id	sid	tid	token	lemma	pos
ejpam-1598	1	1	3_dutta.dvi	3_dutta.dvi	NUM
ejpam-1598	1	2	european	european	ADJ
ejpam-1598	1	3	journal	journal	NOUN
ejpam-1598	1	4	of	of	ADP
ejpam-1598	1	5	pure	pure	ADJ
ejpam-1598	1	6	and	and	CCONJ
ejpam-1598	1	7	applied	apply	VERB
ejpam-1598	1	8	mathematics	mathematic	NOUN
ejpam-1598	1	9	vol	vol	NOUN
ejpam-1598	1	10	.	.	PROPN
ejpam-1598	1	11	5	5	NUM
ejpam-1598	1	12	,	,	PUNCT
ejpam-1598	1	13	no	no	INTJ
ejpam-1598	1	14	.	.	NOUN
ejpam-1598	1	15	2	2	NUM
ejpam-1598	1	16	,	,	PUNCT
ejpam-1598	1	17	2012	2012	NUM
ejpam-1598	1	18	,	,	PUNCT
ejpam-1598	1	19	116	116	NUM
ejpam-1598	1	20	-	-	SYM
ejpam-1598	1	21	128	128	NUM
ejpam-1598	1	22	issn	issn	PROPN
ejpam-1598	1	23	1307	1307	NUM
ejpam-1598	1	24	-	-	SYM
ejpam-1598	1	25	5543	5543	NUM
ejpam-1598	1	26	–	–	PUNCT
ejpam-1598	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-1598	1	28	singular	singular	ADJ
ejpam-1598	1	29	ideals	ideal	NOUN
ejpam-1598	1	30	of	of	ADP
ejpam-1598	1	31	ternary	ternary	ADJ
ejpam-1598	1	32	semirings	semiring	NOUN
ejpam-1598	1	33	tapan	tapan	VERB
ejpam-1598	1	34	k.	k.	PROPN
ejpam-1598	1	35	dutta1	dutta1	PROPN
ejpam-1598	1	36	,	,	PUNCT
ejpam-1598	1	37	kar	kar	PROPN
ejpam-1598	1	38	ping	ping	PROPN
ejpam-1598	1	39	shum	shum	PROPN
ejpam-1598	1	40	2,∗	2,∗	PROPN
ejpam-1598	1	41	,	,	PUNCT
ejpam-1598	1	42	shobhan	shobhan	ADV
ejpam-1598	1	43	mandal3	mandal3	NOUN
ejpam-1598	2	1	1	1	NUM
ejpam-1598	2	2	department	department	NOUN
ejpam-1598	2	3	of	of	ADP
ejpam-1598	2	4	pure	pure	ADJ
ejpam-1598	2	5	mathematics	mathematic	NOUN
ejpam-1598	2	6	,	,	PUNCT
ejpam-1598	2	7	university	university	NOUN
ejpam-1598	2	8	of	of	ADP
ejpam-1598	2	9	calcutta	calcutta	PROPN
ejpam-1598	2	10	,	,	PUNCT
ejpam-1598	2	11	35	35	NUM
ejpam-1598	2	12	,	,	PUNCT
ejpam-1598	2	13	ballygunge	ballygunge	VERB
ejpam-1598	2	14	circular	circular	ADJ
ejpam-1598	2	15	road	road	NOUN
ejpam-1598	2	16	,	,	PUNCT
ejpam-1598	2	17	kolkata710019,india	kolkata710019,india	PROPN
ejpam-1598	2	18	2	2	NUM
ejpam-1598	2	19	institute	institute	PROPN
ejpam-1598	2	20	of	of	ADP
ejpam-1598	2	21	mathematics	mathematics	PROPN
ejpam-1598	2	22	,	,	PUNCT
ejpam-1598	2	23	yunnan	yunnan	PROPN
ejpam-1598	2	24	university	university	PROPN
ejpam-1598	2	25	,	,	PUNCT
ejpam-1598	2	26	kunming	kunming	NOUN
ejpam-1598	2	27	,	,	PUNCT
ejpam-1598	2	28	650091	650091	NUM
ejpam-1598	2	29	,	,	PUNCT
ejpam-1598	2	30	people	people	NOUN
ejpam-1598	2	31	republic	republic	NOUN
ejpam-1598	2	32	of	of	ADP
ejpam-1598	2	33	china	china	PROPN
ejpam-1598	2	34	3	3	PROPN
ejpam-1598	2	35	department	department	NOUN
ejpam-1598	2	36	of	of	ADP
ejpam-1598	2	37	pure	pure	ADJ
ejpam-1598	2	38	mathematics	mathematic	NOUN
ejpam-1598	2	39	,	,	PUNCT
ejpam-1598	2	40	university	university	NOUN
ejpam-1598	2	41	of	of	ADP
ejpam-1598	2	42	calcutta	calcutta	PROPN
ejpam-1598	2	43	,	,	PUNCT
ejpam-1598	2	44	35	35	NUM
ejpam-1598	2	45	,	,	PUNCT
ejpam-1598	2	46	ballygunge	ballygunge	VERB
ejpam-1598	2	47	circular	circular	ADJ
ejpam-1598	2	48	road	road	NOUN
ejpam-1598	2	49	,	,	PUNCT
ejpam-1598	2	50	kolkata710019	kolkata710019	PROPN
ejpam-1598	2	51	,	,	PUNCT
ejpam-1598	2	52	india	india	PROPN
ejpam-1598	2	53	abstract	abstract	NOUN
ejpam-1598	2	54	.	.	PUNCT
ejpam-1598	3	1	the	the	DET
ejpam-1598	3	2	notion	notion	NOUN
ejpam-1598	3	3	of	of	ADP
ejpam-1598	3	4	singular	singular	PROPN
ejpam-1598	3	5	ideal	ideal	NOUN
ejpam-1598	3	6	in	in	ADP
ejpam-1598	3	7	a	a	DET
ejpam-1598	3	8	ternary	ternary	ADJ
ejpam-1598	3	9	semiring	semiring	NOUN
ejpam-1598	3	10	is	be	AUX
ejpam-1598	3	11	introduced	introduce	VERB
ejpam-1598	3	12	.	.	PUNCT
ejpam-1598	4	1	the	the	DET
ejpam-1598	4	2	notions	notion	NOUN
ejpam-1598	4	3	of	of	ADP
ejpam-1598	4	4	singular	singular	ADJ
ejpam-1598	4	5	ternary	ternary	ADJ
ejpam-1598	4	6	semirings	semiring	NOUN
ejpam-1598	4	7	and	and	CCONJ
ejpam-1598	4	8	nonsingular	nonsingular	ADJ
ejpam-1598	4	9	ternary	ternary	ADJ
ejpam-1598	4	10	semirings	semiring	NOUN
ejpam-1598	4	11	are	be	AUX
ejpam-1598	4	12	also	also	ADV
ejpam-1598	4	13	defined	define	VERB
ejpam-1598	4	14	.	.	PUNCT
ejpam-1598	5	1	some	some	DET
ejpam-1598	5	2	properties	property	NOUN
ejpam-1598	5	3	of	of	ADP
ejpam-1598	5	4	singular	singular	ADJ
ejpam-1598	5	5	ideals	ideal	NOUN
ejpam-1598	5	6	in	in	ADP
ejpam-1598	5	7	ternary	ternary	ADJ
ejpam-1598	5	8	semirings	semiring	NOUN
ejpam-1598	5	9	are	be	AUX
ejpam-1598	5	10	given	give	VERB
ejpam-1598	5	11	.	.	PUNCT
ejpam-1598	6	1	our	our	PRON
ejpam-1598	6	2	results	result	NOUN
ejpam-1598	6	3	obtained	obtain	VERB
ejpam-1598	6	4	can	can	AUX
ejpam-1598	6	5	be	be	AUX
ejpam-1598	6	6	used	use	VERB
ejpam-1598	6	7	to	to	PART
ejpam-1598	6	8	study	study	VERB
ejpam-1598	6	9	some	some	DET
ejpam-1598	6	10	radical	radical	ADJ
ejpam-1598	6	11	classes	class	NOUN
ejpam-1598	6	12	related	relate	VERB
ejpam-1598	6	13	to	to	ADP
ejpam-1598	6	14	singular	singular	ADJ
ejpam-1598	6	15	ideals	ideal	NOUN
ejpam-1598	6	16	.	.	PUNCT
ejpam-1598	7	1	2010	2010	NUM
ejpam-1598	7	2	mathematics	mathematic	NOUN
ejpam-1598	7	3	subject	subject	NOUN
ejpam-1598	7	4	classifications	classification	NOUN
ejpam-1598	7	5	:	:	PUNCT
ejpam-1598	7	6	16y60	16y60	NUM
ejpam-1598	7	7	,	,	PUNCT
ejpam-1598	7	8	16y99	16y99	NUM
ejpam-1598	7	9	.	.	PUNCT
ejpam-1598	8	1	key	key	ADJ
ejpam-1598	8	2	words	word	NOUN
ejpam-1598	8	3	and	and	CCONJ
ejpam-1598	8	4	phrases	phrase	NOUN
ejpam-1598	8	5	:	:	PUNCT
ejpam-1598	8	6	singular	singular	PROPN
ejpam-1598	8	7	ideal	ideal	NOUN
ejpam-1598	8	8	,	,	PUNCT
ejpam-1598	8	9	singular	singular	ADJ
ejpam-1598	8	10	ternary	ternary	ADJ
ejpam-1598	8	11	semiring	semiring	NOUN
ejpam-1598	8	12	and	and	CCONJ
ejpam-1598	8	13	non	non	ADJ
ejpam-1598	8	14	-	-	ADJ
ejpam-1598	8	15	singular	singular	ADJ
ejpam-1598	8	16	ternary	ternary	ADJ
ejpam-1598	8	17	semirings	semiring	NOUN
ejpam-1598	8	18	.	.	PUNCT
ejpam-1598	9	1	1	1	X
ejpam-1598	9	2	.	.	X
ejpam-1598	9	3	introduction	introduction	NOUN
ejpam-1598	9	4	it	it	PRON
ejpam-1598	9	5	was	be	AUX
ejpam-1598	9	6	remarked	remark	VERB
ejpam-1598	9	7	by	by	ADP
ejpam-1598	9	8	m.	m.	NOUN
ejpam-1598	9	9	ferrero	ferrero	PROPN
ejpam-1598	9	10	and	and	CCONJ
ejpam-1598	9	11	e.	e.	PROPN
ejpam-1598	9	12	r.	r.	PROPN
ejpam-1598	9	13	puczylowski	puczylowski	VERB
ejpam-1598	9	14	in	in	ADP
ejpam-1598	9	15	[	[	X
ejpam-1598	9	16	10	10	NUM
ejpam-1598	9	17	]	]	PUNCT
ejpam-1598	9	18	“	"	PUNCT
ejpam-1598	9	19	studying	study	VERB
ejpam-1598	9	20	properties	property	NOUN
ejpam-1598	9	21	of	of	ADP
ejpam-1598	9	22	rings	ring	NOUN
ejpam-1598	9	23	one	one	PRON
ejpam-1598	9	24	can	can	AUX
ejpam-1598	9	25	usually	usually	ADV
ejpam-1598	9	26	say	say	VERB
ejpam-1598	9	27	more	more	ADJ
ejpam-1598	9	28	assuming	assume	VERB
ejpam-1598	9	29	that	that	SCONJ
ejpam-1598	9	30	the	the	DET
ejpam-1598	9	31	considered	consider	VERB
ejpam-1598	9	32	rings	ring	NOUN
ejpam-1598	9	33	are	be	AUX
ejpam-1598	9	34	either	either	CCONJ
ejpam-1598	9	35	singular	singular	ADJ
ejpam-1598	9	36	or	or	CCONJ
ejpam-1598	9	37	nonsingular	nonsingular	ADJ
ejpam-1598	9	38	.	.	PUNCT
ejpam-1598	9	39	”	"	PUNCT
ejpam-1598	10	1	the	the	DET
ejpam-1598	10	2	same	same	ADJ
ejpam-1598	10	3	remark	remark	NOUN
ejpam-1598	10	4	is	be	AUX
ejpam-1598	10	5	equally	equally	ADV
ejpam-1598	10	6	true	true	ADJ
ejpam-1598	10	7	in	in	ADP
ejpam-1598	10	8	the	the	DET
ejpam-1598	10	9	case	case	NOUN
ejpam-1598	10	10	of	of	ADP
ejpam-1598	10	11	a	a	DET
ejpam-1598	10	12	ternary	ternary	ADJ
ejpam-1598	10	13	semiring	semiring	NOUN
ejpam-1598	10	14	which	which	PRON
ejpam-1598	10	15	was	be	AUX
ejpam-1598	10	16	first	first	ADV
ejpam-1598	10	17	introduced	introduce	VERB
ejpam-1598	10	18	by	by	ADP
ejpam-1598	10	19	t.	t.	PROPN
ejpam-1598	10	20	k.	k.	PROPN
ejpam-1598	10	21	dutta	dutta	PROPN
ejpam-1598	10	22	and	and	CCONJ
ejpam-1598	10	23	s.	s.	PROPN
ejpam-1598	10	24	kar	kar	PROPN
ejpam-1598	10	25	in	in	ADP
ejpam-1598	10	26	[	[	X
ejpam-1598	10	27	1	1	NUM
ejpam-1598	10	28	]	]	PUNCT
ejpam-1598	10	29	.	.	PUNCT
ejpam-1598	11	1	the	the	DET
ejpam-1598	11	2	notion	notion	NOUN
ejpam-1598	11	3	of	of	ADP
ejpam-1598	11	4	ternary	ternary	ADJ
ejpam-1598	11	5	semiring	semiring	NOUN
ejpam-1598	11	6	was	be	AUX
ejpam-1598	11	7	introduced	introduce	VERB
ejpam-1598	11	8	in	in	ADP
ejpam-1598	11	9	2003	2003	NUM
ejpam-1598	11	10	.	.	PUNCT
ejpam-1598	12	1	the	the	DET
ejpam-1598	12	2	introduction	introduction	NOUN
ejpam-1598	12	3	of	of	ADP
ejpam-1598	12	4	ternary	ternary	ADJ
ejpam-1598	12	5	algebra	algebra	NOUN
ejpam-1598	12	6	dated	date	VERB
ejpam-1598	12	7	back	back	ADV
ejpam-1598	12	8	to	to	ADP
ejpam-1598	12	9	1932	1932	NUM
ejpam-1598	12	10	when	when	SCONJ
ejpam-1598	12	11	lehmer	lehmer	NOUN
ejpam-1598	12	12	[	[	X
ejpam-1598	12	13	13	13	NUM
ejpam-1598	12	14	]	]	PUNCT
ejpam-1598	12	15	studied	study	VERB
ejpam-1598	12	16	certain	certain	ADJ
ejpam-1598	12	17	ternary	ternary	ADJ
ejpam-1598	12	18	system	system	NOUN
ejpam-1598	12	19	called	call	VERB
ejpam-1598	12	20	triplexes	triplexe	NOUN
ejpam-1598	12	21	which	which	PRON
ejpam-1598	12	22	turn	turn	VERB
ejpam-1598	12	23	out	out	ADP
ejpam-1598	12	24	to	to	PART
ejpam-1598	12	25	be	be	AUX
ejpam-1598	12	26	a	a	DET
ejpam-1598	12	27	generalization	generalization	NOUN
ejpam-1598	12	28	of	of	ADP
ejpam-1598	12	29	abelian	abelian	ADJ
ejpam-1598	12	30	groups	group	NOUN
ejpam-1598	12	31	.	.	PUNCT
ejpam-1598	13	1	later	later	ADV
ejpam-1598	13	2	on	on	ADV
ejpam-1598	13	3	,	,	PUNCT
ejpam-1598	13	4	banach	banach	NOUN
ejpam-1598	13	5	[	[	X
ejpam-1598	13	6	cf	cf	NOUN
ejpam-1598	13	7	.	.	PUNCT
ejpam-1598	14	1	los	los	PROPN
ejpam-1598	14	2	15	15	PROPN
ejpam-1598	14	3	]	]	PUNCT
ejpam-1598	14	4	also	also	ADV
ejpam-1598	14	5	studied	study	VERB
ejpam-1598	14	6	such	such	ADJ
ejpam-1598	14	7	algebraic	algebraic	ADJ
ejpam-1598	14	8	structure	structure	NOUN
ejpam-1598	14	9	and	and	CCONJ
ejpam-1598	14	10	gave	give	VERB
ejpam-1598	14	11	some	some	DET
ejpam-1598	14	12	examples	example	NOUN
ejpam-1598	14	13	of	of	ADP
ejpam-1598	14	14	a	a	DET
ejpam-1598	14	15	ternary	ternary	ADJ
ejpam-1598	14	16	semigroup	semigroup	NOUN
ejpam-1598	14	17	which	which	PRON
ejpam-1598	14	18	does	do	AUX
ejpam-1598	14	19	not	not	PART
ejpam-1598	14	20	reduce	reduce	VERB
ejpam-1598	14	21	to	to	ADP
ejpam-1598	14	22	a	a	DET
ejpam-1598	14	23	semigroup	semigroup	NOUN
ejpam-1598	14	24	.	.	PUNCT
ejpam-1598	15	1	in	in	ADP
ejpam-1598	15	2	addition	addition	NOUN
ejpam-1598	15	3	,	,	PUNCT
ejpam-1598	15	4	w.	w.	PROPN
ejpam-1598	15	5	g.	g.	PROPN
ejpam-1598	15	6	lister	lister	PROPN
ejpam-1598	16	1	[	[	X
ejpam-1598	16	2	14	14	NUM
ejpam-1598	16	3	]	]	PUNCT
ejpam-1598	16	4	introduced	introduce	VERB
ejpam-1598	16	5	the	the	DET
ejpam-1598	16	6	notion	notion	NOUN
ejpam-1598	16	7	of	of	ADP
ejpam-1598	16	8	ternary	ternary	ADJ
ejpam-1598	16	9	ring	ring	NOUN
ejpam-1598	16	10	.	.	PUNCT
ejpam-1598	17	1	abstractly	abstractly	ADV
ejpam-1598	17	2	,	,	PUNCT
ejpam-1598	17	3	a	a	DET
ejpam-1598	17	4	ternary	ternary	ADJ
ejpam-1598	17	5	ring	ring	NOUN
ejpam-1598	17	6	t	t	PROPN
ejpam-1598	17	7	is	be	AUX
ejpam-1598	17	8	an	an	DET
ejpam-1598	17	9	abelian	abelian	ADJ
ejpam-1598	17	10	group	group	NOUN
ejpam-1598	17	11	in	in	ADP
ejpam-1598	17	12	which	which	PRON
ejpam-1598	17	13	a	a	DET
ejpam-1598	17	14	ternary	ternary	ADJ
ejpam-1598	17	15	product	product	NOUN
ejpam-1598	17	16	tuv	tuv	PROPN
ejpam-1598	17	17	is	be	AUX
ejpam-1598	17	18	given	give	VERB
ejpam-1598	17	19	which	which	PRON
ejpam-1598	17	20	is	be	AUX
ejpam-1598	17	21	right	right	ADJ
ejpam-1598	17	22	,	,	PUNCT
ejpam-1598	17	23	center	center	NOUN
ejpam-1598	17	24	and	and	CCONJ
ejpam-1598	17	25	left	leave	VERB
ejpam-1598	17	26	distributive	distributive	ADJ
ejpam-1598	17	27	and	and	CCONJ
ejpam-1598	17	28	which	which	PRON
ejpam-1598	17	29	satisfies	satisfy	VERB
ejpam-1598	17	30	(	(	PUNCT
ejpam-1598	17	31	tuv)x	tuv)x	PROPN
ejpam-1598	17	32	y	y	NOUN
ejpam-1598	17	33	=	=	SYM
ejpam-1598	17	34	t(uv	t(uv	PROPN
ejpam-1598	17	35	x)y	x)y	PUNCT
ejpam-1598	18	1	=	=	PUNCT
ejpam-1598	18	2	tu(v	tu(v	X
ejpam-1598	18	3	x	x	SYM
ejpam-1598	18	4	y	y	NOUN
ejpam-1598	18	5	)	)	PUNCT
ejpam-1598	18	6	.	.	PUNCT
ejpam-1598	19	1	in	in	ADP
ejpam-1598	19	2	this	this	DET
ejpam-1598	19	3	paper	paper	NOUN
ejpam-1598	19	4	,	,	PUNCT
ejpam-1598	19	5	our	our	PRON
ejpam-1598	19	6	ternary	ternary	ADJ
ejpam-1598	19	7	semiring	semiring	NOUN
ejpam-1598	19	8	is	be	AUX
ejpam-1598	19	9	a	a	DET
ejpam-1598	19	10	generalized	generalized	ADJ
ejpam-1598	19	11	ternary	ternary	ADJ
ejpam-1598	19	12	ring	ring	NOUN
ejpam-1598	19	13	investigated	investigate	VERB
ejpam-1598	19	14	by	by	ADP
ejpam-1598	19	15	w.	w.	PROPN
ejpam-1598	19	16	g.	g.	PROPN
ejpam-1598	19	17	lister	lister	PROPN
ejpam-1598	19	18	in	in	ADP
ejpam-1598	19	19	1971	1971	NUM
ejpam-1598	19	20	.	.	PUNCT
ejpam-1598	20	1	though	though	SCONJ
ejpam-1598	20	2	the	the	DET
ejpam-1598	20	3	notion	notion	NOUN
ejpam-1598	20	4	of	of	ADP
ejpam-1598	20	5	ternary	ternary	ADJ
ejpam-1598	20	6	semiring	semiring	NOUN
ejpam-1598	20	7	generalizes	generalize	VERB
ejpam-1598	20	8	the	the	DET
ejpam-1598	20	9	notion	notion	NOUN
ejpam-1598	20	10	of	of	ADP
ejpam-1598	20	11	semiring	semiring	NOUN
ejpam-1598	20	12	but	but	CCONJ
ejpam-1598	20	13	it	it	PRON
ejpam-1598	20	14	is	be	AUX
ejpam-1598	20	15	not	not	PART
ejpam-1598	20	16	merely	merely	ADV
ejpam-1598	20	17	a	a	DET
ejpam-1598	20	18	generalization	generalization	NOUN
ejpam-1598	20	19	of	of	ADP
ejpam-1598	20	20	semiring	semiring	NOUN
ejpam-1598	20	21	because	because	SCONJ
ejpam-1598	20	22	there	there	PRON
ejpam-1598	20	23	are	be	VERB
ejpam-1598	20	24	certain	certain	ADJ
ejpam-1598	20	25	notions	notion	NOUN
ejpam-1598	20	26	,	,	PUNCT
ejpam-1598	20	27	for	for	ADP
ejpam-1598	20	28	example	example	NOUN
ejpam-1598	20	29	,	,	PUNCT
ejpam-1598	20	30	the	the	DET
ejpam-1598	20	31	lateral	lateral	ADJ
ejpam-1598	20	32	ideals	ideal	NOUN
ejpam-1598	20	33	which	which	PRON
ejpam-1598	20	34	have	have	VERB
ejpam-1598	20	35	no	no	DET
ejpam-1598	20	36	analogue	analogue	NOUN
ejpam-1598	20	37	in	in	ADP
ejpam-1598	20	38	semirings	semiring	NOUN
ejpam-1598	20	39	.	.	PUNCT
ejpam-1598	21	1	some	some	DET
ejpam-1598	21	2	∗corresponding	∗corresponde	VERB
ejpam-1598	21	3	author	author	NOUN
ejpam-1598	21	4	.	.	PUNCT
ejpam-1598	22	1	email	email	NOUN
ejpam-1598	22	2	addresses	address	NOUN
ejpam-1598	22	3	:	:	PUNCT
ejpam-1598	23	1	duttatapankumar	duttatapankumar	PROPN
ejpam-1598	23	2	�	�	PROPN
ejpam-1598	23	3	yahoo	yahoo	PROPN
ejpam-1598	23	4	.	.	PUNCT
ejpam-1598	24	1	o.in	o.in	PROPN
ejpam-1598	24	2	(	(	PUNCT
ejpam-1598	24	3	t.	t.	PROPN
ejpam-1598	24	4	dutta	dutta	PROPN
ejpam-1598	24	5	)	)	PUNCT
ejpam-1598	24	6	,	,	PUNCT
ejpam-1598	24	7	kpshum�ynu.edu	kpshum�ynu.edu	PROPN
ejpam-1598	24	8	.	.	PROPN
ejpam-1598	25	1	n	n	PROPN
ejpam-1598	25	2	(	(	PUNCT
ejpam-1598	25	3	k.	k.	PROPN
ejpam-1598	25	4	shum),shobhan139	shum),shobhan139	PROPN
ejpam-1598	25	5	�	�	PROPN
ejpam-1598	25	6	gmail	gmail	NOUN
ejpam-1598	25	7	.	.	PUNCT
ejpam-1598	26	1	om	om	PROPN
ejpam-1598	26	2	(	(	PUNCT
ejpam-1598	26	3	s.	s.	PROPN
ejpam-1598	26	4	mandal	mandal	PROPN
ejpam-1598	26	5	)	)	PUNCT
ejpam-1598	26	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1598	27	1	116	116	NUM
ejpam-1598	28	1	c	c	X
ejpam-1598	28	2	©	©	PROPN
ejpam-1598	28	3	2012	2012	NUM
ejpam-1598	28	4	ejpam	ejpam	VERB
ejpam-1598	28	5	all	all	DET
ejpam-1598	28	6	rights	right	NOUN
ejpam-1598	28	7	reserved	reserve	VERB
ejpam-1598	28	8	.	.	PUNCT
ejpam-1598	29	1	t.	t.	PROPN
ejpam-1598	29	2	dutta	dutta	PROPN
ejpam-1598	29	3	,	,	PUNCT
ejpam-1598	29	4	k.	k.	PROPN
ejpam-1598	29	5	shum	shum	PROPN
ejpam-1598	29	6	,	,	PUNCT
ejpam-1598	29	7	s.	s.	PROPN
ejpam-1598	29	8	mandal	mandal	PROPN
ejpam-1598	29	9	/	/	SYM
ejpam-1598	29	10	eur	eur	PROPN
ejpam-1598	29	11	.	.	PUNCT
ejpam-1598	30	1	j.	j.	PROPN
ejpam-1598	30	2	pure	pure	PROPN
ejpam-1598	30	3	appl	appl	PROPN
ejpam-1598	30	4	.	.	PROPN
ejpam-1598	30	5	math	math	PROPN
ejpam-1598	30	6	,	,	PUNCT
ejpam-1598	30	7	5	5	NUM
ejpam-1598	30	8	(	(	PUNCT
ejpam-1598	30	9	2012	2012	NUM
ejpam-1598	30	10	)	)	PUNCT
ejpam-1598	30	11	,	,	PUNCT
ejpam-1598	30	12	116	116	NUM
ejpam-1598	30	13	-	-	SYM
ejpam-1598	30	14	128	128	NUM
ejpam-1598	30	15	117	117	NUM
ejpam-1598	30	16	earlier	early	ADJ
ejpam-1598	30	17	works	work	NOUN
ejpam-1598	30	18	on	on	ADP
ejpam-1598	30	19	ternary	ternary	ADJ
ejpam-1598	30	20	semiring	semiring	NOUN
ejpam-1598	30	21	may	may	AUX
ejpam-1598	30	22	be	be	AUX
ejpam-1598	30	23	found	find	VERB
ejpam-1598	30	24	in	in	ADP
ejpam-1598	30	25	[	[	X
ejpam-1598	30	26	1	1	NUM
ejpam-1598	30	27	,	,	PUNCT
ejpam-1598	30	28	2	2	NUM
ejpam-1598	30	29	,	,	PUNCT
ejpam-1598	30	30	3	3	NUM
ejpam-1598	30	31	,	,	PUNCT
ejpam-1598	30	32	4	4	NUM
ejpam-1598	30	33	,	,	PUNCT
ejpam-1598	30	34	5	5	NUM
ejpam-1598	30	35	,	,	PUNCT
ejpam-1598	30	36	6	6	NUM
ejpam-1598	30	37	,	,	PUNCT
ejpam-1598	30	38	7	7	NUM
ejpam-1598	30	39	,	,	PUNCT
ejpam-1598	30	40	8	8	NUM
ejpam-1598	30	41	,	,	PUNCT
ejpam-1598	30	42	11	11	NUM
ejpam-1598	30	43	]	]	PUNCT
ejpam-1598	30	44	.	.	PUNCT
ejpam-1598	31	1	in	in	ADP
ejpam-1598	31	2	this	this	DET
ejpam-1598	31	3	paper	paper	NOUN
ejpam-1598	31	4	,	,	PUNCT
ejpam-1598	31	5	the	the	DET
ejpam-1598	31	6	singular	singular	ADJ
ejpam-1598	31	7	ideals	ideal	NOUN
ejpam-1598	31	8	in	in	ADP
ejpam-1598	31	9	ternary	ternary	ADJ
ejpam-1598	31	10	semirings	semiring	NOUN
ejpam-1598	31	11	,	,	PUNCT
ejpam-1598	31	12	singular	singular	ADJ
ejpam-1598	31	13	ternary	ternary	ADJ
ejpam-1598	31	14	semirings	semiring	NOUN
ejpam-1598	31	15	and	and	CCONJ
ejpam-1598	31	16	nonsingular	nonsingular	ADJ
ejpam-1598	31	17	ternary	ternary	ADJ
ejpam-1598	31	18	semirings	semiring	NOUN
ejpam-1598	31	19	will	will	AUX
ejpam-1598	31	20	be	be	AUX
ejpam-1598	31	21	considered	consider	VERB
ejpam-1598	31	22	.	.	PUNCT
ejpam-1598	32	1	we	we	PRON
ejpam-1598	32	2	will	will	AUX
ejpam-1598	32	3	investigate	investigate	VERB
ejpam-1598	32	4	the	the	DET
ejpam-1598	32	5	properties	property	NOUN
ejpam-1598	32	6	of	of	ADP
ejpam-1598	32	7	the	the	DET
ejpam-1598	32	8	singular	singular	ADJ
ejpam-1598	32	9	ideals	ideal	NOUN
ejpam-1598	32	10	of	of	ADP
ejpam-1598	32	11	a	a	DET
ejpam-1598	32	12	ternary	ternary	ADJ
ejpam-1598	32	13	semiring	semiring	NOUN
ejpam-1598	32	14	.	.	PUNCT
ejpam-1598	33	1	we	we	PRON
ejpam-1598	33	2	will	will	AUX
ejpam-1598	33	3	give	give	VERB
ejpam-1598	33	4	some	some	DET
ejpam-1598	33	5	examples	example	NOUN
ejpam-1598	33	6	of	of	ADP
ejpam-1598	33	7	singular	singular	ADJ
ejpam-1598	33	8	ternary	ternary	ADJ
ejpam-1598	33	9	semirings	semiring	NOUN
ejpam-1598	33	10	and	and	CCONJ
ejpam-1598	33	11	non	non	ADJ
ejpam-1598	33	12	-	-	ADJ
ejpam-1598	33	13	singular	singular	ADJ
ejpam-1598	33	14	ternary	ternary	ADJ
ejpam-1598	33	15	semirings	semiring	NOUN
ejpam-1598	33	16	.	.	PUNCT
ejpam-1598	34	1	we	we	PRON
ejpam-1598	34	2	first	first	ADV
ejpam-1598	34	3	show	show	VERB
ejpam-1598	34	4	that	that	SCONJ
ejpam-1598	34	5	the	the	DET
ejpam-1598	34	6	class	class	NOUN
ejpam-1598	34	7	of	of	ADP
ejpam-1598	34	8	singular	singular	ADJ
ejpam-1598	34	9	ternary	ternary	ADJ
ejpam-1598	34	10	semirings	semiring	NOUN
ejpam-1598	34	11	with	with	ADP
ejpam-1598	34	12	identity	identity	NOUN
ejpam-1598	34	13	as	as	ADV
ejpam-1598	34	14	well	well	ADV
ejpam-1598	34	15	as	as	ADP
ejpam-1598	34	16	the	the	DET
ejpam-1598	34	17	class	class	NOUN
ejpam-1598	34	18	of	of	ADP
ejpam-1598	34	19	non	non	ADJ
ejpam-1598	34	20	-	-	ADJ
ejpam-1598	34	21	singular	singular	ADJ
ejpam-1598	34	22	ternary	ternary	ADJ
ejpam-1598	34	23	semirings	semiring	NOUN
ejpam-1598	34	24	with	with	ADP
ejpam-1598	34	25	identity	identity	NOUN
ejpam-1598	34	26	is	be	AUX
ejpam-1598	34	27	closed	close	VERB
ejpam-1598	34	28	under	under	ADP
ejpam-1598	34	29	direct	direct	ADJ
ejpam-1598	34	30	products	product	NOUN
ejpam-1598	34	31	and	and	CCONJ
ejpam-1598	34	32	direct	direct	ADJ
ejpam-1598	34	33	sums	sum	NOUN
ejpam-1598	34	34	.	.	PUNCT
ejpam-1598	35	1	then	then	ADV
ejpam-1598	35	2	we	we	PRON
ejpam-1598	35	3	study	study	VERB
ejpam-1598	35	4	the	the	DET
ejpam-1598	35	5	image	image	NOUN
ejpam-1598	35	6	and	and	CCONJ
ejpam-1598	35	7	preimage	preimage	NOUN
ejpam-1598	35	8	of	of	ADP
ejpam-1598	35	9	singular	singular	ADJ
ejpam-1598	35	10	ternary	ternary	ADJ
ejpam-1598	35	11	semiring	semiring	NOUN
ejpam-1598	35	12	and	and	CCONJ
ejpam-1598	35	13	nonsingular	nonsingular	ADJ
ejpam-1598	35	14	ternary	ternary	ADJ
ejpam-1598	35	15	semiring	semiring	NOUN
ejpam-1598	35	16	under	under	ADP
ejpam-1598	35	17	semi	semi	NOUN
ejpam-1598	35	18	-	-	NOUN
ejpam-1598	35	19	isomorphism	isomorphism	ADJ
ejpam-1598	35	20	.	.	PUNCT
ejpam-1598	36	1	finally	finally	ADV
ejpam-1598	36	2	,	,	PUNCT
ejpam-1598	36	3	we	we	PRON
ejpam-1598	36	4	show	show	VERB
ejpam-1598	36	5	that	that	SCONJ
ejpam-1598	36	6	the	the	DET
ejpam-1598	36	7	class	class	NOUN
ejpam-1598	36	8	of	of	ADP
ejpam-1598	36	9	semiprime	semiprime	NOUN
ejpam-1598	36	10	non	non	ADJ
ejpam-1598	36	11	-	-	ADJ
ejpam-1598	36	12	singular	singular	ADJ
ejpam-1598	36	13	ternary	ternary	ADJ
ejpam-1598	36	14	semirings	semiring	NOUN
ejpam-1598	36	15	is	be	AUX
ejpam-1598	36	16	hereditary	hereditary	ADJ
ejpam-1598	36	17	.	.	PUNCT
ejpam-1598	37	1	our	our	PRON
ejpam-1598	37	2	results	result	NOUN
ejpam-1598	37	3	in	in	ADP
ejpam-1598	37	4	this	this	DET
ejpam-1598	37	5	paper	paper	NOUN
ejpam-1598	37	6	can	can	AUX
ejpam-1598	37	7	be	be	AUX
ejpam-1598	37	8	applied	apply	VERB
ejpam-1598	37	9	to	to	PART
ejpam-1598	37	10	study	study	VERB
ejpam-1598	37	11	the	the	DET
ejpam-1598	37	12	special	special	ADJ
ejpam-1598	37	13	radical	radical	ADJ
ejpam-1598	37	14	class	class	NOUN
ejpam-1598	37	15	of	of	ADP
ejpam-1598	37	16	ternary	ternary	ADJ
ejpam-1598	37	17	semirings	semiring	NOUN
ejpam-1598	37	18	and	and	CCONJ
ejpam-1598	37	19	upper	upper	ADJ
ejpam-1598	37	20	radical	radical	ADJ
ejpam-1598	37	21	class	class	NOUN
ejpam-1598	37	22	determined	determine	VERB
ejpam-1598	37	23	by	by	ADP
ejpam-1598	37	24	the	the	DET
ejpam-1598	37	25	above	above	ADJ
ejpam-1598	37	26	radical	radical	ADJ
ejpam-1598	37	27	classes	class	NOUN
ejpam-1598	37	28	which	which	PRON
ejpam-1598	37	29	are	be	AUX
ejpam-1598	37	30	called	call	VERB
ejpam-1598	37	31	the	the	DET
ejpam-1598	37	32	singular	singular	ADJ
ejpam-1598	37	33	radical	radical	ADJ
ejpam-1598	37	34	and	and	CCONJ
ejpam-1598	37	35	special	special	ADJ
ejpam-1598	37	36	singular	singular	ADJ
ejpam-1598	37	37	radical	radical	NOUN
ejpam-1598	37	38	of	of	ADP
ejpam-1598	37	39	ternary	ternary	ADJ
ejpam-1598	37	40	semirings	semiring	NOUN
ejpam-1598	37	41	respectively	respectively	ADV
ejpam-1598	37	42	.	.	PUNCT
ejpam-1598	38	1	for	for	ADP
ejpam-1598	38	2	terminologies	terminology	NOUN
ejpam-1598	38	3	and	and	CCONJ
ejpam-1598	38	4	notions	notion	NOUN
ejpam-1598	38	5	not	not	PART
ejpam-1598	38	6	given	give	VERB
ejpam-1598	38	7	in	in	ADP
ejpam-1598	38	8	this	this	DET
ejpam-1598	38	9	paper	paper	NOUN
ejpam-1598	38	10	,	,	PUNCT
ejpam-1598	38	11	the	the	DET
ejpam-1598	38	12	reader	reader	NOUN
ejpam-1598	38	13	is	be	AUX
ejpam-1598	38	14	referred	refer	VERB
ejpam-1598	38	15	to	to	ADP
ejpam-1598	38	16	w	w	PROPN
ejpam-1598	38	17	.	.	PUNCT
ejpam-1598	39	1	g.	g.	PROPN
ejpam-1598	39	2	lister	lister	PROPN
ejpam-1598	40	1	[	[	X
ejpam-1598	40	2	14	14	NUM
ejpam-1598	40	3	]	]	SYM
ejpam-1598	40	4	.	.	PUNCT
ejpam-1598	41	1	2	2	X
ejpam-1598	41	2	.	.	X
ejpam-1598	41	3	preliminaries	preliminary	NOUN
ejpam-1598	41	4	we	we	PRON
ejpam-1598	41	5	first	first	ADV
ejpam-1598	41	6	give	give	VERB
ejpam-1598	41	7	the	the	DET
ejpam-1598	41	8	following	follow	VERB
ejpam-1598	41	9	definitions	definition	NOUN
ejpam-1598	41	10	.	.	PUNCT
ejpam-1598	42	1	definition	definition	NOUN
ejpam-1598	42	2	1	1	NUM
ejpam-1598	42	3	.	.	PUNCT
ejpam-1598	43	1	[	[	X
ejpam-1598	43	2	1	1	X
ejpam-1598	43	3	]	]	PUNCT
ejpam-1598	43	4	a	a	DET
ejpam-1598	43	5	non	non	ADJ
ejpam-1598	43	6	-	-	ADJ
ejpam-1598	43	7	empty	empty	ADJ
ejpam-1598	43	8	set	set	NOUN
ejpam-1598	43	9	s	s	VERB
ejpam-1598	43	10	together	together	ADV
ejpam-1598	43	11	with	with	ADP
ejpam-1598	43	12	a	a	DET
ejpam-1598	43	13	binary	binary	ADJ
ejpam-1598	43	14	operation	operation	NOUN
ejpam-1598	43	15	,	,	PUNCT
ejpam-1598	43	16	called	call	VERB
ejpam-1598	43	17	addition	addition	NOUN
ejpam-1598	43	18	and	and	CCONJ
ejpam-1598	43	19	a	a	DET
ejpam-1598	43	20	ternary	ternary	ADJ
ejpam-1598	43	21	multiplication	multiplication	NOUN
ejpam-1598	43	22	,	,	PUNCT
ejpam-1598	43	23	denoted	denote	VERB
ejpam-1598	43	24	by	by	ADP
ejpam-1598	43	25	juxtaposition	juxtaposition	NOUN
ejpam-1598	43	26	,	,	PUNCT
ejpam-1598	43	27	is	be	AUX
ejpam-1598	43	28	said	say	VERB
ejpam-1598	43	29	to	to	PART
ejpam-1598	43	30	be	be	AUX
ejpam-1598	43	31	a	a	DET
ejpam-1598	43	32	ternary	ternary	ADJ
ejpam-1598	43	33	semiring	semiring	NOUN
ejpam-1598	43	34	if	if	SCONJ
ejpam-1598	43	35	s	s	VERB
ejpam-1598	43	36	is	be	AUX
ejpam-1598	43	37	an	an	DET
ejpam-1598	43	38	additive	additive	ADJ
ejpam-1598	43	39	commutative	commutative	ADJ
ejpam-1598	43	40	semigroup	semigroup	NOUN
ejpam-1598	43	41	satisfying	satisfy	VERB
ejpam-1598	43	42	the	the	DET
ejpam-1598	43	43	following	follow	VERB
ejpam-1598	43	44	conditions	condition	NOUN
ejpam-1598	43	45	:	:	PUNCT
ejpam-1598	43	46	(	(	PUNCT
ejpam-1598	43	47	i	i	NOUN
ejpam-1598	43	48	)	)	PUNCT
ejpam-1598	43	49	(	(	PUNCT
ejpam-1598	43	50	abc)de	abc)de	NOUN
ejpam-1598	43	51	=	=	SYM
ejpam-1598	43	52	a(bcd)e	a(bcd)e	NOUN
ejpam-1598	43	53	=	=	PUNCT
ejpam-1598	43	54	ab(cde	ab(cde	VERB
ejpam-1598	43	55	)	)	PUNCT
ejpam-1598	43	56	;	;	PUNCT
ejpam-1598	43	57	(	(	PUNCT
ejpam-1598	43	58	ii	ii	NOUN
ejpam-1598	43	59	)	)	PUNCT
ejpam-1598	43	60	(	(	PUNCT
ejpam-1598	43	61	a+	a+	PUNCT
ejpam-1598	43	62	b)cd	b)cd	PROPN
ejpam-1598	43	63	=	=	PUNCT
ejpam-1598	43	64	acd	acd	NOUN
ejpam-1598	43	65	+	+	X
ejpam-1598	43	66	bcd	bcd	PROPN
ejpam-1598	43	67	,	,	PUNCT
ejpam-1598	43	68	(	(	PUNCT
ejpam-1598	43	69	iii	iii	NOUN
ejpam-1598	43	70	)	)	PUNCT
ejpam-1598	44	1	a(b+	a(b+	ADV
ejpam-1598	44	2	c)d	c)d	NOUN
ejpam-1598	44	3	=	=	PUNCT
ejpam-1598	44	4	abd	abd	PROPN
ejpam-1598	44	5	+	+	CCONJ
ejpam-1598	44	6	acd	acd	PROPN
ejpam-1598	44	7	,	,	PUNCT
ejpam-1598	44	8	(	(	PUNCT
ejpam-1598	44	9	iv	iv	X
ejpam-1598	44	10	)	)	PUNCT
ejpam-1598	44	11	ab(c	ab(c	PUNCT
ejpam-1598	44	12	+	+	CCONJ
ejpam-1598	44	13	d	d	X
ejpam-1598	44	14	)	)	PUNCT
ejpam-1598	44	15	=	=	SYM
ejpam-1598	44	16	abc	abc	PROPN
ejpam-1598	44	17	+	+	X
ejpam-1598	44	18	abd	abd	NOUN
ejpam-1598	44	19	for	for	ADP
ejpam-1598	44	20	all	all	DET
ejpam-1598	44	21	a	a	DET
ejpam-1598	44	22	,	,	PUNCT
ejpam-1598	44	23	b	b	NOUN
ejpam-1598	44	24	,	,	PUNCT
ejpam-1598	44	25	c	c	NOUN
ejpam-1598	44	26	,	,	PUNCT
ejpam-1598	44	27	d	d	NOUN
ejpam-1598	44	28	,	,	PUNCT
ejpam-1598	44	29	e	e	PROPN
ejpam-1598	44	30	∈	∈	PROPN
ejpam-1598	44	31	s.	s.	PROPN
ejpam-1598	44	32	definition	definition	NOUN
ejpam-1598	44	33	2	2	NUM
ejpam-1598	44	34	.	.	PUNCT
ejpam-1598	45	1	[	[	X
ejpam-1598	45	2	1	1	X
ejpam-1598	45	3	]	]	PUNCT
ejpam-1598	45	4	let	let	VERB
ejpam-1598	45	5	s	s	PRON
ejpam-1598	45	6	be	be	AUX
ejpam-1598	45	7	a	a	DET
ejpam-1598	45	8	ternary	ternary	ADJ
ejpam-1598	45	9	semiring	semiring	NOUN
ejpam-1598	45	10	.	.	PUNCT
ejpam-1598	46	1	if	if	SCONJ
ejpam-1598	46	2	there	there	PRON
ejpam-1598	46	3	exists	exist	VERB
ejpam-1598	46	4	an	an	DET
ejpam-1598	46	5	element	element	NOUN
ejpam-1598	46	6	0	0	NUM
ejpam-1598	46	7	∈	∈	NOUN
ejpam-1598	46	8	s	s	VERB
ejpam-1598	46	9	such	such	ADJ
ejpam-1598	46	10	that	that	PRON
ejpam-1598	46	11	0+x	0+x	NUM
ejpam-1598	47	1	=	=	SYM
ejpam-1598	47	2	x	x	X
ejpam-1598	47	3	and	and	CCONJ
ejpam-1598	47	4	0x	0x	ADJ
ejpam-1598	47	5	y	y	PROPN
ejpam-1598	47	6	=	=	PUNCT
ejpam-1598	47	7	x0y	x0y	X
ejpam-1598	48	1	=	=	PUNCT
ejpam-1598	48	2	x	x	X
ejpam-1598	48	3	y0=	y0=	NOUN
ejpam-1598	48	4	0	0	NUM
ejpam-1598	48	5	for	for	ADP
ejpam-1598	48	6	all	all	DET
ejpam-1598	48	7	x	x	SYM
ejpam-1598	48	8	,	,	PUNCT
ejpam-1598	48	9	y	y	PROPN
ejpam-1598	48	10	∈	∈	PROPN
ejpam-1598	49	1	s	s	VERB
ejpam-1598	49	2	then	then	ADV
ejpam-1598	49	3	“	"	PUNCT
ejpam-1598	49	4	0	0	NUM
ejpam-1598	49	5	”	"	PUNCT
ejpam-1598	49	6	is	be	AUX
ejpam-1598	49	7	called	call	VERB
ejpam-1598	49	8	the	the	DET
ejpam-1598	49	9	zero	zero	NUM
ejpam-1598	49	10	element	element	NOUN
ejpam-1598	49	11	or	or	CCONJ
ejpam-1598	49	12	simply	simply	ADV
ejpam-1598	49	13	the	the	DET
ejpam-1598	49	14	zero	zero	NUM
ejpam-1598	49	15	of	of	ADP
ejpam-1598	49	16	the	the	DET
ejpam-1598	49	17	ternary	ternary	ADJ
ejpam-1598	49	18	semiring	semire	VERB
ejpam-1598	49	19	s.	s.	PROPN
ejpam-1598	49	20	in	in	ADP
ejpam-1598	49	21	this	this	DET
ejpam-1598	49	22	case	case	NOUN
ejpam-1598	49	23	,	,	PUNCT
ejpam-1598	49	24	s	s	VERB
ejpam-1598	49	25	is	be	AUX
ejpam-1598	49	26	called	call	VERB
ejpam-1598	49	27	a	a	DET
ejpam-1598	49	28	ternary	ternary	ADJ
ejpam-1598	49	29	semiring	semiring	NOUN
ejpam-1598	49	30	with	with	ADP
ejpam-1598	49	31	zero	zero	NUM
ejpam-1598	49	32	.	.	PUNCT
ejpam-1598	50	1	it	it	PRON
ejpam-1598	50	2	is	be	AUX
ejpam-1598	50	3	noted	note	VERB
ejpam-1598	50	4	that	that	SCONJ
ejpam-1598	50	5	a	a	DET
ejpam-1598	50	6	ternary	ternary	ADJ
ejpam-1598	50	7	semiring	semiring	NOUN
ejpam-1598	50	8	does	do	AUX
ejpam-1598	50	9	not	not	PART
ejpam-1598	50	10	necessarily	necessarily	ADV
ejpam-1598	50	11	contain	contain	VERB
ejpam-1598	50	12	an	an	DET
ejpam-1598	50	13	identity	identity	NOUN
ejpam-1598	50	14	but	but	CCONJ
ejpam-1598	50	15	there	there	PRON
ejpam-1598	50	16	are	be	VERB
ejpam-1598	50	17	certain	certain	ADJ
ejpam-1598	50	18	ternary	ternary	ADJ
ejpam-1598	50	19	semirings	semiring	NOUN
ejpam-1598	50	20	which	which	PRON
ejpam-1598	50	21	contain	contain	VERB
ejpam-1598	50	22	generalized	generalized	ADJ
ejpam-1598	50	23	identity	identity	NOUN
ejpam-1598	50	24	in	in	ADP
ejpam-1598	50	25	the	the	DET
ejpam-1598	50	26	sense	sense	NOUN
ejpam-1598	50	27	defined	define	VERB
ejpam-1598	50	28	below	below	ADV
ejpam-1598	50	29	.	.	PUNCT
ejpam-1598	51	1	definition	definition	NOUN
ejpam-1598	51	2	3	3	NUM
ejpam-1598	51	3	.	.	PUNCT
ejpam-1598	52	1	[	[	X
ejpam-1598	52	2	4	4	X
ejpam-1598	52	3	]	]	PUNCT
ejpam-1598	52	4	a	a	DET
ejpam-1598	52	5	ternary	ternary	ADJ
ejpam-1598	52	6	semiring	semiring	NOUN
ejpam-1598	52	7	s	s	VERB
ejpam-1598	52	8	admits	admit	VERB
ejpam-1598	52	9	an	an	DET
ejpam-1598	52	10	identity	identity	NOUN
ejpam-1598	52	11	provided	provide	VERB
ejpam-1598	52	12	that	that	SCONJ
ejpam-1598	52	13	there	there	PRON
ejpam-1598	52	14	exist	exist	VERB
ejpam-1598	52	15	elements	element	NOUN
ejpam-1598	52	16	{	{	PUNCT
ejpam-1598	52	17	(	(	PUNCT
ejpam-1598	52	18	ei	ei	NOUN
ejpam-1598	52	19	,	,	PUNCT
ejpam-1598	52	20	fi	fi	NOUN
ejpam-1598	52	21	)	)	PUNCT
ejpam-1598	52	22	∈	∈	PROPN
ejpam-1598	52	23	s	s	PART
ejpam-1598	52	24	×	×	NOUN
ejpam-1598	52	25	s	s	X
ejpam-1598	52	26	(	(	PUNCT
ejpam-1598	52	27	i	i	NOUN
ejpam-1598	52	28	=	=	SYM
ejpam-1598	52	29	1,2	1,2	NUM
ejpam-1598	52	30	,	,	PUNCT
ejpam-1598	52	31	.	.	PUNCT
ejpam-1598	52	32	.	.	PUNCT
ejpam-1598	53	1	.	.	PUNCT
ejpam-1598	53	2	,	,	PUNCT
ejpam-1598	53	3	n	n	CCONJ
ejpam-1598	53	4	)	)	PUNCT
ejpam-1598	54	1	}	}	PUNCT
ejpam-1598	54	2	such	such	ADJ
ejpam-1598	54	3	that	that	SCONJ
ejpam-1598	54	4	∑n	∑n	PROPN
ejpam-1598	54	5	i=1	i=1	X
ejpam-1598	54	6	ei	ei	NOUN
ejpam-1598	54	7	fi	fi	NOUN
ejpam-1598	55	1	x	x	X
ejpam-1598	56	1	=	=	PUNCT
ejpam-1598	56	2	∑n	∑n	PROPN
ejpam-1598	56	3	i=1	i=1	X
ejpam-1598	56	4	ei	ei	X
ejpam-1598	56	5	x	x	PUNCT
ejpam-1598	56	6	fi	fi	NOUN
ejpam-1598	57	1	=	=	PUNCT
ejpam-1598	57	2	∑n	∑n	PROPN
ejpam-1598	57	3	i=1	i=1	PROPN
ejpam-1598	57	4	xei	xei	PROPN
ejpam-1598	57	5	fi	fi	NOUN
ejpam-1598	58	1	=	=	NOUN
ejpam-1598	58	2	x	x	PROPN
ejpam-1598	58	3	for	for	ADP
ejpam-1598	58	4	all	all	DET
ejpam-1598	58	5	x	x	SYM
ejpam-1598	58	6	∈	∈	PROPN
ejpam-1598	58	7	s.	s.	PROPN
ejpam-1598	58	8	in	in	ADP
ejpam-1598	58	9	this	this	DET
ejpam-1598	58	10	case	case	NOUN
ejpam-1598	58	11	,	,	PUNCT
ejpam-1598	58	12	the	the	DET
ejpam-1598	58	13	ternary	ternary	ADJ
ejpam-1598	58	14	semiring	semiring	NOUN
ejpam-1598	58	15	s	s	X
ejpam-1598	58	16	is	be	AUX
ejpam-1598	58	17	said	say	VERB
ejpam-1598	58	18	to	to	PART
ejpam-1598	58	19	be	be	AUX
ejpam-1598	58	20	a	a	DET
ejpam-1598	58	21	ternary	ternary	ADJ
ejpam-1598	58	22	semiring	semiring	NOUN
ejpam-1598	58	23	with	with	ADP
ejpam-1598	58	24	identity	identity	NOUN
ejpam-1598	58	25	{	{	PUNCT
ejpam-1598	58	26	(	(	PUNCT
ejpam-1598	58	27	ei	ei	NOUN
ejpam-1598	58	28	,	,	PUNCT
ejpam-1598	58	29	fi	fi	NOUN
ejpam-1598	58	30	)	)	PUNCT
ejpam-1598	58	31	:	:	PUNCT
ejpam-1598	59	1	i	i	NOUN
ejpam-1598	59	2	=	=	SYM
ejpam-1598	59	3	1,2	1,2	NUM
ejpam-1598	59	4	,	,	PUNCT
ejpam-1598	59	5	.	.	PUNCT
ejpam-1598	59	6	.	.	PUNCT
ejpam-1598	59	7	.	.	PUNCT
ejpam-1598	59	8	,	,	PUNCT
ejpam-1598	59	9	n	n	CCONJ
ejpam-1598	59	10	}	}	PUNCT
ejpam-1598	59	11	.	.	PUNCT
ejpam-1598	60	1	in	in	ADP
ejpam-1598	60	2	particular	particular	ADJ
ejpam-1598	60	3	,	,	PUNCT
ejpam-1598	60	4	if	if	SCONJ
ejpam-1598	60	5	there	there	PRON
ejpam-1598	60	6	exists	exist	VERB
ejpam-1598	60	7	an	an	DET
ejpam-1598	60	8	element	element	NOUN
ejpam-1598	60	9	e	e	NOUN
ejpam-1598	60	10	∈	∈	NOUN
ejpam-1598	60	11	s	s	VERB
ejpam-1598	60	12	such	such	ADJ
ejpam-1598	60	13	that	that	DET
ejpam-1598	60	14	eex	eex	NOUN
ejpam-1598	60	15	=	=	PROPN
ejpam-1598	60	16	exe	exe	PROPN
ejpam-1598	60	17	=	=	PUNCT
ejpam-1598	60	18	xee	xee	PROPN
ejpam-1598	60	19	=	=	PROPN
ejpam-1598	61	1	x	x	PROPN
ejpam-1598	61	2	for	for	ADP
ejpam-1598	61	3	all	all	DET
ejpam-1598	61	4	x	x	SYM
ejpam-1598	61	5	∈	∈	PROPN
ejpam-1598	61	6	s	s	NOUN
ejpam-1598	61	7	,	,	PUNCT
ejpam-1598	61	8	then	then	ADV
ejpam-1598	61	9	“	"	PUNCT
ejpam-1598	61	10	e	e	X
ejpam-1598	61	11	”	"	PUNCT
ejpam-1598	61	12	is	be	AUX
ejpam-1598	61	13	called	call	VERB
ejpam-1598	61	14	a	a	DET
ejpam-1598	61	15	unital	unital	ADJ
ejpam-1598	61	16	element	element	NOUN
ejpam-1598	61	17	of	of	ADP
ejpam-1598	61	18	a	a	DET
ejpam-1598	61	19	ternary	ternary	ADJ
ejpam-1598	61	20	semiring	semiring	NOUN
ejpam-1598	61	21	s.	s.	PROPN
ejpam-1598	61	22	it	it	PRON
ejpam-1598	61	23	is	be	AUX
ejpam-1598	61	24	easy	easy	ADJ
ejpam-1598	61	25	to	to	PART
ejpam-1598	61	26	see	see	VERB
ejpam-1598	61	27	that	that	SCONJ
ejpam-1598	62	1	x	x	NOUN
ejpam-1598	62	2	ye	ye	NOUN
ejpam-1598	62	3	=	=	PUNCT
ejpam-1598	62	4	(	(	PUNCT
ejpam-1598	62	5	exe)ye	exe)ye	NOUN
ejpam-1598	62	6	=	=	SYM
ejpam-1598	62	7	ex(e	ex(e	NOUN
ejpam-1598	62	8	ye	ye	NOUN
ejpam-1598	62	9	)	)	PUNCT
ejpam-1598	62	10	=	=	SYM
ejpam-1598	62	11	ex	ex	X
ejpam-1598	62	12	y	y	PROPN
ejpam-1598	62	13	and	and	CCONJ
ejpam-1598	62	14	x	x	PUNCT
ejpam-1598	62	15	ye	ye	PROPN
ejpam-1598	62	16	=	=	PUNCT
ejpam-1598	62	17	x(e	x(e	PROPN
ejpam-1598	62	18	ye)e	ye)e	NOUN
ejpam-1598	62	19	=	=	SYM
ejpam-1598	62	20	xe(yee	xe(yee	PROPN
ejpam-1598	62	21	)	)	PUNCT
ejpam-1598	62	22	=	=	SYM
ejpam-1598	63	1	xe	xe	PROPN
ejpam-1598	63	2	y	y	PROPN
ejpam-1598	63	3	,	,	PUNCT
ejpam-1598	63	4	for	for	ADP
ejpam-1598	63	5	all	all	DET
ejpam-1598	63	6	x	x	SYM
ejpam-1598	63	7	,	,	PUNCT
ejpam-1598	63	8	y	y	PROPN
ejpam-1598	63	9	∈	∈	PROPN
ejpam-1598	63	10	s.	s.	PROPN
ejpam-1598	63	11	hence	hence	ADV
ejpam-1598	63	12	,	,	PUNCT
ejpam-1598	63	13	the	the	DET
ejpam-1598	63	14	following	following	ADJ
ejpam-1598	63	15	result	result	NOUN
ejpam-1598	63	16	follows	follow	VERB
ejpam-1598	63	17	.	.	PUNCT
ejpam-1598	64	1	t.	t.	PROPN
ejpam-1598	64	2	dutta	dutta	PROPN
ejpam-1598	64	3	,	,	PUNCT
ejpam-1598	64	4	k.	k.	PROPN
ejpam-1598	64	5	shum	shum	PROPN
ejpam-1598	64	6	,	,	PUNCT
ejpam-1598	64	7	s.	s.	PROPN
ejpam-1598	64	8	mandal	mandal	PROPN
ejpam-1598	64	9	/	/	SYM
ejpam-1598	64	10	eur	eur	PROPN
ejpam-1598	64	11	.	.	PUNCT
ejpam-1598	65	1	j.	j.	PROPN
ejpam-1598	65	2	pure	pure	PROPN
ejpam-1598	65	3	appl	appl	PROPN
ejpam-1598	65	4	.	.	PROPN
ejpam-1598	65	5	math	math	PROPN
ejpam-1598	65	6	,	,	PUNCT
ejpam-1598	65	7	5	5	NUM
ejpam-1598	65	8	(	(	PUNCT
ejpam-1598	65	9	2012	2012	NUM
ejpam-1598	65	10	)	)	PUNCT
ejpam-1598	65	11	,	,	PUNCT
ejpam-1598	65	12	116	116	NUM
ejpam-1598	65	13	-	-	SYM
ejpam-1598	65	14	128	128	NUM
ejpam-1598	65	15	118	118	NUM
ejpam-1598	65	16	proposition	proposition	NOUN
ejpam-1598	65	17	1	1	NUM
ejpam-1598	65	18	.	.	PUNCT
ejpam-1598	66	1	if	if	SCONJ
ejpam-1598	66	2	e	e	PROPN
ejpam-1598	66	3	is	be	AUX
ejpam-1598	66	4	a	a	DET
ejpam-1598	66	5	unital	unital	ADJ
ejpam-1598	66	6	element	element	NOUN
ejpam-1598	66	7	of	of	ADP
ejpam-1598	66	8	a	a	DET
ejpam-1598	66	9	ternary	ternary	ADJ
ejpam-1598	66	10	semiring	semire	VERB
ejpam-1598	66	11	s	s	PART
ejpam-1598	66	12	,	,	PUNCT
ejpam-1598	66	13	then	then	ADV
ejpam-1598	66	14	ex	ex	ADJ
ejpam-1598	66	15	y	y	PROPN
ejpam-1598	66	16	=	=	SYM
ejpam-1598	66	17	xe	xe	PROPN
ejpam-1598	66	18	y	y	PROPN
ejpam-1598	66	19	=	=	PUNCT
ejpam-1598	66	20	x	x	SYM
ejpam-1598	66	21	ye	ye	PROPN
ejpam-1598	66	22	,	,	PUNCT
ejpam-1598	66	23	for	for	ADP
ejpam-1598	66	24	all	all	DET
ejpam-1598	66	25	x	x	SYM
ejpam-1598	66	26	,	,	PUNCT
ejpam-1598	66	27	y	y	PROPN
ejpam-1598	66	28	∈	∈	PROPN
ejpam-1598	66	29	s.	s.	PROPN
ejpam-1598	66	30	we	we	PRON
ejpam-1598	66	31	now	now	ADV
ejpam-1598	66	32	state	state	VERB
ejpam-1598	66	33	the	the	DET
ejpam-1598	66	34	definitions	definition	NOUN
ejpam-1598	66	35	of	of	ADP
ejpam-1598	66	36	ternary	ternary	ADJ
ejpam-1598	66	37	subsemiring	subsemire	VERB
ejpam-1598	66	38	and	and	CCONJ
ejpam-1598	66	39	left	left	ADJ
ejpam-1598	66	40	(	(	PUNCT
ejpam-1598	66	41	right	right	ADJ
ejpam-1598	66	42	,	,	PUNCT
ejpam-1598	66	43	lateral	lateral	ADJ
ejpam-1598	66	44	)	)	PUNCT
ejpam-1598	66	45	ideals	ideal	NOUN
ejpam-1598	66	46	of	of	ADP
ejpam-1598	66	47	a	a	DET
ejpam-1598	66	48	ternary	ternary	ADJ
ejpam-1598	66	49	semiring	semiring	NOUN
ejpam-1598	66	50	.	.	PUNCT
ejpam-1598	67	1	definition	definition	NOUN
ejpam-1598	67	2	4	4	NUM
ejpam-1598	67	3	.	.	PUNCT
ejpam-1598	68	1	[	[	X
ejpam-1598	68	2	1	1	X
ejpam-1598	68	3	]	]	PUNCT
ejpam-1598	68	4	an	an	DET
ejpam-1598	68	5	additive	additive	ADJ
ejpam-1598	68	6	subsemigroup	subsemigroup	PROPN
ejpam-1598	68	7	t	t	PROPN
ejpam-1598	68	8	of	of	ADP
ejpam-1598	68	9	a	a	DET
ejpam-1598	68	10	ternary	ternary	ADJ
ejpam-1598	68	11	semiring	semiring	NOUN
ejpam-1598	68	12	s	s	VERB
ejpam-1598	68	13	is	be	AUX
ejpam-1598	68	14	called	call	VERB
ejpam-1598	68	15	a	a	DET
ejpam-1598	68	16	ternary	ternary	ADJ
ejpam-1598	68	17	subsemiring	subsemire	VERB
ejpam-1598	68	18	if	if	SCONJ
ejpam-1598	68	19	t1	t1	PROPN
ejpam-1598	68	20	t2	t2	PROPN
ejpam-1598	68	21	t3	t3	PROPN
ejpam-1598	68	22	∈	∈	PROPN
ejpam-1598	68	23	t	t	PROPN
ejpam-1598	68	24	for	for	ADP
ejpam-1598	68	25	all	all	DET
ejpam-1598	68	26	t1	t1	NOUN
ejpam-1598	68	27	,	,	PUNCT
ejpam-1598	68	28	t2	t2	NOUN
ejpam-1598	68	29	,	,	PUNCT
ejpam-1598	68	30	t3	t3	PROPN
ejpam-1598	68	31	∈	∈	PROPN
ejpam-1598	68	32	t.	t.	NOUN
ejpam-1598	68	33	definition	definition	NOUN
ejpam-1598	68	34	5	5	NUM
ejpam-1598	68	35	.	.	PUNCT
ejpam-1598	69	1	[	[	X
ejpam-1598	69	2	1	1	X
ejpam-1598	69	3	]	]	PUNCT
ejpam-1598	69	4	an	an	DET
ejpam-1598	69	5	additive	additive	ADJ
ejpam-1598	69	6	subsemigroup	subsemigroup	NOUN
ejpam-1598	70	1	i	i	PRON
ejpam-1598	70	2	of	of	ADP
ejpam-1598	70	3	a	a	DET
ejpam-1598	70	4	ternary	ternary	ADJ
ejpam-1598	70	5	semiring	semiring	NOUN
ejpam-1598	70	6	s	s	VERB
ejpam-1598	70	7	is	be	AUX
ejpam-1598	70	8	called	call	VERB
ejpam-1598	70	9	a	a	DET
ejpam-1598	70	10	left	left	ADJ
ejpam-1598	70	11	(	(	PUNCT
ejpam-1598	70	12	right	right	ADJ
ejpam-1598	70	13	,	,	PUNCT
ejpam-1598	70	14	lateral	lateral	ADJ
ejpam-1598	70	15	)	)	PUNCT
ejpam-1598	70	16	ideal	ideal	NOUN
ejpam-1598	70	17	of	of	ADP
ejpam-1598	70	18	s	s	PRON
ejpam-1598	70	19	if	if	SCONJ
ejpam-1598	70	20	s1s2i	s1s2i	PUNCT
ejpam-1598	70	21	(	(	PUNCT
ejpam-1598	70	22	respectively	respectively	ADV
ejpam-1598	70	23	is1s2	is1s2	PROPN
ejpam-1598	70	24	,	,	PUNCT
ejpam-1598	70	25	s1	s1	PROPN
ejpam-1598	70	26	is2	is2	PROPN
ejpam-1598	70	27	)	)	PUNCT
ejpam-1598	70	28	∈	∈	PROPN
ejpam-1598	70	29	i	i	PRON
ejpam-1598	70	30	for	for	ADP
ejpam-1598	70	31	all	all	DET
ejpam-1598	70	32	s1	s1	NOUN
ejpam-1598	70	33	,	,	PUNCT
ejpam-1598	70	34	s2	s2	NOUN
ejpam-1598	70	35	∈	∈	PROPN
ejpam-1598	70	36	s	s	PART
ejpam-1598	71	1	and	and	CCONJ
ejpam-1598	71	2	i	i	PRON
ejpam-1598	71	3	∈	∈	PROPN
ejpam-1598	72	1	i	i	PRON
ejpam-1598	72	2	.	.	PUNCT
ejpam-1598	73	1	if	if	SCONJ
ejpam-1598	73	2	i	i	PRON
ejpam-1598	73	3	is	be	AUX
ejpam-1598	73	4	a	a	DET
ejpam-1598	73	5	left	left	NOUN
ejpam-1598	73	6	,	,	PUNCT
ejpam-1598	73	7	a	a	DET
ejpam-1598	73	8	right	right	NOUN
ejpam-1598	73	9	and	and	CCONJ
ejpam-1598	73	10	a	a	DET
ejpam-1598	73	11	lateral	lateral	ADJ
ejpam-1598	73	12	ideal	ideal	NOUN
ejpam-1598	73	13	of	of	ADP
ejpam-1598	73	14	s	s	PROPN
ejpam-1598	73	15	,	,	PUNCT
ejpam-1598	73	16	then	then	ADV
ejpam-1598	73	17	i	i	PRON
ejpam-1598	73	18	is	be	AUX
ejpam-1598	73	19	called	call	VERB
ejpam-1598	73	20	an	an	DET
ejpam-1598	73	21	ideal	ideal	NOUN
ejpam-1598	73	22	of	of	ADP
ejpam-1598	73	23	s.	s.	PROPN
ejpam-1598	73	24	in	in	ADP
ejpam-1598	73	25	the	the	DET
ejpam-1598	73	26	following	follow	VERB
ejpam-1598	73	27	proposition	proposition	NOUN
ejpam-1598	73	28	,	,	PUNCT
ejpam-1598	73	29	we	we	PRON
ejpam-1598	73	30	describe	describe	VERB
ejpam-1598	73	31	the	the	DET
ejpam-1598	73	32	left(right	left(right	PROPN
ejpam-1598	73	33	,	,	PUNCT
ejpam-1598	73	34	lateral	lateral	ADJ
ejpam-1598	73	35	)	)	PUNCT
ejpam-1598	73	36	ideal	ideal	NOUN
ejpam-1598	73	37	of	of	ADP
ejpam-1598	73	38	a	a	DET
ejpam-1598	73	39	ternary	ternary	ADJ
ejpam-1598	73	40	semiring	semiring	NOUN
ejpam-1598	73	41	.	.	PUNCT
ejpam-1598	74	1	proposition	proposition	NOUN
ejpam-1598	74	2	2	2	NUM
ejpam-1598	74	3	.	.	PUNCT
ejpam-1598	75	1	[	[	X
ejpam-1598	75	2	1	1	X
ejpam-1598	75	3	]	]	PUNCT
ejpam-1598	75	4	let	let	VERB
ejpam-1598	75	5	s	s	PRON
ejpam-1598	75	6	be	be	AUX
ejpam-1598	75	7	a	a	DET
ejpam-1598	75	8	ternary	ternary	ADJ
ejpam-1598	75	9	semiring	semiring	NOUN
ejpam-1598	75	10	and	and	CCONJ
ejpam-1598	75	11	a	a	DET
ejpam-1598	75	12	∈	∈	NOUN
ejpam-1598	75	13	s.	s.	PROPN
ejpam-1598	75	14	then	then	ADV
ejpam-1598	75	15	the	the	DET
ejpam-1598	75	16	following	follow	VERB
ejpam-1598	75	17	statements	statement	NOUN
ejpam-1598	75	18	hold	hold	VERB
ejpam-1598	75	19	:	:	PUNCT
ejpam-1598	75	20	(	(	PUNCT
ejpam-1598	75	21	i	i	NOUN
ejpam-1598	75	22	)	)	PUNCT
ejpam-1598	75	23	left	leave	VERB
ejpam-1598	75	24	ideal	ideal	NOUN
ejpam-1598	75	25	generated	generate	VERB
ejpam-1598	75	26	by	by	ADP
ejpam-1598	75	27	“	"	PUNCT
ejpam-1598	75	28	a	a	PRON
ejpam-1598	75	29	”	"	PUNCT
ejpam-1598	75	30	is	be	AUX
ejpam-1598	75	31	given	give	VERB
ejpam-1598	75	32	by	by	ADP
ejpam-1598	75	33	〈	〈	PROPN
ejpam-1598	75	34	a〉l	a〉l	NOUN
ejpam-1598	75	35	=	=	PUNCT
ejpam-1598	75	36	ssa+	ssa+	NOUN
ejpam-1598	75	37	na	na	PART
ejpam-1598	75	38	(	(	PUNCT
ejpam-1598	75	39	ii	ii	NOUN
ejpam-1598	75	40	)	)	PUNCT
ejpam-1598	75	41	right	right	ADJ
ejpam-1598	75	42	ideal	ideal	NOUN
ejpam-1598	75	43	generated	generate	VERB
ejpam-1598	75	44	by	by	ADP
ejpam-1598	75	45	“	"	PUNCT
ejpam-1598	75	46	a	a	PRON
ejpam-1598	75	47	”	"	PUNCT
ejpam-1598	75	48	is	be	AUX
ejpam-1598	75	49	given	give	VERB
ejpam-1598	75	50	by	by	ADP
ejpam-1598	75	51	〈	〈	NOUN
ejpam-1598	75	52	a〉r	a〉r	NOUN
ejpam-1598	75	53	=	=	PUNCT
ejpam-1598	75	54	ass	ass	NOUN
ejpam-1598	75	55	+	+	CCONJ
ejpam-1598	75	56	na	na	PART
ejpam-1598	75	57	(	(	PUNCT
ejpam-1598	75	58	iii	iii	NOUN
ejpam-1598	75	59	)	)	PUNCT
ejpam-1598	75	60	two	two	NUM
ejpam-1598	75	61	-	-	PUNCT
ejpam-1598	75	62	sided	sided	ADJ
ejpam-1598	75	63	ideal	ideal	NOUN
ejpam-1598	75	64	generated	generate	VERB
ejpam-1598	75	65	by	by	ADP
ejpam-1598	75	66	“	"	PUNCT
ejpam-1598	75	67	a	a	PRON
ejpam-1598	75	68	”	"	PUNCT
ejpam-1598	75	69	is	be	AUX
ejpam-1598	75	70	given	give	VERB
ejpam-1598	75	71	by	by	ADP
ejpam-1598	75	72	〈	〈	NOUN
ejpam-1598	75	73	a〉t	a〉t	NOUN
ejpam-1598	75	74	=	=	PUNCT
ejpam-1598	75	75	ssa+	ssa+	NOUN
ejpam-1598	75	76	ass	ass	NOUN
ejpam-1598	75	77	+	+	CCONJ
ejpam-1598	75	78	ssass	ssass	VERB
ejpam-1598	75	79	+	+	CCONJ
ejpam-1598	75	80	na	na	PART
ejpam-1598	75	81	(	(	PUNCT
ejpam-1598	75	82	iv	iv	NOUN
ejpam-1598	75	83	)	)	PUNCT
ejpam-1598	75	84	lateral	lateral	ADJ
ejpam-1598	75	85	ideal	ideal	NOUN
ejpam-1598	75	86	generated	generate	VERB
ejpam-1598	75	87	by	by	ADP
ejpam-1598	75	88	“	"	PUNCT
ejpam-1598	75	89	a	a	PRON
ejpam-1598	75	90	”	"	PUNCT
ejpam-1598	75	91	is	be	AUX
ejpam-1598	75	92	given	give	VERB
ejpam-1598	75	93	by	by	ADP
ejpam-1598	75	94	〈	〈	NOUN
ejpam-1598	75	95	a〉m	a〉m	X
ejpam-1598	75	96	=	=	X
ejpam-1598	75	97	sas	sas	X
ejpam-1598	75	98	+	+	X
ejpam-1598	75	99	ssass+	ssass+	ADV
ejpam-1598	75	100	na	na	ADP
ejpam-1598	75	101	(	(	PUNCT
ejpam-1598	75	102	v	v	NOUN
ejpam-1598	75	103	)	)	PUNCT
ejpam-1598	75	104	ideal	ideal	NOUN
ejpam-1598	75	105	generated	generate	VERB
ejpam-1598	75	106	by	by	ADP
ejpam-1598	75	107	“	"	PUNCT
ejpam-1598	75	108	a	a	PRON
ejpam-1598	75	109	”	"	PUNCT
ejpam-1598	75	110	is	be	AUX
ejpam-1598	75	111	given	give	VERB
ejpam-1598	75	112	by	by	ADP
ejpam-1598	75	113	〈	〈	PROPN
ejpam-1598	75	114	a	a	DET
ejpam-1598	75	115	〉	〉	NOUN
ejpam-1598	75	116	=	=	SYM
ejpam-1598	75	117	ssa+	ssa+	NOUN
ejpam-1598	76	1	ass+	ass+	PROPN
ejpam-1598	76	2	sas	sa	NOUN
ejpam-1598	76	3	+	+	CCONJ
ejpam-1598	76	4	ssass	ssass	VERB
ejpam-1598	76	5	+	+	CCONJ
ejpam-1598	76	6	na	na	ADP
ejpam-1598	76	7	,	,	PUNCT
ejpam-1598	76	8	where	where	SCONJ
ejpam-1598	76	9	n	n	X
ejpam-1598	76	10	∈	∈	PROPN
ejpam-1598	76	11	z+	z+	NUM
ejpam-1598	76	12	0	0	NUM
ejpam-1598	76	13	(	(	PUNCT
ejpam-1598	76	14	set	set	VERB
ejpam-1598	76	15	of	of	ADP
ejpam-1598	76	16	all	all	DET
ejpam-1598	76	17	positive	positive	ADJ
ejpam-1598	76	18	integers	integer	NOUN
ejpam-1598	76	19	with	with	ADP
ejpam-1598	76	20	zero	zero	NUM
ejpam-1598	76	21	)	)	PUNCT
ejpam-1598	76	22	.	.	PUNCT
ejpam-1598	77	1	the	the	DET
ejpam-1598	77	2	following	follow	VERB
ejpam-1598	77	3	definitions	definition	NOUN
ejpam-1598	77	4	are	be	AUX
ejpam-1598	77	5	useful	useful	ADJ
ejpam-1598	77	6	in	in	ADP
ejpam-1598	77	7	the	the	DET
ejpam-1598	77	8	study	study	NOUN
ejpam-1598	77	9	of	of	ADP
ejpam-1598	77	10	ternary	ternary	ADJ
ejpam-1598	77	11	semirings	semiring	NOUN
ejpam-1598	77	12	.	.	PUNCT
ejpam-1598	78	1	definition	definition	NOUN
ejpam-1598	78	2	6	6	NUM
ejpam-1598	78	3	.	.	PUNCT
ejpam-1598	79	1	an	an	DET
ejpam-1598	79	2	ideal	ideal	ADJ
ejpam-1598	79	3	i	i	PRON
ejpam-1598	79	4	of	of	ADP
ejpam-1598	79	5	a	a	DET
ejpam-1598	79	6	ternary	ternary	ADJ
ejpam-1598	79	7	semiring	semiring	NOUN
ejpam-1598	79	8	s	s	NOUN
ejpam-1598	79	9	is	be	AUX
ejpam-1598	79	10	said	say	VERB
ejpam-1598	79	11	to	to	PART
ejpam-1598	79	12	be	be	AUX
ejpam-1598	79	13	a	a	DET
ejpam-1598	79	14	k	k	NOUN
ejpam-1598	79	15	-	-	NOUN
ejpam-1598	79	16	ideal	ideal	NOUN
ejpam-1598	79	17	if	if	SCONJ
ejpam-1598	79	18	x	x	PROPN
ejpam-1598	80	1	+	+	NUM
ejpam-1598	80	2	y	y	PROPN
ejpam-1598	80	3	∈	∈	PROPN
ejpam-1598	81	1	i	i	PRON
ejpam-1598	81	2	;	;	PUNCT
ejpam-1598	81	3	x	x	SYM
ejpam-1598	81	4	∈	∈	PROPN
ejpam-1598	81	5	s	s	PROPN
ejpam-1598	81	6	,	,	PUNCT
ejpam-1598	81	7	y	y	PROPN
ejpam-1598	81	8	∈	∈	PROPN
ejpam-1598	82	1	i	i	PRON
ejpam-1598	82	2	imply	imply	VERB
ejpam-1598	82	3	x	x	VERB
ejpam-1598	82	4	∈	∈	PROPN
ejpam-1598	82	5	i	i	PRON
ejpam-1598	82	6	.	.	PUNCT
ejpam-1598	83	1	definition	definition	NOUN
ejpam-1598	83	2	7	7	NUM
ejpam-1598	83	3	.	.	PUNCT
ejpam-1598	84	1	[	[	X
ejpam-1598	84	2	4	4	X
ejpam-1598	84	3	]	]	PUNCT
ejpam-1598	84	4	a	a	DET
ejpam-1598	84	5	ternary	ternary	ADJ
ejpam-1598	84	6	semiring	semiring	NOUN
ejpam-1598	84	7	(	(	PUNCT
ejpam-1598	84	8	ring	ring	NOUN
ejpam-1598	84	9	)	)	PUNCT
ejpam-1598	84	10	s	s	VERB
ejpam-1598	84	11	is	be	AUX
ejpam-1598	84	12	said	say	VERB
ejpam-1598	84	13	to	to	PART
ejpam-1598	84	14	be	be	AUX
ejpam-1598	84	15	zero	zero	NUM
ejpam-1598	84	16	divisor	divisor	NOUN
ejpam-1598	84	17	free	free	ADJ
ejpam-1598	84	18	(	(	PUNCT
ejpam-1598	84	19	zdf	zdf	PROPN
ejpam-1598	84	20	)	)	PUNCT
ejpam-1598	84	21	if	if	SCONJ
ejpam-1598	84	22	for	for	ADP
ejpam-1598	84	23	a	a	DET
ejpam-1598	84	24	,	,	PUNCT
ejpam-1598	84	25	b	b	NOUN
ejpam-1598	84	26	,	,	PUNCT
ejpam-1598	84	27	c	c	PROPN
ejpam-1598	84	28	∈	∈	PROPN
ejpam-1598	84	29	s	s	PART
ejpam-1598	84	30	,	,	PUNCT
ejpam-1598	84	31	abc	abc	PROPN
ejpam-1598	84	32	=	=	SYM
ejpam-1598	84	33	0	0	NUM
ejpam-1598	84	34	implies	imply	VERB
ejpam-1598	84	35	a	a	DET
ejpam-1598	84	36	=	=	SYM
ejpam-1598	84	37	0	0	NUM
ejpam-1598	84	38	or	or	CCONJ
ejpam-1598	84	39	b	b	X
ejpam-1598	84	40	=	=	SYM
ejpam-1598	84	41	0	0	NUM
ejpam-1598	84	42	or	or	CCONJ
ejpam-1598	84	43	c	c	NOUN
ejpam-1598	84	44	=	=	SYM
ejpam-1598	84	45	0	0	PROPN
ejpam-1598	84	46	.	.	PUNCT
ejpam-1598	84	47	definition	definition	NOUN
ejpam-1598	84	48	8	8	NUM
ejpam-1598	84	49	.	.	PUNCT
ejpam-1598	85	1	[	[	X
ejpam-1598	85	2	4	4	X
ejpam-1598	85	3	]	]	PUNCT
ejpam-1598	85	4	a	a	DET
ejpam-1598	85	5	ternary	ternary	ADJ
ejpam-1598	85	6	semiring	semiring	NOUN
ejpam-1598	85	7	s	s	X
ejpam-1598	85	8	is	be	AUX
ejpam-1598	85	9	said	say	VERB
ejpam-1598	85	10	to	to	PART
ejpam-1598	85	11	be	be	AUX
ejpam-1598	85	12	commutative	commutative	ADJ
ejpam-1598	85	13	if	if	SCONJ
ejpam-1598	85	14	abc	abc	PROPN
ejpam-1598	85	15	=	=	SYM
ejpam-1598	85	16	bac	bac	PROPN
ejpam-1598	85	17	=	=	PUNCT
ejpam-1598	85	18	bca	bca	PROPN
ejpam-1598	85	19	for	for	ADP
ejpam-1598	85	20	all	all	DET
ejpam-1598	85	21	a	a	DET
ejpam-1598	85	22	,	,	PUNCT
ejpam-1598	85	23	b	b	NOUN
ejpam-1598	85	24	,	,	PUNCT
ejpam-1598	85	25	c	c	PROPN
ejpam-1598	85	26	∈	∈	PROPN
ejpam-1598	85	27	s.	s.	PROPN
ejpam-1598	85	28	definition	definition	NOUN
ejpam-1598	85	29	9	9	NUM
ejpam-1598	85	30	.	.	PUNCT
ejpam-1598	86	1	[	[	X
ejpam-1598	86	2	4	4	X
ejpam-1598	86	3	]	]	PUNCT
ejpam-1598	86	4	a	a	DET
ejpam-1598	86	5	commutative	commutative	ADJ
ejpam-1598	86	6	ternary	ternary	ADJ
ejpam-1598	86	7	semiring	semiring	NOUN
ejpam-1598	86	8	(	(	PUNCT
ejpam-1598	86	9	ring	ring	NOUN
ejpam-1598	86	10	)	)	PUNCT
ejpam-1598	86	11	is	be	AUX
ejpam-1598	86	12	called	call	VERB
ejpam-1598	86	13	a	a	DET
ejpam-1598	86	14	ternary	ternary	ADJ
ejpam-1598	86	15	semi	semi	ADJ
ejpam-1598	86	16	-	-	ADJ
ejpam-1598	86	17	integral	integral	ADJ
ejpam-1598	86	18	(	(	PUNCT
ejpam-1598	86	19	resp	resp	NOUN
ejpam-1598	86	20	.	.	PUNCT
ejpam-1598	86	21	integral	integral	ADJ
ejpam-1598	86	22	)	)	PUNCT
ejpam-1598	86	23	domain	domain	NOUN
ejpam-1598	86	24	if	if	SCONJ
ejpam-1598	86	25	it	it	PRON
ejpam-1598	86	26	is	be	AUX
ejpam-1598	86	27	zero	zero	NUM
ejpam-1598	86	28	divisor	divisor	NOUN
ejpam-1598	86	29	free	free	ADJ
ejpam-1598	86	30	(	(	PUNCT
ejpam-1598	86	31	zdf	zdf	PROPN
ejpam-1598	86	32	)	)	PUNCT
ejpam-1598	86	33	.	.	PUNCT
ejpam-1598	87	1	definition	definition	NOUN
ejpam-1598	87	2	10	10	NUM
ejpam-1598	87	3	.	.	PUNCT
ejpam-1598	88	1	[	[	X
ejpam-1598	88	2	2	2	X
ejpam-1598	88	3	]	]	PUNCT
ejpam-1598	88	4	a	a	DET
ejpam-1598	88	5	proper	proper	ADJ
ejpam-1598	88	6	ideal	ideal	NOUN
ejpam-1598	88	7	p	p	NOUN
ejpam-1598	88	8	of	of	ADP
ejpam-1598	88	9	a	a	DET
ejpam-1598	88	10	ternary	ternary	ADJ
ejpam-1598	88	11	semiring	semiring	NOUN
ejpam-1598	88	12	s	s	VERB
ejpam-1598	88	13	is	be	AUX
ejpam-1598	88	14	called	call	VERB
ejpam-1598	88	15	a	a	DET
ejpam-1598	88	16	prime	prime	ADJ
ejpam-1598	88	17	ideal	ideal	NOUN
ejpam-1598	88	18	of	of	ADP
ejpam-1598	88	19	s	s	PRON
ejpam-1598	88	20	if	if	SCONJ
ejpam-1598	88	21	for	for	ADP
ejpam-1598	88	22	any	any	DET
ejpam-1598	88	23	three	three	NUM
ejpam-1598	88	24	ideals	ideal	NOUN
ejpam-1598	88	25	a	a	DET
ejpam-1598	88	26	,	,	PUNCT
ejpam-1598	88	27	b	b	NOUN
ejpam-1598	88	28	,	,	PUNCT
ejpam-1598	88	29	c	c	NOUN
ejpam-1598	88	30	of	of	ADP
ejpam-1598	88	31	s	s	PROPN
ejpam-1598	88	32	;	;	PUNCT
ejpam-1598	88	33	abc	abc	PROPN
ejpam-1598	88	34	⊆	⊆	NUM
ejpam-1598	88	35	p	p	PROPN
ejpam-1598	88	36	implies	imply	VERB
ejpam-1598	88	37	a⊆	a⊆	PROPN
ejpam-1598	88	38	p	p	NOUN
ejpam-1598	88	39	or	or	CCONJ
ejpam-1598	88	40	b	b	NOUN
ejpam-1598	88	41	⊆	⊆	NUM
ejpam-1598	88	42	p	p	NOUN
ejpam-1598	88	43	or	or	CCONJ
ejpam-1598	88	44	c	c	NOUN
ejpam-1598	88	45	⊆	⊆	NUM
ejpam-1598	88	46	p.	p.	NOUN
ejpam-1598	88	47	definition	definition	NOUN
ejpam-1598	88	48	11	11	NUM
ejpam-1598	88	49	.	.	PUNCT
ejpam-1598	89	1	[	[	X
ejpam-1598	89	2	2	2	X
ejpam-1598	89	3	]	]	PUNCT
ejpam-1598	89	4	a	a	DET
ejpam-1598	89	5	ternary	ternary	ADJ
ejpam-1598	89	6	semiring	semiring	NOUN
ejpam-1598	89	7	s	s	VERB
ejpam-1598	89	8	is	be	AUX
ejpam-1598	89	9	called	call	VERB
ejpam-1598	89	10	a	a	DET
ejpam-1598	89	11	prime	prime	ADJ
ejpam-1598	89	12	ternary	ternary	NOUN
ejpam-1598	89	13	semiring	semiring	NOUN
ejpam-1598	89	14	if	if	SCONJ
ejpam-1598	89	15	the	the	DET
ejpam-1598	89	16	zero	zero	NUM
ejpam-1598	89	17	ideal	ideal	NOUN
ejpam-1598	89	18	{	{	PUNCT
ejpam-1598	89	19	0	0	NUM
ejpam-1598	89	20	}	}	PUNCT
ejpam-1598	89	21	is	be	AUX
ejpam-1598	89	22	a	a	DET
ejpam-1598	89	23	prime	prime	ADJ
ejpam-1598	89	24	ideal	ideal	NOUN
ejpam-1598	89	25	of	of	ADP
ejpam-1598	89	26	s.	s.	PROPN
ejpam-1598	89	27	definition	definition	NOUN
ejpam-1598	89	28	12	12	NUM
ejpam-1598	89	29	.	.	PUNCT
ejpam-1598	90	1	[	[	X
ejpam-1598	90	2	3	3	X
ejpam-1598	90	3	]	]	X
ejpam-1598	90	4	a	a	DET
ejpam-1598	90	5	proper	proper	ADJ
ejpam-1598	90	6	ideal	ideal	NOUN
ejpam-1598	90	7	p	p	NOUN
ejpam-1598	90	8	of	of	ADP
ejpam-1598	90	9	a	a	DET
ejpam-1598	90	10	ternary	ternary	ADJ
ejpam-1598	90	11	semiring	semiring	NOUN
ejpam-1598	90	12	s	s	VERB
ejpam-1598	90	13	is	be	AUX
ejpam-1598	90	14	called	call	VERB
ejpam-1598	90	15	a	a	DET
ejpam-1598	90	16	semiprime	semiprime	NOUN
ejpam-1598	90	17	of	of	ADP
ejpam-1598	90	18	s	s	PRON
ejpam-1598	90	19	if	if	SCONJ
ejpam-1598	90	20	for	for	ADP
ejpam-1598	90	21	any	any	DET
ejpam-1598	90	22	ideal	ideal	NOUN
ejpam-1598	90	23	a	a	PRON
ejpam-1598	90	24	of	of	ADP
ejpam-1598	90	25	s	s	NOUN
ejpam-1598	90	26	;	;	PUNCT
ejpam-1598	90	27	a3	a3	VERB
ejpam-1598	90	28	⊆	⊆	PROPN
ejpam-1598	90	29	p	p	NOUN
ejpam-1598	90	30	implies	imply	VERB
ejpam-1598	90	31	a⊆	a⊆	PROPN
ejpam-1598	90	32	p.	p.	NOUN
ejpam-1598	90	33	t.	t.	PROPN
ejpam-1598	90	34	dutta	dutta	PROPN
ejpam-1598	90	35	,	,	PUNCT
ejpam-1598	90	36	k.	k.	PROPN
ejpam-1598	90	37	shum	shum	PROPN
ejpam-1598	90	38	,	,	PUNCT
ejpam-1598	90	39	s.	s.	PROPN
ejpam-1598	90	40	mandal	mandal	PROPN
ejpam-1598	90	41	/	/	SYM
ejpam-1598	90	42	eur	eur	PROPN
ejpam-1598	90	43	.	.	PUNCT
ejpam-1598	91	1	j.	j.	PROPN
ejpam-1598	91	2	pure	pure	PROPN
ejpam-1598	91	3	appl	appl	PROPN
ejpam-1598	91	4	.	.	PROPN
ejpam-1598	91	5	math	math	PROPN
ejpam-1598	91	6	,	,	PUNCT
ejpam-1598	91	7	5	5	NUM
ejpam-1598	91	8	(	(	PUNCT
ejpam-1598	91	9	2012	2012	NUM
ejpam-1598	91	10	)	)	PUNCT
ejpam-1598	91	11	,	,	PUNCT
ejpam-1598	91	12	116	116	NUM
ejpam-1598	91	13	-	-	SYM
ejpam-1598	91	14	128	128	NUM
ejpam-1598	91	15	119	119	NUM
ejpam-1598	91	16	definition	definition	NOUN
ejpam-1598	91	17	13	13	NUM
ejpam-1598	91	18	.	.	PUNCT
ejpam-1598	92	1	[	[	X
ejpam-1598	92	2	3	3	X
ejpam-1598	92	3	]	]	PUNCT
ejpam-1598	92	4	a	a	DET
ejpam-1598	92	5	ternary	ternary	ADJ
ejpam-1598	92	6	semiring	semiring	NOUN
ejpam-1598	92	7	s	s	VERB
ejpam-1598	92	8	is	be	AUX
ejpam-1598	92	9	called	call	VERB
ejpam-1598	92	10	a	a	DET
ejpam-1598	92	11	semiprime	semiprime	NOUN
ejpam-1598	92	12	ternary	ternary	NOUN
ejpam-1598	92	13	semiring	semiring	NOUN
ejpam-1598	92	14	if	if	SCONJ
ejpam-1598	92	15	the	the	DET
ejpam-1598	92	16	zero	zero	NUM
ejpam-1598	92	17	ideal	ideal	NOUN
ejpam-1598	92	18	(	(	PUNCT
ejpam-1598	92	19	0	0	NUM
ejpam-1598	92	20	)	)	PUNCT
ejpam-1598	92	21	is	be	AUX
ejpam-1598	92	22	a	a	DET
ejpam-1598	92	23	semiprime	semiprime	NOUN
ejpam-1598	92	24	ideal	ideal	NOUN
ejpam-1598	92	25	of	of	ADP
ejpam-1598	92	26	s.	s.	PROPN
ejpam-1598	92	27	definition	definition	NOUN
ejpam-1598	92	28	14	14	NUM
ejpam-1598	92	29	.	.	PUNCT
ejpam-1598	93	1	[	[	X
ejpam-1598	93	2	5	5	NUM
ejpam-1598	93	3	]	]	PUNCT
ejpam-1598	93	4	an	an	DET
ejpam-1598	93	5	additive	additive	ADJ
ejpam-1598	93	6	commutative	commutative	ADJ
ejpam-1598	93	7	semigroup	semigroup	NOUN
ejpam-1598	93	8	m	m	VERB
ejpam-1598	93	9	with	with	ADP
ejpam-1598	93	10	a	a	DET
ejpam-1598	93	11	zero	zero	NUM
ejpam-1598	93	12	element	element	NOUN
ejpam-1598	93	13	0	0	NUM
ejpam-1598	93	14	m	m	NOUN
ejpam-1598	93	15	is	be	AUX
ejpam-1598	93	16	called	call	VERB
ejpam-1598	93	17	a	a	DET
ejpam-1598	93	18	right	right	ADJ
ejpam-1598	93	19	ternary	ternary	ADJ
ejpam-1598	93	20	semimodule	semimodule	NOUN
ejpam-1598	93	21	over	over	ADP
ejpam-1598	93	22	a	a	DET
ejpam-1598	93	23	ternary	ternary	ADJ
ejpam-1598	93	24	semiring	semiring	NOUN
ejpam-1598	93	25	s	s	X
ejpam-1598	93	26	or	or	CCONJ
ejpam-1598	93	27	simply	simply	ADV
ejpam-1598	93	28	a	a	DET
ejpam-1598	93	29	right	right	ADJ
ejpam-1598	93	30	ternary	ternary	ADJ
ejpam-1598	93	31	s	s	NOUN
ejpam-1598	93	32	-	-	PUNCT
ejpam-1598	93	33	semimodule	semimodule	NOUN
ejpam-1598	93	34	if	if	SCONJ
ejpam-1598	93	35	there	there	PRON
ejpam-1598	93	36	exists	exist	VERB
ejpam-1598	93	37	a	a	DET
ejpam-1598	93	38	mapping	mapping	NOUN
ejpam-1598	93	39	m	m	NOUN
ejpam-1598	93	40	×	×	NOUN
ejpam-1598	93	41	s	s	PART
ejpam-1598	93	42	×	×	NOUN
ejpam-1598	93	43	s	s	PART
ejpam-1598	93	44	−→	−→	NOUN
ejpam-1598	93	45	m	m	NOUN
ejpam-1598	93	46	(	(	PUNCT
ejpam-1598	93	47	images	image	NOUN
ejpam-1598	93	48	to	to	PART
ejpam-1598	93	49	be	be	AUX
ejpam-1598	93	50	denoted	denote	VERB
ejpam-1598	93	51	by	by	ADP
ejpam-1598	93	52	ms1s2	ms1s2	PROPN
ejpam-1598	93	53	for	for	ADP
ejpam-1598	93	54	all	all	DET
ejpam-1598	93	55	m	m	NOUN
ejpam-1598	93	56	∈	∈	ADJ
ejpam-1598	93	57	m	m	NOUN
ejpam-1598	93	58	and	and	CCONJ
ejpam-1598	93	59	s1	s1	NOUN
ejpam-1598	93	60	,	,	PUNCT
ejpam-1598	93	61	s2	s2	NOUN
ejpam-1598	93	62	∈	∈	PROPN
ejpam-1598	93	63	s	s	PART
ejpam-1598	93	64	)	)	PUNCT
ejpam-1598	93	65	satisfying	satisfy	VERB
ejpam-1598	93	66	the	the	DET
ejpam-1598	93	67	following	follow	VERB
ejpam-1598	93	68	conditions	condition	NOUN
ejpam-1598	93	69	:	:	PUNCT
ejpam-1598	93	70	(	(	PUNCT
ejpam-1598	93	71	i	i	NOUN
ejpam-1598	93	72	)	)	PUNCT
ejpam-1598	93	73	(	(	PUNCT
ejpam-1598	93	74	m1	m1	PROPN
ejpam-1598	93	75	+	+	NOUN
ejpam-1598	93	76	m2)s1s2	m2)s1s2	PROPN
ejpam-1598	93	77	=	=	PUNCT
ejpam-1598	94	1	m1s1s2	m1s1s2	NOUN
ejpam-1598	94	2	+	+	PROPN
ejpam-1598	94	3	m2s1s2	m2s1s2	PROPN
ejpam-1598	94	4	(	(	PUNCT
ejpam-1598	94	5	ii	ii	NOUN
ejpam-1598	94	6	)	)	PUNCT
ejpam-1598	94	7	m1s1(s2	m1s1(s2	X
ejpam-1598	95	1	+	+	CCONJ
ejpam-1598	95	2	s3	s3	PROPN
ejpam-1598	95	3	)	)	PUNCT
ejpam-1598	95	4	=	=	SYM
ejpam-1598	96	1	m1s1s2	m1s1s2	PROPN
ejpam-1598	96	2	+	+	NUM
ejpam-1598	96	3	m1s1s3	m1s1s3	PROPN
ejpam-1598	96	4	(	(	PUNCT
ejpam-1598	96	5	iii	iii	NOUN
ejpam-1598	96	6	)	)	PUNCT
ejpam-1598	96	7	m1(s1	m1(s1	NOUN
ejpam-1598	96	8	+	+	CCONJ
ejpam-1598	96	9	s2)s3	s2)s3	ADJ
ejpam-1598	96	10	=	=	PUNCT
ejpam-1598	96	11	m1s1s3	m1s1s3	PROPN
ejpam-1598	96	12	+	+	PROPN
ejpam-1598	96	13	m1s2s3	m1s2s3	PROPN
ejpam-1598	96	14	(	(	PUNCT
ejpam-1598	96	15	iv	iv	NUM
ejpam-1598	96	16	)	)	PUNCT
ejpam-1598	96	17	(	(	PUNCT
ejpam-1598	96	18	m1s1s2)s3s4	m1s1s2)s3s4	NOUN
ejpam-1598	96	19	=	=	SYM
ejpam-1598	96	20	m1(s1s2s3)s4	m1(s1s2s3)s4	NOUN
ejpam-1598	96	21	=	=	SYM
ejpam-1598	96	22	m1s1(s2s3s4	m1s1(s2s3s4	PROPN
ejpam-1598	96	23	)	)	PUNCT
ejpam-1598	96	24	(	(	PUNCT
ejpam-1598	96	25	v	v	NOUN
ejpam-1598	96	26	)	)	PUNCT
ejpam-1598	96	27	0	0	NUM
ejpam-1598	96	28	m	m	VERB
ejpam-1598	96	29	s1s2	s1s2	NOUN
ejpam-1598	97	1	=	=	VERB
ejpam-1598	97	2	0	0	NUM
ejpam-1598	97	3	m	m	NOUN
ejpam-1598	97	4	=	=	SYM
ejpam-1598	97	5	m1s10s	m1s10s	PROPN
ejpam-1598	97	6	=	=	SYM
ejpam-1598	97	7	m10ss2	m10ss2	PROPN
ejpam-1598	97	8	for	for	ADP
ejpam-1598	97	9	all	all	DET
ejpam-1598	97	10	m1	m1	NOUN
ejpam-1598	97	11	,	,	PUNCT
ejpam-1598	97	12	m2	m2	PROPN
ejpam-1598	97	13	∈	∈	PROPN
ejpam-1598	97	14	m	m	PROPN
ejpam-1598	97	15	and	and	CCONJ
ejpam-1598	97	16	for	for	ADP
ejpam-1598	97	17	all	all	DET
ejpam-1598	97	18	s1	s1	NOUN
ejpam-1598	97	19	,	,	PUNCT
ejpam-1598	97	20	s2	s2	PROPN
ejpam-1598	97	21	,	,	PUNCT
ejpam-1598	97	22	s3	s3	PROPN
ejpam-1598	97	23	,	,	PUNCT
ejpam-1598	97	24	s4	s4	PROPN
ejpam-1598	97	25	∈	∈	PROPN
ejpam-1598	97	26	s.	s.	PROPN
ejpam-1598	97	27	a	a	DET
ejpam-1598	97	28	left	left	ADJ
ejpam-1598	97	29	ternary	ternary	ADJ
ejpam-1598	97	30	s	s	NOUN
ejpam-1598	97	31	-	-	PUNCT
ejpam-1598	97	32	semimodule	semimodule	NOUN
ejpam-1598	97	33	can	can	AUX
ejpam-1598	97	34	be	be	AUX
ejpam-1598	97	35	similarly	similarly	ADV
ejpam-1598	97	36	defined	define	VERB
ejpam-1598	97	37	.	.	PUNCT
ejpam-1598	98	1	definition	definition	NOUN
ejpam-1598	98	2	15	15	NUM
ejpam-1598	98	3	.	.	PUNCT
ejpam-1598	99	1	[	[	X
ejpam-1598	99	2	5	5	NUM
ejpam-1598	99	3	]	]	PUNCT
ejpam-1598	99	4	a	a	DET
ejpam-1598	99	5	non	non	ADJ
ejpam-1598	99	6	-	-	ADJ
ejpam-1598	99	7	empty	empty	ADJ
ejpam-1598	99	8	subset	subset	NOUN
ejpam-1598	99	9	n	n	PROPN
ejpam-1598	99	10	of	of	ADP
ejpam-1598	99	11	a	a	DET
ejpam-1598	99	12	right	right	ADJ
ejpam-1598	99	13	ternary	ternary	ADJ
ejpam-1598	99	14	s	s	NOUN
ejpam-1598	99	15	-	-	PUNCT
ejpam-1598	99	16	semimodule	semimodule	NOUN
ejpam-1598	99	17	m	m	NOUN
ejpam-1598	99	18	is	be	AUX
ejpam-1598	99	19	said	say	VERB
ejpam-1598	99	20	to	to	PART
ejpam-1598	99	21	be	be	AUX
ejpam-1598	99	22	a	a	DET
ejpam-1598	99	23	ternary	ternary	ADJ
ejpam-1598	99	24	subsemimodule	subsemimodule	NOUN
ejpam-1598	99	25	of	of	ADP
ejpam-1598	99	26	m	m	PRON
ejpam-1598	99	27	if	if	SCONJ
ejpam-1598	99	28	(	(	PUNCT
ejpam-1598	99	29	i	i	NOUN
ejpam-1598	99	30	)	)	PUNCT
ejpam-1598	99	31	a+	a+	PUNCT
ejpam-1598	100	1	b	b	X
ejpam-1598	100	2	∈	∈	PROPN
ejpam-1598	100	3	n	n	NOUN
ejpam-1598	100	4	and	and	CCONJ
ejpam-1598	100	5	(	(	PUNCT
ejpam-1598	100	6	ii	ii	NOUN
ejpam-1598	100	7	)	)	PUNCT
ejpam-1598	100	8	ast	ast	NOUN
ejpam-1598	100	9	∈	∈	PROPN
ejpam-1598	100	10	n	n	CCONJ
ejpam-1598	100	11	for	for	ADP
ejpam-1598	100	12	all	all	DET
ejpam-1598	100	13	a	a	PRON
ejpam-1598	100	14	,	,	PUNCT
ejpam-1598	100	15	b	b	X
ejpam-1598	100	16	∈	∈	PROPN
ejpam-1598	100	17	n	n	NOUN
ejpam-1598	100	18	and	and	CCONJ
ejpam-1598	100	19	s	s	PROPN
ejpam-1598	100	20	,	,	PUNCT
ejpam-1598	100	21	t	t	PROPN
ejpam-1598	100	22	∈	∈	PROPN
ejpam-1598	100	23	s.	s.	PROPN
ejpam-1598	100	24	definition	definition	NOUN
ejpam-1598	100	25	16	16	NUM
ejpam-1598	100	26	.	.	PUNCT
ejpam-1598	101	1	a	a	DET
ejpam-1598	101	2	nonzero	nonzero	ADJ
ejpam-1598	101	3	right	right	ADJ
ejpam-1598	101	4	ideal	ideal	NOUN
ejpam-1598	101	5	i	i	PRON
ejpam-1598	101	6	of	of	ADP
ejpam-1598	101	7	a	a	DET
ejpam-1598	101	8	ternary	ternary	ADJ
ejpam-1598	101	9	semiring	semiring	NOUN
ejpam-1598	101	10	s	s	NOUN
ejpam-1598	101	11	with	with	ADP
ejpam-1598	101	12	zero	zero	NUM
ejpam-1598	101	13	is	be	AUX
ejpam-1598	101	14	called	call	VERB
ejpam-1598	101	15	an	an	DET
ejpam-1598	101	16	essential	essential	ADJ
ejpam-1598	101	17	right	right	ADJ
ejpam-1598	101	18	ideal	ideal	NOUN
ejpam-1598	101	19	of	of	ADP
ejpam-1598	101	20	s	s	PRON
ejpam-1598	101	21	if	if	SCONJ
ejpam-1598	101	22	for	for	ADP
ejpam-1598	101	23	any	any	DET
ejpam-1598	101	24	nonzero	nonzero	NOUN
ejpam-1598	101	25	right	right	ADV
ejpam-1598	101	26	ideal	ideal	NOUN
ejpam-1598	101	27	j	j	PROPN
ejpam-1598	101	28	of	of	ADP
ejpam-1598	101	29	s	s	PROPN
ejpam-1598	101	30	,	,	PUNCT
ejpam-1598	101	31	i	i	PROPN
ejpam-1598	101	32	∩	∩	PROPN
ejpam-1598	101	33	j	j	PROPN
ejpam-1598	101	34	6=	6=	SYM
ejpam-1598	101	35	(	(	PUNCT
ejpam-1598	101	36	0	0	NUM
ejpam-1598	101	37	)	)	PUNCT
ejpam-1598	101	38	.	.	PUNCT
ejpam-1598	102	1	definition	definition	NOUN
ejpam-1598	102	2	17	17	NUM
ejpam-1598	102	3	.	.	PUNCT
ejpam-1598	103	1	a	a	DET
ejpam-1598	103	2	class	class	NOUN
ejpam-1598	103	3	ρ	ρ	NOUN
ejpam-1598	103	4	of	of	ADP
ejpam-1598	103	5	ternary	ternary	ADJ
ejpam-1598	103	6	semirings	semiring	NOUN
ejpam-1598	103	7	is	be	AUX
ejpam-1598	103	8	called	call	VERB
ejpam-1598	103	9	hereditary	hereditary	ADJ
ejpam-1598	103	10	if	if	SCONJ
ejpam-1598	103	11	i	i	PRON
ejpam-1598	103	12	is	be	AUX
ejpam-1598	103	13	an	an	DET
ejpam-1598	103	14	ideal	ideal	NOUN
ejpam-1598	103	15	of	of	ADP
ejpam-1598	103	16	a	a	DET
ejpam-1598	103	17	ternary	ternary	ADJ
ejpam-1598	103	18	semiring	semiring	NOUN
ejpam-1598	103	19	s	s	X
ejpam-1598	103	20	and	and	CCONJ
ejpam-1598	103	21	s	s	PROPN
ejpam-1598	103	22	∈	∈	NOUN
ejpam-1598	103	23	ρ	ρ	NOUN
ejpam-1598	103	24	implies	imply	VERB
ejpam-1598	103	25	i	i	PROPN
ejpam-1598	103	26	∈	∈	PROPN
ejpam-1598	103	27	ρ	ρ	PROPN
ejpam-1598	103	28	.	.	PUNCT
ejpam-1598	103	29	definition	definition	NOUN
ejpam-1598	103	30	18	18	NUM
ejpam-1598	103	31	.	.	PUNCT
ejpam-1598	104	1	[	[	X
ejpam-1598	104	2	1	1	X
ejpam-1598	104	3	]	]	PUNCT
ejpam-1598	104	4	let	let	VERB
ejpam-1598	104	5	s	s	PRON
ejpam-1598	104	6	and	and	CCONJ
ejpam-1598	104	7	t	t	PROPN
ejpam-1598	104	8	be	be	AUX
ejpam-1598	104	9	two	two	NUM
ejpam-1598	104	10	ternary	ternary	ADJ
ejpam-1598	104	11	semirings	semiring	NOUN
ejpam-1598	104	12	and	and	CCONJ
ejpam-1598	104	13	f	f	PROPN
ejpam-1598	104	14	be	be	AUX
ejpam-1598	104	15	a	a	DET
ejpam-1598	104	16	mapping	mapping	NOUN
ejpam-1598	104	17	which	which	PRON
ejpam-1598	104	18	maps	map	VERB
ejpam-1598	104	19	s	s	VERB
ejpam-1598	104	20	into	into	ADP
ejpam-1598	104	21	t	t	PROPN
ejpam-1598	104	22	.	.	PUNCT
ejpam-1598	105	1	then	then	ADV
ejpam-1598	105	2	the	the	DET
ejpam-1598	105	3	mapping	mapping	NOUN
ejpam-1598	105	4	f	f	X
ejpam-1598	105	5	:	:	PUNCT
ejpam-1598	105	6	s	s	AUX
ejpam-1598	105	7	−→	−→	NOUN
ejpam-1598	105	8	t	t	PROPN
ejpam-1598	105	9	is	be	AUX
ejpam-1598	105	10	called	call	VERB
ejpam-1598	105	11	a	a	DET
ejpam-1598	105	12	homomorphism	homomorphism	NOUN
ejpam-1598	105	13	of	of	ADP
ejpam-1598	105	14	s	s	PROPN
ejpam-1598	105	15	into	into	ADP
ejpam-1598	105	16	t	t	PROPN
ejpam-1598	105	17	if	if	SCONJ
ejpam-1598	105	18	the	the	DET
ejpam-1598	105	19	following	follow	VERB
ejpam-1598	105	20	conditions	condition	NOUN
ejpam-1598	105	21	hold	hold	VERB
ejpam-1598	105	22	:	:	PUNCT
ejpam-1598	105	23	(	(	PUNCT
ejpam-1598	105	24	i	i	NOUN
ejpam-1598	105	25	)	)	PUNCT
ejpam-1598	105	26	f	f	PROPN
ejpam-1598	105	27	(	(	PUNCT
ejpam-1598	105	28	a+	a+	NOUN
ejpam-1598	105	29	b	b	NOUN
ejpam-1598	105	30	)	)	PUNCT
ejpam-1598	106	1	=	=	SYM
ejpam-1598	106	2	f	f	PROPN
ejpam-1598	106	3	(	(	PUNCT
ejpam-1598	106	4	a)+	a)+	NOUN
ejpam-1598	106	5	f	f	X
ejpam-1598	106	6	(	(	PUNCT
ejpam-1598	106	7	b	b	NOUN
ejpam-1598	106	8	)	)	PUNCT
ejpam-1598	106	9	and	and	CCONJ
ejpam-1598	106	10	(	(	PUNCT
ejpam-1598	106	11	ii	ii	PROPN
ejpam-1598	106	12	)	)	PUNCT
ejpam-1598	106	13	f	f	PROPN
ejpam-1598	106	14	(	(	PUNCT
ejpam-1598	106	15	abc	abc	PROPN
ejpam-1598	106	16	)	)	PUNCT
ejpam-1598	107	1	=	=	SYM
ejpam-1598	107	2	f	f	PROPN
ejpam-1598	107	3	(	(	PUNCT
ejpam-1598	107	4	a	a	NOUN
ejpam-1598	107	5	)	)	PUNCT
ejpam-1598	107	6	f	f	NOUN
ejpam-1598	107	7	(	(	PUNCT
ejpam-1598	107	8	b	b	NOUN
ejpam-1598	107	9	)	)	PUNCT
ejpam-1598	107	10	f	f	NOUN
ejpam-1598	107	11	(	(	PUNCT
ejpam-1598	107	12	c	c	NOUN
ejpam-1598	107	13	)	)	PUNCT
ejpam-1598	107	14	for	for	ADP
ejpam-1598	107	15	all	all	DET
ejpam-1598	107	16	a	a	DET
ejpam-1598	107	17	,	,	PUNCT
ejpam-1598	107	18	b	b	NOUN
ejpam-1598	107	19	,	,	PUNCT
ejpam-1598	107	20	c	c	PROPN
ejpam-1598	107	21	∈	∈	PROPN
ejpam-1598	107	22	s.	s.	PROPN
ejpam-1598	108	1	moreover	moreover	ADV
ejpam-1598	108	2	,	,	PUNCT
ejpam-1598	108	3	if	if	SCONJ
ejpam-1598	108	4	both	both	CCONJ
ejpam-1598	108	5	the	the	DET
ejpam-1598	108	6	ternary	ternary	ADJ
ejpam-1598	108	7	semirings	semiring	NOUN
ejpam-1598	108	8	s	s	PART
ejpam-1598	108	9	and	and	CCONJ
ejpam-1598	108	10	t	t	PROPN
ejpam-1598	108	11	have	have	VERB
ejpam-1598	108	12	zeros	zero	NOUN
ejpam-1598	108	13	0s	0s	NOUN
ejpam-1598	108	14	and	and	CCONJ
ejpam-1598	108	15	0	0	NUM
ejpam-1598	108	16	t	t	NOUN
ejpam-1598	108	17	,	,	PUNCT
ejpam-1598	108	18	respectively	respectively	ADV
ejpam-1598	108	19	,	,	PUNCT
ejpam-1598	108	20	then	then	ADV
ejpam-1598	108	21	the	the	DET
ejpam-1598	108	22	following	follow	VERB
ejpam-1598	108	23	condition	condition	NOUN
ejpam-1598	108	24	:	:	PUNCT
ejpam-1598	108	25	(	(	PUNCT
ejpam-1598	108	26	iii	iii	X
ejpam-1598	108	27	)	)	PUNCT
ejpam-1598	108	28	f	f	NOUN
ejpam-1598	108	29	(	(	PUNCT
ejpam-1598	108	30	0s	0s	NUM
ejpam-1598	108	31	)	)	PUNCT
ejpam-1598	108	32	=	=	SYM
ejpam-1598	109	1	0	0	NUM
ejpam-1598	109	2	t	t	NOUN
ejpam-1598	109	3	also	also	ADV
ejpam-1598	109	4	holds	hold	VERB
ejpam-1598	109	5	.	.	PUNCT
ejpam-1598	110	1	we	we	PRON
ejpam-1598	110	2	define	define	VERB
ejpam-1598	110	3	ker	ker	PROPN
ejpam-1598	111	1	f	f	PROPN
ejpam-1598	111	2	=	=	PRON
ejpam-1598	111	3	{	{	PUNCT
ejpam-1598	111	4	x	x	PUNCT
ejpam-1598	111	5	∈	∈	PROPN
ejpam-1598	111	6	s	s	PART
ejpam-1598	111	7	:	:	PUNCT
ejpam-1598	111	8	f	f	PROPN
ejpam-1598	111	9	(	(	PUNCT
ejpam-1598	111	10	x	x	X
ejpam-1598	111	11	)	)	PUNCT
ejpam-1598	111	12	=	=	SYM
ejpam-1598	111	13	0	0	NUM
ejpam-1598	111	14	t	t	NOUN
ejpam-1598	111	15	}	}	PUNCT
ejpam-1598	111	16	.	.	PUNCT
ejpam-1598	112	1	throughout	throughout	ADP
ejpam-1598	112	2	this	this	DET
ejpam-1598	112	3	paper	paper	NOUN
ejpam-1598	112	4	,	,	PUNCT
ejpam-1598	112	5	we	we	PRON
ejpam-1598	112	6	use	use	VERB
ejpam-1598	112	7	s	s	VERB
ejpam-1598	112	8	to	to	PART
ejpam-1598	112	9	denote	denote	VERB
ejpam-1598	112	10	a	a	DET
ejpam-1598	112	11	ternary	ternary	ADJ
ejpam-1598	112	12	semiring	semiring	NOUN
ejpam-1598	112	13	with	with	ADP
ejpam-1598	112	14	zero	zero	NUM
ejpam-1598	112	15	and	and	CCONJ
ejpam-1598	112	16	s∗	s∗	PROPN
ejpam-1598	112	17	=	=	SYM
ejpam-1598	112	18	s	s	PART
ejpam-1598	112	19	\	\	X
ejpam-1598	112	20	{	{	PUNCT
ejpam-1598	112	21	0	0	NUM
ejpam-1598	112	22	}	}	PUNCT
ejpam-1598	112	23	.	.	PUNCT
ejpam-1598	113	1	we	we	PRON
ejpam-1598	113	2	now	now	ADV
ejpam-1598	113	3	use	use	VERB
ejpam-1598	113	4	m	m	PRON
ejpam-1598	113	5	to	to	PART
ejpam-1598	113	6	denote	denote	VERB
ejpam-1598	113	7	a	a	DET
ejpam-1598	113	8	right	right	ADJ
ejpam-1598	113	9	ternary	ternary	ADJ
ejpam-1598	113	10	s	s	NOUN
ejpam-1598	113	11	-	-	PUNCT
ejpam-1598	113	12	semimodule	semimodule	NOUN
ejpam-1598	113	13	with	with	ADP
ejpam-1598	113	14	zero	zero	NUM
ejpam-1598	113	15	.	.	PUNCT
ejpam-1598	114	1	3	3	X
ejpam-1598	114	2	.	.	NOUN
ejpam-1598	114	3	singular	singular	ADJ
ejpam-1598	114	4	ideals	ideal	NOUN
ejpam-1598	114	5	in	in	ADP
ejpam-1598	114	6	this	this	DET
ejpam-1598	114	7	section	section	NOUN
ejpam-1598	114	8	,	,	PUNCT
ejpam-1598	114	9	s	s	PART
ejpam-1598	114	10	is	be	AUX
ejpam-1598	114	11	a	a	DET
ejpam-1598	114	12	ternary	ternary	ADJ
ejpam-1598	114	13	semiring	semiring	NOUN
ejpam-1598	114	14	and	and	CCONJ
ejpam-1598	114	15	m	m	VERB
ejpam-1598	114	16	a	a	DET
ejpam-1598	114	17	right	right	ADJ
ejpam-1598	114	18	ternary	ternary	ADJ
ejpam-1598	114	19	s	s	NOUN
ejpam-1598	114	20	-	-	NOUN
ejpam-1598	114	21	semimodule	semimodule	NOUN
ejpam-1598	114	22	.	.	PUNCT
ejpam-1598	115	1	let	let	VERB
ejpam-1598	115	2	m	m	PRON
ejpam-1598	115	3	∈	∈	VERB
ejpam-1598	115	4	m	m	NOUN
ejpam-1598	115	5	.	.	PUNCT
ejpam-1598	116	1	then	then	ADV
ejpam-1598	116	2	the	the	DET
ejpam-1598	116	3	right	right	ADJ
ejpam-1598	116	4	annihilator	annihilator	NOUN
ejpam-1598	116	5	of	of	ADP
ejpam-1598	116	6	m	m	PROPN
ejpam-1598	116	7	in	in	ADP
ejpam-1598	116	8	s	s	PROPN
ejpam-1598	116	9	,	,	PUNCT
ejpam-1598	116	10	denoted	denote	VERB
ejpam-1598	116	11	by	by	ADP
ejpam-1598	116	12	rs(m	rs(m	PROPN
ejpam-1598	116	13	)	)	PUNCT
ejpam-1598	116	14	,	,	PUNCT
ejpam-1598	116	15	is	be	AUX
ejpam-1598	116	16	defined	define	VERB
ejpam-1598	116	17	by	by	ADP
ejpam-1598	116	18	{	{	PUNCT
ejpam-1598	116	19	x	x	SYM
ejpam-1598	116	20	∈	∈	PROPN
ejpam-1598	116	21	s	s	PART
ejpam-1598	116	22	:	:	PUNCT
ejpam-1598	116	23	mxs	mxs	PROPN
ejpam-1598	116	24	=	=	SYM
ejpam-1598	116	25	0	0	NUM
ejpam-1598	116	26	m	m	VERB
ejpam-1598	116	27	for	for	ADP
ejpam-1598	116	28	all	all	PRON
ejpam-1598	116	29	s	s	PART
ejpam-1598	116	30	∈	∈	NOUN
ejpam-1598	116	31	s	s	PART
ejpam-1598	116	32	}	}	PUNCT
ejpam-1598	116	33	.	.	PUNCT
ejpam-1598	117	1	the	the	DET
ejpam-1598	117	2	right	right	ADJ
ejpam-1598	117	3	annihilator	annihilator	PROPN
ejpam-1598	117	4	rs(m	rs(m	PROPN
ejpam-1598	117	5	)	)	PUNCT
ejpam-1598	117	6	has	have	VERB
ejpam-1598	117	7	the	the	DET
ejpam-1598	117	8	following	follow	VERB
ejpam-1598	117	9	property	property	NOUN
ejpam-1598	117	10	.	.	PUNCT
ejpam-1598	118	1	t.	t.	PROPN
ejpam-1598	118	2	dutta	dutta	PROPN
ejpam-1598	118	3	,	,	PUNCT
ejpam-1598	118	4	k.	k.	PROPN
ejpam-1598	118	5	shum	shum	PROPN
ejpam-1598	118	6	,	,	PUNCT
ejpam-1598	118	7	s.	s.	PROPN
ejpam-1598	118	8	mandal	mandal	PROPN
ejpam-1598	118	9	/	/	SYM
ejpam-1598	118	10	eur	eur	PROPN
ejpam-1598	118	11	.	.	PUNCT
ejpam-1598	119	1	j.	j.	PROPN
ejpam-1598	119	2	pure	pure	PROPN
ejpam-1598	119	3	appl	appl	PROPN
ejpam-1598	119	4	.	.	PROPN
ejpam-1598	119	5	math	math	PROPN
ejpam-1598	119	6	,	,	PUNCT
ejpam-1598	119	7	5	5	NUM
ejpam-1598	119	8	(	(	PUNCT
ejpam-1598	119	9	2012	2012	NUM
ejpam-1598	119	10	)	)	PUNCT
ejpam-1598	119	11	,	,	PUNCT
ejpam-1598	119	12	116	116	NUM
ejpam-1598	119	13	-	-	SYM
ejpam-1598	119	14	128	128	NUM
ejpam-1598	119	15	120	120	NUM
ejpam-1598	119	16	proposition	proposition	NOUN
ejpam-1598	119	17	3	3	NUM
ejpam-1598	119	18	.	.	PUNCT
ejpam-1598	120	1	let	let	VERB
ejpam-1598	120	2	s	s	PRON
ejpam-1598	120	3	be	be	AUX
ejpam-1598	120	4	a	a	DET
ejpam-1598	120	5	ternary	ternary	ADJ
ejpam-1598	120	6	semiring	semiring	NOUN
ejpam-1598	120	7	and	and	CCONJ
ejpam-1598	120	8	m	m	VERB
ejpam-1598	120	9	a	a	DET
ejpam-1598	120	10	right	right	ADJ
ejpam-1598	120	11	ternary	ternary	ADJ
ejpam-1598	120	12	s	s	NOUN
ejpam-1598	120	13	-	-	NOUN
ejpam-1598	120	14	semimodule	semimodule	NOUN
ejpam-1598	120	15	.	.	PUNCT
ejpam-1598	121	1	then	then	ADV
ejpam-1598	121	2	rs(m	rs(m	NUM
ejpam-1598	121	3	)	)	PUNCT
ejpam-1598	121	4	is	be	AUX
ejpam-1598	121	5	a	a	DET
ejpam-1598	121	6	right	right	ADJ
ejpam-1598	121	7	k	k	NOUN
ejpam-1598	121	8	-	-	NOUN
ejpam-1598	121	9	ideal	ideal	NOUN
ejpam-1598	121	10	of	of	ADP
ejpam-1598	121	11	the	the	DET
ejpam-1598	121	12	ternary	ternary	ADJ
ejpam-1598	121	13	semiring	semire	VERB
ejpam-1598	121	14	s.	s.	PROPN
ejpam-1598	121	15	proof	proof	PROPN
ejpam-1598	121	16	.	.	PUNCT
ejpam-1598	122	1	obviously	obviously	ADV
ejpam-1598	122	2	,	,	PUNCT
ejpam-1598	122	3	the	the	DET
ejpam-1598	122	4	set	set	NOUN
ejpam-1598	122	5	rs(m	rs(m	NOUN
ejpam-1598	122	6	)	)	PUNCT
ejpam-1598	122	7	is	be	AUX
ejpam-1598	122	8	non	non	ADJ
ejpam-1598	122	9	-	-	ADJ
ejpam-1598	122	10	empty	empty	ADJ
ejpam-1598	122	11	,	,	PUNCT
ejpam-1598	122	12	since	since	SCONJ
ejpam-1598	122	13	0s	0s	NOUN
ejpam-1598	122	14	∈	∈	PROPN
ejpam-1598	122	15	rs(m	rs(m	PROPN
ejpam-1598	122	16	)	)	PUNCT
ejpam-1598	122	17	.	.	PUNCT
ejpam-1598	123	1	let	let	VERB
ejpam-1598	123	2	a	a	DET
ejpam-1598	123	3	,	,	PUNCT
ejpam-1598	123	4	b	b	PROPN
ejpam-1598	123	5	∈	∈	PROPN
ejpam-1598	123	6	rs(m	rs(m	NOUN
ejpam-1598	123	7	)	)	PUNCT
ejpam-1598	123	8	.	.	PUNCT
ejpam-1598	124	1	then	then	ADV
ejpam-1598	124	2	a+b	a+b	NUM
ejpam-1598	124	3	∈	∈	PROPN
ejpam-1598	124	4	rs(m	rs(m	NOUN
ejpam-1598	124	5	)	)	PUNCT
ejpam-1598	124	6	.	.	PUNCT
ejpam-1598	125	1	again	again	ADV
ejpam-1598	125	2	let	let	VERB
ejpam-1598	125	3	a	a	DET
ejpam-1598	125	4	∈	∈	PROPN
ejpam-1598	125	5	rs(m	rs(m	NOUN
ejpam-1598	125	6	)	)	PUNCT
ejpam-1598	125	7	and	and	CCONJ
ejpam-1598	125	8	s1	s1	NOUN
ejpam-1598	125	9	,	,	PUNCT
ejpam-1598	125	10	s2	s2	PROPN
ejpam-1598	125	11	∈	∈	PROPN
ejpam-1598	125	12	s.	s.	PROPN
ejpam-1598	125	13	then	then	ADV
ejpam-1598	125	14	mas	mas	PROPN
ejpam-1598	125	15	=	=	PROPN
ejpam-1598	125	16	0	0	NUM
ejpam-1598	125	17	m	m	VERB
ejpam-1598	125	18	for	for	ADP
ejpam-1598	125	19	all	all	PRON
ejpam-1598	125	20	s	s	PART
ejpam-1598	125	21	∈	∈	NOUN
ejpam-1598	125	22	s⇒	s⇒	NOUN
ejpam-1598	125	23	mas1s2s	mas1s2s	NOUN
ejpam-1598	126	1	=	=	PUNCT
ejpam-1598	126	2	0	0	NUM
ejpam-1598	126	3	m	m	VERB
ejpam-1598	126	4	for	for	ADP
ejpam-1598	126	5	all	all	DET
ejpam-1598	126	6	s	s	PROPN
ejpam-1598	126	7	∈	∈	PROPN
ejpam-1598	126	8	s.	s.	PROPN
ejpam-1598	126	9	this	this	PRON
ejpam-1598	126	10	leads	lead	VERB
ejpam-1598	126	11	to	to	ADP
ejpam-1598	126	12	as1s2	as1s2	PROPN
ejpam-1598	126	13	∈	∈	PROPN
ejpam-1598	126	14	rs(m	rs(m	NOUN
ejpam-1598	126	15	)	)	PUNCT
ejpam-1598	126	16	.	.	PUNCT
ejpam-1598	127	1	hence	hence	ADV
ejpam-1598	127	2	,	,	PUNCT
ejpam-1598	127	3	rs(m	rs(m	PROPN
ejpam-1598	127	4	)	)	PUNCT
ejpam-1598	127	5	is	be	AUX
ejpam-1598	127	6	a	a	DET
ejpam-1598	127	7	right	right	ADJ
ejpam-1598	127	8	ideal	ideal	NOUN
ejpam-1598	127	9	of	of	ADP
ejpam-1598	127	10	s.	s.	PROPN
ejpam-1598	127	11	let	let	VERB
ejpam-1598	127	12	a	a	DET
ejpam-1598	127	13	,	,	PUNCT
ejpam-1598	127	14	a+	a+	PRON
ejpam-1598	127	15	b	b	PROPN
ejpam-1598	127	16	∈	∈	PROPN
ejpam-1598	127	17	rs(m	rs(m	NOUN
ejpam-1598	127	18	)	)	PUNCT
ejpam-1598	127	19	.	.	PUNCT
ejpam-1598	128	1	then	then	ADV
ejpam-1598	128	2	mas	mas	PROPN
ejpam-1598	128	3	=	=	PROPN
ejpam-1598	128	4	0	0	NUM
ejpam-1598	128	5	m	m	NOUN
ejpam-1598	128	6	=	=	PUNCT
ejpam-1598	128	7	m(a+	m(a+	NOUN
ejpam-1598	128	8	b)s	b)s	X
ejpam-1598	128	9	for	for	ADP
ejpam-1598	128	10	all	all	DET
ejpam-1598	128	11	s	s	PROPN
ejpam-1598	128	12	∈	∈	PROPN
ejpam-1598	128	13	s.	s.	PROPN
ejpam-1598	128	14	this	this	PRON
ejpam-1598	128	15	implies	imply	VERB
ejpam-1598	128	16	mbs	mb	NOUN
ejpam-1598	128	17	=	=	SYM
ejpam-1598	128	18	0	0	NUM
ejpam-1598	128	19	m	m	VERB
ejpam-1598	128	20	for	for	ADP
ejpam-1598	128	21	all	all	DET
ejpam-1598	128	22	s	s	PROPN
ejpam-1598	128	23	∈	∈	PROPN
ejpam-1598	128	24	s.	s.	PROPN
ejpam-1598	128	25	this	this	PRON
ejpam-1598	128	26	shows	show	VERB
ejpam-1598	128	27	that	that	SCONJ
ejpam-1598	128	28	b	b	PROPN
ejpam-1598	128	29	∈	∈	PROPN
ejpam-1598	128	30	rs(m	rs(m	NOUN
ejpam-1598	128	31	)	)	PUNCT
ejpam-1598	128	32	.	.	PUNCT
ejpam-1598	129	1	hence	hence	ADV
ejpam-1598	129	2	,	,	PUNCT
ejpam-1598	129	3	rs(m	rs(m	PROPN
ejpam-1598	129	4	)	)	PUNCT
ejpam-1598	129	5	is	be	AUX
ejpam-1598	129	6	a	a	DET
ejpam-1598	129	7	right	right	ADJ
ejpam-1598	129	8	k	k	NOUN
ejpam-1598	129	9	-	-	NOUN
ejpam-1598	129	10	ideal	ideal	NOUN
ejpam-1598	129	11	of	of	ADP
ejpam-1598	129	12	s.	s.	PROPN
ejpam-1598	129	13	let	let	VERB
ejpam-1598	129	14	s	s	PRON
ejpam-1598	129	15	be	be	AUX
ejpam-1598	129	16	a	a	DET
ejpam-1598	129	17	ternary	ternary	ADJ
ejpam-1598	129	18	semiring	semiring	NOUN
ejpam-1598	129	19	and	and	CCONJ
ejpam-1598	129	20	m	m	VERB
ejpam-1598	129	21	a	a	DET
ejpam-1598	129	22	right	right	ADJ
ejpam-1598	129	23	ternary	ternary	ADJ
ejpam-1598	129	24	s	s	NOUN
ejpam-1598	129	25	-	-	NOUN
ejpam-1598	129	26	semimodule	semimodule	NOUN
ejpam-1598	129	27	.	.	PUNCT
ejpam-1598	130	1	we	we	PRON
ejpam-1598	130	2	define	define	VERB
ejpam-1598	130	3	zs(m	zs(m	NUM
ejpam-1598	130	4	)	)	PUNCT
ejpam-1598	130	5	as	as	SCONJ
ejpam-1598	130	6	follows	follow	VERB
ejpam-1598	130	7	:	:	PUNCT
ejpam-1598	130	8	zs(m	zs(m	X
ejpam-1598	130	9	)	)	PUNCT
ejpam-1598	130	10	=	=	SYM
ejpam-1598	130	11	{	{	PUNCT
ejpam-1598	130	12	m	m	VERB
ejpam-1598	130	13	∈	∈	ADJ
ejpam-1598	130	14	m	m	VERB
ejpam-1598	130	15	:	:	PUNCT
ejpam-1598	130	16	rs(m	rs(m	X
ejpam-1598	130	17	)	)	PUNCT
ejpam-1598	130	18	is	be	AUX
ejpam-1598	130	19	an	an	DET
ejpam-1598	130	20	essential	essential	ADJ
ejpam-1598	130	21	right	right	ADJ
ejpam-1598	130	22	ideal	ideal	NOUN
ejpam-1598	130	23	of	of	ADP
ejpam-1598	130	24	s	s	NOUN
ejpam-1598	130	25	}	}	PUNCT
ejpam-1598	130	26	proposition	proposition	NOUN
ejpam-1598	130	27	4	4	NUM
ejpam-1598	130	28	.	.	NUM
ejpam-1598	130	29	zs(m	zs(m	NUM
ejpam-1598	130	30	)	)	PUNCT
ejpam-1598	130	31	is	be	AUX
ejpam-1598	130	32	a	a	DET
ejpam-1598	130	33	ternary	ternary	ADJ
ejpam-1598	130	34	subsemimodule	subsemimodule	NOUN
ejpam-1598	130	35	of	of	ADP
ejpam-1598	130	36	m.	m.	NOUN
ejpam-1598	130	37	proof	proof	NOUN
ejpam-1598	130	38	.	.	PUNCT
ejpam-1598	131	1	clearly	clearly	ADV
ejpam-1598	131	2	,	,	PUNCT
ejpam-1598	131	3	0	0	NUM
ejpam-1598	131	4	m	m	VERB
ejpam-1598	131	5	∈	∈	NOUN
ejpam-1598	131	6	zs(m	zs(m	NUM
ejpam-1598	131	7	)	)	PUNCT
ejpam-1598	131	8	as	as	ADP
ejpam-1598	131	9	rs(0	rs(0	NOUN
ejpam-1598	131	10	m	m	PROPN
ejpam-1598	131	11	)	)	PUNCT
ejpam-1598	132	1	=	=	PUNCT
ejpam-1598	132	2	s	s	VERB
ejpam-1598	132	3	is	be	AUX
ejpam-1598	132	4	an	an	DET
ejpam-1598	132	5	essential	essential	ADJ
ejpam-1598	132	6	right	right	ADJ
ejpam-1598	132	7	ideal	ideal	NOUN
ejpam-1598	132	8	of	of	ADP
ejpam-1598	132	9	s.	s.	PROPN
ejpam-1598	132	10	so	so	ADV
ejpam-1598	132	11	zs(m	zs(m	NUM
ejpam-1598	132	12	)	)	PUNCT
ejpam-1598	132	13	is	be	AUX
ejpam-1598	132	14	a	a	DET
ejpam-1598	132	15	non	non	X
ejpam-1598	132	16	empty	empty	ADJ
ejpam-1598	132	17	subset	subset	NOUN
ejpam-1598	132	18	of	of	ADP
ejpam-1598	132	19	m	m	PROPN
ejpam-1598	132	20	.	.	PUNCT
ejpam-1598	133	1	let	let	VERB
ejpam-1598	133	2	m1	m1	PROPN
ejpam-1598	133	3	,	,	PUNCT
ejpam-1598	133	4	m2	m2	PROPN
ejpam-1598	133	5	∈	∈	PROPN
ejpam-1598	133	6	zs(m	zs(m	NUM
ejpam-1598	133	7	)	)	PUNCT
ejpam-1598	133	8	.	.	PUNCT
ejpam-1598	134	1	then	then	ADV
ejpam-1598	134	2	,	,	PUNCT
ejpam-1598	134	3	it	it	PRON
ejpam-1598	134	4	follows	follow	VERB
ejpam-1598	134	5	that	that	SCONJ
ejpam-1598	134	6	rs(m1	rs(m1	NOUN
ejpam-1598	134	7	)	)	PUNCT
ejpam-1598	134	8	and	and	CCONJ
ejpam-1598	134	9	rs(m2	rs(m2	X
ejpam-1598	134	10	)	)	PUNCT
ejpam-1598	134	11	are	be	AUX
ejpam-1598	134	12	two	two	NUM
ejpam-1598	134	13	essential	essential	ADJ
ejpam-1598	134	14	right	right	ADJ
ejpam-1598	134	15	ideals	ideal	NOUN
ejpam-1598	134	16	of	of	ADP
ejpam-1598	134	17	s.	s.	PROPN
ejpam-1598	134	18	therefore	therefore	ADV
ejpam-1598	134	19	,	,	PUNCT
ejpam-1598	134	20	rs(m1	rs(m1	NOUN
ejpam-1598	134	21	)	)	PUNCT
ejpam-1598	134	22	∩	∩	NOUN
ejpam-1598	134	23	rs(m2	rs(m2	X
ejpam-1598	134	24	)	)	PUNCT
ejpam-1598	134	25	is	be	AUX
ejpam-1598	134	26	also	also	ADV
ejpam-1598	134	27	an	an	DET
ejpam-1598	134	28	essential	essential	ADJ
ejpam-1598	134	29	right	right	ADJ
ejpam-1598	134	30	ideal	ideal	NOUN
ejpam-1598	134	31	of	of	ADP
ejpam-1598	134	32	s.	s.	PROPN
ejpam-1598	134	33	let	let	VERB
ejpam-1598	134	34	h	h	NOUN
ejpam-1598	134	35	be	be	AUX
ejpam-1598	134	36	a	a	DET
ejpam-1598	134	37	nonzero	nonzero	ADJ
ejpam-1598	134	38	right	right	ADJ
ejpam-1598	134	39	ideal	ideal	NOUN
ejpam-1598	134	40	of	of	ADP
ejpam-1598	134	41	s.	s.	PROPN
ejpam-1598	134	42	now	now	ADV
ejpam-1598	134	43	rs(m1	rs(m1	VERB
ejpam-1598	134	44	)	)	PUNCT
ejpam-1598	134	45	∩	∩	NOUN
ejpam-1598	134	46	rs(m2	rs(m2	NOUN
ejpam-1598	134	47	)	)	PUNCT
ejpam-1598	134	48	∩	∩	NOUN
ejpam-1598	134	49	h	h	NOUN
ejpam-1598	134	50	⊆	⊆	NUM
ejpam-1598	134	51	rs(m1	rs(m1	NOUN
ejpam-1598	134	52	+	+	SYM
ejpam-1598	134	53	m2	m2	NOUN
ejpam-1598	134	54	)	)	PUNCT
ejpam-1598	134	55	∩	∩	PROPN
ejpam-1598	134	56	h	h	NOUN
ejpam-1598	134	57	since	since	SCONJ
ejpam-1598	134	58	rs(m1)∩	rs(m1)∩	NOUN
ejpam-1598	134	59	rs(m2)⊆	rs(m2)⊆	ADP
ejpam-1598	134	60	rs(m1+m2	rs(m1+m2	NOUN
ejpam-1598	134	61	)	)	PUNCT
ejpam-1598	134	62	.	.	PUNCT
ejpam-1598	135	1	therefore	therefore	ADV
ejpam-1598	135	2	,	,	PUNCT
ejpam-1598	135	3	rs(m1+m2	rs(m1+m2	NOUN
ejpam-1598	135	4	)	)	PUNCT
ejpam-1598	135	5	is	be	AUX
ejpam-1598	135	6	an	an	DET
ejpam-1598	135	7	essential	essential	ADJ
ejpam-1598	135	8	right	right	ADJ
ejpam-1598	135	9	ideal	ideal	NOUN
ejpam-1598	135	10	of	of	ADP
ejpam-1598	135	11	s.	s.	PROPN
ejpam-1598	135	12	hence	hence	PROPN
ejpam-1598	135	13	,	,	PUNCT
ejpam-1598	135	14	we	we	PRON
ejpam-1598	135	15	have	have	VERB
ejpam-1598	135	16	m1	m1	PROPN
ejpam-1598	135	17	+	+	PROPN
ejpam-1598	135	18	m2	m2	PROPN
ejpam-1598	135	19	∈	∈	PROPN
ejpam-1598	135	20	zs(m	zs(m	NUM
ejpam-1598	135	21	)	)	PUNCT
ejpam-1598	135	22	.	.	PUNCT
ejpam-1598	136	1	let	let	VERB
ejpam-1598	136	2	m	m	PRON
ejpam-1598	136	3	∈	∈	PROPN
ejpam-1598	136	4	zs(m	zs(m	NUM
ejpam-1598	136	5	)	)	PUNCT
ejpam-1598	136	6	and	and	CCONJ
ejpam-1598	136	7	s1	s1	NOUN
ejpam-1598	136	8	,	,	PUNCT
ejpam-1598	136	9	s2	s2	PROPN
ejpam-1598	136	10	∈	∈	PROPN
ejpam-1598	136	11	s.	s.	PROPN
ejpam-1598	136	12	then	then	ADV
ejpam-1598	136	13	,	,	PUNCT
ejpam-1598	136	14	rs(m	rs(m	PROPN
ejpam-1598	136	15	)	)	PUNCT
ejpam-1598	136	16	is	be	AUX
ejpam-1598	136	17	an	an	DET
ejpam-1598	136	18	essential	essential	ADJ
ejpam-1598	136	19	right	right	ADJ
ejpam-1598	136	20	ideal	ideal	NOUN
ejpam-1598	136	21	of	of	ADP
ejpam-1598	136	22	s.	s.	PROPN
ejpam-1598	136	23	let	let	VERB
ejpam-1598	136	24	h	h	NOUN
ejpam-1598	136	25	be	be	AUX
ejpam-1598	136	26	any	any	DET
ejpam-1598	136	27	nonzero	nonzero	ADJ
ejpam-1598	136	28	right	right	ADJ
ejpam-1598	136	29	ideal	ideal	NOUN
ejpam-1598	136	30	of	of	ADP
ejpam-1598	136	31	s.	s.	PROPN
ejpam-1598	136	32	now	now	ADV
ejpam-1598	136	33	if	if	SCONJ
ejpam-1598	136	34	s1s2h	s1s2h	NUM
ejpam-1598	136	35	=	=	SYM
ejpam-1598	136	36	0	0	NUM
ejpam-1598	136	37	,	,	PUNCT
ejpam-1598	136	38	then	then	ADV
ejpam-1598	136	39	ms1s2hs	ms1s2hs	PROPN
ejpam-1598	136	40	=	=	AUX
ejpam-1598	136	41	0	0	NUM
ejpam-1598	136	42	m	m	VERB
ejpam-1598	136	43	for	for	ADP
ejpam-1598	136	44	all	all	DET
ejpam-1598	136	45	s	s	PROPN
ejpam-1598	136	46	∈	∈	PROPN
ejpam-1598	136	47	s.	s.	PROPN
ejpam-1598	136	48	this	this	PRON
ejpam-1598	136	49	leads	lead	VERB
ejpam-1598	136	50	to	to	ADP
ejpam-1598	136	51	h	h	NOUN
ejpam-1598	136	52	⊆	⊆	NUM
ejpam-1598	136	53	rs(ms1s2	rs(ms1s2	NUM
ejpam-1598	136	54	)	)	PUNCT
ejpam-1598	136	55	.	.	PUNCT
ejpam-1598	137	1	hence	hence	ADV
ejpam-1598	137	2	,	,	PUNCT
ejpam-1598	137	3	rs(ms1s2)∩h	rs(ms1s2)∩h	NOUN
ejpam-1598	137	4	=	=	PUNCT
ejpam-1598	137	5	h	h	PROPN
ejpam-1598	137	6	6=	6=	NOUN
ejpam-1598	137	7	0	0	X
ejpam-1598	137	8	.	.	PUNCT
ejpam-1598	138	1	if	if	SCONJ
ejpam-1598	138	2	s1s2h	s1s2h	NUM
ejpam-1598	138	3	is	be	AUX
ejpam-1598	138	4	a	a	DET
ejpam-1598	138	5	nonzero	nonzero	ADJ
ejpam-1598	138	6	right	right	ADJ
ejpam-1598	138	7	ideal	ideal	NOUN
ejpam-1598	138	8	of	of	ADP
ejpam-1598	138	9	s	s	PROPN
ejpam-1598	138	10	,	,	PUNCT
ejpam-1598	138	11	then	then	ADV
ejpam-1598	138	12	rs(m	rs(m	NOUN
ejpam-1598	138	13	)	)	PUNCT
ejpam-1598	138	14	∩	∩	NOUN
ejpam-1598	138	15	s1s2h	s1s2h	PRON
ejpam-1598	138	16	6=	6=	ADP
ejpam-1598	138	17	0	0	X
ejpam-1598	138	18	.	.	PUNCT
ejpam-1598	139	1	let	let	VERB
ejpam-1598	139	2	s1s2h1	s1s2h1	NOUN
ejpam-1598	139	3	(	(	PUNCT
ejpam-1598	139	4	6=	6=	NOUN
ejpam-1598	139	5	0	0	NUM
ejpam-1598	139	6	)	)	PUNCT
ejpam-1598	139	7	∈	∈	PROPN
ejpam-1598	139	8	rs(m	rs(m	NOUN
ejpam-1598	139	9	)	)	PUNCT
ejpam-1598	139	10	∩	∩	NOUN
ejpam-1598	139	11	s1s2h	s1s2h	PUNCT
ejpam-1598	139	12	for	for	ADP
ejpam-1598	139	13	some	some	DET
ejpam-1598	139	14	h1	h1	NOUN
ejpam-1598	139	15	(	(	PUNCT
ejpam-1598	139	16	6=	6=	NOUN
ejpam-1598	139	17	0	0	NUM
ejpam-1598	139	18	)	)	PUNCT
ejpam-1598	139	19	∈	∈	PROPN
ejpam-1598	139	20	h.	h.	NOUN
ejpam-1598	139	21	then	then	ADV
ejpam-1598	139	22	,	,	PUNCT
ejpam-1598	139	23	ms1s2h1s	ms1s2h1s	X
ejpam-1598	139	24	=	=	PUNCT
ejpam-1598	139	25	0	0	NUM
ejpam-1598	139	26	m	m	VERB
ejpam-1598	139	27	,	,	PUNCT
ejpam-1598	139	28	for	for	ADP
ejpam-1598	139	29	all	all	PRON
ejpam-1598	139	30	s	s	PROPN
ejpam-1598	139	31	∈	∈	PROPN
ejpam-1598	139	32	s.	s.	PROPN
ejpam-1598	139	33	therefore	therefore	ADV
ejpam-1598	139	34	,	,	PUNCT
ejpam-1598	139	35	h1	h1	PROPN
ejpam-1598	139	36	∈	∈	PROPN
ejpam-1598	139	37	rs(ms1s2)∩h	rs(ms1s2)∩h	NOUN
ejpam-1598	139	38	,	,	PUNCT
ejpam-1598	139	39	and	and	CCONJ
ejpam-1598	139	40	so	so	ADV
ejpam-1598	139	41	rs(ms1s2)∩	rs(ms1s2)∩	ADJ
ejpam-1598	139	42	h	h	NOUN
ejpam-1598	139	43	6=	6=	PROPN
ejpam-1598	139	44	0	0	NUM
ejpam-1598	139	45	.	.	PUNCT
ejpam-1598	140	1	thus	thus	ADV
ejpam-1598	140	2	,	,	PUNCT
ejpam-1598	140	3	in	in	ADP
ejpam-1598	140	4	any	any	DET
ejpam-1598	140	5	case	case	NOUN
ejpam-1598	140	6	ms1s2	ms1s2	PROPN
ejpam-1598	140	7	∈	∈	PROPN
ejpam-1598	140	8	zs(m	zs(m	NUM
ejpam-1598	140	9	)	)	PUNCT
ejpam-1598	140	10	.	.	PUNCT
ejpam-1598	141	1	this	this	PRON
ejpam-1598	141	2	shows	show	VERB
ejpam-1598	141	3	that	that	SCONJ
ejpam-1598	141	4	zs(m	zs(m	NUM
ejpam-1598	141	5	)	)	PUNCT
ejpam-1598	141	6	is	be	AUX
ejpam-1598	141	7	a	a	DET
ejpam-1598	141	8	subsemimodule	subsemimodule	NOUN
ejpam-1598	141	9	of	of	ADP
ejpam-1598	141	10	m.	m.	NOUN
ejpam-1598	141	11	we	we	PRON
ejpam-1598	141	12	now	now	ADV
ejpam-1598	141	13	give	give	VERB
ejpam-1598	141	14	the	the	DET
ejpam-1598	141	15	definition	definition	NOUN
ejpam-1598	141	16	of	of	ADP
ejpam-1598	141	17	a	a	DET
ejpam-1598	141	18	singular	singular	ADJ
ejpam-1598	141	19	subsemimodule	subsemimodule	NOUN
ejpam-1598	141	20	over	over	ADP
ejpam-1598	141	21	a	a	DET
ejpam-1598	141	22	ternary	ternary	ADJ
ejpam-1598	141	23	semiring	semiring	NOUN
ejpam-1598	141	24	.	.	PUNCT
ejpam-1598	142	1	definition	definition	NOUN
ejpam-1598	142	2	19	19	NUM
ejpam-1598	142	3	.	.	PUNCT
ejpam-1598	143	1	the	the	DET
ejpam-1598	143	2	ternary	ternary	ADJ
ejpam-1598	143	3	subsemimodule	subsemimodule	NOUN
ejpam-1598	143	4	zs(m	zs(m	NUM
ejpam-1598	143	5	)	)	PUNCT
ejpam-1598	143	6	of	of	ADP
ejpam-1598	143	7	m	m	PROPN
ejpam-1598	143	8	is	be	AUX
ejpam-1598	143	9	called	call	VERB
ejpam-1598	143	10	a	a	DET
ejpam-1598	143	11	singular	singular	ADJ
ejpam-1598	143	12	ternary	ternary	ADJ
ejpam-1598	143	13	subsemimodule	subsemimodule	NOUN
ejpam-1598	143	14	of	of	ADP
ejpam-1598	143	15	the	the	DET
ejpam-1598	143	16	right	right	ADJ
ejpam-1598	143	17	ternary	ternary	ADJ
ejpam-1598	143	18	s	s	NOUN
ejpam-1598	143	19	-	-	PUNCT
ejpam-1598	143	20	semimodule	semimodule	NOUN
ejpam-1598	143	21	m.	m.	NOUN
ejpam-1598	143	22	the	the	DET
ejpam-1598	143	23	singular	singular	ADJ
ejpam-1598	143	24	ternary	ternary	ADJ
ejpam-1598	143	25	subsemimodule	subsemimodule	NOUN
ejpam-1598	143	26	zs(s	zs(s	NUM
ejpam-1598	143	27	)	)	PUNCT
ejpam-1598	143	28	is	be	AUX
ejpam-1598	143	29	a	a	DET
ejpam-1598	143	30	right	right	ADJ
ejpam-1598	143	31	ideal	ideal	NOUN
ejpam-1598	143	32	of	of	ADP
ejpam-1598	143	33	the	the	DET
ejpam-1598	143	34	ternary	ternary	ADJ
ejpam-1598	143	35	semiring	semiring	NOUN
ejpam-1598	143	36	s	s	PRON
ejpam-1598	143	37	and	and	CCONJ
ejpam-1598	143	38	is	be	AUX
ejpam-1598	143	39	called	call	VERB
ejpam-1598	143	40	the	the	DET
ejpam-1598	143	41	(	(	PUNCT
ejpam-1598	143	42	right	right	ADJ
ejpam-1598	143	43	)	)	PUNCT
ejpam-1598	143	44	singular	singular	PROPN
ejpam-1598	143	45	ideal	ideal	NOUN
ejpam-1598	143	46	of	of	ADP
ejpam-1598	143	47	the	the	DET
ejpam-1598	143	48	ternary	ternary	ADJ
ejpam-1598	143	49	semiring	semiring	NOUN
ejpam-1598	143	50	s	s	X
ejpam-1598	143	51	which	which	PRON
ejpam-1598	143	52	is	be	AUX
ejpam-1598	143	53	denoted	denote	VERB
ejpam-1598	143	54	by	by	ADP
ejpam-1598	143	55	z(s	z(s	PROPN
ejpam-1598	143	56	)	)	PUNCT
ejpam-1598	143	57	,	,	PUNCT
ejpam-1598	143	58	i.e.	i.e.	X
ejpam-1598	143	59	z(s	z(s	NOUN
ejpam-1598	143	60	)	)	PUNCT
ejpam-1598	143	61	=	=	PRON
ejpam-1598	144	1	{	{	PUNCT
ejpam-1598	144	2	t	t	PROPN
ejpam-1598	144	3	∈	∈	PROPN
ejpam-1598	144	4	s	s	PART
ejpam-1598	144	5	:	:	PUNCT
ejpam-1598	144	6	rs(t	rs(t	X
ejpam-1598	144	7	)	)	PUNCT
ejpam-1598	144	8	is	be	AUX
ejpam-1598	144	9	an	an	DET
ejpam-1598	144	10	essential	essential	ADJ
ejpam-1598	144	11	right	right	ADJ
ejpam-1598	144	12	ideal	ideal	NOUN
ejpam-1598	144	13	of	of	ADP
ejpam-1598	144	14	s	s	NOUN
ejpam-1598	144	15	}	}	PUNCT
ejpam-1598	144	16	.	.	PUNCT
ejpam-1598	145	1	a	a	DET
ejpam-1598	145	2	ternary	ternary	ADJ
ejpam-1598	145	3	semiring	semiring	NOUN
ejpam-1598	145	4	s	s	NOUN
ejpam-1598	145	5	is	be	AUX
ejpam-1598	145	6	said	say	VERB
ejpam-1598	145	7	to	to	PART
ejpam-1598	145	8	satisfy	satisfy	VERB
ejpam-1598	145	9	the	the	DET
ejpam-1598	145	10	condition	condition	NOUN
ejpam-1598	145	11	α	α	NOUN
ejpam-1598	145	12	if	if	SCONJ
ejpam-1598	145	13	for	for	ADP
ejpam-1598	145	14	any	any	DET
ejpam-1598	145	15	nonzero	nonzero	NOUN
ejpam-1598	145	16	element	element	NOUN
ejpam-1598	145	17	a	a	PRON
ejpam-1598	145	18	in	in	ADP
ejpam-1598	145	19	s	s	NOUN
ejpam-1598	145	20	,	,	PUNCT
ejpam-1598	145	21	rs(a	rs(a	NOUN
ejpam-1598	145	22	)	)	PUNCT
ejpam-1598	145	23	6=	6=	SYM
ejpam-1598	145	24	s	s	VERB
ejpam-1598	145	25	or	or	CCONJ
ejpam-1598	145	26	equivalently	equivalently	ADV
ejpam-1598	145	27	ass	ass	NOUN
ejpam-1598	145	28	=	=	SYM
ejpam-1598	145	29	0	0	NUM
ejpam-1598	145	30	implies	imply	VERB
ejpam-1598	145	31	a	a	DET
ejpam-1598	145	32	=	=	NOUN
ejpam-1598	145	33	0	0	NUM
ejpam-1598	145	34	.	.	PUNCT
ejpam-1598	146	1	we	we	PRON
ejpam-1598	146	2	now	now	ADV
ejpam-1598	146	3	give	give	VERB
ejpam-1598	146	4	some	some	DET
ejpam-1598	146	5	examples	example	NOUN
ejpam-1598	146	6	of	of	ADP
ejpam-1598	146	7	ternary	ternary	ADJ
ejpam-1598	146	8	semirings	semiring	NOUN
ejpam-1598	146	9	satisfying	satisfy	VERB
ejpam-1598	146	10	the	the	DET
ejpam-1598	146	11	condition	condition	NOUN
ejpam-1598	146	12	α	α	NOUN
ejpam-1598	146	13	.	.	PUNCT
ejpam-1598	146	14	example	example	NOUN
ejpam-1598	147	1	1	1	NUM
ejpam-1598	147	2	.	.	PUNCT
ejpam-1598	148	1	let	let	VERB
ejpam-1598	148	2	s	s	PRON
ejpam-1598	148	3	be	be	AUX
ejpam-1598	148	4	a	a	DET
ejpam-1598	148	5	ternary	ternary	ADJ
ejpam-1598	148	6	semi	semi	ADJ
ejpam-1598	148	7	-	-	ADJ
ejpam-1598	148	8	integral	integral	ADJ
ejpam-1598	148	9	domain	domain	NOUN
ejpam-1598	148	10	.	.	PUNCT
ejpam-1598	149	1	then	then	ADV
ejpam-1598	149	2	s	s	VERB
ejpam-1598	149	3	satisfies	satisfie	NOUN
ejpam-1598	149	4	the	the	DET
ejpam-1598	149	5	condition	condition	NOUN
ejpam-1598	149	6	α	α	NOUN
ejpam-1598	149	7	.	.	PUNCT
ejpam-1598	149	8	example	example	NOUN
ejpam-1598	150	1	2	2	NUM
ejpam-1598	150	2	.	.	PUNCT
ejpam-1598	150	3	let	let	VERB
ejpam-1598	150	4	s	s	PRON
ejpam-1598	150	5	be	be	AUX
ejpam-1598	150	6	a	a	DET
ejpam-1598	150	7	prime	prime	ADJ
ejpam-1598	150	8	(	(	PUNCT
ejpam-1598	150	9	semiprime	semiprime	NOUN
ejpam-1598	150	10	)	)	PUNCT
ejpam-1598	150	11	ternary	ternary	ADJ
ejpam-1598	150	12	semiring	semiring	NOUN
ejpam-1598	150	13	.	.	PUNCT
ejpam-1598	151	1	then	then	ADV
ejpam-1598	151	2	s	s	VERB
ejpam-1598	151	3	satisfies	satisfie	NOUN
ejpam-1598	151	4	the	the	DET
ejpam-1598	151	5	condition	condition	NOUN
ejpam-1598	151	6	α	α	NOUN
ejpam-1598	151	7	.	.	PUNCT
ejpam-1598	151	8	example	example	NOUN
ejpam-1598	152	1	3	3	X
ejpam-1598	152	2	.	.	PUNCT
ejpam-1598	152	3	let	let	VERB
ejpam-1598	152	4	s	s	PRON
ejpam-1598	152	5	be	be	AUX
ejpam-1598	152	6	a	a	DET
ejpam-1598	152	7	ternary	ternary	ADJ
ejpam-1598	152	8	semiring	semiring	NOUN
ejpam-1598	152	9	with	with	ADP
ejpam-1598	152	10	a	a	DET
ejpam-1598	152	11	unital	unital	ADJ
ejpam-1598	152	12	element	element	NOUN
ejpam-1598	152	13	e.	e.	PROPN
ejpam-1598	152	14	then	then	ADV
ejpam-1598	152	15	s	s	VERB
ejpam-1598	152	16	satisfies	satisfie	NOUN
ejpam-1598	152	17	the	the	DET
ejpam-1598	152	18	condition	condition	NOUN
ejpam-1598	152	19	α	α	NOUN
ejpam-1598	152	20	.	.	PUNCT
ejpam-1598	152	21	example	example	NOUN
ejpam-1598	153	1	4	4	NUM
ejpam-1598	153	2	.	.	PUNCT
ejpam-1598	153	3	let	let	VERB
ejpam-1598	153	4	s	s	PRON
ejpam-1598	153	5	be	be	AUX
ejpam-1598	153	6	a	a	DET
ejpam-1598	153	7	ternary	ternary	ADJ
ejpam-1598	153	8	semiring	semiring	NOUN
ejpam-1598	153	9	with	with	ADP
ejpam-1598	153	10	identity	identity	NOUN
ejpam-1598	153	11	.	.	PUNCT
ejpam-1598	154	1	then	then	ADV
ejpam-1598	154	2	s	s	VERB
ejpam-1598	154	3	satisfies	satisfie	NOUN
ejpam-1598	154	4	the	the	DET
ejpam-1598	154	5	condition	condition	NOUN
ejpam-1598	154	6	α	α	NOUN
ejpam-1598	154	7	.	.	PUNCT
ejpam-1598	155	1	t.	t.	PROPN
ejpam-1598	155	2	dutta	dutta	PROPN
ejpam-1598	155	3	,	,	PUNCT
ejpam-1598	155	4	k.	k.	PROPN
ejpam-1598	155	5	shum	shum	PROPN
ejpam-1598	155	6	,	,	PUNCT
ejpam-1598	155	7	s.	s.	PROPN
ejpam-1598	155	8	mandal	mandal	PROPN
ejpam-1598	155	9	/	/	SYM
ejpam-1598	155	10	eur	eur	PROPN
ejpam-1598	155	11	.	.	PUNCT
ejpam-1598	156	1	j.	j.	PROPN
ejpam-1598	156	2	pure	pure	PROPN
ejpam-1598	156	3	appl	appl	PROPN
ejpam-1598	156	4	.	.	PROPN
ejpam-1598	156	5	math	math	PROPN
ejpam-1598	156	6	,	,	PUNCT
ejpam-1598	156	7	5	5	NUM
ejpam-1598	156	8	(	(	PUNCT
ejpam-1598	156	9	2012	2012	NUM
ejpam-1598	156	10	)	)	PUNCT
ejpam-1598	156	11	,	,	PUNCT
ejpam-1598	156	12	116	116	NUM
ejpam-1598	156	13	-	-	SYM
ejpam-1598	156	14	128	128	NUM
ejpam-1598	156	15	121	121	NUM
ejpam-1598	156	16	in	in	ADP
ejpam-1598	156	17	the	the	DET
ejpam-1598	156	18	following	follow	VERB
ejpam-1598	156	19	proposition	proposition	NOUN
ejpam-1598	156	20	,	,	PUNCT
ejpam-1598	156	21	we	we	PRON
ejpam-1598	156	22	study	study	VERB
ejpam-1598	156	23	the	the	DET
ejpam-1598	156	24	ternary	ternary	ADJ
ejpam-1598	156	25	semirings	semiring	NOUN
ejpam-1598	156	26	with	with	ADP
ejpam-1598	156	27	condition	condition	NOUN
ejpam-1598	156	28	α	α	NOUN
ejpam-1598	156	29	.	.	PUNCT
ejpam-1598	157	1	proposition	proposition	NOUN
ejpam-1598	157	2	5	5	NUM
ejpam-1598	157	3	.	.	PUNCT
ejpam-1598	158	1	let	let	VERB
ejpam-1598	158	2	s	s	PRON
ejpam-1598	158	3	be	be	AUX
ejpam-1598	158	4	a	a	DET
ejpam-1598	158	5	ternary	ternary	ADJ
ejpam-1598	158	6	semiring	semiring	NOUN
ejpam-1598	158	7	with	with	ADP
ejpam-1598	158	8	condition	condition	NOUN
ejpam-1598	158	9	α	α	NOUN
ejpam-1598	158	10	.	.	PUNCT
ejpam-1598	159	1	then	then	ADV
ejpam-1598	159	2	the	the	DET
ejpam-1598	159	3	(	(	PUNCT
ejpam-1598	159	4	right	right	ADJ
ejpam-1598	159	5	)	)	PUNCT
ejpam-1598	159	6	singular	singular	PROPN
ejpam-1598	159	7	ideal	ideal	PROPN
ejpam-1598	159	8	z(s	z(s	PROPN
ejpam-1598	159	9	)	)	PUNCT
ejpam-1598	159	10	is	be	AUX
ejpam-1598	159	11	a	a	DET
ejpam-1598	159	12	k	k	NOUN
ejpam-1598	159	13	-	-	NOUN
ejpam-1598	159	14	ideal	ideal	NOUN
ejpam-1598	159	15	of	of	ADP
ejpam-1598	159	16	s.	s.	PROPN
ejpam-1598	159	17	proof	proof	PROPN
ejpam-1598	159	18	.	.	PUNCT
ejpam-1598	160	1	from	from	ADP
ejpam-1598	160	2	proposition	proposition	NOUN
ejpam-1598	160	3	4	4	NUM
ejpam-1598	160	4	,	,	PUNCT
ejpam-1598	160	5	it	it	PRON
ejpam-1598	160	6	follows	follow	VERB
ejpam-1598	160	7	that	that	SCONJ
ejpam-1598	160	8	z(s	z(s	PROPN
ejpam-1598	160	9	)	)	PUNCT
ejpam-1598	160	10	is	be	AUX
ejpam-1598	160	11	a	a	DET
ejpam-1598	160	12	right	right	ADJ
ejpam-1598	160	13	ideal	ideal	NOUN
ejpam-1598	160	14	of	of	ADP
ejpam-1598	160	15	s.	s.	PROPN
ejpam-1598	160	16	now	now	ADV
ejpam-1598	160	17	let	let	VERB
ejpam-1598	160	18	a	a	DET
ejpam-1598	160	19	∈	∈	PROPN
ejpam-1598	160	20	z(s	z(s	PROPN
ejpam-1598	160	21	)	)	PUNCT
ejpam-1598	160	22	,	,	PUNCT
ejpam-1598	160	23	s1	s1	NOUN
ejpam-1598	160	24	,	,	PUNCT
ejpam-1598	160	25	s2	s2	PROPN
ejpam-1598	160	26	∈	∈	PROPN
ejpam-1598	160	27	s.	s.	PROPN
ejpam-1598	160	28	then	then	ADV
ejpam-1598	160	29	rs(a	rs(a	NOUN
ejpam-1598	160	30	)	)	PUNCT
ejpam-1598	160	31	is	be	AUX
ejpam-1598	160	32	an	an	DET
ejpam-1598	160	33	essential	essential	ADJ
ejpam-1598	160	34	right	right	ADJ
ejpam-1598	160	35	ideal	ideal	NOUN
ejpam-1598	160	36	of	of	ADP
ejpam-1598	160	37	s.	s.	PROPN
ejpam-1598	160	38	and	and	CCONJ
ejpam-1598	161	1	so	so	ADV
ejpam-1598	161	2	rs(a	rs(a	X
ejpam-1598	161	3	)	)	PUNCT
ejpam-1598	161	4	∩	∩	ADJ
ejpam-1598	161	5	h	h	NOUN
ejpam-1598	161	6	6=	6=	ADP
ejpam-1598	161	7	0	0	NUM
ejpam-1598	161	8	for	for	ADP
ejpam-1598	161	9	any	any	DET
ejpam-1598	161	10	nonzero	nonzero	NOUN
ejpam-1598	161	11	right	right	ADJ
ejpam-1598	161	12	ideal	ideal	ADJ
ejpam-1598	161	13	h	h	PROPN
ejpam-1598	161	14	of	of	ADP
ejpam-1598	161	15	s.	s.	PROPN
ejpam-1598	161	16	now	now	ADV
ejpam-1598	161	17	rs(a	rs(a	VERB
ejpam-1598	161	18	)	)	PUNCT
ejpam-1598	161	19	⊆	⊆	NUM
ejpam-1598	161	20	rs(s1s2a	rs(s1s2a	NOUN
ejpam-1598	161	21	)	)	PUNCT
ejpam-1598	161	22	.	.	PUNCT
ejpam-1598	162	1	it	it	PRON
ejpam-1598	162	2	is	be	AUX
ejpam-1598	162	3	clear	clear	ADJ
ejpam-1598	162	4	that	that	SCONJ
ejpam-1598	162	5	(	(	PUNCT
ejpam-1598	162	6	0	0	NUM
ejpam-1598	162	7	)	)	PUNCT
ejpam-1598	162	8	6=	6=	ADP
ejpam-1598	162	9	rs(a	rs(a	NOUN
ejpam-1598	162	10	)	)	PUNCT
ejpam-1598	162	11	∩	∩	NOUN
ejpam-1598	162	12	h	h	NOUN
ejpam-1598	162	13	⊆	⊆	NUM
ejpam-1598	162	14	rs(s1s2a	rs(s1s2a	NOUN
ejpam-1598	162	15	)	)	PUNCT
ejpam-1598	162	16	∩	∩	NOUN
ejpam-1598	162	17	h.	h.	PROPN
ejpam-1598	162	18	therefore	therefore	ADV
ejpam-1598	162	19	,	,	PUNCT
ejpam-1598	162	20	rs(s1s2a)∩h	rs(s1s2a)∩h	PROPN
ejpam-1598	162	21	6=	6=	ADP
ejpam-1598	162	22	0	0	NUM
ejpam-1598	162	23	.	.	PUNCT
ejpam-1598	163	1	thus	thus	ADV
ejpam-1598	163	2	,	,	PUNCT
ejpam-1598	163	3	s1s2a	s1s2a	PUNCT
ejpam-1598	163	4	∈	∈	PROPN
ejpam-1598	163	5	z(s	z(s	PROPN
ejpam-1598	163	6	)	)	PUNCT
ejpam-1598	163	7	.	.	PUNCT
ejpam-1598	164	1	this	this	PRON
ejpam-1598	164	2	shows	show	VERB
ejpam-1598	164	3	that	that	SCONJ
ejpam-1598	164	4	z(s	z(s	PROPN
ejpam-1598	164	5	)	)	PUNCT
ejpam-1598	164	6	is	be	AUX
ejpam-1598	164	7	a	a	DET
ejpam-1598	164	8	left	left	ADJ
ejpam-1598	164	9	ideal	ideal	NOUN
ejpam-1598	164	10	of	of	ADP
ejpam-1598	164	11	s.	s.	PROPN
ejpam-1598	164	12	next	next	ADV
ejpam-1598	164	13	,	,	PUNCT
ejpam-1598	164	14	let	let	VERB
ejpam-1598	164	15	s1	s1	NOUN
ejpam-1598	164	16	,	,	PUNCT
ejpam-1598	164	17	s2	s2	NOUN
ejpam-1598	164	18	∈	∈	PROPN
ejpam-1598	164	19	s	s	PART
ejpam-1598	164	20	and	and	CCONJ
ejpam-1598	164	21	a	a	DET
ejpam-1598	164	22	∈	∈	PROPN
ejpam-1598	164	23	z(s	z(s	PROPN
ejpam-1598	164	24	)	)	PUNCT
ejpam-1598	164	25	.	.	PUNCT
ejpam-1598	165	1	now	now	ADV
ejpam-1598	165	2	,	,	PUNCT
ejpam-1598	165	3	if	if	SCONJ
ejpam-1598	165	4	s2hs	s2hs	ADP
ejpam-1598	165	5	=	=	SYM
ejpam-1598	165	6	(	(	PUNCT
ejpam-1598	165	7	0	0	NUM
ejpam-1598	165	8	)	)	PUNCT
ejpam-1598	165	9	,	,	PUNCT
ejpam-1598	165	10	then	then	ADV
ejpam-1598	165	11	h	h	NOUN
ejpam-1598	165	12	⊆	⊆	NUM
ejpam-1598	165	13	rs(s2	rs(s2	NOUN
ejpam-1598	165	14	)	)	PUNCT
ejpam-1598	165	15	⊆	⊆	NUM
ejpam-1598	165	16	rs(s1as2	rs(s1as2	NOUN
ejpam-1598	165	17	)	)	PUNCT
ejpam-1598	165	18	.	.	PUNCT
ejpam-1598	166	1	thus	thus	ADV
ejpam-1598	166	2	rs(s1as2	rs(s1as2	NOUN
ejpam-1598	166	3	)	)	PUNCT
ejpam-1598	166	4	∩	∩	ADJ
ejpam-1598	166	5	h	h	NOUN
ejpam-1598	166	6	=	=	SYM
ejpam-1598	166	7	h	h	PROPN
ejpam-1598	166	8	6=	6=	PROPN
ejpam-1598	166	9	(	(	PUNCT
ejpam-1598	166	10	0	0	NUM
ejpam-1598	166	11	)	)	PUNCT
ejpam-1598	166	12	.	.	PUNCT
ejpam-1598	167	1	if	if	SCONJ
ejpam-1598	167	2	s2hs	s2hs	PROPN
ejpam-1598	167	3	6=	6=	X
ejpam-1598	167	4	(	(	PUNCT
ejpam-1598	167	5	0	0	NUM
ejpam-1598	167	6	)	)	PUNCT
ejpam-1598	167	7	,	,	PUNCT
ejpam-1598	167	8	then	then	ADV
ejpam-1598	167	9	s2hs	s2hs	PUNCT
ejpam-1598	167	10	is	be	AUX
ejpam-1598	167	11	a	a	DET
ejpam-1598	167	12	nonzero	nonzero	ADJ
ejpam-1598	167	13	right	right	ADJ
ejpam-1598	167	14	ideal	ideal	NOUN
ejpam-1598	167	15	of	of	ADP
ejpam-1598	167	16	s.	s.	PROPN
ejpam-1598	167	17	so	so	ADV
ejpam-1598	167	18	rs(a	rs(a	NOUN
ejpam-1598	167	19	)	)	PUNCT
ejpam-1598	167	20	∩	∩	NOUN
ejpam-1598	167	21	s2hs	s2hs	PUNCT
ejpam-1598	167	22	6=	6=	ADP
ejpam-1598	167	23	0	0	X
ejpam-1598	167	24	.	.	PUNCT
ejpam-1598	168	1	let	let	VERB
ejpam-1598	168	2	s2h1s3	s2h1s3	X
ejpam-1598	168	3	(	(	PUNCT
ejpam-1598	168	4	6=	6=	NOUN
ejpam-1598	168	5	0	0	NUM
ejpam-1598	168	6	)	)	PUNCT
ejpam-1598	168	7	∈	∈	PROPN
ejpam-1598	168	8	rs(a	rs(a	NOUN
ejpam-1598	168	9	)	)	PUNCT
ejpam-1598	168	10	∩	∩	NOUN
ejpam-1598	168	11	s2hs	s2hs	PUNCT
ejpam-1598	168	12	for	for	ADP
ejpam-1598	168	13	some	some	DET
ejpam-1598	168	14	h1	h1	NOUN
ejpam-1598	168	15	(	(	PUNCT
ejpam-1598	168	16	6=	6=	NOUN
ejpam-1598	168	17	0	0	NUM
ejpam-1598	168	18	)	)	PUNCT
ejpam-1598	168	19	∈	∈	PROPN
ejpam-1598	168	20	h	h	NOUN
ejpam-1598	168	21	and	and	CCONJ
ejpam-1598	168	22	s3	s3	PROPN
ejpam-1598	168	23	(	(	PUNCT
ejpam-1598	168	24	6=	6=	NOUN
ejpam-1598	168	25	0	0	NUM
ejpam-1598	168	26	)	)	PUNCT
ejpam-1598	168	27	∈	∈	PROPN
ejpam-1598	168	28	s.	s.	PROPN
ejpam-1598	169	1	so	so	ADV
ejpam-1598	169	2	as2h1s3s	as2h1s3s	PROPN
ejpam-1598	169	3	=	=	PROPN
ejpam-1598	169	4	0	0	NUM
ejpam-1598	169	5	for	for	ADP
ejpam-1598	169	6	all	all	DET
ejpam-1598	169	7	s	s	PART
ejpam-1598	169	8	∈	∈	PROPN
ejpam-1598	169	9	s	s	PART
ejpam-1598	169	10	⇒	⇒	NOUN
ejpam-1598	169	11	s1as2h1s3st	s1as2h1s3st	PROPN
ejpam-1598	169	12	=	=	SYM
ejpam-1598	169	13	0	0	NUM
ejpam-1598	169	14	for	for	ADP
ejpam-1598	169	15	all	all	DET
ejpam-1598	169	16	s	s	PROPN
ejpam-1598	169	17	,	,	PUNCT
ejpam-1598	169	18	t	t	PROPN
ejpam-1598	169	19	∈	∈	PROPN
ejpam-1598	169	20	s.	s.	PROPN
ejpam-1598	169	21	as	as	ADP
ejpam-1598	169	22	h1s3s	h1s3s	NOUN
ejpam-1598	169	23	∈	∈	NOUN
ejpam-1598	169	24	h	h	NOUN
ejpam-1598	169	25	for	for	ADP
ejpam-1598	169	26	all	all	PRON
ejpam-1598	169	27	s	s	PART
ejpam-1598	169	28	∈	∈	PROPN
ejpam-1598	169	29	s	s	NOUN
ejpam-1598	169	30	,	,	PUNCT
ejpam-1598	169	31	h1s3s	h1s3s	NUM
ejpam-1598	169	32	∈	∈	PROPN
ejpam-1598	169	33	rs(s1as2	rs(s1as2	PROPN
ejpam-1598	169	34	)	)	PUNCT
ejpam-1598	169	35	∩	∩	PROPN
ejpam-1598	169	36	h	h	NOUN
ejpam-1598	169	37	for	for	ADP
ejpam-1598	169	38	all	all	PRON
ejpam-1598	169	39	s	s	PROPN
ejpam-1598	169	40	∈	∈	PROPN
ejpam-1598	169	41	s.	s.	PROPN
ejpam-1598	169	42	now	now	ADV
ejpam-1598	169	43	,	,	PUNCT
ejpam-1598	169	44	h1s3s	h1s3s	PUNCT
ejpam-1598	169	45	=	=	SYM
ejpam-1598	169	46	0	0	NUM
ejpam-1598	169	47	for	for	ADP
ejpam-1598	169	48	all	all	DET
ejpam-1598	169	49	s	s	PART
ejpam-1598	169	50	∈	∈	NOUN
ejpam-1598	169	51	s	s	PART
ejpam-1598	169	52	⇒	⇒	NOUN
ejpam-1598	169	53	s2h1s3st	s2h1s3st	PROPN
ejpam-1598	169	54	=	=	SYM
ejpam-1598	169	55	0	0	NUM
ejpam-1598	169	56	for	for	ADP
ejpam-1598	169	57	all	all	DET
ejpam-1598	169	58	s	s	PROPN
ejpam-1598	169	59	,	,	PUNCT
ejpam-1598	169	60	t	t	PROPN
ejpam-1598	169	61	∈	∈	PROPN
ejpam-1598	169	62	s	s	PART
ejpam-1598	169	63	⇒	⇒	NOUN
ejpam-1598	169	64	s	s	PART
ejpam-1598	169	65	⊆	⊆	NUM
ejpam-1598	169	66	rs(s2h1s3	rs(s2h1s3	NOUN
ejpam-1598	169	67	)	)	PUNCT
ejpam-1598	169	68	i.e.	i.e.	X
ejpam-1598	169	69	rs(s2h1s3	rs(s2h1s3	X
ejpam-1598	169	70	)	)	PUNCT
ejpam-1598	169	71	=	=	VERB
ejpam-1598	170	1	s.	s.	PROPN
ejpam-1598	170	2	now	now	ADV
ejpam-1598	170	3	by	by	ADP
ejpam-1598	170	4	condition	condition	NOUN
ejpam-1598	170	5	(	(	PUNCT
ejpam-1598	170	6	α	α	X
ejpam-1598	170	7	)	)	PUNCT
ejpam-1598	170	8	this	this	PRON
ejpam-1598	170	9	implies	imply	VERB
ejpam-1598	170	10	that	that	SCONJ
ejpam-1598	170	11	s2h1s3	s2h1s3	PROPN
ejpam-1598	170	12	=	=	SYM
ejpam-1598	170	13	0	0	PROPN
ejpam-1598	170	14	,	,	PUNCT
ejpam-1598	170	15	a	a	DET
ejpam-1598	170	16	contradiction	contradiction	NOUN
ejpam-1598	170	17	.	.	PUNCT
ejpam-1598	171	1	hence	hence	ADV
ejpam-1598	171	2	h1s3s	h1s3s	NOUN
ejpam-1598	171	3	6=	6=	ADP
ejpam-1598	171	4	0	0	NUM
ejpam-1598	171	5	for	for	ADP
ejpam-1598	171	6	some	some	DET
ejpam-1598	171	7	s	s	ADP
ejpam-1598	171	8	∈	∈	PROPN
ejpam-1598	171	9	s.	s.	PROPN
ejpam-1598	171	10	thus	thus	ADV
ejpam-1598	171	11	rs(s1as2	rs(s1as2	PROPN
ejpam-1598	171	12	)	)	PUNCT
ejpam-1598	171	13	∩	∩	PROPN
ejpam-1598	171	14	h	h	PROPN
ejpam-1598	171	15	6=	6=	X
ejpam-1598	171	16	(	(	PUNCT
ejpam-1598	171	17	0	0	NUM
ejpam-1598	171	18	)	)	PUNCT
ejpam-1598	171	19	.	.	PUNCT
ejpam-1598	172	1	therefore	therefore	ADV
ejpam-1598	172	2	in	in	ADP
ejpam-1598	172	3	any	any	DET
ejpam-1598	172	4	case	case	NOUN
ejpam-1598	172	5	,	,	PUNCT
ejpam-1598	172	6	s1as2	s1as2	PROPN
ejpam-1598	172	7	∈	∈	PROPN
ejpam-1598	172	8	z(s	z(s	PROPN
ejpam-1598	172	9	)	)	PUNCT
ejpam-1598	172	10	.	.	PUNCT
ejpam-1598	173	1	consequently	consequently	ADV
ejpam-1598	173	2	,	,	PUNCT
ejpam-1598	173	3	z(s	z(s	PROPN
ejpam-1598	173	4	)	)	PUNCT
ejpam-1598	173	5	is	be	AUX
ejpam-1598	173	6	a	a	DET
ejpam-1598	173	7	lateral	lateral	ADJ
ejpam-1598	173	8	ideal	ideal	NOUN
ejpam-1598	173	9	of	of	ADP
ejpam-1598	173	10	s	s	NOUN
ejpam-1598	173	11	and	and	CCONJ
ejpam-1598	173	12	whence	whence	NOUN
ejpam-1598	173	13	,	,	PUNCT
ejpam-1598	173	14	z(s	z(s	PROPN
ejpam-1598	173	15	)	)	PUNCT
ejpam-1598	173	16	is	be	AUX
ejpam-1598	173	17	an	an	DET
ejpam-1598	173	18	ideal	ideal	NOUN
ejpam-1598	173	19	of	of	ADP
ejpam-1598	173	20	s.	s.	PROPN
ejpam-1598	173	21	finally	finally	ADV
ejpam-1598	173	22	,	,	PUNCT
ejpam-1598	173	23	let	let	VERB
ejpam-1598	173	24	a+	a+	PUNCT
ejpam-1598	173	25	b	b	NOUN
ejpam-1598	173	26	,	,	PUNCT
ejpam-1598	173	27	a	a	DET
ejpam-1598	173	28	∈	∈	PROPN
ejpam-1598	173	29	z(s	z(s	PROPN
ejpam-1598	173	30	)	)	PUNCT
ejpam-1598	173	31	.	.	PUNCT
ejpam-1598	174	1	then	then	ADV
ejpam-1598	174	2	rs(a	rs(a	PUNCT
ejpam-1598	174	3	)	)	PUNCT
ejpam-1598	174	4	and	and	CCONJ
ejpam-1598	174	5	rs(a+	rs(a+	PROPN
ejpam-1598	174	6	b	b	X
ejpam-1598	174	7	)	)	PUNCT
ejpam-1598	174	8	are	be	AUX
ejpam-1598	174	9	both	both	PRON
ejpam-1598	174	10	essential	essential	ADJ
ejpam-1598	174	11	right	right	ADJ
ejpam-1598	174	12	ideals	ideal	NOUN
ejpam-1598	174	13	of	of	ADP
ejpam-1598	174	14	s.	s.	PROPN
ejpam-1598	174	15	therefore	therefore	ADV
ejpam-1598	174	16	,	,	PUNCT
ejpam-1598	174	17	rs(a+	rs(a+	NOUN
ejpam-1598	174	18	b)∩	b)∩	NOUN
ejpam-1598	174	19	rs(a)∩h	rs(a)∩h	PROPN
ejpam-1598	174	20	6=	6=	ADP
ejpam-1598	174	21	0	0	NUM
ejpam-1598	174	22	for	for	ADP
ejpam-1598	174	23	any	any	DET
ejpam-1598	174	24	nonzero	nonzero	NOUN
ejpam-1598	174	25	right	right	ADJ
ejpam-1598	174	26	ideal	ideal	ADJ
ejpam-1598	174	27	h	h	PROPN
ejpam-1598	174	28	of	of	ADP
ejpam-1598	174	29	s.	s.	PROPN
ejpam-1598	174	30	suppose	suppose	VERB
ejpam-1598	174	31	that	that	SCONJ
ejpam-1598	174	32	p	p	X
ejpam-1598	174	33	(	(	PUNCT
ejpam-1598	174	34	6=	6=	NOUN
ejpam-1598	174	35	0	0	NUM
ejpam-1598	174	36	)	)	PUNCT
ejpam-1598	174	37	∈	∈	PROPN
ejpam-1598	174	38	rs(a+	rs(a+	NOUN
ejpam-1598	174	39	b)∩	b)∩	NOUN
ejpam-1598	174	40	rs(a)∩	rs(a)∩	X
ejpam-1598	174	41	h.	h.	PROPN
ejpam-1598	174	42	then	then	ADV
ejpam-1598	174	43	,	,	PUNCT
ejpam-1598	174	44	(	(	PUNCT
ejpam-1598	174	45	a+	a+	PUNCT
ejpam-1598	174	46	b)ps	b)ps	PROPN
ejpam-1598	174	47	=	=	SYM
ejpam-1598	174	48	0=	0=	NUM
ejpam-1598	174	49	aps	ap	NOUN
ejpam-1598	174	50	,	,	PUNCT
ejpam-1598	174	51	for	for	ADP
ejpam-1598	174	52	all	all	DET
ejpam-1598	174	53	s	s	PART
ejpam-1598	174	54	∈	∈	NOUN
ejpam-1598	174	55	s	s	NOUN
ejpam-1598	174	56	and	and	CCONJ
ejpam-1598	174	57	p	p	PROPN
ejpam-1598	174	58	∈	∈	PROPN
ejpam-1598	174	59	h	h	NOUN
ejpam-1598	174	60	,	,	PUNCT
ejpam-1598	174	61	which	which	PRON
ejpam-1598	174	62	implies	imply	VERB
ejpam-1598	174	63	bps	bps	NOUN
ejpam-1598	174	64	=	=	SYM
ejpam-1598	174	65	0	0	NUM
ejpam-1598	174	66	for	for	ADP
ejpam-1598	174	67	all	all	DET
ejpam-1598	174	68	s	s	PROPN
ejpam-1598	174	69	∈	∈	PROPN
ejpam-1598	174	70	s.	s.	PROPN
ejpam-1598	174	71	this	this	PRON
ejpam-1598	174	72	leads	lead	VERB
ejpam-1598	174	73	to	to	ADP
ejpam-1598	174	74	p	p	PROPN
ejpam-1598	174	75	∈	∈	PROPN
ejpam-1598	174	76	rs(b	rs(b	NOUN
ejpam-1598	174	77	)	)	PUNCT
ejpam-1598	175	1	∩	∩	ADJ
ejpam-1598	175	2	h	h	NOUN
ejpam-1598	175	3	i.e.	i.e.	X
ejpam-1598	175	4	rs(b	rs(b	NOUN
ejpam-1598	175	5	)	)	PUNCT
ejpam-1598	175	6	∩	∩	PROPN
ejpam-1598	175	7	h	h	PROPN
ejpam-1598	175	8	6=	6=	PROPN
ejpam-1598	175	9	0	0	NUM
ejpam-1598	175	10	.	.	PUNCT
ejpam-1598	176	1	hence	hence	ADV
ejpam-1598	176	2	,	,	PUNCT
ejpam-1598	176	3	b	b	PROPN
ejpam-1598	176	4	∈	∈	PROPN
ejpam-1598	176	5	z(s	z(s	PROPN
ejpam-1598	176	6	)	)	PUNCT
ejpam-1598	176	7	.	.	PUNCT
ejpam-1598	177	1	therefore	therefore	ADV
ejpam-1598	177	2	,	,	PUNCT
ejpam-1598	177	3	z(s	z(s	PROPN
ejpam-1598	177	4	)	)	PUNCT
ejpam-1598	177	5	is	be	AUX
ejpam-1598	177	6	a	a	DET
ejpam-1598	177	7	k	k	NOUN
ejpam-1598	177	8	-	-	NOUN
ejpam-1598	177	9	ideal	ideal	NOUN
ejpam-1598	177	10	of	of	ADP
ejpam-1598	177	11	s.	s.	PROPN
ejpam-1598	177	12	remark	remark	PROPN
ejpam-1598	177	13	1	1	NUM
ejpam-1598	177	14	.	.	PUNCT
ejpam-1598	178	1	the	the	DET
ejpam-1598	178	2	condition	condition	NOUN
ejpam-1598	178	3	“	"	PUNCT
ejpam-1598	178	4	α	α	X
ejpam-1598	178	5	”	"	PUNCT
ejpam-1598	178	6	is	be	AUX
ejpam-1598	178	7	assumed	assume	VERB
ejpam-1598	178	8	only	only	ADV
ejpam-1598	178	9	to	to	PART
ejpam-1598	178	10	show	show	VERB
ejpam-1598	178	11	that	that	SCONJ
ejpam-1598	178	12	z(s	z(s	PROPN
ejpam-1598	178	13	)	)	PUNCT
ejpam-1598	178	14	is	be	AUX
ejpam-1598	178	15	a	a	DET
ejpam-1598	178	16	lateral	lateral	ADJ
ejpam-1598	178	17	ideal	ideal	NOUN
ejpam-1598	178	18	of	of	ADP
ejpam-1598	178	19	s.	s.	PROPN
ejpam-1598	178	20	in	in	ADP
ejpam-1598	178	21	order	order	NOUN
ejpam-1598	178	22	to	to	PART
ejpam-1598	178	23	show	show	VERB
ejpam-1598	178	24	that	that	SCONJ
ejpam-1598	178	25	z(s	z(s	PROPN
ejpam-1598	178	26	)	)	PUNCT
ejpam-1598	178	27	is	be	AUX
ejpam-1598	178	28	an	an	DET
ejpam-1598	178	29	ideal	ideal	NOUN
ejpam-1598	178	30	of	of	ADP
ejpam-1598	178	31	a	a	DET
ejpam-1598	178	32	commutative	commutative	ADJ
ejpam-1598	178	33	ternary	ternary	ADJ
ejpam-1598	178	34	semiring	semiring	NOUN
ejpam-1598	178	35	,	,	PUNCT
ejpam-1598	178	36	it	it	PRON
ejpam-1598	178	37	is	be	AUX
ejpam-1598	178	38	not	not	PART
ejpam-1598	178	39	necessary	necessary	ADJ
ejpam-1598	178	40	to	to	PART
ejpam-1598	178	41	assume	assume	VERB
ejpam-1598	178	42	the	the	DET
ejpam-1598	178	43	condition	condition	NOUN
ejpam-1598	178	44	“	"	PUNCT
ejpam-1598	178	45	α	α	NOUN
ejpam-1598	178	46	”	"	PUNCT
ejpam-1598	178	47	.	.	PUNCT
ejpam-1598	178	48	remark	remark	PROPN
ejpam-1598	178	49	2	2	NUM
ejpam-1598	178	50	.	.	PUNCT
ejpam-1598	179	1	on	on	ADP
ejpam-1598	179	2	assuming	assume	VERB
ejpam-1598	179	3	the	the	DET
ejpam-1598	179	4	condition	condition	NOUN
ejpam-1598	179	5	“	"	PUNCT
ejpam-1598	179	6	α	α	X
ejpam-1598	179	7	”	"	PUNCT
ejpam-1598	179	8	,	,	PUNCT
ejpam-1598	179	9	we	we	PRON
ejpam-1598	179	10	do	do	AUX
ejpam-1598	179	11	not	not	PART
ejpam-1598	179	12	lose	lose	VERB
ejpam-1598	179	13	the	the	DET
ejpam-1598	179	14	generality	generality	NOUN
ejpam-1598	179	15	because	because	SCONJ
ejpam-1598	179	16	our	our	PRON
ejpam-1598	179	17	aim	aim	NOUN
ejpam-1598	179	18	is	be	AUX
ejpam-1598	179	19	to	to	PART
ejpam-1598	179	20	study	study	VERB
ejpam-1598	179	21	the	the	DET
ejpam-1598	179	22	weakly	weakly	ADJ
ejpam-1598	179	23	special	special	ADJ
ejpam-1598	179	24	radical	radical	ADJ
ejpam-1598	179	25	class	class	NOUN
ejpam-1598	179	26	and	and	CCONJ
ejpam-1598	179	27	the	the	DET
ejpam-1598	179	28	special	special	ADJ
ejpam-1598	179	29	radical	radical	ADJ
ejpam-1598	179	30	class	class	NOUN
ejpam-1598	179	31	of	of	ADP
ejpam-1598	179	32	ternary	ternary	ADJ
ejpam-1598	179	33	semiring	semiring	NOUN
ejpam-1598	179	34	which	which	PRON
ejpam-1598	179	35	are	be	AUX
ejpam-1598	179	36	the	the	DET
ejpam-1598	179	37	radical	radical	ADJ
ejpam-1598	179	38	classes	class	NOUN
ejpam-1598	179	39	of	of	ADP
ejpam-1598	179	40	nonsingular	nonsingular	ADJ
ejpam-1598	179	41	semiprime	semiprime	NOUN
ejpam-1598	179	42	ternary	ternary	ADJ
ejpam-1598	179	43	semirings	semiring	NOUN
ejpam-1598	179	44	and	and	CCONJ
ejpam-1598	179	45	nonsingular	nonsingular	ADJ
ejpam-1598	179	46	prime	prime	ADJ
ejpam-1598	179	47	ternary	ternary	ADJ
ejpam-1598	179	48	semirings	semiring	NOUN
ejpam-1598	179	49	respectively	respectively	ADV
ejpam-1598	179	50	in	in	ADP
ejpam-1598	179	51	which	which	PRON
ejpam-1598	179	52	the	the	DET
ejpam-1598	179	53	condition	condition	NOUN
ejpam-1598	179	54	α	α	PROPN
ejpam-1598	179	55	holds	hold	VERB
ejpam-1598	179	56	which	which	PRON
ejpam-1598	179	57	is	be	AUX
ejpam-1598	179	58	evident	evident	ADJ
ejpam-1598	179	59	from	from	ADP
ejpam-1598	179	60	the	the	DET
ejpam-1598	179	61	above	above	ADJ
ejpam-1598	179	62	example	example	NOUN
ejpam-1598	179	63	2	2	NUM
ejpam-1598	179	64	.	.	X
ejpam-1598	179	65	proposition	proposition	NOUN
ejpam-1598	179	66	6	6	NUM
ejpam-1598	179	67	.	.	PUNCT
ejpam-1598	180	1	let	let	VERB
ejpam-1598	180	2	s	s	PRON
ejpam-1598	180	3	be	be	AUX
ejpam-1598	180	4	a	a	DET
ejpam-1598	180	5	ternary	ternary	ADJ
ejpam-1598	180	6	semiring	semiring	NOUN
ejpam-1598	180	7	.	.	PUNCT
ejpam-1598	181	1	then	then	ADV
ejpam-1598	181	2	z(s	z(s	NUM
ejpam-1598	181	3	)	)	PUNCT
ejpam-1598	181	4	=	=	PRON
ejpam-1598	182	1	{	{	PUNCT
ejpam-1598	182	2	x	x	PUNCT
ejpam-1598	182	3	∈	∈	NOUN
ejpam-1598	182	4	s	s	PART
ejpam-1598	182	5	:	:	PUNCT
ejpam-1598	182	6	x	x	X
ejpam-1598	182	7	is	be	AUX
ejpam-1598	182	8	=	=	NOUN
ejpam-1598	182	9	0	0	NUM
ejpam-1598	182	10	for	for	ADP
ejpam-1598	182	11	some	some	DET
ejpam-1598	182	12	essential	essential	ADJ
ejpam-1598	182	13	right	right	ADJ
ejpam-1598	182	14	ideal	ideal	NOUN
ejpam-1598	183	1	i	i	PRON
ejpam-1598	183	2	of	of	ADP
ejpam-1598	183	3	s	s	NOUN
ejpam-1598	183	4	}	}	PUNCT
ejpam-1598	183	5	proof	proof	NOUN
ejpam-1598	183	6	.	.	PUNCT
ejpam-1598	184	1	let	let	VERB
ejpam-1598	184	2	z	z	PROPN
ejpam-1598	184	3	′(s	′(s	VERB
ejpam-1598	184	4	)	)	PUNCT
ejpam-1598	185	1	=	=	PRON
ejpam-1598	186	1	{	{	PUNCT
ejpam-1598	186	2	x	x	PUNCT
ejpam-1598	186	3	∈	∈	NOUN
ejpam-1598	186	4	s	s	PART
ejpam-1598	186	5	:	:	PUNCT
ejpam-1598	186	6	x	x	X
ejpam-1598	186	7	is	be	AUX
ejpam-1598	186	8	=	=	NOUN
ejpam-1598	186	9	0	0	NUM
ejpam-1598	186	10	for	for	ADP
ejpam-1598	186	11	some	some	DET
ejpam-1598	186	12	essential	essential	ADJ
ejpam-1598	186	13	right	right	ADJ
ejpam-1598	186	14	ideal	ideal	NOUN
ejpam-1598	186	15	i	i	PRON
ejpam-1598	186	16	of	of	ADP
ejpam-1598	186	17	s	s	PROPN
ejpam-1598	186	18	}	}	PUNCT
ejpam-1598	186	19	.	.	PUNCT
ejpam-1598	187	1	suppose	suppose	VERB
ejpam-1598	187	2	x	x	X
ejpam-1598	187	3	∈	∈	PROPN
ejpam-1598	187	4	z(s	z(s	PROPN
ejpam-1598	187	5	)	)	PUNCT
ejpam-1598	187	6	and	and	CCONJ
ejpam-1598	187	7	j	j	PROPN
ejpam-1598	187	8	=	=	SYM
ejpam-1598	187	9	rs(x	rs(x	X
ejpam-1598	187	10	)	)	PUNCT
ejpam-1598	187	11	.	.	PUNCT
ejpam-1598	188	1	then	then	ADV
ejpam-1598	188	2	j	j	PROPN
ejpam-1598	188	3	is	be	AUX
ejpam-1598	188	4	an	an	DET
ejpam-1598	188	5	essential	essential	ADJ
ejpam-1598	188	6	right	right	ADJ
ejpam-1598	188	7	ideal	ideal	NOUN
ejpam-1598	188	8	of	of	ADP
ejpam-1598	188	9	s.	s.	PROPN
ejpam-1598	188	10	also	also	ADV
ejpam-1598	188	11	xjs	xjs	PROPN
ejpam-1598	189	1	=	=	SYM
ejpam-1598	190	1	0	0	PROPN
ejpam-1598	190	2	.	.	PUNCT
ejpam-1598	191	1	hence	hence	ADV
ejpam-1598	191	2	,	,	PUNCT
ejpam-1598	191	3	x	x	PUNCT
ejpam-1598	191	4	∈	∈	PROPN
ejpam-1598	191	5	z	z	PROPN
ejpam-1598	191	6	′(s	′(s	NOUN
ejpam-1598	191	7	)	)	PUNCT
ejpam-1598	191	8	.	.	PUNCT
ejpam-1598	192	1	conversely	conversely	ADV
ejpam-1598	192	2	,	,	PUNCT
ejpam-1598	192	3	let	let	VERB
ejpam-1598	192	4	x	x	X
ejpam-1598	192	5	∈	∈	PROPN
ejpam-1598	192	6	z	z	PROPN
ejpam-1598	192	7	′(s	′(s	NOUN
ejpam-1598	192	8	)	)	PUNCT
ejpam-1598	192	9	.	.	PUNCT
ejpam-1598	193	1	then	then	ADV
ejpam-1598	193	2	x	x	X
ejpam-1598	193	3	is	be	AUX
ejpam-1598	193	4	=	=	NOUN
ejpam-1598	193	5	0	0	NUM
ejpam-1598	193	6	for	for	ADP
ejpam-1598	193	7	some	some	DET
ejpam-1598	193	8	essential	essential	ADJ
ejpam-1598	193	9	right	right	ADJ
ejpam-1598	193	10	ideal	ideal	NOUN
ejpam-1598	193	11	i	i	PRON
ejpam-1598	193	12	of	of	ADP
ejpam-1598	193	13	s.	s.	PROPN
ejpam-1598	193	14	consequently	consequently	ADV
ejpam-1598	193	15	,	,	PUNCT
ejpam-1598	193	16	i	i	PROPN
ejpam-1598	193	17	⊆	⊆	NUM
ejpam-1598	193	18	rs(x	rs(x	X
ejpam-1598	193	19	)	)	PUNCT
ejpam-1598	193	20	.	.	PUNCT
ejpam-1598	194	1	let	let	VERB
ejpam-1598	194	2	h	h	PRON
ejpam-1598	194	3	be	be	AUX
ejpam-1598	194	4	any	any	DET
ejpam-1598	194	5	nonzero	nonzero	ADJ
ejpam-1598	194	6	right	right	ADJ
ejpam-1598	194	7	ideal	ideal	NOUN
ejpam-1598	194	8	of	of	ADP
ejpam-1598	194	9	s.	s.	PROPN
ejpam-1598	194	10	then	then	ADV
ejpam-1598	194	11	0	0	NUM
ejpam-1598	195	1	6=	6=	NUM
ejpam-1598	195	2	i	i	PROPN
ejpam-1598	195	3	∩	∩	NOUN
ejpam-1598	195	4	h	h	PRON
ejpam-1598	195	5	⊆	⊆	NUM
ejpam-1598	195	6	rs(x)∩	rs(x)∩	SYM
ejpam-1598	195	7	h	h	NOUN
ejpam-1598	195	8	which	which	PRON
ejpam-1598	195	9	implies	imply	VERB
ejpam-1598	195	10	x	x	X
ejpam-1598	195	11	∈	∈	PROPN
ejpam-1598	195	12	z(s	z(s	PROPN
ejpam-1598	195	13	)	)	PUNCT
ejpam-1598	195	14	and	and	CCONJ
ejpam-1598	195	15	hence	hence	ADV
ejpam-1598	195	16	the	the	DET
ejpam-1598	195	17	result	result	NOUN
ejpam-1598	195	18	.	.	PUNCT
ejpam-1598	196	1	proposition	proposition	NOUN
ejpam-1598	196	2	7	7	NUM
ejpam-1598	196	3	.	.	PUNCT
ejpam-1598	197	1	let	let	VERB
ejpam-1598	197	2	s	s	PRON
ejpam-1598	197	3	be	be	AUX
ejpam-1598	197	4	a	a	DET
ejpam-1598	197	5	ternary	ternary	ADJ
ejpam-1598	197	6	semiring	semiring	NOUN
ejpam-1598	197	7	.	.	PUNCT
ejpam-1598	198	1	then	then	ADV
ejpam-1598	198	2	z(s	z(s	NUM
ejpam-1598	198	3	)	)	PUNCT
ejpam-1598	198	4	=	=	PRON
ejpam-1598	199	1	{	{	PUNCT
ejpam-1598	199	2	x	x	PUNCT
ejpam-1598	199	3	∈	∈	NOUN
ejpam-1598	199	4	s	s	PART
ejpam-1598	199	5	:	:	PUNCT
ejpam-1598	199	6	x	x	X
ejpam-1598	199	7	is	be	AUX
ejpam-1598	199	8	=	=	NOUN
ejpam-1598	199	9	0	0	NUM
ejpam-1598	199	10	for	for	ADP
ejpam-1598	199	11	some	some	DET
ejpam-1598	199	12	essential	essential	ADJ
ejpam-1598	199	13	right	right	ADJ
ejpam-1598	200	1	k	k	NOUN
ejpam-1598	200	2	-	-	NOUN
ejpam-1598	200	3	ideal	ideal	ADJ
ejpam-1598	200	4	i	i	PRON
ejpam-1598	200	5	of	of	ADP
ejpam-1598	200	6	s	s	NOUN
ejpam-1598	200	7	}	}	PUNCT
ejpam-1598	200	8	t.	t.	PROPN
ejpam-1598	200	9	dutta	dutta	PROPN
ejpam-1598	200	10	,	,	PUNCT
ejpam-1598	200	11	k.	k.	PROPN
ejpam-1598	200	12	shum	shum	PROPN
ejpam-1598	200	13	,	,	PUNCT
ejpam-1598	200	14	s.	s.	PROPN
ejpam-1598	200	15	mandal	mandal	PROPN
ejpam-1598	200	16	/	/	SYM
ejpam-1598	200	17	eur	eur	PROPN
ejpam-1598	200	18	.	.	PUNCT
ejpam-1598	201	1	j.	j.	PROPN
ejpam-1598	201	2	pure	pure	PROPN
ejpam-1598	201	3	appl	appl	PROPN
ejpam-1598	201	4	.	.	PROPN
ejpam-1598	201	5	math	math	PROPN
ejpam-1598	201	6	,	,	PUNCT
ejpam-1598	201	7	5	5	NUM
ejpam-1598	201	8	(	(	PUNCT
ejpam-1598	201	9	2012	2012	NUM
ejpam-1598	201	10	)	)	PUNCT
ejpam-1598	201	11	,	,	PUNCT
ejpam-1598	201	12	116	116	NUM
ejpam-1598	201	13	-	-	SYM
ejpam-1598	201	14	128	128	NUM
ejpam-1598	201	15	122	122	NUM
ejpam-1598	201	16	proof	proof	NOUN
ejpam-1598	201	17	.	.	PUNCT
ejpam-1598	202	1	let	let	VERB
ejpam-1598	202	2	z	z	PROPN
ejpam-1598	202	3	′′(s	′′(s	VERB
ejpam-1598	202	4	)	)	PUNCT
ejpam-1598	202	5	=	=	PRON
ejpam-1598	203	1	{	{	PUNCT
ejpam-1598	203	2	x	x	PUNCT
ejpam-1598	203	3	∈	∈	NOUN
ejpam-1598	203	4	s	s	PART
ejpam-1598	203	5	:	:	PUNCT
ejpam-1598	203	6	x	x	X
ejpam-1598	203	7	is	be	AUX
ejpam-1598	203	8	=	=	NOUN
ejpam-1598	203	9	0	0	NUM
ejpam-1598	203	10	for	for	ADP
ejpam-1598	203	11	some	some	DET
ejpam-1598	203	12	essential	essential	ADJ
ejpam-1598	203	13	right	right	ADJ
ejpam-1598	204	1	k	k	NOUN
ejpam-1598	204	2	-	-	NOUN
ejpam-1598	204	3	ideal	ideal	ADJ
ejpam-1598	204	4	i	i	PRON
ejpam-1598	204	5	of	of	ADP
ejpam-1598	204	6	s	s	PROPN
ejpam-1598	204	7	}	}	PUNCT
ejpam-1598	204	8	.	.	PUNCT
ejpam-1598	205	1	from	from	ADP
ejpam-1598	205	2	proposition	proposition	NOUN
ejpam-1598	205	3	6	6	NUM
ejpam-1598	205	4	,	,	PUNCT
ejpam-1598	205	5	it	it	PRON
ejpam-1598	205	6	follows	follow	VERB
ejpam-1598	205	7	that	that	SCONJ
ejpam-1598	205	8	z	z	PROPN
ejpam-1598	205	9	′′(s)⊆	′′(s)⊆	PROPN
ejpam-1598	205	10	z(s	z(s	PROPN
ejpam-1598	205	11	)	)	PUNCT
ejpam-1598	205	12	.	.	PUNCT
ejpam-1598	206	1	conversely	conversely	ADV
ejpam-1598	206	2	,	,	PUNCT
ejpam-1598	206	3	let	let	VERB
ejpam-1598	206	4	x	x	X
ejpam-1598	206	5	∈	∈	PROPN
ejpam-1598	206	6	z(s	z(s	PROPN
ejpam-1598	206	7	)	)	PUNCT
ejpam-1598	206	8	and	and	CCONJ
ejpam-1598	206	9	j	j	PROPN
ejpam-1598	206	10	=	=	SYM
ejpam-1598	206	11	rs(x	rs(x	X
ejpam-1598	206	12	)	)	PUNCT
ejpam-1598	206	13	.	.	PUNCT
ejpam-1598	207	1	then	then	ADV
ejpam-1598	207	2	j	j	PROPN
ejpam-1598	207	3	is	be	AUX
ejpam-1598	207	4	an	an	DET
ejpam-1598	207	5	essential	essential	ADJ
ejpam-1598	207	6	right	right	ADJ
ejpam-1598	207	7	ideal	ideal	NOUN
ejpam-1598	207	8	of	of	ADP
ejpam-1598	207	9	s.	s.	PROPN
ejpam-1598	207	10	now	now	ADV
ejpam-1598	207	11	let	let	VERB
ejpam-1598	207	12	a	a	DET
ejpam-1598	207	13	,	,	PUNCT
ejpam-1598	207	14	a	a	DET
ejpam-1598	207	15	+	+	X
ejpam-1598	207	16	b	b	NOUN
ejpam-1598	207	17	∈	∈	ADJ
ejpam-1598	207	18	j	j	PROPN
ejpam-1598	207	19	.	.	PUNCT
ejpam-1598	208	1	then	then	ADV
ejpam-1598	208	2	xas	xas	PROPN
ejpam-1598	209	1	=	=	SYM
ejpam-1598	209	2	0	0	PUNCT
ejpam-1598	210	1	=	=	SYM
ejpam-1598	210	2	x(a+	x(a+	PROPN
ejpam-1598	210	3	b)s	b)s	X
ejpam-1598	210	4	for	for	ADP
ejpam-1598	210	5	all	all	DET
ejpam-1598	210	6	s	s	PART
ejpam-1598	210	7	∈	∈	PROPN
ejpam-1598	210	8	s	s	NOUN
ejpam-1598	210	9	,	,	PUNCT
ejpam-1598	210	10	this	this	PRON
ejpam-1598	210	11	implies	imply	VERB
ejpam-1598	210	12	x	x	PUNCT
ejpam-1598	210	13	bs	bs	NOUN
ejpam-1598	210	14	=	=	NOUN
ejpam-1598	210	15	0	0	NUM
ejpam-1598	210	16	for	for	ADP
ejpam-1598	210	17	all	all	DET
ejpam-1598	210	18	s	s	PART
ejpam-1598	210	19	∈	∈	PROPN
ejpam-1598	210	20	s	s	NOUN
ejpam-1598	210	21	,	,	PUNCT
ejpam-1598	210	22	and	and	CCONJ
ejpam-1598	210	23	so	so	ADV
ejpam-1598	210	24	b	b	PROPN
ejpam-1598	210	25	∈	∈	PROPN
ejpam-1598	210	26	rs(x	rs(x	X
ejpam-1598	210	27	)	)	PUNCT
ejpam-1598	211	1	=	=	SYM
ejpam-1598	211	2	j	j	PROPN
ejpam-1598	211	3	.	.	PUNCT
ejpam-1598	212	1	thus	thus	ADV
ejpam-1598	212	2	,	,	PUNCT
ejpam-1598	212	3	j	j	PROPN
ejpam-1598	212	4	is	be	AUX
ejpam-1598	212	5	a	a	DET
ejpam-1598	212	6	right	right	ADJ
ejpam-1598	212	7	k	k	NOUN
ejpam-1598	212	8	-	-	NOUN
ejpam-1598	212	9	ideal	ideal	NOUN
ejpam-1598	212	10	of	of	ADP
ejpam-1598	212	11	s.	s.	PROPN
ejpam-1598	212	12	also	also	ADV
ejpam-1598	212	13	,	,	PUNCT
ejpam-1598	212	14	xjs	xjs	PROPN
ejpam-1598	212	15	=	=	SYM
ejpam-1598	212	16	0	0	PROPN
ejpam-1598	212	17	.	.	PUNCT
ejpam-1598	213	1	this	this	PRON
ejpam-1598	213	2	leads	lead	VERB
ejpam-1598	213	3	to	to	ADP
ejpam-1598	213	4	x	x	PUNCT
ejpam-1598	213	5	∈	∈	PROPN
ejpam-1598	213	6	z	z	PROPN
ejpam-1598	213	7	′′(s	′′(s	PROPN
ejpam-1598	213	8	)	)	PUNCT
ejpam-1598	213	9	and	and	CCONJ
ejpam-1598	213	10	hence	hence	ADV
ejpam-1598	213	11	z(s)⊆	z(s)⊆	PROPN
ejpam-1598	213	12	z	z	PROPN
ejpam-1598	213	13	′′(s	′′(s	PROPN
ejpam-1598	213	14	)	)	PUNCT
ejpam-1598	213	15	.	.	PUNCT
ejpam-1598	214	1	the	the	DET
ejpam-1598	214	2	proposition	proposition	NOUN
ejpam-1598	214	3	is	be	AUX
ejpam-1598	214	4	hence	hence	ADV
ejpam-1598	214	5	proved	prove	VERB
ejpam-1598	214	6	.	.	PUNCT
ejpam-1598	215	1	definition	definition	NOUN
ejpam-1598	215	2	20	20	NUM
ejpam-1598	215	3	.	.	PUNCT
ejpam-1598	216	1	a	a	DET
ejpam-1598	216	2	ternary	ternary	ADJ
ejpam-1598	216	3	semiring	semiring	NOUN
ejpam-1598	216	4	s	s	NOUN
ejpam-1598	216	5	is	be	AUX
ejpam-1598	216	6	said	say	VERB
ejpam-1598	216	7	to	to	PART
ejpam-1598	216	8	be	be	AUX
ejpam-1598	216	9	singular	singular	ADJ
ejpam-1598	216	10	if	if	SCONJ
ejpam-1598	216	11	z(s	z(s	PROPN
ejpam-1598	216	12	)	)	PUNCT
ejpam-1598	217	1	=	=	SYM
ejpam-1598	217	2	s	s	VERB
ejpam-1598	217	3	and	and	CCONJ
ejpam-1598	217	4	is	be	AUX
ejpam-1598	217	5	said	say	VERB
ejpam-1598	217	6	to	to	PART
ejpam-1598	217	7	be	be	AUX
ejpam-1598	217	8	nonsingular	nonsingular	ADJ
ejpam-1598	217	9	if	if	SCONJ
ejpam-1598	217	10	z(s	z(s	PROPN
ejpam-1598	217	11	)	)	PUNCT
ejpam-1598	218	1	=	=	PUNCT
ejpam-1598	219	1	0	0	X
ejpam-1598	219	2	.	.	PUNCT
ejpam-1598	220	1	we	we	PRON
ejpam-1598	220	2	now	now	ADV
ejpam-1598	220	3	study	study	VERB
ejpam-1598	220	4	the	the	DET
ejpam-1598	220	5	singular	singular	ADJ
ejpam-1598	220	6	ternary	ternary	ADJ
ejpam-1598	220	7	semirings	semiring	NOUN
ejpam-1598	220	8	.	.	PUNCT
ejpam-1598	221	1	proposition	proposition	NOUN
ejpam-1598	221	2	8	8	NUM
ejpam-1598	221	3	.	.	PUNCT
ejpam-1598	222	1	let	let	VERB
ejpam-1598	222	2	s	s	PRON
ejpam-1598	222	3	be	be	AUX
ejpam-1598	222	4	a	a	DET
ejpam-1598	222	5	non	non	ADJ
ejpam-1598	222	6	-	-	ADJ
ejpam-1598	222	7	singular	singular	ADJ
ejpam-1598	222	8	ternary	ternary	ADJ
ejpam-1598	222	9	semiring	semiring	NOUN
ejpam-1598	222	10	,	,	PUNCT
ejpam-1598	222	11	then	then	ADV
ejpam-1598	222	12	s	s	AUX
ejpam-1598	222	13	satisfies	satisfie	NOUN
ejpam-1598	222	14	the	the	DET
ejpam-1598	222	15	condition	condition	NOUN
ejpam-1598	222	16	α	α	NOUN
ejpam-1598	222	17	.	.	PUNCT
ejpam-1598	223	1	proof	proof	NOUN
ejpam-1598	223	2	.	.	PUNCT
ejpam-1598	224	1	let	let	VERB
ejpam-1598	224	2	ass	ass	NOUN
ejpam-1598	224	3	=	=	SYM
ejpam-1598	224	4	0	0	PROPN
ejpam-1598	224	5	.	.	PUNCT
ejpam-1598	225	1	then	then	ADV
ejpam-1598	225	2	rs(a	rs(a	PUNCT
ejpam-1598	225	3	)	)	PUNCT
ejpam-1598	226	1	=	=	PUNCT
ejpam-1598	226	2	s	s	VERB
ejpam-1598	226	3	is	be	AUX
ejpam-1598	226	4	an	an	DET
ejpam-1598	226	5	essential	essential	ADJ
ejpam-1598	226	6	right	right	ADJ
ejpam-1598	226	7	ideal	ideal	NOUN
ejpam-1598	226	8	of	of	ADP
ejpam-1598	226	9	s.	s.	PROPN
ejpam-1598	226	10	hence	hence	ADV
ejpam-1598	226	11	a	a	DET
ejpam-1598	226	12	∈	∈	PROPN
ejpam-1598	226	13	z(s	z(s	PROPN
ejpam-1598	226	14	)	)	PUNCT
ejpam-1598	226	15	.	.	PUNCT
ejpam-1598	227	1	since	since	SCONJ
ejpam-1598	227	2	s	s	PROPN
ejpam-1598	227	3	is	be	AUX
ejpam-1598	227	4	non	non	ADJ
ejpam-1598	227	5	-	-	ADJ
ejpam-1598	227	6	singular	singular	ADJ
ejpam-1598	227	7	,	,	PUNCT
ejpam-1598	227	8	z(s	z(s	PROPN
ejpam-1598	227	9	)	)	PUNCT
ejpam-1598	228	1	=	=	SYM
ejpam-1598	228	2	0	0	NUM
ejpam-1598	229	1	and	and	CCONJ
ejpam-1598	229	2	so	so	ADV
ejpam-1598	229	3	a	a	DET
ejpam-1598	229	4	=	=	ADJ
ejpam-1598	229	5	0	0	NUM
ejpam-1598	229	6	.	.	PUNCT
ejpam-1598	230	1	thus	thus	ADV
ejpam-1598	230	2	,	,	PUNCT
ejpam-1598	230	3	s	s	VERB
ejpam-1598	230	4	satisfies	satisfie	NOUN
ejpam-1598	230	5	the	the	DET
ejpam-1598	230	6	condition	condition	NOUN
ejpam-1598	230	7	α	α	NOUN
ejpam-1598	230	8	.	.	PUNCT
ejpam-1598	231	1	definition	definition	NOUN
ejpam-1598	231	2	21	21	NUM
ejpam-1598	231	3	.	.	PUNCT
ejpam-1598	232	1	[	[	X
ejpam-1598	232	2	5	5	NUM
ejpam-1598	232	3	]	]	PUNCT
ejpam-1598	232	4	an	an	DET
ejpam-1598	232	5	element	element	NOUN
ejpam-1598	232	6	s	s	NOUN
ejpam-1598	232	7	of	of	ADP
ejpam-1598	232	8	a	a	DET
ejpam-1598	232	9	ternary	ternary	ADJ
ejpam-1598	232	10	semiring	semiring	NOUN
ejpam-1598	232	11	s	s	NOUN
ejpam-1598	232	12	is	be	AUX
ejpam-1598	232	13	said	say	VERB
ejpam-1598	232	14	to	to	PART
ejpam-1598	232	15	be	be	AUX
ejpam-1598	232	16	nilpotent	nilpotent	ADJ
ejpam-1598	232	17	if	if	SCONJ
ejpam-1598	232	18	for	for	ADP
ejpam-1598	232	19	each	each	DET
ejpam-1598	232	20	t	t	NOUN
ejpam-1598	232	21	∈	∈	PROPN
ejpam-1598	232	22	s	s	X
ejpam-1598	232	23	,	,	PUNCT
ejpam-1598	232	24	there	there	PRON
ejpam-1598	232	25	exists	exist	VERB
ejpam-1598	232	26	a	a	DET
ejpam-1598	232	27	positive	positive	ADJ
ejpam-1598	232	28	integer	integer	NOUN
ejpam-1598	232	29	n(depending	n(depende	VERB
ejpam-1598	232	30	on	on	ADP
ejpam-1598	232	31	t	t	PROPN
ejpam-1598	232	32	)	)	PUNCT
ejpam-1598	232	33	such	such	ADJ
ejpam-1598	232	34	that	that	SCONJ
ejpam-1598	232	35	(	(	PUNCT
ejpam-1598	232	36	st)ns	st)ns	X
ejpam-1598	232	37	=	=	SYM
ejpam-1598	232	38	0	0	X
ejpam-1598	232	39	.	.	PUNCT
ejpam-1598	233	1	definition	definition	NOUN
ejpam-1598	233	2	22	22	NUM
ejpam-1598	233	3	.	.	PUNCT
ejpam-1598	234	1	a	a	DET
ejpam-1598	234	2	ternary	ternary	ADJ
ejpam-1598	234	3	semiring	semiring	NOUN
ejpam-1598	234	4	s	s	NOUN
ejpam-1598	234	5	is	be	AUX
ejpam-1598	234	6	said	say	VERB
ejpam-1598	234	7	to	to	PART
ejpam-1598	234	8	be	be	AUX
ejpam-1598	234	9	a	a	DET
ejpam-1598	234	10	nil	nil	ADJ
ejpam-1598	234	11	ternary	ternary	NOUN
ejpam-1598	234	12	semiring	semiring	NOUN
ejpam-1598	234	13	if	if	SCONJ
ejpam-1598	234	14	each	each	DET
ejpam-1598	234	15	a	a	PRON
ejpam-1598	234	16	in	in	ADP
ejpam-1598	234	17	s	s	PROPN
ejpam-1598	234	18	is	be	AUX
ejpam-1598	234	19	nilpotent	nilpotent	ADJ
ejpam-1598	234	20	.	.	PUNCT
ejpam-1598	235	1	as	as	ADP
ejpam-1598	235	2	examples	example	NOUN
ejpam-1598	235	3	of	of	ADP
ejpam-1598	235	4	singular	singular	ADJ
ejpam-1598	235	5	ternary	ternary	ADJ
ejpam-1598	235	6	semirings	semiring	NOUN
ejpam-1598	235	7	and	and	CCONJ
ejpam-1598	235	8	non	non	ADJ
ejpam-1598	235	9	-	-	ADJ
ejpam-1598	235	10	singular	singular	ADJ
ejpam-1598	235	11	ternary	ternary	ADJ
ejpam-1598	235	12	semirings	semiring	NOUN
ejpam-1598	235	13	we	we	PRON
ejpam-1598	235	14	have	have	VERB
ejpam-1598	235	15	the	the	DET
ejpam-1598	235	16	following	follow	VERB
ejpam-1598	235	17	propositions	proposition	NOUN
ejpam-1598	235	18	.	.	PUNCT
ejpam-1598	236	1	proposition	proposition	NOUN
ejpam-1598	236	2	9	9	NUM
ejpam-1598	236	3	.	.	PUNCT
ejpam-1598	237	1	every	every	DET
ejpam-1598	237	2	commutative	commutative	ADJ
ejpam-1598	237	3	nil	nil	ADJ
ejpam-1598	237	4	ternary	ternary	ADJ
ejpam-1598	237	5	semiring	semiring	NOUN
ejpam-1598	237	6	s	s	NOUN
ejpam-1598	237	7	with	with	ADP
ejpam-1598	237	8	a	a	DET
ejpam-1598	237	9	unital	unital	ADJ
ejpam-1598	237	10	element	element	NOUN
ejpam-1598	237	11	e	e	NOUN
ejpam-1598	237	12	is	be	AUX
ejpam-1598	237	13	singular	singular	ADJ
ejpam-1598	237	14	.	.	PUNCT
ejpam-1598	238	1	proof	proof	NOUN
ejpam-1598	238	2	.	.	PUNCT
ejpam-1598	239	1	pick	pick	VERB
ejpam-1598	239	2	a	a	DET
ejpam-1598	239	3	(	(	PUNCT
ejpam-1598	239	4	6=	6=	NOUN
ejpam-1598	239	5	0	0	NUM
ejpam-1598	239	6	)	)	PUNCT
ejpam-1598	239	7	∈	∈	PROPN
ejpam-1598	239	8	s.	s.	PROPN
ejpam-1598	239	9	let	let	VERB
ejpam-1598	239	10	h	h	NOUN
ejpam-1598	239	11	be	be	AUX
ejpam-1598	239	12	a	a	DET
ejpam-1598	239	13	nonzero	nonzero	ADJ
ejpam-1598	239	14	right	right	ADJ
ejpam-1598	239	15	ideal	ideal	NOUN
ejpam-1598	239	16	of	of	ADP
ejpam-1598	239	17	s	s	PROPN
ejpam-1598	239	18	and	and	CCONJ
ejpam-1598	239	19	b	b	PROPN
ejpam-1598	239	20	(	(	PUNCT
ejpam-1598	239	21	6=	6=	NOUN
ejpam-1598	239	22	0	0	NUM
ejpam-1598	239	23	)	)	PUNCT
ejpam-1598	239	24	∈	∈	PROPN
ejpam-1598	239	25	h.	h.	PROPN
ejpam-1598	239	26	as	as	SCONJ
ejpam-1598	239	27	s	s	PROPN
ejpam-1598	239	28	is	be	AUX
ejpam-1598	239	29	nil	nil	ADJ
ejpam-1598	239	30	,	,	PUNCT
ejpam-1598	239	31	a	a	PRON
ejpam-1598	239	32	is	be	AUX
ejpam-1598	239	33	clearly	clearly	ADV
ejpam-1598	239	34	nilpotent	nilpotent	ADJ
ejpam-1598	239	35	.	.	PUNCT
ejpam-1598	240	1	then	then	ADV
ejpam-1598	240	2	,	,	PUNCT
ejpam-1598	240	3	for	for	ADP
ejpam-1598	240	4	each	each	DET
ejpam-1598	240	5	t	t	NOUN
ejpam-1598	240	6	in	in	ADP
ejpam-1598	240	7	s	s	PROPN
ejpam-1598	240	8	,	,	PUNCT
ejpam-1598	240	9	there	there	PRON
ejpam-1598	240	10	exists	exist	VERB
ejpam-1598	240	11	a	a	DET
ejpam-1598	240	12	positive	positive	ADJ
ejpam-1598	240	13	integer	integer	NOUN
ejpam-1598	240	14	n(depending	n(depende	VERB
ejpam-1598	240	15	on	on	ADP
ejpam-1598	240	16	t	t	PROPN
ejpam-1598	240	17	)	)	PUNCT
ejpam-1598	240	18	such	such	ADJ
ejpam-1598	240	19	that	that	SCONJ
ejpam-1598	240	20	(	(	PUNCT
ejpam-1598	240	21	at)na	at)na	NOUN
ejpam-1598	240	22	=	=	NOUN
ejpam-1598	240	23	0	0	NUM
ejpam-1598	240	24	,	,	PUNCT
ejpam-1598	240	25	or	or	CCONJ
ejpam-1598	240	26	equivalently	equivalently	ADV
ejpam-1598	240	27	,	,	PUNCT
ejpam-1598	240	28	we	we	PRON
ejpam-1598	240	29	have	have	VERB
ejpam-1598	240	30	a(ta)n	a(ta)n	NOUN
ejpam-1598	240	31	=	=	SYM
ejpam-1598	240	32	0	0	NUM
ejpam-1598	240	33	.	.	PUNCT
ejpam-1598	241	1	thus	thus	ADV
ejpam-1598	241	2	,	,	PUNCT
ejpam-1598	241	3	in	in	ADP
ejpam-1598	241	4	particular	particular	ADJ
ejpam-1598	241	5	,	,	PUNCT
ejpam-1598	241	6	a(ea)n	a(ea)n	PROPN
ejpam-1598	241	7	=	=	NOUN
ejpam-1598	241	8	0	0	NUM
ejpam-1598	241	9	for	for	ADP
ejpam-1598	241	10	some	some	DET
ejpam-1598	241	11	positive	positive	ADJ
ejpam-1598	241	12	integer	integer	NOUN
ejpam-1598	241	13	n.	n.	NOUN
ejpam-1598	241	14	this	this	PRON
ejpam-1598	241	15	implies	imply	VERB
ejpam-1598	241	16	a(ea)n	a(ea)n	PROPN
ejpam-1598	241	17	bs	bs	NOUN
ejpam-1598	241	18	=	=	SYM
ejpam-1598	241	19	0	0	X
ejpam-1598	241	20	.	.	PUNCT
ejpam-1598	242	1	now	now	ADV
ejpam-1598	242	2	let	let	VERB
ejpam-1598	242	3	m	m	PRON
ejpam-1598	242	4	be	be	AUX
ejpam-1598	242	5	the	the	DET
ejpam-1598	242	6	least	least	ADV
ejpam-1598	242	7	positive	positive	ADJ
ejpam-1598	242	8	integer	integer	NOUN
ejpam-1598	242	9	such	such	ADJ
ejpam-1598	242	10	that	that	SCONJ
ejpam-1598	242	11	a(ea)mbs	a(ea)mb	NOUN
ejpam-1598	242	12	=	=	SYM
ejpam-1598	242	13	0	0	NUM
ejpam-1598	242	14	.	.	PUNCT
ejpam-1598	243	1	then	then	ADV
ejpam-1598	243	2	(	(	PUNCT
ejpam-1598	243	3	ea)mb	ea)mb	PROPN
ejpam-1598	243	4	∈	∈	NOUN
ejpam-1598	243	5	rs(a	rs(a	NOUN
ejpam-1598	243	6	)	)	PUNCT
ejpam-1598	243	7	.	.	PUNCT
ejpam-1598	244	1	also	also	ADV
ejpam-1598	244	2	(	(	PUNCT
ejpam-1598	244	3	ea)mb	ea)mb	PROPN
ejpam-1598	244	4	∈	∈	PROPN
ejpam-1598	244	5	h.	h.	NOUN
ejpam-1598	244	6	now	now	ADV
ejpam-1598	244	7	(	(	PUNCT
ejpam-1598	244	8	ea)m	ea)m	PROPN
ejpam-1598	244	9	b	b	PROPN
ejpam-1598	244	10	=	=	NOUN
ejpam-1598	244	11	a(ea)m−1	a(ea)m−1	NOUN
ejpam-1598	244	12	be	be	AUX
ejpam-1598	244	13	6=	6=	ADP
ejpam-1598	244	14	0	0	NUM
ejpam-1598	244	15	by	by	ADP
ejpam-1598	244	16	minimality	minimality	NOUN
ejpam-1598	244	17	of	of	ADP
ejpam-1598	244	18	m	m	PROPN
ejpam-1598	244	19	,	,	PUNCT
ejpam-1598	244	20	for	for	ADP
ejpam-1598	244	21	a(ea)m−1	a(ea)m−1	NOUN
ejpam-1598	244	22	be	be	AUX
ejpam-1598	244	23	=	=	PUNCT
ejpam-1598	244	24	0⇒	0⇒	PROPN
ejpam-1598	244	25	a(ea)m−1	a(ea)m−1	NOUN
ejpam-1598	244	26	bese	bese	NOUN
ejpam-1598	244	27	=	=	NOUN
ejpam-1598	244	28	0	0	NUM
ejpam-1598	244	29	for	for	ADP
ejpam-1598	244	30	all	all	DET
ejpam-1598	244	31	s	s	PART
ejpam-1598	244	32	∈	∈	PROPN
ejpam-1598	244	33	s	s	NOUN
ejpam-1598	244	34	,	,	PUNCT
ejpam-1598	244	35	that	that	ADV
ejpam-1598	244	36	is	is	ADV
ejpam-1598	244	37	,	,	PUNCT
ejpam-1598	244	38	a(ea)m−1	a(ea)m−1	NOUN
ejpam-1598	244	39	bs	bs	NOUN
ejpam-1598	244	40	=	=	SYM
ejpam-1598	244	41	0	0	PROPN
ejpam-1598	244	42	,	,	PUNCT
ejpam-1598	244	43	a	a	DET
ejpam-1598	244	44	contradiction	contradiction	NOUN
ejpam-1598	244	45	.	.	PUNCT
ejpam-1598	245	1	hence	hence	ADV
ejpam-1598	245	2	,	,	PUNCT
ejpam-1598	245	3	rs(a	rs(a	NOUN
ejpam-1598	245	4	)	)	PUNCT
ejpam-1598	245	5	∩	∩	NOUN
ejpam-1598	245	6	h	h	NOUN
ejpam-1598	245	7	6=	6=	NOUN
ejpam-1598	245	8	0	0	NUM
ejpam-1598	246	1	and	and	CCONJ
ejpam-1598	246	2	so	so	ADV
ejpam-1598	246	3	a	a	DET
ejpam-1598	246	4	∈	∈	PROPN
ejpam-1598	246	5	z(s	z(s	PROPN
ejpam-1598	246	6	)	)	PUNCT
ejpam-1598	246	7	.	.	PUNCT
ejpam-1598	247	1	this	this	PRON
ejpam-1598	247	2	proves	prove	VERB
ejpam-1598	247	3	that	that	SCONJ
ejpam-1598	247	4	z(s	z(s	NOUN
ejpam-1598	247	5	)	)	PUNCT
ejpam-1598	247	6	=	=	PUNCT
ejpam-1598	248	1	s.	s.	PROPN
ejpam-1598	248	2	we	we	PRON
ejpam-1598	248	3	give	give	VERB
ejpam-1598	248	4	below	below	ADP
ejpam-1598	248	5	the	the	DET
ejpam-1598	248	6	definition	definition	NOUN
ejpam-1598	248	7	of	of	ADP
ejpam-1598	248	8	a	a	DET
ejpam-1598	248	9	right	right	ADV
ejpam-1598	248	10	strongly	strongly	ADV
ejpam-1598	248	11	prime	prime	ADJ
ejpam-1598	248	12	ternary	ternary	ADJ
ejpam-1598	248	13	semiring	semiring	NOUN
ejpam-1598	248	14	.	.	PUNCT
ejpam-1598	249	1	definition	definition	NOUN
ejpam-1598	249	2	23	23	NUM
ejpam-1598	249	3	.	.	PUNCT
ejpam-1598	250	1	[	[	X
ejpam-1598	250	2	12	12	NUM
ejpam-1598	250	3	]	]	PUNCT
ejpam-1598	250	4	a	a	DET
ejpam-1598	250	5	ternary	ternary	ADJ
ejpam-1598	250	6	semiring	semiring	NOUN
ejpam-1598	250	7	s	s	VERB
ejpam-1598	250	8	is	be	AUX
ejpam-1598	250	9	called	call	VERB
ejpam-1598	250	10	right	right	ADV
ejpam-1598	250	11	strongly	strongly	ADV
ejpam-1598	250	12	prime	prime	ADJ
ejpam-1598	250	13	if	if	SCONJ
ejpam-1598	250	14	for	for	ADP
ejpam-1598	250	15	every	every	DET
ejpam-1598	250	16	nonzero	nonzero	NOUN
ejpam-1598	250	17	element	element	NOUN
ejpam-1598	250	18	a	a	DET
ejpam-1598	250	19	∈	∈	PROPN
ejpam-1598	250	20	s	s	NOUN
ejpam-1598	250	21	,	,	PUNCT
ejpam-1598	250	22	there	there	PRON
ejpam-1598	250	23	exist	exist	VERB
ejpam-1598	250	24	some	some	DET
ejpam-1598	250	25	finite	finite	ADJ
ejpam-1598	250	26	subsets	subset	NOUN
ejpam-1598	250	27	f1	f1	NOUN
ejpam-1598	250	28	,	,	PUNCT
ejpam-1598	250	29	f2	f2	PROPN
ejpam-1598	250	30	,	,	PUNCT
ejpam-1598	250	31	f3	f3	PROPN
ejpam-1598	250	32	of	of	ADP
ejpam-1598	250	33	s	s	PRON
ejpam-1598	250	34	such	such	ADJ
ejpam-1598	250	35	that	that	SCONJ
ejpam-1598	250	36	af1f2f3	af1f2f3	ADP
ejpam-1598	250	37	y	y	PROPN
ejpam-1598	250	38	=	=	SYM
ejpam-1598	250	39	{	{	PUNCT
ejpam-1598	250	40	0	0	NUM
ejpam-1598	250	41	}	}	PUNCT
ejpam-1598	250	42	⇒	⇒	NOUN
ejpam-1598	250	43	y	y	PROPN
ejpam-1598	250	44	=	=	PUNCT
ejpam-1598	250	45	0	0	NUM
ejpam-1598	250	46	for	for	ADP
ejpam-1598	250	47	all	all	DET
ejpam-1598	250	48	y	y	PROPN
ejpam-1598	250	49	∈	∈	PROPN
ejpam-1598	250	50	s.	s.	PROPN
ejpam-1598	250	51	we	we	PRON
ejpam-1598	250	52	study	study	VERB
ejpam-1598	250	53	below	below	ADP
ejpam-1598	250	54	the	the	DET
ejpam-1598	250	55	properties	property	NOUN
ejpam-1598	250	56	of	of	ADP
ejpam-1598	250	57	a	a	DET
ejpam-1598	250	58	right	right	ADV
ejpam-1598	250	59	strongly	strongly	ADV
ejpam-1598	250	60	prime	prime	ADJ
ejpam-1598	250	61	ternary	ternary	ADJ
ejpam-1598	250	62	semiring	semiring	NOUN
ejpam-1598	250	63	.	.	PUNCT
ejpam-1598	251	1	lemma	lemma	PROPN
ejpam-1598	251	2	1	1	X
ejpam-1598	251	3	.	.	PUNCT
ejpam-1598	252	1	let	let	VERB
ejpam-1598	252	2	s	s	PRON
ejpam-1598	252	3	be	be	AUX
ejpam-1598	252	4	a	a	DET
ejpam-1598	252	5	right	right	ADV
ejpam-1598	252	6	strongly	strongly	ADV
ejpam-1598	252	7	prime	prime	ADJ
ejpam-1598	252	8	ternary	ternary	ADJ
ejpam-1598	252	9	semiring	semiring	NOUN
ejpam-1598	252	10	with	with	ADP
ejpam-1598	252	11	identity	identity	NOUN
ejpam-1598	252	12	.	.	PUNCT
ejpam-1598	253	1	then	then	ADV
ejpam-1598	253	2	every	every	DET
ejpam-1598	253	3	nonzero	nonzero	NOUN
ejpam-1598	253	4	ideal	ideal	NOUN
ejpam-1598	253	5	of	of	ADP
ejpam-1598	253	6	s	s	PRON
ejpam-1598	253	7	obtains	obtain	VERB
ejpam-1598	253	8	a	a	DET
ejpam-1598	253	9	finite	finite	NOUN
ejpam-1598	253	10	subset	subset	NOUN
ejpam-1598	253	11	g	g	PROPN
ejpam-1598	253	12	such	such	ADJ
ejpam-1598	253	13	that	that	PRON
ejpam-1598	253	14	rs(g	rs(g	PUNCT
ejpam-1598	253	15	)	)	PUNCT
ejpam-1598	253	16	=	=	SYM
ejpam-1598	253	17	0	0	NUM
ejpam-1598	253	18	,	,	PUNCT
ejpam-1598	253	19	where	where	SCONJ
ejpam-1598	253	20	rs(g	rs(g	PUNCT
ejpam-1598	253	21	)	)	PUNCT
ejpam-1598	253	22	=	=	SYM
ejpam-1598	254	1	{	{	PUNCT
ejpam-1598	254	2	t	t	PROPN
ejpam-1598	254	3	∈	∈	PROPN
ejpam-1598	254	4	s	s	PART
ejpam-1598	254	5	:	:	PUNCT
ejpam-1598	254	6	gts	gts	NOUN
ejpam-1598	254	7	=	=	NOUN
ejpam-1598	254	8	0	0	NUM
ejpam-1598	254	9	for	for	ADP
ejpam-1598	254	10	all	all	DET
ejpam-1598	254	11	s	s	PART
ejpam-1598	254	12	∈	∈	NOUN
ejpam-1598	254	13	s	s	PART
ejpam-1598	254	14	}	}	PUNCT
ejpam-1598	254	15	.	.	PUNCT
ejpam-1598	255	1	t.	t.	PROPN
ejpam-1598	255	2	dutta	dutta	PROPN
ejpam-1598	255	3	,	,	PUNCT
ejpam-1598	255	4	k.	k.	PROPN
ejpam-1598	255	5	shum	shum	PROPN
ejpam-1598	255	6	,	,	PUNCT
ejpam-1598	255	7	s.	s.	PROPN
ejpam-1598	255	8	mandal	mandal	PROPN
ejpam-1598	255	9	/	/	SYM
ejpam-1598	255	10	eur	eur	PROPN
ejpam-1598	255	11	.	.	PUNCT
ejpam-1598	256	1	j.	j.	PROPN
ejpam-1598	256	2	pure	pure	PROPN
ejpam-1598	256	3	appl	appl	PROPN
ejpam-1598	256	4	.	.	PROPN
ejpam-1598	256	5	math	math	PROPN
ejpam-1598	256	6	,	,	PUNCT
ejpam-1598	256	7	5	5	NUM
ejpam-1598	256	8	(	(	PUNCT
ejpam-1598	256	9	2012	2012	NUM
ejpam-1598	256	10	)	)	PUNCT
ejpam-1598	256	11	,	,	PUNCT
ejpam-1598	256	12	116	116	NUM
ejpam-1598	256	13	-	-	SYM
ejpam-1598	256	14	128	128	NUM
ejpam-1598	256	15	123	123	NUM
ejpam-1598	256	16	proof	proof	NOUN
ejpam-1598	256	17	.	.	PUNCT
ejpam-1598	257	1	since	since	SCONJ
ejpam-1598	257	2	s	s	PROPN
ejpam-1598	257	3	admits	admit	VERB
ejpam-1598	257	4	identity	identity	NOUN
ejpam-1598	257	5	,	,	PUNCT
ejpam-1598	257	6	there	there	PRON
ejpam-1598	257	7	exist	exist	VERB
ejpam-1598	257	8	elements	element	NOUN
ejpam-1598	257	9	{	{	PUNCT
ejpam-1598	257	10	(	(	PUNCT
ejpam-1598	257	11	ei	ei	NOUN
ejpam-1598	257	12	,	,	PUNCT
ejpam-1598	257	13	fi	fi	NOUN
ejpam-1598	257	14	)	)	PUNCT
ejpam-1598	257	15	∈	∈	PROPN
ejpam-1598	257	16	s	s	PART
ejpam-1598	257	17	×	×	NOUN
ejpam-1598	257	18	s	s	X
ejpam-1598	257	19	(	(	PUNCT
ejpam-1598	257	20	i	i	NOUN
ejpam-1598	257	21	=	=	SYM
ejpam-1598	257	22	1,2	1,2	NUM
ejpam-1598	257	23	,	,	PUNCT
ejpam-1598	257	24	.	.	PUNCT
ejpam-1598	257	25	.	.	PUNCT
ejpam-1598	258	1	.	.	PUNCT
ejpam-1598	258	2	,	,	PUNCT
ejpam-1598	258	3	n	n	CCONJ
ejpam-1598	258	4	)	)	PUNCT
ejpam-1598	259	1	}	}	PUNCT
ejpam-1598	259	2	such	such	ADJ
ejpam-1598	259	3	that	that	SCONJ
ejpam-1598	259	4	∑n	∑n	PROPN
ejpam-1598	259	5	i=1	i=1	X
ejpam-1598	259	6	ei	ei	NOUN
ejpam-1598	259	7	fi	fi	NOUN
ejpam-1598	260	1	x	x	X
ejpam-1598	261	1	=	=	PUNCT
ejpam-1598	261	2	∑n	∑n	PROPN
ejpam-1598	261	3	i=1	i=1	X
ejpam-1598	261	4	ei	ei	X
ejpam-1598	261	5	x	x	PUNCT
ejpam-1598	261	6	fi	fi	NOUN
ejpam-1598	262	1	=	=	PUNCT
ejpam-1598	262	2	∑n	∑n	PROPN
ejpam-1598	262	3	i=1	i=1	PROPN
ejpam-1598	262	4	xei	xei	PROPN
ejpam-1598	262	5	fi	fi	NOUN
ejpam-1598	263	1	=	=	NOUN
ejpam-1598	263	2	x	x	PROPN
ejpam-1598	263	3	for	for	ADP
ejpam-1598	263	4	all	all	PRON
ejpam-1598	263	5	x	x	SYM
ejpam-1598	263	6	∈	∈	PROPN
ejpam-1598	263	7	s.	s.	PROPN
ejpam-1598	263	8	suppose	suppose	VERB
ejpam-1598	263	9	i	i	PRON
ejpam-1598	263	10	is	be	AUX
ejpam-1598	263	11	any	any	DET
ejpam-1598	263	12	nonzero	nonzero	ADJ
ejpam-1598	263	13	ideal	ideal	NOUN
ejpam-1598	263	14	of	of	ADP
ejpam-1598	263	15	s.	s.	PROPN
ejpam-1598	263	16	let	let	VERB
ejpam-1598	263	17	a	a	DET
ejpam-1598	263	18	(	(	PUNCT
ejpam-1598	263	19	6=	6=	NOUN
ejpam-1598	263	20	0	0	NUM
ejpam-1598	263	21	)	)	PUNCT
ejpam-1598	263	22	∈	∈	PROPN
ejpam-1598	264	1	i	i	PRON
ejpam-1598	264	2	.	.	PUNCT
ejpam-1598	265	1	since	since	SCONJ
ejpam-1598	265	2	s	s	PROPN
ejpam-1598	265	3	is	be	AUX
ejpam-1598	265	4	a	a	DET
ejpam-1598	265	5	right	right	ADV
ejpam-1598	265	6	strongly	strongly	ADV
ejpam-1598	265	7	prime	prime	ADJ
ejpam-1598	265	8	ternary	ternary	ADJ
ejpam-1598	265	9	semiring	semiring	NOUN
ejpam-1598	265	10	,	,	PUNCT
ejpam-1598	265	11	there	there	PRON
ejpam-1598	265	12	exist	exist	VERB
ejpam-1598	265	13	finite	finite	ADJ
ejpam-1598	265	14	subsets	subset	NOUN
ejpam-1598	265	15	f1	f1	NOUN
ejpam-1598	265	16	,	,	PUNCT
ejpam-1598	265	17	f2	f2	PROPN
ejpam-1598	265	18	,	,	PUNCT
ejpam-1598	265	19	f3	f3	PROPN
ejpam-1598	265	20	of	of	ADP
ejpam-1598	265	21	s	s	PRON
ejpam-1598	265	22	such	such	ADJ
ejpam-1598	265	23	that	that	SCONJ
ejpam-1598	265	24	af1f2f3	af1f2f3	ADP
ejpam-1598	265	25	y	y	PROPN
ejpam-1598	265	26	=	=	SYM
ejpam-1598	265	27	{	{	PUNCT
ejpam-1598	265	28	0	0	NUM
ejpam-1598	265	29	}	}	PUNCT
ejpam-1598	265	30	⇒	⇒	NOUN
ejpam-1598	265	31	y	y	PROPN
ejpam-1598	265	32	=	=	PUNCT
ejpam-1598	265	33	0	0	NUM
ejpam-1598	265	34	for	for	ADP
ejpam-1598	265	35	all	all	DET
ejpam-1598	265	36	y	y	PROPN
ejpam-1598	265	37	∈	∈	PROPN
ejpam-1598	265	38	s.	s.	PROPN
ejpam-1598	265	39	now	now	ADV
ejpam-1598	265	40	,	,	PUNCT
ejpam-1598	265	41	set	set	VERB
ejpam-1598	265	42	e	e	NOUN
ejpam-1598	265	43	=	=	PUNCT
ejpam-1598	265	44	{	{	PUNCT
ejpam-1598	265	45	ei	ei	X
ejpam-1598	265	46	:	:	PUNCT
ejpam-1598	265	47	i	i	NOUN
ejpam-1598	265	48	=	=	SYM
ejpam-1598	265	49	1,2	1,2	NUM
ejpam-1598	265	50	,	,	PUNCT
ejpam-1598	265	51	.	.	PUNCT
ejpam-1598	265	52	.	.	PUNCT
ejpam-1598	266	1	.	.	PUNCT
ejpam-1598	267	1	,	,	PUNCT
ejpam-1598	267	2	n	n	CCONJ
ejpam-1598	267	3	}	}	PUNCT
ejpam-1598	268	1	and	and	CCONJ
ejpam-1598	268	2	f	f	X
ejpam-1598	268	3	=	=	PRON
ejpam-1598	268	4	{	{	PUNCT
ejpam-1598	268	5	fi	fi	NOUN
ejpam-1598	268	6	:	:	PUNCT
ejpam-1598	268	7	i	i	NOUN
ejpam-1598	268	8	=	=	SYM
ejpam-1598	268	9	1,2	1,2	NUM
ejpam-1598	268	10	,	,	PUNCT
ejpam-1598	268	11	.	.	PUNCT
ejpam-1598	268	12	.	.	PUNCT
ejpam-1598	269	1	.	.	PUNCT
ejpam-1598	269	2	,	,	PUNCT
ejpam-1598	269	3	n	n	CCONJ
ejpam-1598	269	4	}	}	PUNCT
ejpam-1598	269	5	.	.	PUNCT
ejpam-1598	270	1	let	let	VERB
ejpam-1598	270	2	g	g	PROPN
ejpam-1598	270	3	=	=	PROPN
ejpam-1598	270	4	af1f2f3e	af1f2f3e	PROPN
ejpam-1598	270	5	.	.	PUNCT
ejpam-1598	271	1	then	then	ADV
ejpam-1598	271	2	g	g	PROPN
ejpam-1598	271	3	is	be	AUX
ejpam-1598	271	4	a	a	DET
ejpam-1598	271	5	finite	finite	NOUN
ejpam-1598	271	6	subset	subset	NOUN
ejpam-1598	271	7	of	of	ADP
ejpam-1598	271	8	i	i	PRON
ejpam-1598	271	9	.	.	PUNCT
ejpam-1598	272	1	it	it	PRON
ejpam-1598	272	2	is	be	AUX
ejpam-1598	272	3	clear	clear	ADJ
ejpam-1598	272	4	that	that	SCONJ
ejpam-1598	272	5	y	y	PROPN
ejpam-1598	272	6	∈	∈	PROPN
ejpam-1598	272	7	rs(g)⇒	rs(g)⇒	PROPN
ejpam-1598	272	8	af1f2f3e	af1f2f3e	PROPN
ejpam-1598	272	9	ys	ys	NOUN
ejpam-1598	272	10	=	=	NOUN
ejpam-1598	272	11	0	0	PROPN
ejpam-1598	272	12	for	for	ADP
ejpam-1598	272	13	all	all	DET
ejpam-1598	272	14	s	s	PART
ejpam-1598	272	15	∈	∈	NOUN
ejpam-1598	272	16	s	s	PART
ejpam-1598	272	17	⇒	⇒	NOUN
ejpam-1598	272	18	af1f2f3e	af1f2f3e	PROPN
ejpam-1598	272	19	yf	yf	NOUN
ejpam-1598	273	1	=	=	SYM
ejpam-1598	273	2	0	0	NUM
ejpam-1598	273	3	⇒	⇒	NOUN
ejpam-1598	273	4	af1f2f3	af1f2f3	ADP
ejpam-1598	273	5	∑n	∑n	PROPN
ejpam-1598	273	6	i=1	i=1	X
ejpam-1598	273	7	ei	ei	X
ejpam-1598	273	8	y	y	PROPN
ejpam-1598	273	9	fi	fi	NOUN
ejpam-1598	273	10	=	=	SYM
ejpam-1598	273	11	0	0	NUM
ejpam-1598	273	12	⇒	⇒	NOUN
ejpam-1598	273	13	af1f2f3	af1f2f3	INTJ
ejpam-1598	273	14	y	y	PROPN
ejpam-1598	273	15	=	=	SYM
ejpam-1598	273	16	0	0	PROPN
ejpam-1598	273	17	⇒	⇒	NOUN
ejpam-1598	273	18	y	y	PROPN
ejpam-1598	273	19	=	=	SYM
ejpam-1598	273	20	0	0	PROPN
ejpam-1598	273	21	.	.	PUNCT
ejpam-1598	274	1	thus	thus	ADV
ejpam-1598	274	2	,	,	PUNCT
ejpam-1598	274	3	rs(g	rs(g	X
ejpam-1598	274	4	)	)	PUNCT
ejpam-1598	274	5	=	=	SYM
ejpam-1598	274	6	0	0	X
ejpam-1598	274	7	.	.	PUNCT
ejpam-1598	274	8	proposition	proposition	NOUN
ejpam-1598	274	9	10	10	NUM
ejpam-1598	274	10	.	.	PUNCT
ejpam-1598	275	1	every	every	DET
ejpam-1598	275	2	right	right	ADV
ejpam-1598	275	3	strongly	strongly	ADV
ejpam-1598	275	4	prime	prime	ADJ
ejpam-1598	275	5	ternary	ternary	ADJ
ejpam-1598	275	6	semiring	semiring	NOUN
ejpam-1598	275	7	s	s	NOUN
ejpam-1598	275	8	with	with	ADP
ejpam-1598	275	9	identity	identity	NOUN
ejpam-1598	275	10	is	be	AUX
ejpam-1598	275	11	non	non	ADJ
ejpam-1598	275	12	-	-	ADJ
ejpam-1598	275	13	singular	singular	ADJ
ejpam-1598	275	14	.	.	PUNCT
ejpam-1598	276	1	proof	proof	NOUN
ejpam-1598	276	2	.	.	PUNCT
ejpam-1598	277	1	if	if	SCONJ
ejpam-1598	277	2	possible	possible	ADJ
ejpam-1598	277	3	,	,	PUNCT
ejpam-1598	277	4	let	let	VERB
ejpam-1598	277	5	z(s	z(s	NOUN
ejpam-1598	277	6	)	)	PUNCT
ejpam-1598	277	7	6=	6=	ADP
ejpam-1598	277	8	0	0	X
ejpam-1598	277	9	.	.	PUNCT
ejpam-1598	278	1	then	then	ADV
ejpam-1598	278	2	z(s	z(s	PROPN
ejpam-1598	278	3	)	)	PUNCT
ejpam-1598	278	4	is	be	AUX
ejpam-1598	278	5	a	a	DET
ejpam-1598	278	6	nonzero	nonzero	NOUN
ejpam-1598	278	7	ideal	ideal	NOUN
ejpam-1598	278	8	of	of	ADP
ejpam-1598	278	9	s.	s.	PROPN
ejpam-1598	278	10	since	since	SCONJ
ejpam-1598	278	11	s	s	PROPN
ejpam-1598	278	12	is	be	AUX
ejpam-1598	278	13	a	a	DET
ejpam-1598	278	14	right	right	ADV
ejpam-1598	278	15	strongly	strongly	ADV
ejpam-1598	278	16	prime	prime	ADJ
ejpam-1598	278	17	ternary	ternary	ADJ
ejpam-1598	278	18	semiring	semiring	NOUN
ejpam-1598	278	19	,	,	PUNCT
ejpam-1598	278	20	by	by	ADP
ejpam-1598	278	21	lemma	lemma	PROPN
ejpam-1598	278	22	1	1	NUM
ejpam-1598	278	23	,	,	PUNCT
ejpam-1598	278	24	there	there	PRON
ejpam-1598	278	25	exists	exist	VERB
ejpam-1598	278	26	a	a	DET
ejpam-1598	278	27	finite	finite	NOUN
ejpam-1598	278	28	subset	subset	NOUN
ejpam-1598	278	29	f	f	PROPN
ejpam-1598	278	30	=	=	PRON
ejpam-1598	278	31	{	{	PUNCT
ejpam-1598	278	32	t1	t1	NOUN
ejpam-1598	278	33	,	,	PUNCT
ejpam-1598	278	34	t2	t2	NOUN
ejpam-1598	278	35	,	,	PUNCT
ejpam-1598	278	36	.	.	PUNCT
ejpam-1598	278	37	.	.	PUNCT
ejpam-1598	279	1	.	.	PUNCT
ejpam-1598	280	1	,	,	PUNCT
ejpam-1598	280	2	tk	tk	PROPN
ejpam-1598	280	3	}	}	PUNCT
ejpam-1598	280	4	of	of	ADP
ejpam-1598	280	5	z(s	z(s	PROPN
ejpam-1598	280	6	)	)	PUNCT
ejpam-1598	281	1	such	such	ADJ
ejpam-1598	281	2	that	that	PRON
ejpam-1598	281	3	rs(f	rs(f	PUNCT
ejpam-1598	281	4	)	)	PUNCT
ejpam-1598	281	5	=	=	SYM
ejpam-1598	281	6	0	0	X
ejpam-1598	281	7	i.e.	i.e.	X
ejpam-1598	281	8	rs(t1)∩	rs(t1)∩	ADJ
ejpam-1598	281	9	rs(t2)∩	rs(t2)∩	NOUN
ejpam-1598	281	10	.	.	PUNCT
ejpam-1598	281	11	.	.	PUNCT
ejpam-1598	282	1	.	.	PUNCT
ejpam-1598	283	1	,	,	PUNCT
ejpam-1598	283	2	∩rs(tk	∩rs(tk	NOUN
ejpam-1598	283	3	)	)	PUNCT
ejpam-1598	284	1	=	=	SYM
ejpam-1598	284	2	0	0	X
ejpam-1598	284	3	.	.	PUNCT
ejpam-1598	285	1	now	now	ADV
ejpam-1598	285	2	t	t	VERB
ejpam-1598	285	3	i	i	PRON
ejpam-1598	285	4	∈	∈	PROPN
ejpam-1598	285	5	z(s	z(s	PROPN
ejpam-1598	285	6	)	)	PUNCT
ejpam-1598	285	7	implies	imply	VERB
ejpam-1598	285	8	that	that	SCONJ
ejpam-1598	285	9	rs(t	rs(t	PUNCT
ejpam-1598	285	10	i	i	PRON
ejpam-1598	285	11	)	)	PUNCT
ejpam-1598	285	12	is	be	AUX
ejpam-1598	285	13	an	an	DET
ejpam-1598	285	14	essential	essential	ADJ
ejpam-1598	285	15	right	right	ADJ
ejpam-1598	285	16	ideal	ideal	NOUN
ejpam-1598	285	17	of	of	ADP
ejpam-1598	285	18	s	s	PRON
ejpam-1598	285	19	for	for	ADP
ejpam-1598	285	20	i	i	PROPN
ejpam-1598	285	21	=	=	SYM
ejpam-1598	285	22	1,2	1,2	NUM
ejpam-1598	285	23	,	,	PUNCT
ejpam-1598	285	24	.	.	PUNCT
ejpam-1598	285	25	.	.	PUNCT
ejpam-1598	286	1	.	.	PUNCT
ejpam-1598	287	1	,	,	PUNCT
ejpam-1598	287	2	k.	k.	PROPN
ejpam-1598	287	3	therefore	therefore	ADV
ejpam-1598	287	4	rs(f	rs(f	NUM
ejpam-1598	287	5	)	)	PUNCT
ejpam-1598	287	6	is	be	AUX
ejpam-1598	287	7	an	an	DET
ejpam-1598	287	8	essential	essential	ADJ
ejpam-1598	287	9	right	right	ADJ
ejpam-1598	287	10	ideal	ideal	NOUN
ejpam-1598	287	11	of	of	ADP
ejpam-1598	287	12	s.	s.	PROPN
ejpam-1598	287	13	consequently	consequently	ADV
ejpam-1598	287	14	,	,	PUNCT
ejpam-1598	287	15	rs(f	rs(f	NUM
ejpam-1598	287	16	)	)	PUNCT
ejpam-1598	287	17	6=	6=	ADP
ejpam-1598	287	18	0	0	NUM
ejpam-1598	287	19	,	,	PUNCT
ejpam-1598	287	20	a	a	DET
ejpam-1598	287	21	contradiction	contradiction	NOUN
ejpam-1598	287	22	.	.	PUNCT
ejpam-1598	288	1	the	the	DET
ejpam-1598	288	2	proposition	proposition	NOUN
ejpam-1598	288	3	is	be	AUX
ejpam-1598	288	4	hence	hence	ADV
ejpam-1598	288	5	proved	prove	VERB
ejpam-1598	288	6	.	.	PUNCT
ejpam-1598	289	1	we	we	PRON
ejpam-1598	289	2	now	now	ADV
ejpam-1598	289	3	give	give	VERB
ejpam-1598	289	4	the	the	DET
ejpam-1598	289	5	definition	definition	NOUN
ejpam-1598	289	6	of	of	ADP
ejpam-1598	289	7	a	a	DET
ejpam-1598	289	8	strongly	strongly	ADV
ejpam-1598	289	9	nilpotent	nilpotent	ADJ
ejpam-1598	289	10	element	element	NOUN
ejpam-1598	289	11	in	in	ADP
ejpam-1598	289	12	a	a	DET
ejpam-1598	289	13	ternary	ternary	ADJ
ejpam-1598	289	14	semiring	semiring	NOUN
ejpam-1598	289	15	.	.	PUNCT
ejpam-1598	290	1	definition	definition	NOUN
ejpam-1598	290	2	24	24	NUM
ejpam-1598	290	3	.	.	PUNCT
ejpam-1598	291	1	an	an	DET
ejpam-1598	291	2	element	element	NOUN
ejpam-1598	291	3	a	a	PRON
ejpam-1598	291	4	in	in	ADP
ejpam-1598	291	5	a	a	DET
ejpam-1598	291	6	ternary	ternary	ADJ
ejpam-1598	291	7	semiring	semiring	NOUN
ejpam-1598	291	8	s	s	NOUN
ejpam-1598	291	9	is	be	AUX
ejpam-1598	291	10	said	say	VERB
ejpam-1598	291	11	to	to	PART
ejpam-1598	291	12	be	be	AUX
ejpam-1598	291	13	strongly	strongly	ADV
ejpam-1598	291	14	nilpotent	nilpotent	ADJ
ejpam-1598	291	15	if	if	SCONJ
ejpam-1598	291	16	there	there	PRON
ejpam-1598	291	17	exists	exist	VERB
ejpam-1598	291	18	a	a	DET
ejpam-1598	291	19	positive	positive	ADJ
ejpam-1598	291	20	integer	integer	NOUN
ejpam-1598	291	21	n	n	CCONJ
ejpam-1598	291	22	such	such	ADJ
ejpam-1598	291	23	that	that	DET
ejpam-1598	291	24	a2n+1	a2n+1	PROPN
ejpam-1598	292	1	=	=	SYM
ejpam-1598	292	2	0	0	X
ejpam-1598	292	3	.	.	PUNCT
ejpam-1598	293	1	definition	definition	NOUN
ejpam-1598	293	2	25	25	NUM
ejpam-1598	293	3	.	.	PUNCT
ejpam-1598	294	1	a	a	DET
ejpam-1598	294	2	ternary	ternary	ADJ
ejpam-1598	294	3	semiring	semiring	NOUN
ejpam-1598	294	4	s	s	NOUN
ejpam-1598	294	5	is	be	AUX
ejpam-1598	294	6	said	say	VERB
ejpam-1598	294	7	to	to	PART
ejpam-1598	294	8	be	be	AUX
ejpam-1598	294	9	a	a	DET
ejpam-1598	294	10	reduced	reduced	ADJ
ejpam-1598	294	11	ternary	ternary	ADJ
ejpam-1598	294	12	semiring	semiring	NOUN
ejpam-1598	294	13	if	if	SCONJ
ejpam-1598	294	14	it	it	PRON
ejpam-1598	294	15	does	do	AUX
ejpam-1598	294	16	not	not	PART
ejpam-1598	294	17	contain	contain	VERB
ejpam-1598	294	18	any	any	DET
ejpam-1598	294	19	nonzero	nonzero	NOUN
ejpam-1598	294	20	strongly	strongly	ADV
ejpam-1598	294	21	nilpotent	nilpotent	ADJ
ejpam-1598	294	22	elements	element	NOUN
ejpam-1598	294	23	.	.	PUNCT
ejpam-1598	295	1	in	in	ADP
ejpam-1598	295	2	the	the	DET
ejpam-1598	295	3	following	follow	VERB
ejpam-1598	295	4	proposition	proposition	NOUN
ejpam-1598	295	5	,	,	PUNCT
ejpam-1598	295	6	we	we	PRON
ejpam-1598	295	7	describe	describe	VERB
ejpam-1598	295	8	the	the	DET
ejpam-1598	295	9	reduced	reduce	VERB
ejpam-1598	295	10	ternary	ternary	ADJ
ejpam-1598	295	11	semirings	semiring	NOUN
ejpam-1598	295	12	.	.	PUNCT
ejpam-1598	296	1	proposition	proposition	NOUN
ejpam-1598	296	2	11	11	NUM
ejpam-1598	296	3	.	.	PUNCT
ejpam-1598	297	1	every	every	DET
ejpam-1598	297	2	reduced	reduce	VERB
ejpam-1598	297	3	ternary	ternary	ADJ
ejpam-1598	297	4	semiring	semiring	NOUN
ejpam-1598	297	5	s	s	NOUN
ejpam-1598	297	6	with	with	ADP
ejpam-1598	297	7	identity	identity	NOUN
ejpam-1598	297	8	is	be	AUX
ejpam-1598	297	9	non	non	ADJ
ejpam-1598	297	10	-	-	ADJ
ejpam-1598	297	11	singular	singular	ADJ
ejpam-1598	297	12	.	.	PUNCT
ejpam-1598	298	1	proof	proof	NOUN
ejpam-1598	298	2	.	.	PUNCT
ejpam-1598	299	1	take	take	VERB
ejpam-1598	299	2	any	any	DET
ejpam-1598	299	3	a	a	DET
ejpam-1598	299	4	∈	∈	NOUN
ejpam-1598	299	5	s∗.	s∗.	ADJ
ejpam-1598	299	6	if	if	SCONJ
ejpam-1598	299	7	x	x	SYM
ejpam-1598	299	8	∈	∈	PROPN
ejpam-1598	299	9	rs(a	rs(a	NOUN
ejpam-1598	299	10	)	)	PUNCT
ejpam-1598	299	11	∩	∩	ADJ
ejpam-1598	299	12	ass	ass	NOUN
ejpam-1598	299	13	,	,	PUNCT
ejpam-1598	299	14	then	then	ADV
ejpam-1598	299	15	for	for	ADP
ejpam-1598	299	16	some	some	DET
ejpam-1598	299	17	y1	y1	NOUN
ejpam-1598	299	18	,	,	PUNCT
ejpam-1598	299	19	y2	y2	PROPN
ejpam-1598	299	20	∈	∈	PROPN
ejpam-1598	299	21	s	s	NOUN
ejpam-1598	299	22	,	,	PUNCT
ejpam-1598	299	23	x	x	PUNCT
ejpam-1598	299	24	=	=	PUNCT
ejpam-1598	299	25	a	a	DET
ejpam-1598	299	26	y1	y1	ADJ
ejpam-1598	299	27	y2	y2	NOUN
ejpam-1598	299	28	and	and	CCONJ
ejpam-1598	299	29	a2	a2	PROPN
ejpam-1598	299	30	y1	y1	PROPN
ejpam-1598	299	31	y2s	y2s	NOUN
ejpam-1598	299	32	=	=	PUNCT
ejpam-1598	300	1	axs	axs	PROPN
ejpam-1598	300	2	=	=	PUNCT
ejpam-1598	300	3	0	0	NUM
ejpam-1598	300	4	for	for	ADP
ejpam-1598	300	5	all	all	DET
ejpam-1598	300	6	s	s	PROPN
ejpam-1598	300	7	∈	∈	PROPN
ejpam-1598	300	8	s.	s.	PROPN
ejpam-1598	300	9	this	this	PRON
ejpam-1598	300	10	implies	imply	VERB
ejpam-1598	300	11	that	that	SCONJ
ejpam-1598	300	12	(	(	PUNCT
ejpam-1598	300	13	a	a	DET
ejpam-1598	300	14	y1	y1	NOUN
ejpam-1598	300	15	y2sa)3	y2sa)3	NOUN
ejpam-1598	300	16	=	=	SYM
ejpam-1598	300	17	0	0	X
ejpam-1598	300	18	.	.	PUNCT
ejpam-1598	301	1	hence	hence	ADV
ejpam-1598	301	2	,	,	PUNCT
ejpam-1598	301	3	since	since	SCONJ
ejpam-1598	301	4	s	s	PRON
ejpam-1598	301	5	is	be	AUX
ejpam-1598	301	6	reduced	reduce	VERB
ejpam-1598	301	7	,	,	PUNCT
ejpam-1598	301	8	a	a	DET
ejpam-1598	301	9	y1	y1	NOUN
ejpam-1598	301	10	y2sa	y2sa	PUNCT
ejpam-1598	302	1	=	=	SYM
ejpam-1598	302	2	0	0	X
ejpam-1598	302	3	.	.	PUNCT
ejpam-1598	303	1	consequently	consequently	ADV
ejpam-1598	303	2	,	,	PUNCT
ejpam-1598	303	3	xsx	xsx	PROPN
ejpam-1598	303	4	=	=	SYM
ejpam-1598	303	5	a	a	DET
ejpam-1598	303	6	y1	y1	NOUN
ejpam-1598	303	7	y2sa	y2sa	NUM
ejpam-1598	303	8	y1	y1	NOUN
ejpam-1598	303	9	y2	y2	NOUN
ejpam-1598	303	10	=	=	NOUN
ejpam-1598	303	11	0	0	NUM
ejpam-1598	303	12	.	.	PUNCT
ejpam-1598	304	1	thus	thus	ADV
ejpam-1598	304	2	,	,	PUNCT
ejpam-1598	304	3	xsxsx	xsxsx	PROPN
ejpam-1598	304	4	=	=	SYM
ejpam-1598	304	5	0	0	PROPN
ejpam-1598	304	6	,	,	PUNCT
ejpam-1598	304	7	xssxsxs	xssxsxs	PROPN
ejpam-1598	304	8	=	=	SYM
ejpam-1598	304	9	0	0	NUM
ejpam-1598	304	10	and	and	CCONJ
ejpam-1598	304	11	sxsxssx	sxsxssx	NOUN
ejpam-1598	304	12	=	=	NOUN
ejpam-1598	304	13	0	0	X
ejpam-1598	304	14	.	.	PUNCT
ejpam-1598	305	1	also	also	ADV
ejpam-1598	305	2	,	,	PUNCT
ejpam-1598	305	3	(	(	PUNCT
ejpam-1598	305	4	xssxssa)3	xssxssa)3	X
ejpam-1598	305	5	=	=	NOUN
ejpam-1598	305	6	0	0	X
ejpam-1598	305	7	.	.	PUNCT
ejpam-1598	306	1	since	since	SCONJ
ejpam-1598	306	2	s	s	NOUN
ejpam-1598	306	3	is	be	AUX
ejpam-1598	306	4	reduced	reduce	VERB
ejpam-1598	306	5	,	,	PUNCT
ejpam-1598	306	6	xssxssa	xssxssa	PROPN
ejpam-1598	306	7	=	=	SYM
ejpam-1598	307	1	0	0	NUM
ejpam-1598	307	2	.	.	PUNCT
ejpam-1598	308	1	thus	thus	ADV
ejpam-1598	308	2	,	,	PUNCT
ejpam-1598	308	3	xssxssx	xssxssx	PROPN
ejpam-1598	308	4	=	=	SYM
ejpam-1598	308	5	xssxssa	xssxssa	PROPN
ejpam-1598	308	6	y1	y1	PROPN
ejpam-1598	309	1	y2	y2	NOUN
ejpam-1598	310	1	=	=	NOUN
ejpam-1598	311	1	0	0	X
ejpam-1598	311	2	.	.	PUNCT
ejpam-1598	312	1	however	however	ADV
ejpam-1598	312	2	,	,	PUNCT
ejpam-1598	312	3	the	the	DET
ejpam-1598	312	4	ternary	ternary	ADJ
ejpam-1598	312	5	semiring	semiring	NOUN
ejpam-1598	312	6	s	s	PRON
ejpam-1598	312	7	,	,	PUNCT
ejpam-1598	312	8	being	be	AUX
ejpam-1598	312	9	reduced	reduce	VERB
ejpam-1598	312	10	,	,	PUNCT
ejpam-1598	312	11	is	be	AUX
ejpam-1598	312	12	semiprime	semiprime	NOUN
ejpam-1598	312	13	,	,	PUNCT
ejpam-1598	312	14	and	and	CCONJ
ejpam-1598	312	15	so	so	ADV
ejpam-1598	312	16	x	x	X
ejpam-1598	313	1	=	=	SYM
ejpam-1598	313	2	0	0	X
ejpam-1598	313	3	.	.	PUNCT
ejpam-1598	314	1	consequently	consequently	ADV
ejpam-1598	314	2	,	,	PUNCT
ejpam-1598	314	3	rs(a)∩	rs(a)∩	PROPN
ejpam-1598	314	4	ass	ass	NOUN
ejpam-1598	314	5	=	=	SYM
ejpam-1598	314	6	0	0	X
ejpam-1598	314	7	.	.	PUNCT
ejpam-1598	315	1	now	now	ADV
ejpam-1598	315	2	,	,	PUNCT
ejpam-1598	315	3	because	because	SCONJ
ejpam-1598	315	4	s	s	PRON
ejpam-1598	315	5	admits	admit	VERB
ejpam-1598	315	6	an	an	DET
ejpam-1598	315	7	identity	identity	NOUN
ejpam-1598	315	8	,	,	PUNCT
ejpam-1598	315	9	ass	ass	NOUN
ejpam-1598	315	10	is	be	AUX
ejpam-1598	315	11	a	a	DET
ejpam-1598	315	12	nonzero	nonzero	ADJ
ejpam-1598	315	13	right	right	ADJ
ejpam-1598	315	14	ideal	ideal	NOUN
ejpam-1598	315	15	of	of	ADP
ejpam-1598	315	16	s.	s.	PROPN
ejpam-1598	315	17	this	this	PRON
ejpam-1598	315	18	means	mean	VERB
ejpam-1598	315	19	that	that	SCONJ
ejpam-1598	315	20	rs(a	rs(a	NOUN
ejpam-1598	315	21	)	)	PUNCT
ejpam-1598	315	22	is	be	AUX
ejpam-1598	315	23	not	not	PART
ejpam-1598	315	24	an	an	DET
ejpam-1598	315	25	essential	essential	ADJ
ejpam-1598	315	26	right	right	ADJ
ejpam-1598	315	27	ideal	ideal	NOUN
ejpam-1598	315	28	of	of	ADP
ejpam-1598	315	29	s	s	PROPN
ejpam-1598	315	30	,	,	PUNCT
ejpam-1598	315	31	for	for	ADP
ejpam-1598	315	32	any	any	DET
ejpam-1598	315	33	a	a	DET
ejpam-1598	315	34	∈	∈	NOUN
ejpam-1598	315	35	s∗.	s∗.	ADJ
ejpam-1598	315	36	this	this	PRON
ejpam-1598	315	37	implies	imply	VERB
ejpam-1598	315	38	that	that	SCONJ
ejpam-1598	315	39	z(s	z(s	NOUN
ejpam-1598	315	40	)	)	PUNCT
ejpam-1598	316	1	=	=	SYM
ejpam-1598	316	2	0	0	X
ejpam-1598	316	3	.	.	PUNCT
ejpam-1598	317	1	lemma	lemma	PROPN
ejpam-1598	317	2	2	2	X
ejpam-1598	317	3	.	.	PUNCT
ejpam-1598	318	1	let	let	VERB
ejpam-1598	318	2	{	{	PUNCT
ejpam-1598	318	3	sα	sα	ADV
ejpam-1598	318	4	:	:	PUNCT
ejpam-1598	318	5	α	α	PROPN
ejpam-1598	318	6	∈	∈	PROPN
ejpam-1598	318	7	λ	λ	PROPN
ejpam-1598	318	8	,	,	PUNCT
ejpam-1598	318	9	where	where	SCONJ
ejpam-1598	318	10	λ	λ	PROPN
ejpam-1598	318	11	is	be	AUX
ejpam-1598	318	12	a	a	DET
ejpam-1598	318	13	index	index	NOUN
ejpam-1598	318	14	set	set	NOUN
ejpam-1598	318	15	}	}	PUNCT
ejpam-1598	318	16	be	be	AUX
ejpam-1598	318	17	a	a	DET
ejpam-1598	318	18	family	family	NOUN
ejpam-1598	318	19	of	of	ADP
ejpam-1598	318	20	ternary	ternary	ADJ
ejpam-1598	318	21	semirings	semiring	NOUN
ejpam-1598	318	22	.	.	PUNCT
ejpam-1598	319	1	then	then	ADV
ejpam-1598	319	2	,	,	PUNCT
ejpam-1598	319	3	r∏	r∏	PROPN
ejpam-1598	319	4	α∈λ	α∈λ	NOUN
ejpam-1598	319	5	sα	sα	ADV
ejpam-1598	319	6	(	(	PUNCT
ejpam-1598	319	7	(	(	PUNCT
ejpam-1598	319	8	aα)α∈λ	aα)α∈λ	NOUN
ejpam-1598	319	9	)	)	PUNCT
ejpam-1598	319	10	=	=	SYM
ejpam-1598	319	11	∏	∏	PROPN
ejpam-1598	319	12	α∈λ	α∈λ	NOUN
ejpam-1598	319	13	rsα	rsα	NOUN
ejpam-1598	319	14	(	(	PUNCT
ejpam-1598	319	15	aα	aα	NOUN
ejpam-1598	319	16	)	)	PUNCT
ejpam-1598	319	17	.	.	PUNCT
ejpam-1598	320	1	proof	proof	NOUN
ejpam-1598	320	2	.	.	PUNCT
ejpam-1598	321	1	let	let	VERB
ejpam-1598	321	2	(	(	PUNCT
ejpam-1598	321	3	xα)α∈λ	xα)α∈λ	PROPN
ejpam-1598	321	4	∈	∈	PROPN
ejpam-1598	321	5	r∏	r∏	PROPN
ejpam-1598	321	6	α∈λ	α∈λ	NOUN
ejpam-1598	321	7	sα	sα	ADV
ejpam-1598	321	8	(	(	PUNCT
ejpam-1598	321	9	(	(	PUNCT
ejpam-1598	321	10	aα)α∈λ	aα)α∈λ	PROPN
ejpam-1598	321	11	)	)	PUNCT
ejpam-1598	321	12	⇔	⇔	X
ejpam-1598	321	13	(	(	PUNCT
ejpam-1598	321	14	aα)α∈λ(xα)α∈λ(sα)α∈λ	aα)α∈λ(xα)α∈λ(sα)α∈λ	PROPN
ejpam-1598	321	15	=	=	SYM
ejpam-1598	321	16	(	(	PUNCT
ejpam-1598	321	17	0α)α∈λ	0α)α∈λ	NUM
ejpam-1598	321	18	for	for	ADP
ejpam-1598	321	19	all	all	DET
ejpam-1598	321	20	(	(	PUNCT
ejpam-1598	321	21	sα)α∈λ	sα)α∈λ	NUM
ejpam-1598	321	22	∈	∈	PROPN
ejpam-1598	321	23	∏	∏	PROPN
ejpam-1598	321	24	α∈λ	α∈λ	NOUN
ejpam-1598	321	25	sα	sα	PROPN
ejpam-1598	321	26	⇔	⇔	PROPN
ejpam-1598	321	27	(	(	PUNCT
ejpam-1598	321	28	aαxαsα)α∈λ	aαxαsα)α∈λ	NOUN
ejpam-1598	321	29	=	=	SYM
ejpam-1598	321	30	(	(	PUNCT
ejpam-1598	321	31	0α)α∈λ	0α)α∈λ	NUM
ejpam-1598	321	32	for	for	ADP
ejpam-1598	321	33	all	all	DET
ejpam-1598	321	34	sα	sα	PROPN
ejpam-1598	321	35	∈	∈	PROPN
ejpam-1598	321	36	sα	sα	PROPN
ejpam-1598	321	37	,	,	PUNCT
ejpam-1598	321	38	α	α	PROPN
ejpam-1598	321	39	∈	∈	PROPN
ejpam-1598	321	40	λ⇔	λ⇔	PROPN
ejpam-1598	321	41	aαxαsα	aαxαsα	NOUN
ejpam-1598	321	42	=	=	PROPN
ejpam-1598	321	43	0α	0α	PROPN
ejpam-1598	321	44	for	for	ADP
ejpam-1598	321	45	all	all	DET
ejpam-1598	321	46	sα	sα	PROPN
ejpam-1598	321	47	∈	∈	PROPN
ejpam-1598	321	48	sα	sα	PROPN
ejpam-1598	321	49	,	,	PUNCT
ejpam-1598	321	50	α	α	PROPN
ejpam-1598	321	51	∈	∈	PROPN
ejpam-1598	322	1	λ⇔	λ⇔	PROPN
ejpam-1598	322	2	xα	xα	PUNCT
ejpam-1598	323	1	∈	∈	PROPN
ejpam-1598	323	2	rsα	rsα	NOUN
ejpam-1598	323	3	(	(	PUNCT
ejpam-1598	323	4	aα	aα	NOUN
ejpam-1598	323	5	)	)	PUNCT
ejpam-1598	323	6	for	for	ADP
ejpam-1598	323	7	each	each	DET
ejpam-1598	323	8	α	α	PRON
ejpam-1598	323	9	∈	∈	PROPN
ejpam-1598	323	10	λ	λ	PROPN
ejpam-1598	323	11	.	.	PUNCT
ejpam-1598	324	1	hence	hence	ADV
ejpam-1598	324	2	,	,	PUNCT
ejpam-1598	324	3	the	the	DET
ejpam-1598	324	4	result	result	NOUN
ejpam-1598	324	5	.	.	PUNCT
ejpam-1598	325	1	t.	t.	PROPN
ejpam-1598	325	2	dutta	dutta	PROPN
ejpam-1598	325	3	,	,	PUNCT
ejpam-1598	325	4	k.	k.	PROPN
ejpam-1598	325	5	shum	shum	PROPN
ejpam-1598	325	6	,	,	PUNCT
ejpam-1598	325	7	s.	s.	PROPN
ejpam-1598	325	8	mandal	mandal	PROPN
ejpam-1598	325	9	/	/	SYM
ejpam-1598	325	10	eur	eur	PROPN
ejpam-1598	325	11	.	.	PUNCT
ejpam-1598	326	1	j.	j.	PROPN
ejpam-1598	326	2	pure	pure	PROPN
ejpam-1598	326	3	appl	appl	PROPN
ejpam-1598	326	4	.	.	PROPN
ejpam-1598	326	5	math	math	PROPN
ejpam-1598	326	6	,	,	PUNCT
ejpam-1598	326	7	5	5	NUM
ejpam-1598	326	8	(	(	PUNCT
ejpam-1598	326	9	2012	2012	NUM
ejpam-1598	326	10	)	)	PUNCT
ejpam-1598	326	11	,	,	PUNCT
ejpam-1598	326	12	116	116	NUM
ejpam-1598	326	13	-	-	SYM
ejpam-1598	326	14	128	128	NUM
ejpam-1598	326	15	124	124	NUM
ejpam-1598	326	16	lemma	lemma	PROPN
ejpam-1598	326	17	3	3	X
ejpam-1598	326	18	.	.	PUNCT
ejpam-1598	327	1	let	let	VERB
ejpam-1598	327	2	{	{	PUNCT
ejpam-1598	327	3	sα	sα	ADV
ejpam-1598	327	4	:	:	PUNCT
ejpam-1598	327	5	α	α	PROPN
ejpam-1598	327	6	∈	∈	PROPN
ejpam-1598	327	7	λ	λ	X
ejpam-1598	327	8	where	where	SCONJ
ejpam-1598	327	9	λ	λ	PROPN
ejpam-1598	327	10	is	be	AUX
ejpam-1598	327	11	an	an	DET
ejpam-1598	327	12	index	index	NOUN
ejpam-1598	327	13	set	set	NOUN
ejpam-1598	327	14	}	}	PUNCT
ejpam-1598	327	15	be	be	AUX
ejpam-1598	327	16	a	a	DET
ejpam-1598	327	17	family	family	NOUN
ejpam-1598	327	18	of	of	ADP
ejpam-1598	327	19	ternary	ternary	ADJ
ejpam-1598	327	20	semirings	semiring	NOUN
ejpam-1598	327	21	with	with	ADP
ejpam-1598	327	22	identity	identity	NOUN
ejpam-1598	327	23	.	.	PUNCT
ejpam-1598	328	1	let	let	VERB
ejpam-1598	328	2	p	p	PRON
ejpam-1598	328	3	be	be	AUX
ejpam-1598	328	4	a	a	DET
ejpam-1598	328	5	right	right	ADJ
ejpam-1598	328	6	ideal	ideal	NOUN
ejpam-1598	328	7	of	of	ADP
ejpam-1598	328	8	∏	∏	PROPN
ejpam-1598	328	9	α∈λ	α∈λ	NOUN
ejpam-1598	328	10	sα	sα	NOUN
ejpam-1598	328	11	.	.	PUNCT
ejpam-1598	329	1	let	let	VERB
ejpam-1598	329	2	πα	πα	VERB
ejpam-1598	329	3	:	:	PUNCT
ejpam-1598	329	4	∏	∏	NUM
ejpam-1598	329	5	α∈λ	α∈λ	NOUN
ejpam-1598	329	6	sα	sα	NOUN
ejpam-1598	329	7	→	→	X
ejpam-1598	329	8	sα	sα	ADV
ejpam-1598	329	9	be	be	AUX
ejpam-1598	329	10	the	the	DET
ejpam-1598	329	11	projection	projection	NOUN
ejpam-1598	329	12	map	map	NOUN
ejpam-1598	329	13	.	.	PUNCT
ejpam-1598	330	1	suppose	suppose	VERB
ejpam-1598	330	2	pα	pα	INTJ
ejpam-1598	330	3	=	=	PUNCT
ejpam-1598	330	4	πα(p	πα(p	NUM
ejpam-1598	330	5	)	)	PUNCT
ejpam-1598	330	6	for	for	ADP
ejpam-1598	330	7	each	each	DET
ejpam-1598	330	8	α	α	NOUN
ejpam-1598	330	9	∈	∈	PROPN
ejpam-1598	330	10	λ	λ	PROPN
ejpam-1598	330	11	.	.	PUNCT
ejpam-1598	331	1	then	then	ADV
ejpam-1598	331	2	p	p	PROPN
ejpam-1598	331	3	=	=	SYM
ejpam-1598	331	4	∏	∏	PROPN
ejpam-1598	331	5	α∈λ	α∈λ	NOUN
ejpam-1598	331	6	pα	pα	NOUN
ejpam-1598	331	7	.	.	PUNCT
ejpam-1598	332	1	proof	proof	NOUN
ejpam-1598	332	2	.	.	PUNCT
ejpam-1598	333	1	obviously	obviously	ADV
ejpam-1598	333	2	,	,	PUNCT
ejpam-1598	333	3	p	p	ADJ
ejpam-1598	333	4	⊆	⊆	NUM
ejpam-1598	333	5	∏	∏	NUM
ejpam-1598	333	6	α∈λ	α∈λ	NOUN
ejpam-1598	333	7	pα	pα	NOUN
ejpam-1598	333	8	.	.	PUNCT
ejpam-1598	334	1	now	now	ADV
ejpam-1598	334	2	let	let	VERB
ejpam-1598	334	3	(	(	PUNCT
ejpam-1598	334	4	aα)α∈λ	aα)α∈λ	PROPN
ejpam-1598	334	5	∈	∈	PROPN
ejpam-1598	334	6	∏	∏	PROPN
ejpam-1598	334	7	α∈λ	α∈λ	NOUN
ejpam-1598	334	8	pα	pα	NOUN
ejpam-1598	334	9	.	.	PUNCT
ejpam-1598	335	1	then	then	ADV
ejpam-1598	335	2	,	,	PUNCT
ejpam-1598	335	3	we	we	PRON
ejpam-1598	335	4	have	have	VERB
ejpam-1598	335	5	aα	aα	NOUN
ejpam-1598	335	6	∈	∈	NOUN
ejpam-1598	335	7	pα	pα	NOUN
ejpam-1598	335	8	.	.	PUNCT
ejpam-1598	336	1	as	as	SCONJ
ejpam-1598	336	2	pα	pα	NOUN
ejpam-1598	336	3	is	be	AUX
ejpam-1598	336	4	the	the	DET
ejpam-1598	336	5	αth	αth	PROPN
ejpam-1598	336	6	projection	projection	NOUN
ejpam-1598	336	7	map	map	NOUN
ejpam-1598	336	8	,	,	PUNCT
ejpam-1598	336	9	there	there	PRON
ejpam-1598	336	10	exists	exist	VERB
ejpam-1598	336	11	a	a	DET
ejpam-1598	336	12	(	(	PUNCT
ejpam-1598	336	13	bγ)γ∈λ−{α	bγ)γ∈λ−{α	NOUN
ejpam-1598	336	14	}	}	PUNCT
ejpam-1598	336	15	∈	∈	PROPN
ejpam-1598	336	16	∏	∏	PROPN
ejpam-1598	336	17	α∈λ−{α	α∈λ−{α	PROPN
ejpam-1598	336	18	}	}	PUNCT
ejpam-1598	336	19	sγ	sγ	NOUN
ejpam-1598	336	20	such	such	ADJ
ejpam-1598	336	21	that	that	PRON
ejpam-1598	336	22	(	(	PUNCT
ejpam-1598	336	23	aα	aα	NOUN
ejpam-1598	337	1	,	,	PUNCT
ejpam-1598	337	2	bγ)γ∈λ−{α	bγ)γ∈λ−{α	AUX
ejpam-1598	337	3	}	}	PUNCT
ejpam-1598	337	4	∈	∈	PROPN
ejpam-1598	337	5	p.	p.	NOUN
ejpam-1598	337	6	let	let	VERB
ejpam-1598	337	7	{	{	PUNCT
ejpam-1598	337	8	(	(	PUNCT
ejpam-1598	337	9	eiα	eiα	NOUN
ejpam-1598	337	10	,	,	PUNCT
ejpam-1598	337	11	fiα	fiα	PROPN
ejpam-1598	337	12	)	)	PUNCT
ejpam-1598	337	13	∈	∈	PROPN
ejpam-1598	337	14	sα×	sα×	AUX
ejpam-1598	337	15	sα	sα	VERB
ejpam-1598	337	16	(	(	PUNCT
ejpam-1598	337	17	iα	iα	INTJ
ejpam-1598	337	18	=	=	SYM
ejpam-1598	337	19	1,2	1,2	NUM
ejpam-1598	337	20	,	,	PUNCT
ejpam-1598	337	21	.	.	PUNCT
ejpam-1598	337	22	.	.	PUNCT
ejpam-1598	337	23	.	.	PUNCT
ejpam-1598	338	1	,	,	PUNCT
ejpam-1598	338	2	nα	nα	NOUN
ejpam-1598	338	3	)	)	PUNCT
ejpam-1598	338	4	}	}	PUNCT
ejpam-1598	338	5	be	be	AUX
ejpam-1598	338	6	the	the	DET
ejpam-1598	338	7	identity	identity	NOUN
ejpam-1598	338	8	of	of	ADP
ejpam-1598	338	9	sα	sα	PROPN
ejpam-1598	338	10	.	.	PUNCT
ejpam-1598	339	1	now	now	ADV
ejpam-1598	339	2	,	,	PUNCT
ejpam-1598	339	3	we	we	PRON
ejpam-1598	339	4	consider	consider	VERB
ejpam-1598	339	5	the	the	DET
ejpam-1598	339	6	sum	sum	NOUN
ejpam-1598	339	7	of	of	ADP
ejpam-1598	339	8	products	product	NOUN
ejpam-1598	339	9	nα∑	nα∑	ADV
ejpam-1598	339	10	i=1	i=1	PROPN
ejpam-1598	339	11	(	(	PUNCT
ejpam-1598	339	12	aα	aα	NOUN
ejpam-1598	339	13	,	,	PUNCT
ejpam-1598	339	14	bγ)γ∈λ−{α}(eiα	bγ)γ∈λ−{α}(eiα	NOUN
ejpam-1598	339	15	,	,	PUNCT
ejpam-1598	339	16	0γ)γ∈λ−{α	0γ)γ∈λ−{α	NUM
ejpam-1598	339	17	}	}	PUNCT
ejpam-1598	339	18	(	(	PUNCT
ejpam-1598	339	19	fiα	fiα	INTJ
ejpam-1598	339	20	,	,	PUNCT
ejpam-1598	339	21	0γ)γ∈λ−{α	0γ)γ∈λ−{α	NUM
ejpam-1598	339	22	}	}	PUNCT
ejpam-1598	339	23	=	=	SYM
ejpam-1598	339	24	nα∑	nα∑	ADV
ejpam-1598	339	25	i=1	i=1	X
ejpam-1598	339	26	(	(	PUNCT
ejpam-1598	339	27	aαeiα	aαeiα	PROPN
ejpam-1598	339	28	fiα	fiα	PROPN
ejpam-1598	339	29	,	,	PUNCT
ejpam-1598	339	30	0γ)γ∈λ−{α	0γ)γ∈λ−{α	NUM
ejpam-1598	339	31	}	}	PUNCT
ejpam-1598	339	32	=	=	SYM
ejpam-1598	339	33	(	(	PUNCT
ejpam-1598	339	34	aα	aα	NOUN
ejpam-1598	339	35	,	,	PUNCT
ejpam-1598	339	36	0γ)γ∈λ−{α	0γ)γ∈λ−{α	NUM
ejpam-1598	339	37	}	}	PUNCT
ejpam-1598	339	38	∈	∈	PROPN
ejpam-1598	339	39	p	p	NOUN
ejpam-1598	339	40	as	as	SCONJ
ejpam-1598	339	41	p	p	PROPN
ejpam-1598	339	42	is	be	AUX
ejpam-1598	339	43	a	a	DET
ejpam-1598	339	44	right	right	ADJ
ejpam-1598	339	45	ideal	ideal	NOUN
ejpam-1598	339	46	of	of	ADP
ejpam-1598	339	47	∏	∏	PROPN
ejpam-1598	339	48	α∈λ	α∈λ	NOUN
ejpam-1598	339	49	sα	sα	NOUN
ejpam-1598	339	50	.	.	PUNCT
ejpam-1598	340	1	thus	thus	ADV
ejpam-1598	340	2	,	,	PUNCT
ejpam-1598	340	3	(	(	PUNCT
ejpam-1598	340	4	aα	aα	NOUN
ejpam-1598	340	5	,	,	PUNCT
ejpam-1598	340	6	0γ)γ∈λ−{α	0γ)γ∈λ−{α	NUM
ejpam-1598	340	7	}	}	PUNCT
ejpam-1598	340	8	∈	∈	PROPN
ejpam-1598	340	9	p	p	NOUN
ejpam-1598	340	10	for	for	ADP
ejpam-1598	340	11	each	each	DET
ejpam-1598	340	12	α	α	PRON
ejpam-1598	340	13	∈	∈	PROPN
ejpam-1598	340	14	λ	λ	PROPN
ejpam-1598	340	15	.	.	PUNCT
ejpam-1598	341	1	hence	hence	ADV
ejpam-1598	341	2	,	,	PUNCT
ejpam-1598	341	3	∑	∑	PROPN
ejpam-1598	341	4	α∈λ(aα	α∈λ(aα	ADJ
ejpam-1598	341	5	,	,	PUNCT
ejpam-1598	341	6	0γ)γ∈λ−{α	0γ)γ∈λ−{α	NUM
ejpam-1598	341	7	}	}	PUNCT
ejpam-1598	341	8	=	=	SYM
ejpam-1598	341	9	(	(	PUNCT
ejpam-1598	341	10	aα)α∈λ	aα)α∈λ	PROPN
ejpam-1598	341	11	∈	∈	PROPN
ejpam-1598	341	12	p	p	PROPN
ejpam-1598	341	13	as	as	SCONJ
ejpam-1598	341	14	p	p	PROPN
ejpam-1598	341	15	is	be	AUX
ejpam-1598	341	16	a	a	DET
ejpam-1598	341	17	right	right	ADJ
ejpam-1598	341	18	ideal	ideal	NOUN
ejpam-1598	341	19	of	of	ADP
ejpam-1598	341	20	∏	∏	PROPN
ejpam-1598	341	21	α∈λ	α∈λ	NOUN
ejpam-1598	341	22	sα	sα	NOUN
ejpam-1598	341	23	.	.	PUNCT
ejpam-1598	342	1	hence	hence	ADV
ejpam-1598	342	2	p	p	PROPN
ejpam-1598	342	3	=	=	SYM
ejpam-1598	342	4	∏	∏	PROPN
ejpam-1598	342	5	α∈λ	α∈λ	NOUN
ejpam-1598	342	6	sα	sα	NOUN
ejpam-1598	342	7	.	.	PUNCT
ejpam-1598	342	8	proposition	proposition	NOUN
ejpam-1598	342	9	12	12	NUM
ejpam-1598	342	10	.	.	PUNCT
ejpam-1598	343	1	for	for	ADP
ejpam-1598	343	2	every	every	DET
ejpam-1598	343	3	family	family	NOUN
ejpam-1598	343	4	{	{	PUNCT
ejpam-1598	343	5	sα	sα	ADV
ejpam-1598	343	6	:	:	PUNCT
ejpam-1598	343	7	α	α	PROPN
ejpam-1598	343	8	∈	∈	PROPN
ejpam-1598	343	9	λ	λ	X
ejpam-1598	343	10	where	where	SCONJ
ejpam-1598	343	11	λ	λ	PROPN
ejpam-1598	343	12	is	be	AUX
ejpam-1598	343	13	a	a	DET
ejpam-1598	343	14	index	index	NOUN
ejpam-1598	343	15	set	set	NOUN
ejpam-1598	343	16	}	}	PUNCT
ejpam-1598	343	17	of	of	ADP
ejpam-1598	343	18	ternary	ternary	ADJ
ejpam-1598	343	19	semirings	semiring	NOUN
ejpam-1598	343	20	with	with	ADP
ejpam-1598	343	21	identity	identity	NOUN
ejpam-1598	343	22	z	z	PROPN
ejpam-1598	343	23	(	(	PUNCT
ejpam-1598	343	24	∏	∏	PROPN
ejpam-1598	343	25	α∈λ	α∈λ	NOUN
ejpam-1598	343	26	sα	sα	NOUN
ejpam-1598	343	27	)	)	PUNCT
ejpam-1598	343	28	=	=	SYM
ejpam-1598	343	29	∏	∏	PROPN
ejpam-1598	343	30	α∈λ	α∈λ	NOUN
ejpam-1598	343	31	z(sα	z(sα	PROPN
ejpam-1598	343	32	)	)	PUNCT
ejpam-1598	343	33	and	and	CCONJ
ejpam-1598	343	34	z	z	PROPN
ejpam-1598	343	35	(	(	PUNCT
ejpam-1598	343	36	⊕	⊕	PROPN
ejpam-1598	343	37	α∈λ	α∈λ	NOUN
ejpam-1598	343	38	sα	sα	NOUN
ejpam-1598	343	39	)	)	PUNCT
ejpam-1598	343	40	=	=	SYM
ejpam-1598	343	41	⊕	⊕	PROPN
ejpam-1598	343	42	α∈λ	α∈λ	NOUN
ejpam-1598	343	43	z(sα	z(sα	PROPN
ejpam-1598	343	44	)	)	PUNCT
ejpam-1598	343	45	.	.	PUNCT
ejpam-1598	344	1	proof	proof	NOUN
ejpam-1598	344	2	.	.	PUNCT
ejpam-1598	345	1	suppose	suppose	VERB
ejpam-1598	345	2	that	that	SCONJ
ejpam-1598	345	3	s	s	AUX
ejpam-1598	345	4	=	=	SYM
ejpam-1598	345	5	∏	∏	PROPN
ejpam-1598	345	6	α∈λ	α∈λ	NOUN
ejpam-1598	345	7	sα	sα	NOUN
ejpam-1598	345	8	.	.	PUNCT
ejpam-1598	346	1	let	let	VERB
ejpam-1598	346	2	(	(	PUNCT
ejpam-1598	346	3	xα)α∈λ	xα)α∈λ	PROPN
ejpam-1598	346	4	∈	∈	PROPN
ejpam-1598	346	5	z(s	z(s	PROPN
ejpam-1598	346	6	)	)	PUNCT
ejpam-1598	346	7	.	.	PUNCT
ejpam-1598	347	1	then	then	ADV
ejpam-1598	347	2	,	,	PUNCT
ejpam-1598	347	3	rs((xα)α∈λ	rs((xα)α∈λ	PROPN
ejpam-1598	347	4	is	be	AUX
ejpam-1598	347	5	an	an	DET
ejpam-1598	347	6	essential	essential	ADJ
ejpam-1598	347	7	right	right	ADJ
ejpam-1598	347	8	ideal	ideal	NOUN
ejpam-1598	347	9	of	of	ADP
ejpam-1598	347	10	s.	s.	PROPN
ejpam-1598	347	11	let	let	VERB
ejpam-1598	347	12	pα	pα	AUX
ejpam-1598	347	13	be	be	AUX
ejpam-1598	347	14	a	a	DET
ejpam-1598	347	15	nonzero	nonzero	ADJ
ejpam-1598	347	16	right	right	ADJ
ejpam-1598	347	17	ideal	ideal	NOUN
ejpam-1598	347	18	of	of	ADP
ejpam-1598	347	19	sα	sα	PROPN
ejpam-1598	347	20	,	,	PUNCT
ejpam-1598	347	21	α	α	PROPN
ejpam-1598	347	22	∈	∈	PROPN
ejpam-1598	347	23	λ	λ	PROPN
ejpam-1598	347	24	.	.	PUNCT
ejpam-1598	348	1	then	then	ADV
ejpam-1598	348	2	∏	∏	NUM
ejpam-1598	348	3	α∈λ	α∈λ	NOUN
ejpam-1598	348	4	pα	pα	NOUN
ejpam-1598	348	5	is	be	AUX
ejpam-1598	348	6	a	a	DET
ejpam-1598	348	7	nonzero	nonzero	ADJ
ejpam-1598	348	8	right	right	ADJ
ejpam-1598	348	9	ideal	ideal	NOUN
ejpam-1598	348	10	of	of	ADP
ejpam-1598	348	11	s.	s.	PROPN
ejpam-1598	348	12	this	this	PRON
ejpam-1598	348	13	shows	show	VERB
ejpam-1598	348	14	that	that	SCONJ
ejpam-1598	348	15	rs((xα)α∈λ)∩	rs((xα)α∈λ)∩	PROPN
ejpam-1598	348	16	∏	∏	PROPN
ejpam-1598	348	17	α∈λ	α∈λ	NOUN
ejpam-1598	348	18	pα	pα	NOUN
ejpam-1598	348	19	6=	6=	PROPN
ejpam-1598	348	20	(	(	PUNCT
ejpam-1598	348	21	0α)α∈λ⇒	0α)α∈λ⇒	NUM
ejpam-1598	348	22	∏	∏	PROPN
ejpam-1598	348	23	α∈λ	α∈λ	NOUN
ejpam-1598	348	24	rsα	rsα	NOUN
ejpam-1598	348	25	(	(	PUNCT
ejpam-1598	348	26	xα)∩	xα)∩	PROPN
ejpam-1598	348	27	∏	∏	PROPN
ejpam-1598	348	28	α∈λ	α∈λ	NOUN
ejpam-1598	348	29	pα	pα	NOUN
ejpam-1598	348	30	6=	6=	PROPN
ejpam-1598	348	31	(	(	PUNCT
ejpam-1598	348	32	0α)α∈λ	0α)α∈λ	NUM
ejpam-1598	348	33	,	,	PUNCT
ejpam-1598	348	34	by	by	ADP
ejpam-1598	348	35	lemma	lemma	PROPN
ejpam-1598	348	36	2	2	NUM
ejpam-1598	348	37	.	.	X
ejpam-1598	349	1	we	we	PRON
ejpam-1598	349	2	claim	claim	VERB
ejpam-1598	349	3	that	that	SCONJ
ejpam-1598	349	4	rsα	rsα	NOUN
ejpam-1598	349	5	(	(	PUNCT
ejpam-1598	349	6	xα	xα	ADJ
ejpam-1598	349	7	)	)	PUNCT
ejpam-1598	349	8	∩	∩	PROPN
ejpam-1598	349	9	pα	pα	PROPN
ejpam-1598	349	10	6=	6=	PROPN
ejpam-1598	349	11	0α	0α	PROPN
ejpam-1598	349	12	,	,	PUNCT
ejpam-1598	349	13	for	for	ADP
ejpam-1598	349	14	each	each	DET
ejpam-1598	349	15	α	α	NOUN
ejpam-1598	349	16	∈	∈	PROPN
ejpam-1598	349	17	λ	λ	PROPN
ejpam-1598	349	18	.	.	PUNCT
ejpam-1598	350	1	if	if	SCONJ
ejpam-1598	350	2	the	the	DET
ejpam-1598	350	3	assertion	assertion	NOUN
ejpam-1598	350	4	is	be	AUX
ejpam-1598	350	5	not	not	PART
ejpam-1598	350	6	true	true	ADJ
ejpam-1598	350	7	.	.	PUNCT
ejpam-1598	351	1	then	then	ADV
ejpam-1598	351	2	there	there	PRON
ejpam-1598	351	3	exists	exist	VERB
ejpam-1598	351	4	i	i	PRON
ejpam-1598	351	5	⊆	⊆	NUM
ejpam-1598	351	6	λ	λ	NOUN
ejpam-1598	351	7	such	such	ADJ
ejpam-1598	351	8	that	that	DET
ejpam-1598	351	9	rsα	rsα	NOUN
ejpam-1598	351	10	(	(	PUNCT
ejpam-1598	351	11	xα)∩	xα)∩	PROPN
ejpam-1598	351	12	pα	pα	PROPN
ejpam-1598	351	13	=	=	PUNCT
ejpam-1598	351	14	0α	0α	PROPN
ejpam-1598	351	15	for	for	ADP
ejpam-1598	351	16	each	each	DET
ejpam-1598	351	17	α	α	NOUN
ejpam-1598	351	18	∈	∈	PROPN
ejpam-1598	352	1	i	i	PRON
ejpam-1598	352	2	and	and	CCONJ
ejpam-1598	352	3	rsα	rsα	NOUN
ejpam-1598	352	4	(	(	PUNCT
ejpam-1598	352	5	xα)∩	xα)∩	PROPN
ejpam-1598	352	6	pα	pα	PROPN
ejpam-1598	352	7	6=	6=	PUNCT
ejpam-1598	352	8	0α	0α	PROPN
ejpam-1598	352	9	,	,	PUNCT
ejpam-1598	352	10	for	for	ADP
ejpam-1598	352	11	each	each	DET
ejpam-1598	352	12	α	α	NOUN
ejpam-1598	352	13	∈	∈	PROPN
ejpam-1598	352	14	λ−	λ−	PROPN
ejpam-1598	353	1	i	i	PRON
ejpam-1598	353	2	.	.	PUNCT
ejpam-1598	354	1	now	now	ADV
ejpam-1598	354	2	∏	∏	NUM
ejpam-1598	354	3	α∈λ−i(0α)×	α∈λ−i(0α)×	NOUN
ejpam-1598	354	4	∏	∏	NUM
ejpam-1598	354	5	α∈i	α∈i	NUM
ejpam-1598	354	6	pα	pα	NOUN
ejpam-1598	354	7	is	be	AUX
ejpam-1598	354	8	a	a	DET
ejpam-1598	354	9	nonzero	nonzero	ADJ
ejpam-1598	354	10	right	right	ADJ
ejpam-1598	354	11	ideal	ideal	NOUN
ejpam-1598	354	12	of	of	ADP
ejpam-1598	354	13	s	s	NOUN
ejpam-1598	354	14	but	but	CCONJ
ejpam-1598	354	15	rs((xα)α∈λ)∩	rs((xα)α∈λ)∩	PROPN
ejpam-1598	354	16	(	(	PUNCT
ejpam-1598	354	17	∏	∏	PROPN
ejpam-1598	354	18	α∈λ−i	α∈λ−i	PROPN
ejpam-1598	354	19	(	(	PUNCT
ejpam-1598	354	20	0α)×	0α)×	NOUN
ejpam-1598	354	21	∏	∏	NUM
ejpam-1598	354	22	α∈i	α∈i	NUM
ejpam-1598	354	23	pα	pα	NOUN
ejpam-1598	354	24	)	)	PUNCT
ejpam-1598	354	25	=	=	PUNCT
ejpam-1598	355	1	[	[	PUNCT
ejpam-1598	355	2	∏	∏	NUM
ejpam-1598	355	3	α∈λ−i	α∈λ−i	PROPN
ejpam-1598	355	4	rsα	rsα	NOUN
ejpam-1598	355	5	(	(	PUNCT
ejpam-1598	355	6	xα)∩	xα)∩	PUNCT
ejpam-1598	355	7	(	(	PUNCT
ejpam-1598	355	8	0α)]×	0α)]×	SYM
ejpam-1598	355	9	[	[	PUNCT
ejpam-1598	355	10	∏	∏	NUM
ejpam-1598	355	11	α∈i	α∈i	NUM
ejpam-1598	355	12	rsα	rsα	NOUN
ejpam-1598	355	13	(	(	PUNCT
ejpam-1598	355	14	xα)∩	xα)∩	PROPN
ejpam-1598	355	15	pα	pα	PROPN
ejpam-1598	355	16	]	]	X
ejpam-1598	355	17	=	=	SYM
ejpam-1598	355	18	(	(	PUNCT
ejpam-1598	355	19	0α)α∈λ	0α)α∈λ	NUM
ejpam-1598	355	20	,	,	PUNCT
ejpam-1598	355	21	a	a	DET
ejpam-1598	355	22	contradiction	contradiction	NOUN
ejpam-1598	355	23	since	since	SCONJ
ejpam-1598	355	24	(	(	PUNCT
ejpam-1598	355	25	xα)α∈λ	xα)α∈λ	PROPN
ejpam-1598	355	26	∈	∈	PROPN
ejpam-1598	355	27	z(s	z(s	PROPN
ejpam-1598	355	28	)	)	PUNCT
ejpam-1598	355	29	.	.	PUNCT
ejpam-1598	356	1	thus	thus	ADV
ejpam-1598	356	2	,	,	PUNCT
ejpam-1598	356	3	xα	xα	PROPN
ejpam-1598	356	4	∈	∈	PROPN
ejpam-1598	356	5	z(sα	z(sα	PROPN
ejpam-1598	356	6	)	)	PUNCT
ejpam-1598	356	7	for	for	ADP
ejpam-1598	356	8	each	each	DET
ejpam-1598	356	9	α	α	NOUN
ejpam-1598	356	10	∈	∈	PROPN
ejpam-1598	356	11	λ⇒	λ⇒	X
ejpam-1598	356	12	(	(	PUNCT
ejpam-1598	356	13	xα)α∈λ	xα)α∈λ	PROPN
ejpam-1598	356	14	∈	∈	PROPN
ejpam-1598	356	15	∏	∏	PROPN
ejpam-1598	356	16	α∈λ	α∈λ	NOUN
ejpam-1598	356	17	z(sα	z(sα	PROPN
ejpam-1598	356	18	)	)	PUNCT
ejpam-1598	356	19	.	.	PUNCT
ejpam-1598	357	1	conversely	conversely	ADV
ejpam-1598	357	2	,	,	PUNCT
ejpam-1598	357	3	let	let	VERB
ejpam-1598	357	4	(	(	PUNCT
ejpam-1598	357	5	xα)α∈λ	xα)α∈λ	PROPN
ejpam-1598	357	6	∈	∈	PROPN
ejpam-1598	357	7	∏	∏	PROPN
ejpam-1598	357	8	α∈λ	α∈λ	NOUN
ejpam-1598	357	9	z(sα)⇒	z(sα)⇒	PROPN
ejpam-1598	357	10	xα	xα	PUNCT
ejpam-1598	358	1	∈	∈	PROPN
ejpam-1598	358	2	z(sα	z(sα	PROPN
ejpam-1598	358	3	)	)	PUNCT
ejpam-1598	358	4	for	for	ADP
ejpam-1598	358	5	each	each	DET
ejpam-1598	358	6	α	α	NOUN
ejpam-1598	358	7	∈	∈	PROPN
ejpam-1598	358	8	λ	λ	PROPN
ejpam-1598	358	9	.	.	PUNCT
ejpam-1598	359	1	let	let	VERB
ejpam-1598	359	2	p	p	PRON
ejpam-1598	359	3	be	be	AUX
ejpam-1598	359	4	a	a	DET
ejpam-1598	359	5	nonzero	nonzero	ADJ
ejpam-1598	359	6	right	right	ADJ
ejpam-1598	359	7	ideal	ideal	NOUN
ejpam-1598	359	8	of	of	ADP
ejpam-1598	359	9	s.	s.	PROPN
ejpam-1598	359	10	now	now	ADV
ejpam-1598	359	11	,	,	PUNCT
ejpam-1598	359	12	let	let	VERB
ejpam-1598	359	13	πα	πα	VERB
ejpam-1598	359	14	:	:	PUNCT
ejpam-1598	359	15	s→	s→	PRON
ejpam-1598	359	16	sα	sα	ADV
ejpam-1598	359	17	be	be	AUX
ejpam-1598	359	18	the	the	DET
ejpam-1598	359	19	projection	projection	NOUN
ejpam-1598	359	20	map	map	NOUN
ejpam-1598	359	21	.	.	PUNCT
ejpam-1598	360	1	then	then	ADV
ejpam-1598	360	2	,	,	PUNCT
ejpam-1598	360	3	pα	pα	INTJ
ejpam-1598	360	4	=	=	PUNCT
ejpam-1598	360	5	πα(p	πα(p	NUM
ejpam-1598	360	6	)	)	PUNCT
ejpam-1598	360	7	is	be	AUX
ejpam-1598	360	8	a	a	DET
ejpam-1598	360	9	nonzero	nonzero	ADJ
ejpam-1598	360	10	right	right	ADJ
ejpam-1598	360	11	ideal	ideal	NOUN
ejpam-1598	360	12	of	of	ADP
ejpam-1598	360	13	sα	sα	ADV
ejpam-1598	360	14	for	for	ADP
ejpam-1598	360	15	at	at	ADV
ejpam-1598	360	16	least	least	ADV
ejpam-1598	360	17	one	one	NUM
ejpam-1598	360	18	α	α	NOUN
ejpam-1598	360	19	.	.	PUNCT
ejpam-1598	361	1	this	this	PRON
ejpam-1598	361	2	shows	show	VERB
ejpam-1598	361	3	that	that	SCONJ
ejpam-1598	361	4	p	p	PROPN
ejpam-1598	361	5	=	=	SYM
ejpam-1598	361	6	∏	∏	PROPN
ejpam-1598	361	7	α∈λ	α∈λ	NOUN
ejpam-1598	361	8	pα	pα	NOUN
ejpam-1598	361	9	by	by	ADP
ejpam-1598	361	10	lemma	lemma	PROPN
ejpam-1598	361	11	3	3	NUM
ejpam-1598	361	12	.	.	PUNCT
ejpam-1598	362	1	now	now	ADV
ejpam-1598	362	2	,	,	PUNCT
ejpam-1598	362	3	rsα	rsα	NOUN
ejpam-1598	362	4	(	(	PUNCT
ejpam-1598	362	5	xα)∩	xα)∩	PROPN
ejpam-1598	362	6	pα	pα	PROPN
ejpam-1598	362	7	6=	6=	NUM
ejpam-1598	362	8	0α	0α	PROPN
ejpam-1598	362	9	for	for	ADP
ejpam-1598	362	10	at	at	ADV
ejpam-1598	362	11	least	least	ADV
ejpam-1598	362	12	one	one	NUM
ejpam-1598	362	13	α⇒	α⇒	SYM
ejpam-1598	362	14	∏	∏	NOUN
ejpam-1598	362	15	α∈λ	α∈λ	NOUN
ejpam-1598	362	16	rsα	rsα	NOUN
ejpam-1598	362	17	(	(	PUNCT
ejpam-1598	362	18	xα)∩	xα)∩	PROPN
ejpam-1598	362	19	∏	∏	PROPN
ejpam-1598	362	20	α∈λ	α∈λ	NOUN
ejpam-1598	362	21	pα	pα	NOUN
ejpam-1598	362	22	6=	6=	PROPN
ejpam-1598	362	23	(	(	PUNCT
ejpam-1598	362	24	0α)α∈λ⇒	0α)α∈λ⇒	NUM
ejpam-1598	362	25	rs((xα)α∈λ)∩	rs((xα)α∈λ)∩	PROPN
ejpam-1598	362	26	p	p	NOUN
ejpam-1598	362	27	6=	6=	PROPN
ejpam-1598	362	28	(	(	PUNCT
ejpam-1598	362	29	0α)α∈λ⇒	0α)α∈λ⇒	NUM
ejpam-1598	362	30	(	(	PUNCT
ejpam-1598	362	31	xα)α∈λ	xα)α∈λ	PROPN
ejpam-1598	362	32	∈	∈	PROPN
ejpam-1598	362	33	z(s	z(s	PROPN
ejpam-1598	362	34	)	)	PUNCT
ejpam-1598	362	35	.	.	PUNCT
ejpam-1598	363	1	t.	t.	PROPN
ejpam-1598	363	2	dutta	dutta	PROPN
ejpam-1598	363	3	,	,	PUNCT
ejpam-1598	363	4	k.	k.	PROPN
ejpam-1598	363	5	shum	shum	PROPN
ejpam-1598	363	6	,	,	PUNCT
ejpam-1598	363	7	s.	s.	PROPN
ejpam-1598	363	8	mandal	mandal	PROPN
ejpam-1598	363	9	/	/	SYM
ejpam-1598	363	10	eur	eur	PROPN
ejpam-1598	363	11	.	.	PUNCT
ejpam-1598	364	1	j.	j.	PROPN
ejpam-1598	364	2	pure	pure	PROPN
ejpam-1598	364	3	appl	appl	PROPN
ejpam-1598	364	4	.	.	PROPN
ejpam-1598	364	5	math	math	PROPN
ejpam-1598	364	6	,	,	PUNCT
ejpam-1598	364	7	5	5	NUM
ejpam-1598	364	8	(	(	PUNCT
ejpam-1598	364	9	2012	2012	NUM
ejpam-1598	364	10	)	)	PUNCT
ejpam-1598	364	11	,	,	PUNCT
ejpam-1598	364	12	116	116	NUM
ejpam-1598	364	13	-	-	SYM
ejpam-1598	364	14	128	128	NUM
ejpam-1598	364	15	125	125	NUM
ejpam-1598	364	16	thus	thus	ADV
ejpam-1598	364	17	,	,	PUNCT
ejpam-1598	364	18	z(s	z(s	PROPN
ejpam-1598	364	19	)	)	PUNCT
ejpam-1598	364	20	=	=	SYM
ejpam-1598	364	21	∏	∏	PROPN
ejpam-1598	364	22	α∈λ	α∈λ	NOUN
ejpam-1598	364	23	z(sα	z(sα	PROPN
ejpam-1598	364	24	)	)	PUNCT
ejpam-1598	364	25	.	.	PUNCT
ejpam-1598	365	1	similarly	similarly	ADV
ejpam-1598	365	2	,	,	PUNCT
ejpam-1598	365	3	we	we	PRON
ejpam-1598	365	4	can	can	AUX
ejpam-1598	365	5	prove	prove	VERB
ejpam-1598	365	6	that	that	SCONJ
ejpam-1598	365	7	z	z	PROPN
ejpam-1598	365	8	(	(	PUNCT
ejpam-1598	365	9	⊕	⊕	PROPN
ejpam-1598	365	10	α∈λ	α∈λ	NOUN
ejpam-1598	365	11	sα	sα	NOUN
ejpam-1598	365	12	)	)	PUNCT
ejpam-1598	365	13	=	=	SYM
ejpam-1598	365	14	⊕	⊕	PROPN
ejpam-1598	365	15	α∈λ	α∈λ	NOUN
ejpam-1598	365	16	z(sα	z(sα	PROPN
ejpam-1598	365	17	)	)	PUNCT
ejpam-1598	365	18	.	.	PUNCT
ejpam-1598	366	1	the	the	DET
ejpam-1598	366	2	following	follow	VERB
ejpam-1598	366	3	theorem	theorem	NOUN
ejpam-1598	366	4	is	be	AUX
ejpam-1598	366	5	a	a	DET
ejpam-1598	366	6	main	main	ADJ
ejpam-1598	366	7	theorem	theorem	NOUN
ejpam-1598	366	8	of	of	ADP
ejpam-1598	366	9	this	this	DET
ejpam-1598	366	10	paper	paper	NOUN
ejpam-1598	366	11	.	.	PUNCT
ejpam-1598	367	1	theorem	theorem	NOUN
ejpam-1598	367	2	1	1	NUM
ejpam-1598	367	3	.	.	PUNCT
ejpam-1598	368	1	the	the	DET
ejpam-1598	368	2	class	class	NOUN
ejpam-1598	368	3	of	of	ADP
ejpam-1598	368	4	singular	singular	ADJ
ejpam-1598	368	5	ternary	ternary	ADJ
ejpam-1598	368	6	semirings	semiring	NOUN
ejpam-1598	368	7	with	with	ADP
ejpam-1598	368	8	identity	identity	NOUN
ejpam-1598	368	9	as	as	ADV
ejpam-1598	368	10	well	well	ADV
ejpam-1598	368	11	as	as	ADP
ejpam-1598	368	12	the	the	DET
ejpam-1598	368	13	class	class	NOUN
ejpam-1598	368	14	of	of	ADP
ejpam-1598	368	15	nonsingular	nonsingular	ADJ
ejpam-1598	368	16	ternary	ternary	ADJ
ejpam-1598	368	17	semirings	semiring	NOUN
ejpam-1598	368	18	with	with	ADP
ejpam-1598	368	19	identity	identity	NOUN
ejpam-1598	368	20	is	be	AUX
ejpam-1598	368	21	closed	close	VERB
ejpam-1598	368	22	under	under	ADP
ejpam-1598	368	23	product	product	NOUN
ejpam-1598	368	24	and	and	CCONJ
ejpam-1598	368	25	direct	direct	ADJ
ejpam-1598	368	26	sum	sum	NOUN
ejpam-1598	368	27	.	.	PUNCT
ejpam-1598	369	1	definition	definition	NOUN
ejpam-1598	369	2	26	26	NUM
ejpam-1598	369	3	.	.	PUNCT
ejpam-1598	370	1	a	a	DET
ejpam-1598	370	2	surjective	surjective	ADJ
ejpam-1598	370	3	homomorphism	homomorphism	NOUN
ejpam-1598	370	4	of	of	ADP
ejpam-1598	370	5	ternary	ternary	ADJ
ejpam-1598	370	6	semirings	semiring	NOUN
ejpam-1598	370	7	γ	γ	NOUN
ejpam-1598	370	8	:	:	PUNCT
ejpam-1598	370	9	r	r	NOUN
ejpam-1598	370	10	→	→	SYM
ejpam-1598	370	11	s	s	PART
ejpam-1598	370	12	is	be	AUX
ejpam-1598	370	13	called	call	VERB
ejpam-1598	370	14	a	a	DET
ejpam-1598	370	15	ternary	ternary	ADJ
ejpam-1598	370	16	semi	semi	NOUN
ejpam-1598	370	17	-	-	NOUN
ejpam-1598	370	18	isomorphism	isomorphism	ADJ
ejpam-1598	370	19	if	if	SCONJ
ejpam-1598	370	20	kerγ	kerγ	VERB
ejpam-1598	370	21	=	=	NOUN
ejpam-1598	370	22	0	0	X
ejpam-1598	370	23	.	.	PUNCT
ejpam-1598	371	1	lemma	lemma	PROPN
ejpam-1598	371	2	4	4	X
ejpam-1598	371	3	.	.	PUNCT
ejpam-1598	372	1	if	if	SCONJ
ejpam-1598	372	2	γ	γ	X
ejpam-1598	372	3	:	:	PUNCT
ejpam-1598	372	4	s→	s→	PROPN
ejpam-1598	372	5	s′	s′	PROPN
ejpam-1598	372	6	is	be	AUX
ejpam-1598	372	7	a	a	DET
ejpam-1598	372	8	ternary	ternary	ADJ
ejpam-1598	372	9	semi	semi	NOUN
ejpam-1598	372	10	-	-	ADJ
ejpam-1598	372	11	isomorphism	isomorphism	NOUN
ejpam-1598	372	12	and	and	CCONJ
ejpam-1598	372	13	z(s	z(s	NUM
ejpam-1598	372	14	)	)	PUNCT
ejpam-1598	373	1	=	=	SYM
ejpam-1598	373	2	s	s	VERB
ejpam-1598	373	3	then	then	ADV
ejpam-1598	373	4	z(s′	z(s′	ADJ
ejpam-1598	373	5	)	)	PUNCT
ejpam-1598	373	6	=	=	PUNCT
ejpam-1598	374	1	s′.	s′.	PROPN
ejpam-1598	374	2	proof	proof	NOUN
ejpam-1598	374	3	.	.	PUNCT
ejpam-1598	375	1	if	if	SCONJ
ejpam-1598	375	2	possible	possible	ADJ
ejpam-1598	375	3	,	,	PUNCT
ejpam-1598	375	4	let	let	VERB
ejpam-1598	375	5	z(s′	z(s′	NUM
ejpam-1598	375	6	)	)	PUNCT
ejpam-1598	376	1	⊂	⊂	PROPN
ejpam-1598	376	2	s′.	s′.	PROPN
ejpam-1598	376	3	then	then	ADV
ejpam-1598	376	4	there	there	PRON
ejpam-1598	376	5	exists	exist	VERB
ejpam-1598	376	6	s′	s′	ADJ
ejpam-1598	376	7	∈	∈	PROPN
ejpam-1598	376	8	s′	s′	NUM
ejpam-1598	376	9	but	but	CCONJ
ejpam-1598	376	10	s′	s′	ADJ
ejpam-1598	376	11	6∈	6∈	PROPN
ejpam-1598	376	12	z(s′	z(s′	NUM
ejpam-1598	376	13	)	)	PUNCT
ejpam-1598	376	14	.	.	PUNCT
ejpam-1598	377	1	let	let	VERB
ejpam-1598	377	2	s	s	PRON
ejpam-1598	377	3	∈	∈	NOUN
ejpam-1598	377	4	s	s	VERB
ejpam-1598	377	5	such	such	ADJ
ejpam-1598	377	6	that	that	SCONJ
ejpam-1598	377	7	γ(s	γ(	NOUN
ejpam-1598	377	8	)	)	PUNCT
ejpam-1598	378	1	=	=	PUNCT
ejpam-1598	378	2	s′.	s′.	PROPN
ejpam-1598	378	3	since	since	SCONJ
ejpam-1598	378	4	s′	s′	ADJ
ejpam-1598	378	5	6∈	6∈	PROPN
ejpam-1598	378	6	z(s′	z(s′	NUM
ejpam-1598	378	7	)	)	PUNCT
ejpam-1598	378	8	,	,	PUNCT
ejpam-1598	378	9	there	there	PRON
ejpam-1598	378	10	exists	exist	VERB
ejpam-1598	378	11	a	a	DET
ejpam-1598	378	12	nonzero	nonzero	ADJ
ejpam-1598	378	13	right	right	ADJ
ejpam-1598	378	14	ideal	ideal	ADJ
ejpam-1598	378	15	h	h	NOUN
ejpam-1598	378	16	′	′	NOUN
ejpam-1598	378	17	of	of	ADP
ejpam-1598	378	18	s′	s′	NUM
ejpam-1598	378	19	such	such	ADJ
ejpam-1598	378	20	that	that	SCONJ
ejpam-1598	378	21	rs′(s	rs′(s	PROPN
ejpam-1598	378	22	′	′	NOUN
ejpam-1598	378	23	)	)	PUNCT
ejpam-1598	378	24	∩	∩	ADJ
ejpam-1598	378	25	h	h	NOUN
ejpam-1598	378	26	′	′	NUM
ejpam-1598	379	1	=	=	NOUN
ejpam-1598	379	2	0	0	X
ejpam-1598	379	3	.	.	PUNCT
ejpam-1598	380	1	let	let	VERB
ejpam-1598	380	2	h	h	NOUN
ejpam-1598	380	3	=	=	PRON
ejpam-1598	381	1	{	{	PUNCT
ejpam-1598	381	2	b	b	X
ejpam-1598	381	3	∈	∈	PROPN
ejpam-1598	381	4	s	s	PART
ejpam-1598	381	5	:	:	PUNCT
ejpam-1598	381	6	γ(b	γ(b	X
ejpam-1598	381	7	)	)	PUNCT
ejpam-1598	381	8	∈	∈	PROPN
ejpam-1598	381	9	h	h	NOUN
ejpam-1598	381	10	′	′	NOUN
ejpam-1598	381	11	}	}	PUNCT
ejpam-1598	381	12	.	.	PUNCT
ejpam-1598	382	1	then	then	ADV
ejpam-1598	382	2	h	h	PROPN
ejpam-1598	382	3	is	be	AUX
ejpam-1598	382	4	a	a	DET
ejpam-1598	382	5	nonzero	nonzero	ADJ
ejpam-1598	382	6	right	right	ADJ
ejpam-1598	382	7	ideal	ideal	NOUN
ejpam-1598	382	8	of	of	ADP
ejpam-1598	382	9	s.	s.	PROPN
ejpam-1598	383	1	so	so	SCONJ
ejpam-1598	383	2	rs(s)∩h	rs(s)∩h	PROPN
ejpam-1598	383	3	6=	6=	ADP
ejpam-1598	383	4	0	0	NUM
ejpam-1598	383	5	.	.	PUNCT
ejpam-1598	384	1	therefore	therefore	ADV
ejpam-1598	384	2	there	there	PRON
ejpam-1598	384	3	exists	exist	VERB
ejpam-1598	384	4	a	a	DET
ejpam-1598	384	5	nonzero	nonzero	PROPN
ejpam-1598	384	6	element	element	NOUN
ejpam-1598	384	7	h	h	NOUN
ejpam-1598	384	8	in	in	ADP
ejpam-1598	384	9	h	h	NOUN
ejpam-1598	384	10	such	such	ADJ
ejpam-1598	384	11	that	that	DET
ejpam-1598	384	12	sht	sht	NOUN
ejpam-1598	385	1	=	=	NOUN
ejpam-1598	385	2	0	0	NUM
ejpam-1598	385	3	for	for	ADP
ejpam-1598	385	4	all	all	DET
ejpam-1598	385	5	t	t	PROPN
ejpam-1598	385	6	∈	∈	PROPN
ejpam-1598	385	7	s.	s.	PROPN
ejpam-1598	385	8	consequently	consequently	ADV
ejpam-1598	385	9	γ(sht	γ(sht	VERB
ejpam-1598	385	10	)	)	PUNCT
ejpam-1598	385	11	=	=	SYM
ejpam-1598	385	12	γ(0	γ(0	PROPN
ejpam-1598	385	13	)	)	PUNCT
ejpam-1598	385	14	for	for	ADP
ejpam-1598	385	15	all	all	DET
ejpam-1598	385	16	t	t	NOUN
ejpam-1598	385	17	∈	∈	NOUN
ejpam-1598	385	18	s	s	PART
ejpam-1598	385	19	⇒	⇒	NOUN
ejpam-1598	385	20	γ(s)γ(h)γ(t	γ(s)γ(h)γ(t	PUNCT
ejpam-1598	385	21	)	)	PUNCT
ejpam-1598	385	22	=	=	SYM
ejpam-1598	385	23	0	0	NUM
ejpam-1598	385	24	for	for	ADP
ejpam-1598	385	25	all	all	DET
ejpam-1598	385	26	t	t	NOUN
ejpam-1598	385	27	∈	∈	PROPN
ejpam-1598	385	28	s	s	PART
ejpam-1598	385	29	⇒	⇒	NOUN
ejpam-1598	385	30	s′γ(h)t′	s′γ(h)t′	NOUN
ejpam-1598	385	31	=	=	NOUN
ejpam-1598	385	32	0	0	NUM
ejpam-1598	385	33	for	for	ADP
ejpam-1598	385	34	all	all	DET
ejpam-1598	385	35	t′	t′	NUM
ejpam-1598	385	36	∈	∈	NOUN
ejpam-1598	385	37	s′	s′	PUNCT
ejpam-1598	385	38	as	as	SCONJ
ejpam-1598	385	39	γ	γ	X
ejpam-1598	385	40	is	be	AUX
ejpam-1598	385	41	surjective	surjective	ADJ
ejpam-1598	385	42	.	.	PUNCT
ejpam-1598	386	1	but	but	CCONJ
ejpam-1598	386	2	γ(h	γ(h	PROPN
ejpam-1598	386	3	)	)	PUNCT
ejpam-1598	386	4	∈	∈	PROPN
ejpam-1598	386	5	h	h	NOUN
ejpam-1598	386	6	′.	′.	PROPN
ejpam-1598	386	7	therefore	therefore	ADV
ejpam-1598	386	8	γ(h	γ(h	PROPN
ejpam-1598	386	9	)	)	PUNCT
ejpam-1598	386	10	∈	∈	PROPN
ejpam-1598	386	11	rs′(s	rs′(s	PROPN
ejpam-1598	386	12	′	′	NOUN
ejpam-1598	386	13	)	)	PUNCT
ejpam-1598	386	14	∩	∩	ADJ
ejpam-1598	386	15	h	h	NOUN
ejpam-1598	386	16	′	′	NOUN
ejpam-1598	386	17	which	which	PRON
ejpam-1598	386	18	implies	imply	VERB
ejpam-1598	386	19	that	that	SCONJ
ejpam-1598	386	20	γ(h	γ(h	NOUN
ejpam-1598	386	21	)	)	PUNCT
ejpam-1598	386	22	=	=	PUNCT
ejpam-1598	387	1	0	0	X
ejpam-1598	387	2	.	.	PUNCT
ejpam-1598	388	1	thus	thus	ADV
ejpam-1598	388	2	h	h	NOUN
ejpam-1598	388	3	∈	∈	NOUN
ejpam-1598	388	4	kerγ	kerγ	NOUN
ejpam-1598	388	5	=	=	PUNCT
ejpam-1598	388	6	0	0	X
ejpam-1598	388	7	.	.	PUNCT
ejpam-1598	389	1	so	so	ADV
ejpam-1598	389	2	h=	h=	X
ejpam-1598	389	3	0	0	NUM
ejpam-1598	389	4	,	,	PUNCT
ejpam-1598	389	5	a	a	DET
ejpam-1598	389	6	contradiction	contradiction	NOUN
ejpam-1598	389	7	.	.	PUNCT
ejpam-1598	390	1	hence	hence	ADV
ejpam-1598	390	2	z(s′	z(s′	NUM
ejpam-1598	390	3	)	)	PUNCT
ejpam-1598	390	4	=	=	SYM
ejpam-1598	391	1	s′.	s′.	PROPN
ejpam-1598	391	2	lemma	lemma	PROPN
ejpam-1598	391	3	5	5	NUM
ejpam-1598	391	4	.	.	PUNCT
ejpam-1598	392	1	if	if	SCONJ
ejpam-1598	392	2	γ	γ	X
ejpam-1598	392	3	:	:	PUNCT
ejpam-1598	392	4	s→	s→	PROPN
ejpam-1598	392	5	s′	s′	PROPN
ejpam-1598	392	6	is	be	AUX
ejpam-1598	392	7	a	a	DET
ejpam-1598	392	8	semi	semi	NOUN
ejpam-1598	392	9	-	-	NOUN
ejpam-1598	392	10	isomorphism	isomorphism	ADJ
ejpam-1598	392	11	and	and	CCONJ
ejpam-1598	392	12	z(s′	z(s′	NOUN
ejpam-1598	392	13	)	)	PUNCT
ejpam-1598	392	14	=	=	VERB
ejpam-1598	393	1	s′	s′	ADJ
ejpam-1598	393	2	then	then	ADV
ejpam-1598	393	3	z(s	z(s	NUM
ejpam-1598	393	4	)	)	PUNCT
ejpam-1598	394	1	=	=	PUNCT
ejpam-1598	394	2	s.	s.	PROPN
ejpam-1598	394	3	proof	proof	NOUN
ejpam-1598	394	4	.	.	PUNCT
ejpam-1598	395	1	if	if	SCONJ
ejpam-1598	395	2	possible	possible	ADJ
ejpam-1598	395	3	,	,	PUNCT
ejpam-1598	395	4	let	let	VERB
ejpam-1598	395	5	z(s	z(s	PRON
ejpam-1598	395	6	)	)	PUNCT
ejpam-1598	396	1	⊂	⊂	PROPN
ejpam-1598	396	2	s.	s.	PROPN
ejpam-1598	396	3	so	so	ADV
ejpam-1598	396	4	there	there	PRON
ejpam-1598	396	5	exists	exist	VERB
ejpam-1598	396	6	s	s	NOUN
ejpam-1598	396	7	(	(	PUNCT
ejpam-1598	396	8	6=	6=	NOUN
ejpam-1598	396	9	0	0	NUM
ejpam-1598	396	10	)	)	PUNCT
ejpam-1598	396	11	∈	∈	PROPN
ejpam-1598	396	12	s	s	PART
ejpam-1598	396	13	but	but	CCONJ
ejpam-1598	396	14	s	s	PROPN
ejpam-1598	396	15	6∈	6∈	PROPN
ejpam-1598	396	16	z(s	z(s	PROPN
ejpam-1598	396	17	)	)	PUNCT
ejpam-1598	396	18	.	.	PUNCT
ejpam-1598	397	1	therefore	therefore	ADV
ejpam-1598	397	2	there	there	PRON
ejpam-1598	397	3	exists	exist	VERB
ejpam-1598	397	4	a	a	DET
ejpam-1598	397	5	nonzero	nonzero	ADJ
ejpam-1598	397	6	right	right	ADJ
ejpam-1598	397	7	ideal	ideal	ADJ
ejpam-1598	397	8	h	h	NOUN
ejpam-1598	397	9	of	of	ADP
ejpam-1598	397	10	s	s	PRON
ejpam-1598	397	11	such	such	ADJ
ejpam-1598	397	12	that	that	SCONJ
ejpam-1598	397	13	rs(s)∩h	rs(s)∩h	NOUN
ejpam-1598	397	14	=	=	SYM
ejpam-1598	397	15	0	0	X
ejpam-1598	397	16	.	.	PUNCT
ejpam-1598	398	1	now	now	ADV
ejpam-1598	398	2	γ(h	γ(h	NOUN
ejpam-1598	398	3	)	)	PUNCT
ejpam-1598	398	4	is	be	AUX
ejpam-1598	398	5	a	a	DET
ejpam-1598	398	6	nonzero	nonzero	ADJ
ejpam-1598	398	7	right	right	ADJ
ejpam-1598	398	8	ideal	ideal	NOUN
ejpam-1598	398	9	of	of	ADP
ejpam-1598	398	10	s′.	s′.	PROPN
ejpam-1598	398	11	now	now	ADV
ejpam-1598	398	12	γ(s	γ(s	PROPN
ejpam-1598	398	13	)	)	PUNCT
ejpam-1598	399	1	6=	6=	ADP
ejpam-1598	399	2	0	0	NUM
ejpam-1598	399	3	because	because	SCONJ
ejpam-1598	399	4	γ(s	γ(	NOUN
ejpam-1598	399	5	)	)	PUNCT
ejpam-1598	400	1	=	=	SYM
ejpam-1598	401	1	0⇒	0⇒	PROPN
ejpam-1598	401	2	s	s	NOUN
ejpam-1598	401	3	∈	∈	NOUN
ejpam-1598	401	4	kerγ	kerγ	NOUN
ejpam-1598	401	5	=	=	PUNCT
ejpam-1598	401	6	0⇒	0⇒	PROPN
ejpam-1598	401	7	s	s	PART
ejpam-1598	401	8	=	=	NOUN
ejpam-1598	401	9	0	0	X
ejpam-1598	401	10	.	.	PUNCT
ejpam-1598	401	11	let	let	VERB
ejpam-1598	401	12	γ(s	γ(s	PRON
ejpam-1598	401	13	)	)	PUNCT
ejpam-1598	402	1	=	=	PUNCT
ejpam-1598	402	2	s′.	s′.	X
ejpam-1598	402	3	we	we	PRON
ejpam-1598	402	4	now	now	ADV
ejpam-1598	402	5	prove	prove	VERB
ejpam-1598	402	6	that	that	SCONJ
ejpam-1598	402	7	,	,	PUNCT
ejpam-1598	402	8	rs′(s	rs′(s	PROPN
ejpam-1598	402	9	′)∩γ(h	′)∩γ(h	NOUN
ejpam-1598	402	10	)	)	PUNCT
ejpam-1598	402	11	=	=	SYM
ejpam-1598	403	1	0	0	X
ejpam-1598	403	2	.	.	PUNCT
ejpam-1598	404	1	if	if	SCONJ
ejpam-1598	404	2	possible	possible	ADJ
ejpam-1598	404	3	,	,	PUNCT
ejpam-1598	404	4	let	let	VERB
ejpam-1598	404	5	rs′(s	rs′(s	PROPN
ejpam-1598	404	6	′)∩γ(h	′)∩γ(h	NOUN
ejpam-1598	404	7	)	)	PUNCT
ejpam-1598	404	8	6=	6=	ADP
ejpam-1598	404	9	0	0	NUM
ejpam-1598	404	10	and	and	CCONJ
ejpam-1598	404	11	h′	h′	PROPN
ejpam-1598	404	12	(	(	PUNCT
ejpam-1598	404	13	6=	6=	NOUN
ejpam-1598	404	14	0	0	NUM
ejpam-1598	404	15	)	)	PUNCT
ejpam-1598	404	16	∈	∈	PROPN
ejpam-1598	404	17	rs′(s	rs′(s	PROPN
ejpam-1598	404	18	′)∩γ(h	′)∩γ(h	NOUN
ejpam-1598	404	19	)	)	PUNCT
ejpam-1598	404	20	.	.	PUNCT
ejpam-1598	405	1	let	let	VERB
ejpam-1598	405	2	γ(h	γ(h	NOUN
ejpam-1598	405	3	)	)	PUNCT
ejpam-1598	406	1	=	=	SYM
ejpam-1598	406	2	h′	h′	PROPN
ejpam-1598	406	3	,	,	PUNCT
ejpam-1598	406	4	then	then	ADV
ejpam-1598	406	5	h	h	PROPN
ejpam-1598	406	6	6=	6=	PROPN
ejpam-1598	406	7	0	0	X
ejpam-1598	406	8	.	.	PUNCT
ejpam-1598	407	1	now	now	ADV
ejpam-1598	407	2	s′h′	s′h′	VERB
ejpam-1598	407	3	t′	t′	X
ejpam-1598	407	4	=	=	SYM
ejpam-1598	407	5	0	0	NUM
ejpam-1598	407	6	for	for	ADP
ejpam-1598	407	7	all	all	DET
ejpam-1598	407	8	t′	t′	NUM
ejpam-1598	407	9	∈	∈	NOUN
ejpam-1598	407	10	s′	s′	PUNCT
ejpam-1598	407	11	and	and	CCONJ
ejpam-1598	407	12	h′	h′	PROPN
ejpam-1598	407	13	∈	∈	PROPN
ejpam-1598	407	14	γ(h	γ(h	PROPN
ejpam-1598	407	15	)	)	PUNCT
ejpam-1598	407	16	.	.	PUNCT
ejpam-1598	408	1	so	so	ADV
ejpam-1598	408	2	γ(sht	γ(sht	ADP
ejpam-1598	408	3	)	)	PUNCT
ejpam-1598	408	4	=	=	PUNCT
ejpam-1598	409	1	γ(s)γ(h)γ(t	γ(s)γ(h)γ(t	PROPN
ejpam-1598	409	2	)	)	PUNCT
ejpam-1598	409	3	=	=	PUNCT
ejpam-1598	409	4	s′h′	s′h′	VERB
ejpam-1598	409	5	t′	t′	NUM
ejpam-1598	409	6	=	=	SYM
ejpam-1598	409	7	0	0	NUM
ejpam-1598	409	8	for	for	ADP
ejpam-1598	409	9	all	all	DET
ejpam-1598	409	10	t	t	PROPN
ejpam-1598	409	11	∈	∈	PROPN
ejpam-1598	409	12	s.	s.	PROPN
ejpam-1598	409	13	therefore	therefore	ADV
ejpam-1598	409	14	sht	sht	VERB
ejpam-1598	409	15	∈	∈	PROPN
ejpam-1598	409	16	kerγ	kerγ	NOUN
ejpam-1598	409	17	=	=	NOUN
ejpam-1598	409	18	0	0	NUM
ejpam-1598	409	19	for	for	ADP
ejpam-1598	409	20	all	all	DET
ejpam-1598	409	21	t	t	NOUN
ejpam-1598	409	22	∈	∈	PROPN
ejpam-1598	409	23	s.	s.	PROPN
ejpam-1598	410	1	so	so	ADV
ejpam-1598	410	2	sht	sht	VERB
ejpam-1598	410	3	=	=	NOUN
ejpam-1598	410	4	0	0	NUM
ejpam-1598	410	5	for	for	ADP
ejpam-1598	410	6	all	all	DET
ejpam-1598	410	7	t	t	NOUN
ejpam-1598	410	8	∈	∈	PROPN
ejpam-1598	410	9	s.	s.	PROPN
ejpam-1598	410	10	thus	thus	ADV
ejpam-1598	410	11	h	h	PROPN
ejpam-1598	410	12	∈	∈	PROPN
ejpam-1598	410	13	rs(s)∩h	rs(s)∩h	PROPN
ejpam-1598	410	14	,	,	PUNCT
ejpam-1598	410	15	a	a	DET
ejpam-1598	410	16	contradiction	contradiction	NOUN
ejpam-1598	410	17	.	.	PUNCT
ejpam-1598	411	1	therefore	therefore	ADV
ejpam-1598	411	2	rs′(s	rs′(s	PROPN
ejpam-1598	411	3	′)∩	′)∩	NOUN
ejpam-1598	411	4	γ(h	γ(h	NOUN
ejpam-1598	411	5	)	)	PUNCT
ejpam-1598	411	6	=	=	PUNCT
ejpam-1598	412	1	0	0	X
ejpam-1598	412	2	.	.	PUNCT
ejpam-1598	413	1	so	so	ADV
ejpam-1598	413	2	s′	s′	ADJ
ejpam-1598	413	3	6∈	6∈	PROPN
ejpam-1598	413	4	z(s′	z(s′	NUM
ejpam-1598	413	5	)	)	PUNCT
ejpam-1598	413	6	,	,	PUNCT
ejpam-1598	413	7	a	a	DET
ejpam-1598	413	8	contradiction	contradiction	NOUN
ejpam-1598	413	9	.	.	PUNCT
ejpam-1598	414	1	hence	hence	ADV
ejpam-1598	414	2	z(s	z(s	PROPN
ejpam-1598	414	3	)	)	PUNCT
ejpam-1598	415	1	=	=	PUNCT
ejpam-1598	416	1	s.	s.	PROPN
ejpam-1598	416	2	lemma	lemma	PROPN
ejpam-1598	416	3	6	6	NUM
ejpam-1598	416	4	.	.	PUNCT
ejpam-1598	417	1	if	if	SCONJ
ejpam-1598	417	2	γ	γ	X
ejpam-1598	417	3	:	:	PUNCT
ejpam-1598	417	4	s→	s→	PROPN
ejpam-1598	417	5	s′	s′	PROPN
ejpam-1598	417	6	is	be	AUX
ejpam-1598	417	7	a	a	DET
ejpam-1598	417	8	semi	semi	NOUN
ejpam-1598	417	9	-	-	NOUN
ejpam-1598	417	10	isomorphism	isomorphism	ADJ
ejpam-1598	417	11	and	and	CCONJ
ejpam-1598	417	12	z(s′	z(s′	NOUN
ejpam-1598	417	13	)	)	PUNCT
ejpam-1598	417	14	=	=	SYM
ejpam-1598	417	15	0	0	PUNCT
ejpam-1598	417	16	then	then	ADV
ejpam-1598	417	17	z(s	z(s	NUM
ejpam-1598	417	18	)	)	PUNCT
ejpam-1598	418	1	=	=	SYM
ejpam-1598	418	2	0	0	X
ejpam-1598	418	3	.	.	PUNCT
ejpam-1598	419	1	proof	proof	NOUN
ejpam-1598	419	2	.	.	PUNCT
ejpam-1598	420	1	if	if	SCONJ
ejpam-1598	420	2	possible	possible	ADJ
ejpam-1598	420	3	,	,	PUNCT
ejpam-1598	420	4	let	let	VERB
ejpam-1598	420	5	z(s	z(s	NOUN
ejpam-1598	420	6	)	)	PUNCT
ejpam-1598	420	7	6=	6=	ADP
ejpam-1598	420	8	0	0	X
ejpam-1598	420	9	.	.	PUNCT
ejpam-1598	421	1	so	so	ADV
ejpam-1598	421	2	there	there	PRON
ejpam-1598	421	3	exists	exist	VERB
ejpam-1598	421	4	s	s	NOUN
ejpam-1598	421	5	(	(	PUNCT
ejpam-1598	421	6	6=	6=	NOUN
ejpam-1598	421	7	0	0	NUM
ejpam-1598	421	8	)	)	PUNCT
ejpam-1598	421	9	∈	∈	PROPN
ejpam-1598	421	10	s	s	VERB
ejpam-1598	421	11	such	such	ADJ
ejpam-1598	421	12	that	that	DET
ejpam-1598	421	13	s	s	PROPN
ejpam-1598	421	14	∈	∈	PROPN
ejpam-1598	421	15	z(s	z(s	PROPN
ejpam-1598	421	16	)	)	PUNCT
ejpam-1598	421	17	.	.	PUNCT
ejpam-1598	422	1	now	now	ADV
ejpam-1598	422	2	γ(s	γ(	NOUN
ejpam-1598	422	3	)	)	PUNCT
ejpam-1598	423	1	6=	6=	ADP
ejpam-1598	423	2	0	0	NUM
ejpam-1598	423	3	because	because	SCONJ
ejpam-1598	423	4	γ(s	γ(	NOUN
ejpam-1598	423	5	)	)	PUNCT
ejpam-1598	424	1	=	=	SYM
ejpam-1598	425	1	0⇒	0⇒	PROPN
ejpam-1598	425	2	s	s	NOUN
ejpam-1598	425	3	∈	∈	NOUN
ejpam-1598	425	4	kerγ	kerγ	NOUN
ejpam-1598	425	5	=	=	PUNCT
ejpam-1598	425	6	0⇒	0⇒	PROPN
ejpam-1598	425	7	s	s	PART
ejpam-1598	425	8	=	=	NOUN
ejpam-1598	425	9	0	0	PROPN
ejpam-1598	425	10	.	.	PUNCT
ejpam-1598	426	1	since	since	SCONJ
ejpam-1598	426	2	z(s′	z(s′	NUM
ejpam-1598	426	3	)	)	PUNCT
ejpam-1598	426	4	=	=	SYM
ejpam-1598	426	5	0	0	NUM
ejpam-1598	426	6	,	,	PUNCT
ejpam-1598	426	7	γ(s	γ(s	PROPN
ejpam-1598	426	8	)	)	PUNCT
ejpam-1598	426	9	6∈	6∈	PROPN
ejpam-1598	426	10	z(s′	z(s′	PROPN
ejpam-1598	426	11	)	)	PUNCT
ejpam-1598	426	12	.	.	PUNCT
ejpam-1598	427	1	hence	hence	ADV
ejpam-1598	427	2	there	there	PRON
ejpam-1598	427	3	exists	exist	VERB
ejpam-1598	427	4	a	a	DET
ejpam-1598	427	5	nonzero	nonzero	ADJ
ejpam-1598	427	6	right	right	ADJ
ejpam-1598	427	7	ideal	ideal	ADJ
ejpam-1598	427	8	h	h	NOUN
ejpam-1598	427	9	′	′	NOUN
ejpam-1598	427	10	of	of	ADP
ejpam-1598	427	11	s′	s′	ADJ
ejpam-1598	427	12	such	such	ADJ
ejpam-1598	427	13	that	that	SCONJ
ejpam-1598	427	14	rs′(γ(s	rs′(γ(s	PROPN
ejpam-1598	427	15	)	)	PUNCT
ejpam-1598	427	16	)	)	PUNCT
ejpam-1598	427	17	∩	∩	NOUN
ejpam-1598	427	18	h	h	NOUN
ejpam-1598	428	1	′	′	NUM
ejpam-1598	429	1	=	=	NOUN
ejpam-1598	430	1	0	0	X
ejpam-1598	430	2	.	.	PUNCT
ejpam-1598	430	3	let	let	VERB
ejpam-1598	430	4	h	h	NOUN
ejpam-1598	430	5	=	=	PRON
ejpam-1598	430	6	{	{	PUNCT
ejpam-1598	430	7	s	s	NOUN
ejpam-1598	430	8	∈	∈	NOUN
ejpam-1598	430	9	s	s	PART
ejpam-1598	430	10	:	:	PUNCT
ejpam-1598	430	11	γ(s	γ(	NOUN
ejpam-1598	430	12	)	)	PUNCT
ejpam-1598	431	1	∈	∈	PROPN
ejpam-1598	431	2	h	h	NOUN
ejpam-1598	431	3	′	′	NOUN
ejpam-1598	431	4	}	}	PUNCT
ejpam-1598	431	5	.	.	PUNCT
ejpam-1598	432	1	then	then	ADV
ejpam-1598	432	2	h	h	PROPN
ejpam-1598	432	3	is	be	AUX
ejpam-1598	432	4	a	a	DET
ejpam-1598	432	5	nonzero	nonzero	ADJ
ejpam-1598	432	6	right	right	ADJ
ejpam-1598	432	7	ideal	ideal	NOUN
ejpam-1598	432	8	of	of	ADP
ejpam-1598	432	9	s.	s.	PROPN
ejpam-1598	432	10	therefore	therefore	ADV
ejpam-1598	432	11	rs(s	rs(s	PUNCT
ejpam-1598	432	12	)	)	PUNCT
ejpam-1598	432	13	∩	∩	PROPN
ejpam-1598	432	14	h	h	PROPN
ejpam-1598	432	15	6=	6=	PROPN
ejpam-1598	432	16	0	0	NUM
ejpam-1598	432	17	.	.	PUNCT
ejpam-1598	433	1	therefore	therefore	ADV
ejpam-1598	433	2	there	there	PRON
ejpam-1598	433	3	exists	exist	VERB
ejpam-1598	433	4	a	a	DET
ejpam-1598	433	5	nonzero	nonzero	PROPN
ejpam-1598	433	6	element	element	NOUN
ejpam-1598	433	7	h	h	NOUN
ejpam-1598	433	8	in	in	ADP
ejpam-1598	433	9	h	h	NOUN
ejpam-1598	433	10	such	such	ADJ
ejpam-1598	433	11	that	that	DET
ejpam-1598	433	12	sht	sht	NOUN
ejpam-1598	434	1	=	=	NOUN
ejpam-1598	434	2	0	0	NUM
ejpam-1598	434	3	for	for	ADP
ejpam-1598	434	4	all	all	DET
ejpam-1598	434	5	t	t	PROPN
ejpam-1598	434	6	∈	∈	PROPN
ejpam-1598	434	7	s.	s.	PROPN
ejpam-1598	434	8	consequently	consequently	ADV
ejpam-1598	434	9	γ(sht	γ(sht	VERB
ejpam-1598	434	10	)	)	PUNCT
ejpam-1598	434	11	=	=	SYM
ejpam-1598	434	12	γ(0	γ(0	PROPN
ejpam-1598	434	13	)	)	PUNCT
ejpam-1598	434	14	for	for	ADP
ejpam-1598	434	15	all	all	DET
ejpam-1598	434	16	t	t	NOUN
ejpam-1598	434	17	∈	∈	NOUN
ejpam-1598	434	18	s	s	PART
ejpam-1598	434	19	⇒	⇒	NOUN
ejpam-1598	434	20	γ(s)γ(h)γ(t	γ(s)γ(h)γ(t	PUNCT
ejpam-1598	434	21	)	)	PUNCT
ejpam-1598	434	22	=	=	SYM
ejpam-1598	434	23	0	0	NUM
ejpam-1598	434	24	for	for	ADP
ejpam-1598	434	25	all	all	DET
ejpam-1598	434	26	t	t	NOUN
ejpam-1598	434	27	∈	∈	NOUN
ejpam-1598	434	28	s	s	PART
ejpam-1598	434	29	⇒	⇒	NOUN
ejpam-1598	434	30	γ(s)γ(h)t′	γ(s)γ(h)t′	PROPN
ejpam-1598	435	1	=	=	NOUN
ejpam-1598	435	2	0	0	NUM
ejpam-1598	435	3	for	for	ADP
ejpam-1598	435	4	all	all	PRON
ejpam-1598	435	5	t′	t′	NUM
ejpam-1598	435	6	∈	∈	NOUN
ejpam-1598	435	7	s′	s′	PUNCT
ejpam-1598	435	8	as	as	SCONJ
ejpam-1598	435	9	γ	γ	X
ejpam-1598	435	10	is	be	AUX
ejpam-1598	435	11	surjective	surjective	ADJ
ejpam-1598	435	12	⇒	⇒	PROPN
ejpam-1598	435	13	γ(h	γ(h	PROPN
ejpam-1598	435	14	)	)	PUNCT
ejpam-1598	435	15	∈	∈	PROPN
ejpam-1598	436	1	rs′(γ(s	rs′(γ(s	PROPN
ejpam-1598	436	2	)	)	PUNCT
ejpam-1598	436	3	)	)	PUNCT
ejpam-1598	437	1	∩	∩	ADJ
ejpam-1598	437	2	h	h	NOUN
ejpam-1598	437	3	′	′	NUM
ejpam-1598	438	1	=	=	SYM
ejpam-1598	438	2	(	(	PUNCT
ejpam-1598	438	3	0	0	NUM
ejpam-1598	438	4	)	)	PUNCT
ejpam-1598	438	5	,	,	PUNCT
ejpam-1598	438	6	which	which	PRON
ejpam-1598	438	7	implies	imply	VERB
ejpam-1598	438	8	that	that	SCONJ
ejpam-1598	438	9	γ(h	γ(h	NOUN
ejpam-1598	438	10	)	)	PUNCT
ejpam-1598	438	11	=	=	PUNCT
ejpam-1598	438	12	0	0	X
ejpam-1598	438	13	.	.	PUNCT
ejpam-1598	439	1	thus	thus	ADV
ejpam-1598	439	2	h	h	NOUN
ejpam-1598	439	3	∈	∈	NOUN
ejpam-1598	439	4	kerγ	kerγ	NOUN
ejpam-1598	439	5	=	=	PUNCT
ejpam-1598	439	6	0	0	X
ejpam-1598	439	7	.	.	PUNCT
ejpam-1598	440	1	so	so	ADV
ejpam-1598	440	2	h	h	NOUN
ejpam-1598	440	3	=	=	SYM
ejpam-1598	440	4	0	0	PROPN
ejpam-1598	440	5	,	,	PUNCT
ejpam-1598	440	6	a	a	DET
ejpam-1598	440	7	contradiction	contradiction	NOUN
ejpam-1598	440	8	.	.	PUNCT
ejpam-1598	441	1	hence	hence	ADV
ejpam-1598	441	2	z(s	z(s	NUM
ejpam-1598	441	3	)	)	PUNCT
ejpam-1598	442	1	=	=	SYM
ejpam-1598	442	2	0	0	X
ejpam-1598	442	3	.	.	PUNCT
ejpam-1598	443	1	lemma	lemma	PROPN
ejpam-1598	443	2	7	7	X
ejpam-1598	443	3	.	.	PUNCT
ejpam-1598	444	1	if	if	SCONJ
ejpam-1598	444	2	γ	γ	X
ejpam-1598	444	3	:	:	PUNCT
ejpam-1598	444	4	s→	s→	PROPN
ejpam-1598	444	5	s′	s′	PROPN
ejpam-1598	444	6	is	be	AUX
ejpam-1598	444	7	a	a	DET
ejpam-1598	444	8	ternary	ternary	ADJ
ejpam-1598	444	9	semi	semi	NOUN
ejpam-1598	444	10	-	-	ADJ
ejpam-1598	444	11	isomorphism	isomorphism	NOUN
ejpam-1598	444	12	and	and	CCONJ
ejpam-1598	444	13	z(s	z(s	NUM
ejpam-1598	444	14	)	)	PUNCT
ejpam-1598	445	1	=	=	SYM
ejpam-1598	445	2	0	0	PUNCT
ejpam-1598	445	3	then	then	ADV
ejpam-1598	445	4	z(s′	z(s′	VERB
ejpam-1598	445	5	)	)	PUNCT
ejpam-1598	445	6	=	=	SYM
ejpam-1598	446	1	0	0	X
ejpam-1598	446	2	.	.	PUNCT
ejpam-1598	447	1	t.	t.	PROPN
ejpam-1598	447	2	dutta	dutta	PROPN
ejpam-1598	447	3	,	,	PUNCT
ejpam-1598	447	4	k.	k.	PROPN
ejpam-1598	447	5	shum	shum	PROPN
ejpam-1598	447	6	,	,	PUNCT
ejpam-1598	447	7	s.	s.	PROPN
ejpam-1598	447	8	mandal	mandal	PROPN
ejpam-1598	447	9	/	/	SYM
ejpam-1598	447	10	eur	eur	PROPN
ejpam-1598	447	11	.	.	PUNCT
ejpam-1598	448	1	j.	j.	PROPN
ejpam-1598	448	2	pure	pure	PROPN
ejpam-1598	448	3	appl	appl	PROPN
ejpam-1598	448	4	.	.	PROPN
ejpam-1598	448	5	math	math	PROPN
ejpam-1598	448	6	,	,	PUNCT
ejpam-1598	448	7	5	5	NUM
ejpam-1598	448	8	(	(	PUNCT
ejpam-1598	448	9	2012	2012	NUM
ejpam-1598	448	10	)	)	PUNCT
ejpam-1598	448	11	,	,	PUNCT
ejpam-1598	448	12	116	116	NUM
ejpam-1598	448	13	-	-	SYM
ejpam-1598	448	14	128	128	NUM
ejpam-1598	448	15	126	126	NUM
ejpam-1598	448	16	proof	proof	NOUN
ejpam-1598	448	17	.	.	PUNCT
ejpam-1598	449	1	if	if	SCONJ
ejpam-1598	449	2	possible	possible	ADJ
ejpam-1598	449	3	,	,	PUNCT
ejpam-1598	449	4	let	let	VERB
ejpam-1598	449	5	z(s′	z(s′	NUM
ejpam-1598	449	6	)	)	PUNCT
ejpam-1598	449	7	6=	6=	ADP
ejpam-1598	449	8	0	0	NUM
ejpam-1598	449	9	and	and	CCONJ
ejpam-1598	449	10	s′	s′	X
ejpam-1598	449	11	(	(	PUNCT
ejpam-1598	449	12	6=	6=	NOUN
ejpam-1598	449	13	0	0	NUM
ejpam-1598	449	14	)	)	PUNCT
ejpam-1598	449	15	∈	∈	PROPN
ejpam-1598	449	16	z(s′	z(s′	NUM
ejpam-1598	449	17	)	)	PUNCT
ejpam-1598	449	18	.	.	PUNCT
ejpam-1598	450	1	since	since	SCONJ
ejpam-1598	450	2	γ	γ	PROPN
ejpam-1598	450	3	is	be	AUX
ejpam-1598	450	4	surjective	surjective	ADJ
ejpam-1598	450	5	,	,	PUNCT
ejpam-1598	450	6	there	there	PRON
ejpam-1598	450	7	exists	exist	VERB
ejpam-1598	450	8	a	a	DET
ejpam-1598	450	9	nonzero	nonzero	PROPN
ejpam-1598	450	10	element	element	NOUN
ejpam-1598	450	11	s	s	PART
ejpam-1598	450	12	∈	∈	NOUN
ejpam-1598	450	13	s	s	VERB
ejpam-1598	450	14	such	such	ADJ
ejpam-1598	450	15	that	that	SCONJ
ejpam-1598	450	16	γ(s	γ(	NOUN
ejpam-1598	450	17	)	)	PUNCT
ejpam-1598	451	1	=	=	PUNCT
ejpam-1598	451	2	s′.	s′.	PROPN
ejpam-1598	451	3	since	since	SCONJ
ejpam-1598	451	4	z(s	z(s	PROPN
ejpam-1598	451	5	)	)	PUNCT
ejpam-1598	451	6	=	=	PUNCT
ejpam-1598	452	1	0	0	NUM
ejpam-1598	452	2	,	,	PUNCT
ejpam-1598	452	3	so	so	SCONJ
ejpam-1598	452	4	s	s	PROPN
ejpam-1598	452	5	6∈	6∈	PROPN
ejpam-1598	452	6	z(s	z(s	PROPN
ejpam-1598	452	7	)	)	PUNCT
ejpam-1598	452	8	.	.	PUNCT
ejpam-1598	453	1	therefore	therefore	ADV
ejpam-1598	453	2	there	there	PRON
ejpam-1598	453	3	exists	exist	VERB
ejpam-1598	453	4	a	a	DET
ejpam-1598	453	5	nonzero	nonzero	ADJ
ejpam-1598	453	6	right	right	ADJ
ejpam-1598	453	7	ideal	ideal	ADJ
ejpam-1598	453	8	h	h	NOUN
ejpam-1598	453	9	of	of	ADP
ejpam-1598	453	10	s	s	PRON
ejpam-1598	453	11	such	such	ADJ
ejpam-1598	453	12	that	that	SCONJ
ejpam-1598	453	13	rs(s	rs(	VERB
ejpam-1598	453	14	)	)	PUNCT
ejpam-1598	453	15	∩	∩	ADJ
ejpam-1598	453	16	h	h	NOUN
ejpam-1598	453	17	=	=	SYM
ejpam-1598	453	18	0	0	X
ejpam-1598	453	19	.	.	PUNCT
ejpam-1598	454	1	now	now	ADV
ejpam-1598	454	2	γ(h	γ(h	NOUN
ejpam-1598	454	3	)	)	PUNCT
ejpam-1598	454	4	is	be	AUX
ejpam-1598	454	5	a	a	DET
ejpam-1598	454	6	nonzero	nonzero	ADJ
ejpam-1598	454	7	right	right	ADJ
ejpam-1598	454	8	ideal	ideal	NOUN
ejpam-1598	454	9	of	of	ADP
ejpam-1598	454	10	s′.	s′.	PROPN
ejpam-1598	454	11	we	we	PRON
ejpam-1598	454	12	now	now	ADV
ejpam-1598	454	13	prove	prove	VERB
ejpam-1598	454	14	that	that	SCONJ
ejpam-1598	454	15	,	,	PUNCT
ejpam-1598	454	16	rs′(s	rs′(s	PROPN
ejpam-1598	454	17	′	′	NOUN
ejpam-1598	454	18	)	)	PUNCT
ejpam-1598	454	19	∩	∩	ADJ
ejpam-1598	454	20	γ(h	γ(h	NOUN
ejpam-1598	454	21	)	)	PUNCT
ejpam-1598	454	22	=	=	SYM
ejpam-1598	455	1	0	0	X
ejpam-1598	455	2	.	.	PUNCT
ejpam-1598	456	1	if	if	SCONJ
ejpam-1598	456	2	possible	possible	ADJ
ejpam-1598	456	3	,	,	PUNCT
ejpam-1598	456	4	let	let	VERB
ejpam-1598	456	5	rs′(s	rs′(s	PROPN
ejpam-1598	456	6	′	′	NOUN
ejpam-1598	456	7	)	)	PUNCT
ejpam-1598	456	8	∩	∩	ADJ
ejpam-1598	456	9	γ(h	γ(h	NOUN
ejpam-1598	456	10	)	)	PUNCT
ejpam-1598	456	11	6=	6=	ADP
ejpam-1598	456	12	0	0	NUM
ejpam-1598	456	13	and	and	CCONJ
ejpam-1598	456	14	h′	h′	PROPN
ejpam-1598	456	15	(	(	PUNCT
ejpam-1598	456	16	6=	6=	NOUN
ejpam-1598	456	17	0	0	NUM
ejpam-1598	456	18	)	)	PUNCT
ejpam-1598	456	19	∈	∈	PROPN
ejpam-1598	456	20	rs′(s	rs′(s	PROPN
ejpam-1598	456	21	′)∩γ(h	′)∩γ(h	NOUN
ejpam-1598	456	22	)	)	PUNCT
ejpam-1598	456	23	.	.	PUNCT
ejpam-1598	457	1	let	let	VERB
ejpam-1598	457	2	γ(h	γ(h	NOUN
ejpam-1598	457	3	)	)	PUNCT
ejpam-1598	458	1	=	=	SYM
ejpam-1598	458	2	h′	h′	PROPN
ejpam-1598	458	3	,	,	PUNCT
ejpam-1598	458	4	then	then	ADV
ejpam-1598	458	5	h	h	PROPN
ejpam-1598	458	6	6=	6=	PROPN
ejpam-1598	458	7	0	0	X
ejpam-1598	458	8	.	.	PUNCT
ejpam-1598	459	1	now	now	ADV
ejpam-1598	459	2	s′h′	s′h′	VERB
ejpam-1598	459	3	t′	t′	X
ejpam-1598	459	4	=	=	SYM
ejpam-1598	459	5	0	0	NUM
ejpam-1598	459	6	for	for	ADP
ejpam-1598	459	7	all	all	DET
ejpam-1598	459	8	t′	t′	NUM
ejpam-1598	459	9	∈	∈	NOUN
ejpam-1598	459	10	s′	s′	PUNCT
ejpam-1598	459	11	and	and	CCONJ
ejpam-1598	459	12	h′	h′	PROPN
ejpam-1598	459	13	∈	∈	PROPN
ejpam-1598	459	14	γ(h	γ(h	PROPN
ejpam-1598	459	15	)	)	PUNCT
ejpam-1598	459	16	.	.	PUNCT
ejpam-1598	460	1	so	so	ADV
ejpam-1598	460	2	γ(sht	γ(sht	ADP
ejpam-1598	460	3	)	)	PUNCT
ejpam-1598	460	4	=	=	PUNCT
ejpam-1598	461	1	γ(s)γ(h)γ(t	γ(s)γ(h)γ(t	PROPN
ejpam-1598	461	2	)	)	PUNCT
ejpam-1598	461	3	=	=	PUNCT
ejpam-1598	461	4	s′h′	s′h′	VERB
ejpam-1598	461	5	t′	t′	NUM
ejpam-1598	461	6	=	=	SYM
ejpam-1598	461	7	0	0	NUM
ejpam-1598	461	8	for	for	ADP
ejpam-1598	461	9	all	all	DET
ejpam-1598	461	10	t	t	PROPN
ejpam-1598	461	11	∈	∈	PROPN
ejpam-1598	461	12	s.	s.	PROPN
ejpam-1598	461	13	therefore	therefore	ADV
ejpam-1598	461	14	sht	sht	VERB
ejpam-1598	461	15	∈	∈	PROPN
ejpam-1598	461	16	kerγ	kerγ	NOUN
ejpam-1598	461	17	=	=	NOUN
ejpam-1598	461	18	0	0	NUM
ejpam-1598	461	19	for	for	ADP
ejpam-1598	461	20	all	all	DET
ejpam-1598	461	21	t	t	NOUN
ejpam-1598	461	22	∈	∈	PROPN
ejpam-1598	461	23	s.	s.	PROPN
ejpam-1598	462	1	so	so	ADV
ejpam-1598	462	2	sht	sht	VERB
ejpam-1598	462	3	=	=	NOUN
ejpam-1598	462	4	0	0	NUM
ejpam-1598	462	5	for	for	ADP
ejpam-1598	462	6	all	all	DET
ejpam-1598	462	7	t	t	NOUN
ejpam-1598	462	8	∈	∈	PROPN
ejpam-1598	462	9	s.	s.	PROPN
ejpam-1598	462	10	thus	thus	ADV
ejpam-1598	462	11	h	h	PROPN
ejpam-1598	462	12	∈	∈	PROPN
ejpam-1598	462	13	rs(s	rs(s	PUNCT
ejpam-1598	462	14	)	)	PUNCT
ejpam-1598	462	15	∩	∩	ADJ
ejpam-1598	462	16	h	h	NOUN
ejpam-1598	462	17	,	,	PUNCT
ejpam-1598	462	18	a	a	DET
ejpam-1598	462	19	contradiction	contradiction	NOUN
ejpam-1598	462	20	.	.	PUNCT
ejpam-1598	463	1	therefore	therefore	ADV
ejpam-1598	463	2	rs′(s	rs′(s	PROPN
ejpam-1598	463	3	′	′	NOUN
ejpam-1598	463	4	)	)	PUNCT
ejpam-1598	463	5	∩	∩	ADJ
ejpam-1598	463	6	γ(h	γ(h	NOUN
ejpam-1598	463	7	)	)	PUNCT
ejpam-1598	463	8	=	=	SYM
ejpam-1598	464	1	0	0	X
ejpam-1598	464	2	.	.	PUNCT
ejpam-1598	465	1	so	so	ADV
ejpam-1598	465	2	s′	s′	ADJ
ejpam-1598	465	3	6∈	6∈	PROPN
ejpam-1598	465	4	z(s′	z(s′	NUM
ejpam-1598	465	5	)	)	PUNCT
ejpam-1598	465	6	,	,	PUNCT
ejpam-1598	465	7	a	a	DET
ejpam-1598	465	8	contradiction	contradiction	NOUN
ejpam-1598	465	9	.	.	PUNCT
ejpam-1598	466	1	hence	hence	ADV
ejpam-1598	466	2	z(s′	z(s′	NUM
ejpam-1598	466	3	)	)	PUNCT
ejpam-1598	466	4	=	=	SYM
ejpam-1598	467	1	0	0	X
ejpam-1598	467	2	.	.	PUNCT
ejpam-1598	467	3	theorem	theorem	NOUN
ejpam-1598	467	4	2	2	NUM
ejpam-1598	467	5	.	.	PUNCT
ejpam-1598	468	1	if	if	SCONJ
ejpam-1598	468	2	s	s	PRON
ejpam-1598	468	3	and	and	CCONJ
ejpam-1598	468	4	s′	s′	ADJ
ejpam-1598	468	5	be	be	VERB
ejpam-1598	468	6	two	two	NUM
ejpam-1598	468	7	semi	semi	ADJ
ejpam-1598	468	8	-	-	ADJ
ejpam-1598	468	9	isomorphic	isomorphic	ADJ
ejpam-1598	468	10	ternary	ternary	ADJ
ejpam-1598	468	11	semirings	semiring	NOUN
ejpam-1598	468	12	.	.	PUNCT
ejpam-1598	469	1	then	then	ADV
ejpam-1598	469	2	s	s	VERB
ejpam-1598	469	3	is	be	AUX
ejpam-1598	469	4	singular	singular	ADJ
ejpam-1598	469	5	(	(	PUNCT
ejpam-1598	469	6	nonsingular	nonsingular	ADJ
ejpam-1598	469	7	)	)	PUNCT
ejpam-1598	469	8	iff	iff	PROPN
ejpam-1598	469	9	s′	s′	PROPN
ejpam-1598	469	10	is	be	AUX
ejpam-1598	469	11	singular	singular	ADJ
ejpam-1598	469	12	(	(	PUNCT
ejpam-1598	469	13	resp	resp	NOUN
ejpam-1598	469	14	.	.	PUNCT
ejpam-1598	470	1	nonsingular	nonsingular	ADJ
ejpam-1598	470	2	)	)	PUNCT
ejpam-1598	470	3	.	.	PUNCT
ejpam-1598	471	1	proposition	proposition	NOUN
ejpam-1598	471	2	13	13	NUM
ejpam-1598	471	3	.	.	PUNCT
ejpam-1598	472	1	if	if	SCONJ
ejpam-1598	472	2	i	i	PRON
ejpam-1598	472	3	is	be	AUX
ejpam-1598	472	4	an	an	DET
ejpam-1598	472	5	ideal	ideal	NOUN
ejpam-1598	472	6	of	of	ADP
ejpam-1598	472	7	a	a	DET
ejpam-1598	472	8	ternary	ternary	ADJ
ejpam-1598	472	9	semiring	semiring	NOUN
ejpam-1598	472	10	s	s	X
ejpam-1598	472	11	and	and	CCONJ
ejpam-1598	472	12	as	as	ADP
ejpam-1598	472	13	a	a	DET
ejpam-1598	472	14	ternary	ternary	ADJ
ejpam-1598	472	15	semiring	semiring	NOUN
ejpam-1598	472	16	,	,	PUNCT
ejpam-1598	472	17	i	i	PRON
ejpam-1598	472	18	is	be	AUX
ejpam-1598	472	19	semiprime	semiprime	NOUN
ejpam-1598	472	20	,	,	PUNCT
ejpam-1598	472	21	then	then	ADV
ejpam-1598	472	22	z(i	z(i	NUM
ejpam-1598	472	23	)	)	PUNCT
ejpam-1598	473	1	=	=	PUNCT
ejpam-1598	473	2	i	i	PRON
ejpam-1598	473	3	∩	∩	ADJ
ejpam-1598	473	4	z(s	z(s	PROPN
ejpam-1598	473	5	)	)	PUNCT
ejpam-1598	473	6	.	.	PUNCT
ejpam-1598	474	1	proof	proof	NOUN
ejpam-1598	474	2	.	.	PUNCT
ejpam-1598	475	1	let	let	VERB
ejpam-1598	475	2	x	x	PUNCT
ejpam-1598	475	3	∈	∈	PROPN
ejpam-1598	475	4	z(i	z(i	PROPN
ejpam-1598	475	5	)	)	PUNCT
ejpam-1598	475	6	and	and	CCONJ
ejpam-1598	475	7	p	p	NOUN
ejpam-1598	475	8	be	be	AUX
ejpam-1598	475	9	a	a	DET
ejpam-1598	475	10	nonzero	nonzero	ADJ
ejpam-1598	475	11	right	right	ADJ
ejpam-1598	475	12	ideal	ideal	NOUN
ejpam-1598	475	13	of	of	ADP
ejpam-1598	475	14	s.	s.	PROPN
ejpam-1598	475	15	if	if	SCONJ
ejpam-1598	475	16	psi	psi	NOUN
ejpam-1598	475	17	=	=	SYM
ejpam-1598	475	18	0	0	NUM
ejpam-1598	475	19	,	,	PUNCT
ejpam-1598	475	20	then	then	ADV
ejpam-1598	475	21	(	(	PUNCT
ejpam-1598	475	22	i	i	PRON
ejpam-1598	475	23	ps)3	ps)3	VERB
ejpam-1598	475	24	=	=	PRON
ejpam-1598	475	25	(	(	PUNCT
ejpam-1598	475	26	i	i	PRON
ejpam-1598	475	27	ps)(i	ps)(i	PROPN
ejpam-1598	475	28	ps)(i	ps)(i	PROPN
ejpam-1598	475	29	ps	ps	NOUN
ejpam-1598	475	30	)	)	PUNCT
ejpam-1598	475	31	=	=	SYM
ejpam-1598	475	32	i(psi)(psi)ps	i(psi)(psi)ps	NOUN
ejpam-1598	475	33	=	=	SYM
ejpam-1598	475	34	0	0	X
ejpam-1598	475	35	.	.	PUNCT
ejpam-1598	476	1	now	now	ADV
ejpam-1598	476	2	,	,	PUNCT
ejpam-1598	476	3	i	i	PRON
ejpam-1598	476	4	is	be	AUX
ejpam-1598	476	5	a	a	DET
ejpam-1598	476	6	semiprime	semiprime	NOUN
ejpam-1598	476	7	ternary	ternary	ADJ
ejpam-1598	476	8	semiring	semiring	NOUN
ejpam-1598	477	1	and	and	CCONJ
ejpam-1598	477	2	i	i	PRON
ejpam-1598	477	3	ps	ps	VERB
ejpam-1598	477	4	is	be	AUX
ejpam-1598	477	5	an	an	DET
ejpam-1598	477	6	ideal	ideal	NOUN
ejpam-1598	477	7	of	of	ADP
ejpam-1598	477	8	i	i	PRON
ejpam-1598	477	9	.	.	PUNCT
ejpam-1598	478	1	and	and	CCONJ
ejpam-1598	478	2	so	so	ADV
ejpam-1598	478	3	,	,	PUNCT
ejpam-1598	478	4	i	i	PRON
ejpam-1598	478	5	ps	ps	VERB
ejpam-1598	478	6	=	=	SYM
ejpam-1598	478	7	0	0	PROPN
ejpam-1598	478	8	.	.	PUNCT
ejpam-1598	479	1	consequently	consequently	ADV
ejpam-1598	479	2	p	p	ADP
ejpam-1598	479	3	⊆	⊆	NUM
ejpam-1598	479	4	rs(i	rs(i	NUM
ejpam-1598	479	5	)	)	PUNCT
ejpam-1598	479	6	⊆	⊆	NUM
ejpam-1598	479	7	rs(x	rs(x	X
ejpam-1598	479	8	)	)	PUNCT
ejpam-1598	479	9	,	,	PUNCT
ejpam-1598	479	10	since	since	SCONJ
ejpam-1598	479	11	x	x	PROPN
ejpam-1598	479	12	∈	∈	PROPN
ejpam-1598	479	13	z(i)⊆	z(i)⊆	PROPN
ejpam-1598	479	14	i	i	INTJ
ejpam-1598	479	15	.	.	PUNCT
ejpam-1598	480	1	so	so	ADV
ejpam-1598	480	2	p	p	X
ejpam-1598	480	3	∩	∩	NOUN
ejpam-1598	480	4	rs(x	rs(x	X
ejpam-1598	480	5	)	)	PUNCT
ejpam-1598	481	1	=	=	PUNCT
ejpam-1598	482	1	p	p	NOUN
ejpam-1598	482	2	6=	6=	ADP
ejpam-1598	482	3	0	0	NUM
ejpam-1598	482	4	.	.	PUNCT
ejpam-1598	483	1	if	if	SCONJ
ejpam-1598	483	2	psi	psi	ADJ
ejpam-1598	483	3	6=	6=	PROPN
ejpam-1598	483	4	0	0	NUM
ejpam-1598	483	5	,	,	PUNCT
ejpam-1598	483	6	then	then	ADV
ejpam-1598	483	7	psi	psi	NOUN
ejpam-1598	483	8	is	be	AUX
ejpam-1598	483	9	a	a	DET
ejpam-1598	483	10	nonzero	nonzero	ADJ
ejpam-1598	483	11	right	right	ADJ
ejpam-1598	483	12	ideal	ideal	NOUN
ejpam-1598	483	13	of	of	ADP
ejpam-1598	483	14	i	i	PRON
ejpam-1598	483	15	.	.	PUNCT
ejpam-1598	484	1	so	so	ADV
ejpam-1598	484	2	ri(x)∩	ri(x)∩	X
ejpam-1598	484	3	psi	psi	PROPN
ejpam-1598	484	4	6=	6=	PROPN
ejpam-1598	484	5	0	0	X
ejpam-1598	484	6	.	.	PUNCT
ejpam-1598	485	1	let	let	VERB
ejpam-1598	485	2	0	0	NUM
ejpam-1598	485	3	6=	6=	ADP
ejpam-1598	485	4	a	a	DET
ejpam-1598	485	5	∈	∈	NOUN
ejpam-1598	485	6	ri(x	ri(x	NUM
ejpam-1598	485	7	)	)	PUNCT
ejpam-1598	485	8	∩	∩	ADJ
ejpam-1598	485	9	psi	psi	NOUN
ejpam-1598	485	10	.	.	PUNCT
ejpam-1598	486	1	hence	hence	ADV
ejpam-1598	486	2	,	,	PUNCT
ejpam-1598	486	3	xai	xai	PROPN
ejpam-1598	486	4	=	=	PROPN
ejpam-1598	486	5	0	0	PROPN
ejpam-1598	486	6	for	for	ADP
ejpam-1598	486	7	all	all	PRON
ejpam-1598	486	8	i	i	PRON
ejpam-1598	486	9	∈	∈	ADV
ejpam-1598	486	10	i	i	PRON
ejpam-1598	486	11	⇒	⇒	VERB
ejpam-1598	486	12	xaii1s	xaii1s	X
ejpam-1598	487	1	=	=	SYM
ejpam-1598	487	2	0	0	NUM
ejpam-1598	487	3	for	for	ADP
ejpam-1598	487	4	all	all	DET
ejpam-1598	487	5	s	s	PART
ejpam-1598	487	6	∈	∈	NOUN
ejpam-1598	487	7	s	s	NOUN
ejpam-1598	487	8	and	and	CCONJ
ejpam-1598	487	9	for	for	ADP
ejpam-1598	487	10	all	all	DET
ejpam-1598	487	11	i	i	PROPN
ejpam-1598	487	12	,	,	PUNCT
ejpam-1598	488	1	i1	i1	PROPN
ejpam-1598	488	2	∈	∈	PROPN
ejpam-1598	489	1	i	i	PRON
ejpam-1598	489	2	.	.	PUNCT
ejpam-1598	490	1	now	now	ADV
ejpam-1598	490	2	if	if	SCONJ
ejpam-1598	490	3	aii1	aii1	NOUN
ejpam-1598	490	4	=	=	NOUN
ejpam-1598	490	5	0	0	NUM
ejpam-1598	490	6	for	for	ADP
ejpam-1598	490	7	all	all	DET
ejpam-1598	490	8	i	i	PROPN
ejpam-1598	490	9	,	,	PUNCT
ejpam-1598	490	10	i1	i1	PROPN
ejpam-1598	490	11	∈	∈	PROPN
ejpam-1598	491	1	i	i	PRON
ejpam-1598	491	2	,	,	PUNCT
ejpam-1598	491	3	then	then	ADV
ejpam-1598	491	4	ai	ai	VERB
ejpam-1598	491	5	i	i	PROPN
ejpam-1598	491	6	=	=	NOUN
ejpam-1598	491	7	0	0	NUM
ejpam-1598	491	8	,	,	PUNCT
ejpam-1598	491	9	aia	aia	X
ejpam-1598	491	10	=	=	SYM
ejpam-1598	491	11	0	0	PROPN
ejpam-1598	491	12	and	and	CCONJ
ejpam-1598	491	13	a3	a3	NOUN
ejpam-1598	491	14	=	=	SYM
ejpam-1598	491	15	0	0	PUNCT
ejpam-1598	492	1	as	as	ADP
ejpam-1598	492	2	a	a	DET
ejpam-1598	492	3	∈	∈	NOUN
ejpam-1598	492	4	i	i	PRON
ejpam-1598	492	5	.	.	PUNCT
ejpam-1598	493	1	now	now	ADV
ejpam-1598	493	2	〈	〈	X
ejpam-1598	493	3	a〉3	a〉3	NOUN
ejpam-1598	493	4	=	=	SYM
ejpam-1598	493	5	(	(	PUNCT
ejpam-1598	493	6	ai	ai	VERB
ejpam-1598	493	7	i	i	PRON
ejpam-1598	494	1	+	+	NUM
ejpam-1598	494	2	i	i	PROPN
ejpam-1598	494	3	ia	ia	PROPN
ejpam-1598	495	1	+	+	CCONJ
ejpam-1598	495	2	iai	iai	VERB
ejpam-1598	496	1	+	+	CCONJ
ejpam-1598	496	2	i	i	PRON
ejpam-1598	496	3	iai	iai	VERB
ejpam-1598	496	4	i	i	NOUN
ejpam-1598	496	5	+	+	CCONJ
ejpam-1598	496	6	na)3	na)3	ADJ
ejpam-1598	496	7	=	=	SYM
ejpam-1598	496	8	(	(	PUNCT
ejpam-1598	496	9	0)⇒	0)⇒	X
ejpam-1598	496	10	a	a	DET
ejpam-1598	496	11	=	=	NOUN
ejpam-1598	496	12	0	0	PUNCT
ejpam-1598	496	13	as	as	SCONJ
ejpam-1598	496	14	i	i	PRON
ejpam-1598	496	15	is	be	AUX
ejpam-1598	496	16	semiprime	semiprime	ADJ
ejpam-1598	496	17	.	.	PUNCT
ejpam-1598	497	1	this	this	PRON
ejpam-1598	497	2	arrives	arrive	VERB
ejpam-1598	497	3	a	a	DET
ejpam-1598	497	4	contradiction	contradiction	NOUN
ejpam-1598	497	5	as	as	ADP
ejpam-1598	497	6	a	a	DET
ejpam-1598	497	7	6=	6=	NUM
ejpam-1598	497	8	0	0	NUM
ejpam-1598	497	9	.	.	PUNCT
ejpam-1598	498	1	thus	thus	ADV
ejpam-1598	498	2	,	,	PUNCT
ejpam-1598	498	3	aii1	aii1	NOUN
ejpam-1598	498	4	6=	6=	ADP
ejpam-1598	498	5	0	0	NUM
ejpam-1598	498	6	for	for	ADP
ejpam-1598	498	7	some	some	DET
ejpam-1598	498	8	i	i	PROPN
ejpam-1598	498	9	,	,	PUNCT
ejpam-1598	498	10	i1	i1	PROPN
ejpam-1598	498	11	∈	∈	PROPN
ejpam-1598	498	12	i	i	PRON
ejpam-1598	498	13	⇒	⇒	VERB
ejpam-1598	498	14	0	0	NUM
ejpam-1598	498	15	6=	6=	ADP
ejpam-1598	498	16	aii1	aii1	NOUN
ejpam-1598	498	17	∈	∈	PROPN
ejpam-1598	498	18	rs(x	rs(x	X
ejpam-1598	498	19	)	)	PUNCT
ejpam-1598	498	20	for	for	ADP
ejpam-1598	498	21	some	some	DET
ejpam-1598	498	22	i	i	PROPN
ejpam-1598	498	23	,	,	PUNCT
ejpam-1598	498	24	i1	i1	PROPN
ejpam-1598	498	25	∈	∈	PROPN
ejpam-1598	499	1	i	i	PRON
ejpam-1598	499	2	.	.	PUNCT
ejpam-1598	500	1	also	also	ADV
ejpam-1598	500	2	aii1	aii1	PROPN
ejpam-1598	500	3	∈	∈	PROPN
ejpam-1598	500	4	psi	psi	VERB
ejpam-1598	500	5	⊆	⊆	NUM
ejpam-1598	500	6	p	p	NOUN
ejpam-1598	500	7	for	for	ADP
ejpam-1598	500	8	all	all	DET
ejpam-1598	500	9	i	i	PROPN
ejpam-1598	500	10	,	,	PUNCT
ejpam-1598	500	11	i1	i1	PROPN
ejpam-1598	500	12	∈	∈	PROPN
ejpam-1598	501	1	i	i	PRON
ejpam-1598	501	2	.	.	PUNCT
ejpam-1598	502	1	hence	hence	ADV
ejpam-1598	502	2	,	,	PUNCT
ejpam-1598	502	3	0	0	PUNCT
ejpam-1598	502	4	6=	6=	ADP
ejpam-1598	502	5	aii1	aii1	NOUN
ejpam-1598	502	6	∈	∈	NOUN
ejpam-1598	502	7	rs(x)∩	rs(x)∩	X
ejpam-1598	503	1	p.	p.	NOUN
ejpam-1598	503	2	this	this	PRON
ejpam-1598	503	3	leads	lead	VERB
ejpam-1598	503	4	to	to	ADP
ejpam-1598	503	5	rs(x)∩	rs(x)∩	X
ejpam-1598	503	6	p	p	NOUN
ejpam-1598	503	7	6=	6=	PROPN
ejpam-1598	503	8	(	(	PUNCT
ejpam-1598	503	9	0	0	NUM
ejpam-1598	503	10	)	)	PUNCT
ejpam-1598	503	11	.	.	PUNCT
ejpam-1598	504	1	thus	thus	ADV
ejpam-1598	504	2	,	,	PUNCT
ejpam-1598	504	3	in	in	ADP
ejpam-1598	504	4	any	any	DET
ejpam-1598	504	5	case	case	NOUN
ejpam-1598	504	6	,	,	PUNCT
ejpam-1598	504	7	rs(x	rs(x	X
ejpam-1598	504	8	)	)	PUNCT
ejpam-1598	504	9	is	be	AUX
ejpam-1598	504	10	an	an	DET
ejpam-1598	504	11	essential	essential	ADJ
ejpam-1598	504	12	right	right	ADJ
ejpam-1598	504	13	ideal	ideal	NOUN
ejpam-1598	504	14	of	of	ADP
ejpam-1598	504	15	s.	s.	PROPN
ejpam-1598	504	16	hence	hence	ADV
ejpam-1598	504	17	,	,	PUNCT
ejpam-1598	504	18	x	x	PROPN
ejpam-1598	504	19	∈	∈	PROPN
ejpam-1598	504	20	z(s	z(s	PROPN
ejpam-1598	504	21	)	)	PUNCT
ejpam-1598	504	22	.	.	PUNCT
ejpam-1598	505	1	also	also	ADV
ejpam-1598	505	2	,	,	PUNCT
ejpam-1598	505	3	x	x	PUNCT
ejpam-1598	505	4	∈	∈	PROPN
ejpam-1598	506	1	i	i	PRON
ejpam-1598	506	2	.	.	PUNCT
ejpam-1598	507	1	this	this	PRON
ejpam-1598	507	2	shows	show	VERB
ejpam-1598	507	3	that	that	SCONJ
ejpam-1598	507	4	x	x	SYM
ejpam-1598	507	5	∈	∈	PROPN
ejpam-1598	507	6	i	i	PRON
ejpam-1598	507	7	∩	∩	ADJ
ejpam-1598	507	8	z(s	z(s	PROPN
ejpam-1598	507	9	)	)	PUNCT
ejpam-1598	507	10	.	.	PUNCT
ejpam-1598	508	1	thus	thus	ADV
ejpam-1598	508	2	,	,	PUNCT
ejpam-1598	508	3	z(i	z(i	NUM
ejpam-1598	508	4	)	)	PUNCT
ejpam-1598	508	5	⊆	⊆	NUM
ejpam-1598	508	6	i	i	PROPN
ejpam-1598	508	7	∩	∩	ADJ
ejpam-1598	508	8	z(s	z(s	PROPN
ejpam-1598	508	9	)	)	PUNCT
ejpam-1598	508	10	.	.	PUNCT
ejpam-1598	509	1	conversely	conversely	ADV
ejpam-1598	509	2	,	,	PUNCT
ejpam-1598	509	3	let	let	VERB
ejpam-1598	509	4	x	x	SYM
ejpam-1598	509	5	∈	∈	PROPN
ejpam-1598	509	6	i	i	PROPN
ejpam-1598	509	7	∩z(s	∩z(s	PROPN
ejpam-1598	509	8	)	)	PUNCT
ejpam-1598	509	9	and	and	CCONJ
ejpam-1598	509	10	p	p	NOUN
ejpam-1598	509	11	be	be	AUX
ejpam-1598	509	12	a	a	DET
ejpam-1598	509	13	nonzero	nonzero	ADJ
ejpam-1598	509	14	right	right	ADJ
ejpam-1598	509	15	ideal	ideal	NOUN
ejpam-1598	509	16	of	of	ADP
ejpam-1598	509	17	i	i	PRON
ejpam-1598	509	18	.	.	PUNCT
ejpam-1598	510	1	since	since	SCONJ
ejpam-1598	510	2	the	the	DET
ejpam-1598	510	3	ternary	ternary	ADJ
ejpam-1598	510	4	semiring	semiring	NOUN
ejpam-1598	510	5	i	i	PRON
ejpam-1598	510	6	is	be	AUX
ejpam-1598	510	7	semiprime	semiprime	NOUN
ejpam-1598	510	8	,	,	PUNCT
ejpam-1598	510	9	pi	pi	NOUN
ejpam-1598	510	10	i	i	PRON
ejpam-1598	510	11	6=	6=	NUM
ejpam-1598	510	12	0	0	PUNCT
ejpam-1598	510	13	as	as	ADP
ejpam-1598	510	14	pi	pi	NOUN
ejpam-1598	510	15	i	i	NOUN
ejpam-1598	510	16	=	=	PUNCT
ejpam-1598	510	17	(	(	PUNCT
ejpam-1598	510	18	0	0	X
ejpam-1598	510	19	)	)	PUNCT
ejpam-1598	510	20	⇒	⇒	NOUN
ejpam-1598	510	21	p3	p3	PROPN
ejpam-1598	510	22	⊆	⊆	NUM
ejpam-1598	510	23	pi	pi	NOUN
ejpam-1598	510	24	i	i	NOUN
ejpam-1598	510	25	=	=	PUNCT
ejpam-1598	510	26	(	(	PUNCT
ejpam-1598	510	27	0)⇒	0)⇒	X
ejpam-1598	510	28	p	p	NOUN
ejpam-1598	510	29	=	=	X
ejpam-1598	510	30	(	(	PUNCT
ejpam-1598	510	31	0	0	NUM
ejpam-1598	510	32	)	)	PUNCT
ejpam-1598	510	33	,	,	PUNCT
ejpam-1598	510	34	a	a	DET
ejpam-1598	510	35	contradiction	contradiction	NOUN
ejpam-1598	510	36	.	.	PUNCT
ejpam-1598	511	1	thus	thus	ADV
ejpam-1598	511	2	,	,	PUNCT
ejpam-1598	511	3	pi	pi	NOUN
ejpam-1598	511	4	i	i	PRON
ejpam-1598	511	5	is	be	AUX
ejpam-1598	511	6	a	a	DET
ejpam-1598	511	7	nonzero	nonzero	ADJ
ejpam-1598	511	8	right	right	ADJ
ejpam-1598	511	9	ideal	ideal	NOUN
ejpam-1598	511	10	of	of	ADP
ejpam-1598	511	11	s.	s.	PROPN
ejpam-1598	511	12	hence	hence	ADV
ejpam-1598	511	13	,	,	PUNCT
ejpam-1598	511	14	rs(x)∩	rs(x)∩	X
ejpam-1598	511	15	pi	pi	NOUN
ejpam-1598	511	16	i	i	PRON
ejpam-1598	511	17	6=	6=	PROPN
ejpam-1598	511	18	(	(	PUNCT
ejpam-1598	511	19	0	0	NUM
ejpam-1598	511	20	)	)	PUNCT
ejpam-1598	511	21	.	.	PUNCT
ejpam-1598	512	1	however	however	ADV
ejpam-1598	512	2	,	,	PUNCT
ejpam-1598	512	3	pi	pi	NOUN
ejpam-1598	512	4	i	i	NOUN
ejpam-1598	512	5	⊆	⊆	NUM
ejpam-1598	512	6	p	p	NOUN
ejpam-1598	512	7	and	and	CCONJ
ejpam-1598	512	8	whence	whence	NOUN
ejpam-1598	512	9	,	,	PUNCT
ejpam-1598	512	10	rs(x)∩	rs(x)∩	X
ejpam-1598	512	11	p	p	NOUN
ejpam-1598	512	12	6=	6=	PROPN
ejpam-1598	512	13	(	(	PUNCT
ejpam-1598	512	14	0	0	NUM
ejpam-1598	512	15	)	)	PUNCT
ejpam-1598	512	16	.	.	PUNCT
ejpam-1598	513	1	consequently	consequently	ADV
ejpam-1598	513	2	,	,	PUNCT
ejpam-1598	513	3	we	we	PRON
ejpam-1598	513	4	have	have	VERB
ejpam-1598	513	5	0	0	NUM
ejpam-1598	514	1	6=	6=	ADP
ejpam-1598	514	2	rs(x)∩	rs(x)∩	X
ejpam-1598	514	3	p	p	ADP
ejpam-1598	514	4	⊆	⊆	NUM
ejpam-1598	514	5	ri	ri	X
ejpam-1598	514	6	(	(	PUNCT
ejpam-1598	514	7	x)∩	x)∩	PROPN
ejpam-1598	514	8	p.	p.	PROPN
ejpam-1598	514	9	thus	thus	ADV
ejpam-1598	514	10	,	,	PUNCT
ejpam-1598	514	11	x	x	PROPN
ejpam-1598	514	12	∈	∈	PROPN
ejpam-1598	514	13	z(i	z(i	NUM
ejpam-1598	514	14	)	)	PUNCT
ejpam-1598	514	15	.	.	PUNCT
ejpam-1598	515	1	hence	hence	ADV
ejpam-1598	515	2	,	,	PUNCT
ejpam-1598	515	3	i	i	PRON
ejpam-1598	515	4	∩	∩	VERB
ejpam-1598	515	5	z(s)⊆	z(s)⊆	PROPN
ejpam-1598	515	6	z(i	z(i	PROPN
ejpam-1598	515	7	)	)	PUNCT
ejpam-1598	515	8	.	.	PUNCT
ejpam-1598	516	1	thus	thus	ADV
ejpam-1598	516	2	,	,	PUNCT
ejpam-1598	516	3	z(i	z(i	NUM
ejpam-1598	516	4	)	)	PUNCT
ejpam-1598	516	5	=	=	PUNCT
ejpam-1598	516	6	i	i	PRON
ejpam-1598	516	7	∩	∩	ADJ
ejpam-1598	516	8	z(s	z(s	PROPN
ejpam-1598	516	9	)	)	PUNCT
ejpam-1598	516	10	.	.	PUNCT
ejpam-1598	517	1	theorem	theorem	NOUN
ejpam-1598	517	2	3	3	NUM
ejpam-1598	517	3	.	.	PUNCT
ejpam-1598	518	1	the	the	DET
ejpam-1598	518	2	class	class	NOUN
ejpam-1598	518	3	of	of	ADP
ejpam-1598	518	4	semiprime	semiprime	NOUN
ejpam-1598	518	5	non	non	ADJ
ejpam-1598	518	6	-	-	ADJ
ejpam-1598	518	7	singular(singular	singular(singular	ADJ
ejpam-1598	518	8	)	)	PUNCT
ejpam-1598	518	9	ternary	ternary	ADJ
ejpam-1598	518	10	semirings	semiring	NOUN
ejpam-1598	518	11	is	be	AUX
ejpam-1598	518	12	hereditary	hereditary	ADJ
ejpam-1598	518	13	.	.	PUNCT
ejpam-1598	519	1	proof	proof	NOUN
ejpam-1598	519	2	.	.	PUNCT
ejpam-1598	520	1	the	the	DET
ejpam-1598	520	2	proof	proof	NOUN
ejpam-1598	520	3	follows	follow	VERB
ejpam-1598	520	4	from	from	ADP
ejpam-1598	520	5	proposition	proposition	NOUN
ejpam-1598	520	6	13	13	NUM
ejpam-1598	520	7	.	.	PUNCT
ejpam-1598	521	1	corollary	corollary	ADJ
ejpam-1598	521	2	1	1	NUM
ejpam-1598	521	3	.	.	PUNCT
ejpam-1598	522	1	the	the	DET
ejpam-1598	522	2	class	class	NOUN
ejpam-1598	522	3	of	of	ADP
ejpam-1598	522	4	prime	prime	ADJ
ejpam-1598	522	5	non	non	ADJ
ejpam-1598	522	6	-	-	ADJ
ejpam-1598	522	7	singular	singular	ADJ
ejpam-1598	522	8	(	(	PUNCT
ejpam-1598	522	9	singular	singular	NOUN
ejpam-1598	522	10	)	)	PUNCT
ejpam-1598	522	11	ternary	ternary	ADJ
ejpam-1598	522	12	semirings	semiring	NOUN
ejpam-1598	522	13	is	be	AUX
ejpam-1598	522	14	hereditary	hereditary	ADJ
ejpam-1598	522	15	.	.	PUNCT
ejpam-1598	523	1	proof	proof	NOUN
ejpam-1598	523	2	.	.	PUNCT
ejpam-1598	524	1	the	the	DET
ejpam-1598	524	2	proof	proof	NOUN
ejpam-1598	524	3	of	of	ADP
ejpam-1598	524	4	this	this	DET
ejpam-1598	524	5	corollary	corollary	NOUN
ejpam-1598	524	6	is	be	AUX
ejpam-1598	524	7	an	an	DET
ejpam-1598	524	8	immediate	immediate	ADJ
ejpam-1598	524	9	consequence	consequence	NOUN
ejpam-1598	524	10	of	of	ADP
ejpam-1598	524	11	theorem	theorem	NOUN
ejpam-1598	524	12	3	3	X
ejpam-1598	524	13	.	.	PUNCT
ejpam-1598	525	1	finally	finally	ADV
ejpam-1598	525	2	,	,	PUNCT
ejpam-1598	525	3	we	we	PRON
ejpam-1598	525	4	state	state	VERB
ejpam-1598	525	5	the	the	DET
ejpam-1598	525	6	following	follow	VERB
ejpam-1598	525	7	propositions	proposition	NOUN
ejpam-1598	525	8	:	:	PUNCT
ejpam-1598	525	9	proposition	proposition	NOUN
ejpam-1598	525	10	14	14	NUM
ejpam-1598	525	11	.	.	PUNCT
ejpam-1598	526	1	[	[	X
ejpam-1598	526	2	7	7	X
ejpam-1598	526	3	]	]	X
ejpam-1598	526	4	let	let	VERB
ejpam-1598	526	5	s	s	PRON
ejpam-1598	526	6	be	be	AUX
ejpam-1598	526	7	a	a	DET
ejpam-1598	526	8	ternary	ternary	ADJ
ejpam-1598	526	9	semiring	semiring	NOUN
ejpam-1598	526	10	.	.	PUNCT
ejpam-1598	527	1	if	if	SCONJ
ejpam-1598	527	2	q	q	NOUN
ejpam-1598	527	3	is	be	AUX
ejpam-1598	527	4	a	a	DET
ejpam-1598	527	5	semiprime	semiprime	NOUN
ejpam-1598	527	6	ideal	ideal	NOUN
ejpam-1598	527	7	of	of	ADP
ejpam-1598	527	8	s	s	PRON
ejpam-1598	528	1	and	and	CCONJ
ejpam-1598	528	2	i	i	PRON
ejpam-1598	528	3	is	be	AUX
ejpam-1598	528	4	an	an	DET
ejpam-1598	528	5	ideal	ideal	NOUN
ejpam-1598	528	6	of	of	ADP
ejpam-1598	528	7	s	s	PRON
ejpam-1598	528	8	then	then	ADV
ejpam-1598	528	9	q	q	PROPN
ejpam-1598	528	10	∩	∩	NOUN
ejpam-1598	528	11	i	i	PRON
ejpam-1598	528	12	is	be	AUX
ejpam-1598	528	13	a	a	DET
ejpam-1598	528	14	semiprime	semiprime	NOUN
ejpam-1598	528	15	ideal	ideal	NOUN
ejpam-1598	528	16	of	of	ADP
ejpam-1598	528	17	i.	i.	PROPN
ejpam-1598	528	18	in	in	ADP
ejpam-1598	528	19	closing	close	VERB
ejpam-1598	528	20	this	this	DET
ejpam-1598	528	21	paper	paper	NOUN
ejpam-1598	528	22	,	,	PUNCT
ejpam-1598	528	23	we	we	PRON
ejpam-1598	528	24	give	give	VERB
ejpam-1598	528	25	the	the	DET
ejpam-1598	528	26	following	follow	VERB
ejpam-1598	528	27	theorem	theorem	NOUN
ejpam-1598	528	28	of	of	ADP
ejpam-1598	528	29	semiprime	semiprime	NOUN
ejpam-1598	528	30	non	non	ADJ
ejpam-1598	528	31	-	-	ADJ
ejpam-1598	528	32	singular	singular	ADJ
ejpam-1598	528	33	ternary	ternary	ADJ
ejpam-1598	528	34	semirings	semiring	NOUN
ejpam-1598	528	35	.	.	PUNCT
ejpam-1598	529	1	references	reference	NOUN
ejpam-1598	529	2	127	127	NUM
ejpam-1598	529	3	theorem	theorem	NOUN
ejpam-1598	529	4	4	4	NUM
ejpam-1598	529	5	.	.	PUNCT
ejpam-1598	530	1	the	the	DET
ejpam-1598	530	2	class	class	NOUN
ejpam-1598	530	3	of	of	ADP
ejpam-1598	530	4	homomorphic	homomorphic	ADJ
ejpam-1598	530	5	images	image	NOUN
ejpam-1598	530	6	of	of	ADP
ejpam-1598	530	7	semiprime	semiprime	NOUN
ejpam-1598	530	8	non	non	ADJ
ejpam-1598	530	9	-	-	ADJ
ejpam-1598	530	10	singular	singular	ADJ
ejpam-1598	530	11	ternary	ternary	ADJ
ejpam-1598	530	12	semirings	semiring	NOUN
ejpam-1598	530	13	is	be	AUX
ejpam-1598	530	14	hereditary	hereditary	ADJ
ejpam-1598	530	15	.	.	PUNCT
ejpam-1598	531	1	proof	proof	NOUN
ejpam-1598	531	2	.	.	PUNCT
ejpam-1598	532	1	let	let	VERB
ejpam-1598	532	2	a	a	PRON
ejpam-1598	532	3	=	=	X
ejpam-1598	532	4	{	{	PUNCT
ejpam-1598	532	5	φ(s	φ(s	NOUN
ejpam-1598	532	6	)	)	PUNCT
ejpam-1598	532	7	:	:	PUNCT
ejpam-1598	532	8	s	s	AUX
ejpam-1598	532	9	be	be	AUX
ejpam-1598	532	10	a	a	DET
ejpam-1598	532	11	semiprime	semiprime	NOUN
ejpam-1598	532	12	nonsingular	nonsingular	ADJ
ejpam-1598	532	13	ternary	ternary	ADJ
ejpam-1598	532	14	semiring	semiring	NOUN
ejpam-1598	532	15	}	}	PUNCT
ejpam-1598	532	16	.	.	PUNCT
ejpam-1598	533	1	let	let	VERB
ejpam-1598	533	2	φ(s	φ(s	NOUN
ejpam-1598	533	3	)	)	PUNCT
ejpam-1598	533	4	∈	∈	PROPN
ejpam-1598	533	5	a	a	PRON
ejpam-1598	533	6	and	and	CCONJ
ejpam-1598	533	7	j	j	PROPN
ejpam-1598	533	8	a	a	DET
ejpam-1598	533	9	nonzero	nonzero	NOUN
ejpam-1598	533	10	ideal	ideal	NOUN
ejpam-1598	533	11	of	of	ADP
ejpam-1598	533	12	φ(s	φ(s	NOUN
ejpam-1598	533	13	)	)	PUNCT
ejpam-1598	533	14	.	.	PUNCT
ejpam-1598	534	1	then	then	ADV
ejpam-1598	534	2	,	,	PUNCT
ejpam-1598	534	3	there	there	PRON
ejpam-1598	534	4	exists	exist	VERB
ejpam-1598	534	5	a	a	DET
ejpam-1598	534	6	nonzero	nonzero	NOUN
ejpam-1598	534	7	ideal	ideal	NOUN
ejpam-1598	534	8	i	i	PRON
ejpam-1598	534	9	of	of	ADP
ejpam-1598	534	10	s	s	PRON
ejpam-1598	534	11	such	such	ADJ
ejpam-1598	534	12	that	that	SCONJ
ejpam-1598	534	13	φ(i	φ(i	PROPN
ejpam-1598	534	14	)	)	PUNCT
ejpam-1598	535	1	=	=	SYM
ejpam-1598	535	2	j	j	PROPN
ejpam-1598	535	3	.	.	PUNCT
ejpam-1598	536	1	now	now	ADV
ejpam-1598	536	2	by	by	ADP
ejpam-1598	536	3	proposition	proposition	NOUN
ejpam-1598	536	4	14	14	NUM
ejpam-1598	536	5	,	,	PUNCT
ejpam-1598	536	6	i	i	PRON
ejpam-1598	536	7	is	be	AUX
ejpam-1598	536	8	semiprime	semiprime	NOUN
ejpam-1598	536	9	as	as	SCONJ
ejpam-1598	536	10	s	s	NOUN
ejpam-1598	536	11	is	be	AUX
ejpam-1598	536	12	semiprime	semiprime	NOUN
ejpam-1598	536	13	.	.	PUNCT
ejpam-1598	537	1	also	also	ADV
ejpam-1598	537	2	by	by	ADP
ejpam-1598	537	3	proposition	proposition	NOUN
ejpam-1598	537	4	13	13	NUM
ejpam-1598	537	5	,	,	PUNCT
ejpam-1598	537	6	i	i	PRON
ejpam-1598	537	7	is	be	AUX
ejpam-1598	537	8	nonsingular	nonsingular	ADJ
ejpam-1598	537	9	.	.	PUNCT
ejpam-1598	538	1	hence	hence	ADV
ejpam-1598	538	2	,	,	PUNCT
ejpam-1598	538	3	φ(i	φ(i	PROPN
ejpam-1598	538	4	)	)	PUNCT
ejpam-1598	538	5	∈	∈	PROPN
ejpam-1598	538	6	a	a	PRON
ejpam-1598	538	7	,	,	PUNCT
ejpam-1598	538	8	that	that	ADV
ejpam-1598	538	9	is	is	ADV
ejpam-1598	538	10	,	,	PUNCT
ejpam-1598	538	11	j	j	PROPN
ejpam-1598	538	12	∈	∈	PROPN
ejpam-1598	538	13	a	a	PRON
ejpam-1598	538	14	.	.	PUNCT
ejpam-1598	539	1	thus	thus	ADV
ejpam-1598	539	2	,	,	PUNCT
ejpam-1598	539	3	the	the	DET
ejpam-1598	539	4	class	class	NOUN
ejpam-1598	539	5	of	of	ADP
ejpam-1598	539	6	homomorphic	homomorphic	ADJ
ejpam-1598	539	7	images	image	NOUN
ejpam-1598	539	8	of	of	ADP
ejpam-1598	539	9	semiprime	semiprime	NOUN
ejpam-1598	539	10	non	non	ADJ
ejpam-1598	539	11	-	-	ADJ
ejpam-1598	539	12	singular	singular	ADJ
ejpam-1598	539	13	ternary	ternary	ADJ
ejpam-1598	539	14	semirings	semiring	NOUN
ejpam-1598	539	15	is	be	AUX
ejpam-1598	539	16	hereditary	hereditary	ADJ
ejpam-1598	539	17	.	.	PUNCT
ejpam-1598	540	1	remark	remark	NOUN
ejpam-1598	540	2	3	3	NUM
ejpam-1598	540	3	.	.	PUNCT
ejpam-1598	541	1	we	we	PRON
ejpam-1598	541	2	make	make	VERB
ejpam-1598	541	3	the	the	DET
ejpam-1598	541	4	following	follow	VERB
ejpam-1598	541	5	observations	observation	NOUN
ejpam-1598	541	6	.	.	PUNCT
ejpam-1598	542	1	(	(	PUNCT
ejpam-1598	542	2	i	i	NOUN
ejpam-1598	542	3	)	)	PUNCT
ejpam-1598	542	4	the	the	DET
ejpam-1598	542	5	ternary	ternary	ADJ
ejpam-1598	542	6	subsemiring	subsemiring	NOUN
ejpam-1598	542	7	of	of	ADP
ejpam-1598	542	8	a	a	DET
ejpam-1598	542	9	semiprime	semiprime	NOUN
ejpam-1598	542	10	non	non	ADJ
ejpam-1598	542	11	-	-	ADJ
ejpam-1598	542	12	singular	singular	ADJ
ejpam-1598	542	13	ternary	ternary	ADJ
ejpam-1598	542	14	semiring	semiring	NOUN
ejpam-1598	542	15	is	be	AUX
ejpam-1598	542	16	not	not	PART
ejpam-1598	542	17	necessarily	necessarily	ADV
ejpam-1598	542	18	semiprime	semiprime	ADJ
ejpam-1598	542	19	non	non	ADJ
ejpam-1598	542	20	-	-	ADJ
ejpam-1598	542	21	singular	singular	ADJ
ejpam-1598	542	22	.	.	PUNCT
ejpam-1598	543	1	if	if	SCONJ
ejpam-1598	543	2	the	the	DET
ejpam-1598	543	3	ring	ring	NOUN
ejpam-1598	543	4	is	be	AUX
ejpam-1598	543	5	commutative	commutative	ADJ
ejpam-1598	543	6	then	then	ADV
ejpam-1598	543	7	the	the	DET
ejpam-1598	543	8	subring	subring	NOUN
ejpam-1598	543	9	of	of	ADP
ejpam-1598	543	10	a	a	DET
ejpam-1598	543	11	semiprime	semiprime	NOUN
ejpam-1598	543	12	ring	ring	NOUN
ejpam-1598	543	13	is	be	AUX
ejpam-1598	543	14	always	always	ADV
ejpam-1598	543	15	semiprime	semiprime	NOUN
ejpam-1598	543	16	.	.	PUNCT
ejpam-1598	544	1	similar	similar	ADJ
ejpam-1598	544	2	result	result	NOUN
ejpam-1598	544	3	holds	hold	VERB
ejpam-1598	544	4	for	for	ADP
ejpam-1598	544	5	ternary	ternary	ADJ
ejpam-1598	544	6	semirings	semiring	NOUN
ejpam-1598	544	7	.	.	PUNCT
ejpam-1598	545	1	(	(	PUNCT
ejpam-1598	545	2	ii	ii	X
ejpam-1598	545	3	)	)	PUNCT
ejpam-1598	545	4	it	it	PRON
ejpam-1598	545	5	has	have	AUX
ejpam-1598	545	6	been	be	AUX
ejpam-1598	545	7	proved	prove	VERB
ejpam-1598	545	8	in	in	ADP
ejpam-1598	545	9	proposition	proposition	NOUN
ejpam-1598	545	10	13	13	NUM
ejpam-1598	545	11	that	that	SCONJ
ejpam-1598	545	12	if	if	SCONJ
ejpam-1598	545	13	i	i	PRON
ejpam-1598	545	14	is	be	AUX
ejpam-1598	545	15	an	an	DET
ejpam-1598	545	16	ideal	ideal	NOUN
ejpam-1598	545	17	of	of	ADP
ejpam-1598	545	18	a	a	DET
ejpam-1598	545	19	ternary	ternary	ADJ
ejpam-1598	545	20	semiring	semiring	NOUN
ejpam-1598	545	21	s	s	X
ejpam-1598	546	1	and	and	CCONJ
ejpam-1598	546	2	i	i	PRON
ejpam-1598	546	3	a	a	DET
ejpam-1598	546	4	semiprime	semiprime	NOUN
ejpam-1598	546	5	ideal	ideal	NOUN
ejpam-1598	546	6	as	as	ADP
ejpam-1598	546	7	a	a	DET
ejpam-1598	546	8	ternary	ternary	ADJ
ejpam-1598	546	9	semiring	semiring	NOUN
ejpam-1598	546	10	,	,	PUNCT
ejpam-1598	546	11	then	then	ADV
ejpam-1598	546	12	z(i	z(i	NUM
ejpam-1598	546	13	)	)	PUNCT
ejpam-1598	547	1	=	=	PUNCT
ejpam-1598	547	2	i	i	PRON
ejpam-1598	547	3	∩	∩	ADJ
ejpam-1598	547	4	z(s	z(s	PROPN
ejpam-1598	547	5	)	)	PUNCT
ejpam-1598	547	6	.thus	.thus	ADV
ejpam-1598	547	7	,	,	PUNCT
ejpam-1598	547	8	by	by	ADP
ejpam-1598	547	9	the	the	DET
ejpam-1598	547	10	above	above	ADJ
ejpam-1598	547	11	result	result	NOUN
ejpam-1598	547	12	,	,	PUNCT
ejpam-1598	547	13	we	we	PRON
ejpam-1598	547	14	can	can	AUX
ejpam-1598	547	15	easily	easily	ADV
ejpam-1598	547	16	deduce	deduce	VERB
ejpam-1598	547	17	that	that	SCONJ
ejpam-1598	547	18	if	if	SCONJ
ejpam-1598	547	19	s	s	NOUN
ejpam-1598	547	20	is	be	AUX
ejpam-1598	547	21	a	a	DET
ejpam-1598	547	22	non	non	ADJ
ejpam-1598	547	23	-	-	ADJ
ejpam-1598	547	24	singular	singular	ADJ
ejpam-1598	547	25	ternary	ternary	ADJ
ejpam-1598	547	26	semiprime	semiprime	NOUN
ejpam-1598	547	27	semiring	semiring	NOUN
ejpam-1598	547	28	,	,	PUNCT
ejpam-1598	547	29	then	then	ADV
ejpam-1598	547	30	i	i	PRON
ejpam-1598	547	31	is	be	AUX
ejpam-1598	547	32	nonsingular	nonsingular	ADJ
ejpam-1598	547	33	.	.	PUNCT
ejpam-1598	548	1	similar	similar	ADJ
ejpam-1598	548	2	situation	situation	NOUN
ejpam-1598	548	3	has	have	AUX
ejpam-1598	548	4	been	be	AUX
ejpam-1598	548	5	investigated	investigate	VERB
ejpam-1598	548	6	by	by	ADP
ejpam-1598	548	7	t.k	t.k	PROPN
ejpam-1598	548	8	.	.	PROPN
ejpam-1598	548	9	dutta	dutta	PROPN
ejpam-1598	548	10	and	and	CCONJ
ejpam-1598	548	11	m.	m.	PROPN
ejpam-1598	548	12	l.	l.	PROPN
ejpam-1598	548	13	das	das	PROPN
ejpam-1598	548	14	in	in	ADP
ejpam-1598	548	15	[	[	X
ejpam-1598	548	16	9	9	NUM
ejpam-1598	548	17	]	]	PUNCT
ejpam-1598	548	18	in	in	ADP
ejpam-1598	548	19	the	the	DET
ejpam-1598	548	20	case	case	NOUN
ejpam-1598	548	21	of	of	ADP
ejpam-1598	548	22	a	a	DET
ejpam-1598	548	23	semiring	semiring	NOUN
ejpam-1598	548	24	.	.	PUNCT
ejpam-1598	549	1	in	in	ADP
ejpam-1598	549	2	the	the	DET
ejpam-1598	549	3	proof	proof	NOUN
ejpam-1598	549	4	of	of	ADP
ejpam-1598	549	5	this	this	DET
ejpam-1598	549	6	result	result	NOUN
ejpam-1598	549	7	,	,	PUNCT
ejpam-1598	549	8	the	the	DET
ejpam-1598	549	9	properties	property	NOUN
ejpam-1598	549	10	of	of	ADP
ejpam-1598	549	11	ideals	ideal	NOUN
ejpam-1598	549	12	have	have	AUX
ejpam-1598	549	13	been	be	AUX
ejpam-1598	549	14	used	use	VERB
ejpam-1598	549	15	.	.	PUNCT
ejpam-1598	550	1	no	no	DET
ejpam-1598	550	2	result	result	NOUN
ejpam-1598	550	3	has	have	AUX
ejpam-1598	550	4	been	be	AUX
ejpam-1598	550	5	proved	prove	VERB
ejpam-1598	550	6	so	so	ADV
ejpam-1598	550	7	far	far	ADV
ejpam-1598	550	8	for	for	ADP
ejpam-1598	550	9	subsemiring	subsemire	VERB
ejpam-1598	550	10	or	or	CCONJ
ejpam-1598	550	11	subring	subring	NOUN
ejpam-1598	550	12	and	and	CCONJ
ejpam-1598	550	13	hence	hence	ADV
ejpam-1598	550	14	,	,	PUNCT
ejpam-1598	550	15	in	in	ADP
ejpam-1598	550	16	the	the	DET
ejpam-1598	550	17	case	case	NOUN
ejpam-1598	550	18	of	of	ADP
ejpam-1598	550	19	ternary	ternary	ADJ
ejpam-1598	550	20	semirings	semiring	NOUN
ejpam-1598	550	21	,	,	PUNCT
ejpam-1598	550	22	the	the	DET
ejpam-1598	550	23	above	above	ADJ
ejpam-1598	550	24	result	result	NOUN
ejpam-1598	550	25	may	may	AUX
ejpam-1598	550	26	not	not	PART
ejpam-1598	550	27	be	be	AUX
ejpam-1598	550	28	true	true	ADJ
ejpam-1598	550	29	.	.	PUNCT
ejpam-1598	551	1	the	the	DET
ejpam-1598	551	2	reader	reader	NOUN
ejpam-1598	551	3	is	be	AUX
ejpam-1598	551	4	invited	invite	VERB
ejpam-1598	551	5	to	to	PART
ejpam-1598	551	6	provide	provide	VERB
ejpam-1598	551	7	a	a	DET
ejpam-1598	551	8	counter	counter	ADJ
ejpam-1598	551	9	example	example	NOUN
ejpam-1598	551	10	.	.	PUNCT
ejpam-1598	552	1	(	(	PUNCT
ejpam-1598	552	2	iii	iii	X
ejpam-1598	552	3	)	)	PUNCT
ejpam-1598	552	4	it	it	PRON
ejpam-1598	552	5	is	be	AUX
ejpam-1598	552	6	noted	note	VERB
ejpam-1598	552	7	that	that	SCONJ
ejpam-1598	552	8	the	the	DET
ejpam-1598	552	9	direct	direct	ADJ
ejpam-1598	552	10	product	product	NOUN
ejpam-1598	552	11	or	or	CCONJ
ejpam-1598	552	12	direct	direct	ADJ
ejpam-1598	552	13	sum	sum	NOUN
ejpam-1598	552	14	of	of	ADP
ejpam-1598	552	15	semiprime	semiprime	NOUN
ejpam-1598	552	16	non	non	ADJ
ejpam-1598	552	17	-	-	ADJ
ejpam-1598	552	18	singular	singular	ADJ
ejpam-1598	552	19	ternary	ternary	ADJ
ejpam-1598	552	20	semirings	semiring	NOUN
ejpam-1598	552	21	with	with	ADP
ejpam-1598	552	22	identity	identity	NOUN
ejpam-1598	552	23	is	be	AUX
ejpam-1598	552	24	semiprime	semiprime	NOUN
ejpam-1598	552	25	and	and	CCONJ
ejpam-1598	552	26	non	non	ADJ
ejpam-1598	552	27	-	-	ADJ
ejpam-1598	552	28	singular	singular	ADJ
ejpam-1598	552	29	.	.	PUNCT
ejpam-1598	553	1	in	in	ADP
ejpam-1598	553	2	the	the	DET
ejpam-1598	553	3	theory	theory	NOUN
ejpam-1598	553	4	of	of	ADP
ejpam-1598	553	5	finite	finite	ADJ
ejpam-1598	553	6	groups	group	NOUN
ejpam-1598	553	7	and	and	CCONJ
ejpam-1598	553	8	semigroups	semigroup	NOUN
ejpam-1598	553	9	,	,	PUNCT
ejpam-1598	553	10	the	the	DET
ejpam-1598	553	11	notion	notion	NOUN
ejpam-1598	553	12	of	of	ADP
ejpam-1598	553	13	the	the	DET
ejpam-1598	553	14	formation	formation	NOUN
ejpam-1598	553	15	and	and	CCONJ
ejpam-1598	553	16	the	the	DET
ejpam-1598	553	17	hereditary	hereditary	ADJ
ejpam-1598	553	18	classes	class	NOUN
ejpam-1598	553	19	of	of	ADP
ejpam-1598	553	20	groups	group	NOUN
ejpam-1598	553	21	and	and	CCONJ
ejpam-1598	553	22	semigroups	semigroup	NOUN
ejpam-1598	553	23	have	have	AUX
ejpam-1598	553	24	been	be	AUX
ejpam-1598	553	25	studied	study	VERB
ejpam-1598	553	26	by	by	ADP
ejpam-1598	553	27	l.	l.	PROPN
ejpam-1598	553	28	a.	a.	NOUN
ejpam-1598	553	29	shemetkov	shemetkov	PROPN
ejpam-1598	553	30	in	in	ADP
ejpam-1598	553	31	[	[	X
ejpam-1598	553	32	16	16	NUM
ejpam-1598	553	33	]	]	PUNCT
ejpam-1598	553	34	.	.	PUNCT
ejpam-1598	554	1	in	in	ADP
ejpam-1598	554	2	view	view	NOUN
ejpam-1598	554	3	of	of	ADP
ejpam-1598	554	4	the	the	DET
ejpam-1598	554	5	above	above	ADJ
ejpam-1598	554	6	observations	observation	NOUN
ejpam-1598	554	7	and	and	CCONJ
ejpam-1598	554	8	the	the	DET
ejpam-1598	554	9	terminology	terminology	NOUN
ejpam-1598	554	10	of	of	ADP
ejpam-1598	554	11	formation	formation	NOUN
ejpam-1598	554	12	given	give	VERB
ejpam-1598	554	13	by	by	ADP
ejpam-1598	554	14	l.a	l.a	PROPN
ejpam-1598	554	15	.	.	PROPN
ejpam-1598	554	16	shemetkov	shemetkov	PROPN
ejpam-1598	554	17	and	and	CCONJ
ejpam-1598	554	18	a.	a.	PROPN
ejpam-1598	554	19	n.	n.	PROPN
ejpam-1598	554	20	skiba	skiba	PROPN
ejpam-1598	555	1	[	[	X
ejpam-1598	555	2	17	17	NUM
ejpam-1598	555	3	]	]	PUNCT
ejpam-1598	555	4	,	,	PUNCT
ejpam-1598	555	5	we	we	PRON
ejpam-1598	555	6	propose	propose	VERB
ejpam-1598	555	7	the	the	DET
ejpam-1598	555	8	following	following	ADJ
ejpam-1598	555	9	conjecture	conjecture	NOUN
ejpam-1598	555	10	.	.	PUNCT
ejpam-1598	556	1	the	the	DET
ejpam-1598	556	2	hereditary	hereditary	ADJ
ejpam-1598	556	3	semiprime	semiprime	NOUN
ejpam-1598	556	4	non	non	ADJ
ejpam-1598	556	5	-	-	ADJ
ejpam-1598	556	6	singular	singular	ADJ
ejpam-1598	556	7	ternary	ternary	ADJ
ejpam-1598	556	8	semirings	semiring	NOUN
ejpam-1598	556	9	form	form	VERB
ejpam-1598	556	10	a	a	DET
ejpam-1598	556	11	hereditary	hereditary	ADJ
ejpam-1598	556	12	formation	formation	NOUN
ejpam-1598	556	13	of	of	ADP
ejpam-1598	556	14	ternary	ternary	ADJ
ejpam-1598	556	15	semirings	semiring	NOUN
ejpam-1598	556	16	.	.	PUNCT
ejpam-1598	557	1	acknowledgements	acknowledgement	NOUN
ejpam-1598	557	2	the	the	DET
ejpam-1598	557	3	third	third	ADJ
ejpam-1598	557	4	author	author	NOUN
ejpam-1598	557	5	is	be	AUX
ejpam-1598	557	6	thankful	thankful	ADJ
ejpam-1598	557	7	to	to	ADP
ejpam-1598	557	8	csir	csir	PROPN
ejpam-1598	557	9	,	,	PUNCT
ejpam-1598	557	10	india	india	PROPN
ejpam-1598	557	11	for	for	ADP
ejpam-1598	557	12	financial	financial	ADJ
ejpam-1598	557	13	assistance	assistance	NOUN
ejpam-1598	557	14	.	.	PUNCT
ejpam-1598	558	1	references	reference	NOUN
ejpam-1598	558	2	[	[	X
ejpam-1598	558	3	1	1	X
ejpam-1598	558	4	]	]	PUNCT
ejpam-1598	558	5	t.	t.	PROPN
ejpam-1598	558	6	k.	k.	PROPN
ejpam-1598	558	7	dutta	dutta	PROPN
ejpam-1598	558	8	and	and	CCONJ
ejpam-1598	558	9	s.	s.	PROPN
ejpam-1598	558	10	kar	kar	PROPN
ejpam-1598	558	11	.	.	PUNCT
ejpam-1598	559	1	on	on	ADP
ejpam-1598	559	2	regular	regular	ADJ
ejpam-1598	559	3	ternary	ternary	ADJ
ejpam-1598	559	4	semirings	semiring	NOUN
ejpam-1598	559	5	;	;	PUNCT
ejpam-1598	559	6	advances	advance	NOUN
ejpam-1598	559	7	in	in	ADP
ejpam-1598	559	8	algebra	algebra	NOUN
ejpam-1598	559	9	,	,	PUNCT
ejpam-1598	559	10	proceedings	proceeding	NOUN
ejpam-1598	559	11	of	of	ADP
ejpam-1598	559	12	the	the	DET
ejpam-1598	559	13	icm	icm	PROPN
ejpam-1598	559	14	satellite	satellite	PROPN
ejpam-1598	559	15	conference	conference	NOUN
ejpam-1598	559	16	in	in	ADP
ejpam-1598	559	17	algebra	algebra	PROPN
ejpam-1598	559	18	and	and	CCONJ
ejpam-1598	559	19	related	related	ADJ
ejpam-1598	559	20	topics	topic	NOUN
ejpam-1598	559	21	,	,	PUNCT
ejpam-1598	559	22	world	world	NOUN
ejpam-1598	559	23	scientific	scientific	ADJ
ejpam-1598	559	24	,	,	PUNCT
ejpam-1598	559	25	343	343	NUM
ejpam-1598	559	26	355	355	NUM
ejpam-1598	559	27	.	.	PUNCT
ejpam-1598	559	28	2003	2003	NUM
ejpam-1598	559	29	.	.	PUNCT
ejpam-1598	560	1	[	[	X
ejpam-1598	560	2	2	2	X
ejpam-1598	560	3	]	]	PUNCT
ejpam-1598	560	4	t.	t.	PROPN
ejpam-1598	560	5	k.	k.	PROPN
ejpam-1598	560	6	dutta	dutta	PROPN
ejpam-1598	560	7	and	and	CCONJ
ejpam-1598	560	8	s.	s.	PROPN
ejpam-1598	560	9	kar	kar	PROPN
ejpam-1598	560	10	.	.	PUNCT
ejpam-1598	561	1	on	on	ADP
ejpam-1598	561	2	prime	prime	ADJ
ejpam-1598	561	3	ideals	ideal	NOUN
ejpam-1598	561	4	and	and	CCONJ
ejpam-1598	561	5	prime	prime	ADJ
ejpam-1598	561	6	radical	radical	ADJ
ejpam-1598	561	7	of	of	ADP
ejpam-1598	561	8	ternary	ternary	ADJ
ejpam-1598	561	9	semirings	semiring	NOUN
ejpam-1598	561	10	;	;	PUNCT
ejpam-1598	561	11	bull	bull	NOUN
ejpam-1598	561	12	.	.	PUNCT
ejpam-1598	562	1	cal	cal	PROPN
ejpam-1598	562	2	.	.	PUNCT
ejpam-1598	563	1	math	math	NOUN
ejpam-1598	563	2	.	.	PUNCT
ejpam-1598	564	1	soc	soc	PROPN
ejpam-1598	564	2	.	.	PUNCT
ejpam-1598	564	3	,	,	PUNCT
ejpam-1598	564	4	vol	vol	NOUN
ejpam-1598	564	5	.	.	PROPN
ejpam-1598	565	1	97	97	NUM
ejpam-1598	565	2	,	,	PUNCT
ejpam-1598	565	3	no	no	INTJ
ejpam-1598	565	4	.	.	NOUN
ejpam-1598	566	1	5	5	NUM
ejpam-1598	566	2	.	.	X
ejpam-1598	566	3	445	445	NUM
ejpam-1598	566	4	454	454	NUM
ejpam-1598	566	5	.	.	PUNCT
ejpam-1598	567	1	2005	2005	NUM
ejpam-1598	567	2	.	.	PUNCT
ejpam-1598	568	1	references	reference	NOUN
ejpam-1598	568	2	128	128	NUM
ejpam-1598	568	3	[	[	X
ejpam-1598	568	4	3	3	NUM
ejpam-1598	568	5	]	]	X
ejpam-1598	568	6	t.k	t.k	PROPN
ejpam-1598	568	7	.	.	PROPN
ejpam-1598	568	8	dutta	dutta	PROPN
ejpam-1598	568	9	and	and	CCONJ
ejpam-1598	568	10	s.	s.	PROPN
ejpam-1598	568	11	mandal	mandal	PROPN
ejpam-1598	568	12	.	.	PUNCT
ejpam-1598	569	1	a	a	DET
ejpam-1598	569	2	general	general	ADJ
ejpam-1598	569	3	type	type	NOUN
ejpam-1598	569	4	of	of	ADP
ejpam-1598	569	5	regular	regular	ADJ
ejpam-1598	569	6	and	and	CCONJ
ejpam-1598	569	7	semiprime	semiprime	NOUN
ejpam-1598	569	8	ideals	ideal	NOUN
ejpam-1598	569	9	in	in	ADP
ejpam-1598	569	10	ternary	ternary	ADJ
ejpam-1598	569	11	semirings	semiring	NOUN
ejpam-1598	569	12	,	,	PUNCT
ejpam-1598	569	13	advances	advance	NOUN
ejpam-1598	569	14	in	in	ADP
ejpam-1598	569	15	algebraic	algebraic	ADJ
ejpam-1598	569	16	structure	structure	NOUN
ejpam-1598	569	17	,	,	PUNCT
ejpam-1598	569	18	proceedings	proceeding	NOUN
ejpam-1598	569	19	of	of	ADP
ejpam-1598	569	20	international	international	ADJ
ejpam-1598	569	21	conference	conference	NOUN
ejpam-1598	569	22	in	in	ADP
ejpam-1598	569	23	algebra	algebra	PROPN
ejpam-1598	569	24	,	,	PUNCT
ejpam-1598	569	25	world	world	NOUN
ejpam-1598	569	26	scientific	scientific	PROPN
ejpam-1598	569	27	,	,	PUNCT
ejpam-1598	569	28	inc	inc	PROPN
ejpam-1598	569	29	.	.	PROPN
ejpam-1598	569	30	,	,	PUNCT
ejpam-1598	569	31	219	219	NUM
ejpam-1598	569	32	-	-	SYM
ejpam-1598	569	33	232	232	NUM
ejpam-1598	569	34	.	.	PUNCT
ejpam-1598	569	35	2010	2010	NUM
ejpam-1598	569	36	.	.	PUNCT
ejpam-1598	570	1	[	[	X
ejpam-1598	570	2	4	4	X
ejpam-1598	570	3	]	]	PUNCT
ejpam-1598	570	4	t.	t.	PROPN
ejpam-1598	570	5	k.	k.	PROPN
ejpam-1598	570	6	dutta	dutta	PROPN
ejpam-1598	570	7	and	and	CCONJ
ejpam-1598	570	8	s.	s.	PROPN
ejpam-1598	570	9	kar	kar	PROPN
ejpam-1598	570	10	.	.	PUNCT
ejpam-1598	571	1	on	on	ADP
ejpam-1598	571	2	semiprime	semiprime	NOUN
ejpam-1598	571	3	ideals	ideal	NOUN
ejpam-1598	571	4	and	and	CCONJ
ejpam-1598	571	5	irreducible	irreducible	ADJ
ejpam-1598	571	6	ideals	ideal	NOUN
ejpam-1598	571	7	of	of	ADP
ejpam-1598	571	8	ternary	ternary	ADJ
ejpam-1598	571	9	semirings	semiring	NOUN
ejpam-1598	571	10	;	;	PUNCT
ejpam-1598	571	11	bull	bull	NOUN
ejpam-1598	571	12	.	.	PUNCT
ejpam-1598	572	1	cal	cal	PROPN
ejpam-1598	572	2	.	.	PUNCT
ejpam-1598	573	1	math	math	NOUN
ejpam-1598	573	2	.	.	PUNCT
ejpam-1598	574	1	soc	soc	PROPN
ejpam-1598	574	2	.	.	PUNCT
ejpam-1598	574	3	,	,	PUNCT
ejpam-1598	574	4	vol	vol	NOUN
ejpam-1598	574	5	.	.	PROPN
ejpam-1598	575	1	97	97	NUM
ejpam-1598	575	2	,	,	PUNCT
ejpam-1598	575	3	no	no	INTJ
ejpam-1598	575	4	.	.	NOUN
ejpam-1598	575	5	5	5	NUM
ejpam-1598	575	6	,	,	PUNCT
ejpam-1598	575	7	467	467	NUM
ejpam-1598	575	8	476	476	NUM
ejpam-1598	575	9	.	.	PUNCT
ejpam-1598	576	1	2005	2005	NUM
ejpam-1598	576	2	.	.	PUNCT
ejpam-1598	577	1	[	[	X
ejpam-1598	577	2	5	5	X
ejpam-1598	577	3	]	]	PUNCT
ejpam-1598	577	4	t.	t.	PROPN
ejpam-1598	577	5	k.	k.	PROPN
ejpam-1598	577	6	dutta	dutta	PROPN
ejpam-1598	577	7	and	and	CCONJ
ejpam-1598	577	8	s.	s.	PROPN
ejpam-1598	577	9	kar	kar	PROPN
ejpam-1598	577	10	.	.	PUNCT
ejpam-1598	578	1	on	on	ADP
ejpam-1598	578	2	ternary	ternary	ADJ
ejpam-1598	578	3	semifields	semifield	NOUN
ejpam-1598	578	4	;	;	PUNCT
ejpam-1598	578	5	discussiones	discussione	NOUN
ejpam-1598	578	6	mathematicae	mathematicae	VERB
ejpam-1598	578	7	general	general	ADJ
ejpam-1598	578	8	algebra	algebra	PROPN
ejpam-1598	578	9	and	and	CCONJ
ejpam-1598	578	10	applications	application	NOUN
ejpam-1598	578	11	,	,	PUNCT
ejpam-1598	578	12	vol	vol	NOUN
ejpam-1598	578	13	24	24	NUM
ejpam-1598	578	14	,	,	PUNCT
ejpam-1598	578	15	no.2	no.2	PROPN
ejpam-1598	578	16	,	,	PUNCT
ejpam-1598	578	17	185	185	NUM
ejpam-1598	578	18	198	198	NUM
ejpam-1598	578	19	.	.	PUNCT
ejpam-1598	578	20	2004	2004	NUM
ejpam-1598	578	21	.	.	PUNCT
ejpam-1598	579	1	[	[	X
ejpam-1598	579	2	6	6	NUM
ejpam-1598	579	3	]	]	PUNCT
ejpam-1598	579	4	t.	t.	PROPN
ejpam-1598	579	5	k.	k.	PROPN
ejpam-1598	579	6	dutta	dutta	PROPN
ejpam-1598	579	7	and	and	CCONJ
ejpam-1598	579	8	s.	s.	PROPN
ejpam-1598	579	9	kar	kar	PROPN
ejpam-1598	579	10	.	.	PUNCT
ejpam-1598	580	1	on	on	ADP
ejpam-1598	580	2	the	the	DET
ejpam-1598	580	3	jacobson	jacobson	PROPN
ejpam-1598	580	4	radical	radical	PROPN
ejpam-1598	580	5	of	of	ADP
ejpam-1598	580	6	a	a	DET
ejpam-1598	580	7	ternary	ternary	ADJ
ejpam-1598	580	8	semiring	semiring	NOUN
ejpam-1598	580	9	;	;	PUNCT
ejpam-1598	580	10	southeast	southeast	ADJ
ejpam-1598	580	11	asian	asian	PROPN
ejpam-1598	580	12	bull.of	bull.of	X
ejpam-1598	580	13	math	math	NOUN
ejpam-1598	580	14	.	.	PUNCT
ejpam-1598	581	1	,	,	PUNCT
ejpam-1598	581	2	vol	vol	NOUN
ejpam-1598	581	3	.	.	PROPN
ejpam-1598	582	1	28	28	NUM
ejpam-1598	582	2	,	,	PUNCT
ejpam-1598	582	3	no	no	INTJ
ejpam-1598	582	4	.	.	NOUN
ejpam-1598	582	5	1	1	NUM
ejpam-1598	582	6	,	,	PUNCT
ejpam-1598	582	7	1	1	NUM
ejpam-1598	582	8	13	13	NUM
ejpam-1598	582	9	.	.	PUNCT
ejpam-1598	582	10	2004	2004	NUM
ejpam-1598	582	11	.	.	PUNCT
ejpam-1598	583	1	[	[	X
ejpam-1598	583	2	7	7	X
ejpam-1598	583	3	]	]	X
ejpam-1598	583	4	t.k	t.k	PROPN
ejpam-1598	583	5	.	.	PROPN
ejpam-1598	583	6	dutta	dutta	PROPN
ejpam-1598	583	7	,	,	PUNCT
ejpam-1598	583	8	k.	k.	PROPN
ejpam-1598	583	9	p.	p.	PROPN
ejpam-1598	583	10	shum	shum	PROPN
ejpam-1598	583	11	and	and	CCONJ
ejpam-1598	583	12	s.	s.	PROPN
ejpam-1598	583	13	mandal	mandal	PROPN
ejpam-1598	583	14	.	.	PUNCT
ejpam-1598	584	1	weakly	weakly	ADJ
ejpam-1598	584	2	special	special	ADJ
ejpam-1598	584	3	radical	radical	ADJ
ejpam-1598	584	4	class	class	NOUN
ejpam-1598	584	5	and	and	CCONJ
ejpam-1598	584	6	special	special	ADJ
ejpam-1598	584	7	radical	radical	ADJ
ejpam-1598	584	8	class	class	NOUN
ejpam-1598	584	9	of	of	ADP
ejpam-1598	584	10	ternary	ternary	ADJ
ejpam-1598	584	11	semirings	semiring	NOUN
ejpam-1598	584	12	;	;	PUNCT
ejpam-1598	584	13	journal	journal	NOUN
ejpam-1598	584	14	of	of	ADP
ejpam-1598	584	15	international	international	ADJ
ejpam-1598	584	16	mathematics	mathematic	NOUN
ejpam-1598	584	17	and	and	CCONJ
ejpam-1598	584	18	mathematical	mathematical	ADJ
ejpam-1598	584	19	science	science	NOUN
ejpam-1598	584	20	.	.	PUNCT
ejpam-1598	585	1	2012	2012	NUM
ejpam-1598	585	2	(	(	PUNCT
ejpam-1598	585	3	to	to	PART
ejpam-1598	585	4	appear	appear	VERB
ejpam-1598	585	5	)	)	PUNCT
ejpam-1598	585	6	.	.	PUNCT
ejpam-1598	586	1	[	[	X
ejpam-1598	586	2	8	8	X
ejpam-1598	586	3	]	]	PUNCT
ejpam-1598	586	4	t.	t.	PROPN
ejpam-1598	586	5	k.	k.	PROPN
ejpam-1598	586	6	dutta	dutta	PROPN
ejpam-1598	586	7	and	and	CCONJ
ejpam-1598	586	8	s.	s.	PROPN
ejpam-1598	586	9	kar	kar	PROPN
ejpam-1598	586	10	.	.	PUNCT
ejpam-1598	587	1	two	two	NUM
ejpam-1598	587	2	types	type	NOUN
ejpam-1598	587	3	of	of	ADP
ejpam-1598	587	4	jacobson	jacobson	PROPN
ejpam-1598	587	5	radicals	radical	NOUN
ejpam-1598	587	6	of	of	ADP
ejpam-1598	587	7	ternary	ternary	ADJ
ejpam-1598	587	8	semirings	semiring	NOUN
ejpam-1598	587	9	;	;	PUNCT
ejpam-1598	587	10	southeast	southeast	ADJ
ejpam-1598	587	11	asian	asian	ADJ
ejpam-1598	587	12	bull	bull	PROPN
ejpam-1598	587	13	.	.	PUNCT
ejpam-1598	588	1	math	math	NOUN
ejpam-1598	588	2	.	.	PUNCT
ejpam-1598	589	1	29	29	NUM
ejpam-1598	589	2	,	,	PUNCT
ejpam-1598	589	3	no	no	INTJ
ejpam-1598	589	4	.	.	NOUN
ejpam-1598	589	5	4	4	NUM
ejpam-1598	589	6	,	,	PUNCT
ejpam-1598	589	7	677	677	NUM
ejpam-1598	589	8	687	687	NUM
ejpam-1598	589	9	.	.	PUNCT
ejpam-1598	590	1	2005	2005	NUM
ejpam-1598	590	2	.	.	PUNCT
ejpam-1598	591	1	[	[	X
ejpam-1598	591	2	9	9	NUM
ejpam-1598	591	3	]	]	PUNCT
ejpam-1598	591	4	t.	t.	PROPN
ejpam-1598	591	5	k.	k.	PROPN
ejpam-1598	591	6	dutta	dutta	PROPN
ejpam-1598	591	7	and	and	CCONJ
ejpam-1598	591	8	m.	m.	PROPN
ejpam-1598	591	9	l.	l.	PROPN
ejpam-1598	591	10	das	das	PROPN
ejpam-1598	591	11	.	.	PROPN
ejpam-1598	591	12	singular	singular	PROPN
ejpam-1598	591	13	radicals	radical	NOUN
ejpam-1598	591	14	in	in	ADP
ejpam-1598	591	15	semiring	semiring	NOUN
ejpam-1598	591	16	;	;	PUNCT
ejpam-1598	591	17	southeast	southeast	ADJ
ejpam-1598	591	18	asian	asian	ADJ
ejpam-1598	591	19	bull	bull	NOUN
ejpam-1598	591	20	.	.	PUNCT
ejpam-1598	592	1	of	of	ADP
ejpam-1598	592	2	math	math	NOUN
ejpam-1598	592	3	.	.	PUNCT
ejpam-1598	593	1	,	,	PUNCT
ejpam-1598	593	2	vol	vol	NOUN
ejpam-1598	593	3	.	.	PROPN
ejpam-1598	594	1	34	34	NUM
ejpam-1598	594	2	,	,	PUNCT
ejpam-1598	594	3	405	405	NUM
ejpam-1598	594	4	-	-	SYM
ejpam-1598	594	5	416	416	NUM
ejpam-1598	594	6	.	.	PUNCT
ejpam-1598	595	1	2010	2010	NUM
ejpam-1598	595	2	.	.	PUNCT
ejpam-1598	596	1	[	[	X
ejpam-1598	596	2	10	10	NUM
ejpam-1598	596	3	]	]	PUNCT
ejpam-1598	596	4	m.	m.	NOUN
ejpam-1598	596	5	ferrero	ferrero	PROPN
ejpam-1598	596	6	and	and	CCONJ
ejpam-1598	596	7	e.	e.	PROPN
ejpam-1598	596	8	r.	r.	PROPN
ejpam-1598	596	9	puczylowski	puczylowski	PROPN
ejpam-1598	596	10	.	.	PUNCT
ejpam-1598	597	1	the	the	DET
ejpam-1598	597	2	singular	singular	PROPN
ejpam-1598	597	3	ideal	ideal	NOUN
ejpam-1598	597	4	and	and	CCONJ
ejpam-1598	597	5	radicals	radical	NOUN
ejpam-1598	597	6	,	,	PUNCT
ejpam-1598	597	7	j.	j.	PROPN
ejpam-1598	597	8	austral	austral	PROPN
ejpam-1598	597	9	.	.	PUNCT
ejpam-1598	598	1	math	math	NOUN
ejpam-1598	598	2	.	.	PUNCT
ejpam-1598	599	1	soc.(series	soc.(serie	NOUN
ejpam-1598	600	1	a	a	PRON
ejpam-1598	600	2	)	)	PUNCT
ejpam-1598	600	3	64	64	NUM
ejpam-1598	600	4	,	,	PUNCT
ejpam-1598	600	5	195	195	NUM
ejpam-1598	600	6	-	-	SYM
ejpam-1598	600	7	209	209	NUM
ejpam-1598	600	8	.	.	PUNCT
ejpam-1598	601	1	1998	1998	NUM
ejpam-1598	601	2	.	.	PUNCT
ejpam-1598	602	1	[	[	X
ejpam-1598	602	2	11	11	NUM
ejpam-1598	602	3	]	]	PUNCT
ejpam-1598	602	4	s.	s.	PROPN
ejpam-1598	602	5	kar	kar	PROPN
ejpam-1598	602	6	.	.	PUNCT
ejpam-1598	603	1	on	on	ADP
ejpam-1598	603	2	structure	structure	NOUN
ejpam-1598	603	3	space	space	NOUN
ejpam-1598	603	4	of	of	ADP
ejpam-1598	603	5	ternary	ternary	ADJ
ejpam-1598	603	6	semirings	semiring	NOUN
ejpam-1598	603	7	;	;	PUNCT
ejpam-1598	603	8	southeast	southeast	ADJ
ejpam-1598	603	9	asian	asian	ADJ
ejpam-1598	603	10	bull	bull	PROPN
ejpam-1598	603	11	.	.	PUNCT
ejpam-1598	604	1	math	math	NOUN
ejpam-1598	604	2	.	.	PUNCT
ejpam-1598	605	1	31	31	NUM
ejpam-1598	605	2	,	,	PUNCT
ejpam-1598	605	3	no	no	INTJ
ejpam-1598	605	4	.	.	NOUN
ejpam-1598	605	5	3	3	NUM
ejpam-1598	605	6	,	,	PUNCT
ejpam-1598	605	7	537	537	NUM
ejpam-1598	605	8	545	545	NUM
ejpam-1598	605	9	.	.	PUNCT
ejpam-1598	606	1	2007	2007	NUM
ejpam-1598	606	2	.	.	PUNCT
ejpam-1598	607	1	[	[	X
ejpam-1598	607	2	12	12	NUM
ejpam-1598	607	3	]	]	PUNCT
ejpam-1598	607	4	s.	s.	PROPN
ejpam-1598	607	5	kar	kar	PROPN
ejpam-1598	607	6	,	,	PUNCT
ejpam-1598	607	7	j.	j.	PROPN
ejpam-1598	607	8	sircar	sircar	PROPN
ejpam-1598	607	9	and	and	CCONJ
ejpam-1598	607	10	s.	s.	PROPN
ejpam-1598	607	11	mandal	mandal	PROPN
ejpam-1598	607	12	.	.	PUNCT
ejpam-1598	608	1	on	on	ADP
ejpam-1598	608	2	right	right	ADV
ejpam-1598	608	3	strongly	strongly	ADV
ejpam-1598	608	4	prime	prime	ADJ
ejpam-1598	608	5	ternary	ternary	ADJ
ejpam-1598	608	6	semirings	semiring	NOUN
ejpam-1598	608	7	;	;	PUNCT
ejpam-1598	608	8	east	east	PROPN
ejpam-1598	608	9	-	-	PUNCT
ejpam-1598	608	10	west	west	PROPN
ejpam-1598	608	11	j.	j.	PROPN
ejpam-1598	608	12	of	of	ADP
ejpam-1598	608	13	mathematics	mathematics	PROPN
ejpam-1598	608	14	,	,	PUNCT
ejpam-1598	608	15	vol	vol	NOUN
ejpam-1598	608	16	.	.	PROPN
ejpam-1598	608	17	12	12	NUM
ejpam-1598	608	18	,	,	PUNCT
ejpam-1598	608	19	no	no	INTJ
ejpam-1598	608	20	.	.	NOUN
ejpam-1598	608	21	1	1	NUM
ejpam-1598	608	22	,	,	PUNCT
ejpam-1598	608	23	pp	pp	ADJ
ejpam-1598	608	24	.	.	PUNCT
ejpam-1598	608	25	59	59	NUM
ejpam-1598	608	26	68	68	NUM
ejpam-1598	608	27	.	.	PUNCT
ejpam-1598	608	28	2010	2010	NUM
ejpam-1598	608	29	.	.	PUNCT
ejpam-1598	609	1	[	[	X
ejpam-1598	609	2	13	13	NUM
ejpam-1598	609	3	]	]	X
ejpam-1598	609	4	d.	d.	PROPN
ejpam-1598	609	5	h.	h.	PROPN
ejpam-1598	609	6	lehmer	lehmer	PROPN
ejpam-1598	609	7	.	.	PUNCT
ejpam-1598	610	1	a	a	DET
ejpam-1598	610	2	ternary	ternary	ADJ
ejpam-1598	610	3	analogue	analogue	NOUN
ejpam-1598	610	4	of	of	ADP
ejpam-1598	610	5	abelian	abelian	ADJ
ejpam-1598	610	6	groups	groups	PROPN
ejpam-1598	610	7	,	,	PUNCT
ejpam-1598	610	8	amer	amer	PROPN
ejpam-1598	610	9	.	.	PUNCT
ejpam-1598	611	1	j.	j.	PROPN
ejpam-1598	611	2	math	math	PROPN
ejpam-1598	611	3	,	,	PUNCT
ejpam-1598	611	4	54(2	54(2	NUM
ejpam-1598	611	5	)	)	PUNCT
ejpam-1598	611	6	,	,	PUNCT
ejpam-1598	611	7	329	329	NUM
ejpam-1598	611	8	-	-	SYM
ejpam-1598	611	9	338	338	NUM
ejpam-1598	611	10	.	.	PUNCT
ejpam-1598	611	11	1932	1932	NUM
ejpam-1598	611	12	.	.	PUNCT
ejpam-1598	612	1	[	[	X
ejpam-1598	612	2	14	14	NUM
ejpam-1598	612	3	]	]	X
ejpam-1598	612	4	w.	w.	PROPN
ejpam-1598	612	5	g.	g.	PROPN
ejpam-1598	612	6	lister	lister	PROPN
ejpam-1598	612	7	.	.	PUNCT
ejpam-1598	613	1	ternary	ternary	ADJ
ejpam-1598	613	2	rings	ring	NOUN
ejpam-1598	613	3	;	;	PUNCT
ejpam-1598	613	4	trans	trans	PROPN
ejpam-1598	613	5	.	.	PROPN
ejpam-1598	614	1	amer	amer	PROPN
ejpam-1598	614	2	.	.	PUNCT
ejpam-1598	614	3	math	math	PROPN
ejpam-1598	614	4	.	.	PUNCT
ejpam-1598	615	1	soc	soc	PROPN
ejpam-1598	615	2	.	.	PUNCT
ejpam-1598	616	1	154	154	NUM
ejpam-1598	616	2	,	,	PUNCT
ejpam-1598	616	3	37	37	NUM
ejpam-1598	616	4	-	-	SYM
ejpam-1598	616	5	55	55	NUM
ejpam-1598	616	6	.	.	PUNCT
ejpam-1598	617	1	1971	1971	NUM
ejpam-1598	617	2	.	.	PUNCT
ejpam-1598	618	1	[	[	X
ejpam-1598	618	2	15	15	NUM
ejpam-1598	618	3	]	]	X
ejpam-1598	619	1	s.	s.	PROPN
ejpam-1598	619	2	j.	j.	PROPN
ejpam-1598	619	3	lö	lö	PROPN
ejpam-1598	619	4	.	.	PROPN
ejpam-1598	620	1	on	on	ADP
ejpam-1598	620	2	the	the	DET
ejpam-1598	620	3	extending	extend	VERB
ejpam-1598	620	4	models	model	NOUN
ejpam-1598	620	5	i	i	PRON
ejpam-1598	620	6	;	;	PUNCT
ejpam-1598	620	7	fundamenta	fundamenta	PROPN
ejpam-1598	620	8	mathematicae	mathematicae	PROPN
ejpam-1598	620	9	.	.	PUNCT
ejpam-1598	621	1	,42	,42	NOUN
ejpam-1598	621	2	,	,	PUNCT
ejpam-1598	621	3	38	38	NUM
ejpam-1598	621	4	-	-	SYM
ejpam-1598	621	5	54	54	NUM
ejpam-1598	621	6	.	.	PUNCT
ejpam-1598	622	1	1955	1955	NUM
ejpam-1598	622	2	.	.	PUNCT
ejpam-1598	623	1	[	[	X
ejpam-1598	623	2	16	16	NUM
ejpam-1598	623	3	]	]	X
ejpam-1598	623	4	l.	l.	PROPN
ejpam-1598	623	5	a.	a.	PROPN
ejpam-1598	623	6	shemetkov	shemetkov	PROPN
ejpam-1598	623	7	.	.	PUNCT
ejpam-1598	624	1	new	new	ADJ
ejpam-1598	624	2	ideas	idea	NOUN
ejpam-1598	624	3	and	and	CCONJ
ejpam-1598	624	4	results	result	NOUN
ejpam-1598	624	5	in	in	ADP
ejpam-1598	624	6	the	the	DET
ejpam-1598	624	7	theory	theory	NOUN
ejpam-1598	624	8	of	of	ADP
ejpam-1598	624	9	formations(russian	formations(russian	PROPN
ejpam-1598	624	10	)	)	PUNCT
ejpam-1598	624	11	,	,	PUNCT
ejpam-1598	624	12	problems	problem	NOUN
ejpam-1598	624	13	in	in	ADP
ejpam-1598	624	14	algebra	algebra	NOUN
ejpam-1598	624	15	,	,	PUNCT
ejpam-1598	624	16	no.4	no.4	PROPN
ejpam-1598	624	17	(	(	PUNCT
ejpam-1598	624	18	russian	russian	PROPN
ejpam-1598	624	19	)	)	PUNCT
ejpam-1598	624	20	,	,	PUNCT
ejpam-1598	624	21	(	(	PUNCT
ejpam-1598	624	22	gomel	gomel	PROPN
ejpam-1598	624	23	)	)	PUNCT
ejpam-1598	624	24	,	,	PUNCT
ejpam-1598	624	25	65	65	NUM
ejpam-1598	624	26	-	-	SYM
ejpam-1598	624	27	67	67	NUM
ejpam-1598	624	28	,	,	PUNCT
ejpam-1598	624	29	problems	problem	NOUN
ejpam-1598	624	30	in	in	ADP
ejpam-1598	624	31	algebra	algebra	PROPN
ejpam-1598	624	32	,	,	PUNCT
ejpam-1598	624	33	(	(	PUNCT
ejpam-1598	624	34	russian	russian	NOUN
ejpam-1598	624	35	)	)	PUNCT
ejpam-1598	624	36	,	,	PUNCT
ejpam-1598	624	37	in	in	ADP
ejpam-1598	624	38	universitet.skoe	universitet.skoe	PROPN
ejpam-1598	624	39	,	,	PUNCT
ejpam-1598	624	40	minsk	minsk	NOUN
ejpam-1598	624	41	,	,	PUNCT
ejpam-1598	624	42	1986,1989	1986,1989	NUM
ejpam-1598	624	43	.	.	PUNCT
ejpam-1598	625	1	[	[	X
ejpam-1598	625	2	17	17	NUM
ejpam-1598	625	3	]	]	PUNCT
ejpam-1598	625	4	a.	a.	PROPN
ejpam-1598	625	5	n.	n.	PROPN
ejpam-1598	625	6	skiba	skiba	PROPN
ejpam-1598	625	7	and	and	CCONJ
ejpam-1598	625	8	l.	l.	PROPN
ejpam-1598	625	9	a.	a.	PROPN
ejpam-1598	625	10	shemetkov	shemetkov	PROPN
ejpam-1598	625	11	.	.	PUNCT
ejpam-1598	626	1	hereditarily	hereditarily	ADV
ejpam-1598	626	2	indecomposable	indecomposable	ADJ
ejpam-1598	626	3	formations	formation	NOUN
ejpam-1598	626	4	of	of	ADP
ejpam-1598	626	5	groups(russian	groups(russian	PROPN
ejpam-1598	626	6	)	)	PUNCT
ejpam-1598	626	7	,	,	PUNCT
ejpam-1598	626	8	dokl	dokl	NOUN
ejpam-1598	626	9	.	.	PUNCT
ejpam-1598	626	10	akad	akad	PROPN
ejpam-1598	626	11	.	.	PUNCT
ejpam-1598	627	1	nauk	nauk	PROPN
ejpam-1598	627	2	bssr	bssr	PROPN
ejpam-1598	627	3	33	33	NUM
ejpam-1598	627	4	,	,	PUNCT
ejpam-1598	627	5	no	no	INTJ
ejpam-1598	627	6	.	.	NOUN
ejpam-1598	627	7	7	7	NUM
ejpam-1598	627	8	,	,	PUNCT
ejpam-1598	627	9	581	581	NUM
ejpam-1598	627	10	-	-	SYM
ejpam-1598	627	11	582	582	NUM
ejpam-1598	627	12	.	.	NUM
ejpam-1598	627	13	1989	1989	NUM
ejpam-1598	627	14	.	.	PUNCT