarily	necessarily	ADV
ejpam-1797	336	38	a	a	DET
ejpam-1797	336	39	k	k	NOUN
ejpam-1797	336	40	-	-	NOUN
ejpam-1797	336	41	ideal	ideal	NOUN
ejpam-1797	336	42	.	.	PUNCT
ejpam-1797	337	1	however	however	ADV
ejpam-1797	337	2	,	,	PUNCT
ejpam-1797	337	3	this	this	PRON
ejpam-1797	337	4	is	be	AUX
ejpam-1797	337	5	impossible	impossible	ADJ
ejpam-1797	337	6	because	because	SCONJ
ejpam-1797	337	7	s	s	VERB
ejpam-1797	337	8	∈	∈	PROPN
ejpam-1797	337	9	um	um	INTJ
ejpam-1797	337	10	.	.	PUNCT
ejpam-1797	338	1	hence	hence	ADV
ejpam-1797	338	2	,	,	PUNCT
ejpam-1797	338	3	we	we	PRON
ejpam-1797	338	4	deduce	deduce	VERB
ejpam-1797	338	5	that	that	SCONJ
ejpam-1797	338	6	j	j	PROPN
ejpam-1797	338	7	∈	∈	PROPN
ejpam-1797	338	8	um	um	INTJ
ejpam-1797	338	9	and	and	CCONJ
ejpam-1797	338	10	hence	hence	ADV
ejpam-1797	338	11	,	,	PUNCT
ejpam-1797	338	12	we	we	PRON
ejpam-1797	338	13	have	have	AUX
ejpam-1797	338	14	proved	prove	VERB
ejpam-1797	338	15	that	that	SCONJ
ejpam-1797	338	16	um	um	INTJ
ejpam-1797	338	17	is	be	AUX
ejpam-1797	338	18	hereditary	hereditary	ADJ
ejpam-1797	338	19	.	.	PUNCT
ejpam-1797	339	1	finally	finally	ADV
ejpam-1797	339	2	,	,	PUNCT
ejpam-1797	339	3	if	if	SCONJ
ejpam-1797	339	4	s	s	VERB
ejpam-1797	339	5	is	be	AUX
ejpam-1797	339	6	any	any	DET
ejpam-1797	339	7	nilpotent	nilpotent	ADJ
ejpam-1797	339	8	ternary	ternary	ADJ
ejpam-1797	339	9	semiring	semiring	NOUN
ejpam-1797	339	10	,	,	PUNCT
ejpam-1797	339	11	then	then	ADV
ejpam-1797	339	12	φ(s	φ(s	NOUN
ejpam-1797	339	13	)	)	PUNCT
ejpam-1797	339	14	is	be	AUX
ejpam-1797	339	15	nilpotent	nilpotent	ADJ
ejpam-1797	339	16	for	for	ADP
ejpam-1797	339	17	any	any	DET
ejpam-1797	339	18	nonzero	nonzero	NOUN
ejpam-1797	339	19	homomorphism	homomorphism	PROPN
ejpam-1797	339	20	φ	φ	NUM
ejpam-1797	339	21	,	,	PUNCT
ejpam-1797	339	22	and	and	CCONJ
ejpam-1797	339	23	hence	hence	ADV
ejpam-1797	339	24	φ(s	φ(s	NOUN
ejpam-1797	339	25	)	)	PUNCT
ejpam-1797	339	26	6∈	6∈	PROPN
ejpam-1797	339	27	m	m	NOUN
ejpam-1797	339	28	.	.	PUNCT
ejpam-1797	340	1	thus	thus	ADV
ejpam-1797	340	2	,	,	PUNCT
ejpam-1797	340	3	s	s	VERB
ejpam-1797	340	4	∈	∈	X
ejpam-1797	340	5	um	um	INTJ
ejpam-1797	340	6	and	and	CCONJ
ejpam-1797	340	7	hence	hence	ADV
ejpam-1797	340	8	,	,	PUNCT
ejpam-1797	340	9	um	um	INTJ
ejpam-1797	340	10	is	be	AUX
ejpam-1797	340	11	supernilpotent	supernilpotent	NOUN
ejpam-1797	340	12	.	.	PUNCT
ejpam-1797	341	1	we	we	PRON
ejpam-1797	341	2	here	here	ADV
ejpam-1797	341	3	call	call	VERB
ejpam-1797	341	4	um	um	INTJ
ejpam-1797	341	5	the	the	DET
ejpam-1797	341	6	upper	upper	ADJ
ejpam-1797	341	7	radical	radical	ADJ
ejpam-1797	341	8	class	class	NOUN
ejpam-1797	341	9	determined	determine	VERB
ejpam-1797	341	10	by	by	ADP
ejpam-1797	341	11	the	the	DET
ejpam-1797	341	12	classm	classm	PROPN
ejpam-1797	341	13	.	.	PUNCT
ejpam-1797	342	1	proposition	proposition	NOUN
ejpam-1797	342	2	8	8	NUM
ejpam-1797	342	3	.	.	PUNCT
ejpam-1797	342	4	s	s	PART
ejpam-1797	343	1	=	=	X
ejpam-1797	343	2	{	{	PUNCT
ejpam-1797	343	3	s	s	X
ejpam-1797	343	4	:	:	PUNCT
ejpam-1797	343	5	s	s	VERB
ejpam-1797	343	6	is	be	AUX
ejpam-1797	343	7	a	a	DET
ejpam-1797	343	8	ternary	ternary	ADJ
ejpam-1797	343	9	semiring	semiring	NOUN
ejpam-1797	343	10	such	such	ADJ
ejpam-1797	343	11	that	that	SCONJ
ejpam-1797	343	12	every	every	DET
ejpam-1797	343	13	nonzero	nonzero	ADJ
ejpam-1797	343	14	homomorphic	homomorphic	ADJ
ejpam-1797	343	15	image	image	NOUN
ejpam-1797	343	16	s′	s′	NUM
ejpam-1797	343	17	of	of	ADP
ejpam-1797	343	18	s	s	PROPN
ejpam-1797	343	19	contains	contain	VERB
ejpam-1797	343	20	a	a	DET
ejpam-1797	343	21	nonzero	nonzero	PROPN
ejpam-1797	343	22	ideal	ideal	NOUN
ejpam-1797	343	23	which	which	PRON
ejpam-1797	343	24	is	be	AUX
ejpam-1797	343	25	singular	singular	ADJ
ejpam-1797	343	26	as	as	ADP
ejpam-1797	343	27	a	a	DET
ejpam-1797	343	28	ternary	ternary	ADJ
ejpam-1797	343	29	semiring	semiring	NOUN
ejpam-1797	343	30	}	}	PUNCT
ejpam-1797	343	31	is	be	AUX
ejpam-1797	343	32	an	an	DET
ejpam-1797	343	33	upper	upper	ADJ
ejpam-1797	343	34	radical	radical	ADJ
ejpam-1797	343	35	class	class	NOUN
ejpam-1797	343	36	determined	determine	VERB
ejpam-1797	343	37	by	by	ADP
ejpam-1797	343	38	the	the	DET
ejpam-1797	343	39	class	class	NOUN
ejpam-1797	343	40	℘	℘	PROPN
ejpam-1797	343	41	of	of	ADP
ejpam-1797	343	42	semiprime	semiprime	NOUN
ejpam-1797	343	43	non	non	ADJ
ejpam-1797	343	44	-	-	ADJ
ejpam-1797	343	45	singular	singular	ADJ
ejpam-1797	343	46	ternary	ternary	ADJ
ejpam-1797	343	47	semirings	semiring	NOUN
ejpam-1797	343	48	.	.	PUNCT
ejpam-1797	344	1	proof	proof	NOUN
ejpam-1797	344	2	.	.	PUNCT
ejpam-1797	345	1	now	now	ADV
ejpam-1797	345	2	u℘	u℘	PUNCT
ejpam-1797	346	1	=	=	PUNCT
ejpam-1797	346	2	{	{	PUNCT
ejpam-1797	346	3	s	s	X
ejpam-1797	346	4	:	:	PUNCT
ejpam-1797	346	5	no	no	DET
ejpam-1797	346	6	nonzero	nonzero	ADJ
ejpam-1797	346	7	homomorphic	homomorphic	ADJ
ejpam-1797	346	8	image	image	NOUN
ejpam-1797	346	9	of	of	ADP
ejpam-1797	346	10	s	s	NOUN
ejpam-1797	346	11	is	be	AUX
ejpam-1797	346	12	in	in	ADP
ejpam-1797	346	13	℘	℘	NOUN
ejpam-1797	346	14	}	}	PUNCT
ejpam-1797	346	15	=	=	PUNCT
ejpam-1797	346	16	{	{	PUNCT
ejpam-1797	346	17	s	s	X
ejpam-1797	346	18	:	:	PUNCT
ejpam-1797	346	19	for	for	ADP
ejpam-1797	346	20	every	every	DET
ejpam-1797	346	21	nonzero	nonzero	ADJ
ejpam-1797	346	22	homomorphic	homomorphic	ADJ
ejpam-1797	346	23	image	image	NOUN
ejpam-1797	346	24	s1	s1	NOUN
ejpam-1797	346	25	of	of	ADP
ejpam-1797	346	26	s	s	PRON
ejpam-1797	346	27	either	either	CCONJ
ejpam-1797	346	28	β(s1	β(s1	NOUN
ejpam-1797	346	29	)	)	PUNCT
ejpam-1797	346	30	6=	6=	ADP
ejpam-1797	346	31	0	0	NUM
ejpam-1797	346	32	or	or	CCONJ
ejpam-1797	346	33	z(s1	z(s1	NOUN
ejpam-1797	346	34	)	)	PUNCT
ejpam-1797	347	1	6=	6=	ADP
ejpam-1797	347	2	0	0	NUM
ejpam-1797	347	3	}	}	PUNCT
ejpam-1797	347	4	=	=	SYM
ejpam-1797	347	5	s	s	X
ejpam-1797	347	6	(	(	PUNCT
ejpam-1797	347	7	by	by	ADP
ejpam-1797	347	8	proposition	proposition	NOUN
ejpam-1797	347	9	6	6	NUM
ejpam-1797	347	10	,	,	PUNCT
ejpam-1797	347	11	since	since	SCONJ
ejpam-1797	347	12	s1	s1	PROPN
ejpam-1797	347	13	∈	∈	PROPN
ejpam-1797	347	14	℘	℘	PROPN
ejpam-1797	347	15	implies	imply	VERB
ejpam-1797	347	16	β(s1	β(s1	NOUN
ejpam-1797	347	17	)	)	PUNCT
ejpam-1797	347	18	=	=	SYM
ejpam-1797	347	19	0	0	NUM
ejpam-1797	347	20	and	and	CCONJ
ejpam-1797	347	21	z(s1	z(s1	NOUN
ejpam-1797	347	22	)	)	PUNCT
ejpam-1797	347	23	=	=	PUNCT
ejpam-1797	348	1	0	0	NUM
ejpam-1797	348	2	)	)	PUNCT
ejpam-1797	348	3	.	.	PUNCT
ejpam-1797	349	1	t.	t.	PROPN
ejpam-1797	349	2	dutta	dutta	PROPN
ejpam-1797	349	3	,	,	PUNCT
ejpam-1797	349	4	k.	k.	PROPN
ejpam-1797	349	5	shum	shum	PROPN
ejpam-1797	349	6	,	,	PUNCT
ejpam-1797	349	7	s.	s.	PROPN
ejpam-1797	349	8	mandal	mandal	PROPN
ejpam-1797	349	9	/	/	SYM
ejpam-1797	349	10	eur	eur	PROPN
ejpam-1797	349	11	.	.	PUNCT
ejpam-1797	350	1	j.	j.	PROPN
ejpam-1797	350	2	pure	pure	PROPN
ejpam-1797	350	3	appl	appl	PROPN
ejpam-1797	350	4	.	.	PROPN
ejpam-1797	350	5	math	math	PROPN
ejpam-1797	350	6	,	,	PUNCT
ejpam-1797	350	7	5	5	NUM
ejpam-1797	350	8	(	(	PUNCT
ejpam-1797	350	9	2012	2012	NUM
ejpam-1797	350	10	)	)	PUNCT
ejpam-1797	350	11	,	,	PUNCT
ejpam-1797	350	12	401	401	NUM
ejpam-1797	350	13	-	-	SYM
ejpam-1797	350	14	413	413	NUM
ejpam-1797	350	15	410	410	NUM
ejpam-1797	350	16	we	we	PRON
ejpam-1797	350	17	now	now	ADV
ejpam-1797	350	18	simply	simply	ADV
ejpam-1797	350	19	call	call	VERB
ejpam-1797	350	20	s	s	PRON
ejpam-1797	350	21	the	the	DET
ejpam-1797	350	22	singular	singular	ADJ
ejpam-1797	350	23	radical	radical	NOUN
ejpam-1797	350	24	.	.	PUNCT
ejpam-1797	351	1	following	follow	VERB
ejpam-1797	351	2	d.	d.	PROPN
ejpam-1797	351	3	m.	m.	PROPN
ejpam-1797	351	4	olson	olson	PROPN
ejpam-1797	351	5	and	and	CCONJ
ejpam-1797	351	6	a.	a.	PROPN
ejpam-1797	351	7	c.	c.	PROPN
ejpam-1797	351	8	nance	nance	PROPN
ejpam-1797	351	9	[	[	X
ejpam-1797	351	10	27	27	NUM
ejpam-1797	351	11	]	]	PUNCT
ejpam-1797	351	12	,	,	PUNCT
ejpam-1797	351	13	we	we	PRON
ejpam-1797	351	14	define	define	VERB
ejpam-1797	351	15	the	the	DET
ejpam-1797	351	16	special	special	ADJ
ejpam-1797	351	17	radical	radical	ADJ
ejpam-1797	351	18	class	class	NOUN
ejpam-1797	351	19	of	of	ADP
ejpam-1797	351	20	a	a	DET
ejpam-1797	351	21	ternary	ternary	ADJ
ejpam-1797	351	22	semiring	semiring	NOUN
ejpam-1797	351	23	as	as	SCONJ
ejpam-1797	351	24	follows	follow	VERB
ejpam-1797	351	25	:	:	PUNCT
ejpam-1797	351	26	definition	definition	NOUN
ejpam-1797	351	27	11	11	NUM
ejpam-1797	351	28	.	.	PUNCT
ejpam-1797	352	1	a	a	DET
ejpam-1797	352	2	classm	classm	NOUN
ejpam-1797	352	3	of	of	ADP
ejpam-1797	352	4	ternary	ternary	ADJ
ejpam-1797	352	5	semirings	semiring	NOUN
ejpam-1797	352	6	is	be	AUX
ejpam-1797	352	7	called	call	VERB
ejpam-1797	352	8	a	a	DET
ejpam-1797	352	9	special	special	ADJ
ejpam-1797	352	10	radical	radical	ADJ
ejpam-1797	352	11	class	class	NOUN
ejpam-1797	352	12	ifm	ifm	NOUN
ejpam-1797	352	13	is	be	AUX
ejpam-1797	352	14	a	a	DET
ejpam-1797	352	15	hereditary	hereditary	ADJ
ejpam-1797	352	16	class	class	NOUN
ejpam-1797	352	17	of	of	ADP
ejpam-1797	352	18	prime	prime	ADJ
ejpam-1797	352	19	ternary	ternary	ADJ
ejpam-1797	352	20	semiring	semiring	NOUN
ejpam-1797	352	21	satisfying	satisfy	VERB
ejpam-1797	352	22	the	the	DET
ejpam-1797	352	23	following	follow	VERB
ejpam-1797	352	24	conditions	condition	NOUN
ejpam-1797	352	25	:	:	PUNCT
ejpam-1797	352	26	(	(	PUNCT
ejpam-1797	352	27	1	1	X
ejpam-1797	352	28	)	)	PUNCT
ejpam-1797	352	29	if	if	SCONJ
ejpam-1797	352	30	s	s	NOUN
ejpam-1797	352	31	is	be	AUX
ejpam-1797	352	32	ternary	ternary	ADJ
ejpam-1797	352	33	semi	semi	ADJ
ejpam-1797	352	34	-	-	ADJ
ejpam-1797	352	35	isomorphic	isomorphic	ADJ
ejpam-1797	352	36	to	to	ADP
ejpam-1797	352	37	t	t	PROPN
ejpam-1797	352	38	and	and	CCONJ
ejpam-1797	352	39	t	t	PROPN
ejpam-1797	352	40	∈m	∈m	NOUN
ejpam-1797	352	41	,	,	PUNCT
ejpam-1797	352	42	then	then	ADV
ejpam-1797	352	43	s	s	VERB
ejpam-1797	352	44	∈m	∈m	NOUN
ejpam-1797	352	45	.	.	PUNCT
ejpam-1797	353	1	(	(	PUNCT
ejpam-1797	353	2	2	2	X
ejpam-1797	353	3	)	)	PUNCT
ejpam-1797	353	4	if	if	SCONJ
ejpam-1797	353	5	i	i	PRON
ejpam-1797	353	6	∈m	∈m	VERB
ejpam-1797	354	1	and	and	CCONJ
ejpam-1797	354	2	i	i	PRON
ejpam-1797	354	3	is	be	AUX
ejpam-1797	354	4	an	an	DET
ejpam-1797	354	5	ideal	ideal	NOUN
ejpam-1797	354	6	of	of	ADP
ejpam-1797	354	7	a	a	DET
ejpam-1797	354	8	ternary	ternary	ADJ
ejpam-1797	354	9	semiring	semire	VERB
ejpam-1797	354	10	s	s	NOUN
ejpam-1797	354	11	,	,	PUNCT
ejpam-1797	354	12	then	then	ADV
ejpam-1797	354	13	s	s	PROPN
ejpam-1797	354	14	/	/	SYM
ejpam-1797	354	15	anns(i	anns(i	NOUN
ejpam-1797	354	16	)	)	PUNCT
ejpam-1797	354	17	∈m	∈m	NOUN
ejpam-1797	354	18	.	.	PUNCT
ejpam-1797	355	1	for	for	ADP
ejpam-1797	355	2	prime	prime	ADJ
ejpam-1797	355	3	ternary	ternary	ADJ
ejpam-1797	355	4	semirings	semiring	NOUN
ejpam-1797	355	5	,	,	PUNCT
ejpam-1797	355	6	we	we	PRON
ejpam-1797	355	7	have	have	VERB
ejpam-1797	355	8	he	he	PRON
ejpam-1797	355	9	following	follow	VERB
ejpam-1797	355	10	lemmas	lemmas	PROPN
ejpam-1797	355	11	.	.	PUNCT
ejpam-1797	356	1	lemma	lemma	PROPN
ejpam-1797	356	2	6	6	NUM
ejpam-1797	356	3	.	.	PUNCT
ejpam-1797	357	1	if	if	SCONJ
ejpam-1797	357	2	the	the	DET
ejpam-1797	357	3	ternary	ternary	ADJ
ejpam-1797	357	4	semirings	semiring	NOUN
ejpam-1797	357	5	s	s	PART
ejpam-1797	357	6	≃	≃	PROPN
ejpam-1797	357	7	t	t	PROPN
ejpam-1797	357	8	and	and	CCONJ
ejpam-1797	357	9	t	t	PROPN
ejpam-1797	357	10	is	be	AUX
ejpam-1797	357	11	prime	prime	ADJ
ejpam-1797	357	12	,	,	PUNCT
ejpam-1797	357	13	then	then	ADV
ejpam-1797	357	14	s	s	VERB
ejpam-1797	357	15	is	be	AUX
ejpam-1797	357	16	prime	prime	ADJ
ejpam-1797	357	17	.	.	PUNCT
ejpam-1797	358	1	proof	proof	NOUN
ejpam-1797	358	2	.	.	PUNCT
ejpam-1797	359	1	suppose	suppose	VERB
ejpam-1797	359	2	that	that	SCONJ
ejpam-1797	359	3	φ	φ	PROPN
ejpam-1797	359	4	is	be	AUX
ejpam-1797	359	5	a	a	DET
ejpam-1797	359	6	semi	semi	NOUN
ejpam-1797	359	7	-	-	NOUN
ejpam-1797	359	8	isomorphism	isomorphism	NOUN
ejpam-1797	359	9	and	and	CCONJ
ejpam-1797	359	10	a	a	DET
ejpam-1797	359	11	,	,	PUNCT
ejpam-1797	359	12	b	b	NOUN
ejpam-1797	359	13	,	,	PUNCT
ejpam-1797	359	14	c	c	PROPN
ejpam-1797	359	15	are	be	AUX
ejpam-1797	359	16	ideals	ideal	NOUN
ejpam-1797	359	17	of	of	ADP
ejpam-1797	359	18	s	s	PRON
ejpam-1797	359	19	such	such	ADJ
ejpam-1797	359	20	that	that	SCONJ
ejpam-1797	359	21	abc	abc	PROPN
ejpam-1797	359	22	=	=	SYM
ejpam-1797	359	23	(	(	PUNCT
ejpam-1797	359	24	0	0	NUM
ejpam-1797	359	25	)	)	PUNCT
ejpam-1797	359	26	.	.	PUNCT
ejpam-1797	360	1	then	then	ADV
ejpam-1797	360	2	φ(a),φ(b),φ(c	φ(a),φ(b),φ(c	NUM
ejpam-1797	360	3	)	)	PUNCT
ejpam-1797	360	4	are	be	AUX
ejpam-1797	360	5	ideals	ideal	NOUN
ejpam-1797	360	6	of	of	ADP
ejpam-1797	360	7	t	t	PROPN
ejpam-1797	360	8	and	and	CCONJ
ejpam-1797	360	9	φ(abc	φ(abc	PROPN
ejpam-1797	360	10	)	)	PUNCT
ejpam-1797	361	1	=	=	PUNCT
ejpam-1797	361	2	(	(	PUNCT
ejpam-1797	361	3	0	0	X
ejpam-1797	361	4	)	)	PUNCT
ejpam-1797	361	5	⇒	⇒	NOUN
ejpam-1797	361	6	φ(a)φ(b)φ(c	φ(a)φ(b)φ(c	NUM
ejpam-1797	361	7	)	)	PUNCT
ejpam-1797	361	8	=	=	SYM
ejpam-1797	361	9	(	(	PUNCT
ejpam-1797	361	10	0	0	NUM
ejpam-1797	361	11	)	)	PUNCT
ejpam-1797	361	12	.	.	PUNCT
ejpam-1797	362	1	since	since	SCONJ
ejpam-1797	362	2	t	t	PROPN
ejpam-1797	362	3	is	be	AUX
ejpam-1797	362	4	prime	prime	ADJ
ejpam-1797	362	5	,	,	PUNCT
ejpam-1797	362	6	φ(a	φ(a	ADJ
ejpam-1797	362	7	)	)	PUNCT
ejpam-1797	362	8	=	=	SYM
ejpam-1797	362	9	(	(	PUNCT
ejpam-1797	362	10	0	0	NUM
ejpam-1797	362	11	)	)	PUNCT
ejpam-1797	362	12	or	or	CCONJ
ejpam-1797	362	13	,	,	PUNCT
ejpam-1797	362	14	φ(b	φ(b	PROPN
ejpam-1797	362	15	)	)	PUNCT
ejpam-1797	362	16	=	=	SYM
ejpam-1797	362	17	(	(	PUNCT
ejpam-1797	362	18	0	0	NUM
ejpam-1797	362	19	)	)	PUNCT
ejpam-1797	362	20	or	or	CCONJ
ejpam-1797	362	21	,	,	PUNCT
ejpam-1797	362	22	φ(c	φ(c	NOUN
ejpam-1797	362	23	)	)	PUNCT
ejpam-1797	362	24	=	=	SYM
ejpam-1797	363	1	(	(	PUNCT
ejpam-1797	363	2	0	0	NUM
ejpam-1797	363	3	)	)	PUNCT
ejpam-1797	363	4	which	which	PRON
ejpam-1797	363	5	implies	imply	VERB
ejpam-1797	363	6	that	that	SCONJ
ejpam-1797	363	7	a	a	DET
ejpam-1797	363	8	⊆	⊆	NUM
ejpam-1797	363	9	kerφ	kerφ	NOUN
ejpam-1797	363	10	=	=	SYM
ejpam-1797	363	11	(	(	PUNCT
ejpam-1797	363	12	0	0	X
ejpam-1797	363	13	)	)	PUNCT
ejpam-1797	363	14	⇒	⇒	VERB
ejpam-1797	364	1	a	a	DET
ejpam-1797	364	2	=	=	X
ejpam-1797	364	3	(	(	PUNCT
ejpam-1797	364	4	0	0	NUM
ejpam-1797	364	5	)	)	PUNCT
ejpam-1797	364	6	or	or	CCONJ
ejpam-1797	364	7	,	,	PUNCT
ejpam-1797	364	8	b	b	PROPN
ejpam-1797	364	9	⊆	⊆	NUM
ejpam-1797	364	10	kerφ	kerφ	NOUN
ejpam-1797	364	11	=	=	SYM
ejpam-1797	364	12	(	(	PUNCT
ejpam-1797	364	13	0	0	X
ejpam-1797	364	14	)	)	PUNCT
ejpam-1797	364	15	⇒	⇒	NOUN
ejpam-1797	364	16	b	b	NOUN
ejpam-1797	365	1	=	=	SYM
ejpam-1797	366	1	(	(	PUNCT
ejpam-1797	366	2	0	0	NUM
ejpam-1797	366	3	)	)	PUNCT
ejpam-1797	366	4	,	,	PUNCT
ejpam-1797	366	5	or	or	CCONJ
ejpam-1797	366	6	,	,	PUNCT
ejpam-1797	366	7	c	c	PROPN
ejpam-1797	366	8	⊆	⊆	NUM
ejpam-1797	366	9	kerφ	kerφ	NOUN
ejpam-1797	366	10	=	=	PUNCT
ejpam-1797	366	11	(	(	PUNCT
ejpam-1797	366	12	0)⇒	0)⇒	X
ejpam-1797	366	13	a=	a=	X
ejpam-1797	366	14	(	(	PUNCT
ejpam-1797	366	15	0	0	NUM
ejpam-1797	366	16	)	)	PUNCT
ejpam-1797	366	17	.	.	PUNCT
ejpam-1797	367	1	thus	thus	ADV
ejpam-1797	367	2	,	,	PUNCT
ejpam-1797	367	3	we	we	PRON
ejpam-1797	367	4	have	have	VERB
ejpam-1797	367	5	either	either	CCONJ
ejpam-1797	367	6	a=(0	a=(0	ADJ
ejpam-1797	367	7	)	)	PUNCT
ejpam-1797	367	8	or	or	CCONJ
ejpam-1797	367	9	b=(0	b=(0	NOUN
ejpam-1797	367	10	)	)	PUNCT
ejpam-1797	367	11	or	or	CCONJ
ejpam-1797	367	12	c=(0	c=(0	NUM
ejpam-1797	367	13	)	)	PUNCT
ejpam-1797	367	14	.	.	PUNCT
ejpam-1797	368	1	hence	hence	ADV
ejpam-1797	368	2	,	,	PUNCT
ejpam-1797	368	3	s	s	VERB
ejpam-1797	368	4	is	be	AUX
ejpam-1797	368	5	prime	prime	ADJ
ejpam-1797	368	6	ternary	ternary	ADJ
ejpam-1797	368	7	semiring	semiring	NOUN
ejpam-1797	368	8	.	.	PUNCT
ejpam-1797	369	1	in	in	ADP
ejpam-1797	369	2	the	the	DET
ejpam-1797	369	3	following	follow	VERB
ejpam-1797	369	4	lemma	lemma	PROPN
ejpam-1797	369	5	,	,	PUNCT
ejpam-1797	369	6	we	we	PRON
ejpam-1797	369	7	study	study	VERB
ejpam-1797	369	8	the	the	DET
ejpam-1797	369	9	hereditary	hereditary	ADJ
ejpam-1797	369	10	radical	radical	ADJ
ejpam-1797	369	11	class	class	NOUN
ejpam-1797	369	12	of	of	ADP
ejpam-1797	369	13	ternary	ternary	ADJ
ejpam-1797	369	14	semirings	semiring	NOUN
ejpam-1797	369	15	.	.	PUNCT
ejpam-1797	370	1	lemma	lemma	PROPN
ejpam-1797	370	2	7	7	X
ejpam-1797	370	3	.	.	PUNCT
ejpam-1797	371	1	let	let	VERB
ejpam-1797	371	2	m	m	PRON
ejpam-1797	371	3	be	be	AUX
ejpam-1797	371	4	a	a	DET
ejpam-1797	371	5	hereditary	hereditary	ADJ
ejpam-1797	371	6	radical	radical	ADJ
ejpam-1797	371	7	class	class	NOUN
ejpam-1797	371	8	of	of	ADP
ejpam-1797	371	9	prime	prime	ADJ
ejpam-1797	371	10	ternary	ternary	ADJ
ejpam-1797	371	11	semirings	semiring	NOUN
ejpam-1797	371	12	which	which	PRON
ejpam-1797	371	13	satisfies	satisfy	VERB
ejpam-1797	371	14	the	the	DET
ejpam-1797	371	15	following	follow	VERB
ejpam-1797	371	16	properties	property	NOUN
ejpam-1797	371	17	:	:	PUNCT
ejpam-1797	371	18	“	"	PUNCT
ejpam-1797	371	19	if	if	SCONJ
ejpam-1797	371	20	s	s	VERB
ejpam-1797	371	21	∈m	∈m	NOUN
ejpam-1797	371	22	and	and	CCONJ
ejpam-1797	371	23	s	s	NOUN
ejpam-1797	371	24	is	be	AUX
ejpam-1797	371	25	semi	semi	ADJ
ejpam-1797	371	26	-	-	ADJ
ejpam-1797	371	27	isomorphic	isomorphic	ADJ
ejpam-1797	371	28	to	to	ADP
ejpam-1797	371	29	t	t	PROPN
ejpam-1797	371	30	then	then	ADV
ejpam-1797	371	31	t	t	PROPN
ejpam-1797	371	32	∈m	∈m	NOUN
ejpam-1797	371	33	”	"	PUNCT
ejpam-1797	371	34	.	.	PUNCT
ejpam-1797	372	1	then	then	ADV
ejpam-1797	372	2	the	the	DET
ejpam-1797	372	3	following	follow	VERB
ejpam-1797	372	4	conditions	condition	NOUN
ejpam-1797	372	5	are	be	AUX
ejpam-1797	372	6	equivalent	equivalent	ADJ
ejpam-1797	372	7	:	:	PUNCT
ejpam-1797	372	8	(	(	PUNCT
ejpam-1797	372	9	1	1	X
ejpam-1797	372	10	)	)	PUNCT
ejpam-1797	372	11	if	if	SCONJ
ejpam-1797	372	12	a∈m	a∈m	NOUN
ejpam-1797	372	13	and	and	CCONJ
ejpam-1797	372	14	a	a	PRON
ejpam-1797	372	15	is	be	AUX
ejpam-1797	372	16	an	an	DET
ejpam-1797	372	17	ideal	ideal	NOUN
ejpam-1797	372	18	of	of	ADP
ejpam-1797	372	19	s	s	PROPN
ejpam-1797	372	20	,	,	PUNCT
ejpam-1797	372	21	then	then	ADV
ejpam-1797	372	22	s	s	PROPN
ejpam-1797	372	23	/	/	SYM
ejpam-1797	372	24	anns(a	anns(a	NOUN
ejpam-1797	372	25	)	)	PUNCT
ejpam-1797	372	26	∈m	∈m	NOUN
ejpam-1797	372	27	;	;	PUNCT
ejpam-1797	372	28	(	(	PUNCT
ejpam-1797	372	29	2	2	X
ejpam-1797	372	30	)	)	PUNCT
ejpam-1797	372	31	if	if	SCONJ
ejpam-1797	372	32	a∈m	a∈m	NOUN
ejpam-1797	372	33	with	with	ADP
ejpam-1797	372	34	a	a	DET
ejpam-1797	372	35	an	an	DET
ejpam-1797	372	36	ideal	ideal	NOUN
ejpam-1797	372	37	of	of	ADP
ejpam-1797	372	38	s	s	NOUN
ejpam-1797	372	39	and	and	CCONJ
ejpam-1797	372	40	anns(a	anns(a	NOUN
ejpam-1797	372	41	)	)	PUNCT
ejpam-1797	372	42	=	=	SYM
ejpam-1797	373	1	0	0	NUM
ejpam-1797	373	2	,	,	PUNCT
ejpam-1797	373	3	then	then	ADV
ejpam-1797	373	4	s	s	VERB
ejpam-1797	373	5	∈m	∈m	NOUN
ejpam-1797	373	6	.	.	PUNCT
ejpam-1797	374	1	(	(	PUNCT
ejpam-1797	374	2	3	3	X
ejpam-1797	374	3	)	)	PUNCT
ejpam-1797	374	4	if	if	SCONJ
ejpam-1797	374	5	a∈m	a∈m	NOUN
ejpam-1797	374	6	and	and	CCONJ
ejpam-1797	374	7	a	a	PRON
ejpam-1797	374	8	is	be	AUX
ejpam-1797	374	9	an	an	DET
ejpam-1797	374	10	essential	essential	ADJ
ejpam-1797	374	11	ideal	ideal	NOUN
ejpam-1797	374	12	of	of	ADP
ejpam-1797	374	13	s	s	PROPN
ejpam-1797	374	14	,	,	PUNCT
ejpam-1797	374	15	then	then	ADV
ejpam-1797	374	16	s	s	VERB
ejpam-1797	374	17	∈m	∈m	NOUN
ejpam-1797	374	18	.	.	PUNCT
ejpam-1797	375	1	proof	proof	NOUN
ejpam-1797	375	2	.	.	PUNCT
ejpam-1797	376	1	the	the	DET
ejpam-1797	376	2	proof	proof	NOUN
ejpam-1797	376	3	follows	follow	VERB
ejpam-1797	376	4	from	from	ADP
ejpam-1797	376	5	lemma	lemma	PROPN
ejpam-1797	376	6	6	6	NUM
ejpam-1797	376	7	.	.	PUNCT
ejpam-1797	376	8	proposition	proposition	NOUN
ejpam-1797	376	9	9	9	NUM
ejpam-1797	376	10	.	.	PUNCT
ejpam-1797	377	1	the	the	DET
ejpam-1797	377	2	class	class	NOUN
ejpam-1797	377	3	℘′	℘′	ADV
ejpam-1797	377	4	of	of	ADP
ejpam-1797	377	5	prime	prime	ADJ
ejpam-1797	377	6	nonsingular	nonsingular	ADJ
ejpam-1797	377	7	ternary	ternary	ADJ
ejpam-1797	377	8	semirings	semiring	NOUN
ejpam-1797	377	9	is	be	AUX
ejpam-1797	377	10	a	a	DET
ejpam-1797	377	11	special	special	ADJ
ejpam-1797	377	12	radical	radical	ADJ
ejpam-1797	377	13	class	class	NOUN
ejpam-1797	377	14	.	.	PUNCT
ejpam-1797	378	1	proof	proof	NOUN
ejpam-1797	378	2	.	.	PUNCT
ejpam-1797	379	1	let	let	VERB
ejpam-1797	379	2	s	s	PRON
ejpam-1797	379	3	∈	∈	NOUN
ejpam-1797	379	4	℘′	℘′	NOUN
ejpam-1797	379	5	and	and	CCONJ
ejpam-1797	379	6	i	i	PRON
ejpam-1797	379	7	be	be	VERB
ejpam-1797	379	8	a	a	DET
ejpam-1797	379	9	nonzero	nonzero	NOUN
ejpam-1797	379	10	ideal	ideal	NOUN
ejpam-1797	379	11	of	of	ADP
ejpam-1797	379	12	s.	s.	PROPN
ejpam-1797	379	13	then	then	ADV
ejpam-1797	379	14	by	by	ADP
ejpam-1797	379	15	proposition	proposition	NOUN
ejpam-1797	379	16	5	5	NUM
ejpam-1797	379	17	,	,	PUNCT
ejpam-1797	379	18	i	i	PRON
ejpam-1797	379	19	is	be	AUX
ejpam-1797	379	20	a	a	DET
ejpam-1797	379	21	prime	prime	ADJ
ejpam-1797	379	22	nonsingular	nonsingular	ADJ
ejpam-1797	379	23	ternary	ternary	ADJ
ejpam-1797	379	24	semiring	semiring	NOUN
ejpam-1797	379	25	.	.	PUNCT
ejpam-1797	380	1	since	since	SCONJ
ejpam-1797	380	2	every	every	DET
ejpam-1797	380	3	prime	prime	ADJ
ejpam-1797	380	4	ternary	ternary	ADJ
ejpam-1797	380	5	semiring	semiring	NOUN
ejpam-1797	380	6	is	be	AUX
ejpam-1797	380	7	semiprime	semiprime	NOUN
ejpam-1797	380	8	,	,	PUNCT
ejpam-1797	380	9	by	by	ADP
ejpam-1797	380	10	proposition	proposition	NOUN
ejpam-1797	380	11	3	3	NUM
ejpam-1797	380	12	,	,	PUNCT
ejpam-1797	380	13	z(i	z(i	NUM
ejpam-1797	380	14	)	)	PUNCT
ejpam-1797	381	1	=	=	SYM
ejpam-1797	381	2	i	i	PRON
ejpam-1797	381	3	∩	∩	ADJ
ejpam-1797	381	4	z(s	z(s	PROPN
ejpam-1797	381	5	)	)	PUNCT
ejpam-1797	381	6	=	=	PUNCT
ejpam-1797	381	7	(	(	PUNCT
ejpam-1797	381	8	0	0	NUM
ejpam-1797	381	9	)	)	PUNCT
ejpam-1797	381	10	since	since	SCONJ
ejpam-1797	381	11	z(s	z(s	PROPN
ejpam-1797	381	12	)	)	PUNCT
ejpam-1797	381	13	=	=	SYM
ejpam-1797	381	14	0	0	PUNCT
ejpam-1797	381	15	as	as	SCONJ
ejpam-1797	381	16	s	s	PRON
ejpam-1797	381	17	is	be	AUX
ejpam-1797	381	18	nonsingular	nonsingular	ADJ
ejpam-1797	381	19	.	.	PUNCT
ejpam-1797	382	1	hence	hence	ADV
ejpam-1797	382	2	,	,	PUNCT
ejpam-1797	382	3	we	we	PRON
ejpam-1797	382	4	have	have	VERB
ejpam-1797	382	5	i	i	PROPN
ejpam-1797	382	6	∈	∈	PROPN
ejpam-1797	382	7	℘′.	℘′.	PROPN
ejpam-1797	382	8	thus	thus	ADV
ejpam-1797	382	9	,	,	PUNCT
ejpam-1797	382	10	the	the	DET
ejpam-1797	382	11	class	class	NOUN
ejpam-1797	382	12	℘′	℘′	ADV
ejpam-1797	382	13	is	be	AUX
ejpam-1797	382	14	a	a	DET
ejpam-1797	382	15	hereditary	hereditary	ADJ
ejpam-1797	382	16	class	class	NOUN
ejpam-1797	382	17	of	of	ADP
ejpam-1797	382	18	prime	prime	ADJ
ejpam-1797	382	19	nonsingular	nonsingular	ADJ
ejpam-1797	382	20	ternary	ternary	ADJ
ejpam-1797	382	21	semirings	semiring	NOUN
ejpam-1797	382	22	.	.	PUNCT
ejpam-1797	383	1	by	by	ADP
ejpam-1797	383	2	theorem	theorem	NOUN
ejpam-1797	383	3	1	1	NUM
ejpam-1797	383	4	and	and	CCONJ
ejpam-1797	383	5	by	by	ADP
ejpam-1797	383	6	lemma	lemma	PROPN
ejpam-1797	383	7	6	6	NUM
ejpam-1797	383	8	,	,	PUNCT
ejpam-1797	383	9	the	the	DET
ejpam-1797	383	10	class	class	NOUN
ejpam-1797	383	11	℘′	℘′	ADV
ejpam-1797	383	12	satisfies	satisfy	VERB
ejpam-1797	383	13	the	the	DET
ejpam-1797	383	14	property	property	NOUN
ejpam-1797	383	15	(	(	PUNCT
ejpam-1797	383	16	1	1	NUM
ejpam-1797	383	17	)	)	PUNCT
ejpam-1797	383	18	of	of	ADP
ejpam-1797	383	19	the	the	DET
ejpam-1797	383	20	definition	definition	NOUN
ejpam-1797	383	21	of	of	ADP
ejpam-1797	383	22	special	special	ADJ
ejpam-1797	383	23	radical	radical	ADJ
ejpam-1797	383	24	class	class	NOUN
ejpam-1797	383	25	.	.	PUNCT
ejpam-1797	384	1	we	we	PRON
ejpam-1797	384	2	now	now	ADV
ejpam-1797	384	3	proceed	proceed	VERB
ejpam-1797	384	4	to	to	PART
ejpam-1797	384	5	prove	prove	VERB
ejpam-1797	384	6	that	that	SCONJ
ejpam-1797	384	7	the	the	DET
ejpam-1797	384	8	class	class	NOUN
ejpam-1797	384	9	℘′	℘′	ADV
ejpam-1797	384	10	is	be	AUX
ejpam-1797	384	11	a	a	DET
ejpam-1797	384	12	special	special	ADJ
ejpam-1797	384	13	radical	radical	ADJ
ejpam-1797	384	14	class	class	NOUN
ejpam-1797	384	15	.	.	PUNCT
ejpam-1797	385	1	in	in	ADP
ejpam-1797	385	2	view	view	NOUN
ejpam-1797	385	3	of	of	ADP
ejpam-1797	385	4	theorem	theorem	NOUN
ejpam-1797	385	5	1	1	NUM
ejpam-1797	385	6	and	and	CCONJ
ejpam-1797	385	7	lemma	lemma	PROPN
ejpam-1797	385	8	6	6	NUM
ejpam-1797	385	9	,	,	PUNCT
ejpam-1797	385	10	it	it	PRON
ejpam-1797	385	11	suffices	suffice	VERB
ejpam-1797	385	12	to	to	PART
ejpam-1797	385	13	prove	prove	VERB
ejpam-1797	385	14	that	that	SCONJ
ejpam-1797	385	15	if	if	SCONJ
ejpam-1797	385	16	i	i	PRON
ejpam-1797	385	17	∈	∈	VERB
ejpam-1797	385	18	℘′	℘′	ADV
ejpam-1797	386	1	and	and	CCONJ
ejpam-1797	386	2	i	i	PRON
ejpam-1797	386	3	is	be	AUX
ejpam-1797	386	4	an	an	DET
ejpam-1797	386	5	essential	essential	ADJ
ejpam-1797	386	6	ideal	ideal	NOUN
ejpam-1797	386	7	of	of	ADP
ejpam-1797	386	8	a	a	DET
ejpam-1797	386	9	ternary	ternary	ADJ
ejpam-1797	386	10	semiring	semiring	NOUN
ejpam-1797	386	11	s	s	PART
ejpam-1797	386	12	,	,	PUNCT
ejpam-1797	386	13	then	then	ADV
ejpam-1797	386	14	s	s	VERB
ejpam-1797	386	15	∈	∈	NOUN
ejpam-1797	386	16	℘′.in	℘′.in	NOUN
ejpam-1797	386	17	order	order	NOUN
ejpam-1797	386	18	to	to	PART
ejpam-1797	386	19	prove	prove	VERB
ejpam-1797	386	20	that	that	SCONJ
ejpam-1797	386	21	s	s	VERB
ejpam-1797	386	22	is	be	AUX
ejpam-1797	386	23	a	a	DET
ejpam-1797	386	24	prime	prime	ADJ
ejpam-1797	386	25	ternary	ternary	ADJ
ejpam-1797	386	26	semiring	semiring	NOUN
ejpam-1797	386	27	,	,	PUNCT
ejpam-1797	386	28	we	we	PRON
ejpam-1797	386	29	let	let	VERB
ejpam-1797	386	30	abc	abc	PROPN
ejpam-1797	386	31	=	=	SYM
ejpam-1797	386	32	0	0	PROPN
ejpam-1797	386	33	,	,	PUNCT
ejpam-1797	386	34	where	where	SCONJ
ejpam-1797	386	35	a	a	DET
ejpam-1797	386	36	,	,	PUNCT
ejpam-1797	386	37	b	b	NOUN
ejpam-1797	386	38	and	and	CCONJ
ejpam-1797	386	39	c	c	PROPN
ejpam-1797	386	40	are	be	AUX
ejpam-1797	386	41	three	three	NUM
ejpam-1797	386	42	ideals	ideal	NOUN
ejpam-1797	386	43	of	of	ADP
ejpam-1797	386	44	s.	s.	PROPN
ejpam-1797	386	45	suppose	suppose	VERB
ejpam-1797	386	46	that	that	SCONJ
ejpam-1797	386	47	a	a	DET
ejpam-1797	386	48	6=	6=	NUM
ejpam-1797	386	49	0	0	NUM
ejpam-1797	386	50	,	,	PUNCT
ejpam-1797	386	51	b	b	PROPN
ejpam-1797	386	52	6=	6=	ADP
ejpam-1797	386	53	0	0	NUM
ejpam-1797	386	54	and	and	CCONJ
ejpam-1797	386	55	c	c	X
ejpam-1797	386	56	6=	6=	PROPN
ejpam-1797	386	57	0	0	X
ejpam-1797	386	58	.	.	PUNCT
ejpam-1797	387	1	let	let	VERB
ejpam-1797	387	2	a′	a′	NOUN
ejpam-1797	387	3	=	=	PUNCT
ejpam-1797	387	4	a∩	a∩	PROPN
ejpam-1797	388	1	i	i	PRON
ejpam-1797	388	2	,	,	PUNCT
ejpam-1797	388	3	b′	b′	NUM
ejpam-1797	388	4	=	=	SYM
ejpam-1797	388	5	b	b	NOUN
ejpam-1797	388	6	∩	∩	NOUN
ejpam-1797	388	7	i	i	PRON
ejpam-1797	388	8	t.	t.	PROPN
ejpam-1797	388	9	dutta	dutta	PROPN
ejpam-1797	388	10	,	,	PUNCT
ejpam-1797	388	11	k.	k.	PROPN
ejpam-1797	388	12	shum	shum	PROPN
ejpam-1797	388	13	,	,	PUNCT
ejpam-1797	388	14	s.	s.	PROPN
ejpam-1797	388	15	mandal	mandal	PROPN
ejpam-1797	388	16	/	/	SYM
ejpam-1797	388	17	eur	eur	PROPN
ejpam-1797	388	18	.	.	PUNCT
ejpam-1797	389	1	j.	j.	PROPN
ejpam-1797	389	2	pure	pure	PROPN
ejpam-1797	389	3	appl	appl	PROPN
ejpam-1797	389	4	.	.	PROPN
ejpam-1797	389	5	math	math	PROPN
ejpam-1797	389	6	,	,	PUNCT
ejpam-1797	389	7	5	5	NUM
ejpam-1797	389	8	(	(	PUNCT
ejpam-1797	389	9	2012	2012	NUM
ejpam-1797	389	10	)	)	PUNCT
ejpam-1797	389	11	,	,	PUNCT
ejpam-1797	389	12	401	401	NUM
ejpam-1797	389	13	-	-	SYM
ejpam-1797	389	14	413	413	NUM
ejpam-1797	389	15	411	411	NUM
ejpam-1797	389	16	and	and	CCONJ
ejpam-1797	389	17	c	c	NOUN
ejpam-1797	389	18	′	′	NUM
ejpam-1797	390	1	=	=	PUNCT
ejpam-1797	390	2	c	c	NOUN
ejpam-1797	390	3	∩	∩	X
ejpam-1797	390	4	i	i	PRON
ejpam-1797	390	5	.	.	PUNCT
ejpam-1797	391	1	then	then	ADV
ejpam-1797	391	2	,	,	PUNCT
ejpam-1797	391	3	a′	a′	NOUN
ejpam-1797	391	4	,	,	PUNCT
ejpam-1797	391	5	b′	b′	NUM
ejpam-1797	391	6	and	and	CCONJ
ejpam-1797	391	7	c	c	NOUN
ejpam-1797	391	8	′	′	NOUN
ejpam-1797	391	9	are	be	AUX
ejpam-1797	391	10	nonzero	nonzero	ADJ
ejpam-1797	391	11	ideals	ideal	NOUN
ejpam-1797	391	12	of	of	ADP
ejpam-1797	391	13	i	i	PRON
ejpam-1797	391	14	,	,	PUNCT
ejpam-1797	391	15	as	as	SCONJ
ejpam-1797	391	16	i	i	PRON
ejpam-1797	391	17	is	be	AUX
ejpam-1797	391	18	an	an	DET
ejpam-1797	391	19	essential	essential	ADJ
ejpam-1797	391	20	ideal	ideal	NOUN
ejpam-1797	391	21	of	of	ADP
ejpam-1797	391	22	s.	s.	PROPN
ejpam-1797	391	23	now	now	ADV
ejpam-1797	391	24	a′b′c	a′b′c	VERB
ejpam-1797	391	25	′	′	NUM
ejpam-1797	391	26	⊆	⊆	NUM
ejpam-1797	391	27	abc	abc	PROPN
ejpam-1797	391	28	=	=	SYM
ejpam-1797	391	29	0	0	PROPN
ejpam-1797	391	30	.	.	PUNCT
ejpam-1797	392	1	since	since	SCONJ
ejpam-1797	392	2	i	i	PRON
ejpam-1797	392	3	is	be	AUX
ejpam-1797	392	4	a	a	DET
ejpam-1797	392	5	prime	prime	ADJ
ejpam-1797	392	6	ternary	ternary	ADJ
ejpam-1797	392	7	semiring	semiring	NOUN
ejpam-1797	392	8	,	,	PUNCT
ejpam-1797	392	9	a′b′c	a′b′c	PROPN
ejpam-1797	392	10	′	′	NUM
ejpam-1797	392	11	=	=	SYM
ejpam-1797	392	12	0	0	NUM
ejpam-1797	392	13	implies	imply	VERB
ejpam-1797	392	14	either	either	CCONJ
ejpam-1797	392	15	a′	a′	PROPN
ejpam-1797	392	16	=	=	SYM
ejpam-1797	392	17	0	0	NUM
ejpam-1797	392	18	or	or	CCONJ
ejpam-1797	392	19	b′	b′	NUM
ejpam-1797	392	20	=	=	SYM
ejpam-1797	392	21	0	0	NUM
ejpam-1797	392	22	or	or	CCONJ
ejpam-1797	392	23	c	c	NOUN
ejpam-1797	392	24	′	′	NUM
ejpam-1797	392	25	=	=	SYM
ejpam-1797	392	26	0	0	NUM
ejpam-1797	392	27	,	,	PUNCT
ejpam-1797	392	28	a	a	DET
ejpam-1797	392	29	contradiction	contradiction	NOUN
ejpam-1797	392	30	.	.	PUNCT
ejpam-1797	393	1	this	this	PRON
ejpam-1797	393	2	shows	show	VERB
ejpam-1797	393	3	that	that	SCONJ
ejpam-1797	393	4	s	s	VERB
ejpam-1797	393	5	is	be	AUX
ejpam-1797	393	6	a	a	DET
ejpam-1797	393	7	prime	prime	ADJ
ejpam-1797	393	8	ternary	ternary	ADJ
ejpam-1797	393	9	semiring	semiring	NOUN
ejpam-1797	393	10	.	.	PUNCT
ejpam-1797	394	1	again	again	ADV
ejpam-1797	394	2	,	,	PUNCT
ejpam-1797	394	3	suppose	suppose	VERB
ejpam-1797	394	4	that	that	SCONJ
ejpam-1797	394	5	s	s	VERB
ejpam-1797	394	6	is	be	AUX
ejpam-1797	394	7	not	not	PART
ejpam-1797	394	8	a	a	DET
ejpam-1797	394	9	nonsingular	nonsingular	ADJ
ejpam-1797	394	10	ternary	ternary	ADJ
ejpam-1797	394	11	semiring	semiring	NOUN
ejpam-1797	394	12	.	.	PUNCT
ejpam-1797	395	1	then	then	ADV
ejpam-1797	395	2	,	,	PUNCT
ejpam-1797	395	3	by	by	ADP
ejpam-1797	395	4	example	example	NOUN
ejpam-1797	395	5	1	1	NUM
ejpam-1797	395	6	and	and	CCONJ
ejpam-1797	395	7	proposition	proposition	NOUN
ejpam-1797	395	8	2	2	NUM
ejpam-1797	395	9	,	,	PUNCT
ejpam-1797	395	10	z(s	z(s	PROPN
ejpam-1797	395	11	)	)	PUNCT
ejpam-1797	395	12	is	be	AUX
ejpam-1797	395	13	a	a	DET
ejpam-1797	395	14	nonzero	nonzero	NOUN
ejpam-1797	395	15	ideal	ideal	NOUN
ejpam-1797	395	16	of	of	ADP
ejpam-1797	395	17	s	s	PRON
ejpam-1797	395	18	and	and	CCONJ
ejpam-1797	395	19	i	i	PROPN
ejpam-1797	395	20	∩	∩	ADJ
ejpam-1797	395	21	z(s	z(s	PROPN
ejpam-1797	395	22	)	)	PUNCT
ejpam-1797	395	23	6=	6=	ADP
ejpam-1797	395	24	0	0	NUM
ejpam-1797	395	25	,	,	PUNCT
ejpam-1797	395	26	since	since	SCONJ
ejpam-1797	395	27	i	i	PRON
ejpam-1797	395	28	is	be	AUX
ejpam-1797	395	29	an	an	DET
ejpam-1797	395	30	essential	essential	ADJ
ejpam-1797	395	31	ideal	ideal	NOUN
ejpam-1797	395	32	of	of	ADP
ejpam-1797	395	33	s.	s.	PROPN
ejpam-1797	395	34	by	by	ADP
ejpam-1797	395	35	prop	prop	PROPN
ejpam-1797	395	36	3	3	NUM
ejpam-1797	395	37	,	,	PUNCT
ejpam-1797	395	38	z(i	z(i	NUM
ejpam-1797	395	39	)	)	PUNCT
ejpam-1797	395	40	6=	6=	ADP
ejpam-1797	395	41	0	0	NUM
ejpam-1797	395	42	,	,	PUNCT
ejpam-1797	395	43	which	which	PRON
ejpam-1797	395	44	is	be	AUX
ejpam-1797	395	45	a	a	DET
ejpam-1797	395	46	contradiction	contradiction	NOUN
ejpam-1797	395	47	.	.	PUNCT
ejpam-1797	396	1	therefore	therefore	ADV
ejpam-1797	396	2	,	,	PUNCT
ejpam-1797	396	3	s	s	VERB
ejpam-1797	396	4	is	be	AUX
ejpam-1797	396	5	nonsingular	nonsingular	ADJ
ejpam-1797	396	6	.	.	PUNCT
ejpam-1797	397	1	thus	thus	ADV
ejpam-1797	397	2	,	,	PUNCT
ejpam-1797	397	3	s	s	VERB
ejpam-1797	397	4	∈	∈	PROPN
ejpam-1797	397	5	℘′.	℘′.	ADV
ejpam-1797	397	6	this	this	PRON
ejpam-1797	397	7	proves	prove	VERB
ejpam-1797	397	8	that	that	SCONJ
ejpam-1797	397	9	℘′	℘′	ADV
ejpam-1797	397	10	is	be	AUX
ejpam-1797	397	11	a	a	DET
ejpam-1797	397	12	special	special	ADJ
ejpam-1797	397	13	radical	radical	ADJ
ejpam-1797	397	14	class	class	NOUN
ejpam-1797	397	15	.	.	PUNCT
ejpam-1797	398	1	finally	finally	ADV
ejpam-1797	398	2	,	,	PUNCT
ejpam-1797	398	3	we	we	PRON
ejpam-1797	398	4	state	state	VERB
ejpam-1797	398	5	a	a	DET
ejpam-1797	398	6	theorem	theorem	NOUN
ejpam-1797	398	7	of	of	ADP
ejpam-1797	398	8	the	the	DET
ejpam-1797	398	9	special	special	ADJ
ejpam-1797	398	10	radical	radical	ADJ
ejpam-1797	398	11	classes	class	NOUN
ejpam-1797	398	12	of	of	ADP
ejpam-1797	398	13	a	a	DET
ejpam-1797	398	14	ternary	ternary	ADJ
ejpam-1797	398	15	ring	ring	NOUN
ejpam-1797	398	16	s.	s.	PROPN
ejpam-1797	398	17	theorem	theorem	VERB
ejpam-1797	398	18	9	9	NUM
ejpam-1797	398	19	.	.	PUNCT
ejpam-1797	399	1	let	let	VERB
ejpam-1797	399	2	s	s	PRON
ejpam-1797	399	3	be	be	AUX
ejpam-1797	399	4	a	a	DET
ejpam-1797	399	5	ternary	ternary	ADJ
ejpam-1797	399	6	ring	ring	NOUN
ejpam-1797	399	7	.	.	PUNCT
ejpam-1797	400	1	if	if	SCONJ
ejpam-1797	400	2	m	m	NOUN
ejpam-1797	400	3	is	be	AUX
ejpam-1797	400	4	a	a	DET
ejpam-1797	400	5	special	special	ADJ
ejpam-1797	400	6	radical	radical	ADJ
ejpam-1797	400	7	class	class	NOUN
ejpam-1797	400	8	of	of	ADP
ejpam-1797	400	9	s	s	PROPN
ejpam-1797	400	10	,	,	PUNCT
ejpam-1797	400	11	then	then	ADV
ejpam-1797	400	12	um	um	INTJ
ejpam-1797	400	13	is	be	AUX
ejpam-1797	400	14	a	a	DET
ejpam-1797	400	15	supernilpotent	supernilpotent	ADJ
ejpam-1797	400	16	radical	radical	ADJ
ejpam-1797	400	17	class	class	NOUN
ejpam-1797	400	18	of	of	ADP
ejpam-1797	400	19	s.	s.	PROPN
ejpam-1797	400	20	proof	proof	PROPN
ejpam-1797	400	21	.	.	PUNCT
ejpam-1797	401	1	in	in	ADP
ejpam-1797	401	2	order	order	NOUN
ejpam-1797	401	3	to	to	PART
ejpam-1797	401	4	show	show	VERB
ejpam-1797	401	5	that	that	SCONJ
ejpam-1797	401	6	um	um	INTJ
ejpam-1797	401	7	is	be	AUX
ejpam-1797	401	8	hereditary	hereditary	ADJ
ejpam-1797	401	9	,	,	PUNCT
ejpam-1797	401	10	let	let	VERB
ejpam-1797	401	11	s	s	PRON
ejpam-1797	401	12	∈	∈	VERB
ejpam-1797	401	13	um	um	INTJ
ejpam-1797	401	14	and	and	CCONJ
ejpam-1797	401	15	j	j	PROPN
ejpam-1797	401	16	be	be	AUX
ejpam-1797	401	17	a	a	DET
ejpam-1797	401	18	nonzero	nonzero	NOUN
ejpam-1797	401	19	ideal	ideal	NOUN
ejpam-1797	401	20	of	of	ADP
ejpam-1797	401	21	s.	s.	PROPN
ejpam-1797	401	22	if	if	SCONJ
ejpam-1797	401	23	j	j	PROPN
ejpam-1797	401	24	6∈	6∈	PROPN
ejpam-1797	401	25	um	um	INTJ
ejpam-1797	401	26	then	then	ADV
ejpam-1797	401	27	there	there	PRON
ejpam-1797	401	28	is	be	VERB
ejpam-1797	401	29	a	a	DET
ejpam-1797	401	30	nonzero	nonzero	ADJ
ejpam-1797	401	31	homomorphic	homomorphic	ADJ
ejpam-1797	401	32	image	image	NOUN
ejpam-1797	401	33	φ(j	φ(j	PROPN
ejpam-1797	401	34	)	)	PUNCT
ejpam-1797	401	35	of	of	ADP
ejpam-1797	401	36	j	j	PROPN
ejpam-1797	401	37	inm	inm	PROPN
ejpam-1797	401	38	.	.	PUNCT
ejpam-1797	402	1	let	let	VERB
ejpam-1797	402	2	k	k	PROPN
ejpam-1797	402	3	=	=	SYM
ejpam-1797	402	4	kerφ	kerφ	PROPN
ejpam-1797	402	5	.	.	PUNCT
ejpam-1797	403	1	then	then	ADV
ejpam-1797	403	2	k	k	PROPN
ejpam-1797	403	3	is	be	AUX
ejpam-1797	403	4	a	a	DET
ejpam-1797	403	5	k	k	NOUN
ejpam-1797	403	6	-	-	NOUN
ejpam-1797	403	7	ideal	ideal	NOUN
ejpam-1797	403	8	of	of	ADP
ejpam-1797	403	9	j	j	PROPN
ejpam-1797	403	10	and	and	CCONJ
ejpam-1797	403	11	we	we	PRON
ejpam-1797	403	12	have	have	VERB
ejpam-1797	403	13	j	j	PROPN
ejpam-1797	403	14	/	/	SYM
ejpam-1797	403	15	k	k	PROPN
ejpam-1797	403	16	≃	≃	ADJ
ejpam-1797	403	17	φ(j	φ(j	PROPN
ejpam-1797	403	18	)	)	PUNCT
ejpam-1797	403	19	.	.	PUNCT
ejpam-1797	404	1	but	but	CCONJ
ejpam-1797	404	2	φ(j	φ(j	PROPN
ejpam-1797	404	3	)	)	PUNCT
ejpam-1797	404	4	is	be	AUX
ejpam-1797	404	5	prime	prime	ADJ
ejpam-1797	404	6	because	because	SCONJ
ejpam-1797	404	7	it	it	PRON
ejpam-1797	404	8	is	be	AUX
ejpam-1797	404	9	inm	inm	PROPN
ejpam-1797	404	10	,	,	PUNCT
ejpam-1797	404	11	and	and	CCONJ
ejpam-1797	404	12	by	by	ADP
ejpam-1797	404	13	lemma	lemma	PROPN
ejpam-1797	404	14	6	6	NUM
ejpam-1797	404	15	,	,	PUNCT
ejpam-1797	404	16	j	j	PROPN
ejpam-1797	404	17	/	/	SYM
ejpam-1797	404	18	k	k	PROPN
ejpam-1797	404	19	is	be	AUX
ejpam-1797	404	20	prime	prime	ADJ
ejpam-1797	404	21	.	.	PUNCT
ejpam-1797	405	1	now	now	ADV
ejpam-1797	405	2	φ(j	φ(j	PROPN
ejpam-1797	405	3	)	)	PUNCT
ejpam-1797	405	4	is	be	AUX
ejpam-1797	405	5	nonzero	nonzero	ADJ
ejpam-1797	405	6	,	,	PUNCT
ejpam-1797	405	7	hence	hence	ADV
ejpam-1797	405	8	j	j	PROPN
ejpam-1797	405	9	/	/	SYM
ejpam-1797	405	10	k	k	PROPN
ejpam-1797	405	11	6=	6=	PROPN
ejpam-1797	405	12	(	(	PUNCT
ejpam-1797	405	13	0	0	NUM
ejpam-1797	405	14	)	)	PUNCT
ejpam-1797	405	15	.	.	PUNCT
ejpam-1797	406	1	since	since	SCONJ
ejpam-1797	406	2	k	k	PROPN
ejpam-1797	406	3	is	be	AUX
ejpam-1797	406	4	a	a	DET
ejpam-1797	406	5	k	k	NOUN
ejpam-1797	406	6	-	-	NOUN
ejpam-1797	406	7	ideal	ideal	ADJ
ejpam-1797	406	8	,	,	PUNCT
ejpam-1797	406	9	k	k	PROPN
ejpam-1797	406	10	is	be	AUX
ejpam-1797	406	11	a	a	DET
ejpam-1797	406	12	prime	prime	ADJ
ejpam-1797	406	13	ideal	ideal	NOUN
ejpam-1797	406	14	of	of	ADP
ejpam-1797	406	15	j	j	PROPN
ejpam-1797	406	16	and	and	CCONJ
ejpam-1797	406	17	,	,	PUNCT
ejpam-1797	406	18	hence	hence	ADV
ejpam-1797	406	19	,	,	PUNCT
ejpam-1797	406	20	k	k	PROPN
ejpam-1797	406	21	is	be	AUX
ejpam-1797	406	22	an	an	DET
ejpam-1797	406	23	ideal	ideal	NOUN
ejpam-1797	406	24	of	of	ADP
ejpam-1797	406	25	s	s	PRON
ejpam-1797	406	26	by	by	ADP
ejpam-1797	406	27	lemma	lemma	PROPN
ejpam-1797	406	28	1	1	NUM
ejpam-1797	406	29	.	.	PUNCT
ejpam-1797	407	1	now	now	ADV
ejpam-1797	407	2	j	j	PROPN
ejpam-1797	407	3	/	/	SYM
ejpam-1797	407	4	k	k	PROPN
ejpam-1797	407	5	is	be	AUX
ejpam-1797	407	6	a	a	DET
ejpam-1797	407	7	nonzero	nonzero	ADJ
ejpam-1797	407	8	ideal	ideal	NOUN
ejpam-1797	407	9	of	of	ADP
ejpam-1797	407	10	s	s	PROPN
ejpam-1797	407	11	/	/	SYM
ejpam-1797	407	12	k	k	NOUN
ejpam-1797	407	13	,	,	PUNCT
ejpam-1797	407	14	and	and	CCONJ
ejpam-1797	407	15	hence	hence	ADV
ejpam-1797	407	16	by	by	ADP
ejpam-1797	407	17	the	the	DET
ejpam-1797	407	18	property	property	NOUN
ejpam-1797	407	19	(	(	PUNCT
ejpam-1797	407	20	z	z	NOUN
ejpam-1797	407	21	)	)	PUNCT
ejpam-1797	407	22	,	,	PUNCT
ejpam-1797	407	23	we	we	PRON
ejpam-1797	407	24	have	have	VERB
ejpam-1797	407	25	(	(	PUNCT
ejpam-1797	407	26	s	s	NOUN
ejpam-1797	407	27	/	/	SYM
ejpam-1797	407	28	k)/	k)/	PROPN
ejpam-1797	407	29	anns(j	anns(j	PROPN
ejpam-1797	407	30	/	/	SYM
ejpam-1797	407	31	k	k	NOUN
ejpam-1797	407	32	)	)	PUNCT
ejpam-1797	407	33	∈	∈	PROPN
ejpam-1797	408	1	m	m	VERB
ejpam-1797	408	2	,	,	PUNCT
ejpam-1797	408	3	since	since	SCONJ
ejpam-1797	408	4	by	by	ADP
ejpam-1797	408	5	the	the	DET
ejpam-1797	408	6	property	property	NOUN
ejpam-1797	408	7	(	(	PUNCT
ejpam-1797	408	8	x	x	NOUN
ejpam-1797	408	9	)	)	PUNCT
ejpam-1797	408	10	,	,	PUNCT
ejpam-1797	408	11	j	j	PROPN
ejpam-1797	408	12	/	/	SYM
ejpam-1797	408	13	k	k	PROPN
ejpam-1797	408	14	∈	∈	PROPN
ejpam-1797	408	15	m	m	VERB
ejpam-1797	408	16	.	.	PUNCT
ejpam-1797	409	1	if	if	SCONJ
ejpam-1797	409	2	s	s	PROPN
ejpam-1797	409	3	/	/	SYM
ejpam-1797	409	4	k	k	PROPN
ejpam-1797	409	5	⊆	⊆	NUM
ejpam-1797	409	6	anns(j	anns(j	PROPN
ejpam-1797	409	7	/	/	SYM
ejpam-1797	409	8	k	k	NOUN
ejpam-1797	409	9	)	)	PUNCT
ejpam-1797	409	10	,	,	PUNCT
ejpam-1797	409	11	then	then	ADV
ejpam-1797	409	12	we	we	PRON
ejpam-1797	409	13	have	have	VERB
ejpam-1797	409	14	(	(	PUNCT
ejpam-1797	409	15	j	j	NOUN
ejpam-1797	409	16	/	/	SYM
ejpam-1797	409	17	k)3	k)3	PROPN
ejpam-1797	409	18	=	=	SYM
ejpam-1797	409	19	(	(	PUNCT
ejpam-1797	409	20	0	0	NUM
ejpam-1797	409	21	)	)	PUNCT
ejpam-1797	409	22	which	which	PRON
ejpam-1797	409	23	can	can	AUX
ejpam-1797	409	24	not	not	PART
ejpam-1797	409	25	happen	happen	VERB
ejpam-1797	409	26	because	because	SCONJ
ejpam-1797	409	27	j	j	PROPN
ejpam-1797	409	28	/	/	SYM
ejpam-1797	409	29	k	k	PROPN
ejpam-1797	409	30	is	be	AUX
ejpam-1797	409	31	a	a	DET
ejpam-1797	409	32	prime	prime	ADJ
ejpam-1797	409	33	ternary	ternary	ADJ
ejpam-1797	409	34	semiring	semiring	NOUN
ejpam-1797	409	35	.	.	PUNCT
ejpam-1797	410	1	hence	hence	ADV
ejpam-1797	410	2	,	,	PUNCT
ejpam-1797	410	3	anns(j	anns(j	PROPN
ejpam-1797	410	4	/	/	SYM
ejpam-1797	410	5	k	k	NOUN
ejpam-1797	410	6	)	)	PUNCT
ejpam-1797	410	7	6=	6=	ADP
ejpam-1797	410	8	s	s	X
ejpam-1797	410	9	/	/	SYM
ejpam-1797	410	10	k	k	PROPN
ejpam-1797	410	11	.	.	PUNCT
ejpam-1797	411	1	thus	thus	ADV
ejpam-1797	411	2	,	,	PUNCT
ejpam-1797	411	3	(	(	PUNCT
ejpam-1797	411	4	s	s	X
ejpam-1797	411	5	/	/	SYM
ejpam-1797	411	6	k)/anns(j	k)/anns(j	NOUN
ejpam-1797	411	7	/	/	SYM
ejpam-1797	411	8	k	k	NOUN
ejpam-1797	411	9	)	)	PUNCT
ejpam-1797	411	10	is	be	AUX
ejpam-1797	411	11	a	a	DET
ejpam-1797	411	12	homomorphic	homomorphic	ADJ
ejpam-1797	411	13	image	image	NOUN
ejpam-1797	411	14	of	of	ADP
ejpam-1797	411	15	s	s	PRON
ejpam-1797	411	16	which	which	PRON
ejpam-1797	411	17	is	be	AUX
ejpam-1797	411	18	in	in	ADP
ejpam-1797	411	19	m	m	PROPN
ejpam-1797	411	20	and	and	CCONJ
ejpam-1797	411	21	(	(	PUNCT
ejpam-1797	411	22	s	s	X
ejpam-1797	411	23	/	/	SYM
ejpam-1797	411	24	k)/anns(j	k)/anns(j	NOUN
ejpam-1797	411	25	/	/	SYM
ejpam-1797	411	26	k	k	NOUN
ejpam-1797	411	27	)	)	PUNCT
ejpam-1797	411	28	6=	6=	ADP
ejpam-1797	411	29	(	(	PUNCT
ejpam-1797	411	30	0	0	NUM
ejpam-1797	411	31	)	)	PUNCT
ejpam-1797	411	32	as	as	ADP
ejpam-1797	411	33	an	an	DET
ejpam-1797	411	34	annihilator	annihilator	NOUN
ejpam-1797	411	35	ideal	ideal	NOUN
ejpam-1797	411	36	is	be	AUX
ejpam-1797	411	37	necessarily	necessarily	ADV
ejpam-1797	411	38	a	a	DET
ejpam-1797	411	39	k	k	NOUN
ejpam-1797	411	40	-	-	NOUN
ejpam-1797	411	41	ideal	ideal	NOUN
ejpam-1797	411	42	.	.	PUNCT
ejpam-1797	412	1	however	however	ADV
ejpam-1797	412	2	,	,	PUNCT
ejpam-1797	412	3	this	this	PRON
ejpam-1797	412	4	is	be	AUX
ejpam-1797	412	5	impossible	impossible	ADJ
ejpam-1797	412	6	as	as	SCONJ
ejpam-1797	412	7	s	s	NOUN
ejpam-1797	412	8	∈	∈	X
ejpam-1797	412	9	um	um	INTJ
ejpam-1797	412	10	.	.	PUNCT
ejpam-1797	413	1	hence	hence	ADV
ejpam-1797	413	2	,	,	PUNCT
ejpam-1797	413	3	j	j	PROPN
ejpam-1797	413	4	∈	∈	PROPN
ejpam-1797	413	5	um	um	INTJ
ejpam-1797	413	6	and	and	CCONJ
ejpam-1797	413	7	we	we	PRON
ejpam-1797	413	8	have	have	AUX
ejpam-1797	413	9	shown	show	VERB
ejpam-1797	413	10	that	that	SCONJ
ejpam-1797	413	11	um	um	INTJ
ejpam-1797	413	12	is	be	AUX
ejpam-1797	413	13	a	a	DET
ejpam-1797	413	14	hereditary	hereditary	ADJ
ejpam-1797	413	15	radical	radical	ADJ
ejpam-1797	413	16	class	class	NOUN
ejpam-1797	413	17	.	.	PUNCT
ejpam-1797	414	1	finally	finally	ADV
ejpam-1797	414	2	,	,	PUNCT
ejpam-1797	414	3	if	if	SCONJ
ejpam-1797	414	4	s	s	VERB
ejpam-1797	414	5	is	be	AUX
ejpam-1797	414	6	an	an	DET
ejpam-1797	414	7	nilpotent	nilpotent	ADJ
ejpam-1797	414	8	ternary	ternary	ADJ
ejpam-1797	414	9	semiring	semiring	NOUN
ejpam-1797	414	10	,	,	PUNCT
ejpam-1797	414	11	then	then	ADV
ejpam-1797	414	12	φ(s	φ(s	NOUN
ejpam-1797	414	13	)	)	PUNCT
ejpam-1797	414	14	is	be	AUX
ejpam-1797	414	15	nilpotent	nilpotent	ADJ
ejpam-1797	414	16	for	for	ADP
ejpam-1797	414	17	any	any	DET
ejpam-1797	414	18	nonzero	nonzero	NOUN
ejpam-1797	414	19	homomorphism	homomorphism	PROPN
ejpam-1797	414	20	φ	φ	NUM
ejpam-1797	414	21	,	,	PUNCT
ejpam-1797	414	22	and	and	CCONJ
ejpam-1797	414	23	hence	hence	ADV
ejpam-1797	414	24	φ(s	φ(s	NOUN
ejpam-1797	414	25	)	)	PUNCT
ejpam-1797	414	26	6∈	6∈	PROPN
ejpam-1797	415	1	m	m	VERB
ejpam-1797	415	2	has	have	VERB
ejpam-1797	415	3	no	no	DET
ejpam-1797	415	4	nonzero	nonzero	ADJ
ejpam-1797	415	5	nilpotent	nilpotent	ADJ
ejpam-1797	415	6	ternary	ternary	ADJ
ejpam-1797	415	7	semiring	semiring	NOUN
ejpam-1797	415	8	can	can	AUX
ejpam-1797	415	9	be	be	AUX
ejpam-1797	415	10	prime.thus	prime.thu	NOUN
ejpam-1797	415	11	,	,	PUNCT
ejpam-1797	415	12	s	s	PART
ejpam-1797	415	13	∈	∈	PROPN
ejpam-1797	415	14	um	um	INTJ
ejpam-1797	415	15	and	and	CCONJ
ejpam-1797	415	16	hence	hence	ADV
ejpam-1797	415	17	,	,	PUNCT
ejpam-1797	415	18	um	um	INTJ
ejpam-1797	415	19	is	be	AUX
ejpam-1797	415	20	supernilpotent	supernilpotent	NOUN
ejpam-1797	415	21	.	.	PUNCT
ejpam-1797	416	1	finally	finally	ADV
ejpam-1797	416	2	,	,	PUNCT
ejpam-1797	416	3	we	we	PRON
ejpam-1797	416	4	state	state	VERB
ejpam-1797	416	5	a	a	DET
ejpam-1797	416	6	theorem	theorem	NOUN
ejpam-1797	416	7	for	for	ADP
ejpam-1797	416	8	the	the	DET
ejpam-1797	416	9	upper	upper	ADJ
ejpam-1797	416	10	radical	radical	ADJ
ejpam-1797	416	11	class	class	NOUN
ejpam-1797	416	12	of	of	ADP
ejpam-1797	416	13	ternary	ternary	ADJ
ejpam-1797	416	14	semirings	semiring	NOUN
ejpam-1797	416	15	.	.	PUNCT
ejpam-1797	417	1	theorem	theorem	VERB
ejpam-1797	417	2	10	10	NUM
ejpam-1797	417	3	.	.	PUNCT
ejpam-1797	418	1	the	the	DET
ejpam-1797	418	2	upper	upper	ADJ
ejpam-1797	418	3	radical	radical	ADJ
ejpam-1797	418	4	class	class	NOUN
ejpam-1797	418	5	determined	determine	VERB
ejpam-1797	418	6	by	by	ADP
ejpam-1797	418	7	the	the	DET
ejpam-1797	418	8	class	class	NOUN
ejpam-1797	418	9	℘′	℘′	ADV
ejpam-1797	418	10	of	of	ADP
ejpam-1797	418	11	prime	prime	ADJ
ejpam-1797	418	12	nonsingular	nonsingular	ADJ
ejpam-1797	418	13	ternary	ternary	ADJ
ejpam-1797	418	14	semirings	semiring	NOUN
ejpam-1797	418	15	is	be	AUX
ejpam-1797	418	16	a	a	DET
ejpam-1797	418	17	supernilpotent	supernilpotent	ADJ
ejpam-1797	418	18	radical	radical	ADJ
ejpam-1797	418	19	class	class	NOUN
ejpam-1797	418	20	.	.	PUNCT
ejpam-1797	419	1	proof	proof	NOUN
ejpam-1797	419	2	.	.	PUNCT
ejpam-1797	420	1	the	the	DET
ejpam-1797	420	2	proof	proof	NOUN
ejpam-1797	420	3	of	of	ADP
ejpam-1797	420	4	the	the	DET
ejpam-1797	420	5	above	above	ADJ
ejpam-1797	420	6	theorem	theorem	NOUN
ejpam-1797	420	7	follows	follow	VERB
ejpam-1797	420	8	immediately	immediately	ADV
ejpam-1797	420	9	from	from	ADP
ejpam-1797	420	10	proposition	proposition	NOUN
ejpam-1797	420	11	9	9	NUM
ejpam-1797	420	12	and	and	CCONJ
ejpam-1797	420	13	by	by	ADP
ejpam-1797	420	14	theorem	theorem	NOUN
ejpam-1797	420	15	9	9	NUM
ejpam-1797	420	16	.	.	NOUN
ejpam-1797	420	17	remark	remark	NOUN
ejpam-1797	420	18	1	1	NUM
ejpam-1797	420	19	.	.	PUNCT
ejpam-1797	421	1	we	we	PRON
ejpam-1797	421	2	call	call	VERB
ejpam-1797	421	3	the	the	DET
ejpam-1797	421	4	upper	upper	ADJ
ejpam-1797	421	5	radical	radical	ADJ
ejpam-1797	421	6	class	class	NOUN
ejpam-1797	421	7	determined	determine	VERB
ejpam-1797	421	8	by	by	ADP
ejpam-1797	421	9	the	the	DET
ejpam-1797	421	10	class	class	NOUN
ejpam-1797	421	11	℘′	℘′	ADV
ejpam-1797	421	12	the	the	DET
ejpam-1797	421	13	special	special	ADJ
ejpam-1797	421	14	singular	singular	ADJ
ejpam-1797	421	15	radical	radical	ADJ
ejpam-1797	421	16	class	class	NOUN
ejpam-1797	421	17	.	.	PUNCT
ejpam-1797	422	1	it	it	PRON
ejpam-1797	422	2	is	be	AUX
ejpam-1797	422	3	clear	clear	ADJ
ejpam-1797	422	4	that	that	SCONJ
ejpam-1797	422	5	above	above	ADP
ejpam-1797	422	6	upper	upper	ADJ
ejpam-1797	422	7	radical	radical	ADJ
ejpam-1797	422	8	class	class	NOUN
ejpam-1797	422	9	is	be	AUX
ejpam-1797	422	10	contained	contain	VERB
ejpam-1797	422	11	in	in	ADP
ejpam-1797	422	12	s	s	PRON
ejpam-1797	422	13	.	.	PUNCT
ejpam-1797	423	1	in	in	ADP
ejpam-1797	423	2	closing	close	VERB
ejpam-1797	423	3	this	this	DET
ejpam-1797	423	4	paper	paper	NOUN
ejpam-1797	423	5	,	,	PUNCT
ejpam-1797	423	6	we	we	PRON
ejpam-1797	423	7	notice	notice	VERB
ejpam-1797	423	8	that	that	SCONJ
ejpam-1797	423	9	in	in	ADP
ejpam-1797	423	10	the	the	DET
ejpam-1797	423	11	1983	1983	NUM
ejpam-1797	423	12	paper	paper	NOUN
ejpam-1797	423	13	of	of	ADP
ejpam-1797	423	14	d.	d.	PROPN
ejpam-1797	423	15	m.	m.	PROPN
ejpam-1797	423	16	olson	olson	PROPN
ejpam-1797	423	17	and	and	CCONJ
ejpam-1797	423	18	t.	t.	PROPN
ejpam-1797	423	19	l	l	PROPN
ejpam-1797	423	20	jenkins	jenkin	NOUN
ejpam-1797	424	1	[	[	X
ejpam-1797	424	2	28	28	NUM
ejpam-1797	424	3	]	]	PUNCT
ejpam-1797	424	4	on	on	ADP
ejpam-1797	424	5	radical	radical	ADJ
ejpam-1797	424	6	theorems	theorem	NOUN
ejpam-1797	424	7	for	for	ADP
ejpam-1797	424	8	hemirings	hemiring	NOUN
ejpam-1797	424	9	,	,	PUNCT
ejpam-1797	424	10	they	they	PRON
ejpam-1797	424	11	asked	ask	VERB
ejpam-1797	424	12	an	an	DET
ejpam-1797	424	13	open	open	ADJ
ejpam-1797	424	14	problem	problem	NOUN
ejpam-1797	424	15	.	.	PUNCT
ejpam-1797	425	1	is	be	AUX
ejpam-1797	425	2	the	the	DET
ejpam-1797	425	3	class	class	NOUN
ejpam-1797	425	4	of	of	ADP
ejpam-1797	425	5	nil	nil	ADJ
ejpam-1797	425	6	hemirings	hemiring	NOUN
ejpam-1797	425	7	a	a	DET
ejpam-1797	425	8	radical	radical	ADJ
ejpam-1797	425	9	class	class	NOUN
ejpam-1797	425	10	?	?	PUNCT
ejpam-1797	426	1	it	it	PRON
ejpam-1797	426	2	seems	seem	VERB
ejpam-1797	426	3	that	that	SCONJ
ejpam-1797	426	4	this	this	DET
ejpam-1797	426	5	open	open	ADJ
ejpam-1797	426	6	problem	problem	NOUN
ejpam-1797	426	7	of	of	ADP
ejpam-1797	426	8	olson	olson	NOUN
ejpam-1797	426	9	-	-	PUNCT
ejpam-1797	426	10	jenkins	jenkins	PROPN
ejpam-1797	426	11	has	have	AUX
ejpam-1797	426	12	not	not	PART
ejpam-1797	426	13	yet	yet	ADV
ejpam-1797	426	14	been	be	AUX
ejpam-1797	426	15	answered	answer	VERB
ejpam-1797	426	16	in	in	ADP
ejpam-1797	426	17	the	the	DET
ejpam-1797	426	18	literature	literature	NOUN
ejpam-1797	426	19	.	.	PUNCT
ejpam-1797	427	1	naturally	naturally	ADV
ejpam-1797	427	2	,	,	PUNCT
ejpam-1797	427	3	we	we	PRON
ejpam-1797	427	4	ask	ask	VERB
ejpam-1797	427	5	a	a	DET
ejpam-1797	427	6	new	new	ADJ
ejpam-1797	427	7	open	open	ADJ
ejpam-1797	427	8	problem	problem	NOUN
ejpam-1797	427	9	:	:	PUNCT
ejpam-1797	427	10	is	be	AUX
ejpam-1797	427	11	the	the	DET
ejpam-1797	427	12	class	class	NOUN
ejpam-1797	427	13	of	of	ADP
ejpam-1797	427	14	all	all	DET
ejpam-1797	427	15	nil	nil	ADJ
ejpam-1797	427	16	singular	singular	ADJ
ejpam-1797	427	17	ternary	ternary	ADJ
ejpam-1797	427	18	semirings	semiring	NOUN
ejpam-1797	427	19	also	also	ADV
ejpam-1797	427	20	a	a	DET
ejpam-1797	427	21	radical	radical	ADJ
ejpam-1797	427	22	class	class	NOUN
ejpam-1797	427	23	?	?	PUNCT
ejpam-1797	428	1	references	reference	NOUN
ejpam-1797	428	2	412	412	NUM
ejpam-1797	428	3	acknowledgements	acknowledgement	NOUN
ejpam-1797	428	4	the	the	DET
ejpam-1797	428	5	third	third	ADJ
ejpam-1797	428	6	author	author	NOUN
ejpam-1797	428	7	is	be	AUX
ejpam-1797	428	8	thankful	thankful	ADJ
ejpam-1797	428	9	to	to	ADP
ejpam-1797	428	10	csir	csir	PROPN
ejpam-1797	428	11	,	,	PUNCT
ejpam-1797	428	12	india	india	PROPN
ejpam-1797	428	13	for	for	ADP
ejpam-1797	428	14	financial	financial	ADJ
ejpam-1797	428	15	assistance	assistance	NOUN
ejpam-1797	428	16	.	.	PUNCT
ejpam-1797	429	1	references	reference	NOUN
ejpam-1797	429	2	[	[	X
ejpam-1797	429	3	1	1	X
ejpam-1797	429	4	]	]	PUNCT
ejpam-1797	429	5	t.	t.	PROPN
ejpam-1797	429	6	anderson	anderson	PROPN
ejpam-1797	429	7	,	,	PUNCT
ejpam-1797	429	8	n.	n.	PROPN
ejpam-1797	429	9	j.	j.	PROPN
ejpam-1797	429	10	divinsky	divinsky	PROPN
ejpam-1797	429	11	and	and	CCONJ
ejpam-1797	429	12	a.	a.	NOUN
ejpam-1797	429	13	suliski	suliski	PROPN
ejpam-1797	429	14	.	.	PUNCT
ejpam-1797	430	1	hereditary	hereditary	ADJ
ejpam-1797	430	2	radicals	radical	NOUN
ejpam-1797	430	3	in	in	ADP
ejpam-1797	430	4	associative	associative	ADJ
ejpam-1797	430	5	and	and	CCONJ
ejpam-1797	430	6	alternative	alternative	ADJ
ejpam-1797	430	7	rings	ring	NOUN
ejpam-1797	430	8	,	,	PUNCT
ejpam-1797	430	9	canad	canad	PROPN
ejpam-1797	430	10	.	.	PUNCT
ejpam-1797	431	1	j.	j.	PROPN
ejpam-1797	431	2	math	math	PROPN
ejpam-1797	431	3	.	.	PUNCT
ejpam-1797	432	1	17	17	NUM
ejpam-1797	432	2	,	,	PUNCT
ejpam-1797	432	3	594	594	NUM
ejpam-1797	432	4	-	-	SYM
ejpam-1797	432	5	603	603	NUM
ejpam-1797	432	6	.	.	PUNCT
ejpam-1797	433	1	1965	1965	NUM
ejpam-1797	433	2	.	.	PUNCT
ejpam-1797	434	1	[	[	X
ejpam-1797	434	2	2	2	X
ejpam-1797	434	3	]	]	PUNCT
ejpam-1797	434	4	j.	j.	PROPN
ejpam-1797	434	5	m.	m.	PROPN
ejpam-1797	434	6	chaudhari	chaudhari	PROPN
ejpam-1797	434	7	and	and	CCONJ
ejpam-1797	434	8	k.	k.	PUNCT
ejpam-1797	434	9	j.ingale	j.ingale	PROPN
ejpam-1797	434	10	.	.	PUNCT
ejpam-1797	435	1	on	on	ADP
ejpam-1797	435	2	partitioning	partition	VERB
ejpam-1797	435	3	and	and	CCONJ
ejpam-1797	435	4	subtractive	subtractive	ADJ
ejpam-1797	435	5	ideals	ideal	NOUN
ejpam-1797	435	6	of	of	ADP
ejpam-1797	435	7	ternary	ternary	ADJ
ejpam-1797	435	8	semirings	semiring	NOUN
ejpam-1797	435	9	;	;	PUNCT
ejpam-1797	435	10	kyungpook	kyungpook	PROPN
ejpam-1797	435	11	math	math	PROPN
ejpam-1797	435	12	.	.	PUNCT
ejpam-1797	436	1	j.	j.	PROPN
ejpam-1797	436	2	51	51	NUM
ejpam-1797	436	3	,	,	PUNCT
ejpam-1797	436	4	69	69	NUM
ejpam-1797	436	5	-	-	SYM
ejpam-1797	436	6	76	76	NUM
ejpam-1797	436	7	.	.	PUNCT
ejpam-1797	436	8	2011	2011	NUM
ejpam-1797	436	9	.	.	PUNCT
ejpam-1797	437	1	[	[	X
ejpam-1797	437	2	3	3	X
ejpam-1797	437	3	]	]	PUNCT
ejpam-1797	437	4	t.	t.	PROPN
ejpam-1797	437	5	k.	k.	PROPN
ejpam-1797	437	6	dutta	dutta	PROPN
ejpam-1797	437	7	and	and	CCONJ
ejpam-1797	437	8	s.	s.	PROPN
ejpam-1797	437	9	kar	kar	PROPN
ejpam-1797	437	10	.	.	PUNCT
ejpam-1797	438	1	on	on	ADP
ejpam-1797	438	2	regular	regular	ADJ
ejpam-1797	438	3	ternary	ternary	ADJ
ejpam-1797	438	4	semirings	semiring	NOUN
ejpam-1797	438	5	,	,	PUNCT
ejpam-1797	438	6	advances	advance	NOUN
ejpam-1797	438	7	in	in	ADP
ejpam-1797	438	8	algebra	algebra	NOUN
ejpam-1797	438	9	,	,	PUNCT
ejpam-1797	438	10	proceedings	proceeding	NOUN
ejpam-1797	438	11	of	of	ADP
ejpam-1797	438	12	the	the	DET
ejpam-1797	438	13	icm	icm	PROPN
ejpam-1797	438	14	satellite	satellite	PROPN
ejpam-1797	438	15	conference	conference	NOUN
ejpam-1797	438	16	in	in	ADP
ejpam-1797	438	17	algebra	algebra	PROPN
ejpam-1797	438	18	and	and	CCONJ
ejpam-1797	438	19	related	related	ADJ
ejpam-1797	438	20	topics	topic	NOUN
ejpam-1797	438	21	,	,	PUNCT
ejpam-1797	438	22	world	world	NOUN
ejpam-1797	438	23	scientific	scientific	ADJ
ejpam-1797	438	24	,	,	PUNCT
ejpam-1797	438	25	343	343	NUM
ejpam-1797	438	26	355	355	NUM
ejpam-1797	438	27	.	.	PUNCT
ejpam-1797	438	28	2003	2003	NUM
ejpam-1797	438	29	.	.	PUNCT
ejpam-1797	439	1	[	[	X
ejpam-1797	439	2	4	4	X
ejpam-1797	439	3	]	]	PUNCT
ejpam-1797	439	4	t.	t.	PROPN
ejpam-1797	439	5	k.	k.	PROPN
ejpam-1797	439	6	dutta	dutta	PROPN
ejpam-1797	439	7	and	and	CCONJ
ejpam-1797	439	8	s.kar	s.kar	NOUN
ejpam-1797	439	9	.	.	PUNCT
ejpam-1797	440	1	a	a	DET
ejpam-1797	440	2	note	note	NOUN
ejpam-1797	440	3	on	on	ADP
ejpam-1797	440	4	regular	regular	ADJ
ejpam-1797	440	5	ternary	ternary	ADJ
ejpam-1797	440	6	semirings	semiring	NOUN
ejpam-1797	440	7	,	,	PUNCT
ejpam-1797	440	8	kyungpook	kyungpook	PROPN
ejpam-1797	440	9	mathematical	mathematical	ADJ
ejpam-1797	440	10	journal	journal	PROPN
ejpam-1797	440	11	,	,	PUNCT
ejpam-1797	440	12	vol	vol	NOUN
ejpam-1797	440	13	.	.	PROPN
ejpam-1797	440	14	46	46	NUM
ejpam-1797	440	15	,	,	PUNCT
ejpam-1797	440	16	no	no	INTJ
ejpam-1797	440	17	.	.	NOUN
ejpam-1797	440	18	3	3	NUM
ejpam-1797	440	19	,	,	PUNCT
ejpam-1797	440	20	357	357	NUM
ejpam-1797	440	21	365	365	NUM
ejpam-1797	440	22	.	.	PUNCT
ejpam-1797	441	1	2006	2006	NUM
ejpam-1797	441	2	.	.	PUNCT
ejpam-1797	442	1	[	[	X
ejpam-1797	442	2	5	5	X
ejpam-1797	442	3	]	]	PUNCT
ejpam-1797	442	4	t.	t.	PROPN
ejpam-1797	442	5	k.	k.	PROPN
ejpam-1797	442	6	dutta	dutta	PROPN
ejpam-1797	442	7	and	and	CCONJ
ejpam-1797	442	8	s.	s.	PROPN
ejpam-1797	442	9	kar	kar	PROPN
ejpam-1797	442	10	.	.	PUNCT
ejpam-1797	443	1	on	on	ADP
ejpam-1797	443	2	prime	prime	ADJ
ejpam-1797	443	3	ideals	ideal	NOUN
ejpam-1797	443	4	and	and	CCONJ
ejpam-1797	443	5	prime	prime	ADJ
ejpam-1797	443	6	radical	radical	ADJ
ejpam-1797	443	7	of	of	ADP
ejpam-1797	443	8	ternary	ternary	ADJ
ejpam-1797	443	9	semirings	semiring	NOUN
ejpam-1797	443	10	,	,	PUNCT
ejpam-1797	443	11	bull	bull	NOUN
ejpam-1797	443	12	.	.	PUNCT
ejpam-1797	444	1	cal	cal	PROPN
ejpam-1797	444	2	.	.	PUNCT
ejpam-1797	445	1	math	math	NOUN
ejpam-1797	445	2	.	.	PUNCT
ejpam-1797	446	1	soc	soc	PROPN
ejpam-1797	446	2	.	.	PUNCT
ejpam-1797	446	3	,	,	PUNCT
ejpam-1797	446	4	vol	vol	NOUN
ejpam-1797	446	5	.	.	PROPN
ejpam-1797	447	1	97	97	NUM
ejpam-1797	447	2	,	,	PUNCT
ejpam-1797	447	3	no	no	INTJ
ejpam-1797	447	4	.	.	NOUN
ejpam-1797	447	5	5	5	NUM
ejpam-1797	447	6	,	,	PUNCT
ejpam-1797	447	7	445	445	NUM
ejpam-1797	447	8	454	454	NUM
ejpam-1797	447	9	.	.	PUNCT
ejpam-1797	448	1	2005	2005	NUM
ejpam-1797	448	2	.	.	PUNCT
ejpam-1797	449	1	[	[	X
ejpam-1797	449	2	6	6	NUM
ejpam-1797	449	3	]	]	PUNCT
ejpam-1797	449	4	t.	t.	PROPN
ejpam-1797	449	5	k.	k.	PROPN
ejpam-1797	449	6	dutta	dutta	PROPN
ejpam-1797	449	7	and	and	CCONJ
ejpam-1797	449	8	s.	s.	PROPN
ejpam-1797	449	9	kar	kar	PROPN
ejpam-1797	449	10	.	.	PUNCT
ejpam-1797	450	1	on	on	ADP
ejpam-1797	450	2	semiprime	semiprime	NOUN
ejpam-1797	450	3	ideals	ideal	NOUN
ejpam-1797	450	4	and	and	CCONJ
ejpam-1797	450	5	irreducible	irreducible	ADJ
ejpam-1797	450	6	ideals	ideal	NOUN
ejpam-1797	450	7	of	of	ADP
ejpam-1797	450	8	ternary	ternary	ADJ
ejpam-1797	450	9	semirings	semiring	NOUN
ejpam-1797	450	10	,	,	PUNCT
ejpam-1797	450	11	bull	bull	NOUN
ejpam-1797	450	12	.	.	PUNCT
ejpam-1797	451	1	cal	cal	PROPN
ejpam-1797	451	2	.	.	PUNCT
ejpam-1797	452	1	math	math	NOUN
ejpam-1797	452	2	.	.	PUNCT
ejpam-1797	453	1	soc	soc	PROPN
ejpam-1797	453	2	.	.	PUNCT
ejpam-1797	453	3	,	,	PUNCT
ejpam-1797	453	4	vol	vol	NOUN
ejpam-1797	453	5	.	.	PROPN
ejpam-1797	454	1	97	97	NUM
ejpam-1797	454	2	,	,	PUNCT
ejpam-1797	454	3	no	no	INTJ
ejpam-1797	454	4	.	.	NOUN
ejpam-1797	454	5	5	5	NUM
ejpam-1797	454	6	,	,	PUNCT
ejpam-1797	454	7	467	467	NUM
ejpam-1797	454	8	476	476	NUM
ejpam-1797	454	9	.	.	PUNCT
ejpam-1797	455	1	2005	2005	NUM
ejpam-1797	455	2	.	.	PUNCT
ejpam-1797	456	1	[	[	X
ejpam-1797	456	2	7	7	X
ejpam-1797	456	3	]	]	PUNCT
ejpam-1797	456	4	t.	t.	PROPN
ejpam-1797	456	5	k.	k.	PROPN
ejpam-1797	456	6	dutta	dutta	PROPN
ejpam-1797	456	7	and	and	CCONJ
ejpam-1797	456	8	s.	s.	PROPN
ejpam-1797	456	9	kar	kar	PROPN
ejpam-1797	456	10	.	.	PUNCT
ejpam-1797	457	1	on	on	ADP
ejpam-1797	457	2	ternary	ternary	ADJ
ejpam-1797	457	3	semifields	semifield	NOUN
ejpam-1797	457	4	,	,	PUNCT
ejpam-1797	457	5	discussiones	discussione	NOUN
ejpam-1797	457	6	mathematicae	mathematicae	VERB
ejpam-1797	457	7	general	general	ADJ
ejpam-1797	457	8	algebra	algebra	PROPN
ejpam-1797	457	9	and	and	CCONJ
ejpam-1797	457	10	applications	application	NOUN
ejpam-1797	457	11	,	,	PUNCT
ejpam-1797	457	12	vol	vol	NOUN
ejpam-1797	457	13	.	.	PROPN
ejpam-1797	458	1	24	24	NUM
ejpam-1797	458	2	,	,	PUNCT
ejpam-1797	458	3	no.2	no.2	PROPN
ejpam-1797	458	4	,	,	PUNCT
ejpam-1797	458	5	185	185	NUM
ejpam-1797	458	6	198	198	NUM
ejpam-1797	458	7	.	.	PUNCT
ejpam-1797	459	1	2004	2004	NUM
ejpam-1797	459	2	.	.	PUNCT
ejpam-1797	460	1	[	[	X
ejpam-1797	460	2	8	8	X
ejpam-1797	460	3	]	]	PUNCT
ejpam-1797	460	4	t.	t.	PROPN
ejpam-1797	460	5	k.	k.	PROPN
ejpam-1797	460	6	dutta	dutta	PROPN
ejpam-1797	460	7	and	and	CCONJ
ejpam-1797	460	8	s.	s.	PROPN
ejpam-1797	460	9	kar	kar	PROPN
ejpam-1797	460	10	.	.	PUNCT
ejpam-1797	461	1	on	on	ADP
ejpam-1797	461	2	the	the	DET
ejpam-1797	461	3	jacobson	jacobson	PROPN
ejpam-1797	461	4	radical	radical	PROPN
ejpam-1797	461	5	of	of	ADP
ejpam-1797	461	6	a	a	DET
ejpam-1797	461	7	ternary	ternary	ADJ
ejpam-1797	461	8	semiring	semiring	NOUN
ejpam-1797	461	9	,	,	PUNCT
ejpam-1797	461	10	southeast	southeast	ADJ
ejpam-1797	461	11	asian	asian	ADJ
ejpam-1797	461	12	bull	bull	NOUN
ejpam-1797	461	13	.	.	PUNCT
ejpam-1797	462	1	of	of	ADP
ejpam-1797	462	2	math	math	NOUN
ejpam-1797	462	3	.	.	PUNCT
ejpam-1797	463	1	28	28	NUM
ejpam-1797	463	2	no.1	no.1	ADJ
ejpam-1797	463	3	,	,	PUNCT
ejpam-1797	463	4	1	1	NUM
ejpam-1797	463	5	13	13	NUM
ejpam-1797	463	6	.	.	PUNCT
ejpam-1797	464	1	2004	2004	NUM
ejpam-1797	464	2	.	.	PUNCT
ejpam-1797	465	1	[	[	X
ejpam-1797	465	2	9	9	NUM
ejpam-1797	465	3	]	]	PUNCT
ejpam-1797	465	4	t.	t.	PROPN
ejpam-1797	465	5	k.	k.	PROPN
ejpam-1797	465	6	dutta	dutta	PROPN
ejpam-1797	465	7	and	and	CCONJ
ejpam-1797	465	8	s.	s.	PROPN
ejpam-1797	465	9	kar	kar	PROPN
ejpam-1797	465	10	.	.	PUNCT
ejpam-1797	466	1	a	a	DET
ejpam-1797	466	2	note	note	NOUN
ejpam-1797	466	3	on	on	ADP
ejpam-1797	466	4	the	the	DET
ejpam-1797	466	5	jacobson	jacobson	PROPN
ejpam-1797	466	6	radical	radical	PROPN
ejpam-1797	466	7	of	of	ADP
ejpam-1797	466	8	a	a	DET
ejpam-1797	466	9	ternary	ternary	ADJ
ejpam-1797	466	10	semiring	semiring	NOUN
ejpam-1797	466	11	,	,	PUNCT
ejpam-1797	466	12	southeast	southeast	ADJ
ejpam-1797	466	13	asian	asian	ADJ
ejpam-1797	466	14	bull	bull	NOUN
ejpam-1797	466	15	.	.	PUNCT
ejpam-1797	467	1	math	math	NOUN
ejpam-1797	467	2	.	.	PUNCT
ejpam-1797	468	1	,	,	PUNCT
ejpam-1797	468	2	29	29	NUM
ejpam-1797	468	3	no.2	no.2	PROPN
ejpam-1797	468	4	,	,	PUNCT
ejpam-1797	468	5	321	321	NUM
ejpam-1797	468	6	331	331	NUM
ejpam-1797	468	7	.	.	PUNCT
ejpam-1797	469	1	2005	2005	NUM
ejpam-1797	469	2	.	.	PUNCT
ejpam-1797	470	1	[	[	X
ejpam-1797	470	2	10	10	NUM
ejpam-1797	470	3	]	]	X
ejpam-1797	470	4	t.k	t.k	PROPN
ejpam-1797	470	5	.	.	PROPN
ejpam-1797	470	6	dutta	dutta	PROPN
ejpam-1797	470	7	and	and	CCONJ
ejpam-1797	470	8	s.	s.	PROPN
ejpam-1797	470	9	kar	kar	PROPN
ejpam-1797	470	10	.	.	PUNCT
ejpam-1797	471	1	two	two	NUM
ejpam-1797	471	2	types	type	NOUN
ejpam-1797	471	3	of	of	ADP
ejpam-1797	471	4	jacobson	jacobson	PROPN
ejpam-1797	471	5	radicals	radical	NOUN
ejpam-1797	471	6	of	of	ADP
ejpam-1797	471	7	ternary	ternary	ADJ
ejpam-1797	471	8	semirings	semiring	NOUN
ejpam-1797	471	9	,	,	PUNCT
ejpam-1797	471	10	southeast	southeast	ADJ
ejpam-1797	471	11	asian	asian	ADJ
ejpam-1797	471	12	bull	bull	NOUN
ejpam-1797	471	13	.	.	PUNCT
ejpam-1797	472	1	math	math	NOUN
ejpam-1797	472	2	.	.	PUNCT
ejpam-1797	473	1	,	,	PUNCT
ejpam-1797	473	2	29	29	NUM
ejpam-1797	473	3	no	no	NOUN
ejpam-1797	473	4	.	.	NOUN
ejpam-1797	473	5	4	4	NUM
ejpam-1797	473	6	,	,	PUNCT
ejpam-1797	473	7	677	677	NUM
ejpam-1797	473	8	687	687	NUM
ejpam-1797	473	9	.	.	PUNCT
ejpam-1797	473	10	2005	2005	NUM
ejpam-1797	473	11	.	.	PUNCT
ejpam-1797	474	1	[	[	X
ejpam-1797	474	2	11	11	NUM
ejpam-1797	474	3	]	]	PUNCT
ejpam-1797	474	4	t.	t.	PROPN
ejpam-1797	474	5	k.	k.	PROPN
ejpam-1797	474	6	dutta	dutta	PROPN
ejpam-1797	474	7	and	and	CCONJ
ejpam-1797	474	8	s.	s.	PROPN
ejpam-1797	474	9	kar	kar	PROPN
ejpam-1797	474	10	.	.	PUNCT
ejpam-1797	475	1	on	on	ADP
ejpam-1797	475	2	matrix	matrix	NOUN
ejpam-1797	475	3	ternary	ternary	ADJ
ejpam-1797	475	4	semirings	semiring	NOUN
ejpam-1797	475	5	;	;	PUNCT
ejpam-1797	475	6	international	international	ADJ
ejpam-1797	475	7	journal	journal	NOUN
ejpam-1797	475	8	of	of	ADP
ejpam-1797	475	9	mathematics	mathematic	NOUN
ejpam-1797	475	10	and	and	CCONJ
ejpam-1797	475	11	analysis	analysis	NOUN
ejpam-1797	475	12	,	,	PUNCT
ejpam-1797	475	13	vol	vol	NOUN
ejpam-1797	475	14	.	.	PROPN
ejpam-1797	475	15	1	1	NUM
ejpam-1797	475	16	,	,	PUNCT
ejpam-1797	475	17	no	no	INTJ
ejpam-1797	475	18	.	.	NOUN
ejpam-1797	475	19	1	1	NUM
ejpam-1797	475	20	,	,	PUNCT
ejpam-1797	475	21	81	81	NUM
ejpam-1797	475	22	95	95	NUM
ejpam-1797	475	23	.	.	PUNCT
ejpam-1797	476	1	2006	2006	NUM
ejpam-1797	476	2	.	.	PUNCT
ejpam-1797	477	1	[	[	X
ejpam-1797	477	2	12	12	NUM
ejpam-1797	477	3	]	]	PUNCT
ejpam-1797	477	4	t.	t.	PROPN
ejpam-1797	477	5	k.	k.	PROPN
ejpam-1797	477	6	dutta	dutta	PROPN
ejpam-1797	477	7	and	and	CCONJ
ejpam-1797	477	8	m.	m.	PROPN
ejpam-1797	477	9	l.	l.	PROPN
ejpam-1797	477	10	das	das	PROPN
ejpam-1797	477	11	.	.	PROPN
ejpam-1797	477	12	singular	singular	PROPN
ejpam-1797	477	13	radicals	radical	NOUN
ejpam-1797	477	14	in	in	ADP
ejpam-1797	477	15	semiring	semiring	NOUN
ejpam-1797	477	16	;	;	PUNCT
ejpam-1797	477	17	southeast	southeast	ADJ
ejpam-1797	477	18	asian	asian	ADJ
ejpam-1797	477	19	bull	bull	PROPN
ejpam-1797	477	20	.	.	PUNCT
ejpam-1797	478	1	math	math	NOUN
ejpam-1797	478	2	.	.	PUNCT
ejpam-1797	479	1	,	,	PUNCT
ejpam-1797	479	2	34	34	NUM
ejpam-1797	479	3	no.3	no.3	PROPN
ejpam-1797	479	4	,	,	PUNCT
ejpam-1797	479	5	405	405	NUM
ejpam-1797	479	6	-	-	SYM
ejpam-1797	479	7	416	416	NUM
ejpam-1797	479	8	.	.	PUNCT
ejpam-1797	479	9	2010	2010	NUM
ejpam-1797	479	10	.	.	PUNCT
ejpam-1797	480	1	[	[	X
ejpam-1797	480	2	13	13	NUM
ejpam-1797	480	3	]	]	PUNCT
ejpam-1797	480	4	t.	t.	PROPN
ejpam-1797	480	5	k.	k.	PROPN
ejpam-1797	480	6	dutta	dutta	PROPN
ejpam-1797	480	7	and	and	CCONJ
ejpam-1797	480	8	s.	s.	PROPN
ejpam-1797	480	9	mandal	mandal	PROPN
ejpam-1797	480	10	.	.	PUNCT
ejpam-1797	481	1	general	general	ADJ
ejpam-1797	481	2	type	type	NOUN
ejpam-1797	481	3	of	of	ADP
ejpam-1797	481	4	regularity	regularity	NOUN
ejpam-1797	481	5	and	and	CCONJ
ejpam-1797	481	6	semiprime	semiprime	NOUN
ejpam-1797	481	7	ideals	ideal	NOUN
ejpam-1797	481	8	in	in	ADP
ejpam-1797	481	9	ternary	ternary	ADJ
ejpam-1797	481	10	semiring	semiring	NOUN
ejpam-1797	481	11	;	;	PUNCT
ejpam-1797	481	12	advances	advance	NOUN
ejpam-1797	481	13	in	in	ADP
ejpam-1797	481	14	algebraic	algebraic	ADJ
ejpam-1797	481	15	structure	structure	NOUN
ejpam-1797	481	16	,	,	PUNCT
ejpam-1797	481	17	proceedings	proceeding	NOUN
ejpam-1797	481	18	of	of	ADP
ejpam-1797	481	19	international	international	ADJ
ejpam-1797	481	20	conference	conference	NOUN
ejpam-1797	481	21	in	in	ADP
ejpam-1797	481	22	algebra	algebra	PROPN
ejpam-1797	481	23	,	,	PUNCT
ejpam-1797	481	24	world	world	PROPN
ejpam-1797	481	25	sci	sci	PROPN
ejpam-1797	481	26	,	,	PUNCT
ejpam-1797	481	27	publ	publ	PROPN
ejpam-1797	481	28	.	.	PUNCT
ejpam-1797	482	1	hackensack	hackensack	PROPN
ejpam-1797	482	2	,	,	PUNCT
ejpam-1797	482	3	n.j	n.j	PROPN
ejpam-1797	482	4	.	.	PROPN
ejpam-1797	482	5	,19	,19	PROPN
ejpam-1797	482	6	-	-	PUNCT
ejpam-1797	482	7	232	232	NUM
ejpam-1797	482	8	,	,	PUNCT
ejpam-1797	482	9	2010	2010	NUM
ejpam-1797	482	10	.	.	PUNCT
ejpam-1797	483	1	[	[	X
ejpam-1797	483	2	14	14	NUM
ejpam-1797	483	3	]	]	X
ejpam-1797	483	4	t.k	t.k	PROPN
ejpam-1797	483	5	.	.	PROPN
ejpam-1797	483	6	dutta	dutta	PROPN
ejpam-1797	483	7	and	and	CCONJ
ejpam-1797	483	8	s.	s.	PROPN
ejpam-1797	483	9	mandal	mandal	PROPN
ejpam-1797	483	10	and	and	CCONJ
ejpam-1797	483	11	j.	j.	PROPN
ejpam-1797	483	12	sircar	sircar	PROPN
ejpam-1797	483	13	.	.	PUNCT
ejpam-1797	484	1	on	on	ADP
ejpam-1797	484	2	right	right	ADV
ejpam-1797	484	3	strongly	strongly	ADV
ejpam-1797	484	4	prime	prime	ADJ
ejpam-1797	484	5	ternary	ternary	ADJ
ejpam-1797	484	6	semirings	semirings	PROPN
ejpam-1797	484	7	ii	ii	PROPN
ejpam-1797	484	8	,	,	PUNCT
ejpam-1797	484	9	international	international	PROPN
ejpam-1797	484	10	j.	j.	PROPN
ejpam-1797	484	11	of	of	ADP
ejpam-1797	484	12	mathematics	mathematics	PROPN
ejpam-1797	484	13	and	and	CCONJ
ejpam-1797	484	14	applications	application	NOUN
ejpam-1797	484	15	;	;	PUNCT
ejpam-1797	484	16	vol	vol	NOUN
ejpam-1797	484	17	.	.	NOUN
ejpam-1797	484	18	4	4	NUM
ejpam-1797	484	19	,	,	PUNCT
ejpam-1797	484	20	no	no	INTJ
ejpam-1797	484	21	.	.	NOUN
ejpam-1797	484	22	2,197	2,197	NUM
ejpam-1797	484	23	-	-	SYM
ejpam-1797	484	24	205	205	NUM
ejpam-1797	484	25	.	.	PUNCT
ejpam-1797	484	26	2011	2011	NUM
ejpam-1797	484	27	.	.	PUNCT
ejpam-1797	485	1	references	reference	NOUN
ejpam-1797	485	2	413	413	NUM
ejpam-1797	486	1	[	[	X
ejpam-1797	486	2	15	15	NUM
ejpam-1797	486	3	]	]	X
ejpam-1797	486	4	t.k	t.k	PROPN
ejpam-1797	486	5	.	.	PROPN
ejpam-1797	486	6	dutta	dutta	PROPN
ejpam-1797	486	7	and	and	CCONJ
ejpam-1797	486	8	s.	s.	PROPN
ejpam-1797	486	9	mandal	mandal	PROPN
ejpam-1797	486	10	and	and	CCONJ
ejpam-1797	486	11	j.	j.	PROPN
ejpam-1797	486	12	sircar	sircar	PROPN
ejpam-1797	486	13	.	.	PUNCT
ejpam-1797	487	1	on	on	ADP
ejpam-1797	487	2	uniformly	uniformly	ADV
ejpam-1797	487	3	strongly	strongly	ADV
ejpam-1797	487	4	prime	prime	ADJ
ejpam-1797	487	5	ternary	ternary	ADJ
ejpam-1797	487	6	semirings	semiring	NOUN
ejpam-1797	487	7	,	,	PUNCT
ejpam-1797	487	8	southeast	southeast	ADJ
ejpam-1797	487	9	asian	asian	ADJ
ejpam-1797	487	10	bull	bull	NOUN
ejpam-1797	487	11	.	.	PUNCT
ejpam-1797	487	12	math	math	NOUN
ejpam-1797	487	13	.	.	PUNCT
ejpam-1797	487	14	,	,	PUNCT
ejpam-1797	487	15	36	36	NUM
ejpam-1797	487	16	no.1	no.1	NOUN
ejpam-1797	487	17	,	,	PUNCT
ejpam-1797	487	18	43	43	NUM
ejpam-1797	487	19	-	-	SYM
ejpam-1797	487	20	56	56	NUM
ejpam-1797	487	21	.	.	NOUN
ejpam-1797	487	22	2012	2012	NUM
ejpam-1797	487	23	.	.	PUNCT
ejpam-1797	488	1	[	[	X
ejpam-1797	488	2	16	16	NUM
ejpam-1797	488	3	]	]	PUNCT
ejpam-1797	488	4	t.	t.	PROPN
ejpam-1797	488	5	k.	k.	PROPN
ejpam-1797	488	6	dutta	dutta	PROPN
ejpam-1797	488	7	,	,	PUNCT
ejpam-1797	488	8	k.	k.	PROPN
ejpam-1797	488	9	p.	p.	PROPN
ejpam-1797	488	10	shum	shum	PROPN
ejpam-1797	488	11	and	and	CCONJ
ejpam-1797	488	12	s.	s.	PROPN
ejpam-1797	488	13	mandal	mandal	PROPN
ejpam-1797	488	14	.	.	PUNCT
ejpam-1797	489	1	singular	singular	PROPN
ejpam-1797	489	2	ideals	ideal	NOUN
ejpam-1797	489	3	of	of	ADP
ejpam-1797	489	4	ternary	ternary	ADJ
ejpam-1797	489	5	semirings	semiring	NOUN
ejpam-1797	489	6	,	,	PUNCT
ejpam-1797	489	7	european	european	PROPN
ejpam-1797	489	8	journal	journal	PROPN
ejpam-1797	489	9	of	of	ADP
ejpam-1797	489	10	pure	pure	ADJ
ejpam-1797	489	11	and	and	CCONJ
ejpam-1797	489	12	applied	applied	ADJ
ejpam-1797	489	13	mathematics	mathematic	NOUN
ejpam-1797	489	14	,	,	PUNCT
ejpam-1797	489	15	vol	vol	NOUN
ejpam-1797	489	16	.	.	PROPN
ejpam-1797	490	1	5	5	NUM
ejpam-1797	490	2	,	,	PUNCT
ejpam-1797	490	3	no	no	INTJ
ejpam-1797	490	4	.	.	NOUN
ejpam-1797	490	5	2	2	NUM
ejpam-1797	490	6	,	,	PUNCT
ejpam-1797	490	7	116	116	NUM
ejpam-1797	490	8	-	-	SYM
ejpam-1797	490	9	128	128	NUM
ejpam-1797	490	10	.	.	PUNCT
ejpam-1797	491	1	2012	2012	NUM
ejpam-1797	491	2	.	.	PUNCT
ejpam-1797	492	1	[	[	X
ejpam-1797	492	2	17	17	NUM
ejpam-1797	492	3	]	]	X
ejpam-1797	492	4	n.	n.	PROPN
ejpam-1797	492	5	j.	j.	PROPN
ejpam-1797	492	6	divinsky	divinsky	PROPN
ejpam-1797	492	7	.	.	PUNCT
ejpam-1797	493	1	rings	ring	NOUN
ejpam-1797	493	2	and	and	CCONJ
ejpam-1797	493	3	radicals	radical	NOUN
ejpam-1797	493	4	,	,	PUNCT
ejpam-1797	493	5	allen	allen	NOUN
ejpam-1797	493	6	and	and	CCONJ
ejpam-1797	493	7	unwin	unwin	PROPN
ejpam-1797	493	8	,	,	PUNCT
ejpam-1797	493	9	1965	1965	NUM
ejpam-1797	493	10	.	.	PUNCT
ejpam-1797	494	1	[	[	X
ejpam-1797	494	2	18	18	NUM
ejpam-1797	494	3	]	]	PUNCT
ejpam-1797	494	4	k.	k.	PROPN
ejpam-1797	494	5	r.	r.	PROPN
ejpam-1797	494	6	goodearl	goodearl	PROPN
ejpam-1797	494	7	.	.	PROPN
ejpam-1797	495	1	ring	ring	PROPN
ejpam-1797	495	2	theory	theory	PROPN
ejpam-1797	495	3	,	,	PUNCT
ejpam-1797	495	4	nonsingular	nonsingular	ADJ
ejpam-1797	495	5	rings	ring	NOUN
ejpam-1797	495	6	and	and	CCONJ
ejpam-1797	495	7	modules	module	NOUN
ejpam-1797	495	8	,	,	PUNCT
ejpam-1797	495	9	(	(	PUNCT
ejpam-1797	495	10	marcel	marcel	PROPN
ejpam-1797	495	11	dekker	dekker	PROPN
ejpam-1797	495	12	,	,	PUNCT
ejpam-1797	495	13	new	new	PROPN
ejpam-1797	495	14	york	york	PROPN
ejpam-1797	495	15	)	)	PUNCT
ejpam-1797	495	16	.	.	PUNCT
ejpam-1797	496	1	1976	1976	NUM
ejpam-1797	496	2	.	.	PUNCT
ejpam-1797	497	1	[	[	X
ejpam-1797	497	2	19	19	NUM
ejpam-1797	497	3	]	]	X
ejpam-1797	497	4	d.	d.	PROPN
ejpam-1797	497	5	handelman	handelman	PROPN
ejpam-1797	497	6	and	and	CCONJ
ejpam-1797	497	7	j.	j.	PROPN
ejpam-1797	497	8	lawrence	lawrence	PROPN
ejpam-1797	497	9	.	.	PUNCT
ejpam-1797	498	1	strongly	strongly	ADV
ejpam-1797	498	2	prime	prime	ADJ
ejpam-1797	498	3	rings	ring	NOUN
ejpam-1797	498	4	,	,	PUNCT
ejpam-1797	498	5	trans	trans	PROPN
ejpam-1797	498	6	.	.	PROPN
ejpam-1797	499	1	amer	amer	PROPN
ejpam-1797	499	2	.	.	PUNCT
ejpam-1797	499	3	math	math	PROPN
ejpam-1797	499	4	.	.	PUNCT
ejpam-1797	500	1	soc	soc	PROPN
ejpam-1797	500	2	.	.	PUNCT
ejpam-1797	501	1	211	211	NUM
ejpam-1797	501	2	,	,	PUNCT
ejpam-1797	501	3	209	209	NUM
ejpam-1797	501	4	223	223	NUM
ejpam-1797	501	5	.	.	PUNCT
ejpam-1797	502	1	1975	1975	NUM
ejpam-1797	502	2	.	.	PUNCT
ejpam-1797	503	1	[	[	X
ejpam-1797	503	2	20	20	NUM
ejpam-1797	503	3	]	]	PUNCT
ejpam-1797	503	4	s.	s.	PROPN
ejpam-1797	503	5	kar	kar	PROPN
ejpam-1797	503	6	.	.	PUNCT
ejpam-1797	504	1	on	on	ADP
ejpam-1797	504	2	quasi	quasi	NOUN
ejpam-1797	504	3	-	-	NOUN
ejpam-1797	504	4	ideals	ideal	NOUN
ejpam-1797	504	5	and	and	CCONJ
ejpam-1797	504	6	bi	bi	NOUN
ejpam-1797	504	7	-	-	NOUN
ejpam-1797	504	8	ideals	ideal	NOUN
ejpam-1797	504	9	of	of	ADP
ejpam-1797	504	10	ternary	ternary	ADJ
ejpam-1797	504	11	semirings	semiring	NOUN
ejpam-1797	504	12	,	,	PUNCT
ejpam-1797	504	13	international	international	ADJ
ejpam-1797	504	14	journal	journal	NOUN
ejpam-1797	504	15	of	of	ADP
ejpam-1797	504	16	mathematics	mathematics	PROPN
ejpam-1797	504	17	and	and	CCONJ
ejpam-1797	504	18	mathematical	mathematical	ADJ
ejpam-1797	504	19	sciences	science	NOUN
ejpam-1797	504	20	;	;	PUNCT
ejpam-1797	504	21	vol	vol	NOUN
ejpam-1797	504	22	.	.	NOUN
ejpam-1797	504	23	2005	2005	NUM
ejpam-1797	504	24	,	,	PUNCT
ejpam-1797	504	25	issue	issue	NOUN
ejpam-1797	504	26	18	18	NUM
ejpam-1797	504	27	,	,	PUNCT
ejpam-1797	504	28	3015	3015	NUM
ejpam-1797	504	29	3023	3023	NUM
ejpam-1797	504	30	.	.	PUNCT
ejpam-1797	505	1	2005	2005	NUM
ejpam-1797	505	2	.	.	PUNCT
ejpam-1797	506	1	[	[	X
ejpam-1797	506	2	21	21	NUM
ejpam-1797	506	3	]	]	PUNCT
ejpam-1797	506	4	s.	s.	PROPN
ejpam-1797	506	5	kar	kar	PROPN
ejpam-1797	506	6	.	.	PUNCT
ejpam-1797	507	1	on	on	ADP
ejpam-1797	507	2	structure	structure	NOUN
ejpam-1797	507	3	space	space	NOUN
ejpam-1797	507	4	of	of	ADP
ejpam-1797	507	5	ternary	ternary	ADJ
ejpam-1797	507	6	semirings	semiring	NOUN
ejpam-1797	507	7	,	,	PUNCT
ejpam-1797	507	8	southeast	southeast	ADJ
ejpam-1797	507	9	asian	asian	PROPN
ejpam-1797	507	10	bull.math.31	bull.math.31	PROPN
ejpam-1797	507	11	,	,	PUNCT
ejpam-1797	507	12	no	no	INTJ
ejpam-1797	507	13	.	.	NOUN
ejpam-1797	507	14	3	3	NUM
ejpam-1797	507	15	,	,	PUNCT
ejpam-1797	507	16	537	537	NUM
ejpam-1797	507	17	545	545	NUM
ejpam-1797	507	18	.	.	PUNCT
ejpam-1797	507	19	2007	2007	NUM
ejpam-1797	507	20	.	.	PUNCT
ejpam-1797	508	1	[	[	X
ejpam-1797	508	2	22	22	NUM
ejpam-1797	508	3	]	]	PUNCT
ejpam-1797	508	4	s.	s.	PROPN
ejpam-1797	508	5	kar	kar	PROPN
ejpam-1797	508	6	and	and	CCONJ
ejpam-1797	508	7	s.	s.	PROPN
ejpam-1797	508	8	bhunia	bhunia	PROPN
ejpam-1797	508	9	.	.	PUNCT
ejpam-1797	509	1	a	a	DET
ejpam-1797	509	2	characterization	characterization	NOUN
ejpam-1797	509	3	of	of	ADP
ejpam-1797	509	4	ternary	ternary	ADJ
ejpam-1797	509	5	semiring	semiring	NOUN
ejpam-1797	509	6	,	,	PUNCT
ejpam-1797	509	7	international	international	ADJ
ejpam-1797	509	8	journal	journal	NOUN
ejpam-1797	509	9	of	of	ADP
ejpam-1797	509	10	algebra	algebra	PROPN
ejpam-1797	509	11	,	,	PUNCT
ejpam-1797	509	12	number	number	NOUN
ejpam-1797	509	13	theory	theory	NOUN
ejpam-1797	509	14	and	and	CCONJ
ejpam-1797	509	15	applications	application	NOUN
ejpam-1797	509	16	,	,	PUNCT
ejpam-1797	509	17	vol	vol	NOUN
ejpam-1797	509	18	.	.	PROPN
ejpam-1797	509	19	1	1	NUM
ejpam-1797	509	20	,	,	PUNCT
ejpam-1797	509	21	no	no	INTJ
ejpam-1797	509	22	.	.	NOUN
ejpam-1797	509	23	1	1	NUM
ejpam-1797	509	24	,	,	PUNCT
ejpam-1797	509	25	43	43	NUM
ejpam-1797	509	26	51	51	NUM
ejpam-1797	509	27	.	.	PUNCT
ejpam-1797	509	28	2009	2009	NUM
ejpam-1797	509	29	.	.	PUNCT
ejpam-1797	510	1	[	[	X
ejpam-1797	510	2	23	23	NUM
ejpam-1797	510	3	]	]	PUNCT
ejpam-1797	510	4	s.	s.	PROPN
ejpam-1797	510	5	kar	kar	PROPN
ejpam-1797	510	6	,	,	PUNCT
ejpam-1797	510	7	j.	j.	PROPN
ejpam-1797	510	8	sircar	sircar	PROPN
ejpam-1797	510	9	,	,	PUNCT
ejpam-1797	510	10	j.	j.	PROPN
ejpam-1797	510	11	and	and	CCONJ
ejpam-1797	510	12	s.	s.	PROPN
ejpam-1797	510	13	mandal	mandal	PROPN
ejpam-1797	510	14	.	.	PUNCT
ejpam-1797	511	1	on	on	ADP
ejpam-1797	511	2	right	right	ADV
ejpam-1797	511	3	strongly	strongly	ADV
ejpam-1797	511	4	prime	prime	ADJ
ejpam-1797	511	5	ternary	ternary	ADJ
ejpam-1797	511	6	semirings	semiring	NOUN
ejpam-1797	511	7	,	,	PUNCT
ejpam-1797	511	8	east	east	PROPN
ejpam-1797	511	9	-	-	PUNCT
ejpam-1797	511	10	west	west	PROPN
ejpam-1797	511	11	journal	journal	PROPN
ejpam-1797	511	12	of	of	ADP
ejpam-1797	511	13	mathematics	mathematics	PROPN
ejpam-1797	511	14	,	,	PUNCT
ejpam-1797	511	15	vol	vol	NOUN
ejpam-1797	511	16	.	.	PROPN
ejpam-1797	511	17	12	12	NUM
ejpam-1797	511	18	,	,	PUNCT
ejpam-1797	511	19	no	no	INTJ
ejpam-1797	511	20	.	.	NOUN
ejpam-1797	511	21	1	1	NUM
ejpam-1797	511	22	,	,	PUNCT
ejpam-1797	511	23	pp	pp	ADJ
ejpam-1797	511	24	.	.	PUNCT
ejpam-1797	511	25	59	59	NUM
ejpam-1797	511	26	68	68	NUM
ejpam-1797	511	27	.	.	PUNCT
ejpam-1797	511	28	2010	2010	NUM
ejpam-1797	511	29	.	.	PUNCT
ejpam-1797	512	1	[	[	X
ejpam-1797	512	2	24	24	NUM
ejpam-1797	512	3	]	]	PUNCT
ejpam-1797	512	4	m.	m.	NOUN
ejpam-1797	512	5	ferrero	ferrero	PROPN
ejpam-1797	512	6	and	and	CCONJ
ejpam-1797	512	7	e.	e.	PROPN
ejpam-1797	512	8	r.	r.	PROPN
ejpam-1797	512	9	puczylowski	puczylowski	PROPN
ejpam-1797	512	10	.	.	PUNCT
ejpam-1797	513	1	the	the	DET
ejpam-1797	513	2	singular	singular	PROPN
ejpam-1797	513	3	ideal	ideal	NOUN
ejpam-1797	513	4	and	and	CCONJ
ejpam-1797	513	5	radicals	radical	NOUN
ejpam-1797	513	6	,	,	PUNCT
ejpam-1797	513	7	j.	j.	PROPN
ejpam-1797	513	8	austral	austral	PROPN
ejpam-1797	513	9	.	.	PUNCT
ejpam-1797	514	1	math	math	NOUN
ejpam-1797	514	2	.	.	PUNCT
ejpam-1797	515	1	soc.(series	soc.(serie	NOUN
ejpam-1797	515	2	a	a	PRON
ejpam-1797	515	3	)	)	PUNCT
ejpam-1797	515	4	64	64	NUM
ejpam-1797	515	5	195	195	NUM
ejpam-1797	515	6	-	-	PUNCT
ejpam-1797	515	7	209	209	NUM
ejpam-1797	515	8	.	.	PUNCT
ejpam-1797	516	1	1998	1998	NUM
ejpam-1797	516	2	.	.	PUNCT
ejpam-1797	517	1	[	[	X
ejpam-1797	517	2	25	25	NUM
ejpam-1797	517	3	]	]	PUNCT
ejpam-1797	517	4	w.	w.	PROPN
ejpam-1797	517	5	j.	j.	PROPN
ejpam-1797	517	6	lister	lister	PROPN
ejpam-1797	517	7	.	.	PUNCT
ejpam-1797	518	1	ternary	ternary	ADJ
ejpam-1797	518	2	rings	ring	NOUN
ejpam-1797	518	3	,	,	PUNCT
ejpam-1797	518	4	trans	trans	PROPN
ejpam-1797	518	5	.	.	PROPN
ejpam-1797	519	1	amer	amer	PROPN
ejpam-1797	519	2	.	.	PUNCT
ejpam-1797	519	3	math	math	PROPN
ejpam-1797	519	4	.	.	PUNCT
ejpam-1797	520	1	soc	soc	PROPN
ejpam-1797	520	2	.	.	PUNCT
ejpam-1797	521	1	154	154	NUM
ejpam-1797	521	2	,	,	PUNCT
ejpam-1797	521	3	37	37	NUM
ejpam-1797	521	4	-	-	SYM
ejpam-1797	521	5	55	55	NUM
ejpam-1797	521	6	.	.	PUNCT
ejpam-1797	522	1	1971	1971	NUM
ejpam-1797	522	2	.	.	PUNCT
ejpam-1797	523	1	[	[	X
ejpam-1797	523	2	26	26	NUM
ejpam-1797	523	3	]	]	X
ejpam-1797	523	4	d.	d.	PROPN
ejpam-1797	523	5	m.	m.	PROPN
ejpam-1797	523	6	olson	olson	PROPN
ejpam-1797	523	7	,	,	PUNCT
ejpam-1797	523	8	g.	g.	PROPN
ejpam-1797	523	9	a.	a.	PROPN
ejpam-1797	523	10	p.	p.	PROPN
ejpam-1797	523	11	heyman	heyman	PROPN
ejpam-1797	523	12	and	and	CCONJ
ejpam-1797	523	13	h.	h.	PROPN
ejpam-1797	523	14	j.	j.	PROPN
ejpam-1797	523	15	l.	l.	PROPN
ejpam-1797	523	16	roux	roux	PROPN
ejpam-1797	523	17	.	.	PUNCT
ejpam-1797	524	1	weakly	weakly	ADJ
ejpam-1797	524	2	special	special	ADJ
ejpam-1797	524	3	classes	class	NOUN
ejpam-1797	524	4	of	of	ADP
ejpam-1797	524	5	hemirings	hemiring	NOUN
ejpam-1797	524	6	,	,	PUNCT
ejpam-1797	524	7	quaestiones	quaestione	NOUN
ejpam-1797	524	8	mathematicae	mathematicae	PROPN
ejpam-1797	524	9	,	,	PUNCT
ejpam-1797	524	10	15	15	NUM
ejpam-1797	524	11	(	(	PUNCT
ejpam-1797	524	12	2	2	NUM
ejpam-1797	524	13	)	)	PUNCT
ejpam-1797	524	14	119	119	NUM
ejpam-1797	524	15	-	-	SYM
ejpam-1797	524	16	126	126	NUM
ejpam-1797	524	17	.	.	PUNCT
ejpam-1797	524	18	1992	1992	NUM
ejpam-1797	524	19	.	.	PUNCT
ejpam-1797	525	1	[	[	X
ejpam-1797	525	2	27	27	NUM
ejpam-1797	525	3	]	]	X
ejpam-1797	525	4	d.	d.	PROPN
ejpam-1797	525	5	m.	m.	PROPN
ejpam-1797	525	6	olson	olson	PROPN
ejpam-1797	525	7	and	and	CCONJ
ejpam-1797	525	8	a.	a.	PROPN
ejpam-1797	525	9	c.	c.	PROPN
ejpam-1797	525	10	nance	nance	PROPN
ejpam-1797	525	11	.	.	PUNCT
ejpam-1797	526	1	a	a	DET
ejpam-1797	526	2	note	note	NOUN
ejpam-1797	526	3	on	on	ADP
ejpam-1797	526	4	radicals	radical	NOUN
ejpam-1797	526	5	for	for	ADP
ejpam-1797	526	6	hemirings	hemiring	NOUN
ejpam-1797	526	7	,	,	PUNCT
ejpam-1797	526	8	quaestiones	quaestione	NOUN
ejpam-1797	526	9	mathematicae	mathematicae	VERB
ejpam-1797	526	10	12	12	NUM
ejpam-1797	526	11	(	(	PUNCT
ejpam-1797	526	12	3	3	NUM
ejpam-1797	526	13	)	)	PUNCT
ejpam-1797	526	14	307	307	NUM
ejpam-1797	526	15	-	-	SYM
ejpam-1797	526	16	314	314	NUM
ejpam-1797	526	17	.	.	PUNCT
ejpam-1797	526	18	1989	1989	NUM
ejpam-1797	526	19	.	.	PUNCT
ejpam-1797	527	1	[	[	X
ejpam-1797	527	2	28	28	NUM
ejpam-1797	527	3	]	]	X
ejpam-1797	527	4	d.	d.	PROPN
ejpam-1797	527	5	m.	m.	PROPN
ejpam-1797	527	6	olson	olson	PROPN
ejpam-1797	527	7	and	and	CCONJ
ejpam-1797	527	8	t.	t.	PROPN
ejpam-1797	527	9	l.	l.	PROPN
ejpam-1797	527	10	jenkins	jenkins	PROPN
ejpam-1797	527	11	.	.	PUNCT
ejpam-1797	528	1	radical	radical	ADJ
ejpam-1797	528	2	theory	theory	NOUN
ejpam-1797	528	3	for	for	ADP
ejpam-1797	528	4	hemirings	hemiring	NOUN
ejpam-1797	528	5	,	,	PUNCT
ejpam-1797	528	6	j.	j.	PROPN
ejpam-1797	528	7	nat	nat	PROPN
ejpam-1797	528	8	.	.	PUNCT
ejpam-1797	529	1	sci	sci	PROPN
ejpam-1797	529	2	.	.	PROPN
ejpam-1797	529	3	and	and	CCONJ
ejpam-1797	529	4	math	math	NOUN
ejpam-1797	529	5	.	.	PUNCT
ejpam-1797	530	1	23	23	NUM
ejpam-1797	530	2	(	(	PUNCT
ejpam-1797	530	3	1	1	NUM
ejpam-1797	530	4	)	)	PUNCT
ejpam-1797	530	5	23	23	NUM
ejpam-1797	530	6	-	-	SYM
ejpam-1797	530	7	32	32	NUM
ejpam-1797	530	8	.	.	PUNCT
ejpam-1797	531	1	1983	1983	NUM
ejpam-1797	531	2	.	.	PUNCT
ejpam-1797	532	1	[	[	X
ejpam-1797	532	2	29	29	NUM
ejpam-1797	532	3	]	]	PUNCT
ejpam-1797	532	4	s.	s.	PROPN
ejpam-1797	532	5	k.	k.	PROPN
ejpam-1797	532	6	sardar	sardar	PROPN
ejpam-1797	532	7	,	,	PUNCT
ejpam-1797	532	8	s.	s.	PROPN
ejpam-1797	532	9	b.	b.	PROPN
ejpam-1797	532	10	chandra	chandra	PROPN
ejpam-1797	532	11	and	and	CCONJ
ejpam-1797	532	12	k.	k.	PROPN
ejpam-1797	532	13	p.	p.	PROPN
ejpam-1797	532	14	shum	shum	PROPN
ejpam-1797	532	15	.	.	PUNCT
ejpam-1797	533	1	h	h	NOUN
ejpam-1797	533	2	-	-	PUNCT
ejpam-1797	533	3	prime	prime	ADJ
ejpam-1797	533	4	and	and	CCONJ
ejpam-1797	533	5	h	h	NOUN
ejpam-1797	533	6	-	-	PUNCT
ejpam-1797	533	7	semiprime	semiprime	NOUN
ejpam-1797	533	8	ideals	ideal	NOUN
ejpam-1797	533	9	in	in	ADP
ejpam-1797	533	10	semirings	semiring	NOUN
ejpam-1797	533	11	and	and	CCONJ
ejpam-1797	533	12	γ	γ	NOUN
ejpam-1797	533	13	-	-	NOUN
ejpam-1797	533	14	semirings	semiring	NOUN
ejpam-1797	533	15	,	,	PUNCT
ejpam-1797	533	16	int	int	NOUN
ejpam-1797	533	17	.	.	PUNCT
ejpam-1797	534	1	j.	j.	PROPN
ejpam-1797	534	2	algebra	algebra	PROPN
ejpam-1797	534	3	4	4	NUM
ejpam-1797	534	4	,	,	PUNCT
ejpam-1797	534	5	no.5	no.5	NOUN
ejpam-1797	534	6	-	-	PUNCT
ejpam-1797	534	7	8	8	NUM
ejpam-1797	534	8	,	,	PUNCT
ejpam-1797	534	9	209	209	NUM
ejpam-1797	534	10	-	-	SYM
ejpam-1797	534	11	220	220	NUM
ejpam-1797	534	12	.	.	PUNCT
ejpam-1797	534	13	2010	2010	NUM
ejpam-1797	534	14	.	.	PUNCT