id	sid	tid	token	lemma	pos
ejpam-6153	1	1	european	european	PROPN
ejpam-6153	1	2	journal	journal	PROPN
ejpam-6153	1	3	of	of	ADP
ejpam-6153	1	4	pure	pure	ADJ
ejpam-6153	1	5	and	and	CCONJ
ejpam-6153	1	6	applied	applied	ADJ
ejpam-6153	1	7	mathematics	mathematic	NOUN
ejpam-6153	1	8	2025	2025	NUM
ejpam-6153	1	9	,	,	PUNCT
ejpam-6153	1	10	vol	vol	NOUN
ejpam-6153	1	11	.	.	PROPN
ejpam-6153	1	12	18	18	NUM
ejpam-6153	1	13	,	,	PUNCT
ejpam-6153	1	14	issue	issue	NOUN
ejpam-6153	1	15	3	3	NUM
ejpam-6153	1	16	,	,	PUNCT
ejpam-6153	1	17	article	article	NOUN
ejpam-6153	1	18	number	number	NOUN
ejpam-6153	1	19	6153	6153	NUM
ejpam-6153	1	20	issn	issn	PROPN
ejpam-6153	1	21	1307	1307	NUM
ejpam-6153	1	22	-	-	SYM
ejpam-6153	1	23	5543	5543	NUM
ejpam-6153	1	24	–	–	PUNCT
ejpam-6153	1	25	ejpam.com	ejpam.com	X
ejpam-6153	1	26	published	publish	VERB
ejpam-6153	1	27	by	by	ADP
ejpam-6153	1	28	new	new	PROPN
ejpam-6153	1	29	york	york	PROPN
ejpam-6153	1	30	business	business	PROPN
ejpam-6153	1	31	global	global	ADJ
ejpam-6153	1	32	1	1	NUM
ejpam-6153	1	33	a	a	DET
ejpam-6153	1	34	study	study	NOUN
ejpam-6153	1	35	on	on	ADP
ejpam-6153	1	36	essential	essential	ADJ
ejpam-6153	1	37	fuzzy	fuzzy	ADJ
ejpam-6153	1	38	ideals	ideal	NOUN
ejpam-6153	1	39	of	of	ADP
ejpam-6153	1	40	semirings2	semirings2	PROPN
ejpam-6153	1	41	ronnason	ronnason	PROPN
ejpam-6153	1	42	chinram1	chinram1	PROPN
ejpam-6153	1	43	,	,	PUNCT
ejpam-6153	1	44	saranya	saranya	PROPN
ejpam-6153	1	45	hangsawat2,∗3	hangsawat2,∗3	PROPN
ejpam-6153	1	46	1	1	NUM
ejpam-6153	1	47	division	division	NOUN
ejpam-6153	1	48	of	of	ADP
ejpam-6153	1	49	computational	computational	ADJ
ejpam-6153	1	50	science	science	NOUN
ejpam-6153	1	51	,	,	PUNCT
ejpam-6153	1	52	faculty	faculty	NOUN
ejpam-6153	1	53	of	of	ADP
ejpam-6153	1	54	science	science	NOUN
ejpam-6153	1	55	,	,	PUNCT
ejpam-6153	1	56	prince	prince	NOUN
ejpam-6153	1	57	of	of	ADP
ejpam-6153	1	58	songkla	songkla	PROPN
ejpam-6153	1	59	university,4	university,4	PROPN
ejpam-6153	1	60	hat	hat	PROPN
ejpam-6153	1	61	yai	yai	PROPN
ejpam-6153	1	62	,	,	PUNCT
ejpam-6153	1	63	songkhla	songkhla	VERB
ejpam-6153	1	64	90110	90110	NUM
ejpam-6153	1	65	,	,	PUNCT
ejpam-6153	1	66	thailand5	thailand5	PROPN
ejpam-6153	1	67	2	2	NUM
ejpam-6153	1	68	mathematics	mathematics	NOUN
ejpam-6153	1	69	program	program	NOUN
ejpam-6153	1	70	,	,	PUNCT
ejpam-6153	1	71	faculty	faculty	NOUN
ejpam-6153	1	72	of	of	ADP
ejpam-6153	1	73	science	science	NOUN
ejpam-6153	1	74	and	and	CCONJ
ejpam-6153	1	75	technology	technology	NOUN
ejpam-6153	1	76	,	,	PUNCT
ejpam-6153	1	77	songkhla	songkhla	VERB
ejpam-6153	1	78	rajabhat	rajabhat	PRON
ejpam-6153	1	79	university,6	university,6	ADV
ejpam-6153	1	80	songkhla	songkhla	VERB
ejpam-6153	1	81	90000	90000	NUM
ejpam-6153	1	82	,	,	PUNCT
ejpam-6153	1	83	thailand7	thailand7	NOUN
ejpam-6153	1	84	8	8	NUM
ejpam-6153	1	85	abstract	abstract	NOUN
ejpam-6153	1	86	.	.	PUNCT
ejpam-6153	2	1	in	in	ADP
ejpam-6153	2	2	this	this	DET
ejpam-6153	2	3	paper	paper	NOUN
ejpam-6153	2	4	,	,	PUNCT
ejpam-6153	2	5	we	we	PRON
ejpam-6153	2	6	define	define	VERB
ejpam-6153	2	7	essential	essential	ADJ
ejpam-6153	2	8	fuzzy	fuzzy	ADJ
ejpam-6153	2	9	ideals	ideal	NOUN
ejpam-6153	2	10	of	of	ADP
ejpam-6153	2	11	semirings	semiring	NOUN
ejpam-6153	2	12	and	and	CCONJ
ejpam-6153	2	13	investigate	investigate	VERB
ejpam-6153	2	14	some	some	DET
ejpam-6153	2	15	properties	property	NOUN
ejpam-6153	2	16	of	of	ADP
ejpam-6153	2	17	them	they	PRON
ejpam-6153	2	18	.	.	PUNCT
ejpam-6153	3	1	we	we	PRON
ejpam-6153	3	2	give	give	VERB
ejpam-6153	3	3	some	some	DET
ejpam-6153	3	4	properties	property	NOUN
ejpam-6153	3	5	of	of	ADP
ejpam-6153	3	6	essential	essential	ADJ
ejpam-6153	3	7	ideals	ideal	NOUN
ejpam-6153	3	8	and	and	CCONJ
ejpam-6153	3	9	essential	essential	ADJ
ejpam-6153	3	10	fuzzy	fuzzy	ADJ
ejpam-6153	3	11	ideals	ideal	NOUN
ejpam-6153	3	12	.	.	PUNCT
ejpam-6153	4	1	moreover	moreover	ADV
ejpam-6153	4	2	,	,	PUNCT
ejpam-6153	4	3	we	we	PRON
ejpam-6153	4	4	show	show	VERB
ejpam-6153	4	5	relationships	relationship	NOUN
ejpam-6153	4	6	between	between	ADP
ejpam-6153	4	7	essential	essential	ADJ
ejpam-6153	4	8	ideals	ideal	NOUN
ejpam-6153	4	9	and	and	CCONJ
ejpam-6153	4	10	essential	essential	ADJ
ejpam-6153	4	11	fuzzy	fuzzy	ADJ
ejpam-6153	4	12	ideals	ideal	NOUN
ejpam-6153	4	13	.	.	PUNCT
ejpam-6153	5	1	2020	2020	NUM
ejpam-6153	5	2	mathematics	mathematic	NOUN
ejpam-6153	5	3	subject	subject	NOUN
ejpam-6153	5	4	classifications	classification	NOUN
ejpam-6153	5	5	:	:	PUNCT
ejpam-6153	5	6	16y60	16y60	NUM
ejpam-6153	5	7	,	,	PUNCT
ejpam-6153	5	8	03e729	03e729	VERB
ejpam-6153	5	9	key	key	ADJ
ejpam-6153	5	10	words	word	NOUN
ejpam-6153	5	11	and	and	CCONJ
ejpam-6153	5	12	phrases	phrase	NOUN
ejpam-6153	5	13	:	:	PUNCT
ejpam-6153	5	14	essential	essential	ADJ
ejpam-6153	5	15	ideals	ideal	NOUN
ejpam-6153	5	16	,	,	PUNCT
ejpam-6153	5	17	essential	essential	ADJ
ejpam-6153	5	18	fuzzy	fuzzy	ADJ
ejpam-6153	5	19	ideals	ideal	NOUN
ejpam-6153	5	20	,	,	PUNCT
ejpam-6153	5	21	minimal	minimal	ADJ
ejpam-6153	5	22	,	,	PUNCT
ejpam-6153	5	23	prime	prime	ADJ
ejpam-6153	5	24	,	,	PUNCT
ejpam-6153	5	25	semiprime10	semiprime10	X
ejpam-6153	5	26	11	11	NUM
ejpam-6153	5	27	1	1	NUM
ejpam-6153	5	28	.	.	PUNCT
ejpam-6153	5	29	introduction12	introduction12	VERB
ejpam-6153	5	30	a	a	DET
ejpam-6153	5	31	semiring	semiring	NOUN
ejpam-6153	5	32	as	as	ADP
ejpam-6153	5	33	the	the	DET
ejpam-6153	5	34	algebraic	algebraic	ADJ
ejpam-6153	5	35	structure	structure	NOUN
ejpam-6153	5	36	,	,	PUNCT
ejpam-6153	5	37	is	be	AUX
ejpam-6153	5	38	definitely	definitely	ADV
ejpam-6153	5	39	a	a	DET
ejpam-6153	5	40	one	one	NUM
ejpam-6153	5	41	of	of	ADP
ejpam-6153	5	42	generalizations	generalization	NOUN
ejpam-6153	5	43	of	of	ADP
ejpam-6153	5	44	rings	ring	NOUN
ejpam-6153	5	45	.	.	PUNCT
ejpam-6153	6	1	a13	a13	NOUN
ejpam-6153	6	2	semiring	semire	VERB
ejpam-6153	6	3	was	be	AUX
ejpam-6153	6	4	appropriate	appropriate	ADJ
ejpam-6153	6	5	to	to	PART
ejpam-6153	6	6	ask	ask	VERB
ejpam-6153	6	7	which	which	PRON
ejpam-6153	6	8	properties	property	NOUN
ejpam-6153	6	9	of	of	ADP
ejpam-6153	6	10	rings	ring	NOUN
ejpam-6153	6	11	can	can	AUX
ejpam-6153	6	12	be	be	AUX
ejpam-6153	6	13	extended	extend	VERB
ejpam-6153	6	14	to	to	ADP
ejpam-6153	6	15	semirings.14	semirings.14	VERB
ejpam-6153	6	16	the	the	DET
ejpam-6153	6	17	concept	concept	NOUN
ejpam-6153	6	18	of	of	ADP
ejpam-6153	6	19	semirings	semiring	NOUN
ejpam-6153	6	20	was	be	AUX
ejpam-6153	6	21	introduced	introduce	VERB
ejpam-6153	6	22	by	by	ADP
ejpam-6153	6	23	vandiver	vandiver	NOUN
ejpam-6153	6	24	in	in	ADP
ejpam-6153	6	25	1935	1935	NUM
ejpam-6153	6	26	.	.	PUNCT
ejpam-6153	7	1	one	one	PRON
ejpam-6153	7	2	may	may	AUX
ejpam-6153	7	3	expect	expect	VERB
ejpam-6153	7	4	semirings15	semirings15	NOUN
ejpam-6153	7	5	always	always	ADV
ejpam-6153	7	6	to	to	PART
ejpam-6153	7	7	be	be	AUX
ejpam-6153	7	8	extended	extend	VERB
ejpam-6153	7	9	to	to	ADP
ejpam-6153	7	10	rings	ring	NOUN
ejpam-6153	7	11	,	,	PUNCT
ejpam-6153	7	12	however	however	ADV
ejpam-6153	7	13	,	,	PUNCT
ejpam-6153	7	14	vandiver	vandiver	VERB
ejpam-6153	7	15	[	[	X
ejpam-6153	7	16	1	1	NUM
ejpam-6153	7	17	]	]	PUNCT
ejpam-6153	7	18	gave	give	VERB
ejpam-6153	7	19	examples	example	NOUN
ejpam-6153	7	20	of	of	ADP
ejpam-6153	7	21	semirings	semiring	NOUN
ejpam-6153	7	22	that16	that16	NOUN
ejpam-6153	7	23	can	can	AUX
ejpam-6153	7	24	not	not	PART
ejpam-6153	7	25	be	be	AUX
ejpam-6153	7	26	embedded	embed	VERB
ejpam-6153	7	27	in	in	ADP
ejpam-6153	7	28	rings	ring	NOUN
ejpam-6153	7	29	.	.	PUNCT
ejpam-6153	8	1	ideal	ideal	PROPN
ejpam-6153	8	2	theory	theory	NOUN
ejpam-6153	8	3	is	be	AUX
ejpam-6153	8	4	the	the	DET
ejpam-6153	8	5	main	main	ADJ
ejpam-6153	8	6	of	of	ADP
ejpam-6153	8	7	researching	researching	NOUN
ejpam-6153	8	8	of	of	ADP
ejpam-6153	8	9	ring	ring	NOUN
ejpam-6153	8	10	theory17	theory17	NOUN
ejpam-6153	8	11	and	and	CCONJ
ejpam-6153	8	12	also	also	ADV
ejpam-6153	8	13	in	in	ADP
ejpam-6153	8	14	semiring	semire	VERB
ejpam-6153	8	15	theory	theory	NOUN
ejpam-6153	8	16	.	.	PUNCT
ejpam-6153	9	1	a	a	DET
ejpam-6153	9	2	proper	proper	ADJ
ejpam-6153	9	3	ideal	ideal	NOUN
ejpam-6153	9	4	of	of	ADP
ejpam-6153	9	5	a	a	DET
ejpam-6153	9	6	ring	ring	NOUN
ejpam-6153	9	7	is	be	AUX
ejpam-6153	9	8	called	call	VERB
ejpam-6153	9	9	essential	essential	ADJ
ejpam-6153	9	10	if	if	SCONJ
ejpam-6153	9	11	it	it	PRON
ejpam-6153	9	12	has	have	VERB
ejpam-6153	9	13	nonzero18	nonzero18	NOUN
ejpam-6153	9	14	intersection	intersection	NOUN
ejpam-6153	9	15	with	with	ADP
ejpam-6153	9	16	each	each	DET
ejpam-6153	9	17	nonzero	nonzero	NOUN
ejpam-6153	9	18	ideal	ideal	NOUN
ejpam-6153	9	19	.	.	PUNCT
ejpam-6153	10	1	some	some	DET
ejpam-6153	10	2	properties	property	NOUN
ejpam-6153	10	3	of	of	ADP
ejpam-6153	10	4	essential	essential	ADJ
ejpam-6153	10	5	ideals	ideal	NOUN
ejpam-6153	10	6	of	of	ADP
ejpam-6153	10	7	rings	ring	NOUN
ejpam-6153	10	8	can	can	AUX
ejpam-6153	10	9	see19	see19	VERB
ejpam-6153	10	10	in	in	ADP
ejpam-6153	10	11	[	[	X
ejpam-6153	10	12	2	2	NUM
ejpam-6153	10	13	]	]	PUNCT
ejpam-6153	10	14	.	.	PUNCT
ejpam-6153	11	1	similar	similar	ADJ
ejpam-6153	11	2	to	to	ADP
ejpam-6153	11	3	ring	ring	NOUN
ejpam-6153	11	4	theory	theory	NOUN
ejpam-6153	11	5	,	,	PUNCT
ejpam-6153	11	6	an	an	DET
ejpam-6153	11	7	essential	essential	ADJ
ejpam-6153	11	8	ideal	ideal	NOUN
ejpam-6153	11	9	of	of	ADP
ejpam-6153	11	10	a	a	DET
ejpam-6153	11	11	semiring	semiring	NOUN
ejpam-6153	11	12	was	be	AUX
ejpam-6153	11	13	similar	similar	ADJ
ejpam-6153	11	14	defined	define	VERB
ejpam-6153	11	15	in	in	ADP
ejpam-6153	11	16	[	[	NOUN
ejpam-6153	11	17	3].20	3].20	NUM
ejpam-6153	11	18	the	the	DET
ejpam-6153	11	19	notion	notion	NOUN
ejpam-6153	11	20	of	of	ADP
ejpam-6153	11	21	fuzzy	fuzzy	ADJ
ejpam-6153	11	22	sets	set	NOUN
ejpam-6153	11	23	as	as	SCONJ
ejpam-6153	11	24	the	the	DET
ejpam-6153	11	25	extension	extension	NOUN
ejpam-6153	11	26	of	of	ADP
ejpam-6153	11	27	classical	classical	ADJ
ejpam-6153	11	28	sets	set	NOUN
ejpam-6153	11	29	was	be	AUX
ejpam-6153	11	30	introduced	introduce	VERB
ejpam-6153	11	31	by	by	ADP
ejpam-6153	11	32	zadeh	zadeh	PROPN
ejpam-6153	12	1	[	[	X
ejpam-6153	12	2	4	4	NUM
ejpam-6153	12	3	]	]	PUNCT
ejpam-6153	12	4	in21	in21	PROPN
ejpam-6153	12	5	1965	1965	NUM
ejpam-6153	12	6	.	.	PUNCT
ejpam-6153	13	1	fuzzy	fuzzy	ADJ
ejpam-6153	13	2	sets	set	NOUN
ejpam-6153	13	3	permitted	permit	VERB
ejpam-6153	13	4	the	the	DET
ejpam-6153	13	5	gradual	gradual	ADJ
ejpam-6153	13	6	assessment	assessment	NOUN
ejpam-6153	13	7	of	of	ADP
ejpam-6153	13	8	the	the	DET
ejpam-6153	13	9	membership	membership	NOUN
ejpam-6153	13	10	of	of	ADP
ejpam-6153	13	11	elements	element	NOUN
ejpam-6153	13	12	and22	and22	NOUN
ejpam-6153	13	13	described	describe	VERB
ejpam-6153	13	14	with	with	ADP
ejpam-6153	13	15	the	the	DET
ejpam-6153	13	16	membership	membership	NOUN
ejpam-6153	13	17	valued	value	VERB
ejpam-6153	13	18	function	function	NOUN
ejpam-6153	13	19	of	of	ADP
ejpam-6153	13	20	for	for	ADP
ejpam-6153	13	21	each	each	DET
ejpam-6153	13	22	element	element	NOUN
ejpam-6153	13	23	in	in	ADP
ejpam-6153	13	24	set	set	NOUN
ejpam-6153	13	25	to	to	ADP
ejpam-6153	13	26	the	the	DET
ejpam-6153	13	27	value	value	NOUN
ejpam-6153	13	28	in23	in23	PROPN
ejpam-6153	13	29	the	the	DET
ejpam-6153	13	30	unit	unit	NOUN
ejpam-6153	13	31	closed	close	VERB
ejpam-6153	13	32	interval	interval	NOUN
ejpam-6153	13	33	[	[	X
ejpam-6153	13	34	0	0	NUM
ejpam-6153	13	35	,	,	PUNCT
ejpam-6153	13	36	1	1	NUM
ejpam-6153	13	37	]	]	PUNCT
ejpam-6153	13	38	.	.	PUNCT
ejpam-6153	14	1	fuzzy	fuzzy	ADJ
ejpam-6153	14	2	sets	set	NOUN
ejpam-6153	14	3	have	have	AUX
ejpam-6153	14	4	been	be	AUX
ejpam-6153	14	5	applied	apply	VERB
ejpam-6153	14	6	to	to	ADP
ejpam-6153	14	7	many	many	ADJ
ejpam-6153	14	8	algebraic	algebraic	ADJ
ejpam-6153	14	9	structures24	structures24	NOUN
ejpam-6153	14	10	like	like	ADP
ejpam-6153	14	11	groups	group	NOUN
ejpam-6153	14	12	,	,	PUNCT
ejpam-6153	14	13	semigroups	semigroup	NOUN
ejpam-6153	14	14	,	,	PUNCT
ejpam-6153	14	15	rings	ring	NOUN
ejpam-6153	14	16	,	,	PUNCT
ejpam-6153	14	17	modules	module	NOUN
ejpam-6153	14	18	and	and	CCONJ
ejpam-6153	14	19	so	so	ADV
ejpam-6153	14	20	on	on	ADV
ejpam-6153	14	21	.	.	PUNCT
ejpam-6153	15	1	similar	similar	ADJ
ejpam-6153	15	2	to	to	ADP
ejpam-6153	15	3	many	many	ADJ
ejpam-6153	15	4	algebraic	algebraic	ADJ
ejpam-6153	15	5	structures,25	structures,25	ADJ
ejpam-6153	15	6	fuzzy	fuzzy	ADJ
ejpam-6153	15	7	semirings	semiring	NOUN
ejpam-6153	15	8	were	be	AUX
ejpam-6153	15	9	studied	study	VERB
ejpam-6153	15	10	in	in	ADP
ejpam-6153	15	11	[	[	X
ejpam-6153	15	12	5	5	NUM
ejpam-6153	15	13	]	]	PUNCT
ejpam-6153	15	14	.	.	PUNCT
ejpam-6153	16	1	the	the	DET
ejpam-6153	16	2	concepts	concept	NOUN
ejpam-6153	16	3	of	of	ADP
ejpam-6153	16	4	essential	essential	ADJ
ejpam-6153	16	5	ideals	ideal	NOUN
ejpam-6153	16	6	were	be	AUX
ejpam-6153	16	7	defined	define	VERB
ejpam-6153	16	8	and26	and26	ADJ
ejpam-6153	16	9	studied	study	VERB
ejpam-6153	16	10	in	in	ADP
ejpam-6153	16	11	semigroups	semigroup	NOUN
ejpam-6153	16	12	[	[	X
ejpam-6153	16	13	6	6	NUM
ejpam-6153	16	14	]	]	PUNCT
ejpam-6153	16	15	.	.	PUNCT
ejpam-6153	17	1	moreover	moreover	ADV
ejpam-6153	17	2	,	,	PUNCT
ejpam-6153	17	3	their	their	PRON
ejpam-6153	17	4	various	various	ADJ
ejpam-6153	17	5	kinds	kind	NOUN
ejpam-6153	17	6	of	of	ADP
ejpam-6153	17	7	fuzzifications	fuzzification	NOUN
ejpam-6153	17	8	of	of	ADP
ejpam-6153	17	9	essential	essential	ADJ
ejpam-6153	17	10	ideals27	ideals27	NOUN
ejpam-6153	17	11	of	of	ADP
ejpam-6153	17	12	semigroups	semigroup	NOUN
ejpam-6153	17	13	were	be	AUX
ejpam-6153	17	14	also	also	ADV
ejpam-6153	17	15	defined	define	VERB
ejpam-6153	17	16	and	and	CCONJ
ejpam-6153	17	17	studied	study	VERB
ejpam-6153	17	18	[	[	PUNCT
ejpam-6153	17	19	6–9	6–9	NOUN
ejpam-6153	17	20	]	]	PUNCT
ejpam-6153	17	21	.	.	PUNCT
ejpam-6153	18	1	fuzzy	fuzzy	ADJ
ejpam-6153	18	2	essential	essential	ADJ
ejpam-6153	18	3	ideals	ideal	NOUN
ejpam-6153	18	4	in	in	ADP
ejpam-6153	18	5	rings	ring	NOUN
ejpam-6153	18	6	were	be	AUX
ejpam-6153	18	7	also28	also28	NOUN
ejpam-6153	18	8	studied	study	VERB
ejpam-6153	18	9	in	in	ADP
ejpam-6153	18	10	[	[	X
ejpam-6153	18	11	10].29	10].29	NUM
ejpam-6153	18	12	∗corresponding	∗corresponde	VERB
ejpam-6153	18	13	author	author	NOUN
ejpam-6153	18	14	.	.	PUNCT
ejpam-6153	19	1	doi	doi	NOUN
ejpam-6153	19	2	:	:	PUNCT
ejpam-6153	19	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6153	https://doi.org/10.29020/nybg.ejpam.v18i3.6153	ADJ
ejpam-6153	19	4	email	email	NOUN
ejpam-6153	19	5	addresses	address	NOUN
ejpam-6153	19	6	:	:	PUNCT
ejpam-6153	19	7	ronnason.c@psu.ac.th	ronnason.c@psu.ac.th	PROPN
ejpam-6153	19	8	(	(	PUNCT
ejpam-6153	19	9	r.	r.	PROPN
ejpam-6153	19	10	chinram	chinram	PROPN
ejpam-6153	19	11	)	)	PUNCT
ejpam-6153	19	12	,	,	PUNCT
ejpam-6153	19	13	saranya.nu@skru.ac.th	saranya.nu@skru.ac.th	PROPN
ejpam-6153	19	14	(	(	PUNCT
ejpam-6153	19	15	s.	s.	PROPN
ejpam-6153	19	16	hangsawat	hangsawat	PROPN
ejpam-6153	19	17	)	)	PUNCT
ejpam-6153	19	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6153	20	1	1	1	NUM
ejpam-6153	20	2	copyright	copyright	NOUN
ejpam-6153	20	3	:	:	PUNCT
ejpam-6153	20	4	©	©	PROPN
ejpam-6153	20	5	2025	2025	NUM
ejpam-6153	20	6	the	the	DET
ejpam-6153	20	7	author(s	author(s	NOUN
ejpam-6153	20	8	)	)	PUNCT
ejpam-6153	20	9	.	.	PUNCT
ejpam-6153	21	1	(	(	PUNCT
ejpam-6153	21	2	cc	cc	NOUN
ejpam-6153	21	3	by	by	ADP
ejpam-6153	21	4	-	-	PUNCT
ejpam-6153	21	5	nc	nc	PROPN
ejpam-6153	21	6	4.0	4.0	NUM
ejpam-6153	21	7	)	)	PUNCT
ejpam-6153	21	8	r.	r.	PROPN
ejpam-6153	21	9	chinram	chinram	PROPN
ejpam-6153	21	10	,	,	PUNCT
ejpam-6153	21	11	s.	s.	PROPN
ejpam-6153	21	12	hangsawat	hangsawat	PROPN
ejpam-6153	21	13	/	/	SYM
ejpam-6153	21	14	eur	eur	PROPN
ejpam-6153	21	15	.	.	PUNCT
ejpam-6153	22	1	j.	j.	PROPN
ejpam-6153	22	2	pure	pure	PROPN
ejpam-6153	22	3	appl	appl	PROPN
ejpam-6153	22	4	.	.	PROPN
ejpam-6153	22	5	math	math	PROPN
ejpam-6153	22	6	,	,	PUNCT
ejpam-6153	22	7	18	18	NUM
ejpam-6153	22	8	(	(	PUNCT
ejpam-6153	22	9	3	3	NUM
ejpam-6153	22	10	)	)	PUNCT
ejpam-6153	22	11	(	(	PUNCT
ejpam-6153	22	12	2025	2025	NUM
ejpam-6153	22	13	)	)	PUNCT
ejpam-6153	22	14	,	,	PUNCT
ejpam-6153	22	15	6153	6153	NUM
ejpam-6153	22	16	2	2	NUM
ejpam-6153	22	17	of	of	ADP
ejpam-6153	22	18	7	7	NUM
ejpam-6153	22	19	the	the	DET
ejpam-6153	22	20	purpose	purpose	NOUN
ejpam-6153	22	21	of	of	ADP
ejpam-6153	22	22	this	this	DET
ejpam-6153	22	23	paper	paper	NOUN
ejpam-6153	22	24	is	be	AUX
ejpam-6153	22	25	to	to	PART
ejpam-6153	22	26	define	define	VERB
ejpam-6153	22	27	essential	essential	ADJ
ejpam-6153	22	28	fuzzy	fuzzy	ADJ
ejpam-6153	22	29	ideals	ideal	NOUN
ejpam-6153	22	30	of	of	ADP
ejpam-6153	22	31	semirings	semiring	NOUN
ejpam-6153	22	32	.	.	PUNCT
ejpam-6153	23	1	moreover	moreover	ADV
ejpam-6153	23	2	,	,	PUNCT
ejpam-6153	23	3	we30	we30	PROPN
ejpam-6153	23	4	show	show	VERB
ejpam-6153	23	5	some	some	DET
ejpam-6153	23	6	relationships	relationship	NOUN
ejpam-6153	23	7	between	between	ADP
ejpam-6153	23	8	essential	essential	ADJ
ejpam-6153	23	9	ideals	ideal	NOUN
ejpam-6153	23	10	and	and	CCONJ
ejpam-6153	23	11	their	their	PRON
ejpam-6153	23	12	fuzzifications.31	fuzzifications.31	PROPN
ejpam-6153	23	13	2	2	NUM
ejpam-6153	23	14	.	.	PUNCT
ejpam-6153	23	15	preliminaries32	preliminaries32	NOUN
ejpam-6153	23	16	in	in	ADP
ejpam-6153	23	17	this	this	DET
ejpam-6153	23	18	section	section	NOUN
ejpam-6153	23	19	,	,	PUNCT
ejpam-6153	23	20	we	we	PRON
ejpam-6153	23	21	will	will	AUX
ejpam-6153	23	22	recall	recall	VERB
ejpam-6153	23	23	in	in	ADP
ejpam-6153	23	24	the	the	DET
ejpam-6153	23	25	basic	basic	ADJ
ejpam-6153	23	26	concepts	concept	NOUN
ejpam-6153	23	27	of	of	ADP
ejpam-6153	23	28	semirings	semiring	NOUN
ejpam-6153	23	29	,	,	PUNCT
ejpam-6153	23	30	fuzzy	fuzzy	ADJ
ejpam-6153	23	31	sets	set	NOUN
ejpam-6153	23	32	and	and	CCONJ
ejpam-6153	23	33	fuzzy33	fuzzy33	ADJ
ejpam-6153	23	34	ideals	ideal	NOUN
ejpam-6153	23	35	of	of	ADP
ejpam-6153	23	36	semirings.34	semirings.34	NOUN
ejpam-6153	23	37	2.1	2.1	NUM
ejpam-6153	23	38	.	.	PUNCT
ejpam-6153	23	39	semirings35	semirings35	NOUN
ejpam-6153	23	40	by	by	ADP
ejpam-6153	23	41	a	a	DET
ejpam-6153	23	42	semiring	semiring	NOUN
ejpam-6153	23	43	we	we	PRON
ejpam-6153	23	44	shall	shall	AUX
ejpam-6153	23	45	mean	mean	VERB
ejpam-6153	23	46	a	a	DET
ejpam-6153	23	47	nonempty	nonempty	ADJ
ejpam-6153	23	48	set	set	VERB
ejpam-6153	23	49	r	r	NOUN
ejpam-6153	23	50	endowed	endow	VERB
ejpam-6153	23	51	with	with	ADP
ejpam-6153	23	52	two	two	NUM
ejpam-6153	23	53	binary	binary	ADJ
ejpam-6153	23	54	operations36	operations36	NOUN
ejpam-6153	23	55	called	call	VERB
ejpam-6153	23	56	the	the	DET
ejpam-6153	23	57	addition	addition	NOUN
ejpam-6153	23	58	+	+	CCONJ
ejpam-6153	23	59	and	and	CCONJ
ejpam-6153	23	60	multiplication	multiplication	NOUN
ejpam-6153	23	61	·	·	PUNCT
ejpam-6153	23	62	satisfying	satisfy	VERB
ejpam-6153	23	63	the	the	DET
ejpam-6153	23	64	following	follow	VERB
ejpam-6153	23	65	conditions.37	conditions.37	NOUN
ejpam-6153	23	66	(	(	PUNCT
ejpam-6153	23	67	1	1	NUM
ejpam-6153	23	68	)	)	PUNCT
ejpam-6153	23	69	(	(	PUNCT
ejpam-6153	23	70	r,+	r,+	NUM
ejpam-6153	23	71	)	)	PUNCT
ejpam-6153	23	72	is	be	AUX
ejpam-6153	23	73	a	a	DET
ejpam-6153	23	74	commutative	commutative	ADJ
ejpam-6153	23	75	semigroup.38	semigroup.38	NOUN
ejpam-6153	23	76	(	(	PUNCT
ejpam-6153	23	77	2	2	NUM
ejpam-6153	23	78	)	)	PUNCT
ejpam-6153	23	79	(	(	PUNCT
ejpam-6153	23	80	r	r	NOUN
ejpam-6153	23	81	,	,	PUNCT
ejpam-6153	23	82	·	·	PUNCT
ejpam-6153	23	83	)	)	PUNCT
ejpam-6153	23	84	is	be	AUX
ejpam-6153	23	85	a	a	DET
ejpam-6153	23	86	semigroup.39	semigroup.39	NOUN
ejpam-6153	23	87	(	(	PUNCT
ejpam-6153	23	88	3	3	NUM
ejpam-6153	23	89	)	)	PUNCT
ejpam-6153	23	90	the	the	DET
ejpam-6153	23	91	multiplication	multiplication	NOUN
ejpam-6153	23	92	distributes	distribute	VERB
ejpam-6153	23	93	over	over	ADP
ejpam-6153	23	94	the	the	DET
ejpam-6153	23	95	addition	addition	NOUN
ejpam-6153	23	96	both	both	CCONJ
ejpam-6153	23	97	from	from	ADP
ejpam-6153	23	98	the	the	DET
ejpam-6153	23	99	left	left	NOUN
ejpam-6153	23	100	and	and	CCONJ
ejpam-6153	23	101	from	from	ADP
ejpam-6153	23	102	the40	the40	ADV
ejpam-6153	23	103	right	right	ADJ
ejpam-6153	23	104	,	,	PUNCT
ejpam-6153	23	105	that	that	ADV
ejpam-6153	23	106	is	is	ADV
ejpam-6153	23	107	,	,	PUNCT
ejpam-6153	23	108	(	(	PUNCT
ejpam-6153	23	109	a+	a+	X
ejpam-6153	23	110	b)c	b)c	X
ejpam-6153	23	111	=	=	NUM
ejpam-6153	23	112	ac+	ac+	PROPN
ejpam-6153	23	113	bc	bc	PROPN
ejpam-6153	23	114	and	and	CCONJ
ejpam-6153	23	115	c(a+	c(a+	PROPN
ejpam-6153	23	116	b	b	X
ejpam-6153	23	117	)	)	PUNCT
ejpam-6153	23	118	=	=	SYM
ejpam-6153	23	119	ca+	ca+	NOUN
ejpam-6153	23	120	cb	cb	NOUN
ejpam-6153	23	121	for	for	ADP
ejpam-6153	23	122	all	all	DET
ejpam-6153	23	123	a	a	DET
ejpam-6153	23	124	,	,	PUNCT
ejpam-6153	23	125	b	b	NOUN
ejpam-6153	23	126	,	,	PUNCT
ejpam-6153	23	127	c	c	PROPN
ejpam-6153	23	128	∈	∈	PROPN
ejpam-6153	23	129	r.41	r.41	ADP
ejpam-6153	23	130	a	a	DET
ejpam-6153	23	131	semiring	semiring	NOUN
ejpam-6153	23	132	r	r	NOUN
ejpam-6153	23	133	is	be	AUX
ejpam-6153	23	134	called	call	VERB
ejpam-6153	23	135	commutative	commutative	ADJ
ejpam-6153	23	136	if	if	SCONJ
ejpam-6153	23	137	ab	ab	PROPN
ejpam-6153	23	138	=	=	SYM
ejpam-6153	23	139	ba	ba	PROPN
ejpam-6153	23	140	for	for	ADP
ejpam-6153	23	141	all	all	DET
ejpam-6153	23	142	a	a	PRON
ejpam-6153	23	143	,	,	PUNCT
ejpam-6153	23	144	b	b	PROPN
ejpam-6153	23	145	∈	∈	PROPN
ejpam-6153	23	146	r.	r.	NOUN
ejpam-6153	23	147	an	an	DET
ejpam-6153	23	148	element	element	NOUN
ejpam-6153	23	149	0	0	NUM
ejpam-6153	23	150	of	of	ADP
ejpam-6153	23	151	a	a	DET
ejpam-6153	23	152	semiring42	semiring42	NOUN
ejpam-6153	23	153	r	r	NOUN
ejpam-6153	23	154	is	be	AUX
ejpam-6153	23	155	called	call	VERB
ejpam-6153	23	156	a	a	DET
ejpam-6153	23	157	zero	zero	NUM
ejpam-6153	23	158	of	of	ADP
ejpam-6153	23	159	s	s	PRON
ejpam-6153	23	160	if	if	SCONJ
ejpam-6153	23	161	0	0	NUM
ejpam-6153	24	1	+	+	NUM
ejpam-6153	24	2	x	x	SYM
ejpam-6153	24	3	=	=	SYM
ejpam-6153	24	4	x	x	X
ejpam-6153	24	5	and	and	CCONJ
ejpam-6153	24	6	x0	x0	PROPN
ejpam-6153	24	7	=	=	PUNCT
ejpam-6153	24	8	0x	0x	NOUN
ejpam-6153	24	9	=	=	SYM
ejpam-6153	24	10	0	0	NUM
ejpam-6153	24	11	for	for	ADP
ejpam-6153	24	12	all	all	DET
ejpam-6153	24	13	x	x	PROPN
ejpam-6153	24	14	∈	∈	PROPN
ejpam-6153	24	15	r.	r.	NOUN
ejpam-6153	24	16	a	a	DET
ejpam-6153	24	17	semiring	semiring	NOUN
ejpam-6153	24	18	which43	which43	NOUN
ejpam-6153	24	19	contains	contain	VERB
ejpam-6153	24	20	a	a	DET
ejpam-6153	24	21	zero	zero	NUM
ejpam-6153	24	22	is	be	AUX
ejpam-6153	24	23	called	call	VERB
ejpam-6153	24	24	a	a	DET
ejpam-6153	24	25	semiring	semiring	NOUN
ejpam-6153	24	26	with	with	ADP
ejpam-6153	24	27	zero	zero	NUM
ejpam-6153	24	28	.	.	PUNCT
ejpam-6153	25	1	a	a	DET
ejpam-6153	25	2	non	non	ADJ
ejpam-6153	25	3	-	-	ADJ
ejpam-6153	25	4	empty	empty	ADJ
ejpam-6153	25	5	subset	subset	NOUN
ejpam-6153	25	6	i	i	PRON
ejpam-6153	25	7	of	of	ADP
ejpam-6153	25	8	a	a	DET
ejpam-6153	25	9	semiring	semire	VERB
ejpam-6153	25	10	r	r	PROPN
ejpam-6153	25	11	is44	is44	PROPN
ejpam-6153	25	12	called	call	VERB
ejpam-6153	25	13	a	a	DET
ejpam-6153	25	14	left	left	ADJ
ejpam-6153	25	15	(	(	PUNCT
ejpam-6153	25	16	resp	resp	NOUN
ejpam-6153	25	17	.	.	PUNCT
ejpam-6153	26	1	right	right	ADJ
ejpam-6153	26	2	)	)	PUNCT
ejpam-6153	26	3	ideal	ideal	NOUN
ejpam-6153	26	4	of	of	ADP
ejpam-6153	26	5	r	r	NOUN
ejpam-6153	26	6	if	if	SCONJ
ejpam-6153	26	7	for	for	ADP
ejpam-6153	26	8	x	x	X
ejpam-6153	26	9	,	,	PUNCT
ejpam-6153	26	10	y	y	PROPN
ejpam-6153	26	11	∈	∈	PROPN
ejpam-6153	27	1	i	i	PRON
ejpam-6153	27	2	and	and	CCONJ
ejpam-6153	27	3	r	r	NOUN
ejpam-6153	27	4	∈	∈	NOUN
ejpam-6153	27	5	r	r	NOUN
ejpam-6153	27	6	imply	imply	VERB
ejpam-6153	27	7	that	that	SCONJ
ejpam-6153	27	8	x	x	X
ejpam-6153	28	1	+	+	CCONJ
ejpam-6153	28	2	y	y	PROPN
ejpam-6153	28	3	∈	∈	PROPN
ejpam-6153	28	4	i	i	PRON
ejpam-6153	28	5	and45	and45	VERB
ejpam-6153	28	6	rx	rx	VERB
ejpam-6153	28	7	∈	∈	PROPN
ejpam-6153	29	1	i	i	PRON
ejpam-6153	29	2	(	(	PUNCT
ejpam-6153	29	3	resp	resp	PROPN
ejpam-6153	29	4	.	.	PUNCT
ejpam-6153	30	1	xr	xr	PROPN
ejpam-6153	30	2	∈	∈	PROPN
ejpam-6153	31	1	i	i	PROPN
ejpam-6153	31	2	)	)	PUNCT
ejpam-6153	31	3	.	.	PUNCT
ejpam-6153	32	1	if	if	SCONJ
ejpam-6153	32	2	i	i	PRON
ejpam-6153	32	3	is	be	AUX
ejpam-6153	32	4	both	both	CCONJ
ejpam-6153	32	5	a	a	DET
ejpam-6153	32	6	left	left	ADJ
ejpam-6153	32	7	and	and	CCONJ
ejpam-6153	32	8	right	right	ADJ
ejpam-6153	32	9	ideal	ideal	NOUN
ejpam-6153	32	10	of	of	ADP
ejpam-6153	32	11	r	r	NOUN
ejpam-6153	32	12	,	,	PUNCT
ejpam-6153	32	13	we	we	PRON
ejpam-6153	32	14	say	say	VERB
ejpam-6153	32	15	i	i	PRON
ejpam-6153	32	16	is	be	AUX
ejpam-6153	32	17	a	a	DET
ejpam-6153	32	18	two	two	NUM
ejpam-6153	32	19	-	-	PUNCT
ejpam-6153	32	20	sided	sided	ADJ
ejpam-6153	32	21	ideal,46	ideal,46	NOUN
ejpam-6153	32	22	or	or	CCONJ
ejpam-6153	32	23	simply	simply	ADV
ejpam-6153	32	24	,	,	PUNCT
ejpam-6153	32	25	an	an	DET
ejpam-6153	32	26	ideal	ideal	NOUN
ejpam-6153	32	27	of	of	ADP
ejpam-6153	32	28	r.	r.	PROPN
ejpam-6153	32	29	for	for	ADP
ejpam-6153	32	30	x	x	PROPN
ejpam-6153	32	31	∈	∈	PROPN
ejpam-6153	32	32	r	r	NOUN
ejpam-6153	32	33	,	,	PUNCT
ejpam-6153	32	34	we	we	PRON
ejpam-6153	32	35	let	let	VERB
ejpam-6153	32	36	⟨x⟩	⟨x⟩	PUNCT
ejpam-6153	32	37	be	be	AUX
ejpam-6153	32	38	the	the	DET
ejpam-6153	32	39	smallest	small	ADJ
ejpam-6153	32	40	ideal	ideal	NOUN
ejpam-6153	32	41	of	of	ADP
ejpam-6153	32	42	r	r	NOUN
ejpam-6153	32	43	generated	generate	VERB
ejpam-6153	32	44	by	by	ADP
ejpam-6153	32	45	x.47	x.47	X
ejpam-6153	32	46	we	we	PRON
ejpam-6153	32	47	have	have	VERB
ejpam-6153	32	48	that	that	PRON
ejpam-6153	32	49	⟨x⟩	⟨x⟩	PUNCT
ejpam-6153	33	1	=	=	PRON
ejpam-6153	33	2	{	{	PUNCT
ejpam-6153	33	3	nx+	nx+	ADJ
ejpam-6153	33	4	ax+	ax+	PROPN
ejpam-6153	33	5	xb	xb	PROPN
ejpam-6153	34	1	|	|	ADV
ejpam-6153	34	2	n	n	CCONJ
ejpam-6153	34	3	∈	∈	PROPN
ejpam-6153	34	4	n0	n0	NOUN
ejpam-6153	34	5	and	and	CCONJ
ejpam-6153	34	6	a	a	DET
ejpam-6153	34	7	,	,	PUNCT
ejpam-6153	34	8	b	b	PROPN
ejpam-6153	34	9	∈	∈	PROPN
ejpam-6153	34	10	r}.48	r}.48	PROPN
ejpam-6153	34	11	let	let	VERB
ejpam-6153	34	12	r	r	PRON
ejpam-6153	34	13	be	be	AUX
ejpam-6153	34	14	a	a	DET
ejpam-6153	34	15	commutative	commutative	ADJ
ejpam-6153	34	16	semiring	semiring	NOUN
ejpam-6153	34	17	with	with	ADP
ejpam-6153	34	18	zero	zero	NUM
ejpam-6153	34	19	0	0	NUM
ejpam-6153	34	20	.	.	PUNCT
ejpam-6153	35	1	let	let	VERB
ejpam-6153	35	2	x	x	PUNCT
ejpam-6153	35	3	∈	∈	PROPN
ejpam-6153	35	4	r	r	NOUN
ejpam-6153	35	5	\	\	PUNCT
ejpam-6153	35	6	{	{	PUNCT
ejpam-6153	35	7	0	0	NUM
ejpam-6153	35	8	}	}	PUNCT
ejpam-6153	35	9	.	.	PUNCT
ejpam-6153	36	1	if	if	SCONJ
ejpam-6153	36	2	xy	xy	PROPN
ejpam-6153	36	3	=	=	NOUN
ejpam-6153	36	4	0	0	NUM
ejpam-6153	37	1	for	for	ADP
ejpam-6153	37	2	some49	some49	NOUN
ejpam-6153	37	3	y	y	PROPN
ejpam-6153	37	4	∈	∈	PROPN
ejpam-6153	37	5	r	r	NOUN
ejpam-6153	37	6	\	\	PUNCT
ejpam-6153	37	7	{	{	PUNCT
ejpam-6153	37	8	0	0	NUM
ejpam-6153	37	9	}	}	PUNCT
ejpam-6153	37	10	,	,	PUNCT
ejpam-6153	37	11	then	then	ADV
ejpam-6153	37	12	x	x	PUNCT
ejpam-6153	37	13	is	be	AUX
ejpam-6153	37	14	called	call	VERB
ejpam-6153	37	15	a	a	DET
ejpam-6153	37	16	zero	zero	NUM
ejpam-6153	37	17	divisor	divisor	NOUN
ejpam-6153	37	18	of	of	ADP
ejpam-6153	37	19	r.50	r.50	X
ejpam-6153	37	20	2.2	2.2	NUM
ejpam-6153	37	21	.	.	PUNCT
ejpam-6153	38	1	fuzzy	fuzzy	ADJ
ejpam-6153	38	2	sets51	sets51	NOUN
ejpam-6153	38	3	a	a	DET
ejpam-6153	38	4	fuzzy	fuzzy	ADJ
ejpam-6153	38	5	subset	subset	NOUN
ejpam-6153	38	6	of	of	ADP
ejpam-6153	38	7	a	a	DET
ejpam-6153	38	8	set	set	NOUN
ejpam-6153	38	9	s	s	PART
ejpam-6153	38	10	is	be	AUX
ejpam-6153	38	11	a	a	DET
ejpam-6153	38	12	function	function	NOUN
ejpam-6153	38	13	from	from	ADP
ejpam-6153	38	14	s	s	PRON
ejpam-6153	38	15	into	into	ADP
ejpam-6153	38	16	the	the	DET
ejpam-6153	38	17	closed	closed	ADJ
ejpam-6153	38	18	interval	interval	NOUN
ejpam-6153	39	1	[	[	X
ejpam-6153	39	2	0	0	NUM
ejpam-6153	39	3	,	,	PUNCT
ejpam-6153	39	4	1	1	NUM
ejpam-6153	39	5	]	]	PUNCT
ejpam-6153	39	6	.	.	PUNCT
ejpam-6153	40	1	let	let	VERB
ejpam-6153	40	2	f	f	PROPN
ejpam-6153	40	3	and52	and52	VERB
ejpam-6153	40	4	g	g	PROPN
ejpam-6153	40	5	be	be	AUX
ejpam-6153	40	6	any	any	DET
ejpam-6153	40	7	two	two	NUM
ejpam-6153	40	8	fuzzy	fuzzy	ADJ
ejpam-6153	40	9	subsets	subset	NOUN
ejpam-6153	40	10	of	of	ADP
ejpam-6153	40	11	a	a	DET
ejpam-6153	40	12	set	set	NOUN
ejpam-6153	40	13	s.53	s.53	NOUN
ejpam-6153	40	14	1	1	NUM
ejpam-6153	40	15	.	.	PUNCT
ejpam-6153	41	1	the	the	DET
ejpam-6153	41	2	intersection	intersection	NOUN
ejpam-6153	41	3	of	of	ADP
ejpam-6153	41	4	f	f	PROPN
ejpam-6153	41	5	and	and	CCONJ
ejpam-6153	41	6	g	g	PROPN
ejpam-6153	41	7	is	be	AUX
ejpam-6153	41	8	a	a	DET
ejpam-6153	41	9	fuzzy	fuzzy	ADJ
ejpam-6153	41	10	subset	subset	NOUN
ejpam-6153	41	11	f	f	PROPN
ejpam-6153	41	12	∩	∩	PROPN
ejpam-6153	41	13	g	g	PROPN
ejpam-6153	41	14	of	of	ADP
ejpam-6153	41	15	s	s	PRON
ejpam-6153	41	16	defined	define	VERB
ejpam-6153	41	17	for	for	ADP
ejpam-6153	41	18	all	all	DET
ejpam-6153	41	19	x	x	SYM
ejpam-6153	41	20	∈	∈	PROPN
ejpam-6153	41	21	s	s	PART
ejpam-6153	41	22	by54	by54	PROPN
ejpam-6153	41	23	(	(	PUNCT
ejpam-6153	41	24	f	f	PROPN
ejpam-6153	41	25	∩	∩	PROPN
ejpam-6153	41	26	g)(x	g)(x	PROPN
ejpam-6153	41	27	)	)	PUNCT
ejpam-6153	41	28	=	=	SYM
ejpam-6153	41	29	min{f(x	min{f(x	PROPN
ejpam-6153	41	30	)	)	PUNCT
ejpam-6153	41	31	,	,	PUNCT
ejpam-6153	41	32	g(x	g(x	NOUN
ejpam-6153	41	33	)	)	PUNCT
ejpam-6153	41	34	}	}	PUNCT
ejpam-6153	41	35	.	.	PUNCT
ejpam-6153	42	1	2	2	X
ejpam-6153	42	2	.	.	X
ejpam-6153	42	3	the	the	DET
ejpam-6153	42	4	union	union	NOUN
ejpam-6153	42	5	of	of	ADP
ejpam-6153	42	6	f	f	PROPN
ejpam-6153	42	7	and	and	CCONJ
ejpam-6153	42	8	g	g	PROPN
ejpam-6153	42	9	is	be	AUX
ejpam-6153	42	10	a	a	DET
ejpam-6153	42	11	fuzzy	fuzzy	ADJ
ejpam-6153	42	12	subset	subset	NOUN
ejpam-6153	42	13	f	f	PROPN
ejpam-6153	42	14	∪	∪	ADP
ejpam-6153	42	15	g	g	NOUN
ejpam-6153	42	16	of	of	ADP
ejpam-6153	42	17	s	s	PRON
ejpam-6153	42	18	defined	define	VERB
ejpam-6153	42	19	for	for	ADP
ejpam-6153	42	20	all	all	DET
ejpam-6153	42	21	x	x	PART
ejpam-6153	42	22	∈	∈	PROPN
ejpam-6153	42	23	s	s	PART
ejpam-6153	42	24	by55	by55	PROPN
ejpam-6153	42	25	(	(	PUNCT
ejpam-6153	42	26	f	f	PROPN
ejpam-6153	42	27	∪	∪	PROPN
ejpam-6153	42	28	g)(x	g)(x	PROPN
ejpam-6153	42	29	)	)	PUNCT
ejpam-6153	42	30	=	=	SYM
ejpam-6153	42	31	max{f(x	max{f(x	PROPN
ejpam-6153	42	32	)	)	PUNCT
ejpam-6153	42	33	,	,	PUNCT
ejpam-6153	42	34	g(x	g(x	NOUN
ejpam-6153	42	35	)	)	PUNCT
ejpam-6153	42	36	}	}	PUNCT
ejpam-6153	42	37	.	.	PUNCT
ejpam-6153	43	1	3	3	X
ejpam-6153	43	2	.	.	X
ejpam-6153	43	3	if	if	SCONJ
ejpam-6153	43	4	f(x	f(x	PROPN
ejpam-6153	43	5	)	)	PUNCT
ejpam-6153	43	6	≤	≤	PUNCT
ejpam-6153	43	7	g(x	g(x	NOUN
ejpam-6153	43	8	)	)	PUNCT
ejpam-6153	43	9	for	for	ADP
ejpam-6153	43	10	all	all	DET
ejpam-6153	43	11	x	x	SYM
ejpam-6153	43	12	∈	∈	PROPN
ejpam-6153	43	13	s	s	X
ejpam-6153	43	14	,	,	PUNCT
ejpam-6153	43	15	we	we	PRON
ejpam-6153	43	16	say	say	VERB
ejpam-6153	43	17	that	that	SCONJ
ejpam-6153	43	18	f	f	PROPN
ejpam-6153	43	19	is	be	AUX
ejpam-6153	43	20	a	a	DET
ejpam-6153	43	21	subset	subset	NOUN
ejpam-6153	43	22	of	of	ADP
ejpam-6153	43	23	g	g	NOUN
ejpam-6153	43	24	and	and	CCONJ
ejpam-6153	43	25	use	use	VERB
ejpam-6153	43	26	the	the	DET
ejpam-6153	43	27	notation56	notation56	NOUN
ejpam-6153	43	28	f	f	PROPN
ejpam-6153	43	29	⊆	⊆	PROPN
ejpam-6153	43	30	g.57	g.57	PROPN
ejpam-6153	43	31	r.	r.	PROPN
ejpam-6153	43	32	chinram	chinram	PROPN
ejpam-6153	43	33	,	,	PUNCT
ejpam-6153	43	34	s.	s.	PROPN
ejpam-6153	43	35	hangsawat	hangsawat	PROPN
ejpam-6153	43	36	/	/	SYM
ejpam-6153	43	37	eur	eur	PROPN
ejpam-6153	43	38	.	.	PUNCT
ejpam-6153	44	1	j.	j.	PROPN
ejpam-6153	44	2	pure	pure	PROPN
ejpam-6153	44	3	appl	appl	PROPN
ejpam-6153	44	4	.	.	PROPN
ejpam-6153	44	5	math	math	PROPN
ejpam-6153	44	6	,	,	PUNCT
ejpam-6153	44	7	18	18	NUM
ejpam-6153	44	8	(	(	PUNCT
ejpam-6153	44	9	3	3	NUM
ejpam-6153	44	10	)	)	PUNCT
ejpam-6153	44	11	(	(	PUNCT
ejpam-6153	44	12	2025	2025	NUM
ejpam-6153	44	13	)	)	PUNCT
ejpam-6153	44	14	,	,	PUNCT
ejpam-6153	44	15	6153	6153	NUM
ejpam-6153	44	16	3	3	NUM
ejpam-6153	44	17	of	of	ADP
ejpam-6153	44	18	7	7	NUM
ejpam-6153	44	19	the	the	DET
ejpam-6153	44	20	support	support	NOUN
ejpam-6153	44	21	of	of	ADP
ejpam-6153	44	22	a	a	DET
ejpam-6153	44	23	fuzzy	fuzzy	ADJ
ejpam-6153	44	24	subset	subset	NOUN
ejpam-6153	44	25	f	f	PROPN
ejpam-6153	44	26	of	of	ADP
ejpam-6153	44	27	a	a	DET
ejpam-6153	44	28	set	set	NOUN
ejpam-6153	44	29	s	s	PART
ejpam-6153	44	30	is	be	AUX
ejpam-6153	44	31	defined	define	VERB
ejpam-6153	44	32	by	by	ADP
ejpam-6153	44	33	supp(f	supp(f	PROPN
ejpam-6153	44	34	)	)	PUNCT
ejpam-6153	44	35	=	=	PRON
ejpam-6153	45	1	{	{	PUNCT
ejpam-6153	45	2	x	x	PUNCT
ejpam-6153	45	3	∈	∈	PROPN
ejpam-6153	45	4	s	s	VERB
ejpam-6153	45	5	|	|	ADV
ejpam-6153	45	6	f(x	f(x	NOUN
ejpam-6153	45	7	)	)	PUNCT
ejpam-6153	46	1	̸=	̸=	PROPN
ejpam-6153	46	2	0}.58	0}.58	NUM
ejpam-6153	46	3	the	the	DET
ejpam-6153	46	4	characteristic	characteristic	ADJ
ejpam-6153	46	5	mapping	mapping	NOUN
ejpam-6153	46	6	of	of	ADP
ejpam-6153	46	7	a	a	DET
ejpam-6153	46	8	subset	subset	NOUN
ejpam-6153	46	9	a	a	PRON
ejpam-6153	46	10	of	of	ADP
ejpam-6153	46	11	a	a	DET
ejpam-6153	46	12	set	set	NOUN
ejpam-6153	46	13	s	s	PART
ejpam-6153	46	14	is	be	AUX
ejpam-6153	46	15	a	a	DET
ejpam-6153	46	16	fuzzy	fuzzy	ADJ
ejpam-6153	46	17	subset	subset	NOUN
ejpam-6153	46	18	of	of	ADP
ejpam-6153	46	19	s	s	PRON
ejpam-6153	46	20	defined	define	VERB
ejpam-6153	46	21	by59	by59	PROPN
ejpam-6153	46	22	ca(x	ca(x	NOUN
ejpam-6153	46	23	)	)	PUNCT
ejpam-6153	46	24	=	=	PRON
ejpam-6153	46	25	{	{	PUNCT
ejpam-6153	46	26	1	1	NUM
ejpam-6153	46	27	if	if	SCONJ
ejpam-6153	46	28	x	x	PROPN
ejpam-6153	46	29	∈	∈	PROPN
ejpam-6153	46	30	a	a	PRON
ejpam-6153	46	31	,	,	PUNCT
ejpam-6153	46	32	0	0	PUNCT
ejpam-6153	46	33	if	if	SCONJ
ejpam-6153	46	34	x	x	X
ejpam-6153	46	35	/∈	/∈	VERB
ejpam-6153	46	36	a.	a.	NOUN
ejpam-6153	46	37	for	for	ADP
ejpam-6153	46	38	any	any	DET
ejpam-6153	46	39	two	two	NUM
ejpam-6153	46	40	subsets	subset	NOUN
ejpam-6153	46	41	a	a	PRON
ejpam-6153	46	42	and	and	CCONJ
ejpam-6153	46	43	b	b	NOUN
ejpam-6153	46	44	of	of	ADP
ejpam-6153	46	45	s	s	PROPN
ejpam-6153	46	46	,	,	PUNCT
ejpam-6153	46	47	we	we	PRON
ejpam-6153	46	48	have	have	VERB
ejpam-6153	46	49	that	that	DET
ejpam-6153	46	50	ca∩b	ca∩b	PROPN
ejpam-6153	47	1	=	=	PUNCT
ejpam-6153	47	2	ca	can	AUX
ejpam-6153	47	3	∩cb	∩cb	VERB
ejpam-6153	47	4	and	and	CCONJ
ejpam-6153	47	5	ca∪b	ca∪b	NOUN
ejpam-6153	47	6	=	=	SYM
ejpam-6153	47	7	ca	can	AUX
ejpam-6153	47	8	∪cb.60	∪cb.60	PROPN
ejpam-6153	47	9	a	a	DET
ejpam-6153	47	10	fuzzy	fuzzy	ADJ
ejpam-6153	47	11	set	set	NOUN
ejpam-6153	47	12	f	f	PROPN
ejpam-6153	47	13	of	of	ADP
ejpam-6153	47	14	a	a	DET
ejpam-6153	47	15	semiring	semiring	NOUN
ejpam-6153	47	16	r	r	NOUN
ejpam-6153	47	17	is	be	AUX
ejpam-6153	47	18	called	call	VERB
ejpam-6153	47	19	a	a	DET
ejpam-6153	47	20	fuzzy	fuzzy	ADJ
ejpam-6153	47	21	ideal	ideal	NOUN
ejpam-6153	47	22	of	of	ADP
ejpam-6153	47	23	r	r	NOUN
ejpam-6153	47	24	if	if	SCONJ
ejpam-6153	47	25	for	for	ADP
ejpam-6153	47	26	all	all	DET
ejpam-6153	47	27	x	x	NOUN
ejpam-6153	47	28	,	,	PUNCT
ejpam-6153	47	29	y	y	PROPN
ejpam-6153	47	30	∈	∈	PROPN
ejpam-6153	47	31	r	r	NOUN
ejpam-6153	47	32	,	,	PUNCT
ejpam-6153	47	33	we	we	PRON
ejpam-6153	47	34	have61	have61	ADJ
ejpam-6153	47	35	(	(	PUNCT
ejpam-6153	47	36	1	1	X
ejpam-6153	47	37	)	)	PUNCT
ejpam-6153	47	38	f(x+	f(x+	NOUN
ejpam-6153	47	39	y	y	NUM
ejpam-6153	47	40	)	)	PUNCT
ejpam-6153	47	41	≥	≥	NOUN
ejpam-6153	47	42	min{f(x	min{f(x	NOUN
ejpam-6153	47	43	)	)	PUNCT
ejpam-6153	47	44	,	,	PUNCT
ejpam-6153	47	45	f(y)},62	f(y)},62	X
ejpam-6153	47	46	(	(	PUNCT
ejpam-6153	47	47	2	2	NUM
ejpam-6153	47	48	)	)	PUNCT
ejpam-6153	47	49	f(xy	f(xy	NUM
ejpam-6153	47	50	)	)	PUNCT
ejpam-6153	47	51	≥	≥	NOUN
ejpam-6153	47	52	max{f(x	max{f(x	PROPN
ejpam-6153	47	53	)	)	PUNCT
ejpam-6153	47	54	,	,	PUNCT
ejpam-6153	47	55	f(y)}.63	f(y)}.63	PROPN
ejpam-6153	47	56	proposition	proposition	NOUN
ejpam-6153	47	57	1	1	NUM
ejpam-6153	47	58	.	.	PUNCT
ejpam-6153	48	1	a	a	DET
ejpam-6153	48	2	nonempty	nonempty	NOUN
ejpam-6153	48	3	subset	subset	VERB
ejpam-6153	48	4	a	a	PRON
ejpam-6153	48	5	of	of	ADP
ejpam-6153	48	6	a	a	DET
ejpam-6153	48	7	semiring	semire	VERB
ejpam-6153	48	8	r	r	NOUN
ejpam-6153	48	9	is	be	AUX
ejpam-6153	48	10	an	an	DET
ejpam-6153	48	11	ideal	ideal	NOUN
ejpam-6153	48	12	of	of	ADP
ejpam-6153	48	13	s	s	PRON
ejpam-6153	48	14	if	if	SCONJ
ejpam-6153	48	15	and	and	CCONJ
ejpam-6153	48	16	only	only	ADV
ejpam-6153	48	17	if	if	SCONJ
ejpam-6153	48	18	ca64	ca64	PROPN
ejpam-6153	48	19	is	be	AUX
ejpam-6153	48	20	a	a	DET
ejpam-6153	48	21	fuzzy	fuzzy	ADJ
ejpam-6153	48	22	ideal	ideal	NOUN
ejpam-6153	48	23	of	of	ADP
ejpam-6153	48	24	r.65	r.65	ADP
ejpam-6153	48	25	proposition	proposition	NOUN
ejpam-6153	48	26	2	2	NUM
ejpam-6153	48	27	.	.	PUNCT
ejpam-6153	49	1	let	let	VERB
ejpam-6153	49	2	f	f	PRON
ejpam-6153	49	3	be	be	AUX
ejpam-6153	49	4	a	a	DET
ejpam-6153	49	5	nonzero	nonzero	ADJ
ejpam-6153	49	6	fuzzy	fuzzy	ADJ
ejpam-6153	49	7	ideal	ideal	NOUN
ejpam-6153	49	8	of	of	ADP
ejpam-6153	49	9	a	a	DET
ejpam-6153	49	10	semiring	semire	VERB
ejpam-6153	49	11	r.	r.	PROPN
ejpam-6153	49	12	then	then	ADV
ejpam-6153	49	13	supp(f	supp(f	PROPN
ejpam-6153	49	14	)	)	PUNCT
ejpam-6153	49	15	is	be	AUX
ejpam-6153	49	16	an	an	DET
ejpam-6153	49	17	ideal66	ideal66	NOUN
ejpam-6153	49	18	of	of	ADP
ejpam-6153	49	19	r.67	r.67	NOUN
ejpam-6153	49	20	3	3	NUM
ejpam-6153	49	21	.	.	PUNCT
ejpam-6153	49	22	main	main	ADJ
ejpam-6153	49	23	results68	results68	NOUN
ejpam-6153	49	24	throughout	throughout	ADP
ejpam-6153	49	25	of	of	ADP
ejpam-6153	49	26	this	this	DET
ejpam-6153	49	27	section	section	NOUN
ejpam-6153	49	28	,	,	PUNCT
ejpam-6153	49	29	we	we	PRON
ejpam-6153	49	30	let	let	VERB
ejpam-6153	49	31	r	r	PRON
ejpam-6153	49	32	be	be	AUX
ejpam-6153	49	33	a	a	DET
ejpam-6153	49	34	semiring	semiring	NOUN
ejpam-6153	49	35	with	with	ADP
ejpam-6153	49	36	zero	zero	NUM
ejpam-6153	49	37	0	0	NUM
ejpam-6153	49	38	.	.	PUNCT
ejpam-6153	50	1	first	first	ADV
ejpam-6153	50	2	,	,	PUNCT
ejpam-6153	50	3	we	we	PRON
ejpam-6153	50	4	recall	recall	VERB
ejpam-6153	50	5	the69	the69	ADV
ejpam-6153	50	6	definition	definition	NOUN
ejpam-6153	50	7	of	of	ADP
ejpam-6153	50	8	essential	essential	ADJ
ejpam-6153	50	9	ideals	ideal	NOUN
ejpam-6153	50	10	of	of	ADP
ejpam-6153	50	11	r	r	NOUN
ejpam-6153	50	12	as	as	SCONJ
ejpam-6153	50	13	follows:70	follows:70	X
ejpam-6153	50	14	definition	definition	NOUN
ejpam-6153	50	15	1	1	NUM
ejpam-6153	50	16	.	.	PUNCT
ejpam-6153	51	1	[	[	X
ejpam-6153	51	2	3	3	X
ejpam-6153	51	3	]	]	X
ejpam-6153	51	4	an	an	DET
ejpam-6153	51	5	ideal	ideal	NOUN
ejpam-6153	51	6	i	i	PRON
ejpam-6153	51	7	of	of	ADP
ejpam-6153	51	8	r	r	NOUN
ejpam-6153	51	9	is	be	AUX
ejpam-6153	51	10	called	call	VERB
ejpam-6153	51	11	an	an	DET
ejpam-6153	51	12	essential	essential	ADJ
ejpam-6153	51	13	ideal	ideal	NOUN
ejpam-6153	51	14	of	of	ADP
ejpam-6153	51	15	r	r	NOUN
ejpam-6153	51	16	if	if	SCONJ
ejpam-6153	51	17	i	i	PRON
ejpam-6153	51	18	∩k	∩k	VERB
ejpam-6153	51	19	̸=	̸=	PROPN
ejpam-6153	51	20	{	{	PUNCT
ejpam-6153	51	21	0	0	NUM
ejpam-6153	51	22	}	}	PUNCT
ejpam-6153	51	23	for	for	ADP
ejpam-6153	51	24	every71	every71	ADJ
ejpam-6153	51	25	nonzero	nonzero	PROPN
ejpam-6153	51	26	ideal	ideal	PROPN
ejpam-6153	51	27	k	k	PROPN
ejpam-6153	51	28	of	of	ADP
ejpam-6153	51	29	r.72	r.72	NOUN
ejpam-6153	51	30	example	example	NOUN
ejpam-6153	51	31	1	1	X
ejpam-6153	51	32	.	.	PUNCT
ejpam-6153	52	1	we	we	PRON
ejpam-6153	52	2	consider	consider	VERB
ejpam-6153	52	3	the	the	DET
ejpam-6153	52	4	semiring	semire	VERB
ejpam-6153	52	5	z6	z6	PROPN
ejpam-6153	52	6	under	under	ADP
ejpam-6153	52	7	the	the	DET
ejpam-6153	52	8	usual	usual	ADJ
ejpam-6153	52	9	addition	addition	NOUN
ejpam-6153	52	10	and	and	CCONJ
ejpam-6153	52	11	multiplication	multiplication	NOUN
ejpam-6153	52	12	of73	of73	PROPN
ejpam-6153	52	13	integers	integer	NOUN
ejpam-6153	52	14	modulo	modulo	VERB
ejpam-6153	52	15	6	6	NUM
ejpam-6153	52	16	.	.	PUNCT
ejpam-6153	53	1	let	let	VERB
ejpam-6153	53	2	i	i	PRON
ejpam-6153	53	3	=	=	PUNCT
ejpam-6153	53	4	⟨2⟩	⟨2⟩	PROPN
ejpam-6153	53	5	=	=	PUNCT
ejpam-6153	53	6	{	{	PUNCT
ejpam-6153	53	7	0	0	NUM
ejpam-6153	53	8	,	,	PUNCT
ejpam-6153	53	9	2	2	NUM
ejpam-6153	53	10	,	,	PUNCT
ejpam-6153	53	11	4	4	NUM
ejpam-6153	53	12	}	}	PUNCT
ejpam-6153	53	13	and	and	CCONJ
ejpam-6153	53	14	j	j	PROPN
ejpam-6153	53	15	=	=	SYM
ejpam-6153	53	16	⟨3⟩	⟨3⟩	PROPN
ejpam-6153	53	17	=	=	SYM
ejpam-6153	53	18	{	{	PUNCT
ejpam-6153	53	19	0	0	NUM
ejpam-6153	53	20	,	,	PUNCT
ejpam-6153	53	21	3	3	NUM
ejpam-6153	53	22	}	}	PUNCT
ejpam-6153	53	23	.	.	PUNCT
ejpam-6153	54	1	we	we	PRON
ejpam-6153	54	2	see	see	VERB
ejpam-6153	54	3	that	that	SCONJ
ejpam-6153	54	4	i	i	PRON
ejpam-6153	54	5	∩	∩	VERB
ejpam-6153	54	6	j	j	PROPN
ejpam-6153	54	7	=	=	PRON
ejpam-6153	54	8	{	{	PUNCT
ejpam-6153	54	9	0}.74	0}.74	NUM
ejpam-6153	54	10	hence	hence	ADV
ejpam-6153	54	11	i	i	PROPN
ejpam-6153	54	12	and	and	CCONJ
ejpam-6153	54	13	j	j	PROPN
ejpam-6153	54	14	are	be	AUX
ejpam-6153	54	15	not	not	PART
ejpam-6153	54	16	essential	essential	ADJ
ejpam-6153	54	17	ideals	ideal	NOUN
ejpam-6153	54	18	of	of	ADP
ejpam-6153	54	19	z6.75	z6.75	PROPN
ejpam-6153	54	20	proposition	proposition	NOUN
ejpam-6153	54	21	3	3	X
ejpam-6153	54	22	.	.	PUNCT
ejpam-6153	55	1	if	if	SCONJ
ejpam-6153	55	2	r	r	NOUN
ejpam-6153	55	3	is	be	AUX
ejpam-6153	55	4	commutative	commutative	ADJ
ejpam-6153	55	5	and	and	CCONJ
ejpam-6153	55	6	i	i	PRON
ejpam-6153	55	7	is	be	AUX
ejpam-6153	55	8	an	an	DET
ejpam-6153	55	9	ideal	ideal	NOUN
ejpam-6153	55	10	containing	contain	VERB
ejpam-6153	55	11	a	a	DET
ejpam-6153	55	12	non	non	ADJ
ejpam-6153	55	13	zero	zero	NUM
ejpam-6153	55	14	divisor	divisor	NOUN
ejpam-6153	55	15	of76	of76	PROPN
ejpam-6153	55	16	r	r	NOUN
ejpam-6153	55	17	,	,	PUNCT
ejpam-6153	55	18	then	then	ADV
ejpam-6153	55	19	i	i	PRON
ejpam-6153	55	20	is	be	AUX
ejpam-6153	55	21	an	an	DET
ejpam-6153	55	22	essential	essential	ADJ
ejpam-6153	55	23	ideal	ideal	NOUN
ejpam-6153	55	24	of	of	ADP
ejpam-6153	55	25	r.77	r.77	PROPN
ejpam-6153	55	26	proof	proof	NOUN
ejpam-6153	55	27	.	.	PUNCT
ejpam-6153	56	1	suppose	suppose	VERB
ejpam-6153	56	2	that	that	SCONJ
ejpam-6153	56	3	i	i	PRON
ejpam-6153	56	4	is	be	AUX
ejpam-6153	56	5	not	not	PART
ejpam-6153	56	6	an	an	DET
ejpam-6153	56	7	essential	essential	ADJ
ejpam-6153	56	8	ideal	ideal	NOUN
ejpam-6153	56	9	of	of	ADP
ejpam-6153	56	10	r.	r.	PROPN
ejpam-6153	56	11	then	then	ADV
ejpam-6153	56	12	there	there	PRON
ejpam-6153	56	13	exists	exist	VERB
ejpam-6153	56	14	a	a	DET
ejpam-6153	56	15	nonzero	nonzero	ADJ
ejpam-6153	56	16	ideal78	ideal78	NOUN
ejpam-6153	56	17	k	k	PROPN
ejpam-6153	56	18	of	of	ADP
ejpam-6153	56	19	r	r	NOUN
ejpam-6153	56	20	such	such	ADJ
ejpam-6153	56	21	that	that	SCONJ
ejpam-6153	56	22	i	i	PRON
ejpam-6153	56	23	∩k	∩k	VERB
ejpam-6153	56	24	=	=	PUNCT
ejpam-6153	56	25	{	{	PUNCT
ejpam-6153	56	26	0	0	NUM
ejpam-6153	56	27	}	}	PUNCT
ejpam-6153	56	28	.	.	PUNCT
ejpam-6153	57	1	let	let	VERB
ejpam-6153	57	2	x	x	PRON
ejpam-6153	57	3	is	be	AUX
ejpam-6153	57	4	be	be	AUX
ejpam-6153	57	5	a	a	DET
ejpam-6153	57	6	non	non	ADJ
ejpam-6153	57	7	zero	zero	NUM
ejpam-6153	57	8	divisor	divisor	NOUN
ejpam-6153	57	9	of	of	ADP
ejpam-6153	57	10	i	i	PRON
ejpam-6153	57	11	and	and	CCONJ
ejpam-6153	57	12	y	y	PROPN
ejpam-6153	57	13	∈	∈	PROPN
ejpam-6153	58	1	k	k	X
ejpam-6153	58	2	\	\	PROPN
ejpam-6153	58	3	{	{	PUNCT
ejpam-6153	58	4	0	0	NUM
ejpam-6153	58	5	}	}	PUNCT
ejpam-6153	58	6	.	.	PUNCT
ejpam-6153	59	1	then79	then79	NOUN
ejpam-6153	59	2	xy	xy	PROPN
ejpam-6153	60	1	̸=	̸=	PROPN
ejpam-6153	60	2	0	0	NUM
ejpam-6153	61	1	and	and	CCONJ
ejpam-6153	61	2	xy	xy	PROPN
ejpam-6153	61	3	∈	∈	PROPN
ejpam-6153	61	4	i	i	PRON
ejpam-6153	61	5	∩k	∩k	PROPN
ejpam-6153	61	6	,	,	PUNCT
ejpam-6153	61	7	this	this	PRON
ejpam-6153	61	8	is	be	AUX
ejpam-6153	61	9	a	a	DET
ejpam-6153	61	10	contradiction	contradiction	NOUN
ejpam-6153	61	11	.	.	PUNCT
ejpam-6153	62	1	then	then	ADV
ejpam-6153	62	2	i	i	PRON
ejpam-6153	62	3	is	be	AUX
ejpam-6153	62	4	an	an	DET
ejpam-6153	62	5	essential	essential	ADJ
ejpam-6153	62	6	ideal	ideal	NOUN
ejpam-6153	62	7	of	of	ADP
ejpam-6153	62	8	r.80	r.80	NOUN
ejpam-6153	62	9	example	example	NOUN
ejpam-6153	63	1	2	2	X
ejpam-6153	63	2	.	.	X
ejpam-6153	63	3	we	we	PRON
ejpam-6153	63	4	have	have	VERB
ejpam-6153	63	5	that	that	SCONJ
ejpam-6153	63	6	the	the	DET
ejpam-6153	63	7	semiring	semire	VERB
ejpam-6153	63	8	n0	n0	NUM
ejpam-6153	63	9	:	:	PUNCT
ejpam-6153	63	10	=	=	NOUN
ejpam-6153	63	11	n	n	CCONJ
ejpam-6153	63	12	∪	∪	X
ejpam-6153	63	13	{	{	PUNCT
ejpam-6153	63	14	0	0	NUM
ejpam-6153	63	15	}	}	PUNCT
ejpam-6153	63	16	under	under	ADP
ejpam-6153	63	17	the	the	DET
ejpam-6153	63	18	usual	usual	ADJ
ejpam-6153	63	19	addition	addition	NOUN
ejpam-6153	63	20	and81	and81	PROPN
ejpam-6153	63	21	multiplication	multiplication	NOUN
ejpam-6153	63	22	of	of	ADP
ejpam-6153	63	23	integers	integer	NOUN
ejpam-6153	63	24	has	have	VERB
ejpam-6153	63	25	no	no	DET
ejpam-6153	63	26	a	a	DET
ejpam-6153	63	27	zero	zero	NUM
ejpam-6153	63	28	divisor	divisor	NOUN
ejpam-6153	63	29	.	.	PUNCT
ejpam-6153	64	1	hence	hence	ADV
ejpam-6153	64	2	,	,	PUNCT
ejpam-6153	64	3	by	by	ADP
ejpam-6153	64	4	proposition	proposition	NOUN
ejpam-6153	64	5	3	3	NUM
ejpam-6153	64	6	,	,	PUNCT
ejpam-6153	64	7	every	every	DET
ejpam-6153	64	8	nonzero82	nonzero82	NOUN
ejpam-6153	64	9	ideal	ideal	NOUN
ejpam-6153	64	10	of	of	ADP
ejpam-6153	64	11	n0	n0	PROPN
ejpam-6153	64	12	is	be	AUX
ejpam-6153	64	13	essential.83	essential.83	ADV
ejpam-6153	64	14	the	the	DET
ejpam-6153	64	15	following	follow	VERB
ejpam-6153	64	16	corollary	corollary	NOUN
ejpam-6153	64	17	follows	follow	VERB
ejpam-6153	64	18	from	from	ADP
ejpam-6153	64	19	proposition	proposition	NOUN
ejpam-6153	64	20	3.84	3.84	NUM
ejpam-6153	64	21	corollary	corollary	ADJ
ejpam-6153	64	22	1	1	NUM
ejpam-6153	64	23	.	.	PUNCT
ejpam-6153	65	1	if	if	SCONJ
ejpam-6153	65	2	r	r	NOUN
ejpam-6153	65	3	is	be	AUX
ejpam-6153	65	4	commutative	commutative	ADJ
ejpam-6153	65	5	and	and	CCONJ
ejpam-6153	65	6	i	i	PRON
ejpam-6153	65	7	is	be	AUX
ejpam-6153	65	8	not	not	PART
ejpam-6153	65	9	an	an	DET
ejpam-6153	65	10	essential	essential	ADJ
ejpam-6153	65	11	ideal	ideal	NOUN
ejpam-6153	65	12	of	of	ADP
ejpam-6153	65	13	r	r	NOUN
ejpam-6153	65	14	,	,	PUNCT
ejpam-6153	65	15	then	then	ADV
ejpam-6153	65	16	every	every	DET
ejpam-6153	65	17	nonzero85	nonzero85	ADJ
ejpam-6153	65	18	element	element	NOUN
ejpam-6153	65	19	in	in	ADP
ejpam-6153	65	20	i	i	PRON
ejpam-6153	65	21	is	be	AUX
ejpam-6153	65	22	a	a	DET
ejpam-6153	65	23	zero	zero	NUM
ejpam-6153	65	24	divisor	divisor	NOUN
ejpam-6153	65	25	of	of	ADP
ejpam-6153	65	26	r.86	r.86	PROPN
ejpam-6153	65	27	r.	r.	PROPN
ejpam-6153	65	28	chinram	chinram	PROPN
ejpam-6153	65	29	,	,	PUNCT
ejpam-6153	65	30	s.	s.	PROPN
ejpam-6153	65	31	hangsawat	hangsawat	PROPN
ejpam-6153	65	32	/	/	SYM
ejpam-6153	65	33	eur	eur	PROPN
ejpam-6153	65	34	.	.	PUNCT
ejpam-6153	66	1	j.	j.	PROPN
ejpam-6153	66	2	pure	pure	PROPN
ejpam-6153	66	3	appl	appl	PROPN
ejpam-6153	66	4	.	.	PROPN
ejpam-6153	66	5	math	math	PROPN
ejpam-6153	66	6	,	,	PUNCT
ejpam-6153	66	7	18	18	NUM
ejpam-6153	66	8	(	(	PUNCT
ejpam-6153	66	9	3	3	NUM
ejpam-6153	66	10	)	)	PUNCT
ejpam-6153	66	11	(	(	PUNCT
ejpam-6153	66	12	2025	2025	NUM
ejpam-6153	66	13	)	)	PUNCT
ejpam-6153	66	14	,	,	PUNCT
ejpam-6153	66	15	6153	6153	NUM
ejpam-6153	66	16	4	4	NUM
ejpam-6153	66	17	of	of	ADP
ejpam-6153	66	18	7	7	NUM
ejpam-6153	66	19	however	however	ADV
ejpam-6153	66	20	,	,	PUNCT
ejpam-6153	66	21	the	the	DET
ejpam-6153	66	22	converse	converse	NOUN
ejpam-6153	66	23	of	of	ADP
ejpam-6153	66	24	corollary	corollary	ADJ
ejpam-6153	66	25	1	1	NUM
ejpam-6153	66	26	is	be	AUX
ejpam-6153	66	27	not	not	PART
ejpam-6153	66	28	generally	generally	ADV
ejpam-6153	66	29	true.87	true.87	VERB
ejpam-6153	66	30	example	example	NOUN
ejpam-6153	67	1	3	3	X
ejpam-6153	67	2	.	.	X
ejpam-6153	67	3	we	we	PRON
ejpam-6153	67	4	consider	consider	VERB
ejpam-6153	67	5	the	the	DET
ejpam-6153	67	6	semiring	semire	VERB
ejpam-6153	67	7	z4	z4	PROPN
ejpam-6153	67	8	under	under	ADP
ejpam-6153	67	9	the	the	DET
ejpam-6153	67	10	usual	usual	ADJ
ejpam-6153	67	11	addition	addition	NOUN
ejpam-6153	67	12	and	and	CCONJ
ejpam-6153	67	13	multiplication	multiplication	NOUN
ejpam-6153	67	14	of88	of88	PROPN
ejpam-6153	67	15	integers	integer	NOUN
ejpam-6153	67	16	modulo	modulo	VERB
ejpam-6153	67	17	4	4	NUM
ejpam-6153	67	18	.	.	PUNCT
ejpam-6153	67	19	let	let	VERB
ejpam-6153	67	20	i	i	PRON
ejpam-6153	67	21	=	=	PUNCT
ejpam-6153	67	22	⟨2⟩	⟨2⟩	PROPN
ejpam-6153	67	23	=	=	PUNCT
ejpam-6153	67	24	{	{	PUNCT
ejpam-6153	67	25	0	0	NUM
ejpam-6153	67	26	,	,	PUNCT
ejpam-6153	67	27	2	2	NUM
ejpam-6153	67	28	}	}	PUNCT
ejpam-6153	67	29	.	.	PUNCT
ejpam-6153	68	1	we	we	PRON
ejpam-6153	68	2	see	see	VERB
ejpam-6153	68	3	that	that	SCONJ
ejpam-6153	68	4	i	i	PRON
ejpam-6153	68	5	is	be	AUX
ejpam-6153	68	6	an	an	DET
ejpam-6153	68	7	essential	essential	ADJ
ejpam-6153	68	8	ideal	ideal	NOUN
ejpam-6153	68	9	of	of	ADP
ejpam-6153	68	10	z4	z4	PROPN
ejpam-6153	68	11	and89	and89	PROPN
ejpam-6153	68	12	every	every	DET
ejpam-6153	68	13	nonzero	nonzero	PROPN
ejpam-6153	68	14	element	element	NOUN
ejpam-6153	68	15	in	in	ADP
ejpam-6153	68	16	i	i	PRON
ejpam-6153	68	17	is	be	AUX
ejpam-6153	68	18	a	a	DET
ejpam-6153	68	19	zero	zero	NUM
ejpam-6153	68	20	divisor	divisor	NOUN
ejpam-6153	68	21	of	of	ADP
ejpam-6153	68	22	r	r	NOUN
ejpam-6153	68	23	because	because	SCONJ
ejpam-6153	68	24	2	2	NUM
ejpam-6153	68	25	·	·	SYM
ejpam-6153	68	26	2	2	NUM
ejpam-6153	68	27	=	=	SYM
ejpam-6153	68	28	0.90	0.90	NUM
ejpam-6153	68	29	the	the	DET
ejpam-6153	68	30	two	two	NUM
ejpam-6153	68	31	following	follow	VERB
ejpam-6153	68	32	propositions	proposition	NOUN
ejpam-6153	68	33	are	be	AUX
ejpam-6153	68	34	some	some	DET
ejpam-6153	68	35	properties	property	NOUN
ejpam-6153	68	36	of	of	ADP
ejpam-6153	68	37	essential	essential	ADJ
ejpam-6153	68	38	ideals	ideal	NOUN
ejpam-6153	68	39	of	of	ADP
ejpam-6153	68	40	r.91	r.91	X
ejpam-6153	68	41	proposition	proposition	NOUN
ejpam-6153	68	42	4	4	X
ejpam-6153	68	43	.	.	PUNCT
ejpam-6153	69	1	let	let	VERB
ejpam-6153	69	2	i	i	PRON
ejpam-6153	69	3	be	be	AUX
ejpam-6153	69	4	an	an	DET
ejpam-6153	69	5	essential	essential	ADJ
ejpam-6153	69	6	ideal	ideal	NOUN
ejpam-6153	69	7	of	of	ADP
ejpam-6153	69	8	r.	r.	PROPN
ejpam-6153	69	9	if	if	SCONJ
ejpam-6153	69	10	j	j	PROPN
ejpam-6153	69	11	is	be	AUX
ejpam-6153	69	12	an	an	DET
ejpam-6153	69	13	ideal	ideal	NOUN
ejpam-6153	69	14	of	of	ADP
ejpam-6153	69	15	r	r	NOUN
ejpam-6153	69	16	containing	contain	VERB
ejpam-6153	69	17	i	i	PRON
ejpam-6153	69	18	,	,	PUNCT
ejpam-6153	69	19	then92	then92	PROPN
ejpam-6153	69	20	j	j	PROPN
ejpam-6153	69	21	is	be	AUX
ejpam-6153	69	22	also	also	ADV
ejpam-6153	69	23	an	an	DET
ejpam-6153	69	24	essential	essential	ADJ
ejpam-6153	69	25	ideal	ideal	NOUN
ejpam-6153	69	26	of	of	ADP
ejpam-6153	69	27	r.93	r.93	NOUN
ejpam-6153	69	28	proof	proof	NOUN
ejpam-6153	69	29	.	.	PUNCT
ejpam-6153	70	1	assume	assume	VERB
ejpam-6153	70	2	that	that	SCONJ
ejpam-6153	70	3	i	i	PRON
ejpam-6153	70	4	is	be	AUX
ejpam-6153	70	5	an	an	DET
ejpam-6153	70	6	essential	essential	ADJ
ejpam-6153	70	7	ideal	ideal	NOUN
ejpam-6153	70	8	of	of	ADP
ejpam-6153	70	9	r	r	NOUN
ejpam-6153	70	10	and	and	CCONJ
ejpam-6153	70	11	j	j	PROPN
ejpam-6153	70	12	is	be	AUX
ejpam-6153	70	13	an	an	DET
ejpam-6153	70	14	ideal	ideal	NOUN
ejpam-6153	70	15	of	of	ADP
ejpam-6153	70	16	r	r	NOUN
ejpam-6153	70	17	such	such	ADJ
ejpam-6153	70	18	that	that	SCONJ
ejpam-6153	70	19	i	i	PRON
ejpam-6153	70	20	⊆	⊆	NUM
ejpam-6153	70	21	j	j	PROPN
ejpam-6153	70	22	.94	.94	NUM
ejpam-6153	70	23	let	let	VERB
ejpam-6153	70	24	k	k	PROPN
ejpam-6153	70	25	be	be	AUX
ejpam-6153	70	26	any	any	DET
ejpam-6153	70	27	nonzero	nonzero	NOUN
ejpam-6153	70	28	ideal	ideal	NOUN
ejpam-6153	70	29	of	of	ADP
ejpam-6153	70	30	r.	r.	PROPN
ejpam-6153	70	31	thus	thus	ADV
ejpam-6153	70	32	i	i	PRON
ejpam-6153	70	33	∩k	∩k	VERB
ejpam-6153	70	34	̸=	̸=	PROPN
ejpam-6153	70	35	{	{	PUNCT
ejpam-6153	70	36	0	0	NUM
ejpam-6153	70	37	}	}	PUNCT
ejpam-6153	70	38	.	.	PUNCT
ejpam-6153	71	1	since	since	SCONJ
ejpam-6153	71	2	i	i	PRON
ejpam-6153	71	3	∩k	∩k	VERB
ejpam-6153	71	4	⊆	⊆	NUM
ejpam-6153	71	5	j	j	PROPN
ejpam-6153	71	6	∩k	∩k	NOUN
ejpam-6153	71	7	,	,	PUNCT
ejpam-6153	71	8	this	this	PRON
ejpam-6153	71	9	implies95	implies95	NOUN
ejpam-6153	71	10	that	that	SCONJ
ejpam-6153	71	11	j	j	PROPN
ejpam-6153	71	12	∩k	∩k	PROPN
ejpam-6153	71	13	̸=	̸=	PROPN
ejpam-6153	71	14	{	{	PUNCT
ejpam-6153	71	15	0	0	NUM
ejpam-6153	71	16	}	}	PUNCT
ejpam-6153	71	17	.	.	PUNCT
ejpam-6153	72	1	hence	hence	ADV
ejpam-6153	72	2	,	,	PUNCT
ejpam-6153	72	3	j	j	PROPN
ejpam-6153	72	4	is	be	AUX
ejpam-6153	72	5	an	an	DET
ejpam-6153	72	6	essential	essential	ADJ
ejpam-6153	72	7	ideal	ideal	NOUN
ejpam-6153	72	8	of	of	ADP
ejpam-6153	72	9	r.96	r.96	PROPN
ejpam-6153	72	10	proposition	proposition	NOUN
ejpam-6153	72	11	5	5	NUM
ejpam-6153	72	12	.	.	PUNCT
ejpam-6153	72	13	assume	assume	VERB
ejpam-6153	72	14	that	that	SCONJ
ejpam-6153	72	15	r	r	NOUN
ejpam-6153	72	16	is	be	AUX
ejpam-6153	72	17	commutative	commutative	ADJ
ejpam-6153	72	18	.	.	PUNCT
ejpam-6153	73	1	let	let	VERB
ejpam-6153	73	2	i	i	PRON
ejpam-6153	73	3	and	and	CCONJ
ejpam-6153	73	4	j	j	PROPN
ejpam-6153	73	5	be	be	VERB
ejpam-6153	73	6	essential	essential	ADJ
ejpam-6153	73	7	ideals	ideal	NOUN
ejpam-6153	73	8	of	of	ADP
ejpam-6153	73	9	r.	r.	PROPN
ejpam-6153	73	10	if97	if97	PROPN
ejpam-6153	73	11	r	r	NOUN
ejpam-6153	73	12	has	have	VERB
ejpam-6153	73	13	no	no	DET
ejpam-6153	73	14	a	a	DET
ejpam-6153	73	15	zero	zero	NUM
ejpam-6153	73	16	divisor	divisor	NOUN
ejpam-6153	73	17	,	,	PUNCT
ejpam-6153	73	18	then	then	ADV
ejpam-6153	73	19	i	i	PRON
ejpam-6153	73	20	∩	∩	PROPN
ejpam-6153	73	21	j	j	PROPN
ejpam-6153	73	22	is	be	AUX
ejpam-6153	73	23	also	also	ADV
ejpam-6153	73	24	an	an	DET
ejpam-6153	73	25	essential	essential	ADJ
ejpam-6153	73	26	ideal	ideal	NOUN
ejpam-6153	73	27	of	of	ADP
ejpam-6153	73	28	r.98	r.98	ADJ
ejpam-6153	73	29	proof	proof	NOUN
ejpam-6153	73	30	.	.	PUNCT
ejpam-6153	74	1	since	since	SCONJ
ejpam-6153	74	2	i	i	PRON
ejpam-6153	74	3	and	and	CCONJ
ejpam-6153	74	4	j	j	PROPN
ejpam-6153	74	5	are	be	AUX
ejpam-6153	74	6	ideals	ideal	NOUN
ejpam-6153	74	7	of	of	ADP
ejpam-6153	74	8	r	r	NOUN
ejpam-6153	74	9	,	,	PUNCT
ejpam-6153	74	10	we	we	PRON
ejpam-6153	74	11	have	have	VERB
ejpam-6153	74	12	i	i	PROPN
ejpam-6153	74	13	∩	∩	PROPN
ejpam-6153	74	14	j	j	PROPN
ejpam-6153	74	15	is	be	AUX
ejpam-6153	74	16	also	also	ADV
ejpam-6153	74	17	an	an	DET
ejpam-6153	74	18	ideal	ideal	NOUN
ejpam-6153	74	19	of	of	ADP
ejpam-6153	74	20	r.	r.	PROPN
ejpam-6153	74	21	let	let	VERB
ejpam-6153	74	22	k	k	PROPN
ejpam-6153	74	23	be99	be99	PROPN
ejpam-6153	74	24	any	any	DET
ejpam-6153	74	25	nonzero	nonzero	NOUN
ejpam-6153	74	26	ideal	ideal	NOUN
ejpam-6153	74	27	of	of	ADP
ejpam-6153	74	28	r.	r.	PROPN
ejpam-6153	74	29	thus	thus	ADV
ejpam-6153	74	30	i	i	PRON
ejpam-6153	74	31	∩k	∩k	VERB
ejpam-6153	74	32	̸=	̸=	PROPN
ejpam-6153	74	33	{	{	PUNCT
ejpam-6153	74	34	0	0	NUM
ejpam-6153	74	35	}	}	PUNCT
ejpam-6153	74	36	.	.	PUNCT
ejpam-6153	75	1	so	so	ADV
ejpam-6153	75	2	there	there	PRON
ejpam-6153	75	3	exists	exist	VERB
ejpam-6153	75	4	a	a	DET
ejpam-6153	75	5	nonzero	nonzero	NOUN
ejpam-6153	75	6	element	element	NOUN
ejpam-6153	75	7	x	x	SYM
ejpam-6153	75	8	∈	∈	PROPN
ejpam-6153	75	9	i	i	PRON
ejpam-6153	75	10	∩k.100	∩k.100	VERB
ejpam-6153	76	1	let	let	VERB
ejpam-6153	76	2	y	y	PROPN
ejpam-6153	76	3	∈	∈	PROPN
ejpam-6153	76	4	j	j	PROPN
ejpam-6153	76	5	∖	∖	X
ejpam-6153	76	6	{	{	PUNCT
ejpam-6153	76	7	0	0	NUM
ejpam-6153	76	8	}	}	PUNCT
ejpam-6153	76	9	.	.	PUNCT
ejpam-6153	77	1	then	then	ADV
ejpam-6153	77	2	xy	xy	PROPN
ejpam-6153	77	3	∈	∈	PROPN
ejpam-6153	77	4	(	(	PUNCT
ejpam-6153	77	5	i	i	PROPN
ejpam-6153	77	6	∩	∩	ADJ
ejpam-6153	77	7	j	j	PROPN
ejpam-6153	77	8	)	)	PUNCT
ejpam-6153	77	9	∩	∩	PROPN
ejpam-6153	77	10	k.	k.	PROPN
ejpam-6153	77	11	by	by	ADP
ejpam-6153	77	12	assumption	assumption	NOUN
ejpam-6153	77	13	,	,	PUNCT
ejpam-6153	77	14	we	we	PRON
ejpam-6153	77	15	have	have	VERB
ejpam-6153	77	16	xy	xy	PROPN
ejpam-6153	77	17	̸=	̸=	PROPN
ejpam-6153	77	18	0	0	NUM
ejpam-6153	77	19	.	.	PUNCT
ejpam-6153	78	1	thus101	thus101	PROPN
ejpam-6153	78	2	(	(	PUNCT
ejpam-6153	78	3	i	i	PROPN
ejpam-6153	78	4	∩	∩	ADJ
ejpam-6153	78	5	j	j	PROPN
ejpam-6153	78	6	)	)	PUNCT
ejpam-6153	78	7	∩k	∩k	PROPN
ejpam-6153	78	8	̸=	̸=	PROPN
ejpam-6153	78	9	{	{	PUNCT
ejpam-6153	78	10	0	0	NUM
ejpam-6153	78	11	}	}	PUNCT
ejpam-6153	78	12	.	.	PUNCT
ejpam-6153	79	1	hence	hence	ADV
ejpam-6153	79	2	,	,	PUNCT
ejpam-6153	79	3	i	i	PROPN
ejpam-6153	79	4	∩	∩	PROPN
ejpam-6153	79	5	j	j	PROPN
ejpam-6153	79	6	is	be	AUX
ejpam-6153	79	7	an	an	DET
ejpam-6153	79	8	essential	essential	ADJ
ejpam-6153	79	9	ideal	ideal	NOUN
ejpam-6153	79	10	of	of	ADP
ejpam-6153	79	11	r.102	r.102	PROPN
ejpam-6153	79	12	definition	definition	NOUN
ejpam-6153	79	13	2	2	NUM
ejpam-6153	79	14	.	.	PUNCT
ejpam-6153	79	15	a	a	DET
ejpam-6153	79	16	fuzzy	fuzzy	ADJ
ejpam-6153	79	17	ideal	ideal	NOUN
ejpam-6153	79	18	f	f	PROPN
ejpam-6153	79	19	of	of	ADP
ejpam-6153	79	20	r	r	NOUN
ejpam-6153	79	21	is	be	AUX
ejpam-6153	79	22	called	call	VERB
ejpam-6153	79	23	a	a	DET
ejpam-6153	79	24	nontrivial	nontrivial	ADJ
ejpam-6153	79	25	fuzzy	fuzzy	ADJ
ejpam-6153	79	26	ideal	ideal	NOUN
ejpam-6153	79	27	of	of	ADP
ejpam-6153	79	28	r	r	NOUN
ejpam-6153	79	29	if	if	SCONJ
ejpam-6153	79	30	there	there	PRON
ejpam-6153	79	31	exists	exist	VERB
ejpam-6153	79	32	a103	a103	X
ejpam-6153	79	33	nonzero	nonzero	NOUN
ejpam-6153	79	34	element	element	NOUN
ejpam-6153	79	35	x	x	SYM
ejpam-6153	79	36	∈	∈	NOUN
ejpam-6153	79	37	r	r	NOUN
ejpam-6153	79	38	such	such	ADJ
ejpam-6153	79	39	that	that	SCONJ
ejpam-6153	79	40	f(x	f(x	NOUN
ejpam-6153	79	41	)	)	PUNCT
ejpam-6153	80	1	̸=	̸=	PROPN
ejpam-6153	80	2	0.104	0.104	NUM
ejpam-6153	80	3	by	by	ADP
ejpam-6153	80	4	definition	definition	NOUN
ejpam-6153	80	5	of	of	ADP
ejpam-6153	80	6	nontrivial	nontrivial	ADJ
ejpam-6153	80	7	fuzzy	fuzzy	ADJ
ejpam-6153	80	8	ideals	ideal	NOUN
ejpam-6153	80	9	,	,	PUNCT
ejpam-6153	80	10	the	the	DET
ejpam-6153	80	11	following	follow	VERB
ejpam-6153	80	12	lemma	lemma	PROPN
ejpam-6153	80	13	is	be	AUX
ejpam-6153	80	14	obvious.105	obvious.105	ADV
ejpam-6153	80	15	lemma	lemma	PROPN
ejpam-6153	80	16	1	1	X
ejpam-6153	80	17	.	.	PUNCT
ejpam-6153	81	1	let	let	VERB
ejpam-6153	81	2	g	g	PRON
ejpam-6153	81	3	be	be	AUX
ejpam-6153	81	4	a	a	DET
ejpam-6153	81	5	fuzzy	fuzzy	ADJ
ejpam-6153	81	6	ideal	ideal	NOUN
ejpam-6153	81	7	of	of	ADP
ejpam-6153	81	8	r.	r.	PROPN
ejpam-6153	81	9	then	then	ADV
ejpam-6153	81	10	g	g	PROPN
ejpam-6153	81	11	is	be	AUX
ejpam-6153	81	12	a	a	DET
ejpam-6153	81	13	nontrivial	nontrivial	ADJ
ejpam-6153	81	14	fuzzy	fuzzy	ADJ
ejpam-6153	81	15	ideal	ideal	NOUN
ejpam-6153	81	16	of	of	ADP
ejpam-6153	81	17	r	r	NOUN
ejpam-6153	81	18	if	if	SCONJ
ejpam-6153	81	19	and	and	CCONJ
ejpam-6153	81	20	only106	only106	PROPN
ejpam-6153	81	21	if	if	SCONJ
ejpam-6153	81	22	supp(g	supp(g	NOUN
ejpam-6153	81	23	)	)	PUNCT
ejpam-6153	81	24	̸=	̸=	PROPN
ejpam-6153	81	25	{	{	PUNCT
ejpam-6153	81	26	0}.107	0}.107	NOUN
ejpam-6153	81	27	we	we	PRON
ejpam-6153	81	28	define	define	VERB
ejpam-6153	81	29	the	the	DET
ejpam-6153	81	30	definition	definition	NOUN
ejpam-6153	81	31	of	of	ADP
ejpam-6153	81	32	essential	essential	ADJ
ejpam-6153	81	33	fuzzy	fuzzy	ADJ
ejpam-6153	81	34	ideals	ideal	NOUN
ejpam-6153	81	35	of	of	ADP
ejpam-6153	81	36	r	r	NOUN
ejpam-6153	81	37	as	as	ADP
ejpam-6153	81	38	follows:108	follows:108	NUM
ejpam-6153	81	39	definition	definition	NOUN
ejpam-6153	81	40	3	3	NUM
ejpam-6153	81	41	.	.	PUNCT
ejpam-6153	82	1	a	a	DET
ejpam-6153	82	2	fuzzy	fuzzy	ADJ
ejpam-6153	82	3	ideal	ideal	NOUN
ejpam-6153	82	4	f	f	PROPN
ejpam-6153	82	5	ofr	ofr	PROPN
ejpam-6153	82	6	is	be	AUX
ejpam-6153	82	7	called	call	VERB
ejpam-6153	82	8	an	an	DET
ejpam-6153	82	9	essential	essential	ADJ
ejpam-6153	82	10	fuzzy	fuzzy	ADJ
ejpam-6153	82	11	ideal	ideal	ADJ
ejpam-6153	82	12	ofr	ofr	PROPN
ejpam-6153	82	13	if	if	SCONJ
ejpam-6153	82	14	supp(f∩g	supp(f∩g	NOUN
ejpam-6153	82	15	)	)	PUNCT
ejpam-6153	82	16	̸=	̸=	PROPN
ejpam-6153	82	17	{	{	PUNCT
ejpam-6153	82	18	0}109	0}109	VERB
ejpam-6153	82	19	for	for	ADP
ejpam-6153	82	20	every	every	DET
ejpam-6153	82	21	nontrivial	nontrivial	ADJ
ejpam-6153	82	22	fuzzy	fuzzy	ADJ
ejpam-6153	82	23	ideal	ideal	NOUN
ejpam-6153	82	24	g	g	PROPN
ejpam-6153	82	25	of	of	ADP
ejpam-6153	82	26	s.110	s.110	PROPN
ejpam-6153	82	27	example	example	NOUN
ejpam-6153	82	28	4	4	X
ejpam-6153	82	29	.	.	PUNCT
ejpam-6153	83	1	we	we	PRON
ejpam-6153	83	2	consider	consider	VERB
ejpam-6153	83	3	the	the	DET
ejpam-6153	83	4	semiring	semiring	NOUN
ejpam-6153	83	5	n0	n0	NOUN
ejpam-6153	83	6	under	under	ADP
ejpam-6153	83	7	the	the	DET
ejpam-6153	83	8	usual	usual	ADJ
ejpam-6153	83	9	addition	addition	NOUN
ejpam-6153	83	10	and	and	CCONJ
ejpam-6153	83	11	multiplication	multiplication	NOUN
ejpam-6153	83	12	of111	of111	PROPN
ejpam-6153	83	13	integers	integer	NOUN
ejpam-6153	83	14	.	.	PUNCT
ejpam-6153	84	1	define	define	VERB
ejpam-6153	84	2	a	a	DET
ejpam-6153	84	3	fuzzy	fuzzy	ADJ
ejpam-6153	84	4	subset	subset	NOUN
ejpam-6153	84	5	f	f	PROPN
ejpam-6153	84	6	of	of	ADP
ejpam-6153	84	7	n0	n0	NUM
ejpam-6153	84	8	by	by	ADP
ejpam-6153	84	9	f(0	f(0	NOUN
ejpam-6153	84	10	)	)	PUNCT
ejpam-6153	84	11	=	=	SYM
ejpam-6153	84	12	1	1	NUM
ejpam-6153	84	13	and	and	CCONJ
ejpam-6153	84	14	f(n	f(n	PROPN
ejpam-6153	84	15	)	)	PUNCT
ejpam-6153	85	1	=	=	PUNCT
ejpam-6153	85	2	n−	n−	NOUN
ejpam-6153	85	3	1	1	NUM
ejpam-6153	85	4	n	n	NOUN
ejpam-6153	85	5	for	for	ADP
ejpam-6153	85	6	all	all	PRON
ejpam-6153	85	7	n	n	DET
ejpam-6153	85	8	∈	∈	PROPN
ejpam-6153	85	9	n.	n.	NOUN
ejpam-6153	85	10	we112	we112	ADV
ejpam-6153	85	11	have	have	VERB
ejpam-6153	85	12	that	that	SCONJ
ejpam-6153	85	13	f	f	PROPN
ejpam-6153	85	14	is	be	AUX
ejpam-6153	85	15	an	an	DET
ejpam-6153	85	16	essential	essential	ADJ
ejpam-6153	85	17	fuzzy	fuzzy	ADJ
ejpam-6153	85	18	ideal	ideal	NOUN
ejpam-6153	85	19	of	of	ADP
ejpam-6153	85	20	n0.113	n0.113	ADJ
ejpam-6153	85	21	proposition	proposition	NOUN
ejpam-6153	85	22	6	6	NUM
ejpam-6153	85	23	.	.	PUNCT
ejpam-6153	86	1	let	let	VERB
ejpam-6153	86	2	f	f	PRON
ejpam-6153	86	3	be	be	AUX
ejpam-6153	86	4	an	an	DET
ejpam-6153	86	5	essential	essential	ADJ
ejpam-6153	86	6	fuzzy	fuzzy	ADJ
ejpam-6153	86	7	ideal	ideal	NOUN
ejpam-6153	86	8	of	of	ADP
ejpam-6153	86	9	r.	r.	PROPN
ejpam-6153	86	10	if	if	SCONJ
ejpam-6153	86	11	h	h	NOUN
ejpam-6153	86	12	is	be	AUX
ejpam-6153	86	13	a	a	DET
ejpam-6153	86	14	fuzzy	fuzzy	ADJ
ejpam-6153	86	15	ideal	ideal	NOUN
ejpam-6153	86	16	of	of	ADP
ejpam-6153	86	17	r	r	NOUN
ejpam-6153	86	18	such	such	ADJ
ejpam-6153	86	19	that114	that114	PROPN
ejpam-6153	86	20	f	f	PROPN
ejpam-6153	86	21	⊆	⊆	NUM
ejpam-6153	86	22	h	h	NOUN
ejpam-6153	86	23	,	,	PUNCT
ejpam-6153	86	24	then	then	ADV
ejpam-6153	86	25	h	h	NOUN
ejpam-6153	86	26	is	be	AUX
ejpam-6153	86	27	also	also	ADV
ejpam-6153	86	28	essential.115	essential.115	NOUN
ejpam-6153	86	29	proof	proof	NOUN
ejpam-6153	86	30	.	.	PUNCT
ejpam-6153	87	1	assume	assume	VERB
ejpam-6153	87	2	that	that	SCONJ
ejpam-6153	87	3	f	f	PROPN
ejpam-6153	87	4	is	be	AUX
ejpam-6153	87	5	an	an	DET
ejpam-6153	87	6	essential	essential	ADJ
ejpam-6153	87	7	fuzzy	fuzzy	ADJ
ejpam-6153	87	8	ideal	ideal	NOUN
ejpam-6153	87	9	of	of	ADP
ejpam-6153	87	10	r	r	NOUN
ejpam-6153	87	11	and	and	CCONJ
ejpam-6153	87	12	let	let	VERB
ejpam-6153	87	13	h	h	PRON
ejpam-6153	87	14	be	be	AUX
ejpam-6153	87	15	a	a	DET
ejpam-6153	87	16	fuzzy	fuzzy	ADJ
ejpam-6153	87	17	ideal	ideal	NOUN
ejpam-6153	87	18	of	of	ADP
ejpam-6153	87	19	s116	s116	PROPN
ejpam-6153	87	20	such	such	ADJ
ejpam-6153	87	21	that	that	SCONJ
ejpam-6153	87	22	f	f	PROPN
ejpam-6153	87	23	⊆	⊆	NUM
ejpam-6153	87	24	h.	h.	PROPN
ejpam-6153	87	25	let	let	VERB
ejpam-6153	87	26	g	g	NOUN
ejpam-6153	87	27	be	be	AUX
ejpam-6153	87	28	any	any	DET
ejpam-6153	87	29	nontrivial	nontrivial	ADJ
ejpam-6153	87	30	fuzzy	fuzzy	ADJ
ejpam-6153	87	31	ideal	ideal	NOUN
ejpam-6153	87	32	of	of	ADP
ejpam-6153	87	33	r.	r.	PROPN
ejpam-6153	87	34	thus	thus	ADV
ejpam-6153	87	35	supp(f	supp(f	PROPN
ejpam-6153	87	36	∩	∩	PROPN
ejpam-6153	87	37	g	g	NOUN
ejpam-6153	87	38	)	)	PUNCT
ejpam-6153	87	39	̸=	̸=	PROPN
ejpam-6153	87	40	{	{	PUNCT
ejpam-6153	87	41	0	0	NUM
ejpam-6153	87	42	}	}	PUNCT
ejpam-6153	87	43	.	.	PUNCT
ejpam-6153	88	1	so117	so117	NOUN
ejpam-6153	88	2	supp(h	supp(h	ADP
ejpam-6153	88	3	∩	∩	PROPN
ejpam-6153	88	4	g	g	NOUN
ejpam-6153	88	5	)	)	PUNCT
ejpam-6153	88	6	̸=	̸=	PROPN
ejpam-6153	88	7	{	{	PUNCT
ejpam-6153	88	8	0	0	NUM
ejpam-6153	88	9	}	}	PUNCT
ejpam-6153	88	10	.	.	PUNCT
ejpam-6153	89	1	hence	hence	ADV
ejpam-6153	89	2	,	,	PUNCT
ejpam-6153	89	3	h	h	NOUN
ejpam-6153	89	4	is	be	AUX
ejpam-6153	89	5	also	also	ADV
ejpam-6153	89	6	an	an	DET
ejpam-6153	89	7	essential	essential	ADJ
ejpam-6153	89	8	fuzzy	fuzzy	ADJ
ejpam-6153	89	9	ideal	ideal	NOUN
ejpam-6153	89	10	of	of	ADP
ejpam-6153	89	11	r.118	r.118	NUM
ejpam-6153	89	12	the	the	DET
ejpam-6153	89	13	two	two	NUM
ejpam-6153	89	14	following	follow	VERB
ejpam-6153	89	15	theorems	theorem	NOUN
ejpam-6153	89	16	show	show	VERB
ejpam-6153	89	17	relationships	relationship	NOUN
ejpam-6153	89	18	between	between	ADP
ejpam-6153	89	19	essential	essential	ADJ
ejpam-6153	89	20	ideals	ideal	NOUN
ejpam-6153	89	21	and	and	CCONJ
ejpam-6153	89	22	essential119	essential119	PROPN
ejpam-6153	89	23	fuzzy	fuzzy	ADJ
ejpam-6153	89	24	ideals	ideal	NOUN
ejpam-6153	89	25	of	of	ADP
ejpam-6153	89	26	r.120	r.120	PROPN
ejpam-6153	89	27	r.	r.	PROPN
ejpam-6153	89	28	chinram	chinram	PROPN
ejpam-6153	89	29	,	,	PUNCT
ejpam-6153	89	30	s.	s.	PROPN
ejpam-6153	89	31	hangsawat	hangsawat	PROPN
ejpam-6153	89	32	/	/	SYM
ejpam-6153	89	33	eur	eur	PROPN
ejpam-6153	89	34	.	.	PUNCT
ejpam-6153	90	1	j.	j.	PROPN
ejpam-6153	90	2	pure	pure	PROPN
ejpam-6153	90	3	appl	appl	PROPN
ejpam-6153	90	4	.	.	PROPN
ejpam-6153	90	5	math	math	PROPN
ejpam-6153	90	6	,	,	PUNCT
ejpam-6153	90	7	18	18	NUM
ejpam-6153	90	8	(	(	PUNCT
ejpam-6153	90	9	3	3	NUM
ejpam-6153	90	10	)	)	PUNCT
ejpam-6153	90	11	(	(	PUNCT
ejpam-6153	90	12	2025	2025	NUM
ejpam-6153	90	13	)	)	PUNCT
ejpam-6153	90	14	,	,	PUNCT
ejpam-6153	90	15	6153	6153	NUM
ejpam-6153	90	16	5	5	NUM
ejpam-6153	90	17	of	of	ADP
ejpam-6153	90	18	7	7	NUM
ejpam-6153	90	19	theorem	theorem	NOUN
ejpam-6153	90	20	1	1	NUM
ejpam-6153	90	21	.	.	PUNCT
ejpam-6153	91	1	a	a	DET
ejpam-6153	91	2	nonzero	nonzero	NOUN
ejpam-6153	91	3	ideal	ideal	NOUN
ejpam-6153	91	4	i	i	PRON
ejpam-6153	91	5	of	of	ADP
ejpam-6153	91	6	r	r	NOUN
ejpam-6153	91	7	is	be	AUX
ejpam-6153	91	8	essential	essential	ADJ
ejpam-6153	91	9	if	if	SCONJ
ejpam-6153	91	10	and	and	CCONJ
ejpam-6153	91	11	only	only	ADV
ejpam-6153	91	12	if	if	SCONJ
ejpam-6153	91	13	ci	ci	PROPN
ejpam-6153	91	14	is	be	AUX
ejpam-6153	91	15	an	an	DET
ejpam-6153	91	16	essential	essential	ADJ
ejpam-6153	91	17	fuzzy121	fuzzy121	PROPN
ejpam-6153	91	18	ideal	ideal	NOUN
ejpam-6153	91	19	of	of	ADP
ejpam-6153	91	20	r.122	r.122	NUM
ejpam-6153	91	21	proof	proof	NOUN
ejpam-6153	91	22	.	.	PUNCT
ejpam-6153	92	1	assume	assume	VERB
ejpam-6153	92	2	that	that	SCONJ
ejpam-6153	92	3	i	i	PRON
ejpam-6153	92	4	is	be	AUX
ejpam-6153	92	5	an	an	DET
ejpam-6153	92	6	essential	essential	ADJ
ejpam-6153	92	7	ideal	ideal	NOUN
ejpam-6153	92	8	of	of	ADP
ejpam-6153	92	9	r.	r.	PROPN
ejpam-6153	92	10	by	by	ADP
ejpam-6153	92	11	proposition	proposition	NOUN
ejpam-6153	92	12	1	1	NUM
ejpam-6153	92	13	,	,	PUNCT
ejpam-6153	92	14	ci	ci	PROPN
ejpam-6153	92	15	is	be	AUX
ejpam-6153	92	16	a	a	DET
ejpam-6153	92	17	fuzzy	fuzzy	ADJ
ejpam-6153	92	18	ideal123	ideal123	PROPN
ejpam-6153	92	19	of	of	ADP
ejpam-6153	92	20	r.	r.	PROPN
ejpam-6153	92	21	we	we	PRON
ejpam-6153	92	22	let	let	VERB
ejpam-6153	92	23	g	g	NOUN
ejpam-6153	92	24	be	be	AUX
ejpam-6153	92	25	any	any	DET
ejpam-6153	92	26	nontrivial	nontrivial	ADJ
ejpam-6153	92	27	fuzzy	fuzzy	ADJ
ejpam-6153	92	28	ideal	ideal	NOUN
ejpam-6153	92	29	of	of	ADP
ejpam-6153	92	30	r.	r.	PROPN
ejpam-6153	92	31	by	by	ADP
ejpam-6153	92	32	lemma	lemma	PROPN
ejpam-6153	92	33	1	1	NUM
ejpam-6153	92	34	and	and	CCONJ
ejpam-6153	92	35	proposition	proposition	NOUN
ejpam-6153	92	36	2	2	NUM
ejpam-6153	92	37	,	,	PUNCT
ejpam-6153	92	38	we	we	PRON
ejpam-6153	92	39	have124	have124	PROPN
ejpam-6153	92	40	supp(g	supp(g	NOUN
ejpam-6153	92	41	)	)	PUNCT
ejpam-6153	92	42	is	be	AUX
ejpam-6153	92	43	a	a	DET
ejpam-6153	92	44	nonzero	nonzero	NOUN
ejpam-6153	92	45	ideal	ideal	NOUN
ejpam-6153	92	46	of	of	ADP
ejpam-6153	92	47	s.	s.	PROPN
ejpam-6153	92	48	so	so	SCONJ
ejpam-6153	92	49	i∩supp(g	i∩supp(g	PROPN
ejpam-6153	92	50	)	)	PUNCT
ejpam-6153	92	51	̸=	̸=	PROPN
ejpam-6153	92	52	{	{	PUNCT
ejpam-6153	92	53	0	0	NUM
ejpam-6153	92	54	}	}	PUNCT
ejpam-6153	92	55	.	.	PUNCT
ejpam-6153	93	1	then	then	ADV
ejpam-6153	93	2	there	there	PRON
ejpam-6153	93	3	exists	exist	VERB
ejpam-6153	93	4	a	a	DET
ejpam-6153	93	5	nonzero	nonzero	NOUN
ejpam-6153	93	6	element125	element125	X
ejpam-6153	93	7	x	x	SYM
ejpam-6153	93	8	such	such	ADJ
ejpam-6153	93	9	that	that	SCONJ
ejpam-6153	93	10	x	x	SYM
ejpam-6153	93	11	∈	∈	PROPN
ejpam-6153	93	12	i	i	PRON
ejpam-6153	93	13	∩supp(g	∩supp(g	NOUN
ejpam-6153	93	14	)	)	PUNCT
ejpam-6153	93	15	,	,	PUNCT
ejpam-6153	93	16	this	this	PRON
ejpam-6153	93	17	implies	imply	VERB
ejpam-6153	93	18	that	that	SCONJ
ejpam-6153	93	19	(	(	PUNCT
ejpam-6153	93	20	ci	ci	PROPN
ejpam-6153	93	21	∩g)(x	∩g)(x	PROPN
ejpam-6153	93	22	)	)	PUNCT
ejpam-6153	93	23	̸=	̸=	PROPN
ejpam-6153	93	24	0	0	NUM
ejpam-6153	93	25	.	.	PUNCT
ejpam-6153	94	1	therefore	therefore	ADV
ejpam-6153	94	2	x	x	X
ejpam-6153	94	3	∈	∈	PROPN
ejpam-6153	94	4	supp(ci	supp(ci	PROPN
ejpam-6153	94	5	∩g).126	∩g).126	NOUN
ejpam-6153	94	6	thus	thus	ADV
ejpam-6153	94	7	supp(ci	supp(ci	ADJ
ejpam-6153	94	8	∩g	∩g	ADJ
ejpam-6153	94	9	)	)	PUNCT
ejpam-6153	94	10	̸=	̸=	PROPN
ejpam-6153	94	11	{	{	PUNCT
ejpam-6153	94	12	0	0	NUM
ejpam-6153	94	13	}	}	PUNCT
ejpam-6153	94	14	.	.	PUNCT
ejpam-6153	95	1	hence	hence	ADV
ejpam-6153	95	2	ci	ci	PROPN
ejpam-6153	95	3	is	be	AUX
ejpam-6153	95	4	an	an	DET
ejpam-6153	95	5	essential	essential	ADJ
ejpam-6153	95	6	fuzzy	fuzzy	ADJ
ejpam-6153	95	7	ideal	ideal	NOUN
ejpam-6153	95	8	of	of	ADP
ejpam-6153	95	9	r.	r.	PROPN
ejpam-6153	95	10	to	to	PART
ejpam-6153	95	11	prove	prove	VERB
ejpam-6153	95	12	the	the	DET
ejpam-6153	95	13	converse,127	converse,127	NOUN
ejpam-6153	95	14	we	we	PRON
ejpam-6153	95	15	assume	assume	VERB
ejpam-6153	95	16	that	that	SCONJ
ejpam-6153	95	17	ci	ci	PROPN
ejpam-6153	95	18	is	be	AUX
ejpam-6153	95	19	an	an	DET
ejpam-6153	95	20	essential	essential	ADJ
ejpam-6153	95	21	fuzzy	fuzzy	ADJ
ejpam-6153	95	22	ideal	ideal	NOUN
ejpam-6153	95	23	of	of	ADP
ejpam-6153	95	24	r.	r.	PROPN
ejpam-6153	95	25	let	let	VERB
ejpam-6153	95	26	k	k	PROPN
ejpam-6153	95	27	be	be	AUX
ejpam-6153	95	28	a	a	DET
ejpam-6153	95	29	nonzero	nonzero	NOUN
ejpam-6153	95	30	ideal	ideal	NOUN
ejpam-6153	95	31	of	of	ADP
ejpam-6153	95	32	r.	r.	PROPN
ejpam-6153	95	33	by128	by128	PROPN
ejpam-6153	95	34	lemma	lemma	PROPN
ejpam-6153	95	35	1	1	NUM
ejpam-6153	95	36	,	,	PUNCT
ejpam-6153	95	37	we	we	PRON
ejpam-6153	95	38	have	have	VERB
ejpam-6153	95	39	that	that	PRON
ejpam-6153	95	40	ck	ck	PROPN
ejpam-6153	95	41	is	be	AUX
ejpam-6153	95	42	a	a	DET
ejpam-6153	95	43	nontrivial	nontrivial	ADJ
ejpam-6153	95	44	fuzzy	fuzzy	ADJ
ejpam-6153	95	45	ideal	ideal	NOUN
ejpam-6153	95	46	of	of	ADP
ejpam-6153	95	47	r.	r.	PROPN
ejpam-6153	95	48	thus	thus	ADV
ejpam-6153	95	49	supp(ci	supp(ci	ADJ
ejpam-6153	95	50	∩	∩	NOUN
ejpam-6153	95	51	ck	ck	NOUN
ejpam-6153	95	52	)	)	PUNCT
ejpam-6153	95	53	̸=	̸=	PROPN
ejpam-6153	95	54	{	{	PUNCT
ejpam-6153	95	55	0}.129	0}.129	NOUN
ejpam-6153	95	56	thus	thus	ADV
ejpam-6153	95	57	ci∩k	ci∩k	NOUN
ejpam-6153	95	58	̸=	̸=	PROPN
ejpam-6153	95	59	c{0	c{0	PROPN
ejpam-6153	95	60	}	}	PUNCT
ejpam-6153	95	61	,	,	PUNCT
ejpam-6153	95	62	this	this	PRON
ejpam-6153	95	63	implies	imply	VERB
ejpam-6153	95	64	that	that	SCONJ
ejpam-6153	95	65	i	i	PRON
ejpam-6153	95	66	∩k	∩k	VERB
ejpam-6153	95	67	̸=	̸=	PROPN
ejpam-6153	95	68	{	{	PUNCT
ejpam-6153	95	69	0	0	NUM
ejpam-6153	95	70	}	}	PUNCT
ejpam-6153	95	71	.	.	PUNCT
ejpam-6153	96	1	hence	hence	ADV
ejpam-6153	96	2	i	i	PRON
ejpam-6153	96	3	is	be	AUX
ejpam-6153	96	4	essential.130	essential.130	ADV
ejpam-6153	96	5	theorem	theorem	ADJ
ejpam-6153	96	6	2	2	NUM
ejpam-6153	96	7	.	.	PUNCT
ejpam-6153	97	1	a	a	DET
ejpam-6153	97	2	nontrivial	nontrivial	ADJ
ejpam-6153	97	3	fuzzy	fuzzy	ADJ
ejpam-6153	97	4	ideal	ideal	NOUN
ejpam-6153	97	5	f	f	PROPN
ejpam-6153	97	6	of	of	ADP
ejpam-6153	97	7	r	r	NOUN
ejpam-6153	97	8	is	be	AUX
ejpam-6153	97	9	essential	essential	ADJ
ejpam-6153	97	10	if	if	SCONJ
ejpam-6153	97	11	and	and	CCONJ
ejpam-6153	97	12	only	only	ADV
ejpam-6153	97	13	if	if	SCONJ
ejpam-6153	97	14	supp(f	supp(f	PROPN
ejpam-6153	97	15	)	)	PUNCT
ejpam-6153	97	16	is	be	AUX
ejpam-6153	97	17	an131	an131	PROPN
ejpam-6153	97	18	essential	essential	ADJ
ejpam-6153	97	19	ideal	ideal	NOUN
ejpam-6153	97	20	of	of	ADP
ejpam-6153	97	21	r.132	r.132	NOUN
ejpam-6153	97	22	proof	proof	NOUN
ejpam-6153	97	23	.	.	PUNCT
ejpam-6153	98	1	assume	assume	VERB
ejpam-6153	98	2	that	that	SCONJ
ejpam-6153	98	3	f	f	PROPN
ejpam-6153	98	4	is	be	AUX
ejpam-6153	98	5	an	an	DET
ejpam-6153	98	6	fuzzy	fuzzy	ADJ
ejpam-6153	98	7	essential	essential	ADJ
ejpam-6153	98	8	ideal	ideal	NOUN
ejpam-6153	98	9	of	of	ADP
ejpam-6153	98	10	r.	r.	PROPN
ejpam-6153	98	11	since	since	SCONJ
ejpam-6153	98	12	f	f	PROPN
ejpam-6153	98	13	is	be	AUX
ejpam-6153	98	14	a	a	DET
ejpam-6153	98	15	fuzzy	fuzzy	ADJ
ejpam-6153	98	16	ideal	ideal	NOUN
ejpam-6153	98	17	of	of	ADP
ejpam-6153	98	18	r,133	r,133	PROPN
ejpam-6153	98	19	supp(f	supp(f	PROPN
ejpam-6153	98	20	)	)	PUNCT
ejpam-6153	98	21	is	be	AUX
ejpam-6153	98	22	an	an	DET
ejpam-6153	98	23	ideal	ideal	NOUN
ejpam-6153	98	24	of	of	ADP
ejpam-6153	98	25	r	r	NOUN
ejpam-6153	98	26	by	by	ADP
ejpam-6153	98	27	proposition	proposition	NOUN
ejpam-6153	98	28	2	2	NUM
ejpam-6153	98	29	.	.	PUNCT
ejpam-6153	99	1	let	let	VERB
ejpam-6153	99	2	j	j	PROPN
ejpam-6153	99	3	be	be	AUX
ejpam-6153	99	4	any	any	DET
ejpam-6153	99	5	nonzero	nonzero	NOUN
ejpam-6153	99	6	ideal	ideal	NOUN
ejpam-6153	99	7	of	of	ADP
ejpam-6153	99	8	r.	r.	PROPN
ejpam-6153	99	9	by	by	ADP
ejpam-6153	99	10	lemma	lemma	PROPN
ejpam-6153	99	11	1,134	1,134	NUM
ejpam-6153	99	12	cj	cj	PROPN
ejpam-6153	99	13	is	be	AUX
ejpam-6153	99	14	a	a	DET
ejpam-6153	99	15	nontrivial	nontrivial	ADJ
ejpam-6153	99	16	fuzzy	fuzzy	ADJ
ejpam-6153	99	17	ideal	ideal	NOUN
ejpam-6153	99	18	of	of	ADP
ejpam-6153	99	19	r.	r.	PROPN
ejpam-6153	99	20	since	since	SCONJ
ejpam-6153	99	21	f	f	PROPN
ejpam-6153	99	22	is	be	AUX
ejpam-6153	99	23	essential	essential	ADJ
ejpam-6153	99	24	,	,	PUNCT
ejpam-6153	99	25	supp(f	supp(f	PROPN
ejpam-6153	99	26	∩	∩	X
ejpam-6153	99	27	cj	cj	NOUN
ejpam-6153	99	28	)	)	PUNCT
ejpam-6153	99	29	̸=	̸=	PROPN
ejpam-6153	99	30	{	{	PUNCT
ejpam-6153	99	31	0	0	NUM
ejpam-6153	99	32	}	}	PUNCT
ejpam-6153	99	33	.	.	PUNCT
ejpam-6153	100	1	thus	thus	ADV
ejpam-6153	100	2	there135	there135	PRON
ejpam-6153	100	3	exists	exist	VERB
ejpam-6153	100	4	a	a	DET
ejpam-6153	100	5	nonzero	nonzero	NOUN
ejpam-6153	100	6	element	element	NOUN
ejpam-6153	100	7	x	x	PUNCT
ejpam-6153	100	8	in	in	ADP
ejpam-6153	100	9	r	r	NOUN
ejpam-6153	100	10	such	such	ADJ
ejpam-6153	100	11	that	that	PRON
ejpam-6153	100	12	(	(	PUNCT
ejpam-6153	100	13	f	f	X
ejpam-6153	100	14	∩cj)(x	∩cj)(x	NOUN
ejpam-6153	100	15	)	)	PUNCT
ejpam-6153	100	16	̸=	̸=	PROPN
ejpam-6153	100	17	0	0	NUM
ejpam-6153	100	18	.	.	PUNCT
ejpam-6153	101	1	thus	thus	ADV
ejpam-6153	101	2	f(x	f(x	PROPN
ejpam-6153	101	3	)	)	PUNCT
ejpam-6153	101	4	̸=	̸=	PROPN
ejpam-6153	101	5	0	0	NUM
ejpam-6153	101	6	and	and	CCONJ
ejpam-6153	101	7	cj(x	cj(x	PUNCT
ejpam-6153	101	8	)	)	PUNCT
ejpam-6153	102	1	̸=	̸=	NOUN
ejpam-6153	102	2	0.136	0.136	NUM
ejpam-6153	102	3	hence	hence	ADV
ejpam-6153	102	4	x	x	ADP
ejpam-6153	102	5	∈	∈	PROPN
ejpam-6153	102	6	supp(f)∩	supp(f)∩	ADP
ejpam-6153	102	7	j	j	PROPN
ejpam-6153	102	8	.	.	PUNCT
ejpam-6153	103	1	then	then	ADV
ejpam-6153	103	2	supp(f)∩	supp(f)∩	VERB
ejpam-6153	103	3	j	j	PROPN
ejpam-6153	103	4	̸=	̸=	PROPN
ejpam-6153	103	5	{	{	PUNCT
ejpam-6153	103	6	0	0	NUM
ejpam-6153	103	7	}	}	PUNCT
ejpam-6153	103	8	,	,	PUNCT
ejpam-6153	103	9	this	this	PRON
ejpam-6153	103	10	implies	imply	VERB
ejpam-6153	103	11	that	that	SCONJ
ejpam-6153	103	12	supp(f	supp(f	PROPN
ejpam-6153	103	13	)	)	PUNCT
ejpam-6153	103	14	is	be	AUX
ejpam-6153	103	15	an	an	DET
ejpam-6153	103	16	essential137	essential137	PROPN
ejpam-6153	103	17	ideal	ideal	NOUN
ejpam-6153	103	18	of	of	ADP
ejpam-6153	103	19	r.	r.	PROPN
ejpam-6153	103	20	conversely	conversely	ADV
ejpam-6153	103	21	,	,	PUNCT
ejpam-6153	103	22	assume	assume	VERB
ejpam-6153	103	23	that	that	SCONJ
ejpam-6153	103	24	supp(f	supp(f	PROPN
ejpam-6153	103	25	)	)	PUNCT
ejpam-6153	103	26	is	be	AUX
ejpam-6153	103	27	an	an	DET
ejpam-6153	103	28	essential	essential	ADJ
ejpam-6153	103	29	ideal	ideal	NOUN
ejpam-6153	103	30	of	of	ADP
ejpam-6153	103	31	r.	r.	PROPN
ejpam-6153	103	32	let	let	VERB
ejpam-6153	103	33	g	g	PROPN
ejpam-6153	103	34	be	be	AUX
ejpam-6153	103	35	a	a	DET
ejpam-6153	103	36	nontrivial138	nontrivial138	PROPN
ejpam-6153	103	37	fuzzy	fuzzy	ADJ
ejpam-6153	103	38	ideal	ideal	NOUN
ejpam-6153	103	39	of	of	ADP
ejpam-6153	103	40	r.	r.	PROPN
ejpam-6153	103	41	thus	thus	ADV
ejpam-6153	103	42	supp(g	supp(g	NOUN
ejpam-6153	103	43	)	)	PUNCT
ejpam-6153	103	44	is	be	AUX
ejpam-6153	103	45	a	a	DET
ejpam-6153	103	46	nonzero	nonzero	ADJ
ejpam-6153	103	47	ideal	ideal	NOUN
ejpam-6153	103	48	of	of	ADP
ejpam-6153	103	49	r.	r.	PROPN
ejpam-6153	103	50	so	so	ADV
ejpam-6153	103	51	supp(f)∩	supp(f)∩	VERB
ejpam-6153	103	52	supp(g	supp(g	NOUN
ejpam-6153	103	53	)	)	PUNCT
ejpam-6153	103	54	̸=	̸=	PROPN
ejpam-6153	103	55	{	{	PUNCT
ejpam-6153	103	56	0	0	NUM
ejpam-6153	103	57	}	}	PUNCT
ejpam-6153	103	58	.	.	PUNCT
ejpam-6153	104	1	thus139	thus139	PART
ejpam-6153	104	2	there	there	PRON
ejpam-6153	104	3	exists	exist	VERB
ejpam-6153	104	4	a	a	DET
ejpam-6153	104	5	nonzero	nonzero	NOUN
ejpam-6153	104	6	element	element	NOUN
ejpam-6153	104	7	x	x	PUNCT
ejpam-6153	104	8	in	in	ADP
ejpam-6153	104	9	r	r	NOUN
ejpam-6153	105	1	such	such	ADJ
ejpam-6153	105	2	that	that	SCONJ
ejpam-6153	105	3	x	x	SYM
ejpam-6153	105	4	∈	∈	PROPN
ejpam-6153	105	5	supp(f	supp(f	PROPN
ejpam-6153	105	6	)	)	PUNCT
ejpam-6153	105	7	∩	∩	ADJ
ejpam-6153	105	8	supp(g	supp(g	NOUN
ejpam-6153	105	9	)	)	PUNCT
ejpam-6153	105	10	,	,	PUNCT
ejpam-6153	105	11	so	so	SCONJ
ejpam-6153	105	12	f(x	f(x	PROPN
ejpam-6153	105	13	)	)	PUNCT
ejpam-6153	105	14	̸=	̸=	PROPN
ejpam-6153	105	15	0	0	NUM
ejpam-6153	105	16	and140	and140	PROPN
ejpam-6153	105	17	g(x	g(x	NOUN
ejpam-6153	105	18	)	)	PUNCT
ejpam-6153	105	19	̸=	̸=	PROPN
ejpam-6153	105	20	0	0	NUM
ejpam-6153	105	21	.	.	PUNCT
ejpam-6153	106	1	therefore	therefore	ADV
ejpam-6153	106	2	,	,	PUNCT
ejpam-6153	106	3	(	(	PUNCT
ejpam-6153	106	4	f	f	PROPN
ejpam-6153	106	5	∩	∩	PROPN
ejpam-6153	106	6	g)(x	g)(x	PROPN
ejpam-6153	106	7	)	)	PUNCT
ejpam-6153	106	8	̸=	̸=	PROPN
ejpam-6153	106	9	0	0	NUM
ejpam-6153	106	10	.	.	PUNCT
ejpam-6153	107	1	hence	hence	ADV
ejpam-6153	107	2	,	,	PUNCT
ejpam-6153	107	3	supp(f	supp(f	PROPN
ejpam-6153	107	4	∩	∩	ADJ
ejpam-6153	107	5	g	g	NOUN
ejpam-6153	107	6	)	)	PUNCT
ejpam-6153	107	7	̸=	̸=	PROPN
ejpam-6153	107	8	{	{	PUNCT
ejpam-6153	107	9	0	0	NUM
ejpam-6153	107	10	}	}	PUNCT
ejpam-6153	107	11	.	.	PUNCT
ejpam-6153	108	1	this	this	PRON
ejpam-6153	108	2	implies	imply	VERB
ejpam-6153	108	3	that	that	SCONJ
ejpam-6153	108	4	f	f	PROPN
ejpam-6153	108	5	is141	is141	NOUN
ejpam-6153	108	6	essential.142	essential.142	VERB
ejpam-6153	108	7	in	in	ADP
ejpam-6153	108	8	the	the	DET
ejpam-6153	108	9	remainder	remainder	NOUN
ejpam-6153	108	10	of	of	ADP
ejpam-6153	108	11	this	this	DET
ejpam-6153	108	12	section	section	NOUN
ejpam-6153	108	13	,	,	PUNCT
ejpam-6153	108	14	we	we	PRON
ejpam-6153	108	15	will	will	AUX
ejpam-6153	108	16	investigate	investigate	VERB
ejpam-6153	108	17	relationships	relationship	NOUN
ejpam-6153	108	18	between	between	ADP
ejpam-6153	108	19	(	(	PUNCT
ejpam-6153	108	20	minimal,143	minimal,143	NOUN
ejpam-6153	108	21	prime	prime	ADJ
ejpam-6153	108	22	,	,	PUNCT
ejpam-6153	108	23	semiprime	semiprime	NOUN
ejpam-6153	108	24	)	)	PUNCT
ejpam-6153	108	25	essential	essential	ADJ
ejpam-6153	108	26	ideals	ideal	NOUN
ejpam-6153	108	27	and	and	CCONJ
ejpam-6153	108	28	(	(	PUNCT
ejpam-6153	108	29	minimal	minimal	ADJ
ejpam-6153	108	30	,	,	PUNCT
ejpam-6153	108	31	prime	prime	ADJ
ejpam-6153	108	32	,	,	PUNCT
ejpam-6153	108	33	semiprime	semiprime	NOUN
ejpam-6153	108	34	)	)	PUNCT
ejpam-6153	108	35	essential	essential	ADJ
ejpam-6153	108	36	fuzzy	fuzzy	NOUN
ejpam-6153	108	37	ideals.144	ideals.144	PROPN
ejpam-6153	108	38	an	an	DET
ejpam-6153	108	39	essential	essential	ADJ
ejpam-6153	108	40	ideal	ideal	NOUN
ejpam-6153	109	1	i	i	PRON
ejpam-6153	109	2	of	of	ADP
ejpam-6153	109	3	r	r	NOUN
ejpam-6153	109	4	is	be	AUX
ejpam-6153	109	5	called	call	VERB
ejpam-6153	109	6	minimal	minimal	ADJ
ejpam-6153	109	7	if	if	SCONJ
ejpam-6153	109	8	for	for	ADP
ejpam-6153	109	9	every	every	DET
ejpam-6153	109	10	essential	essential	ADJ
ejpam-6153	109	11	ideal	ideal	ADJ
ejpam-6153	109	12	j	j	PROPN
ejpam-6153	109	13	of	of	ADP
ejpam-6153	109	14	r	r	PROPN
ejpam-6153	109	15	such145	such145	PROPN
ejpam-6153	109	16	that	that	SCONJ
ejpam-6153	109	17	j	j	PROPN
ejpam-6153	109	18	⊆	⊆	NUM
ejpam-6153	109	19	i	i	PROPN
ejpam-6153	109	20	,	,	PUNCT
ejpam-6153	109	21	we	we	PRON
ejpam-6153	109	22	have	have	VERB
ejpam-6153	109	23	j	j	PROPN
ejpam-6153	109	24	=	=	PROPN
ejpam-6153	109	25	i.	i.	PROPN
ejpam-6153	109	26	an	an	DET
ejpam-6153	109	27	essential	essential	ADJ
ejpam-6153	109	28	fuzzy	fuzzy	ADJ
ejpam-6153	109	29	ideal	ideal	NOUN
ejpam-6153	109	30	f	f	PROPN
ejpam-6153	109	31	of	of	ADP
ejpam-6153	109	32	r	r	NOUN
ejpam-6153	109	33	is	be	AUX
ejpam-6153	109	34	called	call	VERB
ejpam-6153	109	35	minimal	minimal	ADJ
ejpam-6153	109	36	if	if	SCONJ
ejpam-6153	109	37	for	for	ADP
ejpam-6153	109	38	every146	every146	PROPN
ejpam-6153	109	39	essential	essential	ADJ
ejpam-6153	109	40	fuzzy	fuzzy	ADJ
ejpam-6153	109	41	ideal	ideal	NOUN
ejpam-6153	109	42	g	g	NOUN
ejpam-6153	109	43	of	of	ADP
ejpam-6153	109	44	r	r	NOUN
ejpam-6153	109	45	such	such	ADJ
ejpam-6153	109	46	that	that	SCONJ
ejpam-6153	109	47	g	g	PROPN
ejpam-6153	109	48	⊆	⊆	NUM
ejpam-6153	109	49	f	f	NOUN
ejpam-6153	109	50	,	,	PUNCT
ejpam-6153	109	51	we	we	PRON
ejpam-6153	109	52	have	have	VERB
ejpam-6153	109	53	supp(f	supp(f	PROPN
ejpam-6153	109	54	)	)	PUNCT
ejpam-6153	110	1	=	=	SYM
ejpam-6153	110	2	supp(g).147	supp(g).147	NOUN
ejpam-6153	110	3	theorem	theorem	VERB
ejpam-6153	110	4	3	3	NUM
ejpam-6153	110	5	.	.	PUNCT
ejpam-6153	111	1	a	a	DET
ejpam-6153	111	2	nonempty	nonempty	NOUN
ejpam-6153	111	3	subset	subset	VERB
ejpam-6153	111	4	s	s	NOUN
ejpam-6153	111	5	of	of	ADP
ejpam-6153	111	6	r	r	NOUN
ejpam-6153	111	7	is	be	AUX
ejpam-6153	111	8	a	a	DET
ejpam-6153	111	9	minimal	minimal	ADJ
ejpam-6153	111	10	essential	essential	ADJ
ejpam-6153	111	11	ideal	ideal	NOUN
ejpam-6153	111	12	of	of	ADP
ejpam-6153	111	13	r	r	NOUN
ejpam-6153	111	14	if	if	SCONJ
ejpam-6153	111	15	and	and	CCONJ
ejpam-6153	111	16	only	only	ADV
ejpam-6153	111	17	if148	if148	PROPN
ejpam-6153	111	18	cs	cs	PROPN
ejpam-6153	111	19	is	be	AUX
ejpam-6153	111	20	a	a	DET
ejpam-6153	111	21	minimal	minimal	ADJ
ejpam-6153	111	22	essential	essential	ADJ
ejpam-6153	111	23	fuzzy	fuzzy	ADJ
ejpam-6153	111	24	ideal	ideal	NOUN
ejpam-6153	111	25	of	of	ADP
ejpam-6153	111	26	r.149	r.149	NOUN
ejpam-6153	111	27	proof	proof	NOUN
ejpam-6153	111	28	.	.	PUNCT
ejpam-6153	112	1	assume	assume	VERB
ejpam-6153	112	2	that	that	SCONJ
ejpam-6153	112	3	s	s	VERB
ejpam-6153	112	4	is	be	AUX
ejpam-6153	112	5	a	a	DET
ejpam-6153	112	6	minimal	minimal	ADJ
ejpam-6153	112	7	essential	essential	ADJ
ejpam-6153	112	8	ideal	ideal	NOUN
ejpam-6153	112	9	of	of	ADP
ejpam-6153	112	10	r.	r.	PROPN
ejpam-6153	112	11	since	since	SCONJ
ejpam-6153	112	12	s	s	PROPN
ejpam-6153	112	13	is	be	AUX
ejpam-6153	112	14	an	an	DET
ejpam-6153	112	15	essential	essential	ADJ
ejpam-6153	112	16	ideal150	ideal150	PROPN
ejpam-6153	112	17	of	of	ADP
ejpam-6153	112	18	r	r	PROPN
ejpam-6153	112	19	,	,	PUNCT
ejpam-6153	112	20	we	we	PRON
ejpam-6153	112	21	have	have	VERB
ejpam-6153	112	22	cs	cs	PROPN
ejpam-6153	112	23	is	be	AUX
ejpam-6153	112	24	an	an	DET
ejpam-6153	112	25	essential	essential	ADJ
ejpam-6153	112	26	fuzzy	fuzzy	ADJ
ejpam-6153	112	27	ideal	ideal	NOUN
ejpam-6153	112	28	of	of	ADP
ejpam-6153	112	29	r	r	NOUN
ejpam-6153	112	30	by	by	ADP
ejpam-6153	112	31	theorem	theorem	NOUN
ejpam-6153	112	32	1	1	NUM
ejpam-6153	112	33	.	.	PUNCT
ejpam-6153	113	1	let	let	VERB
ejpam-6153	113	2	g	g	NOUN
ejpam-6153	113	3	be	be	AUX
ejpam-6153	113	4	any	any	DET
ejpam-6153	113	5	essential151	essential151	PROPN
ejpam-6153	113	6	fuzzy	fuzzy	ADJ
ejpam-6153	113	7	ideal	ideal	NOUN
ejpam-6153	113	8	of	of	ADP
ejpam-6153	113	9	r	r	NOUN
ejpam-6153	113	10	such	such	ADJ
ejpam-6153	113	11	that	that	SCONJ
ejpam-6153	113	12	g	g	PROPN
ejpam-6153	113	13	⊆	⊆	NUM
ejpam-6153	113	14	cs	cs	PROPN
ejpam-6153	113	15	.	.	PUNCT
ejpam-6153	114	1	thus	thus	ADV
ejpam-6153	114	2	supp(g	supp(g	NUM
ejpam-6153	114	3	)	)	PUNCT
ejpam-6153	114	4	⊆	⊆	NUM
ejpam-6153	114	5	supp(cs	supp(cs	NOUN
ejpam-6153	114	6	)	)	PUNCT
ejpam-6153	114	7	=	=	VERB
ejpam-6153	115	1	s.	s.	PROPN
ejpam-6153	115	2	since	since	SCONJ
ejpam-6153	115	3	g	g	PROPN
ejpam-6153	115	4	is	be	AUX
ejpam-6153	115	5	an	an	DET
ejpam-6153	115	6	essential152	essential152	PROPN
ejpam-6153	115	7	fuzzy	fuzzy	ADJ
ejpam-6153	115	8	ideal	ideal	NOUN
ejpam-6153	115	9	of	of	ADP
ejpam-6153	115	10	r	r	NOUN
ejpam-6153	115	11	,	,	PUNCT
ejpam-6153	115	12	by	by	ADP
ejpam-6153	115	13	theorem	theorem	NOUN
ejpam-6153	115	14	2	2	NUM
ejpam-6153	115	15	,	,	PUNCT
ejpam-6153	115	16	supp(g	supp(g	NOUN
ejpam-6153	115	17	)	)	PUNCT
ejpam-6153	115	18	is	be	AUX
ejpam-6153	115	19	an	an	DET
ejpam-6153	115	20	essential	essential	ADJ
ejpam-6153	115	21	ideal	ideal	NOUN
ejpam-6153	115	22	of	of	ADP
ejpam-6153	115	23	r.	r.	PROPN
ejpam-6153	115	24	therefore	therefore	ADV
ejpam-6153	115	25	supp(g	supp(g	PROPN
ejpam-6153	115	26	)	)	PUNCT
ejpam-6153	115	27	=	=	PUNCT
ejpam-6153	116	1	s153	s153	VERB
ejpam-6153	116	2	because	because	SCONJ
ejpam-6153	116	3	s	s	NOUN
ejpam-6153	116	4	is	be	AUX
ejpam-6153	116	5	minimal	minimal	ADJ
ejpam-6153	116	6	.	.	PUNCT
ejpam-6153	117	1	this	this	PRON
ejpam-6153	117	2	implies	imply	VERB
ejpam-6153	117	3	that	that	SCONJ
ejpam-6153	117	4	supp(g	supp(g	NOUN
ejpam-6153	117	5	)	)	PUNCT
ejpam-6153	117	6	=	=	SYM
ejpam-6153	117	7	supp(cs	supp(cs	NOUN
ejpam-6153	117	8	)	)	PUNCT
ejpam-6153	117	9	.	.	PUNCT
ejpam-6153	118	1	hence	hence	ADV
ejpam-6153	118	2	,	,	PUNCT
ejpam-6153	118	3	cs	cs	PROPN
ejpam-6153	118	4	is	be	AUX
ejpam-6153	118	5	minimal.154	minimal.154	VERB
ejpam-6153	118	6	to	to	PART
ejpam-6153	118	7	prove	prove	VERB
ejpam-6153	118	8	the	the	DET
ejpam-6153	118	9	converse	converse	NOUN
ejpam-6153	118	10	,	,	PUNCT
ejpam-6153	118	11	we	we	PRON
ejpam-6153	118	12	suppose	suppose	VERB
ejpam-6153	118	13	that	that	SCONJ
ejpam-6153	118	14	cs	cs	PROPN
ejpam-6153	118	15	is	be	AUX
ejpam-6153	118	16	a	a	DET
ejpam-6153	118	17	minimal	minimal	ADJ
ejpam-6153	118	18	essential	essential	ADJ
ejpam-6153	118	19	fuzzy	fuzzy	ADJ
ejpam-6153	118	20	ideal	ideal	NOUN
ejpam-6153	118	21	of	of	ADP
ejpam-6153	118	22	r155	r155	PROPN
ejpam-6153	118	23	and	and	CCONJ
ejpam-6153	118	24	let	let	VERB
ejpam-6153	118	25	i	i	PRON
ejpam-6153	118	26	be	be	AUX
ejpam-6153	118	27	an	an	DET
ejpam-6153	118	28	essential	essential	ADJ
ejpam-6153	118	29	fuzzy	fuzzy	ADJ
ejpam-6153	118	30	ideal	ideal	NOUN
ejpam-6153	118	31	of	of	ADP
ejpam-6153	118	32	r	r	NOUN
ejpam-6153	118	33	such	such	ADJ
ejpam-6153	118	34	that	that	SCONJ
ejpam-6153	118	35	i	i	PRON
ejpam-6153	118	36	⊆	⊆	NUM
ejpam-6153	118	37	s.	s.	PROPN
ejpam-6153	118	38	this	this	PRON
ejpam-6153	118	39	implies	imply	VERB
ejpam-6153	118	40	that	that	SCONJ
ejpam-6153	118	41	ci	ci	PROPN
ejpam-6153	118	42	is	be	AUX
ejpam-6153	118	43	an156	an156	PROPN
ejpam-6153	118	44	essential	essential	ADJ
ejpam-6153	118	45	fuzzy	fuzzy	ADJ
ejpam-6153	118	46	ideal	ideal	NOUN
ejpam-6153	118	47	of	of	ADP
ejpam-6153	118	48	r	r	NOUN
ejpam-6153	118	49	such	such	ADJ
ejpam-6153	118	50	that	that	DET
ejpam-6153	118	51	ci	ci	PROPN
ejpam-6153	118	52	⊆	⊆	NUM
ejpam-6153	118	53	cs	cs	PROPN
ejpam-6153	118	54	.	.	PUNCT
ejpam-6153	119	1	since	since	SCONJ
ejpam-6153	119	2	cs	cs	PROPN
ejpam-6153	119	3	is	be	AUX
ejpam-6153	119	4	minimal	minimal	ADJ
ejpam-6153	119	5	,	,	PUNCT
ejpam-6153	119	6	supp(ci	supp(ci	NOUN
ejpam-6153	119	7	)	)	PUNCT
ejpam-6153	119	8	=	=	SYM
ejpam-6153	119	9	supp(cs).157	supp(cs).157	PROPN
ejpam-6153	119	10	therefore	therefore	ADV
ejpam-6153	119	11	,	,	PUNCT
ejpam-6153	119	12	i	i	PROPN
ejpam-6153	119	13	=	=	SYM
ejpam-6153	119	14	supp(ci	supp(ci	ADJ
ejpam-6153	119	15	)	)	PUNCT
ejpam-6153	119	16	=	=	SYM
ejpam-6153	119	17	supp(cs	supp(cs	NOUN
ejpam-6153	119	18	)	)	PUNCT
ejpam-6153	119	19	=	=	SYM
ejpam-6153	120	1	s.	s.	PROPN
ejpam-6153	120	2	hence	hence	ADV
ejpam-6153	120	3	,	,	PUNCT
ejpam-6153	120	4	s	s	VERB
ejpam-6153	120	5	is	be	AUX
ejpam-6153	120	6	minimal.158	minimal.158	NOUN
ejpam-6153	120	7	an	an	DET
ejpam-6153	120	8	essential	essential	ADJ
ejpam-6153	120	9	ideal	ideal	NOUN
ejpam-6153	120	10	i	i	PRON
ejpam-6153	120	11	of	of	ADP
ejpam-6153	120	12	r	r	NOUN
ejpam-6153	120	13	is	be	AUX
ejpam-6153	120	14	called	call	VERB
ejpam-6153	120	15	prime	prime	ADJ
ejpam-6153	120	16	if	if	SCONJ
ejpam-6153	120	17	rs	rs	NOUN
ejpam-6153	120	18	∈	∈	PROPN
ejpam-6153	120	19	i	i	PRON
ejpam-6153	120	20	implies	imply	VERB
ejpam-6153	120	21	r	r	NOUN
ejpam-6153	120	22	∈	∈	PROPN
ejpam-6153	120	23	i	i	PRON
ejpam-6153	120	24	or	or	CCONJ
ejpam-6153	120	25	s	s	NOUN
ejpam-6153	120	26	∈	∈	PROPN
ejpam-6153	120	27	i	i	PRON
ejpam-6153	120	28	for	for	ADP
ejpam-6153	120	29	all	all	DET
ejpam-6153	120	30	r	r	NOUN
ejpam-6153	120	31	,	,	PUNCT
ejpam-6153	120	32	s	s	PART
ejpam-6153	120	33	∈	∈	PROPN
ejpam-6153	120	34	r.159	r.159	VERB
ejpam-6153	120	35	an	an	DET
ejpam-6153	120	36	essential	essential	ADJ
ejpam-6153	120	37	fuzzy	fuzzy	ADJ
ejpam-6153	120	38	ideal	ideal	NOUN
ejpam-6153	120	39	f	f	PROPN
ejpam-6153	120	40	of	of	ADP
ejpam-6153	120	41	r	r	NOUN
ejpam-6153	120	42	is	be	AUX
ejpam-6153	120	43	called	call	VERB
ejpam-6153	120	44	prime	prime	ADJ
ejpam-6153	120	45	if	if	SCONJ
ejpam-6153	120	46	f(rs	f(r	NOUN
ejpam-6153	120	47	)	)	PUNCT
ejpam-6153	120	48	≤	≤	NUM
ejpam-6153	120	49	max{f(r	max{f(r	PROPN
ejpam-6153	120	50	)	)	PUNCT
ejpam-6153	120	51	,	,	PUNCT
ejpam-6153	120	52	f(s	f(s	ADV
ejpam-6153	120	53	)	)	PUNCT
ejpam-6153	120	54	}	}	PUNCT
ejpam-6153	120	55	for	for	ADP
ejpam-6153	120	56	all	all	DET
ejpam-6153	120	57	r	r	NOUN
ejpam-6153	120	58	,	,	PUNCT
ejpam-6153	120	59	s	s	PART
ejpam-6153	120	60	∈	∈	PROPN
ejpam-6153	120	61	r.160	r.160	NOUN
ejpam-6153	120	62	r.	r.	PROPN
ejpam-6153	120	63	chinram	chinram	PROPN
ejpam-6153	120	64	,	,	PUNCT
ejpam-6153	120	65	s.	s.	PROPN
ejpam-6153	120	66	hangsawat	hangsawat	PROPN
ejpam-6153	120	67	/	/	SYM
ejpam-6153	120	68	eur	eur	PROPN
ejpam-6153	120	69	.	.	PUNCT
ejpam-6153	121	1	j.	j.	PROPN
ejpam-6153	121	2	pure	pure	PROPN
ejpam-6153	121	3	appl	appl	PROPN
ejpam-6153	121	4	.	.	PROPN
ejpam-6153	121	5	math	math	PROPN
ejpam-6153	121	6	,	,	PUNCT
ejpam-6153	121	7	18	18	NUM
ejpam-6153	121	8	(	(	PUNCT
ejpam-6153	121	9	3	3	NUM
ejpam-6153	121	10	)	)	PUNCT
ejpam-6153	121	11	(	(	PUNCT
ejpam-6153	121	12	2025	2025	NUM
ejpam-6153	121	13	)	)	PUNCT
ejpam-6153	121	14	,	,	PUNCT
ejpam-6153	121	15	6153	6153	NUM
ejpam-6153	121	16	6	6	NUM
ejpam-6153	121	17	of	of	ADP
ejpam-6153	121	18	7	7	NUM
ejpam-6153	121	19	theorem	theorem	ADJ
ejpam-6153	121	20	4	4	NUM
ejpam-6153	121	21	.	.	PUNCT
ejpam-6153	122	1	a	a	DET
ejpam-6153	122	2	nonempty	nonempty	NOUN
ejpam-6153	122	3	subset	subset	VERB
ejpam-6153	122	4	s	s	NOUN
ejpam-6153	122	5	of	of	ADP
ejpam-6153	122	6	r	r	NOUN
ejpam-6153	122	7	is	be	AUX
ejpam-6153	122	8	a	a	DET
ejpam-6153	122	9	prime	prime	ADJ
ejpam-6153	122	10	essential	essential	ADJ
ejpam-6153	122	11	ideal	ideal	NOUN
ejpam-6153	122	12	of	of	ADP
ejpam-6153	122	13	r	r	NOUN
ejpam-6153	122	14	if	if	SCONJ
ejpam-6153	123	1	and	and	CCONJ
ejpam-6153	123	2	only	only	ADV
ejpam-6153	123	3	if	if	SCONJ
ejpam-6153	123	4	cs161	cs161	ADJ
ejpam-6153	123	5	is	be	AUX
ejpam-6153	123	6	a	a	DET
ejpam-6153	123	7	prime	prime	ADJ
ejpam-6153	123	8	essential	essential	ADJ
ejpam-6153	123	9	fuzzy	fuzzy	ADJ
ejpam-6153	123	10	ideal	ideal	NOUN
ejpam-6153	123	11	of	of	ADP
ejpam-6153	123	12	r.162	r.162	NOUN
ejpam-6153	123	13	proof	proof	NOUN
ejpam-6153	123	14	.	.	PUNCT
ejpam-6153	124	1	let	let	VERB
ejpam-6153	124	2	s	s	PRON
ejpam-6153	124	3	be	be	AUX
ejpam-6153	124	4	a	a	DET
ejpam-6153	124	5	prime	prime	ADJ
ejpam-6153	124	6	essential	essential	ADJ
ejpam-6153	124	7	ideal	ideal	NOUN
ejpam-6153	124	8	of	of	ADP
ejpam-6153	124	9	r.	r.	PROPN
ejpam-6153	124	10	by	by	ADP
ejpam-6153	124	11	theorem	theorem	NOUN
ejpam-6153	124	12	1	1	NUM
ejpam-6153	124	13	,	,	PUNCT
ejpam-6153	124	14	cs	cs	PROPN
ejpam-6153	124	15	is	be	AUX
ejpam-6153	124	16	an	an	DET
ejpam-6153	124	17	essential	essential	ADJ
ejpam-6153	124	18	fuzzy163	fuzzy163	PROPN
ejpam-6153	124	19	ideal	ideal	NOUN
ejpam-6153	124	20	of	of	ADP
ejpam-6153	124	21	r.	r.	PROPN
ejpam-6153	124	22	let	let	VERB
ejpam-6153	124	23	r	r	NOUN
ejpam-6153	124	24	and	and	CCONJ
ejpam-6153	124	25	s	s	VERB
ejpam-6153	124	26	be	be	AUX
ejpam-6153	124	27	any	any	DET
ejpam-6153	124	28	two	two	NUM
ejpam-6153	124	29	elements	element	NOUN
ejpam-6153	124	30	of	of	ADP
ejpam-6153	124	31	r.	r.	PROPN
ejpam-6153	124	32	if	if	SCONJ
ejpam-6153	124	33	rs	rs	PROPN
ejpam-6153	124	34	∈	∈	PROPN
ejpam-6153	124	35	s	s	PART
ejpam-6153	124	36	,	,	PUNCT
ejpam-6153	124	37	then	then	ADV
ejpam-6153	124	38	we	we	PRON
ejpam-6153	124	39	have	have	VERB
ejpam-6153	124	40	that	that	PRON
ejpam-6153	124	41	r	r	NOUN
ejpam-6153	124	42	∈	∈	NOUN
ejpam-6153	124	43	s	s	PART
ejpam-6153	124	44	or164	or164	NOUN
ejpam-6153	124	45	s	s	PART
ejpam-6153	124	46	∈	∈	NOUN
ejpam-6153	124	47	s	s	X
ejpam-6153	124	48	because	because	SCONJ
ejpam-6153	124	49	s	s	NOUN
ejpam-6153	124	50	is	be	AUX
ejpam-6153	124	51	prime	prime	ADJ
ejpam-6153	124	52	.	.	PUNCT
ejpam-6153	125	1	thus	thus	ADV
ejpam-6153	125	2	max{cs(r	max{cs(r	PROPN
ejpam-6153	125	3	)	)	PUNCT
ejpam-6153	125	4	,	,	PUNCT
ejpam-6153	125	5	cs(s	cs(s	NUM
ejpam-6153	125	6	)	)	PUNCT
ejpam-6153	125	7	}	}	PUNCT
ejpam-6153	125	8	=	=	SYM
ejpam-6153	125	9	1	1	NUM
ejpam-6153	125	10	≥	≥	NOUN
ejpam-6153	125	11	cs(rs	cs(rs	PROPN
ejpam-6153	125	12	)	)	PUNCT
ejpam-6153	125	13	.	.	PUNCT
ejpam-6153	126	1	otherwise	otherwise	ADV
ejpam-6153	126	2	,	,	PUNCT
ejpam-6153	126	3	if	if	SCONJ
ejpam-6153	126	4	rs	rs	PROPN
ejpam-6153	126	5	̸∈	̸∈	PROPN
ejpam-6153	126	6	s,165	s,165	VERB
ejpam-6153	126	7	then	then	ADV
ejpam-6153	126	8	cs(rs	cs(rs	NOUN
ejpam-6153	126	9	)	)	PUNCT
ejpam-6153	126	10	=	=	SYM
ejpam-6153	126	11	0	0	NUM
ejpam-6153	126	12	≤	≤	NUM
ejpam-6153	126	13	max{cs(r	max{cs(r	PROPN
ejpam-6153	126	14	)	)	PUNCT
ejpam-6153	126	15	,	,	PUNCT
ejpam-6153	126	16	cs(s	cs(s	NOUN
ejpam-6153	126	17	)	)	PUNCT
ejpam-6153	126	18	}	}	PUNCT
ejpam-6153	126	19	.	.	PUNCT
ejpam-6153	127	1	by	by	ADP
ejpam-6153	127	2	both	both	DET
ejpam-6153	127	3	two	two	NUM
ejpam-6153	127	4	cases	case	NOUN
ejpam-6153	127	5	,	,	PUNCT
ejpam-6153	127	6	we	we	PRON
ejpam-6153	127	7	conclude	conclude	VERB
ejpam-6153	127	8	that	that	SCONJ
ejpam-6153	127	9	cs	cs	PROPN
ejpam-6153	127	10	is	be	AUX
ejpam-6153	127	11	a166	a166	PROPN
ejpam-6153	127	12	prime	prime	ADJ
ejpam-6153	127	13	essential	essential	ADJ
ejpam-6153	127	14	fuzzy	fuzzy	ADJ
ejpam-6153	127	15	ideal	ideal	NOUN
ejpam-6153	127	16	of	of	ADP
ejpam-6153	127	17	r.	r.	PROPN
ejpam-6153	127	18	conversely	conversely	ADV
ejpam-6153	127	19	,	,	PUNCT
ejpam-6153	127	20	we	we	PRON
ejpam-6153	127	21	assume	assume	VERB
ejpam-6153	127	22	that	that	SCONJ
ejpam-6153	127	23	cs	cs	PROPN
ejpam-6153	127	24	is	be	AUX
ejpam-6153	127	25	a	a	DET
ejpam-6153	127	26	prime	prime	ADJ
ejpam-6153	127	27	essential	essential	ADJ
ejpam-6153	127	28	fuzzy167	fuzzy167	PROPN
ejpam-6153	127	29	ideal	ideal	NOUN
ejpam-6153	127	30	of	of	ADP
ejpam-6153	127	31	r.	r.	PROPN
ejpam-6153	127	32	it	it	PRON
ejpam-6153	127	33	follows	follow	VERB
ejpam-6153	127	34	by	by	ADP
ejpam-6153	127	35	theorem	theorem	NOUN
ejpam-6153	127	36	1	1	NUM
ejpam-6153	127	37	that	that	PRON
ejpam-6153	127	38	s	s	VERB
ejpam-6153	127	39	is	be	AUX
ejpam-6153	127	40	an	an	DET
ejpam-6153	127	41	essential	essential	ADJ
ejpam-6153	127	42	ideal	ideal	NOUN
ejpam-6153	127	43	of	of	ADP
ejpam-6153	127	44	r.	r.	PROPN
ejpam-6153	127	45	let	let	VERB
ejpam-6153	127	46	r	r	NOUN
ejpam-6153	127	47	and	and	CCONJ
ejpam-6153	127	48	s	s	PROPN
ejpam-6153	127	49	be168	be168	PROPN
ejpam-6153	127	50	two	two	NUM
ejpam-6153	127	51	elements	element	NOUN
ejpam-6153	127	52	of	of	ADP
ejpam-6153	127	53	r	r	NOUN
ejpam-6153	127	54	such	such	ADJ
ejpam-6153	127	55	that	that	DET
ejpam-6153	127	56	rs	rs	PROPN
ejpam-6153	127	57	∈	∈	PROPN
ejpam-6153	127	58	s.	s.	PROPN
ejpam-6153	127	59	this	this	PRON
ejpam-6153	127	60	implies	imply	VERB
ejpam-6153	127	61	that	that	SCONJ
ejpam-6153	127	62	cs(rs	cs(rs	NOUN
ejpam-6153	127	63	)	)	PUNCT
ejpam-6153	127	64	=	=	SYM
ejpam-6153	128	1	1	1	X
ejpam-6153	128	2	.	.	PUNCT
ejpam-6153	128	3	since	since	SCONJ
ejpam-6153	128	4	cs	cs	PROPN
ejpam-6153	128	5	is	be	AUX
ejpam-6153	128	6	prime,169	prime,169	PROPN
ejpam-6153	128	7	cs(rs	cs(rs	PROPN
ejpam-6153	128	8	)	)	PUNCT
ejpam-6153	128	9	≤	≤	NOUN
ejpam-6153	128	10	max{cs(r	max{cs(r	PROPN
ejpam-6153	128	11	)	)	PUNCT
ejpam-6153	128	12	,	,	PUNCT
ejpam-6153	128	13	cs(s	cs(s	NOUN
ejpam-6153	128	14	)	)	PUNCT
ejpam-6153	128	15	}	}	PUNCT
ejpam-6153	128	16	.	.	PUNCT
ejpam-6153	129	1	then	then	ADV
ejpam-6153	129	2	max{cs(r	max{cs(r	PROPN
ejpam-6153	129	3	)	)	PUNCT
ejpam-6153	129	4	,	,	PUNCT
ejpam-6153	129	5	cs(s	cs(s	NUM
ejpam-6153	129	6	)	)	PUNCT
ejpam-6153	129	7	}	}	PUNCT
ejpam-6153	129	8	must	must	AUX
ejpam-6153	129	9	be	be	AUX
ejpam-6153	129	10	equal	equal	ADJ
ejpam-6153	129	11	to	to	ADP
ejpam-6153	129	12	1	1	NUM
ejpam-6153	129	13	and	and	CCONJ
ejpam-6153	129	14	so	so	ADV
ejpam-6153	129	15	r	r	NOUN
ejpam-6153	129	16	∈	∈	PROPN
ejpam-6153	129	17	s170	s170	NUM
ejpam-6153	129	18	or	or	CCONJ
ejpam-6153	129	19	s	s	PROPN
ejpam-6153	129	20	∈	∈	PROPN
ejpam-6153	129	21	s.	s.	PROPN
ejpam-6153	129	22	hence	hence	ADV
ejpam-6153	129	23	,	,	PUNCT
ejpam-6153	129	24	s	s	VERB
ejpam-6153	129	25	is	be	AUX
ejpam-6153	129	26	a	a	DET
ejpam-6153	129	27	prime	prime	ADJ
ejpam-6153	129	28	essential	essential	ADJ
ejpam-6153	129	29	ideal	ideal	NOUN
ejpam-6153	129	30	of	of	ADP
ejpam-6153	129	31	r.171	r.171	NOUN
ejpam-6153	129	32	an	an	DET
ejpam-6153	129	33	essential	essential	ADJ
ejpam-6153	129	34	ideal	ideal	NOUN
ejpam-6153	130	1	i	i	PRON
ejpam-6153	130	2	of	of	ADP
ejpam-6153	130	3	r	r	NOUN
ejpam-6153	130	4	is	be	AUX
ejpam-6153	130	5	called	call	VERB
ejpam-6153	130	6	semiprime	semiprime	NOUN
ejpam-6153	130	7	if	if	SCONJ
ejpam-6153	130	8	for	for	ADP
ejpam-6153	130	9	all	all	DET
ejpam-6153	130	10	r	r	NOUN
ejpam-6153	130	11	∈	∈	NOUN
ejpam-6153	130	12	r	r	NOUN
ejpam-6153	130	13	,	,	PUNCT
ejpam-6153	130	14	r2	r2	PROPN
ejpam-6153	130	15	∈	∈	PROPN
ejpam-6153	130	16	i	i	PRON
ejpam-6153	130	17	implies	imply	VERB
ejpam-6153	130	18	r	r	PROPN
ejpam-6153	130	19	∈	∈	PROPN
ejpam-6153	130	20	i.	i.	NOUN
ejpam-6153	130	21	an172	an172	PROPN
ejpam-6153	130	22	essential	essential	ADJ
ejpam-6153	130	23	fuzzy	fuzzy	ADJ
ejpam-6153	130	24	ideal	ideal	ADJ
ejpam-6153	130	25	f	f	PROPN
ejpam-6153	130	26	of	of	ADP
ejpam-6153	130	27	r	r	NOUN
ejpam-6153	130	28	is	be	AUX
ejpam-6153	130	29	called	call	VERB
ejpam-6153	130	30	semiprime	semiprime	NOUN
ejpam-6153	130	31	if	if	SCONJ
ejpam-6153	130	32	for	for	ADP
ejpam-6153	130	33	all	all	DET
ejpam-6153	130	34	r	r	NOUN
ejpam-6153	130	35	∈	∈	NOUN
ejpam-6153	130	36	r	r	NOUN
ejpam-6153	130	37	,	,	PUNCT
ejpam-6153	130	38	f(r2	f(r2	ADJ
ejpam-6153	130	39	)	)	PUNCT
ejpam-6153	130	40	≤	≤	NOUN
ejpam-6153	130	41	f(r).173	f(r).173	NOUN
ejpam-6153	130	42	theorem	theorem	VERB
ejpam-6153	130	43	5	5	NUM
ejpam-6153	130	44	.	.	PUNCT
ejpam-6153	131	1	a	a	DET
ejpam-6153	131	2	nonempty	nonempty	NOUN
ejpam-6153	131	3	subset	subset	VERB
ejpam-6153	131	4	s	s	NOUN
ejpam-6153	131	5	of	of	ADP
ejpam-6153	131	6	r	r	NOUN
ejpam-6153	131	7	is	be	AUX
ejpam-6153	131	8	a	a	DET
ejpam-6153	131	9	semiprime	semiprime	NOUN
ejpam-6153	131	10	essential	essential	ADJ
ejpam-6153	131	11	ideal	ideal	NOUN
ejpam-6153	131	12	of	of	ADP
ejpam-6153	131	13	r	r	NOUN
ejpam-6153	131	14	if	if	SCONJ
ejpam-6153	131	15	and	and	CCONJ
ejpam-6153	131	16	only174	only174	ADV
ejpam-6153	131	17	if	if	SCONJ
ejpam-6153	131	18	cs	cs	PROPN
ejpam-6153	131	19	is	be	AUX
ejpam-6153	131	20	a	a	DET
ejpam-6153	131	21	semiprime	semiprime	NOUN
ejpam-6153	131	22	essential	essential	ADJ
ejpam-6153	131	23	fuzzy	fuzzy	ADJ
ejpam-6153	131	24	ideal	ideal	NOUN
ejpam-6153	131	25	of	of	ADP
ejpam-6153	131	26	r.175	r.175	PROPN
ejpam-6153	131	27	proof	proof	NOUN
ejpam-6153	131	28	.	.	PUNCT
ejpam-6153	132	1	let	let	VERB
ejpam-6153	132	2	s	s	PRON
ejpam-6153	132	3	be	be	AUX
ejpam-6153	132	4	a	a	DET
ejpam-6153	132	5	semiprime	semiprime	NOUN
ejpam-6153	132	6	essential	essential	ADJ
ejpam-6153	132	7	ideal	ideal	NOUN
ejpam-6153	132	8	of	of	ADP
ejpam-6153	132	9	r.	r.	PROPN
ejpam-6153	132	10	by	by	ADP
ejpam-6153	132	11	theorem	theorem	NOUN
ejpam-6153	132	12	1	1	NUM
ejpam-6153	132	13	,	,	PUNCT
ejpam-6153	132	14	cs	cs	PROPN
ejpam-6153	132	15	is	be	AUX
ejpam-6153	132	16	an	an	DET
ejpam-6153	132	17	essential176	essential176	PROPN
ejpam-6153	132	18	fuzzy	fuzzy	ADJ
ejpam-6153	132	19	ideal	ideal	NOUN
ejpam-6153	132	20	of	of	ADP
ejpam-6153	132	21	r.	r.	PROPN
ejpam-6153	132	22	let	let	VERB
ejpam-6153	132	23	r	r	NOUN
ejpam-6153	132	24	be	be	AUX
ejpam-6153	132	25	any	any	DET
ejpam-6153	132	26	element	element	NOUN
ejpam-6153	132	27	in	in	ADP
ejpam-6153	132	28	r.	r.	PROPN
ejpam-6153	132	29	if	if	SCONJ
ejpam-6153	132	30	r2	r2	PROPN
ejpam-6153	132	31	∈	∈	PROPN
ejpam-6153	132	32	s	s	PART
ejpam-6153	132	33	,	,	PUNCT
ejpam-6153	132	34	then	then	ADV
ejpam-6153	132	35	we	we	PRON
ejpam-6153	132	36	have	have	VERB
ejpam-6153	132	37	r	r	NOUN
ejpam-6153	132	38	∈	∈	NOUN
ejpam-6153	132	39	s	s	NOUN
ejpam-6153	132	40	because	because	SCONJ
ejpam-6153	132	41	s	s	PART
ejpam-6153	132	42	is177	is177	ADJ
ejpam-6153	132	43	semiprime	semiprime	NOUN
ejpam-6153	132	44	.	.	PUNCT
ejpam-6153	133	1	this	this	PRON
ejpam-6153	133	2	implies	imply	VERB
ejpam-6153	133	3	that	that	PRON
ejpam-6153	133	4	cs(r	cs(r	PUNCT
ejpam-6153	133	5	)	)	PUNCT
ejpam-6153	133	6	=	=	SYM
ejpam-6153	133	7	1	1	X
ejpam-6153	133	8	.	.	PUNCT
ejpam-6153	133	9	hence	hence	ADV
ejpam-6153	133	10	,	,	PUNCT
ejpam-6153	133	11	cs(r	cs(r	PUNCT
ejpam-6153	133	12	)	)	PUNCT
ejpam-6153	133	13	≥	≥	NOUN
ejpam-6153	133	14	cs(r	cs(r	PUNCT
ejpam-6153	133	15	2	2	NUM
ejpam-6153	133	16	)	)	PUNCT
ejpam-6153	133	17	.	.	PUNCT
ejpam-6153	134	1	otherwise	otherwise	ADV
ejpam-6153	134	2	,	,	PUNCT
ejpam-6153	134	3	if	if	SCONJ
ejpam-6153	134	4	r2	r2	PROPN
ejpam-6153	134	5	̸∈	̸∈	PROPN
ejpam-6153	134	6	s,178	s,178	PROPN
ejpam-6153	134	7	then	then	ADV
ejpam-6153	134	8	cs(r	cs(r	PUNCT
ejpam-6153	134	9	2	2	X
ejpam-6153	134	10	)	)	PUNCT
ejpam-6153	134	11	=	=	SYM
ejpam-6153	134	12	0	0	NUM
ejpam-6153	134	13	≤	≤	NUM
ejpam-6153	134	14	cs(r	cs(r	NUM
ejpam-6153	134	15	)	)	PUNCT
ejpam-6153	134	16	.	.	PUNCT
ejpam-6153	135	1	by	by	ADP
ejpam-6153	135	2	both	both	DET
ejpam-6153	135	3	two	two	NUM
ejpam-6153	135	4	cases	case	NOUN
ejpam-6153	135	5	,	,	PUNCT
ejpam-6153	135	6	we	we	PRON
ejpam-6153	135	7	conclude	conclude	VERB
ejpam-6153	135	8	that	that	SCONJ
ejpam-6153	135	9	cs	cs	PROPN
ejpam-6153	135	10	is	be	AUX
ejpam-6153	135	11	a	a	DET
ejpam-6153	135	12	semiprime	semiprime	NOUN
ejpam-6153	135	13	essential179	essential179	PROPN
ejpam-6153	135	14	fuzzy	fuzzy	ADJ
ejpam-6153	135	15	ideal	ideal	NOUN
ejpam-6153	135	16	of	of	ADP
ejpam-6153	135	17	r.	r.	PROPN
ejpam-6153	135	18	to	to	PART
ejpam-6153	135	19	prove	prove	VERB
ejpam-6153	135	20	the	the	DET
ejpam-6153	135	21	converse	converse	NOUN
ejpam-6153	135	22	,	,	PUNCT
ejpam-6153	135	23	assume	assume	VERB
ejpam-6153	135	24	that	that	SCONJ
ejpam-6153	135	25	cs	cs	PROPN
ejpam-6153	135	26	is	be	AUX
ejpam-6153	135	27	a	a	DET
ejpam-6153	135	28	semiprime	semiprime	NOUN
ejpam-6153	135	29	essential	essential	ADJ
ejpam-6153	135	30	fuzzy180	fuzzy180	PROPN
ejpam-6153	135	31	ideal	ideal	NOUN
ejpam-6153	135	32	of	of	ADP
ejpam-6153	135	33	r.	r.	PROPN
ejpam-6153	135	34	by	by	ADP
ejpam-6153	135	35	theorem	theorem	NOUN
ejpam-6153	135	36	1	1	NUM
ejpam-6153	135	37	,	,	PUNCT
ejpam-6153	135	38	we	we	PRON
ejpam-6153	135	39	have	have	VERB
ejpam-6153	135	40	that	that	DET
ejpam-6153	135	41	s	s	NOUN
ejpam-6153	135	42	is	be	AUX
ejpam-6153	135	43	an	an	DET
ejpam-6153	135	44	essential	essential	ADJ
ejpam-6153	135	45	ideal	ideal	NOUN
ejpam-6153	135	46	of	of	ADP
ejpam-6153	135	47	r.	r.	PROPN
ejpam-6153	135	48	let	let	VERB
ejpam-6153	135	49	r	r	NOUN
ejpam-6153	135	50	∈	∈	NOUN
ejpam-6153	135	51	r	r	NOUN
ejpam-6153	136	1	such	such	ADJ
ejpam-6153	136	2	that181	that181	PROPN
ejpam-6153	136	3	r2	r2	PROPN
ejpam-6153	136	4	∈	∈	PROPN
ejpam-6153	136	5	s.	s.	PROPN
ejpam-6153	137	1	so	so	ADV
ejpam-6153	137	2	cs(r	cs(r	PUNCT
ejpam-6153	137	3	2	2	X
ejpam-6153	137	4	)	)	PUNCT
ejpam-6153	137	5	=	=	SYM
ejpam-6153	137	6	1	1	X
ejpam-6153	137	7	.	.	PUNCT
ejpam-6153	137	8	since	since	SCONJ
ejpam-6153	137	9	cs	cs	PROPN
ejpam-6153	137	10	is	be	AUX
ejpam-6153	137	11	semiprime	semiprime	NOUN
ejpam-6153	137	12	,	,	PUNCT
ejpam-6153	137	13	cs(r	cs(r	PUNCT
ejpam-6153	137	14	2	2	X
ejpam-6153	137	15	)	)	PUNCT
ejpam-6153	137	16	≤	≤	NOUN
ejpam-6153	137	17	cs(r	cs(r	NUM
ejpam-6153	137	18	)	)	PUNCT
ejpam-6153	137	19	.	.	PUNCT
ejpam-6153	138	1	since	since	SCONJ
ejpam-6153	138	2	cs(r	cs(r	PUNCT
ejpam-6153	138	3	2	2	X
ejpam-6153	138	4	)	)	PUNCT
ejpam-6153	138	5	=	=	SYM
ejpam-6153	138	6	1	1	NUM
ejpam-6153	138	7	,	,	PUNCT
ejpam-6153	138	8	cs(r)182	cs(r)182	NOUN
ejpam-6153	138	9	must	must	AUX
ejpam-6153	138	10	be	be	AUX
ejpam-6153	138	11	equal	equal	ADJ
ejpam-6153	138	12	1	1	NUM
ejpam-6153	138	13	.	.	PUNCT
ejpam-6153	139	1	therefore	therefore	ADV
ejpam-6153	139	2	,	,	PUNCT
ejpam-6153	139	3	r	r	PROPN
ejpam-6153	139	4	∈	∈	PROPN
ejpam-6153	139	5	s.	s.	PROPN
ejpam-6153	139	6	hence	hence	ADV
ejpam-6153	139	7	,	,	PUNCT
ejpam-6153	139	8	s	s	VERB
ejpam-6153	139	9	is	be	AUX
ejpam-6153	139	10	a	a	DET
ejpam-6153	139	11	semiprime	semiprime	NOUN
ejpam-6153	139	12	essential	essential	ADJ
ejpam-6153	139	13	ideal	ideal	NOUN
ejpam-6153	139	14	of	of	ADP
ejpam-6153	139	15	r.183	r.183	PROPN
ejpam-6153	139	16	4	4	NUM
ejpam-6153	139	17	.	.	PUNCT
ejpam-6153	140	1	conclusion184	conclusion184	AUX
ejpam-6153	140	2	let	let	VERB
ejpam-6153	140	3	r	r	NOUN
ejpam-6153	140	4	be	be	AUX
ejpam-6153	140	5	a	a	DET
ejpam-6153	140	6	semiring	semiring	NOUN
ejpam-6153	140	7	with	with	ADP
ejpam-6153	140	8	zero	zero	NUM
ejpam-6153	140	9	.	.	PUNCT
ejpam-6153	141	1	in	in	ADP
ejpam-6153	141	2	this	this	DET
ejpam-6153	141	3	paper	paper	NOUN
ejpam-6153	141	4	,	,	PUNCT
ejpam-6153	141	5	we	we	PRON
ejpam-6153	141	6	first	first	ADV
ejpam-6153	141	7	show	show	VERB
ejpam-6153	141	8	some	some	DET
ejpam-6153	141	9	properties	property	NOUN
ejpam-6153	141	10	of	of	ADP
ejpam-6153	141	11	essential185	essential185	PROPN
ejpam-6153	141	12	ideals	ideal	NOUN
ejpam-6153	141	13	of	of	ADP
ejpam-6153	141	14	r	r	NOUN
ejpam-6153	141	15	(	(	PUNCT
ejpam-6153	141	16	proposition	proposition	NOUN
ejpam-6153	141	17	3	3	NUM
ejpam-6153	141	18	-	-	SYM
ejpam-6153	141	19	5	5	NUM
ejpam-6153	141	20	and	and	CCONJ
ejpam-6153	141	21	corollary	corollary	ADJ
ejpam-6153	141	22	1	1	NUM
ejpam-6153	141	23	)	)	PUNCT
ejpam-6153	141	24	.	.	PUNCT
ejpam-6153	142	1	next	next	ADV
ejpam-6153	142	2	,	,	PUNCT
ejpam-6153	142	3	we	we	PRON
ejpam-6153	142	4	introduce	introduce	VERB
ejpam-6153	142	5	the	the	DET
ejpam-6153	142	6	definition	definition	NOUN
ejpam-6153	142	7	of	of	ADP
ejpam-6153	142	8	essential186	essential186	PROPN
ejpam-6153	142	9	fuzzy	fuzzy	ADJ
ejpam-6153	142	10	ideals	ideal	NOUN
ejpam-6153	142	11	of	of	ADP
ejpam-6153	142	12	r	r	NOUN
ejpam-6153	142	13	(	(	PUNCT
ejpam-6153	142	14	definition	definition	NOUN
ejpam-6153	142	15	3	3	NUM
ejpam-6153	142	16	)	)	PUNCT
ejpam-6153	142	17	.	.	PUNCT
ejpam-6153	143	1	finally	finally	ADV
ejpam-6153	143	2	,	,	PUNCT
ejpam-6153	143	3	we	we	PRON
ejpam-6153	143	4	show	show	VERB
ejpam-6153	143	5	relationships	relationship	NOUN
ejpam-6153	143	6	between	between	ADP
ejpam-6153	143	7	essential	essential	ADJ
ejpam-6153	143	8	ideals187	ideals187	PROPN
ejpam-6153	143	9	and	and	CCONJ
ejpam-6153	143	10	essential	essential	ADJ
ejpam-6153	143	11	fuzzy	fuzzy	ADJ
ejpam-6153	143	12	ideals	ideal	NOUN
ejpam-6153	143	13	of	of	ADP
ejpam-6153	143	14	r	r	NOUN
ejpam-6153	143	15	in	in	ADP
ejpam-6153	143	16	theorem	theorem	ADJ
ejpam-6153	143	17	1	1	NUM
ejpam-6153	143	18	-	-	SYM
ejpam-6153	143	19	5.188	5.188	NUM
ejpam-6153	143	20	in	in	ADP
ejpam-6153	143	21	the	the	DET
ejpam-6153	143	22	future	future	ADJ
ejpam-6153	143	23	work	work	NOUN
ejpam-6153	143	24	,	,	PUNCT
ejpam-6153	143	25	we	we	PRON
ejpam-6153	143	26	can	can	AUX
ejpam-6153	143	27	define	define	VERB
ejpam-6153	143	28	an	an	DET
ejpam-6153	143	29	essential	essential	ADJ
ejpam-6153	143	30	ideal	ideal	NOUN
ejpam-6153	143	31	of	of	ADP
ejpam-6153	143	32	other	other	ADJ
ejpam-6153	143	33	algebraic	algebraic	ADJ
ejpam-6153	143	34	structures	structure	NOUN
ejpam-6153	143	35	and189	and189	PROPN
ejpam-6153	143	36	show	show	VERB
ejpam-6153	143	37	their	their	PRON
ejpam-6153	143	38	properties	property	NOUN
ejpam-6153	143	39	.	.	PUNCT
ejpam-6153	144	1	moreover	moreover	ADV
ejpam-6153	144	2	,	,	PUNCT
ejpam-6153	144	3	we	we	PRON
ejpam-6153	144	4	will	will	AUX
ejpam-6153	144	5	define	define	VERB
ejpam-6153	144	6	their	their	PRON
ejpam-6153	144	7	fuzzifications	fuzzification	NOUN
ejpam-6153	144	8	of	of	ADP
ejpam-6153	144	9	essential	essential	ADJ
ejpam-6153	144	10	ideals	ideal	NOUN
ejpam-6153	144	11	and190	and190	NOUN
ejpam-6153	144	12	show	show	VERB
ejpam-6153	144	13	some	some	DET
ejpam-6153	144	14	relationships	relationship	NOUN
ejpam-6153	144	15	between	between	ADP
ejpam-6153	144	16	essential	essential	ADJ
ejpam-6153	144	17	ideals	ideal	NOUN
ejpam-6153	144	18	and	and	CCONJ
ejpam-6153	144	19	essential	essential	ADJ
ejpam-6153	144	20	fuzzy	fuzzy	ADJ
ejpam-6153	144	21	ideals.191	ideals.191	NOUN
ejpam-6153	144	22	acknowledgements192	acknowledgements192	VERB
ejpam-6153	144	23	we	we	PRON
ejpam-6153	144	24	sincerely	sincerely	ADV
ejpam-6153	144	25	appreciate	appreciate	VERB
ejpam-6153	144	26	all	all	DET
ejpam-6153	144	27	valuable	valuable	ADJ
ejpam-6153	144	28	comments	comment	NOUN
ejpam-6153	144	29	and	and	CCONJ
ejpam-6153	144	30	suggestions	suggestion	NOUN
ejpam-6153	144	31	of	of	ADP
ejpam-6153	144	32	reviewers	reviewer	NOUN
ejpam-6153	144	33	,	,	PUNCT
ejpam-6153	144	34	which193	which193	PROPN
ejpam-6153	144	35	helped	help	VERB
ejpam-6153	144	36	us	we	PRON
ejpam-6153	144	37	to	to	PART
ejpam-6153	144	38	improve	improve	VERB
ejpam-6153	144	39	the	the	DET
ejpam-6153	144	40	quality	quality	NOUN
ejpam-6153	144	41	of	of	ADP
ejpam-6153	144	42	the	the	DET
ejpam-6153	144	43	manuscript.194	manuscript.194	PROPN
ejpam-6153	144	44	r.	r.	PROPN
ejpam-6153	144	45	chinram	chinram	PROPN
ejpam-6153	144	46	,	,	PUNCT
ejpam-6153	144	47	s.	s.	PROPN
ejpam-6153	144	48	hangsawat	hangsawat	PROPN
ejpam-6153	144	49	/	/	SYM
ejpam-6153	144	50	eur	eur	PROPN
ejpam-6153	144	51	.	.	PUNCT
ejpam-6153	145	1	j.	j.	PROPN
ejpam-6153	145	2	pure	pure	PROPN
ejpam-6153	145	3	appl	appl	PROPN
ejpam-6153	145	4	.	.	PROPN
ejpam-6153	145	5	math	math	PROPN
ejpam-6153	145	6	,	,	PUNCT
ejpam-6153	145	7	18	18	NUM
ejpam-6153	145	8	(	(	PUNCT
ejpam-6153	145	9	3	3	NUM
ejpam-6153	145	10	)	)	PUNCT
ejpam-6153	145	11	(	(	PUNCT
ejpam-6153	145	12	2025	2025	NUM
ejpam-6153	145	13	)	)	PUNCT
ejpam-6153	145	14	,	,	PUNCT
ejpam-6153	145	15	6153	6153	NUM
ejpam-6153	145	16	7	7	NUM
ejpam-6153	145	17	of	of	ADP
ejpam-6153	145	18	7	7	NUM
ejpam-6153	145	19	references195	references195	PROPN
ejpam-6153	145	20	[	[	X
ejpam-6153	145	21	1	1	X
ejpam-6153	145	22	]	]	PUNCT
ejpam-6153	145	23	h.	h.	PROPN
ejpam-6153	145	24	s.	s.	PROPN
ejpam-6153	145	25	vandiver	vandiver	PROPN
ejpam-6153	145	26	.	.	PUNCT
ejpam-6153	146	1	on	on	ADP
ejpam-6153	146	2	some	some	DET
ejpam-6153	146	3	simple	simple	ADJ
ejpam-6153	146	4	types	type	NOUN
ejpam-6153	146	5	of	of	ADP
ejpam-6153	146	6	semi	semi	NOUN
ejpam-6153	146	7	-	-	NOUN
ejpam-6153	146	8	rings	ring	NOUN
ejpam-6153	146	9	.	.	PUNCT
ejpam-6153	147	1	am	be	AUX
ejpam-6153	147	2	.	.	PUNCT
ejpam-6153	148	1	math	math	NOUN
ejpam-6153	148	2	.	.	PUNCT
ejpam-6153	149	1	mon	mon	PROPN
ejpam-6153	149	2	.	.	PROPN
ejpam-6153	149	3	,	,	PUNCT
ejpam-6153	149	4	46:22–26,196	46:22–26,196	NUM
ejpam-6153	149	5	1939.197	1939.197	NUM
ejpam-6153	150	1	[	[	X
ejpam-6153	150	2	2	2	X
ejpam-6153	150	3	]	]	PUNCT
ejpam-6153	150	4	d.	d.	PROPN
ejpam-6153	150	5	m.	m.	PROPN
ejpam-6153	150	6	olson	olson	PROPN
ejpam-6153	150	7	.	.	PUNCT
ejpam-6153	151	1	upper	upper	ADJ
ejpam-6153	151	2	radicals	radical	NOUN
ejpam-6153	151	3	and	and	CCONJ
ejpam-6153	151	4	essential	essential	ADJ
ejpam-6153	151	5	ideals	ideal	NOUN
ejpam-6153	151	6	.	.	PUNCT
ejpam-6153	152	1	j.	j.	PROPN
ejpam-6153	152	2	australian	australian	PROPN
ejpam-6153	152	3	math	math	PROPN
ejpam-6153	152	4	.	.	PUNCT
ejpam-6153	153	1	soc	soc	PROPN
ejpam-6153	153	2	.	.	PUNCT
ejpam-6153	153	3	,	,	PUNCT
ejpam-6153	153	4	30(4):385–198	30(4):385–198	PROPN
ejpam-6153	153	5	389	389	NUM
ejpam-6153	153	6	,	,	PUNCT
ejpam-6153	153	7	1981.199	1981.199	NUM
ejpam-6153	153	8	[	[	X
ejpam-6153	153	9	3	3	X
ejpam-6153	153	10	]	]	PUNCT
ejpam-6153	153	11	k.	k.	PROPN
ejpam-6153	153	12	f.	f.	PROPN
ejpam-6153	153	13	pawar	pawar	PROPN
ejpam-6153	153	14	.	.	PUNCT
ejpam-6153	154	1	on	on	ADP
ejpam-6153	154	2	essential	essential	ADJ
ejpam-6153	154	3	ideal	ideal	ADJ
ejpam-6153	154	4	and	and	CCONJ
ejpam-6153	154	5	radical	radical	ADJ
ejpam-6153	154	6	class	class	NOUN
ejpam-6153	154	7	.	.	PUNCT
ejpam-6153	155	1	int	int	NOUN
ejpam-6153	155	2	.	.	PUNCT
ejpam-6153	156	1	j.	j.	PROPN
ejpam-6153	156	2	pure	pure	PROPN
ejpam-6153	156	3	appl	appl	PROPN
ejpam-6153	156	4	.	.	PUNCT
ejpam-6153	156	5	math	math	PROPN
ejpam-6153	156	6	.	.	PUNCT
ejpam-6153	157	1	sci	sci	PROPN
ejpam-6153	157	2	.	.	PROPN
ejpam-6153	157	3	,200	,200	PROPN
ejpam-6153	158	1	5:1–5	5:1–5	NUM
ejpam-6153	158	2	,	,	PUNCT
ejpam-6153	158	3	2012.201	2012.201	NUM
ejpam-6153	158	4	[	[	X
ejpam-6153	158	5	4	4	NUM
ejpam-6153	158	6	]	]	PUNCT
ejpam-6153	158	7	l.	l.	PROPN
ejpam-6153	158	8	a.	a.	PROPN
ejpam-6153	158	9	zadeh	zadeh	PROPN
ejpam-6153	158	10	.	.	PUNCT
ejpam-6153	158	11	fuzzy	fuzzy	ADJ
ejpam-6153	158	12	sets	set	NOUN
ejpam-6153	158	13	.	.	PUNCT
ejpam-6153	159	1	inform	inform	NOUN
ejpam-6153	159	2	.	.	PUNCT
ejpam-6153	160	1	control	control	NOUN
ejpam-6153	160	2	,	,	PUNCT
ejpam-6153	160	3	8:338–353	8:338–353	NUM
ejpam-6153	160	4	,	,	PUNCT
ejpam-6153	160	5	1965.202	1965.202	PROPN
ejpam-6153	161	1	[	[	X
ejpam-6153	161	2	5	5	X
ejpam-6153	161	3	]	]	PUNCT
ejpam-6153	161	4	j.	j.	PROPN
ejpam-6153	161	5	ahsan	ahsan	PROPN
ejpam-6153	161	6	k.	k.	PROPN
ejpam-6153	161	7	saifullah	saifullah	PROPN
ejpam-6153	161	8	and	and	CCONJ
ejpam-6153	161	9	m.	m.	NOUN
ejpam-6153	161	10	f.khan	f.khan	ADV
ejpam-6153	161	11	.	.	PUNCT
ejpam-6153	162	1	fuzzy	fuzzy	ADJ
ejpam-6153	162	2	semirings	semiring	NOUN
ejpam-6153	162	3	.	.	PUNCT
ejpam-6153	163	1	fuzzy	fuzzy	ADJ
ejpam-6153	163	2	sets	set	VERB
ejpam-6153	163	3	syst	syst	PROPN
ejpam-6153	163	4	.	.	PUNCT
ejpam-6153	163	5	,	,	PUNCT
ejpam-6153	163	6	60(3):302–203	60(3):302–203	NOUN
ejpam-6153	163	7	309	309	NUM
ejpam-6153	163	8	,	,	PUNCT
ejpam-6153	163	9	1993.204	1993.204	NUM
ejpam-6153	164	1	[	[	X
ejpam-6153	164	2	6	6	NUM
ejpam-6153	164	3	]	]	PUNCT
ejpam-6153	164	4	s.	s.	PROPN
ejpam-6153	164	5	baupradist	baupradist	PROPN
ejpam-6153	164	6	b.	b.	PROPN
ejpam-6153	164	7	chemat	chemat	PROPN
ejpam-6153	164	8	k.	k.	PROPN
ejpam-6153	164	9	palanivel	palanivel	PROPN
ejpam-6153	164	10	and	and	CCONJ
ejpam-6153	164	11	r.	r.	PROPN
ejpam-6153	164	12	chinram	chinram	PROPN
ejpam-6153	164	13	.	.	PUNCT
ejpam-6153	165	1	essential	essential	ADJ
ejpam-6153	165	2	ideals	ideal	NOUN
ejpam-6153	165	3	and	and	CCONJ
ejpam-6153	165	4	essential205	essential205	VERB
ejpam-6153	165	5	fuzzy	fuzzy	ADJ
ejpam-6153	165	6	ideals	ideal	NOUN
ejpam-6153	165	7	in	in	ADP
ejpam-6153	165	8	semigroups	semigroup	NOUN
ejpam-6153	165	9	.	.	PUNCT
ejpam-6153	166	1	j.	j.	PROPN
ejpam-6153	166	2	discrete	discrete	PROPN
ejpam-6153	166	3	math	math	PROPN
ejpam-6153	166	4	.	.	PUNCT
ejpam-6153	167	1	sci	sci	PROPN
ejpam-6153	167	2	.	.	PUNCT
ejpam-6153	167	3	crypt	crypt	PROPN
ejpam-6153	167	4	.	.	PUNCT
ejpam-6153	167	5	,	,	PUNCT
ejpam-6153	167	6	24(1):223–233	24(1):223–233	NUM
ejpam-6153	167	7	,	,	PUNCT
ejpam-6153	168	1	2021.206	2021.206	NUM
ejpam-6153	168	2	[	[	X
ejpam-6153	168	3	7	7	X
ejpam-6153	168	4	]	]	PUNCT
ejpam-6153	168	5	p.	p.	NOUN
ejpam-6153	168	6	khamrot	khamrot	PROPN
ejpam-6153	168	7	a.	a.	NOUN
ejpam-6153	168	8	iampan	iampan	PROPN
ejpam-6153	168	9	and	and	CCONJ
ejpam-6153	168	10	t.	t.	PROPN
ejpam-6153	168	11	gaketem	gaketem	PROPN
ejpam-6153	168	12	.	.	PUNCT
ejpam-6153	169	1	essential	essential	ADJ
ejpam-6153	169	2	intuitionistic	intuitionistic	ADJ
ejpam-6153	169	3	fuzzy	fuzzy	ADJ
ejpam-6153	169	4	ideals	ideal	NOUN
ejpam-6153	169	5	in	in	ADP
ejpam-6153	169	6	semi-207	semi-207	NOUN
ejpam-6153	169	7	groups	group	NOUN
ejpam-6153	169	8	.	.	PUNCT
ejpam-6153	170	1	iaeng	iaeng	PROPN
ejpam-6153	170	2	int	int	PROPN
ejpam-6153	170	3	.	.	PUNCT
ejpam-6153	171	1	j.	j.	PROPN
ejpam-6153	171	2	comp	comp	PROPN
ejpam-6153	171	3	.	.	PUNCT
ejpam-6153	172	1	sci	sci	PROPN
ejpam-6153	172	2	.	.	PROPN
ejpam-6153	172	3	,	,	PUNCT
ejpam-6153	172	4	51(12):2026–2033	51(12):2026–2033	NUM
ejpam-6153	172	5	,	,	PUNCT
ejpam-6153	172	6	2024.208	2024.208	NUM
ejpam-6153	172	7	[	[	SYM
ejpam-6153	172	8	8	8	NUM
ejpam-6153	172	9	]	]	PUNCT
ejpam-6153	172	10	p.	p.	NOUN
ejpam-6153	172	11	khamrot	khamrot	PROPN
ejpam-6153	172	12	and	and	CCONJ
ejpam-6153	172	13	t.	t.	PROPN
ejpam-6153	172	14	gaketem	gaketem	PROPN
ejpam-6153	172	15	.	.	PUNCT
ejpam-6153	173	1	essential	essential	ADJ
ejpam-6153	173	2	bipolar	bipolar	ADJ
ejpam-6153	173	3	fuzzy	fuzzy	ADJ
ejpam-6153	173	4	ideals	ideal	NOUN
ejpam-6153	173	5	in	in	ADP
ejpam-6153	173	6	semigroups	semigroup	NOUN
ejpam-6153	173	7	.	.	PUNCT
ejpam-6153	174	1	int	int	NOUN
ejpam-6153	174	2	.	.	PUNCT
ejpam-6153	175	1	j.209	j.209	PROPN
ejpam-6153	175	2	anal	anal	PROPN
ejpam-6153	175	3	.	.	PUNCT
ejpam-6153	176	1	appl	appl	PROPN
ejpam-6153	176	2	.	.	PROPN
ejpam-6153	176	3	,	,	PUNCT
ejpam-6153	176	4	21:1	21:1	NUM
ejpam-6153	176	5	,	,	PUNCT
ejpam-6153	176	6	2023.210	2023.210	NUM
ejpam-6153	176	7	[	[	X
ejpam-6153	176	8	9	9	NUM
ejpam-6153	176	9	]	]	PUNCT
ejpam-6153	176	10	p.	p.	NOUN
ejpam-6153	176	11	khamrot	khamrot	PROPN
ejpam-6153	176	12	and	and	CCONJ
ejpam-6153	176	13	t.	t.	PROPN
ejpam-6153	176	14	gaketem	gaketem	PROPN
ejpam-6153	176	15	.	.	PUNCT
ejpam-6153	177	1	essential	essential	ADJ
ejpam-6153	177	2	interval	interval	NOUN
ejpam-6153	177	3	valued	value	VERB
ejpam-6153	177	4	fuzzy	fuzzy	ADJ
ejpam-6153	177	5	ideals	ideal	NOUN
ejpam-6153	177	6	in	in	ADP
ejpam-6153	177	7	semigroups.211	semigroups.211	PRON
ejpam-6153	177	8	icic	icic	NOUN
ejpam-6153	177	9	express	express	ADJ
ejpam-6153	177	10	letters	letter	NOUN
ejpam-6153	177	11	,	,	PUNCT
ejpam-6153	177	12	17(9):989–996	17(9):989–996	NOUN
ejpam-6153	177	13	,	,	PUNCT
ejpam-6153	177	14	2023.212	2023.212	NUM
ejpam-6153	178	1	[	[	X
ejpam-6153	178	2	10	10	NUM
ejpam-6153	178	3	]	]	PUNCT
ejpam-6153	178	4	a.	a.	NOUN
ejpam-6153	178	5	z.	z.	PROPN
ejpam-6153	178	6	m.	m.	PROPN
ejpam-6153	178	7	alkatib	alkatib	PROPN
ejpam-6153	178	8	r.	r.	PROPN
ejpam-6153	178	9	d.	d.	PROPN
ejpam-6153	178	10	mahmood	mahmood	PROPN
ejpam-6153	178	11	and	and	CCONJ
ejpam-6153	178	12	m.	m.	PROPN
ejpam-6153	178	13	m.	m.	PROPN
ejpam-6153	178	14	f.	f.	PROPN
ejpam-6153	178	15	qublan	qublan	PROPN
ejpam-6153	178	16	.	.	PUNCT
ejpam-6153	179	1	on	on	ADP
ejpam-6153	179	2	fuzzy	fuzzy	ADJ
ejpam-6153	179	3	essential	essential	ADJ
ejpam-6153	179	4	ideals	ideal	NOUN
ejpam-6153	179	5	of213	of213	PROPN
ejpam-6153	179	6	rings	ring	NOUN
ejpam-6153	179	7	.	.	PUNCT
ejpam-6153	180	1	int	int	NOUN
ejpam-6153	180	2	.	.	PUNCT
ejpam-6153	181	1	math	math	NOUN
ejpam-6153	181	2	.	.	PUNCT
ejpam-6153	182	1	forum	forum	PROPN
ejpam-6153	182	2	,	,	PUNCT
ejpam-6153	182	3	8:383–391	8:383–391	NUM
ejpam-6153	182	4	,	,	PUNCT
ejpam-6153	182	5	2020.214	2020.214	NUM