id	sid	tid	token	lemma	pos
ejpam-6096	1	1	european	european	PROPN
ejpam-6096	1	2	journal	journal	PROPN
ejpam-6096	1	3	of	of	ADP
ejpam-6096	1	4	pure	pure	ADJ
ejpam-6096	1	5	and	and	CCONJ
ejpam-6096	1	6	applied	applied	ADJ
ejpam-6096	1	7	mathematics	mathematic	NOUN
ejpam-6096	1	8	2025	2025	NUM
ejpam-6096	1	9	,	,	PUNCT
ejpam-6096	1	10	vol	vol	NOUN
ejpam-6096	1	11	.	.	PROPN
ejpam-6096	1	12	18	18	NUM
ejpam-6096	1	13	,	,	PUNCT
ejpam-6096	1	14	issue	issue	NOUN
ejpam-6096	1	15	2	2	NUM
ejpam-6096	1	16	,	,	PUNCT
ejpam-6096	1	17	article	article	NOUN
ejpam-6096	1	18	number	number	NOUN
ejpam-6096	1	19	6096	6096	NUM
ejpam-6096	1	20	issn	issn	PROPN
ejpam-6096	1	21	1307	1307	NUM
ejpam-6096	1	22	-	-	SYM
ejpam-6096	1	23	5543	5543	NUM
ejpam-6096	1	24	–	–	PUNCT
ejpam-6096	1	25	ejpam.com	ejpam.com	X
ejpam-6096	1	26	published	publish	VERB
ejpam-6096	1	27	by	by	ADP
ejpam-6096	1	28	new	new	PROPN
ejpam-6096	1	29	york	york	PROPN
ejpam-6096	1	30	business	business	PROPN
ejpam-6096	1	31	global	global	PROPN
ejpam-6096	1	32	a	a	DET
ejpam-6096	1	33	note	note	NOUN
ejpam-6096	1	34	on	on	ADP
ejpam-6096	1	35	k	k	NOUN
ejpam-6096	1	36	-	-	NOUN
ejpam-6096	1	37	ideals	ideal	NOUN
ejpam-6096	1	38	in	in	ADP
ejpam-6096	1	39	ternary	ternary	ADJ
ejpam-6096	1	40	semirings	semiring	NOUN
ejpam-6096	1	41	arbaz	arbaz	PROPN
ejpam-6096	1	42	jehan	jehan	PROPN
ejpam-6096	1	43	khan1	khan1	PROPN
ejpam-6096	1	44	,	,	PUNCT
ejpam-6096	1	45	montakarn	montakarn	PROPN
ejpam-6096	1	46	petapirak1	petapirak1	PROPN
ejpam-6096	1	47	,	,	PUNCT
ejpam-6096	1	48	ronnason	ronnason	NOUN
ejpam-6096	1	49	chinram1,∗	chinram1,∗	PROPN
ejpam-6096	1	50	1division	1division	NUM
ejpam-6096	1	51	of	of	ADP
ejpam-6096	1	52	computational	computational	ADJ
ejpam-6096	1	53	science	science	NOUN
ejpam-6096	1	54	,	,	PUNCT
ejpam-6096	1	55	faculty	faculty	NOUN
ejpam-6096	1	56	of	of	ADP
ejpam-6096	1	57	science	science	NOUN
ejpam-6096	1	58	,	,	PUNCT
ejpam-6096	1	59	prince	prince	NOUN
ejpam-6096	1	60	of	of	ADP
ejpam-6096	1	61	songkla	songkla	PROPN
ejpam-6096	1	62	university	university	PROPN
ejpam-6096	1	63	,	,	PUNCT
ejpam-6096	1	64	hat	hat	PROPN
ejpam-6096	1	65	yai	yai	PROPN
ejpam-6096	1	66	,	,	PUNCT
ejpam-6096	1	67	songkhla	songkhla	VERB
ejpam-6096	1	68	90110	90110	NUM
ejpam-6096	1	69	,	,	PUNCT
ejpam-6096	1	70	thailand	thailand	PROPN
ejpam-6096	1	71	abstract	abstract	NOUN
ejpam-6096	1	72	.	.	PUNCT
ejpam-6096	2	1	in	in	ADP
ejpam-6096	2	2	this	this	DET
ejpam-6096	2	3	paper	paper	NOUN
ejpam-6096	2	4	,	,	PUNCT
ejpam-6096	2	5	we	we	PRON
ejpam-6096	2	6	give	give	VERB
ejpam-6096	2	7	some	some	DET
ejpam-6096	2	8	examples	example	NOUN
ejpam-6096	2	9	of	of	ADP
ejpam-6096	2	10	k	k	NOUN
ejpam-6096	2	11	-	-	NOUN
ejpam-6096	2	12	ideals	ideal	NOUN
ejpam-6096	2	13	of	of	ADP
ejpam-6096	2	14	ternary	ternary	ADJ
ejpam-6096	2	15	semirings	semiring	NOUN
ejpam-6096	2	16	.	.	PUNCT
ejpam-6096	3	1	we	we	PRON
ejpam-6096	3	2	examine	examine	VERB
ejpam-6096	3	3	the	the	DET
ejpam-6096	3	4	results	result	NOUN
ejpam-6096	3	5	of	of	ADP
ejpam-6096	3	6	k	k	NOUN
ejpam-6096	3	7	-	-	NOUN
ejpam-6096	3	8	ideals	ideal	NOUN
ejpam-6096	3	9	by	by	ADP
ejpam-6096	3	10	distinguished	distinguished	ADJ
ejpam-6096	3	11	classes	class	NOUN
ejpam-6096	3	12	in	in	ADP
ejpam-6096	3	13	ternary	ternary	ADJ
ejpam-6096	3	14	semirings	semiring	NOUN
ejpam-6096	3	15	,	,	PUNCT
ejpam-6096	3	16	which	which	PRON
ejpam-6096	3	17	include	include	VERB
ejpam-6096	3	18	k	k	X
ejpam-6096	3	19	-	-	ADJ
ejpam-6096	3	20	maximal	maximal	ADJ
ejpam-6096	3	21	,	,	PUNCT
ejpam-6096	3	22	k	k	NOUN
ejpam-6096	3	23	-	-	ADJ
ejpam-6096	3	24	prime	prime	ADJ
ejpam-6096	3	25	and	and	CCONJ
ejpam-6096	3	26	k	k	NOUN
ejpam-6096	3	27	-	-	NOUN
ejpam-6096	3	28	semiprime	semiprime	NOUN
ejpam-6096	3	29	.	.	PUNCT
ejpam-6096	4	1	2020	2020	NUM
ejpam-6096	4	2	mathematics	mathematic	NOUN
ejpam-6096	4	3	subject	subject	NOUN
ejpam-6096	4	4	classifications	classification	NOUN
ejpam-6096	4	5	:	:	PUNCT
ejpam-6096	4	6	16y60,16y99	16y60,16y99	NUM
ejpam-6096	4	7	key	key	ADJ
ejpam-6096	4	8	words	word	NOUN
ejpam-6096	4	9	and	and	CCONJ
ejpam-6096	4	10	phrases	phrase	NOUN
ejpam-6096	4	11	:	:	PUNCT
ejpam-6096	4	12	ternary	ternary	ADJ
ejpam-6096	4	13	semirings	semiring	NOUN
ejpam-6096	4	14	,	,	PUNCT
ejpam-6096	4	15	k	k	NOUN
ejpam-6096	4	16	-	-	NOUN
ejpam-6096	4	17	ideals	ideal	NOUN
ejpam-6096	4	18	,	,	PUNCT
ejpam-6096	4	19	k	k	NOUN
ejpam-6096	4	20	-	-	ADJ
ejpam-6096	4	21	maximal	maximal	ADJ
ejpam-6096	4	22	,	,	PUNCT
ejpam-6096	4	23	k	k	NOUN
ejpam-6096	4	24	-	-	NOUN
ejpam-6096	4	25	prime	prime	ADJ
ejpam-6096	4	26	,	,	PUNCT
ejpam-6096	4	27	k	k	NOUN
ejpam-6096	4	28	-	-	ADJ
ejpam-6096	4	29	semiprime	semiprime	ADJ
ejpam-6096	4	30	1	1	NUM
ejpam-6096	4	31	.	.	PUNCT
ejpam-6096	5	1	introduction	introduction	NOUN
ejpam-6096	5	2	lehmer	lehmer	NOUN
ejpam-6096	5	3	[	[	X
ejpam-6096	5	4	1	1	NUM
ejpam-6096	5	5	]	]	PUNCT
ejpam-6096	5	6	introduced	introduce	VERB
ejpam-6096	5	7	ternary	ternary	ADJ
ejpam-6096	5	8	algebra	algebra	NOUN
ejpam-6096	5	9	in	in	ADP
ejpam-6096	5	10	1932	1932	NUM
ejpam-6096	5	11	and	and	CCONJ
ejpam-6096	5	12	investigated	investigate	VERB
ejpam-6096	5	13	certain	certain	ADJ
ejpam-6096	5	14	algebraic	algebraic	ADJ
ejpam-6096	5	15	systems	system	NOUN
ejpam-6096	5	16	called	call	VERB
ejpam-6096	5	17	triplexes	triplexe	NOUN
ejpam-6096	5	18	,	,	PUNCT
ejpam-6096	5	19	which	which	PRON
ejpam-6096	5	20	are	be	AUX
ejpam-6096	5	21	commutative	commutative	ADJ
ejpam-6096	5	22	ternary	ternary	ADJ
ejpam-6096	5	23	groups	group	NOUN
ejpam-6096	5	24	.	.	PUNCT
ejpam-6096	6	1	later	later	ADV
ejpam-6096	6	2	,	,	PUNCT
ejpam-6096	6	3	banach	banach	NOUN
ejpam-6096	6	4	also	also	ADV
ejpam-6096	6	5	studied	study	VERB
ejpam-6096	6	6	these	these	DET
ejpam-6096	6	7	algebraic	algebraic	ADJ
ejpam-6096	6	8	structures	structure	NOUN
ejpam-6096	6	9	and	and	CCONJ
ejpam-6096	6	10	provided	provide	VERB
ejpam-6096	6	11	examples	example	NOUN
ejpam-6096	6	12	of	of	ADP
ejpam-6096	6	13	a	a	DET
ejpam-6096	6	14	ternary	ternary	ADJ
ejpam-6096	6	15	semigroup	semigroup	NOUN
ejpam-6096	6	16	that	that	PRON
ejpam-6096	6	17	does	do	AUX
ejpam-6096	6	18	not	not	PART
ejpam-6096	6	19	reduce	reduce	VERB
ejpam-6096	6	20	to	to	ADP
ejpam-6096	6	21	a	a	DET
ejpam-6096	6	22	semigroup	semigroup	NOUN
ejpam-6096	6	23	.	.	PUNCT
ejpam-6096	7	1	additionally	additionally	ADV
ejpam-6096	7	2	,	,	PUNCT
ejpam-6096	7	3	lister	lister	PROPN
ejpam-6096	7	4	[	[	X
ejpam-6096	7	5	2	2	X
ejpam-6096	7	6	]	]	PUNCT
ejpam-6096	7	7	introduced	introduce	VERB
ejpam-6096	7	8	the	the	DET
ejpam-6096	7	9	concept	concept	NOUN
ejpam-6096	7	10	of	of	ADP
ejpam-6096	7	11	a	a	DET
ejpam-6096	7	12	ternary	ternary	ADJ
ejpam-6096	7	13	ring	ring	NOUN
ejpam-6096	7	14	.	.	PUNCT
ejpam-6096	8	1	the	the	DET
ejpam-6096	8	2	concept	concept	NOUN
ejpam-6096	8	3	of	of	ADP
ejpam-6096	8	4	ternary	ternary	ADJ
ejpam-6096	8	5	semirings	semiring	NOUN
ejpam-6096	8	6	was	be	AUX
ejpam-6096	8	7	first	first	ADV
ejpam-6096	8	8	introduced	introduce	VERB
ejpam-6096	8	9	by	by	ADP
ejpam-6096	8	10	dutta	dutta	PROPN
ejpam-6096	8	11	and	and	CCONJ
ejpam-6096	8	12	kar	kar	NOUN
ejpam-6096	9	1	[	[	X
ejpam-6096	9	2	3	3	X
ejpam-6096	9	3	]	]	PUNCT
ejpam-6096	9	4	in	in	ADP
ejpam-6096	9	5	2003	2003	NUM
ejpam-6096	9	6	.	.	PUNCT
ejpam-6096	10	1	moreover	moreover	ADV
ejpam-6096	10	2	,	,	PUNCT
ejpam-6096	10	3	dutta	dutta	PROPN
ejpam-6096	10	4	and	and	CCONJ
ejpam-6096	10	5	kar	kar	PROPN
ejpam-6096	10	6	investigated	investigate	VERB
ejpam-6096	10	7	some	some	DET
ejpam-6096	10	8	basic	basic	ADJ
ejpam-6096	10	9	concepts	concept	NOUN
ejpam-6096	10	10	of	of	ADP
ejpam-6096	10	11	prime	prime	ADJ
ejpam-6096	10	12	ideals	ideal	NOUN
ejpam-6096	10	13	and	and	CCONJ
ejpam-6096	10	14	semiprime	semiprime	NOUN
ejpam-6096	10	15	ideals	ideal	NOUN
ejpam-6096	10	16	of	of	ADP
ejpam-6096	10	17	ternary	ternary	ADJ
ejpam-6096	10	18	semirings	semiring	NOUN
ejpam-6096	10	19	in	in	ADP
ejpam-6096	10	20	[	[	X
ejpam-6096	10	21	4	4	NUM
ejpam-6096	10	22	]	]	PUNCT
ejpam-6096	10	23	and	and	CCONJ
ejpam-6096	10	24	[	[	X
ejpam-6096	10	25	5	5	NUM
ejpam-6096	10	26	]	]	PUNCT
ejpam-6096	10	27	,	,	PUNCT
ejpam-6096	10	28	respectively	respectively	ADV
ejpam-6096	10	29	.	.	PUNCT
ejpam-6096	11	1	the	the	DET
ejpam-6096	11	2	concept	concept	NOUN
ejpam-6096	11	3	of	of	ADP
ejpam-6096	11	4	ternary	ternary	ADJ
ejpam-6096	11	5	semirings	semiring	NOUN
ejpam-6096	11	6	arises	arise	VERB
ejpam-6096	11	7	from	from	ADP
ejpam-6096	11	8	the	the	DET
ejpam-6096	11	9	study	study	NOUN
ejpam-6096	11	10	of	of	ADP
ejpam-6096	11	11	algebraic	algebraic	ADJ
ejpam-6096	11	12	structures	structure	NOUN
ejpam-6096	11	13	that	that	PRON
ejpam-6096	11	14	extend	extend	VERB
ejpam-6096	11	15	semirings	semiring	NOUN
ejpam-6096	11	16	.	.	PUNCT
ejpam-6096	12	1	ternary	ternary	ADJ
ejpam-6096	12	2	semirings	semiring	NOUN
ejpam-6096	12	3	consist	consist	VERB
ejpam-6096	12	4	of	of	ADP
ejpam-6096	12	5	two	two	NUM
ejpam-6096	12	6	operations	operation	NOUN
ejpam-6096	12	7	,	,	PUNCT
ejpam-6096	12	8	typically	typically	ADV
ejpam-6096	12	9	the	the	DET
ejpam-6096	12	10	addition	addition	NOUN
ejpam-6096	12	11	and	and	CCONJ
ejpam-6096	12	12	the	the	DET
ejpam-6096	12	13	ternary	ternary	ADJ
ejpam-6096	12	14	multiplication	multiplication	NOUN
ejpam-6096	12	15	.	.	PUNCT
ejpam-6096	13	1	by	by	ADP
ejpam-6096	13	2	using	use	VERB
ejpam-6096	13	3	a	a	DET
ejpam-6096	13	4	ternary	ternary	ADJ
ejpam-6096	13	5	multiplication	multiplication	NOUN
ejpam-6096	13	6	instead	instead	ADV
ejpam-6096	13	7	of	of	ADP
ejpam-6096	13	8	a	a	DET
ejpam-6096	13	9	binary	binary	ADJ
ejpam-6096	13	10	multiplication	multiplication	NOUN
ejpam-6096	13	11	,	,	PUNCT
ejpam-6096	13	12	every	every	DET
ejpam-6096	13	13	semiring	semiring	NOUN
ejpam-6096	13	14	can	can	AUX
ejpam-6096	13	15	be	be	AUX
ejpam-6096	13	16	turned	turn	VERB
ejpam-6096	13	17	to	to	ADP
ejpam-6096	13	18	a	a	DET
ejpam-6096	13	19	ternary	ternary	ADJ
ejpam-6096	13	20	semiring	semiring	NOUN
ejpam-6096	13	21	.	.	PUNCT
ejpam-6096	14	1	however	however	ADV
ejpam-6096	14	2	,	,	PUNCT
ejpam-6096	14	3	a	a	DET
ejpam-6096	14	4	ternary	ternary	ADJ
ejpam-6096	14	5	semiring	semiring	NOUN
ejpam-6096	14	6	does	do	AUX
ejpam-6096	14	7	not	not	PART
ejpam-6096	14	8	necessarily	necessarily	ADV
ejpam-6096	14	9	reduce	reduce	VERB
ejpam-6096	14	10	to	to	ADP
ejpam-6096	14	11	a	a	DET
ejpam-6096	14	12	semiring	semiring	NOUN
ejpam-6096	14	13	.	.	PUNCT
ejpam-6096	15	1	although	although	SCONJ
ejpam-6096	15	2	ternary	ternary	ADJ
ejpam-6096	15	3	semirings	semiring	NOUN
ejpam-6096	15	4	generalize	generalize	VERB
ejpam-6096	15	5	the	the	DET
ejpam-6096	15	6	notion	notion	NOUN
ejpam-6096	15	7	of	of	ADP
ejpam-6096	15	8	a	a	DET
ejpam-6096	15	9	semiring	semiring	NOUN
ejpam-6096	15	10	,	,	PUNCT
ejpam-6096	15	11	they	they	PRON
ejpam-6096	15	12	are	be	AUX
ejpam-6096	15	13	not	not	PART
ejpam-6096	15	14	just	just	ADV
ejpam-6096	15	15	a	a	DET
ejpam-6096	15	16	generalization	generalization	NOUN
ejpam-6096	15	17	because	because	SCONJ
ejpam-6096	15	18	some	some	DET
ejpam-6096	15	19	certain	certain	ADJ
ejpam-6096	15	20	notions	notion	NOUN
ejpam-6096	15	21	,	,	PUNCT
ejpam-6096	15	22	such	such	ADJ
ejpam-6096	15	23	as	as	ADP
ejpam-6096	15	24	lateral	lateral	ADJ
ejpam-6096	15	25	ideals	ideal	NOUN
ejpam-6096	15	26	,	,	PUNCT
ejpam-6096	15	27	lack	lack	VERB
ejpam-6096	15	28	an	an	DET
ejpam-6096	15	29	analog	analog	NOUN
ejpam-6096	15	30	in	in	ADP
ejpam-6096	15	31	a	a	DET
ejpam-6096	15	32	semiring	semiring	NOUN
ejpam-6096	15	33	.	.	PUNCT
ejpam-6096	16	1	many	many	ADJ
ejpam-6096	16	2	concepts	concept	NOUN
ejpam-6096	16	3	from	from	ADP
ejpam-6096	16	4	semiring	semire	VERB
ejpam-6096	16	5	theory	theory	NOUN
ejpam-6096	16	6	were	be	AUX
ejpam-6096	16	7	extended	extend	VERB
ejpam-6096	16	8	to	to	ADP
ejpam-6096	16	9	the	the	DET
ejpam-6096	16	10	study	study	NOUN
ejpam-6096	16	11	of	of	ADP
ejpam-6096	16	12	ternary	ternary	ADJ
ejpam-6096	16	13	semirings	semiring	NOUN
ejpam-6096	16	14	.	.	PUNCT
ejpam-6096	17	1	ideal	ideal	PROPN
ejpam-6096	17	2	theory	theory	NOUN
ejpam-6096	17	3	is	be	AUX
ejpam-6096	17	4	the	the	DET
ejpam-6096	17	5	main	main	ADJ
ejpam-6096	17	6	area	area	NOUN
ejpam-6096	17	7	of	of	ADP
ejpam-6096	17	8	research	research	NOUN
ejpam-6096	17	9	in	in	ADP
ejpam-6096	17	10	the	the	DET
ejpam-6096	17	11	study	study	NOUN
ejpam-6096	17	12	of	of	ADP
ejpam-6096	17	13	many	many	ADJ
ejpam-6096	17	14	algebraic	algebraic	ADJ
ejpam-6096	17	15	structures	structure	NOUN
ejpam-6096	17	16	.	.	PUNCT
ejpam-6096	18	1	in	in	ADP
ejpam-6096	18	2	2005	2005	NUM
ejpam-6096	18	3	,	,	PUNCT
ejpam-6096	18	4	kar	kar	X
ejpam-6096	19	1	[	[	X
ejpam-6096	19	2	6	6	NUM
ejpam-6096	19	3	]	]	PUNCT
ejpam-6096	19	4	introduced	introduce	VERB
ejpam-6096	19	5	the	the	DET
ejpam-6096	19	6	notions	notion	NOUN
ejpam-6096	19	7	of	of	ADP
ejpam-6096	19	8	quasi	quasi	NOUN
ejpam-6096	19	9	-	-	NOUN
ejpam-6096	19	10	ideals	ideal	NOUN
ejpam-6096	19	11	and	and	CCONJ
ejpam-6096	19	12	bi	bi	NOUN
ejpam-6096	19	13	-	-	NOUN
ejpam-6096	19	14	ideals	ideal	NOUN
ejpam-6096	19	15	in	in	ADP
ejpam-6096	19	16	ternary	ternary	ADJ
ejpam-6096	19	17	semirings	semiring	NOUN
ejpam-6096	19	18	and	and	CCONJ
ejpam-6096	19	19	characterized	characterize	VERB
ejpam-6096	19	20	regular	regular	ADJ
ejpam-6096	19	21	ternary	ternary	ADJ
ejpam-6096	19	22	semirings	semiring	NOUN
ejpam-6096	19	23	in	in	ADP
ejpam-6096	19	24	terms	term	NOUN
ejpam-6096	19	25	of	of	ADP
ejpam-6096	19	26	quasi	quasi	NOUN
ejpam-6096	19	27	-	-	NOUN
ejpam-6096	19	28	ideals	ideal	NOUN
ejpam-6096	19	29	and	and	CCONJ
ejpam-6096	19	30	bi	bi	NOUN
ejpam-6096	19	31	-	-	NOUN
ejpam-6096	19	32	ideals	ideal	NOUN
ejpam-6096	19	33	.	.	PUNCT
ejpam-6096	20	1	in	in	ADP
ejpam-6096	20	2	2010	2010	NUM
ejpam-6096	20	3	,	,	PUNCT
ejpam-6096	20	4	malee	malee	ADJ
ejpam-6096	20	5	and	and	CCONJ
ejpam-6096	20	6	chinram	chinram	NOUN
ejpam-6096	20	7	studied	study	VERB
ejpam-6096	20	8	fuzzifications	fuzzification	NOUN
ejpam-6096	20	9	of	of	ADP
ejpam-6096	20	10	some	some	DET
ejpam-6096	20	11	ideals	ideal	NOUN
ejpam-6096	20	12	in	in	ADP
ejpam-6096	20	13	ternary	ternary	ADJ
ejpam-6096	20	14	semirings	semiring	NOUN
ejpam-6096	20	15	[	[	X
ejpam-6096	20	16	7	7	X
ejpam-6096	20	17	]	]	PUNCT
ejpam-6096	20	18	and	and	CCONJ
ejpam-6096	20	19	[	[	X
ejpam-6096	20	20	8	8	NUM
ejpam-6096	20	21	]	]	PUNCT
ejpam-6096	20	22	.	.	PUNCT
ejpam-6096	21	1	∗corresponding	∗corresponde	VERB
ejpam-6096	21	2	author	author	NOUN
ejpam-6096	21	3	.	.	PUNCT
ejpam-6096	22	1	doi	doi	NOUN
ejpam-6096	22	2	:	:	PUNCT
ejpam-6096	22	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6096	https://doi.org/10.29020/nybg.ejpam.v18i2.6096	PRON
ejpam-6096	22	4	email	email	NOUN
ejpam-6096	22	5	addresses	address	NOUN
ejpam-6096	22	6	:	:	PUNCT
ejpam-6096	22	7	arbazjehankhan@gmail.com	arbazjehankhan@gmail.com	X
ejpam-6096	22	8	(	(	PUNCT
ejpam-6096	22	9	a.	a.	PROPN
ejpam-6096	22	10	j.	j.	PROPN
ejpam-6096	22	11	khan	khan	PROPN
ejpam-6096	22	12	)	)	PUNCT
ejpam-6096	22	13	,	,	PUNCT
ejpam-6096	22	14	montakarn.p@psu.ac.th	montakarn.p@psu.ac.th	PROPN
ejpam-6096	22	15	(	(	PUNCT
ejpam-6096	22	16	m.	m.	NOUN
ejpam-6096	22	17	petapirak	petapirak	PROPN
ejpam-6096	22	18	)	)	PUNCT
ejpam-6096	22	19	,	,	PUNCT
ejpam-6096	22	20	ronnason.c@psu.ac.th	ronnason.c@psu.ac.th	PROPN
ejpam-6096	22	21	(	(	PUNCT
ejpam-6096	22	22	r.	r.	PROPN
ejpam-6096	22	23	chinram	chinram	PROPN
ejpam-6096	22	24	)	)	PUNCT
ejpam-6096	22	25	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6096	23	1	1	1	NUM
ejpam-6096	23	2	copyright	copyright	NOUN
ejpam-6096	23	3	:	:	PUNCT
ejpam-6096	23	4	©	©	PROPN
ejpam-6096	23	5	2025	2025	NUM
ejpam-6096	23	6	the	the	DET
ejpam-6096	23	7	author(s	author(s	NOUN
ejpam-6096	23	8	)	)	PUNCT
ejpam-6096	23	9	.	.	PUNCT
ejpam-6096	24	1	(	(	PUNCT
ejpam-6096	24	2	cc	cc	NOUN
ejpam-6096	24	3	by	by	ADP
ejpam-6096	24	4	-	-	PUNCT
ejpam-6096	24	5	nc	nc	PROPN
ejpam-6096	24	6	4.0	4.0	NUM
ejpam-6096	24	7	)	)	PUNCT
ejpam-6096	24	8	a.	a.	NOUN
ejpam-6096	24	9	j.	j.	PROPN
ejpam-6096	24	10	khan	khan	PROPN
ejpam-6096	24	11	,	,	PUNCT
ejpam-6096	24	12	m.	m.	NOUN
ejpam-6096	24	13	petapirak	petapirak	PROPN
ejpam-6096	24	14	,	,	PUNCT
ejpam-6096	24	15	r.	r.	PROPN
ejpam-6096	24	16	chinram	chinram	PROPN
ejpam-6096	24	17	/	/	SYM
ejpam-6096	24	18	eur	eur	PROPN
ejpam-6096	24	19	.	.	PUNCT
ejpam-6096	25	1	j.	j.	PROPN
ejpam-6096	25	2	pure	pure	PROPN
ejpam-6096	25	3	appl	appl	PROPN
ejpam-6096	25	4	.	.	PROPN
ejpam-6096	25	5	math	math	PROPN
ejpam-6096	25	6	,	,	PUNCT
ejpam-6096	25	7	18	18	NUM
ejpam-6096	25	8	(	(	PUNCT
ejpam-6096	25	9	2	2	NUM
ejpam-6096	25	10	)	)	PUNCT
ejpam-6096	25	11	(	(	PUNCT
ejpam-6096	25	12	2025	2025	NUM
ejpam-6096	25	13	)	)	PUNCT
ejpam-6096	25	14	,	,	PUNCT
ejpam-6096	25	15	6096	6096	NUM
ejpam-6096	25	16	2	2	NUM
ejpam-6096	25	17	of	of	ADP
ejpam-6096	25	18	10	10	NUM
ejpam-6096	25	19	the	the	DET
ejpam-6096	25	20	next	next	ADJ
ejpam-6096	25	21	year	year	NOUN
ejpam-6096	25	22	,	,	PUNCT
ejpam-6096	25	23	dubey	dubey	PROPN
ejpam-6096	26	1	[	[	X
ejpam-6096	26	2	9	9	NUM
ejpam-6096	26	3	]	]	PUNCT
ejpam-6096	26	4	investigated	investigate	VERB
ejpam-6096	26	5	quasi	quasi	ADJ
ejpam-6096	26	6	k	k	NOUN
ejpam-6096	26	7	-	-	NOUN
ejpam-6096	26	8	ideals	ideal	NOUN
ejpam-6096	26	9	and	and	CCONJ
ejpam-6096	26	10	bi	bi	ADJ
ejpam-6096	26	11	k	k	NOUN
ejpam-6096	26	12	-	-	NOUN
ejpam-6096	26	13	ideals	ideal	NOUN
ejpam-6096	26	14	in	in	ADP
ejpam-6096	26	15	ternary	ternary	ADJ
ejpam-6096	26	16	semirings	semiring	NOUN
ejpam-6096	26	17	.	.	PUNCT
ejpam-6096	27	1	in	in	ADP
ejpam-6096	27	2	the	the	DET
ejpam-6096	27	3	same	same	ADJ
ejpam-6096	27	4	year	year	NOUN
ejpam-6096	27	5	,	,	PUNCT
ejpam-6096	27	6	chaudhari	chaudhari	NOUN
ejpam-6096	27	7	and	and	CCONJ
ejpam-6096	27	8	ingale	ingale	VERB
ejpam-6096	27	9	[	[	X
ejpam-6096	27	10	10	10	NUM
ejpam-6096	27	11	]	]	PUNCT
ejpam-6096	27	12	introduced	introduce	VERB
ejpam-6096	27	13	a	a	DET
ejpam-6096	27	14	partitioning	partitioning	ADJ
ejpam-6096	27	15	ideal	ideal	NOUN
ejpam-6096	27	16	of	of	ADP
ejpam-6096	27	17	a	a	DET
ejpam-6096	27	18	ternary	ternary	ADJ
ejpam-6096	27	19	semiring	semiring	NOUN
ejpam-6096	27	20	and	and	CCONJ
ejpam-6096	27	21	sunitha	sunitha	VERB
ejpam-6096	27	22	et	et	PROPN
ejpam-6096	27	23	al	al	PROPN
ejpam-6096	27	24	.	.	PUNCT
ejpam-6096	28	1	[	[	X
ejpam-6096	28	2	11	11	NUM
ejpam-6096	28	3	]	]	PUNCT
ejpam-6096	28	4	provided	provide	VERB
ejpam-6096	28	5	the	the	DET
ejpam-6096	28	6	characterization	characterization	NOUN
ejpam-6096	28	7	of	of	ADP
ejpam-6096	28	8	full	full	ADJ
ejpam-6096	28	9	k	k	NOUN
ejpam-6096	28	10	-	-	NOUN
ejpam-6096	28	11	ideals	ideal	NOUN
ejpam-6096	28	12	in	in	ADP
ejpam-6096	28	13	ternary	ternary	ADJ
ejpam-6096	28	14	semirings	semiring	NOUN
ejpam-6096	28	15	.	.	PUNCT
ejpam-6096	29	1	partitioning	partition	VERB
ejpam-6096	29	2	ideals	ideal	NOUN
ejpam-6096	29	3	are	be	AUX
ejpam-6096	29	4	useful	useful	ADJ
ejpam-6096	29	5	to	to	PART
ejpam-6096	29	6	develop	develop	VERB
ejpam-6096	29	7	the	the	DET
ejpam-6096	29	8	quotient	quotient	NOUN
ejpam-6096	29	9	structures	structure	NOUN
ejpam-6096	29	10	of	of	ADP
ejpam-6096	29	11	ternary	ternary	ADJ
ejpam-6096	29	12	semirings	semiring	NOUN
ejpam-6096	29	13	.	.	PUNCT
ejpam-6096	30	1	singular	singular	PROPN
ejpam-6096	30	2	ideals	ideal	NOUN
ejpam-6096	30	3	were	be	AUX
ejpam-6096	30	4	introduced	introduce	VERB
ejpam-6096	30	5	in	in	ADP
ejpam-6096	30	6	[	[	X
ejpam-6096	30	7	12	12	NUM
ejpam-6096	30	8	]	]	PUNCT
ejpam-6096	30	9	in	in	ADP
ejpam-6096	30	10	2012	2012	NUM
ejpam-6096	30	11	.	.	PUNCT
ejpam-6096	31	1	in	in	ADP
ejpam-6096	31	2	2016	2016	NUM
ejpam-6096	31	3	,	,	PUNCT
ejpam-6096	31	4	the	the	DET
ejpam-6096	31	5	concept	concept	NOUN
ejpam-6096	31	6	of	of	ADP
ejpam-6096	31	7	the	the	DET
ejpam-6096	31	8	subtractive	subtractive	NOUN
ejpam-6096	31	9	extension	extension	NOUN
ejpam-6096	31	10	of	of	ADP
ejpam-6096	31	11	an	an	DET
ejpam-6096	31	12	ideal	ideal	NOUN
ejpam-6096	31	13	of	of	ADP
ejpam-6096	31	14	a	a	DET
ejpam-6096	31	15	ternary	ternary	ADJ
ejpam-6096	31	16	semiring	semiring	NOUN
ejpam-6096	31	17	was	be	AUX
ejpam-6096	31	18	introduced	introduce	VERB
ejpam-6096	31	19	in	in	ADP
ejpam-6096	31	20	[	[	X
ejpam-6096	31	21	13	13	NUM
ejpam-6096	31	22	]	]	PUNCT
ejpam-6096	31	23	.	.	PUNCT
ejpam-6096	32	1	in	in	ADP
ejpam-6096	32	2	2024	2024	NUM
ejpam-6096	32	3	,	,	PUNCT
ejpam-6096	32	4	luangchaisri	luangchaisri	VERB
ejpam-6096	32	5	and	and	CCONJ
ejpam-6096	32	6	changphas	changphas	ADJ
ejpam-6096	33	1	[	[	X
ejpam-6096	33	2	14	14	NUM
ejpam-6096	33	3	]	]	PUNCT
ejpam-6096	33	4	studied	study	VERB
ejpam-6096	33	5	right	right	ADJ
ejpam-6096	33	6	weakly	weakly	ADV
ejpam-6096	33	7	regular	regular	ADJ
ejpam-6096	33	8	ternary	ternary	ADJ
ejpam-6096	33	9	semirings	semiring	NOUN
ejpam-6096	33	10	and	and	CCONJ
ejpam-6096	33	11	fully	fully	ADV
ejpam-6096	33	12	prime	prime	ADJ
ejpam-6096	33	13	right	right	ADJ
ejpam-6096	33	14	ternary	ternary	ADJ
ejpam-6096	33	15	semirings	semiring	NOUN
ejpam-6096	33	16	.	.	PUNCT
ejpam-6096	34	1	in	in	ADP
ejpam-6096	34	2	the	the	DET
ejpam-6096	34	3	same	same	ADJ
ejpam-6096	34	4	year	year	NOUN
ejpam-6096	34	5	,	,	PUNCT
ejpam-6096	34	6	goswami	goswami	NOUN
ejpam-6096	34	7	and	and	CCONJ
ejpam-6096	34	8	dube	dube	PROPN
ejpam-6096	34	9	[	[	X
ejpam-6096	34	10	15	15	NUM
ejpam-6096	34	11	]	]	PUNCT
ejpam-6096	34	12	investigated	investigate	VERB
ejpam-6096	34	13	some	some	DET
ejpam-6096	34	14	aspects	aspect	NOUN
ejpam-6096	34	15	of	of	ADP
ejpam-6096	34	16	k	k	NOUN
ejpam-6096	34	17	-	-	NOUN
ejpam-6096	34	18	ideals	ideal	NOUN
ejpam-6096	34	19	of	of	ADP
ejpam-6096	34	20	semirings	semiring	NOUN
ejpam-6096	34	21	.	.	PUNCT
ejpam-6096	35	1	this	this	DET
ejpam-6096	35	2	paper	paper	NOUN
ejpam-6096	35	3	inspired	inspire	VERB
ejpam-6096	35	4	by	by	ADP
ejpam-6096	35	5	their	their	PRON
ejpam-6096	35	6	work	work	NOUN
ejpam-6096	35	7	.	.	PUNCT
ejpam-6096	36	1	our	our	PRON
ejpam-6096	36	2	goal	goal	NOUN
ejpam-6096	36	3	is	be	AUX
ejpam-6096	36	4	to	to	PART
ejpam-6096	36	5	explore	explore	VERB
ejpam-6096	36	6	various	various	ADJ
ejpam-6096	36	7	aspects	aspect	NOUN
ejpam-6096	36	8	of	of	ADP
ejpam-6096	36	9	k	k	NOUN
ejpam-6096	36	10	-	-	NOUN
ejpam-6096	36	11	ideals	ideal	NOUN
ejpam-6096	36	12	in	in	ADP
ejpam-6096	36	13	ternary	ternary	ADJ
ejpam-6096	36	14	semirings	semiring	NOUN
ejpam-6096	36	15	.	.	PUNCT
ejpam-6096	37	1	2	2	X
ejpam-6096	37	2	.	.	X
ejpam-6096	37	3	preliminaries	preliminary	NOUN
ejpam-6096	37	4	in	in	ADP
ejpam-6096	37	5	this	this	DET
ejpam-6096	37	6	section	section	NOUN
ejpam-6096	37	7	,	,	PUNCT
ejpam-6096	37	8	we	we	PRON
ejpam-6096	37	9	will	will	AUX
ejpam-6096	37	10	recall	recall	VERB
ejpam-6096	37	11	some	some	DET
ejpam-6096	37	12	basic	basic	ADJ
ejpam-6096	37	13	definitions	definition	NOUN
ejpam-6096	37	14	of	of	ADP
ejpam-6096	37	15	ternary	ternary	ADJ
ejpam-6096	37	16	semirings	semiring	NOUN
ejpam-6096	37	17	.	.	PUNCT
ejpam-6096	38	1	definition	definition	NOUN
ejpam-6096	38	2	1	1	NUM
ejpam-6096	38	3	(	(	PUNCT
ejpam-6096	38	4	[	[	X
ejpam-6096	38	5	3	3	NUM
ejpam-6096	38	6	]	]	NUM
ejpam-6096	38	7	)	)	PUNCT
ejpam-6096	38	8	.	.	PUNCT
ejpam-6096	39	1	a	a	DET
ejpam-6096	39	2	nonempty	nonempty	ADV
ejpam-6096	39	3	set	set	VERB
ejpam-6096	39	4	r	r	NOUN
ejpam-6096	39	5	together	together	ADV
ejpam-6096	39	6	with	with	ADP
ejpam-6096	39	7	a	a	DET
ejpam-6096	39	8	binary	binary	ADJ
ejpam-6096	39	9	operation	operation	NOUN
ejpam-6096	39	10	,	,	PUNCT
ejpam-6096	39	11	called	call	VERB
ejpam-6096	39	12	the	the	DET
ejpam-6096	39	13	addition	addition	NOUN
ejpam-6096	39	14	,	,	PUNCT
ejpam-6096	39	15	and	and	CCONJ
ejpam-6096	39	16	the	the	DET
ejpam-6096	39	17	ternary	ternary	ADJ
ejpam-6096	39	18	multiplication	multiplication	NOUN
ejpam-6096	39	19	,	,	PUNCT
ejpam-6096	39	20	denoted	denote	VERB
ejpam-6096	39	21	by	by	ADP
ejpam-6096	39	22	juxtaposition	juxtaposition	NOUN
ejpam-6096	39	23	,	,	PUNCT
ejpam-6096	39	24	is	be	AUX
ejpam-6096	39	25	said	say	VERB
ejpam-6096	39	26	to	to	PART
ejpam-6096	39	27	be	be	AUX
ejpam-6096	39	28	a	a	DET
ejpam-6096	39	29	ternary	ternary	ADJ
ejpam-6096	39	30	semiring	semiring	NOUN
ejpam-6096	39	31	if	if	SCONJ
ejpam-6096	39	32	r	r	NOUN
ejpam-6096	39	33	is	be	AUX
ejpam-6096	39	34	an	an	DET
ejpam-6096	39	35	additive	additive	ADJ
ejpam-6096	39	36	commutative	commutative	ADJ
ejpam-6096	39	37	semigroup	semigroup	NOUN
ejpam-6096	39	38	satisfying	satisfy	VERB
ejpam-6096	39	39	the	the	DET
ejpam-6096	39	40	following	follow	VERB
ejpam-6096	39	41	conditions	condition	NOUN
ejpam-6096	39	42	:	:	PUNCT
ejpam-6096	39	43	(	(	PUNCT
ejpam-6096	39	44	i	i	NOUN
ejpam-6096	39	45	)	)	PUNCT
ejpam-6096	39	46	(	(	PUNCT
ejpam-6096	39	47	abc)de	abc)de	NOUN
ejpam-6096	39	48	=	=	SYM
ejpam-6096	39	49	a(bcd)e	a(bcd)e	NOUN
ejpam-6096	39	50	=	=	PUNCT
ejpam-6096	39	51	ab(cde	ab(cde	PROPN
ejpam-6096	39	52	)	)	PUNCT
ejpam-6096	39	53	,	,	PUNCT
ejpam-6096	39	54	(	(	PUNCT
ejpam-6096	39	55	ii	ii	NOUN
ejpam-6096	39	56	)	)	PUNCT
ejpam-6096	39	57	(	(	PUNCT
ejpam-6096	39	58	a+	a+	PUNCT
ejpam-6096	39	59	b)cd	b)cd	PROPN
ejpam-6096	39	60	=	=	SYM
ejpam-6096	39	61	acd+	acd+	NOUN
ejpam-6096	39	62	bcd	bcd	PROPN
ejpam-6096	39	63	,	,	PUNCT
ejpam-6096	39	64	(	(	PUNCT
ejpam-6096	39	65	iii	iii	NOUN
ejpam-6096	39	66	)	)	PUNCT
ejpam-6096	39	67	a(b+	a(b+	ADV
ejpam-6096	39	68	c)d	c)d	NOUN
ejpam-6096	39	69	=	=	SYM
ejpam-6096	39	70	abd+	abd+	ADV
ejpam-6096	39	71	acd	acd	PROPN
ejpam-6096	39	72	,	,	PUNCT
ejpam-6096	39	73	(	(	PUNCT
ejpam-6096	39	74	iv	iv	X
ejpam-6096	39	75	)	)	PUNCT
ejpam-6096	39	76	ab(c+	ab(c+	PROPN
ejpam-6096	40	1	d	d	X
ejpam-6096	40	2	)	)	PUNCT
ejpam-6096	40	3	=	=	SYM
ejpam-6096	40	4	abc+	abc+	PROPN
ejpam-6096	40	5	abd	abd	PROPN
ejpam-6096	40	6	,	,	PUNCT
ejpam-6096	40	7	for	for	ADP
ejpam-6096	40	8	all	all	DET
ejpam-6096	40	9	a	a	DET
ejpam-6096	40	10	,	,	PUNCT
ejpam-6096	40	11	b	b	NOUN
ejpam-6096	40	12	,	,	PUNCT
ejpam-6096	40	13	c	c	NOUN
ejpam-6096	40	14	,	,	PUNCT
ejpam-6096	40	15	d	d	NOUN
ejpam-6096	40	16	,	,	PUNCT
ejpam-6096	40	17	e	e	PROPN
ejpam-6096	40	18	∈	∈	PROPN
ejpam-6096	40	19	r.	r.	PROPN
ejpam-6096	40	20	example	example	NOUN
ejpam-6096	41	1	1	1	NUM
ejpam-6096	41	2	.	.	PUNCT
ejpam-6096	41	3	(	(	PUNCT
ejpam-6096	41	4	1	1	X
ejpam-6096	41	5	)	)	PUNCT
ejpam-6096	41	6	every	every	DET
ejpam-6096	41	7	semiring	semiring	NOUN
ejpam-6096	41	8	can	can	AUX
ejpam-6096	41	9	be	be	AUX
ejpam-6096	41	10	considered	consider	VERB
ejpam-6096	41	11	a	a	DET
ejpam-6096	41	12	ternary	ternary	ADJ
ejpam-6096	41	13	semiring	semiring	NOUN
ejpam-6096	41	14	under	under	ADP
ejpam-6096	41	15	the	the	DET
ejpam-6096	41	16	ordinary	ordinary	ADJ
ejpam-6096	41	17	addition	addition	NOUN
ejpam-6096	41	18	and	and	CCONJ
ejpam-6096	41	19	ternary	ternary	ADJ
ejpam-6096	41	20	multiplication	multiplication	NOUN
ejpam-6096	41	21	of	of	ADP
ejpam-6096	41	22	semirings	semiring	NOUN
ejpam-6096	41	23	.	.	PUNCT
ejpam-6096	42	1	(	(	PUNCT
ejpam-6096	42	2	2	2	X
ejpam-6096	42	3	)	)	PUNCT
ejpam-6096	42	4	z−	z−	PROPN
ejpam-6096	42	5	is	be	AUX
ejpam-6096	42	6	a	a	DET
ejpam-6096	42	7	ternary	ternary	ADJ
ejpam-6096	42	8	semiring	semiring	NOUN
ejpam-6096	42	9	under	under	ADP
ejpam-6096	42	10	the	the	DET
ejpam-6096	42	11	usual	usual	ADJ
ejpam-6096	42	12	addition	addition	NOUN
ejpam-6096	42	13	and	and	CCONJ
ejpam-6096	42	14	ternary	ternary	ADJ
ejpam-6096	42	15	multiplication	multiplication	NOUN
ejpam-6096	42	16	of	of	ADP
ejpam-6096	42	17	integers	integer	NOUN
ejpam-6096	42	18	,	,	PUNCT
ejpam-6096	42	19	but	but	CCONJ
ejpam-6096	42	20	it	it	PRON
ejpam-6096	42	21	is	be	AUX
ejpam-6096	42	22	not	not	PART
ejpam-6096	42	23	a	a	DET
ejpam-6096	42	24	semiring	semiring	NOUN
ejpam-6096	42	25	under	under	ADP
ejpam-6096	42	26	the	the	DET
ejpam-6096	42	27	usual	usual	ADJ
ejpam-6096	42	28	addition	addition	NOUN
ejpam-6096	42	29	and	and	CCONJ
ejpam-6096	42	30	binary	binary	ADJ
ejpam-6096	42	31	multiplication	multiplication	NOUN
ejpam-6096	42	32	of	of	ADP
ejpam-6096	42	33	integers	integer	NOUN
ejpam-6096	42	34	.	.	PUNCT
ejpam-6096	43	1	definition	definition	NOUN
ejpam-6096	43	2	2	2	NUM
ejpam-6096	43	3	(	(	PUNCT
ejpam-6096	43	4	[	[	X
ejpam-6096	43	5	3	3	NUM
ejpam-6096	43	6	]	]	PUNCT
ejpam-6096	43	7	)	)	PUNCT
ejpam-6096	43	8	.	.	PUNCT
ejpam-6096	44	1	let	let	VERB
ejpam-6096	44	2	r	r	PRON
ejpam-6096	44	3	be	be	AUX
ejpam-6096	44	4	a	a	DET
ejpam-6096	44	5	ternary	ternary	ADJ
ejpam-6096	44	6	semiring	semiring	NOUN
ejpam-6096	44	7	.	.	PUNCT
ejpam-6096	45	1	if	if	SCONJ
ejpam-6096	45	2	0	0	NUM
ejpam-6096	45	3	∈	∈	NOUN
ejpam-6096	45	4	r	r	NOUN
ejpam-6096	45	5	such	such	ADJ
ejpam-6096	45	6	that	that	PRON
ejpam-6096	45	7	0	0	NUM
ejpam-6096	46	1	+	+	NUM
ejpam-6096	46	2	x	x	SYM
ejpam-6096	46	3	=	=	SYM
ejpam-6096	46	4	x	x	X
ejpam-6096	46	5	and	and	CCONJ
ejpam-6096	46	6	0xy	0xy	NOUN
ejpam-6096	46	7	=	=	PUNCT
ejpam-6096	46	8	x0y	x0y	PUNCT
ejpam-6096	47	1	=	=	PUNCT
ejpam-6096	47	2	xy0	xy0	X
ejpam-6096	48	1	=	=	SYM
ejpam-6096	48	2	0	0	NUM
ejpam-6096	48	3	for	for	ADP
ejpam-6096	48	4	all	all	DET
ejpam-6096	48	5	x	x	NOUN
ejpam-6096	48	6	,	,	PUNCT
ejpam-6096	48	7	y	y	PROPN
ejpam-6096	48	8	∈	∈	PROPN
ejpam-6096	48	9	r	r	NOUN
ejpam-6096	48	10	,	,	PUNCT
ejpam-6096	48	11	then	then	ADV
ejpam-6096	48	12	0	0	NUM
ejpam-6096	48	13	is	be	AUX
ejpam-6096	48	14	called	call	VERB
ejpam-6096	48	15	a	a	DET
ejpam-6096	48	16	zero	zero	NUM
ejpam-6096	48	17	element	element	NOUN
ejpam-6096	48	18	.	.	PUNCT
ejpam-6096	49	1	in	in	ADP
ejpam-6096	49	2	this	this	DET
ejpam-6096	49	3	case	case	NOUN
ejpam-6096	49	4	,	,	PUNCT
ejpam-6096	49	5	r	r	NOUN
ejpam-6096	49	6	is	be	AUX
ejpam-6096	49	7	called	call	VERB
ejpam-6096	49	8	a	a	DET
ejpam-6096	49	9	ternary	ternary	ADJ
ejpam-6096	49	10	semiring	semiring	NOUN
ejpam-6096	49	11	with	with	ADP
ejpam-6096	49	12	zero	zero	NUM
ejpam-6096	49	13	.	.	PUNCT
ejpam-6096	50	1	definition	definition	NOUN
ejpam-6096	50	2	3	3	NUM
ejpam-6096	50	3	.	.	PUNCT
ejpam-6096	51	1	a	a	DET
ejpam-6096	51	2	ternary	ternary	ADJ
ejpam-6096	51	3	semiring	semiring	NOUN
ejpam-6096	51	4	r	r	NOUN
ejpam-6096	51	5	is	be	AUX
ejpam-6096	51	6	said	say	VERB
ejpam-6096	51	7	to	to	PART
ejpam-6096	51	8	be	be	AUX
ejpam-6096	51	9	commutative	commutative	ADJ
ejpam-6096	51	10	if	if	SCONJ
ejpam-6096	51	11	abc	abc	PROPN
ejpam-6096	51	12	=	=	SYM
ejpam-6096	51	13	bca	bca	PROPN
ejpam-6096	51	14	=	=	SYM
ejpam-6096	51	15	cab	cab	NOUN
ejpam-6096	51	16	=	=	SYM
ejpam-6096	51	17	bac	bac	NOUN
ejpam-6096	51	18	=	=	PROPN
ejpam-6096	51	19	cba	cba	PROPN
ejpam-6096	51	20	=	=	PUNCT
ejpam-6096	51	21	acb	acb	VERB
ejpam-6096	51	22	for	for	ADP
ejpam-6096	51	23	all	all	DET
ejpam-6096	51	24	a	a	DET
ejpam-6096	51	25	,	,	PUNCT
ejpam-6096	51	26	b	b	NOUN
ejpam-6096	51	27	,	,	PUNCT
ejpam-6096	51	28	c	c	PROPN
ejpam-6096	51	29	∈	∈	PROPN
ejpam-6096	51	30	r.	r.	PROPN
ejpam-6096	51	31	definition	definition	NOUN
ejpam-6096	51	32	4	4	NUM
ejpam-6096	51	33	(	(	PUNCT
ejpam-6096	51	34	[	[	X
ejpam-6096	51	35	3	3	NUM
ejpam-6096	51	36	]	]	NUM
ejpam-6096	51	37	)	)	PUNCT
ejpam-6096	51	38	.	.	PUNCT
ejpam-6096	52	1	an	an	DET
ejpam-6096	52	2	additive	additive	ADJ
ejpam-6096	52	3	semigroup	semigroup	NOUN
ejpam-6096	52	4	s	s	PROPN
ejpam-6096	52	5	of	of	ADP
ejpam-6096	52	6	a	a	DET
ejpam-6096	52	7	ternary	ternary	ADJ
ejpam-6096	52	8	semiring	semiring	NOUN
ejpam-6096	52	9	r	r	NOUN
ejpam-6096	52	10	is	be	AUX
ejpam-6096	52	11	called	call	VERB
ejpam-6096	52	12	a	a	DET
ejpam-6096	52	13	ternary	ternary	ADJ
ejpam-6096	52	14	subsemiring	subsemiring	NOUN
ejpam-6096	52	15	of	of	ADP
ejpam-6096	52	16	r	r	NOUN
ejpam-6096	52	17	if	if	SCONJ
ejpam-6096	52	18	s1s2s3	s1s2s3	PROPN
ejpam-6096	52	19	∈	∈	PROPN
ejpam-6096	52	20	s	s	X
ejpam-6096	52	21	for	for	ADP
ejpam-6096	52	22	all	all	DET
ejpam-6096	52	23	s1	s1	NOUN
ejpam-6096	52	24	,	,	PUNCT
ejpam-6096	52	25	s2	s2	PROPN
ejpam-6096	52	26	,	,	PUNCT
ejpam-6096	52	27	s3	s3	PROPN
ejpam-6096	52	28	∈	∈	PROPN
ejpam-6096	52	29	s.	s.	PROPN
ejpam-6096	52	30	a.	a.	PROPN
ejpam-6096	52	31	j.	j.	PROPN
ejpam-6096	52	32	khan	khan	PROPN
ejpam-6096	52	33	,	,	PUNCT
ejpam-6096	52	34	m.	m.	NOUN
ejpam-6096	52	35	petapirak	petapirak	PROPN
ejpam-6096	52	36	,	,	PUNCT
ejpam-6096	52	37	r.	r.	PROPN
ejpam-6096	52	38	chinram	chinram	PROPN
ejpam-6096	52	39	/	/	SYM
ejpam-6096	52	40	eur	eur	PROPN
ejpam-6096	52	41	.	.	PUNCT
ejpam-6096	53	1	j.	j.	PROPN
ejpam-6096	53	2	pure	pure	PROPN
ejpam-6096	53	3	appl	appl	PROPN
ejpam-6096	53	4	.	.	PROPN
ejpam-6096	53	5	math	math	PROPN
ejpam-6096	53	6	,	,	PUNCT
ejpam-6096	53	7	18	18	NUM
ejpam-6096	53	8	(	(	PUNCT
ejpam-6096	53	9	2	2	NUM
ejpam-6096	53	10	)	)	PUNCT
ejpam-6096	53	11	(	(	PUNCT
ejpam-6096	53	12	2025	2025	NUM
ejpam-6096	53	13	)	)	PUNCT
ejpam-6096	53	14	,	,	PUNCT
ejpam-6096	53	15	6096	6096	NUM
ejpam-6096	53	16	3	3	NUM
ejpam-6096	53	17	of	of	ADP
ejpam-6096	53	18	10	10	NUM
ejpam-6096	53	19	definition	definition	NOUN
ejpam-6096	53	20	5	5	NUM
ejpam-6096	53	21	.	.	PUNCT
ejpam-6096	54	1	an	an	DET
ejpam-6096	54	2	additive	additive	ADJ
ejpam-6096	54	3	subsemigroup	subsemigroup	NOUN
ejpam-6096	54	4	i	i	PRON
ejpam-6096	54	5	of	of	ADP
ejpam-6096	54	6	a	a	DET
ejpam-6096	54	7	ternary	ternary	ADJ
ejpam-6096	54	8	semiring	semiring	NOUN
ejpam-6096	54	9	r	r	NOUN
ejpam-6096	54	10	is	be	AUX
ejpam-6096	54	11	called	call	VERB
ejpam-6096	54	12	(	(	PUNCT
ejpam-6096	54	13	1	1	NUM
ejpam-6096	54	14	)	)	PUNCT
ejpam-6096	54	15	a	a	DET
ejpam-6096	54	16	left	left	ADJ
ejpam-6096	54	17	ideal	ideal	NOUN
ejpam-6096	54	18	of	of	ADP
ejpam-6096	54	19	r	r	NOUN
ejpam-6096	54	20	if	if	SCONJ
ejpam-6096	54	21	r1r2a	r1r2a	PUNCT
ejpam-6096	54	22	∈	∈	PROPN
ejpam-6096	54	23	i	i	PRON
ejpam-6096	54	24	for	for	ADP
ejpam-6096	54	25	all	all	DET
ejpam-6096	54	26	r1	r1	NOUN
ejpam-6096	54	27	,	,	PUNCT
ejpam-6096	54	28	r2	r2	PROPN
ejpam-6096	54	29	∈	∈	PROPN
ejpam-6096	54	30	r	r	NOUN
ejpam-6096	54	31	and	and	CCONJ
ejpam-6096	54	32	a	a	DET
ejpam-6096	54	33	∈	∈	NOUN
ejpam-6096	54	34	i	i	PRON
ejpam-6096	54	35	,	,	PUNCT
ejpam-6096	54	36	(	(	PUNCT
ejpam-6096	54	37	2	2	X
ejpam-6096	54	38	)	)	PUNCT
ejpam-6096	54	39	a	a	DET
ejpam-6096	54	40	right	right	ADJ
ejpam-6096	54	41	ideal	ideal	NOUN
ejpam-6096	54	42	of	of	ADP
ejpam-6096	54	43	r	r	NOUN
ejpam-6096	54	44	if	if	SCONJ
ejpam-6096	54	45	ar1r2	ar1r2	PROPN
ejpam-6096	54	46	∈	∈	PROPN
ejpam-6096	54	47	i	i	PRON
ejpam-6096	54	48	for	for	ADP
ejpam-6096	54	49	all	all	DET
ejpam-6096	54	50	r1	r1	NOUN
ejpam-6096	54	51	,	,	PUNCT
ejpam-6096	54	52	r2	r2	PROPN
ejpam-6096	54	53	∈	∈	PROPN
ejpam-6096	54	54	r	r	NOUN
ejpam-6096	54	55	and	and	CCONJ
ejpam-6096	54	56	a	a	DET
ejpam-6096	54	57	∈	∈	NOUN
ejpam-6096	54	58	i	i	PRON
ejpam-6096	54	59	,	,	PUNCT
ejpam-6096	54	60	(	(	PUNCT
ejpam-6096	54	61	3	3	X
ejpam-6096	54	62	)	)	PUNCT
ejpam-6096	54	63	a	a	DET
ejpam-6096	54	64	lateral	lateral	ADJ
ejpam-6096	54	65	ideal	ideal	NOUN
ejpam-6096	54	66	of	of	ADP
ejpam-6096	54	67	r	r	NOUN
ejpam-6096	54	68	if	if	SCONJ
ejpam-6096	54	69	r1ar2	r1ar2	VERB
ejpam-6096	54	70	∈	∈	PROPN
ejpam-6096	54	71	i	i	PRON
ejpam-6096	54	72	for	for	ADP
ejpam-6096	54	73	all	all	DET
ejpam-6096	54	74	r1	r1	NOUN
ejpam-6096	54	75	,	,	PUNCT
ejpam-6096	54	76	r2	r2	PROPN
ejpam-6096	54	77	∈	∈	PROPN
ejpam-6096	54	78	r	r	NOUN
ejpam-6096	54	79	and	and	CCONJ
ejpam-6096	54	80	a	a	DET
ejpam-6096	54	81	∈	∈	NOUN
ejpam-6096	54	82	i	i	PRON
ejpam-6096	54	83	,	,	PUNCT
ejpam-6096	54	84	(	(	PUNCT
ejpam-6096	54	85	4	4	X
ejpam-6096	54	86	)	)	PUNCT
ejpam-6096	54	87	an	an	DET
ejpam-6096	54	88	ideal	ideal	NOUN
ejpam-6096	54	89	of	of	ADP
ejpam-6096	54	90	r	r	NOUN
ejpam-6096	54	91	if	if	SCONJ
ejpam-6096	54	92	i	i	PRON
ejpam-6096	54	93	is	be	AUX
ejpam-6096	54	94	a	a	DET
ejpam-6096	54	95	left	left	ADJ
ejpam-6096	54	96	ideal	ideal	NOUN
ejpam-6096	54	97	,	,	PUNCT
ejpam-6096	54	98	a	a	DET
ejpam-6096	54	99	right	right	ADJ
ejpam-6096	54	100	ideal	ideal	NOUN
ejpam-6096	54	101	,	,	PUNCT
ejpam-6096	54	102	and	and	CCONJ
ejpam-6096	54	103	a	a	DET
ejpam-6096	54	104	lateral	lateral	ADJ
ejpam-6096	54	105	ideal	ideal	NOUN
ejpam-6096	54	106	of	of	ADP
ejpam-6096	54	107	r.	r.	PROPN
ejpam-6096	54	108	an	an	DET
ejpam-6096	54	109	ideal	ideal	NOUN
ejpam-6096	54	110	i	i	PRON
ejpam-6096	54	111	of	of	ADP
ejpam-6096	54	112	r	r	NOUN
ejpam-6096	54	113	is	be	AUX
ejpam-6096	54	114	called	call	VERB
ejpam-6096	54	115	a	a	DET
ejpam-6096	54	116	proper	proper	ADJ
ejpam-6096	54	117	ideal	ideal	NOUN
ejpam-6096	54	118	if	if	SCONJ
ejpam-6096	54	119	i	i	PRON
ejpam-6096	54	120	̸=	̸=	PROPN
ejpam-6096	54	121	r.	r.	NOUN
ejpam-6096	54	122	proposition	proposition	NOUN
ejpam-6096	54	123	1	1	NUM
ejpam-6096	54	124	(	(	PUNCT
ejpam-6096	54	125	[	[	X
ejpam-6096	54	126	3	3	NUM
ejpam-6096	54	127	]	]	PUNCT
ejpam-6096	54	128	)	)	PUNCT
ejpam-6096	54	129	.	.	PUNCT
ejpam-6096	55	1	let	let	VERB
ejpam-6096	55	2	r	r	PRON
ejpam-6096	55	3	be	be	AUX
ejpam-6096	55	4	a	a	DET
ejpam-6096	55	5	ternary	ternary	ADJ
ejpam-6096	55	6	semiring	semiring	NOUN
ejpam-6096	55	7	and	and	CCONJ
ejpam-6096	55	8	a	a	DET
ejpam-6096	55	9	∈	∈	PROPN
ejpam-6096	55	10	r.	r.	NOUN
ejpam-6096	55	11	then	then	ADV
ejpam-6096	55	12	the	the	DET
ejpam-6096	55	13	following	follow	VERB
ejpam-6096	55	14	statements	statement	NOUN
ejpam-6096	55	15	hold	hold	VERB
ejpam-6096	55	16	.	.	PUNCT
ejpam-6096	56	1	(	(	PUNCT
ejpam-6096	56	2	1	1	X
ejpam-6096	56	3	)	)	PUNCT
ejpam-6096	56	4	the	the	DET
ejpam-6096	56	5	principal	principal	NOUN
ejpam-6096	56	6	left	leave	VERB
ejpam-6096	56	7	ideal	ideal	NOUN
ejpam-6096	56	8	generated	generate	VERB
ejpam-6096	56	9	by	by	ADP
ejpam-6096	56	10	a	a	PRON
ejpam-6096	56	11	is	be	AUX
ejpam-6096	56	12	given	give	VERB
ejpam-6096	56	13	by	by	ADP
ejpam-6096	56	14	⟨a⟩l	⟨a⟩l	PROPN
ejpam-6096	56	15	=	=	SYM
ejpam-6096	56	16	n0a+rra	n0a+rra	X
ejpam-6096	56	17	.	.	PUNCT
ejpam-6096	57	1	(	(	PUNCT
ejpam-6096	57	2	2	2	X
ejpam-6096	57	3	)	)	PUNCT
ejpam-6096	57	4	the	the	DET
ejpam-6096	57	5	principal	principal	ADJ
ejpam-6096	57	6	right	right	ADJ
ejpam-6096	57	7	ideal	ideal	NOUN
ejpam-6096	57	8	generated	generate	VERB
ejpam-6096	57	9	by	by	ADP
ejpam-6096	57	10	a	a	PRON
ejpam-6096	57	11	is	be	AUX
ejpam-6096	57	12	given	give	VERB
ejpam-6096	57	13	by	by	ADP
ejpam-6096	57	14	⟨a⟩r	⟨a⟩r	PROPN
ejpam-6096	57	15	=	=	SYM
ejpam-6096	57	16	n0a+	n0a+	X
ejpam-6096	57	17	arr	arr	PROPN
ejpam-6096	57	18	.	.	PUNCT
ejpam-6096	58	1	(	(	PUNCT
ejpam-6096	58	2	3	3	X
ejpam-6096	58	3	)	)	PUNCT
ejpam-6096	58	4	the	the	DET
ejpam-6096	58	5	principal	principal	ADJ
ejpam-6096	58	6	lateral	lateral	ADJ
ejpam-6096	58	7	ideal	ideal	NOUN
ejpam-6096	58	8	generated	generate	VERB
ejpam-6096	58	9	by	by	ADP
ejpam-6096	58	10	a	a	PRON
ejpam-6096	58	11	is	be	AUX
ejpam-6096	58	12	given	give	VERB
ejpam-6096	58	13	by	by	ADP
ejpam-6096	58	14	⟨a⟩m	⟨a⟩m	NOUN
ejpam-6096	58	15	=	=	SYM
ejpam-6096	58	16	n0a+rar+rrarr	n0a+rar+rrarr	NOUN
ejpam-6096	58	17	.	.	PUNCT
ejpam-6096	59	1	(	(	PUNCT
ejpam-6096	59	2	4	4	X
ejpam-6096	59	3	)	)	PUNCT
ejpam-6096	59	4	the	the	DET
ejpam-6096	59	5	principal	principal	ADJ
ejpam-6096	59	6	ideal	ideal	NOUN
ejpam-6096	59	7	generated	generate	VERB
ejpam-6096	59	8	by	by	ADP
ejpam-6096	59	9	a	a	PRON
ejpam-6096	59	10	is	be	AUX
ejpam-6096	59	11	given	give	VERB
ejpam-6096	59	12	by	by	ADP
ejpam-6096	59	13	⟨a⟩	⟨a⟩	NOUN
ejpam-6096	59	14	=	=	PUNCT
ejpam-6096	59	15	n0a	n0a	PROPN
ejpam-6096	59	16	+	+	NUM
ejpam-6096	59	17	rra	rra	NOUN
ejpam-6096	59	18	+	+	CCONJ
ejpam-6096	59	19	arr	arr	NOUN
ejpam-6096	59	20	+	+	CCONJ
ejpam-6096	59	21	rar	rar	NOUN
ejpam-6096	59	22	+	+	X
ejpam-6096	59	23	rrarr	rrarr	ADJ
ejpam-6096	59	24	.	.	PUNCT
ejpam-6096	60	1	if	if	SCONJ
ejpam-6096	60	2	r	r	NOUN
ejpam-6096	60	3	is	be	AUX
ejpam-6096	60	4	commutative	commutative	ADJ
ejpam-6096	60	5	,	,	PUNCT
ejpam-6096	60	6	we	we	PRON
ejpam-6096	60	7	note	note	VERB
ejpam-6096	60	8	that	that	SCONJ
ejpam-6096	60	9	⟨a⟩	⟨a⟩	PRON
ejpam-6096	60	10	=	=	SYM
ejpam-6096	60	11	n0a+rra	n0a+rra	NUM
ejpam-6096	60	12	.	.	PUNCT
ejpam-6096	61	1	definition	definition	NOUN
ejpam-6096	61	2	6	6	NUM
ejpam-6096	61	3	.	.	PUNCT
ejpam-6096	62	1	an	an	DET
ejpam-6096	62	2	ideal	ideal	ADJ
ejpam-6096	62	3	i	i	PRON
ejpam-6096	62	4	of	of	ADP
ejpam-6096	62	5	a	a	DET
ejpam-6096	62	6	ternary	ternary	ADJ
ejpam-6096	62	7	semiring	semiring	NOUN
ejpam-6096	62	8	r	r	NOUN
ejpam-6096	62	9	is	be	AUX
ejpam-6096	62	10	called	call	VERB
ejpam-6096	62	11	a	a	DET
ejpam-6096	62	12	k	k	NOUN
ejpam-6096	62	13	-	-	NOUN
ejpam-6096	62	14	ideal	ideal	NOUN
ejpam-6096	62	15	if	if	SCONJ
ejpam-6096	62	16	,	,	PUNCT
ejpam-6096	62	17	for	for	ADP
ejpam-6096	62	18	all	all	DET
ejpam-6096	62	19	x	x	NOUN
ejpam-6096	62	20	,	,	PUNCT
ejpam-6096	62	21	y	y	PROPN
ejpam-6096	62	22	∈	∈	PROPN
ejpam-6096	62	23	r	r	NOUN
ejpam-6096	62	24	,	,	PUNCT
ejpam-6096	62	25	x	x	SYM
ejpam-6096	62	26	∈	∈	NOUN
ejpam-6096	62	27	i	i	PRON
ejpam-6096	62	28	and	and	CCONJ
ejpam-6096	62	29	x+	x+	ADJ
ejpam-6096	62	30	y	y	PROPN
ejpam-6096	62	31	∈	∈	PROPN
ejpam-6096	63	1	i	i	PRON
ejpam-6096	63	2	imply	imply	VERB
ejpam-6096	63	3	y	y	PROPN
ejpam-6096	63	4	∈	∈	PROPN
ejpam-6096	63	5	i.	i.	NOUN
ejpam-6096	63	6	definition	definition	NOUN
ejpam-6096	63	7	7	7	NUM
ejpam-6096	63	8	.	.	PUNCT
ejpam-6096	64	1	a	a	DET
ejpam-6096	64	2	proper	proper	ADJ
ejpam-6096	64	3	ideal	ideal	NOUN
ejpam-6096	64	4	of	of	ADP
ejpam-6096	64	5	a	a	DET
ejpam-6096	64	6	ternary	ternary	ADJ
ejpam-6096	64	7	semiring	semiring	NOUN
ejpam-6096	64	8	r	r	NOUN
ejpam-6096	64	9	is	be	AUX
ejpam-6096	64	10	called	call	VERB
ejpam-6096	64	11	maximal	maximal	ADJ
ejpam-6096	64	12	if	if	SCONJ
ejpam-6096	64	13	it	it	PRON
ejpam-6096	64	14	is	be	AUX
ejpam-6096	64	15	not	not	PART
ejpam-6096	64	16	properly	properly	ADV
ejpam-6096	64	17	contained	contain	VERB
ejpam-6096	64	18	in	in	ADP
ejpam-6096	64	19	any	any	DET
ejpam-6096	64	20	other	other	ADJ
ejpam-6096	64	21	proper	proper	ADJ
ejpam-6096	64	22	ideal	ideal	NOUN
ejpam-6096	64	23	of	of	ADP
ejpam-6096	64	24	r.	r.	PROPN
ejpam-6096	64	25	definition	definition	NOUN
ejpam-6096	64	26	8	8	NUM
ejpam-6096	64	27	.	.	PUNCT
ejpam-6096	65	1	(	(	PUNCT
ejpam-6096	65	2	[	[	X
ejpam-6096	65	3	4	4	NUM
ejpam-6096	65	4	]	]	PUNCT
ejpam-6096	65	5	)	)	PUNCT
ejpam-6096	65	6	a	a	DET
ejpam-6096	65	7	proper	proper	ADJ
ejpam-6096	65	8	ideal	ideal	NOUN
ejpam-6096	65	9	p	p	NOUN
ejpam-6096	65	10	of	of	ADP
ejpam-6096	65	11	a	a	DET
ejpam-6096	65	12	ternary	ternary	ADJ
ejpam-6096	65	13	semiring	semiring	NOUN
ejpam-6096	65	14	r	r	NOUN
ejpam-6096	65	15	is	be	AUX
ejpam-6096	65	16	called	call	VERB
ejpam-6096	65	17	prime	prime	ADJ
ejpam-6096	65	18	if	if	SCONJ
ejpam-6096	65	19	ijk	ijk	PROPN
ejpam-6096	65	20	⊆	⊆	PROPN
ejpam-6096	65	21	p	p	PROPN
ejpam-6096	65	22	implies	imply	VERB
ejpam-6096	65	23	i	i	PRON
ejpam-6096	65	24	⊆	⊆	NUM
ejpam-6096	65	25	p	p	NOUN
ejpam-6096	65	26	,	,	PUNCT
ejpam-6096	65	27	j	j	PROPN
ejpam-6096	65	28	⊆	⊆	NUM
ejpam-6096	65	29	p	p	NOUN
ejpam-6096	65	30	,	,	PUNCT
ejpam-6096	65	31	or	or	CCONJ
ejpam-6096	65	32	k	k	PROPN
ejpam-6096	66	1	⊆	⊆	NUM
ejpam-6096	66	2	p	p	NOUN
ejpam-6096	66	3	for	for	ADP
ejpam-6096	66	4	all	all	DET
ejpam-6096	66	5	ideals	ideal	NOUN
ejpam-6096	66	6	i	i	PRON
ejpam-6096	66	7	,	,	PUNCT
ejpam-6096	66	8	j	j	PROPN
ejpam-6096	66	9	,	,	PUNCT
ejpam-6096	66	10	k	k	PROPN
ejpam-6096	66	11	of	of	ADP
ejpam-6096	66	12	r.	r.	PROPN
ejpam-6096	66	13	definition	definition	NOUN
ejpam-6096	66	14	9	9	NUM
ejpam-6096	66	15	.	.	PUNCT
ejpam-6096	67	1	(	(	PUNCT
ejpam-6096	67	2	[	[	X
ejpam-6096	67	3	5	5	NUM
ejpam-6096	67	4	]	]	PUNCT
ejpam-6096	67	5	)	)	PUNCT
ejpam-6096	67	6	a	a	DET
ejpam-6096	67	7	proper	proper	ADJ
ejpam-6096	67	8	ideal	ideal	NOUN
ejpam-6096	67	9	p	p	NOUN
ejpam-6096	67	10	of	of	ADP
ejpam-6096	67	11	a	a	DET
ejpam-6096	67	12	ternary	ternary	ADJ
ejpam-6096	67	13	semiring	semiring	NOUN
ejpam-6096	67	14	r	r	NOUN
ejpam-6096	67	15	is	be	AUX
ejpam-6096	67	16	called	call	VERB
ejpam-6096	67	17	semiprime	semiprime	NOUN
ejpam-6096	67	18	if	if	SCONJ
ejpam-6096	67	19	i3	i3	VERB
ejpam-6096	67	20	⊆	⊆	PROPN
ejpam-6096	67	21	p	p	NOUN
ejpam-6096	67	22	implies	imply	VERB
ejpam-6096	67	23	i	i	PRON
ejpam-6096	67	24	⊆	⊆	NUM
ejpam-6096	67	25	p	p	NOUN
ejpam-6096	67	26	for	for	ADP
ejpam-6096	67	27	every	every	DET
ejpam-6096	67	28	ideal	ideal	NOUN
ejpam-6096	67	29	i	i	PRON
ejpam-6096	67	30	of	of	ADP
ejpam-6096	67	31	r.	r.	PROPN
ejpam-6096	67	32	3	3	NUM
ejpam-6096	67	33	.	.	PUNCT
ejpam-6096	68	1	main	main	ADJ
ejpam-6096	68	2	results	result	NOUN
ejpam-6096	68	3	let	let	VERB
ejpam-6096	68	4	j	j	PROPN
ejpam-6096	68	5	(	(	PUNCT
ejpam-6096	68	6	r	r	NOUN
ejpam-6096	68	7	)	)	PUNCT
ejpam-6096	68	8	denote	denote	NOUN
ejpam-6096	68	9	the	the	DET
ejpam-6096	68	10	set	set	NOUN
ejpam-6096	68	11	of	of	ADP
ejpam-6096	68	12	all	all	DET
ejpam-6096	68	13	ideals	ideal	NOUN
ejpam-6096	68	14	of	of	ADP
ejpam-6096	68	15	a	a	DET
ejpam-6096	68	16	ternary	ternary	ADJ
ejpam-6096	68	17	semiring	semiring	NOUN
ejpam-6096	68	18	r	r	NOUN
ejpam-6096	68	19	and	and	CCONJ
ejpam-6096	68	20	jk(r	jk(r	PROPN
ejpam-6096	68	21	)	)	PUNCT
ejpam-6096	68	22	denote	denote	VERB
ejpam-6096	68	23	the	the	DET
ejpam-6096	68	24	set	set	NOUN
ejpam-6096	68	25	of	of	ADP
ejpam-6096	68	26	all	all	DET
ejpam-6096	68	27	k	k	NOUN
ejpam-6096	68	28	-	-	NOUN
ejpam-6096	68	29	ideals	ideal	NOUN
ejpam-6096	68	30	of	of	ADP
ejpam-6096	68	31	r.	r.	PROPN
ejpam-6096	68	32	suppose	suppose	VERB
ejpam-6096	68	33	r	r	NOUN
ejpam-6096	68	34	is	be	AUX
ejpam-6096	68	35	a	a	DET
ejpam-6096	68	36	commutative	commutative	ADJ
ejpam-6096	68	37	ternary	ternary	NOUN
ejpam-6096	68	38	semiring	semiring	NOUN
ejpam-6096	68	39	with	with	ADP
ejpam-6096	68	40	zero	zero	NUM
ejpam-6096	68	41	0	0	NUM
ejpam-6096	68	42	.	.	PUNCT
ejpam-6096	69	1	if	if	SCONJ
ejpam-6096	69	2	a	a	PRON
ejpam-6096	69	3	is	be	AUX
ejpam-6096	69	4	a	a	DET
ejpam-6096	69	5	nonempty	nonempty	ADJ
ejpam-6096	69	6	subset	subset	NOUN
ejpam-6096	69	7	of	of	ADP
ejpam-6096	69	8	r	r	NOUN
ejpam-6096	69	9	,	,	PUNCT
ejpam-6096	69	10	then	then	ADV
ejpam-6096	69	11	the	the	DET
ejpam-6096	69	12	annihilator	annihilator	NOUN
ejpam-6096	69	13	of	of	ADP
ejpam-6096	69	14	a	a	PRON
ejpam-6096	69	15	is	be	AUX
ejpam-6096	69	16	defined	define	VERB
ejpam-6096	69	17	by	by	ADP
ejpam-6096	69	18	annr(a	annr(a	NOUN
ejpam-6096	69	19	)	)	PUNCT
ejpam-6096	69	20	=	=	PRON
ejpam-6096	70	1	{	{	PUNCT
ejpam-6096	70	2	r	r	NOUN
ejpam-6096	70	3	∈	∈	NOUN
ejpam-6096	70	4	r	r	NOUN
ejpam-6096	70	5	|	|	ADV
ejpam-6096	70	6	rxy	rxy	ADV
ejpam-6096	70	7	=	=	NOUN
ejpam-6096	70	8	0	0	NUM
ejpam-6096	70	9	for	for	ADP
ejpam-6096	70	10	all	all	DET
ejpam-6096	70	11	x	x	NOUN
ejpam-6096	70	12	,	,	PUNCT
ejpam-6096	70	13	y	y	PROPN
ejpam-6096	70	14	∈	∈	PROPN
ejpam-6096	70	15	a	a	PRON
ejpam-6096	70	16	}	}	PUNCT
ejpam-6096	70	17	.	.	PUNCT
ejpam-6096	71	1	proposition	proposition	NOUN
ejpam-6096	71	2	2	2	NUM
ejpam-6096	71	3	.	.	PUNCT
ejpam-6096	72	1	let	let	VERB
ejpam-6096	72	2	r	r	PRON
ejpam-6096	72	3	be	be	AUX
ejpam-6096	72	4	a	a	DET
ejpam-6096	72	5	commutative	commutative	ADJ
ejpam-6096	72	6	ternary	ternary	NOUN
ejpam-6096	72	7	semiring	semiring	NOUN
ejpam-6096	72	8	with	with	ADP
ejpam-6096	72	9	zero	zero	NUM
ejpam-6096	72	10	0	0	NUM
ejpam-6096	72	11	.	.	PUNCT
ejpam-6096	73	1	for	for	ADP
ejpam-6096	73	2	any	any	DET
ejpam-6096	73	3	two	two	NUM
ejpam-6096	73	4	nonempty	nonempty	ADJ
ejpam-6096	73	5	subsets	subset	NOUN
ejpam-6096	73	6	a	a	PRON
ejpam-6096	73	7	and	and	CCONJ
ejpam-6096	73	8	b	b	NOUN
ejpam-6096	73	9	of	of	ADP
ejpam-6096	73	10	r	r	NOUN
ejpam-6096	73	11	,	,	PUNCT
ejpam-6096	73	12	if	if	SCONJ
ejpam-6096	73	13	a	a	DET
ejpam-6096	73	14	⊆	⊆	NUM
ejpam-6096	73	15	b	b	NOUN
ejpam-6096	73	16	,	,	PUNCT
ejpam-6096	73	17	then	then	ADV
ejpam-6096	73	18	annr(b	annr(b	PROPN
ejpam-6096	73	19	)	)	PUNCT
ejpam-6096	73	20	⊆	⊆	NUM
ejpam-6096	73	21	annr(a	annr(a	NOUN
ejpam-6096	73	22	)	)	PUNCT
ejpam-6096	73	23	.	.	PUNCT
ejpam-6096	73	24	a.	a.	PROPN
ejpam-6096	73	25	j.	j.	PROPN
ejpam-6096	73	26	khan	khan	PROPN
ejpam-6096	73	27	,	,	PUNCT
ejpam-6096	73	28	m.	m.	NOUN
ejpam-6096	73	29	petapirak	petapirak	PROPN
ejpam-6096	73	30	,	,	PUNCT
ejpam-6096	73	31	r.	r.	PROPN
ejpam-6096	73	32	chinram	chinram	PROPN
ejpam-6096	73	33	/	/	SYM
ejpam-6096	73	34	eur	eur	PROPN
ejpam-6096	73	35	.	.	PUNCT
ejpam-6096	74	1	j.	j.	PROPN
ejpam-6096	74	2	pure	pure	PROPN
ejpam-6096	74	3	appl	appl	PROPN
ejpam-6096	74	4	.	.	PROPN
ejpam-6096	74	5	math	math	PROPN
ejpam-6096	74	6	,	,	PUNCT
ejpam-6096	74	7	18	18	NUM
ejpam-6096	74	8	(	(	PUNCT
ejpam-6096	74	9	2	2	NUM
ejpam-6096	74	10	)	)	PUNCT
ejpam-6096	74	11	(	(	PUNCT
ejpam-6096	74	12	2025	2025	NUM
ejpam-6096	74	13	)	)	PUNCT
ejpam-6096	74	14	,	,	PUNCT
ejpam-6096	74	15	6096	6096	NUM
ejpam-6096	74	16	4	4	NUM
ejpam-6096	74	17	of	of	ADP
ejpam-6096	74	18	10	10	NUM
ejpam-6096	74	19	proof	proof	NOUN
ejpam-6096	74	20	.	.	PUNCT
ejpam-6096	75	1	let	let	VERB
ejpam-6096	75	2	r	r	NOUN
ejpam-6096	75	3	∈	∈	PROPN
ejpam-6096	75	4	annr(b	annr(b	PROPN
ejpam-6096	75	5	)	)	PUNCT
ejpam-6096	75	6	.	.	PUNCT
ejpam-6096	76	1	so	so	ADV
ejpam-6096	76	2	rxy	rxy	ADJ
ejpam-6096	76	3	=	=	NOUN
ejpam-6096	76	4	0	0	NUM
ejpam-6096	76	5	for	for	ADP
ejpam-6096	76	6	all	all	DET
ejpam-6096	76	7	x	x	NOUN
ejpam-6096	76	8	,	,	PUNCT
ejpam-6096	76	9	y	y	PROPN
ejpam-6096	76	10	∈	∈	PROPN
ejpam-6096	76	11	b.	b.	PROPN
ejpam-6096	77	1	this	this	PRON
ejpam-6096	77	2	implies	imply	VERB
ejpam-6096	77	3	rxy	rxy	ADJ
ejpam-6096	77	4	=	=	SYM
ejpam-6096	77	5	0	0	NUM
ejpam-6096	77	6	for	for	ADP
ejpam-6096	77	7	all	all	DET
ejpam-6096	77	8	x	x	NOUN
ejpam-6096	77	9	,	,	PUNCT
ejpam-6096	77	10	y	y	PROPN
ejpam-6096	77	11	∈	∈	PROPN
ejpam-6096	77	12	a.	a.	NOUN
ejpam-6096	77	13	hence	hence	ADV
ejpam-6096	77	14	r	r	NOUN
ejpam-6096	77	15	∈	∈	PROPN
ejpam-6096	77	16	annr(a	annr(a	NOUN
ejpam-6096	77	17	)	)	PUNCT
ejpam-6096	77	18	.	.	PUNCT
ejpam-6096	78	1	example	example	NOUN
ejpam-6096	79	1	2	2	NUM
ejpam-6096	79	2	.	.	PUNCT
ejpam-6096	79	3	let	let	VERB
ejpam-6096	79	4	r	r	NOUN
ejpam-6096	79	5	=	=	SYM
ejpam-6096	79	6	{	{	PUNCT
ejpam-6096	79	7	0	0	NUM
ejpam-6096	79	8	,	,	PUNCT
ejpam-6096	79	9	2	2	NUM
ejpam-6096	79	10	,	,	PUNCT
ejpam-6096	79	11	4	4	NUM
ejpam-6096	79	12	,	,	PUNCT
ejpam-6096	79	13	6	6	NUM
ejpam-6096	79	14	,	,	PUNCT
ejpam-6096	79	15	8	8	NUM
ejpam-6096	79	16	,	,	PUNCT
ejpam-6096	79	17	10	10	NUM
ejpam-6096	79	18	,	,	PUNCT
ejpam-6096	79	19	12	12	NUM
ejpam-6096	79	20	,	,	PUNCT
ejpam-6096	79	21	14	14	NUM
ejpam-6096	79	22	}	}	SYM
ejpam-6096	79	23	⊆	⊆	NUM
ejpam-6096	79	24	z16	z16	NOUN
ejpam-6096	79	25	.	.	PUNCT
ejpam-6096	80	1	we	we	PRON
ejpam-6096	80	2	have	have	VERB
ejpam-6096	80	3	that	that	PRON
ejpam-6096	80	4	r	r	NOUN
ejpam-6096	80	5	is	be	AUX
ejpam-6096	80	6	a	a	DET
ejpam-6096	80	7	commutative	commutative	ADJ
ejpam-6096	80	8	ternary	ternary	ADJ
ejpam-6096	80	9	semiring	semiring	NOUN
ejpam-6096	80	10	under	under	ADP
ejpam-6096	80	11	the	the	DET
ejpam-6096	80	12	usual	usual	ADJ
ejpam-6096	80	13	addition	addition	NOUN
ejpam-6096	80	14	and	and	CCONJ
ejpam-6096	80	15	ternary	ternary	ADJ
ejpam-6096	80	16	multiplication	multiplication	NOUN
ejpam-6096	80	17	of	of	ADP
ejpam-6096	80	18	integers	integer	NOUN
ejpam-6096	80	19	modulo	modulo	VERB
ejpam-6096	80	20	16	16	NUM
ejpam-6096	80	21	.	.	PUNCT
ejpam-6096	81	1	the	the	DET
ejpam-6096	81	2	element	element	NOUN
ejpam-6096	81	3	0	0	NUM
ejpam-6096	81	4	is	be	AUX
ejpam-6096	81	5	a	a	DET
ejpam-6096	81	6	zero	zero	NUM
ejpam-6096	81	7	of	of	ADP
ejpam-6096	81	8	r.	r.	PROPN
ejpam-6096	81	9	(	(	PUNCT
ejpam-6096	81	10	1	1	X
ejpam-6096	81	11	)	)	PUNCT
ejpam-6096	81	12	annr(r	annr(r	NOUN
ejpam-6096	81	13	)	)	PUNCT
ejpam-6096	81	14	=	=	PUNCT
ejpam-6096	82	1	{	{	PUNCT
ejpam-6096	82	2	0	0	NUM
ejpam-6096	82	3	,	,	PUNCT
ejpam-6096	82	4	4	4	NUM
ejpam-6096	82	5	,	,	PUNCT
ejpam-6096	82	6	8	8	NUM
ejpam-6096	82	7	,	,	PUNCT
ejpam-6096	82	8	12	12	NUM
ejpam-6096	82	9	}	}	PUNCT
ejpam-6096	82	10	.	.	PUNCT
ejpam-6096	83	1	(	(	PUNCT
ejpam-6096	83	2	2	2	X
ejpam-6096	83	3	)	)	PUNCT
ejpam-6096	83	4	if	if	SCONJ
ejpam-6096	83	5	a	a	PRON
ejpam-6096	83	6	=	=	X
ejpam-6096	83	7	{	{	PUNCT
ejpam-6096	83	8	4	4	NUM
ejpam-6096	83	9	}	}	PUNCT
ejpam-6096	83	10	,	,	PUNCT
ejpam-6096	83	11	then	then	ADV
ejpam-6096	83	12	annr(a	annr(a	ADJ
ejpam-6096	83	13	)	)	PUNCT
ejpam-6096	83	14	=	=	SYM
ejpam-6096	84	1	r.	r.	NOUN
ejpam-6096	84	2	(	(	PUNCT
ejpam-6096	84	3	3	3	X
ejpam-6096	84	4	)	)	PUNCT
ejpam-6096	84	5	if	if	SCONJ
ejpam-6096	84	6	a	a	PRON
ejpam-6096	84	7	=	=	X
ejpam-6096	84	8	{	{	PUNCT
ejpam-6096	84	9	2	2	NUM
ejpam-6096	84	10	}	}	PUNCT
ejpam-6096	84	11	,	,	PUNCT
ejpam-6096	84	12	then	then	ADV
ejpam-6096	84	13	annr(a	annr(a	ADJ
ejpam-6096	84	14	)	)	PUNCT
ejpam-6096	84	15	=	=	PUNCT
ejpam-6096	84	16	{	{	PUNCT
ejpam-6096	84	17	0	0	NUM
ejpam-6096	84	18	,	,	PUNCT
ejpam-6096	84	19	4	4	NUM
ejpam-6096	84	20	,	,	PUNCT
ejpam-6096	84	21	8	8	NUM
ejpam-6096	84	22	,	,	PUNCT
ejpam-6096	84	23	12	12	NUM
ejpam-6096	84	24	}	}	PUNCT
ejpam-6096	84	25	.	.	PUNCT
ejpam-6096	85	1	proposition	proposition	NOUN
ejpam-6096	85	2	3	3	X
ejpam-6096	85	3	.	.	PUNCT
ejpam-6096	86	1	let	let	VERB
ejpam-6096	86	2	r	r	PRON
ejpam-6096	86	3	be	be	AUX
ejpam-6096	86	4	a	a	DET
ejpam-6096	86	5	commutative	commutative	ADJ
ejpam-6096	86	6	ternary	ternary	NOUN
ejpam-6096	86	7	semiring	semiring	NOUN
ejpam-6096	86	8	with	with	ADP
ejpam-6096	86	9	zero	zero	NUM
ejpam-6096	86	10	0	0	NUM
ejpam-6096	86	11	and	and	CCONJ
ejpam-6096	86	12	a	a	DET
ejpam-6096	86	13	∈	∈	PROPN
ejpam-6096	86	14	r.	r.	NOUN
ejpam-6096	86	15	then	then	ADV
ejpam-6096	86	16	annr({a	annr({a	PROPN
ejpam-6096	86	17	}	}	PUNCT
ejpam-6096	86	18	)	)	PUNCT
ejpam-6096	87	1	=	=	SYM
ejpam-6096	87	2	annr(⟨a⟩	annr(⟨a⟩	PROPN
ejpam-6096	87	3	)	)	PUNCT
ejpam-6096	87	4	.	.	PUNCT
ejpam-6096	88	1	proof	proof	NOUN
ejpam-6096	88	2	.	.	PUNCT
ejpam-6096	89	1	by	by	ADP
ejpam-6096	89	2	proposition	proposition	NOUN
ejpam-6096	89	3	2	2	NUM
ejpam-6096	89	4	,	,	PUNCT
ejpam-6096	89	5	we	we	PRON
ejpam-6096	89	6	have	have	VERB
ejpam-6096	89	7	annr(⟨a⟩	annr(⟨a⟩	NOUN
ejpam-6096	89	8	)	)	PUNCT
ejpam-6096	89	9	⊆	⊆	NUM
ejpam-6096	89	10	annr({a	annr({a	PROPN
ejpam-6096	89	11	}	}	PUNCT
ejpam-6096	89	12	)	)	PUNCT
ejpam-6096	89	13	.	.	PUNCT
ejpam-6096	90	1	let	let	VERB
ejpam-6096	90	2	r	r	NOUN
ejpam-6096	90	3	∈	∈	NOUN
ejpam-6096	90	4	annr({a	annr({a	PROPN
ejpam-6096	90	5	}	}	PUNCT
ejpam-6096	90	6	)	)	PUNCT
ejpam-6096	90	7	.	.	PUNCT
ejpam-6096	91	1	then	then	ADV
ejpam-6096	91	2	raa	raa	VERB
ejpam-6096	91	3	=	=	NOUN
ejpam-6096	91	4	0	0	X
ejpam-6096	91	5	.	.	PUNCT
ejpam-6096	92	1	next	next	ADV
ejpam-6096	92	2	,	,	PUNCT
ejpam-6096	92	3	let	let	VERB
ejpam-6096	92	4	x	x	PRON
ejpam-6096	92	5	,	,	PUNCT
ejpam-6096	92	6	y	y	PROPN
ejpam-6096	92	7	∈	∈	PROPN
ejpam-6096	92	8	⟨a⟩.	⟨a⟩.	PROPN
ejpam-6096	93	1	so	so	ADJ
ejpam-6096	93	2	x	x	NOUN
ejpam-6096	93	3	=	=	PUNCT
ejpam-6096	93	4	ka	ka	PROPN
ejpam-6096	94	1	+	+	CCONJ
ejpam-6096	94	2	∑	∑	PROPN
ejpam-6096	94	3	sis	sis	PROPN
ejpam-6096	94	4	′	′	PROPN
ejpam-6096	94	5	ia	ia	PROPN
ejpam-6096	94	6	and	and	CCONJ
ejpam-6096	94	7	y	y	PROPN
ejpam-6096	95	1	=	=	SYM
ejpam-6096	95	2	k′a	k′a	X
ejpam-6096	95	3	+	+	CCONJ
ejpam-6096	95	4	∑	∑	PUNCT
ejpam-6096	95	5	rjr	rjr	PROPN
ejpam-6096	95	6	′	′	NUM
ejpam-6096	95	7	ja	ja	PROPN
ejpam-6096	95	8	for	for	ADP
ejpam-6096	95	9	some	some	DET
ejpam-6096	95	10	k	k	NOUN
ejpam-6096	95	11	,	,	PUNCT
ejpam-6096	95	12	k′	k′	PROPN
ejpam-6096	95	13	∈	∈	PROPN
ejpam-6096	95	14	n0	n0	PROPN
ejpam-6096	95	15	and	and	CCONJ
ejpam-6096	95	16	si	si	PROPN
ejpam-6096	95	17	,	,	PUNCT
ejpam-6096	95	18	s	s	PART
ejpam-6096	95	19	′	′	NOUN
ejpam-6096	96	1	i	i	PROPN
ejpam-6096	96	2	,	,	PUNCT
ejpam-6096	96	3	rj	rj	PROPN
ejpam-6096	96	4	,	,	PUNCT
ejpam-6096	96	5	r	r	NOUN
ejpam-6096	96	6	′	′	NUM
ejpam-6096	96	7	j	j	PROPN
ejpam-6096	96	8	∈	∈	PROPN
ejpam-6096	96	9	r.	r.	PROPN
ejpam-6096	96	10	since	since	SCONJ
ejpam-6096	96	11	r	r	NOUN
ejpam-6096	96	12	is	be	AUX
ejpam-6096	96	13	commutative	commutative	ADJ
ejpam-6096	96	14	,	,	PUNCT
ejpam-6096	96	15	it	it	PRON
ejpam-6096	96	16	is	be	AUX
ejpam-6096	96	17	easy	easy	ADJ
ejpam-6096	96	18	to	to	PART
ejpam-6096	96	19	see	see	VERB
ejpam-6096	96	20	that	that	DET
ejpam-6096	96	21	rxy	rxy	ADJ
ejpam-6096	96	22	=	=	SYM
ejpam-6096	96	23	0	0	X
ejpam-6096	96	24	.	.	PUNCT
ejpam-6096	97	1	hence	hence	ADV
ejpam-6096	97	2	annr({a	annr({a	PROPN
ejpam-6096	97	3	}	}	PUNCT
ejpam-6096	97	4	)	)	PUNCT
ejpam-6096	98	1	=	=	SYM
ejpam-6096	98	2	annr(⟨a⟩	annr(⟨a⟩	PROPN
ejpam-6096	98	3	)	)	PUNCT
ejpam-6096	98	4	.	.	PUNCT
ejpam-6096	99	1	proposition	proposition	NOUN
ejpam-6096	99	2	4	4	NUM
ejpam-6096	99	3	.	.	PUNCT
ejpam-6096	100	1	let	let	VERB
ejpam-6096	100	2	r	r	PRON
ejpam-6096	100	3	be	be	AUX
ejpam-6096	100	4	a	a	DET
ejpam-6096	100	5	commutative	commutative	ADJ
ejpam-6096	100	6	ternary	ternary	NOUN
ejpam-6096	100	7	semiring	semiring	NOUN
ejpam-6096	100	8	with	with	ADP
ejpam-6096	100	9	zero	zero	NUM
ejpam-6096	100	10	0	0	NUM
ejpam-6096	100	11	and	and	CCONJ
ejpam-6096	100	12	a	a	DET
ejpam-6096	100	13	be	be	NOUN
ejpam-6096	100	14	any	any	DET
ejpam-6096	100	15	nonempty	nonempty	NOUN
ejpam-6096	100	16	subset	subset	NOUN
ejpam-6096	100	17	of	of	ADP
ejpam-6096	100	18	r.	r.	PROPN
ejpam-6096	100	19	then	then	ADV
ejpam-6096	100	20	annr(a	annr(a	ADJ
ejpam-6096	100	21	)	)	PUNCT
ejpam-6096	100	22	is	be	AUX
ejpam-6096	100	23	a	a	DET
ejpam-6096	100	24	k	k	NOUN
ejpam-6096	100	25	-	-	NOUN
ejpam-6096	100	26	ideal	ideal	NOUN
ejpam-6096	100	27	of	of	ADP
ejpam-6096	100	28	r.	r.	PROPN
ejpam-6096	100	29	proof	proof	NOUN
ejpam-6096	100	30	.	.	PUNCT
ejpam-6096	101	1	clearly	clearly	ADV
ejpam-6096	101	2	,	,	PUNCT
ejpam-6096	101	3	0	0	NUM
ejpam-6096	101	4	∈	∈	PROPN
ejpam-6096	101	5	annr(a	annr(a	NOUN
ejpam-6096	101	6	)	)	PUNCT
ejpam-6096	101	7	,	,	PUNCT
ejpam-6096	101	8	which	which	PRON
ejpam-6096	101	9	implies	imply	VERB
ejpam-6096	101	10	that	that	SCONJ
ejpam-6096	101	11	annr(a	annr(a	ADJ
ejpam-6096	101	12	)	)	PUNCT
ejpam-6096	101	13	̸=	̸=	PROPN
ejpam-6096	101	14	∅.	∅.	ADV
ejpam-6096	101	15	let	let	VERB
ejpam-6096	101	16	r	r	NOUN
ejpam-6096	101	17	,	,	PUNCT
ejpam-6096	101	18	s	s	PART
ejpam-6096	101	19	∈	∈	PROPN
ejpam-6096	101	20	annr(a	annr(a	NOUN
ejpam-6096	101	21	)	)	PUNCT
ejpam-6096	101	22	and	and	CCONJ
ejpam-6096	101	23	x	x	X
ejpam-6096	101	24	,	,	PUNCT
ejpam-6096	101	25	y	y	PROPN
ejpam-6096	101	26	∈	∈	PROPN
ejpam-6096	101	27	r.	r.	PROPN
ejpam-6096	101	28	then	then	ADV
ejpam-6096	101	29	rab	rab	PROPN
ejpam-6096	101	30	=	=	PROPN
ejpam-6096	101	31	0	0	PROPN
ejpam-6096	101	32	and	and	CCONJ
ejpam-6096	101	33	sab	sab	ADJ
ejpam-6096	101	34	=	=	SYM
ejpam-6096	101	35	0	0	PROPN
ejpam-6096	101	36	for	for	ADP
ejpam-6096	101	37	all	all	DET
ejpam-6096	101	38	a	a	PRON
ejpam-6096	101	39	,	,	PUNCT
ejpam-6096	101	40	b	b	X
ejpam-6096	101	41	∈	∈	PROPN
ejpam-6096	101	42	a	a	PRON
ejpam-6096	101	43	,	,	PUNCT
ejpam-6096	101	44	so	so	CCONJ
ejpam-6096	101	45	(	(	PUNCT
ejpam-6096	101	46	r+	r+	PUNCT
ejpam-6096	101	47	s)ab	s)ab	PROPN
ejpam-6096	101	48	=	=	SYM
ejpam-6096	101	49	0	0	PUNCT
ejpam-6096	102	1	and	and	CCONJ
ejpam-6096	102	2	(	(	PUNCT
ejpam-6096	102	3	rxy)ab	rxy)ab	X
ejpam-6096	102	4	=	=	NOUN
ejpam-6096	102	5	0	0	NUM
ejpam-6096	102	6	for	for	ADP
ejpam-6096	102	7	all	all	DET
ejpam-6096	102	8	a	a	DET
ejpam-6096	102	9	,	,	PUNCT
ejpam-6096	102	10	b	b	X
ejpam-6096	102	11	∈	∈	PROPN
ejpam-6096	102	12	a.	a.	NOUN
ejpam-6096	102	13	then	then	ADV
ejpam-6096	102	14	r	r	VERB
ejpam-6096	102	15	+	+	SYM
ejpam-6096	102	16	s	s	NOUN
ejpam-6096	102	17	∈	∈	NOUN
ejpam-6096	102	18	annr(a	annr(a	NOUN
ejpam-6096	102	19	)	)	PUNCT
ejpam-6096	102	20	and	and	CCONJ
ejpam-6096	102	21	rxy	rxy	PROPN
ejpam-6096	102	22	∈	∈	PROPN
ejpam-6096	102	23	annr(a	annr(a	NOUN
ejpam-6096	102	24	)	)	PUNCT
ejpam-6096	102	25	.	.	PUNCT
ejpam-6096	103	1	this	this	PRON
ejpam-6096	103	2	implies	imply	VERB
ejpam-6096	103	3	that	that	SCONJ
ejpam-6096	103	4	annr(a	annr(a	NOUN
ejpam-6096	103	5	)	)	PUNCT
ejpam-6096	103	6	is	be	AUX
ejpam-6096	103	7	an	an	DET
ejpam-6096	103	8	ideal	ideal	NOUN
ejpam-6096	103	9	of	of	ADP
ejpam-6096	103	10	r.	r.	PROPN
ejpam-6096	103	11	let	let	VERB
ejpam-6096	103	12	r	r	VERB
ejpam-6096	103	13	,	,	PUNCT
ejpam-6096	103	14	x	x	SYM
ejpam-6096	103	15	∈	∈	NOUN
ejpam-6096	103	16	r	r	NOUN
ejpam-6096	103	17	be	be	VERB
ejpam-6096	103	18	such	such	ADJ
ejpam-6096	103	19	that	that	SCONJ
ejpam-6096	103	20	r	r	NOUN
ejpam-6096	103	21	+	+	NOUN
ejpam-6096	103	22	x	x	SYM
ejpam-6096	103	23	∈	∈	PROPN
ejpam-6096	103	24	annr(a	annr(a	NOUN
ejpam-6096	103	25	)	)	PUNCT
ejpam-6096	103	26	and	and	CCONJ
ejpam-6096	103	27	r	r	NOUN
ejpam-6096	103	28	∈	∈	PROPN
ejpam-6096	103	29	annr(a	annr(a	NOUN
ejpam-6096	103	30	)	)	PUNCT
ejpam-6096	103	31	.	.	PUNCT
ejpam-6096	104	1	so	so	ADV
ejpam-6096	104	2	(	(	PUNCT
ejpam-6096	104	3	r+x)ab	r+x)ab	NOUN
ejpam-6096	104	4	=	=	SYM
ejpam-6096	104	5	0	0	NUM
ejpam-6096	104	6	and	and	CCONJ
ejpam-6096	104	7	rab	rab	PROPN
ejpam-6096	105	1	=	=	NOUN
ejpam-6096	105	2	0	0	PROPN
ejpam-6096	105	3	for	for	ADP
ejpam-6096	105	4	all	all	DET
ejpam-6096	105	5	a	a	PRON
ejpam-6096	105	6	,	,	PUNCT
ejpam-6096	105	7	b	b	X
ejpam-6096	105	8	∈	∈	PROPN
ejpam-6096	105	9	a.	a.	NOUN
ejpam-6096	105	10	then	then	ADV
ejpam-6096	105	11	xab	xab	PROPN
ejpam-6096	105	12	=	=	SYM
ejpam-6096	105	13	0+xab	0+xab	ADJ
ejpam-6096	105	14	=	=	ADJ
ejpam-6096	105	15	rab+xab	rab+xab	NOUN
ejpam-6096	105	16	=	=	SYM
ejpam-6096	105	17	(	(	PUNCT
ejpam-6096	105	18	r+x)ab	r+x)ab	NOUN
ejpam-6096	105	19	=	=	SYM
ejpam-6096	105	20	0	0	NUM
ejpam-6096	105	21	.	.	PUNCT
ejpam-6096	106	1	this	this	PRON
ejpam-6096	106	2	implies	imply	VERB
ejpam-6096	106	3	that	that	SCONJ
ejpam-6096	106	4	x	x	PROPN
ejpam-6096	106	5	∈	∈	PROPN
ejpam-6096	106	6	annr(a	annr(a	NOUN
ejpam-6096	106	7	)	)	PUNCT
ejpam-6096	106	8	,	,	PUNCT
ejpam-6096	106	9	and	and	CCONJ
ejpam-6096	106	10	hence	hence	ADV
ejpam-6096	106	11	annr(a	annr(a	ADJ
ejpam-6096	106	12	)	)	PUNCT
ejpam-6096	106	13	is	be	AUX
ejpam-6096	106	14	a	a	DET
ejpam-6096	106	15	k	k	NOUN
ejpam-6096	106	16	-	-	NOUN
ejpam-6096	106	17	ideal	ideal	NOUN
ejpam-6096	106	18	of	of	ADP
ejpam-6096	106	19	r.	r.	PROPN
ejpam-6096	106	20	let	let	VERB
ejpam-6096	106	21	r	r	NOUN
ejpam-6096	106	22	be	be	AUX
ejpam-6096	106	23	any	any	DET
ejpam-6096	106	24	ternary	ternary	ADJ
ejpam-6096	106	25	semiring	semiring	NOUN
ejpam-6096	106	26	.	.	PUNCT
ejpam-6096	107	1	the	the	DET
ejpam-6096	107	2	k	k	ADJ
ejpam-6096	107	3	-	-	PUNCT
ejpam-6096	107	4	closure	closure	NOUN
ejpam-6096	107	5	operation	operation	NOUN
ejpam-6096	107	6	on	on	ADP
ejpam-6096	107	7	j	j	PROPN
ejpam-6096	107	8	(	(	PUNCT
ejpam-6096	107	9	r	r	NOUN
ejpam-6096	107	10	)	)	PUNCT
ejpam-6096	107	11	is	be	AUX
ejpam-6096	107	12	defined	define	VERB
ejpam-6096	107	13	for	for	ADP
ejpam-6096	107	14	an	an	DET
ejpam-6096	107	15	ideal	ideal	ADJ
ejpam-6096	107	16	i	i	PRON
ejpam-6096	107	17	of	of	ADP
ejpam-6096	107	18	r	r	NOUN
ejpam-6096	107	19	by	by	ADP
ejpam-6096	107	20	ck(i	ck(i	NOUN
ejpam-6096	107	21	)	)	PUNCT
ejpam-6096	107	22	=	=	PRON
ejpam-6096	108	1	{	{	PUNCT
ejpam-6096	108	2	r	r	NOUN
ejpam-6096	108	3	∈	∈	NOUN
ejpam-6096	108	4	r	r	NOUN
ejpam-6096	108	5	|	|	NOUN
ejpam-6096	109	1	r	r	NOUN
ejpam-6096	110	1	+	+	NOUN
ejpam-6096	111	1	x	x	SYM
ejpam-6096	111	2	∈	∈	NOUN
ejpam-6096	111	3	i	i	PRON
ejpam-6096	111	4	for	for	ADP
ejpam-6096	111	5	some	some	DET
ejpam-6096	111	6	x	x	SYM
ejpam-6096	111	7	∈	∈	PROPN
ejpam-6096	111	8	i	i	X
ejpam-6096	111	9	}	}	PUNCT
ejpam-6096	111	10	.	.	PUNCT
ejpam-6096	112	1	example	example	NOUN
ejpam-6096	113	1	3	3	X
ejpam-6096	113	2	.	.	X
ejpam-6096	113	3	we	we	PRON
ejpam-6096	113	4	consider	consider	VERB
ejpam-6096	113	5	a	a	DET
ejpam-6096	113	6	ternary	ternary	ADJ
ejpam-6096	113	7	semiring	semire	VERB
ejpam-6096	113	8	z−	z−	PROPN
ejpam-6096	113	9	0	0	PUNCT
ejpam-6096	114	1	under	under	ADP
ejpam-6096	114	2	the	the	DET
ejpam-6096	114	3	usual	usual	ADJ
ejpam-6096	114	4	addition	addition	NOUN
ejpam-6096	114	5	and	and	CCONJ
ejpam-6096	114	6	ternary	ternary	ADJ
ejpam-6096	114	7	multiplication	multiplication	NOUN
ejpam-6096	114	8	of	of	ADP
ejpam-6096	114	9	integers	integer	NOUN
ejpam-6096	114	10	.	.	PUNCT
ejpam-6096	115	1	(	(	PUNCT
ejpam-6096	115	2	1	1	X
ejpam-6096	115	3	)	)	PUNCT
ejpam-6096	115	4	let	let	VERB
ejpam-6096	115	5	i	i	PRON
ejpam-6096	115	6	=	=	SYM
ejpam-6096	116	1	2z−	2z−	PROPN
ejpam-6096	116	2	0	0	NUM
ejpam-6096	116	3	∖	∖	X
ejpam-6096	116	4	{	{	PUNCT
ejpam-6096	116	5	−2	−2	NOUN
ejpam-6096	116	6	}	}	PUNCT
ejpam-6096	116	7	=	=	SYM
ejpam-6096	116	8	{	{	PUNCT
ejpam-6096	116	9	0,−4,−6,−8,−10	0,−4,−6,−8,−10	NUM
ejpam-6096	116	10	,	,	PUNCT
ejpam-6096	116	11	.	.	PUNCT
ejpam-6096	116	12	.	.	PUNCT
ejpam-6096	116	13	.	.	PUNCT
ejpam-6096	116	14	}	}	PUNCT
ejpam-6096	116	15	.	.	PUNCT
ejpam-6096	117	1	it	it	PRON
ejpam-6096	117	2	is	be	AUX
ejpam-6096	117	3	easy	easy	ADJ
ejpam-6096	117	4	to	to	PART
ejpam-6096	117	5	show	show	VERB
ejpam-6096	117	6	that	that	SCONJ
ejpam-6096	117	7	i	i	PRON
ejpam-6096	117	8	is	be	AUX
ejpam-6096	117	9	an	an	DET
ejpam-6096	117	10	ideal	ideal	NOUN
ejpam-6096	117	11	of	of	ADP
ejpam-6096	117	12	z−	z−	PROPN
ejpam-6096	117	13	0	0	NUM
ejpam-6096	117	14	.	.	PUNCT
ejpam-6096	118	1	we	we	PRON
ejpam-6096	118	2	have	have	VERB
ejpam-6096	118	3	ck(i	ck(i	PUNCT
ejpam-6096	118	4	)	)	PUNCT
ejpam-6096	118	5	=	=	PRON
ejpam-6096	118	6	{	{	PUNCT
ejpam-6096	118	7	0,−2,−4,−6,−8	0,−2,−4,−6,−8	NUM
ejpam-6096	118	8	,	,	PUNCT
ejpam-6096	118	9	.	.	PUNCT
ejpam-6096	118	10	.	.	PUNCT
ejpam-6096	119	1	.	.	PUNCT
ejpam-6096	119	2	}	}	PUNCT
ejpam-6096	120	1	=	=	PUNCT
ejpam-6096	121	1	2z−	2z−	NUM
ejpam-6096	121	2	0	0	NUM
ejpam-6096	121	3	.	.	PUNCT
ejpam-6096	122	1	(	(	PUNCT
ejpam-6096	122	2	2	2	X
ejpam-6096	122	3	)	)	PUNCT
ejpam-6096	122	4	let	let	VERB
ejpam-6096	122	5	i	i	PRON
ejpam-6096	122	6	=	=	PUNCT
ejpam-6096	122	7	z−	z−	X
ejpam-6096	122	8	0	0	NUM
ejpam-6096	122	9	∖	∖	X
ejpam-6096	122	10	{	{	PUNCT
ejpam-6096	122	11	−1	−1	NOUN
ejpam-6096	122	12	}	}	PUNCT
ejpam-6096	122	13	=	=	PUNCT
ejpam-6096	122	14	{	{	PUNCT
ejpam-6096	122	15	0,−2,−3,−4,−5	0,−2,−3,−4,−5	NUM
ejpam-6096	122	16	,	,	PUNCT
ejpam-6096	122	17	.	.	PUNCT
ejpam-6096	122	18	.	.	PUNCT
ejpam-6096	123	1	.	.	PUNCT
ejpam-6096	123	2	}	}	PUNCT
ejpam-6096	123	3	.	.	PUNCT
ejpam-6096	124	1	it	it	PRON
ejpam-6096	124	2	is	be	AUX
ejpam-6096	124	3	easy	easy	ADJ
ejpam-6096	124	4	to	to	PART
ejpam-6096	124	5	prove	prove	VERB
ejpam-6096	124	6	that	that	SCONJ
ejpam-6096	124	7	i	i	PRON
ejpam-6096	124	8	is	be	AUX
ejpam-6096	124	9	an	an	DET
ejpam-6096	124	10	ideal	ideal	NOUN
ejpam-6096	124	11	of	of	ADP
ejpam-6096	124	12	z−	z−	PROPN
ejpam-6096	124	13	0	0	NUM
ejpam-6096	124	14	.	.	PUNCT
ejpam-6096	125	1	we	we	PRON
ejpam-6096	125	2	have	have	VERB
ejpam-6096	125	3	ck(i	ck(i	PUNCT
ejpam-6096	125	4	)	)	PUNCT
ejpam-6096	126	1	=	=	SYM
ejpam-6096	126	2	z−	z−	NOUN
ejpam-6096	126	3	0	0	PUNCT
ejpam-6096	126	4	.	.	PUNCT
ejpam-6096	127	1	proposition	proposition	NOUN
ejpam-6096	127	2	5	5	NUM
ejpam-6096	127	3	.	.	PUNCT
ejpam-6096	128	1	let	let	VERB
ejpam-6096	128	2	r	r	PRON
ejpam-6096	128	3	be	be	AUX
ejpam-6096	128	4	a	a	DET
ejpam-6096	128	5	ternary	ternary	ADJ
ejpam-6096	128	6	semiring	semiring	NOUN
ejpam-6096	128	7	and	and	CCONJ
ejpam-6096	128	8	i	i	PRON
ejpam-6096	128	9	be	be	VERB
ejpam-6096	128	10	an	an	DET
ejpam-6096	128	11	ideal	ideal	NOUN
ejpam-6096	128	12	of	of	ADP
ejpam-6096	128	13	r.	r.	PROPN
ejpam-6096	128	14	then	then	ADV
ejpam-6096	128	15	ck(i	ck(i	PUNCT
ejpam-6096	128	16	)	)	PUNCT
ejpam-6096	128	17	is	be	AUX
ejpam-6096	128	18	the	the	DET
ejpam-6096	128	19	smallest	small	ADJ
ejpam-6096	128	20	k	k	ADJ
ejpam-6096	128	21	-	-	ADJ
ejpam-6096	128	22	ideal	ideal	ADJ
ejpam-6096	128	23	containing	contain	VERB
ejpam-6096	128	24	i.	i.	PROPN
ejpam-6096	128	25	a.	a.	PROPN
ejpam-6096	128	26	j.	j.	PROPN
ejpam-6096	128	27	khan	khan	PROPN
ejpam-6096	128	28	,	,	PUNCT
ejpam-6096	128	29	m.	m.	NOUN
ejpam-6096	128	30	petapirak	petapirak	PROPN
ejpam-6096	128	31	,	,	PUNCT
ejpam-6096	128	32	r.	r.	PROPN
ejpam-6096	128	33	chinram	chinram	PROPN
ejpam-6096	128	34	/	/	SYM
ejpam-6096	128	35	eur	eur	PROPN
ejpam-6096	128	36	.	.	PUNCT
ejpam-6096	129	1	j.	j.	PROPN
ejpam-6096	129	2	pure	pure	PROPN
ejpam-6096	129	3	appl	appl	PROPN
ejpam-6096	129	4	.	.	PROPN
ejpam-6096	129	5	math	math	PROPN
ejpam-6096	129	6	,	,	PUNCT
ejpam-6096	129	7	18	18	NUM
ejpam-6096	129	8	(	(	PUNCT
ejpam-6096	129	9	2	2	NUM
ejpam-6096	129	10	)	)	PUNCT
ejpam-6096	129	11	(	(	PUNCT
ejpam-6096	129	12	2025	2025	NUM
ejpam-6096	129	13	)	)	PUNCT
ejpam-6096	129	14	,	,	PUNCT
ejpam-6096	129	15	6096	6096	NUM
ejpam-6096	129	16	5	5	NUM
ejpam-6096	129	17	of	of	ADP
ejpam-6096	129	18	10	10	NUM
ejpam-6096	129	19	proof	proof	NOUN
ejpam-6096	129	20	.	.	PUNCT
ejpam-6096	130	1	let	let	VERB
ejpam-6096	130	2	x	x	X
ejpam-6096	130	3	∈	∈	PROPN
ejpam-6096	130	4	i.	i.	NOUN
ejpam-6096	130	5	so	so	ADV
ejpam-6096	130	6	x	x	PUNCT
ejpam-6096	131	1	+	+	CCONJ
ejpam-6096	131	2	x	x	SYM
ejpam-6096	131	3	∈	∈	PROPN
ejpam-6096	131	4	i.	i.	NOUN
ejpam-6096	131	5	then	then	ADV
ejpam-6096	131	6	x	x	SYM
ejpam-6096	131	7	∈	∈	PROPN
ejpam-6096	131	8	ck(i	ck(i	PUNCT
ejpam-6096	131	9	)	)	PUNCT
ejpam-6096	131	10	.	.	PUNCT
ejpam-6096	132	1	so	so	ADV
ejpam-6096	132	2	i	i	PRON
ejpam-6096	132	3	⊆	⊆	NUM
ejpam-6096	132	4	ck(i	ck(i	NUM
ejpam-6096	132	5	)	)	PUNCT
ejpam-6096	132	6	.	.	PUNCT
ejpam-6096	133	1	let	let	VERB
ejpam-6096	133	2	r	r	NOUN
ejpam-6096	133	3	,	,	PUNCT
ejpam-6096	133	4	s	s	NOUN
ejpam-6096	133	5	∈	∈	NOUN
ejpam-6096	133	6	ck(i	ck(i	PUNCT
ejpam-6096	133	7	)	)	PUNCT
ejpam-6096	133	8	.	.	PUNCT
ejpam-6096	134	1	then	then	ADV
ejpam-6096	134	2	there	there	PRON
ejpam-6096	134	3	exist	exist	VERB
ejpam-6096	134	4	x	x	NOUN
ejpam-6096	134	5	,	,	PUNCT
ejpam-6096	134	6	y	y	PROPN
ejpam-6096	134	7	∈	∈	PROPN
ejpam-6096	135	1	i	i	PRON
ejpam-6096	135	2	such	such	ADJ
ejpam-6096	135	3	that	that	SCONJ
ejpam-6096	135	4	r	r	NOUN
ejpam-6096	135	5	+	+	NOUN
ejpam-6096	135	6	x	x	SYM
ejpam-6096	135	7	∈	∈	NOUN
ejpam-6096	136	1	i	i	PRON
ejpam-6096	136	2	and	and	CCONJ
ejpam-6096	136	3	s	s	PART
ejpam-6096	136	4	+	+	CCONJ
ejpam-6096	136	5	y	y	PROPN
ejpam-6096	136	6	∈	∈	PROPN
ejpam-6096	136	7	i.	i.	NOUN
ejpam-6096	136	8	then	then	ADV
ejpam-6096	136	9	r	r	PROPN
ejpam-6096	136	10	+	+	SYM
ejpam-6096	136	11	s	s	PART
ejpam-6096	136	12	+	+	NOUN
ejpam-6096	136	13	x	x	SYM
ejpam-6096	137	1	+	+	CCONJ
ejpam-6096	137	2	y	y	PROPN
ejpam-6096	137	3	∈	∈	PROPN
ejpam-6096	137	4	i.	i.	NOUN
ejpam-6096	137	5	this	this	PRON
ejpam-6096	137	6	implies	imply	VERB
ejpam-6096	137	7	that	that	SCONJ
ejpam-6096	137	8	r	r	NOUN
ejpam-6096	137	9	+	+	SYM
ejpam-6096	137	10	s	s	NOUN
ejpam-6096	137	11	∈	∈	NOUN
ejpam-6096	137	12	ck(i	ck(i	PUNCT
ejpam-6096	137	13	)	)	PUNCT
ejpam-6096	137	14	.	.	PUNCT
ejpam-6096	138	1	next	next	ADV
ejpam-6096	138	2	,	,	PUNCT
ejpam-6096	138	3	let	let	VERB
ejpam-6096	138	4	r	r	NOUN
ejpam-6096	138	5	∈	∈	PROPN
ejpam-6096	138	6	ck(i	ck(i	PRON
ejpam-6096	138	7	)	)	PUNCT
ejpam-6096	138	8	and	and	CCONJ
ejpam-6096	138	9	a	a	DET
ejpam-6096	138	10	,	,	PUNCT
ejpam-6096	138	11	b	b	PROPN
ejpam-6096	138	12	∈	∈	PROPN
ejpam-6096	138	13	r.	r.	PROPN
ejpam-6096	138	14	thus	thus	ADV
ejpam-6096	138	15	there	there	PRON
ejpam-6096	138	16	exists	exist	VERB
ejpam-6096	138	17	x	x	X
ejpam-6096	138	18	∈	∈	PROPN
ejpam-6096	138	19	i	i	PRON
ejpam-6096	138	20	such	such	ADJ
ejpam-6096	139	1	that	that	SCONJ
ejpam-6096	139	2	r	r	NOUN
ejpam-6096	139	3	+	+	NOUN
ejpam-6096	139	4	x	x	SYM
ejpam-6096	139	5	∈	∈	PROPN
ejpam-6096	139	6	i.	i.	NOUN
ejpam-6096	139	7	then	then	ADV
ejpam-6096	139	8	rab	rab	PROPN
ejpam-6096	140	1	+	+	CCONJ
ejpam-6096	140	2	xab	xab	PROPN
ejpam-6096	140	3	=	=	SYM
ejpam-6096	140	4	(	(	PUNCT
ejpam-6096	140	5	r	r	NOUN
ejpam-6096	140	6	+	+	PROPN
ejpam-6096	141	1	x)ab	x)ab	PROPN
ejpam-6096	141	2	∈	∈	PROPN
ejpam-6096	142	1	i	i	PRON
ejpam-6096	142	2	and	and	CCONJ
ejpam-6096	142	3	xab	xab	PROPN
ejpam-6096	142	4	∈	∈	PROPN
ejpam-6096	142	5	i.	i.	NOUN
ejpam-6096	142	6	this	this	PRON
ejpam-6096	142	7	shows	show	VERB
ejpam-6096	142	8	that	that	SCONJ
ejpam-6096	142	9	rab	rab	PROPN
ejpam-6096	142	10	∈	∈	PROPN
ejpam-6096	142	11	ck(i	ck(i	PUNCT
ejpam-6096	142	12	)	)	PUNCT
ejpam-6096	142	13	.	.	PUNCT
ejpam-6096	143	1	similarly	similarly	ADV
ejpam-6096	143	2	,	,	PUNCT
ejpam-6096	143	3	arb	arb	PROPN
ejpam-6096	143	4	,	,	PUNCT
ejpam-6096	143	5	abr	abr	NOUN
ejpam-6096	143	6	∈	∈	PROPN
ejpam-6096	143	7	ck(i	ck(i	PUNCT
ejpam-6096	143	8	)	)	PUNCT
ejpam-6096	143	9	.	.	PUNCT
ejpam-6096	144	1	then	then	ADV
ejpam-6096	144	2	ck(i	ck(i	PUNCT
ejpam-6096	144	3	)	)	PUNCT
ejpam-6096	144	4	is	be	AUX
ejpam-6096	144	5	an	an	DET
ejpam-6096	144	6	ideal	ideal	NOUN
ejpam-6096	144	7	of	of	ADP
ejpam-6096	144	8	r.	r.	PROPN
ejpam-6096	144	9	next	next	ADV
ejpam-6096	144	10	,	,	PUNCT
ejpam-6096	144	11	assume	assume	VERB
ejpam-6096	144	12	that	that	SCONJ
ejpam-6096	144	13	r	r	NOUN
ejpam-6096	145	1	+	+	NUM
ejpam-6096	145	2	x	x	NOUN
ejpam-6096	145	3	,	,	PUNCT
ejpam-6096	145	4	x	x	SYM
ejpam-6096	145	5	∈	∈	NOUN
ejpam-6096	145	6	ck(i	ck(i	PUNCT
ejpam-6096	145	7	)	)	PUNCT
ejpam-6096	145	8	.	.	PUNCT
ejpam-6096	146	1	there	there	PRON
ejpam-6096	146	2	exist	exist	VERB
ejpam-6096	146	3	a	a	DET
ejpam-6096	146	4	,	,	PUNCT
ejpam-6096	146	5	b	b	X
ejpam-6096	146	6	∈	∈	NOUN
ejpam-6096	146	7	i	i	PRON
ejpam-6096	147	1	such	such	ADJ
ejpam-6096	147	2	that	that	SCONJ
ejpam-6096	147	3	r	r	NOUN
ejpam-6096	147	4	+	+	NOUN
ejpam-6096	147	5	x	x	X
ejpam-6096	147	6	+	+	CCONJ
ejpam-6096	147	7	a	a	X
ejpam-6096	147	8	,	,	PUNCT
ejpam-6096	147	9	x	x	PUNCT
ejpam-6096	148	1	+	+	SYM
ejpam-6096	148	2	b	b	X
ejpam-6096	148	3	∈	∈	PROPN
ejpam-6096	148	4	i.	i.	NOUN
ejpam-6096	148	5	then	then	ADV
ejpam-6096	148	6	x	x	X
ejpam-6096	149	1	+	+	CCONJ
ejpam-6096	149	2	a	a	DET
ejpam-6096	149	3	+	+	NOUN
ejpam-6096	149	4	b	b	NOUN
ejpam-6096	149	5	∈	∈	NOUN
ejpam-6096	150	1	i	i	PRON
ejpam-6096	150	2	and	and	CCONJ
ejpam-6096	150	3	r	r	NOUN
ejpam-6096	150	4	+	+	CCONJ
ejpam-6096	150	5	x+	x+	ADJ
ejpam-6096	150	6	a+	a+	PRON
ejpam-6096	150	7	b	b	X
ejpam-6096	150	8	∈	∈	PROPN
ejpam-6096	150	9	i.	i.	NOUN
ejpam-6096	150	10	hence	hence	ADV
ejpam-6096	150	11	r	r	NOUN
ejpam-6096	150	12	∈	∈	NOUN
ejpam-6096	150	13	ck(i	ck(i	PUNCT
ejpam-6096	150	14	)	)	PUNCT
ejpam-6096	150	15	.	.	PUNCT
ejpam-6096	151	1	we	we	PRON
ejpam-6096	151	2	can	can	AUX
ejpam-6096	151	3	conclude	conclude	VERB
ejpam-6096	151	4	that	that	PRON
ejpam-6096	151	5	ck(i	ck(i	PUNCT
ejpam-6096	151	6	)	)	PUNCT
ejpam-6096	151	7	is	be	AUX
ejpam-6096	151	8	a	a	DET
ejpam-6096	151	9	k	k	NOUN
ejpam-6096	151	10	-	-	NOUN
ejpam-6096	151	11	ideal	ideal	NOUN
ejpam-6096	151	12	of	of	ADP
ejpam-6096	151	13	r.	r.	PROPN
ejpam-6096	151	14	for	for	ADP
ejpam-6096	151	15	the	the	DET
ejpam-6096	151	16	next	next	ADJ
ejpam-6096	151	17	step	step	NOUN
ejpam-6096	151	18	,	,	PUNCT
ejpam-6096	151	19	let	let	VERB
ejpam-6096	151	20	j	j	PROPN
ejpam-6096	151	21	be	be	AUX
ejpam-6096	151	22	any	any	DET
ejpam-6096	151	23	k	k	NOUN
ejpam-6096	151	24	-	-	PUNCT
ejpam-6096	151	25	ideal	ideal	ADJ
ejpam-6096	151	26	containing	contain	VERB
ejpam-6096	151	27	i.	i.	NOUN
ejpam-6096	151	28	we	we	PRON
ejpam-6096	151	29	need	need	VERB
ejpam-6096	151	30	to	to	PART
ejpam-6096	151	31	show	show	VERB
ejpam-6096	151	32	that	that	PRON
ejpam-6096	151	33	ck(i	ck(i	PUNCT
ejpam-6096	151	34	)	)	PUNCT
ejpam-6096	151	35	⊆	⊆	NUM
ejpam-6096	151	36	j	j	PROPN
ejpam-6096	151	37	.	.	PUNCT
ejpam-6096	152	1	let	let	VERB
ejpam-6096	152	2	r	r	NOUN
ejpam-6096	152	3	∈	∈	PROPN
ejpam-6096	152	4	ck(i	ck(i	PUNCT
ejpam-6096	152	5	)	)	PUNCT
ejpam-6096	152	6	.	.	PUNCT
ejpam-6096	153	1	then	then	ADV
ejpam-6096	153	2	there	there	PRON
ejpam-6096	153	3	exists	exist	VERB
ejpam-6096	153	4	x	x	X
ejpam-6096	153	5	∈	∈	PROPN
ejpam-6096	153	6	i	i	PRON
ejpam-6096	153	7	such	such	ADJ
ejpam-6096	153	8	that	that	SCONJ
ejpam-6096	153	9	r	r	NOUN
ejpam-6096	153	10	+	+	NOUN
ejpam-6096	153	11	x	x	SYM
ejpam-6096	153	12	∈	∈	PROPN
ejpam-6096	153	13	i.	i.	NOUN
ejpam-6096	153	14	since	since	SCONJ
ejpam-6096	153	15	i	i	PROPN
ejpam-6096	153	16	⊆	⊆	NUM
ejpam-6096	153	17	j	j	PROPN
ejpam-6096	153	18	,	,	PUNCT
ejpam-6096	153	19	r	r	NOUN
ejpam-6096	153	20	+	+	NOUN
ejpam-6096	153	21	x	x	SYM
ejpam-6096	153	22	∈	∈	PROPN
ejpam-6096	153	23	j	j	PROPN
ejpam-6096	153	24	and	and	CCONJ
ejpam-6096	153	25	x	x	PROPN
ejpam-6096	153	26	∈	∈	PROPN
ejpam-6096	153	27	j	j	PROPN
ejpam-6096	153	28	.	.	PUNCT
ejpam-6096	154	1	since	since	SCONJ
ejpam-6096	154	2	j	j	PROPN
ejpam-6096	154	3	is	be	AUX
ejpam-6096	154	4	a	a	DET
ejpam-6096	154	5	k	k	NOUN
ejpam-6096	154	6	-	-	NOUN
ejpam-6096	154	7	ideal	ideal	NOUN
ejpam-6096	154	8	such	such	ADJ
ejpam-6096	154	9	that	that	SCONJ
ejpam-6096	154	10	r	r	NOUN
ejpam-6096	154	11	+	+	NOUN
ejpam-6096	154	12	x	x	SYM
ejpam-6096	154	13	∈	∈	PROPN
ejpam-6096	154	14	j	j	PROPN
ejpam-6096	154	15	and	and	CCONJ
ejpam-6096	154	16	x	x	PROPN
ejpam-6096	154	17	∈	∈	PROPN
ejpam-6096	154	18	j	j	PROPN
ejpam-6096	154	19	,	,	PUNCT
ejpam-6096	154	20	it	it	PRON
ejpam-6096	154	21	follows	follow	VERB
ejpam-6096	154	22	that	that	SCONJ
ejpam-6096	154	23	r	r	PROPN
ejpam-6096	154	24	∈	∈	PROPN
ejpam-6096	154	25	j	j	PROPN
ejpam-6096	154	26	.	.	PUNCT
ejpam-6096	155	1	thus	thus	ADV
ejpam-6096	155	2	ck(i	ck(i	PUNCT
ejpam-6096	155	3	)	)	PUNCT
ejpam-6096	155	4	⊆	⊆	NUM
ejpam-6096	155	5	j	j	PROPN
ejpam-6096	155	6	.	.	PUNCT
ejpam-6096	156	1	therefore	therefore	ADV
ejpam-6096	156	2	ck(i	ck(i	PUNCT
ejpam-6096	156	3	)	)	PUNCT
ejpam-6096	156	4	is	be	AUX
ejpam-6096	156	5	the	the	DET
ejpam-6096	156	6	smallest	small	ADJ
ejpam-6096	156	7	k	k	ADJ
ejpam-6096	156	8	-	-	ADJ
ejpam-6096	156	9	ideal	ideal	ADJ
ejpam-6096	156	10	containing	contain	VERB
ejpam-6096	156	11	i.	i.	NOUN
ejpam-6096	156	12	proposition	proposition	NOUN
ejpam-6096	156	13	6	6	NUM
ejpam-6096	156	14	.	.	PUNCT
ejpam-6096	157	1	let	let	VERB
ejpam-6096	157	2	r	r	PRON
ejpam-6096	157	3	be	be	AUX
ejpam-6096	157	4	a	a	DET
ejpam-6096	157	5	ternary	ternary	ADJ
ejpam-6096	157	6	semiring	semiring	NOUN
ejpam-6096	157	7	.	.	PUNCT
ejpam-6096	158	1	the	the	DET
ejpam-6096	158	2	following	follow	VERB
ejpam-6096	158	3	statements	statement	NOUN
ejpam-6096	158	4	hold	hold	VERB
ejpam-6096	158	5	.	.	PUNCT
ejpam-6096	159	1	(	(	PUNCT
ejpam-6096	159	2	1	1	X
ejpam-6096	159	3	)	)	PUNCT
ejpam-6096	159	4	ck(r	ck(r	NOUN
ejpam-6096	159	5	)	)	PUNCT
ejpam-6096	160	1	=	=	SYM
ejpam-6096	160	2	r.	r.	NOUN
ejpam-6096	160	3	(	(	PUNCT
ejpam-6096	160	4	2	2	NUM
ejpam-6096	160	5	)	)	PUNCT
ejpam-6096	160	6	if	if	SCONJ
ejpam-6096	160	7	r	r	NOUN
ejpam-6096	160	8	has	have	VERB
ejpam-6096	160	9	a	a	DET
ejpam-6096	160	10	zero	zero	NUM
ejpam-6096	160	11	,	,	PUNCT
ejpam-6096	160	12	then	then	ADV
ejpam-6096	160	13	ck({0	ck({0	NOUN
ejpam-6096	160	14	}	}	PUNCT
ejpam-6096	160	15	)	)	PUNCT
ejpam-6096	160	16	=	=	PUNCT
ejpam-6096	160	17	{	{	PUNCT
ejpam-6096	160	18	0	0	NUM
ejpam-6096	160	19	}	}	PUNCT
ejpam-6096	160	20	.	.	PUNCT
ejpam-6096	161	1	proof	proof	NOUN
ejpam-6096	161	2	.	.	PUNCT
ejpam-6096	162	1	(	(	PUNCT
ejpam-6096	162	2	1	1	X
ejpam-6096	162	3	)	)	PUNCT
ejpam-6096	162	4	by	by	ADP
ejpam-6096	162	5	proposition	proposition	NOUN
ejpam-6096	162	6	5	5	NUM
ejpam-6096	162	7	,	,	PUNCT
ejpam-6096	162	8	we	we	PRON
ejpam-6096	162	9	have	have	VERB
ejpam-6096	162	10	r	r	NOUN
ejpam-6096	162	11	⊆	⊆	NUM
ejpam-6096	162	12	ck(r	ck(r	NOUN
ejpam-6096	162	13	)	)	PUNCT
ejpam-6096	162	14	.	.	PUNCT
ejpam-6096	163	1	then	then	ADV
ejpam-6096	163	2	ck(r	ck(r	VERB
ejpam-6096	163	3	)	)	PUNCT
ejpam-6096	164	1	=	=	SYM
ejpam-6096	164	2	r.	r.	NOUN
ejpam-6096	164	3	(	(	PUNCT
ejpam-6096	164	4	2	2	X
ejpam-6096	164	5	)	)	PUNCT
ejpam-6096	164	6	let	let	VERB
ejpam-6096	164	7	x	x	PUNCT
ejpam-6096	164	8	∈	∈	PROPN
ejpam-6096	164	9	ck({0	ck({0	NOUN
ejpam-6096	164	10	}	}	PUNCT
ejpam-6096	164	11	)	)	PUNCT
ejpam-6096	164	12	.	.	PUNCT
ejpam-6096	165	1	so	so	ADV
ejpam-6096	165	2	x+	x+	ADJ
ejpam-6096	165	3	0	0	NUM
ejpam-6096	165	4	∈	∈	NOUN
ejpam-6096	165	5	{	{	PUNCT
ejpam-6096	165	6	0	0	NUM
ejpam-6096	165	7	}	}	PUNCT
ejpam-6096	165	8	.	.	PUNCT
ejpam-6096	166	1	this	this	PRON
ejpam-6096	166	2	implies	imply	VERB
ejpam-6096	166	3	that	that	SCONJ
ejpam-6096	166	4	x	x	X
ejpam-6096	166	5	=	=	SYM
ejpam-6096	166	6	0	0	NUM
ejpam-6096	166	7	and	and	CCONJ
ejpam-6096	166	8	ck({0	ck({0	NOUN
ejpam-6096	166	9	}	}	PUNCT
ejpam-6096	166	10	)	)	PUNCT
ejpam-6096	166	11	=	=	PUNCT
ejpam-6096	166	12	{	{	PUNCT
ejpam-6096	166	13	0	0	NUM
ejpam-6096	166	14	}	}	PUNCT
ejpam-6096	166	15	.	.	PUNCT
ejpam-6096	167	1	proposition	proposition	NOUN
ejpam-6096	167	2	7	7	NUM
ejpam-6096	167	3	.	.	PUNCT
ejpam-6096	168	1	let	let	VERB
ejpam-6096	168	2	r	r	PRON
ejpam-6096	168	3	be	be	AUX
ejpam-6096	168	4	a	a	DET
ejpam-6096	168	5	ternary	ternary	ADJ
ejpam-6096	168	6	semiring	semiring	NOUN
ejpam-6096	168	7	.	.	PUNCT
ejpam-6096	169	1	if	if	SCONJ
ejpam-6096	169	2	i	i	PRON
ejpam-6096	169	3	and	and	CCONJ
ejpam-6096	169	4	j	j	PROPN
ejpam-6096	169	5	are	be	AUX
ejpam-6096	169	6	any	any	DET
ejpam-6096	169	7	ideals	ideal	NOUN
ejpam-6096	169	8	of	of	ADP
ejpam-6096	169	9	r	r	NOUN
ejpam-6096	169	10	such	such	ADJ
ejpam-6096	169	11	that	that	SCONJ
ejpam-6096	169	12	i	i	PRON
ejpam-6096	169	13	⊆	⊆	NUM
ejpam-6096	169	14	j	j	PROPN
ejpam-6096	169	15	,	,	PUNCT
ejpam-6096	169	16	then	then	ADV
ejpam-6096	169	17	ck(i	ck(i	PUNCT
ejpam-6096	169	18	)	)	PUNCT
ejpam-6096	169	19	⊆	⊆	NUM
ejpam-6096	169	20	ck(j	ck(j	NUM
ejpam-6096	169	21	)	)	PUNCT
ejpam-6096	169	22	.	.	PUNCT
ejpam-6096	170	1	proof	proof	NOUN
ejpam-6096	170	2	.	.	PUNCT
ejpam-6096	171	1	let	let	VERB
ejpam-6096	171	2	r	r	NOUN
ejpam-6096	171	3	∈	∈	PROPN
ejpam-6096	171	4	ck(i	ck(i	PUNCT
ejpam-6096	171	5	)	)	PUNCT
ejpam-6096	171	6	.	.	PUNCT
ejpam-6096	172	1	then	then	ADV
ejpam-6096	172	2	by	by	ADP
ejpam-6096	172	3	definition	definition	NOUN
ejpam-6096	172	4	of	of	ADP
ejpam-6096	172	5	ck(i	ck(i	PUNCT
ejpam-6096	172	6	)	)	PUNCT
ejpam-6096	172	7	,	,	PUNCT
ejpam-6096	172	8	we	we	PRON
ejpam-6096	172	9	have	have	VERB
ejpam-6096	172	10	r	r	NOUN
ejpam-6096	172	11	+	+	NOUN
ejpam-6096	172	12	x	x	SYM
ejpam-6096	172	13	∈	∈	NOUN
ejpam-6096	172	14	i	i	PRON
ejpam-6096	172	15	for	for	ADP
ejpam-6096	172	16	some	some	DET
ejpam-6096	172	17	x	x	SYM
ejpam-6096	172	18	∈	∈	PROPN
ejpam-6096	172	19	i.	i.	NOUN
ejpam-6096	172	20	since	since	SCONJ
ejpam-6096	172	21	i	i	PROPN
ejpam-6096	172	22	⊆	⊆	NUM
ejpam-6096	172	23	j	j	PROPN
ejpam-6096	172	24	,	,	PUNCT
ejpam-6096	172	25	r	r	NOUN
ejpam-6096	172	26	+	+	NOUN
ejpam-6096	172	27	x	x	SYM
ejpam-6096	172	28	∈	∈	PROPN
ejpam-6096	172	29	j	j	PROPN
ejpam-6096	172	30	and	and	CCONJ
ejpam-6096	172	31	x	x	PROPN
ejpam-6096	172	32	∈	∈	PROPN
ejpam-6096	172	33	j	j	PROPN
ejpam-6096	172	34	.	.	PUNCT
ejpam-6096	173	1	thus	thus	ADV
ejpam-6096	173	2	r	r	NOUN
ejpam-6096	173	3	∈	∈	NOUN
ejpam-6096	173	4	ck(j	ck(j	NOUN
ejpam-6096	173	5	)	)	PUNCT
ejpam-6096	173	6	.	.	PUNCT
ejpam-6096	174	1	therefore	therefore	ADV
ejpam-6096	174	2	ck(i	ck(i	PUNCT
ejpam-6096	174	3	)	)	PUNCT
ejpam-6096	174	4	⊆	⊆	NUM
ejpam-6096	174	5	ck(j	ck(j	NUM
ejpam-6096	174	6	)	)	PUNCT
ejpam-6096	174	7	.	.	PUNCT
ejpam-6096	174	8	theorem	theorem	NOUN
ejpam-6096	174	9	1	1	NUM
ejpam-6096	174	10	.	.	PUNCT
ejpam-6096	175	1	let	let	VERB
ejpam-6096	175	2	r	r	PRON
ejpam-6096	175	3	be	be	AUX
ejpam-6096	175	4	a	a	DET
ejpam-6096	175	5	ternary	ternary	ADJ
ejpam-6096	175	6	semiring	semiring	NOUN
ejpam-6096	175	7	and	and	CCONJ
ejpam-6096	175	8	i	i	PRON
ejpam-6096	175	9	be	be	VERB
ejpam-6096	175	10	an	an	DET
ejpam-6096	175	11	ideal	ideal	NOUN
ejpam-6096	175	12	of	of	ADP
ejpam-6096	175	13	r.	r.	PROPN
ejpam-6096	175	14	then	then	ADV
ejpam-6096	175	15	i	i	PRON
ejpam-6096	175	16	is	be	AUX
ejpam-6096	175	17	a	a	DET
ejpam-6096	175	18	k	k	NOUN
ejpam-6096	175	19	-	-	NOUN
ejpam-6096	175	20	ideal	ideal	NOUN
ejpam-6096	175	21	of	of	ADP
ejpam-6096	175	22	r	r	NOUN
ejpam-6096	175	23	if	if	SCONJ
ejpam-6096	176	1	and	and	CCONJ
ejpam-6096	176	2	only	only	ADV
ejpam-6096	176	3	if	if	SCONJ
ejpam-6096	176	4	i	i	PRON
ejpam-6096	176	5	=	=	PUNCT
ejpam-6096	176	6	ck(i	ck(i	PRON
ejpam-6096	176	7	)	)	PUNCT
ejpam-6096	176	8	.	.	PUNCT
ejpam-6096	177	1	proof	proof	NOUN
ejpam-6096	177	2	.	.	PUNCT
ejpam-6096	178	1	let	let	VERB
ejpam-6096	178	2	i	i	PRON
ejpam-6096	178	3	be	be	AUX
ejpam-6096	178	4	a	a	DET
ejpam-6096	178	5	k	k	NOUN
ejpam-6096	178	6	-	-	NOUN
ejpam-6096	178	7	ideal	ideal	NOUN
ejpam-6096	178	8	of	of	ADP
ejpam-6096	178	9	r.	r.	PROPN
ejpam-6096	178	10	by	by	ADP
ejpam-6096	178	11	proposition	proposition	NOUN
ejpam-6096	178	12	5	5	NUM
ejpam-6096	178	13	,	,	PUNCT
ejpam-6096	178	14	we	we	PRON
ejpam-6096	178	15	have	have	VERB
ejpam-6096	178	16	i	i	PRON
ejpam-6096	178	17	⊆	⊆	NUM
ejpam-6096	178	18	ck(i	ck(i	NUM
ejpam-6096	178	19	)	)	PUNCT
ejpam-6096	178	20	.	.	PUNCT
ejpam-6096	179	1	let	let	VERB
ejpam-6096	179	2	r	r	NOUN
ejpam-6096	179	3	∈	∈	PROPN
ejpam-6096	179	4	ck(i	ck(i	PUNCT
ejpam-6096	179	5	)	)	PUNCT
ejpam-6096	179	6	.	.	PUNCT
ejpam-6096	180	1	then	then	ADV
ejpam-6096	180	2	there	there	PRON
ejpam-6096	180	3	exists	exist	VERB
ejpam-6096	180	4	x	x	X
ejpam-6096	180	5	∈	∈	NOUN
ejpam-6096	180	6	i	i	PRON
ejpam-6096	180	7	such	such	ADJ
ejpam-6096	180	8	that	that	DET
ejpam-6096	180	9	r+	r+	NOUN
ejpam-6096	180	10	x	x	X
ejpam-6096	180	11	∈	∈	PROPN
ejpam-6096	180	12	i.	i.	NOUN
ejpam-6096	180	13	since	since	SCONJ
ejpam-6096	180	14	i	i	PRON
ejpam-6096	180	15	is	be	AUX
ejpam-6096	180	16	a	a	DET
ejpam-6096	180	17	k	k	NOUN
ejpam-6096	180	18	-	-	NOUN
ejpam-6096	180	19	ideal	ideal	NOUN
ejpam-6096	180	20	of	of	ADP
ejpam-6096	180	21	r	r	NOUN
ejpam-6096	180	22	and	and	CCONJ
ejpam-6096	180	23	r+	r+	NOUN
ejpam-6096	180	24	x	x	X
ejpam-6096	180	25	,	,	PUNCT
ejpam-6096	180	26	x	x	SYM
ejpam-6096	180	27	∈	∈	PROPN
ejpam-6096	181	1	i	i	PRON
ejpam-6096	181	2	,	,	PUNCT
ejpam-6096	181	3	we	we	PRON
ejpam-6096	181	4	obtain	obtain	VERB
ejpam-6096	181	5	r	r	NOUN
ejpam-6096	181	6	∈	∈	NOUN
ejpam-6096	181	7	i.	i.	NOUN
ejpam-6096	181	8	thus	thus	ADV
ejpam-6096	181	9	ck(i	ck(i	PUNCT
ejpam-6096	181	10	)	)	PUNCT
ejpam-6096	182	1	⊆	⊆	NUM
ejpam-6096	182	2	i.	i.	NOUN
ejpam-6096	182	3	we	we	PRON
ejpam-6096	182	4	conclude	conclude	VERB
ejpam-6096	182	5	that	that	PRON
ejpam-6096	182	6	ck(i	ck(i	PUNCT
ejpam-6096	182	7	)	)	PUNCT
ejpam-6096	182	8	=	=	SYM
ejpam-6096	183	1	i.	i.	NOUN
ejpam-6096	183	2	conversely	conversely	ADV
ejpam-6096	183	3	,	,	PUNCT
ejpam-6096	183	4	we	we	PRON
ejpam-6096	183	5	assume	assume	VERB
ejpam-6096	183	6	that	that	SCONJ
ejpam-6096	183	7	ck(i	ck(i	PUNCT
ejpam-6096	183	8	)	)	PUNCT
ejpam-6096	183	9	=	=	SYM
ejpam-6096	183	10	i.	i.	NOUN
ejpam-6096	183	11	by	by	ADP
ejpam-6096	183	12	proposition	proposition	NOUN
ejpam-6096	183	13	5	5	NUM
ejpam-6096	183	14	,	,	PUNCT
ejpam-6096	183	15	i	i	PRON
ejpam-6096	183	16	is	be	AUX
ejpam-6096	183	17	a	a	DET
ejpam-6096	183	18	k	k	NOUN
ejpam-6096	183	19	-	-	NOUN
ejpam-6096	183	20	ideal	ideal	NOUN
ejpam-6096	183	21	of	of	ADP
ejpam-6096	183	22	r.	r.	PROPN
ejpam-6096	183	23	proposition	proposition	PROPN
ejpam-6096	183	24	8	8	NUM
ejpam-6096	183	25	.	.	PUNCT
ejpam-6096	184	1	let	let	VERB
ejpam-6096	184	2	i	i	PRON
ejpam-6096	184	3	be	be	AUX
ejpam-6096	184	4	an	an	DET
ejpam-6096	184	5	ideal	ideal	NOUN
ejpam-6096	184	6	of	of	ADP
ejpam-6096	184	7	a	a	DET
ejpam-6096	184	8	ternary	ternary	ADJ
ejpam-6096	184	9	semiring	semire	VERB
ejpam-6096	184	10	r.	r.	PROPN
ejpam-6096	184	11	then	then	ADV
ejpam-6096	184	12	ck(ck(i	ck(ck(i	NOUN
ejpam-6096	184	13	)	)	PUNCT
ejpam-6096	184	14	)	)	PUNCT
ejpam-6096	185	1	=	=	PUNCT
ejpam-6096	185	2	ck(i	ck(i	PUNCT
ejpam-6096	185	3	)	)	PUNCT
ejpam-6096	185	4	.	.	PUNCT
ejpam-6096	186	1	proof	proof	NOUN
ejpam-6096	186	2	.	.	PUNCT
ejpam-6096	187	1	by	by	ADP
ejpam-6096	187	2	proposition	proposition	NOUN
ejpam-6096	187	3	5	5	NUM
ejpam-6096	187	4	,	,	PUNCT
ejpam-6096	187	5	we	we	PRON
ejpam-6096	187	6	know	know	VERB
ejpam-6096	187	7	that	that	PRON
ejpam-6096	187	8	ck(i	ck(i	PUNCT
ejpam-6096	187	9	)	)	PUNCT
ejpam-6096	187	10	is	be	AUX
ejpam-6096	187	11	a	a	DET
ejpam-6096	187	12	k	k	NOUN
ejpam-6096	187	13	-	-	NOUN
ejpam-6096	187	14	ideal	ideal	NOUN
ejpam-6096	187	15	of	of	ADP
ejpam-6096	187	16	r.	r.	PROPN
ejpam-6096	187	17	by	by	ADP
ejpam-6096	187	18	theorem	theorem	NOUN
ejpam-6096	187	19	1	1	NUM
ejpam-6096	187	20	,	,	PUNCT
ejpam-6096	187	21	we	we	PRON
ejpam-6096	187	22	have	have	VERB
ejpam-6096	187	23	that	that	PRON
ejpam-6096	187	24	ck(ck(i	ck(ck(i	NOUN
ejpam-6096	187	25	)	)	PUNCT
ejpam-6096	187	26	)	)	PUNCT
ejpam-6096	188	1	=	=	PUNCT
ejpam-6096	188	2	ck(i	ck(i	PUNCT
ejpam-6096	188	3	)	)	PUNCT
ejpam-6096	188	4	.	.	PUNCT
ejpam-6096	189	1	proposition	proposition	NOUN
ejpam-6096	189	2	9	9	NUM
ejpam-6096	189	3	.	.	PUNCT
ejpam-6096	190	1	let	let	VERB
ejpam-6096	190	2	i	i	PRON
ejpam-6096	190	3	,	,	PUNCT
ejpam-6096	190	4	j	j	PROPN
ejpam-6096	190	5	and	and	CCONJ
ejpam-6096	190	6	k	k	PROPN
ejpam-6096	190	7	be	be	VERB
ejpam-6096	190	8	ideals	ideal	NOUN
ejpam-6096	190	9	of	of	ADP
ejpam-6096	190	10	a	a	DET
ejpam-6096	190	11	ternary	ternary	ADJ
ejpam-6096	190	12	semiring	semire	VERB
ejpam-6096	190	13	r.	r.	PROPN
ejpam-6096	190	14	then	then	ADV
ejpam-6096	190	15	ck(i)ck(j)ck(k	ck(i)ck(j)ck(k	PROPN
ejpam-6096	190	16	)	)	PUNCT
ejpam-6096	190	17	⊆	⊆	NUM
ejpam-6096	190	18	ck(ijk	ck(ijk	X
ejpam-6096	190	19	)	)	PUNCT
ejpam-6096	190	20	.	.	PUNCT
ejpam-6096	191	1	a.	a.	PROPN
ejpam-6096	191	2	j.	j.	PROPN
ejpam-6096	191	3	khan	khan	PROPN
ejpam-6096	191	4	,	,	PUNCT
ejpam-6096	191	5	m.	m.	NOUN
ejpam-6096	191	6	petapirak	petapirak	PROPN
ejpam-6096	191	7	,	,	PUNCT
ejpam-6096	191	8	r.	r.	PROPN
ejpam-6096	191	9	chinram	chinram	PROPN
ejpam-6096	191	10	/	/	SYM
ejpam-6096	191	11	eur	eur	PROPN
ejpam-6096	191	12	.	.	PUNCT
ejpam-6096	192	1	j.	j.	PROPN
ejpam-6096	192	2	pure	pure	PROPN
ejpam-6096	192	3	appl	appl	PROPN
ejpam-6096	192	4	.	.	PROPN
ejpam-6096	192	5	math	math	PROPN
ejpam-6096	192	6	,	,	PUNCT
ejpam-6096	192	7	18	18	NUM
ejpam-6096	192	8	(	(	PUNCT
ejpam-6096	192	9	2	2	NUM
ejpam-6096	192	10	)	)	PUNCT
ejpam-6096	192	11	(	(	PUNCT
ejpam-6096	192	12	2025	2025	NUM
ejpam-6096	192	13	)	)	PUNCT
ejpam-6096	192	14	,	,	PUNCT
ejpam-6096	192	15	6096	6096	NUM
ejpam-6096	192	16	6	6	NUM
ejpam-6096	192	17	of	of	ADP
ejpam-6096	192	18	10	10	NUM
ejpam-6096	192	19	proof	proof	NOUN
ejpam-6096	192	20	.	.	PUNCT
ejpam-6096	193	1	let	let	VERB
ejpam-6096	193	2	x	x	SYM
ejpam-6096	193	3	∈	∈	PROPN
ejpam-6096	193	4	ck(i	ck(i	PUNCT
ejpam-6096	193	5	)	)	PUNCT
ejpam-6096	193	6	,	,	PUNCT
ejpam-6096	193	7	y	y	PROPN
ejpam-6096	193	8	∈	∈	PROPN
ejpam-6096	193	9	ck(j	ck(j	NUM
ejpam-6096	193	10	)	)	PUNCT
ejpam-6096	193	11	and	and	CCONJ
ejpam-6096	193	12	z	z	PROPN
ejpam-6096	193	13	∈	∈	PROPN
ejpam-6096	193	14	ck(k	ck(k	NOUN
ejpam-6096	193	15	)	)	PUNCT
ejpam-6096	193	16	.	.	PUNCT
ejpam-6096	194	1	then	then	ADV
ejpam-6096	194	2	there	there	PRON
ejpam-6096	194	3	exist	exist	VERB
ejpam-6096	194	4	a	a	DET
ejpam-6096	194	5	∈	∈	PROPN
ejpam-6096	194	6	i	i	NOUN
ejpam-6096	194	7	,	,	PUNCT
ejpam-6096	194	8	b	b	PROPN
ejpam-6096	194	9	∈	∈	PROPN
ejpam-6096	194	10	j	j	PROPN
ejpam-6096	194	11	and	and	CCONJ
ejpam-6096	194	12	c	c	NOUN
ejpam-6096	194	13	∈	∈	PROPN
ejpam-6096	194	14	k	k	PRON
ejpam-6096	194	15	such	such	ADJ
ejpam-6096	194	16	that	that	PRON
ejpam-6096	194	17	x	x	X
ejpam-6096	195	1	+	+	CCONJ
ejpam-6096	195	2	a	a	DET
ejpam-6096	195	3	∈	∈	NOUN
ejpam-6096	195	4	i	i	PRON
ejpam-6096	195	5	,	,	PUNCT
ejpam-6096	195	6	y	y	PROPN
ejpam-6096	195	7	+	+	PROPN
ejpam-6096	195	8	b	b	PROPN
ejpam-6096	195	9	∈	∈	PROPN
ejpam-6096	195	10	j	j	PROPN
ejpam-6096	195	11	and	and	CCONJ
ejpam-6096	195	12	z	z	PROPN
ejpam-6096	196	1	+	+	NOUN
ejpam-6096	196	2	c	c	PROPN
ejpam-6096	196	3	∈	∈	PROPN
ejpam-6096	196	4	k.	k.	PROPN
ejpam-6096	197	1	so	so	ADV
ejpam-6096	197	2	abc	abc	PROPN
ejpam-6096	197	3	∈	∈	PROPN
ejpam-6096	197	4	ijk	ijk	PROPN
ejpam-6096	197	5	and	and	CCONJ
ejpam-6096	197	6	each	each	PRON
ejpam-6096	197	7	of	of	ADP
ejpam-6096	197	8	the	the	DET
ejpam-6096	197	9	following	following	NOUN
ejpam-6096	197	10	is	be	AUX
ejpam-6096	197	11	also	also	ADV
ejpam-6096	197	12	a	a	DET
ejpam-6096	197	13	member	member	NOUN
ejpam-6096	197	14	of	of	ADP
ejpam-6096	197	15	ijk	ijk	PROPN
ejpam-6096	197	16	:	:	PUNCT
ejpam-6096	197	17	xbc+	xbc+	PROPN
ejpam-6096	197	18	abc	abc	PROPN
ejpam-6096	197	19	=	=	SYM
ejpam-6096	197	20	(	(	PUNCT
ejpam-6096	197	21	x+	x+	PROPN
ejpam-6096	197	22	a)bc	a)bc	PROPN
ejpam-6096	197	23	∈	∈	PROPN
ejpam-6096	197	24	ijk	ijk	PROPN
ejpam-6096	197	25	,	,	PUNCT
ejpam-6096	197	26	(	(	PUNCT
ejpam-6096	197	27	1	1	X
ejpam-6096	197	28	)	)	PUNCT
ejpam-6096	197	29	ayc+	ayc+	ADJ
ejpam-6096	197	30	abc	abc	NOUN
ejpam-6096	197	31	=	=	PUNCT
ejpam-6096	197	32	a(y	a(y	PROPN
ejpam-6096	197	33	+	+	CCONJ
ejpam-6096	197	34	b)c	b)c	X
ejpam-6096	197	35	∈	∈	PROPN
ejpam-6096	197	36	ijk	ijk	X
ejpam-6096	197	37	,	,	PUNCT
ejpam-6096	197	38	(	(	PUNCT
ejpam-6096	197	39	2	2	X
ejpam-6096	197	40	)	)	PUNCT
ejpam-6096	197	41	abz	abz	NOUN
ejpam-6096	197	42	+	+	CCONJ
ejpam-6096	197	43	abc	abc	PROPN
ejpam-6096	197	44	=	=	PUNCT
ejpam-6096	197	45	ab(z	ab(z	PROPN
ejpam-6096	197	46	+	+	SYM
ejpam-6096	197	47	c	c	X
ejpam-6096	197	48	)	)	PUNCT
ejpam-6096	197	49	∈	∈	PROPN
ejpam-6096	197	50	ijk	ijk	PROPN
ejpam-6096	197	51	,	,	PUNCT
ejpam-6096	197	52	(	(	PUNCT
ejpam-6096	197	53	3	3	X
ejpam-6096	197	54	)	)	PUNCT
ejpam-6096	197	55	xyc+	xyc+	PROPN
ejpam-6096	198	1	ayc+	ayc+	PROPN
ejpam-6096	198	2	xbc+	xbc+	PROPN
ejpam-6096	198	3	abc	abc	PROPN
ejpam-6096	198	4	=	=	SYM
ejpam-6096	198	5	(	(	PUNCT
ejpam-6096	198	6	x+	x+	X
ejpam-6096	198	7	a)(y	a)(y	PROPN
ejpam-6096	198	8	+	+	CCONJ
ejpam-6096	198	9	b)c	b)c	X
ejpam-6096	198	10	∈	∈	PROPN
ejpam-6096	198	11	ijk	ijk	X
ejpam-6096	198	12	,	,	PUNCT
ejpam-6096	198	13	(	(	PUNCT
ejpam-6096	198	14	4	4	NUM
ejpam-6096	198	15	)	)	PUNCT
ejpam-6096	198	16	ayz	ayz	NOUN
ejpam-6096	198	17	+	+	CCONJ
ejpam-6096	198	18	abz	abz	ADJ
ejpam-6096	198	19	+	+	CCONJ
ejpam-6096	198	20	ayc+	ayc+	ADJ
ejpam-6096	198	21	abc	abc	NOUN
ejpam-6096	198	22	=	=	PUNCT
ejpam-6096	198	23	a(y	a(y	PROPN
ejpam-6096	198	24	+	+	CCONJ
ejpam-6096	198	25	b)(z	b)(z	NUM
ejpam-6096	198	26	+	+	CCONJ
ejpam-6096	198	27	c	c	X
ejpam-6096	198	28	)	)	PUNCT
ejpam-6096	198	29	∈	∈	PROPN
ejpam-6096	198	30	ijk	ijk	PROPN
ejpam-6096	198	31	,	,	PUNCT
ejpam-6096	198	32	(	(	PUNCT
ejpam-6096	198	33	5	5	NUM
ejpam-6096	198	34	)	)	PUNCT
ejpam-6096	198	35	xbz	xbz	PUNCT
ejpam-6096	199	1	+	+	CCONJ
ejpam-6096	199	2	xbc+	xbc+	PROPN
ejpam-6096	199	3	abz	abz	PROPN
ejpam-6096	199	4	+	+	CCONJ
ejpam-6096	199	5	abc	abc	PROPN
ejpam-6096	199	6	=	=	SYM
ejpam-6096	199	7	(	(	PUNCT
ejpam-6096	199	8	x+	x+	X
ejpam-6096	199	9	a)b(z	a)b(z	PROPN
ejpam-6096	199	10	+	+	CCONJ
ejpam-6096	199	11	c	c	X
ejpam-6096	199	12	)	)	PUNCT
ejpam-6096	199	13	∈	∈	PROPN
ejpam-6096	199	14	ijk	ijk	PROPN
ejpam-6096	199	15	,	,	PUNCT
ejpam-6096	199	16	(	(	PUNCT
ejpam-6096	199	17	6	6	NUM
ejpam-6096	199	18	)	)	PUNCT
ejpam-6096	199	19	and	and	CCONJ
ejpam-6096	199	20	xyz	xyz	NOUN
ejpam-6096	199	21	+	+	CCONJ
ejpam-6096	200	1	xyc+	xyc+	ADJ
ejpam-6096	200	2	xbz	xbz	PUNCT
ejpam-6096	201	1	+	+	NOUN
ejpam-6096	201	2	xbc+	xbc+	PROPN
ejpam-6096	201	3	ayz	ayz	PROPN
ejpam-6096	201	4	+	+	CCONJ
ejpam-6096	201	5	ayc+	ayc+	ADJ
ejpam-6096	201	6	abz	abz	NOUN
ejpam-6096	201	7	+	+	CCONJ
ejpam-6096	201	8	abc	abc	PROPN
ejpam-6096	201	9	=	=	SYM
ejpam-6096	201	10	(	(	PUNCT
ejpam-6096	201	11	x+	x+	X
ejpam-6096	201	12	a)(y	a)(y	PROPN
ejpam-6096	201	13	+	+	CCONJ
ejpam-6096	201	14	b)(z	b)(z	NUM
ejpam-6096	201	15	+	+	CCONJ
ejpam-6096	201	16	c	c	X
ejpam-6096	201	17	)	)	PUNCT
ejpam-6096	201	18	∈	∈	PROPN
ejpam-6096	201	19	ijk	ijk	PROPN
ejpam-6096	201	20	.	.	PUNCT
ejpam-6096	202	1	(	(	PUNCT
ejpam-6096	202	2	7	7	NUM
ejpam-6096	202	3	)	)	PUNCT
ejpam-6096	202	4	since	since	SCONJ
ejpam-6096	202	5	abc	abc	PROPN
ejpam-6096	202	6	∈	∈	PROPN
ejpam-6096	202	7	ijk	ijk	PROPN
ejpam-6096	202	8	and	and	CCONJ
ejpam-6096	202	9	xbc+abc	xbc+abc	PROPN
ejpam-6096	202	10	∈	∈	PROPN
ejpam-6096	202	11	ijk	ijk	PROPN
ejpam-6096	202	12	,	,	PUNCT
ejpam-6096	202	13	we	we	PRON
ejpam-6096	202	14	obtain	obtain	VERB
ejpam-6096	202	15	xbc	xbc	PROPN
ejpam-6096	202	16	∈	∈	PROPN
ejpam-6096	202	17	ck(ijk	ck(ijk	PROPN
ejpam-6096	202	18	)	)	PUNCT
ejpam-6096	202	19	.	.	PUNCT
ejpam-6096	203	1	by	by	ADP
ejpam-6096	203	2	the	the	DET
ejpam-6096	203	3	same	same	ADJ
ejpam-6096	203	4	argument	argument	NOUN
ejpam-6096	203	5	,	,	PUNCT
ejpam-6096	203	6	from	from	ADP
ejpam-6096	203	7	(	(	PUNCT
ejpam-6096	203	8	2	2	NUM
ejpam-6096	203	9	)	)	PUNCT
ejpam-6096	203	10	and	and	CCONJ
ejpam-6096	203	11	(	(	PUNCT
ejpam-6096	203	12	3	3	NUM
ejpam-6096	203	13	)	)	PUNCT
ejpam-6096	203	14	,	,	PUNCT
ejpam-6096	203	15	we	we	PRON
ejpam-6096	203	16	also	also	ADV
ejpam-6096	203	17	obtain	obtain	VERB
ejpam-6096	203	18	ayc	ayc	NOUN
ejpam-6096	203	19	∈	∈	PROPN
ejpam-6096	203	20	ck(ijk	ck(ijk	X
ejpam-6096	203	21	)	)	PUNCT
ejpam-6096	203	22	and	and	CCONJ
ejpam-6096	203	23	abz	abz	PROPN
ejpam-6096	203	24	∈	∈	PROPN
ejpam-6096	203	25	ck(ijk	ck(ijk	PROPN
ejpam-6096	203	26	)	)	PUNCT
ejpam-6096	203	27	,	,	PUNCT
ejpam-6096	203	28	respectively	respectively	ADV
ejpam-6096	203	29	.	.	PUNCT
ejpam-6096	204	1	again	again	ADV
ejpam-6096	204	2	,	,	PUNCT
ejpam-6096	204	3	since	since	SCONJ
ejpam-6096	204	4	abc	abc	PROPN
ejpam-6096	204	5	∈	∈	PROPN
ejpam-6096	204	6	ijk	ijk	PROPN
ejpam-6096	204	7	and	and	CCONJ
ejpam-6096	204	8	(	(	PUNCT
ejpam-6096	204	9	4	4	X
ejpam-6096	204	10	)	)	PUNCT
ejpam-6096	204	11	holds	hold	VERB
ejpam-6096	204	12	,	,	PUNCT
ejpam-6096	204	13	we	we	PRON
ejpam-6096	204	14	get	get	VERB
ejpam-6096	204	15	xyc+	xyc+	PROPN
ejpam-6096	204	16	ayc+	ayc+	PROPN
ejpam-6096	204	17	xbc	xbc	PROPN
ejpam-6096	204	18	∈	∈	PROPN
ejpam-6096	204	19	ck(ijk	ck(ijk	PROPN
ejpam-6096	204	20	)	)	PUNCT
ejpam-6096	204	21	.	.	PUNCT
ejpam-6096	205	1	by	by	ADP
ejpam-6096	205	2	proposition	proposition	NOUN
ejpam-6096	205	3	5	5	NUM
ejpam-6096	205	4	,	,	PUNCT
ejpam-6096	205	5	we	we	PRON
ejpam-6096	205	6	know	know	VERB
ejpam-6096	205	7	that	that	SCONJ
ejpam-6096	205	8	ck(ijk	ck(ijk	PROPN
ejpam-6096	205	9	)	)	PUNCT
ejpam-6096	205	10	is	be	AUX
ejpam-6096	205	11	a	a	DET
ejpam-6096	205	12	k	k	NOUN
ejpam-6096	205	13	-	-	NOUN
ejpam-6096	205	14	ideal	ideal	NOUN
ejpam-6096	205	15	of	of	ADP
ejpam-6096	205	16	r	r	NOUN
ejpam-6096	205	17	,	,	PUNCT
ejpam-6096	205	18	which	which	PRON
ejpam-6096	205	19	implies	imply	VERB
ejpam-6096	205	20	that	that	SCONJ
ejpam-6096	205	21	xyc	xyc	PROPN
ejpam-6096	205	22	∈	∈	PROPN
ejpam-6096	205	23	ck(ijk	ck(ijk	PROPN
ejpam-6096	205	24	)	)	PUNCT
ejpam-6096	205	25	because	because	SCONJ
ejpam-6096	205	26	xbc	xbc	PROPN
ejpam-6096	205	27	,	,	PUNCT
ejpam-6096	205	28	ayc	ayc	NOUN
ejpam-6096	205	29	∈	∈	PROPN
ejpam-6096	205	30	ck(ijk	ck(ijk	PROPN
ejpam-6096	205	31	)	)	PUNCT
ejpam-6096	205	32	,	,	PUNCT
ejpam-6096	205	33	applying	apply	VERB
ejpam-6096	205	34	the	the	DET
ejpam-6096	205	35	same	same	ADJ
ejpam-6096	205	36	process	process	NOUN
ejpam-6096	205	37	to	to	ADP
ejpam-6096	205	38	(	(	PUNCT
ejpam-6096	205	39	5	5	NUM
ejpam-6096	205	40	)	)	PUNCT
ejpam-6096	205	41	and	and	CCONJ
ejpam-6096	205	42	(	(	PUNCT
ejpam-6096	205	43	6	6	NUM
ejpam-6096	205	44	)	)	PUNCT
ejpam-6096	205	45	,	,	PUNCT
ejpam-6096	205	46	we	we	PRON
ejpam-6096	205	47	conclude	conclude	VERB
ejpam-6096	205	48	that	that	DET
ejpam-6096	205	49	ayz	ayz	NOUN
ejpam-6096	205	50	,	,	PUNCT
ejpam-6096	205	51	xbz	xbz	PROPN
ejpam-6096	205	52	∈	∈	PROPN
ejpam-6096	205	53	ck(ijk	ck(ijk	PROPN
ejpam-6096	205	54	)	)	PUNCT
ejpam-6096	205	55	,	,	PUNCT
ejpam-6096	205	56	respectively	respectively	ADV
ejpam-6096	205	57	.	.	PUNCT
ejpam-6096	206	1	once	once	ADV
ejpam-6096	206	2	again	again	ADV
ejpam-6096	206	3	,	,	PUNCT
ejpam-6096	206	4	we	we	PRON
ejpam-6096	206	5	see	see	VERB
ejpam-6096	206	6	that	that	SCONJ
ejpam-6096	206	7	(	(	PUNCT
ejpam-6096	206	8	7	7	X
ejpam-6096	206	9	)	)	PUNCT
ejpam-6096	206	10	implies	imply	VERB
ejpam-6096	206	11	xyz	xyz	PROPN
ejpam-6096	206	12	+	+	CCONJ
ejpam-6096	206	13	xyc+	xyc+	ADJ
ejpam-6096	206	14	xbz	xbz	PUNCT
ejpam-6096	207	1	+	+	NOUN
ejpam-6096	207	2	xbc+	xbc+	PROPN
ejpam-6096	207	3	ayz	ayz	PROPN
ejpam-6096	207	4	+	+	PROPN
ejpam-6096	207	5	ayc+	ayc+	ADJ
ejpam-6096	207	6	abz	abz	ADJ
ejpam-6096	207	7	∈	∈	PROPN
ejpam-6096	207	8	ck(ijk	ck(ijk	PROPN
ejpam-6096	207	9	)	)	PUNCT
ejpam-6096	207	10	.	.	PUNCT
ejpam-6096	208	1	since	since	SCONJ
ejpam-6096	208	2	xbc	xbc	PROPN
ejpam-6096	208	3	,	,	PUNCT
ejpam-6096	208	4	ayc	ayc	PROPN
ejpam-6096	208	5	,	,	PUNCT
ejpam-6096	208	6	abz	abz	PROPN
ejpam-6096	208	7	,	,	PUNCT
ejpam-6096	208	8	xyc	xyc	PROPN
ejpam-6096	208	9	,	,	PUNCT
ejpam-6096	208	10	ayz	ayz	PROPN
ejpam-6096	208	11	and	and	CCONJ
ejpam-6096	208	12	xbz	xbz	PROPN
ejpam-6096	208	13	are	be	AUX
ejpam-6096	208	14	all	all	PRON
ejpam-6096	208	15	members	member	NOUN
ejpam-6096	208	16	of	of	ADP
ejpam-6096	208	17	ck(ijk	ck(ijk	PROPN
ejpam-6096	208	18	)	)	PUNCT
ejpam-6096	208	19	,	,	PUNCT
ejpam-6096	208	20	we	we	PRON
ejpam-6096	208	21	eventually	eventually	ADV
ejpam-6096	208	22	obtain	obtain	VERB
ejpam-6096	208	23	xyz	xyz	PROPN
ejpam-6096	208	24	∈	∈	PROPN
ejpam-6096	208	25	ck(ijk	ck(ijk	PROPN
ejpam-6096	208	26	)	)	PUNCT
ejpam-6096	208	27	.	.	PUNCT
ejpam-6096	209	1	therefore	therefore	ADV
ejpam-6096	209	2	,	,	PUNCT
ejpam-6096	209	3	ck(i)ck(j)ck(k	ck(i)ck(j)ck(k	PROPN
ejpam-6096	209	4	)	)	PUNCT
ejpam-6096	209	5	⊆	⊆	NUM
ejpam-6096	209	6	ck(ijk	ck(ijk	X
ejpam-6096	209	7	)	)	PUNCT
ejpam-6096	209	8	.	.	PUNCT
ejpam-6096	209	9	suppose	suppose	VERB
ejpam-6096	209	10	that	that	SCONJ
ejpam-6096	209	11	i	i	PRON
ejpam-6096	209	12	,	,	PUNCT
ejpam-6096	209	13	j	j	PROPN
ejpam-6096	209	14	and	and	CCONJ
ejpam-6096	209	15	k	k	PROPN
ejpam-6096	209	16	are	be	AUX
ejpam-6096	209	17	ideals	ideal	NOUN
ejpam-6096	209	18	of	of	ADP
ejpam-6096	209	19	a	a	DET
ejpam-6096	209	20	ternary	ternary	ADJ
ejpam-6096	209	21	semiring	semire	VERB
ejpam-6096	209	22	r.	r.	PROPN
ejpam-6096	209	23	then	then	ADV
ejpam-6096	209	24	the	the	DET
ejpam-6096	209	25	ideal	ideal	ADJ
ejpam-6096	209	26	quotient	quotient	NOUN
ejpam-6096	209	27	of	of	ADP
ejpam-6096	209	28	i	i	PRON
ejpam-6096	209	29	over	over	ADP
ejpam-6096	209	30	j	j	PROPN
ejpam-6096	209	31	and	and	CCONJ
ejpam-6096	209	32	k	k	PROPN
ejpam-6096	209	33	is	be	AUX
ejpam-6096	209	34	defined	define	VERB
ejpam-6096	209	35	by	by	ADP
ejpam-6096	209	36	(	(	PUNCT
ejpam-6096	209	37	i	i	PRON
ejpam-6096	209	38	:	:	PUNCT
ejpam-6096	209	39	j	j	PROPN
ejpam-6096	209	40	,	,	PUNCT
ejpam-6096	209	41	k	k	PROPN
ejpam-6096	209	42	)	)	PUNCT
ejpam-6096	209	43	=	=	PRON
ejpam-6096	209	44	{	{	PUNCT
ejpam-6096	209	45	a	a	PRON
ejpam-6096	209	46	∈	∈	PROPN
ejpam-6096	209	47	r	r	NOUN
ejpam-6096	209	48	|	|	NOUN
ejpam-6096	209	49	ajk	ajk	PROPN
ejpam-6096	209	50	⊆	⊆	NUM
ejpam-6096	209	51	i	i	PROPN
ejpam-6096	209	52	}	}	PUNCT
ejpam-6096	209	53	.	.	PUNCT
ejpam-6096	210	1	example	example	NOUN
ejpam-6096	210	2	4	4	X
ejpam-6096	210	3	.	.	X
ejpam-6096	211	1	consider	consider	VERB
ejpam-6096	211	2	r	r	NOUN
ejpam-6096	211	3	=	=	SYM
ejpam-6096	211	4	{	{	PUNCT
ejpam-6096	211	5	0	0	NUM
ejpam-6096	211	6	,	,	PUNCT
ejpam-6096	211	7	2	2	NUM
ejpam-6096	211	8	,	,	PUNCT
ejpam-6096	211	9	4	4	NUM
ejpam-6096	211	10	,	,	PUNCT
ejpam-6096	211	11	6	6	NUM
ejpam-6096	211	12	,	,	PUNCT
ejpam-6096	211	13	8	8	NUM
ejpam-6096	211	14	,	,	PUNCT
ejpam-6096	211	15	10	10	NUM
ejpam-6096	211	16	,	,	PUNCT
ejpam-6096	211	17	12	12	NUM
ejpam-6096	211	18	,	,	PUNCT
ejpam-6096	211	19	14	14	NUM
ejpam-6096	211	20	}	}	SYM
ejpam-6096	211	21	⊆	⊆	NUM
ejpam-6096	211	22	z16	z16	NOUN
ejpam-6096	211	23	which	which	PRON
ejpam-6096	211	24	is	be	AUX
ejpam-6096	211	25	a	a	DET
ejpam-6096	211	26	ternary	ternary	ADJ
ejpam-6096	211	27	semiring	semiring	NOUN
ejpam-6096	211	28	under	under	ADP
ejpam-6096	211	29	the	the	DET
ejpam-6096	211	30	usual	usual	ADJ
ejpam-6096	211	31	addition	addition	NOUN
ejpam-6096	211	32	and	and	CCONJ
ejpam-6096	211	33	ternary	ternary	ADJ
ejpam-6096	211	34	multiplication	multiplication	NOUN
ejpam-6096	211	35	of	of	ADP
ejpam-6096	211	36	integers	integer	NOUN
ejpam-6096	211	37	modulo	modulo	VERB
ejpam-6096	211	38	16	16	NUM
ejpam-6096	211	39	.	.	PUNCT
ejpam-6096	212	1	let	let	VERB
ejpam-6096	212	2	i	i	PRON
ejpam-6096	212	3	=	=	PUNCT
ejpam-6096	212	4	{	{	PUNCT
ejpam-6096	212	5	0	0	NUM
ejpam-6096	212	6	,	,	PUNCT
ejpam-6096	212	7	8	8	NUM
ejpam-6096	212	8	}	}	PUNCT
ejpam-6096	212	9	.	.	PUNCT
ejpam-6096	213	1	we	we	PRON
ejpam-6096	213	2	have	have	VERB
ejpam-6096	213	3	that	that	PRON
ejpam-6096	213	4	(	(	PUNCT
ejpam-6096	213	5	i	i	PRON
ejpam-6096	213	6	:	:	PUNCT
ejpam-6096	213	7	j	j	PROPN
ejpam-6096	213	8	,	,	PUNCT
ejpam-6096	213	9	k	k	PROPN
ejpam-6096	213	10	)	)	PUNCT
ejpam-6096	214	1	=	=	SYM
ejpam-6096	214	2	r	r	NOUN
ejpam-6096	214	3	for	for	ADP
ejpam-6096	214	4	all	all	DET
ejpam-6096	214	5	ideals	ideal	NOUN
ejpam-6096	214	6	j	j	PROPN
ejpam-6096	214	7	and	and	CCONJ
ejpam-6096	214	8	k	k	PROPN
ejpam-6096	214	9	of	of	ADP
ejpam-6096	214	10	r.	r.	PROPN
ejpam-6096	214	11	proposition	proposition	PROPN
ejpam-6096	214	12	10	10	NUM
ejpam-6096	214	13	.	.	PUNCT
ejpam-6096	215	1	let	let	VERB
ejpam-6096	215	2	i	i	PRON
ejpam-6096	215	3	,	,	PUNCT
ejpam-6096	215	4	j	j	PROPN
ejpam-6096	215	5	and	and	CCONJ
ejpam-6096	215	6	k	k	PROPN
ejpam-6096	215	7	be	be	VERB
ejpam-6096	215	8	ideals	ideal	NOUN
ejpam-6096	215	9	of	of	ADP
ejpam-6096	215	10	a	a	DET
ejpam-6096	215	11	ternary	ternary	ADJ
ejpam-6096	215	12	semiring	semire	VERB
ejpam-6096	215	13	r.	r.	PROPN
ejpam-6096	215	14	then	then	ADV
ejpam-6096	215	15	i	i	PRON
ejpam-6096	215	16	⊆	⊆	NUM
ejpam-6096	215	17	(	(	PUNCT
ejpam-6096	215	18	i	i	PRON
ejpam-6096	215	19	:	:	PUNCT
ejpam-6096	215	20	j	j	PROPN
ejpam-6096	215	21	,	,	PUNCT
ejpam-6096	215	22	k	k	PROPN
ejpam-6096	215	23	)	)	PUNCT
ejpam-6096	215	24	.	.	PUNCT
ejpam-6096	216	1	proof	proof	NOUN
ejpam-6096	216	2	.	.	PUNCT
ejpam-6096	217	1	let	let	VERB
ejpam-6096	217	2	a	a	DET
ejpam-6096	217	3	∈	∈	PROPN
ejpam-6096	217	4	i.	i.	NOUN
ejpam-6096	217	5	since	since	SCONJ
ejpam-6096	217	6	i	i	PRON
ejpam-6096	217	7	is	be	AUX
ejpam-6096	217	8	an	an	DET
ejpam-6096	217	9	ideal	ideal	NOUN
ejpam-6096	217	10	of	of	ADP
ejpam-6096	217	11	r	r	NOUN
ejpam-6096	217	12	and	and	CCONJ
ejpam-6096	217	13	j	j	PROPN
ejpam-6096	217	14	,	,	PUNCT
ejpam-6096	217	15	k	k	PROPN
ejpam-6096	217	16	⊆	⊆	NUM
ejpam-6096	217	17	r	r	NOUN
ejpam-6096	217	18	,	,	PUNCT
ejpam-6096	217	19	we	we	PRON
ejpam-6096	217	20	have	have	VERB
ejpam-6096	217	21	that	that	PRON
ejpam-6096	217	22	ajk	ajk	PROPN
ejpam-6096	217	23	⊆	⊆	NUM
ejpam-6096	217	24	i.	i.	NOUN
ejpam-6096	217	25	this	this	PRON
ejpam-6096	217	26	implies	imply	VERB
ejpam-6096	217	27	that	that	SCONJ
ejpam-6096	217	28	i	i	PRON
ejpam-6096	217	29	⊆	⊆	NUM
ejpam-6096	217	30	(	(	PUNCT
ejpam-6096	217	31	i	i	PRON
ejpam-6096	217	32	:	:	PUNCT
ejpam-6096	217	33	j	j	PROPN
ejpam-6096	217	34	,	,	PUNCT
ejpam-6096	217	35	k	k	PROPN
ejpam-6096	217	36	)	)	PUNCT
ejpam-6096	217	37	.	.	PUNCT
ejpam-6096	218	1	proposition	proposition	NOUN
ejpam-6096	218	2	11	11	NUM
ejpam-6096	218	3	.	.	PUNCT
ejpam-6096	219	1	let	let	VERB
ejpam-6096	219	2	r	r	PRON
ejpam-6096	219	3	be	be	AUX
ejpam-6096	219	4	a	a	DET
ejpam-6096	219	5	commutative	commutative	ADJ
ejpam-6096	219	6	ternary	ternary	ADJ
ejpam-6096	219	7	semiring	semiring	NOUN
ejpam-6096	219	8	.	.	PUNCT
ejpam-6096	220	1	if	if	SCONJ
ejpam-6096	220	2	i	i	PRON
ejpam-6096	220	3	is	be	AUX
ejpam-6096	220	4	a	a	DET
ejpam-6096	220	5	k	k	NOUN
ejpam-6096	220	6	-	-	PUNCT
ejpam-6096	220	7	ideal	ideal	NOUN
ejpam-6096	220	8	and	and	CCONJ
ejpam-6096	220	9	j	j	PROPN
ejpam-6096	220	10	,	,	PUNCT
ejpam-6096	220	11	k	k	PROPN
ejpam-6096	220	12	are	be	AUX
ejpam-6096	220	13	ideals	ideal	NOUN
ejpam-6096	220	14	of	of	ADP
ejpam-6096	220	15	r	r	NOUN
ejpam-6096	220	16	,	,	PUNCT
ejpam-6096	220	17	then	then	ADV
ejpam-6096	220	18	(	(	PUNCT
ejpam-6096	220	19	i	i	PRON
ejpam-6096	220	20	:	:	PUNCT
ejpam-6096	220	21	j	j	PROPN
ejpam-6096	220	22	,	,	PUNCT
ejpam-6096	220	23	k	k	PROPN
ejpam-6096	220	24	)	)	PUNCT
ejpam-6096	220	25	is	be	AUX
ejpam-6096	220	26	a	a	DET
ejpam-6096	220	27	k	k	NOUN
ejpam-6096	220	28	-	-	NOUN
ejpam-6096	220	29	ideal	ideal	NOUN
ejpam-6096	220	30	of	of	ADP
ejpam-6096	220	31	r.	r.	PROPN
ejpam-6096	220	32	a.	a.	PROPN
ejpam-6096	220	33	j.	j.	PROPN
ejpam-6096	220	34	khan	khan	PROPN
ejpam-6096	220	35	,	,	PUNCT
ejpam-6096	220	36	m.	m.	NOUN
ejpam-6096	220	37	petapirak	petapirak	PROPN
ejpam-6096	220	38	,	,	PUNCT
ejpam-6096	220	39	r.	r.	PROPN
ejpam-6096	220	40	chinram	chinram	PROPN
ejpam-6096	220	41	/	/	SYM
ejpam-6096	220	42	eur	eur	PROPN
ejpam-6096	220	43	.	.	PUNCT
ejpam-6096	221	1	j.	j.	PROPN
ejpam-6096	221	2	pure	pure	PROPN
ejpam-6096	221	3	appl	appl	PROPN
ejpam-6096	221	4	.	.	PROPN
ejpam-6096	221	5	math	math	PROPN
ejpam-6096	221	6	,	,	PUNCT
ejpam-6096	221	7	18	18	NUM
ejpam-6096	221	8	(	(	PUNCT
ejpam-6096	221	9	2	2	NUM
ejpam-6096	221	10	)	)	PUNCT
ejpam-6096	221	11	(	(	PUNCT
ejpam-6096	221	12	2025	2025	NUM
ejpam-6096	221	13	)	)	PUNCT
ejpam-6096	221	14	,	,	PUNCT
ejpam-6096	221	15	6096	6096	NUM
ejpam-6096	221	16	7	7	NUM
ejpam-6096	221	17	of	of	ADP
ejpam-6096	221	18	10	10	NUM
ejpam-6096	221	19	proof	proof	NOUN
ejpam-6096	221	20	.	.	PUNCT
ejpam-6096	222	1	let	let	VERB
ejpam-6096	222	2	a	a	DET
ejpam-6096	222	3	,	,	PUNCT
ejpam-6096	222	4	b	b	X
ejpam-6096	222	5	∈	∈	PROPN
ejpam-6096	222	6	(	(	PUNCT
ejpam-6096	222	7	i	i	PRON
ejpam-6096	222	8	:	:	PUNCT
ejpam-6096	222	9	j	j	PROPN
ejpam-6096	222	10	,	,	PUNCT
ejpam-6096	222	11	k	k	PROPN
ejpam-6096	222	12	)	)	PUNCT
ejpam-6096	222	13	and	and	CCONJ
ejpam-6096	222	14	r	r	NOUN
ejpam-6096	222	15	,	,	PUNCT
ejpam-6096	222	16	s	s	PROPN
ejpam-6096	222	17	∈	∈	PROPN
ejpam-6096	222	18	r.	r.	PROPN
ejpam-6096	222	19	then	then	ADV
ejpam-6096	222	20	ajk	ajk	PROPN
ejpam-6096	222	21	⊆	⊆	NUM
ejpam-6096	222	22	i	i	PROPN
ejpam-6096	222	23	and	and	CCONJ
ejpam-6096	222	24	bjk	bjk	PROPN
ejpam-6096	222	25	⊆	⊆	NUM
ejpam-6096	222	26	i.	i.	NOUN
ejpam-6096	222	27	thus	thus	ADV
ejpam-6096	222	28	(	(	PUNCT
ejpam-6096	222	29	a	a	DET
ejpam-6096	222	30	+	+	PROPN
ejpam-6096	222	31	b)jk	b)jk	PROPN
ejpam-6096	222	32	⊆	⊆	NUM
ejpam-6096	222	33	i	i	PROPN
ejpam-6096	222	34	and	and	CCONJ
ejpam-6096	222	35	(	(	PUNCT
ejpam-6096	222	36	ars)jk	ars)jk	PROPN
ejpam-6096	222	37	⊆	⊆	NUM
ejpam-6096	222	38	a(rsj)k	a(rsj)k	NOUN
ejpam-6096	222	39	⊆	⊆	NUM
ejpam-6096	222	40	ajk	ajk	PROPN
ejpam-6096	222	41	⊆	⊆	NUM
ejpam-6096	222	42	i.	i.	NOUN
ejpam-6096	223	1	so	so	SCONJ
ejpam-6096	223	2	a	a	DET
ejpam-6096	223	3	+	+	NOUN
ejpam-6096	223	4	b	b	NOUN
ejpam-6096	223	5	∈	∈	NOUN
ejpam-6096	223	6	(	(	PUNCT
ejpam-6096	223	7	i	i	PRON
ejpam-6096	223	8	:	:	PUNCT
ejpam-6096	223	9	j	j	PROPN
ejpam-6096	223	10	,	,	PUNCT
ejpam-6096	223	11	k	k	PROPN
ejpam-6096	223	12	)	)	PUNCT
ejpam-6096	224	1	and	and	CCONJ
ejpam-6096	224	2	ars	ar	VERB
ejpam-6096	224	3	∈	∈	PROPN
ejpam-6096	224	4	(	(	PUNCT
ejpam-6096	224	5	i	i	PRON
ejpam-6096	224	6	:	:	PUNCT
ejpam-6096	224	7	j	j	PROPN
ejpam-6096	224	8	,	,	PUNCT
ejpam-6096	224	9	k	k	PROPN
ejpam-6096	224	10	)	)	PUNCT
ejpam-6096	224	11	.	.	PUNCT
ejpam-6096	225	1	hence	hence	ADV
ejpam-6096	225	2	(	(	PUNCT
ejpam-6096	225	3	i	i	PRON
ejpam-6096	225	4	:	:	PUNCT
ejpam-6096	225	5	j	j	PROPN
ejpam-6096	225	6	,	,	PUNCT
ejpam-6096	225	7	k	k	PROPN
ejpam-6096	225	8	)	)	PUNCT
ejpam-6096	225	9	is	be	AUX
ejpam-6096	225	10	an	an	DET
ejpam-6096	225	11	ideal	ideal	NOUN
ejpam-6096	225	12	of	of	ADP
ejpam-6096	225	13	r.	r.	PROPN
ejpam-6096	225	14	next	next	ADV
ejpam-6096	225	15	,	,	PUNCT
ejpam-6096	225	16	let	let	VERB
ejpam-6096	225	17	a	a	PRON
ejpam-6096	225	18	,	,	PUNCT
ejpam-6096	225	19	b	b	X
ejpam-6096	225	20	∈	∈	NOUN
ejpam-6096	225	21	r	r	NOUN
ejpam-6096	225	22	be	be	VERB
ejpam-6096	225	23	such	such	ADJ
ejpam-6096	225	24	that	that	SCONJ
ejpam-6096	225	25	a	a	DET
ejpam-6096	225	26	∈	∈	NOUN
ejpam-6096	225	27	(	(	PUNCT
ejpam-6096	225	28	i	i	PRON
ejpam-6096	225	29	:	:	PUNCT
ejpam-6096	225	30	j	j	PROPN
ejpam-6096	225	31	,	,	PUNCT
ejpam-6096	225	32	k	k	PROPN
ejpam-6096	225	33	)	)	PUNCT
ejpam-6096	225	34	and	and	CCONJ
ejpam-6096	225	35	a+	a+	PUNCT
ejpam-6096	225	36	b	b	X
ejpam-6096	225	37	∈	∈	PROPN
ejpam-6096	225	38	(	(	PUNCT
ejpam-6096	225	39	i	i	PRON
ejpam-6096	225	40	:	:	PUNCT
ejpam-6096	225	41	j	j	PROPN
ejpam-6096	225	42	,	,	PUNCT
ejpam-6096	225	43	k	k	PROPN
ejpam-6096	225	44	)	)	PUNCT
ejpam-6096	225	45	.	.	PUNCT
ejpam-6096	226	1	this	this	PRON
ejpam-6096	226	2	implies	imply	VERB
ejpam-6096	226	3	that	that	SCONJ
ejpam-6096	226	4	ajk	ajk	PROPN
ejpam-6096	226	5	+	+	CCONJ
ejpam-6096	226	6	bjk	bjk	NOUN
ejpam-6096	226	7	=	=	PUNCT
ejpam-6096	226	8	(	(	PUNCT
ejpam-6096	226	9	a	a	DET
ejpam-6096	226	10	+	+	PROPN
ejpam-6096	226	11	b)jk	b)jk	PROPN
ejpam-6096	226	12	⊆	⊆	NUM
ejpam-6096	226	13	i	i	PROPN
ejpam-6096	226	14	and	and	CCONJ
ejpam-6096	226	15	ajk	ajk	PROPN
ejpam-6096	226	16	⊆	⊆	NUM
ejpam-6096	226	17	i.	i.	NOUN
ejpam-6096	226	18	let	let	VERB
ejpam-6096	226	19	x	x	X
ejpam-6096	226	20	∈	∈	PROPN
ejpam-6096	226	21	bjk	bjk	PROPN
ejpam-6096	226	22	and	and	CCONJ
ejpam-6096	226	23	y	y	PROPN
ejpam-6096	226	24	∈	∈	PROPN
ejpam-6096	226	25	ajk	ajk	PROPN
ejpam-6096	226	26	.	.	PUNCT
ejpam-6096	227	1	so	so	ADV
ejpam-6096	227	2	x	x	X
ejpam-6096	228	1	+	+	CCONJ
ejpam-6096	228	2	y	y	PROPN
ejpam-6096	228	3	∈	∈	PROPN
ejpam-6096	229	1	i	i	PRON
ejpam-6096	229	2	and	and	CCONJ
ejpam-6096	229	3	y	y	PROPN
ejpam-6096	229	4	∈	∈	PROPN
ejpam-6096	229	5	i.	i.	NOUN
ejpam-6096	229	6	since	since	SCONJ
ejpam-6096	229	7	i	i	PRON
ejpam-6096	229	8	is	be	AUX
ejpam-6096	229	9	a	a	DET
ejpam-6096	229	10	k	k	NOUN
ejpam-6096	229	11	-	-	NOUN
ejpam-6096	229	12	ideal	ideal	NOUN
ejpam-6096	229	13	of	of	ADP
ejpam-6096	229	14	r	r	NOUN
ejpam-6096	229	15	,	,	PUNCT
ejpam-6096	229	16	we	we	PRON
ejpam-6096	229	17	have	have	VERB
ejpam-6096	229	18	x	x	PART
ejpam-6096	229	19	∈	∈	PROPN
ejpam-6096	229	20	i.	i.	NOUN
ejpam-6096	229	21	this	this	PRON
ejpam-6096	229	22	implies	imply	VERB
ejpam-6096	229	23	that	that	SCONJ
ejpam-6096	229	24	bjk	bjk	PROPN
ejpam-6096	229	25	⊆	⊆	NUM
ejpam-6096	229	26	i	i	PRON
ejpam-6096	229	27	,	,	PUNCT
ejpam-6096	229	28	so	so	PROPN
ejpam-6096	229	29	b	b	X
ejpam-6096	229	30	∈	∈	PROPN
ejpam-6096	229	31	(	(	PUNCT
ejpam-6096	229	32	i	i	PRON
ejpam-6096	229	33	:	:	PUNCT
ejpam-6096	229	34	j	j	PROPN
ejpam-6096	229	35	,	,	PUNCT
ejpam-6096	229	36	k	k	PROPN
ejpam-6096	229	37	)	)	PUNCT
ejpam-6096	229	38	.	.	PUNCT
ejpam-6096	230	1	hence	hence	ADV
ejpam-6096	230	2	(	(	PUNCT
ejpam-6096	230	3	i	i	PRON
ejpam-6096	230	4	:	:	PUNCT
ejpam-6096	230	5	j	j	PROPN
ejpam-6096	230	6	,	,	PUNCT
ejpam-6096	230	7	k	k	PROPN
ejpam-6096	230	8	)	)	PUNCT
ejpam-6096	230	9	is	be	AUX
ejpam-6096	230	10	a	a	DET
ejpam-6096	230	11	k	k	NOUN
ejpam-6096	230	12	-	-	NOUN
ejpam-6096	230	13	ideal	ideal	NOUN
ejpam-6096	230	14	of	of	ADP
ejpam-6096	230	15	r.	r.	PROPN
ejpam-6096	230	16	a	a	DET
ejpam-6096	230	17	proper	proper	ADJ
ejpam-6096	230	18	ideal	ideal	NOUN
ejpam-6096	230	19	of	of	ADP
ejpam-6096	230	20	r	r	NOUN
ejpam-6096	230	21	is	be	AUX
ejpam-6096	230	22	called	call	VERB
ejpam-6096	230	23	k	k	ADV
ejpam-6096	230	24	-	-	ADJ
ejpam-6096	230	25	maximal	maximal	ADJ
ejpam-6096	230	26	if	if	SCONJ
ejpam-6096	230	27	it	it	PRON
ejpam-6096	230	28	is	be	AUX
ejpam-6096	230	29	not	not	PART
ejpam-6096	230	30	properly	properly	ADV
ejpam-6096	230	31	contained	contain	VERB
ejpam-6096	230	32	in	in	ADP
ejpam-6096	230	33	any	any	DET
ejpam-6096	230	34	other	other	ADJ
ejpam-6096	230	35	proper	proper	ADJ
ejpam-6096	230	36	k	k	NOUN
ejpam-6096	230	37	-	-	NOUN
ejpam-6096	230	38	ideal	ideal	NOUN
ejpam-6096	230	39	of	of	ADP
ejpam-6096	230	40	r.	r.	PROPN
ejpam-6096	230	41	example	example	NOUN
ejpam-6096	231	1	5	5	X
ejpam-6096	231	2	.	.	X
ejpam-6096	231	3	we	we	PRON
ejpam-6096	231	4	consider	consider	VERB
ejpam-6096	231	5	a	a	DET
ejpam-6096	231	6	ternary	ternary	ADJ
ejpam-6096	231	7	semiring	semiring	NOUN
ejpam-6096	231	8	r	r	NOUN
ejpam-6096	231	9	=	=	PUNCT
ejpam-6096	231	10	z−	z−	X
ejpam-6096	231	11	0	0	PUNCT
ejpam-6096	232	1	under	under	ADP
ejpam-6096	232	2	the	the	DET
ejpam-6096	232	3	usual	usual	ADJ
ejpam-6096	232	4	addition	addition	NOUN
ejpam-6096	232	5	and	and	CCONJ
ejpam-6096	232	6	ternary	ternary	ADJ
ejpam-6096	232	7	multiplication	multiplication	NOUN
ejpam-6096	232	8	of	of	ADP
ejpam-6096	232	9	integers	integer	NOUN
ejpam-6096	232	10	.	.	PUNCT
ejpam-6096	233	1	(	(	PUNCT
ejpam-6096	233	2	1	1	X
ejpam-6096	233	3	)	)	PUNCT
ejpam-6096	233	4	let	let	VERB
ejpam-6096	233	5	i	i	PRON
ejpam-6096	233	6	=	=	PUNCT
ejpam-6096	234	1	2z−	2z−	NUM
ejpam-6096	234	2	0	0	NUM
ejpam-6096	234	3	=	=	SYM
ejpam-6096	234	4	{	{	PUNCT
ejpam-6096	234	5	0,−2,−4,−6,−8,−10	0,−2,−4,−6,−8,−10	NOUN
ejpam-6096	234	6	,	,	PUNCT
ejpam-6096	234	7	.	.	PUNCT
ejpam-6096	234	8	.	.	PUNCT
ejpam-6096	234	9	.	.	PUNCT
ejpam-6096	234	10	}	}	PUNCT
ejpam-6096	234	11	.	.	PUNCT
ejpam-6096	235	1	then	then	ADV
ejpam-6096	235	2	i	i	PRON
ejpam-6096	235	3	is	be	AUX
ejpam-6096	235	4	a	a	DET
ejpam-6096	235	5	k	k	NOUN
ejpam-6096	235	6	-	-	NOUN
ejpam-6096	235	7	ideal	ideal	NOUN
ejpam-6096	235	8	of	of	ADP
ejpam-6096	235	9	r.	r.	PROPN
ejpam-6096	235	10	we	we	PRON
ejpam-6096	235	11	have	have	VERB
ejpam-6096	235	12	that	that	SCONJ
ejpam-6096	235	13	i	i	PRON
ejpam-6096	235	14	is	be	AUX
ejpam-6096	235	15	k	k	ADV
ejpam-6096	235	16	-	-	ADJ
ejpam-6096	235	17	maximal	maximal	ADJ
ejpam-6096	235	18	but	but	CCONJ
ejpam-6096	235	19	not	not	PART
ejpam-6096	235	20	maximal	maximal	ADJ
ejpam-6096	235	21	.	.	PUNCT
ejpam-6096	236	1	(	(	PUNCT
ejpam-6096	236	2	2	2	X
ejpam-6096	236	3	)	)	PUNCT
ejpam-6096	236	4	let	let	VERB
ejpam-6096	236	5	i	i	PRON
ejpam-6096	236	6	=	=	PUNCT
ejpam-6096	236	7	z−	z−	X
ejpam-6096	236	8	0	0	NUM
ejpam-6096	236	9	∖	∖	X
ejpam-6096	236	10	{	{	PUNCT
ejpam-6096	236	11	−1	−1	NOUN
ejpam-6096	236	12	}	}	PUNCT
ejpam-6096	236	13	=	=	SYM
ejpam-6096	236	14	{	{	PUNCT
ejpam-6096	236	15	0,−2,−3,−4,−5,−6	0,−2,−3,−4,−5,−6	NOUN
ejpam-6096	236	16	,	,	PUNCT
ejpam-6096	236	17	.	.	PUNCT
ejpam-6096	236	18	.	.	PUNCT
ejpam-6096	237	1	.	.	PUNCT
ejpam-6096	237	2	}	}	PUNCT
ejpam-6096	237	3	.	.	PUNCT
ejpam-6096	238	1	then	then	ADV
ejpam-6096	238	2	i	i	PRON
ejpam-6096	238	3	is	be	AUX
ejpam-6096	238	4	an	an	DET
ejpam-6096	238	5	ideal	ideal	NOUN
ejpam-6096	238	6	of	of	ADP
ejpam-6096	238	7	r	r	NOUN
ejpam-6096	238	8	but	but	CCONJ
ejpam-6096	238	9	not	not	PART
ejpam-6096	238	10	a	a	DET
ejpam-6096	238	11	k	k	NOUN
ejpam-6096	238	12	-	-	NOUN
ejpam-6096	238	13	ideal	ideal	ADJ
ejpam-6096	238	14	.	.	PUNCT
ejpam-6096	239	1	we	we	PRON
ejpam-6096	239	2	have	have	VERB
ejpam-6096	239	3	that	that	SCONJ
ejpam-6096	239	4	i	i	PRON
ejpam-6096	239	5	is	be	AUX
ejpam-6096	239	6	both	both	PRON
ejpam-6096	239	7	maximal	maximal	ADJ
ejpam-6096	239	8	and	and	CCONJ
ejpam-6096	239	9	k	k	ADV
ejpam-6096	239	10	-	-	ADJ
ejpam-6096	239	11	maximal	maximal	ADJ
ejpam-6096	239	12	.	.	PUNCT
ejpam-6096	240	1	theorem	theorem	NOUN
ejpam-6096	240	2	2	2	NUM
ejpam-6096	240	3	.	.	PUNCT
ejpam-6096	241	1	let	let	VERB
ejpam-6096	241	2	r	r	PRON
ejpam-6096	241	3	be	be	AUX
ejpam-6096	241	4	a	a	DET
ejpam-6096	241	5	ternary	ternary	ADJ
ejpam-6096	241	6	semiring	semiring	NOUN
ejpam-6096	241	7	and	and	CCONJ
ejpam-6096	241	8	let	let	VERB
ejpam-6096	241	9	i	i	PRON
ejpam-6096	241	10	be	be	AUX
ejpam-6096	241	11	a	a	DET
ejpam-6096	241	12	proper	proper	ADJ
ejpam-6096	241	13	ideal	ideal	NOUN
ejpam-6096	241	14	of	of	ADP
ejpam-6096	241	15	r.	r.	PROPN
ejpam-6096	241	16	(	(	PUNCT
ejpam-6096	241	17	1	1	NUM
ejpam-6096	241	18	)	)	PUNCT
ejpam-6096	241	19	if	if	SCONJ
ejpam-6096	241	20	i	i	PRON
ejpam-6096	241	21	is	be	AUX
ejpam-6096	241	22	k	k	ADV
ejpam-6096	241	23	-	-	ADJ
ejpam-6096	241	24	maximal	maximal	ADJ
ejpam-6096	241	25	,	,	PUNCT
ejpam-6096	241	26	then	then	ADV
ejpam-6096	241	27	i	i	PRON
ejpam-6096	241	28	is	be	AUX
ejpam-6096	241	29	a	a	DET
ejpam-6096	241	30	k	k	NOUN
ejpam-6096	241	31	-	-	PUNCT
ejpam-6096	241	32	ideal	ideal	NOUN
ejpam-6096	241	33	or	or	CCONJ
ejpam-6096	241	34	ck(i	ck(i	PUNCT
ejpam-6096	241	35	)	)	PUNCT
ejpam-6096	242	1	=	=	SYM
ejpam-6096	242	2	r.	r.	NOUN
ejpam-6096	242	3	(	(	PUNCT
ejpam-6096	242	4	2	2	NUM
ejpam-6096	242	5	)	)	PUNCT
ejpam-6096	242	6	if	if	SCONJ
ejpam-6096	242	7	i	i	PRON
ejpam-6096	242	8	is	be	AUX
ejpam-6096	242	9	a	a	DET
ejpam-6096	242	10	k	k	NOUN
ejpam-6096	242	11	-	-	NOUN
ejpam-6096	242	12	ideal	ideal	NOUN
ejpam-6096	242	13	and	and	CCONJ
ejpam-6096	242	14	a	a	DET
ejpam-6096	242	15	maximal	maximal	ADJ
ejpam-6096	242	16	ideal	ideal	NOUN
ejpam-6096	242	17	,	,	PUNCT
ejpam-6096	242	18	then	then	ADV
ejpam-6096	242	19	i	i	PRON
ejpam-6096	242	20	is	be	AUX
ejpam-6096	242	21	k	k	ADV
ejpam-6096	242	22	-	-	ADJ
ejpam-6096	242	23	maximal	maximal	ADJ
ejpam-6096	242	24	.	.	PUNCT
ejpam-6096	243	1	proof	proof	NOUN
ejpam-6096	243	2	.	.	PUNCT
ejpam-6096	244	1	(	(	PUNCT
ejpam-6096	244	2	1	1	X
ejpam-6096	244	3	)	)	PUNCT
ejpam-6096	244	4	suppose	suppose	VERB
ejpam-6096	244	5	i	i	PRON
ejpam-6096	244	6	is	be	AUX
ejpam-6096	244	7	k	k	ADV
ejpam-6096	244	8	-	-	ADJ
ejpam-6096	244	9	maximal	maximal	ADJ
ejpam-6096	244	10	.	.	PUNCT
ejpam-6096	245	1	by	by	ADP
ejpam-6096	245	2	proposition	proposition	NOUN
ejpam-6096	245	3	5	5	NUM
ejpam-6096	245	4	,	,	PUNCT
ejpam-6096	245	5	i	i	PRON
ejpam-6096	245	6	⊆	⊆	NUM
ejpam-6096	245	7	ck(i	ck(i	PUNCT
ejpam-6096	245	8	)	)	PUNCT
ejpam-6096	245	9	⊆	⊆	NUM
ejpam-6096	245	10	r	r	NOUN
ejpam-6096	245	11	and	and	CCONJ
ejpam-6096	245	12	ck(i	ck(i	NUM
ejpam-6096	245	13	)	)	PUNCT
ejpam-6096	245	14	is	be	AUX
ejpam-6096	245	15	a	a	DET
ejpam-6096	245	16	k	k	NOUN
ejpam-6096	245	17	-	-	NOUN
ejpam-6096	245	18	ideal	ideal	NOUN
ejpam-6096	245	19	of	of	ADP
ejpam-6096	245	20	r.	r.	PROPN
ejpam-6096	245	21	this	this	PRON
ejpam-6096	245	22	implies	imply	VERB
ejpam-6096	245	23	that	that	SCONJ
ejpam-6096	245	24	ck(i	ck(i	PUNCT
ejpam-6096	245	25	)	)	PUNCT
ejpam-6096	246	1	=	=	SYM
ejpam-6096	246	2	i	i	PRON
ejpam-6096	246	3	or	or	CCONJ
ejpam-6096	246	4	ck(i	ck(i	PUNCT
ejpam-6096	246	5	)	)	PUNCT
ejpam-6096	246	6	=	=	VERB
ejpam-6096	247	1	r.	r.	PROPN
ejpam-6096	247	2	hence	hence	ADV
ejpam-6096	247	3	i	i	PRON
ejpam-6096	247	4	is	be	AUX
ejpam-6096	247	5	a	a	DET
ejpam-6096	247	6	k	k	NOUN
ejpam-6096	247	7	-	-	PUNCT
ejpam-6096	247	8	ideal	ideal	NOUN
ejpam-6096	247	9	or	or	CCONJ
ejpam-6096	247	10	ck(i	ck(i	PUNCT
ejpam-6096	247	11	)	)	PUNCT
ejpam-6096	248	1	=	=	SYM
ejpam-6096	248	2	r.	r.	NOUN
ejpam-6096	248	3	(	(	PUNCT
ejpam-6096	248	4	2	2	X
ejpam-6096	248	5	)	)	PUNCT
ejpam-6096	248	6	let	let	VERB
ejpam-6096	248	7	i	i	PRON
ejpam-6096	248	8	be	be	AUX
ejpam-6096	248	9	a	a	DET
ejpam-6096	248	10	k	k	NOUN
ejpam-6096	248	11	-	-	NOUN
ejpam-6096	248	12	ideal	ideal	NOUN
ejpam-6096	248	13	and	and	CCONJ
ejpam-6096	248	14	a	a	DET
ejpam-6096	248	15	maximal	maximal	ADJ
ejpam-6096	248	16	ideal	ideal	NOUN
ejpam-6096	248	17	.	.	PUNCT
ejpam-6096	249	1	suppose	suppose	VERB
ejpam-6096	249	2	to	to	ADP
ejpam-6096	249	3	the	the	DET
ejpam-6096	249	4	contrary	contrary	NOUN
ejpam-6096	249	5	,	,	PUNCT
ejpam-6096	249	6	that	that	SCONJ
ejpam-6096	249	7	i	i	PRON
ejpam-6096	249	8	is	be	AUX
ejpam-6096	249	9	not	not	PART
ejpam-6096	249	10	k	k	ADV
ejpam-6096	249	11	-	-	ADJ
ejpam-6096	249	12	maximal	maximal	ADJ
ejpam-6096	249	13	.	.	PUNCT
ejpam-6096	250	1	then	then	ADV
ejpam-6096	250	2	there	there	PRON
ejpam-6096	250	3	exists	exist	VERB
ejpam-6096	250	4	a	a	DET
ejpam-6096	250	5	k	k	ADJ
ejpam-6096	250	6	-	-	PUNCT
ejpam-6096	250	7	ideal	ideal	ADJ
ejpam-6096	250	8	j	j	PROPN
ejpam-6096	250	9	of	of	ADP
ejpam-6096	250	10	r	r	NOUN
ejpam-6096	250	11	such	such	ADJ
ejpam-6096	250	12	that	that	SCONJ
ejpam-6096	250	13	i	i	PRON
ejpam-6096	250	14	⊊	⊊	VERB
ejpam-6096	250	15	j	j	PROPN
ejpam-6096	250	16	⊊	⊊	VERB
ejpam-6096	250	17	r.	r.	PROPN
ejpam-6096	250	18	however	however	ADV
ejpam-6096	250	19	,	,	PUNCT
ejpam-6096	250	20	since	since	SCONJ
ejpam-6096	250	21	i	i	PRON
ejpam-6096	250	22	is	be	AUX
ejpam-6096	250	23	a	a	DET
ejpam-6096	250	24	maximal	maximal	ADJ
ejpam-6096	250	25	ideal	ideal	NOUN
ejpam-6096	250	26	and	and	CCONJ
ejpam-6096	250	27	j	j	PROPN
ejpam-6096	250	28	is	be	AUX
ejpam-6096	250	29	an	an	DET
ejpam-6096	250	30	ideal	ideal	NOUN
ejpam-6096	250	31	of	of	ADP
ejpam-6096	250	32	r	r	NOUN
ejpam-6096	250	33	,	,	PUNCT
ejpam-6096	250	34	no	no	DET
ejpam-6096	250	35	such	such	DET
ejpam-6096	250	36	an	an	DET
ejpam-6096	250	37	ideal	ideal	ADJ
ejpam-6096	250	38	j	j	PROPN
ejpam-6096	250	39	exists	exist	VERB
ejpam-6096	250	40	unless	unless	SCONJ
ejpam-6096	250	41	j	j	PROPN
ejpam-6096	250	42	=	=	SYM
ejpam-6096	250	43	r	r	PROPN
ejpam-6096	250	44	,	,	PUNCT
ejpam-6096	250	45	which	which	PRON
ejpam-6096	250	46	contradicts	contradict	VERB
ejpam-6096	250	47	the	the	DET
ejpam-6096	250	48	fact	fact	NOUN
ejpam-6096	250	49	that	that	SCONJ
ejpam-6096	250	50	j	j	PROPN
ejpam-6096	250	51	⊊	⊊	VERB
ejpam-6096	250	52	r.	r.	PROPN
ejpam-6096	250	53	therefore	therefore	ADV
ejpam-6096	250	54	i	i	PRON
ejpam-6096	250	55	must	must	AUX
ejpam-6096	250	56	be	be	AUX
ejpam-6096	250	57	k	k	ADJ
ejpam-6096	250	58	-	-	ADJ
ejpam-6096	250	59	maximal	maximal	ADJ
ejpam-6096	250	60	.	.	PUNCT
ejpam-6096	251	1	a	a	DET
ejpam-6096	251	2	proper	proper	ADJ
ejpam-6096	251	3	ideal	ideal	NOUN
ejpam-6096	251	4	p	p	NOUN
ejpam-6096	251	5	of	of	ADP
ejpam-6096	251	6	a	a	DET
ejpam-6096	251	7	ternary	ternary	ADJ
ejpam-6096	251	8	semiring	semiring	NOUN
ejpam-6096	251	9	r	r	NOUN
ejpam-6096	251	10	is	be	AUX
ejpam-6096	251	11	called	call	VERB
ejpam-6096	251	12	k	k	NOUN
ejpam-6096	251	13	-	-	NOUN
ejpam-6096	251	14	prime	prime	ADJ
ejpam-6096	251	15	if	if	SCONJ
ejpam-6096	251	16	ijk	ijk	PROPN
ejpam-6096	251	17	⊆	⊆	PROPN
ejpam-6096	251	18	p	p	PROPN
ejpam-6096	251	19	implies	imply	VERB
ejpam-6096	251	20	i	i	PRON
ejpam-6096	251	21	⊆	⊆	NUM
ejpam-6096	251	22	p	p	NOUN
ejpam-6096	251	23	,	,	PUNCT
ejpam-6096	251	24	j	j	PROPN
ejpam-6096	251	25	⊆	⊆	NUM
ejpam-6096	251	26	p	p	NOUN
ejpam-6096	251	27	,	,	PUNCT
ejpam-6096	251	28	or	or	CCONJ
ejpam-6096	251	29	k	k	PROPN
ejpam-6096	252	1	⊆	⊆	NUM
ejpam-6096	252	2	p	p	NOUN
ejpam-6096	252	3	for	for	ADP
ejpam-6096	252	4	all	all	DET
ejpam-6096	252	5	k	k	NOUN
ejpam-6096	252	6	-	-	NOUN
ejpam-6096	252	7	ideals	ideal	NOUN
ejpam-6096	252	8	i	i	PRON
ejpam-6096	252	9	,	,	PUNCT
ejpam-6096	252	10	j	j	PROPN
ejpam-6096	252	11	,	,	PUNCT
ejpam-6096	252	12	k	k	PROPN
ejpam-6096	252	13	of	of	ADP
ejpam-6096	252	14	r.	r.	PROPN
ejpam-6096	252	15	every	every	DET
ejpam-6096	252	16	prime	prime	ADJ
ejpam-6096	252	17	ideal	ideal	NOUN
ejpam-6096	252	18	of	of	ADP
ejpam-6096	252	19	r	r	NOUN
ejpam-6096	252	20	is	be	AUX
ejpam-6096	252	21	obviously	obviously	ADV
ejpam-6096	252	22	k	k	NOUN
ejpam-6096	252	23	-	-	NOUN
ejpam-6096	252	24	prime	prime	NOUN
ejpam-6096	252	25	.	.	PUNCT
ejpam-6096	253	1	we	we	PRON
ejpam-6096	253	2	describe	describe	VERB
ejpam-6096	253	3	in	in	ADP
ejpam-6096	253	4	the	the	DET
ejpam-6096	253	5	following	following	NOUN
ejpam-6096	253	6	theorem	theorem	VERB
ejpam-6096	253	7	,	,	PUNCT
ejpam-6096	253	8	where	where	SCONJ
ejpam-6096	253	9	being	be	AUX
ejpam-6096	253	10	prime	prime	ADJ
ejpam-6096	253	11	and	and	CCONJ
ejpam-6096	253	12	k	k	NOUN
ejpam-6096	253	13	-	-	NOUN
ejpam-6096	253	14	prime	prime	NOUN
ejpam-6096	253	15	of	of	ADP
ejpam-6096	253	16	an	an	DET
ejpam-6096	253	17	ideal	ideal	NOUN
ejpam-6096	253	18	always	always	ADV
ejpam-6096	253	19	imply	imply	VERB
ejpam-6096	253	20	each	each	DET
ejpam-6096	253	21	other	other	ADJ
ejpam-6096	253	22	.	.	PUNCT
ejpam-6096	254	1	theorem	theorem	NOUN
ejpam-6096	254	2	3	3	X
ejpam-6096	254	3	.	.	PUNCT
ejpam-6096	255	1	let	let	VERB
ejpam-6096	255	2	r	r	PRON
ejpam-6096	255	3	be	be	AUX
ejpam-6096	255	4	a	a	DET
ejpam-6096	255	5	ternary	ternary	ADJ
ejpam-6096	255	6	semiring	semiring	NOUN
ejpam-6096	255	7	and	and	CCONJ
ejpam-6096	255	8	p	p	NOUN
ejpam-6096	255	9	be	be	AUX
ejpam-6096	255	10	a	a	DET
ejpam-6096	255	11	k	k	NOUN
ejpam-6096	255	12	-	-	NOUN
ejpam-6096	255	13	ideal	ideal	NOUN
ejpam-6096	255	14	of	of	ADP
ejpam-6096	255	15	r.	r.	PROPN
ejpam-6096	255	16	then	then	ADV
ejpam-6096	255	17	p	p	PROPN
ejpam-6096	255	18	is	be	AUX
ejpam-6096	255	19	k	k	NOUN
ejpam-6096	255	20	-	-	ADJ
ejpam-6096	255	21	prime	prime	ADJ
ejpam-6096	255	22	if	if	SCONJ
ejpam-6096	256	1	and	and	CCONJ
ejpam-6096	256	2	only	only	ADV
ejpam-6096	256	3	if	if	SCONJ
ejpam-6096	256	4	p	p	NOUN
ejpam-6096	256	5	is	be	AUX
ejpam-6096	256	6	prime	prime	ADJ
ejpam-6096	256	7	.	.	PUNCT
ejpam-6096	257	1	proof	proof	NOUN
ejpam-6096	257	2	.	.	PUNCT
ejpam-6096	258	1	if	if	SCONJ
ejpam-6096	258	2	p	p	NOUN
ejpam-6096	258	3	is	be	AUX
ejpam-6096	258	4	prime	prime	ADJ
ejpam-6096	258	5	,	,	PUNCT
ejpam-6096	258	6	then	then	ADV
ejpam-6096	258	7	it	it	PRON
ejpam-6096	258	8	is	be	AUX
ejpam-6096	258	9	clear	clear	ADJ
ejpam-6096	258	10	that	that	SCONJ
ejpam-6096	258	11	p	p	PROPN
ejpam-6096	258	12	is	be	AUX
ejpam-6096	258	13	k	k	NOUN
ejpam-6096	258	14	-	-	ADJ
ejpam-6096	258	15	prime	prime	NOUN
ejpam-6096	258	16	.	.	PUNCT
ejpam-6096	259	1	assume	assume	VERB
ejpam-6096	259	2	that	that	SCONJ
ejpam-6096	259	3	p	p	NOUN
ejpam-6096	259	4	is	be	AUX
ejpam-6096	259	5	a	a	DET
ejpam-6096	259	6	k	k	ADJ
ejpam-6096	259	7	-	-	ADJ
ejpam-6096	259	8	prime	prime	ADJ
ejpam-6096	259	9	k	k	NOUN
ejpam-6096	259	10	-	-	NOUN
ejpam-6096	259	11	ideal	ideal	NOUN
ejpam-6096	259	12	of	of	ADP
ejpam-6096	259	13	r.	r.	PROPN
ejpam-6096	259	14	let	let	VERB
ejpam-6096	259	15	i	i	PRON
ejpam-6096	259	16	,	,	PUNCT
ejpam-6096	259	17	j	j	PROPN
ejpam-6096	259	18	and	and	CCONJ
ejpam-6096	259	19	k	k	PROPN
ejpam-6096	259	20	be	be	VERB
ejpam-6096	259	21	ideals	ideal	NOUN
ejpam-6096	259	22	of	of	ADP
ejpam-6096	259	23	r	r	NOUN
ejpam-6096	259	24	such	such	ADJ
ejpam-6096	259	25	that	that	SCONJ
ejpam-6096	259	26	ijk	ijk	PROPN
ejpam-6096	259	27	⊆	⊆	NUM
ejpam-6096	259	28	p	p	NOUN
ejpam-6096	259	29	.	.	PUNCT
ejpam-6096	260	1	by	by	ADP
ejpam-6096	260	2	proposition	proposition	NOUN
ejpam-6096	260	3	9	9	NUM
ejpam-6096	260	4	and	and	CCONJ
ejpam-6096	260	5	theorem	theorem	VERB
ejpam-6096	260	6	1	1	NUM
ejpam-6096	260	7	,	,	PUNCT
ejpam-6096	260	8	we	we	PRON
ejpam-6096	260	9	have	have	VERB
ejpam-6096	260	10	ck(i)ck(j)ck(k	ck(i)ck(j)ck(k	NOUN
ejpam-6096	260	11	)	)	PUNCT
ejpam-6096	260	12	⊆	⊆	NUM
ejpam-6096	260	13	ck(ijk	ck(ijk	X
ejpam-6096	260	14	)	)	PUNCT
ejpam-6096	260	15	⊆	⊆	NUM
ejpam-6096	260	16	ck(p	ck(p	NUM
ejpam-6096	260	17	)	)	PUNCT
ejpam-6096	260	18	=	=	VERB
ejpam-6096	261	1	p.	p.	NOUN
ejpam-6096	261	2	this	this	PRON
ejpam-6096	261	3	implies	imply	VERB
ejpam-6096	261	4	that	that	SCONJ
ejpam-6096	261	5	ck(i)ck(j)ck(k	ck(i)ck(j)ck(k	NOUN
ejpam-6096	261	6	)	)	PUNCT
ejpam-6096	261	7	⊆	⊆	NUM
ejpam-6096	261	8	p.	p.	NOUN
ejpam-6096	261	9	since	since	SCONJ
ejpam-6096	261	10	p	p	PROPN
ejpam-6096	261	11	is	be	AUX
ejpam-6096	261	12	k	k	NOUN
ejpam-6096	261	13	-	-	ADJ
ejpam-6096	261	14	prime	prime	NOUN
ejpam-6096	261	15	and	and	CCONJ
ejpam-6096	261	16	ck(i	ck(i	NUM
ejpam-6096	261	17	)	)	PUNCT
ejpam-6096	261	18	,	,	PUNCT
ejpam-6096	261	19	ck(j	ck(j	NOUN
ejpam-6096	261	20	)	)	PUNCT
ejpam-6096	261	21	and	and	CCONJ
ejpam-6096	261	22	ck(k	ck(k	PUNCT
ejpam-6096	261	23	)	)	PUNCT
ejpam-6096	261	24	are	be	AUX
ejpam-6096	261	25	the	the	DET
ejpam-6096	261	26	smallest	small	ADJ
ejpam-6096	261	27	k	k	ADJ
ejpam-6096	261	28	-	-	PUNCT
ejpam-6096	261	29	ideals	ideal	NOUN
ejpam-6096	261	30	containing	contain	VERB
ejpam-6096	261	31	i	i	PRON
ejpam-6096	261	32	,	,	PUNCT
ejpam-6096	261	33	j	j	PROPN
ejpam-6096	261	34	andk	andk	NOUN
ejpam-6096	261	35	,	,	PUNCT
ejpam-6096	261	36	respectively	respectively	ADV
ejpam-6096	261	37	,	,	PUNCT
ejpam-6096	261	38	we	we	PRON
ejpam-6096	261	39	obtain	obtain	VERB
ejpam-6096	261	40	that	that	SCONJ
ejpam-6096	261	41	i	i	PRON
ejpam-6096	261	42	⊆	⊆	NUM
ejpam-6096	261	43	ck(i	ck(i	PUNCT
ejpam-6096	261	44	)	)	PUNCT
ejpam-6096	261	45	⊆	⊆	NUM
ejpam-6096	261	46	p	p	NOUN
ejpam-6096	261	47	,	,	PUNCT
ejpam-6096	261	48	j	j	PROPN
ejpam-6096	261	49	⊆	⊆	NUM
ejpam-6096	261	50	ck(j	ck(j	NUM
ejpam-6096	261	51	)	)	PUNCT
ejpam-6096	261	52	⊆	⊆	NUM
ejpam-6096	261	53	p	p	NOUN
ejpam-6096	261	54	,	,	PUNCT
ejpam-6096	261	55	or	or	CCONJ
ejpam-6096	261	56	k	k	PROPN
ejpam-6096	261	57	⊆	⊆	NUM
ejpam-6096	261	58	ck(k	ck(k	NUM
ejpam-6096	261	59	)	)	PUNCT
ejpam-6096	261	60	⊆	⊆	NUM
ejpam-6096	261	61	p	p	NOUN
ejpam-6096	261	62	.	.	PUNCT
ejpam-6096	262	1	therefore	therefore	ADV
ejpam-6096	262	2	,	,	PUNCT
ejpam-6096	262	3	p	p	NOUN
ejpam-6096	262	4	is	be	AUX
ejpam-6096	262	5	prime	prime	ADJ
ejpam-6096	262	6	.	.	PUNCT
ejpam-6096	263	1	a.	a.	PROPN
ejpam-6096	263	2	j.	j.	PROPN
ejpam-6096	263	3	khan	khan	PROPN
ejpam-6096	263	4	,	,	PUNCT
ejpam-6096	263	5	m.	m.	NOUN
ejpam-6096	263	6	petapirak	petapirak	PROPN
ejpam-6096	263	7	,	,	PUNCT
ejpam-6096	263	8	r.	r.	PROPN
ejpam-6096	263	9	chinram	chinram	PROPN
ejpam-6096	263	10	/	/	SYM
ejpam-6096	263	11	eur	eur	PROPN
ejpam-6096	263	12	.	.	PUNCT
ejpam-6096	264	1	j.	j.	PROPN
ejpam-6096	264	2	pure	pure	PROPN
ejpam-6096	264	3	appl	appl	PROPN
ejpam-6096	264	4	.	.	PROPN
ejpam-6096	264	5	math	math	PROPN
ejpam-6096	264	6	,	,	PUNCT
ejpam-6096	264	7	18	18	NUM
ejpam-6096	264	8	(	(	PUNCT
ejpam-6096	264	9	2	2	NUM
ejpam-6096	264	10	)	)	PUNCT
ejpam-6096	264	11	(	(	PUNCT
ejpam-6096	264	12	2025	2025	NUM
ejpam-6096	264	13	)	)	PUNCT
ejpam-6096	264	14	,	,	PUNCT
ejpam-6096	264	15	6096	6096	NUM
ejpam-6096	264	16	8	8	NUM
ejpam-6096	264	17	of	of	ADP
ejpam-6096	264	18	10	10	NUM
ejpam-6096	264	19	proposition	proposition	NOUN
ejpam-6096	264	20	12	12	NUM
ejpam-6096	264	21	.	.	PUNCT
ejpam-6096	265	1	let	let	VERB
ejpam-6096	265	2	r	r	PRON
ejpam-6096	265	3	be	be	AUX
ejpam-6096	265	4	a	a	DET
ejpam-6096	265	5	commutative	commutative	ADJ
ejpam-6096	265	6	ternary	ternary	ADJ
ejpam-6096	265	7	semiring	semiring	NOUN
ejpam-6096	265	8	and	and	CCONJ
ejpam-6096	265	9	p	p	NOUN
ejpam-6096	265	10	be	be	AUX
ejpam-6096	265	11	a	a	DET
ejpam-6096	265	12	proper	proper	ADJ
ejpam-6096	265	13	ideal	ideal	NOUN
ejpam-6096	265	14	of	of	ADP
ejpam-6096	265	15	r.	r.	PROPN
ejpam-6096	265	16	if	if	SCONJ
ejpam-6096	265	17	p	p	NOUN
ejpam-6096	265	18	is	be	AUX
ejpam-6096	265	19	prime	prime	ADJ
ejpam-6096	265	20	,	,	PUNCT
ejpam-6096	265	21	then	then	ADV
ejpam-6096	265	22	p	p	NOUN
ejpam-6096	265	23	=	=	PUNCT
ejpam-6096	265	24	(	(	PUNCT
ejpam-6096	265	25	p	p	X
ejpam-6096	265	26	:	:	PUNCT
ejpam-6096	265	27	r	r	NOUN
ejpam-6096	265	28	,	,	PUNCT
ejpam-6096	265	29	r	r	NOUN
ejpam-6096	265	30	)	)	PUNCT
ejpam-6096	265	31	.	.	PUNCT
ejpam-6096	266	1	proof	proof	NOUN
ejpam-6096	266	2	.	.	PUNCT
ejpam-6096	267	1	for	for	ADP
ejpam-6096	267	2	any	any	DET
ejpam-6096	267	3	prime	prime	ADJ
ejpam-6096	267	4	ideal	ideal	NOUN
ejpam-6096	267	5	p	p	NOUN
ejpam-6096	267	6	of	of	ADP
ejpam-6096	267	7	r	r	NOUN
ejpam-6096	267	8	,	,	PUNCT
ejpam-6096	267	9	by	by	ADP
ejpam-6096	267	10	proposition	proposition	NOUN
ejpam-6096	267	11	10	10	NUM
ejpam-6096	267	12	,	,	PUNCT
ejpam-6096	267	13	we	we	PRON
ejpam-6096	267	14	have	have	VERB
ejpam-6096	267	15	that	that	PRON
ejpam-6096	267	16	p	p	VERB
ejpam-6096	267	17	⊆	⊆	NUM
ejpam-6096	267	18	(	(	PUNCT
ejpam-6096	267	19	p	p	X
ejpam-6096	267	20	:	:	PUNCT
ejpam-6096	267	21	r	r	NOUN
ejpam-6096	267	22	,	,	PUNCT
ejpam-6096	267	23	r	r	NOUN
ejpam-6096	267	24	)	)	PUNCT
ejpam-6096	267	25	.	.	PUNCT
ejpam-6096	268	1	assume	assume	VERB
ejpam-6096	268	2	that	that	SCONJ
ejpam-6096	268	3	p	p	PROPN
ejpam-6096	268	4	̸=	̸=	PROPN
ejpam-6096	268	5	(	(	PUNCT
ejpam-6096	268	6	p	p	X
ejpam-6096	268	7	:	:	PUNCT
ejpam-6096	268	8	r	r	NOUN
ejpam-6096	268	9	,	,	PUNCT
ejpam-6096	268	10	r	r	NOUN
ejpam-6096	268	11	)	)	PUNCT
ejpam-6096	268	12	.	.	PUNCT
ejpam-6096	269	1	then	then	ADV
ejpam-6096	269	2	there	there	PRON
ejpam-6096	269	3	exists	exist	VERB
ejpam-6096	269	4	x	x	X
ejpam-6096	269	5	∈	∈	PROPN
ejpam-6096	269	6	(	(	PUNCT
ejpam-6096	269	7	p	p	X
ejpam-6096	269	8	:	:	PUNCT
ejpam-6096	269	9	r	r	NOUN
ejpam-6096	269	10	,	,	PUNCT
ejpam-6096	269	11	r	r	NOUN
ejpam-6096	269	12	)	)	PUNCT
ejpam-6096	269	13	but	but	CCONJ
ejpam-6096	269	14	x	x	X
ejpam-6096	269	15	/∈	/∈	PUNCT
ejpam-6096	269	16	p	p	X
ejpam-6096	269	17	.	.	PUNCT
ejpam-6096	270	1	so	so	ADV
ejpam-6096	270	2	xrr	xrr	PROPN
ejpam-6096	271	1	⊆	⊆	NUM
ejpam-6096	271	2	p	p	NOUN
ejpam-6096	271	3	.	.	PUNCT
ejpam-6096	272	1	this	this	PRON
ejpam-6096	272	2	implies	imply	VERB
ejpam-6096	272	3	that	that	SCONJ
ejpam-6096	272	4	⟨x⟩rr	⟨x⟩rr	NOUN
ejpam-6096	272	5	⊆	⊆	NUM
ejpam-6096	272	6	p	p	NOUN
ejpam-6096	272	7	.	.	PUNCT
ejpam-6096	273	1	however	however	ADV
ejpam-6096	273	2	,	,	PUNCT
ejpam-6096	273	3	since	since	SCONJ
ejpam-6096	273	4	x	x	PROPN
ejpam-6096	273	5	/∈	/∈	PROPN
ejpam-6096	273	6	p	p	NOUN
ejpam-6096	273	7	,	,	PUNCT
ejpam-6096	273	8	we	we	PRON
ejpam-6096	273	9	have	have	VERB
ejpam-6096	273	10	that	that	PRON
ejpam-6096	273	11	⟨x⟩	⟨x⟩	PRON
ejpam-6096	273	12	⊈	⊈	PROPN
ejpam-6096	274	1	p	p	X
ejpam-6096	274	2	which	which	PRON
ejpam-6096	274	3	is	be	AUX
ejpam-6096	274	4	a	a	DET
ejpam-6096	274	5	contradiction	contradiction	NOUN
ejpam-6096	274	6	because	because	SCONJ
ejpam-6096	274	7	p	p	NOUN
ejpam-6096	274	8	is	be	AUX
ejpam-6096	274	9	prime	prime	ADJ
ejpam-6096	274	10	.	.	PUNCT
ejpam-6096	275	1	this	this	DET
ejpam-6096	275	2	forces	force	NOUN
ejpam-6096	275	3	p	p	NOUN
ejpam-6096	275	4	=	=	PUNCT
ejpam-6096	275	5	(	(	PUNCT
ejpam-6096	275	6	p	p	X
ejpam-6096	275	7	:	:	PUNCT
ejpam-6096	275	8	r	r	NOUN
ejpam-6096	275	9	,	,	PUNCT
ejpam-6096	275	10	r	r	NOUN
ejpam-6096	275	11	)	)	PUNCT
ejpam-6096	275	12	.	.	PUNCT
ejpam-6096	276	1	proposition	proposition	NOUN
ejpam-6096	276	2	13	13	NUM
ejpam-6096	276	3	.	.	PUNCT
ejpam-6096	277	1	let	let	VERB
ejpam-6096	277	2	r	r	PRON
ejpam-6096	277	3	be	be	AUX
ejpam-6096	277	4	a	a	DET
ejpam-6096	277	5	commutative	commutative	ADJ
ejpam-6096	277	6	ternary	ternary	ADJ
ejpam-6096	277	7	semiring	semiring	NOUN
ejpam-6096	277	8	and	and	CCONJ
ejpam-6096	277	9	p	p	NOUN
ejpam-6096	277	10	be	be	AUX
ejpam-6096	277	11	a	a	DET
ejpam-6096	277	12	proper	proper	ADJ
ejpam-6096	277	13	ideal	ideal	NOUN
ejpam-6096	277	14	of	of	ADP
ejpam-6096	277	15	r.	r.	PROPN
ejpam-6096	277	16	if	if	SCONJ
ejpam-6096	277	17	p	p	PROPN
ejpam-6096	277	18	is	be	AUX
ejpam-6096	277	19	k	k	NOUN
ejpam-6096	277	20	-	-	NOUN
ejpam-6096	277	21	prime	prime	NOUN
ejpam-6096	277	22	,	,	PUNCT
ejpam-6096	277	23	then	then	ADV
ejpam-6096	277	24	p	p	NOUN
ejpam-6096	277	25	=	=	PUNCT
ejpam-6096	277	26	(	(	PUNCT
ejpam-6096	277	27	p	p	X
ejpam-6096	277	28	:	:	PUNCT
ejpam-6096	277	29	r	r	NOUN
ejpam-6096	277	30	,	,	PUNCT
ejpam-6096	277	31	r	r	NOUN
ejpam-6096	277	32	)	)	PUNCT
ejpam-6096	277	33	.	.	PUNCT
ejpam-6096	278	1	proof	proof	NOUN
ejpam-6096	278	2	.	.	PUNCT
ejpam-6096	279	1	it	it	PRON
ejpam-6096	279	2	is	be	AUX
ejpam-6096	279	3	similar	similar	ADJ
ejpam-6096	279	4	to	to	ADP
ejpam-6096	279	5	proposition	proposition	NOUN
ejpam-6096	279	6	12	12	NUM
ejpam-6096	279	7	.	.	PUNCT
ejpam-6096	280	1	a	a	DET
ejpam-6096	280	2	subset	subset	NOUN
ejpam-6096	280	3	a	a	PRON
ejpam-6096	280	4	of	of	ADP
ejpam-6096	280	5	r	r	NOUN
ejpam-6096	280	6	is	be	AUX
ejpam-6096	280	7	called	call	VERB
ejpam-6096	280	8	a	a	DET
ejpam-6096	280	9	multiplicatively	multiplicatively	ADV
ejpam-6096	280	10	closed	close	VERB
ejpam-6096	280	11	set	set	VERB
ejpam-6096	280	12	if	if	SCONJ
ejpam-6096	280	13	abc	abc	PROPN
ejpam-6096	280	14	∈	∈	VERB
ejpam-6096	280	15	a	a	PRON
ejpam-6096	280	16	for	for	ADP
ejpam-6096	280	17	all	all	DET
ejpam-6096	280	18	a	a	DET
ejpam-6096	280	19	,	,	PUNCT
ejpam-6096	280	20	b	b	NOUN
ejpam-6096	280	21	,	,	PUNCT
ejpam-6096	280	22	c	c	PROPN
ejpam-6096	280	23	∈	∈	PROPN
ejpam-6096	280	24	a.	a.	NOUN
ejpam-6096	280	25	theorem	theorem	NOUN
ejpam-6096	280	26	4	4	X
ejpam-6096	280	27	.	.	PUNCT
ejpam-6096	281	1	let	let	VERB
ejpam-6096	281	2	p	p	PRON
ejpam-6096	281	3	be	be	AUX
ejpam-6096	281	4	a	a	DET
ejpam-6096	281	5	proper	proper	ADJ
ejpam-6096	281	6	k	k	NOUN
ejpam-6096	281	7	-	-	NOUN
ejpam-6096	281	8	ideal	ideal	NOUN
ejpam-6096	281	9	of	of	ADP
ejpam-6096	281	10	a	a	DET
ejpam-6096	281	11	ternary	ternary	ADJ
ejpam-6096	281	12	semiring	semire	VERB
ejpam-6096	281	13	r.	r.	PROPN
ejpam-6096	281	14	then	then	ADV
ejpam-6096	281	15	p	p	PROPN
ejpam-6096	281	16	is	be	AUX
ejpam-6096	281	17	k	k	NOUN
ejpam-6096	281	18	-	-	ADJ
ejpam-6096	281	19	prime	prime	ADJ
ejpam-6096	281	20	if	if	SCONJ
ejpam-6096	282	1	and	and	CCONJ
ejpam-6096	282	2	only	only	ADV
ejpam-6096	282	3	if	if	SCONJ
ejpam-6096	282	4	r∖	r∖	NUM
ejpam-6096	282	5	p	p	NOUN
ejpam-6096	282	6	is	be	AUX
ejpam-6096	282	7	a	a	DET
ejpam-6096	282	8	multiplicatively	multiplicatively	ADV
ejpam-6096	282	9	closed	close	VERB
ejpam-6096	282	10	set	set	NOUN
ejpam-6096	282	11	.	.	PUNCT
ejpam-6096	283	1	proof	proof	NOUN
ejpam-6096	283	2	.	.	PUNCT
ejpam-6096	284	1	assume	assume	VERB
ejpam-6096	284	2	that	that	SCONJ
ejpam-6096	284	3	p	p	PROPN
ejpam-6096	284	4	is	be	AUX
ejpam-6096	284	5	k	k	NOUN
ejpam-6096	284	6	-	-	NOUN
ejpam-6096	284	7	prime	prime	NOUN
ejpam-6096	284	8	.	.	PUNCT
ejpam-6096	285	1	first	first	ADV
ejpam-6096	285	2	of	of	ADP
ejpam-6096	285	3	all	all	PRON
ejpam-6096	285	4	,	,	PUNCT
ejpam-6096	285	5	we	we	PRON
ejpam-6096	285	6	will	will	AUX
ejpam-6096	285	7	show	show	VERB
ejpam-6096	285	8	thatr∖p	thatr∖p	PROPN
ejpam-6096	285	9	is	be	AUX
ejpam-6096	285	10	multiplicatively	multiplicatively	ADV
ejpam-6096	285	11	closed	closed	ADJ
ejpam-6096	285	12	,	,	PUNCT
ejpam-6096	285	13	we	we	PRON
ejpam-6096	285	14	thus	thus	ADV
ejpam-6096	285	15	let	let	VERB
ejpam-6096	285	16	a	a	DET
ejpam-6096	285	17	,	,	PUNCT
ejpam-6096	285	18	b	b	NOUN
ejpam-6096	285	19	,	,	PUNCT
ejpam-6096	285	20	c	c	PROPN
ejpam-6096	285	21	∈	∈	PROPN
ejpam-6096	285	22	r∖p	r∖p	PROPN
ejpam-6096	285	23	.	.	PUNCT
ejpam-6096	286	1	to	to	PART
ejpam-6096	286	2	show	show	VERB
ejpam-6096	286	3	that	that	SCONJ
ejpam-6096	286	4	abc	abc	PROPN
ejpam-6096	286	5	∈	∈	PROPN
ejpam-6096	286	6	r∖p	r∖p	PROPN
ejpam-6096	286	7	,	,	PUNCT
ejpam-6096	286	8	we	we	PRON
ejpam-6096	286	9	suppose	suppose	VERB
ejpam-6096	286	10	for	for	ADP
ejpam-6096	286	11	contradiction	contradiction	NOUN
ejpam-6096	286	12	,	,	PUNCT
ejpam-6096	286	13	that	that	SCONJ
ejpam-6096	286	14	abc	abc	PROPN
ejpam-6096	286	15	∈	∈	PROPN
ejpam-6096	286	16	p	p	PROPN
ejpam-6096	286	17	.	.	PUNCT
ejpam-6096	287	1	since	since	SCONJ
ejpam-6096	287	2	p	p	PROPN
ejpam-6096	287	3	is	be	AUX
ejpam-6096	287	4	k	k	NOUN
ejpam-6096	287	5	-	-	NOUN
ejpam-6096	287	6	prime	prime	NOUN
ejpam-6096	287	7	,	,	PUNCT
ejpam-6096	287	8	this	this	PRON
ejpam-6096	287	9	implies	imply	VERB
ejpam-6096	287	10	a	a	DET
ejpam-6096	287	11	∈	∈	PROPN
ejpam-6096	287	12	p	p	NOUN
ejpam-6096	287	13	,	,	PUNCT
ejpam-6096	287	14	b	b	PROPN
ejpam-6096	287	15	∈	∈	PROPN
ejpam-6096	287	16	p	p	X
ejpam-6096	287	17	,	,	PUNCT
ejpam-6096	287	18	or	or	CCONJ
ejpam-6096	287	19	c	c	NOUN
ejpam-6096	287	20	∈	∈	PROPN
ejpam-6096	287	21	p	p	NOUN
ejpam-6096	287	22	.	.	PUNCT
ejpam-6096	288	1	this	this	PRON
ejpam-6096	288	2	contradicts	contradict	VERB
ejpam-6096	288	3	the	the	DET
ejpam-6096	288	4	fact	fact	NOUN
ejpam-6096	288	5	that	that	SCONJ
ejpam-6096	288	6	a	a	DET
ejpam-6096	288	7	,	,	PUNCT
ejpam-6096	288	8	b	b	NOUN
ejpam-6096	288	9	,	,	PUNCT
ejpam-6096	288	10	c	c	PROPN
ejpam-6096	288	11	∈	∈	PROPN
ejpam-6096	288	12	r	r	NOUN
ejpam-6096	288	13	∖	∖	X
ejpam-6096	288	14	p	p	NOUN
ejpam-6096	288	15	.	.	PUNCT
ejpam-6096	289	1	thus	thus	ADV
ejpam-6096	289	2	abc	abc	PROPN
ejpam-6096	289	3	∈	∈	PROPN
ejpam-6096	289	4	r	r	NOUN
ejpam-6096	289	5	∖	∖	X
ejpam-6096	290	1	p	p	NOUN
ejpam-6096	290	2	and	and	CCONJ
ejpam-6096	290	3	r	r	NOUN
ejpam-6096	290	4	∖	∖	NOUN
ejpam-6096	291	1	p	p	NOUN
ejpam-6096	291	2	is	be	AUX
ejpam-6096	291	3	multiplicatively	multiplicatively	ADV
ejpam-6096	291	4	closed	close	VERB
ejpam-6096	291	5	.	.	PUNCT
ejpam-6096	292	1	conversely	conversely	ADV
ejpam-6096	292	2	,	,	PUNCT
ejpam-6096	292	3	assume	assume	VERB
ejpam-6096	292	4	that	that	SCONJ
ejpam-6096	292	5	r	r	NOUN
ejpam-6096	292	6	∖	∖	NOUN
ejpam-6096	292	7	p	p	NOUN
ejpam-6096	292	8	is	be	AUX
ejpam-6096	292	9	multiplicatively	multiplicatively	ADV
ejpam-6096	292	10	closed	closed	ADJ
ejpam-6096	292	11	.	.	PUNCT
ejpam-6096	293	1	we	we	PRON
ejpam-6096	293	2	need	need	VERB
ejpam-6096	293	3	to	to	PART
ejpam-6096	293	4	show	show	VERB
ejpam-6096	293	5	that	that	SCONJ
ejpam-6096	293	6	p	p	PROPN
ejpam-6096	293	7	is	be	AUX
ejpam-6096	293	8	k	k	NOUN
ejpam-6096	293	9	-	-	NOUN
ejpam-6096	293	10	prime	prime	NOUN
ejpam-6096	293	11	.	.	PUNCT
ejpam-6096	294	1	let	let	VERB
ejpam-6096	294	2	i	i	PRON
ejpam-6096	294	3	,	,	PUNCT
ejpam-6096	294	4	j	j	PROPN
ejpam-6096	294	5	,	,	PUNCT
ejpam-6096	294	6	k	k	PROPN
ejpam-6096	294	7	be	be	VERB
ejpam-6096	294	8	k	k	NOUN
ejpam-6096	294	9	-	-	NOUN
ejpam-6096	294	10	ideals	ideal	NOUN
ejpam-6096	294	11	of	of	ADP
ejpam-6096	294	12	r	r	NOUN
ejpam-6096	294	13	such	such	ADJ
ejpam-6096	294	14	that	that	SCONJ
ejpam-6096	294	15	ijk	ijk	PROPN
ejpam-6096	294	16	⊆	⊆	NUM
ejpam-6096	294	17	p	p	NOUN
ejpam-6096	294	18	.	.	PUNCT
ejpam-6096	295	1	we	we	PRON
ejpam-6096	295	2	need	need	VERB
ejpam-6096	295	3	to	to	PART
ejpam-6096	295	4	show	show	VERB
ejpam-6096	295	5	that	that	SCONJ
ejpam-6096	295	6	i	i	PRON
ejpam-6096	295	7	⊆	⊆	NUM
ejpam-6096	295	8	p	p	NOUN
ejpam-6096	295	9	,	,	PUNCT
ejpam-6096	295	10	j	j	PROPN
ejpam-6096	295	11	⊆	⊆	NUM
ejpam-6096	295	12	p	p	NOUN
ejpam-6096	295	13	,	,	PUNCT
ejpam-6096	295	14	or	or	CCONJ
ejpam-6096	295	15	k	k	PROPN
ejpam-6096	296	1	⊆	⊆	NUM
ejpam-6096	296	2	p	p	NOUN
ejpam-6096	296	3	.	.	PUNCT
ejpam-6096	296	4	suppose	suppose	VERB
ejpam-6096	296	5	to	to	ADP
ejpam-6096	296	6	the	the	DET
ejpam-6096	296	7	contrary	contrary	NOUN
ejpam-6096	296	8	that	that	SCONJ
ejpam-6096	296	9	there	there	PRON
ejpam-6096	296	10	exist	exist	VERB
ejpam-6096	296	11	a	a	DET
ejpam-6096	296	12	∈	∈	PROPN
ejpam-6096	296	13	i	i	NOUN
ejpam-6096	296	14	,	,	PUNCT
ejpam-6096	296	15	b	b	PROPN
ejpam-6096	296	16	∈	∈	PROPN
ejpam-6096	296	17	j	j	PROPN
ejpam-6096	296	18	and	and	CCONJ
ejpam-6096	296	19	c	c	NOUN
ejpam-6096	296	20	∈	∈	PROPN
ejpam-6096	297	1	k	k	PRON
ejpam-6096	297	2	such	such	ADJ
ejpam-6096	297	3	that	that	SCONJ
ejpam-6096	297	4	a	a	DET
ejpam-6096	297	5	,	,	PUNCT
ejpam-6096	297	6	b	b	NOUN
ejpam-6096	297	7	and	and	CCONJ
ejpam-6096	297	8	c	c	PROPN
ejpam-6096	297	9	are	be	AUX
ejpam-6096	297	10	not	not	PART
ejpam-6096	297	11	members	member	NOUN
ejpam-6096	297	12	in	in	ADP
ejpam-6096	297	13	p	p	PROPN
ejpam-6096	297	14	.	.	PUNCT
ejpam-6096	298	1	then	then	ADV
ejpam-6096	298	2	we	we	PRON
ejpam-6096	298	3	obtain	obtain	VERB
ejpam-6096	298	4	that	that	SCONJ
ejpam-6096	298	5	abc	abc	PROPN
ejpam-6096	298	6	∈	∈	PROPN
ejpam-6096	298	7	ijk	ijk	PROPN
ejpam-6096	298	8	and	and	CCONJ
ejpam-6096	298	9	a	a	DET
ejpam-6096	298	10	,	,	PUNCT
ejpam-6096	298	11	b	b	NOUN
ejpam-6096	298	12	,	,	PUNCT
ejpam-6096	298	13	c	c	PROPN
ejpam-6096	298	14	∈	∈	PROPN
ejpam-6096	298	15	r∖p	r∖p	PROPN
ejpam-6096	298	16	.	.	PUNCT
ejpam-6096	299	1	since	since	SCONJ
ejpam-6096	299	2	r	r	NOUN
ejpam-6096	299	3	∖	∖	NOUN
ejpam-6096	300	1	p	p	NOUN
ejpam-6096	300	2	is	be	AUX
ejpam-6096	300	3	multiplicatively	multiplicatively	ADV
ejpam-6096	300	4	closed	closed	ADJ
ejpam-6096	300	5	,	,	PUNCT
ejpam-6096	300	6	we	we	PRON
ejpam-6096	300	7	then	then	ADV
ejpam-6096	300	8	obtain	obtain	VERB
ejpam-6096	300	9	abc	abc	PROPN
ejpam-6096	300	10	∈	∈	PROPN
ejpam-6096	300	11	r	r	NOUN
ejpam-6096	300	12	∖	∖	X
ejpam-6096	300	13	p	p	NOUN
ejpam-6096	300	14	.	.	PUNCT
ejpam-6096	301	1	this	this	PRON
ejpam-6096	301	2	contradicts	contradict	VERB
ejpam-6096	301	3	abc	abc	PROPN
ejpam-6096	301	4	∈	∈	PROPN
ejpam-6096	301	5	ijk	ijk	PROPN
ejpam-6096	301	6	⊆	⊆	NUM
ejpam-6096	301	7	p	p	NOUN
ejpam-6096	301	8	.	.	PUNCT
ejpam-6096	302	1	hence	hence	ADV
ejpam-6096	302	2	a	a	DET
ejpam-6096	302	3	∈	∈	PROPN
ejpam-6096	302	4	p	p	NOUN
ejpam-6096	302	5	,	,	PUNCT
ejpam-6096	302	6	b	b	PROPN
ejpam-6096	302	7	∈	∈	PROPN
ejpam-6096	302	8	p	p	X
ejpam-6096	302	9	,	,	PUNCT
ejpam-6096	302	10	or	or	CCONJ
ejpam-6096	302	11	c	c	NOUN
ejpam-6096	302	12	∈	∈	PROPN
ejpam-6096	302	13	p	p	NOUN
ejpam-6096	302	14	.	.	PUNCT
ejpam-6096	303	1	so	so	ADV
ejpam-6096	303	2	p	p	PROPN
ejpam-6096	303	3	is	be	AUX
ejpam-6096	303	4	k	k	NOUN
ejpam-6096	303	5	-	-	NOUN
ejpam-6096	303	6	prime	prime	NOUN
ejpam-6096	303	7	.	.	PUNCT
ejpam-6096	304	1	a	a	DET
ejpam-6096	304	2	proper	proper	ADJ
ejpam-6096	304	3	ideal	ideal	NOUN
ejpam-6096	304	4	q	q	NOUN
ejpam-6096	304	5	of	of	ADP
ejpam-6096	304	6	r	r	NOUN
ejpam-6096	304	7	is	be	AUX
ejpam-6096	304	8	called	call	VERB
ejpam-6096	304	9	k	k	NOUN
ejpam-6096	304	10	-	-	NOUN
ejpam-6096	304	11	semiprime	semiprime	NOUN
ejpam-6096	304	12	if	if	SCONJ
ejpam-6096	304	13	i3	i3	NOUN
ejpam-6096	304	14	⊆	⊆	X
ejpam-6096	304	15	q	q	NOUN
ejpam-6096	304	16	implies	imply	VERB
ejpam-6096	304	17	i	i	PRON
ejpam-6096	304	18	⊆	⊆	NUM
ejpam-6096	304	19	q	q	NOUN
ejpam-6096	304	20	for	for	ADP
ejpam-6096	304	21	every	every	DET
ejpam-6096	304	22	k	k	NOUN
ejpam-6096	304	23	-	-	NOUN
ejpam-6096	304	24	ideal	ideal	ADJ
ejpam-6096	304	25	i	i	PRON
ejpam-6096	304	26	of	of	ADP
ejpam-6096	304	27	r.	r.	PROPN
ejpam-6096	304	28	we	we	PRON
ejpam-6096	304	29	have	have	VERB
ejpam-6096	304	30	some	some	DET
ejpam-6096	304	31	remarks	remark	NOUN
ejpam-6096	304	32	as	as	ADP
ejpam-6096	304	33	the	the	DET
ejpam-6096	304	34	following	following	NOUN
ejpam-6096	304	35	.	.	PUNCT
ejpam-6096	305	1	(	(	PUNCT
ejpam-6096	305	2	1	1	X
ejpam-6096	305	3	)	)	PUNCT
ejpam-6096	305	4	every	every	DET
ejpam-6096	305	5	prime	prime	ADJ
ejpam-6096	305	6	ideal	ideal	NOUN
ejpam-6096	305	7	is	be	AUX
ejpam-6096	305	8	semiprime	semiprime	NOUN
ejpam-6096	305	9	.	.	PUNCT
ejpam-6096	306	1	(	(	PUNCT
ejpam-6096	306	2	2	2	X
ejpam-6096	306	3	)	)	PUNCT
ejpam-6096	306	4	every	every	DET
ejpam-6096	306	5	k	k	ADJ
ejpam-6096	306	6	-	-	ADJ
ejpam-6096	306	7	prime	prime	ADJ
ejpam-6096	306	8	ideal	ideal	NOUN
ejpam-6096	306	9	is	be	AUX
ejpam-6096	306	10	k	k	NOUN
ejpam-6096	306	11	-	-	NOUN
ejpam-6096	306	12	semiprime	semiprime	NOUN
ejpam-6096	306	13	.	.	PUNCT
ejpam-6096	307	1	(	(	PUNCT
ejpam-6096	307	2	3	3	X
ejpam-6096	307	3	)	)	PUNCT
ejpam-6096	307	4	every	every	DET
ejpam-6096	307	5	semiprime	semiprime	NOUN
ejpam-6096	307	6	ideal	ideal	NOUN
ejpam-6096	307	7	is	be	AUX
ejpam-6096	307	8	k	k	NOUN
ejpam-6096	307	9	-	-	NOUN
ejpam-6096	307	10	semiprime	semiprime	NOUN
ejpam-6096	307	11	.	.	PUNCT
ejpam-6096	308	1	theorem	theorem	NOUN
ejpam-6096	308	2	5	5	NUM
ejpam-6096	308	3	.	.	PUNCT
ejpam-6096	309	1	let	let	VERB
ejpam-6096	309	2	r	r	PRON
ejpam-6096	309	3	be	be	AUX
ejpam-6096	309	4	a	a	DET
ejpam-6096	309	5	ternary	ternary	ADJ
ejpam-6096	309	6	semiring	semiring	NOUN
ejpam-6096	309	7	and	and	CCONJ
ejpam-6096	309	8	q	q	AUX
ejpam-6096	309	9	be	be	AUX
ejpam-6096	309	10	a	a	DET
ejpam-6096	309	11	k	k	NOUN
ejpam-6096	309	12	-	-	NOUN
ejpam-6096	309	13	ideal	ideal	NOUN
ejpam-6096	309	14	of	of	ADP
ejpam-6096	309	15	r.	r.	PROPN
ejpam-6096	309	16	then	then	ADV
ejpam-6096	309	17	q	q	PROPN
ejpam-6096	309	18	is	be	AUX
ejpam-6096	309	19	k	k	NOUN
ejpam-6096	309	20	-	-	ADJ
ejpam-6096	309	21	semiprime	semiprime	NOUN
ejpam-6096	310	1	if	if	SCONJ
ejpam-6096	310	2	and	and	CCONJ
ejpam-6096	310	3	only	only	ADV
ejpam-6096	310	4	if	if	SCONJ
ejpam-6096	310	5	q	q	NOUN
ejpam-6096	310	6	is	be	AUX
ejpam-6096	310	7	semiprime	semiprime	NOUN
ejpam-6096	310	8	.	.	PUNCT
ejpam-6096	311	1	proof	proof	NOUN
ejpam-6096	311	2	.	.	PUNCT
ejpam-6096	312	1	if	if	SCONJ
ejpam-6096	312	2	q	q	NOUN
ejpam-6096	312	3	is	be	AUX
ejpam-6096	312	4	semiprime	semiprime	NOUN
ejpam-6096	312	5	,	,	PUNCT
ejpam-6096	312	6	then	then	ADV
ejpam-6096	312	7	q	q	X
ejpam-6096	312	8	is	be	AUX
ejpam-6096	312	9	clearly	clearly	ADV
ejpam-6096	312	10	k	k	NOUN
ejpam-6096	312	11	-	-	NOUN
ejpam-6096	312	12	semiprime	semiprime	NOUN
ejpam-6096	312	13	.	.	PUNCT
ejpam-6096	313	1	let	let	VERB
ejpam-6096	313	2	q	q	PRON
ejpam-6096	313	3	be	be	AUX
ejpam-6096	313	4	a	a	DET
ejpam-6096	313	5	k	k	ADJ
ejpam-6096	313	6	-	-	ADJ
ejpam-6096	313	7	semiprime	semiprime	ADJ
ejpam-6096	313	8	k	k	NOUN
ejpam-6096	313	9	-	-	NOUN
ejpam-6096	313	10	ideal	ideal	NOUN
ejpam-6096	313	11	of	of	ADP
ejpam-6096	313	12	r	r	NOUN
ejpam-6096	313	13	and	and	CCONJ
ejpam-6096	313	14	assume	assume	VERB
ejpam-6096	313	15	that	that	SCONJ
ejpam-6096	313	16	i3	i3	VERB
ejpam-6096	313	17	⊆	⊆	X
ejpam-6096	313	18	q	q	NOUN
ejpam-6096	313	19	for	for	ADP
ejpam-6096	313	20	an	an	DET
ejpam-6096	313	21	ideal	ideal	NOUN
ejpam-6096	313	22	i	i	PRON
ejpam-6096	313	23	of	of	ADP
ejpam-6096	313	24	r.	r.	PROPN
ejpam-6096	313	25	then	then	ADV
ejpam-6096	313	26	ck(i)ck(i)ck(i	ck(i)ck(i)ck(i	PROPN
ejpam-6096	313	27	)	)	PUNCT
ejpam-6096	313	28	⊆	⊆	NUM
ejpam-6096	313	29	ck(i3	ck(i3	NOUN
ejpam-6096	313	30	)	)	PUNCT
ejpam-6096	313	31	⊆	⊆	NUM
ejpam-6096	313	32	ck(q	ck(q	NUM
ejpam-6096	313	33	)	)	PUNCT
ejpam-6096	313	34	=	=	SYM
ejpam-6096	314	1	q.	q.	PROPN
ejpam-6096	314	2	then	then	ADV
ejpam-6096	314	3	ck(i)ck(i)ck(i	ck(i)ck(i)ck(i	PROPN
ejpam-6096	314	4	)	)	PUNCT
ejpam-6096	314	5	⊆	⊆	NUM
ejpam-6096	314	6	q.	q.	NOUN
ejpam-6096	314	7	since	since	SCONJ
ejpam-6096	314	8	q	q	PROPN
ejpam-6096	314	9	is	be	AUX
ejpam-6096	314	10	k	k	NOUN
ejpam-6096	314	11	-	-	ADJ
ejpam-6096	314	12	semiprime	semiprime	NOUN
ejpam-6096	314	13	and	and	CCONJ
ejpam-6096	314	14	ck(i	ck(i	PUNCT
ejpam-6096	314	15	)	)	PUNCT
ejpam-6096	314	16	is	be	AUX
ejpam-6096	314	17	the	the	DET
ejpam-6096	314	18	smallest	small	ADJ
ejpam-6096	314	19	k	k	ADJ
ejpam-6096	314	20	-	-	ADJ
ejpam-6096	314	21	ideal	ideal	NOUN
ejpam-6096	314	22	containing	contain	VERB
ejpam-6096	314	23	i	i	PRON
ejpam-6096	314	24	,	,	PUNCT
ejpam-6096	314	25	we	we	PRON
ejpam-6096	314	26	get	get	VERB
ejpam-6096	314	27	i	i	PRON
ejpam-6096	314	28	⊆	⊆	NUM
ejpam-6096	314	29	q.	q.	PROPN
ejpam-6096	314	30	a.	a.	PROPN
ejpam-6096	314	31	j.	j.	PROPN
ejpam-6096	314	32	khan	khan	PROPN
ejpam-6096	314	33	,	,	PUNCT
ejpam-6096	314	34	m.	m.	NOUN
ejpam-6096	314	35	petapirak	petapirak	PROPN
ejpam-6096	314	36	,	,	PUNCT
ejpam-6096	314	37	r.	r.	PROPN
ejpam-6096	314	38	chinram	chinram	PROPN
ejpam-6096	314	39	/	/	SYM
ejpam-6096	314	40	eur	eur	PROPN
ejpam-6096	314	41	.	.	PUNCT
ejpam-6096	315	1	j.	j.	PROPN
ejpam-6096	315	2	pure	pure	PROPN
ejpam-6096	315	3	appl	appl	PROPN
ejpam-6096	315	4	.	.	PROPN
ejpam-6096	315	5	math	math	PROPN
ejpam-6096	315	6	,	,	PUNCT
ejpam-6096	315	7	18	18	NUM
ejpam-6096	315	8	(	(	PUNCT
ejpam-6096	315	9	2	2	NUM
ejpam-6096	315	10	)	)	PUNCT
ejpam-6096	315	11	(	(	PUNCT
ejpam-6096	315	12	2025	2025	NUM
ejpam-6096	315	13	)	)	PUNCT
ejpam-6096	315	14	,	,	PUNCT
ejpam-6096	315	15	6096	6096	NUM
ejpam-6096	315	16	9	9	NUM
ejpam-6096	315	17	of	of	ADP
ejpam-6096	315	18	10	10	NUM
ejpam-6096	315	19	4	4	NUM
ejpam-6096	315	20	.	.	PUNCT
ejpam-6096	315	21	conclusion	conclusion	NOUN
ejpam-6096	315	22	in	in	ADP
ejpam-6096	315	23	this	this	DET
ejpam-6096	315	24	paper	paper	NOUN
ejpam-6096	315	25	,	,	PUNCT
ejpam-6096	315	26	we	we	PRON
ejpam-6096	315	27	focus	focus	VERB
ejpam-6096	315	28	on	on	ADP
ejpam-6096	315	29	various	various	ADJ
ejpam-6096	315	30	properties	property	NOUN
ejpam-6096	315	31	of	of	ADP
ejpam-6096	315	32	k	k	NOUN
ejpam-6096	315	33	-	-	NOUN
ejpam-6096	315	34	ideals	ideal	NOUN
ejpam-6096	315	35	of	of	ADP
ejpam-6096	315	36	a	a	DET
ejpam-6096	315	37	ternary	ternary	ADJ
ejpam-6096	315	38	semiring	semire	VERB
ejpam-6096	315	39	r.	r.	NOUN
ejpam-6096	315	40	we	we	PRON
ejpam-6096	315	41	show	show	VERB
ejpam-6096	315	42	that	that	SCONJ
ejpam-6096	315	43	annr(a	annr(a	NOUN
ejpam-6096	315	44	)	)	PUNCT
ejpam-6096	315	45	is	be	AUX
ejpam-6096	315	46	a	a	DET
ejpam-6096	315	47	k	k	NOUN
ejpam-6096	315	48	-	-	NOUN
ejpam-6096	315	49	ideal	ideal	NOUN
ejpam-6096	315	50	of	of	ADP
ejpam-6096	315	51	r	r	NOUN
ejpam-6096	315	52	for	for	ADP
ejpam-6096	315	53	every	every	DET
ejpam-6096	315	54	nonempty	nonempty	NOUN
ejpam-6096	315	55	subset	subset	VERB
ejpam-6096	315	56	a	a	PRON
ejpam-6096	315	57	of	of	ADP
ejpam-6096	315	58	r.	r.	NOUN
ejpam-6096	315	59	for	for	ADP
ejpam-6096	315	60	an	an	DET
ejpam-6096	315	61	ideal	ideal	ADJ
ejpam-6096	315	62	i	i	PRON
ejpam-6096	315	63	of	of	ADP
ejpam-6096	315	64	r	r	NOUN
ejpam-6096	315	65	,	,	PUNCT
ejpam-6096	315	66	we	we	PRON
ejpam-6096	315	67	prove	prove	VERB
ejpam-6096	315	68	that	that	SCONJ
ejpam-6096	315	69	ck(i	ck(i	PUNCT
ejpam-6096	315	70	)	)	PUNCT
ejpam-6096	315	71	is	be	AUX
ejpam-6096	315	72	the	the	DET
ejpam-6096	315	73	smallest	small	ADJ
ejpam-6096	315	74	k	k	ADJ
ejpam-6096	315	75	-	-	ADJ
ejpam-6096	315	76	ideal	ideal	ADJ
ejpam-6096	315	77	containing	contain	VERB
ejpam-6096	315	78	i.	i.	NOUN
ejpam-6096	315	79	in	in	ADP
ejpam-6096	315	80	particular	particular	ADJ
ejpam-6096	315	81	,	,	PUNCT
ejpam-6096	315	82	for	for	ADP
ejpam-6096	315	83	a	a	DET
ejpam-6096	315	84	k	k	NOUN
ejpam-6096	315	85	-	-	NOUN
ejpam-6096	315	86	ideal	ideal	ADJ
ejpam-6096	315	87	i	i	PRON
ejpam-6096	315	88	and	and	CCONJ
ejpam-6096	315	89	ideals	ideal	VERB
ejpam-6096	315	90	j	j	PROPN
ejpam-6096	315	91	,	,	PUNCT
ejpam-6096	315	92	k	k	PROPN
ejpam-6096	315	93	of	of	ADP
ejpam-6096	315	94	a	a	DET
ejpam-6096	315	95	commutative	commutative	ADJ
ejpam-6096	315	96	ternary	ternary	ADJ
ejpam-6096	315	97	semiring	semiring	NOUN
ejpam-6096	315	98	r	r	NOUN
ejpam-6096	315	99	,	,	PUNCT
ejpam-6096	315	100	we	we	PRON
ejpam-6096	315	101	demonstrate	demonstrate	VERB
ejpam-6096	315	102	that	that	SCONJ
ejpam-6096	315	103	(	(	PUNCT
ejpam-6096	315	104	i	i	PRON
ejpam-6096	315	105	:	:	PUNCT
ejpam-6096	315	106	j	j	PROPN
ejpam-6096	315	107	,	,	PUNCT
ejpam-6096	315	108	k	k	PROPN
ejpam-6096	315	109	)	)	PUNCT
ejpam-6096	315	110	is	be	AUX
ejpam-6096	315	111	a	a	DET
ejpam-6096	315	112	k	k	NOUN
ejpam-6096	315	113	-	-	NOUN
ejpam-6096	315	114	ideal	ideal	NOUN
ejpam-6096	315	115	of	of	ADP
ejpam-6096	315	116	r.	r.	PROPN
ejpam-6096	315	117	finally	finally	ADV
ejpam-6096	315	118	,	,	PUNCT
ejpam-6096	315	119	we	we	PRON
ejpam-6096	315	120	explore	explore	VERB
ejpam-6096	315	121	the	the	DET
ejpam-6096	315	122	relationship	relationship	NOUN
ejpam-6096	315	123	between	between	ADP
ejpam-6096	315	124	k	k	NOUN
ejpam-6096	315	125	-	-	NOUN
ejpam-6096	315	126	ideals	ideal	NOUN
ejpam-6096	315	127	and	and	CCONJ
ejpam-6096	315	128	k	k	NOUN
ejpam-6096	315	129	-	-	ADJ
ejpam-6096	315	130	maximal	maximal	ADJ
ejpam-6096	315	131	,	,	PUNCT
ejpam-6096	315	132	k	k	NOUN
ejpam-6096	315	133	-	-	ADJ
ejpam-6096	315	134	prime	prime	ADJ
ejpam-6096	315	135	and	and	CCONJ
ejpam-6096	315	136	k	k	ADJ
ejpam-6096	315	137	-	-	PUNCT
ejpam-6096	315	138	semiprime	semiprime	ADJ
ejpam-6096	315	139	ideals	ideal	NOUN
ejpam-6096	315	140	.	.	PUNCT
ejpam-6096	316	1	in	in	ADP
ejpam-6096	316	2	future	future	ADJ
ejpam-6096	316	3	work	work	NOUN
ejpam-6096	316	4	,	,	PUNCT
ejpam-6096	316	5	we	we	PRON
ejpam-6096	316	6	can	can	AUX
ejpam-6096	316	7	study	study	VERB
ejpam-6096	316	8	other	other	ADJ
ejpam-6096	316	9	types	type	NOUN
ejpam-6096	316	10	of	of	ADP
ejpam-6096	316	11	ideals	ideal	NOUN
ejpam-6096	316	12	and	and	CCONJ
ejpam-6096	316	13	their	their	PRON
ejpam-6096	316	14	properties	property	NOUN
ejpam-6096	316	15	in	in	ADP
ejpam-6096	316	16	ternary	ternary	ADJ
ejpam-6096	316	17	semirings	semiring	NOUN
ejpam-6096	316	18	.	.	PUNCT
ejpam-6096	317	1	acknowledgements	acknowledgement	NOUN
ejpam-6096	317	2	we	we	PRON
ejpam-6096	317	3	sincerely	sincerely	ADV
ejpam-6096	317	4	appreciate	appreciate	VERB
ejpam-6096	317	5	all	all	DET
ejpam-6096	317	6	valuable	valuable	ADJ
ejpam-6096	317	7	comments	comment	NOUN
ejpam-6096	317	8	and	and	CCONJ
ejpam-6096	317	9	suggestions	suggestion	NOUN
ejpam-6096	317	10	of	of	ADP
ejpam-6096	317	11	reviewers	reviewer	NOUN
ejpam-6096	317	12	,	,	PUNCT
ejpam-6096	317	13	which	which	PRON
ejpam-6096	317	14	helped	help	VERB
ejpam-6096	317	15	us	we	PRON
ejpam-6096	317	16	to	to	PART
ejpam-6096	317	17	improve	improve	VERB
ejpam-6096	317	18	the	the	DET
ejpam-6096	317	19	quality	quality	NOUN
ejpam-6096	317	20	of	of	ADP
ejpam-6096	317	21	the	the	DET
ejpam-6096	317	22	manuscript	manuscript	NOUN
ejpam-6096	317	23	.	.	PUNCT
ejpam-6096	318	1	this	this	DET
ejpam-6096	318	2	work	work	NOUN
ejpam-6096	318	3	was	be	AUX
ejpam-6096	318	4	supported	support	VERB
ejpam-6096	318	5	in	in	ADP
ejpam-6096	318	6	part	part	NOUN
ejpam-6096	318	7	by	by	ADP
ejpam-6096	318	8	the	the	DET
ejpam-6096	318	9	psu	psu	NOUN
ejpam-6096	318	10	-	-	ADJ
ejpam-6096	318	11	tuyf	tuyf	ADJ
ejpam-6096	318	12	charitable	charitable	ADJ
ejpam-6096	318	13	trust	trust	NOUN
ejpam-6096	318	14	fund	fund	NOUN
ejpam-6096	318	15	,	,	PUNCT
ejpam-6096	318	16	prince	prince	NOUN
ejpam-6096	318	17	of	of	ADP
ejpam-6096	318	18	songkla	songkla	PROPN
ejpam-6096	318	19	university	university	PROPN
ejpam-6096	318	20	,	,	PUNCT
ejpam-6096	318	21	contract	contract	NOUN
ejpam-6096	318	22	no.1	no.1	NOUN
ejpam-6096	318	23	-	-	SYM
ejpam-6096	318	24	2567	2567	NUM
ejpam-6096	318	25	-	-	SYM
ejpam-6096	318	26	01	01	NUM
ejpam-6096	318	27	.	.	PUNCT
ejpam-6096	319	1	references	reference	NOUN
ejpam-6096	319	2	[	[	X
ejpam-6096	319	3	1	1	X
ejpam-6096	319	4	]	]	PUNCT
ejpam-6096	319	5	d.	d.	PROPN
ejpam-6096	319	6	h.	h.	PROPN
ejpam-6096	319	7	lehmer	lehmer	PROPN
ejpam-6096	319	8	.	.	PUNCT
ejpam-6096	320	1	a	a	DET
ejpam-6096	320	2	ternary	ternary	ADJ
ejpam-6096	320	3	analogue	analogue	NOUN
ejpam-6096	320	4	of	of	ADP
ejpam-6096	320	5	abelian	abelian	ADJ
ejpam-6096	320	6	groups	group	NOUN
ejpam-6096	320	7	.	.	PUNCT
ejpam-6096	321	1	amer	amer	PROPN
ejpam-6096	321	2	.	.	PUNCT
ejpam-6096	322	1	j.	j.	PROPN
ejpam-6096	322	2	math	math	PROPN
ejpam-6096	322	3	.	.	PUNCT
ejpam-6096	322	4	,	,	PUNCT
ejpam-6096	322	5	54(2):329–338	54(2):329–338	PROPN
ejpam-6096	322	6	,	,	PUNCT
ejpam-6096	322	7	1932	1932	NUM
ejpam-6096	322	8	.	.	PUNCT
ejpam-6096	323	1	[	[	X
ejpam-6096	323	2	2	2	X
ejpam-6096	323	3	]	]	PUNCT
ejpam-6096	323	4	w.	w.	PROPN
ejpam-6096	323	5	g.	g.	PROPN
ejpam-6096	323	6	lister	lister	PROPN
ejpam-6096	323	7	.	.	PUNCT
ejpam-6096	324	1	ternary	ternary	ADJ
ejpam-6096	324	2	rings	ring	NOUN
ejpam-6096	324	3	.	.	PUNCT
ejpam-6096	325	1	trans	trans	PROPN
ejpam-6096	325	2	.	.	PUNCT
ejpam-6096	326	1	amer	amer	PROPN
ejpam-6096	326	2	.	.	PUNCT
ejpam-6096	326	3	math	math	PROPN
ejpam-6096	326	4	.	.	PUNCT
ejpam-6096	327	1	soc	soc	PROPN
ejpam-6096	327	2	.	.	PUNCT
ejpam-6096	327	3	,	,	PUNCT
ejpam-6096	327	4	154:37–55	154:37–55	NUM
ejpam-6096	327	5	,	,	PUNCT
ejpam-6096	327	6	1971	1971	NUM
ejpam-6096	327	7	.	.	PUNCT
ejpam-6096	328	1	[	[	X
ejpam-6096	328	2	3	3	X
ejpam-6096	328	3	]	]	PUNCT
ejpam-6096	328	4	t.	t.	PROPN
ejpam-6096	328	5	k.	k.	PROPN
ejpam-6096	328	6	dutta	dutta	PROPN
ejpam-6096	328	7	and	and	CCONJ
ejpam-6096	328	8	s.	s.	PROPN
ejpam-6096	328	9	kar	kar	PROPN
ejpam-6096	328	10	.	.	PUNCT
ejpam-6096	329	1	on	on	ADP
ejpam-6096	329	2	regular	regular	ADJ
ejpam-6096	329	3	ternary	ternary	ADJ
ejpam-6096	329	4	semirings	semiring	NOUN
ejpam-6096	329	5	.	.	PUNCT
ejpam-6096	330	1	advances	advance	NOUN
ejpam-6096	330	2	in	in	ADP
ejpam-6096	330	3	algebra	algebra	NOUN
ejpam-6096	330	4	,	,	PUNCT
ejpam-6096	330	5	proceedings	proceeding	NOUN
ejpam-6096	330	6	of	of	ADP
ejpam-6096	330	7	the	the	DET
ejpam-6096	330	8	icm	icm	PROPN
ejpam-6096	330	9	satellite	satellite	PROPN
ejpam-6096	330	10	conference	conference	NOUN
ejpam-6096	330	11	in	in	ADP
ejpam-6096	330	12	algebra	algebra	PROPN
ejpam-6096	330	13	and	and	CCONJ
ejpam-6096	330	14	related	related	ADJ
ejpam-6096	330	15	topics	topic	NOUN
ejpam-6096	330	16	,	,	PUNCT
ejpam-6096	330	17	world	world	NOUN
ejpam-6096	330	18	scientific	scientific	ADJ
ejpam-6096	330	19	,	,	PUNCT
ejpam-6096	330	20	pages	page	NOUN
ejpam-6096	330	21	343–355	343–355	NUM
ejpam-6096	330	22	,	,	PUNCT
ejpam-6096	330	23	2003	2003	NUM
ejpam-6096	330	24	.	.	PUNCT
ejpam-6096	331	1	[	[	X
ejpam-6096	331	2	4	4	X
ejpam-6096	331	3	]	]	PUNCT
ejpam-6096	331	4	t.	t.	PROPN
ejpam-6096	331	5	k.	k.	PROPN
ejpam-6096	331	6	dutta	dutta	PROPN
ejpam-6096	331	7	and	and	CCONJ
ejpam-6096	331	8	s.	s.	PROPN
ejpam-6096	331	9	kar	kar	PROPN
ejpam-6096	331	10	.	.	PUNCT
ejpam-6096	332	1	on	on	ADP
ejpam-6096	332	2	prime	prime	ADJ
ejpam-6096	332	3	ideals	ideal	NOUN
ejpam-6096	332	4	and	and	CCONJ
ejpam-6096	332	5	prime	prime	ADJ
ejpam-6096	332	6	radical	radical	ADJ
ejpam-6096	332	7	of	of	ADP
ejpam-6096	332	8	ternary	ternary	ADJ
ejpam-6096	332	9	semirings	semiring	NOUN
ejpam-6096	332	10	.	.	PUNCT
ejpam-6096	333	1	bull	bull	NOUN
ejpam-6096	333	2	.	.	PUNCT
ejpam-6096	334	1	cal	cal	PROPN
ejpam-6096	334	2	.	.	PUNCT
ejpam-6096	335	1	math	math	NOUN
ejpam-6096	335	2	.	.	PUNCT
ejpam-6096	336	1	soc	soc	PROPN
ejpam-6096	336	2	.	.	PUNCT
ejpam-6096	336	3	,	,	PUNCT
ejpam-6096	336	4	97(5):445–454	97(5):445–454	NOUN
ejpam-6096	336	5	,	,	PUNCT
ejpam-6096	336	6	2005	2005	NUM
ejpam-6096	336	7	.	.	PUNCT
ejpam-6096	337	1	[	[	X
ejpam-6096	337	2	5	5	X
ejpam-6096	337	3	]	]	PUNCT
ejpam-6096	337	4	t.	t.	PROPN
ejpam-6096	337	5	k.	k.	PROPN
ejpam-6096	337	6	dutta	dutta	PROPN
ejpam-6096	337	7	and	and	CCONJ
ejpam-6096	337	8	s.	s.	PROPN
ejpam-6096	337	9	kar	kar	PROPN
ejpam-6096	337	10	.	.	PUNCT
ejpam-6096	338	1	on	on	ADP
ejpam-6096	338	2	semiprime	semiprime	NOUN
ejpam-6096	338	3	ideals	ideal	NOUN
ejpam-6096	338	4	and	and	CCONJ
ejpam-6096	338	5	irreducible	irreducible	ADJ
ejpam-6096	338	6	ideals	ideal	NOUN
ejpam-6096	338	7	of	of	ADP
ejpam-6096	338	8	ternary	ternary	ADJ
ejpam-6096	338	9	semirings	semiring	NOUN
ejpam-6096	338	10	.	.	PUNCT
ejpam-6096	339	1	bull	bull	NOUN
ejpam-6096	339	2	.	.	PUNCT
ejpam-6096	340	1	cal	cal	PROPN
ejpam-6096	340	2	.	.	PUNCT
ejpam-6096	341	1	math	math	NOUN
ejpam-6096	341	2	.	.	PUNCT
ejpam-6096	342	1	soc	soc	PROPN
ejpam-6096	342	2	.	.	PUNCT
ejpam-6096	342	3	,	,	PUNCT
ejpam-6096	342	4	97(5):467–476	97(5):467–476	NOUN
ejpam-6096	342	5	,	,	PUNCT
ejpam-6096	342	6	2005	2005	NUM
ejpam-6096	342	7	.	.	PUNCT
ejpam-6096	343	1	[	[	X
ejpam-6096	343	2	6	6	NUM
ejpam-6096	343	3	]	]	PUNCT
ejpam-6096	343	4	s.	s.	PROPN
ejpam-6096	343	5	kar	kar	PROPN
ejpam-6096	343	6	.	.	PUNCT
ejpam-6096	344	1	on	on	ADP
ejpam-6096	344	2	quasi	quasi	NOUN
ejpam-6096	344	3	-	-	NOUN
ejpam-6096	344	4	ideals	ideal	NOUN
ejpam-6096	344	5	and	and	CCONJ
ejpam-6096	344	6	bi	bi	NOUN
ejpam-6096	344	7	-	-	NOUN
ejpam-6096	344	8	ideals	ideal	NOUN
ejpam-6096	344	9	in	in	ADP
ejpam-6096	344	10	ternary	ternary	ADJ
ejpam-6096	344	11	semirings	semiring	NOUN
ejpam-6096	344	12	.	.	PUNCT
ejpam-6096	345	1	int	int	NOUN
ejpam-6096	345	2	.	.	PUNCT
ejpam-6096	346	1	j.	j.	PROPN
ejpam-6096	346	2	math	math	PROPN
ejpam-6096	346	3	.	.	PUNCT
ejpam-6096	347	1	math	math	NOUN
ejpam-6096	347	2	.	.	PUNCT
ejpam-6096	348	1	sci	sci	PROPN
ejpam-6096	348	2	.	.	PROPN
ejpam-6096	348	3	,	,	PUNCT
ejpam-6096	348	4	2005:3015–3023	2005:3015–3023	NUM
ejpam-6096	348	5	,	,	PUNCT
ejpam-6096	348	6	2005	2005	NUM
ejpam-6096	348	7	.	.	PUNCT
ejpam-6096	349	1	[	[	X
ejpam-6096	349	2	7	7	X
ejpam-6096	349	3	]	]	X
ejpam-6096	349	4	r.	r.	PROPN
ejpam-6096	349	5	chinram	chinram	PROPN
ejpam-6096	349	6	and	and	CCONJ
ejpam-6096	349	7	s.	s.	PROPN
ejpam-6096	349	8	malee	malee	PROPN
ejpam-6096	349	9	.	.	PUNCT
ejpam-6096	350	1	l	l	ADJ
ejpam-6096	350	2	-	-	ADJ
ejpam-6096	350	3	fuzzy	fuzzy	ADJ
ejpam-6096	350	4	ternary	ternary	ADJ
ejpam-6096	350	5	subsemirings	subsemiring	NOUN
ejpam-6096	350	6	and	and	CCONJ
ejpam-6096	350	7	l	l	ADJ
ejpam-6096	350	8	-	-	ADJ
ejpam-6096	350	9	fuzzy	fuzzy	ADJ
ejpam-6096	350	10	ideals	ideal	NOUN
ejpam-6096	350	11	in	in	ADP
ejpam-6096	350	12	ternary	ternary	ADJ
ejpam-6096	350	13	semirings	semiring	NOUN
ejpam-6096	350	14	.	.	PUNCT
ejpam-6096	351	1	iaeng	iaeng	PROPN
ejpam-6096	351	2	int	int	PROPN
ejpam-6096	351	3	.	.	PUNCT
ejpam-6096	352	1	j.	j.	PROPN
ejpam-6096	352	2	appl	appl	PROPN
ejpam-6096	352	3	.	.	PROPN
ejpam-6096	352	4	math	math	PROPN
ejpam-6096	352	5	.	.	PUNCT
ejpam-6096	353	1	,	,	PUNCT
ejpam-6096	353	2	40(3):40	40(3):40	NUM
ejpam-6096	353	3	3	3	NUM
ejpam-6096	353	4	03	03	NUM
ejpam-6096	353	5	,	,	PUNCT
ejpam-6096	353	6	2010	2010	NUM
ejpam-6096	353	7	.	.	PUNCT
ejpam-6096	354	1	[	[	X
ejpam-6096	354	2	8	8	X
ejpam-6096	354	3	]	]	PUNCT
ejpam-6096	354	4	s.	s.	PROPN
ejpam-6096	354	5	malee	malee	PROPN
ejpam-6096	354	6	and	and	CCONJ
ejpam-6096	354	7	r.	r.	PROPN
ejpam-6096	354	8	chinram	chinram	PROPN
ejpam-6096	354	9	.	.	PUNCT
ejpam-6096	355	1	k	k	ADJ
ejpam-6096	355	2	-	-	ADJ
ejpam-6096	355	3	fuzzy	fuzzy	ADJ
ejpam-6096	355	4	ideals	ideal	NOUN
ejpam-6096	355	5	of	of	ADP
ejpam-6096	355	6	ternary	ternary	ADJ
ejpam-6096	355	7	semirings	semiring	NOUN
ejpam-6096	355	8	.	.	PUNCT
ejpam-6096	356	1	int	int	NOUN
ejpam-6096	356	2	.	.	PUNCT
ejpam-6096	357	1	j.	j.	PROPN
ejpam-6096	357	2	comp	comp	PROPN
ejpam-6096	357	3	.	.	PUNCT
ejpam-6096	358	1	math	math	PROPN
ejpam-6096	358	2	.	.	PUNCT
ejpam-6096	359	1	sci	sci	PROPN
ejpam-6096	359	2	.	.	PROPN
ejpam-6096	359	3	,	,	PUNCT
ejpam-6096	359	4	4(4):206–210	4(4):206–210	NUM
ejpam-6096	359	5	,	,	PUNCT
ejpam-6096	359	6	2010	2010	NUM
ejpam-6096	359	7	.	.	PUNCT
ejpam-6096	360	1	[	[	X
ejpam-6096	360	2	9	9	NUM
ejpam-6096	360	3	]	]	PUNCT
ejpam-6096	360	4	m.	m.	NOUN
ejpam-6096	360	5	k.	k.	PROPN
ejpam-6096	360	6	dubey	dubey	PROPN
ejpam-6096	360	7	.	.	PUNCT
ejpam-6096	361	1	a	a	DET
ejpam-6096	361	2	note	note	NOUN
ejpam-6096	361	3	on	on	ADP
ejpam-6096	361	4	quasi	quasi	ADJ
ejpam-6096	361	5	k	k	NOUN
ejpam-6096	361	6	-	-	NOUN
ejpam-6096	361	7	ideals	ideal	NOUN
ejpam-6096	361	8	and	and	CCONJ
ejpam-6096	361	9	bi	bi	ADJ
ejpam-6096	361	10	k	k	NOUN
ejpam-6096	361	11	-	-	NOUN
ejpam-6096	361	12	ideals	ideal	NOUN
ejpam-6096	361	13	in	in	ADP
ejpam-6096	361	14	ternary	ternary	ADJ
ejpam-6096	361	15	semirings	semiring	NOUN
ejpam-6096	361	16	.	.	PUNCT
ejpam-6096	362	1	italian	italian	ADJ
ejpam-6096	362	2	j.	j.	PROPN
ejpam-6096	362	3	pure	pure	PROPN
ejpam-6096	362	4	appl	appl	PROPN
ejpam-6096	362	5	.	.	PUNCT
ejpam-6096	362	6	math	math	PROPN
ejpam-6096	362	7	.	.	PUNCT
ejpam-6096	362	8	,	,	PUNCT
ejpam-6096	363	1	28:143–150	28:143–150	PROPN
ejpam-6096	363	2	,	,	PUNCT
ejpam-6096	363	3	2011	2011	NUM
ejpam-6096	363	4	.	.	PUNCT
ejpam-6096	364	1	[	[	X
ejpam-6096	364	2	10	10	NUM
ejpam-6096	364	3	]	]	PUNCT
ejpam-6096	364	4	j.	j.	PROPN
ejpam-6096	364	5	n.	n.	PROPN
ejpam-6096	364	6	chaudhari	chaudhari	PROPN
ejpam-6096	364	7	and	and	CCONJ
ejpam-6096	364	8	k.	k.	PROPN
ejpam-6096	364	9	j	j	PROPN
ejpam-6096	364	10	.ingale	.ingale	PROPN
ejpam-6096	364	11	.	.	PUNCT
ejpam-6096	365	1	on	on	ADP
ejpam-6096	365	2	partitioning	partition	VERB
ejpam-6096	365	3	and	and	CCONJ
ejpam-6096	365	4	subtractive	subtractive	ADJ
ejpam-6096	365	5	ideals	ideal	NOUN
ejpam-6096	365	6	of	of	ADP
ejpam-6096	365	7	ternary	ternary	ADJ
ejpam-6096	365	8	semirings	semiring	NOUN
ejpam-6096	365	9	.	.	PUNCT
ejpam-6096	366	1	kyungpook	kyungpook	PROPN
ejpam-6096	366	2	.	.	PUNCT
ejpam-6096	367	1	math	math	NOUN
ejpam-6096	367	2	.	.	PUNCT
ejpam-6096	368	1	j.	j.	PROPN
ejpam-6096	368	2	,	,	PUNCT
ejpam-6096	368	3	51(1):69–76	51(1):69–76	NUM
ejpam-6096	368	4	,	,	PUNCT
ejpam-6096	368	5	2011	2011	NUM
ejpam-6096	368	6	.	.	PUNCT
ejpam-6096	369	1	[	[	X
ejpam-6096	369	2	11	11	NUM
ejpam-6096	369	3	]	]	PUNCT
ejpam-6096	369	4	t.	t.	PROPN
ejpam-6096	369	5	sunitha	sunitha	PROPN
ejpam-6096	369	6	;	;	PUNCT
ejpam-6096	369	7	u.	u.	PROPN
ejpam-6096	369	8	nagi	nagi	PROPN
ejpam-6096	369	9	reddy	reddy	PROPN
ejpam-6096	369	10	and	and	CCONJ
ejpam-6096	369	11	g.	g.	PROPN
ejpam-6096	369	12	shobhalatha	shobhalatha	PROPN
ejpam-6096	369	13	.	.	PUNCT
ejpam-6096	370	1	a	a	DET
ejpam-6096	370	2	note	note	NOUN
ejpam-6096	370	3	on	on	ADP
ejpam-6096	370	4	full	full	ADJ
ejpam-6096	370	5	k	k	NOUN
ejpam-6096	370	6	-	-	NOUN
ejpam-6096	370	7	ideals	ideal	NOUN
ejpam-6096	370	8	in	in	ADP
ejpam-6096	370	9	ternary	ternary	ADJ
ejpam-6096	370	10	semirings	semiring	NOUN
ejpam-6096	370	11	.	.	PUNCT
ejpam-6096	371	1	indian	indian	PROPN
ejpam-6096	371	2	j.	j.	PROPN
ejpam-6096	371	3	sci	sci	PROPN
ejpam-6096	371	4	.	.	PROPN
ejpam-6096	371	5	techno	techno	PROPN
ejpam-6096	371	6	.	.	PUNCT
ejpam-6096	371	7	,	,	PUNCT
ejpam-6096	371	8	14(21):1786–1790	14(21):1786–1790	NUM
ejpam-6096	371	9	,	,	PUNCT
ejpam-6096	371	10	2021	2021	NUM
ejpam-6096	371	11	.	.	PUNCT
ejpam-6096	372	1	a.	a.	PROPN
ejpam-6096	372	2	j.	j.	PROPN
ejpam-6096	372	3	khan	khan	PROPN
ejpam-6096	372	4	,	,	PUNCT
ejpam-6096	372	5	m.	m.	NOUN
ejpam-6096	372	6	petapirak	petapirak	PROPN
ejpam-6096	372	7	,	,	PUNCT
ejpam-6096	372	8	r.	r.	PROPN
ejpam-6096	372	9	chinram	chinram	PROPN
ejpam-6096	372	10	/	/	SYM
ejpam-6096	372	11	eur	eur	PROPN
ejpam-6096	372	12	.	.	PUNCT
ejpam-6096	373	1	j.	j.	PROPN
ejpam-6096	373	2	pure	pure	PROPN
ejpam-6096	373	3	appl	appl	PROPN
ejpam-6096	373	4	.	.	PROPN
ejpam-6096	373	5	math	math	PROPN
ejpam-6096	373	6	,	,	PUNCT
ejpam-6096	373	7	18	18	NUM
ejpam-6096	373	8	(	(	PUNCT
ejpam-6096	373	9	2	2	NUM
ejpam-6096	373	10	)	)	PUNCT
ejpam-6096	373	11	(	(	PUNCT
ejpam-6096	373	12	2025	2025	NUM
ejpam-6096	373	13	)	)	PUNCT
ejpam-6096	373	14	,	,	PUNCT
ejpam-6096	373	15	6096	6096	NUM
ejpam-6096	373	16	10	10	NUM
ejpam-6096	373	17	of	of	ADP
ejpam-6096	373	18	10	10	NUM
ejpam-6096	374	1	[	[	X
ejpam-6096	374	2	12	12	NUM
ejpam-6096	374	3	]	]	PUNCT
ejpam-6096	374	4	t.	t.	PROPN
ejpam-6096	374	5	k.	k.	PROPN
ejpam-6096	374	6	dutta	dutta	PROPN
ejpam-6096	374	7	;	;	PUNCT
ejpam-6096	374	8	k.	k.	PROPN
ejpam-6096	375	1	p.	p.	PROPN
ejpam-6096	375	2	shum	shum	PROPN
ejpam-6096	375	3	and	and	CCONJ
ejpam-6096	375	4	s.	s.	PROPN
ejpam-6096	375	5	mandal	mandal	PROPN
ejpam-6096	375	6	.	.	PUNCT
ejpam-6096	376	1	singular	singular	PROPN
ejpam-6096	376	2	ideals	ideal	NOUN
ejpam-6096	376	3	of	of	ADP
ejpam-6096	376	4	ternary	ternary	ADJ
ejpam-6096	376	5	semirings	semiring	NOUN
ejpam-6096	376	6	.	.	PUNCT
ejpam-6096	377	1	eur	eur	PROPN
ejpam-6096	377	2	.	.	PUNCT
ejpam-6096	378	1	j.	j.	PROPN
ejpam-6096	378	2	pure	pure	PROPN
ejpam-6096	378	3	appl	appl	PROPN
ejpam-6096	378	4	.	.	PUNCT
ejpam-6096	378	5	math	math	PROPN
ejpam-6096	378	6	.	.	PUNCT
ejpam-6096	378	7	,	,	PUNCT
ejpam-6096	378	8	5(2):116–128	5(2):116–128	NOUN
ejpam-6096	378	9	,	,	PUNCT
ejpam-6096	378	10	2012	2012	NUM
ejpam-6096	378	11	.	.	PUNCT
ejpam-6096	379	1	[	[	X
ejpam-6096	379	2	13	13	NUM
ejpam-6096	379	3	]	]	PUNCT
ejpam-6096	379	4	j.	j.	PROPN
ejpam-6096	379	5	n.	n.	PROPN
ejpam-6096	379	6	chaudhari	chaudhari	PROPN
ejpam-6096	379	7	and	and	CCONJ
ejpam-6096	379	8	k.	k.	PROPN
ejpam-6096	379	9	j.	j.	PROPN
ejpam-6096	379	10	ingale	ingale	PROPN
ejpam-6096	379	11	.	.	PUNCT
ejpam-6096	380	1	subtractive	subtractive	NOUN
ejpam-6096	380	2	extension	extension	NOUN
ejpam-6096	380	3	of	of	ADP
ejpam-6096	380	4	ideals	ideal	NOUN
ejpam-6096	380	5	in	in	ADP
ejpam-6096	380	6	ternary	ternary	ADJ
ejpam-6096	380	7	semirings	semiring	NOUN
ejpam-6096	380	8	.	.	PUNCT
ejpam-6096	381	1	thai	thai	PROPN
ejpam-6096	381	2	j.	j.	PROPN
ejpam-6096	381	3	math	math	PROPN
ejpam-6096	381	4	.	.	PUNCT
ejpam-6096	381	5	,	,	PUNCT
ejpam-6096	381	6	14(3):615–625	14(3):615–625	NOUN
ejpam-6096	381	7	,	,	PUNCT
ejpam-6096	381	8	2016	2016	NUM
ejpam-6096	381	9	.	.	PUNCT
ejpam-6096	382	1	[	[	X
ejpam-6096	382	2	14	14	NUM
ejpam-6096	382	3	]	]	PUNCT
ejpam-6096	382	4	p.	p.	NOUN
ejpam-6096	382	5	luangchaisri	luangchaisri	VERB
ejpam-6096	382	6	and	and	CCONJ
ejpam-6096	382	7	t.	t.	PROPN
ejpam-6096	382	8	changphas	changphas	PROPN
ejpam-6096	382	9	.	.	PUNCT
ejpam-6096	383	1	prime	prime	ADJ
ejpam-6096	383	2	one	one	NUM
ejpam-6096	383	3	-	-	PUNCT
ejpam-6096	383	4	sided	sided	ADJ
ejpam-6096	383	5	ideals	ideal	NOUN
ejpam-6096	383	6	in	in	ADP
ejpam-6096	383	7	ternary	ternary	ADJ
ejpam-6096	383	8	semirings	semiring	NOUN
ejpam-6096	383	9	.	.	PUNCT
ejpam-6096	384	1	int	int	NOUN
ejpam-6096	384	2	.	.	PUNCT
ejpam-6096	385	1	j.	j.	PROPN
ejpam-6096	385	2	math	math	PROPN
ejpam-6096	385	3	.	.	PUNCT
ejpam-6096	386	1	comp	comp	PROPN
ejpam-6096	386	2	.	.	PUNCT
ejpam-6096	387	1	sci	sci	PROPN
ejpam-6096	387	2	.	.	PROPN
ejpam-6096	387	3	,	,	PUNCT
ejpam-6096	387	4	19(2):403–409	19(2):403–409	NUM
ejpam-6096	387	5	,	,	PUNCT
ejpam-6096	387	6	2024	2024	NUM
ejpam-6096	387	7	.	.	PUNCT
ejpam-6096	388	1	[	[	X
ejpam-6096	388	2	15	15	NUM
ejpam-6096	388	3	]	]	X
ejpam-6096	388	4	a.	a.	NOUN
ejpam-6096	388	5	goswami	goswami	NOUN
ejpam-6096	388	6	and	and	CCONJ
ejpam-6096	388	7	t.	t.	PROPN
ejpam-6096	388	8	dube	dube	PROPN
ejpam-6096	388	9	.	.	PUNCT
ejpam-6096	389	1	some	some	DET
ejpam-6096	389	2	aspects	aspect	NOUN
ejpam-6096	389	3	of	of	ADP
ejpam-6096	389	4	k	k	NOUN
ejpam-6096	389	5	-	-	NOUN
ejpam-6096	389	6	ideals	ideal	NOUN
ejpam-6096	389	7	of	of	ADP
ejpam-6096	389	8	semirings	semiring	NOUN
ejpam-6096	389	9	.	.	PUNCT
ejpam-6096	390	1	rend	rend	VERB
ejpam-6096	390	2	.	.	PUNCT
ejpam-6096	391	1	circ	circ	PROPN
ejpam-6096	391	2	.	.	PUNCT
ejpam-6096	392	1	mat	mat	PROPN
ejpam-6096	392	2	.	.	PUNCT
ejpam-6096	392	3	palermo	palermo	PROPN
ejpam-6096	392	4	(	(	PUNCT
ejpam-6096	392	5	2	2	NUM
ejpam-6096	392	6	)	)	PUNCT
ejpam-6096	392	7	,	,	PUNCT
ejpam-6096	392	8	73:3105–3117	73:3105–3117	NUM
ejpam-6096	392	9	,	,	PUNCT
ejpam-6096	392	10	2024	2024	NUM
ejpam-6096	392	11	.	.	PUNCT